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Smith, Seaton E., Jr., Ed.; Backman, Carl A., Ed.Teacher-Made Aids for Elementary School Mathematics.Readings from the ARITHMETIC TEACHER.National Council of Teachers of Mathematics, Inc.,Washington, D.C.74192p.National Council of Teachers of Mathematics, 1906Association Drive, Reston, Virginia 22091 ($3.00)

EDRS PRICE MF-$0.65 HC Not Available from EDRS.DESCRIPTORS Activity Learning; *Elementary School Mathematics;

Fractions; Geometric Concepts; *InstructionalMaterials; *Manipulative Materials; Number Concepts;Numbers; *Teacher Developed Materials

IDENTIFIERS *National Council of Teachers of Mathematics; NCTM

ABSTRACTA collection of articles from the ARITHMETIC TEACHER

is presented which are about practical, classroom-tested ideas forthe instruction and use of teacher-made instructional aids. Theseentries deal only with manipulative type aids. They have beenselected for the clarity of purpose and relationship to contemporarytopics in the elementary school mathematics curriculum. The articlesprovide sufficient information and specifications so that teacherscan construct the aid and include directions or examples relative tousing the aid for instruction. The organization of topics is based onmajor strands of elementary school mathematics: whole numbers,numeration, integers, rational numbers, geometry, and measurement.(JP)

ac. eGAMOI 111111b

U S DEPARTMENT OF HEALTH,EOUCATION WELFARENATIONAL INSTITUTE OF

EOUCATION,HIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIGINATING IT POINTS OF VIEW OR OPINIONSSTATED DO NOT NECESSARILY REPRESENT OFFICIAL NATIONAL INSTITUTE OFEDUCATION POSITION OR POLICY'

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Teacher-made Aids forElementary SchoolMathematics

Readings from theArithmetic Teacher

edited by

SEATON E. SMITH, JR.CARL A. BACKMAN

Faculty of Elementary EducationThe University of West FloridaPensacola, Florida

National Council of Teachers of Mathematics

PERMISSION TO RE PRODUCE THISCOPYRIGHTED MATERIAL BY MICRO.FICHE ONLY HAS BEEN GRANTED BY

(140 ERIC AND ORGANIZATIONS OfERATNO UNDER AGREEMENTS WITH THE NA

TIONAL INSTITUTE OF EDUCATIONFURTHER REPRODUCTION OUTSIDETHE ERIC SYSTEM REOLPRES PERMISSION OF THE COPYRIL,HT OWNER"

The special contents of this book are

copyright©1974 by

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, INC.

1906 Association Drive, Reston, Virginia 22091

All rights reserved

Library of Congress Cataloging in Publication Data:

Smith, Seaton E comp.Teacher-made aids for elementary school mathematics.

Includes bibliographical references.1. MathematicsStudy and teaching (Elementary)

2. TeachingAids and devices. I. Backman. Carl A.,joint comp. II. The Arithmetic teacher. III. Title.QA135.5.556 1974 372.7'028 73-21581

Printed in the United States of America

Contents

Introduction

1. Teaching Aids: What, How, and WhyConsiderations for Teachers Using

Manipulative MaterialsDecember 1971

Use or Manipulative Devices inTeaching Mathematics

May 1963

Models and MathematicsDecember 1972

Robert E. Revs

Allen L. Bernstein

Elizabeth H. hnnema

1

3

5

13

17

2. Whole Numbers 23

The Hundred-Board Marvin C. Volpe' 25

December 1959

Kindergarten Mathematics Lucille Fitzsimons 32

January 1964

Our Number Clothesline Sylvia Orans 35

December 1964

A Picture Line Can Be Fun Katherine Patterson 36

December 1969

Simple Materials for Teaching EarlyNumber Concepts to Trainable- Jenny R. Armstrong andLevel Mentally Retarded Pupils Harold Schmidt 39

February 1972

Yardstick Number-Line Balance Jerome S. Borgen andMarch 1971 John B. Wood 44

Elevator Numbers Stanley Becker 45

October 1971

IV CONTENTS

Presenting Multiplication of CountingNumbers on an Array Matrix Merry Schrage 48

December 1969

Multiplication Mastery via theTape Recorder Dale M. Shafer 50

November 1970

Problem Solving with Number-Picture Problem Situations Juliet Sharif

March 1962

3. Numeration 57

Eight-Ring Circus: A Variation inthe Teaching of Counting andPlace Value Ethel Rinker 58

March 1972

Understanding Place Value Shirley S. Ziesche 66December 1970

Placo---a Number-Place Game Robert C. Cairo 68May 1968

Making a Counting Abacus George C. Cunningham 70. February 1967

Concepts to Enhance the Studyof Multiplication Margaret Haines 72

February 1963

Building "Computers" forNondecimal Number Systems Frederick R. Rabinottit: 75

October 1966

Let's Use Our Checkers andCheckerboards to Teach NumberBases Lucile LaGanke

November 1967

Discovering the Mathematics ofa Slide Rule J. Gregory Martin, Jr. 80

January 1968

77

4. IntegersClimb the Ladder Albert H. Mauthe 84

May 1969

Play Shuffleboard with NegativeNumbers Charlotte Frank 87

May 1969

Number Line: Versatility Esther Milne 90December 1968

A Teaching Aid for SignedNumbers Edna M. Pratt 91

November 1966

83

5. Rational NumbersIf the Hands Can Do It the

Head Can FollowNovember 1972

Fraction BingoMarch 1970

FRIO, or FRactions In OrderDecember 1966

Make a Whole -a Game UsingSimple Fractions

February 1971

Concrete Material for TeachingPercentage

December 1954

A Percentage BoardApril 1955

A Fraction CircleApril 1956

Fun with Fractions for SpecialEducation

October 1971

DiVision by a Fraction -a NewMethod

March 1962

A Game with Fraction NumbersJanuary 1970

6. Geometry and MeasurementCreating Mathematics with a

GeoboardApril 1970

A "Limited" Approach to the Sumof the Angles of a Triangle

February 1972

Activities with Easy-to-MakeTriangle Models

February 1972

The Angle Mirror OutdoorsApril 1970

Adapting the Area of a Circle tothe Area of a Rectangle

May 1972

A Pythagorean PuzzleJanuary 1972

Easy-to-Paste SolidsOctober 1965

Israel Jacobs

Nancy Cook

Rowena Dri:igacker

Joann Rode

Eldon Hauck

Inez Bailer

Donald B. Lyrers

Ruth S. Jacobson

George H. McM een

CONTENTS

93

94

101

103

105

Richard Thomas Zytkowski

Peter Wells

James V. Braid

Carl A. Backman andSeaton E. Smith. Jr.

Lauren G. Woodhr

Mary T. Colter

Gary D. Hall

M. Stoessel Wahl

115

118

123

125

127

130

133

135

138

142

145

vi CONTEN'IS

A Permanent Soap-Bubble GeometryApril 1972

Cleo's ClockJanuary 1973

We Made It and It Works! TheClassroom Construction ofSundials

April 1970

M. Swessel Wahl

Cleo Kaprocki

M. Swessel Wahl

149

151

153

7. Multipurpose Aids 157

The Peg Boarda Useful Aidin Teaching Mathematics Alan A. Fisher 159

April 1961

Fostering Enthusiasm throughChild-created Games Sarah R. Golden 162

February 1970

Blast-otT Mathematics Elton E. Beougher 167

April 1971

The Function of Charts in theArithmetic Program Catherine M. Williams 174

October 1955

Bulletin Boards for ElementarySchool Arithmetic Donald E. Hall 179

February 1964

A Mathematics Laboratoryfrom Patricia S. Davidson andDream to Reality Arlene W. Fair 181

February 1970

Introduction

The Arithmetic' Teacherhas been noted through the years as a professionaljournal that presents many practical, classroom-tested ideas for the con-sideration of its readers. Among these ideas are many valuable suggestionsfor the construction and use of teacher-made instructional aids.

The initial screening of articles for possible inclusion in this book of read-ings revealed that there are three closely related categories of instructionalideas: (I) manipulative aids, (2) activities, and (3) games and puzzles. Inthis book, we have limited our selections to those articles that relate pri-marily to the first category. For our purposes, a manipulative aid is anobject that contributes to the learning experience of a pupil by providinghim an opportunity to be physically and mentally involved.

Other factors that were significant in selection were (I) that the articlehave a clear purpose and be related to a contemporary topic in the elemen-tary school mathematics curriculum: (2) that the article provide sufficientinformation and specifications so that a teacher can construct the aid;and (3) that the article provide directions or examples relative to using theaid for instruction.

In this book, we have organized our selections on the basis of majorstrands of elementary school mathematicswhole numbers, numeration,integers, rational numbers, geometry and measurement. We hope you willenvision multiple applications of each teaching aid. With an open mindand a little creative effort, you will be able to think of many ways to fdaptthe suggested ideas or improve on them.

It is our hope that this publication will serve as a meaningful and usefulresource book to teachers of elementary school mathematics.

Teaching Aids:What, How, and Why

The articles in this opening chapter offer the reader much food for thoughtin the appropriate selection and use of teaching aids in the mathematics

classroom.In the opening article, Reys provides an excellent discussion of criteria

that should be considered when selecting manipulative materials for teach-

ing mathematics. Ile considers both the pedagogical and the physicalaspects associated with the selection of aids. Reys summarizes the use ofmanipulative materials and then provides a stimulating discussion of some

dos and don'ts for teachers who plan to use manipulative materials in the

teaching of mathematics.Bernstein also offers a set of principles for the selection and use of teaching

aids for elementary school mathematics. Each of his principles is amplified

with examples of actual teaching aids.In the concluding article of this chapter, Feline= examines the why of

mathematics teaching aids. The role of concrete models at various levels of

the learnimg process is discussed, and much of the relevant research is

summarized.

3

Considerations for teachersusing manipulative materialsROBERT E. REYSAn associate professor of mathetnctics education at

the University of MissouriColubia. Robert Reys is involved in developingmathematics laboratory activities as an integral part of mathematics preparatoryprograms for both preservice and in-service elementary teachers.

Classroom teachers of mathematics arewitnessing an unprecedented period of pro-liferation in manipulative materials. Com-merical catalogs list a great %'ariety.of avail -able materials; professional journals carrymany advertisements claiming that thisdevice or that aid will provide a panaceafor learning a certain mathematics topic;and professional meetings are frequentlyinundated with exhibits displaying newmanipulative materials. This influx of newlyavailable materials has precipitated manyproblems. The wide rang,: of quality foundamong various materials has made theproblem of selection much more difficult.It has made it impossible to list all availablematerials and discuss the meritor lackof meritof each. It has created doubts insome teachers' minds about the educationalvalue of the materials. It has raised addi-tional teacher-oriented questions such as"What are some guidelines for selectingmanipulative materials?" "What materialsshould he used?" What are some dos anddon'ts of using them?"

During the decade of the sixties severalfine articles appeared discussing considera-tions in the selection of learning materials(Berger and Johnson 1959; Bernstein1963; Davidson 1968; Hamilton 1966;Spross 1964). The present article is limitedto a discussion of manipulative materialsas opposed to other teaching aids. Further-more, it is addressed specifically to class-room teachers in an effort to provide somecurrent rationale, as well as guidelines, forthe selection and use of manipulative ma-terials.

5

What are manipulative materials?

The use of the term manipulative ma-terials raises one fundamental question,namely. "Just what are manipulative mate-rials?" In this context manipulative. mate-rials are objects or things that the pupil is"able to feel, touch, handle, and move.They may be real objects which have socialapplication in our everyday affairs, or theymay he objects which are used to representan idea" (Grossnickle, Junge, and Metzner1951, p. 162). Hence, not all teaching aidsor instructional materials arc manipulativematerials. Suffice it to say here that mani-pulative materials appeal to several sensesand are characterized by a physical in-volvement of pupils in an active learningsituation.

Rationale

In teaching mathematics we are pri-marily concerned with concept formationas opposed to the memorization of facts.The mental processes involved in conceptformation are much more complex thanthose associated with the memorization ofa mass of isolated details. Thcrc is little dis-agreement among contemporary psycholo-gists regarding the role of concept for-mation in the learning of mathematics.However, there are several existing theoriesabout how to best foster proper conceptformation. The results of recent psycholo-gical investigations into the ways childrenlearn mathematics by men such as JeromeBruner, Zolton Dienes, Robert Gagne, JcanPiagct, and Richard Skemp are beginning

6 TEACHER-MADE AIDS

to have an influence on mathematical ped-agogy. In short, more is known today abo..:uthe way children learn mathematics, andthe general nature of the mathematics theyare Capable of learning at various stages,than has previously been known. Ironically,it is still not known precisely how childrenlearn. but the efforts of researchers are con-tinually providing new evidence to suggestand oftentimes refute) various learning

theories. Since learning is an individualmatter and invariably dependent on numer-ous factors, some of which are quite elu-sive. it is highly unlikely that a comprehen-sive learning theory that is completelysatisfactory to all people wily evolve in theforeseeable future.

A comparison of prominent learningtheories will not be made here, but it seemsappropriate to identify the following state-ments. subscribed to by most learning psy-chologists:

1. Concept formation is the essence oflearning mathematics.

Learning is based on experience.1.

3. Sensory learning is the foundation of allexperience and thus the heart of learn-ing.

4. Learning is a growth process and is de-velopmental in nature.

5. Learning is characterized by distinct.developmental stages.

6. Learning is enhanced by motivation.

7. Learning proceeds from the concrete tothe abstract.

8. Learning requires active participationby the learner.

9. Formulation of a mathematical abstrac-tion is a long process.

This list is not exhaustive, nor are thestatements independent. In fact, they arcclosely interrelated. Suffice it to say thatthe above statements form the basic foun-dation underlying the rationale for min::manipulative materials in learning mathe-matics.

Many prominent mathematics educatorshave strongly urged greater use of manip-ulative materials in teaching mathematics.The rationale for this emphasis seems edu-cationally sound. Unfortunately, researchstudies in this area have "not been con-clusive in either supporting or refuting thevalue of manipulative aids" ( Mouldier1967, p. 31) . Most of the questions citedby Brown and Abell ( 1965, p. 548), suchas "Are there certain manipulative devicesthat lend themselves better to differentmethods of instruction?" and "Will a devicehelp one child and hinder another?" areyet to he answered. One can only hope thatquality research focused on manipulativematerials and mathematics learning willprovide sonic objective evidence relevant tothe issues. In the meantime, classroomteachers are still faced with the problem ofselecting and using manipulative materialsin their classroom.

Selection criteria

The rapid increase in available commer-cial materials has made the job of selectionnot only difficult but also more crucial asthe market is flooded with products. Thereare many criteria to consider in developingand procuring manipulative materials. Inorder to simplify this discussion, only im-portant criteria in two basic categories,namely, pedagogical and physical. will beconsidered. 11:AC proposed criteria are notexhaustive, nor is any hierarchy of im-portance suggested by the order in whichthey are discussed. Although some con-siderations arc more significant than others,the relative importance attached to eachcriterion should be determined by theteacher. Any final evaluation of manipula-tive materials should weigh strengths andweaknesses against the educational poten-tial.

Pedagogically there are many criteria toconsider in selecting manipulative mate-rink. One of the most. important considera-tions is whether or not the materials servethe purpose for which they are intended.

Furthermore, do these materials do some-thing that could no be done as well or bet-ter without them? Since mathematics is

mental, do the .materials develop the de-sired mental imagery?

The following criteria should be includedin any list purporting to identify peda-gogical considerations in the selection ofmanipulative materials:

I. The materials should provide a trueembodiment of the mathematical conceptor ideas being explored. The materials areintended to provide concrete representa-tions of mathematical principles. There-fore it is important that, above all else, thematerials he mathematically appropriate.

2. The materials .should clearly representthe mathematical concept. Concepts areembedded so deeply in some materials thatfew, if any, pupils extract relevant ideasfrom their experience with the materials.This problem is further compounded bymaterials that have extraneous distractors,such as bright colors, which actually serveas a hindrance to concept formation. Theseexperiences result in an "I can't see theforest for the trees" complex. This is, ofcourse, not all had, as it requires pupils tocull out extraneous data, yet in many casessuch materials serve more as a deterrentto correct concept formation than as an aid.

3. The materials should be motivating.There are many factors that ultimately con-tribute to motivation. Several of these, suchas attractiveness and simplicity, will he

discussed later. Materials with favorablephysical characteristics will frequently stim-ulate the pupil's imagination and interest.

4. The materials should be multipurposeif possible. That is, they should be appro-priate for use in several grade levels as wellas for different levels of concept formation.Ideally. the materials should also he usefulin developing more than a single con-cept. Such wide applicability is frequentlyachieved by using a portion or subset ofmaterials. For example, logic or attributeblocks have much multiapplicahility throughthe careful selection and use of pieces.

TEACHING AIDS 7

This requirement should not precludethe procurement and use of materials de-signed exclusively for embodying one con-cept. In fact, if use of certain materialsresults in concept formation that is other-wise impossible, then such items should beconsidered. In other disciplines, such asscience and physical education, consider-able funds are spent on devices that teacha single concept. Shouldn't mathematicsteachers have a similar opportunity? Be-sides, using materials (even those designedfor one specific function) often suggestsadditional topics or concepts that might heexplored.

5. The materials should provide a basisfor abstraction. This underscores the im-portance of the requirement that materialscorrectly embody the concept. In addition,caution should he exercised to ensure thatthe concept being developed is commensu-rate with the level of abstraction neededto form the mental image. Care must alsobe taken to ensure that the level of ab-straction is commensurate with the abilityof the student to abstract.

6. The materials should provide for indi-vidual manipulation. That is, each pupilshould have ample opportunity to physi-cally handle the materials. This may hedone individually or within a group, ascircumstances dictate. Such manipulationutilizes several senses, including visual,aural, tactile, and kinesthetic. In general,the materials should exploit as many sensesas possible. Compliance with this general-ization is particularly important withyounger children.

Physical criteria are important, sincemany sources of information available toteachers, such as commercial catalogs andbrochures, describe physical features of thematerials. A careful scrutiny of physicalcriteria would be helpful in initally screen-ing manipulative materials. Among thephysical characteristics to consider in se-lecting manipulative materials are the fol-lowing:

8 TEACHER-MADE AIDS

1. DurabilityThe device must be strong enough to

withstand normal use and handling bychildren. If and when maintenance isneeded, it should be readily available ata reasonable cost.

2. AttractivenessThe materials should appeal to the

child's natural curiosity and his desirefor action. Materials in themselvesshould not divert attention away fromthe central concepts being developed.Nevertheless there are certain qualitiessuch as aesthetically pleasing design;precision of construction; durable,smooth, and perhaps colorful finishthat are desirable. Nothing can be moredistracting than pieces of a tangrampuzzle that do not fit properly or a bal-ance beam that doesn't quite balance.

3. Simplicity

The degree of complexity is of coursea function of the concept being devel-oped and of the children involved, butgenerally the materials should be simpleto operate and manipulate. Although thematerials may lend themselves to a hostof complex and challenging ideasforexample, the attribute or logic blocksthey should be simple to use. .In an ef-fort to construct and use simple devices,there is the inherent danger of over-simplifying or misrepresenting the con-cept. In all cases care must be taken toensure that the device properly em-bodies the mathematical concept. Inaddition, the design of materials shouldnot require time-consuming, mundanechores such as distributing, collecting,and keeping an extensive inventory rec-ord of a large number of items.

4. Size

The materials should be designed toaccommodate children's physical com-petencies and thus be easily manipulated.Storage is an important consideration di-rectly related to sizeno device shouldtake up more than a reasonable amount

of storage space. Suitability of size isalso important in preventing misconcep-tions or distorted mental images withinthe child's mind.

5. CostThe index used to assess the worth of

materials must ultimately weigh theiruse against cost. In this context, cost isused in a broad sense. Thus cost mustinclude the initial expenditure and main-tenance and replacement charges basedon the life expectancy of materials underaormal classroom use. The teacher -re-1att: a cost, a function of the time requiredto learn how to use the materials effec-tively, is an item of utmost importance.It is not uncommon for someone otherthan a classroom teacher to order math-ematics materials, but without properplanning, orientation, and preparation,it is ludicrous to expect teachers to usenew materials effectively with their pu-pils. Therefore, any purchase of newmaterials should be accompanied by aplanned program designed to familiarizethe teachers with these materials. As aresult, any cost estimate for manipula-tive materials should reflect the teacher-education phase as well as the expendi-ture for materials.

Teachers are often confronted with thedilemma of whether to use commercial orhomemade manipulative materials. Manymanipulative materials are relatively easyto make and can often be produced by theteacher and/or pupils. There are manypriceless, intangible by-products, such as ad-ditional mathematical insight and increasedmotivation, that result directly from class-room projects. Nevertheless, one shouldweigh production costs for homemade ma-terials, including labor, materials, and soon, against the cost of similar commercialproducts. Quality, of course, is anotherconsideration. Frequently there is a markeddifference in quality between homemadeand commercially produced materials:

It would be ideal if manipulative mate-rials could meet all the aforementioned

criteria. Finding such materials would betantamount to finding a "fish that runs fastand flies high." Consequently the searchcontinues. It is hoped, however, that thesecriteria will provide teachers with someguidelines for both the selection and theuse of manipulatiVe materials.

Using manipulative materials

There have been several fine lists sum-marizing uses and functions of teachingaids. Many such lists apply specifically tomanipulative materials. Among the mostcommon uses of manipulative materials arethe following:

1. To vary instructional activities

To provide experiences in actual prob-lem-solving situations

3. To provide concrete representations ofabstract ideas

4. To provide a basis for analyzing sensorydata, so necessary in concept formation

5. To provide an opportunity for studentsto discover relationships and formulategeneralizations

6. To provide active participation by pupils

7. To provide for individual differences8. To increase motivation related, not to

a single mathematics topic, but to learn-ing in general

From this list it should be evident thatmanipulative materials may be used in -avariety of ways. It should also be notedthat the mere use of manipulative materialsdoes not ensure that they are being usedproperly. Manipulative materials must beused at the right time and in the right wayif they arc to he effective. Failure to selectappropriate manipulative materials and

failure to use them properly can destroytheir effectiveness. Some specific dos anddon'ts for teachers who plan to use manip-ulative materials follow:

1. Do consider pedagogical and physicalcriteria in selecting manipulative materials.A prerequisite for effective use of manip-ulative materials is their appropriateness.

TEACHING AIDS 9

The physical criteria for manipulative ma-terials as well as the pedagogical considera-tions should not be taken lightly.

2. Do construct activities that providemultiple embodiment of the concept. It isdifficult, and often foolhardy, to abstract orgeneralize from a single experience. Thusthe pupil should be presented with differentsituations manifesting the concept or struc-ture to be learned. For example, in devel-oping the concept of three, children mightexamine sets with three elements for oneactivity. The number line, balance beam,and Minnebars might also be used to pro-vide different embodiments for the sameconcept. The case for multiple embodimenthas been ably presented by Dienes. Al-though the idea is pedagogically sound, ithas yet to receive widespread use by class-room teachers.

3. Do prepare in advance for the activity.Be sure you, as the teacher, use the manip-ulative materials in the complete activitybefore they are used by pupils. As youmake this trial run, you should considerquestions such as: What prerequisite skillsare needed More these manipulative ma-terials are introduced? Are the directionsclear, and can they be easily followed? Arethere an adequate number of leading ques-tions? Are the manipulative materials com-mensurate with the level of the pupils andappropriate for the mathematical concept?What are some potential problem areas,and how might they be alleviated?

4. Do prepare the pupils. The type ofpreparation depends on both the manipula-tive materials being used and the age of thepupils. Above all else, be sure the pupilsare ready to profit from experience withthe materials. Care should be taken to pro-vide the necessary directions for beginningthe activity. One must guard against tellingpupils precisely what to do with the mate-rials, as this might sterilize the learning ex-perience. On the other hand, sufficient di-rection should be provided to prevent massconfusion, which may quickly lead to dis-cipline problems.

10 TEACHER-MADE AIDS

5. Do prepare the classroom. Check toensure that all required materials are onhand. Also be sure they are operative, ac-cessible. and available in sufficient quantity.The arrangement of the classroom furnitureshould be examined to ensure that it is

suitable for the planned activities.

6. Do encourage pupils to think forthemselves. The use of manipulative ma-terials in an informal situation provides anide,411 climate for creativity, imagination,and individual exploration. This atmosphereencourages pupils to think for therriselves.However, in order to get students to beginand then continue to think for themselves,it is imperative that the teacher provide en-couragement of and show respect for pupils'ideas. A teacher's dismissal of a student'sidea as being trivial, incorrect, worthless,and so on,' will repress future ideas.

7. Do encourage group interaction. Dis-cussion within, as well as among, groupscan be intellectually stimulating. Encouragestudents to communicate with their peersand teacher. The importance of having thisopportunity to tell one's thoughts, observa-tions, and ideas cannot be overestimated.As pupils grow older this freedom to ex-press personal ideas is accompanied by aresponsibility to defend or at least supporta position, should the need arise. Someteachers fear that one student will dominatea group of peers. This may sometimes hap-pen; however, the carefulselection of groupmembership can keep this problem at aminimum.

8. Do ask pupils 'questions. It is oftenessential that certain.points be called to thepupils' attention. Sometimes big ideas aremissed completely. Other times one childmay divert group attention to some minoror obscure point. In either ease, you, asthe teacher, must be prepared to ask perti-nent leading questions.

9. Do allow children to make errors.Some may view this as heresy. However,children must have an opportunity to hewrong or to make a mistake. Often greaterlearning and more lively discussion follow

an incorrect answer than a correct one.Besides, the natural learning process is

characterized byimuch trial-and-error learn-ing. To do otherWise, that is, to attempt toeliminate incorrect answers or faulty spec-ulation,: is to create a highly artificiallearning situation.

10. Do provide follow-up activities. Dis-cussion, correlated readings, reports, andprojects, as well as replications of activi-ties, enhance the prospects of learning.Searching questions forcing pupils to fur-ther analyze and synthesize their resultscan be very productive, as they encouragestudents to "pull together the loose ends."They might be followed by additional ques-tions that require extrapolation from theseactivities and encourage speculation on theoutcome of other related events.

1 I. Do evaluate the effectiveness of ma-terials after using them. Immediately uponthe completion of an activity, it can be veryhelpful to note particular problem areas,strengths, weaknesses, and suggestions andto define areas of needed improvement aswell as possible areas of modification. Acontinuous reevaluation of manipulativematerials ultimately results in better mate-rials as well as more effective use of them.

12.. Do exchange ideas with colleagues.Many new functions of manipulative mate-rials result from actual classroom use.Sometimes pupils either consciously or un-consciously propose additional uses. Attimes, informal exploration with manipula-tive materials by either teacher or childrensuggests new activities, which adds to thereservoir of potential uses for this set ofmanipulative materials. A mutual exchangeof ideas among teachers allows each toprofit from the experience of the others.Perhaps you remember the fable: If I havea dollar and you have a dollar and we ex-change dollars, we both still have a dollar.However, if I have an idea and you havean idea and we exchange ideas we bothnow have two ideas.

Now for some teacher don'ts!

1. Don't use manipulative materials in-discriminately. Care must be taken to en-sure that these materials properly embodythe mathematical conccpt being developed.Be sure the materials and concept arc com-mensurate with your objectives and thcpupils' level of development.

2. Don't make excessive use of manip-ulative materials. They should be used onlywhen thcy represcnt an integral part of thcinstructional program and when the pro-gram could not be achieved bctter withoutthe materials. One cxccption to this mightbe manipulative materials that are directedmore toward recreation. There are instanceswhcrc thc traditional curriculum fails toreach many pupils. Often the .recreationalaspect of manipulative materials has at-tracted thc attention of thcse youngstersand eventually paved the way to more aca-demically oriented activities. Somc teachersfear that excessive use of manipulative ma-terials will lead to overdependence onphysical representations. There are caseswhere the manipulative materials are usedas "crutches." However, most pupils willgradually stop using thc materials whenthey have reached a higher level of devel-opment. Signs of boredom from thc chil-dren may indicate excessive use of manip-ulative materials; or may suggest the necdfor raising additional questions or extendingthe concepts being explored with the ma-nipulative materials.

3. Don't hurry thc activity. Once thcconcept has bccn developed, most childrenare eager to explore other ideas. HoweVer,every pupil should have ample opportunityto use thc manipulative materials, thcrcbyconvincing himself of the principle or for-mulating the concept. Hurrying through theactivity may impose unnecessary pressurcon some pupils as well as creating a veryartificial learning situation. Few individualslearn well when they arc rushed. Somcchildren may formulate thc concept withinminutes, whereas other children may re-quire several days or perhaps months.Rushing children as thcy use manipulative

TEACHING AIDS 11

materials does not solve thc problem butrather compounds it.

4. Don't rush from the concrete to theabstract level. This is a sequel to the pre-vious suggestion. Perhaps the most frequenterror in using manipulative materials is thespeed at which children are rushed from theconcrete stagc to the symbolic level. Thereseems to be some myth that you can't learnmathematics unless you are actually writingsomething, that is, working with symbols.This is, of course, nonsense! Most goodmathematics at the primary level is donewithout symbolization. In fact, if seriousconsideration were given to Piaget's re-search, nearly all mathematics in the pri-mary grades would be at the concrete stage.It must be notcd that symbolization occursquite late in concept formation. Symbolsare reserved for describing or making arecord of the conccpt or mathematicalprinciple. Hencc, they can only be properlyused after the concept has been abstracted.Since the process of learning a mathe-matical abstraction is time consuming, itis ludicrous (at least with most elementarychildren) to use manipulative materials forone or two days and then move directly tothc symbolic level. The wrong kind of ex-periencc may result in thc children's view-ing manipulative materials as toys or enter-tainment, in no way related to mathematics.

5. Don't provide all the answers. Inworking with manipulative materials pupilsacquire experiencc in abstracting from a setof phenomena or a body of data. As eachchild is actively involved in this process,conflicts frequently arise. One pupil has oneanswer, another child has a somewhat dif-ferent result. Often the first reaction of thetcacher is to settle the issue by providingthe correct answer. It is difficult to resistthe temptation to tell thc correct answer,but resist the teacher must! To do other-wise is to discourage individual thought,squash natural curiosity to search for othersolutions, promote dependence on thetcacher rather than independence, and pre-clude further discussion of thc problem, as

.


everyone now knows the correct answer.On the other hand, you may decide to asksome -leading questions; you may have thepupils explain their solution; you may wishto have them replicate the activity usingthe manipulative materials; or you maypursue some other alternative. Regardlessof the option selected, the teacher mustrefrain from serving as the purveyor oftruth and source of all knowledge. Remem-ber that to children and adults alike, "theart of being a bore consists in telling every-thing."

Conclusion

Pcrhaps the best one can do is identifythose materials that best meet the criteriaand then concentrate on developing effec-tive ways of using them. This requires sev-eral. steps. First, the desired learning mustbe clearly identified. Then manipulativematerials that will aid in the learning proc-ess need to be selected. The third step re-quires that these materials be integratedinto an organized learning sequence, sothat pupils progress from the simple andconcrete to the complex and abstract. Onlyin this way can manipulative materials bean integral part of the mathematics educa-tion program.

Remember that manpulative materialsarc not to be considered a substitute forteaching, i.e., something one uses in lieu ofteaching. There is not now, never has been,and, it is hoped, never will he a genuinesubstitute for a good teacher who knowshow and what children nccd to learn andwhen they need to learn it!

References

Beougher, Elton E. The Review of the Literatureand Research Related to the Use of Manipu-lative Aids in the Teaching of Mathematics.Pontiac, Mich.: Special Publication of Divisionof Instruction, Oakland Schools, 1967.

Berger, Emil J., and Donovan A. Johnson. AGuide to the Use and Procurement of Teach-ing Aids for Mathematics. Washington, D.C.:National Council of Teachers of Mathematics,1959.

Bernstein, Allen L. "Use of Manipulative De-vices in Teaching Mathematics." ARITHMETICTEACHER 10 (May 1963):280 -83.

Brown, Kenneth E., and Theodore L. Abell."Research in the Teaching of ElementarySchool Mathematics." ARITHMETIC TEACHER12 (November 1965):547 -49.

Copeland, Richard W. How Children LearnMathematics: Teaching Implications of Piaget'sResearch. New York: Macmillan Co., 1970.

Davidson, Patricia S. "An Annotated Bibliog-raphy of Suggested Manipulative Devices."ARITHMETIC TEACHER 15 (October 1968):509-24.

Dienes, Zolton P. Building Up Mathematics.London: Hutchinson Education, 1960.

Hamilton, E. W. "Manipulative Devices." ARITH-METIC TEACHER 13 (October 1966):461 -67.

Grossnickle, Foster E., Charlotte hinge, andWilliam Metzner. "Instructional Materials forTeaching Arithmetic." In The Teaching ofArithmetic, Fiftieth Yearbook of the NationalSociety for the Study of Education, pt. 2.

Chicago: University of Chicago Press, 1951.Multi-Sensory Aids in the Teaching of Mathe-

matics. Eighteenth Yearbook of the NationalCouncil of Teachers of Mathematics. NewYork: Bureau of Publications, Teachers Col-lege, Columbia University, 1945.

Spross, Patricia. "Considerations in the Selectionof Learning Aids." ARITHMETIC TEACHER 11(May 1964) :350-53.

The author is grateful to the students in theclasses in which this paper was originally a handoutas well as to Lois Knowles, University of Missouri,and Donovan A. Johnson, University of Minnesota,for -reading and reacting to earlier drafts of thepaper. Many.of their suggestions are reflected in thisarticle.

Use of manipulative devicesin teaching mathematics

ALLEN L. BERNSTEINWayne County Public Schools, D6iroit, Michigan

Mr. Bernstein is consultant in secondary education for the Wayne County Public Schools.

The use of manipulative and pictorialmaterial in the teaching of mathematicshas been an accepted procedure for manyyears. The modern teacher is surroundedwith methods books discussing how tomake these devices, and with commercialcatalogs offering a tremendous variety ofthese materials. How does one navigatethis maze and make wise decisions con-cerning the expenditure of professionaltime and energy and of school funds? Webelieve that from the experience of theprofession certain principles regardingthe selection and use of such material haveevolved.

There should be a direct correlation be-tween tne operations which are carried onwith a device and the operations which arecarried on. in doing the same mathematicswith paper and pencil.

For example, consider the sketch of theabacus shown in Figure 1, illustrating the

13

sum of 23 and 34. Note that the effect inthe columns of the abacus correlatesdirectly with the effect in addition with thesymbols representing the numbers underconsideration.

Now consider the sketch (Fig. 2) andthe accompanying example (Fig. 3). Inthis example we are showing 17 plus 74.Note that in the sum of the first column itis necessary to "carry" because the sumis greater than 10. The operation as shownon the abacus is precisely the same as thatshown by paper and pencil, that of ex-changing ten beads in the ones column forone in the tens column. By contrast oneversion of the ancient Chinese abacuscalls for five beads in each column, eachwith a value of one, and two beads abovea divider, each with a value of five. Thesketches (Figs. 4 and 5) illustrate thesame example, 17 plus 74, added on such

Figure 2

SUM I I


1

1

r-01J

ONETEN

SUM 9 I

Figure 3

cans of soup), would be excellent illustra-tions of this principle. The child then hasthe opportunity to move these itemsaround on his desk, and the motion effecthelps to make the problem more concrete

TEN ----than a simple pictorial representation of

74+ 17

91

aja' abacus. The lack of relationship be-'tween the abacus operation and the paper

and pencil operation will lead to confusion.The use of the manipulative aid should

involve some moving part or parts, or shouldbe something which is moved in the processof illustrating the mathematical principlesinvolved.

The grouping of objects in the earlygrades with the use of bottle caps, tonguedepressors, or the myriad other innova-tions which have assisted teachers in thepast, or the use of the same materials inthe intermediate grades to act as symbolicrepresentations of the elements of wordproblems (i.e., 4 bottle caps represents 4

SUM

(54- 4)9

Figure 4. To add 4 in the ones commn, it isnecessary to deduct one and add a beadrepresenting 5.

the example. The current emphasis onand discussion of discovery methods inteaching reveals another facet of thisimportant principle. The availability ofmanipulative objects in less structuredlearning situations gives the child theopportunity to explore relationships anddiscover principles by manipulating theparts of the aid. Mathematics teachersoften help students to discover that thesame problem can be approached andsolved in a variety of ways. When this istrue of an example most of the differentapproaches can be illustrated with varyingmanipulations of the appropriate aid.

The use of manipulative and pictorialmaterials should exploit as many senses aspossible.

We know that human beings learnthrough all the senses and that among themost important of these in the learningprocess are the visual, aural, tactual, andkinesthetic senses. We believe that thekinesthetic and tactual senses in particularare too often ignored in current practice.When a first-grade child uses an abacus,counting frame, or other similar aid tosolve a problem, he is often observedmoving his fingers over the beads or

Figure 5. Two beads representing five onesapiece are exchanged for one ten bead.

touching the objects as he counts them.This is an important part of the learningprocess. It is therefore desirable that in theuse of such materials the teacher haveenough available so that all the children.,,,have an opportunity to handle. andelate it. It is also desirable that the ma-terials be made in bright colors, with con-trasting colors to stimulate the visualsense where appropriate. There is somedanger of color dependency, which is to beavoided.

A student should have his own aid, orample opportunity to use one many times.

Consider the situation where the teacherdemonstrates a principle with a manipula-tive device, goes through the explanationpossibly a second time, and then assignsthe children a series of seat-work problemsall of which are done with paper, pencil,and symbolic representations only. Con-trast this situation with one in which eachchild has the opportunity to handle an aid.and use it as he does his seat work. Insuch a situation, thechild may manipulatethe aid for many dozens, even hundreds, ofexamples so that the principle illustratedjust once by the teacher becomes increas-ingly clear in the child's mind.

In general, learning may proceed fromusing physical models, to using pictures, tousing only symbols.

Let us consider the example of threecans of soup, under the following headings:

Physical model Picture Symbol

3 actual cans A photograph The printedof soup of 3 cans words, "three

of soup cans of soup"or 3 empty or A sale- or "3 cans of

soup cans matic dia- soup"gram, viz.,

or 3 cylinders or "38"

The use of the physical or manipulativeaid should be permissive for the child ratherthan mandatory.

Some children are able, after brief study,to operate on "an abstract level," andforced use of multisensory aid in their

TEACHING AIDS 15

mathematical work will only slow themdown and may create bordom or resent-ment. On the other hand, denying such chil-dren the availability of the ..aid becausetho.da not.need it defeats opportunity forexplorat ion and discovery of the new pat-terns mentioned above. Other childrenmay need to use the concrete assistancerendered by the aid for a period of timelonger than one would expect for the so-called "average" child. Teaching experi-ence has indicated that students tend togive up the use of manipulative deviceswhen they no longer need them.

A device should be flexible and have manyuses.

The hundred board is a ease in point.This familiar device has been used in manyclassrooms to illustrate some features ofnumber structure for our number systemwith base ten. When some parts of theboard are covered it may be used to illus-trate number systems to other bases. Theboard has also been used to illustrate someof the principles involved in decimal andpercent calculations. Cost is always amajor factor in the choice of devices. (Ourdefinition of cost includes the time thatteachers and children may put into con-structing home-made devices.) Thereforethe finished aid should be convenient touse and store, and should be fairly durable.Obviously such materials as are consumedin use are not included in this principle.

The principles elicited indicate thatmanipulative devices need not. be compli-cated, and apply in a variety of teachingformats, i.e., large group or small groupinstruction, individual work, prescribedexercises, cooperatively planned projects,independent discovery work, and so forth.Students may be involved in producing theaid, or gathering material for it. Theteacher must decide whether the invest-ment of such time or energy is worthwhilerelative to the gains achieved in terms ofcarefully chosen goals.

. While it is obviously impossible to ap-ply all of these criteria to all possible teach-ing aids in all situations, they do provide


some sensible guide lines enabling teachers school financial resources. It is hoped thatto decide which materials are worthy of these criteria will be helpful in the profes-the investment of professional energy and sional decisions facing us in this area.

Models and mathematics

ELIZABETH H. FENNEMA

A lecturer in the Department of Curriculum and Instructionat the University of Wisconsin--Madison, Elizabeth Fennemateaches undergraduate and graduate courses in elementary school

mathematics education, both preservice and in-service.

She often serves as a consultant in mathematics education tovarious public school districts.

Although there is reasonable agreementabout which mathematical ideas should betaught in the elementary school (Beg le1966) and that these ideas should betaught meaningfully (Dawson and Rud-dell [a] 1955), there is little agreement onhow learning environments should be struc-tured to facilitate this learning. One reasonfor this lack of agreement is an inadequaterecognition of the role that concrete andsymbolic models can and should play infacilitating the learning of mathematicalideas.

Most, if not all, mathematical ideas thatare taught in the elementary school canbe represented by at least two types ofmodels that aid in making the ideas mean-ingful to the learner.' A concrete modelrepresents a mathematical idea by meansof three-dimensional objects. A symbolicmodel represents a mathematical idea bymeans of commonly accepted numeralsand signs that denote mathematical opera-tions or relationships. To understand how

1. A third type of model, the pictorial, can re-present mathematical ideas and is commonly usedin many elementary school mathematics textbooks.It shares attributes of both concrete and symbolicmodels. Because the theoretical rationale and em-pirical evidence regarding the usefulness of pictorialmodels in improving learning are less clear-cut thanthose of the other two models, pictorial models arenot a direct concern of this paper. However, inas-much as pictorial models provide a bridge or inter-mediate step between concrete and symbolic models,as the roles of the latter two are made clearer therole of the pictorial model will be clarified also.

17

the same idea can be represented by thesetwo diverse models, consider multiplida-tion of whole numbers (interpreted as theunion of equivalent disjoint sets). A con-crete' model that demonstrates a specificinstance of the idea 4 X 3 = 12 wouldbe 4 groups of 3 countersbuttons, stones,pencilsthe total of which is 12. Thecounters can be manipulated, grouped, andhandled in a variety of ways to show that4 groups of objects with 3 in each groupalways have a total of 12 objects. Meaningcomes directly through manipulation of theobjects. A symbolic model that representsthe same idea is 3 + 3 + 3 + 3 = 12.The threes can be added to obtain twelve,but 3, +, =, and 12 are merely marks(symbols), and when such arbitrarilychosen symbols are used, their only rela-tion to reality is what the child has learnedbefore. If this model is meaningful, it isbecause the learner is able to attach somemeaning to the symbols.

The ability to use symbols to solveproblems and the ability to learn moreadvanced principles are instructional ob-jectives common to all elementary schoolmathematics programs, although these ob-jectives often are not explicitly stated.Symbols help in generalizing ideas so thatthey apply to diverse situations, and theypromote the transfer of learning (Bruneret al. 1966). However, the use of symbolsis not innate; it must be carefully developed


by using both concrete and symbolicmodels in a carefully controlled mannerthroughout the elementary school years.The route to developing this ability is nota direct one because children of seventhrough eleven years of age are able touse symbols effectively only after they haveexperienced the ideas to be learned throughactionthrough the manipulation of con-crete models. This means that most chil-dren in the elementary school years oftenrequire the use of concrete models to makesymbolic models meaningful.

The idea that experiences with concretemodels should precede experiences withsymbolic models is neither new nor unique.Van Engen stated in 1949 (1949, p. 397)that the "meaning of words cannot bethrown back on the meaning of otherwords. When the child has seen the actionand performed the act for himself, he isready for the symbol for the act." How-ever, this idea is meeting increasinglygreater acceptance. Major theoretical sup-port for the use of concrete models beforesymbolic models in teaching mathematicalideas is provided by Piaget, who has pro-posed a comprehensive theory of cognitivedevelopment that encompasses individualgrowth from birth to maturity. Accordingto Piaget's theory, schemas (mental struc-tures) are formed by a continual process ofaccommodation to and assimilation of theindividual's environment. This adaptation(accommodation and assimilation) is pos-sible because of the actions performed bythe individual upon his environment. Theseactions change in character and progressfrom overt, sensory actions done almostcompletely outside the individual to par-tially internalized actions that can bedone with symbols representing previousactions, to completely abstract thought doneentirely with symbols. Thus developmentin cognitive growth involves first the use ofphysical actions to form schemas and thenthe use of symbols to form schemas. Learn-ers change from a predominant relianceon physical action to a predominant re-liance on symbols. It should be noted, how-

ever, that except for the very early monthsof a child's life, learning involves bothphysical actions and symbols that representpreviously performed actions.

Translation of this cognitive-develop-ment theory to instructional practice indi-cates that learning environments for chil-dren at various developmental levels shouldinclude both concrete and symbolic modelsof the ideas to be learned, with specialattention given to ensure a major em-phasis on those kinds of experiences thatrepresent the predominant type of orienta-tion (concrete or symbolic) most appro-priate to the development of schemas at thevarious developmental levels. It followsthat the learning of children who are re-latively more dependent on physical action(at an early level of cognitive develop-ment) is made meaningful by a learningenvironment with a predominance of phys-ical experience (concrete models) and thatthe learning of children at a more advancedlevel of development is made meaningfulby a learning environment with a pre-dominance of symbolic experience (sym-bolic models).

Although empirical evidence concerningthe use of the two models in learning spe-cific ideas is meager and often inconclusive(Fennema 1969), there is a set of studiesthat support the theoretical position setforth in this paper. The largest subset ofthis group of studies concern one particu-lar set of concrete models, the Cuisenairerods and their prescribed teaching method(Gattegno 1964). Most of these studieswere similar in design: two groups wereequated in some ways; the control groupwas taught in a traditional way with tradi-tional materials (often defined no betterthan this) and the experimental group wastaught with the Cuisenaire materials as pre-scribed; and learning by the two groups wasevaluated in some way. Table 1 is a sum-mary of these studies.

