Math 3200 Curriculum Guide - Education and Early Childhood

Math 3200 Curriculum Guide - Education and Early Childhood
Mathematics
Advanced Mathematics 3200
Interim Edition
Curriculum Guide
2013
CONTENTS
Contents
Acknowledgements
.................................................................................................. iii
Introduction
....................................................................................................................1
Background .............................................................................................................................1
Beliefs About Students and Mathematics ...............................................................................1
Affective Domain .....................................................................................................................2
Goals For Students ..................................................................................................................2
Conceptual Framework for 10-12 Mathematics
..................................3
Mathematical Processes .........................................................................................................3
Nature of Mathematics ............................................................................................................7
Essential Graduation Learnings .............................................................................................10
Outcomes and Achievement Indicators ................................................................................. 11
Program Organization ............................................................................................................12
Summary ..............................................................................................................................12
Assessment and Evaluation .. .............................................................................13
Assessment Strategies .........................................................................................................15
Instructional Focus
Planning for Instruction ........................................................................................................17
Teaching Sequence ..............................................................................................................17
Instruction Time Per Unit ......................................................................................................17
Resources ............................................................................................................................17
General and Specific Outcomes
...................................................................18
Outcomes with Achievement Indicators
Unit 1: Polynomial Functions ................................................................................................19
Unit 2: Function Transformations ...........................................................................................39
Unit 3: Radical Functions .......................................................................................................69
Unit 4: Trigonometry and the Unit Circle ...............................................................................81
Unit 5: Trigonometric Functions and Graphs .......................................................................103
Unit 6: Trigonometric Identities ........................................................................................... 117
Unit 7: Exponential Functions .............................................................................................139
Unit 8: Logarithmic Functions .............................................................................................157
Unit 9: Permutations, Combinations and the Binomial Theorem ........................................179
Appendix
Outcomes with Achievement Indicators Organized by Topic ..............................................207
References .......................................................................................................................221
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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CONTENTS
ii
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
ACKNOWLEDGEMENTS
Acknowledgements
The Department of Education would like to thank the Western and Northern Canadian Protocol (WNCP)
for Collaboration in Education, The Common Curriculum Framework for K-9 Mathematics - May 2006
and The Common Curriculum Framework for Grades 10-12 - January 2008, reproduced and/or adapted by
permission. All rights reserved.
We would also like to thank the provincial Grade 12 Mathematics curriculum committee and the following
people for their contribution:
Joanne Hogan, Program Development Specialist – Mathematics, Division of Program
Development, Department of Education
Deanne Lynch, Program Development Specialist – Mathematics, Division of Program
Development, Department of Education
John-Kevin Flynn, Test Development Specialist – Mathematics/Science, Division of
Evaluation and Research, Department of Education
Nicole Bishop, Mathematics Teacher – Prince of Wales Collegiate, St. John’s
Jason Counsel, Mathematics Teacher – Laval High, Placentia
Sheldon Critch, Mathematics Teacher – Botwood Collegiate, Botwood
Dennis Ivany, Mathematics Teacher – CDLI, Grand Falls-Windsor
Renee Jesso, Mathematics Teacher – Piccadilly High, Piccadilly
William MacLellan, Mathematics Teacher – Gander Collegiate, Gander
Jonathan Payne, Mathematics Teacher – Corner Brook Regional High, Corner Brook
Heather Peddle, Numeracy Support Teacher – Eastern School District
Blaine Priddle, Mathematics Teacher – Templeton Academy, Meadows
Paul Thomas, Mathematics Teacher – Carbonear Collegiate, Carbonear
Michael Vivian, Mathematics Teacher – Marystown Central High, Marystown
Jaime Webster, Mathematics Teacher – St. James Regional High, Port aux Basques
Every effort has been made to acknowledge all sources that contributed to the development of this document.
Any omissions or errors will be amended in future printings.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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ACKNOWLEDGEMENTS
iv
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
INTRODUCTION
INTRODUCTION
Background
The curriculum guide
communicates high
expectations for students.
The Mathematics curriculum guides for Newfoundland and Labrador
have been derived from The Common Curriculum Framework for 1012 Mathematics: Western and Northern Canadian Protocol, January
2008. These guides incorporate the conceptual framework for Grades
10 to 12 Mathematics and the general outcomes, specific outcomes
and achievement indicators established in the common curriculum
framework. They also include suggestions for teaching and learning,
suggested assessment strategies, and an identification of the associated
resource match between the curriculum and authorized, as well as
recommended, resource materials.
Beliefs About
Students and
Mathematics
Students are curious, active learners with individual interests, abilities
and needs. They come to classrooms with varying knowledge, life
experiences and backgrounds. A key component in developing
mathematical literacy is making connections to these backgrounds and
experiences.
Mathematical
understanding is fostered
when students build on
their own experiences and
prior knowledge.
Students learn by attaching meaning to what they do, and they need
to construct their own meaning of mathematics. This meaning is best
developed when learners encounter mathematical experiences that
proceed from the simple to the complex and from the concrete to the
abstract. Through the use of manipulatives and a variety of pedagogical
approaches, teachers can address the diverse learning styles, cultural
backgrounds and developmental stages of students, and enhance
within them the formation of sound, transferable mathematical
understandings. Students at all levels benefit from working with a
variety of materials, tools and contexts when constructing meaning
about new mathematical ideas. Meaningful student discussions provide
essential links among concrete, pictorial and symbolic representations
of mathematical concepts.
The learning environment should value and respect the diversity
of students’ experiences and ways of thinking, so that students feel
comfortable taking intellectual risks, asking questions and posing
conjectures. Students need to explore problem-solving situations in
order to develop personal strategies and become mathematically literate.
They must come to understand that it is acceptable to solve problems
in a variety of ways and that a variety of solutions may be acceptable.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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INTRODUCTION
Affective Domain
To experience success,
students must learn to set
achievable goals and assess
themselves as they work
toward these goals.
A positive attitude is an important aspect of the affective domain and
has a profound impact on learning. Environments that create a sense of
belonging, encourage risk taking and provide opportunities for success
help develop and maintain positive attitudes and self-confidence within
students. Students with positive attitudes toward learning mathematics
are likely to be motivated and prepared to learn, participate willingly
in classroom activities, persist in challenging situations and engage in
reflective practices.
Teachers, students and parents need to recognize the relationship
between the affective and cognitive domains, and attempt to nurture
those aspects of the affective domain that contribute to positive
attitudes. To experience success, students must learn to set achievable
goals and assess themselves as they work toward these goals.
Striving toward success and becoming autonomous and responsible
learners are ongoing, reflective processes that involve revisiting,
asssessing and revising personal goals.
Goals For
Students
Mathematics education
must prepare students
to use mathematics
confidently to solve
problems.
The main goals of mathematics education are to prepare students to:
•
use mathematics confidently to solve problems
•
communicate and reason mathematically
•
appreciate and value mathematics
•
make connections between mathematics and its applications
•
commit themselves to lifelong learning
•
become mathematically literate adults, using mathematics to
contribute to society.
Students who have met these goals will:
2
•
gain understanding and appreciation of the contributions of
mathematics as a science, philosophy and art
•
exhibit a positive attitude toward mathematics
•
engage and persevere in mathematical tasks and projects
•
contribute to mathematical discussions
•
take risks in performing mathematical tasks
•
exhibit curiosity.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
MATHEMATICAL PROCESSES
CONCEPTUAL
FRAMEWORK
FOR 10-12
MATHEMATICS
Mathematical
Processes
•
Communication [C]
•
Connections [CN]
•
Mental Mathematics
and Estimation [ME]
•
Problem Solving [PS]
•
The chart below provides an overview of how mathematical processes
and the nature of mathematics influence learning outcomes.
There are critical components that students must encounter in a
mathematics program in order to achieve the goals of mathematics
education and embrace lifelong learning in mathematics.
Students are expected to:
• communicate in order to learn and express their understanding
•
connect mathematical ideas to other concepts in mathematics, to
everyday experiences and to other disciplines
•
demonstrate fluency with mental mathematics and estimation
•
develop and apply new mathematical knowledge through problem
solving
Reasoning [R]
•
develop mathematical reasoning
•
Technology [T]
•
•
Visualization [V]
select and use technologies as tools for learning and for solving
problems
•
develop visualization skills to assist in processing information,
making connections and solving problems.
This curriculum guide incorporates these seven interrelated
mathematical processes that are intended to permeate teaching and
learning.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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MATHEMATICAL PROCESSES
Communication [C]
Students need opportunities to read about, represent, view, write about,
listen to and discuss mathematical ideas. These opportunities allow
students to create links between their own language and ideas, and the
formal language and symbols of mathematics.
Students must be able to
communicate mathematical
ideas in a variety of ways
and contexts.
Communication is important in clarifying, reinforcing and modifying
ideas, attitudes and beliefs about mathematics. Students should be
encouraged to use a variety of forms of communication while learning
mathematics. Students also need to communicate their learning using
mathematical terminology.
Communication helps students make connections among concrete,
pictorial, symbolic, oral, written and mental representations of
mathematical ideas.
Connections [CN]
Through connections,
students begin to view
mathematics as useful and
relevant.
Contextualization and making connections to the experiences
of learners are powerful processes in developing mathematical
understanding. When mathematical ideas are connected to each other
or to real-world phenomena, students begin to view mathematics as
useful, relevant and integrated.
Learning mathematics within contexts and making connections relevant
to learners can validate past experiences and increase student willingness
to participate and be actively engaged.
The brain is constantly looking for and making connections. “Because
the learner is constantly searching for connections on many levels,
educators need to orchestrate the experiences from which learners extract
understanding … Brain research establishes and confirms that multiple
complex and concrete experiences are essential for meaningful learning
and teaching” (Caine and Caine, 1991, p.5).
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
MATHEMATICAL PROCESSES
Mental Mathematics
and Estimation [ME]
Mental mathematics and
estimation are fundamental
components of number sense.
Mental mathematics is a combination of cognitive strategies that
enhance flexible thinking and number sense. It is calculating mentally
without the use of external memory aids.
Mental mathematics enables students to determine answers without
paper and pencil. It improves computational fluency by developing
efficiency, accuracy and flexibility.
“Even more important than performing computational procedures or
using calculators is the greater facility that students need—more than
ever before—with estimation and mental math” (National Council of
Teachers of Mathematics, May 2005).
Students proficient with mental mathematics “... become liberated from
calculator dependence, build confidence in doing mathematics, become
more flexible thinkers and are more able to use multiple approaches to
problem solving” (Rubenstein, 2001, p. 442).
Mental mathematics “... provides the cornerstone for all estimation
processes, offering a variety of alternative algorithms and nonstandard
techniques for finding answers” (Hope, 1988, p. v).
Estimation is used for determining approximate values or quantities or
for determining the reasonableness of calculated values. It often uses
benchmarks or referents. Students need to know when to estimate, how
to estimate and what strategy to use.
Estimation assists individuals in making mathematical judgements and
in developing useful, efficient strategies for dealing with situations in
daily life.
Problem Solving [PS]
Learning through problem
solving should be the focus
of mathematics at all grade
levels.
Learning through problem solving should be the focus of mathematics
at all grade levels. When students encounter new situations and respond
to questions of the type, “How would you know?” or “How could
you ...?”, the problem-solving approach is being modelled. Students
develop their own problem-solving strategies by listening to, discussing
and trying different strategies.
A problem-solving activity requires students to determine a way to get
from what is known to what is unknown. If students have already been
given steps to solve the problem, it is not a problem, but practice. A
true problem requires students to use prior learning in new ways and
contexts. Problem solving requires and builds depth of conceptual
understanding and student engagement.
Problem solving is a powerful teaching tool that fosters multiple,
creative and innovative solutions. Creating an environment where
students openly seek and engage in a variety of strategies for solving
problems empowers students to explore alternatives and develops
confident, cognitive mathematical risk takers.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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MATHEMATICAL PROCESSES
Reasoning [R]
Mathematical reasoning
helps students think
logically and make sense of
mathematics.
Mathematical reasoning helps students think logically and make sense
of mathematics. Students need to develop confidence in their abilities
to reason and justify their mathematical thinking. High-order questions
challenge students to think and develop a sense of wonder about
mathematics.
Mathematical experiences in and out of the classroom provide
opportunities for students to develop their ability to reason. Students
can explore and record results, analyze observations, make and test
generalizations from patterns, and reach new conclusions by building
upon what is already known or assumed to be true.
Reasoning skills allow students to use a logical process to analyze a
problem, reach a conclusion and justify or defend that conclusion.
Technology [T]
Technology contributes
to the learning of a wide
range of mathematical
outcomes and enables
students to explore
and create patterns,
examine relationships,
test conjectures and solve
problems.
Technology contributes to the learning of a wide range of mathematical
outcomes and enables students to explore and create patterns, examine
relationships, test conjectures and solve problems.
Technology can be used to:
•
explore and demonstrate mathematical relationships and patterns
•
organize and display data
•
extrapolate and interpolate
•
assist with calculation procedures as part of solving problems
•
decrease the time spent on computations when other mathematical
learning is the focus
•
reinforce the learning of basic facts
•
develop personal procedures for mathematical operations
•
create geometric patterns
•
simulate situations
•
develop number sense.
Technology contributes to a learning environment in which the growing
curiosity of students can lead to rich mathematical discoveries at all
grade levels.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
NATURE OF MATHEMATICS
Visualization [V]
Visualization “involves thinking in pictures and images, and the ability
to perceive, transform and recreate different aspects of the visual-spatial
world” (Armstrong, 1993, p. 10). The use of visualization in the study
of mathematics provides students with opportunities to understand
mathematical concepts and make connections among them.
Visualization is fostered
through the use of concrete
materials, technology
and a variety of visual
representations.
Visual images and visual reasoning are important components of
number, spatial and measurement sense. Number visualization occurs
when students create mental representations of numbers.
Being able to create, interpret and describe a visual representation is
part of spatial sense and spatial reasoning. Spatial visualization and
reasoning enable students to describe the relationships among and
between 3-D objects and 2-D shapes.
Measurement visualization goes beyond the acquisition of specific
measurement skills. Measurement sense includes the ability to
determine when to measure, when to estimate and which estimation
strategies to use (Shaw and Cliatt, 1989).
Nature of
Mathematics
•
Change
•
Constancy
•
Number Sense
•
Patterns
•
Relationships
•
Spatial Sense
•
Uncertainty
Mathematics is one way of trying to understand, interpret and describe
our world. There are a number of components that define the nature
of mathematics and these are woven throughout this curiculum guide.
The components are change, constancy, number sense, patterns,
relationships, spatial sense and uncertainty.
Change
It is important for students to understand that mathematics is dynamic
and not static. As a result, recognizing change is a key component in
understanding and developing mathematics.
Change is an integral part
of mathematics and the
learning of mathematics.
Within mathematics, students encounter conditions of change and are
required to search for explanations of that change. To make predictions,
students need to describe and quantify their observations, look for
patterns, and describe those quantities that remain fixed and those that
change. For example, the sequence 4, 6, 8, 10, 12, … can be described
as:
•
the number of a specific colour of beads in each row of a beaded
design
•
skip counting by 2s, starting from 4
•
an arithmetic sequence, with first term 4 and a common difference
of 2
•
a linear function with a discrete domain
(Steen, 1990, p. 184).
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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NATURE OF MATHEMATICS
Constancy
Constancy is described by the
terms stability, conservation,
equilibrium, steady state and
symmetry.
Different aspects of constancy are described by the terms stability,
conservation, equilibrium, steady state and symmetry (AAASBenchmarks, 1993, p.270). Many important properties in mathematics
and science relate to properties that do not change when outside
conditions change. Examples of constancy include the following:
•
The ratio of the circumference of a teepee to its diameter is the
same regardless of the length of the teepee poles.
•
The sum of the interior angles of any triangle is 180°.
•
The theoretical probability of flipping a coin and getting heads is
0.5.
Some problems in mathematics require students to focus on properties
that remain constant. The recognition of constancy enables students to
solve problems involving constant rates of change, lines with constant
slope, direct variation situations or the angle sums of polygons.
Number Sense
An intuition about number
is the most important
foundation numeracy.
Number sense, which can be thought of as intuition about numbers,
is the most important foundation of numeracy (British Columbia
Ministry of Education, 2000, p.146).
A true sense of number goes well beyond the skills of simply counting,
memorizing facts and the situational rote use of algorithms. Mastery
of number facts is expected to be attained by students as they develop
their number sense. This mastery allows for facility with more
complex computations but should not be attained at the expense of an
understanding of number.
Number sense develops when students connect numbers to their own
real-life experiences and when students use benchmarks and referents.
This results in students who are computationally fluent and flexible with
numbers and who have intuition about numbers. The evolving number
sense typically comes as a by product of learning rather than through
direct instruction. It can be developed by providing rich mathematical
tasks that allow students to make connections to their own expereinces
and their previous learning.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
NATURE OF MATHEMATICS
Patterns
Mathematics is about
recognizing, describing and
working with numerical
and non-numerical
patterns.
Mathematics is about recognizing, describing and working with
numerical and non-numerical patterns. Patterns exist in all strands of
mathematics.
Working with patterns enables students to make connections within
and beyond mathematics. These skills contribute to students’
interaction with, and understanding of, their environment.
Patterns may be represented in concrete, visual or symbolic form.
Students should develop fluency in moving from one representation to
another.
Students must learn to recognize, extend, create and use mathematical
patterns. Patterns allow students to make predictions and justify their
reasoning when solving routine and non-routine problems.
Learning to work with patterns in the early grades helps students
develop algebraic thinking, which is foundational for working with
more abstract mathematics.
Relationships
Mathematics is used to
describe and explain
relationships.
Spatial Sense
Spatial sense offers a way to
interpret and reflect on the
physical environment.
Mathematics is one way to describe interconnectedness in a holistic
worldview. Mathematics is used to describe and explain relationships. As
part of the study of mathematics, students look for relationships among
numbers, sets, shapes, objects and concepts. The search for possible
relationships involves collecting and analyzing data and describing
relationships visually, symbolically, orally or in written form.
Spatial sense involves visualization, mental imagery and spatial
reasoning. These skills are central to the understanding of mathematics.
Spatial sense is developed through a variety of experiences and
interactions within the environment. The development of spatial sense
enables students to solve problems involving 3-D objects and 2-D
shapes and to interpret and reflect on the physical environment and its
3-D or 2-D representations.
Some problems involve attaching numerals and appropriate units
(measurement) to dimensions of shapes and objects. Spatial sense
allows students to make predictions about the results of changing these
dimensions; e.g., doubling the length of the side of a square increases
the area by a factor of four. Ultimately, spatial sense enables students
to communicate about shapes and objects and to create their own
representations.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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ESSENTIAL GRADUATION LEARNINGS
Uncertainty
Uncertainty is an inherent
part of making predictions.
In mathematics, interpretations of data and the predictions made from
data may lack certainty.
Events and experiments generate statistical data that can be used to
make predictions. It is important to recognize that these predictions
(interpolations and extrapolations) are based upon patterns that have a
degree of uncertainty.
The quality of the interpretation is directly related to the quality of
the data. An awareness of uncertainty allows students to assess the
reliability of data and data interpretation.
Chance addresses the predictability of the occurrence of an outcome.
As students develop their understanding of probability, the language
of mathematics becomes more specific and describes the degree of
uncertainty more accurately.
Essential
Graduation
Learnings
Essential graduation learnings are statements describing the knowledge,
skills and attitudes expected of all students who graduate from high
school. Essential graduation learnings are cross-curricular in nature
and comprise different areas of learning: aesthetic expression, citizenship,
communication, personal development, problem solving, technological
competence and spiritual and moral development.
Aesthetic Expression
Graduates will be able to respond with critical awareness to various forms of
the arts and be able to express themselves through the arts.
Citizenship
Graduates will be able to assess social, cultural, economic and
environmental interdependence in a local and global context.
Communication
Graduates will be able to use the listening, viewing, speaking, reading and
writing modes of language(s) and mathematical and scientific concepts and
symbols to think, learn and communicate effectively.
Personal Development
Graduates will be able to continue to learn and to pursue an active, healthy
lifestyle.
Problem Solving
Graduates will be able to use the strategies and processes needed to solve
a wide variety of problems, including those requiring language and
mathematical and scientific concepts.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
OUTCOMES
Technological
Competence
Graduates will be able to use a variety of technologies, demonstrate an
understanding of technological applications, and apply appropriate
technologies for solving problems.
Spiritual and Moral
Development
Graduates will be able to demonstrate an understanding and appreciation
for the place of belief systems in shaping the development of moral values and
ethical conduct.
See Foundations for the Atlantic Canada Mathematics Curriculum, pages
4-6.
The mathematics curriculum is designed to make a significant
contribution towards students’ meeting each of the essential graduation
learnings (EGLs), with the communication, problem-solving and
technological competence EGLs relating particularly well to the
mathematical processes.
Outcomes and
Achievement
Indicators
The curriculum is stated in terms of general outcomes, specific
outcomes and achievement indicators.
General Outcomes
General outcomes are overarching statements about what students are
expected to learn in each course.
Specific Outcomes
Specific outcomes are statements that identify the specific skills,
understanding and knowledge that students are required to attain by the
end of a given course.
In the specific outcomes, the word including indicates that any ensuing
items must be addressed to fully meet the learning outcome. The phrase
such as indicates that the ensuing items are provided for illustrative
purposes or clarification, and are not requirements that must be
addressed to fully meet the learning outcome.
Achievement Indicators
Achievement indicators are samples of how students may demonstrate
their achievement of the goals of a specific outcome. The range of
samples provided is meant to reflect the scope of the specific outcome.
Specific curriculum outcomes represent the means by which students
work toward accomplishing the general curriculum outcomes and
ultimately, the essential graduation learnings.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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PROGRAM ORGANIZATION
Program
Organization
Program Level
Course 1
Course 2
Course 3
Course 4
Advanced
Academic
Applied
Mathematics
1201
Mathematics 1202
Mathematics 2200
Mathematics 2201
Mathematics 2202
Mathematics 3200
Mathematics 3201
Mathematics 3202
Mathematics 3208
The applied program is designed to provide students with the
mathematical understandings and critical-thinking skills identified
for entry into the majority of trades and for direct entry into the
workforce.
The academic and advanced programs are designed to provide students
with the mathematical understandings and critical-thinking skills
identified for entry into post-secondary programs. Students who
complete the advanced program will be better prepared for programs
that require the study of calculus.
The programs aim to prepare students to make connections between
mathematics and its applications and to become numerate adults, using
mathematics to contribute to society.
Summary
12
The conceptual framework for Grades 10-12 Mathematics (p. 3)
describes the nature of mathematics, mathematical processes and
the mathematical concepts to be addressed. The components are not
meant to stand alone. Activities that take place in the mathematics
classroom should result from a problem-solving approach, be based on
mathematical processes and lead students to an understanding of the
nature of mathematics through specific knowledge, skills and attitudes
among and between topics.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
ASSESSMENT
ASSESSMENT AND
EVALUATION
Purposes of Assessment
What learning is assessed and evaluated, how it is assessed and
evaluated, and how results are communicated send clear messages to
students and others about what is really valued.
Assessment techniques are used to gather information for evaluation.
Information gathered through assessment helps teachers determine
students’ strengths and needs in their achievement of mathematics and
guides future instructional approaches.
Teachers are encouraged to be flexible in assessing the learning success
of all students and to seek diverse ways in which students might
demonstrate what they know and are able to do.
Evaluation involves the weighing of the assessment information against
a standard in order to make an evaluation or judgment about student
achievement.
Assessment has three interrelated purposes:
Assessment for
Learning
•
assessment for learning to guide and inform instruction;
•
assessment as learning to involve students in self-assessment and
setting goals for their own learning; and
•
assessment of learning to make judgements about student
performance in relation to curriculum outcomes.
Assessment for learning involves frequent, interactive assessments
designed to make student understanding visible. This enables teachers
to identify learning needs and adjust teaching accordingly. It is an
ongoing process of teaching and learning.
Assessment for learning:
•
requires the collection of data from a range of assessments as
investigative tools to find out as mush as possible about what
students know
•
provides descriptive, specific and instructive feedback to students
and parents regarding the next stage of learning
•
actively engages students in their own learning as they assess
themselves and understand how to improve performance.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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ASSESSMENT
Assessment as Learning
Assessment as learning actively involves students’ reflection on their
learning and monitoring of their own progress. It focuses on the role of
the student as the critical connector between assessment and learning,
thereby developing and supporting metacognition in students.
Assessment as learning:
Assessment of Learning
•
supports students in critically analysing their learning related to
learning outcomes
•
prompts students to consider how they can continue to improve
their learning
•
enables students to use information gathered to make adaptations
to their learning processes and to develop new understandings.
Assessment of learning involves strategies to confirm what students
know, demonstrate whether or not they have met curriculum
outcomes, or to certify proficiency and make decisions about students’
future learning needs. Assessment of learning occurs at the end of a
learning experience that contributes directly to reported results.
Traditionally, teachers relied on this type of assessment to make
judgments about student performance by measuring learning after
the fact and then reporting it to others. Used in conjunction with the
other assessment processes previously outlined, however, assessment of
learning is strengthened.
Assessment of learning:
•
provides opportunities to report evidence to date of student
achievement in relation to learning outcomes, to parents/guardians
and other stakeholders
•
confirms what students know and can do
•
occurs at the end of a learning experience using a variety of tools.
Because the consequences of assessment of learning are often farreaching, teachers have the responsibility of reporting student learning
accurately and fairly, based on evidence obtained from a variety of
contexts and applications.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
ASSESSMENT
Assessment
Strategies
Assessment techniques should match the style of learning and
instruction employed. Several options are suggested in this curriculum
guide from which teachers may choose, depending on the curriculum
outcomes, the class and school/district policies.
Observation
This technique provides a way of gathering information fairly quickly
while a lesson is in progress. When used formally, the student(s) would
be aware of the observation and the criteria being assessed. Informally,
it could be a frequent, but brief, check on a given criterion. Observation
may offer information about the participation level of a student for a
given task, use of a concrete model or application of a given process.
The results may be recorded in the form of checklists, rating scales or
brief written notes. It is important to plan in order that specific criteria
are identified, suitable recording forms are ready, and all students are
observed within a reasonable period of time.
(formal or informal)
Performance
This curriculum encourages learning through active participation.
Many of the curriculum outcomes promote skills and their applications.
In order for students to appreciate the importance of skill development,
it is important that assessment provide feedback on the various skills.
These may be the correct manner in which to use a manipulative, the
ability to interpret and follow instructions, or to research, organize and
present information. Assessing performance is most often achieved
through observing the process.
Paper and Pencil
These techniques can be formative or summative. Whether as part of
learning, or a final statement, students should know the expectations
for the exercise and how it will be assessed. Written assignments and
tests can be used to assess knowledge, understanding and application of
concepts. They are less successful at assessing processes and attitudes.
The purpose of the assessment should determine what form of paper
and pencil exercise is used.
Journal
Journals provide an opportunity for students to express thoughts
and ideas in a reflective way. By recording feelings, perceptions of
success, and responses to new concepts, a student may be helped to
identify his or her most effective learning style. Knowing how to learn
in an effective way is powerful information. Journal entries also give
indicators of developing attitudes to mathematical concepts, processes
and skills, and how these may be applied in the context of society. Selfassessment, through a journal, permits a student to consider strengths
and weaknesses, attitudes, interests and new ideas. Developing patterns
may help in career decisions and choices of further study.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
15
ASSESSMENT
Interview
This curriculum promotes understanding and applying mathematics
concepts. Interviewing a student allows the teacher to confirm that
learning has taken place beyond simple factual recall. Discussion
allows a student to display an ability to use information and clarify
understanding. Interviews may be a brief discussion between teacher
and student or they may be more extensive. Such conferences allow
students to be proactive in displaying understanding. It is helpful for
students to know which criteria will be used to assess formal interviews.
This assessment technique provides an opportunity to students whose
verbal presentation skills are stronger than their written skills.
Presentation
The curriculum includes outcomes that require students to analyze and
interpret information, to be able to work in teams, and to communicate
information. These activities are best displayed and assessed through
presentations. These can be given orally, in written/pictorial form,
by project summary, or by using electronic systems such as video or
computer software. Whatever the level of complexity, or format used, it
is important to consider the curriculum outcomes as a guide to assessing
the presentation. The outcomes indicate the process, concepts and
context for which a presentation is made.
Portfolio
Portfolios offer another option for assessing student progress in meeting
curriculum outcomes over a more extended period of time. This form
of assessment allows the student to be central to the process. There are
decisions about the portfolio, and its contents, which can be made by
the student. What is placed in the portfolio, the criteria for selection,
how the portfolio is used, how and where it is stored, and how it is
evaluated are some of the questions to consider when planning to collect
and display student work in this way. The portfolio should provide a
long-term record of growth in learning and skills. This record of growth
is important for individual reflection and self-assessment, but it is also
important to share with others. For all students, it is exciting to review a
portfolio and see the record of development over time.
16
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
INSTRUCTIONAL FOCUS
INSTRUCTIONAL
FOCUS
Planning for Instruction
Consider the following when planning for instruction:
• Integration of the mathematical processes within each topic is
expected.
• By decreasing emphasis on rote calculation, drill and practice, and
the size of numbers used in paper and pencil calculations, more
time is available for concept development.
• Problem solving, reasoning and connections are vital to increasing
mathematical fluency and must be integrated throughout the
program.
• There should be a balance among mental mathematics and
estimation, paper and pencil exercises, and the use of technology,
including calculators and computers. Concepts should be
introduced using manipulatives and be developed concretely,
pictorially and symbolically.
• Students bring a diversity of learning styles and cultural
backgrounds to the classroom. They will be at varying
developmental stages.
Teaching Sequence
The curriculum guide for Advanced Mathematics 3200 is organized by
units. This is only a suggested teaching order for the course. There are a
number of combinations of sequences that would be appropriate.
Each two page spread lists the topic, general outcome, and specific
outcome.
Instruction Time Per Unit
The suggested number of hours of instruction per unit is listed in the
guide at the beginning of each unit. The number of suggested hours
includes time for completing assessment activities, reviewing and
evaluating. The timelines at the beginning of each unit are provided
to assist in planning. The use of these timelines is not mandatory.
However, it is mandatory that all outcomes are taught during the
school year, so a long term plan is advised. Teaching of the outcomes is
ongoing, and may be revisited as necessary.
Resources
The authorized resource for Newfoundland and Labrador students and
teachers is Pre-Calculus 12 (McGraw-Hill Ryerson). Column four of the
curriculum guide references Pre-Calculus 12 for this reason.
Teachers may use any other resource, or combination of resources, to
meet the required specific outcomes.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
17
GENERAL AND SPECIFIC OUTCOMES
GENERAL
AND SPECIFIC
OUTCOMES
GENERAL AND SPECIFIC OUTCOMES WITH ACHIEVEMENT
INDICATORS (pages 19-206)
This section presents general and specific outcomes with corresponding
achievement indicators and is organized by unit. The list of indicators
contained in this section is not intended to be exhaustive but rather to
provide teachers with examples of evidence of understanding that may
be used to determine whether or not students have achieved a given
specific outcome. Teachers may use any number of these indicators or
choose to use other indicators as evidence that the desired learning has
been achieved. Achievement indicators should also help teachers form a
clear picture of the intent and scope of each specific outcome.
Advanced Mathematics 3200 is organized into nine units: Polynomial
Functions, Function Transformations, Radical Functions, Trigonometry
and the Unit Circle, Trigonometric Functions and Graphs, Trigonometric
Identities, Exponential Functions, Logarithmic Functions, and
Permutations, Combinations and the Binomial Theorem.
18
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Polynomial Functions
Suggested Time: 14 Hours
POLYNOMIAL FUNCTIONS
Unit Overview
Focus and Context
Previous work with quadratic functions will be extended in this unit to
include polynomial functions of degree ≤ 5.
Students will apply the integral zero theorem, synthetic division, and
factoring techniques to determine the zeros of cubic, quartic and
quintic functions. They will relate the zeros to the x-intercepts of the
graph and then graph and analyze the polynomial functions.
Outcomes Framework
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF11
Graph and analyze polynomial
functions (limited to polynomial
functions of degree ≤ 5).
SCO RF10
Demonstrate an understanding
of factoring polynomials of
degree greater than 2 (limited to
polynomials of degree ≤ 5 with
integral coefficients).
20
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
RF1. Factor polynomial expressions of
the form:
Relations and Functions
AN5. Demonstrate an understanding
of common factoring and trinomial
factoring, concretely, pictorially and
symbolically.
[C, CN, R, V]
RF1. Interpret and explain the
relationships among data, graphs and
situations.
[C, CN, R, T, V]
RF2. Demonstrate an understanding
of relations and functions.
•
ax2 + bx + c, a ≠ 0
•
a2x2 - b2y2, a ≠ 0, b ≠ 0
RF10. Demonstrate an
understanding of factoring
polynomials of degree greater than 2
(limited to polynomials of degree ≤ 5
with integral coefficients).
