Course: Calculus and Vectors

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Calculus and Vectors

This course builds on students’ previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three-dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who choose to pursue careers in fields such as science, engineering, economics, and some areas of business, including those students who will be required to take a university-level calculus, linear algebra, or physics course.

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  • Department: Math
  • Course Developer: The Educators Academy
  • Development Date:
  • Revision Date: 2021
  • Course Title: Calculus and Vectors
  • Course Reviser: Meenakhshi Shah
  • Grade: Grade 12
  • Course Type: University Preparation
  • Ministry Course Code: University Preparation
  • Credit Value: 01
  • Prerequisite: The new Advanced Functions course (MHF4U) must be taken prior to or concurrently with Calculus and Vectors (MCV4U).
  • Ministry Curriculum Policy Document: The Ontario Curriculum, grades 11 and 12, 2007 (Revised)

Overall Curriculum Expectations

Rate of Change

    i. develop, select, apply, and compare a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding; ii. develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, iii. assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments; iv. demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions); v. select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems; vi. make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports); vii. create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems; viii. communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.

Derivatives and Their Applications

    i. demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit; ii. graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between the numeric, graphical, and algebraic representations of a function and its derivative; iii. verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.

Geometry and Algebra of Vectors

    i. make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching; ii. solve problems, including optimization problems, that require the use of the concepts and procedures associated with the derivative, including problems arising from real-world applications and involving the development of mathematical models.

Unit Outline

# Unit Approx. Time
1 Rate of Change 36 Hours
2 Derivatives and their Applications 35 Hours
3 Geometry and Algebra of Vectors 36 Hours
4 Final Examination 03 Hours
Total 110 Hours

Unit Description

Rate of Change

In this unit, students will demonstrate an understanding of examples of real-world applications of rates of change, represented in a variety of ways (e.g., in words, numerically, graphically, algebraically). They will also describe connections between the average rate of change of a function that is smooth (i.e., continuous with no corners) over an interval and the slope of the corresponding secant by making connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function.

Derivatives and Their Applications

The concept of a derivative is, in essence, a way of creating a short cut to determine the tangent line slope function that would normally require the concept of a limit. Once patterns are seen from the evaluation of limits, rules can be established to simplify what must be done to determine this slope function. This unit begins by examining those rules including: the power rule, the product rule, the quotient rule and the chain rule followed by a study of the derivatives of composite functions. The next section is dedicated to finding the derivative of relations that cannot be written explicitly in terms of one variable. Next students will simply apply the rules they have already developed to find higher order derivatives. As students saw earlier, if given a position function, they can find the associated velocity function by determining the derivative of the position function. They can also take the second derivative of the position function and create a rate of change of velocity function that is more commonly referred to as the acceleration function which is where this unit ends.

Geometry and Algebra

In this unit, students will demonstrate an understanding by recognizing a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors. They will also represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways and algebraically and recognize vectors with the same magnitude and direction but different positions as equal vectors. Students will also describe and understand the trigonometric relations.

Program Considerations

Assessment and Evaluation

The primary purpose of assessment and evaluation is to improve student learning. Information gathered through assessment helps teachers to determine students’ strengths and weaknesses in their achievement of the curriculum expectations in each course. This information also serves to guide teachers in adapting curriculum and instructional approaches to students’ needs and in assessing the overall effectiveness of programs and classroom practices.
 
For assessment and evaluation, we follow the Ministry of Education's Growing Success document, and by doing so will benefit the students both in the present and future. We designed assessments in such a way as to make it possible to gather and show evidence of learning in a variety of ways to gradually release responsibility to the students, and to give multiple and varied opportunities to reflect on learning and receive detailed feedback.
 
Assessment and evaluation will be based on the provincial curriculum expectations and the achievement levels outlined in this document. Growing Success articulates the vision the Ministry has for the purpose and structure of assessment and evaluation techniques. 
 
