# Mathematics - Department of Education and Early Childhood ```Mathematics
Interim Edition
Curriculum Guide
2012
CONTENTS
Contents
Acknowledgements
................................................................................................... iii
Introduction
.....................................................................................................................1
Background ..............................................................................................................................1
Affective Domain......................................................................................................................2
Goals For Students...................................................................................................................2
Conceptual Framework for 10-12 Mathematics
...................................3
Mathematical Processes..........................................................................................................3
Nature of Mathematics.............................................................................................................7
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: Sequences and Series ...............................................................................................19
Unit 2: Trigonometry...............................................................................................................51
Unit 5: Radical Expressions and Equations . ........................................................................ 119
Unit 6: Rational Expressions and Equations.........................................................................151
Unit 7: Absolute Value and Reciprocal Functions..................................................................181
Unit 8: Systems of Equations............................................................................................... 211
Unit 9: Linear and Quadratic Inequalities..............................................................................229
References .......................................................................................................................251
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
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CONTENTS
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ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
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
We would also like to thank the provincial Grade 11 Advanced Mathematics curriculum committee and the
following people for their contribution:
Deanne Lynch, Program Development Specialist - Mathematics Division of Program
Development, Department of Education
Joanne Hogan, 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
Mary Ellen Giles, Teacher - Mealy Mountain Collegiate, Happy Valley-Goose Bay
Dean Holloway, Teacher - Heritage Collegiate, Lethbridge
Kelly House, Teacher - Holy Heart of Mary High, St.John’s
Brent Hughes, Teacher - Exploits Valley High, Grand Falls-Windsor
Jennifer Lawlor, Teacher - Baltimore School, Ferryland
Jim Penney, Teacher - Stephenville High School, Stephenville
Clarence White, Teacher - Elwood Regional High, Deer Lake
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 2200 CURRICULUM GUIDE - INTERIM 2012
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ACKNOWLEDGEMENTS
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INTRODUCTION
INTRODUCTION
Background
The curriculum guide
communicates high
expectations for students.
Students and
Mathematics
Mathematical
understanding is fostered
when students build on
their own experiences and
prior knowledge.
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.
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.
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 2200 CURRICULUM GUIDE - INTERIM 2012
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INTRODUCTION
Program Design and Components
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
The main goals of mathematics education are to prepare students to:
• use mathematics confidently to solve problems
• communicate and reason mathematically
Mathematics education
must prepare students
to use mathematics
confidently to solve
problems.
• 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:
• 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.
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ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
MATHEMATICAL PROCESSES
CONCEPTUAL
FRAMEWORK
FOR 10-12
MATHEMATICS
The chart below provides an overview of how mathematical processes
and the nature of mathematics influence learning outcomes.
Mathematical
Processes
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
• Communication [C]
• Connections [CN]
• connect mathematical ideas to other concepts in mathematics, to
everyday experiences and to other disciplines
• Mental Mathematics
and Estimation [ME]
• demonstrate fluency with mental mathematics and estimation
• Problem Solving [PS]
• develop and apply new mathematical knowledge through problem
solving
• Reasoning [R]
• develop mathematical reasoning
• Technology [T]
• select and use technologies as tools for learning and for solving
problems
• Visualization [V]
• 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 2200 CURRICULUM GUIDE - INTERIM 2012
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MATHEMATICAL PROCESSES
Communication [C]
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 2200 CURRICULUM GUIDE - INTERIM 2012
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
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 2200 CURRICULUM GUIDE - INTERIM 2012
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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
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ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
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
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.
• Number Sense
• Patterns
• Relationships
• Spatial Sense
• 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 2200 CURRICULUM GUIDE - INTERIM 2012
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NATURE OF MATHEMATICS
Constancy
Constancy is described by the
terms stability, conservation,
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
is the most important
foundation of a numerate
child.
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 2200 CURRICULUM GUIDE - INTERIM 2012
NATURE OF MATHEMATICS
Patterns
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 2200 CURRICULUM GUIDE - INTERIM 2012
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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
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 2200 CURRICULUM GUIDE - INTERIM 2012
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
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
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PROGRAM ORGANIZATION
Program
Organization
Program Level
Course 1
Course 2
Course 3
Course 4
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.
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 2200 CURRICULUM GUIDE - INTERIM 2012
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 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
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 2200 CURRICULUM GUIDE - INTERIM 2012
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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:
• 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
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 2200 CURRICULUM GUIDE - INTERIM 2012
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
(formal or informal)
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.
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 2200 CURRICULUM GUIDE - INTERIM 2012
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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 2200 CURRICULUM GUIDE - INTERIM 2012
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 2200 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 11 (McGraw-Hill Ryerson). Column four of the
curriculum guide references Pre-Calculus 11 for this reason.
Teachers may use any other resource, or combination of resources, to
meet the required specific outcomes.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
17
GENERAL AND SPECIFIC OUTCOMES
GENERAL
AND SPECIFIC
OUTCOMES
GENERAL AND SPECIFIC OUTCOMES WITH ACHIEVEMENT
INDICATORS (pages 19-214)
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 2200 is organized into nine units: Sequences and
Expressions and Equations, Rational Expressions and Equations, Absolute
Value and Reciprocal Functions, Systems of Equations and Linear and
18
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Sequences and Series
Suggested Time: 13 Hours
SEQUENCES AND SERIES
Unit Overview
Focus and Context
Outcomes
Framework
In this unit, students will differentiate between finite and infinite
sequences. They will then explore and develop a formula for the
general term of arithmetic and geometric sequences, and progress to
arithmetic and geometric series. Students will determine if a given series
is convergent or divergent. They will then write the general equation for
the sum of an infinite geometric series. The concepts of sequences and
series will be applied to solve problems.
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF9
Analyze arithmetic sequences
and series to solve problems.
SCO RF10
Analyze geometric sequences
and series to solve problems.
20
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
Process Standards
Key
[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
RF9 Analyze arithmetic sequences
and series to solve problems.
[CN, PS, R]
RF10 Analyze geometric
sequences and series to solve
problems.
[PS, R]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
21
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Analyze arithmetic
sequences and series to solve
problems.
[CN, PS, R]
Achievement Indicators:
RF9.1 Identify the assumption(s)
arithmetic sequence or series.
RF9.2 Provide and justify
an example of an arithmetic
sequence.
In Grade 9, students were exposed to linear patterns (9PR1). In
Mathematics 1201, they determined the equation of a linear relation
(RF7). In this unit, students will explore the concept of arithmetic
sequences and series. They will work with patterns where consecutive
terms produce a common difference. Students will write a formula for
the general term of an arithmetic sequence and solve for any missing
values.
Once sequences have been explored, students will be introduced to an
arithmetic series. They will make the connection that an arithmetic
series is the sum of an arithmetic sequence, derive a formula and apply it
in a variety of problem solving situations.
Students should investigate the concept of a sequence by observing a
variety of number patterns to develop an understanding of the notation,
symbols and domain associated with arithmetic sequences (i.e. terms
t1, t2, finite sequence, infinite sequence, common difference d, and the
general term tn).
In Mathematics 1201, students were exposed to the concept of
domain as it relates to linear functions (RF1). Provide students with an
arithmetic sequence and ask them to graph the terms of the sequence.
Students should make the connection that the terms of the sequence
are the ordered pairs of the graph. Ask them what the x and y-values of
the ordered pairs represent and what number system they belong to.
They should recognize the domain of the sequence is the set of natural
numbers. Encourage students to provide their own examples of arithmetic
sequences and justify, finding the difference between consecutive
terms, why such a sequence is arithmetic. They should not assume
every sequence that has an increasing or decreasing pattern in addition
is arithmetic. Students should conclude, however, that if a sequence
is arithmetic the same number is being added to each term in the
sequence.
RF9.3 Derive a rule for
determining the general term of
an arithmetic sequence.
Use a numerical example to give students an opportunity to identify
patterns and a possible rule for an arithmetic sequence. Consider a
sequence such as 5, 7, 9, 11, 13,...... and the following questions to
guide student discussion.
•
What is the common difference?
•
Can this sequence be rewritten to show the pattern of the first
term and the common difference?
•
Can you predict the formula for the general term of an arithmetic
sequence based on the pattern 5, 5 + 2, 5 + 2(2), 5 + 3(2),.....?
•
22
Can you write the pattern in general terms for any first term and
common difference?
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to determine the first five terms for each of the
sequences.
(i) tn = 2n - 1
2
(ii) t n = 4 n − 4 n + 1
2n 2 +n −1
(iii) t n = n +1
Ask them if each sequence is arithmetic since the first five terms
result in {1, 3, 5, 7, 9}. They should explain their reasoning.
(RF9.2)
Observation
•
Ask students to explore the following sequences to determine
whether each is arithmetic. They should justify their conclusions.
1.1 Arithmetic Sequences
Student Book (SB): pp.6-21
Teacher Resource (TR): pp.10-18
Blackline Master (BLM): 1-3, 1-4
(i) 5.3, 5.9, 6.5
(ii) x - 1, x + 1, x + 3
(iii) x1, x2, x3,
(iv) 2x + 5, 4x + 5, 6x + 5
(RF9.2)
Journal
•
Ask students to explain the term “sequence” in their own words.
Ask them to compare this to the mathematical definition found at
www.en.wikipedia.org/wiki/Sequence. They should explain how the
two definitions are similar and how they are different.
(RF9.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
23
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicators:
RF9.2, RF9.3 Continued
Students should notice that the terms of the arithmetic sequence with
first term t1 and a common difference d can be written as
t1 = t1, t2 = t1 + d, t3 = t1 + 2d, t4 = t1 + 3d... Using this pattern, they can
write the formula for any arithmetic sequence tn = t1 + (n - 1)d.
RF9.4 Determine t1 , d, n, or
tn in a problem that involves an
arithmetic sequence.
Give students an opportunity to write their own examples of arithmetic
sequences and to write the appropriate formula. They should use
numerical and algebraic examples. Encourage them to list t1, d, and n,
and then write the equation for tn.
RF9.5 Solve a problem that
involves an arithmetic sequence
or series.
When solving problems involving arithmetic sequences consider a
variety of examples such as, but not limited to, the following:
•
Find the general term tn using tables, charts, graphs or an
equation.
•
Find the number of terms in a finite arithmetic sequence when
given the value of the nth term.
•
Find the common difference when provided with algebraic
expressions representing the value of terms for the sequence.
When provided with the general term, students sometimes have
difficulty differentiating between the term and the term number. When
they are asked to find the tenth term, for example, they may be unsure
whether to write t10 or tn = 10. It is important to reinforce that t10
represents the tenth term while tn = 10 represents the nth term having a
value of 10.
RF9.6 Describe the relationship
between arithmetic sequences and
linear functions.
24
Students will make the connection that arithmetic sequences, when
graphed, form linear graphs. Students may need to review graphing
linear functions using a table of values and/or the slope y-intercept
method. They should observe the following features when comparing an
arithmetic sequence to a linear function:
•
The slope is the common difference.
•
The y-intercept is the initial value minus the common difference.
•
The domain of a sequence is the set of natural numbers, while the
domain of a linear function, based on the context, can be the set
of real numbers.
•
The graph of an arithmetic sequence consists of discrete data,
while the graph of a linear function may be discrete or continuous.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
•
Ask students to create two arithmetic sequences and find the
general term tn. Ask them to determine the sum and then the
product of the two sequences. They should explain if these new
sequences created are arithmetic and justify their reasoning.
(RF9.2, RF9.4)
Pre-Calculus 11
1.1 Arithmetic Sequences
SB: pp.6-21
TR: pp.10-18
BLM: 1-3, 1-4
For each arithmetic sequence, ask students to write a formula for
tn and use it to find the indicated term.
(i) -4, 1, 6, 11, … , t13
(ii) 9, 1, -7, -15, …, t46
(RF9.4, RF9.5)
•
Consecutive terms of an arithmetic sequence are (5 + x), 8, and
(1 + 2x). Ask students to determine the value of x.
(RF9.4, RF9.5)
•
The diagram shows a pattern of positive integers in five columns.
If the pattern is continued, ask students to determine the columns
in which these numbers will appear.
(iii) 301
Col 1 Col 2 Col 3 Col 4 Col 5
1
2
3
5
4
(iv) 8725
6
(i) 49
(ii) 117
7
10
11
8
9
12
15
13
14
(RF9.4, RF9.5)
Interview
• Ask students to consider the following problem and explain if the
calculation is correct. They should justify their reasoning.
Paul must determine the 50th term of an arithmetic sequence
beginning with 5 and having a common difference of 9. He
calculates (50 x 9) + 5 = 455.
(RF9.4, RF9.5)
•
Zachary is having trouble remembering the formula
tn = t1 + (n - 1)d correctly. He thinks the formula should be
tn = t1 + nd. Ask students how they would explain to Zachary that
he should use (n - 1)d rather than nd in the formula.
(RF9.4, RF9.5)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
25
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicators:
The relationship bewtween an arithmetic sequence and a linear function
can be highlighted through a discussion such as the following:
n
tn
RF9.6 Continued
1
3
2
5
3
7
4
9
5
11
•
Find the general term tn . Compare this equation to a linear
equation y = mx + b. Are they similar?
•
Graph the sequence. What do you notice?
•
What is the relationship between the slope of the line and the
common difference?
•
Evaluate the first term minus the common difference of the
sequence. What does this value represent in the linear equation?
The relationship between arithmetic sequences and linear functions
could also be developed using a symbolic approach.
RF9.7 Derive a rule for
determining the sum of n terms of
an arithmetic series.
tn = t1 + (n - 1)d
tn = t1 + dn - d
tn = dn + t1 - d
tn = dn + (t1 - d) vs y = mx + b
where d = m and t1 - d = b
Students will write the sum of the first n terms for any arithmetic
sequence. They should work with a numerical example and based on
their observations, write a formula for the general case. The following
example provides an opportunity to review arithmetic sequences and
introduce arithmetic series.
•
on day 1, four more friends on day 2 and four more friends each
day for a total of 25 days. How many friends would you have in
total after the 25th day?
Use the following questions to guide students:
•
What is the arithmetic sequence?
•
What is the first term?
•
What is the rule for tn? What is the last term?
•
What do the sum of the terms in the sequence represent?
•
How many terms are we summing up?
•
How can the value of S25 be determined?
Although some students may just add up all the terms in the sequence,
they should realize this is not the most efficient method.
26
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
1.1 Arithmetic Sequences
Ask students to complete the following activity using technology:
n
tn
1
1
2
-1
3
-3
4
-5
5
-7
SB: pp.6-21
TR: pp.10-18
BLM: 1-3, 1-4
(i) Find a formula for the general term tn and record your
findings.
(ii) Graph the discrete data.
(iii) Draw a line through the points and record the equation.
(iv) Compare your results from steps (i) and (iii). How does the
common difference relate to the slope?
(v) Subtract d from the first term of the sequence. Compare this value to the y-intercept from the line of best fit. What do
you notice?
(RF9.6)
1.2 Arithmetic Series
SB: pp.22-31
TR: pp.19-25
BLM: 1-3, 1-5
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
27
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicator:
RF9.7 Continued
Introduce students to Gauss’s method of determining the sum of
the first 100 positive integers. Use this method to determine S25 in
the previous example. Ask students to write the series twice, once in
ascending order and the other in descending order. Then, sum the two
series.
S25 = 1 + 5 + 9 +.......... + 89 + 93 + 97
+S25 = 97 + 93 + 89 +.......... + 9 + 5 + 1
2S25 = 98 + 98 + 98 +.......... + 98 + 98 + 98
2S25 = 25(98)
25
S25 = (98)
2
Encourage students to try to determine where the different parts of
that formula come from. When they are trying to make a conjecture
about the general case, it is best to leave the original values rather than
simplifying the expression. Ask students what they think the values
25 and 98 represent. They should recognize that 25 is the number of
terms and 98 is the sum of 1 and 97, representing t1 and t25. Therefore,
S25 = 25
2 ( t 1 + t 25 ) .
The algebraic approach should also be explored to derive the formula for
the sum of the general arithmetic series, Sn = n2 ( t 1 + t n ). The sum of n
terms of an arithmetic series is represented by
Sn = t1 + t1 + d + .... t1 + (n - 2)d + t1 + (n - 1)d . Using Gauss’s method,
guide students through the following derivation:
Sn =
+
t1
Sn = t 1 + ( n − 1)d
t1 + d
+ ..... t 1 + ( n − 2)d
+ t 1 + ( n − 2)d
+ .....
2Sn = 2t 1 + ( n − 1)d + 2t 1 + ( n − 1)d
+ .....
2 Sn
t1 + d
2t 1 + ( n − 1)d
+ t 1 + ( n − 1)d
+
t1
+ 2t 1 + ( n − 1)d
= n(2t 1 + ( n − 1)d )
2Sn = n( t 1 + t 1 + ( n − 1)d )
tn
2Sn = n( t 1 + t n )
Sn =
28
n ( t 1 +t n )
2
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
Ask students to use Gauss’s method to determine the sum of the
first 12 positive integers.
(RF9.7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Pre-Calculus 11
1.2 Arithmetic Series
SB: pp.22-31
TR: pp.19-25
BLM: 1-3, 1-5
29
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF9 Continued ...
Achievement Indicators:
RF9.8 Determine t1, d, n, or Sn
in a problem that involves an
arithmetic series.
RF9.5 Continued
Students will evaluate and/or manipulate the given arithmetic series to
determine the first term, the common difference, the number of terms
in the sequence, the nth term or the sum. They should solve problems
that involve an arithmetic series within a context. Consider the
following example:
•
An auditorium has 20 seats on the first row, 24 seats on the second
row, 28 seats on the third row, and so on, and has 30 rows of seats.
How many seats are in the theatre?
To solve this problem, students need to ask themselves the following
questions:
30
•
•
What is the pattern? Does it represent an arithmetic sequence?
•
What information do we need to find?
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
• A round robin tournament is to be played by 3 teams. Ask students
to determine the total number of games required to guarantee that
each team plays the other exactly once. Repeat the same activity
for a tournament consisting of 4 teams. Ask students to use their
findings from above to find the total number of games for a
national round robin tournament with all ten provinces and three
territories participating. They should explain how their answer is
related to an arithmetic series.
Pre-Calculus 11
1.2 Arithmetic Series
SB: pp.22-31
TR: pp.19-25
BLM: 1-3, 1-5
(RF9.5, RF9.8)
•
Ask students to find the sum of 20 + 14 + 8 + … + (-70).
(RF9.8)
•
A theater has 60 seats in the first row, 68 seats in the second row,
76 seats in the third row, and so on in the same increasing pattern.
If the theater has 20 rows of seats, how many seats are in the
theater?
(RF9.5, RF9.8)
•
In an arithmetic series t1 = 6 and S9 = 108. Ask students to find
the common difference and the sum of the first 20 terms.
(RF9.8)
•
For three months in the summer (12 weeks), Job A pays \$325 per
month with a monthly raise of \$100. Job B pays \$50 per week
with a weekly raise of \$10. Ask students which is the better paying
job and why.
(RF9.5, RF9.8)
•
Ask students to show that 1 + 3 + 5 + … + (2n - 1) = n2.
(RF9.5, RF9.8)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
31
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF10 Analyze geometric
sequences and series to solve
problems.
[PS, R]
Students will continue to explore patterns with a focus shifting to
geometric sequences and series. They will explore examples of geometric
sequences having a “common ratio” pattern resulting in a specific
formula. Students will then differentiate between a geometric sequence
and a geometric series.
Achievement Indicators:
RF10.1 Identify assumptions
geometric sequence or series.
RF10.2 Provide and justify an
example of a geometric sequence.
RF10.3 Derive a rule for
determining the general term of a
geometric sequence.
RF10.4 Determine t1, r, n or
tn in a problem that involves a
geometric sequence.
Similar to an arithmetic sequence, students will continue to use the
same notation when working with the terms of a geometric sequence.
The symbol r will represent the common ratio.
Students should provide their own examples of geometric sequences and
illustrate the concept of a common ratio by determining the quotient
t
of pairs of consecutive terms (i.e., tn+1 ), demonstrating why any such
n
sequence is geometric.
It is important for students to think about what makes a sequence
geometric as opposed to arithmetic. They should differentiate between
a common difference and a common ratio. To develop the formula for a
geometric sequence, ask students to consider a sequence such as
1, 3, 9, 27, 81,...... Use the following questions to guide them through
the process:
•
What is the common ratio?
•
Can this sequence be rewitten to show the pattern of the first term
and the common ratio?
•
Can you predict the formula of a geometric sequence based on the
pattern 1, 1(3)1, 1(3)2, 1(3)3, 1(3)4.....?
•
Can you write the pattern in general terms for any first term and
common ratio ?
Students should notice that the terms of the geometric sequence with
first term t1 and a common ratio r can be written in general as
t1= t1, t2 = t1(r)1, t3 = t1(r)2, t4 = t1(r)3... They should recognize the formula
for an geometric sequence is tn = t1 (rn - 1).
Ask students to draw a sketch of the geometric sequence where the
x-axis represents the number of the term and the y-axis represents the
term. They should notice the terms of the geometric sequence are
not linear. Exposure to the graph of an exponential function may be
warranted here since the graph of a geometric sequence is an exponential
function with a domain belonging to the natural numbers. Students will
study exponential functions and their graphs in Mathematics 3200.
Provide numerical and algebraic examples for students to evaluate and/
or manipulate a given geometric sequence to determine the first term,
the common ratio, the number of terms in the finite sequence, or the
general rule.
32
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Provide students with pieces of square paper of varying side
lengths, n. Ask students to fold the paper in half repeatedly. After
each fold, the area of the top surface is measured and recorded.
Ask them what type of sequence is formed and to explain their
reasoning.
(RF10.2)
1.3 Geometric Sequences
SB: pp.32-45
TR: pp.26-33
BLM: 1-3, 1-6
Paper and Pencil
•
Ask students to determine the first three terms of each of the
following sequences: tn = 3n and tn = 3(3n - 1). Ask them what they
notice about the value of t1, t2 and t3 and the type of sequences
created. They should explain their reasoning.
(RF10.1, RF10.2)
•
The formula for a geometric sequence is given by tn = t1 (r n - 1). Ask
students to create examples of various geometric sequences in this
form and explain what happens if n = 1. Ask them what type of a
sequence they will continually produce if n = 1.
(RF10.4)
•
You are hired to complete a job for a month that offers two
different payment options. In plan A, the payment begins with \$3
on day 1, \$6 on day 2, \$9 on day 3, etc. In plan B, the payment
begins with \$0.01 on day 1, \$0.02 on day 2, \$0.04 on day 3, etc.
Ask students to determine which payment plan would be more
feasible. They should explain why an employer might or might
not offer a payment such as plan B.
(RF10.4)
Interview
•
A classmate is having trouble with the formula tn = t1 (r n - 1). He
thinks the formula should be tn = t1( r n) . Ask students how they
would explain to him that he should use r n - 1 rather than r n in the
formula.
(RF10.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
33
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF10 Continued ...
Achievement Indicators:
RF10.5 Derive a rule for
determining the sum of n terms of
a geometric series.
Students will first work with finite geometric sereis. With teacher
support, students will derive a nformula for the sum of the first n terms
t ( r −1)
for a geometric series, Sn = 1 r −1 . Encourage students to test the
formula using the following numerical example. This would give them
an opportunity to review geometric sequences while being introduced to
geometric series. Consider the example:
•
A student is constructing a family tree. She is hoping to trace back
through 10 generations to calculate the total number of ancestors
he has. Determine the total number of ancestors after the 10th
generation.
Students should recognize every person has 2 parents, 4 grandparents,
8 great-grandparents, and so on. Therefore, the number of ancestors
through ten generations is 2 + 4 + 8+ 16 + 32 + 64 + 128 + 256 + 512 +
1024. Let S represent the sum of this series. Ask students to multiply it
by the common ratio, 2.
S10=
2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024
2S10 =
4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048
2S10-1S10 = -2
+ 2048
Students calculate the sum of the first ten terms of the series to be 2046.
Going back through the generations, each person has 2046 ancestors.
RF10.6 Determine t1 , r, n or
Sn in a problem that involves a
geometric series.
RF10.7 Solve a problem that
involves a geometric sequence or
series.
Students will evaluate and/or manipulate the given geometric series to
determine the first term, the common ratio, the number of terms in the
sequence, the nth term or the finite sum. It is also important to solve
contextual problems that involves a geometric series. Consider a ball
that is dropped from a height of 38.28 m and bounces back up 60% of
the original height. Ask students to find the total distance travelled by
the ball by the time it hits the ground for the tenth time.
When working through the problems, remind students of the following
guidelines:
•
•
•
•
•
•
34
Draw a diagram if necessary.
Write out the terms of the sequence.
Determine if the problem is a sequence or series.
Determine if the problem is arithmetic or geometric.
Construct the formula using the given information in the
problem.
Solve the problem.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Interview
•
Ask students to respond to the following questions:
(i) Explain why there can be no infinite geometric series with a first term of 12 and a sum of 5.
(ii) Explain why the sum of an infinite geometric series is
positive if and only if the first term is positive.
(RF10.7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Resources/Notes
Authorized Resource
Pre-Calculus 11
1.4 Geometric Series
SB: pp.46-57
TR: pp.34-40
BLM: 1-3, 1-7
35
SEQUENCES AND SERIES
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF10 Continued ...
Achievement Indicators:
RF10.8 Explain why a geometric
series is convergent or divergent.
RF10.9 Generalize, using
inductive reasoning, a rule
for determining the sum of an
infinite geometric series.
To find the sum of an infinite geometric series, Sn = 1t−1r where
r ≠ 1, students will first need to differentiate between a convergent
and divergent series. To develop the idea of convergence or divergence,
expose students to a variety of geometric sequences where the common
ratio is different for each. Ask students to determine the partial sum
by adding the first two terms, the first three terms and so on. They
will check to see if the partial sum approaches a particular value as the
number of terms get larger.
1
2
Geometric Series
Partial Sum
2 + 4 + 8 +16+...
(r > 1)
S1 = 2, S2 = 6, S3 = 14
+ 41 + 81 + 161 + ....
S1 =
1
2
S2 =
,
3
4
,
S3 = 87
Convergent/
Divergent
diverges
converges to 1
(-1 < r < 1)
-1 + 1+ -1 + 1+ ....
(r = -1)
S1 = -1, S2 = 0, S3 = -1 S4 = 0
diverges
1 + 1 + 1 + 1 +...
(r = 1)
S1 = 1, S2 = 2, S3 = 3, S4 = 4
diverges
Once students determine the common ratio for each geometric series,
they should state their conclusions about convergence or divergence
based on the value of r. Ask students why the value of r cannot equal 0.
Students could use a concrete representation to help with their
understanding of an infinite geometric series. Using a string, 1 metre
long, students will cut it in half and place one of the halves stretched out
on a table. Using the remaining half, they will then cut it in half so that
1
they have two quarters. Place one of the quarters at the end of the 2
string on the table. They now have 43 of the string on the table. Halve
the remaining quarter string so they have two eighths and place one of
the eighths at the end of the string on the table. They now have 87 of the
string on the table, and so on. The goal is for students to recognize that
this infinite series represents a sequence
of partial sums. Students can
1
2
verify, using the formula Sn = 1− 1 , that the sum should equal 1.
2
36
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SEQUENCES AND SERIES
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to determine whether the following sequences
converge or diverge. They should explain your reasoning.
(i) 8, 4, 2, 1, 0.5, ....
