Mathematics

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The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

CONTENTSINTRODUCTION 3Secondary Schools for the Twenty-first Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3The Importance of Mathematics in the Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Roles and Responsibilities in Mathematics Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5THE PROGRAM IN MATHEMATICS 7Overview of the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Curriculum Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Courses and Strands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12THE MATHEMATICAL PROCESSES 17Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Reasoning and Proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Reflecting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Selecting Tools and Computational Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Connecting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Representing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Communicating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22ASSESSMENT AND EVALUATION OFSTUDENT ACHIEVEMENT 23Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23The Achievement Chart for Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Evaluation and Reporting of Student Achievement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Reporting on Demonstrated Learning Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27SOME CONSIDERATIONS FORPROGRAM PLANNING IN MATHEMATICS 30Instructional Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Planning Mathematics Programs for Students WithSpecial Education Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Program Considerations for English Language Learners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Une publication équivalente est disponible en français sous le titre suivant :Le curriculum de l’Ontario, 11 e et 12 e année – Mathématiques, 2007.This publication is available on the Ministry of Education’swebsite, at www.edu.gov.on.ca.


Antidiscrimination Education in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Literacy and Inquiry/Research Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36The Role of Information and Communication Technologyin Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Career Education in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37The Ontario Skills Passport and Essential Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Cooperative Education and Other Forms of Experiential Learning . . . . . . . . . . . . . . . . . . . 38Planning Program Pathways and Programs Leading to aSpecialist High-Skills Major . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Health and Safety in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39COURSES 41Grade 11Functions, University Preparation (MCR3U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Functions and Applications, University/College Preparation (MCF3M) . . . . . . . . . . . . . . 57Foundations for College Mathematics, College Preparation (MBF3C) . . . . . . . . . . . . . . . 67Mathematics for Work and Everyday Life, Workplace Preparation (MEL3E) . . . . . . . . . 77Grade 12Advanced Functions, University Preparation (MHF4U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Calculus and Vectors, University Preparation (MCV4U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Mathematics of Data Management, University Preparation (MDM4U) . . . . . . . . . . . . 111Mathematics for College Technology, College Preparation (MCT4C) . . . . . . . . . . . . . . . 123Foundations for College Mathematics, College Preparation (MAP4C) . . . . . . . . . . . . . . 135Mathematics for Work and Everyday Life, Workplace Preparation (MEL4E) . . . . . . . 147


INTRODUCTIONThis document replaces The Ontario Curriculum, Grade 11: Mathematics, 2006, and theGrade 12 courses in The Ontario Curriculum, Grades 11 and 12: Mathematics, 2000.Beginning in September 2007, all Grade 11 and Grade 12 mathematics courses will bebased on the expectations outlined in this document.SECONDARY SCHOOLS FOR THE TWENTY-FIRST CENTURYThe goal of Ontario secondary schools is to support high-quality learning while givingindividual students the opportunity to choose programs that suit their skills and interests.The updated Ontario curriculum, in combination with a broader range of learningoptions outside traditional classroom instruction, will enable students to better customizetheir high school education and improve their prospects for success in school and in life.THE IMPORTANCE OF MATHEMATICS IN THE CURRICULUMThis document provides a framework outlining what students are expected to know andbe able to do by the end of each of the courses in the Grade 11–12 mathematics curriculum.The required knowledge and skills include not only important mathematical facts andprocedures but also the mathematical concepts students need to understand and themathematical processes they must learn to apply.The principles underlying this curriculum are shared by educators dedicated to the successof all students in learning mathematics. Those principles can be stated as follows: 1Curriculum expectations must be coherent, focused, and well-articulated acrossthe grades.Learning mathematics involves the meaningful acquisition of concepts, skills, andprocesses and the active involvement of students in building new knowledge fromprior knowledge and experience.Learning tools such as manipulatives and technologies are important supports forteaching and learning mathematics.Effective teaching of mathematics requires that the teacher understand the mathematicalconcepts, procedures, and processes that students need to learn, and use avariety of instructional strategies to support meaningful learning.Assessment and evaluation must support learning, recognizing that students learnand demonstrate learning in various ways.1. Adapted from Principles and Standards for School Mathematics, developed by the National Council of Teachers ofMathematics (Reston, VA: NCTM, 2000).


Equity of opportunity for student success in mathematics involves meeting thediverse learning needs of students and promoting excellence for all students.Equity is achieved when curriculum expectations are grade- and destinationappropriate,when teaching and learning strategies meet a broad range of studentneeds, and when a variety of pathways through the mathematics curriculum aremade available to students.The Ontario mathematics curriculum must serve a number of purposes. It must engageall students in mathematics and equip them to thrive in a society where mathematics isincreasingly relevant in the workplace. It must engage and motivate as broad a group ofstudents as possible, because early abandonment of the study of mathematics cutsstudents off from many career paths and postsecondary options.The unprecedented changes that are taking place in today’s world will profoundly affectthe future of today’s students. To meet the demands of the world in which they live, studentswill need to adapt to changing conditions and to learn independently. They willrequire the ability to use technology effectively and the skills for processing large amountsof quantitative information. Today’s mathematics curriculum must prepare students fortheir future roles in society. It must equip them with an understanding of importantmathematical ideas; essential mathematical knowledge and skills; skills of reasoning,problem solving, and communication; and, most importantly, the ability and the incentiveto continue learning on their own. This curriculum provides a framework foraccomplishing these goals.The development of mathematical knowledge is a gradual process. A coherent and continuousprogram is necessary to help students see the “big pictures”, or underlying principles,of mathematics. The fundamentals of important skills, concepts, processes, andattitudes are initiated in the primary grades and fostered throughout elementary school.The links between Grade 8 and Grade 9 and the transition from elementary school mathematicsto secondary school mathematics are very important in developing the student’sconfidence and competence.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsThe secondary courses are based on principles that are consistent with those that underpinthe elementary program, facilitating the transition from elementary school. Thesecourses reflect the belief that students learn mathematics effectively when they are givenopportunities to investigate new ideas and concepts, make connections between newlearning and prior knowledge, and develop an understanding of the abstract mathematicsinvolved. Skill acquisition is an important part of the learning; skills are embeddedin the contexts offered by various topics in the mathematics program and should beintroduced as they are needed. The mathematics courses in this curriculum recognize theimportance of not only focusing on content, but also of developing the thinking processesthat underlie mathematics. By studying mathematics, students learn how to reason logically,think critically, and solve problems – key skills for success in today’s workplaces.Mathematical knowledge becomes meaningful and powerful in application. This curriculumembeds the learning of mathematics in the solving of problems based on real-lifesituations. Other disciplines are a ready source of effective contexts for the study of mathematics.Rich problem-solving situations can be drawn from related disciplines, such ascomputer science, business, recreation, tourism, biology, physics, and technology, as wellas from subjects historically thought of as distant from mathematics, such as geography4


and art. It is important that these links between disciplines be carefully explored, analysed,and discussed to emphasize for students the pervasiveness of mathematical concepts andmathematical thinking in all subject areas.The choice of specific concepts and skills to be taught must take into consideration newapplications and new ways of doing mathematics. The development of sophisticated yeteasy-to-use calculators and computers is changing the role of procedure and technique inmathematics. Operations that were an essential part of a procedures-focused curriculumfor decades can now be accomplished quickly and effectively using technology, so thatstudents can now solve problems that were previously too time-consuming to attempt,and can focus on underlying concepts. “In an effective mathematics program, studentslearn in the presence of technology. Technology should influence the mathematics contenttaught and how it is taught. Powerful assistive and enabling computer and handheldtechnologies should be used seamlessly in teaching, learning, and assessment.” 2 Thiscurriculum integrates appropriate technologies into the learning and doing of mathematics,while recognizing the continuing importance of students’ mastering essentialnumeric and algebraic skills.ROLES AND RESPONSIBILITIES IN MATHEMATICS PROGRAMSStudentsStudents have many responsibilities with regard to their learning. Students who makethe effort required to succeed in school and who are able to apply themselves will soondiscover that there is a direct relationship between this effort and their achievement, andwill therefore be more motivated to work. There will be some students, however, whowill find it more difficult to take responsibility for their learning because of special challengesthey face. The attention, patience, and encouragement of teachers and family canbe extremely important to these students’ success. However, taking responsibility fortheir own progress and learning is an important part of education for all students,regardless of their circumstances.Mastery of concepts and skills in mathematics requires a sincere commitment to workand study. Students are expected to develop strategies and processes that facilitate learningand understanding in mathematics. Students should also be encouraged to activelypursue opportunities to apply their problem-solving skills outside the classroom and toextend and enrich their understanding of mathematics.ParentsParents 3 have an important role to play in supporting student learning. Studies show thatstudents perform better in school if their parents are involved in their education. Bybecoming familiar with the curriculum, parents can find out what is being taught in thecourses their children are taking and what their children are expected to learn. Thisawareness will enhance parents’ ability to discuss their children’s work with them, tocommunicate with teachers, and to ask relevant questions about their children’s progress.2. Expert Panel on Student Success in Ontario, Leading Math Success: Mathematical Literacy, Grades 7–12 – The Report ofthe Expert Panel on Student Success in Ontario, 2004 (Toronto: Ontario Ministry of Education, 2004), p. 47. (Referred tohereafter as Leading Math Success.)3. The word parents is used throughout this document to stand for parent(s) and guardian(s).INTRODUCTION5


Knowledge of the expectations in the various courses also helps parents to interpretteachers’ comments on student progress and to work with them to improve studentlearning.Effective ways for parents to support their children’s learning include attending parentteacherinterviews, participating in parent workshops, becoming involved in school councilactivities (including becoming a school council member), and encouraging their childrento complete their assignments at home.The mathematics curriculum promotes lifelong learning. In addition to supporting regularschool activities, parents can encourage their children to apply their problem-solvingskills to other disciplines and to real-world situations.TeachersTeachers and students have complementary responsibilities. Teachers are responsible fordeveloping appropriate instructional strategies to help students achieve the curriculumexpectations for their courses, as well as for developing appropriate methods for assessingand evaluating student learning. Teachers also support students in developing thereading, writing, and oral communication skills needed for success in their mathematicscourses. Teachers bring enthusiasm and varied teaching and assessment approaches tothe classroom, addressing different student needs and ensuring sound learning opportunitiesfor every student.Recognizing that students need a solid conceptual foundation in mathematics in order tofurther develop and apply their knowledge effectively, teachers endeavour to create aclassroom environment that engages students’ interest and helps them arrive at theunderstanding of mathematics that is critical to further learning.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsUsing a variety of instructional, assessment, and evaluation strategies, teachers providenumerous opportunities for students to develop skills of inquiry, problem solving, andcommunication as they investigate and learn fundamental concepts. The activities offeredshould enable students not only to make connections among these concepts throughoutthe course but also to relate and apply them to relevant societal, environmental, andeconomic contexts. Opportunities to relate knowledge and skills to these wider contexts –to the goals and concerns of the world in which they live – will motivate students to learnand to become lifelong learners.PrincipalsThe principal works in partnership with teachers and parents to ensure that each studenthas access to the best possible educational experience. To support student learning, principalsensure that the Ontario curriculum is being properly implemented in all classroomsthrough the use of a variety of instructional approaches. They also ensure that appropriateresources are made available for teachers and students. To enhance teaching and learningin all subjects, including mathematics, principals promote learning teams and work withteachers to facilitate participation in professional-development activities.Principals are also responsible for ensuring that every student who has an IndividualEducation Plan (IEP) is receiving the modifications and/or accommodations describedin his or her plan – in other words, for ensuring that the IEP is properly developed,implemented, and monitored.6


THE PROGRAM INMATHEMATICSOVERVIEW OF THE PROGRAMThe senior mathematics courses build on the Grade 9 and 10 program, relying on thesame fundamental principles on which that program was based. Both are founded on thepremise that students learn mathematics most effectively when they build a thoroughunderstanding of mathematical concepts and procedures. Such understanding is achievedwhen mathematical concepts and procedures are introduced through an investigativeapproach and connected to students’ prior knowledge in meaningful ways. This curriculumis designed to help students prepare for university, college, or the workplace bybuilding a solid conceptual foundation in mathematics that will enable them to applytheir knowledge and skills in a variety of ways and further their learning successfully.An important part of every course in the mathematics program is the process of inquiry,in which students develop methods for exploring new problems or unfamiliar situations.Knowing how to learn mathematics is the underlying expectation that every student inevery course needs to achieve. An important part of the inquiry process is that of takingthe conditions of a real-world situation and representing them in mathematical form. Amathematical representation can take many different forms – for example, it can be aphysical model, a diagram, a graph, a table of values, an equation, or a computer simulation.It is important that students recognize various mathematical representations ofgiven relationships and that they become familiar with increasingly sophisticated representationsas they progress through secondary school.The prevalence in today’s society and classrooms of sophisticated yet easy-to-use calculatorsand computer software accounts in part for the inclusion of certain concepts andskills in this curriculum. The curriculum has been designed to integrate appropriatetechnologies into the learning and doing of mathematics, while equipping students withthe manipulation skills necessary to understand other aspects of the mathematics thatthey are learning, to solve meaningful problems, and to continue to learn mathematicswith success in the future. Technology is not used to replace skill acquisition; rather, itis treated as a learning tool that helps students explore concepts. Technology is requiredwhen its use represents either the only way or the most effective way to achieve anexpectation.Like the earlier curriculum experienced by students, the senior secondary curriculumadopts a strong focus on the processes that best enable students to understand mathematicalconcepts and learn related skills. Attention to the mathematical processes is


considered to be essential to a balanced mathematics program. The seven mathematicalprocesses identified in this curriculum are problem solving, reasoning and proving, reflecting,selecting tools and computational strategies, connecting, representing, and communicating.Each of the senior mathematics courses includes a set of expectations – referred to in thisdocument as the “mathematical process expectations” – that outline the knowledge andskills involved in these essential processes. The mathematical processes apply to studentlearning in all areas of a mathematics course.A balanced mathematics program at the secondary level also includes the development ofalgebraic skills. This curriculum has been designed to equip students with the algebraicskills needed to solve meaningful problems, to understand the mathematical conceptsthey are learning, and to successfully continue their study of mathematics in the future.The algebraic skills required in each course have been carefully chosen to support thetopics included in the course. Calculators and other appropriate technologies will be usedwhen the primary purpose of a given activity is the development of concepts or the solvingof problems, or when situations arise in which computation or symbolic manipulationis of secondary importance.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsCourses in Grade 11 and Grade 12Four types of courses are offered in the senior mathematics program: university preparation,university/college preparation, college preparation, and workplace preparation. Studentschoose course types on the basis of their interests, achievement, and postsecondary goals.The course types are defined as follows:University preparation courses are designed to equip students with the knowledge andskills they need to meet the entrance requirements for university programs.University/college preparation courses are designed to equip students with the knowledgeand skills they need to meet the entrance requirements for specific programs offered atuniversities and colleges.College preparation courses are designed to equip students with the knowledge andskills they need to meet the requirements for entrance to most college programs or foradmission to specific apprenticeship or other training programs.Workplace preparation courses are designed to equip students with the knowledge andskills they need to meet the expectations of employers, if they plan to enter the workplacedirectly after graduation, or the requirements for admission to many apprenticeship orother training programs.8


Courses in Mathematics, Grades 11 and 12Grade Course Name Course Type Course Code Prerequisite11 Functions University MCR3U Grade 10 Principles of Mathematics, Academic11 Functions andApplicationsUniversity/CollegeMCF3MGrade 10 Principles of Mathematics, Academic,or Grade 10 Foundations of Mathematics, Applied11 Foundationsfor CollegeMathematics11 Mathematicsfor Work andEveryday Life12 AdvancedFunctions12 Calculusand Vectors12 Mathematicsof DataManagement12 Mathematicsfor CollegeTechnology12 Foundationsfor CollegeMathematics12 Mathematicsfor Work andEveryday LifeCollege MBF3C Grade 10 Foundations of Mathematics, AppliedMathematicsWorkplace MEL3E Grade 9 Principles of Mathematics, Academic, orGrade 9 Foundations of Mathematics, Applied, ora Grade 10 Mathematics LDCC (locally developedcompulsory credit) courseUniversity MHF4U Grade 11 Functions, UniversityUniversity MCV4U Grade 12 Advanced Functions, University, must betaken prior to or concurrently with Calculus andVectors.University MDM4U Grade 11 Functions, University, or Grade 11Functions and Applications, University/CollegeCollege MCT4C Grade 11 Functions and Applications,University/College, or Grade 11 Functions,UniversityCollege MAP4C Grade 11 Foundations for CollegeMathematics, College, or Grade 11 Functionsand Applications, University/CollegeWorkplace MEL4E Grade 11 Mathematics for Work and EverydayLife, WorkplaceNote: Each of the courses listed above is worth one credit.THE PROGRAM IN MATHEMATICS9


Prerequisite Chart for Mathematics, Grades 9–12This chart maps out all the courses in the discipline and shows the links between courses and the possibleprerequisites for them. It does not attempt to depict all possible movements from course to course.Note: Advanced Functions (MHF4U) mustbe taken prior to or concurrently withCalculus and Vectors (MCV4U).Calculus andVectorsMCV4UGrade 12UniversityAdvancedFunctionsMHF4UGrade 12UniversityPrinciplesof MathematicsMPM1DPrinciplesof MathematicsMPM2DFunctionsMCR3UGrade 11UniversityMathematics ofData ManagementMDM4UGrade 12UniversityGrade 9AcademicGrade 10AcademicTFunctions andApplicationsMCF3MGrade 11University/CollegeMathematics forCollege TechnologyMCT4CGrade 12CollegeTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics10Foundationsof MathematicsMFM1PGrade 9AppliedMathematicsLDCCGrade 9Foundationsof MathematicsMFM2PGrade 10AppliedMathematicsLDCCGrade 10Foundations forCollegeMathematicsMBF3CGrade 11CollegeMathematics forWork and EverydayLifeMEL3EGrade 11WorkplaceNotes:• T – transfer course• LDCC – locally developed compulsory credit course (LDCC courses are not outlined in this document.)Foundations forCollegeMathematicsMAP4CGrade 12CollegeMathematics forWork and EverydayLifeMEL4EGrade 12Workplace


Half-Credit CoursesThe courses outlined in this document are designed to be offered as full-credit courses.However, with the exception of the Grade 12 university preparation courses, they may also bedelivered as half-credit courses.Half-credit courses, which require a minimum of fifty-five hours of scheduled instructionaltime, must adhere to the following conditions:The two half-credit courses created from a full course must together contain allof the expectations of the full course. The expectations for each half-credit coursemust be divided in a manner that best enables students to achieve the requiredknowledge and skills in the allotted time.A course that is a prerequisite for another course in the secondary curriculum maybe offered as two half-credit courses, but students must successfully complete bothparts of the course to fulfil the prerequisite. (Students are not required to completeboth parts unless the course is a prerequisite for another course they wish to take.)The title of each half-credit course must include the designation Part 1 or Part 2.A half credit (0.5) will be recorded in the credit-value column of both the reportcard and the Ontario Student Transcript.Boards will ensure that all half-credit courses comply with the conditions described above,and will report all half-credit courses to the ministry annually in the School OctoberReport.CURRICULUM EXPECTATIONSThe expectations identified for each course describe the knowledge and skills that studentsare expected to acquire, demonstrate, and apply in their class work, on tests, and invarious other activities on which their achievement is assessed and evaluated.Two sets of expectations are listed for each strand, or broad curriculum area, of each course.The overall expectations describe in general terms the knowledge and skills that studentsare expected to demonstrate by the end of each course.The specific expectations describe the expected knowledge and skills in greaterdetail. The specific expectations are arranged under numbered subheadings thatrelate to the overall expectations and that may serve as a guide for teachers as theyplan learning activities for their students. The specific expectations are also numberedto indicate the overall expectation to which they relate (e.g., specific expectation3.2 is related to overall expectation 3 in a given strand). The organization ofexpectations in subgroupings is not meant to imply that the expectations in anysubgroup are achieved independently of the expectations in the other subgroups.The subheadings are used merely to help teachers focus on particular aspects ofknowledge and skills as they develop and use various lessons and learning activitieswith their students.In addition to the expectations outlined within each strand, a list of seven “mathematicalprocess expectations” precedes the strands in all mathematics courses. These specificexpectations describe the knowledge and skills that constitute processes essential to theeffective study of mathematics. These processes apply to all areas of course content, andTHE PROGRAM IN MATHEMATICS11


students’ proficiency in applying them must be developed in all strands of a mathematicscourse. Teachers should ensure that students develop their ability to apply these processesin appropriate ways as they work towards meeting the expectations outlined in thestrands.When developing detailed courses of study from this document, teachers are expected toweave together related expectations from different strands, as well as the relevant processexpectations, in order to create an overall program that integrates and balances conceptdevelopment, skill acquisition, the use of processes, and applications.Many of the specific expectations are accompanied by examples and/or sample problems.These examples and sample problems are meant to illustrate the kind of skill, the specificarea of learning, the depth of learning, and/or the level of complexity that the expectationentails. Some examples and sample problems may also be used to emphasize the importanceof diversity or multiple perspectives. The examples and sample problems areintended only as suggestions for teachers. Teachers may incorporate the examples andsample problems into their lessons, or they may choose other topics, approaches, orproblems that are relevant to the expectation.COURSES AND STRANDSThe courses in the Grade 11–12 mathematics curriculum are briefly described below, bycourse type. The strands in each course are listed in the graphic provided in each section,and their focus is discussed in the following text.University Preparation CoursesGrade 12ADVANCED FUNCTIONS(MHF4U)Grade 12CALCULUS AND VECTORS(MCV4U)THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsGrade 11FUNCTIONS (MCR3U)A. Characteristics ofFunctionsB. Exponential FunctionsC. Discrete FunctionsD. Trigonometric FunctionsA. Exponential andLogarithmic FunctionsB. Trigonometric FunctionsC. Polynomial and RationalFunctionsD. Characteristics ofFunctionsGrade 12MATHEMATICS OF DATAMANAGEMENT (MDM4U)A. Counting and ProbabilityB. Probability DistributionsC. Organization of Data forAnalysisD. Statistical AnalysisE. Culminating DataManagementInvestigationA. Rate of ChangeB. Derivatives and TheirApplicationsC. Geometry and Algebra ofVectors12


The Grade 11 university preparation course, Functions, builds on the concepts and skillsdeveloped in the Grade 9 and 10 academic mathematics courses. The course is designedto prepare students for Grade 12 mathematics courses that lead to one of many universityprograms, including science, engineering, social sciences, liberal arts, and education.The concept of functions is introduced in the Characteristics of Functions strand of thiscourse and extended through the investigation of two new types of relationships in theExponential Functions and Trigonometric Functions strands. The Discrete Functionsstrand allows students, through the study of different representations of sequences andseries, to revisit patterning and algebra concepts introduced in elementary school andmake connections to financial applications involving compound interest and ordinarysimple annuities.The Grade 12 university preparation course Advanced Functions satisfies the mathematicalprerequisite for some universities in areas that include business, social science,and health science programs. The strands in this course help students deepen theirunderstanding of functions by revisiting the exponential and trigonometric functionsintroduced in Grade 11 to address related concepts such as radian measure and logarithmicfunctions and by extending prior knowledge of quadratic functions to explore polynomialand rational functions. The Characteristics of Functions strand addresses someof the general features of functions through the examination of rates of change andmethods of combining functions.The Grade 12 university preparation course Calculus and Vectors is designed to preparestudents for university programs, such as science, engineering, and economics, thatinclude a calculus or linear algebra course in the first year. Calculus is introduced in theRate of Change strand by extending the numeric and graphical representation of rates ofchange introduced in the Advanced Functions course to include more abstract algebraicrepresentations. The Derivatives and Their Applications strand provides students withthe opportunity to develop the algebraic and problem-solving skills needed to solveproblems associated with rates of change. Prior knowledge of geometry and trigonometryis used in the Geometry and Algebra of Vectors strand to develop vector concepts thatcan be used to solve interesting problems, including those arising from real-worldapplications.The Grade 12 university preparation course Mathematics of Data Management isdesigned to satisfy the prerequisites for a number of university programs that mayinclude statistics courses, such as those found in the social sciences and the humanities.The expectations in the strands of this course require students to apply mathematicalprocess skills developed in prerequisite courses, such as problem solving, reasoning, andcommunication, to the study of probability and statistics. The Counting and Probabilitystrand extends the basic probability concepts learned in the elementary school programand introduces counting techniques such as the use of permutations and combinations;these techniques are applied to both counting and probability problems. The ProbabilityDistributions strand introduces the concept of probability distributions; these include thenormal distribution, which is important in the study of statistics. In the Organization ofData for Analysis strand, students examine, use, and develop methods for organizinglarge amounts of data, while in the Statistical Analysis strand, students investigate anddevelop an understanding of powerful concepts used to analyse and interpret largeamounts of data. These concepts are developed with the use of technological tools suchTHE PROGRAM IN MATHEMATICS13


as spreadsheets and Fathom, a ministry-licensed dynamic statistical program. TheCulminating Data Management Investigation strand requires students to undertakea culminating investigation dealing with a significant issue that will require theapplication of the skills from the other strands of the course.University/College Preparation and College Preparation CoursesGrade 11FUNCTIONS ANDAPPLICATIONS (MCF3M)A. Quadratic FunctionsB. Exponential FunctionsC. Trigonometric FunctionsGrade 12MATHEMATICS FORCOLLEGETECHNOLOGY (MCT4C)A. Exponential FunctionsB. Polynomial FunctionsC. Trigonometric FunctionsD. Applications of GeometryThe Grade 11 university/college preparation course, Functions and Applications, providespreparation for students who plan to pursue technology-related programs in college, whilealso leaving the option open for some students to pursue postsecondary programs thatrequire the Grade 12 university preparation course Mathematics of Data Management.The Functions and Applications course explores functions by revisiting key concepts fromthe Grade 10 mathematics curriculum and by using a more applied approach with lessemphasis on abstract concepts than in the Grade 11 university preparation course,Functions. The first strand, Quadratic Functions, extends knowledge and skills relatedto quadratics for students who completed the Grade 10 applied mathematics course andreviews this topic for students entering from the Grade 10 academic course. The strandalso introduces some of the properties of functions. The other two strands, ExponentialFunctions and Trigonometric Functions, emphasize real-world applications and helpstudents develop the knowledge and skills needed to solve problems related to theseapplications.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsThe Grade 12 college preparation course Mathematics for College Technology providesexcellent preparation for success in technology-related programs at the college level. Itextends the understanding of functions developed in the Grade 11 university/collegepreparation course, Functions and Applications, using a more applied approach, and mayhelp students who decide to pursue certain university programs to prepare for theGrade 12 university preparation course Advanced Functions. Exponential and trigonometricfunctions are revisited, developing algebraic skills needed to solve problemsinvolving exponential equations and extending the skills associated with graphical representationsof trigonometric functions. The Polynomial Functions strand extends to polynomialfunctions concepts that connect graphs and equations of quadratic functions.Finally, students apply geometric relationships to solve problems involving compositeshapes and figures and investigate the properties of circles and their applications.14


Grade 11FOUNDATIONS FORCOLLEGEMATHEMATICS (MBF3C)A. Mathematical ModelsB. Personal FinanceC. Geometry andTrigonometryD. Data ManagementGrade 12FOUNDATIONS FORCOLLEGEMATHEMATICS (MAP4C)A. Mathematical ModelsB. Personal FinanceC. Geometry andTrigonometryD. Data ManagementThe Grade 11 college preparation course, Foundations for College Mathematics, includesa blend of topics needed by students who plan to pursue one of a broad range of collegeprograms. The course has been designed with four strands that address different areas ofmathematics. The Mathematical Models strand uses the concepts connected to linear andquadratic relations developed in the Grade 9 and 10 applied mathematics courses torevisit quadratic relations and introduce exponential relations. The Personal Financestrand focuses on compound interest and applications related to investing and borrowingmoney and owning and operating a vehicle. Applications requiring spatial reasoning areaddressed in the Geometry and Trigonometry strand. The fourth strand, DataManagement, explores practical applications of one-variable statistics and probability.The Grade 12 college preparation course Foundations for College Mathematics satisfiesthe mathematical prerequisites for many college programs, including programs in business,human services, hospitality and tourism, and some of the health sciences. The fourstrands of this course focus on the same areas of mathematics addressed in the Grade 11college preparation course, Foundations for College Mathematics. The MathematicalModels strand extends the concepts and skills that related to exponential relationsintroduced in Grade 11 and provides students with an opportunity to revisit all of therelations they have studied in the secondary mathematics program by using a graphicaland algebraic approach. The Personal Finance strand focuses on annuities and mortgages,renting or owning accommodation, and designing budgets. Problem solving in theGeometry and Trigonometry strand reinforces the application of relationships associatedwith a variety of shapes and figures. The fourth strand, Data Management, addressespractical applications of two-variable statistics and examines applications of datamanagement.Workplace Preparation CoursesGrade 11MATHEMATICS FOR WORKAND EVERYDAY LIFE(MEL3E)A. Earning and PurchasingB. Saving, Investing, andBorrowingC. Transportation and TravelGrade 12MATHEMATICS FOR WORKAND EVERYDAY LIFE(MEL4E)A. Reasoning With DataB. Personal FinanceC. Applications ofMeasurementTHE PROGRAM IN MATHEMATICS15


The Grade 11 workplace preparation course, Mathematics for Work and Everyday Life, isdesigned to help students consolidate the basic knowledge and skills of mathematics usedin the workplace and in everyday life. This course is ideal for students who would like totake the Grade 12 workplace preparation course before graduating from high school andentering the workplace. The course also meets the needs of students who wish to fulfillthe senior mathematics graduation requirement but do not plan to take any furthercourses in mathematics. All three strands, Earning and Purchasing; Saving, Investing,and Borrowing; and Transportation and Travel, provide students with the opportunityto use proportional reasoning to solve a variety of problems.The Grade 12 workplace preparation course, Mathematics for Work and Everyday Life,extends the knowledge and skills developed in Grade 11. The gathering, interpretation,and display of one-variable data and the investigation of probability concepts are the maincomponents of the Reasoning With Data strand. Topics in the Personal Finance strandaddress owning or renting accommodation, designing a budget, and filing an income taxreturn. A variety of problems involving metric and imperial measurement are presentedin the Applications of Measurement strand. The expectations support the use of hands-onprojects and other experiences that make the mathematics more meaningful for students.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics16


THE MATHEMATICALPROCESSESPresented at the start of every course in this curriculum document are seven mathematicalprocess expectations that describe a set of skills that support lifelong learning in mathematicsand that students need to develop on an ongoing basis, as they work to achievethe expectations outlined within each course. In the 2000 mathematics curriculum, expectationsthat addressed the mathematical processes were present within individual strandsto varying degrees. Here, the mathematical processes are highlighted in each course toensure that students are actively engaged in developing their skills to apply themthroughout the course, rather than only in specific strands.The mathematical processes are as follows:problem solvingreasoning and provingreflectingselecting tools and computational strategiesconnectingrepresentingcommunicatingEach course presents students with rich problem-solving experiences through a variety ofapproaches, including investigation. These experiences provide students with opportunitiesto develop and apply the mathematical processes.The mathematical processes are interconnected. Problem solving and communicatinghave strong links to all the other processes. The problem-solving process can be thoughtof as the motor that drives the development of the other processes. It allows students tomake conjectures and to reason as they pursue a solution or a new understanding.Problem solving provides students with the opportunity to make connections to theirprior learning and to make decisions about the representations, tools, and computationalstrategies needed to solve the problem. Teachers should encourage students to justify theirsolutions, communicate them orally and in writing, and reflect on alternative solutions.By seeing how others solve a problem, students can begin to think about their ownthinking (metacognition) and the thinking of others, and to consciously adjust their ownstrategies in order to make their solutions as efficient and accurate as possible.


The mathematical processes cannot be separated from the knowledge and skills that studentsacquire throughout the course. Students who problem solve, communicate, reason,reflect, and so on, as they learn mathematics, will develop the knowledge, the understandingof concepts, and the skills required in the course in a more meaningful way.PROBLEM SOLVINGProblem solving is central to learning mathematics. It forms the basis of effective mathematicsprograms and should be the mainstay of mathematical instruction. It is consideredan essential process through which students are able to achieve the expectations in mathematics,and is an integral part of the mathematics curriculum in Ontario, for the followingreasons. Problem solving:helps students become more confident mathematicians;allows students to use the knowledge they bring to school and helps them connectmathematics with situations outside the classroom;helps students develop mathematical understanding and gives meaning to skillsand concepts in all strands;allows students to reason, communicate ideas, make connections, and applyknowledge and skills;offers excellent opportunities for assessing students’ understanding of concepts,ability to solve problems, ability to apply concepts and procedures, and ability tocommunicate ideas;promotes collaborative sharing of ideas and strategies, and promotes talking aboutmathematics;helps students find enjoyment in mathematics;increases opportunities for the use of critical-thinking skills (e.g., estimating,classifying, assuming, recognizing relationships, hypothesizing, offering opinionswith reasons, evaluating results, and making judgements).THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics18Not all mathematics instruction, however, can take place in a problem-solving context.Certain aspects of mathematics must be explicitly taught. Conventions, including the useof mathematical symbols and terms, are one such aspect, and they should be introducedto students as needed, to enable them to use the symbolic language of mathematics.Selecting Problem-Solving StrategiesProblem-solving strategies are methods that can be used to solve various types of problems.Common problem-solving strategies include: making a model, picture, or diagram; lookingfor a pattern; guessing and checking; making assumptions; creating an organized list;making a table or chart; solving a simpler problem; working backwards; and using logicalreasoning.Teachers who use problem solving as a focus of their mathematics teaching help studentsdevelop and extend a repertoire of strategies and methods that they can apply whensolving various kinds of problems – instructional problems, routine problems, and nonroutineproblems. Students develop this repertoire over time, as their problem-solvingskills mature. By secondary school, students will have learned many problem-solvingstrategies that they can flexibly use to investigate mathematical concepts or can applywhen faced with unfamiliar problem-solving situations.


REASONING AND PROVINGReasoning helps students make sense of mathematics. Classroom instruction in mathematicsshould foster critical thinking – that is, an organized, analytical, well-reasonedapproach to learning mathematical concepts and processes and to solving problems.As students investigate and make conjectures about mathematical concepts and relationships,they learn to employ inductive reasoning, making generalizations based on specificfindings from their investigations. Students also learn to use counter-examples to disproveconjectures. Students can use deductive reasoning to assess the validity of conjectures andto formulate proofs.REFLECTINGGood problem-solvers regularly and consciously reflect on and monitor their own thoughtprocesses. By doing so, they are able to recognize when the technique they are using isnot fruitful, and to make a conscious decision to switch to a different strategy, rethink theproblem, search for related content knowledge that may be helpful, and so forth. Students’problem-solving skills are enhanced when they reflect on alternative ways to perform atask even if they have successfully completed it. Reflecting on the reasonableness of ananswer by considering the original question or problem is another way in which studentscan improve their ability to make sense of problems.SELECTING TOOLS AND COMPUTATIONAL STRATEGIESThe primary role of learning tools such as calculators, manipulatives, graphing technologies,computer algebra systems, dynamic geometry software, and dynamic statistical softwareis to help students develop a deeper understanding of mathematics through theuse of a variety of tools and strategies. Students need to develop the ability to select theappropriate learning tools and computational strategies to perform particular mathematicaltasks, to investigate mathematical ideas, and to solve problems.Calculators, Computers, Communications TechnologyVarious types of technology are useful in learning and doing mathematics. Students canuse calculators and computers to extend their capacity to investigate and analyse mathematicalconcepts and to reduce the time they might otherwise spend on purely mechanicalactivities.Technology helps students perform operations, make graphs, manipulate algebraicexpressions, and organize and display data that are lengthier or more complex than thoseaddressed in curriculum expectations suited to a paper-and-pencil approach. It can beused to investigate number and graphing patterns, geometric relationships, and differentrepresentations; to simulate situations; and to extend problem solving. Students also needto recognize when it is appropriate to apply their mental computation, reasoning, andestimation skills to predict results and check answers.THE MATHEMATICAL PROCESSES19


Technologies must be seen as important problem-solving tools. Computers and calculatorsare tools of mathematicians, and students should be given opportunities to select anduse the learning tools that may be helpful to them as they search for their own solutionsto problems.It is important that teachers introduce the use of technology in ways that build students’confidence and contribute to their understanding of the concepts being investigated,especially when students may not be familiar with the use of some of the technologiessuggested in the curriculum. Students’ use of technology should not be laborious orrestricted to inputting and learning algorithmic steps. For example, when using spreadsheetsand statistical software (e.g., Fathom), teachers could supply students with prepareddata sets, and when using dynamic geometry software (e.g., The Geometer’s Sketchpad),pre-made sketches could be used to ensure that students focus on the important mathematicalrelationships, and not just on the inputting of data or on the construction of thesketch.Whenever appropriate, students should be encouraged to select and use the communicationstechnology that would best support and communicate their learning. Computersoftware programs can help students collect, organize, and sort the data they gather, andwrite, edit, and present reports on their findings. Students, working individually or ingroups, can use Internet websites to gain access to Statistics Canada, mathematics organizations,and other valuable sources of mathematical information around the world.ManipulativesAlthough technologies are the most common learning tools used by students studyingsenior level mathematics, students should still be encouraged, when appropriate, to selectand use concrete learning tools to make models of mathematical ideas. Students need tounderstand that making their own models is a powerful means of building understandingand explaining their thinking to others.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics20Representation of mathematical ideas using manipulatives 4 helps students to:see patterns and relationships;make connections between the concrete and the abstract;test, revise, and confirm their reasoning;remember how they solved a problem;communicate their reasoning to others.Computational StrategiesProblem solving often requires students to select an appropriate computational strategysuch as applying a standard algorithm, using technology, or applying strategies related tomental computation and estimation. Developing the ability to perform mental computationand to estimate is an important aspect of student learning in mathematics. Knowingwhen to apply such skills is equally important.4. See the Instructional Approaches section, on page 30 of this document, for additional information about the use ofmanipulatives in mathematics instruction.


