mathematics 8 and 9 - british columbia · for mathematics 8 and 9. the development of this irp has...

Integrated Resource Package 2001

MATHEMATICS 8 AND 9

Ministry of Education

IRP 113

Copyright © 2001 Ministry of Education, Province of British Columbia.

Copyright Notice

No part of the content of this document may be reproduced in any form or by any means, including electronic storage,reproduction, execution or transmission without the prior written permission of the Province.

Proprietary Notice

This document contains information that is proprietary and confidential to the Province. Any reproduction,disclosure or other use of this document is expressly prohibited except as the Province may authorize in writing.

Limited Exception to Non-reproduction

Permission to copy and use this publication in part, or in its entirety, for non-profit educational purposes withinBritish Columbia and the Yukon, is granted to all staff of B.C. school board trustees, including teachers andadministrators; organizations comprising the Educational Advisory Council as identified by Ministerial Order; andother parties providing direct or indirect education programs to entitled students as identified by the School Act or theIndependent School Act.

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TABLE OF CONTENTS

PREFACE: USING THIS INTEGRATED RESOURCE PACKAGE

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III

INTRODUCTION TO MATHEMATICS 8 AND 9

The Development of this Integrated Resource Package . . . . . . . . . . . . . . . . . . . . . . . . . 1Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1B.C. Secondary Mathematics Courses Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Organization of the Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Suggested Instructional Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Integration of Cross-Curricular Interests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Suggested Assessment Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

THE MATHEMATICS 8 AND 9 CURRICULUM

Mathematics 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Mathematics 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

MATHEMATICS 8 AND 9 APPENDICES

Appendix A: Prescribed Learning Outcomes Mathematics 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3 Mathematics 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-7

Appendix B: Learning Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3Appendix C: Assessment and Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3

Assessment and Evaluation Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . C-5Assessment Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-19

MATHEMATICS 8 AND 9 APPENDICES

Appendix D: Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3Appendix E: Illustrated Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3Appendix F: Illustrated Examples

Mathematics 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-3 Mathematics 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-35

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TABLE OF CONTENTS

III

his Integrated Resource Package(IRP) provides the basic informationthat teachers will require to

implement the curriculum. It is also on theMinistry of Education’s Web site atwww.bced.gov.ca/irp/irp.htm

THE INTRODUCTION

The Introduction provides general informationabout Mathematics 8 and 9, including specialfeatures and requirements. It also provides arationale for the subject—why mathematicsis taught in BC schools—and an explanationof the curriculum organizers.

THE MATHEMATICS 8 AND 9 CURRICULUM

The provincially prescribed curriculumfor Mathematics 8 and 9 is structured interms of curriculum organizers. The mainbody of this IRP consists of four columnsof information for each organizer. Thesecolumns describe:

• provincially prescribed learning outcomestatements for Mathematics 8 and 9

• suggested instructional strategies forachieving the outcomes

• suggested assessment strategies fordetermining how well students areachieving the outcomes

• provincially recommended learningresources

Prescribed Learning Outcomes

Learning outcome statements are contentstandards for the provincial educationsystem. Learning outcomes set out theknowledge, enduring ideas, issues,concepts, skills, and attitudes for each


subject. They are statements of what studentsare expected to know and be able to do ineach grade. Learning outcomes are clearlystated and expressed in measurable terms.All learning outcomes complete this stem:“It is expected that students will. . . . ”Outcome statements have been written toenable teachers to use their experience andprofessional judgment when planning andevaluating. The outcomes are benchmarksthat will permit the use of criterion-referencedperformance standards. It is expected thatactual student performance will vary.Evaluation, reporting, and student placementwith respect to these outcomes dependson the professional judgment of teachers,guided by provincial policy.

Suggested Instructional Strategies

Instruction involves the use of techniques,activities, and methods that can be employedto meet diverse student needs and to deliverthe prescribed curriculum. Teachers are freeto adapt the suggested instructional strategiesor substitute others that will enable theirstudents to achieve the prescribed outcomes.These strategies have been developed byspecialist and generalist teachers to assisttheir colleagues; they are suggestions only.

Suggested Assessment Strategies

The assessment strategies suggest a varietyof ways to gather information about studentperformance. Some assessment strategiesrelate to specific activities; others are general.These strategies have been developed byspecialist and generalist teachers to assisttheir colleagues; they are suggestions only.

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Provincially Recommended LearningResources

Provincially recommended learning resourcesare materials that have been reviewed andevaluated by British Columbia teachers incollaboration with the Ministry of Educationaccording to a stringent set of criteria. Theyare typically materials suitable for studentuse, but they may also include informationprimarily intended for teachers. Teachersand school districts are encouraged to selectthose resources that they find most relevantand useful for their students, and to supple-ment these with locally approved materialsand resources to meet specific local needs.The recommended resources listed in the mainbody of this IRP are those that have a com-prehensive coverage of significant portionsof the curriculum, or those that provide aunique support to a specific segment of thecurriculum. Appendix B contains a completelisting of provincially recommended learningresources to support this curriculum.

THE APPENDICES

A series of appendices provides additionalinformation about the curriculum andfurther support for the teacher.

• Appendix A contains a listing of theprescribed learning outcomes foreach grade.

• Appendix B contains a comprehensivelisting of the provincially recommendedlearning resources for this curriculum.As new resources are evaluated, thisappendix will be updated.

• Appendix C contains assistance forteachers related to provincial evaluationand reporting policy. Curriculumoutcomes have been used as the sourcefor examples of criterion-referencedevaluations.

• Appendix D acknowledges the manypeople and organizations that have beeninvolved in the development of this IRP.

• Appendix E contains an illustratedglossary of mathematical terms.

• Appendix F contains illustrated examples,which are intended to indicate to thereader the competencies an averagestudent is expected to demonstrate foreach of the prescribed learning outcomes.

V

SUGGESTED INSTRUCTIONAL STRATEGIESPRESCRIBED LEARNING OUTCOMES

the

SUGGESTED ASSESSMENT STRATEGIES RECOMMENDED LEARNING RESOURCES

MATHEMATICS 8 • Patterns and Relations (Patterns)


Suggested AssessmentStrategies

The SuggestedAssessment Strategies

offer a wide range ofdifferent assessmentapproaches useful in

evaluating the PrescribedLearning Outcomes.

Teachers should considerthese as examples they

might modify to suit theirown needs and theinstructional goals.

Suggested InstructionalStrategies

The SuggestedInstructional Strategiescolumn of this IRPsuggests a variety ofinstructional approachesthat include group work,problem solving, and theuse of technology. Teachersshould consider these asexamples that they mightmodify to suit thedevelopmental levels oftheir students.

Recommended LearningResources

The RecommendedLearning Resourcescomponent of this IRP is acompilation of provinciallyrecommended resourcesthat support the PrescribedLearning Outcomes. Acomplete list including ashort description of theresource, its media type,and distributor isincluded in Appendix Bof this IRP.

Prescribed LearningOutcomes

The Prescribed LearningOutcomes column of this

IRP lists the specificlearning outcomes for

each curriculumorganizer or sub-

organizer. These aid theteacher in day-to-day

planning.

Grade Curriculum Organizerand Suborganizer

Grade


Curriculum Organizerand Suborganizer

SUGGESTED EXTENSIONS

Suggested Extensions

The SuggestedExtensions are not

provincial curriculum.The Suggested

Extensions provide extratopics and enrichment to

add breadth and depth totopics under study.

In order to prepare students to use patterns, variablesand expressions, and graphs to solve problems, it isexpected that students will:

• substitute numbers for variables in expressionsand graph and analyse the relation

• translate between an oral or written expressionand an equivalent algebraic expression

• generalize a pattern from a problem-solvingcontext

To extend students’ understanding of patterns, theycould:

• represent a pattern using mathematicalexpressions and equations, and verify bysubstitution


Looking for patterns and making generalizations from them aremathematical skills useful in investigating and solving real-world problems. The variables, expressions, and equations usedto describe patterns and relationships are the basis of students’study of algebra.

• Have students work individually or collaboratively toexplore patterns by:- working at concrete activities using simple cases (e.g.,

dividing a circle by lines to create patterns)- developing a pattern using, for example, two or more

shapes, colours, or textures- differentiating between growing and repeating patterns- changing a growing pattern to a repeating pattern- changing a given pattern to a new one- determining the formula to extend the patternEncourage students to use concrete materials where possible(e.g., algebra tiles, algebra lab gear, two-colour counters).Remind them that there are many possible ways ofdescribing patterns.

• Ask students to bring in examples or pictures of patternsfrom their surroundings (e.g., flower petals, architecture,needles on a tree, ploughed fields). Invite them to makeconjectures about the patterns. Ask students:- How did these patterns occur?- How would you go about finding or creating a pattern?

• Display a table of ordered pairs. Have students work withpartners to determine the rule that was used to generate eachpair. Students could then create patterns using their ownrules and challenge their partners to generate the rules fromthe patterns.

• Have students work in groups to examine graphs from avariety of sources and attempt to interpret their meaning.

• Brainstorm terms that have the same meaning, such as sum,

difference, product, and quotient. Have students developposters of these terms, which can be put up in the classroom.

• Ask students to suggest one number each, and respond totheir numbers with others based on a pattern or equation.Ask students to plot each pair of numbers. Continue untilstudents can guess what the pattern is.

Students use higher-order thinking skills to identifypatterns and to generalize. Assessment should provideopportunities for students to demonstrate their problem-solving skills.

Question

• As students solve problems using patterns, variables,expressions, equations, and graphs, ask them toexplain the methods and processes they are using.Provide feedback to students concerning theirapplication of problem-solving approaches.

Collect

• Ask students to annotate their work to describe theprocesses they use to solve problems. Alternatively,students could provide brief descriptions of whichprocesses did and did not work.

Self-Assessment

• Work with students to generate a set of criteria thatcan be used to evaluate problem-solving skills. Usethe criteria to create a rating scale that students canuse to evaluate their own skills. Criteria shoulddemonstrate:- a willingness to persevere to solve difficult

problems- flexibility in trying different approaches

• Note: Refer to Suggested Assessment Strategies In

Problem Solving section.

Print Materials

• Interactions (Level 8)• MATHPOWER 8, Western Edition

Multimedia

• The Learning Equation Mathematics 8 (TLE)• Math Tools• Mathematics 8 (Distance Education Package)• Minds on Math 8, Revised Edition• Understanding Math Series

CD-ROM

• Mathville VIP

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INTRODUCTION

This Integrated Resource Package (IRP) setsout the provincially prescribed curriculumfor Mathematics 8 and 9. The developmentof this IRP has been guided by the principlesof learning:

• Learning requires the active participationof the student.

• People learn in a variety of ways and atdifferent rates.

• Learning is both an individual and agroup process.

THE DEVELOPMENT OF THIS INTEGRATED

RESOURCE PACKAGE

A variety of resources were used in thedevelopment of this IRP:

• The prescribed learning outcomes,suggested instructional strategies,suggested assessment strategies, andillustrative examples for Mathematics 8and 9 were developed with reference toThe Common Curriculum Framework for K to9 Mathematics (Western Canadian Protocolfor Collaboration in Basic Education,1995).

• Other resources used include:- Principles and Standards for School

Mathematics (National Council ofTeachers of Mathematics, 2000)

- Guidelines for Student Reporting- provincial reference sets Evaluating

Problem Solving Across Curriculum andEvaluating Mathematical DevelopmentAcross Curriculum

- BC Performance Standards, Numeracy (2000)- Assessment Handbook Series- Summary of Responses to the Mathematics

8 and 9 IRP- Report of the Mathematics Task Force

(June 1999)

This IRP represents the ongoing effort of theprovince to provide education programs thatplace importance on high standards ineducation while providing equity and accessfor all learners.

RATIONALE

Mathematics is increasingly important in ourtechnological society. Students today requirethe ability to reason and communicate, tosolve problems, and to understand and usemathematics. Development of these skillshelps students become numerate.

Numeracy can be defined as the combinationof mathematical knowledge, problemsolving, and communication skills requiredby all persons to function successfully withinour technological world. Numeracy is morethan knowing about numbers and numberoperations. (BC Association of MathematicsTeachers, 1998)

Being numerate involves the application ofmathematical understanding in dailyactivities at school, at home, at work, and inthe community. To become numerate,students need to develop the ability toconjecture, reason logically, employquantitative and spatial information, andapply a variety of mathematical methods tosolve problems and make decisionsconfidently and independently.

The ability to recognize the mathematicaldemands and possibilities in a situation is animportant aspect of numeracy. Numeracy isbased on mathematical foundations andrequires the application of concepts andskills related to the formal aspects of thediscipline of mathematics.

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To ensure that students are prepared for thedemands of further education and theworkplace, the mathematics curriculum:• emphasizes the development of the

knowledge, skills, and attitudes relevant tothe development of numeracy

• supports the creative and aesthetic aspectsof mathematics by exploring the connectionsbetween mathematics, art, and design

• promotes the development of positiveattitudes, problem solving,communication, applications, reasoning,and the use of technology

Developing Positive Attitudes

Research, including provincial assessments,consistently indicates that there is a positivecorrelation between student attitudes andperformance. A classroom climate thatpromotes positive attitudes about the valueand relevance of mathematics to students’lives gives them the desire to succeed andthe confidence that their efforts areworthwhile. Classroom activities that allowstudents to apply their learning to practicalsituations help them see that mathematics isapplicable to daily living and valuable tofuture education and employment.

Becoming Mathematical Problem Solvers

Problem solving is the cornerstone ofmathematics instruction. Students workingalone and in groups must learn the skills ofeffective problem solving, which include theability to:• read and analyse a problem• identify the significant elements of a

problem• select an appropriate strategy to solve a

problem• verify and judge the reasonableness of an

answer• communicate solutions

Acquiring these skills can help studentsbecome reasoning individuals able tocontribute to society.

As students move through the grades, thecurriculum presents them with increasinglydiverse and complex mathematical problemsto solve. To encourage students’ abilities tocommunicate, explore, create, adjust tochanges, and actively acquire newknowledge throughout their lives,mathematical problem solving shouldevolve naturally out of their experiences andbe an integral part of all mathematicalactivity. Effective problem solving consists ofmore than being able to solve many differenttypes of problems. Students need to be able tosolve mathematical problems that arise in anysubject area and to draw upon skills developedin more than one area of mathematics.Becoming a mathematical problem solverrequires a willingness to take risks andpersevere when faced with problems that donot have an immediately apparent solution.

Communicating Mathematically

Mathematics is a language—a way ofcommunicating ideas. Communication playsan important role in helping students buildlinks between their informal, intuitivenotions and the abstract language andsymbolism of mathematics. Communicationalso plays a key role in helping studentsmake important connections amongphysical, pictorial, graphic, symbolic, verbal,descriptive, and mental representations ofmathematical ideas. All activities that helpstudents explore, explain, investigate,describe, and justify their decisions promotethe development of communication skills.

The Kindergarten to Grade 12 Mathematicscurriculum emphasizes discussing, writing,and representing mathematical thinking invarious ways.

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Connecting and Applying Mathematical Ideas

Learning activities should help studentsunderstand that mathematics is a changingand evolving domain to which manycultural groups have contributed. Studentsbecome aware of the usefulness ofmathematics when mathematical ideas areconnected to everyday experiences. Learningactivities should help students relatemathematical concepts to realistic situationsand allow them to see how one mathematicalidea can help them understand others.Mathematics helps students solve problems,describe and model real-world phenomena,and communicate complex thoughts andinformation with conciseness and precision.

Reasoning Mathematically

Mathematics instruction should helpstudents develop confidence in their abilitiesto reason and to justify their thinking.Students should understand thatmathematics is not simply memorizing rules.Mathematics should make sense, be logical,and also be enjoyable. The ability to reasonlogically usually develops on a continuumfrom concrete to formal to abstract. Studentsuse inductive reasoning when they makeconjectures by generalizing from a pattern ofobservations; they use deductive reasoningwhen they test those conjectures. To developmathematical reasoning skills, studentsrequire the freedom to explore, conjecture,and validate and to convince others. It isimportant that their ability to reason isvalued as much as their ability to findcorrect answers.

Using Technology

The Grades 8 and 9 Mathematics curriculumrequires students to be proficient in usingtechnology as a problem-solving tool.

Computers and calculators are tools forlearning. They are powerful aids to problemsolving. The ability to compute rapidly,analyse data in various ways, and graphmathematical relationships instantly canhelp students explore mathematical conceptsand relationships in greater depth. Studentsmust have access to and use the mostappropriate tool or method for a calculation.

It is important to recognize that calculatorsassist in rote calculation or algorithms, butpeople accomplish the work at hand. At thesame time, access to technology does noteliminate the need for students to learn basicfacts and algorithms.

Estimation and Mental Math

Mathematics involves more than exactness.Estimation strategies help students deal witheveryday quantitative situations. Estimationskills also help them gain confidence andenable them to determine if something ismathematically reasonable. Although theymay have access to calculators fromKindergarten to Grade 12, students need touse reasoning, judgment, and decision-making strategies when estimating.Instruction should therefore emphasize therole that these strategies play.

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Essentials ofMathematics 12

B.C. SECONDARY MATHEMATICS COURSE STRUCTURE

Mathematics 8

Mathematics 9



Principles ofMathematics 10

Principles of Mathematics 11

Applications ofMathematics 10


Principles of Mathematics 12

Calculus 12


Note: The diagram above does not show all possible student transitions between theApplications of Mathematics pathway, the Essentials of Mathematics Pathway, and thePrinciples of Mathematics pathway.

The course structure for secondary mathematics was designed with the expectationthat approximately 50% of students would be registered in Applications of Mathematics,20% of students in Essentials of Mathematics, and 30% in Principles of Mathematics.

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BC SECONDARY MATHEMATICS COURSE

STRUCTURE

The mathematics curriculum for Grades 8 to12 offers students a choice of routes throughthe different mathematics courses offered.

Mathematics 8 and 9

The Mathematics 8 and 9 curriculumprovides all students with the opportunity todevelop the knowledge, skills, and attitudesnecessary to be numerate.

The prescribed learning outcomes for theMathematics 8 and 9 curriculum form thebasis of preparing students for Applicationsof Mathematics 10, Essentials of Mathematics10, and Principles of Mathematics 10.

At the bottom of each column 1 in the bodyof this IRP, Suggested Extensions areidentified. The Suggested Extensions are notprovincial curriculum but are provided toassist teachers in developing programs ofstudy that go beyond the provincialcurriculum. Using the suggested extensionscan provide students with opportunities forenrichment, giving them the ability toexplore additional topics and add greaterbreadth and depth to the topics under study.

Students intending to take Principles ofmathematics 10 should explore most of thesuggested extensions. Successful completionof most of the suggested extensions willimprove students’ preparation for Principlesof Mathematics 10. It is the responsibility ofteachers to determine which, if any, of thesuggested extensions their students shouldstudy as teachers are in the best position tomake these decisions.After completion of Mathematics 9, studentsmust choose one of three provinciallydeveloped mathematics programs of study:

• Applications of Mathematics• Essentials of Mathematics

• Principles of Mathematics

Both Applications of Mathematics 12 andPrinciples of Mathematics 12 have aprovincial examination component. Studentswho successfully complete Applications ofMathematics 11, Essentials of Mathematics11, or Principles of Mathematics 11 will meetBritish Columbia’s graduation requirements.

In all three pathways, teachers are expectedto adapt instruction, learning resources, time,setting, or any other factors to support theirstudents in successfully meeting theoutcomes of the curriculum.

The Applications of Mathematics Pathway

The Applications of Mathematics pathwayprovides a practical focus that encouragesstudents to develop their mathematicalknowledge, skills, and attitudes in thecontext of their lives and possible careers.The instructional approaches usedemphasize concrete activities and modelling,with less emphasis on symbol manipulation.

Students following the Applications ofMathematics pathway will be prepared formany postsecondary programs that do notrequire calculus as part of the program ofstudy. The Applications of Mathematicscurriculum is intended to prepare studentsfor entrance into some university degree,certificate, diploma, continuing education,trades, and technical programs, none ofwhich require calculus.

The Essentials of Mathematics Pathway

The Essentials of Mathematics pathwayfocusses on the development of a sense ofnumeracy that will help students understandhow mathematical concepts permeate dailylife, business, industry, and government. Theinstructional approaches used to develop therequired numeracy knowledge, skills, andattitudes emphasize concrete activities andmodelling.

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Students following this pathway will beprepared to use mathematics in theirpersonal lives as citizens and consumers andwill be prepared to enter a limited number ofpostsecondary programs such as trades andvocational programs.

The Principles of Mathematics Pathway

Students following the Principles ofMathematics pathway will focus ondeveloping an understanding of theoreticalmathematics concepts such as algebra,trigonometry, functions, statistics, andprobability.

One of the primary purposes of Principles ofMathematics is to develop the formalismstudents will need to pursue a wide range ofpostsecondary programs, particularly thosethat require the study of calculus, such asmathematics, science, and engineering.

Calculus 12: This course is intended forstudents who have completed (or are takingconcurrently) Principles of Mathematics 12or who have completed an equivalent collegepreparatory course that includes algebra,geometry, and trigonometry.

Students taking Calculus 12 should beprepared to write the UBC, SFU, UVic,UNBC Challenge Examination if they chooseto do so. For more information concerningthe Challenge Examination, contact themathematics department at one of the above-mentioned universities.

Some schools may choose to developarticulation agreements with their localcolleges. Students under these agreementsmay receive credit for first-term calculus(depending upon the particular agreement).

ORGANIZATION OF THE CURRICULUM

The prescribed learning outcomes for thecourses described in this Integrated ResourcePackage are grouped under five curriculumorganizers.

• Problem Solving• Number• Patterns and Relations• Shape and Space• Statistics and Probability

These curriculum organizers reflect the mainareas of mathematics that students areexpected to address. They form theframework of the curriculum and act asconnecting threads across all grade levels foreach pathway. The organizers are notequivalent in terms of number of outcomesor the time that students will require in orderto achieve these outcomes. Suggestions forestimated instructional times have beenincluded in this IRP. Teachers are expected toadjust these estimated instructional times tomeet their students’ diverse needs.

A broad statement of the associated generalgoal for mathematics introduces each set ofprescribed learning outcomes. (Materialrelated to the general goals or suborganizersis not addressed in every course.)

The ordering of organizers and outcomes inthe Grades 8 and 9 Mathematics curriculumdoes not imply an order of instruction. Theorder in which various outcomes and topicsare addressed is left to the professionaljudgment of teachers.

SUGGESTED INSTRUCTIONAL STRATEGIES

Instructional strategies have been includedfor each curriculum organizer (orsuborganizer) and grade level. Thesestrategies are suggestions only, designed to

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provide guidance for generalist andspecialist teachers planning instruction tomeet the prescribed learning outcomes.Some links to other subjects are indicated.

The strategies may be teacher-directed,student-directed, or both. There is notnecessarily a one-to-one relationshipbetween learning outcomes and instructionalstrategies, nor is this organization intendedto prescribe a linear approach to coursedelivery. It is expected that teachers willadapt, modify, combine, and organizeinstructional strategies to meet the needs ofstudents and respond to local requirements.

Context Statements

Each set of instructional strategies starts witha context statement followed by severalexamples of learning activities. The contextstatement links the prescribed learningoutcomes with instruction. It also states whythese outcomes are important for thestudent’s mathematical development.

Instructional Activities

The mathematics curriculum is designedto emphasize the skills needed in theworkplace, including those involving theuse of probability and statistics, reasoning,communicating, measuring, andproblem solving.

Additional emphasis is given to strategiesand activities that:

• foster the development of positive attitudes

Students should be exposed to experiencesthat encourage them to enjoy and valuemathematics, develop mathematical habits ofmind, and understand and appreciate therole of mathematics in human affairs. Theyshould be encouraged to explore, take risks,

exhibit curiosity, and make and correct errorsso that they gain confidence in their abilitiesto solve complex problems. The assessmentof attitudes is indirect and is based oninferences drawn from students’ behaviour.We can see what students do and hear whatthey say and, from these observations, makeinferences and draw conclusions about theirattitudes.

• apply mathematics

For students to view mathematics as relevantand useful, they must see how it can beapplied to a wide variety of real-worldapplications. Mathematics helps studentsunderstand and interpret their world andsolve problems that occur in their daily lives.

• use manipulatives

Using manipulatives is an effective way toactively involve students in mathematics.Manipulatives encourage students toexplore, develop, estimate, test, and applymathematical ideas in relation to the physicalworld. Manipulatives range from commercialmaterials to simple collections of materialssuch as boxes, cans, or cards. They can beused to introduce new concepts or to providea visual model of a mathematical concept.

• use technology

The use of technology in our society isincreasing. Technological skills are becomingmandatory in the workplace. Instruction andassessment strategies that use a range oftechnologies such as calculators, computers,CD-ROMs, and videos will help studentsrelate mathematics to their personal livesand prepare them for the future. The use oftechnology in developing mathematicalconcepts and as an aid in solving complexproblems is encouraged to a greater extent asthe student moves from grade to grade.

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• require problem solving

For students to develop decision-making andproblem-solving skills, they need learningexperiences that challenge them to recognizeproblems and actively try to solve them, todevelop and use various strategies, and tolearn to represent solutions in waysappropriate to their purposes. Problems thatoccur within students’ environment can beused as the vehicle or context for students toachieve the learning outcomes in any of thecurriculum organizers.

Instructional Focus

The Grades 8 and 9 Mathematics courses arestructured by a number of organizers,including Problem Solving. Fluency withnumber facts is essential to success inmathematics. However, decreasing emphasison rote calculation, drill and practice, andthe size of numbers used in paper-and-pencilcalculations allows more time for conceptdevelopment.

In addition to problem solving, other criticalthinking processes—reasoning and makingconnections—are vital to increasing students’mathematical power and must be integratedthroughout the program. A minimum of halfthe available time within all organizersshould be dedicated to activities related tothese processes.

Instruction should provide a balancebetween estimation and mental mathematics,paper-and-pencil exercises, and theappropriate use of technology, includingcalculators and computers. (It is assumedthat all students have regular access toappropriate technology such as calculatorsor computers.) Concepts should beintroduced using manipulatives andgradually developed from the concrete to thepictorial to the symbolic.

INTEGRATION OF CROSS-CURRICULAR

INTERESTS

Throughout the curriculum developmentand revision process, the development teamhas done its best to ensure that relevance,equity, and accessibility issues are addressedin this IRP. Wherever appropriate for thesubject, these issues have been integratedinto the learning outcomes, suggestedinstructional strategies, and suggestedassessment strategies. Although anexhaustive list of such issues is neitherpractical nor possible, teachers areencouraged to continue to ensure thatclassroom activities and resources alsoincorporate appropriate role portrayals,relevant issues, and exemplars of themessuch as inclusion and acceptance.

The ministry, in consultation withexperienced teachers and other educators,has developed a set of criteria to be used toevaluate learning resources. Althoughneither exhaustive nor prescriptive, most ofthese criteria can be usefully applied toinstructional and assessment activities aswell as learning resources. Brief descriptionsof these criteria, grouped under the headingsof Content, Instructional Design, TechnicalDesign, and Social Considerations, may befound on pages 30 through 45 of the ministrydocument Evaluating, Selecting, and ManagingLearning Resources (2000). This document hasbeen distributed to all schools. Additionalcopies may be ordered from Office ProductsCentre (telephone 1-800-282-7955) by citingdocument number RB0065.

Gender Issues in Mathematics

The education system is committed tohelping both male and female studentssucceed equally well. In British Columbia,significant progress has been made in

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improving the participation and success rateof female students in secondary mathcourses. They now take about the samenumber of secondary math courses as males.There continues, however, to be a relativelylow rate of female participation in math-related careers and education. Positiveattitudes toward the practice of mathematics,as well as skill in mathematics, are essentialto the workplace and to everyone’s ability toparticipate fully in society. Teaching,assessment materials, learning activities, andclassroom environments should place valueon the mathematical experiences andcontributions of both men and women andpeople of diverse cultures.

Research regarding gender and mathematicshas raised a number of important issues thatteachers should consider when teachingmathematics. These include the diversity oflearning styles, gender bias in learningresources, and unintentional gender bias inteaching. The following instructional strategiesare suggested to help the teacher deliver agender-sensitive mathematics curriculum.

As guest speakers or subjects of study in theclassroom, feature both females and maleswho are mathematicians or who makeextensive use of mathematics in their careers.

• Design instruction to acknowledgedifferences in experiences and interestsbetween young women and young men.

• Demonstrate the relevance of mathematicsto a variety of careers and to everyday lifein ways that are apt to appeal to particularstudents in the class or school. Successfullinks include biology, environmentalissues, and current topics in mass media.

• Explore not only the practical applicationsof mathematics but also the humanelements, such as ways in which ideas

have changed throughout history and thesocial and moral implications ofmathematics.

• Explore ways of approaching mathematicsthat will appeal to a wide variety ofstudents. Use co-operative rather thancompetitive instructional strategies. Focuson concept development, encouragingstudents to question until they can say“I’ve got it.” Include a wide variety ofapplications that demonstrate the role ofmath in the social fabric of our world.Varying approaches appeal to a widervariety of students.

• Emphasize that ordinary people with avariety of interests and responsibilities usemathematics.

• Allow for informal social interaction withsuccessful “math-using” members of thecommunity to help change the negativestereotypes of mathematicians and theirsocial style.

• Provide opportunities for visual andhands-on activities, which most studentsenjoy. Experiments, demonstrations, fieldtrips, and exercises that provideopportunities to explore the relevance ofmathematics are particularly important.

Adapting Instruction for Diverse StudentNeeds

Teachers will need to adapt theirinstructional approach to meet the diverselearning needs of their students. This mayinclude English-as-a-second-languagestudents (ESL), students with special needs,or students from a variety of cultural andsocial backgrounds. For example, teachingESL students in any subject area includingmathematics should involve a focus onlanguage development.

10


The following strategies may help ESLstudents and students with special needssucceed in mathematics:

• Adapt the Environment- Arrange the classroom to foster

attention and interaction.- Make use of co-operative grouping.- Reinforce important mathematics ideas

with visual representations.

• Adapt Presentations- Provide students with advance

organizers of the key mathematicalconcepts.

- Demonstrate or model new conceptsand provide practice using functional,everyday contexts such as cooking.

- Adapt the pace of activities as required.- Reword questions so they match the

level of understanding of the students.

• Adapt Materials- Use techniques such as colour-coding

the steps to solving a problem, to makethe organization of activities moreexplicit.

- Use manipulatives such as large dice,cards, and dominoes.

- Use large-print charts such as a 100schart or a times-table chart.

- Provide students with a talkingcalculator or a calculator with a largekeypad.

- Use large print on activity sheets.- Use a variety of resources that represent

various levels of complexity.- Highlight key points on activity sheets.

• Adapt Methods of Assistance- Have peers or volunteers assist students

with special needs.- Have students with special needs help

younger students learn mathematics.

- Have teacher assistants work withindividuals and small groups ofstudents with special needs.

- Work with consultants and supportteachers to develop problem-solvingactivities and strategies for mathematicsinstruction for students with specialneeds.

- Collaborate with specialist staff such asESL or Learning Assistance teachers tosupport students.

• Adapt Methods of Assessment- Allow students to demonstrate their

understanding of mathematicalconcepts in a variety of ways, such asmurals, displays, models, puzzles, gameboards, mobiles, and tape recordings.

- Modify assessment tools to matchstudent needs. For example, oral tests,open-book tests, and tests with no timelimit may allow students to betterdemonstrate their learning than atraditional timed paper-and-pencil test.

- Set achievable goals.- Use computer programs that provide

opportunities for students to practisemathematics as well as record and tracktheir results.

- Require fewer tasks to demonstratelearning. Focus on quality of learningand mastery rather than on volume ofwork done.

When students with special needs areexpected to achieve or surpass the learningoutcomes set out in the mathematicscurriculum, regular grading practices andreporting procedures are followed.Adaptations to the environment,presentations, materials, methods ofassistance, and methods of assessment canbe done while still using regular reportingpractices. When students with special needsare not expected to achieve the learning

11


outcomes and individual goals and objectivesare set for them, it is called modification.Modifications must be noted in an IndividualEducation Plan (IEP), and reporting should bebased on these modified goals.

SUGGESTED ASSESSMENT STRATEGIES

Teachers determine the best assessmentmethods for their students. The assessmentstrategies in this document describe a varietyof ideas and methods for gathering evidenceof student performance. The assessmentstrategies column for a particular organizeralways includes specific examples ofassessment strategies. Some strategies relateto particular activities, while others aregeneral and could apply to any activity.These specific strategies may be introducedby a context statement that explains howstudents at this age can demonstrate theirlearning, what teachers can look for, andhow this information can be used to adaptfurther instruction.

About Assessment in General

Assessment is the systematic process ofgathering information about students’learning in order to describe what theyknow, are able to do, and are workingtoward. From the evidence and informationcollected in assessments, teachers describeeach student’s learning and performance.They use this information to providestudents with ongoing feedback, plan furtherinstructional and learning activities, setsubsequent learning goals, and determineareas requiring diagnostic teaching andintervention. Teachers base their evaluationof a student’s performance on theinformation collected through assessment.

Teachers determine the purpose, aspects, orattributes of learning on which to focus theassessment; when to collect the evidence;and the assessment methods, tools, ortechniques most appropriate to use.Assessment focusses on the critical orsignificant aspects of the learning to bedemonstrated by the student.

The assessment of student performance isbased on a wide variety of methods andtools, ranging from portfolio assessment topencil-and-paper tests to the use of the BCPerformance Standards for Numeracy.Appendix C includes a more detaileddiscussion of assessment and evaluation.

12


MATHEMATICS CURRICULUM STRUCTURE

Number ConceptsStudents:• use numbers to describe quantities• represent numbers in multiple ways

Number OperationsStudents:• demonstrate an understanding of

and proficiency with calculations• decide which arithmetic operation

or operations can be used to solvea problem then solve the problem

PatternsStudents:• use patterns to describe the world

around them and solve problems

Variables and EquationsStudents:• represent algebraic expressions in

multiple ways

Relations and FunctionsStudents:• use algebraic and graphical models

to generalize patterns, makepredictions, and solve problems

MeasurementStudents:• describe and compare real-world

phenomena using either direct orindirect measurement

3-D Objects and 2-D ShapesStudents:• describe the characteristics of 3-D

objects and 2-D shapes and analysethe relationships among them

TransformationsStudents:• perform, analyse, and create

transformations

Data AnalysisStudents:• collect, display, and analyse data to

make predictions about apopulation

Chance and UncertaintyStudents:• use experimental or theoretical

probability to represent and solveproblems involving uncertainty

Grade 7 Grade 8 Grade 9Problem Solving

Number

Statistics and Probability

Patterns and Relations

In Grade 7, problem solving isintegrated into the learningoutcomes of all organizers.

• demonstrate a number sense fordecimal fractions and integers(including whole numbers)

• apply arithmetic operations ondecimal fractions and integersand illustrate their use in solvingproblems. Illustrate the use ofratios, rates, percentages, anddecimal numbers in solvingproblems

• express patterns in terms ofvariables and use expressionscontaining variables to makepredictions

• use variables and equations toexpress, summarize, and applyrelationships as problem-solvingtools in a restricted range ofcontexts

Learning outcomes commence inGrade 10.

• solve problems involving theproperties of circles and theirrelationships to angles and timezones

• link angle measurements to theproperties of parallel lines

• create and analyse patterns anddesigns using congruence,symmetry, translation, rotation,and reflection

• develop and implement a plan forthe collection, display, and analysisof data using measures ofvariability and central tendency

• create and solve problems usingprobability

• use a variety of methods to solvereal-life, practical, technical, andtheoretical problems

• demonstrate a number sense forrational numbers, includingcommon fractions, integers, andwhole numbers

• apply arithmetic operations onrational numbers to solveproblems

• apply the concepts of ratio, rate,percent, and proportion to solveproblems in meaningful contexts

• use patterns, variables andexpressions, and graphs to solveproblems

• solve and verify one-step linearequations with rational numbersolutions


• apply indirect measurementprocedures to solve problems

• generalize measurement patternsand procedures and solve problemsinvolving area and perimeter

• link angle measures and theproperties of parallel lines to theclassification and properties ofquadrilaterals

• analyse design problems andarchitectural drawings using theproperties of scaling, proportion,and networks

• develop and implement a plan forthe collection, display, and analysis ofdata, using technology as required

• evaluate and use measures ofcentral tendency and variability

• compare theoretical andexperimental probability ofindependent events


• develop a number sense ofpowers with integral exponentsand variable and rational bases

• use a scientific calculator or acomputer to solve problemsinvolving rational numbers

• generalize, design, and justifymathematical procedures usingappropriate patterns, models, andtechnology

• evaluate, solve, and verify linearequations in one variable

• generalize arithmetic operationsfrom the set of rational numbersto the set of polynomials


• use trigonometric ratios to solveproblems involving right triangles

• use spatial problem solving inbuilding, describing, and analysinggeometric shapes

• specify conditions under whichtriangles may be similar orcongruent and use theseconditions to solve problems

• collect data and analyseexperimental results expressed intwo variables, using technology asrequired

• explain the use of probability andstatistics in the solution ofproblems

Shape and Space

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Number Concepts

Number Operations

Patterns

Variables and Equations

Relations and Functions

Measurement

3-D Objects and 2-DShapes

Data Analysis

Chance and Uncertainty

Essentials of Mathematics 10Problem Solving

Problem Solving

Number


• prepare bank forms includingcheques, deposit slips,chequebook activity record andreconcilation statements

• solve problems involving wage,salaries and expenses

• design and use a spreadsheet tomake and justify decisions

• apply the concepts of rate, ratioand proportion to solve problems

• demonstrate an understanding ofratio and proportion and applythese concepts in solving triangles

• complete a project that includes a2-D plan and a 3-D model ofsome physical structure

• develop and implement a plan forthe collection, display and analysisof data using technology asrequired

Statistics and Probability

Applications of Math 10


• analyse the numerical data in atable for trends, patterns, andinterrelationships

• use basic arithmetic operationson real numbers to solveproblems

• describe and apply arithmeticoperations on tables to solveproblems, using technology asrequired

• examine the nature of relationswith an emphasis on functions

• represent data, using functionmodels

• demonstrate an understanding ofscale factors and theirinterrelationship with thedimensions of similar shapes andobjects

• solve problems involving triangles,including those found in 3-D and2-D applications

• solve coordinate geometryproblems involving lines and linesegments

• implement and analyse samplingprocedures and draw appropriateinferences from the datacollected

Principles of Math 10


• analyse the numerical data in atable for trends, patterns, andinterrelationships

• explain and illustrate the structureand the interrelationships of thesets of numbers within the realnumber system

• use basic arithmetic operations onreal numbers to solve problems

• describe and apply arithmeticoperations on tables to solveproblems, using technology asrequired

• generate and analyse numberpatterns

• generalize operations onpolynomials to include rationalexpressions

• examine the nature of relationswith an emphasis on functions

• represent data, using linearfunction models

• demonstrate an understanding ofscale factors and theirinterrelationship with thedimensions of similar shapes andobjects

• solve problems involving triangles,including those found in 3-D and2-D applications

• solve coordinate geometryproblems involving lines and linesegments

• implement and analyse samplingprocedures and draw appropriateinferences from the datacollected

Rate, Ratio and Proportion

Probability and Sampling

Personal Banking

Wages, Salaries and Expenses

Spreadsheets

Patterns and Relations

TrigonometryShape and Space

Geometry Project

RECOMMENDED LEARNING RESOURCESSUGGESTED ASSESSMENT STRATEGIES

15

CURRICULUMMathematics 8

PRESCRIBED LEARNING OUTCOMES SUGGESTED INSTRUCTIONAL STRATEGIES

16


17

ESTIMATED INSTRUCTIONAL TIME

Mathematics 8 has been developed assuming that teachers have 100 instructional hours available tothem. The following chart shows the estimated instructional time for each curriculum suborganizer,expressed as a percentage of total time available to teach the course.

