Course
Overview
Functions,
Grade 11, University/College Preparation, MCF3M
Prerequisite: Principles of Mathematics, Grade 10, Academic
Course Description
This
course extends student experiences with functions and trigonometry and
introduces some financial applications of mathematics. Many of the expectations
of this course are based on direct extensions of concepts introduced in Grades
9 and 10. Having previously explored linear and quadratic relationships,
students study various polynomial and rational functions, and investigate the
relationship of functions and their inverses. Students not only consolidate
their previous study of trigonometry but also discover new properties and
contexts to which they can be applied. Prior graphing and algebraic skills are
consolidated and extended in this course. Identifying connections between the
algebraic and graphical representations of functions continues to be an
important skill.
Successful
completion of MCF3M Functions will prepare students for the two Grade 12
University Preparation courses (Advanced Functions and Introductory Calculus,
MCB4U and Mathematics of Data Management, MDM4U) and for the two Grade 12
College Preparation courses (Mathematics for College Technology, MCT4C and
College and Apprenticeship Mathematics, MAP4C).
The
majority of university-bound students and students planning to study technology
or apprenticeship programs at college are expected to take the MCF3M Functions
course. In the delivery of the program emphasis must be placed on helping the
students to build solid foundations so that they will keep open doors to their
own futures. Emphasis must also be placed on:
·
practice
and consolidation of skills;
·
applications that address a broad scope of scenarios;
·
diagnosis, identification, and remediation of skill
weaknesses;
·
reflection on and summary of new learning;
·
explorations and activities that help students become
more independent problem solvers.
Because of the intended destination of students
enrolled in MCF3M, the contextual examples and activities should be drawn from
a wide variety of areas with minimal emphasis on areas that are generally
mathematically intensive (e.g., engineering, computer science, pure
mathematics, etc.).
In the Financial Applications of Sequences and Series
strand, students acquire the tools required to make sound personal financial
decisions. Students investigate and solve problems involving applications of
sequences related to compound interest, annuities, and financial
decision-making. In the Trigonometric Functions strand, students investigate
and apply properties of the primary trigonometric functions and develop a
competency for the manipulation of these functions. This strand has been
divided into two units: Trigonometry (including radian measure and the sine and
cosine laws) and Applied Trigonometry (with emphasis on the study of sinusoidal
functions). The Tools for Operating and Communicating with Functions strand,
which has been divided into two units: Algebraic Manipulation of Functions and
Function Notation, Inverses and Transformations. This allows students to
develop skills in operating with various algebraic expressions and to develop
facility in using function notation and in communicating reasoning.
An emphasis on technology allows students to
investigate efficiently and effectively. Appropriate technology enables
students to more easily visualize concepts and also allows for more time for
consolidation, practice and the necessary remediation of skills.
How This Course Supports
the Ontario Catholic School
This
course encourages the Catholic learner to develop his/her God-given gifts and
abilities to promote growth toward personal responsibility in preparation for a
chosen career path. Throughout this course, emphasis should be placed on moral,
ethical, and realistic decision-making in an effort to build responsible citizenship.
The classroom environment should instill a spirit of cooperation, rather than
competition amongst students, and foster a collaborative sense of community.
This course provides many opportunities for students to work effectively as
interdependent team members and to acknowledge and respect others for their
opinions.
Course Notes
This course profile builds on previous mathematics
course profiles written for Grades 9 and 10. The Grade 11 course profiles
produced by the Catholic and Public systems represent a collaborative effort
between the two writing teams. Due to the common core of learning expectations
with the MCR3U course, a common unit breakdown has been suggested, and four
different sample units have been developed. Thus, the MCR3U and MCF3M course
profiles can also be used as complementary resources. With appropriate
adjustments to the complexity of problems, timing, and the need for abstract
thinking, a teacher of either course can use activities from either profile. It
should be noted that at this level, it is appropriate in certain situations for
mathematics itself to provide the context for new concepts, while in this MCF3M
profile it would be more appropriate to provide other contexts as well. The
sample units provided in this (MCF3M) profile are Function Notation, Inverses
and Transformations and Financial Applications of Sequences and Series while
the sample units provided in the MCR3U profile are Algebraic Manipulations of
Functions and Applied Trigonometry. In addition to these four complete model
units, a less detailed Unit Overview Chart offers a recommended clustering of
expectations for the remaining units, which can provide a starting point from
which teachers can develop their own, individualized units.
The recommended sequence of units in this course
begins with Exploring Functions: Connecting Algebra and Geometry, which
continues the investigative approach to relationships undertaken in Grades 9
and 10, and lays the groundwork for a deeper algebraic and geometric
understanding of functions, which is the fundamental theme of the course.
Activity 1.4 of the detailed unit in the MCR3U Course Profile has been removed
from this profile since it is based on an expectation exclusive to the MCR3U
course. The time allotted for this activity has been realigned to two other
activities. In order to provide a more contextual framework and then a need for
skill development, Activity 3.9 for the MPM2D Public Course Profile can be used
as a springboard for Activity 1.1 of this Course Profile. Similarly, Activity
1.7 of the MPM1D Public Course Profile can be used to provide a contextual
framework for Activity 1.6 of this Course Profile. This unit is followed by the
Function Notation, Inverses and Transformations unit, which includes a number
of important concepts consolidated and extended in the later two Trigonometry
units. The Financial Applications of Sequences and Series unit, which follows,
will also use concepts from the previous units.
For some students, mathematics is perceived to be a
collection of isolated and complex topics, each requiring skills that may soon
be forgotten. The mathematics teacher must address these perceptions by
creating a context in which students can learn and connect concepts and skills.
Students must be exposed to a variety of teaching, learning, and problem
solving techniques to best synthesize the information presented by the
curriculum, and should be provided applications and context to bring meaning to
their learning.
The activities in this profile are designed to both
introduce and consolidate skills necessary for success in this course and the
appropriate destination math courses. These activities can be used in
conjunction with or independently of one another. Alternate teaching strategies
and suggestions for technological tools are included to help teachers present
the lessons contained in the activities.