Mental computation involves calculations done in the mind, with little or no use of paperand pencil. Students who have developed the ability to calculate mentally can select fromand use a variety of procedures that take advantage of their knowledge and understandingof numbers, the operations, and their properties. Using knowledge of the distributiveproperty, for example, students can mentally compute 70% of 22 by first considering 70%of 20 and then adding 70% of 2. Used effectively, mental computation can encouragestudents to think more deeply about numbers and number relationships.Knowing how to estimate and recognizing when it is useful to estimate and when it isnecessary to have an exact answer are important mathematical skills. Estimation is auseful tool for judging the reasonableness of a solution and for guiding students in theiruse of calculators. The ability to estimate depends on a well-developed sense of numberand an understanding of place value. It can be a complex skill that requires decomposingnumbers, compensating for errors, and perhaps even restructuring the problem. Estimationshould not be taught as an isolated skill or a set of isolated rules and techniques.Recognizing calculations that are easy to perform and developing fluency in performingbasic operations contribute to successful estimation.CONNECTINGExperiences that allow students to make more connections – to see, for example, howconcepts and skills from one strand of mathematics are related to those from another orhow a mathematical concept can be applied in the real world – will help them developdeeper mathematical understanding. As they continue to make such connections, studentsbegin to see mathematics more as a study of relationships rather than a series ofisolated skills and concepts. Making connections not only deepens understanding, butalso helps students develop the ability to use learning from one area of mathematics tounderstand another.Making connections between the mathematics being studied and its applications in thereal world helps convince students of the usefulness and relevance of mathematicsbeyond the classroom.REPRESENTINGIn the senior mathematics curriculum, representing mathematical ideas and modellingsituations generally involve concrete, numeric, graphical, and algebraic representations.Pictorial, geometric representations as well as representations using dynamic softwarecan also be very helpful. Students should be able to recognize the connections betweenrepresentations, translate one representation into another, and use the different representationsappropriately and as needed to solve problems. Knowing the different ways inwhich a mathematical idea can be represented helps students develop a better understandingof mathematical concepts and relationships; communicate their thinking andunderstanding; recognize connections among related mathematical concepts; and modeland interpret mathematical, physical, and social phenomena. When students are able torepresent concepts in various ways, they develop flexibility in their thinking about thoseconcepts. They are not inclined to perceive any single representation as “the math”; rather,they understand that it is just one of many representations that help them understanda concept.THE MATHEMATICAL PROCESSES21


COMMUNICATINGCommunication is the process of expressing mathematical ideas and understandingsorally, visually, and in writing, using numbers, symbols, pictures, graphs, diagrams, andwords. Providing effective explanations and using correct mathematical notation whendeveloping and presenting mathematical ideas and solutions are key aspects of effectivecommunication in mathematics. Students communicate for various purposes and fordifferent audiences, such as the teacher, a peer, a group of students, or the whole class.Communication is an essential process in learning mathematics. Through communication,students are able to reflect upon and clarify ideas, relationships, and mathematicalarguments.Many opportunities exist for teachers to help students develop their ability to communicatemathematically. For example, teachers can:model proper use of symbols, vocabulary, and notations in oral and written form;expect correct use of mathematical symbols and conventions in student work;ensure that students are exposed to and use new mathematical vocabulary as it isintroduced (e.g., as they gather and interpret information; by providing opportunitiesto read, question, and discuss);provide feedback to students on their use of terminology and conventions;ask clarifying and extending questions and encourage students to ask themselvessimilar kinds of questions;ask students open-ended questions relating to specific topics or information;model ways in which various kinds of questions can be answered.Effective classroom communication requires a supportive and respectful environmentthat makes all members of the class comfortable when they speak and when they question,react to, and elaborate on the statements of their classmates and the teacher.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics22


ASSESSMENTAND EVALUATIONOF STUDENTACHIEVEMENTBASIC CONSIDERATIONSThe primary purpose of assessment and evaluation is to improve student learning.Information gathered through assessment helps teachers to determine students’ strengthsand weaknesses in their achievement of the curriculum expectations in each course. Thisinformation also serves to guide teachers in adapting curriculum and instructionalapproaches to students’ needs and in assessing the overall effectiveness of programsand classroom practices.Assessment is the process of gathering information from a variety of sources (includingassignments, demonstrations, projects, performances, and tests) that accurately reflectshow well a student is achieving the curriculum expectations in a course. As part ofassessment, teachers provide students with descriptive feedback that guides their effortstowards improvement. Evaluation refers to the process of judging the quality of studentwork on the basis of established criteria, and assigning a value to represent that quality.Assessment and evaluation will be based on the provincial curriculum expectations andthe achievement levels outlined in this document.In order to ensure that assessment and evaluation are valid and reliable, and that theylead to the improvement of student learning, teachers must use assessment and evaluationstrategies that:address both what students learn and how well they learn;are based both on the categories of knowledge and skills and on the achievementlevel descriptions given in the achievement chart on pages 28–29;are varied in nature, administered over a period of time, and designed to provideopportunities for students to demonstrate the full range of their learning;are appropriate for the learning activities used, the purposes of instruction, andthe needs and experiences of the students;


are fair to all students;accommodate students with special education needs, consistent with the strategiesoutlined in their Individual Education Plan;accommodate the needs of students who are learning the language of instruction(English or French);ensure that each student is given clear directions for improvement;promote students’ ability to assess their own learning and to set specific goals;include the use of samples that provide evidence of their achievement;are communicated clearly to students and parents at the beginning of the course orthe school term and at other appropriate points throughout the school year.All curriculum expectations must be accounted for in instruction, but evaluation focuseson students’ achievement of the overall expectations. A student’s achievement of theoverall expectations is evaluated on the basis of his or her achievement of related specificexpectations (including the process expectations). The overall expectations are broad innature, and the specific expectations define the particular content or scope of the knowledgeand skills referred to in the overall expectations. Teachers will use their professionaljudgement to determine which specific expectations should be used to evaluate achievementof the overall expectations, and which ones will be covered in instruction andassessment (e.g., through direct observation) but not necessarily evaluated.The characteristics given in the achievement chart (pages 28–29) for level 3 represent the“provincial standard” for achievement of the expectations in a course. A complete pictureof overall achievement at level 3 in a course in mathematics can be constructed by readingfrom top to bottom in the shaded column of the achievement chart, headed “70–79%(Level 3)”. Parents of students achieving at level 3 can be confident that their children willbe prepared for work in subsequent courses.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsLevel 1 identifies achievement that falls much below the provincial standard, while stillreflecting a passing grade. Level 2 identifies achievement that approaches the standard.Level 4 identifies achievement that surpasses the standard. It should be noted thatachievement at level 4 does not mean that the student has achieved expectations beyondthose specified for a particular course. It indicates that the student has achieved all oralmost all of the expectations for that course, and that he or she demonstrates the abilityto use the specified knowledge and skills in more sophisticated ways than a studentachieving at level 3.24


THE ACHIEVEMENT CHART FOR MATHEMATICSThe achievement chart for mathematics (see pages 28−29) identifies four categories ofknowledge and skills. The achievement chart is a standard province-wide guide to beused by teachers. It enables teachers to make judgements about student work that arebased on clear performance standards and on a body of evidence collected over time.The purpose of the achievement chart is to:provide a common framework that encompasses the curriculum expectations for allcourses outlined in this document;guide the development of quality assessment tasks and tools (including rubrics);help teachers to plan instruction for learning;assist teachers in providing meaningful feedback to students;provide various categories and criteria with which to assess and evaluate studentlearning.Categories of Knowledge and SkillsThe categories, defined by clear criteria, represent four broad areas of knowledge andskills within which the expectations for any given mathematics course are organized. Thefour categories should be considered as interrelated, reflecting the wholeness and interconnectednessof learning.The categories of knowledge and skills are described as follows:Knowledge and Understanding. Subject-specific content acquired in each course (knowledge),and the comprehension of its meaning and significance (understanding).Thinking. The use of critical and creative thinking skills and/or processes, 5 as follows:planning skills (e.g., understanding the problem, making a plan for solving theproblem)processing skills (e.g., carrying out a plan, looking back at the solution)critical/creative thinking processes (e.g., inquiry, problem solving)Communication. The conveying of meaning through various oral, written, and visualforms (e.g., providing explanations of reasoning or justification of results orally or inwriting; communicating mathematical ideas and solutions in writing, using numbersand algebraic symbols, and visually, using pictures, diagrams, charts, tables, graphs,and concrete materials).Application. The use of knowledge and skills to make connections within and betweenvarious contexts.Teachers will ensure that student work is assessed and/or evaluated in a balanced mannerwith respect to the four categories, and that achievement of particular expectations isconsidered within the appropriate categories.5. See the footnote on page 28, pertaining to the mathematical processes.ASSESSMENT AND EVALUATION OF STUDENT ACHIEVEMENT25


CriteriaWithin each category in the achievement chart, criteria are provided that are subsets ofthe knowledge and skills that define each category. For example, in Knowledge andUnderstanding, the criteria are “knowledge of content (e.g., facts, terms, procedural skills,use of tools)” and “understanding of mathematical concepts”. The criteria identify theaspects of student performance that are assessed and/or evaluated, and serve as guidesto what to look for.DescriptorsA “descriptor” indicates the characteristic of the student’s performance, with respect to aparticular criterion, on which assessment or evaluation is focused. In the achievementchart, effectiveness is the descriptor used for each criterion in the Thinking, Communication,and Application categories. What constitutes effectiveness in any given performancetask will vary with the particular criterion being considered. Assessment of effectivenessmay therefore focus on a quality such as appropriateness, clarity, accuracy, precision, logic,relevance, significance, fluency, flexibility, depth, or breadth, as appropriate for the particularcriterion. For example, in the Thinking category, assessment of effectiveness mightfocus on the degree of relevance or depth apparent in an analysis; in the Communicationcategory, on clarity of expression or logical organization of information and ideas; or in theApplication category, on appropriateness or breadth in the making of connections. Similarly,in the Knowledge and Understanding category, assessment of knowledge might focus onaccuracy, and assessment of understanding might focus on the depth of an explanation.Descriptors help teachers to focus their assessment and evaluation on specific knowledgeand skills for each category and criterion, and help students to better understand exactlywhat is being assessed and evaluated.QualifiersA specific “qualifier” is used to define each of the four levels of achievement – that is,limited for level 1, some for level 2, considerable for level 3, and a high degree or thoroughfor level 4. A qualifier is used along with a descriptor to produce a description of performanceat a particular level. For example, the description of a student’s performance atlevel 3 with respect to the first criterion in the Thinking category would be: “the studentuses planning skills with considerable effectiveness”.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsThe descriptions of the levels of achievement given in the chart should be used to identifythe level at which the student has achieved the expectations. In all of their courses, studentsshould be provided with numerous and varied opportunities to demonstrate the fullextent of their achievement of the curriculum expectations, across all four categories ofknowledge and skills.EVALUATION AND REPORTING OF STUDENT ACHIEVEMENTStudent achievement must be communicated formally to students and parents by meansof the Provincial Report Card, Grades 9–12. The report card provides a record of thestudent’s achievement of the curriculum expectations in every course, at particular pointsin the school year or semester, in the form of a percentage grade. The percentage graderepresents the quality of the student’s overall achievement of the expectations for the26


course and reflects the corresponding level of achievement as described in the achievementchart for the discipline.A final grade is recorded for every course, and a credit is granted and recorded for everycourse in which the student’s grade is 50% or higher. The final grade for each course inGrades 9–12 will be determined as follows:Seventy per cent of the grade will be based on evaluations conducted throughoutthe course. This portion of the grade should reflect the student’s most consistentlevel of achievement throughout the course, although special consideration shouldbe given to more recent evidence of achievement.Thirty per cent of the grade will be based on a final evaluation in the form of anexamination, performance, essay, and/or other method of evaluation suitable to thecourse content and administered towards the end of the course.REPORTING ON DEMONSTRATED LEARNING SKILLSThe report card provides a record of the learning skills demonstrated by the student inevery course, in the following five categories: Works Independently, Teamwork, Organization,Work Habits, and Initiative. The learning skills are evaluated using a four-point scale(E-Excellent, G-Good, S-Satisfactory, N-Needs Improvement). The separate evaluationand reporting of the learning skills in these five areas reflect their critical role in students’achievement of the curriculum expectations. To the extent possible, the evaluation oflearning skills, apart from any that may be included as part of a curriculum expectationin a course, should not be considered in the determination of percentage grades.ASSESSMENT AND EVALUATION OF STUDENT ACHIEVEMENT27


ACHIEVEMENT CHART: MATHEMATICS, GRADES 9–12Categories 50−59%(Level 1)60−69%(Level 2)70−79%(Level 3)80−100%(Level 4)Knowledge and Understanding – Subject-specific content acquired in each course (knowledge), andthe comprehension of its meaning and significance (understanding)The student:Knowledge of content(e.g., facts, terms, proceduralskills, use of tools)demonstrateslimited knowledgeof contentdemonstratessome knowledgeof contentdemonstratesconsiderableknowledge ofcontentdemonstratesthorough knowledgeof contentUnderstanding ofmathematical conceptsdemonstrateslimited understandingofconceptsdemonstratessome understandingofconceptsdemonstratesconsiderableunderstandingof conceptsdemonstratesthorough understandingofconceptsThinking – The use of critical and creative thinking skills and/or processes*The student:Use of planning skills− understanding theproblem (e.g., formulatingand interpreting theproblem, makingconjectures)uses planningskills with limitedeffectivenessuses planningskills with someeffectivenessuses planningskills withconsiderableeffectivenessuses planningskills with ahigh degree ofeffectiveness− making a plan for solvingthe problemTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics28Use of processing skills− carrying out a plan (e.g.,collecting data, questioning,testing, revising,modelling, solving, inferring,forming conclusions)− looking back at thesolution (e.g., evaluatingreasonableness, makingconvincing arguments,reasoning, justifying,proving, reflecting)Use of critical/creativethinking processes(e.g., problem solving,inquiry)uses processingskills with limitedeffectivenessuses critical/creative thinkingprocesseswith limitedeffectivenessuses processingskills with someeffectivenessuses critical/creative thinkingprocesseswith someeffectivenessuses processingskills withconsiderableeffectivenessuses critical/creative thinkingprocesses withconsiderableeffectivenessuses processingskills with ahigh degree ofeffectivenessuses critical/creative thinkingprocesses with ahigh degree ofeffectiveness* The processing skills and critical/creative thinking processes in the Thinking category include some but not all aspects of the mathematicalprocesses described on pages 17−22 of this document. Some aspects of the mathematical processes relate to the other categories of theachievement chart.


Categories 50−59%(Level 1)60−69%(Level 2)70−79%(Level 3)80−100%(Level 4)Communication – The conveying of meaning through various formsThe student:Expression and organizationof ideas and mathematicalthinking (e.g.,clarity of expression, logicalorganization), using oral,visual, and written forms(e.g., pictorial, graphic,dynamic, numeric, algebraicforms; concrete materials)expresses andorganizes mathematicalthinkingwith limitedeffectivenessexpresses andorganizes mathematicalthinkingwith someeffectivenessexpresses andorganizes mathematicalthinkingwith considerableeffectivenessexpresses andorganizes mathematicalthinkingwith a highdegree of effectivenessCommunication fordifferent audiences(e.g., peers, teachers) andpurposes (e.g., to presentdata, justify a solution,express a mathematicalargument) in oral, visual,and written formscommunicates fordifferent audiencesand purposeswith limited effectivenesscommunicates fordifferent audiencesand purposeswith someeffectivenesscommunicates fordifferent audiencesand purposeswith considerableeffectivenesscommunicates fordifferent audiencesand purposeswith a highdegree ofeffectivenessUse of conventions,vocabulary, and terminologyof the discipline (e.g.,terms, symbols) in oral,visual, and written formsuses conventions,vocabulary, andterminology ofthe disciplinewith limitedeffectivenessuses conventions,vocabulary, andterminology ofthe disciplinewith someeffectivenessuses conventions,vocabulary, andterminology ofthe disciplinewith considerableeffectivenessuses conventions,vocabulary, andterminology ofthe discipline witha high degree ofeffectivenessApplication – The use of knowledge and skills to make connections within and between various contextsThe student:Application of knowledgeand skills in familiar contextsTransfer of knowledgeand skills to new contextsMaking connections withinand between various contexts(e.g., connectionsbetween concepts, representations,and forms withinmathematics; connectionsinvolving use of prior knowledgeand experience; connectionsbetween mathematics,other disciplines,and the real world)applies knowledgeand skills in familiarcontexts with limitedeffectivenesstransfers knowledgeand skillsto new contextswith limitedeffectivenessmakes connectionswithin and betweenvarious contextswith limitedeffectivenessapplies knowledgeand skills in familiarcontexts with someeffectivenesstransfers knowledgeand skillsto new contextswith someeffectivenessmakes connectionswithin and betweenvarious contextswith someeffectivenessapplies knowledgeand skills in familiarcontexts withconsiderableeffectivenesstransfers knowledgeand skillsto new contextswith considerableeffectivenessmakes connectionswithin and betweenvarious contextswith considerableeffectivenessapplies knowledgeand skills in familiarcontexts with ahigh degree ofeffectivenesstransfers knowledgeand skillsto new contextswith a high degreeof effectivenessmakes connectionswithin and betweenvarious contextswith a high degreeof effectivenessASSESSMENT AND EVALUATION OF STUDENT ACHIEVEMENTNote: A student whose achievement is below 50% at the end of a course will not obtain a credit for the course.29


SOMECONSIDERATIONS FORPROGRAM PLANNINGIN MATHEMATICSTeachers who are planning a program in mathematics must take into account considerationsin a number of important areas, including those discussed below.INSTRUCTIONAL APPROACHESTo make new learning more accessible to students, teachers build new learning upon theknowledge and skills students have acquired in previous years – in other words, theyhelp activate prior knowledge. It is important to assess where students are in their mathematicalgrowth and to bring them forward in their learning.In order to apply their knowledge effectively and to continue to learn, students must havea solid conceptual foundation in mathematics. Successful classroom practices engagestudents in activities that require higher-order thinking, with an emphasis on problemsolving. 6 Learning experienced in the primary, junior, and intermediate divisions shouldhave provided students with a good grounding in the investigative approach to learningnew mathematical concepts, including inquiry models of problem solving, and thisapproach continues to be important in the senior mathematics program.Students in a mathematics class typically demonstrate diversity in the ways they learnbest. It is important, therefore, that students have opportunities to learn in a variety ofways – individually, cooperatively, independently, with teacher direction, through investigationinvolving hands-on experience, and through examples followed by practice. Inmathematics, students are required to learn concepts, acquire procedures and skills, andapply processes with the aid of the instructional and learning strategies best suited to theparticular type of learning.6. See the resource document Targeted Implementation & Planning Supports for Revised Mathematics (TIPS4RM):Grade 7, 8, 9 Applied and 10 Applied (Toronto: Queen’s Printer for Ontario, 2005) for helpful informationabout problem solving.


The approaches and strategies used in the classroom to help students meet the expectationsof this curriculum will vary according to the object of the learning and the needs ofthe students. For example, even at the secondary level, manipulatives can be importanttools for supporting the effective learning of mathematics. These concrete learning tools,such as connecting cubes, measurement tools, algebra tiles, and number cubes, invite studentsto explore and represent abstract mathematical ideas in varied, concrete, tactile, andvisually rich ways. 7 Other representations, including graphical and algebraic representations,are also a valuable aid to teachers. By analysing students’ representations of mathematicalconcepts and listening carefully to their reasoning, teachers can gain usefulinsights into students’ thinking and provide supports to help enhance their thinking.All learning, especially new learning, should be embedded in well-chosen contexts forlearning – that is, contexts that are broad enough to allow students to investigate initialunderstandings, identify and develop relevant supporting skills, and gain experiencewith varied and interesting applications of the new knowledge. Such rich contexts forlearning open the door for students to see the “big ideas” of mathematics – that is, themajor underlying principles or relationships that will enable and encourage students toreason mathematically throughout their lives.Promoting Positive Attitudes Towards Learning MathematicsStudents’ attitudes have a significant effect on how students approach problem solvingand how well they succeed in mathematics. Students who enjoy mathematics tend toperform well in their mathematics course work and are more likely to enrol in the moreadvanced mathematics courses.Students develop positive attitudes when they are engaged in making mathematicalconjectures, when they experience breakthroughs as they solve problems, when they seeconnections between important ideas, and when they observe an enthusiasm for mathematicson the part of their teachers. 8 With a positive attitude towards mathematics, studentsare able to make more sense of the mathematics they are working on, and to viewthemselves as effective learners of mathematics. They are also more likely to perceivemathematics as both useful and worthwhile, and to develop the belief that steady effortin learning mathematics pays off.It is common for people to feel inadequate or anxious when they cannot solve problemsquickly and easily, or in the right way. To gain confidence, students need to recognizethat, for some mathematics problems, there may be several ways to arrive at a solution.They also need to understand that problem solving of almost any kind often requires aconsiderable expenditure of time and energy and a good deal of perseverance. To counteractthe frustration they may feel when they are not making progress towards solvinga problem, they need to believe that they are capable of finding solutions. Teachers canencourage students to develop a willingness to persist, to investigate, to reason, to explorealternative solutions, to view challenges as opportunities to extend their learning, and totake the risks necessary to become successful problem solvers. They can help studentsdevelop confidence and reduce anxiety and frustration by providing them with problemsthat are challenging but not beyond their ability to solve. Problems at a developmentallyappropriate level help students to learn while establishing a norm of perseverance forsuccessful problem solving.7. A list of manipulatives appropriate for use in intermediate and senior mathematics classrooms is provided inLeading Math Success, pp. 48–49.8. Leading Math Success, p. 42SOME CONSIDERATIONS FOR PROGRAM PLANNING IN MATHEMATICS31


Collaborative learning enhances students’ understanding of mathematics. Working cooperativelyin groups reduces isolation and provides students with opportunities to shareideas and communicate their thinking in a supportive environment as they work togethertowards a common goal. Communication and the connections among ideas that emergeas students interact with one another enhance the quality of student learning. 9PLANNING MATHEMATICS PROGRAMS FOR STUDENTS WITH SPECIALEDUCATION NEEDSClassroom teachers are the key educators of students who have special education needs.They have a responsibility to help all students learn, and they work collaboratively withspecial education teachers, where appropriate, to achieve this goal. Special EducationTransformation: The Report of the Co-Chairs with the Recommendations of the Working Tableon Special Education, 2006 endorses a set of beliefs that should guide program planningfor students with special education needs in all disciplines. Those beliefs are as follows:All students can succeed.Universal design and differentiated instruction are effective and interconnectedmeans of meeting the learning or productivity needs of any group of students.Successful instructional practices are founded on evidence-based research,tempered by experience.Classroom teachers are key educators for a student’s literacy and numeracydevelopment.Each student has his or her own unique patterns of learning.Classroom teachers need the support of the larger community to create a learningenvironment that supports students with special education needs.Fairness is not sameness.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics32In any given classroom, students may demonstrate a wide range of learning styles andneeds. Teachers plan programs that recognize this diversity and give students performancetasks that respect their particular abilities so that all students can derive the greatestpossible benefit from the teaching and learning process. The use of flexible groupings forinstruction and the provision of ongoing assessment are important elements of programsthat accommodate a diversity of learning needs.In planning mathematics courses for students with special education needs, teachersshould begin by examining the current achievement level of the individual student, thestrengths and learning needs of the student, and the knowledge and skills that all studentsare expected to demonstrate at the end of the course in order to determine which ofthe following options is appropriate for the student:no accommodations 10 or modifications; oraccommodations only; ormodified expectations, with the possibility of accommodations; oralternative expectations, which are not derived from the curriculum expectationsfor a course and which constitute alternative programs and/or courses.9. Leading Math Success, p. 4210. “Accommodations” refers to individualized teaching and assessment strategies, human supports, and/orindividualized equipment.


If the student requires either accommodations or modified expectations, or both, the relevantinformation, as described in the following paragraphs, must be recorded in his orher Individual Education Plan (IEP). More detailed information about planning programsfor students with special education needs, including students who require alternativeprograms and/or courses, can be found in The Individual Education Plan (IEP): A ResourceGuide, 2004 (referred to hereafter as the IEP Resource Guide, 2004). For a detailed discussionof the ministry’s requirements for IEPs, see Individual Education Plans: Standards forDevelopment, Program Planning, and Implementation, 2000 (referred to hereafter as IEPStandards, 2000). (Both documents are available at http://www.edu.gov.on.ca.)Students Requiring Accommodations OnlySome students are able, with certain accommodations, to participate in the regular coursecurriculum and to demonstrate learning independently. Accommodations allow access tothe course without any changes to the knowledge and skills the student is expected todemonstrate. The accommodations required to facilitate the student’s learning must beidentified in his or her IEP (see IEP Standards, 2000, page 11). A student’s IEP is likely toreflect the same accommodations for many, or all, subjects or courses.Providing accommodations to students with special education needs should be the firstoption considered in program planning. Instruction based on principles of universaldesign and differentiated instruction focuses on the provision of accommodations to meetthe diverse needs of learners.There are three types of accommodations:Instructional accommodations are changes in teaching strategies, including styles ofpresentation, methods of organization, or use of technology and multimedia.Environmental accommodations are changes that the student may require in the classroomand/or school environment, such as preferential seating or special lighting.Assessment accommodations are changes in assessment procedures that enable thestudent to demonstrate his or her learning, such as allowing additional time tocomplete tests or assignments or permitting oral responses to test questions (seepage 29 of the IEP Resource Guide, 2004, for more examples).If a student requires “accommodations only” in mathematics courses, assessment andevaluation of his or her achievement will be based on the appropriate course curriculumexpectations and the achievement levels outlined in this document. The IEP box on thestudent’s Provincial Report Card will not be checked, and no information on the provisionof accommodations will be included.Students Requiring Modified ExpectationsSome students will require modified expectations, which differ from the regular courseexpectations. For most students, modified expectations will be based on the regularcourse curriculum, with changes in the number and/or complexity of the expectations.Modified expectations represent specific, realistic, observable, and measurable achievementsand describe specific knowledge and/or skills that the student can demonstrateindependently, given the appropriate assessment accommodations.SOME CONSIDERATIONS FOR PROGRAM PLANNING IN MATHEMATICS33


It is important to monitor, and to reflect clearly in the student’s IEP, the extent to whichexpectations have been modified. As noted in Section 7.12 of the ministry’s policy documentOntario Secondary Schools, Grades 9 to 12: Program and Diploma Requirements, 1999,the principal will determine whether achievement of the modified expectations constitutessuccessful completion of the course, and will decide whether the student is eligibleto receive a credit for the course. This decision must be communicated to the parents andthe student.When a student is expected to achieve most of the curriculum expectations for the course,the modified expectations should identify how the required knowledge and skills differ fromthose identified in the course expectations. When modifications are so extensive that achievementof the learning expectations (knowledge, skills, and performance tasks) is not likelyto result in a credit, the expectations should specify the precise requirements or tasks onwhich the student’s performance will be evaluated and which will be used to generate thecourse mark recorded on the Provincial Report Card.Modified expectations indicate the knowledge and/or skills the student is expected todemonstrate and have assessed in each reporting period (IEP Standards, 2000, pages 10and 11). The student’s learning expectations must be reviewed in relation to the student’sprogress at least once every reporting period, and must be updated as necessary (IEPStandards, 2000, page 11).If a student requires modified expectations in mathematics courses, assessment and evaluationof his or her achievement will be based on the learning expectations identified inthe IEP and on the achievement levels outlined in this document. If some of the student’slearning expectations for a course are modified but the student is working towards acredit for the course, it is sufficient simply to check the IEP box on the Provincial ReportCard. If, however, the student’s learning expectations are modified to such an extent thatthe principal deems that a credit will not be granted for the course, the IEP box must bechecked and the appropriate statement from the Guide to the Provincial Report Card,Grades 9–12, 1999 (page 8) must be inserted. The teacher’s comments should include relevantinformation on the student’s demonstrated learning of the modified expectations,as well as next steps for the student’s learning in the course.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsPROGRAM CONSIDERATIONS FOR ENGLISH LANGUAGE LEARNERSYoung people whose first language is not English enter Ontario secondary schools withdiverse linguistic and cultural backgrounds. Some English language learners may haveexperience of highly sophisticated educational systems, while others may have comefrom regions where access to formal schooling was limited. All of these students bring arich array of background knowledge and experience to the classroom, and all teachersmust share in the responsibility for their English-language development.Teachers of mathematics must incorporate appropriate adaptations and strategies forinstruction and assessment to facilitate the success of the English language learners intheir classrooms. These adaptations and strategies include:modification of some or all of the course expectations so that they are challengingbut attainable for the learner at his or her present level of English proficiency, giventhe necessary support from the teacher;34


use of a variety of instructional strategies (e.g., extensive use of visual cues, scaffolding,manipulatives, pictures, diagrams, graphic organizers; attention to clarityof instructions);modelling of preferred ways of working in mathematics; previewing of textbooks;pre-teaching of key vocabulary; peer tutoring; strategic use of students’ first languages);use of a variety of learning resources (e.g., visual material, simplified text, bilingualdictionaries, materials that reflect cultural diversity);use of assessment accommodations (e.g., granting of extra time; simplification oflanguage used in problems and instructions; use of oral interviews, learning logs,portfolios, demonstrations, visual representations, and tasks requiring completionof graphic organizers or cloze sentences instead of tasks that depend heavily onproficiency in English).When learning expectations in any course are modified for English language learners(whether or not the students are enrolled in an ESL or ELD course), this must be clearlyindicated on the student’s report card.Although the degree of program adaptation required will decrease over time, studentswho are no longer receiving ESL or ELD support may still need some program adaptationsto be successful.For further information on supporting English language learners, refer to The OntarioCurriculum, Grades 9 to 12: English As a Second Language and English Literacy Development,2007 and the resource guide Many Roots Many Voices: Supporting English LanguageLearners in Every Classroom (Ministry of Education, 2005).ANTIDISCRIMINATION EDUCATION IN MATHEMATICSTo ensure that all students in the province have an equal opportunity to achieve their fullpotential, the curriculum must be free from bias, and all students must be provided witha safe and secure environment, characterized by respect for others, that allows them toparticipate fully and responsibly in the educational experience.Learning activities and resources used to implement the curriculum should be inclusivein nature, reflecting the range of experiences of students with varying backgrounds, abilities,interests, and learning styles. They should enable students to become more sensitiveto the diverse cultures and perceptions of others, including Aboriginal peoples. By discussingaspects of the history of mathematics, teachers can help make students aware ofthe various cultural groups that have contributed to the evolution of mathematics overthe centuries. Finally, students need to recognize that ordinary people use mathematics ina variety of everyday contexts, both at work and in their daily lives.Connecting mathematical ideas to real-world situations through learning activities canenhance students’ appreciation of the role of mathematics in human affairs, in areasincluding health, science, and the environment. Students can be made aware of the use ofmathematics in contexts such as sampling and surveying and the use of statistics toanalyse trends. Recognizing the importance of mathematics in such areas helps motivatestudents to learn and also provides a foundation for informed, responsible citizenship.SOME CONSIDERATIONS FOR PROGRAM PLANNING IN MATHEMATICS35


Teachers should have high expectations for all students. To achieve their mathematicalpotential, however, different students may need different kinds of support. Some boys,for example, may need additional support in developing their literacy skills in order tocomplete mathematical tasks effectively. For some girls, additional encouragement toenvision themselves in careers involving mathematics may be beneficial. For example,teachers might consider providing strong role models in the form of female guest speakerswho are mathematicians or who use mathematics in their careers.LITERACY AND INQUIRY/RESEARCH SKILLSLiteracy skills can play an important role in student success in mathematics courses. Manyof the activities and tasks students undertake in mathematics courses involve the use ofwritten, oral, and visual communication skills. For example, students use language torecord their observations, to explain their reasoning when solving problems, to describetheir inquiries in both informal and formal contexts, and to justify their results in smallgroupconversations, oral presentations, and written reports. The language of mathematicsincludes special terminology. The study of mathematics consequently encourages studentsto use language with greater care and precision and enhances their ability to communicateeffectively.The Ministry of Education has facilitated the development of materials to support literacyinstruction across the curriculum. Helpful advice for integrating literacy instruction inmathematics courses may be found in the following resource documents:Think Literacy: Cross-Curricular Approaches, Grades 7–12, 2003Think Literacy: Cross-Curricular Approaches, Grades 7–12 – Mathematics: Subject-Specific Examples, Grades 10–12, 2005THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsIn all courses in mathematics, students will develop their ability to ask questions and toplan investigations to answer those questions and to solve related problems. Studentsneed to learn a variety of research methods and inquiry approaches in order to carry outthese investigations and to solve problems, and they need to be able to select the methodsthat are most appropriate for a particular inquiry. Students learn how to locate relevantinformation from a variety of sources, such as statistical databases, newspapers, andreports. As they advance through the grades, students will be expected to use suchsources with increasing sophistication. They will also be expected to distinguish betweenprimary and secondary sources, to determine their validity and relevance, and to usethem in appropriate ways.36


THE ROLE OF INFORMATION AND COMMUNICATION TECHNOLOGYIN MATHEMATICSInformation and communication technologies (ICT) provide a range of tools that cansignificantly extend and enrich teachers’ instructional strategies and support students’learning in mathematics. Teachers can use ICT tools and resources both for whole-classinstruction and to design programs that meet diverse student needs. Technology can helpto reduce the time spent on routine mathematical tasks, allowing students to devotemore of their efforts to thinking and concept development. Useful ICT tools includesimulations, multimedia resources, databases, sites that give access to large amounts ofstatistical data, and computer-assisted learning modules.Applications such as databases, spreadsheets, dynamic geometry software, dynamic statisticalsoftware, graphing software, computer algebra systems (CAS), word-processingsoftware, and presentation software can be used to support various methods of inquiryin mathematics. Technology also makes possible simulations of complex systems thatcan be useful for problem-solving purposes or when field studies on a particular topicare not feasible.Information and communications technologies can be used in the classroom to connectstudents to other schools, at home and abroad, and to bring the global community intothe local classroom.Although the Internet is a powerful electronic learning tool, there are potential risksattached to its use. All students must be made aware of issues of Internet privacy, safety,and responsible use, as well as of the ways in which this technology is being abused –for example, when it is used to promote hatred.Teachers, too, will find the various ICT tools useful in their teaching practice, both forwhole class instruction and for the design of curriculum units that contain variedapproaches to learning to meet diverse student needs.CAREER EDUCATION IN MATHEMATICSTeachers can promote students’ awareness of careers involving mathematics by exploringapplications of concepts and providing opportunities for career-related project work. Suchactivities allow students the opportunity to investigate mathematics-related careers compatiblewith their interests, aspirations, and abilities.Students should be made aware that mathematical literacy and problem solving are valuableassets in an ever-widening range of jobs and careers in today’s society. The knowledgeand skills students acquire in mathematics courses are useful in fields such as science,business, engineering, and computer studies; in the hospitality, recreation, and tourismindustries; and in the technical trades.SOME CONSIDERATIONS FOR PROGRAM PLANNING IN MATHEMATICS37


THE ONTARIO SKILLS PASSPORT AND ESSENTIAL SKILLSTeachers planning programs in mathematics need to be aware of the purpose and benefitsof the Ontario Skills Passport (OSP).The OSP is a bilingual web-based resource that enhancesthe relevancy of classroom learning for students and strengthens school-work connections.The OSP provides clear descriptions of Essential Skills such as Reading Text, Writing,Computer Use, Measurement and Calculation, and Problem Solving and includes anextensive database of occupation-specific workplace tasks that illustrate how workersuse these skills on the job. The Essential Skills are transferable, in that they are used invirtually all occupations. The OSP also includes descriptions of important work habits,such as working safely, being reliable, and providing excellent customer service. The OSPis designed to help employers assess and record students’ demonstration of these skillsand work habits during their cooperative education placements. Students can use theOSP to identify the skills and work habits they already have, plan further skill development,and show employers what they can do.The skills described in the OSP are the Essential Skills that the Government of Canadaand other national and international agencies have identified and validated, throughextensive research, as the skills needed for work, learning, and life. These Essential Skillsprovide the foundation for learning all other skills and enable people to evolve with theirjobs and adapt to workplace change. For further information on the OSP and the EssentialSkills, visit: http://skills.edu.gov.on.ca.COOPERATIVE EDUCATION AND OTHER FORMS OF EXPERIENTIAL LEARNINGCooperative education and other workplace experiences, such as job shadowing, fieldtrips, and work experience, enable students to apply the skills they have developed in theclassroom to real-life activities. Cooperative education and other workplace experiencesalso help to broaden students’ knowledge of employment opportunities in a wide rangeof fields, including science and technology, research in the social sciences and humanities,and many forms of business administration. In addition, students develop their understandingof workplace practices, certifications, and the nature of employer-employeerelationships.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsCooperative education teachers can support students taking mathematics courses bymaintaining links with community-based businesses and organizations, and with collegesand universities, to ensure students studying mathematics have access to hands-on experiencesthat will reinforce the knowledge and skills they have gained in school. Teachers ofmathematics can support their students’ learning by providing opportunities for experientiallearning that will reinforce the knowledge and skills they have gained in school.Health and safety issues must be addressed when learning involves cooperative educationand other workplace experiences. Teachers who provide support for students inworkplace learning placements need to assess placements for safety and ensure studentsunderstand the importance of issues relating to health and safety in the workplace. Beforetaking part in workplace learning experiences, students must acquire the knowledge andskills needed for safe participation. Students must understand their rights to privacy andconfidentiality as outlined in the Freedom of Information and Protection of Privacy Act.They have the right to function in an environment free from abuse and harassment, and38


they need to be aware of harassment and abuse issues in establishing boundaries for theirown personal safety. They should be informed about school and community resourcesand school policies and reporting procedures with regard to all forms of abuse andharassment.Policy/Program Memorandum No. 76A, “Workplace Safety and Insurance Coverage forStudents in Work Education Programs” (September 2000), outlines procedures for ensuringthe provision of Health and Safety Insurance Board coverage for students who areat least 14 years of age and are on placements of more than one day. (A one-day jobshadowingor job-twinning experience is treated as a field trip.) Teachers should also beaware of the minimum age requirements outlined in the Occupational Health and SafetyAct for persons to be in or to be working in specific workplace settings.All cooperative education and other workplace experiences will be provided in accordancewith the ministry’s policy document entitled Cooperative Education and Other Forms ofExperiential Learning: Policies and Procedures for Ontario Secondary Schools, 2000.PLANNING PROGRAM PATHWAYS AND PROGRAMS LEADING TO A SPECIALISTHIGH-SKILLS MAJORMathematics courses are well suited for inclusion in programs leading to a SpecialistHigh-Skills Major (SHSM) or in programs designed to provide pathways to particularapprenticeship or workplace destinations. In an SHSM program, mathematics courses canbe bundled with other courses to provide the academic knowledge and skills important toparticular industry sectors and required for success in the workplace and postsecondaryeducation, including apprenticeship. Mathematics courses may also be combined withcooperative education credits to provide the workplace experience required for SHSMprograms and for various program pathways to apprenticeship and workplace destinations.(SHSM programs would also include sector-specific learning opportunities offeredby employers, skills-training centres, colleges, and community organizations.)HEALTH AND SAFETY IN MATHEMATICSAlthough health and safety issues are not normally associated with mathematics, theymay be important when learning involves fieldwork or investigations based on experimentation.Out-of-school fieldwork can provide an exciting and authentic dimension tostudents’ learning experiences. It also takes the teacher and students out of the predictableclassroom environment and into unfamiliar settings. Teachers must preview and planactivities and expeditions carefully to protect students’ health and safety.SOME CONSIDERATIONS FOR PROGRAM PLANNING IN MATHEMATICS39


