DOCUMENT RESUME SE 046 043 McCardle, Lisa, Ed. · DOCUMENT RESUME ED 261 911 SE 046 043 AUTHOR McCardle, Lisa, Ed. TITLE Mathematics 10-20-30 Curriculum Guide 1983. INSTITUTION Alberta

DOCUMENT RESUME

ED 261 911 SE 046 043

AUTHOR McCardle, Lisa, Ed.TITLE Mathematics 10-20-30 Curriculum Guide 1983.INSTITUTION Alberta Dept. of Education, Edmonton.PUB DATE 83NOTE, 178p.PUB TYPE Guides - Classroom Use Guides (For Teachers) (052)

EDRS PRICE MF01/PC08 Plus Postage.DESCRIPTORS *Course Descriptions; Curriculum Guides; *Educational

Objectives; Elective Courses; *MathematicsCurriculum; *Mathematics Instruction; MinimumCompetencies; Secondary Education; *Secondary SchoolMathematics ;. State Departments of Education

IDENTIFIERS *Alberta

ABSTRACTThe Mathematics 10-20-30 program consists of core and

elective components, both mandatory, for three courses. The corerepresents the common set of minimum educational objectivesprescribed for all students taking the program; the electivecomponent allows for variety and flexibility in the choice of topics.Emphasis, is placed on the theoretical development of mathematicsconcepts and relationships through deductive reasoning. Problemsolving, applications, and statistics are strong facets of theprogram. Suggested time allocations are included. Objectives,comments, applications, and textbook references are listed for eachcourse. Objectives and comments or activilies for the electives arealso included. Appendices contain the Alberta metrication policy,problem-solving strategies, and recommendations for schoolmathematics of the 1980s. (MNS)

************************************************************************ Reproductions supplied by EDRS are the best that can be made *

* from the original document. *

***********************************************************************

NATIONAL INSTITUTe OF EDUCATIONE0 CATIONAL RESOURCES INFORMATION

VCENTER (ERICI,

This document has been reproduced asreceived from the person or organizationoriginating itMinor changes have been made to improvereproduction quality

Points ot view or opinions stated in this documeet do not necessarily represent official MEpositron or policy.

"PERMISSION TO REPRODUCE THISMATERIAL HAS BEEN GRANTED BY

S WO& kt) MI11

aTO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)."

g MATHEMATICS 10-20-30al CURRICULUM GUIDE 19834114111.411VAIVANIACIZIV //

a

TABLE OF CONTENTS

PROGRAM RATIONALE 3

PROGRAM STRUCTURE

CoreElective Format / 5

Common Core Content / 5

Independent Core Content / 5Electives / 5

PROGRAM EMPHASIS

Problem Solving / 6Applications / 6Statistics / 7

PROGRAM PLANNING 8

THE ELECTIVE COMPONENT 9

Structure of the Elective / 9Guidelines / 9Suggested Time Allocations / 10Suggested Outlines / 10Provision for the Academically Talented / 11

-MATHEMATICS PROGRAM OVERVIEW 12

LEARNING RESOURCES 15

PROGRAM OF STUDIES 17

Mathematics 10 Core / 19

Mathematics 20 Core /Mathematics 20 Electives / 55Mathematics 30 Core / 67Mathematics 30 Electives / 91

APPENDICES

Appendix A: Metrication Policy / 104Appendix B: Strategies of Problem Solving / 106Appendix C: Recommendations for School

Mathematics of the 1980's / 110

103

NOTE: This publication is a service document. The advice anddirection offered is suggestive except where it duplicatesor paraphrases the contents of the Program of Studies. Inthese instances, the content isinthe same distinctivemanner as this notice so that,-the reader may readily identifyall prescriptive statements or segments of the document.

1

ACKNOWLEDGEMENTS

Alberta Education acknowledges with appreciation thecontribution of the following members of the Committee on SeniorEigh School Mathematics operating under the direction of theMathematics Coordinating Committee and the Curriculum PoliciesCommittee.

COMMITTEE ON SENIOR HIGH MATHEMATICS

Mr. V. Anand - Teacher, Paul Kane High School, St. Albert

Mr. G. Darbellay

Dr. E. Enns

Mr. A. Horovitch

Ms. C. McCabp

Mr. A. Miele;

Mr. A. Peddicord

Mr. G.

Mr. E. Probrt.

Dr. S. Sigurdson

Dr. S. Willard

- Vice Principal, Crescent Heights HighSchool, Calgary

Chairman, Division of Statistics,University of Calgary

Teacher, Vauxhall High School, Vauxhall

Teacher, Spruce Grove Composite High School,Spruce Grove

- Vice Principal, Bishop Carroll High School,Calgary

Consultant, Alberta Education, Chairman

- Associate Director of Curriculum,Alberta Education

Principal, Eastglen High School, Edmonton

Professor, Faculty of Education,University of Alberta

- Associate Professor, Department of MathematicsUniversity of Alberta

Alberta Education also acknowledges the valuable contributionof many senior high school teachers throughout the province inProviding assistance in the development of the program.

Special thanks to the secretarial staff of the EducationRegional Office for many hours of typing and assistance in production.

CREDITS: Lisa McCardle EditorShih-Chien Chen ArtistEsther Pang TypistsLesley BarthRod. E. McConnell Learning Resource Officer

- Computer Graphics

2 4

PROGRAM RATIONALE

The ability to quantify information and to perform mathematicaloperations is an important component of public education and one'sability to function effectively in today's society. Mathematicsplays a significant role in developing mathematical skills andunderstandings needed by students as they progress through theirschool years and when later, as adults, they prepare themselves toenter the career field.

Mathematics as a formal study of mathematical concepts andrelationships must be in balance with applied use of mathematicsin everyday situations. It. is in this regard that the develop-ment of problem solving becomes an important part of the schoolingprocess as students learn to apply mathematical solutions to thediversity of problems encountered in daily living.

Mathematics programs must provide a strong knowledge base andat the same time assist in the development of mathematical reasoning.Pedagogically, the program should be ,based on a solid foundationof concrete experiences in the early school years, leading toprogressively increasing levels of cognitive development. Pcogramsshould be flexible and take into account students' individualabilities and rates of learning. Topics traditionally covered inthe past may become less important as new demands require greateruse of microelectronic application in processing information.Mathematics as a part of the basic education of students at alllevels must ensure continuity and yet be responsive to emerginginfluences that are characteristic of our changing times.

LA

It is toward these ends that the high school mathematicsprogram has been developed and organized. Three program streams(Mathematics 10-20-30; 13-23-33) are available to students in

. recognition of the need to provide differentiated programs toaccommodate varying mathematical abilities and the background

\required for specific career patterns. Each program route variesin the mathematics subject matter presented, the instructionalmphasis, and time spent on concept development.

\The program has been organized in a "core-elective" format

w th mandatory components of mathematical subject matter prescribedfo all students. This provides a measure of commonality in thein truction and mathematical background of students taking apar icular program route. Flexibility is provided in the electivecom onent which allows for the selection of topics to meet studentneed. or as a means of addressing emerging applications or mathe-mati s in new fields of human endeavor.

The influence of microelectronic technology on mathematicsprogramming is only teginning to be felt. The integration anduse of calculators and computers in the mathematics program isstrongly encouraged in the program and in the instructional process.A number of elective topics and suggestions in the Comments andApplications section of this gu de are designed to facilitatetheir use.

The senior high school mathematics program has also recognizedthe major recommendations of the National Council of Teachers ofMathematics. These recommendations have been developed to serveas "an agenda for action" for mathematics education in'the 1980's.The emphases placed on problem solving, integration of microelectronictechnology, applications, and the flexibility needed within acurriculum are incorporated in the overall design and content of theAlberta program.

PROGRAM STRUCTURE

Core Elective Format

The Mathematics 10 -20 -30 program consists of core and elective

components. The core component of 'the programtepresents thecommon set of minimum educational ObYrectives prescribed for allstudentS to ing the program. The elective component allows for

Y

,,-

variety an 4 flexibility in the choice of topics to be covered withinthe guidelines outlined on page 7 of this document. The core andelective are both mandatory requirements of the senior high schoolmathematics program.

Common Core Content

Topics and objectives common to both the Mathematics 10-20-30and 13-23-33 programs have been identified as content Dasic to eachof the program streams (see chart, page 11). While the contentstatements (objectives) are identical, it is intended that the timeand approach, instructionally, would vary for each program. Mathe-matics 10-20-30 should emphasize the theoretical development ofmathematics concepts and relationships through deductive reasoning.Mathematics 13-23-33 places greater emphasis on inductive, experimentalapproaches to the understanding and assimilation of mathematics subjectmatter.

Independent Core Content

The independent core component serves to differentiate betweenthe two program streams by identifying specific concepts and skillsunique to each program. The distinction is further enhanced by wayof the depth to which concepts are covered and the emphasis placedon abstract/theoretical concept development. Independent core topicscovered at lower grade levels in Mathematics 10-20-30 are repeated,in some instances, in the Mathematics 13-23-33 program at higher gradelevels. Again, the approach and depth should vary for each programstream.

Electives

Elective topics for Mathematics 10-20-30 have been provided forat the Grade 11 and 12 levels only. The Grade -10 courseexcludes theelective component to ensure sufficient time for developing a strongmathematics foundation fdr sequent courses. For further information,regarding the electives, please refef to 'page 7 of this guide.

5

7

PROGRAM EMPHASIS

Problem Solving

A major emphasis in mathematics education has traditionally beenplaced on the ability of students to solve problems. The skills attainedthrough experiences that are mathematics related and relevant to everydaysituations assume greater significance and importance in preparing studentsto function in a changing world. Hence, a major emphasis of the seniorhigh school mathematics program is the maintenance and further developmentof problem solving skills developed in earlier grades. A number ofapplied problem situations are suggested toward the achievement of thisgoal, in the applications column of the Objectives Statement sectionsbeginning on page 20 (Mathematics 10 Core), page 38 (Mathematics 20 Core),and page 68 (Mathematics 30 Core).

Typically, the problems presented to students in the past requirednothing more than a repetition of operations involving a series of computa-tions and, in many cases, number transfer in stated formulas. The solutionprocess was largely mechanical in nature once the steps needed to arriveat an answer were known (or stated) and subsequently practiced by thestudent. Problems necessitating the use of,a variety of problem-solvingtechniques were seldom presented, and the need to reach "the right answer"was stressed as the most important outcome, rather than the developmentof skills and processes that could be internalized and applied to morethan one type or set, of problems. The senior high school mathematicsprogram recognizes the importance of obtaining the 'right answer" but alsostresses the development of problem solving skills which can be appliedto almost any problem situation.. Appendix B on page 106 outlines a four-stage model as a means of initiating and developing problem-solvingskills.

Applications

Making the study of mathematics both meaningful and relevant tostudents is regarded as an important function or mathematics educationat all levels. The senior high school mathematics program has made aserious attempt to do so by suggesting interesting and relevant situationswhere mathematical concepts and skills may be applied to real lifesituations. These suggestions appear in the Comments and Applicationscolumns of the Objectives Statement sections referred to in paragraphone, above. Teachers are encouraged to draw probleMs from a variety ofcontexts as concepts are developed in the instructional process.

Applications, like problem solving, should be integrated intothe overall program rather than be dealt with as an independent unit.Whenever possible, integration and coordination with other subjectareas should be encouraged. When applications require extensivecomputations or data storage, the use of a calculator and microcomputer

should be encourgaed.

StatisticsFor a number of years, statistics has played an increasing role in

public information. The public is continually presented with statisticalinformation, such as the consumer price index, baseball averages, weatherforecasting, election polls and stock market indices in addition to

those being used extensively in research and industry.

In any enterprise where large amounts of data need to be handled

and processed, and where predictions are to be made, statistical methods

play a vital role. Statistical techniques are also used in designingtelephone exchanges; optimally setting traffic lights; designinginsurance policies; determining the reliability of a particular makeand model of car; criminal detection, and market research, to mentiononly a few.

The introduction of statistics is intended to familiarize the

student with the elementary descriptive measures that form the basis of

any further work with large data sets. In most cases the data should

be collected by the students, preferably different data for each student.

The subsequent analysis then becomes more meaningful and hopefullyprovides not, only statistical insight, but some new knowledge about our

surroundings. In order to understand and intelligently discuss much of

the information to which we are daily subjected, some knowledge of the

terminology and underlying assumptions of statistics is necessary.

Statistics is an applied science. To motivate this subject, especially

at the introductory level, one should concentrate on the experimental

approach. This strand provides the opportunity for field trips to collect

data. By displaying this data in various forms via graphs, it is possible

to make some basic inferences about the data source, and it is important

to make some inference in order to justify collection of data. It is

equally important to ask what further information would be worth knowing,

and to discuss how students might set about discovering it.

7

O

PROGRAM PLANNING

The topics and objectives for each course are not meant to befollowed in any particular sequence and will be largely influencedbyteacher preference and the order of presentation in the prescribedand recommended '_earning resources. Teachers are encouraged todevelop long range plans and to consider the time allocationssuggested for each topic.

8 1.0

THE ELECTIVE COMPONENT

The elective component is a mandatory part, of the senior highschool program at the Grade 11 and 12 levels. The elective componentprimarily offers an opportunity for students to spend time oninteresting and useful areas of mathematics not necessarily containedin the core and independent core components.

The time allotment for the elective component should beapproximately 15 hours at the grade 11 level and 20 hours at thegrade 12 level. It is suggested that the 15 or 20 hours of electivesbe determined according to the plans of the individual teacher.

Structure of the Elective

The elective component of the program may be:

1) a content area not prescribed as a core or independent coretopic

OR2) a locally developed unit as determined by the teacher or

school systemOR

3) an extension of the subject matter in any of the core orindependent core topics to provide students with enrichment.

Guidelines

Teachers should keep the following guidelines in mind:

1) The topics are open-ended so that the interests and abilitiesof students may be taken into account

2) Student initiated projects may be considered as an elective

3) A teacher should make use of any appropriate resources

4)- Electives should he included in the course throughout the year,wherever appropriate. These should not necessarily be taughtduring a 15 or 20 hour block. Where more than one elective isincluded, the time for each elective does not necessarily haveto be the same

9

11

5) The elective component of the program should be included inthe evaluation of the students

6) Prerequisite core'material may be required before some electivesare attempted. Particular elective topics have been recommendedfor the different courses. (see program chart, page 11)

Suggested Time Allocations

The time allocated to topics in the elective component of thecourse may vary according to the topic(s) chosen and instructionalprferences of the teacher. A total of 15 or 20 hours may be devotedto cover one elective topic, or alternatively, two or more topics.The time allOtted to elective topic(s) is at the discretion of theteacher.

Example 1:

Example 2:

Linear Programming covering all 15 hours.Probability covering all 20 hours.

Linear,Programming and Inequalities covering15 hours.

Matrices and Vectors covering 20 hours.

Example 3: Absolute Value/Inequalities/Complex Numberscovering 15 hours.

History of Math/Induction/Matricescovering 20 hours.

Suggested Elective Outlines

The elective component is designed to be an interesting andmotivational aspect of the mathematics program. The outlines(p.55 & p.91) for elective topics are intended to act as guidelinesfor individual teachers. Teachers may wish to fdllow the suggestedoutlines or incorporate their own ideas for the elective component.The prescribed references provide useful material for many of theelective units.