•
a(f(x))2 + b(f(x)) + c, a ≠ 0
[C, CN, ME]
•
a (f(x)) - b (g(y)) , a ≠ 0, b ≠ 0
2
2
2
2
where a, b and c are rational numbers.
[CN, ME, R]
RF3. Analyze quadratic functions
of the form y = a(x - p)2 + q and
determine the:
•
vertex
•
domain and range
•
direction of opening
RF6. Relate linear relations expressed
in:
•
axis of symmetry
•
x- and y-intercepts. [CN, R, T, V]
•
slope-intercept form y = mx + b
•
general form Ax + By + C = 0
•
slope-point form
y - y1 = m(x - x1)
RF4. Analyze quadratic functions of
the form y = ax2 + bx + c to identify
characteristics of the corresponding
graph, including:
[C, R, V]
to their graphs.
[CN, R, T, V]
•
vertex
•
domain and range
•
direction of opening
•
axis of symmetry
•
x- and y-intercepts
RF11. Graph and analyze polynomial
functions (limited to polynomial
functions of degree ≤ 5).
[C, CN, T, V]
and to solve problems. [CN, PS, R,
T, V]
RF5. Solve problems that involve
quadratic equations. [C, CN, PS, R,
T, V]
RF7. Solve problems that involve
linear and quadratic inequalities in
two variables. [C, PS, T, V]
RF8. Solve problems that involve
quadratic inequalities in one variable.
[CN, PS, V]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
21
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Graph and analyze
polynomial functions (limited to
polynomial functions of degree
≤ 5).
[C, CN, T, V]
In Mathematics 1201, students related linear relations expressed in
slope-intercept form, general form and slope-point form to their graphs
(RF6). In Mathematics 2200, they analyzed quadratic functions to
identify the characteristics of the corresponding graph (RF3, RF4). In
this unit, students graph and analyze polynomial functions of degree 5
or less.
Achievement Indicators:
RF11.1 Identify the polynomial
functions in a set of functions,
and explain the reasoning.
RF11.2 Explain the role of
the constant term and leading
coefficient in the equation of a
polynomial function with respect
to the graph of the function.
Linear and quadratic functions are examples of polynomial functions
that students have already studied. They will now extend their study
of polynomials to include cubic, quartic and quintic functions. A
function of the form anxn + an-1xn-1 + ... + a2x2 + a1x + a0, where an ≠ 0, is
a polynomial of degree n, and the number an is the leading coefficient.
Students have previously sketched the graph of polynomial functions of
degree 0, 1 and 2. They should recognize the following:
Function
f(x) = a
f(x) = ax + b
f(x) = ax2 +bx + c
Degree
0
1
2
Type of Function
constant
linear
quadratic
Graph
horizontal line
line with slope a
parabola
They will now be introduced to some of the basic features of the graphs
of polynomial functions of degree greater than 2:
• The graph of a polynomial function is continuous.
• The graph of a polynomial function has only smooth turns. A
function of degree n has at most n − 1 turns.
• If the leading coefficient of the polynomial function is positive, then
the graph rises to the right. If the leading coefficient is negative, then
the graph falls to the right.
• The constant term is the y-intercept of the graph.
The intent at this point is that students learn to recognize these basic
features. Later in the unit, they will use these features, point-plotting,
and intercepts to make reasonably accurate sketches.
To examine the basic features, students could graph polynomials with
technology, such as graphing calculators, or other graphing software,
such as FX Draw and WinPlot. Graphing software apps available for
students’ mobile devices could also be utilized.
22
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Students should determine whether or not each of the following is a
polynomial function and explain the reasoning.
Pre-Calculus 12
(i)
f (x ) = x − x
(ii)
f (x ) = 3 x 3 + 4
2
3.1 Characteristics of Polynomial
Functions
Student Book (SB): pp. 106-117
(RF11.1)
Teacher’s Resource (TR): pp. 6266
• Ask students to identify the features of the graph related to the
function f(x) = −3x2 + 9x + x5.
(RF11.2)
Interview
• Ask students to answer the following:
(i)
How many turns can the graph of a polynomial function of
degree 5 have? Explain.
(ii) Describe the characteristics of the graphs of cubic and quartic
functions with the largest possible number of terms.
(RF11.2)
Performance
• Students can work in groups to match the equations with the
appropriate graph. They should explain the features that guided the
selection of the appropriate graph.
Functions:
Graphs:
y
f(x) = x3 − 5x
f(x) = x5 − 3x3 + 2
f(x) = −x4 + x + 4
y
5
4
3
2
1
-1 1 2 3 4 5 x
-5-4-3-2-1
-2
-3
-4
-5
5
4
3
2
1
-1 1 2 3 4 5 x
-5-4-3-2-1
-2
-3
-4
-5
y
5
4
3
2
1
x
-1
-5-4-3-2-1
-2 1 2 3 4 5
-3
-4
-5
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
(RF11.2)
23
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Continued ...
Achievement Indicator:
RF11.3 Generalize rules for
graphing polynomial functions of
odd or even degree.
Students should explore the graphs of various polynomials with even
degree and odd degree. The polynomial functions that have the simplest
graphs are the monomial functions f(x) = anxn. When n is even, the
graph is similar to the graph of f(x) = x2. When n is odd, the graph is
similar to the graph of f(x) = x3. The greater the value of n, the flatter the
graph of a monomial is on the interval −1 ≤ x ≤ 1.
y
y=x
y
4
y=x
4
2
-2
-1
-2
5
y=x
3
4
2
-3
y=x
2
1
2
3 x
-2
-1
-4
-2
1
2 x
-4
Through exploration, students should see that if the degree of a
polynomial function is even, then its graph has the same behaviour
to the left and right. The graph of f(x) = x4, for example, rises to the
right and rises to the left. It extends up into Quadrant II and up into
Quadrant I. If the degree is odd, the graph has opposite behaviours to
the right and left. The graph of f(x) = -x3, for example, falls to the right
and rises to the left. It extends up into Quadrant II and down into
Quadrant IV.
Students could use graphing technology to determine any similarities
and differences between polynomials such as the following:
• f(x) = 2x + 1
• f(x) = x2 + 2x − 3
• f(x) = x3 + 2x2 − x − 2
• f(x) = x4 + 5x3 + 5x2 − 5x − 6
• f(x) = 0.2x5 − x4 − 2x3 + 10x2 + 1.4x − 9
They could then graph each of these with a negative leading coefficient.
From this, they should identify a pattern in the graphs of odd and even
degree functions. They should note patterns in the end behaviour, the
constant term and the number of real x-intercepts. Examples should be
limited to polynomials with real x-intercepts to allow students to easily
identify the patterns.
24
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to create a foldable or a graphic organizer to summarize
the rules for graphing polynomial functions of odd or even degree.
Pre-Calculus 12
(RF11.3)
3.1 Characteristics of Polynomial
Functions
SB: pp. 106-117
TR: pp. 62-66
Performance
• The activity Commit and Toss gives students an opportunity to
anonymously commit to an answer and provide a justification for
the answer they selected. Provide students with a selected response
question, as shown below. Students write their answer, crumble
their solutions into a ball, and toss the papers into a basket. Once all
papers are in the basket, ask students to reach in and take one out.
They then move to the corner of the room designated to match the
selected response on the paper they have taken. In their respective
corners, they should discuss the similarities and differences in the
explanation provided and report back to the class.
Which of the following is the graph of an even degree function?
(A)
(B)
y
y
4
3
3
2
2
1
1
-4 -3 -2 -1
1
-1
2
3
4 x
-3
-2
-1
1
2
3 x
-1
-2
-2
-3
-4
-3
(C)
(D)
y
y
4
3
3
2
2
1
1
-3
-2
-1
-2
1
2
3 x
-1
1
2 x
-1
-1
-2
-2
-3
Explain your reasoning:
(RF11.3)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
25
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF10 Demonstrate an
understanding of factoring
polynomials of degree greater
than 2 (limited to polynomials
of degree ≤ 5 with integral
coefficients).
Students were introduced to factoring techniques for quadratic functions
in Mathematics 1201 (AN5). They used common factors and trinomial
factoring to express quadratics in factored form. In Mathematics 2200,
these factoring techniques were extended (RF1) to factor expressions
with rational coefficients of the forms:
•
ax2 + bx + c, a ≠ 0
[C, CN, ME]
•
a2x2 − b2y2, a ≠ 0, b ≠ 0
•
a(f(x))2 + b(f(x)) + c, a ≠ 0
•
a2(f(x))2 − b2(g(y))2, a ≠ 0, b ≠ 0.
Achievement Indicators:
RF10.1 Explain how long
division of a polynomial
expression by a binomial
expression of the form x - a, a∈ Ι,
is related to synthetic division.
RF10.2 Divide a polynomial
expression by a binomial
expression of the form x - a,
a∈ I using long division or
synthetic division.
In this unit, these techniques are used to factor cubic, quartic and
quintic polynomials.
To factor higher order polynomials, long division or synthetic division
is used in combination with factoring techniques. Students should be
introduced to synthetic division in terms of it’s connection to long
division. They should observe that long division with polynomials
is similar to division with numerical expressions. To determine the
quotient for the expression (x3 − 2x2 + 3x − 4) ÷ (x + 2), for example,
either long division or synthetic division could be used. Students should
see the connection between the divisor and dividend in long division
and the root and coefficients in synthetic division. As a result, synthetic
division requires fewer calculations.
Long Division:
x + 2 x 3 − 2 x 2 + 3x − 4
Synthetic Division:
−2 1 − 2 3 − 4
− 2 8 − 22
1 − 4 11 −26
Teachers should work through both processes to allow students to make
the connection between each type of division. Synthetic division can
be regarded as a more efficient method than doing long division of two
polynomials when the divisor is a linear function of the form (x − a).
Although synthetic division can be completed using either the addition
or subtraction operation, it is recommended that the addition operation
be used. This helps make a connection between synthetic division and
the remainder theorem.
Students should see that there may be a remainder at the end of the
process. In the example above,
(x 3 − 2 x 2 + 3x − 4 )÷ (x + 2 ) = (x 2 − 4 x + 11)+ x−+262 .
Students should also be encouraged to include any restrictions that may
exist. In this particular case, x ≠ −2. They determined non-permissible
values for rational expressions in Mathematics 2200 (AN4).
26
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
• Ask students to explain the connection between long
division and synthetic division, using the example
(2x3 − x2 − 13x − 6) ÷ (x + 2).
Pre-Calculus 12
3.2 The Remainder Theorem
(RF10.1)
SB: pp. 118-125
TR: pp. 67-70
Paper and Pencil
• Ask students to write an incorrect, but plausible, solution for the
division statement (x4 − x3 − 8x2 + 8) ÷ (x + 2). They should then
exchange solutions with a partner and find the errors in the solution.
(RF10.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
27
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF10 Continued ...
Achievement Indicators:
RF10.2 Continued
Encourage students to carefully examine the polynomial when setting
up synthetic division. A zero must be included for each missing
term in the dividend. A common error occurs in questions such as
(x 4 − 10x 2 + 2 x + 3 )÷ (x − 3 ) when students do not include zero for
the missing cubic term. Students should check the result by multiplying
the quotient and the divisor. This can help inform them if the division is
done correctly.
RF10.3 Explain the relationship
between the remainder when a
polynomial expression is divided
by x - a, a ∈ I, and the value of
the polynomial expression at x = a
(remainder theorem).
Students should be introduced to the remainder theorem which says
that if (x – a) is a linear divisor of a polynomial function p(x), then p(a)
is the remainder. Initially, students may not see a need for this theorem
since they can obtain remainders by using synthetic division. Exposure
to polynomials such as x9 - 1 or x100 + 1 should help them see that the
remainder theorem is more efficient in some cases.
RF10.4 Explain and apply
the factor theorem to express
a polynomial expression as a
product of factors.
The factor theorem, which states that a polynomial, P(x), has a factor
x - a if and only if P(a) = 0, can be used to determine the factors
of a polynomial expression. Students use the integral zero theorem
to relate the factors of a polynomial and the constant term of the
polynomial. The factors of the constant term indicate possible factors
of the polynomial. They then verify using the factor theorem. For
the polynomial f (x ) = x 3 − 3 x 2 − 4 x + 12 , for example, the possible
integral zeros are the factors of 12. Remind students to test both the
positive and negative factors. Since f(2) = 0, f(−2) = 0 and f(3) = 0, the
factors of the polynomial are (x - 2), (x + 2) and (x - 3). This method is
restricted to polynomial functions with distinct integral zeros only.
RF10.5 Explain the relationship
between the linear factors of
a polynomial expression and
the zeros of the corresponding
polynomial function.
Students should work with polynomial functions that also have noninteger zeros. The factor theorem can also be used in conjunction with
synthetic division to factor a polynomial. Students should use the
integral zero theorem to determine possible factors and verify one of
the factors using the factor theorem. Synthetic division is then applied,
resulting in a polynomial to be factored further. For the polynomial
f (x ) = 4 x 3 − 12 x 2 + 5 x + 6 , f (2) = 0. Using synthetic division gives:
2 4 −12 5
6
8 −8 −6
4
− 4 −3
0
This results in (x - 2)(4x2 - 4x - 3), which can be further factored to
(x -2)(2x - 3)(2x + 1).
28
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to determine the remainder for the function
f(x)= x9 + 3x4 − 5x + 1 if it is divided by x – 3.
(RF10.3)
• The volume of a triangular prism is given by
V = x3 + 18x2 + 80x + 96. Ask students to determine the missing
dimension(s) if two of the dimensions are (x + 2) and (x + 12).
They should identify any restrictions on the variable x.
(RF10.4)
Authorized Resource
Pre-Calculus 12
3.2 The Remainder Theorem
SB: pp. 118-125
TR: pp. 67-70
Interview
• Ask students to explain why the remainder theorem is an efficient
way to find the remainder when a polynomial expression is divided
by x – a.
(RF10.3)
3.3 The Factor Theorem
Performance
SB: pp. 126-135
• Ask students to explain how to determine the remainder when
10x4 − 11x3 − 8x2 + 7x + 9 is divided by 2x − 3.
TR: pp. 71-74
(RF10.3)
• Ask students to explain and demonstrate, using the integral zero
theorem, the factor theorem and synthetic division, how they
would determine the values of k that make (x – k) a factor of
f(x) = x3 − 4x2 − 11x + 30.
(RF10.4, RF10.2)
• Ask students to create a polynomial of degree ≥ 3 using linear
factors. They should exchange with other students and explain the
relationship between linear factors and the corresponding zeros of
the function.
(RF10.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
29
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF10 Continued ...
Achievement Indicators:
RF10.4, RF10.5 Continued
Students should also be able to use alternative methods to factor
polynomials, such as grouping. They should have been introduced to
grouping when they factored quadratics using decomposition. This can
be extended to polynomials such as f(x) = x3 - 3x2 - 4x + 12:
f (x ) = x 3 − 3 x 2 − 4 x + 12
f (x ) = x 2 (x − 3 ) − 4 (x − 3 )
f (x ) = (x − 3 )(x 2 − 4 )
f (x ) = (x − 3 )(x + 2 )(x − 2 )
Once a polynomial has been factored, students apply the zero product
property to determine the zeros. This is a natural extension of the work
done with solving quadratic equations in Mathematics 2200.
RF11 Continued ...
RF11.4 Explain the relationship
among the following:
•
the zeros of a polynomial
function
•
the roots of the corresponding
polynomial equation
•
the x-intercepts of the graph
of the polynomial function.
Students were introduced to the basic features of the graphs of
polynomial functions of degree greater than 2 at the beginning of this
unit. They will now use these features, along with the intercepts, to
graph polynomials of degree 5 or less.
Students were introduced to the zeros of a quadratic function, the
roots of the quadratic equation, and the x-intercepts of the graph in
Mathematics 2200 (RF5). It is important that they distinguish between
the terms zeros, roots and x-intercepts, and use the correct terms in
a given situation. Students could be asked to find the roots of the
equation 3 x 3 − 10 x 2 − 23 x − 10 = 0 , find the zeros of the function
f (x ) = 3 x 3 − 10 x 2 − 23 x − 10 , or determine the x-intercepts of
the graph of f (x ) = 3 x 3 − 10 x 2 − 23 x − 10 . In each case, they are
identifying the factors of the polynomial and solving to arrive at the
solution x = −1, x = − 23 , x = 5.
Students should realize that the degree of a polynomial indicates the
maximum number of x-intercepts for its graph. For each real x-intercept
there is a linear factor and a zero for the polynomial function.
30
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Given the equation and the graph below, ask students to identify the
zeros, roots and x-intercepts and explain how they are determined.
f(x) = x(x − 1)(x + 1)(x − 2)
Authorized Resource
Pre-Calculus 12
3.3 The Factor Theorem
y
SB: pp. 126-135
5
TR: pp. 71-74
4
3
2
1
-4
-3
-2
-1
-1
1
2
3
4
x
-2
(RF11.4)
• Ask students to answer the following questions for the polynomial
f(x) = x3 − 4x2 + x + 6.
(i) Algebraically or graphically determine the linear factors of f(x).
(ii) Explain the relationship between the linear factors and any xintercepts.
(RF11.4, RF10.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
3.4 Equations and Graphs of
Polynomial Functions
SB: pp. 136-152
TR: pp. 75-79
31
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Continued ...
Achievement Indicator:
RF11.5 Explain how the
multiplicity of a zero of a
polynomial function affects the
graph.
Students have solved polynomials to find distinct, real zeros that
correspond to distinct, real x-intercepts. They need to be aware that
the zeros will not always be distinct. Some polynomial functions may
have a multiplicity of a zero (i.e., double, triple, etc.), also referred
to as the order of a zero. Students should graph polynomials such as
f (x) = (x −1)(x − 1)(x + 2) or f (x) = (x + 1)(x + 1)(x + 1)(x − 1)to see
the effect of multiplicity of roots.
y
5
4
3
2
1
-1 1 2 3 4 5 x
-5-4-3-2-1
-2
-3
-4
-5
f(x) = (x – 1)(x – 1)(x + 2)
y
5
4
3
2
1
-1 1 2 3 4 5 x
-5-4-3-2-1
-2
-3
-4
-5
f(x) = (x + 1)(x + 1)(x + 1)(x – 1)
Ask students questions such as:
• What would happen to the graph if there was a zero of
multiplicity 4?
• What would be the effect on the graph if there are two double roots?
• What effect does a triple root have on the graph? a double root? a
single root?
Students should be encouraged to check other possibilities for
multiplicity of zeros for polynomials of degree ≤ 5 and the effect on
their respective graphs.
32
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Interview
• Ask students to explain, using an example, how a zero of multiplicity
5 affects the graph of a polynomial.
(RF11.5)
Performance
• Ask students to examine the following graphs for the possibility of
y
multiplicity of zeros.
9
8
7
6
5
4
3
2
1
-1
-5 -4 -3 -2 -1-2
-3
-4
-5
-6
-7
-8
-9
-10
Authorized Resource
Pre-Calculus 12
3.4 Equations and Graphs of
Polynomial Functions
SB: pp. 136-152
TR: pp. 75-79
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
x
y
20
18
16
14
12
10
8
6
4
2
-5 -4 -3 -2 -1-2
x
y
4
2
-5 -4 -3 -2 -1
x
-2
-4
Using Think-Pair-Share, give individual students time to think about
the similarities and differences among the graphs with respect to
multiplicity of zeros. Students then pair up with a partner to discuss
their ideas. After pairs discuss, students share their ideas in a smallgroup or whole-class discussion.
(RF11.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
33
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Continued ...
Achievement Indicator:
RF11.6 Sketch, with or without
technology, the graph of a
polynomial function.
Students should realize that in order to sketch a graph of any polynomial
without graphing technology they have to identify points such as the
x- and y-intercepts. They use the value of the leading coefficient to
determine the end behaviour and consider how multiplicity of zeros
affects the graph of the function.
Students should also be able to identify when the function is positive
and when it is negative. A table of intervals or a sign diagram consisting
of a number line, roots and test points can help with this. They
should realize that the function is neither positive nor negative at
the x-intercepts. The intervals should be expressed as set or interval
notation. Students should be familiar with both types of notation from
Mathematics 1201 (RF1).
Ask students to determine the intervals where the graph represented by
the function f(x) = (x + 4)(x + 1)(x − 1) is positive or negative.
They could use a table to determine the intervals:
x < −4
Interval
−4 < x < −1
+
−
Sign
−1 < x < 1
x>1
−
+
Sign lines were introduced in Mathematics 2200 when students graphed
absolute value functions (RF2). Relating the sign diagram to the x-axis
of the graph, students can substitute an x-value from each interval into
the function to determine where the function is positive or negative.
–
-5
+
-4
-3
–
-2
-1
0
+
1
2
3
4
5
Remind students of the effect of a multiplicity of a zero on the graph
of a polynomial function. Discuss how this would appear on a sign
diagram. Students should realize that for a zero of odd multiplicity
(e.g., a single root or a triple root), the sign of the function changes. If
a function has a zero of even multiplicity (e.g., a double root), the sign
does not change.
34
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Without using graphing technology, students could sketch the graph
of a polynomial such as f(x) = x3 − 2x2 − 4x + 8. They should be
exposed to examples where the x-intercepts can be determined using
grouping or other factoring techniques.
(RF11.6)
Authorized Resource
Pre-Calculus 12
3.4 Equations and Graphs of
Polynomial Functions
SB: pp. 136-152
Observation
• Observe as students sketch the graph of f(x) = −(x + 2)3(x − 4)
and verify using graphing technology. Ask them to explain what
characteristics of the polynomial (e.g., intercepts, multiplicity of
zeros, leading coefficient, positive or negative intervals) helped them
sketch the graph.
(RF11.6)
Performance
• Give Me Five is an activity that provides students with an
opportunity to individually and publicly reflect on their learning.
Ask students: What was the most significant thing you learned about
graphing polynomials?
TR: pp. 75-79
Note
SB: pp. 143-144
Students are not expected to
use transformations to graph
polynomial functions.
Give time for students to quietly reflect before asking for five
volunteers to share their reflections. A show of hands can also
indicate how many students had a similar thought each time a
student shares his or her reflection.
(RF11.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
35
POLYNOMIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Continued ...
Achievement Indicator:
RF11.7 Solve a problem by
modeling a given situation with a
polynomial function.
In Mathematics 2200, students solved problems by determining and
analyzing a quadratic equation (RF5). This will now be extended to
polynomial functions. When solving a problem, it is often necessary
to simplify the problem and express the problem with mathematical
language or symbols to make a mathematical model. It is an expectation
that students will apply skills developed in this unit to solve problems in
various contexts. For example, they should answer questions based on
area, volume and numbers, such as:
• Three consecutive integers have a product of −720. What are the
integers?
Students could model this situation with a polynomial function and
solve the equation to determine the integers.
• An open box is to be made from a 10-in. by 12-in. piece of cardboard
by cutting x-in. squares from each corner and folding up the sides. If
the volume of the box is 72 in.3, what are the dimensions?
It is necessary here to place restrictions on the independent variable.
Ask students why, in this case, the value of x is restricted to 0 < x < 5.
It is important for students to consider the possibility of inadmissable
roots in the context of the problem. They should realize that time,
length, width and height, for example, cannot be negative values.
RF11.8 Determine the equation
of a polynomial function given its
graph.
36
Students should also be able to analyze a graph and create a polynomial
equation that models the graph. They should be given a variety of
graphs that have both distinct roots and multiplicity of roots and asked
to determine the equation of the polynomial function that represents
the graph. In cases where a graph may possibly represent a polynomial
of either degree 3 or 5, or of either degree 2 or 4, students would need
to be provided with the degree of the resulting equation.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
POLYNOMIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Performance
• Students can work in groups of two for the activity Pass the Problem.
Each pair gets a problem that involves a situation to be modelled
with a polynomial function. Ask one student to write the first line
of the solution and then pass it to the second student. The second
student verifies the workings and checks for errors. If there is an
error, students should discuss what the error is and why it occurred.
The student then writes the second line of the solution and passes it
to the partner. This process continues until the solution is complete.
Authorized Resource
Pre-Calculus 12
3.4 Equations and Graphs of
Polynomial Functions
Sample Problem:
SB: pp. 136-152
The length, width, and height of a rectangular box are x cm,
(x – 4) cm, and (x + 5) cm, respectively. Find the dimensions of
the box if the volume is 132 cm3.
TR: pp. 75-79
(RF12.7)
Paper and Pencil
• Ask students to answer the following:
(i)
An open-topped box with a volume of 900 cm3 is made from
a rectangular piece of cardboard by cutting equal squares from
four corners and folding up the sides.
(a)
If the original dimensions of the cardboard are 30 cm
by 40 cm, find the side length of the square that is cut
from each corner.
(b)
Calculate the surface area.
(ii) The actual and projected number, C (in millions), of computers
sold for a region between 2010 and 2020 can be modelled
by C = 0.0092(t3 + 8t2 + 40t + 400) where t = 0 represents
2010. During which year are 8.51 million computers projected
to be sold?
(RF11.7)
• Ask students to model the polynomial function represented by the
graph as both a cubic function and a quintic function.
y
4
2
-4
-2
2
4
x
-2
-4
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
(RF11.8)
37
POLYNOMIAL FUNCTIONS
38
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Function Transformations
Suggested Time: 11 Hours
FUNCTION TRANSFORMATIONS
Unit Overview
Focus and Context
The concept of functions is one of the most important mathematical
ideas students will study. Functions are used in essentially every
branch of mathematics because they are an efficient and powerful
way to organize and manage a variety of mathematical concepts and
relationships.
In this unit, students explore the effects of horizontal and vertical
translations and stretches, and reflections in the x-axis, the y-axis,
and the line y = x on the graphs of general functions and their related
equations. In later units, they will apply these transformations to
specific functions, including radical, sinusoidal, exponential, and
logarithmic functions.
Students are also introduced to inverses. They identify inverse relations
and inverse functions and verify that two functions are inverses of each
other.
Outcomes Framework
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF2
SCO RF1
Demonstrate an understanding
of the effects of horizontal and
vertical translations on the graphs of
functions and their related equations.
SCO RF4
SCO RF3
Demonstrate an understanding
of the effects of reflections on the
graphs of functions and their related
equations, including reflections
through the:
40
•
x-axis
•
y-axis
•
line y = x.
Demonstrate an understanding of
the effects of horizontal and vertical
stretches on the graphs of functions
and their related equations.
Apply translations and stretches
to the graphs and equations of
functions.
SCO RF5
Demonstrate an understanding of
inverses of relations.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
Relations and Functions
RF2. Demonstrate an understanding
of relations and functions.
RF3. Analyze quadratics of the form
y = a(x - p)2 + q and determine the:
[C, R, V]
•
vertex
RF1. Demonstrate an understanding
of the effects of horizontal and
vertical translations on the graphs of
functions and their related equations.
•
domain and range
[C, CN, R, V]
•
direction of opening
•
axis of symmetry
•
x- and y-intercepts.
[CN, R, T, V]
RF4. Analyze quadratic functions of
the form y = ax2 + bx + c to identify
characteristics of the corresponding
graph, including:
•
vertex
•
domain and range
•
direction of opening
•
axis of symmetry
•
x- and y-intercepts
and to solve problems.
[CN, PS, R, T, V]
RF4. Demonstrate an understanding
of the effects of reflections on the
graphs of functions and their related
equations, including reflections
through the:
•
x-axis
•
y-axis
•
line y = x.
[C, CN, R, V]
RF2. Demonstrate an understanding
of the effects of horizontal and
vertical stretches on the graphs of
functions and their related equations.
[C, CN, R, V]
RF3. Apply translations and stretches
to the graphs and equations of
functions.
[C, CN, R, V]
RF11. Graph and analyze reciprocal
functions (limited to the reciprocal of
linear and quadratic functions).
RF5. Demonstrate an understanding
of inverses of relations.
[CN, R, T, V]
[C, CN, R, V]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
41
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF1 Demonstrate an
understanding of the effects
of horizontal and vertical
translations on the graphs of
functions and their related
equations.
[C, CN, R, V]
In Mathematics 2200, students were introduced to quadratic functions
in vertex form y = a(x − h)2 + k (RF3). They discovered that h and k
translated the graph horizontally and vertically. These transformations
allow students to identify the vertex directly from the quadratic
equation. Students will now look at these translations for general
functions and compare how the graph and table of values of y = f (x)
compares to y – k = f(x – h).
Achievement Indicators:
RF1.1 Compare the graphs of
a set of functions of the form
y - k = f(x) to the graph of y = f(x)
and generalize, using inductive
reasoning, a rule about the effect
of k.
Students should investigate the effect of changing the value of k by
comparing functions of the form y – k = f (x) or y = f (x) + k to the graph
of y = f (x). Provide students with a base graph, such as the graph of
y = f (x) below:
y=f(x)
y
5
RF1.2 Compare the graphs of
a set of functions of the form
y = f (x - h) to the graph of
y = f (x), and generalize, using
inductive reasoning, a rule about
the effect of h.
RF1.3 Compare the graphs of
a set of functions of the form
y - k = f (x - h) to the graph of
y = f (x), and generalize, using
inductive reasoning, a rule about
the effect of h and k.
RF1.4 Sketch the graph of
y − k = f ( x ) , y = f ( x − h ) or
y − k = f ( x − h ) for given values
of h and k, given a sketch of the
function y = f (x), where the
equation of y = f (x) is not given.
42
-5
5
x
-5
Ask students to identify the key points of y = f (x) and create a
table of values. They should then create new tables of values and
respective graphs for functions such as y + 3 = f (x), y = f (x) + 2, and
y - 1 = f (x), which have a vertical translation, k. Discuss with students
how this vertical translation affects the position but not the shape
nor the orientation of the graph. Students should then use mapping
notation (x, y) → (x, y + k) to describe the applicable vertical translation
for each graph.
In a similar fashion, students should then investigate the effect of the
parameter h on a function in the form y = f (x − h), and apply the
corresponding mapping notation (x, y) → (x + h, y).
Students are also expected to work with functions that have been
translated both horizontally and vertically when compared to a base
function and to generalize those transformations using the mapping
rule (x, y) → (x + h, y + k). They should be exposed to both forms of the
transformed function, y - k = f (x - h) and y = f (x - h) + k.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Interview
Authorized Resource
• Ask students to describe the similarities and differences among the
graphs of the following functions if they are transformations of the
base function y = f (x):
Pre-Calculus 12
(i) y - 10 = f (x)
(ii) y = f (x + 7)
Student Book (SB): pp. 6-15
(RF1.1, RF1.2)
• Ask students to describe the translations of each function when
compared to y = f (x):
(i)
(ii)
(iii)
(iv)
1.1 Horizontal and Vertical
Translations
Teacher’s Resource (TR): pp. 8-12
y = f (x - 2)
y = f (x) - 9
y = f (x + 3) - 7
y - 12 = f (x + 4)
(RF1.1, RF1.2, RF1.3)
Performance
• Provide each student with a piece of graph paper and a sticky note.
On the graph paper, ask students to construct a function consisting
of at least four points. On the sticky note, they should write an
equation in the form y – k = f(x) or y = f(x) + k. Students should trade
sticky notes and apply the transformation on the new sticky note to
their own graphs.
(RF1.1)
Journal
• A friend phones for help with her homework. She has a function
with four key points and wants to graph y = f (x + 5) - 7. Ask
students to describe two ways that she can create this graph.
(RF1.3, RF1.4)
Paper and Pencil
• Given the graph of y = f (x), ask students to create a mapping rule
and a table of values for each of the transformations below and graph
the transformed functions.
y = f(x)
y
(i) y + 2 = f (x - 6)
10
8
(ii) y = f (x + 2) + 5
6
4
(iii)y = f (x - 4) - 7
2
-10
-8
-6
-4
-2
-2
2
4
6
8
10
x
-4
-6
-8
(RF1.4)
-10
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
43
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF1 Continued ...
Achievement Indicator:
RF1.5 Write the equation of a
function whose graph is a vertical
and/or horizontal translation of
the graph of the function y = f (x).
Given the graph of a base function y = f (x) and the graph of
y − k = f (x − h), students should identify the horizontal and vertical
translations, and write the equation of the translated function. Students
could be given the two graphs below and asked to describe the
horizontal and vertical translations. They should then write the equation
for the translated function as y − 3 = f (x + 2) or y = f (x + 2) + 3.
y
y=f(x)
8
6
4
2
-8
-6
-4
-2
2
4
6
x
8
-2
-4
-6
-8
y
y=f(x-h)+k
8
6
4
2
-8
-6
-4
-2
2
4
6
8
x
-2
-4
-6
-8
44
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Performance
• Students could work in small groups for this activity. Provide each
group with a base function y = f (x) and ask each student to create
their own translated graph of y = f (x). They should then trade
their graphs within their group and determine the equation of the
function for each graph.