COURSES


Functions, Grade 11University PreparationMCR3UThis course introduces the mathematical concept of the function by extending students’experiences with linear and quadratic relations. Students will investigate properties ofdiscrete and continuous functions, including trigonometric and exponential functions;represent functions numerically, algebraically, and graphically; solve problems involvingapplications of functions; investigate inverse functions; and develop facility in determiningequivalent algebraic expressions. Students will reason mathematically and communicatetheir thinking as they solve multi-step problems.Prerequisite: Principles of Mathematics, Grade 10, Academic43


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.44


A. CHARACTERISTICS OF FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of functions, their representations, and their inverses, and makeconnections between the algebraic and graphical representations of functions using transformations;2. determine the zeros and the maximum or minimum of a quadratic function, and solve problemsinvolving quadratic functions, including problems arising from real-world applications;3. demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, andrational expressions.FunctionsSPECIFIC EXPECTATIONS1. Representing FunctionsBy the end of this course, students will:1.1 explain the meaning of the term function, anddistinguish a function from a relation that isnot a function, through investigation of linearand quadratic relations using a variety of representations(i.e., tables of values, mapping diagrams,graphs, function machines, equations)and strategies (e.g., identifying a one-to-oneor many-to-one mapping; using the verticallinetest)Sample problem: Investigate, using numericand graphical representations, whether the2relation x = y is a function, and justify yourreasoning.1.2 represent linear and quadratic functions usingfunction notation, given their equations, tablesof values, or graphs, and substitute into andevaluate functions [e.g., evaluate f (1 ) , given22f(x) = 2x + 3x – 1]1.3 explain the meanings of the terms domainand range, through investigation using numeric,graphical, and algebraic representations of2the functions f(x) = x, f(x) = x , f(x) = √x,1and f(x) = ; describe the domain and range ofx2a function appropriately (e.g., for y = x + 1,the domain is the set of real numbers, and theSample problem: A quadratic function representsthe relationship between the heightrange is y ≥ 1); and explain any restrictions onthe domain and range in contexts arising fromof a ball and the time elapsed since the ballreal-world applicationswas thrown. What physical factors willSample problem: A quadratic function representsthe relationship between the heightof a ball and the time elapsed since the ballwas thrown. What physical factors willrestrict the domain and range of the quadraticfunction?1.4 relate the process of determining the inverseof a function to their understanding ofreverse processes (e.g., applying inverseoperations)1.5 determine the numeric or graphical representationof the inverse of a linear or quadraticfunction, given the numeric, graphical, oralgebraic representation of the function, andmake connections, through investigationusing a variety of tools (e.g., graphing technology,Mira, tracing paper), between thegraph of a function and the graph of itsinverse (e.g., the graph of the inverse is thereflection of the graph of the function in theline y = x)Sample problem: Given a graph and a table ofvalues representing population over time,produce a table of values for the inverse andgraph the inverse on a new set of axes.1.6 determine, through investigation, the relationshipbetween the domain and range of a functionand the domain and range of the inverserelation, and determine whether or not theinverse relation is a functionSample problem: Given the graph of f(x) = x ,graph the inverse relation. Compare the domainand range of the function with the domain2MCR3UCHARACTERISTICS OF FUNCTIONS45


Grade 11, University PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematicsand range of the inverse relation, and investigateconnections to the domain and range ofthe functions g(x) = √x and h(x) = –√x.1.7 determine, using function notation whenappropriate, the algebraic representation ofthe inverse of a linear or quadratic function,given the algebraic representation of the2function [e.g., f(x) = (x – 2) – 5], and makeconnections, through investigation using avariety of tools (e.g., graphing technology,Mira, tracing paper), between the algebraicrepresentations of a function and its inverse(e.g., the inverse of a linear function involvesapplying the inverse operations in the reverseorder)Sample problem: Given the equations ofseveral linear functions, graph the functionsand their inverses, determine the equationsof the inverses, and look for patterns thatconnect the equation of each linear functionwith the equation of the inverse.1.8 determine, through investigation usingtechnology, the roles of the parametersa, k, d, and c in functions of the formy = af(k(x – d)) + c, and describe these rolesin terms of transformations on the graphs2of f(x) = x, f(x) = x , f(x) = √x, and1f(x) = (i.e., translations; reflections in thexaxes; vertical and horizontal stretches andcompressions to and from the x- and y-axes)Sample problem: Investigate the graph2f(x) = 3(x – d) + 5 for various values of d,using technology, and describe the effects ofchanging d in terms of a transformation.1.9 sketch graphs of y = af(k(x – d)) + cby applying one or more transformations2to the graphs of f(x) = x, f(x) = x , f(x) = √x,1and f(x) = , and state the domain andxrange of the transformed functionsSample problem: Transform the graph of f(x)to sketch g(x), and state the domain andrange of each function, for the following:1f(x) = √x, g(x) = √x – 4; f(x) = ,x1g(x) = – .x + 12. Solving Problems InvolvingQuadratic FunctionsBy the end of this course, students will:2.1 determine the number of zeros (i.e.,x-intercepts) of a quadratic function, usinga variety of strategies (e.g., inspecting graphs;factoring; calculating the discriminant)Sample problem: Investigate, using graphingtechnology and algebraic techniques, thetransformations that affect the number ofzeros for a given quadratic function.2.2 determine the maximum or minimum valueof a quadratic function whose equation is2given in the form f(x) = ax + bx + c, usingan algebraic method (e.g., completing thesquare; factoring to determine the zeros andaveraging the zeros)Sample problem: Explain how partially2factoring f(x) = 3x – 6x + 5 into the formf(x) = 3x(x – 2) + 5 helps you determine theminimum of the function.2.3 solve problems involving quadratic functionsarising from real-world applications andrepresented using function notationSample problem: The profit, P(x), of a videocompany, in thousands of dollars, is given by2P(x) = – 5x + 550x – 5000, where x is theamount spent on advertising, in thousandsof dollars. Determine the maximum profitthat the company can make, and the amountsspent on advertising that will result in aprofit and that will result in a profit of atleast $4 000 000.2.4 determine, through investigation, the transformationalrelationship among the family ofquadratic functions that have the same zeros,and determine the algebraic representation ofa quadratic function, given the real roots ofthe corresponding quadratic equation and apoint on the functionSample problem: Determine the equation ofthe quadratic function that passes through(2, 5) if the roots of the correspondingquadratic equation are 1 + √5 and 1 – √5.46


2.5 solve problems involving the intersection ofa linear function and a quadratic functiongraphically and algebraically (e.g., determinethe time when two identical cylindrical watertanks contain equal volumes of water, if onetank is being filled at a constant rate and theother is being emptied through a hole in thebottom)Sample problem: Determine, through investigation,the equations of the lines that have aslope of 2 and that intersect the quadraticfunction f(x) = x(6 – x) once; twice; never.3. Determining Equivalent AlgebraicExpressions*By the end of this course, students will:3.1 simplify polynomial expressions by adding,subtracting, and multiplyingSample problem: Write and simplify anexpression for the volume of a cube withedge length 2x + 1.3.2 verify, through investigation with andwithout technology, that √ab = √a x √b,a ≥ 0, b ≥ 0, and use this relationship tosimplify radicals (e.g., √24) and radicalexpressions obtained by adding, subtracting,and multiplying [e.g., (2 + √6)(3 – √12)]3.3 simplify rational expressions by adding,subtracting, multiplying, and dividing, andstate the restrictions on the variable valuesSample problem: Simplify2x 3– , and state the24x + 6x 2x + 3restrictions on the variable.3.4 determine if two given algebraic expressionsare equivalent (i.e., by simplifying; bysubstituting values)Sample problem: Determine if the expressions22x – 4x – 62and 8x – 2x(4x – 1) – 6 arex + 1equivalent.FunctionsMCR3U*The knowledge and skills described in the expectations in this section are to be introduced as needed, and applied andconsolidated, as appropriate, in solving problems throughout the course.CHARACTERISTICS OF FUNCTIONS47


B. EXPONENTIAL FUNCTIONSGrade 11, University PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsOVERALL EXPECTATIONSBy the end of this course, students will:1. evaluate powers with rational exponents, simplify expressions containing exponents, and describeproperties of exponential functions represented in a variety of ways;2. make connections between the numeric, graphical, and algebraic representations of exponentialfunctions;3. identify and represent exponential functions, and solve problems involving exponential functions,including problems arising from real-world applications.SPECIFIC EXPECTATIONS1. Representing Exponential FunctionsBy the end of this course, students will:1.1 graph, with and without technology, an exponentialrelation, given its equation in the formxy = a (a > 0, a ≠ 1), define this relation as thexfunction f(x) = a , and explain why it is afunction1.2 determine, through investigation using avariety of tools (e.g., calculator, paper andpencil, graphing technology) and strategies(e.g., patterning; finding values from a graph;interpreting the exponent laws), the value ofma power with a rational exponent (i.e., x n ,where x > 0 and m and n are integers)Sample problem: The exponent laws suggest1 1that 42x 42 1= 4 . What value would you1to 273? Explain your reasoning. Extend yourreasoning to make a generalization about the1meaning of x n, where x > 0 and n is a naturalnumber.1.3 simplify algebraic expressions containinginteger and rational exponents [e.g.,13 6 31assign to 42? What value would you assign1(x ) ÷ ( x 2 ) , (x y) 3 ], and evaluate numericexpressions containing integer and rationalexponents and rational bases1–3 3[e.g., 2 , (–6) , 42 120, 1.01 ]1.4 determine, through investigation, anddescribe key properties relating to domainand range, intercepts, increasing/decreasingintervals, and asymptotes (e.g., the domainis the set of real numbers; the range is theset of positive real numbers; the functioneither increases or decreases throughout itsdomain) for exponential functions representedin a variety of ways [e.g., tables of values,mapping diagrams, graphs, equations of thexform f(x) = a (a > 0, a ≠ 1), functionmachines]xxSample problem: Graph f(x) = 2,g(x) = 3,xand h(x) = 0.5 on the same set of axes. Makecomparisons between the graphs, and explainthe relationship between the y-intercepts.2. Connecting Graphs and Equationsof Exponential FunctionsBy the end of this course, students will:2.1 distinguish exponential functions from linearand quadratic functions by making comparisonsin a variety of ways (e.g., comparingrates of change using finite differences intables of values; identifying a constant ratio ina table of values; inspecting graphs; comparingequations)Sample problem: Explain in a variety of wayshow you can distinguish the exponentialxfunction f(x) = 2 from the quadratic function2f(x) = x and the linear function f(x) = 2x.48


2.2 determine, through investigation using technology,the roles of the parameters a, k, d, andc in functions of the form y = af(k(x – d)) + c,and describe these roles in terms of transformationson the graph of f(x) = a (a > 0, a ≠ 1)x(i.e., translations; reflections in the axes; verticaland horizontal stretches and compressionsto and from the x- and y-axes)Sample problem: Investigate the graph ofx – df(x) = 3 – 5 for various values of d,using technology, and describe the effectsof changing d in terms of a transformation.2.3 sketch graphs of y = af(k(x – d)) + c byapplying one or more transformationsxto the graph of f(x) = a (a > 0, a ≠ 1),and state the domain and range of thetransformed functionsSample problem: Transform the graph ofx– (xf(x) = 3 to sketch g(x) = 3 + 1)– 2, and statethe domain and range of each function.2.4 determine, through investigation using technology,that the equation of a given exponentialfunction can be expressed using different basesx2x[e.g., f(x) = 9 can be expressed as f(x) = 3 ],and explain the connections between theequivalent forms in a variety of ways (e.g.,comparing graphs; using transformations;using the exponent laws)2.5 represent an exponential function with anequation, given its graph or its propertiesSample problem: Write two equations to representthe same exponential function with ay-intercept of 5 and an asymptote at y = 3.Investigate whether other exponential functionshave the same properties. Use transformationsto explain your observations.Sample problem: Collect data and graph thecooling curve representing the relationshipbetween temperature and time for hot watercooling in a porcelain mug. Predict the shapeof the cooling curve when hot water cools inan insulated mug. Test your prediction.3.2 identify exponential functions, includingthose that arise from real-world applicationsinvolving growth and decay (e.g., radioactivedecay, population growth, cooling rates,pressure in a leaking tire), given variousrepresentations (i.e., tables of values, graphs,equations), and explain any restrictions thatthe context places on the domain and range(e.g., ambient temperature limits the rangefor a cooling curve)Sample problem: Using data from StatisticsCanada, investigate to determine if there wasa period of time over which the increase inCanada’s national debt could be modelledusing an exponential function.3.3 solve problems using given graphs orequations of exponential functions arisingfrom a variety of real-world applications(e.g., radioactive decay, population growth,height of a bouncing ball, compound interest)by interpreting the graphs or by substitutingvalues for the exponent into the equationsSample problem: The temperature of acooling liquid over time can be modelledby the exponential functionx1T(x) = 60 ( )30+ 20, where T(x) is the2temperature, in degrees Celsius, and x is theelapsed time, in minutes. Graph the functionand determine how long it takes for the temperatureto reach 28ºC.FunctionsMCR3U3. Solving Problems InvolvingExponential FunctionsBy the end of this course, students will:3.1 collect data that can be modelled as an exponentialfunction, through investigation withand without technology, from primary sources,using a variety of tools (e.g., concrete materialssuch as number cubes, coins; measurementtools such as electronic probes), or fromsecondary sources (e.g., websites such asStatistics Canada, E-STAT), and graphthe dataEXPONENTIAL FUNCTIONS49


C. DISCRETE FUNCTIONSTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 11, University PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of recursive sequences, represent recursive sequences in a variety ofways, and make connections to Pascal’s triangle;2. demonstrate an understanding of the relationships involved in arithmetic and geometric sequencesand series, and solve related problems;3. make connections between sequences, series, and financial applications, and solve problems involvingcompound interest and ordinary annuities.SPECIFIC EXPECTATIONS1. Representing SequencesBy the end of this course, students will:1.1 make connections between sequences anddiscrete functions, represent sequences usingfunction notation, and distinguish between adiscrete function and a continuous function[e.g., f(x) = 2x, where the domain is the set ofnatural numbers, is a discrete linear functionand its graph is a set of equally spaced points;f(x) = 2x, where the domain is the set of realnumbers, is a continuous linear function andits graph is a straight line]1.2 determine and describe (e.g., in words; usingflow charts) a recursive procedure for generatinga sequence, given the initial terms(e.g., 1, 3, 6, 10, 15, 21, …), and representsequences as discrete functions in a varietyof ways (e.g., tables of values, graphs)1.3 connect the formula for the nth term of asequence to the representation in functionnotation, and write terms of a sequence givenone of these representations or a recursionformula1.4 represent a sequence algebraically using arecursion formula, function notation, or theformula for the nth term [e.g., represent 2, 4,8, 16, 32, 64, … as t 1 = 2; t n = 2t n – 1 , asnn1 2 3f(n) = 2 , or as t n = 2 , or represent , , ,2 3 44 5 611, , , … as t 1 = ; t n = t n – 1 + ,5 6 72n(n + 1)nnas f(n) = , or as t n = , where nn + 1n + 1is a natural number], and describe the informationthat can be obtained by inspectingeach representation (e.g., function notationor the formula for the nth term may showthe type of function; a recursion formulashows the relationship between terms)Sample problem: Represent the sequence0, 3, 8, 15, 24, 35, … using a recursionformula, function notation, and the formulafor the nth term. Explain why this sequencecan be described as a discrete quadraticfunction. Explore how to identify a sequenceas a discrete quadratic function by inspectingthe recursion formula.1.5 determine, through investigation, recursivepatterns in the Fibonacci sequence, in relatedsequences, and in Pascal’s triangle, andrepresent the patterns in a variety of ways(e.g., tables of values, algebraic notation)1.6 determine, through investigation, anddescribe the relationship between Pascal’striangle and the expansion of binomials,and apply the relationship to expand binomialsraised to whole-number exponents4 5 6 2 5[e.g., (1 + x) ,(2x–1) , (2x – y), (x + 1) ]50


2. Investigating Arithmetic andGeometric Sequences and SeriesBy the end of this course, students will:2.1 identify sequences as arithmetic, geometric,or neither, given a numeric or algebraicrepresentation2.2 determine the formula for the generalterm of an arithmetic sequence [i.e.,t n = a + (n –1)d ] or geometric sequence(i.e., t n = ar n –1 ), through investigationusing a variety of tools (e.g., linking cubes,algebra tiles, diagrams, calculators) andstrategies (e.g., patterning; connecting thesteps in a numerical example to the steps inthe algebraic development), and apply theformula to calculate any term in a sequence2.3 determine the formula for the sum of anarithmetic or geometric series, through investigationusing a variety of tools (e.g., linkingcubes, algebra tiles, diagrams, calculators)and strategies (e.g., patterning; connectingthe steps in a numerical example to the stepsin the algebraic development), and applythe formula to calculate the sum of a givennumber of consecutive termsSample problem: Given the following arraybuilt with grey and white connecting cubes,investigate how different ways of determiningthe total number of grey cubes can beused to evaluate the sum of the arithmeticseries 1 + 2 + 3 + 4 + 5. Extend the series,use patterning to make generalizations forfinding the sum, and test the generalizationsfor other arithmetic series.3. Solving Problems InvolvingFinancial ApplicationsBy the end of this course, students will:3.1 make and describe connections betweensimple interest, arithmetic sequences, andlinear growth, through investigation withtechnology (e.g., use a spreadsheet orgraphing calculator to make simple interestcalculations, determine first differences inthe amounts over time, and graph amountversus time)Sample problem: Describe an investmentthat could be represented by the functionf(x) = 500(1 + 0.05x).3.2 make and describe connections betweencompound interest, geometric sequences,and exponential growth, through investigationwith technology (e.g., use a spreadsheetto make compound interest calculations,determine finite differences in the amountsover time, and graph amount versus time)Sample problem: Describe an investmentthat could be represented by the functionxf(x) = 500(1.05) .3.3 solve problems, using a scientific calculator,that involve the calculation of the amount,A (also referred to as future value, FV ),the principal, P (also referred to aspresent value, PV ), or the interest rateper compounding period, i, using thecompound interest formula in the formnnA = P(1 + i) [or FV = PV(1 + i) ]Sample problem: Two investments areavailable, one at 6% compounded annuallyand the other at 6% compounded monthly.Investigate graphically the growth of eachinvestment, and determine the interestearned from depositing $1000 in eachinvestment for 10 years.FunctionsMCR3U2.4 solve problems involving arithmetic and geometricsequences and series, including thosearising from real-world applications3.4 determine, through investigation usingtechnology (e.g., scientific calculator, theTVM Solver on a graphing calculator, onlinetools), the number of compounding periods, n,using the compound interest formula in thennform A = P(1 + i) [or FV = PV(1 + i) ];describe strategies (e.g., guessing and checking;using the power of a power rule forexponents; using graphs) for calculating thisnumber; and solve related problemsDISCRETE FUNCTIONS51


Grade 11, University Preparation3.5 explain the meaning of the term annuity, anddetermine the relationships between ordinarysimple annuities (i.e., annuities in which paymentsare made at the end of each period, andcompounding and payment periods are thesame), geometric series, and exponentialgrowth, through investigation with technology(e.g., use a spreadsheet to determine andgraph the future value of an ordinary simpleannuity for varying numbers of compoundingperiods; investigate how the contributions ofeach payment to the future value of an ordinarysimple annuity are related to the termsof a geometric series)3.6 determine, through investigation usingtechnology (e.g., the TVM Solver on a graphingcalculator, online tools), the effects ofchanging the conditions (i.e., the payments,the frequency of the payments, the interestrate, the compounding period) of ordinarysimple annuities (e.g., long-term savingsplans, loans)Sample problem: Compare the amounts atage 65 that would result from making anannual deposit of $1000 starting at age 20,or from making an annual deposit of $3000starting at age 50, to an RRSP that earns 6%interest per annum, compounded annually.What is the total of the deposits in eachsituation?3.7 solve problems, using technology (e.g., scientificcalculator, spreadsheet, graphing calculator),that involve the amount, the presentvalue, and the regular payment of an ordinarysimple annuity (e.g., calculate the totalinterest paid over the life of a loan, using aspreadsheet, and compare the total interestwith the original principal of the loan)THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics52


D. TRIGONOMETRIC FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometricidentities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;2. demonstrate an understanding of periodic relationships and sinusoidal functions, and makeconnections between the numeric, graphical, and algebraic representations of sinusoidal functions;3. identify and represent sinusoidal functions, and solve problems involving sinusoidal functions,including problems arising from real-world applications.SPECIFIC EXPECTATIONS1. Determining and ApplyingTrigonometric RatiosBy the end of this course, students will:1.1 determine the exact values of the sine, cosine,and tangent of the special angles: 0º, 30º, 45º,60º, and 90º1.2 determine the values of the sine, cosine, andtangent of angles from 0º to 360º, throughinvestigation using a variety of tools (e.g.,dynamic geometry software, graphing tools)and strategies (e.g., applying the unit circle;examining angles related to special angles)1.3 determine the measures of two angles from0º to 360º for which the value of a giventrigonometric ratio is the same1.4 define the secant, cosecant, and cotangentratios for angles in a right triangle interms of the sides of the triangle (e.g.,hypotenusesec A =), and relate these ratiosadjacentto the cosine, sine, and tangent ratios (e.g.,1sec A = )cos A1.5 prove simple trigonometric identities, using2 2the Pythagorean identity sin x + cos x = 1;sinxthe quotient identity tanx = cosx ; and1the reciprocal identities secx = ,cosx1cscx = , and cotx = 1sinxtanxSample problem: Prove that21 – cos x = sinxcosxtanx.1.6 pose problems involving righttriangles and oblique triangles in twodimensionalsettings, and solve these andother such problems using the primarytrigonometric ratios, the cosine law, andthe sine law (including the ambiguous case)1.7 pose problems involving right triangles andoblique triangles in three-dimensional settings,and solve these and other such problemsusing the primary trigonometric ratios,the cosine law, and the sine lawSample problem: Explain how a surveyorcould find the height of a vertical cliff thatis on the other side of a raging river, usinga measuring tape, a theodolite, and sometrigonometry. Determine what the surveyormight measure, and use hypothetical valuesfor these data to calculate the height of thecliff.2. Connecting Graphs and Equationsof Sinusoidal FunctionsBy the end of this course, students will:2.1 describe key properties (e.g., cycle, amplitude,period) of periodic functions arising fromreal-world applications (e.g., natural gasconsumption in Ontario, tides in the Bayof Fundy), given a numeric or graphicalrepresentationMathematics for Functions Work and Everyday LifeMCR3UTRIGONOMETRIC FUNCTIONS53


Grade 11, University PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics2.2 predict, by extrapolating, the future behaviourof a relationship modelled using a numeric orgraphical representation of a periodic function(e.g., predicting hours of daylight on a particulardate from previous measurements; predictingnatural gas consumption in Ontario fromprevious consumption)2.3 make connections between the sine ratio andthe sine function and between the cosine ratioand the cosine function by graphing therelationship between angles from 0º to 360ºand the corresponding sine ratios or cosineratios, with or without technology (e.g., bygenerating a table of values using a calculator;by unwrapping the unit circle), defining thisrelationship as the function f(x) = sinx orf(x) = cosx, and explaining why the relationshipis a function2.4 sketch the graphs of f(x) = sinx andf(x) = cosx for angle measures expressedin degrees, and determine and describetheir key properties (i.e., cycle, domain, range,intercepts, amplitude, period, maximumand minimum values, increasing/decreasingintervals)2.5 determine, through investigation using technology,the roles of the parameters a, k, d, andc in functions of the form y = af(k(x – d)) + c,where f(x) = sinx or f(x) = cosx with anglesexpressed in degrees, and describe these rolesin terms of transformations on the graphs off(x) = sinx and f(x) = cosx (i.e., translations;reflections in the axes; vertical and horizontalstretches and compressions to and from thex- and y-axes)Sample problem: Investigate the graphf(x) = 2sin(x – d) + 10 for various values of d,using technology, and describe the effects ofchanging d in terms of a transformation.2.6 determine the amplitude, period, phaseshift, domain, and range of sinusoidalfunctions whose equations are given inthe form f(x) = asin(k(x – d)) + c orf(x) = acos(k(x – d)) + c2.7 sketch graphs of y = af(k(x – d)) + c byapplying one or more transformations to thegraphs of f(x) = sinx and f(x) = cosx, and statethe domain and range of the transformedfunctionsSample problem: Transform the graph off(x) = cos x to sketch g(x) = 3cos2x – 1, andstate the domain and range of each function.2.8 represent a sinusoidal function with anequation, given its graph or its propertiesSample problem: A sinusoidal function has anamplitude of 2 units, a period of 180º, and amaximum at (0, 3). Represent the function withan equation in two different ways.3. Solving Problems InvolvingSinusoidal FunctionsBy the end of this course, students will:3.1 collect data that can be modelled as a sinusoidalfunction (e.g., voltage in an AC circuit,sound waves), through investigation withand without technology, from primarysources, using a variety of tools (e.g., concretematerials, measurement tools such as motionsensors), or from secondary sources (e.g.,websites such as Statistics Canada, E-STAT),and graph the dataSample problem: Measure and recorddistance−time data for a swinging pendulum,using a motion sensor or other measurementtools, and graph the data.3.2 identify periodic and sinusoidal functions,including those that arise from real-worldapplications involving periodic phenomena,given various representations (i.e., tables ofvalues, graphs, equations), and explain anyrestrictions that the context places on thedomain and rangeSample problem: Using data from StatisticsCanada, investigate to determine if there wasa period of time over which changes in thepopulation of Canadians aged 20–24 could bemodelled using a sinusoidal function.3.3 determine, through investigation, how sinusoidalfunctions can be used to model periodicphenomena that do not involve anglesSample problem: Investigate, using graphingtechnology in degree mode, and explain howthe function h(t) = 5sin(30(t + 3)) approximatelymodels the relationship between theheight and the time of day for a tide with anamplitude of 5 m, if high tide is at midnight.3.4 predict the effects on a mathematical model(i.e., graph, equation) of an applicationinvolving periodic phenomena when theconditions in the application are varied(e.g., varying the conditions, such as speedand direction, when walking in a circle infront of a motion sensor)54


Sample problem: The relationship betweenthe height above the ground of a person ridinga Ferris wheel and time can be modelledusing a sinusoidal function. Describe theeffect on this function if the platform fromwhich the person enters the ride is raised by1 m and if the Ferris wheel turns twice as fast.3.5 pose problems based on applications involvinga sinusoidal function, and solve these andother such problems by using a given graphor a graph generated with technology froma table of values or from its equationSample problem: The height above theground of a rider on a Ferris wheel can bemodelled by the sinusoidal functionh(t) = 25 sin(3(t – 30)) + 27, where h(t) isthe height, in metres, and t is the time, inseconds. Graph the function, using graphingtechnology in degree mode, and determinethe maximum and minimum heights of therider, the height after 30 s, and the timerequired to complete one revolution.Mathematics for Functions Work and Everyday LifeMCR3UTRIGONOMETRIC FUNCTIONS55


Functions and Applications,Grade 11University/College PreparationMCF3MThis course introduces basic features of the function by extending students’ experienceswith quadratic relations. It focuses on quadratic, trigonometric, and exponential functionsand their use in modelling real-world situations. Students will represent functionsnumerically, graphically, and algebraically; simplify expressions; solve equations; andsolve problems relating to applications. Students will reason mathematically andcommunicate their thinking as they solve multi-step problems.Prerequisite: Principles of Mathematics, Grade 10, Academic, or Foundations ofMathematics, Grade 10, Applied57


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.58


A. QUADRATIC FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of aquadratic equation to the corresponding graph;2. demonstrate an understanding of functions, and make connections between the numeric, graphical,and algebraic representations of quadratic functions;3. solve problems involving quadratic functions, including problems arising from real-world applications.SPECIFIC EXPECTATIONS1. Solving Quadratic EquationsBy the end of this course, students will:1.1 pose problems involving quadratic relationsarising from real-world applications andrepresented by tables of values and graphs,and solve these and other such problems (e.g.,“From the graph of the height of a ball versustime, can you tell me how high the ball wasthrown and the time when it hit the ground?”)1.2 represent situations (e.g., the area of a pictureframe of variable width) using quadraticexpressions in one variable, and expandand simplify quadratic expressions in one2variable [e.g., 2x(x + 4) – (x + 3) ] *1.3 factor quadratic expressions in one variable,including those for which a ≠ 1 (e.g.,23x + 13x – 10), differences of squares2(e.g., 4x – 25), and perfect square trinomials2(e.g., 9x + 24x + 16), by selecting andapplying an appropriate strategy*Sample problem: Factor 2x – 12x + 10.1.4 solve quadratic equations by selecting andapplying a factoring strategy1.5 determine, through investigation, and describethe connection between the factors usedin solving a quadratic equation and thex-intercepts of the graph of the correspondingquadratic relation2Sample problem: The profit, P, of a videocompany, in thousands of dollars, is given2by P = –5x + 550x – 5000, where x is theamount spent on advertising, in thousands ofdollars. Determine, by factoring and bygraphing, the amount spent on advertisingthat will result in a profit of $0. Describe theconnection between the two strategies.1.6 explore the algebraic development of thequadratic formula (e.g., given the algebraicdevelopment, connect the steps to a numericexample; follow a demonstration of thealgebraic development, with technology,such as computer algebra systems, or withouttechnology [student reproduction of thedevelopment of the general case is notrequired]), and apply the formula to solvequadratic equations, using technology1.7 relate the real roots of a quadratic equation tothe x-intercepts of the corresponding graph,and connect the number of real roots to thevalue of the discriminant (e.g., there are no2real roots and no x-intercepts if b – 4ac < 0)1.8 determine the real roots of a variety of quadraticequations (e.g., 100x = 115x + 35), and2describe the advantages and disadvantages ofeach strategy (i.e., graphing; factoring; usingthe quadratic formula)Sample problem: Generate 10 quadratic equationsby randomly selecting integer values2for a, b, and c in ax + bx + c = 0. Solve the*The knowledge and skills described in this expectation may initially require the use of a variety of learning tools (e.g., computeralgebra systems, algebra tiles, grid paper).Functions and ApplicationsMCF3MQUADRATIC FUNCTIONS59


Grade 11, University/College PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics60equations using the quadratic formula. Howmany of the equations could you solve byfactoring?2. Connecting Graphs and Equationsof Quadratic FunctionsBy the end of this course, students will:2.1 explain the meaning of the term function, anddistinguish a function from a relation that isnot a function, through investigation of linearand quadratic relations using a variety ofrepresentations (i.e., tables of values, mappingdiagrams, graphs, function machines, equations)and strategies (e.g., using the verticallinetest)Sample problem: Investigate, using numericand graphical representations, whether the2relation x = y is a function, and justifyyour reasoning.2.2 substitute into and evaluate linear andquadratic functions represented using1function notation [e.g., evaluate f ( 2 ) , given2f(x) = 2x + 3x – 1], including functionsarising from real-world applicationsSample problem: The relationship betweenthe selling price of a sleeping bag, s dollars,and the revenue at that selling price,r(s) dollars, is represented by the function2r (s) = –10s + 1500s. Evaluate, interpret, andcompare r (29.95), r (60.00), r (75.00), r(90.00),and r (130.00).2.3 explain the meanings of the terms domain andrange, through investigation using numeric,graphical, and algebraic representations of linearand quadratic functions, and describe thedomain and range of a function appropriately2(e.g., for y = x + 1, the domain is the set ofreal numbers, and the range is y ≥ 1)2.4 explain any restrictions on the domain andthe range of a quadratic function in contextsarising from real-world applicationsSample problem: A quadratic function representsthe relationship between the height of aball and the time elapsed since the ball wasthrown. What physical factors will restrict thedomain and range of the quadratic function?2.5 determine, through investigation usingtechnology, the roles of a, h, and k in quadratic2functions of the form f(x) = a(x – h) + k, anddescribe these roles in terms of transformationson the graph of f(x) = x (i.e.,2translations;reflections in the x-axis; vertical stretches andcompressions to and from the x-axis)Sample problem: Investigate the graph2f(x) = 3(x – h) + 5 for various valuesof h, using technology, and describe theeffects of changing h in terms of atransformation.22.6 sketch graphs of g(x) = a(x – h) + k byapplying one or more transformations to2the graph of f(x) = xSample problem: Transform the graph of22f(x) = x to sketch the graphs of g(x) = x – 42and h(x) = – 2(x + 1) .2.7 express the equation of a quadratic function2in the standard form f(x) = ax + bx + c, given2the vertex form f(x) = a(x – h) + k, and verify,using graphing technology, that these formsare equivalent representationsSample problem: Given the vertex form2f(x) = 3(x – 1) + 4, express the equation instandard form. Use technology to comparethe graphs of these two forms of the equation.2.8 express the equation of a quadratic function2in the vertex form f(x) = a(x – h) + k, given2the standard form f(x) = ax + bx + c, bycompleting the square (e.g., using algebratiles or diagrams; algebraically), includingbcases where is a simple rational numbera(e.g.,1, 0.75), and verify, using graphing2technology, that these forms are equivalentrepresentations2.9 sketch graphs of quadratic functions in thefactored form f(x) = a(x – r )(x – s) by usingthe x-intercepts to determine the vertex2.10 describe the information (e.g., maximum,intercepts) that can be obtained by inspecting2the standard form f(x) = ax + bx + c, the2vertex form f(x) = a(x – h) + k, and thefactored form f(x) = a(x – r)(x – s) of aquadratic function2.11 sketch the graph of a quadratic functionwhose equation is given in the standard2form f(x) = ax + bx + c by using a suitablestrategy (e.g., completing the square andfinding the vertex; factoring, if possible, tolocate the x-intercepts), and identify the keyfeatures of the graph (e.g., the vertex, thex- and y-intercepts, the equation of the axisof symmetry, the intervals where the functionis positive or negative, the intervals wherethe function is increasing or decreasing)


3. Solving Problems InvolvingQuadratic FunctionsBy the end of this course, students will:3.1 collect data that can be modelled as a quadraticfunction, through investigation with andwithout technology, from primary sources,using a variety of tools (e.g., concrete materials;measurement tools such as measuringtapes, electronic probes, motion sensors), orfrom secondary sources (e.g., websites suchas Statistics Canada, E-STAT), and graphthe dataSample problem: When a 3 x 3 x 3 cube madeup of 1 x 1 x 1 cubes is dipped into red paint,6 of the smaller cubes will have 1 face painted.Investigate the number of smaller cubeswith 1 face painted as a function of the edgelength of the larger cube, and graph thefunction.3.2 determine, through investigation using a varietyof strategies (e.g., applying properties ofquadratic functions such as the x-interceptsand the vertex; using transformations), theequation of the quadratic function that bestmodels a suitable data set graphed on ascatter plot, and compare this equation to theequation of a curve of best fit generated withtechnology (e.g., graphing software, graphingcalculator)3.3 solve problems arising from real-world applications,given the algebraic representation of aquadratic function (e.g., given the equation ofa quadratic function representing the heightof a ball over elapsed time, answer questionsthat involve the maximum height of the ball,the length of time needed for the ball to touchthe ground, and the time interval when theball is higher than a given measurement)Sample problem: In the following DC electricalcircuit, the relationship between thepower used by a device, P (in watts, W), theelectric potential difference (voltage), V (involts, V), the current, I (in amperes, A), andthe resistance, R (in ohms, Ω ), is represented2by the formula P = IV – I R. Representgraphically and algebraically the relationshipbetween the power and the current when theelectric potential difference is 24 V and theresistance is 1.5 Ω. Determine the currentneeded in order for the device to use themaximum amount of power.VRIDeviceFunctions and ApplicationsMCF3MQUADRATIC FUNCTIONS61


Grade 11, University/College PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsB. EXPONENTIAL FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. simplify and evaluate numerical expressions involving exponents, and make connections betweenthe numeric, graphical, and algebraic representations of exponential functions;2. identify and represent exponential functions, and solve problems involving exponential functions,including problems arising from real-world applications;3. demonstrate an understanding of compound interest and annuities, and solve related problems.SPECIFIC EXPECTATIONS1. Connecting Graphs and Equationsof Exponential FunctionsBy the end of this course, students will:1.1 determine, through investigation using avariety of tools (e.g., calculator, paper andpencil, graphing technology) and strategies(e.g., patterning; finding values from a graph;interpreting the exponent laws), the valueof a power with a rational exponent (i.e., x ,where x > 0 and m and n are integers)Sample problem: The exponent laws suggest1 1that 42x 42 1= 4 . What value would you1assign to 42? What value would you assign1to 273? Explain your reasoning. Extend yourreasoning to make a generalization about the1meaning of x n, where x > 0 and n is a naturalnumber.1.2 evaluate, with and without technology,numerical expressions containing integerand rational exponents and rational bases1–3 3[e.g., 2 , (–6) , 42 120, 1.01 ]1.3 graph, with and without technology, an exponentialrelation, given its equation in the formxy = a (a > 0, a ≠ 1), define this relation as thexfunction f(x) = a , and explain why it is afunction1.4 determine, through investigation, and describekey properties relating to domain and range,intercepts, increasing/decreasing intervals,mn2and asymptotes (e.g., the domain is the set ofreal numbers; the range is the set of positivereal numbers; the function either increasesor decreases throughout its domain) forexponential functions represented in avariety of ways [e.g., tables of values, mappingdiagrams, graphs, equations of the formxf(x) = a (a > 0, a ≠ 1), function machines]Sample problem: Graph f(x) = 2, g(x) = 3,xand h(x) = 0.5 on the same set of axes.Make comparisons between the graphs,and explain the relationship between they-intercepts.1.5 determine, through investigation (e.g., bypatterning with and without a calculator),the exponent rules for multiplying anddividing numeric expressions involving3 2exponents [e.g., (1 ) x (1 ) ], and the2 2exponent rule for simplifying numericalexpressions involving a power of a power3[e.g., ( 5 ) ], and use the rules to simplifynumerical expressions containing integer3 5 8exponents [e.g., (2 )(2 ) = 2]1.6 distinguish exponential functions fromlinear and quadratic functions by makingcomparisons in a variety of ways (e.g.,comparing rates of change using finitedifferences in tables of values; identifying aconstant ratio in a table of values; inspectinggraphs; comparing equations), within thesame context when possible (e.g., simpleinterest and compound interest, populationgrowth)xx62