10 12

Provision for the Academically Talented

Topics within the elective component of the course should beutilized to challenge the academically talented students. Such topicsas Binomial Theorem and Topology:for example, may be used to providea challenge to students of this calibre-, Teachers may also extendcore topics and concepts to a higher level of complexity to meet theneeds of the stronger academic student.

L)'tic

MATHEMATICS PROGRAM OVERVIEW

ELECTIVE INDEPENDENT CORE I CORE I INDEPENDENT CORE ELECTIVE

Arrangements and SelectiOnsHistory of MathMath InductionMatricesProbabilityTopologyVectorsBinomial Theorem

MATH 30

Sequences, Series andLimits

TrigonometryQuadratic Relations

(Conics)Polynomial Functions

Presentationof Data andDescriptiveStatistics

Lo:1rithms

Trigonometry

MATH 33

Relations and functionsQuadratic :actions,

Equat I m and, Applia,ions

Absolute ValueComplex Numbers

Consumer MathematicsHistory of Math

Linear ProgrammingPrGbabili'y

TopologyVectors

Absolute Value

Complex Numbers

Inequalities

Linear Programming

Math Art

TransiormatIoL,l C,comett,

MATH 20Coordinate Geometry MATH 23

Relations and Functions -Systems of Equatigns RadicalsPolynomials

Systems of Equations Radicals ,Variation GeometryQuadratic Functions,

Equations and Trigonometry Geometry PolynomialsApplications

Presentation of Data Descr,otivoRadlcalsStatisticsGeometr,,

Polynomials

Area awl

Consumer Math

Inequalities

Math Art

iransformational Geometry

14

I/I

Number Systems Variation TrigonometryExponents and RadicalsGeometry (Deductive) Exponents Presentation of Data Geometry (Inductive)

Polynomials equations and Graphing and Descriptive

lEquations and Graphing Statistics

IGeometry

MATH 10 MATH 13Number Systems Polynomials

0

BEST COPY AVAILABLE0 15

SUGGESTED TIME ALLOCATIONSMathematics 10TOPIC

1

NUMBER OF HOURS

Number Systems 10

Equations and Graphing 16

Presentation of ata and Descriptive Statistics 10

Variation 7

Exponents and Radicals 20

Polynomials 25

Trigonometry 12

Geometry 25

TOTAL HOURS 125

Mathematics 20TOPIC NUMBER OF HOURS

Radicals 7

Polynomials 10

Coordinate Geometry 12

Presentation of Data and Descriptive Statistics 6

Relations and Functions 12

Quadratic Functions, Equations and Applications 20

Systems of Equations 12

Geometry 15

Trigonometry 10

Variation 6

Electives 15

TOTAL HOURS 125

Mathematics 30TOPIC

Trigonometry

Quadratic Relations (Conic Sections)

Logarithms

Sequences, Series, Limits

Statistics

Polynomial Functions

Electives

NUMBER OF HOURS

26

23

10

20

16

JO

20

TOTAL HOURS 125

14 17

LEARNING RESOURCES

Mathematics 10

Prescribed Deference

Prescribed Reference:

Recommended Reference:

Mathematics 20



Recommended Reference:

HOLT. liATHEMATICS 4

M. P. Bye, T. J. Griffiths, A. P. Hanwell

Toronto: Holt, Rinehart & Winston, 1980.

MATH IS 4

F. Ebos, B. Tut

Don Mills: Thomas Nelson & Sons (Canada)

Limited, 1979.

FOUNDATIONS OF MATHEMATICS FOR TOMORROW:

INTRODUCTION

D. Dottori, R. McVean, G. Knill, J. Seymour

Toronto: McGraw-Hill Ryerson Ltd., 1974.

MATH IS 5

F. Ebos, B. Tuck

Nelson/Canada Ltd., 1980.

HOLT MATHEMATICS 5

K. D. Fryer, R. G. Dunkley, H. A. Elliot,

N. J. Hill, R. J. MacKay

Holt, Rinehart & Winston of Canada Ltd., 1980.

FOUNDATIONS OF MATHEMATICS FOR TOMORROW:

INTERMEDIATE

D. Dottori, G. Knill, J. Stewart

McGraw-Hill Ryerson Ltd., 1978.

15

18

Mathematics 3)3

Prescribed Reference:\\AFH IS 6

F>.Ebos,,... B. Tuc

Nelson/Canada Ltd., 1982.

Prescribed Reference: FOUNDATIONS OF MATHEMATICS FOR TOMORROW: SENIOR

D. Dottori, G. Knill, J. Stewart

McGraw7-Hill Ryerson Ltd., 1979.

Recommended. Reference: HOLT MATHEMATICS 6

K. D. Fryer, R. G. Dunkley, H. A. Elliott,

N. J. Hill, R. J. MacKay.

Holt, Rinehart & Winston of Canada Ltd., 1981.

Learning Resource Approvals

In terms of provincial policy, learning resources are those print,nonprint and electronic courseware materials used by teachers or studentsto facilitate teaching and learning.

PRESCRIBED LEARNING RESOURCES are those. learning resources approved bythe Minister as being most appropriate for meeting the majority of oalsand objectives for courses, or substantial components of courses, outlinedin provincial Programs of Study.

RECOMMENDED LEARNING RESOURCES are those learning resources approved byAlberta Education because they complement Prescribed Learning Resourcesby making an important contribution to the attainment of one or more ofthe major goals of courses outlined in the provincial Programs of Study.

SUPPLEMENTARY LEARNING RESOURCES are those additional learning resourcesidentified by teacheis, school boards or Alberta Education to supportcourses outlined in the provincial Programs of Study by reinforcing orenriching the learning experience.

1916

PROGRAM OF STUDIES

GOALS OF THE SENIOR HIGH SCHOOLMATHNEMATICS PROGRAM

Altho ,11 the different courses of the senior high school mathematicsprogram have 'ifferent specific objectives, t..-:e.e goals of the senior highmathematics p ogram are set,forth in relation to three main expectationsand needs: .those of the individual, those el' the discipline of mathematicsand thes o society at large. They are listed as follow=4:

Student Developmenta) To develop in each student a positive attitude towards mathematics

b) To develop an appreciation of the contribution of mathematics to theprogress of ciiilization.

c) To develop the ability to utilize mathematical concepts, skills andprocesses.

d) To develop the powers of logical analysis and inquiry.

e) To develop an ability to communicate mathematical ideas clearly andcorrectly to others.

Discipline of Mathematics

a) To provide an understanding that mathematics is a language usingcarefully defined terms and concise symbolic representations.

b) To provide an undAstanding of the concepts, skills and processes ofmathematics.

c) To provide an understanding of the common unifying structure inmathematics.

d) To furnish a mode of reasoning and problem Solving with a capabilityof using mathematics and mathematical reasoning in practical situations.

Societal Needs

a) To develop a mathematical competedce in students in order to functionas citizens in today's society.

b) To develop an appreciation of the importance and relevance ofmathematics as part of the cultural heritage that assists people toutilize relationships that influence their environment.

c) To develop an appreciation of the role of mathematics in man'stotal environment.

18 21

MATHEMATICS 10CORE

"ft hi11k

LtZ:6,,,,``1/4,..4 L.N.4:+111. NI "11111,1:1,111111j11110 III

/ii,11,11:11iiii,i11,11,1

Nol i il!ttil . ,II II i I '

7111r,,,,aIII 01,111111 li

Ilifiltti;q1.111111i i in 'ilFulligard:

,11.11,11111.1 . Illi

16"4"-&-491-1....iNT:ed,1:1,1 ,111! .1. ij

ir.r.lecf:r311 uarlearpul L 1111111i,11

__HouthonumautuuluEll I11111,1 111

r-rurjrarr4r.firez.iourrditi_morroili°11i11.1:11tililuitlilil i,1111115111;

riritil rillillIIHIIIIP#1111411 I ifilli tr11/11111

fil je II

sr'

[11

,r,r4-11.4

li ll!iliititt17""urmil'a____."-al1 1 111111 111510r=r111

"Illra;411,1"71-71,_.___,

t II ES wIlht, u616i614, Li,.

'O'NPuNtill'utsQInSil":0%,,..,

i, ki:°141:\ths

10

22

OBJECTIVES

Mathematics 10

COMMENTS APPLICATIONSMathIs 4

HoltMath 4

A. NUMBER SYSTEMS

mommup

1. Identify numbers as natural,whole, integral and rational.

Although students enteringhigh school mathematicsshould have a good knowledgeof how the various numbersystems are related, avariety of mathematics

activities which will reviewand maintain previouslydeveloped basic skills willbe necessary.

10-17 4-7

2. Add, subtract, multiply and dividerational numbers.

3. Convert a rational number fromdecimal form to fraction forma and vice versa.

14-17 21-23

1;

4. Apply percentage to consumerrelated problems:

The calculator should be usedto facilitate the exploration

100,'101,68

24, 25,182, 185,

a) simple interestb) discounts and mark-upsc) commissions

of consumer-related problems. 191, 199,386, 393,395

)

5. Apply percentage to 'the calculationof compound interest.

Students should become awareof the difference in interestearned using simple interestand be aware of differentcompounding periods. Thecalculator can be used inthis section.

Calculate the interest earned onan investment of $400.00 for 5years if interest is compoundedsemi-annually or monthly.

100, 101 24, 25,

386, 393

6. Identify rationale as:

a) infinite repeating decimalsb) terminating decimals

Show the density. of rationaleon a number line.

14-17 21-23, 31

4

OBJECTIVES

Mathematics 10


HaltMath 4

7. Identify irrationals as:

a) infinite non-repeatingdecimals

b) square roots of numberswhich are not perfectsquares

c) special cases such as W

46, 47 31-34

8. Represent the relation betweennatural .7,imbers, whole numbers,integers, rationale, irrationalsand reels by a pictorial diagram.

The relationship may bedemonstrated using a Venndiagram, tree diagram orflow chart.

50 6-7

B. EQUATIONS AND GRAPHING

1. Maintain skills in solving firstdegree equations with rationalcoefficients.

Equations with variables inthe denominator will becovered with polynominals.

69-77

ASOM

39-42

-2. Solve word problems whose solutions

are based ou first degree equationswith rational coefficients.

Several practical problemsexist in the sciences,business and technical fieldsthat require the developmentof equations and theirrespective algebraic andgraphical solutions.

,

78-86,97

177-185

3. Identify and use the terms:quadrant, origin, axis,coordinate, ordered pair,

abscissa and ordinate.

(-;

Latitude and longitude; grid

plans of cities and laying outof new townsites; telephonerates based on a grid system;grid organization of an oilfield.

102-106 115-118

25 26

27

N.)N.)

OBJECTIVES

Mathematics 10


Holt

Math 4

4. Recognize and graph ordered pairs. Came approaches, such as"Battleship" and pictureplotting, can be used toreinforce graphing concept .

Computer games often involvegrids and the plotting ofpoints.

103-108 116-118

5. Interpret graphs of straight lines.

o

Students should be able toread the values of a variablefrom a graph.

Several examples such as supply-cost graphs, Celsius-Fahrenheitgraphs, distance-time graphs canbe used.

126-130

6. Graph linear equations using

a) ordered pairsb) intercepts

7. Find the intercepts of a line

a) by examining its graphb) from its equation

To find the x-intercept thestudent must understand thatalong the x-ax5.s, y=0. Byputting y=0 in the equationone can get the x-intercept.Similarly, by putting x=0 onecan get the y-intercept.

188-189 128-130

8. Apply the skills of graphingto practical problems involvingtwo linear equations.

116-117

28

OBJECTIVES

Mathematics 10


HoltMath 4

. _

C. PRESENTATION OF DATA ANDDESCRIPTIVE STATISTICS

1. Organize data by:

a) collecting various types ofdata

b) grouping data into classesc) determining frequency of

each classd) defining class width

(interval), class boundaries,and class marks

As an extension to thegraphing work previouslydone, students should be able

to collect and organize dataso that proper decisions,inferences, and predictionscan be made. A thoroughknowledge of the basicconcepts_of_statistics will__

stimulate problem solving.Grouping data into classes isused here to construct ahistogram.

Statistical applicationsinvolving the students themselvesare plentiful. Some examples

are: their heights to thenearest cm, lengths of theirforearms and sizes of theirwrists.

375-382 317-339,342-344

2. Graph data using bar graphs,circle graphs, histograms andfrequency polygons.

3. Calculate the mean, median andmode for ungrouped (raw) data.

4. Select the most suitable of threetype of averages for a givenset of ungrouped data.

5. Select a suitable sample from agiven population.

2J

OBJECTIVES

Mathematics .10


HoltMath 4

D. VARIATION=. N

1. Identify direct variation..

.

Illustrate graphically thedifferent types of variation.

323-329 140-147

2. Identify inverse variation.

3. Identify partial variation.

4. Solve problems based on direct.inverse and partial variation.

Have the students developpractical problems that arefamiliar to them, such aswage earnings, consumerproblems and scienceproblems.

325,

327-329

323-327

140-147

140-147:5. Find the constant of

proportionality for a givenvariation.

Additional explanations maybe required if the problemsencountered involve variablesother than the first degree.

S

OBJECTIVES

Mathematics 10COMMENTS APPLICATIONS

MathIs 4

Holt

Math 4

E. EXPONENTS AND RADICALS

1. Utilize the following laws ofexponents:

Where a, b, e I; x, y E R;x 0 0, y 0 0

xa xb . xa+b

xa .2. xb - xa-b

(xa)b . xab

(x9)11'' xaYa

xa

r Fa-

x° = 1

x'a = 1

iii.

Initially, the values of "a"and "b" should be specifiedintegers while the values ofx and y should.be eitherspecific numbers orvariables.

Students should be able toto recognize the parts ofa power from an example.

e.g., x3I x3 is the powerx Is the base3 is the exponent

Every attempt should be madeto help students understandthe reasons why these lawsexist. Demonstrate that:

)

5(2 `).' (2) --- (2. 2.) (2.. 2. 2) =.-

and then short cut to the law

(2)2 . (2)3 = 22 + 3 - 25

20-24,60-63,

64

10-14

2. Transform a number in decimal formto scientific (standard) notationand vice versa.

64 19-20

3. Perform the operations ofmultiplication and division on

,numbers expressed in scientific(standard) notation.

64 19-20

33 34

OBJECTIVES

Mathematics 10


HoltMath 4

4. Use the laws of exponents,the exponents are 1/2

,

whereand 1/3.

.

The exponents are extended torational values as a means ofrelating rational exponent'sto radicals to show relation-ships between radicals andpowers.

'Demonstrate: tif /2 =

and 211

212

.

Therefore, 21/2 must equal /1

61, 62 104-107

5. Identify and use tradicand and radic

;/

e terms radical,I sign.

Students should be able toidentify each part of aradical expression from agiven example.

\,g. 3/J: /Ts the radical sign3 is the indexx4 is the radicand

3vrt-4 is the radical

33 31-39

6. Utilize the definition

.11 1)471. (1.))61-11,)x

where b .., 2, 3, x e It

Multiplication of radicalexpressions of the type belowshould be avoided:

3475 4/0 240

60-63 104

7YX-/7 2

YX/7

7. Simplify radical expressions ofthe formbii, where b 2

included in this objective ischanging a radical from mixedto entire form and viceversa.

