Mathematics MAT421A - Prince Edward Island

Transcription

ACKNOWLEDGMENTS
Acknowledgments
The Department of Education and Early Childhood Development of Prince Edward Island gratefully
acknowledges the contributions of the following groups and individuals toward the development of the
Prince Edward Island MAT421A Mathematics Curriculum Guide:

The following specialist from the Prince Edward Island Department of Education and Early
Childhood Development:
J. Blaine Bernard,
Secondary Mathematics Specialist,
Department of Education and
Early Childhood Development

The 2010-2011 MAT421A pilot teachers:
Kirk Cutcliffe,
Colonel Gray Senior High School
Bethany MacLeod,
Montague Regional High School
Alden DeRoche,
Karen Malone,
Lianne Garland,
Alan McAlduff,
Westisle Composite High School
Tobey Kielly,
Glenda McInnis,
Suzanne Lee,
Heidi Morgan,
Mary Ellen Lowther,
Laura Rogers,
Rob MacDonald,
Scott Wicksted,

The Western and Northern Canadian Protocol (WNCP) for Collaboration in Education

Alberta Education
PRINCE EDWARD ISLAND MAT421A MATHEMATICS CURRICULUM GUIDE
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TABLE OF CONTENTS
Table of Contents
Background and Rationale............................................................................................... 1
Essential Graduation Learnings ................................................................ 1
Curriculum Focus ...................................................................................... 2
Connections across the Curriculum .......................................................... 2
Conceptual Framework for 10-12 Mathematics ............................................................. 3
Pathways and Topics ................................................................................ 4
Mathematical Processes ........................................................................... 5
The Nature of Mathematics ....................................................................... 8
Contexts for Learning and Teaching ............................................................................ 11
Homework ............................................................................................... 11
Diversity of Student Needs ..................................................................... 12
Gender and Cultural Diversity ................................................................. 12
Mathematics for EAL Learners ............................................................... 12
Education for Sustainable Development ................................................. 13
Inquiry-Based Learning and Project Based Learning ............................. 13
Assessment and Evaluation .......................................................................................... 14
Assessment............................................................................................. 14
Evaluation ............................................................................................... 16
Reporting................................................................................................. 16
Guiding Principles ................................................................................... 16
Structure and Design of the Curriculum Guide ........................................................... 18
Specific Curriculum Outcomes...................................................................................... 20
Measurement .......................................................................................... 20
Algebra and Number ............................................................................... 30
Relations and Functions ......................................................................... 42
Curriculum Guide Supplement ...................................................................................... 63
Unit Plans ......................................................................................................................... 65
Chapter 1 Measurement ..................................................................... 65
Chapter 2 Trigonometry ...................................................................... 73
Chapter 3 Factors and Products ......................................................... 81
Chapter 4 Roots and Powers .............................................................. 91
Chapter 5 Relations and Functions .................................................... 99
Chapter 6 Linear Functions .............................................................. 107
Chapter 7 Systems of Linear Equations ........................................... 115
Glossary of Mathematical Terms................................................................................. 123
Solutions to Possible Assessment Strategies ........................................................... 137
Mathematics Research Project .................................................................................... 145
References ..................................................................................................................... 155
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BACKGROUND AND RATIONALE
Background and Rationale
The development of an effective mathematics curriculum has encompassed a solid research base.
Developers have examined the curriculum proposed throughout Canada and secured the latest research
in the teaching of mathematics, and the result is a curriculum that should enable students to understand
and use mathematics.
The Western and Northern Canadian Protocol (WNCP) Common Curriculum Framework for Grades 1012 Mathematics (2008) has been adopted as the basis for a revised high school mathematics curriculum
in Prince Edward Island. The Common Curriculum Framework was developed by the seven Canadian
western and northern ministries of education (British Columbia, Alberta, Saskatchewan, Manitoba, Yukon
Territory, Northwest Territories, and Nunavut) in collaboration with teachers, administrators, parents,
business representatives, post-secondary educators, and others. The framework identifies beliefs about
mathematics, general and specific student outcomes, and achievement indicators agreed upon by the
seven jurisdictions. This document is based on both national and international research by the WNCP,
and on the Principles and Standards for School Mathematics (2000), published by the National Council of
Teachers of Mathematics (NCTM).
 Essential Graduation Learnings
Essential Graduation Learnings (EGLs) are statements describing the knowledge, skills, and attitudes
expected of all students who graduate from high school. Achievement of the essential graduation
learnings will prepare students to continue to learn throughout their lives. These learnings describe
expectations not in terms of individual school subjects but in terms of knowledge, skills, and attitudes
developed throughout the curriculum. They confirm that students need to make connections and develop
abilities across subject boundaries if they are to be ready to meet the shifting and ongoing demands of
life, work, and study today and in the future. Essential graduation learnings are cross curricular, and
curriculum in all subject areas is focused to enable students to achieve these learnings. Essential
graduation learnings serve as a framework for the curriculum development process.
Specifically, graduates from the public schools of Prince Edward Island will demonstrate knowledge,
skills, and attitudes expressed as essential graduation learnings, and will be expected to
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respond with critical awareness to various forms of the arts, and be able to express
themselves through the arts;
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assess social, cultural, economic, and environmental interdependence in a local and
global context;
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use the listening, viewing, speaking, and writing modes of language(s), and mathematical
and scientific concepts and symbols, to think, learn, and communicate effectively;
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continue to learn and to pursue an active, healthy lifestyle;
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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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use a variety of technologies, demonstrate an understanding of technological
applications, and apply appropriate technologies for solving problems.
More specifically, curriculum outcome statements articulate what students are expected to know and be
able to do in particular subject areas. Through the achievement of curriculum outcomes, students
demonstrate the essential graduation learnings.
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BACKGROUND AND RATIONALE
 Curriculum Focus
There is an emphasis in the Prince Edward Island mathematics curriculum on particular key concepts at
each grade which will result in greater depth of understanding. There is also more emphasis on number
sense and operations in the early grades to ensure students develop a solid foundation in numeracy. The
intent of this document is to clearly communicate to all educational partners high expectations for
students in mathematics education. Students must learn mathematics with understanding, actively
building new knowledge from experience and prior knowledge (NCTM, Principles and Standards for
School Mathematics, 2000).
The main goals of mathematics education are to prepare students to
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use mathematics confidently to solve problems;
communicate and reason mathematically;
appreciate and value mathematics;
make connections between mathematics and its applications;
commit themselves to lifelong learning;
become mathematically literate adults, using mathematics to contribute to society.
Students who have met these goals will
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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.
 Connections across the Curriculum
The teacher should take advantage of the various opportunities available to integrate mathematics and
other subjects. This integration not only serves to show students how mathematics is used in daily life,
but it helps strengthen the understanding of mathematical concepts by students and provides them with
opportunities to practise mathematical skills. There are many possibilities for integrating mathematics in
literacy, science, social studies, music, art, physical education, and other subject areas. Efforts should be
made to make connections and use examples drawn from a variety of disciplines.
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CONCEPTUAL FRAMEWORK FOR 10-12 MATHEMATICS
Conceptual Framework for 10-12 Mathematics
The chart below provides an overview of how mathematical processes and the nature of mathematics
influence learning outcomes.
GRADE
10
11
12
TOPICS
The topics vary in the courses
from grades ten to twelve. These
topics include:
Algebra
Calculus
Financial Mathematics
Geometry
Logical Reasoning
Mathematics Research
Project
Measurement
Number
Permutations,
Combinations, and the
Binomial Theorem
Probability
Relations and Functions
Statistics
Trigonometry
MATHEMATICAL PROCESSES
GENERAL
CURRICULUM OUTCOMES (GCOs)
SPECIFIC
CURRICULUM OUTCOMES (SCOs)
ACHIEVEMENT INDICATORS
NATURE
OF
MATHEMATICS
Change
Constancy
Number Sense
Patterns
Relationships
Spatial Sense
Uncertainty
Communication, Connections,
Reasoning, Mental Mathematics
and Estimation, Problem Solving,
Technology, Visualization
The mathematics curriculum describes the nature of mathematics, as well as the mathematical processes
and the mathematical concepts to be addressed. This curriculum is arranged into a number of topics, as
described above. These topics are not intended to be discrete units of instruction. The integration of
outcomes across topics makes mathematical experiences meaningful. Students should make the
connections among concepts both within and across topics. Consider the following when planning for
instruction:
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Integration of the mathematical processes within each topic is expected.
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Decreasing emphasis on rote calculation, drill, and practice, and the size of numbers
used in paper and pencil calculations makes more time available for concept
development.
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Problem solving, reasoning, and connections are vital to increasing mathematical fluency,
and must be integrated throughout the program.
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There is to 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 models and gradually developed from the concrete to the
pictorial to the symbolic.
 Pathways and Topics
The Prince Edward Island 10-12 mathematics curriculum includes pathways with corresponding topics
rather than strands, which are found in the Prince Edward Island K-9 mathematics curriculum. Three
pathways are available: Apprenticeship and Workplace Mathematics, Foundations of Mathematics,
and Pre-Calculus. A common grade ten course (Foundations of Mathematics and Pre-Calculus, Grade
10) is the starting point for the Foundations of Mathematics pathway and the Pre-Calculus pathway.
Each topic area requires that students develop a conceptual knowledge base and skill set that will be
useful to whatever pathway they have chosen. The topics covered within a pathway are meant to build
upon previous knowledge and to progress from simple to more complex conceptual understandings.
These pathways are illustrated in the diagram below:
K to 9
GRADE
10
MAT431A
Apprenticeship and
Workplace
Mathematics 10
GRADE
11
MAT531A
Apprenticeship and
Workplace
Mathematics 11
MAT421A
Common Grade 10 Course
Foundations of Mathematics &
Pre-Calculus
MAT521A
Foundations
of
Mathematics 11
MAT521B
Pre-Calculus 11
MAT621A
Foundations
of
Mathematics 12
MAT621B
Pre-Calculus 12
MAT521E
Pre-Calculus
Elective
MAT801A
Applied
Mathematics
GRADE
12
MAT631A
Apprenticeship and
Workplace
Mathematics 12
Please note that MAT801A may be taken by any student in
grade eleven or twelve from any pathway, as it is an open
course.
MAT611B
Advanced
Mathematics and
Calculus
The goals of all three pathways are to provide the prerequisite knowledge, skills, understandings, and
attitudes for specific post-secondary programs or direct entry into the work force. All three pathways
provide students with mathematical understandings and critical-thinking skills. It is the choice of topics
through which those understandings and skills are developed that varies among pathways. Each
pathway is designed to provide students with the mathematical understandings, rigour and criticalthinking skills that have been identified for specific post-secondary programs of study or for direct entry
into the work force. When choosing a pathway, students should consider their interests, both current and
future. Students, parents and educators are encouraged to research the admission requirements for
post-secondary programs of study as they vary by institution and by year.
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Apprenticeship and Workplace Mathematics
This pathway 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 work force. Topics include
algebra, geometry, measurement, number, statistics, and probability.
Foundations of Mathematics
skills identified for post-secondary studies in programs that do not require the study of theoretical
calculus. Topics include financial mathematics, geometry, measurement, algebra and number, logical
reasoning, relations and functions, statistics, probability, and a mathematics research project.
Pre-Calculus
skills identified for entry into post-secondary programs that require the study of theoretical calculus.
Topics include algebra and number, measurement, relations and functions, trigonometry, combinatorics,
and introductory calculus.
 Mathematical Processes
There are critical components that students must encounter in a mathematics program in order to achieve
the goals of mathematics education and encourage lifelong learning in mathematics. The Prince Edward
Island mathematics curriculum incorporates the following seven interrelated mathematical processes that
are intended to permeate teaching and learning. These unifying concepts serve to link the content to
methodology.
Students are expected to
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communicate in order to learn and express their understanding of mathematics;
[Communications: C]
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connect mathematical ideas to other concepts in mathematics, to everyday experiences,
and to other disciplines; [Connections: CN]
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demonstrate fluency with mental mathematics and estimation; [Mental Mathematics and
Estimation: ME]
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develop and apply new mathematical knowledge through problem solving; [Problem
Solving: PS]
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develop mathematical reasoning; [Reasoning: R]
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select and use technologies as tools for learning and solving problems; [Technology: T]
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develop visualization skills to assist in processing information, making connections, and
solving problems. [Visualization: V]
Communication [C]
Students need opportunities to read about, represent, view, write about, listen to, and discuss
mathematical ideas. These opportunities allow students to create links between their own language and
ideas, and the formal language and symbols of mathematics. Communication is important in clarifying,
reinforcing, and modifying ideas, knowledge, 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 can help students
make connections among concrete, pictorial, symbolic, verbal, written, and mental representations of
mathematical ideas.
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Connections [CN]
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 can 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.
For instance, opportunities should be created frequently to link mathematics and career opportunities.
Students need to become aware of the importance of mathematics and the need for mathematics in many
career paths. This realization will help maximize the number of students who strive to develop and
maintain the mathematical abilities required for success in further areas of study.
Mental Mathematics and Estimation [ME]
Mental mathematics is a combination of cognitive strategies that enhance flexible thinking and number
sense. It involves calculation 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 mathematics (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). Mental mathematics “provides a cornerstone for all estimation
processes offering a variety of alternate algorithms and non-standard techniques for finding answers”
(Hope, 1988).
Estimation is a strategy for determining approximate values or quantities, usually by referring to
benchmarks or using referents, or for determining the reasonableness of calculated values. Students
need to know when to estimate, what strategy to use, and how to use it. Estimation is used to make
mathematical judgments and develop useful, efficient strategies for dealing with situations in daily life.
Students need to develop both mental mathematics and estimation skills through context and not in
isolation so they are able to apply them to solve problems. Whenever a problem requires a calculation,
students should follow the decision-making process described below:
Problem Situation
Calculation Required
Approximate
Answer Appropriate
Exact Answer
Needed
Use Mental
Calculation
Use Paper
and Pencil
Use a
Calculator/Computer
Estimate
(NCTM)
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Problem Solving [PS]
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. . . ?" or “How could
you. . . ?” the problem-solving approach is being modelled. Students develop their own problem-solving
strategies by being open to listening, discussing, and trying different strategies.
In order for an activity to be problem-solving based, it must ask students to determine a way to get from
what is known to what is sought. If students have already been given ways 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 also a powerful teaching tool that fosters multiple, creative, and innovative solutions.
Creating an environment where students openly look for and engage in finding a variety of strategies for
solving problems empowers students to explore alternatives and develops confident and cognitive
mathematical risk takers.
Over time, numerous problem-solving strategies should be modelled for students, and students should be
encouraged to employ various strategies in many problem-solving situations. While choices with respect
to the timing of the introduction of any given strategy will vary, the following strategies should all become
familiar to students:
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using estimation
guessing and checking
looking for a pattern
making an organized list or table
using a model
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working backwards
using a formula
using a graph, diagram, or flow chart
solving a simpler problem
using algebra.
Reasoning [R]
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 inductive and deductive reasoning.
Inductive reasoning occurs when students explore and record results, analyse observations, make
generalizations from patterns, and test these generalizations. Deductive reasoning occurs when students
reach new conclusions based upon what is already known or assumed to be true.
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.
Calculators and computers can be used to
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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 and test properties;
develop personal procedures for mathematical operations;
create geometric displays;
simulate situations;
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
develop number sense.
Technology contributes to a learning environment in which the growing curiosity of students can lead to
rich mathematical discoveries at all grade levels. While technology can be used in K-3 to enrich learning,
it is expected that students will meet all outcomes without the use of technology.
Visualization [V]
Visualization involves thinking in pictures and images, and the ability to perceive, transform, and recreate
different aspects of the visual-spatial world. The use of visualization in the study of mathematics provides
students with opportunities to understand mathematical concepts and make connections among them.
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 and when to estimate, and knowledge of several
estimation strategies (Shaw & Cliatt, 1989).
Visualization is fostered through the use of concrete materials, technology, and a variety of visual
representations.
 The Nature of Mathematics
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 which are woven throughout this document. These
components include change, constancy, number sense, patterns, relationships, spatial sense, and
uncertainty.
Change
It is important for students to understand that mathematics is dynamic and not static. As a result,
recognizing change is a key component in understanding and developing mathematics. 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
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skip counting by 2s, starting from 4;
an arithmetic sequence, with first term 4 and a common difference of 2; or
a linear function with a discrete domain.
Constancy
Different aspects of constancy are described by the terms stability, conservation, equilibrium, steady
state, and symmetry (AAAS–Benchmarks, 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:
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The area of a rectangular region is the same regardless of the methods used to
determine the solution.
The sum of the interior angles of any triangle is 1800.
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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
Number sense, which can be thought of as intuition about numbers, is the most important foundation of
numeracy (The Primary Program, B.C., 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. Number sense
develops when students connect numbers to real-life experiences, and use benchmarks and referents.
This results in students who are computationally fluent, and flexible and intuitive with numbers. The
evolving number sense typically comes as a by-product of learning rather than through direct instruction.
However, number sense can be developed by providing rich mathematical tasks that allow students to
make connections.
Patterns
Mathematics is about recognizing, describing, and working with numerical and non-numerical patterns.
Patterns exist in all topics in mathematics and it is important that connections are made among topics.
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 develop students’
algebraic thinking that is foundational for working with more abstract mathematics in higher grades.
Relationships
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 the collecting and analysing of data, and describing relationships visually,
symbolically, orally, or in written form.
Spatial Sense
Spatial sense involves visualization, mental imagery, and spatial reasoning. These skills are central to
the understanding of mathematics. Spatial sense enables students to interpret representations of 2-D
shapes and 3-D objects, and identify relationships to mathematical topics. 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 2-D shapes and 3-D objects.
Spatial sense offers a way 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 objects. Spatial sense allows students to use dimensions and make predictions about the
results of changing dimensions.
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Knowing the dimensions of an object enables students to communicate about the object
and create representations.
The volume of a rectangular solid can be calculated from given dimensions.
Doubling the length of the side of a square increases the area by a factor of four.
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Uncertainty
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.
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CONTEXTS FOR LEARNING AND TEACHING
Contexts for Learning and Teaching
The Prince Edward Island mathematics curriculum is based upon several key assumptions or beliefs
about mathematics learning which have grown out of research and practice:
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Mathematics learning is an active and constructive process.
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Learners are individuals who bring a wide range of prior knowledge and experiences, and
who learn via various styles and at different rates.
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Learning is most likely to occur in meaningful contexts and in an environment that
supports exploration, risk taking, and critical thinking, and that nurtures positive attitudes
and sustained effort.
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Learning is most effective when standards of expectation are made clear with ongoing
assessment and feedback.
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 successfully
developing numeracy is making connections to these backgrounds and experiences.
Young children develop a variety of mathematical ideas before they enter school. They make sense of
their environment through observations and interactions at home and in the community. Their
mathematics learning is embedded in everyday activities, such as playing, reading, storytelling, and
helping around the home. Such activities can contribute to the development of number and spatial sense
in children. Initial problem solving and reasoning skills are fostered when children are engaged in
activities such as comparing quantities, searching for patterns, sorting objects, ordering objects, creating
designs, building with blocks, and talking about these activities. Positive early experiences in
mathematics are as critical to child development as are early literacy 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. The use of models and a
variety of pedagogical approaches can address the diversity of learning styles and developmental stages
of students, and enhance the formation of sound, transferable, mathematical concepts. At all levels,
students benefit from working with a variety of materials, tools, and contexts when constructing meaning
about new mathematical ideas. Meaningful discussions can provide essential links among concrete,
pictorial, and symbolic representations of mathematics.
The learning environment should value and respect the experiences and ways of thinking of all students,
so that learners are 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. Learners must be encouraged that it is acceptable to solve problems in different
ways and realize that solutions may vary.
 Homework
Homework is an essential component of the mathematics program, as it extends the opportunity for
students to think mathematically and to reflect on ideas explored during class time. The provision of this
additional time for reflection and practice plays a valuable role in helping students to consolidate their
learning.
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Traditionally, homework has meant completing ten to twenty drill and practice questions relating to the
procedure taught in a given day. With the increased emphasis on problem solving, conceptual
understanding, and mathematical reasoning, however, it is important that homework assignments change
accordingly. More assignments involving problem solving, mathematical investigations, written
explanations and reflections, and data collection should replace some of the basic practice exercises
given in isolation. In fact, a good problem can sometimes accomplish more than many drill-oriented
exercises on a topic.
As is the case in designing all types of homework, the needs of the students and the purpose of the
assignment will dictate the nature of the questions included. Homework need not be limited to reinforcing
learning; it provides an excellent opportunity to revisit topics explored previously and to introduce new
topics before teaching them in the classroom. Homework provides an effective way to communicate with
parents and provides parents an opportunity to be actively involved in their child’s learning. By ensuring
that assignments model classroom instruction and sometimes require parental input, a teacher can give a
parent clearer understanding of the mathematics curriculum and of the child’s progress in relationship to
it. As Van de Walle (1994, p. 454) suggests, homework can serve as a parent’s window to the
classroom.
 Diversity in Student Needs
Every class has students at many different cognitive levels. Rather than choosing a certain level at which
to teach, a teacher is responsible for tailoring instruction to reach as many of these students as possible.
In general, this may be accomplished by assigning different tasks to different students or assigning the
same open-ended task to most students. Sometimes it is appropriate for a teacher to group students by
interest or ability, assigning them different tasks in order to best meet their needs. These groupings may
last anywhere from minutes to semesters, but should be designed to help all students (whether strong,
weak or average) to reach their highest potential. There are other times when an appropriately openended task can be valuable to a broad spectrum of students. For example, asking students to make up
an equation for which the answer is 5 allows some students to make up very simple equations while
others can design more complex ones. The different equations constructed can become the basis for a
very rich lesson from which all students come away with a better understanding of what the solution to an
equation really means.
 Gender and Cultural Equity
The mathematics curriculum and mathematics instruction must be designed to equally empower both
male and female students, as well as members of all cultural backgrounds. Ultimately, this should mean
not only that enrolments of students of both genders and various cultural backgrounds in public school
mathematics courses should reflect numbers in society, but also that representative numbers of both
genders and the various cultural backgrounds should move on to successful post-secondary studies and
careers in mathematics and mathematics-related areas.
 Mathematics for EAL Learners
The Prince Edward Island mathematics curriculum is committed to the principle that learners of English as
an additional language (EAL) should be full participants in all aspects of mathematics education. English
deficiencies and cultural differences must not be barriers to full participation. All students should study a
comprehensive mathematics curriculum with high-quality instruction and co-ordinated assessment.
The Principles and Standards for School Mathematics (NCTM, 2000) emphasizes communication “as an
essential part of mathematics and mathematics education” (p.60). The Standards elaborate that all
Page 12
students, and EAL learners in particular, need to have opportunities and be given encouragement and
support for speaking, writing, reading, and listening in mathematics classes. Such efforts have the
potential to help EAL learners overcome barriers and will facilitate “communicating to learn mathematics
and learning to communicate mathematically” (NCTM, p.60).
To this end,

schools should provide EAL learners with support in their dominant language and English
language while learning mathematics;

teachers, counsellors, and other professionals should consider the English-language
proficiency level of EAL learners as well as their prior course work in mathematics;

the mathematics proficiency level of EAL learners should be solely based on their prior
academic record and not on other factors;

mathematics teaching, curriculum, and assessment strategies should be based on best
practices and build on the prior knowledge and experiences of students and on their
cultural heritage;

the importance of mathematics and the nature of the mathematics program should be
communicated with appropriate language support to both students and parents;

to verify that barriers have been removed, educators should monitor enrolment and
achievement data to determine whether EAL learners have gained access to, and are
succeeding in, mathematics courses.
 Education for Sustainable Development
Education for sustainable development (ESD) involves incorporating the key themes of sustainable
development - such as poverty alleviation, human rights, health, environmental protection, and climate
change - into the education system. ESD is a complex and evolving concept and requires learning about
these key themes from a social, cultural, environmental, and economic perspective, and exploring how
those factors are interrelated and interdependent.
With this in mind, it is important that all teachers, including mathematics teachers, attempt to incorporate
these key themes in their subject areas. One tool that can be used is the searchable on-line database
Resources for Rethinking, found at http://r4r.ca/en. It provides teachers with access to materials that
integrate ecological, social, and economic spheres through active, relevant, interdisciplinary learning.
 Inquiry-Based Learning and Project Based Learning
Inquiry-based learning (IBL) allows students to explore, investigate, and construct new meaning from
prior knowledge and from new information that is retrieved from other sources. It is not linear in nature,
but promotes a continual looping back and forth throughout the process as students gather and process
new information, redirect their inquiries, and continue through the process. Mathematical inquiry will
require students to practise and refine their critical and creative-thinking skills. The terms inquiry and
research are often used interchangeably within an educational context. While research often becomes
the end-result of an inquiry process, it is the process itself that should be emphasized within an
educational context. More information regarding the development of a mathematics research project is
included in the appendix at the end of this document.
Page 13
ASSESSMENT AND EVALUATION
Assessment and Evaluation
Assessment and evaluation are essential components of teaching and learning in mathematics. The
basic principles of assessment and evaluation are as follows:

Effective assessment and evaluation are essential to improving student learning.

