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'AUTHOR Smith, Seaton E., Jr.; Backman, Carl A., Ed. TITLE Games and Puzzles for Elementary and Middle School Mathematics. Readings from the ARITHMETIC TEACHER. INSTITUTION National Council of Teachers of,,Mathematics, Inc., Washington, D.C. . t

PUB DATE 75 1, NOTE 287p. AVAILABLE FROM National Council of Teachers of Matheiatics, Inc., ------1906 Association Drive,, Reston, Virginia 22091 ,

1 EDRS PRICE MP-40.76 PLUS POSTAGE. HC Not Available from EDRS.,. DEStRIPTORS Activity learning; Educational tames; Eleikentary' Education; *Elementary School Mathematics; *games; *Instruction; laboratory Procedures; *Learnihg Activities; *Kathematics Education; Middle Schools; StudentDevelopedMaterials; Teacher Developed Material's 4 N ABSTRACT . ''Ttilis,_,bakcontains ,s105 articles dealing with games for use in the el -1.1tary or middle school matkematics classroom.. All the articles orig nall ppeared in the "Arithmetic Teacher" between 1956 and 1914. IniiiitXTIis volume the papers are arranged in nine categories'(1) using games and puzzles,(2) whole,numbers, (3) numeration,(4) integers,*(5) :rational numbers, (6) number theory and patterns,(7) geometry and measurement, (8)reasonihg and logic, and (9),multipurpote games and puzzles. While some,of the games are aimed at review or practice on basic skills others were designed with multiple,objecti'Ves in mind. There is a great variety'sin the forms of the games; board games, crossword -type' puzzle=s, paper and pencil games, manipulative puzzles,for individual solution, and many others are included. Some of the games were developed by students; many others were designed by teachers. (SD)

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TO ERIC AND ORGANIZATIONS OPERAT ING UNDER AGREEMENTS WITH THE NA TIONAL INSTITUTE OfEDUCATION FURTHER RFPROOUCTION OUTSIDE THE ERIC SYSTEM REOUIRES PERMIS SION Or THE COPYRIGHT OWNER " Games and Puzzles for Elementary and Middle Sehool Mathematics

Readings from the Arithmetic Teacher

edited by

SEATON E. ,SMITH, JR. CARL A. BACK MAN

Faculty of Elementary Education The University of WestFlorida Pensacola, Florida

National Council of Teachers of Mathematics Copyright .© 1075 by THE NATIONAL couNciE OF TEACHERS OF mAuEmATIcs. 1906 Association Drive. Reston, Virginia :14.1091 All rights reserred

LibraryofConaressCataloging m Publication Data:

Main entry under.title: Games and puzzles for elementary and middle school mathematics.

1. Mathematics--Study and teaching (Elemen- tary)--Addresses, essays, lectures. 2. Mathe- maticsStudy and teaching (Secondary)- - Addresses, essays, lectures. 3. Mathematical recreations--Addresses, essays, lectures. T. Smith, Seaton E. II. Backman, Carl A. III. The Arithmetic teacher. 'QA135.5.G26 510'.7'8 75-16349

Printed in the United StatesofAmerica 1

Contents Introduction 1 1. Using Games and Puzzles in Mathematics Instruction 3

Commercial Games for the Arithmetic Class Donovan A Johnson 5 March 1958

The Role of Games, Punks, and Riddles in Elementary Mathematics Dora Dohler 7 November 1963

Fostering Enthusiasm through Child-created Games Sarah R. Golden 10 February 1970

Mathematics Games in the Classroom Donald R. Kerr. Jr. 15 March 1974

Sparking Interest in the Mathematics Classroom Donald L. Bradfield 19 it- March 1970

3 Payoff in Increased Instructional Time and Enrichment Activities Robert E Rea and Jat.,mes French 23 December 1972 \,

2. Whole Numbers, 29

Games for the Early Grades Edwina Deans 31 --- February 1966 ----

Creative Drill with Pictures Richard Crouse and Elir.abeth Rinehart 33 April 1973

Secret Number Sentence William L. Swart 36, Fehruary 1969

Making Drill More Interesting W. IiPatterson;Jr. 38 February 1974.

Go Shopping! Problem-solving Activities for the Primary Grades with Provisions for Individualisation Sylvia Grans 41 November 1970

They All Add Up ,. Al.Jane Heckman 43 April 1974

Mastering the Basic Facts with Dice Howard Y Goman 46 May 1973

Let's Play Computer Elaine V. Bartel 48 March 1974 1 Multi-bet Arthur K. Milo 50 December 1973

Arithmetic Games ClydeA. Dille). and Walter E. Rucker 52 February 1972 uIII

k - t V ----- CONTENTS .

TIAVitch's Best Game C. Winston Smith. Jr. 54 December 1966

A Game for Reviewing Basic Facts of Arithmetic , Edward E. Arnsdorf 56 November 1972

"Contig": A Game to Practice and Sharpen . Skills and Facts inthe Four Fundamental 58 , Operations ,:s Frank W. Broadbent May 1972

Cardematics 1Using Playing Cards as Reinforcers and Motivators in Basic Operations Seymour Metzner and Richard M. Sharp 61 May 1974

Green Chimneys Poker T. Ristorcelli 64 May 1974 \ Diffy Herbert Wills 65 October 1971

Magic TrianglesPractice in Skills and Thinking Donald L. Zalewski 69 October 1974

Easy Construction of Magic Squares for Classroom Use John Cappon, Sr. 73 February 1965 1 3. Numeration 79 A Place-Value Game fo First Gradersa, Teacher-made Device _- Irene R. MacRae 80 November 1957

,Plano a Number-Place Game Robert C. Calm 82 May 1968

The Game of Five Harold J. Shurlo w, 84 May 1963

From Second Base to Third Base Michael Alfonso. Richard Baker. November 1973 and Paul Hartung 86

A Game Introduction to the Binary Numeration System John Niman 89 December 1971

Tantaliling Ternary Ernest R. Ranucci 91 December 1968 4. Integers 97 Integer "Football" Virginia C. Demchik 98 October 1973

Subtraction of IntegersDiscovered 'through a Game Esther Milne 100 February 1969 I j CONTENTS V

Climb the Ladder Albert H. Mauthe 102 May 1969

Play Shufflebbard with Negative Numbers Charlotte Frank 105 May 1969

Disguised Practice for MbIttplication and Addition of Directed Numbers Esther Milne 108 May 1969

A Student-constructed Game for Drill with Integers Merle Mae Carillon. Doris Homan, November 1972 and Barbara Stone 110' Grisly Grids William G. Mehl and David W. Mehl 113 May 1969

Editorial Feedback Mary Helen Bean 116 . December 1970 1 t 5. Rational Numbers 119

A Game of Fractions Charlotte W. Junge 121 October 1966

Crazy Fractions: An Equivalence Game Earnest Carlisle 122 April 1973

Fracto Marie Molinoski 123 April 1974

The Conversion Game Beth Ellen Lazerick 124 January 1971

"Fradecent" a game Using Equivalent Fractions, Decimals, and Percents Charles Armstrong 126 March 1972

An Adventure in Division Lois Stephens 128 May 1968

Introduction to Ratio and Proportion Roland L. Brousseau, Thomas A. Brown, February 1969 and Peter J. Johnson 131

Make a Wholea Game Using Simple Fractions Joann Rode 133 February 1971 I

A Game with Fraction Numbers Richard Thomas Zytkowski 135 January 1970

6. Number Theory and Patterns 137

Fun Can Be Mathematics Audrey Kopp and Robert Hamada 139 November 1969

Bang, Buzz, Buzz-Bang, and Prime Ernest E. Allen 141 October 1969 . ) Vi CONTENTS Prime: A Drill to the Recognition of Prime and Composite Numbers Gregory Holdall / February 1969

The Factor Game J. B. Harbin and D. S. Martin November 1973

Five "Nontrivial" NuMber Games Terre! Trotter. Jr. 148 November 1972

Magic Squares: Extensions into Mathematics Daiyo Sawada .151 March 1974

"Interest Getters" Karl G. Zahn 157 April 1968

Just for Fun . J. D Caldwell / 160 May 1968 7. 'Geometry and Measurement 161 Don't Miss the Train Corinna Pother 163 February 1973

Procedures for Designing Your Own Metric Games for Pupil Involvement Cecil R. Trueblood and Michael S:abo 167 May 1974

Geo-Gin Janis A. Berman 172 April 1974

Paper Folding and Cutting a Set of Tangram Pieces Steven S. Dickoff 174 April 1971

A Game with Shapes Da sy Gogan' 177 April 1969

An Example of Informal Geometry: Mirror Cards Marion Walter 179 October 1966

Popsicle Sticks and Flying Polygons Charles Lund 184 October 1970

The Try-Angle Puzzle George Janicki 185 November 1956

A Pythagorean Puzzle Gary D. Hall 186 January 19/2

Tic-Tac-Toea Mathematical Game for Gr4es 4 through 9 Robert A. Timmons 189 Os.tober 1967

Hide-a-Region N > 2 Can Play Jean S. Overholser 192 October 1969

Car:esian Coordinates and Battleship William R. Bell 194 May 1974

Holiday Plot-Dots Gary A. Deatsman 195 February 1974 ;J CONTENTS vii

Stick Puzzle Anonymous 197 October 1959

A Game of Squares George Janicki 198 November 1956

Cross-Figure Puzzle, Measures George Janwki 198 February 1956

8. Reasoning and Logic 199

Rainy-Day Games Robert C. Gessel. Carolyn Johnson. April 1972 Marty Boren. and Charles Smith 201

N u- Tic -Tac -Toe Harry D. Ruderman 203 November 1965

Computtrs Teach Math Judy Ackerman 205 May 1968

KALAHan Ancient Game of Mathematical Skill John B. Haggerty 207 May 1964

A Project for the Low-Budget Mathematics LabOratory: The Game of Kalah Randall L. Brill 212 December 1974

Drill SOme Fun into Your Mathematics Class Marie Masse 215 Ma y 1974

Jupiter Horse Race Elinor J. Writ: 217 Janpary 1973

A Coin Trick Anonymous 219 March 1956

Paper, Pencil, and Book Henry Lulli 219 Dectber 1973

9. Multipurpose Games and Puzzles 221 Active Gaines: An Approach to Teaching Mathematical Skills to the Educable Mentally Retarded George R. Taylor and Susan T. Watkins 222 December 1974

