acknowledgments · the 2010-2011 mat421a pilot teachers: kirk cutcliffe, bethany macleod, colonel...

ACKNOWLEDGMENTS

PRINCE EDWARD ISLAND MAT421A MATHEMATICS CURRICULUM GUIDE i

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, Bethany MacLeod, Colonel Gray Senior High School Montague Regional High School

Alden DeRoche, Karen Malone, Colonel Gray Senior High School Montague Regional High School

Lianne Garland, Alan McAlduff, Montague Regional High School Westisle Composite High School

Tobey Kielly, Glenda McInnis, Colonel Gray Senior High School Montague Regional High School

Suzanne Lee, Heidi Morgan, Colonel Gray Senior High School Westisle Composite High School

Mary Ellen Lowther, Laura Rogers, Montague Regional High School Westisle Composite High School

Rob MacDonald, Scott Wicksted, Colonel Gray Senior High School Westisle Composite High School

The Western and Northern Canadian Protocol (WNCP) for Collaboration in Education

Alberta Education

TABLE OF CONTENTS

PRINCE EDWARD ISLAND MAT421A MATHEMATICS CURRICULUM GUIDE ii

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

BACKGROUND AND RATIONALE

PRINCE EDWARD ISLAND MAT421A MATHEMATICS CURRICULUM GUIDE Page 1

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 10-12 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

respond with critical awareness to various forms of the arts, and be able to express themselves through the arts;

assess social, cultural, economic, and environmental interdependence in a local and global context;

use the listening, viewing, speaking, and writing modes of language(s), and mathematical and scientific concepts and symbols, to think, learn, and communicate effectively;

continue to learn and to pursue an active, healthy lifestyle;

use the strategies and processes needed to solve a wide variety of problems, including those requiring language, and mathematical and scientific concepts;

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.

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

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

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.

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 TOPICS

10 11 12

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

GENERAL CURRICULUM OUTCOMES (GCOs)

SPECIFIC

CURRICULUM OUTCOMES (SCOs)

ACHIEVEMENT INDICATORS

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:

Integration of the mathematical processes within each topic is expected.

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.

MATHEMATICAL PROCESSES Communication, Connections, Reasoning, Mental Mathematics

and Estimation, Problem Solving, Technology, Visualization

NATURE OF

MATHEMATICS

Change Constancy Number Sense Patterns Relationships Spatial Sense Uncertainty



Problem solving, reasoning, and connections are vital to increasing mathematical fluency, and must be integrated throughout the program.

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:

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 critical-thinking 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.

K to 9

MAT611B Advanced

Mathematics and Calculus

MAT631A Apprenticeship and

Workplace Mathematics 12

MAT621A Foundations

of Mathematics 12

MAT621B

Pre-Calculus 12

GRADE 12

MAT801A Applied

Mathematics



MAT521A Foundations

of Mathematics 11

MAT521B

Pre-Calculus 11

MAT521E Pre-Calculus

Elective

GRADE 11



MAT421A Common Grade 10 Course

Foundations of Mathematics & Pre-Calculus

GRADE 10

Please note that MAT801A may be taken by any student in grade eleven or twelve from any pathway, as it is an open

course.



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

This pathway is designed to provide students with the mathematical understandings and critical-thinking 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

This pathway is designed to provide students with the mathematical understandings and critical-thinking 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

communicate in order to learn and express their understanding of mathematics; [Communications: C]

connect mathematical ideas to other concepts in mathematics, to everyday experiences, and to other disciplines; [Connections: CN]

demonstrate fluency with mental mathematics and estimation; [Mental Mathematics and Estimation: ME]

develop and apply new mathematical knowledge through problem solving; [Problem Solving: PS]

develop mathematical reasoning; [Reasoning: R]

select and use technologies as tools for learning and solving problems; [Technology: T]

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.



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)



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:

using estimation working backwards guessing and checking using a formula looking for a pattern using a graph, diagram, or flow chart making an organized list or table solving a simpler problem using a model 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

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;



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

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:

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.



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.

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.



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.

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:

Mathematics learning is an active and constructive process.

Learners are individuals who bring a wide range of prior knowledge and experiences, and who learn via various styles and at different rates.

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.

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.



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 open-ended 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



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.

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 portfolios work samples learning journals anecdotal records questioning conferences performance assessment teacher-made and other tests peer- and self-assessment.

This balanced approach for assessing mathematics development is illustrated in the diagram on the next page.



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;

Assessing Mathematics Development in a Balanced

Manner

Work Samples math journals portfolios drawings, charts, tables, and graphs individual and classroom assessment pencil and paper tests

Rubrics constructed response generic rubrics task-specific rubrics questioning

Surveys attitude interest parent questionnaires

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



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;



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.

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.

Measurement (M)

Develop spatial sense and proportional reasoning. (Foundations of Mathematics and Pre-Calculus)

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;

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.

SPECIFIC CURRICULUM OUTCOMES


MEASUREMENT




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.



MAT421A – Topic: Measurement (M) GCO: Develop spatial sense and proportional reasoning.

GRADE 9 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

SCO: M1 – Solve problems that involve linear measurement, using: SI and imperial units of measure; estimation strategies; measurement strategies.

[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 [ME] Mental Mathematics [PS] Problem Solving [T] Technology [CN] Connections and Estimation [R] Reasoning [V] Visualization



SCO: M1 – Solve problems that involve linear measurement, using: SI and imperial units of measure; estimation strategies; measurement strategies.

[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

nearest 1

in.16

An imperial caliper can measure to the nearest 1

in.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.





M2 Apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.

MAT521A

M1 Solve problems that involve the application of rates.

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)




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.





SS2 Determine the surface area of composite 3-D objects to solve problems.

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.

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

SCO: M3 – Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D objects, including:


[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)




SCO: M3 – Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D objects, including:


[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

2 2 2 SA lw lh wh 22 2 SA r rh

V lwh 2 V r h

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

1 2 SA lw ls ws 2 SA r rs

1

3V lwh 21

3 V r h

The surface area and the volume of a sphere both depend on the only the radius.

r

24 SA r

34

3 V r





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.

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)




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 A C A adjacent opposite

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 tan .

length of side adjacent to

AA

A

The short form for the sine ratio of angle A is sin A. It is defined as

length of side opposite sin .

length of hypotenuse

AA

The short form for the cosine ratio of angle A is cos A. It is defined as

length of side adjacent to cos .


