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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)
Text: McGraw-Hill Ryerson, McGraw-Hill Ryerson Calculus and Vectors 12,
© 2008
ISBN-13: 9780070126596
Department: Mathematics
Course Developer: Jenny Li
Development Date: May 2013
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.
Overall Expectations: MCV4U
By the end
of this course, students will:
RATE OF CHANGE
|
Overall Expectations
|
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;
|
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;
|
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
|
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;
|
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
|
demonstrate an understanding of
vectors in two-space and three-space by representing them algebraically and
geometrically and by recognizing their applications;
|
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;
|
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;
|
represent lines and planes using
scalar, vector, and parametric equations, and solve problems involving
distances and intersections.
|
Unit
details:
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 10
|
Trig Differentiation and
Application
Determine, through investigation using technology, the
graph of the derivative and of a given
sinusoidal function. 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
combinations of functions. 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
|
10 hours
|
|
Final Evaluation
The final assessment task is a
proctored two hour exam worth 20% of the student's final mark.
|
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 or other suitable assessment of learning such as
projects. 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 the process of gathering information that
accurately reflects how well a student is achieving the curriculum expectations
in a subject or course. The primary purpose of assessment is to improve student
learning. Assessment for the purpose of improving student learning is seen as
both “assessment for learning” and “assessment as learning”. As
part of assessment for learning, teachers provide students with
descriptive feedback and coaching for improvement. Teachers engage in
assessment as learning by helping all students develop their capacity to
be independent, autonomous learners who are able to set individual goals,
monitor their own progress, determine next steps, and reflect on their thinking
and learning.
Teachers will obtain assessment information through a variety of
means, which may include formal and informal observations, discussions,
learning conversations, questioning, conferences, homework, tasks done in
groups, demonstrations, projects, portfolios, developmental continua,
performances, peer and self-assessments, self-reflections, essays, and tests.
As essential steps in assessment for learning and as learning,
teachers need to:
• plan assessment concurrently and integrate it seamlessly with
instruction;
• share learning goals and success criteria with students at the
outset of learning to ensure that students and teachers have a common and
shared understanding of these goals and criteria as learning progresses;
• gather information about student learning before, during, and
at or near the end of a period of instruction, using a variety of assessment
strategies and tools;
• use assessment to inform instruction, guide next steps, and
help students monitor their progress towards achieving their learning goals;
• analyse and interpret evidence of learning;
• give and receive specific and timely descriptive feedback
about student learning;
• help students to develop skills of peer and self-assessment.
Teachers will also ensure that they assess students’ development
of learning skills and work habits, using the assessment approaches described
above to gather information and provide feedback to students.
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 and assessments of learning 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. All assessments of learning will be based on evaluations
developed from the four categories of the Achievement Chart for the
course.
- 30% of the
grade will be based on a final evaluation administered at the end of the
course and may be comprised of one or more strategies including tests and
projects.. This final evaluation will be based on an evaluation developed
from all four categories of the Achievement Chart for the course and of
expectations from all units of the course. The weighting of the four
categories of the Achievement Chart for the entire course including the
final evaluation will be as follows.
Knowledge
& Understanding
|
Thinking,
Inquiry & Problem Solving
|
Application
|
Communication
|
30%
|
25%
|
25%
|
20%
|
Evaluation:
l 70% for assessment of learning throughout the course
ü 5 Tests: 35%=5 * 7%
ü 2 Assignments: 14 %=2 * 7%
ü 3 Projects: 21% = 3* 7%
l 30% for final evaluations conducted near/at the end of the course
ü Project= 10%
ü Final exam= 20%
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.
|
Program
Planning Considerations for Mathematics:
Teachers
who are planning a program in Mathematics must take into account considerations
in a number of important areas.
The
areas of concern to all teachers that are outlined include the following:
- Teaching
Approaches
- Program
Considerations for English Language Learners
- Literacy
and Inquiry/Research Skills
- The Role of
Information and Communication Technology in Mathematics
- Career
Education 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.
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.
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.
Resources:
McGraw-Hill
Ryerson, McGraw-Hill Ryerson Calculus
and Vectors 12, © 2008
ISBN-13: 9780070126596
Nelson
Education Ltd., Nelson Calculus and
Vectors, © 2009
ISBN-13: 9780176374440
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