MCV4U - Calculus and Vectors

COURSE OUTLINE

Course Title: Calculus and Vectors
Course Code: MCV4U
Grade: 12
Course Type: University Preparation
Credit Value: 1
Prerequisite: MHF4U (Note: MHF4U may be take concurrently)
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2007 (Revised)
Department: Mathematics
Course Developer: Virtual High School (Ontario)
Development Date: 2008
Revision Date: 2009

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Course Description:

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.


Unit

Titles and Descriptions

Time and Sequence

Part One

The Geometry and Algebra of Vectors



Unit 1

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 2

Linear Dependence and Coplanarity

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. Students investigate the concepts of linear dependence and independence, and collinearity and coplanarity of vectors.

10 hours

Unit 3

Vector Applications

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.

10 hours

Unit 4

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.

12 hours

Part Two

Calculus and Rates of Change



Unit 5

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.

12 hours

Unit 6

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.

13 hours
Unit 7

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.

12 hours
Unit 8

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.

9 hours
Unit 9

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.

10 hours
Unit 9

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.

10 hours
 

Final Evaluation

2 hours

 

Total

110 hours

Teaching / Learning Strategies:

Students will follow a similar pattern of instructions in all units. To begin students will be involved in the exploration of an investigation of a concept. Then they will apply what they have learned in several real life scenarios or applications of the concept. Students will see solutions to applications after they try to solve them for themselves. Then students will complete assignments where no solutions are provided and submit these for assessment. Finally the unit ends with a test. Since the over-riding aim of this course is to help students use the language of mathematics skillfully, confidently and flexibly, a wide variety of instructional strategies are used to provide learning opportunities to accommodate a variety of learning styles, interests and ability levels.

Seven mathematical processes will form the heart of the teaching and learning strategies used.

Communicating: To improve student success there will be several opportunities for students to share their understanding both in oral as well as written form.

Problem solving: Scaffolding of knowledge, detecting patterns, making and justifying conjectures, guiding students as they apply their chosen strategy, directing students to use multiple strategies to solve the same problem, when appropriate, recognizing, encouraging, and applauding perseverance, discussing the relative merits of different strategies for specific types of problems.

Reasoning and proving: Asking questions that get students to hypothesize, providing students with one or more numerical examples that parallel these with the generalization and describing their thinking in more detail.

Reflecting: Modeling the reflective process, asking students how they know.

Selecting Tools and Computational Strategies: Modeling the use of tools and having students use technology to help solve problems.

Connecting: Activating prior knowledge when introducing a new concept in order to make a smooth connection between previous learning and new concepts, and introducing skills in context to make connections between particular manipulations and problems that require them.

Representing: Modeling various ways to demonstrate understanding, posing questions that require students to use different representations as they are working at each level of conceptual development - concrete, visual or symbolic, allowing individual students the time they need to solidify their understanding at each conceptual stage.

Other strategies used include; Guided Exploration, Problem Solving, Graphing, Visuals, Direct Instruction, Independent Reading, Independent Study, Ideal Problem Solving, Model analysis, Logical Mathematical Intelligence, Graphing Applications, and Problem Posing.

Assessment and Evaluation Strategies of Student Performance:

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.

Overall Expectations: MCV4U

RATE OF CHANGE
Overall Expectations
110.050.01.01 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;
110.050.01.02 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;
110.050.01.03 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.
DERIVATIVES AND THEIR APPLICATIONS
Overall Expectations
110.050.02.01 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;
110.050.02.02 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.
GEOMETRY AND ALGEBRA OF VECTORS
Overall Expectations
110.050.03.01 demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;
110.050.03.02 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;
110.050.03.03 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;
110.050.03.04 represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.

The Final Grade:

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. 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 achievement.


  • 30% of the grade will be based on a final evaluation administered at the end of the course. This final evaluation will be based on an evaluation of achievement from all four categories of the Achievement Chart for the course and of expectations from all units of the course.

The Report Card:

The report card will focus on two distinct but related aspects of student achievement; the achievement of curriculum expectations and the development of learning skills. The report card will contain separate sections for the reporting of these two aspects.

A Summary Description of Achievement in Each Percentage Grade Range
and Corresponding Level of Achievement

Percentage Grade Range

Achievement Level

Summary Description

80-100%

Level 4

A very high to outstanding level of achievement. Achievement is above the provincial standard.

70-79%

Level 3

A high level of achievement. Achievement is at the provincial standard.

60-69%

Level 2

A moderate level of achievement. Achievement is below, but approaching, the provincial standard.

50-59%

Level 1

A passable level of achievement. Achievement is below the provincial standard.

below 50%

Level R

Insufficient achievement of curriculum expectations. A credit will not be granted.

