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TITLE Mathematics Curriculum Guide, Grades 7-12.oNorth Carolina State Dept. of Public Instruction,Raleigh. Div. of Mathematics. 1

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58p.

RP-$0.83 HC-$3.50 PluS Postage.Course Content; *Course Descripticns; *Curriculum;*Instruction; *Objectiies; Secondary Education;*Secondary School Mathematics; *State CurriculumGuides

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ABSTRACTThe global objectives and content outlines of

mathemat' courses for grades ',through 12 are listed and theoverall$iDnale and goal of the courses are discussed. Severalsugges014p are made for changes in the curriculum., The contentsinclude; ik;,C-nrriculum of Choices, Grades 7 and 8, GeneralMathematics, Introductory Algebra, Algebra I, Geometry, AppliedVocational Mathematics, Algebra II, Consumer Mathematics, AdvancedMathematics, Calculus, and Additional Curriculum Considerations.(MP)

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mathematicCURRICULUM GUIDE, GROVES 7 -12DIVISION OF MATHEMATICS/ NORTH CAROLINA DEPARTMENT OF PUBLIC INSTRUCTION

4presii- d-,

he mathematics= curriculdm in the secondary schools cannot be,'described as a fixe'd body of knowledge that is never changing: Instead, the.curriculum is forever changing, slowly at times and more rapidly on other oc-\'casions. These changes in the curriculum are- often the result of societalpressures 'or technological advances. As the result of such pressure; and \advances during the late 1950's and early 1960's, a major restructuring of thesecondary mathematics curriculum was undertaken.

Another way changes odcur in the curriculum is as the-result of the ef-forts of teachers in a particular school. These teachers identify a need for achange in the curriculum. They develop alternatives and experiment withthem. Then, if their efforts 'are successful, the courses they develop ere soonadopted by other schools.

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The developers of this guide have made an efforTtiianalyze objectivelythe' status of the curriculum that exists in many schools across North%.Carolina. As a result of this analysis, several suggestions are includedin this'guide for changes in the curriculum. Hopefully, some of the suggeitions in-

/ cluded in t is publication will provide teachers with the impetus tcitry them intheir §ch&ols and share their findings with others.

Th: high school mathematics curriculum should be broad enough topro vi. : a full sequence.of elective courses for non - college -pound studentsas wall as college-bourfd students. This publication is an attempt to provideguidance to those SChohl systems wishing toexpand their course offerings inm thematics fa-- the extent that appropriate coursesare ,available for each

udent,at every grade.The primary purpose of this guide for secondary school mathematics is

to provide an aid for local personnel involved in curriculum planning and ex-perimentation. This glide provides a general orientation and frameworupon which a more detailed plan which meets the needs of the communitycan be built.

Two objectives of this guide are:To propose a pattern of mathematics offerings that is flexible. anddiverse enough to accommodate many local innovations andinterpretations.

:st,To provide descriptions of different mathematics /courses thatschools can use in building a curriculum that insure theavailability of a substantive mathematics program for their students,who upon graduation, might enter the work force, technical school orcollege

A. Craig PhillipsState Superintendent Public Instruction

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he direct involvement of teachers in development of curriculumpp lications is a practice of .long standing in the Department of Public In-4truction. We value the services we receive from them and are .particularly

/ grateful to the following members of the North Carolina Mathematics Ad-z' visory Council for their assistance in the preparation of this publication:

Mrs, Margaret Callahan, Shelby Junior High SchoolMr. Richard Cutler, Farmville Middle SchoolMrs. Geraldine Deans; Wilson County Schools

( Dr. Joseph Dodson, Winston-Salem/Forsyth County SchoolsMrs. Ann Hooker, Leroy Martin. Junior High. SchoolMrs. Kathryn Kirkpatrick, Haywood County SchoolsDr.' John Kolb, N. C. State UniversityMr. Jim Murphy, Graham High SchoolDr. Alvin Myrick, Greensboro City SchoolsDr. William Palmer, Catawba CollegeDr.. William Paul, Appalachian State UniversityMrs.- Mary Prescott, Buncombe County SchoolsMrs. Shirley Raper, Wayne Community CollegeDr. Joseph Schell, UNC-CharlotteMiss Jean Taylor, John T. Hoggard High SchoolMrs. Elizabeth Thonlas, High Point City SchoolsDr. William Waters, N. C. State UniversityMrs. Rebecca Weatherford, St. Augustine's CollegeMr. Paul White, Morehead High SchoolMrs. Betty Williamson, Richmond Senior High School

Two members of this committee, Dr. John Kolb and Dr. William Waters,prepared a reaction paper on the secondary mathematics curriculum. Manyof their ideas helped form the substance of this publication.

Robert R. Jones, DirectorDivision of Mathematics

Introduction

A Curriculum of Choices

Grades 7 and 8Mathematics 7Mathematics 8 , f \

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General Matl-i# tics . 10\ ',1:

Introductory Alkebra ,' 13Part I i 14/Part Ii 15

Algebra I 17

Geometry . / 20

Applied Vocational MatheMatics !' 24

Al9etira II - . / 27

Consumer Mathemati A 31

Advanced Mathema s/ \ 35

Advanced Alger ra 38Analytic Georrtry 39Trigonometry/ 40Probability and Statistics 42

Calculus 45

Additional CuOiculum Considerations 48Role of the Computer in Education 48Algebr II with Computer Programming 48Engi ering Concepts Curriculum Project 49Alge ra: An Applied Approach 50Ge try: An Applied Approach 50Teçihnical Mathematics 51

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The major portion of this publication isdevoted to lists of glObarobjectives and contentoutlines of mathematics courses for grades 7through 12. Each Outline is only one of the manypossible. It is hoped.that individual school unitswill develop their own outlines, taking. into ac-count. the specific needs of the pupils theyserve.-

Mathematics teachers must consider theinterests, aspirations,and abilities of individualstudents as they develop mathematics orb-grams. It is not enough to teach students themathematics they must know today; they must,if possible, be given .the foundation that willserve them in their world of tomorrow. Thissuggests that attempts must be made todevelop within students the ability to contin etheir studies under their own directions tenable them to learn the mathematics they willneed to survive in a changing world. Provisions,must 6e made for adequate pre-professionaland vocational backgrounds, simultaneously,emphasizing the role of mathematics as a vitalpart of general education.. The role of mathe-matics as an independent discipline must be

shown, at the same time not neglecting its manyapplications. The teacher should introduce stu-.

Pr dents to current developments in mathematicsso as to make students,awareof the fact thatmathematids is d subject that is alive and everchanging. This enrichment may get students in-volved in individdal and small interest-groupresearch projects beyond the regular coursecontent of the curriculum.

The mathematics program of the elemen-,tary school must be considered along with thatof the secondary schools since mathematics ofthe junior and senior high schools cannot beisolated. k sequential and well-coordinatedprogram should be devel?)ped. In planning sucha program, representatives from all educationallevels should be involved.

The mathematics program described inthis publication reflects the view that the basiccomputational skills introduced in the elemen-tary grades require further development. Theextension, refinement, and maintenance ofarithmetiC processes and concepts are impor-tant components to be stressed in the secon-dary school mathematics program.

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Shown in Figure 1 is a proposedframework for planning secondary schoolmathematics programs. Beneath the myriad ofchoices is the basic change of providing asequenCe of mathematics courses that arealternatives to but co-equal in mathematicalvalue with the traditional Algebra r, Georrietr4;Algebra II, and Aelvanced Mathematics se-quence.

Courses s as Algebra I, Algebra II,Geometry and Advanced Mathematics have areasonably well-defined core of content that isbasically the same from class to class, schoOl tpschool, and state to state. The teacher has tomake provisions for adapting the material to theindividual student through well, planned teach-,ing strategies. On the other hand, ConsumerMathematics, Applied Vocational Mathematicsand General Mathematics do nOt contain a bodyof fixed subject matter. Instead the subject matter can be drawn from any part of mathematicsand its fringes or related fields. The course con-tent cannot be fully prescribed until the stu-dents and their inputs are considered.

It should be emphasized that ConsumerMathematics and Applied Vocational Mathe-matici' are different from each other and fromother courses. In particular, these. are NOTcourses in remedial arithmetic. The only com-putation taught in these courses is for main-tenance and extension of existing skills, notcompensatgry instruction for lack of skills. Anyremedial instruction in arithmetic required bystudents at this level .shbuld be provided in anovel nbn-traditional way. Students who aftereight years have not mastered computationalskills often have problems so serious they can-not be solved in the trailneed the specialized intenan individualized diagnoproach in a remedial clini

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ional manner. Theyve care provided bytic prescriptive ap-

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Students who complete two or more ye s"study in "college preparatory" mathema csbeyond Geometry Should have an utidestanding of the non - optional topics contained irr'the syllabus for Algebra II ,plus one or more ofthe modules listedsted under Advanced Mathe-maticsAt-the trigonometry topics are ihcludedin Algebra II, this course can be followed by Ad-ydnced Mathematics (self-contained or :inmodules). If, however, the trigonometry topics',areoot included in Algebra II, then this courseshould be followed by one or more of themodules from ..Advanced .Mathematics, one ofwhich must be Trigonometry. ,

Several course outlines include sugges-tions for optional 'topics. It is hoped that themathematically, talented students will beprovided the opportunity to study many of thesetopics. -

For advanced students, a third yearbeyond Geometry could be offered by combin-ing additional modules from Advanced',Mathematics or by teaching CaleulUsi. Schoolsoffering Calculus are encouraged to irivestigatethe feasibility of teaching Advanced PlacementMathematics. This program enables students tocomplete college-levelstudies while still in highschool. Information on the Advanced Place-ment Program can be obtained by writing:College Entrance Examination Board, Box2815, PrincetOn, New Jerzsey 08540.

The grade level under which a course islisted indicates where it is ordinarily taught.However, this listing should be used only as asuggestion. With careful planning, any courselisted might be taught at a different grade level.

The italicized course titles in Figure 1 arediscussed in the section of this guide titled "Ad-ditional Curriculum Considerations."

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Figure 1

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e Mathematics Prcieem Sequences, ,Grades '1,,12

Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12

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REGULAR SEQUENCE

Mathematics 7 Mathematics 8 Gitrel MathematicsMathematicsMathematics

Vocati

Algebra I *BuoinesS Mathematics

Consumer Mathematics

Introductory IntroductoryGeometry , Algebra II

Algebra (Part I) Algebro.(Part II) ..,

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Algebra I Geomairy Algebra II' "Advonced MathOmallii

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ACCELERATED SEQUENCE

\Whom tics 7, Mathematics 8, Algebra I Geometry Algebra!' "" Advanced Mathematics Caldulus

,COMPUTER RELATED SEQUENCE ,

Algebra I Geometry

AlgIbra IIEngineering Concepts

WithCurriculum Poled

Compaer Programming

Mathematics 7

POSSIBLE SEQUENCE FOR APPLIED MATHEMATICS

Algebra*. An Geometry,* tiMathematics 8,

Applied Approach Applied Approach

.v ,0 ITechnical Mathertiitics

Business Mathematics is described in the curriculum guide For Business Education.

** Advanced Mathematics can be taught as a survey course or as a series of !local courses which could include Trigonometry,

Analytic Geometry, Advanced Algebra, and Probability and Statistics.

The tonics of major importance at this levelare, numbers, operationS, algebraic concepts,problem solving, probability and statistics, in-tuitive geometry, measurement, and, graphsand, scale drawings.'The notion of a set shouldalso be studied, not as an isolated topic\but as itis 'used to unify mathematical concepts'.

NUMBERSOne of the major goals of the mathematics

program at this level is to provide students withan understanding of the rational numbersystem, its structure and the properties that itpossesses. Emphasis should be placed on howthese properties can often make computationeasier. Once students undeastand these ideasand principles, they can begin to investigate theintegers as another number system. A brief in-troduction to the irrational numbers can also bemade as,tiine permits.

The number line could prove to be a usefulmodel for illustrating the numbers in each num-ber system studied as well as the relationshipsbetween them. Models frorn the real worldshould be used whenever they are appropriate.

Concepts that shoUld be, studied in con-junction with number systems are square roots,multiples, prime numbers, composite, numbers,scientific notation, and ratio and proportion.

OPERATIONSThe ability to work with numbers is a

necessity if individual's are going' to havemathematics serve them in their daily living. It isexpected that the computational skills devel-oped in earlier grades with whole numbers andfractional numbers will be maintained and ex-tended here. It is important that students learnthe inversenature of the operations of additionand subtraction; and that division is theoperational inverse of multiplication.

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The number line is an effective model forillustrating the operations of addition, subtrac-tion, multiplication, and division with the-in-tegers. These operations should also be relatedto real life situations. Efforts should be made todevelop an understanding of how operationswith the integers differ from operations in othernumber systems. In particular, students shouldinvestigate how the structural properties thatwere not true for Other number systems may betrue for the integers.

One of the most useful mathematical ideasis that of ratio and proportion. Students shouldexplore many of the properties of a prdportionas well as how to solve problems using theproportion. Since the solution of problems in-volving proportion requires the use of m ethan one operation, studerits should be given. agreat deal of practice in this area.