Several conclusions can be drawn fromthese studies. The Cuisenaire materialswere more effective than traditional ma-

TEACHING AIDS 19

Table .1Summary of some studies comparing the effectiveness of Cuisenaire andtraditional materials

Author' Grade level Test used

Significant difference inloo?. of Cuisenaire ortraditional materials Mathematical content

Aurich First Standardized Cuisenaire treatment Total range of(1963) achievement first-grade work

Hollis First Standardized Cuisenaire treatment Total range of(1964) achievement first-grade work

Crowder First Standardized Cuisenaire treatment Total range of(1965) achievement first-grade work

Nasca Second Standardized Neither treatment Total range of(1966) achievement second-grade work

Cuisenaireachievement

Cuisenaire treatment

Passy Third Standardized Traditional treatment Computation and(1963) achievement arithmetic reasoning

Haynes(1963)

Third Authorconstructed

Neither treatment Multiplication

Standardizedachievement

Neither treatment

Lucow Third Author Neither treatment Multiplication and(1963) constructed division

t Omitted from this group of studies is Brownell's study (1966). It does not fit the paradigm of the other studies, and the results were ambiguous.

terials at the first-grade level (Aurich1963; Hollis 1964; Crowder 1965). Witholder children the results are ambiguous.Passy (1963) found that the traditionalmethod was more effective than the Cui -.senaire method, whereas Nasca (1966),using a test that included more advancedprinciples than those in standardized tests,found that the Cuisenaire method wasmore effective. In two studies (Haynes1963 and Lucow 1963), significant dif-ferences were not detected. In spite of theinadequacies of these studies, there is someindication that children learn better whenthe learning environment includes a pre-dominance of experiences with modelssuited to the children's level of cognitivedevelopment.

Additional studies involving other con-crete models have been reported. In thesestudies, summarized in table 2, traditionalteaching (usually undefined) was com-pared with teaching supplemented by theuse of concrete models.

Here again, first-grade children (Lucas1966) appear to benefit when learning isaided with concrete models representingmathematical ideas. Some studies (Daw-son and Ruddell [b] 1955; Norman 1955;Howard 1950) indicated that children inthe middle grades learned better whenlearning was facilitated with concretemodels. The use of concrete models witholder children (Mott 1959; Spross 1962;Price 1950; Anderson 1957) appears toneither improve nor hamper learning ofmathematical ideas.

Collectively, these data tend to supportthe hypothesis that a learning environmentembodying representational models suitedto the developmental level of the learnerfacilitates learning better than a learningenvironment that ignores the develop-mental level of the learner. Specifically,learning of mathematical ideas is likely tobe facilitated by a predominance of con-crete models in the early grades and agradually increasing proportion of symbolic


Table 2Summary of studies comparing effectiveness of learningfacilitated by concrete or symbolic models

mathematical ideas

GradeAuthor level Concrete model Test used

Significant differencein favor of:

Mathematicalcontent

Lucas First(1966)

Dienes AttributeBlocks

Standardizedachievementand authorconstructed

Dienes treatment forconservation of numberand conceptualization ofmathematical principles;traditional for computa-tion and solving of verbalproblems

Identified inPiagetian terms:multiplicationof relationsand addition-subtraction,relations

Ekman Third(1966)

Counters Authorconstructed

Neither treatment Addition andat end of instructional subtractionperiod; concrete-model algorithmsgroup on a retention test

Dawson & FourthRuddell([111955)

Many diversemodels

Authorconstructed

Concrete-model group Division ofwhole numbers

Norman Third(1955)

Concrete andsemiconcretemodels (numberlines, drawings,and counters)

Authorconstructed

Neither treatmentat the end of instruction;concrete and semi-concrete-model groupat the end of two weeks

Division ofwhole numbers

Howard Fifth &(1950) Sixth

Concrete andsemiconcretemodels

No informationavailable

Neither treatment atthe end of instruction;concrete-model groupthree months later

No informationavailable

Mott Fifth & Many multi- Standardized Neither treatment(1959) Sixth sensory aids achievement

Spross Fifth & Concrete aids that Standardized Neither treatment(1962) Sixth had cultural achievement

significance

Measurement

Total range offifth- & sixth-grade work

Price Fifth & Multisensory No information Neither treatment(1950) Sixth aids available

Anderson Eighth Various visual- Author Neither treatment(1957) tactual devices constructed

Division offractions

Area, volume,& Pythagoreantheorem

models as children move through the ele-mentary school.

Although the evidence, both theoreticaland empirical, appears to indicate that theratio of concrete to symbolic models usedto convey mathematical ideas should reflectthe developmental level of the learner, itshould not be inferred that either modelalone can suffice at any level in the elemen-tary school. Piaget placed children up totwelve years of age in the concrete-opera-tional stage of cognitive development.Children in this stage are capable of learn-ing with symbols but only if those symbolsrepresent actions the learners have donepreviously. The function of the classroomlearning environment is to ensure that the

children perform or have performed thoseactions that represent all the mathematicalideas that are to be taught in the elemen-tary school. This will mean that as childrenenter elementary school, the learning ofmost mathematical ideas will need to befacilitated through concrete representationboth because the developmental level ofthe children indicates that this is the appro-priate learning style and because the ex-periential background of the children ismeager. At upper levels the reverse will betrue. The learners' experiential backgroundwill be much richer and their learningorientation will indicate that their use ofsymbols is more adequate. However, whennew ideas are introduced, regardless of the

grade level, care must be taken to ensurethat appropriate actions (experiential back-ground) have taken place.

A study reported by Fennema (1969)emphasized the importance of this point.Eight randomly assigned groups of second-grade children were introduced to multi-plication. Half the groups learned with asymbolic model that reflected actions prev-iously done. That is, 2 + 2 + 2 = 6 wasused to represent 3 x 2 = 6. The otherhalf of the groups learned with a concretemodel, the Cuisenaire rods. Three two-rodsthat measured the same length as one six-rod represented 3 X 2 = 6. At the end ofa three-week instructional period, the learn-ing transfer of children who used the sym-bolic model measured at a higher levelthan the learning transfer of children whohad used the concrete. These children wereat a developmental level in which a pre-dominance of concrete models would nor-mally be more effective. However, becausethe children had previously performed theactions necessary for the understanding ofmultiplication they learned better withsymbolic models that permitted generaliza-tion of learned ideas to unlearned ones.

The best indicator of which representa-tional model should be used with a certainchild to explicate a specific idea is theperformance of 'the child. If children aregiven freedom, they will select the modelthat makes the idea most meaningful tothem. Concrete models when first used areintriguing, and children prefer using thembecause of their novelty and because theymake mathematical ideas meaningful. How-ever, concrete models are inefficient andcumbersome to use in problem solving.Children soon become aware of this andprefer to use symbolic models as a meansof learning when symbols are truly mean-ingful. Therefore an effective teacher willcarefully observe children and attempt todetermine which models are most meaning-ful and acceptable to the children con-cerned. At the same time, children shouldhave freedom to choose from alternative

TEACHING AIDS 21

types of models whenever they are calledupon to solve a problem.

If concrete and symbolic models areused in the way suggested, meaningfullearning of mathematical ideas is morelikely. Children will be better able to applytheir learning to new situations, and com-prehension of the highly abstract contentof advanced mathematics will be moreeasily acquire.

References

Anderson, G. L. "Visual-Tactual Devices andTheir Efficacy: An Experiment in GradeEight." ARITHMETIC TEACHER 4 (November1957 ) : 196-203.

Aurich, Sister M. R. "A Comparative Study toDetermine the Effectiveness of the CuisenaireMethod of Arithmetic Instruction with Chil-dren of the First Grade Level." Master's thesis,Catholic University of America, 1963. Re-produced by the Cuisenaire Company ofAmerica.

Begle, E. G. "Curriculum Research in Mathe-matics." Paper based on a talk given at theWisconsin Research and Development Centerfor Cognitive Learning, University of Wiscon-sinMadison, 1966.

Brownell, W. A. "Conceptual Maturity in Arith-metic under Differing Systems of Instruction."Elementary School Journal 69 (1968): 151-63.

Bruner, J. S., et al. Studies in Cognitive Growth.New York: John Wiley & Sons, 1966.

Crowder, A. B. "A Comparative Study of TwoMethods of Teaching Arithmetic in the FirstGrade." Doctoral dissertation, North TexasState University, 1965.

Dawson, D. T., and A. K. Ruddell [at "TheCase for Meaning Theory in Teaching Arith-metic." Elementary School Journal 55 (1955):393-99.

[b]. "An Experimental Approach to theDivision Idea." ARITHMETIC TEACHER 2 (Feb-ruary 1955): 6-9.

Ekman, L. G. "A Comparison of the Effective-ness of Different Approaches to the Teachingof Addition and Subtraction Algorithms in theThird Grade." Doctoral dissertation, Universityof Minnesota, 1966.

Fennema, E. A Study of the Relative Effective-ness of a Meaningful Concrete and a Meaning-ful Symbolic Model in Learning a SelectedMathematical Principle. Technical report no.101. Madison: Wisconsin Research and De-velopment Center for Cognitive Learning,1969.

Gattegno, C. For the Teaching of Elementary


Mathematics. New Rochelle, N.Y.: CuisenaireCo. of America, 1964.

Haynes, J. D. "Cuisenaire Rods and the Teachingof Multiplication to Third Grade Children."Doctoral dissertation, Florida State University,1963.

Hollis, L. Y. "A Study to Compare the Effectsof Teaching First Grade Mathematics by theCuisenaire-Gattegno Method with the Tradi-tional Method." Doctoral dissertation, TexasTechnical College, 1964.

Howard, C. F. "Three Methods of TeachingArithmetic." California Journal o/ EducationalResearch 1 (January 1950): 25-29.

Lucas, J. S. "The Effect of Attribute-Block Train-ing on Children's Development of Arithmetic."Doctoral dissertation, University of CaliforniaBerkeley, 1966.

Lucow, W. H. "Testing the Cuisenaire Method."ARITHMETIC TEACHER 10 (November 1963):435-38.

Mott, E. R. "An Experimental Study Testing theValue of Using Multisensory Experiences inthe Teaching of Measurement Units on theFifth and Sixth Grade Level." Doctoral disser-tation, Pennsylvania State University, 1959.

Nasca, D. "Comparative Merits of a Manipula-

tive Approach to Second-Grade Arithmetic."ARITHMETIC TEACHER 13 (March 1966):221-26.

Norman, M. "Three Methods of Teaching BasicDivision Facts." Doctoral dissertation, StateUniversity of Iowa, 1955.

Passy, R. A. "The Effect of Cuisenaire Materialson Reasoning and Computation." ARITHMETICTEACHER 10 (November 1963): 439-40.

Price, R. D. "An Experimental Evaluation of theRelative Effectiveness of the Use of CertainMulti-Sensory Aids in Instruction in the Divi-sion of Fractions." Doctoral Dissertation, Uni-versity of Minnesota, 1950.

Spross, P. M. "A Study of the Effect of a Tan-gible and Conceptualized Presentation of Arith-metic on Achievement in the Fifth and SixthGrades." Doctoral dissertation, Michigan StateUniversity, 1962.

Van Engen, H. [a] "The Formation of Concepts."In The Learning of Mathematics: Its Theoryand Practice. Twenty-first Yearbook of the Na-tional Council of Teachers of Mathematics.Washington, D.C.: The Council, 1953.

[b]. "Analysis of Meaning in Arithmetic."Elementary School Journal 49 (FebruaryMarch 1949): 321-29; 395-400.

Whole Numbers

The study of whole numbers extends throughout the mathematics cur-riculum of the elementary school from initial experiences in counting andfinding the cardinal number of a set through work with the algorithms foraddition, subtraction, multiplication, and division. Intrzrent in much of thisstudy is the development of an understanding of the characteristics of ourdecimal system of numeration. Since this work with decimal bases andplace-value concepts is of such importance, we have included in a separatesection (chapter 3, "Numeration") the articles dealing primarily with theseideas.

As children's mathematical concepts are developed along the concrete-abstract continuum, we note that the hundred-board is an excellent manipu-lative aid that has a broad range of applications. Volpel offers a compre-hensive and practical discussion of this device and its many uses. This ar-ticle should provide a stimulus for the reader to explore multiple uses of themany teaching aids that are presented in this book of readings.

The cardinal and ordinal uses of number are among the early numberconcepts taught to young children. Fitzsimons provides some interestingideas for teaching the cardinal number of a set, and Patterson suggests someactivities with a picture line which involve pupils informally with bothcardinal and ordinal number work. Experiences with the picture line shouldserve as excellent preparation for future work with the number line. A brief,but clever, suggestion for using a "number clothesline" to teach basic addi-tion facts is offered by Orans.

23


The numeral-number association is another very important instructionaltask in the primary grades. Armstrong and Schmidt give a good descriptionof a set of inexpensive materials that they constructed for the purpose ofteaching the numeral-quantity association to trainable-level mentally re-tarded pupils. Many of these materials could also be used effectively toteach the same concepts to any child at the early stages of his mathematicalawareness.

Some early experiences with basic addition facts and missing addendsmay be provided by work with the number-line balance suggested byBorgen and Wood. Becker's "elevator numbers" combine addition practiceand fun in a punk format.

For those children who have difficulty understanding a table of basicmultiplication facts, Schrage suggests the construction of an array matrix.This should be very helpful for the average-to-slow pupils. Shafer con-tributes some interesting thoughts about the possibilities of using a tape re-corder to provide some of that much-needed individualized drill and prac-tice in building skill with the multiplication facts.

Teaching children to apply their mathematical knowledge continues tobe one of the trouble spots for teachers. The concluding article in this chap-ter focuses on one technique for teaching problem solving. Shari suggeststhe use of number-picture problem situations that involve the pupils in thecreation of problem situations as well as in finding solutions. She has somevery interesting ideas and illustrations.

The Hundred-BoardMARVIN C. VOLPEL

State Teachers College at Towson, Maryland

EVERY TEACHER OF ARITHMETIC should

have a hundred-board. The writer be-lieves it is one of the most efficient teachingaids for use in the development of under-standings of concepts and operations withnumber quantities. If a teacher could selectbut one gadget for use in the arithmeticclassroom he should choose the hundred-board because of its many uses with regardto both the scope and grade placement ofmaterial.

The hundred-board can take one of sev-eral forms. The simplest home -made boardconsists of a square of one-half inch plywoodlarge enough to be ruled off into 100smaller squares each 2 inches by 2 inches,with a 2 inch bordcr around the outside. Afinishing nail should bc driven at the centerof the top line of each small square. Thefinished product should then be staincd andequipped with a handle to facilitate han-dling, hanging, and storing. In this form itcan be used on a table, stand, easel, tripod,or in a chalk tray.

1111111111111111111111111111110111M111111=111111111111E

MOM Mu1111111111111111111

11111111111111111M111111111111111

Variations of the above can be effectedby use of pegs or hooks plaCed at the exactccntcr of each small square, by increasing..its Size so that it can be set on the floor for

easy handling by small children, or can bcconstructed with feet so that it can be usedwhile resting in a horizontal position on adesk, or -. table.

The teacher should have a set of numbersfrom 1 to 100 and a set of one hundred un-numbered objects which can be hung on theboard. The numbers can be purchased com-mercially or prepared by numbering oak-tag labels, department store price tags, orpoker chips punched for hanging. Theteacher should use objects which permiteasy handling and which at the same timeare attractive, such as spools, curtain rings,rubber washers, identification tags, one-inch pieces of plastic hose, and similar ob-jects.

1:310.2011 AMAMI=imumaim11m1111111111

III NUMMI.=1 111M MENII11111

In the paragraphs which follow the writerwill show a few uses of the hundred-board.

NumberThe hundred-board, initially, can be

used to show the cardinal aspect of num-bers. To show that two represents 1 morethan one, that three represents I more thantwo, etc. we present the pictures of these ob-jects. Thus when a child learns to say "4"he should associate that name with thatthings. This is "fourness." The meaning ofthe numbcr expressions from one to tenshould be shown on the board as follows:

25


0

00 2000 3000000000 5000000 60000000 700000000 8000000000 90000000000 10

Now after a child has learned the mean-ing of "ten" he can count tens. Referenceto a hundred-board filled with numbersclearly reveals groups of tens and the childshould he taught to count them as tens. Itshould he as easy for him to comprehend 1ten, 2 tens, 3 tens, 4 tens, etc. as it is for himto learn to count 1 marble, 2 marbles. 3marbles, . . or 1 case, 2 cases, 3 cases, etc.

Next we introduce the "teen'' numberswith the help of a second hundred-board.Each row of the hundred-board representsa ten, so if a filled hundred-board is placedbeside the one pictured above the childought to visualize number quantities liketen and 1, ten and 2, ten and 3 and theother numbers up to ten and 1(1 or 2 tens.If we encourage the child to continue tocount in this manner, in terms of groups oftens and ones, he will readily learn and, wehope, comprehend the meaning of the num-bers up to 10 tens or one hundred. In thisway the child will learn to say expressionlike "3 ,ens and 5," "3 tens and 6," "3 tens

'SI0:° 0 6oo-000 ooOlotioo o o

and

tom mad 1tin sad 2tem sad )

7" before he is told that such numberexpressions can he shortened to thirty-five(35), thirty-six (36), and thirty-seven (37).

When the child knows that four hill rows ofdiscs reprccm 1 grutips of ten discs he canhe >110\\'I1 110\V to express this quantity sym-.bulirally as 40. An extension of this conceptshould enable him to replace all the discswith others numbered to 100.

We can use the numbers on the hundred-board for teaching the child to count bytwos. 1Vc 'with) bY having all of the numbersturned over (face down!. We ask the childto skip the first disc. number 1. and turn overthe next disc. We skip one again and turnover the next one, we skip one again and"Bice -up" another. 11 we continue doingthis until we reach 5D or 60 the child shouldobserve the pattern of endings and shouldobserve that full columns are turned clownand full columns are exposed. Those num-bers exposed are called "even numbers andall of them end in. either 2. 4. 6. 8, or 0.

The same procedure might be followedwhen teaci. ing children to count by threes,fours. or fives. If all of the discs are turneddown, then we skip two and turn over one,skip two and turn over one. etc. when count-ing by threes. The numbers exposed will be3, 6. 9, 12, 15. 18, 21, 24, 27, 30, etc. Thepattern of endings does not begin to repeatuntil after we have shown ten threes.

0 2 0 4 0 6 0 8 0 10 0 0 3 0 0 6 0 0 9 00 12 0 1.4 0 16 0 18 0 20 012 0 03.5 0 018 0 00 22 0 24 0 26 21. 0 0 24 0 0 27 0 0 30

Another technique to follow in learningto count by twos or threes is to stack all thenumbers in an ordered row or pile; takethem one by one, place them in rows of 2or 3.as the case may be. One .does not haveto replace all 100 numbers on the board butenough to enable him to understand theconcept of counting b groups and to learnthe respective patterns. The number boardthen will look like this:

1 2 21. 22 1 2 3 31 32 333 4 23 24 4 5 6 34 35 365 6 25 26 7 8 9 37 38 397 8 10 11 129 10 etc. 13 1.4 15 etc.

11. 12 16 17 1813 114 19 20 2115 16 22 23 2417 18 25 26 2719 20 28 29 30

The hundred-board is useful for picturingthe various patterns of number quantities.The quantity 6, for instance, may be shownon the number board in several patterns. Ifchildren re-arrange discs (a constant num-ber of discs) they will discoVer numerousfacts concerning the given number. This ac-tivity develops insight into number tuan-tities and number relationships.

Groups of different sizes may be picturedon the board and children challenged to de-termine the largest group. If they see groupscontaining 4, 6, and 7 spools they ought tobe able to deduce that 6 contains (2) morethan 4, that 7 contains (1) more than 6, andthat 7 contains (3) more than 4. Converselythey will learn that 4 and 6 are smaller than7 by 3 and by 1 'respectively. The compari-son phase of counting can be taught throughthe use of the hundred-board.

c3 0 Eo1111111M111111111111111I

o EMMENOolol 01 _L_LO101MENNEEMINOCIWIN01111E 01111111111111110 HEIM o o MID 011111

1111110111111 111111M111111111111M 1111111111110MOM0UM= EU1111N11EENCIOCI000

Groups of 6

MUM=Groups of 4, 6, and 7

AdditionThe hundred-board is very useful in help-

ing children discover basic addition facts.Since addition is a "putting together" op-eration we want to know how many we havealtogether if we put 3 discs with a group of 4discs. We show 4 discs on the top row and 3discs on the second row and then combinethem into one group. We discover thatwhen we put 1 (of the 3) with 4 we will have5, another 1 will make 6, and the last 1 will

WHOLE NUMBERS 27

make 7. Thus 4 discs and 3 discs are 7 discs.Children will discover that there are manyaddition facts whose sums are less than 10the combining of two small sets of discs willoften leave the top row incomplete. Also,in learning the meanings of the numbers 1to 10 they will have discovered hcw manymore are needed to complete the row of 10.For instance, the last row of discs needs 0,the next to last row needs 1 so 9+1 =10, therow containing 8 needs 2 so 8+2=10, etc.In working with the complete set of 10,children ought to learn the sub-sets whichadd to 10. The board showing the sum of 4discs and 3 discs should gradually have thefollowing appearance:

0000 becomes 00000 then 000000 and then 0000000.000 00 0

Manipulations of this type suggest to thestudents that addition represents an in-crease, a movement to the righta forwardmotion.

12

+ 9+ 8

10= 10

oncomacxxoomoccxxxx00 OXXXX)00t 3 + 7 - 100000=000C 4 + 6 - 100000 OX100tX 5 + 5 1000000 OXX:10C 6 + 4 - 100000000)0C 7 + 300000000XX 8 + 2 10000000000X 9 + 1 100000000000 10 +0 - 10

When children have mastered the com-binations which total 10 they should havelittle difficulty recognizing sums whichtotal more than 10. Since 7+3=10, then7+4, 7+5, 7+6, etc. must be greater than10. Let us show the addition of 7 discs and 5discs on the hundred-board, The first rowshould show 7 discs. The second row shouldshow 5 discs. The discs from the set of 5should be moved one by one until we havecompleted the first row. We need 3 of these,therefore there should be 2 left on the secondrow. Thus the sum of 7 discs and 5 discs is thesame as the sum of 10 discs and 2 discs. Since10 and 2 represent 12, then the sum of 7discs and 5 discs is 12 discs.

In adding 6+7 (representing spools)children will visualize that they must take 4


of the 7 to put with the 6 to make a 10therefore 6+7 is the same as 10 and 3. Ad-dition with the hundred-board is similar toaddition performed along a number line;the hundred-board is a number line decom-posed into ten sections of ten where the con-cept of bridging is equally prominent. Toadd 6 to 7 on the number line one begins atthe point marked 6 and moves forward (tothe right) 4 places and then 3 more arrivingat 13. Thus 6+7 =13.

When showing the addition of 3 or morequantities on the hundred-board children

0

ought to sense the expediency of filling upgroups of 10 so as to state the final sum asgroups of tens plus ones. Thus 8+6+9when combined on the hundred-board be-comes first 10+4 (2 discs have been movedfrom the group of 6 to the 8 group making acomplete 10). Then 6 of the 9 are moved tothe second row completing another 10.Since there-are 3 left on the third row thesum of 8+6+9 is 2 tens plus 3 or 23. Astudent might move 2 of the. 9 into the toprow and 4 of the 9 into the second row toobtain the same result.

0 0 0 0 0 0 0 0 0 00

0

The rationale of carrying is evident whenone demonstrates the addition of 2 or moretwo-digit numbers with the aid of the hun-dred-board. To show the addition of 27 and38 we set up 2 rows of tens and 7 ones andthen 3 rows of tens and 8 ones. To find thecomposite sum we use some of the 8 ones tofill in the row containing the 7 ones. Since3 are needed to complete the row the finalresult shows 2 groups of ten, and 3 groupsof ten (which we had in the beginning), and1 more group of ten, and 5 ones. The sum,clearly visible on the hundred-board is 65.There were enough ones to make anothercomplete 10.

41.0,110

0O

O

mmEMRioomME1

mamamom

27+ 38

65

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0

Two boards can be used effectively show-ing the tens on one board and the ones onthe other. Then some of the ones can beused to make a ten. This procedure will en-able children to "discover" that 2 tens and 7plus 3 tens and 8 is the same as 6 tens and 5.

SubtractionSubtraction is an operation which sep-

arates a given group into 2 or more sub-groups and is usually taught first as a "take-away" operation, the opposite of the addi-tion operation. If we wish to find the num-ber of discs remaining after we have taken3 discs from a group of 9 discs we first pic-ture 9 discs on the hundred-board. When 3of these are removed, one at a time, studentsshould be encouraged to do so from right toleft removing first the ninth one, then theeighth one, and then the seventh one. Thisbackward operation will suggest a move-ment to the left along a number track . . .

and that's the basic idea of "take-away sub-traction."

Should we want to know the remainderwhen 5 things arc taken from a group of 12things we picture first the 12 things on theboard. Children will see that the takingaway operation (the 5 steps backward) willresult in a remainder which is less than 10They will first remove the 2 things on thesecond row and then move into the top rowand take some from the group of 10.12 5 .?

The 'same reasoning is applicable whenwe perform compound subtraction. Toillustrate further let us subtract 8 from 23.First we fill in 2 complete rows and place 3items on the third row to make 23. We startremoving 8 of these one-by-onewe takeoff the 3 and then move into the group of10 to get 5 more (3+5=8). There will beonly 15 left.

The solution of more difficult compoundsubtraction examples involving two digitsillustrates the principle of reading and think-ing from "left to right." Should we want toshow the example

7235

we would first show 72 items on the hun-dred-board. From these we arc to remove35. We will first take off 3 groups of 10 andthen will take 5 ones. We take off the 2single items and then take 3 more (enoughto make 5) from one of the groups of 10.The final result is 37 and the algorism whichwill illustrate the thought pattern is as fol-lows:

72 10 10 10 10 10 10 10 + 2- 35 10 10 10 -5

10 10 10 7

MultiplicationIf the addends of an addition example are

equal then the example can be solved bythe operation called multiplication. Shouldwe wish to add three groups of 2 we picturethem as follows on the hundred-board:

WHOLE NUMBERS 29

X XX XX X

We discover that when we regroup theseitems they will fill up the first 6 places. Thus3 twos equal 6. These can also be shownwith numbers after the result has been ob-tainedthe numbered discs can be placedtwo on each row so that the numbers show1 and 2, 3 and 4, and 5 and 6 on the threerows.

To show that 4 threes are 12 we arrangefour rows of three discs and then regroupthem to form sets of 10, if possible. Therewill be one set of 10 with 2 remaining. Thus4 X3=12. When numbered discs are usedto show the representation of 4 threes thenumbers will form the following pattern:

Children should become aware of the factthat the numbers in the last column are the"threes' facts" representing the products inthe table of threes.

To show that 3 sixes are 18 we first put 6discs on each of the first 3 rows. Now we re-group them to form sets of 10. In doing thiswe discover that there will be one set of 10with 8 remaining. Thus 3 X6 =18. When thesingle discs are moved to fill in rows thereare many ways in which this can be donethere is no particular "best" procedure.Again, when these discs are shown as num-bered discs the numbers in the last columnrepresent the products when 6 is the multi-plicand. 1 X 6 = 6, 2 X 6 = 12, 3 X 6 = 18, etc.

30

0

TEACHER-MADE AIDS

A hundred-board, with the addition of afew duplicate numbers, can be used to de-velop and display all of the multiplicationfacts and at the same time provide the op-portunity for children to learn how to readtables. To do this, leave the first block ofthe hundred-board empty and hang all thenumbers from I to 9 in the first row (9 willbe in the place usually occupied by 10).Also, still leaving the first block empty,hang numbers from 1 to 9 in the first col-umn. Now in the cells formed by the rowsand columns hang numbers which repre-sent the product of the two numbers whichhead the row and column. Thus in the cellof the row headed 4 and the column headed7 put the number 28. Likewise in the cell ofthe row headed 7 and the column headed 4put another 28. No other 28s are needed inthe table. However, there are some num-bers which appear more than twice as prod-ucts within the table, namely, 8, 16, and 24.

1 2 3 h 5 6 -7 8 9123

285

67 28

9

DivisionInasmuch as we have already shown ways

to solve subtraction examples on the hundred-board, then division, which is also aseparating or subtracting operation can alsobe illustrated thereon. There are two con-cepts of the division process, the measure-ment concept and the part-taking concept.We will first illustrate the measurementconcept which asks one to find the numberof groups when we know the size of thegroup (the multiplicand). An illustration ofthis idea is the question which asks how

many groups of 2 can be formed from agroup of 8. We begin with the dividend 8and show that many things on the board.Now we remove 2 of these things and placethem in a group elsewhere on the board.Then we repeat the operation with 2 more,and again 2 more until they have all beenregrouped into sets of 2. We have discoveredthat 4 groups of 2 can be formed from agroup of 8.

0 0 0 0 0 0 0ENMESH=

NNE NM linMEI0 Clain NM IIIIIIC113 MINIMEN I.ME

0 01211:111111

The example 16 --3 asks us to find thenumber of groups of 3 objects which can beformed from a group of 16 objects. We il-lustrate 16 objects on the board and thenpick them off in sets of 3. We discover 5 setsof 3 objects with I object left over or re-maining.

The example 97+8 asks us to find thenumber of groups of 8 which can be formedfrom 97. Since the size of the sub-set is quitelarge and would require much tedious han-dling of objects we suggest the use of rubberbands, string, ribbons, etc. to encircle thegroup. We circle 8, then circle 8 more, etc.until all possible groups of 8 have beenformed. We will discover that there are 12groups of 8 in 97 and 1 item remaining.

The partitioning aspect of division asks usto find the size of each sub-group when thenumber of groups is known. For instance,6+3 -might ask us to find, the number of

items in each group if 6 items are dividedinto 3 equal groups. We show 6 items on theboard and then remove 1 at a time placing 1in each of 3 different groups. "We put 1here, I there, and 1 over there." Then we re-move 1 more and I more and I more untilthey are all used up. Since we are trying toregroup into sets of the same size we mustalways put the same number in each of thesub-groups. In 6 there are 2 in each of 3groups.

17+5 asks us to distribute 17 thingsequally among 5 people. We arrange 17items on the hundred-board and then pickthem off 1 at a time putting one in each of 5different places. Now we do the same thingagain putting one in each of 5 differentplaces, then we do it again. Now we seethat there are not enough to go around againso we have learned that there will be 3 ineach of the 5 groups with 2 items remaining.

After some experience with the simplerproblems students will see that it is possibleto remove more than 1 at a timetheycould put 5 in each sub-group if they had todivide 37 by 6 and they could put as manyas 10 in each sub-set if they were to divide 59into 5 equal parts.

FractionsThe relationships inherent in both com-

mon and decimal fractions can be exWbitedon the hundred-board. By the use of elasticribbons it is easy to show that 1/2 =2/4,that 1/2=50/100, that 25/100 =1/4, that1/5=20/100, etc. Also that 1/10 is equiva-

WHOLE NUMBERS 31

lent to 10/100 and that 3/10 is equivalent to30/100. If the whole board representsunity or 100% then each square represents Iout of the 100 or 1% and is useful in show-ing the common fraction and per cent equiv-alents. The board will be useful for reveal-ing that 1/8 of 100 blocks represents 121little blocks. Since each block represents 1out of 100 or 1% of the whole, then 1/8 isequivalent to 121%.

One column represents 1/10 or .1 of thewhole board and 6 rows or 6 columns repre-sents 6/10 of the board or .6 of it. Forty-three of the little blocks represent 43 of the100 blocks or 43 hundredths of the whole.Thus were we to compare with .43 we seethat the .43 is larger than .4 by 3 littleblocks or by .03 (3 hundredths). Theboard is a helpful adjunct in teaching themeaning of and the relative sizes of commonand decimal fractions whose values lie be-tween 0 and 1.

There are numerous othcr uses for thehundred-board which will be discovered bythe teacher who uses one.

EDITOR'S NOTE. Dr. Volpel's "hundred-board"has many uses. Many people have used such a de-vice for percentage but have not explored all theideas he has presented. Perhaps he is right when hesays it is the most valuable piece of equipment forenhancing the teaching of arithmetic. As with, otherdevices, it is the way in which an item is used tofoster discovery, understanding, and learning thatwakes it valuable. Teachers will see possible varia-tions of design and construction and more valueswill be discovered as the use of the board becomesmore common. Devices are valuable in leadingchildren to sense the objective background for thelearnings which later should become more abstract.

Kindergarten mathematics

LUCILLE FITZSIMONSAppleton Public Schools, Appleton, WisconsinMiss Fitzsimons teaches kindergarten in the Lincoln Elementary School in Appleton.She also represents the kindergarten teachers on a city-wide mathematics study committee.

In our eagerness to develop mathematicsconcepts for today's five- and six-year-olds, are we mixing up our number ideasand merely confusing the children, or arewe preparing thoughtful programs Whichhelp to develop number concepts at thisparticular age level? Clarity in planningand identifying activities which developspecific mathematics understandings isessential.

An effective kindergarten mathematicsprogram has two approachesboth im-portant. Many of our mathematics ex-periences develop from our everydayschool living. This provides for opportuneteaching. The teacher is continually alertto situations which provide the oppor-tunities to emphasize simple number con-cepts. But this "catch technique," whilemeaningful, does not entirely satisfy themathematical appetite of today's kinder-garteners. A planned program of activitieswhich challenges the children's abilitiesmay also be woven into the day's Jiving.

In developing a planned program, wemay include the continued development ofrational counting, the identification ofnumerals, the development of the cardinalconcept of number, an understanding ofgreater than and less than (included inone-to-one correspondence) and the in-stantaneous recognition of small groups orsets. Materials suitable for the extensionof these understandings for young childrenare scarce; therefore, the teacher neces-sarily becomes the manufacturer of manymathematical materials.

The kindergarten surprise box whichfrequently holds little surprise treasures

32

such as a bright new puzzle or a soft fluffytoy bunny may also be the hidden spotfor a numeral and the corresponding num-ber of objects. When developing the 'con-cept of one, our surprise boxwhich isapproximately 8" square and gaily deco-rated with colored paper circles, squares,rectangles and trianglescontained a cutout numeral 1 pasted on a heavy 7"0'cardboard with a back support. With itwas one small block, one plastic spoon, andone toy milk bottle. We discussed each ob-ject as we took it out of the box, theteacher commenting, "What do you callthis numeral? How many blocks did wefind? How many spoons? And how manytoy milk bottles?"

We then directed our search about theroom to discover other things of which wehad only one. Typical responses were

"Look, we have one piano.""And there is just one teacher's desk.""We have only one round table.""There's one punching bag.""I have just one nose and one head."

And so it continued.Several days later the surprise box pro-

duced the numeral 2,.and.with it were twosmall blocks, two plastic spoons, and twotoy milk bottles. Where else could we findtwo of something? Our search was againdirected to ourselves and our immediatesurroundings.

"I have two shoes.""We have two clocks -the rsal one up

there and our play one.""I have two eyes and two hands.""We have two dolls in the doll house and

two doll buggies."

We were on our planned way. It wasvery natural then to find "pairs" and alsointroduce "duets" at singing time. Teacherjudgment guided the appearance of thenext numeral with its accompanying ob-jects. In fairly like manner, we developedthe cardinal concept of all numbersthrough ten, and also the ability to iden-tify the numerals.

Our garage game has fascinated thechildren. We painted red numerals 1 to 10on ten blocks which were approximately6" X3" X11". We cut and folded coloredpaper to represent the roofs of the garagesand taped them on top of the blocks. Infront of each garage we placed a paperwith a heavy dividing line. A bag ofminiature plastic automobiles purchasedat the ten-cent store completed the ma-terials which we needed. Using the numberof miniature autos equivalent to the sumof the numerals on the garages, we wereready to play the garage game. One childplays at a time, moving the cars into the"double garage areas." In the beginning,the children arranged the cars with littleregard for the dividing line, but as theycontinued experimenting with the gamethey discovered many possibilities. Whenrunning the cars into garage 6, they couldput three cars in each side of the garage orFive in one side and only one in the other orfour in one side and two in the other side.Accordingly, the children made discoverieswith each of the garages. We varied thegame by changing the garages and byusing only a limited number at a time. The

WHOLE NUMBERS 33

number of autos used was necessarilychanged too, so that all garages were filledand no cars left on the street at the end ofeach game.

These children were identifying nu-merals, doing rational counting, buildingtheir own little sets or groups of cars andexperiencing a clear understanding of thecardinal concept of number. The watchfulteacher would frequently inquire aboutthe car arrangements at the end of eachgame, thereby helping the Child to de-velop mathematical sentences, At times itmay be necessary to add a car or takeaway a car from some garage areas.

Our domino game proved to be enthusi-astically received by the children. Square9" linoleum tiles were cut in half, andpaper circles, arranged domino style, werepasted on them. We used various coloredcircles. The ones were red, the twos blue,the threes green, the fours yellow, thefives orange, and the sixes black.

With the children standing in circleformation, twenty of the children (thenumber of dominoes prepared) were givena domino which they held in both hands sothat all the children in the circle couldview them.

The teacher started the game by placinga pair of fives on the floor in the center ofthe room. She then called for a domino onwhich there is a five and something else.This is played at one side of her pair. Thenshe called for a five and something elseto be played at the other side of her pair.As each new number appeared, the teachercalled for its pair and then a matching


domino. Calling the dominoes for alternat-ing sides of the pair of fives providedchallenge and excitement for the children.

Mathematically, the children were de-veloping an instantaneous recognition ofgroups or sets, as well as a clear under-standing of the cardinal number.

When the children say again and againwith spirited enthusiasm, "Let's play thatGiant Domino game today," or "May I bethe garage man today?" we may feelreasonably confident that these games are--bringing a pleasant exeitement and intel-

I.. 41. 00

lectual stimulation to beginning mathe-matics.

Whether children are counting beads ona counting hoard, fishing out numeral-marked fish from an artificial pond andmatching them to appropriate schools offish, scrambling and unscrambling nu-merals or matching children with chairs inpreparation for a game, it is imperativethat the teacher be acutely aware of herpurpose and specific objectives in order todevelop an effective and continuing kinder-garten mathematics program.

EDITORIAL COMMENT. The surprise-box idea can be expanded in many ways. For exam-

ple, the box may contain a card like the following one and a set of objects for the children

to place in the appropriate places on the card.

one two three

I 2 3

Likewise the garage-game idea may be adapted for use with boats, airplanes, farm ani-mals, and so forth.

While making the suggested dominoes, the teacher may wish to construct some set cardsfor the numbers 1 through 10. These cards may then be used to provide experiences withorder. For example, the cards may be harted out to the children to start the activity. Theteacher asks for the child with the "I" card to place his card in the chalktray. Then theteacher asks for the child who has the card with the "next" number to place his in thechalktray, and so on. For another activity, the teacher may select a child with a numbercard such as the "3" card. The teacher may then ask for all children who have a numberless than 3 to step forward or she may ask for all who have number greater than 3 to standup

Our number clothesline

SYLVIA ORANSParkway School, Plainview, New York

On a large, oblong piece of corrugatedboard I mounted two long strips of wood.From each strip I suspended a "line"(string) and hung doll clothes on it withclothespins. (The doll clothes are paperdoll clothes mounted on pieces of corru-gated boardwhen more clothes areneeded, the children make them.)

Alongside one line I placed the words,"on the first line" ; next to the otherclothesline, the words "on the secondline." A space large enough to displayremovable numeral cards was left between

6

9

ON THE FIRST LINE

ON THE SECOND LINE

IN ALL

the line and the words. Cards are hung oncuphooks.

The clothesline is used for addition, sub-traction, sets, and sets within sets. And,best of all, it is used by the children.

EDP ORIAL COMMENT.Clothespins or other items may be substituted for doll clothes inthese activities. Experiences with equivalent sets and with "greater than" and "less than"may be provided on the clotheslines.

35

A picture line can be fun!

KATHERINE PATTERSON

Katherine Patterson is an assistant principal nt the

Laytonsville Elementary School in Montgomery County, Maryland.

She has been a classroom teacher, a student - teacher sponsor, and

a resource teacher in elementary mathematics.

Wen the young child enters school, hehas many opportunities to become involvedwith mathematics. Among the varied class-room activities that the writer has used, thewalk-on picture line is one that delightschildren. This activity uses movement,builds on the child's environment, and de-velops oral language. The pictures may bechanged according to the interest of thegroup. The activities with the picture lineare informal in nature, and they are fun!Many mathematical concepts can be intro-duced, such as the cardinal and ordinal useof numbers; idea of a starting point; useof a precise vocabulary such as before,after, between, how many, which one, andwhich place; directions on a line; counting;one-to-one correspondence; addition andsubtraction.

To work with a picture line, start with aset of five pictures. As the child's conceptsof number values from one to ten increase,the set may be enlarged to ten pictures.The pictures should be large cutout pic-tures that can be taped to the floor. A12" x 15" picture is a good size.

Pictures are cut out to show the shapecf the object to add to the interest level. Atfirst, the set can include a variety of objectssuch as cat, ball, top, tree, and kite. Afterthe set is assembled, the teacher can holdup one picture at a time, and the childrencan have a chance to identify the picture.As a child identifies a picture, he can holdthe picture and stand in line with the otherchildren side by side. Also, the class can

36

work on spatial relationships as they placethe pictures on the floor. Questions can beasked such as: "How can we place ourpictures in a straight line?" "Is there thesame space between each picture?"

After the pictures are taped in place onthe floor, the teacher and the class canestablish some rules for the games. A recallof outdoor games in which a starting placeis needed will serve as a basis for decidingon the rules.

1. A starting place is needed. (We needa place to wait for directions. This placeindicates that we have not taken any steps.A chalk mark or a piece of tape can beused for this purpose.)

2. Let each picture represent a step.3. Always start on the starting line un-

less told otherwise.4. Proceed from left to right unless

given special directions.

Example:

rt

Then the teacher and children play thegame.

Types of questions anc: activities

Where will you start?Take two steps. On which picture are

you standing? Take five steps, etc.Walk to the tree. How many steps did it

take to get to the tree?

How many steps is the ball from thekite? The cat from the tree?

Which picture is two steps away fromthe ball?

Take two steps and one more step. Onwhich picture are you standing? How manysteps from the starting place is it?

Take four steps. Go back two steps. Onwhich picture are you standing? How manysteps from the starting place is it?

Show a pattern. Explain that whenarrows are used, each arrow represents onestep. The arrow also shows the directionof the step. Therefore, means one stepto.the right. means two steps to theright. means three steps to the right.The three arrows may be 'arranged intodifferent patterns such as

or

Four may be arranged as

or

Show a pattern. Example:.^9

Take this many steps plus one more step.

Show a pattern. Example:

Take the number of steps that is one lessthan what the pattern shows.

Show two patterns. Example:

and

Take the number of steps that is the smallernumber of steps in the picture; the largernumber.

Take one step. On which picture areyou standing? Which place is the picture?

How many steps to the third picture?Which picture is it?

Walk to the last picture, first, third, etc.

Walk to the picture just before the tree,kite, etc.

Walk to the picture just after the ball,Walk to the picture between the top and

the kite; the ball and the tree.

WHOLE NUMBERS 37

Walk to the picture just before the thirdpicture; just after the second picture; be-tween the first and third picture, etc.

Walk to the second picture. Take threemore steps. Where are you? How manysteps is it altogether from the startingplace?

Walk to the picture of something thatbounces.

Have children close eyes and change theorder of the pictures. (This is one reasonwhy it is a good idea to tape the picturesrather than have them painted on thefloor.) Then ask the same questions as be-fore.

Have children change the fourth picturewith the last picture; the second with thefourth.

After the children have had many ex-periences walking the picture line and areready for paper work, each child can makehis individual picture line. The child's pic-ture line could be made on adding-machinepaper and rolled up when not in use. Thesame type of questions as before may beasked, and the child could point to thecorrect picture or could cover up the pic-ture with a counter. After much work withthe individual picture line, a follow-up ac-tivity such as the following could be pro-vided.

means one step to the right; 4-- meansone step to the left. means to takeone step to the right and turn around andgo back one step to the left.

(--

means go three steps to the right and atthat point go one step to the left.

Finish the chart on the next page.

When two parts of the chart are given,the child should discover the third partfor himself. A different column can beleft blank each time. This provides variety.

After many experiences with a pictureline, the transition to the number line isquite logical. Have a picture line made andhave another line with points marked.


v..

tiv

Finish thechart:

4

START NUMBER OF grErs sToP

--

ProcedureThe position on the pictureline that indicated the child was waitingfor direction and hadn't taken any stepswas symbolized by the word "start." There-fore the symbol that could be used on thenumber line to indicate no steps have beentaken, and the child is waiting to begin,is zero.

0

On the picture line, it took one step toget from "start" to the . On thenumber line it takes one step to get from"0" to the next point that can be labeled"1." To get from "0" to the point after "1,"it takes two steps that we can label "2," etc.

Children can see order and sequence onthe number line. Likewise, they can see thereason why we start on '0" and not "1"when taking steps.

There are many more activities andgames that can be done with the numberline. Be creative. Try some and have fun!

Simple materials for teachingearly number conceptsto trainable-levelmentally retarded pupils

JENNY R. ARMSTRONG andHAROLD SCHMIDT

The project described here was carried out at the Universityof Wisconsin, where Jenny Armstrong is director of research andevaluation for the Special Education Instructional Materials Center.She is also a lecturer in the Department of Studiesin Behavioral Disabilities.

Harold Schmidt was a graduate student during the course of the projectand is currently serving in the United States Army.

As a part of a project designed to teachearly number concepts to trainable-levelmentally retarded pupils, a step-by-stepseries of materials was developed. The ma-terials were characterized by being (1)sequential, (2) easily utilized by pupils oflow-level motoric, cognitive, and verbalskills, and (3) easily constructed with in-expensive materials that are readily avail-able to most classroom teachers.