Effective assessment and evaluation are aligned with the curriculum outcomes.

A variety of tasks in an appropriate balance gives students multiple opportunities to
demonstrate their knowledge and skills.

Effective evaluation requires multiple sources of assessment information to inform
judgments and decisions about the quality of student learning.

Meaningful assessment data can demonstrate student understanding of mathematical
ideas, student proficiency in mathematical procedures, and student beliefs and attitudes
about mathematics.
Without effective assessment and evaluation it is impossible to know whether students have learned,
teaching has been effective, or how best to address student learning needs. The quality of assessment
and evaluation in the educational process has a profound and well-established link to student
performance. Research consistently shows that regular monitoring and feedback are essential to
improving student learning. What is assessed and evaluated, how it is assessed and evaluated, and how
results are communicated send clear messages to students and others.
 Assessment
Assessment is the systematic process of gathering information on student learning. To determine how
well students are learning, assessment strategies have to be designed to systematically gather
information on the achievement of the curriculum outcomes. Teacher-developed assessments have a
wide variety of uses, such as







providing feedback to improve student learning;
determining if curriculum outcomes have been achieved;
certifying that students have achieved certain levels of performance;
setting goals for future student learning;
communicating with parents about their children’s learning;
providing information to teachers on the effectiveness of their teaching, the program, and
the learning environment;
meeting the needs of guidance and administration.
A broad assessment plan for mathematics ensures a balanced approach to summarizing and reporting. It
should consider evidence from a variety of sources, including





formal and informal observations
work samples
anecdotal records
conferences
teacher-made and other tests





portfolios
learning journals
questioning
performance assessment
peer- and self-assessment.
This balanced approach for assessing mathematics development is illustrated in the diagram on the next
page.
Page 14
Work Samples
 math journals
 portfolios
 drawings, charts, tables, and graphs
 individual and classroom assessment
 pencil and paper tests
Surveys
 attitude
 interest
 parent questionnaires
Rubrics
 constructed response
 generic rubrics
 task-specific rubrics
 questioning
Assessing Mathematics
Development in a Balanced
Manner
Self-Assessment
 personal reflection/evaluation
Math Conferences
 individual
 group
 teacher-initiated
 child-initiated
Observations
 planned (formal)
 unplanned (informal)
 read aloud
 shared and guided math activities
 performance tasks
 individual conferences
 anecdotal records
 checklists
 interactive activities
There are three interrelated purposes for classroom assessment: assessment as learning, assessment
for learning, and assessment of learning. Characteristics of each type of assessment are highlighted
below.
Assessment as learning is used




to engage students in their own learning and self-assessment;
to help students understand what is important in the mathematical concepts and
particular tasks they encounter;
to develop effective habits of metacognition and self-coaching;
to help students understand themselves as learners - how they learn as well as what they
learn - and to provide strategies for reflecting on and adjusting their learning.
Assessment for learning is used




to gather and use ongoing information in relation to curriculum outcomes in order to
adjust instruction and determine next steps for individual learners and groups;
to identify students who are at risk, and to develop insight into particular needs in order to
differentiate learning and provide the scaffolding needed;
to provide feedback to students about how they are doing and how they might improve;
to provide feedback to other professionals and to parents about how to support students’
learning.
Assessment of learning is used


to determine the level of proficiency that a student has demonstrated in terms of the
designated learning outcomes for a unit or group of units;
to facilitate reporting;
Page 15

to provide the basis for sound decision-making about next steps in a student’s learning.
 Evaluation
Evaluation is the process of analysing, reflecting upon, and summarizing assessment information, and
making judgments or decisions based upon the information gathered. Evaluation involves teachers and
others in analysing and reflecting upon information about student learning gathered in a variety of ways.
This process requires




developing clear criteria and guidelines for assigning marks or grades to student work;
synthesizing information from multiple sources;
weighing and balancing all available information;
using a high level of professional judgment in making decisions based upon that
information.
 Reporting
Reporting on student learning should focus on the extent to which students have achieved the curriculum
outcomes. Reporting involves communicating the summary and interpretation of information about
student learning to various audiences who require it. Teachers have a special responsibility to explain
accurately what progress students have made in their learning and to respond to parent and student
inquiries about learning. Narrative reports on progress and achievement can provide information on
student learning which letter or number grades alone cannot. Such reports might, for example, suggest
ways in which students can improve their learning and identify ways in which teachers and parents can
best provide support. Effective communication with parents regarding their children’s progress is
essential in fostering successful home-school partnerships. The report card is one means of reporting
individual student progress. Other means include the use of conferences, notes, phone calls, and
electronic methods.
 Guiding Principles
In order to provide accurate, useful information about the achievement and instructional needs of
students, certain guiding principles for the development, administration, and use of assessments must be
followed. The document Principles for Fair Student Assessment Practices for Education in Canada
(1993) articulates five fundamental assessment principles, as follows:

Assessment methods should be appropriate for and compatible with the purpose and
context of the assessment.

Students should be provided with sufficient opportunity to demonstrate the knowledge,
skills, attitudes, or behaviours being assessed.

Procedures for judging or scoring student performance should be appropriate for the
assessment method used and be consistently applied and monitored.

Procedures for summarizing and interpreting assessment results should yield accurate
and informative representations of a student’s performance in relation to the curriculum
outcomes for the reporting period.

Assessment reports should be clear, accurate, and of practical value to the audience for
whom they are intended.
These principles highlight the need for assessment which ensures that

the best interests of the student are paramount;
Page 16



assessment informs teaching and promotes learning;
assessment is an integral and ongoing part of the learning process and is clearly related
to the curriculum outcomes;
assessment is fair and equitable to all students and involves multiple sources of
information.
While assessments may be used for different purposes and audiences, all assessments must give each
student optimal opportunity to demonstrate what he or she knows and can do.
Page 17
STRUCTURE AND DESIGN OF THE CURRICULUM GUIDE
Structure and Design of the Curriculum Guide
The learning outcomes in the Prince Edward Island high school mathematics curriculum are organized
into a number of topics across the grades from ten to twelve. Each topic has associated with it a general
curriculum outcome (GCO). They are overarching statements about what students are expected to learn
in each topic from grades ten to twelve.
Topic
General Curriculum Outcome (GCO)
Algebra (A)
Develop algebraic reasoning.
Algebra and Number (AN)
Develop algebraic reasoning and number sense.
Calculus (C)
Develop introductory calculus reasoning.
Financial Mathematics (FM)
Develop number sense in financial applications.
Geometry (G)
Develop spatial sense.
Logical Reasoning (LR)
Develop logical reasoning.
Mathematics Research Project
(MRP)
Develop an appreciation of the role of mathematics in society.
Develop spatial sense and proportional reasoning.
(Foundations of Mathematics and Pre-Calculus)
Measurement (M)
Develop spatial sense through direct and indirect measurement.
(Apprenticeship and Workplace Mathematics)
Number (N)
Develop number sense and critical thinking skills.
Permutations, Combinations and
Binomial Theorem (PC)
Develop algebraic and numeric reasoning that involves
combinatorics.
Probability (P)
Develop critical thinking skills related to uncertainty.
Relations and Functions (RF)
Develop algebraic and graphical reasoning through the study of
relations.
Statistics (S)
Develop statistical reasoning.
Trigonometry (T)
Develop trigonometric reasoning.
Each general curriculum outcome is then subdivided into a number of specific curriculum outcomes
(SCOs). Specific curriculum outcomes are statements that identify the specific skills, understandings, and
knowledge students are required to attain by the end of a given grade.
Finally, each specific curriculum outcome has a list of achievement indicators that are used to determine
whether students have met the corresponding specific curriculum outcome.
In this curriculum guide, each specific curriculum outcome (SCO) is presented in a two-page format, and
includes the following information:



its corresponding topic and general curriculum outcome;
the scope and sequence of the specific curriculum outcome(s) from grades nine to eleven
which correspond to this SCO;
the specific curriculum outcome, with a list of achievement indicators;
Page 18
STRUCTURE AND DESIGN OF THE CURRICULUM GUIDE


a list of the sections in Foundations and Pre-Calculus Mathematics 10 which address the
SCO, with specific achievement indicators highlighted in brackets;
an elaboration for the SCO.
In the second half of this document, a curriculum guide supplement is presented which follows the
primary resource, Foundations and Pre-Calculus Mathematics 10. As well, an appendix is included which
outlines the steps to follow in the development of an effective mathematics research project.
Page 19
SPECIFIC CURRICULUM OUTCOMES
MEASUREMENT
Page 20
M1 – Solve problems that involve linear measurement, using:

SI and imperial units of measure;

estimation strategies;

measurement strategies.
M2 – Apply proportional reasoning to problems that involve conversions between SI and
imperial units of measure.
M3 – Solve problems, using SI and imperial units, that involve the surface area and volume
of 3-D objects, including:

right cones;

right cylinders;

right prisms;

right pyramids;

spheres.
M4 – Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve
problems that involve right triangles.
Page 21
MAT421A – Topic: Measurement (M)
GCO:
GRADE 9
SCO:
GRADE 10 – MAT421A
GRADE 11
M1 Solve problems that involve
linear measurement, using:
 SI and imperial units of measure;
 estimation strategies;
 measurement strategies.
MAT521A
MAT521B



[ME, PS, V]
Students who have achieved this outcome should be able to:
A.
Provide referents for linear measurements, including millimetre, centimetre, metre, kilometre,
inch, foot, yard and mile, and explain the choices.
B.
Compare SI and imperial units, using referents.
C.
Estimate a linear measure, using a referent, and explain the process used.
D.
Justify the choice of units used for determining a measurement in a problem-solving context.
E.
Solve problems that involve linear measure, using instruments such as rulers, calipers or tape
measures.
F.
Describe and explain a personal strategy used to determine a linear measurement; e.g.,
circumference of a bottle, length of a curve, perimeter of the base of an irregular 3-D object.
Section(s) in Foundations and Pre-Calculus Mathematics 10 text that address the specific
curriculum outcome with relevant Achievement Indicators in brackets:
1.1 (A C D)
1.2 (A C D E F)
1.3 (B)
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
[PS] Problem Solving
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 22
SCO:



[ME, PS, V]
Elaboration
Canada’s official measurement system is SI (Système International d’Unités). Some SI units for linear
measurement are listed in the following table.
UNIT
ABBREVIATION
MULTIPLYING
FACTOR
kilometre
km
1000
hectometre
hm
100
decametre
dam
10
metre
m
1
decimetre
dm
0.1
centimetre
cm
0.01
millimetre
mm
0.001
Note that each unit in the SI measuring system is based on a power of 10. All linear measurements are derived
from the metre. The most common units are the kilometre (km), metre (m), centimetre (cm), and the millimetre (mm).
The kilometre is a large unit (1 km  1000 m) and is suitable for measuring large distances. The millimetre is a small
unit (1 mm  0.001 m) and is suitable for measuring small distances. Time will have to be spent on converting
between metric units, as this concept is not covered as a specific curriculum outcome prior to grade ten.
The imperial system of measurement is widely used in the United States for measuring distances. Even though
SI is Canada’s official measurement system, some Canadian industries still use imperial units. The following units
are the basic imperial units used for measuring distances.
inch (in)
foot (ft)
1 ft  12 in
yard (yd)
1 yd  3 ft  36 in
mile (mi)
1 mi  1760 yd  5280 ft
Various measuring instruments allow accurate measurement of distances in standard units. Personal referents
can also be developed when estimating measurements. SI rulers, metre sticks, and measuring tapes give
measurements to the nearest millimetre, or 0.1 cm. An SI caliper can accurately measure to the nearest tenth of a
millimetre, or 0.01 cm, depending on its scales. An imperial ruler or measuring tape can measure distances to the
1
1
nearest
in. An imperial caliper can measure to the nearest
in.
16
1000
A non-standard measuring unit can be used as a personal referent. Referents can help individuals estimate in
standard units, such as SI or imperial units. For example, suppose that the width of a fingernail is used to
approximate 1 cm. Then, something appears to be as wide as four fingernails can be estimated as being 4 cm wide.
Page 23
GCO:
GRADE 9
GRADE 11
M2 Apply proportional reasoning to
problems that involve conversions
between SI and imperial units of
measure.
M1 Solve problems that involve the
application of rates.
MAT521A
M2 Solve problems that involve
scale diagrams, using proportional
reasoning.
MAT521B
SCO:
M2 – Apply proportional reasoning to problems that involve conversions between SI and imperial
units of measure. [C, ME, PS]
A.
Explain how proportional reasoning can be used to convert a measurement within, or between,
SI and imperial systems.
B.
Solve a problem that involves the conversion of units within, or between, SI and imperial
systems.
C.
Verify, using unit analysis, a conversion within, or between, SI and imperial systems, and
explain the conversion.
D.
Justify, using mental mathematics, the reasonableness of a solution to a conversion problem.
1.1 (A B C)
1.3 (A B C D)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 24
SCO:
M2 – Apply proportional reasoning to problems that involve conversions between SI and imperial
units of measure. [C, ME, PS]
Elaboration
When solving problems involving measurement, it is crucial to work with the same units. It may be necessary to
convert units within one measurement system (for example, inches to feet), or between imperial and SI units (for
example, inches to centimetres).
To convert from one measurement system to another, it is necessary to understand the relationships among the
units of length in each system. All conversions involve proportional reasoning and unit analysis. Conversions
between measurement systems may be approximate or exact. For example, the inch in the imperial system has
been defined as exactly 2.54 cm.
The following are some common conversions:
Exact Conversions
1 in  2.54 cm
1 ft  30.48 cm
1 yd  0.9144 m
Approximate Conversions
Rounded to four significant digits
1 mm  0.03937 in
1 cm  0.3937 in
1 m  1.094 yd
1 m  3.281 ft
1 km  0.6214 mi
1 mi  1.609 km
Please note that these conversions should be used when solving problems, as the answers in the textbook do
not reflect the approximate conversions that are presented on page 18 in the textbook.
Page 25
GCO:
GRADE 9
SS2 Determine the surface area of
composite 3-D objects to solve
problems.
SCO:
M3 Solve problems, using SI and
imperial units, that involve the
surface area and volume of 3-D
objects, including:
 right cones;
 right cylinders;
 right prisms;
 right pyramids;
 spheres.
GRADE 11
MAT521A
M3 Demonstrate an understanding
of the relationships among scale
factors, areas, surface areas and
volumes of similar 2-D shapes and
3-D objects.
MAT521B
M3 – Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D
objects, including:

right cones;

right cylinders;

right prisms;

right pyramids;

spheres.
[CN, PS, R, V]
A.
Sketch a diagram to represent a problem that involves surface area or volume.
B.
Determine the surface area of a right cone, right cylinder, right prism, right pyramid or sphere,
using an object or its labelled diagram.
C.
Determine the volume of a right cone, right cylinder, right prism, right pyramid or sphere, using
an object or its labelled diagram.
D.
Determine an unknown dimension of a right cone, right cylinder, right prism, right pyramid or
sphere, given the object’s surface area or volume and the remaining dimensions.
E.
Solve a problem that involves surface area or volume, given a diagram of a composite 3-D
object.
F.
Describe the relationship between the volumes of:

right cones and right cylinders with the same base and height;

right pyramids and right prisms with the same base and height.
1.4 (A B D)
1.5 (A C D F)
1.6 (B C D)
1.7 (A D E)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 26
SCO:
M3 – Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D
objects, including:

right cones;

right cylinders;

right prisms;

right pyramids;

spheres.
[CN, PS, R, V]
Elaboration
The surface area of a right rectangular prism and of a right cylinder can be calculated using the area of the bases
(top and bottom) plus the lateral area. The volume of a right rectangular prism and a right cylinder can be calculated
by multiplying the area of the base by the height.
r
h
h
w
l
SA  2lw  2lh  2wh
SA  2r 2  2rh
V  lwh
V  r 2h
The surface area of a right rectangular pyramid and of a right cone can be calculated using the area of the base
plus the lateral area. The volume of a right rectangular pyramid is found by calculating one-third of the volume of its
related right prism. The volume of a right cone is found by calculating one-third of the volume of its related right
cylinder.
s1
s2
h
s
l
r
w
SA  lw  ls1  ws2
V 
SA  r 2  rs
1
lwh
3
V 
1 2
r h
3
The surface area and the volume of a sphere both depend on the only the radius.
r
SA  4r 2
V 
4 3
r
3
Page 27
GCO:
GRADE 9
SS3 Demonstrate an
understanding of similarity of
polygons.
M4 Develop and apply the primary
trigonometric ratios (sine, cosine,
tangent) to solve problems that
involve right triangles.
GRADE 11
MAT521A
G1 Derive proofs that involve the
properties of angles and triangles.
G2 Solve problems that involve the
properties of angles and triangles.
G3 Solve problems that involve the
cosine law and the sine law,
including the ambiguous case.
MAT521B
T1 Demonstrate an understanding
of angles in standard position (00 to
3600).
T2 Solve problems, using the three
primary trigonometric ratios for
angles from 00 to 3600 in standard
position.
T3 Solve problems, using the
cosine law and sine law, including
the ambiguous case.
SCO:
M4 – Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems
that involve right triangles. [C, CN, PS, R, T, V]
A.
Explain the relationships between similar right triangles and the definitions of the primary
trigonometric ratios.
B.
Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given
acute triangle in the triangle.
C.
Solve right triangles, with or without technology.
D.
Solve a problem that involves one or more right triangles by applying the primary trigonometric
ratios or the Pythagorean theorem.
E.
Solve a problem that involves indirect and direct measurement, using the trigonometric ratios,
the Pythagorean theorem and measurement instruments such as a clinometer or a metre stick.
2.1 (A B D)
2.2 (B D)
2.3 (E)
2.4 (A B D)
2.5 (B D)
2.6 (B C D)
2.7 (B D)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 28
SCO:
M4 – Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems
that involve right triangles. [C, CN, PS, R, T, V]
Elaboration
In similar triangles, corresponding angles are equal, and corresponding sides are in proportion. Therefore, the
ratios of the lengths of corresponding sides are equal. The sides of a right triangle are labelled according to a
reference angle, θ.
B
B

opposite
hypotenuse
adjacent
hypotenuse

C
adjacent
A
C
opposite
A
A trigonometric ratio is a ratio of the measures of two sides of a right triangle. The three primary trigonometric
ratios are tangent, sine, and cosine. The short form for the tangent ratio of angle A is tan A. It is defined as
length of side opposite A
tan A 
. The short form for the sine ratio of angle A is sin A. It is defined as
length of side adjacent to A
sin A 
. The short form for the cosine ratio of angle A is cos A. It is defined as
length of hypotenuse
cos A 
. A mnemonic device such as SOH – CAH – TOA can be used to remember
the three primary trigonometric ratios.
Students will also use the Pythagorean Theorem extensively when solving trigonometric problems. The
Pythagorean Theorem may need to be reviewed, as it was originally taught in grade eight. Also, the concept of
similar triangles, which was taught in grade nine, will be used as well.
A line of sight is an invisible line from one person or object to another person or object. Some applications of
trigonometry involve an angle of elevation or an angle of depression. An angle of elevation is the angle formed by the
horizontal and a line of sight above the horizontal. An angle of depression is the angle formed by the horizontal and a
line of sight below the horizontal. As the diagram below shows, the angle of elevation and the angle of depression
along the same line of sight are equal.
angle of
depression
angle of
elevation
Page 29
ALGEBRA AND NUMBER
Page 30
AN1 – Demonstrate an understanding of factors of whole numbers by determining the:

prime factors;

greatest common factor;

least common multiple;

square root;

cube root.
AN2 – Demonstrate an understanding of irrational numbers by:

representing, identifying and simplifying irrational numbers;

ordering irrational numbers.
AN3 – Demonstrate an understanding of powers with integral and rational exponents.
AN4 – Demonstrate an understanding of the multiplication of polynomial expressions
(limited to monomials, binomials and trinomials), concretely, pictorially and symbolically.
AN5 – Demonstrate an understanding of common factors and trinomial factoring,
concretely, pictorially and symbolically.
Page 31
MAT421A – Topic: Algebra and Number (AN)
GCO:
GRADE 9
N5 Determine the square root of
positive rational numbers that are
perfect squares.
N6 Determine an approximate
square root of positive rational
numbers that are non-perfect
squares.
SCO:
AN1 Demonstrate an
understanding of factors of whole
numbers by determining the:
 prime factors;
 greatest common factor;
 least common multiple;
 square root;
 cube root.
GRADE 11
MAT521A
MAT521B
AN2 Solve problems that involve
operations on radicals and radical
expressions with numerical and
variable radicands.

prime factors;



square root;

cube root.
[CN, ME, R]
A.
Determine the prime factors of a whole number.
B.
Explain why the numbers 0 and 1 have no prime factors.
C. Determine, using a variety of strategies, the greatest common factor or the least common
multiple of a set of whole numbers, and explain the process.
D.
Determine, concretely, whether a given whole number is a perfect square, a perfect cube or
neither.
E.
Determine, using a variety of strategies, the square root of a perfect square, and explain the
process.
F.
Determine, using a variety of strategies, the cube root of a perfect cube, and explain the
process.
G. Solve problems that involve prime factors, greatest common factors, least common multiples,
square roots or cube roots.
3.1 (A B C G)
3.2 (D E F G)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 32
SCO:

prime factors;



square root;

cube root.
[CN, ME, R]
Elaboration
A number is prime is it has exactly two distinct factors. All natural numbers greater than one can be expressed
as a unique product of prime factors, called the prime factorization of that number. The prime factorization can be
used to find the greatest common factor and the least common multiple of a set of natural numbers. Please note that
the concepts of greatest common factor and least common multiple are not fully developed previous to grade ten.
Perfect squares and square roots are linked to each other. The number 25 is a perfect square. It is formed by
multiplying two factors of 5 together, that is,  5  5   25 or 52  25. The square root of 25 is 5, or
25  5. Perfect
cubes and cube roots are also linked to each other. The number 27 is a perfect cube. It is formed by multiplying
three factors of 3 together, that is  3  3  3   27 or 33  27. The cube root of 27 is 3, or
3
27  3.
To help students familiarize themselves with perfect squares and cubes, they should develop a table which
includes the first twenty perfect squares and the first ten perfect cubes.
Perfect squares and perfect cubes can be modelled using manipulatives. For example, the perfect square 25
can be modelled as a square with side length 5, and the perfect cube 27 can be modelled as a cube with side length
3, as shown below.
Page 33
GCO:
GRADE 9
N3 Demonstrate an understanding
of rational numbers by:
 comparing and ordering rational
numbers;
 solving problems that involve
arithmetic operations on rational
numbers.
AN2 Demonstrate an
understanding of irrational numbers
by:
 representing, identifying and
simplifying irrational numbers;
 ordering irrational numbers.
SCO:
GRADE 11
MAT521A
MAT521B