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ALGEBRAIC CONCEPTSMany of the concepts of arithmetic can be

made much easier by the use of the equation:percentage, for example. The mathematicalsentence should be introduced early In a still.dent's study of mathematics. Associated withthe mathematical sentence is the concept of avariable (a symbol whiCh represents an un-specified member of a set) and the meaning ofequality (identity).

Along with the equation, the following con-cepts should be introduced: inequalities, for-mulae, and properties of equations in one,andtwo unknowns. A study of the Cartesian coor-dirvta system and its use in plotting a pointwhen the_coordinates are given and in findingthe coordinates of a given point should be in-cluded. These ides are to be developed as abasis for the study'bf graphs as pictures of func-tions in Algebra 1.

PROBLEM. SOLVING The' properties Of parallelograms. canrtiessums of vari pbeAs mathematical skills and concepts are developed, and the angle dUsoly-

developed, they are fixed by applying, them in : gons can be examined. The circle, its rdlus,

spOrts, transportation, and money management and the measure relationships existinl.betweenieltican h

existing beiproblem solving situations. Problems relating to 4 -diameter, and circumference,

can be used. Ideas and data from current them Can be analyzed. , ,provide material for other. meaningful prob- constructor's withcostrnippUalastsedc;nOnbdeitdions using

Constructrons,biological, physical, and chemical experiments includingBasic geometric

!ems. Puzzle problems may ,ISo be used. These straight edge and :re I:opieicids.

spective drawings might fbe.

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introduced.course, thus avoiding their presentation asproblems may be interspersed throughout the

isolated topics.

The construction of theshould be considered. some scale

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student should understand the difference bet,;tween a plane figure and the region Which it enr.:

PROBABILITY AND STATISTICS ' closes. Formulae should be developed and "'.used to compute perirbeters,Thinking, planning, and decision making , areas, and

are often Influenced by one's ability to interpret volumes, including the area of a sector of a pir-numerical intprrnation. Success or failure in an ` cle. A study of special triangles should,individual's life can sometimes be affected by (equilateral, isbsceles, scalene, a'ilif. right).

beliVide

statistics. The ,increasing importance of theseAlong with ,applications of `the pythagoreanrelationship,;'other concepts in geometry that

her or his knowledgepand USe of probability and

two mathematical ideas in toLlay's world makes Should receive aftentian here include theit necessary, fOr students to become more geometry on the coordinate grid, symmetry,knowledgeable of them, In particular, they similarity, and ratio problems involving similarshould know how data.are collected, presented - figures.(charts, tables, and graphs), read and inter-preted, and the meanings of central tendency(mean, median, and mode). They should also MEASUREMENTknow'the meaning of a probability ratio and be Second to computation, measurement

of measure. (length, area,

able to distinguish between theoretical and ex- probably the most frequentlyperimental probability. mathematics. 'Students should gain an under-

standing of measurement andcapacity,

INTUITIVE GEOMETRY . mass, and temperature) in both the metric

to workStudents should be ifilFoduced to some Stu -

geometric concepts at this level. They shouldsystem and the U.S. Customary System. Stu-dents should also ledrn with . angle

know what a simple closed curve 'is and should measure and be provided with opportunities tobe familiar with the various common polygons, estimate the measure of objects.circles, and ellipses. They should also, become During the teaching of measurement thereacquainted with certain solids (spheres, cubes,*

..is no place for conversions between the ti.S.

cones, and cylinders). system.Following this recognition and identifica-

Customary System and the metricHowever, comparison of units through estima-

tion level, the analytical stage shauld begin.There should be a discussion Or undefined

tion is both practical and example of goodanpedagogy in mathematics. As measurement

terms (point, line, and plane). The students applications are being learn ed, students shouldshould understand that diagrams on paper are be taught now to*convert, but not to Convert. Inmerely representations ofivthe,ideas of point, specialized cases, it may be necessary to teachline, and plane. They should have some,con- students specific, conversion techniquescept of the following: a simple closed figure,

,whichmhill emphasize the use of, tables and calculators.

parallelism, a polygon, angle, surface, face,edge, and vertex. Through exploratorY, activi-ties, they should be led to discover the idea ofcongruence and the basic relatid1ships Otjstittgbetween angles formed by intersectiVig -linesand between -angles formed by twaparaffel

4lines cut by a transversal. atol.z

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GRAPHS AND SCALEIDRAWINGSGraph are used extensively td display

'date. They are effectiveway to commdnicate'information quickly and clearly. Attentionshould be4given to the bar, circle, and broken-line graphs. Some time shou,{d also bVdevotedto the graph of- pointS on &number line, andpoints on a rectangular coordihate system. Notonly do Students need to learn how to displaydata in a graph, they als0 need to learn how tointerpret data presented in a graph and tp drawinferences from the data presented there.

The use of scale drawings is almost com-monplace in today's world. Students should betaught to interpret and -construct scale draw-ings. Maps, charts, and house plans are some

of scale drgwings that should beetudied. The construction of scale drawingsallows students to make use of some of theknowledg.": they have .gained in the .study ofgeometitY and measurement. .For example, tomake a scale drawing of some object of shape,the student has to draw angles the same size asthe original and either increase or decrease thelengths of the corresponding sides in a fixedratio.

Mathematics 7OBJECTIVES

1. Oeview or develop computational skills involving, the whole, fractional, and decimal numbers.2. Explore some of the basic concepts of the geometry. of two and three dimensions.3. Use some of the properties of number theory as ameans of developing an understanding of the,

algorithms involving whole, fractional, and decimal numbers. '`4. Provide opportunities for students to learn to estimate the answers to mathematical problems.5. Examine the concept Of measurement and then apply this to situations both real and contrived.

Both the U.S. Customary and metric systems should be examined; however2,emphasis shouldbe on the metric system.

6. Investigate problem solving techniques and then make use of these throughout the year.7.. Explore same of the elementary notions of probability and statistics.8. Provide students experiences in reading and constructing graphs gathering data and recording

this graphically.

ON ENTI. NUMBER CONCEPTS

A. Readingand writing numeralsB. Historic numeration systemsQ. Pffice value systemsD. Rules for rounding numbers .

E. Pooiers and roots

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II. MATHEMATICAL SENTENCESA. True sentences, false sentences, and open sentences

VariablesC. NumberAine representations of solution sets

NATURAL NUMBERS AND ZEROA. Basic principles of addition and multiplicationB. Algorithms of addition and multiplicationC. Inverse operationsD. Addition, subtraction, multiplication, and division algorithmsE. Estimation and,approximation

IV. FACTORS, P,PMEAND COMPOSITE NUMBERSA. Tests for divisibilityB. Review and definitionsC. Greatest common factor and least common multiple

V. NON-NEGATIVE RATIONAL;igUMBERS(Fractions)A. Changing to higher, terAilsB. Changing to lower terms and simplifyingC. Finding the least common denominatorD. Addition, subtraction, multiplication, and division (vertical and horizontal notation)E. Finding the multiplicative inverse (reciprocal)F. Comparing fractions (equal Cr not equal, greater than or less than)G. Ratio and proportionH. PercentI. Using fractions to solve problemsJ. Estimation and approximation"K. Properties of operations on fractional numbers

(Decimals)A. Reading and writing (to four decimal places)B. Adding, subtracting, multiplying, and dividing (vertical and horizontal notation;

expanded notation)C. Changing fractions to decimals and decimals to fractions Oerminating and repeating

decimals)D. Rounding numbers (nearest thousandth, hundredth', tenth, one, ten, hundred, etc.)E. Multiplying and dividing by powers of tenF. UsiN decimals to solve problemsG. Estimating and approximating

VI. GEOMETRYA. Sets of points in spaceB. Measurement of segments and anglesC. Sketchs of geometric figuresD. Constructions

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VII. MEASUREMENTA.cPerimeters, areas, and volumesB. Formulas in measurementC. Standard units of measure emphasizing the metric systemD. Appropriate units of measureE. Estimation

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V111. INTEGERS (Optional).A. Negative integersB. Addition of integers&C. Properties of additiM -D. Multiplication of integers .

E. Propettles ofmultiplicationF. Subtraction and division of integersG. Number sentences,and pictorial representationsH. Estimation and,approximation

I-X. MATHEMATICAL SYSTEMS (Optional)A. Properties of binary operationsB. Systems with rational numbers for elementsd. Mathematical.systems from physical modelsD. Abstract systems

Mathematics. 8OBJECTIVES

1. Develop proficiency with the oparations involving integers.2. Maintain previously learhed .computational skills with whole, fractional, and decimal numbers.3 Explore some of the structural Properties of the whole, fractional, and decimal number systems.

Particular emphasis should be placeds on the usefulness of the e properties in simplifying com-putations.

4.. Extend previously gained knowledge about geometry to coprdinate geometry.Investigate problem solving techniques and mak'e use of these in solving mathematicalproblems.

6. Provide experiences which will enable student§ to estimate answers with a reasonable degree ofaccuracy.

7. Develop techniques for solving lineIir equations and inequalities with one variable.8. Study, on an informal basis, some of the ideas of probability and statistics.9. Provide students experiences in reading graphs, gathering data and recording this graphically.

CONTENTI. NUMBER SYSTEMS

A. Natural Numbers1. Prime and composite numbers2. Place value3. Order of numbers4. Scientific notation5. Greatest common factor6. Least common multiple

B. Integers1. Negative integers2. Addition of integers3. Properties of addition4. Multiplication of integers5. Properties of multiplication6. Subtraction and division of integers7. Number sentences and pictorial representations

C. Rational Numbers1. Fractional numbers2. Decimal numbers8

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3. Equivalent and non!equivalent4. Order relationship5. Terminating and non-terminating6. Estimation and approximation

D. Real Numbers1. Rational numbers2. Irrational numbers

E. Structural Properties of Various Number Systems1. Closure2. Commutativity3. Associativity4. Distributivity5. Identity elements: 0 for addition and 1 for multiplication

II. OPERATIONS AND NUMBER SENTENCESA. Addition, subtraction, multiplication, and division of whole numbersj'ational numbers,

and integersB., Linear equations with one variableC. PercentD. Ratio and proportionE. Powers and rootsF. Estimation and approximation

III. ,GEOMETRYA. Language of one, two, and three dimensionsB. Informal experiences in one, two and three dimensionsC. Coordinate geometryD. Pythagorean theorem

IV. MEASUREMENTA. Standard units emphasizing the metric systemB. Approximation and estimationC. Direct and indirectD. Experiences with length, area, volume, and temperature

V. GRAPHS AND SCALE DRAWINGSA. Bar, circle, and broken-line graphsB. Scale drawings

VI. ALGEBRA OF POLYNOMIALS (Optional)A. Polynomial expressionsB. Operations and polynomial elementsC. Equalities and inequalities

VII. INFDRMAt PROBABILITY AND STATISTICSA. Mean, median, and modeB. Certain and uncertain events (optional)C. Meaning of a probability ratio and oddsD. Random and biased sample (optional)E. Equally likely events (optional)F. Frequency distributionG. HistogramH. Theoretical versus experimental probability (optional)

TRIGONOMETRY (OptionallA. Trigonometric ratios - sines cosine, and tangentB. Using the tables for trigonometric ratios

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COURSE NUMBER 2030

The purposes of this course are (1) toteach or reteach computational skills involvingwhole, fractional, and decimal numbers; (2) tosurvey some of the other areas of mathematicslike geometry, measurement, probability,fstatis-tias, and elementary algebra; and (3) to ex-amine some of the applications of mathe-matics. The approach for teaching com-pCitational skills needs to be novel, as it is un-likely that a person who has previously been un-successful in learning computation will sud-denly learn )t in another year of similar contentand methodology. Games, contests, puzzleSTlaboratory activities, outdoor mathematics andthe hand-held or desk calculator could be usedto motivate students to want to learn how to docomputations.

One way of organizing this course isaround topics. Efforts should be made to in-clude experimuts, games, surveys, and com-puting devices k the study of as many topics as

feasible. The major concern should be that stu-dents discover mathethatical ideas. The go.,tivities in the course should teach the studentshow to learn and add meaning to theirunderstanding of mathematical ideas. In theprocess students would perform computationsas needed. The course could be developedaround various combinations of the followingtopics: elementary Istatistics; patterns; flowcharts; calculating devices; mathematicalrecreations; rational numbers; measurement;experimental probability; equations and rela-tions; ratio, proportion, and percent; graphs,formulas, and patterns; mathematical reason-ing; metric and motion geometry; and creativegeometric constructions.

If General Mathematics is to be limited toremedial arithmetic, then major emphasisshould be placed on the first three headings inthe content outline.

OBJECTIVES1. Provide experiences which will enable students to become proficient in the basic computational

skills involving whole numberS, fractions, and decimals.2. Explore some of the basic concepts of the geometr'y of two and three dimensions.3. Make use of some basic concepts of number theory as a means of developing an understanding

of the important properties of the whole numbers.4. Provide experiences which will enable students to become proficient in estimating answers and

doing mental arithmetic.5. Present experiences in ratio and proportion and then_ use these in solving problems involving

percentage, similar triangles, and num ical trigonometry.6. Explore some of the elementary princt les of probability and statistics.7. Extend the students' knowledge of numbers to the integers. Then explore the addition, subtrac-

tion, multiplication, and division of integers.8. Develop the ability to solve linear equations and inequalities.9. Present problems, both real and contrived, which can be solved by means of equation solving.