Numeral-quantity associationtaught in the enactive mode

In a research study designed to comparethe early number learning of trainable-level mentally retarded when taught in anenactive mode of representation (usingmanipulative materials) with the learning

The authors wish to acknowledge the fund-ing of the project by the Bureau of theHandicapped, USOE, Department of Health,Education, and Welfare, under grant num-bers 32-59-0500-1002 and OEG-0-8-080568-4598 ( 032) .

39

acquired through a pictorial mode of repre-sentation (using nonmanipulative mate-rials) following the information-processingtheory of Bruner (1964), it became ap-parent that there were very few manipula-tive materials available for teaching theconcepts related to numeral-quantity asso-ciation (Armstrong 1969). Therefore, asa part of the project a series of simplestep-by-step materials was developed.

One of the major criteria of the ma-terials was that they be manipulative incharacterthat is, in order to use the ma-terial the pupil must manipulate objects.Another criterion of the materials was thatthey be specifically designed to teach anumeral-quantityassociation concept. Im-plicit, therefore, in the use of each materialdeveloped was manipulation and focus on anumeral-quantityassociation concept.

Materials design with concernfor sequential development

A series of materials structured in a veryrigorous developmental sequence was then


designed. Since the pupils for whom thematerials were originally designed were atthe very beginning stage of number learn-ing (that is, several had not even learnedto count by rote, let alone associate anymeaningful concept of quantity with agiven numeral), the first set of materialsin the sequence was designed to providea simple manipulative aid to the oral ver-balization of the numerals from 1 through5.

This first set of materials in the sequenceconsisted of block-chain counters (see fig.1). Pupils were given the chains of blocksand instructed to count the blocks andverbally tell the number of blocks on eachchain. This activity was repeated severaltimes until the pupils could give verballythe numeral that expressed the number ofblocks on each of the block chains. Thepupils were highly motivated by this ma-

4

Fig. 1. Block chains

terial, and in quite a short time they wereable to achieve the instructional goal.

A second set of materials that was de-veloped for this same instructional goalwas not as successful. This set of materialsconsisted of glass jelly jars containingvarying numbers of lima beans (see fig. 2).Because of the low level of the motoricskills of this group of children, the jarswere frequently dropped. This was espe-cially hazardous as the jars were made of

Fig. 2. Bean jars

glass. Also, the lids were much too difficultto get off the jars.

The next set of materials in the sequencewas designed to aid the pupils in associat-ing the number symbols visually andtactilely to quantities of objects. This setof materials consisted of a series of plas-ticene molds that had insets for differentnumbers of candles and imprints of the as-sociated numerals, from 0 through 5 (seefig. 3). Pupils were given one "candlecounter" at a time and instructed to countthe candles in the holder, to verbally givethe number that told the number of candlesin the holder, and then to use a finger tofeel the shape of the number symbol; theywere then told to look at the shape of thenumber symbol, verbalize it, and retrace itwith a finger. The pupils enjoyed theirwork with this set of materials and wereaided in getting one step closer to readingthe number symbol. This material also

Fig. 3. Clay candle counters

provided readiness for the actual writingof the number symbol associated with aparticular quantity.

The next set of materials in the sequencealso allowed for both visual and tactileexperiences with the number symbols aswell as for quantity association with thesymbols (see fig. 4). Each board had twonumerals in sequence made from sand-paper to provide a tactile sensation as thepupil ran his finger over the numeral to geta feeling of the shape as well as how itwould be to write the numeral. A smallwell in the board by each numeral provided

WHOLE NUMBERS 41

well in accomplishing its goal, when con-structing the material for future use onemight wish to separate the single numeralsand wells by cutting each board in half. Inthis way there would be only one numeralon each board (that is, 2 and 3 would notbe confused with 23).

The next set of materials in the sequencewas designed to provide the first experi-ences with writing the numerals with someguidance (see fig. 5). Each board pro-vided a stencil for the numeral concernedand removable blocks on small pegs toillustrate the quantity. Pupils were pro-

Fig. 4. Sandpaper numeral boards

a place for the pupil to place the numberof beans, rocks, or other types of countersthat illustrated in quantity what the nu-meral meant.

In using this set of materials, the pupilwas instructed to work with each numberindividually. First he was asked to verballyidentify the numeral. Next, he was askedto count out the number of beans that thenumeral represented and place them in thewell associated with the numeral. Then hewas asked to run his finger over the nu-meral several times to get an idea of itsshape as he repeated the number name.Although this set of materials worked quite

vided with a sack of blocks, paper, andfelt tip marking pens. They were instructedto place the number of blocks the numeral.suggested on the board and then using thestencil portion of the board to write thenumeral. After writing the numeral usingthe stencil, pupils were asked to write thenumeral without using the stencil, using thestenciled numeral as a pattern.

The next set of materials in the sequenceconsisted of egg-carton counters (see fig.6). Since the major focus was on quantityrather than on size or shape, this materialserved two functions. First, it served itsgoal as the next step in the developmental


I

Fig. 5. Numeral stencil boards

learning sequence, and second, it servedto emphasize the dominant characteristicof concern. The size of the carton did notdirectly relate to the number of balls thecarton held (for example, the one cartonand the six carton were the same size butrepresented quite different numbers of ob-jects). This material was designed to take

4_

the pupils one step further in the sequence.Each pupil was provided with a can oftable-tennis balls and a carton. He wasinstructed to fill the carton with "eggs" andthen count to determine whether or not thenumeral shown in the lid of the carton toldthe number of eggs in the carton. Pupilswere asked in this case to look at a pat-

Fig. 6. Egg-carton counters

lt"fli*4(5,,

1

t

-17 "

Fig. 7. Magic slate boards

tern of the numeral on the inside of thecarton lid at the same time as they lookedat the number of eggs (table-tennis balls)the carton would hold. Next, the pupilswere instructed to close the lid of the car-too and write the numeral on a tablet pro-vided on top of the carton. After writingthe numeral, they were instructed to re-move from the tablet the sheet of paper onwhich they had written, open the cartonlid and compare the numeral they wrotewith the numeral pattern in the lid. Thismaterial worked quite well except for theproblem of tennis-ball attrition.

The final set of materials in the sequenceconsisted of magic-slate tablets (see fig. 7).Each pupil was provided with a magicslate with a certain number of clothespinsclipped to the top. The pupils were in-structed to count to determine the numberof clothespins on their slate. Then theywere instructed to write on the magic slatethe numeral that told the number ofclothespins pinned to the slate. This ma-terial worked very well in accomplishingthe desired goal. It was very flexible in itsutility for the provision of individual dif-

WHOLE NUMBERS 43

/11111i

ferences. One needed only to add or deleteclothespins to provide for pupils at differ-ent levels on the numeral scale.

Summary

The purpose of this paper was to de-scribe in detail a series of materialsdesigned to teach numeral-quantity ass,scia-tion to pupils at very early stages of cogni-tive and mathematical growth. The ma-terials were developed to provide verysmall steps in the learning pattern. In thisway, the learning of pupils with learningproblems is most particularly facilitated.These materials can be inexpensively re-produced for classroom use and have beenshown to be quite successful with at leastone sample of pupils with severe learningproblems.

References

Armstrong, J. R. "Teaching: An OngoingProcess of Assessing, Selecting, Developing,Generalizing, Applying, and Reassessing," Edu-cation and Training of the Mentally Retarded4 (1969): 168-76.

Bruner, J. S. "The course of cognitive growth,"American Psychologist 19 (1964): 1-15.

Yardstick number-line balanceJEROME S. BORGEN andJOHN B. WOOD

-Jerome Borger is principal at North Hill Elementary Schoolin Minot, North Dakota.John Wood is an assistant professor in the New School of Nun

University of North Dakota.

Abalance made from a yardstick cangive a visual interpretation of the balancesituations expressed by equations.

Materials. One yardstick, a knife orfine-tooth saw to cut it with, glue, a pieceof rubber band 1/4 inch in width andabout 11/4 inches long. 21 pieces of paperabout 1/2 inch across, and from three tonine 1/2-inch metal washers.

Directions. Cut the yardstick at 9.5,11, 12.5, and 14 inches. The 22-inchpiece is the beam of the balance; thethree 1.5-inch pieces are glued to formthe fulcrum (see fig. 1 ); and the 9.5-inchpiece is excess. Glue the small pieces ofpaper over the numerals (not the mark-ings) of the beam, and relabel 25 as 0, 24and 26 as 1 and 15 and 35 as 10.The piece of rubber band glued along thetop of the fulcrum helps to keep the beam

Fulcrum foryartistick balance

Fig. I

44

from sliding on it. If the beam does notbalance right at zero, trim off the heavy.end of the yardstick or stick some tapeunder the light end.

Some activities. Place one or two metalwashers anywhere on the balance beam:find where one more washer must beplaced to make it balance; find where twomore washers must be placed to make itbalance. Children can use equations torecord balance situations, with the equalsign representing the fulcrum. The balanceis especially a good interpretation for prob-lems like ____ + 6 = 9 and 7 + = 10.

EDITOR'S NOTE. Is it possible that childrenmight move from balancing equations as sug-gested by the authors to discovery of prin-ciples of the lever? As long as the beam restsfreely on the fulcrum and is not fixed 'in po-sition, children-will experiment and find that( I ) a heavier weight can he balanced by alighter weight merely by moving one closerto the fulcrum, and (2) when the beam itselfis moved, a "long" lever arm can he balancedby placing weight only on the "short" leverarm (the principle of the cantilever). I'm sureour authors didn't have this in mind in writ-ing this article, but given free discovery anda movable lever arm, even six-year-olds willfind the three principles of the lever. Olderchildren can express these "discoveries" in theform of rules or formulas. Relate science andmathematics wherever necessary to the under-standing of science!CHARLOTTE W. JUNGE.

Elevator numbers

STANLEY BECKER

The principal of P.S. 191, Manhattan, a prekindergarten throughthird grade school in New York, Stanley Becker has taught collegecourses in mathematics and education and has given in-service

courses for teachers.

All teachers are constantly on the lookout for games and devices that will stimulate interestin mathematics. The following description of a device called "Elevator Numbers" readilyfills that need. It enables children to practice adding numbers to produce sums from 10to 20 or more.

The idea is that children manipulate vertical strips of numbers to produce two rows ofaddends that must total the same stated amount when added across in either row. Eachchild receives a holder that allows two numbers from each column to be viewed at one time(fig. 1). He must move the strips up and down until he sees that the addends in each rowtotal the same sum.

Strips of addends

Openings through which

the addends are visible

3

Fig. 1

45

Holder


Start the strips

To prepare this device, cut four or fivestrips, each one an inch wide. Use oaktagor some other material that is strong andflexible. Then mark each strip at one-inchintervals. Decide on the sum that you wishto appear in both rows on the card andput appropriate addends on the strips first.

In the case shown in figure 2, a sum of16 was decided upon and the 6-3-6-1 com-bination (indicated by check marks) wasput on the strips first. The circled numbers,5-8-3-0, were then put on the strips byskipping one space beneath each of theoriginal numbers. Skipping a space is neces-sary because the holder that the strips arewoven through exposes only two numeralson each strip and these numerals are onespace apart.

Fig.

Finish the strips

Now complete the strips by placing anumber in each space on every strip (fig.

3). Any numbers are fine. Your card nowhas at least one correct solution and mayhave more. The order in which the childrenplace the stripS in the card is of no im-portance. Also, the strips can be of anyconvenient length. Longer strips with morenumerals make the game more challenging.

6

3

s

3

3

3

1/.3

5

.3

I

Fig. 3

6

3

0

6

Make the holder

Oaktag may be used for the holder aswell. It should have four slits, one inchapart, that permit the strips to be woventhrough the holder in such a way that twoof the numbers on each strip will be visibleat one time (fig. 4). For ease of construc-tion, place a piece of file folder or oaktagunderneath this page along with a piece ofcarbon paper and trace the form. Thencut along the dotted lines. Using the com-pleted holder as a model, make as manyholders as desired.

WHOLE NUMBERS 47

Fig. 4

Children will spend a great deal of timeplaying with these machines. Two secondgraders spent two hours finding the rowsthat totaled 16 and checking the strips tosee if they could find any other ways to

manipulate the strips so that the same sumwould appear in both rows of exposedaddends. They seemed to enjoy it con-siderably more than an equivalent amountof drill with the addition facts.

EDITORIAL COMMENT.The students may enjoy challenging their colleagues with "elevatornumbers" devices that they have made themselves. The number of strips is a variable andmay be increased or decreased to satisfy the desired level of difficulty. The universe of num-bers may be changed from whole numbers to integers or rational numbers in fractionalor decimal form for students who enjoy a greater challenge.

-Presenting multiplicationof counting numbers onan array matrix

MERRY SCHRAGEapp

Merry Selvage is a classroom teacher of grade three at

West view Elementary School in Miami, Florida. She is alsoa graduate student at Barry College,

Multiplication tends to imply an avenueof excitement and adventure for children.During my ten years as a grade threeteacher, I have yet to find a year whensome eager child hasn't asked the question,"When are we going to learn multiplica-tion?" Prior to third grade the children areexposed to readiness activities in prepara-tion for multiplication. However, the realdrive to master facts and algorisms is inthe third-grade mathematics sequence inDade County, Florida.

A visual technique was developed to beused primarily during the introductorystages of multiplication. With the use ofa matrix and individual cards showing thevarious arrays, it is possible to present andstrengthen a number of basic understand-ings peculiar to multiplication.

The basic construction of the matrixcan vary in size. I use a large one (36" x36") because it is utilized and displayedfor a period of time. Figure 1 shows theblank matrix chart in preparation for themultiplication facts through seven timesseven.' A set of separate cards, whichfit in the blanks of the matrix, can beeasily made by using a magic marker,gummed circles, stars, or any other con-venient item. The cards can be attachedto the matrix with tape or other types of

X / 02. 3 ' / 6" 6 7/

.2_

3#,s-

6

7Fm. 1.Blank matrix

affixing material. Making the arrays onseparate cards enables development of thematrix chart in a sequential manner ac-cording to readiness, properties, and facts.Different colors may be used for the arrays,making the commutative facts the same incolor. Figure 2 shows how the array cardslook.

I P.

I The matrix has not included the zero facts since.the array would show only blank cards.

48

.

Fm. 2..--Spmple array cards

The primary concept that needs to bedeveloped prior to using the matrix is themeaning and use of the array. Visual ac-tivities should be provided that enable thechildren to experience the manipulatioc.and development of arrays. After activitiesof this type have been accomplished, thematrix chart can be introduced as a meansof recording our findings in an oiderlyfashion.

During the initial introduction of thematrix, the children will readily recall us-ing a matrix for the purpose of recordingaddition and subtraction facts. It is usuallyenough to indicate that we can use thematrix to find out some important ideasthat the arrays show us in multiplication.Point out that the numerals across the topand along the left side of the matrix relatedirectly to the definition of an array. Passout a few selected array cards to the class.Ask someone to develop a specific arrayon the chalkboard. "Does anyone have anarray which shows the same array Johnhas made for us on the board?" There willbe two children with array cards that matchthe one made on the chalkboard unless theteacher has selected an array showing theproduct of two identical factors. Also, ifsome type of color scheme has been utilizedin making the array cards, the two cardsfor the specific array will be the same color.

The next question that needs to be dis-cussed is, "Where should these array cardsbe placed on the matrix ?" Lead the chil-dren to a consideration of the definitionof an array and how it relates to the num-erals on the matrix. The children shouldalso be encouraged to see that a quarterturn of one of the array cards producesthe commutative fact.

The arrays to be developed should beapproached in the manner just briefly out-

WHOLE NUMBERS 49

lined. Of course, there are many morequestions that can be asked to stimulatediscovery on the part of the children. Theability of the class will also be indicativeof the duration that this activity will take.As the pattern of the matrix emergesthrough the development of the facts interms of arrays, the matrix will lend itselfto a visual understanding of the identityelement of one, the commutative propertyand the multiplicative property of zero.Figure 3 shows an array-matrix that hasbeen completed through the facts of five.

X / 02_ 3 1i1 5/I . . Of e

.4 0000002a

. s . OOOOO

.0 .

e a 06

= 1111.. ,v

3.Completed array-matrix through factsof five

Through the use of the array-matrix,the children readily develop an under-standing of the process of multiplicationand its properties. This technique is suc-cessful with the less capable child becauseit provides an opportunity for manipula-tion and visualization of the concepts. Themore capable child is greatly stimulatedtoward independent discovery of additionalpatterns and concepts.

EDITORIAL COMMENT.Students may make their own array cards by using index cards anda hole punch. A Concentration type of game activity may be organized to provide addi-tional experience with the multiplication fact-array relationship. One board holds the mul-tiplication fact cards in random positions, and another board holds the array cards posi-tioned randomly on the board. Neither the facts nor the arrays are visible until a studentattempts to make a match.

Multiplication masteryvia the tape recorder

DALE M. SHAFER

Dale Shafei is professor of mathematics at

Indiana University of Pennsylvania,

This article results from his previous experience as

coordinator of a K-12 mathematics program.

Most instructors who have had the re-sponsibility of teaching children the multi-plication facts will readily admit that atseveral intervals preceding student masterya certain amount of drill is necessary. Theword drill should suggest nineteenth-cen-tury pedagogy only to those individualswho do not know how this teaching tech-nique is being imaginatively employed inelementary classrooms today.

Teachers generally view drilling themultiplication facts as time-consuming,boring, and frustrating. This is certainlythe case if the teacher stands in front ofthe room with flash cards and methodicallymakes the rounds of the class. The studentknows that in a class of twenty-five pupilshe will be actively involved as a participantOnly -IA:, of the time. On the one hand, ifhe has mastered the multiplication facts,time drags. On the other hand, if he is aslow student, he sits in class assuring him-self that when his turn to participate doe:.arrive he will surely be asked to recite oneof the multiplication facts about which heis still not confident.

In this article, let us sec how a taperecorder can be used to stimulate, both instudents and in teachers, a more positiveattitude toward mastery of the multiplica-tion facts. The approach here suggestedinvolves use of a reel-to-reel or cassettetape recorder, earphones, a student re-sponse sheet, and a teacher-made prere-corded tape.

Initially, the author wrote on slips ofpaper the 144 multiplication facts thatwould appear in a 12-row, 12-columnmultiplication table. Then one slip at atime was selected from this set so thatthe entries on the audio tape would bemade in a random order.

The tape begins with a motivating state-ment encouraging the student to win thecoveted "Master of Multiplication Award."This award is a large orange X placed be-side his name on a chart when he has mas-tered the multiplication facts. The chartalso provides space to record a student'sprogress through an entire series of pre-recorded arithmetic tapes.

The tape then gives instructions abouthow the student is to record his answerson the student response sheet. The studentresponse sheet for the multiplication factscontains 6 columns with 24 consecutivelynumbered answer spaces in each column.The problems now begin"Number 1,five times eight [PAUSE]; number 2, ninetimes six [PAUSE]; . . ."and continuethrough number 144.

By this time you, the reader, are prob-ably thinking of the Herculean task in-volved in evaluating the response sheets.This task is greatly simplified by using ahomemade correction mask constructedfrom half of a manila folder. Twelve differ-ent masks are used to correct a multiplica-tion-facts response sheet.

To construct the "sevens mask" indi-

50

cated in figure 1, a piece of carbon paperand a student response sheet were placedon the manila-folder material. Then a rec-tangular box was drawn around each areaon the response sheet where a "sevensfact" was to be recorded. The carbon-paper impression was then cut out of themanila-folder material with a penknife.Above each of the rectangular holes in themask was written the problem that corre-sponded to this position on the student re-sponse sheet. Finally, a straight line wasused to join those boxes whose multiplica-tion facts were related by the commutativelaw of multiplication.

Multiplication: 7

8x7 =5677x10=70

9x7=63

7x8=56INE7x7=49

7x7=49

7x12=84

12x7=847x4=28

7x9=63

10x7=70

I x 7=7

2x7-I4

5x7=35

7xH=777x5=35

7x6=42

7 x3=214x7=28

Hx7=77

3x7 =21

FIGURE 1

When the correction mask is laid on astudent response sheet, the teacher can tellat a glance if the student has mastered orhas retained mastery of the sevens table.If he has missed 7 x 9, it is quite easy tocheck whether he has also missed 9 x 7.The facts that need to be reinforced or re-taught to a particular student are apparentthrough the boxes in the correction mask.

One can notice that the correction maskis labeled along the horizontal and vertical

WHOLE NUMBERS 51

edges. This labeling provides for easy iden-tification when the mask is being used forcorrection purposes and for easy locationwhen it is stored in a file folder.

When recording a tape, a stopwatch isused to time the length of a pause betweeneach problem. By keeping these time in-tervals constant throughout the tape, a

rhythm of presentation is established forthe student.

When the teacher experiments with thisidea, he will probably want to have severaltapes available on which the length of thepause interval between problems is variedto account for differing ability levels. Hewill also want to have available some tapescontaining only a subset of the 144 multi-plication facts. Whatever the case may be,he need record only one master tape. Byconnecting two tape recorders in series andmanipulating the pause lever on eithermachine he can extract from the mastertape whatever data he desires to record onthe second tape and vary the pause intervalbetween problems to suit his particularneeds.

Therefore, by recording on tape themultiplication tables, a teacher can ascer-tain, whenever he desires to do so, whethera student has mastered these facts and/orwhether he has continued to maintain thesebasic skills. When a student or a group ofstudents place the earphones on their headsand work on the same or different tapes,the teacher can be working with the re-mainder of the class on some entirely dif-ferent material. This approach is certainlyconducive to total student participationthroughout the entire teaching period.

The multiplication facts constitute onlyone of many areas where your instructioncan be imaginatively aided by using a taperecorder with one person, a group, or theentire class. A teacher will find as he buildshis library of teacher-made tapes that hisapproach to teaching becomes more flexibleand that he has moved one step closer toindividualization of instruction.

Problem solving with number-pictureproblem situations

JULIET SHARFF Arlington, VirginiaAirs. Sharff is a fourth-grade leacher at Waller Reed Elementary School,Arlington, Virginia.

The class was inspired by the weather todevelop its first picture problem situa-tion. The teacher sketched at the chalk-board in response to children's suggestionsand guided them so that basic grade-levelnumber concepts were included. For ex-ample, the first cooperative class sketchfeatured a snowy hill and boys and girlswith sleds. All data are not pictured; someare provided as factual information. Thesketch (Fig. 1) and some of the resultingnumber problems were similiar to the fol-lowing.

Johnny and Tommy go sledding

Johnny lived at the foot of the hill onone side. Tommy lived at the foot of the

JOHNNY'SHOUSE

Data not pictured:18 boys14 girls15 sleds

Figure 1

TOMMY'SHOUSE

52

hill on the other side. They met at the topof the hill. Tommy invited Johnny to hishouse for hot chocolate. Johnny had to gohome and ask his mother if he could go.How many feet was Johnny's trip homeand back to the top of the hill? (2 X70=140)

How far did the 2 boys walk togetherwhen they went to Tommy's house for thehot chocolate? (2 X55 =110)

How many feet did Johnny walk whenhe went home from Tommy's house?(55+70=125)

How many feet longer was the hill onJohnny's side than on Tommy's side?(70 55 =15 longer)

It took the boys 5 minutes to pull theirsleds from Tommy's house to the top ofthe hill and 1 minute to ride down. In 10trips how many feet did they walk?(10X55 =550 for each boy or 1100 for2 boys)

How many minutes did they spend inpulling the sled? (10 X 5 = 55)

How many minutes did they ride?(10 X 1 = 10)

If they had to walk 10 minutes to ride 1minute, how far would they have to walkto ride 2 minutes? (20 minutes) 3 minutes?(30 minutes) 4 minutes? (40 minutes)6 minutes? (60 minutes or an hour)

L1800

WHOLE NUMBERS 53

J

ri m"551-4"f"

Data not pictured:3 squirrels in the tree6 fish in the lake100 nuts in the nut pile2 adults and 3 children in the house5 baby birds and 1 mother bird in a nest

Figure 2

As they stood on the top of the hill,Johnny counted the children on one sideand Tommy counted the children on theother side. Johnny counted 16 and Tommycounted 14. How many children did theycount in all including themselves?(14+16+2=32)

If 18 of the total number of children onthe hill were boys, how many were girls?(32 18 =14)

If every child had on a pair of mittens,how many pairs of mittens were there?(32 children, 32 pairs of mittens; one-to-one correspondence between children andpairs of mittens.) How many mittens?.(32 X2 = 64)

Nine of the 15 sleds belonged to boysand the rest to girls. How many belongedto girls? (15-9=6)

Three children_rode on _a 4d-together--and they took turns pulling the sIEd-1557ck-up- the hill. After 18 rides down the hill,

how many times had each child pulled thesled back up the hill? (18 ÷3 =6)

The squirrel family

The picture problem situation about the.T_iuirrel family (Fig. 2) was first developedon the chalkboard in a similar mannerwith children offering suggestions on whatto include, approximating distances fromone place to another, and giving informa-tion not pictured to be used in problemsolving. The children took turns makingup and solving problems from the pictureand from the information given. After theclass experience the teacher duplicated thepicture with some typical and appropriateproblems in order to provide each childwith opportunities for problem solvingand to evaluate the effectiveness of theclass learning experience.

Materials were distributed and the dataat the top of the page was reviewed. Theteacher explained that only measurementspictured by broken lines could be used inproblem solving. As they completed theirwork the children were instructed:

1 To make up other problems about thepicture.


2 To plan a picture of their own, composeproblems about it, solve problems on aseparate sheet of paper, and keep thesolutions.

3 Exchange pictureS and problems with aneighbor, solve problems, and checkanswers with the owner's key.

It will be noted that the level of diffi-culty of the problems which follow werevaried to meet the needs of children withdifferent abilities.

1 If the squirrel and a friend were goingto store 1 the nuts, how many nutswould be left in the pile? (3 of 100 is50)

2 All the squirrels went to the lake tolook at the fish. All the fish came up tolook at the squirrels. How many squir-rel eyes were looking at the fish?(2 X3 = 6) How many fish eyes werelooking at the squirrels? (2 X6=12)How many eyes were there altogether?(6+12 = 18)

3 The whole family walked to the nutpile and back home. How many feetdid the 2 adults travel? (2 X=140 or 1 round trip; 2X140 =280or 2 round trips.) How many feet didall the children travel? (3 X140=420) How many feet did all 5 peopletravel? (280+420 = 700)

.1 Twenty of the nuts are walnuts. Therest are pecans. How many pecans arein the pile? (100-20=80)

5 The squirrel on the ground climbed tothe lowest branch of the tree and calledall his friends from the highest branchto come down and help him. When allthe squirrels reached the nut pile, whatwas the total number of feet all hadtraveled? (25+50 =75distance fromhighest branch to the ground; 50X2=100distance traveled by squirrelon the ground; 7.5 X3 = 225distancetraveled by three squirrels; 225+100=325distance traveled by all thesquirrels.) How many yards did all thesquirrels travel? (325 ± 3 =1081 or 108yards and 1 foot)

6 A squirrel went from the nut pile to thetree stump and ran I of the way. Howmany feet did he run? (1 of 25 = 5) Ifhe jumped from the nut pile to thestump with 5' jumps, how many jumpsdid he take? (25+5=5)

7 Father walked from the house to thefire tower and halfway back beforesitting down to rest. How many feethad he walked by then? How manyyards? (1800+900=2700; 2700+3= 900)

8 The squirrels were carrying the nutsto the tree behind the house. Squirrel 1took the shortest route. Squirrel .2took the long way. How many feetfarther did squirrel 2 travel? (70+30=100distance traveled by squirrel1; 25+30 =55 =110distance trav-eled by squirrel 2; 110-100 =10;squirrel 2 traveled 10 feet farther.)

9 The little squirrel stored s of the nuts.How many nuts did he store? (5 of100 =20) How many nuts were left inthe pile? (100-20=80 nuts)

10 The mother bird found 27 worms. Shedivided them evenly with the rest ofthe family, without cutting any. Sheate the leftover worms. How manyworms did she eat? (27 =5 =5; 5worms were eaten with 2 left over;mother bird ate 2 worms.)

The baseball game

The picture (Fig. 3) -and series of prob-lems about baseball by Daniel Chao il-lustrates the individual work clone bymembers of the class.

1 One fourth of the 980 people at. the ball-game were in reserved seats; the restwere in the bleachers. How many werein reserved seats? (I of 980 is 245) Howmany were in the bleachers? (1- X980= 3X245 =735)

2 If a pitcher throws 8 balls from thepitcher's mound to the plate, how manyfeet in all is this distance? (80 X8 = 640')

3 If 15 balls were used, how many werenot used? (32 15 =17)

I 2. I

Data not pictured:17 players in the dugout9 players in the field32 balls, 15 bats for each team980 people15 vendors24 hot dogs50 cokes

Figure 3

WHOLE NUMBERS 55

4 If there were 4 double plays, how manyfeet would these four players run?(180 =1 [double play], 180 X4 =720 or 4double plays.)

5 If there were 5 homers, how many feetwould be covered in all? (90 X4 X360or 1 homer; 360 X5 =1800 or 5 homers.)

The shopping center

The picture problem situations increasedin difficulty with experience until the chil-dren could handle much more complicatedproblems and computations, such as arerequired for the shopping center picture(Fig. 4) and problems.

1 Mother went from the house to thegrocery store and had to make a secondtrip to carry all the groceries. Howmany feet did she walk in the 2 trips?How many yards?

Data not pictured:24 buildings in the shopping center2 adults, 5 children, 1 dog, and 2 cats in the houseNote: The house is 5' from the sidewalk. All streets and all sidewalks are the same width. All doors areexactly in the middle of each building.

Figure 4


5 (distance from house to side-walk)

6 (width of sidewalk in front ofhouse)

25 (width of street)6 (width of sidewalk beside stores)

25 (across front of parking lot)25 (across front of drugstore)15 (to door of grocery store)

107

X4428 (4 trips or 2 round trips)

(428 +.3 =1421 or 142 yardsand 2 feet)

2 Susie left the house to go to the postoffice. She was daydreaming and reachedthe doors of the bank before sherealized it. How many feet too far hadshe gone? (15+30+8 = 53)

3 A man painted doubled crosswalk linesfrom sidewalk to sidewalk on all sides ofthe intersection. How many feet did hepaint? (4X25 =100; 100X2=200, or8 X25 =200, or 50X4 =200)

4 If the dime-store owner owns of thebuildings in the shopping center howmany did he own? a of 24 is 8)

5 The parking lot holds 50 cars and thecharge is 25 cents an hour. If the lot ishalf full for 1 hour, how much will theowner collect? (25 X25=625 or $6.25

for 25 cars)6 John is going to hop from curb to curb

on all four sides of the intersection. If he

hops 2' at a time, how many hops will hetake? (25X4=100; 100+2=50 or 50

hops.)7 Jackie put on his skates at his own

curb. He skated from there to the end ofthe bank and back to his curb. Howmany. feet did he skate? How manyyards?

25

6

50

(across street)(width of sidewalk beside stores)(across parking lot and drug-store)

46 (across grocery store and postoffice)

60 (across dime store and bank)187 (1 trip)X2374 (round trip)

(374+3 X124f or 124 yardsand 2 feet)

Planning for a picture often involvedresearch to find the correct statisticaldata, as in the case of Daniel's baseballpicture. Other members of the class drewupon their imaginations to draw parkscenes with fountains, park benches,painters at work, fish in the lake, flowerbeds, etc. Vigorous discussions with regardto the reasonableness of distances werefrequent as perspective was considered.The teacher felt that the students enjoyedand profited from their experiences withclass and individual number-problem pic-tures.

EDITORIAL COMMENT.-By involving students in the design of number-picture problemsituations, the teacher may also capitalize on the interrelationship of many areas of thecurriculum. Problems may be generated from the sciences, social studies, music, art, phy-sical education, and so on. Doing the research required to validate the problem data is amost worthwhile experience.

Numeration

The articles in this chapter are organized in two major categoriesdecimaland nondecimal numeration. The significance of the base and place-valueconcepts is easily perceived. An understanding of our standard operationalalgorithms for whole numbers is heavily dependent on the student's knowl-edge of these concepts.

Rinker's ideas for an "eight-ring circus" should be well received if one islooking for a way to provide a wide range of activities that focus on thestudy of counting and place value in the primary grades. She provides acomprehensive set of directions for constructing the manipulative materialsused in each of the eight rings.

Other varied ideas for working with place value are provided in the nextthree articles. The expanding place-value cards suggested by Ziesche shouldbe most helpful in teaching the "numbers have many names" concept re-lated to place value. The Placo game created by Calvo will generate manykinesthetic place-value experiences for the children. The flexibility of thecounting abacus described by Cunningham allows the number of beads oneach wire to be changed, thus providing multiple uses.

The role of place value in understanding the multiplication algorithm isemphasized by Haines in her illustrations with Napier's rods. She also pro-vides directions for making and using the rods.

Several ideas for working with nondecimal numeration are given in thelast three articles of this chapter. Rabinowitz provides some directions toassist selected students in constructing various types of "computers" as anenrichment activity. Teachers are encouraged by LaGanke to bring theircheckers and checkerboards to school as an aid in teaching various numberbases to children in the intermediate grades. Several interesting examplesare provided for the purpose of illustrating her ideas. Finally, the slide ruleis brought into focus by Martin in his suggestion to have junior high schoolmathematics students make their own slide rules for use in adding and sub-tracting numbers represented in various bases. Martin also provides sug-gestions for several other experiences that he feels will aid students in under-standing the operation and construction of various types of slide rules.

57

Eight-ring circus: A variation in theteaching of counting and place value

ETHEL RINKERNow teaching in the primary grades in the Balsz School District,Phoenix, Arizona, Ethel Rinker has a background of experienceand professional education in New Jersey and Arizona.

This article describes a technique used byone third-grade teacher to provide enjoy-able learning experiences in the study ofcounting and place value. The techniquemay be used in other grades with appropri-ate adjustments for the grade level and theneeds of the individual classes.

"Eight-Ring Circus" begins at the verybeginning of the study of counting andplace value, thereby giving each child de-tailed instruction and experience with allmaterials and aids.

The various materials and aids areplaced in the room in suitable sectionsthat we call "circus rings." At least onechild performs in each ring. My third grad-ers and I have found that eight rings aredesirable for us. When we feel that the timehas arrived that it may be too expensive orspace consuming to continue performancein a ring, as may be the case in Ring Onewith the use of the tongue depressors, we

ig; NA

merely place a sign at the ring or make amental note that this particular ring is notperforming at the present time. However,the class may describe and discuss the re-sults possible if the ring were continuingperformance with the others.

The following describes the first step ofperformance in each ring:

1. All eyes are on the child in Ring One.He raises the first stick. When this signal isgiven and the stick is placed in a slot of theones box, then all other circus rings go intoaction.

2. A child in Ring Two places one card-board square in a slot of the ones box.

3. Each child in Ring Three records thenumeral one in the ones place on his pre-pared paper.

4. A child in Ring Four brings over onebead on the wire in the ones place on thecommercial place-value board. Another

": it 'is rums ettitsit TOMIN-11111TO

':jTi

Entire Scene of Eight-Ring Circus

58

41.a-,

child brings over one ring on the wire inthe ones place on the handmade, pipe-cleaner place-value aid. A third childplaces one ring on the clip in the ones placeon the handmade, paper-clip place-valueaid.

5. Each child in Ring Five pushes thelever once on his small hand counter orodometer (which has been set at zero) todisplay the numeral one in the ones place.

6. A child in Ring Six exposes the num-eral one in the ones place on the large,handmadepaper-counter odometer aid.Another child exposes the numeral one inthe ones place on the small, handmade-counter place-value aid.

7. A child in Ring Seven records thenumeral one in the ones place on the chalk-board, where place-value columns havebeen prepared.

8. A child in Ring Eight places onepenny in the ones compartment of themoney place-value aid.

The foregoing procedure is continued byadding one more each time and placing orrecording it in the proper place. Whencomprehension is good and a speedup ofthe process is appropriate, 10, 100, ormore may be added each time instead ofone.

The children and I make discoveries aswe go along. The relationship between anumber and a numeral and the advantages

twou.--7

000°.°821.11.

NUMERATION 59

of the place-value system are studied andexperienced in detail.

We continue with this circus-ring tech-nique during a large part of the schoolyear and frequently use the aids in corre-lation with the work in the mathematicstextbook. The place-value columns arekept on the chalkboard for a long time. Thelines for the columns are drawn on thechalkboard with a special white wax pen-cil, so these usually do not interfere withother chalkboard use. They can be erasedover many times before they wear away.The labels for the names of the periods andplace values are on a strip of paper thatis easily removed and returned as needed.We refer to and make use of these columnsin our daily study of mathematics. Addi-tion and subtraction and other problemsare solved in the proper place-value col-umns. A large supply of the children's dit-toed papers, like those used in Ring Three,is kept on hand for the same purpose.

The following are scenes in our class-room that show most rings at the count of68,973. Ring One is at 1,973, with thechild available for discussions.

The following materials and aids areused for the circus rings:

1. Ring OneApproximately 2,000 sticks (tongue de-

pressors)A box with nine slots for supporting

'At;*%-


single sticksA box with nine holes for supporting

bundles of sticks, ten in each bundleRubber bands for bundling the sticks

2. Ring Two

44,

Approximately 2,000 cardboard squares(the weight of tagboard), 17/8-by-1Va inches

A ones box with nine slots for support-ing single cardboard squares

74311

\\Fig. 1

Fig. 2

Back

Bottom

Section A Section B

A tens box with nine slots suitablefor supporting packs of cardboardsquares, ten squares in each pack

Rubber bands for packing the card-board squares

Half-gallon milk cartons to be trans-formed into oblong boxes, approxi-mately 73/4 -by-33A -by-17/a inches

When ten of the tens packs are accumu-lated, they are packed together and secured

Section A Section B

Fig. 4with a rubber band and placed in a thirdplace, the hundreds place. The hundredspacks stand on the table without additionalsupport.

34

311

71-

Fig. 5

When ten of the hundreds packs are ac-cumulated, they are arranged in two rowsin the oblong box, as shown in figure 6.

The oblong box containing the 1,000cardboard squares is placed in the thou-sands place on the table. The first oblongbox is left open so that the contents canbe seen at all times. Boxes representing1,000 can be closed by fitting the two sec-tions A and B of the divided milk cartonsinto each other so that all sides are closed.

NUMERATION 61

The boxes may or may not be filled. Theclosed boxes are tied and covered, if de-sired, with foil or paper to look more at-tractive. At this stage the child may becapable of visualizing 1,000 cardboardsquares inside each closed box, but he mayrefer at the same time to the box that isopen, which contains the 1,000 cardboardsquares.

Fig. 6. Oblong box containing 1,000 cardboardsquares

When the ten-thousands place is needed.ten of the oblong thousands boxes are tiedtogether to represent one 10,000, as shownin figure 7. We can go on like this, as faras we wish to go.

Fig. 7

5

62 : TEACHER-MADE AIDS

;.111111.11111111. ;

to . f C. A ;-.7,1

r."

3. Ring ThreeDittoed papers showing ruled columns

for place values and names of theperiods, similar to what is shown onthe chalkboard for ring seven

-

4. Ring FourCommercial place-value board to bil-

lionsA handmade, pipe-cleaner place-value

aid made with various colored pipecleaners bent in a U shape, with ninerings on each pipe cleaner and eachpipe cleaner glued to a base as shownin figure 8 Fig. 8.

A handmade, paper-clip place-value aidmade with giant paper clips, eachpulled apart to form a horizontal baseand a vertical pole, so that the de-sired number of bent paper clips canbe glued in a row on a strip of card-board as shown in figure 9

NUMERATION 63

Fig. 10

discarded odometers as shown in fig-ure 10

6. Ring SixLarge, handmade, paper counter-odom-

Fig. 9. eter aid to hang from a chart rackas shown in figures 11 and 12

Ringsfrom curtains or tin cans ormade from pipe cleaners, nine foreach placeto be used with theabove, paper-clip place-value aid

5. Ring FiveSmall, commercial hand counters and

Fig. 11

Fig. 12


6 8 9 7 3-Am

Small, handmade counterplace-valueaid as shown in figure 13

7. Ring SevenPlace-value columns drawn on the

chalkboard with white wax pencil

-

OP r,4!.Z4 VAS 54g&

-,_ ..,,., t,..,,,,- .241::-;',1r. -.., .,. g4 big;

Fig. 13

Strip of paper labeled with the namesof the periods to go across the topof the columns on the chalkboard

,.

8. Ring EightHandmade, money place-value aid

made from quart milk cartonsstapled side by side after the flap andone side have been cut away fromeach carton

Real or play money:(1) Pennies for the ones place(2) Dimes for the tens place(3) One-dollar bills for the hundreds

place(4) Tcn-dollar bills for the thousands

place(5) One-hundred-dollar bills for the

ten-thousands place(6) One-thousand-dollar bills for the

hundred-thousands placeTen-thousand-dollar bills for themillions place

(8) And so on

Figure 14 shows a play bill that can bemade from green butcher paper.

(7)

NUMERATION 65

t;1

DOLLARS 1 0 0 0

PENNIES 1 0 0 0 -0 0

Fig. 14

Through the years, my teaching of count-ing and place value has evolved to the pres-cnt method, Eight-Ring Circus. It hasproved to be a considerable improvementover the comparatively vague old ways.We began with just a few circus rings, andas equipment was accumulated more ringswere added. The whole idea grew from theapparent needs of the children for moredetailed experience for understanding inthis learning situation.

Eight-Ring Circus has been truly an en-joyable experience in learning through theuse of concrete teaching materials.

Understanding place value

SHIRLEY S. ZIESCHEGulf Gate School, Sarasota, Florida

FIGURE 1

ave your pupils ever had difficulty un-derstanding place value? Perhaps my dis-covery could help you, too.

A few years back I was trying to explainto my second graders how the numeral 29meant 20 + 9, 2 tens and 9 ones. Thebrighter ones in the class could renamemost numerals accurately but, I felt, with-out really understanding the meaning, eventhough we had used various concrete ma-terials. I took a plain strip of paper andfolded it accordion fashion twice (see fig.1). -I wrote a "2" close to the fold on theleft and the "9" on the right. Within thefold I put the word "tens" and a "+" signso that when the paper was pulled open,the children saw 2 tens -I- 9. I made aduplicate, only this time. I put a "0" insteadof the word. "tens." This one opened toread 20 + 9.

I faced the class again and asked themto rename 29. As the answer was given, Iexpanded the paper showing 2 tens + 9.

66

Then I asked how much 2 tens would be.At the response, "20" I showed the otherpaper saying, "Then 2 tens + 9 is the sameas 20 + 9, 20 + 9 names the same num-ber as 29."

The reaction of sudden understandingthat I saw reflected on the children's facesled me to pursue my discovery. I madecardboard expansion cards with pocketsand sets of number symbols from 1 to 9on smaller cards to fit the pockets. (Thezero was placed on the expansion carditself.) Some expansion cards had "tens +";others, "0 +" on the inside. Then I gavethem to the children to experiment withrenaming of numbers. Later I made expan-sion cards to include hundreds and thou-sands places. These were used to help thepupils understand that regrouping does notchange the value of the numeral; e.g., thenumeral 9483 could be expanded to 9thousands + 4 hundreds + 8 tens + 3ones; or it could be partially expanded to

94 hundreds + 8 tens + 3 ones; or to 948tens + 3 ones (see fig. 2).

FIGURE 2

For my fourth grade, I made a set ofcards with hundreds, tens, and ones thatwere fitted into larger pockets of millions,thousands, and units (see fig. 3).

I found that most of my pupils masteredplace-value concepts much more quicklywith these expansion cards. Even seventh

NUMERATION 67

and eighth graders found expanded nota-tion much easier after experimenting withthese expansion cards.

lialla 8611-111110 aillMillions Thousands Units

FIGURE 3

Try making a set for your class. Or bet-ter still, have each child make his own set.Then, relax and have fun with your mathe-matics lesson.

EDITORIAL. COMMENT.Another version of the expanding place-value card is shown below.

295= 2 hurwireds 9 tens 5 ones

In order to provide flex bility, the numeral card and the digit cards are laminated or cov-ered with clear contact paper so that the numeral and digits may then be written on thecards with a grease pencil. Thus the expanding place-value card may be used over and overagain with different numerals.

Materials needed:1 card 6" x 3" (white--laminated or covered with clear contact paper)3 cards 2" x 3" (whitelaminated or covered with clear contact paper)6 cards 2" x 3" (yellowassembled in pairs with taped hinge)1 piece clear contact paper 6" x 3"3 pieces clear contact paper 2" x 3"tapemarking pen

The cards are assembled as shown by making tape hinges on the back at each joint tofacilitate folding.

Placoa number-place game

ROBERT C. C A L V O Woodland Hills, California

This game emphasizes fun and the learn-ing of positional and place-value aspects ofour number system. Place, (pronouncedplay-so) is a number game that boys andgirls can play to good advantage in theclassroom. The children blindfold eachother, one at a time, and then attempt tofit the rings on the highest place-value pole(see diagram). One value of the game isthe manipulative aspect. The children en-joy handling the rings, fitting them on thedowels, and then computing to find theamount they "win."

While Placo is chiefly a place-valuegame, in actual classroom practice it con-tributes to other operations in arithmeticas well. It provides stimulating practice incomputation and enables 7hUren to havefun while they play the g r.e. It can beused with success in Grades 2 through 6and can be refined to challenge even thebrightest students. It may be played by asfew as two children or as many as thewhole class.

Three colored dowels are glued on abase, as shown. These dowels are station-ary. The dowel to the left is highest invalue, the one to the right is least in value.They may be painted blue, red, and whitein that order. They are labeled 100's, 10's,and l's. Thus 100's (in a whole-numbergame) are blue, 10's are red, and l's white.