More Games for the Early Grades Edwina beans 226 March 1966

A Deck of Cards, a Bunch of Kids, and Thou Peter K. Gurau 229 February 1969

Arithmetic Card Games Martin 11. Hunt 231 December 1968

Arithmetic Football Earl A. Karau 234 November 1956

Television Games Adapted for Use in Junior High Mathematics Classes Doris Homan 236 March 1913 ..../ t viii coNTIN l s

The Concentration (lame Homer F. Hampton 240 Januar 1972

The Match Game Larry lloltkantp 243 March I9?2

A Cross-Number Punk for St. Patrick's Day Margaret FWillerding 244 March 1957

Dominoes in the Mathematics Classroom TOME. Masco' 245 January 1971

A Cross-Number Punk for Valentine's Day Margaret F. Willerding 246 February 1957

A CrosszNumber Punk for Junior High School Margaret F. Willerding 247 November 1957

A Cross-Number Punk for Intermediate Grades - Margaret F. Willerding 248 November 1957

A Cross-Numberfunfor Primary Grades Margaret F. Willerding 249 November 195.7-

Mathematicalostern Sally Mathison 250 January 1969,

Editorial Feedback Verne G. Jeffers 252 December 1969

Two Mathematical Games with Dice Ronald G Good 254 January 1974

Three Games Bruce F. God .tave 257 May 1971

Take a Mathematical Holiday Elaine McCalla 260 December 1968

Classroom Experiences with Recreational Arithmetic Ruth It Nies 262 April.1956

"Arithmecode" Punk David F. Winick 266 February 1968

Arithmetical Brain,- Teasers for the Young Henry Winthrop 268 January 1967

"Parallelograms": A Simple Answer to Drill Motivation and Individualized Instruction Benny F. Tucker 270 November 1971

Take a Chance with the Wheel of Fortune Barbara Roster 274 November 1970

Using Functional Bulletin Boards in Elementary Mathematics Williant E. Schall 276 October 1972 Introduction

During the past decade, theuse of games and puzzles in the teaching of mathematics has increased dramatically. This current interest ingaroes and puzzles is evidenced by the growing number of presentationson the subject at NCTM regional and national meetings in recentyears. It is also highly visible in the new products offered by commercial producers of books andmaterials related to the teaching of mathematics. Although most teachers are well aware of the importance ofgoodplanning in all instruction, it seems worthy to note thatgames and puzzles should be used primarily within the framework ofa well-planned sequence of instruction. Wheq selecting a game or puzzle for classroomuse, the teacher should make is or her selection because the gameor puzzle can serve a particular purpose related to the mathematical content to be studied. It is also important to note that mathematicalgames and puzzles per se are not the ultimate solution for all our attempts to individualize instruction,to provide enrichment, or to provide experiences ina mathematics laboratory. It is true, however, that games and puzzles may be used effectivelyas one of many components in each of these areas. One of the general purposes for using mathematicalgames and puzzles is stimulate interest and develop favorable attitudes toward mathematics. In addition,013 most games and puzzles can be keyed toone or more of the following instructional purposes: v. 1. To develop concepts 2. To provide drill and reinforcement experiences 3. To develop perceptual abilities 4. To provide opportunities for logical thinkingor problem solving A most important consideration in the effective use ofgames and puzzles in mathematics education is preparing the students to play thegames properly. Instruction in the use of games and puzzles should be given thesame careful attention that is given to any other aspects of a lesson. Children shouldbe conscious of the fact that a particular game or puzzle has been selectedbe- cause it will help them learn something relative to mathematics and because it 2 GAS1FS 1tiPUZZ1 1.S is hoped that they wilt ve fun while: they are learning.They should also realize that if the game loses[ its mathematical focus and becomessimply a recreational activity. they can expect it either to be broughtback into focus or ter- minated. Through the years since its conception in 19f-4, theArithmetic Teacherhas served as a forum for sharing ideas about tetiaingmathematics in the elementary and middle schools. Many excellentarticles on games and puzzles have been published during this time. For this bookof readings more 'than one hundred of these articles havebeen selected and organized on the basis of the following major strands of elementary and middleschool mathematics: whole numbers. numeration, integers, rational numbers,number theory and patterns. geometry and measurement, reasoning,and logic. As you read these articles. try and envision multiple adaptations andimprovements for the many games and puzzles so thatthey may be used more effectively for your own purposes. We hope that you will find this publication a meaningfuland useful resource and that you will be stimulated toshare some of your ideas for mathematical games and puzzles with your fellowtolichers in the future, perhaps .through theArithmetic Teacher.

r Using Games and Puzzles in Mcthematics Instruction

Mathematicalgames and puzzles can serve many functions in the elementary school classroom. In the opening article, Donovan Johnson provides an excellent discdssion of some of these functions as well as a set of guidelines for using mathematical games "at the right time, for the right purpose, in the right, way," Additional support for the idea of using mathematical games, puzzles, and riddles is offered by Dohler. who illustrates her ideas with sample activities. Golden suggests that child-made games can foster enthusiasm in the classroom andecon tribute to learning in a number of ways. She supports her point of view with a discussion of many interesting games that were created by first- and second-grade students. Kerr provides a refreshing discussion about the involvement of a group of students with a mathematical game that resulted in some excellent analytical thinking and ultimately in a much im proved version of the orig;nal game. Bradfield prOposes to enliven the mathematics classroom by providing more comedy, mystery, and drama. He then' classifies and analyzes a number of enrichment problems accordingly and suggests that a "teacher is limited only by his own ingenuity and creativity." In the concluding article of this chapter, Rea and French provide some evidence from a small-scale research study to support the idea that mental computation, number puzzles, and enrichment activities can produce an increase in mathematics achievement as measured by standardized tests. They are also careful to note some of the limitations of their study, and they describe and illustrate some of the mental computation and enrichment activities that were employed with sixth-gradestudents.

. 3 rL ,\5 CoMmercial Games for the Arithmetic Class

DONOVAN A. JOHNSON University of Minneksta, Minneapolis

1 AND PLEASURE ought to becal competence that business, science, indus- Acombined with instruction in ordertry, and education are demanding. Arith- w inaltethe subject more interesting. Theremetic games are ideally suited for a. syste- should be games of various kinds such as amatic program of practice. However, games game played with different kinds of coinscan serve a variety of functions in your class- mixed together. There should also be prob-room, such as: lems connected with boxing and wrestling 43uild desirable attitudes towardsarith- matches. These thibgs make a pupil useful / metic 4 0 to himself and more wide awake."Plato Provide for individUal differences Even as Plato proclaimed, .ivine learning Provide appropriate "hoineW;ork" for 01 arithmetic can be accomplished through parent-child activity games. Games are usually considered a Make practice periods pleasant and suc- rt.creational activity. They are a means for cessful releasing boredom or tens:on. Games break dim n harriers bens een strangers and quickly Right now there are a great variety,of 1 stablish a friendly group spirit. At the samearithmetic gannet available from commer- lime games may be a way to learn to followcial publishers. Most of these games arc miles, to be cooperative, to be observant, oravailable at school supply stores or toy io practice diligently. Games also give par-stores. They are usuaily attractive and dur- ticipants a chance to develop restraint, toable with complete playing instructions in- contribute as a leader and follower, to ac-cluded. Of course you can make up a tept responsibility for individual as well asvariety of games with flash cards or number grttp action. However, learning arithmeticcards. However, the convenience of com- is not all a game. Other types of individualmercial cards makes them highly desirable. and independent group activities are obvi-Try them sometime. You will be pleasantly ously essential. But the variety and activitysurprised with the vigor pupils will exhibit of a game may break tip the monotony soin learning combinations in order to win a frequently present in practice or drill lessons.game instead of working problems to win UnquestiOnably the key to learning arith-the teachers favor. meticis through meaningful experiences and practical applications. However, the skills Using Arithmetic Games in the of computation need to be nurtured by a Classroom %ariety of systematic practice and drill. Thus the total approach to learning arithmetic The success of an arithmetic g me, like uses meaning, practice, and application. any classroom material ur tecl nique,is 'Hid meaning of numbers, the understand- highly dependent on how itis Used. If an ing of a process, the trathematical structure arithmetic game is to serve a real function iin uked precedes the practice. Practice-then it should be used at the right time, for the is the part of the learning process which right purpose, in the right way. Wilds accuracy, efficiency, and retention. 1. Select a game according to the needs of the Pur\poseful practice in the right amount at class. The basic criteria is that the game the right time will help.,build' the arithirteti- make a unique contribution to learn- .. ,`,1 5 6 GAMES AND 14.17.7.1.1S ing of arithmetic that cannot be at- and should be working on material ac- tained as well or better by any other cording to their ability. material or technique. The material of 4_ Plan and organize the game carefully so that the game should be closely related to the informality and tintement of the setting that of the regular classwork, What- does nog defeatits purpose. 'Peach the ever game is selectedshould involve playing of games in a planned, organ- important skills and concepts. Major ized way as you would present other emphasis should be on the learning of activities. Have all the materials at concepts or skills rather than onthe' hand so, that the game can proceed in pleasure of playing the game itself. an orderly fashion. t \lien a new, complex game is being played, start with a 2. Use the game at the proper time. If games few essential rules and then add other are to make themaximum contribu- rules as they arc needed during the tion to the learning of arithmeticthey game. Use a few practice plays to help should be used at the time when the get started. Then expect good work, ideas or skills are being taught or reas neat and accurate as regular class- viewed. How e%tr, many teachers pre- work. Pupils n ay referee the competi- fer to use games aftercompleting a tion as well as play the game. Before topic, on the clay before a vacation, or beginning the game the participants during the pupils free time at noon,in should be instructed on the purpose of study periods or homerooms.Others the game, the rules of the game, and use than (luring claysof heavy absence the way to participate. Often the pu- due to storms, concerts, orexcursions. pils can establish ground rules, so that Some teachers use them as rewardsfor eeryone (including theteacher) may work well done while others use them enjoy the acti hies. "Coaching" or for remedial work.Usually games "kibitizing" should never be allowed. should be played arelatively short The loss of points for the breaking of time so that pupils do not lose interest. rules will usually be sufficient to main- ribnge the game situation so that.ILL tain appropriate behavior. Avoid the 3. choosing of team members by pupil pupilsill be patiiapattng in every play. Even though only one person is work- captains so that low ability pupils will last ing on a certain problem at a given notbe embarrassed by being time every team member must be CC- choice. responsibility of learning sponsilAc for its solution too. Came.. 5. Emphasize th something from the game. Follow-up ac- must also avoid embarrassingthe per- tivities such as discussions, readings, or son who cannot solve a problem.Keep tests will emphasize thisresponsibility. commentspositive, commend good work rather dam making unfavorable As the teacher, evaluate the results by comparisMis. Whew: \ er possible, pu- asking yourself how successful the game learning. pils should compete with their peers, was in promoting desired