Since this
course has been designed to prepare students for further studies at the
university or college level, which requires a substantial level of expertise in
mathematics, the specific nature of the learning activities should reflect
these destinations. In particular, students in this course should routinely be
challenged with investigations and problems that require a high degree of
independent effort. Students destined for university or college should have the
opportunity to develop and demonstrate their problem-solving abilities.
Students with learning disabilities will need specific
guidance in order to benefit from the investigative approach presented in this
profile. Review of prerequisite skills and instructions in the use of
technology, and in particular graphing calculators, will be necessary before
commencing activities in this profile. Clear and precise instructions with
examples will need to be provided.
The Achievement Chart for Mathematics is the basis of
assessment and evaluation for this course. In the Principles of Mathematics -
Academic (Grade 10) Course Profile (p.11) includes charts suggesting strategies
that can be used for the assessment and evaluation of all categories of the
Achievement Chart. In addition, a chart outlining the component actions that
are needed for successful inquiry and problem solving in particular is also
included (p.12). These charts provide an excellent base with which to begin the
implementation of these strategies and for teachers of this course to extend
depending on their degree of readiness. Another excellent resource is the
Concerning Assessment and Reflective Evaluation (CARE) package of materials,
available for free download at - http://www.oame.on.ca. Among the resources
included in the package are generic rubrics for Communication and Thinking,
Inquiry and Problem Solving skills, along with suggested applications of these
instruments.
Units: Titles and Time
|
Unit 1 |
Exploring
Functions: Connecting Algebra and Geometry |
20 hours |
|
* Unit 2 |
Function
Notation, Inverses and Transformations |
20 hours |
|
* Unit 3 |
Financial
Applications of Sequences and Series |
25 hours |
|
Unit 4 |
Trigonometry |
15 hours |
|
Unit 5 |
Trigonometric
Functions |
20 hours |
|
Unit 6 |
Final Summative
Assessment |
10 hours |
* These units are fully developed in this Course Profile.
Unit Overviews
Unit 1: Exploring Functions: Connecting Algebra and
Geometry
Time: 20 hours
Unit
Description
Students investigate quadratic functions and related
concepts from algebraic and geometric perspectives, in order to deepen their
understanding and prepare them for further explorations of functions. A winter
recreation theme is loosely woven throughout selected activities in the unit,
providing a contextual framework for students to solve problems, both with and
without the use of graphing technology. Students solve first-degree
inequalities and graph their solutions on number lines. Skills involving
operations with polynomials and rational expressions are consolidated, and then
extended to the complex number system, which is introduced in this unit.
Students apply the method of completing the square in order to solve
maximum/minimum problems involving quadratic functions. Algebraic and graphical
methods are used in to determine the roots of quadratic equations. The exponent
laws are applied to expressions, which have powers containing integer and
rational exponents. Students discover the nature of exponential functions and
solve exponential equations.
Unit Overview Chart
|
Activity |
Time |
Expectations |
Assessment |
Tasks |
|
1.1 A |
150 min |
OCV.01, OCV.03,
OC1.01, OC1.02, OC3.04, OC3.05 |
Knowledge
Communication |
Solve and graph
inequalities; Perform operations with polynomials |
|
1.2 Ski-Jumping
to the Max! |
150 min |
OCV.01, OCV.03,
OC1.03, OC1.05, OC3.01, OC3.05 |
Knowledge Inquiry Communication
Application |
Complete the
square; Investigate the graphs of quadratic functions |
|
1.3 Rooting
Around the Parabola |
150 min |
OCV.01, OCV.03,
OC1.04, OC1.05, OC3.02, OC3.05 |
Knowledge Inquiry Communication
Application |
Determine
real/complex roots of quadratic equations; Relate roots to x-intercepts
of quadratic functions |
|
1.4 Can We
Please be Rational?! |
150 min |
OCV.01, OCV.03,
OC1.06, OC3.01, OC3.03, OC3.04, OC3.05 |
Knowledge
Communication |
Perform
algebraic operations on rational expressions; State restrictions on the
variables |
|
1.5 Power Play |
150 min |
OCV.01, OCV.03,
OC1.07, OC3.01, OC3.05 |
Inquiry Communication
Application |
Explore powers with
rational exponents; Apply exponent laws to powers containing integer and
rational exponents |
|
1.6 Its Snowing
Cats and Dogs! |
150 min |
OCV.01, OCV.03,
OC1.08, OC.3.03, OC3.05 |
Knowledge Application |
Explore exponential
relations; Solve exponential equations |
|
1.7 Summative
Assessment |
75 min |
All Overall
Expectations within unit |
Knowledge Inquiry Communication
Application |
Summative
assessment |
Note: Of the 20 hours in this unit, 3.75 hours has
been allotted for remediation and consolidation of skills to be used at the
discretion of the teacher for needs of the students.
Unit 2: Function Notation, Inverses and
Transformations
Time: 20 hours
Unit
Description
Students, through authentic models, are introduced to
the definition of a function and the associated notation. Students use graphing
technology and paper-and-pencil tasks to investigate the properties of
functions, their inverses and transformations of functions. The investigations
are used to introduce and extend the use of function notation to inverses and
transformations. Students explore the domain and range of functions, inverses
and transformations.
Unit Overview Chart
|
Activity |
Time |
Expectations |
Assessment |
Tasks |
|
2.1. Wrap Around Follow up Skills |
150 min 75 min |
OCV.02, OCV.03,
OC2.01, OC2.02, OC3.01, OC3.02, OC3.04, CGE2c, CGE3c, CGE4f, CGE5e, CGE5g |
Knowledge
Inquiry Application Communication |
Investigate the concept
of function; formalize the definition of function; apply function notation |
|
2.2. Home on the
Range Follow up Skills |
150 min 75 min |
OCV.02, OCV.03,
OC2.02, OC2.03, OC3.03, OC3.05, CGE4f, CGE5a |
Knowledge
Communication Application |
Explore through
the use of graphing technology the properties of various functions.