68

PLACO

0 Rings

Pegs

One player is blindfolded and given somerings (9-18) of which he may put notmore than nine on any one dowel. Thisblindfolded player then tries to place therings on the dowels. The other players(split in teams) watch in glee. The objectof the game is to get the largest score byputting the rings on the highest-valuedowel. After all rings have been placed, theblindfold is removed, and the whole classor a small group computes the amount.The other players then take their turn.Players can play with a goal of 10,000 orwith a time limit, the highest score winning.In playing decimal-fraction Placo, the bluedowel represents ones; the red dowel,tenths; and the white dowel, hundredths.

Placo cards may be developed to usewith the game. These cards are used asfollows: After each turn, a player selects

one of the cards and follows the directionsgiven. They say, "Double your score" or"Cut your score in half" or "Subtract threetens," etc. This serves to provide varietyin the game. These cards can be developedto reinforce specific skills needed by theclass or to provide the class with neededdrill. Cards might give directions to ado,subtract, multiply, or divide. In addition todramatizing and creating added, interest,they serve to equalize scores of fast andslow learners without penalizing the fastones.

A word here about the use of coloreddowels. While some experts say that place-value instruction should not depend oncolor, we can't rule it out completely atearly levels of learning. In place value, itis important that the child knows that "thisring is in the place to the right" or "thatpeg is in the units column," but that doesn'tmean color can't be brought in to augmentthese instructions. It merely means colorshould not be used as a sole means ofidentifying number-place. If color canhelp the children gain a concept or canmake a game more attractive, then itsuse is defensible.

Other uses can be found for this game.One possibility is to use it with sight-sav-ing classes by cutting niches in the baseto identify the corresponding number-value. Computation could be done men-tally. Another potential use would be toroll special'clice and compute the score inwhatever number base the dice show whenrolled. The dice could be made so onlycertain number bases would result.

Materials list for Placo

1. Piece of wood 12 inches long, 4inches high, and 6 inches wide

..1

NUMERATION 69

2. Doweling (also known as "closet pol-ing"), 1 inches in diameter, three piecesof 7 inches each

3. Drapery "cafe curtain" rings, 11inches inside diameter, 12 to 20

4. Cards and tacks5. Sandpaper6. Paintwhite, red, and blue high

gloss

7. Carpentry tools, including drill andsaw

8. For variation noted, 3-by-5 cards9. Large handkerchief or clean rag for

blinder10. Oak tag for power chart

Building the game

1. Drill three holes equally distant fromeach other in the 12-by-6-by-4 inch piece ofwood, making these holes 1 inch deep and1* inches in diameter.

2. Sand one end of each 7-inch piece ofdoweling smooth. Put some white glue onthe bottom inch of the dowels and the in-side of the drilled holes. Slip the dowels inand allow to dry overnight.

3. Purchase cafe curtain rings of thesize specified in the materials list.

4. Paint the game three different colors,as mentioned previously, making thedowels and the base of matching colors.

5. You may want to put two smallnails about 3 inches from the top of eachdowel (exact placement would dependon the thickness of the curtain rings). Inthis way, players could put only 9 rings oneach pega 10th would have nothing tokeep it from falling off. This style of count-ing uses only 9 units. When 1 more isadded, it becomes the number on the left.

Making a counting abacus*GEORGE C. CUNNINGHAMWalton High School, Walton, West Virginia

Use of the counting abacus (Fig. I) may 3 shirt cardboardsbegin in the first grade and continue 3 " diameter wooden beads (less

4$1.50 for 100)

1

FIGURE 1

through high school level in general mathcourses. It is a very effective tool in show-ing the action of addition, subtraction, andregrouping. By changing the number ofbeads used, a teacher may show thesesame actions in other base systems.

The cost of constructing the countingabacus will be less than $2.00. You willneed the following list of materials:

Board 6" x 8" x 3" (less than $.50)Electric drill and 7 " bit or hammer

64and a nail about 7 " diameter

644 all-metal coat hangers2 pairs of pliers

Developed nt modern math workshop, West Vir-ginia' State College: Miss Lynch, instructor.

70

than

1The board must first have a cut2=-

inch deep down its center length (Fig. 2).

cut deep

1

6"FIGURE 2

Ask the lumber company to make thiscut for you if you do not have a circularsaw. Most companies will do this for you

1" 2"

8"FIGURE 3

6"

free. Draw a line 1 inch from each longedge. Then, beginning at one end, measure1 inch along this line and mark a point.Measure 2 inches from this point and

put another point. Do this until youhave 4 marks. Repeat the same prOcesson the other edge (Fig. 3). With the

7electric drill, drill holes 5 to inches8

deep at each point (Fig. 4). The hammer

holes Siff " -1 /e" deep

I

81./.

FIGURE 4

and nail may be substituted to make theseholes if you do not have an electric drill.For a more attractive board you maysand it or sand and paint it. Put it aside.

The coat hangers must first be separatedat the neck. To do this take a pair ofpliers and twist the neck of the coathangers in a clockwise motion until itseparates. Using both pairs of pliers, bendthe coat hangers into a straight line. Thenbreak off each end so that you have awire about 32 inches long. With both handsgrasp the coat hanger in the middle andbend it into a semicircle with a diameterof 4 inches (Fig. 5). The beads are nowready to be put on the wire.

<----- 13"4" dia.

FIGURE 5

The use of 18 beads on the first threeplaces and 9 beads on the last place isrecommended. This will allow you to showhow to regroup after borrowing and carry-ing.

The third part in making the countingabacus is making the cardboards. Mostcardboards are 8" wide but must be cutto 10 -I" high. (Cut 1 cardboard 8 x 10.)

2Draw horizontal lines on the larger cards.On the first line, behind each place value,

NUMERATION 71

write the name of that place value. Forbase five write ones, fives, twenty-fives,125's. On the next line above, write theplace value in exponential form: 5", 5', etc.On the next line write the place value inexpanded notation: 1, 5, 5 x 5, etc. Use

1the two 8-x-102

pieces for this, with thesmooth surfaces showing. Tape the topsof these two cards together. This givesthem added stability. Use both sides andeach end to put 4 more base systems onthe 8-x-10 card. This card is to be insertedbetween the two longer cards for ease ofstorage. When in use it can be placedin front of the other cards to show thebase system being discussed.

Now you must put all of these differentparts together. Place the beads on eachwire, insert the wires into the holes drilledin the boards, put the cardboards in theslot, and the counting abacus is ready touse.

If a board cannot be obtained, a tem-porary base may be made from a blockof styrofoam. Use a knife or razor bladeto cut the slot, and use only one cardboard.The wires may be punched into the styro-foam without any problem. The teacher canuse this type of counting abacus to showthe class; but it cannot be used very long.by the children, because the holes willsoon become enlarged and the wires willcome out.

A simplified "open-end" version can bemade with 4 straight sections of coat-hanger wire. Prepare the board as before,

holes forcoat hangers

FIGURE 6

Cardboards

14" section ofcoat hanger

6 x 8 x 11/2base

but mark and drill the holes on one sideof the board only. Cut a 14" section from

the bottom of the coat hangers. Insert thewires and cards into the board (Fig. 6).The beads should be kept in a small box

or bag to prevent their being spilled andlost.

Concepts to enhance the studyof multiplication

MARGARET HAINESFairview Elementary School, Elizabethtown, Pennsylvania

The use of a system called Napier's Rodswill make multiplication an interestingstudy for both fast and slow learners.These were invented about 1617 by a Scot-tish mathematician named John Napierfor performing multiplications.-These can

be made and used for other number bases,but for the present let's consider onlybase ten or our decimal system.

For best results and proper manipula-tion these rods should be cut from heavycardboard or thin plywood about nine

1 2 3 4 5 6 7 8 6

24 li 8

I

0

1 1

4

1

6

1

8

:3

6 9

I

2

1

8

2

I

2 9

4 8I

9I

6 02

42

83

2

:3

6

5I

0

1

5

9 2

5

:3 3

5

4

0 5

(I wally 5

4

71

4

2

5

4

6

6

3

I

(i

2

4

:3

2

4

0

4

8

1

8 7

3

6

4

5 4

6

3

7

2

E

1

Figure 1

72

inches long and one inch wide. Four stripsare needed if one set of answers is placedon each side of the strip. However, in ourfifth-grade room we like to show all theanswers at once in order to make moredifficult problems so we used eight strips.All one-inch blocks should be divideddiagonally to take care of two placeanswers. We used a woodburner pento make our numbers show more dis-tinctly. The answers on the rods representmultiples from 1 through 9. When all therods are finished a coat of shellac will pre-serve the printed matter. (See Figure 1.)

1 8 7 9

6 4 S

9 23

1 7

3

4, 6

:3 4

55

5/4

. 5 6

7

9 3

6 5 78

6 2

7 69

:3 1

Figure 2

NUMERATION 73

The strips are movable and can be ar-ranged to form any multiplicand. Thestrip representing the multiplier is placedat the left-hand side. The product isfound by arranging the answers found onthe Napier's rods in a certain orderanorder that corresponds to the order of thedigits in the multiplicand. Please notethat these preceding sentences prove to bevery valuable in teaching the meaning ofmultiplication terms that often seem elu-sive to many pupils. Pupils have to masterthese terms in order to use the rodssomething they need not do in ordinarywritten multiplication.

Suppose the example to be multipliedis 879X6. Arrange rods 8, 7, and 9 to formthe multiplicand. Place the rods in theorder which corresponds with the order ofthe digits in the multiplicand. Place thestrip representing tite multiplier at the leftof the three striiis you have arranged.(See Figure 2.)

To find the answer, find the multiplier,6, at the far left. Start at the right of thisrow and read the answers following thediagonal lines. The answer for the onesplace (6X9) is 4; the answers for the tensplace (6 X7) is 5 to carry plus 2 or 7; theanswers for the hundreds place (6X8) is4 to carry plus 8; then a thousands placeis needed for 1 to carry plus 4 or 5. Theanswer, therefore, is 5274.

6

5

thousands

8 7 9

2

hundreds

7

tens

4

Check: 879X6

5274

ones


The example may require multiplica-tion by two numbers, such as

879X36

In this case the answer for multiplicationby 6 is found and recorded as indicated.Then the row across from 3, the tens partof the multiplier is found. It must be re-membered that multiplication is by 30;therefore, 7 becomes 7 tens or 70. Otherdigits in the answer are 26370.

5274 (879 X 6)26370 (879 X 30)

31644 (product)

Check: 879X 36

52742637031644

My pupils check these without my re-minding them for they feel that just onceone must be wrong.

Halving and doublingAnother method of multiplication to

erase the need for memorization of allmultiplication tables (save the twos) hasbeen devised. In fact, there is sufficientevidence to establish that it was knownto the ancient Egyptians as early as 2000B.C. The principle is simple enough thatit may be introduced in the fifth grade and

will serve as an introduction to more diffi-cult mathematics since it introduces dou-bling and halving of numbers.

Suppose one has two sets of numbers tobe multiplied and suddenly forgets allmultiplication facts except the two's. Ifone number is halved, it matters notwhich one, and the other doubled until thehalving reaches a quotient of "1" theproduct will be the same as that of theoriginal number. There is only one smallitem to remembercross out all evennumbers which are opposite even numbersin the halved column. While halving num-bers discard all remainders.

Halving Doubling27 3413 686-3 272

544

918

Example: 27 X34 = ? Six is even so omitfrom both columns before adding remain-ing figures in doubling column.

Now add and check using the regularmethod:

27X34

10881

918

Children need not be coaxed to checkthis type of multiplication and I foundmy brighter pupils loved it.

EDITORIAL COMMENT.-It is also possible for children to make their own sets of Napier'srods by using tongue depressors. The actual construction of these rods by elementaryschool children in grades 4 through 6 may accomplish several objectives. For some chil-dren this activity may provide another vehicle to review the multiplication facts, may bea stimulating enrichment experience, or may be an interesting link with history. For others,the rods may simply be a curiosity to be explored and used in performing the multiplica-tion algorithm.

Building "computers" fornondecimal number systems

FREDERICK R. RABINOWITZMathematics Consultant, Philadelphia, Pennsylvania

A. rash of comic-strip entries provided anunexpected form of motivation for aPhiladelphia sixth-grade class. Mrs. Kath-arine McFadden's class at the John H.Webster School used them as a springboardfor studying various systems of numera-tion. The class had been introduced tothe binary system early in the year, hadachieved very well, and had enjoyed thehours working with it. But when the car-toons on "new mathematics" appeared laterin the year, and interest again became high,the pupils sought additional outlets fortheir interest in numeration systems.

First, in connection with their workwith the binary system the children con-structed a binary "computer," which con-sisted of a black box, dry-cell batteries,switches, and the necessary wiring forlighting five bulbs, which represented theones, twos, fours, eights, and sixteensplaces.

FIGURE

75

A particular base-two number could benamed, and then an operator would choosethe correct combination of switches to lightthat number on the "computer." Ofcourse, it did not "compute," but it didserve as an effective teaching aid in base-two numeration.

The boys made individual computers ofvarying degrees of sophistication. Some-how, the girls felt that this was a "manly"manifestation and so did not engage in thistype of construction. With outside con-sultant help, Mrs. McFadden set up meansfor the girls, and the boys, to constructless complicated "computers" and adaptthem for different bases.

Wire coat hangers were used for thebinary system. Ten strips of manila oaktag paper, each 2" x 21/2", were cut. Thenumeral 0 was inked on five of them witha felt-tip marker and the numeral I inkedon the remaining five. A "one" and a"zero" were stapled together around thebase of the hanger and thus operated asone unit when they were rotated. The sizeof the hanger and the size of the cardsallowed for five places from the right inthe binary system. Consequently, thegreatest number that could be computedwas 1 i 111 , or 31 in our base ten.


A diagrammatic example of the use ofthis binary "counter" is shown in Figure 2,

FIGURE 2

10110. two = 22 ten

This device worked well for the binarysystem, but what about the ternary sys-tem? Here, again, no obstacle was met,since a coat hanger and the same-sized tagstrips were used. Now, however, one morenumeral was needed in the base-three sys-tem; so three strips were stapled around thehanger, providing the three numerals nec-essary for the ternary system, thus:

FIGURE 3

Admittedly, these computers were a bitmore difficult to manipulate than the onesdesigned for base two; yet they were ef-fective. The greatest base-three numberpossible on this computer was 22222 three,or 242 in base ten.

Four pieces of oak tag were used foreach place in base four, as in Figure 4.It was possible to turn the "dials" here to33333. This meant that 333331. =1023 ten.

FIGURE 4

This still did not satisfy the more ad-venturous minds, so they tackled base sixin a different way. Their computer wasmodeled after an odometer in an automo-bile. Six cylinders were made to revolvearound a horizontal axis consisting of adowel stick suspended on two wire hingersupports (Fig. 5).

FIGURE 5

A strip of paper was pasted over thelateral face of each cylinder. On eachstrip had been written the six digits nec-essary for base six-0, 1, 2, 3, 4, 5. Apupil would merely turn the dial to selectthe proper base number. In Figure 5, thecenter row of digits would produce2053415; = 16765 ten.

Not every sixth-grade class in theauthor's scope of responsibility could dealwith number bases in this manner. Theseactivities are for enrichment and must beidentified as such from the outset.

However, a great deal of mathematicallearning as well as enjoyment can resultfrom these aivities. The children reapedboth of these harvests.

Let's use our checkers andcheckerboards to teachnumber bases

L U C I L E L A G A N K E Wilmington College, Wilmington, Ohio

In this day of teaching modern mathe-matics, one notices the tremendous num-bers of manipulative materials beingmanufactured for the sole purpose ofteaching number concepts. However, theteacher might find interesting math ma-terials at home.

Why shouldn't the teacher and hispupils bring their checkers and checker-boards to school, to use in teaching theidea of various number bases to fifth andsixth graders.

A checkerboard has sixty-four squaresand twenty-four checkers. The pupil, bydrawing four rows of eight squares, thesize of those on his checkerboard, on anappropriate size of paper, can change thecheckerboard into an arithmetic tool.

The squared piece of paper can be placedat the top of the checkerboard. The first

row should be marked from right to leftwith the number base being used, eachnumber having the proper exponent abovethe base number, to indicate its value.The second row should have the numeri-cal equivalent of each base number, writ-ten in base ten. Then, two rows could beused for manipulating the checkers.

A pupil could place checkers on eachspace of the fourth row, and then pushthe checkers into the proper spaces in thethird row, to build numbers. This is reallyusing the checkerboard as an abacus.

Figure 1 is a sample of what a paperguide forbase two or the binary systemwould look like.

To express the base-ten number 21, inbase two a pupil would push the checkersinto the proper squares in the third row.

If checkerboards would take up too

27 26 26 2 23 27 21 20

128 64 32 16 8 4 2 1

FIGURE 1

77


Base 3

32 36 35 34 33 32 31 3o

2,187 729 243 81 27 9 3 1

Base 5

52 56 55 5' 53 52 51 50

78,125 15,625 3,125 625 125 25 5 1

Base 2

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2

Base 10

107 106 105 10' 103 102 101 100

FIGURE 2

much space in the classroom, the teachercould make hectograph checkerboardsheets, and as each number base wasstudied, the pupil could work out his ownnumber bases and write the correct num-bers in the proper squares. (See Fig. 2.)Cardboard or paper disks could be usedin lieu of checkers.

These number boards could be used tobuild numbers, to add numbers, to sub-tract numbers, and to compare how acertain number appears in different num-ber bases. Drills might be given to see

how quickly the children have learned toperceive the component parts of a number.

Adding several numbers in a differentnumber base should be fun, for as soonas a pupil obtained the maximum numberof checkers permissible in each square inthe number base being studied, he wouldthen discard the excess number of check-ers, and move one checker into the nextcolumn to the left. He would really be"carrying." (See Fig. 3.)

It would probably be wise for the childto start with base ten, and then learn to

The Number 27 in Four Bases

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

: 1

;

,...,

i t

;

<-4,

I, E--I ki 't(--; :

v

FIGURE 3

adding

17 binary+ 23 system

40

basetwo

17

23

40

work in base two, base three, and basefive. The number of symbols or checkersthat could be used in each number systemcould be found out. by the "discovery"method. Other bases might be presentedin the junior high grades.

NUMERATION 79

There are many possibilities in teachingvarious aspects of number bases, by usinga checkerboard and checkers. Probably allthe members of the family will join inwith the children in playing checkers, amath game for the twentieth century.

EDITOR/M. COMMENT.The checkerboard may be used in a variety of ways as a teachingaid in mathematics. For example, to provide a "practice experience" -ith basic multiplica-tion facts. the teacher may affix a piece of masking tape on top of each checker and write amultiplication fact on the tape.

Two students may then be permitted to play a regular game. of checkers with the follow-ing modifications to the rules: Before a player can move a checker, he must correctly statethe product for that checker (the other player checks to see.that he gives the correct prod-uct): before a player can make jumps, he must give the correct product for every checkerhe jumps: before a player can accept a crown, he must give the correct product for at leastone of the checkers available for use as a crown.

Another variation for a practice activity with checkers is to affix a pi,:te of tape to eachblack square on the checkerboard as well as to each checker. Then write a numeral on

--each_pie.c.e of -tTiTe to prOyide for the desired practice. For example, if addition of tensis the desired practice experience:the teacher writes two-digit numerals such as 10, 20. 30,and so forth on the pieces of tape. Then as two students play a game of checkers, they arerequired to give the correct sums generated from the two addendsone on the checkerbeing moved and the other on the checkerboard square where he lands.

Discovering the mathematics ofa slide rule

J. GREGORY MARTIN, JR. Ithaca, New York

Mr. Martin has been a junior high school teacher. He isnow in his second year of study on a doctoral program in mathematicseducation at Cornell University.

Like mathematics teachers in most juniorand senior high schools, I find myselflooking for ways to refresh my. own en-thusiasm as well as new ways to channelthe enthusiasm of my students. One methodI find to be effective is to introduce atopic that is not in the text, and at thesame time is a topic I have not presentedbefore.

Thus, a few years ago, when severalstudents asked about learning to use aslide rule, a couple of my fellow mathteachers jumped at the chance to capitalizeon the student's enthusiasm. In fact, forthe past three years, the school bookstorehas maintained a large inventory of sliderules. The pros and cons of incorporatingthe use of a slide rule into any math class,and in particular into the seventh andeighth grades, were discussed, quite nat-urally, among us teachers.

Initially I could see no justification forslide-rule drill in any math class; however,one of our teachers, a man who had cometo us from several years of engineering,altered my thinking. To him, practice withthe slide rule, like practice with any im-plement, gives to the user the skill andconfidence necessary to render the tooluseful. Thus, a little time given each yearwill bring this needed confidence and skillto the student for the time when he mayneed to make use of it in physics andchemistry.

While I conceded the point, I could notbring myself to tolerate the sight of the

"things" in any class of mine that went bythe name of mathematics. But my preju-dices not withstanding, we soon had an in-fectious rash of two-dollar K.E. slide rulesgrowing to epidemic proportions. Thesymptoms of the disease became manifestin my own classroom. I inoculated all mystudents against the slide-rule disease bywarning that I intended to confiscate anycommercial, slide rule that I might see inmy class.

Of course, it would have been a mistakenot to utilize the. force of this infectiousand wide-spreading enthusiasm. Thus, oneday I announced that we would take upthe topic of why a slide rule works. Sev-eral students wanted to know if this werenow the signal to rush forth to the bookstorethe quarantine period seemed over.Yes, we were now going to have slide rules,but not commercial ones; instead, each stu-dent would make his own. This is veryeasily done with cardboard and scissors.(See Fig. 1.)

I I I III

FIGURE 1

What follows is an outline of ideas whichcan be.successfully presented to junior highschool students, or which could be profit-ably used in conjunction with topics onexponents at any level.

By numbering the marks of the sticks'

80.

with consecutive integers as shown in Fig-ure2, we are able to obtain answers to ad-

2 + 3 s 11

FIGURE 2

dition combinations. In the figure, we areadding

(2 -f- 3 11)base four.

The slide rule in Figure 2 is also set toadd the combinations

2 + 0, 2 + 1, 2 + 10, etc.It should be clear what is happening in theillustration. The outside stick contributes2, the inside stick contributes 3, and theanswer is read back on the outside stick.Using slide rules marked in different basesis of great interest to students; hoWever, itis probably best to use base-ten numeralsfirst. Later, if it is appropriate, the left endof the stick could begin with negative num-bers.

If other teachers in a building have al-roady devoted time to slide rules, anti evenif they have not, the students will very soontire of just adding. They will be impatientto multiply. But the teacher should, in myopinion, counter any student suggestions tomultiply with a statement that the slide rulecan do nothing but add and subtract. It willprobably already have occurred to someonethat a subtraction problem takes place onthe slide rule for each addition problem.Thus, in Figure 2, the slide rule also gives

(11 3 =-. 2)bas- four

or it could be regarded as2 0, 3 1, 10 2, etc.

Slide rules are generally not constructed

NUMERATION 81

to supply answers to addition combinations.This is because the combinations are wellknown, or very easy to determine. How-ever, the student-constructed slide rule canbe of aid when addition is done in a numer-ation system with a negative base, for thenit is not quite so easy to determine sums.(An article entitled "Using a negative basefor number notation" by Chauncey H.Wells, Jr., appeared in THE MATHEMATICSTEACHER, February 1963.)

After the students have spent some timeconstructing and using slide rules that addin different bases, they will be ready toconsider multiplication. At this point I dis-play a commercial demonstration slide rule.(I always expect some student to tell me Iam breaking my own rule about commer-cial slide rules, but this response has nevercome. apparently the students arc too ab-sorbed in learning.) Using the large sliderule, I demonstrate a couple of multiplica-tions. "See, it works just like your addingslide rulesbut because the numbers arearranged in a strange manner, the answeris the product instead of the sum."

Students and teachers will derive maxi-mum satisfaction if the class is challengedto find the way in which the numbers areplaced on the slide rule. Of course, the stu-dents should be warned that they will

not be able to find a scheme whichwill locate every number. .I tell my juniorhigh school students that it would be a verysmart twelfth grader who could devise asystem for placing every number and, infact, most twelfth graders probably couldnot place any number..

In some years, T have allowed the chal-lenge to drag on all year, while I periodi-cally gave hints. Discovering the scheme isnot an easy task, and any student who canindependently find 'the solution is indeed'perceptive. For the teacher's part, he willmost likely find that several hints will be'needed before the students are able to dis-cover the method.

Before the students can find the methodfor placing a few numbers, or before theywill be able to understand an explanation


of this, they must have some experiencewith exponents. When such examples as

= 32+ 5

are familiar to the students, it can be ex-plained that we are using addition to finda product, a slide rule will only add, but aslide rule will find a product. Even' this ex-ample will not allow every student to dis-cover the method for finding products. Ofcourse, the student must know that re = 1,with n not zero since 0' is undefined.

Once the method for locating numbers is'known, it is a good exercise for the studentsto construct a slide rule like the one shownin Figure 3. The numbers on the upper

\. 1 6- 8 E) 31\

2 4 8 \61 la 3

0 / 2 3 4

FIGURE 3

32

5

halves of the sticks are just powers of 2.Thus, on the top scales we have the sliderule set for the multiplication,

2 x 4 = 8,or

(2 x 2, 2 x 8, 2 x 16, 2 x 32).

On the lowerthe addition,

Thus,

is for the slide

halves of the slides we have

1 + 2 = 3.

2' + 22 = 23rule just the same ast + 2 = 3.

After the pattern of numbering the sliderule is known, the student will be quick tofollow up with numberings other thanthose that use 2 as the base.

A natural extension, and a most profit-able one, is the use, of the slide rule withnegative exponents. Here, concrete mean-ing and lasting understanding can come tothe student in a very easy and natural way.Another obvious extension would be frac-tional exponents. This additional way ofviewing the concept is very helpful andmeaningful, though this may not be as in-herently clear as the previous notions men-tioned.

A cardboard slide rule is not a practicalslide rule, and it v-Duld be a mistake to as-sume that the student is acquiring any de-gree of skill in manipulation. However,using a homemade slide rule allows thestudent the opportunity for mathematicalunderstanding, and it gives him a chance to"see" why it works.

Integers

The system of integers is b,:ginning to play an increasingly important rolein the mathematics curriculum of the intermediate grades. As elementaryschool teachers. we shall be seeking more and more ways to make the set ofintegers and their operations meaningful to our students. The four briefarticles in this section are a start in that direction.

Mauthe describes a game in which upward motions on a ladder cor-respond to positive integers and downward motions correspond to negativeintegers. The composition of movements corresponds to addition. Imaginethe glee of students as they reach the top of the ladder or as their opponentsdrop off the kidder.

Frank's article provides a model for addition and subtraction of integers.A."take away" interpretation of subtraction is given for positive and nega-tive integers. Frank relates several conceptsintegers, number lines, andoperationswhilc incorporating much fun in her shuffleboard game. ..

Very often our students come away with. the impression that number linesare either horizontal or vertical and that positive is always up or to theright. Milne describes a number-line model that not only preserves the or,Bering of the integers and their readability but also-shows students that thenumber line may have many orientations.

The sign rules for multiplication. of 'integers scent to bailie some of ourstudents. Some can follow the abstract arguments and are willing to acceptthe rules: other students need more tangible illustrations. Pratt's flip cardsoffer students an interesting way to visualize and correlate these rules.

83

Climb the ladder

ALBERT H. MAUTHE

Albert Mauthe is chairman of the department of mathematicsat A. D. Eisenhower. High School at Norristown, Pennsylvania.

He is on leave from his position this year to participate in the

Experienced Teacher Fellowship Program in Mathematics for

Prospective Supervisors at Teachers College, Columbia University.

Here is a new game. It is called "Climbthe Ladder." The rules of the game arevery easy to learn, and you will have lotsof fun playing Climb the Ladder.

To play the game, you will have to makea ladder and a dial-spinner. (These will beshown later.)

As many players who wish to play thegame together may do so. The object ofthe game is to go above the *12 step onthe ladder. Anyone who falls below the -12step on the ladder has fallen off the ladder,and he is out of the game. Thus a playercan win the game in either of two ways:

1. if he is the first person to go abovethe *12 step on the ladder, he wins thegame; or

2. if he is the last person remaining onthe ladder after all the other players havefallen off the ladder (by falling below the-12 step), he wins the game.

Everyone starts by putting his marker at"Zero", the "0" step. We will call thenumbers above 0 "positive numbers," andwe will call the numbers below 0 "negativenumbers." We will not call 0 either "posi-tive" or "negative"; 0 will be called "zero."

The first person spins the spinner. If thespinner stops on a 1, '2, *3, or '4 space,the player moves his marker up the ladderthe number of steps indicated (1, 2, 3, or4). We will call the numbers 1 , '2, '3, and*4 on the spinner-dial "positive numbers

If the spinner stops on a -1, -2, -3, orspace, the player moves his marker down

84

the ladder the number of steps indicated(1, 2, 3, or 4). We will call the numbers-1, -2, -3, -4 on the spinner-dial "negativenumbers."

If the spinner stops on the 0 space, theplayer does not move up or down theladder. We will call this number "zero."

Each player takes one turn at a timeuntil he 'has fallen off the ladder or someplayer has been declared the winner.

Let us watch Betty, Bob, and Dave playthe game. We will watch each player'sprogress up the ladder on the demonstra-tion ladder. At the same time, we will keepa record cf each player's steps at the rightside of this

Betty begins. She starts BETTYat Zero, 0. Start 0

She spins a +3. Thus, shegoes 3 steps up the ladder. Add

Sheis,nOw on step +3. Step

+3

+3

Bob and Dave each take their first turns.Bob lands on -2, Dave on *4. The secondturns for Betty, Bob, and Dave look likethis:

Betty was at +3.She spins a +1. Thus, she

goes-1 step up the ladder.

She is now on step +4.

Bob was at -2.He spins a +3. Thus, he

goes 3 steps up the ladder.

He is now on step 41.

BETTYStart +3

Add +1

Step +4

BOBStart -2

Add +3

Step +1

Dave was at -4.He spins a -2. Thus, he goes

2 steps down the ladder.

He is now on step -6.

Bob spins a -2 on his thirdturn. He was at +1.

Thus, he goes 2 steps downthe ladder.

He is now at -1.

Bob spins a 0 on his fourthturn. He was at -1.

Thus, he does not move eitherup or down the ladder.

He is still at -1,

DAVEStart -4

Add -2

Step -6

BOB

Start +1

Add -2

Step -1

BOB

Start -1

Add 0

Step -1

The game continues until one playerpasses '12 or all but one player passes -12.

Make your own game

Materials needed.We'll want to playthis game in class. But before we beginplaying, we'll want to make our own gameto keep. To make the game, each personin the room will make the game from thesematerials:

1. A square piece of heavy cardboard,about 15 centimeters on a side (15centimeters is about 53/4, inches).

2. A piece of heavy paper or cardboard,approximately 8 centimeters by 28 cen-timeters (that is, about 3 inches by11 inches).

3. A paper clip.4. A thumbtack.5. Several small pieces of cardboard or

other objects to serve as markers.6. A ruler, preferably one with centimeter

markings.

7. A protractor.8. Compasses (to draw a circle).

The dial-spinner.--On the heavy card-board, draw a circle, with center in theMiddle of the cardboard, with a radius of5 centimeters (about 2 inches).

Study the spinner-dial shown in Figure

INTEGERS 85

1. Note that there are nine different sec-tors, four for the numbers *1, +2, +3, and '4,four for the numbers -1, -2, -3, and -4, andone for 0.

Dial

FIGURE 1

Do you remember that there are 360degrees in one complete turn around thecenter of a circle? How many degrees mustthere be in each of the angles around thecircle if we use 9 sectors that have equalangles?

, Draw a 1,'-adius from the center of thecircle you have drawn to any point on thecircle. Measuring with your protractor,draw another radius so that the two radiimake an angle of 40 degrees at the centerof the circle.

Continue making 40-degree angles untilyou have nine equally spaced radii, andnine sectors of the same size around thecenter of the circle. Label these sectors asshown in Figure 1.

Take a paper clip and bend the lastloop out so that it looks like the one shownin Figure 2.

Spinner

FIGURE 2

Place a thumbtack through the ovalstill remaining in the paper clip, and placethe thumbtack through the center of thecircle you have drawn. Your spinner-dial


should look like the one shown in Figure 3.

Flo. 3. Thumbtack through oval of paperclip.

Make sure the paper clip spins freelyaround the dial.

The Ladder.On a heavy piece ofpaper (or cardboard), draw a straight linesegment 24 centimeters long (or 9 inches,if you do not have a ruler marked incentimeters). Make a mark at the bottomof the line segment. Then mark off eachcentimeter (use 3/a inch, if you used a 9-inch line segment), until you get to thetop of the line segment.

Label each step from -12 to '1 2, as wasdone with the ladder in Figure 4.

Let's play!

Although any number of players canplay together, let's divide into groups ofabout three players each and try playingour new game. Each group will use oneladder; each player will use his ownmarker.

With each turn, be sure to move yourmarker up or down the ladder. And besure to keep score, as we did for Betty,Bob, and Dave.

If your game finishes before the others,play another game. Keep score for thisgame also. If time runs out before anyonehas won the game, the player highest onthe ladder will be the winner.

Have fun!

+12

+H

+10

+9+8'

+7ed +6ACA +4

...I +3+2

IA +10

~ -1m -2

-3-4

O -5-6-7-8'-9-10

/1

12

Ladder

LZERO-

FIGURE 4

+6+5 a.+4+3+2+1

0-123-4-5

6

EDITORIAL. COMMENT. YOU may wish to have the student. write a number sentence to de-scribe each of his moves on the ladder. This will provide the connecting link between themanipulations and movements on the ladder and the symbolic representation of thesemovements. Many times the student fails to realize that these game experiences have anyrelationship with the symbolic language of mathematics.

Play shuffleboard withnegative numbersCHARLOTTE FRANKAlbert Einstein Intermediate School 131, Bronx, New York

Shuffleboard, anyone?" Can you imaginehow a student-filled math classroom wouldrespond to that invitation? Compare thatwith the reaction to a teacher announcing,"Students, today we are going to learn thefour fundamental operations of the set ofintegers." By the time the teacher hadfinished saying the word "fundamental"his pupils would have tuned out both himand his lesson. Negative Shuffleboard canbe used to introduce the four basic opera-ticns of negative integers to the elementaryschool youngster in a fun setting.

For the few who have never played,regular Shuffleboard is a game in whichplayers use Icing-handled cues' to shovewooden disks into scoring beds of a dia-

O -4 QO

QO

0

FIGURE

gram. Each player tries to push his owndisk into one of the scoring areas or toknock his opponent's disk out of a scoringarea and hopefully into a penalty area.

By adding positive and negative signs,the playing field (Fig. 1) can easily betransformed into Negative Shuffleboard(Fig. 2).

Other adjustments necessary to fit ournew game to the facilities of a classsroomare the following:

1. Target diagrams are drawn on papersheeting large enough to cover the stu-dent's table.

2. Red and black checker-size disksreplace the larger disks normally used.

87


3. A pencil, pen, or ruler is used as apropelling; instrument.

4. The ratio of the playing field to thedisk is increased in order to allow formore scoring activity.

5. A number line is used to keep arunning score.

When Anne and Harley played the game,their mathematical computations developedthis way:

ROUND 1

410

+8 +8

+7 +7

Zo -10 c710

Anne (C) disks) -10 -1- -10 = -20.or 2 x -10 = -20.

Harley ( disks) +8 -1- '7 + -10 =

Round 2

+10

0+80

0'+7

+8

+7

-10 10Anne (0 disks) 7 + +8 + +7 = +22.Harley ( disks: +10 + -10 = 0.

Round 3

Anne (0 disks) (3 x +7) + 10 = 11.Harley ( disks) 4 x 10 = -40.

'INS game continues until a previouslyagreed upon winning score is reached.

Some of the computations at anothertable, where Matthew and Sidney wereseated, were as follows:

Situation 1. Matthew had +7, +8, +10and one disk out of limits, giving him atotal of +25. When Sidney's turn came hesuccessfully knocked Matthew's disk outof the +8 box and into the -10 box. Mat-thew's +8 was taken away; written math-ematically, it was +8. This number sen-tence describes the last play:

+8 +8 = 0.Situation 2. Sidney successfully knocked

his own disk out of the -10 penalty box.Translated into our language of numbers,Sidney had a -10. When he took away the-10, it gave him a 0 for the box. Thisnumber sentence is written:

-10 -10 = 0.Situation 3. After Round 4 Matthew

had 30 points more than Sidney had. Withthree disks still to shoot, Sidney told thegroup watching that he could win if Mat-thew were unlucky enough to end theround with a -30. That would be writtenby the following number sentence:

-30 ÷ 3 = -10.The target diagrams vary from table to

table to meet the needs of the individual

FIGURE 3

INTEGERS 89

children. Some suggestions for a few pos-sible changes to match the learning stylesof your students are indicated in Figure 3.

EDITORIAL COMMENT:Many variations of this activity are possible. A beanbag gameformat could be used in a manner similar to that suggested for the shuffleboard activities.The game arrangement will actively ivolve the children and provide an interesting formatfor skill development. If the students can be encouraged to record their scores on a numberline and write equations to represent the operations, they will gain additional insight intothe operations with directed numbers.

Number line: Versatility

ESTHER MILNETucson Public Schools, Tucson. Arizona

How can you show a number line at in-numerable positions? How can you, at thesame time, show that the positive integersmay be on either side of the origin?there a simple device you can constructto illustrate all possibilities?

Such was the problem confronting a.in-service group in our school district.While writing a unit on intuitive ideas foran introduction to integers, the groupwanted to indicate all possible varieties of

a number line. The use of a stick about ameter long, a dozen or so thumb tacks,and a like number of small tags, each witha hole punched off - 'enter (like key tags)?rovided the solution. The tags, markedwith names for positive integers, zero, andnegative integers in order of value, aretacked on the stick so that they rotatefreely. As the stick is turned to any posi-tion the numerals appear right side up.

EDITORIAL CONINIENT.The benefits to be derived from this teaching aid could be increasedby having each student make his own model. A variation of thisdevice could be made fromclothesline with the numeral cards tied to the clothesline at intervals with string.

90

A teaching aid for signed numbers

EDNA M. PRATT

Chairman, Mathematics Department, Monmouth Regional High School,New Shrewsbury, New Jersey

Each year the time comes when it isnecessary to illustrate the multiplication ofsigned numbers. There are many ap-proaches to helping the student develop in-sight and skill. Some of the more commonare multiplying two negative numbers us-ing the number line,l* an inductive ap-proach,2 the modern mathematical proofapproach,3 the statement approach,4 or themodel approach,3 but some young mathe-maticians still have a hard time under-standing these ideas. The following is avery effective device which may be usedto help students discover and then reinforcethe operation of multiplication of signednumbers.

Each student is given a pack of twenty3-inch-by-5-inch -plain filing cards whichhave been cut in half so that they becometwo packs of 3-inch-by-21/2-inch cards.On the first card of the first pack, thestudent proceeds to draw a picture ofa road and a small car at the top leftof the card. On the second card, thesame road is illustrated but the car isadvanced one-eighth inch to the right; onthe third card the car is advanced anotherone-eighth inch to the right. The process iscarried on until the last card in the packshows the car at the right hand side of the

"Footnote references are given at the end of the article.

91

card or at the end of the road. Thus PalkNo. 1 shows the car on the road going for-ward, step by step.

A similar procedure is followed with thesecond pack of cards, except that the firstcard shows the car to the extreme right andeach succeeding card in turn is drawn sothat the car is one-eighth inch to the left.The last card shows the car to the extremeleft. Pack No. 2, therefore, shows the cargoing backward, step by step.

First card of Pack No. 1or last card of Pack No. 2

t=a

Bound with glue and tape

Last card of Pack No. 1or first card of Pack No. 2

FIGURE 1

Figure 1 shows the first card of PackNo. 1 or the last card of Pack No. 2, andthe last card of Pack No. 1 or the first cardof Pack No. 2.

Each pack of cards is bound together atthe bottom with glue and tape so that it isconvenient to spread the top of the packwith the thumb and see each card in turn.

The cards are used to illustrate the multi-plication of signed numbers in the follow-ing manner:


1. To illustrate (+)(+) = (+), PackNo. 1 is used. The cards (one by one)show the car going along the road in a for-ward direction (+). If the cards areflipped forward (+), the result is that thecar appears to go in a forward direction(+).

2. Again using Pack No. 1, (+)( ) =( ) may be illustrated. The cards showthe car going along the road in a forwarddirection (+). If the cards are flippedbackward ( ), the result is that the carappears to go in a backward direction ( ).

3. To illustrate ( ) (+) = ( ), PackNo. 2 is used. The cards (one by one)show the car going in a backward direction( ). If the cards are flipped forward (+),the result is that the car appears to gobackward ( ).

4. Again using Pack No. 2, ( )( )= (+) may be illustrated. The cards showthe car going along the road in a back-ward direction ( ). If the cards areflipped backward ( ), the result is thatthe car appears to go in a forward direc-tion (+). By regulating the spacing of thecar and the rate of flipping of the cards, thequantity values may be demonstrated.

After carrying out the above process,each student is given another set of cardsand allowed to demonstrate the multiplica-tion of signed numbers in his own way.Some of the motions illustrated are as fol-lows:

1. A boy swinging upward (+ ) anddownward ( ).

2. Piling up rocks (+) and removingrocks ( ).

3. Notes on a musical scale: up (+)and down ( ).

4. Climbing a ladder (+) and descend-ing a ladder ( ).

5. Ship sailing forward (+) and sailingbackward ( ).

6. Airplane flying eastward (+) andwestward ( ).

7. Kite going up (+) and a kite beingpulled in ( ).

8. Clock going in a clockwise direction(+) and in a counterclockwise direction()

9. Pail being filled with sand (+) andemptied ( ).

10. MONMOUTH REGIONAL HIGH .spelled forward (+) and backward ( ).

There seemed to be no end to the imagi-nation of the students. This approach re-inforced the traditional approaches and inaddition was enjoyed.

References

1. Adam Pawlowski, "Multiplication ofIntegers," THE ARITHMETIC TEACHER,XII (January 1965), 64.

2. William G. Annett, "Products ofSigned Nambers, an Inductive Approach,"THE MATHEMATICS TEACHER, LIV (March1961), 169-70.

3. John Rechzek, "The Product of TwoNegative Numbers," New Jersey Mathe-matics Teacher, XIV (October 1961), 26-27.

4. Joseph P. Marra, "Minus Times aMinus," New Jersey Mathematics Teacher,XVIII (March 1961), 10.

5. National Council of Teachers ofMathematics, Eighteenth Yearbook, Multi-sensory Aids in the Teaching of Mathe-matics (New York: Columbia University,1945), pp. 139-45 and 376-77. (EDITOR'SNOTE.Teachers using this device mayfind it necessary to establish a forward andbackward direction on the card. Not allstudents will understand readily that as thecar moves to the left this is called "back-wards" and is represented as 1, 2, 3,etc.CHARLOTTE W. JUNGE. )

Rational Numbers

Exploration of the system of rational numbers begins early in the ele-mentary school program. Children in the primary grades work with "partsof wholes" and "parts of sets." The fraction symbol is introduced early. Inlater grades students learn to work with different names for the same frac-tional number. The notations include fractions, ratios, decimals, and per-cents, Whole number operations are reinterpreted in the system of rationalnumbers, and computational algorithms for the different notations areformulated.

The title of Jacobs's article echoes the theme of this collectionif chil-dren are given the opportunity to manipulate concrete materials, they willmore likely be able to reason with the abstractions these materials repre-sent.lacobs describes how the familiar egg carton can be used to generateequivalent fractions and to find products of rational numbers.

The game of Bingo, as developed by Cook, provides a medium for work-ing with equivalent fractions. The game is more than simple practice inrecognizing equivalent fractions, since the children manipulate sets of ob-jects to discover a rule for converting a fraction to one of its equivalentforms.

The article by Drizigacker and the one by Rode describe aids that areuseful in showing the relative sizes of rational numbers. In Drizigacker'sgame, the emphasis is on putting rational numbers in order by using proper-ties of the fraction symbol. Rode's activity involves matching fraction piecesto fraction symbols and deciding whether the piece is too small or too largeto "make a whole."

Hauck provides a three-dimensional model for visualizing percentagesand their relationship to common fractions and decimals. He shows how touse the model to explain how percents function in problem situations. Theshort article that follows describes a two-dimensional percentage board sug-gested by Bailey.

Lyvers provides directions for constructing a fraction circle kit that isused to make rational number operations more meaningful to children. Thefraction strips described by Jacobson can be used for similar purposes. Notethat these strips may also be used with children other than the special edu-cation groups suggested by the title.

Division of rational numbers seems to be one of the most difficult opera-tions to make understandable to children. Many students do not alwayscomprehend rationalizations that involve transformations of complex frac-tions. McMeen suggests using a set of fraction wheels that have been care-fully constructed to demonstrate certain key division problems. Perceivingthe division relationship in a manipulative setting will open the door forgreater comprehension at a more abstract level.

The last article in the section describes a game used to review skills incomputing with fractions. Zytkowski's game board has the added advan-tage of showing students that substituting different names for a given rationalnumber in a problem does not change the final answer to that problem.

93

If the hands can do itthe head can follow

ISRAEL JACOBS

A teacher in the mathematics department at Aiming S. PrallJunior High School in Staten Island, New York, Israel Jacobsreports here on a mathematics laboratory project with seventh-grade students.

Maureen looked down at the egg cartons.Tentatively she separated the singles, thenshe took the three sets of four out of thegroup and put them aside. Hesitatingly shecontinued the process of separating the car-tons until the whole set of egg cartonswas apart. She pursed her lips andstared down at the disassembled cartons. Apencil and paper lay nearby to record com-ments. She reread the activity card:

Materials: One set of nested egg cartons (onetwelve, two sixes, three fours, six twos, andtwelve singles).Report any pattern you find.Comment: What do you think of all this?

