MHF4U - Grade 12 Advanced Functions
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MHF4U - Grade 12 Advanced Functions
Course Code: | MHF4U |
Course Type: | University Preparation |
Credit Value: | 1.0 |
Prerequisite: | Functions, Grade 11, University Preparation or Mathematics for College Technology for College Technology |
Tuition Fee (CAD): | $574 |
Course Description For MHF4U Grade 12 Advanced Functions Online Course
MHF4U Advanced Functions: 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. Contact us to know more.
Overview of Units and Timelines For Grade 12 Advanced Functions MHF4U
Here’s the suggested sequence for delivering course units, along with the recommended hours needed to complete each one. For a detailed breakdown of the specific expectations and activities included in each unit, refer to the Unit Overviews provided in the MHF4U course profile.
Unit
Titles and Descriptions
Time and Sequence
Unit 1
Concepts of Calculus
A variety of mathematical operations with functions are needed in order to do the calculus of this course. This unit begins with students developing a better understanding of these essential concepts. Students will then deal with rates of change problems and the limit concept. While the concept of a limit involves getting close to a value but never getting to the value, often the limit of a function can be determined by substituting the value of interest for the variable in the function. Students will work with several examples of this concept. The indeterminate form of a limit involving factoring, rationalization, change of variables and one-sided limits are all included in the exercises undertaken next in this unit. To further investigate the concept of a limit, the unit briefly looks at the relationship between a secant line and a tangent line to a curve. To this point in the course students have been given a fixed point and have been asked to find the tangent slope at that value, in this section of the unit students will determine a tangent slope function similar to what they had done with a secant slope function. Sketching the graph of a derivative function is the final skill and topic.
15 hours
Unit 2
Derivatives
The concept of a derivative is, in essence, a way of creating a short cut to determine the tangent line slope function that would normally require the concept of a limit. Once patterns are seen from the evaluation of limits, rules can be established to simplify what must be done to determine this slope function. This unit begins by examining those rules including: the power rule, the product rule, the quotient rule and the chain rule followed by a study of the derivatives of composite functions. The next section is dedicated to finding the derivative of relations that cannot be written explicitly in terms of one variable. Next students will simply apply the rules they have already developed to find higher order derivatives. As students saw earlier, if given a position function, they can find the associated velocity function by determining the derivative of the position function. They can also take the second derivative of the position function and create a rate of change of velocity function that is more commonly referred to as the acceleration function which is where this unit ends.
16 hours
Unit 3
Curve Sketching
In previous math courses, functions were graphed by developing a table of values and smooth sketching between the values generated. This technique often hides key detail of the graph and produces a dramatically incorrect picture of the function. These missing pieces of the puzzle can be found by the techniques of calculus learned thus far in this course. The key features of a properly sketched curve are all reviewed separately before putting them all together into a full sketch of a curve.
6 hours
Unit 4
Derivative Applications and Related Rates
A variety of types of problems exist in this unit and are generally grouped into the following categories: Pythagorean Theorem Problems (these include ladder and intersection problems), Volume Problems (these usually involve a 3-D shape being filled or emptied), Trough Problems, Shadow problems and General Rate Problems. During this unit students will look at each of these types of problems individually.
8 hours
Unit 5
Derivative of Exponents and Log Functions-Exponential Functions
This unit begins with examples and exercises involving exponential and logarithmic functions using Euler’s number (e). But as students have already seen, many other bases exist for exponential and logarithmic functions. Students will now look at how they can use their established rules to find the derivatives of such functions. The next topic should be familiar as the steps involved in sketching a curve that contains an exponential or logarithmic function are identical to those taken in the curve sketching unit studied earlier in the course. Because the derivatives of some functions cannot be determined using the rules established so far in the course, students will need to use a technique called logarithmic differentiation which is introduced next.
