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Course
Profile Geometry and Discrete Mathematics (MGA4U), Grade 12, University
Preparation, Catholic
Course Overview
Policy
Document: The Ontario
Prerequisite:
Functions and Relations, Grade 11, University
Preparation, MCR3U
Course Description
This
course enables students to broaden mathematical knowledge and skills related to
abstract mathematical topics and to the solving of complex problems. Students
will solve problems involving geometric and Cartesian vectors, and
intersections of lines and planes in three-space. They also develop an
understanding of proof, using deductive, algebraic, vector, and indirect
methods. Students will solve problems involving counting techniques and prove
results using mathematical induction.
This
course is designed for students planning to study university programs that are
highly focused on mathematics, including engineering, pure mathematics,
computer science, and the physical sciences. Contextual examples and activities
should thus be drawn largely from these fields. Because of the academic demands
of these programs, the expectations of this course require students to
consistently demonstrate the ability to:
·
research,
investigate, and construct mathematical concepts independently;
·
conjecture
and, through inquiry, test a hypothesis in a variety of ways including using
technology;
·
generate
multiple types to solutions to complex problems which may cross strands and
require abstract thinking;
·
analyse
and design proofs from multiple perspectives.
This
course is comprised of three strands: Proof and Problem Solving, Geometry, and
Discrete Mathematics. These strands have been divided into a total of six
units. Unit 1 – Deductive Geometry, encompasses the Proof and Problem Solving
strand, in which students apply deductive, algebraic, and vector methods to
demonstrate and prove properties of plane figures. The rigors of this
curriculum require students to expand the depth of their understanding of the
comprehensiveness of problem solving techniques. Independent work skills are
stressed in this unit. The Geometry strand is divided into three units. In Unit
2 – Vectors and Unit 3 – Vector Applications, students investigate, manipulate,
and apply geometric vectors. In Unit 4 – Intersections of Lines and Planes,
students determine equations of lines and planes, solve systems of equations
using matrices, and determine intersections of lines and planes. In the
Discrete Mathematics strand, students solve problems involving counting
techniques. Students use mathematical induction to prove the binomial theorem
and the formulas for the sums of series. The Discrete Mathematics strand is
also divided into two units. Unit 5 – Mathematical Induction and Combinatorics
introduces sequences and series, mathematical induction, and counting
techniques. Students further explore these concepts and apply their learning in
Unit 6 – Application of Counting Techniques. A final summative unit contains
activities designed to assess the students’ skills across the four categories
of learning using a variety of instruments.
How This Course Supports
the Ontario Catholic School
The mind and its capacity for
rational analysis are seen as gifts from the Creator to be used and enjoyed.
The training of the intellect ensures that all knowledge can be scrutinized and
the divine and the human more deeply understood. In this sense, knowledge is
illuminated with the light of faith. In Catholic education, the commitment to
maturity of mind and academic excellence is always in reference to God’s
revealed wisdom as to life’s purpose and meaning. One of the fundamental
characteristics of our humanity is to discern meaning within our experience.
That meaning, however tentative, once embraced, seeks expression in such a way
that it becomes a
Before
each Unit Overview Chart, several Ontario Catholic School Graduate Expectations
have been listed. These expectations can be easily addressed by the activities
in each unit, as demonstrated in Unit 5 – Mathematical Induction and
Combinatorics, one of the two units which has been fully developed in this
Course Profile.
Course Notes
The
Grade 12 Course Profiles produced by the Catholic and Public systems represent
a collaborative effort between the two writing teams. The completed units
provided in this profile are Unit 3 – Vector Applications, and Unit 5 –
Mathematical Induction and Combinatorics. In addition to these two complete
“sample” units, a less-detailed Unit Overview Chart offers a recommended
clustering of expectations for each of the remaining units, and provides a
starting point from which teachers can plan the delivery of the curriculum.
In
this course, students use prerequisite skills in analytic geometry and extend
their skills in verifying properties of plane figures to those of crafting
formal proofs of the same properties. Deductive geometry provides a solid
platform from which students can recognize and form efficient strategies for
proof and problem solving using vector methods.
Displacement,
velocity, force, work, and torque lend context to the introduction of vector
properties, vector projections and the dot and cross products. As these
applications are commonly associated with the natural sciences (and physics in
particular), teaching strategies and investigations may be coordinated with the
physics teacher. The notion that inductive reasoning and the ability to make
informed decisions both involve an element of uncertainty should be reinforced
for students destined for programs involving the application of mathematical
principles to design problems. Wherever possible, dynamic geometry software
should be used as a tool for recognizing patterns and suggesting conjectures.