While Maureen was absorbed in her ac-tivity, the other three girls in the group keptup a running commentary:

"These hold only six eggs.""And two of these make one whole.""What about these little ones?""How many are there of them?""One, two . . . how many of them have

you got? Eleven . . . no, twelve of them.""Twelve makes one whole, too!"Maureen sucked the eraser of her pencil

a moment and then wrote: "They all fit

into each other." She paused and thenadded, "It has to do with fractions."

94

The pencil and paper were put aside. Thefour girls started to rearrange the pieces,fitting them into various groupings, nest-ing them into each other to make onewhole.

Maureen and her.group were involved inan experimental project at IntermediateSchool 27 (Anning S. Prall) on StatenIsland, N.Y. It was a mathematics labora-tory program that started on May 21, 1971and continued every Friday thereafter untilthe end of June. Some one hundred twentychildren in the seventh grade were involved.Within each class, the children were dividedinto teams of four or five of their ownchooSing. The teams were seated arounddouble desks that had been turned so thatthe youngsters could face each other. Ex-cept for two pilot programs, none of thesechildren had ever participated in such anactivity before.

A mathematics laboratory approach isnot new. In England, the Nuffield Mathe-matics Project was set up in 1964. TheMadison Project promotes an individualizedlaboratory approach. The ARITHMETICTEACHER has included reports of a numberof mathematics laboratory projects. Thepurpose of a mathematics laboratory is to

involve the hands of the youngstersanapproach that will, we hope, lead to theunderstanding of concepts.

The program at Prall was not funded.All games were constructed by the authorwith the help of the youngsters. Many ofthe activities were culled from the mathe-matics laboratory course for teachers con-ducted by Professor Judith Jacobs atRichmond College, Staten Island, N.Y.Suggestions for other activities came fromthe pages of the ARITHMETIC TEACHER andfrom various textbooks.

In the mathematics laboratory, the roleof the teacher was to hand out preparedactivity cards that provided instructions forthe pupils to follow and to give limitedassistance if it was needed. At first theteacher's function was to walk around,smile, and to each anguished questionanswer, "i don't know. What does the cardsay?" If a group was completely stalled,then the attitude of the teacher changed. Aleading question from the teacher usuallyprecipitated meaningful action.

. Maureen and her group were playingwith egg cartons that the children had col-lected and brought in. Twenty groups wereinvolved in this activity. A majority of thegroups did not relate this game to fractionsimmediately or discover any kind of rela-tionships among the various pieces beyonddiscovering the parts of twelve. One groupturned the cartons into a tossing game.They used the 12- section piece as a recep-tacle and tossed the singles at it. The win-ner was the one who scored the highestnumber of "ins." Other groups used thecartons as erector sets to build various.

-shapes.

RATIONAL NUMBERS 95

The second activity with egg cartons in-cluded specific questions:

How does the smallest carton relate to the largestcarton?How does the next smallest carton relate to thelargest carton?What other relationships do you find?

The response to this activity was positive.Answers to specific questions were wellwithin the students' experiences. It was onthis activity that a group of advanced stu-dents came up with some interesting re-sults.

AA

Two teams in the advancedmoved their desks together andgroup of eight. Their work area wascovered with egg cartons, bottle caps, andsheets of scribbled paper. A debate was inprogress. Sue placed bottle caps in the eggreceptacles, counted them, made notationson a sheet of paper, and handed the paperover to Glenn. Seven heads bent over asthey checked it. They looked up and Glennsaid, "It works!"

class hadformed a

MMI1110.

ti

"What works?" I asked."Our computer. A computer to multiply

fractions!" said Sue.

96 1111 TEACHER-MADE AIDS

"How does it operate?"A single 12 egg carton was set out and

Sue explained. "Suppose you want to showa fraction of 5/6 on the computer. Place abottle cap in ten of the openings. . . ."

I held up a hand. "Let's backtrack justa little bit. Ten? Why ten?"

She sorted through the sheets on thedesk and pulled out a diagram. (See figure1.)

I I I I I I

6 6 6 6 6 6

Fig. I

"Each double up and down row. . .""Vertical.""Each vertical row is 1/6 of the egg

carton. If we want 5/6," she explained,"we place a large bottle cap in ten of theholes. Now our computer reads 5/6."

"But," I objected, "you have ten capsin the computer. How does that come out5/6?"

"Look," said Glenn, pointing to anotherdiagram. (See figure 2.)

I I I I I I

66 6 6 6 6

00O000 0 0 0 O

Fig. 2

"This computer is a 12-hole computer.So we double the 5/6. That's why thiscomputer shows ten bottle caps in a 12-hole egg carton. Suppose we want to multi-ply 1/2 times 5/6. Now we place smallbottle caps over the large bottle caps, butonly on half of them like this." (See figure3.)

"Hmm. How would you show 1/7 onyour computer?"

"Easy!" they chorused.A 2-section egg carton was added to

OO

00 00Fig. 3

the original set and figure 4 was the result-ing diagram.

I I I I I I 1

7 7 7 7 7 7 7

Fig. 4

"What would 1/2 of 3/7 be?" I asked."Let one show it now," said Sue, "I

found it first.""Go right ahead," said Glenn, "but I

figured out the 14-hole computer to startwith."

Sue went on to explain. "Since all theholes in the egg cartons total fourteen wecall this a 14-hole computer. Now for 3/7,all I have to do is put six large caps in theopenings, because each vertical row has tobe filled completely." She went on, "For1/2 of 3/7, all we do is put three smallcaps over the large ones, and the result-3/14!" (See figure 5.)III III I

7 7 7 7 7

O O O0 0

Fig. 5

7

"That's all very nice," I conceded. "Theegg cartons are conveniently set up in setsof pairs. Suppose you want your computerto show 1/3 of 3/7?" With that I walkedaway.

In a little while, Sue and Glenn signaledfrantically. The egg cartons had been ar-ranged as in the diagram in figure 6.

"Now," explained Glenn triumphantly,"we have a 21-hole computer."

3

7 7 7 7 7 7

OO OO

30 O3000

Fig. 6

7

"Why 21?""Because," explained Sue, "if you want

1/3 of 3/7 we need a "whole" into whichboth 3 and 7 can go evenly. And that is21."

Glenn interposed. "In this 21-hole com-puter, 3/7 means nine large caps in nineholes. To take 1/3 of 3/7, we put threesmall caps over the three large ones, andthe computer now reads 3/21."

"Very good! Write it up." I walkedaway.

It is possible that either Sue or Glennmay have come across fraction diagramsbefore and applied this knowledge to theegg cartons. Whether this exercise in

fraction multiplication was a discovery orjust reinforcement, it was a worthwhileexperience for the whole group.

From the front of the room, left, a boywas screaming, "I won! I won!"

He was in a group of four playing thespinner fraction game. Two of this samegroup had been active in constructing thegame. Round cardboards, eight and one-half inches in diameter and in variouscolors, had been cut up into pie slices of

RATIONAL NUMBERS 97

1/2, 1/4, 1/8, 1/16, and 1/32. (Seefigure 7.) In cutting up these slices, theyoungsters quickly learned that it tooktwo 1/32-pie slices to make one 1/16-pieslice.

(Red)

(Green)

011_

(Yel low) 32 Black playing board

Fig. 7

(Blue)

16(White)

The spinner was made in the schoolshop. The sections on the spinner board


Spinner

Fig. 8

itself are labeled 1/2, 1/4, 1/8, 1/16, asshown in figure 8.

The wooden arrow, pegged loosely tothe board, is spun by each player in turn.Each player has to select from the generalpile of pie slices a piece corresponding tothe fraction the arrow on the spinnerpoints to, and he places the piece on hisplaying board (the black one). To win,all fractions on the playing board musttotal up to one whole. If the spinner selectsa fraction that does not fit on his board,then the player must return an equalamount to the general pile.

Figure 9 shows the playing boards offour players (A, B, C, and D) in progress.On his next turn, C gets 1/4 on the

Player A

Player C

Fig. 9

32 Player B

Player D

spinner. Since this cannot be used on C'sboard, he has to return a 1/4 slice to thegeneral pile so that C's board then lookslike figure 10.

C's Board after he lost the "*" sliceFig. 10

After he lost the 1/4-pie slice, C had justtwo slices left, 1/2 and 1/16.

Suppose C had had no 1/4-slice on hisboard. He would have had to change his1/2 to two 1/4s and give up one of them.Such changes were checked carefully byeach player. The students learned to addand subtract these fractions without theuse of pencil and paper.

Spinner fractions was the most populargame, and all the youngsters were anxiousto play with it. It was the spinner thatintrigued them. At one time when thespinner was out of action for a while, cardswith fractions marked on them were sub-stituted. The cards were not as much funfor the students as the spinner.

The second activity with the spinnerfraction game added something new. Itwas not enough to tabulate the number ofwins and the ways in which a game couldbe won. Now the goal was to determinewhich fraction on the board came up mostfrequently. The students were to try 100,500, and 1000 spins at a time.

One group in each class worked on thisactivity. All of the groups that were in-volved concluded that the "winningest"fraction was 1/2. One boy simply wrote"The more we spin the more it's 1/2."No one group came up with any specificreason why this was so, although one ofthe boys did comment that he could makea better shaped arrow!

"Does the shape of the arrow have any-thing to do with 1/2 corning up mostoften?" I asked.

He did not know.Another activity card read as follows:

SCALE MEASUREMENTMaterials: road map, string, 12-inch ruler,Use string to follow along the main roads be-tween selected cities.How many miles are there between the selectedcities?

Interest in this game varied. Somegroups were absorbed all period. Theywere the ones who had worked out themeaning of the scale at the bottom cornerof the map. Other groups, those who couldnot figure out the meaning of the mileagescale, tired of this game very quickly andwent to something else. Still others justsat and gossiped until the scale idea wasexplained.

Another activity card gave the followinginstructions:

MEASUREMENT IN INCHES, FEET.AND YARDS

Materials: piece of string. 12 -inch rule.Measure as many objects in this room as possi-ble.Record results in feet and yard..:. Compare re-sults with other groups for accuracy.

The group that showed the most in-terest in this activity (and had the mostfun) was in a class of low achievers. Theyhad a wonderful time measuring eachother's heights, including the height of theteacher. They measured desks, chairs, thelength of the floor, and even the lengthof a girl's blond hair. They did discoverthat different measurers came up with dif-ferent results. for the same object meas-ured. Errors were quickly recognized. Theboy who translated 72 inches into 5 feetwas corrected by everyone in his group.Generally this activity evoked such en-thusiasm as to involve everyone in thegroup.

From the youngsters' point of view, themathematics laboratory was a huge suc-cess. There was always laughter, and attimes, the joyous noise level was so high

RATIONAL NUMBERS 99

that the place did not sound like a class-room at all. No one was cracking theobvious learning whip. There were notests to take, no homework to worry about,no notes to copy. It was a period of free-dom. On Fridays the students sought theauthor out on the stairways, in the hall-ways, and in the lunchroom; or theystopped by his room to ask only one ques-tion, "Are we having math lab today?"

Two obvious questions can be raised.What did this experiment prove? Is a

mathematics laboratory a worthwhile pro-gram of learning on an extended basis?

The experiment was implementedtoward the end of the school year, and itwas of short duration. Generally the ac-tivities were haphazard; the pupils couldtry one game or another, visit, or gossip.Learning was casual, self-oriented, anduntested; but it was there, whether we callit discovery or reinforcement. Wheneveran activity card was deliberately vague andallowed the youngsters a choice of direc-tion, the usual response at first was a deadstop. As the program progressed, the chil-dren caught on to the new idea: they wereto proceed on their own.

The strict regimen in mathematics is

all too familiar. It does not reach all ofthe youngsters. The phrase under achieverwas coined for that group of students whofind mathematics too difficult to compre-hend. A mathematics laboratory could beprogrammed once a week and it could becoordinated with the week's lessons. Itshould make more use of manipulativeactivities than pencil and paper activities.In our experimental mathematics lab-oratory program, the youngsters who phys-ically worked with inches, feet, millimeters,and centimeters learned about measure-ment more quickly (and had a great dealof fun with it) than those who learnedwith pencil and paper exercises and tablememorization. On an extended basis,

learning with the hands can be fun, andit can become more meaningful to thestudent.

The role of the teacher has to be more


than to just stand aside and hope thatJohnny will make a great discovery or get:started on his own. Usually Johnny does.neither. It is in a laboratory setting, withproper guidance, that Johnny can learn toget started on his own and perhaps go ina direction of his own choosing and stillhave fun doing it. But he must be placed insuch a position again and again before helearns that he is allowed to wander off onhis own and that this wandering has value.

During the final Friday program, some ofthe youngsters brought in wood dice they

had made in shop."Why can't we use these in the games?""You can. How true are they?""Ti cy re pretty good. I made them.'.'"1-16»v would you test them out?""Use them, f guess.""Would you like to try it out and bring

in a report?""How many times would I have to roll

them?""100, 500, maybe 1,000 times.""Boy! It'll take all summer!""Not as long as that. Is it worth a try?""O.K., 1,000 times!"

EDITORIAL. CONINILNT,It should be apparent that the egg carton pieces may be used toidentify equivalent fractions. Therefore, we can also use the egg cartons- for adding andsubtracting rational numbers. For example, suppose we wish to add 2/3 and 1/4. First wetry to find a number of pieces from which we can easily make both thirds and fourths (thatis, we find a common denominator). This turns out to be twelve.

thirds fourths

To find the sum we consider:

2/3 + 1/48/12 + 3/12

11/12

Fraction bingoNANCY COOKrile Evergreen Schott/, Seattle, Washington

To almost all fourth graders, the conceptof one-half is meaningful and well under-stood. Many. fourth graders also know thatto-fourths names the same fractional num-ber, but why it does is frequently not un-derstood. To help my class of fourth, fifth,and sixth graders to better understand andto discover for theniselves how to findother names for a fractional number, Imade the game that they named "FractionBingo." .

Playing cards were made of lightweightposter board, using many bright colors.These cards consisted of three rows ofcommon fractions with three fractions ina row, the center space being a free space(sec fig. I ). Various names were usedfor the more frequently used fractionalnumbers, such as 1/2, 2/4, and 3/6. Frac-tions that could also be written as decimalfractions, such as 5/10, 25/100, and 2/10,were also used.

The game is played like Bingo, The"caller" picks a card from the "caller's

PLAYING CARD

Freespa 6

15 ; (.0 9

25 . 2 10i 100 I 10 I 12

FIG. I. Playing Card.

pile," and anyone who has on his cardsome name for that numeral covers it witha disc. "Three-in-a-row" wins.

Originally the playing cards were fiveby five, with a free space in the center;but to search a card of twenty-four frac-tions for a particular value proved dis-couraging to the novice as it took too longand the game moved too slowly. Whenthe game was changed to the three-by-three cards described above, even the be-ginner could manage the search amongeight fractions. The game now movedquickly and the students did not lose in-terest.

There are many ways you could startto play this game. 1 chose to use as "call-ers" the simple fractions and made smallcards with these names and illustrationsto be used as the "caller's pile." (See fig.2.)

"CALLERS'

223

I

Fic. 2. "Callers."

When I was explaining the game to theclass, someone asked, "What do you mean'another name' for the same fractionalnumber?" I put a pile of plastic discs onthe table and, taking two discs, said thatthese two discs could be considered to bea whole, a unitsay, one whole candy bar(an easy concept for the pupils since manycandy bars now come as doubles, two smallcandy bars in one big wrapper). I thenasked how to divide the candy bar intotwo equally sized (congruent) pieces. Theyall knew how it could be done. What dowe call one of the two congruent pieces?One-half, of course; and we reviewed thefraction 1/2 and discussed what the dc-

101


nominator and numerator represent. Then,taking three discs. I called the three discsmy whole candy bar. How can it he dividedinto two congruent pieces? The class agreedthat one disc could be broken into twocongruent pieces. I explained that I didn'twant to ruin (break, cut, or tear) any ofthe discs and asked if there were anotherway? They agreed that there wasn't, butsomeone remarked that we could still writedown another name for 1/2: 11/2/3. Ithen took four discs and called the fourmy candy bar. Everyone agreed that twoof the four discs would represent one-halfof the new candy bar. I asked what one-fourth would be. One disc. What wouldtwo-fourths be? Two discs. Hadn't we justgiven another name for two of the fourdiscs? Yes, one-half. And they agreed thatone-half and two-fourths were two differentnames for the same value. Now they wereready to play Fraction Bingo.

We started the game. I called one-fourth,and the search was on. Everyone was mani-pulating discs in an effort to find othernames for one-fourth, but not everyonewas using the discs in the same way. Onechild took four discs, lined them in a row,and separated one somewhat from theother three. He said the one disc was one-fourth of the whole. In another row di-rectly under the first row he put four morediscs, making eight discs. He now hadtwo discs separated from the other six andsaid the two discs represented two-eighths

of the whole. He continued in this fashion,each time checking his card for the newname he found. Another child took fourdiscs, separating one a little from the others.He first let each disc stand for one (leadingto one-fourth). Without taking any morediscs he let each of the four discs standfor two (leading to two-eighths), and hecontinued in this way, checking his cardfor the new names.

By the end of the first day two studentshad discovered that multiplying the nu-merator and denominator by the samecounting number led to another name forthe original rational number. But not every-body was that quick. By thc next day afew students had thcir "favorite cards"memorized and they spent their extra timehelping friends. On the third day one boysaid to me, "I know that eight twenty-fourths is another name for one-third.figured it out in bed last night!"

By the fourth day everyone had somesystem other than discs alone to find othernames for the rational numbers, systemsthat they themselves had worked out, sys-tems that were meaningful to them. Manyhad discovered "the rule" of multiplyingthe numerator and denominator by thesame counting number, and we spent sometime finding out why that rule worked.Many already knew why it worked justfrom the work done in discovering it, andthe rest had laid thc groundwork neces-sary for complete understanding of it.

EDITORIAL COMMENT.The bingo type of game can be used in many areas of the mathe-matics program, especially where drill and practice are required. Care must be taken, how-ever, that individual games do not become too long and drawn nut.

The cards can be used to review basic facts. Sums and products, for example, can bewritten on the spaces. The caller would give the two addends or factors, and the playerswould cover the appropriate sums or products. For variation, the caller could name thesum with the addition problem written in the space on the card.

The bingo cards may also be used for identifying numerals in other bases or in ancientnumeration systems. Pictures of geometric shapes may be drwn on the card with thecaller giving the name of the figure.

FRIO, or FRactions In Order

ROWENA DRIZIGACKER, Scottsdale, Arizona

FRIO is a game that will help studentsunderstand the relative value of commonfractions. It can be played and enjoyedby third graders as well as the upper-grade students and even by their parents.It can be made as simple or as difficultas desired.

For a learning deck you will needtwenty-six cards. On these write the nu-merals that represent the halves, thirds,fourths, fifths, eighths, and tenths. Withthis small deck two players may play. Latermore fractions may be added, and moreplayers may play at once. Also, when thegame is being learned, each player is givenfive cards. Later each may be given up toten cards, depending on the number playingand the number of cards being used in thedeck.

The cards are placed face up in front ofthe player in the order in Which they aredealt to him. Each player in turn drawsfrom the pile that is left face down, or hemay take the last card to be discarded. Ifthis new card will help him to get his cardsin order from low value to high, he ex-changes it for one of his; if not, he dis-cards it.

The object of the game is to be the firstone to arrange his cards in order beginning

with the smallest value. When a player hasdone this, he yells, "Frio!"

Now for a sample hand. Suppose aplayer is dealt the following cards:

5

2

3

7

8 10

On his first turn he draws

places the 2

3with it.

3

4

2

8and re-

When his next turn comes, there is a 1

8

on the discard pile, but he decides to draw

Next he draws

with it.

5

2

8

24

and replaces the

2

4 10

3

4

78

Now all he needs is a card to replace the

1

10that is between

4and 3

4. As soon

as he has done this, he has a FRIO andwins the game. If he draws a card that willnot help, he discards it.

Score may be kept in various ways. Arecord may be kept of the number of wins.(This is the simplest way and is suggested

103


for beginners.) A win may count 50 points,and play may continue until one playerreaches 500 or to any desired goal. Bonuspoints may be given if all the fractions arein "lowest terms" (basic fractions) or ifall have the same denominator.

Students and teachers will think of manyvariations, and these will add to the valueof the game. It is surprising how muchthis game can contribute to the students'understanding of fractions.

To prepare the children to play FRIO,it is helpful to practice arranging the frac-tions in order. Each pupil should make hisown list and perhaps refer to it while he islearning. Such a list would include

1 1 1 2 1 2 3 1 3 2 4 1 2- or , or n r or Or

108 5 10 4 8 10 3 8 5 10 2 4

4 5 6 3 5 2 7 6 3 8 4 7

or -, - or -, or -, -, and8 10 10 5 8 3 10 8 4 10 5 8

9

10

According to the background and abilityof the students, different methods of deter-mining the above order may be used. Frac-tions may be represented as parts of circlesor pictured on a number line and thencompared to establish relative size. Com-mon denominators may be :7nund or, forolder players, decimal fraction equivalentsmay be considered. The game is a goodincentive for learning the decimal equiva-lents.

EDITORIAL COMMENT. Many card games may be adapted to mathematical drill and prac-tice. Rummy is easily converted to drill on basic facts. Ten or more sets of four cards areprepared with different names for the same number on each of the four cards (for example:12, 7 + 5, 8 + 4, 9 + 3). To play, students must match corresponding number names.

Old Maid can be adapted to drill with fraction equivalences. Pairs of cards namingequivalent fractions are used. One "monster" card is inserted in the deck. Students playout the game by matching pairs of equivalent fractions.

"Crazy eights" can be converted to "crazy ones" for fraction recognition. Suits can bemade from fraction denominators. In the play of the cards, the suit can be changed byplaying a "crazy one" card.

Make a whole a gameusing simple fractions

JOANN RODEUniversity of West Florida, Pensacola, Florida

The game called "Make a Whole" helpsdevelop the concept of fractional num-bers by using concrete examples; it givesreinforcing experience with the use of frac-tions and their equivalents. It is suggestedfor use at third-grade level and above, de-.pending on the achievement of the stu-dents.

The material required for the game ispictured in figure 1 and described morecompletely in a later section. The sectionsare made in four different colors, equallydistributed; that is, of 8 sections represent-ing 1,4, 2 could be red, 2 blue, 2 green,and 2 yellow. The area for the "Whole"is also prepared in four different colors.The faces of the die are marked with a

105

star and the following fractions: 1/2, 1/2,

1/4, %, and 1/2.

Rules of play

The object of the game is to use thefractional sections to construct a multi-colored disk, the "Whole," in the blackmat frame. The Whole may be constructedwith any of the differently colored sectionsso long as they fit together to be equivalentto a whole.

Two to eight children may play thegame in individual competition (in whichcase the color of the area for the Wholemakes no difference), or they may playon color teams as determined by the colorof that area.


Multicolored Sections Die

Fig. 1

The fractional sections are placed in abox, from which they are to be selectedby the children as they roll the die, inclockwise order.

To determine who goes first, each childrolls the die seen in figure 1. The childrolling the star or the fraction with thegreatest value goes first.

At each turn the child must decidewhether he can use one of the sectionsrepresented by the fraction showing ontop of the die.

If the player cannot use the section, hepasses:

If he decides to take the section and canuse it, he places the section in his mat.(If the section taken completes the Wholefor that player, then the game is over andthat player has won. If it does not, theroll of the die passes to the next player.)

If he decides to take the section andcannot use it, he loses his next turn.

If he passes up a section he could haveused, he loses his next turn.

If the star is rolled, the player maychoose any section he can use.

The winner is the first to construct amulticolored Whole.

The game may be played in variousother ways, according to rules designed bythe teacher. For example, a child mightbe asked to construct five Wholes with nomore than one of a kind of fractional sec-tion (IA, %, etc.) to go into each. For

advanced students, the numeral identifica-tion may be omitted from the sections.

The materials may also be used by theteacher to show equivalent fractions inclassroom demonstrations and by the indi-vidual student in working with equivalencesat his desk.

Construction of game materials

A complete list of pieces follows, thecolor of the mat disks and sections beingequally divided among whatever four col-ors are used.

Mats, 81/2 sections,% sections, 121/4 sections, 161/6 sections, 24% sections, 32Die, 1

The material used for the black matframe and the fractional sections is inex-pensive tagboard (poster board) in differ-ent colors. Because the Wholes constructedby the students are to be multicolored andbecause the fractional sections come inequal numbers of the different colors, thecolors give no clues to value.

The mats were made first by cutting 8eight-inch squares from black tagboard.Disks six inches in diameter were cutfrom the center of these squares, and theframes were then glued to other eight-

inch squares, 2 of each of the four colorschosen.

Five disks in each of the four colorswere then constructed, with a diameter ofsix inches. A compass was used for ac-curacy. The disks were then divided, withthe aid of a protractor, into sections cor-responding to %, %, 1/4, %, and Va. Thecentral-angle measurements for the fivesections are 180°, 120°, 90°, 60°, and45°, respectively. Depending on the levelof the children, the sections may or maynot be marked with the fractions to whichthey correspond, as seen in figure 1.

RATIONAL NUMBERS 107

The die was made from a cube cut at alumber yard so that each edge had ameasurement of three-fourths of an inch.The five fractions were painted, one to aface on each of five faces of the cube.A star was placed on the sixth face. Thecube was sprayed with a clear varnish toprevent smudging.

EDITOR'S NOTE. Here is a device that canbe used for independent learning activity. Idare say the game could be varied by theconstruction of rectangular sections and mats,so that children would view fractional partsin a broader spectrum. CHARLOTTE W. JUNGE.

EDITORIAL COMMENT.The game may be extended to include shapes other than circles.The game may also be played on a number line with the draw pile being pieces of paperwith fraction names. A student draws a piece from the pile and marks on his number line asegment representing that fraction.

0 I

Laminated number line Grease pencil

Draw pile

Concrete Materials for Teaching PercentageELDON HAUCK

Seattle, Washington

MANY PROBLEMS AND DIFFICULTIES inelementary school arithmetic can be

resolved through: (1) better instructionalmethods, (2) more instructional materials,(3) better instructional materials, (4) con-crete materials for each basic phase of thesubject, and most important (5) the knowhow in using materials in the classroom.Concrete materials for teaching and learn-ing should have: (1) the property of beinghandled in such a way as to conduct alearning situation to the handler or ob-server, (2) the inherent structure to pre-sent to the learner a sequential patternleading to the abstract, and (3) a specificrelation and application to the 'subject be-ing treated which, at the same time, givesan immediate experience into reality. Con-crete materials' will enable the child toparticipate in discovery and developorganization and insight.

Leading into Percentage

Understanding of problems and abilityto solve problems using common anddecimal fractions provide the pupil with abackground which will facilitate his under-standing of percentage. A thorough reviewand reteaehing of these fundamentals isdesirable prior to introducing percentage.Stories about the inventionnf per cent andits use will arouse interest. Newspaper ad-vertisements showing per cents may bebrought to school and displayed and dis-cussed, a store or bank may be visited, andpupils may talk with their parents andothers about the use of percentage. Theseactivities should pave the way to a readi-ness for a study of percentage. All pupilswho bring things to the classroom shouldhave an opportunity to present and dis-cuss them with the class.

Our goal is an understanding of per cent

together with an ability to solve problems.The factors which lead to an understand-ing of the thought processes of solutionsare:

(1) Changing fractions to decimals andper cent.

(2) Changing decimals to fractions andper cent.Changing per cent to decimals andfractions.

(4) Per cent means "hundredths."(5) Of means times ( X).(6) Per cent is a simpler way of working

with fractions.To find a per cent of something isthe same as finding a part.

There are two chief difficulties encoun-tered in the teaching of per cent. These are:(l) the inability to understand the relation-ship of per cent to fractions and decimalsand (2) that 100% means the whole ofanything. A third difficulty is encounteredin that, if one whole thing equals 100%then two whole things (like objects) equal200%, etc.

An understanding of the above listedfactors and a doing away with the difficul-ties encountered may be achieved whollyor in part with the use of the "PercentageBox" as illustrated on page ten.

Figure 1 shows the entire box as beingequal to 1 or 100% or 1.00; Figure 4 (sec-tion A) is equal to or 50% or .5 andfigure 2 (section B) is equal to 4 or 25%or .25.

A combination of A and B equals or75% or .75.

An introductory procedure of explaining.the box and showing the above; with dis-cussion and questions would show a de-.finite relationship between fractions, percent and decimals.

(3)

(7)

108

Remove section C from the percentagebox (hide sections A and B) and point outthat you have something in your hand,section C (figure 3) and that you hold allof this something. Ask what per cent itequals (it is important to keel) the insidesection printing hidden from the class). Inmost cases they will answer; 25%. Repeatthat you have the whole thing and nothingless. They will soon deduce that you have100%. Then show them the printing you


out of hiding; place all together again andgo over the procedure. Point out to them(their questions will call for answers) thatall of anything is equal to 100%, but apart of anything can also be equal to 100%of that part.

Then separate section C from the boxand separate its parts. Holding Cl in yourhand, state that you have a whole thing inyour hand and ask them what per cent itequals. They'll answer, 100%. Place C'2

THE PERCENTAGE Box

Fro. I

The box in entirety is a cube. Its di-mensions are flexible. It is held togetherby dowels in section A fitting into theholes of sections B and C.

A

SO 70.5

es)laelde

FIG. 4

Section C is held together in the samemanner: Cl having the dowel and C4having the hole with 02 and C3 eachhaving a dowel on one side and a holeon the opposite side.

have been hiding from them. Let themdecide what each part and combinationsof two or more parts equal in per cent,decimals or fractions. e.g., C2 equals whatper cent? What fraction, what decimal?Cl and C2 equal what per cent? Whatdecimal?, what fraction?, etc.

Now bring the other sections, A and B,

Ma. 2 'FIG. 3

beside Cl and say, "If Cl equals 100%,what do both of these, Cl and C2, equal?"If they do not come up with the correct200%, repeat the process. They will! Thenadd C3 and C4 to the group for an even-tual 400%.. Work hack through the proc-esses and let the students demonstrateand .ask questions about the percentage.box.

The second and third difficulties men-tioned above have now been surmounted.Repetition of the procedure will help thegreat majority of pupil's through theseobstacles.

To change fractions to decimals and percent show the whole box. and ask, "Howmany hundredths in one whole thing ?""To find how many hundredths in One-halfa whole thing, what would you do?" "lfyou wanted to find how many hundredthsare in one-fourth of a whole thing whatwould you do?" "Per cent is just anothername for hundredths. If you know how.many hundredths are in something; canyou tell what per cent it equals ?" etc.


To change decimals to fractions and percent, "You know how many hundredths arein a whole thing, how would you find whatpart of a whole thing fifty-hundredthsequal?" "Five-tenths?" "HoW many hun-dredths do five-tenths equal?" "Howwouldyou find what part of a whole thingtwenty-five hundredths equals?" "Fiftyper cent?" "Twenty-five per cent?" etc.

To change per cent to decimals and frac-tions, "What is the difference between100% and 1.00, 25% and .25, 50% and.5?" "What would you have to do to the percent to make it like the decimals ?" "Youknow how to change per cent to decimals,can you find the fractions they equal?""These are called equivalents."

The above procedures have broughtabout the fact that per cent is just anotherway of saying hundredths (accompanied byunderstanding).

Ming forth the percentage box once.more. "Let us assume that this box isequal to one hundred cows." "If the wholething is equal to one hundred cows, howmany cows would one-half of it. equal?50%? 25%? 1? 75%? 1?" etc.

To find a per cent of a number raise thesequestions: "Fifty per cent of one hundred.cows equals how many cows?" "25% of 100.COws'.equals how many cows? One hundedper cent of 100 cows equals? Two hundredper cent of 100 cows equals?" etc.

Now show them the process of workingthe problem out on paper by changing theper cent to hundredths (decimal) or a com-mon fraction and multiplying. And laterwhen the above has become known to themajority, give them the following step :

To find .what per cent one number is ofanother"If the whole box equals 100cows and one-half the box equals fiftycows, how would you find what per centfifty cows is of one hundred cows?" "Arefifty cows a part (fraction) of 100 cows?""What part?" "This part equals what percent?"

"Twenty -five cows equals what per centof 100 cows?," etc., etc. If 100 cows equal

one hundred per cent, what per cent would200 cows equal?" "Two hundred cows areequal to what per cent of 100 cows?"etc.

Now show the problem solving proc-esses: "find the part, change this fractionto a decimal, change the decimal to a percent." And later:

To find a number when its per cent of thewhole is known:

"The whole box is equal to one hundredcows. Twenty-five cows are 25% of howmany cows? Seventy-five cows are 75%of how many cows?" etc. "Now let's for-get about the box and the hundred cows!Ten cows are twenty-five per cent of howmany cows? Twenty-five cows are fiftyper cent of how many? Fifty cows are100% of how many cows?" etc.

The cows need not, be used entirely, butthe cow is a good example because a cow,when divided, is no longer a cow but abeef.

And now the problem working processcan be brought in: finding the answer byeither first determining 1 per cent and then100 per cent, or by dividing the statednumbef by its per cent expressed as adecimal.

The above exercises should allow eachstudent the knowledge that he can workthe problems in his head and thereforemost certainly can do them on paper. Thepractice involved will now be more of apart of his makeup than if the methodshad been taught to him by role! His under-standing of all fractions should be morecomplete and his work habits in percent-age should he of the desirable nature.

Money may also be used to illustratefurther the above points. A silver dollar,50 cents, 25 cents, dime, nickel, and apenny. The only drawback is the inabilityto divide the silver dollar. In its favorrests the children's familiarity with money.Once they get the idea from the percentagebox, money can be used to good advan,tage. Whenever a difficulty arises, refer tothe percentage box again and again.

Conclusion

Much has been said about and manymaterials supplied for learning certainparts of arithmetic such as the numbelsystem, fractions, and measures but per-centages has been neglected. There is needfor an interchange of ideas on teachingmethods that teachers have found bene-ficial. These ideas should properly comefrom teachers experienCed in working withchildren. Better teaching will help the in-dividual child growing up with a feeling ofcourage, accomplishment, and under-standing. Lack of understanding is thebasis of fear and the average adult today

has a fear of mathematics brought aboutchiefly through school experiences in thissubject. The average adult holds the sub-ject as one of extreme importance in get-ting along in life and berates his child,hoping the child will learn better than theparent.

Good learning of arithmetic dependsupon good guidance and teaching. A goodteacher needs not only how to deal withchildren but also he needs a good back-ground in the materials he is teaching. Inmany cases this calls for in-service educa-tion of teachers in the Klection and use ofinstructional methods and materials. Somenew policies in requirements in the teltcher,training colleges are also desired.

A PERCENTAGE BOARDMiss Inez Bailey of Rawlins, Wyoming

likes to use a large square divided into 100small squares so that each small squarerepresents 1% and the large square 100%or a whole.. When a per cent is mentioned,-the children count as many squares as arenecessary to represent the given per cent.They then find what part this is of thelarge square. This teaches them per centand fractional equivalents in a meaningfulway. When a part of a per cent is intro-duced, they again study their percentagesquare and see that only a part of one ofthe small squares is meant. This helpschildren to visualize and understand thedifference between a %, for example, and 1or 75%. It is a simple but very helpful de-vide.

Such a percentage board can be made ofmaterials such as oilcloth., muslin, slatedblackboard cloth, or any suitable material.

A Fraction CircleDONALD B. LYVERS

Parley Coburn School, Elmira, New York

OF ALL THE TOPICS presented in the 7thand 8th grade mathematics courses,

those dealing with fractions, decimals,and percentage seem to be the mosttroublesome and the least understood bythe student. Often meaningless rules havebeen committed to memory by the stu-dent, but little comprehension has takenplace. Occasionally there are students in aclass who may add denominators, or insistthat 1/4 times 1/2 does not give the sameresult as 1/2 times 1/4, or become con-fused that 1/4 divided by 1/8 is 2. Somestudents do not know or are not surethat 1/3 is larger than 1/4. These are justa few examples of problems common tomany classrooms. I believe that many ofthese problems have their roots in oneor more of the following:

1. The basic idea of number and thenature of our decimal system hasnever been thoroughly understood.

2. The student has been a victim ofthe ". . invert the 2nd term andmultiply . . . do as I say" type oftraining.

3. The student has not had the oppor-tunity to discover for himself whata fraction really is and some of therelationships which exist among frac-tions.

With this third idea in mind, I havemade up a device which I hope will en-able students to appreciate and under-stand this business of fractions to agreater degree.

A circle is drawn on a piece of paper orcardboarda 6" circle seems to work verynicely. The circle is then divided into 24equal parts. I found that by dividing eachof these parts into thirds would increasethe versatility for certain combinations

of fractions. Since I wanted to use thecircle for halves, quarters, thirds, sixths,and eighths, it was necessary to also di-vide the circle into 32 equal parts tohandle the multiplication of fourths andeighths. The halves, quarters, thirds,sixths, and eighths are labeled with theproper symbol. The 24ths are representedby small whole numbers (numerators), asare the 32nds, but with a different color.The 72nds do not need to be labeled. Agood protractor is needed in order to doan accurate job, preferably one which isgraduated in half-degrees.

Next, five 6" circles are cue frOtri colored6-ply cardboard or construction paper.It is convenient to have a different colorfor each circle to facilitate recognition.Then each circle is cut up into its respec-tive unit fraction parts. This should pro-duce 23 parts which are labeled with theproper fraction symbol on both sides. Itis also necessary to put on each part 1/2,1/3, and 1/4 marks on both sides. Thiscan easily be accomplished by placingthe pieces on the completed fraction circleand marking them accordingly. Thiscompletes the construction of the kit.

Arbitrarily, I chose the "9 o'clock"position on the fraction circle as thestarting point for the operations of addi-tion, multiplication, and division. Thefraction pieces are placed on the circle ina clockwise direction beginning at theradius OA. Suppose we want to add 1/3and 1/4. The proper fraction pieces areselected and placed on the circle. The twoparts added together reach the 14 markon the circle which is 14/24 or 7/12.More than two fractions with unlikedenominators can be added in the samemanner, such as 1/3, 1/4, and 1/6.

Subtraction is performed by locating

112


the minuend on the circle and laying thefraction piece(s) representing the sub-trahend in a counterclockwise directionfrom the radius of the minuend. For ex-ample, if we wish to subtract 3/8 from 5/6,place three of the 1/8 pieces on the circlein a counterclockwise direction beginningat the 5/6 mark. The result we read offthe circle as 11/24.

Division can be accomplished by placingthe number of fraction pieces equal to thedivisor on the circle in a clockwise direc-tion until the fraction represented by thedividend is reached or passed. The numberof pieces needed to reach the dividend(it may be a mixed number) is the quo-

tient. Perhaps the problem is to divide 5/6by 1/3. This requires placing all three 1/3pieces on the circle. We notice that ittakes all of two of the thirds and part ofthe third piece. We see that 5/6 whichis our dividend coincides with the 1/2mark on the third piece so that our an-swer is 2 1/2.

Multiplication is a bit different in thatthe fraction pieces representing either ofthe two fractions to be multiplied canbe placed on the circle, but sometimes oneworks much better than the other. In thecase of 1/6 times 3/4, a piece representing1/6 can be placed on the circle and thenlocating the fraction on the circle which


coincides with the 3/4 mark on the piecewill give us the answer. Also, placingthree 1/4 pieces on the circle and notingthat half of the first piece represents 1/6of the three 1/4 pieces, the fraction coin-ciding with this mark must also give usthe correct solu'

With a little experimentation andimagination, it is possible to handle manycombinations in each of the four opera-tions. One of the pleasant features is thatmost answers can be read off directly,and even when adding and subtractingfractions with unlike denominators, thereis no need for finding a common denom-inator. At most, one may occasionallyhave to reduce to simplest form.

Provisions for the use of fifths may beadded if desired. This could probably bestbe accomplished by making up anotherfraction circle divided into fifths, tenths,20ths and 40ths instead of including themon the circle described above.

It would be ideal for each student in theclass to make his own kit, either in partor entirely. All 24 pieces can easily bemade from colored 6-ply cardboard whichis durable, easily cut, and is readily avail-able at a low cost. The parts can be kept inindividual envelopes which would fit in anotebook.

The device described here may be moreelaborate than necessary for students whoare just beginning the study of fractions.

The use of halves, quarters, and eighthsmay be sufficient. For those who havenot yet learned to use compasses andprotractor, it may be necessary for theteacher to make up mimeograph or dittocopies of the fraction circle and let thestudent fill in the fraction symbols. Alsothe teacher may have to prepare the card-board or construction paper with thecircles and parts already drawn on them,but the student can cut them out. For themore advanced student, the entire kitcan probably be made with only a littleassistance from the teacher. Just havingthe student computing the number ofdegrees required for each of the parts ofthe circle will prove to be a valuablelearning experience for him.

I believe that this project would pro-vide the student with the much needed"discovery" type of learning. The studentis able to manipulate fractions physicallyrather than struggle with meaninglessabstract symbols. He can readily "see"that 1/5 is larger than 1/8, that 1/6 ishalf of 1/3, and that 7/8 divided by 1/6means how many sixths are contained in7/8? It must be emphasized that howmuch a student "discovers" is dependentupon how skillful the teacher is in askingthe kind of questions which will fosterdiscovery. Even though this is not in-tended to be a toy, the "game element"can be used to advantage to stimulateinterest.

Fun with fractions for specialeducation

RUTH S. JACOBSON

Ruth Jacobson is assistant in the Department of Pupil Personnel Services,Brookline Public Schools, Brookline, Massachusetts. She has also been a

demonstration teacher in the Institute on Learning Disorders at BostonUniversity School of Education, where she is now in the doctoral program.

Adevice to aid the learning of additionand subtraction of fractional numbersnamed by like and unlike fractions hasbeen developed in a class for children withspecial learning disabilities at Boston Uni-versity School of Education. It is a manip-ulative device that provides immediatefeedback on the correctness of the child'sanswer to a problem.

This method may be used as preparationor reinforcement for addition and subtrac-tion of fractional numbers when differentdenominators are involved. Although theexample in this article involves halves,fourths, and eighths, the device may also bepresented in other fractional values, suchas fifths and tenths. This device appearsenjoyable to use and, most important, givesthe child a method of attack when workingwith the L difficult concepts of additionand subtraction of fractional numbers.

The aid consists of an eleven-by-eight-inch acetate projectual and correspondingfractional parts (fig. 1). Acetate is used be-cause (1) it can be used on the overheadprojector by the instructor and (2) thestudent can see through the material to theline diagram underneath. The fractionalparts are made from a second projectual ofanother color. This second acetate is cutinto pieces that represent fractions. Thus

The author is grateful to John J. Callahan,Boston State College, for assistance in thepreparation of this article.

the child's aid has two components, theacetate and the matching pieces, the latterbeing kept in an envelope.

The procedure for the understanding ofaddition is as follows:

1. Place the acetate projectual (fig. 1) onthe desk.

2. Gather the pieces that represent thefractional numbers to be added, forexample, 1/2 + % (fig. 2).

3. Join the pieces and place them on theportion of the acetate marked "1 whole."

4. Slide the pieces down until the right endmeets a linefor this exal,.?le, 3/4 (fig.3).

5. Look above the line to the name of thepoint. This represents the sum of thetwo fractional numbers.

At this time in the lesson some childrenmight discover that 1,./.t + % also equals% or 1%6. This would be a suitable timeto discuss the fact that 3/4, %, cad 1%6name the same number. When our classdiscovered that many seemingly differentresults are sometimes found by using thisprocedure, we developed a rule statingthat although all answers are correct, thebest result is the solution on the line clos-est to the line labeled "1 whole," thesimplest form of the fractional number.

115


0

I whole

0 1 2 3, 416 16 16 16 16

5 6 7 8 9 10 11 12 13 14 15 1616 16 16 16 16 16 16 16 16'. t6 16 16

III 116IFig. 1

For subtraction the first and secondsteps are the same, but in step three theform is different.

In the example IA 1/8, the piece thatrepresents I/8 is placed on top of and onthe right hand side of the piece that rep-resents 12. The child starts at the topand moves the pieces down until the leftside of the Vs piece is aligned with a point

8

named by a fractional numeral, for ex-ample, Vii, 1/8 3/8 (g. 4).

Many problems with fractional numbersless than one were pursued in our class andthe answers were derived with the help ofthis aid. Further, after using this device fora short period of time many of the childrencould solve similar problems with paperand pencil only.


Fig. 3

Fig. 4

EDITORIAL CONIMENT.Similar devices may be used for whole-number operations. Piecesare made to represent the whole numbers from one to ten. For example, a ten-inch stripcan represent ten units. The pieces may be used to find basic sums as well as to show thestructure of the "teen" numbers, as in 12 = 10 + 2.

This type of aid may also be used for decimal fractions and percents. Making the unitstrip ten inches long allows for easy division into tenths. If an inch ruler with I/10-inchsubdivisions is available, it is easy to subdivide the unit into hundredths, Or, in keepingwith trends toward metrication, make the unit strip one decimeter long. Tenths and hun-dredths will then correspond to the centimeter and millimeter marks on the ruler.

Division by a fractiona new method

GEORGE H. McMEENCalifornia State Polytechnic College, San Luis Obispo, CaliforniaDr. McAteen is professor of elementary-school mathematics at CaliforniaState Polytechnic College. This year he has been granted a leave of absenceto serve as Special Consultant in Mathematics to the California StateDepartment of Education.

Editor's Note

Presented at the annual meeting of theNational Council of Teachers of Mathe-matics at Chicago, Illinois, April, 1961.