Sample problem: Explain in a variety ofways how you can distinguish the exponentialfunction f(x) = 2 from the quadraticx2function f(x) = x and the linear functionf (x) = 2x.2. Solving Problems InvolvingExponential FunctionsBy the end of this course, students will:2.1 collect data that can be modelled as anexponential function, through investigationwith and without technology, from primarysources, using a variety of tools (e.g., concretematerials such as number cubes, coins; measurementtools such as electronic probes), orfrom secondary sources (e.g., websites suchas Statistics Canada, E-STAT), and graphthe dataSample problem: Collect data and graph thecooling curve representing the relationshipbetween temperature and time for hot watercooling in a porcelain mug. Predict the shapeof the cooling curve when hot water cools inan insulated mug. Test your prediction.2.2 identify exponential functions, includingthose that arise from real-world applicationsinvolving growth and decay (e.g., radioactivedecay, population growth, cooling rates,pressure in a leaking tire), given variousrepresentations (i.e., tables of values, graphs,equations), and explain any restrictions thatthe context places on the domain and range(e.g., ambient temperature limits the rangefor a cooling curve)2.3 solve problems using given graphs orequations of exponential functions arisingfrom a variety of real-world applications(e.g., radioactive decay, population growth,height of a bouncing ball, compound interest)by interpreting the graphs or by substitutingvalues for the exponent into the equationsSample problem: The temperature of acooling liquid over time can be modelled byxthe exponential function T(x) = 60 (1)30+ 20,2where T(x) is the temperature, in degreesCelsius, and x is the elapsed time, in minutes.Graph the function and determine how long ittakes for the temperature to reach 28ºC.3. Solving Financial ProblemsInvolving Exponential FunctionsBy the end of this course, students will:3.1 compare, using a table of values and graphs,the simple and compound interest earned fora given principal (i.e., investment) and a fixedinterest rate over timeSample problem: Compare, using tables ofvalues and graphs, the amounts after each ofthe first five years for a $1000 investment at5% simple interest per annum and a $1000investment at 5% interest per annum, compoundedannually.3.2 solve problems, using a scientific calculator,that involve the calculation of the amount, A(also referred to as future value, FV ), and theprincipal, P (also referred to as present value,PV ), using the compound interest formula innnthe form A = P(1 + i) [or FV = PV(1 + i) ]Sample problem: Calculate the amount if$1000 is invested for three years at 6% perannum, compounded quarterly.3.3 determine, through investigation (e.g., usingspreadsheets and graphs), that compoundinterest is an example of exponentialgrowth [e.g., the formulas for compoundninterest, A = P(1 + i) , and present value,–nPV = A(1 + i) , are exponential functions,where the number of compounding periods,n, varies]Sample problem: Describe an investmentthat could be represented by the functionxf(x) = 500(1.01) .3.4 solve problems, using a TVM Solver on agraphing calculator or on a website, thatinvolve the calculation of the interest rateper compounding period, i, or the numberof compounding periods, n, in thencompound interest formula A = P(1 + i)n[or FV = PV(1 + i) ]Sample problem: Use the TVM Solver in agraphing calculator to determine the time ittakes to double an investment in an accountthat pays interest of 4% per annum, compoundedsemi-annually.3.5 explain the meaning of the term annuity,through investigation of numeric andgraphical representations using technologyFunctions and ApplicationsMCF3MEXPONENTIAL FUNCTIONS63


Grade 11, University/College Preparation3.6 determine, through investigation usingtechnology (e.g., the TVM Solver on a graphingcalculator, online tools), the effects ofchanging the conditions (i.e., the payments,the frequency of the payments, the interestrate, the compounding period) of ordinarysimple annuities (i.e., annuities in which paymentsare made at the end of each period, andthe compounding period and the paymentperiod are the same) (e.g., long-term savingsplans, loans)Sample problem: Compare the amounts atage 65 that would result from making anannual deposit of $1000 starting at age 20,or from making an annual deposit of $3000starting at age 50, to an RRSP that earns6% interest per annum, compoundedannually. What is the total of the depositsin each situation?3.7 solve problems, using technology (e.g., scientificcalculator, spreadsheet, graphing calculator),that involve the amount, the presentvalue, and the regular payment of an ordinarysimple annuity (e.g., calculate the totalinterest paid over the life of a loan, using aspreadsheet, and compare the total interestwith the original principal of the loan)THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics64


C. TRIGONOMETRIC FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. solve problems involving trigonometry in acute triangles using the sine law and the cosine law,including problems arising from real-world applications;2. demonstrate an understanding of periodic relationships and the sine function, and make connectionsbetween the numeric, graphical, and algebraic representations of sine functions;3. identify and represent sine functions, and solve problems involving sine functions, includingproblems arising from real-world applications.SPECIFIC EXPECTATIONSFunctions and Applications1. Applying the Sine Law and theCosine Law in Acute TrianglesBy the end of this course, students will:1.1 solve problems, including those that arisefrom real-world applications (e.g., surveying,navigation), by determining the measures ofthe sides and angles of right triangles usingthe primary trigonometric ratios1.2 solve problems involving two right trianglesin two dimensionsSample problem: A helicopter hovers 500 mabove a long straight road. Ahead of the helicopteron the road are two trucks. The anglesof depression of the two trucks from thehelicopter are 60° and 20°. How far apart arethe two trucks?1.3 verify, through investigation using technology(e.g., dynamic geometry software,spreadsheet), the sine law and the cosine law(e.g., compare, using dynamic geometrya b csoftware, the ratios , , andsinA sinB sinCin triangle ABC while dragging one of thevertices)1.4 describe conditions that guide when it isappropriate to use the sine law or the cosinelaw, and use these laws to calculate sides andangles in acute triangles1.5 solve problems that require the use of thesine law or the cosine law in acute triangles,including problems arising from real-worldapplications (e.g., surveying, navigation,building construction)2. Connecting Graphs and Equationsof Sine FunctionsBy the end of this course, students will:2.1 describe key properties (e.g., cycle, amplitude,period) of periodic functions arising fromreal-world applications (e.g., natural gasconsumption in Ontario, tides in the Bayof Fundy), given a numeric or graphicalrepresentation2.2 predict, by extrapolating, the future behaviourof a relationship modelled using a numericor graphical representation of a periodicfunction (e.g., predicting hours of daylighton a particular date from previous measurements;predicting natural gas consumptionin Ontario from previous consumption)2.3 make connections between the sine ratio andthe sine function by graphing the relationshipbetween angles from 0º to 360º and thecorresponding sine ratios, with or withouttechnology (e.g., by generating a table ofvalues using a calculator; by unwrappingthe unit circle), defining this relationshipas the function f(x) = sinx, and explainingwhy the relationship is a functionMCF3MTRIGONOMETRIC FUNCTIONS65


Grade 11, University/College PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics2.4 sketch the graph of f(x) = sinx for anglemeasures expressed in degrees, and determineand describe its key properties (i.e., cycle,domain, range, intercepts, amplitude, period,maximum and minimum values, increasing/decreasing intervals)2.5 make connections, through investigation withtechnology, between changes in a real-worldsituation that can be modelled using a periodicfunction and transformations of the correspondinggraph (e.g., investigate the connectionbetween variables for a swimmer swimminglengths of a pool and transformations of thegraph of distance from the starting pointversus time)Sample problem: Generate the graph of aperiodic function by walking a circle of2-m diameter in front of a motion sensor.Describe how the following changes in themotion change the graph: starting at a differentpoint on the circle; starting a greaterdistance from the motion sensor; changingdirection; increasing the radius of the circle.2.6 determine, through investigation usingtechnology, the roles of the parameters a, c,and d in functions in the form f(x) = a sinx,f(x) = sinx + c, and f(x) = sin(x – d), anddescribe these roles in terms of transformationson the graph of f(x) = sinx with anglesexpressed in degrees (i.e., translations;reflections in the x-axis; vertical stretches andcompressions to and from the x-axis)2.7 sketch graphs of f(x) = a sinx, f(x) = sinx + c,and f(x) = sin(x – d) by applying transformationsto the graph of f(x) = sinx, and statethe domain and range of the transformedfunctionsSample problem: Transform the graphof f(x) = sinx to sketch the graphs ofg(x) = –2sinx and h(x) = sin(x – 180°),and state the domain and range of eachfunction.3. Solving Problems Involving SineFunctionsBy the end of this course, students will:3.1 collect data that can be modelled as a sinefunction (e.g., voltage in an AC circuit, soundwaves), through investigation with andwithout technology, from primary sources,using a variety of tools (e.g., concretematerials, measurement tools such as motionsensors), or from secondary sources (e.g.,websites such as Statistics Canada, E-STAT),and graph the dataSample problem: Measure and record distance−time data for a swinging pendulum, using amotion sensor or other measurement tools,and graph the data.3.2 identify periodic and sinusoidal functions,including those that arise from real-worldapplications involving periodic phenomena,given various representations (i.e., tables ofvalues, graphs, equations), and explain anyrestrictions that the context places on thedomain and range3.3 pose problems based on applications involvinga sine function, and solve these and othersuch problems by using a given graph or agraph generated with technology from a tableof values or from its equationSample problem: The height above the groundof a rider on a Ferris wheel can be modelled bythe sine function h(x) = 25 sin(x – 90˚) + 27,where h(x) is the height, in metres, and x isthe angle, in degrees, that the radius from thecentre of the ferris wheel to the rider makeswith the horizontal. Graph the function,using graphing technology in degree mode,and determine the maximum and minimumheights of the rider and the measures of theangle when the height of the rider is 40 m.66


Foundations for CollegeMathematics, Grade 11College PreparationMBF3CThis course enables students to broaden their understanding of mathematics as a problemsolvingtool in the real world. Students will extend their understanding of quadraticrelations; investigate situations involving exponential growth; solve problems involvingcompound interest; solve financial problems connected with vehicle ownership; developtheir ability to reason by collecting, analysing, and evaluating data involving one variable;connect probability and statistics; and solve problems in geometry and trigonometry.Students will consolidate their mathematical skills as they solve problems andcommunicate their thinking.Prerequisite: Foundations of Mathematics, Grade 10, Applied67


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.68


A. MATHEMATICAL MODELSOVERALL EXPECTATIONSBy the end of this course, students will:1. make connections between the numeric, graphical, and algebraic representations of quadraticrelations, and use the connections to solve problems;2. demonstrate an understanding of exponents, and make connections between the numeric, graphical,and algebraic representations of exponential relations;3. describe and represent exponential relations, and solve problems involving exponential relationsarising from real-world applications.SPECIFIC EXPECTATIONS1. Connecting Graphs and Equationsof Quadratic RelationsBy the end of this course, students will:1.4 sketch graphs of quadratic relations representedby the equation y = a(x – h) + k (e.g.,2using the vertex and at least one point oneach side of the vertex; applying one or more2transformations to the graph of y = x )Foundations for College MathematicsMBF3C1.1 construct tables of values and graph quadraticrelations arising from real-world applications(e.g., dropping a ball from a givenheight; varying the edge length of a cubeand observing the effect on the surface areaof the cube)1.2 determine and interpret meaningful valuesof the variables, given a graph of a quadraticrelation arising from a real-world applicationSample problem: Under certain conditions,there is a quadratic relation between theprofit of a manufacturing company and thenumber of items it produces. Explain howyou could interpret a graph of the relationto determine the numbers of items producedfor which the company makes a profit and todetermine the maximum profit the companycan make.1.3 determine, through investigation usingtechnology, the roles of a, h, and k in quadratic2relations of the form y = a(x – h) + k, anddescribe these roles in terms of transformationson the graph of y = x (i.e., translations;2reflections in the x-axis; vertical stretches andcompressions to and from the x-axis)Sample problem: Investigate the graph2y = 3(x – h) + 5 for various values of h,using technology, and describe the effects ofchanging h in terms of a transformation.1.5 expand and simplify quadratic expressions inone variable involving multiplying binomials1[e.g., ( x + 12 ) (3x – 2)] or squaring a binomial2[e.g., 5(3x – 1) ], using a variety of tools (e.g.,paper and pencil, algebra tiles, computeralgebra systems)1.6 express the equation of a quadratic relation in2the standard form y = ax + bx + c, given the2vertex form y = a(x – h) + k, and verify, usinggraphing technology, that these forms areequivalent representationsSample problem: Given the vertex form2y = 3(x – 1) + 4, express the equation instandard form. Use technology to comparethe graphs of these two forms of theequation.21.7 factor trinomials of the form ax + bx + c,where a = 1 or where a is the common factor,by various methods1.8 determine, through investigation, anddescribe the connection between the factorsof a quadratic expression and the x-interceptsof the graph of the corresponding quadraticrelationSample problem: Investigate the relationship2between the factored form of 3x + 15x + 122and the x-intercepts of y = 3x + 15x + 12.MATHEMATICAL MODELS69


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 11, College Preparation701.9 solve problems, using an appropriate strategy(i.e., factoring, graphing), given equations ofquadratic relations, including those that arisefrom real-world applications (e.g., break-evenpoint)Sample problem: On planet X, the height,h metres, of an object fired upward from theground at 48 m/s is described by the equation2h = 48t – 16t , where t seconds is the timesince the object was fired upward. Determinethe maximum height of the object, thetimes at which the object is 32 m above theground, and the time at which the object hitsthe ground.2. Connecting Graphs and Equationsof Exponential RelationsBy the end of this course, students will:2.1 determine, through investigation using avariety of tools and strategies (e.g., graphingwith technology; looking for patterns in tablesof values), and describe the meaning of negativeexponents and of zero as an exponent2.2 evaluate, with and without technology,numeric expressions containing integer–3 3exponents and rational bases (e.g., 2 , 6 ,0 103456 , 1.03 )2.3 determine, through investigation (e.g., bypatterning with and without a calculator),the exponent rules for multiplying anddividing numerical expressions involvingexponents [e.g., (13) x (12) ], and the2 2exponent rule for simplifying numericalexpressions involving a power of a power23[e.g., ( 5 ) ]2.4 graph simple exponential relations, usingpaper and pencil, given their equationsxxx[e.g., y = 2, y = 10 , y = (1) ]22.5 make and describe connections betweenrepresentations of an exponential relation(i.e., numeric in a table of values; graphical;algebraic)2.6 distinguish exponential relations from linearand quadratic relations by making comparisonsin a variety of ways (e.g., comparingrates of change using finite differences intables of values; inspecting graphs; comparingequations), within the same context whenpossible (e.g., simple interest and compoundinterest, population growth)Sample problem: Explain in a variety of wayshow you can distinguish exponential growthxrepresented by y = 2 from quadratic growth2represented by y = x and linear growth representedby y = 2x.3. Solving Problems InvolvingExponential RelationsBy the end of this course, students will:3.1 collect data that can be modelled as anexponential relation, through investigationwith and without technology, from primarysources, using a variety of tools (e.g., concretematerials such as number cubes, coins; measurementtools such as electronic probes), orfrom secondary sources (e.g., websites suchas Statistics Canada, E-STAT), and graphthe dataSample problem: Collect data and graph thecooling curve representing the relationshipbetween temperature and time for hot watercooling in a porcelain mug. Predict the shapeof the cooling curve when hot water cools inan insulated mug. Test your prediction.3.2 describe some characteristics of exponentialrelations arising from real-world applications(e.g., bacterial growth, drug absorption) byusing tables of values (e.g., to show a constantratio, or multiplicative growth or decay) andgraphs (e.g., to show, with technology, thatthere is no maximum or minimum value)3.3 pose problems involving exponential relationsarising from a variety of real-world applications(e.g., population growth, radioactivedecay, compound interest), and solve theseand other such problems by using a givengraph or a graph generated with technologyfrom a given table of values or a given equationSample problem: Given a graph of thepopulation of a bacterial colony versustime, determine the change in populationin the first hour.3.4 solve problems using given equations ofexponential relations arising from a varietyof real-world applications (e.g., radioactivedecay, population growth, height of a bouncingball, compound interest) by substitutingvalues for the exponent into the equationsSample problem: The height, h metres, of aball after n bounces is given by the equationnh = 2(0.6) . Determine the height of the ballafter 3 bounces.


B. PERSONAL FINANCEOVERALL EXPECTATIONSBy the end of this course, students will:1. compare simple and compound interest, relate compound interest to exponential growth, and solveproblems involving compound interest;2. compare services available from financial institutions, and solve problems involving the cost of makingpurchases on credit;3. interpret information about owning and operating a vehicle, and solve problems involving theassociated costs.SPECIFIC EXPECTATIONS1. Solving Problems InvolvingCompound InterestBy the end of this course, students will:1.1 determine, through investigation using technology,the compound interest for a given investment,using repeated calculations of simpleinterest, and compare, using a table of valuesand graphs, the simple and compound interestearned for a given principal (i.e., investment)and a fixed interest rate over timeSample problem: Compare, using tables ofvalues and graphs, the amounts after each ofthe first five years for a $1000 investment at5% simple interest per annum and a $1000investment at 5% interest per annum,compounded annually.1.2 determine, through investigation (e.g., usingspreadsheets and graphs), and describe therelationship between compound interest andexponential growth1.3 solve problems, using a scientific calculator,that involve the calculation of the amount, A(also referred to as future value, FV ), and theprincipal, P (also referred to as present value,PV ), using the compound interest formula innnthe form A = P(1 + i ) [or FV = PV (1 + i )]Sample problem: Calculate the amountif $1000 is invested for 3 years at 6% perannum, compounded quarterly.1.4 calculate the total interest earned on an investmentor paid on a loan by determining thedifference between the amount and the principal[e.g., using I = A – P (or I = FV – PV )]1.5 solve problems, using a TVM Solver on agraphing calculator or on a website, thatinvolve the calculation of the interest rate percompounding period, i, or the number of compoundingperiods, n, in the compound interestnnformula A = P(1 + i ) [or FV = PV (1 + i )]Sample problem: Use the TVM Solver on agraphing calculator to determine the time ittakes to double an investment in an accountthat pays interest of 4% per annum, compoundedsemi-annually.1.6 determine, through investigation usingtechnology (e.g., a TVM Solver on a graphingcalculator or on a website), the effect on thefuture value of a compound interest investmentor loan of changing the total length oftime, the interest rate, or the compoundingperiodSample problem: Investigate whether doublingthe interest rate will halve the time ittakes for an investment to double.Foundations for College MathematicsMBF3CPERSONAL FINANCE71


2. Comparing Financial Services 3. Owning and Operating a VehicleTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 11, College PreparationBy the end of this course, students will:2.1 gather, interpret, and compare informationabout the various savings alternatives commonlyavailable from financial institutions(e.g., savings and chequing accounts, terminvestments), the related costs (e.g., cost ofcheques, monthly statement fees, early withdrawalpenalties), and possible ways ofreducing the costs (e.g., maintaining aminimum balance in a savings account;paying a monthly flat fee for a packageof services)2.2 gather and interpret information about investmentalternatives (e.g., stocks, mutual funds,real estate, GICs, savings accounts), and comparethe alternatives by considering the riskand the rate of return2.3 gather, interpret, and compare informationabout the costs (e.g., user fees, annual fees,service charges, interest charges on overduebalances) and incentives (e.g., loyalty rewards;philanthropic incentives, such as support forOlympic athletes or a Red Cross disaster relieffund) associated with various credit cards anddebit cards2.4 gather, interpret, and compare informationabout current credit card interest rates andregulations, and determine, through investigationusing technology, the effects of delayedpayments on a credit card balance2.5 solve problems involving applications of thecompound interest formula to determine thecost of making a purchase on creditSample problem: Using information gatheredabout the interest rates and regulations fortwo different credit cards, compare the costsof purchasing a $1500 computer with eachcard if the full amount is paid 55 days later.By the end of this course, students will:3.1 gather and interpret information about theprocedures and costs involved in insuring avehicle (e.g., car, motorcycle, snowmobile)and the factors affecting insurance rates (e.g.,gender, age, driving record, model of vehicle,use of vehicle), and compare the insurancecosts for different categories of drivers andfor different vehiclesSample problem: Use automobile insurancewebsites to investigate the degree to whichthe type of car and the age and gender of thedriver affect insurance rates.3.2 gather, interpret, and compare informationabout the procedures and costs (e.g., monthlypayments, insurance, depreciation, maintenance,miscellaneous expenses) involved inbuying or leasing a new vehicle or buying aused vehicleSample problem: Compare the costs ofbuying a new car, leasing the same car, andbuying an older model of the same car.3.3 solve problems, using technology (e.g., calculator,spreadsheet), that involve the fixed costs(e.g., licence fee, insurance) and variable costs(e.g., maintenance, fuel) of owning and operatinga vehicleSample problem: The rate at which a car consumesgasoline depends on the speed of thecar. Use a given graph of gasoline consumption,in litres per 100 km, versus speed, inkilometres per hour, to determine how muchgasoline is used to drive 500 km at speeds of80 km/h, 100 km/h, and 120 km/h. Use thecurrent price of gasoline to calculate the costof driving 500 km at each of these speeds.72


C. GEOMETRY AND TRIGONOMETRYOVERALL EXPECTATIONSBy the end of this course, students will:1. represent, in a variety of ways, two-dimensional shapes and three-dimensional figures arising fromreal-world applications, and solve design problems;2. solve problems involving trigonometry in acute triangles using the sine law and the cosine law,including problems arising from real-world applications.SPECIFIC EXPECTATIONS1. Representing Two-DimensionalShapes and Three-DimensionalFiguresBy the end of this course, students will:1.1 recognize and describe real-world applicationsof geometric shapes and figures, throughinvestigation (e.g., by importing digital photosinto dynamic geometry software), in a varietyof contexts (e.g., product design, architecture,fashion), and explain these applications (e.g.,one reason that sewer covers are round is toprevent them from falling into the sewerduring removal and replacement)Sample problem: Explain why rectangularprisms are often used for packaging.1.2 represent three-dimensional objects, usingconcrete materials and design or drawingsoftware, in a variety of ways (e.g., orthographicprojections [i.e., front, side, and topviews], perspective isometric drawings, scalemodels)1.3 create nets, plans, and patterns from physicalmodels arising from a variety of real-worldapplications (e.g., fashion design, interior decorating,building construction), by applyingthe metric and imperial systems and usingdesign or drawing software1.4 solve design problems that satisfy given constraints(e.g., design a rectangular berm thatwould contain all the oil that could leak froma cylindrical storage tank of a given heightand radius), using physical models (e.g., builtfrom popsicle sticks, cardboard, duct tape) ordrawings (e.g., made using design or drawingsoftware), and state any assumptions madeSample problem: Design and construct amodel boat that can carry the most pennies,using one sheet of 8.5 in. x 11 in. card stock,no more than five popsicle sticks, and someadhesive tape or glue.2. Applying the Sine Law and theCosine Law in Acute TrianglesBy the end of this course, students will:2.1 solve problems, including those that arisefrom real-world applications (e.g., surveying,navigation), by determining the measures of thesides and angles of right triangles using theprimary trigonometric ratios2.2 verify, through investigation using technology(e.g., dynamic geometry software, spreadsheet),the sine law and the cosine law (e.g.,compare, using dynamic geometry software,a bcthe ratios , , and insin A sinB sin Ctriangle ABC while dragging one of thevertices);2.3 describe conditions that guide when it isappropriate to use the sine law or the cosinelaw, and use these laws to calculate sides andangles in acute triangles2.4 solve problems that arise from real-worldapplications involving metric and imperialmeasurements and that require the use of thesine law or the cosine law in acute trianglesFoundations for College MathematicsMBF3CGEOMETRY AND TRIGONOMETRY73


Grade 11, Grade University/College 11, Preparation PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics74D. DATA MANAGEMENTOVERALL EXPECTATIONSBy the end of this course, students will:1. solve problems involving one-variable data by collecting, organizing, analysing, and evaluating data;2. determine and represent probability, and identify and interpret its applications.SPECIFIC EXPECTATIONS1. Working With One-Variable DataBy the end of this course, students will:1.1 identify situations involving one-variabledata (i.e., data about the frequency of a givenoccurrence), and design questionnaires (e.g.,for a store to determine which CDs to stock,for a radio station to choose which music toplay) or experiments (e.g., counting, takingmeasurements) for gathering one-variabledata, giving consideration to ethics, privacy,the need for honest responses, and possiblesources of biasSample problem: One lane of a three-lanehighway is being restricted to vehicles withat least two passengers to reduce trafficcongestion. Design an experiment to collectone-variable data to decide whether trafficcongestion is actually reduced.1.2 collect one-variable data from secondarysources (e.g., Internet databases), and organizeand store the data using a variety of tools(e.g., spreadsheets, dynamic statisticalsoftware)1.3 explain the distinction between the termspopulation and sample, describe the characteristicsof a good sample, and explain whysampling is necessary (e.g., time, cost, orphysical constraints)Sample problem: Explain the terms sampleand population by giving examples withinyour school and your community.1.4 describe and compare sampling techniques(e.g., random, stratified, clustered, convenience,voluntary); collect one-variable datafrom primary sources, using appropriatesampling techniques in a variety of real-worldsituations; and organize and store the data1.5 identify different types of one-variable data(i.e., categorical, discrete, continuous), andrepresent the data, with and without technology,in appropriate graphical forms (e.g.,histograms, bar graphs, circle graphs,pictographs)1.6 identify and describe properties associatedwith common distributions of data (e.g.,normal, bimodal, skewed)1.7 calculate, using formulas and/or technology(e.g., dynamic statistical software, spreadsheet,graphing calculator), and interpretmeasures of central tendency (i.e., mean,median, mode) and measures of spread(i.e., range, standard deviation)1.8 explain the appropriate use of measuresof central tendency (i.e., mean, median, mode)and measures of spread (i.e., range, standarddeviation)Sample problem: Explain whether the meanor the median of your course marks wouldbe the more appropriate representation ofyour achievement. Describe the additionalinformation that the standard deviationof your course marks would provide.1.9 compare two or more sets of one-variabledata, using measures of central tendency andmeasures of spreadSample problem: Use measures of centraltendency and measures of spread to comparedata that show the lifetime of an economylight bulb with data that show the lifetime ofa long-life light bulb.1.10 solve problems by interpreting and analysingone-variable data collected from secondarysources


2. Applying ProbabilityBy the end of this course, students will:2.1 identify examples of the use of probability inthe media and various ways in which probabilityis represented (e.g., as a fraction, as apercent, as a decimal in the range 0 to 1)2.2 determine the theoretical probability ofan event (i.e., the ratio of the number offavourable outcomes to the total number ofpossible outcomes, where all outcomes areequally likely), and represent the probabilityin a variety of ways (e.g., as a fraction, as apercent, as a decimal in the range 0 to 1)2.3 perform a probability experiment (e.g., tossinga coin several times), represent the resultsusing a frequency distribution, and use thedistribution to determine the experimentalprobability of an event2.4 compare, through investigation, the theoreticalprobability of an event with the experimentalprobability, and explain why theymight differSample problem: If you toss 10 coins repeatedly,explain why 5 heads are unlikely toresult from every toss.2.5 determine, through investigation using classgenerateddata and technology-based simulationmodels (e.g., using a random-numbergenerator on a spreadsheet or on a graphingcalculator), the tendency of experimentalprobability to approach theoretical probabilityas the number of trials in an experimentincreases (e.g., “If I simulate tossing a coin1000 times using technology, the experimentalprobability that I calculate for tossing tails islikely to be closer to the theoretical probabilitythan if I simulate tossing the coin only10 times”)Sample problem: Calculate the theoreticalprobability of rolling a 2 on a number cube.Simulate rolling a number cube, and use thesimulation to calculate the experimentalprobability of rolling a 2 over 10, 20, 30, …,200 trials. Graph the experimental probabilityversus the number of trials, and describe anytrend.2.6 interpret information involving the use ofprobability and statistics in the media, andmake connections between probability andstatistics (e.g., statistics can be used togenerate probabilities)Foundations for College MathematicsMBF3CDATA MANAGEMENT75


Mathematics for Work andEveryday Life, Grade 11Workplace PreparationMEL3EThis course enables students to broaden their understanding of mathematics as it isapplied in the workplace and daily life. Students will solve problems associated withearning money, paying taxes, and making purchases; apply calculations of simple andcompound interest in saving, investing, and borrowing; and calculate the costs oftransportation and travel in a variety of situations. Students will consolidate theirmathematical skills as they solve problems and communicate their thinking.Prerequisite: Principles of Mathematics, Grade 9, Academic, or Foundations ofMathematics, Grade 9, Applied, or a ministry-approved locally developedGrade 10 mathematics course77


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.78


A. EARNING AND PURCHASINGOVERALL EXPECTATIONSBy the end of this course, students will:1. interpret information about different types of remuneration, and solve problems and make decisionsinvolving different remuneration methods;2. demonstrate an understanding of payroll deductions and their impact on purchasing power;3. demonstrate an understanding of the factors and methods involved in making and justifying informedpurchasing decisions.SPECIFIC EXPECTATIONS1. Earning2. Describing Purchasing PowerMathematics for Work and Everyday LifeBy the end of this course, students will:1.1 gather, interpret, and compare informationabout the components of total earnings (e.g.,salary, benefits, vacation pay, profit-sharing)in different occupations1.2 gather, interpret, and describe informationabout different remuneration methods (e.g.,hourly rate, overtime rate, job or project rate,commission, salary, gratuities) and remunerationschedules (e.g., weekly, biweekly, semimonthly,monthly)1.3 describe the effects of different remunerationmethods and schedules on decisions relatedto personal spending habits (e.g., the timing ofa major purchase, the scheduling of mortgagepayments and other bill payments)1.4 solve problems, using technology (e.g., calculator,spreadsheet), and make decisionsinvolving different remuneration methodsand schedulesSample problem: Two sales positions areavailable in sportswear stores. One pays anhourly rate of $11.25 for 40 h per week. Theother pays a weekly salary of $375 for thesame number of hours, plus a commission of5% of sales. Under what conditions wouldeach position be the better choice?By the end of this course, students will:2.1 gather, interpret, and describe informationabout government payroll deductions(i.e., CPP, EI, income tax) and other payrolldeductions (e.g., contributions to pensionplans other than CPP; union dues; charitabledonations; benefit-plan contributions)2.2 estimate and compare, using currentsecondary data (e.g., federal tax tables),the percent of total earnings deductedthrough government payroll deductionsfor various benchmarks (e.g., $15 000,$20 000, $25 000)Sample problem: Compare the percentage oftotal earnings deducted through governmentpayroll deductions for total earnings of$15 000 and $45 000.2.3 describe the relationship between gross pay,net pay, and payroll deductions (i.e., net payis gross pay less government payroll deductionsand any other payroll deductions), andestimate net pay in various situations2.4 describe and compare the purchasing powerand living standards associated with relevantoccupations of interestMEL3EEARNING AND PURCHASING79


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 11, Workplace Preparation3. PurchasingBy the end of this course, students will:3.1 identify and describe various incentives inmaking purchasing decisions (e.g., 20% off;1off; buy 3 get 1 free; loyalty rewards;3coupons; 0% financing)3.2 estimate the sale price before taxes whenmaking a purchase (e.g., estimate 25% off1of $38.99 as 25% or off of $40, giving4a discount of about $10 and a sale price ofapproximately $30; alternatively, estimate3the same sale price as about of $40)43.3 describe and compare a variety of strategies forestimating sales tax (e.g., estimate the salestax on most purchases in Ontario by estimating10% of the purchase price and addingabout a third of this estimate, rather than estimatingthe PST and GST separately), and usea chosen strategy to estimate the after-tax costof common itemsSample problem: You purchase three itemsfor $8.99 each and one item for $4.99.Estimate the after-tax total.3.4 calculate discounts, sale prices, and after-taxcosts, using technology3.5 identify forms of taxation built into the cost ofan item or service (e.g., gasoline tax, tire tax)3.6 estimate the change from an amount offeredto pay a chargeSample problem: Estimate the change fromthe $20 offered to pay a charge of $13.87.3.7 make the correct change from an amountoffered to pay a charge, using currencymanipulativesSample problem: Use currency manipulativesto explain why someone might offer $15.02,rather than $15.00, to pay a charge of $13.87.3.8 compare the unit prices of related items tohelp determine the best buySample problem: Investigate whether or notpurchasing larger quantities always results ina lower unit price.3.9 describe and compare, for different types oftransactions, the extra costs that may be associatedwith making purchases (e.g., interestcosts, exchange rates, shipping and handlingcosts, customs duty, insurance)Sample problem: What are the various costsincluded in the final total for purchasing adigital audio player online from an Americansource? Using an online calculator, calculatethe final cost, and describe how it compareswith the cost of the purchase from a majorretailer in Ontario.3.10 make and justify a decision regarding thepurchase of an item, using various criteria(e.g., extra costs, such as shipping costsand transaction fees; quality and quantityof the item; shelf life of the item; method ofpurchase, such as online versus local) undervarious circumstances (e.g., not having accessto a vehicle; living in a remote community;having limited storage space)Sample problem: I have to take 100 mL of aliquid vitamin supplement every morning.I can buy a 100 mL size for $6.50 or a 500 mLsize for $25.00. If the supplement keeps inthe refrigerator for only 72 h, investigatewhich size is the better buy. Explain yourreasoning.80


B. SAVING, INVESTING, ANDBORROWINGOVERALL EXPECTATIONSBy the end of this course, students will:1. describe and compare services available from financial institutions;2. demonstrate an understanding of simple and compound interest, and solve problems involvingrelated applications;3. interpret information about different ways of borrowing and their associated costs, and make andjustify informed borrowing decisions.SPECIFIC EXPECTATIONSMathematics for Work and Everyday Life1. Comparing Financial ServicesBy the end of this course, students will:1.1 gather, interpret, and compare informationabout the various savings alternatives commonlyavailable from financial institutions(e.g., savings and chequing accounts, terminvestments), the related costs (e.g., costof cheques, monthly statement fees, earlywithdrawal penalties), and possible waysof reducing the costs (e.g., maintaining aminimum balance in a savings account;paying a monthly flat fee for a packageof services)1.2 gather, interpret, and compare informationabout the costs (e.g., user fees, annual fees,service charges, interest charges on overduebalances) and incentives (e.g., loyalty rewards;philanthropic incentives, such as support forOlympic athletes or a Red Cross disaster relieffund) associated with various credit cards anddebit cards1.3 read and interpret transaction codes andentries from various financial statements(e.g., bank statement, credit card statement,passbook, automated banking machineprintout, online banking statement, accountactivity report), and explain ways of usingthe information to manage personal financesSample problem: Examine a credit cardstatement and a bank statement for oneindividual, and comment on the individual’sfinancial situation.2. Saving and InvestingBy the end of this course, students will:2.1 determine, through investigation using technology(e.g., calculator, spreadsheet), theeffect on simple interest of changes in theprincipal, interest rate, or time, and solveproblems involving applications of simpleinterest2.2 determine, through investigation usingtechnology, the compound interest for a giveninvestment, using repeated calculations ofsimple interest for no more than 6 compoundingperiodsSample problem: Someone deposits $5000 at4% interest per annum, compounded semiannually.How much interest accumulates in3 years?2.3 describe the relationship between simpleinterest and compound interest in variousways (i.e., orally, in writing, using tablesand graphs)MEL3ESAVING, INVESTING, AND BORROWING81


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 11, Workplace Preparation2.4 determine, through investigation usingtechnology (e.g., a TVM Solver on a graphingcalculator or on a website), the effect on thefuture value of a compound interest investmentof changing the total length of time,the interest rate, or the compounding periodSample problem: Compare the results atage 40 of making a deposit of $1000 atage 20 or a deposit of $2000 at age 30, if bothinvestments pay 6% interest per annum,compounded monthly.2.5 solve problems, using technology, that involveapplications of compound interest to savingand investing3. BorrowingBy the end of this course, students will:3.1 gather, interpret, and compare informationabout the effects of carrying an outstandingbalance on a credit card at current interestratesSample problem: Describe ways of minimizingthe cost of carrying an outstanding balanceon a credit card.3.2 gather, interpret, and compare informationdescribing the features (e.g., interest rates,flexibility) and conditions (e.g., eligibility,required collateral) of various personal loans(e.g., student loan, car loan, “no interest”deferred-payment loan, loan to consolidatedebt, loan drawn on a line of credit, paydayor bridging loan)3.3 calculate, using technology (e.g., calculator,spreadsheet), the total interest paid over thelife of a personal loan, given the principal, thelength of the loan, and the periodic payments,and use the calculations to justify the choiceof a personal loan3.4 determine, using a variety of tools (e.g.,spreadsheet template, online amortizationtables), the effect of the length of time taken torepay a loan on the principal and interestcomponents of a personal loan repayment3.5 compare, using a variety of tools (e.g., spreadsheettemplate, online amortization tables),the effects of various payment periods (e.g.,monthly, biweekly) on the length of timetaken to repay a loan and on the total interestpaid3.6 gather and interpret information about creditratings, and describe the factors used to determinecredit ratings and the consequences of agood or bad rating3.7 make and justify a decision to borrow, usingvarious criteria (e.g., income, cost of borrowing,availability of an item, need for an item)under various circumstances (e.g., having alarge existing debt, wanting to pursue aneducation or training opportunity, needingtransportation to a new job, wanting to setup a business)82