When delivering the prescribed curriculum, teachers may freely adjust the instructional time tomeet their students’ diverse needs. These estimated instructional times have been recommended bythe IRP writers to assist their colleagues. They are suggestions only.

Organizer (Suborganizer) % of Time

Problem Solving Integrated Throughout

Number (Number Concepts) 10 - 15

Number (Number Operations) 20 - 25

Patterns and Relations (Patterns) 5 - 15

Patterns and Relations (Variables and Equations) 10 - 15

Shape and Space (Measurement) 5 - 10

Shape and Space (3-D Objects and 2-D Shapes) ≈5

Shape and Space (Transformations) ≈5

Statistics and Probability (Data Analysis) 5 - 10

Statistics and Probability (Chance and Uncertainty) 10 - 15

MATHEMATICS 8


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MATHEMATICS 8 • Problem Solving

In order to prepare students to use a variety ofmethods to solve real-life, practical, technical,and theoretical problems, it is expected thatstudents will:

• solve problems that involve a specific contentarea (e.g., geometry, algebra, statistics,probability)

• solve problems that involve more than onecontent area within mathematics

• solve problems that involve mathematicswithin other disciplines

• analyse problems and identify the significantelements

• develop specific skills in selecting and usingan appropriate problem-solving strategy orcombination of strategies chosen from, but notrestricted to, the following:- guess and check- identify patterns and use a systematic list- make and use a drawing or model- eliminate possibilities- work backward- simplify the original problem- select and use appropriate technology to

assist in problem solving- analyse keywords

• solve problems individually and co-operatively

• determine that solutions to problems arecorrect and reasonable

• clearly and logically communicate a solutionto a problem and the process used to solve it

• evaluate the efficiency of the processes used• use appropriate technology to assist in

problem solving

Problem solving is a key aspect of any mathematicscourse. Working on problems in mathematics can givestudents a sense of the excitement involved in creativeand logical thinking. It can also help students developtransferable real-life skills and attitudes. Multi-strandand interdisciplinary problems should be includedthroughout Mathematics 8.

• Reinforce the concept that problem solving is morethan just word problems and includes aspects ofmathematics other than algebra (e.g., geometry,statistics, probability).

• Introduce new problems directly to students(without demonstration) and play the role offacilitator as they attempt to solve the problems.

• Recognize when students use a variety of approaches(e.g., algebraic or geometric). Avoid becomingprescriptive about approaches to problem solving.

• Emphasize that problems might not be solved inone sitting and that “playing around” with theproblem—revisiting it and trying again—issometimes needed.

• Assign students a set of problems that fit aparticular strategy. After sufficient time, studentsdraw a number to find which solutions to presentto the class. After about five minutes (to givestudents a chance to complete all problems), beginthe presentations. Tell students that they may usethe presentations of others to complete their work.

• Ask directed questions such as:- What are you being asked to find out?- What do you already know?- Do you need additional information?- Have you ever seen similar problems?- What else can you try?

• Once students have arrived at solutions toparticular problems, encourage them to generalizeor extend the problem situation.

• Encourage students to maintain mathematicsjournals for recording the things they are learningand any difficulties they may be having.

• Note: See Appendix F for examples of multi-strand andinterdisciplinary problems that most students should beable to solve. These problems are indicated with anasterisk (*).


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Students analyse problems and solve them using avariety of approaches. Assessment of problem-solving skills is made over time, based onobservations of many situations.

Observe• Have students present solutions to the class

individually, in pairs, or in small groups. Note theextent to which they are able to:- articulate and clarify problems- describe the processes used- describe what worked and what did not- identify ways to get additional information as

needed- find alternative methods- link mathematics to new situations

Question• To check the approaches students use when solving

problems, ask questions that prompt them to:- paraphrase or describe problems in their own

words- explain the processes used to derive answers- describe alternative methods to solve the

problems- relate the strategies used in new situations- link mathematics to other subjects and to the

world of work

Collect• On selected problems, have students annotate

their work to describe the processes they used.Alternatively, have them provide briefdescriptions of problems that worked and thosethat did not.

Self-Assessment• Ask students to keep journals of problems they are

working on, describing where they find problemsand reflecting on the processes they used indealing with problems. Have students describestrategies that worked and those that did not.

• Develop with students a set of criteria to assesstheir own problem solving. The reference setEvaluating Problem Solving Across Curriculum maybe helpful in identifying such criteria.

The Western Canada Protocol Learning ResourceEvaluation Process identified numerous teacherresources and professional references. These aregenerally cross-grade planning resources thatinclude ideas for a variety of activities andexercises.

These resources, while not part of the GradeCollections, have Provincially Recommendedstatus.

Appendix B includes an annotated bibliographyof these resources for ordering convenience.


20

MATHEMATICS 8 • Number (Number Concepts)

In order to prepare students to demonstrate anumber sense for rational numbers, includingcommon fractions, integers, and whole numbers,it is expected that students will:

• define, identify, compare, and order anyrational numbers

• express two-term ratios in equivalent forms• represent and apply fractional percents and

percents greater than 100 in fraction ordecimal form and vice versa

• represent square roots concretely, pictorially,and symbolically

• distinguish between a square root and itsdecimal approximation as it appears on acalculator

• express rates in equivalent forms• represent any number in scientific notation


To extend students’ understanding of numberconcepts, they could:

• express three-term ratios in equivalent forms• demonstrate and explain the meaning of a

negative exponent, using patterns (limit tobase 10)

• demonstrate concretely, pictorially, andsymbolically that the product of reciprocals isequal to 1

The development of students’ number sense requiresthat they incorporate new types of numbers,including common fractions, integers, and wholenumbers, into their understanding of the numbersystem. Linking new concepts to prior understandingand working from concrete to abstractrepresentations are essential in developing numeracy.

• Invite students to work in pairs to examineproblems requiring operations with very large orvery small numbers. Demonstrate how torepresent these numbers in decimal (“normal”)notation and in scientific notation. Discuss thebenefits of each method.

• Ask students to consider the following strategies:- divide any number by numbers progressing

closer to zero and record the results- divide by zero and use multiplication to check

the solutions.

- Example: 11—0

= 0 therefore

0 • 0 = 11 false

11—0

= 11 therefore

0 • 11 = 11 false

• Ask students to represent the same number (e.g.,5) in as many ways as possible (e.g., ratio, fraction,percent, decimal). Have them identify and labelthe form for each representation.

• Bring in examples of ratios, percents, decimals,and fractions from newspapers or trade magazinesor by searching the Internet. Discuss with the classwhy the form might have been chosen in eachcase. Does one form tend to be used more often inparticular types of media?

• Give groups of students the same set of rationalnumbers to order on a number line. As each grouppresents their number lines to the class, have themjustify the placement of each number.

• Have students calculate the square root of a numberusing a calculator and round the results to:- no decimal places- one decimal place- two decimal places, etc.Have students explain in writing how their resultsdiffer in each situation and how their results mayaffect further calculations.


21


By demonstrating an understanding of the rationalnumber system, students can show they possess thebasic knowledge needed to use mathematics to solveproblems. Assessment focusses on determiningstudents’ abilities to work with rational numbers.

Observe• To what extent are students able to represent a

simple number (e.g., a percent, a decimal, afraction)? Determine the degree of accuracy in thelabeling.

Collect• Give students lists of fractions and decimals to

arrange in numerical order and place on a numberline. Have students develop lists of rules based ontheir work.- Can students correctly order and place

numbers?- Do they make conversions correctly?- Do the rules they identify accurately reflect the

process?

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MATHEMATICS 8 • Number (Number Operations)

In order to prepare students to apply arithmeticoperations on rational numbers to solveproblems and apply the concepts of rate, ratio,percent, and proportion to solve problems inmeaningful contexts, it is expected that studentswill:

• add, subtract, multiply, and divide fractionsconcretely, pictorially, and symbolically

• estimate, compute, and verify the sum,difference, product, and quotient of rationalnumbers

• estimate, compute (using a calculator), andverify approximate square roots of wholenumbers

• use concepts of rate, ratio, proportion, andpercent to solve problems in meaningfulcontexts

• derive and apply unit rates• express rates and ratios in equivalent forms


To extend students’ understanding of numberoperations, they could:

• calculate combined percents in a variety ofmeaningful contexts

• estimate, compute (using a calculator), andverify approximate square roots of decimals

Students’ understanding of the number system isexpanded to include rational numbers. They practisefamiliar arithmetic operations in new contexts.

• Ask students to solve basic operations withfractions using student-constructed orcommercially available manipulatives (e.g.,fraction tiles, Cuisenaire rods, attribute blocks,grid paper).

• Invite students to play games and to create theirown games that incorporate operations withrational numbers (e.g., mathematics crosswordor “cross-number” puzzles, mathematics bingo,Who Am I?)

• Demonstrate an iteration strategy (e.g., guess,check, modify) by having students developmethods of finding an accurate square rootapproximation to a number that is not a perfectsquare (e.g., Newton’ s method).

Newton’ s method: 18 is between 4 and 5 since

18 is between 16 and 25• Have students consider open-ended problems

using the context of “consumers.” Then askstudents to display solutions to the followingtypes of questions and, if appropriate, substitutefigures for those in the newspaper ads:- Compare newspaper ads or create examples

where the same product is advertised “on sale”at more than one location (e.g., 20% off, 15% offwith more free options, $5 off). Which is thebest deal?

- What is the final price of a $5 item marked30% off?

- What is the percent discount of a $10 itemreduced by $2?

- Compare similar products to determine whichis the best buy.

• Pose, or have students pose, problems involvingratio, rate, proportion, or percent selected fromthe media. Discuss the mathematical validity ofeach example and the message it communicates tothe reader.


23


Students demonstrate their understanding of newnumber concepts by performing arithmeticoperations using these numbers. Assessment focusseson students’ understanding of the processes, as wellas their accuracy in performing the operations.

Observe• Watch students as they interact with other

students to play mathematics games.- Does a lack of participation indicate a need for

additional instruction?- Do the games students create reflect an

understanding of concepts?• Observe students’ work with manipulatives.

Have them explain how they use manipulatives torepresent various operations. Do they use correctterminology?

• When students solve problems, look for evidence thatthey understand the operations they are using by:- correctly performing calculations- identifying the correct ratios- performing calculations in the correct orderAsk questions like:- Why did you use this operation?- What would happen if you changed the order

of calculations?

Question• Ask students to explain the steps involved in

calculating a percent of a number.• Have students describe two different ways they

can verify their work.• Ask students to explain the processes they use to

estimate and verify square roots.

Collect• Have students develop written or oral descriptions

of the steps they follow to add, subtract, multiply,and divide rational numbers, providing examplesthat illustrate each step. Check their work todetermine where students are having problems.

Peer Assessment• Ask students to develop numerical examples of

operations and challenge other students torepresent them using manipulatives. Havestudents identify and correct errors in each other’srepresentations.

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In order to prepare students to use patterns,variables and expressions, and graphs to solveproblems, it is expected that students will:

• substitute numbers for variables inexpressions and graph and analyse therelation

• translate between an oral or writtenexpression and an equivalent algebraicexpression

• generalize a pattern from a problem-solvingcontext


To extend students’ understanding of patterns,they could:



Looking for patterns and making generalizationsfrom them are mathematical skills useful ininvestigating and solving real-world problems. Thevariables, expressions, and equations used todescribe patterns and relationships are the basis ofstudents’ study of algebra.

• Have students work individually orcollaboratively to explore patterns by:- working at concrete activities using simple cases

(e.g., dividing a circle by lines to create patterns)- developing a pattern using, for example, two or

more shapes, colours, or textures- differentiating between growing and repeating

patterns- changing a growing pattern to a repeating pattern- changing a given pattern to a new one- determining the formula to extend the patternEncourage students to use concrete materialswhere possible (e.g., algebra tiles, algebra lab gear,two-colour counters). Remind them that there aremany possible ways of describing patterns.

• Ask students to bring in examples or pictures ofpatterns from their surroundings (e.g., flowerpetals, architecture, needles on a tree, ploughedfields). Invite them to make conjectures about thepatterns. Ask students:- How did these patterns occur?- How would you go about finding or creating a

pattern?• Display a table of ordered pairs. Have students

work with partners to determine the rule that wasused to generate each pair. Students could thencreate patterns using their own rules andchallenge their partners to generate the rules fromthe patterns.

• Have students work in groups to examine graphsfrom a variety of sources and attempt to interprettheir meaning.

• Brainstorm terms that have the same meaning,such as sum, difference, product, and quotient. Havestudents develop posters of these terms, which canbe put up in the classroom.

• Ask students to suggest one number each, andrespond to their numbers with others based on apattern or equation. Ask students to plot each pairof numbers. Continue until students can guesswhat the pattern is.


25


Students use higher-order thinking skills to identifypatterns and to generalize. Assessment shouldprovide opportunities for students to demonstratetheir problem-solving skills.

Question• As students solve problems using patterns,

variables, expressions, equations, and graphs, askthem to explain the methods and processes theyare using. Provide feedback to studentsconcerning their application of problem-solvingapproaches.

Collect• Ask students to annotate their work to describe

the processes they use to solve problems.Alternatively, students could provide briefdescriptions of which processes did and did notwork.

Self-Assessment• Work with students to generate a set of criteria

that can be used to evaluate problem-solvingskills. Use the criteria to create a rating scale thatstudents can use to evaluate their own skills.Criteria should demonstrate:- a willingness to persevere to solve difficult

problems- flexibility in trying different approaches

• Note: Refer to Suggested Assessment Strategies InProblem Solving section.

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MATHEMATICS 8 • Patterns and Relations (Variables and Equations)

In order to prepare students to solve and verifyone-step linear equations with rational numbersolutions, it is expected that students will:

• illustrate the solution process for a one-step,single-variable, first-degree equation, usingconcrete materials or diagrams

• solve and verify one-step, first-degreeequations of the form:- x + a = b- ax = b

- x—a

= b

where a and b are integers

• solve problems involving one-step, first-degree equations


To extend students’ understanding of variablesand equations, they could:

• illustrate the solution process for a two-step,single-variable, first-degree equation, usingconcrete materials or diagrams

• solve and verify two-step, first-degreeequations of the form:- ax + b = c

- x—a

+ b = c

where a, b, and c are integers

• solve problems involving two-step, first-degree equations

Variables and equations provide students with themathematical tools needed to solve complex, real-world problems. Learning is facilitated whenstudents are given opportunities to use manipulativesand to describe equations in their own words.

• Have students use manipulatives or models toexplore the concept of balance or equality, usingexamples such as:- a weight balance- a function machine- algebra tiles

• Give students opportunities to solve equations invarious ways (e.g., manipulatives, computersoftware, games).

• Suggest that students work in pairs to constructflow charts of equation-solving procedures, thenswitch with other pairs and attempt to apply thesteps to a given problem. Is there more than oneway to solve the problem? Discuss as a class.

• Discuss with students the value of algebraicoperations for solving problems. Have themconsider longer, more complicated equations thatcan be solved easily and quickly using algebra butwould take much longer to solve using arithmetic.

• Use student research (e.g., from the Internet) tocreate a timeline of events in the history of the useand development of algebra.


27


Assessment focusses on students’ understanding ofthe basics of solving equations: demonstrating anunderstanding of the solution processes, maintainingthe equality of both sides of algebraic expressions,and verifying solutions.

Observe• Review students’ work as they solve first-degree

equations. Look for common misconceptions andpresent them to the class for discussion.

Question• Ask students to explain or demonstrate the

processes they use to verify their solutions.

Collect• Have students create flow charts showing the

processes used to solve and verify the listed formsof first-degree equations. Evaluate students’ workusing criteria such as the following:- defines the processes clearly, logically, and

accurately- shows all necessary steps in the processes- maintains equality throughout the solution

process- accurately illustrates the process for verifying

solutions- method of display effectively communicates

informationGive students the feedback necessary to correcttheir work.

• Ask students to solve two-step linear equations,explaining in writing how to do each step in theprocess and showing their work. Note the level ofunderstanding demonstrated by students’explanations. Do they maintain the equality ofexpressions on both sides of the equations?

Peer Assessment• Have students solve equations, showing all their

work. Ask them to exchange solutions with otherstudents and use keys to mark each other’ s work.Students can identify the errors in their peers’solutions, explain how to fix them, and usepartners’ feedback to correct their own work.

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MATHEMATICS 8 • Shape and Space (Measurement)

In order to prepare students to apply indirectmeasurement procedures to solve problemsgeneralize measurement patterns and proceduresand solve problems involving area andperimeter, it is expected that students will:

• use the Pythagorean relationship to calculatethe measure of the third side of a righttriangle, given the other two sides in 2-Dapplications

• describe patterns and generalize therelationships by determining the areas andperimeters of quadrilaterals and the areas andcircumferences of circles

• estimate and calculate the area of compositefigures


To extend students’ understanding ofmeasurement, they could:

• estimate, measure, and calculate the surfacearea and volume of any right prism orcylinder

• estimate, measure, and calculate the surfaceareas of composite 3-D objects

• estimate, measure, and calculate the volume ofcomposite 3-D objects

The Pythagorean relationship is pervasive inmathematics and critical to the concept of indirectmeasurement. To facilitate students’ understanding,use hands-on activities with physical objects. Theability to conceptualize and apply measurementformulae for 2-D shapes is best developed usinghands-on activities.

• Demonstrate the Pythagorean relationship usingconcrete examples such as dissection puzzles, areaconstructions, and student-generated illustrations.

• Divide a piece of rope or string into 12 equallengths by placing tape on 11 spots. Ask threestudents to hold the rope in a way that constructsa triangle with a right angle (each studentrepresents a vertex; one student holds the twoends of the rope together). Students shoulddiscover that there is only one solution. Discuss:- Can any other right triangle be made?- Does 3, 4, 5 always result in a right triangle?- What are the applications?- What is the history of this method?

• Encourage student to research the Pythagoreanrelationship in a variety of contexts, such as:- the history of the relationship in various

cultures- how trades people determine right angles in

their trade- how Aboriginal people ensured that their

longhouses were square• Provide students with a variety of composite

shapes and have them estimate the perimeter andarea of each. Then have students check theirguesses using suitable software such as dynamicgeometry software, Computer Assisted Drafting(CAD), or Geographic Information Systems (GIS).

• Ask students to estimate the perimeters, areas,surface areas, and volumes of composite shapesand objects (e.g., concrete building blocks, Romanwindows, desks) and then measure, calculate, andcompare their estimates. Have students write intheir journals to reflect on the differences betweentheir estimates and the actual measurements. Weretheir estimates closer for one type of measurementthan for another?


29


By demonstrating an ability to apply the Pythagoreanrelationship, students show that they have thefoundation for learning many of the procedures andrelationships they will encounter in algebra andgeometry. Assessment also focusses on students’abilities to develop ideas about measurement andapply them to practical situations.

Question• Ask students to illustrate the Pythagorean

relationship with concrete materials and diagrams.Are their illustrations accurate? Can they explainthe relationship?

Collect• Have students create simple blueprints of their

dream homes, specifying the measurements ofeach room and calculating the areas andperimeters. Can students explain why it isimportant to know these measurements if they aregoing to build a house? Ask students to exchangetheir projects with other students and evaluateeach other’ s work using the following criteria:- Are the measurements realistic?- Based on the measurements students have

supplied, are the calculations of area andperimeter accurate?

Collect and review students’ blueprints and thecorrections and comments made by their peers,and provide feedback.

Observe• As students make estimates and measurements of

the area, surface area, perimeter, and volume ofcomposite shapes or objects, observe:- Do students select the most appropriate

measurement scale?- Do students select the appropriate units for the

answer?- Do their estimates approximate the

measurements?- Do they identify the various shapes that make

up the composite figures before they make theirestimates?

- Do they recognize that the non-included spacemust be subtracted?

- Can they identify reasons why their estimatesmight be off?

Print Materials


Multimedia

• The Geometer’s Sketchpad• The Learning Equation Mathematics 8 (TLE)• Math Tools• Mathematics 8 (Distance Education Package)• Minds on Math 8, Revised Edition• Understanding Math Series

CD-ROM

• Geometry Blaster• Mathville VIP• Mirror Symmetry


30

MATHEMATICS 8 • Shape and Space (3-D Objects and 2-D Shapes)

In order to prepare students to link anglemeasures and the properties of parallel lines tothe classification and properties of quadrilaterals,it is expected that students will:

• identify, investigate, and classifyquadrilaterals, regular polygons, and circlesaccording to their properties


To extend students’ understanding of 3-D objectsand 2-D shapes, they could:

• build 3-D objects from a variety ofrepresentations (e.g., nets, skeletons)

Disciplines such as architecture, engineering, graphicdesign, cartography, drawing, and sculpting allincorporate geometric principles. An understandingof linear and spatial visualization is valuable forstudents to develop their artistic and aestheticexpression.

• Display examples of Escher art on the overhead(or as a handout). Have students discuss therepresentations of 3-D space on 2-D space and thetransformations through tessellations. Askstudents to research other patterns of the sametype (e.g., in Greek, Moorish, or Islamic mosaics).

• Using large sheets of graph paper, have studentsdraw and cut out a rhombus, square, kite,parallelogram, rectangle, trapezoid, and dart. Byfolding the shapes, students can discover andchart the properties of diagonals, sides, andangles.

• Have students create a chart or concept maporganizing and displaying the variousquadrilaterals classified by their properties.

• Ask students to create “find-the-polygon” puzzles(like “find-the-word” puzzles), in which they hidepolygons within matrixes of lines and provide“lists” of the polygons for other students to find.

• Have students work with origami to discovervarious ways to manipulate 2-D shapes and toconsider their applications. (e.g., What is the resultwhen you fold a square diagonally?) Invitestudents to organize a contest for tessellationartwork for an upcoming school event or holiday.

• Ask students to research BC Aboriginal beading orweaving designs and represent them on grids.They could then design and apply their ownpatterns based on the designs researched (usingactual textiles or computer simulations).

• Ask students to investigate the relationshipbetween tessellations and quilting. (e.g., MarjorieRice applied her knowledge of quilting to solve adifficult tessellation problem.)

• Have students construct 3-D models from nets andskeletons (e.g., using straws, balsa wood,toothpicks, construction paper, clay,marshmallows, or jujubes).


31


An understanding of related terminology and theability to recognize and classify geometric shapeshelp students use their understanding of shape andspace to solve problems and describe the worldaround them.

Question• Ask students to explain the reasons behind their

classification of quadrilaterals, circles, andpolygons.

Collect• Have students design covers for their mathematics

textbooks, portfolios, or displays on bulletinboards, incorporating all types of quadrilaterals,polygons, and circles. Do they use all possibletypes? Can they identify the types when asked?Does it follow a theme?

• Ask students to prepare notes for a friend who hasbeen out of town and had to miss class. Studentsshould describe the properties of different types ofquadrilaterals, regular polygons, and circles.Diagrams can be used to clarify their descriptions.Use the following criteria to evaluate students’work. Check for:- clarity and logic of descriptions- accuracy of descriptions- appropriate classification of quadrilaterals,

regular polygons, and circles- effective use of examples and diagramsProvide feedback to students to help them correcttheir mistakes.

• Provide students with a 6 x 6 square with 36points. Have them connect the points in as manyways as possible to produce as many differentquadrilaterals as they can. Have students classifytheir results by looking for patterns of similarity.Can they justify their classifications? As studentswork, circulate through the classroom and askquestions about lines of symmetry andclassification.

• Ask students to draw nets (and cut them out) ofvarious 3-D shapes and have them explain whycertain nets will not work, even though they havethe requisite faces.

Print Materials


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32

In order to prepare students to analyse designproblems and architectural drawings using theproperties of scaling and proportion, it is expectedthat students will:

• represent, analyse, and describe enlargementsand reductions

• draw and interpret scale diagrams


To extend students’ understanding oftransformations, they could:

• represent, analyse, and describe colouringproblems

• describe, analyse, and solve network problems(e.g., bus routes, a telephone exchange)

Knowledge of transformational geometry is essentialfor students to understand much of what they see ingraphic representations.

• Using dynamic geometry software, demonstratethe rate of change as scaling is applied to 2-D and3-D objects.

• Have students compare the difference in areas ofsmall, medium, and large pizzas. Ask them toconstruct a graph to represent the relationshipsbetween the diameter and the area for each size.What generalizations can they make for theserelationships?

• Divide the class into groups. Have each groupinvestigate a real-world application ofenlargement and reduction (e.g., how a knitting orsewing pattern is adjusted for size, map scales)and report to the class.

• Provide students copies of maps of BC thatidentify Aboriginal language groups. Havestudents colour the regions so that they use theminimum number of colours and adjoiningregions do not use the same colour.

• Suggest that students find examples of bus andairplane routes, truck routes, and telephoneroutes. Have them work in pairs to create andanswer questions related to the networks or todesign their own networks showing the mostefficient paper route in their neighbourhood.

MATHEMATICS 8 • Shape and Space (Transformations)


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MATHEMATICS 8 • Shape and Space (Transformations)

Assessment is based on students’ ability todemonstrate their understanding of the ideas andprocedures they have learned by drawing, building,and discussing.

Question• Determine if students can identify places outside

school where enlargements, reductions, and scalediagrams (and networks) are used.

Collect• Give students a scale drawing of a 2-D or 3-D

object. Ask them to produce a second scaleddrawing by doubling the dimensions of the firstand then construct a model of the object that isfour times its original dimensions. Ask students todescribe the effect that the enlargements have hadon surface area and volume. Use the same activityto have students demonstrate their understandingof reductions.- Are students’ enlargements and reductions

made to scale and accurate?- Can students describe the methods they use to

enlarge or reduce the original drawings?- Can students move between 2-D and 3-D

representations?• Determine if students can predict the most

efficient method of getting from point A to point Bin their school. Give students a map of the schooland a list of Grade 8 teachers, specifying theirroom locations. Ask students to predict the mostand least efficient timetables possible for movingfrom room to room in terms of the distancetravelled in the school.

• Ask students to bring material about networksand use it to design questions for other students tosolve. Evaluate the complexity of the questionsand the accuracy of students’ solutions to theirown and other’ s questions.

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34

MATHEMATICS 8 • Statistics and Probability (Data Analysis)

In order to prepare students to develop andimplement a plan for the collection, display andanalysis of data, using technology as requiredand to evaluate and use measures of centraltendency and variability, it is expected that studentswill:

• formulate questions for investigation, usingexisting data

• select, defend, and use appropriate methods ofcollecting data:- designing and using surveys- research, using electronic media

• display data by hand or by computer in avariety of ways

• determine and use the most appropriatemeasure of central tendency in a given context


To extend students’ understanding of dataanalysis, they could:

• describe the variability of data sets, using suchtechniques as range and box-and-whiskerplots

• construct sets of data given measures ofcentral tendency and variability

• determine the effect on the mean, median,and/or mode when:- a constant is added or subtracted from each

value- each value is multiplied or divided by the

same constant- a significantly different value is included

The science of statistics is a powerful tool to helpconvey information. It can also misrepresent thetruth. As consumers, students must develop anunderstanding of data analysis if they are to makeinformed decisions.

• Assign a survey project requiring each student to:- formulate a question- decide on a suitable sample- record data- display the results (using available technology)- calculate the mean, median, and mode where

appropriate- make and justify conclusions based on the

survey resultsEncourage students to design projects based onlearning in another subject area (e.g., career goals,cultures of origin, musical preferences).

• Ask students to select survey or research projectsreported in the media and evaluate them in termsof methods of display and conclusions made. Theycould then write letters to the editor reporting anyquestions or problems they have with the data orits representation. Where possible, students coulddesign and conduct their own surveys on the sameissue.

• Discuss with the class the use of mode, median,and mean with different sets of data and fordifferent purposes (e.g., ice cream flavours, gradeson a test, bowling scores, population density).

• Ask students to select sets of data andintentionally bias the information by manipulatingthe way they display it. Have them present theirdisplays to other students, who attempt to identifythe bias in each case.

• Present students with a box-and-whisker plot.Have them record in their own words whatinformation is being presented and share theirresponses with partners.


35


Students can demonstrate a broad range of data-analysis skills by planning, implementing, andanalysing their own research projects.

Observe• Have students describe possible sources of bias in

a collection and display of data, and explain ordemonstrate how bias can be avoided.

• As students work with box-and-whisker plots,note whether they can identify the components.Do they correctly determine the extremes,quartiles, and median?

Collect• Ask students to bring in articles from the media

that present statistical information. Then havethem analyse the data and graphs and draw theirown conclusions. Compare conclusions drawn bystudents to those drawn by the media. Discussdifferences. Are students’ conclusions similar tothose of the media? If not, can students defendtheir conclusions?

Self-/Peer Assessment• Work with students to develop criteria for

evaluating their research projects. For example:- appropriateness of research questions- effectiveness of data-collection methods- adequacy of survey design- effectiveness of the method used to record data- appropriateness of the graph scale and type- accurate calculation of mode, range, median,

and mean- validity of conclusions- organization and clarity of presentation- students’ ability to justify their conclusionsAsk students to identify strengths and weaknessesin their work and briefly summarize howidentified problems might be corrected in futureprojects.

• Have students determine the mode, median, andmean for different sets of data. Students shouldidentify the best methods for describing differentkinds of data and provide a rationale to supporttheir choices. Check the accuracy of students’calculations and provide feedback regarding thevalidity of their conclusions.

Print Materials

• Interactions (Level 8)• MATHPOWER 8, Western Edition• Triple ‘A’ Mathematics Program: Data Management & Probability

Multimedia

• Hot Dog Stand: The Works• The Learning Equation Mathematics 8 (TLE)• Math Tools• Mathematics 8 (Distance Education Package)• Minds on Math 8, Revised Edition• Statistics Workshop

CD-ROM

• Mathville VIP


36

MATHEMATICS 8 • Statistics and Probability (Chance and Uncertainty)

In order to prepare students to comparetheoretical and experimental probability ofindependent events, it is expected that studentswill:

• use various data-collection techniques(including computers) to simulate and solveprobability problems

• recognize that if n events are equally likely, theprobability of any one of them occurring is 1—

n

• determine the probability of two independentevents where the combined sample space has52 or fewer elements

• predict population characteristics from sampledata

An understanding of probability has direct relevanceto students’ future careers and lives. Today’ spolitics, tomorrow’ s weather report, and next year’ sautomobile insurance premiums all depend onprobability theory.

• Invite a guest speaker from an insurance companyto discuss how life expectancy tables are used inthe industry. Then have students use their journalsto reflect on how their own lifestyle choices canaffect their longevity, based on the probabilitiesrepresented in the life expectancy tables.

• Discuss the difference between favourableoutcomes and possible outcomes. Ask students towork in groups to design probability experiments(e.g., dice rolling, coin tossing), collect data, andgenerate summaries of the results. Studentsshould:- list possible outcomes- calculate the probability of events happening- describe the experiments and summarize

results- have students create a generic formula to

predict the odds of winningDiscuss with students the relationship betweentheir experimental results and the theoreticalexpectations of these events.

• Ask each student to research a particular game ofchance and prepare a presentation of the findings,including the rules of the game, how probability isinvolved, where the game is played, and whatequipment is needed. Students can then presenttheir findings in the form of “warnings togamblers.” As an extension, students couldinvestigate games of chance in a range of cultures(e.g., the Aboriginal bone game Lhal). Inviterepresentatives of these cultures to come to class todemonstrate the games.

• Have the class brainstorm and discuss decisionsstudents have made that were based (or couldhave been based) on probability.

• Access E-Stat (www.statcan.ca) for sample data ona topic of interest to your students. Use this data topredict population characteristics for your town.


37


By demonstrating an understanding of probabilityand chance, students show that they have basicknowledge and skills needed to make manyimportant life decisions. Students can demonstratetheir under-standing while participating in enjoyableactivities.

Observe• As students work with probability, check to see

that they understand and differentiate betweenfavourable and possible outcomes. Check to seethat students understand that the probability is0 ≤ p ≤ 1

Question• Ask students to identify situations in their lives

and society in which they might encounter chanceand uncertainty. Can they explain why it isimportant to have strategies for dealing withuncertainty?

Collect• Ask students working in small groups to develop

probability experiments such as simple games ofchance. Review these to determine if studentshave:- identified all possible outcomes- performed the experiments a sufficient number

of times- accurately calculated the probability of

specified events- calculated probability of winning- clearly described their experiments- accurately summarized their results (including

fairness of the game)- created a generic formula

Peer Assessment• Have the class develop criteria for students to

evaluate each other’ s presentations on games ofchance from different cultures. Criteria mightinclude:- depth of research- clarity and organization of information- effective use of visuals- level of detail- mathematical accuracy

Print Materials


Multimedia

• The Learning Equation Mathematics 8 (TLE)• Math Tools• Mathematics 8 (Distance Education Package)• Minds on Math 8, Revised Edition

CD-ROM

• Mathville VIP


39

CURRICULUMMathematics 9


40


41

ESTIMATED INSTRUCTIONAL TIME

Mathematics 9 has been developed assuming that teachers have 100 instructional hours available tothem. The following chart shows the estimated instructional time for each curriculum suborganizer,expressed as a percentage of total time available to teach the course.

When delivering the prescribed curriculum, teachers may freely adjust the instructional time tomeet their students’ diverse needs. These estimated instructional times have been recommended bythe IRP writers to assist their colleagues. They are suggestions only.

Organizer (Suborganizer) % of Time

Problem Solving Integrated Throughout

Number (Number Concepts) 5 - 10

Number (Number Operations) 10 - 15

Patterns and Relations (Patterns) 5 - 10

Patterns and Relations (Variables and Equations) 15 - 20

Shape and Space (Measurement) 10 - 20

Shape and Space (3-D Objects and 2-D Shapes) 15 - 20

Statistics and Probability (Data Analysis) 10 - 15

Statistics and Probability (Chance and Uncertainty) 5 - 15

MATHEMATICS 9


42


In order to prepare students to use a variety ofmethods to solve real-life, practical, technical,and theoretical problems, it is expected thatstudents will:

• solve problems that involve a specific contentarea (e.g., geometry, algebra, statistics,probability)

• solve problems that involve more than onecontent area within mathematics


• analyse problems and identify the significantelements

• develop specific skills in selecting and usingan appropriate problem-solving strategy orcombination of strategies chosen from, but notrestricted to, the following:- guess and check- identify patterns and use a systematic list- make and use a drawing or model- eliminate possibilities- work backward- simplify the original problem- select and use appropriate technology to

assist in problem solving- analyse keywords



• clearly and logically communicate a solutionto a problem and the process used to solve it

• evaluate the efficiency of the processes used• use appropriate technology to assist in

problem solving

Problem solving is a key aspect of any mathematicscourse. Working on problems in mathematics can givestudents a sense of the excitement involved in creativeand logical thinking. It can also help students developtransferable real-life skills and attitudes. Multi-strandand interdisciplinary problems should be includedthroughout Mathematics 9.

• Reinforce the concept that problem solving is morethan just word problems and includes aspects ofmathematics other than algebra (e.g., geometry,statistics, probability).

• Introduce new problems directly to students(without demonstration) and play the role offacilitator as they attempt to solve the problems.

• Recognize when students use a variety ofapproaches (e.g., algebraic or geometric). Avoidbecoming prescriptive about approaches toproblem solving.

• Emphasize that problems might not be solved inone sitting and that “playing around” with theproblem—revisiting it and trying again—issometimes needed.

• Assign students a set of problems that fit aparticular strategy. After sufficient time, studentsdraw a number to find which solutions to presentto the class. After about five minutes (to givestudents a chance to complete all problems), beginthe presentations. Tell students that they may usethe presentations of others to complete their work.

• Ask directed questions such as:- What are you being asked to find out?- What do you already know?- Do you need additional information?- Have you ever seen similar problems?- What else can you try?

• Once students have arrived at solutions toparticular problems, encourage them to generalizeor extend the problem situation.

• Encourage students to maintain mathematicsjournals for recording the things they are learningand any difficulties they may be having.

• Note: See Appendix F for examples of multi-strand andinterdisciplinary problems that most students should beable to solve. These problems are indicated with anasterisk (*).


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Students’ work and what they say about it revealstheir thinking while solving problems. Assessment ofstudents’ problem-solving skills should be carriedout over time, based on observations of manysituations.

Observe• Have students present solutions to the class

individually, in pairs, or in small groups. Note theextent to which they are able to:- articulate and clarify problems- describe the processes used- describe what worked and what did not- identify ways to get additional information as

needed- find alternative methods- link mathematics to new situations

Question• To check the approaches students use when

solving problems, ask questions that prompt themto:- paraphrase or describe problems in their own

words- explain the processes used to derive answers- describe alternative methods to solve the

problems- relate the strategies used in new situations- link mathematics to other subjects and to the

world of work

Collect• On selected problems, have students annotate

their work to describe the processes they used.Alternatively, have them provide briefdescriptions of problems that worked and thosethat did not.

Self-Assessment• Ask students to keep journals of problems they are

working on, describing where they find problemsand reflecting on the processes they used indealing with problems. Have students describestrategies that worked and those that did not.

• Develop with students a set of criteria to assesstheir own problem solving. The reference setEvaluating Problem Solving Across Curriculum maybe helpful in identifying such criteria.

The Western Canada Protocol Learning ResourceEvaluation Process identified numerous teacherresources and professional references. These aregenerally cross-grade planning resources thatinclude ideas for a variety of activities andexercises.

These resources, while not part of the GradeCollections, have Provincially Recommendedstatus.

Appendix B includes an annotated bibliographyof these resources for ordering convenience.


44


In order to prepare students to develop a numbersense of powers with integral exponents andvariable and rational bases, it is expected thatstudents will:

• give examples of situations where answerswould involve the positive (principal) squareroot or both positive and negative square rootsof a number

• illustrate power, base, coefficient, andexponent using rational numbers or variablesas bases or coefficients


To extend students’ understanding of numberconcepts, they could:

• give examples of numbers that satisfy theconditions of natural, whole, integral, andrational numbers, and show that thesenumbers comprise the rational number system

• describe, orally and in writing, whether or nota number is rational

• explain and apply the exponent laws forpowers with integral exponents

0,1

0,1

0,

)(

)(

0

≠=

≠=

≠=

==

=÷=•

−

−

+

xx

x

xx

yy

x

y

x

yxxy

xx

xxx

xxx

nn

n

nn

mmm

mnnm

nmnm

nmnm

• determine the value of powers with integralexponents, using the exponent laws

As students extend their knowledge of numbers toinclude the integral exponents, they become exposedto more difficult yet “real-world” equations andformulae.

• Give examples of situations in which answerswould involve the positive (principal) square rootor both positive and negative square roots of anumber. Pose as a problem: Why does theequation X2 = 9 have two solutions [± 3], but thearea of a square with an area of 9cm2 is foundusing the principal square root [3cm]?Ask students to consider the concept in anhistorical context, noting the difficulty that havingtwo answers posed for mathematics.

• Ask students to use a mathematics glossary orencyclopedia (print or electronic) to define theterms power, base, coefficient, and exponent.

• Use cubes and diagrams to represent and explainthe difference between two numbers (e.g., 32 and 23.)

• Ask students to practise translating formulae withpowers into formulae without powers and viceversa.