EDITORI IA. CONIMEN t. Although this article originally appeared in 1958, the message may be even more to the point today, when mathematical games areriding a crest of popularity. Donovan Johnson has noted that much value can befound in good mathematical games. He has also noted that games alone will not suffice to teachmathematics. Much depends on how a game is used. z

The role of games, puzzles; and riddles in elementary mathematics

DORA DOH LER Owatonna Public Schools, Owatonna, Minnesota

Whatcan we do to make math more in- Carefulanalysisofso-called"trick viting for our students? How can we lead riddles" can show fallacies in thinking and them intotheinteresting experiences bring out the necessity of careful study be- which are so much a part of mathematics? fore drawing conclusions. This al- I have found that puzzles, riddles, and so can help concept formation Ebyby pointing games can not only arouse interest, but out the "whys." A thorough understanding they can also help further mathematics of mathematics is necessary if a child is to knowledge and understanding. tell why a certain incorrect solution was

7 8 GAMES AND PUZZLES find another point equidistantfrom the xeb xtit 4,,_ second point. They could also plot two other points which would be the same tb Ok \ 4) ,xv distance apart as the first two points.In 0 0)4t) this way children naturally workinto a study of fraCtional numbers. Addition-subtractionandmultiplication-division "wheels" provide practice (0 ,110 o) x A x with the various operations as well as 0j. cb showing the inverse relationshipbetween multiplication-division and addition-sub- the num- f* De traction. The sum (or product) of 10 bers recorded in the two inner circles 17ef9 oig,01 4, equals the numbers recorded in the outer 1,0 0 0c circle (see Fig. 2).

6red 8 yellow 10-- green 12-blue.15black Figure I

obtained or why a certain problem always "works out." Much meaningful practice can be presented in an exciting way through the use of games. Dot-to-dot puzzles can be con- strUcted where the sums or products (depending upon the grade level) are the numerals to be connected. This is good practice wif the mathematics facts. Another game for practice with the various operationsisa squared sheet where each triangle is to be colored to correspond to the given number repre- sentihg the simplest name for the expression .11 the triangle (see Fig. 1). TI e children consider work with a num- berinc as a real "fun" experience. We grasshoppers and can can imagine we arc Fig ire 2 juMp a given number of units. This offers practice ini skip-counting, addition, subtraction, ormultiplication. The grass- Puzzles can offei. an early introduction hopper may jump thirteen units begin- to geometry. For example, thechildren nil* at 7. Where will he be after 4 jumps can find the numberoftriangles in.objects of!thirteenunits each? The children may such as these: , axirive at the solution in many different ways, which they shouldbe encouraged to share with the rest of the class. Two points, A and B, may be plotted on a number line. The childrenfind a point Judi way between the two points, or they USING GAMES AND PUZZLES 9 or the number of squares in an object: 1,469,201 0 469,201 1i+i Q 1-1-i+i+i (The 0 serves as a placeholder for the equals sign or for the signs of inequality.) (4+5)X2 0 18 (6+3)+9 0 6X3 4X9 0 (5X6)+(4 X6) 2,462,910 0 4,262,910 Equations with operation symbols to be supplied by the pupils prove quite chal- A good puzzle for that spare moment is lenging to many children besides supplying the oral "string problem" where we (An a means of practicing computation. Ex- attempt to solve a complex equation in- amples are: volving many varied operations such as in the following illustration: Take 7*ulti- 6 4 = 8 10 ply by 3; add 4; divide by 5; multiply by (The /\serves as a placeholder for 9; add 3; divide by 6; multiply by 4; add the operational sign.) 4; divide by 9. The answer is 4. Depending 9 3= (4 (5 3) upon the ability of the class, we can \speed up or slow down the rate ofpresenting the (Distributive Property of Multiplica- operations. Success depends upon kriotw- tion) ing the mathematics facts, an ability `co Number sequences are a good method of solve equations mentally, and rapid think- developing skill and accuracy of compu- ing. tation. The students find the pattern and Children enjoy reconstruction problems supply the missing terms. This can be if a careful developmental sequence is pre- easily adapted to any grade level or con- sented. One may begin using examples cept being developed. Examples are: such as the following. a. 1, 3, 5, 11,

2+0= 6 b. 5, 10, 15,. +3=4 ,40, ,30 10+= (7+3)+4 20+ =17+4 c. 71, 9, 101, , 4 X 6 = 0 X4=12 d ,301, 25, 191, (X7)+6=20 e. 2:00, 1:00, 12:40, 2-1=(0X6)+(2 X6) 70 X8= X10 f. 50, 350, , $1.55 f+ =1} g. 1, 3, 6, 10, 71=51 , 55 $1 ,294+0=$4 ,927 $1,090 OXi=1 These and other puzzles and riddles de- pt.+ 0 qt.= 1 gal. veloped by both the teacher and members 1,692,046 +0 =7,298,102 of the class supply a little extra "spark" to 4fX =27 begin or end a class period, ari enjoyable way to practice skills, an effective enrich- Work with inequalities and equalities ment or remedial lesson, and an interest-. gives experience with the operations. Ex- catcher for our mathematics program. amples: Sul) ly the sign =, < (less thiv.1), They offer the dessert for a mathematics or > (greateran). meal that can be enjoyed by all. Fostering enthusiasm through child-created games

SARAH R. GOLDEN

Sarah Golden is a qasstoom teacher of a combination first- and seeond-grade class at lit:Wrest School in Chula Vista City School District, Chula Vista, California. Mathematics is a favorite subject for her and for her class.

Howcan the young child develop en- Mathematics, Science, reading, and spell- thusiasm for mathematics, maintain needed ing were suggested by the children as the skills, gain mathematical insight, and work contentareas. The mathematics games at his individual level in a relaxed class- might include, they said, go ahead and room climate of success? go back, counting, addition, subtraction, One of the possible ways developed as and multiplication. Dismissal time arrived a result of a recent visit by Donald Cohen, all too quickly. Madison Project Resident Coordinator in Traci (Grade 2) New York City, to my combination first- Object: Get to finish line. second class. He introduced an original Rules. Use spinner or die to indicate number of moves. Correct response, stay on that box. Guess the Rules board game involving Incorrect response. back to previous box.' movement of imaginary traffic. Following this experience, one boy tried to make up 7FINISH his own Guess the Rules game while the 1219 + 0DJ other, children played with commercial :end 970-1-0 _j teacher-made games. I noted that in play- 468 + ing some of the games, variations of the original rules were used spontaneously by I 0_1 ,small groups to increase total scores possi- 0+0 12 ble. For example, the chalkboard scores 0=a 8+ 12: for Ring Toss ran into the thousands, al- 15 "',L+0 though the indicated score for each toss 0-2-15 was only 5 or 15. "Oh," said Jeff, sitting on the floor with one foot extended, "When 3 8 he rings my foot, it counts for one hundred. 9`"Z1 When he rings Robert's foot, that's five LI -10-20 hundred." Excitement ran high. Elt0 The next day we discussed the possibility 0=15-30 of each child creating his own board game. 030:010+300=41 The following criteria were established: Moo .0 +0 1. The game must be fun to play. START 2. The game must have rules. FIG. I. Number _Game 3. The game should enable childrento play together and become friends. I.It was surprising that Traci deliberately planned to have negative numbers as responses in her equa- 4. The children must learn from the game. tions.

10 USING GAMES AND PUZZLES i i The following morning one girl, sub- In making games that involved rounded mitted a sketch of her game which used shapes, the children encountered the prob- dice with three golf tees as pawns. Her lem of dividing the rounded portions into explanations were quickly grasped by the congruent regions. They then learned how class. As she proceeded to make a more to use simple protractors to construct such durable copy of her game, children used regions. Several lessons on curves followed, the following vocabulary to describe the includingsimple,complex,open,and situation: Traci's game, Traci's pattern, closed curves. Traci's copy, Traci's original, Traci's guide, Upon completion of the games the chil- Traci's directions, Traci's instructions. dren were given repeated opportunities to explain the object and rules of their games and to play them with their classmates. Scores were tallied and comparisons such as "I beat him by 35 points" were made. Deborah's Streets and Numbers game led to a discussion of odd and even numbers. We took a walk in the neighborhood to 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- v. main there until next spin. Wrong answer makes player go back to his preceding posilion.2 Eager to start on his own, each child that developed a game on a sheet of chip- board about 12 by 22 inches. Dice, spinners, or number cubes were used to indicate direction and number of moves. In some cases, answers on cards were to be matched withthe corresponding problem inthe game. The games showed a wide variation in difficulty, ranging from simple counting to. the use of negative numbers. Danny asked, "What if you landed on 3 and your card said, 'Go back 6'? Where would you be?- Off of the board?" He went on, "I guess that would be a negative number, and you'd have to get 'Go ahead 6' to get back to where you were on 3." Danny thought about this and incorporated negative numbers in his Bang Bang Chitty Chitty game by using a starting line with negatiVe numbers behind it leading to the Repair Shop. The twenty-seven games includedsuch Sql1+4+4 names asTry and Guess; The 'Happy Game; Go Ahead, Go Back; Treasure FIG. 2, Snoopy Game Island; The Whacky Racer; Streets\ and . - Numbers; The Counting Game; and Bang 2. Robert, a first grader, was interested in using Bang Chitty Chitty. \ three addends. -1 I

12 GAMES AND PUZZLES

111

Each child made a map of his street and to explain how numbers were assigned to the houses on it and numbered the houses. new homes in the city. Deborah revised We invited a member of the Chula Vista her game to reflect her new 'earnings in City Engineering Staff to come to school this areaodd numbers on one side of the street, even numbers on the opposite side. For one or two players. Kathy (Grade I ) Object: Put the greatest number of cards where The Bang Bang Chitty Chitty board they belong in the counting ladder. game was devised by four boys as an out- Rules: Players taketurns picking top card come of -theirinterestinmathematics, from a shuffled pack. Cards show numerals 6, 12, 14, etc. Player puts the card on his side of the model cars, races, and the story of Chitty ladder. If incorrect, player loses turn. Each player Chitty Bang Bang. Speed limits were dis- counts his correct responses at end of game to cussed. The boys consulted Sports Ency- determine winner. clopedia, almanacs, and other books for START information on speed records and reason- Player1 2 Player 2 ableness of posted speed signs. Of course, puts his 4 puts his the intricate speedway had to be made wide 6 cards on cards on enough to accommodate the toy racing cars this side 8 this side 10 used. Much experience in measuring re- I I sulted.