AA

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



ALGEBRA AND NUMBER




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.



MAT421A – Topic: Algebra and Number (AN) GCO: Develop algebraic reasoning and number sense.


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.

AN1 Demonstrate an understanding of factors of whole numbers by determining the: prime factors; greatest common factor; least common multiple; square root; cube root.

MAT521A

MAT521B

AN2 Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands.

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.

[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)




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.

[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 25 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.





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.

MAT521A

MAT521B

SCO: AN2 – Demonstrate an understanding of irrational numbers by: representing, identifying and simplifying irrational numbers; ordering irrational numbers.

[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)




SCO: AN2 – Demonstrate an understanding of irrational numbers by: representing, identifying and simplifying irrational numbers; ordering irrational numbers.

[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., 3

,4

–5, 17.2

Irrational numbers – the set of all numbers that cannot be expressed as a fraction, e.g., 2, , 3 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





N1 Demonstrate an understanding 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.

N2 Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents.

AN3 Demonstrate an understanding of powers with integral and rational exponents.

MAT521A

MAT521B

SCO: AN3 – Demonstrate an understanding of powers with integral and rational exponents. [C, CN, PS, R]


A. Explain, using patterns, why 1

, 0.nn

a aa

B. Explain, using patterns, why 1 , 0.n na a n

C. Apply the exponent laws:

;m n m na a a

, 0;m n m na a a a

;nm mna a

;m m mab a b

, 0n n

n

a ab

b b

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)




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:

1nn

aa

and 1

,nn

aa as long as 0a

A power with a fractional exponent can be written as a radical using the following principles:

nm n ma a or mm n na a

These principles can be applied to the exponent laws previously learned in grade nine, as stated below.

EXPONENT NAME EXPONENT LAW

Product of Powers m n m na a a

Quotient of Powers , 0m

m nn

aa a

a

Power of a Power nm mna a

Power of a Product m m mab a b

Power of a Quotient , 0n n

n

a ab

b b

Zero Exponent 0 1, 0a a





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)




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 2 4 5 3 4 5 2 4 5x x x x x

212 15 8 10x x x

212 7 10x x

2 2 23 4 6 4 6 3 4 6c c c c c c c c

3 2 24 6 12 3 18c c c c c

3 24 13 9 18c c c

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.





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.

MAT521A

MAT521B

RF1 Factor polynomials of the form:

2 , 0;ax bx c a

2 2 2 2, 0, 0;a x b y a b

2 2, 0;a f x b f x c a

2 22 2 ,a f x b g y

0, 0a b

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 0.b

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)




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, 3 2 2 4 2 32 8 12 2 4 6 .m n m n mn mn m n m n A

common factor can be any polynomial. For example, 2 2 2 .a x b x x a b

To factor a trinomial of the form 2 ,x 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 2 12 27,x x 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 3 9 .x x

To factor a trinomial of the form 2 ,ax 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 22 5 3,x x 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 22 6 3.x x x After grouping and removing common

factors, we get 2 1 3 2 1 ,x x x which factors to 2 1 3 .x x

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, 2 2 .a b 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, 2 22a ab b

2.a b



RELATIONS AND FUNCTIONS




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 0 ; Ax By C

slope-point form 1 1y y m x x

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.



MAT421A – Topic: Relations and Functions (RF) GCO: Develop algebraic and graphical reasoning through the study of relations.


RF1 Interpret and explain the relationships among data, graphs and situations.

MAT521A

MAT521B

SCO: 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)




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 0

Constant Rate Rate of Change No Rate of Change Is Not Constant of Change





RF2 Demonstrate an understanding of relations and functions.

MAT521A

MAT521B

SCO: 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)




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.





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.

MAT521A

MAT521B

SCO: RF3 – Demonstrate an understanding of slope with respect to: rise and run; line segments and lines; rate of change; parallel lines; perpendicular 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)




rise

runm

SCO: RF3 – Demonstrate an understanding of slope with respect to: rise and run; line segments and lines; rate of change; parallel lines; perpendicular 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

run

x

The slope of a line can be determined using two points on the line, 1 1,x y and 2 2, ,x y by using the formula

2 11 2

2 1

, .y y

m x xx x

The slope indicates the rate of change of a linear relation.

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

x

Positive Slope

y

x

Zero Slope

y

x

Negative Slope

y

x

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.






RF4 Describe and represent linear relations, using: words; ordered pairs; tables of values; graphs; equations.

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: RF4 – Describe and represent linear relations, using: 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)




SCO: RF4 – Describe and represent linear relations, using: 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: 3f e

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 e 2 4 6

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.






RF5 Determine the characteristics of the graphs of linear relations, including the: intercepts; slope; domain; range.

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 2y a x p q and

determine the: vertex; domain and range; direction of opening; axis of symmetry; x- and y-intercepts.

RF4 Analyse quadratic functions of

the form 2y ax bx c to identify

characteristics of the corresponding graph, including: vertex; domain and range; direction of opening; axis of symmetry; x- and y-intercepts and to solve problems.

SCO: RF5 – Determine the characteristics of the graphs of linear relations, including the: intercepts; slope; domain; range. [CN, PS, R, V]

Students who have achieved this outcome should be able to: 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.

Section(s) in Foundations and Pre-Calculus Mathematics 10 text that address the specific curriculum outcome with relevant Achievement Indicators in brackets: 5.7 (A C E G) 6.1 (B D) 6.4 (E F G)




SCO: RF5 – Determine the characteristics of the graphs of linear relations, including the: 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 0y 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 0x 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 3

Set Notation: 2 3, integern n

List: { 2, 1, 0, 1, 2, 3}





RF6 Relate linear relations expressed in: slope-intercept form

;y mx b

general form 0 ;Ax By C

slope-point form

1 1y y m x x

to their graphs.

MAT521A

MAT521B

SCO: RF6 – Relate linear relations expressed in:




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 slope-point 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)




SCO: RF6 – Relate linear relations expressed in:




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 0,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.

For a non-vertical line through the point 1 1,x y with slope m, the equation of the line can be written in slope-

point form as 1 1 .y y m x x 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.





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.

MAT521A

MAT521B

SCO: 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. [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)




SCO: 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. [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

is ,y mx b where m represents the slope rise

run

and b represents the y-intercept.

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.






RF8 Represent a linear function using function notation.

