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Course Profile Mathematics for College Technology (MCT4C),
Grade 12, College Preparation, Combined
Course Overview
Policy Document: The
Ontario Curriculum, Grades 11 and 12, Mathematics, 2000.
Prerequisite: Functions, Grade 11,
University/College Preparation, or
Functions
and Relations, Grade 11, University Preparation
Course Description
This course
equips students with the mathematical knowledge and skills needed for entry
into college technology programs. Students will investigate and apply
properties of polynomial, exponential, and logarithmic functions; solve
problems involving inverse proportionality; and explore the properties of
reciprocal functions. They will also analyse models of a variety of functions,
solve problems involving piecewise-defined functions, solve linear-quadratic
systems, and consolidate key manipulation and communication skills.
Students entering
mathematics-focused programs at the college level benefit from MCT4C. This
course enables students to consolidate and expand many pre-calculus concepts
explored in previous mathematics courses. Contextual applications and
technological tools are integrated throughout to support the development of new
skills and the exploration of a variety of mathematical models.
How This Course Supports the Ontario
Catholic School Graduate Expectations
Through the
investigation and exploration of various mathematical models, students are
challenged to discover a deeper understanding and appreciation of the mathematical
nature of the world in which we live. These underlying mathematical principles
provide the student another academic or intellectual lens for addressing
possible problems and investigating potential solutions to situations around
them. Students should have the opportunity to communicate these discoveries to
others within a classroom environment that emphasizes collaboration rather than
competition.
Course Notes
Making a Connection to Students’
Career Paths
Providing a
real-life context for the student in this course is important. Reference to
students’ Annual Education Plans (AEPs) assists the teacher in making possible
connections with postsecondary programs, while aiding in the transition to
students’ new programs of study. Making this connection between career paths
and curriculum also helps foster student interest.
Using Technology as a Tool for
Learning
Students may use
technology in discovering relationships and behaviours. Technology use should
enhance and extend the understanding and communication of fundamental concepts
presented in this course. Appropriate technology includes, but is not limited
to scientific calculators, graphing calculators, and spreadsheet/graphing
software.
A Focus on Mathematical Models
Mathematical models
provide a realistic context for many of the concepts explored in this course.
By the creation and examination of these models, students achieve an underlying
understanding of the problem while working towards potential solutions. A focus
on mathematical modelling provides the student with an opportunity to frame
his/her skill development with potential applications to real-life problems.
Course Development
The organization of
this course follows a logical progression of concepts. In Unit 1, students
study the behaviour of polynomial functions of various degrees through the
investigation of their graphs on the Cartesian plane. Unit 2 further explores
polynomial graphs and equips the student with the algebraic skills required to
understand and to manipulate polynomial functions. Unit 3 focuses on
mathematical modelling and problem solving with both inverse and reciprocal
functions. Unit 4 examines exponential and logarithmic functions and relates
these functions to a variety of contexts. In Unit 5, students continue to investigate
special functions classified as piecewise defined functions. Unit 6 focuses on
linear and quadratic systems and extends students’ knowledge of trigonometry.
Unit 7 is a summative unit that allows students to demonstrate and apply their
acquired skills and concepts of function behaviour.
Units: Titles and Time
|
Unit 1 |
Key Features of
Polynomial Functions |
17 hours |
|
Unit 2 |
Exploring
Polynomial Functions: Connecting Algebra and Geometry |
18 hours |
|
* Unit 3 |
Exploring
Reciprocal and Inverse Functions |
15 hours |
|
* Unit 4 |
Exponential and
Logarithmic Functions |
20 hours |
|
Unit 5 |
Piecing It
Together |
16 hours |
|
Unit 6 |
Linear Functions,
Quadratic Functions, and Trigonometry |
16 hours |
|
Unit 7 |
Mathematical
Modelling – Summative Assessment |
8 hours |
* These
units are fully developed in this Course Profile.
Any additional time
can be allocated for remediation and consolidation of skills at the discretion
of the teacher, depending on the needs of students.