* * *

Great progress has been made in the pastdecade in making the various operations ofarithmetic intelligible for the learner.However, a few arithmetic operations con-tinue to challenge our best efforts in thisdirection. Of these more challenging op-erations, I can think of none more difficultto rationalize than the operation of divi-sion by a fraction in which we "invert thedivisor and multiply." My experienceswith hundreds of college education majorshave led me to observe that very few ofthese students come to college understand-ing the rationale of the operation. This isunderstandable when we consider themechanical way in which the operationhas been presented in most elementaryschools.

In the past decade, marked by renewedinterest in the mathematical rationale ofarithmetic, we have seen some arithmetictexts attempting to present division by afraction intelligibly. On the other hand,many authors continue to present the op-eration largely in a mechanical fashion andjustify their treatment on the groundsthat (1) the operation is too difficult forpupils to comprehend fully other than inthe simplest case of division by a unitfraction, (2) an understanding of the op-eration is relatively unimportant, and (3)

118

the operation is seldom encountered bymost people in everyday life, and hence itshould be accorded only minor attention.These authors usually continue to clarifytheir position by stating that the reallyimportant things with regard to the opera-tion are for the student to understand (4)when to divide, (5) how to judge whetherthe obtained quotient is reasonable, and(6) how to verify in simple cases that the"invert and multiply" process reallyworks.

While agreeing at least partially withthe latter statements, I wish to take excep-tion to the first three. It is my positionthat the operation is not too difficult formost pupils to understand if presentedproperly and that an understanding of it isvery important, is rounding out a clearunderstanding of the four fundamentaloperations and their interrelationshipswith respect to whole numbers and frac-tions. While continuing to stress the socialaim of arithmetic instruction, contempo-rary mathematical programs have beenplacing increased emphasis upon soundmathematical development. In keepingwith this trend it is important that weleave no stone unturned in our efforts tofind improved ways of rationalizing divi-sion by a fraction even though it be truethat the operation is used infrequently bymany people in everyday life.

New method provides rationale

For the past decade I have been search-ing for a better way to rationalize division

by a fraction. I now believe that I havediscovered such a method which I encour-age the reader to try. Not only does mymethod lead to a clearer understanding ofthe operation, but it promotes a deeperunderstanding of the role of a reciprocal orinverse ratio. When in the past I havemeasured the effectiveness of my collegeinstruction using conventional procedures,I have usually found a substantial portionof my students still unable to supply anintelligent explanation of why we multiplythe reciprocal of the divisor by the divi-dend in dividing by a fraction. I cherishthe hope that usage of my method willgreatly alleviate this situation. Recent ex-periences with my college students and twoclasses of sixth graders have been very en-couraging in support of my method.

My method provides no easy road tounderstanding for those students incapa-ble of good thinking, and I am convincedthat a demonstration of the method bythe teacher is not alone sufficient. Sincethe method follows the pattern of manyscientific experiments, it should be pre-sented in a laboratory approach throughwhich the students make discoveries us-ing their own laboratory materials andrecord their findings as they progress fromone learning sequence to another. Dem-onstration of principles and generaliza-tions in front of the class using a demon-stration set of materials should follow butnot precede individual experimentation.When approached in this way, I am con-vinced that a majority of sixth graders orteachers college students are capable ofachieving a mature understanding of thedivision operation.

My method employs number wheels toillustrate the well-known concept of "meas-urement division" and enables the studentto see easily how many times the divisor iscontained in the dividend. Originally Iused sticks of various lengths to illustrate"measurement division" and found themquite effective. However I found .that stu-dents using she sticks wee prone to makea certain type of mistake. For example, in


measuring a stick one foot long (A) witha stick three quarters of a foot long (B)they were apt to state that the three-quarter foot stick was contained one and aquarter times in the one-foot stick, notclearly perceiving that the remaining one-quarter foot was one-third of anothermeasure with the divisor.

BII ft.

.......... .........

A B = I = 11 =

However, when I shifted from sticks tonumber wheels to portray the role of vari-ous divisors, this difficulty vanished. Thisis due primarily to the ease of determiningthe number of turns dlr., number wheelsmake as they are rolled along the numberline by means of the markings on the num-ber wheels. The diagram below illustrateshow easy this is. Familiarity with the

Start -I turn I turn I turn 4 turns

1 ft.

many uses of wheels in our modern worldalso assists students to grasp their sig-nificance quickly in this new division set-ting.

Construction and useof pupil materialsPrepare a stencil, and mimeograph on

long paper four circles and a paper rulerfor each pupil as shown on the followingpage.

The drawings in this article are reducedin scale, but the pupil materials should be


In. 11 21 31 41 61 6 7[ 81 9 10$ 111 12Ft. . II 1. 4

Ft. 41 i 4

full size to prevent confusion. To get thetwelve-inch ruler on a stencil it will benecessary to draw it lengthwise on thestencil. To draw the four circles on thestencil having circumferences of 3 .in., 4

in., 8 in., and 9 in., set your compass toradii of 23/48 in., 31/48 in., 1 13/48 in.,and 1 21/48 in. respectively. Constructingcircles having the correct circumferencesrequires some skill and the teacher is ad-vised to first experiment with a compassand make the circles before cutting thestencil. Since any error in the radius pro-duces about three times as much error inthe circumference, the construction radiiabove are given to the nearest forty-eighth of an inch. A mechanical drawingruler usually includes this scale.

After the pupils have been given mimeo-graphed sheets showing the four circlesand ruler they can cut them out with theirscissors. The circles should be pasted oncardboard cut to the same shape to makethem rigid. When a common pin is in-serted through the center of the numberwheel, the wheel may then be rotatedabout its axle. In cutting out the circles,the pupils should be instructed to makethem a trifle large, for after they have beentested on the ruler for exactness they canbe trimmed down to the correct size. Havethe pupils print the inch measure on oneside and the foot measure on the otherside of their number wheels.

Now that the individual materials havebeen constructed, the pupils are ready tobegin experimenting with them. It wouldbe a gocid idea for the teacher to mimeo-

graph a guide sheet as indicated below.The guide sheet will not only direct thepupils from one learning sequence to an-other, but it will provide appropriateplaces for them to record their findingsand guide them in making generalizations.The teacher should illustrate briefly withher large demonstration materials how toroll the number wheels along the appro-priate scale on the number ruler andcount how many times the wheels turn intraveling a given distance. She shouldlead her pupils to observe that a numberwheel dramatizes the role of the divisor,the number ruler indicates the dividend,and the number of turns of a number wheelindicates the quotient.

The tables in the Guide Sheet belowprovide experiences with both wholenumbers (inch measure) and fractions(foot measure). This procedure not onlyhelps to relate the two measures, but itgives the pupils more confidence in work-ing with fractions since they can see howthe same principle works with whole num-bers. The numerals in parentheses indi-cate the answers pupils should obtain us-ing their experimental materials. Pupilsshould be instructed to first insert the miss-ing quotients in each table and then tocomplete the generalizations which follow.Pupils should not be forced to use theirmaterials if they are sure they can operateon a mental level without them.

Pupil guide and data sheet(with answers)

See Table 1 on p. 125. The wheel in Ex-periment 1-B had (3) times as large a cir-cumference as the wheel in Experiment1-A. Hence the nine-inch wheel turnedaround only (1/3) as many times as thethree-inch wheel in traveling twelve inches.

The divisor in Experiment 1-D was(3) times as large as the divisor in Ex-periment 1-C. Hence the divisor in 1-Dwas contained in one foot only (1/3) asmany times as the divisor in 1-C.

Suppose that Bicycle B has wheels three

Table 1


Experiment 1-A 1-B 1-C 1-D

Dividend (length of ruler) 12 in. 12 in. 1 ft. 1 ft.Divisor (wheel of circumference) 3 in. 9 in. 1/4 ft. 3/4 ft.Quotient (turns of wheel) (4) (1 1/3) (4) (1 1/3)Quotient written as an improper fraction (4/1) (4/3) (4/1) (4/3)

times the circumference of those of Bicy-cle A. If both bicycles travel the same dis-tance, the wheels of B will turn aroundonly (1/3) as many times as those of A.

We know that 1 =1/4 = (4), and since3/4 is (3) times as much as 1/4, there-fore 1+3/4 = (1/3) of 4 =1 1/3 or (4/3).

See Table 2. The wheel in Experiment2-B had (2) times as large a circumferenceas the wheel in Experiment 2-A. Hence theeight-inch wheel turned around only (1/2)as many times as the four-inch wheel intraveling twelve inches.

The divisor in Experiment 2-D was (2)times as large as the divisor in Experiment2-C. Hence the divisor in 2-D was con-tained in one foot only (1/2) as manytimes as the divisor in 2-C.

Suppose that American Car B haswheels two times the circumference ofthose of Foreign Car A. If both cars travelthe same distance, the wheels of B willturn around only (1/2) as many, times asthose in A.

We know that 1+1/3= (3), and since2/3 is (2) times as much as 1/3, therefore1 +2/3 = (1/2) of 3=1 1/2 or (3/2).

Study the quotients of 1-C, 1-D, 2-C,and 2-D when expressed as improper frac-tions. Now look at the divisors. Could youpredict how many times the fractionaldivisor is contained in one whole by writ-ing the divisor upside down for the quo-tient? This is called inverting the divisor.We call 4/3 the reciprocal of 3/4.

Table 2

1 +-1/5 = (5/4) 1+5/6= (6/5)

1 + X/ Y = (Y/X)

The reciprocal of the divisor tells howmany times a fraction is contained in (1)whole. The truth of this generalizationmay be verified by proving the foregoingdivisions by multiplication:

(5/4) X4/5 =1 (6/5) of the 5/6ths =1

There are (Y/X) of the X/ Y's in 1.

The pupils should now be led to extendtheir thinking to distances which aremultiples of 1 foot. Table 3 on the fol-lowing page illustrates one such ap-proach, and the teacher may wish to con-struct other tables of a similar nature. Thedividend may be portrayed by laying sev-eral number rulers end to end. An equallygood procedure is simply to measure againand again on a single ruler according tothe size of the dividend. Many pupils bythis time probably will be able to omit thisexperimental stage and proceed immedi-ately to the level of abstract thinking in-volved in completing the table.

Since the divisor in Experiment 3-A iscontained (3/2) times in one whole, in Ex-periment 3-B we found that the samedivisor is contained in two wholes, (2)times as much or (2) X (3/2) times, and inExperiment 3-C we found that the samedivisor is contained in five wholes, (5)times as much or (5) X (3/2) times.

Experiment 2-A 2-B 2-C 2-D .

Dividend (length of ruler) 12 in. 12 in. 1 ft. 1 ft.Divisor (wheel of circumference) 4 in. 8 in. 1/3 ft. 2/3 ft.Quotient (turns of wheel) (3) (1 1/2) (3) (1 1/2)Quotient written as an improper fraction (3/1) (3/2) (3/1) (3/2)


Table 3

Experiment 3-A 3-B 3-C 3-D 3-E

Dividend (length of ruler) 1 ft. 2 ft. 5 ft. 1/3 ft. 4/3 ft.Divisor (wheel of circumference) 2/3 ft, 2/3 ft. 2/3 ft. 2/3 ft. 2/3 ft.

.. Quotient (turns of wheel) (3/2) (2) X3/2 (5) X3/2 (1/3) X3/2 (4/3) X3/2

If 1 4.2/3 =3/2,

then 1/3 2/3 = (1/:3) X3/2

and 4/3 4-2/3 = (4/3) X3/2,

M-i- Y/Z= (M) X (Z/ Y),

Y/Z= N)X(Z /3').

To divide any number by a fractioninvert the divisor and multiply it by thenumber.

Construction and useof demonstration materials

In order that the demonstration ma-terials be clearly visible in front of theclass, they should be much larger andmade of more durable materials. Multi-plying the dimensions of the pupil ma-terials by three provides a convenient sizeand at the same time results in a changefrom foot measure to yard measure, avaluable variation. Anyone possessingminimum woodworking skills can easilygroove a board 36 in. long to guide thewheels as they are turned. Inch measuremay be printed on the top of the board,thirds of a yard on one edge, and fourthsof a yard on the other edge. When workingwith fractional parts of a yard, the appro-priate edge can be turned so as to face theclass. A minimum of four number wheelsshould be constructed of plywood, mason-ite, or plastic. The four number wheelsshould have circumferences of 9 in., 12in., 24 in., and 27 in. Their constructionradii to the nearest forty-eighth of an inchare respectively 1 21/48 in., 1 44/48 in.,3 39/48 in., and 4 11 in. Print the cir-cumference in ::th,;,..; on one side andyards on the side. Holes shouldbe bored through the v-,nters of the wheelsso that a nail or other object suitable forturning the wheels may be inserted. The

number wheels and number board may beused in exactly the same way as the pupilmaterials, keeping in mind that they .arein yard measure instead of foot measure.If desired, still larger wheels whose circum-ferences are multiples of one yard mightbe constructed. This would permit theteacher to relate division by a fraction todivision by a whole number. Thus, for ex-ample, it would be easy for the pupils tosee that a wheel having a circumference oftwo yards will turn only one-half time intraveling one yard. This type of measure-ment division often has little meaning forpupils in spite of its simplicity.

Summary

Division by a fraction has long been re-garded as one of the most difficult algo-risms, if not the most difficult, to rational-ize with pupils. Preliminary testing of myexperimental approach to the operationindicates that encouraging numbers ofpupils are capable of developing a matureunderstanding of the operation. Theunique feature of the method is the em-ployment of number wheels of varioussizes to portray the role of the divisor. Us-ing their easily constructed materialspupils first discover that a unit fraction,1/Z, is contained Z times in one whole.Comparing this quotient with the one ob-tained when dividing by a nonunit frac-tion, Y/Z, they deduce that since Y/Z isY times as large as 1/Z it will be con-

,tained,only 1/Y as many times as 1/Z.Hence they generalize that 1+ Y/Z=Z/Y.And finally, the pupils are led to discoverthat when the dividend is a multiple orfraction of one whole, the quotient will bethat number times the reciprocal of thedivisor.

A game with fraction numbersRICHARD THOMAS ZYTKOWSKIUniversity Heights, Ohio

The game described below is a uniqueway to provide practice in the addition,subtraction, multiplication, and division offractions. The game can also be used toteach ratios as well as the sequence ofnumbers.

Use a 2-by-2 ft. piece of oaktag or card-board to make a playing board. Divide itinto spaces as shown below, then write thefractions as figure 1 shows.

A

13

C

D

E

F

Start 2 4 : 10 11225

46

69

812

10 112IT 18

1.`4

A8

.112

12.16

.11 M.20 24

45 81010

12

15

1620

20 12425 30

5 1

61012

1518

2024

25 30301:36I24 3028 35

*Win-ner

67

12

141821

FIGURE I

On some small pieces of tagboard 31/,in. by 8 in. write the following symbolsand numerals: '/2; +1/2; x1 /4; -:-%;

+13

xi4FIGURE 2

+Y6; X Vi; -Y8; X IA; + 11/2; x22/3;+3%; +4%; x5%; 6%; +7%; andx84,.

In figure 2, examples are shown of howthe front side of the small pieces of tag-board would be labeled.

On the back of each small piece of tag-board write the answers for the problemthat is on the front. There will only be sixpossible answers, since there are only sixrows: A, B, C, D, E, F; the rest of thefractions in the row are equivalent to thefirst fraction in that row. Figure 3 showshow the back side of 1/2, +1/3, x v4, +.1A,would look respectively.

Two to six students can easily play thisgame, or the class may be divided intogroups.

123

124

_1 1

A 0

13 -h-

C *

1.) it

E iFa

TEACHER-MADE

1

AIDS

1X4A- 1

8

B*C iD iE11

F}

. 1

A 2iB 31-

C 34

D5

E46F4?

A tB I

iC 1 T-2-

D I 2;5

E I*F 1j

FIGURI 3

The rules of the game are as follows:

Each player will begin with the first numeralon the large tagboard. which is the friction 1/2.

He will, without looking, pick a small pieceof tagboard with a symbol and fraction on it.

He will then perform the mathematical opera-tion in order to arrive at the correct answer.

If the player gets the answer correct, he movesone space. If he is wrong, he remains where heis. (The answer can he verified by checking theback of the card used by the player in perform-ing the mathematical operation.)

The first student to reach "winner" is the"mathematical champ" or "whiz kid."

All answers must be reduced to their lowestterms. Improper fractions must he changed tomixed numbers.

Each player is given only one chance percard, then it is the next player's turn.

The fraction on the small piece of tagboardor cardboard should always be considered torepresent the second numeral in the problem;thus, it will either he the second addend, theminuend, the multiplier, or the divisor.

Example:To begin the game, player A, without

looking, picks a card. The card states x1/4 , Since all players begin with the first

square on the large oaktag board (1/2 ), theplayer must multiply 1/2 x 1/4 .

Player A's computations:x

Player B checks the back of the cardbeing used to see if the answer is correct.The answer is correct, so player A movesto the next space, "a. It is now player B'sturn. Player B misses his problem. There-fore, he must remain at the same placeuntil it is his turn again.

If there are no more than two players,it will be player A's turn again. Only thistime instead of being at 1/2, he has nowmoved to %. He draws another card; thecard states 1/2. Player A must subtract

1/2 fromPlayer A's computations:

Player B checks his answer by looking atthe back side of 1/2 to find the answer.He sees that the answer is correct, so playerA moves to the next space, :M. It is nowplayer B's turn again. Player B shouldalways get his turn last, since player Awas first.

If the game is used by the teacher inrow competition a few times, the pupilsshould be led to see the relationship be-tween numerals. The student should beasked what numeral would follow thesequence in each row; then this can beexpanded to lead the pupils to see therelationship between the numerator andthe denominator of each fraction.

Geometry andMeasurement

Geometry in the elementary school is treated informally and intuitively.The teaching of geometry centers more on the use of manipulative aids thanany other topic in elementary school mathematics. A child is introduced tospace concepts by the physical objects for which geometric ideas are theabstractions.

The NCTM publication Readings in Geometry from the ArithmeticTeacher contains a number of articles that describe manipulative aids forteaching geometry. We have not attempted to duplicate the materials con-tained in that collection.

The lead article in this section describes an activity with the geoboard.Wells uses a nine-pin board, but it is possible to construct boards with agreater number of pins. The article by Fisher on pegboards in the last chap-ter of this book also describes activities suitable for the geoboard.

Bruni describes two methods for justifying the sum of the measures of theangles of a triangle. The first is the well-known experiment in which the"three corners" are rearranged to suggest a straight angle. The second usesan intuitive notion of limits. Backman and Smith suggest models for in-vestigating the classification of triangles by properties of their sides and byproperties of their angles.

The outdoors is the laboratory in which Woodby suggests activities withthe angle mirror. This is a device for establishing right angles over largedistances. It is an easy step from the angle mirror to kaleidoscopes and toinvestigations of other reflective properties of mirrors.

125


Students usually have little difficulty accepting the product relationshipfor finding the area of rectangular regions. Colter describes a manipulativeaid that helps students rationalize the area formula for a circular region,The aid relates the circle area to formulas for the area of a rectangularregion and the circumference of a circle.

Hall's article on the Pythagorean theorem does more than provide stu-dents with a visualization of the relationship of the squares of the sides of aright triangleit also explores relationships between dissimilar figures withequal areas.

Children are usually enthusiastic about constructing models of geometricsolids, especially when they use the models for holiday decorations or con-temporary mobiles. Wahl provides us with a set of patterns as well as direc-tions for the construction of a tetrahedron, a hexahedron, an octahedron, adecahedron, and an icosahedron.

Using material originally designed for crafting artificial flowers from aviscous plastic solution, Wahl dips wire frames for various solids and comesout with some very creative surfaces. One of the surfaces generated by thistechnique is the Mobius strip.

Kaprocki shares a method for wrapping a number line around the face ofa clock, thus giving children a better idea of the hour scale on a clock. Onecould go another step and fashion a similar number line for the minutescale.

The last article shows how to integrate many facets of geometry and mea-surement into the construction of a very ancient devicethe sundial. Wahlhas simplified the construction so that elementary school children can makea reasonable sundial that actually works.

Creating mathematicswith a geoboard

PETER WELLSShell Center for Mathematical EducationUniversity of Nottingham, Nottingham, England

.A nine-pin geoboard is easily constructedfrom ;/e -inch plywood and 5/a-inch panelpins, the pins being about two inches apart.A large quantity of colored elastic handsis also needed.

A piece of wood, some pins and someelastic bands! What is this contraption for?What shall we do with it? The geoboard isessentially a piece of material for a childto work creatively with; it gives rise tomany activities that lead to discussions ofan open-ended nature. The nine-pin boardhas the advantage of being so simple thatthe child believes that any mathematicsthat may arise can soon be sorted out. Re-sults are quickly and easily recorded onsquared or dotted paper; this provides adefinite advantage to those who find writ-ten communication of their mathematicsmore difficult that the mathematics itself.

To illustrate the mathematical flexibilityof the geoboard, this article will discusssome lessons experienced with a group ofchildren. One board and a pile of elastic

127

bands between two children is idealapartner is there to encourage, correct, andargue with. Initially the children just putthe bands on the board, and the girls soonneeded more bands of particular colors tocontinue the patterns they were making(possible work on symmetry here?). Dur-ing this free activity stage, the teacher iswatching for situations that particularlyinterest the children and that are suitablefor development and extension.

Some children became interested in thesmallest triangles.

How many triangles the same as this onecan you make on the board? A problemarose immediately. Is

the same as


Some children said yes; others, no. Somewanted to include this one.

What do we mean by "the same"? Aftermuch argument most of the childrenseemed happy with counting the first twoonly as the same, agreeing with Michaelwho said, "If I cut one out of cardboard Ican fit it on top of the other." Argumentsof this type challenge mathematical in-terpretation. Definitions, classification, andcommunication are all involved. Ideas ofsets and equivalence relations are behindthe thinking, and the teacher should beaware of this although the children maynot be. The children might not have classi-fied ; \ and /1 as "the same," and theproblem would then have developed dif-ferently; but once a field of discourse hasbeen agreed upon, mathematics will de-velop.

How many of the smallest triangles arethere then? All the children except onereached for elastic bands and the boardand set out to make all of them. Theysoon found recording was essential as theboards became overcrowded. James, on theother hand, thought for a moment andcame over and said, "There are sixteen."His explanation was "Four fit into a smallsquare, and there are four small squares."When asked to find how many of these

there are, he went away and began puttingbands on a board.

The discussion moved on to larger tri-angles. How many different triangles (inMichael's sense) can be made on a nine-pin board? This can be played as a gamebetween two children, alternately makingand recording triangles different from pre-

vious ones. An extension to the differentquadrilaterals is more challenging althoughboth games should result in a draw. Someof the sixteen different quadrilaterals arenot easy to find. Moving the elastic bandson the board illustrates how figures arerelated dynamically, a different kind of ac-tivity from drawing quadrilaterals onsquared paper.

I list some other starting points on theclassification of shapes on a nine-pin boardwhich a teacher could draw upon if theopportunity arose.

I. How many different squares can bemade on the board? How many ofeach type?

2. Consider the four-sided shapes.Which ones have an axis of sym-metry? How many can you give afamiliar name to?

3. Look at the triangles you can make.How many are isosceles?. How manyare right-angled? How many of eachtype are there on the board?

4. Make shapes with more than foursides. What is the largest number ofsides you can get?

5. How many straight line segments ofdifferent length can be made?

6. What shapes cannot be made on theboard?

Although the teacher should be aware ofthese possibilities, I consider it preferableto follow up the children's own suggestions.It is essential to listen to the children andto allow them to agree on a method ofclassifying, which may not be the teacher'sown method.

In discussing point 4 one of the girls pro-duced

and said it had six sides. The class soondivided itself into those who said it hadfour sides and those who said it had six.

The argument was settled for this class byone boy who said, "If you think that

has six sides, then,

has four."On larger geoboards there are so many

of the various shapes that it is not practicalto go through all these questions again, butif the question is more definite (e.g. makean isosceles trapezium or a right-angledtriangle that is not isosceles) the activitycould be worthwhile. The children are ableto creat: the shapes themselves withoutthe diffic: (ties of ruler and pencil. Discus-sion of various points arising can followwith the whole class (if necessary) veryquickly. Teacher: "Make a shape like mineon your board." Useful ideas are develop-ing as the child moves the band to thecorrect position before the discussion ofthe particular point even begins.

In this article emphasis has been uponclassification and counting, but there aremany other mathematical situations thatmay develop from a geoboard. Chapter 8of Notes on Mathematics in PrimarySchools, Association of Teachers of Math-ematics; Cambridge University Press, is aninvaluable source of ideas and problemsfor both children and teachers.

Let us end with a problem for theteacher. For a nine-pin geoboard the larg-est number of sides a simple closed polygoncan have is seven. One solution:

GEOMETRY AND MEASUREMENT 129

What about a sixteen-pin board, twenty-five pin board, n2-pin board?

EDITORIAL COMMENT. The geoboard maybe used to display the three motions oftransformation geometrytranslations (orslides), reflections (or flips), and rotations(or turns). Rubber bands of two colorscould be used, one color to represent theoriginal figure and the second color for theimage.

TranslationMove each vertex up two pins.

Z11

Image

Move each vertexup two nails.

ReflectionIdentify a flip line and find the op-posite location for each vertex.

Image

Flip lineRotationRotate a guide line and then build thefigure on this new line.

- - -0- -Nr

Guide line

Image

Guide linerotated 90°

A "limited" approach to thesum of the angles of a triangle

JAMES V. BRUNI

An assistant professor of education at Herbert H. LehmanCollege, City University of New York, Dr. Bruni teachesmathematics education courses for elementary school teachers.

0 ne of the exciting discoveries childrencan make about triangles is that the sumof the measures of the angles (in degrees)of any triangle always equals 180. Teach-ers often lead them to "discover" thisphenomenon by having them make allsorts of triangles, use a protractor to mea-sure each of the angles, record the mea-sure of each angle for individual triangles,and look for a pattern or rule. This methodgives the child the opportunity to practiceusing the protractor for measuring angles.

Another approach frequently used is tohave students take a piece of cardboardcut into any triangular shape and ask themto perform the following experiment:

Make a black border around the triangularshape. This will serve as a model for atriangle. Label the insides of the angles1, 2, and 3 respectively, as in figure 1. Nowtear or cut the model into three parts andrearrange the pieces as in figure 2. You

Fig. 1

130

have now moved the angles so that theyform one large angle. That large angle isa straight angle and contains 180 degrees.Since combining the angles is like findingthe sum of their measures, this procedureshows that the sum of the number of de-grees in the three angles is 180.

Fig. 2

The following approach to the discoveryof the sum-of-the-angles property of tri-angles is less well known.' It is a very use-ful one because it is an intuitive demon-stration that employs the concept of amathematical limit as the student examinesthis property of the angles of a triangle.

You will need to construct a very simplemodel using a small piece of wood orheavy cardboard, two nails or brass fasten-ers, and a rubber band. Hammer two nails

1. This approach is introduced in Emma Castel-nuovo's La via della maternaticala geometria(Florence: La Nuova Italia, 1970).

almost entirely into the wood at points Aand B as illustrated in figure 3. Stretch

large elastic band around the nails, form-ing a figure that looks like a line segment.At about the midpoint of that "line seg-ment," pull upward on part. of the elasticband so that you form vertex C and, conse-quently, the model of a triangle (fig. 4).As you pull at point C you make differentisosceles triangles. Ask students to examinethe angles of these triangles.

Fig. 3

At one point the sides are all the samelength, forming an equilateral triangle. Allthe angles of an equilateral triangle arecongruent and each contains 60 degrees.The sum of the measures of the three anglesin degrees is 180.

Fig. 4


Next, pull upward at C and form othertriangles. Ask students, "How is the sizeof LC changing as I pull upward?" and"How are the, sizes of L A and L B chang-ing?" As you pull the elastic upward at C,L C gets smaller and smaller while L Aand LB get larger and larger (fig. 5). Askstudents to imagine the elastic stretchingfarther and farther toward an extreme case(before which the band would break!),with L C getting closer and closer to 0 de-grees in measure and LA and L B lookingmore and more like right angles. Againthe sum of the measures of the angles indegrees approaches 180 (0 + 90 + 90).

Now slowly release the elastic banddownward and have students examine thetriangles formed. They should note whathappens to the angles now: LC gets largerand larger while both LA and LB getsmaller. L C increases in size until it looksalmost like a straight angle of 180 degrees(fig. 6); meanwhile, / A and. LB seem tobe getting closer and closer to 0 degreesin measure. Once more the sum of themeasures of the angles of the trianglesformed seems to be fixed at 180 (180 +0 + 0).

Although the preceding experience doesnot prove that the sum of the measures ofthe angles of a triangle is always 180 (indegrees), it suggests that this conclusionis reasonable. Actually, only isosceles tri-

Fig. 5


1 Vigil \ I

Fig. 6

angles are formed, and the student uses hisintuition to discover what is happening tothe angles as the model is transformed.

This experience is a valuable way oftaking another look at the sum-of-the-angles property of triangles because it in-volves thinking in terms of limits, a funda-mental concept in mathematics. As the

elastic band is stretched upward the baseangles are increasing in size. They eachapproach 90 degrees in measure while thethird angle, or vertex angle, approacheszero degrees. The extreme case is neverreached, since no triangle would exist inthat extreme case (you could not have oneangle of zero degrees in a triangle). Yetthe important point is that the base anglesget as close to 90 degrees as you want tomake them while the vertex angle gets asclose to zero degrees as you wish.

Similarly, in releasing the band down-ward, you can make the base angles asclose to zero degrees as you wish whilethe vertex angle increases toward 180 de-grees. Again, no triangle would exist in

the extreme case, since you would end upwith a line segment. Yet you can get asclose to the extreme case as you wish.

This intuitive demonstration of the spe-cial property of the sum of the angles of atriangle can help students develop an earlyfoundation for an understanding of the con-cept of a mathematical limit.

EDITORIAL COMMENT.A similar board may be used to explore the effect of changes in thealtitude of a triangle on the area of the triangular region. On the board shown in figure 3of the article, rule off a square-grid system. As the vertex C (in fig. 4) is drawn upward,count the number of square regions enclosed by the figure. If you establish a given positionof C as the starting position, you can then investigate areas for altitudes that are givenmultiples of the original altitude (such as twice as high, half as high, and so on).

A second board may be constructed by starting with the board shown in figure 3 andruling off the square-grid system. Now fasten a piece of clothes hanger parallel to the linedetermined by vertices A and B. Slip a rubber band over the wire before fastening. Nowvertex C can move along the wire. What happens to the area of triangular region ABC asvertex C is moved along the wire?

A

zClothes hangerwire

Rubber band

Activities with easy-to-maketriangle modelsCARL A. BACKMAN andSEATON E. SMITH, JR.

Faculty of Elementary Education, University of West Florida,Pensacola, Florida

Strips of colored tagboard and brassfasteners can be used to make a set ofmodels for demonstrating a variety ofproperties of triangles. The models are in-expensive, durable, and easy to store, andthey can be made by the classroom teacherfor use in classroom demonstrations. Theyalso are easily assembled by pupils foruse in mathematics-laboratory activities.

Construction

Specifications and directions for a set ofsix models illustrating the two classifica-tion systems for triangles (sidesscalene,isosceles, equilateraland anglesacute,right, obtuse) are given below.

Each model is made from three half-inch-wide strips of colored tagboard(poster board) and three short brassfasteners. Use a paper cutter to cut thestrips from sheets of colored tagboard.Holes for the brass fasteners are punchedhalf an inch from each end of each strip.(A ticket punch makes a hole of accept-able size.) After assembling the strips,

133

trim the corners to make them pointed.(See fig. 1.)

Listed below are lengths for the tag-board strips which have been found satis-factory for each of the six models.

SIDE CLASSIFICATION SIDE 1 SIDE 2 SIDE 31. Scalene (no two sides

congruent)2. Isosceles (two sides

congruent)3. Equilateral (three sides

congruent)

9/1

9"

9" 911

ANGLE CLASSIFICATION SIDE 1 SIDE 2 SIDE 3

I. Acute (three acuteangles)

pr

2r Trim off2 shaded portions

Fig. 1

Brassfasteners

9,, 8" .7,


2. Right (one right angle) 11" 9,, 7,,

3. Obtuse (one obtuseangle) 10" 7,, 6"

Suggested activities1. Detecting differences between models

Students are given the set of three modelsillustrating the side-classification system(or angle classification). Assuming thatthe students have not as yet been intro-duced to the classification system, they areasked to state ways in which the threemodels differ from each other.

2. Sorting the models by typeThe models are gathered together. Stu-

dents are asked to select a model, identifyits type, and list its characteristics. (Forsorting activities, all models should bemade from the same color of tagboard.Otherwise, there is a strong possibility thatstudents may learn to identify the modelsby color rather than by the desired char-acteristics.)

3. Constructing the modelsAssuming that the students have learned

the classification systems, they are given

a set of prepunched tagboard strips, brassfasteners, a ruler, and a protractor andare asked to assemble models for each ofthe six basic types of triangles.

Students may also be asked to constructmodels for certain classification combina-tions. For example, they might be askedto construct an isosceles right triangle(possible) or an obtuse equilateral triangle(impossible).

4. Designing new models

Students are asked to create a set ofmodels that illustrate the basic types oftriangles using different strip lengths thanthose given above.

5. Developing perimeter formulas

The models for the side-classificationset are constructed so that only strips ofthe same length are the same color. Stu-dents may then use the color cues to de-velop the three formulas for the perimeterof a triangle: (a) scalene, p = a + b + c;(b)isosceles, p = 2a + b; and (c) equi-lateral, p = 3a.

EDITORIAL COMMENT.The model construction suggested in the article is easily extendedto polygons of more than three sides. These models, however, will not be rigid. Studentsmay be asked to "stabilize" the models, thus introducing the engineering function ofdiagonals and exploring the characteristics of triangle rigidity.

The angle mirror outdoors

LAUREN G. WOODBYMichigan State University, East Lansing, Michigan

Lauren Woodby is professor of mathematics at Michigan State University.During the current academic year he is on a Post-Doctoral Fellowship.at New York University and is studying the education of elementary teachersin mathematics. He is on the Board of Directors of NCTMand is chairman of the Committee on Meetings.

Measurement activities outside the class-room have a special appeal to children ifthey use some optical device. A simple anduseful optical instrument can be made byfastening two small mirrors so that theirfaces form a dihedral angle of 45°. Theinstrument is sometimes called an "opticalsquare" because it can be used to establishperpendicular lines.

How to make the angle mirror

Two mirrors, some small pieces of wood,and some glue are all that is needed. Thesize of the mirrors is unimportant. How-ever, since tiny ones work just as well aslarge ones, and a small instrument is easierto handle, mirrors about one inch squareare suggested. One way to mount them isto glue a small rectangular block to theback of each mirror and then glue the twoblocks to a flat piece of wood. First glueone block so that the attached' mirror isverticcil to the base, then glue the othermirror so that the angle between the planesof the two mirrors is 45°. The reflectingsurfaces should face each other. See fig-ure 1.

The simplest way to obtain a 45° anglefor fixing the position of the mirrors is tofold the square corner of a sheet of paper.Of course, a draftsman's 45° trianglecould be used if one is available. Anotherway to mount the mirrors is to cut a wedge -shaped piece of wood with a miter saw,and then glue the mirrors directly to thefaces of the wedge.

Flo. 1. Top view of the angle mirror

How to use the angle mirror

Basically, the angle mirror just describedhas only one capabilityit can be used toestablish lines that form a right angle. Thelines are lines of sight between objects. Infigure 2, the observer at 0 looks in the di-rection of an object A. He then brings theangle mirror into the position shown, andwhile still looking toward A sees object Bby double reflection in the mirror. Thus Bappears to be in line with A. When thissituation occurs, angle AMB is twice thesize of the angle at P formed by the mirrors.Since ais angle between the mirrors is

135

BFlo. 2. Sighting with the angle mirror

Y

z.

Q


45°, angle AMB is 90°. (For an analysisof why this is true, see the article by Hartthat will be published in one of the fallissues of the journal.)

Since the angle mirror is used out-of-doors, the distances are large comparedwith the dimensions of the mirror. So themirror (held close to the observer's eye),is considered a single point, M. A plumbbob on a string fastened to the angle mirrorcan be used to locate a point on the grounddirectly below point M. It takes some prac-tice for the observer to see the correctimage and to determine when the imageof the image of 8 is lined up with A. Ithelps if the objects A and 8 are peoplewho can move as directed, so the observercan visualize the lines involved. In learninghow to use the angle mirror, the observershould move slowly while looking towardA and notice the movement of the imageof B.

The hardest part is to discover what youare trying to line up. Remember that youare trying to determine two lines at rightangles to each other, and the mirror willbe the vertex of the right angle.

After making your angle mirror, go out-side with two friends and find two linesthat are perpendicular to each other, on thetennis court or playground. Stand at thecorner so that the angle mirror is abovethe vertex of the right angle and facetoward A, Have one friend stand at Afacing you.

Without using the angle mirror, you seefriend A. Now have your other friendstand at B, facing you, but you keep lookingstraight at A. So you cannot see friend B.But now hold the angle mirror up in yourline of sight between you and A, and rotateit until, by looking in mirror 2, you see thereflection of B's reflection in mirror 1. Nowyour eye, the image of the image of friendB, and friend A are all lined up, and theangle AMB is a right angle.

Now try constructing a rectangle. Beginby fixing two points P and Q and stretch-ing a string between these two points toget one side of the rectangle.

S

P

FIG. 3. Constructing a rectangle

The observer then stands at P and sightstoward a friend standing at Q. In the mir-ror, the observer sees the image of anotherfriend at X. He directs friend X to moveuntil X is lined up in the mirror with friendQ. Friend X then marks his position andstretches a string from P to X so that angleQPX is a right angle. The observer thenmoves to point Q and in the same mannerdirects a person Y to move so angle PQYis a right angle. Stretch a string from Q toY. Now several alternatives are possiblefor completing the rectangle. For example,the observer could nick any point R on theray QY and locate Z so that QRZ is a rightangle, and stretch a. string from R to Z. Sis the intersection of PX and RZ.

Notice that no distances were measuredin constructing the rectangle PQRS. Onlythree right angles were established, andthe fourth angle turns out to be a rightangle, since the sum of the angles of aquadrilateral is equal to four right angles.Another method would be to mark off thesame distance on the rays PX and QY toestablish points S and R. In the specialcase when the distance chosen is the dis-tance PQ, the rectangle would be a square.

Another basic construction with theangle mirror begins with a line (again astretched string between A and 8) and agiven point P on that line, as shown infigure 4. The observer at P positions a per-son at X so that APX is a right angle. Hecan check the accuracy of the mirror bychecking the angle BPX. If BPX is not alsoa right angle, as judged by the image in themirror, then there is an error in the con-struction of the mirror. This error can beovercome by finding the two points X and

Flo. 4. Establishing perpendicular lines

13

Y equidistant from P that apparently de-termine right angles APX and BPY. Thenmark point Z midway between X and Y.Stretch a string from Z to W so that thestring passes through P, and ZW, will beperpendicular to AB. From this basic con-struction a rhombus can be made by mark-ing M.and N so that PM PN. If PNPR = PS, MRNS will be a square. (If thediagonals of a quadrilateral bisect eachother and are perpendicular, the quadrilat-eral is a rhombus. If, in addition, the dia-gonals are equal, the figure is a square.)

C4

Mo. 5. Locus of the vertex of a right angle


A special application of the angle mirrorallows children to discover for themselvesthat the locus of the vertex of the rightangle of a triangle with hypotenuse fixedis a circle. Let two fixed points A and Bdetermine the hypotenuse. Sec figure 5above. Stakes are set at A and B, and astudent with an angle mirror is asked tolocate point C so that ACB is a rightangle. He marks his point C and standsthere. Another student finds another pointC2 such that ACB is a right angle. Nowevery student is asked to use his anglemirror and locate himself anywhere hechooses so that he is at the vertex of aright angle, with rays through A and B.Students see at once that the locus is asemicircle by observing where other stu-dents are standing. Usually someone willtry out the other side of line AB. Anotherway to dramatize the locus is to have astudent, one who is skillful with the anglemirror, walk wherever he can and still keepthe stakes lined up in the angle mirror.This technique gives a spectacular result ifthe ground is covered by freshly fallensnow.

When students have mastered the useof the angle mirror, they can use it in avariety of applications. For example, theycan realign (or establish) the baseballdiamond or the football field. Outdoor con-struction with stretched strings for lines isparticularly appealing. Children who con-struct a rectangle or a rhombus on theplayground, given only a ball of twine andan angle mirror, experience somethingquite different from a similar task done attheir desks using ruler, compass, pencil,and paper.

EDITORIAL COMMENT. Students are fascinated by the reflective properties of mirrors. You

might consider constructing kaleidoscopes or studying parabolic and spherical mirrors.The instrument described in this article is designed to construct right angles. Students

may wish to investigate other methods of making right angles, such as the Egyptian knotted

ropes or paper-folding techniques.

Adapting the area of a circleto the area of a rectangle

MARY T. COLTER

Presently employed by the Department of Defense on the islandof Taiwan, Mary Colter is teaching military and civiliandependents. Her previous experience includes teaching Navajo

Indians in a New Mexico high school.

One of the most difficult problems fora student studying the areas of regionsbounded by geometric figures, regardless ofwhether he is in the fifth or ninth levelin mathematics, is the area of a circularregion. The student may memorize theformula, know where to plug in the values,consistently arrive at the correct answer,but never comprehend and master com-pletely the problem. The teacher assumesJohnnie knows what the area of a circularregion means by hearing him repeat some"fed facts."

The areas of regions bounded by squares,rectangles, triangles, or trapezoids are rela-tively easy for the student to visualize.Often teachers show by blocks, cutouts,three-dimensional objects, and so on, howto find the area of a region, especially if itcan be expressed as a certain number ofsquare units in the base row multiplied bya number of rows, or "base times height."The area of a circle is not as easy to illus-trate, and by the time the class has reachedthe study of circular regions, they may betired of deriving formulas. Sometimes theteacher, unsure of himself, approaches thistopic as a memorization lesson.

Working with the fifth to the ninth levels,an easy way to teach the area of a circularregion is to make the circular region "fit"a rectangular region. This also providesan excellent way to estimate the area of acircular region, thus allowing the student

to see if his answer is reasonable. The onlymaterials needed for the class are construc-tion paper, scissors, paste, and a ruler.

The study of areas logically begins withrectangles, squares, parallelograms, trape-zoids, and triangles. After careful develop-ment of the formula for the area of a rec-tangle, when the time comes for the studyof circles and circular regions, the classmay ask whether the area of a circularregion can be adapted to the area of arectangular region.

The study of circles begins with experi-ments to see how the circumference of acircle compares with the diameter. It maytake a couple of class periods and half ahundred circular objects, or each studentmay do about five experiments at homeand report the results in class. The mea-surements are tabulated, and the classlooks for a pattern. Experiments continueuntil it is evident that in every instancethe circumference of a circle is approxi-mately three times the diameter of thecircle. The ratio of the circumference tothe diameter is identified as pi, and thisleads to the formula for the circumferenceof a circle, c = /rd.

The circle has been defined as the set ofpoints equidistant from a given pointcalled the center of the circle. The distancefrom the center to any point on the circleis identified as the radius, and a few meas-urements of diameters and radii confirm

138

the fact that d = 2r. The class is now readyfor experiments with the areas of circularregions.

Circles of many different radii are drawnon construction paper, and the many circu-lar regions are cut out. The center of eachcircle should be clearly labeled. Each stu-dent, is given one of these circular regionsto work with. Thus each student can pro-ceed through the following steps:

1. Draw a diameter. Mark the two radii onthe diameter, and mark one-half thecircumference (c/2) on each semicircleas shown in figure 1.

Fig. 1

2. Cut along the diameter. You now havetwo semicircular regions. Put these twosemicircular pieces together so the edgescoincide.

3. Beginning at the center each time, cutalong many radii, coming very, veryclose to the edge of the circular regionon each radius but not cutting the edge.This results in a string of pie-shapedwedges as shown in figure 2. (Generallyspeakingthe size of the circle will af-fect thiseach semicircular regionshould have about ten wedges.)

2Fig. 2

4. Paste one string of pie-shaped wedgeson a different piece of colored paper in


such a way as to suggest a rectangularregion as shown in figure 3.

2Fig. 3

5. Note that the length of this rectangularregion is one-half the circumference ofthe circle (c/2) and the height is theradius of the circle (r). Half the rec-tangular region is covered with the semi-circular region.

6. Paste the second string of pie-shapedwedges on the first as shown in figure4 so that the rectangular region is cov-ered and the wedges are not overlap-ping.

r

Fig. 4

7. Note that the circular region has nowbeen adapted to a rectangular region,and it is possible to estimate the area ofthe circle. Since the area of a rectangu-lar region is "length times width," thearea of the circular region is one-halfthe circumference times the radius.

After practice in estimating areas ofcircular regions, the class is ready for thedevelopment of the formula for the areaof a circular region.

Through their earlier experiences withmeasuring the circumferences and diame-ters of many circular objects, studentslearned that c = 7rd, or c = 27rr. The circu-lar region was cut into two parts, thus eachpart included one-half the circumference,or c/2. Using the formula 1/2 x c = 1/2 x(27r), or in simpler form, c/2 = ,rr, the


length and the width of the rectangularregion are given in terms of the radius ofthe circle. (See fig. 5.) With a little helpand direction, the students are able to seethat the area of the rectangular region(I x w) in terms of the circle is (irr) x r.Thus the formula for the area, of a circu-lar region is A = irr2.

Practice in finding the areas of circularregions follows the development of theformula. In each problem, students are en-couraged to estimate the area using theidea c/2 x r before using the formula it-self in calculating the area of the circularregion.

It is amazing how much confidence stu-

111111112 =

Fig. 5

dents will develop in a problem that wasonce dreadedhow to find the area of acircular region. Students are also intriguedwhen they find that the area of a circularregion can be made to fit a rectangularregion.

