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.
By the end of this course, students develop the following skills in these different areas:
| 1. Characteristics of Functions | |
| 1.1 | demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations; |
| 1.2 | determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including those arising from real-world applications; |
| 1.3 | demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions. |
| 2. Exponential Functions | |
| 2.1 | evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways; |
| 2.2 | make connections between the numeric, graphical, and algebraic representations of exponential functions; |
| 2.3 | identify and represent exponential functions, and solve problems involving exponential functions, including those arising from real-world applications. |
| 3. Trigonometric Functions | |
| 3.1 | determine the values of the trigonometric ratios for angles less than 360[degree sign]; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law; |
| 3.2 | demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions; |
| 3.3 | identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including those arising from real-world applications. |
| 4. Discrete Functions | |
| 4.1 | determine the values of the trigonometric ratios for angles less than 360[degree sign]; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law; |
| 4.2 | demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions; |
| 4.3 | identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including those arising from real-world applications. |
| Time Allocated | Online/Offline Component | |
|---|---|---|
| 1. Unit 1: Skill Building | ||
This unit will build a foundation of skills for the rest of the course. Students will be introduced into function notation and domain and range. They will learn to evaluate and simplify expressions with rational exponents and radicals and they will learn about algebraic and geometric sequences and series. Students will review trigonometry and learn about special triangles and angles in the Cartesian plane. | 20 hours | Online |
| 2. Unit 2: Graphing | ||
The focus of the second unit will be making connections between the algebraic and graphical representation of functions using transformations. Domain and range of continuous functions will be explored as well as graphical representations of inverse functions. Transformations will be applied to many different functions. Sinusoidal functions and their transformations will be introduced. The solutions of linear-quadratic systems are investigated graphically. | 18 hours | Online |
| 3. Unit 3: Simplifying Rational Expressions | ||
Students will extend algebraic skills to simplify complex algebraic expressions including operations with rational expressions. These algebraic skills will be applied to proving trigonometric identities. Students will apply Pascal’s triangle to binomial expansion. | 15 hours | Online |
| 4. Unit 4: Solving Quadratic, Exponential and Discrete Functions | ||
With a strong foundation of skills and understanding of functions, students will learn to solve equations with quadratic, exponential, and discrete functions. They will determine the zeros and vertex of quadratic functions and the solutions of linear-quadratic systems. | 15 hours | Online |
| 5. Unit 5: Solving Sinusoidal and Trigonometric Equations | ||
Students will revisit special triangles and angles in the Cartesian plane while solving sinusoidal equations. They will use trigonometric ratios, the Sine Law, and the Cosine Law to solve triangles. Students will investigate the Ambiguous Case of the Sine Law, understanding how to determine the number of possible solutions and solve a triangle when provided with two sides and an angle opposite one of those sides. | 5 hours | Online |
| 6. Unit 6: Applications | ||
Students will apply the skills obtained solving equations to real world problems. Quadratic functions can be applied to many situations to solve and determine optimum values. Exponential functions will be used in growth and decay problems and sinusoidal functions will be applied to periodic problems. Trigonometry will be used to solve applications involving triangles. | 20 hours | Online |
| 7. Unit 7: Discrete and Financial Applications | ||
Sequences and Series will be applied to real world situations and solved. Students will solve financial problems with simple and compound interest and apply series to problems involving loans and annuities. | 15 hours | Online |
| 8. Final Evaluation | ||
Final Exam | 3 hours | Online |
| Total | 111 Hours | |
This course is organized into a semester format. Lessons and activities will be presented to students via the internet. Synchronous lessons will be provided though live online teaching and student to student discussion forums.
A variety of strategies will be used in the online delivery of this course. Instructional strategies will include but are not limited to:
Learning goals will be discussed at the beginning of each lesson and success criteria be provided to students. The success criteria are used to develop the assessment tools in this course, including rubrics and checklists.
The overriding 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: Develop, select, apply, compare, and adapt a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding.
Reasoning and Proving: Develop and apply reasoning skills (e.g., use of inductive reasoning, deductive reasoning, and counter-examples; construction of proofs) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments.
Reflecting: 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).
Connecting: 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).
Representing: 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.
Selecting Tools and Computational Strategies: Select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems.
Communicating: Communicate mathematical thinking orally, visually, and in writing, using precise mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
A variety of assessment and evaluation methods, strategies and tools are required as appropriate to the expectation being assessed. These include diagnostic, formative and summative within the course and within each unit. We strictly follow the Ministry of Education’s Growing Success document.
Assessment FOR Learning and Assessment AS Learning is obtained through a variety of means, including the following:
Evidence of student achievement (assessment OF learning) is collected through ongoing observations of most consistent work, with consideration given to most recent work from various sources.
Assessment and evaluation in this course will be based on the provincial Ontario Curriculum expectations. Students will be provided with numerous and varied opportunities to demonstrate the full extent of their achievement. Categories of assessment and breakdowns are as follows:
A final grade will be determined as follows:
Students with special needs and English Language Learners will be provided with accommodation, including additional time, assistive technology and scribe where available. Teachers who are planning a program in this subject 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.
Learning Skills listed below are key to student success. Learning Skills are assessed independently of achievement and are determined through observation and participation. A checklist and student conference will be used to determine the level in each category.
$549.00
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