In order to ensure that assessment and evaluation are valid and reliable and that they lead to the improvement of students’ learning, The Educators Academy’s assessment and evaluation strategies focus on:
 
·         Address both what students learn and how well they learn;
·         Are varied in nature, administered over a period of time, and designed to provide opportunities for students to demonstrate the full range of their learning;
·         Are appropriate for the learning activities used, the purposes of instruction, and the needs and experiences of the students;
·         Are fair to all students;
·         Accommodate students with special education needs, consistent with the strategies outlined in their Individual Education Plan; and those 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 of students’ work that provide evidence of their achievement;
·         Are communicated clearly to students and parents at the beginning of the school year and at other appropriate points (Parent Teacher Nights) throughout the school year.
 
The overall expectations are broad in nature, and the specific expectations define the particular content or scope of the knowledge and skills referred to in the overall expectations. Our teachers use their professional judgment to determine which specific expectations should be used to evaluate achievement of the overall expectations, and which ones will be covered in instruction and assessment (e.g., through direct observation) but not necessarily evaluated.
 
Three different types of Assessments are used for this course: Assessment of Learning, Assessment for Learning and Assessment as Learning.
 
For Assessment of Learning,
  • Projects
  • Assignments
  • Tests
  • Classroom Discussions
  • Questions and Answers during Investigation
  • Presentations
  • Final Exam
  • Worksheets
 
For Assessment for Learning,
  • Group Discussions
  • Investigations
  • Homework
  • Practice Worksheets
  • Pre-Tests
  • Portfolios
 
For Assessment of Learning,
  • Self Evaluations
  • Exit Cards
  • Conversations
  • Checklists
  • Rubrics
 
Assessment Strands:
 
The Educators Academy will ensure that student work is assessed and/or evaluated in a balanced manner with respect to the four categories, and that achievement of particular expectations is considered within the appropriate categories.
 
Knowledge and Understanding (K/U)                                                                                                    
 
Thinking and Inquiry (T/I)                                                                                                                                           
 
Communication (C)                                                                                                                                                        
 
Application (A)
 
The purpose of the achievement chart is to:
 
  • provide a common framework that encompasses the curriculum expectations for all courses 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 student learning.                                                                                                                                               
Evaluation and Reporting of Students’ Achievements by Report Cards
 
Student achievement is communicated formally to students and parents by means of the Provincial Report Card. The report card provides a record of the student’s achievement of the curriculum expectations in every course, at particular points in the school year or semester, in the form of a percentage grade. Report cards are issued upon completion of the course. Each report card will focus on related aspects of student achievement. The percentage grade will represent the quality of the student’s overall achievement of the expectations for the course and will reflect the corresponding level of achievement. The Educators Academy will record a final grade for every course, and a credit is granted for the course in which the student’s grade is 50% or higher.
 
  • Seventy per cent of the grade will be based on evaluations conducted throughout the course. This portion of the grade should reflect the student’s most consistent level of achievement throughout the course, although special consideration should be given to more recent evidence of achievement.
 
  • Thirty per cent of the grade will be based on a final evaluation in the form of an examination, performance, essay, and/or other method of evaluation suitable to the course content and administered towards the end of the course.
 
Ø  Term work will account 70% of the course work
 
Ø  Final Exam would be a value of 30%
 
Final Assessment and Evaluation = 100%
 
The teacher will also provide written comments concerning the student's strengths, areas for improvement, and next steps (E–Excellent, G–Good, S–Satisfactory, N–Needs Improvement). The report card will indicate whether an OSSD credit has been earned or not. Upon completion of a course, The Educators Academy will send a copy of the report card back to the student's home school where the course will be added to the ongoing list of courses on the student's Ontario Student Transcript. The report card will also be sent to the student's home address for parents’ communication.
 
Evaluation Instruments/ Strategies:
 
*      Rubrics                                                                                                                                                 Observation
*      Checklist                                                                                                                                              Project Work
*      Peer                                                                                                                                                       Interviewing
*      Self                                                                                                                                                        Researching
*      Group                                                                                                                                                Conferencing
 
A Summary Description of Achievement in Each Percentage Grade Range
and Corresponding Level of Achievement
Percentage Grade Range
Achievement Level
Summary Description
80-100%
Level 4
A very high to outstanding level of achievement. Achievement is above the provincial standard.
70-79%
Level 3
A high level of achievement. Achievement is at the provincial standard.
60-69%
Level 2
A moderate level of achievement. Achievement is below, but approaching, the provincial standard.
50-59%
Level 1
A passable level of achievement. Achievement is below the provincial standard.
below 50%
Level R
Insufficient achievement of curriculum expectations. A credit will not be granted.
 