Achievement Chart: Mathematics, Grades 9-12

Categories 50-59%
(Level 1)
60-69%
(Level 2)
70-79%
(Level 3)
80-100%
(Level 4)
Knowledge and Understanding - Subject-specific content acquired in each course (knowledge), and the comprehension of its meaning and significance (understanding)

The student:
Knowledge of content (e.g., facts, terms, definitions) demonstrates limited knowledge of content demonstrates some knowledge of content demonstrates considerable knowledge of content demonstrates thorough knowledge of content
Understanding of content (e.g., concepts, ideas, theories, procedures, processes, methodologies, and/or technologies) demonstrates limited understanding of content demonstrates some understanding of content demonstrates considerable understanding of content demonstrates thorough and insightful understanding of content
Thinking - The use of critical and creative thinking skills and/or processes

The student:
Use of planning skills (e.g., focusing research, gathering information, organizing an inquiry, asking questions, setting goals) uses planning skills with limited effectiveness uses planning skills with moderate effectiveness uses planning skills with considerable effectiveness uses planning skills with a high degree of effectiveness
Use of processing skills (e.g., inquiry process, problem-solving process, decision-making process, research process) uses processing skills with limited effectiveness uses processing skills with some effectiveness uses processing skills with considerable effectiveness uses processing skills with a high degree of effectiveness
Use of critical/creative thinking processes (e.g., oral discourse, research, critical analysis, critical literacy, metacognition, creative process) uses critical / creative thinking processes with limited effectiveness uses critical / creative thinking processes with some effectiveness uses critical / creative thinking processes with considerable effectiveness uses critical / creative thinking processes with a high degree of effectiveness
Communication - The conveying of meaning through various forms

The student:
Expression and organization of ideas and information (e.g., clear expression, logical organization) in oral, graphic, and written forms, including media forms expresses and organizes ideas and information with limited effectiveness expresses and organizes ideas and information with some effectiveness expresses and organizes ideas and information with considerable effectiveness expresses and organizes ideas and information with a high degree of effectiveness
Communication for different audiences (e.g., peers, adults) and purposes (e.g., to inform, to persuade) in oral, written, and visual forms communicates for different audiences and purposes with limited effectiveness communicates for different audiences and purposes with some effectiveness communicates for different audiences and purposes with considerable effectiveness communicates for different audiences and purposes with a high degree of effectiveness
Use of conventions (e.g., conventions of form, map conventions), vocabulary, and terminology of the discipline in oral, written, and visual forms uses conventions, vocabulary, and terminology of the discipline with limited effectiveness uses conventions, vocabulary, and terminology of the discipline with some effectiveness uses conventions, vocabulary, and terminology of the discipline with considerable effectiveness uses conventions, vocabulary, and terminology of the discipline with a high degree of effectiveness
Application - The use of knowledge and skills to make connections within and between various contexts

The student:
Application of knowledge and skills (e.g., concepts, procedures, processes, and/or technologies) in familiar contexts applies knowledge and skills in familiar contexts with limited effectiveness applies knowledge and skills in familiar contexts with some effectiveness applies knowledge and skills in familiar contexts with considerable effectiveness applies knowledge and skills in familiar contexts with a high degree of effectiveness
Transfer of knowledge and skills (e.g., concepts, procedures, methodologies, technologies) to new contexts transfers knowledge and skills to new contexts with limited effectiveness transfers knowledge and skills to new contexts with some effectiveness transfers knowledge and skills to new contexts with considerable effectiveness transfers knowledge and skills to new contexts with a high degree of effectiveness
Making connections within and between various contexts (e.g., past, present, and future; environmental; social; cultural; spatial; personal; multidisciplinary) makes connections within and between various contexts with limited effectiveness makes connections within and between various contexts with some effectiveness makes connections within and between various contexts with considerable effectiveness makes connections within and between various contexts with a high degree of effectiveness

Resources required by the student:

  • MCV4U online course of study
  • Access to a scanner or digital camera

Reference Texts:

Note: This course is entirely online and does not require or rely on any textbook. Should students wish to seek additional information we would recommend these texts:

  • Calculus and Vectors 12, McGraw-Hill Ryerson, 2008.
  • Calculus and Vectors, Nelson Education Ltd., 2009.

Program Planning Considerations for Mathematics:

Teachers who are planning a program in Mathematics must take into account considerations in a number of important areas. Essential information that pertains to all disciplines is provided in the companion piece to this document, The Ontario Curriculum, Grades 9 to 12: Program Planning and Assessment, 2005 (Revised). The areas of concern to all teachers that are outlined there include the following:

  • Teaching Approaches
  • Planning Mathematics Programs for Exceptional Students
  • Program Considerations for English Language Learners
  • Antidiscrimination Education in Mathematics
  • Literacy and Inquiry/Research Skills
  • The Role of Information and Communication Technology in Mathematics
  • Career Education in Mathematics
  • The Ontario Skills Passport and Essential Skills
  • Cooperative Education and Other Forms of Experiental Learning
  • Health and Safety in Mathematics

Considerations relating to the areas listed above that have particular relevance for teachers planning programs in Mathematics:

Teaching Approaches. To make learning accessible to students, teachers must draw upon the prior knowledge and skills possessed by students. Students must have a solid conceptual foundation in mathematics. Students must be provided with the opportunity to learn the expectations of their mathematical curriculum in diverse ways. Teachers should make use of manipulatives in their teaching of mathematics which allow students to represent abstract ideas of math in concrete ways. Teachers will provide a rich math curriculum which will allow students to investigate and identify thus gaining experience with applications of the new math curriculum. Teachers need to promote attitudes conducive to the learning of math by showing students multiple ways of solving problems so that they gain confidence in problem solving.

