Functions, University Preparation
This course introduces the mathematical concept of the function by extending students’ experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
Register now- Department: Math
- Course Developer: Educators Academy
- Development Date:
- Revision Date: 2021
- Course Title: Functions, University Preparation
- Course Reviser: Hersimran Kaur
- Grade: Grade 11
- Course Type: University
- Ministry Course Code: MCR3U
- Credit Value: 01
- Prerequisite: Principles of Mathematics MPM2D Grade 10
- Ministry Curriculum Policy Document: The Ontario Curriculum, grades 11 and 12, 2007 (Revised)
Overall Curriculum Expectations
Mathematical Process Expectations
-
i. develop, select, apply, and compare a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;
ii. develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures,
iii. assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;
iv. demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);
v. select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
vi. make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
vii. create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
viii. communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
Characteristics of Functions
-
i. demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations;
ii. determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications;
iii. demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.
Exponential Function
-
i. evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways;
ii. make connections between the numeric, graphical, and algebraic representations of exponential functions;
iii. identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.
Discrete Functions
-
i. demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal’s triangle;
ii. demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems;
iii. make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.
Trigonometric Functions
-
i. determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;
ii. demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;
iii. identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.
Unit Outline
# | Unit | Approx. Time |
---|---|---|
1 | Characteristics of Functions | 27 Hours |
2 | Exponential Functions | 27 Hours |
3 | Discrete Functions | 27 Hours |
4 | Trigonometric Functions | 27 Hours |
5 | Final Examination | 02 Hours |
Total | 110 Hours |
Unit Description
Characteristics of Functions
Students will explore functions in this unit, their representations, and their inverses, and how to make connections between the algebraic and graphical representations of functions using transformations. Students will learn how to determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications. By the end of the unit students will be able to demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.
Exponential Functions
This unit will explore several topics including evaluating powers with rational exponents, simplifying expressions containing exponents, and describing properties of exponential functions represented in a variety of ways. The emphasis will be on problem solving using these concepts.
Discrete Functions
This unit will explore several topics including evaluating powers with rational exponents, simplifying expressions containing exponents, and describing properties of exponential functions represented in a variety of ways. The emphasis will be on problem solving using these concepts.
Trigonometric Functions
This unit concentrates students' attention on determining the values of the trigonometric ratios for angles less than 360º; proving simple trigonometric identities and solving problems using the primary trigonometric ratios. The sine law and the cosine law are developed. Students will learn to demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions while solving problems involving sinusoidal functions, including problems arising from real-world applications.
Program Considerations
Assessment and Evaluation
- Projects
- Assignments
- Tests
- Classroom Discussions
- Questions and Answers during Investigation
- Presentations
- Final Exam
- Worksheets
- Group Discussions
- Investigations
- Homework
- Practice Worksheets
- Pre-Tests
- Portfolios
- Self Evaluations
- Exit Cards
- Conversations
- Checklists
- Rubrics
- provide a common framework that encompasses the curriculum expectations
for all courses outlined in this document;
- guide the development of quality assessment tasks and tools
(including rubrics);
- help teachers to plan instruction for learning;
- assist teachers in providing meaningful feedback to students;
- Seventy per cent of the grade will be based on
evaluations conducted throughout the course. This portion of the grade
should reflect the student’s most consistent level of achievement
throughout the course, although special consideration should be given to
more recent evidence of achievement.
- 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 suitable to the course content and administered
towards the end of the course.
A Summary Description of Achievement
in Each Percentage Grade Range
and Corresponding Level of Achievement |
||
Percentage Grade
Range
|
Achievement Level
|
Summary
Description
|
80-100%
|
Level 4
|
A very high to outstanding level of achievement.
Achievement is above the provincial standard.
|
70-79%
|
Level 3
|
A high level of achievement. Achievement is at the
provincial standard.
|
60-69%
|
Level 2
|
A moderate level of achievement. Achievement
is below, but approaching, the provincial standard.
|
50-59%
|
Level 1
|
A passable level of achievement. Achievement
is below the provincial standard.
|
below 50%
|
Level R
|
Insufficient achievement of curriculum
expectations. A credit will not be granted.