Investigate domain and range |
|
2.3. Follow The
Bouncing Ball |
105 min |
OCV.02, OCV.03,
OC2.02, OC3.03, OC3.04, OC3.05, CGE2b, CGE3c, CGE5a |
Application Knowledge
Communication |
Develop a model
with the use of graphing technology; apply appropriate function notation with
the model; investigate properties of the function/model |
|
2.4. Lets
Switch Seats! Follow up Skills |
105 min 75 min |
OCV.02, OCV.03,
OC2.04, OC3.03, OC3.05, CGE3c, CGE4f |
Application
Inquiry |
Investigate the
properties of inverse functions; discover algebraic approaches for finding
the inverse of functions. |
|
2.5. On the Move |
150 min |
OCV.02, OCV.03, OC2.06,
OC2.07, OC2.08, OC3.02, OC3.04, CGE2c, CGE5a |
Knowledge
Application Communication |
Investigate the
effect of transformations on mathematical functions; apply appropriate
function notation to transformation(s) of functions |
|
2.6. Be My Valentine |
105 min |
OCV.02, OCV.03,
OC2.06, OC2.07, OC2.08, OC3.01, OC3.05, CGE2b, CGE2f |
Knowledge
Application Communication Inquiry |
Apply and
analyse transformations and functions |
|
2.7.
Consolidating and Connecting |
105 min |
OCV.01, OCV.02,
OCV.03, OC1.03, OC2.02, OC2.06, OC2.07, OC3.02, OC3.03, CGE3c, CGE4f, CGE5a |
Knowledge
Communication Application Inquiry |
Summative
assessment |
Note: Suggested Follow-up Skills and Extension Activities
are listed in the activities that can be used to supplement the activities.
Along with these, the teacher can use appropriate remediation and consolidation
of skills to account for the total of 20 hours in the unit.
Unit 3: Financial Applications of Sequences and
Series
Time: 25 hours
Unit
Description
Students investigate arithmetic and geometric
sequences and series. This knowledge serves as the basis for applications of
personal finance. Students develop the formula for compound interest and solve
problems related to compound interest and annuities. As skills are developed,
students use spreadsheets to investigate the cost of borrowing when interest
rates, compound periods, lending terms, etc., are varied. The activities are
designed to reflect the type of decisions that students are likely to face in
the future. Students apply skills with linear and exponential functions.
Unit
Overview Chart
|
Activity |
Time |
Expectations |
Assessment |
Tasks |
|
3.1
Investigating Attributes of Sequences |
150 min |
FAV.01, FA1.01,
FA1.02 |
Inquiry,
Communication |
Investigate
sequences |
|
3.2 Summing Up: Arithmetic
Sequences and Series |
150 min |
FAV.01, FAV.02,
FA1.04, FA1.05, FA2.04 |
Knowledge,
Application, Inquiry |
Develop formula
for arithmetic sequences and series |
|
3.3 Compound
Interest: Exploring Geometric Sequences |
150 min |
FAV.01, FAV.02, FA1.03,
FA1.04, FA2.01, FA2.02, FA2.05 |
Knowledge,
Application, Communication |
Use a financial
application to investigate geometric sequences. Develop the compound interest
formula |
|
3.4 Applications:
Finding the Amount and the Present Value of a Long Term Investment |
75 min |
FAV.02, FAV.03,
FA2.02, FA3.01, FA3.02 |
Knowledge,
Application |
Extend knowledge
of the compound interest formula and introduce financial applications of a
graphing calculator |
|
3.5 Introduction
to Geometric Series |
75 min |
FAV.01, FA1.05 |
Knowledge,
Application |
Develop the
formula for the sum of a geometric series |
|
3.6
Applications: Finding the Amount and the Present Value of an Annuity |
150 min |
FAV.02, FA2.02,
FA2.03, FA3.05 |
Inquiry,
Communication |
Apply geometric
sequences and series to finding the amount and present value of an annuity |
|
3.7 What Happens
When? : Changing the Time, Rate and Amount |
150 min |
FAV.03, FA3.01,
FA3.05 |
All |
Use financial
applications of a graphing calculator to investigate the effect of changing
conditions when borrowing and saving |
|
3.8 Mortgages: How They Work |
150 min |
FAV.03, FA3.02,
FA3.03, FA3.04, FA3.05 |
Application,
Communication |
Use technology
to generate amortization tables |
|
3.9 Financial
Decision Making: A Case Study |
300 min |
All |
All |
Summative
Assessment Activity |
Note: In order to account for the 25 hours in this unit,
2.5 hours has been allotted to practise essential skills, review and to do
other assessments.
Unit 4: Trigonometry
Time: 15 hours
Unit
Description
Students consolidate and extend triganomic concepts
first introduced in Grade 10. Students use the primary trigonometric ratios,
the sine law, and the cosine law to model and solve two- and three-dimensional
problems involving acute, right, and oblique triangles. Students investigate
the relationship between degree and radian measure, and explore the use of the
unit circle and special triangles to determine selected values of the primary
trigonometric ratios. Methods of proof are introduced and applied to verify
trigonometric identities. Students develop the skills to manipulate and solve
trigonometric equations.