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. For example,
properties of plane figures (from Unit 1) and the dot and cross products (from
Unit 3) should be applied to problems involving the intersection of lines and
planes (from Unit 4). Students must be exposed to a variety of teaching,
learning, proving, and problem-solving techniques to best synthesize the
information presented by the curriculum, and should be provided with
applications and context to bring meaning to their learning.
The
requisite spatial skills for the three-dimensional visualization demanded by
this course and in many postsecondary destinations may be enhanced through the
use of geometry software and through the construction of physical models.
The function of proof has
traditionally been viewed as a verification of the a
Although
the expectations contained in the MGA4U (Geometry and Discrete Mathematics)
course are fewer in number than in the Grade 12 University Preparation courses
MCB4U (Advanced Functions and Introductory Calculus) and MDM4U (Mathematics of
Data Management), they are broad in scope, allowing for the exploration and
extension of a wide range of topics at the teacher’s discretion. This breadth
enables the teacher to use a wide variety of teaching strategies, learning
materials, and resources, as befits a course of this academic rigour.
Of
particular prominence in this course is the specific expectation PS3.03,
requiring students to demonstrate significant learning and the effective use of
skills in tasks such as solving challenging problems, researching problems, applying
mathematics, creating proofs, using technology effectively, and presenting
course topics or extensions of course topics. The teacher may wish to consider
the use of an independent project as a means of addressing this expectation in
part. Students would be required to research new contexts in which
problem-solving and proving techniques may be applied. In order to revisit this
expectation throughout the course, the teacher may wish to introduce this
project early in the year. The students’ research would then comprise a portion
of the final summative evaluation.
The
activities in this profile are designed to both introduce and consolidate
skills necessary for su
Because
this course has been designed to prepare students for further mathematical
studies at university, the specific nature of the learning activities should
reflect this destination. In particular, students in this course should
routinely be challenged with investigations and problems which require
sustained, independent effort. Students destined for university should have the
opportunity to develop and demonstrate a high level of complex problem-solving
ability.
Several
of the activities presented in this profile include extensions of the required
content, which can be used to meet the need to challenge gifted students.
Units: Titles and Time
|
Unit 1 |
Deductive
Geometry |
18.5
hours |
|
Unit 2 |
Vectors |
12.5
hours |
|
* Unit
3 |
Vector
Applications |
15 hours |
|
Unit 4 |
Intersections
of Lines and Planes |
19 hours |
|
* Unit
5 |
Mathematical
Induction and Combinatorics |
19 hours |
|
Unit 6 |
Applications
of Counting Techniques |
16 hours |
|
Unit 7 |
Final
Summative Assessment |
10 hours |
* These
units are fully developed in this Course Profile.
Unit Overviews
Unit 1: Deductive Geometry
Time: 18.5
hours
Unit
Description
The
language of deductive geometry, e.g., properties, theorems, etc., is introduced.
Geometric properties studied in previous grades are revisited using dynamic
geometry software. Further properties of plane figures are investigated and
linked to applicable theorems. Students are introduced to the structure of
formal deductive proofs. Strategies for establishing proof are explored through
cooperative and individual learning. Indirect methods of proof are used to
prove some properties of plane geometry.
Unit
Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
PSV.01,
PSV.03, PS1.01, PS3.03 |
Knowledge/Understanding |
Introduce
the language of proofs, e.g., proposition, converse, property, axiom,
theorem, if, if and only if. |
|
2 |
PSV.01,
PSV.02, PS1.01, PS1.06, PS2.02, PS2.03 |
Knowledge/Understanding |
Investigate
the properties of plane figures and the theorems that pertain to them using
dynamic geometry software. |
|
3 |
PSV.01,
PSV.02, PSV.03, PS1.01, PS1.02, PS1.06, PS2.01, PS2.02, PS2.03, PS2.04,
PS3.03 |
Knowledge/Understanding |
Test
conjectures and prove properties of plane figures using deduction. |
|
4 |
PSV.01,
PSV.02, PSV.03, PS1.01, PS1.05, PS2.01, PS2.02, PS3.03 |
Knowledge/Understanding |
Prove
some properties of plane figures using indirect methods. |
|
5 |
All within unit |
Knowledge/Understanding |
Summative
Assessment |
Unit 2: Vectors
Time: 12.5
hours
Unit
Description
Students are introduced to the
concept of vectors as directed line segments. The addition, subtraction and
scalar multiplication of geometric vectors are investigated. The algebraic and
geometric properties of vectors are applied to proofs of properties of plane
figures and to velocity and force problems. Dynamic geometry software is used
as a tool to make conjectures and verify vector properties. Students will use
vector methods to prove some of the same properties of plane figures that were
explored in Unit 1 – Deductive Geometry.