Planning Mathematics Programs Education for Exceptional Students. In planning courses in Mathematics, teachers should take into account the needs of exceptional students as set out in their Individual Education Plan. All Mathematics courses reflect the real world very closely, which offers a vast array of opportunities for exceptional students. Students who use alternative techniques for communication may find a venue for their talents in this online mathematics course as they go about researching the nature of their world. Both environmental and assessment accommodations will be made for students with identified needs. However, the student must still participate in the regular course curriculum and the assessment and evaluation of the student's achievement will be based on the appropriate course curriculum expectations and achievement levels indicated in this course outline. If exceptional students require modified curriculum expectations in this mathematics course, then these must be indicated in the student's Individual Education Plan (IEP Standards, 2000). The assessment and evaluation of the student's achievement of these identified modified expectations will proceed based upon the achievement levels put forward in this course outline.

Program Considerations for English Language Learners. This Mathematics course can provide a wide range of options to address the needs of ESL/ELD students. Assessment and evaluation exercises will help ESL students in mastering the English language and all of its idiosyncrasies. In addition, since all occupations require employees with a wide range of English skills and abilities, many students will learn how the operation of their own physical world can contribute to their success in their social world. Assessment and evaluation accommodations, as well as other program accommodations can and will be made to facilitate the success of the ESL or ELD students.

Antidiscrimination Education in Mathematics. The mathematics curriculum in this online course attempts to be unbiased with respect to culture, experiences, interests and learning styles. Where possible, the content should reflect a diverse range of cultures and backgrounds. Attempts are made to make to embrace all students in the learning process. In addition, girls are encouraged to consider careers involving mathematics and boys are encouraged to become involved in the learning of mathematics through the inclusion of content that would be of high interest for boys.Connecting the world of math to real world situations should motivate students to learn.

Literacy and Inquiry/Research Skills. Communication skills are fundamental to the development of mathematical literacy. Fostering students' communication skills is an important part of the teacher's role in the math curriculum. When reading in mathematics, students use a different set of skills than they do when reading fiction or general non-fiction. They need to understand vocabulary and terminology that are unique to mathematics, and must be able to interpret symbols, charts, diagrams, and graphs. In all math courses, students are expected to use appropriate and correct terminology, and are encouraged to use language with care and precision in order to communicate effectively. Math courses also encourage students to communicate with precision in order to communicate effectively. Students are encouraged throughout their online mathematics course to ask questions to their peers and teacher and, as well, to become proactive in the solving of their own questions through investigations.

The Role of Information and Communication Technology in Mathematics. Information and communication technology (ICT) is considered a learning tool that must be accessed by Mathematics students when the situation is appropriate. As a result, students will develop transferable skills through their experience with word processing, internet research, presentation software, and equation editors as might be expected in any environment. By using ICT tools, the students will be able to reduce the time required to perform mundane or repetitive tasks thus creating more time to be spent on higher order tasks such as thinking or concept development. The nature of the online course itself, with students enrolled from all over the world, brings the global community into the classroom.

Career Education in Mathematics. Mathematics definitely helps prepare students for employment in a huge number of diverse areas - Engineering, Science, Business, etc. The skills, knowledge and creativity that students acquire through this course are essential for a wide range of careers. Being able to express oneself in a clear concise manner without ambiguity, solve problems, make connections between this Mathematics course and the larger world, etc., would be an overall intention of this Mathematics course, as it helps students prepare for success in their working lives.

The Ontario Skills Passport and Essential Skills. Teachers planning Mathematics programs are aware of the purpose and benefits of the Ontario Skills Passport (OSP). The OSP is a bilingual, web-based resource that enhances the relevance of classroom learning for students and strengthens school-work connections. The OSP provides clear descriptions of Essential Skills such as Reading Text, Writing, Computer Use, Measurement and Calculation, and Problem Solving and includes an extensive database of occupation-specific workplace tasks that illustrate how workers use these skills on the job. The Essential Skills are transferable, in that they are used in virtually all occupations. For further information on the OSP and the Essential Skills, visit Ontario Skills Passport.

Cooperative Education and Other Forms of Experiental Learning. By applying the skills they have developed, students will readily connect their classroom learning to real-life activities in the world in which they live. Cooperative education and other workplace experiences will broaden their knowledge of employment opportunities in a wide range of fields.

Health and Safety in Mathematics. The Mathematics program provides the reading and analytical skills for the student to be able to explore the variety of concepts relating to health and safety in the workplace. Teachers who provide support for students in workplace learning placements need to assess placements for safety and ensure that students can read and understand the importance of issues relating to health and safety in the workplace.

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