Unit
Overview Chart
|
Cluster |
Expectations |
Assessment |
Focus |
|
1 |
TFV.02, TF2.01, TF2.02,
TF2.03, TF2.07 |
Knowledge Inquiry Application |
Define terms and
concepts; Convert degrees to radians; Apply radian
measure |
|
2 |
TFV.01, TF1.02,
TF2.03 |
Knowledge Inquiry Communication Application |
Review the primary
trigonometric ratios; Review the sine and cosine laws for acute triangles; Explore the sine
and cosine laws for oblique triangles; Investigate the
ambiguous case of the sine law |
|
3 |
TFV.02, TF2.03,
TF2.04 |
Knowledge Inquiry Communication Application |
Investigate
special triangles |
|
4 |
TFV.02, TF1.01,
TF2.05 |
Knowledge Inquiry Communication Application |
Derive the unit
circle; Use the
Pythagorean theorem to prove identities; Discuss the use
of proof |
|
5 |
TFV.02, TF2.03,
TF2.06, TF2.07 |
Knowledge Inquiry |
Solve linear and
quadratic trigonometric equations |
|
6 |
All expectations
within unit |
Knowledge Inquiry Communication Application |
Summative
assessment |
Unit 5: Trigonometric Functions
Time: 20 hours
Unit
Description
Students investigate
the periodic nature and graphical properties of the primary trigonometric
functions. Using technology, students explore the effects of simple
transformations on their graphs and equations. Students apply these concepts to
model authentic problems.
Unit
Overview Chart
|
Activity |
Time |
Expectations |
Assessment |
Tasks |
|
5.1 Surfs Up!
Lets Catch the Sine Wave |
75 min |
TFV.02, TFV.03,
TF2.07, TF3.01, CGE3c |
Knowledge |
Derive y
= sin x and y = cos x from the unit circle |
|
5.2
Transformations: More Than Meets the Eye Follow-up Skills |
75 min 30 min |
TFV.02, TFV.03,
OCV.02, TF2.07, TF3.01, TF3.02, TF3.03, TF3.04, TF3.05, OC2.06, OC2.07,
OC2.08, CGE3c, CGE5a, CGE5e, CGE5f |
Knowledge
Inquiry Communication |
Investigate the
properties of various transformations of |
|
Give Me a Sine Follow-up Skills |
75 min 45 min |
|||
|
5.3 Dont Go Off
On A Tangent Follow-up Skills |
75 min 30 min |
TFV.03, TF3.01,
TF3.06, CGE3c, CGE 4b, CGE 4f |
Knowledge
Inquiry Communication |
Investigate the
properties of |
|
5.4a Its a
Spring Thing |
150 min |
TFV.03, TFV.04,
OCV.02, TF3.02, TF3.03, TF3.04, TF3.05, TF4.01, TF4.02, OC2.06, OC2.08,
CGE2c, CGE3c, CGE4b, CGE5a, CGE5g |
Knowledge
Inquiry Communication Application |
Model the motion
of a mass on a spring |
|
5.4b Ferris Fair
Follow-up Skills |
150 min 75 min |
Model the height
of a gondola on a Ferris wheel |
||
|
5.4c Let the
Sine Shine In Follow-up Skills |
150 min 60 min |
Model an hours
of daylight function |
||
|
5.5 Summative
Assessment |
150 min |
All expectations
within unit CGE2b, CGE3c,
CGE3e |
Knowledge
Inquiry Communication Application |
Apply concepts
in familiar and unfamiliar contexts |
Note: Additional time can be allocated for remediation
and consolidation of skills (to be used at the discretion of the teacher),
depending on the needs of the students, to account for the total of 20 hours in
the unit.
Unit 6: Final Summative Assessment
Time: 10 hours
Overall Expectations:
All
Unit
Description
Summative assessment should be designed to provide the
opportunity for students to demonstrate comprehensive learning in each of the
four achievement categories. Some ideas are suggested in the chart that
follows, however any of the various assessment tools mentioned in the
Assessment Strategies section could be used. A short paper-and-pencil task
would review key terms and concepts. Investigations comparing the buying and
leasing of a car yield a wide variety of applications pertaining to both
personal finance and the modeling of functions. An assignment exploring
trigonometric inverses (for example, the arcsine function) would serve to
review the concepts introduced in the functions and trigonometry units. This
topic also provides students with an exposure to a subject further explored in
Grade 12. These topics are suggested as one possible way to revisit the expectations
in a new mathematical context. Accordingly, students are to be assessed solely
on the expectations of this course, and not on the extension topics themselves.
Due to the particular emphasis of cumulative tests and examinations in
university and college programs, a formal examination should be a prominent
role in the final summative assessment of the student.
Unit
Overview Chart
|
Cluster |
Expectations (by |
Assessment |
Focus |
|
1 |
All Strands |
Knowledge Application |
Review Key Concepts
and Terms |
|
2 |
· Financial Applications of Sequences and Series · Tools for Operating and Communicating with Functions |
Knowledge Inquiry Communication Application |
Examine
Financial Commitments of Owning a Car: Buying vs. Leasing |
|
3 |
· Trigonometric Functions · Tools for Operating and Communicating with Functions |
Knowledge Inquiry Communication Application |
Explore
Trigonometric Inverses |
|
4 |
All Strands |
Knowledge Inquiry Communication Application |
Final
Examination |
Note: Time should also be spent helping students
prepare for the final assessment tasks and especially for the formal final
examination (i.e., identify areas of weakness and provide remediation in these
areas).
Teaching/Learning
Strategies
In order to address the wide range of expectations in
this course, a variety of teaching, learning, and assessment strategies and
tools need to be used. Teachers should assume a variety of roles (including
guide, facilitator, consultant, and instructor), and should employ a variety of
strategies including:
·
a balance of whole-class, small group, mixed-ability
structured group, and individual instruction through student-centred and
teacher-directed activities (group work should be carefully structured along
cooperative learning principles to be effective);
·
the use of rich contextual problems which engage
students and provide them with opportunities to demonstrate learning;
·
the prompting, supporting, and challenging of
individual students as well as the class as a whole;
·
approaches that will accommodate multiple learning
styles (e.g., the provision of verbal and written instructions, the inclusion
of hands-on activities, etc.);
·
the use of technological tools and software (e.g.,
graphing software, dynamic geometry software, the Internet, spreadsheets, and
multimedia) in activities, demonstrations, and investigations to facilitate the
exploration and understanding of mathematical concepts;
·
the use of learning/performance tasks that are
designed to link several expectations and give the students occasion to
demonstrate their optimal levels of achievement through the demonstration of
skill acquisition, the communication of results, the ability to pose extending
questions following an inquiry, and the determination of a solution to unfamiliar
problems;
·
the use of accommodations, remediation, and/or
extension activities, where necessary, to meet the needs of exceptional
students;
·
the provision of opportunities for students to
practise and extend their skills and knowledge outside of the classroom.