1.5 Infinite Geometric Series
SB: pp.58-65
TR: pp.41-46
BLM: 1-3, 1-8
(ii) 3, 73 , 53 ,1, 13 ,.....
(iii) 5-3, 5-2, 5-1, 50, ...
(iv) t1 + d, t1 + 2d, t1 + 3d, t1 + 4d, ...
(RF10.8)
•
The midpoints of a square with sides 1 m long are joined to form
another square. Then the midpoints of the sides of the second
square are joined to form a third square. This process is continued
indefinitely to form an infinite set of smaller and smaller squares
converging on the center of the original square. Ask students to
determine the total length of the segments forming the sides of all
the squares.
(RF10.9)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
37
SEQUENCES AND SERIES
38
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Trigonometry
Suggested Time: 14 Hours
TRIGONOMETRY
Unit Overview
Focus and Context
In this unit, students will be exposed to angles in standard position
where 0 °≤ θ ≤ 360°. They will also be introduced to reference angles
and quadrantal angles. Students will determine the exact values of the
primary trigonometric ratios for angles between 0° and 360°.
Students will use the sine law and the cosine law to determine unknown
side lengths and angle measures in oblique triangles. They will be asked
to explain their thinking about which law to use.
Outcomes
Framework
GCO
Develop trigonometric reasoning.
SCO T1
Demonstrate an understanding
of angles in standard position
[0° to 360°].
SCO T2
Solve problems, using the three
primary trigonometric ratios
for angles from 0° to 360° in
standard position.
SCO T3
Solve problems, using the
cosine law and sine law,
including the ambiguous case.
40
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
Process Standards
Key
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Measurement
M4 Develop and apply the
primary trigonometric ratios
(sine, cosine, tangent) to solve
problems that involve right
triangles.
[C, CN, PS, R, T, V]
Mathematics 2200
Mathematics 3200
Trigonometry
Trigonometry
T1 Demonstrate an understanding T1 Demonstrate an understanding
of angles in standard position [0°
of angles in standard position,
to 360°].
[CN, ME, R, V]
[R, V]
T2 Develop and apply the equation
of the unit circle.
T2 Solve problems, using the three [CN, R, V]
primary trigonometric ratios for
angles from 0° to 360° in standard T3 Solve problems, using the six
trigonometric ratios for angles
position.
[C, ME, PS, R, T, V]
[ME, PS, R, T, V]
T3 Solve problems, using the
cosine law and sine law, including
the ambigous case.
[C, CN, PS, R, T]
T4 Graph and analyze the
trigonometric functions sine, cosine
and tangent to solve problems.
[CN, PS, T, V]
T5 Solve, algebraically and
graphically, first and seond degree
trigonometric equations with the
domain expressed in degrees and
[CN, PS, R, T, V]
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)
[R, T, V]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
41
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T1 Demonstrate an
understanding of angles in
standard position [0° to 360°].
[R, V]
In Mathematics 1201, students used the Pythagorean theorem and the
primary trigonometric ratios to find missing side lengths and angle
measures in right triangles. They also investigated and worked with
angles of elevation and depression (M4). In this unit, students will
evaluate the primary trigonometric ratios for angles from 0° to 360°
using the coordinate plane. They will also use the primary trigonometric
ratios to solve problems involving oblique triangles, using the Law of
Sines (including the ambiguous case) and the Law of Cosines.
Achievement Indicators:
T1.1 Sketch an angle in standard
position, given the measure of the
angle.
In this unit, students will solve problems involving angles with measures
between 0° and 360°. Negative rotational angles and co-terminal angles
are not part of this outcome. They will be addressed in Mathematics
3200.
which a given angle in standard
position terminates.
Introduce students to terminology such as the initial arm, terminal
arm, vertex and standard position. They will sketch angles in standard
position on the coordinate plane and identify the quadrant where the
terminal arm lies.
T1.3 Determine the reference
angle for an angle in standard
position.
An understanding of reference angles will be critical when students
compare the trigonometic ratio of an angle in standard position to the
trigonometric ratio of its reference angle. It will also be useful when
determining the exact trigonometric ratios for angles in standard
position that are multiples of 30°, 45°and 60°, and solving equations of
the form sin θ = a or cos θ = a.
T1.4 Explain, using examples,
how to determine the angles from
0° to 360° that have the same
reference angle as a given angle.
T1.5 Illustrate, using examples,
that any angle from 90° to 360°
is the reflection in the x-axis
and/or the y-axis of its reference
angle.
To help students visualize the relationship between an angle sketched
in standard position and its corresponding reference angle, provide
students, individually or in small groups, with a copy of a circle sketched
on grid paper. Give them a colored cut out of the right angle showing
the corresponding angle sketched in standard position.
60°
60°
Guide students using the following directions and questions:
•
42
Place the right triangle over the angle on the circle. What do you
notice about the two angles? They should verify the 60° angle is in
standard position.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Observation
•
Teachers create a model of the coordinate grid with an initial arm
and terminal arm that can be moved physically. Ask students to
demonstrate the placement of the terminal arm when given the
measure of an angle between 0° and 360°. They should explore
acute, obtuse, right, straight and reflex angles .
Pre-Calculus 11
(T1.1)
Paper and Pencil
• Provide students with angles sketched in standard position from
Quadrants II, III and IV, such as 150°, 210°, and 330°. Ask them
to determine the related reference angle and explain why the
reference angle for all three is the same.
(T1.1, T1.2, T1.3, T1.4)
•
2.1 Angles in Standard Position
Student Book (SB): pp. 74-87
Teacher Resource (TR): pp. 57-64
Blackline Master (BLM): 2-3, 2-4
Ask students to determine the reference angle for each of the
following angles:
(i) 100°
(ii) 250°
(iii) 315°
(T1.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
43
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T1 Continued ...
Achievement Indicators:
T1.3, T1.4, T1.5 Continued
•
Reflect the triangle across the y-axis into Quadrant II.
•
Determine the measure of the Quadrant II angle that is formed
with the positive x-axis and the hypotenuse (terminal arm) of this
triangle.
120°
60°
•
60°
Explain how the two angles are related.
Students should recognize that 120° = 180° - 60° or θR = 180° - θ°.
•
to reflect the original angle across the x-axis into Quadrant IV.
Encourage students to explain their observations.
Students should observe that in Quadrant III, the two angles have the
relationship 240° = 180° + 60° or θR= 180° + θ°. In Quadrant IV, the
angles are 300° = 360°- 60° or θR = 360° - θ°.
Now that students have explored all quadrants, this would be a good
opportunity to question them about the general properties of reference
angles. Students should notice that the reference angle is always positive
and measures between 0° and 90°.
T1.6 Draw an angle in standard
position given any point P(x, y)
on the terminal arm of the angle.
T1.7 Illustrate, using examples,
that the points P(x, y), P(-x, y),
P(-x, -y )and P(x, -y) are points
on the terminal sides of angles in
standard position that have the
same reference angle.
44
Once students have sketched an angle in standard position and
determined the quadrant in which it terminates, it is a natural extension
to draw an angle given a point on its terminal arm. Students will
determine which quadrant the terminal arm of the angle is located based
on the point P(x, y) given.
Students should be given an opportunity to graph a point and reflect it
on the y-axis, on the x-axis and across the origin. The various reflections
will illustrate how a point on the terminal arm of (1, 2), for example,
reflects to become (-1, 2) in quadrant II, (-1, -2) in quadrant III and
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Interview
Authorized Resource
•
Pre-Calculus 11
Provide students with examples representing the endpoint of
the terminal arm for angles sketched in standard position from
Quadrants II, III and IV (e.g., (-3, 4), (-3, -4) and (3, -4) ). Ask
them to explain why the related reference angle for all three is the
same.
(T1.1, T1.2, T1.3, T1.4)
2.1 Angles in Standard Position
SB: pp. 74-87
TR: pp. 57-64
BLM: 2-3, 2-4
Observation
•
Ask students to explore the endpoint of the terminal arm of θ if it
lies on the axis (ie., (1, 0)(0, 1), (-1, 0) and (0, -1)).
(T1.6, T1.7)
Performance
•
Ask students to create a “Four Door Book” foldable. On the front
of the foldable, they should write the quadrants. On the inside of
the foldable, students write the general “rule” for generating the
reference angle in each quadrant and provide an example.
Front
Inside
θR
θR = 180° - θ
θR = 180° + θ
The following site provides
information on how to create
different types of foldables.
θR = 360° - θ
http://mathnmind.com/PDF%20
Files/Pre-Algebra/dzf.pdf
(T1.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
45
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T2 Solve problems, using the
three primary trigonometric
ratios for angles from 0° to 360°
in standard position.
[C, ME, PS, R, T, V]
In Mathematics 1201, students applied the primary trigonometric ratios
to angles between 0° and 90°. They will now explore angles between 0°
and 360° using coordinates and reference angles.
If θ is an angle in standard position, and point P(x, y) is a point on the
terminal arm of angle θ, students will use the Pythagorean theorem
to determine the length of the hypotenuse r. The three primary
trigonometric ratios will be defined in terms of x, y and r.
sin θ =
opposite side
hypotenuse
=
y
r
cosθ =
hypotenuse
=
x
r
tan θ =
opposite side
=
y
x
Students will work with exact trigonometric ratios which may require
finding exact values for the hypotenuse. Special angles formed by
the intersection of the x-axis and y-axis will also be investigated
(i.e., quadrantal angles). Students will be expected to solve 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.
Achievement Indicators:
T2.1 Determine, using the
Pythagorean theorem, the distance
from the origin to a point
P(x, y) on the terminal arm of an
angle.
An activity such as the following provides an opportunity for students
to explore the trigonometric ratios given any point on the terminal arm
of angle θ. Begin with the point (-4, -5). Continue the activity to ensure
that endpoints from all quadrants are used. Use the following directions
and questions to guide students:
•
T2.2 Determine the value of
sin θ, cos θ, or tan θ given any
point P(x, y) on the terminal arm
of angle θ .
Plot the given point P(-4, -5). What quadrant does this point lie
in?
•
Construct the corresponding angle in standard position.
•
T2.3 Determine the sign of a
given trigonmetric ratio for a
given angle, without the use of
technology, and explain.
Drop a perpendicular to the x-axis creating a right triangle. Which
value represents the adjacent side? Which value represents the
opposite side?
•
How can you determine the exact length of the hypotenuse?
•
State the cosine, sine and tangent ratios associated with the angle.
•
What determines the sign of the ratio? Explain your reasoning.
T2.4 Sketch a diagram to
represent a problem.
If θ is an angle in standard position, students should be able to
summarize in which quadrant(s) the terminal arm of θ will lie if:
(i) tan θ > 0 or tan θ < 0
(ii) sin θ > 0 or sin θ < 0
(iii) cos θ > 0 or cos θ < 0
46
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to complete the following table and suggest strategies
for determining the sign (±) of the various ratios in each of the
quadrants. They should share and compare their findings with the
rest of the class.
cos θ
sin θ
tan θ
I
II
III
IV
(T2.4)
2.2 Trigonometric Ratios of Any
Angle
SB: pp. 88-99
TR: pp. 65-72
BLM: 2-3, 2-5
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
47
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T2 Continued...
Achievement Indicators:
T2.5 Determine, without the use
of technology, the value of sin θ,
cos θ, or tan θ given any point
P(x, y) on the terminal arm of
angle θ, where θ = 0 °, 90 °,
180 °, 270 °or 360 ° .
Trigonometric ratios for angles whose measurements are 0°, 90°,
180°, 270° or 360° will now be explored. Ask students to choose
an angle and then sketch it on the coordinate plane. Ask them the
following questions using the 90° angle as a guide:
T2.4 Continued
•
Where is the terminal arm?
•
Does this angle have a corresponding reference angle?
•
Can a reference triangle be drawn? Explain.
Students should recognize that every angle drawn in standard position
has a corresponding reference angle, except for quadrantal angles.
While the unit circle is not formally introduced until Mathematics
3200, this would be an effective strategy to use when discussing
quadrantal angles. The unit circle, where the radius equals one and the
centre is (0, 0), can help students understand facts about the sine and
cosine function.
P(0, 1)
P(x, y)
1
P(-1, 0)
θ
x
y
P(1, 0)
P(0, -1)
Encourage students to use the diagram to help them write their
definitions for sine, cosine and tangent for each quadrantal angle. Since
y
y
cosθ = xr = x1 = x and sin θ = r = 1 = y then P(x, y) = (cos θ, sin θ).
Challenge students to further explore this activity using the points
(3, 0), (0, 3), (-3, 0) and (0, -3). Ask them how the length of the
terminal arm affects the values of the cosine and sine ratios.
48
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Observation
Authorized Resource
•
Pre-Calculus 11
Ask students to plot the points (1,0), (0,1), (-1,0) and (0,-1), and
evaluate, using technology, the values of the sine and cosine ratio
for 0°, 90°, 180°, 270° or 360°. They should explain any patterns
or relationships that appear relating the x and y coordinate of each
endpoint to the value of each trigonometric ratio.
(T2.4, T2.5)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
2.2 Trigonometric Ratios of Any
Angle
SB: pp. 88-99
TR: pp. 65-72
BLM: 2-3, 2-5
49
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T2 Continued ...
Achievement Indicators:
T2.6 Solve, for all values of θ,
an equation of the form
sin θ = a or cos θ = a, where
−1 ≤ a ≤ 1, and an equation of
the form tan θ = a, where a is a
real number.
Previously, students were given an angle from 0° to 360° and asked to
find the trigonometric value. They will now work backwards to find the
missing angle when given an equation of the form sin θ = a or
cos θ = a, where −1 ≤ a ≤ 1, and the equation of the form tan θ = a,
where a is a real number. Use the following questions to guide them
through the solution of an equation such as cos θ = 0.25: •
Determine the quadrants which contain solutions. Are there any
restrictions?
•
Determine the reference angle for the given value of a = 0.25.
•
Determine the measure of the related angles in standard position
where 0° ≤ θ ≤ 360°.
Students will be exposed to problems where they will have to rearrange
the equation before they solve it. Ask them how they might solve an
equation such as -2cos θ - 1 = 0 by comparing it to the linear equation
-2x - 1 = 0. They should explain how the process is similar and how it is
different.
T2.7 Determine the exact value
of the sine, cosine or tangent of a
given angle with a reference angle
of 30°, 45°or 60°.
T2.4 Continued
Students have calculated the trigonometric values for angles between 0°
and 360°. They have also discovered relationships that exist between the
angle and its reference angle. For example, sin θ = - sin(θR) where
θ is an angle in the third quadrant. Reference angles of 30°, 45° or 60°
occur frequently in problems. It is equally important for students to
understand where the sine, cosine or tangent value of these angles come
from.
Geometric properties of right triangles containing 30°, 45° and 60° can
be used to obtain the trigonometric values. To demonstrate the exact
trigonometric value for θ = 45°, students can construct an isosceles
triangle with the smallest side measuring 1 unit. They can then use the
Pythagorean theorem to determine the length of the hypotenuse, 2 .
50
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
Ask students to solve the equations for 0° ≤ θ ≤ 360°.
Pre-Calculus 11
(i) sin θ = 13
3
(ii) cosθ = − 7
2.2 Trigonometric Ratios of Any
Angle
(iii) tan θ = 3
SB: pp. 88-99
TR: pp. 65-72
BLM: 2-3, 2-5
(iv) cos θ = 0.8660
(v) sin θ = 0.7071
(T2.6)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
51
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T2 Continued ...
Achievement Indicators:
T2.7, T2.4 Continued
Consider the following guiding questions:
•
What is the exact value of sin 45°?
•
What do you notice if you evaluate sin 45° with a calculator?
•
What do you notice about the values of sin 45° and cos 45°. Can
you explain why they have the same value?
•
What is the value of tan 45°? Can you determine the value of
tan 90° using the right triangle? Explain your reasoning.
•
What is the same and what is different about sin 45° and sin 225°?
•
What is the same and what is different about cos 45° and cos
135°?
•
What is the same and what is different about tan 45° and tan
135°?
When determining the exact value of the trigonometric ratios of a given
angle, students will sometimes result in a value where the denominator
contains a radical. Consider the example, cos 45°= 12 . Students will not
be expected to rationalize the denominator since the Radicals Unit is not
done until later in this course.
A similar strategy to demonstrate the exact trigonometric values for
θ = 30° or θ = 60° is for students to construct an equilateral triangle
with a side length of 2. Drawing an altitude from vertex A, students
can then use the Pythagorean theorem to determine the length of the
altitude, 3 .
A
30°
C
52
2
60°
1
B
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Divide the class into two groups. Individual students in one group
will be given a card containing trigonometric ratios of special
angles (i.e., sin 60°, cos 120°, etc). In the other group, students
should be given associated exact values (ie., 23 , − 23 , etc). Ask
students to find a partner to form a matching pair.
(T2.7, T2.4)
Journal
•
When evaluating sin 45°, ask students to explain why
an exact value while 0.707 is approximate.
Interview
•
1
2
2.2 Trigonometric Ratios of Any
Angle
SB: pp. 88-99
TR: pp. 65-72
BLM: 2-3, 2-5
is called
(T2.7, T2.4)
Ask students to respond to the following questions when working
with a 30°-60°-90° triangle.
(i) Calculate the exact value of sin 30° and cos 30°.
(ii) Calculate the exact value of sin 60° and cos 60°.
(iii) Determine sin 30°, cos 30°, sin 60° and cos 60° using a
calculator. What do you notice?
(iv) Which is greater, sin 60° or sin 30°? Why?
Note: The ratios are often left in
irrational form in the resource.
(i.e., sin 45° = 12 )
(v) What is the same and what is different about sin 30° and
sin 150°?
(vi) What is the same and what is different about cos 30° and
cos 150°?
(vii) What do you notice when you compare the exact value of
tan 30° and tan 60° using the right triangle?
(viii)What is the same and what is different about tan 30° and
tan 330°?
(T2.7, T2.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
53
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T2 Continued ...
Achievement Indicators:
T2.8 Describe patterns in and
among the values of the sine,
cosine and tangent ratios for
angles from to 0° to 360°.
T2.9 Solve a contextual problem,
using trigonometric ratios.
T2.4 Continued
Encourage students to use mental math to determine the trigonometric
function values whenever the terminal side makes a 30°, 45° and 60°
angle with the x-axis.
Students should be given an opportunity to explore the patterns in the
sine, cosine and tangent ratios. They could use graphing technology and
a table to record the points. Remind students to choose an appropriate
increment for θ when graphing y = sin θ, y = cos θ and y = tan θ from
0° to 360°.
Students will be expected to solve problems using the trigonometric
ratios. Encourage students to draw a sketch of a diagram to help them
gain a visual understanding of the problem. Consider an example such
as the following:
•
54
The arm of a crane used for lifting very heavy objects can move so
that it has a minimum angle of inclination of 30° and a maximum
of 60°. Use exact values to find an expression for the change in the
vertical displacement of the end of the arm, in terms of the length
of the arm, a.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Observation
•
Observe students graphing y = sin θ, y = cos θ and y = tan θ from
(i) What is the maximum value of sin θ, cos θ and tan θ? the
minimum value?
(ii) Where does the maximum value occur? the minimum value?
(iii) What are the x-intercepts and y-intercepts?
Resources/Notes
Authorized Resource
Pre-Calculus 11
2.2 Trigonometric Ratios of Any
Angle
SB: pp. 88-99
TR: pp. 65-72
BLM: 2-3, 2-5
(iv) For what values of θ is the function positive? For what values
is it negative?
(v) What comparisons can you make between the sine and
cosine function?
(T2.8)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
55
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Solve problems, using the
cosine law and the sine law,
including the ambiguous case.
[C, CN, PS, R, T]
In Mathematics 1201, students used the three primary trigonometric
ratios to determine the side lengths and angle measures in right triangles
(M4). Solving problems involving trigonometry is rarely limited to right
triangles. In this unit, students will derive the Law of Sines and the Law
of Cosines and utilize them in a number of problem-solving situations.
Achievement Indicators:
T3.1 Sketch a diagram to
represent a problem that involves
a triangle without a right angle.
Students have been exposed to right-triangle trigonometry to solve
problems involving right triangles. They will now solve oblique
triangles. Encourage students to draw a diagram to help them gain a
visual understanding of the problem. Consider the following example:
A
10
T3.2 Solve, using primary
trigonometric ratios, a triangle
that is not a right triangle.
66°
44°
B
x
C
Ask students if this triangle can be divided into two right triangles and
what strategies can be applied to find the indicated side length. They
should recognize that this requires a multi-step solution. A strategy must
be developed before a solution is attempted. Drawing an altitude from
vertex A, students can use the primary trigonometric ratios and the
Pythagorean theorem to solve for the unknown value.
A
10
y
cos44° = 10 ( y = 7.2)
10 2 = 7.2 2 + h 2 ( h = 6.9)
h
66°
44°
B
y
w
x
C
tan 66° = 6.9
( w = 3.1)
w
x = y+w
Students will be introduced to methods that are more efficient when
solving an oblique triangle, namely the sine law and the cosine law.
56
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to determine the length of AC in the following
diagram:
2.3 The Sine Law
SB: pp. 100-113
TR: pp. 73-80
BLM: 2-3, 2-7
(T3.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
57
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Continued ...
Achievement Indicator:
T3.3 Explain the steps in a given
proof of the sine law and cosine
law.
Students will prove the sine law and then use the law to solve triangles.
The Law of Sines can be derived using the area formula of a right
triangle. Consider the following diagram:
B
c
h
C
A
b
Triangle ABC is not a right triangle. Therefore, students will draw
an altitude from vertex B. The area formula for a right triangle is
Area = 21 (base)(height). Ask students to write an expression for the
height (h) using the sine ratio. Since sin A = hc , then h = c(sinA).
Therefore, Area = 21 bc sin A . Ask students to repeat this procedure by
drawing an altitude from the other vertices and writing an equation for
the area of triangle ABC.
B
B
a
c
c
A
a
h
h
C
A
b
C
They should conclude that Area = 21 ac sin B and Area = 21 ab sin C
Promote student discussion as to why the three area expressions are
equivalent. Since 21 bc sin A = 21 ac sin B = 21 ab sin C , students can divide
each expression by 21 abc to obtain the sine law.
sin A sin B sin C
=
=
a
b
c
In other words, the sine law is a proportion that compares the ratio of
each side of a triangle to its included angle.
58
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Provide students with a triangle and have them measure the side
lengths and angles using a ruler and protractor.
2.3 The Sine Law
A
SB: pp. 100-113
TR: pp. 73-80
BLM: 2-3, 2-7
B
C
Measure
Measure
Calculate
∠A
side a
sin A
a
∠B
side b
sin B
b
∠C
side c
sinC
c
(i) What can you conclude regarding the ratios calculated
above?
(ii) Would your conclusion be valid if you were to use the
reciprocal of the ratios?
(T3.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
59
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Continued ...
Achievement Indicator:
T3.4 Sketch a diagram and solve
a problem, using the sine law.
Students will apply the sine law to determine unknown lengths and
angle measures in triangles. Encourage them to draw diagrams with both
the given and unknown information marked.
Ask students what information is needed to solve problems using the
sine law. The Law of Sines involves a ratio of the sine of an angle to the
length of its opposite side. Students should recognize that it will not
work if no angle of the triangle is known or if one angle and its opposite
side is not given.
When working with the sine law, students sometimes incorrectly
identify side and opposite angle pairs. To avoid this error, encourage
them to use arrows on the diagram when identifying the angle and its
opposite side. They could also encounter problems when they cross
inA = sin 30° , for example, students
multiply the ratios. When solving s 12
4
may incorrectly write 4sinA = sin 360°. Teachers should emphasize
the use of brackets and write 4(sinA)= 12(sin 30°). Another common
student error occurs when students try to solve a triangle given two
angles and an included side and mistakenly think there is not enough
information to use the sine law. Consider an example such as the
following:
C
b
A
43°
71°
B
c = 12
Students can use the property that the sum of the angles in a triangle
is 180°. Therefore, the measure of ∠ C is 66°. They can then proceed
to use the sine law to find the length of side AC. Encourage students
to check the reasonableness of their answer. For example, since ∠ C is
a little smaller than ∠ A, we expect the length of side AB to be a little
shorter than the length of side CB. Students should also consider asking
questions such as: Is the shortest side opposite the smallest angle? Is the
longest side opposite the largest angle?
60
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Observation
Authorized Resource
•
Pre-Calculus 11
Provide students with several practice problems using the sine law.
As teachers observe students working through the problems, ask
them the following questions:
(i) What is the unknown? Is it an angle or a side?
(ii) How can you isolate the unknown?
2.3 The Sine Law
SB: pp. 100-113
TR: pp. 73-80
BLM: 2-3, 2-7
(iii) How can you complete the calculations?
(iv) How do you know whether to determine the sine or the
inverse of the sine?
(T3.4)
Interview
•
A surveyor is located on one side of a river that is impossible to
cross and only has a 100 m measuring tape and a sextant (used to
measure angles) in his possession. Ask students to explain how the
surveyor could use only these two tools and the Law of Sines to
find the distance from point A to point C.
(T3.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
61
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Continued ...
Achievement Indicator:
T3.5 Describe and explain
situations in which a problem
may have no solution, one
solution or two solutions.
When using the sine law, students will be exposed to the following
situations:
(i) Provided with two angles and an included side (ASA)
(ii) Provided with two angles and a non-included side (AAS)
(iii) Provided with two sides and a non-included angle (SSA) – the
ambiguous case
The ambiguous case may cause difficulty for some students. Provided
with two sides and an angle opposite one of those sides, students
generally expect that one triangle will result. A numerical example,
outlined below, could help students further investigate the Law of Sines
and observe special situations where one, two or no triangle is in fact
possible.
Students are familar with the 30°-60°-90° triangle.
C
2
a=1
30°
A
B
Using C as the center of the circle, ask students to construct ∠ A = 30°
and length AC = 10. Students should first notice that one triangle is
possible where ∠ B =90° and a = 5. They can verify this using the Law
of Sines.
sin 30°
5
C
10
a=5
30°
A
B
= sin10B
sin B =
10(sin 30° )
5
sin B = 1
∠B = 90°
Students should then consider the case where a < 5. Using geometry,
draw an arc length a = 4.
sin 30°
4
= sin10B
sin B =
10(sin 30° )
4
sin B = 1.25
Students should notice that no triangle is physically possible. If they do
not attempt this approach, they may use the Law of Sines to try and
solve this problem. This is an opportunity for students to review the
maximum and minimum values of y = sin θ. Students should recognize
that no triangle exists since the value of sin B > 1.
62
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to solve each of the following triangles using the Law
of Sines (if possible) indicating which results in one triangle, two
triangles or no triangle.
(i)
∠ A = 45°, ∠ B = 60°, and a = 14
(ii) ∠ B = 40°, a = 12, b = 6
2.3 The Sine Law
SB: pp. 100-113
TR: pp. 73-80
BLM: 2-3, 2-6, 2-7
(iii) ∠ A = 25°, a = 2, b = 3
(T3.5)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
63
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Continued ...
Achievement Indicators:
T3.5 Continued
T3.3 Continued
Ask students to repeat this activity where a > 5. Using an arc length of
a = 8, for example, the Law of Sines will yield two possible values for
angle B (i.e., ∠B = 53.1° or ∠B = 126.9° ) and therefore, two possible
triangles exist.