06 hours
Unit 6
Trig Differentiation and Application
A brief trigonometry review kicks off this unit. Then students turn their attention to special angles and the CAST rule which has been developed to identify which of the basic trigonometric ratios is positive and negative in the four quadrants. Students will then solve trigonometry equations using the CAST rule to locate other solutions. Two fundamental trigonometric limits are investigated for the concepts of trigonometric calculus to be fully understood. The unit ends, as in all other units in the course, with an assignment and a unit quiz.
08 hours
Unit 7
Vectors
There are four main topics pursued in this initial unit of the course. These topics are: an introduction to vectors and scalars, vector properties, vector operations and plane figure properties. Students will tell the difference between a scalar and vector quantity, they will represent vectors as directed line segments and perform the operations of addition, subtraction, and scalar multiplication on geometric vectors with and without dynamic geometry software. Students will conclude the first half of the unit by proving some properties of plane figures, using vector methods and by modeling and solving problems involving force and velocity. Next students learn to represent vectors as directed line segments and to perform the operations of addition, subtraction, and scalar multiplication on geometric vectors with and without dynamic geometry software. The final topic involves students in proving some properties of plane figures using vector methods.
12 hours
Unit 8
Vector Applications
Cartesian vectors are represented in two-space and three- space as ordered pairs and triples, respectively. The addition, subtraction, and scalar multiplication of Cartesian vectors are all investigated in this unit. Applications involving work and torque are used to introduce and lend context to the dot and cross products of Cartesian vectors. The vector and scalar projections of Cartesian vectors are written in terms of the dot product. The properties of vector products are investigated and proven. These vector products will be revisited to predict characteristics of the solutions of systems of lines and planes in the intersections of lines and planes.
16 hours
Unit 9
Intersection of Lines and Planes
This unit begins with students determining the vector, parametric and symmetric equations of lines in R2 and R3. Students will go on to determine the vector, parametric, symmetric and scalar equations of planes in 3-space. The intersections of lines in 3-space and the intersections of a line and a plane in 3-space are then taught. Students will learn to determine the intersections of two or three planes by setting up and solving a system of linear equations in three unknowns. Students will interpret a system of two linear equations in two unknowns geometrically, and relate the geometrical properties to the type of solution set the system of equations possesses. Solving problems involving the intersections of lines and planes, and presenting the solutions with clarity and justification forms the next challenge. As work with matrices continues students will define the terms related to matrices while adding, subtracting, and multiplying them. Students will solve systems of linear equations involving up to three unknowns, using row reduction of matrices, with and without the aid of technology and interpreting row reduction of matrices as the creation of new linear systems equivalent to the original constitute the final two new topics of this important unit.
18 hours
Unit 10
Final Evaluation
The final assessment task is a three hour exam worth 30% of the student’s final mark.
3 hours
Total
110 hours
Throughout this course students will:
Problem solve:Â by developing, selecting, applying, and adapting a variety of problem-solving strategies
Reason and prove:Â by developing and applying reasoning skills to make mathematical conjectures, assess conjectures, and justify conclusions, plan and construct mathematical arguments;
Reflect:Â by monitoring their thinking to help clarify understanding as they complete an investigation or problem;
Select tools and computational strategies:Â by selecting and using a variety of concrete, visual, and electronic learning tools and computational strategies;
Connect:Â by relating mathematical ideas to situations or phenomena drawn from other contexts;
Represent:Â by making representations (e.g. Numeric, geometric, algebraic, graphical, pictorial and onscreen);
Communicate:Â by thinking orally, visually and in writing using precise mathematical vocabulary and conventions. Teachers will employ guided exploration, visuals, model analysis, direct instruction, problem posing and self-assessment to enable these student strategies.
Assessment is a systematic process of collecting information or evidence about a student’s progress towards meeting the learning expectations. Assessment is embedded in the instructional activities throughout a unit. The expectations for the assessment tasks are clearly articulated and the learning activity is planned to make that demonstration possible. This process of beginning with the end in mind helps to keep focus on the expectations of the course. The purpose of assessment is to gather the data or evidence and to provide meaningful feedback to the student about how to improve or sustain the performance in the course. Scaled criteria designed as rubrics are often used to help the student to recognize their level of achievement and to provide guidance on how to achieve the next level. Although assessment information can be gathered from a number of sources (the student himself, the student’s course mates, the teacher), evaluation is the responsibility of only the teacher. For evaluation is the process of making a judgment about the assessment information and determining the percentage grade or level.