(RF1.5)
Authorized Resource
Pre-Calculus 12
1.1 Horizontal and Vertical
Translations
SB: pp. 6-15
Paper and Pencil
TR: pp. 8-12
• Ask students to determine the values of h and k and write the
equation for the translated graph for each of the following:
y
(i)
10
8
6
4
y=f(x)
2
-10 -8 -6 -4 -2
-2
2
4
-4
6
8 10 x
y-k=f(x-h)
-6
-8
-10
y
(ii)
y=f(x-h)+k
10
8
6
4
y=f(x)
2
-10 -8 -6 -4 -2
-2
2
4
6
8
10
x
-4
-6
-8
-10
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
(RF1.5)
45
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Demonstrate an
understanding of the effects
of reflections on the graphs
of functions and their related
equations, including reflections
through the:
•
x-axis
•
y-axis
•
line y = x.
In Mathematics 2200, students were introduced to parabolas that were
reflected through the x-axis (RF3). They will now progress to reflecting
functions in both the x-axis and y-axis. The focus of this outcome is
not to combine reflections with translations, but to work only with the
horizontal and vertical reflections. Reflecting a point through the line
y = x will be addressed when students study inverses.
[C, CN, R, V]
Achievement Indicators:
RF4.1 Generalize the relationship
between the coordinates of an
ordered pair and the coordinates
of the corrseponding ordered
pair that results from a reflection
through the x-axis or the y-axis.
RF4.2 Sketch the reflection of
the graph of a function y = f (x)
through the x-axis or the y-axis,
given the graph of the function
y = f (x), where the equation of
y = f (x) is not given.
RF4.3 Generalize, using
inductive reasoning, and explain
rules for the reflection of the graph
of the function y = f (x) through
the x-axis or the y-axis.
RF4.4 Sketch the graphs of the
functions y = -f (x) and y = f (-x),
given the graph of the function
y = f (x), where the equation of
y = f (x) is not given.
Students should explore the effects on the coordinates of a point when
it is reflected through the x-axis or the y-axis. Ask them to plot a variety
of points. As they reflect the points in the x-axis, they should see that the
mapping rule (x, y) → (x, −y) applies. Similarly, they should conclude
that the reflection of points in the y-axis results in the mapping rule
(x, y) → (−x, y). Students then progress to sketching the reflection of a
function by treating the graph as a series of points. Discuss with them
how reflections, like translations, do not affect the shape of the graph.
However, reflections may change the orientation of the graph.
It is important for students to recognize how the equation of f (x)
changes when reflections in the x-axis or y-axis occur.
•
y = −f (x) produces a reflection in the x-axis.
•
y = f (−x) produces a reflection in the y-axis.
Given the graph of a function y = f (x), such as the one below, students
should graph y = −f (x) and y = f (−x) using a mapping rule or
y
transformations.
y=f(x)
8
6
4
2
-8 -6 -4 -2
2
4
6
8 x
-2
-4
RF4.5 Write the equation of a
function, given its graph which
is a reflection of the graph of the
function y = f (x) through the xaxis or the y-axis.
46
-6
Similarly, given the graph of a function and a reflected graph, they
should determine whether the equation of the reflected graph is of the
form y = f (−x) or y = −f (x).
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Interview
Authorized Resource
• Ask students which axis the first point was reflected through to get
the second point:
Pre-Calculus 12
(i) (6, 7) and (−6, 7)
(ii) (−2, −7) and (−2, 7)
(iii) (5, 0) and (−5, 0)
1.2 Reflections and Stretches
SB: pp. 16-31
TR: pp. 13-18
(RF4.1)
• Given the graph of a base function and a reflected graph, ask
students how they can use key points to determine if the graph has
been reflected in the x-axis or y-axis.
(RF4.3)
Journal
• Ask students to explain if it is possible for the coordinates to remain
the same after a point has been reflected in an axis.
(RF4.1)
Performance
• Ask students to create the graph of a function y = f (x) using at least
four key points. They should trade their graphs with a partner and
reflect the base function:
(i) in the x-axis
(ii) in the y-axis
Ask students to write the equation of the new function.
(RF4.2, RF4.4, RF4.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
47
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Demonstrate an
understanding of the effects of
horizontal and vertical stretches
on the graphs of functions and
their related equations.
[C, CN, R, V]
In Mathematics 2200, students explored the effect of a vertical stretch
on quadratic functions (RF3). They will now work with both vertical
and horizontal stretches for general functions. Students should describe
how these stretches change the shape of the graph and determine
these stretches when given the graphs of the base function and the
transformed function. This outcome focuses on horizontal and vertical
stretches, so at this point, students should not encounter functions that
have been both stretched and translated.
Achievement Indicator:
RF2.1 Compare the graphs of
a set of functions of the form
y = af (x) to the graph of y = f (x),
and generalize, using inductive
reasoning, a rule about the effect
of a.
Students should recognize the effect of a as a vertical stretch of the
y
function y = f (x) in the form y = af (x) or a = f (x ) . They could be
given a graph of f (x) as shown.
y=f(x)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8
x
Using key points, ask them to generate the table of values for y = 2f (x)
and y = 12 f (x ) .
Students should notice that a changes the shape of the function by
stretching the graph vertically. They should compare the key points
of the original function to the points of the transformed function to
generalize the mapping rule (x, y) → (x, ay).
When creating the table of values and graph of the transformed
function, students may notice that some points did not change even
after a vertical stretch is applied. Points that do not change after a
transformation has been applied are called invariant points. Students
were exposed to these points in Mathematics 2200 when they worked
with reciprocal functions (RF11).
Students should also work with values of a that are negative. It should
be noted that a negative a value will cause a reflection in the x-axis,
and that the vertical stretch is positive; that is, the vertical stretch is |a|.
Students should compare relations where |a| > 1 to those where |a| < 1
to see how these values of a influence the graph.
48
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Interview
Authorized Resource
• Ask students to explain why the vertical stretch factor for the
function y = af (x) is determined using |a|. The following questions
could also be posed to lead to further elaboration: “Is it possible to
have a negative vertical stretch factor? If stretches are always positive,
what does a negative value for a indicate about the graph?”
Pre-Calculus 12
1.2 Reflections and Stretches
SB: pp. 16-31
TR: pp. 13-18
(RF2.1, RF4.1, RF4.2, RF4.3)
Journal
• Given the graph of the function below, ask students to explain which
points are invariant points and why they do not change after the
application of a vertical stretch.
y
10
5
-10
-5
5
10 x
-5
-10
(RF2.1)
• Ask students to explain what the effects will be on the graph of a
function when |a| > 1 or |a| < 1.
(RF2.1)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
49
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Continued ...
Achievement Indicator:
RF2.2 Compare the graphs of
a set of functions of the form
y = f (bx) to the graph of y = f (x)
and generalize, using inductive
reasoning, a rule about the effect
of b.
This will be students’ first exposure to the concept of horizontal stretch.
Graphing technology could be used to display a base function such as
y = sin(x) with a restricted domain of [0˚, 360˚]. It is not necessary for
students to know the equation of the base function.
( )
Providing students with the graphs of y = f (3x) and y = f 12 x , they
can examine the effect on the graph and the table of values. Students
should discuss the general effects of a horizontal stretch on the graph
of the base function and the table of values. The general mapping
(x , y ) → b1 x , y from the base function to the transformed function
can be inferred from the example. Students should compare relations
where |b| < 1 to those for which |b| >1. It should also be noted that if
b < 0 , the graph will be stretched, as well as reflected in the y-axis. As
with vertical stretches, horizontal stretches are always positive.
( )
50
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Interview
• Ask students to explain why the horizontal stretch factor is given
by 1 for the function y = f (bx).
b
(RF2.2)
Authorized Resource
Pre-Calculus 12
1.2 Reflections and Stretches
Journal
• Given the graph of the function below, ask students to explain which
points are invariant points and why they do not change after the
application of a horizontal stretch.
SB: pp. 16-31
TR: pp. 13-18
(RF2.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
51
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Continued ...
Achievement Indicator:
RF2.3 Compare the graphs of
a set of functions of the form
y = a f(bx) to the graph of
y = f (x), and generalize, using
inductive reasoning, a rule about
the effects of a and b.
Students should also work with functions that have both a vertical and
horizontal stretch. By comparing two graphs, they should be able to
determine the effects that the values of a and b have on the graph of
y = af (bx) when compared to y = f (x). Students could be given the two
graphs below, for example, and asked the following questions:
• What effect did the 2 have on the graph?
• What effect did the 2 have on the table of values?
• What effect did the 13 have on the graph?
• What effect did the 13 have on the table of values?
Students should then generalize a mapping rule for functions that have a
vertical stretch of |a| and a horizontal stretch of 1 as (x , y ) → bx , ay .
b
It should be noted that the focus on vertical and horizontal stretch
should be with functions that are bounded.
(
)
Students could be introduced to unbounded functions such as y = x2
and y = |x| and explore transformations of these functions with both the
vertical and horizontal stretch. For many of the unbounded functions
that students have already seen, stretches can be described as either a
horizontal stretch or a vertical stretch. Using a function such as y = 4x2,
which has a vertical stretch of 4, students can compare the effect of a
vertical stretch and a horizontal stretch. If this function is written as
y = (2x)2, there is a horizontal stretch of 21 . Encourage students to create
a mapping rule for both functions and produce a table of values and
graph from the base function y = x2 to verify that these stretches produce
the same effect.
52
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Performance
• Using technology, ask students to demonstrate the effects of
stretching an image for different values of a and b in various
combinations.
Authorized Resource
(RF2.3)
Pre-Calculus 12
1.2 Reflections and Stretches
SB: pp. 16-31
TR: pp. 13-18
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
53
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Continued ...
Achievement Indicators:
RF2.4 Sketch the graph of
y = af (x), y = f (bx) or y = af(bx)
for given values of a and b, given
a sketch of the function y = f (x),
where the equation of y = f (x) is
not given.
RF2.5 Write the equation of a
function, given its graph which
is a vertical and/or horizontal
stretch of the graph of the function
y = f (x).
Given the graph of a function, students should apply a horizontal
stretch and/or a vertical stretch to produce the graph of the transformed
function y = af (bx). Two possible ways to produce the graph are:
• to identify key points and use a mapping rule to create a new table of
values
• to use knowledge of stretches to transform the graph directly.
Given the graph of a transformed function and a base function to
compare it with, students should determine the horizontal and vertical
stretch, and then write the equation of the transformed function in
y
terms of y = af (bx) or a = f (bx ) . The focus of this outcome is not to
determine the specific equation for the base function or transformed
y
function such as 7 = 4 x , but to work with general equations like
y
7 = f (4 x ) .
The stretches can be determined by comparing the domain and range
of the functions. Students should be exposed to graphs such as the
following:
The domain of y = f (x) is [−4, 8], which has a span of 12 units. The
domain of y = af (bx) is [−2, 4], which has a span of 6 units. Students
should realize that the horizontal stretch is 21 , which means |b| = 2.
Similarly, the range of y = f (x) is [0, 6], a span of 6 units, and the range
of y = af (bx) is [0, 18], a span of 18 units. Students should realize
that the vertical stretch is 3, which means |a| = 3. Since there are no
reflections, they can conclude that the equation of the transformed
graph is y = 3f (2x).
54
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to write the general equation, in terms of f (x) for the
stretched image shown on the same set of axes, by determining the
values of a and b from the graph.
Authorized Resource
Pre-Calculus 12
1.2 Reflections and Stretches
SB: pp. 16-31
TR: pp. 13-18
(RF2.5)
Journal
• Ask students to explain when the x-intercepts are invariant for a
stretch, and when the y-intercepts are invariant for a stretch. They
could also be asked to describe the circumstances under which a
point would be invariant for both types of stretches.
(RF2.4)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
55
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF3 Apply translations and
stretches to the graphs and
equations of functions.
[C, CN, R, V]
Students have worked with translations, reflections in the x- and
y-axis, and stretches. For the most part, these transformations have been
addressed independently of one another. They will now extend their
work to functions and graphs that have all types of transformations.
Achievement Indicator:
RF3.1 Sketch the graph of the
function y - k = af (b(x - h)) for
given values of a, b, h and k,
given the graph of the function
y = f (x), where the equation of
y = f (x) is not given.
Given the graph of the function y = f (x), students should graph the
function y − k = af (b(x − h)) for different values of a, b, h, and k.
The graph could be created using transformations. Using the graph
of a function, such as y = f (x) shown here, ask students to graph a
transformed function such as y = −2 f (3 (x − 1)) + 4 .
y=f(x)
y
2
2 4 6 8 10 x
-10 -8 -6 -4 -2
-2
-4
-6
Remind them of the importance of the order of operations. Since
stretches and reflections are the result of multiplication and translations
are the result of addition, the stretches and reflections are applied first.
They apply the following transformations to each point to produce the
transformed graph:
• horizontal stretch of 13
• vertical stretch of 2
• reflection in the x-axis
• horizontal translation of 1 unit right
• vertical translation of 4 units up
Students should note that the stretches and reflections can be applied in
any order, as long as it is before the translations. Similarly, the order in
which the translations are applied is not important, as long as they are
applied after the stretches and reflections.
Students could also generate a table of values based on the key points,
and use the mapping (x , y ) → 13 x + 1, −2 y + 4 to generate a new table
of values.
(
)
It is sometimes necessary to rewrite a function before it can be graphed.
Before graphing y - 6 = 3f (4x - 8), for example, students should write
the function as y - 6 = 3f (4(x - 2)). This will help them correctly
identify the value of h as 2, rather than 8.
56
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Given the graph of the function y = f (x) shown, ask students to
sketch the graph of y = 2 f ( −3( x + 1)) − 2 .
Pre-Calculus 12
y
1.3 Combining Transformations
SB: pp. 32-43
10
TR: pp. 19-24
8
6
4
2
-10 -8
-6
-4
-2
-2
2
4
6
8 10 x
y=f(x)
-4
-6
-8
-10
(RF3.1)
Journal
• Ask students to describe the steps in graphing a function of the form
y = af (b(x - h)) + k, where the shape of the graph of f (x) is wellknown.
(RF3.1)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
57
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF3 Continued ...
Achievement Indicator:
RF3.2 Write the equation of a
function, given its graph which is
a translation and/or stretch of the
graph of the function y = f (x).
Students should compare the graph of a base function with the graph
of a transformed function, identify all transformations, and state the
equation of the transformed function. The focus should be on functions
that have a bounded domain and range. In an example such as the
following students should find the equation of g(x) as a transformation
of f (x):
The following prompts could be used to initiate discussion:
• What is the domain of each function? How will this help in
identifying the horizontal stretch?
Students should recognize that the domain of f (x) is [−3, 4] which
has a span of 7 units, and that the domain of g(x) is [−9, 5] which
has a span of 14 units. Therefore, g(x) has a horizontal stretch of 2.
• What is the range of each function? How will this help in identifying
the vertical stretch?
By comparing the ranges in a similar fashion, students should
recognize the vertical stretch is 3.
• Is the graph reflected in the y-axis? Explain your reasoning.
Student findings should result in a = 3 and b = − 21 . Applying these
transformations to f (x) produces the graph:
As students compare the key points, they should notice that the graph
must be shifted 1 unit left and 4 units up to produce g(x), resulting in
the function y = 3 f (− 12 (x + 1))+ 4.
58
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Using a graph such as the one shown, ask students to determine
the specific equation for the image of y = f (x) in the form
y = af (b( x − h )) + k .
Authorized Resource
Pre-Calculus 12
1.3 Combining Transformations
SB: pp. 32-43
TR: pp. 19-24
(RF3.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
59
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF5 Demonstrate an
understanding of inverses of
relations.
[C, CN, R, V]
As students begin work with inverse relations, they will explore the
relationship between the graph of a relation and its inverse, and
determine whether a relation and its inverse are functions. Students
produce the graph of an inverse from the graph of the original relation,
restrict the domain of a function so that its inverse is also a function,
and determine the equation for f -1 given the equation for f.
RF4 Continued ...
Achievement Indicators:
RF4.6 Generalize the relationship
between the coordinates of an
ordered pair and the coordinates
of the corresponding ordered
pair that results from a reflection
through the line y = x.
RF5.1 Explain how the
transformation (x, y) → (y, x) can
be used to sketch the inverse of a
relation.
Discuss with students how an inverse of a relation ‘undoes’ whatever the
original relation did. Using examples from non-mathematical situations
may provide a better understanding of a topic that is otherwise very
abstract. The inverse of opening a door is closing the door; the inverse
of wrapping a gift is unwrapping a gift. From a mathematics perspective,
encourage students to think of inverse relations as undoing all of the
mathematical operations. The following table helps to illustrate this
concept:
Function
f (x ) = x + 2
f
f (x ) = 5 x
Inverse
(x ) = x − 2
(x ) = 5x
f −1 (x ) = x
f
f (x ) = x 2
RF5.2 Explain the relationship
between the domains and ranges
of a relation and its inverse.
−1
−1
Students should compare a function to its inverse using a table of values.
Examining a table of values for y = x + 4 and its inverse, y = x – 4, shows
that the x and y values are interchanged.
x
−2
−1
0
1
2
y
2
3
4
5
6
x
2
3
4
5
6
y
−2
−1
0
1
2
From this, students should see the mapping notation (x, y) → (y, x)
as a reversal of the x and y values in order to represent an undoing
of a process. The input of the function is the output of the inverse,
and vice versa. This leads to the relationship between the domains
and ranges of a relation and its inverse. Students may have difficulty
generalizing this relationship for linear functions, as the domain and
range are all real numbers. Use the graph of a quadratic function,
2
such as y = (x − 2 ) + 1 , and the graph of its inverse to highlight the
relationship. The original function has all real numbers in its domain
and the range is {y y ≥ 1, y ∈ R}. They should note that the domain and
range are interchanged for the inverse.
60
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
• Students respond to three reflective prompts that describe what they
learned about inverse relations.
Pre-Calculus 12
Example of a 3-2-1 reflection sheet:
1.4 Inverse of a Relation
SB: pp. 44-55
3 new things I learned
TR: pp. 25-30
1.
2.
3.
2 things I am still struggling with
1.
2.
1 thing that will help me tomorrow
1.
Provide students with a copy of the reflection sheet and time to
complete their reflections. They could also be paired up to share their
3-2-1 reflections.
(RF4.6, RF5.1, RF5.2, RF5.3)
• Ask students to explain what it means in theory for one function to
be the inverse of another in terms of their domains and ranges.
(RF5.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
61
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Continued ...
RF5 Continued ...
Achievement Indicators:
RF5.3 Explain how the graph
of the line y = x can be used to
sketch the inverse of a relation.
RF5.4 Sketch the graph of the
inverse relation, given the graph
of a relation.
RF4.7 Sketch the reflection of
the graph of a function y = f (x)
through the line y = x, given the
graph of the function y = f (x),
where the equation of y = f (x) is
not given.
RF4.8 Generalize, using
inductive reasoning, and explain
rules for the reflection of the graph
of the function y = f (x) through
the line y = x.
By having students use (x, y) → (y, x) to create a table of values for the
inverse of a function, they should quickly realize that there is reflective
symmetry when the graphs of f and f -1 are sketched on the same set of
axes. The axis of symmetry becomes apparent when they draw the line
y = x. As an example, a graph such as the following could be presented:
Based on the graph, ask students to complete the following:
• Create a table of values for the function using the ordered pairs for
the four key points shown.
• Transform the table by applying the mapping (x, y) → (y, x).
• Graph the relation represented by the new table.
• Look for a relationship between the graphs of f and f -1.
• Generalize the pattern to a relationship between graphs of functions
and their inverses in general.
After seeing the graphical relationship between a relation and its inverse,
the goal is for students to take a graph of a relation and produce a graph
of its inverse on the same set of axes without having to go through the
process of producing tables of values.
It should be noted that a common error occurs in the notation used for
inverse functions. Students may incorrectly write f -1(x) as f 1x .
()
62
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• For each graph shown, ask students to sketch the inverse relation on
the same set of axes.
(i)
Authorized Resource
Pre-Calculus 12
1.4 Inverse of a Relation
SB: pp. 44-55
TR: pp. 25-30
(ii)
(RF5.4)
Journal
• Students should discuss how they could sketch the graph of
the inverse of the relation graphed below, using both indicated
approaches:
(i) the line y = x
(ii) the mapping rule (x, y) → (y, x)
y
2
y = 2x + 1
4
2
-4
-2
2
4
x
-2
-4
(RF5.1, RF5.3, RF4.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
63
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF5 Continued ...
Achievement Indicator:
RF5.5 Determine if a relation
and its inverse are functions.
In Mathematics 1201, students differentiated between a function and
a relation (RF2). They also used the vertical line test to determine if a
relation is also a function. This content could be quickly reviewed to
begin a discussion on determining if the inverse of a function is also a
function.
Students could be given a graph such as the following, and asked to
complete the tasks below.
• Use a vertical line test to determine if y = f (x) is a function.
• Construct the graph of y = f -1(x) by reflecting the graph of y = f (x)
in the line y = x.
• Use the vertical line test to determine if y = f -1(x) is a function.
• What kind of line test could be used with the graph of y = f (x) to
determine if its inverse would be a function?
Students should conclude that the horizontal line test is used to
determine if an inverse relation is a function.
64
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Journal
• Ask students to explain the difference between a vertical line test and
a horizontal line test in terms of what each one is used to determine,
and why they work.
(RF5.5)
Authorized Resource
Pre-Calculus 12
1.4 Inverse of a Relation
Paper and Pencil
• Ask students to determine whether the inverse of each relation
graphed here is a function, without actually sketching it.
SB: pp. 44-55
TR: pp. 25-30
(i)
(ii)
(RF5.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
65
FUNCTION TRANSFORMATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Continued ...
RF5 Continued ...
Achievement Indicators:
RF5.6 Determine restrictions on
the domain of a function in order
for its inverse to be a function.
RF5.7 Determine the equation
and sketch the graph of the inverse
relation, given the equation of a
linear or quadratic relation.
RF4.9 Write the equation of a
function, given its graph which
is a reflection of the graph of the
function y = f (x) through the line
y = x.
Students are required to restrict the domain of functions to ensure that
the inverse is also a function. Examples should be limited to linear
and quadratic functions. Students should realize that the only linear
functions whose inverses are not functions are those that have graphs
which are horizontal lines.
In Mathematics 2200, students converted the equations of quadratic
functions from standard form, y = ax2 + bx + c, to vertex form,
y = a(x - h)2 + k, by completing the square (RF4) and graphed quadratic
functions in vertex form (RF3). Students continue to convert forms of
quadratic equations, in order to identify the vertex of the function so
that its domain may be restricted. Students could be guided through
the following process for the quadratic function represented by
f (x) = 2x2 - 8x + 11:
• write the equation in vertex form, and sketch its graph.
• use the graph of the quadratic to sketch its inverse on the same set of
axes.
• determine if the inverse relation is also a function. If not, consider
what would need to change about the graph of the original quadratic
in order for its inverse to be a function.
• restrict the domain of the function y = f (x) so that y = f -1 (x) is also a
function.
Restricting the domain of a parabola so that x > h produces a function
whose inverse is also a function. Restricting the domain so that x < h is
also acceptable, although not as common.
Students should then be shown the process by which interchanging
the x and y variables in the equation and then rearranging to isolate y
produces the equation for y = f -1 (x).
RF5.8 Determine, algebraically
or graphically, if two functions are
inverses of each other.
When presented with two functions, students should also determine
whether or not they are inverses of one another. This can be done on
a graph that displays both functions by sketching the line y = x and
deciding if the functions are mirror images of one another.
Algebraically, students should be given the equation representing each
function. They determine the equation of the inverse of one of the given
functions, and then decide if it is equivalent to the other given function.
66
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
FUNCTION TRANSFORMATIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to determine the equation of the inverse for each of the
following relations:
(i) y = 5x
(ii) y = 3x - 4
(iii) y = −2x + 5
Authorized Resource
Pre-Calculus 12
(RF5.7)
• Ask students to state the restricted domain for each of the following
relations so that the inverse relation is a function, and write the
equation of the inverse:
1.4 Inverse of a Relation
SB: pp. 44-55
TR: pp. 25-30
2
(i) y = x - 6x + 10
(ii) y = 5x2 + 20x - 9
(iii) y = 2x2 - 8x + 1
(RF5.6, RF5.7)
• Ask students to determine if the inverses of the following functions
are functions. If the inverse is not a function, they should describe
how they can be modified to become functions.
(i)
f (x ) = 3 x 2
(ii)
f (x ) = x 2 + 2 x
Note:
(iii) f (x ) = 2 x 2 + 4
(RF5.5, RF5.6)
• Students match each of the equations from the first list with its
inverse in the second list:
Function
y = 4x - 1
y = x2 + 8x + 2, x ≥ -4
y = 3x2 - 12x + 15, x ≥ 2
Inverse
y = x −416
y=
x +1
4
The examples in the student
book on pp. 49-50 are limited to
quadratics of the form y = ax2 + c.
Students should also be exposed
to equations of the form
y = ax2 + bx + c, as on p. 53 - #10
and p. 54 - #12.
y = x + 14 − 4
y=
x −3
3
+2
(RF5.8)
Performance
• Create square sheets of paper with graphs of pairs of functions.
Create the pairs in such a way that some are inverses of one another
while others are not. Construct the grid on each graph so that the
scales are identical on each axis, and so that the x- and y-axes have
the same limits. Distribute one square to each student and ask them
to fold their papers in a way that would determine whether or not
the graphs are inverses of one another. Ask them to explain their
method.
(RF5.8)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
67
FUNCTION TRANSFORMATIONS
68
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Radical Functions
Suggested Time: 8 Hours
RADICAL FUNCTIONS
Unit Overview
Focus and Context
Work with function transformations continues in this unit with a
specific focus on radical functions. Students sketch graphs of radical
functions by applying translations, stretches and reflections to the graph
of y = x . They analyze transformations to identify the domain and
range of radical functions.
In this unit, the graph of y = f ( x ) is related to the graph of y = f (x)
and the domain and range for each function are compared. Finding
approximate solutions to radical equations is also addressed graphically.
Outcomes Framework
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF12
Graph and analyze radical functions
(limited to functions involving one
radical).
70
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
RADICAL FUNCTIONS
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
Relations and Functions
AN2. Demonstrate an understanding
of irrational numbers by:
•
•
representing, identifying and
simplifying irrational numbers
ordering irrational numbers.
AN2. Solve problems that involve
operations on radicals and radical
expressions with numerical and
variable radicands.
RF12. Graph and analyze radical
functions (limited to functions
involving one radical).
[CN, R, T, V]
[CN, ME, PS, R]
[CN, ME, R, V]
RF6. Relate linear relations expressed
in:
•
slope-intercept form y = mx + b
•
general form Ax + By + C = 0
•
slope-point form
y - y1 = m(x - x1)
to their graphs.
[CN, R, T, V]
AN3. Solve problems that involve
radical equations (limited to square
roots).
[C, PS, R]
RF3. Analyze quadratics of the form
y = a(x - p)2 + q and determine the:
•
vertex
•
domain and range
•
direction of opening
•
axis of symmetry
•
x- and y-intercepts
and to solve problems.
[CN, R, T, V]
RF5. Solve problems that involve
quadratic equations.
[C, CN, PS, R, T, V]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
71
RADICAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF12 Graph and analyze radical
functions (limited to functions
involving one radical).
[CN, R, T, V]
In Mathematics 2200, students worked with radicals and radical
expressions with numerical and variable radicands (AN2), and solved
equations and problems dealing with radical equations (AN3). Students
now use their knowledge of transformations from the previous unit
(RF4) to graph radical functions. They also determine the domain and
range of radical functions.
Achievement Indicators:
RF12.1 Sketch the graph of the
function y = x , using a table of
values, and state the domain and
range.
RF12.2 Sketch the graph of the
function y − k = a b (x − h ) by
applying transformations to the
graph of the function y = x ,
and state the domain and range.
While students have been introduced to radical equations (AN3),
they may or may not have been exposed to what the graphs of radical
functions look like. They should be introduced to the graphs by
creating the graph of y = x using a table of values. Once the graph
is produced, the domain and range of y = x should be discussed.
Students should note that because it is not possible to take the square
root of negative numbers in the real number system, the radicand must
be greater than or equal to zero, resulting in a domain of x ∈ [0, ∞).
Technology could also be used to create the table of values and the
graph.
Once students are familiar with the characteristics of the graph of
y = x , the characteristics of a transformed radical function of the form
y − k = a b (x − h ) or y = a b (x − h ) + k should be examined. The
graphs can be produced by:
• applying the transformations
• creating a mapping rule, deriving a table of values, and plotting the
points.
Students have already been exposed to these graphing techniques using
general functions (RF4). They should now apply these techniques to the
specific function y = x .
Students could explore a radical function such as y − 1 = −3
Using transformations, they should see that:
1
2
(x + 6 ).
• a = −3, so all points have a vertical stretch of 3
• Since a < 0, all points are reflected in the x-axis
• b = 2, so all points have a horizontal stretch of 2
• h = −6, so all points have a horizontal translation of 6 units left
• k = 1, so all points have a vertical translation of 1 unit up.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
RADICAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
• Ask students to describe the similarities and differences between
the graphs of y = x and y = 13 −2 (x + 1) − 4. They should
discuss the domain and range of each function.
Pre-Calculus 12
(RF12.1, RF12.2)
2.1 Radical Functions and
Transformations
Student Book (SB): pp. 62-77
Teacher’s Resource (TR): pp. 3843
Paper and Pencil
• Ask students to create the graph of y − 3 = − 12 (x − 1) ,
describing all transformations and stating the domain and range.
(RF12.2)
Performance
• Create several pairs of cards where one card contains the equation
of a radical function, and the second card contains the range.
Distribute the cards among the students and have them find their
partner by matching the radical function with its range. Once
students have found their partners, they should create the graph of
the radical function and find its domain.
(RF12.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
73
RADICAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF12 Continued ...
Achievement Indicator:
RF12.2 Continued
When students create the graph by applying transformations, remind
them that they must apply stretches and reflections first, and translations
last.
y
10
8
6
4
y= x
2
-10 -8
-6
-4
-2
-2
2
4
6
8
10
x
-4
-6
-8
-10
y – 1 = -3
1
(x
2
+ 6)
Students could also write the mapping rule for the function, transform
the points of y = x , and plot these points to create the graph.
Students should determine the domain and range of the function from
the graph. After they have worked through a number of examples,
they should see a pattern that allows them to determine the domain
and range of a radical expression such as y = 4 −2 (x + 5 ) + 7
without creating an accurate graph. This can be done by examining the
reflections and translations. They should realize that stretches do not
affect the domain and range of radical functions.
It is important to note that students are not expected to determine the
equation of a radical function given its graph or points on the graph.
74
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
RADICAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to sketch the graphs of f (x ) = x − 3 and
g (x ) = − x − 3 on the same coordinate plane. They should notice
that the two graphs together form a parabola. Ask them to explain
why a quadratic equation cannot be used to define y as a function of
x for the resulting graph.
Pre-Calculus 12
(RF12.2)
2.1 Radical Functions and
Transformations
SB: pp. 62-77
TR: pp. 38-43
Note
SB: p. 68 - Example 3, p. 74 - #10
Students are not expected to
determine a radical function from
a graph.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
75
RADICAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF12 Continued ...
Achievement Indicators:
RF12.3 Sketch the graph of the
function y = f ( x ) , given the
equation or graph of the function
y = f(x), and explain the strategies
used.
Given the graph of y = f (x), students should graph y =
them to consider, for example, the graph of y = x2 - 1.
f ( x ) . Ask
2
y=x –1
y
4
3
2
RF12.4 Compare the domain
and range of the function
y = f ( x ) , to the domain and
range of the function y = f(x),
and explain why the domains and
ranges may differ.
1
-4
-2
2
x
4
-1
-2
-3
-4
One method to produce the graph of y = x 2 − 1 is to first generate
a table of values for y = x 2 − 1 from the given graph. Then, to graph
y = x 2 − 1 , students take the square root of the y-values.
x
f (x )
−2
−1
0
1
3
0
−1
0
2
3
f (x )
3
0
undefined
0
2
y=x –1
y
2
y= x –1
4
3
2
1
-4
-3
-2
-1
-1
1
2
3
4
x
-2
3
-3
Ask students to think about the following:
-4
• Why is the graph undefined from x ∈ (−1, 1) ?
• Are there any invariant points? If so, what are they?
• Where is the graph of y =
f ( x ) above y = f (x)? Below y = f (x)?
Students should realize that y = f ( x ) is undefined where f(x) < 0, that
the invariant points occur where f(x) = 0 or f(x) = 1, and that f (x )
is above the graph of y = f(x) where 0 < f(x) < 1. These results, along
with other key points, can then be used to help create the graph of
y = f ( x ) when y = f(x) is given.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
RADICAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Without graphing, ask students to determine the domain and range
of each function:
Pre-Calculus 12
(i)
y = x 2 + 4x − 5
(ii)
y = − 12 x 2 + x + 12
2.2 Square Root of a Function
SB: pp. 78-89
TR: pp. 44-49
(RF12.4)
• Given the graph of f(x) below, students should create the graph of
y = f (x ) by examining the intercepts and the invariant points:
y
10
8
6
4
2
-10 -8 -6 -4 -2-2
-4
-6
-8
-10
2
4
6
8 10 x
y=f(x)
(RF12.3)
Journal
• Ask students to explain why the domain of y = x 2 + 4 is
{x | x ∈ R}, but the domain of y = x 2 − 4 is
{x | x ≤ −2 or x ≥ 2, x ∈ R}.