Unit Overviews
Unit 1: Key Features of Polynomial Functions
Time: 17
hours
Unit Description
Students are
introduced to the main concepts of graphing polynomial functions in order to
explore them later in the course. Students examine the type and number of
intercepts, the effects of changing numerical coefficients, the existence of
symmetry, and the degree in relation to the shape of the function. Using skills
from previous years, students explore curve sketching from a factored form.
Unit Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
PFV.01, PFV.02,
PF1.01, PF1.03, PF2.02 CGE2b, 5b |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Communication |
Identifying
properties of polynomial functions using technology |
|
2 |
PFV.01, PF1.02,
PF1.04 CGE2d |
Knowledge/Understanding Application Communication |
Using table of
values to assist in the construction of graphs Using finite
differences to describe the nature of function change |
|
3 |
PFV.01, ACV.04, PF1.05, AC4.01 CGE2e, 5b |
Knowledge/Understanding Thinking/Inquiry/Problem Solving Application |
Determining the equation of various graphs |
|
4 |
PFV.01, PFV.02, PF1.01, PF2.08 CGE3c |
Knowledge/Understanding Application Communication |
Solving non-factorable polynomial inequalities using technology |
Unit 2: Exploring Polynomial Functions: Connecting Algebra and Geometry
Time: 18 hours
Unit Description
Students explore
polynomial equations and inequalities. Real and complex roots of both
factorable and non-factorable polynomials are determined through graphical
investigations and algebraic manipulation. Students use the remainder theorem,
the factor theorem, and absolute-value notation.
Unit Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
PFV.01, PFV.02,
ACV.04, PF1.04, PF2.01, PF2.02, AC4.01 CGE2b, 2c |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Communication |
Investigating
factors and graphs of polynomials |
|
2 |
PFV.02, ACV.04,
PF2.02, PF2.03, PF2.05, AC4.01, AC4.04 CGE2d, 3c |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Application |
Exploring real and
complex roots |
|
3 |
PFV.02, PF2.04 CGE2e, 4f |
Thinking/Inquiry/Problem
Solving Communication |
Solving
non-factorable polynomial equations using graphing technology |
|
4 |
PFV.02, PF2.02,
PF2.07, PF2.08 CGE2b, 2c, 2e |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Application |
Solving
inequalities algebraically and using graphing technology |
|
5 |
PFV.02, PF2.06 CGE2b, 2c |
Knowledge/Understanding Communication Application |
Applying absolute
value |
Unit 3: Exploring Reciprocal and Inverse Functions
Time: 15 hours
Unit Description
Students explore the
behaviour of reciprocal and inverse functions. Students apply their acquired
knowledge to create mathematical models derived from realistic
inverse-proportional relationships. Using extrapolation, students predict
future behaviour and pose questions related to the generated models of these
functions. Key features of linear and quadratic reciprocal functions are
investigated through curve sketching. Technological aids, such as a graphing
calculator, assist students in understanding key concepts of reciprocal and
inverse functions.
Unit Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
ACV.01, ACV.04,
PFV.03, AC1.01, AC1.02, AC4.02, PF3.01 CGE2b, 3c, 3f |
Knowledge/Understanding Application Thinking/Inquiry/Problem
Solving |
Exploring inverse
proportionality |
|
2 |
ACV.01, PFV.03,
AC1.03, AC1.04, PF3.01, PF3.02 CGE2a, 2d, 2e |
Communication Application Thinking/Inquiry/Problem
Solving |
Problem solving
inverse proportionality with technology |
|
3 |
PFV.04, PF4.01 CGE2c, 5g |
Knowledge/Understanding Communication Thinking/Inquiry/Problem
Solving |
Sketching
reciprocal functions |
|
4 |
PFV.04, PF4.02,
PF4.03 CGE2e, 5a |
Communication Thinking/Inquiry/Problem
Solving |
Investigating
properties of reciprocal functions with technology |
|
5 |
ACV.01, PFV.03,
AC1.01, AC1.02, AC1.03, PF3.01, PF3.02 CGE3c, 5a, 5g, 7b |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Application Communication |
Summative
assessment |
Unit 4: Exponential and Logarithmic Functions
Time: 20 hours
Unit Description
Students investigate
properties of exponential and logarithmic functions. The relationship between
exponential and logarithmic functions is explored both graphically and
algebraically. Students use the laws of logarithms to simplify and evaluate
logarithmic expressions, and to solve problems. A variety of exponential and
logarithmic applications and models is examined.