Achievement Chart – Mathematics MPM1D, Grades 9–10
 
Categories
50–59%
(Level 1)
60–69%
(Level 2)
70–79%
(Level 3)
80–100%
(Level 4)
Knowledge and Understanding
The student:
 
 
 
Knowledge of content
(e.g., facts, terms,
procedural skills, use
of tools)
Understanding of
mathematical concepts
– demonstrates limited
knowledge of content
– demonstrates limited
understanding of
concepts
– demonstrates some
knowledge of content
– demonstrates some
understanding of
concepts
– demonstrates
considerable knowledge
of content
– demonstrates
considerable understanding
of concepts
– demonstrates
thorough knowledge
of content
– demonstrates
thorough understanding
of concepts
Understanding of content (e.g., concepts, ideas, theories, principles,
procedures, processes)
demonstrates
limited
understanding
of content
demonstrates
some
understanding
of content
demonstrates
considerable
understanding
of content
demonstrates
thorough
understanding
of content
 
Categories
50–59%
(Level 1)
60–69%
(Level 2)
70–79%
(Level 3)
80–100%
(Level 4)
Thinking/ Inquiry
The student:
 
 
 
Use of planning skills
– understanding the
problem (e.g., formulating
and interpreting
the problem, making
conjectures)
– making a plan for solving
the problem)
– uses planning
skills with limited
effectiveness
uses planning
skills with some
effectiveness
uses planning skills
with considerable
effectiveness
– uses planning skills
with a high degree
of effectiveness
Use of processing skills
– carrying out a plan (e.g., collecting data, questioning, testing, revising, modelling,
solving, inferring, forming conclusions)
– looking back at the solution (e.g., evaluating
reasonableness,
making convincing
arguments, reasoning,
justifying, proving,reflecting)
uses processing
skills and
strategies with
limited
effectiveness
uses processing
skills with
some
effectiveness
uses processing
skills with
considerable
effectiveness
uses processing
skills with a
high degree of
effectiveness
Use of critical/creative
thinking processes (e.g.,
problem solving, inquiry)
uses critical/
creative thinking
processes with limited
effectiveness
uses critical/
creative thinking
processes with some
effectiveness
uses critical/
creative thinking
processes, with considerable
effectiveness
uses critical/
creative thinking
processes with a high degree of
effectiveness
 
Categories
50–59%
(Level 1)
60–69%
(Level 2)
70–79%
(Level 3)
80–100%
(Level 4)
Communication
The student:
 
 
 
Expression and organization of ideas and mathematical thinking (e.g.,
clarity of expression, logical
organization), using oral, visual, and written forms (e.g., pictorial, graphic, dynamic, numeric, algebraic
forms; concrete materials)
expresses and organizes
mathematical
thinking with limited
effectiveness
expresses and organizes
mathematical
thinking with some
effectiveness
expresses and organizes
mathematical
thinking with considerable
effectiveness
expresses and organizes
mathematical
thinking with a high degree of
effectiveness
Communication for different
audiences (e.g., peers, teachers) and purposes (e.g., to present data, justify a solution, express a mathematical argument) in oral, visual, and written formsin oral, visual, and/ or written forms
communicates for
different audiences and
purposes with
limited effectiveness
communicates for
different
audiences and
purposes with
some
effectiveness
communicates
for different
audiences and
purposes with
considerable
effectiveness
communicates
for different
audiences and
purposes with a
high degree of
effectiveness
Use of conventions,
vocabulary, and terminology
of the discipline (e.g.,
terms, symbols) in oral,
visual, and written forms
uses conventions,
vocabulary, and
terminology of the discipline with limited
effectiveness
uses conventions,
vocabulary, and
terminology of
the discipline
with some
effectiveness
uses conventions,
vocabulary,and terminology of
the discipline
with considerable
effectiveness
uses conventions,
vocabulary, and
terminology of
the discipline
with a high
degree of
effectiveness
 