(RF12.4)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
77
RADICAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF12 Continued ...
Achievement Indicators:
RF12.3, RF12.4 Continued
Students could also be given the equation y = f(x) and use it to generate
a table of values. From this, they can then graph y = f ( x ) . The graph
and table of values of y = f(x) can also be generated with the use of
technology.
The graphs of y = f(x) are limited to linear and quadratic functions.
Students can use the graphs of y = f(x) and y = f ( x ) to determine the
domain and range of both functions. When equations are given for the
functions, they can first graph each function and then determine the
domain and range.
An alternative method to determine domain and range of y = f(x) and
y = f ( x ) involves analyzing key points. Given y = −x2 + 6x − 5,
for example, students can use knowledge of quadratic functions from
Mathematics 2200 to determine the x- and y-intercepts of the parabola
(RF5), and the vertex of the parabola (RF3). This information can then
be used to identify key points on the graph of y = − x 2 + 6 x − 5 :
Function
x -intercepts
y -intercepts
Maximum value
Minimum value
y = −x 2 + 6x − 5
1 and 5
−5
(3,4 )
none
y = −x 2 + 6x − 5
1 and 5
none
(3,2 )
0
From the above information, students should determine that the
domain of y = − x 2 + 6 x − 5 is x ∈ [1, 5]and the range is y ∈ [0, 2].
This should lead to the generalization that the domain of y = f (x )
consists of all values where f(x) ≥ 0, and the range consists of the square
roots of all of the values in the range of f(x) for which f(x) is defined.
RF12.5 Describe the relationship
between the roots of a radical
equation and the x-intercepts of
the graph of the corresponding
radical function.
RF12.6 Determine, graphically,
an approximate solution of a
radical equation.
78
The relationship between the roots of a general function and the xintercepts of the corresponding graph has been established earlier in this
course (RF12). This is now extended to radical functions.
Initially, students should be provided with a graph of a radical function,
or they should use graphing technology to create the graph, in order
to approximate the solution. Students should only be expected to
graph without technology radical functions that include simple
transformations.
In Mathematics 2200, students solved radical equations algebraically
(AN3). The intent here is to solve equations graphically.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
RADICAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to respond to the following:
Pre-Calculus 12
(i)
In general, in what ways does the graph of y = f ( x )
resemble that of y = f(x)? In what ways does it differ?
2.2 Square Root of a Function
SB: pp. 78-89
(RF12.3)
TR: pp. 44-49
(ii) Explain why the function f ( x ) = 2 x + 5 has a restricted
domain while the function f ( x ) = 2 x 2 + 5 has no
restrictions.
(RF12.4)
• Ask students to sketch the graphs of the radical functions below and
approximate their solutions graphically:
(i)
x +5 = 4
(ii)
x2 −1 = x + 3
(RF12.6)
2.3 Solving Radical Equations
Graphically
SB: pp. 90-98
TR: pp. 50-54
Note
Solving radical equations
algebraically is a review from
Mathematics 2200. It is not the
focus of this outcome.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
79
RADICAL FUNCTIONS
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Trigonometry and the Unit Circle
Suggested Time: 12 Hours
TRIGONOMETRY AND THE UNIT CIRCLE
Unit Overview
Focus and Context
In this unit, students are introduced to radian measure for angles.
They also continue to work with angles in degree measure and convert
between radians and degrees. They sketch angles in standard position
and determine measures of coterminal angles. Students work with
problems involving arc lengths, central angles, and the radius of a circle.
The equation of the unit circle, x2 + y2 = 1, is introduced and then
generalized to a circle with centre (0, 0) and radius r. Students locate
the coordinates on the unit circle, and identify the measure of an angle
given a point on the unit circle.
Students then relate the trigonometric ratios to the coordinates
of points on the unit circle. They are introduced to the reciprocal
trigonometric ratios.
Students algebraically solve first-degree and second-degree
trigonometric equations in radians and degrees. This will be continued
throughout the remaining work with trigonometry.
Outcomes Framework
GCO
Develop trigonometric reasoning
SCO T1
Demonstrate an understanding of
angles in standard position, expressed
in degrees and radians.
SCO T2
Develop and apply the equation of
the unit circle.
82
SCO T3
Solve problems, using the six
trigonometric ratios for angles
expressed in radians and degrees.
SCO T5
Solve, algebraically and graphically,
first and second degree trigonometric
equations with the domain expressed
in degrees and radians.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
Trigonometry
M4. Develop and apply the primary
trigonometric ratios (sine, cosine,
tangent) to solve problems that
involve right triangles.
[C, CN, PS, R, T, V]
M2. Apply proportional reasoning
to problems that involve conversions
between SI and imperial units of
measure.
[C, ME, PS]
T1. Demonstrate an understanding
of angles in standard position [0° to
360°].
T1. Demonstrate an understanding
of angles in standard position,
expressed in degrees and radians.
[R, V]
[CN, ME, R, V]
T2. Solve problems, using the three
T2. Develop and apply the equation
primary trigonometric ratios for angles of the unit circle.
from 0° to 360° in standard position.
[CN, R, V]
[C, ME, PS, R, T, V]
T3. Solve problems, using the cosine
law and sine law, including the
ambiguous case.
T3. Solve problems, using the six
trigonometric ratios for angles
expressed in radians and degrees.
[ME, PS, R, T, V]
[C, CN, PS, R, T]
T5. Solve, algebraically and
graphically, first and second degree
trigonometric equations with the
domain expressed in degrees and
radians.
[CN, PS, R, T, V]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
83
TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T1 Demonstrate an
understanding of angles in
standard position, expressed in
degrees and radians.
[CN, ME, R, V]
In Mathematics 2200, students were introduced to angles in standard
position between 0° and 360°(Τ1). This will be their first exposure to
negative rotational angles and to rotational angles greater than 360˚.
Up to this point, students have only worked with angles in degree
measure. They will be introduced to radian measure and explore the
relationship between radians and degrees. They will also be introduced
to coterminal angles and arc length.
Achievement Indicators:
T1.1 Sketch, in standard
position, an angle (positive or
negative) when the measure is
given in degrees.
Revisit sketching positive angles given in degree measure. A review
of terminology such as initial arm, terminal arm, vertex, sector and
standard position may be required. Students should then be introduced
to clockwise versus counterclockwise as it pertains to angle rotations
from standard position. Some students may have difficulty equating
counterclockwise with positive and clockwise with negative. It may be
helpful to suggest that a positive rotation opens upward from standard
position, whereas a negative angle opens downward. The x-axis could
also be considered the “west(−) - east(+)” line and the y-axis the
“south(−) - north(+)” line. Counterclockwise rotation from the x-axis
can then be considered naturally positive.
T1.2 Sketch, in standard
position, an angle with a measure
of 1 radian.
To introduce students to radian measure, they could picture a 90° angle
in standard position in the coordinate plane. This angle subtends an arc
equal to one-fourth the circumference of any circle centred at the origin.
Students should observe that:
T1.3 Describe the relationship
between radian measure and
degree measure.
• In a unit circle with radius 1, a 90° angle subtends an arc ʌ2 units
long since C = 2π and 14 of 2π is ʌ2 .
• In a circle of radius 5, the circumference is 10π, so a 90° angle
subtends an arc length of 10ʌ 52ʌ .
4
Students should examine the ratio of the arc length subtended by a
90° angle to the radius of the circle for various size circles. They should
conclude that these ratios are all equal to ʌ2 . The fact that the ratio is
constant is the basis for the radian measure for angles.
radian measure =
arc length
radius
Since the radian measure of an angle tells how many times the circle’s
radius is contained in the length of the subtended arc, students should
conclude that the radian measure of an angle will be 1 if the length of
the subtended arc is equal to the radius.
r
r
1
r
84
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Observation
Authorized Resource
• Ask students to draw a circle and show approximate angle measures
of 1 radian, 2 radians, 3 radians, 4 radians, 5 radians, and 6 radians.
Pre-Calculus 12
(T1.2)
4.1 Angles and Angle Measure
Student Book (SB): pp. 166-179
Teacher’s Resource (TR): pp. 9296
Interview
• Ask students to explain how to determine, with the aid of a diagram,
whether 2.8 radians or 180˚ is a greater measure.
(T1.3)
• Ask students to explain which is greater: 4.2 radians or 1.4π radians.
(T1.3)
Journal
• Ask students to respond to the following:
Your classmate has missed the introduction to radian measure.
Describe, with the aid of a diagram, how to sketch an angle with a
measure of 2 radians.
(T1.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
85
TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T1 Continued ...
Achievement Indicators:
T1.3 Continued
After an introduction to radian measure using a circle of radius 1 unit,
students should understand that 360° is equivalent to 2π in radian
measure. Therefore, 180° is equivalent to π. There are certain radian
measures that occur frequently. Students should become familiar with
2ʌ, ʌ, ʌ2 , ʌ3 , ʌ4 , ʌ6 and also with the multiples of each.
T1.4 Sketch, in standard
position, an angle with a measure
expressed in the form kπ radians,
where k ε Q.
When sketching an angle with a measure of kπ radians, students
should visualize angles as a fraction of π or 2π. For example, since
π is equivalent to half a rotation and ʌ2 is 12 of π, ʌ2 is a quarter of
a rotation. For students having trouble visualizing angles in radian
measure, it could be suggested to convert to degree measure to check
the accuracy of their sketch. The goal, however, is to work with radians
without having to convert to degree measure first.
T1.5 Express the measure of an
angle in radians (exact value or
decimal approximation), given its
measure in degrees.
Once the relationship π = 180° has been developed, students can
convert between degree and radian measure. To convert from radians to
degrees, they can solve this equation in terms of radians.
π rad = 180°
q
∴ 1 rad = 180
ʌ
T1.6 Express the measure of an
angle in degrees (exact value or
decimal approximation), given its
measure in radians.
To rewrite 92ʌ radians as degrees, for example, they multiply by 1
radian, or 180q .
ʌ
Similarly, to convert to radians they should solve the equation in
ʌ . Students should become
terms of degrees: 180° = π rad, so 1° = 180
comfortable expressing radian measure in both exact and approximate
values.
A common error occurs when students invert the conversion factor.
This may be avoided through unit analysis which was worked with in
Mathematics 1201 (M2).
Students should note that any angle measure given without a degree
symbol is assumed to be in radians.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to sketch the following angles in standard position:
(i)
ʌ
4
(ii)
2ʌ
3
(iii)
7ʌ
6
Authorized Resource
Pre-Calculus 12
4.1 Angles and Angle Measure
SB: pp. 166-179
(iv) 32ʌ
TR: pp. 92-96
(T1.4)
• Ask students to express the measures of the following angles in radian
measure:
(i)
(ii)
(iii)
(iv)
(v)
60˚
150˚
−225˚
−144˚
214.5˚
(T1.5)
• Ask students to express the following radian measures in degrees:
(i)
2ʌ
3
7ʌ
(ii) 4
ʌ
(iii) 11
12
(iv) −5
(T1.6)
Performance
• For Five-Minute Review, allow teams five minutes to review how
to sketch angles in standard position and the relationship between
degrees and radians. Students in their groups can ask a clarifying
question to the other members or answer questions of others.
(T1.1, T1.4, T1.5, T1.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T1 Continued ...
Achievement Indicators:
T1.7 Determine the measures, in
degrees or radians, of all angles
in a given domain that are
coterminal with a given angle in
standard position.
T1.8 Determine the general
form of the measures, in degrees
or radians, of all angles that are
coterminal with a given angle in
standard position.
T1.9 Explain the relationship
between the radian measure of an
angle in standard position and the
length of the arc cut on a circle of
radius r, and solve problems based
upon that relationship.
The concept of coterminal angles is new to students. Coterminal
angles are angles in standard position with the same terminal arms and
can be measured in degrees or radians. Examples should include both
positive and negative coterminal angles, found by adding or subtracting
multiples of 360° or 2π. This leads to developing the general form,
expressed as θ ± (360°)n or θ ± 2πn where n ε W. The general solution
could also be expressed as θ + (360°)n or θ + 2πn, n ∈ Ι. When
introducing coterminal angles, encourage students to sketch the angles.
This highlights the fact that coterminal angles share a terminal arm.
Based on the definition of a radian, the relationship between a central
angle θ and the length of the arc cut on a circle of radius r can be
developed.
arc length = θ × radius, where θ is the central angle in radians
Students should be able to determine any variable in the relationship,
given the measure of the other two. Given an arc length of 20 cm cut on
a circle of radius 5.4 cm, for example, they could be asked to determine
the measure of the central angle in radians or degrees. They have the
choice of rearranging the equation in terms of θ first, or substituting the
known values before solving for θ.
arc length = θ × radius
20 = θ × 5.4
θ ≈ 3.7 radians or 212°
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to determine a positive and a negative angle measure
that is coterminal to the following measures:
(i) 120˚
Authorized Resource
(ii) −515˚
Pre-Calculus 12
4.1 Angles and Angle Measure
(iii) 3ʌ
4
SB: pp. 166-179
(iv) 9ʌ
TR: pp. 92-96
(T1.7)
• Ask students to determine the measure of all angles that are
coterminal with:
(i) 310˚
(ii) 3ʌ
4
(T1.8)
• Ask students to determine the measures of the arc length subtended
by the angles and radii below:
(i)
Central angle of 23ʌ with radius 10 cm
(ii) Central angle of 2.6 rad with radius 4.9 cm
(T1.9)
• Ask students to determine the measure of the radius of a circle if an
arc length of 42 ft. is subtended by an angle of 7 ʌ .
4
(T1.9)
• Ask students to answer the following:
During a family vacation, you go to dinner at the Seattle Space
Needle. There is a rotating restaurant at the top of the needle that is
circular and has a radius of 40 feet. It makes one rotation per hour.
At 6:42 p.m., you take a seat at a window table. You finish dinner at
8:28 p.m. Through what angle did your position rotate during your
stay? How many feet did your position revolve?
(T1.9)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T2 Develop and apply the
equation of the unit circle.
[CN, R, V]
In Mathematics 2200, students determined the exact values of the
primary trigonometric ratios for angles between 0° and 360° using the
relationship between the sides of special right triangles. Students may
have been introduced to the unit circle as a strategy for finding exact
values; however, it was not a direct outcome. Now students will be
formally introduced to the unit circle and use its equation to generalize
the equation of any circle centred at the origin.
Achievement Indicators:
T2.1 Derive the equation of the
unit circle from the Pythagorean
theorem.
T2.2 Generalize the equation
of a circle with centre (0,0) and
radius r.
Students can find the equation of the unit circle using the Pythagorean
theorem. Using a circle of radius 1 unit centred at the origin, they
should mark a point P on the circle and draw a right triangle. Ask them
to consider why the absolute value of the y-coordinate represents the
distance from a point to the x-axis.
P(x,y)
Applying the Pythagorean theorem results in:
1
x2 + y2 = 12
2
O
y
x
2
x +y =1
Ask students how the equation would differ if the radius was r instead of
1. From this, they should generalize the equation of a circle with centre
(0,0) and radius r to be x2 + y2 = r2.
Given an angle θ in standard position, expressed in degrees or radians,
students should determine the coordinates of the corresponding point
on the unit circle. Conversely, they should determine an angle in
standard position that corresponds to a given point on the unit circle.
T2.3 Describe the six
trigonometric ratios, using a point
P(x, y) that is the intersection of
the terminal arm of an angle and
the unit circle.
In Mathematics 1201 and 2200, students worked with the three
primary trigonometric ratios. This is their first exposure to the reciprocal
ratios: csc θ, sec θ and cot θ.
y
Revisiting the unit circle above, students should observe that sin θ = 1 ,
y
cos θ = 1x , and tan θ = x . They should also see that tan θ = sin ș . Once
cos ș
students have been introduced to the reciprocal ratios as csc θ = 1y ,
sec θ = 1x , and cot θ = xy , it follows as a natural extension to discuss the
following relationships:
csc θ =
1
sinș
sec θ =
1
cosș
cot θ =
1
tanș
or
cos ș
sin ș
Students should be exposed to the restrictions on θ for tan θ, csc θ, sec θ
and cot θ.
90
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to determine the equation of a circle whose centre is at
(0, 0) with a radius of:
Pre-Calculus 12
(i)
(ii)
4 units
4.2 The Unit Circle
SB: pp. 180-190
34 units
TR: pp. 97-101
(iii) 2 5 units
(T2.2)
• Ask students to determine whether the given point is on the circle
whose equation is given.
(i) P(−2, 6); x2 + y2 = 49
(
)
(ii) P 12 , − 23 ; the unit circle
(T2.2)
• Ask students to determine the x- or y- coordinate on the unit circle
given the other coordinate. Use examples such as:
(i)
⎛1 ⎞
P⎜ , y ⎟ in Quadrant I
⎝5 ⎠
(ii)
P (x , − 38 ) in Quadrant III
(T2.3)
4.3 Trigonometric Ratios
SB: pp. 191-205
TR: pp. 102-107
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Solve problems, using the six
trigonometric ratios for angles
expressed in radians and degrees.
[ME, PS, R, T, V]
In Mathematics 2200, students solved problems using the three primary
trigonometric ratios (T2). Angles were expressed in degrees. This will
now be extended to include the reciprocal ratios. Students will work
with angles expressed in both degrees and radians.
Achievement Indicators:
T3.1 Determine, with technology,
the approximate value of a
trigonometric ratio for any angle
with a measure expressed in either
degrees or radians.
T3.2 Determine, using a unit
circle or reference triangle, the
exact value of a trigonometric
ratio for angles expressed in
degrees that are multiples of 0°,
30°, 45°, 60° or 90°, or for
angles expressed in radians that
are multiples of 0, S6 , S4 , S3 , or S2 ,
and explain the strategy.
Students should understand how to properly use a scientific or graphing
calculator to evaluate all six of the trigonometric ratios. They should be
able to efficiently use their calculators in both degree and radian mode,
being careful to check for the appropriate mode in all calculations.
Students have had previous experience determining reference angles for
positive rotational angles. This can also be applied to negative rotational
angles. Students can also use their previous knowledge of reference
triangles and/or the unit circle to determine the exact value of the six
trigonometric ratios. From the sketch below, for example, they can
determine the value of the trigonometric ratios for 150°.
T3.3 Sketch a diagram to
represent a problem that involves
trigonometric ratios.
cot150° =
x
y
=
−1
2
3
2
=−
3
3
Students should also simplify expressions such as
cos§¨ ʌ ·¸ sin ʌ ©6¹
.
tan 30q
Expressions requiring rationalizing are limited to those with monomial
denominators. From Mathematics 2200, students are familiar with
performing operations on rational expressions (AN5).
Students could use a calculator to verify their answers. The emphasis
here, however, is on finding exact values using the unit circle, reference
triangles, and mental math strategies.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to find the exact value of the following expressions:
Pre-Calculus 12
(i)
sin 23ʌ cos 2 116ʌ (ii)
csc ʌ3 cot 114ʌ (iii) cot 2 76ʌ
4.3 Trigonometric Ratios
SB: pp. 191-205
TR: pp. 102-107
ʌ
11ʌ
(iv) cot 3 cos 3
csc 240q (T3.2, T3.3)
Performance
• For the Four Corners activity, each student is given a card containing
a different trigonometric ratio. The four corrners of the room
correspond to the four quadrants. Students decide which quadrant
the angle on the card lies in and they move to the appropriate corner.
The small group should then verify that the angles are appropriately
placed in the quadrant, and evaluate each of the trigonoemtric ratios.
Cards should include a mixture of angles in degree and radian
measure, positive and negative angles, and those that have exact and
approximate values.
Sample Ratios:
cos 43S
tan 210q csc 114S
sin 310q cot 53S
sec 315q
sin123q
sec 3.6
(T3.1, T3.2, T3.3)
• In groups, students can play the Carousel game. Stations are set up
around the room, with completed expressions posted on the wall.
Students identify the error(s) and write the correct solution.
(T3.2, T3.3)
• Students can play a Match game. One set of cards contains
trigonometric expressions and the second set contains the completed
solutions. Teachers can vary the set-up, whereby they can have the
cards face up or face down.
Note
Simplifying trigonometric
expressions involving rationalizing
requires supplementary practice
questions.
(T3.2, T3.3)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Continued ...
Achievement Indicators:
T3.4 Determine, with or without
technology, the measures, in
degrees or radians, of the angles
in a specified domain, given the
value of a trigonometric ratio.
Students determined the measures of angles in degrees for the three
primary trigonometric ratios. They have also used reference triangles
and the unit circle to evaluate the six trigonometric ratios. Determining
the measure of angles for all six trigonometric ratios in both radians
and degrees should be a natural extension. Students could be asked
to determine the value of θ, for example, when sec θ = − 2 for
the domain −2π ≤ θ ≤ 2π. To determine the reference angle, they
could think about a triangle with hypotenuse 2 and adjacent side 1.
Alternatively, they could apply the reciprocal ratio cos θ = − 12 . Once
the reference angle is determined, they identify the quadrants where the
secant ratio is negative. QII:
QIII:
ʌ ʌ4
ʌ ʌ4
3ʌ
4
5ʌ
4
The final step focuses on identifying all possible values within the given
domain. Remind students of earlier work with coterminal angles.
ș
T3.5 Describe the six
trigonometric ratios, using a point
P(x, y) that is the intersection of
the terminal arm of an angle and
the unit circle.
^ 54ʌ , 34ʌ , 34ʌ , 54ʌ `
Given the coordinates of a point on the terminal arm of an angle in
standard position, the six trigonometric ratios can be determined. To
determine the ratios given that the point P(−3, −4) lies on the terminal
arm of the angle, for example, students should first sketch a diagram.
Point out that the right triangle is always made with the x-axis.
y
T3.6 Determine the measures of
the angles in a specified domain
in degrees or radians, given a
point on the terminal arm of an
angle in standard position.
x
P( -3 , -4 )
Encourage them to note that because the angle is in the third quadrant,
sine, cosine and their reciprocals are negative, and tangent and
cotangent are positive. This can be used to check the reasonableness of
their ratios. Students should also be asked to determine the measure of
the given angle in radians or degrees.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to solve the following trigonometric equations:
(i) sin θ = − 23 ; 0 ≤ θ < 2π
(ii) csc θ = −2; 0˚ ≤ θ < 360˚
(iii) sec θ = − 2 33 ; -2π ≤ θ < 2π
Authorized Resource
Pre-Calculus 12
(T3.4)
• Given that each of the following points lie at the intersection of the
unit circle and the terminal arm of an angle in standard position, ask
students to:
4.3 Trigonometric Ratios
SB: pp. 191-205
TR: pp. 102-107
(i) sketch the diagram
(ii) determine the values of the six trigonometric ratios
(iii) determine the angle of rotation from standard position
5
(a) P 13
, − 12
13
(
(
)
(b) P − 23 , − 12
)
(c) P (−5, 4 )
(T3.5, T3.6)
Journal
• Ask students to respond to the following:
Give an example of a trigonometric equation that does not have a
solution. Explain why.
(T3.4)
Performance
• Ask students to summarize what they have learned about the location
of positive and negative trigonometric ratios in the four quadrants.
They should use a graphic organizer such as:
Quadrant Signs
y
Quadrant II
Sine +
sin x
Quadrant I
All +
sin x
cos x
tan x
x
tan x
cos x
Tangent +
Quadrant III
Cosine +
Quadrant IV
Students could also include the reciprocal ratios.
(T3)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Continued ...
Achievement Indicators:
T3.7 Determine the exact
values of the other trigonometric
ratios, given the value of one
trigonometric ratio in a specified
domain.
Students have had previous experience finding the other two primary
trigonometric ratios given one of them. This is now extended to include
5
the reciprocal trigonometric ratios. Given cos x = 13
, for example, they
could be asked to determine the value of csc x for 0° ≤ x ≤ 90°. They
should be encouraged to draw a sketch similar to the one below to help
y
visualize a solution.
13
12
x
5
x
Students should be exposed to cases with a variety of domains, including
negative values.
T3.8 Solve a problem, using
trigonometric ratios.
Thorough understanding of the trigonometric ratios and work with
angles in standard position (T1) could be determined by asking students
to find the distance between two given points on a circle. Consider
using an example such as the following:
Students could be asked to find the arc length between points A(−6, 8)
y
and B(−8,6).
A( -6 , 8 )
B(-8,6)
D
O
C
Using A(−6,8): tan ∠AOC = 8
6
Using B(−8, 6): tan ∠BOD =
6
8
x
∴∠AOD 53°
∴∠BOD 37°
This results in ∠AOB 16° , and thus a curved arc length between
points A and B of 2.8 units.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Students could answer questions such as:
(i)
Given sin T 12 , 180˚ ≤ θ ≤ 270˚, determine the value of
cot θ.
(ii) Given sec x = 5.5, where
3ʌ
2
d x d 2ʌ, determine tan x.
Pre-Calculus 12
(T3.7)
(iii) A regular dodecagon (12-sided figure) is inscribed in the unit
circle. If one vertex is at (1,0), what are the exact coordinates
of the other vertices? Explain your reasoning.
(T3.8)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Authorized Resource
4.3 Trigonometric Ratios
SB: pp. 191-205
TR: pp. 102-107
97
TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T5 Solve, algebraically and
graphically, first and second
degree trigonometric equations
with the domain expressed in
degrees and radians.
[CN, PS, R, T, V]
In Mathematics 2200, students solved simple trigonometric equations of
the form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1, and tan θ = a, where
a is a real number (T2). They worked with angles in degree measure.
This will now be extended to include trigonometric equations with
all six trigonometric ratios. Students will solve first and second degree
trigonometric equations with the domain expressed in degrees and
radians.
Achievement Indicators:
T5.1 Determine, algebraically,
the solution of a trigonometric
equation, stating the solution in
exact form when possible.
T5.2 Determine, using
technology, the approximate
solution of a trigonometric
equation.
T5.3 Verify, with or without
technology, that a given value
is a solution to a trigonometric
equation.
When solving first degree equations, rearrangement will sometimes be
necessary to isolate the trigonometric ratio. When solving an equation
such as sec θ = 2 , students can consider which reference angle results
when the hypotenuse is 2 and the adjacent side is 1. Alternatively, they
can think about secant as the reciprocal of cosine and determine the
reference angle for the equation cos θ = 12 . Knowledge of exact values
of the sine, cosine or tangent of a 30°, 45° or 60° angle, as well as their
corresponding radian measures, and an understanding of the unit circle
continue to be important when solving trigonometric equations.
Students also solve second degree equations through techniques such
as factoring (e.g., sin2 θ - 3sinθ + 2 = 0, for all θ) or isolation and square
root principles (e.g., tan2 θ - 3 = 0, 0 ≤ θ < 2π). Students sometimes
mistakenly use only the principal square root when both negative
and positive should be considered. In the equation tan2 θ - 3 = 0,
0 ≤ θ < 2π, isolating the trigonometric ratio results in tan θ = ± 3
Students should realize that using only the principal square root in this
equation causes a loss of roots. Another common error occurs when
students do not find all solutions for the given domain. Remind them to
focus on the given domain. In the example above, the reference angle is
ʌ
3 and since there are two cases to consider (tangent being negative and
positive), there are solutions in all four quadrants.
Students should be encouraged to check all solutions with a calculator
or using the unit circle where appropriate. When solving equations
containing reciprocal ratios, students should also check that the
solutions are defined for the domain of the reciprocal ratios.
In the next unit, Trigonometric Functions and Graphs, students
will solve trigonometric equations for which the argument may
include a horizontal stretch or horizontal translation, such as
cos 2ș ʌ 12 .
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to solve the following trigonometric equations:
Pre-Calculus 12
(i)
(ii)
(iii)
(iv)
(v)
4.4 Introduction to Trigonometric
Equations
2 cos x − 1 = 0; x ∈ [−2π, 2π]
4 cot θ + 3 = -2 cot θ - 8; θ ∈ (0, 360˚)
2 csc2θ - 8 = 0; for all θ in radians.
5 sec2x = 1- sec x; for all x in radians.
2 sin2x + 5 sin x - 3 = 0; x ∈ (−π, 2π)
SB: pp. 206-214
TR: pp. 108-111
(T5.1, T5.2)
• Given g(θ) = cos2θ - 3 and p(θ) = 2cosθ, ask students to determine the
values of θ such that g(θ) = p(θ), where θ ∈ [0, 4π].
(T5.1)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRY AND THE UNIT CIRCLE
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T5 Continued ...
Achievement Indicator:
T5.4 Identify and correct errors
in a solution for a trigonometric
equation.
Students have had exposure to identifying and correcting errors
throughout the mathematics curriculum. This approach is continued in
the context of solving trigonometric equations.
Students should be encouraged to check the entire given solution for
errors and not stop checking once they have encountered the first error.
The following solution, for example, contains two errors: the first error
stems from improperly applying the zero product principle and the
second error occurs when all solutions for the given domain are not
found.
2 cos 2 ș cos ș 1 0, 0 b ș 360n
2 cos ș 1
cos ș 1
0
cos ș 12 , cos ș 1
ș cos1 12 , ș cos1 1
= ș 60n, 180n
100
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRY AND THE UNIT CIRCLE
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Observation
• Set up centres containing examples of trigonometric equations that
have been solved incorrectly. Students should move around the
centres to identify and correct the errors. Samples are shown below:
2
(i) 4 sin x 2 0, for all values of x in radians
sin 2 x
sin x
sin x
2
2
ʌ
4
ref ‘
Quadrant I:
4.4 Introduction to Trigonometric
Equations
SB: pp. 206-214
TR: pp. 108-111
ʌ
4
Quadrant II: ʌ ʌ4
x
Pre-Calculus 12
1
2
1
2
Authorized Resource
3ʌ
4
­° ʌ4 2ʌn, n  I
® 3ʌ
°̄ 4 2ʌn, n  I
2
(ii) 2 tan x 5tan x 3 0, 360q d x d 360q
tan x 3 2 tan x 1
tan x
3
0
tan x
12
ref ‘ 71.6q
ref ‘ 26.6q
QuadrantI: 71.6q
Quadrant II: 153.4q
Quadrant III: 251.6q
Quadrant IV: 333.4q
x 71.6q, 153.4q, 251.6q, 333.4q
(T5.4)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
101
TRIGONOMETRY AND THE UNIT CIRCLE
102
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Trigonometric Functions and Graphs
Suggested Time: 12 Hours
TRIGONOMETRIC FUNCTIONS AND GRAPHS
Unit Overview
Focus and Context
In this unit, students sketch the graphs of y = sin x and y = cos x and
determine characteristics such as the period, amplitude, maximum and
minimum values, intercepts, and domain and range. They also explore
the effects of transformations on these graphs.
Once each of these functions has been explored, trigonometric
functions are determined to model real-world situations and solve
problems.
Students sketch the graph of y = tan x and identify the domain and
range, period, asymptotes and intercepts.
Outcomes Framework
GCO
Develop trigonometric reasoning.
SCO T4
Graph and analyze the trigonometric
functions sine, cosine and tangent to
solve problems.
SCO T5
Solve, algebraically and graphically,
first and second degree trigonometric
equations with the domain expressed
in degrees and radians.
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TRIGONOMETRIC FUNCTIONS AND GRAPHS
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
Trigonometry
M4. Develop and apply the primary
trigonometric ratios (sine, cosine,
tangent) to solve problems that
involve right triangles.
T1. Demonstrate an understanding
of angles in standard position [0° to
360°].
T4. Graph and analyze the
trigonometric functions sine, cosine
and tangent to solve problems.
[R, V]
[CN, PS., T, V]
[C, CN, PS, R, T, V]
T5. Solve, algebraically and
T2. Solve problems, using the three
primary trigonometric ratios for angles graphically, first and second degree
from 0° to 360° in standard position. trigonometric equations with the
domain expressed in degrees and
[C, ME, PS, R, T, V]
radians.
[CN, PS, R, T, V]
T3. Solve problems, using the cosine
law and sine law, including the
ambiguous case.
[C, CN, PS, R, T]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRIC FUNCTIONS AND GRAPHS
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T4 Graph and analyze the
trigonometric functions sine,
cosine and tangent to solve
problems.
[CN, PS, T, V]
In Mathematics 1201 students first encountered and applied the three
primary trigonometric ratios (M4). In Mathematics 2200, students
developed an understanding of angles as rotations, and solved simple
trigonometric equations in degrees only. They also graphed reciprocal
functions (limited to the reciprocal of linear and quadratic functions)
and developed an understanding of asymptotes. Earlier in Mathematics
3200, students were exposed to work in radian measure, the unit circle,
and solved more complex trigonometric equations (T1, T2, T3, T5).
Achievement Indicators:
T4.1 Sketch, with or without
technology, the graphs of y = sin x
and y = cos x.
T4.2 Determine the
characteristics (amplitude,
domain, period, range and zeros)
of the graphs of y = sin x and
y = cos x.