16 Summary Child-made games foster enthusiasm and contributeto learning inseveral ways. Children learn through a pleasutable medium of their own creation. The varying 26 degrees of complexity in the games are commensurate with a child's mathematical concepts and interests.In devising the 34 games, the children must try out their ideas 36 and pursue them to a reasonable outcome. --- Language development is advanced as the FINISH children are highlt;motivated to explain Fin. 3. Counting Game the games clearly to others. There is an USING GAMES AND PUZZLES 13

Leslie (Grade 1) Object:Get tohome. Rules:Use die to,indicate number of boxes to advance. Then follow directions in box.

ediG3o Back GoGio G; Go GoBack 'auk 0 6

Fin. 4. Happy Game

exchange of information between players, levels of thinking. For example, it was sur- especially when more than one correct re- prising to note some children's use of diffi- sponse is possible. The teacher gains infor- cult combinations and negative numbers mation concerning individuals and their and their free use of commutative and as-

For two players. Danny (Grade 2), Chris (Grade 1), Bober (Grade 1), Bobby (Grade2) Object: Getto finish line. Rules:From a stack of cards, each player takes a card, which tells him what to do.3

mph 310

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[Go topitstop niNi [33Go ahead 4 ari.ows Go-3.1 IrIGo back 2 arrows Go to' 90 mph (Go ahead(Means stay F-1 Go ahead 0 where you are) +2 Flo. 5. Bang Bang Chitty Chitty

3. Numbers back of starting line were negative numbers and led back to the repair shop with ih grease rack, etc. 14 GAMES AND PUZZLES sociative properties of addition in comput- Chris (Grade 1) This game is played similarly to the Snoopy ing total scores. The games provideteach- Game (fig. 2). This first grader included multi- able moments for the teacher tointroduce plication and subtraction in his game. Object: Get to finish line. pertinent material in greater depth. Rules: Use number cube to indicate number of boxes for move. Correct answer allows player to Bobby (Grade 2) nextspin, Wrong answer Object; Get to treasure at end. i. remain thereuntil Rules: Use spinnerto indicate number of makes player go back to his preceding position. moves. If answer is correct, stayuntil next spin. If player lands on "Go back," he must go back If answer is not correct, go back to preceding the number of places indicated in the box. place.4 .4 START r- 1 x/0.f4:2r i Crintsdim i'v 10)0°0 \D.- 6+4+100 -A Go ahead -0 00ci-, 1"--- 1 2 X 5 2 3 ,x Go back 84+84 4 6.5 500 +70 +8 IX_ 3 r- I 36+3 [420+420 49-27f I 1X35+15 r-- Go hackl Co ahead . _ 80-60 . I X 0 IPond ahead Go back t_ 13+24 I 210+210 47 X 0+47 frrossbridgei 7+7 60+60 3X6

43+8 Go ahead (r-ha71 10+2 43C careful 1 +0_ _ 9 +2 40 + 40 6X4 +10007) mire] 1 34-3 FINISH rr Fin. 6. Treasure Island Game )6> 4. Each child has a marker of a differentcolor - or of a different kind. Flo. 7. Try and Guess Game Mathematicsgames in the classroom

DONALD R. KERR, JR.

The Indiana. University Mathematics Eduration Development Center is currently at work on a program that combines mathematics content. mathematics methods, and public school experience for prospective elementary school teachers. Donald Kerr is assistant director of the Center.

Games are fun, anditisimportant to have fun. Mathematical games in school are good because, in playing them, children have fun associated with a topic that is not always considered to be enjoyable. Itisneither possible nor desirableto organize the bulk of mathematics instruction around games, butitis sometimes Fig. 1 possible to develop games in such a way that they complement the. regular mathe- 1 to 20 gave rise to using other intervals, matics instruction and thereby justify more to using longintervals, and to discussing classroom lime. This article chronicles one the effect of longer intervals on the pact classroom experience of theauthorin of the game. which a simple game was introduced to' The inevitable cry of "He wins all the a fifth-grade class on a Friday afternoon. time," led to a discussion of which'. num- Our first' game-playing involved a ver- bers were thebestto choose. A child sion of the well-known game Multo. The kept a tally on the frequency of called children were given copies of three-by- numbers, and this led to a discussion of three arrays of squares (see fig.I) and why some numbers seemed to be called were told to put some number between more frequently than others. At this point, I and 20 in each of the nine sqmies. caller-player competitionstartedto de- They were also asked to tear up bits of %clop. Some callers tried to call numbers paper to use to cover the squaret Every that were low on the frequency chart; child in the class knew how to play bingo, some players clearly understood the greater so no other instructions were needed ex- likelihood of composites over primes. cept to say that numbers would be called The pupils were quite content to con-, by giving a multiplied pairtwo times tinue to play the sortie game with minor four, three times five, and so on. The modifications during last period on Friday author called off the numbers for the first afternoons, but the author decided that it game, and the winner thereafter served was time (if you'll pardon the pun) to as caller. add a new dimension to the game; _So Class impatience with repeated numbers three-dimensional Multo was introduced. and a few disputes over winners soon The children were given copies of three dictated that the caner keep track of the three-by-three arrays of squares (see fig. numbers he had called. Disenchantment 2) and were instructed to fill them in with with the limitation to the numbers from numbers betweenIand 30. I 15 - 16 GAMES ANI) RIZZI ES

i I

Top Middle , Bottqm t Fig. 2

Since none of the pupils was familiar what was not a ,winner. For examplcin withthethree-dimensionalversions of figure 4 the Us, Xs, or I's provide a win- bingo or tic-tac-toe, some explanation was 'ner, while the Zs and Ws do not.Some needed. cpildren got the three-dimensional per- spective immediately, but others never did. F.munately members of the latter group were still able to win since atwo-dimen- Top sional winner, such as with the Us in figure 4, was possible. Atthispointthe idea of repeating --embers came up. The children wanted to be able to use the same number more than once in filling in their squares. They . decided totryusing repeated numbers Middle but soon found that this led to quick wins with many tics, so they went back to using r numbers only once. The three-dimensional form of the game had the obvious advantage of encouraging spatial visualization and making a three- Yet this . /.//1--- to tv/o-dimensional "projection." / 1 Bottom form had not enriched the numerical ex- -:--,, ,"/ / perience inherent in the game so the au- ./ l V thor decided to irttroduce another modifi- Fig. 3 cation Of, the game. The children were instructed topUt 1, With a sketch like figure 3 on the board, 10, and 100 respectively under the three the pupils and the author went through squares onl the three-dimensional\ Multo a number of examplesof what was and sheet(see fig.5) and then to number

X 1 X f I 0 i I el Middle Bottom Fig. 4 I , ,...ti USING GA\Ms AND PUZZLES 17 ,------7- , * 1 1 1 LL ] i 1 100 10 1

Fig. 5

the squares as before. This time the caller squares were given values between 1 and was to call off Ave numbers (as multiplied 9, as shown in figure 6. The children then pairs). The winner would be the pupil filled in their Multo numbers as before, with the highest total, where each riiimber. and the game pioceeded in the same fash- called in 41 square counted for the number ion. So, for example, the Xs in figure 6 of points under the square. This modifica- would yield a score of tion, while abandoning the three-dimensional aspect, had the advantage that each (4 + 1) X 100 ± (7 -Fs 4)X 10+ 5 X 1 pupil was required to add up hisscore = 6 X ido + I X 10 + 5 X 1 to see if he had won. For example, the pupil who had the Xs in 'figure 5 would = 615 receive a score of 212. Quick ways of adding scores to take ad- The pupils evolved a streamlined sys- vantage of the cc.apatibility of the format tem for deciding who had won. Any child with the base-ten numeration systemwere who felt he had a good score wouldan- discussed. nounceit.Someone whose score was There are clearly many additional direc- higher would announce his, and soon. tions we could have moved in.Other Thus the winner was quickly identified. operations and combinations of otherop- The children kilSO engaged in discussions erations could have been used for naming on which scores were possible and from points. The squares could have been given these the next versio'n of the game arose. values corresponding to other bases-1, The next and final form of the game 2, 4 or 1, 3, 9. Many more probability emerged from an effort 'y the author and and statistics problems could have been the,pupils to fill in the gaps between the ,pursued. And basic games other than possible scores in the previous form. The Multu could have been used as a point small squares in each of the three big of depa lure.

r 91 9 8 07 1 91 8 71

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, 3 31 2 I 31 1 2 j 1 , L 1 10

Fig. 6 18 GAMES AND PUZZI ES The following points are significant lit The particular game and itsvariations the -sequence of activities that have been are not important as such.More important described: is the idea of involving children in the evolution of a game and of directing that (a) the skill reinforcement in the game. evolution in order to supplement and ex- (b) the pupil involvement in the evolu- tend the regular mathematics program. tion of the game, (c) the teacher involvement in providing direction to the evolution AUTHORS NOTI : The author wishes to express and inseeing that many pupilshis- appieciation to teachers Katheryn Vaughan andBeth Williams, landtoPrincipal Edwin participated, and Smith for giving him the opportunity to have introduCtion of new mathe- i, (d) the theexperiences ,describedhereinfifth-and matical concepts such as elemenl sixth -gradeclassesatTempleton Schoolin tary statistics and probability. - Bloomington, Indiana. ...