Categories
50–59%
(Level 1)
60–69%
(Level 2)
70–79%
(Level 3)
80–100%
(Level 4)
Application
The student:
 
 
 
Application of knowledge
and skills in familiar
contexts
applies
knowledge and
skills in familiar
contexts with
limited
effectiveness
applies
knowledge and
skills in familiar
contexts with
some
effectiveness
applies
knowledge and
skills in familiar
contexts with
considerable
effectiveness
applies
knowledge and
skills in familiar
contexts with a
high degree of
effectiveness
Transfer of knowledge and
skills to new contexts
transfers
knowledge and
skills to unfamiliar
contexts with
limited
effectiveness
transfers
knowledge and
skills to unfamiliar
contexts with
some effectiveness
transfers
knowledge and
skills to unfamiliar
contexts with
considerable
effectiveness
transfers
knowledge and
skills to unfamiliar
contexts with a
high degree of
effectiveness
use of equipment, materials and technology
uses equipment, materials and technology safely and correctly only with supervision
uses equipment, materials and technology safely and correctly with some supervision
uses equipment, materials and technology safely and correctly
demonstrates and promotes the safe and correct
uses of  equipment, materials and technology
Making connections within
and between various contexts (e.g., connections
between concepts, representations, and forms
within mathematics; connections involving use of
prior knowledge and experience; connections between mathematics, other disciplines, and the real world)
makes connections
within and between
various contexts with
limited effectiveness
makes connections
within and between
various contexts with
some effectiveness
makes connections
within and between
various contexts
with considerable
effectiveness
makes connections
within and between
various contexts
with a high degree
of effectiveness
Submission of Assignments
 
*      All assignments should be submitted for grading on the stated due date.
*      Any late assignments may be subjected to a 15% penalty.
*      Work not submitted within 5 school days after the stated due date will be assigned a mark of 0.
*      If a student is ill or away for a documented reason, all assignments must be submitted upon return to class, unless arrangements are negotiated with the teacher.
*      It is vital that the student realize the potential consequences of incomplete work and absences, including failure to gain the credit for the course. It is the responsibility of the student to catch up on all work missed from being absent.
 
Program Planning Considerations
 
Teachers who are planning a program in this subject will make an effort to take into account considerations for program planning that align with the Ontario Ministry of Education policy and initiatives in a number of important areas.
 
Planning Mathematics Programs for Exceptional Students
 
The Educators Academy believes that classroom teachers are the key educators of students who have special education needs. They have a responsibility to help all students to learn and work collaboratively with special education resource teachers, where appropriate, to achieve this goal.
 
In planning mathematics courses for exceptional students, The Educators Academy teachers’ begin by examining both the curriculum expectations for the course and the needs of the individual student to determine which of the following options is appropriate for the student:
 
  • no accommodations or modifications; or
  • accommodations only; or
  • modified expectations, with the possibility of accommodations.
 
If the student requires either accommodations or modified expectations, or both, our teachers record the relevant information in his or her Individual Education Plan (IEP).
 
Students Require Accommodations Only:
 
In The Educators Academy, with the aid of accommodations, some exceptional students are able to participate in the regular course curriculum and to demonstrate their learning independently. We believe on these three types of accommodations, Instructional accommodations by which changes are in teaching strategies, including styles of presentation, methods of organization, or use of technology and multimedia, Environmental accommodations by which changes are that the student may require in the classroom such as preferential seating or special lighting and Assessment accommodations by which changes are  in assessment procedures that enable the student to demonstrate his or her learning, such as allowing additional time to complete tests or assignments or permitting oral responses to test questions.
 
The Educators Academy is committed to ensuring that all students, especially those with special education needs, are provided with the learning opportunities and supports they require to gain the knowledge, skills, and confidence needed to succeed in a rapidly changing society. The context of special education and the provision of special education programs and services for exceptional students in Ontario are constantly evolving.
 
