Calculus and Vectors, Grade 12,MCV4U

Unit 1: Rates of Change

In this unit, students will describe examples of real-world applications of rates of change, represented in a variety of ways. Connections between the average rate of change of a function that is smooth over an interval and the slope of the corresponding secant, and between the instantaneous rate of change of a smooth function at a point and the slope of the tangent at that point, will also be learned. Students will recognize, through investigation, graphical and numerical examples of limits, and explain the reasoning involved. They application of the difference quotient to determine average rate of change and the application of the limit of the difference quotient to determine instantaneous rate of change will also be taught. Lastly, students will learn to determine the derivatives of polynomial functions using the First Principles Definition of the Derivative.

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.

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.

Unit 4: Derivatives of Sinusoidal Functions

In this unit, students will learn to determine the graph of the derivative of a given sinusoidal function. They will also solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions, radical functions, and other simple combination of functions. Lastly, they will solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results.

Unit 5: Exponential and Logarithmic 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.

Unit 6: Geometric Vectors

There are three main topics pursued in this initial unit of the course. These topics are: an introduction to vectors and scalars, vector properties, and vector operations. 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.

Unit 7: Cartesian Vectors

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. 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 also all investigated in this unit. Students will investigate the concepts of linear dependence and independence, and collinearity and co-planarity of vectors.

Unit 8: Lines and Planes

This unit begins with students determining the vector, parametric and symmetric equations of lines in two-dimensions and three-dimensions. 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.

Unit 9:Course Review and Final Exam

Students will be given time near the end of the course to review all the concepts they have learned throughout the course in preparation for the final exam that covers all overall expectations.

Resources required by the student:

· A scanner, smart phone camera, or similar device to digitize handwritten or hand-drawn work,

· A non-programmable, non-graphing, scientific calculator.

· Word processing software (e.g. Microsoft Word^TM, Mac Pages^TM, or equivalent)

· Microphone and audio recording software

Types of Assessment

Assessmentsforandaslearning will have a diagnostic and formative purpose; their role is to check for students’ understanding. Assessments that serve this purpose will usually manifest themselves in the form of short, daily quizzes, teacher checking of homework, and conversations about progress. The purpose these quizzes serve is to encourage students to review daily and to alert students when there is a specific expectation they have not yet achieved. Quizzes are effective simply because they provide immediate feedback for the student.

Assessmentsoflearning have a summative purpose and are given at strategic instances- for example, after a critical body of information/set of overall or specific expectations has been covered. “This type of assessment collects evidence for evaluating the student’s achievement of the curriculum expectations and for reporting to students and parents/guardians” (Growing Success- assessment, evaluation, and reporting: improving student learning, pg. 1-ii). Assessments of learning consider product, observation, and conversation as sources of evidence.

Final Mark Calculation

Calculation of the Term Mark will be based upon theCategories of theAchievement Chart. This chart is meant to assist teachers in planning instruction and learning activities for the achievement of the curriculum expectations. It is also used in designing assessment and evaluation tasks and tools and in providing feedback to students. Each mathematical topic will contain each category in the chart due to the integrated nature of the discipline in mathematics. Final marks will be calculated as follows:

Term Work: 70%Final Summative Evaluation: 30%

CategoryWeightTaskWeight

Knowledge and Understanding: 25%

Thinking and Inquiry: 25% Final Exam 30%

Application: 25%

Communication: 25%

Final Summative Evaluation: 30%

CategoryWeight

Knowledge and Understanding: 25%

Thinking and Inquiry: 25%

Application: 25%

Communication: 25%

The Final Summative Evaluation will take the form of a proctored 3 hours exam worth 30% of the students final mark. It is important to note that the Final Summative Evaluation will share the same weighting for each category of the achievement chart as the term work. This means that K/T/A/C will have a 25%/25%/25%/25% split on the final examination in accordance with the rest of the term work.

Teaching and Learning Strategies:

A variety of strategies are used to allow students many opportunities to attain the necessary skills for success in this course and at university. The teacher uses a variety of whole class, small group and individual activities to facilitate learning in MCV 4U. The following mathematical processes will form the heart of the teaching and learning strategies used:

Dynamic teaching: Students will have access to materials and support from the teacher when needed. Dynamic examples will be used to help the students learn to expand their knowledge and connect examples to real life. Different types of activities will be used to help the students learn where they struggle and excel.

Communication: Teacher support will allow students to engage on a need basis and provide more individual support if needed. Student will be able to discuss their ideas and thoughts in a variety of methods such as virtual office hours, discussions board and many more example.

Reflection: A variety of tools such as exit cards, formative quizzes, and other activities will allow students to engage and self reflect on their skills and identify their own weakness and strength to further improve and adapt their knowledge in their math journey. Student will also engage in constant feedback between peers and teachers to be able to further enhance their skills thought free flowing conversation and interactive class examples.

Technology:  The course utilizes a variety of many tools that will allow students to learn and develop technological prowess to solve example and problems through a variety of means. Student will learn how to use graphing software and utilize technology such as drop boxes and other mean to help prepare for the future.

Accessibility : Student will be able to engage the course at their own pace and be provided support by the teacher as they see fit. Students will be able to hand in materials and do online assessment based on their own convince.