• Have the class brainstorm examples of commonmisconceptions or errors that occur whenapplying the exponent laws. For example:

X • X = X

33

2 • 2 = 2

= 1 or 1 or 3

= M = 1

4

128 3

2 4 8

4

M 2

M -2

0

7 28

• Have students work in co-operative learninggroups to develop the rules of exponents andcreate visual representations of these roles.Display their work.

-

-

-

-

-

-

-

-

-

-

-


45


By demonstrating an understanding of exponents,students show they possess the basic knowledgeneeded to use mathematics to solve problems.Assessment focusses on the understanding ofmeaning and procedures.

Question• Ask students to illustrate the difference between 23

and 32. Do their diagrams clearly differentiatebetween a 2-D and a 3-D representation?

• To determine if students understand the differencebetween the square root of a number and theprincipal square root, ask questions such as:- What numbers squared equal 25? Students

should recognize that both 5 and -5, whensquared, equal 25.

- Students should recognize 9 has one solution,3. The equation X2 – 9 = 0 has two solutions, ± 3.

- Do students know when it is appropriate to useonly the principal (positive) square root?

Collect• Have students demonstrate their understanding of

the meaning of negative exponents by giving thema pattern of powers such as 24, 23, 22, 21, 20,2-1, 2-2, 2-3, 2-4. Have them predict the rulesx0 = 1, x0 and justify their predictions to the rest ofthe class. Provide feedback to the students to helpthem clarify their thinking. To what extent dostudents:- predict rules correctly- clearly support their predictions- continue the pattern when asked

• After students have studied exponent laws forpowers with integral exponents, give themproblems in which the solutions contain errors.Ask students to use their knowledge of the laws toidentify the errors, describe them, and makecorrections where appropriate.

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46


In order to prepare students to use a scientificcalculator or a computer to solve problemsinvolving rational numbers, it is expected thatstudents will:

• document and explain the calculator keyingsequences used to perform calculationsinvolving rational numbers

• solve problems, using rational numbers inmeaningful contexts

• evaluate exponential expressions withnumerical bases


To extend students’ understanding of numberoperations, they could:

• use the exponent laws to simplify expressionswith variable bases

• use a calculator to perform calculationsinvolving scientific notation and exponentlaws

Proficiency and efficiency in number operations is animportant skill for the workplace, further education,and informed citizenship. Students can improvenumber operation skills and fluency throughestimation, pencil-and-paper calculations, andappropriate calculator use.

• Create an activity sheet of questions thatdemonstrate the limitations of the calculator.Include questions that require the use of bracketsand calculator memory buttons. For example:

- 5 x 32

- 21000

- -32 and (-3)2

Ask students to develop their own questions thatcan test the appropriate use of their calculators.

• Have students use spreadsheet software to:- automate a chequebook- track homework time versus TV-watching time- keep sports statistics on school teams or

individual student athletes.• Ask students to identify resource people in the

community who might use formulae withexponents in their work (e.g., foresters, plumbers,engineers, farmers). Have students interview aresource person about a formula they use and thenreturn to the class to teach their peers about theformula, what the values are, and whatinformation it provides to the person interviewed.Discuss as a class how real-life problems can beexplained using formulae.

• Use mathematics games to give students practicein working with exponents (e.g., a game in whichstudents are dealt a single card each, withnumbers expressed in exponents). Have studentshold their cards face out, so that they can seeeveryone else’s but not their own, and bid basedon how big they think their numbers are.

• Calculate with examples from the natural andsocial sciences for practice in scientific notation(e.g., number of micro-organisms in a samplepopulation, distance between stars).

• Have students estimate some large numbers foruse in operations (e.g., number of seconds sincethe “big bang” divided by the number of wordsever spoken). Have them attempt to perform theoperation using both long division and scientificnotation, and discuss the relative merits of each.


47


Number operations are the tools students use to solveproblems. Assessment focusses on students’ ability touse these tools accurately and appropriately in variouscontexts.

Collect• Make up fraction “twisters” and have students

determine whether they are true. For example: If afifth of a fifth is less than a fourth of a fourth, then afourth of a fifth must be less than a fifth of a fourth, ofcourse. Is a third of a third more than a third and athird or is a third of a third less that a third cut in athird, or is it all absurd? Ask students to create theirown twisters and challenge each other. Check foraccuracy and understanding.

• At the start of instruction, identify outcomes thatstudents are expected to achieve at the end ofinstruction. Have students organize examples of theirown work into portfolios as evidence that they haveattained the desired outcomes. Examples mightinclude homework assignments, graded quizzes,personal summary sheets of learning, or anythingelse that shows students have acquired the intendedlearning.

Observe• Examine students’ work as they use the exponent

laws to simplify and evaluate expressions. Watch tosee that they are performing the appropriateoperations. Provide feedback to help studentsidentify and correct their errors.

Question• To what extent can students:

- estimate solutions to teacher-supplied problems- use their calculators or other appropriate

technology to solve the problems- compare their estimates to the solutions they

obtain using their calculators- determine possible reasons for any large

differences between their estimates and calculatorsolutions.

Peer Assessment• Have students work in pairs to make up questions to

challenge their partners’ skills with a calculator. Lookfor use of brackets, exponents, and negativecoefficients. Confirm that the authors of the questionscan solve them and explain the solutions to theirpartners.

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48


In order to prepare students to generalize,design, and justify mathematical proceduresusing appropriate patterns, models, andtechnology, it is expected that students will:

• model situations that can be represented byfirst-degree expressions

• write equivalent forms of algebraicexpressions, or equations, with integralcoefficients


To extend students’ understanding of patterns,they could:

• use logic to present mathematical argumentsin solving problems

• write equivalent forms of algebraicexpressions, or equations, with rationalcoefficients.

By searching for patterns, describing them in variousways (including algebraically), and recreating them,students are able to extend patterns as a part of theirpersonal expression.

• Demonstrate to the class problem-solvingtechniques that may be employed to solve difficultproblems. For example:- Using dots to represent the triangular numbers

(1,3,6,10,...) predict the 20th, 100th, and nthterm.

- Find the total number of squares on achessboard (all squares, not just 1 x 1), andextend the pattern to an n x n board.

• Have students collect graphs and formulae fromscience, social studies, and other subjects, andcategorize each as either linear or non-linear

• Give students an algebraic expression followed byfour choices (a, b, c, and d). Have students selectwhich of the four choices are equivalent to theoriginal and explain their selections. This activitycan be repeated until most of the students are ableto select correctly.

• Have students work in pairs to measure twovariables (e.g., length of their arms and theirheight or the circumference and diameter ofcircular objects). Then have students post theirmeasures on the board as pairs of numbers.Challenge students to search for relationshipsbetween the numbers.

• Challenge students to determine an algebraicexpression for the sum of n consecutive naturalnumbers and the sum of n consecutive squarenumbers. Have students share results and discussthem as a class. Did they find more than onemethod?

• Present an open-ended question such as thehandshake model (i.e., given n people in the room,how many handshakes can take place?). In groups,students could determine solutions and thenrecord and report their methods. Encourage themto use hands-on equations, algebra tiles, anddiagrams to model the problem. Discuss eachmethod.- How well would it work if there were 50 people

in the room? 100?- Are all methods equally powerful?


49


Assessment of this suborganizer focusses onstudents’ understanding of skills related to theintroduction of algebra: accuracy and the use ofappropriate procedures.

Question• As students work independently and in small

groups to find equivalent forms, ask them toexplain the processes they are using. Note theclarity of students’ explanations and theirreasoning. Provide feedback to studentsconcerning their clarity and reasoning.

Collect• From the strategies students develop for searching

for relationships, have them generate and verifyformulae. Check the accuracy of students’ workand provide feedback.

• On selected problems, have students annotatetheir work to describe the processes they used tosolve the problems. Alternatively, have studentsprovide brief descriptions of what worked andwhat did not work as they attempted to solveparticular problems.

Peer Assessment• Have students justify to classmates their methods

for determining whether equations are linear.Work with students to develop criteria to use tohelp them decide whether the justifications areconvincing.

• Having discussed with the class what generallyconstitutes a good problem (e.g., clarity, is solvableby peers), ask students to work in small groups togenerate problems for other groups to solve. Notethe complexity of the problems generated as wellas the success of the groups in solving them.

Print Materials


Multimedia



50


In order to prepare students to evaluate, solve, andverify linear equations in one variable and togeneralize arithmetic operations from the set ofrational numbers to the set of polynomials. It isexpected that students will:

• illustrate the solutions process for a first-degree,single-variable equation, using concretematerials or diagrams

• solve and verify first-degree, single-variableequations of forms such as: ax = b + cx; a(x+b)=c;

ax + b = cx + d where a, b, c, and d are integers,and use equations of this type to model andsolve problems

• identify constant terms, coefficients, andvariables in polynomial expressions

• evaluate polynomial expressions, given thevalue(s) of the variable(s)


To extend students’ understanding of variables andequations, they could:

• solve and verify first-degree, single-variableequations of forms such as: a(bx + c) = d(ex + f);

a—x = b where a, b, c, d, e, and f are rational

numbers• solve, algebraically, first-degree inequalities in

one variable, display the solutions on a numberline, and test the solutions

• perform the operations of addition andsubtraction on polynomial expressions

• represent multiplication, division, and factoringof monomials, binomials, and trinomials of theform x2 + bx + c using concrete materials anddiagrams

• find the product of two monomials, a monomialand a polynomial, and two binomials

• determine equivalent forms of algebraicexpressions by identifying common factors andfactoring of the form x2 + bx + c

• find the quotient when a polynomial is dividedby a monomial

Skills in manipulating algebraic expressions can bedeveloped by connecting new ideas and operationsto arithmetic skills and concrete representations.Extending algebra beyond the linear relationshipsenhances students’ understanding of the 2-D and 3-Dworld. Drill and practice techniques help students tointernalize the abstractions and improve their skillsin the language of algebra.

• Have students substitute numbers in expressionsinvolving different powers and the same variablebase, to verify that the terms are not “like terms.”Students can also verify their factoring bysubstitution of numbers.

• Give students abstract forms of algebraicexpressions or equations and ask them to createnumerical examples. Then reverse the activity,challenging them to write the abstract forms ofnumerical examples.

• Ask students to use a mathematics glossary orencyclopedia (either print or electronic) to definethe terms constant, coefficient, and variable.

• Invite students to explore ways of solvingequations with their programmable or graphingcalculators. Alternatively, have them use computerspreadsheet applications to practise“programming” standard equations, using themto perform multiple calculations for a range ofdata.

• Ask students to work individually to create a first-degree, single-variable equation and determine itssolution. Then have them work in pairs to solveeach other’s equations.

• Display an inequality on the board or overhead(e.g., 3x + 2 > 8). Ask students to determine a valuefor x that satisfies this inequality. Plot andcompare students’ answers. Discuss as a class whythere are many solutions. Can students display allsolutions efficiently?

• Use algebra tiles, algebra lab gear, or diagrams todemonstrate expansion of polynomials. Forexample, (2x + y)2 ≠ 4x2 + y2 may not be readilyapparent without tiles.


51


Students should have an understanding of variousforms of linear equations and how to use them tomodel and solve problems.

Observe• As students use algebra tiles or diagrams to

model equations concretely, check their work andprovide feedback. Determine the extent to whichthey:- use tiles or diagrams to represent various

operations- are willing to modify their efforts based on

their experiences

Question• As students solve and verify linear equations in

one variable, have them explain the processesthey are using.

• As students work individually and in smallgroups to perform operations on polynomialsand factor binomials and trinomials, discuss theirwork with them to determine the extent to whichthey:- use correct terminology to identify constants,

coefficients, and variables in polynomialexpressions

- persist in their efforts to solve the difficultproblems

- use a variety of resources such as textbooks,other students, and technology to solveequations requiring them to performoperations

Collect• Give students study sheets containing examples

of monomials, binomials, and polynomials thathave been factored, some correctly and someincorrectly. Ask students to identify the incorrectexamples, identify the mistakes, and correctthem.

Peer Assessment• Have students develop algebraic expressions and

challenge each other to represent them usingalgebra tiles or diagrams. Have students identifyand correct errors in each other’s work.

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Multimedia



52


In order to prepare students to use trigonometricratios to solve problems involving right triangles,it is expected that students will:

• explain the meaning of sine, cosine, andtangent ratios in right triangles

• demonstrate the use of trigonometric ratios(sine, cosine, and tangent) in solving righttriangles

• calculate an unknown side or an unknownangle in a right triangle, using appropriatetechnology

• model and then solve given problemsituations involving only one right triangle


To extend students’ understanding ofmeasurement, they could:

• relate expressions for volumes of pyramids tovolumes of prisms, and volumes of cones tovolumes of cylinders

• calculate and apply the ratio of area toperimeter to solve design problems in twodimensions

• calculate and apply the ratio of volume tosurface area to solve design problems in threedimensions

The ability to describe the world using direct orindirect measurement is an important skill forvarious careers. Students develop trigonometricconcepts through hands-on measurement activities.

• Have students use protractors and straight edges,or dynamic geometry software, to construct righttriangles (with given angles) that just fit on letter-size pieces of paper. Then have them measure thelengths of the opposite and adjacent sides of oneangle and record the tangent ratio in tables. Thenintroduce the calculator and the definition oftangent ratio and compare students’ measurementswith the values shown on the calculator display.Repeat with the sine and cosine ratios.

• Have students use trigonometry to measure theheights of several trees to determine how tall atotem pole they could make.

• Have students solve right triangle problems usingappropriate software (e.g., spreadsheet program,dynamic geometry software) or a graphingcalculator.

• Ask students to estimate the volumes of models ofcones, cylinders, prisms, and pyramids. Then havethem fill the models (e.g., with sand, water,macaroni) and compare volumes. Have studentswork with different sizes of models to comparetheir estimates, strategies, and conclusions.

• Provide groups of students with a variety ofvolume-to-surface-area and area-to-perimeterproblems such as:- minimum-maximum problems (e.g., maximum

volume for given surface area, minimumsurface area for given volume, maximum areafor given perimeter, packing smaller boxes intolarger ones)

- relationships among volume, surface area,radius, and height of cones and cylinders and/or pyramids and prismsAssign different problems to different groups.Ask students to design several solutions to theproblems and identify the ramifications of eachsolution (e.g., cost-effectiveness, aesthetics).Where appropriate, students could use nets andmodels to represent the situations. Havestudents present their findings to the class forevaluation and discussion.


53


Students’ ability to correctly relate the informationfrom a right triangle to the trigonometric ratios isessential for the solution of a triangle. Assessmentfocusses on solving problems using their knowledgeof right triangle properties.

Observe• As students use trigonometric ratios to solve right

triangles, look for patterns of errors that indicate aneed for additional instruction. Ask the students to:- identify the adjacent and opposite sides and the

hypotenuse, given a specific angle- identify the trigonometric relationship, given

any two sides

Collect• Have students develop charts that compare

similarities and differences between the volumesand surface areas of pyramids and prisms andbetween the volumes and surface areas of conesand cylinders. Ask students to give examples,using drawings to illustrate their work andidentifying related formulae.

Peer Assessment• Have students use the criteria to evaluate their

own or each other’s presentations. Comparestudent and teacher ratings and discussdifferences.

• Work with students to develop criteria to evaluatethe presentations of their solutions to volume andsurface area problems. Criteria might include:- development or correct application of the

appropriate formula- logic of conclusions reached- effective organization of data- clarity of visual display- accuracy of calculations and measurements- accuracy of conversions among metric units, if

appropriate- accuracy of nets (if used in problem)

Print Materials


Multimedia


CD-ROM

• Geometry Blaster• Mirror Symmetry


54


In order to prepare students to use spatialproblem solving in building, describing, andanalysing geometric shapes and specifyconditions under which triangles may be similaror congruent and use these conditions to solveproblems, it is expected that students will:

• draw the plan and elevation of a 3-D objectfrom sketches and models

• sketch or build a 3-D object, given its plan andelevation views

• recognize when, and explain why, twotriangles are similar, and use the properties ofsimilar triangles to solve problems

• recognize when, and explain why, twotriangles are congruent, and use the propertiesof congruent triangles to solve problems


To extend students’ understanding of 3-D objectsand 2-D shapes, they could:

• recognize and draw the locus of points insolving practical problems

• relate congruence to similarity in the contextof triangles

• draw the image of a 2-D shape as a result of:- a single transformation- a dilatation- a combination of translation and/or

reflections• identify the single transformation that

connects a shape with its image• demonstrate that a triangle and its dilatation

image are similar• demonstrate the congruence of a triangle

with its:- translation image- rotation image- reflection image

Developing an understanding of triangle congruencycan strengthen reasoning and problem-solving skills.Visualization skills and spatial relations abilities aredeveloped when students move back and forthbetween a 3-D object and its 2-D representation.

• Display a triangle on the overhead. Move theoverhead closer to and farther from the screen anddiscuss what changes occur.

• Have students use grid paper or cardboardmodels to solve problems involving similar andcongruent triangles related to stair-stringer, truss,and rafter designs.

• Invite students to investigate the role of thetriangle in engineering, architecture, andconstruction. Discuss:- Are some shapes used more often than others?- Are particular shapes used to accomplish

particular purposes?- What part does congruence play in the use of

triangles for construction?- Why are rectangles and squares not so

common?• Use dynamic geometry software to construct

similar or congruent triangles and then use thesoftware to explore the conditions necessary forsimilarity and congruence.

• Arrange to have the class co-ordinate withtechnology education students and teachers toexplore software for creating design plans (e.g.,drawing programs, CAD).

• Have students work in pairs to consider locusproblems such as the following:- design of a safety fence around a camel’s cage,

given the locus and range of the camel’s spit- efficiency of a kitchen in terms of accessibility

to sink, stove, and refrigerator- where to place shade-loving plants around a

house, given the locus of the sun’s shadow- area around a supermarket cashier, in terms of

accessibility for both employee and customerAsk students to suggest additional problems andapplications of locus (e.g., design of car consolesor airplane cockpits in a range of cars or aircraft).Where possible, conduct field studies to observethe application of their suggestions.


55


To solve many geometric and physical problems, anunderstanding of the properties of triangles and theirapplication is essential. It is often helpful to be able tomove between a 3-D shape and its 2-D representationto correctly analyse and solve problems as well.

Collect• Ask students to imagine they are going to give

classmates directions over the telephone fordrawing simple 3-D objects. Because the otherstudents cannot see the objects, students mustrecord or write out the instructions to be sure theyare specific and clear. Check students’ recordedinstructions for accuracy, clarity, precision, andproper use of terminology. Alternatively, have twostudents, sitting back-to-back, take turns readingtheir written instructions to each other (onestudent reading while the other attempts to sketchor build the described 3-D object) to see if thedirections are accurate. Have students modify theirinstructions before submitting them to the teacher.

• Have students draw plans and elevations for 3-Dobjects. Students could exchange plans with otherstudents, who could sketch or build the objectsand provide feedback to allow for corrections.When they are satisfied with their plans, havestudents submit them to the teacher for comments.

• Give students problems (see examples inAppendix F) that require them to recognize anddraw the locus of points in solving practicalproblems. Ask students to present their solutionsto the class and explain their reasoning. Assess theaccuracy of students’ answers, the clarity and logicof their presentations, and the reasonableness oftheir conclusions.

Peer Assessment• Have students work individually to solve

problems requiring the application of theproperties of similar and congruent triangles.Then have students exchange their work withother students, check each other’s answers, andresolve any differences. Ask students to work withdifferent classmates until everyone has agreed onthe same answers. Give the class the correctanswers and have them compare results.

Print Materials


Multimedia


CD-ROM

• Geometry Blaster• Mirror Symmetry


56


In order to prepare students to collect andanalyse experimental results expressed in twovariables, using technology as required, it isexpected that students will:

• design, conduct, and report on an experimentto investigate a relationship between twovariables

• create scatter plots• interpret a scatter plot to determine if there is

an apparent linear relationship• determine the lines of best fit from a scatter

plot for an apparent linear relationship by:- inspection- using technology (equations are not

expected)• draw and justify conclusions from the line of

best fit


To extend students’ understanding of dataanalysis, they could:

• assess the strengths, weaknesses, and biases ofsamples and data-collection methods

• critique ways in which statistical informationand conclusions are presented by the mediaand other sources

Collecting, displaying, and interpreting data can helpstudents see the relevance of statistical analysis.Statistical analysis allows us to make sense of thedata in the world around us, which is an essentialpart of being an informed citizen.

• Have students design, conduct, and report on aninvestigation into a selected relationship, using ascatter plot to investigate that relationship. Askstudents to estimate and draw the lines that bestfit their scatter plot and then discuss the followingquestions with a partner:- Is it appropriate to connect the dots (e.g., is the

relationship linear)?- Can the line be used to make predictions?- Would any point that lies on the line have

meaning with respect to the two variables?Why or why not?

• Give each student a scatter plot based on a linearrelationship, and have them:- determine the line of best fit manually- determine the line of best fit using technology- identify conclusions that can be drawn from the

data- describe the relationship

• Ask students to collect and report on datacomparing the relationship between twoseemingly unrelated variables (e.g., height versusperformance in mathematics class, headcircumference versus grade in English). Discuss asa class:- Is there a direct relationship between the two

factors, or are there associated factors (e.g.,relationships between race and crime rates)?

- How may this type of information be used ormisused in society (e.g., statistics correlatingperformance to race or gender)?

- Do statistics provide the answers to allquestions?

• Use population or immigration statistics over timeto initiate a discussion about the applications ofdata analysis. For example, looking at populationstatistics of Aboriginal peoples over the last twocenturies raises questions about why and whenthe population declined. A good source of data is E-stat (http://www.statcan.ca)


57


Research projects can provide students withmeaningful experiences in collecting, analysing, anddisplaying information, as well as using scatter plotsto draw and justify conclusions derived from theresearch.

Collect• Help students design individual experiments to

investigate relationships between two variables.Have them gather data, create scatter plots, anddetermine lines of best fit. Ask them to analysetheir findings and draw conclusions. Then havestudents describe their experiments and data-collection methods to the class, present theirfindings, and justify their conclusions. A preparedlist of questions may be used to help studentsorganize their thoughts when doing evaluations ofeach other’s work. Evaluate presentations usingcriteria such as:- relevancy of the experiment- appropriateness of data-collection methods- validity of conclusions and student’s ability to

justify conclusions- organization and clarity of the presentation- accuracy of development of scatter plot and

identification of line of best fit• Provide articles from the media that present

statistical information and conclusions. Askstudents to write simple evaluations, analysing theway the data were gathered and presented and theconclusions that were drawn. Review students’evaluations and provide feedback. Evaluationsshould address questions such as:- How were the samples selected? Why were

they selected in this manner? Are they biassed?- Were the data-collection methods appropriate

for the data and the issues?- Is the data presented clearly and honestly?- Do conclusions follow logically from the data?- What questions were left unanswered? Was this

done on purpose?

Self-Assessment• Ask students to compare lines of best fit they

create through inspection to those they createusing technology. Are they relatively close?

Print Materials


Multimedia

• Hot Dog Stand: The Works• The Learning Equation Mathematics 9 (TLE)• Math Tools• Mathematics 9 (Distance Education Package)• Minds on Math 9, Revised Edition• Statistics Workshop


58


In order to prepare students to explain the use ofprobability and statistics in the solution ofproblems, it is expected that students will:

• recognize that decisions based on probabilitymay be a combination of theoreticalcalculations, experimental results, andsubjective judgments

• demonstrate an understanding of the role ofprobability and statistics in society

• solve problems involving the probability ofindependent events

Important life decisions often involve an element ofchance. Using probability in decision makinginvolves theoretical calculation, empirical estimation,and informed judgment. Using data from their ownexperiments can help students see the relevance ofthis important branch of mathematics.

• Ask students to create tables or diagrams torepresent data from events and interpret theresults in terms of independent events.

• Have students work in groups to preparearguments concerning the independence ordependence of pairs of events such as thefollowing:- A: Jennifer will get an A on her next

mathematics quiz.B: Jennifer got an A on her last mathematicsquiz.

- A: It will snow tonight.B: Jasdev’s school bus will be late tomorrowmorning.

Bring the class back together to discuss thefindings.

• Suggest that students use dictionaries to look upand, or, dependence, and independence. How do thedictionary definitions relate to the mathematicaluses? Work with the class to create mathematicaldefinitions for these terms.

• Discuss with the class the use of technology ingenerating random numbers. Have studentsexperiment with this type of application.

• Have students work in groups to design gamesthat use predicting probability as the basis formaking decisions. Have groups share their gameswith other groups.


59


Assessment should probe students’ understanding ofprobability and challenge them to demonstrate theirknowledge of and skills with chance and uncertainty.

Observe• Have students test the claim that no more than

25% of M&Ms are brown. Observe students asthey work, noting how they approach the problemand their success in correctly applying what theyknow about probability.

Question• Ask students to explain and give examples of

independent events. Are their explanationsaccurate? Are the examples appropriate?

Collect• Ask groups of students to design and conduct

probability experiments. Each experiment shouldinvolve a pair of independent events. Havestudents describe their experiments andsummarize the results. Note the following:- Do students identify all possible outcomes?- Were the experiments conducted in an

appropriate manner?- Did students accurately calculate the

probability of specified events?- Can they clearly describe their experiments?- Did they effectively summarize the results?

• Have students use sketches to illustrate thedifference between the use of AND and OR inmathematical sentences. (e.g., Of the localpopulation, 45% see a certain advertisement onTV, while 30% hear it on the radio. Only 15% seethe ad on television AND hear it on the radio.) Arestudents’ illustrations accurate? Have they choseneffective methods for illustrating the relationships?

Print Materials


Multimedia

• The Learning Equation Mathematics 9 (TLE)• Math Tools• Mathematics 9 (Distance Education Package)• Minds on Math 9, Revised Edition

APPENDICESMathematics 8 and 9

APPENDIX APrescribed Learning Outcomes

A-3


▼▼ NUMBER

(Number Concepts)

In order to preparestudents to demonstrate anumber sense for rationalnumbers, includingcommon fractions,integers, and wholenumbers, it is expected thatstudents will:

• solve problems that involve a specific content area (e.g., geometry,algebra, statistics, probability)

• solve problems that involve more than one content area withinmathematics

• solve problems that involve mathematics within other disciplines• analyse problems and identify the significant elements• develop specific skills in selecting and using an appropriate

problem-solving strategy or combination of strategies chosen from,but not restricted to, the following:- guess and check- identify patterns and use a systematic list- make and use a drawing or model- eliminate possibilities- work backward- simplify the original problem- select and use appropriate technology to assist in problem solving- analyse keywords

• solve problems individually and co-operatively• determine that solutions to problems are correct and reasonable• clearly and logically communicate a solution to a problem and the

process used to solve it• evaluate the efficiency of the processes used• use appropriate technology to assist in problem solving

PROBLEM SOLVING

In order to preparestudents to use a varietyof methods to solve real-life, practical, technical,and theoretical problems,it is expected that studentswill:

• define, identify, compare, and order any rational numbers• express two-term ratios in equivalent forms• represent and apply fractional percents and percents greater than 100 in

fraction or decimal form and vice versa• represent square roots concretely, pictorially, and symbolically• distinguish between a square root and its decimal approximation as it

appears on a calculator• express rates in equivalent forms• represent any number in scientific notation

APPENDIX A: PRESCRIBED LEARNING OUTCOMES • Mathematics 8

A-4



▼ NUMBER

(Number Operations)

In order to preparestudents to applyarithmetic operations onrational numbers to solveproblems and apply theconcepts of rate, ratio,percent, and proportion tosolve problems inmeaningful contexts, it isexpected that students will:

• add, subtract, multiply, and divide fractions concretely, pictorially,and symbolically

• estimate, compute, and verify the sum, difference, product, andquotient of rational numbers

• estimate, compute (using a calculator), and verify approximate squareroots of whole numbers

• use concepts of rate, ratio, proportion, and percent to solve problemsin meaningful contexts

• derive and apply unit rates• express rates and ratios in equivalent forms

▼ PATTERNS AND

RELATIONS

(Patterns)

In order to preparestudents to use patterns,variables andexpressions, and graphsto solve problems, it isexpected that students will:

• substitute numbers for variables in expressions and graph and analysethe relation

• translate between an oral or written expression and an equivalentalgebraic expression

• generalize a pattern from a problem-solving context

PATTERNS AND

RELATIONS

(Variables andEquations)

In order to preparestudents to solve andverify one-step linearequations with rationalnumber solutions, it isexpected that students will:

• illustrate the solution process for a one-step, single-variable, first-degreeequation, using concrete materials or diagrams

• solve and verify one-step, first-degree equations of the form:- x + a = b- ax = b

- x—a

= b

where a and b are integers• solve problems involving one-step, first-degree equations

▼

A-5



▼ SHAPE AND SPACE

(Measurement)

In order to preparestudents to applyindirect measurementprocedures to solveproblems generalizemeasurement patternsand procedures andsolve problemsinvolving area andperimeter, it is expectedthat students will:

• use the Pythagorean relationship to calculate the measure of the third sideof a right triangle, given the other two sides in 2-D applications

• describe patterns and generalize the relationships by determining theareas and perimeters of quadrilaterals and the areas and circumferencesof circles

• estimate and calculate the area of composite figures

▼ SHAPE AND SPACE

(3-D Objects and2-D Shapes)

In order to preparestudents to link anglemeasures and theproperties of parallellines to the classificationand properties ofquadrilaterals, it isexpected that studentswill:

• identify, investigate, and classify quadrilaterals, regular polygons, andcircles according to their properties

▼ SHAPE AND SPACE

(Transformations)

In order to preparestudents to analysedesign problems andarchitectural drawingsusing the properties ofscaling and proportionit is expected that studentswill:

• represent, analyse, and describe enlargements and reductions• draw and interpret scale diagrams

A-6



▼ STATISTICS AND

PROBABILITY

(Data Analysis)

In order to preparestudents to develop andimplement a plan forthe collection, displayand analysis of data,using technology asrequired and to evaluateand use measures ofcentral tendency andvariability, it is expectedthat students will:

• formulate questions for investigation, using existing data• select, defend, and use appropriate methods of collecting data:

- designing and using surveys- research, using electronic media

• display data by hand or by computer in a variety of ways• determine and use the most appropriate measure of central tendency in a

given context

▼ STATISTICS AND

PROBABILITY

(Chance andUncertainty)

In order to preparestudents to comparetheoretical andexperimentalprobability ofindependent events, it isexpected that studentswill:

• use various data-collection techniques (including computers) to simulateand solve probability problems

• recognize that if n events are equally likely, the probability of any one ofthem occurring is 1—

n

• determine the probability of two independent events where thecombined sample space has 52 or fewer elements

• predict population characteristics from sample data

A-7



▼ PROBLEM SOLVING

In order to preparestudents to use a varietyof methods to solvereal-life, practical,technical, andtheoretical problems, itis expected that studentswill:

• solve problems that involve a specific content area (e.g., geometry,algebra, statistics, probability)

• solve problems that involve more than one content area withinmathematics

• solve problems that involve mathematics within other disciplines• analyse problems and identify the significant elements• develop specific skills in selecting and using an appropriate problem-

solving strategy or combination of strategies chosen from, but notrestricted to, the following:- guess and check- identify patterns and use a systematic list- make and use a drawing or model- eliminate possibilities- work backward- simplify the original problem- select and use appropriate technology to assist in problem solving- analyse keywords

• solve problems individually and co-operatively• determine that solutions to problems are correct and reasonable• clearly and logically communicate a solution to a problem and the

process used to solve it• evaluate the efficiency of the processes used• use appropriate technology to assist in problem solving

▼ NUMBER

(Number Concepts)

In order to preparestudents to develop anumber sense of powerswith integral exponentsand variable and rationalbases, it is expected thatstudents will:

• give examples of situations where answers would involve the positive(principal) square root or both positive and negative square roots of anumber

• illustrate power, base, coefficient, and exponent using rational numbersor variables as bases or coefficients

A-8


▼ NUMBER

(Number Operations)

In order to preparestudents to use a scientificcalculator or a computerto solve problemsinvolving rationalnumbers, it is expected thatstudents will:

• document and explain the calculator keying sequences used toperform calculations involving rational numbers

• solve problems, using rational numbers in meaningful contexts• evaluate exponential expressions with numerical bases

▼ PATTERNS AND

RELATIONS

(Patterns)

In order to preparestudents to generalize,design, and justifymathematical proceduresusing appropriatepatterns, models, andtechnology, it is expectedthat students will:

• model situations that can be represented by first-degree expressions• write equivalent forms of algebraic expressions, or equations, with

integral coefficients

PATTERNS AND

RELATIONS

(Variables andEquations)

In order to preparestudents to evaluate,solve, and verify linearequations in one variableand generalize arithmeticoperations from the setof rational numbers tothe set of polynomials, itis expected that studentswill:

• illustrate the solutions process for a first-degree, single-variable equation,using concrete materials or diagrams

• solve and verify first-degree, single-variable equations of forms such as:ax = b + cx; a(x+b)=c; ax + b = cx + d where a, b, c, and d are integers, and useequations of this type to model and solve problems

• identify constant terms, coefficients, and variables in polynomialexpressions

• evaluate polynomial expressions, given the value(s) of the variable(s)

▼


A-9



▼ SHAPE AND SPACE

(Measurement)

In order to preparestudents to usetrigonometric ratios tosolve problemsinvolving righttriangles, it is expectedthat students will:

• explain the meaning of sine, cosine, and tangent ratios in right triangles• demonstrate the use of trigonometric ratios (sine, cosine, and tangent) in

solving right triangles• calculate an unknown side or an unknown angle in a right triangle, using

appropriate technology• model and then solve given problem situations involving only one right

triangle

▼ SHAPE AND SPACE

(3-D Objects and2-D Shapes)

In order to preparestudents to use spatialproblem solving inbuilding, describing,and analysing geometricshapes and specifyconditions under whichtriangles may be similaror congruent and usethese conditions to solveproblems, it is expectedthat students will:

• draw the plan and elevation of a 3-D object from sketches and models• sketch or build a 3-D object, given its plan and elevation views• recognize when, and explain why, two triangles are similar, and use the

properties of similar triangles to solve problems• recognize when, and explain why, two triangles are congruent, and use

the properties of congruent triangles to solve problems

▼ STATISTICS AND

PROBABILITY

(Data Analysis)

In order to preparestudents to collect andanalyse experimentalresults expressed in twovariables, usingtechnology as required,it is expected that studentswill:

• design, conduct, and report on an experiment to investigate a relationshipbetween two variables

• create scatter plots• interpret a scatter plot to determine if there is an apparent linear

relationship• determine the lines of best fit from a scatter plot for an apparent linear

relationship by:- inspection- using technology (equations are not expected)

• draw and justify conclusions from the line of best fit

A-10


▼ STATISTICS AND

PROBABILITY

(Chance andUncertainty)

In order to preparestudents to explain theuse of probability andstatistics in the solutionof problems, it isexpected that studentswill:

• recognize that decisions based on probability may be a combination oftheoretical calculations, experimental results, and subjective judgments

• demonstrate an understanding of the role of probability and statistics insociety

• solve problems involving the probability of independent events


APPENDIX BLearning Resources

B-2

APPENDIX B: LEARNING RESOURCES • Mathematics 8 and 9

B-3


WHAT IS APPENDIX B?

Appendix B for this IRP includes Grade Collection resources for Grades 8 and 9 Mathematicscourses selected in 1999. These resources are all provincially recommended. It is the ministry’sintention to add additional resources to these Grade Collections as they are evaluated.

Supplier: McGraw-Hill Ryerson Ltd. (Ontario)

300 Water StreetWhitby, ON

L1N 9B6

Tel: 1-800-565-5758 (orders only)Fax: (905) 430-5020

Price: Student Book: $33.90Blackline Masters: $89.25Teacher's Edition: $105.00

ISBN/Order No: Student Book: 0075526506Blackline Masters: 0075526514Teacher's Edition: 0075526522

Copyright Year: 1996

Year Recommended for Grade Collections: 2001

Curriculum Organizer(s): NumberPatterns and RelationsShape and SpaceStatistics and Probability

Author(s): Knill, G.; et al.

General Description: Student book, blackline masters, andteacher's edition support the learning of mathematics invarious ways. They present concepts and skills in problem-solving contexts. Questions help students express their ideasabout math orally and in writing. Uses manipulatives for theunderstanding of abstract concepts and integrates uses ofappropriate technology. manipulatives such as algebra titles,fraction circles, linking cubes, geoboards, and suitablesoftware are necessary for using the resource effectively; theseare not supplied with the resource.

Cautions: There is excessive emphasis on routine drill in thenumber strand.

Audience: General

Category: Student, Teacher Resource

1. General Description

2. Media Format

▲

▲

▲

▲

▲

▲▲

▲

What information does an annotation provide?

5. Supplier3. Author(s)

4. Curriculum Organizers

6. Ordering Information

7. Category 8. Audience 9. Caution(s)

MATHPOWER 8,Western Edition

▲

B-4


1. General Description: This section providesan overview of the resource.

2. Media Format: This part is represented byan icon next to the title. Possible iconsinclude:

Audio Cassette

CD-ROM

Film

Games/Manipulatives

Laserdisc/Videodisc

Multimedia

Music CD

Print Materials

Record

Slides

Software

Video

3. Author(s): Author and editor information isprovided where it might be of use to theteacher.

4. Curriculum Organizers: This category helpsteachers make links between the resourceand the curriculum.

5. Supplier: The name and address of thesupplier are included in this category. Pricesshown here are approximate and subject tochange. Prices should be verified with thesupplier.

6. Ordering Information: This category isused to give the ISBN or publisher orderinginformation.

7. Category: This section indicates whether itis a student and teacher resource, teacherresource, or professional reference.

8. Audience: This category indicates thesuitability of the resource for different typesof students. Possible student audiencesinclude the following:

• general• English as a second language (ESL)• Students who are:

- gifted- blind or have visual impairments- deaf or hard of hearing

• Students with:- severe behaviour disorders- dependent handicaps- physical disabilities- autism- learning disabilities (LD)- mild intellectual disabilities (ID-mild)- moderate to severe/profound disabilities

(ID-moderate to severe/profound)

9. Cautions: This category is used to alertteachers about potentially sensitive issues.

B-5


SELECTING LEARNING RESOURCES FOR THE

CLASSROOM

Selecting a learning resource for Grades 8 and9 Mathematics means choosing locallyappropriate materials from the GradeCollection or other lists of evaluatedresources. The process of selection involvesmany of the same considerations as theprocess of evaluation, though not to the samelevel of detail. Content, instructional design,technical design, and social considerationsmay be included in the decision-makingprocess, along with a number of other criteria.

The selection of learning resources should bean ongoing process to ensure a constant flowof new materials into the classroom. It ismost effective as an exercise in group decisionmaking, coordinated at the school, district,and ministry levels. To function efficientlyand realize the maximum benefit from finiteresources, the process should operate inconjunction with an overall district andschool learning resource implementationplan.

Teachers may choose to use provinciallyrecommended resources to support provincialor locally developed curricula; chooseresources that are not on the ministry’s list; orchoose to develop their own resources.Resources that are not on the provinciallyrecommended list must be evaluated througha local, board-approved process.

CRITERIA FOR SELECTION

There are a number of factors to considerwhen selecting learning resources.

Content

The foremost consideration for selection is thecurriculum to be taught. Prospectiveresources must adequately support the

particular learning outcomes that the teacherwants to address. Teachers will determinewhether a resource will effectively supportany given learning outcomes within acurriculum organizer. This can only be doneby examining descriptive informationregarding that resource; acquiring additionalinformation about the material from thesupplier, published reviews, or colleagues;and by examining the resource first-hand.