The Educators Academy believes on that:
·         All students can succeed.
·         Universal design and differentiated instruction are effective and interconnected means 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 numeracy development.
·         Each student has his or her own unique patterns of learning.
·         Classroom teachers need the support of the larger community to create a learning environment that supports students with special education needs.
·         Fairness is not sameness.
Students Require Modifications Only:
The Educators Academy provide comprehensive procedures for the identification of exceptional pupils, for the placement of those pupils in educational settings where the special education programs and services appropriate to their needs can be delivered, and for the review of the identification of exceptional pupils and their placement. If the student requires either accommodations or modified expectations, or both, then we will take into account these needs of exceptional students as they are set out in the students' Individual Education Plan. Our courses offer a vast array of opportunities for students with special educations needs to acquire the knowledge and skills required for our evolving society.
The Educators Academy realizes that some exceptional students will require modified expectations, which are different from the regular course expectations. For most of these students, modified expectations will be based on the regular course curriculum, with changes in the number and/or complexity of the expectations. We carefully monitor that these are reflected clearly in the student’s IEP, the extent to which expectations have been modified. This decision must be communicated to the parents and the students that
v  accommodations only; or
v  modified expectations, with the possibility of accommodations; or
v  alternative expectations, which are not derived from the curriculum expectations for a course and which constitute alternative programs and/or courses.
In The Educators Academy, if a student requires modified expectations in mathematics courses, assessment and evaluation of his or her achievement will be based on the learning expectations identified in the IEP and on the achievement levels outlined in the Growing Success document.
 
Program Consideration for English Language Learners
 
Ontario schools have some of the most multilingual student populations in the world. The first language of approximately 20 per cent of the students in Ontario’s English-language schools is a language other than English. Ontario’s linguistic heritage includes several Aboriginal languages and many African, Asian, and European languages. It also includes some varieties of English – also referred to as dialects – that differ significantly from the English required for success in Ontario schools.
 
The Educators Academy provides a number of strategies to address the needs of ESL students. This course must be flexible in order to accommodate the needs of students who require instruction in English as a second language or English literacy development. Our teachers consider it to be their responsibility to help students to develop their ability to use the English language properly. Appropriate accommodations affecting the teaching, learning, and evaluation strategies in this course may be made in order to help students gain proficiency in English, since students taking English as a second language at the secondary level have limited time in which to develop this proficiency.
 
Our mathematics teachers incorporate appropriate strategies for instruction and assessment to facilitate the success of the ESL students in their classrooms. These strategies include:
·         modification of some or all of the course expectations, based on the student’s level of English proficiency;
·         use of a variety of instructional strategies (e.g., extensive use of visual cues, manipulatives, pictures, diagrams, graphic organizers; attention to clarity of instructions; modelling of preferred ways of working in mathematics; previewing of textbooks; pre-teaching of key specialized vocabulary; encouragement of peer tutoring and class discussion; strategic use of students’ first languages);
·         use of a variety of learning resources (e.g., visual material, simplified text, bilingual dictionaries, culturally diverse materials);
·         use of assessment accommodations (e.g., granting of extra time; use of alternative forms of assessment, such as oral interviews, learning logs, or portfolios; simplification of language used in problems and instructions).
Our students, who are no longer taking ESL courses, sometimes may still need program adaptations to be successful. If any of the students require modified expectations or accommodations in a mathematics course, a checkmark is placed in the ESL box on the student’s report card.
 
The Educators Academy determines the student's level of proficiency in the English Language upon registration. This information is communicated to the teacher of the Mathematics course following the registration and the teacher then invokes a number of strategies and resources to support the student in the course. The Educators Academy has created course content to enrich the student's learning experience. Many occupations in Canada require employees with capabilities in the English language. Enabling students to learn English language skills will contribute to their success in the larger world. With exposure to the English language in a supportive learning environment, most young children will develop oral fluency quite quickly, making connections between concepts and skills acquired in their first language and similar concepts and skills presented in English. Also, a glossary of key mathematical terms will be used to enhance the mathematical skills of students.
 
Environmental Education
 
Helping students become environmentally responsible is a role assumed by The Educators Academy. We work on different aspects like to promote learning about environmental issues and solutions, to engage students in practicing and promoting environmental stewardship in their community and to focus on the importance of the education system providing leadership by implementing and promoting responsible environmental practices so that all stakeholders become dedicated to living more sustainably.
 