Students will prove and use the cosine law to solve triangles. Ask
students to consider the triangle ABC with side lengths a, b and c . Draw
an altitude, h, from vertex C and let D be the intersection of AB and
the altitude as shown in the figure below. If x is the length of AD, then
students should recognize that BD = c - x. Guide students through the
following process:
C
b
A
x
a
h
D
c-x
B
c
•
Use the Pythagorean theorem in ∆BCD : a2 = h2 + (c - x)2
•
Expand the binomial: a2 = h2 + c2 - 2cx + x2
•
Apply the Pythagorean theorem in ∆ADC (x2 + h2 = b2). What do
the expanded binomial yield if b2 - x2 is substituted for h2. They
should recognize the equation will result in a2 = b2 + c2- 2cx.
•
What primary trigonometric ratio can be used to determine
the altitude? They should recognize that in ∆ACD, cos A = xb ,
resulting in x = bcosA.
•
Substitute x = bcosA into the equation a2 = b2 + c2- 2cx. What do
you notice? (a2 = b2 + c2 - 2cbcosA)
Students can then express the formula in different forms to find the
lengths of the other sides of the triangles.
64
b2 = a2 +c2 -2accosB
c2 = a2 +b2 -2abcosC
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
•
Pre-Calculus 11
Ask students to explain how the Law of Cosines validates the
Pythagorean theorem if the included angle is 90°.
(T3.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
2.4 The Cosine Law
SB: pp. 114-125
TR: pp. 81-87
BLM: 2-3, 2-8
65
TRIGONOMETRY
Trigonometry
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
T3 Continued ...
Achievement Indicator:
T3.6 Sketch a diagram and solve
a problem, using the cosine law.
The cosine law can be used to determine an unknown side or angle
measure in a triangle. Continue to encourage students to draw diagrams
with both the given and unknown information marked when solving
problems. Ask them what information is needed to use the cosine law.
They should consider why the cosine law is the only option to find the
unknown angle if three sides are known or if two sides and the included
angle are known.
When three sides of a triangle are known, students will use the cosine
law to find one of the angles. Some students may rearrange the equation
to solve for a particular angle. Others may substitute the unknown
values into the cosine law and then rearrange the equation to find the
angle. It is important for them to recognize they have a choice when
trying to find the second angle. They can either use the cosine law or the
sine law. Students should notice the third angle can then be determined
using the sum of the angles in a triangle.
When solving triangles, encourage students to consider the following
questions:
•
What is the given information?
•
What am I trying to solve for?
•
With the given information, should I use the sine law or the
cosine law? Is there a choice?
• Which form of the cosine law do I use to solve for an unknown
side? Which form do I use to solve for an unknown angle?
If students know two sides and a non-included angle, they can use the
cosine law in conjunction with the sine law to find the other side. As
an alternative, they could apply the sine law twice. Students have to be
exposed to numerous examples to find the method that works best for
them.
When working with the cosine law, students sometimes incorrectly
apply the order of operations. When asked to simplify
a2 = 365 - 360cos70°, for example, they often write a2 = 5cos70°. To
avoid this error, teachers should emphasize that multiplication is to be
completed before subtraction.
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ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
TRIGONOMETRY
General Outcome: Develop trigonometric reasoning.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
In the activity, Four Corners, students have to think about which
method they would use to solve a triangle. Post four signs, one in
each corner labelled sine law, cosine law, Pythagorean theorem,
trigonometric ratios. Provide each student with one triangle.
Instruct the students to make a decision as to which method they
would use to find the missing angle or side and to stand in the
corner where it is labelled. Once students are all placed, ask them
to discuss why their triangle(s) would be best solved using that
particular method. Sample triangles are given below:
B
a
a
A
A
66°
C
C
A
26 cm
B
22 cm
31 cm
x°
88°
32°
33 cm
21 cm
B
72°
x°
C
B
A
C
www.k12pl.nl.ca/seniorhigh/
introduction/math2201/
classroomclips.html
35 cm
C
92 cm
In the Trigonometry clip, students
participate in the activity Four
Corners. They discuss the strategy
to solve for the missing side
lengths and angles of a given
triangle.
22 cm
14 cm
A
B
2.4 The Cosine Law
SB: pp. 114-125
TR: pp. 81-87
BLM: 2-3, 2-8
65 cm
c
67 cm
B
A
x°
C
42 cm
(T3.4, T3.6)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
67
TRIGONOMETRY
68
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Suggested Time: 12 Hours
Unit Overview
Focus and Context
In this unit, students will be introduced to standard form and vertex
form of a quadratic function. They will investigate the characteristics
of a quadratic function using features such as x-and y-intercepts, vertex,
axis of symmetry, domain and range.
Students will solve problems involving situations that can be modelled
Outcomes
Framework
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF3
y = a(x - p)2 + q and determine the:
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x-and y-intercepts.
SCO 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.
70
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Process Standards
Key
[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
Algebra and Number
Relations and Functions
Relations and Functions
AN4 Demonstrate an
understanding of the
multiplication of polynomial
expressions (limited to
monomials, binomials and
trinomials) concretely, pictorially
and symbolically.
the form y = a(x - p)2 + q and
determine the:
[CN, R, V]
AN5 Demonstrate an
understanding of common
factors and trinomial factoring,
concretely, pictorially and
symbolically.
[C, CN, R, V]
•
vertex
•
domain and range
•
direction of opening
•
axis of symmetry
•
x- and y-intercepts.
[CN, R,T,V]
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
RF11 Demonstrate an
understanding of factoring
polynomials of degree greater than 2
(limited to polynomials of degree
≤ 5 with integral coefficients).
[C, CN, ME]
RF12 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]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
71
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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.
[CN, R,T, V]
Achievement Indicator:
RF3.1 Explain why a function
given in the form y = a(x - p)2 + q
In Mathematics 1201, students were introduced to functional notation
through work with linear functions. They were introduced to the terms
“relation” and “function” and determined if a relation was a function
(RF2). They also determined the domain value of a linear function
given a range value and vice versa (RF8). The domain and range of a
graph was written using interval notation or set notation (RF1). In this
unit, students will be introduced to quadratic functions. They will first
examine quadratic functions expressed in vertex form, y = a(x - p)2 + q,
and determine the x-and y-intercepts, vertex, axis of symmetry, direction
of opening, and domain and range. They will also sketch graphs using
the characteristics above as well as transformations. Using inductive
reasoning, students will study the effect of changing the parameters a, p
and q on the shape of the graph and on the number of x-intercepts.
Before students are exposed to the vertex form of a quadratic, they need
to become familar with the shape of a quadratic function and how to
new to students. This will be their first exposure to functions that are
non-linear.
Students should have an opportunity to investigate what makes a
quadratic function. Ask studets to turn to a partner to multiply two
linear equations or square a binomial of the form ax + b. Consider the
examples y = (x + 1)(x - 4) and y = (3x - 2)2. Ask students what they
notice in terms of the degree of the polynomial and if this is similar to
expanding functions of the form y = a(x - p)2 + q.
Projectile motion can be used to explain the path of a baseball or a skier
in flight. To help students visualize the motion of a projectile, toss a ball
to a student. Ask students to describe the path of the ball to a partner
and sketch the path of the height of the ball over time (the independent
axis represents time and the dependent axis represents height of the
ball). Encourage them to share their graphs with other students. Ask
them to think of other examples that might fit the diagrams of parabolas
that open upward or downward. They should explain why they think
Characteristics of the resulting parabola, such as the vertex and axis of
symmetry, should be discussed. It is important for them to recognize
that each point on a parabola has a corresponding point on its mirror
image, resulting in a parabolic shape.
Students may conclude that the graph resembles a U-shape. It is
important to note, however, that obtaining a U-shaped graph does
not guarantee the function is quadratic. Other functions may produce
a similarly shaped graph. The graph of y = x4, for example, may be
misinterpreted as being parabolic if not examined closely.
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ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
Provide the following table with a set of statements. Ask students
to describe the reasoning they used to decide whether each
statement is true or false.
Polynomial
Classification True or False Explain/Justify
Function
y = 5(x + 3)
Linear
2
y = 5(x + 3)
2
y = 5 (x + 3)
y = 5x(x + 3)
Linear
y = (5x + 1)(x + 3) Quadratic
y = 5(x + 3)2 + 2
(RF3.1)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Pre-Calculus 11
Functions in Vertex Form
Student Book (SB): pp.142-162
Teacher Resource (TR): pp.103112
BlackLine Master (BLM): 3-3, 3-4
73
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF3 Continued ...
Achievement Indicators:
RF3.2 Compare the graphs of a
set of functions of the form
y = ax2 to the graph of y = x2,
and generalize, using inductive
reasoning, a rule about the effect
of a.
As students begin varying the parameters of a quadratic function, they
will first explore quadratic functions in vertex form. The characteristics
of the quadratic function, y = a(x - p)2 + q, should be developed through
an investigation where the parameters a, p and q are manipulated
individually. Technology, such as graphing calculators, FX Draw,
Graphmatica, Geometer’s Sketchpad, Winplot, Geogebra, or other
suitable graphing software, should be used, as students’ graphing abilities
for quadratics would be limited to the use of a table of values at this
point.
Students will first investigate the effect of changing the value of a by
comparing quadratic functions y = x2 and y = ax2. As they compare the
graphs of y = 2x2 and y = -2x2 to the graph of y = x2, use prompts such as
the following to promote student discussion:
•
What happens to the direction of opening of the quadratic if a < 0 and a > 0?
• Is the shape of the parabola affected by the parameter a?
•
Are some graphs wider or narrower when compared to the graph
of y = x2 ?
•
What is the impact on the graph if a = 0?
•
What effect does varying a have on the vertex?
Encourage students to pay particular attention to the points and how
they change. For example, the point (2,4) on the graph of y = x2 changes
to the point (2,8) on the graph of y = 2x2.
2
Similarly, students can compare the graphs of y = 21 x and y = − 21 x 2 to
the graph of y = x2.
RF3.3 Compare the graphs of a
set of functions of the form
y = (x - p)2 to the graph of y = x2,
and generalize, using inductive
reasoning, a rule about the effect
of p.
RF3.4 Compare the graphs of a
set of functions of the form
y = x2+ q to the graph of y = x2,
and generalize, using inductive
reasoning, a rule about the effect
of q.
74
Use graphing technologies to examine the effects of manipulating the
values of p. Students will investigate the effect of changing the value of
p by comparing quadratic functions y = x2 and y = (x - p)2. They could
describe how the graphs of y = (x - 3)2 and y = (x + 3)2 compare to the
graph of y = x2. Ask questions such as:
•
How does each graph change when compared to y = x2?
•
What effect does varying p have on the vertex?
Similarly, students will examine the effects of manipulating the value
of q. They will investigate the effect of changing the value of q by
comparing quadratic functions y = x2 and y = x2 + q. They could
describe how the graph of y = x2 + 2 and y = x2 - 2 compares to the graph
of y = x2. Ask students what effect does varying q have on the vertex.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
• Pre-Calculus 11
Ask students to draw a tile from a bag containing various values
for the parameter a. They will then proceed to the board or
on the wall of a classroom to complete a table, such as the one
shown below, indicating the effect of a on the graph of y = x2.
Ask students to discuss the results. Be sure to include positive and
negative values for a, as well as integers and rational numbers.
Value of a
Function
a=2
y = 2x2
a = − 12
y = − 21 x 2
Opening Up
or Down
up
down
Functions in Vertex Form
SB: pp.142-162
TR: pp.103-112
BLM: 3-3, 3-4
Narrower or
Wider
narrower
wider
(RF3.2)
Paper and Pencil
•
Ask students to complete the webbing to describe the effects of
the parameters a, p and q on the quadratic function
y = a(x - p)2 + q.
y = a ( x − p) + q
2
a
p
q
(RF3.2, RF3.3, RF3.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
75
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF3 Continued ...
Achievement Indicators:
RF3.5 Determine the coordinates
of the vertex for a quadratic
function of the form
y = a(x - p)2 + q, and verify with
or without technology.
RF3.6 Generalize, using
inductive reasoning, a rule for
determining the coordinates of the
the form y = a(x - p)2 + q.
In Grade 9, students graphed linear relations using a table of values
(9PR2). They also investigated line symmetry (9SS5). In Mathematics
1201, students used technology and/or a table of values to graph linear
functions (RF6). They will now extend these concepts to determine the
vertex and axis of symmetry of the graph of a quadratic function. When
graphing the quadratic function y = 3(x + 2)2 - 1, for example, students
can create the following table of values by hand or through the use of
graphing technology.
x
y
-4
11
-3
2
-2
-1
-1
2
0
11
Promote student discussion by asking questions such as the following:
•
What is the connection between the axis of symmetry of the graph and the vertex?
• What is the equation of the axis of symmetry?
•
What is the connection between the maximum or minimum point on the graph and the vertex?
•
How can the vertex be obtained directly from the vertex form of
The goal is for students to recognize that in a table of values, the vertex
will be the point with the unique y-value. It is the point at which the
graph of the quadratic changes direction, and it is the maximum or
minimum point on the graph. Whether students use a graph or a table
of values, they should be able to identify the symmetry of the graph
since the points on the parabola that share the same y-coordinate are
equidistant from the vertex. Students should also realize that the axis
of symmetry of a parabola is the vertical line of symmetry that passes
through the vertex and has equation x = p.
76
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Observation
Authorized Resource
•
Ask students to graph y = 3(x - 2)2 + 1 using a table of values
or graphing technology. They should compare the given graph
to each of the following graphs and discuss the similarities and
differences between the two.
Pre-Calculus 11
(i) y = -3(x - 2)2 + 1
SB: pp.142-162
TR: pp.103-112
BLM: 3-3, 3-4
(ii) y = 3(x - 3)2+ 1
2
1
(iii) y = − 3 ( x − 2) + 4
Functions in Vertex Form
2
1
(iv) y = 3 ( x − 2) + 1
(v)
y = 21 ( x + 3)2 + 2
(vi) y = -2x2 + 1
(vii) y = 3x2 + 1
(RF3.1, RF3.2, RF3.3, RF3.4, RF3.5)
Journal
•
Your friend has missed the class on determining the vertex for any
quadratic function of the form y = a(x - p)2 + q and asks you to
explain how this is done. Ask students to write a paragraph, with
examples, explaining how this is done.
(RF3.6)
The following site provides a
simulation that models changes in
a, p, and q. (Note: the variables h
and k are used in place of p and q).
http://www.mathopenref.com
Explorer)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
77
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF3 Continued ...
Achievement Indicator:
RF3.7 Sketch the graph of
y = a(x - p)2 + q, using
transformations, and identify
the vertex, domain and range,
direction of opening, axis of
symmetry and x- and y-intercepts.
Students previously investigated the effects on the graph of changing
the values of a, p, and q in the vertex form of a quadratic function. They
also found the vertex directly from the quadratic function.
In Mathematics 1201, students graphed linear relations by plotting the
x-and y-intercepts and found the domain and range of various relations
(RF8 and RF1). They should recognize that all non-contextual quadratic
functions have a domain of {x| x Î Â}, whereas the range depends on
their observations as they graph and analyze quadratic functions. They
should consider the following:
•
Why is the domain the set of all real numbers when only some
points are plotted from the table of values?
•
How is the range related to the direction of opening?
Students should realize that a negative value of a will indicate that the
range is less than or equal to q while a positive value of a will indicate
that the range is greater than or equal to q.
Students can consolidate their learning by sketching the graph of
y = a(x - p)2 + q using transformations for various values of a, p, and q.
They can apply the change in width using the value of a by selecting the
vertex and two other points on the graph of y = x2. They can then use
the values of p and q to translate the graph. Consider the function
y = 3(x - 5)2 + 1. The a value, 3, results in a narrower graph. The points
(0, 0), (-1, 1) and (1, 1) on the graph y = x2 will transform to (0, 0),
(-1, 3), (1, 3) on the graph of y = 3x2. Students will then translate
the graph using the horizontal translation (5 units) and the vertical
translation (1 unit). The points transform to (5, 1), (4, 4), and (6, 4).
Students should note the vertex and symmetry of the graph and other
features such as the domain, range and intercepts.
2
y
y=x
2
2
y = 3x
y = 3(x – 5) + 1
8
6
4
2
-2
2
4
6
8
x
-2
78
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Observation
• Students can work in pairs to construct the graphs of a number
of quadratic functions of the form y = a(x - p)2 + q, using
transformations for various values of a, p and q. Each group is
provided with a handout containing a grid and the function. Place
the handout inside a sheet protector so each group can graph
a different function using an erasable marker. Ask questions to
the group related to the characteristics of the quadratic function
(vertex, direction of opening, x-and y-intercepts, domain and
range). When completed, ask students to erase the function and
pass it along to another group. Repeat the activity, providing
students the opportunity to construct a number of graphs.
y
10
8
6
4
2
•
2
4
6
Pre-Calculus 11
Functions in Vertex Form
SB: pp.142-162
TR: pp.103-112
BLM: 3-3, 3-4
www.k12pl.nl.ca/seniorhigh/
introduction/math2200/
classroomclips.html
2
y = 2( x – 1) + 3
-10 -8 -6 -4 -2-2
-4
-6
-8
-10
Authorized Resource
x
8 10
(RF3.7)
The Frayer Model clip related
Functions demonstrates
students finding the different
function and completing a puzzle.
Students can work in pairs to complete the following quadratic
puzzle investigating the characteristics and graphs of various
quadratic functions of the form y = a(x - p)2 + q. They should
work with 20 puzzle pieces (4 complete puzzles consisting of a
function and four related characteristics) to correctly match the
characteristics with each function. A sample is shown below.
Vertex
(2,4)
{ y | y ≤ 4}
2
y = -3( x – 2) + 4
y
10
8
Axis of
Symmetry
x=2
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8
10
x
-4
-6
-8
-10
(RF3.5, RF3.6, RF3.7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
79
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF3 Continued...
Achievement Indicators:
RF3.8 Explain, using examples,
how the values of a and q may
be used to determine whether a
or two x-intercepts.
in the form
y = a(x - p)2 + q for a given
graph or a set of characteristics of
a graph.
Teachers could use a variety of examples to demonstrate the six different
combinations of a and q shown in the table below. The use of graphing
technologies or simulation software is recommended here.
Value of a
positive
positive
positive
negative
negative
negative
Value of q
positive
negative
0
0
positive
negative
Number of x-intercepts
0
2
1
1
2
0
Given a graph of a quadratic function, students will determine the
quadratic function in the form y = a(x - p)2 + q. They can determine the
vertex directly from the graph and hence, the values of p and q. When
determining the value of a, students can substitute a point
(x, y) that is on the parabola into the quadratic function and solve for a.
Sometimes a description of the characteristics of the graph are given
rather than the graph. The vertex is given, as a maximum or minimum
value, and at least one other point. Using this information, students will
determine the quadratic function in vertex form.
80
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
Ask students to participate in the following card game. Place three
headings labelled “No x-intercepts”, “One x-intercept”, and
“Two x-intercepts” on the classroom wall. Make a set of cards for
the values of a and q. For example,
a = 3, -8, 2.67,
3
4
q = 0, -10, 11,
−24
7
Pre-Calculus 11
Functions in Vertex Form
SB: pp.142-162
TR: pp.103-112
BLM: 3-3, 3-4
One student will select a card from the “a” deck and a second
student will select a card from the “q” deck. The two students
then collaborate to determine under which heading the two
numbers should be placed. At the end of the activity, they should
determine if any of the pairs of cards are misplaced.
Paper and Pencil
•
function from the given table of values.
The Describe and Draw clip
related to Characteristics of
function by listening to the
information presented by their
function given the following information.
partner.
(i) Range is {y / y ≤ 3, y∈ \ } and the x-intercepts are -2 and 4.
x
y
•
www.k12pl.nl.ca/seniorhigh/
introduction/math2201/
classroomclips.html
-6
1
-5
3
-4
1
-3
-5
-2
-15
-1
-29
0
-47
(ii) Equation of the axis of symmetry is x = 2, the minimum
value of y is -5, and the y-intercept is 3.
(RF3.9)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
81
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
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.
In this unit, students will examine quadratic functions expressed in
standard form y = ax2 + bx + c and determine the x- and y-intercepts,
vertex (as a maximum or minimum point), axis of symmetry, direction
of opening, and domain and range. They will also sketch the graph of a
quadratic function using the characteristics above. Students will study
the effect of changing the parameters a, b, and c of the equation on the
shape of the graph. They will also solve problems involving quadratic
functions.
[CN, PS, R, T, V]
Achievement Indicators:
RF4.1 Determine the
function given in the form
y = ax2 + bx + c, and explain the
strategy used.
RF4.2 Sketch the graph of a
form y = ax2 + bx + c .
The characteristics of the quadratic function, y = ax2 + bx + c, should
be developed through an investigation where the parameters a, b, and
c are manipulated individually. This should be completed through the
use of technology. Ask students what parameters affect the shape of the
graph, the direction of opening, and the y-intercept. Using the graph as
a visual, ask students to discuss the following:
•
•
•
•
maximum or minimum value
domain and range
equation of the axis of symmetry
x-intercepts
Students should first be exposed to quadratics for which the vertex has
integer coordinates to make it easier to identify characteristics precisely.
Students may use the formula x = − 2ba to find the equation of the axis
of symmetry for quadratic functions of the form y = ax2 + bx + c. This
value can then be used to determine the x-coordinate of the vertex. The
formula should be developed through a student-based activity similar to
the one outlined below.
Provide students with a table containing quadratic functions written in
standard form and the vertex of each corresponding parabola. Students
should complete the table for the values of a, b and c along with the
value of x = − 2ba .
Function
Vertex
y = x2 - 4x + 7
(2, 3)
Equation
of Axis of
Symmetry
a
b
c
− 2ba
y = -2x2 - 16x - 34 (-4, -2)
y = 3x2 - 6x + 10
82
(1, 7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
Ask students to work in groups for this activity. Each group
should be given a pair of dice (or they can create their own). On
the first die, there will be 2 a values, 2 b values, and 2 c values. On
the second die, there will be various rational numbers. Students
should roll the dice and state the effect on y = x2 of changing the
given parameter to the given number.
(RF4.1, RF4.2)
Pre-Calculus 11
Functions in Standard Form
SB: pp.163-179
TR: pp.113-124
BLM: 3-3, 3-5
Interview
•
Ask students to explain how they could determine whether a
quadratic function has either a maximum or minimum value
without graphing.
Note: In Mathematics 1204,
standard form would have been
referred to as general form.
(RF4.1)
Paper and Pencil
•
The following site provides an
animation of how the shape of the
graph is affected by the parameters
a, b, and c.
Ask students to complete the following webbing to describe the
effects of the parameters a, b and c on the quadratic function
y = ax2 + bx + c.
y = ax 2 + bx + c
a
b
http://www.livephysics.com/
c
The following is an interactive
tutorial where students can vary
the values of a, b, and c and its
impact on the shape of the graph.
http://www.analyzemath.com/
(RF4.1, RF4.2)
•
Ask students to write an email message explaining how changing
the parameter a will affect the graph of the function
y = ax2 + bx + c .
(RF4.1, RF4.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
83
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Continued ...
Achievement Indicators:
RF4.2 Continued
RF4.3 Explain the reasoning
for the process of completing
the square as shown in a given
example.
Students should discover that the value of − 2ba is the x-coordinate
of the vertex and its connection should be made to the equation of
the axis of symmetry. The y-coordinate of the vertex can be found by
substituting the x-coordinate into the quadratic function. Note that
the y-coordinate of the vertex can also be found using the formula
2
y = 4 ac4−ab . This formula could be developed through an activity
similar to the one described previously.
In Mathematics 1201, students represented a quadratic expression using
algebra tiles. They also demonstrated, through the use of algebra tiles, an
understanding of the multiplication of polynomial expressions (limited
to monomials, binomials and trinomials), common factoring and
trinomial factoring (AN4, AN5).
Students should be given an opportunity to understand the process of
creating perfect square trinomials and the patterns formed. The method
of completing the square is one method students will use when they
rewrite a quadratic equation from standard form to vertex form. The
perfect square binomial (x - p)2 is part of the quadratic function in
vertex form.
Algebra tiles can be used to visualize how a perfect square trinomial can
be formed. Students should first be exposed to examples where a = 1 and
c = 0. Consider the following example:
Ask students to model x2 + 8x. The goal is to find a number c to create
a perfect square trinomial x2 + 8x + c. Students will use the algebra tiles
to create a square. Ask students why the number of tiles must be split
evenly.
Continue to use leading questions, such as the following, to promote
discussion:
•
What tiles must be added to complete the square?
16 unit tiles
84
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Observation
•
Ask students to graph the following using a table of values or
graphing technology.
(i)
(ii)
(iii)
(iv)
(v)
y = x2
y = x2 + 2
y = x2 + 5
y = x2 - 3
y = x2 - 4
Functions in Standard Form
SB: pp.163-179
TR: pp.113-124
BLM: 3-3, 3-5
As teachers observe students’ work, ask them to explain the
changes in the parabola as the parameter c is manipulated.
(RF4.1, RF4.2)
Interview
•
Pre-Calculus 11
3.3 Completing the Square
SB: pp.180-200
TR: pp.125-133
BLM: 3-3, 3-6
Using algebra tiles, ask students to model and explain the process
of completing the square. The teacher photographs the group as
they participate in the activity. After a few classes, give students
their photograph and ask them to describe what they were doing
in the picture. They should write about the activity under the
photograph, describing what they were doing and what they
learned as a result.
(RF4.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
85
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Continued ...
Achievement Indicator:
RF4.3 Continued
•
What is the expression that represents the new completed square?
•
What is the relationship between the coefficient of the linear term
and the constant term?
•
What is the trinomial written as the square of a binomial?
Students should realize 16 unit tiles have been added. Since the side
length of the square is represented by x + 4, the area is (x + 4)(x + 4).
This perfect square trinomial x2 + 8x + 16, can be rewritten as (x + 4)2.
Continue to work with various examples, such as x2 + 2x, and x2 + 6x,
to give students an opportunity to describe the pattern that exists. The
algebra tile method illustrates the constant term is half the coefficent of
the linear term squared.
When modelling with algebra tiles, it is important for teachers to choose
a value of b that is even and relatively small. Before students move from
the concrete representation to the symbolic method give students an
opportunity to work with expressions where a > 0 and c = 0. Consider
the example:
Ask students to model 2x2 + 8x using algebra tiles and why the tiles
should be arrange into two equal parts.
x2 + 4x
x2 + 4x
4 unit
tiles
4 unit
tiles
Students will write the expression that represents the new completed
square. They should recognize that 4 unit tiles were added to each
diagram creating two perfect square trinomials, each of which can be
written as x2 + 4x + 4 or (x + 2)2. Since there are two squares in the
model, the total area is 2(x2 + 4x + 4) or 2(x + 2)2.
This visual allows students to gain a better understanding as to why the
value of b must be divided by a before the process of completing the
square takes place. That is, 2x2 + 8x = 2(x2 + 4x).
86
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
•
Ask students to complete the square for y = ax2 + bx + c to show
2
that the coordinates of the vertex are − 2ba , 4 ac4−a b .
)
(
(RF4.1, RF4.3)
Pre-Calculus 11
3.3 Completing the Square
SB: pp.180-200
TR: pp.125-133
BLM: 3-3, 3-6
Ask students to complete a table investigating the relationship
between the value of b and c when completing the square. They
should share their findings with the class. A sample is shown
below.
Binomial of
the form
x2 + bx
x2 + 6x
to complete the
square?
9
Modeled with
algera tiles
Perfect square
x2 + 6x + 9
Factored and
expanded form
Value of
coeffcicent
c
How are
b and c
related?