MAT521A

MAT521B

SCO: 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

3 2,f x x determine 1 .f

D. Determine the related domain value, given a range value, for a linear function; e.g., if

7 ,g t t determine t so that 15.g t

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)




SCO: RF8 – Represent a linear function using function notation. [CN, ME, V]

Elaboration

Functions can be written using function notation. For example, the function 4 1y x can be written as

4 1.f x x The name of the function is f, with a variable name of x. In this example, 4 1x 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 29.8 ,v t t for the velocity, in metres per second, of a dropped object after t seconds and 2,A r r

for the area of a circle with radius r.

Function notation highlights the input-output aspect of a function. The function 4 1f x x takes any input

value for x, multiplies it by 4, and adds 1 to give the result. For example, if 2x is the input, then 2 9f is the

output, since 2 4 2 1 9.f Therefore, the point (2,9) is a on the graph of the function 4 1.f x x





PR3 Model and solve problems using linear equations of the form: ;ax b

, 0;x

b aa

;ax b c

, 0;x

b c aa

;ax b cx

;a x b c

;ax b cx d

;a bx c d ex f and

, 0a

b xx

where a, b, c, d, e and f are rational numbers.

RF9 Solve problems that involve 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

AN3 Solve problems that involve radical equations (limited to square roots).

AN6 Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials).

RF5 Solve problems that involve quadratic equations.

RF6 Solve, algebraically and graphically, problems that involve systems of linear-quadratic and quadratic-quadratic equations in two variables.

SCO: RF9 – Solve problems that involve systems of linear equations in two variables, graphically and 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)




SCO: RF9 – Solve problems that involve systems of linear equations in two variables, graphically and 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

y

y x

different slopes

y-intercepts can be the same or different

No solution

y

x

same slope

different y-intercepts

An infinite number of solutions

y

x

same slope

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.

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.

UNIT PLANS


CHAPTER 1

MEASUREMENT

SUGGESTED TIME

12 classes

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

IMPERIAL UNITS OF MEASURE

IMPERIAL UNIT

ABBREVIATION REFERENT RELATIONSHIP

BETWEEN UNITS

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 Pre-Calculus Mathematics 10:

pp. 11-12: #1-18

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.

For each inch of the frame, 1

32

inches of sinew

are needed. The diameter of the frame is 1

14

feet.

What length of sinew is needed? Express your answer to the nearest quarter of a foot.

UNIT PLANS


Section 1.2 – Math Lab: Measuring Length and Distance (pp. 13-15)



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

UNIT PLANS


Section 1.3 – Relating SI and Imperial Units (pp. 16-23)




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 m 3.281 ft

1 yd 91.44 cm

1 yd 0.9144 m

1 km 0.6214 mi 1 mi 1.609 km


pp. 22-23: #1-14


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

another news report referred to as 1

4 mile long.

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.

\

UNIT PLANS


Section 1.4 – Surface Areas of Right Pyramids and Right Cones (pp. 26-35)




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


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

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) 1 2SA lw ls ws

Right Pyramid (Regular Polygon

Base)

1Perimeter of base

2SA s

base area

Right Cone 2SA r rs


pp. 34-35: #1-18


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

value, such as 3.14 or 22

.7

Encourage students to

estimate to check that their solutions are reasonable.


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?

UNIT PLANS


Section 1.5 – Volumes of Right Pyramids and Right Cones (pp. 36-44)




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


volume – the amount of space occupied by an object

capacity – the amount a container can hold

FORMULAS FOR VOLUME

Right Prism V Ah

Right Pyramid 1

3V Ah

Rectangular Prism V lwh

Right Rectangular Pyramid

1

3V lwh

Right Cylinder 2V r h

Right Cone 21

3V r h


pp. 41-44: #1-19


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.

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.



.7

As well, the height

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.

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?

UNIT PLANS


Section 1.6 – Surface Area and Volume of a Sphere (pp. 45-52)




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


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

hemisphere – one half of a sphere

FORMULAS INVOLVING SPHERES

Surface Area 24SA r

Volume 34

3V r


pp. 50-52: #1-20


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.



.7

Encourage students to

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.


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.

UNIT PLANS


Section 1.7 – Solving Problems Involving Objects (pp. 55-61)




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

After this lesson, students should understand the following concept:

composite object – the result of combining two or more objects to make a new object


pp. 59-61: #1-11


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.


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 ice-cream 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? Round off the answer to one decimal place.

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.

UNIT PLANS


CHAPTER 2

TRIGONOMETRY

SUGGESTED TIME

11 classes

UNIT PLANS


Section 2.1 – The Tangent Ratio (pp. 70-77)




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


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

length of side opposite tan


AA

A


pp. 74-77: #1-20


Remind students that a diagram showing all of the given information should be the first step in 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.


For the given triangle, determine the value of each trigonometric ratio.

a. tan L L

b. tan N

12 13

M N

5

Complete each of the following tables. Round off all values to four decimal places and all angles to one decimal place, where necessary.

tan

270

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?

UNIT PLANS


Section 2.2 – Using the Tangent Ratio to Calculate Lengths (pp. 78-83)




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


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


pp. 81-83: #1-14



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.


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?

UNIT PLANS


Section 2.3 – Math Lab: Measuring an Inaccessible Height (pp. 84-86)



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

UNIT PLANS


Section 2.4 – The Sine and Cosine Ratios (pp. 89-96)




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


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

length of side opposite sin


AA

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

length of side adjacent to cos


AA

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


pp. 94-96: #1-16



Some students get confused between the sine and cosine ratios. Mnemonic devices can be used to help students such as Canadian Amateur Hockey

for adjacent

cos hypotenuse

and Ships Of Halifax for

oppositesin .

hypotenuse


For the given triangle, determine the value of each trigonometric ratio.

a. sin L L

b. sin N

c. cos L

d. cos N 12 13

M N

5

Evaluate each trigonometric ratio, to four decimal places.

a. 0sin 60

b. 0sin 30

c. 0sin 45

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.

UNIT PLANS


Section 2.5 – Using the Sine and Cosine Ratios to Calculate Lengths (pp. 97-102)




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


pp. 101-102: #1-12



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.


Find the length of x. Round off the answer to one decimal place.

10 cm 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.

UNIT PLANS


Section 2.6 – Applying the Trigonometric Ratios (pp. 105-112)




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


solving a triangle – determining the measure of each angle in a triangle and the length of each side of the triangle


pp. 110-112: #1-14



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.