50-51 31-33

OBJECTIVES

Mathematics 10


Holt

Math 4

8. Perform the four basic operationson radicals of the form

b

v/r; where b ... 2

Addition and subtraction ofradicals can effectively beexplained using the distribu-tive law for algebraicexpressions.

e.g.: 2x + 3x = (2+3)x =

for radicals:

2/75-+ 3/-5-= (2+3)/ =5f

51-59 33-38

9. Rationalize radical denominatorsthat are monomials.,

0

Demonstrate the reason forrationalizing. It is easierto write /TO

2

as a decimal than theunrationalized form of/

Tir

which requires division by anapproximate decimal.

36-38

P. "POLYNOMIALS

1. Identify and use thefollowing terms:

a) algebraic expressionb) termc) '*actor

d) Monomial-e) binomialf) trinomialg) polynomialh) coefficienti) degree

Like and unlike terms shouldbe clearly explained.Considerable time should bespent here on thelanguage of mathematics.Order of operations shouldalso be reviewed. Stressthe difference between termand factor.

18, 43,24,

230-233

49-101

3 38

39

OBJECTIVES

Mathematics 10

COMMENTS APPLICATIONSMath

Is 4

Holt

Math 4

2 Evaluate a polynomial for givenvalues of the variables.

Substitute values for thevariables in some of the morecommon formulae used inscience such as area, volumeand measurement.

65 49-101

3. Add and subtract polynomials. 18-20

4. Multiply: Numerical examples 24-28,[(6 + 2) (6 2)] can be 43-45,

a) monomial z monomial used to introduce algebraic 231b) monomial z binomial multiplication ofc) monomial z trinomiald) binomial z binomial

(x + 2) (x - 2).'

5. Write the expansions of Illustrations of. (p + q)2 Show the biological implications 240-241

(P + Q)2, (P - Q)2 and

(P ", Q) (P + Q)

can be presented.

p + q

in heredity such as a pea havinga dominant wrinkled gene (W)and a recessive smooth gene (w)crossed with game (W + w).

and recognize them as general cases.P

+q

P2

Pq

(W + w) (W + w) = W2 + 2Ww + w

2

Pq q2

Dominant two recessivewrinkled hater- smoothpea ogeneous pea

(W2)

wrinkled (w2)

peas(2Ww)

6. Recognize and factor a polynomialwith a common factor where thecommon factor is:

a) a monimialb) a binomial

.

230-233

,em...

OBJECTIVES

Mathematics 10


HoltMath '.

-.:

7. Factor a trinomial of the form:

ax2 + bx + c

where a, b, c e I, a 4 0

233-237 49-101

8. Factor polynomials of the form:

p2 - Q2

P and Q should be limited tomonomials and binomials andshould not require grouping

240-243

to get a perfect square.

9. Factor polynomials by,using anycombination of the methods out-lined in objectives 6 through 8.

.:,

10. Divide a polynomial by al 24, 28,

a) monomialb) binomial

235

11. Simplify rational expressions Review numerical rational 235, 237by factoring. ,) number operations before

proceeding with the rationalalgebraic number operations.

244, 245

Include rational expressionsof the form:

a - b a # bb a

4142

OBJECTIVES

Mathematics 10


HoltMath 4

12. Perform the operations ofmultiplication and divisionwith rational expressions.

244, 245,

253

49-101

'.-

13. Perform the operations ofaddition and subtraction ofrational expressions with:

a) the same denominatorb) different denominators

246-248

14. Determine permissible and non-permissible values of thevariables in rationalexpressions.

235, 244246

15. Determine the zeros of apolynomial of one variable byfactoring.

237-239

16. Solve equations involvingrational expressions.

f

....

248-250

OBJECTIVES

Mathematics 10


HoltMath 4

IMMMINIWG.

11111MMINIIIIIMIONEMMINEIMOMIEW

1.

TRIGONOMETRY

Find the unknown sides in similartriangles.

It is worthtrigonometrytriangles sincestudents practicein terms ofrather thanment of thetriangle. Thiscentral to righttrigonometry:given angletriangle, regardlessabsolute measurementsides.

Example:

200B

introducingby using similar

this givesin thinking

ratios of sides

Problemstrigonometrytrianglesthe school.include:a) height

shadowsb) height

trigonometryangleprotractor)

c) distance(realtrigonometrytriangles.

INDIRECTSIMILAR

1. HeightA

tree

B

By

involving right triangleand similar

can be generated aroundSome examples

of a tree by using

of a building using(measurement of

by simple gravity

across a riveror fictitious) using

or similar

MEASUREMENT USINGTRIANGLES

of a tree

D

personC E F

329-337 305-315

.

2. Apply similar triangles topractical problems.

absolute measure-sides of a

concept istriangle

the ratio of ain a right

of the

of the

A D

,411141,

3. Define sine, cosine and tangentratios for right angle triangles.

4. Find the trigonometric ratios ofthe acute angles in a righttriangle when the sides are given.

5. Determine the trigonometric ratiosof any given acute angle.

E

AC = DF 200

AS DE

Outdoor measurement problemsshould be limited to simpleright triangle problems.Instruments for dititance andangle measurement Lan be madefrom very simple materials.

Students should becomefamiliar with trigonometrictables and how to use thembut should be allowed to usethe calculator.

6. Determine the measure of anyacute angle given one of itstrigonometric ratios.

shadow shadow

measuring, find BC,

DE,

EF.

Calculate AB.

AABC - A DEF. Why?7. Solve problems based on righttriangles usidi trigonometricratios.

y

4546

OBJECTIVES

Mathematics 10

COMMENTS APPLICA':IONSMathIs 4

HoltMath 4

2.

E

Pointscollinear.arbitrarilyE socollinear.

Distance across

A

a riverpick a point ormark with a stake

river

find BC,DC,DE.

Calculate AB.

and B areand C are

Choose pointand A are

329-337 305-315

By measuring,

D and CD

chosen.that E, C

H. GEOMETRY

1. Recognize and use the followingterms:vertex, side(ray), degree,straight angle, right angle,acute angle, obtuse angle,reflex Ingle, adjacent angle,complementary angle andsupplementary angle.

Using diagrams, the studentsshould be able to describeeach of these terms. To moststudents this is a reviewsection.

264-267 209-212

2. Recognize and use the followingterms associated with triangles:equilateral, equiangular,isosceles, scalene and righttriangles; hypotenuse. o

Where applicable, measurements could be assigned todifferent types of trianglesand polygons so thatperimeters and areas canalso be calculated

OBJECTIVES

Mathematics 10


HoltMath 4

3. Use the Pythagoras theorem tosolve right triangles andassociated problems.

I

If a 2000 mete rail ay trackexpands 2 metres, howl, highwill the track rise, assumingthe ends are fixed?

37-42,

313, 315308-315

042

1000 m 1000 m

4. Recognise and use the following 261, 265 210, 211terms associated with polygons: . 312, 313quadrilaterial, trapezoid,parallelogram, rectangle,rhombus, square, regularpolygon, and diagonal.

,

5. Recognize and use the following Students should be encouraged Building Construction: Doors '296-299 223-226,terms associated with parallel to find examples of parallel and door frames, windows, etc. 237-240lines: transversal, lines in the world around us.corresponding angles, alternateangles and interior angles.

6. Recognize and use the following Using parallel lines in the same 264, 241-249,terms: congruency, similarity, plane (e.g., exercise book) have 271-274, 214

perpendicular, bisector and students draw transversals and 367,

perpendicular bisector. discuss measurement of angles 329-333fofmed. Skew lines could bediscussed.

49 50

OBJECTIVES

Mathematics 10


HoltMath 4

7. Measure an angle with aprotractor.

Use the face of a clock with theminute hand on 12 and variouspositions of the hour hand (guessmeasurement first).

209

8. Construct an angle congruent toa given angle.

255 213-214

9. Construct the bisector of agiven angle.

,

10. Construct a perpendicular toa given line segment:

a) at a given point on asegment

b) through a point not on thesame megment.

11. Construct the right bisector ofa segment. .

12. Construct a line parallel to agiven line.

..,,,

OBJECTIVES

Mathematics 10


HoltMath 4

S3. Recognize that a formalaxiomatic development requires:

Differentiate betweeninductive and deductivereasoning.

262-263 217

221-222224a) undefined terms

b) definitionsc) assumptions (postulates

Or axioms)d) theorems

In order to develop anorderly structure we mustagree on the "rules of thegame." After we have agreedto certain definitions andrules of procedure, we cannotchange them; otherwise ourdiscussion would beconfusing.

258-259

Different texts use differentprocedures and in order to beconsistent as to order ofpresentation one basic textshould be used.

_

'....

14. State, prove and apply these270, 288 216-217

basic theorems of geometry:296-300 241-249

a) vertically opposite angle301-304,306

250-252237-240

theorem

b) congruence of trianglesusing SAS, ASA, SSS

c) isosceles triangle theorem

d) parallel line theorems

e) the sum of the measure ofthe interior angles of atriangle is 180.

. .

229-230

15. Apply the basic theorems tosolve problems involvingnumerical application.

302-304 261-263

53 54

MATHEMATICS 20CORE

itv '0111:1111,,,' 110' j 11

aim . iinilli 'nulliii'lluir:Iii1111,:"111,11131:.110PI'l 1 ',',1 111111 , t"'

v,1,.1,11,,, 1,1Ini Iii!pillIgcl I I( i

.__........._ ...n...,..--nn.in, I 1 Iii,"111"11 ;

r......ulnuuti.nniuntrun:161

' l ir

ttlion 1;111110, dill

,96.,,g111,Linillir11,4 111.A11)1.41111'

11.11, 111111), .0111! if jihni.' Alt!, 11'

11,111: 'T.1"11

-"

.enr""'"::::fira",.5:Fi'llrf:17,01:Ii , 1,4111.1, 11

dr ourei 101111,' 11 11111

110,,11

irty4,11,, 114,4

ii

loliPt'or'Stivilej.,,,o'etu,

',nonI 1411p'

"

lr gill illifilate=r111

1 "191Inan

II q ,1,0.1;n1 ,nti [iirJuinitthni, `1"561lilt-11111111 .

104116 191 j411144 +11n,,1111H11,111,'IL"'16-n

11111:1101 `Innn 11;1111,1

In hi U 1111 11 1 °nun

L

i d 41111 11%,,o'N1.1n .1111

'I IN 114

r oli- u n11 111li

uu" 111 0-1111.1,1111;1

els 1110 to- ur`1114- r

0.0 _nite llu. en lir r

ii 1ç

I1ll

ne/ jnil"

Ii

I

in

20

5i

Lk)

OBJECTIVES

Mathematics 20

COMMENTS APPLICATIONSMATHHOLT 5

MATH

IS/5

111111111 _

A. RADICALS

. . L .

1. Maintain previous skills. Review of concepts fromMath 10; refer to commentson "Exponents and Radicals"in Math 10.

34-46 92-116

-

_

2. Simplify radical expressions ofthe form WE, b .., 2, 3

Restrict to radicals ofthe form b

/xm yn

where b = 2 or 3 and in and nare positive integers.

3. Perform the four basic operations, on radicals.

, .

.

,

Same type of radicals asabove; addition andsubtraction can be explainedby using the distributive lawfor algebraic expressions:

2x + 3x (2 + 3)x 5x

for radicals

2 ir5 + 3 (2+3)/5-= ST'S'

4. Rationalize radical denominatorsthat are monomials and b nomials.

-.-

\

Demonstrate the reason forfor rationalizing. It is

easier to write /152

as a decimal than theunrationalized form ofvrT

J7rwhich requires division of anapproximate decimal.

, .

.

.

7

OBJECTIVES

Mathematics 20

COMMENTS APPLICATIONSMATH MATH

HOLT 5 IS/5

5. Solve radical equationscontaining one radical inone variable.

.

Emphasize isolation ofradical on one side of theequation before squaring.

Radical equations can be appliedto determine:

a) the braking distance of a car,knowing its speed and type ofroad.

Use:

a /5515, where s m speed,f = coefficient of friction

and,

d - braking distance (skidmarks).

The formula can also be usedto find speed by measuring theskid marks and thus determin-ing whether the driver wasspeeding.

b) the duration of a hurricane ortornado. Use:

vT1t m 2 216 or

2d /7 2d 16d- d i67d

34-46 92-116

t ' f' 36 ' 18

in simplified form, wheret m time in hours and d -diameter of storm in km.

6.

2

SOlve radical equations containingtwo radicals in one variable.

Before squaring, ensure thatone radical appears on eachside of the equation.

58 5'9

(7)

OBJECTIVES

Mathematics 20


MATHIS/5

B. POLYNOMIALS4

I. Maintain previous skillsin algebraic operations.

This includes working withpolynomials and rationalexpressions.

How many games are necessary ifeach team in an "n" team leagueis to play each other twice, onceat home and once away?

Ans. (n)(n-1)

Suppose there are 90 games, howmany teams are there?

1-6,

18-2142-55

2. Maintain previous skills offactoring.

7-12

3. Maintain previous skills of solvinglinear equations with one unknown.

4. Factor polynomials of the'form

p3 A. Q3

Polynomials arise from manyproblem situations. For example:

a) Distance objects fall during agiven time.

b) The lead distance necessaryfor a runner on the outsidelane of a racetrack.

c) Problems involvingsimplifying calculations.

17, 18 65-68

5. Factor polynomials which areincomplete squares. 22, 23 55

6. Factor polynomials'by thegrouping method.

This should include groupingto obtain a common factor andgrouping to obtain thedifference of squares.

7-10 47,48,53

C. COORDINATE GEOMETRY

1. Maintain previous skills relatedto the following: quadrants,axes, origin, ordered pairs andintercepts.

The usefulness of mathematicsincreased greatly whenmathematics applied algebraicmethods to geometry (in the17th century) to form asubject called coordinategeometry.

Telephone long distance ratesare determined by a rectangulargrid system. Contact yourlocal telephone business office.

OBJECTIVES

Mathematics 20

COMMENTS APPLICATIONSMATH

HOLT 5MATHIS/5

2. Determine the distance betweentwo points.

Using two points on a graph,say P(1,2) and Q(5,4) derivethe distance formula usingthe Pythagorean Theorembefore generalizing the

103-106 227-233

formula. -.NN

3. Determine the coordinates of themidpoint of a line segment.

By taking two points on agraph, the students canpredict the midpoint beforederiving the midpointformula.

214,215

4. Determine the elope of a linepassing through two given points.

Since coordinate geometry ishighly visual, students

ould be encouraged to doneat work so that they canbetter predict the slope ofline, state relationships

b tween slopes of lines,gra h lines from Ax + By + C

If the slope of a roof is 1:4 andthe span of the roof is 8 metres,calculate the rise of the roofand the length of the roofrafters.

61-75 197-200

.

=0 ry= mx + b, andinterpret graphs of straightlines.

5. State the relationship betweenslopes of:

A staircase is to be built witha rise of 7 and a ,. n of 12.

a) parallel linesb) perpendicular lines

Determine a method of using acarpenter square to cut the mainsupport for risers and treads.