[CN, ME, R, V]
A.
Sort a set of numbers into rational and irrational numbers.
B.
Determine an approximate value of a given irrational number.
C.
Approximate the locations of irrational numbers on a number line, using a variety of strategies,
and explain the reasoning.
D.
Order a set of irrational numbers on a number line.
E.
Express a radical as a mixed radical in simplest form (limited to numerical radicands).
F.
Express a mixed radical as an entire radical (limited to numerical radicands).
G. Explain, using examples, the meaning of the index of a radical.
H.
Represent, using a graphic organizer, the relationship among the subsets of the real numbers
(natural, whole, integer, rational, irrational).
4.1 (B G)
4.2 (A C D H)
4.3 (E F)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 34
SCO:


[CN, ME, R, V]
Elaboration
It is important that students understand each of the following sets of numbers:

Natural numbers – the set of counting numbers {1, 2, 3, 4, 5, ...}

Whole numbers – the set of counting numbers, together with zero {0, 1, 2, 3, 4, ...}

Integers – the set of whole numbers and their opposites {..., –3, –2, –1, 0, 1, 2, 3, ...}

Rational numbers – the set of all numbers that can be expressed as a fraction, e.g.,

Irrational numbers – the set of all numbers that cannot be expressed as a fraction, e.g.,
3

3
, –5, 17.2
4
2, ,
17
3
Real numbers – the combined set of rational and irrational numbers
The following diagram shows the relationships among all of these sets of numbers:
Real Numbers
Rational Numbers
Integers
Whole Numbers
Natural
Numbers
Irrational Numbers
Page 35
GCO:
GRADE 9
of powers with integral bases
(excluding base 0) and whole
number exponents by:
 representing repeated
multiplication using powers;
 using patterns to show that a
power with an exponent of zero is
equal to one;
 solving problems involving
powers.
GRADE 11
AN3 Demonstrate an
understanding of powers with
integral and rational exponents.
MAT521A
MAT521B
of operations on powers with
integral bases (excluding base 0)
and whole number exponents.
SCO:
AN3 – Demonstrate an understanding of powers with integral and rational exponents. [C, CN, PS, R]
A.
Explain, using patterns, why a  n 
1
, a  0.
an
B. Explain, using patterns, why a1 n  n a , n  0.
C. Apply the exponent laws:

 a  a   a

a m  a n  a m  n , a  0;

a 

 ab 

an
a
 b   bn , b  0
 
m
n
m
mn
n
 a mn ;
m
 ambm ;
;
n
to expressions with rational and variable bases, and integral and rational exponents, and
explain the reasoning.
D.
Express powers with rational exponents as radicals and vice versa.
E.
Solve a problem that involves exponent laws or radicals.
F.
Identify and correct errors in a simplification of an expression that involves powers.
4.4 (B D E F)
4.5 (A E F)
4.6 (C E F)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 36
SCO:
AN3 – Demonstrate an understanding of powers with integral and rational exponents. [C, CN, PS, R]
Elaboration
This outcome continues that work that was done with whole number exponents in grade nine.
A power with a negative exponent can be written as a power with a positive exponent using the following
principles:
an 
1
1
and  n  a n , as long as a  0
an
a
A power with a fractional exponent can be written as a radical using the following principles:
am n  n a m or am n 
 a
n
m
These principles can be applied to the exponent laws previously learned in grade nine, as stated below.
EXPONENT NAME
EXPONENT LAW
Product of Powers
 a  a   a
Quotient of Powers
am
 am n , a  0
an
Power of a Power
Power of a Product
m
n
a 
m
 ab 
m
n
mn
 a mn
 ambm
n
Power of a Quotient
an
a
   n,b0
b
b
Zero Exponent
a0  1, a  0
Page 37
GCO:
GRADE 9
GRADE 11
PR7 Model, record and explain the
operations of multiplication and
division of polynomial expressions
(limited to polynomials of degree
less than or equal to 2) by
monomials, concretely, pictorially
and symbolically.
AN4 Demonstrate an
understanding of the multiplication
of polynomial expressions (limited to
monomials, binomials and
trinomials), concretely, pictorially
and symbolically.
MAT521A
MAT521B
AN4 Determine equivalent forms of
rational expressions (limited to
numerators and denominators that
are monomials, binomials or
trinomials).
AN5 Perform operations on rational
expressions (limited to numerators
and denominators that are
monomials, binomials or trinomials).
SCO:
AN4 – Demonstrate an understanding of the multiplication of polynomial expressions (limited to
monomials, binomials and trinomials), concretely, pictorially and symbolically. [CN, R, V]
A.
Model the multiplication of two given binomials, concretely and pictorially, and record the
process symbolically.
B.
Relate the multiplication of two binomial expressions to an area model.
C.
Explain, using examples, the relationship between the multiplication of binomials and the
multiplication of two-digit numbers.
D. Verify a polynomial product by substituting numbers for the variables.
E.
Multiply two polynomials symbolically, and combine like terms in the product.
F.
Generalize and explain a strategy for multiplication of polynomials.
G. Identify and explain errors in a solution for polynomial multiplication.
Note: It is intended that the emphasis of this outcome be on binomial by binomial multiplication, with extension
to polynomial by polynomial to establish a general pattern for multiplication.
3.5 (A B C)
3.6 (A B E)
3.7 (D E F G)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 38
SCO:
AN4 – Demonstrate an understanding of the multiplication of polynomial expressions (limited to
monomials, binomials and trinomials), concretely, pictorially and symbolically. [CN, R, V]
Elaboration
The distributive property is used to multiply polynomials together by multiplying each term in the first polynomial
by each term in the second polynomial, and then collecting like terms. For example:
 3 x  2 4 x  5    3 x  4 x  5   2  4 x  5 
 12 x 2  15 x  8 x  10
 12 x 2  7 x  10
 c  3   4c 2  c  6   c  4c 2  c  6   3  4c 2  c  6 
 4c 3  c 2  6c  12c 2  3c  18
 4c 3  13c 2  9c  18
When both polynomials are binomials, the acronym FOIL can be used to remember the order in which the terms
are multiplied. The acronym reminds us to multiply the First terms in each binomial, next the Outside terms, then the
Inside terms, and then finally the Last terms in each binomial.
Please note that it may be necessary to review the operations of addition and multiplication of polynomials, which
were taught in grade nine, before teaching this outcome.
Page 39
GCO:
GRADE 9
PR7 Model, record and explain the
operations of multiplication and
division of polynomial expressions
(limited to polynomials of degree
less than or equal to 2) by
monomials, concretely, pictorially
and symbolically.
AN5 Demonstrate an
understanding of common factors
and trinomial factoring, concretely,
pictorially and symbolically.
GRADE 11
MAT521A
MAT521B
RF1 Factor polynomials of the
form:
 ax 2  bx  c, a  0;
 a 2 x 2  b 2 y 2 , a  0, b  0;
 a f  x    b f  x    c, a  0;
2
2
 a 2 f  x    b 2 g  y   ,
2
2
a  0, b  0
where a, b and c are rational
numbers.
SCO:
AN5 – Demonstrate an understanding of common factors and trinomial factoring, concretely,
pictorially and symbolically. [C, CN, R, V]
A.
Determine the common factors in the terms of a polynomial, and express the polynomial in
factored form.
B.
Model the factoring of a trinomial, concretely or pictorially, and record the process symbolically.
C.
Factor a polynomial that is a difference of squares, and explain why it is a special case of
trinomial factoring where b  0.
D.
Identify and explain errors in a polynomial factorization.
E.
Factor a polynomial, and verify by multiplying the factors.
F.
Explain, using examples, the relationship between multiplication and factoring of polynomials.
G. Generalize and explain strategies used to factor a trinomial.
H.
Express a polynomial as a product of its factors.
3.3 (A B D E)
3.4 (B F)
3.5 (B D E F H)
3.6 (B D E F G H)
3.8 (C E H)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 40
SCO:
AN5 – Demonstrate an understanding of common factors and trinomial factoring, concretely,
pictorially and symbolically. [C, CN, R, V]
Elaboration
Students should understand that factoring an algebraic expression is the reverse of multiplying polynomials,
which is a process similar to division. This will be the first time that students will have factored polynomials.
To find the GCF of a polynomial, find the GCF of the coefficients and the variables separately. To find the GCF
of the variables, select the lowest exponent that appears in the polynomial for each variable. Then, to factor the GCF
from the polynomial, simply divide each term by the GCF. The factored polynomial will then be a product of the GCF


and the sum or difference of the remaining factors. For example, 2m3 n 2  8m 2n  12mn 4  2mn m 2n  4m  6n 3 . A
common factor can be any polynomial. For example, a  x  2   b  x  2    x  2  a  b  .
To factor a trinomial of the form x 2  bx  c, find two integers with a product of c and a sum of b. Use these two
integers to write the factored form, which will be the product of x plus one integer, by x plus the other integer. For
example, to factor x 2  12 x  27, find two integers with a product of 27 and a sum of 12. Since these integers are 3
and 9, the factored form of the trinomial is  x  3  x  9  .
To factor a trinomial of the form ax 2  bx  c, find two integers with a product of ac and a sum of b. Then,
expand the middle term as a sum of these two integers. Finally, factor by grouping and removing common factors.
For example, to factor 2 x 2  5 x  3, find two integers with a product of  2  3  , or –6, and a sum of 5. Since these
integers are –1 and 6, we can expand the trinomial to 2 x 2  x  6 x  3. After grouping and removing common
factors, we get x  2 x  1  3  2 x  1 , which factors to  2 x  1 x  3  .
Students should be aware that there are many trinomial expressions that do not factor. As well, algebra tiles can
be particularly helpful for students who have difficulty factoring trinomial expressions.
Some polynomials are a result of special products. When factoring, use the patterns that formed these products.

Difference of Squares
The expression is a binomial where both terms are perfect squares and the operation between the
terms is subtraction. In general, a 2  b 2   a  b  a  b  .

Perfect Square Trinomial
The expression is a trinomial where the first and last terms are perfect squares and the middle term
is twice the product of the square root of the first and last terms. In general, a 2  2ab  b 2 
a  b 
2
.
Page 41
RELATIONS AND FUNCTIONS
Page 42
RF1 – Interpret and explain the relationships among data, graphs and situations.
RF2 – Demonstrate an understanding of relations and functions.
RF3 – Demonstrate an understanding of slope with respect to:

rise and run;

line segments and lines;

rate of change;

parallel lines;

perpendicular lines.
RF4 – Describe and represent linear relations, using:

words;

ordered pairs;

tables of values;

graphs;

equations.
RF5 – Determine the characteristics of the graphs of linear relations, including the:

intercepts;

slope;

domain;

range.
RF6 – Relate linear relations expressed in:

slope-intercept form  y  mx  b  ;

general form  Ax  By  C  0 ;
slope-point form  y  y 1  m  x  x1  
to their graphs.

RF7 – Determine the equation of a linear relation, given:

a graph;

a point and the slope;

two points;

a point and the equation of a parallel or perpendicular line
to solve problems.
RF8 – Represent a linear function using function notation.
RF9 – Solve problems that involve systems of linear equations in two variables, graphically and
algebraically.
Page 43
MAT421A – Topic: Relations and Functions (RF)
GCO:
Develop algebraic and graphical reasoning through the study of relations.
GRADE 9
GRADE 11
RF1 Interpret and explain the
relationships among data, graphs
and situations.
SCO:
MAT521A
MAT521B
RF1 – Interpret and explain the relationships among data, graphs and situations. [C, CN, R, T, V]
A.
Graph, with or without technology, a set of data, and determine the restrictions on the domain
and range.
B.
Explain why data points should or should not be connected on the graph for a situation.
C.
Describe a possible situation for a given graph.
D.
Sketch a possible graph for a given situation.
E.
Determine, and express in a variety of ways, the domain and range of a graph, a set of ordered
pairs, or a table of values.
5.1 (E)
5.2 (E)
5.3 (C D)
5.4 (A B E)
5.5 (B E)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 44
SCO:
RF1 – Interpret and explain the relationships among data, graphs and situations. [C, CN, R, T, V]
Elaboration
A graph is an effective way to show the relationship between two quantities. A constant rate of change is
represented graphically by a straight line. The steepness of the line indicates the rate at which one quantity is
changing in relation to the other.
Not all relationships are represented by a straight line. A curve shows that the rate of change is not constant. A
horizontal line means that there is no rate of change, since every value on the horizontal axis is related to the same
value on the vertical axis.
0
0
Constant Rate
of Change
0
Rate of Change
Is Not Constant
No Rate
of Change
Page 45
GCO:
GRADE 9
GRADE 11
RF2 Demonstrate an
understanding of relations and
functions.
SCO:
MAT521A
MAT521B
RF2 – Demonstrate an understanding of relations and functions. [C, R, V]
A.
Explain, using examples, why some relations are not functions but all functions are relations.
B.
Determine if a set of ordered pairs represents a function.
C.
Sort a set of graphs as functions or non-functions.
D.
Generalize and explain rules for determining whether graphs and sets of ordered pairs
represent functions.
5.2 (A B D)
5.4 (D)
5.5 (C D)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 46
SCO:
RF2 – Demonstrate an understanding of relations and functions. [C, R, V]
Elaboration
A relation is any set of ordered pairs. A function is a relation where each value in the domain corresponds to
exactly one value in the range. As a result, all functions are relations, but not all relations are functions.
For example, the relation which relates a person with his or her age would be a function, since each person has
a unique age at any given time. However the relation which relates a person to one of his or her biological parents
would not be a function, since each person has two biological parents.
Page 47
GCO:
GRADE 9
PR2 Graph linear relations, analyse
the graph and interpolate or
extrapolate to solve problems.
RF3 Demonstrate an
understanding of slope with respect
to:
 rise and run;
 line segments and lines;
 rate of change;
 parallel lines;
 perpendicular lines.
SCO:
GRADE 11
MAT521A
MAT521B

rise and run;


rate of change;

parallel lines;

[PS, R, V]
A.
Determine the slope of a line segment by measuring or calculating the rise and run.
B.
Classify lines in a given set as having positive or negative slopes.
C.
Explain the meaning of the slope of a horizontal or vertical line.
D.
Explain why the slope of a line can be determined by using any two points on that line.
E.
Explain, using examples, slope as a rate of change.
F.
Draw a line, given its slope and a point on the line.
G. Determine another point on a line, given the slope and a point on the line.
H.
Generalize and apply a rule for determining whether two lines are parallel or perpendicular.
I.
Solve a contextual problem involving slope.
6.1 (A B C D E F G I)
6.2 (H I)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 48
SCO:

rise and run;


rate of change;

parallel lines;

[PS, R, V]
Elaboration
The slope of a line or line segment indicates how steep the line is. The slope of a line is the ratio of the rise to
the run.
y
rise
m
run
rise
run
x
The slope of a line can be determined using two points on the line,  x1, y1  and  x2 , y 2  , by using the formula
m
y 2  y1
, x1  x2 . The slope indicates the rate of change of a linear relation.
x2  x1
The sign of the slope indicates the direction of the line. A line or line segment that rises from the left to right has
a positive slope. A line or line segment that falls from left to right has a negative slope. Also, horizontal lines have
zero slope and vertical lines have undefined slopes.
y
y
x
Positive Slope
x
Negative Slope
y
y
x
x
Zero Slope
Undefined Slope
Parallel lines have the same slopes but different intercepts. All horizontal lines, which have zero slope, are
parallel to each other, and all vertical lines, which have undefined slope, are parallel to each other.
The slopes of oblique perpendicular lines are negative reciprocals of each other. The product of negative
reciprocals is –1. A vertical line, which has an undefined slope, and a horizontal line, which has zero slope, are also
perpendicular to each other.
Page 49
GCO:
GRADE 9
RF4 Describe and represent linear
relations, using:
 words;
 ordered pairs;
 tables of values;
 graphs;
 equations.
GRADE 11
MAT521A
MAT521B
RF2 Graph and analyse absolute
value functions (limited to linear and
quadratic functions) to solve
problems.
RF7 Solve problems that involve
linear and quadratic inequalities in
two variables.
RF11 Graph and analyse reciprocal
functions (limited to the reciprocal of
linear and quadratic functions).
SCO:

words;

ordered pairs;

tables of values;

graphs;

equations.
[C, CN, R, V]
A.
Identify independent and dependent variables in a given context.
B.
Determine whether a situation represents a linear relation, and explain why or why not.
C.
Determine whether a graph represents a linear relation, and explain why or why not.
D.
Determine whether a table of values or a set of ordered pairs represents a linear relation, and
explain why or why not.
E.
Draw a graph from a set of ordered pairs within a given situation, and determine whether the
relationship between the variables is linear.
F.
Determine whether an equation represents a linear relation, and explain why or why not.
G. Match corresponding representations of linear relations.
5.2 (A)
5.5 (A)
5.6 (A B C D E F G)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 50
SCO:

words;

ordered pairs;

tables of values;

graphs;

equations.
[C, CN, R, V]
Elaboration
A relation can be presented in a variety of ways. For example:
Words:
Three times the length of your ear, e, is equal to the length of your face, f, from chin to
hairline.
Equation:
f  3e
Ordered Pairs:
 4,12 ,  4.5,13.5  ,  5,15  ,  5.5,16.5  ,  6,18  ,  6.5,19.5 
Table of Values:
EAR LENGTH, e
(cm)
FACE LENGTH, f
(cm)
4
12
4.5
13.5
5
15
5.5
16.5
6
18
6.5
19.5
Graph:
f
16
8
0
2
4
6
e
There are a number of ways to determine whether a relation is a linear relation or a non-linear relation:

Linear relations have graphs that are straight lines.

In the table of values of a linear relation, values of y increase or decrease by a constant amount as
values of x increase or decrease by a constant amount.

When a linear relation is written as an equation, it will contain one or two variables and there will be
no term whose degree is higher than one.
Page 51
GCO:
GRADE 9
RF5 Determine the characteristics
of the graphs of linear relations,
including the:
 intercepts;
 slope;
 domain;
 range.
GRADE 11
MAT521A
RF2 Demonstrate an
understanding of the characteristics
of quadratic functions, including:
 vertex;
 intercepts;
 domain and range;
 axis of symmetry.
MAT521B
RF3 Analyse quadratic functions of
the form y  a  x  p   q and
2
determine the:
 vertex;
 direction of opening;
 axis of symmetry;
 x- and y-intercepts.
RF4 Analyse quadratic functions of
the form y  ax 2  bx  c to identify
characteristics of the corresponding
graph, including:
 vertex;
 direction of opening;
 axis of symmetry;
 x- and y-intercepts
and to solve problems.
SCO:

intercepts;

slope;

domain;

range.
[CN, PS, R, V]
A. Determine the intercepts of the graph of a linear relation, and state the intercepts as values or
ordered pairs.
B. Determine the slope of the graph of a linear relation.
C. Determine the domain and range of the graph of a linear relation.
D. Sketch a linear relation that has one intercept, two intercepts or an infinite number of
intercepts.
E. Identify the graph that corresponds to a given slope and y-intercept.
F. Identify the slope and y-intercept that correspond to a given graph.
G. Solve a contextual problem that involves intercepts, slope, domain or range of a linear relation.
5.7 (A C E G)
6.1 (B D)
6.4 (E F G)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 52
SCO:

intercepts;

slope;

domain;

range.
[CN, PS, R, V]
Elaboration
The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. It is found by
substituting y  0 into the equation of the line and solving for x. The y-intercept of a line is the y-coordinate of the
point where the line crosses the y-axis. It is found by substituting x  0 into the equation of the line and solving for y.
The slope of a line is found by writing the equation of the line in slope-intercept form, y  mx  b. The slope, m,
will be the coefficient of the linear term, mx.
When comparing two quantities, the words domain and range are used to describe the values that are
appropriate for the relation. In a set of ordered pairs, the domain is the set of the first elements of each pair, and the
range is the set of the second elements. On a graph, values of the domain are plotted against the horizontal axis.
Values of the range are plotted against the vertical axis.
There are a variety of ways to express the domain and the range of a relation:
Words:
all integers greater than or equal to –2, and less than or equal to 3
Number Line:
–3 –2 –1 0
1
2
Set Notation:
2  n  3, n  integer
List:
{2,  1, 0, 1, 2, 3}
3
Page 53
GCO:
GRADE 9
GRADE 11
RF6 Relate linear relations
expressed in:
 slope-intercept form
 y  mx  b;
MAT521A
 general form  Ax  By  C  0 ;
MAT521B
 slope-point form
 y  y1  m  x  x1  
to their graphs.
SCO:


to their graphs. [CN, R, T, V]

A.
Express a linear relation in different forms, and compare the graphs.
B.
Rewrite a linear relation in either slope-intercept or general form.
C. Generalize and explain strategies for graphing a linear relation in slope-intercept, general or
slope-point form.
D.
Graph, with and without technology, a linear relation given in slope-intercept, general or slopepoint form, and explain the strategy used to create the graph.
E.
Identify equivalent linear relations from a list of linear relations.
F.
Match a set of linear relations to their graphs.
6.3 (C D)
6.4 (C D F)
6.5 (B C D F)
6.6 (A B C D E F)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 54
SCO:


to their graphs. [CN, R, T, V]

Elaboration
The slope-intercept form of a linear equation is y  mx  b, where m represents the slope and b represents the
y-intercept. This form is obtained by solving the given linear equation for y.
The standard form of a linear equation is Ax  By  C, where A, B and C are integers, and A and B are not both
zero. By convention, A is a whole number.
The general form of a linear equation is Ax  By  C  0, where A, B and C are integers, and A and B are not
both zero. By convention, A is a whole number.
For a non-vertical line through the point  x1, y1  with slope m, the equation of the line can be written in slopepoint form as y  y1  m  x  x1  . Any point on the line can be used when determining the equation of the line in
slope-point form.
Linear equations can be converted from one form to another by applying the rules of algebra.
Page 55
GCO:
GRADE 9
GRADE 11
RF7 Determine the equation of a
linear relation, given:
 a graph;
 a point and the slope;
 two points;
 a point and the equation of a
parallel or perpendicular line
to solve problems.
SCO:
MAT521A
MAT521B

a graph;


two points;

to solve problems. [CN, PS, R, V]
A.
Determine the slope and y-intercept of a given linear relation from its graph, and write the
equation in the form y  mx  b.
B.
Write the equation of a linear relation, given its slope and the coordinates of a point on the line,
and explain the reasoning.
C.
Write the equation of a linear relation, given the coordinates of two points on the line, and
explain the reasoning.
D.
Write the equation of a linear relation, given the coordinates of a point on the line and the
equation of a parallel or perpendicular line, and explain the reasoning.
E.
Graph linear data generated from a context, and write the equation of the resulting line.
F.
Solve a problem, using the equation of a linear relation.
6.4 (A F)
6.5 (B C D F)
6.6 (E F)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 56
SCO:

a graph;


two points;

to solve problems. [CN, PS, R, V]
Elaboration
To write the equation of a straight-line graph, use the following two constraints:

the rate of change or slope, m;

the y-intercept. If  0, b  is the point where the line crosses the y-axis, then b is the y-intercept.
The equation of a non-vertical straight line graph can be written in slope-intercept form. The form of the equation
 rise 
is y  mx  b, where m represents the slope 
 and b represents the y-intercept.
 run 
To determine an equation of a line that is parallel or perpendicular to a given line, use the properties of the
slopes of parallel and perpendicular lines, along with the slope-intercept form of the line.
Page 57
GCO:
GRADE 9
SCO:
GRADE 11
RF8 Represent a linear function
using function notation.
MAT521A
MAT521B
RF8 – Represent a linear function using function notation. [CN, ME, V]
A.
Express the equation of a linear function in two variables, using function notation.
B. Express an equation given in function notation as a linear function in two variables.
C. Determine the related range value, given a domain value, for a linear function; e.g., if
f  x   3 x  2, determine f  1 .
D. Determine the related domain value, given a range value, for a linear function; e.g., if
g  t   7  t , determine t so that g  t   15.
E.
Sketch the graph of a linear function expressed in function notation.
5.2 (A B C D)
5.5 (C D)
5.7 (E)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 58
SCO:
RF8 – Represent a linear function using function notation. [CN, ME, V]
Elaboration
Functions can be written using function notation. For example, the function y  4 x  1 can be written as
f  x   4 x  1. The name of the function is f, with a variable name of x. In this example, 4 x  1 is the rule that
assigns a unique value to y for each value of x. Any letter may be used to name a function. Two other examples of
functions are v  t   9.8t 2 , for the velocity, in metres per second, of a dropped object after t seconds and A  r   r 2 ,
for the area of a circle with radius r.
Function notation highlights the input-output aspect of a function. The function f  x   4 x  1 takes any input
value for x, multiplies it by 4, and adds 1 to give the result. For example, if x  2 is the input, then f  2   9 is the
output, since f  2   4  2   1  9. Therefore, the point (2,9) is a on the graph of the function f  x   4 x  1.
Page 59
GCO:
GRADE 9
PR3 Model and solve problems
using linear equations of the form:
 ax  b;
x
 b, a  0;
a
 ax  b  c;

systems of linear equations in two
variables, graphically and
algebraically.
MAT521A
RF1 Model and solve problems that
involve systems of linear
inequalities in two variables.
MAT521B
radical equations (limited to square
roots).
x
 b  c, a  0;
a
 ax  b  cx;

rational equations (limited to
numerators and denominators that
are monomials, binomials or
trinomials).
 a  x  b   c;
 ax  b  cx  d ;
 a  bx  c   d  ex  f  ; and
quadratic equations.
a
 b, x  0
x
where a, b, c, d, e and f are rational
numbers.