.1 r- if Sparking interest in the mathematics classroom DONALD L. BRADFIELD

Donald Brad field is a classroom leacher in Sand Springs, Oh lahoma.

From the glamorous stars of the enter- most mathematics classrooms. One way to tainment world come the clues for sparking stimulate the atmosphere is to use recrea- interestinthemathematics classroom. tional mathematics materials. Basically, these stars make use of three kinds of entertainmentcomedy, mystery, The comedy of mathematics and drama. f hrough proper use of these To entertainisto amuse. To many methods of entertainment, the mathematics pupils, there is nothing funny about mathe- classroom can come alite with ditersions matics. However, in the world of mathemat- and excitement. The intent of this article ics, catchy problems and tricky problems is to illustiate how to spark interest in the exemplify types of comedy that portray mathematics classroom byutilizing types humorous situations and unexpected solu- of recreational m ithonlatics that are cate- tions which, consequently, are funny. Con gorized under three entertainment media skier the following two examples: headingsSuggestions will be made as to how to assimilate each type into a regular Monkey Problem program of mathematical instruction. A monkey is at the bottom of a 30-foot Before a discussion of mathematical en- well. Every day, he climbs up three feet tertainment, the need for sparking interest and slides back two feet. When does he in the mathematics classroom should be reach the top? stressed. Many teachers of mathematics are unbelievably boring in their approach to teact"g. C,)nsequently. many poten- Cigarette Problem tially strong students never dc.Lelo; any great liking for mathematics. Weak stu- If a tramp can make one cigarette from dents are disenchanted quickly with the six butts, how many cigarettes can he course, and elect to avoid all courses in smoke from 36 butts? mathematics. This loss of interestis an educational wasteboth to the indit idual Thesok/donto the monkey problem is 28 and to society as a wnole. Thertiou, a days, since on the 28th day he would climb more stimulating atmosphere is needed in and not slip back. The solution to the

19 L.) 20 GAMES AN!) PUZZLES cigarette problem is seven cigarettes, since Pythagoras Coded Message after making six cigarettes and smoking them he would have six more butts from Use the following key to decipher the which another cigarette could be made. message below: These comedy problems capture attention and appeal to a sense of humo?. Similar problems from thefieldof recreational n.o,p,q,r,s.t,u,v,w,x,y,z mathematics may be labeled as "Rat Prob- Pythagoras says: lem," "Squirrel Problem," "Coin Problem," "Egg Problem," "Barrel Problem," "Mar- 0 7 !445 ble Problem," 'April Fool Problem," etc. 4 704 These ties attract the attention of children 6 7 4 and ignite sparks of interest withinthe classroom. By solving these comedy prob- 7 0 88445 4 lems, children develop a pleasant keling., toward problem solving that should transfer to the regular curriculum of mathe- Salary Progression matics. If you were offered a salary of one p...nny the first day, two pennies on the The mystery of mathematics second day, four pennies on the third To entertainisto mystify. To many Say and so on for thirty days, would pupils, all of mathematics is a great my- you accept? stery. However, in the field of mathemat- Days: Salary ics, magic problems, puzzle problems, code 1 .01 problems, and progression problems illus- 2 .02 trate special types of mysteries that will 3 .04 fascinate many students. The following four 4 .08 examples provide mystery and surprise. 5 .16

Faded Document Puzzle

121 30 Total What is the total salary?

The quotient in the faded document puz- Reproduce the numbers on the docu- zle is 90809. The coded message of Pytha- ment. goras says, "Numbers rule the universe." The total salary in the progression problem $10,737,418.23. These problems Age and Family Magic' and their solutions provoke amazement, stir curiosity, and challenge the intellect. Write down''your age. Double it. Multi- Similar problems in the field of recreational ply by 5. Add 15. Add the number of mathematics may be labeled as "Age and members of your family less than ten. CoinMagic,""CryptorhythmPuzzle," Subtract 15. The first two digits in the "Wiggle Operation Puzzle," "Gauss's Sec- result is your age. The last digit is the ret Message," "Notebook Paper Progres- number in the family. sion Problem," etc. These kinds of prob- i 4 USING GAMES AND PUZZLES 21 lems catch the eye of children and sp. rk Three Jealous Men and interest in the classroom. By solving these Their Wives Problem mystery problems, children experience success in problem solving. Three jealous men and their wives come to a river to be crossed. Using one boat The drama of mathematics that holds only two people and havihg To entertain is to dramatize. To many no wife ever in the presence of another pupils, mathematics is a dull subject and man unless her own husband is present, has no dramatic appeal. However, in the how can the group cross the river? area of mathematics, many problems can be presented to childien in such a manner as to have dramatic appeal. Consider the The solution to these problems is left for following three examples: the reader 'to find. These kinds of dramatiC ...,. problems arouse emotions, encourage in- Kings Problem genuity, andte:,tperseveranceas the readerwill A king suspected his advisors were half- discover as he attempts to solve these problems. Similar problems in wits. To test the wit of the advisors, he threw them into a dungeon and told the field of recreational mathematics may helabeled "Railroad Problem," "Chris- them that they would lose their heads if tian and Turk Problem," "Bridge Prob- they did not solve the following problem: lem," etc. Such problems have emotional connotations for children, sparking interest in the classroom. By solving these dramatic The problem is to draw six straight lines problems, children discover the human- through the dots without lifting your interest factor in mathematics. pencil or retracing a line. Could you To effectively use recreational mathematics in the classroom, the following pro- have saved your head? cedures are helpful. First, make color posters of problems similar to those' listed in Josephus Problem the previous topics of this article and place them on the walls and bulletin boards When the Romans invaded their town, within the classroom. These posters create Josephus and forty other men took rean exciting mathematical environment and fuge in a cave. Believing that they were can he referred to by the teacher when about to be captured, the group wanted appropriate. Second, allocate time in the to commit suicide except Josephus and schedule of mathematicalinstructionto hisfriend. Fearing the othersinthe study topics from recreational mathemat- group and wanting to save himself and ics. Between units of stud-, and the days his friend, Josephus suggested a plan before vacations are especially good times whereby the killings would take place to provide pupils with a treat thatwill inan orderly manner. He suggested leave them wide a good feeling toward that the group form a circle and begin killing every third man until only two mathematics and make them more willing to tackle new mathematical topics. Third, would be left. These reniaining two per- aJd recreational problems to examinations sons would then supposedly kill them- and give extra pointsforsolving them selves.If Joscphusselectedthefirst correctly. These problems keep fast stu- man in the circle from which to begin dents occupied profitably while slower stu- the counting, where did he place him- tieff and his friend in order that they dents are finishing the main part of a test; the only danger involved in this procedure would keep from getting killed? is that slower students must be cautioned

,..... 22 GanttS ;\t1 Pt_ ZZI ES in the to finish the Main portion ofthe test he- for using recreational mathematics classroom The teacher is limitedon!) b) foreattempting. the bonus problems.A %ariet) of other procedures can he leised hi(mil ingenuit) and creatiit)

EDTORRI CONNE \ T.The author has pro.. tded an interesting classification structure for enrichment problems. Almost eery teacher could find man) opportunities to use afile of puzzles and problems such as these. If )ou are interested in starting or extending .our on set of enrichment actnities.ou will find mans games. puzzles, and problems in theArithmetic Teacher, in the Mathematics Teacher, in anous NCTM publications, in children's magazines. in the magazines provided b) airlines, in the newspaper, inpaperback books on mathematics, in books on methods of teaching mathematics, in the teacher'smanuals for various series of textbooks, and in mans books do oted primarily to games, puzzles. number magic. and so on. A teacher is limited onl) by his on effort and imagination in finding andcreating "interest getters" in mathematics. I

Payoff in increased instructional time and enrichment activities ROBERT E. REA University of Missouri, St. Louis, Missouri JAMI S FRENCH Hazelwood Public Schools, Hazelwood, Missouri

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A. recent research project assessed the Procedure merits of oral instruction in mental computation versus pencil and paper enrich- A sixth-grade class was divided into two ment activities for sixth-grade pupils. An treatment groups that were matched on the outcome of particular interest to teachers bases of age, sex, 1.0., and achievement of elementary mathematics isthat both scores. The matching process produced the treatments, as measured by standardized two groups shown in table 1. The instruc- achievement-test scores, resulted in a sig- tional activities for group 1 were based on nificant growth over a small number of in- Kramer's mental computation series (1965), structional sessions. and the activities for, group 2 were adapted

23 t J 24 GAMES AND PUZZLES from Crescimbenes- collection of enrich- tained for both groups on the first and of theproject,with mentactivities (1965). Care was taken to twenty-fifthdays ensure that he type ofcomputational ex- twenty-four successive instructional periods amples was the same for both groups- intervening. Both groups received their reg- fif- that is, if the mental computation exercises ular mathematics period each day, plus involved multiplying a three-digit number teen minutes of specialactivities. The by a two-digit number, then the game, grade-equivalent scores derived from these the scatl puzzle, or activity taken fromCrescimbeni test administrations are shown in was modified to providesimilar practice. tergrams in figures 1 and 2. SRA achievement -test scores wereob- In both groups, there were individual

Treatment group 1-Mental computation Treatment group 2-Enrichment activities .6 .8 .4 X .6 ,.2 .4 10.0 .2 X 8.0 .8 0 .6 .8 .4 .6 .2 .4 X 9.0 .2 X )(0 V .8 7.0 .8 .6 X .4 .6 00 a .2 S .4 .2 8.0 0 XX .8 '56.0 OX .6 .8 X .6 .4 0 X 0 0 OXO .2 o x .4 X 7.0 .2 0 X .8 5.0 .6 0 .8 0 X .4 .6 .2 0 .4 0 .2 6.0 X 4.0 .8 0 .6 0 .8 X .4 xx .6 5.2 0 0 .4 0 00 3.2 1 2 3 4 3 6 7 8 9 10 11 12 13 Student numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 Student numbers

Grade equivalents derived from table accompanying Grade equivalents derived from table accompanying SRA Achievement Series: Arithmetic, Form E, SRA Achievement Series: Arithmetic, Form E, Blue level, 1971. Blue level, 1971. Pretest grade equivalent Pretest grade equivalent X Posttest grade equivalent X - Postttst grade equivalent Fig. 1 Fig. 2 USING GAMES AND PUZZLES 2S Table 1 Matched data for treatment groups