0 204-208

6. Graph lines whose equations are 121-124in the form iSc + By + C 0 195

or y suc + b

using the slope and y-intercept.

201-204

00

62 63

OBJECTIVES

Mathematics 20

COMMENTS APPLICATIONS

0

MATHHOLT 5

MATHIS/5

7. Use the slope test to determinewhether three points are collinear.

61-75 198-200

8. it-i an equation and draw thegraph of:

a) a vertical lineb) a horizontal line

9. Write an equation of a line and.

draw its graph given the slope anda point on the lice.

.

Again, students should be'encouraged to draw reason-able sketches if they areto obtain the requiredequation or graph.

(Math Is/4, p. 199-201)

NOTE:

Many good examples ofproblems related to thissection are found in variousbooks. A reference to benoted is the recent NCTMbook, A Sourcebook ofApplications of SchoolMathematics (1980).

68

10. Write the equation of a linepassing through a given point and

a) parallel to a given lineb) 'perpendicular to a given line

71-75 214-216

11. Given two points:

a) draw the graph of the linepassing through them

b) write the equation of theline passing through them

(Math Is/4, p. 199-201)

61-69

OBJECTIVES


MATHHOLT 5

MATHIS/5

D. PRESENTATION OP DATA ANDDESCRIPTIVE STATISTICS

IIIIMENIMEMMININIMIIIMMINIa

1. Maintain previous skills oforganizing ungrouped (raw)data.

273-275 264-277

2. Calculate the mean, median andnode for grouped data.

3. Draw histograms and cumulativefrequency histograms (ogivea).

Draw histograms as a reviewof previous material. (Seesuggestions in Math 10applications.) Use thesehistograms to now constructa cumulative frequencyhistogram. Graphically,(see page 383 of Math Is/4)one can now determine anydesired percentile.There are various ways tocalculate percentile. Yourprescribed reference indi-cates one approach.

278-283

286

4. Demonstrate how to obtain andinterpret quartiles andpercentilet (graphically andfro. grouped data).

285,287

E. RELATIONS AND FUNCTIONS

1. Define a relation. Relations may be definedby ordered pairs, graphs oropen sentences. Emphasis atthis level should be onrelating an equation, itsgraph and its domain andrange. The study here shouldconcentrate on the line withsome introduction to circles,

parabolas and ellipses.

According to a Texaco advertise-ment, it costs $200,000 to drillone kilometre deep," 450,000 todrill for two kilometres and$1,000,000 to drill for threekilometres.

49-59 190,441

2. Define the domain and rangeof a relation.

190

6667

OBJECTIVES

Mathematics 20


MATHIS/5

3. Determine the inverse of arelation.

'Family relations actually areexamples which can be used toillustrate mathematicalrelations.

This would be an appropriateplace to introduce theconcepts of absolute valueand inequalities %s a way ofexpressing the domain andrange.

a) Graph these points, sketch acurve through them (not aline, for sure) and estimatethe cost of drilling 2.4kilometres.

b) Extend the curve to crossthe vertical axis and thisestimates the "fixed costs"of drilling in this situation- the cost even if no holeis drilled.

445,446

4. State the relationthip betweenthe domain and range of a relation s

and the domain and range of itsinverse.

5. State the relationship betweenthe graph of a relation and thegraph of its inverse.

446

6. Define a function. A function may be defined as"a relation which has no'fickle pickers'". Thefollowing examples willilaustrate this:

R = {(2,3), (2,4), OM}may be represented as:

2--3 Using the matching\,,, notation

4 2 is a "fickle picker"because

3-4.5 it picks more than onemember of the range.

S = 1(2,3), (4,5), (6,7))may be represented as:

2-'3 Ths:e are no "ficklepickers" so this

4-*5 relation is a function.6-'7

:.-

.

49-59

.:,7

191,441

OBJECTIVES


MATHHOLT 5

MATHIS/5

7; Use the functional notation f(x)in defining a function.'

37

8.. For particular values of x,find f(x).

Imbedded functions of theform f(g(x)) are notrequired,

37,38193

9. Define and graph a linear function. 61-63 194-196

F. QUADRATIC FUNCTIONS, EQUATIONS _

AND APPLICATIONS

-1. Identify and express quadraticfunctions in the fora y ax2+bx4cwhere a, b, c e R, a 0 O.

295-314 412-436

2. Identify and express quadraticequations in the form ax2 + bx + cwhere a, b, c e R, a 0 0.

3. Graph a quadratic function usinga table of values.

Students should becomefamiliar with the graphs ofquadratic fupctions of the

\ forms:

- ax2, - ax2 + bx, andy\7. ax2 + bx + c

717u

OBJECTIVES

Mathematics 20.COMMENTS APPLICATIONS

MATHHOLT 5

MATHIS/5

4. Find the vertex, axis of symmetry,domain, range and maximum orminimum value of a quadraticfunction from its graph.

The importance of the vertexand axis of symmetry can beillustrated using. appropriateproblem solving applications.

A road passes under a railroadoverpass in the form of aparabolic arch 5 metres high(at the apex) and 20 metres wide.The equation representing theparabolic arch is y 5 x2.

To

Find the height of the tallesttruck that4can pass under thearch. Assume the truck is 3metres wide, that it stays on itsown side of the roads and thatthe centreline of the road passesdirectly under the apex of thearch.

295-314 412-436

.

5. Use the formula for vertex and axisof symmetry if the quadraticfunction is given in the formy ,., ax2 + bx + c.

It is suggested that theformula be derived before itis used.

6. State the relationship between thegraph of a quadratic function andthe roots of the correspondingequation.

7. Write quadratic equations in thefora ax2 + bx + c 0 and specifythe value of a, la, c.

8. Use the method of completing thesquare of a quadratic function tofind the vertex, axis and symmetry,range and maximum or minimum value.Draw the graph using this information.

OBJECTIVES


MATHHOLT 5

MATHIS/5

9. Solve problems involving themaximum or minimum value of aquadratic function.

.

,

1

a) Theof

rectangularuse

sidesions

area?

b) Fast$4.00had

nightly.estimatessales

eachticketprofitable(A

usingbeconcept

N

Wishbonesfence

a sideof thewill

Freddie'sper

a full

would$1.00

cost.

simplea table

used toof

400

have

with whichdog

of therun,

give

ticket,

houseThe managerthat

decreaseincrease

Whatpricearithmetic

of

introducemaximum

350

30 metres

to make arun. If they

house as onewhat dimenthe maximum

Theatre chargesand it hasof 400

the ticketby 50 forin the

is the mostto charge?

solutionvalues could

the

value.)

3001 250

295-314 412-436

/...-',.)

C 4 5 6I 7

I 1600 1750 1800 11750

1..

Max.

c) The collision impact (I) of anautomobile with mass (m) andspeed (s) is given by theformula I ,.. kms2. If thespeed is tripled, what happensto the collision impact of a1000 kg car?

10. Compute the real roots of aquadratic equation by:

a) factoringb) using the quadratic formulac) completing the square.

74 75

OBJECTIVES


MATHHOLT 5

MATHIS/5

11. Define and evaluate the discriminantof a quadratic equation. .

295-314 412-436

12. State the nature of the roots byexamining the discriminant.

13. Solve problems whose solutionsbased on quadratic equations.

are a) A club bought a snowmobile for$720.00, planning to dividethe expenses equally among themembers. However, two memberswithdrew from the club,increasing each share 'by

b)

$4.00. How many members -f

were in the club originally?

How wide a strip must besodded around a square court

c)

d)

10 metres on a side, so thatonehalf of the court issodded?

A 6 centimetre square is cutfrom each corner of a squarepiece of sheet metal. Thesides are folded up to forman open box having a volumeof 600 cc. What is the lengthof a side of the originalsquare piece of sheet metal?

Two students agree to sharethe work of cutting the grasson a large lot measuring 80metres by 60 metres. One

student starts by cutting astrip all of the way aroundthe lot and continuing aroundin this way. How wide willthis strip be when half thework is completed?

OBJECTIVES

SMathematics 20

COMMENTS, APPLICATIONSMATHHOLT 5

MATHIS/5

G. SYSTEMS OF EQUATIONS

1. Solve linear systems of equationsboth algebraically and graphically.

c

It would be worth introducingsolutions by graphing beforethe algebraic solutions aretaught. In the graphicalsolution a student has achance to understand howequivalent systems can begenerated and used to arriveat a solution.

'More sophisticated methodssuch as general formulas anddeterminants may be left tobetter students for furtherexploration. In actualpractice, a combination ofaddition-subtraction andsubstitution will probablyprovi:.1 the best method ofsolution.

Problems involving two vehiclestravelling at different speedsand leaving a fixed point atdifferent times can.. be used.Using graphical or algebraicmeans, we can find out when thetwo vehicles meet.

83-88 120 -155

2. Identify systems ash having manysolutions, one sqlfition or nosolution.

Systems can be describedgraphically showing:

a) coincident lines(dependent system)

b) parallel lines(inconsistent)

c) intersecting lines(independent)

Students should be able todetermine the type of systemby examining the slopes ofthe lines.

.

76 79

OBJECTIVES

Mathematics 20


MATHIS/5

3. Solve problems based on thesolution of systems of equations.

Problems involving statementsabout numbers, supplementaryand complementary angles,speeds of two ,yehicles,digits of numbers andinvestments can be used.

93-95 120-155

4. Solve linear-quadratic andtwo-quadratic systems.

' The four conic sections canbe demonstrated using adisectable cone-. Thenstudents can be asked toconsider the number ofpossible intersections of astraight line and each ofthe conic sections. The samecan be done for two-quadraticsystems. This should de-monstrate that, unlike asystem of linear equations,we may have as many as fourpoints of intersection.Students are not requiredto solve these systems bygraphical means.

314-317 437,438

,

',..

H. GEOMETRY.

1. Define and illustrate the folloW=ing terms related to the circle:radius, chord, interior, exterior,arc, semicircle, segment, sector,central angle, inscribed angle,secant line and tangent line.

205-213 29'5,296

OBJECTIVES

. Mathematics 20 \\,


HOLT 5MATHIS/5

2. Apply the following basic theorems:

a) a line containing the centreof a circle that bisects achord, which is not a diameter,is perpendicular to that chord.(Include corollaries.)

b) a tangent line is perpendicularto the radius drawn to the pointof contact.

c) The measure of an inscribedangle i3 half the measure of thecentral anglecsubtended by thesame arc (or oangrueo:t arcs).

d) The sngle,between the tangentand the chord is one-half themeasure of tilt intercepted arc.

Have students draw:

1) The inscrit4d circle ofa triangle.

2) The circumscribed circleof a triangle.

%DiscJss circular forms In nature,in the community, in art, inindustry (e.g., wheels,pulley, gears) and so forth.

Students can produce designsusing circles.

205-218 320,321

,

3. Construct a tangent to a circlefrom a given point.

4. Find the length of an arc giventhe measures of the central angleand the radius.=

250-252

.

< -41

L

5. Find the area of a sector giventhe measures of the central angleand the radius.

For interest, using formulasfor perimeter and area,attempt to prove algebraical-ly that the greatest areathat can be enclosed withinany given perimeter is acircular area.

,

82 8-4

8 4

OBJECTIVES

Mathematics 20


HOLT 5MATHIS/5

6. Solve problems Involving numericalapplications of the above theorems.

250-252 321

'/I. TRIGONOMETRY

1. Maintain previously developedskills in solving right triangles.

*

,-

This section is intended as areview of the trigonometrystudies in Math 10. Morecomplex problems can beintroduced here.

'

a) To find the lenthrough a mountainobtuse tri lgle.

r"a,-,

A

a tunnelusing

.

239-240 80-84391

KnowingaboveB,

calCulated.

b)

Threein

the

ingthe

h (height of mountainAB) and the angles A and

the length of AB can be

a

holes are to ba drilleda rectangular plate. Find

lengths of a and b, knowthe value of x and one ofacute angles.

S

OBJECTIVES

Mathematics 20


MPTH

lei 5

2. Identify cosecant, secant andcotangent ratios as'reciprocalratios of sine, cosine and tangent.

230-232 391

3. Determine the relative measures ofthe sides of:

a) a 30-" 60 - 90 triangleb) a 45 - 45 - 90 triangle

228,229

4. Solve right triangles using any ofthe trigonometric ratios.

242-246 391-396

5. Solve the problems involving righttriangles including:

a) =angles of elevation anddepression

b) three dimensionsc) more than one triangle

Immimmom lismm =maw

Students should understandwhy angles of elevationand depression are equal.

J. VARIATION

1. Maintain previous skills of direct,inverse and partial variation.

Partial variation can beextended to include problemsinvolving two equations intwo unknowns.

177-184 168-183

2. Identify joint variation.

3. Solve problems related to jointvariation.

86 87

MATHEMATICS 20ELECTIVES

14.. \.-4 \

\t\ \11 1///..1:7

-,-1/4's "s \''':\\NI\ All li i r:////re/4... 7-4, N. \ 71_ 4 1\1 I rl I rify/

1.,-4.`t g. -4. S .1 1 t 11 e 0 e

--, '-,-.. %,'....N.'',...'4.":.1' l ik Ur tr e'r.i4'` ''' '

..--4 _ -7-4-4 4?-4. ':41 4'4 `r "1 I ! ". ; .'","..-'`',!--"''..,!"-- ..-- ....,_,........,-...--".-,-..., -,-....__--,,, -,,-,..--;.,-, la ..,, 0, I, .. r i ., f ,rf r-,,,,,,- ,,, ,,.r.,..-,-,...-." .......,..r,--r''''"-.-,-..7.."'-:"'"a.,",...."-:".s.-",-"-`. l'.4 's I.? II IJIIJI? -""or'.07'-4-'

,--..- --"--4.-_74,_,-4-...77-. -74,-,,........ . .4,, ;0 ,. : -:-...--77.---....r.r,-r-r-,- --a. . ---",--.7.--.-.'"7--:".-Cts:N.70- - !;, :. a '' --' --`"..--r-

.7....--r- a

47. -4, 4- - 47-, -I r -o.

11 N,ZT41%y , 4.%t ,

' e-4-'" ole . .F .0 R-1 ,r tee qg

..r J. .A.e. I 1, 1.2...N7.: NI,

I 1A, .4 *it nt

1

'1/411'

20

8i

ABSOLUTE VALUE

In some problems only the magnitude of a value is required. How much did the temperature changefrom 9:00 a.m. to 3:00 p.m. if it were 8°C at 9:00 a.m. and 4°C at 3:00 p.m.?

We would disregard the direction on a number line in order to solve the above. In other words, we woulduse the absolute value of the numbers.

References:

1. Ebos, F., and Tuck, B., Mdth Is 4, Don Mills: Thomas Nelson and Sons (Canada) Limited, 1979.

2. Dottori, D., Knill, G., and Stewart, J., Foundations of Mathematics for Tomorrow: Intermediate, Tr,-onto:McGraw-Hill Ryerson Ltd., 1978.