SCO:
GRADE 11
RF6 Solve, algebraically and
graphically, problems that involve
systems of linear-quadratic and
quadratic-quadratic equations in two
variables.
algebraically. [CN, PS, R, T, V]
A.
Model a situation, using a system of linear equations.
B. Relate a system of linear equations to the context of a problem.
C.
Determine and verify the solution of a system of linear equations graphically, with and without
technology.
D. Explain the meaning of the point of intersection of a system of linear equations.
E.
Determine and verify the solution of a system of linear equations algebraically.
F.
Explain, using examples, why a system of equations may have no solution, one solution or an
infinite number of solutions.
G. Explain a strategy to solve a system of linear equations.
H. Solve a problem that involves a system of linear equations.
7.1 (A B)
7.2 (A C D H)
7.3 (C H)
7.4 (A B E G H)
7.5 (A B E G H)
7.6 (A F)
[C] Communication
[CN] Connections
and Estimation
[R] Reasoning
[T]
[V]
Technology
Visualization
Page 60
SCO:
algebraically. [CN, PS, R, T, V]
Elaboration
A pair of two linear equations is called a system of linear equations. It can be represented graphically in order to
make comparisons or solve problems. The point(s) of intersection of the two lines on a graph represents the solution
to the system of linear equations.
A system of linear equations can have one solution, no solution or an infinite number of solutions. Before
solving, the number of solutions for a linear system can be predicted by comparing the slopes and y-intercepts of the
equations.
INTERSECTING LINES
PARALLEL LINES
COINCIDENT LINES
One solution
No solution
An infinite number of solutions
y
y
y x
y
x
x
different slopes
same slope
same slope
y-intercepts can be the same or
different
different y-intercepts
same y-intercepts
Systems of linear equations can be solved using three methods:

Graphically: Graph both lines on the same coordinate plane. The solution will occur at the point
of intersection.

Substitution Method: Solve one equation for one variable, substitute that expression into the
other equation, and then solve for the remaining variable.

Elimination Method: Add or subtract the equations to eliminate one variable and then solve for
the remaining variable.
Page 61
Page 62
CURRICULUM GUIDE SUPPLEMENT
Curriculum Guide Supplement
This supplement to the Prince Edward Island MAT421A Mathematics Curriculum Guide is designed to
parallel the primary resource, Foundations and Pre-Calculus Mathematics 10.
For each of the chapters in the text, an approximate timeframe is suggested to aid teachers with planning.
The timeframe is based on a total of 80 classes, each with an average length of 75 minutes:
CHAPTER
SUGGESTED TIME
Chapter 1 – Measurement
12 classes
Chapter 2 – Trigonometry
11 classes
Chapter 3 – Factors and Products
12 classes
Chapter 4 – Roots and Powers
9 classes
Chapter 5 – Relations and Functions
14 classes
Chapter 6 – Linear Functions
11 classes
Chapter 7 – Systems of Linear Equations
11 classes
Each chapter of the text is divided into a number of sections. In this document, each section is supported
by a one-page presentation, which includes the following information:







the name and pages of the section in Foundations and Pre-Calculus Mathematics 10;
the specific curriculum outcome(s) and achievement indicator(s) addressed in the section
(see the first half of the curriculum guide for an explanation of symbols);
the student expectations for the section, which are associated with the SCO(s);
the new concepts introduced in the section;
other key ideas developed in the section;
suggested problems in Foundations and Pre-Calculus Mathematics 10;
possible instructional and assessment strategies for the section.
Page 63
Page 64
UNIT PLANS
CHAPTER 1
MEASUREMENT
SUGGESTED TIME
12 classes
Page 65
UNIT PLANS
Section 1.1 – Imperial Measures of Length (pp. 4-12)
ELABORATIONS &
SUGGESTED PROBLEMS
POSSIBLE INSTRUCTIONAL &
ASSESSMENT STRATEGIES
Specific Curriculum Outcome(s) and Achievement
Indicator(s) addressed:

M1 (A C D)

M2 (A B C)
After this lesson, students will be expected to:

choose an appropriate referent for a linear
measure and describe their choice

use a personal referent to estimate a linear
measure and explain the process used

use proportional reasoning to convert between
imperial units

verify a unit conversion using unit analysis

solve problems that involve the conversion of
imperial units
After this lesson, students should understand the
following concepts:

SI system of measures – a system of units based
on powers of 10; the fundamental unit of length is
the metre (m), of mass is the kilogram (kg), and of
time is the second (s)

imperial units – measurement units such as the
mile, yard, foot and inch, commonly used in the
United States and in some industries in Canada

referent – used to estimate a measure; for
example, a referent for a length of 1 mm is the
thickness of a dime

proportional reasoning – the ability to understand
and compare quantities that are related
multiplicatively

unit analysis – a method of converting a measure
in a given unit to a measure in a different unit by
multiplying the measure by a conversion factor

conversion factor – a number used to multiply or
divide a quantity to convert from one unit of
measure to another
Possible Instructional Strategies:

Ask students to list objects at school or at home
that they could use as a referent for one inch, one
foot, and one yard. Describe how to use these
referents to measure the dimensions of an object.

Have rulers, yard sticks and measuring tapes in
imperial units available for students who need
visual assistance with the problems in this section.
Remind students to use estimation strategies to
check the reasonableness of their solutions.

Some students have difficulty multiplying a fraction
by a whole number to when solving a proportion.
Remind them that the whole number can be written
as a fraction with denominator 1. Then, they
simply multiply numerators and denominators.
Possible Assessment Strategies:

The size of a television is measured across the
screen diagonally. A standard television has a
ratio of width to height of 4 : 3, and a widescreen
television has a ratio of width to height of 16 : 9.
What is the difference between the viewing area of
a 46-inch widescreen television and the viewing
area of a 46-inch standard television? Round off
the answer to one decimal place.

A round Inuit drum needs to have its skin
restretched and then lashed into place with sinew.
1
For each inch of the frame, 3 inches of sinew
2
1
are needed. The diameter of the frame is 1 feet.
4
What length of sinew is needed? Express your
answer to the nearest quarter of a foot.
IMPERIAL UNITS OF MEASURE
RELATIONSHIP
BETWEEN
UNITS
IMPERIAL
UNIT
ABBREVIATION
REFERENT
inch
in
thumb
length
foot
ft
foot length
1 ft = 12 in
yard
yd
arm span
1 yd = 3 ft
1 yd = 36 in
mile
mi
distance
walked in
20 min
1 mi = 1760 yd
1 mi = 5280 ft
Suggested Problems in Foundations and PreCalculus Mathematics 10:

pp. 11-12: #1-18
Page 66
UNIT PLANS
Section 1.2 – Math Lab: Measuring Length and Distance (pp. 13-15)
ELABORATIONS &
SUGGESTED PROBLEMS

M1 (A C D E F)

estimate linear measures using referents for both
imperial and SI units

describe the strategy used to determine a linear
measure of an object

use a variety of measuring instruments to
determine the linear dimensions of an object

use the correct units and symbols to record
measurements on a sketch of an object

p. 15: #1-6
Page 67
UNIT PLANS
Section 1.3 – Relating SI and Imperial Units (pp. 16-23)
ELABORATIONS &
SUGGESTED PROBLEMS

M1 (B)

M2 (A B C D)

use proportional reasoning to convert a length from
an imperial unit to an SI unit

use proportional reasoning to convert a length from
an SI unit to an imperial unit

verify a unit conversion using unit analysis

solve problems that involve the conversion
between SI and imperial units

use mental math and estimation to check that a
solution is reasonable
CONVERSIONS BETWEEN SI AND IMPERIAL
UNITS
SI UNITS TO IMPERIAL
UNITS
IMPERIAL UNITS TO SI
UNITS
1 mm  0.03937 in
1 in  2.54 cm
1 cm  0.3937 in
1 ft  30.48 cm
1 m  39.37 in
1 yd  91.44 cm
1 m  3.281 ft
1 yd  0.9144 m
1 km  0.6214 mi
1 mi  1.609 km

Please note that the conversions given in the
curriculum guide should be used, as the answers in
the textbook do not reflect the approximate
conversions that are given in the textbook.

Ask students to research the development of some
measures of length, such as the inch, foot, yard,
mile, centimetre, metre and kilometre, to determine
how each measure was developed. As part of their
research, have them determine which conversions
between SI and imperial units are approximate and
which conversions are exact.

Some students have difficulty in setting up the
correct proportion when converting between units.
Remind them that both pairs of numerators and
denominators should have the same units.

Swimmer Brian Johns of Richmond, BC,
represented Canada at the 2008 Olympics in
Beijing. He finished seventh in a race that one
news report referred to as 400 metres long and
1
mile long.
another news report referred to as
4
Are the two measures equivalent? If not, what is
the difference between them? Round off the
answer to the nearest foot.

Convert 90 km/h into miles per hour. Round off the
answer to the nearest mile per hour.

pp. 22-23: #1-14
\
Page 68
UNIT PLANS
Section 1.4 – Surface Areas of Right Pyramids and Right Cones (pp. 26-35)
ELABORATIONS &
SUGGESTED PROBLEMS

M3 (A B D)

determine the surface area of a right pyramid or
cone

use the Pythagorean theorem to determine the
slant height of a right pyramid or cone

determine an unknown dimension of a right
pyramid or cone given its surface area and
remaining dimensions

solve problems involving the surface area of a right
pyramid or cone

sketch a diagram to represent a problem that
involves the surface area of a right pyramid or cone
following concepts:

Bring in models of right pyramids and right cones to
help explain surface area. Cut them out into their
nets to help show the class how to derive their
formulas.

Remind students that a sketch showing all the
given information should be the first step in the
solution of any problem involving surface area.
Encourage students to think about the reason for
calculating the surface area, then decide which
faces are involved. Students should ask
themselves, “Should the area of the base be
included in the surface area calculation?”

Remind students that to obtain the greatest
possible accuracy, they should always use the 
key on their calculators, and not an approximate
22
value, such as 3.14 or
. Encourage students to
7
estimate to check that their solutions are
reasonable.

right pyramid – an object that has one face that is
a polygon (the base) and other faces that are
triangles with a common vertex; the line through
the vertex and the centre of the base is
perpendicular to the base

apex – the vertex farthest from the base of an
object


slant height – the distance from a point on the
perimeter of the base of a cone to the apex of the
cone; the distance from the midpoint of the base of
one triangular face of a rectangular pyramid to the
apex of the pyramid
Sketch a right cone with diameter 16 cm and slant
height 12 cm. What is its surface area? Round off
the answer to one decimal place.

Sketch a right rectangular pyramid with a square
base measuring 10 cm on each side. The slant
height of each face is 8.5 cm. What is the surface
area of the pyramid?

tetrahedron – a pyramid that has a triangular base

regular tetrahedron – an object with four
congruent equilateral triangular faces; a regular
triangular pyramid

lateral area – the surface area of an object, not
including the area of its bases

right cone – an object with one circular base and
one vertex; the line through the vertex and the
centre of the base is perpendicular to the base
FORMULAS FOR SURFACE AREA
Right Pyramid
(Rectangular
Base)
Right Pyramid
(Regular Polygon
Base)
Right Cone
SA  lw  ls1  ws2
SA 
1
s Perimeter of base  
2
base area 
SA  r 2  rs

pp. 34-35: #1-18
Page 69
UNIT PLANS
Section 1.5 – Volumes of Right Pyramids and Right Cones (pp. 36-44)
ELABORATIONS &
SUGGESTED PROBLEMS


M3 (A C D F)

sketch a diagram to represent a problem that
involves volume

determine the volumes of a right cone, cylinder,
prism and pyramid

describe the relationship between the volumes of a
right cone and a right cylinder, and a right pyramid
and a right prism, with the same base and height

determine an unknown dimension of a right cone,
cylinder, prism or pyramid, given its volume and
remaining dimensions


solve problems involving the volumes of right
prisms, pyramids, cylinders and cones
following concepts:

volume – the amount of space occupied by an
object

capacity – the amount a container can hold
FORMULAS FOR VOLUME
Right Prism
Right Pyramid
Rectangular Prism
Right Rectangular
Pyramid
V  Ah
V 
1
Ah
3

Have students explain and demonstrate to the
class what happens when they fill the cone or
the pyramid and pour the contents into the
cylinder or the prism, respectively.

Ask students to write, in words and using
symbols, the relationship between the volume
of a cone and a cylinder, or of a pyramid and a
prism, which have equal base and height.
22
. As well, the height
7
should be left as a radical expression until the final
step when calculating the volume. If any of these
approximations are rounded, then used in further
calculations, this may result in the volume being
less accurate.

Determine the volume of a right pyramid with a
vertical height of 16 cm and having a rectangular
base with dimensions of 6 cm by 8 cm.

a.
A cylinder has a height of 22 cm and a radius
of 10 cm. What is it volume? Round off the
answer to one decimal place.
b.
What is the volume of its corresponding right
cone with a height of 22 cm and radius 10 cm?
Round off the answer to one decimal place.
V  lwh
V 
1
lwh
3
Right Cylinder
V  r 2h
Right Cone
1
V  r 2h
3


Ask students to make one of the following, using
materials of their own choosing: a cylinder and a
cone which have the same height and base, or a
right prism and a pyramid which have the same
height and base.
pp. 41-44: #1-19
Many of the operating costs of a greenhouse
depend on its volume. Each of the two large
greenhouses at the Muttart Conservatory in
Edmonton, AB is a right pyramid having a square
base measuring 26 m on each side. The apex of
each greenhouse is 24 m high. What is the volume
of each greenhouse?
Page 70
UNIT PLANS
Section 1.6 – Surface Area and Volume of a Sphere (pp. 45-52)
ELABORATIONS &
SUGGESTED PROBLEMS


The volume of a sphere is two-thirds of the volume
of a cylinder with the same radius and a height
equal to the diameter of the sphere. Using the
formula for the volume of a cylinder, ask students
to derive the formula for the volume of a sphere.

22
. Encourage students to
7
estimate to check that their solutions are
reasonable.

Some students believe that the surface area of a
hemisphere is equal to one-half of the surface area
of the corresponding sphere. Remind them that in
a hemisphere, a circular face is exposed. So, the
surface area of a hemisphere is equal to one-half
of the surface area of the sphere plus the area of
the exposed circular face.
M3 (B C D)

calculate the surface area and volume of a sphere

use the surface area of a sphere to determine its
diameter or radius

solve problems involving the surface area or
volume of a sphere
following concepts:

sphere – an object where every point on the
surface of the object is the same distance from the
centre of the object

radius – the distance or line segment from the
centre of a circle or a sphere to any point on the
circle or the sphere, respectively

diameter – the distance across a circle or a
sphere, measured through its centre; or the line
segment that joins two points on the circle or the
sphere and passes through its centre


Calculate the surface area of a beach ball with
radius 15 cm. Round off the answer to one
decimal place.

Determine the volume of a soccer ball whose
diameter is 24 cm. Round off the answer to one
decimal place.

A sphere which is 12 cm in diameter fits exactly
into a cube. Find the surface area and volume of
the cube.

A hot air balloon has a spherical shape with a
diameter of 4 m. If 30 additional cubic metres of air
are pumped into the balloon, what will be the new
diameter? Round off the answer to one decimal
place.

Find the diameter, correct to one decimal place, of
a sphere with volume 500 cm3.
hemisphere – one half of a sphere
FORMULAS INVOLVING SPHERES
Surface Area
Volume
SA  4r 2
V 
4 3
r
3

pp. 50-52: #1-20
Page 71
UNIT PLANS
Section 1.7 – Solving Problems Involving Objects (pp. 55-61)
ELABORATIONS &
SUGGESTED PROBLEMS


The use of concrete materials will promote the
understanding of which surfaces overlap and which
are exposed when objects are combined into
composite 3-D objects.

Some students have difficulty identifying
overlapping faces when presented with a surface
area problem. Suggest that they sketch a net for
each object in the composite object, then colour
the areas of overlap.
M3 (A D E)

recognize the individual objects that make up a
composite object and identify any overlapping
areas

calculate the surface area of a composite object

calculate the volume of a composite object

solve problems involving the surface area and
volume of composite objects

A witch’s hat was made for Halloween from a piece
of heavy cardboard. Sandy decided that, in order
to have enough room to fit the hat over her witch’s
wig, she would need the opening to be 56 cm in
circumference. She wanted the hat to measure
30 cm from the brim to the point at the top of the
cone shape.
a. What was the area of the cardboard that she
needed to cut to form the cone to make the
hat? Round off the answer to the nearest
whole number.
b. The brim of the hat is circular and is 8 cm
wide. What are the radii of the inner and the
outer circle that will need to be cut to make
the brim? Round off the answer to one
decimal place.

A commercially produced icecream treat has the cone filled
with ice cream and has ice cream
on the top. If the ice cream is 2.5
cm above the cone in the shape
of a perfect hemisphere, and the
cone is 5 cm in diameter and 12
cm high, how much ice cream
would be required to make one such ice-cream
treat? Round off the answer to one decimal place.

The Heritage Committee is restoring the old church
in town. The steeple, which is a square pyramid,
will be covered with metal sheeting. What is the
minimum amount of sheeting that will be required
to cover the steeple if the base has an area of 4 m2
and the slant height is 8 m?

The radius of a beach ball is 14 cm. By how much
does its surface area increase if the beach ball is
inflated so that its radius increases by 1 cm?

The Dominion Astrophysical Observatory near
Victoria, BC, has a cylindrical base with a diameter
of 20.1 m and a height of 9.8 m. The dome is half
a sphere with the same diameter as the cylindrical
base. What is the volume of the observatory?
Round off the answer to the nearest whole number.
following concept:

composite object – the result of combining two or
more objects to make a new object

pp. 59-61: #1-11
Page 72
UNIT PLANS
CHAPTER 2
TRIGONOMETRY
SUGGESTED TIME
11 classes
Page 73
UNIT PLANS
Section 2.1 – The Tangent Ratio (pp. 70-77)
ELABORATIONS &
SUGGESTED PROBLEMS


Remind students that a diagram showing all of the
solution of any trigonometric problem. Their
sketches need not be accurate, but reasonable
representations of the given situation can help
them decide on a strategy.

Some students have difficulty identifying the
opposite and adjacent sides for the angle under
consideration. Remind them that the hypotenuse is
the longest side in a right triangle. Then, the
adjacent side will be the leg of the right triangle that
is next to the angle under consideration and the
opposite side will be the leg opposite the angle.
M4 (A B D)

draw a right triangle and identify the hypotenuse,
the side opposite a given angle, and the side
adjacent to that angle

calculate the tangent of an acute angle in a given
right triangle

determine the measure of an acute angle, given
the tangent of that angle

use the tangent ratio to solve problems involving
angles
following concepts:

angle of inclination – the acute angle between the
horizontal and a line or line segment

opposite side – the leg of a right triangle that is
directly across from the reference angle

adjacent side – the leg of a right triangle that is
next to both the reference angle and the right angle

tangent ratio – for an acute angle A in a right
triangle, the ratio of the length of the side opposite
angle A to the length of the side adjacent to angle
A; written tan A
tan A 

a.
tan L
b.
tan N
L
12
13
M
N
5


For the given triangle, determine the value of each
trigonometric ratio.
Complete each of the following tables. Round off
all values to four decimal places and all angles to
one decimal place, where necessary.

tan 
0
27
pp. 74-77: #1-20
450
570

tan 
0.3329
0.5543
1.4653

A radio transmission tower is to be supported by a
guy wire. The wire reaches 30 m up the tower and
is attached to the ground at a horizontal distance of
14 m from the base of the tower. What angle does
the guy wire form with the ground, to the nearest
tenth of a degree?
Page 74
UNIT PLANS
Section 2.2 – Using the Tangent Ratio to Calculate Lengths (pp. 78-83)
ELABORATIONS &
SUGGESTED PROBLEMS



When identifying the sides of a right triangle, it is a
good idea to put hyp next to the hypotenuse, opp
next to the opposite side and adj next to the
adjacent side.
M4 (B D)

draw and label the right triangle described in a
given problem

set up an equation involving the tangent ratio, then
solve the equation to determine the measure of a
leg of a right triangle

use the tangent ratio to solve problems involving
an unknown side length of a right triangle
following concepts:

direct measurement – a measurement made
using a measuring instrument or by counting

indirect measurement – a measurement made
using a ratio, formula or other mathematical
reasoning

A ladder leaning against a wall forms an angle of
630 with the ground. How far up the wall will the
ladder reach if the foot of the ladder is 2 m from the
wall? Round off the answer to one decimal place.

A surveyor wants to determine the width of a river
for a proposed bridge. The distance from the
surveyor to the proposed bridge site on the same
side of the river is 400 m. The surveyor uses a
theodolite to measure angles. The surveyor
measures a 310 angle to the bridge site across the
river. What is the width of the river, to the nearest
tenth of a metre?

pp. 81-83: #1-14
Page 75
UNIT PLANS
Section 2.3 – Math Lab: Measuring an Inaccessible Height (pp. 84-86)
ELABORATIONS &
SUGGESTED PROBLEMS

M4 (E)

read the angle measure from a clinometer

calculate the angle of inclination using a clinometer

sketch and label a triangle to represent the height
of an inaccessible object

use their measurements and the tangent ratio to
determine an inaccessible height

p. 86: #1-3
Page 76
UNIT PLANS
Section 2.4 – The Sine and Cosine Ratios (pp. 89-96)
ELABORATIONS &
SUGGESTED PROBLEMS



Some students get confused between the sine and
cosine ratios. Mnemonic devices can be used to
help students such as Canadian Amateur Hockey
adjacent
for cos  
and Ships Of Halifax for
hypotenuse
M4 (A B D)

identify the hypotenuse in a right triangle, the side
opposite a given acute angle and the side adjacent
to that angle

determine the measure of an acute angle, given
the sine or cosine of that angle

use the sine and cosine ratios to solve problems
involving angle measures in a right triangle
sin  
following concepts:

sine ratio – for an acute angle A in a right triangle,
the ratio of the length of the side opposite angle A
to the length of the hypotenuse; written sin A
sin A 



cosine ratio – for an acute angle A in a right
triangle, the ratio of the length of the side adjacent
to angle A to the length of the hypotenuse; written
cos A
cos A 
For the given triangle, determine the value of each
trigonometric ratio.
a.
sin L
b.
sin N
c.
cos L
d.
cos N
primary trigonometric ratios – three ratios
involving sides in right triangles (cosine, sine and
tangent)

trigonometry – the study of the properties and
applications of triangles

angle of elevation – the angle between the
horizontal, through eye level, and a line of sight to
a point above eye level
L
12
13
M
N
5



opposite
.
hypotenuse
pp. 94-96: #1-16

Evaluate each trigonometric ratio, to four decimal
places.
a.
sin 600
b.
sin 300
c.
sin 450
Determine the measure of each angle, to the
nearest tenth of a degree.
a.
sin   0.4384
b.
cos   0.2079
A guy wire supporting a cell tower is 24 m long. If
the wire is attached at a height of 17 m up the
tower, determine the angle that the guy wire forms
with the ground. Round off the answer to the
nearest tenth.
Page 77
UNIT PLANS
Section 2.5 – Using the Sine and Cosine Ratios to Calculate Lengths (pp. 97-102)
ELABORATIONS &
SUGGESTED PROBLEMS



When solving a problem involving a right triangle,
students should label what is known and what is
not known to help them determine which
trigonometric function to use to find the solution.
M4 (B D)

identify the hypotenuse in a right triangle, the side
opposite a given acute angle and the side adjacent
to that angle

use the sine or cosine ratio to determine the length
of the hypotenuse, given the length of a leg and the
measure of an acute angle

use the sine or cosine ratio to determine the length
of a leg, given the length of the hypotenuse and the
measure of an acute angle


use the sine or cosine ratio to solve problems

Find the length of x. Round off the answer to one
decimal place.
10 cm
pp. 101-102: #1-12
x
250

Determine the height of a kite above the ground if
the kite string extends 480 m from the ground and
makes an angle of 620 with the ground. Express
the answer to the nearest tenth of a metre.