C. TRANSPORTATION AND TRAVELOVERALL EXPECTATIONSBy the end of this course, students will:1. interpret information about owning and operating a vehicle, and solve problems involving theassociated costs;2. plan and justify a route for a trip by automobile, and solve problems involving the associated costs;3. interpret information about different modes of transportation, and solve related problems.SPECIFIC EXPECTATIONS1. Owning and Operating a VehicleBy the end of this course, students will:1.1 gather and interpret information about theprocedures (e.g., in the graduated licensingsystem) and costs (e.g., driver training; licensingfees) involved in obtaining an Ontario driver’slicence, and the privileges and restrictionsassociated with having a driver’s licence1.2 gather and describe information about theprocedures involved in buying or leasing anew vehicle or buying a used vehicle1.3 gather and interpret information about theprocedures and costs involved in insuring avehicle (e.g., car, motorcycle, snowmobile)and the factors affecting insurance rates (e.g.,gender, age, driving record, model of vehicle,use of vehicle), and compare the insurancecosts for different categories of drivers and fordifferent vehiclesSample problem: Use automobile insurancewebsites to investigate the degree to whichthe type of car and the age and gender of thedriver affect insurance rates.1.4 gather and interpret information about the costs(e.g., monthly payments, insurance, depreciation,maintenance, miscellaneous expenses)of purchasing or leasing a new vehicle orpurchasing a used vehicle, and describe theconditions that favour each alternativeSample problem: Compare the costs of buyinga new car, leasing the same car, and buyingan older model of the same car.1.5 describe ways of failing to operate a vehicleresponsibly (e.g., lack of maintenance,careless driving) and possible financial andnon-financial consequences (e.g., legal costs,fines, higher insurance rates, demerit points,loss of driving privileges)1.6 identify and describe costs (e.g., gas consumption,depreciation, insurance, maintenance)and benefits (e.g., convenience, increasedprofit) of owning and operating a vehicle forbusinessSample problem: Your employer pays35 cents/km for you to use your car forwork. Discuss how you would determinewhether or not this is fair compensation.1.7 solve problems, using technology (e.g., calculator,spreadsheet), that involve the fixed costs(e.g., licence fee, insurance) and variablecosts (e.g., maintenance, fuel) of owning andoperating a vehicleSample problem: The rate at which a car consumesgasoline depends on the speed of thecar. Use a given graph of gasoline consumption,in litres per 100 km, versus speed, inkilometres per hour, to determine how muchgasoline is used to drive 500 km at speeds of80 km/h, 100 km/h, and 120 km/h. Use thecurrent price of gasoline to calculate the costof driving 500 km at each of these speeds.Mathematics for Work and Everyday LifeMEL3ETRANSPORTATION AND TRAVEL83


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 11, Workplace Preparation2. Travelling by AutomobileBy the end of this course, students will:2.1 determine distances represented on maps(e.g., provincial road map, local street map,Web-based maps), using given scalesSample problem: Compare the driving distancesbetween two points on the same mapby two different routes.2.2 plan and justify, orally or in writing, a routefor a trip by automobile on the basis of a varietyof factors (e.g., distances involved, thepurpose of the trip, the time of year, the timeof day, probable road conditions, personalpriorities)2.3 report, orally or in writing, on the estimatedcosts (e.g., gasoline, accommodation, food,entertainment, tolls, car rental) involved in atrip by automobile, using information fromavailable sources (e.g., automobile associationtravel books, travel guides, the Internet)2.4 solve problems involving the cost of travellingby automobile for personal or businesspurposesSample problem: Determine and justify acost-effective delivery route for ten deliveriesto be made in a given area over two days.3. Comparing Modes of TransportationBy the end of this course, students will:3.1 gather, interpret, and describe informationabout the impact (e.g., monetary, health,environmental) of daily travel (e.g., to workand/or school), using available means (e.g.,car, taxi, motorcycle, public transportation,bicycle, walking)Sample problem: Discuss the impact if100 students decided to walk the 3-kmdistance to school instead of taking aschool bus.3.2 gather, interpret, and compare informationabout the costs (e.g., insurance, extra chargesbased on distance travelled) and conditions(e.g., one-way or return, drop-off time andlocation, age of the driver, required type ofdriver’s licence) involved in renting a car,truck, or trailer, and use the information tojustify a choice of rental vehicleSample problem: You want to rent a traileror a truck to help you move to a new apartment.Investigate the costs and describe theconditions that favour each option.3.3 gather, interpret, and describe informationregarding routes, schedules, and fares fortravel by airplane, train, or bus3.4 solve problems involving the comparison ofinformation concerning transportation by airplane,train, bus, and automobile in terms ofvarious factors (e.g., cost, time, convenience)Sample problem: Investigate the cost ofshipping a computer from Thunder Bay toWindsor by airplane, train, or bus. Describethe conditions that favour each alternative.84


Advanced Functions,Grade 12University PreparationMHF4UThis course extends students’ experience with functions. Students will investigate theproperties of polynomial, rational, logarithmic, and trigonometric functions; developtechniques for combining functions; broaden their understanding of rates of change; anddevelop facility in applying these concepts and skills. Students will also refine their useof the mathematical processes necessary for success in senior mathematics. This courseis intended both for students taking the Calculus and Vectors course as a prerequisitefor a university program and for those wishing to consolidate their understanding ofmathematics before proceeding to any one of a variety of university programs.Prerequisite: Functions, Grade 11, University Preparation, or Mathematics for CollegeTechnology, Grade 12, College Preparation85


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.86


A. EXPONENTIAL AND LOGARITHMICFUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of the relationship between exponential expressions and logarithmicexpressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;2. identify and describe some key features of the graphs of logarithmic functions, make connectionsamong the numeric, graphical, and algebraic representations of logarithmic functions, and solverelated problems graphically;3. solve exponential and simple logarithmic equations in one variable algebraically, including thosein problems arising from real-world applications.Advanced FunctionsSPECIFIC EXPECTATIONS1. Evaluating Logarithmic ExpressionsBy the end of this course, students will:1.1 recognize the logarithm of a number to agiven base as the exponent to which the basemust be raised to get the number, recognizethe operation of finding the logarithm to bethe inverse operation (i.e., the undoing orreversing) of exponentiation, and evaluatesimple logarithmic expressionsSample problem: Why is it not possible todetermine log10(– 3) or log20? Explain yourreasoning.1.2 determine, with technology, the approximatelogarithm of a number to any base, includingbase 10 (e.g., by reasoning that log329 isbetween 3 and 4 and using systematic trial todetermine that log 29 is approximately 3.07)31.3 make connections between related logarithmicand exponential equations (e.g., log5125 = 33can also be expressed as 5 = 125), and solvesimple exponential equations by rewritingxthem in logarithmic form (e.g., solving 3 = 10by rewriting the equation as log310 = x)1.4 make connections between the laws of exponentsand the laws of logarithms [e.g., usea+b a bthe statement 10 = 10 10 to deduce thatlog10x + log10y = log10(xy)], verify the laws oflogarithms with or without technology (e.g.,use patterning to verify the quotient law forlogarithms by evaluating expressions such aslog101000 – log10100 and then rewriting theanswer as a logarithmic term to the samebase), and use the laws of logarithms tosimplify and evaluate numerical expressions2. Connecting Graphs and Equationsof Logarithmic FunctionsBy the end of this course, students will:2.1 determine, through investigation with technology(e.g., graphing calculator, spreadsheet)and without technology, key features (i.e.,vertical and horizontal asymptotes, domainand range, intercepts, increasing/decreasingbehaviour) of the graphs of logarithmic functionsof the form f(x) = logbx, and make connectionsbetween the algebraic and graphicalrepresentations of these logarithmic functionsSample problem: Compare the key featuresof the graphs of f(x) = log2x, g(x) = log4x,and h(x) = log8x using graphing technology.2.2 recognize the relationship between an exponentialfunction and the corresponding logarithmicfunction to be that of a function andits inverse, deduce that the graph of a logarithmicfunction is the reflection of the graphof the corresponding exponential function inthe line y = x, and verify the deduction usingtechnologyMHF4UEXPONENTIAL AND LOGARITHMIC FUNCTIONS87


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University PreparationSample problem: Give examples to show thatthe inverse of a function is not necessarily afunction. Use the key features of the graphs oflogarithmic and exponential functions to givereasons why the inverse of an exponentialfunction is a function.2.3 determine, through investigation using technology,the roles of the parameters d and c infunctions of the form y = log10(x – d) + c andthe roles of the parameters a and k in functionsof the form y = alog10(kx), and describethese roles in terms of transformations on thegraph of f(x) = log10x (i.e., vertical andhorizontal translations; reflections in the axes;vertical and horizontal stretches andcompressions to and from the x- and y-axes)Sample problem: Investigate the graphs off(x) = log10(x) + c, f(x) = log10(x – d),f(x) = alog10x,and f(x) = log10(kx) forvarious values of c, d, a, and k, using technology,describe the effects of changing theseparameters in terms of transformations, andmake connections to the transformations ofother functions such as polynomial functions,exponential functions, and trigonometricfunctions.2.4 pose problems based on real-world applicationsof exponential and logarithmic functions(e.g., exponential growth and decay, theRichter scale, the pH scale, the decibel scale),and solve these and other such problems byusing a given graph or a graph generatedwith technology from a table of values orfrom its equationSample problem: The pH or acidity of a solutionis given by the equation pH = – logC,+where C is the concentration of [H ] ions inmultiples of M = 1 mol/L. Use graphingsoftware to graph this function. What is thechange in pH if the solution is diluted from aconcentration of 0.1M to a concentration of0.01M? From 0.001M to 0.0001M? Describethe change in pH when the concentration of1any acidic solution is reduced to of its10original concentration. Rearrange the givenequation to determine concentration as afunction of pH.3. Solving Exponential andLogarithmic EquationsBy the end of this course, students will:3.1 recognize equivalent algebraic expressionsinvolving logarithms and exponents, andsimplify expressions of these typesSample problem: Sketch the graphs off(x) = log10(100x) and g(x) = 2 + log10x,compare the graphs, and explain yourfindings algebraically.3.2 solve exponential equations in one variableby determining a common base (e.g., solvex x+34 = 8 by expressing each side as a powerof 2) and by using logarithms (e.g., solvex x+34 = 8 by taking the logarithm base 2of both sides), recognizing that logarithmsbase 10 are commonly used (e.g., solvingx3 = 7 by taking the logarithm base 10 ofboth sides)Sample problem: Solve 300(1.05) = 600 andx+2 x2 – 2 = 12 either by finding a commonbase or by taking logarithms, and explainyour choice of method in each case.3.3 solve simple logarithmic equations in onevariable algebraically [e.g., log3(5x + 6) = 2,log (x + 1) = 1]103.4 solve problems involving exponential andlogarithmic equations algebraically, includingproblems arising from real-worldapplicationsSample problem: The pH or acidity of a solutionis given by the equation pH = – logC,+where C is the concentration of [H ] ions inmultiples of M = 1 mol/L. You are given asolution of hydrochloric acid with a pH of 1.7and asked to increase the pH of the solutionby 1.4. Determine how much you must dilutethe solution. Does your answer differ if youstart with a pH of 2.2?n88


B. TRIGONOMETRIC FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of the meaning and application of radian measure;2. make connections between trigonometric ratios and the graphical and algebraic representations ofthe corresponding trigonometric functions and between trigonometric functions and their reciprocals,and use these connections to solve problems;3. solve problems involving trigonometric equations and prove trigonometric identities.SPECIFIC EXPECTATIONSAdvanced Functions1. Understanding and ApplyingRadian MeasureBy the end of this course, students will:1.1 recognize the radian as an alternative unit tothe degree for angle measurement, define theradian measure of an angle as the length ofthe arc that subtends this angle at the centreof a unit circle, and develop and apply therelationship between radian and degreemeasure1.2 represent radian measure in terms of π (e.g.,πradians, 2π radians) and as a rational number3(e.g., 1.05 radians, 6.28 radians)1.3 determine, with technology, the primarytrigonometric ratios (i.e., sine, cosine, tangent)and the reciprocal trigonometric ratios (i.e.,cosecant, secant, cotangent) of anglesexpressed in radian measure1.4 determine, without technology, the exactvalues of the primary trigonometric ratiosand the reciprocal trigonometric ratios forπ π π πthe special angles 0, , , , , and their6 4 3 2multiples less than or equal to 2π2. Connecting Graphs and Equationsof Trigonometric FunctionsBy the end of this course, students will:2.1 sketch the graphs of f(x) = sin x and f(x) = cos xfor angle measures expressed in radians, anddetermine and describe some key properties(e.g., period of 2π, amplitude of 1) in terms ofradians2.2 make connections between the tangent ratioand the tangent function by using technologyto graph the relationship between angles inradians and their tangent ratios and definingthis relationship as the function f(x) = tan x,and describe key properties of the tangentfunction2.3 graph, with technology and using the primarytrigonometric functions, the reciprocaltrigonometric functions (i.e., cosecant, secant,cotangent) for angle measures expressed inradians, determine and describe key propertiesof the reciprocal functions (e.g., state thedomain, range, and period, and identify andexplain the occurrence of asymptotes), andrecognize notations used to represent thereciprocal functions [e.g., the reciprocal off(x) = sin x can be represented using csc x,1 1–1 –1, or , but not using f (x) or sin x,f(x) sin xwhich represent the inverse function]MHF4UTRIGONOMETRIC FUNCTIONS89


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparation2.4 determine the amplitude, period, and phaseshift of sinusoidal functions whose equationsare given in the form f(x) = a sin (k(x – d)) + cor f(x) = acos(k(x – d)) + c, with anglesexpressed in radians2.5 sketch graphs of y = a sin (k(x – d)) + c andy = acos(k(x – d)) + c by applying transformationsto the graphs of f(x) = sin x andf(x) = cos x with angles expressed in radians,and state the period, amplitude, and phaseshift of the transformed functionsSample problem: Transform the graph off(x) = cos x to sketch g(x) = 3cos(2x) – 1,and state the period, amplitude, and phaseshift of each function.2.6 represent a sinusoidal function with anequation, given its graph or its properties,with angles expressed in radiansSample problem: A sinusoidal function hasan amplitude of 2 units, a period of π, and amaximum at (0, 3). Represent the functionwith an equation in two different ways.2.7 pose problems based on applications involvinga trigonometric function with domainexpressed in radians (e.g., seasonal changes intemperature, heights of tides, hours of daylight,displacements for oscillating springs),and solve these and other such problems byusing a given graph or a graph generatedwith or without technology from a table ofvalues or from its equationSample problem: The population size, P,of owls (predators) in a certain region canbe modelled by the functionP(t) = 1000 + 100 sin (πt) , where t represents12the time in months. The population size, p,of mice (prey) in the same region is given byp(t) = 20 000 + 4000 cos (πt) . Sketch the12graphs of these functions, and pose andsolve problems involving the relationshipsbetween the two populations over time.3. Solving Trigonometric EquationsBy the end of this course, students will:3.1 recognize equivalent trigonometric expressions[e.g., by using the angles in a right triangleto recognize that sin x and cos (π– x ) are2equivalent; by using transformations torecognize that cos (πx + ) and –sin x are2equivalent], and verify equivalence usinggraphing technology3.2 explore the algebraic development of thecompound angle formulas (e.g., verify theformulas in numerical examples, using technology;follow a demonstration of the algebraicdevelopment [student reproduction ofthe development of the general case is notrequired]), and use the formulas to determineexact values of trigonometric ratios [e.g.,determining the exact value of sin (π) by12first rewriting it in terms of special anglesas sin (π π–4 6) ]3.3 recognize that trigonometric identities areequations that are true for every value in thedomain (i.e., a counter-example can be usedto show that an equation is not an identity),prove trigonometric identities through theapplication of reasoning skills, using a varietysin xof relationships (e.g., tan x =cos x;2 2sin x + cos x = 1; the reciprocal identities;the compound angle formulas), and verifyidentities using technologySample problem: Use the compound angleformulas to prove the double angle formulas.3.4 solve linear and quadratic trigonometric equations,with and without graphing technology,for the domain of real values from 0 to 2π,and solve related problemsSample problem: Solve the following trigonometricequations for 0 ≤ x ≤ 2π, and verify bygraphing with technology: 2 sin x + 1 = 0;22 sin x + sin x – 1 = 0; sin x = cos 2x;1cos 2x = .290


C. POLYNOMIAL AND RATIONALFUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. identify and describe some key features of polynomial functions, and make connections between thenumeric, graphical, and algebraic representations of polynomial functions;2. identify and describe some key features of the graphs of rational functions, and represent rationalfunctions graphically;3. solve problems involving polynomial and simple rational equations graphically and algebraically;4. demonstrate an understanding of solving polynomial and simple rational inequalities.Advanced FunctionsSPECIFIC EXPECTATIONS1. Connecting Graphs and Equationsof Polynomial FunctionsBy the end of this course, students will:1.1 recognize a polynomial expression (i.e., aseries of terms where each term is the productof a constant and a power of x with a nonnegativeintegral exponent, such as3 2x – 5x + 2x – 1); recognize the equation ofa polynomial function, give reasons why itis a function, and identify linear and quadraticfunctions as examples of polynomialfunctions1.2 compare, through investigation using graphingtechnology, the numeric, graphical, andalgebraic representations of polynomial (i.e.,linear, quadratic, cubic, quartic) functions(e.g., compare finite differences in tables ofvalues; investigate the effect of the degree of apolynomial function on the shape of its graphand the maximum number of x-intercepts;investigate the effect of varying the sign of theleading coefficient on the end behaviour ofthe function for very large positive or negativex-values)Sample problem: Investigate the maximumnumber of x-intercepts for linear, quadratic,cubic, and quartic functions using graphingtechnology.1.3 describe key features of the graphs of polynomialfunctions (e.g., the domain and range,the shape of the graphs, the end behaviour ofthe functions for very large positive or negativex-values)Sample problem: Describe and compare thekey features of the graphs of the functions2 3 3 2f(x) = x, f(x) = x , f(x) = x , f(x) = x + x ,3and f(x) = x + x.1.4 distinguish polynomial functions fromsinusoidal and exponential functions [e.g.,xf(x) = sin x, g(x) = 2 ], and compare andcontrast the graphs of various polynomialfunctions with the graphs of other types offunctions1.5 make connections, through investigationusing graphing technology (e.g., dynamicgeometry software), between a polynomialfunction given in factored form [e.g.,f(x) = 2(x – 3)(x + 2)(x – 1)] and thex-intercepts of its graph, and sketch thegraph of a polynomial function given infactored form using its key features (e.g.,by determining intercepts and end behaviour;by locating positive and negativeregions using test values between and oneither side of the x-intercepts)Sample problem: Investigate, using graphingtechnology, the x-intercepts and the shapesof the graphs of polynomial functions withMHF4UPOLYNOMIAL AND RATIONAL FUNCTIONS91


Grade 12, University PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematicsone or more repeated factors, for example,f(x) = (x – 2)(x – 3), f(x) = (x – 2)(x – 2)(x – 3),f(x) = (x – 2)(x – 2)(x – 2)(x – 3), andf(x) = (x + 2)(x + 2)(x – 2)(x – 2)(x – 3),by considering whether the factor isrepeated an even or an odd number oftimes. Use your conclusions to sketchf(x) = (x + 1)(x + 1)(x – 3)(x – 3), and verifyusing technology.1.6 determine, through investigation using technology,the roles of the parameters a, k, d, andc in functions of the form y = af(k(x – d)) + c,and describe these roles in terms of transformationson the graphs of f(x) = x and f(x) = x3 4(i.e., vertical and horizontal translations;reflections in the axes; vertical and horizontalstretches and compressions to and from thex- and y-axes)Sample problem: Investigate, using technology,the graph of f(x) = 2(x – d) + c for3various values of d and c, and describethe effects of changing d and c in terms oftransformations.1.7 determine an equation of a polynomial functionthat satisfies a given set of conditions (e.g.,degree of the polynomial, intercepts, pointson the function), using methods appropriateto the situation (e.g., using the x-intercepts ofthe function; using a trial-and-error processwith a graphing calculator or graphing software;using finite differences), and recognizethat there may be more than one polynomialfunction that can satisfy a given set of conditions(e.g., an infinite number of polynomialfunctions satisfy the condition that they havethree given x-intercepts)Sample problem: Determine an equation fora fifth-degree polynomial function that intersectsthe x-axis at only 5, 1, and –5, andsketch the graph of the function.1.8 determine the equation of the family of polynomialfunctions with a given set of zerosand of the member of the family that passesthrough another given point [e.g., a familyof polynomial functions of degree 3 withzeros 5, –3, and –2 is defined by the equationf(x) = k(x – 5)(x + 3)(x + 2), where k is a realnumber, k ≠ 0; the member of the familythat passes through (–1, 24) isf(x) = –2(x – 5)(x + 3)(x + 2)]Sample problem: Investigate, using graphingtechnology, and determine a polynomialfunction that can be used to model the functionf(x) = sin x over the interval 0 ≤ x ≤ 2π.1.9 determine, through investigation, and comparethe properties of even and odd polynomialfunctions [e.g., symmetry about the y-axisor the origin; the power of each term; thenumber of x-intercepts; f(x) = f(– x) orf(– x) = – f(x)], and determine whether a givenpolynomial function is even, odd, or neitherSample problem: Investigate numerically,graphically, and algebraically, with and withouttechnology, the conditions under whichan even function has an even number ofx-intercepts.2. Connecting Graphs and Equationsof Rational FunctionsBy the end of this course, students will:2.1 determine, through investigation with andwithout technology, key features (i.e., verticaland horizontal asymptotes, domain andrange, intercepts, positive/negative intervals,increasing/decreasing intervals) of the graphsof rational functions that are the reciprocals oflinear and quadratic functions, and make connectionsbetween the algebraic and graphicalrepresentations of these rational functions [e.g.,1make connections between f(x) =2x – 4and its graph by using graphing technologyand by reasoning that there are verticalasymptotes at x = 2 and x = –2 and a horizontalasymptote at y = 0 and that the functionmaintains the same sign as f(x) = x –24]Sample problem: Investigate, with technology,the key features of the graphs of families ofrational functions of the form11f(x) = and f(x) = ,2x + nx + nwhere n is an integer, and make connectionsbetween the equations and key features ofthe graphs.2.2 determine, through investigation with andwithout technology, key features (i.e., verticaland horizontal asymptotes, domain andrange, intercepts, positive/negative intervals,increasing/decreasing intervals) of the graphsof rational functions that have linear expressionsin the numerator and denominator2x x – 2[e.g., f(x) = , h(x) = ], andx – 3 3x + 4make connections between the algebraic andgraphical representations of these rationalfunctions92


Sample problem: Investigate, using graphingtechnology, key features of the graphs of thefamily of rational functions of the form8xf(x) = for n = 1, 2, 4, and 8, and makenx + 1connections between the equations and theasymptotes.2.3 sketch the graph of a simple rational functionusing its key features, given the algebraic representationof the function3. Solving Polynomial and RationalEquationsBy the end of this course, students will:3.1 make connections, through investigation usingtechnology (e.g., computer algebra systems),between the polynomial function f(x), thedivisor x – a, the remainder from the divisionf(x), and f(a) to verify the remainder theoremx – aand the factor theoremSample problem: Divide4 3 2f(x) = x + 4x – x – 16x – 14 by x – a forvarious integral values of a using a computeralgebra system. Compare the remainder fromeach division with f(a).3.2 factor polynomial expressions in one variable,of degree no higher than four, by selectingand applying strategies (i.e., common factoring,difference of squares, trinomial factoring,factoring by grouping, remainder theorem,factor theorem)3 2Sample problem: Factor: x + 2x – x – 2;4 3 2x – 6x + 4x + 6x – 5.3.3 determine, through investigation using technology(e.g., graphing calculator, computeralgebra systems), the connection between thereal roots of a polynomial equation and thex-intercepts of the graph of the correspondingpolynomial function, and describe this connection[e.g., the real roots of the equation4 2x – 13x + 36 = 0 are the x-intercepts of the4 2graph of f(x) = x – 13x + 36]Sample problem: Describe the relationshipbetween the x-intercepts of the graphs oflinear and quadratic functions and the realroots of the corresponding equations.Investigate, using technology, whether thisrelationship exists for polynomial functionsof higher degree.3.4 solve polynomial equations in one variable,of degree no higher than four (e.g.,3 22x – 3x + 8x – 12 = 0), by selecting andapplying strategies (i.e., common factoring,difference of squares, trinomial factoring,factoring by grouping, remainder theorem,factor theorem), and verify solutions usingtechnology (e.g., using computer algebrasystems to determine the roots; using graphingtechnology to determine the x-interceptsof the graph of the corresponding polynomialfunction)3.5 determine, through investigation using technology(e.g., graphing calculator, computeralgebra systems), the connection betweenthe real roots of a rational equation and thex-intercepts of the graph of the correspondingrational function, and describe this connectionx – 2[e.g., the real root of the equation = 0x – 3is 2, which is the x-intercept of the functionx – 21f(x) = ; the equation = 0has nox – 3x – 31real roots, and the function f(x) = doesx – 3not intersect the x-axis]3.6 solve simple rational equations in one variablealgebraically, and verify solutions using technology(e.g., using computer algebra systemsto determine the roots; using graphing technologyto determine the x-intercepts of thegraph of the corresponding rational function)3.7 solve problems involving applications ofpolynomial and simple rational functions andequations [e.g., problems involving the factortheorem or remainder theorem, such as determiningthe values of k for which the function3 2f(x) = x + 6x + kx – 4 gives the same remainderwhen divided by x – 1 and x + 2]Sample problem: Use long division to express2x + 3x – 5the given function f(x) =as thex – 1sum of a polynomial function and a rationalAfunction of the form (where A is ax – 1constant), make a conjecture about the relationshipbetween the given function and thepolynomial function for very large positiveand negative x-values, and verify your conjectureusing graphing technology.Advanced FunctionsMHF4UPOLYNOMIAL AND RATIONAL FUNCTIONS93


4. Solving InequalitiesTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University PreparationBy the end of this course, students will:4.1 explain, for polynomial and simple rationalfunctions, the difference between the solutionto an equation in one variable and the solutionto an inequality in one variable, anddemonstrate that given solutions satisfy aninequality (e.g., demonstrate numericallyand graphically that the solution to14< 5 is x < –1 or x > – );x + 154.2 determine solutions to polynomial inequalitiesin one variable [e.g., solve f(x) ≥ 0, where3 2f(x) = x – x + 3x – 9] and to simple rationalinequalities in one variable by graphing thecorresponding functions, using graphing technology,and identifying intervals for which xsatisfies the inequalities4.3 solve linear inequalities and factorable polynomialinequalities in one variable (e.g.,3 2x + x > 0) in a variety of ways (e.g., by determiningintervals using x-intercepts and evaluatingthe corresponding function for a singlex-value within each interval; by factoring thepolynomial and identifying the conditions forwhich the product satisfies the inequality),and represent the solutions on a number lineor algebraically (e.g., for the inequality4 2x – 5x + 4 < 0, the solution representedalgebraically is – 2 < x < –1 or 1 < x < 2)94


D. CHARACTERISTICS OF FUNCTIONSGrade 12, University PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of average and instantaneous rate of change, and determine,numerically and graphically, and interpret the average rate of change of a function over a giveninterval and the instantaneous rate of change of a function at a given point;2. determine functions that result from the addition, subtraction, multiplication, and division of twofunctions and from the composition of two functions, describe some properties of the resultingfunctions, and solve related problems;3. compare the characteristics of functions, and solve problems by modelling and reasoning withfunctions, including problems with solutions that are not accessible by standard algebraic techniques.SPECIFIC EXPECTATIONSAdvanced Functions1. Understanding Rates of ChangeBy the end of this course, students will:1.1 gather, interpret, and describe informationabout real-world applications of rates ofchange, and recognize different ways ofrepresenting rates of change (e.g., in words,numerically, graphically, algebraically)1.2 recognize that the rate of change for a functionis a comparison of changes in the dependentvariable to changes in the independentvariable, and distinguish situations in whichthe rate of change is zero, constant, or changingby examining applications, includingthose arising from real-world situations (e.g.,rate of change of the area of a circle as theradius increases, inflation rates, the risingtrend in graduation rates among Aboriginalyouth, speed of a cruising aircraft, speed of acyclist climbing a hill, infection rates)Sample problem: The population of bacteriain a sample is 250 000 at 1:00 p.m., 500 000 at3:00 p.m., and 1 000 000 at 5:00 p.m. Comparemethods used to calculate the change inthe population and the rate of change in thepopulation between 1:00 p.m. to 5:00 p.m. Isthe rate of change constant? Explain yourreasoning.1.3 sketch a graph that represents a relationshipinvolving rate of change, as described inwords, and verify with technology (e.g.,motion sensor) when possibleSample problem: John rides his bicycle at aconstant cruising speed along a flat road. Hethen decelerates (i.e., decreases speed) as heclimbs a hill. At the top, he accelerates (i.e.,increases speed) on a flat road back to hisconstant cruising speed, and he then acceleratesdown a hill. Finally, he comes to anotherhill and glides to a stop as he starts to climb.Sketch a graph of John’s speed versus timeand a graph of his distance travelled versustime.1.4 calculate and interpret average rates of changeof functions (e.g., linear, quadratic, exponential,sinusoidal) arising from real-world applications(e.g., in the natural, physical, and social sciences),given various representations of the functions(e.g., tables of values, graphs, equations)Sample problem: Fluorine-20 is a radioactivesubstance that decays over time. At time 0,the mass of a sample of the substance is 20 g.The mass decreases to 10 g after 11 s, to 5 gafter 22 s, and to 2.5 g after 33 s. Comparethe average rate of change over the 33-sinterval with the average rate of change overconsecutive 11-s intervals.1.5 recognize examples of instantaneous rates ofchange arising from real-world situations, andmake connections between instantaneousrates of change and average rates of change(e.g., an average rate of change can be used toapproximate an instantaneous rate of change)MHF4UCHARACTERISTICS OF FUNCTIONS95


Grade 12, University PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | MathematicsSample problem: In general, does the speedometerof a car measure instantaneous rate ofchange (i.e., instantaneous speed) or averagerate of change (i.e., average speed)? Describesituations in which the instantaneous speedand the average speed would be the same.1.6 determine, through investigation using variousrepresentations of relationships (e.g., tables ofvalues, graphs, equations), approximate instantaneousrates of change arising from real-worldapplications (e.g., in the natural, physical, andsocial sciences) by using average rates ofchange and reducing the interval over whichthe average rate of change is determinedSample problem: The distance, d metres,travelled by a falling object in t seconds is2represented by d = 5t . When t = 3, theinstantaneous speed of the object is 30 m/s.Compare the average speeds over differenttime intervals starting at t = 3 with theinstantaneous speed when t = 3. Use yourobservations to select an interval that can beused to provide a good approximation of theinstantaneous speed at t = 3.1.7 make connections, through investigation,between the slope of a secant on the graphof a function (e.g., quadratic, exponential,sinusoidal) and the average rate of changeof the function over an interval, and betweenthe slope of the tangent to a point on thegraph of a function and the instantaneousrate of change of the function at that pointSample problem: Use tangents to investigatethe behaviour of a function when the instantaneousrate of change is zero, positive, ornegative.1.8 determine, through investigation using a varietyof tools and strategies (e.g., using a tableof values to calculate slopes of secants orgraphing secants and measuring their slopeswith technology), the approximate slope ofthe tangent to a given point on the graph ofa function (e.g., quadratic, exponential, sinusoidal)by using the slopes of secants throughthe given point (e.g., investigating the slopesof secants that approach the tangent at thatpoint more and more closely), and make connectionsto average and instantaneous ratesof change1.9 solve problems involving average and instantaneousrates of change, including problemsarising from real-world applications, by usingnumerical and graphical methods (e.g., byusing graphing technology to graph a tangentand measure its slope)Sample problem: The height, h metres, of aball above the ground can be modelled by2the function h(t) = – 5t + 20t, where t isthe time in seconds. Use average speeds todetermine the approximate instantaneousspeed at t = 3.2. Combining FunctionsBy the end of this course, students will:2.1 determine, through investigation using graphingtechnology, key features (e.g., domain,range, maximum/minimum points, numberof zeros) of the graphs of functions created byadding, subtracting, multiplying, or dividing–x 2 xfunctions [e.g., f(x) = 2 sin 4x, g(x) = x + 2,sin xh(x) = ], and describe factors that affectcos xthese propertiesSample problem: Investigate the effect ofthe behaviours of f(x) = sin x, f(x) = sin 2x,and f(x) = sin 4x on the shape off(x) = sin x + sin 2x + sin 4x.2.2 recognize real-world applications of combinationsof functions (e.g., the motion of adamped pendulum can be represented by afunction that is the product of a trigonometricfunction and an exponential function; the frequenciesof tones associated with the numberson a telephone involve the addition of twotrigonometric functions), and solve relatedproblems graphicallySample problem: The rate at which a contaminantleaves a storm sewer and enters alake depends on two factors: the concentrationof the contaminant in the water from thesewer and the rate at which the water leavesthe sewer. Both of these factors vary withtime. The concentration of the contaminant,in kilograms per cubic metre of water, is2given by c(t) = t , where t is in seconds. Therate at which water leaves the sewer, in cubic1metres per second, is given by w(t) = .4t + 10Determine the time at which the contaminantleaves the sewer and enters the lake at themaximum rate.96


2.3 determine, through investigation, and explainsome properties (i.e., odd, even, or neither;increasing/decreasing behaviours) of functionsformed by adding, subtracting, multiplying,and dividing general functions [e.g.,f(x) + g(x), f(x)g(x)]Sample problem: Investigate algebraically,and verify numerically and graphically,whether the product of two functions is evenor odd if the two functions are both even orboth odd, or if one function is even and theother is odd.2.4 determine the composition of two functions[i.e., f(g(x))] numerically (i.e., by using a tableof values) and graphically, with technology,for functions represented in a variety of ways(e.g., function machines, graphs, equations),and interpret the composition of two functionsin real-world applicationsSample problem: For a car travelling at a constantspeed, the distance driven, d kilometres,is represented by d(t) = 80t, where t is thetime in hours. The cost of gasoline, in dollars,for the drive is represented by C(d) = 0.09d.Determine numerically and interpret C(d(5)),and describe the relationship represented byC(d(t)).2.5 determine algebraically the composition oftwo functions [i.e., f(g(x))], verify that f(g(x))is not always equal to g( f(x)) [e.g., by determiningf(g(x)) and g( f(x)), given f(x) = x + 1and g(x) = 2x], and state the domain [i.e., bydefining f(g(x)) for those x-values for whichg(x) is defined and for which it is included inthe domain of f(x)] and the range of the compositionof two functionsSample problem: Determine f(g(x)) and g( f(x))given f(x) = cos x and g(x) = 2x + 1, state thedomain and range of f(g(x)) and g( f(x)), comparef(g(x)) with g( f(x)) algebraically, andverify numerically and graphically withtechnology.2.6 solve problems involving the composition oftwo functions, including problems arisingfrom real-world applicationsSample problem: The speed of a car, v kilometresper hour, at a time of t hours is representedby v(t) = 40 + 3t + t . The rate of2gasoline consumption of the car, c litres perkilometre, at a speed of v kilometres per houris represented by c(v) = (v 2– 0.1 ) + 0.15.500Determine algebraically c(v(t)), the rate ofgasoline consumption as a function of time.Determine, using technology, the time whenthe car is running most economically duringa four-hour trip.2.7 demonstrate, by giving examples for functionsrepresented in a variety of ways (e.g.,function machines, graphs, equations), theproperty that the composition of a functionand its inverse function maps a number onto–1 –1itself [i.e., f ( f(x)) = x and f( f (x)) = xdemonstrate that the inverse function is thereverse process of the original function andthat it undoes what the function does]2.8 make connections, through investigationusing technology, between transformations(i.e., vertical and horizontal translations;reflections in the axes; vertical and horizontalstretches and compressions to and from thex- and y-axes) of simple functions f(x) [e.g.,3f(x) = x + 20, f(x) = sin x, f(x) = log x] andthe composition of these functions with alinear function of the form g(x) = A(x + B)Sample problem: Compare the graph of2f(x) = x with the graphs of f(g(x)) andg( f(x)), where g(x) = 2(x – d), for variousvalues of d. Describe the effects of d interms of transformations of f(x).3. Using Function Models to SolveProblemsBy the end of this course, students will:3.1 compare, through investigation using a varietyof tools and strategies (e.g., graphing withtechnology; comparing algebraic representations;comparing finite differences in tables ofvalues) the characteristics (e.g., key features ofthe graphs, forms of the equations) of variousfunctions (i.e., polynomial, rational, trigonometric,exponential, logarithmic)3.2 solve graphically and numerically equationsand inequalities whose solutions are notaccessible by standard algebraic techniques2 xSample problem: Solve: 2x < 2; cosx = x,with x in radians.3.3 solve problems, using a variety of tools andstrategies, including problems arising fromreal-world applications, by reasoning withfunctions and by applying concepts andprocedures involving functions (e.g., byAdvanced FunctionsMHF4UCHARACTERISTICS OF FUNCTIONS97


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparationconstructing a function model from data,using the model to determine mathematicalresults, and interpreting and communicatingthe results within the context of the problem)Sample problem: The pressure of a car tirewith a slow leak is given in the followingtable of values:Time, t (min) Pressure, P (kPa)0 4005 33510 29515 25520 22525 19530 170Use technology to investigate linear, quadratic,and exponential models for the relationshipof the tire pressure and time, and describehow well each model fits the data. Use eachmodel to predict the pressure after 60 min.Which model gives the most realisticanswer?98


Calculus and Vectors,Grade 12University PreparationMCV4UThis course builds on students’ previous experience with functions and their developingunderstanding of rates of change. Students will solve problems involving geometric andalgebraic representations of vectors and representations of lines and planes in threedimensionalspace; broaden their understanding of rates of change to include thederivatives of polynomial, sinusoidal, exponential, rational, and radical functions; andapply these concepts and skills to the modelling of real-world relationships. Studentswill also refine their use of the mathematical processes necessary for success in seniormathematics. This course is intended for students who choose to pursue careers in fieldssuch as science, engineering, economics, and some areas of business, including thosestudents who will be required to take a university-level calculus, linear algebra, orphysics course.Note: The new Advanced Functions course (MHF4U) must be taken prior to orconcurrently with Calculus and Vectors (MCV4U).99


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.100


A. RATE OF CHANGEOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of rate of change by making connections between average rate ofchange over an interval and instantaneous rate of change at a point, using the slopes of secants andtangents and the concept of the limit;2. graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connectionsbetween the numeric, graphical, and algebraic representations of a function and its derivative;3. verify graphically and algebraically the rules for determining derivatives; apply these rules todetermine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions,and simple combinations of functions; and solve related problems.Calculus and VectorsSPECIFIC EXPECTATIONS1. Investigating Instantaneous Rateof Change at a PointBy the end of this course, students will:1.1 describe examples of real-world applicationsof rates of change, represented in a variety ofways (e.g., in words, numerically, graphically,algebraically)1.2 describe connections between the average rateof change of a function that is smooth (i.e.,continuous with no corners) over an intervaland the slope of the corresponding secant,and between the instantaneous rate of changeof a smooth function at a point and the slopeof the tangent at that pointSample problem: Given the graph of f(x)shown below, explain why the instantaneousrate of change of the function cannot bedetermined at point P.−33−3yP3x1.3 make connections, with or without graphingtechnology, between an approximate valueof the instantaneous rate of change at a givenpoint on the graph of a smooth function andaverage rates of change over intervals containingthe point (i.e., by using secants through thegiven point on a smooth curve to approachthe tangent at that point, and determining theslopes of the approaching secants to approximatethe slope of the tangent)1.4 recognize, through investigation with orwithout technology, graphical and numericalexamples of limits, and explain the reasoninginvolved (e.g., the value of a functionapproaching an asymptote, the value of theratio of successive terms in the Fibonaccisequence)Sample problem: Use appropriate technologyto investigate the limiting value of the termsin the sequence (111 + ) , (121 + ) , (131 + ) ,1 2 3( 1 + 14) , …, and the limiting value of the series41 1 1 14 x 1 – 4 x + 4 x – 4 x + 4 x – ….3 5 7 91.5 make connections, for a function that is smoothover the interval a ≤ x ≤ a + h, between theaverage rate of change of the function overthis interval and the value of the expressionf(a + h) – f(a), and between the instantaneoushrate of change of the function at x = a and thef(a + h) – f(a)value of the limit limh→ 0 hMCV4URATE OF CHANGE101


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparation102Sample problem: What does the limitf(4 + h) – f(4)lim= 8 indicate about theh→ 0 h2graph of the function f(x) = x ? The graph ofa general function y = f(x)?1.6 compare, through investigation, the calculationof instantaneous rates of change at apoint (a, f(a)) for polynomial functions2 3[e.g., f(x) = x , f(x) = x ], with and withoutf(a + h) – f(a)simplifying the expressionhbefore substituting values of h that approach2zero [e.g., for f(x) = x at x = 3, by determiningf(3 + 1) – f(3) f(3 + 0.1) – f(3)= 7, = 6.1,10.1f(3 + 0.01) – f(3)= 6.01, and0.01f(3 + 0.001) – f(3)= 6.001, and0.001f(3 + h) – f(3)by first simplifyingash2 2(3 + h) – 3= 6 + h and then substitutinghthe same values of h to give the same results]2. Investigating the Concept of theDerivative FunctionBy the end of this course, students will:2.1 determine numerically and graphically theintervals over which the instantaneous rateof change is positive, negative, or zero for afunction that is smooth over these intervals(e.g., by using graphing technology to examinethe table of values and the slopes of tangentsfor a function whose equation is given;by examining a given graph), and describe thebehaviour of the instantaneous rate of changeat and between local maxima and minimaSample problem: Given a smooth functionfor which the slope of the tangent is alwayspositive, explain how you know that thefunction is increasing. Give an example ofsuch a function.2.2 generate, through investigation using technology,a table of values showing the instantaneousrate of change of a polynomialfunction, f(x), for various values of x (e.g.,construct a tangent to the function, measureits slope, and create a slider or animation tomove the point of tangency), graph theordered pairs, recognize that the graphrepresents a function called the derivative,dyf ’(x) or , and make connections betweendxdythe graphs of f(x) and f ’(x) or y anddx[e.g., when f(x) is linear, f ’(x) is constant;when f(x) is quadratic, f ’(x) is linear; whenf(x) is cubic, f ’(x) is quadratic]Sample problem: Investigate, using patterningstrategies and graphing technology, relationshipsbetween the equation of a polynomialfunction of degree no higher than 3 and theequation of its derivative.2.3 determine the derivatives of polynomial functionsby simplifying the algebraic expressionf(x + h) – f(x)and then taking the limit of thehsimplified expression as h approaches zerolim f(x + h) – f(x)[i.e., determining h→ lim0]h→ h2.4 determine, through investigation using technology,the graph of the derivative f ’(x) ordyof a given sinusoidal function [i.e.,dxf(x) = sin x, f(x) = cos x] (e.g., by generatinga table of values showing the instantaneousrate of change of the function for various valuesof x and graphing the ordered pairs; by usingdynamic geometry software to verify graphicallythat when f(x) = sin x, f ’(x) = cos x, andwhen f(x) = cos x, f ’(x) = – sin x; by usinga motion sensor to compare the displacementand velocity of a pendulum)2.5 determine, through investigation using technology,the graph of the derivative f ’(x) ordyof a given exponential function [i.e.,dxxf(x) = a (a > 0, a ≠ 1)] [e.g., by generating atable of values showing the instantaneous rateof change of the function for various valuesof x and graphing the ordered pairs; by usingdynamic geometry software to verify thatxwhen f(x) = a , f ’(x) = kf(x)], and make connectionsbetween the graphs of f(x) and f ’(x)dyor y and [e.g., f(x) and f ’(x) are bothdxf ’(x)exponential; the ratio is constant, orf(x)f ’(x) = kf(x); f ’(x) is a vertical stretch fromthe x-axis of f(x)]Sample problem: Graph, with technology,xf(x) = a (a > 0, a ≠ 1) and f ’(x) on the sameset of axes for various values of a (e.g., 1.7,2.0, 2.3, 3.0, 3.5). For each value of a,f ’(x)investigate the ratio for various valuesf(x)of x, and explain how you can use this ratioto determine the slopes of tangents to f(x).