Unit
Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
GEV.01,
PSV.02, GE1.01, GE1.02, GE1.04, PS2.02 |
Knowledge/
Understanding |
Define
vectors as directed line segments and use a geometric representation of
addition of displacement vectors to propose various billiards shots. The
distinction between vector and scalar quantities is made. |
|
2 |
GEV.01,
PSV.02, GE1.01, GE1.02, GE1.04, PS2.03 |
Knowledge/
Understanding |
Investigate
the properties of addition, subtraction, and scalar multiplication of
geometric vectors using dynamic geometry software. |
|
3 |
GEV.01,
PSV.01, PSV.02, GE1.01, GE1.02, PS1.04, PS1.06, PS2.01, PS2.02, PS2.03 |
Knowledge/
Understanding |
Prove
properties of plane figures using geometric vectors. |
|
4 |
GEV.01,
PSV.02, GE1.01, GE1.02, GE1.03, GE1.04, PS2.01, PS2.04 |
Thinking/Inquiry/
Problem Solving |
Applications
geometric vectors: force and velocity. Introduce projections of geometric
vectors through the resolution of the gravitational force into components. |
|
5 |
All within unit |
Knowledge/
Understanding |
Summative
assessment |
Unit 3: Vector
Applications
Time: 15 hours
Unit
Description
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
investigated. 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 Unit 4 – Intersections of Lines and Planes.
Unit
Overview Chart
|
Activity |
Time |
Learning Expectations |
Assessment Categories |
Tasks |
|
3.1 |
3.75 hours |
GEV.01,
PSV.01, PSV.02, GE1.01, GE1.06, GE1.07, PS1.06, PS2.01, PS2.02, PS2.03 |
Knowledge/
Understanding |
Introduce
and investigate Cartesian vectors. The properties of addition, subtraction,
and scalar multiplication are applied to Cartesian vectors. |
|
3.2 |
2.5 hours |
GEV.01,
PSV.02, GE1.01, GE1.02, GE1.03, PS2.01, PS2.03 |
Knowledge/
Understanding |
Investigate
the dot product of geometric and Cartesian vectors through the concept of
physical work. |
|
3.3 |
3.75 hours |
GEV.01,
PSV.01, PSV.02, GE1.01, GE1.02, GE1.03, GE1.05, PS1.04, PS1.06, PS2.01,
PS2.02, PS2.03 |
Knowledge/
Understanding |
Investigate
the properties of the dot product of Cartesian vectors. The dot product is
applied to the physical definition of work and to the determination of the geometric
relationships between vectors. |
|
3.4 |
2.5 hours |
GEV.01,
PSV.01, PSV.02, GE1.01, GE1.02, GE1.03, PS1.04, PS2.01, PS2.02 |
Communication |
Investigate
the properties of the cross product of geometric and Cartesian vectors. The
cross product is applied to the physical definition of torque and to the
determination of the geometric relationships between vectors. |
|
3.5 |
2.5 hours |
All within unit |
Knowledge/
Understanding |
Summative
assessment |
Unit 4: Intersections of Lines and Planes
Time: 19 hours
Unit
Description
Students
determine the vector and parametric equations of lines in two-space, the
vector, parametric and symmetric equations of lines in three-space and the
vector, parametric and scalar equations of planes. The intersections of two
lines, a line and a plane, two planes, and three planes are investigated.
Matrices are used to represent systems of up to three unknowns and row
reduction is used to solve the systems. The types of solution sets are related
to the geometric properties of the system.