In the previous unit, students determined the values of sin x and cos x
for various angle measures in degrees and radians. They also explored
the unit circle. In this unit, students develop the graphs for y = sin(x),
y = cos(x), and later y = tan(x) and identify the characteristic features
of each. Transformations of y = sin(x) and y = cos(x) are performed and
are used to model and solve problems. Students are not expected to
transform y = tan(x) or solve problems that involve the tangent function.
Students could use the values from the unit circle to plot (θ, cos θ) and
(θ, sin θ) on the interval x ∈ [0, 2π]. These results can be effectively
verified using graphing technology.
Draw attention to the 5 key points associated with each graph, since
these points help determine the characteristics of the graph (amplitude,
domain, period, range, and zeros) and will also be used to graph
transformations of them.
y = sin(x):
0
S
S
3S
2
2S
1
maximum
0
sinusoidal
axis
-1
minimum
0
sinusoidal
axis
S
S
3S
2
2S
2
0
sinusoidal
axis
y = cos(x):
0
2
1
maximum
0
-1
0
1
sinusoidal minimum
sinusoidal maximum
axis
axis
For each base graph the 5 key points give a skeleton of the graph on the
interval 0 to 2π, (one complete period) with points spaced a quarterperiod apart. Students should develop a proficiency in using patterning
to continue the skeleton and to sketch the graph. Their graphs should
show the periodic and continuous nature of the graphs of sinusoidal
functions. This requires that students show more than one period of
a sinusoidal graph. In addition to the characteristics listed, students
should also identify local maximums and minimums, and the equation
of the sinusoidal axis, also referred to as the horizontal central axis.
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TRIGONOMETRIC FUNCTIONS AND GRAPHS
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
• Ask students to create a Venn diagram to compare the characteristics
of the functions y = sin(x) and y = cos(x).
Pre-Calculus 12
(T4.2)
5.1 Graphing Sine and Cosine
Functions
Student Book (SB): pp. 222-237
Interview
• Ask students to show if the point S , 22 lies on the graph of
4
y = sin(x). Ask them if the same point lies on the graph of y = cos(x).
Teacher’s Resource (TR): pp. 118123
(T4.1)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRIC FUNCTIONS AND GRAPHS
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T4 Continued ...
Achievement Indicators:
T4.3 Determine how varying
the value of a affects the graph of
y = a sin x and y = a cos x.
T4.4 Determine how varying the
value of b affects the graphs of
y = sin bx and y = cos bx.
T4.5 Determine how varying the
value of d affects the graphs of
y = sin x + d and y = cos x + d.
T4.6 Determine how varying
the value of c affects the graphs of
y = sin(x + c) and y = cos(x + c).
T4.7 Sketch, without
technology, graphs of the form
y = a sin b (x − c ) + d and
y = a cos b (x − c ) + d using
transformations, and explain the
strategies.
T4.8 Determine the
characteristics (amplitude,
domain, period, phase shift,
range and zeros) of the graphs
of trigonometric functions of the
form y = a sin b (x − c ) + d and
y = a cos b (x − c ) + d .
Students have developed an understanding of transformations for graphs
of the form y = af (b(x − h)) + k in the Function Transformations unit,
and will now apply that knowledge to sinusoidal graphs, noting that
parameters h and k are now referred to as c and d. Students should
determine the end position of the five key points using transformations,
and then extend the graph appropriately.
Students need to know what characteristics of the graph change with
each parameter. To achieve this, they should investigate how each
parameter change affects the resulting graph one parameter at a time,
and match these changes to the defining characteristics of sinusoidal
graphs. Changing the value of a, for example, affects the amplitude of
the graph, and negative values result in a reflection in the x-axis. These
explorations can be effectively carried out using graphing technology
and could be accomplished through a guided exploration of the
parameters.
Ensure that students work with non-integer values for these parameters
since many real-world applications do not use integer values. They
should be exposed to negative values for a and b, but the focus of the
work should be with positive values.
Students should be able to determine the characteristics of a
trigonometric function using the value of the parameters in the equation
without necessarily having to graph the equation. Linking students’
knowledge of transformations to their understanding of the language of
sinusoidal functions should be emphasized here rather than memorizing
2S
formulas. Rather than only memorizing period = b , for example, the
concept should be developed from an understanding that b affects the
horizontal stretch, and the period of the base graph for sine is 2π.
Students should know from the Function Transformations unit that the
c parameter relates to horizontal translation. With sinusoidal functions
this is referred to as the phase shift, and will determine how the 5 key
points will be translated horizontally.
Students should be encouraged to use appropriate mathematical
language. They are expected to understand the terms phase shift and
vertical displacement, recognizing that they refer to horizontal and
vertical translations, respectively.
Students should also analyze sinusoidal functions that may require factoring. The function y = sin(4x − 6π), for example, should be factored
to y sin 4 x 32 S to correctly identify the phase shift.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC FUNCTIONS AND GRAPHS
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Performance
• For the function y = a sinb(x − c) + d, the parameters a, b, c, and
d will be randomly chosen (cards, dice, etc). Students compete in
teams to be the first to identify the five key points.
Alternatively, they could be asked to determine the period or any of
the other characteristics.
(T4.3 to T4.8)
• For a Jigsaw activity, students begin in a home group and each
member is a given a number (1 to 4). The groups dissolve and the
students with the same numbers form expert groups. Each expert
group explores a different type of transformation for y = sin x
and y = cos x. Once the exploration is complete, students return
to their home groups. Each expert teaches the others about the
transformation they explored.
(T4.3 to T4.6)
Authorized Resource
Pre-Calculus 12
5.1 Graphing Sine and Cosine
Functions
SB: pp. 222-237
TR: pp. 118-123
Web Link
www.desmos.com
Paper and Pencil
• Students are each given the equations of two sinusoidal functions
(e.g., y = 3sin(2(x – 30˚)) + 6 and y = −4cos(2(x – 60˚)) + 6) and
asked to describe how the functions are alike and how they differ.
This online graphing calculator
can be used to graph functions,
plot tables of data, evaluate
equations, and explore
transformations.
(T4.3 to T4.8)
Interview
5.2 Transformations of Sinusoidal
Functions
• Ask students the following questions about the curve
y = 5cos3(x − 30˚) + 2:
SB: pp. 238-255
(i)
(ii)
(iii)
(iv)
What is the period?
What is the amplitude?
What is the range?
Suppose we wanted to write the equation in the form
y = a sinb(x − c) + d. What values could be used for a, b, c and
d?
(v) How could the equation be modified so that the resulting
function will have no x-intercepts?
(T4.3 to T4.8)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TR: pp. 124-129
109
TRIGONOMETRIC FUNCTIONS AND GRAPHS
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T4 Continued ...
Achievement Indicators:
T4.9 Determine the values of
a, b, c and d for functions of
the form y = a sin b(x - c) + d
and y = a cos b(x - c) + d that
correspond to a given graph,
and write the equation of the
function.
To determine the equation of a sinusoidal function, students will have to
determine the key characteristics of the graph, and then link them to the
parameters in the equation. If they determine that the amplitude for the
graph is 4, for example, they should realize that a = 4.
While there are an infinite number of correct choices for the phase shift,
the convention is to use the smallest positive value. Teachers need to be
careful to accept all correct answers, not just the conventional answer
when assessing student work. Students should also be made aware that
more than one correct equation is possible and they should be able to
identify equations that produce the same graph.
Students should be cautioned when using negative values for a and b,
since these would involve reflections. Consequently, when determining
the value of the phase shift, different points on the graph need to be
considered.
T4.10 Solve a given problem
by analyzing the graph of a
trigonometric function.
Trigonometric functions are commonly used to model problems that
are periodic in nature, including circular motion, pistons, tides, climate,
daylight, populations of species, electricity, etc.
T4.11 Explain how the
characteristics of the graph of
a trigonometric function relate
to the conditions in a problem
situation.
Students should be encouraged to draw well-labelled sketches of the
graphs that represent the problems. This will help them more easily
identify the characteristics of the trigonometric function. While
students should be able to write a sine and a cosine equation to model
any application, cosine is more often used since it is easier to identify a
correct phase shift. Students are free to choose any correct equation to
solve the problem.
T4.12 Determine a trigonometric
function that models a situation
to solve a problem.
Teachers need to be mindful of the type of assessment questions used.
Questions that require graphing technology should be explored in a
classroom setting, not on formal assessments.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC FUNCTIONS AND GRAPHS
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Journal
• Students could be given the graph and the equation of a sinusoidal
function. Ask them to determine if the equation is correct for the
graph. If not, they should explain how the equation would need to
be modified.
y
Authorized Resource
2
Example:
Pre-Calculus 12
-4S
-2S
2S
4S
x
5.2 Transformations of Sinusoidal
Functions
SB: pp. 238-255
-2
TR: pp. 124-129
-4
-6
y = 2cos(2x) − 3
(T4.8, T4.9)
• Students could be given a graph that models a problem and asked to
identify questions that could be answered using the graph.
Height of Tides since noon
Height (m)
2.8
2.4
2
1.6
1.2
0.8
0.4
6
12
18
(T4.10, T4.11)
Paper and Pencil
• Ask students to determine the values of a, b, c, and d required to
write an equation for the graph.
y
2
-3S
-5S
2
-2S
-3S
2
-S
-S
2
S
2
S
3S
2
2S
5S
2
3S
x
-2
-4
-6
(T4.8, T4.9)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRIC FUNCTIONS AND GRAPHS
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T4 Continued ...
Achievement Indicators:
T4.13 Sketch, with or without
technology, the graph of y = tan x.
T4.14 Determine the
characteristics (asymptotes,
domain, period, range and zeros)
of the graph of y = tan x.
Although students have been exposed to the concept of a tangent line
in Grade 9, they may need reminding before investigating the graph of
y = tan x.
Students should develop the graph of y = tan(x), paying attention to
the characteristics of this graph. As with sine and cosine, 5 key points
(three points and two asymptotes) are conventionally used to graph one
complete period of y = tan(x):
S2
S4
0
S
4
2
asymptote
-1
0
1
asymptote
S
As with sine and cosine, ensure students realize that these key points
help guide the construction of one period of the function and they
should be able to use these key points to produce reasonably accurate
graphs. The graphs should be extended to include more than one
period.
Discuss whether the graph of y = tan(x) has an amplitude. It may be
helpful for students to see that “amplitude” is a characteristic of sine and
cosine graphs and it depends on a maximum and a minimum height.
Since the function y = tan(x) has no maximum or minimum, it cannot
have an amplitude.
A similar discussion could be held to help students understand why the
period of y = tan(x) is π.
Note that transformations of y = tan(x) will not be explored in this
course.
When discussing the behaviour around the asymptotes, it is important
to remember that students have no formal experience with limits, limit
notation, or infinity. Ask students to analyse the slope of the terminal
arm on the unit circle as it rotates from 0˚ to 90˚.
• When is the slope 0?
• What happens as the angle increases?
• What happens to the slope as the angle approaches 90˚?
• What is the slope of the line at 90˚?
Ask students the same questions concerning rotations from 0˚ to −90˚.
Have them compare these to slopes when the terminal arm is rotated
past 90˚. They should see the periodic nature of the curve and the
behaviour around the asymptotes.
112
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC FUNCTIONS AND GRAPHS
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to sketch a graph of y = sin(x) and y = tan(x) on the
same grid.
They should explain how the x-intercepts of y = sin(x) relate to the
x-intercepts of y = tan(x).
(T4.13, T4.14)
• Ask students to sketch the graph of y = cos(x) and y = tan(x) on the
same grid.
They should explain how the x-intercepts of y = cos(x) relate to the
asymptotes of y = tan(x).
Authorized Resource
Pre-Calculus 12
5.3 The Tangent Function
SB: pp. 256-265
TR: pp. 130-133
(T4.13, T4.14)
Interview
• Ask students if x = 3π is part of the domain of y = tan(x).
(T4.14)
Note
SB: p. 263, #5-#7
These questions develop the
sin x
relationship between tan x = cos
x
and the slope of the terminal arm,
which connects nicely with the
Trigonometry and the Unit Circle
unit.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRIC FUNCTIONS AND GRAPHS
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T5 Solve, algebraically and
graphically, first and second
degree trigonometric equations
with the domain expressed in
degrees and radians.
In Mathematics 2200, students solved quadratic equations by graphing
the related quadratic function and determining its x-intercepts (RF3,
RF4). They also solved systems of equations (quadratic-quadratic and
quadratic-linear) graphically and algebraically (RF6).
[CN, PS, R, T, V]
Achievement Indicators:
T5.1 Determine, algebraically,
the solution of a trigonometric
equation, stating the solution in
exact form when possible.
T5.2 Determine, using
technology, the approximate
solution of a trigonometric
equation.
T5.3 Verify, with or without
technology, that a given value
is a solution to a trigonometric
equation.
T5.5 Relate the general solution
of a trigonometric equation to
the zeros of the corresponding
function (restricted to sine and
cosine functions).
In the previous unit, students solved first and second degree
trigonometric equations algebraically. In this unit, they continue to
solve equations algebraically. They also use the graphs of trigonometric
functions to solve equations. For assessment purposes, students should
analyze given graphs to determine solutions.
Students also solve trigonometric equations for which the argument may
include a horizontal stretch or a horizontal translation.
To algebraically determine all solutions in radian measure for
cos ª¬ 4 x ʌ2 º¼ 23 students could proceed as follows:
Ref ‘ : șR
cos 1
3
2
ʌ
6
Quadrant II: 56ʌ
Quadrant III: 76ʌ
5ʌ
­ 2 ʌ n, n  I
? 4 x ʌ2 °® 76ʌ
°̄
x ʌ2
6
2 ʌ n, n  I
­° 524ʌ ʌ2 n, n  I
® 7ʌ ʌ
°̄ 24 2 n, n  I
­° 174ʌ ʌ2 n, n  I
x ® 19ʌ
ʌ
°̄ 4 2 n, n  I
Students should be reminded to check solutions by evaluating each side
of the original equation at all the solution values.
Discuss the effect a horizontal stretch has on the restricted domain.
To solve the equation sin 3x = 1, 0 ≤ x < 2π, for example, students
find all possible solutions for 3x first. The restricted domain would be
0 ≤ 3x < 6π. This results in 3 x ʌ2 , 52ʌ , 92ʌ and x ʌ6 , 56ʌ , 32ʌ .
114
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC FUNCTIONS AND GRAPHS
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Using the graph shown, ask students to determine the general
solution for 2 sin x ʌ2 1 0.
Pre-Calculus 12
5.4 Equations and Graphs of
Trigonometric Functions
y
= 2 sin
−
SB: pp. 266-281
2
TR: pp. 134-138
=1
x
S
S
S
S
(T5.5)
• Ask students to determine how many solutions there are for the
equation 3sin ª¬ 4 x ʌ2 º¼ 2 on the interval x ∈ [−π, 2π].
(T5.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRIC FUNCTIONS AND GRAPHS
116
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Trigonometric Identities
Suggested Time: 15 Hours
TRIGONOMETRIC IDENTITIES
Unit Overview
Focus and Context
In this unit, students will explore trigonometric relationships
that always hold and others that hold only for given values of the
angle. In this context, students will have the opportunity to derive
proofs of trigonometric relationships and find algebraic solutions of
trigonometric equations. Students will also simplify trigonometric
expressions.
Outcomes Framework
GCO
Develop trigonometric reasoning.
SCO T6
Prove trigonometric identities, using:
•
reciprocal identities
•
quotient identities
•
Pythagorean identities
•
sum or difference identities
(restricted to sine, cosine and
tangent)
•
double-angle identities
(restricted to sine, cosine and
tangent).
SCO T5
Solve, algebraically and graphically,
first and second degree trigonometric
equations with the domain expressed
in degrees and radians.
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TRIGONOMETRIC IDENTITIES
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
Trigonometry
M4. Develop and apply the primary
trigonometric ratios (sine, cosine,
tangent) to solve problems that
involve right triangles.
[C, CN, PS, R, T, V]
T1. Demonstrate an understanding
of angles in standard position [0° to
360°].
[R, V]
T6. Prove trigonometric identities,
using:
•
reciprocal identities
•
quotient identities
•
T2. Solve problems, using the three
•
primary trigonometric ratios for angles
from 0° to 360° in standard position.
[C, ME, PS, R, T, V]
T3. Solve problems, using the cosine
law and sine law, including the
ambiguous case.
[C, CN, PS, R, T]
•
Pythagorean identities
sum or difference identities
(restricted to sine, cosine and
tangent)
double-angle identities (restricted
to sine, cosine and tangent).
[R, T, V]
T5. Solve, algebraically and
graphically, first and second degree
trigonometric equations with the
domain expressed in degrees and
radians.
[CN, PS, R, T, V]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T6 Prove trigonometric
identities, using:
•
reciprocal identities
•
quotient identities
•
Pythagorean identities
•
sum or difference identities
(restricted to sine, cosine and
tangent)
•
double-angle identities
(restricted to sine, cosine and
tangent).
Students have had previous exposure to graphing trigonometric
functions (T4) and solving trigonometric equations (T5). They were
also introduced to the reciprocal trigonometric ratios (T3). Students are
now introduced to trigonometric identities as trigonometric equations
that are true for all permissible values of the variable in the expressions
on both sides of the equation. They will verify identities both graphically
and numerically, and prove identities using the Pythagorean identities,
the quotient identities, the reciprocal identities, the sum/difference
identities, and the double-angle identities.
[R, T, V]
Achievement Indicator:
T6.1 Explain the difference
between a trigonometric identity
and a trigonometric equation.
Students should be able to explain the difference between a
trigonometric equation and a trigonometric identity. An identity is true
for all permissible values, whereas an equation is only true for a smaller
subset of the permissible values. This difference can be demonstrated
with the aid of graphing technology.
Students have used the graphs of trigonometric functions to solve
equations. The equation sin x = 12 , for example, can be solved for
0˚ ≤ x < 360˚, using the graphs of y = sin x and y = 12 :
The solutions to sin x = 12 for 0˚ ≤ x < 360˚are x = 30˚ and x = 150˚,
which are the x-values of the intersection points.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
• Ask students to respond to the following:
Pre-Calculus 12
Explain to a friend who missed today’s class the difference between
an identity and an equation.
(T6.1)
6.1 Reciprocal, Quotient, and
Pythagorean Identities
Student Book (SB): pp. 290-298
Teacher’s Resource (TR): pp. 146149
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T6 Continued ...
Achievement Indicators:
T6.1 Continued
T6.2 Determine, graphically,
the potential validity of a
trigonometric identity, using
technology.
Students could then work with the relation sin x = tan x cos x. They
should see that, when graphed, y = sin x and y = tan x cos x are almost
identical:
y
1
y = sinx
60°
T6.3 Determine the nonpermissible values of a
trigonometric identity.
120° 180° 240° 300° 360°
x
-1
y
1
y = tanxcosx
60°
120° 180° 240° 300° 360°
x
-1
The only differences in the graphs occur at the points (90˚, 1)
and (270˚, −1), which are non-permissible values of x. Therefore,
sin x = tan x cos x is an identity since the expressions are equivalent for
all permissible values.
Students should discuss why there are points for which identities are not
equivalent. Non-permissible values for identities occur where one of the
expressions is undefined. In the above example, y = tan x cos x is not
defined when x = 90˚ + 180˚n, n ∈ Ι since y = tan x is undefined at these
values. Students should note that non-permissible values often occur
when a trigonometric expression contains:
• a rational expression, resulting in values that give a denominator of
zero
• tangent, cotangent, secant and cosecant, since these expressions all
have non-permissible values in their domains.
Students should determine non-permissible values both graphically and
algebraically.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to determine graphically if the following are identities.
They should identify the non-permissible values.
(i) sin θ + cos θ tan θ = 2 sin θ
(ii) tan2 θ + 1 = sec2 θ
(iii) cosT sec T
sinT
Authorized Resource
Pre-Calculus 12
6.1 Reciprocal, Quotient, and
Pythagorean Identities
(T6.1, T6.2, T6.3)
SB: pp. 290-298
TR: pp. 146-149
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
123
TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T6 Continued ...
Achievement Indicators:
T6.4 Verify a trigonometric
identity numerically for a given
value in either degrees or radians.
T6.5 Prove, algebraically, that a
trigonometric identity is valid.
Students should also verify numerically that an identity is valid by
substituting numerical values into both sides of the equation. Angles in
both degree and radian measures should be used.
To introduce the Pythagorean identities, students could be given the
expression sin2 θ + cos2 θ and asked to substitute in different values
for θ. They should conclude inductively that sin2 θ + cos2 θ = 1 for all
values of θ. Discuss how this approach is insufficient to conclude that
the equation is an identity because only a limited number of values
were substituted for θ, and there may be a certain group of numbers for
which this identity does not hold. This discussion should lead to the
idea of a proof – a deductive argument that is used to show the validity
of a mathematical statement. To prove sin2 θ + cos2 θ = 1, the unit circle
(T2), the definitions of sin θ and cos θ (T2), and the Pythagorean
theorem can be used:
Encourage students to use the left side and right side notation where
appropriate:
LS = sin2 θ + cos2 θ
= (y)2 + (x)2
= 12
=1
= RS
Using the above identity and the reciprocal and quotient identities,
students should derive the other two Pythagorean identities (1 + tan2 θ
= sec2 θ and cot2 θ + 1 = csc2 θ) and identify the non-permissible values.
They should verify the identities numerically and validate them with
proofs.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Journal
• Ask students to justify whether cos2 θ - sin2 θ = cos 2θ is an identity,
using numerical or graphical evidence.
(T6.1, T6.2, T6.3, T6.4)
Authorized Resource
Pre-Calculus 12
Interview
• Ask students to explain whether or not sin θ + cos θ = 1 given that
sin2 θ + cos2 θ = 1.
(T6.3, T6.4)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
6.1 Reciprocal, Quotient, and
Pythagorean Identities
SB: pp. 290-298
TR: pp. 146-149
125
TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T6 Continued ...
Achievement Indicator:
T6.6 Simplify trigonometric
expressions using trigonometric
identities.
Students should also simplify expressions using the Pythagorean
identities, the reciprocal identities, and the quotient identities. Discuss
specific strategies with students that they might use to begin the
simplifications:
• Replace a “squared” term with a Pythagorean identity
• Write the expression in terms of sine or cosine
• For expressions involving addition or subtraction, it may be necessary
to use a common denominator to simplify a fraction
Remind students to determine any non-permissible values of the
variable in an expression. Students could be asked, for example, to
identify the non-permissible values of θ in sinT cosT cot T , and then
simplify the expression.
Students often find simplifying trigonoemtric expressions more
challenging than proving trigonometric identities because they may
be uncertain of when an expression is simplified as much as possible.
Developing a good foundation with simplifying expressions makes the
transition to proving trigonometric identities easier.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to simplify
values.
1 sin2 T
cosT ,
identifying the non-permissible
(T6.3, T6.6)
Authorized Resource
Journal
Pre-Calculus 12
• Ask students to respond to the following:
6.1 Reciprocal, Quotient, and
Pythagorean Identities
For one of your homework problems, the answer is sec x csc x. You
and your friend get different answers. Which of them is correct?
Explain.
(i)
sec2 x
tan x
SB: pp. 290-298
TR: pp. 146-149
(ii) cot x + tan x
(T6.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
127
TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T6 Prove trigonometric
identities, using:
•
reciprocal identities
•
quotient identities
•
Pythagorean identities
•
sum or difference identities
(restricted to sine, cosine and
tangent)
•
double-angle identities
(restricted to sine, cosine and
tangent).
Frequently, trigonometric relationships involve angle measures that are
related as either the sum or difference of other angles or the double
of other angles. In such cases, students can use formulas to evaluate
trigonometric functions. Once introduced to these, they should realize
that the advantage in using the sum and difference formulas or the
double-angle formulas is that resulting evaluations can be expressed as
exact values rather than as approximate decimal values. The formulas are
also used to simplify trigonometric expressions and verify identities.
[R, T, V]
Achievement Indicators:
T6.7 Determine, using the sum,
difference and double-angle
identities, the exact value of a
trigonometric ratio.
Students should be exposed to the sum and difference identities for the
primary trigonometric ratios:
sin(a + b) = sina cosb + cosa sinb
sin(a - b) = sina cosb - cosa sinb
cos(a + b) = cosa cosb - sina sinb
T6.2, T6.3, T6.4, T6.6
Continued
cos(a - b) = cosa cosb + sina sinb
tan a + tan b
tan (a + b ) =
1 − tan a tan b
tan a − tan b
tan (a − b ) =
1 + tan a tan b
After working with the sum and difference identities, students should
be exposed to the double-angle identities for the primary trigonometric
ratios:
sin2a = 2sina cosa
cos2a = cos2a - sin2a = 1 - 2sin2a = 2cos2a - 1
2 tan a
tan 2 a =
1 − tan 2 a
These identities should be verified numerically. Teachers could expose
students to the derivation of these formulas but the proofs are not
required for assessment.
Students should see that the three formulas for cos 2a are all equivalent
and that they can use whichever one is most convenient in a problem.
A common student error is to express tan 2x as 2tanx or cos2x as 2cosx.
Ensure students realize that these expressions are not equivalent.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to determine the exact value of the following:
23S
(i) tan 12 Pre-Calculus 12
6.2 Sum, Difference and DoubleAngle Identities
(ii) sin(255˚)
(T6.7)
SB: pp. 299-308
TR: pp. 150-153
7 where 270˚ ≤ θ < 360˚, ask students to determine the
• If cos T 25
exact values of tan 2T and sin T 32S .
(T6.7)
Performance
• Ask students to explain whether or not sin2x = 2sinx is an identity.
(T6.7)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
129
TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T6 Continued ...
Achievement Indicators:
T6.2, T6.3, T6.4, T6.5 T6.6,
T6.7 Continued
The sum, difference, and double-angle identities are used to determine
exact values of trigonometric expressions. Using the double-angle
identities, students should determine the exact trigonometric ratios of
angles that are not multiples of 30˚ or 45˚ but are multiples of 15˚.
They could algebraically determine, for example, the exact value of
cos 712S and tan145˚. Students should also use these formulas to find
°+ tan55°
the exact value of expressions such as 1tan80
−tan80° tan55° . They should realize
that the sum and difference identities can be applied in either direction.
They often ignore the fact that the sum and difference formulas are
equally true when read from right to left.
The sum, difference, and double-angle identities are also needed to
x
simplify certain trigonometric expressions. The expression 1−sin2
cos2 x , for
example, can be simplified to cot x, using the appropriate double-angle
formula for cos 2x. Ask students to identify the non-permissible roots
algebraically, and verify the solution numerically and/or graphically.
To find the non-permissible roots algebraically, it is necessary to solve
1 - cos 2x ≠ 0. Students could:
• substitute the appropriate double-angle formula and solve:
1 cos 2 x z 0
1 1 2 sin 2 x z 0
2 sin 2 x z 0
sin 2 x z 0
sin x z 0
x z S n, n  I
• solve cos 2x ≠ 1, noticing that there is a horizontal stretch of
which gives a period of π:
1
2
,
cos 2 x z 1
2 x z 0 2S n, n  I
x z S n, n  I
130
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to use numerical examples to show that
tan(x + y) ≠ tan x + tan y.
(T6.7)
Authorized Resource
Interview
x
• Students can work in pairs to prove that 1sin2
cos2 x tan x in three
ways, using a different identity for cos2x in each proof. Ask them
to discuss which identity they found the easiest to work with, and
identify any strategies that they learned.
The teacher then interviews the student pairs to gain insight into
the level of understanding and ability to put mathematical ideas into
words. Ask questions such as:
Pre-Calculus 12
6.2 Sum, Difference and DoubleAngle Identities
SB: pp. 299-308
TR: pp. 150-153
(i)
Which identity for cos 2x did you use first? Why did you
choose this one?
(ii) Which identity did you find easiest to work with?
(iii) What strategies have you learned that might help you choose
the best identity for cos 2x for future proofs?
Rather than interview each student for each topic, teachers may
decide to select a sample of students to interview, ensuring all
students are included over time.
(T6.5)
Performance
• Prepare cards containing the steps in the proof of a trigonometric
identity. Students work in small groups to decide on a logical
sequence in which to place the cards. As students examine the cards,
they should discuss their ideas about a possible sequence.
(T6.5)
Paper and Pencil
• Ask students to simplify:
(i)
cos2 x + sin2 x
sin2 x
(ii) sin T S2 sin T S2 (T6.6, T6.7)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
131
TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T6 Continued ...
Achievement Indicators:
T6.8 Explain why verifying that
the two sides of a trignometric
identity are equal for given values
is insufficient to conclude that the
identity is valid.
T6.3, T6.5 Continued
Students should use the reciprocal identities, the Pythagorean identities,
the sum and difference identities, and the double-angle identities to
prove other trigonometric identities. To prove identities, they must
understand that one side of the identity is rewritten in terms of the
functions found on the other side. Strategies for validating an identity
include:
• writing expressions in terms of sine and cosine
• expressing the given trigonometric functions in terms of a single
trigonometric function
• factoring expressions, including expressions with common factors,
difference of squares, and trinomials
• writing expressions with a common denominator
• expanding an expression, such as multiplying two binomials together
• writing one fraction as two or more fractions
• multiplying by the conjugate
• multiplying an expression by a fraction equivalent to 1.
Students should also be reminded to identify any non-permissible roots
when proving identities.
For the identity (1 + cot2x)(1 - cos2x) = 2, the non-permissible roots
2x
are given by sin x ≠ 0 or x ≠ πn, n ∈ I, since cot 2 x = cos
. To prove
2
sin x
this identity, encourage them to start with the side that appears more
complicated. This proof uses both a Pythagorean identity and a doubleangle identity. As they work through this proof, ask students how they
decide which form of the double-angle formulas for cosine is most
appropriate. They should see the importance of being able to express the
trigonometric functions in terms of a single function.
Students can verify this numerically, but remind them that showing an
identity is true for certain values of x is not a proof, since there may be
other values of x that do not work in the identity. It does give evidence,
however, that the identity is valid. They can also verify the result
graphically.
Students should be exposed to proofs that require more than one
strategy.
In some cases, both sides of an identity may be independently simplified
to a common expression. Subsequently, the proof is validated by the
transitive property.
132
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to prove that the following identities are valid:
(i)
(ii)
sin S2 T cosT
1 cos 2T sin 2T
1 cos 2T sin 2T
Authorized Resource
Pre-Calculus 12
tan T
6.3 Proving Identities
SB: pp. 309-315
(iii) 1 + sin 2θ = (sin θ + cos θ)2
(T6.8)
TR: pp. 154-157
• Ask students to graphically verify that cos 2T 1tan2 T . They should
1 tan T
determine the non-permissible roots and algebraically prove the
identity.
2
(T6.3,T6.5,T6.8)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
133
TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T5 Solve, algebraically and
graphically, first and second
degree trigonometric equations
with the domain expressed in
degrees and radians.
Students should recall solving trigonometric equations in Mathematics
2200 (T2) and earlier in this course (T5). The identities encountered
earlier in this unit (T6) can now be applied to solve trigonometric
equations.
[CN, PS, R, T, V]
Achievement Indicator:
T5.1 Determine, algebraically,
the solution of a trigonometric
equation, stating the solution in
exact form when possible.
Continue to emphasize the connection between graphical solutions
and algebraic solutions for trigonometric equations. Students could,
for example, be asked to find the solutions of sin 2 x = 3 cos x for
0° ≤ x < 360° . Use the graphs of y = sin2x and y = 3 cos x with
domain 0° ≤ x < 360° to show the intersection at 60˚, 90˚, 120˚ and
270˚.
A double-angle identity is used to solve this same equation algebraically:
sin 2 x = 3 cos x
2 sin x cos x − 3 cos x = 0
(
)
cos x 2 sin x − 3 = 0
cos x = 0
2 sin x − 3 = 0
x = 90°,270°
sin x = 23
x = 60°,120°
Students should see the relationship between the algebraic and graphical
solutions.
Students should also solve trigonometric equations with unrestricted
domains resulting in a general solution, in degrees and radian measure.
Using the previous example with an unrestricted domain, the solutions
in radians are x S2 S n, x S3 2S n, x 23S 2S n, n  I . Here
students see when the domain has no restriction there are an infinite
number of solutions. Extending the graph so that students can see
more intersection points may help consolidate understanding of general
solutions.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to solve each of the following equations for
0° ≤ x ≤ 360° , giving exact solutions where possible.
Pre-Calculus 12
(i)
sin 2 x = 2 cos x
6.4 Solving Trigonometric
Equations Using Identities
(ii)
tan 2 x =
SB: pp. 316-321
2
sec2 x
Ask students to write the general solution for the above equations in
degrees and radian measure.
TR: pp. 158-161
(T5.1, T5.2, T5.4)
• Ask students to solve cos 2 x = sin x graphically, and verify the
answers numerically.
(T5.4, T5.3)
Journal
• Ask students which identity should be used as a substitution for
cos 2x when solving 1 − cos 2 x = cos x . Students should justify their
answers and solve the equation with domain 0 ≤ θ ≤ 2π.
(T5.1, T5.4)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
135
TRIGONOMETRIC IDENTITIES
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T5 Continued ...