2.6 determine, through investigation using technology,the exponential function f(x) = ax(a > 0, a ≠ 1) for which f ’(x) = f(x) (e.g., byusing graphing technology to create a sliderthat varies the value of a in order to determinethe exponential function whose graph isthe same as the graph of its derivative), identifythe number e to be the value of a for whichxxf ’(x) = f(x) [i.e., given f(x) = e , f ’(x) = e ], andrecognize that for the exponential functionxf(x) = e the slope of the tangent at any pointon the function is equal to the value of thefunction at that pointSample problem: Use graphing technology todetermine an approximate value of e by graphingf(x) = a (a > 0, a ≠ 1) for various valuesxof a, comparing the slope of the tangent at apoint with the value of the function at thatpoint, and identifying the value of a for whichthey are equal.2.7 recognize that the natural logarithmic functionf(x) = log e x, also written as f(x) = ln x,is the inverse of the exponential functionxf(x) = e , and make connections betweenxf(x) = ln x and f(x) = e [e.g., f(x) = ln xxreverses what f(x) = e does; their graphs arereflections of each other in the line y = x; thelnx xcomposition of the two functions, e or ln e ,lnxmaps x onto itself, that is, e = x andxln e = x]2.8 verify, using technology (e.g., calculator,graphing technology), that the derivativexof the exponential function f(x) = a isxf ’(x) = a ln a for various values of a [e.g.,xverifying numerically for f(x) = 2 thatxf ’(x) = 2 ln 2 by using a calculator to showh(2 – 1)xthat lim is ln 2 or by graphing f(x) = 2,h→ 0 hdetermining the value of the slope and thevalue of the function for specific x-values, andf ’(x)comparing the ratio with ln 2]f(x)xSample problem: Given f(x) = e , verifynumerically with technology usingx + h x(e – e )limthat f ’(x) = f(x)lne.h→ 0 h3. Investigating the Properties ofDerivativesBy the end of this course, students will:3.1 verify the power rule for functions of the formnf(x) = x , where n is a natural number [e.g., bydetermining the equations of the derivatives2 3of the functions f(x) = x, f(x) = x , f(x) = x,4and f(x) = x algebraically usingf(x + h) – f(x)limand graphically using slopesh→ 0 hof tangents]3.2 verify the constant, constant multiple, sum,and difference rules graphically and numerically[e.g., by using the function g(x) = kf(x)and comparing the graphs of g’(x) and kf’(x);by using a table of values to verify thatf ’(x) + g’(x) = ( f + g)’(x), given f(x) = x andg(x) = 3x], and read and interpret proofsf(x + h) – f(x)involving limof the constant,h→ 0 hconstant multiple, sum, and difference rules(student reproduction of the development ofthe general case is not required)Sample problem: The amounts of water flowinginto two barrels are represented by thefunctions f (t) and g(t). Explain what f ’(t),g’(t), f ’(t) + g’(t), and ( f + g)’(t) represent.Explain how you can use this context to verifythe sum rule, f ’(t) + g’(t) = ( f + g)’(t).3.3 determine algebraically the derivatives ofpolynomial functions, and use these derivativesto determine the instantaneous rate ofchange at a point and to determine point(s)at which a given rate of change occursSample problem: Determine algebraically the3 2derivative of f(x) = 2x + 3x and the point(s)at which the slope of the tangent is 36.3.4 verify that the power rule applies to functionsnof the form f(x) = x , where n is a rationalnumber [e.g., by comparing values of the1slopes of tangents to the function f(x) = x 2with values of the derivative function determinedusing the power rule], and verifyCalculus and VectorsMCV4URATE OF CHANGE103


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparationalgebraically the chain rule using monomialfunctions [e.g., by determining the same3derivative for f(x) = (5x )1 3 by using the chainrule and by differentiating the simplified1form, f(x) = 5 3x] and the product rule usingpolynomial functions [e.g., by determining the2same derivative for f(x) = (3x + 2)(2x – 1) byusing the product rule and by differentiating3 2the expanded form f(x) = 6x + 4x – 3x – 2]Sample problem: Verify the chain rule byusing the product rule to look for patterns in22 2the derivatives of f(x) = x + 1, f(x) = (x + 1) ,2 3 2 4f(x) = (x + 1) , and f(x) = (x + 1) .3.5 solve problems, using the product and chainrules, involving the derivatives of polynomialfunctions, sinusoidal functions, exponentialfunctions, rational functions [e.g., by2x + 1expressing f(x) = as the productx – 12–1f(x) = (x + 1)(x – 1) ], radical functions [e.g.,2by expressing f(x) = √x + 5 as the power21f(x) = (x + 5) 2 ], and other simple combinationssin xof functions [e.g., f(x) = x sin x, f(x) =cos x] **The emphasis of this expectation is on the application of the derivative rules and not on the simplification of resulting complexalgebraic expressions.104


B. DERIVATIVES AND THEIRAPPLICATIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. make connections, graphically and algebraically, between the key features of a function and its firstand second derivatives, and use the connections in curve sketching;2. solve problems, including optimization problems, that require the use of the concepts and proceduresassociated with the derivative, including problems arising from real-world applications and involvingthe development of mathematical models.Calculus and VectorsSPECIFIC EXPECTATIONS1. Connecting Graphs and Equationsof Functions and Their DerivativesBy the end of this course, students will:1.1 sketch the graph of a derivative function,given the graph of a function that is continuousover an interval, and recognize points ofinflection of the given function (i.e., points atwhich the concavity changes)Sample problem: Investigate the effect on thegraph of the derivative of applying verticaland horizontal translations to the graph of agiven function.1.2 recognize the second derivative as the rate ofchange of the rate of change (i.e., the rate ofchange of the slope of the tangent), and sketchthe graphs of the first and second derivatives,given the graph of a smooth function1.3 determine algebraically the equation of thesecond derivative f”(x) of a polynomial orsimple rational function f(x), and makeconnections, through investigation usingtechnology, between the key features of thegraph of the function (e.g., increasing/decreasing intervals, local maxima andminima, points of inflection, intervals of concavity)and corresponding features of thegraphs of its first and second derivatives(e.g., for an increasing interval of the function,the first derivative is positive; for a point ofinflection of the function, the slopes of tangentschange their behaviour from increasing todecreasing or from decreasing to increasing,the first derivative has a maximum or minimum,and the second derivative is zero)Sample problem: Investigate, using graphingtechnology, connections between key properties,such as increasing/decreasing intervals,local maxima and minima, points of inflection,and intervals of concavity, of the functions2f(x) = 4x + 1, f(x) = x + 3x – 10,3 2f(x) = x + 2x – 3x, and4 3 2f(x) = x + 4x – 3x – 18x and the graphsof their first and second derivatives.1.4 describe key features of a polynomial function,given information about its first and/or secondderivatives (e.g., the graph of a derivative,the sign of a derivative over specificintervals, the x-intercepts of a derivative),sketch two or more possible graphs of thefunction that are consistent with the giveninformation, and explain why an infinitenumber of graphs is possibleMCV4UDERIVATIVES AND THEIR APPLICATIONS105


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University PreparationSample problem: The following is the graph ofthe function g(x).2yy = g(x)If g(x) is the derivative of f(x), and f(0) = 0,sketch the graph of f(x). If you are now giventhe function equation g(x) = (x – 1)(x – 3),determine the equation of f”(x) and describesome features of the equation of f(x). Howwould f(x) change graphically and algebraicallyif f(0) = 2?1.5 sketch the graph of a polynomial function,given its equation, by using a variety ofstrategies (e.g., using the sign of the firstderivative; using the sign of the secondderivative; identifying even or odd functions)to determine its key features (e.g., increasing/decreasing intervals, intercepts, local maximaand minima, points of inflection, intervals ofconcavity), and verify using technology2. Solving Problems UsingMathematical Models andDerivativesBy the end of this course, students will:22.1 make connections between the concept ofmotion (i.e., displacement, velocity, acceleration)and the concept of the derivative in avariety of ways (e.g., verbally, numerically,graphically, algebraically)Sample problem: Generate a displacement–time graph by walking in front of a motionsensor connected to a graphing calculator.Use your knowledge of derivatives to sketchthe velocity–time and acceleration–timegraphs. Verify the sketches by displayingthe graphs on the graphing calculator.x2.2 make connections between the graphical oralgebraic representations of derivatives andreal-world applications (e.g., population andrates of population change, prices and inflationrates, volume and rates of flow, heightand growth rates)Sample problem: Given a graph of pricesover time, identify the periods of inflationand deflation, and the time at which themaximum rate of inflation occurred. Explainhow derivatives helped solve the problem.2.3 solve problems, using the derivative, thatinvolve instantaneous rates of change, includingproblems arising from real-world applications(e.g., population growth, radioactivedecay, temperature changes, hours of daylight,heights of tides), given the equationof a function *Sample problem: The size of a population ofbutterflies is given by the function6000P(t) =where t is the time in days.t1 + 49(0.6)Determine the rate of growth in the populationafter 5 days using the derivative, andverify graphically using technology.2.4 solve optimization problems involving polynomial,simple rational, and exponential functionsdrawn from a variety of applications,including those arising from real-worldsituationsSample problem: The number of bus ridersfrom the suburbs to downtown per day is–xrepresented by 1200(1.15), where x is thefare in dollars. What fare will maximize thetotal revenue?2.5 solve problems arising from real-world applicationsby applying a mathematical modeland the concepts and procedures associatedwith the derivative to determine mathematicalresults, and interpret and communicatethe resultsSample problem: A bird is foraging for berries.If it stays too long in any one patch it will bespending valuable foraging time looking forthe hidden berries, but when it leaves it willhave to spend time finding another patch. Amodel for the net amount of food energy in*The emphasis of this expectation is on the application of the derivative rules and not on the simplification of resulting complexalgebraic expressions.106


joules the bird gets if it spends t minutes in a3000tpatch is E = . Suppose the bird takest + 42 min on average to find each new patch, andspends negligible energy doing so. How longshould the bird spend in a patch to maximizeits average rate of energy gain over the timespent flying to a patch and foraging in it?Use and compare numeric, graphical, andalgebraic strategies to solve this problem.Calculus and VectorsMCV4UDERIVATIVES AND THEIR APPLICATIONS107


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparation108C. GEOMETRY AND ALGEBRAOF VECTORSOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of vectors in two-space and three-space by representing themalgebraically and geometrically and by recognizing their applications;2. perform operations on vectors in two-space and three-space, and use the properties of theseoperations to solve problems, including those arising from real-world applications;3. distinguish between the geometric representations of a single linear equation or a system of twolinear equations in two-space and three-space, and determine different geometric configurationsof lines and planes in three-space;4. represent lines and planes using scalar, vector, and parametric equations, and solve problemsinvolving distances and intersections.SPECIFIC EXPECTATIONS1. Representing Vectors Geometricallyand AlgebraicallyBy the end of this course, students will:1.1 recognize a vector as a quantity with bothmagnitude and direction, and identify, gather,and interpret information about real-worldapplications of vectors (e.g., displacement,forces involved in structural design, simpleanimation of computer graphics, velocitydetermined using GPS)Sample problem: Position is representedusing vectors. Explain why knowing thatsomeone is 69 km from Lindsay, Ontario, isnot sufficient to identify their exact position.1.2 represent a vector in two-space geometricallyas a directed line segment, with directions expressedin different ways (e.g., 320º; N 40º W),and algebraically (e.g., using Cartesian coordinates;using polar coordinates), and recognizevectors with the same magnitude and directionbut different positions as equal vectors1.3 determine, using trigonometric relationships–1 y[e.g., x = rcosθ, y = rsinθ, θ = tan ( ) or–1tan (yx)2 2+ 180º, r = √x + y ],xthe Cartesian representation of a vector intwo-space given as a directed line segment, orthe representation as a directed line segmentof a vector in two-space given in Cartesianform [e.g., representing the vector (8, 6) as adirected line segment]Sample problem: Represent the vector with amagnitude of 8 and a direction of 30º anticlockwiseto the positive x-axis in Cartesianform.1.4 recognize that points and vectors in three-spacecan both be represented using Cartesian coordinates,and determine the distance betweentwo points and the magnitude of a vectorusing their Cartesian representations2. Operating With VectorsBy the end of this course, students will:2.1 perform the operations of addition, subtraction,and scalar multiplication on vectorsrepresented as directed line segments in twospace,and on vectors represented in Cartesianform in two-space and three-space2.2 determine, through investigation with andwithout technology, some properties (e.g.,commutative, associative, and distributiveproperties) of the operations of addition,subtraction, and scalar multiplication ofvectors


2.3 solve problems involving the addition, subtraction,and scalar multiplication of vectors,including problems arising from real-worldapplicationsSample problem: A plane on a heading ofN 27° E has an air speed of 375 km/h. Thewind is blowing from the south at 62 km/h.Determine the actual direction of travel of theplane and its ground speed.2.4 perform the operation of dot product on twovectors represented as directed line segments→ → → →(i.e., using a •b =|a||b|cosθ) and in→ →Cartesian form (i.e., using a •b = a 1b 1+ a 2b 2or→ →a •b = a 1b 1+ a 2b 2+ a 3b 3) in two-space andthree-space, and describe applications ofthe dot product (e.g., determining the anglebetween two vectors; determining the projectionof one vector onto another)Sample problem: Describe how the dot productcan be used to compare the work donein pulling a wagon over a given distance ina specific direction using a given force fordifferent positions of the handle.2.5 determine, through investigation, propertiesof the dot product (e.g., investigate whetherit is commutative, distributive, or associative;investigate the dot product of a vector withitself and the dot product of orthogonalvectors)Sample problem: Investigate geometricallyand algebraically the relationship betweenthe dot product of the vectors (1, 0, 1) and(0, 1, – 1) and the dot product of scalar multiplesof these vectors. Does this relationshipapply to any two vectors? Find a vector thatis orthogonal to both the given vectors.2.6 perform the operation of cross producton two vectors represented in Cartesianform in three-space [i.e., using→ →a x b = (a 2b 3– a 3b 2, a 3b 1– a 1b 3, a 1b 2– a 2b 1)],determine the magnitude of the cross product→ → → →(i.e., using|a x b|=|a||b|sin θ ), and describeapplications of the cross product (e.g., determininga vector orthogonal to two given vectors;determining the turning effect [or torque]when a force is applied to a wrench at differentangles)Sample problem: Explain how you maximizethe torque when you use a wrench and howthe inclusion of a ratchet in the design of awrench helps you to maximize the torque.2.7 determine, through investigation, propertiesof the cross product (e.g., investigate whetherit is commutative, distributive, or associative;investigate the cross product of collinearvectors)Sample problem: Investigate algebraically therelationship between the cross product of→→the vectors a = (1, 0, 1) and b = (0, 1, – 1)and the cross product of scalar multiples→ →of a and b. Does this relationship apply toany two vectors?2.8 solve problems involving dot product andcross product (e.g., determining projections,the area of a parallelogram, the volume of aparallelepiped), including problems arisingfrom real-world applications (e.g., determiningwork, torque, ground speed, velocity,force)Sample problem: Investigate the dot products→ → → → → →→a •(a x b) and b •(a x b) for any two vectors a→and b in three-space. What property of the→ →cross product a x b does this verify?3. Describing Lines and Planes UsingLinear EquationsBy the end of this course, students will:3.1 recognize that the solution points (x, y) intwo-space of a single linear equation in twovariables form a line and that the solutionpoints (x, y) in two-space of a system of twolinear equations in two variables determinethe point of intersection of two lines, if thelines are not coincident or parallelSample problem: Describe algebraically thesituations in two-space in which the solutionpoints (x, y) of a system of two linear equationsin two variables do not determine apoint.3.2 determine, through investigation with technology(i.e., 3-D graphing software) and withouttechnology, that the solution points (x, y, z) inthree-space of a single linear equation in threevariables form a plane and that the solutionpoints (x, y, z) in three-space of a system oftwo linear equations in three variables formthe line of intersection of two planes, if theplanes are not coincident or parallelSample problem: Use spatial reasoning tocompare the shapes of the solutions in threespacewith the shapes of the solutions in twospacefor each of the linear equations x = 0,Calculus and VectorsMCV4UGEOMETRY AND ALGEBRA OF VECTORS109


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparation110y = 0, and y = x. For each of the equationsz = 5, y – z = 3, and x + z = 1, describe theshape of the solution points (x, y, z) in threespace.Verify the shapes of the solutions inthree-space using technology.3.3 determine, through investigation using avariety of tools and strategies (e.g., modellingwith cardboard sheets and drinking straws;sketching on isometric graph paper), differentgeometric configurations of combinations ofup to three lines and/or planes in three-space(e.g., two skew lines, three parallel planes,two intersecting planes, an intersecting lineand plane); organize the configurations basedon whether they intersect and, if so, how theyintersect (i.e., in a point, in a line, in a plane)4. Describing Lines and Planes UsingScalar, Vector, and ParametricEquationsBy the end of this course, students will:4.1 recognize a scalar equation for a line intwo-space to be an equation of the formAx + By + C = 0, represent a line intwo-space using a vector equation (i.e.,→ → →r = r 0 + tm) and parametric equations, andmake connections between a scalar equation,a vector equation, and parametric equationsof a line in two-space4.2 recognize that a line in three-space cannotbe represented by a scalar equation, and representa line in three-space using the scalarequations of two intersecting planes andusing vector and parametric equations (e.g.,given a direction vector and a point on theline, or given two points on the line)Sample problem: Represent the line passingthrough (3, 2, – 1) and (0, 2, 1) with the scalarequations of two intersecting planes, witha vector equation, and with parametricequations.4.3 recognize a normal to a plane geometrically(i.e., as a vector perpendicular to the plane)and algebraically [e.g., one normal to theplane 3x + 5y – 2z = 6 is (3, 5, – 2)], and determine,through investigation, some geometricproperties of the plane (e.g., the direction ofany normal to a plane is constant; all scalarmultiples of a normal to a plane are also normalsto that plane; three non-collinear pointsdetermine a plane; the resultant, or sum, ofany two vectors in a plane also lies in theplane)Sample problem: How does the relationship→ → →a •(b x c) = 0 help you determine whetherthree non-parallel planes intersect in a point, if→ →→a, b, and c represent normals to the threeplanes?4.4 recognize a scalar equation for a plane inthree-space to be an equation of the formAx + By + Cz + D = 0 whose solution pointsmake up the plane, determine the intersectionof three planes represented using scalarequations by solving a system of three linearequations in three unknowns algebraically(e.g., by using elimination or substitution),and make connections between the algebraicsolution and the geometric configuration ofthe three planesSample problem: Determine the equationof a plane P 3that intersects the planesP 1, x + y + z = 1, and P 2, x – y + z = 0, ina single point. Determine the equation of aplane P 4that intersects P 1and P 2in morethan one point.4.5 determine, using properties of a plane, thescalar, vector, and parametric equations ofa planeSample problem: Determine the scalar, vector,and parametric equations of the plane thatpasses through the points (3, 2, 5), (0, – 2, 2),and (1, 3, 1).4.6 determine the equation of a plane in its scalar,vector, or parametric form, given another ofthese formsSample problem: Represent the plane→r = (2, 1, 0) + s(1, – 1, 3) + t(2, 0, – 5), wheres and t are real numbers, with a scalarequation.4.7 solve problems relating to lines and planes inthree-space that are represented in a varietyof ways (e.g., scalar, vector, parametric equations)and involving distances (e.g., between apoint and a plane; between two skew lines) orintersections (e.g., of two lines, of a line and aplane), and interpret the result geometricallySample problem: Determine the intersectionof the perpendicular line drawn from thepoint A(– 5, 3, 7) to the plane→v = (0, 0, 2) + t(– 1, 1, 3) + s(2, 0, – 3),and determine the distance frompoint A to the plane.


Mathematics of DataManagement, Grade 12University PreparationMDM4UThis course broadens students’ understanding of mathematics as it relates to managingdata. Students will apply methods for organizing and analysing large amounts ofinformation; solve problems involving probability and statistics; and carry out aculminating investigation that integrates statistical concepts and skills. Students willalso refine their use of the mathematical processes necessary for success in seniormathematics. Students planning to enter university programs in business, the socialsciences, and the humanities will find this course of particular interest.Prerequisite: Functions, Grade 11, University Preparation, or Functions and Applications,Grade 11, University/College Preparation111


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.112


A. COUNTING AND PROBABILITYOVERALL EXPECTATIONSBy the end of this course, students will:1. solve problems involving the probability of an event or a combination of events for discrete samplespaces;2. solve problems involving the application of permutations and combinations to determine theprobability of an event.SPECIFIC EXPECTATIONS1. Solving Probability ProblemsInvolving Discrete Sample SpacesBy the end of this course, students will:1.1 recognize and describe how probabilities areused to represent the likelihood of a result ofan experiment (e.g., spinning spinners; drawingblocks from a bag that contains differentcolouredblocks; playing a game with numbercubes; playing Aboriginal stick-and-stonegames) and the likelihood of a real-worldevent (e.g., that it will rain tomorrow, that anaccident will occur, that a product will bedefective)1.2 describe a sample space as a set that containsall possible outcomes of an experiment, anddistinguish between a discrete sample spaceas one whose outcomes can be counted (e.g.,all possible outcomes of drawing a card ortossing a coin) and a continuous sample spaceas one whose outcomes can be measured (e.g.,all possible outcomes of the time it takes tocomplete a task or the maximum distance aball can be thrown)1.3 determine the theoretical probability, P i (i.e.,a value from 0 to 1), of each outcome of adiscrete sample space (e.g., in situations inwhich all outcomes are equally likely),recognize that the sum of the probabilitiesof the outcomes is 1 (i.e., for n outcomes,P1+ P2+ P3+ … + P n = 1), recognize that theprobabilities Piform the probability distributionassociated with the sample space, andsolve related problemsSample problem: An experiment involvesrolling two number cubes and determiningthe sum. Calculate the theoretical probabilityof each outcome, and verify that the sum ofthe probabilities is 1.1.4 determine, through investigation using classgenerateddata and technology-based simulationmodels (e.g., using a random-numbergenerator on a spreadsheet or on a graphingcalculator; using dynamic statistical softwareto simulate repeated trials in an experiment),the tendency of experimental probability toapproach theoretical probability as the numberof trials in an experiment increases (e.g.,“If I simulate tossing two coins 1000 timesusing technology, the experimental probabilitythat I calculate for getting two tails onthe two tosses is likely to be closer to the1theoretical probability of than if I simulate4tossing the coins only 10 times”)Sample problem: Calculate the theoreticalprobability of rolling a 2 on a single roll of anumber cube. Simulate rolling a numbercube, and use the simulation results to calculatethe experimental probabilities of rollinga 2 over 10, 20, 30, …, 200 trials. Graph theexperimental probabilities versus the numberof trials, and describe any trend.1.5 recognize and describe an event as a set ofoutcomes and as a subset of a sample space,determine the complement of an event, determinewhether two or more events are mutuallyexclusive or non-mutually exclusive (e.g.,the events of getting an even number orMathematics of Data ManagementMDM4UCOUNTING AND PROBABILITY113


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparationgetting an odd number of heads from tossinga coin 5 times are mutually exclusive), andsolve related probability problems [e.g., calculateP(~A), P(A and B), P(A or B)] using avariety of strategies (e.g., Venn diagrams,lists, formulas)1.6 determine whether two events are independentor dependent and whether one event isconditional on another event, and solverelated probability problems [e.g., calculateP(A and B), P(A or B), P(A given B)] using avariety of strategies (e.g., tree diagrams, lists,formulas)2. Solving Problems Using CountingPrinciplesBy the end of this course, students will:2.1 recognize the use of permutations and combinationsas counting techniques with advantagesover other counting techniques (e.g.,making a list; using a tree diagram; making achart; drawing a Venn diagram), distinguishbetween situations that involve the use of permutationsand those that involve the use ofcombinations (e.g., by considering whether ornot order matters), and make connectionsbetween, and calculate, permutations andcombinationsSample problem: An organization with10 members is considering two leadershipmodels. One involves a steering committeewith 4 members of equal standing. The otheris an executive committee consisting of apresident, vice-president, secretary, andtreasurer. Determine the number of ways ofselecting the executive committee from the10 members and, using this number, thenumber of ways of selecting the steeringcommittee from the 10 members. How arethe calculations related? Use the calculationsto explain the relationship between permutationsand combinations.2.2 solve simple problems using techniques forcounting permutations and combinations,where all objects are distinct, and expressthe solutions using standard combinatorialnotation [e.g., n!, P(n, r), (n) ]rSample problem: In many Aboriginal communities,it is common practice for people toshake hands when they gather. Use combinationsto determine the total number of handshakeswhen 7 people gather, and verifyusing a different strategy.2.3 solve introductory counting problems involvingthe additive counting principle (e.g.,determining the number of ways of selecting2 boys or 2 girls from a group of 4 boys and5 girls) and the multiplicative counting principle(e.g., determining the number of ways ofselecting 2 boys and 2 girls from a group of4 boys and 5 girls)2.4 make connections, through investigation,between combinations (i.e., n choose r) andPascal’s triangle [e.g., between (2) andrrow 3 of Pascal’s triangle, between (n) and2diagonal 3 of Pascal’s triangle]Sample problem: A school is 5 blocks westand 3 blocks south of a student’s home.Determine, in a variety of ways (e.g., bydrawing the routes, by using Pascal’s triangle,by using combinations), how many differentroutes the student can take from home to theschool by going west or south at each corner.2.5 solve probability problems using countingprinciples for situations involving equallylikely outcomesSample problem: Two marbles are drawnrandomly from a bag containing 12 greenmarbles and 16 red marbles. What is theprobability that the two marbles are bothgreen if the first marble is replaced? If thefirst marble is not replaced?114


B. PROBABILITY DISTRIBUTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of discrete probability distributions, represent them numerically,graphically, and algebraically, determine expected values, and solve related problems from a varietyof applications;2. demonstrate an understanding of continuous probability distributions, make connections to discreteprobability distributions, determine standard deviations, describe key features of the normaldistribution, and solve related problems from a variety of applications.SPECIFIC EXPECTATIONSMathematics Functionsof Data Management1. Understanding ProbabilityDistributions for Discrete RandomVariablesBy the end of this course, students will:1.1 recognize and identify a discrete random variableX (i.e., a variable that assumes a uniquevalue for each outcome of a discrete samplespace, such as the value x for the outcome ofgetting x heads in 10 tosses of a coin), generatea probability distribution [i.e., a functionthat maps each value x of a random variableX to a corresponding probability, P(X = x)] bycalculating the probabilities associated withall values of a random variable, with andwithout technology, and represent a probabilitydistribution numerically using a table1.2 calculate the expected value for a givenprobability distribution [i.e., usingE(X) = ∑ xP(X = x)], interpret the expectedvalue in applications, and make connectionsbetween the expected value and the weightedmean of the values of the discrete randomvariableSample problem: Of six cases, three eachhold $1, two each hold $1000, and one holds$100 000. Calculate the expected value andinterpret its meaning. Make a conjectureabout what happens to the expected value ifyou add $10 000 to each case or if you multiplythe amount in each case by 10. Verifyyour conjectures.1.3 represent a probability distribution graphicallyusing a probability histogram (i.e., a histogramon which each rectangle has a base ofwidth 1, centred on the value of the discreterandom variable, and a height equal to theprobability associated with the value of therandom variable), and make connectionsbetween the frequency histogram and theprobability histogram (e.g., by comparingtheir shapes)Sample problem: For the situation involvingthe rolling of two number cubes and determiningthe sum, identify the discrete randomvariable and generate the related probabilityhistogram. Determine the total area of thebars in the histogram and explain yourresult.1.4 recognize conditions (e.g., independent trials)that give rise to a random variable that followsa binomial probability distribution, calculatethe probability associated with each value ofthe random variable, represent the distributionnumerically using a table and graphicallyusing a probability histogram, and make connectionsto the algebraic representationP(X = x) = (n)x n – xp (1 – p)xSample problem: A light-bulb manufacturerestimates that 0.5% of the bulbs manufacturedare defective. Generate and graph theprobability distribution for the random variablethat represents the number of defectivebulbs in a set of 4 bulbs.MDM4UPROBABILITY DISTRIBUTIONS115


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparation1.5 recognize conditions (e.g., dependent trials)that give rise to a random variable that followsa hypergeometric probability distribution,calculate the probability associated witheach value of the random variable (e.g., byusing a tree diagram; by using combinations),and represent the distribution numericallyusing a table and graphically using a probabilityhistogram1.6 compare, with technology and using numericand graphical representations, the probabilitydistributions of discrete random variables(e.g., compare binomial distributions withthe same probability of success for increasingnumbers of trials; compare the shapes of ahypergeometric distribution and a binomialdistribution)Sample problem: Compare the probabilitydistributions associated with drawing 0, 1, 2,or 3 face cards when a card is drawn 3 timesfrom a standard deck with replacement (i.e.,the card is replaced after each draw) andwithout replacement (i.e., the card is notreplaced after each draw).1.7 solve problems involving probability distributions(e.g., uniform, binomial, hypergeometric),including problems arising fromreal-world applicationsSample problem: The probability of a businessperson cancelling a reservation at LaPlace Pascal hotel is estimated to be 8%.Generate and graph the probability distributionfor the discrete random variable thatrepresents the number of business peoplecancelling when there are 10 reservations.Use the probability distribution to determinethe probability of at least 4 of the 10 reservationsbeing cancelled.2. Understanding ProbabilityDistributions for ContinuousRandom VariablesBy the end of this course, students will:2.1 recognize and identify a continuous randomvariable (i.e., a variable that assumes valuesfrom the infinite number of possible outcomesin a continuous sample space), and distinguishbetween situations that give rise to discretefrequency distributions (e.g., counting thenumber of outcomes for drawing a card ortossing three coins) and situations that giverise to continuous frequency distributions(e.g., measuring the time taken to completea task or the maximum distance a ball canbe thrown)2.2 recognize standard deviation as a measureof the spread of a distribution, and determine,with and without technology, the mean andstandard deviation of a sample of values ofa continuous random variable2.3 describe challenges associated with determininga continuous frequency distribution (e.g.,the inability to capture all values of the variable,resulting in a need to sample; uncertaintiesin measured values of the variable),and recognize the need for mathematicalmodels to represent continuous frequencydistributions2.4 represent, using intervals, a sample of valuesof a continuous random variable numericallyusing a frequency table and graphically usinga frequency histogram and a frequency polygon,recognize that the frequency polygonapproximates the frequency distribution, anddetermine, through investigation using technology(e.g., dynamic statistical software,graphing calculator), and compare the effectivenessof the frequency polygon as anapproximation of the frequency distributionfor different sizes of the intervals2.5 recognize that theoretical probability for acontinuous random variable is determinedover a range of values (e.g., the probabilitythat the life of a lightbulb is between 90 hoursand 115 hours), that the probability that acontinuous random variable takes any singlevalue is zero, and that the probabilities ofranges of values form the probability distributionassociated with the random variable2.6 recognize that the normal distribution iscommonly used to model the frequency andprobability distributions of continuous randomvariables, describe some properties ofthe normal distribution (e.g., the curve has acentral peak; the curve is symmetric about themean; the mean and median are equal;approximately 68% of the data values arewithin one standard deviation of the meanand approximately 95% of the data values arewithin two standard deviations of the mean),and recognize and describe situations that canbe modelled using the normal distribution(e.g., birth weights of males or of females,household incomes in a neighbourhood,baseball batting averages)116


2.7 make connections, through investigationusing dynamic statistical software, betweenthe normal distribution and the binomial andhypergeometric distributions for increasingnumbers of trials of the discrete distributions(e.g., recognizing that the shape of the hypergeometricdistribution of the number of maleson a 4-person committee selected from agroup of people more closely resembles theshape of a normal distribution as the sizeof the group from which the committee wasdrawn increases)Sample problem: Explain how the total areaof a probability histogram for a binomialdistribution allows you to predict the areaunder a normal probability distributioncurve.2.8 recognize a z-score as the positive or negativenumber of standard deviations from the meanto a value of the continuous random variable,and solve probability problems involvingnormal distributions using a variety of toolsand strategies (e.g., calculating a z-score andreading a probability from a table; using technologyto determine a probability), includingproblems arising from real-world applicationsSample problem: The heights of 16-month-oldmaple seedlings are normally distributedwith a mean of 32 cm and a standard deviationof 10.2 cm. What is the probability thatthe height of a randomly selected seedlingwill be between 24.0 cm and 38.0 cm?Mathematics Functionsof Data ManagementMDM4UPROBABILITY DISTRIBUTIONS117


Grade 12, University PreparationC. ORGANIZATION OF DATA FORANALYSISOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of the role of data in statistical studies and the variability inherent indata, and distinguish different types of data;2. describe the characteristics of a good sample, some sampling techniques, and principles of primarydata collection, and collect and organize data to solve a problem.SPECIFIC EXPECTATIONS1. Understanding Data ConceptsBy the end of this course, students will:1.1 recognize and describe the role of data instatistical studies (e.g., the use of statisticaltechniques to extract or mine knowledge ofrelationships from data), describe examplesof applications of statistical studies (e.g., inmedical research, political decision making,market research), and recognize that conclusionsdrawn from statistical studies of thesame relationship may differ (e.g., conclusionsabout the effect of increasing jail sentences oncrime rates)1.3 distinguish different types of statistical data(i.e., discrete from continuous, qualitativefrom quantitative, categorical from numerical,nominal from ordinal, primary from secondary,experimental from observational, microdatafrom aggregate data) and give examples(e.g., distinguish experimental data used tocompare the effectiveness of medical treatmentsfrom observational data used to examinethe relationship between obesity andtype 2 diabetes or between ethnicity and type 2diabetes)2. Collecting and Organizing DataTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics1181.2 recognize and explain reasons why variabilityis inherent in data (e.g., arising from limitedaccuracy in measurement or from variationsin the conditions of an experiment; arisingfrom differences in samples in a survey), anddistinguish between situations that involveone variable and situations that involve morethan one variableSample problem: Use the Census at Schooldatabase to investigate variability in themedian and mean of, or a proportion estimatedfrom, equal-sized random samplesof data on a topic such as the percentage ofstudents who do not smoke or who walk toschool, or the average height of people of aparticular age. Compare the median andmean of, or a proportion estimated from,samples of increasing size with the medianand mean of the population or the populationproportion.By the end of this course, students will:2.1 determine and describe principles of primarydata collection (e.g., the need for randomization,replication, and control in experimentalstudies; the need for randomization in samplesurveys) and criteria that should be consideredin order to collect reliable primary data(e.g., the appropriateness of survey questions;potential sources of bias; sample size)2.2 explain the distinction between the termspopulation and sample, describe the characteristicsof a good sample, explain why samplingis necessary (e.g., time, cost, or physical constraints),and describe and compare somesampling techniques (e.g., simple random,systematic, stratified, convenience, voluntary)Sample problem: What are some factors thata manufacturer should consider when determiningwhether to test a sample or the entirepopulation to ensure the quality of a product?