Group I Group 2 . N (7 girls, 6 boys) N (7 girls, 6 boys)

Range Afedion Range Median

Age 11.1-12.0 11.65 11.0-12.6 11.5 I.Q. 86-141 102.6 86-133 102.0 Achievement scores' 3.9-9.2 5.2 4.2-6.2 5.2 Data from 5thgrade records. SRA Achievement Senes Anthrnetic. Form C. Blue leel. 1964. Administered September 1970.

youngsters whose scoresincreased,only Mental computation activities slightly (students 3 and 13 in figure 1) and The information given in figures 3 and even certain individuals whose scores de- 4 was reproduced for use in group instfuc- creased (students 4 and 9 in figure 1 and .tion. The instruction began with ttie read- student 2 in figure 2). However, the ma- ing of the first sentence, which contains jority of the students in both groups gained enough information for the solution of so dramatically in achievement scores that the questions that follow it. At this point, the average growth for group 2 was one the children were cautioned not to write full year, and for group 1 was eight months. the problem down on paper, but- rather to think of it in the way that was indicated Discussion inthe instructions. Children were then While this project is subject to many asked to search for other methods of solv- limitations and possible sources of error, ing the problem mentally. Following this, the increase in achievement scores seems a similar set of problems was presented for to be worth pursuing (formally in more mental solution by the method that was rigorously designed experiments and infor- presented, or by an alternative method that mally in classroom situations where a sim- was discovered by thy. class. ilar set of experiences may be desirable). The second example in both figures 3 The following examples of activities and and 4 presents the inverse of the process in suggestions for additional modifications are example one. In all levons, an example offered to elementary teachers who may of one process was followed immediately wish to try a few of them in their own class- by an example of the inveice proCcs.. This rooms., Teachers wishing to use a greater MENTAL COMPUTATION ACTIVITY variety of activities may refer to the sources 1. The Deck family traveled 250 miles on the firsi listed in the references. day of their trip and 160 miles on the second day. How many miles did they travel on the firsi two The examples presented here were well days? received by the youngsters in this project 250 + 160 = 0 Think:.160 is 100 + 60 and can easily, be modified to most topics 250 plus 100 is how much? (350) or levels of computation. The mental coin- 350 plus 60 is how much? putation and enrichment examples that fol- 180+ 160= 160+ 150 = 430+ 170= 140+ 150 = low do not involve the same computational 340+ 120= -- 290+ 120 - skills. However, parallel examples can eas- 2. Mr. Thomas has a debt of $120. If he pays S70 ily be constructed. In fact, constructing of it. how large a debt will he have left? such parallels greatly increases a teach- 120- 70 = 100- 20= 100- 20 = er's understanding and appreciation of the 110'- 40= 130- 40 = 140 - 60= 150- 70 = power of these activities in generating com- 110- 20= - 160- 40 =_ putational practice. Fig. 3

i 26 GAMES AND PUZZLES

MENTAL COMPUTATION ACTIVITY ca a)g 1. In an auditorium there are 10 rows of 14 chairs. 4). + How many chairs are there in all? + na id 10 X 14 = 0 co 4). Think: A number is multiplied by 10 by annexing 11 a zero to the numeral to find the number. 10 X 24 = 10 X 19 = 1 0x6=0 10 X 69 = _.. - 10 X 46 = +11=12 10 X 95 =.___ 10 X 15 z- 10 X 85 = _. 10 X 58 = I 9+17+24=50 2. Ten boys had to carry 110 books to the science 23+ +54=78 room. How many books did each boyhave to carry? 110÷ 10 = 0 7-2=5 27 +I +5 = 33 2404.10= 720_10 = 380÷10=_ 470 ÷10 = 540÷10= 840 ÷10 = 730÷10= _ 640 ÷'10 = co 4:. Fig. 4 iv 4. 4:. al + sequence was followed in the hopeof mak- IT n ing the inverse relationship easier for the children to grasp. +96+34=135 x2I =126 Enrichment activities 9x = 63 Ladder Arithmetic, Draw-a-Trail, Star- 84- =81 Burst, and Brain Teasers are examples of the activities used with group 2. Such activ- +6+9=19 x12=108 ities are a direct attempt to encapsule mathematics reinforcement, practice, and Fig. 5.Ladder Arithmetic enrichment in forms that are appealing and Directions: Fill in the frames with the number enjoyable for the children. The purposes that makes the horizontal sentences true; then determine if these digits complete the vertical of activities of this nature include making sentences. The digits may be in order, reversed, mathematics more enticing, and involvingt. or scrambled. students in situations that stimulate curiosity and quantitative thinking. within a given time limit. The teacher can Ladder Arithmetic has a built-in, self- adjust the level of difficulty of this exercise checking device (see figure 5). The frxnzs by increasing the size of the numbers, or are filled with the numbersthat make the by varying the operations to be performed. horizontal sentences true. These numbers Such variations should be designed to meet provide the digits necessary to solve the the needs of a particular group or indi- vertical sentences. The horizontal sentences vidual. (See figure 6). can be arranged to give the correct se- The examples of Draw-a-Trail selected quence of digits or, if additionalcomplex- here illustrate variations that might be used ity is desired, a scrambled order of digits. from. a simple addition exercise to a Scrambled digits further encourage chil- morecomplicatedmixed-fractionand dren to perform the self-checking feature whole-numberexercise.Oncechildren of the activity. understand the procedure for finding the In Draw-a-Trail, solutions vary from solutions to Draw-a-Trail, they may be student to student depending on the start- encouraged to construct their own exam- ing point and the path that is followed to ples. After constructing a trail, pupils can the bottom box. Another dimension may exchange examples and check one an- be added to the exercise by asking the chil- other's solutions. dren to discover as many paths as possible Star-Burst is an activity that provides

a ;: i ; 0 USING GAMES ANDPUZZLES 27 1/2 X2 1/4 3 1/3 XI/3 3/4 1 1/2 2 1/3 1 2/3 1/8 1/4 2/3 1/2" 1/3 2 4/4 1/3 -3 4 X2 11: X1/3 3/4 3 1/2 2 4 1 2/3 1/8 1/4

7 '6 8 9 5 5 8 9 4 9 '6 7 9 8 8 7 4 8 7 9 8 5 6 7 8 9 8 7 9 9 7 8 7 5 5 .Put the numbers 11 through18 in the outer ring 61 of circles so that each line totals35.

Fig. 6.Draw-A-Trail - Directions: The objective isto draW a trail' that leads to the bottom boxof the puzzle. The trail must be a continuous lineconnecting numerals in the boxes. Itmust end at the numeral in the bottom box, and thenumbers in the trail must add up to the total in thebottom box. All boxes need not be used, buteach box may be used only once.

practice in variousoperations simply by changing the sign ofthe number in the middle circle. The level ofdifficulty of this activity can be adjustedto different groups, and it can'also provide forthe development of specific computationalskills. Two activities are illustrated in figure7. Put the numbers 21 through28 in the outer Brain Teasers,such as the two that fol- ring of circles so that eachline totals 40. low, were interspersed withother activities, Fig, 7.Star-Burst and the pupils seemedunusually motivated to find the correct solutions. collected concerning word-problemskills, John was given $2.00 to buysome school sup- classroom observation providedample tes-, plies for the opening day of school.He went to timony that such practicewas both enjoy- the store and found that pencilscost 100 each; able and beneficial. erasers, 15e each; paper clips, 25? per box;and rubber bands, two for I. He bought 100things. What did he buy? Conclusion and recommendations A man bought a lawn mower for$20.00. He A small-scale study usingactivities in later sold it for $25.00. Still later, he bought the mental computation and selectedenrich- same lawn mower back for $15.00. Didhe lose ment money or make money on the lawn-mowertrade? activities produced almosta full year's increase in mathematics These activities represent achievement a most difficult scores for a sixth-grade class. This result level of word problems,yet allpupils was obtained in a total of sixextra hours achieved an increased degreeof success. of instruction accumulated Although no specific objective over a five- data wereI week period. .) 28 GAMES AND PIZZ1ES actisity.andmathematics There can be littledoubt that these re- instructional achioement. . sults were influencedby factors other than actisities such as: (a)the 2. Exploratory useof such actisities in the instructional classeswhere effect, (that often ac- elementarymathematics Hawthorne. or halo are lagging companies enthusiasticexperimentation); komputational performances (h) the instructionalvariable (since this seems imperative. I project was directedtoward specific computational skills, instructionwas aimed at (c) a References any test thatmeasured such skills); during the pretest Crescimbeni. Joseph eiralimelicEnrichment Ac- testing set established Children. West weeks elapsed between tivities for Elemeniar School (even though five Nyack. N.Y.: Parker PublishingCo.. 1965. there may base been test .administrations, Kramer. Klaas. Menial Computation.bks. AF, Science Research. some retentionof test material). teacher's guide. Chicago. While there is room forfurther specula- Associates. 1965. Achioement Series. two recommendations Science Research Associates. tion on these results, Form E. -Blue Lod Chicago.Science Re- are justifiablehere: search Associates. 197). research is needed tode- NOTE. A more detailed report ofthe research 1. Additional findings reported in this article ismailable from termire more accuratelythe relationships titnc. nature of the the authors am. n.he amount of Whole Numbers .4

Amajor portion ofthe elementary school mathematics curriculum is devoted to a study of the whole numbers and their operations. Many of the games selected for this chapter offer a vagietylof ways to provide the much- 'needed drill and practice on the basic fact:, of addition, subtraction; multiplication, and division. In the opening article, Deans offers several game activities to aid young children in developing their ,counting skills. This article should stimulate the reader to think of ways to adapt various classroom and playground games to aid in mathematics instruction. The idea of using pictures to motivate students to complete a drill sheet is suggested by Crouse and Rinehart. Many students will find these "connect the dot" activities to be most enjoyable, and it is possible to create similar activities without any special artistic ability. The "Secret Number Sentence" game suggeited by Swart will provide students with an opportunity to polish up their skill in asking questions while they work with number sentences at an appropriate level of difficulty. Patterson provides a nice set of directions for constructing drill wheels that may be used in a variety of individual or group activities to practice the basic facts. In the "Go Shopping! ..." activity suggested by Orans. students will have opportunities to role-play a shopping trip to a candy store. This should be a sweet experience for all! The three addition games suggested by Heckman are easy to play, and the needed materials can be readily prepared by the classroom teacher for a very 4. small price. She also suggests variations to provide practice in multiplication. Howard Gosman suggests that dice are "loaded" with activities to help children master basic number facts while enjoying the experience. He then provides instructions for playing at least a half -dn7en gamec 'icing dice. In the next article, Bartel suggeos a "computer" game that also uses dice. She provides several "for instances" to demonstrate how her activity can be varied to suit the needs of children at various grade levels. The dice game "Multi-bet" by Miki will provide practice with the "more difficult" basic nloltiplication facts and appeal to students who have a flair for gambling. For those who need practice in two- and three-digit addition, Dilley and Rucker suggest an interesting game called "Build the Greatest Sum." This game also provides opportunities for students to formulate strategies that will increase the chances of winning. Several other games are also described in this article. An activity for three-digit subtraction is offered by Smith in his "Witch's Best Game." The students will have many opportunities to gain experience with the various regrouping situations that occur in three-digit subtraction problems. The card game described by Arnsdorf should provide the players with opportunities to make strategic decisions while they continually do mental computation in the four basic operations. This should provide an effective and enjoyable review of the basic facts.