Unit
Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
GEV.01,
GEV.02, GE1.06, GE1.07, GE2.01 |
Knowledge/Understanding |
Determine
the equations of lines (vector, parametric, and symmetric) in two and three
dimensions. |
|
2 |
GEV.01,
GEV.02, GE1.06, GE1.07, GE2.03 |
Knowledge/Understanding |
Determine
the vector, parametric, and scalar equations of planes. |
|
3 |
GEV.01,
GEV.02, GE1.06, GE1.07, GE2.02, GE2.04 |
Thinking/Inquiry/Problem
Solving |
Determine
the intersections of two lines, a line, and a plane. |
|
4 |
GEV.02,
GE2.07, GE2.08, GE2.09 |
Knowledge/Understanding |
Solve
systems of linear equations to determine the intersections of two or three
planes. Interpret the solutions geometrically. |
|
5 |
GEV.02,
GE2.05, GE2.06 |
Knowledge/Understanding |
Introduces
matrices, use them to solve systems of equations pertaining to the
intersections of planes. |
|
6 |
All within unit |
Knowledge/Understanding |
Summative
assessment |
Unit 5: Mathematical Induction and Combinatorics
Time: 19 hours
Unit
Description
Students
use pattern recognition to derive formulas for the terms and sums of arithmetic
and geometric sequences and series. Sigma notation is used to express a series
as a sum of related terms. Mathematical induction is introduced as an effective
means of proving sequence- and series-related results. Inductive reasoning
skills are extended in the proof of other algebraic results. Using factorial
notation, permutations, and combinations, students solve introductory counting
problems.
Unit
Overview Chart
|
Activity |
Time |
Learning
Expectations |
Assessment
Categories |
Tasks |
|
5.1 |
|
DMV.02,
PSV.02, PSV.03, DM2.01, DM2.02, PS2.02, PS3.01, PS3.03 |
Knowledge/
Understanding |
Determine
sums of series using pattern recognition and deductive reasoning; Consolidate
understanding of concepts and terms |
|
5.1b |
1.25 hours |
|
|
Use an
informal mathematical induction proof and recursive thinking to determine
formulas for use in the construction of geometric figures |
|
5.2 Follow-Up
Skills |
1 hour |
PSV.02,
PSV.03, DMV.02, PS2.01, PS2.04, PS3.01, PS3.03, DM2.01, DM2.02, DM2.03 |
Knowledge/
Understanding |
Use
inductive reasoning to conjecture formulas based on pattern recognition |
|
5.2b |
1 hour |
|
|
Use
deductive reasoning to verify conjectures |
|
5.2c |
1.25 hours |
|
|
Investigate
the method of proof by mathematical induction |
|
5.3 |
2.5 hours |
DMV.01,
PSV.02, DM1.01, DM1.05, PS2.01, PS2.04 |
Knowledge/
Understanding |
Explore
additive counting techniques; Consolidate understanding of concepts and
terms; Explore multiplicative counting techniques |
|
5.4 |
3.75 hours |
DMV.01, PSV.02, PSV.03, DM1.01, DM1.02,
DM1.03, DM1.04, DM1.05, PS2.01, PS2.03, PS2.04, PS3.01 |
Knowledge/ Understanding |
Define factorial notation; Investigate
permutations and combinations |
|
Summative
Assessment |
2 hours |
PSV.02,
PSV.03, DMV.01, DMV.02, PS2.01, PS2.04, PS3.01, PS3.03, DM1.01, DM1.02,
DM1.03, DM1.05, DM2.01, DM2.02, DM2.03 |
Knowledge/
Understanding |
Consolidate
new learning |
Unit 6: Applications of Counting Techniques
Time: 16 hours
Unit
Description
Students
solve counting problems in diverse contexts. The binomial theorem and its
connection to Pascal’s Triangle are investigated. Students explore and prove properties
of Pascal’s Triangle using inductive reasoning.
Unit
Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
DMV.01,
PSV.02, DM1.04, DM1.05, PS2.03, PS2.04 |
Communication |
Solve
counting problems, including card games, tournaments, shaking of hands,
spread of rumours, poker hands, probability |
|
2 |
DMV.01,
DMV.02, DM1.06, DM1.07, DM2.01, DM2.04, DM2.05 |
Knowledge/Understanding |
Use
sigma notation and combinatorics to express the terms of a binomial
expansion; Introduce Pascal’s Triangle; Prove various results using Pascal’s
Triangle and mathematical induction |
|
3 |
All within unit |
Knowledge/Understanding |
Summative
assessment |
Unit 7: Final Summative Assessment
Time:
10 hours
Ontario Catholic School Graduate Expectations: CGE2b, CGE2c, CGE3c, CGE3e, CGE4a, CGE4b, CGE4f, CGE5a, CGE5g, CGE7b, CGE7j.