Unit Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
ELV.01, EL1.01,
EL1.02, EL1.03 CGE2a, 2d, 5a |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Communication |
Investigating
properties of exponential functions |
|
2 |
ELV.01, ACV.01,
EL1.04, EL1.05, AC1.01, AC1.02, AC1.03, AC1.04 CGE3c, 5g |
Thinking/Inquiry/Problem
Solving Application Communication |
Modelling
exponential growth and decay |
|
3 |
ELV.01, ACV.04,
EL1.05, AC4.04 CGE2b, 2c, 7b |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Application Communication |
Applying
exponential growth and decay |
|
4 |
ELV.01, ELV.02,
ACV.04, EL1.01, EL2.01, EL2.02, AC4.04 CGE2c, 2d |
Knowledge/Understanding Communication |
Determining
connections between exponential functions and logarithmic functions |
|
5 |
ELV.02, ACV.01, EL2.04, AC1.01, AC1.02, AC1.03, AC1.04 CGE3c, 5g |
Thinking/Inquiry/Problem Solving Communication |
Modelling logarithmic functions |
|
6 |
ELV.02, ACV.04,
EL2.03, EL2.04, AC4.04 CGE2b, 2c |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Application Communication |
Exploring laws of
logarithms and applications of logarithmic scales |
|
7 |
ELV.01, ELV.02,
ACV.04, EL1.04, EL1.05, EL2.02, EL2.03, EL2.04, AC4.04 CGE2b, 2c, 5g, 7b |
Knowledge/Understanding Thinking/Inquiry/Problem
Solving Application Communication |
Summative
assessment |
Unit 5: Piecing It Together
Time: 16 hours
Unit Description
Students identify
and interpret different piecewise functions using paper and pencil, a graphing
calculator, and graphing software. Piecewise functions are applied in a variety
of contexts. Students analyse the key properties that set these functions apart
from exponential, trigonometric, and quadratic functions.
Unit Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
ACV.01, ACV.02,
ACV.04, AC1.01, AC2.01, AC4.02 CGE2a, 2b |
Knowledge/Understanding |
Recognizing
piecewise functions |
|
2 |
ACV.02, AC2.02,
AC2.03, AC2.04, AC2.05 CGE2c, 5g |
Application Thinking/Inquiry/Problem
Solving |
Graphing piecewise
functions |
|
3 |
ACV.01, ACV.02,
AC1.03, AC1.04, AC2.04 CGE3c, 5b |
Thinking/Inquiry/Problem
Solving Communication |
Analysing and
interpreting individual models of piecewise functions |
Unit 6: Linear Functions, Quadratic Functions, and Trigonometry
Time: 16 hours
Unit Description
Students investigate
properties of linear functions and quadratic functions. Students connect
elements of a contextual problem to elements of the function representing the
problem. Linear-quadratic systems are solved graphically and algebraically, and
students interpret their solutions contextually. Students extend and apply
their knowledge of trigonometry to solve problems involving right triangles and
oblique triangles.
Unit Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
ACV.03, ACV.04, AC3.01, AC4.04 CGE2b, 2c |
Knowledge/Understanding Application |
Determining the relationship between properties of linear functions
and contextual examples |
|
2 |
ACV.03, ACV.04,
AC3.01, AC4.04 CGE2c, 2d |
Knowledge/Understanding Application |
Determining the
relationship between properties of quadratic functions and contextual
examples |
|
3 |
ACV.03, ACV.04,
AC3.02, AC3.03, AC4.04 CGE2e, 3c, 5g |
Thinking/Inquiry/Problem
Solving Application Communication |
Solving
linear-quadratic systems graphically and within a variety of contexts |
|
4 |
ACV.04, AC4.03 CGE2b, 3c |
Knowledge/Understanding Application Communication |
Using trigonometry
to solve problems involving right triangles and oblique triangles |
Unit 7: Mathematical Modelling
Time: 8 hours
Ontario Catholic
School Graduate Expectations: CGE 2c, 2e, 3c, 5b, 5g.