.., 4 30 GAMES AND PUZZLES

Broadbent's "Contig" game is simple to make and will providemuch practice with the basic facts of addition, subtraction, multiplication,and division. Many students will enjoy playing this game at home withtheir older brothers, sisters, and parents. Six games that help the learning and retaining of a varietyof mathematical skills and understandings arc described by Metzner andSharp. Eachf the activities requires three decks of ordinary playing cards, andseveral teas of two or three students each may play in &game. Inthe next article, Risto cell' suggests "Grecrr Chimneyc Poker" as a game todevelop proficiency with e multiplication tables through twelve. '"Ditty" is a game in which the student himself initiates thedrill activities. The game automatically provides for individual differencesand can be productive academic recreation at almost every grade level. Willsprovides an excellent set,of directions and guidelines for effective use in theclassroom. In his article on "Magic Triangles ...," Zalewski offers anumber of ideas for providing practice in addition while simultaneously stimulatingthe students to practice the searching, guessing, and thinking thatis essential for -problem solving. Similar experiences may be developedfrom the work with magic squares proposed by Cappon.

I- 4 .,

Games for the early grades v

EDWINA DEANS

Gamesare one means of helping young that they use frequently in connection children develop number concepts. with their instructional programs. Good learning situations have been observed as teachers and childrenengage in Kitty-cat games such as the ones described below. It A game for kitukrgarteh and first grade will be noted thatsome of these are The children sit on the floor in a circle. adapted from familiargames with slight One child is chosen to be Kitty-cat. Kitty- variations or adjustments made to high- cat sits in the center of the circle. When all light the mathematics aspect. Perhaps are ready to begin the game, one child readers willsee: -;;tar possibilities in slowly counts to twenty. The children many classroom or playground games forming the circle nave a designatednum- 31 ..,0 1 i 32 GAMES AND PUZZLES ber of small balls which they roll from one A similar game, well adapted to die to another very quickly across thecircle. school room, makes/use of paper blocks As' the balls roll across, theKitty-cat instead of pins. The paper blocks are made catches as many as he can before the from heavy construction paper, using the counter reaches twenty. The numberof familiar sixteen square fold. The paper balls caught is counted, and anotherKdty- blocks are piled. up pyramid fashion (Fig. cat and another counter are chosen. Each child has one turn and scores ac- Any number of children can play this cording to the number of squares knocked game at one time. Childrenforming the down. Since the paper blocks make very circle sit close together so that no gaps little noise as they fall, this game can be will be created through which balls may played while group activities are going on roll to the outside. without causing undue distraction. The number value, of course, comes through the checking. "Ho- w many balls did we have?" "How many balls did Kitty-cat get?" "How many did he fail to get?" The numbeg of balls is changed from Figure 2 time to time. All children should be aware Guessing how many of any change in the number of ballsbefore This simple game is appealing to small the game begins. children. The only materials needed are a Tenpins given number of small objects which can Commercial tenpins can be used but be held in'ime hand such as wooden beads, homemade ones work just as well. Large little pieces of chalk, or pennies. dowel rods or broom handles -sawed into Three to five children may play. The objects to the about four-inch lengths make very good game starts by showing the children asking, "How many do you see?" ones. Paper'cones madefrom construction Suppose the children count and discover paper may also be used.Old golf or tennis balls are ery good for rolling. Two to four that there are six. The leader continues, childa play. The tenpins are placed' in "I have them all in one hand. Now I am going to turn around so you cannot see, the formation shown in Figure 1. and put some in one hand and somein the other. Think about them and see if you 6666 can guess how many Ihave in one hand 66666 and how many in the other." . The children take turns guessing the number in each _liand=cimr and two, five six and none. After cach child leader Fsgwe 1 has had his turn at guessing the opens his hand andshows the sets. The Each child is given two turns to seehow first child to guess correctly gets the next many pins he is able toknock down. Scor- chance to be leader. ing can be done according tothe method Interest is maintained if the material best suited to thd ability of thechildren used and number of objects ischanged who are playing. This game may beplayed often. is with any number of pins. It is oftenplayed An interesting variation of this game also the with five or six at first with anadditional to show children the total and pin being added as the children areable to number in one hand. The children guess how many are in the other hand. cope with largernumbers. t.

Creative drill with pictures RICHARD CROUSE

University of Delaware, Newark, Delawcfre ELIZABETH RINEHART

Krebs School, Newport, Delaware

The word drill has some bad connotations Look forthepointthatis marked2,and for many mathematics educatorsprob- connect the two points by a line segment. Next, la- k for the point marked 3, and connect it ably generated by the recollections of gross by a line segment to the point marked 2. Repeat misuses of drill in the past. However, many this process as far as you can. educators believe that drill as meaningful Some teachers may feel that they do not practice has a definite place in the sequence have enough artistic ability to make these of teachingactivities, and theyaccept drawings. However, the drawings in this axiomatically that drill should occur after article were made from a coloring book. students understand a concept or skill. The picture that is wanted can be placed A key to effective drill seems to be how on top of ditto reproduction paper and itis motivated, and teachers often find traced with a pencil, or preferably a ball- this to be a difficult task. The following point pen. The rest is filled in freehand. activity is one that some teachers will find As an added feature to thisactivity, useful in their own classrooms. pictures could be chosen that would lend The idea is to use pictures to motivate themselves to other activities. For example, students to complete a drill exercise. Fig- after the picture of the sherd', was com- uresIand 2 were used in a third-gradc pleted, the children could make up a story class to review simple addition and sub- about the sheriff and the number of men tractioncombination:,butotherskills he captured. Many of these pictures could could easily be substituted and practiced. tend themselves to blending of subject The directions for this activity are simple: 4reas and perhapmake teaching and Place your pencil at the point where the answer ..'earning more woiwhile and exciting. The is1. (This pointis marked with the star *. teacher is limited only by her imagination. 33 / / 34 GAMES AND PUZZLES

7+9

8+9 9+9, 7+6

10+9' ...-----***7+5

25-4 .10+0

.1 \6_, \/"----- I

8-6 .5-I 3+3 13-10 60+8 4 +3

Fig.1 WHOLE NUMBERS 35

11-6

H-9

6 3+3

Fig. 2 Secret number sentence

WILLIAM L. SWART Central Michigan Univeniiy, MOi4iii Pleasant, Afichigan

William Swart is assistant professor of mathematics at Central Michigan University.

Here is a game for primary arithmetic, 1. Is the numeral in the (square, tri- providing stimulating drill on mathematical angle, blank) equal to...._? sentences and on addition,subtraction, 2.Is the numeral in the (square, tri- multiplication, and division combinations. angle, blank) greater than_.? Player A secretlyfillsin this pattern 3. Is the numeral in the (square, tri- with a number sentence: angle, blank) less than_? 0 L' - 4. Is the operation_.? (addition, sub- Player A must choose his numbers from traction, etc.) suit- some agreed-upon set. In Grade 2, a Each time Player B asks a question about able set would be 0, 1, 2, 3, 4, 5, 6, 7, 8, a position, he is able to eliminate one or 9. If he selects 6 + 3 = 9 as his secret more numerals under that position.For number sentence, he replaces the square the secret sentence 6 + 3 = 9, the ques- and the triangle with 6 and 3, the circle tion, "Is the numeral in the square equal . with the operational sign +, and the blank to 4?" would eliminate\4 as a possibility with 9. It must also be understood which for the square. And the question, "Is the operations are to be used. numeral in the square, g7ter than 7?" Player B puts the following on his paper: would be answered "No,'revealing that 00 el =...,..* 8 and 9 are not possible.`ter these two 0 + 0 0 questions, Player B's paper 'Should look

I ... 1 like this: 1 2 2 = 3 3 3 0 t 0 0 0 4 4 4 + 1 1 s s 5 1 2 2 6 6 6 2 3 3 3 7 7 7 -4.. 4 4 8 8 8 5 5 9 9 9 5 6 6 6 Then Player B proceeds to ask certain 7 q 7 questions of Player A until he discovers -8.. 8 8 what Player A's secret sentence is. The -9... 9 9 object isto discover the secret sentence in as few questions as possible. Player B Ifhis next question were, "Is the nu- can only ask thefollowing' questions: meral in the square less than 5?" he would 36 t3 0 / WHOLE NUMBERS 37 get a response of "No," andcross off 0, ceedinthis manner until 1, 2, and 3. Then one the children or two more ques- know how. to play. However, whilethe tions will reveal that 6 is thenumeral in game ishighly stimulating as a the square. group activity, itis doubtful that the groupap- As the game is played, the pupil should proach will result in significant gainfor become aware that "greater than"and the child who needs it most. The kind "less than" questions usually yield of more thinking promoted by thegame requires information than "equal to" questions. that the child have timeto ponderto When Player B succeeds indiscover- decide which questionsto ask, and what ing Player A's secret sentence, the roles information an answer provideshim. are reversed and Player A becomes the In the upper grades thegame can be questioner. The player who discovers the made more challenging by includingall other player's secret with the fewestques- four fundamental operations and by tions wins. allowing the relationship to include "greater It is advisable to start thegame with than" and "less than"as well as "equals," the teacher in the role of Player A and so that secret sentences such as 4 X 8< the children asking questions, andto pro- 380 may be used.