Assessment is embedded within the instructional process throughout each unit rather than being an isolated event at the end. Often, the learning and assessment tasks are the same, with formative assessment provided throughout the unit. In every case, the desired demonstration of learning is articulated clearly and the learning activity is planned to make that demonstration possible. This process of beginning with the end in mind helps to keep focus on the expectations of the course as stated in the course guideline. The evaluations are expressed as a percentage based upon the levels of achievement.
English Grade 9: A variety of strategies are used to allow students opportunities to attain the necessary skills for success in this course and at the post-secondary level of study. To facilitate learning, the teacher uses a variety of activities engaging the whole class, small groups, and individual students.
The assessment will be based on the following processes that take place in the classroom:
Assessment FOR Learning | Assessment AS Learning | Assessment OF Learning |
---|---|---|
During this process the teacher seeks information from the students in order to decide where the learners are and where they need to go. | During this process the teacher fosters the capacity of the students and establishes individual goals for success with each one of them. | During this process the teacher reports student’s results in accordance to established criteria to inform how well students are learning. |
Conversation | Conversation | Conversation |
Classroom discussion Self-evaluation Peer assessment | Classroom discussion Small group discussion Post-lab conferences | Presentations of research Debates |
Observation | Observation | Observation |
Drama workshops (taking direction) Steps in problem solving | Group discussions | Presentations Group Presentations |
Student Products | Student Products | Student Products |
Reflection journals (to be kept throughout the duration of the course) Check Lists Success Criteria | Practice sheets Socrative quizzes | Projects Poster presentations Tests In Class Presentations |
Some of the approaches to teaching/learning include
Strategy | Who | Assessment Tool |
Class discussion | Teacher | Observation Checklist |
Response Journal | Teacher | Anecdotal Comments |
Student Chosen Song | Teacher | Observation Checklist |
Narrative Poem/Song | Teacher | Rubric and Anecdotal Comments |
Character Sketch | Self | Checklist |
Journal Responses | Self/teacher | Anecdotal comments |
Short Story Analysis | Teacher | Rating scale |
Short Story Outline | Teacher | Rating scale |
Anecdote | Teacher | Direct Observation |
Found poem | Teacher | Direct Observation |
Journal Entries | Teacher | Anecdotal |
Research Notes | Self/Teacher | Checklist |
Non-fiction Report/Presentation | Teacher | Â Rubric |
Presentation to group | Self/Peer | Self and peer assessment rubric |
Sight passage | Teacher | Marking scheme |
Narrative piece | Teacher | Rubric |
English Grade 9: The evaluation of this course is based on the four Ministry of Education achievement categories of knowledge and understanding (25%), thinking (25%), communication (25%), and application (25%). . The evaluation for this course is based on the student’s achievement of curriculum expectations and the demonstrated skills required for effective learning.
The 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.
English Grade 9: A credit is granted and recorded for this course if the student’s grade is 50% or higher. The final grade for this course 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
- 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
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Textbook
- Live Ink: Print and Digital Student Kit A, Karen Hume, Sharon Jeroski, Rich MacPherson, et al Pearson Education
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Potential Resources
- Novel: Skud by Dennis Foon or The Chrysalids by John Wyndham
- OSSLT Practice Materials (www.eqao.com)
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Frequently Asked Questions (FAQ)​​
This course covers polynomial, rational, logarithmic, and trigonometric functions, techniques for combining functions, rates of change, and vector applications.
The course is intended for students preparing for university programs requiring calculus or those consolidating their math knowledge.
Students must have completed Grade 11 Functions, University Preparation or Mathematics for College Technology.
Strategies include problem-solving, reasoning, reflecting, using tools, and connecting mathematical concepts to real-world scenarios.
The final grade consists of 70% coursework and 30% final evaluation, including exams or projects.