Unit Description
Students use their
acquired knowledge to mathematically model real-world situations using the
various functions explored in this course. Previous knowledge of the key
properties that differentiate polynomial, inverse proportional, reciprocal,
exponential, logarithmic, and piecewise functions assists students in deciding
the function type that best approximates the situation under a given set of
circumstances. The use of technology, such as a graphing calculator, assists
students in their exploration of potential solutions.
Unit Overview Chart
|
Cluster |
Learning Expectations |
Assessment Categories |
Focus |
|
1 |
All Strands |
Knowledge/Understanding Application Thinking/Inquiry/Problem
Solving Communication |
Using key
properties of polynomial functions, inverse proportionality, reciprocal
functions, exponential functions, logarithmic functions, and piecewise
functions to mathematically model real-world situations |
Teaching/Learning Strategies
Opportunities
to apply a variety of both teaching and learning strategies are provided
throughout the course. In applying the strategies, the teacher:
·
provides regular,
constructive feedback;
·
integrates
technological tools and software where appropriate;
·
demonstrates the
use of any technological tools or software used in the classroom;
·
uses current and
local information in contextual questions to promote relevance;
·
teaches skills within
a context as often as possible;
·
uses a balance of
whole-class, small-group, and individual instruction through student-centred
and teacher-directed activities;
·
uses a variety of
instructional methods to address a variety of learning styles;
·
uses positive
reinforcement to foster a positive learning environment;
·
provides
opportunities for students to present mathematical results in a variety of
different presentation formats;
·
provides
extension opportunities;
·
provides review
and remediation where appropriate.
In
achieving the expectations of this course, students:
·
investigate and
explore concepts using technology;
·
increase their
proficiency with technology;
·
demonstrate their
knowledge and understanding using a variety of methods and
mathematical/technological tools;
·
communicate their
understanding using a variety of mediums;
·
summarize and
support decisions using a variety of strategies;
·
apply and develop
individual and group learning skills;
·
work individually
and cooperatively;
·
further develop
their problem-solving strategies;
·
develop
responsibility for their own learning and decision-making.
Assessment & Evaluation of
Student Achievement
Assessment
is a systematic process of collecting information or evidence about student
learning.
Assessment
is defined in Ontario Secondary Schools,
Grades 9-12, Program and Diploma Requirements, 1999, as “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 is
used for diagnostic, formative, and summative purposes.
Evaluation
requires that the teacher not simply average marks. Evaluation is defined by Ontario Secondary Schools, Grades 9-12,
Program and Diploma Requirements, 1999, as “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. As part of assessment,
teachers provide students with descriptive feedback that guides their efforts
towards improvement” (p. 31). In forming an evaluative judgement, the teacher
considers the student’s performance in the four categories of the Achievement
Chart.
The purpose of
assessment and evaluation is to improve student learning. Assessment and
evaluation strategies and tools must address the variety of learning styles and
needs. A balanced assessment and evaluation program is based on the provincial
curriculum expectations and the Achievement Chart levels.
The following
can serve as a guide for assessing student achievement.
Knowledge/Understanding
Achievement in this
category reflects the student’s ability to demonstrate understanding of
mathematical concepts and to perform algorithms. The teacher assesses:
·
quizzes;
·
short-answer and
skill-based calculations on unit tests and exams;
·
student-teacher
conferencing;
·
accuracy of
mathematical answers in reports and presentations.
Thinking/Inquiry/Problem-Solving
Achievement in this
category reflects the student’s ability to demonstrate reasoning and to
effectively apply the steps of an inquiry/problem-solving process. Rubrics may
be used due to the open-ended nature of many of the problems. The teacher
assesses:
·
broad-based,
open-ended problems on assessment tasks;
·
rich assessment
tasks and assignments;
·
problem-solving
strategies used in group work, through observation;
·
student-teacher
conferencing;
·
tasks requiring
mathematical reasoning.