In addition to the contribution of the teacher,
students themselves should play an active role in their own learning. In order
to successfully complete the requirements of this course, students are expected
to:
·
develop an increased responsibility for their own
learning;
·
be accountable for pre-requisite skills;
·
participate as active learners;
·
engage in explorations using technology;
·
apply individual and group learning skills;
·
describe the mathematical patterns that emerge
verbally, algebraically, and visually in the course of learning.
Assessment
Strategies
An effective assessment program in mathematics must
include a balance of diagnostic, formative and summative assessment instruments
that incorporate the categories of learning as defined in The Achievement Chart
for Mathematics. One approach is shown below:
|
|
Knowledge/ Understanding |
Thinking/Inquiry Problem Solving |
Communication |
Application |
|
final
examinations |
|
|
|
|
|
journals |
|
|
|
|
|
observations |
|
|
|
|
|
oral
presentations |
|
|
|
|
|
performance
tasks |
|
|
|
|
|
portfolios |
|
|
|
|
|
quizzes |
|
|
|
|
|
reports/assignments |
|
|
|
|
|
student-teacher
conferences |
|
|
|
|
|
unit tests |
|
|
|
|
Note: Assessment tools such as observational
checklists, performance criteria, rubrics, The Achievement Chart for
Mathematics, marking schemes, rating scales, peer evaluation, and
self-evaluation can and should be used to assist in developing objective and
consistent evaluations of student achievement.
Assessment
& Evaluation of Student Achievement
Assessment as defined in the document Ontario
Secondary Schools, Grades 9-12: Program and Diploma Requirements, 1999,
is the process of gathering information from a variety of sources (including
assignments, demonstrations, projects, performances, and tests) that accurately
reflects how well students are achieving the curriculum expectations (p. 31).
Assessment tools should be designed to allow students to demonstrate the full
extent of their learning across the four categories. As teachers will use a
variety of assessment tools, it is necessary to ensure that a consistent
standard is maintained. Thus, these tools should be developed with the learning
expectations of the course as the criteria for this standard. That is, a grade
of 70-79% using an objective marking scheme should be equivalent to a Level 3
performance, as defined by the Achievement Chart. Teachers may find it more
appropriate to use rubrics to assess Thinking, Inquiry, Problem Solving, and
Communication skills, but to use objective scales for Knowledge, Understanding,
and Application skills. High quality assessment can measure individual and
group performance, and individual performance within a group.
The students effective demonstration of communication
skills is an essential component of this course when evaluating achievement.
Students are required to display and convey their knowledge and understanding
of concepts, share their process of thought and inquiry, and justify their
application of concepts in an unfamiliar situation. In addition, their ability
to communicate these skills is also assessed.
It should also be noted that teachers must continue to
expand their own understanding of application skills to include non-routine
problems. This view requires a shift from the specific application of
concepts (i.e., familiar situations), to the general application of
concepts (i.e., unfamiliar situations).
Assessment strategies and tools must address a wide
variety of teaching and learning styles in addition to the criteria established
by the learning expectations. Tests consisting only of questions that ask
students to perform algorithms and apply their knowledge do not necessarily
offer an opportunity for students to demonstrate Level 4 performance. Also, it
is understood that students will meet course expectations at a variety of
performance levels. An effective and well-balanced assessment program will provide
students with several opportunities to demonstrate growth and improvement over
time, across all of the knowledge and skill categories.
Evaluation, as defined by Ontario
Secondary Schools, Grades 9-12: Program and Diploma Requirements, 1999, is
the process of judging the quality of a students work on the basis of
established achievement criteria, and assigning a value to represent that
quality (p. 31). Whereas assessment is the collecting of information about
student performance in a variety of methods, evaluation is the determination of
a quantitative value describing the students overall level of achievement.
Effective assessment, evaluation, and reporting require the teacher to do more
than just average marks. While averaging may be more useful in some Knowledge
and Application skill categories, it is not comprehensive enough for accurate
reporting in the Inquiry and Communication skill categories. The use of rubrics
is a suggested technique for these categories. As students can be expected to
improve their performances over time, particular emphasis should be placed on
their most recent and most consistent level of achievement.
Students
who receive a final performance evaluation of Level 3 or better are believed to
be well prepared for work in any of the following Grade 12 mathematics courses:
Advanced Functions and Introductory Calculus (MCB4U), Mathematics of
Data Management (MDM4U), Mathematics for College Technology (MCT4C),
and College and Apprenticeship Mathematics (MAP4C). Hence students need
to be prepared for university programs that require some degree of mathematical
expertise and for college programs that require a level of mathematics
necessary for success in technology based and apprenticeship programs.
Accordingly, in order to prepare students for the academic reality of
university and college programs, proper attention should be placed on the
effective preparation for a comprehensive final examination. While other rich,
performance-based activities can and should be part of the Final Summative
Assessment unit a formal examination should play a more significant role
in this course.
Seventy per cent of the grade will be based on
assessments and evaluations conducted throughout the course. Thirty per cent of
the grade will be based on a final evaluation in the form of an examination,
performance, essay, and/or other method of evaluation.
Accommodations
For exceptional students teachers should refer to the
students Individual Education Plan (IEP) and use the recommendations to make
any necessary accommodations. Teachers should work in consultation with
resource teachers, ESL/ELD teachers, and parents or guardians to determine
appropriate accommodations as they work through the course.
Accommodations for ESL/ELD
Students
·
Have students work in pairs, with peer tutors, or with
classmates that have the same linguistic background.
·
Use peer conferencing to reinforce instructions or
information.
·
Ask an ESL/ELD teacher to review questions,
assignments, or assessment instruments.