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10. Emphasize problem solving techniques.11.. Present arithmetic concepts in a non-traditional manner' and on a level in keeping with the

maturity of they@ students.12. Examine some o the applications of mathematics which Make use of the mathematics explored .

in this course.

CONTENTI. WHOLE NUMBERS

A. Addition.B. SubtractionC. Multiplication-D. DivisionE. EstimatiOn and approximationF. Raising to a powerG. Finding the root of a numberH. Applications invflving whole numbers

H. FRACTIONAL NUMBERSA. Additi,onB. SubtractionC. MultiplicationD. DivisionE. Estimation and approximationF. Basic operations involving mixed numeralsG. Applications involving fractional numbers

Ill. DECIMALS1

A. AdditionB. SubtractionC. MultiplicationD. DivisionE. Estimation and apprciximationF. Applications involving decimals

IV. INTEGERSA. Meaning and representation of integers on a number lineB. Addition.C. SubtractionD. MultiplicationE. DivisionF. Applications involving integers

V. INFORMAL GEOMETRYA. Geometry of two dimensionsB. Geometry of three d4nensionsC. Applications of geometry

VI. LINEAR EQUATIONS AND INEQUALITIESA. Solving equations by additionB. Solving equations by subtractionC. Solving equations by multiplicationD. Solving equations by divisionE. Solving equations involving more than one operationF. Problem solving involving equationsG. Solving linear inequalities

VII. MEASUREMENTA. LinearB. AreaC. VolumeD. TemperatureE. TimeF. Indirect

PROBABILITY AND STATISTICSA. Gathering data including samplingB., Analyzing dataC: Reporting data including histograms and graphsD. PrObability of an event

IX RATIO AND PROPORTIONA, Using ratios and proportiorts to solve prQblemsB. Scale drawings

\tiC. Percent as a ratio

X. NUMERICAL TRIGONOMETRYA. Sine, cosine, and tangent ratios .

B. Using a table of trigonometric ratios to solve problems

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Essentially, this is an. Algebra I coursetaught over a two-year period. In addition to be-ing taught at a slower pace, the presentation ofthis content should be of .a less abstract naturewith less stress being put on precision of lan-guage and Should include many opportunitiesfor reinforcement.

In addition to algebraic topics, some atten-tion should be devoted to arithmetic, in par-ticular in the sense that some algebraic con-cepts can be thought of as being generalizedarithmetic. It is hoped that students.Will improveand extend their computational skills as theresult of experiences in this two-part course.

Whenever appropriate, applicatrons ofmathematics should be presented: Many stu-dents have a need -to see the usefulness ofalgebra in order for them to become motivatedto study it. In particular, they want to know howthe mathematics they study applies to the worldin which they live.

Some of the elementary concepts ofgeometry should be included so as to enablestudents to acquire an understanding of the in-terrelatedness of geometric and algebraic con-cepts.

s.OBJECTIVES-

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1. To use algebraic ideas as a vehicle for improving computational skills in arithmetic.2. To examine the structure of algebra at an informal level.3. To utilize some of the structural aspects of various number systems in explaining marry of the

algorithms of algebra.4. To develop facility in algebraic-computations.5. To examine several types of applied problems which require algebra ideas for their solution.6. To explore some_of the elementary aspects of geometry with particular emphasis on the in-

terrelat4ness of(geometric and algebraic concepts.1107

4.

19 13

Part ICONTENT

I. \ LANGUAGE OF ALGEBRA'A. VariablesB. Order of operations .

C. Using punctuation marks in numerical phrasesD. Exponents

PROPERTIES OF NUMBER SYSTEMSA. ClosureB. Order (commutative) propertyC. Grouping (associative). propertyD. Inverse elementsE. Identity elements

III. WHOLE NUMBER OPERATIONS

1\

A. Addition, subtraction, multiplication, divisionB. FactorsC. MultiplesD. Prime factorsE. DivisibilityF. Tests of divisibilityG. Least common multiple (LCM)H. Least common divisor (LCD)

OPERATIONS WITH FRACTIONS AND DECI.,SA. Addition, subtraction, multiplication, divis.B. RatioC. ProportionD. PercentE. InterestF. Discount

V. OPEN SENTENCESA. Addition property of equalityB. Subtraction property of equalityC. Multiplickation property of equalityD. Division property of equality

VI. MEASUREMENT INVOLVING U.S. CUSTOMARYON THE METRIC SYSTEM,

A. LengthB. AreaC. VolumeD. Mass (weight)E. Temperature 4

F. Indirect measurement

VII. INTEGERSA. Number lirje,-B. Opposites (additive inverse)C. ComparisonD. Number line addition,E. Number line subtractionF. AdditiOnG. Subtraction

14

AND METRIC UNITS WITH EMPHASIS

we.

,

H. MultiplicationI. Division

VIII. EQUATIONS AND PROBLEM SOLVINGA. Addition property of equalityB. Multiplication property of equalityC. UsiN equations to solve problems

IX. INEQUALITIES OF ONE VARIABLEA. Using addition to solve inequalitiesB. Using multiplication to solve inequalitiesC. Using more than one operation to solve inequalities

X. GEOMETRIC FIGURESA. Points, lines, avd plane figuresB. CurvesC. Angles.D. TrianglesE. Pythagorean TheoremF. Polygons

PartCONTENT

I. REAL NUMBERSA. Locating points on the number line ,

B. Order relationshipsC. AdditionD.' SubtractionE. MultiplicationF. Division

II. SOLVING EQUATIONS AND INEQUALITIES OF ONE VARIABLEA. Graphing solution sets on the number lineB. Using the addition property of equality to solve equations and inequalitiesC. =Using the multiplication property of equality to solve equations and inequalitiesD. Using more than one property to solve equations and inequalities

Ill. PRODUCTS AND FACTORSA. ExponentsB. FactorsC. Prime factorsD. Greatest common factor

IV. POLYNOMIALSA. Adding polynomialsB. Subtracting polynomialsC. Multiplying polynomialsD. Raising polynomials to a powerE. Dividing pblynomialsF. Removing a common monomial factorG. Computing the products of binomialsH. Squaring a binomialI. Fadtoring a perfect square trinomial

rx

15

J. Determining the product of sum and difference of two binomialsK. Factoring the difference of two squaresL. Factoring a trinomialM. Doing complete factoring

ALGEBRAIC FRACTIONSA. Simplifying fractionsB. Adding fractionsC. Subtracting fractionsD. Multiplying fractions'E. Dividing fractions

VI. SOLVING EQUATIONS WITH FRACTIONSA. Equations with fractio-nsB. Algebraic equationsC. Using algebraic equations in problem solving

VII. SYSTEMS OF EQUATIONSA. Equations with two variablesB. Graphing on the coordinate planeC. Graphing equations on the coordinate planeD. Slope of a lineE., Slope-intercept form of a lineF. Solving systems of equations by.graphingG. Solving systems of equations by additionH. Solving systems of equations by substitutionI. Using multiplication in solving systems of equationsJ. Using systems of equatiOns to solve problems

VIII. EXPOK NTS AND ROOTS

K._A. Powers of numbers

`B. Irrational numbersC. Square roots of numbersD. Roots other than square roots of numbersE. Simplifying radicalsF. Multiplication and `division of radicalsG. Addition and subtraction of radiCalsH. Approximations of the values of real numbers

IX. LINEAR AND QUADRATIC FUNCTIONSA. Meaning of a relationB. Meaning of a functionC. Graphing linear ft_inctionsD. Graphing quadratic functionsE. Graphing parabolas (optional)F. Graphing hyperbolas (optional)G. Graphing circles (optional)H. Quadratic equationsI. Solving quadratic equationsJ. Quadratic formulaK. Solving problems involving quadratics

TRIGONOMETRY OF THE RIGHT TRIANGLEA. Similar figuresB. Sine ratioC. Cosine ratioD. Tangent ratioE. Solving problems using trigonometric ratios

/6

I IP I

The major goal of Algebra I, noi'mallydied ir} the ninth grade, is to develop within

each student, skill in certain algebraic tech-niques and some understanding of algebra as alogical structure.

The course might start with a study ofsome elementary concepts and notions concer-ning sets and sentences, such as equality, sub-sets, mappings, the empty set, variables, andopen sentences. This study would be for thepurpose of supplying the student with conceptsand language useful for clarifying fundamenthlideas in mathematics, such as operation, func-tion, and graphs. The amount of time neededfor this aspect of the course would dependupon the nature of the students/ mathematicaltraining in previous courses. For some stu-dents, this part of the course (as well as some ofthe topics mentioned later) would be a review,and could be covered rapidly.

Algebra I should include a careful treat-ment of the real number system and its varioussubsystems: the natural numbers, integers,rational numbers and irrational numbers. Oper-ations of addition and multiplication and the im-portance of the commutative, associative, anddistributive properties goverr.g these opera-tions should be stressed. Justification by meansof these properties of manipulative maneuversof algebra shotild be made where feasible. Theulhique roles played by 0 and 1, the identity ele-ments, should be pointed out, as should be themeanings of inverse elements and inverseoperations.

The number line should be employed to il-luminate such processes as operations and theextension of a number, system to a larger one.Set terminology should be used where bene-

7

ficial in clarifying concepts, but should tbt bestudied as a separate entity.

Solution of simple equations should becarried out by logical steps based on explicitlystated axioms, with attention paid to the rever-sibility of the steps.

The order properties of the rational andreal number systems and the solution of in-,equalities by application of these propertiesshould be treated thoroughly.

increase interest, historical develop-ments and biographical sketches of mathema-ticians should be used where- feasible.

Without being too formal, an attemptshould be made to exhibit structure in algebraby showing that the subjeCt is baSed upon a fewsimple principles and is developed by logicalreasoning from these basic assumptions.

In treating symbols of grouping, the rolesof the distributive, commutative, and associ-,ative properties in simplifying algebraic expres-sions should be stressed.

The properties of polynomials and ratiional.operations with polynomials should be taught.Special prothicts and factoring should betreated by means of the distributive property.Attention should be called to the analogy bet-ween the system of polynomials with coef-ficients in some number field and the system ofintegers concerning such matters as factoriza-tion into primes and the division algorithm. In-sistence upon identification of the numbersystem from which coefficients are taken is im-portant, especially in factorization of poly-nomials.

Rational expressions quotients ofpolynomials may be treated as analogous to

2

, 17

rational numbers. Students need considerablepractice in perforining operations with alge-braic functions.

The concept of a function should be de-fined both as a mapping from ape set to asecond set and as a collection of ordered partsin which nb two pairs have the same first num-ber. The equivalence of ,the two definitionsshould be made clear Relations as generaliza-tions of functions may be included. The terms,domain and range of a function, should be in-troduced and taught using many examples.Special types of functions linear, quadratic,and polynomial should be studied. Modernnotation, such as the use of a single letter todevote a function, shOuld be used.

Graphs may be introduced as 'pictures Offunctions and applied to solving such problemsinvolving functions and relations as determiningzeros or maximum-minimum values.

Solution of systems of linear equations byvarious methods including graphs should bepresented.

An introduction td quadratic equations ndquadraticrtne'qualities should be included in thecourse.

Correct mathematical language and usageshould be introduced at appropriate placps andemployed thereafter.. Such ideas as equationsand inequalities being sentences, the. practiceof denoting numbers by letters, .the' meaning ofintegral exponents, and symbols of groupingare 'examples of mathematical language.

Proofs should be given where feasible,with' the.purpose of introducing to students, thedeductive aspect of mathematics, batVeassiVeformality should be avoided. With each topic,good realistic applications word problemsshould be given where feasible:

OBJECTIVES1. To develop a foundation in algebra which will prepare the student for further mathematics

courses.2. To extend the student's understanding of the real number 'system.3. To further examine the elementary notions Of relations and functions..4. To study in-depth the procedures for Solving equatiOns and inequalities.5. To considesome of the principles used to interpret equations and inequalities geometrically.6. To examine some of the fundamental ideas of trigonometry.7. To emphasize the necessity for using precise mathematical language and symbolism.,6. To introduce students to some of the beginning notions of mathematical proof.9. To emphasize the use-of algebraic techniques to solve problems.