Application
Achievement in this
category reflects the student’s ability to apply concepts and procedures to
familiar and unfamiliar settings. The teacher assesses:
·
appropriate
application of technological tools;
·
rich problems in
unit tests and tasks;
·
application of
mathematical knowledge and understanding in reports and presentations;
·
investigations in
examinations.
Communication
Achievement in this
category reflects the student’s ability to communicate his/her reasoning using
mathematical language, symbols, and conventions. Rubrics are effective,
efficient tools for evaluating presentations and displays. The teacher
assesses:
·
verbal
presentation of homework solutions;
·
appropriate use
of mathematical language and terminology on tests and assignments;
·
visual aids
during presentations;
·
clarity of
written expression in solutions;
·
student
interaction during group work, through observation;
·
clarity of
mathematical reasoning in reports and presentations;
·
mathematical
conventions on all written work.
Evaluation Notes
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.
Learning Skills
Learning skills are
not to be included in the determination of a student’s percentage grade.
Learning skills are assessed and reported separately from the student’s
percentage grade. Students should receive ongoing feedback concerning their
demonstration of learning skills; therefore, learning skills should be tracked
throughout the term. See Appendix A for examples of learning skill indicators.
Assessment
Strategies
Though assessment
strategies are listed for each activity, it is not intended that they all be
used for summative purposes. Teachers use some of the strategies for formative
purposes in order to build capacity and confidence in students in preparation
for the unit summative assessment and evaluation. References to assessment of
Learning Skills assume the understanding that these do not contribute to the
final mark but may provide data for the Learning Skills section of the report
card.
Accommodations
Teachers should
consult individual student IEPs for specific direction on accommodation for
individuals. Teachers work in consultation with resource teachers, where
available, and parents/guardians to determine appropriate accommodations as
students work to achieve the expectations in their IEPs.
Learning
Disabilities
·
Provide students
with an overview of activities to anticipate issues that may arise.
·
Assist with
lesson-specific terminology.
·
Modify handouts
in terms of terminology, content, and font size. Allow plenty of space for
written responses.
·
Allow assignments
to be completed in alternate formats or in longer timelines.
·
Allow students to
work in alternative settings.
·
Provide a list of
terms (possibly simplified) before an activity begins.
·
Provide
manipulatives, grid paper, formula sheets, and other aids.
ESL/ELD
·
Review questions,
assignments, or assessment instruments for language level.
·
Pair written
instructions with verbal instructions.
·
Provide visual or
auditory cues.
·
Provide
opportunities for students to practise oral presentations skills in low-risk
settings.
·
Use visuals to
illustrate definitions.
·
Simplify
instructions and highlight key words and phrases.
·
Have students
work in pairs, with peer tutors, with classmates who share the same linguistic
background, or in co-operative, supportive groups.
·
Use peer
conferencing to reinforce instructions or information.
·
Reinforce main
ideas by using the think/pair/share strategy.
·
Brainstorm in
groups using the student’s first language if their usage of English is limited.
·
Participate in
ongoing student-teacher conferencing.
·
Provide sets of
reference notes; outlines of critical information; and models of charts,
timelines, and diagrams.
Resources
Units in this Course
Profile make reference to the use of specific texts, magazines, films, videos,
and websites. The 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
licence and that this licence 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
licence from an authorized distributor, e.g., Audio Cine Films Inc. The
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 from
the Internet is not allowed without the permission of the owner.
Software (Ministry-Licensed)
Geometer’s
Sketchpad (dynamic geometry)
Maple (word processor/programming)
Math Trek
(concept and skill
development)
Virtual Tiles (algebraic concept and skill development)
Zap-a-Graph (graphing)
Websites
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 student use.
Education
Network of Ontario – www.enoreo.on.ca/
Hewlett-Packard
– www.hp.com/calculators/
Internet Public
Library – www.ipl.org
Math Forum –
http://forum.swarthmore.edu
National
Council of Teachers of Mathematics – www.nctm.org
Ontario
Association of Mathematics Educators – www.oame.on.ca
Texas Instruments –
www.ti.com/calc/docs
Books
Brueningsen,
C., et al. Real-World Math with the CBL
System – 25 Activities Using the CBL and TI-82. Texas Instruments, 1994.
