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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.

Course Credit

1

Course Price

\$ 550.00

Course Developer

My Learning Oasis

Prerequisite(s) (Text)

Functions, Grade 11, University Preparation, or Mathematics for College Technology, Grade 12, College Preparation(MCR3U) or (MCT4C)

Course Code

MHF4U

Course Type

University

Course Development Date

June 10th, 2021

## Course Outline

Exponential and Logarithmic Functions

In this unit, students will study the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numerical expressions. Students will also evaluate the variable in logarithmic equations of various bases within the same equation. They will analyse, identify, and describe some key features of the graphs of logarithmic functions. Students will have to develop exponential and logarithmic equations from real-life situations and solve them.

Expected Hours of Instruction: 30 Hours

Trigonometric Functions

In this unit students will study the different units used to measure angles. In particular, they will relate radians to degrees and to cycles and revolutions. The students will study multi-angle trigonometric functions with phase shifts. They will sketch these trigonometric functions, their reciprocals, and their inverses. In this unit students will learn techniques to prove trigonometric identities. They will use trigonometric identities to solve trigonometric equations and use the graphs to determine the solution set.

Expected Hours of Instruction: 30 Hours

Polynomial and Rational Functions

Students will identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of these. They will identify and describe some key features of the graphs of rational functions, and represent them graphically. Students will be introduced to and solve problems involving polynomial and simple rational equations graphically and algebraically. In this unit they will be introduced to simple rational inequalities.

Expected Hours of Instruction: 25 Hours

Characteristics of Functions

Students will determine the average of an instantaneous rate of change, 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. They will study functions that result from the combination of other functions for instance the addition, subtraction, multiplication, and division of two functions and from the composition of two functions. They will then analyse the properties that result from the composite function. Students will 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.

Expected Hours of Instruction: 22 Hours

Exam

Expected Hours of Instruction: 3 ​Hour

Toal 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. Exponential and Logarithmic Functions

A1 demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numerical expressions;
A2 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;
A3 solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

B. Trigonometric Functions
B1 demonstrate an understanding of the meaning and application of radian measure;
B2 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;
B3 solve problems involving trigonometric equations and prove trigonometric identities.

C. Polynomial and Rational Functions

C1 identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
C2 identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
C3 solve problems involving polynomial and simple rational equations graphically and algebraically;
C4 demonstrate an understanding of solving polynomial and simple rational inequalities.

D. Characteristics of Functions
D1 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;
D2 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;
D3 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.

## 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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