3. Carter, J., Clark, J., Porter, B., and Stouffer, M., Mathematics Alive 3, Toronto: Copp Clark Publishing,1978.

4. Bye, M., and Elliott, H., Math Probe 3, Toronto: Holt, Rinehart and Winston of Canada, Ltd., 1973.

OBJECTIVES COMMENTS/ACTIVITIES

1. Use the absolute value symbol correctly.

2. Use absolute value to indicate the measure ofthe distance between the point and the origin ona number line.

3. Use absolute value to tell the distance betweenany two points on the number line.

4. Solve equations like! 2x 41 =8

5. Find the solution sets of inequalities like

I3xl < 12 and graph the solutiOn set.

COMPLEX NUMBERS

Complex numbers- are a natural outgrowth,of the study f quadratic equations. Develoi.ment of a new numbersystem through the introduction of a new symbol can be ill strated. For example, in expanding th systemNo to I, we merely introduce a new symbol: the minus sign. Similarly for the complex numiers, w introducea new symbol, i, to represent 77 This allot'As us to solve quadratic equations which have no solution in thereal number system.

Although few practical applications of\ cOmplex numbers xist at a level which high scho 1 stud nts canunderstand, complex members are used extenFA,vely by electrical engineers. Several examples cf this, at a verybasic level, are included in reference 3. \

References:

1. Algebra Two and Trigonometry, Vogeli et al, Silver-Burdett\Company.

2. Holt AZgebra 2 With Trigonometry, Nichols et al, Holt, Rine art and Winston, 1974.

3. Using Adoaneed Algebra, Travers et al, Laidlaw Publishers, 1 75.

4. Algebra and Trigonometry: Structure and Method (.5-:)k Dol Tani et al, Houghton, Mifflin, 1.77.

OBJECTIVES OMMENTS/ACTIVITIES

1. Demonstrate the need for a new number systembeyond the real numbers2in order to solveequations of the form x + 1 = 0.

2. (a) Perform simple arithmetic operations withcomplex numbers of the form a = bi.

(b) Evaluate the conjugate and use it indivision of complex numbers.

(c) Solve for unknowns for equations in theform 2 + 3i = x + yi.

91 92


3. Graph complex numbers on an Argand diagram,(rectangular, coordinate system) using a vectorto represent the complex number.

.

4. Evaluate the absolute value (modules) of acomplex number and interpret it as the length ofa vector on an Argand diagram.

5. Solve quadratic equations which have non-realroots.

6. Solve higher degree equations by specialfactoring methods (sum or differences of cubes,differences of squares).

.

7: Find roots of a number algebraically. (e.g.,x + I = 0 can be used to find the 3 cube rootsof -1).

8. Find the roots of a number of graphical means,using an Argand diagram. (All roots representedby vectors with an equal length). ,

9. Solve application problems of compl numbers(impedence of an electrical circuit).

. ,

.

(I :Zv

INEQUALITIES

All or most of our past experience in mathematics has dealt with equalities. We can make use of thisexperience to solve inequalities since inequalities are solved in the same manner as equalities except whenmultiplying or dividing by a negative. In this case the direction of the inequality sign must be changed.

References:

1. Burns, A., Pinkney, R., and Del Grande, J. :&zthenatZ.cc A!odern World: Book 3;: Second Edition,Toronto, Gage Educational PublishingLtd., 1976.

2. Ebos, F., and Tuck, B., Math is 4, Don Mills: Thomas Nelson and Sons (Canada) LiNited, 1979.


1. Solve inequalities.

2. Graph the solution sets of inequalities.

94

LINEAR PROGRAMMING

Linear programming involves the attempt to maximize results and minimize efforts to produce thoseresults. Such attempts originated during World War II when the Allies attempted to maximize production andminimize costs. Manufacturing problems involving numbers of items and various other constraints, such astime required for production, can be analyzed in a systematic way using linear equations and inequalities.

The fundamental assumption of linear programming is: the maximum value of the parameter P,, for therelation P = Ax + By, occurs at one of the vertices of the polygonal region determined by the various con-straints in the problem. The maximum value of P can be found by examining these vertices.

References:

1. Ebos and Tuck, Math is 4, Thomas Nelson and Sons, 1979.

2. Travers, Dalton et al, Using Advanced Algebra, Doubleday, 1977.

cp 3. Wigle, Jenning and Dowling, Mathematical Pursuits Three, Macmillan of Canada, 1377.

4. Dotton, Knill and Seymour, Applied Mathematics for Today, McGraw-Hill Ryerson, 1976-

5. Hanwell, Bye and Griffiths, Holt Mathematics 4 (Second Edition), Holt, Rinehart and Winston, 1980.


1. Define the following terms: parameter, region,constraint, maximum point.

2. Draw the graph of a linear equation in theform Ax + By = C.

3. Draw the graph of a region defined by aninequality.

91,"

OBJECTIVES COMMENTS/ACTIVITIES.

4. Determine a region bounded by severalinequalities.'

5. Find the intersection of 2 linear equations bygraphical or algebraic means.

6. Find the maximum value of a perimeter P definedby P = Ax + By for a given set of constraints.

7. Determine the value of a parameter P at each of thevertices of the polygon which results fromgraphing all the constraint conditions on x and y.

8. Apply linear programming to solving business-oriented problems.

98

MATH ART

This unit is intended to give students an opportunity to apply in an interesting way some of the skillsand concepts learned in Geometry (both Euclidean and Analytic). Once a student learns some of the simpleprinciples of the artwork, he can create his own designs.

References.

1. Art Matii, Billings, Campbell and Schwandt (Action Math Associates, Inc.).

2. Creating Escher Type Drawings, Rannucci and Teeters (Creative Publications).

3. Graph Gallery, Boyle, (Creative Publications).

4. Paper Folding in the Classroom, Johnson (NCTM Publications)

(7, 5. Creative Constructions, Seymour (Creative Publications)

6. Line Designs, Seymour (Creative Publications).

7. Mathematics Teacher (A FTM)

(a) Tangram Mathematics, February, 1977, pp. 143-146(b) The Artist as Mathematician, April, 1977, pp. 298-308(c) Filing (tesselatjons), March, 1978, pp. 199-202.(d) Transformation Geometry and the Artwork of M.C. Escher,(e) Creativity With Colors, March, 1976, pp. 215-218.

December, 1976, pp. 647-652.

8. Geometry, Jacobs, (W. H. Freeman Publishing).

POBJ ECTIVES COMMENTS/ACTIVITIES

99

1. Graphing pictures

(a) Given a set or ordered pairs, locate theseon a coordinate system and join them toform a picture.

100O

OBJECTIVES COMMENTS/ACTIVITIES'

1. Graphing Pictures (continued)

(b) Given a set of linear equations, each with Student writes his name or suitable message in blocka restricted domain, draw the lines on a letters on graph paper, then analyzes each lettercoordinate system to form a picture, to write the equations which represent them

(restrict to straight lines).

2. Line Designs

(a) Draw line designs within one angle. Student creates his own pattern using basic construc-tion techniques for angles. Alternate rectanglescan be colored to form a pattern. The same approachcan be used with colored thread and small nails ona sheet of plywood to create string designs.

3. Geometric Constructions

(a) Perform basic constructions related to line Student analyzes and recreates simple drawings fromsegments, angles, perpendicular and patterns.

parallel lines.

(b) Draw a polygon within a circle usingruler and compass (e.g., hexagon, polygon, Student creates and colors his own designs.square).

4. Tesselatioils

(a) Draw a tesselation based on a rectangle, Students find examples of tesselations; e.g., linoleu ma triangle or other polygons. floor tiles, wallpaper, brick wall, etc. 0

5. Escher Drawings

(a) (An extension of tesselations). Draw a Students create their own Escher type drawings,simple tesselation based on Escher's work Students analyze some of Escher's paintings to findand analyze how it is drawn. geometric contradictions (e.g., "Belvoir").

101102


6. Paper Folding

(a) Demonstrate the following concepts .usingpaper folding:

angle bisectorbisection of a segmentline perpendicular to acentre of a circle

given line

(b) Construct a parabola, hyperbola and ellipseusing paper folding.

7. Tabgrams

(a) Rearrange polygonal shapes to form geometricfigures.

(b) Illustrate basic properties of polygons(e.g., form an.isosceles trapezoid andshow the non-parallel sides are equal bymoving the ,component parts around).

Using the 7 pieces of a tangram square, formrectangles, isosceles triangles, parallelograms andtrapezoids.

Use the tangram pieces to make up figures such as theletter "T", the number "1", a house, a cat, etc.

Q

103104

TRANSFORMATIONAL GEOMETRY

Euclid has been criticized for his use of the super-position argument to prove the familiar side-angle-side (SAS) congruence theorem. Simplified, the argument calls for one to pick up a triangle and attempt tofit it on another. While deficiences have sometimes been pointed out in Euclid's system of axioms, it may benoted that those seeking to evolve mor4 logically satisfactory systems of axioms have generally attempted toby pass the difficulty.

Although it may be necessary to justify every step of .a proof one should not lose theintuitive appeal of Euclid's approach. Most geometry teachers probably explain congruence by placing one tri-angle over another. E.G., by cutting out cardboard triangles, by tracing one triangle over another, etc.

Basically then, we could consider the result of Mapping a figure by some appropriate transformation.

References:

1. Bye, M., Griffiths, T. and Hanwell, A., Holt Math 4, Toronto: Holt, Rinehart and Winston, 1980, pages265-303.

2. Dottori, D., Knill, G., and Stewart, J., Foundations of Mathematics for Tomorrow: An Introduction,Toronto: McGraw-Hill Ryerson Ltd., 1977, pages 320-344.

3. Ebos, F., and Tuck, B., Math is 4, Don Mills: Thomas Nelson and Sons (Canada) Limited, 1979, pages 353-371.

4. he Mathematics Teacher (NC2M): Transformation Geometry and the Artwork of M. C. Escher, December 1976.pages 647-652.


1. To develop and describe the following differenttypes of transformations:

a) Translationsb) Reflectionsc) Rotationsd) Dilations

Graph paper should be used whenever possible.

(For obvious reasons, these are also referred to asslides or glides.)

A transparent plastic reflector called a MIRA wouldbe an excellent tool in teachinc reflections.

The hands of a clock noting its centre is a veryuseful example.

Most students know what similarity (same shape,different size) means. Use examples (photo enlarger,microscope, etc.), to explain dilations.

105106

cc

OBJECTIVES

0

Mathematics 30

COMMENTS APPLICATIONSMATHIS 6 FMT

A. TRIGONOMETRY

1. Maintain previously developed Review terme P'dociated with 251-255 175-178skills. a rectangular.coordinate

system; review the theoremof Pythagoras. Find CJ-ordinates of points on thex-axis and y-axis.

209-210

Note: The relationshipbetween dezrees, minutes anddecimal degrees should bediscussed.

2. Describe, circular paths usingthe initial: point and directeddistance of the path.

203-208

3. Define the unit circlex2 + y2 . 1.

The'unit circle can be thebasis for definition oftrigonometric functions; if

213-216,248

(x,y) is the terminal pointof the circular arc deter- .:

mined by 0, then x = cos 0and y = sin 0.

4. Determine coordinates of pointson the unit circle.

A good starting place wouldbe to consider the-points onthe unit circle where thex-axis and the y -axis inter-sect the unit circle.Continue by determining co-ordinates of pointscorresponding to angles whichare multiples of 30 degreesand 45 degrees.

129

OBJECTIVES

Mathematics 30


5. Define the trigonometric ratiosin terms of coordinates of pointson the unit circle.

Exemplify by determiningexact values of thetrigonometric ratios forangles which are multiplesof 30 degrees and 45 degrees.

213-216248

6. Find the domain and range of thesix trigonometric ratios.

These concepts are usefulwhen discussing the 'amplitudeof a trigonometric functiongraph. For example, y =sin 0 has a range-1 =sin 0 51. Thus itsamplitude is 1.

For y = 3 sin 0, the range is-3=3 sin 0:53. Thus itsamplitude is 3.

226 212-224

7. Solve simple trigonometricequations.

.

Coverage should, includeequations of the linear andquadratic format. Forexample,3 cos 8 + 1 = 2,2 sin20 + sin 0 = 1 and2 tang 20 3 tan 20 1 = 8.

Note that the game argument(e.g., 0, 20,10 should beused throughout the equation.

. .

219

243-246

.

' 9

231-238

8. Given the value of one of thetrigonOmetric ratios, evaluatethe 'other five trigonometricratios.

' .

,:.

Consider both exact andapproximate values.

,.

211-216

110 111

O

OBJECTIVES

Mathematics 30


IS 6 FMT

9. Derive and apply the followingidentities:

a) Quotient relations:

sin 0tan 8

8,cot 8 cos 8

.

L

239-243

.

231-238

..

.

cos 0 sin 0

b) Reciprocal relations:V

91

cscsin 0

81

seccos 0

,.'cot 9 1

tan 61 i

k-

c) Pythagorean relations:

sin20 + cos20 14-. .1

1 + tan20 sec20

1 + cot20 csc20

10. Derive and apply the following-identities:

a) Negative arc formulas:"X

sin (70) -sin ecos ( -8) cos 8tan ( -8) -tan 8

Identities involving extreme-ly long solutions should beavoided. Most identitiesrequire transformation ofthe side with the more com-plicated expression into anew form which is the same asthat appearing on the otherside. In some cases, bothsides will have to be trans-formed into the sameexpression.

220

413-415242-246179-183

.

i,i'tst

OBJECTIVES

Mathematics 30


10. b) Sun formulas:

sin (A+B) sin & cos B + cos A. sin B

cos (A+B) cos A cos B + sin-A. sin B

c) Complementary arc formulas: °

cos( - 8) sin 0

ain(2

- . cos 0

,

220

413-415242-246179-183

11. Draw and identify graphs of the sine,cosine and tangentfunctions.

Consider the effects of pare-meters a, h and c on thegraph of y .. a sin bx + c.Effects of changes in theseparameters can he consideredindividually, and then incombination. Concepts ofphase shift, amplitude andperiod should be introduced.

Students can chart their ownbiorhythms using sine curves.This is a highly motivationalapplication of trigonometricfunction graphs. See MathematicsTeacher, October, 1977.Amplitude, period and frequencyof sine waves can be demonstratedusing an audio generator and anoscilloscope. Assistance fromyour physics department may berequired. Periodic functionswhich are non-trigonometric suchas square-wave and sawtooth mayalso= be demonstrated if appro-priate equipment is available.Several computer graphingexperiments are now available.

226-232 208-217

__

12. Define periodic function andstate the periods of sin 0,cos 0 and tan 0.

1 I.

Mathematics 30

MATHOBJECTIVES . COMMENTS APPLICATIONS IS 6 FMT

)

13. Define radian measure.

l/l/-----

---

Radian measure can be used tofind the, length of an arc along ,agreat circle of a sphere (eg,the earth) by the formula S = r0.This formula requires 0 to be aradian measure. Radian measureis frequently used to describeangular velocity of a rotatingpulley or shaft, such as. is foundin farm machinery or cars.

205-209 208-217

14. Convert degree measure toradian measure and vice versa.

Conversion formulas depend onthe fact that an arc of acircle is proportional to thecentral angle which it intercepts. For example, a semicircle intercepts an angleof 180 degrees.

15. Determine exact values oftrigouGmetric ratios of

0', 30', 45', 60' and 90'.

