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About the course

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.

Functions

Course Credit

1

Course Price

$ 550.00

Course Developer

My Learning Oasis

Prerequisite(s) (Text)

Principles of Mathematics, Grade 10, Academic(MPM2D)

Course Code

Department Head & Contact Information

MCR3U

Course Type

University

Grade Level

Grade 11

Course Development Date

June 10th, 2021

Course Outline

Characteristics of Functions

In this unit students will do a detailed analysis of functions and their inverse then look at their representations. They will then make connections between the algebraic and graphical representations of functions using different kinds of transformations. This unit will do a brief review of the quadratic functions to determine their roots and local maxima and minima and relate them to real life applications. It will also look at techniques to determine equivalence as it relates to simplifying polynomial, radical, and rational expressions.

Expected Hours of Instruction: 25 Hours

Exponential Functions

This unit will expose students to techniques to simplify and manipulate expressions and variables with rational exponents. It explores the connections between the numeric, graphical, and algebraic representations of exponential functions. This unit also indulges in the representation of real life concepts and applications that involve expressions with exponential growth/decay and functions with radical exponents. The exponential function will be used to determine bacterial growth rates, half-life of nuclear wastes, the spread of mis-information on the news etc.

Expected Hours of Instruction: 25 Hours

Discrete Functions

In this unit, students will be introduced to recursive sequences. They will explore ways of representing recursive sequences and how such representations can ease the burden of complex algebraic manipulations, for instance, using Pascal’s Triangle to give the first 5 terms of (ax +b)^10. They will be introduced to geometric and arithmetic series and sequences and solve related problems, as in business applications, engineering applications etc. They will look at the difference between compound and simple interest from a series point of view.

Expected Hours of Instruction: 25 Hours

Trigonometric Functions

In this unit students will review the basic trigonometric ratios sine, cosine, and tangent. They will be exposed to cyclic and periodicity and apply these concepts to the trigonometric ratios. They will be introduced to various trigonometric identities to assist them in solving trigonometric equations for various ranges of values. Real life applications will be introduced from the perspective of trigonometry . They will be introduced to complex trigonometric identities with multiple angles and various exponents. They will then explore techniques for proving these identities.

Expected Hours of Instruction: 32.5 Hours

Exam

This course includes a two and a half hour final exam and is worth 30% of your final grade.

Expected Hours of Instruction: 2.5 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. 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 force 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..

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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