Advanced Functions, Grade 12, University Preparation
| Course Title | Advanced Functions, Grade 12, University Preparation |
| Course Code | MHF4U |
| Grade | 12 |
| Course Type | University Preparation |
| Credit Value | 1.0 |
| Prerequisite | Functions, Grade 11, University Preparation, or Mathematics for College Technology, Grade 12, College Preparation |
| Co requisite | None |
| Department | Mathematics |
| Name of the Department Head | Colleen McMillan (B.Sc., B.Ed., OCT) |
| Course Developer or Teacher | Colleen McMillan (B.Sc., B.Ed., OCT) |
| Curriculum Policy Document | Mathematics, The Ontario Curriculum, Grades 11 & 12, 2018 (Revised) |
| Course development date | September 2019 |
| Most recent revision date | March 2024 |
| Course reviser | Colleen McMillan (B.Sc., B.Ed., OCT) |
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.
By the end of this course, students will develop the following skills in these different areas:
| 1. Exponential and Logarithmic Functions | |
| 1.1 | Demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numerical expressions; |
| 1.2 | Identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically; |
| 1.3 | Solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications. |
| 2. Trigonometric Functions | |
| 2.1 | Demonstrate an understanding of the meaning and application of radian measure; |
| 2.2 | Make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems; |
| 2.3 | Solve problems involving trigonometric equations and prove trigonometric identities. |
| 3. Polynomial and Rational Functions | |
| 3.1 | Identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions; |
| 3.2 | Identify and describe some key features of the graphs of rational functions, and represent rational functions graphically; |
| 3.3 | Solve problems involving polynomial and simple rational equations graphically and algebraically; |
| 3.4 | Demonstrate an understanding of solving polynomial and simple rational inequalities. |
| 4. Characteristics of Functions | |
| 4.1 | Demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point; |
| 4.2 | Determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems; |
| 4.3 | Compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques. |
| Time Allocated | Online/Offline Component | Learning Expectations | |
|---|---|---|---|
| 1. Skill Building | |||
A solid foundation of skills is required to succeed in Advanced Functions. This unit focuses on acquiring and mastering skills such as: dividing polynomials and factoring higher degree polynomials; multiplying and dividing rational functions; writing exponential equations in logarithmic form and applying logarithmic rules. Students will learn to use radians as an alternative unit to the degree for angle measurement, understand the relationship between degrees and radians and apply trigonometry skills with radians. Logarithm rules are applied to evaluate logarithms and exponentials. Proper formatting and communication are emphasized including interval notation. | 15 hours | (6 hrs online / 9 hrs offline) | A1.1-1.4, A3.1, B1.1-1.4, C1.1, C3.1, 3.2 |
| 2. Characteristics and Transformations of Functions | |||
This unit investigates the key properties of parent functions (polynomial, exponential, logarithm, rational, square root) reviewing domain, range, and intercepts, as well as introducing positive/negative intervals, asymptotes, end behaviours and symmetry. Students will learn to interpret piecewise functions and find the inverse of a given function. Students also review transformations of functions and extend their knowledge to higher degree polynomial, rational and logarithmic functions. Graphing software is used to investigate the characteristics of functions. The characteristics of trigonometric functions are investigated leading to the graphing of trigonometric functions using radians. | 12 hours | (6 hrs online / 6 hrs offline) | A2.1-2.4, A3.1, B2.1-2.3, C1.1-1.3, C1.6-1.9, C2.1, C2.2, D2.7, 3.1 |
| 3. Graphing Rational and Polynomial Functions | |||
In this unit, students will use the skills acquired in the previous unit to graph polynomial and rational functions. The key features of a function are analyzed separately before putting them all together into a full sketch of a curve. Domain and range, intercepts, positive/negative intervals, asymptotes, holes and end behaviours used to analyze a given equation and create a graph. Reciprocal functions of linear and quadratic functions are explored to make connections between the properties of the original and reciprocal graphs. Graphing technology is used to explore the graphs of different functions. | 12 hours | (6 hrs online / 6 hrs offline) | C1.3-1.6, C2.1-2.3 |
| 4. Solving Polynomial and Rational Functions | |||
Now that students have an understanding of polynomials and rational functions, we take the next step and use the skills to solve equations and inequalities. Interval charts or number lines are introduced to solve inequalities. The skills obtained in previous units are used to algebraically manipulate polynomial and rational equations in order to solve for unknowns. The relationship between algebraic and graphical solutions is explored using graphing technology. | 10 hours | (4 hrs online / 6 hrs offline) | C3.3-3.7, C4.1-4.3 |
| 5. Solving Trigonometric, Logarithmic and Exponential Functions | |||
Students will build upon the foundational skills and learn to solve trigonometric, logarithmic and exponential functions. The periodicity of trigonometric functions is explored algebraically and graphically to find multiple solutions to trigonometric equations. Students will use their knowledge of trigonometry and solving equations to solve linear and quadratic trigonometric equations. Logarithms and logarithmic rules are applied to exponential equations to solve exactly. Logarithmic rules are also applied to logarithmic equations to create single logarithms on both sides of an equation in order to solve. | 12 hours | (5 hrs online / 7 hrs offline) | A3.1-3.3, B3.4 |
| 6. Applications and Problem Solving | |||