A supporting cable is run from the top of a billboard
to a point 38 m from the base of the billboard. The
cable makes a 350 angle with the ground. Find the
length of the cable. Round off the answer to the
nearest tenth.
Page 78
UNIT PLANS
Section 2.6 – Applying the Trigonometric Ratios (pp. 105-112)
ELABORATIONS &
SUGGESTED PROBLEMS



Ensure that students understand that a side and
the angle opposite to that side in a right triangle
have the same variable name, with the angle name
written in uppercase and the side name written in
lowercase.
M4 (B C D)

sketch and label a right triangle to represent a
problem

identify the trigonometric ratio required to solve a
problem

use trigonometry and the Pythagorean theorem to
solve a triangle by determining the measures of all
unknown sides and angles
following concept:


solving a triangle – determining the measure of
each angle in a triangle and the length of each side
of the triangle
Solve the triangle shown. Round off all measures
to the nearest tenth.
A

22
pp. 110-112: #1-14
420
C

B
Solve the triangle shown. Round off all measures
to the nearest tenth.
F
42
D
31
E

A pilot starts his takeoff and climbs steadily at an
angle of 12.20. Determine the horizontal distance
the plane has travelled when it has climbed 5.4 km
along its flight path. Express the answer to the
nearest tenth of a kilometre.

A surveyor needs to determine the height of a large
grain silo. He positions his transit 65 m from the
silo and records an angle of elevation of 520. If the
height of the transit is 1.7 m, determine the height
of the silo, to the nearest tenth of a metre.
Page 79
UNIT PLANS
Section 2.7 – Solving Problems Involving More Than One Right Triangle
(pp. 113-121)
ELABORATIONS &
SUGGESTED PROBLEMS



When presented with a problem that involves two
right triangles, encourage students to draw the two
triangles separately with the sides and angles
labelled in both triangles.
M4 (B D)

solve problems involving more than one right
triangle
following concept:

angle of depression – the angle between the
a point below eye level


From a height of 50 m in his fire tower near the
lake, a ranger observes the beginnings of two fires.
One fire is due west at an angle of depression of
90. The other fire is due east at an angle of
depression of 70. What is the distance between the
two fires, to the nearest tenth of a metre?

From his hotel window overlooking the street, Ken
observes a bus moving away from the hotel. The
angle of depression to the bus changes from 460 to
220. Determine the distance the bus travels in that
time if Ken’s window is 100 m above street level.

The triangles ABC and BCD have right angles at B
and C, respectively. Calculate the length of side
CD. Round off the answer to one decimal place.
p. 118-121: #1-16
D
C
600
B

600
2 cm
A
Canada’s highest waterfall is Delta Falls on
Vancouver Island, BC. An observer standing at the
same level as the base of the falls views the top of
the falls at an angle of elevation of 580. When an
observer moves 31 m closer to the base of the
falls, the angle of elevation increases to 610. Find
the height of Delta Falls, to the nearest metre.
Page 80
UNIT PLANS
CHAPTER 3
FACTORS AND PRODUCTS
SUGGESTED TIME
12 classes
Page 81
UNIT PLANS
Section 3.1 – Factors and Multiples of Whole Numbers (pp. 134-141)
ELABORATIONS &
SUGGESTED PROBLEMS


Depending on the situation, encourage students to
use a variety of methods for determining the
greatest common factor and the least common
multiple.

Some students will confuse the prime factorization
methods used to determine the greatest common
factor and the least common multiple. Remind
them that them greatest common factor is always
less than or equal to the given numbers and that
the least common multiple is always greater than or
equal to the given numbers.
AN1 (A B C G)

determine the prime factors of a whole number

explain why 0 and 1 have no prime factors

use powers to write a number as a product of its
prime factors

use a variety of strategies to determine the greatest
common factor and the least common multiple of a
set of whole numbers

solve problems that involve prime factors, greatest
common factors or least common multiples
following concepts:

prime factor – a prime number that is a factor of a
number; for example, 5 is a prime factor of 30

prime factorization – writing a number as a
product of its prime factors, for example, the prime

Determine the prime factorization of 2400.

Determine the greatest common factor of 144 and
384.

Determine the least common multiple of 32, 40 and
50.

What is the side length of the smallest square that
could be tiled with rectangles that measure 10 cm
by 12 cm? Assume that the rectangles cannot
overlap or be cut.
factorization of 20 is 2  2  5 or 22  5

prime number – a whole number with exactly two
factors; for example, 2, 3, 5, 7, 11, 29, 31 and 43

composite number – a number with three or more
factors; for example, 8 is a composite number
because its factors are 1, 2, 4 and 8

factor tree – a branching diagram with a number at
the top and its prime factors at the bottom

greatest common factor (GCF) – the greatest
number that divides into each number in a set; for
example, 5 is the greatest common factor of 10 and
15

least common multiple (LCM) – the smallest
number that is a multiple of each number in a set;
for example, the least common multiple of 12 and
21 is 84

pp. 139-141: #1-20
Page 82
UNIT PLANS
Section 3.2 – Perfect Squares, Perfect Cubes, and Their Roots (pp. 142-147)
ELABORATIONS &
SUGGESTED PROBLEMS


Some students may benefit from examples of nonperfect squares and cubes. These
counterexamples may assist students to
understand what a perfect square and a perfect
cube is. Sometimes, it helps to see what
something is not, rather than what it is.

It may be helpful to point out to students that the
side of a square is the square root, and that the
area is the square. Also, the side of a cube is the
cube root and the volume is the cube.

Reinforce the fact that the square root can be
obtained by rearranging the prime factors into two
equal groups, and that the cube root can be
obtained by rearranging the prime factors into three
equal groups.

Students may benefit from creating lists of perfect
squares and perfect cubes up to 1000. They could
refer to these lists as they work through the
exercises.
AN1 (D E F G)

use algebra tiles and linking cubes to identify
perfect squares and perfect cubes

explain why a given whole number is a perfect
square, a perfect cube or neither

use a variety of strategies to determine the square
root of a perfect square, or the cube root of a
perfect cube, and explain the process used

solve problems that involve square roots or cube
roots
following concepts:

perfect square – a number that can be written as
a power with an integer base and exponent 2; for
example, 49  72

perfect cube – a number that can be written as a
power with an integer base and exponent 3; for
example, 8  23

Determine the square root of 4624.

Determine the cube root of 74,088.

A cube has a volume of 2744 cm3. What is the
surface area of the cube?

To determine how far away the horizon is, the

square root – a number which, when multiplied by
itself, results in a given number; for example, 5 is a
square root of 25

cube root – a number which, when raised to the
exponent 3, results in a given number; for example,
5 is the cube root of 125

pp. 146-147: #1-14
formula d  2rh may be used as an
approximation, where r is radius of the earth and h
is the height, both measured in metres. The mean
radius of the earth is 6400 km. Find the distance,
in kilometres, to the horizon for a height of 45 m,
and for a height of 400 m. Round off the answers
to one decimal place, where necessary.
Page 83
UNIT PLANS
Section 3.3 – Common Factors of a Polynomial (pp. 150-156)
ELABORATIONS &
SUGGESTED PROBLEMS


AN5 (A B D E)
Remind students to check their work by using the
inverse processes of expanding and factoring.


use algebra tiles to model the factoring of a
polynomial, then record the process symbolically

write a multiplication sentence to represent an
algebra-tile model

determine the common factors and the greatest
common factor for the terms of a polynomial, then
express the polynomial as a product of its factors

Determine the GCF of each pair of terms.
a.
5m 2n and 15mn 2
b.
48ab 3c and 36a 2b 2c 2
Factor each of the following polynomials.
a.
3x  3
b.
5 x  10
c.
a 2  3a

identify and explain errors in solutions for factoring
polynomials

factor a polynomial, then verify by expanding
d.
3a 2  12a
following concepts:
e.
4 xy 2  8 x 2 y
f.
27r 2s 2  18r 3s 3  36rs 3

factors – numbers or algebraic expressions that
are multiplied together to get a product; for
example, 3 and 7 are factors of 21, and x  1 and

x  2 are factors of x 2  3 x  2

polynomial – one term or the sum of terms whose
variables have whole-number exponents; for
a.
example, x 2  3 xy  2y 2  5 x

factored fully – factoring a polynomial so each
factor cannot be factored further

expanding an expression – writing a product of
polynomial factors as a polynomial

distributive property – the property stating that a
product can be written as a sum or difference of
two products; for example, a  b  c   ab  ac
b.

Create a rectangle using the given algebra tiles,
record the dimensions and the area of each
rectangle, and write a mathematical sentence
involving the dimensions and the area for each
rectangle.
pp. 154-156: #1-20

Write a trinomial with different numerical
coefficients that has a greatest common factor of
2xy. Write the trinomial in both factored and
expanded form.
Page 84
UNIT PLANS
Section 3.4 – Math Lab: Modelling Trinomials as Binomial Products (pp. 157-158)
ELABORATIONS &
SUGGESTED PROBLEMS

AN5 (B F)

use algebra tiles to model the factoring of a
trinomial

relate the binomial factors of a trinomial to the
dimensions of the rectangle formed by the trinomial

write the multiplication sentence represented by a
rectangle made from algebra tiles

determine whether a given trinomial can be
represented by a rectangle, and therefore can be
factored

p. 158: #1-4
Page 85
UNIT PLANS
Section 3.5 – Polynomials of the Form x 2  bx  c (pp. 159-167)
ELABORATIONS &
SUGGESTED PROBLEMS

AN4 (A B C)

AN5 (B D E F H)

Encourage students to check their answers using
multiplication.

Some students think that products such as
 t  4  t  8  and  t  4  t  8  are equivalent.

model the multiplication of two binomials using
algebra tiles or a sketch of a rectangle, and record
the process symbolically

relate the multiplication of two binomials to an area
model

multiply two binomials

model the factoring of a trinomial using algebra
tiles or a sketch of a rectangle, and record the
process symbolically

identify and explain errors in a trinomial
factorization
Demonstrate that they are not equivalent by using
the distributive property to multiply both
expressions. They will see that the t-terms of the
two resulting polynomials have opposite
coefficients, and therefore are not equal.


Draw a rectangle to show the area represented by
each product.
a.
length x  1 , width x  3
b.
length x  2 , width x  1
Determine each product.

factor a trinomial and verify by multiplying the
factors
a.
 x  3  x  5 

explain the relationship between multiplying
binomials and factoring a trinomial
b.
 m  4  m  6 
c.
 n  10  n  10 
following concepts:


descending order – writing a polynomial such that
its terms are written with the largest degree first
and the smallest degree last
ascending order – writing a polynomial such that
its terms are written with the smallest degree first
and the largest degree last


pp. 165-167: #1-21

The area of a rectangle is x 2  6 x  8.
a.
Find the possible dimensions of the rectangle,
if neither dimension can be 1.
b.
Are the other rectangles which can be formed
that have the same area?
c.
For the rectangle(s) that you have formed, find
the perimeter.
Explain why a rectangle cannot be formed with an
area of x 2  3 x  1. What can you conclude about
this trinomial?


Factor each of the following trinomials.
a.
x 2  7x  6
b.
x 2  6x  9
c.
x 2  2x  3
d.
x 2  7 x  10
Determine all possible values of k such that the
trinomial x 2  kx  24 is factorable.
Page 86
UNIT PLANS
Section 3.6 – Polynomials of the Form ax 2  bx  c (pp. 168-178)
ELABORATIONS &
SUGGESTED PROBLEMS

AN4 (A B E)

AN5 (B D E F G H)

multiplication.

When factoring polynomials, encourage students to
try different methods of factoring.

model the multiplication of two binomials using
algebra tiles or a sketch of a rectangle, and record
the process symbolically

relate the multiplication of two binomials to an area
model

multiply two binomials

model the factoring of a trinomial using algebra
tiles or a sketch of a rectangle, and record the
process symbolically

identify and explain errors in a trinomial
factorization

factor a trinomial and verify by multiplying the
factors

explain the relationship between multiplying
binomials and factoring a trinomial



Draw a rectangle to show the area represented by
each product.
a.
length 4 x  2, width 2 x  1
b.
length x  3, width 2 x  1
a.
 4 x  1 x  2 
b.
 x  6  2x  1
c.
 5m  1 2m  6 
Write an expression for the area of the border
around the inner rectangle.
x 9
x4
following concepts:

x
coefficient – the numerical factor of a term; for
x 5
example, in the terms 3x and 3 x 2 , the coefficient is
3


zero principle – the property of addition that states
that adding 0 to a number does not change the
number; for example, 3  0  3

factoring by decomposition – factoring a
trinomial after writing the middle term as the sum of
two terms, then determining a common binomial
factor from the two pairs of terms formed

p. 176-178: #1-20

Factor each of the following trinomials.
a.
2 x 2  11x  15
b.
4 x 2  17 x  15
c.
2 x 2  13 x  15
d.
5x 2  7x  6
e.
6 x 2  11x  3
f.
3x 2  5x  2
Determine all possible values of k such that the
trinomial 4 x 2  kx  3 is factorable.
Page 87
UNIT PLANS
Section 3.7 – Multiplying Polynomials (pp. 182-187)
ELABORATIONS &
SUGGESTED PROBLEMS


AN4 (D E F G)

draw a rectangle diagram to determine the product
of two polynomials

multiply two polynomials symbolically, and combine
like terms in the product

verify a polynomial product by substituting numbers
for the variables

identify and explain errors in a solution for
polynomial multiplication

the product  x  1 x 2  3 x  2 , the following

A rectangle diagram can be used to determine the
product of two polynomials. For example, to find
rectangle diagram can be used:
x2
3x
2
x
x3
3x 2
2x
1
x 2
3 x
2
Combining like terms, we get a product of
x 3  2 x 2  x  2.

pp. 185-187: #1-17

Encourage students to substitute values for the
variables to check their answers.

For students who have difficulty organizing their
work, encourage them to put square brackets
around each polynomial product. Then the student
should then simplify the product within each set of
square brackets. Remind them not to attempt too
many steps at one time.

Expand and simplify each of the following products.
a.
 3 x  4   2x 2  3 x  1
b.
 r  4   3r 2  8r  6 
c.
 5 x  3   2x 2  6 x  12 
d.
 2x  y 
e.
 x  3  5 x  2  4  x  1 2x  5 
f.
2  3 x  2    4 x  7  2 x  5 
3
Page 88
UNIT PLANS
Section 3.8 – Factoring Special Polynomials (pp. 188-195)
ELABORATIONS &
SUGGESTED PROBLEMS


multiplication.

Ask students to verbally explain how to recognize a
difference of squares and a perfect square
trinomial.

Some students have difficulty factoring a trinomial
in two variables. Encourage them to rewrite the
trinomial in one variable, then once it is factored,
the second variable can be simply written in each
binomial adjacent to the constant term.
AN5 (C E H)

factor a trinomial that is a difference of squares

explain why factoring a difference of squares is a
special case of factoring a trinomial

model the factoring of a difference of squares

factor a perfect square trinomial and identify
patterns in the trinomial and its factors

factor a trinomial in two variables and verify by
multiplying the factors

following concepts:

perfect square trinomial – a trinomial of the form
a 2  2ab  b 2 ; it can be factored as
a 2  2ab  b 2   a  b 

2
difference of squares – a binomial of the form

a  b ; it can be factored as
2
2
a 2  b 2   a  b  a  b 

pp. 194-195: #1-18
a.
 x  3  x  3 
b.
 x  5  x  5 
c.
 2x  1 2x  1
d.
 4 x  y  4 x  y 
Factor each of the following polynomials.
a.
4 x 2  20 x  25
b.
x 2  24 xy  144 y 2
c.
y 2  18 y  81
d.
3b 2  24b  48
e.
4 x 2  16
f.
49a 2  25b 2
g.
125 x 2  80 y 2
h.
9 p 2q 2  64
Page 89
UNIT PLANS
Page 90
UNIT PLANS
CHAPTER 4
ROOTS AND POWERS
SUGGESTED TIME
9 classes
Page 91
UNIT PLANS
Section 4.1 – Math Lab: Estimating Roots (pp. 204-206)
ELABORATIONS &
SUGGESTED PROBLEMS

AN2 (B G)

predict whether a root will be exact or approximate

use mental math to calculate roots

estimate a root by using benchmarks and explain
why the estimated root is an underestimate or
overestimate

explain the meaning of the index of a radical
following concepts:

root – the inverse of a power; for example, in
23  8 or

8  2, 2 is the cube root of 8
radical – an expression consisting of the radical
sign,

3
, and a radicand; for example,
radicand – the number under a radical sign; for
example, 81 is the radicand in

81
81
index – in a radical, the number above the radical
symbol that indicates which root is to be taken; for
example, 4 is the index in the radical 4 81; if the
index is not written, it is assumed to be 2

p. 206: #1-6
Page 92
UNIT PLANS
Section 4.2 – Irrational Numbers (pp. 207-212)
ELABORATIONS &
SUGGESTED PROBLEMS


Ask students if there are more rational or irrational
numbers, and to explain their reasoning.

Ask students to explain why 1.112111211112... is
irrational.

For students who have difficulty ordering numbers
on a number line, ask them to convert the numbers
to decimal form. That will make the task much
easier.
AN2 (A C D H)

identify rational numbers as whole numbers,
integers, fractions and decimals

sort a set of numbers into rational and irrational
numbers

locate irrational numbers on a number line, explain
how they did it, and then order the numbers


use a graphic organizer to illustrate the set of real
numbers
Place an X in the correct spaces to indicate that the
number belongs in the number set.
5
following concepts:



7
0
Integers
Rational
Numbers
irrational number – any number that cannot be
m
written in the form
, where m and n are integers,
n
and n  0
pp. 211-212: #1-20
–1.3
Whole
Numbers
Irrational
Numbers
Real
Numbers

Sort the list of numbers 1, 2, 3, 4, ..., 20
into two sets, those that are rational numbers and
those that are irrational numbers. What do all
numbers in each set have in common?

Arrange the following numbers in ascending order:

3
4
Natural
Numbers
rational number – any number that can be written
m
in the form
, where m and n are integers, and
n
n0
real number – any number that is a rational
number or an irrational number; a member of the
set of numbers that have a decimal representation
–2
,
22
, 3.141,
7
10,
9.5

Determine whether each of the following
statements is always true, sometimes true or
never true.
a. All whole numbers are integers.
b. If a number is a rational number, then it is also
an integer.
c. There is a number which is both rational and
irrational.

Complete each statement with one of the following
symbols: <, > or =.
a.
1.732
b.
 6
c.
2
3
 5
1.4142135
Page 93
UNIT PLANS
Section 4.3 – Mixed and Entire Radicals (pp. 213-219)
ELABORATIONS &
SUGGESTED PROBLEMS


Encourage students to memorize the perfect
squares from 1 to 144. This will help them work
with radical expressions much more efficiently.

For students who have difficulty determining
whether a mixed radical and an entire radical are
equivalent, ask them to use a calculator to
determine the approximate values of both.
AN2 (E F)

use the relationships between the lengths of the
hypotenuses of two right isosceles triangles to
explain why an entire radical can be written as a
mixed radical

express an entire radical as a mixed radical in
simplest form


express a mixed radical as an entire radical
following concepts:

mixed radical – a number written as a product of
another number and a radical; for example, 3 5

entire radical – a radical sign and the number
under it; for example,
5
32

MULTIPLICATIVE PROPERTY OF RADICALS
ab  a  b , where n is a natural number, and a
and b are real numbers.
n
n
n
Write each of the following radicals in simplest
form:
a.
24
b.
72
c.
96
d.
45
Express each mixed radical as an entire radical.
a.
5 11
b.
2 10
c.
6 3

pp. 217-219: #1-23
Page 94
UNIT PLANS
Section 4.4 – Fractional Exponents and Radicals (pp. 222-228)
ELABORATIONS &
SUGGESTED PROBLEMS


AN3 (B D E F)
Ensure that students understand how the
numerator and the denominator of a fractional
exponent relate to a radical expression.

use patterns to explain why a1 n  n a , n  0

recognize that, in a rational exponent expressed as
a fraction, the denominator represents the index of
a radical and the numerator represents the power
to which the radical or radicand is raised

Express each power as a radical.
a.
73 5
b.
10241 3
c.
x 
4
38

evaluate a numerical expression containing rational
exponents with and without a calculator

express a power with a rational exponent as a
radical

express a radical as a power with a rational
exponent
b.
3
y5

solve problems involving exponents or radicals
c.
n
272
following concept:


Express each radical as a power.
53
a.

rational exponent – an exponent that is a rational
number
POWERS WITH RATIONAL EXPONENTS
Evaluate each expression.
a.
43 2
b.
322 5
c.
 8 
 
 27 
23
When n is a natural number and x is a rational
number,
x1 n  n x
When m and n are rational numbers, and x is a
rational number,

x m n  x 1/ n

m

 x
n
m
and
x
mn
 
 xm
1/ n
 n xm

pp. 227-228: #1-21
Page 95
UNIT PLANS
Section 4.5 – Negative Exponents and Reciprocals (pp. 229-234)
ELABORATIONS &
SUGGESTED PROBLEMS


Remind students that if a base is positive, a
negative exponent always produces a positive
answer.

Ask students to explain why the product
AN3 (A E F)

use patterns to explain why a power with a
negative exponent is equal to the reciprocal of the
corresponding power with a positive exponent

recognize that a power with a positive exponent is
equivalent to the reciprocal of the corresponding
power with a negative exponent

simplify an expression that contains negative
exponents

explain why the product of two powers with the
same base and opposite exponents is 1
24  24  53  5 3  105  10 4 is easy to calculate
mentally.

Some students have difficulty evaluating a power
with a fractional base and a negative exponent.
Remind them that the negative exponent means to
first take the reciprocal of the base. Then, they can
evaluate the remaining power with a positive
exponent.

Simplify each of the following expressions.
following concepts:
a.
46  44  40

negative exponent – an exponent that is a
negative number
b.
79  77  71

reciprocals – two numbers whose product is 1; for
2
3
and
example,
3
2
c.
1453  1452  145 4
d.
42  42
e.
2 2
   
3 3
f.
 2
3
POWERS WITH NEGATIVE EXPONENTS
When x is any non-zero number and n is a rational


2
4
number, x  n is the reciprocal of x n . That is,
x n 
1
, x0
xn
and
1
 xn, x  0
x n

pp. 233-234; #1-19
Page 96
UNIT PLANS
Section 4.6 – Applying the Exponent Laws (pp. 237-243)
ELABORATIONS &
SUGGESTED PROBLEMS


Review the laws of exponents with students before
beginning this section.