SUGGESTED CONTENTI. . THE LANGUAGE ANQ STRUCTURE OF ALGEBRA

A. A finite number systemB. f'roperties of operations

II. THE REAL NUMBER SYSTEMA. Natural numbersB. IntegersC. Rational numbersD. Irrational numbers

r

Ill. FIRST DEGREE EQUATIONS AND INEQUALITIE.IN ONE VARIABLEA. rrst degree equations in one variableB. Fi st degree'inequalities in orb variable

IV. FIRSA. e Cartesian plane 'B. Ines. in the coordinate planeC First degree equations in two variables

Mathematical relations

18

DEGREE EQUATIONS AND INEQUALITIES IN TWO VARIABLES

V/

ALGEBRAIC EXPRESSIONSA. Expressions and identitiesB. Addition and subtractionC. MuftiptioatiOnD. ExponentsE. Factoring,F. DivisionG. Rational expressions

VI. POLYNOMIALSA. System of polynomialsB. Rational functions

VII. RATIONAL SENTENCES AND FUNCTIONSA. Fractional equations and inequalitiesB. Rational functions

VIII. SYSTEMS OF LINEAR EQUATIONS AND INEdUALITIESA. Graphical solutions of systems of linear equationsB. Algebraic solutions of systems of equationsC. Graphical solutions of systems of linear inequalitiesD. Linear programming (optional)

IX EXPONENTS AND RADICALSA. Rational exponents

. B. Radicals

X QUADRATIC FUNCTIONS AND QUADRATIC EQUATIONSA. Graphs and quadratic functionsB. Quadratic equations and inequalities (optional)C. Systems of equations and inequalities (optional)

Xl. TRIGONOMETRY (Optional)A. Geometric conceptsB. Trigonometric functionsC. Tables of trigonometric functionsD. Applications of trigonometry

XII. STATISTICS AND PROBABILITY (Optional)A. Representing data graphicallyB. Describing dataC. Statistical averagesD. Probability of an eventE. Probability experiments

I

'In planning for a well-coordinated andsequential program of- mathematics, a singlecourse cannot be considered out of the contextof the total program. Mathematics developmentcontinues to break down the barriers betkeenthe traditional compartments of mathematicalthought. No longer is a definitive distinctionmade between algebra and geometry. Neitheris there a need for a separation of the study ofplane geoMetry and solid geometry.

Since a sequential study,of:Mathematics ingrades 7-12 is planned, this guideline forgeometry assumes but does not assure thatstudents haVe dealt with the following areas ofstudy.

1. Vocabulary of sets2. Algebraic development of the basic

properties of the real numbers in-cluding the order relationships andelementary simplification of radicals

3. Definitions and descriptions of elemen-tary geometric figures such as triangles,

' quadrilaterals and circles4, Exploration Of elementary properties of

Physical lines and points5. Basic constructions using compass and

straight edge6. Basic graphing of straight linesAny program must be adapted by the in-

dividual system and teacher to meet the needsand capabilities of the students. There are con-cepts that should permeate the entire courseand should not necessarily be taught asseparate units. There are basic topics that everygeometry course should include; there aretopics for, the accelerated groups. This geo-metry guideline is developed according to thesecategories. The sequence in which the

20

(

materials is to be taught is to be determined bythe individual system or teacher.

CONCEPTS TO PERMEATE THE STUDY OFGEOMETRY

Geometry should be presented as much aspossible as a discovery topic to develop insightinto the basic structure of a mathematicalsystem consisting of undefined terms, assump-tions, definitions, and theorems. It is not impor-tant at this level to prove all statements.However, what is assumed and what is atheoreM should be stressed. That thOre are dif-ferent developments of geometry, what IS atheorem in one might be an assumption iritheother and Vice versa, should be noted. Thestudy of geometry as a formal, rigid body ofmaterials to be memorized should be avoided,with creativity and individual thinking encour-aged.

The course should be thoroughly in-tegrated, not plane geometry and solidgeometry taught as separate courses in thesame year. Algebra and geometry should alsobe integrated wherever possible. However,geometry should not be sacrificed for theteaching of algebra.

Equality, as used mathematiclly, should bediscussed and used. Equality-Means identity inmathematics.

An introduction to the elementary ideas oflogic, especially those pertaining to deductivereasoning, is advisable early in the course.These ideas should be extended and new prin-ciples introduced whenever it is advantageous.A study of logic, per se, is notadvocated.

The history. of mathematics should beutilized throughout geometry lo give the.studenta better appreciation of mathematics. If stu-dents are aware of the processes of develop-ment, thly will 'better appreciate mathematicalthought 'and, structure. History is especiallyuseful in' the topics in which the geometry now

'amends Of deviates from Euclid.

TOPICS BASIC TO A GEOINIETRY COURSEA study of geometry should use the

properties of the real numbers, including" order,.The concept of betweenness can be Oged todefine segment anclsoy. In turn, segment andray can be used q&tudying angles and tri-angles, separation of a plane by a lirle, andseparation of space by a' plane. A distinctionshould be made between interior oleergkirib? of,and the geometric figure itself. The correspon-dence of a teal number to an angle4'`gives ameans of angle :measurement. Perpendicularlines, dihedral angles, and perpendicularplanes are defined in the spiraling process ofdefinitions and development of concepts.

Parallel lines are defined and a statementof a parallel postulate is given. The history ofthis idea is very interesting and informative, asis an introduction to the ideas of non- Euclideangeometries. Basic properties. of parallel linesand planes can then be developed.

The concept, of congruence is used todefine congruent segments and congruentpolygons. The difference between congruenceand equality should be emphasized and used.Congruence theorems in two and three dimen-sions shOuld be studied.

The algebraic idea of ratio should 'be ap-rted to the measures of segments. Similaritymay be developed as a generalization of con-gruence, with similarity of triangles andpolygons in general defined, followed by proofsof similarity theorems in two and three dimen-sions. The study of the trigonometric ratios(sine, cosine, and tangent) as developed fromthe similarity of two right triangles could also beincluded.

Proof of the Pythagorean theorem usingproperties of similarity follows. The applicationto special right triangles such as the 30°- 60 °-90 °, the 45°-45°- 90°, and the 3-4-5, is studied.

The study of geometry extends to polygonsand polyhedrons. The special polygons andpolyhedrons are defined and the history of theregular polyhedrons studied. The properties of

polygons and polyhedrons should be discussedand selected properties proved. .

A circle and a sphere should be Clhned,using the idea of a set of points,-and their basicproperties developed. Angles connected withcircles and spheres should be Otudied, alongwith tangent lines and tangent planes. Therelationships between polygons and circlesshould lead to an introduction of the idea of alimit.

Measurement should not be thought of asa separate wit of study but should be in-tegrated throt.Trout the course, the measure ofsegments, ang es, and arcs being defined andextended to i'rIclude the measurements ofperimeters, circumferences, and calculations ofareas and perim'eters. Applications and basictheorems should be developed.

A separate study could be made on con-structions, with special emphasis on the logicalproperties or proofs of the constructions.

A unit that possibly would be consideredas basic is the study of coordinate geometryand loci. This would provide another method ofproof for use in the remainder of the course.

A study of finite geometries is a goodenrichment topic for a group to foster creativityand mathethatical insight.

The history mentioned throughout thecourse could be extended to a complete unit onthe history of mathematics, as could the logicpermeating the course. This unit would includesome symbolic logic; however, topics such as"truth tables" should not be overemphasized.

21

Other topics for consideration, are: geometricinterpretation of vectors, geometric transforma-tions, and, non-Euclidean geometries.

This course in geometry emphasizes ideasdeveloped in previous courses such as gets,real numbers, and deductive reasoning. The

role of undefined terms, definitions, axioms,_theorems, and the nature of proof is madeclear. Thus .the value Inherent in Euclideangeometry is retained and strengthened. Con-cepts of three dirnensions are introduced alongwith analogous two dimensional concepts.

OBJECTIVES1. To show the inadequacy of common sense reasoning, in establishing matherioatical proof.2. To: teach the process of deductiv reasoning in mathematical and nonmathematical situations.3. To extend the concepts of sets t include sets of points as well as sets of numbers, and to use

sets for purposes of clarity and convenience throughout the course.4. To use the properties of the re I numbers, including order, in the ,study of various geometric

figures.5. To develop and to use the concepts of betweenness and separation.6. To study relationships between lines in a plane and in space, between planes, and between lines

and planes7. To teach the idea of congruences as correspondences and to study congruences of a variety of

geometric figures.8. To distinguish between the concepts of existence and unicrness, and to prove some theorems

involving these concepts.9. To teach the principles of ratio and proportion, and to use these principles in proving similarity

theorems and in solving problems.10. To study regions whose boundaries are rectilinear and to derive formulas for finding the area of

these regions.11. To consider the incidence properties of lines and circles and th analogous properties of planes

and spheres.12. To develop intuitively formulas for the circumference and the area of a circle, for the length of an

arc, for the area of a sector, and for the surface areas and volumes of familiarthree-dimensionalgeometric figures.

CONTENTI. EUCLIDEAN GEOMETRY AS A MATHEMATICAL SYSTEM

A. Undefined terms: point, line, and planeB. Relation: betweennessC. PostillatesD. DefinitionsE. Theorems

II. CONGRUENCEA. Congruence of segments and anglesB. Congruence of triangles

III. INEQUALITIES'A. Inequalities of measures of segments and anglesB. Inequalities of lengths of sizes of angles in a triangle, and also in pairs of triangles

IV. PERPENDICULAR AND PARALLEL LINES AND PLANESA. Perpendicular lines in a plane and lines perpendicular to a planeB. Perpendicular and parallel relationsC. Dihedral angles and perpendicular planes

V. SIMILARITY OF TRIANGLESA. L'inesegments whose measures are proportionalB. Similar triangles

22

VI. letORDINATE GEOMETRYA. One-to-one correspondence between points of a plane and ordered pairs of real numbersB. Midpoint and distance formulasC. The slope of a lineD. Theorems pertaining to the slopes of parallel lines and the slopes of perpendicular linesE. The equation of a lineF. Simple analytic proofs

VII. POLYGONS AND POLYHEDRONSA. Definition of polygon and names of common polygons

. A detailed study of quadrilateralsC. Regular polygonsD. Sum of measures of angles of a polygonE. Ratios of measures of corresponding line segments in polygonsF. PolyhedraG. Polyhedral anglesH. Coordinate systems in space

VIII. CIRCLES, SPHERES, CYLINDERS, AND CONESA. Geometry of the circle and the sphereB. CylindersC. Cones

IX. .LOCI AND CONSTRUCTIONSA. Two and-three-dimensional lociB. Constructions with compasses and straightedgeC. Constructions utilizing the compass alone (Optional)D. Constructions using the straightedge and a fixed circle (Optional)

X. AREAS AND VOLUMESA. Area of polygonal regionsB. Polyhedrons prisms, pyramids, frustums of pyramids and prismsC. Circumferences of circles and lengths of arcsD. Areas of circular regions and sectors of circlesE. Surface area and volume of a sphere, cylinder, cone, and frustum of a cone

XI. A GEOMETRIC INTERPRETATION OF VECTORS (Optional)A: Directed segmentsB. Vectors.

XII. GEOMETRIC TRANSFORMATIONS (Optional)A. Transformations of Euclidean GeometryB. Introduction to Projective Geometry

XIII. NON-EUCLIDEAN GEOMETRIES (Optional)A. Historical background of non-Euclidean GeometriesB. Hyperbolic GeometryC. Elliptic Geometry

23

Id I

t

Many Students tend to reject Studying anycourse. for which they fail to see a practical ap-plication. The purpose of Applied VocationalMathematics is to relate mathematics to a widerange of vocational applications.- Studentsshould be provided with the opportunity to seehow mathematics is used in. several careers.

At times it might be advantageous to teacha mathematical concept or skill to a large groupof students or to the entire class and then allowthe individuals within the group to apply thisknowledge to an application related to their owncareer goals. For example, several studentsmight be taught how to find what percent onenumber is of another, then they could apply thisskill to several areas of interest like the masonrytrades, electrical trades, or health occupations.

MathematicS is a common languageamong many vocations. Very often facility withthe basic computational skills is a job entry re-quirement. Thus,'the teacher will want to workwith students on improving their ability to com-pute with whole, fractional, and decimal num-bers. Attention should also be given to pro-ficiency with operations involving percents.However, care should be taken not tooveremphasize computational skill to the extentthat students never get a chance to see the ap-plications of mathematics.

Students need to learn how to work withsome of the formulas used in the vocations.Techniques for splving formulas related to aspecific vocational area should be studied inmore detailltthose students with an interest inthis area.

Since geometric shapes are used in everyvocational area, students should becomefamiliar with both two and three-dimensionalgeometric figures. Some of the special proper-

zH

ties of geometric figures should be examined.For example, because the triangle is a rigidfigure,. it is widely used in the construction in-dustries.' Time should be taken to review orteach students the techniques for constructingsome of the geometric figures. Many of theideas studied concerning geometric construc-tions will prove useful when students work withtechnical drawings.

The use of technical drawings is wide-spread in many vocations. Students should beprovided with the opportunity to solve mathe-matical problems, which arise from the use ofthese drawings. Efforts should be made to en-sure that the metric system of measurement isincluded on the technical drawings as the stu-dents taking this course will find an increasedusage of this system of measurement by thetime they enter the labor force;

One of the most common uses of mathe-matics is in measurement. Attention should begiven to the study of length, area, volume, ter"n-perature, and mass (weight). Both the U.S.Customary and the metric system of measure-ment should be studied. Some aspects of thestudy of measurement will allow students to ap-ply knowledge they have gained earlier in thecourse. For example, if they are to find the dis-placement of a motorcycle engine, they willneed to work out a formula which will involveseveral mathematical operations.