Solve the triangle shown. Round off all measures to the nearest tenth.

A

22

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.

UNIT PLANS


Section 2.7 – Solving Problems Involving More Than One Right Triangle (pp. 113-121)




M4 (B D)


solve problems involving more than one right triangle


angle of depression – the angle between the horizontal, through eye level, and a line of sight to a point below eye level


p. 118-121: #1-16



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.


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. Round off the answer to one decimal place.

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.

D C

600

600

B 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.

UNIT PLANS


CHAPTER 3

FACTORS AND PRODUCTS

SUGGESTED TIME

12 classes

UNIT PLANS


Section 3.1 – Factors and Multiples of Whole Numbers (pp. 134-141)




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


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

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


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.


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.

UNIT PLANS


Section 3.2 – Perfect Squares, Perfect Cubes, and Their Roots (pp. 142-147)




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


perfect square – a number that can be written as a power with an integer base and exponent 2; for

example, 249 7

perfect cube – a number that can be written as a power with an integer base and exponent 3; for

example, 38 2

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


Some students may benefit from examples of non-perfect 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.


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

formula 2d rh 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.

UNIT PLANS


Section 3.3 – Common Factors of a Polynomial (pp. 150-156)




AN5 (A B D E)


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

identify and explain errors in solutions for factoring polynomials

factor a polynomial, then verify by expanding


factors – numbers or algebraic expressions that are multiplied together to get a product; for example, 3 and 7 are factors of 21, and 1x and

2x are factors of 2 3 2x x polynomial – one term or the sum of terms whose

variables have whole-number exponents; for

example, 2 23 2 5x xy y 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


pp. 154-156: #1-20


Remind students to check their work by using the inverse processes of expanding and factoring.


Determine the GCF of each pair of terms.

a. 25m n and 215mn

b. 348ab c and 2 2 236a b c

Factor each of the following polynomials.

a. 3 3x

b. 5 10x

c. 2 3a a

d. 23 12a a

e. 2 24 8xy x y

f. 2 2 3 3 327 18 36r s r s rs

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.

a.

b.

Write a trinomial with different numerical coefficients that has a greatest common factor of 2xy. Write the trinomial in both factored and expanded form.

UNIT PLANS


Section 3.4 – Math Lab: Modelling Trinomials as Binomial Products (pp. 157-158)



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

UNIT PLANS


Section 3.5 – Polynomials of the Form 2x bx c (pp. 159-167)




AN4 (A B C)

AN5 (B D E F H)


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


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


Encourage students to check their answers using multiplication.

Some students think that products such as

4 8t t and 4 8t t are equivalent.

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 1x , width 3x

b. length 2x , width 1x

Determine each product.

a. 3 5x x

b. 4 6m m

c. 10 10n n

The area of a rectangle is 2 6 8.x x

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 2 3 1.x x What can you conclude about this trinomial?

Factor each of the following trinomials.

a. 2 7 6x x

b. 2 6 9x x

c. 2 2 3x x

d. 2 7 10x x

Determine all possible values of k such that the

trinomial 2 24x kx is factorable.

UNIT PLANS


Section 3.6 – Polynomials of the Form 2ax bx c (pp. 168-178)




AN4 (A B E)

AN5 (B D E F G H)


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


coefficient – the numerical factor of a term; for

example, in the terms 3x and 23 ,x 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



When factoring polynomials, encourage students to try different methods of factoring.


Draw a rectangle to show the area represented by each product.

a. length 4 2,x width 2 1x

b. length 3,x width 2 1x


a. 4 1 2x x

b. 6 2 1x x

c. 5 1 2 6m m

Write an expression for the area of the border around the inner rectangle.

9x

4x

x 5x

Factor each of the following trinomials.

a. 22 11 15x x

b. 24 17 15x x

c. 22 13 15x x

d. 25 7 6x x

e. 26 11 3x x

f. 23 5 2x x

Determine all possible values of k such that the

trinomial 24 3x kx is factorable.

UNIT PLANS


Section 3.7 – Multiplying Polynomials (pp. 182-187)




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


pp. 185-187: #1-17


A rectangle diagram can be used to determine the product of two polynomials. For example, to find

the product 21 3 2 , x x x the following

rectangle diagram can be used:

x 2 x3 2

x 3x 23x 2x

1 2x 3 x 2

Combining like terms, we get a product of

3 22 2. x x x

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. 23 4 2 3 1x x x

b. 24 3 8 6r r r

c. 25 3 2 6 12x x x

d. 32x y

e. 3 5 2 4 1 2 5x x x x

f. 2 3 2 4 7 2 5x x x

UNIT PLANS


Section 3.8 – Factoring Special Polynomials (pp. 188-195)




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


perfect square trinomial – a trinomial of the form 2 22 ;a ab b it can be factored as

22 22a ab b a b

difference of squares – a binomial of the form 2 2;a b it can be factored as

2 2a b a b a b


pp. 194-195: #1-18



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.



a. 3 3x x

b. 5 5x x

c. 2 1 2 1x x

d. 4 4x y x y

Factor each of the following polynomials.

a. 24 20 25x x

b. 2 224 144x xy y

c. 2 18 81y y

d. 23 24 48b b

e. 24 16x

f. 2 249 25a b

g. 2 2125 80x y

h. 2 29 64p q

UNIT PLANS


UNIT PLANS


CHAPTER 4

ROOTS AND POWERS

SUGGESTED TIME

9 classes

UNIT PLANS


Section 4.1 – Math Lab: Estimating Roots (pp. 204-206)



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


root – the inverse of a power; for example, in 32 8 or 3 8 2, 2 is the cube root of 8

radical – an expression consisting of the radical

sign, , and a radicand; for example, 81

radicand – the number under a radical sign; for

example, 81 is the radicand in 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

UNIT PLANS


Section 4.2 – Irrational Numbers (pp. 207-212)




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


rational number – any number that can be written

in the form ,m

n where m and n are integers, and

0n

irrational number – any number that cannot be

written in the form ,m

n where m and n are integers,

and 0n

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


pp. 211-212: #1-20


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.


Place an X in the correct spaces to indicate that the number belongs in the number set.