Brueningsen,
C., et al. Real-World Math with the CBL
System – Activities for the TI-83 and TI-83 Plus. Texas Instruments, 1994.
Bush, W.S.
and A.S. Greer, eds. Mathematics
Assessment – A Practical Handbook for Grades 9-12. Retson, VA: The National
Council of Teachers of Mathematics, 1999.
Garland, T.
and C. Kahn. Math and Music – Harmonious
Connections. Dale Seymour
Publications, 1995.
Gregory, K.,
C. Cameron, and A. Davies. Knowing What
Counts: Setting and Using Criteria. Meriville, BD: Connections Publishing,
1999.
High School Assessment: Balanced Assessment for
the Mathematics Curriculum, Package 1. Dale Seymour
Publications, 2000.
High School Assessment: Balanced Assessment for
the Mathematics Curriculum, Package 2. Dale Seymour Publications, 2000.
National
Council of Teachers of Mathematics. Assessment
Standards for School Mathematics. Reston, VA: National Council of Teachers
of Mathematics, 1997.
O’Connor, K.
The Mindful School: How to Grade for
Learning. Palatine, IL: Skylight Training and Publishing Inc., 1998.
O.S.S.T.F. Quality Assessment. Toronto: Educational
Services Committee, 1999.
Rogers, S.
and S. Graham. The High Performance
Toolbox. Evergreen, CO: Peak Learning Systems, 1997.
Stiggins, R.
Classroom Assessment for Student Success.
Washington, DC: National Education Association of the United States, 1998
Stiggins, R.
Student-Centered Classroom Assessment,
2nd ed. Columbus OH: MacMillan, 1997.
Taggart, G., ed. Rubrics – A Handbook for Construction and
Use. Lancaster, PA: Techonomic Publishing, 1998.
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:
· Catholic Curriculum Cooperative (Central Ontario Region). Blueprints.
· Ontario Catholic School Trustees’ Association. Catholicity Across the Curriculum.
· Institute for Catholic Education. Educating the Soul.
· Institute for Catholic Education. Ontario Catholic Secondary School Graduate Expectations.
· Ontario Conference of Catholic Bishops. This Moment of Promise.
Career Goals/Cooperative
Education Programs:
· Ontario Youth Apprenticeship Program
· Youth Employment Skills Program
Community Partnerships:
·
Refer to local board policies, e.g.,
Relations with Business – Corporate Donations, Sponsorships, and Agreements.
OSS Considerations
The
following resources support many of the Ontario Secondary School policies, as
well as the Ontario Catholic School Graduate Expectations.
Ministry of Education Policy and Reference Documents:
Choices Into Action: Guidance and Career
Education Program Policy, 2000.
Cooperative Education: Policies and Procedures
for Ontario Secondary Schools, 2000.
Individual Education Plans: Standards for
Development, Program Planning, and Implementation, 2000.
The Ontario Curriculum, Grades 9-10, Mathematics,
1999.
The Ontario Curriculum, Grades 11-12,
Mathematics, 2000.
Ontario Schools Code of Conduct.
Ontario Secondary Schools, Grades 9-12, Program
and Diploma Requirements, 1999.
Program Planning and Assessment, Grades 9-12,
2000.
Violence-Free Schools Policy.
Appendix A
Indicators of Learning Skills
Organization
The student:
·
uses a planning
process;
·
brings the
required materials to class;
·
shows
organization in his/her notebook;
·
uses appropriate
resources.
Work Habits
The student:
·
completes class work
and homework;
·
works with
attention to detail;
·
shows thought and
revision in written work;
·
reviews and
studies appropriately;
·
follows
instructions of assigned work;
·
uses class time
effectively and submits work on time.
Teamwork
The student:
·
listens actively;
·
shows respect for
all group members;
·
completes the
appropriate portion of the group’s work;
·
co-operates to
complete the task;
·
uses
conflict-management skills;
·
adopts a variety
of roles in group-work settings;
·
shares ideas;
·
works
constructively towards group goals.