.

A number such as liT is exact.This is a difficult conceptfor students who will readthe decimal equivalent of1.414 and say that it is moreexact. It should beemphaSized that the decimis infinite and has beenrounded off in the tables.

212-225 244-245

.-----/

117Ilk,---- .

OBJECTIVES

Mathematics 30


16. Determine the value. of atrigonometric ratio of any angle.

This may be developed easilyusing the addition formulasfor sin (A + B). For example,sin 1400 -. sin (90° + 500).

216-220223-225

182-207

Alternatively related anglesmay be illustrated bygeometric ideas such ascongruent triangles or reflections. In actualpractice students will findit easier to use the 3stepreduction of angles method.

1. Find the related angle.2. Use the same function as

the given one.3. Determine the sign of the

function.

17. Solve oblique triangles by usingthe Sine law and/or Cosine law.

Sine Law: Ambiguous Case -Problems in which two sidesand an angle opposite one ofthem are given may have morethan one solution depending onthe relattve measure of thesides. Better students maywish to pursue an analysis ofthis case in detail.

257-267

18. Apply the Sine law and Cosine law In aerial navigation, determine 257-267to practical problems. which course to fly to counteractthe effects of wind. In surveying, find inaccessible distances.

274-277

4

118119

14

OBJECTIVES

Mathematics 30

COMMENTS APPLICATIONSMATHIS 6

19. Solve problems involving areasof regular polygons.

Find the length of a side ofa regular polygon inscribedin a circle of known radius;also find the length of thesegment from the centre tohe midpoint of any side.

These quantities can then beused to find the area of thepolygon.

Find dimensions of a hexagonalbolt head, the radius of acircular rod from which-it mustbe cut and the length of eachside of the head.

412

AMOMMEMEMOOF

B. QUADRATIC RELATIONS (CONIC SECTIONS) .

_

1. Maintain previous skills in In addition to 2-variable x - xii 1.a. 402analytic geometry: systems, it is worth ex- 347 b. 402

tending this concept to three 360-365 c. 392a) linear functions and slope 'unknowns. This will be 396b) distance and midpoint formulasc) properties of tangentsd) solution of systems of

equations in two variables

required in some problemsinvolving the equation of acircle. The same techniquesof elimination of variablesby addition-subtraction orsubstitution can be used.

d. 425

Properties of tangents arethose referred to in Math 20.

2. State the definition of the circleand derive the standard form.

The definition involves theconcept of locus (path tracedby a point moving accordingto a certain rule). Formulasshould be derived for thecircle with centre as theorigin and also at some otherpoint off the origin.

370-378 397-408

(x- h)- 2 +(y- k)2 =r2

OBJECTIVES

Mathematics 30


3. Convert the equation of a circlefrom standard to general form.

When this conversion has beencompleted, students should beable to fihd thecentre-andradius by using the formulas.The equation of a circle ingeneral form is:

X2 + Y2 + AX + BY + C 2. O.

370-378 397-408

.

4. -Determine the equation of a circle These problems require one of 375-380and sketch the graph given these several approaches depending 402-404conditions: on the given information:

a) centre and radius a) direct ....:ostitution ofb) centre and a point centre and radius intoc) centre and equation of standard form

tangent line b) calculation of radiusd) three points on a circle using distance formulae) two points and equation of c) use Of tangent at, point

line containing the centre of contact.d) substitution of co-

ordinates.of points intothe general form and solv-ing the resulting system

,

of equations.

Care should be taken inassigning problems of type4(e) as these can be verylong and discouraging tostudents.

5. Define and identify a parabola _

and the terms: focus, vertex,381-382

.axis and directrix.

122 123

rn

OBJECTIVES

Mathematics 30


6. Derive the standard form of theequation of a parabola with ahorizontal or vertical axis ofsymmetry.

-

.

Use the "focus-directrix"definition of a parabola andthe distance formula. Thereare four equations requireddepending on whether theparabola opens upward, down-wa'id, to the left or to theright. This should inclu4parabolas whose vertices areoff the origin.

381-382 397-408

.....

7. Find he focus and directrix fromthe cation of a parabola:

1

,

\

.

if

,..

Use the formula derived inobjective 6 to determine theorientation of the parabola.By comparing the given equa-tion to one of the four .

types, the value of p can bedetermined.

y2 - -32x can be compared toy2 = -4px," hence p 8.-

(p is the distance betweenthe vertex and the focus.)

.

,

382-389406

8.

.

Determine an equation and sketch ,

the parabola,'given;

a) focus and directrixb) vertex and directrixc) vertex and point on the parabola

Determine the orientation ofthe parabola to find which ofcthe four equations should beused. Find the value of p,either from the diagram or bysubstitution of coordinatesof a known point into the

382-395

r,..,

o

formula.

n

124125

OBJECTIVES

Mathematics 30


9. Solve applied problems relatedto the parabola.

.

,

1.

2.

3.

Parabolic reflectors for sound(microphone at the focus) ormicrowave (antenna at thefocus).

Cut away view of an auto headlight.

Suspension bridge supported by

382-3?5

t

419-424434-435

a cable in the form of aparabola-.

Find the length of one of thecables supporting the roadway.

4. Parabolic pool table.

20mm plywood withparabola cut

F

out

6mm masonite,

.

plywood

sideview

masonite

To demonstrate the reflectionproperty of a parabola place arubber ball at F. Using apointer (pool cue) shootanother ball along any line

-parallel to the main axis.When the ball bounces off theparabola, it will hit the ballplaced at F. Conversely, theball placed at F, when shot

126 127

CO

OBJECTIVES

Mathematics 30

COM;ENTS APPLICATIONSMATHIS 6 FMT

9. continued

- k..

5

414,

towards the parabola, Willtravel outward along aparallel line.

5. Height of a parabolic archover a roadway.

_382-395 4-19,424-

434-435

10. Define and identify an ellipseand the terms: foci, major axis,minor axis, vertices, and focalradii.

406-412

11. Derive the standard form of the -equation of an ellipse with focion the x -axis or y-axis.

Use th "constant sum"definition and distanceformula to derive the equa-tion. \ The meanfng of theparameters a, 1-J, and c shouldbe clearly demOnstrated on agraph before the derivationis attempted.

.

12. Given the equation of the ellipse_determine: foci, vertices, majoraxis and minor axis.

Studente should recognize theform of the equation andhence the orientation of thegraph. The parameters a, b, ancc can then be determined.

I

13. Derive the relation between theparameters a, b and c for theellipse. (aZ - b2 + c2)

126 123_

OBJECTIVES

Mathematics 30


\

14. Determine an equation and sketchthe ellipse given:

a) minor axis and distancebetween foci

b) vertices and focic) vertices and a point on

the ellipse

A rough sketch can helpdetermine the required para-meters. In some cases, sub-stit;ition of the coordi-nato of a point on theellipse is required.Ellipses with centre off theorigin are not included.

382-395 406-412

15. Solve applied problems related 1. Orbital path of a Satellite. 419-424to the ellipse. 434-435

2. Whispering Galleries an

elliptical shaped room withtwo listening,pnits (one ateach focus 4: A personwhispefing at one focus willbe heard by a second personat the other focus because ofthe reflective properties ofan ellipse. A person stand-ing between the two foci willnot hear the conversation.

3. Eccentric Gears one turns ata constant rate while theother has a varying speed ofrotation. An ellipticalshaped cam in a car engineoperates in a similar way.

4. An Elliptical Pool Table todemonstrate the reflectiveprOperties of an ellipse.

1'1F.

A

20mm plywood

ball starting at Fi willreflect off the ellipse andpass through F2.

130131

0

co0

.

132_

0

Mathematics 30MATH

OBJECTIVES COMMENTS APPLICATIONS IS 6 FMT

16. Define and identify a hyperbolaand the terms: vertices, focitransverse axis, conjugate axisand asymptotes.

382-1q5

.

.

412-424

17. Derive the standard form of theequation of a hyperbola with focion the xaxis or the yaxis.

Use the "constant difference"definition and distanceformula to derive the equation.. The meaning of parameters a, b and c should beclearly demonstrated beforethe-derivation is attempted.

.

18. From the equation of the hyperbola,find the foci, vertices, transverseaxis, conjugate axis and asymptotes.

Students should recognize theform of the equation and

hence the orientation of thegraph. The parameters a, band c can then be determined.It should be noted that,unlike the ellipse, a is notalways greater than b.

19. Derive the relation between thet parameters a, b and c for the/ hyperbola.

/(a2 4. b2 . c2)

20. Determine an equation and sketchthe hyperbola given:

a) transverse axis andconjugate axis

b) foci and length of one axiscY equation of an asymptote and

a point on the hyperbola.

A rough sketch can helpdetermine the requiredparameters. In some cases,

substitution of coordinatesof a point may be required tofind one of the parameters.Hyperbolas with centres offthe origin are not included.

,0 07 133. .

OBJECTIVES1

Mathematics 30


21. Solve applied problems relatedto the hyperbola.

382-395 419-424434-435

C. LOGARITHMS

1. Maintain previous skill onexponents.

..,

142-202 135-174

2. Identify and graph exponentialfunctions.

This study of logarithmsshould emphasize thefunctions, their graphs andtheir use in applications.

Logarithms are frequently used todescribe rates of growth, ratesof decay, sound intensity, lightintensity and earthquakeintensity.

Sample Questions3. Convert equations from exponentialform to logarithmic form and viceversa. a. Loudness of sound is measured

on a decibel scale accordingto the formula:D .. 10 Log (L),

whereD is the number of decibels ofsound and L is the loudness ofthe sound. How many timeslouder is sound of 52 decibelsthan sound of 37 decibels?

b. A scientist predicts that 2Ugrams of a radioactive sub-stance will decay in such away that after (T) days thenumber of grams remaining(G) may be estimated accordingto the formula

G 20X -T/15

4. Solve logarithmic equations byconverting to exponential form.

-5 27log16x 4 logx T- = -3

1

1°g2 16 x

5. Define the inverse of an exponentialfunction in logarithmic fora.

Emphasize the relationshipbetween exponential andlogarithmic functions.

6. Evaloate expressions and solve ,

equations involving logarithmicform and exponential form.

The following should indicatethe depth to which this topic-should he taught.

log6(x + 3) + log6(x - 2) =

Find x if 2x . 3x = 15

134 135

00

OBJECTIVES

Mathematics 30


7. State and use the basic laws orproperties of logarithms for:

a) productsb) quotientsc) powersd) roots,

Logarithms are no longerneeded to do complexnumerical calculations. How-

ever, familiarity withthe properties of logarithmsis necessary to solve certainexponential equations.Students should be able to

recognize that:

log (2a) = log 2 + log a

log --4a- = log 4 -- log a

log a3 = 3 log a, and vice

versa

Determine to one decimal place,the number of days required forthe substance to reduce to8 g.

c. The length of time to doubleyour money at 12% is found bysolving

2 = (1.12)n for n.

142-202 135-174

,

8. ,Use logarithms to solve practicalproblems.

D. SEQUENCES, SERIES, LIMITS11.111MMINIMEMENIMMIN

434-438

1

254-289

,

o

1. ,Recognize:

a) the difference between a ' '

aequence'and a aeriesb) the difference between

a finite and infinite sequence

2. a) Recognize and define arithmeticsequences and series and statethe common difference (d).

Ben Business, a bright youngcollege graduate, sells popsiclesfor 144 each, while having to%buythem from the wholesaler for only3 1/24 each. As the cooler

439-445

460-465

OBJECTIVES

Mathematics 30

COMMENTS APPLICATIONSMATH1p_ 6 FMT

2. b)

c)

,

Derive and apply:

1. the general term formula

an .. al + (n 1) d

2.Athe sus formula

nSn 2

...--- (al + an)

Apply formulas to problemsinvolving arithmetic sequencesand aeries. -

e

The Nrmula,

Sn ' 7 [2a + (n 1)d]may be more convenient forsome problems and shouldbe derived from

Sn ' 11 [al + an)

.

weather approaches, Ben findshimself reducing the, price ofpopsicles by one cent a week,while his distributor, trying tomaintain his profit, raises thewholesale price by 1/24 a week.If Ben is really smart whenshould he pull out? , (7 weeks)

Quasi Motors Inc. produces-12units of farm machinery a du andsells them the same day to retaildistribution centres across the_country. If production costs arerisinjkat $3 per unit per day,and the-nompetitors are forcingthe sales prime down at $5 perunit per day, when -,should QuasiMotors stop production-q produc-tion costs are now $10 and-theprice of machinery is $150 per --unit? How much profit will theymake in that time? (13,872)

439-445460-465

254-289

3. a)

b)

Recognize and define geometricsequences and series and statethe common ratio (r).

Derive and apply:,

_

i.the general term formula

an ... al rn-1

ii. the sum formulas

al (rn 1)

Sn r i 1

.

If 100 is invested at 9%pounded annually, showamount grows over a termyears.- , . .

.r 01 O" .N' 100 MN' NONNINOIN .N1Nj OIN I

4%NNII.I'll .1

Notice that the amountsends of successive yearsterms of a geometric sequencewhere:

com-how the

of

'. -1ell .1,

at theform

n

Me

,

the

446-456

r

466-474

r - i

0. ran - al ,on r it 1

r - 1

138

O

140

OBJECTIVES

Mathematics 30


c. Apply formulas to problenminvolving geometric sequences,and series, with speciale.Apnasis being given to themathematics of finance:

i. Applications of simpleand compound interest.

ii. Use tables to determineaccumulated and presentvalue accounts involvingcompound interest overdifferent time periods.

iii. Illustrate the various 'z.

annuities by using linediagrams.

iv. Apply geometric series toboth accumulated and presentvalue annuities with bothidentical and differinginterest and payment periods.

Apply concepts of annuitiesto relevant amortizationplans, house mortgages, autofinancing and interest-accumulations.

al = 100 (1.09)r - 1.09and an is the nth term of thesequence.

The amount after n years:

an = airn-1

an = 100 (1.09) (1.09)n-1

= 100 (1.09)n

An insurance policy pays $30,000at age 60 which may be taken asa lump sum or in 30 equal half-yearly payments with interest at8% compounded semi-annually. If

the first payment is made 6months after the 60th birthday,how large is each payment?

466-474 254-289

4. Generate the terms of a series usingsigma notation (E).

41

the general termFor examole1 , 1

for 1 + T +1-6+ . . . can bewritten in E notation byfinding the general term ofthis geometric series.

457-459

141

OBJECTIVES

Mathematics 30


5. Determine the unite of variousfunctions. . , 2

.

.

.

Functions involved should belimited to those which after.appropriate algebraic. simpli-fication can be analyzedusing the fact that

lim 1 0

n 4,.,, n

, 475-480 442-453

6. Recognize the differences betyeeninfinite convergent and divergentsequences.

:1

7. Find the limits of infiniteconvergent sequences.

8.

.

Find the sums of infiniteconvergent series.

9..

Solve problems involvinginfinite geometric series.

,

.

E. PRESENTATION OF DATA ANDDESCRIPTIVE STATISTIC

1 lie.

1.

,.