·
Provide sets of reference notes, outlines, or critical
information, as well as models of charts, timelines or diagrams.
·
Reinforce main ideas by using the think/pair/share
peer-assessment strategy.
·
Pair written instructions with verbal instructions.
·
Use visuals to illustrate definitions for the
students dictionary of terms.
·
Simplify instructions.
·
Highlight key words or phrases.
·
Brainstorm in groups using the students first
language if English is limited.
·
Provide opportunities for students to practice oral
presentation skills.
·
Provide visual or auditory cues.
Accommodations for
Students with Learning Disabilities
·
Provide extensive student-teacher conferencing;
·
Pair students. Due care must be given in the pairing
to provide support, not solutions, for the identified student;
·
Provide a list of terms (possibly simplified) before
an activity begins;
·
Modify handouts in terms of the terminology and
content used, as well as the size and typeface of the selected font. Allow
plenty of space for written responses;
·
Allow assignments to be completed in alternate formats
or using longer timelines;
·
Keep manipulatives, grid paper, formula sheets, and
other aids available for needs that arise;
·
Contact parents or guardians for support and
suggestions;
·
Provide the students with oral pre-planning of
activities.
Resources
This Course Profile has been provided as a resource to
aid the teacher in delivering the curriculum. Through the discretionary use of
other materials, the teacher can enrich, remediate, or otherwise supplement
their students education. The following is a partial list of resources widely
available to the industrious teacher.
Software (Ministry-Licensed)
Geometers Sketchpad (dynamic
geometry)
Maple (word
processor/programming)
Mastering Calculus (concept and skill
development)
Math Trek (concept
and skill development)
Virtual Tiles (algebraic concept
and skill development)
Zap-a-Graph (graphing)
Websites
Note:
The URLs for the
websites have been verified by the writer prior to publication. Given the frequency
with which these designations change, teachers should always verify the
websites prior to assigning them for student use.
Canadian Education on the Web
http://www.oise.on.ca/~mpress/eduweb.html A compendium of Canadian
education-related resources maintained by Marian Press at the Ontario Institute
for Studies in Education/University of Toronto.
Education Network of
Hewlett-Packard http://www.hp.com/calculators/
National Council of Teachers of Mathematics
http://www.nctm.org
Books, Periodicals, etc.
CARE (Concerning Assessment and Reflective Evaluation)
Package (download from
http://www.oame.on.ca)
Stiggins, R. Classroom Assessment for Student
Success. National
Gadanidis, G. MathMania: Adventures in Mathematics.
The Mathematics Teacher.
ISSN 0025-5769
Connecting Mathematics: Addenda Series, Grades
9-12.
Burtz, H.L and K. Marshall. Performance-Based
Learning and Assessment.
O.S.S.T.F. Quality Assessment.
Taggart, G. Rubrics A Handbook for Construction
and Use.
OSS
The following list of
resources will support many of the Ontario Secondary School Policies as well as
the Ontario Catholic Secondary School Graduate Expectations:
Ministry of Education Policy and Reference Documents
·
Choices into Action: Guidance and Career
Education Program Policy
·
Cooperative Education: Policies and Procedures
for Ontario Secondary Schools
·
Individual Education Plans: Standards for
Development, Program Planning, and Implementation, 2000
·
Mathematics, Grades 9-10
·
Mathematics, Grades 11-12
·
Ontario Schools Code of Conduct
·
Ontario Secondary Schools, Grades 9 to 12:
Program and Diploma Requirements
·
Program Planning and Assessment, Grades 9-12
·
Violence-Free Schools Policy
The Ministry of Education has also published several
resource documents, brochures, and policy/program memoranda in support of its
Publications Concerning Faith Development
·
Blueprints (Catholic Curriculum Cooperative -
·
Catholicity Across The Curriculum (
·
Educating the Soul (Institute for Catholic
Education)
·
·
This Moment of Promise (
Career Goals/Cooperative Education Programs
·
·
Youth Employment Skills Program
Community Partnerships
Refer to local board policies (e.g., Relations with
Business - Corporate Donations, Sponsorships and Agreements).
Coded
Expectations, Functions, Grade 11, University/College Preparation, MCF3M
Financial
Applications of Sequences and Series
Overall
Expectations
FAV.01 · solve problems involving arithmetic and
geometric sequences and series;
FAV.02 · solve problems involving compound interest
and annuities;
FAV.03 · solve problems involving financial decision
making, using spreadsheets or other appropriate technology.
Specific
Expectations
Solving Problems Involving Arithmetic and Geometric
Sequences and Series
FA1.01 write terms of a sequence, given the formula
for the nth term;
FA1.02 determine a formula for the nth term
of a given sequence (e.g., the nth term of the sequence
is 
);
FA1.03 identify sequences as arithmetic or
geometric, or neither;
FA1.04 determine the value of any term in an
arithmetic or a geometric sequence, using the formula for the nth term of
the sequence;
FA1.05 determine the sum of the terms of an
arithmetic or a geometric series, using appropriate formulas and techniques.
Solving Problems Involving Compound Interest and
Annuities
FA2.01 derive the formulas for compound interest
and present value, the amount of an ordinary annuity, and the present value of
an ordinary annuity, using the formulas for the nth term of a geometric
sequence and the sum of the first n terms of a geometric series;
FA2.02 solve problems involving compound interest
and present value;
FA2.03 solve problems involving the amount and the
present value of an ordinary annuity;
FA2.04 demonstrate an understanding of the
relationships between simple interest, arithmetic sequences, and linear growth;
FA2.05 demonstrate an understanding of the
relationships between compound interest, geometric sequences, and exponential
growth.