Initiative
The student:
·
actively and
constructively participates in class discussions;
·
takes
responsibility for his/her own learning;
·
demonstrates
classroom leadership;
·
acts to solve
problems;
·
tries new
techniques or approaches to learning;
·
reflects on
his/her own progress and adapts strategies;
·
shows interest in
new learning.
Works Independently
The student:
·
demonstrates
commitment to the task;
·
uses a variety of
problem-solving strategies;
·
accepts
responsibility for his/her own behaviour;
·
plans and
executes tasks with minimal teacher assistance.
Coded Expectations, Mathematics for College Technology,
Grade 12,
College Preparation, MCT4C
Polynomial Functions and Inverse
Proportionality
Overall Expectations
PFV.01 · determine, through investigation, the
characteristics of the graphs of polynomial functions of various degrees;
PFV.02 · demonstrate facility in the algebraic
manipulation of polynomials;
PFV.03 · demonstrate an understanding of inverse
proportionality;
PFV.04 · determine, through investigation, the key
properties of reciprocal functions.
Specific Expectations
Investigating the
Graphs of Polynomial Functions
PF1.01 – determine, through investigation, using
graphing calculators or graphing software, various properties of the graphs of
polynomial functions (e.g., determine the effect of the degree of a polynomial
function on the shape of its graph; the effect of varying the coefficients in
the polynomial function; the type and the number of x-intercepts; the
behaviour near the x-intercepts; the end behaviours; the existence of
symmetry);
PF1.02 – describe the nature of change in polynomial
functions of degree greater than two, using finite differences in tables of
values;
PF1.03 – compare the nature of change observed in
polynomial functions of higher degree with that observed in linear and
quadratic functions;
PF1.04 – sketch the graph of a polynomial function
whose equation is given in factored form;
PF1.05 – determine an equation to represent a given
graph of a polynomial function, using methods appropriate to the situation
(e.g., using the zeros of the function; using a trial-and-error process on a
graphing calculator or graphing software; using finite differences).
Manipulating
Algebraic Expressions
PF2.01 – demonstrate an understanding of the
remainder theorem and the factor theorem;
PF2.02 – factor polynomial expressions of degree
greater than two, using the factor theorem;
PF2.03 – determine, by factoring, the real or complex
roots of polynomial equations of degree greater than two;
PF2.04 – determine the real roots of non-factorable
polynomial equations by interpreting the graphs of the corresponding functions,
using graphing calculators or graphing software;
PF2.05 – write the equation of a family of polynomial
functions, given the real or complex zeros [e.g., a polynomial function having
non-repeated zeros 5, –3, and –2 will be defined by the equation
f(x) = (x – 5)(x + 3)(x + 2), for k 
R;
PF2.06 – describe intervals and distances, using
absolute-value notation;
PF2.07 – solve factorable polynomial inequalities;
PF2.08 – solve non-factorable polynomial inequalities
by graphing the corresponding functions, using graphing calculators or graphing
software and identifying intervals above and below the x-axis.
Understanding
Inverse Proportionality
PF3.01 – construct tables of values, graphs, and
formulas to represent functions of inverse proportionality derived from
descriptions of realistic situations (e.g., the time taken to complete a job
varies inversely as the number of workers; the intensity of light radiating
equally in all directions from a source varies inversely as the square of the
distance between the source and the observer);
PF3.02 – solve problems involving relationships of
inverse proportionality.
Determining the Key
Properties of Reciprocal Functions
PF4.01 – sketch the graph of the reciprocal of a
given linear or quadratic function by considering the implications of the key
features of the original function as predicted from its equation (e.g., such
features as the domain, the range, the intervals where the function is positive
or negative, the intervals where the function is increasing or decreasing, the
zeros of the function);
PF4.02 – describe the behaviour of a graph near a
vertical asymptote;
PF4.03 – identify the horizontal asymptote of the
graph of a reciprocal function by examining the patterns in the values of the
given function.