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Knowledge
and Understanding
|
The student:
|
|
|
|
Knowledge of content
(e.g., facts, terms,
procedural skills, use
of tools)
Understanding of
mathematical concepts
|
– demonstrates limited
knowledge of content
– demonstrates limited
understanding of
concepts
|
– demonstrates some
knowledge of content
– demonstrates some
understanding of
concepts
|
– demonstrates
considerable knowledge
of content
– demonstrates
considerable understanding
of concepts
|
– demonstrates
thorough knowledge
of content
– demonstrates
thorough understanding
of concepts
|
Understanding of content (e.g., concepts, ideas, theories, principles,
procedures,
processes)
|
demonstrates
limited
understanding
of content
|
demonstrates
some
understanding
of content
|
demonstrates
considerable
understanding
of content
|
demonstrates
thorough
understanding
of content
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Thinking/
Inquiry
|
The student:
|
|
|
|
Use of planning skills
– understanding the
problem (e.g., formulating
and interpreting
the problem, making
conjectures)
– making a plan for solving
the problem)
|
– uses planning
skills with limited
effectiveness
|
uses planning
skills with some
effectiveness
|
uses planning skills
with considerable
effectiveness
|
– uses planning skills
with a high degree
of effectiveness
|
Use of processing skills
– carrying out a plan (e.g., collecting data,
questioning, testing, revising, modelling,
solving, inferring, forming conclusions)
– looking back at the solution (e.g.,
evaluating
reasonableness,
making convincing
arguments, reasoning,
justifying, proving,reflecting)
|
uses processing
skills and
strategies with
limited
effectiveness
|
uses processing
skills with
some
effectiveness
|
uses processing
skills with
considerable
effectiveness
|
uses processing
skills with a
high degree of
effectiveness
|
Use of critical/creative
thinking processes (e.g.,
problem solving, inquiry)
|
uses critical/
creative thinking
processes with limited
effectiveness
|
uses critical/
creative thinking
processes with some
effectiveness
|
uses critical/
creative thinking
processes, with considerable
effectiveness
|
uses critical/
creative thinking
processes with a high degree of
effectiveness
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Communication
|
The student:
|
|
|
|
Expression and organization of ideas and
mathematical thinking (e.g.,
clarity of expression, logical
organization), using oral, visual, and written forms
(e.g., pictorial, graphic, dynamic, numeric, algebraic
forms; concrete materials)
|
expresses and organizes
mathematical
thinking with limited
effectiveness
|
expresses and organizes
mathematical
thinking with some
effectiveness
|
expresses and organizes
mathematical
thinking with considerable
effectiveness
|
expresses and organizes
mathematical
thinking with a high degree of
effectiveness
|
Communication for different
audiences (e.g., peers, teachers) and purposes
(e.g., to present data, justify a solution, express a mathematical argument)
in oral, visual, and written formsin oral, visual, and/ or written forms
|
communicates for
different audiences and
purposes with
limited effectiveness
|
communicates for
different
audiences and
purposes with
some
effectiveness
|
communicates
for different
audiences and
purposes with
considerable
effectiveness
|
communicates
for different
audiences and
purposes with a
high degree of
effectiveness
|
Use of conventions,
vocabulary, and terminology
of the discipline (e.g.,
terms, symbols) in oral,
visual, and written forms
|
uses conventions,
vocabulary, and
terminology of the discipline with limited
effectiveness
|
uses conventions,
vocabulary, and
terminology of
the discipline
with some
effectiveness
|
uses conventions,
vocabulary,and terminology of
the discipline
with considerable
effectiveness
|
uses conventions,
vocabulary, and
terminology of
the discipline
with a high
degree of
effectiveness
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Application
|
The student:
|
|
|
|
Application of knowledge
and skills in familiar
contexts
|
applies
knowledge and
skills in familiar
contexts with
limited
effectiveness
|
applies
knowledge and
skills in familiar
contexts with
some
effectiveness
|
applies
knowledge and
skills in familiar
contexts with
considerable
effectiveness
|
applies
knowledge and
skills in familiar
contexts with a
high degree of
effectiveness
|
Transfer of knowledge and
skills to new contexts
|
transfers
knowledge and
skills to unfamiliar
contexts with
limited
effectiveness
|
transfers
knowledge and
skills to unfamiliar
contexts with
some effectiveness
|
transfers
knowledge and
skills to unfamiliar
contexts with
considerable
effectiveness
|
transfers
knowledge and
skills to unfamiliar
contexts with a
high degree of
effectiveness
|
use of equipment, materials and technology
|
uses equipment, materials and technology safely
and correctly only with supervision
|
uses equipment, materials and technology safely
and correctly with some supervision
|
uses equipment, materials and technology safely and
correctly
|
demonstrates and promotes the safe and correct
uses of
equipment, materials and technology
|
Making connections within
and between various contexts (e.g., connections
between concepts, representations, and forms
within mathematics; connections involving use of
prior knowledge and experience; connections between
mathematics, other disciplines, and the real world)
|
makes connections
within and between
various contexts with
limited effectiveness
|
makes connections
within and between
various contexts with
some effectiveness
|
makes connections
within and between
various contexts
with considerable
effectiveness
|
makes connections
within and between
various contexts
with a high degree
of effectiveness
|
- no accommodations or modifications; or
- accommodations only; or
- modified expectations, with the possibility of
accommodations.
Teaching & Learning Strategies
- Communicating: To improve student success there will be several
opportunities for students to share their understanding both in oral as
well as written form.
- 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;
- Learning
Goals and Success Criteria is explained to the students before starting any unit, task or
activity.
- Problem
solving: Scaffolding
of knowledge, detecting patterns, making and justifying conjectures,
guiding students as they apply their chosen strategy, directing students
to use multiple strategies to solve the same problem, when appropriate,
recognizing, encouraging, and applauding perseverance, discussing the
relative merits of different strategies for specific types of problems.
- Reasoning
and proving: Asking
questions that get students to hypothesize, providing students with one or
more numerical examples that parallel these with the generalization and
describing their thinking in more detail.
- Reflecting: Modeling the reflective process, asking
students how they know.
- Selecting
Tools and Computational Strategies: Modeling the use of tools and having
students use technology to help solve problems.
- Connecting: Activating prior knowledge when introducing
a new concept in order to make a smooth connection between previous
learning and new concepts, and introducing skills in context to make
connections between particular manipulations and problems that require
them.
- Representing: Modeling various ways to demonstrate
understanding, posing questions that require students to use different
representations as they are working at each level of conceptual
development - concrete, visual or symbolic, allowing individual students
the time they need to solidify their understanding at each conceptual
stage.
- Group
Work: Working
cooperatively in
groups reduces isolation and provides students with opportunities to share
ideas and communicate
their thinking in a supportive environment as they work together towards a common goal.
- Comparison and
evaluation of written work is very important in this course. This course
focuses on giving many examples of correct work, and helping students
build the skills needed to peer-correct and self correct.
- Oral presentations are a good tool
for learning. Students can learn from one another, and from their teacher.
Charts and graphs are used to present effective learning opportunities of
concepts and skills to different students.