Unit
Description
Summative
assessment should be designed to provide the opportunity for students to
demonstrate comprehensive learning in each of the four Achievement Chart
categories. Several ideas are suggested in the Unit Overview Chart, however any
of the various assessment tools mentioned in the Assessment Strategies section
could be utilized. Due to the emphasis of cumulative tests and examinations in
university programs, a formal final examination should play a prominent role in
the final summative assessment of the student. A short paper-and-pencil task
could be used to assess key terms, skills, and concepts.
In
this course, students are expected to demonstrate significant learning and the
effective use of skills in tasks such as solving challenging problems,
researching problems, applying mathematics, creating proofs, using technology
effectively, and presenting course topics or extensions of course topics
(PS3.03). An independent research project could allow students to revisit the
course expectations in a new mathematical context and at the same time provide
exposure to concepts that they may encounter at university. The context
provides interest and direction, but it is only the tool with which the
expectations are addressed. A
As
summative evaluation is to comprise of 30% of the final grade, it is suggested
that the final examination comprise of the majority of this component.
Unit
Overview Chart
|
Cluster |
Learning Expectations (by Strand) |
Assessment Categories |
Focus |
|
1 |
All Strands |
Knowledge/Understanding |
Assess
key concepts and terms |
|
2 |
All Strands |
Knowledge/Understanding |
Research
and investigate extensions to course topics; |
|
3 |
All Strands |
Knowledge/Understanding |
Final
examination |
Note: Research topics may include Fibona
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 and appreciate the need for new skills;
·
approaches
that will a
·
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 o
·
the
use of a
·
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 play an active role in their learning. In
order to su
·
develop
an increased responsibility for their own learning;
·
be
a
·
participate
as active learners;
·
engage
in explorations using technology;
·
apply
individual and group learning skills;
Assessment Strategies
The
Achievement Chart for Mathematics is the basis of all assessment and evaluation
for this course. 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.
Assessment
tools such as observational checklists, performance criteria, rubrics, marking
schemes, and rating scales 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 a
It should also be noted that teachers must
continue to expand their understanding of application skills to include
non-routine applications. 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. 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 student’s
work on the basis of established achievement criteria, and assigning a value to
represent that quality” (p. 31). Whereas assessment is the collection 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. 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.
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
methods of evaluation.
Students
who receive a final performance evaluation of Level 3 or better are
well-prepared for study at the university level. A
Acc ommodations
Teachers
should consult individual student IEPs for specific direction on a
·
Provide
sets of reference notes, outlines, or critical information, as well as models
of charts, timelines or diagrams.
·
Use
visuals to illustrate definitions for the students’ dictionary of terms.
·
Pair
written instructions with verbal instructions. Provide visual or auditory cues.
·
Provide
opportunities for students to practise oral presentation skills.
·
Provide
extensive student-teacher conferencing.
·
Provide
an interpreter or a scribe to help formalize the students’ reasoning.
·
Provide
a list of terms (possibly simplified) before an activity begins. Include visual
descriptions.
·
Adapt
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.
Provide intermediate deadlines with built-in conferencing at these dates.
·
Keep
technology, manipulatives, grid paper, formula sheets, and other aids available
for needs that arise.
Considerations for
Enrichment
·
Pose
open-ended questions that require higher-level thinking.
·
Model
creative thinking strategies, such as decision-making and evaluation of
problem-solving approaches, and facilitate original and independent problems
and solutions.
·
Encourage
independent investigations and projects.
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, or otherwise supplement their students’ education.