5 –2 3

4 –1.3 7 0

Natural Numbers

Whole Numbers

Integers

Rational 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:

22, , 3.141, 10, 9.5

7

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 3

b. 6 5

c. 2 1.4142135

UNIT PLANS


Section 4.3 – Mixed and Entire Radicals (pp. 213-219)




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


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

,n n nab a b where n is a natural number, and a

and b are real numbers.


pp. 217-219: #1-23


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.


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

UNIT PLANS


Section 4.4 – Fractional Exponents and Radicals (pp. 222-228)




AN3 (B D E F)


use patterns to explain why 1 , 0n na a n

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

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

solve problems involving exponents or radicals


rational exponent – an exponent that is a rational number

POWERS WITH RATIONAL EXPONENTS

When n is a natural number and x is a rational number,

1 n nx x

When m and n are rational numbers, and x is a rational number,

1/mmm n n nx x x

and

1/n nm n m mx x x


pp. 227-228: #1-21


Ensure that students understand how the numerator and the denominator of a fractional exponent relate to a radical expression.


Express each power as a radical.

a. 3 57

b. 1 31024

c. 3 84x

Express each radical as a power.

a. 35

b. 53 y

c. 227n

Evaluate each expression.

a. 3 24

b. 2 532

c. 2 3

8

27

UNIT PLANS


Section 4.5 – Negative Exponents and Reciprocals (pp. 229-234)




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


negative exponent – an exponent that is a negative number

reciprocals – two numbers whose product is 1; for

example, 2

3 and

3

2

POWERS WITH NEGATIVE EXPONENTS

When x is any non-zero number and n is a rational

number, nx is the reciprocal of .nx That is,

1, 0 n

nx x

x

and

1, 0 n

nx x

x


pp. 233-234; #1-19


Remind students that if a base is positive, a negative exponent always produces a positive answer.

Ask students to explain why the product 4 4 3 3 5 42 2 5 5 10 10 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.

a. 6 4 04 4 4

b. 9 7 17 7 7

c. 3 2 4145 145 145

d. 2 24 4

e. 3 2

2 2

3 3

f. 42

UNIT PLANS


Section 4.6 – Applying the Exponent Laws (pp. 237-243)




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 m n m na a a

Quotient of Powers , 0m

m nn

aa a

a

Power of a Power nm mna a

Power of a Product m m mab a b

Power of a Quotient , 0n n

n

a ab

b b


pp. 241-243: #1-20


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.


Write each expression as a power with a single exponent. Write all powers with positive exponents.

a. 1.5 3.5x x

b. 5/4 1/2p p

c. 4/3

1/6

1.5

1.5

Simplify each expression. Write all powers with positive exponents.

a. 2/3627x

b. 94/3 1/3t t

c. 2/33

64

x

Simplify 5 33 2 .x x Write the answer in radical

fo

UNIT PLANS


UNIT PLANS


CHAPTER 5

RELATIONS AND FUNCTIONS

SUGGESTED TIME

14 classes

UNIT PLANS


Section 5.1 – Representing Relations (pp. 256-263)




RF1 (E)


express a relation in words, as a set of ordered pairs, in a table, and using an arrow diagram


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


pp. 261-263: #1-11


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.


This table shows the largest four Prince Edward Island communities with their populations in 2006.

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.

UNIT PLANS


Section 5.2 – Properties of Functions (pp. 264-273)




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


function – a relation where each element in the first set is associated with exactly one element in the second set

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

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


pp. 270-273: #1-19


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.

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 1.8 32F C C is used to convert a

temperature in degrees Celsius (0C) to a temperature in degrees Fahrenheit (0F).

a. Determine 25 .F

b. Determine C so that 68.F C

If 2 5 3,f x x x find 2 .f What is an ordered

pair describing the point having a y-coordinate of

2 ?f

UNIT PLANS


Section 5.3 – Interpreting and Sketching Graphs (pp. 276-283)




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


pp. 281-283: #1-15


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.

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)

UNIT PLANS


Section 5.4 – Math Lab: Graphing Data (pp. 284-286)



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

UNIT PLANS


Section 5.5 – Graphs of Relations and Functions (pp. 287-297)




RF1 (B E)

RF2 (C D)

RF4 (A)

RF8 (C D)


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


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


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


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.


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.

b.

c.

UNIT PLANS


Section 5.6 – Properties of Linear Relations (pp. 300-310)




RF4 (A B C D E F G)



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


linear relation – a relation that has a straight-line graph

rate of change – the change in one quantity with respect to the change in another quantity


pp. 307-310: #1-17


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.


Determine whether each of the following relations is linear or non-linear.

a. 7x

b. 3 2 12 x y

c. 22 6x y

d. 2

53

y x

e. 3xy

Sketch the graph of each of the following linear relations.

a. 1y x

b. 2

23

y x

c. 7 2y x

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. 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?

UNIT PLANS


Section 5.7 – Interpreting Graphs of Linear Functions (pp. 311-323)




RF5 (A C E G)

RF8 (E)


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

identify the graph that corresponds to a given slope and y-intercept

solve a problem that involves intercepts, slope, domain or range of a linear relation

sketch the graph of a linear function expressed in function notation


linear function – a linear relation whose graph is not a vertical line

vertical intercept – the y-intercept; the y-coordinate of a point where a graph intercepts the y-axis

horizontal intercept – the x-intercept; the x-coordinate of a point where a graph intercepts the x-axis


pp. 319-323: #1-17


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 0,x and to find the x-intercept, set 0.y


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

b. the cost of having 100 T-shirts made

c. the cost of having 200 T-shirts made

d. the cost of each extra T-shirt made

Sketch a graph of each linear function.

a. 2 3f x x

b. 32

4f x x

c. 3 1f x x

UNIT PLANS


CHAPTER 6

LINEAR FUNCTIONS

SUGGESTED TIME

11 classes

UNIT PLANS


Section 6.1 – Slope of a Line (pp. 332-343)




RF3 (A B C D E F G I)

RF5 (B D)


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


slope – a measure of how one quantity changes with respect to the other; it can be determined by

calculating rise

run

rise – the vertical distance between two points

run – the horizontal distance between two points

SLOPE OF A LINE

A line passes through 1 1,A x y and 2 2, .B x y The

slope of the line AB is

2 1

1 22 1

y y

AB x xx x


pp. 339-343: #1-28


Remind students that rise

slope .run

To help

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.


Determine the slope, m, of each line segment with the given end points.

a. 3,6S and 5,2T

b. 4,3H and 4,8K

c. 9, 7M and 1, 7N

d. 2, 2W and 5,5S

Determine the slope of each side of the triangle below.

y A

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.