2.3 describe how the use of random samples witha bias (e.g., response bias, measurement bias,non-response bias, sampling bias) or the useof non-random samples can affect the resultsof a study2.4 describe characteristics of an effective survey(e.g., by giving consideration to ethics, privacy,the need for honest responses, and possiblesources of bias, including cultural bias),and design questionnaires (e.g., for determiningif there is a relationship between a person’sage and their hours per week of Internet use,between marks and hours of study, or betweenincome and years of education) or experiments(e.g., growth of plants under different conditions)for gathering dataSample problem: Give examples of concernsthat could arise from an ethical review ofsurveys generated by students in yourschool.2.5 collect data from primary sources, throughexperimentation, or from secondary sources(e.g., by using the Internet to access reliabledata from a well-organized database such asE-STAT; by using print sources such as newspapersand magazines), and organize datawith one or more attributes (e.g., organizedata about a music collection classified byartist, date of recording, and type of musicusing dynamic statistical software or aspreadsheet) to answer a question or solve aproblemMathematics Functionsof Data ManagementMDM4UORGANIZATION OF DATA FOR ANALYSIS119


D. STATISTICAL ANALYSISTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. analyse, interpret, and draw conclusions from one-variable data using numerical and graphicalsummaries;2. analyse, interpret, and draw conclusions from two-variable data using numerical, graphical, andalgebraic summaries;3. demonstrate an understanding of the applications of data management used by the media and theadvertising industry and in various occupations.SPECIFIC EXPECTATIONS1. Analysing One-Variable DataBy the end of this course, students will:1.1 recognize that the analysis of one-variabledata involves the frequencies associated withone attribute, and determine, using technology,the relevant numerical summaries (i.e.,mean, median, mode, range, interquartilerange, variance, and standard deviation)1.2 determine the positions of individual datapoints within a one-variable data set usingquartiles, percentiles, and z-scores, use thenormal distribution to model suitable onevariabledata sets, and recognize theseprocesses as strategies for one-variabledata analysis1.3 generate, using technology, the relevantgraphical summaries of one-variable data(e.g., circle graphs, bar graphs, histograms,stem-and-leaf plots, boxplots) based on thetype of data provided (e.g., categorical,ordinal, quantitative)1.4 interpret, for a normally distributed population,the meaning of a statistic qualified by astatement describing the margin of error andthe confidence level (e.g., the meaning of astatistic that is accurate to within 3 percentagepoints, 19 times out of 20), and make connections,through investigation using technology(e.g., dynamic statistical software), betweenthe sample size, the margin of error, and theconfidence level (e.g., larger sample sizescreate higher confidence levels for a givenmargin of error)Sample problem: Use census data fromStatistics Canada to investigate, usingdynamic statistical software, the minimumsample size such that the proportion of thesample opting for a particular consumer orvoting choice is within 3 percentage points ofthe proportion of the population, 95% of thetime (i.e., 19 times out of 20).1.5 interpret statistical summaries (e.g., graphical,numerical) to describe the characteristics of aone-variable data set and to compare tworelated one-variable data sets (e.g., comparethe lengths of different species of trout;compare annual incomes in Canada and in athird-world country; compare Aboriginal andnon-Aboriginal incomes); describe how statisticalsummaries (e.g., graphs, measures ofcentral tendency) can be used to misrepresentone-variable data; and make inferences, andmake and justify conclusions, from statisticalsummaries of one-variable data orally and inwriting, using convincing arguments2. Analysing Two-Variable DataBy the end of this course, students will:2.1 recognize that the analysis of two-variabledata involves the relationship between twoattributes, recognize the correlation coefficient120


as a measure of the fit of the data to a linearmodel, and determine, using technology, therelevant numerical summaries (e.g., summarytables such as contingency tables; correlationcoefficients)Sample problem: Organize data from StatisticsCanada to analyse gender differences (e.g.,using contingency tables; using correlationcoefficients) related to a specific set of characteristics(e.g., average income, hours ofunpaid housework).2.2 recognize and distinguish different types ofrelationships between two variables that havea mathematical correlation (e.g., the causeand-effectrelationship between the age of atree and its diameter; the common-cause relationshipbetween ice cream sales and forestfires over the course of a year; the accidentalrelationship between the consumer priceindex and the number of known planets inthe universe)2.3 generate, using technology, the relevantgraphical summaries of two-variable data(e.g., scatter plots, side-by-side boxplots)based on the type of data provided (e.g.,categorical, ordinal, quantitative)2.4 determine, by performing a linear regressionusing technology, the equation of a line thatmodels a suitable two-variable data set, determinethe fit of an individual data point to thelinear model (e.g., by using residuals to identifyoutliers), and recognize these processesas strategies for two-variable data analysis2.5 interpret statistical summaries (e.g., scatterplot, equation representing a relationship)to describe the characteristics of a twovariabledata set and to compare two relatedtwo-variable data sets (e.g., compare therelationship between Grade 12 English andmathematics marks with the relationshipbetween Grade 12 science and mathematicsmarks); describe how statistical summaries(e.g., graphs, linear models) can be used tomisrepresent two-variable data; and makeinferences, and make and justify conclusions,from statistical summaries of two-variabledata orally and in writing, using convincingarguments3. Evaluating ValidityBy the end of this course, students will:3.1 interpret statistics presented in the media(e.g., the UN’s finding that 2% of the world’spopulation has more than half the world’swealth, whereas half the world’s populationhas only 1% of the world’s wealth), andexplain how the media, the advertising industry,and others (e.g., marketers, pollsters) useand misuse statistics (e.g., as represented ingraphs) to promote a certain point of view(e.g., by making a general statement based ona weak correlation or an assumed cause-andeffectrelationship; by starting the verticalscale at a value other than zero; by makingstatements using general population statisticswithout reference to data specific to minoritygroups)3.2 assess the validity of conclusions presentedin the media by examining sources of data,including Internet sources (i.e., to determinewhether they are authoritative, reliable,unbiased, and current), methods of datacollection, and possible sources of bias (e.g.,sampling bias, non-response bias, cultural biasin a survey question), and by questioning theanalysis of the data (e.g., whether there is anyindication of the sample size in the analysis)and conclusions drawn from the data (e.g.,whether any assumptions are made aboutcause and effect)Sample problem: The headline that accompaniesthe following graph says “Big Increasein Profits”. Suggest reasons why this headlinemay or may not be true.Profits ($ billions)232221201918172001 2002 2003 2004 2005 2006 2007Year3.3 gather, interpret, and describe informationabout applications of data management inoccupations (e.g., actuary, statistician, businessanalyst, sociologist, medical doctor,psychologist, teacher, community planner),and about university programs that explorethese applicationsMathematics Functionsof Data ManagementMDM4USTATISTICAL ANALYSIS121


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, University Preparation122E. CULMINATING DATA MANAGEMENTINVESTIGATIONOVERALL EXPECTATIONSBy the end of this course, students will:1. design and carry out a culminating investigation* that requires the integration and application of theknowledge and skills related to the expectations of this course;2. communicate the findings of a culminating investigation and provide constructive critiques of theinvestigations of others.SPECIFIC EXPECTATIONS1. Designing and Carrying Out aCulminating InvestigationBy the end of this course, students will:1.1 pose a significant problem of interest thatrequires the organization and analysis of asuitable set of primary or secondary quantitativedata (e.g., primary data collected from astudent-designed game of chance, secondarydata from a reliable source such as E-STAT),and conduct appropriate background researchrelated to the topic being studied1.2 design a plan to study the problem (e.g., identifythe variables and the population; developan ethical survey; establish the procedures forgathering, summarizing, and analysing theprimary or secondary data; consider the samplesize and possible sources of bias)1.3 gather data related to the study of the problem(e.g., by using a survey; by using the Internet;by using a simulation) and organize the data(e.g., by setting up a database; by establishingintervals), with or without technology1.4 interpret, analyse, and summarize data relatedto the study of the problem (e.g., generate andinterpret numerical and graphical statisticalsummaries; recognize and apply a probabilitydistribution model; calculate the expectedvalue of a probability distribution), with orwithout technology1.5 draw conclusions from the analysis of thedata (e.g., determine whether the analysissolves the problem), evaluate the strength ofthe evidence (e.g., by considering factors suchas sample size or bias, or the number of timesa game is played), specify any limitations ofthe conclusions, and suggest follow-up problemsor investigations2. Presenting and Critiquing theCulminating InvestigationBy the end of this course, students will:2.1 compile a clear, well-organized, and detailedreport of the investigation2.2 present a summary of the culminating investigationto an audience of their peers within aspecified length of time, with technology (e.g.presentation software) or without technology2.3 answer questions about the culminating investigationand respond to critiques (e.g., byelaborating on the procedures; by justifyingmathematical reasoning)2.4 critique the mathematical work of others in aconstructive manner*This culminating investigation allows students to demonstrate their knowledge and skills from this course by addressing a singleproblem on probability and statistics or by addressing two smaller problems, one on probability and the other on statistics.


Mathematics for CollegeTechnology, Grade 12College PreparationMCT4CThis course enables students to extend their knowledge of functions. Students willinvestigate and apply properties of polynomial, exponential, and trigonometricfunctions; continue to represent functions numerically, graphically, and algebraically;develop facility in simplifying expressions and solving equations; and solve problemsthat address applications of algebra, trigonometry, vectors, and geometry. Studentswill reason mathematically and communicate their thinking as they solve multi-stepproblems. This course prepares students for a variety of college technology programs.Prerequisite: Functions and Applications, Grade 11, University/College Preparation, orFunctions, Grade 11, University Preparation123


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.124


A. EXPONENTIAL FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. solve problems involving exponential equations graphically, including problems arising fromreal-world applications;2. solve problems involving exponential equations algebraically using common bases and logarithms,including problems arising from real-world applications.SPECIFIC EXPECTATIONS1. Solving Exponential EquationsGraphicallyBy the end of this course, students will:1.1 determine, through investigation with technology,and describe the impact of changingthe base and changing the sign of the exponenton the graph of an exponential function1.2 solve simple exponential equations numericallyand graphically, with technology (e.g.,use systematic trial with a scientific calculatorto determine the solution to the equationx1.05 = 1,276), and recognize that the solutionsmay not be exactSample problem: Use the graph of y = 3 toxsolve the equation 3 = 5.1.3 determine, through investigation using graphingtechnology, the point of intersection ofthe graphs of two exponential functions− x x + 3(e.g., y = 4 and y = 8 ), recognize thex-coordinate of this point to be the solutionto the corresponding exponential equation− x x + 3(e.g., 4 = 8 ), and solve exponentialequations graphically (e.g., solvex + 2 x2 = 2 + 12 by using the intersectionx + 2xof the graphs of y = 2 and y = 2 + 12)x x + 3Sample problem: Solve 0.5 = 3graphically.1.4 pose problems based on real-world applications(e.g., compound interest, populationgrowth) that can be modelled with exponentialequations, and solve these and other suchproblems by using a given graph or a graphgenerated with technology from a table ofvalues or from its equationxSample problem: A tire with a slow punctureloses pressure at the rate of 4%/min. If thetire’s pressure is 300 kPa to begin with, whatis its pressure after 1 min? After 2 min? After10 min? Use graphing technology to determinewhen the tire’s pressure will be 200 kPa.2. Solving Exponential EquationsAlgebraicallyBy the end of this course, students will:2.1 simplify algebraic expressions containinginteger and rational exponents using the laws312 6 12of exponents (e.g., x ÷ x , √x y )3 2 3abcSample problem: Simplify and then√a 2 b 4evaluate for a = 4, b = 9, and c = – 3. Verifyyour answer by evaluating the expression withoutsimplifying first. Which method for evaluatingthe expression do you prefer? Explain.2.2 solve exponential equations in one variablexby determining a common base (e.g., 2 = 32,5x −1 2(x + 11) 5x+ 8 x4 = 2 , 3 = 27 )Sample problem: Solve 3 = 27 bydetermining a common base, verify by substitution,and investigate connections to the5x+ 8 xintersection of y = 3 and y = 27 usinggraphing technology.5x+ 8 x2.3 recognize the logarithm of a number to agiven base as the exponent to which the basemust be raised to get the number, recognizethe operation of finding the logarithm to bethe inverse operation (i.e., the undoing orreversing) of exponentiation, and evaluatesimple logarithmic expressionsMathematics for College TechnologyMCT4CEXPONENTIAL FUNCTIONS125


Grade 12, College PreparationSample problem: Why is it possible todetermine log10(100) but not log10(0) orlog10(– 100)? Explain your reasoning.2.4 determine, with technology, the approximatelogarithm of a number to any base, includingbase 10 [e.g., by recognizing that log10(0.372)can be determined using the LOG key ona calculator; by reasoning that log329 isbetween 3 and 4 and using systematic trial todetermine that log 29 is approximately 3.07]32.5 make connections between related logarithmicand exponential equations (e.g., log5125 = 33can also be expressed as 5 = 125), and solvesimple exponential equations by rewritingxthem in logarithmic form (e.g., solving 3 = 10by rewriting the equation as log310 = x)2.6 pose problems based on real-world applicationsthat can be modelled with given exponentialequations, and solve these and othersuch problems algebraically by rewritingthem in logarithmic formSample problem: When a potato whose temperatureis 20°C is placed in an oven maintainedat 200°C, the relationship between thecore temperature of the potato T, in degreesCelsius, and the cooking time t, in minutes, istmodelled by the equation 200 – T = 180(0.96) .Use logarithms to determine the time whenthe potato’s core temperature reaches 160°C.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics126


B. POLYNOMIAL FUNCTIONSOVERALL EXPECTATIONSBy the end of this course, students will:1. recognize and evaluate polynomial functions, describe key features of their graphs, and solveproblems using graphs of polynomial functions;2. make connections between the numeric, graphical, and algebraic representations of polynomialfunctions;3. solve polynomial equations by factoring, make connections between functions and formulas, andsolve problems involving polynomial expressions arising from a variety of applications.SPECIFIC EXPECTATIONSMathematics for College Technology1. Investigating Graphs of PolynomialFunctionsBy the end of this course, students will:1.1 recognize a polynomial expression (i.e., aseries of terms where each term is the productof a constant and a power of x with a nonnegativeintegral exponent, such as3 2x – 5x + 2x – 1); recognize the equation ofa polynomial function and give reasons whyit is a function, and identify linear and quadraticfunctions as examples of polynomialfunctions1.2 compare, through investigation using graphingtechnology, the graphical and algebraicrepresentations of polynomial (i.e., linear,quadratic, cubic, quartic) functions (e.g., investigatethe effect of the degree of a polynomialfunction on the shape of its graph and themaximum number of x-intercepts; investigatethe effect of varying the sign of the leadingcoefficient on the end behaviour of thefunction for very large positive or negativex-values)Sample problem: Investigate the maximumnumber of x-intercepts for linear, quadratic,cubic, and quartic functions using graphingtechnology.1.3 describe key features of the graphs of polynomialfunctions (e.g., the domain and range,the shape of the graphs, the end behaviourof the functions for very large positive ornegative x-values)Sample problem: Describe and compare thekey features of the graphs of the functions234f(x) = x, f(x) = x , f(x) = x , and f(x) = x .1.4 distinguish polynomial functions fromsinusoidal and exponential functions [e.g.,xf(x) = sin x, f(x) = 2 )], and compare andcontrast the graphs of various polynomialfunctions with the graphs of other typesof functions1.5 substitute into and evaluate polynomial functionsexpressed in function notation, includingfunctions arising from real-world applicationsSample problem: A box with no top is beingmade out of a 20-cm by 30-cm piece ofcardboard by cutting equal squares ofside length x from the corners and foldingup the sides. The volume of the box isV = x(20 – 2x)(30 – 2x). Determine the volumeif the side length of each square is 6 cm. Usethe graph of the polynomial function V(x) todetermine the size of square that should becut from the corners if the required volume3of the box is 1000 cm .1.6 pose problems based on real-world applicationsthat can be modelled with polynomialfunctions, and solve these and other suchproblems by using a given graph or a graphgenerated with technology from a table ofvalues or from its equation1.7 recognize, using graphs, the limitations ofmodelling a real-world relationship using apolynomial function, and identify and explainany restrictions on the domain and range(e.g., restrictions on the height and time for aMCT4CPOLYNOMIAL FUNCTIONS127


Grade 12, College PreparationTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematicspolynomial function that models the relationshipbetween height above the ground andtime for a falling object)Sample problem: The forces acting on a horizontalsupport beam in a house cause it to sagby d centimetres, x metres from one end of thebeam. The relationship between d and x canbe represented by the polynomial function12 3d(x) = x(1000 – 20x + x ). Graph the1850function, using technology, and determinethe domain over which the function modelsthe relationship between d and x. Determinethe length of the beam using the graph, andexplain your reasoning.2. Connecting Graphs and Equationsof Polynomial FunctionsBy the end of this course, students will:2.1 factor polynomial expressions in one variable,of degree no higher than four, by selectingand applying strategies (i.e., common factoring,difference of squares, trinomial factoring)Sample problem: Factor: x – 16; x – 2x – 8x.2.2 make connections, through investigationusing graphing technology (e.g., dynamicgeometry software), between a polynomialfunction given in factored form [e.g.,f(x) = x(x – 1)(x + 1)] and the x-intercepts ofits graph, and sketch the graph of a polynomialfunction given in factored form using itskey features (e.g., by determining interceptsand end behaviour; by locating positive andnegative regions using test values betweenand on either side of the x-intercepts)Sample problem: Sketch the graphs off(x) = – (x – 1)(x + 2)(x – 4) andg(x) = – (x – 1)(x + 2)(x + 2) and comparetheir shapes and the number of x-intercepts.2.3 determine, through investigation using technology(e.g., graphing calculator, computeralgebra systems), and describe the connectionbetween the real roots of a polynomial equationand the x-intercepts of the graph of thecorresponding polynomial function [e.g., the4 2real roots of the equation x – 13x + 36 = 0are the x-intercepts of the graph of4 2f(x) = x – 13x + 36]Sample problem: Describe the relationshipbetween the x-intercepts of the graphs oflinear and quadratic functions and the real432roots of the corresponding equations. Investigate,using technology, whether this relationshipexists for polynomial functions ofhigher degree.3. Solving Problems InvolvingPolynomial EquationsBy the end of this course, students will:3.1 solve polynomial equations in one variable, of2degree no higher than four (e.g., x – 4x = 0,42x – 16 = 0, 3x + 5x + 2 = 0), by selecting andapplying strategies (i.e., common factoring;difference of squares; trinomial factoring), andverify solutions using technology (e.g., usingcomputer algebra systems to determine theroots of the equation; using graphing technologyto determine the x-intercepts of thecorresponding polynomial function)Sample problem: Solve x – 2x – 8x = 0.3.2 solve problems algebraically that involvepolynomial functions and equations of degreeno higher than four, including those arisingfrom real-world applications3.3 identify and explain the roles of constants andvariables in a given formula (e.g., a constantcan refer to a known initial value or a knownfixed rate; a variable changes with varyingconditions)Sample problem: The formula P = P0+ kh isused to determine the pressure, P kilopascals,at a depth of h metres under water, wherek kilopascals per metre is the rate of changeof the pressure as the depth increases, andP0kilopascals is the pressure at the surface.Identify and describe the roles of P, P0, k,and h in this relationship, and explain yourreasoning.3.4 expand and simplify polynomial expressionsinvolving more than one variable [e.g., simplify– 2xy(3x y – 5x y)], including expres-2 3 3 2sions arising from real-world applicationsSample problem: Expand and simplify theexpression π(R + r)(R – r) to explain why itrepresents the area of a ring. Draw a diagramof the ring and identify R and r.n3.5 solve equations of the form x = a using3rational exponents (e.g., solve x = 7 by1raising both sides to the exponent )332128


3.6 determine the value of a variable of degreeno higher than three, using a formula drawnfrom an application, by first substitutingknown values and then solving for the variable,and by first isolating the variable andthen substituting known valuesSample problem: The formula s = ut + 1 2at2relates the distance, s, travelled by an objectto its initial velocity, u, acceleration, a, and theelapsed time, t. Determine the accelerationof a dragster that travels 500 m from rest in15 s, by first isolating a, and then by firstsubstituting known values. Compare andevaluate the two methods.3.7 make connections between formulas and linear,quadratic, and exponential functions [e.g.,recognize that the compound interest formula,nA = P(1 + i) , is an example of an exponentialfunction A(n) when P and i are constant, andof a linear function A(P) when i and n areconstant], using a variety of tools and strategies(e.g., comparing the graphs generatedwith technology when different variables ina formula are set as constants)Sample problem: Which variable(s) in the2formula V = πr hwould you need to set asa constant to generate a linear equation?A quadratic equation?3.8 solve multi-step problems requiring formulasarising from real-world applications (e.g.,determining the cost of two coats of paintfor a large cylindrical tank)3.9 gather, interpret, and describe informationabout applications of mathematical modellingin occupations, and about college programsthat explore these applicationsMathematics for College TechnologyMCT4CPOLYNOMIAL FUNCTIONS129


C. TRIGONOMETRIC FUNCTIONSGrade 12, College PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. determine the values of the trigonometric ratios for angles less than 360º, and solve problems usingthe primary trigonometric ratios, the sine law, and the cosine law;2. make connections between the numeric, graphical, and algebraic representations of sinusoidalfunctions;3. demonstrate an understanding that sinusoidal functions can be used to model some periodicphenomena, and solve related problems, including those arising from real-world applications.SPECIFIC EXPECTATIONSTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics1301. Applying Trigonometric RatiosBy the end of this course, students will:1.1 determine the exact values of the sine, cosine,and tangent of the special angles 0°, 30°, 45°,60°, 90°, and their multiples1.2 determine the values of the sine, cosine, andtangent of angles from 0º to 360º, throughinvestigation using a variety of tools (e.g.,dynamic geometry software, graphing tools)and strategies (e.g., applying the unit circle;examining angles related to the special angles)1.3 determine the measures of two angles from 0ºto 360º for which the value of a given trigonometricratio is the same (e.g., determine oneangle using a calculator and infer the otherangle)Sample problem: Determine the approximatemeasures of the angles from 0º to 360º forwhich the sine is 0.3423.1.4 solve multi-step problems in two and threedimensions, including those that arise fromreal-world applications (e.g., surveying, navigation),by determining the measures of thesides and angles of right triangles using theprimary trigonometric ratiosSample problem: Explain how you could findthe height of an inaccessible antenna on topof a tall building, using a measuring tape, aclinometer, and trigonometry. What wouldyou measure, and how would you use thedata to calculate the height of the antenna?1.5 solve problems involving oblique triangles,including those that arise from real-worldapplications, using the sine law (includingthe ambiguous case) and the cosine lawSample problem: The following diagramrepresents a mechanism in which point B isfixed, point C is a pivot, and a slider A canmove horizontally as angle B changes. Theminimum value of angle B is 35º. How faris it from the extreme left position to theextreme right position of slider A?B28 cm2. Connecting Graphs and Equationsof Sinusoidal FunctionsBy the end of this course, students will:2.1 make connections between the sine ratio andthe sine function and between the cosine ratioand the cosine function by graphing the relationshipbetween angles from 0º to 360º andthe corresponding sine ratios or cosine ratios,with or without technology (e.g., by generatinga table of values using a calculator;by unwrapping the unit circle), definingthis relationship as the function f(x) = sin xor f(x) = cos x, and explaining why the relationshipis a functionC20 cmA


2.2 sketch the graphs of f(x) = sin x and f(x) = cos xfor angle measures expressed in degrees, anddetermine and describe their key properties(i.e., cycle, domain, range, intercepts, amplitude,period, maximum and minimum values,increasing/decreasing intervals)Sample problem: Describe and compare thekey properties of the graphs of f(x) = sin xand f(x) = cos x. Make some connectionsbetween the key properties of the graphsand your understanding of the sine andcosine ratios.2.3 determine, through investigation usingtechnology, the roles of the parameters d and cin functions of the form y = sin (x – d) + c andy = cos (x – d) + c, and describe these roles interms of transformations on the graphs off(x) = sin x and f(x) = cos x with anglesexpressed in degrees (i.e., vertical andhorizontal translations)Sample problem: Investigate the graphf(x) = 2 sin (x – d) + 10 for various values ofd, using technology, and describe the effectsof changing d in terms of a transformation.2.4 determine, through investigation using technology,the roles of the parameters a and k infunctions of the form y = a sin kx andy = a cos kx, and describe these roles interms of transformations on the graphs off(x) = sin x and f(x) = cos x with anglesexpressed in degrees (i.e., reflections in theaxes; vertical and horizontal stretches andcompressions to and from the x- and y-axes)Sample problem: Investigate the graphf(x) = 2 sin kx for various values of k, usingtechnology, and describe the effects of changingk in terms of transformations.2.5 determine the amplitude, period, and phaseshift of sinusoidal functions whose equationsare given in the form f(x) = a sin (k(x – d)) + cor f(x) = a cos (k(x – d)) + c, and sketchgraphs of y = a sin (k(x – d)) + c andy = a cos (k(x – d)) + c by applying transformationsto the graphs of f(x) = sin x andf(x) = cos xSample problem: Transform the graph off(x) = cos x to sketch g(x) = 3cos(x + 90°)and h(x) = cos (2x) – 1, and state the amplitude,period, and phase shift of each function.2.6 represent a sinusoidal function with anequation, given its graph or its propertiesSample problem: A sinusoidal function hasan amplitude of 2 units, a period of 180º, anda maximum at (0, 3). Represent the functionwith an equation in two different ways,using first the sine function and then thecosine function.3. Solving Problems InvolvingSinusoidal FunctionsBy the end of this course, students will:3.1 collect data that can be modelled as a sinusoidalfunction (e.g., voltage in an AC circuit,pressure in sound waves, the height of a tackon a bicycle wheel that is rotating at a fixedspeed), through investigation with and withouttechnology, from primary sources, usinga variety of tools (e.g., concrete materials,measurement tools such as motion sensors),or from secondary sources (e.g., websitessuch as Statistics Canada, E-STAT), andgraph the dataSample problem: Measure and record distance−time data for a swinging pendulum, using amotion sensor or other measurement tools,and graph the data. Describe how the graphwould change if you moved the pendulumfurther away from the motion sensor. Whatwould you do to generate a graph with asmaller amplitude?3.2 identify periodic and sinusoidal functions,including those that arise from real-worldapplications involving periodic phenomena,given various representations (i.e., tables ofvalues, graphs, equations), and explain anyrestrictions that the context places on thedomain and rangeSample problem: The depth, w metres, ofwater in a lake can be modelled by the functionw = 5 sin (31.5n + 63) + 12, where n isthe number of months since January 1, 1995.Identify and explain the restrictions on thedomain and range of this function.3.3 pose problems based on applications involvinga sinusoidal function, and solve these andother such problems by using a given graph ora graph generated with technology, in degreemode, from a table of values or from itsequationMathematics for College TechnologyMCT4CTRIGONOMETRIC FUNCTIONS131


Grade 12, College PreparationSample problem: The height above theground of a rider on a Ferris wheel can bemodelled by the sinusoidal functionh(t) = 25 cos (3 (t – 60)) + 27, where h(t) is theheight in metres and t is the time in seconds.Graph the function, using graphing technologyin degree mode, and determine themaximum and minimum heights of the rider,the height after 30 s, and the time requiredto complete one revolution.THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics132


D. APPLICATIONS OF GEOMETRYOVERALL EXPECTATIONSBy the end of this course, students will:1. represent vectors, add and subtract vectors, and solve problems using vector models, including thosearising from real-world applications;2. solve problems involving two-dimensional shapes and three-dimensional figures and arising fromreal-world applications;3. determine circle properties and solve related problems, including those arising from real-worldapplications.SPECIFIC EXPECTATIONSMathematics for College Technology1. Modelling With VectorsBy the end of this course, students will:1.1 recognize a vector as a quantity with bothmagnitude and direction, and identify, gather,and interpret information about real-worldapplications of vectors (e.g., displacement;forces involved in structural design; simpleanimation of computer graphics; velocitydetermined using GPS)Sample problem: Position is represented usingvectors. Explain why knowing that someoneis 69 km from Lindsay, Ontario, is not sufficientto identify their exact position.1.2 represent a vector as a directed line segment,with directions expressed in different ways(e.g., 320°; N 40° W), and recognize vectorswith the same magnitude and direction butdifferent positions as equal vectors1.3 resolve a vector represented as a directedline segment into its vertical and horizontalcomponentsSample problem: A cable exerts a force of558 N at an angle of 37.2° with the horizontal.Resolve this force into its vertical andhorizontal components.37.2°558 N1.4 represent a vector as a directed line segment,given its vertical and horizontal components(e.g., the displacement of a ship that travels3 km east and 4 km north can be representedby the vector with a magnitude of 5 km anda direction of N 36.9° E)1.5 determine, through investigation using a varietyof tools (e.g., graph paper, technology)and strategies (i.e., head-to-tail method; parallelogrammethod; resolving vectors into theirvertical and horizontal components), the sum(i.e., resultant) or difference of two vectors1.6 solve problems involving the addition andsubtraction of vectors, including problemsarising from real-world applications (e.g.,surveying, statics, orienteering)Sample problem: Two people pull on ropesto haul a truck out of some mud. The firstperson pulls directly forward with a forceof 400 N, while the other person pulls witha force of 600 N at a 50° angle to the firstperson along the horizontal plane. What isthe resultant force used on the truck?2. Solving Problems InvolvingGeometryBy the end of this course, students will:2.1 gather and interpret information about realworldapplications of geometric shapes andfigures in a variety of contexts in technologyrelatedfields (e.g., product design, architecture),and explain these applications (e.g., oneMCT4CAPPLICATIONS OF GEOMETRY133


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, College Preparationreason that sewer covers are round is to preventthem from falling into the sewer duringremoval and replacement)Sample problem: Explain why rectangularprisms are often used for packaging.2.2 perform required conversions between theimperial system and the metric system usinga variety of tools (e.g., tables, calculators,online conversion tools), as necessary withinapplications2.3 solve problems involving the areas of rectangles,parallelograms, trapezoids, triangles,and circles, and of related composite shapes,in situations arising from real-worldapplicationsSample problem: Your company suppliescircular cover plates for pipes. How manyplates with a 1-ft radius can be made froma 4-ft by 8-ft sheet of stainless steel? Whatpercentage of the steel will be available forrecycling?2.4 solve problems involving the volumes andsurface areas of spheres, right prisms, andcylinders, and of related composite figures, insituations arising from real-world applicationsSample problem: For the small factory shownin the following diagram, design specificationsrequire that the air be exchanged every30 min. Would a ventilation system that3exchanges air at a rate of 400 ft /min satisfythe specifications? Explain.45° 45°20 ft50 ft3. Solving Problems InvolvingCircle PropertiesBy the end of this course, students will:10 ft3.1 recognize and describe (i.e., using diagramsand words) arcs, tangents, secants, chords,segments, sectors, central angles, andinscribed angles of circles, and some of theirreal-world applications (e.g., construction ofa medicine wheel)3.2 determine the length of an arc and the area ofa sector or segment of a circle, and solve relatedproblemsSample problem: A circular lake has a diameterof 4 km. Points A and D are on oppositesides of the lake and lie on a straight linethrough the centre of the lake, with eachpoint 5 km from the centre. In the routeABCD, AB and CD are tangents to the lakeand BC is an arc along the shore of the lake.How long is this route?AB4 km3.3 determine, through investigation using a varietyof tools (e.g., dynamic geometry software),properties of the circle associated with chords,central angles, inscribed angles, and tangents(e.g., equal chords or equal arcs subtend equalcentral angles and equal inscribed angles; aradius is perpendicular to a tangent at thepoint of tangency defined by the radius, andto a chord that the radius bisects)Sample problem: Investigate, using dynamicgeometry software, the relationship betweenthe lengths of two tangents drawn to a circlefrom a point outside the circle.3.4 solve problems involving properties of circles,including problems arising from real-worldapplicationsSample problem: A cylindrical metal rod witha diameter of 1.2 cm is supported by a woodenblock, as shown in the following diagram.Determine the distance from the top of theblock to the top of the rod.C5 km 5 km1.0 cm1.0 cm1.2 cm1.0 cmD1.0 cm130°134


Foundations for CollegeMathematics, Grade 12College PreparationMAP4CThis course enables students to broaden their understanding of real-world applicationsof mathematics. Students will analyse data using statistical methods; solve problemsinvolving applications of geometry and trigonometry; solve financial problemsconnected with annuities, budgets, and renting or owning accommodation; simplifyexpressions; and solve equations. Students will reason mathematically and communicatetheir thinking as they solve multi-step problems. This course prepares students forcollege programs in areas such as business, health sciences, and human services, andfor certain skilled trades.Prerequisite: Foundations for College Mathematics, Grade 11, College Preparation, orFunctions and Applications, Grade 11, University/College Preparation135


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.136


A. MATHEMATICAL MODELSOVERALL EXPECTATIONSBy the end of this course, students will:1. evaluate powers with rational exponents, simplify algebraic expressions involving exponents, andsolve problems involving exponential equations graphically and using common bases;2. describe trends based on the interpretation of graphs, compare graphs using initial conditions andrates of change, and solve problems by modelling relationships graphically and algebraically;3. make connections between formulas and linear, quadratic, and exponential relations, solve problemsusing formulas arising from real-world applications, and describe applications of mathematicalmodelling in various occupations.SPECIFIC EXPECTATIONS1. Solving Exponential EquationsBy the end of this course, students will:1.1 determine, through investigation (e.g., byexpanding terms and patterning), the exponentlaws for multiplying and dividing algebraic3 2expressions involving exponents [e.g., (x )(x ),3 5x ÷ x ] and the exponent law for simplifyingalgebraic expressions involving a power of a6 3 2power [e.g. (x y)]1.2 simplify algebraic expressions containing integerexponents using the laws of exponents2 5 5abcSample problem: Simplify and–3 4ab cevaluate for a = 8, b = 2, and c = – 30.1.3 determine, through investigation using avariety of tools (e.g., calculator, paper andpencil, graphing technology) and strategies(e.g., patterning; finding values from a graph;interpreting the exponent laws), the value ofma power with a rational exponent (i.e., x n ,where x > 0 and m and n are integers)Sample problem: The exponent laws suggest1 1that 42x 42 1= 4 . What value would you1assign to 42? What value would you assign1to 273? Explain your reasoning. Extend yourreasoning to make a generalization about the1meaning of x n, where x > 0 and n is a naturalnumber.1.4 evaluate, with or without technology, numericalexpressions involving rational exponents1–3 3and rational bases [e.g., 2 , (–6) , 42 120, 1.01 ] *1.5 solve simple exponential equations numericallyand graphically, with technology (e.g.,use systematic trial with a scientific calculatorto determine the solution to the equationx1.05 = 1.276), and recognize that the solutionsmay not be exactSample problem: Use the graph of y = 3 toxsolve the equation 3 = 5.1.6 solve problems involving exponential equationsarising from real-world applications byusing a graph or table of values generatedwith technology from a given equation [e.g.,nh = 2(0.6) , where h represents the height ofa bouncing ball and n represents the numberof bounces]Sample problem: Dye is injected to test pancreasfunction. The mass, R grams, of dye remainingin a healthy pancreas after t minutestis given by the equation R = I(0.96) , whereI grams is the mass of dye initially injected.If 0.50 g of dye is initially injected into ahealthy pancreas, determine how much timeelapses until 0.35 g remains by using a graphand/or table of values generated withtechnology.*The knowledge and skills described in this expectation are to be introduced as needed, and applied and consolidated, whereappropriate, throughout the course.xFoundations for College MathematicsMAP4CMATHEMATICAL MODELS137