Maintain previously developed skillswith ungrouped data and grouped lista.

a) Frequency diseibutionb) Measures of central tendencyc) Measures of diapers/on

,

v28'0-324

'

336-337

142 143

CCo

144

OBJECTIVES

Mathematics 30


. Develop and apply standarddeviation and z-scores.

X - TEQuality control and guaranteesin industry.

280-324 374-391Standard measure z ..a

may be introduced as ameasure independent of unitsused. This allows easy useof tables involving areasunder the normal curve.

3. Illustrate and develop the normaldistribution. _.

________-1-

,,.

.-----C- omparing the normal curveto histograms of progresssively narrower intervals(or frequency polygons)shows how a smooth curveis obtained from actualdata.

2. Relate measures of centraltendency and dispersion tothe normal curve.

Predictions, life expectancy,mortality tables.

4. Introduce probability using anexperimental approach.

,

Introduction to such terms asoutcome, sample space andevent would be appropriate.

[

356-360

5. Apply probability to theoreticalfrequency distribution.

Relate probability toprevious objectives as atheoretical long term fre-quency. Estimated probabili-ty is the relative frequencyof occurrence of the eventwhen the number is verylarge.,,

Find the probability120 tosses of abetween 40% andheads.

Lt in

fair chin60% will be

1.,_,,

)

l

1

374-391

x = 60

OBJECTIVES

Mathematics 30

COMMENTS APPLICATIONSMATHIS 6 PMT

5. continued

_

Given a -, 5.5

40% of 120 = 4860% of 120 - 72

48 converted to standard

-485.5602.2

280-324 374-391

units = =

78 converted to standard

72 60units .. 5.5 = 2.2

Required probability = areaunder the normal curve betweenz = -2.2 and z 2.2

P = 2(.4861) = .9722

F. POLYNOMIAL FUNCTIONS7

I. Maintain previous skills with ..

polynomials.

O

431 74-76127 -129

2. Classify a polynomial functionaccording to degree.

3. Define integral polynomialfunctions.

146 147

COCO

OBJECTIVES

Mathematics 30


4. Write polynomial functions (indescending order of degree) of thefora

f(x) ... aexn + aixn-I + a2xn-2. . . + a .x + an-i n

ae # 0, n E N

127-129

5. Divide integral polynomialfunctions in one variable bya binomial of the form x - a, ac Iusing long division and syntheticdivision.

-427-431 113-129-

6. Evaluate integral polynomialfunctions-for-given values of thedomain utilizing the UmsinderTheorem.

Students should he given someproblems which involve morethan elementary substitution.Example, ifP(x) - X4+ax3-bx2+28x-24is divided by x-1 the remainderis -4 and if P(x) is divided byk - 3 then the remainder is 6.Find "a" and "b".

,

7. Find factors of integral polynomialfunctions using the Factor Theorem.

,-

Students could be given poly-nomial functions with animbedded quadratic functioncontaining complex zeros andirrational x-intercepts.

146 14,}

OBJECTIVES

Mathematics 30

COMMENTS APPLICATIONSMATHIS 6 EMT

8. Determine the rintercepts ofintegral polynomial functionswhere x EQ.

427-431 113-129

9. Sketch the graph of the integral It would be meaningful topolynomial functions using the students when graphing tointercepts. introduce and explain how

graphs are affected byirrational intercepts andcomplex zeros. For example,polynomials may have

1. Unique xrintercept .2. Two equal xintercepts.3. Three equal xintercepts. _

, t4. Complex zeros.

0)0 ()

. 1r 1rI

150 151

MATHEMATICS 30ELECTIVES

'-1/4., -"-, Nrs'1:i\.1/4

...,___ -....--, , -...,-,-.. NoL.k, .,, I..; 1 Ir. odr.rfx.,,e-z-e;e---,.....,g,,.,2.6,4-.:..,Nytt f .1 re/ e.,":",,,`. o-f-r"---.--..._.--....

-7-........

r) t , 1, 1

rerrry,,,;4:r1,;,--...,,----:-,_._, --_,:_- ...-.....-17c1,-,,,--,g+. 0,,, ,.,,-.5. Er. w ,.4i.r.--,...-.4).:-.7 .,---tr-------r---"'...._.7-':7"--'7--..pF,,I,k.,%S .. z, -: -1.-.-c-.,.7 .-- ---- '. 7-"--.-,..-----.,a-14-. , --:. - . ',.2.--.7:.---

. xV...44.,-,s.,?..,:-.-.,-r---....%-----1. ..4r-:;;;;;;'-'.47- :-77--.."---7..---"...."7".._,.,...mr, "'olv,-W,,,,q14 ..-1-. ,?!:,:ic$4,;71Z-..4--..,

4.,.743,,rket,...,Ais '-',4. 4 - .7-;'7.74,"..n...:-'-...;;: 1 .Vr,:47.:4,*,..,-...,....,__...,---=:=7.-,...--"- ,..:::,_.-' A-i".'`;','Off 1 -,"'-, .,.07," 0-001,/, T

'-',.-7::,.:,-,- '''str,4u .4., CIP'40 ae .1' .P

#

ARRANGEMENTS AND SELECTIONS

References:

1. Nichols, E., Heimer, R., and Garland, E., Modern Intermediate Algebra, Toronto: Holt, Rinehart andWinston of Canada.

2. Johnson, R., Lendsey, L., Slesnick, W., and Bates, G., Algebra and Trigonometry, Don Mills, Ontario:Addison-Wesley Publishing Company.

3. Elliott, H., Fryer, K., and Gardner, Algebraic Structures and. Probability, Toronto: Holt, Rinehartand Winston of Canada.

4. Travers, Dalton et al, Using Advanced Algebra, Toronto: Doubleday Canada Ltd.

5. Dolciani, M., Berman, S., and Wooton, W., Modern Algebra and Trigonometry, Don Mills: Thomas Nelson andSons (Canada) Ltd.


1. Evaluate and/or simplify expressions involvingfactorials and /or IIPI,.

2. Solve word problems involving linear arrangementsof n different objects.

\

3. Solve word problems involving a linear arrange-ment of r objects, given n different objects.

4.. Solve word problems involving linear arrange-ments of one or more objects at a time, given ndifferent objects.

154

HISTORY OF MATHEMATICS

The history of mathematics can provide many interesting discussions and lends itself to many interestingprojects for the ofassroom. A look at historical topics is unlimited and is dependent only upon the creativityof the teacher.

Reference:

1. Historical Topics for the Mathematics Classroom, Thirty-first yearbook, National Council of Teachers ofMathematics, 1969.

NOTE: This yearbook also contains an excellent bibliography of the many books and articles available onhistorical topics in Mathematics.


1. To acquire an appreciation for the historicaldevelopment of mathematics from counting to modernday computers.

Discuss various types of early developments inmathematics.

Many of these can be set in activity stations, asdisplay areas or in written project form.

2. To humanize mathematics by looking at the livesof some of the 'mathematicians. ,

Investigate the lives of the famous mathematiciansby using different methods such as: reporting,writ,ing essays, role-playing, discussions, films.

3. To interrelate mathematics with other subjectareas such as science, music, social studies andart.

4. To familiarize students with thr use of variousmathematical instruments which have beendeveloped throughout the years.

Discuss, demonstrate the use of, or construct thebasic mathematical instruments for:

155156


4. (Continued) (a) Measurementsundial

waterclocktransit

sextantangle mirror

micrometercalipertrundle"wheel

(b) Calculationsslide rulelogarithmscalCulatorsNapier's bonesabacuscomputers

15/

MATHEMATICAL INDUCTION

The name, mathematical induction, is somewhat misleading because the process is deductive in nature,leading to a firm conclusion. It is usually employed in proving the validity of a statement involving allpositive integral values of "n".

For example, one. method of finding a square root on an ordinary computing or adding machine is based onthe fact that the sum of "n" odd integers is equal to n2.

C

1 = 1 = 1

2

1 + 3 = 4 = 22

1 + 3 +5 = 9 = 32

1 + 3 + 5 + 7 = 16 = 42

3 + 5 + + (2n-1) = n2

If this can be proven for (a) specific values of n and, (b) when nth posit'on = k as well as k + 1 whereright side equals left side then the conclusion (c) is verified.

References:

Johnson, R.E. , Lendsey, L.L. , Slesnich, W.E. , Bates, G.E., Ain."1P,7 and Trijonometv, London: Addison-Wesley Publishing Co., 1976.

2. Del Grande, J.J., Duff,,G.F.D., Egsgard. J.C., Mathematics 12 - Third Edition, Toronto: Gage PublishingLtd., 1980.

3. Vance, E.P., :,fatheatical Induct on and Conic Sections, London: Addison-Wesley Publishing Co., 1971.


1. Use mathematical induction to prove variousadditive algorithms.

158 159


2. Use mathematical induction to prove variousdivision algorithms.

3. Use mathematical induction to prove trigonometricalgorithms.

160

-

MATRICES

Tables are very useful ways of arranging information. Data from tables can be set in a more conciseform called a matrix where we write this data in columns and rows.

Reference:

I. Ebos, F., and Tuck, B., Math Is 4, Don Mills: Thomas Nelson and Sons (Canada) Limited, 1979.


1. To recognize and use the matrix as a conciseform of arranging information.

____

,

Construct a 3 x 3 magic square. Then rearrange thenumbers in the cells so that the sum of each new row,column and diagonal is also equal. How many arrange-ments are there if the square is to remain a magicsquare?

Do likewise for a ; 4 and a 5 x 5 magic square.

2. To comprehend the difference between columnsand rows.

Using everyday situations construct matriceswith different orders. E.g., team records.

3. To multiply matrices.

4. To apply matrices in solving problems.

162

161

PROBABILITY

There are numerous books which cover this topic, usually together with some statistics. in a short

elective component, it is important to just get the feel of calculating some interesting probabilities. If

one can experimentally verify or dispi-ove ones' results, so much the better.


The following concepts should be taught asa preamble:

1. The idea of an event E as the outcome of someexperiment of trial.

2. The meaning of P(E), the probability of thisevent occurring. Also the fact that probabil-ities are numbers between 0 and 1.

3. The concept of independence of events.

4. The meaning of mutually exclusive events.

5. If one has two events A and B, then:P(A or B) = P(A) = P(B) P(A and B).

If the above five ideas can be mastered, then one canfollow with some simple combinatorial results beforesome interesting examples can be tried. The studentshould learn about:

a) Factorial notation

b) that )= n! representsthe number of waysk: (n-k)! of arranging k objects among

n locations.

c) the hypergeometric distribution.

One has a box containing N1black marbles and N, white

.,.

marbles for a total ofN = N

1+ N

2marbles. If one now takes

n marbles from the box (at random without replacement)then the probability one obtains nl black and n, whitemarbles is/N

1/n

2

)

/N

7n n. 21 7-z.

e.g., if the box contains 5 black and 3 white marbles,then in drawing 2 marbles the probability one obtains oneof each colour is:

(5) (3\ /8) = 15

1) 11 2 28

16 3

OBJECTIVES. COMMENTS/ACTIVITIES

continued With these minimal tools, one can now tackle manyin'teresting -problems. For example, when dealt a5-card hand from a standard deck; following are thepoker hands one can obtain:

a) nothingb) one pairc) 2 pairsd) 3 of a kind (a triple)

e) full house (one triple, one pair)f) four of a kindg) flush (all in one suit)h) straight (5 cards in sequence)i) straight flush (5 cards in sequence in one suit)j) royal flush (highest straight-flush)

If one considers these 10 events as mutually exclusive, one can calculate the probability of each occurringand hence order, via probabilities, the sequence of winning events.

There is a multitude of variations on this one theme alone, namely what happens if you have a wild card?Does this change the order of winning events? All these can be answered, and if'in doubt, deal a few handsto check your answer.

Reference:

1. Most of these notations can be found in any respectable text. One in which the card hands are discussedand the probabilities calculated is: Basic Probability and Applications, by M. Nosal., Published byW. B. Saunders.

161

TOPOLOGY

Reference:

1. Excursions into Mathematics, by Beck, Bleicher and Crowe. Published by Worth.


1. Euler 's formula. Pages 3 7 11.

2. The number of regularjpolyhedra. Pages 12 16.

3. Tesselation of the plane: Use Euler 's formulato show that triangles, hexagons and squares arethe only regular polygons one can use to tile afloor.

4. A European "football" is made of pieces ofleather in the shape of regular pentagons andregular hexagons. These are sewed together so'that each pentagon is surrounded by hexagons andeach hexagon is surrounded (alternately)-by threepentagons and three hexagon.. Determine thenumber of pentagons and hexagons of such a-fOotball.

5. Deltahedra: This section deals with non-regularpolyhedra, all of whose faces are triangles. It

also illustrates methods of. constructing thesepolyhedra out of cardboard.

NOTE: The objectives listed are allfully illustrated in the-textwith numerous examples that canactually be constructed.

6. If time permits, there is a section of polyhedrawithout diagonals using Euler's formula.

10

VECTORS

Vectors provide a useful tool for analyzing problems that deal with trips, forces, velocities, accelera-tions and displacement. A vector is a quantity that requires both a magnitude and direction to describe it.

References:

1. Bye, M. and Elliott, H., Math. Probe 3, Toronto: Holt, Rinehart and Winston of Canada, Ltd., 1973.

2. Burns, A.G. Pinkney, R.G., and Del Grande, J.J., Mathematics for a Modern World: Book 3, Toronto, GageEducational Publishing Ltd., 1976.

3 Dottori, D. , Knill, G. and Seymour, J. , Applied Alat7zematlos for To1a;J: Tntermediate, Toronto: McGraw-HillRyerson Ltd., 1976.

4. Dottori, D., McVean, R., Knill, G., and Seymour, J., Foundations of Mathematics for Tomorrow: Introduc-tion, Toronto: McGraw-Hill Ryerson Ltd., 1974. Nichole, Modern Intermediate Algebra.

5 Dottori, D., Knill, G., and Seymour, J., Foundations of Mathematics for Tomorrow: Intermediate, Toronto:McGraw-Hill Ryerson Ltd., 1978.

6. Ebos, F., and Tuck, B., Math is 4, Don Mills: Thomas Nelson and Sons (Canada) Limited, 1979.

7. Hilton, A., Vectors, London: Macdonald Educational Colour Units, 1976.


1. Understand those phenomena which possessmagnitude and direction.

situationsDescribe five s.Ituations in which vectors are used.