EDITORIAL COMMENT.--The technique of partitioning a region and rearranging the piecesto form another figure may be used to generate area formulas for other figures.

A parallelogram region can often be rearranged to form a rectangular region.

Cut along an altitude.

Rearrange to form a rectangular regionof equal area with base and altitudeequal to that of the parallelogram.

A triangular region can also be converted to a rectangular region.

Right triangle

Acute triangle

Cut at half-way point of the altitude.

Rearrange to form a rectangular regionof equal area. The base of the triangleis also the base of the rectangle. Therectangle is half as high as the triangle.

Cut at half-way point of altitude.Cut top piece along altitide.

Rearrange to form a rectangular regionof equal area. The base of the triangle-is also the base of the rectangle. Therectangle is half as high as the triangle.

Obtuse triangle


Cut at half-way point of altitude.Cut along altitude.

Rearrange to form a rectangular regionof equal area. The base of the triangleis also the base of the rectangle. Therectangle is half as high as the triangle.

A Pythagorean puzzle

GARY D. HALL North Judson San-Pierre Schools,North Judson, Indiana

This puzzle was constructed as a projectfor a graduate course. It has been used inteaching sixth-grade mathematics.

Purpose

The purpose of this project is to teachthe Pythagorean theorem to children whohave no background in plane geometry.The ideas of squared numbers and theconcept of a 3-4-5 right triangle are alsointroduced.

The child is taught these related con-cepts through the use of a brightly coloredmanipulative puzzle that guides him toform relationships involving area.

Materials

right triangle with sides 6 inches, 8inches, and 10 inches.square (side 6 inches) and 1 parallelo-gram (sides 6 inches and 10 inches; anangle equal to the smaller acute angle

of the triangle described above). Eachof these figures has an area of 36 squareinches. They should be painted the samecolor.

1 square (side 8 inches) and 1 parallelo-gram (sides 8 inches and 10 inches; anangle equal to the larger acute angle ofthe triangle described above). Each ofthese figures has an area of 64 squareinches. They should be painted the samecolor.

142

Fig. 1


1 square (side 10 inches) with an area of100 square inches.

50 squares (side 2 inches), each with anarea of 4 square inches; 9 should bepainted to match the 6-inch square; 16should be painted to match the 8-inchsquare; and 25 should be painted tomatch the 10-inch square.

1 frame like that shown in figure 1.

Empty the puzzle again and give thechild the pieces, substituting he smallerparallelogram for the smallest of the threesquares. (Note: These two pieces arepainted the same color to facilitate the con-clusion that they are the same area.) Thechild should then conclude that the smallparallelogram and the small square are thesame size by constructing figure 3.

After emptying the puzzle the third time,give the child the largest square, thesmallest square, and the triangle and sub-

Fig. 2

Procedure

Empty the puzzle frame and give thechild the basic puzzle pieces (the triangleand the three large squares). He shouldconstruct a figure such as that shown infigure 2.

Fig. 3

Fig. 4

stitute the larger parallelogram for themedium square. (These two are alsopainted the same color and are the samesize.) The child should construct figure 4.

The fourth construction (fig. 5) is

Fig. 5


formed by emptying the puzzle frame andgiving the child the two parallelograms, thesmallest square and the medium square,and the triangle. After making figure 4,the child should conclude that the twoparallelograms are the same size as thelargest square, hence the two smallersquIres are also the same size as the largestsquare.

The next series of constructions is de-signed to reinforce the idea of the Pythago-rean theorem and to introduce the idea ofsquared numbers and the concept of a3-4-5 right triangle.

Empty the puzzle and put in the fiftysmall squares. The nine squares paintedto match the six-inch square go in the smallcompartment, the sixteen squares paintedto match the eight-inch square go in themiddle-sized compartment, and the twenty-five squares painted to match the ten-inchsquare go in the largest compartment. Also,put the triangle in the middle. Have thechild take the squares from the large com-partment and fill up the two smaller com-partments. Then have him take the squaresfrom the two smaller compartments andconstruct various designs, such as those

Fig. 6

shown in figure 6, in the large compart-ment.

Use of the puzzle would not necessarilyrequire a rigid lecture type of presentationsuch as has been outlined. If students aremerely allowed access to the puzzle, manyof them will make interesting discoveriesin their free time.

Easy-to-paste solids

M. STOESSEL WAHL Danbury ,State College, Danbury, Connecticut

Geometry has gained an accepted placein the mathematics curriculum of the ele-mentary school. Elementary teachers havefound that their pupils enjoy using com-passes to construct designs and mathe-matical figures. Some teachers, who havetried the construction of three-dimensionalsolids, have found the pasting techniquedifficult for young fingers when conven-tional patterns were used. Here is a set ofpatterns designed to make the pastingproblem easy enough so that third-gradestudents can succeed with the simplersolids. Teachers have been most enthusi-astic about their pupils' delight and suc-cess in using them.

The construction of these five solidstetrahedron, octahedron, decahedron, ico-sahedron, and hexahedronis similar, sogeneral directions will be given. The basicpattern guides for each solid are included.

The materials needed include a com-pass, a straightedge or ruler, pencil, scis-sors, and an instrument, such as a stylusor table knife, for scoring the lines. Theoak tag used in file folders is a satisfactoryconstruction material. Construction paperis more colorful, but does not hold up aswell. A good glue that dries quickly is im-portant (Elmer's, Uhu, or rubber cement).

The directions given here are quite de-tailed because younger children need very

specific instructions. The construction oftriangular grids with a compass should bepracticed first because accuracy of the gridconstruction is important.

Construction of the triangular grid1 Place the oak tag paper on a newspaper

padding to prevent the compass pointfrom slipping.

2 Using a ruler, carefully draw a straightline across the middle of the paper.

3 Mark a distinct point near the middle ofthe line.

4 Set the compass to any convenientradius. An inch and a half is a goodradius for beginners. The same radiuswill be used throughout the construc-tion.

5 Place the point of the compass on themarked point and swing a completecircle.

6 At the point where the circle intersectsthe straight line, place the point of thecompass and draw another circle. Con-tinue in the same manner across the line.

7 Continue drawing circles by placing thepoint of the compass where two arcsintersect. The paper should be coveredwith intersecting circles.

8 Draw straight lines between the pointsof intersection of the circles so equilat-eral triangles are formed. (It should

145


2

5

6

6

Aik

AIM

segaveggegiAlalepor",woommew

AgrATAVAWATAIMAMAIPAIPMAIMPVIRAIMIMIMA

11111M72.01111,41/

AVAV vgr AtTAY/WAWA v vaNky4. r_T AuTyATAIr */ vATAvAvo A-ik VAYAYAv 41W AvAYAWAAr vmAvAvAvAvAvAv

Figure 1. Triangular grid

look like a piece of graph paper formedby triangles instead of squares. SeeFigure 1.) This is the basic triangulargrid for four of the solid patterns.

Directions for transferring pattern,and scoring, cutting, and folding

1 Transfer the numerals and letters fromthe pattern guides in this article to theconstructed grid. Check to see that nonumeral or letter is omitted. (All pastingedges are supplied in the pattern.)

2 Using a .ruler and a stylus, or semi-. pointed instrument, score each line in

the pattern. This ensures the necessarysharp folds.

3 Cut out the pattern. Also cut whereverbold lines occur.

4 Carefully fold on all lines. (Just one linenot folded can spoil the solid.) .

5 Optional. Fold the solid experimentallyto get the feel of its shape. Some solids,especially the tetrahedron, can betemporarily secured without glueing bytucking in the Y flaps (Fig. 2).

Directions for pastingand finishing the solid

1 Pasting is done so the numerals appearon the inside of the solid.

2 For patterns containing X labels, startby pasting the triangle labeled X5

AVAVAi

NAVAVAVvArmAry

Figure 2. Tetrahedron

directly over the triangle labeled 5, thetriangle X8 over the triangle labeled 8,etc. Paste so the triangles fit togethercarefully and hold long enough for theglue to take a ;irm hold.

3 A string should be added before the lastflap is pasted. It should be attached to asmall button or object large enough notto pull out.

4 After all X flaps are pasted, the Y flapsare pasted to the outside of the solid.(A more experienced person could tuckthe Y triangles inside, but this is toodifficult for the younger children.)

5 The solid can now be admired!

The delight in the child's eyes will morethan compensate for the slightly grubbyresults of the first efforts. It is wise to workwith the simplest solid, the tetrahedron,first. After the child has sensed the im-portance of accuracy in his grid construc-tion, and felt the need for care in scoringand patience in pasting, then it is time totry the octahedron and the decahedron(Figs. 3 and 4).

VA LI WAWA kVAVAVAVAVAVAV

Ble TRAY VFigure 3. Octahedron

seeTAVAVMAYAN

VAWAY

A rdi ftFigure 4. Decahedron

The icosahedron is more difficult. Theteacher may prefer to ditto this pattern onoak tag for the first experience with thistwenty-faced solid (Fig. 5). This may beaccomplished by hand-feeding the oak taginto the ditto machine. Later, the students


DIVA FAYYAVAVAVAITA

WAN SWAVAVAWAWA

Figure 5. hosahedron

who wish can construct their own ico-sahedrons, as they did the previous solids.It is interesting to note that often theslower children are more successful thanothers because they have developed thepatience that brighter children seldomtake time for. It is a good experience forthe slower learner to have the brighterchild look to him in admiration!

11.11 4ini136 M

Figure 6. Hexahedron

FOLD ASUSUAL

FOLDINWARD

---AFTERCUTTINGSOLID

For the cube, or hexahedron, a squaregrid is needed (Fig. C). On the elementarylevel it is preferable to use graph paper forthis pattern. This pattern, too, differs fromthe conventional pattern so that pastingis made easier. In this pattern some linesare folded in and some folded out, so alittle more care is needed in scoring andfolding.

Some teachers like to motivate theirpupils by displaying a solid made by theteacher as evidence of the kind of workthe pupil may be capable of doing in amore advanced grade. Figure 7 is a pat-tern for a sixty-faced solid for the teacherto construct. It is based on the triangulargrid construction and is also easy to paste.


"AVM AryVANATAVVYT4555bATVAVAVilivAVAVATS

ATNAVAvA NAVA&VA1 FAVAVAWAVAVAYA

Y; TUCK INSIDE

NAVA& ATVI

Yo GLUE OUTSIDE

Figure 7. 2ixly-faced solid

DecorationSome pupils cover the solids with glitter,

or spray-paint them. Some find ingeniousand inexpensive ways of decorating theirsolids. The art or home economics depart-ments may cooperate on these projects.The solids arc beautiful when covered withvelvet and decorated with sequins, butquite difficult to make.

At times, these solids are used asChristmas tree decorations, but manystates have new laws forbidding flam-mable tree decorations. The childrencari also make them into attractive mo-

biles by using a bit of wire (cut coathangers) and black string.

Most of these solids were known toPlato and Archimedes, ancient Greekmathematicians who lived in the thirdand fourth centuries B.C. Mobiles werepopularized by the American artistAlexander Calder in our own century.It is interesting that an elementaryschool child can enjoy success in mak-ing a mobile that combines the con-tributions of geniuses who lived morethan twenty-three centuries apart.

EDITORIAL COMMENT.--A number of different materials may be used to construct geo-metric solids. Patterns may be traced onto acetate sheets (leave out the glue flaps). Useplastic model cement to form the edges. These figures are very effective for placement onthe stage of an overhead projector. Students may be asked to identify the three-dimensionalfigure from its two-dimensional projection. Or you may ask the students to place a cubeon the projector so that its projection has a hexagonal outline.

A permanent-soap-bubble geometry

M. STOESSEL WAHLWestern Connecticut State College, Danbury, Connecticut

For years the study of soap-bubble ge-ometry has been a source of surprise andfascination to individual students. How-ever, it has not been a satisfactory tech-nique for classroom study: just as theteacher holds up a good specimen for theclass to admire, the bubble bursts!

Fortunately, because a new art mediumis available, the students and their teachercan now make and study permanent speci-mens. This new medium comes undervarious trade names and can be purchasedin hobby shops that sell the new viscousfluid for making fancy glasslike flowers ofvarious colors. The wire they sell for mak-ing the flowers is suitable for our geometricmodels, but any thin' wire can be used.

The materials needed are wire, a wirecutter (old scissors might do), and at leastone bottle of the purchased liquid. Long-nosed pliers help make neater models, butbending the wires with one's fingers willsuffice.

Students usually begin by fashioning

This paper was presented in a workshop atthe NCTM meeting in Boston, 13 November1971.

149

wire tetrahedra or hexahedra about 1 1/4inches on a side, with convenient handlesabout five inches long (see fig. 1). Thefinished wire model is completely immersedin the liquid and then held up to the airfor a few moments so the thin film candry. Contrary to expectations, the filmdoes not coat the outside of the model.Surface tensions within the liquid force itto form special patterns inside the poly-hedra. When dry, it can be passed fromstudent to student for examination anddiscussion.

Fig. 1


For the inquiring student who wishes tofind out why the surface tension forcedthe liquid into such surprising patterns,there is an excellent paperback book avail-able. It describes lectures on soap bubblesgiven in London in 1889 to a juvenileaudience by Sir Charles Vernon Boys.' Forthe student whose interest or curiosity leadshim further, various pyramid and prismmodels lead to interesting discoveries. Bytwisting a wire in spiral fashion around acylinder and attaching each end to a wirehandle before immersion, a helix is formedthat is beautiful as well as mathematical(fig. 2).

One of the most fascinating forms isfortunately the easiest to make. Here arethe directions:

1. Bend a 1/4-inch hook at one end of a15-inch strand of wire.Make two twists of wire around a

cylindrical pill bottle of about 1 1/4inches in diameter.

3. Fasten the hook from the end of thewire onto the remaining length of wirehandle. Pull it taut and fasten well bytwisting the hook.

4. Pull the bottle out from the wire en-circling it.

5. Gently pull the two loops apart so thereis about half an inch between them.(See figure 3.)

6. Immerse the model in the liquid.7. Pull it out and hold it up to dry.8. Examine it closely. How many sides

does it have? Could a flea traverse theinside and outside without walking overan edge?

Immersion of the wire model producesa Mobius strip.

How far the student can go with thistechnique is limited only by creativity andinterest. Soap-bubble geometry is an open-ended study correlating mathematics andscience. Luckily, it is inexpensive enough

2.

1. C. V. Boys, Soap Bubbles and the Forces WhichMold Them, Science Study Series. Garden City,N.Y.: Doubleday & Co., Anchor Books, 1959.

Fig. 2

for classroom laboratory use. It can alsobe easily utilized as a mathematics andscience station in the newly popular openclassroom.

Cleo's clock

CLEO KAPROCKI

Forest City Elementary School, Forest City, Florida

n my teaching I have developed and suc-cessfully used the idea of associating thenumber line and the clock. Since the chil-dren had had experience with the numberline, my idea was to relate this experienceto the telling of time.

A six-foot folding carpenter's rulethetype with metal jointswas used as thebase for the number line. Starting at oneend of the rule, and at each joint there-after, a 21/2-inch diameter tag was fastenedto the rule. Each tag had one numeral onit, and the tags were ordered from onethrough twelve (1, 2, 3, . . . 12). To avoidconfusion, the rule was painted white, thuseliminating the original numerals on therule from any consideration.

The circular clockface, which was cutfrom 1/4-inch plywood, was 28 inches indiameter. A .handle and stand were cut aspart of the face to facilitate the clock's use(see figure 1). The clock had a backstandso it would be freestanding (see figure 2).

Magnets were attached 2 inches fromthe edge and 6 inches apart. The magnetswere positional where the hour marks ona clock would ordinarily be found (seefigure 1). The face of the clock was paintedwhite, and removable felt eyes, mouth, andmustache were added as motivational fac-tors for first graders.

To use the clock, a student (or teacher)places the end of the rule so that the taglabelled .1 is on the first magnet to theright of top center. The joint with the tagmarked 2 will land on the next magnet asyou move clockwise. By bending the ruleat the "2" joint, the tag marked 3 willland on the next magnet. Continuing thisprocedure, the result will be the "number-

151

28"

Fig. 1

line rule" bent around the face of the clock(see figure 3).

The clock is versatile because the num-ber line can be removed, and individualnumbers can be given to students to place

F10" --*-1

Fig. 2

9"

23"


in the appropriate places. Since the clockis large enough for the students to seeacross the room, children can set the handsof the clock at a particular time and askother students to read the indicated time.The clock can be used to test telling timeby removing all, or some, of the numbers.

This clock was found to be a motiva-tional device that attracted the children'sattention and held their interest. Its useproved to be a problem - solving method forstudents who found telling time difficult.Because of its versatility, this clock canhave other uses in the classroom. Fig. 3

EDITORIAL CdN:IMENT.--Kaprocki suggests relating a number line to the hour function ofa clock. One may also use a number line for the minute sequence. Students can make thetwo number lines simultaneously, thus noting the factor-of-five relationship between mark -.ings on the two scales.

Time telling may also be investigated through the use of a sundial as featured in thenext article. You may also want to construct a "water clock" using paper cups. The col-lecting cup may be calibrated by drawing marks on the cup to show the water level after 30seconds, 60 seconds, 90 seconds, and so on.

Calibrations

CIPour in water

, Small hole

Collecting cup

"We made it and it works!"the classroom constructionof sundialsM. STOESSEL WAHLWestern Connecticut State CollegeDanbury, Connecticut

M. Stoessel Wahl teaches courses in mathematics to teachers of elementary school

at Western Connecticut State College. Formerly, she taught

at Lincoln School, Teachers College, Columbia University, and

at the State University of Iowa.

In the early days, each household hadits own sundial where time was read fromthe slowly moving shadow traced on thedial face by the movement of the sun acrossthe skies. "Dialling"' was a subject in-cluded in the school curriculum. Later, thestandardization of time and the general useof accurate clocks rendered sundials obso-lete. Although there has been renewed in-terest in these time-telling garden orna-ments, many of today's children have neverseen them function.

It is fun to make your own sundial, butmost available instructions are too mathe-matical to be used by the elementary pupil.Fortunately, an old book on sundials,'"printed in 1902, contains an elementaryset of instructions formulated by H. R.Mitchell, Esq., of Philadelphia. Mr. Mitch-ell's rules have been simplified by the pres-ent author; for the past thirty years herstudents (and many of their pupils) havehad the pleasure of making their own dials.

* These materials were presented at a workshop,"We Made It and It Works," at the forty-eighthannual meeting of the NCTM in Philadelphia, April1967. Forty teachers from Florida to Canada con-structed sundials, each one custom-made for his ownlatitude.

I. Dialling is the proper spelling as it was usedin all the old mathematical treatises and texts.

2. Alice Morse Earle, Sundials and Roses ofYesterday (New York: The Macmillan Co., 1902).

153

This set of instructions, written in care-ful detail for elementary pupils, can beeasily adapted to the junior high schoollevel. For instance, where elementarypupils are directed to draw perpendicularsand parallels, the junior high student canbe asked to construct them. Simple over-head transparencies can aid the pupils infollowing the directions.

For the basic construction each pupilneeds an oaktag file folder, a compass, aprotractor and ruler, a sharp pencil, and apair of scissors. A plastic right triangle isuseful. Other equipment can later be usedto make a more permanent dial.

Construction of the gnomon

The gnomon is the part of the dial thatcasts the shadow of the sun. Its namecomes from a word meaning "one whoknows," for the shadow tells the correctsun time. To make a gnomon, one needsto know the latitude of the place where thedial will be used. One also needs to knowthe longitude of the location to correct thereadings to clock time. Both readingsshould be made as precisely as possible.

1. Write your latitude and longitudeas read from the map.

2. Place your file folder so the foldededge is nearest you. With a pencil


mark a point on the fold, one inchin from the left edge. Label it A.

3. With A as a vertex use your pro-tractor to lay off an angle equal toyour latitude.

4. From A measure three inches alongthis line. Mark the point and labelit C.

5. Draw a lightly dotted line, makingC the vertex of a right angle. r..x-tend this line till it intersects thefold. Label the intersection B. (Seefig. 1.)

CLatitude

"-angle

\\B fold

FIGURE I

6. The length of the hypotenusc ofyour gnomon AB will be importantin the construction of the face ofthe dial. Set your compass to ex-actly equal AB as a radius and putyour compass aside.

7. If you wish to make a fancy gnomon,draw a curved line between pointsA and B where you now have adotted line. Leave AB and AC un-changed.

8. On line AC mark off a point E soAE equals 3/4 of an inch. On lineAC mark off a point F so CF equals3/4 of an inch.

9. Draw pasting flaps about one inchwide. (See fig. 2.)

10. Cut out your gnomon, carefully cut-ting through both thicknesses ofcardboard so your gnomon will havea double thickness.

11. Score along line EF on both thick-nesses to make folding easier. Fold

Pastingflaps C

A B fold

FIGURE 2

back on these lines. Your gnomonshould stand upright. Later it willbe inserted in the face of the dial.

Construction of the dial face

I. Near the center of your folder marka point 0 for the center of the dial.

2. With 0 as a center, and AB thelength of the hypotenuse of thegnomon as radius, draw a circle.

3. With 0 as center, and the length AC(three inches) as a radius, draw aninner circle.

4. Draw a diameter through 0 extend-ing through both circles.

5. Place a protractor with its center onthe point 0 on the diameter. Lightlymark 15 degree angles in the upperhalf of the circle. Using a straight-edge, mark points where these 15degree angles intersect the inner andthe outer circle.

6. Label these points on the inner circlewith fetters a through k counterclock-wise. Label the points on the outer

circle with numerals I through 11

counterclockwise. (See fig. 3.)

Note: In steps 7 and 8 draw lines only inthe region between the two circles as infigures 4a and 4b.

FIGURE 4a

7. Through point a draw a line perpen-dicular to the diameter line. In likemanner draw perpendiculars to thediameter from points b through k.

FIGURE 4b

8. Draw lines parallel to the diameterby connecting lines between points1 and 11, 2 and 10, etc.

9. Find the point where the perpendic-ular through point a and the parallelthrough point 1 intersect. Call atten-tion to this important point by draw-ing a small circle around the inter-sections noted by the ordered pairs,(a,1), (b,2), (c,3), (d,4), (e,5),(1,6), (g,7), (h,8), (1,9), (j,10),and (k,11). You will see that thesepoints form a smooth curve. All theconstruction work completed to thispoint was done to locate these pointsaccurately. (See fig. 5.)


FIGURE 5

Marking the hour lines

1. Carefully draw lines connecting eachcircled intersection point with thecenter of the dial 0.

2. Extend the two lower lines into thelower half of the dial. (See fig. 6a.)

(11,k

(9,i

it 6,2)

AIS

FIGURE 6a

3. The diameter is also an hour line forsix in the morning and six in theevening.

4. The hour line perpendicular to thediameter is the noon or twelve o'clockline. The hours can now be markedon the dial face. (See fig. 6b.)

FIGURE 6b


5. On the noon line measure in 3/4 inchfrom 0 and label it. E. On the noonline measure in 3/4 inch from theinner circle and label the point F.

6. Using scissors or a razor blade cuta sharp line through line segmentEF.

7. Into the slash EF insert your gnomonso the A of the gnomon coincideswith the dial center 0 and the Es andthe Fs of the gnomon match those ofthe dial face. The gnomon will beperpendicular to the dial face.

8. Cut out your sundial by cuttingthrough both thicknesses of folder.

9. The mathematical portion of the dialis now completed. It can be placedin the sun with the gnomon pointedtrue north (not magnetic north) tothe location of the north star.

Making the dial more attractive

Once the hour lines have been estab-lished mathematically for your 'ocality, thedial can be enlarged, made smaller, ormounted on fancier shapes such as anoctagon. An art teacher c a shop teachercan help the students make permanentdials of metal cr wood. Some mathematicsteachers without this additional help mustrely on their own ingenuity. It is possibleto sew Roman $Iumerals and hour lines onthe cardboard with heavy carpet thread.The dial face can then be covered with foil(kitchen foil or fancy florists' foil) andgently pressed with the fingers so the nu-merals and hour lines stand out nicely. Theresulting dial is one that the pupil can takehome with pride.

Reading the sundial

With the gnomon pointed true north,the shadow cast by the sun reads true suitime. Hence, it corresponds to our artificialclock time only four times a year. Unlessthe pupil is made aware of that fact hemight be disappointed. Questions raisedby the difference between sun time andstandard time can lead to a study of im-portant scientific facts. The science teachercan be helpful in showing how latitude andlongitude affect these differences. Usingdata from the world almanac one can con-struct a graph (or use an analemma) tofind how many minutes the shadow is "sunfast" or "sun slow" that day. A studentdesiring greater accuracy can make a longi-tudinal correction as well. Since almanactables are given for the edge of time zones,each degree of longitude variation changesthe shadow by four minutes. If :11;s changeis incorporated in the making of the originalgraph the student will be able to read hissundial time to a surprising closeness toour clock time.

Conclusion

The time used for making a sundial inthe classroom can be justified from a mathe-matical standpoint. The student's skills inusing angles, parallels, perpendiculars, andordered pairs are put to a practical use.The student learns to follow directions, andthe subject matter can easily be correlatedwith science, art, and shop, or can be usedby the social studies teacher in the study ofman's measurement of time. The best justi-fication for the mathematical effort lies inthe eyes of the student who announcestriumphantly, "I made it and it works!"

Multipurpose Aids

The articles that follow differ in focus somewhat from those presented inearlier sections, Some of these articles do not describe manipulative aids inthe strict sense of the word. We include them because they illustrate howone manipulative aid can be used in teaching widely divergent concepts orbecause they describe how one kind of construction or technique can be ap-plied in a wide variety of situations.

The first article describes a truly versatile manipulative aidthe peg-board. Anyone who has access to masonite pegboard and a sack of golf teescan equip the class with sufficient pegboards for everyone. Fisher describeshow this aid can be used in teaching perimeter, area, angle measurements ina polygon, coordinate geometry, place value, properties of operations,number theory, and ratiu and proportion. One can hardly ask for more!

Golden's article on games in mathematics is important for two reasons.First, it shows that games can be useful teaching aids. Second, it points upthe fact that all teaching aids are not necessarily made by the teacher or bycommercial distributors. Some of the most successful ones are those createdby your own students.

Too often we complain that not enough is done to show how mathe-matics is related to other areas of human endeavor. Beougher's article givessome very helpful suggestions for relating mathematics and science. Evenmore pertinent to the theme of this collection, Beoughet shows how themodels we use and construct for science teaching have a role to play inteaching mathematics. included among his devices are a gravity model, asolar-system model, and instruments for measuring sun elevation and forcalculating heights of tall objects.

157


Ma lative devices have their role in the earlier stages of concept de-velopni,nt, but how does one make the transition from the concrete to themore abstract or symbolic modes? Williams suggests that classroom chartsdeveloped by teachers and students can help. The charts can be summariesof things experienced firsthand.

The bulletin board can be more than something that lies flat against thewall. It can have three dimensions and come to life. Children can manipu-late the components of the board as they develop the illustrated concepts.Hall gives us some hints for making bulletin boards that are both attractiveand useful teaching aids.

Finally, there is the dream of using all our manipulative aids in an ap-propriate setting and in the proper place in the curriculum. The mathe-matics laboratory provides one medium for integrating many of the phy-sical activities suggested by the use of manipulative aids. Davidson andFair describe how their school transformed a small room into a place inwhich students could carry out a math-lab program. Their success and en-thusiasm should give each of us the incentive to try out the many variedideas in this book and to share with others our own successes.

The peg boarda useful aidin teaching mathematics

ALAN A. FISHER Mt. Tabor School, Portland, Oregon

A. peg board* is an excellent arithmeticteaching aid. It will assist the teacher inpresenting new concepts and help the stu-dents visualize these ideas in a concreteform. It is useful at any elementary leveland is helpful in making algebra and geom-etry understandable. (Fig. 1)

Figure 1

How to use it

Perimeter and area

Colored rubber bands slipped over thepegs make an attractive teaching aid inpresenting areas and perimeters of rec-tangles, squares, parallelograms and tri-angles. An arrangement like Figure 2 canlead to discussions, such as: "What a

* The peg board may be made by gluing wood dowels cutinto one-inch lengths in the holes of the Masonite peg board,leaving two-inch spaces between pegs. The tnaterial can bepurchased at any building-supply store or lumberyard. Thesize of the board ma, vary, but a 2 by 3 foot size with 144 pegshas proved satisfact.ory for me. Most 100 boards have 10 rowswith 10 pegs each but many times a larger number of pegs areneeded, and those pegs not in use can be separated by rubberbands. A white background sets off any display which is ar-ranged on the pegs, so that the display can be seen from anypart of the classroom.

159

square looks like," "Can any smallersquares be made by -.1sing more rubberbands and the same four pegs?" (Fig. 3)"How many triangles can be formed?""What a right triangle is and how to de-termine its area." The perimeter may bemeasured with a dressmaker's cloth tapeand the area determined in square inches.Bisecting angles look similar to Figure 4.The distances AB, BC, CD, and AD are

Figure 2

Figure 3

Figure 4

0


equal, which shows that the figure is asquare. It may also be observed that thediagonals bisect the angles.

Place value and hundred board

The board can be used in developingthe idea of grouping by tens and placevalue. One-inch wooden beads or blockswith holes large enough to slip over thepegs will help with the explanation. Startwith the basic idea of how many blockscan be put on the first row, giving the ideaof ten. If two rows are completed, we havetwenty, etc. This material can also be usedto represent parts of one hundred, to pre-sent per cents greater than one hundred,and also amounts less than 1 per cent, ifwashers are slipped over one peg.

Commutative law for multiplication

The commutative law for multiplicationcan be demonstrated by placing the blocksover the pegs in this design.

After this is done, ask the students if theanswer will be the same by multiplyingthree times four or rotating the board aquarter turn and showing four times three.

The law of compensation, or the idea ofinverse variation, can be demonstrated

Figure 5

d 41-0

'AI

very effectively, too. Start the discussionby using the chalkboard to present6X4 =24, (6÷2)X(2 X4)=24, or (2X6)X (4÷2) =24. Using any of the combina-tions, the answer will be 24. Then dem-onstrate on the peg hoard with the aid ofrubber bands. One may present the prob-lem, "How could we change the dimen-sions of a fence and still have the samearea?" (Fig. 5)

Squares of numbers

To represent squares of numbers, placebeads on the pegs to form squares, fourbeads thus

will represent 22; nine beads thus

will represent 32. This can continue, beinglimited only by the size of the board, pos-sibly 10' or 122.

Equal ratios

Equal ratios are made more interestingby using beads to show all of one ratio andonly half of the proportionate ratio on theadjoining pegs. Students can completethe ratio. See Figure 6.

000-(

Figure 6

-0000

Co-ordinate axes

In the selection of quadrants in teachinggraphing with co-ordinate axes, it is help-ful to start with a number line locating thepositive and negative values on the hori-

O.

X X X1,71IL X XX

Figure 7

zontal axis. This fixes in the student'smind the idea as to which direction tocount for the value of x, or the first of theordered pair. Then, work with the verticalaxis may be begun in order to give thestudents practice in placing the value ofthe second of the ordered pair, or y co-ordinate. When graphing, the position of

MULTIPURPOSE AIDS 161

the rubber bands can be adjusted to suitthe demonstration. (Fig. 7)

Degrees in a polygon

The formula 180° (n-2) is used in de-termining the total number of degrees ofthe interior angles in any polygon and canbe made clear by starting with the conceptof 180° in a triangle. Students will enjoyforming as many triangles as possible bystretching rubber bands from one point toall other points or pegs in the polygon'sperimeter. Interest will be high when theydiscover that the number of trianglesformed is always two less than the numberof sides of the polygon.

As teachers use the peg board, they un-doubtedly will see many other uses forthis device in representing mathematicalideas.

EDITORIAL COMMENT.Teachers may also use the pegboard to create statistical graphs,to explore the distributive property, or to function as an abacus.

To illustrate a simple example of graphing, consider asking each member of your classto identify his favorite colorred, blue, or yellow. On a pegboard identify three columns,one for each color. As each student calls out his color, place a peg in the proper column.The completed board produces a bar graph showing the number of students selecting agiven color.

. ..

. .

Red Blue Yellow

To use the pegboard as an abacus, identify columns with place values. Then establishthe rule that each time a column contains ten pegs, the student should replace them witha single peg in the column to the left.

Remove theten pegs and

'replace withone peg in theten's column.

Fostering enthusiasm Throughchild-created games

SARAH R. GOLDEN

Sarah Golden is a classroom teacher of a combination first- and

second-grade class at Ilalecrest School in Chula Vista City School

Diso'ict, Chula Vista, California. MatheMatics is a favoritesubject for her and for her class.

How can the young child develop en-thusiasm for mathematics, maintain neededskills, gain mathematical insight, and workat his individual level in a relaxed class-room climate of success?

One of the possible ways developed asa result of a recent visit by Donald Cohen,Madison Project Resident Coordinator inNew York City, to my combination first-second class. He introduced an originalGuess the Rules board game involvingmovement of imaginary traffic. Followingthis experience, one boy tried to make uphis own Guess the Rules game while theother children played with commercial vndteacher-made games. I noted that in play-ing some of the games, variations of theoriginal rules were used spontaneously bysmall groups to increase total scores possi-ble. For example, the chalkboard scoresfor Ring Toss ran into the thousands, al-though the indicated score for each tosswas only 5 or 15, "Oh," said Jeff, sitting onthe floor with one foot extended, "Whenhe rings my foot, it counts for one hundred.When he rines Robert's foot, that's fivehundred." Excitement ran high.

The next day we discussed the possibilityof each child creating his own board game.The following criteria were established:

1. The game must be fun to play.2. The game must have rules.3. The game should enable children to

play together and , become friends..4. The children must learn from the game.

Mathematics, science, reading, and spell-ing were suggested by the children as thecontent areas. The mathematics gamesmight include, they said, go ahead andgo back,. counting, addition, subtraction,and multiplication. Dismissal time arrivedall too quickly.

Traci (Grade 2)Object: Get to finish line.Rules: Use spinner or die to indicate number

of moves. Correct response, stay on that box.Incorrect response, back to previous box.'

L FINISH1219 + 0 =0970-1=468 +1=0

+1=1010= +Q

=8-10+ =12

0=8+ 12

0= 2-1516= +0 =3-819= Q+

=10-2018= +

=15-30D=10-3030=0+0100= +

START

FIG. 1. Number Game

1. It was surprising that Traci deliberately plannedto have negative numbers as responses in her equa-tions:

162

The following morning one girl sub-mitted a sketch of her game which useddice with three golf tees as pawns. Herexplanations were quickly grasped by theclass, As she proceeded to make a moredurable copy of her game, children usedthe following vocabulary to describe thesituation; Traci's game, Traci's pattern,Traci's copy, Traci's original, Traci's guide,Traci's directions, Traci's instructions.

Eager to start on his own, each childthen developed a game on a sheet of chip-board about 12 by 22 inches. Dice, ;spin-ners, or number cubes were used to indicatedirection and number of moves. In somecases, answers on cards were to be matchedwith the corresponding problem in thegame. The games showed a wide variationin difficulty, ranging from simple countingto the use of negative numbers. Dannyasked, "What if you landed on 3 and yourcard said, 'Go back 6'? Where would yoube? Off of the board?" He went on, "Iguess that would be a negative number, andyou'd have to get 'Go ahead 6' to get backto where you were on 3," Danny thoughtabout this and incorporated negative num-bers in his Bang Bang Chitty Chitty gameby using a starting line with negative num-bers behind it leading to the Repair Shop.The twenty-seven games included suchnames as Try and Guess; The HappyGame; Go Ahead, Go Back; TreasureIsland; The Whacky Racer; Streets andNumbers; The Counting Game; and BangBang Chitty Chitty.


In making games that involved roundedshapes, the children encountered the prob-lem of dividing the rounded portions intocongruent regions. They then learned howto use simple protractors to construct suchregions. Several lessons on curves followed,including simple, complex, open, andclosed curves.

Upon completion of the games the chil-dren were given repeated opportunities toexplain the object and rules of their gamesand to play them with their classmates.Scores were tallied and comparisons suchas "I beat him by 35 points" were made.

Deborah's Streets and Numbers gameled to a discussion of odd and even num-bers. We took a walk in the neighborhoodto find out how the houses were numbered.

For two players, Robert (Grade I )Object: Get Snoopy from the yard to his dog

house.Rules: Use spinner to show number of boxes

to advance, Correct answer allows player to re-main there until next spin. Wrong answer makesplayer go back to his preceding position.2

FIG. 2, Snoopy Game

2, Robert, a first grader, was interested in usingthree addends.


ez

Each child made a map of his street andthe houses on it and numbered the houses.We invited a member of the Chula VistaCity Engineering Staff to come to school

For one or two players. Kathy (Grade 1)Object: Put the greatest number of cards where

they belong in the counting ladder.Rules: Players take turns picking top card

from a shuffled pack. Cards show numerals 6, 12,14, etc. Player puts the card on his side of theladder. If incorrect, player loses turn. Each playercounts his correct responses at Ind of game todetermine winner,

Player Iputs hiscards on 1:this side

1

START

2

4

810

18

22

26

3436

FINISH

Player 2puts his

cards onthis side

16

FIG. 3.. Counting Game

iQ

to explain how numbers were assigned tonew homes in the city. Deborah revisedher game .to reflect her new learnings inthis areaodd numbers on one side of thestreet, even numbers on the opposite side.

The Bang Bang Chitty Chitty boardgame was devised by four boys as an out-come of their interest in mathematics,model cars, races, and the story of ChittyChitty Bang Bang. Speed limits were dis-cussed. The boys consulted Sports Ency-clopedia, almanacs, and other books forinformation on speed records and reason-ableness of posted speed signs. Of course,the intricate speedway had to be made wideenough to accommodate the toy racing carsused. Much experience in measuring re-sulted.

Summary

Child-made games foster enthusiasm andcontribute to learning in several ways.Children learn through a pleasurable me-dium of their own creation. The varyingdegrees of complexity in the ,games arecommensurate with a child's mathematicalconcepts and interests. In devising tho.

games, the children must try out their ideasand pursue them to a reasonable outcome.Language development is advanced as thechildren are highly motivated to explainthe games clearly to others. There is an


Leslie (Grade 1)Object: Get to home.Rules: Use die to indicate number of boxes to advance. Then follow directions in box.

'START

FIG. 4. Happy Game

exchange of information between players,especially when more than one correct re-sponse is possible. The teacher gains infor-mation concerning individuals and their

levels of thinking. For example, it was sur-prising to note some children's use of diffi-cult combinations and negative numbersand their free use of commutative and as-

For two players. Danny (Grade 2), Chris (Grade 1), Robert (Grade 1), Bobby (Grade 2)Object: Get to finish line.Rules: From a stack of cards, each player takes a card, which tells him what to do.a

10 1

Co to pit stop no.1MGo ahead 4 arrowsMGo back 2 arrows

Go ahead0

(Means staywhere you are)

Go to90mph

Go ahead+2

FIG. 5. Bang Bang Chitty Chitty

3. Numbers back of starting line were negative numbers and led back to the repair shop with its greaserack, etc.


sociative properties of addition in comput-ing total scores. The games provide teach-able moments for the teacher to introducepertinent material in greater depth.

Bobby (Grade 2).Object: Get to treasure at end.Rules: Use spinner to indicate number of

moves. If answer is correct, stay until next spin.If answer is not correct, go back to precedingplace.4

START

1 'ILE sr kLcc]la

500+70+8To treasure

1X 35+15

Pond ahead

47 X 0+47[Coss bridge

43+8Be careful

1000 +1000

FINISH

FIG. 6. Treasure Island Game

4. Each child has a marker of a different coloror of a different kind.

Chris (Grade 1)This game is played similarly to the Snoopy

Game (fig. 2). This first grader included multi-plication and subtraction in his game.

Object: Get to finish line.Rules: Use number cube to indicate number of

boxes for move. Correct answer allows player toremain there until next spin. Wrong answermakes player go back to his preceding position.If player lands on "Go back," he must go backthe number of places indicated in the box.

,c)

c\ 5

300÷20c

2X5

Go back1X3

36+3

80-60

13+24

7+7

Go ahead

1 +0

9 + 2

>--/0X100

Go ahead2X3

84+84

420+420

Go back9-8

210+210

60+60

Go ahead

3+4

40 + 40

65

49 27Go ahead

I X 0

Go back7X1

3X6

10+2

6x4

ui

FIG. 7. Try and Guess Game

Blast-off mathematics

ELTON E.. BEOUGHER

Elton Beougher is an assistant professor of mathematics at FortHays Kansas State College, Hays, Kansas. He teaches courses inmathematics and mathematics education for elementary andsecondary teachers.

A. more appropriate and informativetitle for this discussion might be "Astron-omy, Space Travel, and Arithmetic." Anattempt will be made here to support theclaim that the study of arithmetic and oftopics from space science can be coordi-nated in the elementary school program.Further, it is claimed that this coordinationcan enhance both the mathematics andscience areas and that each can promotelearning in the other. The author hopes toprovide some ideas that teachers mightuse to give variety to their lessons.

A number of specific gains can be madethrough this endeavor. A few that will beillustrated are:

1. Practice using large numbers2. Practice in estimation and rounding of num-

bers3. Provision of numerous examples of the use

of ratios4. Development of skill in measuring and using

angles5. Provision of examples Of scale models and

development of skill in building such models6. Provision of examples of many geometric

figures, both plane and solid7. Practice in use of the function concept, that

is, relationships between varying quantities8. Development of a better understanding of

our system of time measurement9. Drill in the fundamental arithmetic opera-

tions10. Provision of many examples for graphing31. (Trite as it may sound, but still valid) prep-

aration of students to be intelligent citizensof the space age

The reader will note that the advantagesclaimed generally deal with mathematicsunderstandings and skills. This reflects the

167

fact that the author's main interest is inmathematics education.

At this point the reader may say, "Stop!Put the countdown on HOLD. These claimscould be made for many areas of scienceor life, and such advantages could accruein many other areas." This is correct. How-ever, there are distinct advantages in favorof the approach through astronomy andspace travel.

One advantage is that students alreadyhave words in their vocabulary to preparethem for this study: g-force, weightlessness,LM, satellite, zero g, orbital path, apogeeand perigee, solar orbit, extraterrestrial,and so on. (This may not be true of theteacher's vocabulary!)

A second advantage is that a great dealof. motivation is already present for thisstudy. These are truly children of the spaceage with whom we deal. Space travel isas real and acceptable to them as a driveto the next town. If the reader doubts themotivation possibilities, he should take alook at the Saturday morning and after-school shows on television which cater tochildren. Science fiction and space travelhave long been the strong suit of these tel-evision hours. Consider as added evidencethe attraction of children to space toys andgames.

A third, major advantage is that thereare a multitude of aids, booklets, and filmsavailable on the subjects of astronomy andspace travel, many of which are free inclass quantities. At the end of this articleis a short, annotated list of such resourcematerials and sources of aids.


Specific topics

In cider to add credence to the previousclaims, some specific examples of how thisstudy might be accomplished in the ele-mentary school are in order. A more ex-tensive listing of topics and projects is

appended at the end of the article. Eachtopic might be one day's lesson, or, if

enough interest developed, could be ex-panded into a unit.

1. Connection between astronomy andtime measurement. Few people realize thestrong connection between our position inspace and our measurement of time. Ahistorical look at time measurement mightbe revealing in this aspect. It is recordedfact that ancient peoples told time by ob-servance of heavenly bodiesthe .sun,moon, and stars. Our present-day units oftimeyears, months, days, hours, minutes,and secondsall developed out of theseearly methods. To show the arbitrarinessof our time system and its structure, a

study of the length of "days" and "years,"and possibly "months" (if they had moons),of other planets would be very revealing.For example, Mercury's "day" is about 59Earth days and its "year" is about 88Earth days. In view of these facts, a livelydiscussion could ensue from the question,"If we lived on Mercury, how could wemeasure time?"

Pupils of early and middle elementarygrades would be fascinated by the calcula-tion of their ages in Mercury "years" orJupiter "years." They would be surprisedto find they were less than one Jupiter"year" old. (Jupiter's "year" is about 12Earth years.)

2. Study of astronomical navigation.There are two facets of thisearthboundnavigation and navigation in space. Ateacher and his students could study therelationship between earthbound naviga-tion and the angle of elevation of the sunat a given spot on the earth. The measur-ing of this angle could be accomplished byusing a simple instrument whose construc-

tion is discussed later. The study of theconnection between this angle and theideas of longitude and latitude could thenlead to an interesting study of navigation.

Navigation in space is seen to be amuch different and more complex task.This results from at least three facts. Noreference system such as latitude and longi-tude on earth exists in space. On earth,points on land and in the ocean stay fixed.The poles and equator do not move rela-tive to the earth. In space, no such fixedpoints exist. Any reference system is con-stantly moving, since all bodies in spaceare in constant motion. Secondly, twonumbers suffice to fix a position on theearth (latitude and longitude), but inspace three or more numbers are needed.Thus space navigation necessarily wouldbe based in a three-dimensional geometry(or maybe four-dimensional, if time needsto be considered) rather than in a two-di-mensional one. This is a fascinating studyin itself. A third inherent difficulty of spacenavintion is that much more planningand plotting of the course must be donebefore the flight begins. This was evidentin the recent moon flights. Only minorcorrections can be made after such a tripis in progress. Essentially, the flight mustbe completely preplanned down to the mostminute detail.

A study of navigation with your stu-dents could lead to fruitful results for laterwork in algebra, since there is a strongconnection between navigation and graph-ing and graphing plays a vital role in al-gebra.

