About the course
This course builds on students’ previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three-dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who choose to pursue careers in fields such as science, engineering, economics, and some areas of business, including those students who will be required to take a university-level calculus, linear algebra, or physics course.
Calculus and Vectors
Course Credit
1
Course Price
$ 550.00
Course Developer
My Learning Oasis
Prerequisite(s) (Text)
Advanced Functions, Grade 12, University Preparation, must be taken prior to or concurrently with Calculus and Vectors.(MHF4U)
Course Code
Department Head & Contact Information
MCV4U
Ravi Sharma (ravi@mylearningoasis.com)
Course Type
University
Grade Level
Grade 12
Course Development Date
June 10th, 2021
Course Outline
Vectors
This section gives an introduction to vectors, scalars, vector quantities versus scalar quantities, vector operations and properties of a vector. Vector operations will be introduced such as addition and subtraction. Students will be introduced to matrix operations such as addition, subtraction, multiplication, and division (by a scalar). This forms the basis for determining the determinant of a matrix, which then forms the basis for studying the cross- product of two vectors - a topic to be addressed later in the course. Students will learn how to apply matrices to solve a system of linear equations in two and three unknowns.
Expected Hours of Instruction: 17 Hours
Linear Dependence and Coplanarity
In this section, students will learn how to represent a three coordinate point on a 3D axes system. Students will be taught the concepts of linear dependence and independence, collinearity, and coplanarity of vectors.
Expected Hours of Instruction: 17 Hours
Vector Applications
In this section, students will look at the dot and cross product after a brief review of Matrices. Here, they will be used in modelling applications in practical engineering, business, science, and other scenarios. Some typical applications involve torque, work, angular momentum etc.
Expected Hours of Instruction: 15 Hours
Intersection of Lines and Planes
The concept of vectors will be used to develop the equation of a directed line segment and a plane.
In this unit students will determine the parametric, scalar, and vector equations of lines and planes. In R2 and R3. They will determine the shortest distance of a point from a plane, the point of interaction of a line and a plane, and the line of intersection of two flat planes.
Expected Hours of Instruction: 12 Hours
Fundamentals of Calculus
Students will be introduced to rates of change and the rules governing ‘limits’. This deals with rates of change problems and the limit concept. Students will be introduced to right hand and left handed limits. The students will look at the relationship between a secant and a tangent line. Students will be given a curve and by inspection, estimate the slope of the tangent at each point. From this, the student will sketch the derivative curve from the given curve.
Expected Hours of Instruction: 12 Hours
Derivatives, Curve Sketching, Application and Optimization
In this section, the student is introduced to the concept of a ‘derivative function’, and thereafter using the first principle to find that function. The short- cut method of finding the derivative will also be introduced.
Students will use the new tools learnt to sketch a curve and extract various information from it, namely, local maximum, local minimum, roots, equation of axis of parabola.
Students will then apply this concept to solve real life problems relating to rates of change in Science, Engineering, Business, and other areas. Students will learn to optimize designs using derivatives.
Expected Hours of Instruction: 20 Hours
Derivatives of Trigonometric and Transcendental Functions
In this unit students will review the laws of logarithms and trigonometric identities. These laws will play a vital role in determining the derivatives of logarithms and trigonometric functions, and transcendental functions in general.
Expected Hours of Instruction: 15 Hours
Final Exam
This is a proctored exam worth 30% of your final grade.
Expected Hours of Instruction: 2 Hours
Total: 110 Hours
Resources
All notes and assignments will be provided by the teacher.
The students are responsible to have:
● A non-programmable, non-graphing, scientific calculator.
● A note taking device for online students
● Internet connection for online students
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.
Special Accommodations
Only Some students are able, with accommodations, to be part of a regular course curriculum and to demonstrate independent learning. These accommodations allow access to the course without any dilution of the knowledge and skills the student is expected to demonstrate. These required accommodations to facilitate the student’s learning will be identified in his or her IEP (see IEP Standards, 2000, page 11*). It is likely that IEP for many or all courses will reflect the same accommodations. The instructions and accommodations are geared to meet the diverse needs of learners.
The three types of accommodations that are going to be used are:
i) Instructional accommodations - changes in teaching/learning strategies facilitated by different styles of presentation; methods of organization; the use of technology and multimedia.
ii) Environmental accommodations - Certain classroom settings and preferential seating may benefit these students.
iii) Assessment: assessment procedures that enable the student to demonstrate his or her learning, such as Multiple Intelligence Theory, giving more time to complete tasks (see page 29 of the IEP Resource Guide, 2004, for more examples).