The Educators Academy also ensures that the student will have opportunities to acquire the knowledge, skills, perspectives and practices needed to become an environmentally literate citizen. Our courses should provide opportunities for each student to address environmental issues in their home, in their local community, or even at the global level.
 
Anti Discrimination Education
 
The implementation of antidiscrimination principles in education influences all aspects of school life. It promotes a school climate that encourages all students to work to attain high standards, affirms the worth of all students, and helps students strengthen their sense of identity and develop a positive self-image. It encourages staff and students alike to value and show respect for diversity in the school and the wider society. It requires schools to adopt measures to provide a safe environment for learning, free from harassment, violence, and expressions of hate.
 
Antidiscrimination education encourages students to think critically about themselves and others in the world around them in order to promote fairness, healthy relationships, and active, responsible citizenship.
 
The Educators Academy ensures that school–community interaction reflects the diversity in the local community and wider society. We are highly concerned about a variety of strategies for communicating and working with parents and community members from diverse groups, in order to ensure their participation in such school activities as parent teacher nights. It is our policy to encourage the families new to Canada, who may be unfamiliar with the Ontario school system, so they can get special outreach and encouragement in order to feel comfortable in their interactions with The Educators Academy.
 
The Educators Academy believes on that learning activities and resources used to implement the curriculum should be inclusive in nature, reflecting the range of experiences of students with varying backgrounds, abilities, interests, and learning styles. They will enable students to become more sensitive to the diverse cultures and perceptions of others, including the students from multicultural environment. For example, activities are designed to relate concepts in geometry or patterning to the arches and tile work often found in Asian architecture. Our teachers, by discussing aspects of the history of mathematics, help to make students aware of the various cultural groups that have contributed to the evolution of mathematics over the centuries.
 
The Educator Academy realizes that students need to recognize that ordinary people use mathematics in a variety of everyday contexts, both at work and in their daily lives. Our teachers work on connecting mathematical ideas to real-world situations through learning activities which enhance students’ appreciation of the role of mathematics in human affairs, in areas including health, science, and the environment. By this way, our students are aware of the use of mathematics in contexts such as sampling and surveying and the use of statistics to analyse trends. They recognize the importance of mathematics in such areas which motivate them to learn and also provide a foundation for informed, responsible citizenship.
 
The Educators Academy teachers have high expectations for all students. We realize that to achieve mathematical potential, however, different students may need different kinds of support. Some boys, for example, may need additional support in developing their literacy skills in order to complete mathematical tasks effectively. For some girls, additional encouragement to envision themselves in careers involving mathematics may be beneficial. Our teachers consider providing strong role models in the form of female guest speakers who are mathematicians or who use mathematics in their careers.
 
We hope that all these attitudes and attributes provide a foundation on which students can develop their own identity, explore interconnectedness with others, and form and maintain healthy relationships.
 
Literacy and Inquiry Skills in Mathematics
 
Literacy is defined as the ability to use language and images in rich and varied forms to read, write, listen, view, represent, and think critically about ideas. It involves the capacity to access, manage, and evaluate information; to think imaginatively and analytically; and to communicate thoughts and ideas effectively. Literacy includes critical thinking and reasoning to solve problems and make decisions related to issues of fairness, equity, and social justice. Literacy connects individuals and communities and is an essential tool for personal growth and active participation in a cohesive, democratic society. Literacy involves a range of critical-thinking skills and is essential for learning across the curriculum. Literacy instruction takes different forms of emphasis in different subjects, but in all subjects, literacy needs to be explicitly taught. Literacy, mathematical literacy, and inquiry/research skills are critical to students' success in all subjects of the curriculum and in all areas of their lives.
 
The Educators Academy believes that literacy skills can play an important role in student success in mathematics courses. Many of the activities and tasks students undertake in math courses involve the use of written, oral, and visual communication skills. For example, students use language to record their observations, to explain their reasoning when solving problems, to describe their inquiries in both informal and formal contexts, and to justify their results in small-group conversations, oral presentations, and written reports. The language of mathematics includes special terminology. The study of mathematics consequently encourages students to use language with greater care and precision and enhances their ability to communicate effectively.
 