Instructional Design

When selecting learning resources, teachersmust keep in mind the individual learningstyles and abilities of their students, as well asanticipate the students they may have in thefuture. Resources have been recommended tosupport a variety of special audiences,including gifted, learning disabled, mildlyintellectually disabled, and ESL students. Thesuitability of a resource for any of theseaudiences has been noted in the resourceannotation. The instructional design of aresource includes the organization andpresentation techniques; the methods used tointroduce, develop, and summarize concepts;and the vocabulary level. The suitability ofall of these should be considered for theintended audience.

Teachers should also consider their ownteaching styles and select resources that willcomplement them. Lists of provinciallyrecommended resources contain materialsthat range from prescriptive or self-containedresources to open-ended resources thatrequire considerable teacher preparation.There are provincially recommendedmaterials for teachers with varying levels ofexperience with a particular subject, as well asthose that strongly support particularteaching styles.

B-6


Technology Considerations

Teachers are encouraged to embrace a varietyof educational technologies in theirclassrooms. To do so, they will need to ensurethe availability of the necessary equipmentand familiarize themselves with its operation.If the equipment is not currently available,then the need must be incorporated into theschool or district technology plan.

Social Considerations

All resources on the ministry’s provinciallyrecommended lists have been thoroughlyscreened for social concerns from a provincialperspective. However, teachers mustconsider the appropriateness of any resourcefrom the perspective of the local community.

Media

When selecting resources, teachers shouldconsider the advantages of various media.Some topics may be best taught using aspecific medium. For example, video may bethe most appropriate medium when teachinga particular, observable skill, since it providesa visual model that can be played over andover or viewed in slow motion for detailedanalysis. Video can also bring otherwiseunavailable experiences into the classroomand reveal “unseen worlds “ to students.Software may be particularly useful whenstudents are expected to develop critical-thinking skills through the manipulation of asimulation, or where safety or repetition is afactor. Print resources or CD-ROM can bestbe used to provide extensive backgroundinformation on a given topic. Once again,teachers must consider the needs of theirindividual students, some of whom may learnbetter from the use of one medium thananother.

Funding

As part of the selection process, teachersshould determine how much money isavailable to spend on learning resources. Thisrequires an awareness of school and districtpolicies, and procedures for learning resourcefunding. Teachers will need to know howfunding is allocated in their district and howmuch is available for their needs. Learningresource selection should be viewed as anongoing process that requires a determinationof needs, as well as long-term planning to co-ordinate individual goals and local priorities.

Existing Materials

Prior to selecting and purchasing newlearning resources, an inventory of thoseresources that are already available should beestablished through consultation with theschool and district resource centres. In somedistricts, this can be facilitated through theuse of district and school resourcemanagement and tracking systems. Suchsystems usually involve a database to helpkeep track of a multitude of titles. If such asystem is available, then teachers can checkthe availability of a particular resource via acomputer.

SELECTION TOOLS

The Ministry of Education has developed avariety of tools to assist teachers with theselection of learning resources.

These include:• Integrated Resource Packages (IRPs) that

contain curriculum information, teachingand assessment strategies, and provinciallyrecommended learning resources,including Grade Collections

• resource databases on disks or on-line• sets of the most recently recommended

learning resources (provided each year to a

B-7


number of host districts throughout theprovince to allow teachers to examine thematerials first-hand at regional displays)

• sample sets of provincially recommendedresources (available on loan to districts onrequest)

A MODEL SELECTION PROCESS

The following series of steps is one way aschool resource committee might go aboutselecting learning resources:

1. Identify a resource coordinator (forexample, a teacher-librarian).

2. Establish a learning resources committeemade up of department heads or leadteachers.

3. Develop a school vision and approach toresource-based learning.

4. Identify existing learning resource andlibrary materials, personnel, andinfrastructure.

5. Identify the strengths and weaknesses ofexisting systems.

6. Examine the district Learning ResourcesImplementation Plan.

7. Identify resource priorities.8. Apply criteria such as those found in

Evaluating, Selecting, and ManagingLearning Resources: A Guide to shortlistpotential resources.

9. Examine shortlisted resources first-hand ata regional display or at a publishers ‘display, or borrow a set by contactingeither a host district or the CurriculumBranch.

10. Make recommendations for purchase.

FURTHER INFORMATION

For further information on evaluation andselection processes, catalogues, CD-ROMcatalogues, annotation sets, or resourcedatabases, please contact the CurriculumBranch of the Ministry of Education.

APPENDIX BLearning ResourcesGrade Collections

B-10


B-11


MATHEMATICS 8 AND 9 GRADE COLLECTIONS

INTRODUCTION

The majority of the learning resourcesin these Grade Collections have beenevaluated through the WesternCanadian Protocol Learning ResourceEvaluation Process and subsequentlyhave been given Provincially Recommendedstatus by the Ministry of Education.

Grade collections are not prescriptive; theyare intended to provide assistance and adviceonly. Teachers are encouraged to use existingresources that match the learning outcomesand to select additional resources to meettheir specific classroom needs. It isrecommended that teachers use theMathematics 8 and 9 IRP when makingresource decisions.

Resources that are identified through theWestern Canadian Protocol Evergreeningprocess as having strong curriculum matchwill be added to the collections as theybecome available.

Categories of Resources

Learning resources selected for the GradeCollection have been categorized as eithercomprehensive, or additional.

• Comprehensive resources tend to provide abroad coverage of the learning outcomesfor most of the curriculum organizers.

• Additional resources are more topic specificand support individual curriculumorganizers or clusters of outcomes. Theyare recommended as valuable support orextensions for specific topics.

In many cases, Grade Collections providemore than one resource to support specificoutcomes, enabling teachers to selectresources that best match different teachingand learning styles.

Other Recommended Resources

The Western Canadian Protocol LearningResource Evaluation Process has alsoidentified numerous teacher resources andprofessional references. These are generallycross-grade planning resources that includeideas for a variety of activities and exerciseswhich may complement the comprehensiveresources. These resources, while not part ofthe Grade Collections, have provinciallyRecommended status. An alphabetical listindicating the suggested course is included inthis package, along with an annotatedbibliography for ordering convenience.

Appendix B includes annotations for otherRecommended resources not in the GradeCollections. While these resources meet onlya limited number of outcomes, teachers areencouraged to consider them for differentaudience needs, teaching and learning styles,theme development, in-depth research, andso on.

Grade Collection Information

The following pages begin with an overviewof the comprehensive resources for thiscurriculum, then present Grade Collectioncharts for each course. These charts list bothcomprehensive and additional resources foreach curriculum organizer for the course.Each chart is following by an annotatedbibliography. There is an alphabetical list ofthe teacher resources, indicating course. Thislist is followed by an annotated bibliography.Please confirm with suppliers for completeand up-to-date price information. There isalso a chart that shows the alphabetical list ofGrade Collection titles for each course and ablank template that can be used by teachers torecord their individual choices.

B-12


OVERVIEW OF COMPREHENSIVE RESOURCES

FOR MATHEMATICS 8 AND 9

• Interactions (Level 8) andInteractions (Level 9)

Student textbook and teacher’s resource bookwith assessment booklet and blacklinemasters provide opportunities to developmathematical literacy along three dimensions:number sense, spatial sense, and problem-solving. Includes many learning activities.Requires manipulatives such as algebra tiles,integer disks, and suitable software.

• The Learning Equation Mathematics 8and The Learning Equation Mathematics 9(TLE)

Interactive program for Macintosh orWindows consisting of two CD-ROMs, astudent refresher, and a teacher’s guide. Alllessons follow the pattern of an introduction,followed by a tutorial example for students, asummary of concepts, practice problems,extra practice, and a self check. Students areable to monitor their own progress throughthe various components of each lesson. Alsoincluded on the CD-ROMs is a glossary whichstudents can access as a pull-down menu.The student refresher follows the organizationof the CD-ROMs and provides additionalexamples for students as well as answers.The teacher’s manual includes installationnotes, instructional summaries, prerequisiteskills, and cross-references to the CD-ROMs.

System requirements for Macintosh: System7.1 or later, 8 Mb RAM (16 Mb preferred, VGA256-colour 14" monitor, 15 Mb or more of harddrive space, double speed CD-ROM drive, 8bit sound support for audio.

System requirements for Windows 3.1: MPC2compliant computer, 486SX 25 MHz (50 orbetter preferred), 8 Mb RAM (16 Mbpreferred), VGA 256-colour monitor, 40 Mb

hard drive space, double speed CD-ROMdrive, soundblaster or 100% support foraudio.

System requirements for Windows 95/98 orNT: MPC2 compliant computer, 486SX 66MHz (Pentium or better preferred), 16 MbRAM, VGA 256-colour monitor, 40 Mb harddrive space, double speed CD-ROM drive,soundblaster or 100% support for audio.

• Mathematics 8 and 9 (Distance LearningPackage)

This resource package, designed for distanceeducation, consists of six coilbound moduleswith accompanying assignment booklets.One videocassette on key concepts supportsthe print material. A learning facilitator’smanual is also available. The programcorrelates exactly with all specific outcomes ofthe Common Curriculum Framework.

• MATHPOWER 8 and MATHPOWER 9,Western Editions

Student book, teacher’s edition, and blacklinemasters support the learning of mathematicsin various ways. The resources presentconcepts and skills in problem-solvingcontexts. Questions help students expresstheir ideas about math orally and in writing.The resources suggest the use ofmanipulatives for the understanding ofabstract concepts. Technology is integratedthroughout. Manipulatives such as algebratiles and suitable software are necessary forusing the resources effectively; these are notsupplied with the resource.

• Minds on Math 8 and Minds on Math 9,Revised Editions

Student book provides activities, instruction,discussion, exercises, and projects to engagestudents in meaningful, real-worldmathematics. Each of the eight chapters

B-13


begins with activities and exercises that allowstudents to assess their prior knowledgebefore they explore new concepts. Includesteacher’s resource book, template and datadisk package, and activity cards with answerkey. Manipulatives such as algebra tiles andsuitable software are necessary for using theresource effectively; these are not suppliedwith the resource. The software packagerequires either ClarisWorks or MicrosoftWorks in order to run.

• Understanding Math Series

CD-ROM series for Macintosh or Windowsassists students in developing anunderstanding of mathematics conceptsbased on a solid foundation of concreteactivities. Topics include fractions, integers,exponents, algebra, equations, coordinategeometry and graphing, and percent. Foreach topic the program contains conceptsections with explanations, example sections,practice sections, cumulative checks. Theuser is in control through pull-down menusand proceed and go-back buttons. Eachprogram includes a teacher manual.

System requirements for Macintosh: 68020 orfaster processor, including Power Macintosh,Macintosh 7.5.1 or later, 40 Mb hard drivespace or CD-ROM drive.

System requirements for Windows: Windows3.1, Windows 95, Windows NT3.51 or 4.0;486DX2-66 (Pentium recommended); 640x480with 256 colour display (or higher); 40 Mbhard drive space or CD-ROM drive.

APPENDIX BGrade Collections

B-16


B-17


Statistics andP

robability

Number

ConceptsN

umberO

perationsPatterns

Variables &Equations

Relations &Functions

Measurement3-D O

bjects &2-D Shapes

TransformationsData Analysis

Chance &Uncertainty

Mathem

atics 8 Grade C

ollectio

n

Num

berP

atterns and Relatio

nsS

hape and Space

Co

mprehensive R

esources

Interactions (Level 8)

The Learning Equation Mathem

atics 8 (TLE)

Mathem

atics 8 (Distance Education Package)

MATH

POW

ER 8, Western Edition

Minds on M

ath 8, Revised Edition

Triple ‘A’ Mathem

atics Program: D

ataM

anagement &

Probability

The Geom

eter’s Sketchpad

Geom

etry Blaster

Hot D

og Stand: The Works

Math Tools

Mathville VIP

Mirror Sym

metry

Statistics Workshop

Understanding M

ath Series

✔✔

✔✔

✔✔

✔✔

✔

✔✔

✔✔

✔✔

✔✔

✔

✔✔

✔✔

✔✔

✔✔

✔

✔✔

✔✔

✔✔

✔✔

✔

Additio

nal Reso

urces-Print

Additio

nal Reso

urces-So

ftware/C

D-R

OM

✔✔

✔✔

✔

✔✔

✔

✔✔

✔

✔✔

✔✔

✔✔

✔✔

✔

✔✔

✔✔

✔✔

✔✔

✔

✔✔

✔✔

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✔✔

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✔

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Indicates that the curriculum sub-organizer is not included at this grade level

B-18


Sta

tist

ics

and

Pro

babi

lity

Num

ber

Conc

epts

Num

ber

Ope

ratio

nsPa

ttern

sVa

riable

s &Eq

uatio

nsRe

lation

s &Fu

nctio

nsMe

asur

emen

t3-

D O

bjects

&2-

D Sh

apes

Tran

sform

ation

sDa

ta An

alysis

Mat

hem

atic

s 9

Gra

de C

olle

ctio

n

Num

ber

Pat

tern

s an

d R

elat

ions

Sha

pe a

nd S

pace

Co

mpr

ehen

sive

Res

our

ces

Inte

ract

ions

(Le

vel 9

)

The

Lear

ning

Equ

atio

n M

athe

mat

ics 9

(TLE

)

Mat

hem

atics

9 (D

istan

ce E

duca

tion

Pack

age)

MAT

HPO

WER

9, W

este

rn E

ditio

n

Min

ds o

n M

ath

9, R

evise

d Ed

ition

Trip

le ‘A

’ Mat

hem

atics

Pro

gram

: D

ata

Man

agem

ent &

Pro

babi

lity

The

Geo

met

er’s

Sket

chpa

d

Geo

met

ry B

last

er

Hot

Dog

Sta

nd: T

he W

orks

Mat

h To

ols

Mirr

or S

ymm

etry

Stat

istics

Wor

ksho

p

Und

erst

andi

ng M

ath

Serie

s

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

✔✔

Add

itio

nal R

eso

urce

s-P

rint

Add

itio

nal R

eso

urce

s-S

oft

war

e/C

D-R

OM

✔

✔✔

✔

✔✔

✔

✔✔

✔

✔✔

✔✔

✔✔

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✔✔

✔✔

✔✔

✔✔

✔

✔

✔✔

✔✔

✔✔

Indi

cate

s th

at t

he c

urri

culu

m s

ub-o

rgan

izer

is n

ot in

clud

ed a

t th

is g

rade

leve

l

Chan

ce &

Unce

rtaint

y

✔ ✔ ✔ ✔ ✔✔ ✔ ✔

B-19


About Teaching Mathematics: A K-8 Resource

Acces: The Teacher’s Database

Active Learning: Alge-Tiles

Constructing Ideas: Grades 6-8

Cool Math Games

Figures, Facts & Fables: Telling Tales in Science and Math

Fraction Blocks: Grades 5-8

Glencoe Mathematics Professional

Hands-on Math Projects with Real-Life Applications

How to’s in Getting Started with Assessment & Evaluation Using Portfolios

In the Balance

The Intermediate Geoboard

Junior High Probability Jobcards

Just a Minute: Activities for Spare Moments in Class

Learning Mathematics in Elementary and Middle Schools

Math Assessment: Grade 8

Mathematics: What are You teaching My Child?

Mathematics with Polydrons

Mental Math in Junior High

Measure for Measure: Using Portfolios in K-8 Mathematics

The Multicultural Classroom: Bringing in the World

The Nelson Canadian School Mathematics Dictionary

Problem Solving: What you Do When You Don’t Know What To DoSolutions & Extensions for Illustrative Examples from the Western CanadianProtocol: Grade 9

Using the Explorer Plus Calculator

The World’s Most Popular Puzzles and Problems

Writing in Math Class

Mathematics 8 & 9 Teacher Resources and Professional References

Grade LevelTitle 8 9

✔ ✔

✔ ✔

✔ ✔

✔

✔

✔ ✔

✔

✔ ✔

✔ ✔

✔ ✔

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✔

✔

✔ ✔

✔ ✔

✔

✔ ✔

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✔

✔

✔ ✔

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✔ ✔

✔

✔

✔

✔

B-20


Sta

tist

ics

and

Pro

babi

lity

Num

ber

Conc

epts

Num

ber

Ope

ratio

nsPa

ttern

sVa

riable

s &Eq

uatio

nsRe

lation

s &Fu

nctio

nsMe

asur

emen

t3-

D O

bjects

&2-

D Sh

apes

Tran

sform

ation

sDa

ta An

alysis

Mat

hem

atic

s G

rade

Co

llect

ion

Cur

ricu

lum

Org

aniz

er C

hart

Num

ber

Pat

tern

s an

d R

elat

ions

Sha

pe a

nd S

pace

Co

mpr

ehen

sive

Res

our

ces

Chan

ce &

Unce

rtaint

y

Add

itio

nal R

eso

urce

s-P

rint

Add

itio

nal R

eso

urce

s-G

ames

/Man

ipul

ativ

es

Add

itio

nal R

eso

urce

s-S

oft

war

e/C

D-R

OM

B-21


APPENDIX BLearning ResourcesComplete Listings

Mathematics Grade 8 Collection

Student, Teacher ResourceCategory:

General Description: Software package for Macintosh or Windowsconsists of disks, a user's guide and reference manual, teaching notes,a quick reference, and an introductory video. Sketches depictconcrete geometry and emphasize spatial reasoning, while scriptsdescribe constructions verbally and abstractly. Program allows for theconstruction, labelling, measurement, and manipulation of anygeometric figure, as well as the exploration of geometry concepts.The user's guide and reference manual provide adequate instructionfor learning to use the program. The teaching notes with sampleactivities provide suggestions for classroom use. Several sampleinvestigations, explorations, demonstrations, and constructionactivities on a wide range of geometry topics are included.

System requirements for Macintosh: Macintosh Plus or later; 1 Mb ofRAM; System 6 or later.System requirements for Windows: 4 Mb of RAM; 3.5" floppy disk;hard drive; Windows 3.1 in enhanced mode.

Cautions: Program defaults to inches, but it can be set to centimetres.

Audience: General

Shape and SpaceCurriculum Organizer(s):The Geometer's Sketchpad

Supplier:

ISBN/Order No:

Price:

Macintosh: 1-55953-098-7Windows: 1-55953-099-5

Macintosh: $289.95Windows: $289.95

125 Mary St.Aurora, ONL4G 1G3

Tel: (905) 841-0600 Fax: (905) 727-6265

Spectrum Educational Supplies Ltd.


2001Year Recommended for Grade Collection:


General Description: CD-ROM package for Macintosh or Windowsincludes lessons, activities and games, all of which teach andreinforce geometric concepts. It uses a three-level approach,allowing a student to progress at a beginning, an intermediate, or amastery level. The main topics explored are points and lines,triangles, polygons and quadrilaterals, similarity, circles, perimeterand area, solids in 3D, coordinate geometry, transformationalgeometry and logical reasoning and proof. Requires decision makingand creative thinking by students.

System requirements for Macintosh: System 7.0 or later, 8 Mb RAM.System requirements for Windows 3.1: 386/33 MHz, 8 Mb RAM,Pentium processor recommended, 256 super VGA or better colourmonitor, sound and graphics cards, CD-ROM drive, and a mouserequired; printer is recommended.System requirements for Windows 95: 486/50 MHz, 8 Mb RAM,Pentium processor recommended, 256 super VGA or better colourmonitor, sound and graphics cards, CD-ROM drive, and a mouserequired; printer is recommended.

Audience: General

Shape and SpaceCurriculum Organizer(s):Geometry Blaster

Supplier:

ISBN/Order No:

Price:

0-7849-1063-4

$79.95

19840 Pioneer AvenueP.O. Box 2961Torrance, CA90503

Tel: 1-800- 545-7677 Fax: (310) 793-0601

Davidson & Associates, Inc.



B-23



General Description: This software for Macintosh or Windowssimulates running a hot dog stand. Students gather information,keep records, plan inventory, decide on pricing and look at sellingstrategies. There are three levels of operation. At the intermediateand advanced leves, students also deal with the unexpected eventsof running a business, such as weather, unreliable suppliers, andmachine breakdown. Students estimate, calculate, read graphs, andkeep track of their progress in a journal - incorporated as a mini-wordprocessor. In the teacher's guide there are follow-up activitiesrelating to starting and running a business.

System requirements for Macintosh: System 7.0 or later; 68040 or laterprocessor; Mb RAM; 31Mb hard drive space; double speed CD-ROMdrive; and 640x480, 256 colour monitor.System requirements for Windows: Windows 3.1 or Windows 95,8Mb RAM, 30Mb hard drive, double speed CD-ROM drive, 640x480256 colour monitor.

Audience: General

NumberStatistics and Probability

Curriculum Organizer(s):Hot Dog Stand: The Works

Supplier:

ISBN/Order No:

Price:

Not available

$89.95 each version

164 Commander BoulevardScarborough, ONM1S 3C7

Tel: 1-800-667-1115 Fax: (416) 293-9009

Gage Educational Publishing Co. (Scarborough)




General Description: Student textbook and teacher's resource bookwith assessment booklet and blackline masters provide opportunitiesto develop mathematical literacy along three dimensions: numbersense, spatial sense, and problem solving. Includes many learningactivities. Requires manipulatives such as algebra tiles, integer disks,and suitable software.

Cautions: Supplementary material on geometrical transformations isnecessary.

Audience: General

Author(s): Elchuk, L. ... (et al.);

NumberPatterns and RelationsShape and SpaceStatistics and Probability

Curriculum Organizer(s):Interactions (Level 8)

Supplier:

ISBN/Order No:

Price:

Student Textbook: 0132588153Teacher Resource Package: 0132588234

Student Textbook: $32.86Teacher Resource Package: $116.60

26 Prince Andrew PlaceDon Mills, ONM3C 2T8

Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948

Pearson Education Canada



B-24



General Description: Interactive program for Macintosh orWindows consists of two CD-ROMs, a student refresher, and ateacher's guide. All lessons follow the pattern of an introduction,followed by a tutorial example for students, a summary of concepts,practice problems, extra practice, and a self check. Students are ableto monitor their own progress through the various components ofeach lesson. Also included on the CD-ROMs is a glossary whichstudents can access as a pull-down menu. The student refresherfollows the organization of the CD-ROMs and provides additionalexamples for students as well as answers. The teacher's manualincludes installation notes, instructional summaries, prerequisite skills,and cross-references to the CD-ROMs.

System requirements for Macintosh: System 7.1 or later, 8 Mb RAM(16 Mb preferred), VGA 256-colour 14" monitor, 15 Mb or more harddisk space, double speed CD-ROM drive, 8 bit sound support foraudio.System requirements for Windows 3.1: MPC2 compliant computer,486SX 25 Mhz (50 or better preferred), 8 Mb RAM (16 Mb preferred),VGA 256-colour monitor, 40 Mb hard disk space, double speedCD-ROm drive, soundblaster or 100% support for audio.System requirements for Windows 95: MPC2 compliant computer,486SX 66 MHz (Pentium preferred), 16 Mb RAM, VGA 256-colourmonitor, 40 Mb hard disk space, double speed CD-ROM drive,soundblaster or 100% support for audio.System requirements for Windows 98 or NT: MPC2 compliantcomputer, Pentium 100 MHz (Pentium 133 or faster recommended),16 Mb or more RAM, VGA 256 colour monitor, 40 Mb or moe of freehard disk space, double speed CD drive or better, Soundblaster or100% support for audio.Additional requirements for network installations: faster networksprovide more satisfactory performance; to run THE LEARNINGEQUATION: MATHEMATICS 8 from the server hard drive andcopy all the CD-ROM files requires 2.0 GB or more free space; to runTHE LEARNING EQUATION: MATHEMATICS 8 from a centralCD-ROM drive, the drive or jukebox must be connected to thenetwork server.

Audience: General


Curriculum Organizer(s):The Learning Equation: Mathematics 8

Supplier:

ISBN/Order No:

Price:

CD-ROMS 1 and 2 (Version 2.0): 0176155783Student Refresher: 017615681XTeacher's Manual: 0176156828

CD-ROMS 1 and 2 (Version 2.0): $150.00Student Refresher: $8.50Teacher's Manual: $74.00

1120 Birchmount RoadScarborough, ONM1K 5G4

Tel: 1-800-268-2222 Fax: (416) 752-8101

Nelson Thomson Learning



B-25



General Description: This package is a set of Windows softwareutilities that can share data. The tools include spreadsheet,calculation, geometry, graphs, manipulatives, fraction strips, andprobability. A colour linking feature between two or more windowsallows data to be shared and represented in a variety of formats.Includes information on installing and using the software andchapters specific to each math tool. Provides a glossary and index.

System requirements for Windows 3.1: 386/33 MHz, 4Mb of RAM,13" VGA monitor, 2Mb hard disk space, PKZIP.System requirements for Windows 95: 486/50 MHz, 8Mb of RAM,13" VGA monitor, 2Mb hard disk space, PKZIP.

Cautions: The algebra tile manipulatives have limited usage (first-degreevariables only.) The spreadsheet uses radians as units for sine, cosine, andtangent calculations.

Audience: General


Curriculum Organizer(s):Math Tools

Supplier:

ISBN/Order No:

Price:

0134109783

$58.94


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948





General Description: This resource package, designed for distanceeducation, consists of six coilbound modules with accompanyingassignment booklets. Two videocassettes on key concepts support theprint material. A learning facilitator's manual is also available. Theprogram correlates exactly with all specific outcomes of the CommonCurriculum Framework.

Audience: General


Curriculum Organizer(s):Mathematics 8 (Distance Learning Package)

Supplier:

ISBN/Order No:

Price:

Video 1: MA0033Video 2: MA0034

Modules: $16.10 eachVideo 1: $24.00Video 2: $23.00

12360 - 142 StreetEdmonton, ABT5L 4X9

Tel: (780) 427-5775 Fax: (780) 422-9750

Learning Resources Distributing Centre



B-26



General Description: Student book, blackline masters, and teacher'sedition support the learning of mathematics in various ways. Theypresent concepts and skills in problem-solving contexts. Questionshelp students express their ideas about math orally and in writing.Uses manipulatives for the understanding of abstract concepts andintegrates use of appropriate technology. Manipulatives such asalgebra tiles, fraction circles, linking cubes, geoboards, and suitablesoftware are necessary for using the resource effectively; these arenot supplied with the resource.

Cautions: There is excessive emphasis on routine drill in the numberstrand.

Audience: General

Author(s): Knill, G.; et al.


Curriculum Organizer(s):MATHPOWER 8, Western Edition

Supplier:

ISBN/Order No:

Price:

Student Book: 0075526506Blackline Masters: 0075526514Teacher's Edition: 0075526522

Student Book: $33.90Blackline Masters: $89.25Teacher's Edition: $105.00

300 Water StreetWhitby, ONL1N 9B6

Tel: 1-800-565-5758 (orders only) Fax: (905)430-5020

McGraw-Hill Ryerson Ltd. (Ontario)




General Description: Interactive CD-ROM for Macintosh orWindows integrates a broad range of thinking and mathematical skillswith everyday life in the multimedia environment of a virtualvillage. In Mathville each student becomes a Very Important Personby walking, shopping, and playing related to the four strands of themathematics program. Activities are imaginative and engaging forstudents. The program is flexible and noncompetitive and supportsdifferent levels of student learning. It is suitable for review at theGrades 7 to 8 levels and for enrichment of Grade 6, usingproblem-solving skills.

System requirements for Macintosh: System 7 or higher, 640x480colour monitor, 8Mb hard disk space, CD-ROM drive.System requirements for Windows 3.1: 486/50 MHz, 8 Mb RAM,colour monitor, sound and graphics cards, mouse, CD-ROM drive.System requirements for Windows 95: 486/66 MHz, 8 Mb RAM,colour monitor, sound and graphics cards, mouse, CD-ROM drive.

Audience: General


Curriculum Organizer(s):Mathville VIP

Supplier:

ISBN/Order No:

Price:

1-89569154-0

$56.00






B-27



General Description: Student book provides activities, instruction,discussion, and projects to engage students in meaningful, real-worldmathematics. Each of the nine chapters begins with activities andexercises that allow students to assess their prior knowledge beforethey explore new concepts. Includes teacher's resource book,template and data disk package, and activity cards with answer key.Manipulatives such as algebra tiles, integer disks, and suitablesoftware are necessary to use the resource effectively; these are notsupplied with the resource. The software package requires eitherClarisWorks or Microsoft Works in order to run.

Audience: General

Author(s): Alexander, R., et al.

NumberShape and SpacePatterns and RelationsStatistics and Probability

Curriculum Organizer(s):Minds on Math 8, Revised Edition

Supplier:

ISBN/Order No:

Price:

Student Book: 0201426811Teacher's Resource Book: 0201426889Temple and Data Disk Package: 0201426897

Student Book: $35.98Teacher's Resource Book: $111.25Template and Data Disk Package: $111.25


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948




B-28



General Description: CD-ROM for Macintosh or Windows allowsfor an extensive exploration of symmetry and related concepts. Itincludes a wide range of learning activities that span simple tocomplex applications. The resource promotes independent learning,critical thinking, and connections to real-world situations, andencourages mathematical communication oral and written. Teachersupport materials include a teacher guide that has many extensionand assessment tasks, and a computer component that providesflexible class management options.

Sytem requirements for Macintosh: System 7.1 or higher, MacintoshLC II or higher (25 MHz 68030 processor or better), 8 MB RAM (5 MBavailable), 256 colour 13-inch monitor, double speed CD drive,approximately 14 MB free hard disk space, microphone (built-in orexternal).System requirements for Windows 3.1: 486/DX2-66 MHz (Pentiumrecommended), 8 MB installed RAM, SVGA 256 colour compatiblevideo card and monitor, double speed CD drive, approximately 12MB free hard disk space, microphone (built-in or external), SoundBlaster card or equivalent, mouse.System requirements for Windows 95: 486/DX2-66 MHz (Pentiumrecommended), 8 MB installed RAM, SVGA 256 colour compatiblevideo card and monitor, double speed CD drive, approximately 12MB free hard disk space, microphone (built-in or external), SoundBlaster card or equivalent, mouse.

Audience: GeneralESL - provides opportunities for a wide range of student learning needsGifted - provides opportunities for a wide range of student learning needsLD - provides opportunities for a wide range of student learning needsID (Mild) - provides opportunities for a wide range of student learningneeds

Shape and SpaceCurriculum Organizer(s):Mirror Symmetry

Supplier:

ISBN/Order No:

Price:

188861868X

99.95

7554 Haszard StreetBurnaby, BCV5E 3X1

Tel: (604) 983-0922

Gage Educational Publishing Co.




General Description: Macintosh software provides a set ofcomputer tools for understanding and exploring fundamentalconcepts of data analysis. Students can explore the underlyingmeaning of abstract statistical concepts and processes. Provides aneasy-to-use system for entering, manipulating, and displaying data.Includes a binder with support materials.

System requirements: Macintosh Plus or later (Classic or laterrecommended) 1Mb RAM.

Audience: General

Statistics and ProbabilityCurriculum Organizer(s):Statistics Workshop

Supplier:

ISBN/Order No:

Price:

Not available

$129.95


Tel: 1-800-667-1115 Fax: (416) 293-9009




B-29



General Description: Binder of activities, projects, and assessmentitems for data analysis and probability includes extensive teachersupport in terms of suggested teaching strategies, sequencing, andscoring rubrics. Activities include both routine calculation andopen-ended projects. Some activities involve critical thinking.Provides random number tables, but other manipulatives must beprovided by the teacher. There is no discussion of the best-fit line atthe Grade 9 level.

Audience: General

Author(s): Birse ... (et al.)

Statistics and ProbabilityCurriculum Organizer(s):Triple 'A' Mathematics Program: DataManagement & Probability

Supplier:

ISBN/Order No:

Price:

Not available

$33.26

243 Saunders RoadBarrie, ONL4N 9A3

Tel: 1-800-563-1166 Fax: (705) 725-1167

Exclusive Educational Products (Ontario)




General Description: CD-ROM series for Macintosh or Windowsassists students in developing an understanding of mathematicsconcepts based on a solid foundation of concrete activities. Topicsinclude fractions, integers, exponents, algebra, equations, coordinategeometry and graphing, and percent. For each topic the programcontains concept sections with explanation, example sections,practice sections, and cumulative checks. The user is in controlthrough pull-down menus and proceed and go-back buttons. Eachprogram includes a teacher manual.

System requirements for Macintosh: 68020 or faster processor,including Power Macintosh, Macintosh 7.5.1 or later, 40 Mb harddrive space or CD-ROM drive.System requirements for Windows: Windows 3.1, Windows 95,Windows NT3.51 or 4.0; 486DX2-66 (Pentium recommended);640x480 with 256 colour display (or higher); 40 Mb hard drive spaceor CD-ROM drive.

Audience: GeneralLD - can be used for remediation

NumberPatterns and RelationsShape and Space

Curriculum Organizer(s):Understanding Math Series

Supplier:

ISBN/Order No:

Price:

Not available

$79.95 each


Tel: 1-800-667-1115 Fax: (416) 293-9009




B-30



General Description: Software package for Macintosh orWindows consists of disks, a user's guide and reference manual,teaching notes, a quick reference, and an introductory video.Sketches depict concrete geometry and emphasize spatialreasoning, while scripts describe constructions verbally andabstractly. Program allows for the construction, labelling,measurement, and manipulation of any geometric figure, as well asthe exploration of geometry concepts. The user's guide andreference manual provide adequate instruction for learning to usethe program. The teaching notes with sample activities providesuggestions for classroom use. Several sample investigations,explorations, demonstrations, and construction activities on a widerange of geometry topics are included.

System requirements for Macintosh: Macintosh Plus or later; 1 Mbof RAM; System 6 or later.System requirements for Windows: 4 Mb of RAM; 3.5" floppy disk;hard drive; Windows 3.1 in enhanced mode.

Caution: Program defaults to inches, but it can be set to centimetres.

Audience: General

Shape and SpaceCurriculum Organizer(s):The Geometer's Sketchpad

Supplier:

ISBN/Order No:

Price:

Macintosh: 1-55953-098-7Windows: 1-55953-099-5

Macintosh: $289.95Windows: $289.95

125 Mary St.Aurora, ONL4G 1G3

Tel: (905) 841-0600 Fax: (905) 727-6265





General Description: CD-ROM package for Macintosh orWindows includes lessons, activities and games, all of which teachand reinforce geometric concepts. It uses a three-level approach,allowing a student to progress at a beginning, an intermediate, or amastery level. The main topics explored are points and lines,triangles, polygons and quadrilaterals, similarity, circles, perimeterand area, solids in 3D, coordinate geometry, transformationalgeometry and logical reasoning and proof. Requires decisionmaking and creative thinking by students.

System requirements for Macintosh: System 7.0 or later, 8 MbRAM.System requirements for Windows 3.1: 386/33 MHz, 8 Mb RAM,Pentium processor recommended, 256 super VGA or better colourmonitor, sound and graphics cards, CD-ROM drive, and a mouserequired; printer is recommended.System requirements for Windows 95: 486/50 MHz, 8 Mb RAM,Pentium processor recommended, 256 super VGA or better colourmonitor, sound and graphics cards, CD-ROM drive, and a mouserequired; printer is recommended.

Audience: General

Shape and SpaceCurriculum Organizer(s):Geometry Blaster

Supplier:

ISBN/Order No:

Price:

0-7849-1063-4

$79.95

19840 Pioneer AvenueP.O. Box 2961Torrance, CA90503

Tel: 1-800- 545-7677 Fax: (310) 793-0601

Davidson & Associates, Inc.



B-31



General Description: This software for Macintosh or Windowssimulates running a hot dog stand. Students gather information,keep records, plan inventory, decide on pricing and look at sellingstrategies. There are three levels of operation. At the intermediateand advanced leves, students also deal with the unexpectedevents of running a business, such as weather, unreliable suppliers,and machine breakdown. Students estimate, calculate, readgraphs, and keep track of their progress in a journal - incorporatedas a mini-word processor. In the teacher's guide there arefollow-up activities relating to starting and running a business.

System requirements for Macintosh: System 7.0 or later; 68040 orlater processor; Mb RAM; 31Mb hard drive space; double speedCD-ROM drive; and 640x480, 256 colour monitor.System requirements for Windows: Windows 3.1 or Windows 95,8Mb RAM, 30Mb hard drive, double speed CD-ROM drive,640x480 256 colour monitor.

Audience: General

NumberStatistics and Probability

Curriculum Organizer(s):Hot Dog Stand: The Works

Supplier:

ISBN/Order No:

Price:

Not available

$89.95 each version


Tel: 1-800-667-1115 Fax: (416) 293-9009





General Description: Student textbook and teacher's resourcebook with assessment booklet and blackline masters help developmathematical literacy along three dimensions: number sense,spatial sense, and problem solving. Includes many learningactivities. Requires manipulatives such as algebra tiles, integerdisks, and suitable software.

Caution: A four-function calculator, rather than a scientificcalculator, is used in some software references.

Audience: General


Curriculum Organizer(s):Interactions (Level 9)

Supplier:

ISBN/Order No:

Price:

Student Textbook: 0132588315Teacher Resource Package: 0132588498

Student Textbook: $32.86Teacher Resource Package: $116.60


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948




B-32



General Description: Interactive program for Macintosh orWindows consists of four CD-ROMs, a student refresher, and ateacher's guide. All lessons follow the pattern of an introduction,followed by a tutorial example for students, a summary of concepts,practice problems, extra practice, and a self check. Students areable to monitor their own progress through the various componentsof each lesson. Also included on the CD-ROMs is a glossary whichstudents can access as a pull-down menu. The student refresherfollows the organization of the CD-ROMs and provides additionalexamples for students as well as answers. The teacher's manualincludes installation notes, instructional summaries, prerequisiteskills, and cross-references to the CD-ROMs.

System requirements for Macintosh: System 7.1 or later, 12MbRAM (16Mb preferred), VGA 256-colour monitor, 40Mb hard diskspace, double speed CD-ROM drive, 8 bit sound support for audio.System requirements for Windows 3.1: MPC2 compliant computer,486SX 25 Mhz (33 or better preferred), 8Mb RAM, VGA 256-colourmonitor, 40Mb hard disk space, double speed CD-ROm drive,soundblaster or 100& support for audio.System requirements for Windows 95: MPC2 compliant computer,486SX 25 MHz (33 or better preferred), 8Mb RAM, VGA 256-colourmonitor, 40Mb hard disk space, double speed CD-ROM drive,soundblaster or 100% support for audio.

Audience: General


Curriculum Organizer(s):The Learning Equation: Mathematics 9

Supplier:

ISBN/Order No:

Price:

Not available

Not available


Tel: 1-800-268-2222 Fax: (416) 752-8101





General Description: This package is a set of Windows softwareutilities that can share data. The tools include spreadsheet,calculation, geometry, graphs, manipulatives, fraction strips, andprobability. A colour linking feature between two or more windowsallows data to be shared and represented in a variety of formats.Includes information on installing and using the software andchapters specific to each math tool. Provides a glossary and index.

System requirements for Windows 3.1: 386/33 MHz, 4Mb of RAM,13" VGA monitor, 2Mb hard disk space, PKZIP.System requirements for Windows 95: 486/50 MHz, 8Mb of RAM,13" VGA monitor, 2Mb hard disk space, PKZIP.

Caution: The algebra tile manipulatives have limited usage(first-degree variables only.) The spreadsheet uses radians as units forsine, cosine, and tangent calculations.

Audience: General


Curriculum Organizer(s):Math Tools

Supplier:

ISBN/Order No:

Price:

0134109783

$58.94


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948




B-33



General Description: This resource package, designed fordistance education, consists of six coilbound modules withaccompanying assignment booklets. One videocassette on keyconcepts supports the print material. A learning facilitator'smanual is also available. The program correlates exactly with allspecific outcomes of the Common Curriculum Framework.