(x + 3)(x + 3)=
x2 + 6x + 9
9
c = ( 21 b )2
(RF4.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
87
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Continued ...
Achievement Indicator:
given in the form
y = ax2 + bx + c as a quadratic
function in the form
y = a(x - p)2 + q by completing
the square.
This would be an opportunity for students to extend their work with
algebra tiles to convert an equation from standard form to vertex form
using the process of completing the square. They will then convert
between the two forms algebraically. Use algebra tiles to model the
function y = x2 + 6x + 7. Students will continue to split the tiles evenly
and add tiles to form a square leaving the constant term alone.
x2 + 6x
7 unit tiles
It is important for students to notice that by adding nine extra tiles, the
opposite tiles to balance the function.
x2 + 6x+ 9
7-9
The function is y = (x2 + 6x + 9) + (7 - 9). They should recognize when
a number is added to form a perfect square, its opposite value is also
added to keep the original expression unchanged. Students can then
rewrite this as y = (x + 3)2 - 2. This visual representation allows students
to observe patterns and then record their work symbolically.
Ask students why it is to their advantage to write an equation in vertex
form. Students should emphasize that the coordinates of the vertex
(p, q) can be directly determined when written in this form. Hence,
the maximum or minimum point is known. Students should first be
then progress to examples where a ≠ 1.
88
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Ask students to work in pairs for this activity. Give each pair of
rewrite the equation in vertex form. Ask one student to write the
first line of the solution and then pass it to the second student.
The second student will verify the workings to determine if an
error is present. If there is an error present, the student will correct
it and then write the second line of the solution and pass it along
to their partner. This process continues until the solution is
complete.
3.3 Completing the Square
SB: pp.180-200
TR: pp.125-133
BLM: 3-3, 3-6
(RF4.4, RF4.5)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
89
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Continued ...
Achievement Indicators:
RF4.5 Identify, explain and
correct errors in an example of
completing the square.
Common errors occur when converting a quadratic function from
standard form to vertex form. A quadratic function where a ≠ 1,
for example, sometimes causes difficulty for students. Consider the
following example and discuss with students the possible errors that may
occur: y = -3x2 + 18x - 23
y = -3(x2 - 6x) - 23
y = -3(x2 - 6x + 9) - 23 + 27
y = -3(x - 3)2 + 4
•
The common factor (-3) is not factored out from both the
•
There is an incorrect sign on the linear term when a negative
•
The constant term inside the parentheses is doubled instead of
squared.
•
When a perfect square is created, the constant term inside the
parentheses is not multiplied by the common factor to produce
the compensated term.
•
RF4.6 Verify, with or without
function in the form
y = ax2 + bx + c represents
the same function as a given
y = a(x - p)2 + q.
The perfect square trinomial is incorrectly factored.
When converting a quadratic function from vertex form to standard
form, students will expand the perfect square trinomial and then
combine like terms. Students should also verify that a quadratic function
in standard form represents the same function in vertex form. They can
use the method of completing the square to compare the functions or
compare the graphs of both functions. Ask students what features of the
graph must be the same. They must ensure that both functions have the
same vertex and one other point (e.g., the y-intercept) on both graphs.
As an alternative, an algebraic combined with a graphical approach
could be used. Ask students to graph one of the functions. They should
choose three points from the graph and substitute them into the other
form of the function to verify the points satisfy the function. Promote
student discussion as to why three points must be used and not two
points.
90
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
In groups of two, ask students to play the Domino Game. Provide
students with 10 domino cards. One side of the card contains a
quadratic function in standard form, while the other side contains
a quadratic function in vertex form. Ask each group to lay the
dominos out such that the standard form on one card will match
with the correct vertex form on another. They will eventually form
a complete loop, with the first card matching the last card. Some
sample cards are shown below:
2
y = ( x + 5) – 3
2
y = 2( x – 3) + 5
2
y = 2x – 12 x + 23
Pre-Calculus 11
3.3 Completing the Square
SB: pp.180-200
TR: pp.125-133
BLM: 3-3, 3-6
The following site provides
examples where students write the
shapes.
2
y = – x – 2x + 2
http://www.thefutureschannel.com/
algebra/windsail_design.php
2
y = – ( x + 1) + 3
2
y = x + 10 x + 22
(RF4.6)
•
In groups of two, ask students to move around the classroom to
various stations where solutions have been posted outlining the
process of completing the square. At each station the solution
could contain one or more errors that the group has to identify
and then produce correct solutions for each of the problems.
(RF4.5)
APPLIED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
91
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF4 Continued ...
Achievement Indicators:
that models a given situation, and
RF4.8 Solve a problem, with or
without technology, by analyzing
92
Students will be expected to write a quadratic function that models
a situation and then solve the problem. They will be exposed to
problems where they will determine the maximum or minimum value.
Contextual problems could involve maximum revenue, finding the
maximum possible area, the maximum height, or a minimum product.
It is important for students to understand the terminology that is being
used in the problems. They should make the connection between the
maximum or minimum value, and the y-value of the vertex.
MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
(i) A stream of water from a fountain forms a parabolic shape.
Given the spout on the fountain is 5 cm high and the
maximum height reached by the water is 14 cm at a distance
of 6 cm from the spout, what is the height of the water when
it is 8 cm from the spout?
3.3 Completing the Square
SB: pp.180-200
TR: pp.125-133
BLM: 3-3, 3-6
(ii) A student makes and sells necklaces at the beach during the
summer months. The material for each necklace costs her
\$6.00 and she has been selling about 20 per day at \$10.00
each. She has been wondering whether or not to raise the
price, so she takes a survey and finds that for every dollar
increase she would lose two sales a day. What price should
she set for the necklaces to maximize profit?
(iii) Find the dimensions of the rectangle of maximum area that
can be inscribed in an isosceles triangle of altitude 8 and base
6. (Hint: Use similar triangles to express the height of the
rectangle in terms of its base).
8
6
(RF4.7, RF4.8)
APPLIED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
93
94
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Suggested Time: 14 Hours
Unit Overview
Focus and Context
In this unit, students will extend their factoring skills and use a variety
of strategies to determine the roots of quadratic equations. They will
make the connection that the x-intercepts of the graph or the zeros
of the quadratic function correspond to the solutions, or roots, of the
Students will explore various algebraic methods for solving quadratic
equations including factoring, completing the square and the quadratic
formula. They will use the discriminant to determine the number of
Contextual problems will also be solved by modelling a situation with a
Outcomes
Framework
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF1
Factor polynomial expressions of the form:
• ax2 + bx + c, a ≠ 0
• 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
where a, b and c are rational numbers.
SCO RF5
equations.
96
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Process Standards
Key
[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
Algebra and Number
Relations and Functions
Relations and Functions
AN5 Demonstrate an
understanding of common
factors and trinomial factoring,
concretely, pictorially and
symbolically.
[C, CN, R, V]
RF1 Factor polynomial expressions RF10 Solve problems that involve
of the form:
exponential and logarithmic
equations.
2
• ax + bx + c, a ≠ 0
[C, CN, PS, R]
• 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
RF11 Demonstrate an
where a, b and c are rational
understanding of factoring
numbers.
polynomials of degree greater than 2
(limited to polynomials of degree
[CN, ME, R]
≤ 5 with integral coefficients).
RF5 Solve problems that involve
[C, CN, ME]
[C, CN, PS, R, T, V]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
97
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF5 Solve problems that involve
[C, CN, PS, R, T, V]
In Mathematics 1201, students factored differences of squares, perfect
square trinomials and polynomials of the form x2 + bx + c and
ax2 + bx + c (AN5).
In the previous unit, students were introduced to quadratic functions
expressed in standard form and vertex form. They sketched graphs using
characteristics such as x-and y-intercepts, vertex, axis of symmetry and
domain and range. Students also converted a quadratic function in
standard form to vertex form using the process of completing the square.
In this unit, students will solve quadratic equations using various
strategies, including factoring, completing the square, applying the
quadratic formula or using a graph.
Achievement Indicators:
RF5.1 Explain, using examples,
the relationship among the roots
of a quadratic equation, the zeros
function and the x-intercepts
of the graph of the quadratic
function.
Students have been exposed to the graph of a quadratic function and the
points where a parabola crosses the x-axis. They are aware a quadratic
function can have zero, one or two x-intercepts. When solving a
quadratic equation of the form ax2 + bx + c = 0, students can graph the
corresponding quadratic function and determine the x-intercepts. They
can use a table of values or graphing technology to make the connection
between the x-intercepts of the graph and the roots of the quadratic
equation.
of the form ax2 + bx + c = 0 by
using strategies such as:
• determining square roots
• factoring
• completing the square
• graphing its corresponding function.
It is important for students to distinguish between the terms roots,
zeros and x-intercepts, and to use the correct term in a given situation.
The x-intercepts of the graph or the zeros of the quadratic function
correspond to the roots of the quadratic equation. Students could be
asked to find the roots of the equation x2 - 7x + 12 = 0, find the zeros of
f (x) = x2 - 7x + 12, or determine the x-intercepts of y = x2 - 7x + 12. In
each case they are solving x2 - 7x + 12 = 0 and arriving at the solution
x = 3 or x = 4.
98
Discuss with students what it means to solve a quadratic graphically
and the limitations of this method. They will then work with quadratics
algebraically and decide when it is best to factor, complete the square or
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Journal
•
Ask students to write a paragraph explaining the relationship
between the roots of a quadratic equation, the zeros of a quadratic
function and the x-intercepts of the graph of a quadratic function.
Pre-Calculus 11
(RF5.1)
Paper and Pencil
•
(i) Find the zeros of f(x) = 2x2 + 5x - 7.
(ii) Identify the x-intercepts of the graph.
y
4.1 Graphical Solutions of
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
Student Book (SB): pp.206-217
Teacher Resource (TR): pp.143148
BlackLine Master (BLM): 4-3, 4-4
-8
-10
(iii) Find the roots of 2x2 + 5x - 7= 0.
questions?
(RF5.1, RF5.2))
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
99
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF1 Factor polynomial
expressions of the form:
ax2 + bx + c, a ≠ 0
• 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
where a, b and c are rational
numbers.
[CN, ME, R]
Students were first exposed to factoring polynomials in Mathematics
1201 (AN5). They removed the greatest common factor from the terms
of a polynomial and factored polynomials of the form x2 + bx + c and
ax2 + bx + c. As well, students factored perfect square trinomials,
difference of squares and trinomials in two variables.
In this unit, students will extend their knowledge of factoring trinomials
and differences of squares to factoring polynomials of the form
a2x2 - b2y2, a ≠ 0, b ≠ 0, a(f (x))2 + b(f (x)) + c, a ≠ 0 and
a2(f (x))2 - b2(g (y))2, a ≠ 0, b ≠ 0.
Achievement Indicators:
RF1.1 Factor a given polynomial
expression that requires the
identification of common factors.
RF1.2 Factor a given polynomial
expression of the form:
• ax2 + bx + c, a ≠ 0
• a2x2 - b2y2, a ≠ 0, b ≠ 0
RF1.3 Determine whether a
given binomial is a factor for a
given polynomial expression, and
explain why or why not.
RF1.4 Factor a given polynomial
pattern, including:
• a(f (x))2+b(f(x))+c, a ≠ 0
• a2(f (x))2-b2(g(y))2, a ≠ 0, b ≠ 0
100
In Mathematics 1201, students worked extensively with factoring
polynomial expressions. They were introduced to factoring using
concrete and pictorial models and then moved to a symbolic
representation. Most of the previous work dealt with integer coefficients.
Remind students of the strategies used when factoring polynomial
expressions, including the removal of the greatest common factor and
the method of decomposition (AN5). In this course, students will be
expected to be proficient with factoring at a symbolic level, including
expressions with rational coefficients. Students should also be given the
opportunity to apply their own personal strategies.
Students should recognize when factoring expressions such as
x2 + 6x + 8, possible binomial factors would only contain factors of 8.
Therefore, a student should realize that x + 5, for example, would not
be a possible factor of x2 + 6x + 8. This will improve their factoring skills
which will be useful later when using factoring to solve a given quadratic
equation.
Factoring trinomials and differences of squares will be extended to
factoring polynomials of the form a(f (x))2 + b(f (x)) + c, a ≠ 0 and
a2(f (x))2 - b2(g (y))2, a ≠ 0, b ≠ 0. They can attempt these problems using
different methods. Students can treat the expression as a single variable,
continue to factor using their previous skills, and then use substitution
to complete the factoring. Alternatively, they can expand and group like
terms and then factor.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
For the game Three in a Row, students are given a 5 × 5 game
board which consists of 25 squares. Each square will have a
quadratic equation written on it. Ask them to play in pairs and
take turns solving each quadratic equation with the aim to be the
first person to make three in a row.
Pre-Calculus 11
(RF1.1, RF1.2)
Paper and Pencil
•
Ask students to determine two values of n that will allow the
polynomial 25b2 + nb + 49 to be a perfect square trinomial. They
should use them both to factor the trinomial.
(RF1.2)
•
Ask students to list possible binomial factors if the following
expression x2 + bx + 24 can be factored.
(RF1.3)
Equations
SB: pp.218-233
TR: pp.149-154
BLM: 4-3, 4-5
Interview
• Ask students to respond to the following:
(i) Explain why 9x2 − 16y2 can be factored but 9x2 + 16y2 cannot
be factored.
(RF1.2)
(ii) Explain why (2x - 3y)2 ≠ 4x2 - 9y2.
(RF1.2)
•
When Jenny factored x2 - x - 12, she said one of her binomial
factors was x + 5. Without actually factoring the trinomial, ask
students to explain why her response is incorrect.
(RF1.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
101
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF1, RF5 Continued ...
Achievement Indicators:
RF1.4 Continued
Consider the following example to compare the two methods:
Factor: 2(x - 2)2 + 7(x - 2) + 5
Substitute p = x - 2
Expand (x - 2)2
2p2 + 7p + 5
2(x-2)(x-2)+7(x-2)+5
(2p + 5)(p + 1)
2(x2 - 4x + 4) + 7(x - 2) + 5
[2(x - 2) + 5][(x - 2 + 1]
2x2- 8x + 8 + 7x - 14 + 5
(2x - 4 + 5)(x - 2 + 1)
2x2 -1x - 1
(2x + 1)(x - 1)
(2x + 1)(x - 1)
Regardless of the method used, remind students that is is important to
use brackets for all substitution.
RF5.2 Continued
Students solved a quadratic equation graphically by finding the
x-intercepts. They will now solve using algebraic methods. They will
explore different strategies to find values for the variable that make the
function equal to zero.
Students can use factoring to solve equations. Once a quadratic equation
has been factored, they use the zero product property to determine the
roots. If students were asked to solve 5x2 + 14x - 3 = 0, for example,
5x - 1 = 0 or x + 3 = 0. Remind students to substitute the value of
each root into the original equation to verify that the value makes the
equation true.
Provide students with examples where a quadratic equation of the form
ax2 + bx + c = 0 is missing the b or c value. Consider a quadratic equation
where c = 0. When solving x2 - 5x = 0, for example, students might
factor x(x - 5) = 0 and then solve by dividing both sides of the equation
by x, resulting in only one solution. Ask students if this is correct and
whether a root has been eliminated. Encourage students to check the
reasonableness of their answers using the zero product property.
Expose students to quadratic equations similar to x2 - 8 = 0 or
(x - 2)2 - 49 = 0. They can isolate the squared term and take the square
root of both sides of the equation. Remind students there are two
possible solutions to these equations. They should also be asked to solve
equations in which the right hand side of the equation does not equal 0
(i.e., 3x2 - 4 = x).
102
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Interview
•
cannot be factored. They should explain their reasoning.
(RF1.1, RF1.2, RF1.3, RF1.4)
•
After factoring the different types of polynomials, ask students
which ones they found easiest to factor and why. Which ones were
most difficult and why?
(RF1.1, RF1.2, RF1.3, RF1.4)
Pre-Calculus 11
Equations
SB: pp.218-233
TR: pp.149-154
BLM: 4-3, 4-5
Paper and Pencil
•
Ask students to factor 4(x + 2)2 - 9(y + 1)2 using two methods.
They should explain which method they prefer and why.
(RF1.4)
Performance
•
Two sets of different colored cards are required for this activity.
One set will contain quadratic equations and the other set will
have their corresponding solutions. Ask students to lay out the
cards and match the equation card with its corresponding solution
card.
x 2 − 2 x − 24 = 0 x = 5 ; x = 2
2
2 x 2 + 5x + 3 = 0
3
x = ; x = −1
2
2 x 2 − 5x + 3 = 0
2x2 + 7x + 5 = 0
3
x = − ; x =1
2
x = 4 ; x = −6
x 2 + 2 x − 24 = 0
2 x 2 − 9 x + 10 = 0
2x2 − x − 3 = 0
x = −4 ; x = 6
2x2 + x − 3 = 0
5
x = − ; x = −1
2
3
x = ;x =1
2
3
x = − ; x = −1
2
The following is an interactive site
and its graphical representation.
(RF5.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
103
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF5 Continued ...
Achievement Indicators:
RF5.2 Continued
formula, using deductive
reasoning.
When students are exposed to quadratic equations that cannot be
factored, they have the option to complete the square or apply the
quadratic formula. The process of completing the square was introduced
in the previous unit when students converted a quadratic function from
standard form to vertex form. They will now use completing the square
to determine the roots of a quadratic equation in standard form. Once
students solve a quadratic equation using this method, provide students
the graph of the corresponding function to reinforce the connection
between the roots of the equation and the x-intercepts of the graph.
It is important for students to understand how the quadratic formula is
developed before they apply it to quadratic equations in standard form.
They should use a numerical example before moving to the general form
ax2 + bx + c = 0. Ask them to complete the square using an example
similar to 3x2 - 7x + 1 = 0. Assist students as they follow the same
procedure to derive the quadratic formula for ax2 + bx + c = 0. Once the
quadratic formula has been derived it can be used to find the roots of
any quadratic equation in standard form.
3x 2 − 7x + 1 = 0
3( x 2 − 73 x ) + 1 = 0
49
3( x 2 − 73 x + 49
) − 12
+1= 0
36
ax 2 + bx + c = 0
a( x 2 + ba x ) + c = 0
a( x 2 + ba x +
b2
4a2
)−
ab 2
4a 2
+c =0
( x − 76 )2 =
37
36
b )2 − b + c = 0
2a
4a
2
2
a( x + 2ba ) − 4b a + 44aca = 0
2
a( x + 2ba )2 = b 4−a4 ac
2
( x + 2ba )2 = b −42ac
4a
x − 76 = ±
37
6
x+
b
2a
=±
b 2 −4 ac
4a2
x+
b
2a
=±
b 2 −4 ac
2a
x = − 2ba ±
b 2 −4 ac
2a
49
3( x − 76 )2 = 12
−1
49 − 12
3( x − 76 )2 = 12
12
37
3( x − 76 )2 = 12
x = 76 ±
37
6
x = 7± 6 37
2
a( x +
x=
−b ± b 2 −4 ac
2a
2
Students may need assistance when adding rationals ( − 4b a + c ) and
simplifying variable roots ( 4a 2 ). These topics are covered in greater
detail later in this course in the Radical Expressions Unit and the
Rational Expressions Unit.
104
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
For the game Quadratic Bingo, a list of approximately 50 possible
sets of solutions to various quadratic equations will be placed
on the board. Each student is given a blank bingo card. They
will randomly fill in the 24 squares of their card with one set
of solutions. They are now ready to play BINGO. Present one
quadratic equation at a time for students to solve. If the solutions
are present on their card, the square is covered. Continue to
provide equations to be solved until a student has completed a
diagonal, horizontal or vertical line on the card.
Pre-Calculus 11
by Completing the Square
SB: pp.234-243
TR: pp.155-159
BLM: 4-3, 4-6
BINGO
SB: pp.244-256
TR: pp.160-165
BLM: 4-3, 4-7
FREE
(RF5.2)
www.k12pl.nl.ca/seniorhigh/
introduction/math2201/
classroomclips.html
demonstrates students taking turns
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
105
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF5 Continued...
Achievement Indicators:
RF5.4 Identify and correct errors
in a solution to a quadratic
equation.
It is beneficial to have students analyze solutions that contain errors.
Students should be provided with worked solutions of quadratic
equations that may or may not contain errors. If errors are present,
students should identify the error and provide the correct solution
including why/how the error occurred. This reinforces the importance
of recording solution steps rather than only giving a final answer.
Some common errors occur when students are simplifying the quadratic
formula. These include:
•
Apply the quadratic formula without ensuring the equation is
written in standard form.
•
Use x = −b ± b 2−a4 ac rather than the correct form of the
•
Incorrectly produce two possible common errors if the b value is
negative.
2
(i) If b = -2 then -b = -(-2) = -2
(ii) If b = -2 then b2 = -22 = -4
•
Incorrectly simplify when applying the quadratic formula.
(i)
8± 5
2 =4±
5
(ii) 2 + 42 5 = ± 2 5
• Do not recognize that the ± results in two solutions. Suggest that
students work through the solutions separately, showing calculations
for both the positive solution and the negative solution.
RF5.5 Select a method for solving
choice, and verify the solution.
As students work through the various strategies for solving quadratic
equations, they should realize that sometimes one method is more
efficient than another. The method students choose to solve a quadratic
equation will depend on the way the equation is presented.
RF5.6 Explain, using examples,
how the discriminant may be used
equation has two, one, or no
real roots; and relate the number
of zeros to the graph of the
Students have solved quadratic equations graphically and algebraically.
Regardless of their strategy to solve the equation, they should be able
to connect the fact that their algebraic solution is the same as the
x-intercepts of the graph. In the previous unit, when functions were
written in the form y = a(x - p)2 + q , students investigated how the
values of a and q affected the number of x-intercepts. They will now
explore how these situations arise when solving equations using the
106
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to identify the most efficient strategy (i.e.,
determining square roots, factoring, completing the square,
applying the quadratic formula or graphing its corresponding
function) when solving each equation. They should justify their
choice.
(i) (ii)
(iii)
(iv)
(v) (x + 2)2 - 1 = 15
49x2 - 64 = 0
3x2 - 11x + 6 = 0
x2 - 10x - 24 = 0
3x2 + 8x + 7 = 0
SB: pp.244-256
TR: pp.160-165
BLM: 4-3, 4-7
(RF5.5)
•
Given the quadratic equation ax + bx + c = 0, ask students what relationship must be true for the coefficients a, b, and c so that the equation has:
2
(i) two distinct real roots
(ii) two equal real roots
(iii) no real roots
(RF5.6)
Observation
•
Mary, David, and Ron are students in a group. They are given the
equation A = x2 + 3x – 110 where A represents the area of a field
and x represents the width in metres. The students were asked to
find the width if the area was 100 m2. Each student decided to
solve the equation using their own preferred method. Here are
their solutions:
Mary
2
Ron
x + 3x − 110 = 100
x 2 + 3x − 110 = 0
x 2 + 3x − 210 = 0
x 2 + 3x − 10 = 0
(x + 5)(x − 3) = 0
x=
x=
3 ± 9 − (4)(1)( −210)
2
3 ± −831
x=
2
3 ± 28.827
x=
2
x = 15.9 or x = −12.9
width is 15.9 m
David
x + 3x − 110 = 100
2
x = −5 or x = 3
width is 3 m
−3 ± 9 − (4)(1)( −110)
2
−3 ± 9 + 440
x=
2
−3 ± 449
x=
2
−3 ± 21.2
x=
2
x = 9.1 or x = −12.1
width is 9.1 m
Ask students to identify and explain any errors in the students’
work. They should then proceed to write the correct solution.
(RF5.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
107
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF5 Continued...
Achievement Indicators:
RF5.6 Continued
When students use the quadratic formula to find roots of a quadratic
equation they will be exposed to trying to find the square roots of nonperfect squares and negative numbers. It is important for students to
distinguish between exact solutions and approximate solutions.
Although, it is not the intention of this course to introduce students to
the imaginary number system, they need to be aware that there is no
real solution to the square root of a negative number. This will lead to a
discussion regarding the conditions that are necessary for the quadratic
formula to result in two real roots, one real root and no real solutions.
Ask students to solve each of the following using the quadratic formula.
A: 2x2 + 10x + 3 = 0 B: x2 + 6x + 9 = 0 C: 2x2 + 3x + 5 = 0
As teachers observe students’ work, use the following prompts to
promote discussion:
RF5.7 Solve a problem by:
• determining and analyzing a
•
What is the value under the square root in each equation? What does this tell you about the roots of the equation?
•
What connection can be made between the value of b2 - 4ac and
the number of real roots an equation has?
•
Quadratic equations can be used to model a variety of situations such as
projectile motion and geometry based word problems. Students should
be exposed to examples which require them to model the problem using
a quadratic equation, solve the equation and interpret the solution.
Consider the example:
•
A rectangular lawn measuring 8 m by 4 m is surrounded by a
flower bed of uniform width. The combined area of the lawn and
flower bed is 165 m2. What is the width of the flower bed?
It is important for students to recognize that the context of the problem
dictates inadmissible roots. Discuss with students different scenerios
that produce inadmissible roots. For example, time, height and length,
would not make sense if they have a negative numerical value. However,
temperature could be both negative and positive.
108
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
When Chantal was asked to describe the roots of the equation
14x2 – 5x = – 5, she rearranged the equation so that it would
equal zero, then used the quadratic formula to find the roots. Her
workings are shown below. Edward said that she didn’t have to
do all that and he then showed the class his work. Ask students if
they are both correct. They should identify their preferred method
and explain their reasoning.
Chantal
Edward
5 ± 25 − 280
28
5 ± −255
x=
28
x=
no real roots
•
SB: pp.244-256
TR: pp.160-165
BLM: 4-3, 4-7
b 2 − 4ac
= 25 − 280
= −255
no real roots
(RF5.6)
(i) For what values of t does x2 + tx + t + 3 = 0 have one real
root?
(RF5.6)
(ii) Show that if the quadratic equation px2 +(2p + 1)x + p = 0
has two real unequal roots, then 4p + 1 > 0.
(RF5.6)
(iii) Assume a, b and c are real numbers. How many times would
y = ax2 + bx + c, intersect the x-axis if the discriminant of
ax2 + bx + c = 0 is
(a) positive?
(b) zero?
(c) negative?
(RF5.6)
(iv) Create quadratic equations that have two distinct roots, two
equal roots, and no real roots. They should explain their
reasoning.
(RF5.6)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
109
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF5 Continued...
Achievement Indicator:
RF5.7 Continued
When modelling situations for students, emphasize that restrictions
sometimes need to be placed on the independent variable of the
function. If a solution does not lie in the restricted domain, then it
is not a solution to the problem. The following is an example with a
restricted domain:
•
A baseball is thrown from an initial height of 3 m and reaches a
maximum height of 8 m, 2 seconds after it is thrown. At what
time does the ball hit the ground?
In the above example, the quadratic equation only models the path
of the ball from the time it leaves the throwers’ hand to the time it
makes first contact with the ground. This quadratic equation yields two
possible solutions, one of which is negative. This implies that it occurred
before the ball was thrown. The restriction on the domain causes the
negative solution to be inadmissible since time cannot be negative and
only the positive solution is accepted.
110
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
A ball is thrown from a building at an initial height of 11 metres
and reaches a maximum height of 36 metres, 5 seconds after it is
thrown. Ask students to do the following:
Pre-Calculus 11
(i) Write a quadratic equation which models this situation.