The skills of previous units are reviewed and reinforced in problem solving and applications. Students will use critical thinking to explore concepts from different angles, such as using the factor and remainder theorem to solve for an unknown coefficient in a polynomial function. Students learn to interpret real life situations and create representative equations. Polynomial equations and inequalities are used to solve volume, surface area, profit and revenue problems. Work rates and other applications are represented by rational equations and both equalities and equalities are solved. Radians are used to solve arc length and angular velocity problems. Trigonometric equations are created to represent periodic situations such as circular motion, simple harmonic motion, seasonal changes in temperature or daylight hours, etc. Students will use logarithms to solve loudness, Richter and pH scale problems. Exponential applications from grade 11 are expanded upon to solve more complicated growth and decay problems. | 15 hours | (6 hrs online / 9 hrs offline) | A2.4,3.2,3.4, B2.7, C1.7,C3.7, D2.2, 3.3 |
| 7. Trigonometric Equations and Identities | |||
Special triangles and transformations of trigonometric functions are investigated to recognize equivalent trigonometric expressions. Compound angle formulas are developed and used to find exact values for non-special angles. Double Angle formulas are derived from the compound angle formulas. Pattern recognition is emphasized in using formulas to simplify expressions. Students will learn to prove trigonometric identities using the Pythagorean identities as well as compound and double angle formulas. | 12 hours | (5 hrs online / 7 hrs offline) | B3.1-3.3 |
| 8. Combining Functions | |||
Students will determine functions that result from the addition, subtraction, multiplication, division and composition of two functions. Properties of the combined functions will be analyzed such as domain and range, intercepts, positive and negative intervals and asymptotes. Graphing technology will be used to explore the properties of the combined functions. Equations that cannot be solved through standard algebraic methods are solved using a Guess and Improve strategy and graphing technology. | 10 hours | (4 hrs online / 6 hrs offline) | D2.1, 2.3-2.8 |
| 9. Rates of Change & Modelling | |||
Rates of change are explored both algebraically and graphically. Students will gain insight into the relationship between average rate of change and slope of a secant, and instantaneous rate of change and slope of a tangent. Average rate of change will be calculated over an interval and various methods will be used to calculate the instantaneous rate of change at a point, such as centered interval method, preceding and following method and difference quotient. Rates of change will be applied to real life situations such as distance-time problems. | 9 hours | (4 hrs online / 5 hrs offline) | D1.1-1.10, 3.2 |
| 10. Final Examination | |||
This is a proctored exam worth 25% of the final grade. | 3 hours | (3 hrs online) | — |
| Total | 110 Hours | ||
| Online | Offline |
|---|---|
| Communicating with the instructor | Reading materials of the course |
| Contributing to forums | Researching topics on the internet |
| Watching instructional videos | Completing assignments |
| Watching additional resource videos | Completing practice activities |
| Practicing through online quizzes | Reviewing for tests and exams |
| Reviewing peer submissions | Completing online timed exam |
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.
Assessment FOR Learning and Assessment AS Learning is obtained through a variety of means, including the following:
- Ongoing descriptive feedback, including descriptive feedback on students’ practice activities
- Self-assessment
- Peer assessment
- Student/Teacher communication on a regular basis to:
- verbalize observations
- ask questions
- clarify understanding
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 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:
- Knowledge – 30%
- Thinking Inquiry – 25%
- Application – 25%
- Communication – 20%
A final grade will be determined as follows:
- Term Work – 65%
- Tests – 60%
- Assignments – 30%
- Investigations, Discussion Forums and Student Presentations – 10%
- Attendance & Participation – 10%
- Final Examination – 25%
| Students enrolled in this course through CHS’s Hybrid (H) program will participate in synchronous learning via live online teaching sessions, online support material, and student-to-student discussions organized throughout the semester. Conversely, students taking this course as part of CHS’s Self-Paced (S) program will learn asynchronously through recorded video lessons, presentations, online support material, and simulations. While self-paced students have up to one year to complete the course, they are encouraged to finish within five months.
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 and success criteria are listed at the beginning of each lesson. The success criteria are used to develop the assessment tools in this course, including rubrics.
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 | 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 knowledge they have gained to solve problems. |
| Selecting Tools and Computational Strategies | Course models the use of graphing software to familiarize students with available software and resources which will allow them to simplify calculations in order to better understand the characteristics of functions. |
| Connecting | Course uses 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 | 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. investigations, quizzes and online interactives) students receive instantaneous feedback and are able to self-assess their understanding of concepts. |
Students with special needs and English Language Learners will be provided with accommodation, including additional time, assistive technology and scribe where available.
Learning Skills listed below are key to student success. Learning Skills are assessed independently of achievement and are determined through observation and participation. A check list and student-teacher communication will be used to determine the level in each category.
- Responsibility
- Organization
- Independent Work
- Collaboration
- Initiative
- Self-Regulation
- Scientific calculator.
- PowerPoint and video lessons
- Activities
- Supplementary lessons.
- Links to simulations and interactives