This course introduces the mathematical concept of the function by extending students’ experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions, and develop facility in determining equivalent algebraic expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
|Unit Titles and Descriptions||Time Allocated|
Students will explore the concepts of relations and functions in this unit, their representations, their inverses, and how to make connections between the algebraic and graphical representations of functions using transformations. Students will learn how to apply and determine the different transformations that can be applied to functions using transformational parameters, along with learning to apply function and set notations, and graphing methods consistently.
|Characteristics of Functions|
Students will explore properties of functions, in particular by determining the zeros and the maximum or minimum of a quadratic function and solve problems involving quadratic functions, including problems arising from real-world applications. By the end of the unit, students will be able to demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.
This unit will explore several topics including evaluating powers with rational exponents, simplifying expressions containing exponents, and describing properties of exponential functions represented in a variety of ways. The emphasis will be on modeling and problem solving using these concepts.
The unit begins with an exploration of recursive sequences and how to represent them in a variety of ways. Making connections to Pascal’s triangle, demonstrating understanding of the relationships involved in arithmetic and geometric sequences and series, and solving related problems involving compound interest and ordinary annuities will form the rest of the unit.
This unit concentrates students’ attention on determining the values of the trigonometric ratios for angles less than 360° proving simple trigonometric identities and solving problems using the primary trigonometric ratios. The sine law and the cosine law are developed. Students will learn to demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions while solving problems involving sinusoidal functions, including problems arising from real-world applications
|Trigonometric Functions and graphs|
Students will investigate the relationship between the graphs and the equations of sinusoidal functions sketching and describing the graphs and describing their periodic properties.
This course includes a two and a half hour final exam and is worth 30% of your final grade.
Resources required by the student:
- A scanner, smartphone camera, or similar device to digitize handwritten or hand-drawn work,
- A non-programmable, non-graphing, scientific calculator.
- Spreadsheet software (e.g. Microsoft ExcelTM, Mac NumbersTM, or equivalent)
Resources provided by ICE:
- This course is entirely online and does not require or rely on any textbook.
- Video solutions to demonstrate mathematical form and procedure are provided.
Overall Curriculum Expectations
|A. Characteristics of Functions|
|A1||demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations;|
|A2||determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications;|
|A3||demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.|
|B. Exponential Functions|
|B1||evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways;|
|B2||make connections between the numeric, graphical, and algebraic representations of exponential functions;|
|B3||identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.|
|C. Discrete Functions|
|C1||demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal’s triangle;|
|C2||demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems;|
|C3||make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.|
|D. Trigonometric Functions|
|D1||determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;|
|D2||demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;|
|D3||identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.|
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 are used throughout the course as strategies for teaching and learning the concepts presented.
- Problem Solving: This course scaffolds learning by providing students with opportunities to review and activate prior knowledge (e.g. reviewing concepts related to trigonometry from prior mathematics courses), and build off of this knowledge to acquire new skills. The course guides students toward recognizing opportunities to apply the knowledge they have gained to solve problems.
- Selecting Tools and Computational Strategies: This course model the use of spreadsheet software and a TVM solver for personal finance to familiarize students with available software and resources which will allow them to simplify calculations in order to better and more accurately manage money.
- Connecting: This course connects the concepts taught to real-world applications, such as simple harmonic motion and sound or light waves.
- Representing: Through the use of examples, practice problems, and solution videos, the course models various ways to demonstrate understanding, poses questions that require students to use different representations as they are working at each level of conceptual development – concrete, visual or symbolic, and allows individual students the time they need to solidify their understanding at each conceptual stage.
- 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 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.
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 the 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 aligns with the Ontario Ministry of Education policy and initiatives in a number of important areas.