OVERALL EXPECTATIONS

Rate of Change 

• 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; 

• 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; 

• 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. 

Derivatives and their Applications 

• 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; 

• 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. 

Geometry and Algebra of Vectors 

• Demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications; 

• Perform operations on vectors in two-space and three-space, and use the properties of these 

• Distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space; 

• Represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections. 

Course Contents

1. Introduction to Calculus - This unit continues on from the Rates of Change unit in Advanced Functions. --- 13.5 hours

2. Derivatives - This unit will introduce student to shortcuts to taking derivatives, such as the product, quotient and chain rules. --- 11.5 hours

3. Derivatives and their Applications - second derivatives will be discussed, as well as the real world and graphical applications of derivatives. --- 13.5 hours

4. Increasing and Decreasing Functions -The concepts of instantaneous rate of change, tangent lines, and first principles will be discussed. --- 11.5 hours

5. The Derivatives of Exponential Functions - The concept of the derivative function will be worked on and how first principles can be used to develop an equation that models the slope of the tangent. --- 8.5 hours

6. Introduction to Vectors - Students will be introduced to vectors and various applications, such as the dot and cross product. Various forms of equations of lines and planes will be introduced to the student, and the advantages/disadvantages of each will be discussed. ---12.5 hours

7. Vectors as Force Velocity - Students will learn how to determine the nature of intersection between lines and planes in 3D. --- 12.5 hours

8. Equations in R^2 and R^3 - Students will apply concepts from the previous unit to determine angles between planes, orthogonal solutions, and learn to solve and sketch systems of equations. --- 17.5 hours

9. Final Assignment: ISU & Final Exams (2) (30% of final mark) --- 9 hours

Total 110 hours

TEACHING & LEARNING STRATEGIES

• Direct Instruction (teacher-led)

• Class Discussion (teacher facilitated)

• 1:1 Conferencing Teacher & Student

• Silent individual reading

• Independent Work (teacher facilitation)

• Worksheets/Surveys

• Individual or Group Research

• Use of Computers / Internet

• Use of video or audio materials

• Presentations

ASSESSMENT & EVALUATION

Purpose

The primary purpose of assessment is to improve student learning. Assessment relates directly to the expectations for the course.

A variety of assessments for and as learning are conducted on a regular basis to allow

ample opportunities for students to improve and ultimately demonstrate their full range of learning and in order for the teacher to gather information to provide feedback. Assessment tasks relate to the success criteria set out in lesson plans. Success criteria allow students to see what quality looks like.

Evaluation is the process of judging the quality of student work in relation to the achievement chart categories and criteria, and assigning a percentage grade to represent that quality. Evaluation is based on gathering evidence of student achievement through:

• Products

• Observations

• Conversations

Weighting of Categories

Knowledge & Understanding --- 30%

Thinking --- 20%

Communication --- 20%

Application --- 30%

Grading

• The final grade is based on performance in 3 areas: products, observations, conversations.

• 70% of the grade is based on evaluations conducted throughout the course.

• 30% is based on a final evaluation.

  
  

Assessment Tools

Marking schemes / Rubrics / Checklists

Assessment Strategies

Assessment for Learning

Quizzes / Journals / Conferencing / Researching / Problem Solving (process focused) / Debates / Discussions

Assessment as Learning

Reflective Journal / Exit and Entrance Cards / Graphic Organizers / Self/Peer Assessment

Assessment of Learning

Tests / Presentations / Projects / Problem Solving (process focused)

CONSIDERATIONS FOR PROGRAM PLANNING

Instructional Approaches

Teachers in the school are expected to:

• clarify the purpose for learning

• help students activate prior knowledge

• differentiate instruction for individual students and small groups according to need

• explicitly teach and model learning strategies

• encourage students to talk through their thinking and learning processes

• provide many opportunities for students to practise and apply their developing knowledge and skills

• apply effective teaching approaches involve students in the use of higher-level thinking skill

• encourage students to look beyond the literal meaning of texts

Teachers use a variety of instructional and learning strategies best suited to the particular type of learning. Students have opportunities to learn in a variety of ways:

• individually

• cooperatively

• independently with teacher direction

• through investigation involving hands-on experience

• through examples followed by practice

• by using concrete learning tools - manipulatives - in mathematics such as connecting cubes, measurement tools, algebra tiles, and number cubes

• by encouraging students to gain experience with varied and interesting applications of the new knowledge. Rich contexts for learning open the door for students to see the “big ideas” of mathematics that will enable and encourage them to reason mathematically throughout their lives.

Promoting Positive Attitudes Towards Learning Mathematics

Teachers must be careful to build a positive environment in which students may study mathematics; students that enjoy the courses are more likely to do well and enrol in more advanced mathematics courses.

Teachers can set students up for developing positive attitudes by providing opportunities for them to:

• be engaged in making mathematical conjectures

• experience breakthroughs as they solve problems

• see connections between important ideas

• see their teacher’s enthusiasm about teaching mathematics

Teachers must be mindful students developing negative attitudes whether through a feeling of inadequacy or anxiety from not solving problems quickly, easily, or in the correct manner. Students should be able to recognize that:

• There are many correct ways to come to a solution

• Problem solving requires time and effort to learn, and requires perseverance

• With this perseverance comes the ability to move past barriers and overcome the frustration of getting stuck

Teachers can encourage students to keep trying a problem when they are stuck and guide them through routes of thought to arrive at a solution, as well as encourage students to develop this perseverance as being challenged and overcoming barriers are cornerstones of education.