Achievement Indicators:
T5.3 Verify, with or without
technology, that a given value
is a solution to a trigonometric
equation.
T5.2 Determine, using
technology, the approximate
solution of a trigonometric
equation.
Students should verify that a particular value is a solution to a given
trigonometric equation by simply substituting it into the equation and
determining if it satisfies the equation.
Students should solve trigonometric equations for non-special angles as
well. This can be done with the use of a scientific calculator where the
degree or radian measure is found by finding the inverse trigonometric
function of a ratio. Students could solve cos 2x + sin2x = 0.7311, for
example, for the domain 0˚ ≤ x < 360˚.
Students are also required to provide the general solution of a
trigonometric equation. Remind them that the general solution of the
equation above includes all angles that are coterminal with the solutions
already found.
T5.4 Identify and correct errors
in a solution for a trigonometric
equation.
Identifying errors and providing the correct solution is a good technique
for developing analytical skills. Students could be given a particular
example with an error and asked to identify the mistake.
sin 2 x − sin x = 0
sin 2 x = sin x
sin 2 x sin x
=
sin x sin x
sin x = 1
x = 90°
In this case, a solution has been lost as a result of dividing both sides of
the equation by sin x.
136
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
TRIGONOMETRIC IDENTITIES
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• A student’s solution for tan 2 x = sec x tan 2 x for 0 ≤ x < π is shown
below:
tan 2 x sec x tan 2 x
tan 2 x
tan 2 x
1 sec x
x 0, 2S
Ask students to identify and explain the error(s).
(T5.1, T5.4)
• Ask students to solve cos 2x = 0.8179 for 0˚ ≤ x ≤ 360˚. They
should also write the general solution in both degrees and radians.
Authorized Resource
Pre-Calculus 12
6.4 Solving Trigonometric
Equations Using Identities
SB: pp. 316-321
TR: pp. 158-161
(T5.2)
Journal
• Distribute half sheets of paper or index cards and ask students to
describe the “muddiest point” of solving trigonometric equations.
They should jot down any ideas or parts of the lesson that were
difficult to understand. This is a quick monitoring technique that
allows any difficulties to be addressed.
(T5.1, T5.2, T5.3, T5.4)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
137
TRIGONOMETRIC IDENTITIES
138
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Exponential Functions
Suggested Time: 12 Hours
EXPONENTIAL FUNCTIONS
Unit Overview
Focus and Context
In this unit, students are introduced to the graph of an exponential
function y = cx, c > 0, c ≠ 1. They explore the effects of translations,
stretches and reflections on the basic graph and apply these
transformations to graph an exponential function of the form
y = a(c)b(t - h) + k.
Students solve exponential equations by writing both sides as rational
powers of the same base. In cases where this is not possible, they use
systematic trial or graphing technology to approximate a solution.
This will be revisited in the next unit when they are introduced to
logarithms.
Throughout the unit, students are exposed to problems that can be
modelled using exponential functions.
Outcomes Framework
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF8
Graph and analyze exponential and
logarithmic functions.
SCO RF9
Solve problems that involve
exponential and logarithmic
equations.
140
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
EXPONENTIAL FUNCTIONS
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
RF11. Graph and analyze reciprocal
functions (limited to the reciprocal of
linear and quadratic functions).
RF8. Graph and analyze exponential
and logarithmic functions.
Relations and Functions
AN3. Demonstrate an understanding
of powers with integral and rational
exponents.
[C, CN, PS, R]
[C, CN, T, V]
[CN, R, T, V]
RF1. Interpret and explain the
relationships among data, graphs and
situations.
RF9. Solve problems that involve
exponential and logarithmic
equations.
[C, CN, R, T, V]
[C, CN, PS, R]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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EXPONENTIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF8 Graph and analyze
exponential and logarithmic
functions.
[C, CN, T, V]
Achievement Indicators:
RF8.1 Sketch, with or without
technology, a graph of an
exponential function of the form
y = cx, c > 0, c ≠ 1.
In Mathematics 1201, students worked with the laws of exponents,
integral exponents, and rational exponents. They also applied the
exponent laws to expressions with rational and variable bases, as well
as integral and rational exponents (AN3). Additionally, students
graphed, with and without technology, a set of data and determined
the restrictions on the domain and range (RF1). In Mathematics
2200, students graphed and analyzed reciprocal functions (limited to
reciprocals of linear and quadratic functions) of the form y = f 1x ,
()
where they would have been introduced to the concept of asymptotes
(RF11).
In the Function Transformations unit of Mathematics 3200, students
worked with horizontal and vertical translations (RF1), horizontal and
vertical stretches (RF2), and combinations of stretches and translations
(RF3). Mapping rules were also studied in the context of these
transformations. In this unit, they work with exponential functions.
As an introduction to exponential functions, students should create
tables of values and their corresponding graphs for exponential functions
of the form y = cx where c > 1 and 0 < c < 1. They could create tables for
functions such as the following:
•
y = 2x
•
y = (32 )
•
y = 2.8x
•
x
y = (23 )
x
• y = 0.16x
RF8.2 Identify the characteristics
of the graph of an exponential
function of the form y = cx, c > 0,
c ≠ 1, including the domain,
range, horizontal asymptote
and intercepts, and explain the
significance of the horizontal
asymptote.
142
When discussing features of these graphs, the focus should be on:
• x- and y-intercepts
• domain and range
• whether the graph is increasing or decreasing
• equation of the horizontal asymptote
As students think about the significance of the horizontal asymptote,
they should consider in which cases the graph approaches its asymptote
as x-values increase (tend toward positive infinity) and in which cases
the graph approaches its asymptote as x-values decrease (tend toward
negative infinity).
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
EXPONENTIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Interview
Authorized Resource
• For each of the following, ask students to determine whether the
function is increasing or decreasing and explain why.
Pre-Calculus 12
(i) y = 4x
(13 )
y = (52 )
(ii) y =
(iii)
x
7.1 Characteristics of Exponential
Functions
Student Book (SB): pp. 334-345
x
(RF8.2)
Teacher’s Resource (TR): pp. 174180
Journal
• Ask students to explain why an exponential function cannot have a
negative base or a base that equals 0 or 1.
(RF8.2)
• Ask students to explain why graphs of functions of the form y = cx,
c > 0, c ≠ 1:
(i) do not have x-intercepts
(ii) always have y-intercept (0, 1)
(iii) always have the same domain and range
(RF8.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
143
EXPONENTIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF8 Continued ...
Achievement Indicators:
RF8.1, RF8.2 Continued
Teachers should use this opportunity to define whether a function
represents exponential growth (increases) or decay (decreases). Students
should identify which functions are increasing or decreasing according
to whether the base, c, is greater than 1 or between 0 and 1. They should
also identify how the graphs are similar and how they are different.
Students should explore graphs of other exponential functions where c>1
and 0 < c < 1 to confirm the conclusions reached in the introductory
activity. They should also graph y = cx where c = 1 and where c < 0.
When c = 1, a horizontal line is produced. When c is negative, if integer
values of x are chosen the y-values “oscillate” between positive and
negative values; for rational values of x, non-real values of y may be
obtained. To see this, students could complete the table below for the
functions y = (−2)x and y = (1)x.
x
−2
− 32
−1
− 12
0
y
1
2
1
3
2
2
Graphs and tables of values for exponential functions should also be
analyzed by students to determine the corresponding functions.
Problems involving exponential growth and decay can be introduced
here in contexts such as half-lives, bacterial growth/decay, light
intensity, and finance, provided these situations involve functions of
the form y = cx. More extensive work involving functions of the form
b (x − h )
y = a (c )
+ k will be done later in the unit, once students have
applied transformations to y = cx. Students should identify any necessary
restrictions on the domain and range due to the context of the problem.
144
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
EXPONENTIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Interview
• Ask students to determine which situations require a value of c > 1
(growth) and which require a value of 0 < c < 1 (decay). They should
explain the choice.
(i)
The number of neutrons present in a nuclear fission reaction
triples at each stage of progression.
(ii) The volume of ice in a certain region of the arctic ice-cap is
shrinking at a rate of 0.5% per year.
(iii) Lionel receives a pay increase of 2.5% per year.
(RF8.2)
Authorized Resource
Pre-Calculus 12
7.1 Characteristics of Exponential
Functions
SB: pp. 334-345
TR: pp. 174-180
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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EXPONENTIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF8 Continued...
Achievement Indicator:
RF8.3 Sketch the graph of an
exponential function by applying
a set of transformations to the
graph of y = cx, c > 0, c ≠1 and
state the characteristics of the
graph.
Previous work with transformations is now extended to include
exponential functions. Students apply reflections, stretches and
translations to exponential growth and decay curves, and then relate
them to the parameters a, b, h, and k in a function of the form
y = a(c)b(x - h) + k, for a variety of values of c. Students could also write
the mapping rules as they work with the transformations. It may be
beneficial to first explore the types of transformations one at a time.
• Translations
Students could create tables of values and the graphs for functions of
the form f(x) = cx + k. Using functions such as f(x) = 2x, f(x) = 2x + 3,
and f(x) = 2x - 4, they should explore the connection between the value
of k and the vertical translation. Students should then explore the
effects of h for functions of the form f(x) = cx - h, such as f(x) = 2x + 3 and
f(x) = 2x - 4. Further exploration with other bases could be done with the
aid of graphing technology. Students could also write the mapping rules
relating the graph of y = cx to the transformed graphs.
• Stretches
Vertical stretches should be explored with functions of the form
y = a(c)x by creating tables of values for a set of functions such as
x
y = 2(5)x, y = 4(5)x and y = 13 (5 ) . Horizontal stretches can be explored
x
using a set of functions of the form y = cbx such as y = 52x, y = 5 3 and
y = 50.2x.
• Reflections
Reflections across the x-axis should be explored for functions of the
form y = -cx with tables of values for a set of functions such as y = -2x,
x
y = − (13 ) and y = -(4)x.
Reflections across the y-axis for functions of the form y = c -x can be
explored by creating tables of values for a set of functions such as y = 2 − x
−x
and y = (12 ) . Note that students should consider the relationship
x
between y = (1c ) and y = c -x.
Once the effect of each transformation has been explored, students
should work with combinations of transformations for functions of the
form y = a(c)b(x - h) + k, using a variety of values for the parameters a, b,
h, and k. In the Function Transformations unit, students explored the
order in which transformations should be applied. This concept should
be reviewed here.
Applications of exponential growth or decay, such as cooling behaviour
of a liquid, radioactive decay, medications, and light intensity, may be
further addressed at this point.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
EXPONENTIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to graph each of the following, giving the domain,
range, and equation of the horizontal asymptote:
(i)
(ii)
y = − 13 (2 )
x +3
−1
−1
2 (x − 2 )
y = 2 (3 )
Authorized Resource
+5
Pre-Calculus 12
(RF8.3)
• Ask students to identify the value of each parameter (a, b, h, and k)
and its effect on the original graph for the functions:
(i)
y = −2 (
(ii)
y=
1
2
1
2
(3 )
)
3(x −1)
− x +2
+4
7.2 Transformations of
Exponential Functions
SB: pp. 346-357
TR: pp. 181-186
−4
(RF8.3)
Journal
• Ask students to create an exponential decay function that has a
horizontal asymptote at y = -4 and a y-intercept at (0, 2). They
should create the graph of their function and explain why they
selected the values of a, b, h, and k that they did.
(RF8.3)
Observation
• Students can work in pairs to complete puzzles containing the
characteristics and graphs of various exponential functions of the
form y = a(c)b(x - h) + k. They should work with 20 puzzle pieces (4
complete puzzles) to correctly match the characteristics with each
function. Sample puzzles are shown below.
Web Link
https://www.desmos.com/calculator
This online graphing calculator
could be used to explore the effects
of transformations on graphs of
exponential functions.
(RF8.3)
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EXPONENTIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Solve problems that involve
exponential and logarithmic
equations.
The focus here is on problem solving involving exponential equations
using common bases and estimation methods. Problem solving
situations using logarithmic equations will be explored in the next unit.
[C, CN, PS, R]
Achievement Indicators:
RF9.1 Determine the solution
of an exponential equation for
which both sides can be written as
rational powers of the same base.
In Grade 9, students rewrote numbers as powers and solved problems
involving the laws of exponents for whole number exponents (9N1,
9N2). In Mathematics 1201, they worked with negative exponents,
rational exponents, the exponent laws, radicals, and problems involving
laws of exponents and laws of radicals (AN3). They also worked with
variable bases. Students now solve exponential equations with variable
exponents where the bases can both be expressed as rational powers of
the same base, including
radical bases. They will work with equations
3
1
such as 25x = 125
. In this case, both bases can be expressed as
integer powers of 5.
( )
RF9.2 Determine the solution of
an exponential equation in which
the bases are not rational powers
of one another, using a variety of
strategies.
Students should develop estimation skills for the solutions of
exponential equations with variable exponents, including those where
the bases cannot both be expressed as rational powers of the same base.
When solving equations using logarithms, they will be better able to
determine the reasonableness of solutions. Suggested strategies include
systematic trial and graphing technology.
• Systematic Trial
Students could solve 2x = 10, correct to two decimal places, using the
process outlined here:
Since 10 is closer to 23 = 8 than to 24 = 16, they might begin with
x 3.3.
Test value for x
Power
Approximate value
3.3
3.3
2
9.849
3.4
10.556
3.4
2
The value obtained for 23.3 is closer to 10, so the next estimate should
be closer to 3.3 than to 3.4.
3.31
23.31
3.32
23.32
3.33
23.33
They should reason that the best estimate is x
closer to 10 than 10.056 is.
9.918
9.987
10.056
3.32 because 9.987 is
Students should use systematic trial to solve exponential equations with
a variety of bases, including rational bases. Although the precision of
estimates can vary, a minimum of one decimal place is required.
148
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
EXPONENTIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to determine the solution for equations such as the
following:
Pre-Calculus 12
(i)
92 x+1 = 81
(ii) 162 x +1 =
7.3 Solving Exponential Equations
SB: pp. 358-365
(12 )
x −3
TR: pp. 187-192
(iii) 40 = 42 x+3
(RF9.1, RF9.2)
• Ask students to algebraically determine the solution for the following
equations:
(i)
(ii)
(13 ) = (81)
5 (14 ) = 80
2 x −1
3− x
x
5 = 25x −1
(iii)
Notes
(iv) 27 2 x −1 = 3 3
(v)
(vi)
5
8x −1 = 3 16 x
•
Exponential equations with
radical bases are not addressed
in the student book.
•
There is limited treatment in
the student book of solving
exponential equations where
negative exponents are
required.
3 x = 92 x +1
(RF9.1)
Performance
• Using a Quiz-Quiz-Trade activity, students can solve a variety of
exponential equations for which both sides can be written as rational
powers of the same base.
Supplementing is necessary for
these topics.
(i)
Prepare a set of cards that at least matches the number of
students in the class. One side of the card contains the question
and the other side contains the answer.
(ii) Give each student a card. Allow students a minute or two to
become familiar with the question on the card. Then they find
a partner to whom they will ask the question.
(iii) If the partner is not able to answer the question, they should
coach them first before providing the answer.
(iv) After both partners are finished asking and answering the
questions, they switch cards and find a new partner.
(RF9.1)
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EXPONENTIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicator:
RF9.2 Continued
• Graphing Technology
Students could also use graphing technology to determine the solution
for an exponential equation. With graphing technology, the solution can
be found using the graph or a table of values.
To determine the solution to the equation 1.7x = 30, for example, the
graph of y = 1.7x can be used to determine the value of x that makes the
value of the function approximately 30.
Students could also generate the table of values associated with the
equation to find the value of x that makes the value of the function
approximately 30.
Alternatively, they can determine the intersection of the graphs of
y = 1.7x and y = 30 to get the approximate solution.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
EXPONENTIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Interview
• Ask students if it is best to estimate the solution to 53 x −1 = 122 x +1
using systematic trial or graphing technology, and to defend their
choice.
(RF9.2)
Authorized Resource
Pre-Calculus 12
7.3 Solving Exponential Equations
SB: pp. 358-365
Journal
• Ask students to explain when they can solve an exponential equation
algebraically and when they must use an estimation method
(graphing or systematic trial).
TR: pp. 187-192
(RF9.1, RF9.2)
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EXPONENTIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicators:
RF9.3 Solve a problem that
involves exponential growth or
decay.
RF9.4 Solve a problem that
involves the application of
exponential equations to loans,
mortgages and investments.
Students should work with problems that can be modeled by
exponential functions or equations, such as modeling the cooling
behaviour of a liquid, radioactive decay, medications, half-lives,
doubling time, bacterial growth/decay, light intensity, and finance. They
should solve problems in situations where:
• an exponential function or equation is given
• the graph of an exponential function is given
• a situation is given and they have to create an exponential model to
find the solution
Students should answer questions such as the following:
RF9.5 Solve a problem by
modeling a situation with an
exponential or a logarithmic
equation.
1. Shelly initially invests $500 and the value of the investment
increases by 4% annually.
•
Create a function to model the situation.
•
How much money is in Shelly’s investment after 30 years?
•
What amount of time will it take for the investment to double?
Students should solve this problem using systematic trial, a table of
values, or graphing technology.
2. The half-life of Radon 222 is 92 hours. From an initial sample of
48 g, how long would it take to decay to 6 g?
Students could use either of the two methods below to solve this
problem algebraically:
A = 48 (12 )
A = 48 (12 )92
n is the number of 92 hour increments
t is the number of hours
6 = 48 (
6 = 48 (12 )
n
1
2
1
8
)
n
= (12 )
n
(12 ) = (12 )
3
n
3=n
∴ It would take 276 hours.
152
t
t
92
1
8
= (12 )92
t
(12 ) = (12 )
3
3=
t
92
t
92
276 = t
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
EXPONENTIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• The half-life of a certain radioactive isotope is 30 hours. Ask students
to algebraically determine the amount of time it takes for a sample of
1792 mg to decay to 56 mg.
(RF9.3)
Authorized Resource
Pre-Calculus 12
7.3 Solving Exponential Equations
• If a new car, purchased for $20 000, depreciates at a rate of 28%
every two years, ask students to answer the following:
(i) What will be the value of this car after 6 years?
(ii) What amount of time will it take for the car to lose half its
value?
(RF9.3, RF9.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
SB: pp. 358-365
TR: pp. 187-192
153
EXPONENTIAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicators:
RF9.3, RF9.4, RF9.5 Continued
Students are also required to model a situation such as the following,
using an exponential function, when given specific parameters.
A cup of hot chocolate is served at an initial temperature of 80˚C and
then allowed to cool in a stadium with an air temperature of 5˚C. The
difference between the hot chocolate temperature and the temperature
of the room will decrease by 30% every 6 minutes. If T represents
the temperature of the hot chocolate in degrees Celsius, measured as
a function of time, t, in minutes, students can answer the following
questions:
• What is the transformed exponential function in the form
t
T = a(c)b(t - h) + k?
[Solution: T = 75 (0.7 )6 + 5 ]
• What is the temperature at time t = 11 minutes?
• How long does it take the hot chocolate to cool to a temperature of
40˚C?
As a possible method, a graphical solution is shown:
Students are required to model a situation with an exponential function
from a given graph, a table of values, or a description when the equation
of the horizontal asymptote is y = 0. When the asymptote is y = 0,
students are expected to determine all parameters from the given
information. When the asymptote is not y = 0, as in the above example,
students are not required to determine the common ratio from a table of
values or graph. They are required to determine the common ratio from
the problem description.
To solve problems involving finance, students must become familiar
with how annual interest rates are applied. Discuss common
compounding periods. Students should understand that compounding
semi-annually, for example, means interest is calculated twice a year.
Solving problems involving logarithmic equations will be addressed in
the next unit.
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EXPONENTIAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Journal
• Ask students to choose between two investment options and justify
their choice: earning 12% interest per year compounded annually or
12% interest per year compounded monthly.
(RF9.4)
Authorized Resource
Pre-Calculus 12
7.3 Solving Exponential Equations
SB: pp. 358-365
TR: pp. 187-192
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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EXPONENTIAL FUNCTIONS
156
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Logarithmic Functions
Suggested Time: 12 Hours
LOGARITHMIC FUNCTIONS
Unit Overview
Focus and Context
In the previous unit, students worked with exponential equations where
both sides could be written as rational powers of the same base. They
will now solve exponential equations where this is not possible.
Students are introduced to logarithms as inverses of exponential
equations. They graph y = log c x, c ≠ 0, c > 1 and explore the effects of
various transformations on this graph.
Using the laws of logarithms, students simplify and evaluate
expressions. They solve logarithmic equations and problems involving
both exponential and logarithmic equations.
Outcomes Framework
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF6
Demonstrate an understanding of
logarithms.
SCO RF7
Demonstrate an understanding of
the product, quotient and power
laws of logarithms.
SCO RF8
Graph and analyze exponential and
logarithmic functions.
SCO RF9
Solve problems that involve
exponential and logarithmic
equations.
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LOGARITHMIC FUNCTIONS
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
Relations and Functions
RF1. Interpret and explain the
relationships among data, graphs and
situations.
RF6. Demonstrate an understanding
of logarithms.
[CN, ME, R]
[C, CN, R, T, V]
AN3. Demonstrate an understanding
of powers with integral and rational
exponents.
RF7. Demonstrate an understanding
of the product, quotient and power
laws of logarithms.
[C, CN, ME, R, T]
[C, CN, PS, R]
RF8. Graph and analyze
exponential and logarithmic
functions.
[C, CN, T, V]
RF9. Solve problems that involve
exponential and logarithmic
equations.
[C, CN, PS, R]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF6 Demonstrate an
understanding of logarithms.
[CN, ME, R]
In previous units in this course, students worked with inverses (RF5),
and exponential functions and their graphs (RF8, RF9). They now build
upon these concepts to study logarithms and their graphs.
Achievement Indicators:
RF6.1 Explain the relationship
between logarithms and
exponents.
RF6.2 Express a logarithmic
equation as an exponential
equation and vice versa.
RF6.3 Determine, without
technology, the exact value of a
logarithm, such as log28.
Students should be introduced to logarithms as a different form of
an exponential statement. The statement 32 = 9, for example, can be
written as log39 = 2. The base of the exponent is the same as the base of
the logarithm. Students should understand that a logarithm logc x = y is
asking “What exponent, y, is needed so that c y = x?”. Given a statement
in exponential form, they should be able to write it in logarithmic form,
and vice versa. Introduce students to the common logarithm and note
that, in this case, the base is usually not written: log10x = log x.
When the value of a logarithm is a rational number, such as log644 and
log 1 32 , students should be able to determine the exact value without
2
technology. As well, students are required to determine an unknown
value in logarithmic equations. The following examples can be solved by
rewriting in exponential form:
•
log3x = -2
•
81
log x 16
=2
Students should also think about the restrictions on c for the equation
logc x = y. Ask them to rewrite the following equations in exponential
form to determine the value of x:
• log14 = x
• log-28 = x
From this they should see that when c = 1, the resulting exponential
equation cannot be solved. The only exception is an equation with both
c = 1 and x = 1. When c < 0, the resulting exponential equation may
be unsolvable. Therefore, when working with logarithms, the base is
restricted to positive values other than 1; that is, for y = logc x, c > 0,
c ≠ 1.
RF6.4 Estimate the value of a
logarithm, using benchmarks, and
explain the reasoning; e.g., since
log28 = 3 and log216 = 4, log29 is
approximately equal to 3.1.
160
81 = 2 , students often forget that
When solving an equation such as log x 16
81
the base must be positive. Even though the quadratic equation x 2 = 16
9
has two solutions ( x = ± 4 ), the only solution for the logarithmic
equation is 94 .
Students should also use benchmarks to estimate the value of a
logarithm. To estimate log5106, for example, they should notice that
52 = 25 and 53 = 125. Therefore, the answer must be between 2 and 3,
and is closer to 3. Through systematic trial, students should be able to
determine the value accurately to a minimum of one decimal place.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to evaluate the following without using technology:
Pre-Calculus 12
(i) log 1 25
8.1 Understanding Logarithms
5
(ii) log432
Student Book (SB): pp. 372-382
(RF6.2,RF6.3)
• Ask students to determine the value of the following to one decimal
place:
Teacher’s Resource (TR): pp. 200204
(i) log324
(ii) log6200
(iii) log 1 12
2
(RF6.4)
Interview
• Ask students to explain whether or not any positive real number can
be the base of a logarithm.
(RF6.1)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF8 Graph and analyze
exponential and logarithmic
functions.
This outcome was addressed in the previous unit in relation to
exponential functions of the form y = cx, where c > 0, c ≠ 1. In this unit,
students are introduced to the graph of y = log c x, where c > 0, c ≠ 1.
[C, CN, T, V]
Achievement Indicator:
RF8.4 Demonstrate, graphically,
that a logarithmic function and
an exponential function with the
same base are inverses of each
other.
It is important that students understand the inverse relationship
between exponential and logarithmic functions. They can use their
previous knowledge of exponential functions and inverses to explore this
relation. In the Function Transformations unit, students determined
inverse equations by interchanging the x and y variables and solving for
y (RF5). They can apply this procedure to determine the inverse of an
exponential function, such as f (x) = 2x. Writing this function as y = 2x
and switching x and y leads to solving x = 2y for y. This requires writing
the equation in logarithmic form and results in y = log2x. The inverse
function is f -1(x) = log2(x).
Students can use the table of values for f (x) = 2x to generate the table for
f -1(x) = log2x by interchanging the domain and range:
x
−3
−2
−1
0
1
2
3
f (x ) = 2 x
x
1
8
1
4
1
2
1
8
1
4
1
2
1
2
4
8
1
2
4
8
f
−1
(x ) = log 2 x
−3
−2
−1
0
1
2
3
From this, the graphs can be created.
Students should see that the graphs are reflections of each other in the
line y = x. Ask them to identify the relationship between the domain and
range for both functions. This method of graphing can be applied to any
logarithmic function of the form y = logc x.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• The point (361 , −2 ) is on the graph of y = logc x. If the point
(k, 216) is on the graph of the inverse, ask students to determine the
values of c and k.
(RF6.3, RF8.4)
• Given the graph and equation of an exponential function of the form
y = cx, ask students to determine the equation and the domain and
range of the inverse relation.
Authorized Resource
Pre-Calculus 12
8.1 Understanding Logarithms
SB: pp. 372-382
TR: pp. 200-204
(RF8.4)
Journal
• Ask students to explain, using the graph of y = logc x, why they
cannot evaluate logc(-3)and logc(0).
(RF8.4)
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF8 Continued ...
Students have been exposed to graphing using transformations for
general functions (RF1, RF2, RF3, RF4) and for exponential functions
(RF8). These transformations will now be applied to logarithmic
functions. Logarithms will be restricted to bases that are greater than 1.
Achievement Indicators:
RF8.5 Sketch with or without
technology, the graph of a
logarithmic function of the form
y = logc x, c > 1.
RF8.6 Identify the charactersitics
of the graph of a logarithmic
function of the form y = logcx,
c> 1, including the domain,
range, vertical asymptote and
intercepts, and explain the
significance of the vertical
asymptote.
Students can now apply their previous work with transformations to
logarithmic functions. Ask them to graph a logarithmic function, such
as y = log3 x, and identify the domain, range, intercepts, and vertical
asymptote. Students should notice that the domain is restricted by the
vertical asymptote. Next, ask them to graph a transformation of this
logarithmic function, such as y = 2log3(3(x −1))− 6. To graph this,
students could identify the transformations:
• The vertical stretch is 2
• The horizontal stretch is 13
• The horizontal translation is 1
• The vertical translation is −6
The graph can then be created by applying the stretches first, followed
by the translations. Another approach is to create the mapping rule
(x, y) → ( 13 x + 1, 2y − 6) and then produce a table of values for the
transformed function.
RF8.7 Sketch the graph of a
logarithmic function by applying
a set of transformations to the
graph of y = logc x, c > 1, and
state the characteristics of the
graph.
Students should recognize that the horizontal translation determines the
vertical asymptote, which also defines the domain.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to graph each of the following using transformations:
(i)
(ii)
y = 2 log 2 (− x + 1) − 6
y=
− 12 log 3
Authorized Resource
(2 (x + 3))+ 1
Pre-Calculus 12
(RF8.7)
• Ask students to identify the intercepts, vertical asymptote, and
domain of y = 2log6(3(x + 4)) - 4.
(RF8.7)
8.2 Transformations of
Logarithmic Functions
SB: pp. 383-391
TR: pp. 205-208
• Ask students to apply the mapping rule (x , y ) → (− 12 x + 1,3 y − 12 )
to y = log4x. They should write the resulting function and identify
the domain, range, intercepts and vertical asymptote.
(RF8.6)
Interview
• Ask students to explain why all functions of the form y = log c x
intersect at (1, 0).
(RF8.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF8 Continued ...
Achievement Indicators:
RF8.6, RF8.7 Continued
To determine the x-intercept of the transformed logarithmic function,
students should solve the equation for y = 0. They are familiar with this
from work with other types of functions. They can determine the xintercept of y = 2log3(3(x - 1)) - 6, for example, as follows:
0 = 2log3(3(x - 1)) - 6
6 = 2log3(3x - 3)
3 = log3(3x - 3)
27 = 3x - 3
x = 10
Similarly, when determining the y-intercept, students let x = 0. At this
point, there may be cases where it is necessary to use benchmarks to
approximate the intercept. Later, they will be able to determine the
y-intercepts more accurately and quickly with the use of a calculator.
Students should also work with logarithmic functions that do not have
a y-intercept. Evaluating y = 2log3(3(x - 1)) - 6 for x = 0, for example,
results in y = 2log3(-3) - 6. Since the domain of y = log3 x is x ∈ (0, ∞),
this cannot be evaluated.
Students may also have to factor an expression in order to determine the
horizontal stretch. Before y = 3log4(−5x − 5) − 4 can be graphed, for
example, it should be rewritten as y = 3log4(−5(x + 1)) − 4.
Students should graph logarithmic equations with a reflection in the
y-axis to see the effect on the domain. They should recognize that for
y = 3log4(−5(x − 6)) + 1, the vertical asymptote is x = 6, but the domain
is x ∈ (−∞, 6) due to the reflection in the y-axis.
Students are not responsible for determining the equation of a
logarithmic function given the graph. However, given the graph of a
logarithmic function, they should identify the function from a list of
options.
Students should sketch the graph of a given logarithmic function, clearly
showing the asymptote and intercepts, and identify the graph of a given
logarithmic function from a list of options.
166
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Observation
• Students can work in pairs to complete puzzles containing the
characteristics and graphs of various logarithmic functions. Refer to
the observation strategy on page 147.
(RF8.6, RF8.7)
Authorized Resource
Pre-Calculus 12
8.2 Transformations of
Logarithmic Functions
SB: pp. 383-391
TR: pp. 205-208
Note
Example 3 on p. 387 and #6
on p. 390 of the SB are not an
expectation in this course.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
167
LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF7 Demonstrate an
understanding of the product,
quotient and power laws of
logarithms.
[C, CN, ME, R, T]
Achievement Indicators:
RF7.1 Develop and generalize the
laws of logarithms, using numeric
examples and exponent laws.
RF7.2 Derive each law of
logarithms.
In Mathematics 1201, students worked with the laws of exponents,
including integral and rational exponents. They also applied the
exponent laws to expressions with rational and variable bases, as well as
integral and rational exponents (AN3). The laws of logarithms will now
be developed using both numerical examples and the exponent laws.
Students will work with the following laws of logarithms, with the
conditions c > 0, m > 0, n > 0 and c ≠ 1 where c, m, n ∈ R:
•
Product Law:
logcMN = logcM + logcN
•
Quotient Law:
log c
•
Power Law:
logcMP = PlogcM
M
N
= log c M − log c N
The focus here is to use several numerical examples to allow students to
develop the laws of logarithms.
To develop the quotient law, for example, a procedure similar to the
following could be used:
• Begin with the exponential equations: 16 = 42 and 64 = 43
• Convert each equation to logarithmic form:
log416 = 2 and log464 = 3
= 43 − 2
• Divide the equations from Step 1:
64
16
• Rewrite using logarithmic form:
64 = 3 − 2
log 4 16
• Substitute log416 in for 2 and log464 in for 3:
64 = log 64 − log 16
log 4 16
4
4
Students could evaluate each side of the equation to verify that the left
hand side of the equation is indeed the same as the right hand side.
They should then derive the general case for the quotient law. A similar
exercise could be used to develop the product law.
An example such as the following could be used for the power law:
• Start with the exponential equation: 16 = 42 (i.e., log416 = 2)
• Raise both sides of the equation to the power p: 16p = (42)p
• Simplify the equation: 16p = (4)2p
• Convert the equation to logarithmic form: log416p = 2p
• Substitute log416 in for 2: log416p = (log416)(p)
• Rewrite: log416p = plog416
From this, students should derive the general case for the power law.
Students should be exposed to the proofs of the laws of logarithms, but
not required to reproduce these proofs.