Students should gain a good under-standing of ratio ,and proportion and be able toapply this useful mathematical tool in the solu-tion of different types of real life settings. Workwith scale drawings can provide students withadditional experiences with ratio and propor-tion. Provisions should be made to provide stu-dents with hands-on experiences in workingwith scale drawings.

30

GENERAL OBJECTIVES1. To use applications from the vocations as a vehicle for developing and reinforcing students'

backgrounds.2. To present mathematical problems as they exist in a real-life setting with which students can

re e.

3. To ffimiliarize students with some of the technical vocabulary used in various vocations.4. To acquaint students with some of the mathematical concepts and skills needed for selected

vocations.

SPECIFIC OBJECTIVES1. Review and reinforce computational skills involving whole numbers, fractions, and decimals.2. Extend computations involving fractions and decimals to include percents.3. Develop equation and formula solving facility.4. Provide opportunities for measurements with the U.S. Customary and metric systems using

basic formulas toicomputte linear, area, and volume Measures of various geometric figures.5. Develop an understanding of some of the elementary concepts of geometry. Use theseconcepts

to help solve problems related to layout, sketching, and scale drawings.6. Read and interpret bar, line, and circle graphs.7. ConStruct bar, line, and circle graphs as a means of communicating information.8. Apply the ideas of ratio and proportion to help solve real or contrived problems.9. Use the Pythagorean Theorem to solve problems.

10. Read and use mathematical tables.NOTE: As mathematical ideas are presented, opportunities should be provided to apply these to problemsfrom various vocations.

CONTENTI. WHOLE NUMBERS

A. Applications involving whole number operations as used in various trades and industriesB. Estimating answers

II. FRACTIONSA. Applications from various trades and industriesB. Fractions as percentsC. Estimating answers

III. DECIMALSA. Applications from various trades and industriesB. Decimals as percentsC. Estimating answers

IV. EQUATIONSA. Using informal techniques to solve equationsB. Using addition, subtraction, multiplication, and division to solve equationsC. Applications from various trades and industries

V. FORMULASA. Working with formulasB. Using formulas from various vocations

VI. GEOMETRYA. AnglesB. Plane figuresC. Three-dimensional figuresD. ConstructionsE. Applications

25

ti

VII. MEASUREMENT INVOLVING. BOTH THE U.S. CUSTOMARY AND METRIC UNITS'A. LengthB. AngleC. AreaD. VolumeE. TemperatureF. Mass (Weight)G. Applications

VIII.. GRAPHSA. BarB. LineC. CircleD. Interpreting and drawing inferences from graphsE. Constructing graphs A

F. Applications

IX. SCALE DRAWINGSA. RatioB. ProportionC. SimilarityD. Applications

X. APPLICATIONS OF THE PYTHAGOREAN THEOREM

XI. TRIGONOMETRY OF THE RIGHT TRIANGLEA. Sine, cosine and tangent ratios rB. Using trigonometric tables.C. Usirig sines, cosibes, and tangents in problem solving

26

KitcshenDiningBath

EXAMPLE An architect is designing a house which willbe 10 meters long and 8 meters'wide.. A-scale drawingof the floor plan is 30 cm by 24 cm. In the drawing.find ttie dimensions representing a 4.2 m x 5.5 m living

troom.

SOLUTION

Bedroom Using the proportion -

1 m = 100 cm. so 10 m = 1000 cm

LivingRoom Bedroom

v have

actual size 1000drawing size 30

420 1000 and 55,0 100030 30w /

Solving

= 420 30vv

1000

550.30/

1000

=-- 12 6 cm

16 5 cm

Can you find the dimen-sion§ that represent theother rooms in the draw-mg?

I P

The study of the real number system andfunctions begun in Algebr.al I should be con-tinued in Algebra II. The field and order axiomsof the real. number system should be reviewed,and applied to equations and inequalities.Throughout the course, emphasis should beplaced upon understanding the /structure ofalgebra as well as developing ski in algebraictechniques. -

The fUnction concept should be presentedboth as a mapping and as a special set of or-dered pairs, with stress on correct terminologyand notation. This concept should be used ex-tensively- throughout the course. Examplesshould be given of various types of functions.

, A thorough study should be made of linearand quadratic functions, including their zeros.An introduction :to parabolas can be given bymeans of the graph's of quadratic fundtions.

The process of determining the zeros ofquadratic functions (solutions of quadratkgequations) indicates the need for extending thereal 'number system to the complex .numbersystem. Properties of the latter should.- beclearly stated and compared with those of -thereal number system. The relations -of thevarious number systems to each Other shouldbe made clear.

-Ali& complex alimbers have been in-troduced, the problem of solving-quadraticsgeneral may be approaChed- by completirfg thsquare, with the quadratic formula derived.

Equations involving rational algebraic frac-tions and square roots, of simple algebraic. ex-pressions should be solved. It is important toprovide adeqUate review and practice materialfor operations with algebraic functions.

$ysteme of linear equations and systemsof linear inequalities should be studied, withgraphs and set terminology used freely toclarify the meaning of solutions.

Integral exponents should be reviewed andrational fractional .exponents introduced. Rulesof operations with radicals, along with theirlimitations should be established and amplepractice provided. .

The elementary properties a logarithmicand .exponential functions, including theirgraphs, should be studied. Logarithms andtheir use in computation should be included.Applications of 'exponents and logarithms torealistic problems in thet natural and socialsciences is desirable..

33

The more elementary aspects of trigo-nometry should be treated in this course. ,Thetrigonometric functions should be defined forgeneral angles. Measurement of angles, specialangles, simple identities, solution of righttriangles, and the Laws of Sines and Cosinesshould '.be included. In connection with thedefinitions of the functions, polar coordinatesand their relations to rectangular coordinatesmight be pointed out. An introduction to vectorsprovides a good application of elementarytrigonometry.

A treatment should be given of se-quences, including arithmetic and geometricprogressions.

Throughout the course, where possible,meaningful applications drawn from the bio-logical, physical, and social sciences should be

presented. Every suitable occasion should beutilized to develop deductive reasoning. Stu-dents should not be left with the impression. thatalgebra is a subject without form or pattern.

Optional topics to be covered if time per-mits or used as supplementary or enrichmenttopics include matrices" and cieterminants,linear programming, algebra of fu'nctionS,logarithms of bases other than 10, Iogarithrnicinterpolatio,n, rational roots of rational poly-nomials, irrationality of roots, factoring rationalpolynomials, complex roots, trigonometridequations, graphs of trigonometric functions,amplitude and period, inverse trigonometricfunctions, polar coordinates, trigonometricfunctions of complex numbers, vectors inphysics and vector components, and finiteprobability.

OBJECTIVES1. To provide a review of previously introduced algebraic topics as they relate to the structure of the

real number system.2. To consider the role of deductive reasoning in algebra and to apply this in the manipulative tech-

niques used in algebra.3. To use the basic ideas of sets to define and to explain algebraic concepts.4. To review and to extend the concept of open mathematical sentences, and 'systems of sen-

tences, involving inequality.as well as equalitv. .

5. To use one, two, and Three-dimensional coordinate systems to represent graphically v.ariousmathematical relationships.

6. To strengthen understanding of relations and functions and to use these ideas throughout thecourse.

7: To teach in-depth linear and quadratic functions and relations.B. To extend the laws of exponents to include, rational and irrational exponents.9. To develop logarithms from exponents and to develop skill in using logarithmic and exponential

functions.10. To teach the fundamental ideas of numerical trigonometry of the general angle.

CONTENTI. SETS OF NUMBERS

A: Natural numbersB. Integers,C. Rational numbersD. Irrational numbersE. Complex numbers

II. RELATIONSA. Definitions and notationsB. 'Linear and quadratic functionsC. Inverse of a relationD. Direct, inverse, and joint variation

III. POLYNOMIALS. AND RATIONAL FRACTIONAL EXPRESSIONS

28

34

iy, ALGEBRAIC EQUATIONS AND INEQUALITIES INVOLVING ONE VARIABLEA., Solution of quadratic equationsB. Solution of higher degree polynomial equationsC. Solution of special irrational equationsD. Solution of quadratic inequalities

V. SYSTEMS OF EQUATIONS AND INEQUALITIESA. Solutions of systems of linear equations and inequalities by linear combinations,

substitution and graphsB. DeterminantsC. Solutions of systems of linear equations using determinantsD. Solutions of systems involving linear and quadratic equations in two variables

VI. EXPONENTIAL AND LOGARITHMIC FUNCTIONSA. A review of integral and rational exponents and the extension of the concept to include

any reaLexponentsB. Exponential functions and equ'ationsC. The logarithm of a number to a base "a" as the inverse of an exponential functionD. Properties of logarithms and computation using logarithmsE. Solutions of exponential and logarithmic equations and inequalities

VII. COMPLEX NUMBERSA. Sums, products, differences, quotients, pOWers and roots of complex numbers in

rectangular formB. Fundamental operations with complex numbers in exponential form

VIII. TRIGONOMETRIC `UNCTIONSA. Radian MeasureB. Definition of certain quotient functionsC. ,Reference angleD. Periodicity ,

E. Complementary anglesF. Complementary functions -

G. Use of trigonometric tablesH. Graphs of the basic trigonometric functions (cosine, sine, tangent, secant, cosecant,

cotangent) through one complete periodI. Use of trigonometric functions to determine components of vector quantitiesJ. Pythagorean identitiesK. Law of CosinesL. Law of SinesM, Formulas for sums and differences of anglesN. Solutions of trigonometric equations and proof of trigonometric identities0. Solutions of trianglesP. Inverse trigonometric relations and functionsQ. Algorithms with complex numbers using polar formR. DeMbivre's Theorem (proof optional)

IX. SE19.UENCES AND SERIESA. Sequences and SeriesB. itogressions and sums of prbgressions

X. MATHEMATICAL INDUCTION (Optipnal)

PROBABILITY (Optional)A. Permutations and combinationsB. Binomial Theorem and Binomial SeriesC. Probability

4)

29

XII. MATRICES (Optional)A. Symbolism used with matriB. Special kinds of matricesC. Definition of the sum of certairi1c fs of matricesD. Definition of the product of certain kindi of matricesE. Properties of addition and multiplication of matricesF. Applications

3036

I

COURSE NUMBER 2410

The goal of the course in consumermathematics is to provide students with basicmathematical- skills and competenciesnecessary for economic survival in the highlycompetitive society of today. The skills andcompetencies in mathematics of the studentsenrolling in this program should be carefullyassessed. Instruction should then be based onthe findings of such an assessment. If the stu-dent is lacking in basic computational skills,remedial work should. be provided at a level ap-'propriate for the individual student.

The ability to read and interpret intelligent-ly current data and statistical information isneeded in many activities of the consumer. An-understanding of appropriate methods ofselecting data for given purposes, 'desirablesizes of samples, the representativeness ofgiven samples, the comparability of data given,and the relevance of data should be striven for.Methods of presentation of data (various typesof tablesand graphs)ond interpretation of datashould be investigated. The above-namedtopics in elementary descriptive statisticsshould be developed as tools for later topics inthe course, and should be used in discussionsof later work as much as possible. Introductoryexperiences should involve as much concretematerial for illustration purposes as possible. Itis not the purpose of this unit to provide an in-depth study of statistics. Some classes may,however, have students with the interests andability to do further work in statistics. For thesestudents, a beginning study of measures of cen-tral tendency (mean, median, and mode) anddispersion (range, mean deviation, averagedeviation, and standard deviation) may beappropriate.

Almost all if not all adults have'to beconcerned with the earning of a living and the

spending of their money. Many problems inadult life can be traced to the inability to controlthe spending of money. To,10Ip in learning howto curb unnecessary spending and to find betterways of saving and spending, budgeting shouldbe studied. The reasons for,and the purOoses ofbudgets can be developed by students 1n classdiscussion. Income and outgo of money fromand to all sources need to be considered, along,with the types of records possible and the,problems that can exist.

Our American economy has often beendescribed as one built upon credit: Being an in-tegral part of our modern life, credit should beunderstood as actually being a way of borrow-ing money rather than of borrowing time to pay.The dangers and advantages of the variousplans should be known by all who expect to usethem. Short-term charge accounts and long-term installment buyinrthould be comparedand contrasted, along with the various plans ofinstallment buying. The paying' of cash byMeans of borrowing money should be com-pared with these other more usual credit plans.

The members of our society have come'toexpect local, -state, and federal govemments toprovide many comforts, conveniences, and ser-vices. All three forms of government are fi-nanced by taxes, "Usually property taxes at thelocal level, income and sales taxes at the statelevel, and income taxes at the national level. Inthe study-,of income taxes, concern must. begiven to income deductions, exemptions, andcomputation of taxes. The difference, betweenreal and personal property must be consideredwith property taxes, as must such concepts asassessed valpe*and tax rates. Many other taxes

31

also 'exist that are a normal part of our ex-penses: excise taxes, licenses, gasoline takes,inheritance -taxes, and import duties.