UNIT PLANS


Section 6.2 – Slopes of Parallel and Perpendicular Lines (pp. 344-351)




RF3 (H I)


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


parallel lines – lines on the same flat surface that do not intersect

perpendicular lines – lines or line segments that intersect at right angles

oblique line – a line that is neither horizontal nor vertical

negative reciprocals – two numbers whose

product is –1; for example, 3

7 and

7

3 are

negative reciprocals

SLOPES OF PERPENDICULAR LINES

The slopes of two oblique perpendicular lines are negative reciprocals; that is, a line with slope a,

0,a is perpendicular to a line with slope 1

.a


pp. 348-351: #1-20


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).


Determine whether each pair of lines with the given slopes is parallel, perpendicular, or neither.

a. 3, 6

2

b. 7 2

, 2 7

c. 3 4

, 4 3

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. 2, 2 , 3, 3A B and 2,2 , 3,3C D

b. 6,8 , 7,10E F and 0,1 , 2,0G H

c. 6,4 , 0,0W X and 1,1 , 3,4Y Z

UNIT PLANS


Section 6.3 – Math Lab: Investigating Graphs of Linear Functions (pp. 354-356)



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 slope-intercept 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

UNIT PLANS


Section 6.4 – Slope-Intercept Form of the Equation for a Linear Function (pp. 357-364)




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 slope-intercept 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


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


pp. 362-364: #1-21


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 0.y mx

Therefore, 0.b

If the equation is in the form ,y x b remind

students that it can be written as 1 .y x b

Therefore, 1.m


Determine the slope and the y-intercept of each of the following lines.

a. 2 5y x

b. 2y

c. 5 15x y

d. 2 6 12x y

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 12 840 0.x y

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?

A decorator’s fee can be modelled by the equation 75 .F t 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. 1

722 7

y x

y x

b. 2

65

5 2 8

y x

x y

UNIT PLANS


Section 6.5 – Slope-Point Form of the Equation for a Linear Function (pp. 365-374)




RF6 (B C D F)

RF7 (B C D F)


graph a linear relation, given its equation in slope-point form and explain the strategy used to create the graph

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 the equation of a linear relation, given the coordinates of two points on the line

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



slope-point form – the equation of a line in the

form 1 1 ,y y m x x where m is the slope of

the line, and the line passes through the point

1 1,P x y


pp. 371-374: #1-25


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 3 2 1 ,y x the student may rewrite the

equation as 3 2 1 ,y x showing that it

passes through the point (–1,–3).


Write an equation of the line through the point (3,–4) with slope 2.

Write an equation of the line that passes through each pair of points.

a. (1,1) and (3,–3)

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 3 3y x and passes through the point (5,–6).

Two perpendicular lines intersect on the x-axis. The equation of one of the lines is 2 6.y x Find

the equation of the second line.

UNIT PLANS


Section 6.6 – General Form of the Equation for a Linear Relation (pp. 377-385)




RF6 (A B C D E F)

RF7 (E F)


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


graph linear data generated from context and write the equation of the resulting line



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 0,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


pp. 383-385: #1-25


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.


Rewrite the equation 3

24

y x in general form.

Consider the linear equation 4 5 20 0.x y

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?

UNIT PLANS


UNIT PLANS


CHAPTER 7

SYSTEMS OF LINEAR EQUATIONS

SUGGESTED TIME

11 classes

UNIT PLANS


Section 7.1 – Developing Systems of Linear Equations (pp. 394-402)




RF9 (A B)


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


system of linear equations – two equations of linear functions in the same two variables; can also be called a linear system


pp. 400-402: #1-14


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.


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 3 9x y and

2 4.x y

UNIT PLANS


Section 7.2 – Solving a System of Linear Equations Graphically (pp. 403-410)




RF9 (A C D H)



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


point of intersection – the point where two graphs intersect


pp. 408-410: #1-16


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.


Solve each system of equations graphically.

a.

2 3 11

2 3 17

x y

x y

b.

8

3 2 14

x y

x y

UNIT PLANS


Section 7.3 – Math Lab: Using Graphing Technology to Solve a System of Linear Equations (pp. 411-413)



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

UNIT PLANS


Section 7.4 – Using a Substitution Strategy to Solve a System of Linear Equations (pp. 416-427)




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

verify 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


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




pp. 424-427: #1-22


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 5 27

4 16

x y

x

b.

2 5

2 10

x y

x y

c.

3 9

6 2 6

x y

x y

d.

3 4 15

5

x y

x y

e.

2 13

4.5 0.4 16

x y

x y

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?

UNIT PLANS


Section 7.5 – Using an Elimination Strategy to Solve a System of Linear Equations (pp. 428-439)




RF9 (A B E G H)



relate a system of linear equations to the context of a problem

use an elimination strategy to determine the solution of a system of linear equations

verify 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


solving by elimination – adding or subtracting the equations to eliminate one variable and then solving for the other variable


pp. 437-439: #1-21


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:

Line up the equations with the like terms in the same columns.

Choose a variable to eliminate.

To the right of each equation, write down what equation is to be multiplied by.

Write the new equations to the right of the original equations.

Add the equations together and solve for the remaining variable.

Substitute this value into either original equation to find the value of the other variable.


Solve each system of equations using the elimination method.

a. 2 10

2 3 14

x y

x y

b.

2 4

3 16

y x

y x

c.

3 2 10

9 6 16

x y

y y

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.

UNIT PLANS


Section 7.6 – Properties of Systems of Linear Equations (pp. 442-449)




RF9 (A F)


determine the number of solutions of a system of linear equations graphically


determine the number of solutions of a system of linear equations using the slopes and y-intercepts of the lines

explain why a system of linear equations may have no solution, one solution or infinite solutions

write a second equation to create linear systems with different numbers of solutions, given the first equation


infinite – having no boundaries or limits

coincident lines – two lines that appear in the same location; lines that coincide


pp. 447-449: #1-21


Remind students that equations should be written in slope-intercept form when determining how many solutions there are to a system of linear equations.


Predict the number of solutions for each system of linear equations.

a.

2 4

14

2

x y

y x

b.

6 4 6

21

3

y x

y x

c.

3 1

2 1

y x

y x

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

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 10 16 0

5 8 0

x y

x y

b.