(ii) Three targets are placed at different locations on the ground.
One is at (10,0), another at (11,0) and a final target is placed
at (12, 0). Which target does the ball hit? Explain how you
(RF5.7)
SB: pp.244-256
TR: pp.160-165
BLM: 4-3, 4-7
•
Ask students to find two consecutive whole numbers such that the
sum of their squares is 265.
(RF5.7)
• A diver’s path when diving off a platform is given by
d = -5t2 + 10t + 20, where d is the distance above the water (in
feet) and t is the time from the beginning of the dive (in seconds).
(i) How high is the diving platform?
(ii) When is the diver 25 feet above the water?
(iii) When does the diver enter the water?
(RF5.7)
•
notes or group workstations and use it as a guide to create their
own word problem (encourage them to use a real-life situation
that they are interested in). Remember to have them include their
solutions on a separate sheet. Students could give their problem to
another student to solve.
(RF5.7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
111
112
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Suggested Time: 13 Hours
Unit Overview
Focus and Context
In this unit, students will simplify radical expressions and perform
the four operations on these expressions (addition, subtraction,
multiplication and division). When working with division, students
will be expected to rationalize denominators containing radicals. They
will also identify the restrictions on the values of variables in radical
expressions that are real numbers.
Students will solve problems that involve radical equations, limited to
square roots, and identify the extraneous roots.
Outcomes
Framework
GCO
Develop algebraic reasoning and
number sense.
SCO AN2
Solve problems that involve
expressions with numerical and
SCO AN3
Solve problems that involve
square roots).
114
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Process Standards
Key
[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
Algebra and Number
Algebra and Number
Relations and Functions
AN2 Demonstrate an
understanding of irrational
numbers by:
•
representing, identifying
and simplifying irrational
numbers
AN2 Solve problems that involve
expressions with numerical and
functions (limited to functions
[CN, R, T,V]
[CN, ME, PS, R]
•
ordering irrational numbers. AN3 Solve problems that involve
[CN, ME, R, V]
square roots).
[C, PS, R]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
115
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Solve problems that involve
expressions with numerical and
[CN, ME, PS, R]
In Grade 9, students determined the square root of a perfect square and
worked with benchmarks to approximate the square root of non-perfect
square rational numbers (9N5, 9N6). In Mathematics 1201, students
(AN2). The notation n x was introduced, where the index of the radical
was a maximum of 5.
In this unit, students will simplify radical expressions with numerical
expressions. When working with division, students will be responsible
for a monomial and binominal denominator. They will also identify
values of the variable for which the radical expression is defined.
expressions. Solutions will be verified by substitution and extraneous
roots will be explored.
Achievement Indicators:
AN2.1 Compare and order
with a numerical radicand as a
with a numerical radicand as an
In Mathematics 1201, students expressed a radical as either a mixed
students and reinforce that if radicals have the same index, the radicands
can be compared. It is helpful to rearrange the mixed radical as an entire
radical for the purpose of ordering and estimation without the use of
technology. When given two right triangles, for example, ask students to
determine which hypotenuse length is greater, 3 5 or 4 3 . Although
students could use a calculator to approximate the length, the focus
here is to rewrite the numerical radicals in equivalent forms and make a
comparison.
Part of this unit is a review of topics taught in Mathematics 1201, with
the exception of the introduction to secondary roots. Refer to the Roots
and Powers unit in the Mathematics 1201 Curriculum Guide for further
information regarding teaching strategies and common student errors
(AN2.6, AN2.7, AN2.8).
Students should have some exposure to principal and secondary square
roots. They should recognize that every positive number has two roots.
For example, the square root of 49 is 7 since 72 = 49. Likewise
(-7)2 = 49 so -7 is also a square root of 49. The value 49 = 7 is called
the principal square root and − 49 = −7 is the secondary square root.
Although students will compare principal and secondary square roots,
it is equally important for them to understand why it makes sense to
only use the principal square root in certain situations. For example, if
students are using the Pythagorean theorem to calculate the length of
a leg of a right triangle, length is a positive number, so the square root
must also be positive.
116
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Set up a clothesline across the whiteboard to represent a number
line with several benchmarks identified. Each student is given
a card with an expression of mixed or entire radical. Ask them
to pin the card along the number line. They should be able to
explain why they placed the card in that position.
•
•
(AN2.1, AN2.2 AN2.3)
Each group (3-4 students) will be given a deck of cards. Each
card will have a different mixed radical. The group will then work
together to sort the cards from largest to smallest. The first group
with the cards sorted in the correct order wins the competition.
(AN2.1, AN2.2 AN2.3)
Students can play the Radical Matching Game in groups of
two. Give students a deck of cards containing pairs that display
placed face down on the table. The first student turns over 2 of
the cards, looking for a pair. If they get a pair, they remove the
cards and go again. If the overturned cards do not form a pair, it
is the other player’s turn. The player with the most matches at the
end of the game wins.
Student Book (SB): pp.272-281
Teacher Resource (TR): pp.179185
Blackline Master (BLM): 5-3, 5-4
(AN2.1, AN2.2 AN2.3)
The information regarding roots
and powers from the Mathematics
1201 Curriculum Guide can be
found at:
www.ed.gov.nl.ca/edu/k12/
curriculum/guides/index.html
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
117
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Continued ...
Achievement Indicator:
AN2.4 Explain, using examples,
that (-x)2 = x2, x 2 = x and
2
x ≠ ± x ; e.g., 9 ≠ ±3 .
When asked to simplify x 2 , students may initially conclude that
2
x = x . Prompt student discussion using the following questions:
•
Does this happen with all positive values of x?
•
Does this happen with all negative values of x?
Ask students to simplify ( −5)2 . The following is a sample of student
2
2
1
( −5) = ( −5) 2 = ( −5) = −5
2
( −5) =
25 = 5
This is an opportunity for discussion around the correct answer when
the value of x is negative. Some students may challenge the incorrect
solution provided in column one above. It is important for students to
m
recognize that when a is a negative number, a n is not defined because
it is not possible to define such expressions consistently. Students can
1
compare this value when n is even to when it is odd. (i.e., ( −4) 2 = −4
is undefined under the set of real numbers but ( −8) = 3 −8 is defined).
1
3
This is a great opportunity for discussion as to what is the correct answer
when the value of x is negative. Although it is common for students to
replace x 2 with x, students must recognize that this is correct when
x ≥ 0. They should also realize that x 2 is equivalent to -x when x ≤ 0.
Consider the following:
When x = 4: 2
(4 )
2
16
x =
x =
2
2
( −4 )
2
16
x =
x =
2
2
2
x =4
x =4
∴ x
2
When x = -4:
=x
2
x = −x
≠x
Although students will be exposed to the concept of absolute value
later in this course, an introduction to absolute value is important
here. Ensure students understand the absolute value symbol produces a
result that is always positive, which is the principal square root. Hence
x 2 = x . Once this idea has been established, students can assume, for
the sake of simplicity, all variables are positive. Therefore x 2 = x .
∴ x
118
2
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Journal
•
Pre-Calculus 11
Ask students to respond to the following:
2
(i) Your friend stated that since 32 = 9 and (-3)2 = 9, then x is
always ±x. Use specific examples to show whether you agree
or disagree with his statement.
(AN2.4)
1
SB: pp.272-281
TR: pp.179-185
BLM: 5-3, 5-4
1
(ii) Explain why ( −16) 2 is undefined, where as ( −64) 3 is defined
under the set of real numbers.
(AN2.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
119
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Continued ...
Achievement Indicators:
AN2.5 Identify the values of the
variable for which a given radical
expression is defined.
In Mathematics 1201, students simplified radicals with numerical
radicands. They also determined the domain of a variety of functions
(RF1). This is their first exposure to simplifying expressions with
variable radicands. They will write the restrictions on the variable and
then write the expression in its simplest form.
Students should be comfortable recognizing square roots of nonnegative numbers are defined under the set of real numbers. Remind
students of this concept using any number of numerical examples such
as 4 = 2 while −4 is not defined. Expose students to examples
similar to 16 , 4 −81, 3 −27 , 3 8 , 6 −64 . Students should recognize that
if a radical has an even index, the radicand must be non-negative. If a
radical has an odd index, the radicand can be any real number, including
negative numbers.
This could lead into a discussion of what happens if the radicand is
variable in nature such as x . Reinforce the concept that the domain
of a square root function is limited to values for which the function
has meaning. Use examples to allow students to intuitively investigate
variable expressions as radicands and then progress to actually solving
an inequality algebraically. While students solved inequalities in Grade
9(9PR4), it may be necessary to review the various rules used to solve
inequalities.
with a variable radicand as a
with a variable radicand as a
120
Students will convert an entire radical with one variable to a mixed
radical and will then progress to multiple variables. They will also
work backwards and express a mixed radical with a variable as an entire
radical. When simplifying 4 x 7 , for example, students can rewrite as
4
4
4 3
3
x x = x x . For the radical to represent a real number,
x ≥ 0 because the index is a real number. When writing 2 x 3 5 x 2 as an
entire radical, the variable x in the expression 3 2 3 x 3 5x 2 can be any real
number since the index of the radical is an odd number.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Observation
Authorized Resource
•
Pre-Calculus 11
x
Ask students to complete the following chart and make
predictions about the restrictions on each variable.
x
1
x
1
x
x −1
x +1
1
x −1
1
x +1
-3
-2
-1
0
1
2
3
SB: pp.272-281
TR: pp.179-185
BLM: 5-3, 5-4
As teachers observe students’ work, use the following questions to
guide discussion:
(i) What values of x were undefined? What values of x were
defined?
(ii) Is the restriction different if the radical expression is in the
denominator?
(iii) How could solving inequalities help when determining the
restriction?
(AN2.5)
Performance
•
For the activity Sticky Bars, present students with a selected
response question where they could be expected to convert an
versa. The answer is anonymously recorded on a post it note
and submitted to the teacher. The teacher or student volunteer
arranges the sticky notes on the wall or whiteboard as a bar graph
representing the different student responses. Have a discussion
regarding why students may have selected the answers they did.
(AN2.6, AN2.7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
121
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Continued...
Achievement Indicator:
AN2.8 Perform one or more
expressions with numerical or
variable radicands before they move into multiplication and division.
questions such as the following:
•
What process is involved in the addition of monomials?
•
How is this expression similar to 2 7 + 3 7 ?
•
•
How can this process be applied to radicals?
expressions is comparable to combining variable expressions with like
terms. It is necessary for the radical to have the same index and the same
radicand. Students can then apply the same strategies with indices other
than 2.
The distributive property can also be applied when simplifying sums
2x + 3x in another form, namely (2 + 3)x. Similarly, 2 7 + 3 7 can be
expressed as (2 + 3) 7 = 5 7 . Check students’ understanding by asking
them to express −4 3 as the sum of two like radicals.
It is important for students to recognize that, even when adding or
subtracting like radicals, the solution can require further simplifying.
When subtracting the expression 3 8 − 7 8 , for example, the solution
−4 8 can be simplified to −8 2 . It would be good practice to
simplify the radical before like terms are combined. This is especially
beneficial when working with large numerical radicands. Students
should also be exposed to examples, such as, 2 18 + 3 50 − 5 2 or
4 3 54 + 3 −45 − 3 2 , where it is necessary to simplify one or more
to add 4 + 2 3 , for example, they may write 6 3 . Students should
be encouraged to check their answers by expressing the sum/difference
using the distributive property. Another common error occurs when
students incorrectly apply the operations of addition and subtraction
to the radicands. The value of 31 3 + 17 3 , for example, does not
equal 48 6 . This error may occur more often once students have been
introduced to the multiplication of radicals.
122
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
The voltage V required for a circuit is given by V = PR where
P is the power in watts and R is the resistance in ohms. How
many more volts are needed to light a 100-W bulb than a 75-W
bulb if the resistance for both is 100 ohms? They should solve the
problem in exact and approximate form.
(AN2.8)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SB: pp.272-281
TR: pp.179-185
BLM: 5-3, 5-4
123
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Continued...
Achievement Indicator:
AN2.8 Continued
Student learning will continue with addition and subtraction involving
•
similar to adding and subtracting numerical expressions with
radical values? Use examples similar to 8 x 3 − 4 2 x where
x ≥ 0, and −2 3 16x 4 + 5x 3 54 x and 2 45 − 6 20 to explain your
reasoning.
• What strategy did you use?
multiplication property of radicals, reiterate the relationship between a
radical and a power with rational exponents.
1
1
1
1
2
3 2 × 5 2 = (3 × 5) 2 = 15 ⇔ 3 × 5 = 3 × 5 = 15
In Mathematics 1201, students applied the laws of exponents to rational
exponents (AN3).
Encourage students to look for a pattern through the use of several
examples. The multiplicative property of radicals can then be introduced
to students, a × b = ab where a ≥ 0 and b ≥ 0. This property
can also be used to discuss why two radicals with the same index can
be multiplied. Although students were exposed to this property in
Mathematics 1201, it was used exclusively for the purpose of expressing
Students should be given an opportunity to further explore the product
rule for radicals (i.e., c a × d b = cd ab ) and the commutative
property of multiplication. Ask them to rewrite the expression
1
1
2(3 2 ) × 5(6 2 ) using radicals to generalize a pattern. Students’ workings
may vary but the result should be the same. Consider the following
sample:
1
1
2(3 2 ) × 5(6 2 )
2( 3 ) × 5( 6 )
1
1
(2 × 5)(3 2 × 6 2 )
1
(10)(3 × 6) 2
10(18)
1
2
(2 × 5)
(10)
(
(
3× 6
3×6
)
)
10 18
Students will also be exposed to examples where the index is not 2.
They should be introduced to the rule c n a × d n b = cd n ab , where n
is a natural number and c, d, a, and b are real numbers. They should
recoginze that if n is even then a ≥ 0 and b ≥ 0.
124
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
Create centres in the classroom containing worked solutions
of simplifying radical expressions with numerical or variable
radicands. Students will participate in a carousel activity where
they are asked to move throughout the centres to identify and
correct errors. Samples are shown below:
Pre-Calculus 11
SB: pp.272-281
TR: pp.179-185
BLM: 5-3, 5-4
25 5 + 13 5
(i)
= 38 10
(ii)
3
18 x + 2 8 x
3
= 3 2x + 4
= 7 4x
= 14 x
5.2 Multiplying and Dividing
3
2x
3
SB: pp.282-293
TR: pp.186-193
BLM: 5-3, 5-5
6
3
(AN2.8)
Paper and Pencil
•
Ask students to simplify each of the following:
(i) 2 18 + 9 7 − 63
3
3
3
(ii) 6 32 x − 5 8 x + 3 2 x
3
4
(iii) −5 3 256x + 192 x
(AN2.8)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
125
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Continued...
Achievement Indicator:
AN2.8 Continued
Advise students that they are less likely to make simplification errors
if they simplify radicals before multiplying. An example such as the
following could be used to illustrate the two methods:
Multiply first: 80 × 12 = 960 = 8 15
Simplify first: 80 × 12 = 4 5 × 2 3 = 8 15
Ask students which method they prefer and why.
Students will also multiply radicals using the distributive property.
It may be helpful to walk them through the similarities between
multiplying radical expressions and multiplying polynomials.
In order to avoid common errors, this would be a good opportunity to
reinforce the commutative and associative property of multiplication.
When multiplying 5 × 3 , for example, students may write 15 . They
may understand their error if they apply the commutative property to
rewrite the expression as 3 5 . Another error occurs when students are
asked to multiply an expression such as 3 5 × 6 and their result is
90 . This can be avoided if the associative property is used to re-order
the expression as 3 5 × 6 = 3 30 .
(
)
Students will explore how multiplying and dividing algebraic
expressions with radicals is similar to multiplying and dividing
similar to −4 12 × −2 18 and ( −4 x ) × ( −2 x 2 ) where x ≥ 0, and
3
2
( −4 3 x ) × ( −2 x ) to explain their reasoning.
126
•
What strategy did you use?
•
When is it necessary to use the distributive property to multiply
expressions that contain radicals? Create an example and show
the solution.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to simplify the following:
(i) ( 2 +
6)
5.2 Multiplying and Dividing
2
(ii) (3 8 − 4 )(2 + 7 3 )
(iii) ( 20 +
SB: pp.282-293
TR: pp.186-193
BLM: 5-3, 5-5
24 )(3 12 − 5 32 )
(iv) -2 3 12(4 3 2 − 5 3 9 )
(v) ( −3 x )(6 x 3 )
( vi) (3 x + 2)(3 − 5 x )
(AN2.8)
Performance
•
Ask students to participate in Commit and Toss. Provide
students with a selected response problem as shown below. They
anonymously commit to an answer and provide a justification for
the answer they selected. Students crumble their solutions into a
ball and toss the papers into a basket. Once all papers are in the
then move to the corner of the room designated to match their
selected response. In their respective corners, they should discuss
the similarities or differences in the explanations provided and
report back to the class.
www.k12pl.nl.ca/seniorhigh/
introduction/math2201/
classroomclips.html
demonstrates students performing
expression using the activity
Commit and Toss.
Express ( 3 − 2 )2 in simplest form.
(A) 1
(B) 5 − 2 6
(C) 1 − 2 3
(D) 5
(AN2.8)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
127
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Continued...
Achievement Indicator:
AN2.9 Rationalize the
denominator of a rational
expression with monomial or
binomial denominators.
Division of radicals and the rationalization of the denominator are
new concepts for students. The denominator can be a monomial or a
binomial.
The rules of exponents should be integrated when introducing the
( )
4
9
1
2
1
2
= 41 ⇔
92
4
9
( )
8
27
= 4
9
1
3
1
3
8 = 38
= 8 1 ⇔ 3 27
3
27
27 3
examples, reinforce to students that they can only divide radicals that
have the same index. They should recognize the quotient rule of radicals
states that the nth root of a quotient is the quotient of the nth root. In
n
other words, n ba = n ba where n is a natural number, a and b are real
numbers, and b ≠ 0. Remind students if n is even then a ≥ 0 and b > 0.
Students should be exposed to a variety of cases when simplifying
radicals with fractions. Include examples such as the following:
25
25 5
=
=
4
2
4
12
6
=
12
6
=
2
12
5
=
12
5
=
2 3
5
the following questions:
•
Is the denominator a perfect root?
•
Can the numerator and denominator divide into a rational
number?
•
Will the denominator have a radical when simplified?
Students will develop a strategy for converting a fraction that has
in the denominator. Rationalizing the denominator provides a standard
notation for expressing results. Using an example such as 2 3 , ask them
5
what they can multiply the numerator and denominator by that results
in a rational expression.
128
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
In groups of two, ask students to participate in the activity Pass
the Problem. Give each pair a simplification problem that involves
rationalizing the denominator. Ask one student to write the first
line of the solution and pass it to the second student. The second
student will verify the workings and check for errors. If there
is an error present, ask students to discuss the error and why
it occurred. The student will then write the second line of the
solution and pass it their partner. This process continues until the
solution is complete.
Pre-Calculus 11
5.2 Multiplying and Dividing
SB: pp.282-293
TR: pp.186-193
BLM: 5-3, 5-5
(AN2.9)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
129
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN2 Continued...
Achievement Indicator:
AN2.9 Continued
Expose students to examples where the index is greater than 2. For
3
2
example, when students simplify 3 5 , they may initially multiply the
numerator and denominator by the term 3 5 . They should recognize
a radical still remains in the denominator. When rationalizing the
denominator, the root should guide student choices. They multiply
numerator and denominator by a value where the powers of the
2
3
denominator will equal the index of the root. Therefore, 2 ⎛⎜ ( 5 ⎞⎟
3 1
5 ⎝ ( 52 ⎠
results in a rational denominator.
3
3
This strategy can be applied to variable monomial denominators. Given
4
an example, such as 4 7 x2 2 , the index will help students determine
27 x y
what expression they should use to rationalize the denominator. In other
words, 4 7 x ⎛ 3 x y ⎞ will produce a rational denominator.
⎟
4 33 x 2 y 2 ⎜
⎝ 3x y ⎠
4
1
2
2
4
1
2
2
2 3
Examples such as 7 5 , where the denominator is a mixed radical should
be included. Encourage students to think about what they would have
to multiply by to rationalize the denominator. The initial tendency may
be to multiply the numerator and denominator by 7 5 . Although it
is not the most efficient strategy, it is correct. However, the resulting
expression will have to be simplified.
5
They should realize multiplying by 5 will rationalize the denominator.
Students should first be exposed to expressions with entire radicals in
both the numerator and denominator. This should then be extended to
include mixed radicals and examples where there is more than one term
in the numerator. It is the students’ choice whether they simplify before
or after they rationalize the denominator. When working with larger
numbers, however, simplifying first would allow them to work with
smaller numbers.
AN2.10 Describe the relationship
between rationalizing a binomial
denominator of a rational
expression and the product of the
factors of a difference of squares
expression.
130
1
Students will simplify an expression such as 2 + 3 where it is necessary
to rationalize a binomial denominator. Initially, they may think they can
multiply the numerator and denominator by 3 , as in their previous
work with monomial denominators. As they explore this, however, they
should discover that 2 +1 3 × 33 = 2 33+ 3 , and the denominaor still contains
a radical. Prompt them to multiply the numerator and denominator by
the conjugate of 2 + 3 . This is similar to multiplying the factors of
a difference of squares expression. Students should also be exposed to
expressions where both terms in the binomial are irrational. Expressions
involving variable numerators and denominators should be explored in a
similar fashion.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
Pre-Calculus 11
• Ask students to rationalize the denominator for each of the
expressions:
(i)
(ii)
(iii)
(iv)
5.2 Multiplying and Dividing
7
1+
SB: pp.282-293
TR: pp.186-193
BLM: 5-3, 5-5
7
2+3 5
2 5 −4
b
a+
b
x +3
y
y −x
(AN2.9, AN2.10)
Journal
•
When asked to rationalize the denominator in the expression
4
2 + 7 your friend said he could just multiply the expression by
Ask students to explain why this would not work.
7
7.
(AN2.9, AN2.10)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
131
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN3 Solve problems that involve
square roots).
[C, PS, R]
Achievement Indicators:
Students will solve radical equations involving square roots. It is
intended that the equations will have no more than two radicals and the
radicand will contain variables that are first and second degree. Students
will be responsible for solving equations resulting in a linear or quadratic
equation.
AN3.1 Determine any restrictions
on values for the variable in a
Earlier in this unit, students determined restrictions within the real
number system on a variable in a radical that had an even index. They
will continue to determine the restrictions before solving a radical
equation.
AN3.2 Determine the roots of
and explain the process used to
solve the equation.
This would be a good opportunity for students to compare a radical
equation to its graph to develop an understanding of restrictions for the
variable and the points that satisfy the equation. Using a table of values,
ask students to graph y = x .
.
y
AN3.3 Verify, by substitution,
that the values determined
algebraically are roots of the
equation.
AN3.4 Explain why some roots
equation algebraically are
extraneous.
4
(4,2)
2
-4
-2
2
-2
x
4
-4
Focusing on the point (4,2), ask students how they would algebraically
solve the equation given only the y-coordinate 2. When solving 2 = x ,
the value of x can be determined by inspection. Students could also use
the idea that squaring a number is the inverse operation of taking the
square root. This technique may seem straight forward but students
should be exposed to equations where the value is not a solution to the
original equation. Consider the example 2 x − 1 = −3 . The left-hand
side of the equation calls for a positive square root, but the right-hand
side of the equation is negative. Intuitively, there can be no solution.
y
4
y = 2x – 1
2
-4
-2
2
-2
-4
132
4
x
y = -3
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
• Ask students to explain why the domain of 2 x − 5 is x ≥ 52 , while
1
the domain of 2 x −5 is x > 52 . They should illustrate their answers
graphically with technology and then algebraically.
Pre-Calculus 11
•
(AN3.1)
Ask students to state the restrictions, solve and check for
extraneous roots.
(i) 4x = 8
(ii) x +4 =5
SB: pp.294-303
TR: pp.194-199
BLM: 5-3, 5-6
(iii) 2 x − 3 = −2
(AN3.1, AN3.2, AN3.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
133
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN3 Continued...
Achievement Indicators:
AN3.1, AN3.2
AN3.3, AN3.4 Continued
Students should recognize that given the y-coordinate of -3, there is
no possible x-coordinate that satisfies the equation. However, squaring
both sides of the equation results in x = 5. This cannot be correct, as
both substitution and the graph have shown that the equation has no
solution. This is a great lead in to the concept of extraneous roots.
Extraneous roots occur because squaring both sides and solving the
equation may result in roots that do not satisfy the original equation.
As students solve equations, reinforce the importance of checking that
the value is a solution to the original equation. Any extraneous roots are
Another strategy students can use to solve a radical equation involves
applying a power that will eliminate the radical expression on both
sides of the equation. To solve x + 1 = 4 , for example, students would
first rewrite
the radical expression with a rational exponent, resulting
in
1
2
( x + 1) 2 = 4. Using the multiplicative inverse gives ( x + 1)
= (4)2 and
(
1
2
)
Students should be exposed to equations where the radical is not
isolated. In such situations, make a comparison to solving a linear
equation. They should recognize that solving an equation such as
3 + 2 x + 1 = 7 follows a process that is similar to solving 3 + x = 7.
Rather than squaring both sides of a radical equation, students
sometimes mistakenly square the individual terms. When solving
3 + 2 x + 1 = 7, for example, they may not isolate the radical. Squaring
each term results in the incorrect equation 32 + ( 2 x + 1 )2 = 7 2.
The following illustration could be used to reinforce why squaring
individual terms of an equation is not the same as squaring both sides of
the equation.
3+4=7
32 + 42 = 72
9 + 16 = 49
25 ≠ 49
134
3+4=7
(3 + 4)2 = 72
72 = 72
49 = 49
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Ask students to participate in Commit and Toss. Provide
students with a selected response problem as shown below. They
anonymously commit to an answer and provide a justification for
the answer they selected. Students crumble their solutions into a
ball and toss the papers into a basket. Once all papers are in the
then move to the corner of the room designated to match their
selected response. In their respective corners, they should discuss
the similarities or differences in the explanations provided and
report back to the class.
SB: pp.294-303
TR: pp.194-199
BLM: 5-3, 5-6
What is the value of x in the equation −3x + 6 = 6 ?
(A)
(B)
(C)
(D)
-10
-2
0
10
(AN3.1, AN3.2, AN3.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
135
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN3 Continued...
Achievement Indicators:
AN3.1, AN3.2
AN3.3, AN3.4 Continued
The area model can also be used to solve radical equations. In Grade
8, students viewed the area of the square as the perfect square number,
and the side length of the square as the square root (8N1). Recall that
if a square has an area of 4, then its side has a length of 2. Similarly,
if a square has an area of 3, then its side has a length of 3 . Teachers
should prompt discussion about the side length of a square if its area is
x. Consider the following example: Solve x − 7 = 3 .
Using the area model, students label the dimensions of one square as
x − 7 and the dimension of the other square as 3. They can then
determine the area of each square.
x–7
3
A = x–7
A = 9
=
Students should recognize that the goal is to determine the value of x
which results in the same area for both squares. Solving the equation
x - 7 = 9 results in x = 16. This representation helps students visualize
what each equation is describing. Encourage them to check their
answers by substituting the value back into the original equation.
equations. Use an example such as x − 9 = 4 and the following
questions to promote student discussion:
2
•
What type of equation will this radical equation result in?
•
How is the radicand similar yet different to those studied to date?