What is insurance and why is it desirable orecesSary? Is insurance anqinvestment or is it

.':'an operating expense? In the study of perSonaland property insurances, these are just two ofthe many possible questions. The various typeslof life insurance policies should be discussed.Health, accident, and hospitalization insurance's)Are often provided in-many employment situa-tions. The worker should know the features ofthese policies; In North Carolina, automobileowners must by law have automobile liability in-surance. Mahji abuses have occurred in financ-ing this type of insurance because of a lack ofunderstanding. Other property insurancesfire, theft, damage are often needed by theowner. Personal liability insurance can oftenprove of value in protecting a person fromfinancial disaster. Insurance can also providefunds for future retirement. Social security andother forms of annuities are appropriate topicsfor discussion.

If persons making living wages canmanage their money well, they should havemoney beyond that needed for ordinary livingexper4ses, money which perhaps should beconsidered for investment. The characteristicsof a good investment safety, salability, andincome are points to study. Various kinds of-investments should be compared and con-trasted: life insurance, savings accounts, socialsecurity, real estate, and stocks and bonds.

Although many of the previous topics willof necessity have to be considered as isolatedtopics, wherever possible, broad units that will

_involve rnany of these topics should be de-veloped. For example, a study of the problem of11drne, ownership versus rental,; -would entailman y of- these separate areas. 'he Problern of

'proViding adequately for retirement would beanother such broad unit. The teacher shoulddevelop this type bf unit Wherwossible.

The teacher should involve the type of stu-deht taking this course as much as possible inthe actual discussion and development of cori-tent. Much wcirk of a concrete level will benecessary, as will be many specific illustrationsand examples from everyday sources. Theteacher should' not attempt to go excessivelyinto the theories behind these areas,of concernwith the .majoril\of students, bueehbuld at-tempt to provide the4asic survival skills neededby average citizens.

This is a mathematics course, not aneconomics course, and should be taught withthis in mind. Although primary emphasis isupon the applications of mathematics to socialsituations, mathematical theory should pot becompletely. neglected. At all times, whenevercomputatiOnal algorithms are used, effortsshould be trade to continue the development ofthe understanding of the mathematical princi-ples behind them.

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OBJECTIVES1. Investigate the functions of a bank and the services a bank proVides consumers.2. Perform the following tasks: (a) open an account; (b) write a check; (c) determine a balance; (d)

make out a deposit ticket; and (e) r concile a bank statement.3. Investigate the interrelationships of redit and the \station's economy.4. Perform the following tasks: (a) det ine +he amount of interest on credit; (b) compute the rate

of interest; (c)::find the difference between the costs of various credit plans; and (d) computegross, net or disposable income, and discretionary income.

5. Identify variations in housing needs and the factors influencing them.

32

35aka

6. Perform the following tasks: (a) find trie difference between the purchase price of a home and thecost of owning and maintaining a home; (b) use interestables to derive the purchase price of a

rtfouse under different mortgage plans; (C) find variations in mortgage costs as determined by thenumber of payments, .interest rate, and amount of the loan; (d) make a-scale drawing of a floorplan; and (e) estimate The costs of simple home repairs, renovations, and utilities.

7. Identify various forms of insurance along with the purposes, advantages, and disadvantages ofeach.

8. Identify factors which are used in determining premium rates of insurance policies.9. Use tables to determine: (a) the costs of various types of insurance, and (b) the amount of an em-

ployee's current FICA tax contribution.investigate the elementary concepts of probability and statistics that affect the typical consumer,in particular, how sampling techniques and graphs can be used to present a biased picture.

11. Perform activities in probability and statistics which will make the student better aware of theseimportant branches of mathematics.

12. Identify various forms of taxation as well as the derived benefits from tax dollars.13. Perform the following tasks: (a) complete state and federal tax forms; (b) compute direct and in-

direct taxes on specified items; and (c) keep records for making tax reporting easier.14. Investigate the various modes of transportation' and the ,advantages and disadvantages of each.15. Identify the costs other than purchase cost rgiative to automobile ownership, operation, and

maintenance, and determine the aggregate cost.16. Investigate various schemes used to defraud the public and the agencies which can be called

upon to assist a victim of fraudulent practices.

CONTENTI. BANKS AND THEIR SERVICES

A. Types of banksB. Bank services

II. CONSUMER CREDITA. Using creditB. Kinds of credit plansC'. Cost of consumer creditD. The credit contractE. Legal aspects

III. HOUSINGA. Buying a house, condominium, and mobile homeB. Renting a house, apartment, and mobile home

IV. INSURANCEA. Basic kinds of insuranceB. Buying insurance

PROBABILITY AND STATISTICSA. ProbabilityB. Statistical data

1. Collection2. Analysis

VI. MONEY MANAGEMENTA. IncomeB. BudgetsC. Buying practicesD. Consumer protection

r

33

VII. SAVINGS AND INVESTMENTSA. SavingsB. Investments

VIII. TAXESA. Federal income taxB. State income taxC. Property taxeiii S

D. Other taxes: sales, excise, inheritance, gift, use, and

IX. SWINDLES AND GYPSA. Deceptive advertisingB. Deceptive pricing '0'C. BusineSs opportunity schemesD. Home improvement schemesE. Referral scherhesF. Vacation schemesG. Games of chanceH. Charity collectionsI. Self-improvement schemes5. Flim flamK. Worthless stocks and bondsL.. Land salesM. Family swindles

X. TRANSPORTATIONA. Personal transportation

1. Purchasing, leasing, and renting a vehicle2. Continuing costs3. Insuradrice

B. Trip planningC. Specialized recreational vehidies

34

intangible

This course, usually taken in the twelfthgrade, is designed primarily for those studentswho intend to continue their study of mathe-matics beyond the high school level. Comple-tion should prepare the students to begin thestudy of calculus and analytic geometry incollege.

With this objective in view, any topics notcovered in previous courses should be studiedbefore proceeding with the new material of thiscourse. This may include sequences, permuta-tions, probability, and the binomial theorem. Anintroduction to infinite seqUences and seriesshould be made, and summation notation (E)used.

Mathematical induction should be studiedand used to prove a variety of properties, in-cluding formulas for progressions and thebinothial theorem.

The algebra of functions addition, sub-traction, multiplication, division, and composi-tion of functions should be given carefultreatment. This work will provide a good reviewof the concept of a function, considered both asa mapping and as a special set of ordered pairs.Particular attention should be paid to thedomains and ranges of new functions construc-ted by combining two functions. The concept ofthe inverse of a function should be presented.

Polynomials and polynomial functionsshould be studied intensively. This study should

I I I r

11

include the factor and remainder theorems,synthetic division, theorems about roots, andmethods of calculating roots.

Some further study of the exponential andlogarithmic functions may be included. Theslope function may be introduced as a prepara-tion for calculus. Trigonometric (circular) func-tions should be defined as functions of realnumbers and their relations to function's ofangles made clear. Work in trigonometryshould include a study of the periodicity of thetrigonometric functions, graphs, applications touniform circular, motion and waves, identities,equations, addition and related formulas, andinverse trigonometric functions. Careful-atten-tion should be given to the domain and range ofall the functions studied.

Applications of trigonometry may be madeto vectors and the polar form of complex num-bers, using DeMoivre's theorem, to calculationof roots, and to complex numbers.

The remainder of the course should 1).5)devoted to analytic geometry. Lines and linearequations, systems of lines and systems oflinear equations, and conics and second degreeequations constitute the material to be studied.

As many properties of circles, parabolas,ellipses and hyperbolas should be investigatedas time permits.

Optional additional topics includ0 majtrices and determinants.

OBJECTIVES1. To extend previously studied principles of analytic geometry.2. To emphasize the role of logic in deductive systems of mathematics.3. To emphasize the circular functions of real numbers and the analytical aspects of geometry4. To consider some of the principles and applications of permutations, combinations, and

probability.35- ,

5. To develop understanding and skill in expanding binomial expressions.4 6. To introduce mathematical induction as a method of proof and to use this method to prove a

variety of properties, including formulas for progressions, the binomial theorem and DeMoivre'stheorem.

7. To study in depth the properties of polynomial functions and methods for solving polynomialequations.

8. To further examine the exponential and logaritr. iic functions.9. To give the student experience with.an algebraic system different from the real and complex

number systems by introducing the algebra of matrices.10. To use \matrices and determinants to solve systems of linear equations.

CONTENTI. FUNCTIONS

A. The function conceptB. Classificat of functignsC. Algebra of functionsD. Graphs of functionsE. Inverse of a function

II. POLYNOMIALSA. Factor and remainder theoremB. Synthetic divisionC. Fundamental Theorem of AlgebraD. Other theorems concerning existence and number of rootsE. Methods of computing roots

III. EXPONENTIAL AND LOGARITHMIC FUNCTIONSA. Review of rational exponentsB. Exponential functionsC. Continuity of exponential functionsD. Inverse of exponential functionsE. Linear interpolationF. Combination of logarithmic functions and other functions

IV. CIRCULAR (TRIGONOMETRIC) FUNCTIONSA. Functions of real numbersB. PeriodicityC. GraphsD. Applications to circular motion and waves

'E. IdentitiesF. Equations,:G. Addition fah:it-11,1as and similar formulasH. Inverse trigonometric functionsI. Applicatibnsto vectors ancicOM lex numbers

V. ANALYTIC GEOMETRYA. LinesB. Systems of linesC. CirclesD. ParabolasE. EllipsesF. HyperbolasG. The Slope FunctionH. Lines and planes in three space

VI. MATHEMATICAL INDUCTION

360

ti

eVII. SEQUENCES, SERIES, AND LIMITS

A. Arithmetic and geometric sequencesB. Sums of arithmetic and geometric sequencesC. Limits of infinite sequencesD.- Sums of infinite seriesE. Sigma notation

VIII. LIMITSA. Intervals, on the real number line (Neighborhoods)B Limit of a sequenceC. Definition of UnlitD. Convergence of a sequenceE. Divergence of a sequence

IX. MATRICES AND DETERMINANTSA. Matrices and 'determinants of order 2B. Matrices and determiAants of order 3C. Properties of determinantsD. Cramer's RuleE. The sums and products of. matricesF. Multiplicative inverse of a matrixG. Matrix solution of a system of equations

X. PROBABILITY AND STATISTICSA. Basic conceptsB. Counting principles

1. Permutations2. Combinations3. Binomial Theorem

C. Probability of equally likely termsD. Organizing and reporting dataE. Random samplingF. Theoretical distributionsG. Inferential statisticsH. Game theory

Xl. GROUPS, RINGS, AND FIELDS (Optional)

If a school is operating on thi quarter or semester system, it might be more con-venient to teach the course content from Advanced Mathematics as self-containedtopical courses. Three separate courses whichr6ould mike up the course content ofAdvanced Mathematics are Trigonometry, Analytic Geometry, and AdvancedAlgebra. A brief discussion of each course and a course description of each of thesefollows.

r 43

Advanced AlgebraCourse Number 2402

This course Is designed to provide an In-depth look at some of the Important algebraicideas that often receive only a brief treatment InAlgebra II or In some cases, they are not dis-cussed at all. Some of the topics Included In thiscourse are of the nature that they will be used as"stepping stones" to more complex mathemat-ical topics. Yet, others are of the nature 'thatthey can be used in some of the applications ofmathematics.

4

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OBJECTIVES1. To Investigate the nature of polynomial functions of degree n, n > 2, and methods for finding

their zeros whenever they exist.R. To study systems of equations which contain two or more variables and the various methods for

determining Common solutions.3. To examine the nature of combinations and permutations and to solve problems Involving them.

N. To explore the properties of matrices and determinants and how they can be used to solvesystems of equations.

5. To examine the method of mathematical induction and to use this td prove various mathe,'matical statements.

6. To study the characteristics of arithmetic and geometric series al with techniques forrepresenting series.

7. To explore the notion of a limit and some of the theorems about limits.

CONTENTI. POLYNOMIAL FUNCTIONS OF DEGREE n, n > 2

A. Product of two binomialsB. Product of two trinomialsC. Factoring binomials and trinomialsD. Remainder TheoremE. Factor Theorem and its converseF. Synthetic divisionG. Graph of a polynomialH. Locating the rootsI. Number of rootsJ. Bounds of the real rootsK. Descartes' Rule of SignsL. Imaginary rootsM. Approximation of irrational roots

II. SYSTEMS OF EQUATIONS IN TWO OR MORE VARIABLESA. Solution by graphical methodsB. Consistent, inconsistent, and dependent equationsC. Solution by algebraic methodsD. Linear programming

38 44

PERMUTATIONS AND COMBINATIONSA. The Fundamental Counting PrincipleB. Permutations of n things taken-Mt-a timeC. Permutations of n things not all differentD. Cyclic permutationsE. CombinationsF. The sum of certain combinations

VI. MATRICES AND DETERMINANTSA. Matrices and determinants of order 2B. Matrices and determinants of order 3C. Properties of determinantsD. Cramer's RuleE. The sums and products of matricesF. Multiplicative inverse of a matrixG. Matrix solution of a system of equations

V. MATHEMATICAL INDUCTION

VI. SEQUENCES, SERIES AND LIMITSA. Arithmetic and geometric sequencesB. Sums of arithmetic and geometric sequencesC. Limits of infinite sequencesD. Sums of infinite seriesE. Sigma notation

AnalyticGeometry

Course Number 2406Analytic Geometry serves as a bridge be-

tween algebra and geometry. It should help thestudent to .see how one area of mathematics,can help clarify and extend ideas from anotherarea. The examination of coordinate systems ofone and two spaces should enable the studentto begin to generalize coordinate systemsbeyond two dimensions.