Units in
this course profile make reference to the use of specific texts, magazines,
films, videos, and websites. Teachers need to consult their board policies
regarding use of any copyrighted materials. Before reproducing materials for
student use from printed publications, teachers need to ensure that their board
has a Cancopy license and that this license covers the resources they wish to
use. Before screening videos/films with their students, teachers need to ensure
that their board/school has obtained the appropriate public performance
videocassette license from an authorized distributor, e.g., Audio Cine Films
Inc. Teachers are reminded that much of the material on the Internet is
protected by copyright. The copyright is usually owned by the person or
organization that created the work. Reproduction of any work or substantial
part of any work on the Internet is not allowed without the permission of the
owner
Software
(Ministry-Licensed)
The
Geometer’s Sketchpad (dynamic
geometry)
Fathom
(statistical
software)
Maple
(word
processor/programming)
Math
Trek (concept and
skill development)
Zap-a-Graph
(graphing)
Software (Freeware)
Peanuts
(concept and skill
development) (http://math.exeter.edu/rparris)
Internet
sites
The URLs
for the websites were verified by the writers prior to publication. Given the
frequency with which these designations change, teachers should always verify
the websites prior to assigning them for the students’ 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 Ontario (http://www.enoreo.on.ca/)
ENO is a computer communications network for everyone who works in elementary
and secondary education in Ontario. Members have private a
Hewlett-Packard
(http://www.hp.com/calculators/)
National
Council of Teachers of Mathematics (http://www.nctm.org)
Ontario
Association of Mathematics Educators (http://www.oame.on.ca)
Ontario
Curriculum Centre (http://www.curriculum.org)
Texas Instruments (http://www.ti.com/calc/docs)
Print
Concerning Assessment and Reflective Evaluation
(CARE) Package (source – http://www.oame.on.ca)
Connecting
Mathematics: Addenda Series, Grades 9-12. NCTM Reston, VA: National Council of Teachers
of Mathematics (NCTM, 1991. ISBN 0-87353-327-5
de
Villiers, M. Rethinking Proof with The Geometer’s Sketchpad. Emeryville,
CA: Key Curriculum Press, 1999. ISBN 1-55953-390-4
The
Mathematics Teacher.
Reston, VA: National Council of Teachers of Mathematics (NCTM).
ISBN 0025-5769
Ontario
Secondary School Teachers Federation. Quality Assessment. Toronto, ON:
Educational Services Committee, 1999.
Taggart,
G., ed. Rubrics – A Handbook for Construction and Use. Lancaster, PA:
Techonomic Publishing, 1998.
OSS Considerations
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
The
Ontario Curriculum Mathematics, Grades 11-12, 2000
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 OSS policies. They are available
online at the Ministry of Education website,
http://www.edu.gov.on.ca/eng/document/document.html.
Publications Concerning
Faith Development
Blueprints (Catholic Curriculum Cooperative -
Central Ontario Region)
Catholicity
Across The Curriculum
(Ontario Catholic School Trustees’ Association)
Educating
the Soul (Institute
for Catholic Education)
Ontario
Catholic Secondary School Graduate Expectations (Institute for Catholic Education)
This
Moment of Promise
(Ontario Conference of Catholic Bishops)
Creating
Catholic Curriculum
(Eastern Ontario Catholic Curriculum Cooperative)
Community Partnerships
Refer to
local board policies, e.g., Relations with Business - Corporate Donations,
Sponsorships and Agreements.
Coded
Expectations, Geometry and Discrete Mathematics, Grade 12, University
Preparation, MGA4U
Geometry
Overall Expectations
GEV.01 · perform operations with geometric
and Cartesian vectors;
GEV.02 · determine intersections of lines
and planes in three-space.
Specific Expectations
Operating
with Geometric and Cartesian Vectors
GE1.01 – represent vectors as directed
line segments;
GE1.02 – perform the operations of
addition, subtraction, and scalar multiplication on geometric vectors;
GE1.03 – determine the components of a
geometric vector and the projection of a geometric vector;
GE1.04 – model and solve problems
involving velocity and force;
GE1.05 – determine and interpret the dot
product and cross product of geometric vectors;
GE1.06 – represent Cartesian vectors in
two-space and in three-space as ordered pairs or ordered triples;
GE1.07 – perform the operations of
addition, subtraction, scalar multiplication, dot product, and cross product on
Cartesian vectors.
Determining
Intersections of Lines and Planes in Three-Space
GE2.01 – determine the vector and
parametric equations of lines in two-space and the vector, parametric, and
symmetric equations of lines in three-space;
GE2.02 – determine the intersections of
lines in three-space;
GE2.03 – determine the vector, parametric,
and scalar equations of planes;
GE2.04 – determine the intersection of a
line and a plane in three-space;
GE2.05 – solve systems of linear equations
involving up to three unknowns, using row reduction of matrices, with and
without the aid of technology;
GE2.06 – interpret row reduction of
matrices as the creation of a new linear system equivalent to the original;
GE2.07 – determine the intersection of two
or three planes by setting up and solving a system of linear equations in three
unknowns;
GE2.08 – interpret a system of two linear
equations in two unknowns and a system of three linear equations in three
unknowns geometrically, and relate the geometrical properties to the type of
solution set the system of equations possesses;
GE2.09 – solve problems involving the
intersections of lines and planes, and present the solutions with clarity and
justification.