•
How many solutions might this equation have?
Using technology, present the graph of the corresponding function
y = x − 9 for students to analyze.
2
y
10
8
6
y=4
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10
x
-4
From the graph, students should observe that the domain for the
function is x ≥ 3 or x ≤ -3. This will help them determine whether
x = ±5 are the solutions to the equation.
136
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Ask students to respond to the following:
Pre-Calculus 11
The following steps show how a student solved the equation
3 + 2 n + 4 = 5 . Ask students if the final answer is correct and
whether the student should receive full marks for the solution.
SB: pp.294-303
TR: pp.194-199
BLM: 5-3, 5-6
3+2 n +4 =5
5 n+4 =5
n +4 =1
(
n+4
)
2
= 12
n +4 =1
n = −3
(AN3.1, AN3.2, AN3.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
137
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN3 Continued...
Achievement Indicators:
AN3.1, AN3.2
AN3.3, AN3.4 Continued
Students should proceed to solve the radical equation algebraically
by squaring both sides of the equation. They can solve the resulting
quadratic equation using the method of their choice (i.e., graphing,
factoring, completing the square or the quadratic formula).
Students will also solve equations that involve two radical expressions.
As students square both sides in an equation such as x + 7 = x + 1
they should recognize the resulting equation still contains a radical.
Therefore, they will need to repeat the process of isolating the radical
term and squaring both sides of the equation again. Continue to remind
students of the importance of checking for extraneous roots when
AN3.5 Solve problems by
modeling a situation using a
Students will be exposed to application problems where the equation
may contain a radical that is a square root. They will solve for the
unknown variable by squaring both sides of the equation. Provide
questions:
•
138
Collision investigators can approximate the initial velocity, v, in
kilometres per hour, of a car based on the length, l, in metres,
of the skid mark. The formula v = 12.6 l + 8, where l ≥ 0
models the relationship. What length of skid is expected if a
car is travelling 50 km/hr when the brakes are applied? How is
knowledge of radical equations used to solve this problem?
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
Ask students to solve each of the following equations. Remind
them to check for extraneous roots.
(i)
(ii)
3x − 5 − 2 = 3
Pre-Calculus 11
SB: pp.294-303
TR: pp.194-199
BLM: 5-3, 5-6
2
x − 8x = 3
(iii) 2 x + 4 x = 3 − 2 x
(AN3.1, AN3.2, AN3.3)
•
(i) The period T (in seconds) is the time it takes a pendulum to
make one complete swing back and forth. This is modelled
L
by T = 2π 32
, where L is the length of the pendulum in
feet. Ask students to determine the period of the pendulum
if its length is 2 ft.
(AN3.5)
(ii) The radius of a cylinder can be found using the equation
r = πVh where r is the radius, V is the volume, and h is the
height. A cylindrical tank can hold 105.62 m3 of water. If the
height of the tank is 2 m, what is the radius of its base?
(AN3.5)
(iii) The surface area (S) of a sphere with radius r can be found
using the equation S = 4πr2.
(a) Using the given equation, how could you find the radius of a sphere given its surface area? Write the
equation.
(b) The surface area of a ball is 426.2 cm2. What is its radius?
(AN3.5)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
139
140
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Rational Expressions and Equations
Suggested Time: 12 Hours
RATIONAL EXPRESSIONS AND EQUATIONS
Unit Overview
Focus and Context
In this unit, students will simplify a rational expression and determine
the non-permissible values. They will perform the operations on
rational expressions (addition, subtraction, multiplication, division).
Students will solve problems that involve rational equations. They will
determine the solution to a rational equation algebraically and identify
the non-permissible values.
Outcomes
Framework
GCO
Develop algebraic reasoning and number sense.
SCO AN4
Determine equivalent forms of
rational expressions (limited to
numerators and denominators
that are monomials, binomials or
trinomials).
SCO AN5
Perform operations on rational
expressions (limited to
numerators and denominators
that are monomials, binomials or
trinomials).
SCO AN6
Solve problems that involve
rational equations (limited to
numerators and denominators
that are monomials, binomials or
trinomials).
142
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
Process Standards
Key
[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
Algebra and Number
Algebra and Number
Relations and Functions
AN5 Demonstrate
an understanding of
commonfactors and trinomial
factoring,concretely, pictorially
and symbolically.
AN4 Determine equivalent forms
of rational expressions (limited
to numerators and denominators
that are monomials, binomials or
trinomials).
RF14 Graph and analyze
rational functions (limited to
numerators and denominators
that are monomials, binomials or
trinomials).
[C, ME, R]
[CN, R, T, V]
[C, CN, R, V]
AN5 Perform operations on
rational expressions (limited to
numerators and denominators
that are monomials, binomials or
trinomials).
[CN, ME, R]
AN6 Solve problems that involve
rational equations (limited to
numerators and denominators
that are monomials, binomials or
trinomials).
[C, PS, R]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
143
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN4 Determine equivalent forms
of rational expressions (limited
to numerators and denominators
that are monomials, binomials or
trinomials).
In Grade 9, students solved problems that involved arithmetic
operations on rational numbers (9N3). They will now be introduced
to rational expressions limited to numerators and denominators that
are monomials, binomials and trinomials. They will simplify them and
determine the non-permissible values.
[C, ME, R]
Achievement Indicators:
AN4.1 Explain why a given
value is non-permissible for a
given rational expression.
AN4.2 Determine the nonpermissible values for a rational
expression.
A rational expression is any expression that can be witten as the quotient
P( x )
of two polynomials, in the form Q ( x ) where Q(x) ≠ 0. To begin work
with rational expressions, provide
students with several examples of
2
x
expressions, such as 45 , 2yx , xx +−14 , 5 , 2π , 2 y and ask them to identify
and explain why an expression is or is not a rational expression. It should
be pointed out to students that all rational expressions are algebraic
fractions but not all algebraic fractions are rational expressions. In the
above list, for example, 45 , 5 , and 2π are not rational expressions.
Non-permissible values are the values of a variable that make the
denominator of a rational expression equal zero. In Grade 7, students
were introduced to the concept of why a number cannot be divided by
zero (7N1). Students should first find the non-permissible values of a
rational expression where the denominator is a first degree polynomial
and then progress to second degree polynomials. Given the expression,
x
x + 2 , for example, use the following to promote student discussion
around non-permissible values:
•
Using inspection, what value of x would make the denonimator
zero?
•
Explain why this value of x is called a non-permisible value.
•
Fill in the following table of values. What do you notice? What is
the domain of the expression x x+ 2 ?
x
y
-3
-2
-1
0
1
3
•
Write a general rule for determining the non-permissible value for
a first degree denominator.
•
Can the numerator be equal to zero? Explain your reasoning.
Students should notice that the non-permissible value of
x
can be written as x + 2 , x ≠ -2.
144
2
x
x +2
is -2. This
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
Ask students to write a rational expression for the following
non-permissible values of 0, -2 and 3. They should compare their
Pre-Calculus 11
(AN4.2)
Interview
•
Ask students to explain why x = 2 is a non-permissible value for
3x
x −2 .
(AN4.2)
6.1: Rational Expressions
Student Book (SB): pp.310-321
Teacher Resource (TR): pp.209216
BlackLine Master (BLM): 6-4, 6-5
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
145
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN4 Continued ...
Achievement Indicators:
AN4.1, AN4.2 Continued
A common error occurs when students generalize that the nonpermissible value is zero rather than looking at the value(s) of x that
produce a denominator of zero. Encourage them to substitute the
non-permissible value(s) for x back into the denominator to verify the
denominator results in zero.
Students should also be exposed to rational expressions where the
denominator is a second degree polynomial. Discuss an expression, such
3 x −12
•
Using inspection, what value(s) of x would make the denominator
zero?
•
What other strategies can be used to solve the quadratic equation?
•
Are there any rational expressions without non-permissible values?
To solve 3x2-12, students may remove the greatest common factor
(GCF) and apply the zero product property, apply the quadratic formula
or the square root property. Question students as to which method is
more efficient and why.
If students use the square root property, they divide the equation
3x2 = 12 by 3 and solve x2 = 4. Some may mistakenly write the nonpermissible value as x = 2 rather than x = ±2. Another error occurs when
students factor 3x2 - 12 = 0 as 3(x - 2)(x + 2) and include 3 as a nonpermissible value. They may also incorrectly factor the expression in
the denominator, whether it is a binomial or trinomial. Remind them
to verify their work by expanding the product of the factors using the
distributive property.
Students should be exposed to rational expressions with more than one
variable. If the expression 2x - 3y is in the denominator, for example,
the non-permissible values can be found by solving the equation
2x - 3y = 0 for x or y. Students have this option unless it is specifically
stated in the problem to solve for a particular variable.
146
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Ask students to work in groups to participate in the Domino
Game. Provide each group with10 domino cards. One side of the
card should contain a rational expression, while the other side
contains non-permissible values for a different rational expression.
The task is for students to lay the dominos out such that the nonpermissible values on one card will match with the correct rational
expression on another. They will eventually form a complete loop
with the first card matching with the last card. A sample is shown
below:
3x
2 x−1
x ≠ 13
x+2
3 x −1
x ≠ − 12
7x
2 x+1
6.1: Rational Expressions
SB: pp.310-321
TR: pp.209-216
BLM: 6-4, 6-5
x≠2
(AN4.1, AN4.2)
Journal
•
Ask students to respond to the following:
x +3
What are the non-permissible values for x 2 − 16 ?
I think the nonpermissible value
is 4.
I think the nonpermissible
values are – 4
and 4.
I think the nonpermissible
values are – 4,
-3 and 4.
(AN4.1, AN4.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
147
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN4 Continued ...
Achievement Indicators:
AN4.1, AN4.2 Continued
It is important for students to differentiate between non-permissible
functions to model situations. These are values that do not make sense
in a given context. Students will continue to work with inadmissible
values for a variable in a rational expression. If a boat traveled 20 km
with a speed of x km/h, for example, the time taken for the trip would
be represented by . If students 20
are asked to determine the slowest
x
speed the boat can travel, they should recognize that the non-permissible
value is 0 but the inadmissible value is x < 0.
AN4.3 Compare the strategies
for writing equivalent forms
of rational expressions to the
strategies for writing equivalent
forms of rational numbers.
In Grade 7, students developed skills in writing equivalent positive
rational numbers (7N7). Students will apply these strategies to rational
expressions. This concept is essential when adding and subtracting
rational expressions later in this unit.
AN4.4 Determine a rational
expression that is equivalent
to a given rational expression
by multiplying the numerator
and denominator by the same
factor (limited to a monomial
or a binomial), and state the
non-permissible values of the
equivalent rational expression.
Students should recognize that they can multiply or divide a rational
expression by 1 without changing its value. A rational expression is
not equivalent to another rational expression if their restrictions are
different. Guide students through the following activity:
•
Consider the rational number 23 . Ask students to write a rational
number by multiplying both the numerator and denominator by
3, by 5, and by -4? Did the value of their fraction change?
•
Consider the rational expression x4 where x ≠ 0. Ask students to
write a rational expression by multiplying both the numerator and
denominator by 2, by x, and by x + 1. Did any of their expressions
produce a new restriction?
Using substitution, ask students to verify if the expressions are
equivalent. When the expressions x4 , x ≠ 0 and 4( x +1) , x ≠ 0,-1 are
x ( x +1)
compared, they are both undefined at x = 0. When x = -1, however, the
expression x4 simplifies to -4 while the expression x4(( xx ++1)
is undefined.
1)
Since the expressions are not equal for the same value of x, the
expressions are not equivalent.
Although graphing rational functions is not an outcome in this
course, teachers could, as an alternative, prove rational expressions are
equivalent for all permissible values of the variable by showing students
the graphs of these functions.
148
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to complete the following table:
Are the expressions
equivalent?
x +3
x −4
5
x −5
x +2
x −3
and
and
and
Yes
No
SB: pp.310-321
TR: pp.209-216
BLM: 6-4, 6-5
4 x + 12
4 x − 16
5x + 25
2
x − 25
3x + 6
2x − 6
(AN4.3, AN4.4)
Journal
•
6.1: Rational Expressions
x +1)
Your friend thinks the expressions x2−x3 and ( x2−x3)(
are
( x +1)
equivalent. Ask students to explain why these expressions are not
equivalent?
(AN4.3, AN4.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
The resource Pre-Calculus 11
discusses equivalent rational
expressions on p.313. It
is important to note that
7s
7s 2
s − 2 and s ( s − 2 ) are not equivalent
rational expressions since a new
149
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN4 Continued ...
Achievement Indicators:
AN4.3, AN4.4 Continued
When writing an equivalent expression, caution students to use the
x
distributive property appropriately. When simplifying x + 4 × 22 , for
example, students may incorrectly write 2 x2 x+ 4 or x2+x8. To avoid this
error, encourage students to place brackets around the binomial when
multiplying.
AN4.5 Simplify a rational
expression.
Simplifying a rational expression to lowest terms mirrors the process of
simplifying fractions. In both cases, common factors in the numerator
and denominator form a ratio of one and can be simplified.
AN4.6 Explain why the nonpermissible values of a given
rational expression and its
simplified form are the same.
Discuss with students the benefit of simplying rational expressions,
whether it be for evaluating or performing operations. Ask students
2
to evaluate the expression x +x 4 x where x ≠ 0 at x = 2. Then evaluate
the expression x + 4 where x ≠ 0 at x = 2. Students should answer the
following questions:
( )
•
What is the result when substituting the value into the original
unsimplified expression?
•
What is the result when substituting the value into the simplified
expression?
•
Why were the results the same?
•
What is the benefit of simplifying an expression before
substituting values for the variables?
•
Why does the simplified expression include a non-permissible
value?
Students should recognize one of the benefits of simplifying an
expression is to create an equivalent expression that is easier to evaluate.
Ask them why the domain of a rational expression is always determined
before and after the expression is simplified. A simplified rational
expression, for example, may not have any non-permissible values.
However, the simplified expression must retain the non-permissible
values of the original expression for both to be equivalent.
150
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Ask students to participate in the following activities:
(i) Divide the class into two groups. One group will be given
rational expressions and the other group will be given the
associated rational expression in simplest form. Ask students
to find a partner who has a rational expression equivalent to
theirs.
(AN4.5)
(ii) Students create a unique Bingo card for Rational Expression
Bingo. Distribute a blank Bingo card to each student.
Teachers should predetermine various expressions involving
rational expressions they would like students to simplify.
The expressions should be placed in a bag, with simplified
expressions on the board. Ask students to write one of the
simplified expressions in each sqaure. The center square
should remain a "free" space. The teacher pulls an expression
from a bag. Students simplify the expression, find its value
on their card and cross it off. The first person with a straight
line or four corners wins, or the first person with an X or a T
on the Bingo card could win.
(AN4.5)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
6.1: Rational Expressions
SB: pp.310-321
TR: pp.209-216
BLM: 6-4, 6-5
151
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN4 Continued...
Achievement Indicator:
AN4.7 Identify and correct errors
in a given simplification of a
rational expression, and explain
the reasoning.
It is beneficial to have students analyze solutions that contain errors.
Along with providing the correct solutions, they should be able to
identify incorrect solutions, including why errors might have occurred
and how they can be corrected.
When simplifying rational expressions, students may cancel
terms
x 2 +x
x 2 +x
rather than factors. They may simplify, for example, x 2 −1 as x 2 −1
resulting in -x. To help students see this error ask them to make a
8
= 55++73 and
comparison with a numerical rational expression such as 12
8
= 55++73 = 73 . Ask them if 128 is equal to 73 . Students should realize
12
that cancelling a portion of the factor is incorrect. Another error occurs
when students omit a numerator of 1 after the rational expression is
simplified. They mistakenly simplify 6x3 , for example, as 2x. Encourage
students to check the reasonableness of their answer by rewriting the
the expression as 63 i x1 .
152
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Observation
Authorized Resource
•
Pre-Calculus 11
Set up centres containing examples of incorrect simplified rational
expressions and their non-permissible values. Ask students to
move around the centres to identify and correct the errors. A
sample is shown below:
8 x − 12
, x ≠ 0, 23
2
6x − 4 x
4(2 x − 3)
6.1: Rational Expressions
SB: pp.310-321
TR: pp.209-216
BLM: 6-4, 6-5
2 x (3x − 2)
4
(1)
2x
2 x , x ≠ 0, 23
(AN4.7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
153
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN5 Perform operations on
rational expressions (limited to
numerators and denominators
that are monomials, binomials or
trinomials.
In Grade 9, students solved problems involving operations on rational
numbers (9N3). This will now be extended to adding, subtracting,
multiplying and dividing rational expressions with numerators and
denominators limited to monomials, binomials and trinomials.
[CN, ME, R]
Achievement Indicators:
AN5.1 Compare the strategies
for performing a given operation
on rational expressions to the
strategies for performing the same
operation on rational numbers.
AN5.2 Determine the nonpermissible values when
performing operations on rational
expressions.
Multiplying and dividing rational expressions is very similar to the
process students used to multiply and divide rational numbers. Using
2
examples such as 12
× 10 and xx2 −−49x × xx −−43 , ask students to simplify
25 21
and find the product for each. They should think about whether the
strategy for multiplying rational expressions is the same as the strategy
for multiplying rational numbers. Ask them to also consider at what step
the non-permissible values are determined.
It is important for students to recognize the importance of factoring
the numerator and denominator of the rational expression, if possible,
•
AN5.3 Determine, in simplified
form, the product or quotient of
rational expressions.
2
Find the product of x x2 −4 × x x+ 2 using two different strategies.
Which strategy is more efficient? Why?
Reinforce that multiplication of rational expressions follows the same
procedure as multiplying rational numbers, but with the added necessity
of determining the non-permissible values for the variables.
Provide an opportunity for students to compare the division of rational
numbers to division of rational expressions. Students sometimes forget
to identify the non-permissible values for the numerator of the divisor in
a division statement. Reinforce the importance of this step through the
use of examples.
154
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to create an activity sheet where the column on the
left contains operations with rational expressions and the column
on the right contains the non-permissible values (not in the same
order). Students will then exchange their sheets. The task is to
match each expression with its correct non-permissible values. (the
non-permissible values may match more than one expression on
the left and may not match any).
(AN5.2)
•
6.2: Multiplying and Dividing
Ask students to work in groups to complete the following table.
Rational Expressions
Students should explain the similarities between finding the lowest
common denominator (LCM) of two rational numbers versus two SB: pp.322-330
TR: pp.217-223
rational expressions.
BLM: 6-4, 6-6
Rational
LCM
Rational Expres- LCM
Similarities
Number
sion
4
5
1
5
+
−
7
12
3
6
5
2x − 1
7
4x
15
x −3
+
3
8
−
+
2x − 1
5
6x − 18
2
2
−2
x − 36
+
4
3x + 18
(AN5.1)
Journal
•
Sean stated that the permissible values for the quotient and the
2
2
8
product of the expressions x x+2 4−x16+ 3 and xx2 −−72xx+−12
are the same.
Ask students if they agree or disagree with his statement. They
(AN5.2, AN5.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
155
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN5 Continued ...
Achievement Indicators:
AN5.4 Determine, in simplified
form, the sum or difference of
rational expressions with the same
denominator.
AN5.5 Determine, in simplified
form, the sum or difference of
rational expressions in which
the denominators are not the
same and which may or may not
contain common factors.
AN5.2 Continued
In Mathematics 1201, students were introduced to the lowest common
multiple for a set of numbers (AN1.3). Students should compare finding
the lowest common denominator of rational numbers to finding the
lowest common denominator of rational expressions. Allow students
to discover the different situations that occur when finding the lowest
common denominator of two numbers and then compare this to
rational expressions. Consider the following table:
Rational Number Situation
Rational Expression
the denominators are the
3 2
x2
−
− 1
7 7
x +1 x +1
same
one denominator is a
1
3
+5
− 1
12 6
x +5 4 x + 20
multiple of the other
the denominators have no 3 + 4
2 7
+
2x
x −1
3 2
common factors
the denominators have a
7
5
+ 1
+1
x 2 −9 4 x +12
14 6
common factor
expressions:
•
How do you find the lowest common denominator? Why is it
beneficial to simplify the expression before finding the lowest
common denominator?
•
What are the non-permissible values?
•
Can you list other examples that fit each situation?
Similar to rational numbers, rational expressions can be added if they
have common denominators. Once students determine the lowest
common denominator, they should rewrite each rational expression with
that common denominator.
A common student error involves adding or subtracting the numerators
without first writing the fractions with a common denominator. For
example, students mistakenly add x5 + 23 as x 8+ 2 . Remind students to
be careful when subtracting rational expressions. They sometimes forget
to distribute the negative sign when there is more than one term in
3x −2
2 x −4
the numerator. For example, ( x + 2 )( x − 2 ) − ( x + 2 )( x − 2 ) is often written as
3 x − 2 − 2 x −4
( x + 2 )( x − 2 ) . Encourage students to use brackets to help them avoid this
mistake.
156
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to simplify the rational expressions:
(i)
x +7
2 x +14
(ii)
2 x −6
x 2 − x −6
− −35xx− 21 −
Rational Expressions
3 x +12
x 2 + x −12
(AN5.5)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SB: pp.331-340
TR: pp.224-231
BLM: 6-4, 6-7
157
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN5 Continued ...
Achievement Indicator:
AN5.6 Simplify an expression
that involves two or more
operations on rational expressions.
A complex fraction is an example of an expression involving two or
more operations on a rational expression. In order to avoid errors,
students should place brackets appropriately and use the order of
operations correctly.
Provide examples to students illustrating various strategies to simplify
an expression containing a complex fraction. Students may first simplify
both the numerator and denominator, invert and multiply and then
simplify. Consider the example :
1 + 1
x+2
x −2
x
+ x +1 2
x2 −4
=
x −2
( x + 2)( x − 2)
x
( x + 2)( x − 2)
x+2
+ ( x + 2)(
x − 2)
+
x −2
( x + 2)( x − 2)
x − 2)
2x
= ( x + 2)(
× ( x +22)(
=
x − 2)
x−2
2x
2 x −2
=
x
x −1
As an alternative, students can multiply the entire expression by the
common denominator divided by itself. This common denominator is
obtained by considering all existing denominators in the expression.
⎛ x 1+ 2 + x 1−2 ⎞ ( x + 2)( x −2)
=
⎜ x
⎟
⎜ 2 + 1 ⎟ ( x + 2)( x −2)
⎝ x −4 x + 2 ⎠
=
1
1
x + 2 ( x + 2)( x − 2) + x − 2 ( x + 2)( x − 2)
x
1
( x − 2)( x + 2) ( x + 2)( x − 2) + x + 2 ( x + 2)( x − 2)
x − 2 +x + 2
x + x −2
=
2x
2 x −2
=
x
x −1
As students work through the two possible strategies, they should think
about efficiency and what will work best for any complex fraction.
158
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Presentation
Authorized Resource
•
Pre-Calculus 11
Ask students, working in pairs or small groups, to create two
rational expressions. The first rational expression should contain
each of the operations. The other expression would involve two
or more operations on rational expressions. They should solve the
expressions and present their findings to the class.
(AN5.3, AN5.4, AN5.5, AN5.6)
Rational Expressions
SB: pp.331-340
TR: pp.224-231
BLM: 6-4, 6-7
Paper and Pencil
•
(AN5.6)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
159
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN6 Solve problems that involve
rational equations (limited to
numerators and denominators
that are monomials, binomials or
trinomials).
[C, PS, R]
In Grade 9, students solved linear equations (9PR3). Previously in this
roots (RF5). They will now solve equations containing rational
expressions and check if the solutions are permissible values. Roots
that are non-permissible are extraneous. (RF2). It is intended that the
rational equations be those that can be simplified to linear and quadratic
equations.
Achievement Indicators:
AN6.1 Determine the nonpermissible values for the variable
in a rational equation.
AN6.2 Determine the solution to
a rational equation algebraically,
and explain the strategy used to
solve the equation.
AN6.3 Explain why a value
obtained in solving a rational
equation may not be a solution of
the equation.
Students will be exposed to different strategies when solving rational
equations. It would be beneficial to begin with an example that is
easier to visualize before moving on to more complex equations. Some
x
= 52 .
students may use trial and error to solve an equation such as 10
Others may be able to determine the solution by inspection. Encourage
students to discuss their ideas. For example, a student may respond that
in order to get the number 10, the number 5 must be doubled therefore
2 is also doubled resulting in x = 4. This student response is a great
lead into the strategy of creating an equivalent rational equation with
common denominators. Ask students to rewrite the rational equation
x
4
= 10
with a common denominator ( 10
) and then write an equation with
the numerators.
Another strategy involves eliminating the denominators. Use an
x
= 52 , to promote discussion around lowest common
example such as, 10
denominator:
•
What is the lowest common denominator of 10 and 5?
•
What would happen if the lowest common denonimator
was multiplied on both sides of the equation? Why is this
mathematically correct?
•
What is the simplified equation?
•
What is the solution?
Once the strategies have been discussed, students will then be exposed to
solving more complex rational equations.
160
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Provide each student with a rational equation. Ask them to
identify the non-permissible roots and then solve the equation.
Ask students not to write their name on the paper since student
solutions will be collected and then redistributed randomly
around the room. Ask students to verify if the solution is correct.
If the solution is incorrect, they will identify the error and write
the correct solution.
(AN6.1, AN6.2)
6.4: Rational Equations
SB: pp.341-351
TR: pp.232-238
BLM: 6-4, 6-8
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
161
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN6 Continued ...
Achievement Indicators:
AN6.1, AN6.2
AN6.3 Continued
When solving x3 + 27x = 51 , for example, students may find a common
denominator for the left side of the equation and then proceed to solve
13
= 51 . They can also multiply both sides of the equation by the
2x
lowest common denominator (10 x ) x3 + (10 x ) 27x = (10 x ) 51 . It is important,
however, for students to recognize that they can reduce the numbers of
steps by multiplying both sides of the original equation by the lowest
common denominator.
Students can add or subtract the terms on the left hand side or the
right hand side of the equation before they cross multiply. This process,
however, may lead to an equation where the degree of the polynomial is
greater than what they started with. Consider the following example:
2 x 2 +1
x +3
=
x
4
2 x 2 +1
x +3
=
x 2 + 3 x + 20
4( x + 3)
2
+
5
x +3
2
4( 2 x + 1)( x + 3) = ( x + 3x + 20)( x + 3)
This example results in a cubic equation. Students are only familar with
solving quadratic equations at this point. Therefore, multiplying both
sides of the equation by the lowest common denominator would be the
method students would choose.
Caution students that it is necessary to find the non-permissible roots at
the beginning of the solution since some rational equations may lead to
extraneous roots. Consider the equation 2xx++53 + 21 = 2(−x14+5) . Ask students
•
What is the non-permissible root? What does this mean?
•
What is the solution to the resulting linear equation?
•
Why is it important to check the solution by using the original
equation?
Students should recognize solutions that are non-permissible values are
extraneous roots. Therefore, they must be eliminated as a valid solution.
When solving rational equations, the modified equation may result
in either a linear or quadratic equation. Students will have a choice
whether to use the quadratic formula or their factoring skills to solve
the quadratic equation. Remind them to verify their solutions to avoid
extraneous roots.
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ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
•
Pre-Calculus 11
Ask students to reflect on the process of solving a rational
equation. They should respond in writing to three reflective
prompts providing six responses, as shown below, to describe what
they learned.
3 things I understand:
1.
6.4: Rational Equations
SB: pp.341-351
TR: pp.232-238
BLM: 6-4, 6-8
2.