Exponential and Logarithmic Functions
Overall Expectations
ELV.01 · demonstrate an understanding of the nature
of exponential growth and decay;
ELV.02 · define and apply logarithmic functions.
Specific Expectations
Understanding the
Nature of Exponential Growth and Decay
EL1.01 – identify, through investigations, using
graphing calculators or graphing software, the key properties of exponential
functions of the form ax (a > 0, a ¹ 1) and their graphs (e.g., the domain is the
set of the real numbers; the range is the set of the positive real numbers; the
function either increases or decreases throughout its domain; the graph has the
x-axis as an asymptote and has
y-intercept = 1);
EL1.02 – describe the graphical implications of
changes in the parameters a, b, and c in the equation
y = cax + b;
EL1.03 – compare the rates of change of the graphs of
exponential and non-exponential functions (e.g., those with equations y
= 2x, y = x2, y = x
, and y = 2x);
EL1.04 – describe the significance of exponential
growth or decay within the context of applications represented by various
mathematical models (e.g., tables of values, graphs, equations);
EL1.05 – pose and solve problems related to models of
exponential functions drawn from a variety of applications, and communicate the
solutions with clarity and justification.
Defining and
Applying Logarithmic Functions
EL2.01 – define the logarithmic function loga
x (a > 1) as the inverse of the exponential function ax,
and compare the properties of the two functions;
EL2.02 – express logarithmic equations in exponential
form, and vice versa;
EL2.03 – simplify and evaluate expressions containing
logarithms, using the laws of logarithms;
EL2.04 – solve simple problems involving logarithmic
scales (e.g., the Richter scale, the pH scale, the decibel scale).
Applications and Consolidation
Overall Expectations
ACV.01 · analyse models of linear, quadratic,
polynomial, exponential, or trigonometric functions drawn from a variety of
applications;
ACV.02 · analyse and interpret models of
piecewise-defined functions drawn from a variety of applications;
ACV.03 · solve linear-quadratic
systems and interpret their solutions within the contexts of applications;
ACV.04 · demonstrate facility in carrying out and
applying key manipulation skills.
Specific Expectations
Analysing Models of
Functions
AC1.01 – determine the key features of a mathematical
model (e.g., an equation, a table of values, a graph) of a function drawn from
an application;
AC1.02 – compare the key features of a mathematical
model with the features of the application it represents;
AC1.03 – predict future behaviour within an
application by extrapolating from a given model of a function;
AC1.04 – pose questions related to an application and
use a given function model to answer them.
Analysing and
Interpreting Models of Piecewise-Defined Functions
AC2.01 – demonstrate an understanding that some
naturally occurring functions cannot be represented by a single formula (e.g.,
the temperature at a particular location as a function of time);
AC2.02 – graph a piecewise-defined function, by hand
and by using graphing calculators or graphing software;
AC2.03 – analyse and interpret a given mathematical
model of a piecewise-defined function, and relate the key features of the model
to the characteristics of the application it represents;
AC2.04 – make predictions and answer questions about
an application represented by a graph or formula of a piecewise-defined
function;
AC2.05 – determine the effects on the graph and
formula of a piecewise-defined function of changing the conditions in the
situation that the function represents.
Solving
Linear-Quadratic Systems
AC3.01 – determine the key properties of a linear
function or a quadratic function, given the equation of the function, and
interpret the properties within the context of an application;
AC3.02 – solve linear-quadratic systems arising from
the intersections of the graphs of linear and quadratic functions;
AC3.03 – interpret the solution(s) to a linear-
quadratic system within the context of an application.
Consolidating Key
Skills
AC4.01 – perform numerical computations effectively,
using mental mathematics and estimation;
AC4.02 – solve problems involving ratio, rate, and
percent drawn from a variety of applications;
AC4.03 – solve problems involving trigonometric
ratios in right triangles and the sine and cosine laws in oblique triangles;
AC4.04 – demonstrate facility in using manipulation
skills related to solving linear, quadratic, and polynomial equations,
simplifying rational expressions, and operating with exponents.
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 -accepts accountability for
one’s 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 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.