Some students have difficulty determining whether
an algebraic expression has been simplified
correctly. One method of doing this is by
substituting numbers for the variables in both the
original and simplified expression to see whether
the two values are the same.
AN3 (C E F)

use prime factorization and the exponent laws to
simplify expressions

describe different strategies used to simplify
expressions, and explain their reasoning

apply the exponent laws only to powers with the
same base

EXPONENT LAWS
EXPONENT NAME
EXPONENT LAW
Product of Powers
 a  a   a
Quotient of Powers
am
 am n , a  0
an
Power of a Power
Power of a Product
m
n
a 
m
 ab 
m
n
 a mn

 ambm
an
a
 b   bn , b  0
 

pp. 241-243: #1-20
a.
 x  x 
b.
 p  p 
c.
1.5 4 / 3
1.51/ 6
mn
n
Power of a Quotient
Write each expression as a power with a single
exponent. Write all powers with positive
exponents.

1.5
3.5
5 / 4
1/ 2
Simplify each expression. Write all powers with
positive exponents.
a.
 27 x 
b.
 t 4 / 3 t 1/ 3 


c.
 x3 
 
 64 
6
2/ 3
  
Simplify
9
2/3
 x  x  .
5
3
3
2
Write the answer in radical
fo
Page 97
UNIT PLANS
Page 98
UNIT PLANS
CHAPTER 5
RELATIONS AND FUNCTIONS
SUGGESTED TIME
14 classes
Page 99
UNIT PLANS
Section 5.1 – Representing Relations (pp. 256-263)
ELABORATIONS &
SUGGESTED PROBLEMS


Some students have difficulty expressing a relation
in words. Have the student choose one ordered
pair in the relation, then write a sentence involving
the two elements. In the sentence, the first
element in the ordered pair should come before the
second element. For example, (penny, 0.01) could
be interpreted as, “A penny has a value of $0.01.”

Ask each student to create a graph with the axes
labelled x and y and then exchange their graph
with someone else. Each student is to try to
develop a scenario for the given graph.
RF1 (E)

express a relation in words, as a set of ordered
pairs, in a table, and using an arrow diagram
following concepts:

set – a collection of distinct objects

element – an element of a set is one object in the
set

relation – a rule that associates the elements from
one set with the elements of another set

set of ordered pairs – a collection of ordered pairs

arrow diagram – used to represent a relation; the
ovals show the sets, and the arrows associate
elements of the first set with the elements of the
second set

This table shows the largest four Prince Edward
Island communities with their populations in 2006.

pp. 261-263: #1-11
COMMUNITY
POPULATION (2006)
Charlottetown
32,200
Summerside
14,500
Stratford
7,100
Cornwall
4,700
a.
Describe the relation in words.
b.
Represent this relation as a set of ordered
pairs.
c.
Represent this relation as an arrow diagram.
Page 100
UNIT PLANS
Section 5.2 – Properties of Functions (pp. 264-273)
ELABORATIONS &
SUGGESTED PROBLEMS

RF1 (E)

RF2 (A B D)

RF4 (A)

RF8 (A B C D)

determine the domain and range of a set of
ordered pairs and from a table of values

determine if a set of ordered pairs represents a
function

use examples to explain why some relations are
not functions but all functions are relations

identify independent and dependent variables in a
given context

represent a linear function using function notation,
and represent an equation given in function
notation as a linear function in two variables

use function notation to determine values
following concepts:

domain – the set of first elements of a relation

range – the set of second elements associated
with the set of first elements (domain) of a relation

dependent variable – a variable whose value is
determined by the value of another (the
independent) variable


independent variable – a variable whose value is
not determined by the value of another variable,
and whose value determines the value of another
(the dependent) variable
Some students mistakenly think that f  x  means “f
multiplied by x.” Tell them that f  x  is part of
function notation, where f is the name of the
function, x is the input, and f  x  is the output.

Which of the following relations are functions?
Explain your choices.
a.
 2,1 ,  0,0  ,  2,1 ,  2,2
b.

function – a relation where each element in the
first set is associated with exactly one element in
the second set



x
y
1
3
2
3
3
4
4
4
5
4
For each relation, determine whether it is a function
and state the domain and the range.
a.
1,1 ,  2,8  ,  3,27  ,  4,64 
b.
 3, 4  ,  3,5  ,  3,6  ,  3,7 
The function F C   1.8C  32 is used to convert a
temperature in degrees Celsius (0C) to a
temperature in degrees Fahrenheit (0F).

function notation – notation used to show the
independent variable in a function; for example,
f  x  means that the value of the function f
a.
Determine F  25  .
b.
Determine C so that F C   68.
If f  x   x 2  5 x  3, find f  2  . What is an ordered
pair describing the point having a y-coordinate of
f  2 ?
depends on the value of the independent variable x

pp. 270-273: #1-19
Page 101
UNIT PLANS
Section 5.3 – Interpreting and Sketching Graphs (pp. 276-283)
ELABORATIONS &
SUGGESTED PROBLEMS


RF1 (C D)

describe a possible situation for a given graph

sketch a possible graph for a given situation

match a situation to a graph that best represents it

use information provided by a graph to answer
questions

On distance-time or speed-time graphs, some
students will interpret a line segment that goes up
from left to right as a person travelling up a hill.
Remind them that a line segment that goes up to
the right indicates that both variables are
increasing. So, on a distance-time graph, a line
segment like this shows the distance travelled is
increasing as time increases. On a speed-time
graph, it shows that the speed is increasing as time
increases.

The graph shows the speed of a boat that is towing
a water-skier. Describe what the boat is doing.
pp. 281-283: #1-15
Speed
0
Time

Which graph best represents a person’s height as
the person ages? Explain your choice.

Josie leaves her home and walks to the store.
After buying a drink, she slowly jogs to her friend’s
house. Josie visits with her friend for a while and
then jogs directly home. Draw a distance-time
graph that shows Josie’s distance from her home.

Sketch graphs to illustrate the following situations.
a. the cost of renting a car for one day as a
function of the kilometres driven, assuming
that a flat fee is paid for renting
b. the population of Canada as a function of the
year
c. the number of hours of daylight in Prince
Edward Island as a function of the date

A hydrologist studied the relationship between the
pressure on an object and the depth of submersion
in a liquid. The following graph was sketched.
Draw conclusions based upon the sketch.
Pressure
(kPa) 100
Depth (m)
Page 102
UNIT PLANS
Section 5.4 – Math Lab: Graphing Data (pp. 284-286)
ELABORATIONS &
SUGGESTED PROBLEMS

RF1 (A B E)

RF2 (D)

graph, with or without technology, a set of data and
determine the restrictions on the domain and range

explain why data points should or should not be
connected on the graph for a situation

determine whether a graph represents a function

p. 286: #1-2
Page 103
UNIT PLANS
Section 5.5 – Graphs of Relations and Functions (pp. 287-297)
ELABORATIONS &
SUGGESTED PROBLEMS

RF1 (B E)

RF2 (C D)

RF4 (A)

RF8 (C D)

Remind students that when stating domain and
range, it is important to remember whether the
relation is discrete or continuous. When the graph
is continuous, state the domain and range as an
inequality. When it is discrete, simply list the set of
numbers.

explain why data points should or should not be
connected on the graph for a situation


determine the domain and range of a graph

sort a set of graphs as functions or non-functions

generalize and explain rules for determining
whether a graph represents a function

given context

use the graph of a function to determine the range
value for a related domain value, and the domain
value for a related range value
What is the domain and range of the circle in the
graph shown below?
y
x

Determine whether each of these graphs is a
function.
a.
following concept:

vertical line test for a function – a graph
represents a function when no two points on the
graph lie on the same vertical line

pp. 293-297: #1-20
b.
c.
Page 104
UNIT PLANS
Section 5.6 – Properties of Linear Relations (pp. 300-310)
ELABORATIONS &
SUGGESTED PROBLEMS


RF4 (A B C D E F G)

given context

determine whether a situation represents a linear
relation

determine whether a graph, a table of values, a set
of ordered pairs, or an equation represents a linear
relation



draw a graph from a set of ordered pairs from a
given situation, and determine whether the relation
between the variables is linear
match corresponding representations of linear
functions
It is important that students recognize whether a
relation is linear by its equation. Remind students
that a relation will be linear if all of its terms are no
more than degree 1.

following concepts:
Determine whether each of the following relations
is linear or non-linear.
a.
x7
b.
3 x  2y  12
c.
2x  y 2  6
d.
y 
e.
xy  3
2
x5
3
Sketch the graph of each of the following linear
relations.
a.
y  x 1

linear relation – a relation that has a straight-line
graph
b.
y

rate of change – the change in one quantity with
respect to the change in another quantity
c.
y  7  2x


pp. 307-310: #1-17
Henry rented a carpet cleaner for a flat fee of $20
plus an additional $4 per day. Let d represent the
number of days the machine was rented. Let C
represent the cost.
a.
2
x2
3
Make a table of values for 5 days.
b.
Draw a graph of C versus d.
c.
Write an equation of the function.
d.
How much would it cost Harry to rent the
machine for 8 days?
Page 105
UNIT PLANS
Section 5.7 – Interpreting Graphs of Linear Functions (pp. 311-323)
ELABORATIONS &
SUGGESTED PROBLEMS

RF5 (A C E G)

RF8 (E)

Remind students that when finding a particular
intercept to a graph, they set the other variable
equal to zero. Specifically, to find the y-intercept,
set x  0, and to find the x-intercept, set y  0.


determine the intercepts of the graph of a linear
relation, and state the intercepts as values or
ordered pairs

determine the domain and range of the graph of a
linear relation
A school council wants to sell school T-shirts.
There is a charge for having a digital master of the
school logo made. From the graph below, find
a.
the cost of the digital master
identify the graph that corresponds to a given slope
and y-intercept
b.
the cost of having 100 T-shirts made
c.
the cost of having 200 T-shirts made

solve a problem that involves intercepts, slope,
domain or range of a linear relation
d.
the cost of each extra T-shirt made

sketch the graph of a linear function expressed in
function notation

following concepts:

linear function – a linear relation whose graph is
not a vertical line

vertical intercept – the y-intercept; the ycoordinate of a point where a graph intercepts the
y-axis

horizontal intercept – the x-intercept; the xcoordinate of a point where a graph intercepts the
x-axis


pp. 319-323: #1-17
Sketch a graph of each linear function.
a.
f  x   2 x  3
b.
f x 
c.
f  x   3x  1
3
x 2
4
Page 106
UNIT PLANS
CHAPTER 6
LINEAR FUNCTIONS
SUGGESTED TIME
11 classes
Page 107
UNIT PLANS
Section 6.1 – Slope of a Line (pp. 332-343)
ELABORATIONS &
SUGGESTED PROBLEMS

RF3 (A B C D E F G I)

RF5 (B D)

rise
. To help
run
students remember this, remind them that the word
rise occurs before the word run in the dictionary, so
we place rise in the numerator and run in the
denominator. Since rise represents a vertical
distance, it is the change in y. Since run
represents a horizontal distance, it is the change in
x.
Remind students that slope 

determine the slope of a line segment and a line

classify lines in a given set as having positive or
negative slopes

explain the meaning of the slope of a horizontal or
vertical line

draw a line, given its slope and a point on the line


determine the coordinates of a point on a line,
given the slope and another point on the line

solve a problem involving slope

use examples to explain slope as a rate of change

explain why the slope of a line can be determined
using any two points on that line

sketch a linear relation that has one intercept, two
intercepts, or an infinite number of intercepts

Determine the slope, m, of each line segment with
the given end points.
a.
S  3,6  and T  5,2 
b.
H  4,3  and K  4,8 
c.
M  9, 7  and N  1, 7 
d.
W  2, 2  and S  5,5 
Determine the slope of each side of the triangle
below.
following concepts:

y
slope – a measure of how one quantity changes
with respect to the other; it can be determined by
rise
calculating
run

rise – the vertical distance between two points

run – the horizontal distance between two points
x
B
C

If the slope of a line is 6 and the line passes
through the points (2,5) and (1,k), what is the value
of k?

If two points on a line are (4,3) and (6,4), find three
other points on the line.
SLOPE OF A LINE
A line passes through A  x1, y1  and B  x2, y 2  . The
slope of the line AB is
AB 
A
y 2  y1
x1  x2
x2  x1

pp. 339-343: #1-28
Page 108
UNIT PLANS
Section 6.2 – Slopes of Parallel and Perpendicular Lines (pp. 344-351)
ELABORATIONS &
SUGGESTED PROBLEMS


RF3 (H I)
Some students believe that two lines whose slopes
are reciprocals are perpendicular. Remind
students that in order for lines to be perpendicular,
their slopes must have opposite signs as well as
having reciprocal numerical values. This is due to
the fact that, except for horizontal and vertical lines,
one line must be rising from left to right (positive
slope) and one line must be falling from left to right
(negative slope).

identify parallel lines

identify perpendicular lines

identify the slope of a line parallel or perpendicular
to a given line

draw a line parallel or perpendicular to a given line,
given a point on the new line

use slopes to identify some polygons

following concepts:




Determine whether each pair of lines with the given
slopes is parallel, perpendicular, or neither.
6
2
a.
3,
parallel lines – lines on the same flat surface that
do not intersect
b.
perpendicular lines – lines or line segments that
intersect at right angles
7 2
,
2 7
c.
3
4
, 
4
3
oblique line – a line that is neither horizontal nor
vertical

negative reciprocals – two numbers whose
3
7
are
product is –1; for example,  and
7
3
negative reciprocals
SLOPES OF PERPENDICULAR LINES
The slopes of two oblique perpendicular lines are
negative reciprocals; that is, a line with slope a,
1
a  0, is perpendicular to a line with slope  .
a
The coordinates of the endpoints of two line
segments are given below. Are the lines
containing each pair of line segments parallel,
perpendicular, or neither?
a.
A  2, 2  , B  3, 3  and C  2,2  , D  3,3 
b.
E  6,8  , F  7,10  and G  0,1 , H  2,0 
c.
W  6, 4  , X  0,0  and Y 1,1 , Z  3, 4 

pp. 348-351: #1-20
Page 109
UNIT PLANS
Section 6.3 – Math Lab: Investigating Graphs of Linear Functions (pp. 354-356)
ELABORATIONS &
SUGGESTED PROBLEMS

RF6 (C D)

identify the slope and y-intercept of a line from the
equation of the line written in slope-intercept form

graph a linear function whose equation is in slopeintercept form

describe a strategy for graphing a linear function in
slope-intercept form

describe how changing the value of m or b
changes the appearance of a graph

predict the appearance of a graph whose equation
is in slope-intercept form

p. 356: #1-7
Page 110
UNIT PLANS
Section 6.4 – Slope-Intercept Form of the Equation for a Linear Function
(pp. 357-364)
ELABORATIONS &
SUGGESTED PROBLEMS

RF5 (E F G)

RF6 (C D F)

RF7 (A F)


determine the slope and y-intercept from the graph
of a linear relation and write an equation in the form
y  mx  b

identify the graph that corresponds to a given slope
and y-intercept, and vice-versa

solve a contextual problem that involves intercepts
or slope

graph a linear relation, given its equation in slopeintercept form, and explain the strategy used to
create the graph

match the equation of a linear relation to its graph

solve a problem using the equation of a linear
relation




pp. 362-364: #1-21
Determine the slope and the y-intercept of each of
the following lines.
a.
y  2x  5
b.
y 2
c.
x  5 y  15
d.
2 x  6 y  12
Parents of members of the cheerleading squad rent
a hall. They arrange a talent show as a fundraiser.
The relationship between the number of tickets
sold, x, and the profit, y, in dollars, may be
represented by the equation 12 x  y  840  0.
a. What is the slope of the line? What does the
slope represent?
b. Identify the y-intercept. What does it
represent?
c. How many tickets must the parents sell to
reach the break-even point?
slope-intercept form – the equation of a line in the
form y  mx  b, where m is the slope of the line
and b is its y-intercept
If the equation is in the form y  x  b, remind
students that it can be written as y  1x  b.
Therefore, m  1.
following concept:

Some students have difficulty determining the
values of m and b in certain cases.
 If the equation is in the form y  mx, remind
students that it can be written as y  mx  0.
Therefore, b  0.

A decorator’s fee can be modelled by the equation
F  75t  b. In the equation, F represents the fee,
in dollars, t represents time, in hours, and b
represents the cost of the initial consultation, in
dollars.
a. Suppose the decorator spends 4 h for a client
and charges the client $450. Determine the
value of the parameter b.
b. How many hours does the decorator work if a
client is charged $975?

Determine whether each pair of lines is parallel,
perpendicular, or neither.
a.
b.
1
x 7
2
y  2x  7
y
2
x 6
5
5 x  2y  8
y
Page 111
UNIT PLANS
Section 6.5 – Slope-Point Form of the Equation for a Linear Function (pp. 365-374)
ELABORATIONS &
SUGGESTED PROBLEMS

RF6 (B C D F)

RF7 (B C D F)


graph a linear relation, given its equation in slopepoint form and explain the strategy used to create
the graph
Some students will incorrectly identify the
coordinates of a point on the line from an equation
in slope-point form. Remind the student that when
an equation is written in slope-point form, the
operations are subtraction. For an equation such
as y  3  2  x  1 , the student may rewrite the
equation as y   3   2  x   1  , showing that it
passes through the point (–1,–3).

write an equation in slope-point form as an
equation in slope-intercept form



write the equation of a linear relation, given its
slope and the coordinates of a point on the line
Write an equation of the line through the point
(3,–4) with slope 2.


write the equation of a linear relation, given the
coordinates of two points on the line
Write an equation of the line that passes through
each pair of points.


write the equation of a linear relation, given the
coordinates of a point on the line and the equation
of a parallel or perpendicular line


b.
(–1,3) and (4,2)
A spring with no mass attached is 25.2 cm long.
For each 1-g mass attached to the spring, the
spring’s length increases by 4 mm. Find a linear
equation for this relationship.

A jet ski rental operation charges a fixed insurance
premium, plus an hourly rate. The total cost for
2 hours is $50 and for 5 hours is $110. Determine
the fixed insurance premium and the hourly rate to
rent the jet ski.

Write the equation of the line that is parallel to
y  3 x  3 and passes through the point (5,–6).

Two perpendicular lines intersect on the x-axis.
The equation of one of the lines is y  2 x  6. Find
the equation of the second line.
slope-point form – the equation of a line in the
form y  y1  m  x  x1  , where m is the slope of
the line, and the line passes through the point
P  x1, y1 
(1,1) and (3,–3)

relation
following concept:
a.
pp. 371-374: #1-25
Page 112
UNIT PLANS
Section 6.6 – General Form of the Equation for a Linear Relation (pp. 377-385)
ELABORATIONS &
SUGGESTED PROBLEMS

RF6 (A B C D E F)

RF7 (E F)

Some students may need some review
transforming linear equations between their various
forms.

Have students manipulate the general form of a
straight line into the slope-intercept form of a
straight line. Determine the rules that connect A, B
and C to the slope and to the x- and y-intercepts.

write the equation of a linear relation in different
forms

graph a linear relation, given the general form and
explain the strategy used to create the graph

identify equivalent linear relations

Rewrite the equation y 


graph linear data generated from context and write
the equation of the resulting line

Consider the linear equation 4 x  5 y  20  0.

relation
following concepts:



standard form – the equation of a line in the form
Ax  By  C, where A, B and C are integers, and A
and B are not both zero; by convention, A is a
whole number
general form – the equation of a line in the form
Ax  By  C  0, where A, B and C are integers,
and A and B are not both zero; by convention, A is
a whole number

pp. 383-385: #1-25
3
x  2 in general form.
4
a.
What is the x-intercept of a graph of the
equation?
b.
What is the y-intercept?
c.
Use the intercepts to graph the line.
Brooke wants to save $336 to decorate her
bedroom. She has two part-time jobs. On
weekends, she works as a snowboard instructor
and earns $12 per hour. On weeknights, she earns
$16 per hour working as a high school tutor.
a.
Write an equation to represent the number of
hours Brooke needs to work as a snowboard
instructor, S, and as a tutor, T.
b.
What is the S-intercept of a graph of the
equation? What does the S-intercept
represent?
c.
What would the T-intercept be? What does it
represent?
d.
Suppose Brooke works 8 h as a snowboard
instructor. How many hours will she need to
work as a tutor?
Page 113
UNIT PLANS
Page 114
UNIT PLANS
CHAPTER 7
SYSTEMS OF LINEAR EQUATIONS
SUGGESTED TIME
11 classes
Page 115
UNIT PLANS
Section 7.1 – Developing Systems of Linear Equations (pp. 394-402)
ELABORATIONS &
SUGGESTED PROBLEMS


RF9 (A B)
Some students have difficulty creating a problem
that may be modelled by a given linear equation,
especially when some of the coefficients are not 1.
To help them, remind them that the letters used as
variables can be the first letters of the items they
represent.

create a linear system to model a situation

relate a linear system to the context of a problem

verify the solution of a linear system

write a situation that might be modelled by a given
linear system

David earns $40 plus $10 per hour. Carmen earns
$50 plus $8 per hour. Develop a system of linear
equations to represent this situation.

Verify by substitution that (3,–2) is the solution to
the system of linear equations x  3y  9 and
following concept:

system of linear equations – two equations of
linear functions in the same two variables; can also
be called a linear system
2 x  y  4.

pp. 400-402: #1-14
Page 116
UNIT PLANS
Section 7.2 – Solving a System of Linear Equations Graphically (pp. 403-410)
ELABORATIONS &
SUGGESTED PROBLEMS


RF9 (A C D H)
Remind students that after finding the point of
intersection of two lines, they should verify that it is
a solution by substituting it into both equations.


determine and verify the solution of a system of
linear equations graphically, without technology


explain the meaning of the point of intersection of a
system of linear equations

solve a problem by graphing a system of linear
equations
Solve each system of equations graphically.
a.
2 x  3 y  11
2 x  3 y  17
b.
xy 8
3 x  2y  14
following concept:

point of intersection – the point where two graphs
intersect

pp. 408-410: #1-16
Page 117
UNIT PLANS
Section 7.3 – Math Lab: Using Graphing Technology to Solve a System of Linear
Equations (pp. 411-413)
ELABORATIONS &
SUGGESTED PROBLEMS

RF9 (C H)


determine and verify the solution of a system of
linear equations graphically, using technology

verify the solution of a system of linear equations

solve a problem by graphing a system of linear
equations using technology

pp. 412-413: #1-5
Page 118
UNIT PLANS
Section 7.4 – Using a Substitution Strategy to Solve a System of Linear Equations
(pp. 416-427)
ELABORATIONS &
SUGGESTED PROBLEMS


RF9 (A B E G H)

write a system of linear equations to model a
situation

relate a system of linear equations to the context of
a problem

use a substitution strategy to determine the
solution of a system of linear equations

algebraically

explain the steps taken to solve a system of linear
equations

solve a problem that involves a system of linear
equations

following concepts:

solving by substitution – solving one equation for
one variable, substituting that expression into the
other equation, and then solving for the remaining
variable

equivalent linear system – a linear system that
has the same solution as another linear system

distributive property – the property stating that a
product can be written as a sum or difference of

Remind students that once a linear system has
been solved for one variable, that value may be
substituted into either original equation to find the
value of the other variable.
Solve each system of equations using the
substitution method.
a.
3 x  5 y  27
4 x  16
b.
2x  y  5
x  2y  10
c.
3x  y  9
6 x  2y  6
d.
3 x  4 y  15
xy 5
e.
2 x  y  13
4.5 x  0.4 y  16

A principal of $42,000 is invested partly at 7%
simple interest and partly at 9.5% simple interest
for one year. If the interest is $3700, how much is
invested at each interest rate?

An 82-m cable is cut into two pieces. One piece is
18 m longer than the other. What is the length of
each piece?

