This course extends students’ experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.
|Unit Titles and Descriptions||Time Allocated|
|Basic Skills Review|
This unit reviews the foundational concepts that have been covered in prerequisite math courses. Students revisit the definition of a function, function notation, and the key properties of functions. Students also review transformations of functions and inverse functions. The unit assessment evaluates students’ ability to carry out proper communication, formatting, and technical skills in their work, all of which will be important aspects of their assignments in the remainder of the course.
In this unit students learn to identify and describe some key features of polynomial functions and to make connections between the numeric, graphical, and algebraic representations of polynomial functions. These concepts allow students to manipulate functions in a number of ways and apply their skills to solve real-world problems. Strategies will be employed to aid in the connection to an understanding of rates of change.
|Rational Functions and Inequalities|
Students begin this unit by identifying and describing some of the key features of rational functions. Students then learn to represent and manipulate these functions to solve real-life problems, graphically and algebraically. This unit also introduces the idea of inequalities and how they produce different solutions than equations.
|Exponential and Logarithmic Functions|
This unit begins with a review of exponential functions, their properties, and applications. This leads into discussions about a related function, the logarithmic function. From here students learn about logarithmic properties and then apply their knowledge of exponential and logarithmic functions to solve real-world problems.
This unit examines the meaning and application of radian measure. This allows students to solve more complex situations in exact values. Students will make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and use these connections to solve problems involving trigonometric equations and to prove trigonometric identities.
|Trigonometric Functions and Graphs|
This unit develops students understanding of trigonometry by expanding on the functions behind the trigonometric ratios. Students look at trigonometric functions and their reciprocals, examine their key properties and behaviours, and learn how they can be transformed to model a wide range of data.
|Operations and Functions|
Having studied various types of functions and transformations of functions, and understood the significance of differential rates of change in functions, this final unit focuses on the theory and practice of performing arithmetic operations on entire functions, including but not limited to the algebraic, graphical and practical implications of performing those operations.
This is a proctored exam worth 30% of your final grade.
Overall Curriculum Expectations
|A. Rate of Change|
|A1||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;|
|A2||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;|
|A3||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.|
|B. Derivatives and their Applications|
|B1||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;|
|B2||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.|
|C. Geometry and Algebra of Vectors|
|C1||demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;|
|C2||perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications;|
|C3||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;|
|C4||represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.|
Teaching & Learning Strategies:
Teaching and Learning Strategies:
The over-riding 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. The following mathematical processes will form the heart of the teaching and learning strategies used:
- Communicating: This course offers students many opportunities to share their understanding both in oral as well as written form. Students will discuss concepts they have learned through discussion boards, write reports which relate concepts taught to real-world applications, and create presentations to demonstrate understanding of some concepts.
- Problem solving: This course scaffolds student learning by building on prior knowledge and skills. Students will have the opportunity to review prior concepts and will be presented with problems that require them to apply their skills in new ways to solve problems related to real-world applications.
- Reflecting: This course models the reflective process. Through the use of examples and practice exercises, the course demonstrates proper communication to explain intermediate steps and reflect on solutions to determine if they make sense in the given context.
- Selecting Tools and Computational Strategies: This course models the use of graphing software to help solve problems and to familiarize students with technologies that can help make solving problems faster and more accurate.
- Connecting: Students will connect the concepts taught in the course to real-world applications (e.g. concepts related to polynomial functions will be connected to applications in engineering). Students will have opportunities to connect previous concepts to new concepts through posed problems, investigations, and enrichment activities.
- Self-Assessment: Through the use of interactive activities (e.g. multiple choice quizzes, and drag-and-drop activities) students receive instantaneous feedback and are able to self-assess their understanding of concepts.
Assessment, Evaluation and Reporting Strategies of Student Performance:
Our theory of assessment and evaluation follows the Ministry of Education’s Growing Success document, and it is our firm belief that doing so is in the best interests of students. We seek to design assessment 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.
Growing Success articulates the vision the Ministry has for the purpose and structure of assessment and evaluation techniques. There are seven fundamental principles that ensure best practices and procedures of assessment and evaluation by ICE teachers. ICE assessments and evaluations,
- are fair, transparent, and equitable for all students;
- support all students, including those with special education needs, those who are learning the language of instruction (English or French), and those who are First Nation, Métis, or Inuit;
- are carefully planned to relate to the curriculum expectations and learning goals and, as much as possible, to the interests, learning styles and preferences, needs, and experiences of all students;
- are communicated clearly to students and parents at the beginning of the course and at other points throughout the school year or course;
- are ongoing, varied in nature, and administered over a period of time to provide multiple opportunities for students to demonstrate the full range of their learning;
- provide ongoing descriptive feedback that is clear, specific, meaningful, and timely to support improved learning and achievement;
- develop students’ self-assessment skills to enable them to assess their own learning, set specific goals, and plan next steps for their learning.
The Final Grade:
The evaluation for this course is based on the student’s achievement of curriculum expectations and the demonstrated skills required for effective learning. The final percentage grade represents the quality of the student’s overall achievement of the expectations for the course and reflects the corresponding level of achievement as described in the achievement chart for the discipline. A credit is granted and recorded for this course if the student’s grade is 50% or higher. The final grade will be determined as follows:
- 70% of the grade will be based upon evaluations conducted throughout the course. This portion of the grade will reflect the student’s most consistent level of achievement throughout the course, although special consideration will be given to more recent evidence of achievement.
- 30% of the grade will be based on final evaluations administered at the end of the course. The final assessment may be a final exam, a final project, or a combination of both an exam and a project.
The Report Card:
Student achievement will be communicated formally to students via an official report card. Report cards are issued at the midterm point in the course, as well as upon completion of the course. Each report card will focus on two distinct, but related aspects of student achievement. First, the achievement of curriculum expectations is reported as a percentage grade. Additionally, the course median is reported as a percentage. The teacher will also provide written comments concerning the student’s strengths, areas for improvement, and next steps. Second, the learning skills are reported as a letter grade, representing one of four levels of accomplishment. The report card also indicates whether an OSSD credit has been earned. Upon completion of a course, ICE will send a copy of the report card back to the student’s home school (if in Ontario) 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.
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.
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