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, College Preparation1.7 solve exponential equations in one variablexby determining a common base (e.g., 2 = 32,5x − 1 2(x + 11) 5x + 8 x4 = 2 , 3 = 27 )5x + 8Sample problem: Solve 3 = 27 bydetermining a common base, verify bysubstitution, and make connections to the5x + 8xintersection of y = 3 and y = 27 usinggraphing technology.2. Modelling GraphicallyBy the end of this course, students will:2.1 interpret graphs to describe a relationship(e.g., distance travelled depends on drivingtime, pollution increases with traffic volume,maximum profit occurs at a certain sales volume),using language and units appropriateto the context2.2 describe trends based on given graphs, anduse the trends to make predictions or justifydecisions (e.g., given a graph of the men’s100-m world record versus the year, predictthe world record in the year 2050 and stateyour assumptions; given a graph showingthe rising trend in graduation rates amongAboriginal youth, make predictions aboutfuture rates)Sample problem: Given the following graph,describe the trend in Canadian greenhousegas emissions over the time period shown.Describe some factors that may have influencedthese emissions over time. Predict theemissions today, explain your predictionusing the graph and possible factors, andverify using current data.Canadian Greenhouse Gas EmissionsGigatons of CO2equivalent0.80.70.60.50.40.30.20.1Kyoto benchmark 0.01980 1984 1988 1992 1996 2000Source: Environment Canada, Greenhouse Gas Inventory1990-2001, 2003x2.3 recognize that graphs and tables of valuescommunicate information about rate of change,and use a given graph or table of values for arelation to identify the units used to measurerate of change (e.g., for a distance–time graph,the units of rate of change are kilometres perhour; for a table showing earnings over time,the units of rate of change are dollars per hour)2.4 identify when the rate of change is zero,constant, or changing, given a table of valuesor a graph of a relation, and compare twographs by describing rate of change (e.g.,compare distance–time graphs for a car thatis moving at constant speed and a car that isaccelerating)2.5 compare, through investigation with technology,the graphs of pairs of relations (i.e.,linear, quadratic, exponential) by describingthe initial conditions and the behaviour ofthe rates of change (e.g., compare the graphsof amount versus time for equal initialdeposits in simple interest and compoundinterest accounts)Sample problem: In two colonies of bacteria,the population doubles every hour. The initialpopulation of one colony is twice theinitial population of the other. How do thegraphs of population versus time comparefor the two colonies? How would the graphschange if the population tripled every hour,instead of doubling?2.6 recognize that a linear model correspondsto a constant increase or decrease over equalintervals and that an exponential modelcorresponds to a constant percentage increaseor decrease over equal intervals, select amodel (i.e., linear, quadratic, exponential) torepresent the relationship between numericaldata graphically and algebraically, using avariety of tools (e.g., graphing technology)and strategies (e.g., finite differences, regression),and solve related problemsSample problem: Given the data table at thetop of page 139, determine an algebraicmodel to represent the relationship betweenpopulation and time, using technology. Usethe algebraic model to predict the populationin 2015, and describe any assumptions made.138


Years after 1955 Population of Geese0 5 00010 12 00020 26 00030 62 00040 142 00050 260 0003. Modelling AlgebraicallyBy the end of this course, students will:n3.1 solve equations of the form x = a using3rational exponents (e.g., solve x = 7by1raising both sides to the exponent )33.2 determine the value of a variable of degreeno higher than three, using a formula drawnfrom an application, by first substitutingknown values and then solving for the variable,and by first isolating the variable andthen substituting known values4 3Sample problem: Use the formula V = πr3to determine the radius of a sphere with a3volume of 1000 cm .3.3 make connections between formulas and linear,quadratic, and exponential functions [e.g.,recognize that the compound interest formula,nA = P(1 + i) , is an example of an exponentialfunction A(n) when P and i are constant, andof a linear function A(P) when i and n are constant],using a variety of tools and strategies(e.g., comparing the graphs generated withtechnology when different variables in aformula are set as constants)Sample problem: Which variable(s) in the2formula V = πr hwould you need to set asa constant to generate a linear equation?A quadratic equation? Explain why you canexpect the relationship between the volumeand the height to be linear when the radiusis constant.3.4 solve multi-step problems requiring formulasarising from real-world applications (e.g.,determining the cost of two coats of paint fora large cylindrical tank)3.5 gather, interpret, and describe informationabout applications of mathematical modellingin occupations, and about college programsthat explore these applicationsFoundations for College MathematicsMAP4CMATHEMATICAL MODELS139


B. PERSONAL FINANCETHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, College PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. demonstrate an understanding of annuities, including mortgages, and solve related problems usingtechnology;2. gather, interpret, and compare information about owning or renting accommodation, and solveproblems involving the associated costs;3. design, justify, and adjust budgets for individuals and families described in case studies, and describeapplications of the mathematics of personal finance.SPECIFIC EXPECTATIONS1. Understanding AnnuitiesBy the end of this course, students will:1.1 gather and interpret information about annuities,describe the key features of an annuity,and identify real-world applications (e.g.,RRSP, mortgage, RRIF, RESP)1.2 determine, through investigation using technology(e.g., the TVM Solver on a graphingcalculator; online tools), the effects of changingthe conditions (i.e., the payments, thefrequency of the payments, the interest rate,the compounding period) of an ordinarysimple annuity (i.e., an annuity in whichpayments are made at the end of each period,and compounding and payment periods arethe same) (e.g., long-term savings plans,loans)Sample problem: Given an ordinary simpleannuity with semi-annual deposits of $1000,earning 6% interest per year compoundedsemi-annually, over a 20-year term, which ofthe following results in the greatest return:doubling the payments, doubling the interestrate, doubling the frequency of the paymentsand the compounding, or doubling the paymentand compounding period?1.3 solve problems, using technology (e.g., scientificcalculator, spreadsheet, graphing calculator),that involve the amount, the presentvalue, and the regular payment of an ordinarysimple annuitySample problem: Using a spreadsheet, calculatethe total interest paid over the life of a$10 000 loan with monthly repayments over2 years at 8% per year compounded monthly,and compare the total interest with the originalprincipal of the loan.1.4 demonstrate, through investigation usingtechnology (e.g., a TVM Solver), the advantagesof starting deposits earlier when investingin annuities used as long-term savingsplansSample problem: If you want to have a milliondollars at age 65, how much would youhave to contribute monthly into an investmentthat pays 7% per annum, compoundedmonthly, beginning at age 20? At age 35?At age 50?1.5 gather and interpret information about mortgages,describe features associated withmortgages (e.g., mortgages are annuities forwhich the present value is the amount borrowedto purchase a home; the interest on amortgage is compounded semi-annually butoften paid monthly), and compare differenttypes of mortgages (e.g., open mortgage,closed mortgage, variable-rate mortgage)1.6 read and interpret an amortization table fora mortgageSample problem: You purchase a $200 000condominium with a $25 000 down payment,and you mortgage the balance at 6.5% per yearcompounded semi-annually over 25 years,140


payable monthly. Use a given amortizationtable to compare the interest paid in the firstyear of the mortgage with the interest paid inthe 25th year.1.7 generate an amortization table for a mortgage,using a variety of tools and strategies (e.g.,input data into an online mortgage calculator;determine the payments using the TVMSolver on a graphing calculator and generatethe amortization table using a spreadsheet),calculate the total interest paid over the lifeof a mortgage, and compare the total interestwith the original principal of the mortgage1.8 determine, through investigation using technology(e.g., TVM Solver, online tools, financialsoftware), the effects of varying paymentperiods, regular payments, and interest rateson the length of time needed to pay off amortgage and on the total interest paidSample problem: Calculate the interest savedon a $100 000 mortgage with monthly payments,at 6% per annum compounded semiannually,when it is amortized over 20 yearsinstead of 25 years.2. Renting or Owning AccommodationBy the end of this course, students will:2.1 gather and interpret information about theprocedures and costs involved in owning andin renting accommodation (e.g., apartment,condominium, townhouse, detached home)in the local community2.2 compare renting accommodation with owningaccommodation by describing the advantagesand disadvantages of each2.3 solve problems, using technology (e.g., calculator,spreadsheet), that involve the fixed costs(e.g., mortgage, insurance, property tax) andvariable costs (e.g., maintenance, utilities) ofowning or renting accommodationSample problem: Calculate the total of thefixed and variable monthly costs that areassociated with owning a detached housebut that are usually included in the rent forrental accommodation.3. Designing BudgetsBy the end of this course, students will::3.1 gather, interpret, and describe informationabout living costs, and estimate the livingcosts of different households (e.g., a family offour, including two young children; a singleyoung person; a single parent with one child)in the local community3.2 design and present a savings plan to facilitatethe achievement of a long-term goal (e.g.,attending college, purchasing a car, rentingor purchasing a house)3.3 design, explain, and justify a monthly budgetsuitable for an individual or family describedin a given case study that provides the specificsof the situation (e.g., income; personalresponsibilities; costs such as utilities, food,rent/mortgage, entertainment, transportation,charitable contributions; long-term savingsgoals), with technology (e.g., using spreadsheets,budgeting software, online tools)and without technology (e.g., using budgettemplates)3.4 identify and describe the factors to be consideredin determining the affordability ofaccommodation in the local community (e.g.,income, long-term savings, number of dependants,non-discretionary expenses), and considerthe affordability of accommodationunder given circumstancesSample problem: Determine, through investigation,if it is possible to change from rentingto owning accommodation in your communityin five years if you currently earn $30 000per year, pay $900 per month in rent, andhave savings of $20 000.3.5 make adjustments to a budget to accommodatechanges in circumstances (e.g., loss ofhours at work, change of job, change in personalresponsibilities, move to new accommodation,achievement of a long-term goal,major purchase), with technology (e.g.,spreadsheet template, budgeting software)3.6 gather, interpret, and describe informationabout applications of the mathematics of personalfinance in occupations (e.g., selling realestate, bookkeeping, managing a restaurant,financial planning, mortgage brokering), andabout college programs that explore theseapplicationsFoundations for College MathematicsMAP4CPERSONAL FINANCE141


C. GEOMETRY AND TRIGONOMETRYTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, College PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. solve problems involving measurement and geometry and arising from real-world applications;2. explain the significance of optimal dimensions in real-world applications, and determine optimaldimensions of two-dimensional shapes and three-dimensional figures;3. solve problems using primary trigonometric ratios of acute and obtuse angles, the sine law, and thecosine law, including problems arising from real-world applications, and describe applications oftrigonometry in various occupations.SPECIFIC EXPECTATIONS1. Solving Problems InvolvingMeasurement and GeometryBy the end of this course, students will:1.1 perform required conversions between theimperial system and the metric system usinga variety of tools (e.g., tables, calculators,online conversion tools), as necessary withinapplications1.2 solve problems involving the areas of rectangles,triangles, and circles, and of relatedcomposite shapes, in situations arising fromreal-world applicationsSample problem: A car manufacturer wants todisplay three of its compact models in a triangulararrangement on a rotating circular platform.Calculate a reasonable area for this platform,and explain your assumptions and reasoning.1.3 solve problems involving the volumes andsurface areas of rectangular prisms, triangularprisms, and cylinders, and of related compositefigures, in situations arising from realworldapplicationsSample problem: Compare the volumes ofconcrete needed to build three steps that are4 ft wide and that have the cross-sectionsshown below. Explain your assumptions andreasoning.2. Investigating Optimal DimensionsBy the end of this course, students will:2.1 recognize, through investigation using a varietyof tools (e.g., calculators; dynamic geometrysoftware; manipulatives such as tiles, geoboards,toothpicks) and strategies (e.g., modelling;making a table of values; graphing), andexplain the significance of optimal perimeter,area, surface area, and volume in variousapplications (e.g., the minimum amount ofpackaging material, the relationship betweensurface area and heat loss)Sample problem: You are building a deckattached to the second floor of a cottage, asshown below. Investigate how perimetervaries with different dimensions if you buildthe deck using exactly 48 1-m x 1-m deckingsections, and how area varies if you useexactly 30 m of deck railing. Note: the entireoutside edge of the deck will be railed.DeckCottage142


2.2 determine, through investigation using avariety of tools (e.g., calculators, dynamicgeometry software, manipulatives) and strategies(e.g., modelling; making a table of values;graphing), the optimal dimensions of a twodimensionalshape in metric or imperial unitsfor a given constraint (e.g., the dimensionsthat give the minimum perimeter for a givenarea)Sample problem: You are constructing a rectangulardeck against your house. You willuse 32 ft of railing and will leave a 4-ft gapin the railing for access to stairs. Determinethe dimensions that will maximize the areaof the deck.2.3 determine, through investigation using a varietyof tools and strategies (e.g., modelling withmanipulatives; making a table of values;graphing), the optimal dimensions of a rightrectangular prism, a right triangular prism,and a right cylinder in metric or imperial unitsfor a given constraint (e.g., the dimensionsthat give the maximum volume for a givensurface area)Sample problem: Use a table of values and agraph to investigate the dimensions of a rectangularprism, a triangular prism, and a3cylinder that each have a volume of 64 cmand the minimum surface area3. Solving Problems InvolvingTrigonometryBy the end of this course, students will:3.1 solve problems in two dimensions usingmetric or imperial measurements, includingproblems that arise from real-world applications(e.g., surveying, navigation, buildingconstruction), by determining the measuresof the sides and angles of right triangles usingthe primary trigonometric ratios, and of acutetriangles using the sine law and the cosine law3.2 make connections between primary trigonometricratios (i.e., sine, cosine, tangent) ofobtuse angles and of acute angles, throughinvestigation using a variety of tools andstrategies (e.g., using dynamic geometrysoftware to identify an obtuse angle withthe same sine as a given acute angle; usinga circular geoboard to compare congruenttriangles; using a scientific calculator to comparetrigonometric ratios for supplementaryangles)3.3 determine the values of the sine, cosine, andtangent of obtuse angles3.4 solve problems involving oblique triangles,including those that arise from real-worldapplications, using the sine law (in nonambiguouscases only) and the cosine law,and using metric or imperial unitsSample problem: A plumber must cut a pieceof pipe to fit from A to B. Determine thelength of the pipe.A8Pipe115°3.5 gather, interpret, and describe informationabout applications of trigonometry in occupations,and about college programs that explorethese applicationsSample problem: Prepare a presentation toshowcase an occupation that makes use oftrigonometry, to describe the education andtraining needed for the occupation, and tohighlight a particular use of trigonometryin the occupation.5BFoundations for College MathematicsMAP4CGEOMETRY AND TRIGONOMETRY143


D. DATA MANAGEMENTTHE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, College PreparationOVERALL EXPECTATIONSBy the end of this course, students will:1. collect, analyse, and summarize two-variable data using a variety of tools and strategies, andinterpret and draw conclusions from the data;2. demonstrate an understanding of the applications of data management used by the media and theadvertising industry and in various occupations.SPECIFIC EXPECTATIONS1. Working With Two-Variable DataBy the end of this course, students will:1.1 distinguish situations requiring one-variableand two-variable data analysis, describe theassociated numerical summaries (e.g., tallycharts, summary tables) and graphical summaries(e.g., bar graphs, scatter plots), andrecognize questions that each type of analysisaddresses (e.g., What is the frequency of aparticular trait in a population? What is themathematical relationship between twovariables?)Sample problem: Given a table showing shoesize and height for several people, pose aquestion that would require one-variableanalysis and a question that would requiretwo-variable analysis of the data.1.2 describe characteristics of an effective survey(e.g., by giving consideration to ethics, privacy,the need for honest responses, and possiblesources of bias, including cultural bias),and design questionnaires (e.g., for determiningif there is a relationship between age andhours per week of Internet use, betweenmarks and hours of study, or between incomeand years of education) or experiments (e.g.,growth of plants under different conditions)for gathering two-variable data1.3 collect two-variable data from primary sources,through experimentation involving observationor measurement, or from secondarysources (e.g., Internet databases, newspapers,magazines), and organize and store the datausing a variety of tools (e.g., spreadsheets,dynamic statistical software)Sample problem: Download census data fromStatistics Canada on age and average income,store the data using dynamic statistics software,and organize the data in a summarytable.1.4 create a graphical summary of two-variabledata using a scatter plot (e.g., by identifyingand justifying the dependent and independentvariables; by drawing the line of bestfit, when appropriate), with and withouttechnology1.5 determine an algebraic summary of the relationshipbetween two variables that appearto be linearly related (i.e., the equation of theline of best fit of the scatter plot), using avariety of tools (e.g., graphing calculators,graphing software) and strategies (e.g., usingsystematic trials to determine the slope andy-intercept of the line of best fit; using theregression capabilities of a graphing calculator),and solve related problems (e.g., use theequation of the line of best fit to interpolateor extrapolate from the given data set)1.6 describe possible interpretations of the line ofbest fit of a scatter plot (e.g., the variables arelinearly related) and reasons for misinterpretations(e.g., using too small a sample; failingto consider the effect of outliers; interpolatingfrom a weak correlation; extrapolating nonlinearlyrelated data)1.7 determine whether a linear model (i.e., a lineof best fit) is appropriate given a set of twovariabledata, by assessing the correlation144


etween the two variables (i.e., by describingthe type of correlation as positive, negative, ornone; by describing the strength as strong orweak; by examining the context to determinewhether a linear relationship is reasonable)1.8 make conclusions from the analysis of twovariabledata (e.g., by using a correlation tosuggest a possible cause-and-effect relationship),and judge the reasonableness of theconclusions (e.g., by assessing the strength ofthe correlation; by considering if there areenough data)2. Applying Data ManagementBy the end of this course, students will:2.1 recognize and interpret common statisticalterms (e.g., percentile, quartile) and expressions(e.g., accurate 19 times out of 20) usedin the media (e.g., television, Internet, radio,newspapers)2.2 describe examples of indices used by themedia (e.g., consumer price index, S&P/TSXcomposite index, new housing price index)and solve problems by interpreting and usingindices (e.g., by using the consumer priceindex to calculate the annual inflation rate)Sample problem: Use the new housing priceindex on E-STAT to track the cost of purchasinga new home over the past 10 years in theToronto area, and compare with the cost inCalgary, Charlottetown, and Vancouver overthe same period. Predict how much a newhome that today costs $200 000 in each ofthese cities will cost in 5 years.2.3 interpret statistics presented in the media(e.g., the UN’s finding that 2% of the world’spopulation has more than half the world’swealth, whereas half the world’s populationhas only 1% of the world’s wealth), andexplain how the media, the advertising industry,and others (e.g., marketers, pollsters) useand misuse statistics (e.g., as represented ingraphs) to promote a certain point of view(e.g., by making a general statement basedon a weak correlation or an assumed causeand-effectrelationship; by starting the verticalscale on a graph at a value other than zero; bymaking statements using general populationstatistics without reference to data specific tominority groups)2.4 assess the validity of conclusions presentedin the media by examining sources of data,including Internet sources (i.e., to determinewhether they are authoritative, reliable,unbiased, and current), methods of datacollection, and possible sources of bias (e.g.,sampling bias, non-response bias, a bias in asurvey question), and by questioning theanalysis of the data (e.g., whether there is anyindication of the sample size in the analysis)and conclusions drawn from the data (e.g.,whether any assumptions are made aboutcause and effect)Sample problem: The headline that accompaniesthe following graph says “Big Increasein Profits”. Suggest reasons why this headlinemay or may not be true.Profits ($ billions)232221201918172001 2002 2003 2004 2005 2006 2007Year2.5 gather, interpret, and describe informationabout applications of data management inoccupations, and about college programs thatexplore these applicationsFoundations for College MathematicsMAP4CDATA MANAGEMENT145


Mathematics for Work andEveryday Life, Grade 12Workplace PreparationMEL4EThis course enables students to broaden their understanding of mathematics as it isapplied in the workplace and daily life. Students will investigate questions involvingthe use of statistics; apply the concept of probability to solve problems involving familiarsituations; investigate accommodation costs, create household budgets, and prepare apersonal income tax return; use proportional reasoning; estimate and measure; andapply geometric concepts to create designs. Students will consolidate their mathematicalskills as they solve problems and communicate their thinking.Prerequisite: Mathematics for Work and Everyday Life, Grade 11, Workplace Preparation147


MATHEMATICAL PROCESS EXPECTATIONSThe mathematical processes are to be integrated into student learning in all areas of this course.Throughout this course, students will:Problem SolvingReasoning andProvingReflecting• develop, select, apply, compare, and adapt a variety of problem-solvingstrategies as they pose and solve problems and conduct investigations, tohelp deepen their mathematical understanding;• develop and apply reasoning skills (e.g., use of inductive reasoning, deductivereasoning, and counter-examples; construction of proofs) to make mathematicalconjectures, assess conjectures, and justify conclusions, and plan and constructorganized mathematical arguments;• demonstrate that they are reflecting on and monitoring their thinking tohelp clarify their understanding as they complete an investigation or solve aproblem (e.g., by assessing the effectiveness of strategies and processes used,by proposing alternative approaches, by judging the reasonableness of results,by verifying solutions);Selecting Tools andComputationalStrategies• select and use a variety of concrete, visual, and electronic learning tools andappropriate computational strategies to investigate mathematical ideas andto solve problems;ConnectingRepresentingCommunicating• make connections among mathematical concepts and procedures, and relatemathematical ideas to situations or phenomena drawn from other contexts(e.g., other curriculum areas, daily life, current events, art and culture, sports);• create a variety of representations of mathematical ideas (e.g., numeric,geometric, algebraic, graphical, pictorial representations; onscreen dynamicrepresentations), connect and compare them, and select and apply theappropriate representations to solve problems;• communicate mathematical thinking orally, visually, and in writing, using precisemathematical vocabulary and a variety of appropriate representations, andobserving mathematical conventions.148


A. REASONING WITH DATAOVERALL EXPECTATIONSBy the end of this course, students will:1. collect, organize, represent, and make inferences from data using a variety of tools and strategies,and describe related applications;2. determine and represent probability, and identify and interpret its applications.SPECIFIC EXPECTATIONS1. Interpreting and Displaying DataBy the end of this course, students will:1.1 read and interpret graphs (e.g., bar graph,broken-line graph, histogram) obtained fromvarious sources (e.g., newspapers, magazines,Statistics Canada website)1.2 explain the distinction between the termspopulation and sample, describe the characteristicsof a good sample, and explain whysampling is necessary (e.g., time, cost, orphysical constraints)Sample problem: What are some factors thata manufacturer should consider when determiningwhether to test a sample or the entirepopulation to ensure the quality of a product?1.3 collect categorical data from primary sources,through experimentation involving observation(e.g., by tracking food orders in restaurantsoffering healthy food options) or measurement,or from secondary sources (e.g., Internet databases,newspapers, magazines), and organizeand store the data using a variety of tools (e.g.,spreadsheets, dynamic statistical software)Sample problem: Observe cars that passthrough a nearby intersection. Collect data onseatbelt usage or the number of passengersper car.1.4 represent categorical data by constructinggraphs (e.g., bar graph, broken-line graph,circle graph) using a variety of tools (e.g.,dynamic statistical software, graphingcalculator, spreadsheet)1.5 make inferences based on the graphical representationof data (e.g., an inference about asample from the graphical representation ofa population), and justify conclusions orallyor in writing using convincing arguments(e.g., by showing that it is reasonable toassume that a sample is representative ofa population)1.6 make and justify conclusions about a topicof personal interest by collecting, organizing(e.g., using spreadsheets), representing (e.g.,using graphs), and making inferences fromcategorical data from primary sources (i.e.,collected through measurement or observation)or secondary sources (e.g., electronicdata from databases such as E-STAT, datafrom newspapers or magazines)1.7 explain how the media, the advertising industry,and others (e.g., marketers, pollsters) useand misuse statistics (e.g., as represented ingraphs) to promote a certain point of view(e.g., by making general statements based onsmall samples; by making statements usinggeneral population statistics without referenceto data specific to minority groups)Sample problem: The headline that accompaniesthe following graph says “Big Increasein Profits”. Suggest reasons why this headlinemay or may not be true.Profits ($ billions)232221201918172001 2002 2003 2004 2005 2006 2007YearMathematics for Work and Everyday LifeMEL4EREASONING WITH DATA149


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, Workplace Preparation1.8 gather, interpret, and describe informationabout applications of data management in theworkplace and in everyday life2. Investigating ProbabilityBy the end of this course, students will:2.1 determine the theoretical probability of anevent (i.e., the ratio of the number offavourable outcomes to the total number ofpossible outcomes, where all outcomes areequally likely), and represent the probabilityin a variety of ways (e.g., as a fraction, as apercent, as a decimal in the range 0 to 1)2.2 identify examples of the use of probabilityin the media (e.g., the probability of rain, ofwinning a lottery, of wait times for a serviceexceeding specified amounts) and variousways in which probability is represented(e.g., as a fraction, as a percent, as a decimalin the range 0 to 1)2.3 perform simple probability experiments (e.g.,rolling number cubes, spinning spinners, flippingcoins, playing Aboriginal stick-and-stonegames), record the results, and determine theexperimental probability of an event2.4 compare, through investigation, the theoreticalprobability of an event with the experimentalprobability, and describe how uncertaintyexplains why they might differ (e.g., “I knowthat the theoretical probability of getting tailsis 0.5, but that does not mean that I willalways obtain 3 tails when I toss the coin6 times”; “If a lottery has a 1 in 9 chanceof winning, am I certain to win if I buy9 tickets?”)2.5 determine, through investigation using classgenerateddata and technology-based simulationmodels (e.g., using a random-numbergenerator on a spreadsheet or on a graphingcalculator), the tendency of experimentalprobability to approach theoretical probabilityas the number of trials in an experimentincreases (e.g., “If I simulate tossing a coin1000 times using technology, the experimentalprobability that I calculate for getting tails inany one toss is likely to be closer to the theoreticalprobability than if I simulate tossingthe coin only 10 times”)Sample problem: Calculate the theoreticalprobability of rolling a 2 on a number cube.Simulate rolling a number cube, and use thesimulation to calculate the experimentalprobability of rolling a 2 after 10, 20, 30, …,200 trials. Graph the experimental probabilityversus the number of trials, and describe anytrend.2.6 interpret information involving the use ofprobability and statistics in the media, anddescribe how probability and statistics canhelp in making informed decisions in avariety of situations (e.g., weighing the riskof injury when considering different occupations;using a weather forecast to planoutdoor activities; using sales data to stocka clothing store with appropriate stylesand sizes)Sample problem: A recent study on youthgambling suggests that approximately 30%of adolescents gamble on a weekly basis.Investigate and describe the assumptionsthat people make about the probability ofwinning when they gamble. Describe otherfactors that encourage gambling and problemsexperienced by people with a gamblingaddiction.150


B. PERSONAL FINANCEOVERALL EXPECTATIONSBy the end of this course, students will:1. gather, interpret, and compare information about owning or renting accommodation and aboutthe associated costs;2. interpret, design, and adjust budgets for individuals and families described in case studies;3. demonstrate an understanding of the process of filing a personal income tax return, and describeapplications of the mathematics of personal finance.SPECIFIC EXPECTATIONS1. Renting or Owning AccommodationBy the end of this course, students will:1.1 identify the financial implications (e.g., responsibilityfor paying the cost of accommodationand furnishings; greater responsibilityfor financial decision making) and the nonfinancialimplications (e.g., greater freedomto make decisions; the demands of timemanagement or of adapting to a new environment;the possibility of loneliness or of theneed to share responsibilities) associatedwith living independently1.2 gather and compare, through investigation,information about the costs and the advantagesand disadvantages of different types ofrental accommodation in the local community(e.g., renting a room in someone’s house;renting a hotel room; renting or leasing anapartment)1.3 gather and compare, through investigation,information about purchase prices of differenttypes of owned accommodation in the localcommunity (e.g., trailer, condominium, townhouse,detached home)1.4 gather, interpret, and compare informationabout the different types of ongoing livingexpenses associated with renting and owningaccommodation (e.g., hydro, cable, telephone,Internet, heating, parking, laundry, groceries,cleaning supplies, transportation) and relatedcosts1.5 gather, interpret, and describe informationabout the rights and responsibilities of tenantsand landlords1.6 generate a checklist of necessary tasks associatedwith moving (e.g., change of address,set-up of utilities and services, truck rental),and estimate the total cost involved undervarious conditions (e.g., moving out ofprovince; hiring a moving company)2. Designing BudgetsBy the end of this course, students will:2.1 categorize personal expenses as nondiscretionary(e.g., rent, groceries, utilities,loan payments) or discretionary (e.g., entertainment,vacations)2.2 categorize personal non-discretionary expensesas fixed (e.g., rent, cable, car insurance)or variable (e.g., groceries, clothing,vehicle maintenance)2.3 read and interpret prepared individual orfamily budgets, identify and describe the keycomponents of a budget, and describe howbudgets can reflect personal values (e.g., asthey relate to shopping, saving for a longtermgoal, recreational activities, family,community)2.4 design, with technology (e.g., using spreadsheettemplates, budgeting software, onlinetools) and without technology (e.g., usingbudget templates), explain, and justify aMathematics for Work and Everyday LifeMEL4EPERSONAL FINANCE151


Grade 12, Workplace Preparationmonthly budget suitable for an individual orfamily described in a given case study thatprovides the specifics of the situation (e.g.,income; personal responsibilities; expensessuch as utilities, food, rent/mortgage, entertainment,transportation, charitable contributions;long-term savings goals)2.5 identify and describe factors to be consideredin determining the affordability of accommodationin the local community (e.g., income,long-term savings, number of dependants,non-discretionary expenses)2.6 make adjustments to a budget to accommodatechanges in circumstances (e.g., loss ofhours at work, change of job, change in personalresponsibilities, move to new accommodation,achievement of a long-term goal,major purchase), with technology (e.g.,spreadsheet template, budgeting software)3. Filing Income TaxBy the end of this course, students will:3.1 explain why most Canadians are expected tofile a personal income tax return each year,and identify and describe the major parts of apersonal income tax return (i.e., identification,total income, net income, taxable income,refund or balance owing)3.2 gather, interpret, and describe the informationand documents required for filing a personalincome tax return (e.g., CRA guides, forms,and schedules; T4 slips; receipts for charitabledonations), and explain why they are required3.3 gather, interpret, and compare informationabout common tax credits (e.g., tuition fees,medical expenses, charitable donations) andtax deductions (e.g., moving expenses, childcare expenses, union dues)3.4 complete a simple personal income tax return(i.e., forms and schedules), with or withouttax preparation software3.5 gather, interpret, and describe some additionalinformation that a self-employed individualshould provide when filing a personal incometax return (e.g., a statement of business activitiesthat includes business expenses such asinsurance, advertising, and motor-vehicleexpenses)3.6 gather, interpret, and describe informationabout services that will complete a personalincome tax return (e.g., tax preparation service,chartered accountant, voluntary service inthe community) and resources that will helpwith completing a personal income tax return(e.g., forms and publications available on theCanada Revenue Agency website, tax preparationsoftware for which rebates are available),and compare the services and resourceson the basis of the assistance they provide andtheir cost3.7 gather, interpret, and describe informationabout applications of the mathematics ofpersonal finance in the workplace (e.g.,selling real estate, bookkeeping, managinga restaurant)THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics152


C. APPLICATIONS OF MEASUREMENTOVERALL EXPECTATIONSBy the end of this course, students will:1. determine and estimate measurements using the metric and imperial systems, and convert measureswithin and between systems;2. apply measurement concepts and skills to solve problems in measurement and design, to constructscale drawings and scale models, and to budget for a household improvement;3. identify and describe situations that involve proportional relationships and the possible consequencesof errors in proportional reasoning, and solve problems involving proportional reasoning, arising inapplications from work and everyday life.SPECIFIC EXPECTATIONSMathematics for Work and Everyday Life1. Measuring and EstimatingBy the end of this course, students will:1.1 measure, using a variety of tools (e.g., measuringtape, metre or yard stick, measuringcups, graduated cylinders), the lengths ofcommon objects and the capacities of commoncontainers, using the metric system andthe imperial system1.2 estimate lengths, distances, and capacities inmetric units and in imperial units by applyingpersonal referents (e.g., the width of a fingeris approximately 1 cm; the length of a piece ofstandard loose-leaf paper is about 1 ft; thecapacity of a pop bottle is 2 L)Sample problem: Based on an estimate of thelength of your stride, estimate how far it is tothe nearest fire exit from your math classroom,and compare your estimate with themeasurement you get using a pedometer.1.3 estimate quantities (e.g., bricks in a pile, timeto complete a job, people in a crowd), anddescribe the strategies usedSample problem: Look at digital photos thatshow large quantities of items, and estimatethe numbers of items in the photos.1.4 convert measures within systems (e.g., centimetresand metres, kilograms and grams,litres and millilitres, feet and inches, ouncesand pounds), as required within applicationsthat arise from familiar contexts1.5 convert measures between systems (e.g., centimetresand inches, pounds and kilograms,square feet and square metres, litres and U.S.gallons, kilometres and miles, cups and millilitres,millilitres and teaspoons, degreesCelsius and degrees Fahrenheit), as requiredwithin applications that arise from familiarcontextsSample problem: Compare the price of gasolinein your community with the price ofgasoline in a community in the United States.2. Applying Measurement and DesignBy the end of this course, students will:2.1 construct accurate right angles in practicalcontexts (e.g., by using the 3-4-5 triplet toconstruct a region with right-angled cornerson a floor), and explain connections to thePythagorean theorem2.2 apply the concept of perimeter in familiarcontexts (e.g., baseboard, fencing, door andwindow trim)Sample problem: Which room in your homerequired the greatest, and which requiredthe least, amount of baseboard? What is thedifference in the two amounts?MEL4EAPPLICATIONS OF MEASUREMENT153


THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics Grade 12, Workplace Preparation2.3 estimate the areas and volumes of irregularshapes and figures, using a variety of strategies(e.g., counting grid squares; displacingwater)Sample problem: Draw an outline of yourhand and estimate the area.2.4 solve problems involving the areas of rectangles,triangles, and circles, and of relatedcomposite shapes, in situations arising fromreal-world applicationsSample problem: A car manufacturer wantsto display three of its compact models in atriangular arrangement on a rotating circularplatform. Calculate a reasonable area for thisplatform, and explain your assumptions andreasoning.2.5 solve problems involving the volumes andsurface areas of rectangular prisms, triangularprisms, and cylinders, and of related compositefigures, in situations arising from realworldapplicationsSample problem: Compare the volumes ofconcrete needed to build three steps that are4 ft wide and that have the cross-sectionsshown below. Explain your assumptions andreasoning.2.6 construct a two-dimensional scale drawing ofa familiar setting (e.g., classroom, flower bed,playground) on grid paper or using design ordrawing softwareSample problem: Your family is moving to anew house with a living room that is 16 ft by10 ft. Cut out and label simple geometricshapes, drawn to scale, to represent everypiece of furniture in your present livingroom. Place all of your cut-outs on a scaledrawing of the new living room to find outif the furniture will fit appropriately (e.g.,allowing adequate space to move around).2.7 construct, with reasonable accuracy, a threedimensionalscale model of an object or environmentof personal interest (e.g., appliance,room, building, garden, bridge)Sample problem: Design an innovative combinationof two appliances or two other consumerproducts (e.g., a camera and a cellphone,a refrigerator and a television), and constructa three-dimensional scale model.2.8 investigate, plan, design, and prepare a budgetfor a household improvement (e.g., landscapinga property; renovating a room), usingappropriate technologies (e.g., design or decoratingwebsites, design or drawing software,spreadsheet)Sample problem: Plan, design, and prepare abudget for the renovation of a 12-ft by 12-ftbedroom for under $2000. The renovationscould include repainting the walls, replacingthe carpet with hardwood flooring, andrefurnishing the room.3. Solving Measurement ProblemsUsing Proportional ReasoningBy the end of this course, students will:3.1 identify and describe applications of ratio andrate, and recognize and represent equivalentratios (e.g., show that 4:6 represents the sameratio as 2:3 by showing that a ramp with aheight of 4 m and a base of 6 m and a rampwith a height of 2 m and a base of 3 m areequally steep) and equivalent rates (e.g., recognizethat paying $1.25 for 250 mL of tomatosauce is equivalent to paying $3.75 for 750 mLof the same sauce), using a variety of tools(e.g., concrete materials, diagrams, dynamicgeometry software)3.2 identify situations in which it is useful tomake comparisons using unit rates, and solveproblems that involve comparisons of unitratesSample problem: If 500 mL of juice costs$2.29 and 750 mL of the same juice costs$3.59, which size is the better buy? Explainyour reasoning.3.3 identify and describe real-world applicationsof proportional reasoning (e.g., mixing concrete;calculating dosages; converting units;painting walls; calculating fuel consumption;calculating pay; enlarging patterns), distinguishbetween a situation involving a proportionalrelationship (e.g., recipes, where doubling thequantity of each ingredient doubles the numberof servings; long-distance phone calls154


illed at a fixed cost per minute, where talkingfor half as many minutes costs half as much)and a situation involving a non-proportionalrelationship (e.g., cellular phone packages,where doubling the minutes purchased doesnot double the cost of the package; food purchases,where it can be less expensive to buythe same quantity of a product in one largepackage than in two or more small packages;hydro bills, where doubling consumption doesnot double the cost) in a personal and/orworkplace context, and explain their reasoning3.4 identify and describe the possible consequences(e.g., overdoses of medication; seized engines;ruined clothing; cracked or crumbling concrete)of errors in proportional reasoning (e.g., notrecognizing the importance of maintainingproportionality; not correctly calculating theamount of each component in a mixture)Sample problem: Age, gender, body mass,body chemistry, and habits such as smokingare some factors that can influence the effectivenessof a medication. For which of thesefactors might doctors use proportional reasoningto adjust the dosage of medication?What are some possible consequences ofmaking the adjustments incorrectly?3.5 solve problems involving proportional reasoningin everyday life (e.g., applying fertilizers;mixing gasoline and oil for use in smallengines; mixing cement; buying plants forflower beds; using pool or laundry chemicals;doubling recipes; estimating cooking timefrom the time needed per pound; determiningthe fibre content of different sizes of foodservings)Sample problem: Bring the label from a largecan of stew to class. Use the information onthe label to calculate how many calories andhow much fat you would consume if you atethe whole can for dinner. Then search outinformation on a form of exercise you couldchoose for burning all those calories. Forwhat length of time would you need toexercise?3.6 solve problems involving proportional reasoningin work-related situations (e.g., calculatingovertime pay; calculating pay for piecework;mixing concrete for small or large jobs)Sample problem: Coiled pipe from theUnited States is delivered in 200-ft lengths.Your company needs pipe in 3.7-m sections.How many sections can you make from each200-ft length?Mathematics for Work and Everyday LifeMEL4EAPPLICATIONS OF MEASUREMENT155


The Ministry of Education wishes to acknowledgethe contribution of the many individuals, groups, andorganizations that participated in the developmentand refinement of this curriculum policy document.


Printed on recycled paper06-256ISBN 978-1-4249-4699-0 (Print)ISBN 978-1-4249-4700-3 (PDF)ISBN 978-1-4249-4701-0 (TXT)© Queen’s Printer for Ontario, 2007

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