Considerable attention should be given tothe study of the properties of lines, linear equa-tiofis, systems of linear equations, conics andsecond degree equations. This should befollowed by an examination of the properties ofcircles, parabolas, ellipses, and hyperbolas.

OBJECTIVES1. To examine the use of the rect gular coordinate system as a means of illustrating algebraic ex-

pressions and the relationshl s that exist between many of them.2. To investigate the equations of lines with special characteristics.3. To stress techniques for sketching the graphs of any equation of the form AX2 + BY2 + CX + DY

+ E = 0.

3

4. To examine, how to graph .some algebraic equations on a polar coordinate system,5. To introduce students to the use of ,space coordinates in graphing.6- To study how analytical methods can be applied to the proofs of geometric theorems;

CONTENTI. ' LINES

A. Coordinate systemsB. Slope of a lineC. Forms of first degree equationsD. bistances

II. SYSTWMS OF LINESA. parallel linesB. Cfincurrent linesC. Intersecting lines

Ill. PLANESA. Parallel planesB. Intersecting planes

IV. CONIC SECTIONS.At, General equations of a conicB. CircleC. ParabolaD. ElE. H

V. CURVE TF#ACINGA Intercepts

AsymptotesC. SymmetryD. PeriodicityE. Excluded values

V,I. POLAR COORDINATESA. Graphs of polar coordinate equationsB. RelatiOns between polar arNi rectangular coordinatesC. Polar equations of lines and conics

VII. THREE DIMENSIONAL ANALYTIC GEOMETRYA. Space coordinatesB. Cylindrical surfacesC. Figures of revolution

TrigonometryCourse Number 2401

Efforts should be made to show how trigo-norrietry relates to other branches of mathe-

nmatics. The role of functions, both circular andtrigonometric, should be emphasized in- de-veloping . trigonometric concepts. This

emphasis should help the student gain anunderstanding of the ideas associated withangles, triangles, and vectors. In addition, thiscourse should provide the student with oppor-tunities to explore some of the applications oftrigonometric concepts.

46

OBJECTIVES1. To explore the definitions of the six trigonometric functions and to use these in the derivation of

other trigonometric relations.2..To emphasiie the graphs of trigonometri functions as a means of picturing how the domain,', range; period, and amplitude of an equation influence the graph of the equation.3. To examine the definitions and graphs of the inverse trigonometric functions.4. To investigate the solutions of equations irwolving trigonometric and-inverse trigonometric func-

tions.5. To study many of the applications of trigonometry including those which involve the Law of

Sines, Law of Cosines,and right triangle relations.6. To explore the polar coordinate plane and use this as another method of exploring complex

numbers. ,

7. To re-examine logarithmic functions and extend these to include logarithms of some of thetrigonometric functions.

CONTENTI. INTRODUCTION

A. Review pertinent algebraic and geometric concepts: relation's, functions, mappings,domain, range,axioms for real. numbers, exponents and radicals, absolute value, andinequality statements; Pythagorean Theorem, and the distanCe formula.

II. THE WRAPPING FUNCTION AND/OR CIRCULAR FUNCTIONSA. Definition of the six trigonometric functionsB. CofunctionsC. Reciprocal functions.D. Special anglesE. Radians to degrees and degrees to radians

III. GRAPHING TRIGONOMETRIC FUNCTIONSA. AmplitudeB. PeriodC. TranslationD. Converses and inverses

IV. EQUATIONS AND IDENTITIESA. The fundamental identitiesB. Equivalent trigonometric expressions

41.

41

C. Solutions of trigonometric equationsD. Proving identitiesE. Using identities to solve equationsF. Sine, cosine, and tangent of the sum and difference of two anglesG. Functions of twice an angleH. The product and sum formulas

'INVERSE TRIGONOMETRIC FUNCTIONSA. Inverse of a functionB. Inverse,trigonometric functionsC. Inverse trigonometric equations

VI. POLAR. FORM OF COMPLEX NUMBERSA. Polar coordinates,B. Complex numbers: introduction, operations, and trigonometric representationC. DeMpivre's Theorem

VII. SOLVING TRIANGLESA. Right trianglesB. The Lai/ of SinesC. The Law of Cosines

VIII. VECTORSA. Addition and subtractionB. Scalar multiplication and dot productC. Vectors in physicsD. Vector components

IX.- LOGARITHMSA. Exponential and logarithmic functions and their graphsB. Laws of logarithmsC. Common logarithms, operations, and applicationsD. ComputationE. InterpolationF. Logarithms of other number bases

Probability and StatistiCsCourse Number 2408

,The study of probability includes the studyof experiments involving chance events, theoutcomes of such experiments, and the like-lihood that particUlar outcomes will occur. Theprobability bf an outcome for a particular ex-periment is a numerical measure of the like-lihood that the outcome will occur. The theory ofprobability can then be considered to be themethods for assigning probabilities to out-comes for experiments and the study of therelationships among them.

427

Statistics refers to the ways of collecting,organizing, analyzing, interpreting, and sum-marizing numerical data of all kinds.

Recent applications of mathematics to thesocial, biological, and physical sciences haveincreasingly involved probability and statistics.Understanding the predictions of voting pat-terns, the validity of opinion polls, and themerchants' application of opinions of question-naires to decision making requires an under-standing of probability artd statistics. Such an

understanding will better enable students to ap-preciate the extent of certainty of events in theirworld.

It is suggested that many students beencouraged to take this course. Strong em-

pnasis should be put upon laboratory exercisesand individual or group projects that will allowstudents to investigate probability and statisticsat the various levels of their mathematicalabilities.

The Galton BoardHole

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If the balls are poured into thehole, they will form a modelof the normal curve.

Normal curve

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The board was invented by a British mathematician, Sir Francis Galton(1822-1811).

'By permission: Holt, Rinehart and Winston, PubliShers

'ft43

OBJECTIVES1. Interpret data presented in tabular and graphical form.2. Organize and present data in tabular and graphical form.3. Summarize and analyze selected data tij calculating the measures of central tendency (mean, _

median, and mode); discuss their relative merits.4. Calculate measures of dispersion: range, quartile deviation, variance, and standard deviation.5. Compute the mean and standard deviation for large numbers of measurements by grouping

techniques.6. Determine whether outcomes of an experiment are equally likely.7. Determine whether outcomes of an experiment are mutually exclusive.8. Determine the sample spaces of various experiments.9. Express events as subsets of a sample space.

10. Determine whether two events are complematitary.11. Determine whether events are exhaustive.12. Develop and apply the formulas for calculating the number of permutations and combinations of

n objects taken r at a time, and determine whether the problem involves permutations or com-binations.

13. Apply correctly the two basic counting principles to determine the number of outcomes in event""A or B" and event "A and B."

14. Determine the probability of an event in a finite sample space using the classical definition oftheoretical probability.

15. Identify binomial experiments and apply the laws of chance to the binomial distribution.16. Compute the conditional probability of event A given B.17. Apply the appropriate additive and multiplicative theorems to determine the probability of multi-

ple events.18. Distinguish between samples and populations.19. Use elementary sampling theory to perform a random sampling.20. Apply the theory of probability to acceptance sampling.21. Apply the theory of probability to test statistical hypotheses involving normal and binomial

distributions22. Use samples to make estimates of population measures.

CONTENTI. BASIC CONCEPTS

II. COUNTING PRINCIPLESA. PermutationsB. CombinationsC. Binomial Theorem

III. PROBABILITY OF EQUALLY LIKE OUTCOMES

IV. ORGANIZING AND REPORTING DATA

V. RANDOM SAMPLING

VI. THEORETICAL DISTRIBUTIONS

VII. INFERENTIAL STATISTICS

VIII. ?AME THEORY

44

Calculus

COURSE NUMBER 2420

A first course in calculus at the high schoollevel could serve as the culmination of a'mathe-matically talented student's high school pro-gram. It should serve as a course which takes;the mathematics a student has previously learn=ed and applies this in a new setting. The contentshould be presented from a point of view thatmaintains a balance between the theory ofcalculus and its many applications.

The students i.h this course will be at manylevels of intellectual maturity and will have vafty-ing amounts of mathematical motivation. Somestudents will be able to understand the methodsof, proof while others may only be able to graspan intuitive understanding of the notion beingproved. The teachers will have to determine thedegree or rigor and abstraction to be main-tained in the course.

OBJECTIVES1.. Define functions, and state the domain and range of a function. Illustrate various methods of

combining functions.2. Analyze the properties of the conic sections, write the general equations of the conic sectiori us-

ing the definitions or standard forms, and write basic georrietric proofs usinganalytic geometry.3. Write the equation of a lisiieusing the two-point foam, point-slope form, or slope-intercept form;

and express in the stan rd form, AX + BY + C = 0.4. Graph functions and relations using the concepts of symmetry, asymptotes, domain, range,

translation of axes, and rotation of axes.5. Find the angle between two intersecting lines, and find the distance from a point to a line.6. Find the limit at a finite point and at infinity, if either or both exist, of selected algebraic functions.7. Use L'Hospital's rule, if applicable, to find the limit, if it exists, of a function which is discon-

tinuous at a point.8. Apply theorems on limits to find the limit at a finite point or at infinity, if either or both g4ist, of

selected functions.sinN

9. 'Prove: lim = I

0

10. .Use the definition of a derivative to find the derivative of selected algebraic functions,

45

.

11., Use the sum, difference, prot, quotient, and chain rule to find the derivative of algebraic,trigonometric, inverse trigonometric,, exponential, and logarithmic functions.

12. Use the derivatives of f(x) = sin x and g(x) = cos x to derive the derivatives of sec x, csc x, tan x,and cot x.

13. Use the derivative to find the maximum, minimum, and points of inflection of the oraph-of a func-tion.

14. <Use the derivative to find the equation of the tangent and zormal line to a curve at a point on thecurve..

15. Use the derivative to find the equation of the tangent lines from an'external point to a curve._16. Use the derivative to find the velocity and acceleration, and solve related rate problems.17. Use the derivative to approximate roots of an equation.18. Use th derivative to find the angle between the graphs of two curves.19. Find a d apply the derivative of implicit functions and equations written in parametric form.0 Use th methods of substitution,ipartial fractions, and integration by parts to find the integral of

algebr ic, trigonometric, inverse trigonometric, exponential and logarithmic functions.21. Use the definite integral to find the area under a curve and between two curves, length of an arc,

surfaces of revolution, and centroids.22. Use .the definite integral to find volumes of solids of revolution, find volume by the cylindrical

shell and disk methods, and find volume by the method of slicing.23. Apply the integral to selected problems of physics such as hydrostatic force, work, acceleeation,

and velocity.24. Use the trapezoidal rule to approximate selected definite integrals.25. Evaluate integrals which are infinite or discontinuous.26. State and illustrate the basic theorems such as the fundamental theorems of differential and

tegral calculus, mean value theorems of differenflaland integral calculus, and theorems of con-tinuous functions.

CONTENTI. FUNCTIONS

A. RelationsB. Definition of a functionC. Domain and range of a functionD. Combining functionsE. Intercepts, symmetry, and asymptotesF. Composite functionF. Composite functionG. Continuous and discontinuous functionsH. Power functionsI. Inverse functions

II. ELEMENTS OF ANALYTIC GEOMETRYA. The rectangular coordinate planeB. Distance formulaC. Midpoint formulaD.. Slope of a lineE. Equation of a lineF. Parallel and perpendicular linesG. Angles between two linesH. The circleI. The parabolaJ. The ellipseK. The hyperbolaL. Translatiori and rotation of axesM. Conic sections

46

III. LIMITSA. Intuitive view of limitsB. Definition of" limitsC. TheoreMs on limitsD. Limits at infinityE. Limits of sequences

IV. DIFFERENTIATION OF ALGEBRAIC FUNCTIONSA. Polynomial functions and their derivativesB. Inverse functions and their derivativesC. The chain ruleD. The differentials dx and dyE. Higher order differentiation

V APPLICATIONS OF DIFFERENTIATIONA. Curve plottingB. Rolle's theorem J,C, The mean value'theoremD. Maxima and minimaE. Points of inflection

VI. INTEGRATIONA. The indefinite integralB. Area under a curveC. Area between curves

VII. THE DEFINITE INTEGRALA. The- definite integralB. The fundamental theorem of calculusC. Area between curvesD. DistanceE. VolumeF. Area of a surface of retolutionG. Work

VIII. TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS'A. Trigonometric functionsB. Differentiation of trigonometric functionsC.. Integration of trigonometric functionsD. Inverse trigonometric functionsE. Logarithmic functionsF. Exponential functionsG. Differentiation of logarithmic and exponential functions

4

47

Some of the courses (Engineering Con-cepts Curriculum Project and Algebra II withComputer Programming) discussed in this sec-tion have already been .successfully taught in alimited number 'of schools in North Carolina.The others are courses which a school system

11,

might develop locally on an experimental basis.If the result of such an effort is satisfactoiy, thelocal school system should share this with theMathematics Division so that other schoolsmight be encouraged to include the course intheir curriculum.