Proof and Problem Solving
Overall Expectations
PSV.01 · prove properties of plane figures
by deductive, algebraic, and vector methods;
PSV.02 · solve problems, using a variety
of strategies;
PSV.03 · complete significant
problem-solving tasks independently.
Specific Expectations
Proving
Properties of Plane Figures by Deductive, Algebraic, and Vector Methods
PS1.01 – demonstrate an understanding of the
principles of deductive proof (e.g., the role of axioms; the use of “if … then”
statements; the use of “if and only if” statements and the necessity to prove
them in both directions; the fact that the converse of a proposition differs
from the proposition) and of the relationship of deductive proof to inductive
reasoning;
PS1.02 – prove some properties of plane
figures (e.g., circles, parallel lines, congruent triangles, right triangles),
using deduction;
PS1.03 – prove some properties of plane
figures (e.g., the midpoints of the sides of a quadrilateral are the vertices
of a parallelogram; the line segment joining the midpoints of two sides of a
triangle is parallel to the third side) algebraically, using analytic geometry;
PS1.04 – prove some properties of plane
figures, using vector methods;
PS1.05 – prove some properties of plane
figures, using indirect methods;
PS1.06 – demonstrate an understanding of
the relationship between formal proof and the illustration of properties that
is carried out by using dynamic geometry software.
Using a
Variety of Strategies to Solve Problems
PS2.01 – solve problems by effectively
combining a variety of problem-solving strategies (e.g., brainstorming,
considering cases, choosing algebraic/geometric/vector or direct/indirect
approaches, working backwards, visualizing by using concrete materials or
diagrams or software, iterating, varying parameters, creating a model,
introducing a coordinate system);
PS2.02 – generate multiple solutions to
the same problem;
PS2.03 – use technology effectively in
making and testing conjectures;
PS2.04 – solve complex problems and
present the solutions with clarity and justification.
Completing
Significant Problem-Solving Tasks Independently
PS3.01 – solve problems of significance,
working independently, as individuals and in small groups;
PS3.02 – solve problems requiring effort
over extended periods of time;
PS3.03 – demonstrate significant learning
and the effective use of skills in tasks such as solving challenging problems,
researching problems, applying mathematics, creating proofs, using technology
effectively, and presenting course topics or extensions of course topics.
Discrete Mathematics
Overall Expectations
DMV.01 · solve problems, using counting techniques;
DMV.02 · prove results, using mathematical
induction.
Specific Expectations
Using
Counting Techniques
DM1.01 – solve problems, using the
additive and multiplicative counting principles;
DM1.02 – express the answers to
permutation and combination problems, using standard combinatorial symbols
[e.g. 
, P(n, r)];
DM1.03 – evaluate expressions involving factorial
notation, using appropriate methods (e.g., evaluate mentally, by hand, by using
a calculator);
DM1.04 – solve problems involving permutations and
combinations, including problems that require the consideration of cases;
DM1.05 – explain solutions to counting problems with
clarity and precision;
DM1.06 – describe the connections between
Pascal’s triangle, values of 
, and values for the binomial coefficients;
DM1.07 – solve problems, using the
binomial theorem to determine terms in the expansion of a binomial.
Using
Mathematical Induction to Prove Results
DM2.01 – demonstrate an understanding of
the principle of mathematical induction;
DM2.02 – use sigma notation to represent a
series or the sum of a series;
DM2.03 – prove the formulas for the sums
of series, using mathematical induction;
DM2.04 – prove the binomial theorem, using
mathematical induction;
DM2.05 – prove relationships between the
coefficients in Pascal’s triangle, by mathematical induction and directly.
Ontario
Catholic School Graduate Expectations
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 God’s 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 one’s purpose or call in life comes from God and strives to discern
and live out this call throughout life’s 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 Canada’s official languages;
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 one’s personal values, abilities and aspirations influencing
life’s 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
one’s 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 one’s 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 -a
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 world’s peoples and
cultures;
CGE7g -respects
and understands the history, cultural heritage and pluralism of today’s
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.