Audience: General


Curriculum Organizer(s):Mathematics 9 (Distance Learning Package)

Supplier:

ISBN/Order No:

Price:

Modules: N/ALearning Facilitator's Manual: 123456789Video: MA0035

Modules: $16.10Learning Facilitator's Manual: $12.00Video: $23.00

12360 - 142 StreetEdmonton, ABT5L 4X9

Tel: (780) 427-5775 Fax: (780) 422-9750

Learning Resources Distributing Centre




General Description: Student book, teacher's edition, andblackline masters support the learning of mathematics in variousways. They present concepts and skills in problem-solvingcontexts. Questions help students express their ideas about mathorally and in writing. Uses manipulatives for the understanding ofabstract concepts and integrates technology throughout.Manipulatives such as algebra tiles and suitable software arenecessary for using the resource effectively; these are not suppliedwith the resource.

Audience: General


Curriculum Organizer(s):MATHPOWER 9, Western Edition

Supplier:

ISBN/Order No:

Price:

Student Book: 0075526530Blackline Masters: 0075526549Teacher's Edition: 0075526557

Student Book: $35.65Blackline Masters: $89.25Teacher's Edition: $105.00






B-34



General Description: Student book provides activities,instruction, discussion, exercises, and projects to engage students inmeaningful, real-world mathematics. Each of the eight chaptersbegins with activities and exercises that allow students to assesstheir prior knowledge before they explore new concepts. Includesteacher's resource book, template and data disk package, andactivity cards with answer key. Manipulatives such as algebra tilesand suitable software are necessary for using the resourceeffectively; these are not supplied with the resource. The softwarepackage requires either ClarisWorks or Microsoft Works in order torun.

Caution: Body measurements are used in two examples on the activitycards. Teachers should be careful and sensitive before using these cards.

Audience: General


Curriculum Organizer(s):Minds on Math 9, Revised Edition

Supplier:

ISBN/Order No:

Price:

Student Book: 020142682XTeacher's Resource Book: 0201426935Temple and Data Disk Package: 0201677041

Student Book: $37.05Teacher's Resource Book: $111.25Template and Data Disk Package: $111.25


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948




B-35



General Description: CD-ROM for Macintosh or Windows allowsfor an extensive exploration of symmetry and related concepts. Itincludes a wide range of learning activities that span simple tocomplex applications. The resource promotes independentlearning, critical thinking, and connections to real-world situations,and encourages mathematical communication oral and written.Teacher support materials include a teacher guide that has manyextension and assessment tasks, and a computer component thatprovides flexible class management options.

Sytem requirements for Macintosh: System 7.1 or higher,Macintosh LC II or higher (25 MHz 68030 processor or better), 8MB RAM (5 MB available), 256 colour 13-inch monitor, doublespeed CD drive, approximately 14 MB free hard disk space,microphone (built-in or external).System requirements for Windows 3.1: 486/DX2-66 MHz (Pentiumrecommended), 8 MB installed RAM, SVGA 256 colour compatiblevideo card and monitor, double speed CD drive, approximately 12MB free hard disk space, microphone (built-in or external), SoundBlaster card or equivalent, mouse.System requirements for Windows 95: 486/DX2-66 MHz (Pentiumrecommended), 8 MB installed RAM, SVGA 256 colour compatiblevideo card and monitor, double speed CD drive, approximately 12MB free hard disk space, microphone (built-in or external), SoundBlaster card or equivalent, mouse.

Audience: GeneralESL - provides opportunities for a wide range of student learning needsGifted - provides opportunities for a wide range of student learning needsLD - provides opportunities for a wide range of student learning needsID (Mild) - provides opportunities for a wide range of student learningneeds

Shape and SpaceCurriculum Organizer(s):Mirror Symmetry

Supplier:

ISBN/Order No:

Price:

188861868X

$99.95

7554 Haszard StreetBurnaby, BCV5E 3X1

Tel: (604) 983-0922

Gage Educational Publishing Co.




General Description: Macintosh software provides a set ofcomputer tools for understanding and exploring fundamentalconcepts of data analysis. Students can explore the underlyingmeaning of abstract statistical concepts and processes. Provides aneasy-to-use system for entering, manipulating, and displaying data.Includes a binder with support materials.

System requirements: Macintosh Plus or later (Classic or laterrecommended) 1Mb RAM.

Audience: General

Statistics and ProbabilityCurriculum Organizer(s):Statistics Workshop

Supplier:

ISBN/Order No:

Price:

Not available

$129.95


Tel: 1-800-667-1115 Fax: (416) 293-9009




B-36



General Description: Binder of activities, projects, andassessment items for data analysis and probability includesextensive teacher support in terms of suggested teaching strategies,sequencing, and scoring rubrics. Activities include both routinecalculation and open-ended projects. Some activities involvecritical thinking. Provides random number tables, but othermanipulatives must be provided by the teacher. There is nodiscussion of the best-fit line at the Grade 9 level.

Audience: General

Author(s): Birse ... (et al.)

Statistics and ProbabilityCurriculum Organizer(s):Triple 'A' Mathematics Program: DataManagement & Probability

Supplier:

ISBN/Order No:

Price:

Not available

$33.26


Tel: 1-800-563-1166 Fax: (705) 725-1167





General Description: CD-ROM series for Macintosh or Windowsassists students in developing an understanding of mathematicsconcepts based on a solid foundation of concrete activities. Topicsinclude fractions, integers, exponents, algebra, equations,coordinate geometry and graphing, and percent. For each topicthe program contains concept sections with explanation, examplesections, practice sections, and cumulative checks. The user is incontrol through pull-down menus and proceed and go-backbuttons. Each program includes a teacher manual.

System requirements for Macintosh: 68020 or faster processor,including Power Macintosh, Macintosh 7.5.1 or later, 40 Mb harddrive space or CD-ROM drive.System requirements for Windows: Windows 3.1, Windows 95,Windows NT3.51 or 4.0; 486DX2-66 (Pentium recommended);640x480 with 256 colour display (or higher); 40 Mb hard drive spaceor CD-ROM drive.

Audience: GeneralLD - can be used for remediation


Curriculum Organizer(s):Understanding Math Series

Supplier:

ISBN/Order No:

Price:

Not available

$79.95 each


Tel: 1-800-667-1115 Fax: (416) 293-9009




B-37

Recommended Teacher Resources and Professional References

Teacher ResourceCategory:

General Description: Book reflects current ideas aboutmathematics teaching and the importance of developing in studentsthe ability to think, reason, and communicate mathematically. Itdescribes strategies for establishing a supportive classroomenvironment. Includes over 240 activities as well as assessmentstrategies and blackline masters. Contains SI and Imperial measure.

Audience: General

Author(s): Burns, Marilyn

Patterns and RelationsCurriculum Organizer(s):About Teaching Mathematics: A K-8Resource

Supplier:

ISBN/Order No:

Price:

0-941355-05-5/18880

$44.10


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948





General Description: This resource package is a publishing systemthat allows teachers to draw from a collection of thousands of itemsarranged in topics. Most questions are of multiple-choice format, butthese can be recast in open-ended format by removing thedistractors. There is a preponderance of procedural items, but thereare enough conceptual and problem-solving examples to allow theteacher to build an assessment consistent with a more conceptualblueprint. The technology assumed is that of the scientific calculatorrather than the graphing calculator.

System Requirements for Macintosh: System 7.0 or higher, 68030processor (68040 or PowerMac recommended), 4 MB RAM(6 MB recommended).System Requirements for Windows: 486/50 MHz (Pentiumrecommended), disc cache set to at least 512 KB

Cautions: The Windows setup program is not yet released; theDOS/Windows Version 3.32 must be loaded under DOS, even inWindows 3.1 and Windows 95 environments.Windows 3.1 installations require 32-bit extensions before they run.

Audience: General


Curriculum Organizer(s):Access: The Teacher's Database (MacintoshVersion 3.33; DOS/Windows Version 3.32)

Supplier:

ISBN/Order No:

Price:

Not available

Macintosh Version 3.33: $495.00DOS/Windows Version 3.32: $595.00Canadian Math Database (Grades 8-10) (Catalog):$225.00

102 - 3065 Richmond ParkwayRichmond, CA94806

Tel: 1-800-669-9405 Fax: (510) 222-0165

EducAide Software




General Description: This resource provides activities for hands-onexperience using algebra tiles to help students develop anunderstanding of ratios, integers, polynomials, and factoring andsolving equations. It provides ideas for teachers on how to helpstudents use algebra tiles and the accompanying activity sheets.Each topic includes teacher notes, student blackline activity sheets,and answers for all activities. Manipulatives available separately.

Audience: General

Author(s): Scully, J; Scully, B.

NumberPatterns and Relations

Curriculum Organizer(s):Active Learning: Alge-Tiles

Supplier:

ISBN/Order No:

Price:

Not available

$31.46


Tel: 1-800-563-1166 Fax: (705) 725-1167




B-38



General Description: Set of three teacher's resource books focusseson fractions, decimals and percents, and geometry and data analysis.Learning explorations encourage students to think, invent,investigate, and make connections. Lessons are logical and easy tofollow. Reproducible homework pages support in-class explorations.

Audience: General

Author(s): Ward, S.

NumberShape and SpaceStatistics and Probability

Curriculum Organizer(s):Constructing Ideas: Grades 6-8

Supplier:

ISBN/Order No:

Price:

About Fractions ... 1561078123About Geometry: 1561078131About Data Analysis: 1561078115

$28.10 each


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948





General Description: This book of card games deals with aspects offractions, integers, and algebraic equations. The games are clearlyexplained, short, and each addresses a specific outcome. The cards tobe cut out from sheets are clearly labelled, which makes them easy touse. This resource could be used to help students master concepts ina different way.

Audience: General

Author(s): Gadanidis, G.


Curriculum Organizer(s):Cool Math Games

Supplier:

ISBN/Order No:

Price:

Not available

$29.95

3436 West 35th Avenue

Vancouver, BCV6N 2N2

Tel: 1-800-563-1166 Fax: (604) 266-2287

Exclusive Educational Products (B.C.)




General Description: This resource addresses storytelling as a toolfor introducing, exploring, and teaching mathematical concepts. Itincludes helpful advice on how to tell stories, how to teachstorytelling to students and how to apply storytelling to mathematicsand science. Provides an extensive bibliography of resources forstorytelling, stories, and specific topics, along with a collection oforiginal and traditional stories to tell and use as models.

Audience: General

Author(s): Lipke, B.


Curriculum Organizer(s):Figures, Facts & Fables: Telling Tales inScience and Math

Supplier:

ISBN/Order No:

Price:

043507105X

$38.60

325 Humber College Blvd.Toronto, ONM9W 7C3

Tel: 1-800-263-7824 Fax: (416) 798-1384

Irwin Publishing



B-39



General Description: Resource binder provides activity sheets,lessons, teaching strategies, and assessment ideas for hands-onactivities using fraction blocks. Students start with developing anunderstanding of the concept of a fraction and work through to theoperations of addition, subtraction, multiplication, and division offractions. Does not include fraction blocks.

Audience: GeneralESL - promotes vocabulary developmentLD - active use of concrete materials facilitates connections

Author(s): Glanfield, F.; Pallos-Haden, K.

NumberCurriculum Organizer(s):Fraction Blocks: Grades 5-8

Supplier:

ISBN/Order No:

Price:

Not available

$31.46


Tel: 1-800-563-1166 Fax: (705) 725-1167




Professional ReferenceCategory:

General Description: Resource package consists of three books:ALTERNATIVE ASSESSMENT IN THE MATHEMATICSCLASSROOM helps teachers better monitor and measure theperformance of students against curriculum standards;CO-OPERATIVE LEARNING IN THE MATHEMATICSCLASSROOM helps teachers provide opportunities for students towork together in problem-solving situations; and INVOLVINGPARENTS AND THE COMMUNITY IN THE MATHEMATICSCLASSROOM helps teachers communicate with parents aboutmathematics teaching and learning. Some measurement activitiesuse Imperial measure. Grade level references in INVOLVINGPARENTS AND THE COMMUNITY IN THE MATHEMATICSCLASSROOM do not match the Protocol Framework.

Audience: General

Author(s): Foster, A.G.

Curriculum Organizer(s):Glencoe Mathematics Professional

Supplier:

ISBN/Order No:

Price:

Alternative Assessment ...: 0028245466Cooperative Learning ...: 0028245458Involving Parents ...: 0028248201

Alternative Assessment ...: $21.70Cooperative Learning ...: $21.65Involving Parents ...: $19.50







General Description: This resource includes a collection of 60projects, with advice for teachers regarding strategies andassessment. Connections with other subjects is a major focus. Eachproject provides opportunities for learners to demonstrate their skillsand knowledge through various means of communication.

Audience: General

Author(s): Muschla, G. R.; Muschla, J. A

NumberPatterns and RelationsStatistics and ProbabilityShape and Space

Curriculum Organizer(s):Hands-on Math Projects with Real-LifeApplications

Supplier:

ISBN/Order No:

Price:

0876283849

$32.54


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948




B-40



General Description: This resource provides practical suggestionsto help teachers and students use and enjoy portfolios as part of theassessment and evaluation process. It describes how to develop,organize, evaluate and maintain portfolios, and is applicable to anysubject area. Descriptions of classroom activities, related blacklinemasters, information for parents and ideas for student and teacherconferences are provided. Suggestions are not exclusive tomathematics.

Cautions: Some blackline masters may not be age-appropriate for olderstudents.

Audience: General

Author(s): Cross, M.


Curriculum Organizer(s):How to's in Getting Started with Assessment& Evaluation Using Portfolios (Across theCurriculum): Grades 1-10

Supplier:

ISBN/Order No:

Price:

Not available

$31.46


Tel: 1-800-563-1166 Fax: (705) 725-1167





General Description: In the Balance: Algebra Logic Puzzles: Grades4-6 and Grades 7-9 both contain 25 reproducible logic-based numberpuzzles ranging from the simple to the very complex. These areextension activities that support a number of the CommonCurriculum Framework mathematical processes.

Audience: General

Author(s): Kroner, L.


Curriculum Organizer(s):In the Balance

Supplier:

ISBN/Order No:

Price:

Grades 4-6: 0762205512Grades 7-9: 0762205520

$22.05 each level


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948





This resource is designed to assist teachers in using the large geoboardwith students in grades 7-9. It contains teacher notes, studentactivity pages, and follow-up activities on how to use the 10x10 pin,11x11 pin, and linking geoboards. Topics include shapes, symmetry,similarity, congruency, area, perimeter, Pythagorean theorem,fractions, percent, Cartesian coordinates, and analytictransformation. Provides blackline masters and answers for theactivities. Geoboards are not included.

General

Danbrook, C.; Lesage, J.

NumberShape and Space

Curriculum Organizer(s):The Intermediate Geoboard

Supplier:

ISBN/Order No:

Price:

Not available

$31.46


Tel: 1-800-563-1166 Fax: (705) 725-1167




B-41



General Description: Set of 20 activity cards poses questions aboutthe probability of something occurring, then follows up withexperiments in which students can make predictions, gather data,record and display data, and draw conclusions. Students alsodetermine whether the games are fair or unfair. Includes teachernotes, solutions, and four spinners. Does not supply regular dice,icosahedra (20-faced) dice, coins or cubes. Some activities showpictures of American coins.

Audience: General

Author(s): Goodnow, Judy; Hoogeboom, Shirley

Statistics and ProbabilityCurriculum Organizer(s):Junior High Probability Jobcards

Supplier:

ISBN/Order No:

Price:

1561071749

$24.48


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General Description: This resource offers a variety of open-endedmathematics activities based on all strands. It is designed as an activetool to allow students to reflect, review, communicate and solveproblems through such things as drama, debate and games. Allmathematical processes are addressed. The resource uses languagearts activities extensively to help students communicate knowledgeand reasoning in mathematics. The mathematics activities can beused for cooperative small-group review purposes, reinforcement,evaluation or for challenge purposes.

Audience: General

Author(s): Cowan, L

NumberPatterns and RelationsStatistics and ProbabilityShape and Space

Curriculum Organizer(s):Just a Minute: Activities for SpareMoments in Class

Supplier:

ISBN/Order No:

Price:

Not available

$34.95

3436 West 35th Avenue

Vancouver, BCV6N 2N2

Tel: 1-800-563-1166 Fax: (604) 266-2287

Exclusive Educational Products (B.C.)




General Description: Book is for preservice and practisingmathematics teachers in elementary and middle schools. Presentstheories of mathematical development along with practical examplesand suggested organizational structures. Promotes the idea of theteacher as a learner and a researcher. The accompanying instructor’smanual provides direction for inservicing and overheadtransparency masters.

Audience: General

Author(s): Cathcart, G. et al.


Curriculum Organizer(s):Learning Mathematics in Elementary andMiddle Schools

Supplier:

ISBN/Order No:

Price:

Text: 0205266886Instructor's Manual: 0205267548

Text: $48.40Instructor's Manual: $10.00


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948




B-42



General Description: Teacher’s resource book offers assessmentactivity sets correlated to specific learning outcomes, rubrics for fourlevels of performance, and performance indicators and samplestudent responses for each task. Student's booklet contains five setsof activities designed to assess in seven dimensions. Resource is moresuitable for formative, rather than summative assessment. Someactivities on mean and median are more appropriate for Grade 7.

Audience: General

Author(s): Flewelling, G.; et al.


Curriculum Organizer(s):Math Assessment: Grade 8

Supplier:

ISBN/Order No:

Price:

Teacher's Resource Book: 0771533748Student's Booklet: 0771533713

Teacher's Resource Book: $129.95Student's Booklet: $9.95


Tel: 1-800-667-1115 Fax: (416) 293-9009





General Description: Twenty-two-minute American videofocusses on activity-based classrooms in which students from grades3-6 are seen as collaborative problem solvers and effectivecommunicators. It clearly and effectively addresses how and whymathematics teaching has changed and how parents can helpsupport their children's learning. Reference to Imperial measure andAmerican money.

Audience: General


Curriculum Organizer(s):Mathematics: What Are You Teaching MyChild?

Supplier:

ISBN/Order No:

Price:

0-201-87210-2

$49.40


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948





General Description: This resource contains 36 activities involvingthe use of polydrons. Each activity consists of a teacher sheet and astudent activity sheet that can be photocopied.

Audience: General

Author(s): Ansell, B.; Baker, L.; Harris, I.

Shape and SpaceCurriculum Organizer(s):Mathematics with Polydrons

Supplier:

ISBN/Order No:

Price:

0952381508

$49.95

2102 Elspeth PlacePort Coquitlam, BCV3C 1G3

Tel: 1-800-668-0600 Fax: (604) 941-1066




B-43



General Description: This is a professional resource for teacherswho wish to have information about portfolio assessment inmathematics. It describes how teachers can use portfolios andexplain portfolio assessment to students and parents. It also providesinformation on using rubrics for assessing portfolios

Audience: General

Author(s): Kuhs, T. M.

Curriculum Organizer(s):Measure for Measure: Using Portfolios inK-8 Mathematics

Supplier:

ISBN/Order No:

Price:

0435071351

$27.80


Tel: 1-800-263-7824 Fax: (416) 798-1384

Irwin Publishing




General Description: American book of blackline masters presentsstrategies for teaching mental mathematics. It extends earlymethods to more advanced methods of calculating with fractions,decimals, percents, and whole numbers. There are 50 lessons withpractice problems, tests, and answer keys. Some examples useImperial measure and U.S. contexts.

Audience: General

Author(s): Hope, Jack A.

NumberCurriculum Organizer(s):Mental Math in Junior High

Supplier:

ISBN/Order No:

Price:

0-86651-433-3/68385

$17.75


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948





General Description: Book introduces a multicultural perspectiveto elementary and middle grade mathematics, revealing how such aperspective can enrich the learning of all students, whatever theirgender, ethnic/racial heritage, or socio-economic status. Studentslearn that mathematics was created by real people attempting tosolve real problems. The collection of open-ended activitiesaddresses a range of mathematical topics and cultural connections,and promotes problem solving and critical and creative thinking.

Audience: General

Author(s): Zaslavsky, C.


Curriculum Organizer(s):The Multicultural Math Classroom:Bringing in the World

Supplier:

ISBN/Order No:

Price:

0435083732

$36.20


Tel: 1-800-263-7824 Fax: (416) 798-1384

Irwin Publishing



B-44



General Description: Softcover dictionary provides acomprehensive list of mathematical terms. Features includecross-referencing, diagrams, pronunciation, word origins, andguidance on usage. Conceptual level of the definitions varieswidely; this dictionary is most suitable for use as a teacher resource,but may be used with students, as appropriate.

Audience: General

Author(s): Fyfield, J.; Blane, D.


Curriculum Organizer(s):The Nelson Canadian School MathematicsDictionary

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ISBN/Order No:

Price:

17-604800-6

$10.75


Tel: 1-800-268-2222 Fax: (416) 752-8101





General Description: Binder describes the processes and strategiesused to solve problems, including trial and error, making a drawing orsketch, logical thinking, finding a pattern, working backward,elimination, and making a list. Explains each strategy and includesover 200 challenging activities.

Audience: GeneralGifted - opportunities for critical thinking

Author(s): Jones, G.


Curriculum Organizer(s):Problem Solving: What You Do When YouDon't Know What To Do

Supplier:

ISBN/Order No:

Price:

Not available

$31.46


Tel: 1-800-563-1166 Fax: (705) 725-1167





General Description: Resource provides assistance for teachers inimplementing the Protocol framework. Keyed directly to eachIllustrative Example (IE), it provides teachers with solutions,including diagrams, tables, interpretations, and suggestions forconcept development. It highlights the processes that are associatedwith the corresponding Specific Outcomes and is cross-referenced tothe page where the respective Illustrative Example appears in theProtocol framework.

Audience: General


Curriculum Organizer(s):Solutions & Extensions for IllustrativeExamples From the Western CanadianProtocol: Grade 8

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ISBN/Order No:

Price:

Not available

$9.75

315 Dechene RoadEdmonton, ABT6M 1W3

Tel: (780) 481-9913 Fax: (780) 481-9913

Math Alive



B-45



General Description: This resource is specific to the TexasInstruments Explorer Plus calculator. This book was written as afacilitator's manual to conduct workshops on how to use theExplorer Plus calculator in the classroom. It contains 64 pagesintended to be made into transparencies for use in the presentationof workshops. It also contains 26 sample lessons covering the use ofthis calculator in the Number, Shape and Space, and Statistics andProbability strands.

Audience: General

Author(s): Bitter, G. G.; Mikesell, J. L.

NumberShape and SpaceStatistics and Probability

Curriculum Organizer(s):Using the Explorer Plus Calculator

Supplier:

ISBN/Order No:

Price:

1572328533

$21.24


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948





General Description: Collection of activities is designed to enhanceand motivate logical thinking, improve problem-solving strategies,and explore the history of games. Ancient puzzles, such as tangrams,dominoes, tower of Hanoi, and Xs and Os are combined withpentominoes, Nim, toothpick geometries, number puzzles, andpatterns. The resource contains problem-solving and evaluationstrategies for teachers and students. Includes blackline masters andanswers.

Audience: General

Author(s): Jones, G.


Curriculum Organizer(s):The World's Most Popular Puzzles andProblems

Supplier:

ISBN/Order No:

Price:

Not available

$31.46


Tel: 1-800-563-1166 Fax: (705) 725-1167





General Description: This teacher resource presents strategies toassist teachers in making writing a part of their mathematicsinstruction. It explains why students should write in mathematicsclass and offers tips and suggestions for teachers. The sample lessonsprovided encourage students to demonstrate the mathematicalprocesses cited in the Common Curriculum Framework.

Audience: General

Author(s): Burns, M.


Curriculum Organizer(s):Writing in Math Class

Supplier:

ISBN/Order No:

Price:

0941355136

$25.83


Tel: 1-800-361-6128 Fax: (416) 447-2551/443-0948




B-46

APPENDIX CAssessment and Evaluation Samples

C-3

APPENDIX C: ASSESSMENT AND EVALUATION • Samples

The samples in this section show howa teacher might link criteria tolearning outcomes. Each sample is

based on prescribed learning outcomes takenfrom one or more organizers. The samplesprovide background information to explainthe classroom context; suggested instructiontasks and strategies; the tools and methodsused to gather assessment information;and the criteria used to evaluate studentperformance.

HOW THE SAMPLES ARE ORGANIZED

There are six parts to each sample:

• identification of the prescribed learningoutcomes

• unit focus• planning the unit• the unit• defining the criteria• assessing and evaluating student

performance


This part identifies the organizer ororganizers and the specific prescribedlearning outcomes selected for the sample.

Unit Focus

This is a summary of the key features of thesample.

Planning the Unit

This part describes how the teacher preparedfor the unit.

The Unit

This part outlines:

• background information to explain theclassroom context

• instructional tasks• the opportunities that students were given

to practise learning• the feedback and support that was offered

students by the teacher• the ways in which the teacher prepared

students for the assessment

Defining the Criteria

This part illustrates the specific criteria(based on prescribed learning outcomes), theassessment task, and various reference sets.

Assessing and Evaluating StudentPerformance

This part includes:

• assessment tasks or activities• the support that the teacher offered

students• tools and methods used to gather the

assessment information• the way the criteria were used to evaluate

the student performance

EVALUATION SAMPLES

The samples on the following pages illustratehow a teacher might apply criterion-referencedevaluation in Mathematics 8 and 9.

• Sample 1: Mathematics 8Data Analysis(Page C-5)

• Sample 2: Mathematics 9Risk Management(Page C-11)

C-5


▼ SAMPLE 1:MATHEMATICS 8

Topic: Data Analysis

PRESCRIBED LEARNING OUTCOMES:

Problem Solving

It is expected that students will:

• use a variety of methods to solve real-life,practical, technical, and theoreticalproblems


Number (Number Concepts)


• represent and apply fractional percentsand percents greater than 100 in fraction ordecimal form and vice versa

Statistics and Probability (Data Analysis)


• select, defend, and use appropriatemethods of collecting data:- designing and using surveys- research, using electronic media


In addition to these prescribed learningoutcomes, assessment and evaluation alsofocussed on communication skills.

UNIT FOCUS

The teacher’s overall goals for the unit wereto extend students’ knowledge of conceptsrelated to data analysis and to help themunderstand the usefulness and applicabilityof these concepts. Assessment andinstruction were interwoven in the unit sothat one led naturally to and informed the

other. The teacher designed activities toassess students’ problem-solving andcommunication skills and theirunderstanding and application of concepts.

PLANNING THE UNIT

To develop the unit, the teacher:

• determined the overall goals for the unit• identified the related IRP outcomes to be

targeted in the unit• examined the prerequisite knowledge and

skills students needed to achieve theseoutcomes and planned for their review

• looked for ways to connect students’learning to other desirable outcomes, suchas those associated with communicationskills

• planned a variety of integratedinstructional and assessment activities tohelp students achieve identified outcomes

• determined criteria with which to evaluatestudents’ learning

• planned to evaluate students’ learning formarking and reporting purposes and toinform future instruction

THE UNIT

Recalling, Reviewing, and ExtendingRelevant Concepts

• The teacher checked students’ knowledgeof percents, decimals, and fractions,providing instruction and practice asneeded to ensure that they were preparedto complete the activities included in theunit. The teacher also helped studentsrecall what they knew about differenttypes of charts and graphs.

• To relate students’ prior knowledge tomaterial included in the unit, the teachergave pairs of students a set of data shownas decimals and instructed them to convert

C-6


it to percents and then to fractions.Students chose two forms of the data andprepared a graph for each on poster paper,using any types of graphs they wished.Students hung their graphs around theroom and compared them, discussing theeffectiveness of the different types ofgraphs for displaying different types ofdata and the effects of differences inscaling. The class concluded that theproportions shown on the graphs were thesame, regardless of the type of graph usedor the form of the data set graphed. Theteacher helped students discover howgraphs could be misleading and how theycould be used to misrepresent data.

• Formative Assessment—As students workedon this activity, the teacher moved aroundthe classroom checking the accuracy oftheir conversions and graphs, providingassistance as needed. When the graphswere posted, the teacher challengedstudents to identify what errors had beenmade and how to correct them. Duringdiscussions, the teacher took note of thecomprehension shown by students’comments and asked questions tostimulate and direct their thinking.

• The teacher showed students other typesof charts and graphs that could be used fororganizing and presenting data. The classdiscussed the appropriateness of each foruse with different types of data.

Assigning Performance Activity—ResearchProjects

• Students were asked to work in pairs tocomplete one of two research projects. Theteacher gave each pair of students adetailed assignment sheet outlining thesteps required for the completion of theirprojects.

- topic 1: Prepare an itemized budgetshowing the average (mean) per-personcosts associated with a 10-day familyholiday to Disneyland.

- topic 2: Prepare an itemized order sheetfor a school store.

• As a class, students brainstormed budgetcategories for each topic and thencategorized their ideas under majorheadings. For topic 1, these included food,drinks, games, rides, clothes, movies,souvenirs, and so on. For topic 2, theheadings included pop, chips, bubblegum, school supplies, and so on.

• For topic 1, students were required to:- make an estimate of the personal

amounts of spending money requiredfor each category

- make an estimate of the mean budgetper item for the class members doingtopic 1

- collect budget items from at least threeother class members

- organize their personal and classmates’data into tables

• For topic 2, students were required to:- make an initial estimate of the costs of

the initial stock for each type of item- research the net cost of items for the

school store- survey a sample of the school population

to determine the representative average(mode) number of items that the samplewould consume in a school week (Theyalso surveyed the sample to find theaverage—mean and median—the samplewould be willing to spend in the schoolstore each week.)

- organize their data on tables showing allstock items as per-week amounts andthe projected purchasing amounts

C-7


• For topics 1 and 2, students were requiredto:- use at least two different types of charts

or graphs to display their data (Studentschose the forms of the data—e.g.,percents, fractions—they wished todisplay.)

- write a brief summary that describeddata-collection methods, findings, andconclusions

Topic One: Holiday Project

• The class talked about where they couldfind information required for thecompletion of their projects (e.g., travelagents, the Internet, banks, classifieds) andways they might organize and displaytheir data (e.g., stem-and-leaf plots, circleor bar graphs).

• Students presented the results of theirresearch projects to the class for discussionand review. Presentations included adescription of the methods used for datacollection and a summary of findings andconclusions. The teacher worked with theclass to identify errors in students’ work.

FractionEstimated Personal Mean of Total

Expenses

Drinks

Rides

Souvenirs

Clothes

Games

Food

(and so on)

Total

Extensions—Using the Results of thePerformance Activity in Further Instruction

• The teacher worked with the class to createtables for both projects that compiled all ofstudents’ data for selected budget items.This information was then used tointroduce measures of central tendency(mode, median, and mean). The classdiscussed the advantages anddisadvantages of each measure when usedwith different types of data. The teacherused students’ marks on the performanceactivity to reinforce these concepts.Students practised their skills bycompleting a variety of instructionalactivities that required them to determinethe mode, median, and mean for differentsets of data.

• The teacher introduced students tosurveys, discussing their development anduse. Students developed a survey todetermine their schoolmates’ perceptionsof the cost of quitting school and living ontheir own. Students used the surveys tocanvass the general student populationand then compiled and discussed theresults, comparing them to the class’sfindings regarding actual costs.

DEFINING THE CRITERIA

Mathematical Thinking

To what extent did students:

• recognize the relationship betweenpercents, decimals, and fractions

• accurately perform calculations usingrational numbers

• select, use, and defend effective methodsof displaying, organizing, and graphingdifferent types of data

• select, use, and defend appropriatemethods of collecting data

C-8


• effectively present findings to the classand justify conclusions

• recognize sources of error

Communication Skills


• communicate ideas in a clear, logical, andunderstandable fashion to the teacher andother students

• justify and explain their reasoning andconclusions

ASSESSING AND EVALUATION

STUDENT PERFORMANCE

The teacher designed assessment activities toensure that decisions concerning students’learning were based on information from avariety of sources.

Observation and Questioning

The teacher assessed students’understanding and communication skillsinformally throughout instruction by:

• observing students as they participated inclass discussions and worked individuallyand in small groups, taking note of theirflexibility in dealing with challenges, theeffectiveness with which they used variousresources to solve problems, and theirability to communicate and justify theirideas and reasoning in a clear and logicalfashion

• asking questions to determine how wellstudents understood the information theywere learning

• looking for comprehension of the unit’scentral concepts in the questions studentsasked and in the comments they made toother students

The teacher used information obtainedthrough observation and questioning to:

• give students feedback regarding theirprogress

• determine the need for individualassistance

• regulate the pace of instruction• determine the depth of information to

include in the unit• complete the marking scale for the

performance activity

Evaluating the Performance Activity—Research Projects

The teacher and students worked together todetermine the criteria for judging students’research projects and presentations. Theteacher made sure that the relevant definingcriteria were included with the evaluationcriteria for the project. The teacher organizedthe identified project evaluation criteria ontoa rating scale, which was given to studentsbefore their presentations.

C-9


Key: 5 Excellent4-Good3-Average2-Needs improvement1-Needs complete review

Criteria

• used effective and appropriate methods of collecting data

• made accurate conversions among decimals, fractions, and percents

• made accurate calculations

• determined reasonable amounts for each category

• used effective and appropriate graphs for displaying data

• used appropriate scaling for graphs

• clearly and adequately described data-collection methods, findings, andconclusions in the written summary

• clearly described the relationship between estimated and researchedbudget amounts in the written summary

• made valid conclusions that were clearly based on the data

• clearly and logically presented research and findings to the class

• used effective methods to organize and display data and findings for thepresentation

• provided adequate justifications for methods, findings, and conclusions

• communicated ideas in a clear, logical, and understandable fashion

• worked effectively with partner

1 2 3 4 5

Evaluating the Research Projects and Presentations

C-10


Self-/Peer Evaluation

Students used the preceding rating scale as aguide for asking questions and providingfeedback to their peers during theirpresentations. Following their presentations,each pair of students used the same ratingscale to evaluate their own work on thisproject.

Teacher Evaluation

Students submitted their project summaries,data charts, graphs, and calculations to theteacher for marking, along with a descriptionof the role played by each student incompleting the project. The teacher used therating scale to evaluate students’ work. Theteacher compiled the results of the ratingsand calculated a class mean for each item.This information was used to identify areasneeding additional instruction and to modifythe unit for future use.

Improvement Plans

Based on feedback provided by their peersand the results of the self- and teacherevaluations, each pair of students completedan improvement plan. Improvement plansidentified areas of strength and weaknessand described what might have been donedifferently to improve the work on theproject. Improvement plans were markedfor:

• accuracy in identifying weaknesses (5points)

• thoroughness (5 points)• feasibility and effectiveness of proposed

improvements (10 points)

Marks for the improvement plans werecombined with those for the research projectsto determine students’ total marks for theunit.

C-11


▼ SAMPLE 2:MATHEMATICS 9

Topic: Risk Management

PRESCRIBED LEARNING OUTCOMES:

Problem Solving


• use a variety of methods to solve real-life,practical, technical, and theoreticalproblems


Number (Number Operations)



Statistics and Probability (Chanceand Uncertainty)


• recognize that decisions based onprobability may be a combination oftheoretical calculations, experimentalresults, and subjective judgments

• demonstrate an understanding of the roleof probability and statistics in society

• solve problems involving the probabilityof independent events

UNIT FOCUS

The overall goals for the unit were to extendstudents’ knowledge of concepts related toprobability and chance and to help themrecognize the need for developing effectivemethods for managing risk. Both assessmentand instruction focussed on helping studentsunderstand where in their lives they mightencounter probability and chance and on

ensuring that they had the skills and under-standing needed to make good choices andinformed decisions.

PLANNING THE UNIT

To develop the unit, the teacher:

• determined the overall goals of the unit• identified the related IRP outcomes• planned to review the prerequisite

knowledge and skills students needed toachieve the targeted outcomes

• looked for ways to connect students’learning to other desirable outcomes,including those associated with groupskills

• planned a variety of instructional andassessment activities to help studentsachieve identified outcomes

• planned to integrate instruction andassessment so that one informed the other

• established defining criteria to use toevaluate students’ learning and determinetheir marks for the unit

THE UNIT

Recalling, Reviewing, and ExtendingRelevant Concepts

• The teacher used spinners and dice toreview the Grade 8 concepts related tochance and probability. Students predictedthe probable occurrence of specified singleevents. They then determined the numberof times the events occurred duringrepeated trials and compared thosenumbers to their predictions.

• The teacher gave small groups of studentsdecks of cards and asked them to predictthe probability of drawing the seven ofhearts. The students in each group thenpicked a card from the deck to see ifanyone got a seven of hearts. Thisexperiment was repeated several times,

C-12


with students replacing the cards eachtime. The teacher challenged students todevelop a method for determining theprobability of choosing a seven on eachsuccessive trial. Students repeated theexperiment additional times in their smallgroups, keeping track on a simple chart ofthe number of times a seven was drawn.The students discussed the results for eachgroup and then combined their results fora class total. Class totals were compared topredicted probabilities to establish therelationship between theoreticalcalculations of probability and the actualfrequency of occurrence of an event.

• The teacher used similar methods to helpthe class understand how to determine theprobability of two independent events.The class practised their skills using cards,spinners, and pairs of dice.

• Formative Assessment—As studentsparticipated in these activities, the teachermoved through the classroom askingquestions to check for understanding andproviding additional instruction asneeded. The teacher noted students’willingness to participate and to help eachother as they worked in their smallgroups.

• Students also practised their skills byworking individually, showing all theirwork, on a variety of problems compiledby the teacher from sources such astextbooks, worksheets, and the illustratedexamples.

• Peer Assessment—Students exchanged theirresponses to the problems and checkedeach other’s work using a key the teacherdisplayed on the overhead projector.Students were responsible for identifyingthe errors in each other’s work. Inaddition, they had to describe how these

errors could be fixed so the students whohad done the problems could correct theirown mistakes. The teacher collected andreviewed students’ corrected work,looking for patterns of errors that mightindicate a need for re-teaching.

Performance Activity 1—Games of Chance

• The class discussed games of chancestudents may have encountered in theircommunity, such as charity raffles and gamebooths at local fairs. They talked about howthese games were used and what kinds ofthings had to happen for them to serve theirpurposes most effectively.

• Students worked in small groups to designtheir own games of chance. Each groupwas required to:- design a game that involved at least two

independent events- provide a written explanation of the

rules for playing- calculate the theoretical probability of

winning their games- gather empirical data about their games

through a minimum of 50 trials- compare their calculated probabilities of

winning to the results of their trials todetermine the fit

• Self-/Peer Assessment—Students tried outtheir games and directions for playingwith other groups of students and refinedthem based on their own observations andfeedback from their peers.

• The students turned the classroom into amidway and invited parents (and otherstudents) in to play the games they haddeveloped. Each player was given $50 inplay money with which to play the games,and the class developed a payout schemethat was consistent for all games so thatconclusions could be drawn from data

C-13


resulting from the activity. A set fee of $2was established as the cost of playingeach game.

• During the activity, each group of studentskept a running total of the number oftimes their game was played, the amountof money collected, and the amount paidout to winners.

• Teacher Review—The teacher moved amongthe games, evaluating their effectiveness,rating students’ group skills, answeringquestions, and providing assistance asneeded.

Using the Results of the Performance Activityin Further Instruction

• Following the midway activity, each groupof students organized the data for theirgame and developed charts and graphsthat would help them present theirfindings to the class. During thepresentations, the class compared thesuccess of various games and discussedthe relationship between thepredetermined probability of winning, thetotal amount of money paid out, and theaccumulated profits.