For students who require accommodations for only the mathematics courses, the assessment and evaluation of their achievement will be based on the appropriate course curriculum expectations and the achievement levels outlined in this document. The IEP box on the students’ Provincial Report Cards will not be checked, and no information on the provision of accommodations will be included.
* Taken from: Ministry of Education, Ontario. Extracted from The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007; (Pg:- 28-30) Date of extraction: Sunday, March 14, 2021
Program Considerations For English Language Learners
Students from a variety of cultural and linguistic backgrounds. For many of these students, English is not their spoken language. They may be coming from highly sophisticated educational systems, while others may have come from regions where access to formal schooling was limited. These students offer a rich addition to the classroom experience by way of their background knowledge and experience. All teachers will assist with their English-language development. In mathematics the teachers will include appropriate adaptations and strategies in their instructions and assessments to facilitate the success of the English language learners in their classrooms. Some of these strategies and adaptations are: modification of some or all of the course expectations so that they are challenging but attainable for the learner at his or her present level of English proficiency, given the necessary support from the teacher.
Teaching/Learning Strategy
The key learning strategy at My Learning Oasis Elite Private School is Constructivism. This format facilitates learning by many techniques, most or all of which will be adopted in the classroom. The most dominant of these is group learning. The facilitator places students of different backgrounds in the same group so that they can feed off each other. Each may bring to the table a different reasoning strategy to facilitate problem-solving. Now, each student becomes a learner and a teacher at the same time, as he/she has to communicate his/her solution. This builds the students' knowledge base and by default, increases their confidence to speak in a crowd, albeit a small group at the beginning. The famous educationalist, Vygotsky, proved that by placing students in a group they function at the upper level of their zone of proximal development, each one scaffolding the other.
This strategy is further enhanced by the teacher asking leading questions as opposed to giving the answer outright, then allowing for group discussion. The students are encouraged to make connections between what they have learnt and their life experiences, then share with the group. The effect of this strategy is intrinsic motivation and learning. Each student develops an expanded appreciation of the topic at hand by seeing how it applies in different settings around the world by way of listening to their group members.
This Constructivist approach will be further accentuated by implementing “fish-bowling”. There are many ways to implement this technique. The one that will mostly be used will be by dividing up the larger problem (technical, mathematics, science, or otherwise) into smaller bits and having each student thoughtfully master one part. That student then teaches the group and facilitates a discussion reflection about the strategy (computational or otherwise) used in the solution. Each student in turn does this.
The above techniques enable students to reflect on the material learnt, make real life connections, and develop problem solving skills. One important by-product of the technique of Constructivism is that each student develops an appreciation of each other’s culture. This cultivates healthy people’s skill, which is not only important for the professional world but for life itself.
Constructivism lends itself well to students whose first language is not the language of instruction and who is new to the class. While other strategies will be used for students having difficulty with the English Language, this technique will definitely be used to enhance their English skill.
Assessment And Evaluation
At My Learning Oasis, course facilitators do not wait for a quiz or exam to determine how well a student is doing. Here, evaluation is an on-going exercise. The pedagogical techniques (refer to Teaching and Learning Strategies) used at My learning Oasis are perhaps the best techniques suited for on-going assessment, hence, they being an integral part of our delivery methodologies.
Concrete assessments are made through projects and assignments. However, the evaluation is based on “our flavor” of the Mastery Teaching technique. This ensures that the emphasis is on the quality of learning and NOT grading. Students' projects and homework will continuously be evaluated and re-evaluated with appropriate guidance to meet the school’s and Ministry’s expectations. At My Learning Oasis, we will work with the students until the projects meet a minimum of a B-grade, unless in extreme circumstances where the willful negligence of the students force lower grades. While this is a lot more taxing on the facilitator, it does not matter because My Learning Oasis is a Learner-centered institution NOT a Grade-Centered nor a Teacher-Centered institution.
Four categories of knowledge and skills are outlined in the achievement chart - knowledge and understanding, thinking, communication, and application. Student’s work is assessed and evaluated with respect to these categories, and that achievement of particular expectations is considered within the appropriate categories. A final grade will then be recorded for this course and if that grade is 50% or higher, a credit is granted to the student and recorded for this course. The final grade for this course will be determined as follows:
● For material evaluated throughout the course, seventy percent of the grade will be assigned. This portion of the grade should reflect the student's consistency in his/her level of achievement throughout the course, although special consideration should be given to more recent evidence of achievement.
● Thirty percent of the grade will be based on a final evaluation, which is administered towards the end of the course
Final Exam: 30%
Grading for all course work, projects, presentation, participation, interim quizzes and exam: 70%
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