Teachers must be mindful of their students’ confidence in their skills in order for them to continue seeing success and understanding in their studies.

Program Considerations for Students with Special Education Needs

Teachers must incorporate appropriate strategies for instruction and assessment to facilitate the success of students with special educational needs in their classrooms. These strategies stem from the beliefs as laid out in Special Education Transformation: The report of the Co-Chairs with the Recommendations of the Working Table on Special Education, 2006:

• 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 students’ 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.

Teachers must plan their program that recognize the diversity of students’ learning styles, needs, and responses, so students can have performance tasks that respect their abilities so they can derive the greatest possible benefit from the teaching and learning process.

Teachers must be mindful of three types of accommodations for students at Brain Power:

• Instructional Accommodations: changes in teaching strategies, including styles of presentation, methods of organization, or use of technology and multimedia

• Environmental Accommodations: changes that the student may require in the classroom and/or school environment, such as preferential seating or special lighting.

• Assessment accommodations: changes 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

No modifications to course expectations are made at Brain Power.

Program Considerations for English Language Learners

Teachers must incorporate appropriate strategies for instruction and assessment to facilitate the success of the English language learners in their classrooms. These strategies include:

• modification of some or all of the subject expectations depending on the level of English proficiency

• use of a variety of instructional strategies (e.g., extensive use of visual cues, graphic organizers, scaffolding; 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, bilingual dictionaries, and materials that reflect cultural diversity)

• use of assessment accommodations (e.g., granting of extra time; use of oral interviews, demonstrations or visual representations, or tasks requiring completion of graphic organizers and cloze sentences instead of essay questions and other assessment tasks that depend heavily on proficiency in English).

  
  

Antidiscrimination Education

Learning resources reflect students’ interests, backgrounds, cultures, and experiences. Learning materials should:

• enable students to become more sensitive to the diverse cultures and perceptions of others, including Aboriginal peoples

• discuss aspects of the history of mathematics to make students aware of the various cultural groups that have contributed to the evolution of mathematics over the centuries

• illustrate to students that everyday people use mathematics in their everyday lives, both at work and at home

• connect mathematics to real world situations and human affairs such as health, science, environmental studies, trend analysis, and politics.

Literacy and Inquiry/Research Skills

The school emphasizes the importance of the following:

• using clear, concise communication in the classroom involving the use of diagrams, charts, tables, and graphs

• emphasizing students’ ability to interpret and use graphic texts.

• acquiring the skills to locate relevant information from a variety of sources, such as books, newspapers, dictionaries, encyclopaedias, interviews, videos, and the Internet.

• learning that all sources of information have a particular point of view

• learning that the recipient of the information has a responsibility to evaluate it, determine its validity and relevance, and use it in appropriate ways.

Role of Technology

Information and communications technologies (ICT) tools used in many ways:

• Students use multimedia resources, databases, Internet websites, digital cameras, and word-processing programs.

• Students use databases, spreadsheets, dynamic geometry and statistical software, graphing software, computer algebra systems, and so on in order to quickly navigate through complex problems, to see the effect of dynamic data on their values and trends, and to see a graphical representation of data.

• They use technology to collect, organize, and sort the data they gather and to write, edit, and present reports on their findings.

• Students are encouraged to use ICT to support and communicate their learning. For example, students working individually or in

groups can use computer technology and/or Internet websites to gain access to museums and archives in Canada and around the world.

• Students use digital cameras and projectors to design and present the results of their research to their classmates.

• The school plans to use ICT to connect students to other schools and to bring the global community into the classroom.

• Students are made aware of issues of 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.

Career Education

Students are given opportunities to develop career-related skills by:

• applying their skills to work-related situations

• exploring educational and career options

• developing research skills

• developing key essential skills such as reading text, writing, computer use, measurement and calculation, and problem solving

• practising expository writing

• learning strategies for understanding informational reading material

• making oral presentations

• working in small groups with classmates to help students express themselves confidently and work cooperatively with others.

Financial Literacy

The school is emphasizing the importance of ensuring that Ontario students have the opportunity to improve their financial literacy. Financial literacy is defined as “having the knowledge and skills needed to make responsible economic and financial decisions with competence and confidence”. The goal is to help students acquire the knowledge and skills that will enable them to understand and respond to complex issues regarding their own personal finances and the finances of their families, as well as to develop an understanding of local and global effects of world economic forces and the social, environmental, and ethical implications of their own choices as consumers. Thus, an attempt will be made to integrate Financial Literacy in all the school’s courses.

Academic Honesty

Students who present the work of others as their own are guilty of plagiarism and will receive a mark of zero for the work and will have the details of the plagiarism noted in their school records. Students who are guilty of cheating on tests or examinations will receive a mark of zero on the test or examination and have the details of the cheating noted in their school records.

Late Assignments

Students are responsible for providing evidence of their achievement of the overall expectations within the time frame specified by the teacher, and in a form approved by the teacher. There are consequences for not completing assignments for evaluation or for submitting those assignments late.

Resources

Calculus and Vectors, Nelson (2009)

Calculus and Vectors 12, McGraw Hill Ryerson (2008) Moodle Website

Dictionaries, Thesaurus etc.

Various Daily Newspapers, Magazines, and Periodicals (Audio and Video material) CBC, The Fifth Estate, etc. Various Internet Resources:

• OWL English Purdue

• The University of Toronto Library

• The Ontario Ministry of Health and Long Term Care

• The Toronto Star

• The Globe and Mail