• Formative Assessment—The teacherprompted students’ thinking andmeasured the extent of theirunderstanding with questions like:- What characteristics made some games

more profitable than others?- What might have happened if participants

could bet any amount they wished?- Based on this experience, what

conclusions can be drawn about thegames of chance you encounter at fairsand carnivals?

The teacher’s formative assessment wasbased on their responses.

• The teacher helped students relate whatthey had learned from working with theirown games of chance to major games ofchance like a lottery. The class calculatedand reviewed the probability of winning amajor prize in a lottery.

Extending and Relating Learning

• The class discussed how risk andprobability affect different decisionspeople make throughout their lives (e.g.,where to go to college, what job to take, orhow to invest their savings).

• An investment consultant from a localbank was invited to talk to the class abouthow probability and risk relate to variousinvestment options (e.g., purchasingstocks, bonds, mutual funds, lifeinsurance, real estate, or putting moneyinto a savings account).

• Students worked in small groups on abrief, structured assignment to comparepotential profits after 20 years on a $1,000investment if they invested it in specificstocks, bonds, or mutual funds; put themoney into savings; or bought $1,000worth of lottery tickets (based onprobability of winning). To help studentswith their calculations, the teacherprovided tables they could use todetermine interest and demonstrated theuse of relevant computer programs.Students rated each investment option ona scale of 1 to 5 according to the amount ofrisk involved.

• Formative Assessment—Students discussedtheir responses to the assignment in class.The teacher reviewed students’ work andlistened to the discussion to determine ifthey were ready to complete PerformanceActivity 2, or if more preparation wasneeded.

C-14


Performance Activity 2—Investment Project

• Students were told that they had each justinherited $10,000 and that the conditionsfor the inheritance required that theyinvest at least $5,000 of it for 10 years.Students were encouraged to gatherinformation from a variety of sources (e.g.,Internet, parents, school, communitylibraries, investment magazines, localbanks, accountants) before making theirinvestment decisions. Each student wasrequired to:- account for what they would do with

the entire $10,000- identify different investment options

they might consider and specify thebenefits and risks of each, consideringthe probability of return

- develop an investment plan for theamount they decided to invest,specifying the investment choices theyhad made and justifying the reasons forthose choices

- speculate, based on what they hadlearned and on some basic research,how much money they would have after10 years, specifying interest rates andshowing all calculations whereappropriate

- specify where they had obtained theirinformation

DEFINING THE CRITERIA

Mathematical Thinking


• determine the probability of twodependent events

• solve problems involving the probabilityof events

• describe the relationship betweenprobability of winning, payout, and profitin games of chance

• describe how risk and probability relate tovarious investment decisions

• identify benefits and risks of variousinvestment options

• choose and defend a strategy for investinga specified sum of money

• correctly perform calculations usingrational numbers

Group Skills


• make positive contributions to the group• build and elaborate on the ideas of others• help develop the understanding of group

members• suggest positive ways to resolve

differences of opinion among groupmembers

ASSESSING AND EVALUATING STUDENT

PERFORMANCE

The teacher designed assessment activities toensure that decisions concerning students’learning were based on information from avariety of sources.

Observation and Questioning

The teacher assessed students’ mathematicalunderstanding and group skills informallythroughout instruction by:

• observing them as they participated inclass discussions and worked individually

• observing their work in small groups,taking note of their contributions to theirgroups, their success in building andelaborating on the ideas of others, theirwillingness to help other students, andtheir attempts to resolve any differencesamong group members

• asking questions to determine how wellstudents understood the informationincluded in the unit

C-15


• looking for evidence of understanding ofand interest in central concepts in thequestions students asked and in thecomments they made to other students

The teacher used information obtainedthrough observation and questioning to:

• provide feedback to students regardingtheir progress

• determine the need for individualassistance

• regulate the pace of instruction

• regulate the depth of information includedin the unit

• help make decisions regarding students’marks

Evaluating Performance Activity 1—Gamesof Chance

The teacher used a marking guide toevaluate each group of students in relation totheir performance in this activity. Studentswere given a copy of the marking guidebefore they developed their games.

Criteria

• the game involved at least two independent events

• the written explanation of the rules was clear, logical, and easy to follow

• students accurately calculated the theoretical probability of winning theirgame

• students performed at least 50 trials to gather empirical data about theirgame and recorded their observations

• students compared their calculated theoretical probabilities to theresults of their trials and accurately described the fit

• students effectively refined their directions and game based on trialswith other groups of students

• students effectively organized and displayed their game for use in theclassroom midway

• during the midway activity, students kept careful records of the numberof times their game was played, the amount participants paid for eachplay, and the amount paid to winners

• students accurately and effectively displayed their findings using chartsand graphs and presented them to the class

Rating

Evaluating Performance Activity 1—Games of Chance

/3

/5

/5

/3

/5

/5

/5

/5

/5

C-16


The teacher evaluated members of eachgroup individually using the followingcriteria:

• the student makes positivecontributions to the group (3 pointspossible)

• the student builds and elaborates onthe ideas of others (2 points possible)

• the student helps to develop theunderstanding of fellow groupmembers (2 points possible)

• the student suggests positive ways toresolve differences of opinion amonggroup members (2 points possible)

Evaluating Performance Activity 2—Investment Project

• Peer Assessment—Students exchangedtheir investment projects with a partnerand marked each other’s work using amarking guide. Students were expectedto explain the reasons for the marksthey gave in each category and tospecify what they felt their partnersmight have done differently to earnhigher marks.

• Teacher Assessment—The teacher markedstudents’ work using the same markingguide as used in the peer assessment.Students’ final marks for the project werebased on teacher ratings. Students’ finalmarks for the unit were based on theirmarks for the two performance activities.

Student Survey

At the end of the unit, students were askedto complete a survey. The teacher used theresults of the survey as a measure of thesuccess of the unit and to modify activitiesfor future use.

Criteria

• adequately considers risk and probability of return in making investmentdecisions

• considers a variety of investment options

• utilizes a range of resources to gather information

• adequately justifies investment decisions

• provides realistic estimates of potential returns on investments

• shows all work and correctly performs calculations

Rating


/10

/10

/10

/10

/10

/10

C-17


Student Survey


1. What did you like best about the unit?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

2. What did you like least?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

3. What would you change?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

4. What was the most important thing you learned from this unit?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

5. Is there anything you would have liked more information about?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

C-19

APPENDIX C: ASSESSMENT AND EVALUATION • Assessment Practices

APPENDIX CAssessment Practices

C-20


C-21


T eachers should base their assessmentand evaluation of studentperformance on a wide variety of

methods and tools, including observation,student self-assessments, daily practiceassignments, quizzes, work samples, pencil-and-paper tests, holistic rating scales,projects, oral and written reports,performance reviews, and portfolioassessments. Using a variety of assessmentmethods can help teachers to compilecomprehensive profiles of student learning.The Assessment Handbook Series—Performance Assessment, Portfolio Assessment,Student Self-Assessment, Student-CentredConferences, and Numeracy PerformanceStandards—provide useful, detailedinformation about a range of appropriateassessment and evaluation practices. The restof this appendix provides guidance increating classroom tests.

CONSTRUCTING CLASSROOM TESTS

There are two types of evaluations that takeplace in a classroom, each with its owndistinct purpose.

• Formative evaluations are ongoingassessments used to guide learning ratherthan draw final conclusions.

• Summative evaluations refer to evaluativeprocedures (e.g., tests, reports, projects)usually conducted at the end of majorunits, to assess performance againstpredetermined criteria. These normallyrepresent a substantial portion of finalgrades. The classroom tests described inthis section fall under the category ofsummative evaluations.

All tests should be developed using theprinciples of criterion-referenced evaluation.In criterion-referenced evaluation, studentachievement is interpreted in relation to

previously defined levels of achievement,rather than relative to the achievement ofother students. Questions on a criterion-referenced test should be representativeof a clearly defined domain of prescribedlearning outcomes. In this way, scores moreaccurately represent the student’s presentstatus with respect to those outcomes.

A test should measure what it intends tomeasure. For example, if a test requires areading level far above the abilities of manyof the students taking the test, test resultswould measure differences in reading levelsrather than differences in subject knowledge.

STEPS IN CLASSROOM TEST CONSTRUCTION

The following suggests the key points toconsider when developing classroom tests.

C-22


Steps in ClassroomTest Construction Points for Consideration

Plan the Test • Begin the planning process well in advance.• Identify the prescribed learning outcomes to be tested. Learning

outcomes provide a framework for the development of criteria.• Create a table of specifications that covers the learning outcomes

and cognitive levels (e.g., knowledge, understanding, higher mentalprocesses).

• Balance the table of specifications to reflect the topics andcognitive levels in the curriculum.

Write Test Items • Word questions clearly. (e.g., Use “determine a value for x”rather than “find x.”)

• Define the answer format so students understand the form theiranswers must take.

• Don’t repeat questions on one outcome.• When possible, design questions that connect various topics

within the curriculum area and between various curricula.• Create questions that require various forms of answers (e.g.,

explaining, comparing, illustrating, graphing, calculating, solving,justifying).

• Categorize every question according to the criteria.• Don’t phrase distractors in multiple-choice questions so that they

point to the answer.• Review questions for appropriate vocabulary and targetted

reading levels.• Ask a colleague to answer the questions to identify possible

marking problems and time requirements, and to providefeedback.

Format the Test • Use easier questions at the beginning of the test to establishstudents’ confidence.

• Group similar items.• Organize items on the page so they are easy to read and provide

adequate space for responses.• Develop test instructions that are clear and unambiguous.

C-23


Develop a Scoring Key • Mark for processes as well as correct answers.• Allow for possible alternative solutions or different forms of

answers (e.g., format, notation, detail).• Consider a range of scoring methods (e.g., holistic, analytic).

Prepare Students • Establish test criteria with students.• Help students brainstorm topics that are likely to be covered.• Discuss how to approach the test (budget time and the weight

given test results in the final grade).• Give students sufficient time to prepare for the test.• Review terminology used on the test (e.g., evaluate, simplify).

Administer the Test • Allow time for all, or nearly all, students to finish.• Ensure a test location free of distractions.• Ensure all necessary supplies are available.

Score the Tests • Test the effectiveness of the key on a few sample tests. Adjustthe key as necessary. Note exemplar papers.

• Mark tests at one sitting (or one question at one sitting) toensure consistency.

• Return marked tests promptly, and review them with thestudents to help them improve their understanding of theconcepts involved.

Table of Specifications A unit test in mathematics must measure the skills or conceptstaught in the unit. A table of specifications can help the teacher planthe amount of emphasis to give each skill or concept.

A table of specifications is a chart showing the content categoriesand cognitive levels to be tested. The percentage weighting of testitems within any row or column is determined by the amount oftime devoted to the content categories and their degree of difficulty.

C-24


Table of Specifications

Unit # Variables and Equations Mathematics 9

Content Knowledge• recall• conventions• classification• notation

Percentageof Total

Higher MentalProcesses• analysis• synthesis• evaluation

Understanding• application of

theories, ideas,principles, ormethods to a newsituation

Problem Solving 3 questions 4 questions 28%

Algebraic Skills 6 questions 2 questions 52%5 questions

MathematicalReasoning

% of Total 20% 36% 44% 100%

5 questions 20%

APPENDIX DAcknowledgements

D-3

APPENDIX D: ACKNOWLEDGMENTS

Many people contributed their expertise to the document. The project co-ordinator was BruceMcAskill of the Curriculum Branch, working with ministry personnel and our partners ineducation. We would like to thank all who participated in the process.

MATHEMATICS OVERVIEW TEAM

Cathy BockBC Confederation of Parent AdvisoryCouncils

Jack BradshawColleges and Institutes

Russell BreakeyBC Association of Learning Materials andEducational Representatives

Cary ChienBC Teachers’ Federation

Chris EvansBC School Trustees Association

David LeemingUniversities

David LidstoneColleges and Institutes

Tom O’SheaFaculties of Education

David PaulBC Principals’ and Vice-Principals’Association

Garry PhilipsBC Teachers’ Federation

Dennis SemeniukBC School Superintendents Association

Cary ChienSchool District No. 39 (Vancouver)

Keith ChongSchool District No. 37 (Delta)

Delee CowanSchool District No. 23 (Central Okanagan)

WESTERN CANADIAN PROTOCOL: LEARNING OUTCOMES TEAM (BC COMPONENT)

Ivan JohnsonSchool District No. 41 (Burnaby)

Richard MacDonaldSchool District No. 36 (Surrey)

Dhavinder TiwariSchool District No. 68 (Nanaimo-Ladysmith)

APPENDIX D: ACKNOWLEDGMENTS

D-4

INTEGRATED RESOURCE PACKAGE WRITING TEAM

Bob BoykoSchool District No. 68 (Nanaimo-Ladysmith)

David EllisSchool District No. 39 (Vancouver)

Brad EppSchool District No. 73 (Kamloops-Thompson)

Wendy MundieSchool District No. 54 (Bulkley Valley)

Barb WagnerSchool District No. 60 (Peace River North)

Rick WunderlichSchool District No. 83 (North Okanagan–Shuswap)

APPENDIX EGlossary

APPENDIX E: GLOSSARY

E-2

l

E-3


ABOUT APPENDIX F

This appendix provides an illustrated glossary of terms used in this Integrated Resource Package.

The terms and definitions are intended to be used by readers unfamiliar with mathematicalterminology. For a more complete definition of each term, refer to a mathematical dictionarysuch as the Nelson Canadian School Mathematics Dictionary (ISBN 17-604800-6).

absolute value of a numberHow far the number is from 0. Example: the absolute value of both-4.2 and 4.2 is 4.2

absolute value functionThe function f defined by f(x) = |x|, where |x| denotes the absolutevalue of x.

accuracyA measure of how far an estimate is from the true value.

acute angleAn angle whose measure is between 0° and 90°.

algorithmA mechanical method for solving a certain type of problem, often amethod in which one kind of step is repeated a number of times.

alternate interior anglesIn the diagram to the left, the angles labelled a and c are alternateinterior angles, as are the angles b and d.

/_ a=/_ c/_ b=/_ d

a b

d c

y

x

f(x)=|x|

A


E-4

altitude of a triangleA line segment PH, where P is a vertex of the triangle, H lies on the linethrough the other two vertices, and PH is perpendicular to that line.

ambiguous caseTwo sides of a triangle and the angle opposite one of them arespecified, and we want to calculate the remaining angles or side.There may be no solution, exactly one, or exactly two.

amplitude (of a periodic curve)The maximum displacement from a reference level in either apositive or negative direction. That reference level is often chosenhalfway between the biggest and smallest values taken on by the curve.

analytic geometry (coordinate geometry)An approach to geometry in which position is indicated by usingcoordinates, lines, and curves; other objects are represented byequations; and algebraic techniques are used to solve geometricproblems.

angle bisectorA line that divides an angle into two equal parts.

antiderivativeIf f(x) is the derivative of F(x), then F(x) is an antiderivative of f(x).Indefinite integral means the same thing.

antidifferentiationThe process of finding antiderivatives.

arcA connected segment of a circle or curve.

arc sine (of x)The angle (in radians) between -π/2 and π/2 whose sine is x.Notation: sin-1 x or arcsin x.

Given a, b, and /_ A,find length c

Solution(s) c = AB c = AB'

a

a

A

A

B

B'

Cb

b

c

c

P

H

E-5


arc tangentThe angle (in radians) between -π/2 and π/2 whose tangent is x.Notation: tan-1 x or arctan x.

arithmetic operationAddition, subtraction, multiplication, and division.

arithmetic sequence (arithmetic progression)A sequence in which any term is obtained from the preceding termby adding a fixed amount, the common difference.

If a is the first term and d the common difference, then the sequenceis a, a + d, a + 2d, a + 3d, . . . . The n-th term tn is given by the formulatn = a + (n – 1)d.

arithmetic seriesThe sum Sn of the first n terms of an arithmetic sequence. If thesequence has first term a and common difference d, then

Sn = – n[2a + (n – 1)d] = – n (a + l)

where l is a + (n – 1)d, the “last’’ term.

asymptote (to a curve)A line l such that the distance from points P on the curve to lapproaches zero as the distance of P from an origin increaseswithout bound as P travels on a certain path along the curve.

average velocityThe net change in position of a moving object divided by theelapsed time.

axis of symmetry (of a geometric figure)A line such that for any point P of the figure, the mirror image of Pin the line is also in the figure.

y

x

asymptotes

12

12


E-6

Bbar graphA graph using parallel bars (vertical or horizontal) which areproportional in length to the data they represent.

baseIn the expression st, the number or expression s is called the base,and t is the exponent. In the expression loga u, the base is a.

binomialThe sum of two monomials.

binomial distributionThe probabilities associated with the number of “successes’’ whenan experiment is repeated independently a fixed number of times.For example, the number of times a six is obtained when a fair die istossed 100 times has a binomial distribution.

Binomial TheoremA rule for expanding expressions of the form (x + y)n.

bisectTo divide into two equal parts.

broken-line graphA graph using line segments to join the plotted points torepresent data.

CCartesian (rectangular) coordinate systemA coordinate system in which the position of a point is specified byusing its signed distances from two perpendicular reference lines(axes).

E-7


central angleAn angle determined by two radii of a given circle; equivalently, anangle whose vertex is at the center of the circle.

chain ruleA rule for differentiating composite functions.

If h(x) = f(g(x)), then h’(x) = f’(g(x))g’(x).

chordThe line segment that joins two points on a curve, usually a circle.

circumferenceThe boundary of a closed curve, such as a circle; also, the measure(length) of that boundary. Please see perimeter.

circumscribedThe polygon P is circumscribed about the circle C if P is inside C andthe edges of P are tangent to C. The circle C is circumscribed about thepolygon Q if Q is inside C and the vertices of Q are on the boundaryof C. The notion can be extended to other figure and to threedimensions.

clusterA collection of closely grouped data points.

coefficientA numerical or constant multiplier in an algebraic expression. Thecoefficient of x2 in 4x2 – 2axy is 4, and the coefficient of xy is -2a.

collinearLying on the same line.

combinationA set of objects chosen from another set, with no attention paid tothe order in which the objects are listed (see also permutation). Thenumber of possible combinations of r objects selected from a set of ndistinct objects is nCr, or (—

rn ) pronounced “n choose r.’’

Cluster


E-8

common factor (CF)A number that is a factor of two or more numbers. For example, 3 is acommon factor of 6 and 12. Common divisor means the same thing.The term is also used with polynomials. For example, x – 1 is acommon factor of x2 – x and x2 – 2x + 1.

common fractionA number written as – , where the numerator a and the denominatorb are integers, and b is not zero. Examples:

– —— —

compassAn instrument for drawing circles or arcs of circles.

complementary anglesTwo angles that add up to a right angle.

completing the squareRewriting the quadratic polynomial ax2 + bx + c in the forma(x – p)2 + q, perhaps to solve the equation ax2 + bx + c = 0.

complex fractionA fraction in which the numerator or the denominator, or both,contain fractions.

composite functionA function h(x) obtained from two functions f and g by using the ruleh(x) = f(g(x)) (first do g to x, then do f to the result).

composite numberAn integer greater than 1 that is not prime, such as 9 or 14.

compound interestThe interest that accumulates over a given period when eachsuccessive interest payment is added to the principal in order tocalculate the next interest payment.

A = P(I + —)tn

45

-136

31

ab

in

E-9


concave down (or downward)The function f(x) is concave down on an interval if the graph ofy = f(x) lies below its tangent lines on that interval.

concave up (or upward)The function f(x) is concave up on an interval if the graph of y = f(x)lies above its tangent lines on that interval.

conditional probabilityThe probability of an event given that another event has occurred.The (conditional) probability that someone earns more than $200,000a year, given that the person plays in the NHL, is different from theprobability that a randomly chosen person earns more than$200,000.

cone (right circular)The three-dimensional object generated by rotating a right triangleabout one of its legs.

confidence interval(s)An interval that is believed, with a preassigned degree of confidence, toinclude the particular value of some parameter being estimated.

congruentHaving identical shape and size.

conic sectionA curve formed by intersecting a plane and the surface of a doublecone. Apart from degenerate cases, the conic sections are the el-lipses, the parabola, and the hyperbolas.

conjectureA mathematical assertion that is believed, at least by some, to betrue but has not been proved.

constantA fixed quantity or numerical value.

f (x)

xConcave

Down

ConcaveUp


E-10

continuous dataData that can, in principle, take on any real value in some interval.For example, the “exact’’ height of a randomly chosen individualor the exact length of life of a U-235 atom can be modelled by acontinuous distribution.

continuous functionInformally, a function f(x) is continuous at a if f(x) does not make asharp jump at a. More formally, f(x) is continuous at a if f(x)approaches f(a) as x approaches a.

contrapositiveThe contrapositive of “Whenever A is true, B must be true’’ is“Whenever B is false, then A must be false.’’ Any assertion islogically equivalent to its contrapositive, so one strategy for provingan assertion is to prove its contrapositive.

converse (of a theorem)The assertion obtained by interchanging the premise and theconclusion. If the theorem is “Whenever A happens, B musthappen,’’ then its converse is “Whenever B happens, A musthappen.’’ The converse of a theorem need not be true.

coordinate geometryPlease see analytic geometry.

coordinatesNumbers that uniquely identify the position of a point relative to acoordinate system.

correlation coefficientA number (between -1 and 1) that measures the degree to which acollection of data points lies on a line.

corresponding angles and corresponding sidesAngles or sides that have the same relative position in geometricfigures.

A

B C

D

E F

E-11


cosecant (of x)This is ——. Notation: csc x.

cosine law (law of cosines)A formula used for solving triangles in plane geometry.

cosine (function)See primary trigonomic functions.

cotangent (of x)This is ——. Notation: cot x.

coterminal anglesAngles that are rotations between the same two lines, termed theinitial and terminal arms. For example:

20°, -340°, 380° are coterminal angles

critical number (of a function)A number where the function is defined and where the derivative ofthe function is equal to 0 or doesn’t exist.

cyclic (inscribed) quadrilateralA quadrilateral whose vertices all lie on a circle.

Ddecimal fractionIn principle, a fraction a/b where a is an integer and b is a power of10. For example, 1/4 = 25/100, so 1/4 can be expressed as a decimalfraction, usually written as 0.25.

decreasing functionThe function f(x) is decreasing on an interval if, for any numbers sand t in that interval, if t is greater than s, then f(t) is less than f(s).

c2 � a2 � b2 � 2ab cos C

1sin x

1tan x


E-12

deductive reasoningA process by which a conclusion is reached from certainassumptions by the use of logic alone.

degreeThe highest power or sum of powers that occurs in any term of agiven polynomial or polynomial equation. For example, 6x + 17 hasdegree 1, and 2 + x3 + 7x has degree 3, as does 2 + 6x + 7y + xy2.

diagonalA line segment that joins two non-adjacent vertices in a polygon orpolyhedron.

diameterA line segment that joins two points on a circle or sphere and passesthrough the center. All diameters of a circle or sphere have the samelength. That common length is called the diameter.

difference of squaresAn expression of the form A2 – B2, where A and B are numbers,polynomials, or perhaps other mathematical expressions. We canfactor A2 – B2 as (A + B)(A – B).

differentiableA function is differentiable at x = a if, under extremely highmagnification, the graph of the function looks almost like a straightline near a. Most familiar functions are differentiable everywherethat they are defined.

differential equationAn equation that involves only two variable quantities, say x and y,and the first derivative, or higher derivatives, of y with respect to x.

Example: 3y2 –— = ex

differentiate; differentiationTo find the derivative; the process of finding derivatives.

dydx

E-13


direct variationThe quantity Q varies directly with x if Q = ax for some constant a.This can be contrasted with inverse variation, in which Q = – forsome a.

discrete dataData arising from situations in which the possible outcomes lie in afinite or infinite sequence.

discriminantThe discriminant of the quadratic polynomial ax2 + bx + c (or of theequation ax2 + bx + c = 0) is b2 – 4ac and is used to described thenature of the roots.

displacementPosition, as measured from some reference point.

distance formulaThe formula used in coordinate geometry to find the distancebetween two points. If A has coordinates (x1, y1) and B hascoordinates (x2, y2), then the distance from A to B is

(x2 – x1)2 + (y2 – y1)

2

domain (of a function)The set of numbers where the function is defined. For example, if

f(x) = ——— , then the domain of f(x) consists of all real numbersgreater than or equal to 2, except for the number 5.

double bar graphA bar graph that uses bars to represent two sets of data visually.

EedgeThe straight line segment that is formed where two faces of apolyhedron meet.

ax

x – 2x – 5

A B C


E-14

ellipseA closed curve obtained by intersecting the surface of a cone with aplane. Please see conic section.

equationA statement that two mathematical expressions are equal, such as:

3x � y � 7

equidistantHaving equal distances from some specified object, point, or line.

estimatev. To approximate a quantity, perhaps only roughly.n. The result of estimating. Also, an approximation, based onsampling, to some number associated with a population, such as theaverage age.

Euclidean geometryGeometry based on the definitions and axioms set out in Euclid’sElements.

eventA subset of the sample space of all possible outcomes of an experiment.

experimental probabilityAn estimate of the probability of an event obtained by repeating anexperiment many times. If the event occurred in k of the nexperiments, it has experimental probability k/n.

exponentThe number that indicates the power to which the base is raised.For example, the exponent of 34 is 4.

exponential decayA quantity undergoes exponential decay if its rate of decrease at anytime is proportional to its size at the time. Exponential decay modelswell the decay of radioactive substances.

E-15


exponential functionAn exponential function is a function of the form f(x) = ax, wherea > 0 and the variable x occurs as the exponent. The exponentialfunction is the function f(x) = ex, where e is a mathematical constantroughly equal to 2.7182818284.

exponential growthA quantity undergoes exponential growth if its rate of increase at anytime is proportional to its size at the time. Exponential growth modelswell the growth of a population of bacteria under ideal conditions.

exterior angles on the same side of the transversalA transversal of two parallel lines forms two supplementary exteriorangles.

extraneous rootA spurious root obtained by manipulating an equation. For example,if we square both sides of 1 – x = x – 1 and simplify, we obtain(x – 1)(x – 2) = 0, that is, x = 1 or x = 2. Since 2 is not a root of theoriginal equation, it is sometimes called an extraneous root.

extrapolateEstimate the value of a function at a point from values at places onone side of the point only.

extreme valuesThe highest and lowest numbers in a set.

FfaceOne of the plane surfaces of a polyhedron.

y

x

f(x) = a

x


E-16

factorn. A factor of a number n is a number (usually taken to be positive)that divides n exactly. For example, the factors of 18 are 1, 2, 3, 6, 9,and 18. Similarly, a factor of a polynomial P(x) is a polynomial thatdivides P(x) exactly. Thus x and x – 1 are two of the factors of x3 – x.v. To factor a number or polynomial is to express it as a product ofbasic terms. For example, x3 – x factors as x(x – 1)(x + 1).

Factor TheoremIf P(x) is a polynomial and a is a root of the equation P(x) = 0, thenx – a is a factor of P(x).

factor(s)Numbers multiplied to produce a specific product. For example:

2 � 3 � 3 � 18: factors are 2 and 3x2 � x � 2: factors are (x � 2) and (x � 1)

first-hand dataData collected by an individual directly from observations ormeasurements.

flipAnother word for reflection.

fractalA set of points from a recursive formula which creates a figure that,however magnified, has the same shape as the whole.

frequency diagramA diagram used to record the number of times various eventsoccurred.

functionA rule that produces, for any element x of a certain set A, an objectf(x). The set A is the domain of the function; the set of values taken onby f(x) is the range of the function.

More formally, a function is a collection of ordered pairs (x, y) suchthat the second entry y is completely determined by the first entry x.

E-17


function notationIf a quantity y is completely determined by a quantity x, then y iscalled a function of x. For example, the area of a circle of generalradius x might be denoted by A(x) ( pronounced “A of x"). In thiscase, A(x) = π x2.

fundamental counting principleIf an event can happen in x different ways, and for each of theseways a second event can happen in y different ways, then the twoevents can happen in x � y different ways.

Ggeneral polynomial equationAn equation of the form

a0xn � a1x

n-1 � a2xn-2 � . . . � an-1x � an � 0

general term (of a sequence)If n is unspecified, an is called the general term of the sequence a1, a2,a3, . . . Sometimes there is an explicit formula for an in terms of n.

geometric sequence (progression)A sequence in which each term except the first is a fixed multiple ofthe preceding term. If the first term is a and each term is r times theprevious one (r is the common ratio), then the general term tn is givenby tn = arn-1.

geometric seriesThe sum Sn of the first n terms of a geometric sequence. If a is thefirst term, r the common ratio, where r ≠ 1, then

See also infinite geometric series.

greatest common factor (GCF)The largest positive integer that divides two or more given numbers.For example, the GCF of 12 and 18 is 6. The GCF is also called thegreatest common divisor (GCD).

Sn � a(1 � rn)1 � r


E-18

HHeron’s formulaA formula for the area of a triangle in terms of its sides. The area of atriangle with sides of length a, b, and c is equal to

s(s – a)(s – b)(s – c), where s = ————

higher derivativesThe derivative of the derivative of f(x), the derivative of the deriva-tive of the derivative of f(x), and so on.

histogramA bar graph showing the frequency in each class using class intervalsof the same length.

hyperbolaA curve with two branches where a plane and a circular conicalsurface meet. Please see conic section.

hypotenuseThe side opposite the right angle in a right triangle.

hypothesisA statement or condition from which consequences are derived.

a + b + c2

y

x

y

x

__ __x2 y2

a2 b2

-

= -1

or

a2 b2x2 y2__ __– = 1

–

__ __y2 x2

b2 a2 = 1–

E-19


IidentityA statement that two mathematical expressions are equal for allvalues of their variables.

if-then propositionA mathematical statement that asserts that if certain conditions hold,then certain other conditions hold.

Imperial measureThe system of units (foot, pound, and so on) for measuring length,mass, and so on that was once the legal standard in Great Britain.

implicit functionA function y of x defined by a formula of shape H(x, y) = 0. Forexample, y3 – x 2 + 1 = 0 defines y implicitly as a function of x. In thiscase, y is given explicitly by y = (x2 – 1)1/3. But often (example:H(x, y) = y7 + (x2 + 1)(y – 1), it is not possible to give an explicitformula for y.

improper fractionA proper fraction is a fraction whose numerator is less in absolutevalue than its denominator. An improper fraction is a fraction that isnot a proper fraction.

increasing functionThe function f(x) is increasing on an interval if, for any numbers sand t in that interval, if t is greater than s, then f(t) is greater than f(s).

indefinite integralAnother word for antiderivative.

independent eventsTwo events are independent if whether or not one of them occurs hasno effect on the probability that the other occurs.


E-20

inductive reasoningA form of reasoning in which the truth of an assertion in someparticular cases is used to leap to the (tentative) conclusion that theassertion is true in general.

inequalityA mathematical statement that one quantity is greater than or lessthan the other. The statement s > t means that s is greater than t,while s < t means that s is less than t.

infinite geometric seriesThe “sum’’ a + ar + ar2 + . . . + arn-1 + . . . of all of the terms of ageometric sequence. If |r|< 1, then this sum is equal to —– .

inflection pointA point on a curve that separates a part of the curve that is concaveup from one that is concave down.

initial value problemA function is described by specifying a differential equation that itsatisfies, together with the value of the function at some “initial’’point. The problem is to find the function.

inscribed angleThe angle PQR, where P, Q, and R are three points on a curve, inmost cases a circle.

instantaneous velocity (at a particular time)The exact rate at which the position is changing at that time.

integerOne of 0, 1, -1, 2, -2, 3, -3, 4, -4, and so on.

interior angles on the same side of the transversalThe transversal of two parallel lines forms interior supplementaryangles.

a1 – r

f(x)

x

inflection point

P

Q

R

E-21


integrationIn part, the process of finding antiderivatives.

interpolateEstimate the value of a function at a point from values of thefunction at places on both sides of the point.

intersectionThe point or points where two curves meet.

intervalThe set of all real numbers between two given numbers, which may ormay not be included. The set of all real numbers from a given point onor up to a given point is also an interval, as is the set of all real numbers.

inverse (of a function)The function g(x) is the inverse of the function f(x) if f(g(x)) = x andg(f(x)) = x for all x, or more informally if each function “undoes’’what the other did.

inverse operationsOperations that counteract each other. For example, addition andsubtraction are inverse operations.

inverse trigonometric functionsInverses of the six basic trigonometric functions. For the two mostcommonly used, please see arc sine and arc tan.

irrational numberA number that cannot be expressed as a quotient of two integers.For example, 2 , π, and e are irrational numbers.

irregularLacking in symmetry or pattern.

isosceles triangleA triangle that has two or more equal sides. Occasionally defined asa triangle that has exactly two equal sides.

Estimate

Known


E-22

Lleast squaresA criterion used to find the line of best fit, namely that the sum ofthe squares of the differences between “predicted values’’ and actualvalues should be as small as possible.

limitThe limit of f(x) as x approaches a (notation: lim f(x)) is thenumber that f(x) tends to as x moves closer and closer to a. Theremay not be such a number. For example, if x is measured in radians,lim ——— = 1, but lim sin (—) does not exist.

line of best fitFor a collection of points in the plane obtained from an experiment,a line that comes in some sense closest to the points. Please see leastsquares.

linear functionA function f given by a formula of the type f(x) = ax + b, where a andb are specific numbers.

linear programmingFinding the largest or smallest value taken on by a given functiona1 x1 + a2 x2 +. . . + anxn (the objective function) given that x1, x2, . . ., xn

satisfy certain linear constraints. The constraints are inequalities ofthe form b1 x1 + b2 x2 + . . . + bn ≥ c. Many applied problems, such asdesigning the cheapest animal feed that meets given nutritionalgoals, can be formulated as linear programming problems.

local maximumThe function f(x) is said to reach a local maximum at x = a if there isa neighborhood of a kind such that f(x) ≤ f(a) for any x in theneighborhood; informally, (a, f(a)) is at the top of a hill.

local minimumThe function f(x) is said to reach a local minimum at x = a if there is aneighborhood of a kind such that f(x) ≥ f(a) for any x in the neighbor-hood; informally, (a, f(a)) is at the bottom of a valley.

x → a

x → 0

sin xx

1xx → 0

f(x)

x

Local Max.

Local Min.

E-23


logarithmic differentiationThe process of differentiating a product/quotient of functions byfinding the logarithm and then differentiating. For example, lety = (1 + x)2/(1 + 3x). Then

ln y = 2ln(1 + x) – ln(1 + 3x) and — —– = —— – ——–

logarithmic functionLet a be positive and not equal to 1. The logarithm of x to the base ais the number u such that au = x, and is denoted by loga x. Anyfunction of the form f(x) = loga x is called a logarithmic function.

lowest common multiple (LCM)The smallest positive integer that is a multiple of two or more givenpositive integers. For example, the LCM of 3, 4, and 6 is 12. TheLCM is often called the least common multiple.

MmatrixA rectangular array of numbers. For example:

maximum point (or value)The greatest value of a function.

mean (of a sequence of numerical data)A measure of the average value, obtained by adding up the terms ofthe sequence and dividing by the number of items.

median (of a sequence of numerical data)The “middle value’’ when the data are arranged in order of size. Ifthere is an even number of data, then it's the average of the twomiddle values. For example, the median of 5, 3, 7.4, 5, 8 is 5, whilethe median of 5, 7.4, 5, 8 is 6.2.

1y

dydx

21 + x

31 + 3x

y

x

f(x) = logax

3 4-2 5

2 � 2 matrix 3 � 1 matrix

172


E-24

median (of a triangle)The line segment that joins a vertex of the triangle to the midpoint ofthe opposite side.

minimum point (or value)The lowest value of a function.

mixed numberA number that is expressed as the sum of a whole number and afraction. For example:

2 5

modeThe value that occurs most often in a sequence of data.

monomialAn algebraic expression that is a product of variables and constants.For example: 6x2, 1, (—), x2y

multiple (of an integer)The result obtained when the given integer is multiplied by someinteger. Equivalently, an integer that has the given integer as afactor. (Often negative integers are not allowed.)

Nnatural logarithmLogarithm to the base e, where e is a fundamental mathematicalconstant roughly equal to 2.7182818284. The natural logarithm of xis usually written ln x.

natural number (counting number)One of the numbers 1, 2, 3, 4, . . . Positive integer means the samething.

34

3

E-25


netA flat diagram consisting of plane faces arranged so that it may befolded to form a solid.

Newton’s Law of CoolingThe assertion that if a warm object is placed in a cool room, itstemperature decreases at a rate proportional to the difference intemperature between the object and its surroundings.

Newton’s MethodAn often highly efficient iterative method for approximating theroots of f(x) = 0. If rn is the current estimate, then the next estimate isthe x-intercept of the tangent line to y = f(x) at x = rn.

non-differentiable (function)A function f(x) is non-differentiable at x = a if it does not have aderivative there. Example: if f(x) = |x| then f(x) is not differentiableat x = 0, basically because the curve y = |x| has a sharp kink there.

normal distribution curveThe standard normal distribution curve has equation y � e

-x22

2„

( ) .

The general normal is obtained by shifting the standard normal tothe left or right and/or rescaling. These curves are sometimes calledbell-shaped curves. They figure importantly in probability, statistics,and signal processing.

Oobtuse angleAn angle whose measure is between 90° and 180°.

one-sided limitSometimes f(x) exhibits different behaviour depending on whether xapproaches a from the right (through values of x greater than a) orfrom the left. For example, let f(x) = 1/(1 + 21/x). As x approaches 0from the right, f(x) approaches 0 (notation: lim f(x) = 0), while f(x)approaches 1 as x approaches 0 from the left.

x → 0+

y

x 2π


E-26

optimization problemA problem, often of an applied nature, in which we need to find thelargest or smallest possible value of a quantity. Also called a max/min problem.

ordered pairA sequence of length 2. Ordered pairs (x, y) of real numbers are usedto indicate the x and y coordinates of a point in the plane.

ordinal numberA number designating the place occupied by an item in an orderedsequence (e.g., first, second, and third).

originThe point in a coordinate system at the intersection of the axes.

PparabolaThe intersection of a conical surface and a plane parallel to a line onthe surface.

parallel linesTwo lines in the plane are parallel if they do not meet. Inthree-dimensional space, two lines are parallel if they do not meetand there is a plane that contains them both. Alternately, in the planeor in space, two lines are parallel if they stay a constant distanceapart.

parallelogramA quadrilateral such that pairs of opposite sides are parallel.

percentageIn a problem such as “Find 15% of 400,’’ the number 400 is some-times called the base, 15% or 0.15 is called the rate, and the answer 60is sometimes called the percentage.

y

x

parabola

E-27


percent errorThe relative error expressed in parts per hundred. Let A be anestimate of a quantity whose true value is T. Then A – T is the error,and (A – T)/T is the relative error.

percentileThe k-th percentile of a sequence of numerical data is the number xsuch that k percent of the data points are less than or equal to x. (Oftenx is not precisely determined, particularly if the data set is not large.)

perimeterThe length of the boundary of a closed figure.

periodThe interval taken to make one complete oscillation or cycle.

permutationAn ordered arrangement of objects. The number of ways of produc-ing a permutation of r (distinct) objects from a collection of n objectsis nPr , where nPr = n(n – 1)(n - 2) . . . (n – r+1) or nPr = ——— .

perpendicular bisectorA line that intersects a line segment at a right angle and divides theline segment into two equal parts.

perpendicular lineTwo lines that intersect at a right angle.

phase shiftA horizontal translation of a periodic function. For example, thefunction cos 2(x – π/3) is cos 2x with a phase shift of π/3.

pictographA graph that uses pictures or symbols to represent similar data.

n!(n – r)!