2. Add and subtract vectors. Create and solve problems using vectors.

3. Multiply a vector by a scalar.

4. Solve problems using vectors. .

169 170

L

44: 4 1/4'

110,00 eridfie0,' noir, I P

1111 TO ao niA Nolifrli1fr

'141' mill'ibiZmI'71/4";1"'`1/4, '111:1"11lliltIZIN:e,"ur roil?' ,041iii. no.rAili-u_

raki% 1111{11

:1141

11-8. .1' ri 1 11i111 ' 1

LWAI:!"6171*:1/411,11. 1101Ip iLlilli 944111

11 0 rigian:ille:Pit '°.19h 1411111, 1111M "rligaA4111141 givkItt . Niki. ito,,

I/1 lllIbINPIlitp,h1/4hfilll;1/4a414111111., kul4),,tillo alliviinei .apit uk hiiii, ol'lripil..11nn,jh n7714,14 p I N 1411jIlligir11111ni 14 gti

Nr11111 1144111. 11"11"104114111 411'11

;10'1 iloper

"Rn .1111 11 9n9.. h4t.. kit 9en', 'Tn I lalj :1 .T.1,.,11,p11,.,...!tqliLYoio,

Nna '10141r4hill.,,P4111,11141;41. 111,14,,,

,11111iiIii;11,;;!::01nit0411

OroniiiinImp 1.1 41, I I ,ThR!lima.;

47413101119111111.2wiLm:m

410 I ,,,,Jratuili "NR,LnaudlilmII

111f HID lititnaimai

atin h)0,41 malimm"

110,;',!;6411,,-tintiara"'"r

iH191172 th .ir r . i.,,c114 1411,9rann,rint. I. %111'11111,1., n. ! in, 111".1 """R. IW"ni""""nn

n 11.!Hi !J*,q, .1161m,,,2711.n.g,,,,693. ".11tniRITINI*"411941..111"'",:fr i,,Artr,;711,4PlitiPtFoi,HIP4odiFIIRPtYf,;h4r

r'"211151"" "...141..1"4"711""riraT41,1:';'rellr)T"li'irj.-1410Purlrlliii:111111.011111;01r.r11Pliir0111 111111 1[171.

llvtitmtrinpn'holte!.1,.ar7,,,arik r el, el illP.

-1."11411

.7tIPPI4L,Fen,;p11rnirrilv,,,i-e. 11.4%0 illOten

en,,*-41141* P","' n FIR 110

',41.11hh

mrarlir fAntall11 ne011l ',err 01,10 00;;toti'

te' li14intrac'r' linnm4rmiliont"Im

. !PI.'111.1111111''14

;7414'" 'Fr Ala-ir04111f1,1:Ing41"} Virir 411 110:11.1,,,liiii,h,

PPIV rrrri'rrerr" Oen' rrf 11°V 011

rr.....N..1.1"A'rerjneor'r

'SPISnl

tr- peporreipre \\1411n,

rff"" Er'141 erg" Or'

ti

omz

cr)

Alberta METRICATION POLICYEDUCATION

METRICATION POLICY

It is-the policy of Alberta Education that:

1. SI units become the principal system of measure-ment in the curriculum of the schools in theprovince;

2. the change to the use of SI units in schools be suchthat the instructional programs are predominantlymetric by June, 1978;

3. changes in the curriculum of individual schoolsproceed in concert with corresponding changes inindustry and commerce;

4. selected consultative and in-service resources bemade available to teachers for professional prepa-ration in the integration of SI units in their instruc-tional programs;

5. conversion of the schools to the metric system (SI)be carried out on a pre-planned basis with a gradualreplacement and/or modification of measurement-sensitive resources;

6. conversion costs will generally be borne by theresponsibility centre incurring them;

7. conversion will be accomplished by mr ns ofexisting administrative structures, and tl re willbe only a minimum number of short-term special,purpose assignments associated with the change toSI.

EXPLICATION OF THE METRICATION POLICY

1 . . that SI units become the principal system ofmeasurement in the curriculum of the schoolsin the province;

1.1 As part of a world-wide movement to stand-ardize commonly used elements of trade,Canada has committed itself to the use of SIunits by the early 1980's. It is already evidentthat many sections of our society which usemeasures will be predominantly metric before1980.

It has become imperative that the school curric-ulum prepare students to cope with metricmeasures in all facets of their life. In additionmany students Will require a detailed knowledgeof more specilized units used in specific careerfields.

1.2 Some imperial units will be in use for some time;therefore as teachers introduce SI they are

expected to retain selected reference to imperialunits. This teaching of specific imperial unitsshould be related only to those that are relevantto student needs and should be kept to aminimum. Mathematical conversions from onesystem to the other are to be avoided whereverpossible.,

1.3 Resource materials for the classroom shoulduse SI units as the principal measuring systemwith only such references to the imperial or"old" metric systems as are unavoidable.

2 . . . that the change to the use of SI units inschools be such that the instructional programsare predominantly metric by June, 1978.

2.1 In a predominantly metric program the student,depending upon age and ability, is familiar withthe appropriate metric units of length, area,volume, mass and temperature, and uses theseunits in school for making and expressingmeasurements in all areas of the curriculum.

2.2 The date (June, 1978) is a general target bywhich it is expected that curriculum guides andprograms of studies and the various acts andregulations will have been changed to reflectthe new measurement language. Most schoolswill have begun the transition of their classroomprograms by June, 1978; indeed many schoolswill have completed the change before this date.

2.3 It is entirely possible that some of the highschool technical and vocational programs willhave a need to continue teaching the use ofimperial units. It may take a specific techno-logical area some considerable time to make thechange, especially one that uses machinerywith a long working life.

2.4 The correct usage of the SI units will be directed,by the Metric Practice Guide (CSA Z234.11973) or the Metric Style Guide (Council ofMinisters of Education, Canada). A copy of theMetric Style Guide has been made availableto each teacher (in English or French).

3 . . . that changes in the curriculum of individualschools proceed in concert with correspondingchanges in industry and commerce.

3.1 The Canadian Metric Commission under thefederal Ministry of Industry, Trade andCommerce has been coordinating the change tometrics since 1970. Timelines and metricationdates have been established for many sectorsof the economy. Educators should .keep them-selves informed as to progress in both the

104

1 72

IP

technical areas in which they may be teachingand the activities occurring in the community.

3.2 The Department of Education will attempt tokeep schools informed regarding progress in theuse of metric units.

o4 that selected consultative and inserviceresources be made available to teachers forprofessional preparation in the integration of SIunits in their instructional programs.

4.1 Achieving metric conversion by the target dateis dependent upon many factors, not the leastof which is the whole-hearted cooperation byeducators in carrying out their assigned roles.In preparing teachers and administrators tobetter cope with the changes, the Departmentof Education will provide consultative andinservice resources through the RegionalOffices in Grande Prairie, Edmonton, Red Deer,Calgary and. Lethbridge.

4.2 The staff of a school should plan co-operativelyto bring a unified approach to the teaching anduse of measurement. Metrication goes beyondthe formal content of the curriculum. Thechangeover with its many implications willaffect everyone associated with the process ofeducation.

4.3 Since measurement is an activity relatedprocess, it follows that individuals teachersas well as students learn the metric unitswhile measuring. That is, some degree of activeinvolvement in measuring or in using the unitsthat one is learning.

4.4 Very few people will have to know the entiremetric system. That is, when presenting SI, oneshould only attempt to deal with that part of thesystem which is necessary for the task at hand.

5 . . that conversion of school programs to SI becarried out on a pre-planned basis with agradual replacement and/or modification ofmeasurement-sensitive resources.

5.1 Most measurement-sensitive devices will haveto be replaced over the next few years. Items

ranging from inexpensive rulers to costly, metallathes will have to be either replaced ormodified. Exactly when and how much modi-fication versus replacement occurs dependsupon the economics. For a machine that is nearthe end of its useful life, any modifications willhave to be minimal. For a nearly new machinea scale replacement or recalibration may be thebest course of action. In any event, commonsense must prevail.

5.2 Any new acquisitions of machines or toolsshould be those with metric capabilities. Inspite of the best planning and careful budgeting,there is bound to be some frustration assuppliers fail to deliver as promised and aspeople do not respond as predicted.

6 ... that conversion costs will generally be borneby the responsibility centre incurring them.

6.1 The term "responsibility centre" is meant torefer to those parts of the administrative struc-ture which are responsible for budgeting andpurchasing services, materials, and equipmentfor schools.

6.2 By placing responsibility for costs as' close aspossible to the actual use of goods or services,it is hoped that there will be a greater account-ability in the changing over to metrics.

6.3 Implicit in this policy is the idea that thoseexercising responsibility will do so withrestraint.

7 . . . that conversion will be accomplished bymeans of existing administrative structures, andthere will only be a minimum number of short-term special-purpose assignments associatedwith the change to SI.

7.1 In other words, as far as is possible, there will beonly temporary positions associate,d with metri-cation. In addition, administrative costs areto bekept as low as possible, with the additionalwork load being handled on a contract or short-term assignment basis.. In this way, it is hopedthat the money spent on metrication will havemaximum effect on the classroom.

105

173

Strategies of Problem Solving

In the teaching/learning of problem solving, an instructionalapproach should be used which helps students learn and chooseprocedures for solving problems. These procedures are easy tostate and recognize, but they are often quite elusive when teaching.Difficulty%requently exists when teaching problem solving because,unlike the teaching of computational skills or concepts, there isno specific content involved. In problem solving, an individuallyacquired set of processes is brought to bear on a situation thatconfronts the individual.

There are generally four procedures (steps) which appearinherent in problem solving. These'procedures, their descriptionsand associated strategies.have been compiled and .adapted from avariety of sources and authors (George Polya, J.F. LeBlanc, OhioDepartment of Education, Math Resource Project, 1980, NCTM Yearbook)and are listed below:

1

STEPS IN PROBLEM SOLVING

) UNDERSTAND THE PROBLEM'

What is the problem? What are you trying to find? What is

happening? What are you asked to do?

Suggested Strategies:

Paraphrase the problem or question (Restate the problem inyour own words to internalize what the problem entails.)

Identify wanted, given and needed information (Helps

students focus on what is yet to be determined from problemstatement as well as listing information so that they maybetter he able to discover a relationship between what is

known and what is required.)

- Make a drawing (May help to depict the information of aproblem, especially situations involving geometric ideas.)

Act it out (Helps to picture how the problem actions occurand how they are related thereby giving a better understandingof the problem.)

Check for hidden assumptions (What precisely does theproblem say or not say? Are you assuming something thatmay not he implied? Beware of mistaken inferences,)

106

2) DEVISE A PLAN TO SOLVE THE-PROBLEM

What operations should you use? What do you need to do to solvethe problem? How can you obtain more information or data to,seek the solution?

Suggested Strategies:

Solve a simpler (or similar) problem (Momentarily setaside the original problem to wort, J a simpler or similarcase. Hopefully the relationship'of the simpler problemwill point to the solution for the original problem.)

Construct a table (Organizing data in tabular form makes iteasier to establish patterns and to identify-information whichis missing.)

Look for a pattern or trend (woes a pattern continue or exist?In connection with the use of a table, graph, etc., patternsor trends may be more apparent.)

Solve part of the problem (Sometimes .a series of actionseach dependent upoj the preceding one, is required toreach a solution. Similarly it may be that certain initialactions will either produce a solution or uncover additionalinformation to simplify the task of solving the problem.)

Make a graph or numberline (May help organize informationin such a way that it makes the relationship between giveninformation and desired solution more apparent.)

Make a diagram or model (When using the model strategyattempt to select objects or actions to model those fromthe actual problem that represents the situation accuratelyand enables you to relate the simplified problem to theactual problem. May be used in connection with, or in placeof other similar strategies, i.e., acting out'the problem.)

Guess and check (Guessing for a solution should not beassoc:ated with aimless casting about for an answer. Thekey-element to this strategy is the "and check" when theproblem solver checks his guesses against the problemconditions to determine how to improve his guess. Thisprocess is repeated until the answer appears reasonable.An advantage of this "guess and check" strategy is that itgets the individual involved in finding a solution byestablishing a starting point from which he can progress.Used constructively with a table or graph this strategy maybe a valuable tool,)

175107

Work backwards (Frequently, problems are posed in Which.the final conditionS' of an action are given and a conditionis asked for which occurred earlier or which caused thefinal outcome. Under these circumstances working backwardsmay be valuable.)

Change ybur point of view (Some problems require adifferent point of view to be takqp. Often one tends tohave a "mind set" or certain perspective of the problemwhich creates a difficulty in discovering a solution.Frequently, if the first plan adopted is not successful,the tendency is to return to the same point of view andadopt a new plan. This may be productive, but might alsoresult in contiguous failure to obtain a solution.Attempt to discard pr'evious notions of the problem and tryto redefine the problem in a completely different way.)

Write an open sentence or equation (Often in conjunctionwith other strategies using a table, diagram, etc., oneselects appropriate notation and attempts to represent arelationship between given and sought information in anopen sentence.)

3.: CARRY OUT THE PLAN

For some students the ,Strategy/strategies selected may notlend itself/themselves' to a so.ution. If the plan doesjlotwork, the problem solver should revise the plan, review step 1,and/or try another plan or combination of plans from step 2:

4. LOOK BACK AT THE STEPS TAKEN (Consolidating Gains)

IS the result reasonable and correct? Is there anotWr,method of solution? Is there another solution? Is obtainingthe answer the end of the problem?

Generalize (Obtaining an answer is not necessarily the endof a problem, Re-examination of the problem, the resultand the way it was obtained will frequently generate insightsfar more significant than the answer to the specific situation.It may enable the student to solve whole classes of similarand even more difficult problems.)

1 0-8

176

Check the solution (The very length of .a problem er theFact that symbolic notation is used may'tend to make onelose sight of the original problem. Does thd answer

`appear reasonable, does it satisfy all the problemrequirements?)

,- Find another way to solve it (Can you find a better way toconfront and deal with the problem? The goal of problemsolving is to study the processes that lead to solutionsto problems. Once a solution -is discovered, search theproblem for further insights and unsuspected ideas andrelationships.)

Find another solution (Students tend to approach many,problem situation with the expectation of only one'.correct solution, In many practical, daily life situationsthere-may be many answers that are correct and acceptable.)

Study the solution process (Studying the prOcess ofsolution makes the activity of problem solving more thananswer-getting and can expand an individual problem intoa meaningful total view of a family of related problems.)

It must be noted that the four steps Of the above model arec', not necessarily discreet, For example, one may move without

notice into Step 2 while attempting to generate more informationto understand the problem better,

If the 4-step model is used the key is to 5elect anappropriqte strategy or strategies to help answer the questionssuggested byeach step. The strategies listed, and those devisedby students wift hopefully alter the problem information, organizeit, expand it, and make it more easily understood. Strategiesthen may be,thought of as the tools of problem solving and the4-step model, the blueprint.

109 1 7r7

Recommendations for School Mathematics of the 1980s

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

The National Council of Teachers of Mathematics recommends that

1. problem solving be the focus of school mathematics in the1980's;

2. basic skills in mathematics be defined to encompass- more

than computational facility;

3. mathematics programs take full advantage of the power ofcalculators and computers at all grade levels;

4, stringent standar-cis of both effectiveness and efficiency beappl-ied to the teaching of mathematics;

5. the success of mathematics programs and student learningbe evaluated by a-wider range of measures than conventional

testing;

6. more mathematics study be required for all students and aFlexible curriculum with a greater range of options bedesigned to accommodate the diverse needs of the s,:udentpopulation;

7. mathematics teachers demand of themselves and theircolleagues a high level of professionalism;

8. public support for mathematics instruction be raised to alevel commensurate with the importance of mathematicalundqrstanding to individuals and society.

110 178

Documents

DOCUMENT RESUME SE 046 043 McCardle, Lisa, Ed. · DOCUMENT RESUME ED 261 911 SE 046 043 AUTHOR McCardle, Lisa, Ed. TITLE Mathematics 10-20-30 Curriculum Guide 1983. INSTITUTION Alberta