Solving Problems Involving Financial Decision Making
FA3.01 analyse the effects of changing the
conditions in long-term savings plans (e.g., altering the frequency of
deposits, the amount of deposit, the interest rate, the compounding period, or
a combination of these) (Sample problem: Compare the results of making
an annual deposit of $1000 to an RRSP, beginning at age 20, with the results of
making an annual deposit of $3000, beginning at age 50);
FA3.02 describe the manner in which interest is
calculated on a mortgage (i.e., compounded semi-annually but calculated
monthly) and compare this with the method of interest compounded monthly and
calculated monthly;
FA3.03 generate amortization tables for mortgages,
using spreadsheets or other appropriate software;
FA3.04 analyse the effects of changing the
conditions of a mortgage (e.g., the effect on the length of time needed to pay
off the mortgage of changing the payment frequency or the interest rate);
FA3.05 communicate the solutions to problems and
the findings of investigations with clarity and justification.
Trigonometric
Functions
Overall
Expectations
TFV.01 · solve problems involving the sine law and
the cosine law in oblique triangles;
TFV.02 · demonstrate an understanding of the meaning
and application of radian measure;
TFV.03 · determine, through investigation, the relationships
between the graphs and the equations of sinusoidal functions;
TFV.04 · solve problems involving models of
sinusoidal functions drawn from a variety of applications.
Specific
Expectations
Solving Problems Involving the Sine Law and the Cosine
Law in Oblique Triangles
TF1.01 determine the sine, cosine, and tangent of
angles greater than 90°, using a suitable technique (e.g., related angles, the
unit circle), and determine two angles that correspond to a given single
trigonometric function value;
TF1.02 solve problems in two dimensions and three
dimensions involving right triangles and oblique triangles, using the primary
trigonometric ratios, the cosine law, and the sine law (including the ambiguous
case).
Understanding the Meaning and Application of Radian
Measure
TF2.01 define the term radian measure;
TF2.02 describe the relationship between radian
measure and degree measure;
TF2.03 represent, in applications, radian measure
in exact form as an expression involving π (e.g., 
, 2π) and in approximate form as a real number (e.g.,
1.05);
TF2.04 determine the exact values of the sine,
cosine, and tangent of the special angles 0, 
, and their multiples less than or equal to 2π;
TF2.05 prove simple identities, using the
Pythagorean identity, sin2x + cos2x = 1,
and the quotient relation, tan x = 
;
TF2.06 solve linear and quadratic trigonometric
equations (e.g., 6 cos2x sin x 4 = 0) on the
interval
0 £ x £ 2π;
TF2.07 demonstrate facility in the use of radian
measure in solving equations and in graphing.
Investigating the Relationships Between the Graphs and
the Equations of Sinusoidal Functions
TF3.01 sketch the graphs of y = sin x
and y = cos x, and describe their periodic properties;
TF3.02 determine, through investigation, using
graphing calculators or graphing software, the effect of simple transformations
(e.g., translations, reflections, stretches) on the graphs and equations
of y = sin x and y = cos x;
TF3.03 determine the amplitude, period, phase
shift, domain, and range of sinusoidal functions whose equations are given in
the form y = a sin(kx + d) + c or y =
a cos(kx + d) + c;
TF3.04 sketch the graphs of simple sinusoidal
functions [e.g., y = a sin x, y = cos kx, y
= sin(x + d),
y = a cos kx + c];
TF3.05 write the equation of a sinusoidal function,
given its graph and given its properties;
TF3.06 sketch the graph of y = tan x;
identify the period, domain, and range of the function; and explain the
occurrence of asymptotes.
Solving Problems Involving Models of Sinusoidal
Functions
TF4.01
determine, through investigation, the periodic properties of various models
(e.g., the table of values, the graph, the equation) of sinusoidal functions drawn
from a variety of applications;
TF4.02 explain
the relationship between the properties of a sinusoidal function and the
parameters of its equation, within the context of an application, and over a
restricted domain;
TF4.03 predict the effects on the mathematical
model of an application involving sinusoidal functions when the conditions in
the application are varied;
TF4.04 pose and solve problems related to models of
sinusoidal functions drawn from a variety of applications, and communicate the
solutions with clarity and justification, using appropriate mathematical forms.
Tools
for Operating and Communicating with Functions
Overall
Expectations
OCV.01 · demonstrate facility in manipulating
polynomials, rational expressions, and exponential expressions;
OCV.02 · demonstrate an understanding of inverses and
transformations of functions and facility in the use of function notation;
OCV.03 · communicate mathematical reasoning with
precision and clarity throughout the course.
Specific
Expectations
Manipulating Polynomials, Rational Expressions, and
Exponential Expressions
OC1.01 solve first-degree inequalities and
represent the solutions on number lines;
OC1.02 add, subtract, and multiply polynomials;
OC1.03 determine the maximum or minimum value of a
quadratic function whose equation is given in the form y = ax2
+ bx + c, using the algebraic method of completing the square;
OC1.04 identify the structure of the complex number
system and express complex numbers in the form
a + bi, where i2 = 1 (e.g., 4i, 3 2i);
OC1.05 determine the real or complex roots of
quadratic equations, using an appropriate method (e.g., factoring, the
quadratic formula, completing the square), and relate the roots to the x-intercepts
of the graph of the corresponding function;
OC1.06 add, subtract, multiply, and divide rational
expressions, and state the restrictions on the variable values;
OC1.07 simplify and evaluate expressions containing
integer and rational exponents, using the laws of exponents;
OC1.08 solve exponential equations (e.g., 4x
= 8x + 3, 22x 2x
= 12).
Understanding Inverses and Transformations and Using
Function Notation
OC2.01 define the term function;
OC2.02 demonstrate facility in the use of function
notation for substituting into and evaluating functions;
OC2.03 determine, through investigation, the
properties of the functions defined by f(x) = 
[e.g., domain, range, relationship to f(x) = x2]
and f(x) = 
[e.g., domain, range,
relationship to
f(x) = x];
OC2.04 explain the relationship between a function
and its inverse (i.e., symmetry of their graphs in the line y = x;
the interchange of x and y in the equation of the function; the
interchanges of the domain and range), using examples drawn from linear and
quadratic functions, and from the functions
f(x) = and f(x)
= ;
OC2.05 represent inverse functions, using function
notation, where appropriate;
OC2.06 represent transformations (e.g.,
translations, reflections, stretches) of the functions defined by
f(x) = x, f(x) = x2, f(x)
= , f(x) = sin x, and f(x) =
cos x, using function notation;
OC2.07 describe, by interpreting function notation,
the relationship between the graph of a function and its image under one or
more transformations;
OC2.08 state the domain and range of
transformations of the functions defined by
f(x) = x, f(x)
= x2, f(x)
= , f(x) =
sin x, and f(x) = cos x.