168
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to verify using the logarithm laws:
(i) log327 = log39 + log33
(ii) log525 = log5125 - log55
(iii) log264 = 6 log22
Authorized Resource
(RF7.1)
Pre-Calculus 12
8.3 Laws of Logarithms
SB: pp. 392-403
TR: pp. 209-213
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF7 Continued ...
Achievement Indicators:
RF7.3 Determine, using the
laws of logarithms, an equivalent
expression for a logarithmic
expression.
Students are expected to simplify logarithmic expressions involving
both numerical and variable arguments. For expressions with variable
arguments, they should determine the restrictions on the variable(s).
The arguments should be restricted to polynomials of degree 2 or less.
They were introduced to factoring techniques in Mathematics 1201
(AN5). They should now be exposed to problems where factoring is
necessary in order to simplify the expression. Students should also be
exposed to questions where they are required to write a logarithmic
expression as a single logarithm and simplify if necessary:
•
log64 − (log672 +
•
logb2x + 3(logbx − logby)
1
4
log616)
Conversely, they should write a single logarithm as the sum and
difference of multiple logarithms. Applying the laws of logarithms
to expand or condense logarithmic expressions is useful in solving
logarithmic equations.
Students should note that there is no general property of logarithms that
can be used to simplify logc(x + y). They sometimes mistakenly think
that this expression is equal to logcx + logcy. To verify that this is not
true, students could evaluate logarithmic expressions such as:
RF7.4 Determine, with
technology, the approximate value
of a logarithmic expression, such
as log29.
•
log10(2 + 3) = log105 ≈ 0.699
•
log102 + log103 ≈ 0.301 + 0.477 = 0.778
Students were introduced to evaluating logarithmic expressions without
the use of technology by using benchmarks (RF6). They will now extend
this to approximating the solution, using technology. To determine the
approximate value of log29, for example, the equation log29 = x can be
rewritten in exponential form: 2x = 9
log 2x = log 9
x log 2 = log 9
x=
log9
log2
x ≈ 3.17
As students work through this example, they should be able to connect
the solution of the problem to the original equation.
log29 = x → x =
log9
log2
log b
This leads to the property log a b = log a . The logarithmic expression can
be evaluated using a calculator, as the base has been changed to 10.
170
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to answer the following:
(i) A student was asked to simplify log216 - log232 + 2 log24 and
provided the solution:
log 2 16 − log 2 32 + log 2 8
log 2
log 2
Authorized Resource
Pre-Calculus 12
16 + log 8
2
32
1 + log 8
2
2
8.3 Laws of Logarithms
SB: pp. 392-403
log 2 4
2
TR: pp. 209-213
State where the error first occurred and write the correct solution
to the problem.
(RF7.3)
(ii) If P = log38 and Q = log36, write log 3 8 6 in terms of P and Q.
(RF7.3)
• Ask students to evaluate using the laws of logarithms:
(i)
3log6(2) + log6(27)
(ii) log5(2.5) + 2log5(10) – log5(2)
(RF7.3)
Note
log b
The property log a b = log a is
not developed in the SB, but is
referenced in question #19 on p.
402.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Solve problems that involve
exponential and logarithmic
equations.
[C, CN, PS, R]
In the previous unit, students solved exponential equations where the
powers could both be expressed as rational powers of the same base.
In cases where the bases were not the same graphing technology and
systematic trial were used to estimate the value of the variable (RF9).
Students will now use logarithms to solve these equations.
Achievement Indicators:
RF9.6 Determine the solution
of a logarithmic equation, and
verify the solution.
Ask students to solve logarithmic equations that require them to find
solutions for both linear and quadratic equations:
•
log5(x + 1) + log5(x - 2) = log54
•
log2(4x - 1) - log2(2x + 1) = 3
RF9.7 Explain why a value
obtained in solving a logarithmic
equation may be extraneous.
When solving equations, non-permissible values of the variable must be
considered. Solving log2(x + 6) + log2(x + 4) = log28, for example, results
in the roots x = −2 and x = −8. Students should note that x = −2 is
permissible, while x = −8 is extraneous.
RF9.2 Determine the solution of
an exponential equation in which
the bases are not rational powers
of one another, using a variety of
strategies.
Students should now use logarithms to solve exponential equations.
They should be able to give answers as both approximate and exact
values. When solving 3x + 1 = 54x - 3 students can state the answer as
−3log5− log3
x = log3− 4log5 or x ≈ 1.11.
A common error that students may make is not using the distributive
property correctly when multiplying the logarithm and the variable
expression. Encourage them to put brackets around the exponent
portion of the equation when moving it to the front of the logarithm.
They will also have to verify that the solution is not extraneous. Students
should be encouraged to use graphing technology or systematic trial to
check the reasonableness of their solution. Verification can also be done
by substituting the solution back into the original equation.
A common student error occurs when students solve questions similar to
7(23x) = 21. Discuss with students why 7(23x) cannot be written as 143x.
Students should divide both sides of the equation by 7 in order to isolate
the power (23x = 3).
172
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Journal
• Ask students to create a logarithmic equation where the only solution
is x = 4.
(RF9.6)
• Students should explain why the equation
log4(1 − x) + log4(x − 3) = 1 has no roots.
Authorized Resource
Pre-Calculus 12
(RF9.6, RF9.7)
8.4 Logarithmic and Exponential
Equations
SB: pp. 404-415
Interview
TR: pp. 214-218
• Discuss with the student why the solution to
log4(x + 2) + log4(x - 4) = 2 must be in the domain {x > 4, x ∈ R}.
(RF9.7)
Paper and Pencil
• Ask students to identify the error in the given solution and explain
why it is incorrect. They should then write the correct solution.
10 x + 5 = 60
log (10 x + 5 )= log 60
log10 x + log5 = log 60
x log10 = log 60 − log5
x=
log60− log5
log10
(RF9.2)
Performance
Web Link
• Students could work in small groups to put together a jigsaw puzzle
where the expressions on the adjacent sides of the puzzle pieces have
to be equivalent. This activity provides students with the opportunity
to practice work with logarithms, make mathematical arguments
about whether or not pieces fit together, and check and revise their
work.
www.mmlsoft.com/index.
php?option=com_content&task=vie
w&id=11&Itemid=12
Tarsia is a software program for
creating jigsaw puzzles.
(RF7.3, RF7.4, RF9.2, RF9.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicators:
RF9.3 Solve a problem that
involves exponential growth or
decay.
In the previous unit, students solved problems involving exponential
growth and decay, as well as financial applications (RF9). These types
of problems should now be revisited, as the majority of them involve
exponential equations that are solved using logarithms.
Students should solve problems where:
RF9.5 Solve a problem by
modeling a situation with an
exponential or a logarithmic
equation.
RF9.8 Solve a problem that
involves logarithmic scales, such as
the Richter scale and the pH scale.
• the logarithmic or exponential equation is given
• the graph of an exponential function is given
• a situation is given that can be modeled by an exponential or
logarithmic equation.
Students will solve problems involving logarithmic scales such as the
Richter scale (used to measure the magnitude of an earthquake), the pH
scale (used to measure the acidity of a solution), and decibel scale (used
to measure sound level). When dealing with Richter scale, pH scale or
decibel scale problems, students are not expected to develop formulas
but should be given the formula when it is required.
Students may be familiar with pH scales from work in science courses.
The pH scale of a solution is determined using the equation y = −log x,
where x is the concentration of hydrogen ions in moles per litre (mol/L).
The scale ranges from 0 to 14 with the lower numbers being acidic and
the higher numbers being basic. A value of pH = 7 is considered neutral.
The scale is a logarithmic scale with one unit of increase in pH resulting
in a 10 fold decrease in acidity. Another way to look at this would be
that a one unit increase in pH results in a 10 fold increase in basicity.
The magnitude of an earthquake, y, can be determined using
y = log x, where x is the amplitude of the vibrations measured using a
seismograph. An increase in one unit in magnitude results in a 10 fold
increase in the amplitude.
Sound levels are measured in decibels using β = 10(log I + 12), where β
is the sound level in decibels (dB) and I is the sound intensity measured
in watts per metre squared (w/m2). This would be a good opportunity
for students to measure audio volumes in the environment around
them. They could use a smartphone application, for example, to show
the approximate decibel level of their location. Although quite accurate,
the application is mainly a tool for detecting noise level in casual
settings.
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to answer the following:
(i) The magnitude of an earthquake, M, as measured on the
Richter scale is given by M = log I, where I is the intensity of
the earthquake (measured in micrometres from the maximum
amplitude of the wave produced on a seismograph). A town
experiences an earthquake with a magnitude of 4.2 on the Richter
scale. Four years later, the same town experiences an earthquake
that is 5 times as intense as the first earthquake. What is the
magnitude of the second earthquake?
(RF9.8)
Authorized Resource
Pre-Calculus 12
8.4 Logarithmic and Exponential
Equations
SB: pp. 404-415
TR: pp. 214-218
(ii) After taking a cough suppressant, the amount, A, in mg,
remaining in the body is given by A = 10(0.85)t, where t is given
in hours.
(a) What was the initial amount taken?
(b) What percent of the drug leaves the body each hour?
(c) How much of the drug is left in the body 6 hours after the
dose is administered?
(d) How long is it until only 1 mg of the drug remains in the
body?
(RF9.3)
• Ask students to answer the following questions using the table below:
Location and Date
Chernobyl - 1987
Haiti - January 12, 2012
Northern Italy - May 20, 2012
Magnitude
4
7
6
(i)
How many times as intense was the earthquake in Haiti
compared to the one in Chernobyl?
(ii) How many times as intense was the earthquake in Haiti
compared to the one in Northern Italy?
(iii) How many times as intense was the earthquake in Northern
Italy compared to the one in Chernobyl?
(iv) If a recent earthquake was half as intense as the one in Haiti,
what would be the approximate magnitude?
(RF9.8)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicators:
RF9.4 Solve a problem that
involves the application of
exponential equations to loans,
mortgages and investments.
RF9.5 Continued
176
Questions involving finance should also be revisited. In the previous
unit, students were exposed to situations where the bases could be
written the same. They will now work with questions where logarithmic
equations are used. A reminder of the different compounding periods
may be necessary.
The formula A = A0(1 + r)n can be used for finance calculations, where
A0 is the initial value, r is the interest rate per compounding period
and n is the number of compounding periods. For example, if a $1000
deposit is made at a bank that pays 12% interest compounded monthly,
students should be able to determine, using logarithms, how long it will
take for the investment to reach $2000. Students should also be exposed
to situations where it is necessary to determine the initial value, the
interest rate, or the number of compounding periods.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
LOGARITHMIC FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study of
relations.
Suggested Assessment Strategies
Resources/Notes
Journal
• Ask students to create a flowchart of the steps needed to solve
exponential equations and a flowchart of the steps needed to solve
logarithmic equations. Then have them compare the flowcharts for
similarities. Ask them to consider if one flowchart can be designed
that handles both types of equations.
(RF9)
Authorized Resource
Pre-Calculus 12
8.4 Logarithmic and Exponential
Equations
SB: pp. 404-415
Performance
TR: pp. 214-218
• Students can work in groups of two for the activity Pass the Problem.
Each pair gets a problem that involves a situation to be modelled
with an exponential or a logarithmic equation. Ask one student to
write the first line of the solution and then pass it to the second
student. The second student verifies the workings and checks for
errors. If there is an error, students should discuss what the error is
and why it occurred. The student then writes the second line of the
solution and passes it to their partner. This process continues until
the solution is complete.
(RF9.3, RF9.4, RF9.5, RF9.8)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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LOGARITHMIC FUNCTIONS
178
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Permutations, Combinations and
The Binomial Theorem
Suggested Time: 14 Hours
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Unit Overview
Focus and Context
In this unit, students are introduced to the fundamental counting
principle and the concept of counting with permutations, where the
arrangement of elements in the event is important. They are also
introduced to the role and significance of n!. The ability to rewrite
a factorial in several different ways can be helpful in simplifying
expressions with binomial coefficients.
Students also see that the process of counting events where the
arrangement of elements is not important requires a different approach.
For this, they are introduced to counting with combinations.
Finally, they are introduced to the binomial theorem and how it is used
to expand a binomial that is raised to a power. They see that Pascal’s
triangle is a convenient pattern for writing binomial coefficients.
Outcomes Framework
GCO
Develop algebraic and numeric reasoning
that involves combinatorics.
SCO PCBT1
Apply the fundamental counting
principle to solve problems.
SCO PCBT2
Determine the number of
permutations of n elements taken r at
a time to solve problems.
SCO PCBT3
Determine the number of
combinations of n different elements
taken r at a time to solve problems.
SCO PCBT4
Expand powers of a binomial in
a variety of ways, including using
the binomial theorem (restricted to
exponents that are natural numbers).
180
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Mathematical
Processes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Mathematics 3200
Permutations, Combinations and Binomial Theorem
AN4. Demonstrate an understanding
of the multiplication of polynomial
expressions (limited to monomials,
binomials and trinomials), concretely,
pictorially and symbolically.
AN5. Perform operations on rational
PCBT1. Apply the fundamental
expressions (limited to numerators
counting principle to solve problems.
and denominators that are monomials,
[C, PS, R, V]
binomials or trinomials).
[CN, ME, R]
[CN, R, V]
PCBT2. Determine the number of
permutations of n elements taken r at
a time to solve problems.
[C, PS, R, V]
PCBT3. Determine the number of
combinations of n different elements
taken r at a time to solve problems.
[C, PS, R, V]
PCBT4. Expand powers of a
binomial in a variety of ways,
including using the binomial theorem
(restricted to exponents that are
natural numbers).
[CN, R, V]
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
181
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT1 Apply the fundamental
counting principle to solve
problems.
[C, PS, R, V]
Achievement Indicator:
PCBT1.1 Count the total
number of possible choices that
can be made, using graphic
organizers such as lists and tree
diagrams.
The fundamental counting principle is a means of finding the number
of ways of performing two or more operations together. It will be
developed by solving problems through the use of graphic organizers
such as lists and tree diagrams. Later, the fundamental counting
principle will be applied in work with permutations and combinations.
Tree diagrams were used in Grade 7 (7SP5 and 7SP6) and Grade 8
(8SP2) to determine the number of possible outcomes in probability
problems. Students now apply tree diagrams and other graphic
organizers to counting problems. The following example could be used
to activate students’ prior knowledge.:
The school cafeteria advertises that it can serve up to 24 different meals
consisting of one item from each of three categories:
Fruit:
Apples, Bananas, or Cantaloupe
Sandwiches:
Roast Beef or Turkey
Beverages:
Lemonade, Milk, Orange Juice or Pineapple Juice
Meals with
Cantaloupe
Meals with Bananas
Meals with Apples
Is their advertising accurate?
182
Fruits
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
C
C
C
C
C
C
C
C
Sandwiches
R
R
R
R
T
T
T
T
R
R
R
R
T
T
T
T
R
R
R
R
T
T
T
T
Beverages
L
M
O
P
L
M
O
P
L
M
O
P
L
M
O
P
L
M
O
P
L
M
O
P
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Jill is trying to select a new cell phone based on the following:
Pre-Calculus 12
Brands: Ace, Best, Cutest
11.1 Permutations
Colour: Lime, Magenta, Navy, Orange
Student Book (SB): pp. 516-527
Plans: Text, Unlimited Calling
Ask students to construct a tree diagram, a table, and an organized
list to determine the number of ways of selecting Jill’s new cell
phone.
Teacher’s Resource (TR): pp. 280286
(PCBT1.1)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
183
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT1 Continued ...
Achievement Indicator:
PCBT1.1 Continued
Students should be reminded that the possible meals indicated by the
table and tree diagram could also be written as an organized list.
Meals with Apples
ARL
Meals with Bananas
BRL
Meals with Cantaloupe
CRL
ARM
BRM
CRM
ARO
BRO
CRO
ARP
BRP
CRP
.
.
.
.
.
.
.
.
.
Modifying this example to include a fourth category or adding more
options in one of the categories can be used to illustrate limitations
on the practicality of using graphic organizers for counting problems
and offers a good introduction to the fundamental counting principle.
Sometimes the task of listing and counting all the outcomes in a given
situation is unrealistic, since the sample space may be very large. The
fundamental counting principle enables students to find the number of
outcomes without listing and counting each one.
184
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Pre-Calculus 12
11.1 Permutations
SB: pp. 516-527
TR: pp. 280-286
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
185
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT1 Continued ...
Achievement Indicators:
PCBT1.2 Explain, using
examples, why the total number
of possible choices is found
by multiplying rather than
adding the number of ways the
individual choices can be made.
PCBT1.3 Solve a simple
counting problem by applying the
fundamental counting principle.
In the previous example, there are three fruit options. For each of these,
there are two sandwich choices, and for each sandwich choice there are
four beverage choices. Multiplying the number of options from each
category gives possible meal choices, which agrees with the previous
result. This illustrates the fundamental counting principle, which states
that if there are a ways to perform a task, b ways to perform a second
task, c ways to perform a third task, etc., then the number of ways of
performing all the tasks together is a × b × c × …
Point out to students that when choosing a fruit and a sandwich and a
beverage, the word “and” indicates these three selections (operations)
are performed together, so the number of ways of doing each individual
selection are multiplied. If, instead, a fruit or a sandwich or a beverage is
being selected, then the possibilities are:
3 fruit choices + 2 sandwich choices + 4 beverage choices
= 9 possibilities.
In effect, the problem would become the same as if there were only 9
items on the menu and only one of those items was being selected. Ask
students how many different selections would be possible in this case.
The intent of this discussion is instructional in nature and is meant to
help students understand why, when using the fundamental counting
principle, the individual choices are multiplied rather than added. It is
not intended that students be explicitly evaluated on the use of “and”
versus “or”.
186
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Journal
• Ask students to respond to the following:
Your school cafeteria offers three salads, four main courses, two
vegetables, and three desserts.
With the aid of a tree diagram, a table, and/or an organized list,
explain why it makes sense to multiply the options from each
category to determine the total number of possible meals if a salad, a
main course, a vegetable and a dessert is included in the meal.
(PCBT1.2)
Authorized Resource
Pre-Calculus 12
11.1 Permutations
SB: pp. 516-527
TR: pp. 280-286
Paper and Pencil
• Ask students to answer the following:
(i)
Sheldon has a red, a green and a blue shirt. He also has a pair
of brown pants, a pair of beige pants, and a pair of black pants.
His sock drawer contains one pair of black socks and one pair
of grey socks. His shoe rack has one pair of moccasins,
one pair of loafers, a pair of rubber boots, and a pair of
sneakers. Determine the number of ways Sheldon can select an
outfit consisting of one item from each category.
(ii) In Newfoundland and Labrador, a license plate consists of a
letter-letter-letter-digit-digit-digit arrangement such as
CXT 132.
(a) How many license plates are possible?
(b) How many license plates are possible if no letter or
digit can be repeated?
(c) How many license plates are possible if vowels (a, e, i,
o, u) are not allowed?
(iii) Canadian postal codes consist of a letter-digit-letter-digit-letterdigit arrangement.
(a) How many codes are possible, and how does this
compare with the number of license plates in Newfoundland
and Labrador?
(b) In Newfoundland and Labrador, all postal codes begin with
the letter A. How many postal codes are possible?
(PCBT1.3)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
187
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT2 Determine the number
of permutations of n elements
taken r at a time to solve
problems.
The concept of a permutation will be explored through the use of
graphic organizers. A formula will be developed and applied in problem
solving situations, including those that involve permutations with
constraints.
[C, PS, R, V]
Achievement Indicators:
PCBT2.1 Count, using graphic
organizers such as lists and tree
diagrams, the number of ways of
arranging the elements of a set in
a row.
PCBT2.2 Determine, in
factorial notation, the number
of permutations of n different
elements taken n at a time to solve
a problem.
A permutation is an ordered arrangement of all or part of a set. For
example, the possible permutations of the letters A, B and C are ABC,
ACB, BAC, BCA, CAB and CBA. The order of the letters matters.
Students should first be introduced to permutations of n different
elements taken n at a time and will then move to permutations of n
different elements taken r at a time.
Students should learn to recognize and use n! to represent the number
of ways to arrange n distinct objects. For example, how many ways are
there to arrange or permute a group of five people in a line?
1st Person
2nd Person
3rd Person
4th Person
5th Person
5 options
4 options
remaining
3 options
remaining
2 options
remaining
1 option
remaining
By the fundamental counting principle, there are 5 × 4 × 3 × 2 × 1 or
120 ways. This product can be written in compact form as 5! Generally,
n ! = n (n − 1)(n − 2 )(n − 3)... (2 )(1) where n ∈ N .
Note that 0! = 1 will be addressed in the context of permutations after
the formula for nPr is introduced.
In preparation for working with formulas for permutations and
combinations, students should simplify factorial expressions such as:
100! 100 × 99 × 98 × 97!
=
= 100 × 99 × 98
97!
97!
n (n − 1)(n − 2 )!
n!
•
=
= n (n − 1) = n 2 − n
(n − 2 )!
(n − 2 )!
•
•
3!(n + 1)!
2!(n − 2 )!
=
3 × 2!(n + 1)n (n − 1)(n − 2 )!
2!(n − 2 )!
= 3 (n + 1)n (n − 1)
= 3n3 − 3n
188
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
• The code for a lock consists of three numbers selected from 0, 1, 2,
3 with no repeats. For example, the code 1-2-1 would not be allowed
but 3-0-2 would be allowed. Ask students to use a tree diagram or
other graphic organizer to determine the number of possible codes
and explain why this is a permutation problem.
Pre-Calculus 12
11.1 Permutations
SB: pp. 516-527
TR: pp. 280-286
(PCBT2.1)
Journal
• Ask students to explain why a typical so-called “combination lock”
used on school lockers should more properly be called a permutation
lock.
(PCBT2.1)
Paper and Pencil
• Ask students to answer the following:
(i)
In how many different ways can a set of 5 distinct books be
arranged on a shelf?
(ii) In how many different orders can 15 different people stand in a
line?
(iii) Simplify:
1000!
(a)
(b)
998!
1000!
998!1000
(c)
4!(n + 1)!
3!(n − 1)!
(d)
(n + 2 )!
(n + 4 )!
(iv) Determine the number of ways of selecting a president, a
secretary, and a treasurer from a group of 10 people if no person
can hold more than one position.
(PCBT2.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
189
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT2 Continued ...
Achievement Indicators:
PCBT2.3 Determine, using a
variety of strategies, the number
of permutations of n different
elements taken r at a time to solve
a problem.
Solving a simple counting problem can help students develop a formula
for determining the number of permutations of n different elements
taken r at a time, which will be more efficient when working with larger
values of r. Ask students to determine how many ways there are to
arrange any three of a group of five people in a line. They should reason
that there are 5 options for the first position, 4 options remaining for
the second position, and 3 options remaining for the third position.
By the fundamental counting principle, there are 5 × 4 × 3 = 60
permutations. The symbol commonly used to represent this is 5P3 or
P for the number of “n” objects taken “r” at a time. Students should
n r
3×2×1 . This can be rewritten as
notice that 5P3 = 5 × 4 × 3 or 5 P3 = 5×4×2×
1
5!
5!
.
5 P3 = 2! or 5 P3 = ⎛
⎞
⎜
⎝
5−3 ⎟!
⎠
Generally, the number of permutations for n objects taken r at a time is
given by:
P =
n (n − 1)(n − 2 )... (n − r )(n − r − 1)(n − r − 2 )... (3 )(2 )(1)
(n − r )(n − r − 1)(n − r − 2 )... (3 )(2 )(1)
P =
n!
(n − r )!
n r
n r
PCBT2.4 Explain why n must be
greater than or equal to r in the
notation nPr .
Continue to emphasize that, in the notation nPr, n is the number of
elements in a set and r is the number of elements to be selected at any
given time. Students should realize that r cannot be bigger than n since
it is not possible to select more elements than are actually in the set.
When using the formula n Pr = n−n!r ! , if r were greater than n, then the
( )
denominator would contain a factorial of a negative number, which is
undefined.
Students should note that the number of permutations of six people
being arranged in a line is 6! This is also a permutation of a set of 6
objects from a set of 6. Therefore, applying the formula n Pr = n −n!r ! ,
( )
the result is 6 P6 = 6! = 6!
. This means 6! = 6!
, and the only value
0!
0!
6
−
6
!
( )
of 0! that makes sense is 0! = 1.
Evaluating 0! means determining the number of ways there are to count
an empty set. Since there is nothing to count, ask students In how many
ways can one count nothing?. A mathematical answer to this is one.
For n objects taken n at a time, the number of permutations is
n ! = n Pn = n−n!n ! = n0!! . Again, 1 is the only realistic value of 0!.
( )
190
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to answer the following:
(i)
How many two-digit numbers can be formed using
the digits 1, 2, 3, 4, 5, 6 if repetition is allowed?
(PCBT2.3)
(ii) How many distinct arrangements of three letters can be formed
using the letters of the word LOCKERS?
(PCBT2.3)
(iii) The code for a lock consists of four numbers selected from 0,
1, 2, 3 with no repeats. For example, the code 1-2-1-3
would not be allowed but 3-0-2-1 would be allowed. Using the
permutation formula, determine the number of possible codes.
(PCBT2.4)
Authorized Resource
Pre-Calculus 12
11.1 Permutations
SB: pp. 516-527
TR: pp. 280-286
Journal
• Ask students to respond to the following:
(i)
A code consists of three letters chosen from A to Z and three
digits chosen from 0 to 9, with no repetitions of letters or
numbers. Students should explain why the total number of
possible codes can be found using the expression 26P3 × 10P3 .
(PCBT2.3)
(ii) Explain why an error is obtained when trying to calculate 5P7
on a calculator.
(PCBT2.4)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
191
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT2 Continued ...
Achievement Indicator:
PCBT2.5 Given a value of k,
k ε N, solve nPr = k for either n
or r.
Knowledge of permutations can be applied to solve equations of the
form nPr = k. Students should solve equations such as the following,
many of which will involve simplification of rational expressions similar
to work done in Mathematics 2200 (AN5). Once the expression is
simplified such that it no longer contains factorials, they are revisiting
this previous work with rational expressions.
Example 1
P = 30
n 2
n!
(n −2)!
= 30
n (n − 1) = 30
• Students can reason that consecutive integers with a product of 30
are needed. Since 6(6 − 1) = 30, n = 6.
• Alternatively, they can solve the quadratic equation
n2 − n − 30 = 0. This equation has roots 6 and −5. Since n must be
positive, −5 is extraneous.
Example 2
P = 12
n-1 2
(n −1)!
(n −1− 2 ) !
= 12
(n −1)(n −2)(n −3)!
= 12
(n −3)!
(n − 1)(n − 2 ) = 12
• Students could reason that consecutive integers with a product
of 12 will satisfy this equation. Therefore, n = 5 because
(5 − 1)(5 − 2) = 12.
• They could also solve the quadratic equation n2 − 3n + 2 = 12 to
determine that n = 5 (−2 is an extraneous root).
Example 3
P = 5!
5 r
5!
(5 − r ) !
= 5!
(5 − r )! = 1
To solve this equation, students must recall that 1! = 1 and 0! = 1.
Therefore, 5 – r = 1 or 5 − r = 0, resulting in r = 4, r = 5.
192
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to solve the following:
Authorized Resource
P = 42
(i)
n 2
(ii)
8 r
(iii)
n+1 2
(iv)
n+4 3
Pre-Calculus 12
P = 8!
11.1 Permutations
P = 20
SB: pp. 516-527
TR: pp. 280-286
P = 120
• Ask students to answer the following:
Denise has a set of posters to arrange on her bedroom wall. She can
only fit two posters side by side. If there are 72 ways to choose and
arrange two posters, how many posters does she have in total?
(PCBT2.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
193
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT2 Continued ...
Achievement Indicators:
PCBT2.6 Explain, using
examples, the effect on the
total number of permutations
when two or more elements are
identical.
When determining the total number of permutations with two or more
identical elements, students only work with a set of n elements taken n
at a time. They should be asked to solve and explain their solution to a
problem such as the following:
Four beads, one blue, two red and one white, are being placed on a
string. How many different arrangements are possible?
A systematic list is one strategy for determining the total number of
arrangements.
Rbrw
Rrwb
wbrR
wbRr
bRrw
brRw
rbRw
rRwb
Rbwr
Rwrb
wrbR
wRbr
bRwr
brwR
rbwR
rwRb
Rrbw
Rwbr
wrRb
wRrb
bwRr
bwrR
rRbw
rwbR
Students could also recall that there are 4! = 24 different permutations
of four objects. However, there are two identical red beads, making
permutations such as Rbrw and rbRw the same. If the red beads were
different, they could be arranged in 2! ways. Thus, the total number of
permutations with identical beads is 4!
2! or 12.
Generally, in a set of n objects, with a of one kind that are identical, b of
a second kind that are identical, c of a third kind that are identical, etc.,
the entire set of n objects can be arranged in a!bn!c!!… ways. For example,
if there were one blue, two red, one white, and three black beads in the
problem above, then the number of possible arrangements of all seven
7! = 7×6×5×4×3! = 420 .
beads would be 2!3!
2!3!
194
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to answer the following:
(i)
Show that you can form 120 distinct five-letter arrangements
from GREAT but only 60 distinct five-letter arrangements from
GREET.
(PCBT2.6)
(ii) How many distinct arrangements can be formed using all the
letters of STATISTICS?
(PCBT2.6)
(iii) Find the total number of arrangements of the word SILK and
the total number of arrangements of the word SILL. How do
your answers compare? Explain why this relationship exists.
(PCBT2.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Authorized Resource
Pre-Calculus 12
11.1 Permutations
SB: pp. 516-527
TR: pp. 280-286
195
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT2 Continued ...
Achievement Indicator:
PCBT2.7 Solve problems
involving permutations with
constraints.
A variety of constraints can be imposed in situations involving
permutations. In certain situations, for example, objects must be
arranged in a line where:
• two or more objects must be placed together
• two or more objects cannot be placed together
• certain object(s) must be placed in certain positions.
Students should answer questions such as the following:
How many ways can a group of five people be arranged in a line if two
of them are good friends and want to sit together?
If the five people are A, B, C, D, and E, and the friends are A and B:
• The two friends can sit together as either AB or BA. Students should
recognize this to be 2! ways.
• The problem can be simplified somewhat by first considering A and
B as one person. Students can then temporarily think of the problem
as being four different people to arrange, which can be done in 4!
ways.
• By the fundamental counting principle, the five people can be
arranged in 4!2! = 48 ways.
Similarly, if three friends had wanted to sit together, the five people can
be arranged in 3!3! = 36 ways.
196
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to answer the following:
Alice, Beatrice, Colin and Don are to be arranged in a line from left
to right.
(i) How many ways can they be arranged?
(ii) How many ways can they be arranged if Alice and Don cannot
be side by side?
(iii) How many ways can they be arranged if Beatrice and Colin
must be side by side?
(iv) How many ways can they be arranged if Alice must be at one
end of the line?
(PCBT2.7)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
Authorized Resource
Pre-Calculus 12
11.1 Permutations
SB: pp. 516-527
TR: pp. 280-286
197
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT3 Determine the number
of combinations of n different
elements taken r at a time to
solve problems.
In contrast to permutations, combinations are an arrangement of objects
without regard for order. A formula will be developed and applied in
problem solving situations.
[C, PS, R, V]
Achievement Indicators:
PCBT3.1 Explain, using
examples, the differences
between a permutation and a
combination.
PCBT3.2 Determine the number
of combinations of n different
elements taken r at a time to solve
a problem.
To distinguish between permutations and combinations, students
should be given a situation for each where the number of possibilities
can be determined with simple counting methods.
• Adam, Marie and Brian are standing in a line at a banking machine.
In how many ways could they order themselves?
• Paul, Renee and Emily are members of a committee. In how many
ways could two of them be selected to attend a conference?
The essential difference between these two situations needs to be
discussed and emphasized. Students should already recognize the first
problem as a permutation, where order is important. Use the second
problem to introduce combinations, where order is not important.
A situation such as the following could also be used to highlight the
difference between permutations and combinations.
In a lottery, six numbers from 1 to 49 are selected. A winning ticket
must contain the same six numbers but they may be in any order.
If order mattered, the number of permutations would be
49!
49 P6 = 49−6 ! = 10 068 347 520. Since order does not matter, the
( )
number of permutations must be divided by 6! (the number of ways of
arranging the six selected numbers). The number of combinations is
49
C6 =
49!
(49−6)!6!
=
49×48×47×46×45×44 = 13
6!
983 816.
Discuss with students why the number of combinations is less than the
number of permutations.
Generally, given a set of n objects taken r at a time, the number of
possible combinations is n Cr = n −nr! !r! .
( )
⎛n⎞
⎝r⎠
Note that the notation ⎜ ⎟ is sometimes used instead of nCr.
198
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
• Ask students to identify which problem should be done using the
fundamental counting principle, which one using the permutation
formula, and which one with the combination formula. They should
explain their choices.
Pre-Calculus 12
(i)
How many ways can a committee of three people be selected
from a group of 12 people?
(ii) How many ways can three of eight people line up at a ticket
counter?
(iii) How many four-digit numbers are there?