3.
2 things I am still struggling with:
1.
2.
1 thing that I will work on:
1.
(AN6.1, AN6.2, AN6.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
163
RATIONAL EXPRESSIONS AND EQUATIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN6 Continued ...
Achievement Indicator:
AN6.4 Solve problems by
modeling a situation using a
rational equation.
Students will be expected to write an equation to represent a problem.
They should be exposed to examples such as the following:
5
The sum of a number and its reciprocal is 2 . Students can begin this
example using trial and error and discuss possible solutions. They should
then proceed to write the rational equation x + x1 = 52 . Encourage
students to write their own example and share with the class.
It is important for students to recognize that inadmissible roots come
from the context of the problem. Discuss different scenerios that
produce inadmissible roots. A negative numerical value, for example,
would not make sense if referring to time, height and length.
Students may have difficulty interpreting the information from the
word problem and writing the rational equation. Encourage them to use
tables and diagrams to help them break down the information. Consider
the following example:
Sherry mows a lawn in 4 hours. Mary mows the same lawn in 5 hours.
How long would it take both of them working together to mow the
lawn? Pose the following questions to begin a discussion:
•
How much of the lawn would Sherry mow in 1 hour?
•
How much of the lawn would Mary mow in 1 hour?
•
How much of the lawn would both mow together in 1 hour?
Completing a table such as the one below should help students organize
their information.
Sherry
Time to mow lawn
(hours)
4
Fraction of lawn
mowed in 1 hr
Mary
5
1
5
Both
x
1
x
1
4
When solving the equation 41 + 51 = x1 , encourage students to check that
the solutions satisfy the original equation are permissible, and in the case
of a word problem, realistic in the context.
Students should also be exposed to word problems that produce a
rational equation resulting in solving a quadratic equation. In such cases,
there may be an inadmissible value that will need to be rejected in the
context of the problem.
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ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
RATIONAL EXPRESSIONS AND EQUATIONS
General Outcome: Develop algebraic reasoning and number sense.
Suggested Assessment Strategies
Resources/Notes
Performance
Authorized Resource
•
Pre-Calculus 11
Create pairs of cards with word problems and matching equations
to solve the word problems. Distribute the cards amongst the
students and have them find their partner by matching the word
problem with the corresponding equation. Once they have found
their partner, students should work in pairs to solve the equation
and verify their solution.
6.4: Rational Equations
SB: pp.341-351
TR: pp.232-238
BLM: 6-4, 6-8
(AN6.2, AN6.4)
Paper and Pencil
•
A student was given the following word problem:
It takes Mike 9 hours longer to construct a fence than it takes
Jason. If they work together, they can construct the fence in 20
hours. How long would it take Mike to construct the fence alone?
20
The student solved the equation 20
t + t +9 = 1 and stated the
solutions to the word problem were 36 and -5. Ask students to
verify the solution and state whether the student is correct.
(AN6.2, AN6.3, AN6.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
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RATIONAL EXPRESSIONS AND EQUATIONS
166
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Absolute Value and Reciprocal Functions
Suggested Time: 16 Hours
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Unit Overview
Focus and Context
In this unit, students will focus on absolute values, determine the
absolute value of numerical expressions, and solve problems involving
the absolute value. They will graph the absolute value of a linear
function and the absolute value of a quadratic function. Students will
identify characteristics of the graph, including the intercepts and the
domain and range. They will also represent the absolute value function
using piecewise notation. Students will solve absolute value equations
graphically and algebraically.
Students will expore reciprocal functions by comparing the graphs of
a function and its reciprocal. The focus will be on linear and quadratic
functions.
Outcomes
Framework
GCO
GCO
Develop algebraic reasoning
and number sense.
Develop algebraic and graphical
reasoning through the study of
relations.
SCO AN1
Demonstrate an understanding
of the absolute value of real
numbers.
SCO RF2
Graph and analyze absolute value
functions (limited to linear and
problems.
SCO RF11
Graph and analyze reciprocal
functions (limited to the
reciprocal of linear and
168
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Process Standards
Key
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Algebra and Number
AN1 Demonstrate an
understanding of the absolute value
of real numbers.
[R, V]
Relations and Functions
Relations and Functions
RF5 Determine hte characteristics RF2 Graph and analyze absolute
of the graphs of linear relations,
value functions (limited to linear
including the:
and quadratic functions ) to solve
problems.
• intercepts
[C, PS, R, T, V]
• slope
Mathematics 3200
Relations and Functions
RF14 Graph and analyze rational
functions (limited to numerators and
denominators that are monomials,
binomials or trinomials).
[CN, R, T, V]
• domain
• range
[CN, PS, R, V]
RF6 Relate linear equations
expressed in:
RF11 Graph and analyze reciprocal
functions (limited to the reciprocal
[CN, R, T, V]
• 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]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
169
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Algebra and Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
AN1 Demonstrate an
understanding of the absolute
value of real numbers.
[R, V]
Achievement Indicators:
AN1.1 Determine the distance of
two real numbers of the form ±a,
a ∈ \ , from 0 on a number line,
and relate this to the absolute
value of a ( a ).
AN1.2 Determine the absolute
value of a positive or negative real
number.
AN1.3 Explain, using examples,
how distance between two points
on a number line can be expressed
in terms of absolute value.
AN1.4 Determine the absolute
value of a numerical expression.
AN1.5 Compare and order the
absolute values of real numbers in
a given set.
170
The concept of absolute value is new to students. They will determine
the absolute value of numerical expressions and compare and order the
absolute values of real numbers in a given set.
Introduce students to the absolute value of a number as its distance
from zero. Ask them to plot the integers 5 and -5 on a number line.
They should notice, when graphed, that these numbers are are the
same distance from zero. It is important for students to understand that
absolute value only asks “how far?”, not “in which direction?”. Reinforce
that distance is always positive. This leads to the definition of the
absolute value of any real number a:
⎧a , if a ≥ 0
a =⎨
⎩−a , if a < 0
Students will extend the concept of absolute value to include the
distance between any two real numbers. This could be first investigated
using natural numbers and then extended to integers and real numbers.
Completing a table such a the one below should help students recognize
that the distance between a and b can be represented by |a - b| or |b - a|.
a
b
2
-5
2.68
6
-10
5.75
distance
between a and b
4
value of
|a - b|
|2-6|=|-4|=4
value of
|b - a|
|6-2|=|4|=4
In Grade 9, students simplified numerical expressions using the order
of operations (9N4). This is now extended to include expressions
containing absolute value. Ask students to compare the expressions
3 - 4(2) and |3 - 4(2)|. They may have difficulty determining where
the absolute value of a numerical expression occurs in the order of
operations. They will simplify the expression inside the absolute value
symbol using the order of operations and then take the absolute value of
the resulting expression.
Students will compare and order absolute values of real numbers. If the
absolute value contains a fraction, students may find it helpful to change
it to a decimal representation. Once students evaluate and compare the
absolute values in a given set, they can place the values on a number line
to help them order the set.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic reasoning and number sense
Suggested Assessment Strategies
Resources/Notes
Journal
Authorized Resource
• Pre-Calculus 11
Provide students a fictitious town with one main street. This
street should contain at least eight landmarks located to the right
and left of the town square which is representative of the origin.
Each landmark should be given a name (gas station, library, etc.)
Ask students to pose three questions where the distance between
any two landmarks is requested.They should then answer the
questions.
(AN1.3)
7.1 Absolute Value
Student Book (SB): pp.358-367
Teacher Resource (TR): pp.250255
Blackline Master (BLM): 7-3, 7-4
Observation
•
Create a set of cards with a variety of examples involving the
absolute values of real numbers and numerical expressions. In
groups of three, each student will receive five cards. They will
place the cards in postition that produces an ordered set of cards.
When each member of the group is finished, ask each student to
explain their resoning to the other members.
(AN1.2, AN1.4, AN1.5)
•
Ask students to create a human number line. Each student is
given a card containing an absolute value of a real number or
numerical expression. They order themselves into a line based on
the relative size of their number. Ask them to expain why they
chose their position. As an alternative, a skipping rope or piece of
string could be used. Students attach their number to the line in
the appropriate position.
(AN1.2, AN1.4, AN1.5)
Performance
• Divide the class into teams consisting of about five students. Each
team would line up one behind the other. Provide each team with
a list of absloute value problems to simplify. The first student
in each line would write the answer to one of the absolute value
problems using mental math skills. They would then move to the
back of the line and the second student would repeat this with a
different problem. The first team to complete their questions with
the correct answers would be the winners.
(AN1.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
The following website is an
interactive game for ordering the
absolute values of real numbers.
http://www.sheppardsoftware.com
(search: number balls absolute
value)
171
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Graph and analyze absolute
value functions (limited to linear
problems.
[C, PS, R, T, V]
In Grade 9, students solved linear equations (9PR3). In Mathematics
1201, students analyzed the graphs of linear relations (RF6). Earlier in
this course, students analyzed quadratic functions in vertex and standard
form to identify the characteristics of the corresponding graph (RF3,
RF4). They will now graph and analyze absolute value functions, limited
Achievement Indicators:
RF2.1 Create a table of values for
y = |f (x)|, given a table of values
for y = f (x).
RF2.2 Sketch the graph of
y = |f (x)|; state the intercepts,
domain and range; and explain
the strategy used.
Students will compare the graph of a linear function to its corresponding
absolute value function, using a table of values. They should identify
the similarities and differences between y = |ax + b| and y = ax + b. Ask
students to graph y = |x + 3|, for example, using the table of values for
y = x + 3. To get an accurate picture of the absolute value graph, it is
important to include appropriate x-values that will produce negative
y-values.
y
y
6
6
5
5
4
4
3
3
2
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
1 2 3 4 5 6
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
1 2 3 4 5 6
x
Using the graph as a visual aid, students should recognize that the
section of the graph which lies above the x-axis remains the same and
the section which lies below the x-axis is reflected in the x-axis.
Ask students to identify characteristics of the absolute value graph, such
as the intercepts, domain and range. As they look for the similarities and
differences between the graphs, they should consider the following:
•
Is the x-intercept significant?
•
Is the domain dependent on the x-intercept? Explain.
•
Why is the domain of the the function the same as the domain of
the absolute value function? Why is the range different?
Comparing the graphs of y = |ax + b| and y = ax + b, students recognize
the resulting shape of the graph. Using this shape, along with the x-and
y-intercepts, they should be able to draw the graph.
172
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
Ask students to work in groups for this activity. Each group will
be given a different tables of values. They will create a table of
values for y = |f (x)|. Ask students to display their tables around the
classroom. They should observe and analyze the data. Each group
will then present their conclusions to the rest of the class.
Pre-Calculus 11
7.2 Absolute Value Functions
SB: pp.368-379
TR: pp.256-261
BLM: 7-3, 7-5
(RF2.1)
•
Working in small groups, students will play the Absolute Matching
Game. Each group will be given several cards. Half of the cards
will have the graph of y = f (x) while the other half will have the
graph of y = |f (x)|. The object of the game is to be the first group
to pair up each graph with its absolute value graph.
Note: For this game to be most effective, the graphs should have
similar characteristics, such as the same x-intercepts but different
slopes/vertices.
(RF2.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
173
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Continued ...
Achievement Indicators:
RF2.1, RF2.2 Continued
Students will then proceed to work with and analyse the absolute value
of quadratic functions. They will compare the graphs of the absolute
value of a quadratic function to its original graph using a table of values.
Ask students to graph, for example, y = |(x - 2)2 - 4| using the table of
values for y = (x - 2)2 - 4. Remind them that the graph of a quadratic
function can be obtained using vertex and the x-intercepts, as previously
introduced in the Quadratic Functions unit.
y
y
6
6
5
5
4
4
3
3
2
2
1
1
-5
-4
-3
-2
-1
1
-1
2
3
4
x
5
-5
-4
-3
-2
-1
1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
x
Students should notice that the graph of y = |f (x)| reflects the negative
part of the graph of y = f (x) across the x-axis, while the positive part
remains unchanged. Students should again note the similarities and
differences between the two graphs.
RF2.3 Generalize a rule for
writing absolute value functions
in piecewise notation.
Students will develop a rule for writing absolute value functions in
piecewise notation using a graph. Explain to students that piecewise
functions are used to describe functions that contain distinct functions
over different intervals. The graph of a linear absolute value function, for
example, consists of two separate linear functions. The domain for each
interval is dependent on the x-intercept(s). Revisit the graphs of
y = x + 3 and y = |x + 3|.
y
y
6
6
5
5
4
4
3
3
2
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
1 2 3 4 5 6
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
1 2 3 4 5 6
x
Students should notice the turning point on the absolute value graph
is located at the x-intercept. For values of x where y is positive, the
graphs of y = |x + 3| and y = x + 3 are the same. For values of x where y is
negative, the graph of y = |x + 3| is a reflection of the graph of
y = x + 3.
174
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
As a pre-assessment, ask students to complete the following
functions.
Linear
Example
y = 3x − 4
Characteristics
Pre-Calculus 11
7.2 Absolute Value Functions
SB: pp.368-379
TR: pp.256-261
BLM: 7-3, 7-5
Graph
Example
2
Characteristics
Graph
y = ( x − 3) − 4
(RF2.1, RF2.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
175
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Continued ...
Achievement Indicators:
RF2.3 Continued
The absolute value function y = |x + 3| can be written in piecewise
notation as:
⎧x + 3, if x ≥ −3
y =⎨
⎩−( x + 3), if x < −3
This can be generalized to all absolute value functions:
⎧ f ( x ), where f ( x ) ≥ 0
y=⎨
⎩− f ( x ), where f ( x ) < 0
Similarly, students can use the visual representation of the absolute value
of a quadratic function to determine the piecewise function. This would
be a good opportunity to also expose students to an algebraic method
using sign diagrams to analyse where the quadratic function is positive
or negative. Ask students to write, for example, the function
y = |(x - 2)2 - 4| using piecewise notation. They will first use the
quadratic function y = (x - 2)2 - 4 to determine the x-intercepts. They
will then determine the distinct intervals where the function is positive
and where it is negative on a number line.
y
6
5
4
3
2
1
-3
-2
-1
1
-1
2
3
4
5
x
-2
-3
-4
-5
-6
y-values +
-3
-2
–
y-values
-1
0
1
2
y-values +
3
4
5
Using the sign diagram, students can write the piecewise function as:
2
⎪⎧( x − 2) − 4, x ≤ 0 or x ≤ 0
y=⎨
2
⎪⎩−[( x − 2) − 4], 0 < x < 4
This can also be done without the graph as a visual aid. Relating the sign
diagram to the x-axis of the graph, students could substitute an x-value
from each interval into the function to determine where the function is
positive or negative.
When graphing the absolute value of a linear or quadratic function,
they should recognize the shape of the graph, the y-intercept and the
x-intercept(s). Using the information from the sign diagram, they can
then draw the graph of the absolute value function.
176
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Performance
•
In small groups, ask students to participate in a card game,
Piecewise Pairs, involving graphs and piecewise functions. Each
group will be given several cards. Half the cards contain a graph
while the other half will have a piecewise function. The object
of the game is to be the first group to match each graph with its
correct piecewise notation.
Pre-Calculus 11
7.2 Absolute Value Functions
SB: pp.368-379
TR: pp.256-261
BLM: 7-3, 7-5
Note: Graphs should be similar enough to each other so that
students cannot just match them without any thought (e.g., have
several graphs with the same x-intercept so additional analysis is
involved).
(RF2.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
177
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Continued...
Achievement Indicator:
RF2.4 Solve an absolute value
equation graphically, with or
without technology.
Students will explore the solutions to absolute value equations, focusing
first on the graphical representation and then moving to an algebraic
solution. Remind them of the properties of absolute value:
•
For a real number a, the absolute value |a| is the distance from a to
the origin.
•
For two real numbers a and b, |a - b| is the distance between a and
b on the number line.
Students should first work with the absolute value of a linear equation
before moving on to the absolute value of a quadratic equation. Ask
students what it means to solve an equation such as |x - 2| = 6. They
should be looking for points whose distance from 2 is 6. Using a
number line, they should realize that both -4 and 8 are at a distance of
6 from 2. This reasoning will allow students to better understand the
solutions when using a graph.
Ask students to graph the function y = |x - 2| and y = 6 on the same
coordinate plane to determine the points of intersection.
y
10
y = |x – 2|
8
y=6
6
4
2
-10
-8
-6
-4
-2
2
4
6
8
10
x
-2
They should notice the x-coordinates of the points of intersection
are the solutions to the equation. Encourage students to verify the
solution by substituting the values back into the equation. Similarly, this
graphical approach can be used to solve the absolute value of quadratic
equations.
178
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Pre-Calculus 11
7.2 Absolute Value Functions
SB: pp.368-379
TR: pp.256-261
BLM: 7-3, 7-5
7.3 Absolute Value Equations
SB: pp.380-391
TR: pp.262-267
BLM: 7-3, 7-6
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
179
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF2 Continued...
Achievement Indicator:
RF2.5 Solve, algebraically, an
equation with a single absolute
value, and verify the solution.
Students will move from the graphical representation of a solution to an
algebraic method to solve absolute value equations. It is important for
students to begin with simpler absolute value equations and then move
on to more complex equations.
Remind students that the definition of an absolute value consists of two
parts: where f (x) ≥ 0 and f (x) < 0. Consider the previous example
|x - 2| = 6. The equation is split into two possible cases: ± (x - 2) = 6.
Students will solve the equations (x - 2) = 6 and -(x - 2) = 6 to result in
x = -4 and x = 8. Each solution should be checked for extraneous roots.
When students solve absolute value equations involving a quadratic
expression, such as |x2 - 4| = 3x, they may have to solve the resulting
Encourage them to verify the solutions by substituting the x-values back
into the equation and making sure the left hand side of the equation is
equal to the right hand side of the equation.
RF2.6 Explain why the absolute
value equation |f(x)|< 0 has no
solution.
RF2.7 Determine and correct
errors in a solution to an absolute
value equation.
RF2.8 Solve a problem that
involves an absolute value
function.
180
Remind students the absolute value of a number is always greater than
or equal to zero. Students can use inspection to see if this property
applies. Consider the example |x - 2| = -6. Ask students if their is any
possible x value that can be substituted into the equation to result in a
negative value.
Students should be provided with worked solutions containing errors to
a number of absolute value equations and asked to identify and correct
the errors. Common errors include:
• Treating the absolute value sign like parentheses.
•
Multiplying a constant by the expression within the absolute value
sign (e.g., -2|x - 3| = |-2x + 6|).
•
Incorrectly placing the negative in front of the variable rather than
the entire expression (i.e., When solving |x - 3| = 8, students may
wirte -x - 3 = 8 instead of -(x - 3) = 8).
•
Not identifying extraneous roots.
•
Errors when using the quadratic formula.
Problem solving using linear absolute value functions should be
embedded throughout the unit and situated in a variety of contexts. In
some problems, students will be given an absolute value function and
asked to analyze it. In other problems, students will be required to create
the function from the given information. Encourage students to check
their answers and identify extraneous solutions.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
7.3 Absolute Value Equations
Ask students to illustrate with a graph why |x - 2| = -6 has no
solution.
(RF2.6)
•
Place cards on the classroom wall consisting of linear and
two cards, one linear and one quadratic. They should solve the
equations and pass them in as exit cards. These exit cards can then
be used later for students to determine and correct any errors.
(RF2.5, RF2.7)
•
For the activity Pass the Problem, each group of students is given
absolute value equations to solve algebraically. After a specific
amount of time, ask students to swap their problem with another
pair. If the group finished the problem, the other group will check
the solution. If errors are identified the group will correct the
error and then continue to complete the problem. If the group,
however, did not finish answering the problem, the other pair will
check the partially completed solution and pick up from where
the group left off. When they are finished, they should share the
completed responses with each other, defending their reasons
for any changes they made and provide feedback on each other's
thinking.
SB: pp.380-391
TR: pp.262-267
BLM: 7-3, 7-6
The following sites provides
interactive games related to
absolute value.
http://www.crctlessons.com/absolutevalue-equations-game.html
http://www.crctlessons.com/absolutevalue-equations-game.html
(RF2.5, RF2.7)
Presentation
•
Ask groups of students to research a problem that incorporates the
absolute value function. They should present the problem to the
class, including the solution.
(RF2.8)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
181
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Graph and analyze
reciprocal functions (limited
to the reciprocal of linear and
[CN, R, T, V]
Achievement Indicators:
RF11.1 Compare the graph of
y = f (1x ) to the graph of
y = f (x).
RF11.2 Identify, given a
function f(x), values of x for
which y = f (1x ) will have
vertical asymptotes; and describe
their relationship to the nonpermissible values of the related
rational expression.
RF11.3 Graph, with or without
1
technology, y = f ( x ) , given
y = f (x) as a function or a graph,
and explain the strategies used.
Students will compare the graphs of a function and its reciprocal.
Examples will include both linear and quadratic functions.
When graphing a function and its reciprocal, there are invariant points
that remain unchanged. Consider showing students the following table
to highlight the invariant points using a function such as f (x) = 2x + 1.
x
f(x)
1
f (x )
1
3
1
3
0.5
2
1
2
0
-0.5
-1
1
0
-1
1
undefined
-1
Students should notice the invariant points can be located by setting
f (x) = ±1. These points will help students when they graph reciprocal
functions, as these will be common to both functions and are therefore
the points of intersection.
As students explore linear and quadratic reciprocal functions, there are
students to graph f (x) = 2x + 1.
f(x)
f(x) = 2x + 1
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
-2
-3
-4
-5
182
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Presentation
Authorized Resource
• Pre-Calculus 11
Ask students to use graphing technology to analyse and describe
the characteristics of a function and its reciprocal, and vice versa.
As teachers observe this activity they should ask questions such as:
Where do the graphs intersect? What do the vertical asymptotes of
the reciprocal function correspond to in the original function?
(RF11.1, RF11.2, RF11.3, RF11.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
7.4 Reciprocal Functions
SB: pp.392-409
TR: pp.268-276
BLM: 7-3, 7-7
183
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Continued...
Achievement Indicators:
RF11.1, RF11.2
Students should consider the following:
RF11.3 Continued
•
If the point (x, y) is on the graph y = f (x), what do you notice
about the point on the graph of the reciprocal function?
•
Is the sign of the reciprocal function the same as the sign of the
original function?
•
What do you notice about the x-intercept of the linear function,
the non-permissible value of the reciprocal function, and the
location of the vertical asymptote?
•
What is the horizontal asymptote?
•
What are the invariant points?
•
Why do the intervals of decrease on the original function become
intervals of increase on the reciprocal function?
•
Why do the intervals of increase on the original function become
intervals of decrease on the reciprocal function?
•
What do you notice about the behaviour of the reciprocal
function as it approaches the asymptotes?
They should also recognize, from the table and the graph, that the
x-intercept of the function produces a non-permissible value for the
reciprocal function. This is the location of the vertical asymptote. When
analyzing and comparing the graphs of the function and its reciprocal,
it is important to note that as the y-values of one function increase the
y-values of the other decrease and vice versa. This could be demonstrated
by looking at parts of the graph on either side of the invariant points.
As students recognize the shape of the graph along with the intercepts
and the asymptotes, they should be able to sketch the graph of the
reciprocal function.
These ideas will now be extended to the graphs of quadratic functions
and their reciprocals. There are three cases that need to be considered
and explored with students, namely, quadratic functions that have one,
two or no x-intercepts.
184
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Provide the following table with a set of statements. Ask students
to describe the reasoning they used to decide whether each
statement is true or false.
Consider the function
f (x ) =
1
True
False
Why I (we) think so
7.4 Reciprocal Functions
SB: pp.392-409
TR: pp.268-276
BLM: 7-3, 7-7
3x − 4
(1) There is a vertical
asymptote at x = 43
(2) There is a horizontal
asymptote at y = 0.
(3) f (0) =
1
4
(4) The point (4, 8) is on
the graph of y = 3x -4 and
the point (4, -8) is on the
graph of y = 3x1−4 .
(RF11.1, RF11.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
185
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Continued...
Achievement Indicators:
First investigate a quadratic function with one x-intercept. Consider the
graph of y =(x - 2)2 and its reciprocal y = ( x −12 )2 .
RF11.1, RF11.2
y
6
5
4
3
2
1
RF11.3 Continued
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
1
2
3
4
5
6
x
Using the graph as a visual aid, students should observe the following:
•
There is one x-intercept on the graph of y = f (x) located at x = 2.
This corresponds to the location of the vertical asymptote on the
graph of y = f (1x ) .
•
The graphs of y = f (x) and y = f (1x ) intersect where f (x) = ±1(in
this case only at f (x) = 1). These are the invariant points.
•
The graph of y = f (1x ) is also asymptotic to the x-axis. The
horizontal asymptote is located at y = 0.
Similarly, students can investigate a quadratic function with two
x-intercepts. Consider the graph of y = (x - 3)(x + 2) and it’s reciprocal
y = ( x − 3)(1 x + 2 ) .
y
10
9
8
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
1 2 3 4 5 6 7 8 9 10
x
Students should note the following features:
•
The x-intercepts of the graph of y = f (x), x = -2 and x = 3
correspond to the location of the vertical asymptotes on the graph
of y = f (1x ) .
186
1
f ( x ) intersect
•
The graphs of y = f (x) and y =
are the invariant points.
•
The equation of the horizontal asymptote is located at y = 0.
•
The parts of the graph of y = f (x) that were positive remain
positive on the graph of y = f (1x ) . The parts of the graph of y = f
(x) that were negative remain negative on the graph of y = f (1x ) .
at f (x) = ±1. These
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Observation
Authorized Resource
•
Pre-Calculus 11
7.4 Reciprocal Functions
Students can work in pairs for this activity. Give one student the
graph of a quadratic function and its reciprocal on the same set
of axes. Ask him/her to turn to their partner and describe, using
the characteristics of the quadratic function and its reciprocal, the
graph they see. The other student will draw the graph based on
the description from the student. Both students will then check to
see if their graphs match.
SB: pp.392-409
TR: pp.268-276
BLM: 7-3, 7-7
(RF11.1, RF11.2, RF11.3)
Paper and Pencil
•
Provide the following table with a set of statements. Ask students
to describe the reasoning they used to decide whether each
statement is true or false.
Consider the function
f (x ) =
1
True
False
Why I (we) think so
x 2 − 5x + 6
(1) There is a vertical
asymptote at x = 2
(2) There is a vertical
asymptote at x = -3
(3) f (0) =
1
6
(4) y = f (1x ) has
x-intercepts at 2 and 3
(RF11.1, RF11.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
187
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF11 Continued...
Achievement Indicators:
RF11.1, RF11.2
RF11.3 Continued
When graphing a quadratic function with no x-intercepts and its
reciprocal, students should notice that the characteristics are different
from the previous cases. Consider the graph of y = x2 + 3 and y = ( x 21+ 3) .
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
•
The graph of y = f (x) has no x-intercepts. Therefore, the graph of
y = f (1x ) has no vertical asymptotes.
•
The graph of y = f (x) does not have any points at f (x) = ±1.
Therefore, there are no points of intersection between the graphs
(i.e., there are no invariant points).
•
The equation of the horizontal asymptote is located at y = 0.
This investigation helps students recognize the shape of the graph of a
reciprocal function. Using the intercepts and asymptotes, students can
proceed to graph a reciprocal function.
RF11.4 Graph, with or without
technology y= f(x) given
y = f (1x ) as a function or a
graph, and explain the strategies
used.