E-28

plane of symmetryA 2-D flat surface that cuts through a 3-D object, forming two partswhich are mirror images.

polygonA closed curve formed by line segments that do not intersect otherthan at the vertices.

polygonal regionA part of a plane that has a polygon as boundary.

polyhedronA solid bounded by plane polygonal regions.

polynomialA mathematical expression that is a sum of monomials. Examples:

4x3 – 3x – — π x2 + 2πxy – xyz

populationThe items, actual or theoretical, from which a sample is drawn.

powerA power of q is any term of the form qk. Often but not always, k istaken to be a positive integer.

precisionA measure of the estimated degree of repeatability of a measurement,often described by a phrase such as “correct to two decimal places.’’

12

E-29


primary trigonometric functionssine A� a/c � opp/hypcosine A� b/c � adj/hyptangent A� a/b � opp/adjFunctions of angles defined, for an acute angle, as ratios of sidesin a right triangle.

primeA positive integer that is divisible by exactly two positive integers,namely 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13.

prime factorization (of a positive integer)The given integer expressed as a product of primes. For example,2 � 5 � 3 � 2 is a prime factorization of 60. Usually the primes arelisted in increasing order. The standard prime factorization of 60 is22 � 3 � 5.

prismA solid with two parallel and congruent bases in the shape ofpolygons. The other faces are parallelograms.

probability (of an event)A number between 0 and 1 that measures the likelihood that theevent will occur. The probability of the event A is often denoted byP.(A).

productThe product of two or more objects (e.g., numbers, functions) is theresult of multiplying these objects together.

product ruleThe rule for finding the derivative of a product of two functions.

If p(x) = f(x)g(x), then p’(x) = f(x)g’(x) + g(x)f’(x).

pyramidA polyhedron, one of whose faces is an arbitrary polygon (called thebase) and whose remaining faces are triangles with a common vertex(called the apex).

Sine A = __

a

A

B

Cb

c

ac


E-30

Pythagorean theoremIn a right-angled triangle, the sum of the squares of the lengths ofthe sides containing the right angle is equal to the square of thehypotenuse (a2 � b2 � c2).

QquadrantOne of the four regions that the plane is divided into by two perpen-dicular lines. When these lines are the usual coordinate axes, thequadrants are called the first quadrant, the second quadrant, and soon as in the diagram.

quadratic formulaA formula used to determine the roots of a quadratic equation.

quadratic functionA function of the form f(x) = ax2 + bx + c, where a ≠ 0. The graph ofsuch a function is a parabola.

quadrilateralA polygon with four sides.

quartileThe 25th percentile is the first quartile, the 50th percentile is thesecond quartile (or median), and the 75th percentile is the thirdquartile. Please see percentile.

quotientThe result of dividing one object (number, function) by another.

quotient ruleThe rule for differentiating the quotient of two functions.

If q(x) = —— , then q’(x) = —————————.

a

A

B

Cb

c

I

IV

II

III

x � -b � b2 � 4ac2a

f(x)g(x)

g(x) f’(x) – f(x) g’(x)(g(x))2

E-31


RradianEqual to the central angle subtended by an arc of unit length at thecentre of a circle of unit radius.

radicalThe square root or cube root, and so on, of a quantity. For example,the cube root of the quantity Q is the quantity R such that R3 (the

cube of R) is equal to Q. The square root of Q is written Q ( isthe radical sign). The cube root of Q is written 3 Q.

radiusA line segment that joins the centre of a circle or sphere to a point onthe boundary. All radii of a circle or sphere have the same length.That common length is called the radius.

rangeA measure of variability of a sequence of data, defined to be thedifference between the extremes in the sequence. For example, if thedata are 27, 22, 27, 20, 35, and 34, then the range is 15.

range (of a function)The set of values taken on by a function. Please see function.

rank orderingOrdering (of a sample) according to the value of some statisticalcharacteristic.

rateA comparison of two measurements with different units. Forexample, the speed of an object measured in kilometres per hour.

rate of change (of a function at a point)How fast the function is changing. If f(x) is the function, its rate ofchange with respect to x at x = a is the derivative of f(x) at x = a.

r

=1radr

r


E-32

ratioAnother word for quotient. Also an indication of the relative size oftwo quantities. We say that P and Q are in the ratio a:b if the “size’’of A divided by the size of B is a/b.

rational expressionThe quotient of two polynomials.

rational numberA number that can be expressed as a/b, where a and b are integers.

rationalize the denominatorTransform a quotient P/Q where the denominator Q involvesradicals into an equivalent expression with the denominator free ofradicals. For example:

——— = ——————— = ————

real numberAn indicator of location on a line with respect to an origin; aquantity represented by an arbitrary decimal expansion.

reciprocalThe number or expression produced by dividing 1 by a givennumber or expression.

rectangular prismA prism whose bases are congruent rectangles.

recursive definition (of a sequence)A way of defining a sequence by possibly specifying some termsdirectly and giving an algorithm by which any term can beobtained from its predecessors. For example, the Fibonaccisequence is defined recursively by the rules F0 = F1 = 1 andFn= Fn-1 + Fn-2 for n ≥ 2.

4

4 – 7

(4)(4 + 7)

(4 – 7)(4 + 7)

4(4 + 7)9

E-33


reference angleThe acute angle between the ray line and the x-axis. For example,the reference angle of both a 165˚ and of a 195˚ angle is a 15˚ angle.

reflection (in a line)The transformation that takes any 2-D object to the object that issymmetrical to it with respect to the line—that is, to its mirror imagein the line. Flip means the same thing. In 3-D, we can defineanalogously reflection in a plane.

relative maximum, minimumPlease see local maximum, local minimum.

Remainder TheoremIf we divide the polynomial P(x) by x – a, the remainder is equal toP(a).

repeating decimalA decimal expansion that has a block of digits that ultimately cyclesforever. For example, 23/22 has the decimal expansion1.0454545 . . ., with the block 45 ultimately cycling forever. Aterminating decimal like 0.25 is usually viewed as being a repeatingdecimal, indeed in two ways: as 0.25000 . . . and 0.24999 . . .

resultantThe sum of two or more vectors.

right angleAn angle whose measure is 90°.

root of an equation (in one variable)If the equation has the form F(x) = G(x), a root of the equation is anumber a such that F(a) = G(a).


E-34

rotation (in the plane)A transformation in which an object is turned through some angleabout a point. An analogous notion can be defined in threedimensions. There the turn is about a line.

roundingA process to follow when making an approximation to a givennumber by using fewer significant figures.

SsampleA selection from a population.

sample spaceThe set of all possible outcomes of an experiment.

scalarA number. Usually used in contexts where there are also vectorsaround, or functions. Examples of usage:“the length of a vector is a scalar’’“-3 sin x is a scalar multiple of sin x’’

scatter plotIf each item in a sample yields two measurements, such as theheight x and weight y of the individual chosen, the point withcoordinates (x, y) is plotted. If this is repeated for all members of thesample, the resulting collection of points is a scatter plot.

secant (of x)

This is ——–. Notation: sec x.

secant lineA line that passes through two points on a curve.

1cos x

E-35


second derivativeThe second derivative of f(x) is the derivative of the derivative of f(x).Two common notations: f’’(x) and ––––

second derivative testSuppose that f’(a) = 0. The second derivative test gives a way ofchecking whether, at x = a, the function f(x) reaches a local minimumor a local maximum.

second-hand dataData not collected directly by the researcher. For example, encyclopedia.

self-similarHaving the same general appearance under all magnifications.

semicircleA half-circle. Any diameter cuts a circle into two semicircles.

sequenceA finite ordered list t1, t2, t3, . . . , tn of terms (finite sequence) or a listt1, t2, . . . , tn, . . . that goes on forever (an infinite sequence).

seriesAny sum t1 + t2 + . . . + tn of the first n terms of a sequence. The“sum’’ t1 + t2 + . . . + tn + . . . of all the terms of an infinite sequence isan infinite series. The concept of limit is required to define the sum ofinfinitely many terms.

SI measureAbbreviation for Système International d’Unités—InternationalSystem of Units—kilogram, second, ampere, kelvin, candela, mole,radian, and so on.

side (of a polygon)Any of the line segments that make up the boundary of the polygon.

d2 fdx2


E-36

sigma notationThe use of the sign ∑ (Greek capital sigma) to denote sum. Forexample:

∑ ai a1 + a2 + a3 + a4 + a5

simple interestInterest computed only on the original principal of a loan or bankdeposit.

I = Prt

sine (function)Please see primary trigonometric functions.

sine law (law of sines)A formula used for solving triangles in plane trigonometry.

——– = ——– = ——–

skeletonA representation of the edges of a polyhedron.

skip countingCounting by multiples of a number. For example, 2, 4, 6, 8.

slideA transformation of a figure by moving it up and down and/or leftand right without any rotation. The word is a synonym for the morestandard mathematical term translation.

slopeThe slope of a (non-vertical) line is a measure of how fast the line isclimbing. It can be defined as the change in the y-coordinate of apoint on the line when the x-coordinate is increased by 1. If a curvehas a (non-vertical) tangent line at the point, the slope of the curve atthe point is defined to be the slope of that tangent line.

slope-intercept form (of the equation of a line)An equation of the form y = mx + b. All lines in the plane except forvertical lines can be written in this form. The number m is the slope,and b is the y-intercept.

5i=1

a

A

B

Cb

c

sinAa

sinBb

sinCc

E-37


solution (of a differential equation)A function that satisfies the differential equation. For example, forany constant C, the function given by y = (x2 + C)1/3 is a solution ofthe differential equation 3y2 —– = 2x.

sphereA solid whose surface is all points equidistant from a centre point.

square rootThe square root of x is the non-negative number that, whenmultiplied by itself, produces x. For example, 5 is the square root of25. In general the square root of x2 is |x|, the absolute value of x.

standard deviationSample standard deviation is the square root of the sample variance.Population standard deviation is the square root of the populationvariance.

standard formThe usual form of an equation. For example, the standard form ofthe equation of a circle is (x – a)2 + (y – b)2 = r2, because it revealsgeometrically important features, the centre and the radius.

standard position (angle in)The initial arm of the angle is the positive horizontal axis (x-axis).Counterclockwise rotation gives a positive angle.

step functionA function whose graph is flat except at a finite number of points,where it takes a sudden jump.

supplementary anglesTwo angles whose sum is 180°.

dydx

y

x

terminal arm

horizontal axisinitial arm

y

x


E-38

symmetrical (has symmetry)A geometrical figure is symmetrical if there is a rotation reflection,or combination of these, that takes the figure to itself but movessome points. For example, a square has symmetry because it is takento itself by a rotation about its center through 90˚.

system of equationsA set of equations. A solution of the system is an assignment ofvalues to the variables such that all of the equations are (simulta-neously) satisfied. For example, x = 1, y = 2, z = -3 is a solution of thesystem x + y + z = 0, x – y – 4x = 11.

Ttangent (function)Please see primary trigonometric functions.

tangent (to a curve)A line is tangent to a curve at the point P if, under very high magni-fication, the line is nearly indistinguishable from the curve at pointsclose to P. A tangent line to a circle can be thought of as a line thatmeets the circle at only one point.

tangent line approximationIf P is a point on a curve, then close to P the curve can be approxi-mated by the tangent line at P. In symbols, if x is close to a, then f(x)is very closely approximated by f(a) + (x – a)f’(a).

tangramA square cut into seven shapes: two large triangles, one mediumtriangle, two small triangles, one square, and one parallelogram.

termPart of an algebraic expression. For example, x3 and 5x are terms ofthe polynomial x3 + 3x2 + 5x – 1.

terminating decimalA decimal expansion that (ultimately) ends. For example, 3.73.

E-39


tesselationA covering of a surface (usually the entire plane) without overlap orbare spots, by copies of a given geometric figure or of a finite num-ber of given geometric figures. The word comes from tessala, theLatin word for a small tile.

theoretical probability (of an event)A numerical measure of the likelihood that the event will occur,based on a probability model. If, as can happen with dice or coins,an experiment has only a finite number n of possible outcomes, allequally likely, and in k of these the event occurs, then the theoreticalprobability of the event is k/n.

tolerance (interval)The set of numbers that are considered acceptable as the dimensionof an item. For Example, a manufacturer’s tolerance interval for theweight of a “400 gram’’ box of cereal might be from 395 grams to 420grams.

transformationA change in the position of an object and/or a change in size andrelated changes. Also a change in the form of a mathematicalexpression.

translationPlease see slide.

transversalA line that intersects two or more lines at different points.

trapezoidA quadrilateral that has two parallel sides. Some definitions requirethat the remaining two sides not be parallel.

tree diagramA pictorial way of representing the outcomes of an experiment thatinvolves more than one step.


E-40

trigonometryThe branch of mathematics concerned with the properties andapplications of the trigonometric functions, in particular their use in“solving’’ triangles, in surveying, in the study of periodicphenomena, and so on.

trinomialA polynomial that has three terms. For example:

ax2 � bx � c

turnPlease see rotation.

UunbiasedA sampling procedure for estimating a population parameter (likethe proportion of BC teenagers who smoke) is unbiased if, onaverage, it should yield the correct value. At a more informal level, apolling procedure is unbiased if proper randomization proceduresare used to select the sample, the wording of the questions is neu-tral, and so on.

unit circleA circle of radius 1.

unit vectorA vector of length 1.

VvariableA mathematical entity that can stand for any of the members of agiven set.

E-41


varianceSample variance is a measure of the variability of a sample, based onthe sum of the squared deviations of the data values about themean. Population variance is a theoretical measure of the variability ofa population.

vectorA directed line segment (arrow) used to describe a quantity that hasdirection as well as magnitude.

vertex (pl. vertices)In a polygon, a point of intersection of two sides. In a polyhedron, avertex of a face.

vertically opposite anglesOpposite (and equal) angles resulting from the intersection of twolines.

Wwhole numberOne of the counting numbers 0, 1, 2, 3, 4, and so on; a non-negativeinteger.

Xx-intercept(s)The point(s) at which a curve meets the x-axis (horizontal axis).

Yy-intercept(s)The point(s) at which a curve meets the y-axis (vertical axis).


E-42

Zz-scoreIf x is the numerical value of one observation in a sample, the z-scoreof x is (x – x )/s, where x is the sample mean and s is the samplestandard deviation. The z-score measures how far x is from themean.

zero (root) of f(x)Any number a such that f(a) = 0.

APPENDIX FMathematics 8

F-3

APPENDIX F: ILLUSTRATIVE EXAMPLES • Mathematics 8

Prescribed Learning Outcomes Illustrative ExamplesThis appendix contains a series of examples designed to help teachers understand theprescribed learning outcomes and suggested extensions for Mathematics 8 and 9.

The prescribed learning outcomes have been paired with examples that illustrate the types ofactivities an average student should be able to complete for each course.

• With the exception of the Problem Solving organizer, all prescribed learning outcomes for earliercourses are listed.

• In some cases, an illustrative example may encompass more than one learning outcome;conversely, some outcomes have more than one example.

Please note that:

• these illustrative examples are not intended to be used for assessment of student performance

• the suggested extensions are not provincial curriculum – they provide extra topics andenrichment

Examples of multi-strand and interdisciplinary problems that most students should be able to solveare indicated with an asterisk (*).

F-5


Prescribed Learning Outcomes Illustrative Examples

PROBLEM SOLVING

In order to prepare students to use a variety of methods to solve real-life, practical, technical, and theoretical problems,it is expected that students will:

• solve problems that involve a specificcontent area (e.g., geometry, algebra,statistics, probability)

• solve problems that involve more thanone content area within mathematics


• analyse problems and identify thesignificant elements

• develop specific skills in selecting andusing an appropriate problem-solvingstrategy or combination of strategieschosen from, but not restricted to, thefollowing:

- guess and check- identify patterns and use a systematic

list- make and use a drawing or model- eliminate possibilities- work backward- simplify the original problem- select and use appropriate technology

to assist in problem solving- analyse keywords



• clearly and logically communicate asolution to a problem and the processused to solve it

• evaluate the efficiency of the processesused

• use appropriate technology to assist inproblem solving

Examples of multi-strand and interdisciplinary problemsthat most students should be able to solve are indicatedwith an asterisk (*).

F-6



NUMBER (Number Concepts)

In order to prepare students to demonstrate a number sense for rational numbers, including common fractions, integers,and whole numbers, it is expected that students will:

• define, identify, compare, and order anyrational numbers

� Show how you could represent and compare the numbers0.34 and 0.43, using base-10 blocks when:- the cube represents 1- the flat represents 1

Explain why -0.43 is less than -0.34.

� Explain where you would place each of the followingnumbers on the number line.

+1.75, -1.2, - –, + –23

65

• • • • • • •–3 –2 –1 0 +1 +2 +3

• express two-term ratios in equivalentforms

� In Elisapee’s class there are six girls and five boys whileBert’s class has 15 boys. How many girls would there be inBert’s class if the ratio of boys to girls is equivalent in thetwo classes?

• represent and apply fractional percentsand percents greater than 100 in fractionor decimal form and vice versa

� John made a chart to illustrate percents. He started with alarge 10 x 10 grid. He folded it in half and shaded half thesquares. He counted the shaded squares and wrote50100—– = 50%. He then folded the unshaded part in half andshaded the new half a different colour. He counted shadedparts and wrote 25

100—– = 25%. He did this three more times.

Use a large grid to copy and complete John’s work. Use theresults from your work to show 150%, 212%, and 1031

8–%.

� How could you use 10 x 10 grid sheets to represent:

3313–%

16623–%

210%

� The mark-up on Playstation 2 is 150%. Find the actual pricesif the stores pay:- $20.00 for controllers- $24.50 for multi-tap- $47.50 for video games

F-7



• represent square roots concretely,pictorially, and symbolically



• distinguish between a square root and itsdecimal approximation as it appears on acalculator

5}25 = 5

4} 16 = 4

� Shamin used small square tiles to form larger squares as away of finding the square roots of 25 and 16.

Use Shamin’s method to show the square roots of 36, 49, 64,and 100.

� Greg calculates the area of a circle using p = 3.14, whileMary uses the button on her calculator. Show that theiranswers are different and explain why. If the radius of thecircle is 140 cm, how far apart are their answers?

� Hannah used square tiles and grid paper to show that thesquare root of 42 is not a whole number. She made thelargest square possible, using 36 of the 42 tiles, and traced a6 x 6 square on grid paper. She then cut a strip of sixsquares to represent the six leftover tiles. She cut it andplaced it on the grid, as shown below.

Estimate 42 from the diagram.

Compare your estimate with a calculator result.

Use Hannah’s method to estimate the square roots of 56 and130, and explain your solution.

F-8





• express rates in equivalent forms � If Harvey walks at an average speed of 6 km/hr, how farwill he walk in 2.5 hours? How long will it take him to walk21 km?

� Jelly powder was on sale for three packages for $1.68. Makea chart to show the cost of six packages, nine packages, 12packages.

• represent any number in scientificnotation

� The number of visitors to Banff National Park in 1989 was4.032396 x 106, and the number of visitors to KootenayNational Park was 1 555 607. Which park had more visitors?How many more? Give your answer in standard notation.

� If 5.03 x 10-5 was incorrectly written as 5.03 x 105, how manytimes larger is this?

� The diameter of a human hair is 0.00007 m. Write thisnumber in scientific notation, using metres as the unit ofmeasure. What is the diameter in centimetres?


• express three-term ratios in equivalentforms

� A recipe calls for 250 mL of sugar, 500 mL of oatmeal, and750 mL of flour. Write the amounts of ingredients as a ratio.Write another equivalent ratio. Write the equivalent ratio fordoubled and tripled recipes.

• demonstrate and explain the meaning ofa negative exponent, using patterns (limitto base 10)

� Look for a pattern in both the top and bottom numbers.Continue the patterns.

100 000, 10 000, 1000, 100, 10, ___, ___, ___, ___

105, 104, 103, 102, 101, ___, ___, ___, ___

What is the connection between the two patterns? Make arule to explain.

F-9



� Doris has 113– large pizzas left over from a party. At lunch the

next day, her family ate 34– of the leftovers. Doris said they ateone whole pizza in total. Use fraction circles to represent thepizzas to decide if Doris is correct. Explain why or why not.

� Select appropriate Cuisenaire rods or fraction tiles to explainwhy 41– x 14– = 1.

Draw a diagram to show what you did.



• demonstrate concretely, pictorially, andsymbolically that the product ofreciprocals is equal to 1

F-10



NUMBER (Number Operations)

In order to prepare students to apply arithmetic operations on rational numbers to solve problems apply the concepts ofrate, ratio, percent, and proportion to solve problems in meaningful contexts, it is expected that students will:

• add, subtract, multiply, and dividefractions concretely, pictorially, andsymbolically

� Eric ordered several large pizzas for a party, and 112–

pepperoni pizzas and 23– of a pineapple pizza were not eaten.Was there more than one large pizza left over? Explain howyou can estimate the answer. Add 11

2– + 23–, using a pencil-and-paper method, and use paper circles to explain your methodand your answer.

� *Mr. Blair’s gas tank was 78– full when he left home. He used 34–

of a tank of gas on his errands. What fraction of a tank of gaswas left? Explain how you know the answer is less than 14– bysubtracting 78– – 34–, using a pencil-and-paper method, and usefraction strips to explain your method and your answer

� Lisa had 34– of a large candy bar. She gave 13– of what she hadto Shannon. Explain how you know that Shannon got lessthan 13– of a whole bar by:

- multiplying 13– x 34–, using a pencil-and-paper method- explaining your method and your answer by folding a

piece of paper that represents a whole candy bar

� Miko has 212– m of blue cloth. How many pieces 14– m long can

she cut from her piece? Estimate the answer and explain thesolution by:

- dividing 212– � 14–, using a pencil-and-paper method

- using Cuisenaire rods or fraction tiles to explain yourmethod and your answer

� *In the community hall, 14– of the people present are men, 13–

are women and the rest are children. There are 840 people inthe hall. How many children are there?

F-11



• estimate, compute, and verify the sum,difference, product, and quotient ofrational numbers

� *Pam recorded the daily high temperatures for one weekand found the average high temperature for the week to be–4.1°C. If the temperatures from Sunday to Friday were+11.7°C, –17.4°C, 0°C, –23.6°C, –13.9°C, and +9.1°C, whatwas the temperature on Saturday? Explain how you wouldestimate the answer. Calculate the answer and compare itwith your estimate.

� Réné earns $80. He saves 14– of it for school and 12– for a CDplayer, and he owes his dad $5. How much is left?

� The stock prices on a tech stock varied by the followingamount last week:

1/2, -3/4, -1/4, 2/3, -5/8

• estimate, compute (using a calculator), andverify approximate square roots of wholenumbers

� The square root of 30 is closest to what whole number?

� *Steve knew the square root of 30 must be between 5 and 6since 30 is between 25 and 36. He estimated it to be 5.6. Hethen used his calculator to find (5.6)2 = 31.36. He then tried(5.5)2 = 30.25 and (5.4)2 = 29.16. He said 5.5 was the closest.Explain. Use Steve’s method to find the square root of 40 tothe nearest tenth and the square root of 20.5 to the nearesthundredth.

� A domino is two squares side by side. If the area of the topis 882 mm2, what are the dimensions of the domino?



F-12



• use concepts of rate, ratio, proportion,and percent to solve problems inmeaningful contexts

length of middle finger

length of foot

your choice

Body Part Actual Length Length in Lilliput

• derive and apply unit rates

� *Have you read or heard of the book by Jonathan Swiftcalled Gulliver’s Travels? Gulliver, a ship captain, suffers ashipwreck and finds himself in the land of Lilliput. Here hefinds that the heights of the people, plants, and animals arein a 1:12 ratio to the heights of the people, plants, andanimals in his world. Use the measuring tape to measureyourself. Then complete this chart.

Each day the Emperor of Lilliput gave Gulliver the food anddrink necessary to feed about 1728 Lilliputians. How did theEmperor ’s mathematicians arrive at this number? Explainwhy this should be about the right amount.

� Which is the better buy: 1.2 L orange juice for $2.50 or 0.75 Lorange juice for $1.40?

� Walter and Pat have the same ratio of cats to dogs in theirkennels. Walter has three cats for every five dogs. Pat has48 cats and dogs altogether. How many of Pat’s animalsare dogs?

� A class of 25 students has an average mark of 65% on awritten test; a second class, of 21 students, has an average of60%; and a third class, of 23, has an average of 67%. Find theaverage mark for all of the students.

� Jerry bought 3.5 kg of apples for $5.25. What was the cost of1 kg?

� Toothpaste is advertised as 75¢ for a 50 mL tube. A 75 mLtube is priced at $1.09. Which is the better buy? Why?



F-13





• express rates and ratios in equivalentforms

� *Gas usage rate is expressed as the number of litres of gasused per 100 km. On a 225 km trip, Nadia used 20.5 L of gas.Express her usage in terms of the above rate. Why do youthink this type of rate is used?

� In Canada, there are one million curlers registered in 1200clubs. In Scotland, there are 50 000 curlers in 52 clubs, and inSweden there are 9000 curlers in 36 clubs. Write a ratio foreach to compare the number of curlers to the number ofclubs, and arrange these in order of size from least togreatest.


• calculate combined percents in a variety ofmeaningful contexts

� Suits selling regularly for $185.00 were marked down by25%. To further improve sales, the discount price wasreduced by another 15%. What was the final selling price?What was the total percent of discount on the original price?

� A store had a NO GST sale. Darcy purchased a skirt pricedat $39.99. When she paid for it, the clerk first subtracted 7%to get a new price and then added 7% GST to this new price.Is this a fair way to calculate the price? Why would a storeuse this practice?

• estimate, compute (using a calculator),and verify approximate square roots ofdecimals

� Estimate and then calculate the square root of 1.44 and12.25.

� Find the side length of a square with an area of 18.75 m2 tothe nearest tenth.

F-14



PATTERNS AND RELATIONS (Patterns)

In order to prepare students to use patterns, variables and expressions, and graphs to solve problems, it is expected thatstudents will:

• substitute numbers for variables inexpressions and graph and analyse therelation

� Brock started making a chart to show the value of y when xchanges for the expression y = x + 2

xy

02

13

2…4…

Complete Brock’s chart and make a graph to show therelationship. Analyse the graph.

• translate between an oral or writtenexpression and an equivalent algebraicexpression

� Write an algebraic expression for the following:

When a number is doubled and increased by seven, theresult is 20.

� Describe the following algebraic equation in words.

x2– + 5 = 2

� Carl has 30 coins, all dimes and quarters. Write anexpression to represent:- the number of dimes if he has x quarters- the total value of the coins

F-15



• generalize a pattern from a problem-solving context


� *Long-Foi made the following pictures with circles andtriangles.

He started making a chart to show the number of circles andtriangles in each picture. Complete Long-Foi’s chart andlook for a pattern.

Write a mathematics sentence to show the relationshipbetween the number of circles and the number of triangles.

- Make concrete models or pictures to verify your answers.- How many circles would you need in a picture with 12

triangles?- How can you find and verify the answer?- Substitute numbers in your sentence for each picture.

Picture Number of Circles Number of Triangles

2 5 2

1 3 1

3

4

1 2 3 4


In order to prepare students to use patterns, variables and expressions, and graphs to solve problems, it is expected thatstudents will:

F-16



PATTERNS AND RELATIONS (Variables and Equations)

In order to prepare students to solve and verify one-step linear equations with rational number solutions, it is expected thatstudents will:

• illustrate the solution process for a one-step, single-variable, first-degreeequation, using concrete materials ordiagrams

� Kassidy bought five CDs at the same price and paid a totalof $84.45. How much did each CD cost? Write an equationand show how to solve it algebraically. Verify your answerby substituting it in your equation or by using algebra tiles.

• solve and verify one-step, first-degreeequations of the form:

x + a = b

ax = b

x—a

= b

where a and b are integers

� Solve:

x + 7 = 10

4x = 20

a—2

= 2.75

x – 2.1 = 4.7

� *Maria had a length of fabric to make banners. She dividedthe fabric into six equal pieces, and each piece was 2.75 mlong. What was the length of the fabric? Write an equationand show how to solve it algebraically. Verify your answerby substituting it in your equation or by using strips of gridpaper.

• solve problems involving one-step, first-degree equations

� Use the following information to construct a word problem.

It is 300 km from Regina to Gull Lake. About halfwaybetween the two locations is Chaplin. Deleho drives her carthe speed limit on Highway 1. Alain drives his convertible10 km slower than Deleho.

How much longer will it take Alain to make the trip thanDeleho?

F-17




• illustrate the solution process for a two-step, single-variable, first-degree equation,using concrete materials or diagrams

• solve and verify two-step, first-degreeequations of the form:

ax + b = c

x—a

+ b = c

where a, b, and c are integers.

• solve problems involving two-step, first-degree equations


In order to prepare students to solve and verify one-step linear equations with rational number solutions, it is expected thatstudents will:

� Joe had five sports cards. He bought three packs with thesame number of cards in each pack. If he now has 35 cardsin all, how many were in each pack? Write an equation andshow how to solve it algebraically. Verify your answer bysubstituting it in your equation or by using algebra tiles.

� Hans gave Vera half his marbles. She lost seven of themarbles Hans gave her and had 23 left. How many marblesdid Hans have to start with? Write an equation and showhow to solve it algebraically. Verify your answer bysubstituting it in your equation or by using counters.

� Kiotaka made 76 sandwiches for a party. If 29 were left over,how many were eaten? Write an equation and show how tosolve it algebraically. Verify your answer by substituting itin your equation or by using base-10 blocks.

F-18



SHAPE AND SPACE (Measurement)

In order to prepare students to apply indirect measurement procedures to solve problems generalize measurementpatterns and procedures, and solve problems involving area and perimeter, it is expected that students will:

• use the Pythagorean relationship tocalculate the measure of the third side ofa right triangle, given the other two sidesin 2-D applications

� *Tara is investigating the relationship among the three sidesof a right triangle. She drew a right triangle in the middle ofa sheet of paper and then constructed a square on each sideof the triangle. Then she tried to cut the two smaller squaresand fit them on the largest square. Try Tara’s investigation,using right triangles with different shapes. Write a briefparagraph stating your findings.

� Jamie wants to walk from one corner of the rectangularplayground to the opposite corner. The playground is 30 mby 50 m. What is the length of the shortest route he can take?Explain.

� A 5.0 m ladder leans against a wall and is planted on theground 4.2 m out from the base of the wall. How far up thewall does the ladder reach?

• describe patterns and generalize therelationships by determining the areasand perimeters of quadrilaterals and theareas and circumferences of circles

� The dimensions of five decorative gardens are given below.Which garden has the greatest area?- square with sides 10.2 m- rectangle with length 15 m and width 6.9 m- parallelogram with base 14.6 m and height 7.2 m- trapezoid with bases 18.1 m and 10.4 m and height 7.1 m- a circle with radius of 3.7 m

� Create a lake and island board by using the followingdirections:- a rectangular island A with an area of about 100 cm2

- a triangular island B with an area of about 18 cm2

- an irregular-shaped island C with an area of about 50 cm2

- a circular-shaped island D with an area of about 25 cm2

� *You want to paint one wall of your room. The wall is 7.0 mlong and 2.4 m high. It takes one small can of paint to cover9 m2, and the paint sells for $3.99 a can.- What would it cost you if you purchased only paint?- What else do you need to think of?- Make a plan for your trip to the store for supplies for this

painting job.

F-19



3

2.5

1.5

3

4

3.5

4.2

6.5

12

8.75

3

2.5

4

3.5

12

8.75

Base of Parall-elogram

Height of Parall-elogram

Area of Parall-elogram

Base of Rect-angle

Height of Rect-angle

Area of Rect-angle

� Melodie said that to find the perimeter of a triangle, youonly have to measure one side and multiply by 3. Do youagree? Cut straws into several different lengths and make asmany different triangles as you can. Use these strawtriangles to explain your answer. Make a rule to find theperimeter of a triangle.

� Draw a circle (radius 5 cm). Fold it in half four times tomake 16 sectors. Cut out the sectors. Place the sectors in aline, alternating the bases to form a parallelogram. (Seediagram.)

Show that the height is the radius of the circle and the baseis half the circumference. Use this to find a rule for the areaof a circle.

� Aaron sketched some parallelograms on grid paper and cutthem out. Then he cut a piece off one end of eachparallelogram and fit it onto the other side to form arectangle. He made this chart:

Finish Aaron’s chart and look for a pattern. Test yourpattern. Make a rule to find the area of a parallelogram.What other information should Aaron include on his chartto identify a pattern for finding the perimeter of aparallelogram?



F-20



� Yolande drew some triangles in different sizes and shapes.She then cut out two of each triangle. She fit each pair oftriangles together to make a parallelogram. Try Yolande’sinvestigation. Justify that each shape you make is aparallelogram. How can you use this investigation to make arule for finding the area of a triangle? Do the same thingwith trapezoids.

� How many sides of a trapezoid must you measure to findthe perimeter? Explain your answer.

� The perimeter of the square LMNP is 60 cm. Find the:- diameter of the circle- circumference of the circle- area of the circle- area of the shaded region

L P

M N

� Collect some cardboard cylinders that have lids. Cut thecylinders to form nets.- How many faces does each have?- What shape are the faces?- Are any of the faces identical?- Could you find the area of each face?

Use the data you collected to make a rule for finding thesurface area of a cylinder. Use your rule to find the surfaceareas of your cylinders.

� Kelly designed a juice can that holds 600 cm3. The can is 10cm tall. What is the radius of the can?



F-21



� Hugh had a small juice can and some centicubes. He firstestimated how many cubes would fit in the can. Next, hefilled the can with cubes, dumped them out, and countedthem. Is the volume (in cubic centimetres) he gets from thisexperiment larger or smaller than the actual volume?Explain. Hugh decided to find a way to get a more accurateanswer. He traced the base of the juice can on some cm2 gridpaper and counted the number of squares inside the circle.What will this tell him? What else does Hugh need to do tofind the volume of the cylinder? Make a rule for finding thevolume of any cylinder. Test your rule with anothercylinder.

� The areas of the faces of a rectangular box are given in cm2.What is the volume of the box?

80 cm2

60 cm2

48 cm2



F-22



• estimate and calculate the area ofcomposite figures

� First estimate, and then find, the area of the figures below.



5 m

B = 2.7 cmH = 3.1 cm

L = 6.2 cmW = 1/8 cm

R = 12 cm

5 m

2 m 2 m

4 m

7 m

2 m

B = 2 cmH = 2.5 cm

D = 2.8 cm8 cm

8 cm

� How much cardboard does it take to make a cereal box? Cutsome cereal boxes to form nets.

- How many faces does each have?- What shape are the faces?- Are any of the faces the same size?- How could you find the area of each face?

Use the data you collected to make a rule for finding thesurface area of a right prism. Use your rule to find which ofyour cereal boxes has the greatest surface area.

• estimate, measure, and calculate thesurface areas and volume of any rightprism or cylinder


F-23



• estimate, measure, and calculate thesurface area of composite 3-D objects

• estimate, measure, and calculate thevolume of composite 3-D objects

� Thirty unit cubes are stacked in square layers to form atower, as shown below.

Determine the total surface area of the tower of cubes.

Suppose that the number of cubes and height of the towerare increased according to this pattern. Complete the tablebelow for several specific towers.

5 cubes by 5 cubes

8 cubes by 8 cubes

10 cubes by 10 cubes

Bottom Layerof Tower

Total Number of Cubes

Surface Area of Tower

� First estimate, and then find, the volume and the surface areaof the figure below. The figure is a 3 cm X 4 cm X 5 cm solidblock of wood with a 1 cm X 0.5 cm X 4 cm hole cut in it.

0.5 cm

5 cm

4 cm

3 cm1 cm



F-24



SHAPE AND SPACE (3-D Objects and 2-D Shapes)

In order to prepare students to link angle measures and the properties of parallel lines to the classification and propertiesof quadrilaterals, it is expected that students will:

• identify, investigate, and classifyquadrilaterals, regular polygons, andcircles according to their properties

� Investigate and describe the properties of intersections ofdiagonals of any quadrilaterals.

Where possible, use computer software.

� Identify, compare, and debate the merits of shape in presentand past architectural construction methods and decorationfeatures (e.g., golden rectangle).

� Given a variety of cutout polygons and circles, find severalways to sort the figures and identify the characteristics ofthe subsets for each different way (the polygons should beregular and irregular with different numbers of sides andthe quadrilaterals should include irregular shapes,trapezoids, parallelograms, rectangles, rhombuses, squares,and kites).

� Take all the quadrilaterals from the above set. Sort them indifferent ways (e.g., number of parallel sides, number ofright angles, number of congruent sides, number ofcongruent angles). Use sets of nesting boxes to show howthe different kinds of quadrilaterals are related.

� Draw five different rectangles. Devise a numerical measureof squareness that would allow you to rank your rectanglesfrom the one most like a square to the one least like a square.Justify your choice.

� Find two different nets for a cylinder.

� Use toothpicks and molding clay to build prisms andpyramids with various polygons for bases.

• build 3-D objects from a variety ofrepresentations (e.g., nets, skeletons)

� Raymond cut this net for a cube from grid paper. Howmany different nets can you cut that make cubes?


F-25



SHAPE AND SPACE (Transformations)

In order to prepare students to analyse design problems and architectural drawings using the properties of scaling andproportion, it is expected that students will:

• represent, analyse, and describeenlargements and reductions

� If the following figure is drawn on 1 cm grid paper, draw itsenlargement on 2 cm grid paper.

00

2

4

6

8

10

2 4 6 8 10 12 14

� The figure ABC is said to be reduced by 12– to form the imageA’B’C’. Use a series of measurements to show whether or notthis is true.

A

C

B

P

A'

B'

C'

� *Describe some everyday situations in which 2-D and 3-Denlargements and reductions are necessary or useful (e.g.,photocopies, photographs, scale models, statues). Explainhow the enlargement or reduction is the same and how it isdifferent from the original figure or object (e.g., size, shape,proportion).

� Darren had some small unit cubes. He used them to buildlarger cubes. What are the three smallest cubes Darren couldbuild? How much larger is each one than the original unitcube? Explain, using cubes or a diagram.

� Sandra was making squares with toothpicks for sides. Whatare the three smallest squares she can make? How muchlarger is each one than the square with one toothpick oneach side? Explain your answer, using toothpicks.

F-26





• draw and interpret scale diagrams

• represent, analyse, and describe colouringproblems

� *Make a scale diagram of your bedroom or your classroom.In what units will you measure the room? What ratio willyou use for your scale diagram?

� Working in pairs, make a scale drawing of a rectangularsheet of ice (at a curling rink), where the rectangle is 44.5 mby 4.3 m and the scale is 1 cm = 3000 m.

� The four-colour map theorem says that any flat map, nomatter how many separate regions it has, can be colouredusing only four colours, so that no bordering regions are thesame colour. Cover a page with a design like the one belowand test the theorem. Also test the theorem with a real map,such as that of Canada, the United States, or Europe.


F-27



• describe, analyse, and solve networkproblems (e.g., bus routes, a telephoneexchange)

� On a map of Canada, mark the cities Whitehorse, Victoria,Edmonton, Yellowknife, Regina, and Winnipeg. Devise anairplane network so that you can get from any one of thesecities to any other one of them by changing planes, at most,once. Each route can have no more than two stops. Youwant the least number of routes.