In all courses in mathematics, our students will develop their ability to ask questions and to plan investigations to answer those questions and to solve related problems. Students will learn a variety of research methods and inquiry approaches in order to carry out these investigations and to solve problems, and they would be able to select the methods that are most appropriate for a particular inquiry. In this way, our students learn how to locate relevant information from a variety of sources, such as statistical databases, newspapers, and reports.
 
Inquiry and research are at the heart of learning in all subject areas at The Educators Academy. Students are encouraged to develop their ability to ask questions and to explore a variety of possible answers to those questions. As they advance through the grades, they acquire the skills to locate relevant information from a variety of print and electronic sources. The questioning they practiced in the early grades becomes more sophisticated as they learn that all sources of information that have a particular point of view and that the recipient of the information has a responsibility to evaluate it, determine its validity and relevance, and use it in appropriate ways. The ability to locate, question, and validate information allows a student to become an independent, mature and lifelong learner.
 
The Role of a Library
 
The school library program in many schools can help build and transform students' knowledge in order to support lifelong learning in our information- and knowledge-based society. The school library program of these schools supports student success across the curriculum by encouraging students to read widely, teaching them to examine and read many forms of text for understanding and enjoyment, and helping them improve their research skills and effectively use information gathered through research. The Educator Academy teachers assist students in accessing a variety of online resources and collections (e.g., professional articles, image galleries, videos, databases and much more). Our Teachers will also guide students through the concept of ownership of work and the importance of copyright in all forms of media.
 
The Role of Technology in Mathematics
 
Information and communication technology (ICT) provides a range of tools that can significantly extend and enrich teachers’ instructional strategies and support students’ learning in mathematics. The Educators Academy Teachers use ICT tools and resources both for whole-class instruction and to design programs that meet diverse student needs. By this way, they reduce the time spent on routine mathematical tasks and to allow students to devote more of their efforts to thinking and concept development. The useful ICT tools, which are used, are simulations, multimedia resources, databases, sites that gave access to large amounts of statistical data, and computer-assisted learning modules. Applications such as databases, spreadsheets, dynamic geometry software, dynamic statistical software, graphing software, computer algebra systems (CAS), word-processing software, and presentation software are also used to support various methods of inquiry in mathematics. By using these methods, our teachers also make possible simulations of complex systems that can be useful for problem solving. Our teachers also use information and communications technology in the classroom to connect students to other schools, and to bring the global community into the local classroom.
 
As a result, students can develop transferable skills through their experience with word processing, internet research, presentation software, and telecommunication tools, as would be expected in any other course or any business environment. Although the Internet is a powerful learning tool, there are potential risks attached to its use. All students must be made aware of issues related to Internet privacy, safety, and responsible use, as well as of the potential for abuse of this technology, particularly when it is used to promote hatred. Our teachers understand that ICT tools are valuable in their teaching practice, both for whole class instruction and for the design of curriculum units that contain varied approaches to learning to meet diverse student needs.
 
The Ontario Skills Passport and Essential Skills
 
Ontario Skills Passport (OSP) is a bilingual, web-based resource that enhances the relevance of classroom learning for students and strengthens school–work connections. The skills described in the OSP are the Essential Skills that the Government of Canada and other national and international agencies have identified and validated, through extensive research, as the skills needed for work, learning, and life.  The Educators Academy can engage students by using OSP tools and resources to show how what they learn in class can be applied in the workplace and in everyday life.
 
Career Education
 
The Educators Academy teachers promote students’ awareness of careers involving mathematics by exploring applications of concepts and providing opportunities for career-related project work. By following this procedure, students try to investigate mathematics-related careers compatible with their interests, aspirations, and abilities. By the end of course, our students are also fully aware that mathematical literacy and problem solving are valuable assets in an ever-widening range of jobs and careers in today’s society. The knowledge and skills students acquire in mathematics courses in The Educators Academy, are useful in fields such as science, business, engineering, and computer studies; in the hospitality, recreation, and tourism industries; and in the technical trades.
 
The framework of the program is a four-step inquiry process based on four questions linked to four areas of learning:
·         knowing yourself - Who am I?;
·         exploring opportunities - What are my opportunities?;
·         making decisions and setting goals - Who do I want to become?;
·         achieving goals and making transitions - What is my plan for achieving my goals?
 