Rory’s grandmother is 58 years older than Rory. In
5 years, they plan to have a party to celebrate that
their ages have a sum of 100. How old are they
now?
pp. 424-427: #1-22
Page 119
UNIT PLANS
Section 7.5 – Using an Elimination Strategy to Solve a System of Linear
Equations (pp. 428-439)
ELABORATIONS &
SUGGESTED PROBLEMS

Some students have difficulty organizing their work
when solving systems of equations by elimination.
Encourage students use the following steps when
working through a problem:

situation

Line up the equations with the like terms in the
same columns.

relate a system of linear equations to the context of
a problem

Choose a variable to eliminate.

To the right of each equation, write down what
equation is to be multiplied by.

RF9 (A B E G H)

use an elimination strategy to determine the
solution of a system of linear equations


algebraically
Write the new equations to the right of the
original equations.


explain the steps taken to solve a system of linear
equations
Add the equations together and solve for the
remaining variable.


solve a problem that involves a system of linear
equations
Substitute this value into either original
equation to find the value of the other variable.
following concept:


solving by elimination – adding or subtracting the
equations to eliminate one variable and then
solving for the other variable

Solve each system of equations using the
elimination method.
a.
x  2y  10
2 x  3 y  14
b.
y  2 x  4
y  3 x  16
c.
3 x  2y  10
9 y  6 y  16
pp. 437-439: #1-21

A group of people bought tickets for a UPEI
basketball playoff game. Two student tickets and
six adult tickets cost $102. Eight student tickets
and three adult tickets cost $114. What was the
price for a single adult ticket and a single student
ticket?

During lunch, the cafeteria sold a total of 160
muffins and individual yogurts. The price of each
muffin is $1.50 and the price of each individual
yogurt is $2.00. The cafeteria collected $273.50.
Determine the number of muffins and the number
of individual yogurts sold.

A rectangular parking pad for a car has a perimeter
of 12.2 m. The width is 0.7 m shorter than the
length. Use a linear system to determine the
dimensions of the pad.
Page 120
UNIT PLANS
Section 7.6 – Properties of Systems of Linear Equations (pp. 442-449)
ELABORATIONS &
SUGGESTED PROBLEMS


RF9 (A F)

determine the number of solutions of a system of
linear equations graphically

situation

determine the number of solutions of a system of
linear equations using the slopes and y-intercepts
of the lines



x  2y  4
infinite – having no boundaries or limits

coincident lines – two lines that appear in the
same location; lines that coincide
y 
y
y  3x  1
y  2x  1
c.

2
x 1
3
Four vehicles travel in the same direction on a
long, straight stretch of highway. Their current
distances and speeds are shown in the following
table.
CURRENT
DISTANCE
(km)
CURRENT
SPEED
(km/h)
Car
40
90
Minivan
25
90
Truck
30
110
RV
40
90

1
x4
2
6y  4 x  6
b.
write a second equation to create linear systems
with different numbers of solutions, given the first
equation

Predict the number of solutions for each system of
linear equations.
a.
explain why a system of linear equations may have
no solution, one solution or infinite solutions
following concepts:
Remind students that equations should be written
in slope-intercept form when determining how
many solutions there are to a system of linear
equations.
pp. 447-449: #1-21
For each vehicle, represent the distance-time
relationship using a linear equation. Suppose the
vehicles continue at their current speeds. Identify
and interpret the solution to the linear system
represented by each pair of vehicles.
a.

the car and the minivan
b.
the car and the RV
c.
the truck and the RV
Determine whether each linear system has infinite
solutions or no solution. Explain your reasoning.
a.
2 x  10 y  16  0
x  5y  8  0
b.
x  2y  4  0
x  2y  6  0
Page 121
Page 122
GLOSSARY OF MATHEMATICAL TERMS

A

angle of inclination – the acute angle between
the horizontal and a line or line segment
acute angle – an angle measuring less than 900
angle of inclination

acute triangle – a triangle with three acute
angles

apex – the vertex farthest from the base of an
object
apex

adjacent side – the leg of a right triangle that is
next to both the reference angle and the right
angle

adjacent

algebraic expression – a mathematical
expression containing a variable; for example,
6x  4

angle of depression – the angle between the
a point below eye level

approximate – a number close to the exact
value of an expression; the symbol  means “is
approximately equal to”

area – a measure of the number of square units
needed to cover a region

arithmetic operations – the operations of
addition, subtraction, multiplication and division

arrow diagram – used to represent a relation;
the ovals show the sets, and the arrows
associate elements of the first set with the
elements of the second set
f
1
angle of
depression
2
A
3
B
4
angle of
elevation

angle of elevation – the angle between the
a point above eye level

ascending order – writing a polynomial such
that its terms are written with the smallest degree
first and the largest degree last

average – a single number that represents a set
of numbers; see also mean
Page 123
B


circumference – the distance around a circle,
also the perimeter of the circle

clinometer – a tool used to measure an angle
above or below the horizontal

coefficient – the numerical factor of a term; for
bar graph – a graph that displays data by using
horizontal or vertical bars
Average Mark
%
80
40
Boys

Girls
bar notation – the use of a horizontal bar over a
decimal digit to indicate that it repeats; for
example, 1.3 means 1.333333...

example, in the terms 3x and 3 x 2 , the coefficient
is 3

coincident lines – two lines that appear in the
same location; lines that coincide
base – the side of a polygon or the face of an
object from which the height is measured
y
x
base

base of a power – see power

binomial – a polynomial with two terms; for
example, 3 x  8

C

caliper – a tool used to measure the diameter or
thickness of an object

capacity – the amount a container can hold

central angle – an angle whose arms are radii of
a circle
8 x 2 y  4 xy  12y

common multiple – a number that is a multiple
of each number in a set; for example, 6 is a
common multiple of 2 and 3

composite number – a number with three or
more factors; for example, 8 is a composite
number because its factors are 1, 2, 4 and 8

composite object – the result of combining two
or more objects to make a new object

cone – see right cone
central angle
common factor – a number that divides into
each number in a set; for example, 3 is a
common factor of 9, 15, and 21; an expression
that divides into each term of a given polynomial;
for example, 4y is a common factor of
Page 124

congruent – shapes that match exactly, but do
not necessarily have the same orientation

cosine ratio – for an acute angle A in a right
triangle, the ratio of the length of the side
adjacent to angle A to the length of the
hypotenuse; written cos A
cos A 

consecutive numbers – integers that come one
after the other without any integers missing; for
example, 34, 35, 36 are consecutive numbers, so
are –2, –1, 0, 1

constant term – the term in an expression or
equation that does not change; for example, in
the expression 4 x  3, 3 is the constant term

cube – an object with six congruent square faces

cube number – a number that can be written as
a power with an integer base and exponent 3; for
example, 8  23

conversion factor – a number used to multiply
or divide a quantity to convert from one unit of
measure to another

cube root – a number which, when raised to the
exponent 3, results in a given number; for
example, 5 is the cube root of 125

coordinate axes – the horizontal and vertical
axes on a grid

cubic units – units that measure volume
y
D
vertical

decagon – a polygon with 10 sides

denominator – the term below the line in a
fraction

dependent variable – a variable whose value is
determined by the value of another (the
independent) variable

descending order – writing a polynomial such
that its terms are written with the largest degree
first and the smallest degree last

diagonal – a line segment that joins two vertices
of a shape, but is not a side
x
horizontal

coordinates – the numbers in an ordered pair
that locate a point on a coordinate grid; see also
ordered pair, x-coordinate and y-coordinate

corresponding angles – matching angles in
similar polygons; in the diagram below, the two
labelled angles are corresponding angles
a
b

corresponding lengths – matching lengths on
an original diagram and its scale diagram; in the
diagram above, a and b are corresponding
lengths

corresponding sides – matching sides of similar
polygons; in the diagram above, a and b are
corresponding sides
diagonal
Page 125

diameter – the distance across a circle or a
sphere, measured through its centre; or the line
segment that joins two points on the circle or the
sphere and passes through its centre

equilateral triangle – a triangle with three equal
sides

equivalent – having the same value; for
1
2
and , 3 : 4 and 9 : 12
example,
2
4

equivalent linear system – a linear system that
has the same solution as another linear system

estimate – a reasoned guess that is close to the
actual value, without calculating it exactly
diameter

difference of squares – a binomial of the form
a 2  b 2 ; it can be factored as
a 2  b 2   a  b  a  b 

digit – any of the symbols used to write
numerals; the base-ten digits are 0, 1, 2, 3, 4, 5,
6, 7, 8 and 9

dimensions – measurements such as length,
width and height

evaluate – to determine the value of a numerical
expression

direct measurement – a measurement made
using a measuring instrument or by counting

even integer – a number that has 2 as a factor;
for example, 2, 4, 6

displacement – the volume of water moved or
displaced by an object put in the water; the
volume of the object is equal to the volume of
water displaced

expanding an expression – writing a product of
polynomial factors as a polynomial

exponent – see power

exponent laws – the rules that describe how
combinations of powers can be written differently

expression – a mathematical statement made
up of numbers and/or variables connected by
operations

distributive property – the property stating that
a product can be written as a sum or difference of

divisor – the number that divides into another
number

domain – the set of first elements of a relation

E

F
face – a flat surface of an object
edge – two faces of an object meet at an edge
face
edge



element – an element of a set is one object in
the set
entire radical – a radical sign and the number
under it; for example,

factor – to factor means to write as a product; for
example, 20  2  2  5
5
32
equation – a mathematical statement that two
expressions are equal
Page 126

H
factor tree – a branching diagram with a number
at the top and its prime factors at the bottom


factored fully – factoring a polynomial so each
factor cannot be factored further

factoring a polynomial – writing a polynomial as
a product of its factors

factoring by decomposition – factoring a
trinomial after writing the middle term as the sum
of two terms, then determining a common
binomial factor from the two pairs of terms
formed

factors – numbers or algebraic expressions that
are multiplied together to get a product; for
example, 3 and 7 are factors of 21, and x  1 and
height – the perpendicular distance from the
base of a shape to the opposite side or vertex;
the perpendicular distance from the base of an
object to the opposite face or vertex
height

hemisphere – one half of a sphere

hexagon – a polygon with 6 sides

horizontal axis – see x-axis

horizontal intercept – see x-intercept

horizontal line – a line parallel to the horizon

hypotenuse – the side opposite the right angle in
a right triangle
x  2 are factors of x 2  3 x  2

formula – a rule that is expressed as an equation

fraction – an indicated quotient of two quantities

function – a relation where each element in the
first set is associated with exactly one element in
the second set

function notation – notation used to show the
independent variable in a function; for example,
f  x  means that the value of the function f
depends on the value of the independent variable
x
G

general form – the equation of a line in the form
Ax  By  C  0, where A, B and C are integers,
and A and B are not both zero; by convention, A
is a whole number

greatest common factor (GCF) – the greatest
number that divides into each number in a set; for
example, 5 is the greatest common factor of 10
and 15
hypotenuse
Page 127
I

imperial units – measurement units such as the
mile, yard, foot and inch, commonly used in the
United States and in some industries in Canada

independent variable – a variable whose value
is not determined by the value of another
variable, and whose value determines the value
of another (the dependent) variable

index – in a radical, the number above the
radical symbol that indicates which root is to be
taken; for example, 4 is the index in the radical
4
81; if the index is not written, it is assumed to
be 2

indirect measurement – a measurement made
using a ratio, formula or other mathematical
reasoning

infinite – having no boundaries or limits

integers – the set of numbers ..., –3, –2, –1, 0, 1,
2, 3, ...

inverse operation – an operation that reverses
the result of another operation; for example,
subtraction is the inverse of addition, and division
is the inverse of multiplication

irrational number – any number that cannot be
m
written in the form
, where m and n are
n
integers, and n  0


isosceles triangle – a triangle with two equal
sides
K

kite – a quadrilateral with two pairs of adjacent
sides equal
L

lateral area – the surface area of an object, not
including the area of its bases

least common multiple (LCM) – the smallest
number that is a multiple of each number in a set;
for example, the least common multiple of 12 and
21 is 84

legs – the sides of a right triangle that form the
right angle; see also hypotenuse
isometric – equal measure; on isometric dot
paper, the line segments joining two adjacent
dots in any direction are equal
leg

isosceles trapezoid – a trapezoid with two
equal, non-parallel sides
leg

like terms – terms that have the same variables
raised to the same powers; for example, 4x and
–3x are like terms

line segment – the part of a line between two
points on the line
Page 128

linear function – a linear relation whose graph is
not a vertical line

linear relation – a relation that has a straight-line
graph

linear system – see system of linear
equations
M

mass – the amount of matter in an object

mean – the sum of a set of numbers divided by
the number of numbers in the set

midpoint – the point that divides a line segment
into two equal parts

oblique line – a line that is neither horizontal nor
vertical

obtuse triangle – a triangle with one angle
greater than 900

octagon – a polygon with 8 sides

operation – a mathematical process such as
addition, subtraction, multiplication, division or
raising to a power

opposite side – the leg of a right triangle that is
directly across from the reference angle
midpoint

mixed radical – a number written as a product of
another number and a radical; for example, 3 5

monomial – a polynomial with one term; for
example, 14 and 5x 2 are monomials

multiple – the product of a given number and a
natural number; for example, some multiples of 8
are 8, 16, 24, ...
N

natural numbers – the set of numbers 1, 2, 3, 4,
5, ...

negative exponent – an exponent that is a
negative number

negative number – a number less than 0

negative reciprocals – two numbers whose
3
7
product is –1; for example,  and
are
7
3
negative reciprocals

numerator – the term above the line in a fraction

numerical coefficient – see coefficient
opposite


opposites – two numbers with a sum of 0; for
example, 2.4 and –2.4 are opposite numbers

order of operations – the correct sequence of
steps for a calculation: Brackets, Exponents,
Divide and Multiply in order from left to right, Add
and Subtract in order from left to right; the
acronym BEDMAS is often used to remember the
order of operations

ordered pair – two numbers in order, for
example, (2,4); on a coordinate grid, the first
number is the horizontal coordinate of a point and
the second number is the vertical coordinate of
the point
O

object – a solid or shell that has three
dimensions
Page 129

origin – the point where the horizontal and the
vertical axes meet
y

perfect square trinomial – a trinomial of the
form a 2  2ab  b 2 ; it can be factored as
a 2  2ab  b 2   a  b 
origin
x
2

perimeter – the distance around a closed shape

perpendicular lines – lines that intersect at right
angles
right angle
P

parallel lines – lines on the same flat surface
that do not intersect

pi ( ) – the ratio of the circumference of a circle
to its diameter


circumference
diameter
point of intersection – the point where two
graphs intersect
y

parallelogram – a quadrilateral with opposite
sides parallel and opposite angles equal
point of
intersection


polygon – a closed shape that consists of line
segments; for example, triangles and
quadrilaterals are polygons

polyhedron – an object with faces that are
polygons; the plural is polyhedra

polynomial – one term or the sum of terms
whose variables have whole-number exponents;
pentagon – a polygon with 5 sides

percent – the number of parts per 100; the
numerator of a fraction with denominator 100

perfect cube – a number that can be written as a
power with an integer base and exponent 3; for
for example, x 2  3 xy  2y 2  5 x
example, 8  23

perfect square – a number that can be written
as a power with an integer base and exponent 2;
for example, 49  72
x

power – an expression of the form a n , where a
is the base and n is the exponent; when n is a
natural number, it represents a product of equal
factors; for example, 4  4  4 can be written as
43
Page 130

primary trigonometric ratios – three ratios
involving sides in right triangles (cosine, sine and
tangent)

prime factor – a prime number that is a factor of
a number; for example, 5 is a prime factor of 30

prime factorization – writing a number as a
product of its prime factors, for example, the
Q

quadrilateral – a polygon with 4 sides

quotient – the result when one number is divided
by another; the expression of one number divided
by another
prime factorization of 20 is 2  2  5 or 22  5

prime number – a whole number with exactly
two factors; for example, 2, 3, 5, 7, 11, 29, 31 and
43

prism – an object with two bases; see also right
prism
R

radical – an expression consisting of the radical
sign,


product – the result when two or more numbers
are multiplied together; the expression of one
number multiplied by another

proportion – a statement that two ratios are
equal; for example, r : 24  3 : 4

proportional reasoning – the ability to
understand and compare quantities that are
related multiplicatively

Pythagorean theorem – the rule that states that,
for any right triangle, the area of the square on
the hypotenuse is equal to the sum of the areas
of the squares on the legs; from the diagram,
a b c
2
2

b2
81
radius – the distance or line segment from the
centre of a circle or a sphere to any point on the
circle or the sphere, respectively; the plural is
radii
radius

range – the set of second elements associated
with the first elements (domain) of a relation

rate – a comparison of two quantities measured
in different units

rate of change – the change in one quantity with
respect to the change in another quantity; see
also slope

ratio – a comparison of two or more quantities
with the same unit

rational exponent – an exponent that is a
rational number

rational number – any number that can be
m
written in the form
, where m and n are
n
integers, and n  0

real number – any number that is a rational
number or an irrational number; a member of the
set of numbers that have a decimal
representation
a2
81
radicand – the number under a radical sign; for
example, 81 is the radicand in
2
c2
, and a radicand; for example,
Page 131

reciprocals – two numbers whose product is 1;
2
3
for example,
and
3
2

rectangle – a quadrilateral that has four right
angles

rectangular prism – see right rectangular
prism

rectangular pyramid – see right rectangular
pyramid

referent – used to estimate a measure; for
example, a referent for a length of 1 mm is the
thickness of a dime

regular polygon – a polygon that has all sides
equal and all angles equal

regular pyramid – a pyramid with a regular
polygon as a base

regular tetrahedron – an object with four
congruent equilateral triangular faces; a regular
triangular pyramid

relation – a rule that associates the elements
from one set with the elements of another set

repeating decimal – a decimal with a repeating
pattern in the digits to the right of the decimal
point; it is written with a bar above the repeating
digits; for example, 0.3  0.333333...


regular polyhedron – a polyhedron with
congruent faces, each of which is a regular
polygon

rhombus – a parallelogram with four equal sides

right angle – a 900 angle

right cone – an object with one circular base and
one vertex; the line through the vertex and the
centre of the base is perpendicular to the base
regular prism – a prism with regular polygons as
bases; for example, a cube
Page 132


right cylinder – an object with two parallel,
congruent, circular bases; the line through the
centres of the bases is perpendicular to the
bases
right prism – an object that has two congruent
and parallel faces (the bases), and other faces
that are rectangles

right triangle – a triangle that has one right
angle

rise – the vertical distance between two points;
see also slope

root – the inverse of a power; for example, in
23  8 or

3
8  2, 2 is the cube root of 8
run – the horizontal distance between two points;
see also slope
S



right pyramid – an object that has one face that
is a polygon (the base) and other faces that are
triangles with a common vertex; the line through
the vertex and the centre of the base is
perpendicular to the base
right rectangular prism – a prism that has
rectangular faces

scale – the numbers on the axes of a graph

scale diagram – a diagram that is an
enlargement or a reduction of another diagram

scale factor – the ratio of corresponding lengths
of two similar shapes

set – a collection of distinct objects

set of ordered pairs – a collection of ordered
pairs

SI system of measures – a system of units
based on powers of 10; the fundamental unit of
length is the metre (m), of mass is the kilogram
(kg), and of time is the second (s)

similar polygons – polygons with the same
shape; one polygon is an enlargement or a
reduction of the other polygon

simplest form – a ratio with terms that have no
common factors, other than 1; a fraction with
numerator and denominator that have no
common factors, other than 1
right rectangular pyramid – a pyramid that has
a rectangular base; the line through the vertex
and the centre of the base is perpendicular to the
base
Page 133

sine ratio – for an acute angle A in a right
opposite angle A to the length of the hypotenuse;
written sin A
sin A 

square – a rectangle with four equal sides

square number – a number that can be written
as a power with an integer base and exponent 2;
slant height – the distance from a point on the
perimeter of the base of a cone to the apex of the
cone; the distance from the midpoint of the base
of one triangular face of a rectangular pyramid to
the apex of the pyramid; in this diagram, s1 and
s2 represent the slant heights
s1
s2


slope – a measure of how one quantity changes
with respect to the other; it can be determined by
rise
calculating
run

slope-intercept form – the equation of a line in
the form y  mx  b, where m is the slope of the
line and b is its y-intercept

slope-point form – the equation of a line in the
form y  y1  m  x  x1  , where m is the slope of
for example, 49  72

square root – a number which, when multiplied
by itself, results in a given number; for example,
5 is a square root of 25

square units – units that measure area

standard form – the equation of a line in the
form Ax  By  C, where A, B and C are
integers, and A and B are not both zero; by
convention, A is a whole number

substituting into an equation – in an equation
of a linear function, replacing one variable with a
number or an expression

surface area – the total area of the surface of an
object

system of linear equations – two equations of
linear functions in the same two variables; can
also be called a linear system
the line, and the line passes through the point
P  x1, y1 

solving a triangle – determining the measure of
each angle in a triangle and the length of each
side of the triangle

solving by elimination – adding or subtracting
the equations to eliminate one variable and then
solving for the other variable


solving by substitution – solving one equation
for one variable, substituting that expression into
the other equation, and then solving for the
remaining variable
sphere – an object where every point on the
surface of the object is the same distance from
the centre of the object
T

tangent ratio – for an acute angle A in a right
opposite angle A to the length of the side
adjacent to angle A; written tan A
tan A 

term – a number, a variable, or the product of
numbers and variables; for example, –5, y, 7a 2

terminating decimal – a decimal with a certain
number of digits after the decimal point; for
example, 0.125

tetrahedron – a pyramid that has a triangular
base

three-dimensional – having length, width, and
height or depth
Page 134

trapezoid – a quadrilateral with exactly one pair
of parallel sides

triangle – a polygon with three sides

triangular prism – a prism with triangular bases

vertical line – a line perpendicular to the
horizontal

vertical line test for a function – a graph
represents a function when no two points on the
graph lie on the same vertical line

volume – the amount of space occupied by an
object
W


trigonometry – the study of the properties and
applications of triangles

trinomial – a polynomial with three terms; for
example, 3 x 2  5 x  8
X

U

whole numbers – the set of numbers 0, 1, 2, 3,
4, ...
x-axis – the horizontal number line on a
coordinate grid
y-axis
unit analysis – a method of converting a
measure in a given unit to a measure in a
different unit by multiplying the measure by a
conversion factor
x-axis
V

variable – a letter or symbol representing a
quantity that can vary

vertex – the point where two sides of a shape
meet; the point where three or more edges of an
object meet; the plural is vertices

x-coordinate – on a coordinate grid, the first
number in an ordered pair

x-intercept – the x-coordinate of a point where a
graph intersects the x-axis
Y

y-axis – the vertical number line on a coordinate
grid
y-axis
vertex

vertical axis – see y-axis

vertical intercept – the y-intercept; the ycoordinate of a point where a graph intercepts the
y-axis
x-axis
Page 135

y-coordinate – on a coordinate grid, the second
number in an ordered pair

y-intercept – the y-coordinate of a point where a
graph intersects the y-axis
Z

zero principle – the property of addition that
states that adding 0 to a number does not
change the number; for example, 3  0  3
Page 136
SOLUTIONS TO POSSIBLE ASSESSMENT STRATEGIES
SECTION 1.1




111.5 in2
13
3
ft
4
SECTION 1.3

8 ft

56 mph

27
0.5095
450
1
0
SECTION 1.4

2
502.7 cm
57
1.5399

tan 
18.40
0.3329
0
29.0
0.5543
55.70
1.4653
2
270 cm
SECTION 1.5

65.00
SECTION 2.2

256 cm3

a.
6911.5 cm3

3.9 m
b.
3

240.3 m

2303.8 cm
5408 m3
SECTION 2.4
SECTION 1.6
2

2827.4 cm

7238.2 cm3

SA  864 cm2 ; V  1728 cm3

5.0 m

9.8 cm

SECTION 1.7


a.
876 cm2
b.
inner radius  8.9 cm;
outer radius  16.9 cm



3
111.3 cm
a.
5
13
b.
12
13
c.
12
13
d.
5
13
a.
0.8660
b.
0.5
c.
0.7071
a.
26.00
78.00
b.
2
32 m

0
45.1
2

364.4 cm

5236 m3
SECTION 2.5
SECTION 2.1

tan 
0
a.
5
12
b.
12
5

23.7 cm

423.8 m

46.4 m
SECTION 2.6

A  480 , a  16.3, b  14.7
Page 137

E  42.40 , F  47.60 , e  28.3

25.0 km

84.9 m

a.
SECTION 2.7

722.9 m

150.9 m

6 cm

439 m
The dimensions are 2x and x  2, the area is
2 x 2  4 x; 2 x 2  4 x  2 x  x  2 
SECTION 3.1

25  3  52

48

800

60 cm
b.
SECTION 3.2

68
The dimensions are 3 and x  2, the area is

42
1176 cm2
3 x  6; 3 x  6  3  x  2 


24 km; 71.6 km

One possible solution is 6 x 3 y 3  4 x 2 y 2  2 xy ;

SECTION 3.3


a.
5mn
b.
12ab 2c
a.
3  x  1
b.
5  x  2
c.
a a  3
d.
3a  a  4 
e.
4 xy  y  2 x 
f.
9rs 2 3r  2r 2s  4s

Answers may vary.