3. Measuring distance in space. The dis-tances bctwen natural objects in space areso great that ordinary units of measure andnotation for numbers become unwieldy.For example, the distance to the star nearestto us (besides the sun) is approximately25,800,000,000,000 miles. Generally, as-tronomers and other scientists write suchlarge distances in scientific notation. Forexample, the number mentioned would bewritten as 2.58 x 10'3 miles in scientificnotation. Another example would be the

mean distance of the planet Pluto from thesun, 3.67 x 10° miles. This would be3,670,000,000 miles in usual notation.(Note the relationship between the powerof 10 and the number of places that thedecimal is moved in changing from onenotation to the other.) Astronomers alsouse another unit of measurement in namingsuch large distances. This is the light-year.One light-year is the distance that lighttravels in one year at the fantastic rate of186,000 miles per second (approximately).To provide an idea of the enormity of this,there are 86,400 seconds in one day. Ineach of those seconds light would travel186,000 miles. Think now of the 3651/4days per year to get an idea of the size of alight-year. To place it in the familiar frame-work, this is approximately 6,000,000,-000,000 miles. In this system, the closeststar is 4.3 light-years away. (This meansit takes 4.3 years for light to reach us fromthat star.) As mentioned before, this is

25,800,000,000,000, or 2.58 x 1013, miles.Another fascinating study is that of

methods of measuring distances in space.There are a number of sources on this sub-ject of "indirect" measurement, and manyof these are within the grasp of elementaryschool students. Later we will discuss theapplications of some of these methodswhich your students could make.

4. Variation of gravity on the planets. Afruitful study of ratios could accrue fromthis topic. Pupils would be highly motivatedto find what their weight would be on otherplanets and the moon. They would be sur-prised to learn that if they were fairly goodhigh jumpers, on the moon they couldprobably jump over their family's car withease. Furthermore, they could probablypick up their own mother and carry heraround, assuming she weighed.about 100pounds. This is because the moon's gravityis '1/4 the earth's, and objects would weigh1/; their earth weight on the moon. Yourstudents could find the weight of familiarobjects and multiply by IX; to find moonweight so as to test their "moon strength."


However, your students would be as-tonished to learn that on Jupiter they mightnot even be able to lift a medium-sized dog,since the gravity of Jupiter is about 21/2times that of Earth. The gravity of the sunis about 28 times that of the earth. In sucha situation, one of your students could noteven lift his own arithmetic book! The pos-sibilities for practice in multiplication andin the use of ratios are almost limitless.

5. Shapes of orbits. Most of your studentswould be aware from diagrams on televi-sion and in the newspaper that orbitingobjects, whether man-made satellites orplanets, do not travel in perfect circles.The path is an ellipse. You (or your stu-dents) can easily construct an ellipse byusing two pins stuck into a board, a pencil,and a piece of string with its ends tiedtogether to form a closed loop (see fig. 1).

Fig. 1

The pencil point is placed within the closedloop and then the loop is pulled taut bythe point. A closed figure is then drawn onthe board by moving the pencil around,keeping the string taut. Viewed from above,the result is as in figure 2. The closed figuredrawn is the ellipse.

Fig. 2

Ellipse

String

The two points where the pins are placed.are called foci (singular, focus). When aplanet orbits the sun, the sun is at one- ofthese foci. In general, when a small bodyorbits a much larger one under gravitational


influence, the large body is at a focus of theellipse. There are precise mathematical lawsgoverning this motion which would be outof reach of most students' minds, even atthe upper levels of elementary school.However, the motion paths are simpleenough to understand.

The other topics on the list at the endof the article offer many possibilities forstudy. With a little bit of thought and prep-aration and some reading on the part ofthe teacher, each topic could be developedinto an interesting lesson for students. Whatshould be kept in mind is that the purposeof the lessons is to use the subject matter ofastronomy and space travel to illuminateand expand arithmetic lessons and to moti-vate students in arithmetic. The sourceslisted at the end of the article would provevaluable in this light. Some of these wouldbe useful strictly for their content of facts.Others list aids that may serve to make theteacher's task more appealing both to himand to his students.

Models and instruments

One possibility for an interesting activityis the construction of simple measuringinstruments and devices for demonstratingvarious concepts. One such device worksnicely to illustrate number 18 on the listof topics. The device is made using a spool,two weights (large fishing weights from thesporting-goods store will do nicelyexperi-ment to find best sizes), and a piece ofstring about 31/2 feet long. Assemble thesematerials as in Lgure 3. Weight 1 is whirledaround in a circular path by grasping thespool. Weight 2 will hold weight 1 in apath of a certain radius depending on therelative weights of 1 and 2. Weight 2 actsas the force of gravity does. If we increasethe "force" of gravity by increasing theweight of 2, then we note that weight 1

"orbits" closer to the spool. Experimentswill show that it will also orbit faster in thisnew position. Planets 5n orbit around thesun behave in this way. The planets closeto the sunMercury. Venus, Earth, and

Weight I

Fig. 3

Marsmove faster in their orbits than theones farther outJupiter, Saturn, Uranus,Neptune, and Pluto. This is because gravityis stronger close to the sun than it is fartherfrom the sun.

In addition to the mental practice to begained from these studies are the skills tobe learned in planning and carrying outprojects involving mathematics and spacescience: Some such projects are suggestedin the list at the conclusion of the article.This is only a small sample of the possibili-ties.

There is much to be gained from thebuilding of scale models. The topics ofratio, multiplication, division, and scales arejust a few concepts that are involved. Also,pride in the completed project can contri-bute much to its value as an educationaltool. Depending on the age of the child,the teacher might have to offer more or lessguidance in the construction.

Building of a scale model of thesolar system

A word of caution should be offeredhere. It would be impossible to construct ascale model of the solar system all on onescale, that is, with the size of planets scaledthe same as the interplanetary distances,and have it fit in a classroom. If this wereattempted, in order for Mercury to belarge enough to see (say 1/2 inch in diam-eter) the model of Pluto would have to beabout 25 miles away from the model ofthe sun. (This gives some idea of astronom-ical distances). So unless you want your

"scale" model of the solar system to be 50miles across, you'd better use two separatescales! A convenient choice might be Vsinch = 1,000 miles for the size of planetsand 1 inch = 100,000,000 miles for thesize of the solar system. This would makethe solar system model about 6 feet across.A model of the sun could not be made onthe same scale as the planets and have itfit in your model solar system (it would benecessary to make it 9 feet across). Thusit would be best just to indicate its positionat the center of the solar system and nothave a scale model of it in your solar sys-tem.

Projects involving graphing

Projects that could certainly involvegraphing are numbers 3, 5, and 8 on listB. An easily made instrument could beused for measuring the angles in numbers3 and 8. The instrument is constructed byusing a nail, a piece of string, a small weight,and a piece of scrap wood about a footsquare (see fig. 4). The nail is placed atthe center of the arc on which the degreescale is indicated (i.e., at the center of thecircle of which the arc is a part). 'Ill:weight and string are attached to the nailand are used as a plumb line. In use, theinstrument is held so that the string passesthrough the 90-degree mark on the scale.

To measure the angle of elevation of

//7

2 / t\\s' Sun

Nail

20

30-'40

70 fp 90L.

String


the sun, the device is held plumb so that thenail casts a shadow down across the scale,and the angle is read from where theshadow intersects the scale. The anglemeasured in figure 4 is approximately 35degrees. In carrying out number 3 of listB, it would be important to measure theangle at the same time every day (prefer-ably at noon) and to face the instrumentin the same direction. A record of thesemeasurements over a length of time wouldcontribute to an understanding by studentsof the relationship between the changingangle of the sun's rays striking the earthand the change of seasons. A point ofcaution that should be stressed to yourstudents is that they should never lookdirectly at the sun in carrying out any ofthese measurements.

One source indicated that nine-year-oldstudents carried out project 6 with use of akit available for less than two dollars. Thetelescope was powerful enough for one tosee many details of the moon's surfaceand possibly to observe four of Jupiter'smoons. They could be seen with the nakedeye were they not so close to Jupiter as tobe blotted out by its bright light.

A number of the sources listed under Cdescribe how to construct sundials of vari-ous types. This would be instructive againin relation to our system of time measure-ment as well as valuable as a project to becarefully completed.

Indirect measurement and triangulation

To get a feel for the methods of indirectmeasurement used by astronomers, studentscould carry out "earthbound" measure-ments indirectly. Either trigonometry orratio and proportion could be applied. Useof the latter would involve measuring thelength of the shadow cast by the sun of anobject of known height (a student) and,at the same time, measuring the shadowof an object of unknown height (a tree).These two are then compared as follows:

boy's height tree's height

Fig. 4 boy's shadow length tree's shadow length


AWire

14- 12 -0114

Fig. 5

The use of ratio is obvious. This is some-what simpler than the method astronomersuse, but the basic principle is the same-"triangulation."

An easily constructed instrument forfinding measurements indirectly is basedon the principles of triangulation. Materialsneeded are two pieces of scrap lumberabout two inches wide and one foot long.These are fastened at right angles and aone-foot scale is drawn on one as in figure5. A sighting wire is attached at A by useof a small bolt or screw so that the wirecan be rotated around A. To use the in-strument, one sights along the wire at thetop of the object whose height, h, is to bemeasured (fig. 5). Care must be taken sothat the lower edge of the horizontal boardis kept horizontal (a plumb line along thevertical board would facilitate this). Thedistance to the object, d, is then measuredand a ratio is set up as follows:

reading on scale It

12

Again the use of ratios is obvious. Pupils inthe later grades would have had experiencewith ratios, proportion, and similar tri-angles and could easily comprehend themathematics of the instrument.

Conclusion

The versatile teacher can make manyuses of the instruments described previ-ously. Furth:rmore, the, topics, projects,and sources given in the appended listswould provide a wealth of material for thesame teacher.

A. Some suggested topics for studyI. Connection between astronomy and time

measurement

2. Navigation and astronomy3. Units of measure and methods of measuring

distance in space4. Variation of gravity on planets5. Relative size of planets6. Relative distance of planets from the sun7. Length of "years" of planets8. Shapes of orbits of natural and artificial

satellites9. Number of moons of planets as related to size

of planet10. Weightlessness and orbiting1I. g-forces12. Shapes of spaceships and reasons for these

shapes13. Size of rockets as compared to man14. Size of payloads compared to size of rockets15. How rockets work16. Decrease in pull of gravity as distance from

earth increases17. Why planets are spherical18. Speed of orbiting object as related to distance

from object it is orbiting

B. Some suggested projects1. Construct scale models of the sun and the

planets.2. Construct a scale model of the solar system.3. Keep a record of change of the angle of

elevation of the sun over time and study theuse of such a record by ancients to determineseasons.

4. Observe and time an eclipse of the moon.5. Keep a record of the change in the length of

days with the seasons.6. Build a simple telescope and calculate its

power.7. Observe the moon's craters and compare

their size with the size of land masses on theearth.

8. Determine the relationship between the angleof thetilt of the earth's axis with the plane ofits orbit and the seasons.

9. Construct a sundial.10. Measure distances and heights indirectly by

triangulation.I f. Taking the conditions of amount of light,

temperature, gravity, and so on, on a givenplanet, determine the physical characteristics

beings that could live there. Perhaps evenwrite a story about their lives.

C. Sources of content, ideas, and aidsAnthony, James K. "Events That Led to the

Discovery of Pluto." Schoo! Science andMathematics 53 (January 1953): 316-18.

Arey. Charles. Science Experiences for Elemen-tary Schools. New York: Bureau of Publica-tions, Teachers College, Columbia University,1961.

Bernardo, James V. "The Space Age: Its Impacton Education." School Science and Mathe-matics 63 (January 1963): 5-19.

A short history of the space age and NASA.Discusses the role of NASA in elementary andsecondary education and details t'ztt aids andservices available to educators thrr ugh NASA.

Blough, Glenn 0., and Marjorie H. Campbell.Making and Using Classroom Science Ma-terials. New York: Dryden Press, 1954.

The Book of Popular Science. New York:Grolier.

Deason, Hillary J., and Ruth N. Fey. ScienceBook List for Children. Washington, D.C.:American Association for the Advancementof Science, 1961.

Gleason, Walter P. "Real Astronomy." SchoolScience and Mathematics 54 (January 1954):31-38.

Discusses construction and use of someeasily. made instruments that demonstrateastronomical concepts and principles: (1) in-struments for measuring the altitude of thesun, (2) sizes for a scale model of the solarsystem,' (3) a sundial, and (4) a model todemonstrate how gravity keeps a satellite inorbit.

Johnson, LeRoy D. "A Physical Science Demon-stration: The Planets." Sc! Science and.Mathematics 53 (October 19:34: 569.

Jones, Louise M. "A Trip to the Moon." SchoolScience and Mathematics 60 (June 1951):486-87.

Discusses an excellent and promising inter-disciplinary idea: Have youclass plan a tripto the moonplanning food, clothing, jobsthey would take in transit, and so on.

Kambly, Paul, and Winifred ladley.."ElementarySchool Science Library for 1961," SchoolScience and Mathematics 62 (June 1962):419-38.. "Elementary School Science Libraryfor 1962," School Science and Mathematics 63(May 1963): 387-414. "Elementary SchoolScience Library for 1963," School Silence andMathematics 64 (June 1964): 467-82.

Good selection of astronomy, mathematics,,and general science books and books on-rockets, jets,. and space travel. Each book isannotated and the grades 1-6 are given forwhich the material would be most appropriate.

Mattison, George C., and Jacqueline V. Buck.A Bibliography of Reference Books for Ele-mentary Science. Washington, D.C.: NationalEducation Association, National ScienceTeachers Association, 1960.

National Aerospace Education Council. Aero-space Bibliography. Washington, D.C.: Gdv-ernthent Printing Office, 1970.

---. Free and Inexpensive Pictures, Pamphletsand Packets for Air 'Space Education. Wash-ington, D.C.: National Aerospace EducationCouncil.

A list of many free and inexpensive ma-terials, including sources from which thesemay be obtained by teachers.

Nelson, Leslie W., and George C. Lorbeer.Science Activities for Elementary Children.Dubuque, Iowa: Wm. C. Brown Co., 1955.

Norris, Theodore R. "An Inverse Square Rela-tionship in Science." ARITHMETIC TEACHER 15(December 1968): 707 -12.


Report of a sixth grader's experience inexperimenting and drawing conclusions basedon data and observations. .

Pettorf, H. Ronald. "Fundamental Ideas ofSpace Travel." School Science and Mathe-matis 63 (May 1963): 405-10.

Background information for the teacher(to catch you up with your students!).

Ray, William J. "Just for Fun: From Arc toTime and Time to Arc." ARITHMETIC TEACHER14 (December 1967): 671-73.

Reuter, Kathleen. "Sixth Graders ComposeSpace Problems." ARITHMETIC TEACHER 11

(March 1964): 201 -4.Report of fruitful results of discussions in

class which followed some of the first U.S.orbital flights. A. host of student-developedproblems based on _recorded facts about theflights are given. I

Ruchlis, Hy, and Jack Engelhardt. The Story ofMathematics: Geometry for the Young Scien-tist. Irvington-on-Hudson, N.Y.: HarveyHouse, 1958.

Excellent resource for teacher. Practicalproblems in many areas, including spacetravel.

Smith, Bernice. "Sunpaths That Lead to Under-standing." ARITHMETIC TEACHER 14 (Decem-ber 1967): 674-77.

Deals with the problem of helping childrenunderstand the summer and winter solsticesand the equinoxes.

"Space and Oceanography Bibliography." Instructor 79 (January 1970): 68.

A fairly extensive bibliography of audio-visual materials, pamphlets, and books suit-able for elementary school students.

EDITORIAL COMMENT.Have you thoughtabout correlating your studies of ancientnumeration systems with social studies(ancient history)? Or perhaps you mightconsider relating Mathematical languageto the study of other languages. Have yourealized that when you teach latitude andlongitude, you are teaching a mathematicalcoordinate system for a sphere?

The Function of Charts in the Arithmetic ProgramCATHERINE M. WILLIAMS

Ohio State University, Columbus

MANY teachers do an effective job ofusing concrete manipulative ma-

terials to help children with initial stagesof the formation of arithmetic concepts. Alltoo frequently, however, the teachers shiftso rapidly from these concrete, visual ex-periences to abstract verbal symbols thatchildren are unable to breach the gap.Children need intermediary helps paced totheir emerging insights. Charts are onetype of visual aid which can be developedto help children along the road from needfor concrete materials to ability to operatewith abstractions.

Usually children should participate inthe development of the chart. This par-ticipation fosters the habit of organizingone's own ideas and of resourcefully work-ing out one's own self - helps. Then, too,participation in developing a chart servesas insurance against introduction of ma-terial in chart form before the children areready for the organized presentation. Be-cause the pupils understand a chart whichthey have helped to work,out, it becomesthe meaningful reference which they needtemporarily.

One of the common mistakes in theteaching of arithmetic is that charts (thisincludes all types of tables) which appearin arithmetic texts are introduced beforeexperience has made them meaningful.Not even the most simple of organizationalcharts should be introduced until afterconcepts have been developed through richfirst-hand experiences. This seems obviouswhen we remember that the sense, themeaning of any logically organized formis a development process ;which evolves asexperience provides new insights. Since themeaning of any logical_ofganization is anoutgrowth of one's own experiences, chil-

dren should not be denied opportunities tobuild up their own organization. Afterchildren have worked out their organiza-tion, they have basic understanding andare ready critically to study the mathema-tician's mature organization. Only thenwill they be able to appreciate his refinedterms and short-cuts and/or his inclusionof information which they may have over-looked or lacked.

In this article it will be possible to ex-plore only a limited number of situationswherein the teacher-pupil developed chartcan be an important aid to the learning ofarithmetic. Often chart-making situationscan profitably include all members of aclass group. However, when learning ispaced to maturation, teachers will findthat some charts ought to be worked outwith sub-groups to meet their readinessneeds.

Developing Charts

Teachers have discovered that beginnersmaster forming letter symbols more read-ily when the manuscript alphabet is ex-hibited for their reference. Comparableguidance for the formation of figure sym-bols is seldom provided. However, aschildren learn the number symbols,teacher and pupils together might wellwork out a helping chart. For such a chartblack crayon might be used for making thesymbols. Red crayon might well be usedfor arrows which would indicate the paththe pencil should take. Perhaps a dottedline could be used to indicate the part tobe made last, for example.

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. Sometimes charts should not be com-pleted all at once but should evolve cumu-latively as new experiences contributefurther pertinent information. A simplechart of this type might bear the title,Signs We Should Know. Signs of opera-tion, the equal signs and the dollar signare among those which would have a placeon this chart.

When children reach that sbge of ma-turity where familiarity with the terms fordesignating the numbers involved in eachof the four fundamental processes isdesirable, a chart might be developed.This chart would include a simple exampleof each process, accompanied by theproper term for designating each number.This chart would become a useful refer-ence until such time as children are thor-oughly familiar with all of the .terms.

The three charts already mentionedtypify the sort of information which isusually introduced with care, but whichbecomes unnecessarily complicated forchildren because they are not providedwith helps to tide them over until the in-formation is firmly established. The childfor whom the coordination required toform the figures is a struggle should not bedenied adequate reminder concerning thepath the pencil should take. Children whoare uncertain of which operation is indi-cated or those for whom memory fails andterms arc a matter of guesswork cannotdevelop the sort of confidence which en-genders further success. We teachers aresheik- sighted, indeed,- if we fail to.developwith children reference materials so thatthey an confidently and resourcefullyundertake their tasks. GiVen such helpsalong with encouragement to becomeindependent of,these aids, children will de-light in reporting, "I don't have to lookanymore," or "I'm going to try this timeand not look even once."

Occasionally charts should be developedsimultaneously with the initial -learningexperience. A chart, Rectangles, might bedeveloped when children are learning to


recognize and identify the rectangle. Inthis case actual objects in the classroom,the door, the windows, and the like arebeing studied and, if recognized as belong-ing, are listed. The children might nextbring in cut-outs from periodicals andmount them on another chart, More Rec-tangles, thereby extending the number ofobjects recognizable as rectangular. Afterchildren have had similar experience withsquare, circle and other common geo-metric figures, teacher and pupils togethermight well summarize the information ina chart, Shapes Have Names. This chartwould. .include the picture symbols ac-eornpanied by appropriate word symbols.

Another type of chart which shouldclosely follow experience is illustrated bya picture chart of liquid measure. Afteryoung children have used water to fill andempty into one another of the standardmeasures, along with ordinary containerssuch as milk bottles, a simple picture chartLiquid dleasurc would serve to summarizethe information so recently discovered andverified throtigh the experience. Thechart would serve also as a reference untilsuch time as a reminder becomes no longeressential. When children are more mature,the pictures would be unnecessary. How-ever, participation in working out theorganization of tables of measure, follow-ing initial experience, can enrich' thelearning at all levels. A chart regardingthe measurement of time for exampleshould not be developed all at once. Itshonld be an outgrowth of experienceswhich, perhaps, had their origin in the dis-covery that seven days equal a week.When .week, month and year have mean-.ing, their relationship might be workedout into a simple chart. Informationgathered over a feW school years is neces-sary before a complete table of time meas-ures can be meaningfully developed. Theadvantage of pupil participation in or-ganizing their own table-type chart obvi-ously is not that they will develop a uniquechart. The advantages in their participa- .


tion are that the learners think throughwhat belongs and work out where itbelongs in the scheme. When this type ofchart seems complete to the learners, theyshould be encouraged to compare theirresults with standard tables found in text-books designed for their level of maturity.Children who have this sort of experiencehave been spared- rote memorization ofanother's organization of information,they have been guided to organizing theirown information into a table. This latterprocedure affords a very different qualityof e:perience for the learner. .

Let us turn attention to developing in-sight and facility with various ways ofgrouping whole numbers. If purposefullearning experiences do not occur withfrequency and variety, the teacher shouldsupplement them with contrived experi-ences. Experiences should be planned toutilize a variety of concrete materials forchildren to think with, to experiment with,to use for verification (which leads toformulation of generalizations aboutgroups of numbers. When children demon-strate understanding as they work withconcrete materials, they are ready toorganize information in simple chart form.

'An example is a chart, About 8, made toshow the addition facts about eight. Foran easy-to-make chart of this' type darkbackground satisfactorily sets off white,half-inch, self-sticking discs (Dennison'sPres-a-ply Labels). Not only does thischart summarize the various combinationswhich total eight but also it is useful inestablishing the relationship between ad-dition and subtraction.

Charts may well be developed to buildunderstanding of a proceSs, for example,division. A chart of this sort might growout of a situation where children need tofind how many five-cent items can be pur-chased with thirty-five cents. Childrenshould be encouraged each to figure outhis own way of arriving at the solution.One child may 'perform successive sub-tractions. Another may fence off dot pic-

tures in this fashion . . /... . Anothermay use multiplication as an aid. Whensolutions have been arrived at, the teachermight ask each child to demonstrate themeans by which he arrived at seven. Thevarious ways should be studied and, per-haps, become a part of a chart We LearnTo Divide. The various methods of arrivingat solutions can help children to importantdiscoveries. Experimentation will showthat the new method, division, is shortestand fastest. At this point the teacher hasan obligation to help children to realizationof the fact that people lived for manyyears before anyone discovered this faster,shorter method called division. Childrenwill appreciate mathematics if constantlythey are helped to see that number is aningenious system worked out by man ashis insights developed. After several ex-periences with the new process, division,children should be ready to work out whatadditional information they think belongson the chart, We Learn To Divide. Pro-ducing such a chart challenges children toself-reliant thinking and, because itbuilds both confidence and insight, servesas impetus for using the short-cut, di-vision.

A group of charts which will aid in de-veloping insight into the process of divi-sion consists of several helping charts.'Helps for Dividing by 7 would be but oneof the series. Others would be developedby substituting various ones of the digitsfor the seven. To make the seven chart,mark off the paper into seven verticalspaces and ten horizontal spaces. The toprows from left to right would read

1 2 3 4 5 6 7

8 9 10 11 12 13 14.

Use of the chart is facilitated if the figureswhich are multiples of seven are shown incolor. Such a chart helps to build under-standing especially when remainders areinvolved. Confronted by the example,7(45, children will recognize 42 as the

product nearest 45. They will readily ar-rive at the quotient, six, with three re-mainder. Later when, children need totransfer remainders into fractional yarts,this chart enables them to see that tliethree is three toward the next seven orthree-sevenths.

When children are learning to under-stand large numbers both whole and thoseinvolving decimal fractions, they need agreat deal of guided experience designedto build insight into value dependent uponposition. When children experience a largenumber the teacher has both opportunityand obligation to help them to some reali-zation of what such a number really signi-;fies. A case in point occurred in a fourthgrade engaged in a study of prehistoric ani-mals. The earth was found to be about twobillion years old.' The teacher inquired ifanyone could write two billion in numbersand a volunteer did so correctly. Then shesuggested that the group might work to-gether to find out how much two billionis. They worked out a chart This Is TwoBillion, by starting with ten, 10 X10 =100and so on. In this manner the teacherhelped to extend the tens concept and tobuild understanding of position as well asto show children a good way of attempt-ing to grasp the meaning of a large num-ber.

Another type of chart, one showing thecomposition of a number, is also helpfulin developing understapding. For example,the number 630,734 .s partly composed of:

6 hundred thousands 600,0003 ten thousands 30,000

0* thousands 0,000.

Breaking down the number and making atotal for verification provides insight. Thefinished chart may be referred to in check-ing subsequent performances in readingand writing numbers.

A very different kind of chart is oneworked out to indicate ways numbers areWritten and read for special business usage.


This sort of chart would be worked outonly after children had developed consid-erable facility-with_ reading numbers. Forgaining background information for niak-ilig a chart to indicate correct reading ofnumbers in business relationships, childrenmight be asked to investigate how tele-phone numbers, street numbers, licenseplate numbers, dates and the like are read.Experiences will differ widely. Some chil-dren will be ready to cite the difference inthe way 1492 is read if it is a date, a tele-phone number, or, perhaps an amount ofmoney. .0ther children will need time tocollect data..When they have collectedpertinent information, the teacher willguide their activities so that pupils be-come familiar with the variations in eachcategory. For example, at least four typesof telephone numbers are essential. Typi-cal of the four are these: 5081, 6000,7700, 2-2209. After each, the correct read-ing should be indicated.

Production of a time line can becomeanother rich chart-making experience. Ifpupils work out a time line of their own,they discover relationships between fixeddates. With the complete chart for readyreference the scope and sequence of timebecome significant.

The direct teaching chart is anotherimportant type of chart. A simple illustra-tion of this type is one titled Things inPairs. For such a chart children will clipfrom periodicals pictures of items belong-.ing in pairs and before mounting, sort thepictures into those in which the pair is twoseparate items and those in which the"twoness" is apparent:but the two arefastened together as are a pair of eye-glasses. The term pair takes oon meaningand the concept is extended through theactivity. Other concepts such as that ofinstruments of measurement 'can be ex-tended through the process of makingsimple picture charts.

Another illustration of the direct tcacli-ng chart is one designed to show square

measure. On wrapping paper or building


paper of adequate size, children can meas-ure off a square yard, the square feet in asquare yard, and the square inches in asquare foot, -all in actual size. Relation-ships are apparent when they see the realthing instead of only the scale representa-tion which appears in textbooks.

Not all charts will be teacher-pupilmade. At times teachers will want to makeup their own demonstration charts. Theconcept of average, for example, can beclarified by use of a few charts designedfor the purpose. Such charts are effectiveteaching tools if used to stimulate thought-ful examination, followed by developmentof tentative generalizations which are ex-perimentally tested.

Usually we teachers overlook numerousopportunities for pointing out the rolenumber plays in other areas. Children willgain new understanding and appreciationif introduced to a chart showing fractionsin music. They will be interested to dis-cover that relationships between notes,and between the notes and rests in a meas-ure are actually fractional relationships.

The few charts which it has been possi-ble to discuss indicate that charts can befunctional aids to the teaching of arith-

metic. Each teacher will recognize manyinstances wherein the teacher-pupil de-veloped chart will meet the needs of herparticular group.

What is to be done with charts oncetheir initial purpose is served? If placed ina flip-chart rack or in an oversize "book,"the charts will serve as reference so longas they are needed. Also, they will serveas valuable review material and as mile-stones of progress. Since participation. inthe making of the charts is so importanta part of the learning experience, it isobvious that most of the charts should bediscarded at the end of the school year.Because they should be discarded it isneither necessary nor desirable that theybe made of expensive, long-lasting ma-terial. Newsprint size 24" X36", both plainand ruled, is inexpensive paper whichproves satisfactory for most of the chartsmentioned. Because it is large enough formaterial to be made visible from everypoint in the classroom, it is often prefer-able to the more expensive poster boards:usually available only in smaller size.Crayons and brush pens are both satisfac-tory for writing. Both are easy to writewith and both provide easy-to-read mark-ings.

Enrromm. CommENTs.Classes that use the laboratory approach may find student-madecharts-an excellent way to summarize laboratory activities. Teachers may find that "activitycards" or laboratory directions can easily be presented in chart format.

Using charts in the classroom may save many classroom hours often wasted writingmaterial on chalkboards. Once.prepared, the charts may be saved for future classes. Lami-nating them will add many years to their life expectancy.

Bulletin boardsfor elementary school arithmetic

DONALD E. HALLUniversity of Massachusetts, Amherst, Massachusetts

The bulletin board, an often-neglectedtype of visual education in the field of eleL.mentary ariththetic, is a powerful meansfor getting ideas across, an excellent at-.tention-getter and interest-stimulator.

The following suggestions are given forthe construction and maintenance of bul-letin boards in the areas encompass bythe field of number. The manner in whichthese materials and children's specialtiesare displayed indicates the type of teach-ing done and the concern for the use of onemore effective avenue of communicationin arithmetic education.

1 To sharpen a particular point that you

A

rilT ITT

Figure 1

# Figure 2

wish to emphasize, provide a bulletinboard title; often two-toned letters indefinite contrast to background colorare effective.

2 It is not always necessary to place titlesin the center of the .board. Letteringon a slant is permissible, especially inthe upper grades.

3 Attention is drawn to the board by thewording of the title. Sometimes ques-tions such as "How many. sets."' or"Can you fivre this out in the binarysystem?" serve this purpose very well.

179


4 Experiment with three-dimensionalmaterials. Lettering made from rope,sandpaper, etc., often enhances theideas. Wire or staple "things" to thebulletin board. Concs, cubcs, andcylinders, as well as pictures, may beused to illustrate effectively a conceptwhich, without the dimensional ap-proach, might not be understood.

5 Watch carefully the colors chosen formounting pictures and other back-ground material. If the material or ob-ject is predominantly warm in color,it should be counterbalanced with acool color. Try parts of silver and goldgift boxes for mounts and frames.

Figure 3

6 Definite emotional appeal is providedwhen there are distinct contrasts oflight against dark. Dark mounts tendto advance and light ones to recede.

7 The following props are usually avail-able to all teachers: mailing-tubes,wire coat hangers, glass blocks, ping-pong balls, whitewashed stones, seeds,corrugated paper, shoe boxes, coloredcellophane, yarn, rope, twine, sand,cotton, cork, plastic, felt, tiles,gummed paper, balsa wood, straws,dowels, paper milk-bottle caps, metalbottle caps, etc.

Usually, completed bulletin boardsserve their purpose in two or three weeks.The exception is the type of display to

which additions are made around a cen-tral theme or understanding daily orweekly. In a display where Steps One,Two, and Three are used, the final step issometimes left open-ended for the childrento discover.

Figure 4

The type of bulletin board probablyused least of all is the Review Board, yet itis an excellent way of summarizing a par-ticular subject or area in arithmet.'1,

A crowded, monotonous di;. ac-tually worse than no display at all. This isalso true of bulletin boards put up hur-riedly with the use of one thumbtack orpin per picture, or pictures poorly cut andmounted. An uncluttered board is to bedesired, for it is indeed true that too muchmaterial crowded onto a given space givesthe viewer visual indigestion. Generallyspeaking, stapled papers and pictures lackdignity and good taste. Common or plainpins are sometimes less offensive thanthumbtacks; masking tape is more easilycontrolled than transparent tape.

Through the use of convenient materialsand simple procedures, any elementaryclassroom may be revitalized in the area ofarithmetic. It is hoped that these sugges-tions will be used as a springboard for theinvestigation of vast, new possibilities inteaching arithmetic to today's children.

A mathematics laboratory fromdream to reality

PATRICIA S. DAVIDSON andARLENE W. FAIR

Patricia Davidson is coordinator of mathunatics for the Newton PublicSchools of Newton, Massachusetts.

Arlene Fair is an elementary teacher who serves as a mathematics

coordinator at the Oak Hill School in Newton.

Just a year ago last September, the mathe-matics laboratory program at the Oak HillElementary School was still a dream. Afew plans had been made during the previ-ous spring for an initial purchase of mate-rials, and the first job in the fall was tofind "the place." It was decided to converta small storage room on the basement floorinto a mathematics laboratory. By the endof the year, the dream had become arealitythe laboratory approach to learn-ing mathematics had become a vital partof the mathematics program in the wholeschool.

Since visitors to the lab ask so manyof the same questions"How did you or-ganize the room?" "Just how did you be-gin?" "What is the purpose of a mathlab?" "How do you urganize the chil-dren?" "Are the classroom teachers in-volved?" "How does the lab program relateto the class work?" "What materials areneeded?"we should like to share some ofour experiences with those of you who maywish to develop a laboratory program.

Getting started

First of all, we were going to have towait several months to have the roompainted (Does this sound familiar?)so

with the help of the children, we cleaned,washed, and made the room colorful andattractive in other ways. Some sixth-gradeboys, who enjoy this type of work and areparticularly adept at it, took great pride inrepairing and paintingin bright colorssome discarded tables and chairs, whichwere then decorated by the art teacher.(We have been asked several times whereto obtain such delightful furniture!)

Along one of the long walls wk: hunga piece of fabric on which we had sewnpockets to hold individual problem cardsand booklets.

PROJECTS IN MATH

Such a rack can be made any height orlength. To be most serviceable, the back-ground should 'be of a dark color withbright colored pockets adding. to the decor.The pockets can be made in different sizesto accommodate various-sized cards, book-lets, or project sheets, and they can be.

All photographs used in this article were taken by William Winston.

181


labeled if they arc to be used for the samething all thc time. We never have labeledours because we are 'constantly evaluatingthe written materials that we are develop-ing, and with the help of the children weare always adding new ideas.

The other long wall had shelving fromfloor to ceiling which we used for storingthe concrete materials. We organized. andlabeled the shelves in categories so thateach item could be found easily and thechildren could learn to put away their ownthings, Since our particular shelves had sixmajor sections, we used six main headings:'

'Blocks (with subheadings such as AttributeBlocks. Geo-Blocks, Poleidoblocs, LogicalBlocks, Discovery Blocks, . . .); Geometric".aerials (with subheadings such asModels, Mirror Cards, Geoboards, . . ,);Measuring Devices (which included draw-ing tools and all kinds of balances); Nu-merical Games (with subheadings such asCalculators, Chips, Abaci, Kalah, Gil-Wah-Ree, Tuf, Imout, Heads Up, Card Games,Slide Rules, Cuisenaire Rods, . . .); Strat-egy Games (with subheadings such asTower Puzzle, Qubic, Twixt, Jumpin, . , .) ;and Shapes and Tiles (with subheadingssuch as Tangram Puzzles, Pattern Blocks,2Mosaic Shapes, Linjo, Three-D-Domi-noes,

In addition, the room had some bookdisplay racks to hold other kinds of studentinstructional materials and cupboards thatcontain,xl .,abeled boxes of miscellaneousitems like paper, pencils, oak tag, construc-tion paper; scissors, string, rubber bands.needles, thread, buttons, tongue depressors,egg cartons, beans, sticks, and so. forth.

A large sheet of brown paper helped tocover one cracked wall while serving also

1. For a detailed listing and ordering informationfor most of the materials mentioned here, see P. David-son, An Annotated Bibliography of Suggested Manip-ulative Devices," TI117, ARITHMETIC TEACHER, Octo-ber 1968, These six categories were a consolidationof the fifteen used in this article.

2. Not available at the time of the bibliography,Pattern Blocks were developed recently by Elemen-tary Science Study. The materials and Teacher'sGuide are available from Webster Division, Mc-Graw Hill.

as a useful backing for hanging students'workfor example, various geometric de-signs, curve stitching, and graphing proj-ects. Also early in the fall, some Studentsmade coat-hanger mobiles with colorfulthree-dimensional models cut from contactpaper to hang from the ceiling.

We placed a bulletin board on the fourthwall with pictures from magazines headedwith the caption "Guess How Many."Underneath was a small table with jars ofpeas, beans, macaroni, rice, etc., for the"Peas and Particles" unit. A few s:udycarrels made by students out of Tri-Wall"cardboard and a portable blackboard com-pleted that side of.the room.

We started out with one small patch ofrug so that children could work on thefloor, and since that time parents havedonated enough scraps to "cover" the floorpatchwork style, Pupils enjoy working withmaterials on the floor. The rugs help to cutdown thc noise level and also the amountof furniture necessary.

Oak Hill Mathematics Laboratory

The picture wr.. taken in October afterabout a month's effort to transform a dirty,dingy room into a warm environment that"looks like" a Math Lab.

Several teachers who have visited thelab have used many of these ideas to con---

3. For information on Cardboard CAientry writeto Education Development Center, Inc., 55 ChapelStreet, Newton, Massachusetts 02160.

vert a corner of their omi classrooms intoa math lab area.

Working with children

Although we were able to have somesmall groups of children working with ma-terials during Septemberas they helpedus get the room readythe lab reallyopened in October.

During the month of October, our aimwas to have every student come to the labfor at least one 1/2-1 hour session. Inorder to make it possible for each teacherto come to the lab along with her students,we worked with whole classes, which oftenmeant that there were too many childrenfor the small room. But the orientation fora teacher with her own students is verynecessary. In these first sessions, we triedto set a tone-

1. that you can learn math not only withpaper and pencil, but also through the useof manipulative materials.

2. that the math lab approach involvesactive investigating, exploring, hypothesiz-ing, looking for patterns, and "doing"rather than being told or shown.

3. that mathematics is many things; thatthere are often many right answers to aproblem; and that usuallc, you can checkyour hunches yourself by means of thematerials.

4. that although much of the work willseem like fun and games, all of the labexperiences can be related to specific mathconcepts, to problem-solving techniques, orto modes of mathematical thinking.

5. that at the beginning, the lab teacherwill choose what activity you embark on,but once you have pursued enough of thematerials to know what some of the possi-bilities are, you will be given some choice.

6. that care of the materials is the re-sponsibility of each student; that the lossof one piece may mean that the entire setof materials is'unuseable; and that you areresponsible for putting the materials youhave .sed in their proper place.


7. that often projects will be started ormaterials introduced in the lab that will befollowed up in your classroom or at home.

Training of teachers

Simultaneously, the classroom teacherswere trained. We tried to do differentthings with each class during that firstmonth, in order to get most of the materialsin use. Also within each class, groups ofchildren worked with several differentthings. For example, in one third-gradeclass; a little group worked with PatternBlocks; another little group with balances;another little group with geoboards; whilea fourth group did chip trading activities,'

Chip trading for understanding of place value

two students worked with Tangram Puz-zles, and two explored Geo-Blocks. Wheneach class left the lab after their initialsession, the students took one or two thingsfor example, chips and geoboardsbackwith them. The teacher was responsible forreading manuals and getting help from:usto be able to continue the work. Thosechildren who had used these particularmaterials also helped the other children getstarted.

There were also workshops for all or

4. See P. Davidson, "Chip Tradinga Strategy forTeaching Place Value," Professional Growth forTeachers. Third Quarter Issue 1968-69, and "Ex-changes with Chips Teach 'Borrowing' and 'Carry-ing,' " Professional Growth for Teachers, First Quar-ter Issue, 1969-70. Croft Educational Services, Inc.,100 Garfield Avenue, New London, Connecticut06320.


some of the teachers on Tuesday or Thurs-day afternoons throughout the year. (Weare fortunate in Newton in that elementaryteachers have released time every Tuesdayand Thursday afternoon.) Also several

Ifiererk.

Measuring with spring scales

members of the faculty attended citywide,nath workshops, and needless to say, therewere many informal discussions in theteachers' room. Within the building, as wellas citywide, there was a constant sharingof ideas and of written sheets or math labcards.

Expanding the' program

After that first month, the lab really"exploded" to permeate the whole school.As much as possible, a lab schedule wasbuilt around the math periods of the entireschool, so that each class including kinder-garten and special classes could have 1/2-1hour each week. A copy of the schedulewas given to each teacher who was re-sponsible for sending her children tolab on time:

How many children came at on-,:.? Oc-casionally, the children still came b classeswith the teacher so that a -unit With -suchmaterials geoboards, rods, balances, orvarious me Asuring devices' could be intro-duced by the math lab teacher (Mrs. Fair,who had sometime released from other,teaching duties) with the help of the class-\room teacher, and then taken back to theClassroom.

More often a teacher would send one ofthe arithmetic groups, szpi 10 -1 2 children,

for special help in some area. These chil-dren were taught by the math lab teacheror by a capable college student teacher.Sometimes, the classroom teacher arrangedfor coverage of the rest of the class in orderto come too.

Very often, small groups of studentsworked in the lab on anything that themath lab teacher, or a college studentteacher, or a high school student teacher,Or a rent volunteer' wished to introduce,or that they might choose. The gamesbecame very popular, as we constantlychanged the rules to make them moremathematical and more challenging.

Nongraded sessions, with 4-6 childrenfrom three grades at once, or with threechildren from each grade, proved veryfruitful. Sometimes many units of workwere introduced; while at other times, thestudents of various ages worked on thesame thing, 'going as far as ability willallow., (Mast of the materials are so open-ended that they are appropriate for stu-dents, kindergartengrade 6 and higher.)We were in for many surprises as to whatchildren can do when given concrete situ-ationsit became clear that many of thecommon conceptions about age and abilitygrouping need to be challenged seriously.

After the program got started, thc chil-dren often acted as lab assistants. Childrenfrom all grades acted as teachers in an areafor which they had been checked as beingwell versed. Eventually, we accumulatedenough materials in the.lab to start a lganservice to each classroom. -"laity of theteachers established a math lab day onceor twice a week in their classrooms andused children to help.

The buddy system

Another aspect of the program was thebuddy system, in which sixth-grade stu-dents volunteered to work in math with

5. Through math lab sessions with parents andinvitations fcir them to visit the math lab in sessionwith children, several parents became interested inhelping in various ways.

first graders. A workshop (four sessions)was given to these sixth graders in the useof materials, with the stress on the meaningof number and number relationships. Thefirst-grade teachers identified those childrenwho needed help or enrichment, as the casemight be. Each sixth grader was assignedone of these first graders with whom towork throughout the year. They workedfor about one-half hour at a time, usuallythree afternoons a week. The improvementin attitude of all of the children was re-markable. A couple of youngsters who nolonger needed help did not want to give uptheir "buddies," and every first graderwanted to be included. We were never lack-ing in teachers (sixth graders); studentskept volunteering as those involved sharedtheir experiences. The comments of thesixth graders were interesting: "I didn'trealize teaching was so difficult." "It is suchfun to see him begin to understand it." "Ididn't think I could find so many ways toshow 3 + 4 = 7." "It is like having a realsister," (said an only child),

The buddy system gave the sixth gradersa real purpose, a feeling of importance, anda sense of responsibility. The children ini-tiated the system that if a sixth grader wasto be absent, he would call a friend on thephone to arrange for a substitute who wasto report to the lab teacher in the morning.In handling all of the materials and games,the sixth graders really improved their ownbackgrounds"to teach is to really knowit.

The mathematics lab fair .

The year culminated in a Math Lab Fairat which more than 350 children in theschool were at work with materials at thesame time in nine classrooms, the corridors,the gym, and the ninth lab. The picturesillustrate sonic of the activities. In oneevening, over 500 parents came to observeand to work with the children. The Fairwas also held on a Thursday afternoon forteachers from the other 22 elementaryschools in the city and for visitors fromother communities...During another morn-


ing, the Oak Hill children helped groupsof children from other schools in the citywhich were planning for a math lab pro-gram.

How to make a start

Starting a. math lab program in a schoolis no easy adven+.:re. The success of theprogram at Oak Hill is highly dependenton the cooperation and support of theprincipal, Mr. Samuel Turner, and the hardwork and enthusiasm of the entire staff.From a small beginning, the program isnow growing and expanding in ways thatcould not possibly have been ticipated.You must begin at whatever is ap-propriate to your present mathematics pro-gram,

ORIGNIL i`ants

,

flriginal games

In deciding what to purchase initially,consider each large category of manipula-tive materials and get some from each, asmuch as money allows: The materials spe-cifically mentioned in this article have beenparticularly useful for our program. Mate-rials like geoboards. Cuisenaire rods, bal-ances, chips, etc., should be bought in smallquantities, but there is no need to start withmore than one each of numerical games,strategy games, and puzzles. See which onesthe children like before ordering more, andalso use your own ingenuity in makingmaterials. Once you start, you will find thatthrough improvising, you often can addmany improvements that "hand tailor" thematerials to .the needs of your particularstudents. The children can make 'a great


many things and will enjoy creating someof their own games and problems. A mustis adequate space for proper care of themateri

But most of all you need one largebottle of vitamin pills, unlimited vitality

PROBABILITYAND

TISTICS7(

Probability and statistics

and enthusiasm, and the patience of Job,for you will not only be the mathematicscoordinator for your building and the re-source person for every teacher in the

school but you will be expected to know:

every game and hoW to "win"; the answerto every puzzle presented to you by kin- .dergarteners through sixth graders; whereall the missing pieces are; and tow to mendevery broken game.

But when yogi sense the enthusiasm and

The mathematics fair

see the growth in your students, you nodoubt wilkshlife'the feelings of the teachersat Oak Hillthat the laboratory approachto teaching mathematics is exciting andrewar ng.

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document resume ed 087 639 se 017 405 author smith, … · ed 087 639. author title. institution....

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