PLANNING PROGRAM PATHWAYS AND PROGRAMS LEADING TO SPECIALIST HIGH SKILLS MAJOR
 
The Educators Academy courses are well suited for inclusion in Specialist High Skills Majors (SHSM) or in programs designed to provide pathways to particular apprenticeship, college, university, or workplace destinations. In some SHSM programs, courses at The Educators Academy can be bundled with other courses to provide the academic knowledge and skills important to particular economic sectors and required for success in the workplace and postsecondary education, including apprenticeship training.
 
Health and Safety
 
Although health and safety issues are not normally associated with mathematics, but The Educators Academy puts its best efforts to make it safe and secure when the learning involves fieldwork or investigations based on experimentation. We realize that fieldwork can provide an exciting and authentic dimension to students’ learning experiences. It also takes the teacher and students out of the predictable classroom environment and into unfamiliar settings. Our teachers preview and plan their activities and expeditions carefully to protect students’ health and safety.
 
In order to provide a suitable learning environment for The Educators Academy’s staff and students, it is critical that classroom practice and the learning environment complies with relevant federal, provincial, and municipal health and safety legislation and by-laws, including, but not limited to, the Workplace Safety and Insurance Act, the Workplace Hazardous Materials Information System (WHMIS), the Food and Drug Act, the Health Protection and Promotion Act, the Ontario Building Code, and the Occupational Health and Safety Act (OHSA). The OHSA requires all schools to provide a safe and productive learning and work environment for both students and employees.
 
Resources:
 
v  Nelson Textbook Grade 12
v  Simulations and Animations
v  Internet Videos (Khan Academy)
v  A Graphing Calculator
v  Graph Paper
v  Handouts
v  Board
v  Puplemath.com

Teaching & Learning Strategies

To make new learning more accessible to students, The Educators Academy’s teachers draw upon the knowledge and skills students have acquired in previous years – in other words, they help to activate prior mathematical knowledge. It is important to assess where students are in their mathematical growth and to bring them forward in their learning.  The aim of this course is to help students use the language of mathematics skillfully, confidently and flexibly, a wide variety of instructional strategies are used to provide learning opportunities to accommodate a variety of learning styles, interests and ability levels.
 
For mathematical processes, some teaching and learning strategies used are:
  • Communicating: To improve student success there will be several opportunities for students to share their understanding both in oral as well as written form.
  • The use of technological tools and software (e.g., graphing software, dynamic geometry software, the Internet, spreadsheets, and multimedia) in activities, demonstrations, and investigations to facilitate the exploration and understanding of mathematical concepts;
  • Learning Goals and Success Criteria is explained to the students before starting any unit, task or activity.
  • Problem solving: Scaffolding of knowledge, detecting patterns, making and justifying conjectures, guiding students as they apply their chosen strategy, directing students to use multiple strategies to solve the same problem, when appropriate, recognizing, encouraging, and applauding perseverance, discussing the relative merits of different strategies for specific types of problems.
  • Reasoning and proving: Asking questions that get students to hypothesize, providing students with one or more numerical examples that parallel these with the generalization and describing their thinking in more detail.
  • Reflecting: Modeling the reflective process, asking students how they know.
  • Selecting Tools and Computational Strategies: Modeling the use of tools and having students use technology to help solve problems.
  • Connecting: Activating prior knowledge when introducing a new concept in order to make a smooth connection between previous learning and new concepts, and introducing skills in context to make connections between particular manipulations and problems that require them.
  • Representing: Modeling various ways to demonstrate understanding, posing questions that require students to use different representations as they are working at each level of conceptual development - concrete, visual or symbolic, allowing individual students the time they need to solidify their understanding at each conceptual stage.
  • Group Work: Working cooperatively in groups reduces isolation and provides students with opportunities to share ideas and communicate their thinking in a supportive environment as they work together towards a common goal.
  • Comparison and evaluation of written work is very important in this course. This course focuses on giving many examples of correct work, and helping students build the skills needed to peer-correct and self correct.
  • Oral presentations are a good tool for learning. Students can learn from one another, and from their teacher. Charts and graphs are used to present effective learning opportunities of concepts and skills to different students.
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