(PCBT3.1)
11.2 Combinations
SB: pp. 528-536
TR: pp. 287-292
Paper and Pencil
• Ask students to answer the following:
(i)
Alan, Bill, Cathy, David and Evelyn are members of the same
club. Determine the number of ways of selecting:
(a) a two-member committee
(b) a president and a treasurer
(ii) Every member of the 48 students in the graduating class
at a local high school would like to attend a special leadership
conference, but only ten members will be allowed to attend.
How many ways are there to select the lucky ten?
(PCBT3.2)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
199
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT3 Continued ...
Achievement Indicators:
PCBT3.2 Continued
Students should also solve problems such as the following, where they
apply both combinations and the fundamental counting principle.
A baseball team has 5 pitchers, 6 outfielders and 10 infielders. For a
game, the manager needs to field a starting group with 1 pitcher, 3
outfielders and 5 infielders. How many ways can she select the starting
group?
• There are 5C1, or 5, ways to select a pitcher.
• There are 6C3, or 20, ways to select the outfielders.
• There are 10C5, or 252 ways to select the infielders.
Students can apply the fundamental counting principle to determine
that there are 25 200 ways to select the starting group.
PCBT3.3 Explain why n must be
greater than or equal to r in the
notation nCr or ⎛ n ⎞ .
⎜r ⎟
⎝ ⎠
Students should be able to easily verify that the value of n must be
greater than or equal to r in nCr, having explained the reasoning when
working with permutations. If they understand that the notation nCr
means choosing r elements from a set of n elements, it naturally flows
that r cannot be larger than n.
PCBT3.4 Explain, using
examples, why nCr = nCn-r or
⎛n⎞ ⎛ n ⎞
⎜ r ⎟ = ⎜ n − r ⎟.
⎝ ⎠ ⎝
⎠
To explore why nCr = nCn-r, a problem such as the following could be
used:
Suppose the letters A, B, C, D and E are placed in a bag and two letters
are selected at random. How many combinations of letters can be
selected?
Most students will determine the number of combinations of two letters
5! . Prompt them to think of a different way to solve
to be 5 C 2 = 3!2!
the problem. It can be thought of in “reverse”. Each time two letters
are selected, three letters are also being selected to remain in the bag.
5! . Students
The number of combinations of three letters is 5 C3 = 2!3!
should conclude from this that 5C2 = 5C5-2 and generally nCr = nCn-r .
An algebraic proof of this result can be presented but is not required
knowledge for students.
PCBT3.5 Given a value of k,
k ε N, solve nCr = k or ⎛ n ⎞ = k
⎜ ⎟
⎝r ⎠
for either n or r.
200
Students are expected to solve a variety of equations for either n or r.
⎛n⎞
C = 15
C = 15
C = 15
⎜ ⎟ = 15
n 2
6 r
n n-2
⎝2⎠
To solve an equation such as 6Cr = 15, students should realize that
r ≤ 6 since r elements are being chosen from a set of 6 elements.
Checking each of the possibilities results in the solution r = 2, r = 4. For
this type of equation, manageable numbers that make trial-and-error a
reasonable strategy should be used.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Ask students to answer the following:
(i) A set of flash cards consists of 13 red, 13 blue, 13 black and
13 yellow cards. The cards in each colour are numbered from 1
through 13.
(a) How many groups of 5 cards can be selected from the
entire set?
(b) How many groups of 5 cards can be selected from the
red cards?
(c) How many groups of 20 cards can be selected from the
entire set if there must be five of each colour?
Authorized Resource
Pre-Calculus 12
11.2 Combinations
SB: pp. 528-536
TR: pp. 287-292
(ii) How many different sums of money can be made by selecting
four coins out of a set consisting of a nickel, a dime, a quarter, a
dollar coin, and a two-dollar coin?
(PCBT3.2)
• Ask students to solve the following:
⎛ n + 1⎞
(i) ⎜
⎟ = 20
⎝ 1 ⎠
(ii)
7
(iv)
n+1
Cr = 21
Cn − 1 = 6
(PCBT3.5)
Journal
• Ask students to explain why an error is obtained when attempting to
calculate 5C7 on a calculator.
(PCBT3.3)
• Ask students to explain why the number of ways of selecting eight
people from a group of ten is equal to the number of ways of
selecting two people from a group of 10.
Then they should discuss whether the number of ways of selecting
eight or two people would be equal if this was a permutation
problem where the selected people had to be lined up in certain
positions.
(PCBT3.4)
• Ask students to explain how they could solve the equation 6(nC3) = 6
using their understanding of combinations rather than an algebraic
solution.
(PCBT3.5)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
201
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT4 Expand powers of a
binomial in a variety of ways,
including using the binomial
theorem (restricted to exponents
that are natural numbers).
[CN, R, V]
Students expand a binomial using Pascal’s triangle and the binomial
theorem. They see that if Pascal’s Triangle has already been completed
for the desired expansion, the appropriate row can be read with no
computation required. They should also see that a disadvantage in using
Pascal’s triangle is that each of the preceding rows in the display must
be completed before the row needed for an expansion can be obtained.
This leads to the development of the binomial theorem.
Achievement Indicator:
PCBT4.1 Explain the patterns
found in the expanded form of
(x + y)n, n ≤ 4 and n ε N, by
multiplying n factors of (x + y).
In Mathematics 1201, students multiplied polynomial expressions,
including monomials, binomials and trinomials (AN4). They should
perform the following binomial expansions algebraically and record the
coefficients of the terms:
Expression
Coefficients of the Terms
0
(x + y) = 1
(x + y)1 = x + y
(x + y)2 = x2 + 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
Students should look for patterns in the exponents of the terms for each
expansion.
• How is the exponent of the binomial related to the exponents of the
first and last terms in the expansions?
• Moving from left to right in each expansion, what patterns are there
in the exponents for x and y?
They should also look for patterns in the triangle. For example, each
element is the sum of the two elements immediately above it.
PCBT4.2 Explain how to
determine the subsequent row in
Pascal’s triangle, given any row.
202
This activity develops the elements in rows 1 through 5 for Pascal’s
triangle, a triangular array of numbers described by Blaise Pascal in
1653. From the activity, students should be able to explain how to
determine the elements in any row once the preceding row is known.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
• Ask students to describe the pattern that the exponents of x and y
follow when doing an expansion such as (x + y)4.
Pre-Calculus 12
(PCBT4.1)
• Ask students to respond to the following:
Suppose a friend has missed the class in which you learned to
construct Pascal’s triangle. Write a point-form description of the
steps you would tell her to follow in order to construct the triangle
on her own.
11.3 The Binomial Theorem
SB: pp. 537-545
TR: pp. 293-297
(PCBT4.2)
Paper and Pencil
• Based on what was observed for expansions with powers up to
n = 4, ask students to predict the exponents for the 5th term in the
expansion of (x + y)7.
(PCBT4.1)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
203
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
Permutations, Combinations and The Binomial Theorem
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
PCBT4 Continued ...
Achievement Indicators:
PCBT4.3 Relate the coefficients
of the terms in the expansion of
(x + y)n to the (n + 1) row in
Pascal’s triangle.
Using Pascal’s triangle, students should now also be able to find the
expansion for (x + y)n for any value of n where n ≤ 12.
The elements in each row of Pascal’s triangle can be determined using
the formula for nCr, as shown in the diagram.
PCBT4.4 Explain, using
examples, how the coefficients
of the terms in the expansion
of (x + y)n are determined by
combinations.
PCBT4.5 Expand, using the
binomial theorem, (x + y)n.
Using combinations, students should be able to predict the element
or the coefficient of any element in the expansion of (x + y)n. In the
expansion of (x + y)7, for example, the coefficient of x3y4 is 7C4 = 35.
Generally, the expansion of (x + y)n is:
n
C0xny0 +nC1xn - 1y1 + nC2xn - 2y2 + ... + nCn - 1x1yn - 1 + nCnx0yn
This is known as the binomial theorem.
PCBT4.6 Determine a specific
term in a binomial expansion.
Students should now apply the binomial theorem to expansions of other
binomial expressions, such as:
•
(2a + b)7
•
(3x − 2 )
5
(x + x1 )
10
•
204
3
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
General Outcome: Develop algebraic and numeric reasoning that involves
combinatorics.
Suggested Assessment Strategies
Resources/Notes
Journal
• Ask students to respond to the following:
(i) Explain how using Pascal’s triangle makes expanding binomials
easier when using larger exponents.
Authorized Resource
Pre-Calculus 12
(ii)Would there be any disadvantages to using Pascal’s triangle if you
wanted to find a single term in a binomial expansion with a very
large exponent?
(PCBT4.3)
11.3 The Binomial Theorem
SB: pp. 537-545
TR: pp. 293-297
Paper and Pencil
• Ask students to answer the following:
(i) The combination formula is used to find a particular coefficient
in a binomial expansion.
(a) Which row and entry in Pascal’s triangle could be used to
accomplish the same task?
(b) What is the exponent being used to expand the binomial?
(c) For which term in the expansion will this give the coefficient?
(d) What will be the exponents of the variables for this particular
term?
(e) What will be the largest coefficient in this particular
expansion?
(PCBT 4.4)
(ii) Expand and simplify each of the following:
4
(a) (a + 3b )
(b) (1 − 5 y )
3
(c) 2x + 4
(
)
(d) x 2 −
3
x2
(
6
)
12
(PCBT 4.5)
(iii) Determine the middle term in the expansion of (2c 3d)6.
(PCBT 4.6)
(iv) Find the indicated term in each of the following expansions:
(a) The second term of (5 + x)6
(b) The fifth term of (x + 7)7
(c) The third term of (x − 2)6
(d) The third term of (2x + 3y)7
(e) The fourth term of (3x − 7y)5
(PCBT 4.6)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
205
PERMUTATIONS, COMBINATIONS AND BINOMIAL THEOREM
206
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
APPENDIX
Appendix:
Outcomes with Achievement Indicators
Organized by Topic
(With Curriculum Guide Page References)
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
207
APPENDIX
Topic: Relations and Functions
Specific Outcomes
General Outcome: Develop algebraic and graphical reasoning through
the study of relations.
Achievement Indicators
Page
It is expected that students will:
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
RF1.1 Compare the graphs of a set of functions of
RF1. Demonstrate an
the form y − k = f (x) to the graph of y = f (x) and
understanding of the effects of
generalize,
using inductive reasoning, a rule about the
horizontal and vertical translations
effect of k.
on the graphs of functions and
their related equations.
RF1.2 Compare the graphs of a set of functions of
the form y = f (x − h) to the graph of y = f (x) and
[C, CN, R, V]
generalize, using inductive reasoning, a rule about the
effect of h.
RF1.3 Compare the graphs of a set of functions of the
form y − k = f (x − h) to the graph of y = f (x) and
generalize, using inductive reasoning, a rule about the
effect of h and k.
RF1.4 Sketch the graph of y − k = f (x), y = f (x − h)
or y − k = f (x − h) for given values of h and k, given
a sketch of the function y = f (x), where the equation of
y = f (x) is not given.
RF1.5 Write the equation of a function whose graph is
a vertical and/or horizontal translation of the graph of
the function y = f (x).
RF2. Demonstrate an
understanding of the effects of
horizontal and vertical stretches on
the graphs of functions and their
related equations.
[C, CN, R, V]
p. 42
p. 42
p. 42
p. 42
p. 44
RF2.1 Compare the graphs of a set of functions of the
form y = af (x) to the graph of y = f (x), and generalize,
using inductive reasoning, a rule about the effect of a.
p. 48
RF2.2 Compare the graphs of a set of functions of the
form y = f (bx) to the graph of y = f (x), and generalize,
using inductive reasoning, a rule about the effect of b.
p. 50
RF2.3 Compare the graphs of a set of functions of the
form y = af (bx) to the graph of y = f (x), and generalize,
using inductive reasoning, a rule about the effects of a
and b.
208
Reference
p. 52
RF2.4 Sketch the graph of y = af (x), y = f (bx) or
y = af (bx) for given values of a and b, given a sketch of
the function y = f (x), where the equation of y = f (x) is
not given.
p. 54
RF2.5 Write the equation of a function, given its graph
which is a vertical and/or horizontal stretch of the graph
of the function y = f (x).
p. 54
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
APPENDIX
Topic: Relations and Functions
Specific Outcomes
It is expected that students will:
RF3. Apply translations and
stretches to the graphs and
equations of functions.
General Outcome: Develop algebraic and graphical reasoning through
the study of relations.
Achievement Indicators
Page
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
RF3.1 Sketch the graph of the function
y − k = af (b(x − h)) for given values of a, b, h and
k, given the graph of the function y = f (x), where the
equation of y = f (x) is not given.
Reference
p. 56
[C, CN, R, V]
RF4. Demonstrate an
understanding of the effects
of reflections on the graphs
of functions and their related
equations, including reflections
through the:
•
x-axis
•
y-axis
•
line y = x.
[C, CN, R, V]
RF3.2 Write the equation of a function, given its graph
which is a translation and/or stretch of the graph of the
function y = f (x).
RF4.1 Generalize the relationship between the
coordinates of an ordered pair and the coordinates
of the corresponding ordered pair that results from a
reflection through the x-axis or the y-axis.
p. 58
p. 46
RF4.2 Sketch the reflection of the graph of a function
y = f (x) through the x-axis or the y-axis, given the graph
of the function y = f (x), where the equation of y = f (x)
is not given.
p. 46
RF4.3 Generalize, using inductive reasoning, and
explain rules for the reflection of the graph of the
function y = f (x) through the x-axis or the y-axis.
p. 46
RF4.4 Sketch the graphs of the functions y = −f (x)
and y = f (−x), given the graph of the function y = f (x),
where the equation of y = f (x) is not given.
p. 46
RF4.5 Write the equation of a function, given its
graph which is a reflection of the graph of the function
y = f (x) through the x-axis or the y-axis.
p. 46
RF4.6 Generalize the relationship between the
coordinates of an ordered pair and the coordinates
of the corresponding ordered pair that results from a
reflection through the line y = x.
RF4.7 Sketch the reflection of the graph of a function
y = f (x) through the line y = x, given the graph of the
function y = f (x), where the equation of y = f (x) is not
given.
RF4.8 Generalize, using inductive reasoning, and
explain rules for the reflection of the graph of the
function y = f (x) through the line y = x.
RF4.9 Write the equation of a function, given its
graph which is a reflection of the graph of the function
y = f (x) through the line y = x.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
p. 60
p. 62
p. 62
p. 66
209
APPENDIX
Topic: Relations and Functions
Specific Outcomes
It is expected that students will:
RF5. Demonstrate an
understanding of inverses of
relations.
[C, CN, R, V]
RF6. Demonstrate an
understanding of logarithms.
[CN, ME, R]
RF7. Demonstrate an
understanding of the product,
quotient and power laws of
logarithms.
[C, CN, ME, R, T]
210
General Outcome: Develop algebraic and graphical reasoning through
the study of relations.
Achievement Indicators
Page
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
Reference
RF5.1 Explain how the transformation (x, y) → (y, x)
can be used to sketch the inverse of a relation.
p. 60
RF5.2 Explain the relationship between the domains
and ranges of a relation and its inverse.
p. 60
RF5.3 Explain how the graph of the line y = x can be
used to sketch the inverse of a relation.
p. 62
RF5.4 Sketch the graph of the inverse relation, given
the graph of a relation.
p. 62
RF5.5 Determine if a relation and its inverse are
functions.
p. 64
RF5.6 Determine restrictions on the domain of a
function in order for its inverse to be a function.
p. 66
RF5.7 Determine the equation and sketch the graph
of the inverse relation, given the equation of a linear or
quadratic relation.
p. 66
RF5.8 Determine, algebraically or graphically, if two
functions are inverses of each other.
RF6.1 Explain the relationship between logarithms and
exponents.
p. 66
p. 160
RF6.2 Express a logarithmic equation as an exponential
equation and vice versa.
p. 160
RF6.3 Determine, without technology, the exact value
of a logarithm, such as log 2 8.
p. 160
RF6.4 Estimate the value of a logarithm, using
benchmarks, and explain the reasoning; e.g., since
log 2 8 = 3 and log 2 16 = 4, log 2 9 is approximately
equal to 3.1.
RF7.1 Develop and generalize the laws of logarithms,
using numeric examples and exponent laws.
p. 160
p. 168
RF7.2 Derive each law of logarithms.
p. 168
RF7.3 Determine, using the laws of logarithms, an
equivalent expression for a logarithmic expression.
p. 170
RF7.4 Determine, with technology, the approximate
value of a logarithmic expression, such as log 2 9.
p. 170
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
APPENDIX
Topic: Relations and Functions
Specific Outcomes
It is expected that students will:
RF8. Graph and analyze
exponential and logarithmic
functions.
[C, CN, T, V]
General Outcome: Develop algebraic and graphical reasoning through
the study of relations.
Page
Achievement Indicators
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
Reference
RF8.1 Sketch, with or without technology, a graph of
an exponential function of the form y = cx, c > 0, c ≠ 1.
p. 142-144
RF8.2 Identify the characteristics of the graph of an
exponential function of the form y = cx, c > 0, c ≠ 1,
including the domain, range, horizontal asymptote and
intercepts, and explain the significance of the horizontal
asymptote.
RF8.3 Sketch the graph of an exponential function by
applying a set of transformations to the graph of y = cx,
c > 0, c ≠ 1 and state the characteristics of the graph.
p. 142-144
p. 146
RF8.4 Demonstrate, graphically, that a logarithmic
function and an exponential function with the same
base are inverses of each other.
p. 162
RF8.5 Sketch with or without technology, the graph of
a logarithmic function of the form y = log c x, c > 1.
p. 164
RF8.6 Identify the characteristics of the graph of a
logarithmic function of the form y = log c x, c > 1,
including the domain, range, vertical asymptote and
intercepts, and explain the significance of the vertical
asymptote.
RF8.7 Sketch the graph of a logarithmic function by
applying a set of transformations to the graph of
pp. 164-166
pp. 164-166
y = log c x, c > 1, and state the characteristics of the
graph.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
211
APPENDIX
Topic: Relations and Functions
Specific Outcomes
It is expected that students will:
RF9. Solve problems that involve
exponential and logarithmic
equations.
[C, CN, PS, R]
RF10. Demonstrate an
understanding of factoring
polynomials of degree greater
than 2 (limited to polynomials
of degree ≤ 5 with integral
coefficients).
[C, CN, ME]
212
General Outcome: Develop algebraic and graphical reasoning through
the study of relations.
Achievement Indicators
Page
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
RF9.1 Determine the solution of an exponential
equation for which both sides can be written as rational
powers of the same base.
Reference
p. 148
RF9.2 Determine the solution of an exponential
equation in which the bases are not rational powers of
one another, using a variety of strategies.
p. 148-150,
172
RF9.3 Solve a problem that involves exponential
growth or decay.
p. 152-154,
174
RF9.4 Solve a problem that involves the application
of exponential equations to loans, mortgages and
investments.
p. 152-154,
176
RF9.5 Solve a problem by modeling a situation with an
exponential or a logarithmic equation.
p. 152-154,
174-176
RF9.6 Determine the solution of a logarithmic
equation, and verify the solution.
p. 172
RF9.7 Explain why a value obtained in solving a
logarithmic equation may be extraneous.
p. 172
RF9.8 Solve a problem that involves logarithmic scales,
such as the Richter scale and the pH scale.
RF10.1 Explain how long division of a polynomial
expression by a binomial expression of the form x − a,
a ∈ I, is related to synthetic division.
p. 174
RF10.2 Divide a polynomial expression by a binomial
expression of the form x − a, a ∈ I using long division
or synthetic division.
RF10.3 Explain the relationship between the remainder
when a polynomial expression is divided by x − a,
a ∈ I, and the value of the polynomial expression at
x = a (remainder theorem).
p. 26
pp. 26-28
p. 28
RF10.4 Explain and apply the factor theorem to express
a polynomial expression as a product of factors.
pp. 28-30
RF10.5 Explain the relationship between linear
factors of a polynomial expression and the zeros of the
corresponding polynomial function.
pp. 28-30
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
APPENDIX
Topic: Relations and Functions
General Outcome: Develop algebraic and graphical reasoning through
the study of relations..
Achievement Indicators
Page
Specific Outcomes
It is expected that students will:
RF11. Graph and analyze
polynomial functions (limited to
polynomial functions of degree
≤ 5).
[C, CN, T, V]
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
RF11.1 Identify the polynomial functions in a set of
functions, and explain the reasoning.
Reference
p. 22
RF11.2 Explain the role of the constant term and
leading coefficient in the equation of a polynomial
function with respect to the graph of the function.
p. 22
RF11.3 Generalize rules for graphing polynomial
functions of odd or even degree.
p. 24
RF11.4 Explain the relationship among the following:
RF12. Graph and analyze radical
functions (limited to functions
involving one radical).
[CN, R, T, V]
•
the zeros of a polynomial function
•
the roots of the corresponding polynomial equation
•
the x-intercepts of the graph of the polynomial
function.
p. 30
RF11.5 Explain how the multiplicity of a zero of a
polynomial function affects the graph.
p. 32
RF11.6 Sketch, with or without technology, the graph
of a polynomial function.
p. 34
RF11.7 Solve a problem by modeling a given situation
with a polynomial function.
p. 36
RF11.8 Determine the equation of a polynomial
function given its graph.
p. 36
RF12.1 Sketch the graph of the function y = x ,
using a table of values, and state the domain and range.
p. 72
RF12.2 Sketch the graph of the function
y − k = a b (x − h ) by applying transformations to the
graph of the function y = x , and state the domain
and range.
pp. 72-74
RF12.3 Sketch the graph of the function y = f ( x ) ,
given the equation or graph of the function y = f (x),
and explain the strategies used.
pp. 76-78
RF12.4 Compare the domain and range of the function
y = f ( x ) to the domain and range of the function
y = f (x), and explain why the domains and ranges may
differ.
RF12.5 Describe the relationship between the roots of a
radical equation and the x-intercepts of the graph of the
corresponding radical function.
RF12.6 Determine, graphically, an approximate
solution of a radical equation.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
pp. 76-78
p. 78
p. 78
213
APPENDIX
Topic: Trigonometry
Specific Outcomes
It is expected that students will:
T1. Demonstrate an
understanding of angles in
standard position, expressed in
degrees and radians.
[CN, ME, R, V]
General Outcome: Develop trigonometric reasoning.
Achievement Indicators
Page
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
T1.1 Sketch, in standard position, an angle (positive or
negative) when the measure is given in degrees.
T1.2 Sketch, in standard position, an angle with a
measure of 1 radian.
214
p. 84
pp. 84-86
T1.4 Sketch, in standard position, an angle with a
measure expressed in the form kπ radians, where k ∈ Q.
p. 86
p. 86
T1.6 Express the measure of an angle in degrees, (exact
value or decimal approximation) given its measure in
radians.
p. 86
T1.7 Determine the measures, in degrees or radians, of
all angles in a given domain that are coterminal with a
given angle in standard position.
p. 88
T1.8 Determine the general form of the measures, in
degrees or radians, of all angles that are coterminal with
a given angle in standard position.
[CN, R, V]
p. 84
T1.3 Describe the relationship between radian measure
and degree measure.
T1.5 Express the measure of an angle in radians (exact
value or decimal approximation), given its measure in
degrees.
T2. Develop and apply the
equation of the unit circle.
Reference
p. 88
T1.9 Explain the relationship between the radian
measure of an angle in standard position and the length
of the arc cut on a circle of radius r, and solve problems
based upon that relationship.
T2.1 Derive the equation of the unit circle from the
Pythagorean theorem.
p. 88
T2.2 Generalize the equation of a circle with centre
(0, 0) and radius r.
p. 90
T2.3 Describe the six trigonometric ratios, using a
point P(x, y) that is the intersection of the terminal arm
of an angle and the unit circle.
p. 90
p. 90
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
APPENDIX
Topic: Trigonometry
Specific Outcomes
It is expected that students will:
T3. Solve problems, using the six
trigonometric ratios for angles
expressed in radians and degrees.
[ME, PS, R, T, V]
General Outcome: Develop trigonometric reasoning.
Achievement Indicators
Page
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
T3.1 Determine, with technology, the approximate
value of a trigonometric ratio for any angle with a
measure expressed in either degrees or radians.
T3.2 Determine, using a unit circle or reference
triangle, the exact value of a trigonometric ratio for
angles expressed in degrees that are multiples of 0˚, 30˚,
45˚, 60˚ or 90˚, or for angles expressed in radians that
multiples of 0, S , S , S or S and explain the strategy.
6 4 3
Reference
p. 92
p. 92
2
T3.3 Sketch a diagram to represent a problem that
involves trigonometric ratios.
p. 92
T3.4 Determine, with or without technology, the
measures, in degrees or radians, of the angles in a
specified domain, given the value of a trigonoemtric
ratio.
p. 94
T3.5 Describe the six trigonometric ratios, using a
point P(x, y) that is the intersection of the terminal arm
of an angle and the unit circle.
p. 94
T3.6 Determine the measures of the angles in a
specified domain in degrees or radians, given a point on
the terminal arm of an angle in standard position.
p. 94
T3.7 Determine the exact values of the other
trigonometric ratios, given the value of one
trigonometric ratio in a specified domain.
T3.8 Solve a problem, using trigonometric ratios.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
p. 96
p. 96
215
APPENDIX
Topic: Trigonometry
Specific Outcomes
It is expected that students will:
T4. Graph and analyze the
trigonometric functions sine,
cosine and tangent to solve
problems.
[CN, PS, T, V]
216
General Outcome: Develop trigonometric reasoning.
Achievement Indicators
Page
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
Reference
T4.1 Sketch, with or without technology, the graph of
y = sin x and y = cos x.
p. 106
T4.2 Determine the characteristics (amplitude, domain,
period, range and zeros) of the graph of y = sin x and
y = cos x.
p. 106
T4.3 Determine how varying the value of a affects the
graph of y = a sin x and y = a cos x.
p. 108
T4.4 Determine how varying the value of b affects the
graph of y = sin bx and y = cos bx.
p. 108
T4.5 Determine how varying the value of d affects the
graph of y = sin x + d and y = cos x + d.
p. 108
T4.6 Determine how varying the value of c affects the
graph of y = sin(x + c) and y = cos(x + c).
p. 108
T4.7 Sketch, without technology, graphs of the form
y = a sin b(x − c) + d and y = a cos b(x − c) + d using
transformations, and explain the strategies.
p. 108
T4.8 Determine the characteristics (amplitude, domain,
period, phase shift, range and zeros) of the graph of a
trigonometric function of the form
y = a sin b(x − c) + d and y = a cos b(x − c) + d.
p. 108
T4.9 Determine the values of a, b, c and d for functions
of the form y = a sin b(x − c) + d and
y = a cos b(x − c) + d that correspond to a given graph,
and write the equation of the function.
p. 110
T4.10 Solve a given problem by analyzing the graph of
a trigonometric function.
p. 110
T4.11 Explain how the characteristics of the graph of
a trigonometric function relate to the conditions in a
problem situation.
p. 110
T4.12 Determine a trigonometric function that models
a situation to solve a problem.
p. 110
T4.13 Sketch, with or without technology, the graph of
y = tan x.
p. 112
T4.14 Determine the characteristics (asymptotes,
domain, period, range and zeros) of the graph of
y = tan x.
p. 112
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
APPENDIX
Topic: Trigonometry
General Outcome: Develop trigonometric reasoning.
Specific Outcomes
Achievement Indicators
Page
It is expected that students will:
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
Reference
T5. Solve, algebraically and
graphically, first and second degree
trigonometric equations with the
domain expressed in degrees and
radians.
[CN, PS, R, T, V]
T5.1 Determine, algebraically, the solution of a
trigonometric equation, stating the solution in exact
form when possible.
pp. 98, 114,
134
T5.2 Determine, using technology, the approximate
solution of a trigonometric equation.
pp. 98, 114,
136
T5.3 Verify, with or without technology, that a given
value is a solution to a trigonometric equation.
pp. 98, 114,
T5.4 Identify and correct errors in a solution for a
trigonometric equation.
pp. 100, 136
T5.5 Relate the general solution of a trigonometric
equation to the zeros of the corresponding function
(restricted to sine and cosine functions).
T6. Prove trigonometric identities, T6.1 Explain the difference between a trigonometric
identity and a trigonometric equation.
using:
•
reciprocal identities
•
quotient identities
•
Pythagorean identities
•
sum or difference identities
(restricted to sine, cosine and
tangent)
•
double-angle identities
(restricted to sine, cosine and
tangent).
[R, T, V]
T6.2 Determine, graphically, the potential validity of a
trigonometric identity, using technology.
T6.3 Determine the non-permissible values of a
trigonometric identity.
136
p. 114
pp. 120-122
pp. 122, 128130
pp. 122, 128132
T6.4 Verify a trigonometric identity numerically for a
given value in either degrees or radians.
pp. 124, 128130
T6.5 Prove, algebraically, that a trigonometric identity
is valid.
pp. 124, 130132
T6.6 Simplify trigonometric expressions using
trigonometric identities.
pp. 126-130
T6.7 Determine, using the sum, difference and doubleangle identities, the exact value of a trigonometric ratio.
pp. 128-130
T6.8 Explain why verifying that the two sides of a
trigonometric identity are equal for given values is
insufficient to conclude that the identity is valid.
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
p. 132
217
APPENDIX
Topic: Permutations,
Combinations and the Binomial
Theorem
Specific Outcomes
General Outcome: Develop algebraic and numeric reasoning that
involves combinatorics.
Achievement Indicators
Page
It is expected that students will:
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
Reference
PCBT1. Apply the fundamental
counting principle to solve
problems.
PCBT1.1 Count the total number of possible choices
that can be made, using graphic organizers such as lists
and tree diagrams.
pp. 182-184
[C, PS, R, V]
PCBT1.2 Explain, using examples, why the total
number of possible choices is found by multiplying
rather than adding the number of ways the individual
choices can be made.
PCBT2. Determine the number
of permutations of n elements
taken r at a time to solve
problems.
[C, PS, R, V]
PCBT1.3 Solve a simple counting problem by applying
the fundamental counting principle.
PCBT2.1 Count, using graphic organizers such as lists
and tree diagrams, the number of ways of arranging the
elements of a set in a row.
PCBT2.2 Determine, in factorial notation, the number
of permutations of n different elements taken n at a
time to solve a problem.
p. 186
p. 188
p. 188
PCBT2.3 Determine, using a variety of strategies, the
number of permutations of n different elements taken r
at a time to solve a problem.
p. 190
PCBT2.4 Explain why n must be greater than or equal
to r in the notation nPr.
p. 190
PCBT2.5 Given a value of k, k ∈ Ν, solve nPr = k for
either n or r.
p. 192
PCBT2.6 Explain, using examples, the effect on the
total number of permutations when two or more
elements are identical.
PCBT2.7 Solve problems involving permutations with
constraints.
218
p.186
p. 194
p. 196
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
APPENDIX
Topic: Permutations,
Combinations and the Binomial
Theorem
Specific Outcomes
It is expected that students will:
PCBT3. Determine the number
of combinations of n different
elements taken r at a time to solve
problems.
[C, PS, R, V]
General Outcome: Develop algebraic and numeric reasoning that
involves combinatorics.
Achievement Indicators
Page
The following sets of indicators help determine whether
students have met the corresponding specific outcome.
Reference
PCBT3.1 Explain, using examples, the differences
between a permutation and a combination.
PCBT3.2 Determine the number of combinations of n
different elements taken r at a time to solve a problem.
pp. 198-200
PCBT3.3 Explain why n must be greater than or equal
to r in the notation nCr or ⎛ n ⎞ .
⎜r ⎟
⎝ ⎠
p. 200
PCBT3.4 Explain, using examples, why nCr = nCn - r or
⎛n⎞ ⎛ n ⎞ .
⎜ r ⎟ = ⎜n − r ⎟
⎝ ⎠ ⎝
⎠
PCBT4. Expand powers of a
binomial in a variety of ways,
including using the binomial
theorem (restricted to exponents
that are natural numbers).
[CN, R, V]
p. 198
PCBT3.5 Given a value of k, k ∈ N, solve nCr = k or
for either n or r.
⎛n⎞
⎜r ⎟ = k
⎝ ⎠
PCBT4.1 Explain the patterns found in the expanded
form of (x + y)n, n ≤ 4 and n ∈ Ν, by multiplying n
factors of (x + y).
p. 200
p. 200
p. 202
PCBT4.2 Explain how to determine the subsequent
row in Pascal’s triangle, given any row.
p. 202
PCBT4.3 Relate the coefficients of the terms in the
expansion of (x + y)n to the (n + 1) row in Pascal’s
triangle.
p. 204
PCBT4.4 Explain, using examples, how the coefficients
of the terms in the expansion of (x + y)n are determined
by combinations.
p. 204
PCBT4.5 Expand, using the binomial theorem, (x + y)n.
p. 204
PCBT4.6 Determine a specific term in a binomial
expansion.
p. 204
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
219
APPENDIX
220
ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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ADVANCED MATHEMATICS 3200 CURRICULUM GUIDE - INTERIM 2013
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