Communicating Mathematical Reasoning
OC3.01 explain mathematical processes, methods of
solution, and concepts clearly to others;
OC3.02 present problems and their solutions to a
group, and answer questions about the problems and the solutions;
OC3.03 communicate solutions to problems and to
findings of investigations clearly and concisely, orally and in writing, using
an effective integration of essay and mathematical forms;
OC3.04 demonstrate the correct use of mathematical
language, symbols, visuals (e.g., diagrams, graphs), and conventions;
OC3.05 use graphing technology effectively (e.g.,
use appropriate menus and algorithms; set the graph window to display the
appropriate section of a curve).
The graduate is expected to be:
A Discerning Believer Formed in the Catholic Faith
Community who
CGE1a -illustrates a basic understanding of the saving
story of our Christian faith;
CGE1b -participates in the sacramental life
of the church and demonstrates an understanding of the centrality of the
Eucharist to our Catholic story;
CGE1c -actively reflects on Gods Word as
communicated through the Hebrew and Christian scriptures;
CGE1d -develops attitudes and values founded on
Catholic social teaching and acts to promote social responsibility,
human solidarity and the common good;
CGE1e -speaks the language of life...
recognizing that life is an unearned gift and that a person entrusted with
life does not own it but that one is called to protect and cherish it.
(Witnesses to Faith)
CGE1f -seeks intimacy with God and celebrates communion
with God, others and creation through prayer and worship;
CGE1g -understands that ones purpose or call
in life comes from God and strives to discern and live out this call
throughout lifes journey;
CGE1h -respects the faith traditions, world
religions and the life-journeys of all people of good will;
CGE1i -integrates faith with life;
CGE1j -recognizes that sin, human weakness,
conflict and forgiveness are part of the human journey and that the cross, the
ultimate sign of forgiveness is at the heart of redemption. (Witnesses
to Faith)
An Effective Communicator who
CGE2a -listens actively and critically to
understand and learn in light of gospel values;
CGE2b -reads, understands and uses written
materials effectively;
CGE2c -presents information and ideas clearly and
honestly and with sensitivity to others;
CGE2d -writes and speaks fluently one or both of
CGE2e -uses and integrates the Catholic faith
tradition, in the critical analysis of the arts, media, technology and
information systems to enhance the quality of life.
A Reflective and Creative Thinker who
CGE3a -recognizes there is more grace in our world
than sin and that hope is essential in facing all challenges;
CGE3b -creates, adapts, evaluates new ideas in
light of the common good;
CGE3c -thinks reflectively and creatively to
evaluate situations and solve problems;
CGE3d -makes decisions in light of gospel values
with an informed moral conscience;
CGE3e -adopts a holistic approach to life by
integrating learning from various subject areas and experience;
CGE3f -examines, evaluates and applies knowledge
of interdependent systems (physical, political, ethical, socio-economic and
ecological) for the development of a just and compassionate society.
A Self-Directed,
Responsible, Life Long Learner who
CGE4a -demonstrates
a confident and positive sense of self and respect for the dignity and welfare
of others;
CGE4b -demonstrates
flexibility and adaptability;
CGE4c -takes initiative and demonstrates Christian
leadership;
CGE4d -responds to, manages and constructively
influences change in a discerning manner;
CGE4e -sets appropriate goals and priorities in
school, work and personal life;
CGE4f -applies effective communication,
decision-making, problem-solving, time and resource management skills;
CGE4g -examines and reflects on ones personal
values, abilities and aspirations influencing lifes choices and opportunities;
CGE4h -participates in leisure and fitness
activities for a balanced and healthy lifestyle.
A Collaborative Contributor who
CGE5a -works effectively as an interdependent team
member;
CGE5b -thinks critically about the meaning and
purpose of work;
CGE5c -develops ones God-given potential and
makes a meaningful contribution to society;
CGE5d -finds meaning, dignity, fulfillment and
vocation in work which contributes to the common good;
CGE5e -respects the rights, responsibilities and
contributions of self and others;
CGE5f -exercises Christian leadership in the
achievement of individual and group goals;
CGE5g -achieves excellence, originality, and
integrity in ones own work and supports these qualities in the work of others;
CGE5h -applies skills for employability,
self-employment and entrepreneurship relative to Christian vocation.
A Caring Family Member who
CGE6a -relates to family members in a loving,
compassionate and respectful manner;
CGE6b -recognizes human intimacy and sexuality as
God given gifts, to be used as the creator intended;
CGE6c -values and honours the important role of
the family in society;
CGE6d -values and nurtures opportunities for
family prayer;
CGE6e -ministers to the family, school, parish,
and wider community through service.
A Responsible Citizen who
CGE7a -acts morally and legally as a person formed
in Catholic traditions;
CGE7b -accepts accountability for ones own
actions;
CGE7c -seeks and grants forgiveness;
CGE7d -promotes the sacredness of life;
CGE7e -witnesses Catholic social teaching by
promoting equality, democracy, and solidarity for a just, peaceful and
compassionate society;
CGE7f -respects and affirms the diversity and
interdependence of the worlds peoples and cultures;
CGE7g -respects and understands the history,
cultural heritage and pluralism of todays contemporary society;
CGE7h -exercises the rights and responsibilities
of Canadian citizenship;
CGE7i -respects the environment and uses
resources wisely;
CGE7j -contributes to the common good.
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