2 xy 3 x 2 y 2  2 xy  1  6 x 3 y 3  4 x 2 y 2  2 xy
SECTION 3.5


b.


a.
a.
x 2  8 x  15
b.
m 2  2m  24
c.
n 2  100
a.
x  2 by x  4
b.
no
Page 138
c.

4 x  12

Since it not possible to find two factors of 1 that
have a sum of 3, we cannot create a rectangle
with an area of x 2  3 x  1. Therefore, this
polynomial cannot be factored.


 x  3  2x  5 
b.
 x  5  4 x  3 
c.
 x  5  2x  3 
a.
 x  6  x  1
d.
 x  2  5 x  3 
b.
 x  3  x  3 
e.
 2 x  3  3 x  1
c.
 x  1 x  3 
f.
 x  2 3 x  1
d.
 x  5  x  2

k  7,  8,  13
SECTION 3.7
k  10,  11,  14  25
SECTION 3.6

a.

a.
a.
6 x 3  x 2  15 x  4
b.
3r 3  4r 2  38r  24
c.
10 x 3  36 x 2  78 x  36
d.
8 x 3  12 x 2 y  6 xy 2  y 3
e.
13 x 2  25 x  26
f.
8 x 2  12 x  31
SECTION 3.8


b.


a.
4x 2  7x  2
b.
2 x 2  13 x  6
c.
10m 2  32m  6
a.
x2  9
b.
x 2  25
c.
4x 2  4x  1
d.
16 x 2  8 xy  y 2
a.
 2x  5 
b.
 x  12y 
c.
 y  9
d.
3 b  4
e.
4  x  2  x  2 
f.
 7a  5b  7a  5b 
g.
5  5 x  4 y  5 x  4 y 
h.
 3 pq  8  3 pq  8 
2
2
2
2
18 x  45
Page 139
SECTION 4.2

5
3
4
–2
Natural
Numbers
X
Whole
Numbers
X
Integers
X
X
Rational
Numbers
X
X
–1.3


X
X
x3
a.
53/ 2
b.
y 5/3
c.
272/ n
a.
8
b.
4
c.
4
9
SECTION 4.5
X

Real
Numbers
X
X
X
Rational numbers:
1,
Irrational numbers:
10,
11,
12,
13,
X
4,
2,
9,
3,
14,
5,
15,
X
X
16
6,
7,
17,
18,
8,
a.
16
b.
7
c.
145
d.
256
e.
2
3
f.
1
16
19,
20
The rational numbers have radicands that are
perfect squares and the irrational numbers have
radicands that are not perfect squares.



9.5, 3.141, ,
22
,
7
a.
always true
b.
sometimes true
c.
never true
a.
<
b.
<
c.
>
10

a.
2 6
b.
6 2
c.
4 6
d.
3 5
a.
275
b.
40
c.
108
SECTION 4.6


SECTION 4.3

1024
c.
X
Irrational
Numbers

0
7
X
X
3
b.

a.
x5
b.
1
p3 / 4
c.
1.57 / 6
a.
9x 4
b.
t 15
c.
16
x2
15
x 19
SECTION 5.1

a.
In 2006, the population of Charlottetown was
32,200, the population of Summerside was
14,500, the population of Stratford was
7,100, and the population of Cornwall was
4,700.
b.
{(Charlottetown, 32,200),
(Summerside, 14,500), (Stratford, 7,100),
(Cornwall, 4,700)}
SECTION 4.4

a.
5
73
Page 140

c.
a.
Population
CH
SU
ST
CO
32,200
14,500
7,100
4,700
b.
SECTION 5.2


a.
This is not a function, because the domain
value 2 has two range values, 1 and 2.
b.
This is a function, because every domain
value has exactly one range value.
a.
function
c.
D  {1, 2, 3, 4}; R  {1, 8, 27, 64}
b.
not a function
D  {3}; R  {4, 5, 6, 7}


a.
770F
b.
200C
f  2   3; (2,–3)

SECTION 5.3


The boat is starting from rest and accelerating to
its maximum speed. Once it’s at its maximum
speed, it travels at that speed until the water-skier
falls. At that point, the boat slows down until it
stops to pick up the water-skier. Once the waterskier is ready, the boat repeats this process.
However, it takes a bit longer to slow down the
second time than the first time.
At the surface of the water, the pressure is
100 kPa. The deeper an object is submerged in
the water, the more pressure is exerted on it.
SECTION 5.5


D  {1  x  3}; R  { 2  y  2}
a.
function
b.
not a function
c.
not a function
SECTION 5.6
The fourth graph is the best choice, since a
person’s height changes gradually, not suddenly.


Distance
a.
linear
b.
linear
c.
non-linear
d.
linear
e.
non-linear
0
Time
Page 141

a.
b.
b.
c.
C  4d  20
d.
$52
SECTION 5.7


a.
$400
b.
$900
c.
$1400
d.
$5
a.
c.
b.

a.
Days (d)
Cost (C)
1
24
2
28
3
32
4
36
5
40
Page 142
c.


b.
b  840; The y-intercept represents the
cost of renting the hall. The value is
negative because it is the amount that must
be paid to rent the hall, which is subtracted
from the revenue.
c.
70 tickets
a.
$150
b.
11 h
a.
neither
b.
perpendicular
SECTION 6.1


1
2
a.

b.
undefined
c.
0
d.
–1
mAB 
3
1
, mAC  4, mBC  
4
5

k  1

Answers may vary. Possible solutions are (0,1),
(2,2), (8,5).
1
Note: Any point on the line y  x  1 is a
2
solution.
SECTION 6.2


a.
parallel
b.
neither
c.
perpendicular
a.
parallel
b.
neither
c.
perpendicular
SECTION 6.5

y  2 x  10

a.
y  2 x  3
b.
y 
1
14
x
5
5

y  0.4 x  25.2

The fixed insurance premium is $10, and the
hourly rate is $20/h.

y  3 x  9

y 
1
3
x
2
2
SECTION 6.6

3 x  4y  8  0

a.
(5,0)
b.
(0,4)
c.
SECTION 6.4


a.
m  2, b  5
b.
m  0, b  2
c.
m
d.
1
m , b2
3
a.
1
, b  3
5
m  12; The slope represents the price per
ticket.

a.
12S  16T  336
b.
(28,0); The S-intercept represents how
many hours she would have to work as a
snowboard instructor if she did no work as a
high school tutor.
c.
(0,21); The T-intercept represents how
many hours she would have to work as a
high school tutor if she did no work as a
snowboard instructor.
Page 143
d.
15 h
SECTION 7.1

y  10 x  40
y  8 x  50

x  3 y  3  3  2   3  6  9

There were 93 muffins and 67 individual yogurts
sold.

2.7 m by 3.4 m
SECTION 7.6

2 x  y  2  3    2   6  2  4
a.
b.
infinite solutions
c.
one solution

Car: y  90 x  40
SECTION 7.2

Minivan: y  90 x  25
a.
(7,–1)
Truck: y  110 x  30
b.
(6,2)
RV: y  90 x  40
SECTION 7.4

a.
(4,3)
b.
(4,–3)
c.
no solution
d.
(5,0)
e.
(4,5)

$11,600 at 7% and $30,400 at 9.5%

32 m and 50 m

Rory is 16 and his grandmother is 74

a.
This system of equations has no solution,
which means that the car and the minivan
will never meet.
b.
This system of equations has infinite
solutions, which means that the car and the
RV are travelling together
c.
This system of equations has a solution of
(0.5,85), which means that the truck will pass
1
hour at a distance of 85 km.
the RV in
2
a.
Since the first equation is equal to the
second equation multiplied by 2, the lines
are coincident, so there will be infinite
solutions.
b.
If we rewrite each equation in slope-intercept
form, it is easy to see that the lines are
1
parallel, since m   for each equation.
2
Therefore, there will be no solution.
SECTION 7.5


no solution
a.
(–2,6)
b.
(4,4)
c.
no solution
A single adult ticket cost $14 and a single student
ticket cost $9.
Page 144
MATHEMATICS RESEARCH PROJECT
APPENDIX
Page 145
 Introduction
A research project can be a very important part of a mathematics education. Besides the greatly
increased learning intensity that comes from personal involvement with a project, and the chance to show
universities, colleges, and potential employers the ability to initiate and carry out a complex task, it gives
the student an introduction to mathematics as it is – a living and developing intellectual discipline where
progress is achieved by the interplay of individual creativity and collective knowledge. A major research
project must successfully pass through several stages. Over the next few pages, these stages are
highlighted, together with some ideas that students may find useful when working through such a project.
 Creating an Action Plan
As previously mentioned, a major research project must successfully pass though several stages. The
following is an outline for an action plan, with a list of these stages, a suggested time, and space for
students to include a probable time to complete each stage.
STAGE
SUGGESTED TIME
Select the topic to explore.
1 – 3 days
Create the research question to
be answered.
1 – 3 days
Collect the data.
5 – 10 days
Analyse the data.
5 – 10 days
Create an outline for the
presentation.
2 – 4 days
Prepare a first draft.
3 – 10 days
Revise, edit and proofread.
3 – 5 days
Prepare and practise the
presentation.
3 – 5 days
PROBABLE TIME
Completing this action plan will help students organize their time and give them goals and deadlines that
they can manage. The times that are suggested for each stage are only a guide. Students can adjust the
time that they will spend on each stage to match the scope of their projects. For example, a project
based on primary data (data that they collect) will usually require more time than a project based on
secondary data (data that other people have collected and published). A student will also need to
consider his or her personal situation – the issues that each student deals with that may interfere with the
completion of his or her project. Examples of these issues may include:
 a part-time job;
 after-school sports and activities;
 regular homework;
 assignments for other courses;
 tests in other courses;
 time they spend with friends;
 family commitments;
 access to research sources and technology.
Page 146
 Selecting the Research Topic
To decide what to research, a student can start by thinking about a subject and then consider specific
topics. Some examples of subjects and topics may be:
SUBJECT
TOPIC

effects of new electronic
devices

file sharing

doctor and/or nurse
shortages

funding

entry requirements

graduate success

relations among First Nations

immigration
Entertainment
Health care
Post-secondary education
History of Western and Northern
Canada
It is important to take the time to consider several topics carefully before selecting a topic for a research
project. The following questions will help a student determine if a topic that is being considered is
suitable.

Does the topic interest the student?
Students will be more successful they choose a topic that interests them. They will be
more motivated to do research, and they will be more attentive while doing the research.
As well, they will care more about the conclusions they make.

Is the topic practical to research?
If a student decides to use first-hand data, can the data be generated in the time
available, with the resources available? If a student decides to use second-hand data,
are there multiple sources of data? Are the sources reliable, and can they be accessed
in a timely manner?

Is there an important issue related to the topic?
A student should think about the issues related to the topic that he or she has chosen. If
there are many viewpoints on an issue, they may be able to gather data that support
some viewpoints but not others. The data they collect from all viewpoints should enable
them to come to a reasoned conclusion.

Will the audience appreciate the presentation?
The topic should be interesting to the intended audience. Students should avoid topics
that may offend some members of their audience.
 Creating the Research Question or Statement
A well-written research question or statement clarifies exactly what the project is designed to do. It
should have the following characteristics:

The research topic is easily identifiable.
Page 147

The purpose of the research is clear.

The question or statement is focused. The people who are listening to or reading the
question or statement will know what the student is going to be researching.
A good question or statement requires thought and planning. Below are three examples of initial
questions or statements and how they were improved.
UNACCEPTABLE QUESTION
OR STATEMENT
WHY?
ACCEPTABLE QUESTION
OR STATEMENT
Too general
What role has mathematics
played in the development of
computer animation?
Water is a shared resource.
Too general
Homes, farms, ranches, and
businesses east of the Rockies
all use runoff water. When there
is a shortage, that water must be
shared.
Do driver’s education programs
help teenagers parallel park?
Too specific, unless the student
is generating his or her own data
Do driver’s education programs
reduce the incidence of parking
accidents?
Is mathematics used in computer
technology?
The following checklist can be used to determine if the research question or statement is effective.

Does the question or statement clearly identify the main objective of the research? After
the question or statement is read to someone, can they tell what the student will be
researching?

Is the student confident that the question or statement will lead him or her to sufficient
data to reach a conclusion?

Is the question or statement interesting? Does it make the student want to learn more?

Is the topic that the student chose purely factual, or is that student likely to encounter an
issue, with different points of view?
 Carrying Out the Research
As students continue with their projects, they will need to conduct research and collect data. The
strategies that follow will help them in their data collection.
There are two types of data that students will need to consider – primary and secondary. Primary data is
data that the student collects himself or herself using surveys, interviews and direct observations.
Secondary data is data that the student obtains through other sources, such as online publications,
journals, magazines, and newspapers.
Both primary and secondary data have their advantages and disadvantages. Primary data provide
specific information about the research question or statement, but may take time to collect and process.
Secondary data is usually easier to obtain and can be analysed in less time. However, because the data
was originally gathered for other purposes, a student may need to sift through it to find exactly what he or
she is looking for.
Page 148
The type of data chosen can depend on many factors, including the research question, the skills of the
student, and available time and resources. Based on these and other factors, the student may chose to
use primary data, secondary data, or both.
When collecting primary data, the student must ensure the following:

For surveys, the sample size must be reasonably large and the random sampling
technique must be well designed.

For surveys and interviews, the questionnaires must be designed to avoid bias.

For experiments and studies, the data must be free from measurement bias.

The data must be compiled accurately.
When collecting secondary data, the student should explore a variety of resources, such as:

textbooks, and other reference books;

scientific and historical journals, and other expert publications;

the Internet;

library databases.
After collecting the secondary data, the student must ensure that the source of the data is reliable:

If the data is from a report, determine what the author’s credentials are, how recent the
data is, and whether other researchers have cited the same data.

Be aware that data collection is often funded by an organization with an interest in the
outcome or with an agenda that it is trying to further. Knowing which organization has
funded the data collection may help the student decide how reliable the data is, or what
type of bias may have influenced the collection or presentation of the data.

If the data is from the Internet, check it against the following criteria:
o
authority – the credentials of the author should be provided;
o
accuracy – the domain of the web address may help the student determine the
accuracy;
o
currency – the information is probably being accurately managed if pages on a
site are updated regularly and links are valid.
 Analysing the Data
Statistical tools can help a student analyse and interpret the data that is collected. Students need to think
carefully about which statistical tools to use and when to use them, because other people will be
scrutinizing the data. A summary of relevant tools follows.
Measures of central tendency will give information about which values are representative of the entire set
of data. Selecting which measure of central tendency (mean, median, or mode) to use depends on the
distribution of the data. As the researcher, the student must decide which measure most accurately
describes the tendencies of the population. The following criteria should be considered when deciding
upon which measure of central tendency best describes a set of data.

Outliers affect the mean the most. If the data includes outliers, the student should use
the median to avoid misrepresenting the data. If the student chooses to use the mean,
the outliers should be removed before calculating the mean.
Page 149

If the distribution of the data is not symmetrical, but instead strongly skewed, the median
may best represent the set of data.

If the distribution of the data is roughly symmetrical, the mean and median will be close,
so either may be appropriate to use.

If the data is not numeric (for example, colour), or if the frequency of the data is more
important than the values, use the mode.
Both the range and the standard deviation will give the student information about the distribution of data
in a set. The range of a set of data changes considerably because of outliers. The disadvantage of using
the range is that it does not show where most of the data in a set lies – it only shows the spread between
the highest and the lowest values. The range is an informative tool that can be used to supplement other
measures, such as standard deviation, but it is rarely used as the only measure of dispersion.
Standard deviation is the measure of dispersion that is most commonly used in statistical analysis when
the mean is used to calculate central tendency. It measures the spread relative to the mean for most of
the data in the set. Outliers can affect standard deviation significantly. Standard deviation is a very
useful measure of dispersion for symmetrical distributions with no outliers. Standard deviation helps with
comparing the spread of two sets of data that have approximately the same mean. For example, the set
of data with the smaller standard deviation has a narrower spread of measurement around the mean, and
therefore has comparatively fewer high or low scores, than a set of data with a higher standard deviation.
When working with several sets of data that approximate normal distributions, you can use z-scores to
compare the data values. A z-score table enables a student to find the area under a normal distribution
curve with a mean of zero and a standard deviation of one. To determine the z-score for any data value
in a set that is normally distributed, the formula z 
xx
can be used where x is any observed data
s
value in the set, x is the mean of the set, and is s is the standard deviation of the set.
When analysing the results of a survey, a student may need to interpret and explain the significance of
some additional statistics. Most surveys and polls draw their conclusions from a sample of a larger group.
The margin of error and the confidence level indicate how well a sample represents a larger group. For
example, a survey may have a margin of error of plus or minus 3% at a 95% level of confidence. This
means that if the survey were conducted 100 times, the data would be within 3 percent points above or
below the reported results in 95 of the 100 surveys.
The size of the sample that is used for a poll affects the margin of error. If a student is collecting data, he
or she must consider the size of the sample that is needed for a desired margin of error.
 Identifying Controversial Issues
While working on a research project, a student may uncover some issues on which people disagree. To
decide on how to present an issue fairly, he or she should consider some questions to ask as the
research proceeds.

What is the issue about?
The student should identify which type of controversy has been uncovered. Almost all
controversy revolves around one or more of the following:
o
Values – What should be? What is best?
o
Information – What is the truth? What is a reasonable interpretation?
Page 150
o

Concepts – What does this mean? What are the implications?
What positions are being taken on the issue?
The student should determine what is being said and whether there is reasonable
support for the claims being made. Some questions to ask are:

o
Would you like that done to you?
o
Is the claim based on a value that is generally shared?
o
Is there adequate information?
o
Are the claims in the information accurate?
o
Are those taking various positions on the issue all using the same term
definitions?
What is being assumed?
Faulty assumptions reduce legitimacy. The student can ask:

o
What are the assumptions behind an argument?
o
Is the position based on prejudice or an attitude contrary to universally held
human values, such as those set out in the United Nations Declaration of Human
Rights?
o
Is the person who is presenting a position or an opinion an insider or an outsider?
What are the interests of those taking positions?
The student should try to determine the motivations of those taking positions on the
issue. What are their reasons for taking their positions? The degree to which the parties
involved are acting in self-interest could affect the legitimacy of their opinions.
 The Final Product and Presentation
The final presentation should be more than just a factual written report of the information gathered from
the research. To make the most of the student’s hard work, he or she should select a format for the final
presentation that suits his or her strengths, as well as the topic.
To make the presentation interesting, a format should be chosen that suits the student’s style. Some
examples are:

a report on an experiment or an investigation;

a summary of a newspaper article or a case study;

a short story, musical performance, or play;

a web page;

a slide show, multimedia presentation, or video;

a debate;

an advertising campaign or pamphlet;

a demonstration or the teaching of a lesson.
Sometimes, it is also effective to give the audience an executive summary of the presentation. This is a
one-page summary of the presentation that includes the research question and the conclusions that were
made.
Page 151
Before giving the presentation, the student can use these questions to decide if the presentation will be
effective.

Did I define my topic well? What is the best way to define my topic?

Is my presentation focused? Will my classmates find it focused?

Did I organize my information effectively? Is it obvious that I am following a plan in my
presentation?

Am I satisfied with my presentation? What might make it more effective?

What unanswered questions might my audience have?
 Peer Critiquing of Research Projects
After the student has completed his or her research for the question or statement being studied, and the
report and presentation have been delivered, it is time to see and hear the research projects developed
by other students. However, rather than being a passive observer, the student should have an important
role – to provide feedback to his or her peers about their projects and presentations.
Critiquing a project does not involve commenting on what might have been or should have been. It
involves reacting to what is seen and heard. In particular, the student should pay attention to:

strengths and weaknesses of the presentation;

problems or concerns with the presentation.
While observing each presentation, students should consider the content, the organization, and the
delivery. They should take notes during the presentation, using the following rating scales as a guide.
These rating scales can also be used to assess the presentation.
Content
Shows a clear sense of audience and purpose.
1
2
3
4
5
Demonstrates a thorough understanding of the topic.
1
2
3
4
5
Clearly and concisely explains ideas.
1
2
3
4
5
Applies knowledge and skills developed in this course.
1
2
3
4
5
Justifies conclusions with sound reasoning.
1
2
3
4
5
Uses vocabulary, symbols and diagrams correctly.
1
2
3
4
5
Page 152
Organization
Presentation is clearly focused.
1
2
3
4
5
Engaging introduction includes the research question, clearly stated.
1
2
3
4
5
Key ideas and information are logically presented.
1
2
3
4
5
There are effective transitions between ideas and information.
1
2
3
4
5
Conclusion follows logically from the analysis and relates to the question.
1
2
3
4
5
Speaking voice is clear, relaxed, and audible.
1
2
3
4
5
Pacing is appropriate and effective for the allotted time.
1
2
3
4
5
Technology is used effectively.
1
2
3
4
5
Visuals and handouts are easily understood.
1
2
3
4
5
Responses to audience’s questions show a thorough understanding of
the topic.
1
2
3
4
5
Delivery
Page 153
Page 154
REFERENCES
REFERENCES
American Association for the Advancement of Science [AAAS-Benchmarks]. Benchmark for Science Literacy. New
York, NY: Oxford University Press, 1993.
Banks, James A. and Cherry A. McGee Banks. Multicultural Education: Issues and Perspectives. Boston: Allyn and
Bacon, 1993.
Black, Paul and Dylan Wiliam. “Inside the Black Box: Raising Standards Through Classroom Assessment.” Phi
Delta Kappan, 20, October 1998, pp.139-148.
Canavan-McGrath, Cathy, et. al. Foundations of Mathematics 11. Nelson Education, 2012.
British Columbia Ministry of Education. The Primary Program: A Framework for Teaching, 2000.
Davies, Anne. Making Classroom Assessment Work. British Columbia: Classroom Connections International, Inc.,
2000.
Hope, Jack A., et. al. Mental Math in the Primary Grades. Dale Seymour Publications, 1988.
National Council of Teachers of Mathematics. Mathematics Assessment: A Practical Handbook. Reston, VA: NCTM,
2001.
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA:
NCTM, 2000.
Rubenstein, Rheta N. Mental Mathematics Beyond the Middle School: Why? What? How? September 2001, Vol. 94,
Issue 6, p. 442.
Shaw, Jean M. and Mary Jo Puckett Cliatt. “Developing Measurement Sense.” In P.R. Trafton (ed.), New Directions
for Elementary School Mathematics (pp. 149–155). Reston, VA: NCTM, 1989.
Steen, Lynn Arthur (ed.). On the Shoulders of Giants – New Approaches to Numeracy. Washington, DC: National
Research Council, 1990.
Van de Walle, John A. and Louann H. Lovin. Teaching Student-Centered Mathematics, Grades 5-8. Boston:
Pearson Education, Inc. 2006.
Western and Northern Canadian Protocol. Common Curriculum Framework for 10-12 Mathematics, 2008.
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Mathematics MAT421A - Prince Edward Island

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