� A power company must connect Grande Prairie, FortMcMurray, Edmonton, Red Deer, Calgary, Lethbridge, andMedicine Hat into a power grid. The design must use theleast amount of wire possible. The following table shows thekilometres between major cities in Alberta.

Your task is to design a route network and draw the routeson the Alberta map shown on the next page. State theminimum amount of wire needed.

Grande Prairie

Fort McMurray

Edmonton

Red Deer

Calgary

Lethbridge

Medicine Hat

0

720

460

620

760

985

1010

720

0

445

605

745

970

990

460

445

0

160

300

525

500

620

605

160

0

140

375

420

760

745

300

140

0

225

280

985

970

525

375

225

0

170

1010

990

500

420

280

170

0

From/To GrandePraire

FortMcMurray

Edmonton RedDeer

Calgary Lethbridge MedicineHat



F-28



Map of Alberta with cities marked:

Grande Prairie

FortMcMurray

Edmonton

Lethbridge

MedicineHat

Red Deer

Calgary

� A network consists of vertices (points) and arcs that jointhem. A vertex is called even or odd, depending on whetheror not an even or odd number of arcs are connected to it.Kwigah tried to trace each of the networks below withoutlifting his pencil or retracing any arcs. He made a chart of hisfindings. Trace the network and fill in Kwigah’s chart. Canyou find a pattern?



F-29



Number ofEven Vertices

Number ofOdd Vertices

Can the figurebe traced?

Based on the pattern you find, draw a network that can betraced and one that cannot be traced.

� Research the famous problem of the Bridges of Koenigsberg.Make a drawing and describe the problem in words.



F-30



STATISTICS AND PROBABILITY (Data Analysis)

In order to prepare students to develop and implement a plan for the collection, display, and analysis of data, usingtechnology as requiredand evaluate and use measures of central tendency and variability, it is expected that students will:

• formulate questions for investigation,using existing data

� Find some data collected and presented in a local newspaperthat are related to a current civic, regional, or health issue.- Do the data seem to support the conclusions the

newspaper makes?- Are the data presented in a fair, clear, and appropriate

manner?- What questions about the issue are not addressed?

• select, defend, and use appropriatemethods of collecting data:

- designing and using surveys- research, using electronic media

� *How much household garbage is produced in our homes?In the average home in Canada? Design a questionnaire toinvestigate this problem. Justify your questions. Explainhow you will carry out this survey. Could you collect datavia e-mail or the Internet? How can you use a computer torecord, organize, and display your data?

� Survey students on their weekly allowance, put the data in atable, and graph it as a histogram.


� Play a memory game with your class. Write 16 words on theboard or overhead projector. Let everyone look at them fortwo minutes. When the time is up, each person writes asmany words as he or she remembers. Collect the data(number of words remembered). Find the median andquartile scores, and make a box-and-whisker plot. Why isthis method of displaying variability useful?

� Using published data, find the life expectancy for females of20 different countries. Graph the results using a box-and-whisker plot.

• determine and use the most appropriatemeasure of central tendency in a givencontext

� Explain why each of the following people might select themean, median, or mode in a set of data.- A storeowner deciding what sizes of shoes to order- Someone moving to a new city and looking at housing

costs- Reporting the average score on a test

F-31



• describe the variability of data sets, usingsuch techniques as range and box-and-whisker plots

� Explain why each of the following people might select themean, median, or mode in a set of data.- A storeowner deciding what sizes of shoes to order.- Someone moving to a new city and looking at housing

costs.- Reporting the average score on a test.

• construct sets of data given measures ofcentral tendency and variability

� Janice is sales manager in a department store. She mustmaintain average (mean) daily sales of at least $8,500. Salesfor the first four days of the week are $7,530, $8,475, $6,550,and $7155. The store is not open on Sunday. What sales willJanice need to make on Friday and Saturday to come in overthe target? Discuss whether or not it is likely that Janice willachieve her target.

� The mean score on a test was 5. The median was also 5, butthe mode was 6. The 13 scores ranged from 2 to 10.Construct a set of scores that has these measures. Representeach score with centicubes or unifix to show the measuresconcretely. Another score of 15 is added to the data. Howwill this affect each of the measures?

• determine the effect on the mean, median,and/or mode when:

- a constant is added or subtracted fromeach value

- each value is multiplied or divided bythe same constant

- a significantly different value isincluded

� The number of passengers in different buses was recorded.The mean was 46 and the median was 47. If 20 extrapassengers rode on each bus, what would the new meanand median be? If each passenger paid $1.25 to ride the bus,what would be the mean and median amount of moneycollected?


In order to prepare students to develop and implement a plan for the collection, display, and analysis of data, usingtechnology as requiredand evaluate and use measures of central tendency and variability, it is expected that students will:


F-32



STATISTICS AND PROBABILITY (Chance and Uncertainty)

In order to prepare students to compare theoretical and experimental probability of independent events, it is expected thatstudents will:

• use various data-collection techniques(including computers) to simulate andsolve probability problems

� Draw vertical lines on large chart paper exactly twotoothpick-lengths apart. Toss 100 toothpicks randomly.Record any toothpick that touches a line as a “hit.” Calculatethe ratio between the number of tosses and the number ofhits. Compare results. As more trials are attempted, theoutcome will converge on p. Experiment with differingspaces between the lines as well as with different sticks.

� A soft-drink company placed a lucky liner in the caps of halftheir one-litre bottles. Derek said he bought five bottles andthey all had lucky liners. How could you use computer-generated random numbers to simulate the situation andfind the probability of getting the five lucky liners?

� What is the probability of having exactly two boys in afamily of five children? Design a simulation, using coins toanswer the question.

• recognize that if n events are equallylikely, the probability of any one of themoccurring is 1

—n

� If you toss one standard die, what are the possible events?Are they equally likely? Explain. Write the probability ofrolling a 4. If you did the same experiment with a 12-sideddie, what would be the probability of rolling a 4?

� If you draw a card from a deck, what suit could it be? Are allsuits equally likely? What is the probability of drawing aheart?

� Bruce is writing a multiple-choice test. There are fourpossible answers to a question. What is the probability ofgetting the right answer if he guesses? How does theprobability change if Bruce can eliminate two wronganswers?

F-33



• determine the probability of twoindependent events where the combinedsample space has 52 or fewer elements

� *Using 30 people from a representative sample, Annadiscovered that 12 people own a VCR and TV. If Anna’s cityhas a population of 60,000, how many people own a VCRand TV? How many do not? Would you make money if youopened a video store? What other information would youneed in order to make your final decision?

• predict population characteristics fromsample data

� A number spun on Spinner A is multiplied by a numberspun on Spinner B. Calculate the expected probability of theproduct being:- 5 or less- even- a multiple of 5

Draw a diagram or table to help explain your reasoning.

� Players in a dice game threw two dice each. A player wins ifthe total is exactly 11. What is the probability of winning? Ifyou toss two dice and find the sum, what are the possiblesums? Are they equally likely? Explain. Give an example oftwo sums that are equally likely. What sum has the sameprobability as 10?

1

4

2

3

Spinner A Spinner B

1

2

3

4

5


In order to prepare students to compare theoretical and experimental probability of independent events, it is expected thatstudents will:

APPENDIX FMathematics 9

F-37


Prescribed Learning Outcomes Illustrative ExamplesThis appendix contains a series of examples designed to help teachers understand theprescribed learning outcomes and suggested extensions for Mathematics 8 and 9.

The prescribed learning outcomes have been paired with examples that illustrate the types ofactivities an average student should be able to complete for each course.

• With the exception of the Problem Solving organizer, all prescribed learning outcomes for earliercourses are listed.

• In some cases, an illustrative example may encompass more than one learning outcome;conversely, some outcomes have more than one example.

Please note that:

• these illustrative examples are not intended to be used for assessment of student performance

• the suggested extensions are not provincial curriculum – they provide extra topics andenrichment

Examples of multi-strand and interdisciplinary problems that most students should be able to solveare indicated with an asterisk (*).

F-39



PROBLEM SOLVING

In order to prepare students to use a variety of methods to solve real-life, practical, technical, and theoretical problems,it is expected that students will:

• solve problems that involve a specificcontent area (e.g., geometry, algebra,statistics, probability)

• solve problems that involve more thanone content area within mathematics


• analyse problems and identify thesignificant elements

• develop specific skills in selecting andusing an appropriate problem-solvingstrategy or combination of strategieschosen from, but not restricted to, thefollowing:

- guess and check- identify patterns and use a systematic

list- make and use a drawing or model- eliminate possibilities- work backward- simplify the original problem- select and use appropriate technology

to assist in problem solving- analyse keywords

• solve problems individually andco-operatively


• clearly and logically communicate asolution to a problem and the processused to solve it

• evaluate the efficiency of the processesused

• use appropriate technology to assist inproblem solving

Examples of multi-strand and interdisciplinary problemsthat most students should be able to solve are indicated withan asterisk (*).


F-40




In order to prepare students to develop a number sense of powers with integral exponents and variable and rationalbases, it is expected that students will:

• give examples of situations whereanswers would involve the positive(principal) square root or both positiveand negative square roots of a number

� What two values satisfy x2 = 16?

� *If you wanted to find the length of one side of a gardenwhose area is 25 m2, explain why you would use only thepositive square root of 25.

� A square has one corner at (0, 0) and an area of 36 squareunits. Find the possible coordinates of the other vertices.

(0, 0)

• illustrate power, base, coefficient, andexponent, using rational numbers orvariables as bases or coefficients

� What is the value of the coefficient in the expression -x4? x2

5–?

� Use cubes or draw diagrams to represent and explain thedifference between 23 and 32.

� When an object is falling, the relationship between distancetravelled and time is given as d = 12– t2. Identify the power,base, coefficient, and exponent of this formula. If theexponent were changed to 3 and the coefficient to 4, whatwould the new equation look like?

� The surface area of a sphere is given by the formula 4πr2, and

the volume is given by 43–πr3. Compare these with the

formulae for the volume and surface area of a cube of side r.Indicate those parts of the expressions that are coefficients,powers, bases, and exponents. What do the volumeformulae have in common? Is this usually true?

� Which is greater, 2-5 or 5-2? Explain your reasoning. Compareyour answer with your calculator answers.

F-41



• give examples of numbers that satisfy theconditions of natural, whole, integral, andrational numbers, and show that thesenumbers comprise the rational numbersystem

• describe, orally and in writing, whether ornot a number is rational

� Explain why 6 belongs to the natural, whole, integral, andrational numbers. Explain why -4 is a rational number butnot a whole number. Give an example of a number that is aninteger but not a whole number. Explain. Draw four boxesthat nest inside one another. Label each box as naturalnumbers, whole numbers, integers, or rational numbers toshow how the number systems are “nested.”

� The ratio of the circumference to the diameter of any circle isπ. Explain whether or not π is a rational number.

• explain and apply the exponent laws forpowers with integral exponents

xm • xn = xm+n

xm � xn = xm-n

(xm)n = xmn

(xy)m = xm ym

— = –—, y ≠ 0

x0 = 1, x ≠ 0

x -n = —, x ≠ 0

xy( )n xn

xn

yn

1

� Explain, orally and in written form, why 23•25 = 28. Giveother examples of multiplication of powers with the samebase. What is the pattern? Generalize to variable bases andexponents.

� Use the exponent laws and guess and test to find values for n.

� Match examples of exponent laws (1-8) with laws (A-H).

1) 23 � 24 = 27 A) xm • xn = xm+n

2) 30 = 1 B) xm

xn— = xm-n

3) 26 � 23 = 23 C) (xm)n = xmn

4)52

3( )— -5 = 32( )— 5 = 3

2-5— D) (xy)m = xm ym

5) (▲ � ❑ )2 = ▲2 � ❑ 2 E) xy— = xyn— ( )n n

6) (352)3 = 356 F) x1 = x

7) 63( )— 1 = 2 G) x0 = x1

8) 17( )— 2 = 1

75— 5

H) xy— = ( ) y

x— ( )-n n

n4 � n2 = 64 n-5 = —

n5 � n3 = 25

(n2)3 = 729

132





F-42





• determine the value of powers withintegral exponents, using the exponentlaws

� Explain how you could estimate the value of (2 x 3)3.Compare your answer with your calculator answer.

� If the price of a hamburger doubles every two years, whatwould it cost in 100 years? Find an alternative way ofsolving this, using exponents.

� Explore the values generated by 23, 22, 21, 20, 2-1, 2-2, and soon, using a calculator. What would the next number in thesequence be? What is the calculator doing to get this? Howdoes 23 compare with 2-3? What is the meaning of thenegative exponent? Use a similar pattern to explain thedifference between 43 and 4-3.

� Explain why some calculators give a different answer for(-2)4 and -24.

F-43




In order to preare students to use a scientific calculator or a computer to solve problems involving rational numbers, it isexpected that students will:

• document and explain the calculatorkeying sequences used to performcalculations involving rational numbers

� Do the following calculation with as few keystrokes aspossible.

The calculation to be done is 21.612.3 � (14.5 – 7.9)

, which givesthe answer 0.2660754.- Devise one way of obtaining this answer on your

calculator. Write down the keystrokes that you used,both digit and operation keys, and record the number ofkeystrokes.

- Now devise another method. Which method uses fewerkeystrokes? Again, write down the keystrokes that youused, both digit and operation keys, and record thenumber of keystrokes used.

- Explain each keying sequence, and explain why one ofthe sequences uses fewer keystrokes.

� Complete the following calculations with as fewkeystrokes as possible.

3.2 [2.1 + 3 (4.6) 4 6.9] – 15.1 = -2.02

21.6 = 0.2660754

= 9.90625

12.3 � (14.5 – 7.9)

54 – 32

43


� *A swimming pool is filled by means of three pipes. Thefirst pipe can fill the pool in eight hours, the second can fillit in 12 hours, and the third can fill it in 24 hours. When allthree pipes are in use at the same time, how long does ittake to fill the pool?

� Using each of the digits from 1 to 5 only once, write thelargest and smallest power possible.

� What are the last two digits of 11100? Explain how youarrived at your answer.

� A culture of 100 bacteria doubles every half hour. Theformula to determine the size of the population (P) aftertime is given by P = 100 (2n), where n is the number of halfhours. If the bacteria reach toxic levels when the populationis 1.0 x 105, how long does it take before this population isconsidered toxic?


F-44




In order to prepare students to use a scientific calculator or a computer to solve problems involving rational numbers, it isexpected that students will:

• evaluate exponential expressions withnumerical bases

� *The depth of a well in metres can be estimated by droppinga stone into it and timing the number of seconds it takes tohear the splash. Using the equation d = 4.9t2 – 0.015t,calculate the depth of the well if it takes 4.5 seconds to hearthe splash.

� A civil engineer uses a formula to determine how muchforce a square wooden pillar will take before it crushes. Thisis called the crushing load (L) and is determined by thefollowing formula.

L = 51T4

H2 , where L = crushing load (metric tonne), T =thickness (cm), and H = height (cm)

Find the crushing load for a pillar 10 cm thick and 2.5 mhigh.

� Evaluate �53

5246 � 4-2

(42 )2

• use the exponent laws to simplifyexpressions with variable bases

� By correctly applying the exponent laws to evaluateexpressions, replace any incorrect value with the correctvalues. Explain your changes.

21000

2500 = 22

37 • 38 = 915

(5m2n3)2 = 5m4n5

(23)2 = 46

-52 = 25

(-3)2 = 9

� Use the exponent laws to simplify 51x-4y6

17x2y-2 . Leave youranswer in the form axbyc, where a, b, and c are integers. Nowchange each expression so that it has only positive exponents.

• use a calculator to perform calculationsinvolving scientific notation and exponentlaws

� How long has it taken the light from a star at the edge of thegalaxy to reach us if we are near the centre and the galaxyhas a diameter of 760 000 000 000 000 000 000 kilometres?Light travels at approximately 300 000 km/s.


F-45




In order to prepare students to generalize, design, and justify mathematical procedures using appropriate patterns,models, and technology, it is expected that students will:

• model situations that can be representedby first-degree expressions

� Write an expression or equation to represent each situation.

The cost to rent a DVD player is a $25 deposit plus $10 foreach day. How much will it cost to rent a DVD player forfour days? For 10 days? For d days?

� Bruce bought some licorice. It cost $3.75 for the firstkilogram and $3.25 for each additional kilogram. Howmuch would he pay for 3 kg? 10 kg? m kg?

� Write an expression or equation to solve the followingproblem. A mail-order record club allows you to buy 10 CDsfor $1. You then must buy 10 more at $15 each over the next12 months. What is the cost per CD if you fulfil yourobligation?

� Write an expression to represent the following situation.Kim earns $12 per hour and pays 20% income tax. Calculatehis net income for a 40-hour week and for a 25-hour week.

� There is a relationship between mass and height of a person,such that the average mass of a person in kilograms can beestimated by taking three-quarters of his or her height incentimetres and subtracting 72. Does this work for yourmass and height? According to the equation, what shouldyour mass be? What is it? Compare your results with others.Does the rule work?

• write equivalent forms of algebraicexpressions, or equations, with integral

� Which of the following expressions is equivalent to 3x – 2 =4? Justify your choice.

3x = 6 -2 = 12x x = 2 x = -6

� Explain how -5x – 6 = -40, 5x + 6 = 40, and 15x + 18 = 120 arerelated.


F-46



• use logic to present mathematicalarguments in solving problems



� *This figure contains several “upright” triangles. Constructyour own definition of an upright triangle. Using yourdefinition, how many upright triangles are there in a similarfigure with 10 rows?

� Explain how you can use the laws of exponents and acalculator to order the following powers from largest tosmallest: 3666, 4555, 5444, 6333.

� The sum of the interior angles of each polygon is shown.

180° 360°540°

If the pattern were to continue, what would the sum of theangles be for an octagon? For a 15-sided polygon? If thenumber of sides is known, how can the sum of the angles befound? Write an expression to show this.

� In a barnyard there are only chickens and pigs. Emily counts100 legs and 40 heads. How many chickens and how manypigs are there?

� Rearrange these eight rows of three buttons each into 10rows of three buttons each.


F-47



• write equivalent forms of algebraicexpressions, or equations, with rationalcoefficients

� Which of the following expressions is equivalent to x + 32 ?

Justify your choice.

x

x + 3 � 2

2(x + 3)

232

+

� Explain how C = 2πr and r = C2π are related.

� Given that density is mass divided by volume, explain whyvolume is mass divided by density.




F-48




In order to prpeare students to evaluate, solve, and verify linear equations in one variable generalize arithmeticoperations from the set of rational numbers to the set of polynomials, it is expected that students will:

• illustrate the solutions process for a first-degree, single-variable equation, usingconcrete materials or diagrams

� The equation 4x = 4 + 3x has been modelled with algebratiles. Explain how you can use the tiles to justify an algebraicsolution process.

=

� Use algebra tiles to justify an algebraic solution to3x – 7 = -2x + 8

=

• solve and verify first-degree, single-variable equations of forms such as:ax = b + cx; a(x + b) = c; ax + b = cx + dwhere a, b, c, and d are integers, and useequations of this type to model and solveproblems

� A string measuring 50 cm in length is cut into three pieces.One piece is twice as long as the shortest piece, and the otherpiece is 10 cm longer than the shortest piece. Find the lengthof each piece of string.

� Dennis has $25 and can save $2.80 per day. Jeena has $18 andcan save $3.70 per day. Who will be the first to be able to buya $72 tennis racquet?

� Yutaka goes to the record store. Compact disks cost $14 forthe first one and $13 for each additional one. If Yutaka buysM compact disks and spends D dollars, write an equationthat represents the relationship between M and D.

� C represents the number of compact disks, and C + C + 4 +2C = 56. Using this information, write a problem.

� *The cost of installing a fence is calculated by C = 7L+ 15p +80, where L is the total length of the fence in metres, and p isthe number of posts required. You have budgeted $3,025,and the fence has to cover a perimeter of 250 metres. Howmany posts can you afford?

• identify constant terms, coefficients, andvariables in polynomial expressions

� What is the numerical coefficient of -6a4b?

� What is the constant term in the expression 4x – 3 = 2y?

F-49



• evaluate polynomial expressions, giventhe value(s) of the variable(s)

� Evaluate the following expressions for the numbers given.

x-3 + y3 when x = 2 and y = -2

2x + 6x2 – 7 when x = -1

� A ball is thrown in the air. Its height, h (in metres), is given by

h = -4.9t2 + 30t + 2, where t = time (in seconds).

What is the height of the ball as it is thrown (t = 0 seconds)?What does this answer mean?What is the height of the ball after 2 seconds (t = 2 seconds)?Estimate the time the ball will take to hit the ground.

� Explain how the algebra tiles given below can be used tojustify an algebraic process for simplifying:

(4x2 – 3x + 5) + (4x – 2).

+


(4x2 – 3x + 2) – (3 + x – 4x2)

—

� Simplify (3x2 – 2xy + 6y2) + (xy – 7y2).


In order to prepare students to evaluate, solve, and verify linear equations in one variable generalize arithmeticoperations from the set of rational numbers to the set of polynomials, it is expected that students will:


F-50





� Justin used algebra tiles and an area model to explain themultiplication 2x(3y). He set up the model by drawing aframe with dimensions 2x and 3y.

2x

3y

Show how he filled the area model in to get the product.

� Use an area model with algebra tiles to explain youralgebraic solution to the product (4x + 1)(x + 2).

� Natalka modelled the process of factoring x2 + 4x + 4 byusing algebra tiles and forming a square with them.

x + 2

x + 2

What are the factors of x2 + 4x + 4?Use Natalka’s method to factor x2 + 5x + 6.Use algebra tiles to factor x2 – x – 2.

� Illustrate geometrically the quotient of a trinomial and abinomial if a rectangle’s length is x + 7 and its area isx2 + 9x + 14.

x x2 7x

x 7

2 2x 14

� Find the product of -2x – 3 and 3x + 4.

F-51



• solve and verify first-degree, single-variable equations of forms such as:

a(bx + c) = d(ex + f);

a—x

= b where a, b, c, d, e, and f are rationalnumbers

� Explain why the area model with algebra tiles can justify theproduct:

2x(x – 2)

� Simplify the following expressions by combining like terms,identifying any common factors, and completing thefactorization:x2 + 7x + 103x2 + 15x + 186x2 – 3x +x2 – 18x + 75x2 – 11x + 3x2 + 32 – 29x

� Find the quotient: 12x3 – 16x2 + 8x4x

� Solve for x: 2(4x – 5) =3(-2x + 6)

� Explain the steps you would use to solve 12—x

= 6

algebraically.





F-52




In order to prpeare students to evaluate, solve, and verify linear equations in one variable generalize arithmeticoperations from the set of rational numbers to the set of polynomials, it is expected that students will:

• solve, algebraically, first-degreeinequalities in one variable, display thesolutions on a number line, and test thesolutions

� Lillian received 77%, 69%, 81%, and 76% on her mathematicstests. What mark does she need on her fifth test in order toachieve an arithmetic mean (average) of at least 80%?

� Solve the following inequalities and graph each solution ona number line.x – 5 < 12

-2x + 3 > 10

� Explain whether or not each of the following numbers{-3, +4, -7, +7} is a solution to the inequality 2x – 3 > 5.

• perform the operations of addition andsubtraction on polynomial expressions


(4x2 – 3x + 5) + (4x – 2).

+


(4x – 3x + 2) – (3 + x – 4x2)

—

F-53



• represent multiplication, division, andfactoring of monomials, binomials, andtrinomials of the form x2 + bx + c usingconcrete materials and diagrams

• find the product of two monomials, amonomial and a polynomial, and twobinomials

• determine equivalent forms of algebraicexpressions by identifying commonfactors and factoring of the form x2 + bx + c

• find the quotient when a polynomial isdivided by a monomial

� Justin used algebra tiles and an area model to explain themultiplication 2x(3y). He set up the model by drawing aframe with dimensions 2x and 3y.

2x

3y

Show how he filled the area model in to get the product.

� Use an area model with algebra tiles to explain youralgebraic solution to the product (4x + 1)(x + 2).

� Natalka modelled the process of factoring x2 + 4x + 4 byusing algebra tiles and forming a square with them.

x + 2

x + 2

What are the factors of x2 + 4x + 4?Use Natalka’s method to factor x2 + 5x + 6.Use algebra tiles to factor x2 – x – 2.

� Find the product of -2x – 3 and 3x + 4.

� Simplify the following expressions by combining like terms,identifying any common factors, and completing thefactorization.x2 + 7x + 103x2 + 15x + 186x2 – 3x + x2 – 18x + 75x2 – 11x + 3x2 + 32 – 29x

Find the quotient: 12x3 – 16x2 + 8x4x




F-54




In order to prepare students to use trigonometric ratios to solve problems involving right triangles, it is expected thatstudents will:

• explain the meaning of sine, cosine, andtangent ratios in right triangles

� The calculator shows the sine of 32° is equal to 0.5299.Thisimplies that for ∆ABC:

B

CA

c

a

b

32…

a = 0.5299 and c = 1.000a = 5299 and c = 10 000the length of a is 0.5299 of the length of cthe length of c is 1.887 of the length of a

Explain why each of these statements is true.

� Given the following diagram, complete the problems.

B

CA

c

a

b

Using angle A, identify:- the side opposite to A- the side adjacent to A- the hypotenuse

Write the ratio:- sin A- cos A- tan A

°

F-55



• demonstrate the use of trigonometricratios (sine, cosine, and tangent) in solvingright triangles

• calculate an unknown side or an unknownangle in a right triangle, using appropriatetechnology

� Jenna walked across a rectangular schoolyard from onecorner to the opposite corner. If the schoolyard is 40 m by60 m, at what angle, with respect to the longer side, didshe walk?

� A 10-metre ladder is leaning against a building. The anglebetween the ladder and the ground is 65°. How far is the topof the ladder from the ground?

� In the following triangle, if you wanted the length of a,which ratio would you use? Could you use another ratio?What is the length of b?

c = 10 cm

b

a

20°A

B

C




F-56



• model and then solve given problemsituations involving only one righttriangle

� *Explain how a golfer could use trigonometric ratios todetermine whether she could safely drive the ball from thetee by the shortest path over the pond to the green. Themaximum distance the golfer can hit is 220 m.

45°

Tee

Green

150

m

Pond

250 m

The total distance from the tee to the green along the “dogleg” is 400 m. The angle of the shortcut is approximately45∞∞. How far would the golfer need to hit the ball to reachthe green across the pond? If she must reach the green in oneshot to avoid a marsh, should she try the shortcut?

� A carpenter needed to replace a set of steps with a ramp forwheelchair access to a building. The vertical drop of thesteps was 1 m and the building codes require the slope ofthe ramp to be 1 m to 14 m. Determine the angle for the topof the supports. Also determine how much plywood thecarpenter would need if the original steps were 2 m wide.

A

1

14

� You work for a bottling company. You have been asked todesign a cost-effective way to box 24 cans. The requirementsare that the cans must be oriented vertically and the boxshould use a minimal amount of cardboard. If you are ableto determine the dimensions of a pop can, compare differentways of boxing that would involve the least amount ofcardboard to surround 24 cans.

� Cereal is packed in boxes with a volume of 2000 cm3.

What dimensions should the cereal company choose for theboxes?

Explain the reasons for your choice.



F-57



• relate expressions for volumes ofpyramids to volumes of prisms, andvolumes of cones to volumes of cylinders

� Carlos and Marie made nets and constructed a pyramid anda prism with identical heights and congruent triangularbases. They made similar pairs with congruent square bases.They estimated how much greater the volume of the prismwas than the volume of the pyramid for each pair. Thenthey used sand to measure and compare their estimates.- Carry out their investigation and find the relationship

between the volume of a pyramid and the volume of aprism with the same base and height.

- State this relationship in words.- Does the same relationship apply to cylinders and cones

having identical heights and bases?- Explain, using models.

• calculate and apply the ratio of area toperimeter to solve design problems in twodimensions

� Barrie wanted to fence off a rectangular garden area. Thefencing material comes in one-metre lengths that cannot becut. If Barrie has 12 m of fencing, what are the dimensions ofthe largest garden area he can make? Draw a diagram toexplain your reasoning.

� A store owner wants to make a rectangular area for a specialdisplay in one corner of his store. He has 8 m of enclosurerope to block off two sides of the area, using walls for theother two sides. What are the dimensions of the largest areahe could rope off?

� If you had a length of flexible fencing wire that could bendanywhere, how could you find the largest area you couldenclose without measuring? Explain, using differentgeometric shapes.

If you had 16.25 m of the fencing wire, what would thedimensions be?





F-58



• calculate and apply the ratio of volume tosurface area to solve design problems inthree dimensions

� What is the maximum number of boxes measuring 6 cm x 3cm x 2 cm that can be packed into a box measuring 24 cm by8 cm by 11 cm? If each of the dimensions of the largepacking box doubles, how many smaller boxes will fit?

� Create a graph that illustrates height versus surface area forseveral cans with the same radius. Conduct a similarinvestigation to determine how the volumes of the cans arerelated.

� Design three different containers that will hold 12 one-centimetre cubes and determine the most cost-efficientcontainer.

� *Dana and Akira made nets to construct cylinders. Theyboth used the same rectangular piece, but Dana used thelength to form the circumference of the cylinder and Akiraused the width.

Which cylinder will have the greatest surface area? Explain.

Which cylinder will have the greatest volume? Explain.

How would the results of this activity be useful to thecanning industry?



F-59




In order to prepare students to use spatial problem solving in building, describing, and analysing geometric shapes and specify conditionsunder which triangles may be similar or congruent and use these conditions to solve problems, it is expected that students will:

• draw the plan and elevation of a 3-Dobject from sketches and models

� Six cubes were used to build this model.

Using isometric dot paper, draw the overview plan, frontelevation, and the left and right elevation.

� Draw and label the plan, front, right, and left elevations ofthis sketch.

1.5 cm

3 cm9 cm

2 cm

• sketch or build a 3-D object, given its planand elevation views

� Build the object that follows the plan (top) and the front andside (elevation) views.

Plan (Top) Front Side (Elevation)

� Use isometric dot paper to sketch the object illustrated bythe following views.

Overview Plan Front Elevation Side Elevation


F-60



• recognize when, and explain why, twotriangles are similar, and use theproperties of similar triangles to solveproblems

� Given one triangle, magnify two of the sides by a factor of 2.Explore the relationships between the angles and sides ofthe original triangle and the enlarged triangle.

� A person 180 cm tall casts a shadow 45 cm long. A nearbytelephone pole casts a shadow 300 cm long at the same timeof day. What is the height of the pole?

� Sol made a scale drawing of his triangular vegetable gardenso he could plan how to plant it. Two sides of the garden are10 m and 12 m, and they form an angle of 50°. He drew a 50°angle on paper and made a triangle by marking off 20 cmand 24 cm on the sides of the angle and connecting them. Hemeasured this side to be 19 cm. What is the length of thethird side of this garden?

� *Shandra said that two triangles drawn on a page lookedsimilar. How can she find out for sure if they are or are notsimilar? Find two different ways she can do this, and explainyour reasoning.



F-61



• recognize when, and explain why, twotriangles are congruent, and use theproperties of congruent triangles to solveproblems

� *Heidi thought that two triangles looked congruent. Tomake sure, she cut them out and placed one on top of theother. If she couldn’t cut them out, how else could she besure? Find two different ways she could do this, and explainyour reasoning.

� Given the triangle ABC, use your protractor, straight-edge,and a pair of compasses to construct:- a congruent triangle- a triangle that is congruent but not similar- a triangle that is similar but not congruent

Provide reasons for each of your answers

A

B

C

� Pasha is hiding from Quentin behind a large tree. Using thediagram below, find:- the region (field of view) that Quentin can see- the points that Pasha can hide at

Tree




F-62



• recognize and draw the locus of points insolving practical problems

� This is a plan of a backyard with a fence around it. The grassmust be at least 1 m away from the tree and at least 2 m fromthe fence. Shade the area that will be grass.

tree

18 m

7 m

� A monkey can reach out 60 cm from the base of his cage. Hiscage is rectangular and measures 150 cm by 100 cm. Drawthe cage and shade the part of the ground outside the cagewhere the monkey can reach. (The bars go all around thecage.)

� Take two coins and explain why, when the coin on the lefthas rolled around the coin on the right, the design on thecoin is right side up. Should it not be upside down? Explainhow this occurs.

• relate congruence to similarity in thecontext of triangles

� State, giving examples, whether each of the followingstatements is true or false.- All similar triangles are congruent.- All congruent triangles are similar.




F-63



• draw the image of a 2-D shape as aresult of:- a single transformation- a dilatation- a combination of translation and/or

reflections

� Draw a triangle in the first quadrant. Identify thecoordinates.- Perform a translation so that the image is completely in

the fourth quadrant. Identify the coordinates of theimage.

- Perform a reflection of the above image so that its imageis completely in the second quadrant. Identify thecoordinates of this image.

� This image M’ (-4, 3), N’ (-5, 0), P’ (1, -2), Q’ (0, 0), R’ (1, 2)was obtained by subtracting 3 from each x-coordinate of thevertices M, N, P, Q, and R. Draw the original figure.

P'

R'

Q'

M'

y

xN'

� Plot each figure on a set of axes. Draw the reflection imagein the reflection line.

A

BC

0 2 4 6 x

2

4

6

y

0 2 4 6 x

2

4

6

y

R

S T

Reflection Line

Reflection Line




F-64



� Draw a new image by reflecting it in the reflection line, thenrotating it 270˚ counterclockwise.

y

0 2 4 6 x

2

4

6

y

R

TS

A

B C

0 4 6

2

4

6

2

Reflection Line

Reflection Line

x

� In each row, which image is a reflection of the first figure?



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� Identify each image in a row as a translation, reflection, orrotation of the first figure.

� Rectangle ABCD was transformed, and the image lies on topof ABCD.

C

A

D

P

B

What single rotation is required for a rotation about:- point A?- point P?




F-66



� Each of the following diagrams shows a figure and itsreflection image. Copy each diagram onto grid paper. Findthe reflection line.

0

y

6 102 4 8 x

2

6

4

12

0

y

6 102 4 8 x

2

6

4

12

A

BC

A'

B'C'

C

B

B'

A

A'

• identify the single transformation thatconnects a shape with its image

� A triangle ABC has vertices at coordinates A (-3, 1), B (2, 3),and C (5, -2). Draw the figure in a coordinate plane and:- dilate it by a factor of 2- translate it 3 units to the right- reflect it in the line y = 5- rotate it 180˚ about point B

� ABC has vertices A (-4, 1), B (-3, 5), and C (-1, 1).- Plot ∆ ABC on grid paper.- Reflect ∆ ABC in the x-axis.- Reflect ∆ ABC in the y-axis.

• demonstrate that a triangle and itsdilatation image are similar

� Draw a triangle with coordinates (2, 3), (4, 6), and (5, 4).Locate the dilatation image of the triangle with the dilatationcentre at (0, 0) and a scale factor of 2. Explain how you knowthat the triangle and its image are similar.



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• demonstrate the congruence of a trianglewith its:

- translation image- rotation image- reflection image

� Draw a triangle with coordinates (3, 1), (6, 1) and (5, 3).Draw the resulting images for the following:- 90° clockwise rotation with rotation centre at (3, 1)- reflection with y-axis as line of reflection- translation—2 units right and 4 units down.

Explain how each image is the same as and different fromthe original figure.

� Label triangles B, C, and D as reflections, translations, androtations of triangle A.

C

AB

D

Compare the properties of triangles B,C, and D to those oftriangle A, and label the triangles as either congruent,similar, or neither.

Use your answer in the previous question to state a ruleregarding images of single transformations with respect tothe original triangle. Justify your statement.




F-68




In order to prepare students to collect and analyse experimental results expressed in two variables, using technology asrequired, it is expected that students will:

• design, conduct, and report on anexperiment to investigate a relationshipbetween two variables

• create scatter plots

• interpret a scatter plot to determine ifthere is an apparent linear relationship

• determine the lines of best fit from ascatter plot for an apparent linearrelationship by:- inspection- using technology (equations are not

expected)

• draw and justify conclusions from theline of best fit

� *Design, conduct, and report on an investigation into one ofthe following:- spring extension versus mass- mass versus volume for several samples of the same

substance- price in Canadian dollars versus price in US dollars for

books and magazines- temperature versus time of day over a two-day period

(nonlinear)- height versus “arm stretch”—distance between fingertips

with arms fully extended- any other possible relationship you wish to investigate.

� Create a scatter plot to investigate the relationship between:- the distance, in kilometres, that a student lives from

school and the time, in minutes, required to travel toschool each morning

- the number of cars in the school parking lot at 9 a.m. andthe day of the week

Examine your scatter plot to:- describe the patterns of the dots- account for the dots that do not lie on the line- state a relationship in words for your plotUse your ruler. Estimate and draw the line that best fits yourdot pattern. Could your line be used to make predictions?Would any point that lies on the line have meaning withrespect to the two variables?

� Draw a line of best fit. What conclusions can be drawn fromthis data? Describe the relationship between shots made anddistance.

1 2 3 4 5 6 7 8 9 10

25

20

15

10

5

Shot

s M

ade

Distance from Basket (m)

F-69



• assess the strengths, weaknesses, andbiases of samples and data-collectionmethods

• critique ways in which statisticalinformation and conclusions are presentedby the media and other sources

� Collect data presented via newspaper, magazine, radio,or TV.- How were samples for the data selected? Why do you

think they were selected that way? Are they biassed?- Were the data-collection methods appropriate for the

data and the issue?- How would you do it differently? Why?- Are the data presented clearly and honestly?- Do the conclusions follow logically from the data?- What questions are left unanswered? Is this deliberate?

� Use the following scatter plot to determine the line of bestfit. What conclusions can be drawn from this data? Describethe relationship between the shots made and the distancefrom the basket. What predictions could be made from yourline? If available, use a graphing calculator to establish theline of best fit. Does it match your approximation?

1 2 3 4 5 6 7 8 9 10

25

20

15

10

5Sh

ots

Mad

e

Distance from Basket (m)


In order to prepare students to collect and analyse experimental results expressed in two variables, using technology asrequired, it is expected that students will:



F-70




In order to prepare students to explain the use of probability and statistics in the solution of problems, it is expected thatstudents will:

• recognize that decisions based onprobability may be a combination oftheoretical calculations, experimentalresults, and subjective judgments

� Interview some people to find out how they pick lotterynumbers and why they choose particular numbers.

� Jay checked data on how often each number has been drawnin a particular lottery. He chose six numbers that had beendrawn the least often. Do they have a greater probability ofbeing drawn the next time? Explain.

� The weather forecast indicates that the probability ofprecipitation for tomorrow is 60%. Should Sasha go golfing?Explain your answer.

• demonstrate an understanding of the roleof probability and statistics in society

� Find examples from newspapers, radio, TV, or other sourcesthat use probability (e.g., marketing of products andservices, weather forecasting, opinion polls). Are the datavalid? Are they presented in an honest or in a misleadingway? What assumptions are made?

• solve problems involving the probabilityof independent events

� If you toss three pennies, what is the probability that theywill all land heads? What other events are possible? Are allthe events equally likely? Explain. What is the probability ofgetting two heads and one tail? Justify your answer by usingpennies to illustrate all possible outcomes.

� Amanda chose three single digits for her combination lock.What is the probability that someone could make a luckyguess and open her lock? Explain. How could you set up asimulation experiment using the computer to solve thisproblem?

� There are two red, two green, and two blue candies in a bag.What is the probability of drawing a red one? How manywill you have to draw before you are sure of drawing ared one?

mathematics 8 and 9 - british columbia · for mathematics 8 and 9. the development of this irp has...

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