2 4 0

2 6 0

x y

x y

GLOSSARY OF MATHEMATICAL TERMS



A

acute angle – an angle measuring less than 900

acute triangle – a triangle with three acute angles

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, 6 4x

angle of depression – the angle between the horizontal, through eye level, and a line of sight to a point below eye level

angle of elevation – the angle between the horizontal, through eye level, and a line of sight to a point above eye level

angle of inclination – the acute angle between the horizontal and a line or line segment

angle of inclination

apex – the vertex farthest from the base of an object

apex

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

2 A

3 B

4

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

angle of depression

angle of elevation



B

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...

base – the side of a polygon or the face of an object from which the height is measured

base

base of a power – see power

binomial – a polynomial with two terms; for example, 3 8x

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

central angle

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

example, in the terms 3x and 23 ,x the coefficient

is 3

coincident lines – two lines that appear in the same location; lines that coincide

y x

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

28 4 12x y xy y

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



congruent – shapes that match exactly, but do not necessarily have the same orientation

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 3,x 3 is the constant term

conversion factor – a number used to multiply or divide a quantity to convert from one unit of measure to another

coordinate axes – the horizontal and vertical axes on a grid

y vertical 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

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

length of side adjacent to cos


AA

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, 38 2

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

cubic units – units that measure volume

D

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

diagonal



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

diameter

difference of squares – a binomial of the form 2 2;a b it can be factored as

2 2a b 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

direct measurement – a measurement made using a measuring instrument or by counting

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



divisor – the number that divides into another number

domain – the set of first elements of a relation

E

edge – two faces of an object meet at an edge

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, 5 32

equation – a mathematical statement that two expressions are equal

equilateral triangle – a triangle with three equal sides

equivalent – having the same value; for

example, 1

2 and

2,

4 3 : 4 and 9 :12

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

evaluate – to determine the value of a numerical expression

even integer – a number that has 2 as a factor; for example, 2, 4, 6

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

F

face – a flat surface of an object

face

factor – to factor means to write as a product; for example, 20 2 2 5



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 1x and

2x are factors of 2 3 2x x 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 0,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

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

H

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

hypotenuse



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


n where m and n are

integers, and 0n

isometric – equal measure; on isometric dot paper, the line segments joining two adjacent dots in any direction are equal

isosceles trapezoid – a trapezoid with two equal, non-parallel sides

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

leg

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



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

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 25x 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

product is –1; for example, 3

7 and

7

3 are

negative reciprocals

numerator – the term above the line in a fraction

numerical coefficient – see coefficient

O

object – a solid or shell that has three dimensions

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

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



origin – the point where the horizontal and the vertical axes meet

y origin x

P

parallel lines – lines on the same flat surface that do not intersect

parallelogram – a quadrilateral with opposite sides parallel and opposite angles equal

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

example, 38 2

perfect square – a number that can be written as a power with an integer base and exponent 2;

for example, 249 7

perfect square trinomial – a trinomial of the

form 2 22 ;a ab b it can be factored as

22 22a ab b a b

perimeter – the distance around a closed shape

perpendicular lines – lines that intersect at right angles

right angle

pi ( ) – the ratio of the circumference of a circle

to its diameter

circumference

diameter

point of intersection – the point where two graphs intersect

y

point of

intersection x

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;

for example, 2 23 2 5x xy y x

power – an expression of the form ,na 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

34



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

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

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, : 24 3 : 4r

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,

2 2 2a b c

2c

2b

2a

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

R

radical – an expression consisting of the radical

sign, , and a radicand; for example, 81

radicand – the number under a radical sign; for

example, 81 is the radicand in 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


n where m and n are

integers, and 0n

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



reciprocals – two numbers whose product is 1;

for example, 2

3 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 polyhedron – a polyhedron with congruent faces, each of which is a regular polygon

regular prism – a prism with regular polygons as bases; for example, a cube

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...

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



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 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

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

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 32 8 or 3 8 2, 2 is the cube root of 8

run – the horizontal distance between two points; see also slope

S

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



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

length of side opposite sin


AA

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

calculating rise

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 1 1 ,y y m x x where m is the slope of

the line, and the line passes through the point

1 1,P x y

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

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;

for example, 249 7

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

T

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

length of side opposite tan


AA

A

term – a number, a variable, or the product of

numbers and variables; for example, –5, y, 27a

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



trapezoid – a quadrilateral with exactly one pair of parallel sides

triangle – a polygon with three sides

triangular prism – a prism with triangular bases

trigonometry – the study of the properties and applications of triangles

trinomial – a polynomial with three terms; for

example, 23 5 8x x

U

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

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

vertex

vertical axis – see y-axis

vertical intercept – the y-intercept; the y-coordinate of a point where a graph intercepts the y-axis

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

whole numbers – the set of numbers 0, 1, 2, 3, 4, ...

X

x-axis – the horizontal number line on a coordinate grid

y-axis x-axis

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 x-axis



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

SOLUTIONS TO POSSIBLE ASSESSMENT STRATEGIES



SECTION 1.1

111.5 in2

3

134

ft

SECTION 1.3

8 ft

56 mph

SECTION 1.4

502.7 cm2

270 cm2

SECTION 1.5

256 cm3

a. 6911.5 cm3

b. 2303.8 cm3

5408 m3

SECTION 1.6

2827.4 cm2

7238.2 cm3

2 3864 cm ; 1728 cmSA V

5.0 m

9.8 cm

SECTION 1.7

a. 876 cm2

b. inner radius 8.9 cm;

outer radius 16.9 cm

111.3 cm3

32 m2

364.4 cm2

5236 m3

SECTION 2.1

a. 5

12

b. 12

5

tan

270 0.5095

450 1

570 1.5399

tan

18.40 0.3329

29.00 0.5543

55.70 1.4653

65.00

SECTION 2.2

3.9 m

240.3 m

SECTION 2.4

a. 5

13

b. 12

13

c. 12

13

d. 5

13

a. 0.8660

b. 0.5

c. 0.7071

a. 26.00

b. 78.00

45.10

SECTION 2.5

23.7 cm

423.8 m

46.4 m

SECTION 2.6

048 , 16.3, 14.7A a b



0 042.4 , 47.6 , 28.3E F e

25.0 km

84.9 m

SECTION 2.7

722.9 m

150.9 m

6 cm

439 m

SECTION 3.1

5 22 3 5

48

800

60 cm

SECTION 3.2

68

42

1176 cm2

24 km; 71.6 km

SECTION 3.3

a. 5mn

b. 212ab c

a. 3 1x

b. 5 2x

c. 3a a

d. 3 4a a

e. 4 2xy y x

f. 2 29 3 2 4rs r r s s

a.