188
Students will work backwards and sketch the graph of the original
function given the graph of the reciprocal function. They should be able
to determine by inspection whether the original function is linear or
quadratic by analyzing the reciprocal function. It is important to remind
students of the key concepts when working with reciprocal functions.
The vertical asymptote of the reciprocal function is the x-intercept(s) of
the original graph. The point (x,y) on the reciprocal function becomes
(x, 1y ) on the original function. The two functions will intersect when f
(x) = ±1.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Provide students with a reciprocal function. Ask them to sketch
the graph of the original function, y = f (x), and explain the
strategies they used.
(RF11.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
7.4 Reciprocal Functions
SB: pp.392-409
TR: pp.268-276
BLM: 7-3, 7-7
189
ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS
190
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Systems of Equations
Suggested Time: 8 Hours
SYSTEMS OF EQUATIONS
Unit Overview
Focus and Context
Outcomes
Framework
In this unit, students are introduced to linear-quadratic and quadraticquadratic systems of equations. The solutions to these systems are
determined graphically by finding points of intersection. Students
will solve the system of equations algebraically using the methods of
substitution and elimination. They will then solve application problems
involving a system of equations and verify their solutions.
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF6
Solve, algebraically and graphically,
problems that involve systems of
variables.
192
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SYSTEMS OF EQUATIONS
Process Standards
Key
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
SCO Continuum
Mathematics 1201
Mathematics 2200
Relations and Functions
Relations and Functions
RF9 Solve problems that
involve systems of linear
equations in two variables,
graphically and algebraically.
[CN, PS, R, T, V]
Mathematics 3200
RF6 Solve, algebraically and
graphically, problems that
variables.
[CN, PS, R, T, V]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
193
SYSTEMS OF EQUATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF6 Solve, algebraically and
graphically, problems that involve
two variables.
[CN, PS, R, T, V]
In Mathematics 1201, students modelled and solved systems of
linear equations in two variables, graphically and algebraically (RF9).
They created a linear system to model a situation, as well as wrote
a description of a situation that might be modeled by a given linear
system. They solved linear systems graphically, with and without
technology, and progressed to solving linear systems symbolically using
substitution and elimination.
Achievement Indicators:
RF6.1 Explain the meaning
of the points of intersection of
RF6.2 Explain, using examples,
may have zero, one, two or an
infinite number of solutions.
In Mathematics 1201, students created graphs of linear functions
(RF6). They also investigated the meaning of the point of intersection
of a system of linear equations (RF9). Students discovered that the
intersection points of the graphs represented the solution to the system
of equations.
Ask students to sketch a line and a parabola on the same axis. As they
examine their intersection points, students should recognize there are
three situations possible for a linear-quadratic system. It can have zero,
one or two solutions.
y
y
x
y
x
x
Introducing the concepts of a tangent and a secant line may be waranted
here. If a line intersects the parabola more than once, the line is referred
to as a secant line. If the line intersects the parabola at exactly one point,
the line is called a tangent line. Although this is not an outcome in this
course, exposure to the terms will help them with courses to follow.
Ask students to repeat the same activity by drawing two parabolas. Ask
Students should recognize the same results with one exception; the
system can also have an infinite number of solutions (i.e., coincident).
194
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SYSTEMS OF EQUATIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
Pre-Calculus 11
•
Consider the parabola C whose equation is y = x2 - 2x - 3.
(i) What is a possible equation for line L such that C and L
form a system with no solution? one solution? two solutions?
(ii) What is a possible equation for quadratic Q such that C
and Q form a system with no solution? One solution? Two
solutions? Infinite number of solutions?
(RF6.1, RF6.2)
•
Ask students how they can tell by observation which of the
following systems has no solution and which one has an infinite
number of solutions.
(i) −2 x 2 + 3x − y + 4 = 0 ⎫⎪
⎬
−4 x 2 + 6x − 2 y + 8 = 0 ⎪⎭
(ii) y − 5 = 0
⎫⎪
⎬
y = −( x + 1) − 3⎪⎭
8.1 Solving Systems of Equations
Graphically
Student Book (SB): pp.424-439
Teacher Resource (TR): pp.291299
BlackLine Master (BLM): 8-3, 8-4
2
(RF6.1, RF6.2)
Performance
•
Ask students if the following lines intersect the parabola shown
below at zero, one or two points.
y
(i) y = 21 x + 3
(ii) y = 21 x − 4
(iii) y = -2x
(iv) x - 3 = 0
10
8
6
4
2
-2
-2
-4
2
4
6
8
x
(RF6.1, RF6.2)
Journal
•
cannot have an infinite number of solutions whereas a system of
disagree with this statement. They should justify their reasoning.
(RF6.1, RF6.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
195
SYSTEMS OF EQUATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF6 Continued ...
Achievement Indicators:
RF6.3 Determine and verify the
equations graphically, with and
without technology.
Students will be working with the graphs of linear-quadratic and
Students will need to revist the methods used for graphing linear and
quadratic relations. In Mathematics 1201, students were exposed to
graphing linear functions using the slope-intercept method, slopepoint method, and using the x and y-intercept method. In this course,
students explored the characteristics of standard form and vertex form
of a quadratic function. Identifying the form of the equation will help
students decide which method they should choose when graphing the
linear or quadratic function. Graphing technology can also be used to
solve a system of equations. Students can be exposed to, but not limited
to a graphing calculator or graphing Apps.
When students identify the points of intersection, remind them to
verify the solution for both equations. There are limitations to solving a
system by graphing. Non-integral intersection points are possible where
students will have to estimate the coordinates. In such cases, an algebraic
method of solving these systems is more efficient.
RF6.4 Determine and verify the
equations algebraically.
Through their work with solving systems graphically, students should
realize that algebra provides a more efficient means of finding the
points of intersection. In Mathematics 1201, students were exposed
to algebraic methods of substitution and elimination to solve a linear
system. They will apply these methods to systems involving quadratic
equations, including ones with rational coefficients. Provide students an
opportunity to decide which algebraic method is more efficient when
solving a system by focusing on the coefficients of like variables. If
necessary, ask students to rearrange the equations so that like variables
appear in the same position in both equations.
When students solve the system using substitution or elimination, they
will have to use factoring or the quadratic formula to solve the resulting
quadratic equation. After determining the solution to the system, ensure
that students verify the ordered pair satisfies the equations. Discuss with
them why it is important to verify the solution in both equations.
196
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SYSTEMS OF EQUATIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Sam solved the system: 2 x − y = 9 ⎫⎪
⎬
y = x 2 − 4x ⎪
⎭
His solution was (3,-3). Ask students to verify whether his
solution is correct. Ask them to explain how Sam’s results can be
illustrated on a graph.
(RF6.3)
•
8.1 Solving Systems of Equations
Graphically
SB: pp.424-439
TR: pp.291-299
BLM: 8-3, 8-4
⎧ x 2 − x + y = −2
Andrew was asked to solve the system: ⎪⎨
2
⎪⎩2 x − 4 x + 3 y = 0
The beginning of his solution is shown below. Ask students to
finish the solution and verify their answer. Identify and correct
any error(s) you or Andrew may have made.
Andrew’s Solution:
Multiply the first equation by -3 and add the second equation.
x − x + y = −2 ⎯⎯→ −3 x + 3 x − 3 y = −2
× ( −3)
2
2 x − 4 x + 3 y = 0 ⎯⎯
→
2
2
2
−x − x
8.2 Solving Systems of Equations
Algebraically
2
2x − 4x + 3y = 0
=−2
Now, solve -x2 - x = -2.
(RF6.4)
SB: pp.440-456
TR: pp.300-306
BLM: 8-3, 8-5
Observation
• Ask students to participate in the following activity: Cards are
distributed amongst the students. One third of the cards contain
the coordinates of the points of intersection of two graphs.
Another third of the cards contain a system of linear-quadratic
and vertex form. The final third contain the corresponding
graphs for the system of equations. Students move around the
classroom attempting to form a group of three by matching the
cards containing the corresponding systems, graphs and points of
intersections.
(R)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
197
SYSTEMS OF EQUATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF6 Continued ...
Achievement Indicators:
RF6.4 Continued
Encourage students to predict the number of solutions before they solve
the system graphically or algebraically. The discriminant, b2 - 4ac, can be
used to help them with their prediction. Consider the system:
y = 3x + 5 and y = 3x2 - 2x - 4. Ask students to equate the equations and
simplify. Their resulting quadratic equation is 3x2 - 5x - 9 = 0. They
determine the value of the discriminant to be 133. Students should
realize that when the discriminant is greater than zero, there are two x
values indicating two different solutions. This means that the linearquadratic system has two points of intersection. If the discriminant is
equal to zero, there is one solution and a negative discriminant implies
no point of interesection. They would then proceed to solve the system
and the resulting quadratic equation using methods learned earlier in
the square).
also produce an infinite number of solutions. When the functions are
coincident (i.e., identical) students will recognize this results in a system
where the left hand side equals the right hand side (i.e., 0 = 0).
RF6.5 Model a situation, using
equations to the context of a given
problem.
198
Students will model situations using a system of linear-quadratic and
When they can relate their learning to real-life applications it has more
meaning. Paths of thrown or falling objects can be used to model both
quadratic and linear functions. Ensure students understand and define
the variables that are being used to represent the unknown quantities.
Discuss with students that in order to solve a system, the number of
unknowns must match the number of equations.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SYSTEMS OF EQUATIONS
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Authorized Resource
Paper and Pencil
•
Pre-Calculus 11
Ask students to create and solve the system of equations
represented by the graphs below. They should verify that the
solutions obtained algebraically match those found graphically.
y
y
6
4
4
2
2
-2
SB: pp.440-456
TR: pp.300-306
BLM: 8-3, 8-5
6
8
-4
8.2 Solving Systems of Equations
Algebraically
-2
2
-2
4
6
8
2
4
6
x
-2
x
-4
-4
-6
(RF6.3, RF6.4)
•
The price C, in dollars per share, of a high-tech stock has
fluctuated over a twelve-year period and is represented by the
parabola shown. The price C, in dollars per share, of a second
high-tech stock has shown a steady increase during the same time
period.
y
60
50
40
30
20
10
-4
-2
(0, 14)
2
4
6
8
10
12
14
x
-10
(i) Determine the system of equations that models the price
over time.
(ii) Solve the system.
(iii) Determine the values where the two prices are the same.
(RF6.5, RF6.6)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
199
SYSTEMS OF EQUATIONS
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF6 Continued...
Achievement Indicator:
RF6.7 Solve a problem that
equations, and explain the
strategy used.
200
Students will solve problems involving a system of linear-quadratic or
method is more efficient when solving the system of equations and they
should be able to explain why they chose the preferred method.
Explaining the meaning of a solution in a particular context and
verifying the solution is an important part of solving systems. An answer
to a system of equations may not necessarily be a possible solution
in the context of the situation (i.e., inadmissible root). While solving
a system of equations, a student may determine that the solutions
are, for example, (-2, 5) and (5, 8). In the context of the problem the
x-coordinate may represent a measurement such as time or length. The
point (-2, 5), therefore, would have to be rejected as a solution leading
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
SYSTEMS OF EQUATIONS
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 solve the following:
Pre-Calculus 11
(i) A rectangular field has a perimeter of 500 m and an area of
14 400 m2. Find the length of the sides.
(RF6.7)
8.2 Solving Systems of Equations
Algebraically
(ii) Ask students to write a system of equations to represent two
numbers that differ by 4 and whose squares have a sum of
136?
(RF6.7)
SB: pp.440-456
TR: pp.300-306
BLM: 8-3, 8-5
(iii) A right triangle has a hypotenuse 10 cm long. If the
perimeter is 22 cm, find the lengths of the other two sides.
(RF6.7)
(iv) A sky diver jumped from a tower and fell freely for several
seconds before releasing her parachute. Her height, h, in
metres, above the ground at any time is given by:
h = -4.9t2 + 5000 before she released her parachute, and
h = -4t + 4000 after she released her parachute. If t represents time in seconds, how long after jumping did she release her
parachute? How high was she above the ground at that time?
(RF6.7)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
201
SYSTEMS OF EQUATIONS
202
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Suggested Time: 8 Hours
Unit Overview
Focus and Context
Outcomes
Framework
In this unit, students will solve quadratic inequalities in one variable
as well as linear and quadratic inequalities in two variables. They will
investigate finding the solutions using various methods, ideally leading
to a preferred personal strategy. They will apply the various skills
developed to solve a number of real-world problems.
GCO
Develop algebraic and graphical reasoning
through the study of relations.
SCO RF7
Solve problems that involve
inequalities in two variables.
SCO RF8
Solve problems that involve
variable.
204
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
Process Standards
Key
[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
Relations and Functions
Relations and Functions
RF6 Relate linear equations
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]
RF7 Solve problems that involve
in two variables.
RF12 Graph and analyze
polynomial functions (limited to
polynomial functions of degree ≤ 5).
[C, PS, T, V]
[C, CN, T, V]
RF8 Solve problems that involve
variable.
[CN, PS, V]
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
205
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF7 Solve problems that involve
in two variables.
[C, PS, T, V]
In Grade 9, students solved single variable linear inequalities with
rational coefficients and graphed their solutions on a number line
(9PR4). Students are familiar with the inequality symbols, the terms
continuous and discrete data, as well as their effect on a graph. In
Mathematics 1201, students graphed linear relations expressed in slopeintercept form, general form and point-slope form (RF6).
Students are familiar with verifying solutions to linear equations and
linear inequalities using substitution. They also verified solutions by
examining the number line graphs of inequalities. Students have been
exposed to the concept that the solution of a linear inequality consists
of a set of points while the solution of a linear equation has only one
solution.
In this unit, students will solve quadratic inequalities in one variable, as
well as linear and quadratic inequalities in two variables.
Achievement Indicators:
RF7.1 Explain, using examples,
how test points can be used to
determine the solution region that
satisfies a linear inequality.
RF7.2 Explain, using examples,
when a solid or broken line
should be used in the solution for
a linear inequality.
RF7.3 Sketch, with or without
technology, the graph of a linear
inequality.
206
Students will first investigate linear inequalities with two variables. The
solution of a linear inequality in two variables, such as
Ax + By > C, consists of any ordered pair (x, y) that produces a true
statement when the values of x and y are substituted into the inequality.
Since there may be infinitely many solutions, it is not possible for
students to list all of them. Discussion around the difference between
an equation and an inequality should highlight that the solution to
an inequality in two variables is represented by a region. Explain to
students that the boundary line Ax + By = C divides the coordinate plane
into two parts, where one part represents the solutions to the inequality.
To be successful with graphing inequalities in two variables, students
must be proficient with graphing linear equations, with and without
technology. Identifying the form of the equation will help students
decide which method they should choose when graphing the line (i.e.,
slope-intercept, point-slope or x-and y-intercept method). Initially,
students use test points to investigate the region of the plane that
satisfies the inequality which will help guide them to an understanding
of when to shade above or below the boundary. Working through several
examples should help them make the connection between the sign of the
inequality, the shaded region and whether to use a solid or broken line.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
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 complete a table similar to the one below:
Inequality
broken or solid line
y ≤ -2x + 8
x+y>2
(RF7.2, RF7.3)
Pre-Calculus 11
•
A student was asked to graph 3x - 2y > 12. His solution is shown below. Ask stidents to identify and correct his error(s).
Student Solution:
3x - 2y > 12
-2y > -3x + 12
y
10
8
6
4
y > 23 x − 6
2
-10
-8
-6
-4
-2
2
4
6
8
10
x
-2
9.1 Linear Inequalities in Two
Variables
-4
-6
-8
-10
(RF7.3)
Performance
•
Ask students to work in groups of two for this activity. Each group
should be given a deck of cards consisting of 12 with a graph, 12
with an inequality and one “YOU WIN” card.
Student Book (SB): pp.464-475
Teacher Resource (TR): pp.317323
BlackLine Master (BLM): 9-3, 9-4
A dealer is chosen to deal the cards to each student. Students
make pairs consisting of a graph and its matching inequality.
These pairs are set aside face up on the table. Students take turns
selecting a card from their partner and making matches if possible.
Play continues until all matches are made. The students who has
the YOU WIN card at the end is declared the winner.
(RF7.3)
Journal
•
3y + 2x = 4, 3y + 2x > 4, and 3y + 2x ≥ 4. Ask students to write a
response.
(RF7.2)
•
The solution region for the inequality 5x - 3y > 10 is above the
line since it contains a “greater than sign”. Ask students if they
agree or disagree with their statement. They should explain their
reasoning.
(RF7.3)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
207
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF7 Continued ...
Achievement Indicators:
RF7.1, RF7.2, RF7.3 Continued
When solving single variable inequalities, students sometimes forget
to reverse the sign of the inequality when multiplying or dividing both
sides by a negative number. A brief review may be necessary as this
error will also affect the feasible region when solving inequalities in two
variables.
Students should be provided with opportunities to write the equation of
the linear inequality given its graph. They will find the equation of the
boundary line and then use the given shaded region to determine the
correct inequality. Remind students what a broken or solid boundary
line represents.
RF7.4 Solve a problem that
involves a linear inequality.
Students will relate inequalities to real life contexts. Problems that
can be expressed as an inequality in two variables require students
to find two unknown quantities under certain constraints. Students
will translate the word problem into an inequality. Ensure that they
understand and define the variables that are being used to represent the
unknown quantities. They should also realize that the shaded feasible
region represents all possible combinations for the two quantities. An
example such as the following could be used to highlight feasible and
realistic solutions to a problem.
•
With two minutes left in a basketball game, your team is 12 points
behind. What are two different numbers of 2-point and 3-point
shots your team could score to earn at least 12 points?
Students should rcognize that the domain and range contain only
positive values. Ask them to identify realistic points in the feasible region
considering the time remaining in the game.
208
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
(i) Susan plans to spend a maximum of 15 hours reviewing
Math and Biology in preparation for examinations. Draw
a graph showing how much time she could spend studying
each subject.
(RF7.4)
(ii) A contractor has at least one hundred tonnes of soil to be
moved using two trucks. One truck has a 4 ton capacity and the other has a 5 ton capacity. Make a graph to show the
various combinations of loads the two trucks could carry to
complete the job.
(RF7.4)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
9.1 Linear Inequalities in Two
Variables
SB: pp.464-475
TR: pp.317-323
BLM: 9-3, 9-4
209
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF8 Solve problems that involve
variable.
[CN, PS, V]
Achievement Indicators:
RF8.1 Determine the solution
of a quadratic inequality in one
variable, using strategies such as
case analysis, graphing, roots and
test points, or sign analysis; and
explain the strategy used.
RF8.2 Represent and solve a
inequality in one variable.
RF8.3 Interpret the solution to a
inequality in one variable.
In Mathematics 1201, students were introduced to domain and range
(RF1). They used interval notation and set notation to represent
solution sets. This will now be extended to solving quadratic
inequalities.
It is important to have a discussion with students about what it means
to solve a quadratic inequality in one variable. When solving
ax2 + bx + c ≥ 0, for example, students should interpret this as finding
the possible x-values where the corresponding y-values are zero or
positive. In other words, when is the graph of y = ax2 + bx + c on or
above the x-axis. This can be done graphically or algebraically. Students
may have a better understanding of the algebraic method, however,
if it is related back to the graph. Regardless of the technique used,
proficiency in solving quadratic equations is important.
One strategy that can be used to solve a quadratic inequality involves
roots and test points. Solving an inequality such as x2 + 2x - 3 < 0,
students first determine the roots of the quadratic equation
x2 + 2x - 3 = 0. They will then use a sign diagram consisting of a number
line and test points to determine the intervals that satisfy the inequality.
The sign diagram was introduced in the Absolute Value and Reciprocal
Unit.
y
y = (x – 1)(x + 3)
9
6
3
-5
-4
-3
-2
-1
1
2
3
4
5
x
-3
(-1, -4)
-6
-9
-12
y-values -
y-values +
test point
x = -4
-3
test point
x=0
y-values +
1
test point
x=2
x
Relating the sign diagram to the x-axis of the graph, students use test
points to determine if the function is positive or negative. They should
recognize the roots -3 and 1 are not part of the solution. Students can
write the solution set as set notation, {x/ -3 < x < 1, x ∈ \ } or interval
notation (-3, 1).
Students will solve problems that involve a quadratic inequality in one
variable. These types of problems will be similar to solving a quadratic
equation except there is usually a minimum or maximum constraint.
Students should relate their solution to the context of the problem.
210
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
(i) Given A = x2 - 7 and B = -4x + 5, for what values of x is A <
B?
(RF8.1)
Variable
(ii) Use the graphs shown below to state the x-values for which
f (x) > 0.
y
8
7
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
y
y = f(x)
1 2 3 4 5 6
SB: pp.476-487
TR: pp.324-329
BLM: 9-3, 9-5
8
7
6
5
4
3
2
1
x
-5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6
x
y = f(x)
(RF8.1)
(iii) When a projectile is fired into the air, its height h, in metres,
t seconds later is given by the equation h = 11t - 3t2. When is
the projectile at least 6 m above the ground?
(RF8.2, RF8.3)
(iv) When a baseball is hit by a batter, the height of the ball, h(t),
at time t, is determined by the equation h(t) = -16t2 + 64t +
4. For which interval of time is the height of the ball greater
than or equal to 52 feet?
(RF8.2, RF8.3)
(v) The surface area, A, of a cylinder with radius r is given by the
formula A = 2r2 - 5r. What possible radii would result in an
area that is greater than 12 cm2?
(RF8.2, RF8.3)
Journal
•
x2 - 2x - 3 = 0, x2 - 2x - 3 ≥ 0, and x2 - 2x - 3 < 0. Ask students to
write a response.
(RF8.1, RF8.2)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
211
Relations and Functions
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
RF7 Continued...
Achievement Indicators:
RF7.5 Explain, using examples,
how test points can be used to
determine the solution region that
RF7.6 Explain, using examples,
when a solid or broken line
should be used in the solution for
RF7.7 Sketch, with or without
technology, the graph of a
RF7.8 Solve a problem that
212
The strategies used to graph linear inequalities in two variables will
now be extended to quadratic inequalities in two variables. Solving
quadratic inequalities in one variable involved students finding the set
of all x values that made a particular inequality true. Solving a quadratic
inequality in two variables implies finding, graphically, all the coordinate
pairs (x, y) that make a particular inequality true.
The graph of a quadratic function separates the plane into two regions,
one of which contains all the points that satisfy the inequality. Students
will graph the corresponding function that is associated with the
inequality and use test points to see which region should be shaded.
They should continue to ask themselves if the graph should be drawn
with a solid or broken curve.
Students will also write an inequality to describe a graph, given the
function defining its boundary. Ensure students recognize whether
the parabola is a solid or broken curve. Ask them how to identify the
inequality using test points.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
General Outcome: Develop algebraic and graphical reasoning through the study
of relations.
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
Authorized Resource
•
Pre-Calculus 11
Ask students to sketch the feasible region for each of the following
inequalities.
(i) y ≥ x2 + 2x - 3
2
(ii) y < −2( x − 1) + 4
(iii) y < 13 ( x − 3)( x + 3) 9.3 Quadratic Inequalities in Two
Variables
(RF7.7)
•
Give students a number of linear and quadratic inequalities such
as:
SB: pp.488-500
TR: pp.330-336
BLM: 9-3, 9-6
(i) 2x - 3y < 7
(ii) -x +4y ≥ -6
(iii) y > -2(x + 1)2 - 4
(iv) y ≤ -x2 + 1
Students can work in groups. Provide them with a list of 5 points
and have them work together to determine which inequality, if
any, satisfies their point.
(RF7.1, RF7.5)
•
The base of a rectangular bin currently has dimensions 12 m by
5 m. The base is to be enlarged by an equal amount on the width
and length so that the area is more than doubled. Ask students by
how much should the length and width be increased to produce
the desired area?
x
12
5
x
(RF7.8)
•
Ask students to determine an inequality to match each graph.
y
y
6
6
5
5
4
4
3
3
2
2
1
1
-8
-7
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
-6
Inequality:
2
3
4
x
-2
-1
1
2
3
4
5
6
7
8
9
10
x
-1
-2
-3
-4
-5
-6
Inequality:
(RF7.6)
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
213
214
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
REFERENCES
REFERENCES
American Association for the Advancement of Science [AAAS–Benchmarks]. Benchmarks for Science Literacy. New York, NY: Oxford University Press, 1993.
Armstrong, Thomas. 7 Kinds of Smart: Identifying and Developing Your Many Intelligences. New York, NY:
Plume, 1993.
British Columbia Ministry of Education. The Primary Program: A Framework for Teaching. Victoria, BC:
British Columbia Ministry of Education, 2000.
Brown, Richard and David Robbins. Advanced Mathematics - A Precalculus Course. Markham, Ontario:
Caine, Renate Nummela and Geoffrey Caine. Making Connections: Teaching and the Human Brain.
Alexandria, VA: Association for Supervision and Curriculum Development, 1991.
Government of Newfoundland and Labrador. Foundations for the Atlantic Canada Mathematics Curriculum. Halifax, NS: Atlantic Provinces Education Foundation.
Government of Newfoundland and Labrador: Department of Education. Mathematics Grade 10 Interim Edition Curriculum Guide 2011. Retrieved March 2012 from the World Wide Web: http://www.ed.gov.
nl.ca/edu/k12/curriculum/guides/mathematics/index.html.
Hope, Jack A. et al. Mental Math in Junior High. Palo Alto, CA: Dale Seymour Publications, 1988.
Hope, Jack A. et al. Mental Math in the Primary Grades. Palo Alto, CA: Dale Seymour Publications, 1988.
Keeley, Page and Cheryl Rose Tobey. Mathematics Formative Assessment. Thousand Oaks, California: Corwin, 2011.
Kelly, Brendan. et al. Addison-Wesley Mathematics 12. Don Mills, Ontario: Addison-Wesley Publishers
Limited., 1991.
McAskill, Bruce. et al. Pre-Calculus 11. Toronto, ON: McGraw-Hill Ryerson Limited., 2011.
ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012
215
REFERENCES
National Council of Teachers of Mathematics. Computation, Calculators, and Common Sense: A Position of
the National Council of Teachers of Mathematics. May 2005. http://www.nctm.org/about/pdfs/position/
computation.pdf (Accessed February 22, 2007).
Rubenstein, Rheta N. “Mental Mathematics beyond the Middle School: Why? What? How?” Mathematics
Teacher 94, 6 (September 2001), pp. 442–446.
Shaw, J. M. and M. J. P. Cliatt. “Developing Measurement Sense.” In P. R. Trafton (ed.), New Directions
for Elementary School Mathematics: 1989 Yearbook (Reston, VA: National Council of Teachers of
Mathematics, 1989), pp. 149–155.
Small, Marian. Making Math Meaningful to Canadian Students, K-8. Toronto, ON: Nelson Education, 2008.
Steen, L. A., ed. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: Mathematical
Sciences Education Board, National Research Council, 1990.
Travers, Kenneth. et al. Using Advanced Algebra. Toronto: Collier Macmillan Canada, Inc., 1985.
Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmemtally. 4th ed.
Boston, MA: Addison Wesley LAyman, INC., 2001.
Western and Northern Canadian Protocol for Collaboration in Basic Education (Kindergarten to Grade
12). The Common Curriculum Framework for K–9 Mathematics: Western and Northern Canadian Protocol
– May 2006 and The Common Curriculum Framework for Grades 10–12 – January 2008. Reproduced