Advanced Functions
This course extends students’ experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.
Register now- Department: Math
- Course Developer: The Educators Academy
- Development Date:
- Revision Date: 2021
- Course Title: Advanced Functions
- Course Reviser: Hersimran Kaur
- Grade: Grade 12
- Course Type: University
- Ministry Course Code: MHF4U
- Credit Value: 01
- Prerequisite: Functions, Grade 11, University Preparation or Mathematics for College Technology, Grade 12, College Preparation
- 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.
Exponential and Logarithmic Functions
-
i. demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
ii. identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
iii. solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.
Trigonometric Functions
-
i. demonstrate an understanding of the meaning and application of radian measure;
ii. make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;
iii. solve problems involving trigonometric equations and prove trigonometric identities.
Polynomial and Rational Functions
-
i. identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
ii. identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
iii. solve problems involving polynomial and simple rational equations graphically and algebraically; demonstrate an understanding of solving polynomial and simple rational inequalities.
Characteristics of Functions
-
i. demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;
ii. determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;
iii. compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.
Unit Outline
# | Unit | Approx. Time |
---|---|---|
1 | Exponential and Logarithmic Functions | 27 Hours |
2 | Trigonometric Functions | 27 Hours |
3 | Polynomial and Rational Functions | 27 Hours |
4 | Characteristics of Functions | 27 Hours |
5 | Final Examination | 02 Hours |
Total | 110 Hours |
Unit Description
Exponential and Logarithmic Functions
This unit begins with a review of exponential functions, their properties, and applications. This leads into discussions about a related function, the logarithmic function. From here students learn about logarithmic properties and then apply their knowledge of exponential and logarithmic functions to solve real-world problems.
Trigonometric Functions
This unit develops students understanding of trigonometry by expanding on the functions behind the trigonometric ratios. Students look at trigonometric functions and their reciprocals, examine their key properties and behaviours, and learn how they can be transformed to model a wide range of data.
Polynomial and Rational Functions
In this unit students learn to identify and describe some key features of polynomial functions and to make connections between the numeric, graphical, and algebraic representations of polynomial functions. These concepts allow students to manipulate functions in a number of ways and apply their skills to solve real-world problems. Strategies will be employed to aid in the connection to an understanding of rates of change. Students will demonstrate an understanding by identifying and describing some of the key features of rational functions. Students then learn to represent and manipulate these functions to solve real-life problems, graphically and algebraically.
Characteristics of Functions
Having studied various types of functions and transformations of functions, and understood the significance of differential rates of change in functions, this final unit focuses on the theory and practice of performing arithmetic operations on entire functions, including but not limited to the algebraic, graphical and practical implications of performing those operations.
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.