The dimensions are 2x and 2,x the area is

22 4 ;x x 22 4 2 2x x x x

b.

The dimensions are 3 and 2,x the area is

3 6;x 3 6 3 2x x

Answers may vary.

One possible solution is 3 3 2 26 4 2 ;x y x y xy

2 2 3 3 2 22 3 2 1 6 4 2xy x y xy x y x y xy

SECTION 3.5

a.

b.

a. 2 8 15x x

b. 2 2 24m m

c. 2 100n

a. 2x by 4x

b. no



c. 4 12x

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 2 3 1.x x Therefore, this polynomial cannot be factored.

a. 6 1x x

b. 3 3x x

c. 1 3x x

d. 5 2x x

10, 11, 14 25k

SECTION 3.6

a.

b.

a. 24 7 2x x

b. 22 13 6x x

c. 210 32 6m m

18 45x

a. 3 2 5x x

b. 5 4 3x x

c. 5 2 3x x

d. 2 5 3x x

e. 2 3 3 1x x

f. 2 3 1 x x

7, 8, 13k

SECTION 3.7

a. 3 26 15 4x x x

b. 3 23 4 38 24r r r

c. 3 210 36 78 36x x x

d. 3 2 2 38 12 6x x y xy y

e. 213 25 26x x

f. 28 12 31x x

SECTION 3.8

a. 2 9x

b. 2 25x

c. 24 4 1x x

d. 2 216 8x xy y

a. 2

2 5x

b. 2

12x y

c. 2

9y

d. 2

3 4b

e. 4 2 2x x

f. 7 5 7 5a b a b

g. 5 5 4 5 4x y x y

h. 3 8 3 8pq pq



SECTION 4.2

5 –2 3

4 –1.3 7 0

Natural Numbers

X

Whole Numbers

X X

Integers X X X

Rational Numbers

X X X X X

Irrational Numbers X

Real Numbers

X X X X X X

Rational numbers: 1, 4, 9, 16

Irrational numbers: 2, 3, 5, 6, 7, 8,

10, 11, 12, 13, 14, 15, 17, 18, 19,

20

The rational numbers have radicands that are perfect squares and the irrational numbers have radicands that are not perfect squares.

22

9.5, 3.141, , , 107

a. always true

b. sometimes true

c. never true

a. <

b. <

c. >

SECTION 4.3

a. 2 6

b. 6 2

c. 4 6

d. 3 5

a. 275

b. 40

c. 108

SECTION 4.4

a. 5 37

b. 3 1024

c. 3x

a. 3/25

b. 5/3y

c. 2/27 n

a. 8

b. 4

c. 4

9

SECTION 4.5

a. 16

b. 7

c. 145

d. 256

e. 2

3

f. 1

16

SECTION 4.6

a. 5x

b. 3/4

1

p

c. 7/61.5

a. 49x

b. 15t

c. 2

16

x

15 19x

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)}



c.

Population

CH 32,200

SU 14,500

ST 7,100

CO 4,700

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

{1, 2, 3, 4};D {1, 8, 27, 64}R

b. not a function

{3};D {4, 5, 6, 7}R

a. 770F

b. 200C

2 3;f (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 water-skier is ready, the boat repeats this process. However, it takes a bit longer to slow down the second time than the first time.

The fourth graph is the best choice, since a person’s height changes gradually, not suddenly.

Distance

0

Time

a.

b.

c.

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

{ 1 3};D x { 2 2}R y

a. function

b. not a function

c. not a function

SECTION 5.6

a. linear

b. linear

c. non-linear

d. linear

e. non-linear



a.

b.

c.

a.

Days (d) Cost (C)

1 24

2 28

3 32

4 36

5 40

b.

c. 4 20C d

d. $52

SECTION 5.7

a. $400

b. $900

c. $1400

d. $5

a.

b.



c.

SECTION 6.1

a. 1

2

b. undefined

c. 0

d. –1

3 1

, 4, 4 5AB AC BCm m m

1k

Answers may vary. Possible solutions are (0,1), (2,2), (8,5).

Note: Any point on the line 1

12

y x is a

solution.

SECTION 6.2

a. parallel

b. neither

c. perpendicular

a. parallel

b. neither

c. perpendicular

SECTION 6.4

a. 2, 5m b

b. 0, 2m b

c. 1

, 35

m b

d. 1

, 23

m b

a. 12;m The slope represents the price per

ticket.

b. 840;b 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 6.5

2 10y x

a. 2 3y x

b. 1 14

5 5y x

0.4 25.2y x

The fixed insurance premium is $10, and the hourly rate is $20/h.

3 9y x

1 3

2 2y x

SECTION 6.6

3 4 8 0x y

a. (5,0)

b. (0,4)

c.

a. 12 16 336S T

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.



d. 15 h

SECTION 7.1

10 40

8 50

y x

y x

3 3 3 2 3 6 9x y

2 2 3 2 6 2 4x y

SECTION 7.2

a. (7,–1)

b. (6,2)

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

SECTION 7.5

a. (–2,6)

b. (4,4)

c. no solution

A single adult ticket cost $14 and a single student ticket cost $9.

There were 93 muffins and 67 individual yogurts sold.

2.7 m by 3.4 m

SECTION 7.6

a. no solution

b. infinite solutions

c. one solution

Car: 90 40y x

Minivan: 90 25y x

Truck: 110 30y x

RV: 90 40y x

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

the RV in 1

2 hour at a distance of 85 km.

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

parallel, since 1

2 m for each equation.

Therefore, there will be no solution.

MATHEMATICS RESEARCH PROJECT


APPENDIX




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 PROBABLE 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

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.



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

Entertainment effects of new electronic

devices

file sharing

Health care doctor and/or nurse

shortages

funding

Post-secondary education entry requirements

graduate success

History of Western and Northern Canada

relations among First Nations

immigration

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.



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

Is mathematics used in computer technology?

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?

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.



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.



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

x x

zs

can be used where x is any observed data

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?



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.



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



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

Delivery

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

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.

acknowledgments · the 2010-2011 mat421a pilot teachers: kirk cutcliffe, bethany macleod, colonel...

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