The Role Of The Computer In EductionMany recent changes in our society can be

traced to the electronic computer. Automationis often controlled by the computer. Dataprocessing by the computer is changing our ac-counting and record-keeping procedures.

`Scientific research can be done much morequickly than in the past due to the capability ofthe computer to provide an almost instan-taneous analysis of data. The computer is usedas an aid in decision making in government,education, and business. Medical doctors areusing inforMation retrieval by the computer fordiagnostic pdrposes when working with theirpatients. Space travel and commercial ,airlinescheduling are dependent on the computer.Even historians, linguists, and psychologists areusing the computer Ator research in their fields.

P_

qome schools have access to computerfacilities through the use of computer ter-minals. These terminals make it; possible forstudents to use a computer for the solution of avariety of 'mathematical problems. It is increas-ingly evident .that more schools must exploreways of providing computer access to all stu-dents to the extent that they become acquain-ted with the nature of the computer.

The next two courses described in thispublication were designed for use with the com-puter. They haVe been successfully taught inseveral schools across the State.. It is hOped thatmore schools will be able to offer theSe twocourses and other computer related mathe-matics ,courses in the future.

Algebra 11 With Computer ProgrammingCourse Number 2301

One of the objectives of this course is touse computer methods to stimulate interest inmathematics; Most of the topics of Algebra II asdescribed on page 00 are found in this coursealthough the emphasis placed on various topicsmay be different from that normally given. Com-

48

puter techniques are used to support and rein-'force the presentation of the mathematics.

New learning experiences and techniqulik§,_enhance the student's understancling,Ofmatics. These include flowcharting, ''thedevelopment of algorithms, modeling of mathe-matical concepts, and student programming.

054

Experience with the computer in schools isdemonstrating that one of the finest ways for astudent to reinforce his understanding of a topicis to teach the computer that is, write aprogram that requires the computer to handlethat particular. topic.

A3AS11161 language is used throughout.BASIC is easy to learn and at the same time itscapabilities are sufficient for the/problems thatwill be confronted in the course. In addition,BASIC is a language, available to most schoolshaving access to computing.

1020

0- 302 40

5060

LET X=-2PRINT 2*X/2 3*X 5

LET X=X+1IF X>2 THEN 60GOTO 20END

This program can be represented by the following flow

chart (notice the shapes of the boxes)

(Input box)

(Processing box)

(Output box)

(Processing box)

(Dvision box)

There is, no attempt in this course todevelop exceptionally proficient programmers,although this may happen, but rather to intro-duce computer concepts and techniques that

will strengthen the understanding of mathe-matics: This is a mathematics course ratherthan a computer science cours- nd wouldnormally be offered,over two

Engineering Concepts and CurfiCulum Pro = t (ECCP)Cdurse Number 2404

ThpVevelopment of this course grew out ofa coneerff-ever the fact that fewer and fewer stu-dents were electing to take mathematics andphysical science courses at a time when theUnited States was entering "the age of tech-nology." Survival of our civilization depends onour ability to adapt to changes which wouldfollow technological developments and to con-trol these changes. In a democratic society, thiscontrol clan only result from an informed public.The widespread ignorance-regarding almostanything which might be classified as"modern" in the field of technology justifies thedevelopment of this course.

The course is multi-disciplinary and, iftaught as it is intended, the social studies

aspgct ould be very evident as the mathe-matical and scientific techniques arejpresented.The interplay betweerrtechnology and social,economic, political, and psychological forces isa predominant theme.

The "Engineering Concepts CurriculumProject" is intended for students with a back-ground of at least one year of algebra who arecollege bound.

The course deals with that section of tech-nology which is known as information systemsto scientists and engineers. The concern then iswith information, how it is communicated, how itis used to control systems, and how men andmachines interact in a system. The computerseems to be the single most significant develop-ment in modern technology and, as an infor-

49

mation processor, this plays an important partin the course.

The systems approach to decision makingand problem solving is the principal theme. Op-

, timization (linear programming), modeling ofreal world problems on the analog computer(docking of a boat, landing a LEM on the moon,etc.), pollution (noise, air, water), communi-cation (speech, music), feedback (disturbancecontrol, self-regulation), stability (supply anddemand, epidemics), man-machine interface(bio-medical engineering) are topics covered inthe course which are understandable to thehigh school student. Students also study thehistory of the digital computer along with someof its uses.

Discussion of communication with thecomputer leads into an appreciation of the

organization of the computer, of machine code,compilers and 'higher level languages.

The course emphasizes discussion and isa laboratory course which uses speciallydesigned equipment. The cost of such equip-ment, in order to do a satisfactory job, would bebetween approximately $500 and $5,000, rang-ing from demonstrations to a complete set foreach pair of students in a class of 24.

The mathematical content of the coursetouches on many topics (pormally not Includedto any extent in high school) although vital to theworld in which we live. These include queuipg,probability, development of algorithms, gametheory, exponentlakgrowth and decay, linearprogramming, and sinusoidal motioft.

Algebra: An Applied ApproachThe traditional course in Algebra I is not

suitable for many students. Yet; there`are somestudents who need and can profit from thestudy of algebraic ideas and problem-solving inpractical situations. ALGEBRA: AN APPLIEDAPPROACH is a course to meet these needs. Itis built around the conceptof motivating thestudy of algebra; Students will) not have to askjin" hat good is this?" or "whaiis it ued for?" Theu es of algebra will be as conigh-tly before the

udent as the mechanics of algebra. Thinkinga concrete Way will be more emphasized than

the abstract, deductive mathematics of Algebra

It is intended that this course would con-tain enough mastery of algebraic content toserve the needs of students who- must havealgebra for college admission or job qUalifica-tiOns. Students who are more mature in theirdevelopment dn'd need little' motivation, otherthan intellectual stimulation should take Alge-bra I.

The content of this course would consist offour main strands: (1) the meChanics of manip-ulating algebraic entities; (2) the direct applica-tion of each newly developed technique to showits usefulness; (3) the development of formulareasoning and functional, dependence; 4ind (4)the formulation of algebraic models to solveproblems.

The first strand contains the basic alge-braic facts which will be presented withott

50

proofs. The algebraic content is presented asgeneralized arithmetic rather than as an ab-gtract deductive system. Tatics to be coveredmight include:

review of integersusing variablet to write mathematicalexIxessionssolving simple equationsfundamental operations with poly-nomialsspecial products and factoringfundamental operations with rationalexpressionssolving fractional equationsfundamental Operations with powersand radicals

- solving radical equationssolving quadratic equations

The second strand would emphasize orhighlight the use ,of each of4the manipulativeskills listed above. Situations and problemsalready formulated would be used to motivatethexeed for studying a topic. That is, the stu-dents would be shown what they will be able todo as a result of mastering each new aspect ofthe topic. After the topic is developed, it will bepracticed in applied situations where its use cantbe shown. The applications of strand 2 aremuch more, low-level than those of strand 4. Instrand 2, students. substitute into alreadyderived formulas or relationships and interpretthe results within the setting of the problem.

1.

,

SOMe of the word problems in present text-books dealing with physical situations that onlycall for direct translation with. little real formu-lation would fall in this category.' Strand '3 emphasizes:algebra as a subjectthat extends our power to reason. In this phase,soloing, manipulating, and, 'More Importantly,interpreting literal equations would be stressed.Functional' relationships would be stressed aswell as the' ideas of Neklation.

The fourth strand' involves identifying rela-tionships in problems that can be presentedalgebraically or graphically.. The stress in thisstrand is the formulation of a problem Into

mathematical terms. Students would generateformulas or graphical means to representrelationships in sets of ordered pairs, relation-ships In word problems, physical principles orrelationships shown in geometric situations.

ALGEBRA: AN APPLIED APPROACHwould not cover as broad a' range of topics inalgebra as Algebra I nor to the same depth.Rather it would concentrate on the mastery of anarrow band of essential content. The remain-ing time would be spent on motivation of thestudy of algebra, applications of the subject,and how it is used to model and solve realistic,worthwhile problems.

GeOmetry: An Applied. ApproachGEOMETRY: AN APPLIED APPROACH is

a course built around the concept of showingthe uses and usefulness of geometry andgeometric thinking. The course Itself would beorganized around four main strands: (1)geometric facts; (2) meaningful applications ofgeometric facts in practical problems; (3)logical reasoning and deduction; and (4)problem-solving using geometric modeling.

It is intended that this course would con-tain enough geometric content to serve theneeds of students who. must have geometry forcotlege'-admission or job qualifications. Stu-dents who are more mature in their develop-ment of abstract thinking should take theregular geometry course which is less orientedtoward.cohcrete thinking.

They content of the- regular geometrycourse would be transmitted as basic factswithout 'formal proofs. Theorems would bepresented as statements concerning relation-ships in the physical world rather than proposi-tions that can be logically derived from a set ofassumptions, caret Ncloms. Instead, of proofs,there Woi.il.bei VeTIfi:Oations such as measuringto justify that an angle inscribed in a circle isone-half Its Intercepted arc. Or, pouring sand toshow the volume of a right circular cone is one-third the volume of a right circular cylinder withthe same base and height.

As theorems are proved, applications ofthem to the direct solutions of problems wouldbe given. Applications would Include disciplinedrawings, the 'science of perspective and otherdraftsmen and trAdes orientations of geometry.

I

Drawings with ruler and protractor would beutilized as well as constructions with compassand straightedge.

Logical reasoning would be Introducedthrough the use of flowcharts. Students wouldlearn to analyze problems and reduce them to astep-by-step algorithm procedure. From thisbackground, deduction would be introduced aswell as some geometric proofs in flowchartform.

The entire emphasis of the logical reason-ing strand would not be upon proofs of geo-metric theorems, but teaching for under-standing of and transfer of deductive thinkingand logical reasoning to other areas andactivities.

A water tank is in the shape of a large cylinder. Itsbase has an area of 867T square meters, and its altitudemeasures 5' meters. Find the Volume.

Base area is 861r sq. in,(86S cubes per layer)

5 layers of cubes

Volume = no. of cubesper layer

Volume = base areaV = 867T

= 4301r

x no.no oflayers

x length of altitudex 5

Thus, the volume is 430 IT cubic meters. -

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The fourth strand would emphasize solvingproblems using geometric content as mathe-matical models. The stress would be uponproblem formulation, i.e., abstracting fromphysical situations relationships that can berepresented as relationships among points,lines: circles, triangles and angles. The studentswould then solve the mathematical formulationand translate the mathematical solution back tothe physical settings. ,Problems would includeconstruction problems and locus problems.

Another example would be drawing the patternfor a Christmas ball where 5 congruent lines are'to cover a ball with a 5-inch diameter. Anotherexample would be to find the inside diameter ofa pipe that must slip over a triangular rod.

A part of the course would have the spiritof a geometry drawing course for a draftsmanor a skilled machinist. Included also would bethe notion of proof and deduction in a broadersense and problem formulation skills asso-ciated with mathematical modeling.

Technical MathematicsTechnical mathematics provides the need-

ed background for students who wish to enterskilled crafts and trades. Tk)ese students needthe math matics that will enable them to passemploym t and apprenticeship tests or to en-ter a tech Ica' or vocational curriculum for tech-nicians and skilled workers. Algebra, geometryand trigonometry provide the neededbackground for students who wish to becomescientists, engineers, and mathematicians.Business mathematics provides the necessarybackground , for students who will enter thebusiness world or business colleges.

A shortage of craftsmen, technicians, andskilled] workers continues in our society at atime when we have an oversupply of both un-skilled labor and college-trained specialists.The high school curriculum is in a unique posi-tion to offer a program that will both meet the in-terests of many students and also preparethese studentq for entry into the skilled tradeand service areas. Mathematics courses thathave been offered in rriost schools have not

52

provided students with the mathematics theyneed to enter skilled areas immediately ,afterthey leave high school. Technical mathematicsis an effort to rectify this situation.

T e fixed portion of the content of Tech-nical thematics should draw upon algebra,geome y, graphing, measurement, ratio andproporti , and numerical trigonometry. Thefirst step in i entifying appropriate mathemat-ical applications should be to survey local in-dustries so as to determine the skilled trades ofthe area and the mathematical requirements. Alocal technical or trade school could be helpfulin this endeavor as they have records abouttrade and ,industrial needs in the area. Sampleemployment and apprenticeship tests shouldbe used which simulate the type of situation thata prospective employee might experience. Thesample should not be confined to just shop orindustrial' vocations. Vocations should be in-cluded that have been traditionally ,female aswell as those which have been traditionallymale.

t- r