EDITORIAL CONINIFNT. This is one of manygame activities that provide students withop- portunities to gain experience in several importantareas. First, the game will provide desirable practice with the basic facts of arithmetic.Second, and perhaps evenmore important, the activity will provide a framework in which students may learn how to ask the "rightques- tions" to secure needed information. There are many types of games that can aid in the development of questioning skill. Forin- stance, one'Studenrcan select a whole number fromsome specified imerval, such as less than 30, and the second student can try to identify the number in five questionsor fewer. Points can be awarded on the basis of the results of the questioningas follows: Points scored If number is identified in 5 1 question 4 2 questions 3 3 questions 2 4 questions I 5 questions 0 6 questions or more If the students are ready to work with negative integers, the scoring,could be continuedas fol 1 ows: . I point 7 questions 2 points 8 questions 3 points 9 questions

The game could be continued for a specified amount of time, or until one 'student getsa prescribed number of points. I

Making drill more interesting W. H. PATTERSON, JR.

University of Southwestern Louisiana, Lafayette, Louisiana

Looking for ways to make drill more in- to drill sessions. Also, a few, rings will pro-. teresting, entertaining, and even fun? Isn't vide a large supply of drill problems for a everyone? I have developed a little device teacher. For example, 25 numerals printed that may help you provide more interesting on each of two wheels will result in 625 drill sessions for your students. I call it a problems! drill wheel. To make a drill wheel first cut out the The drill wheel is easy and inexpensive mounts for the spinners, using posterboard to make. It consists of a board on which or cardboard. Paste or glue these mounts are mounted two or more spinners. Card- to the rectangular base, which can also be board rings with numerals printed on them made from posterboard or cardboard. are fitted over the spinners,. Thespinners`Next, punch holes through each mount are then twirled and the students add (or and using a fastener, attach the spinners. subtract. or multiply,...) the numbers The spinners can be made of any con- indicated by the spinners. A drill wheel venient, stiff material, such as posterboard, with two spinners is pictured in figure 1. cardboardoor an old photo negative. The Although simple, the device has several number rings can be cut,easily from card- advantages. It is different and will add fun board and labeled with felt pens. Care 0 WI101.1: NUNIBERS 39 should be taken so that the hole in the ring improper fractions on them, sets the op- fits over the spinner mount. Wholenum- eration indicator on +, and goes to work. bers have been printed on the rings shown A large supply of rings for a classroom in figureI, but other types of numbers wheel can be obtained quickly by simply (negative integers, propei- and improper letting the students, either individually or fractions, and so on) can be used. An op. in small groups, make them, eration indicator (see fig. 1) is a nice ad- While larger drill wheels can be used for dition to the drill wheel. regular class drill, they can also be ef- I have made drill wheel models as large fectively used in other ways. For example, as 4 feet wide and as small as a regular a class might be divided into two teams piece of notebook paper. A teacher can that compete in working problems with the build up a file of number rings and, with teacher (or team captains) doing the spin- a large drill wheel hanging in the front of ning. To add interest, some numerals on the classroom, be ready for many types of the rings could be printedindifferent drill. For example, if a class needs drill colors and bonus points given when these in adding improper fractions, the teacher numbers turn up. simply selects two (or more) rings with Smaller models of the drill wheel can be

RING

BASE

OPERATION INDICATOR

Fig.I 40 GAMES AND PUZZLES made and used individually by students. Consider using the drill wheel in your Those students with any artistic inclina- classes. I think that you, too, will find that tions will probably enjoy making their own it is an inexpensive, economical device that spinner wheel and rings. In fact, an inter- can give you an opportunity to vary your ested student can)nake a "drill wheel kit" instructional activities, add interest to your with a pocket for the rings, that will fit in classes, and make your students more en- his notebook. thusiastic about drill.

EDITORIAL. COMMENT.Another activity that you can use tomake drill more interesting is called "Pencil Point Facts." First you need to construct somepractice cards for the related facts you want the students to work on. For'example, amultiplication/division practice card ould appear as follows:

Students may work together in pairs to practicetheir facts. The holder of the card inserts give his pencil point from the back of the cardthrough the hole for 8. The other student must the product for 6 x 8. The holder of thecard can check the answer (48) on the backof the card beside the hole through which heinserted his pencil point. For division, the card is reversed so that"-..-6" shows for the student computing the answers. The holder inserts his pencilfrom the back side of the card as previouslydescribed. If 6. the pencil point is inserted through the holefor 18, the student gives the quotient for 18 The holder can check the answer (3) on theback of the card. Cards may be constructed in a similar mannerfor related addition/subtraction facts. The cards may be used in game activitiesfor pairs of students by setting goals andaward- ing points for correct answers. For example, onestudent may pick a card (from the set identified by the teacher according to need)and challenge his opponent with five"pencil point facts" on the card. One point is awarded foreach correct answer, then the two students reverse roles. Go shopping! Problem-solving activities for the primary grades with provisions for individualization

S Y L V I A ORANS Parkway School, Plainview, New York

Materials Real money (one dime, two nickels, ten pennies) brought from home by each child Real candy (large candy gum drops, several colors) Construction paper for price tags Clear plastic bags (sandwich size) The approach .Problem situations requiring a minimal reading vocabulary of ten words. using real materials and real money in a gamelike atmosphere of playing store. The procedure Our candy-store activities were initiated after much exploration with simple games using real money. Our bulletin board, on which real candies were displayed, was our NOTE: Color names may be omitted. Color of candy "store" (see fig.1). Several candies of is visible. one color were placed in eadh plastic bag. FIGURE 1 The bags, containing orange, yellow, red, green, and black candies, were tacked to children, were changed often to vary the the bulletin board. Price tags, made by the level of difficulty and/or to lend interest.

LA 41 42 GAMES AND PUZZLES

At the beginning, store activities were You TYou End corcly rYou must Your teacher-directed. All the children used the by', Cola 1 PoY Change 15 coin or coins designated at- their desks. 2 green at I 4 One child was selected to "go shopping" rcd 44 I 4 /04 3 yellow 31 _4 _4 for each problem situation. Another child /Of 2 toocK ,44 4 was the storekeeper, and gave change under teacher supervision. As each problem !CURL2a was solved, the shopper was reimbursed with his original coin or coins. The chil- Variation YOv Nov EocbcoNlirovmosr Your dren who were seated at their desks were have buy costs pay cbonseis involved in checking out each problem. 34 _4 if1 The children used their coins; the candy, 04 4 s4 however, remained on the bulletin board --4 of (!) with the promise of its distribution to all when the store dosed, at the end of the FIGe121.2b week. Sonic oral problems follow. Anoth, r variationopen-ended probIegn. I. Go shopping. 1 You You fah cordi,You must Your You have a dime. 1 hay( by costs' pay change How many greencandies canvoubuy" _4 (green-5 cents each) .34 ___ yellow 34 --.4 204 _ black 6 4 ___4 4 2. Go shopping. 204 _ red 53 4 _4 You have a dime. i 1i _____4 ___01 How many candies do you want to buy" s2;50:44 !:,:',e;t 54 ', _4 I What must you pay? __ _..4 , 4 _4 How much change will you get? " These problems were elective 3. Go shopping. FIGUR!.2c You have a nickel. Can you buy black car dy? (black-6 cents each -shopping" on paper. They copied the in- How much more money do you need" formation from the charts on a ditto that 4. You bought five candies of ore color had Inc columns and headings to cor- You had a dime. respond to the chart. The problemshad What color candy did you huy9 become more difficult (* items were elec- 5. You bought one candy. tive), yet the problems were stillwithin You had a dime. the range of the slowest pupil, becausehe You got four cents change. each What color candy did bus" could choose to buy oily one of enjoyed 6. You bought two candies candy. The more capable pupil You had a dime. the challenge of choosing candy that was You got three cents change priced at afigurethat enabled him to Were your candies the same color'' spend of his money or to get the least Could they he the same color" amount of change. Then we went shopping on the chalk- It was possible to involve all thepupils, board. The candies were still on display. regardless of reading level, in these prob- to refer to. and for "show" appeal.A lem-solving actin ities. Because of the vari- chart was set up ( we figs. 2a. 2b. and 2c ). ables of price. number of candies,and Written work followed when the chil- otherariables inthe chart, these store dren showed a readiness to apply the tech- acti% ities prop ided individuali itionfor a niques. Problems suggested in the charts. broad range of abilities and challenged "Vanation" and "Another `variation," were each pupil's unique mathematical poten- used for written work. The pupils went tial. 51 They all addup

M. JANE HECKMAN

Now principal of the Pembroke Elementary School, Birmingham Public Schools, Troy, Michigan, Jane Heckman has also served as an elementary school teacher and a mathematics consultant for grades kindergarten through six.

The games described here have provided Directions: practice in addition for many children in 1. To start the game, each player rolls numerous classes. The games are easy to one of thereddice. The player with the play - and the needed materials can be largest number appearing on the top face readily prepared by a classroom teacher of his thrown die plays first. for a small price. They all add up to addi- 2. On his turn, a player. selects two of tion 'n fun. the four dice and rolls them on the playing area. Thesumof the two numbers appearing on the top faces of the thrown dice is Roll Hut sum covered on the player's gameboard. If the Number of players: 2 sum is already covered, the player loses a Game equipment: chance to cover a square.

Two cubes with the numerals 0, 10, 9, 3. Players take turns rolling the dice. _ 8, 7, 6 in red on the faces of the cubes. 4. Play continues in this manner until Two cubes with the numerals 0,1, 2, one player has either four markers in a 3, 4, 5 in green on the faces of the cubes. vertical row or five markers in a horizon- Two ROLL THE sum gameboards. (Fig. tal row. 1.) 5. The game can be varied by having Markersbeans, corn,buttons. disks players (at least 40). : cover the entire board with markers. cover four corners. cover a "T" (top horizontal row and middle vertical row). - A ---t.'_-=---- Author's note.By giving each player chance to pick his dice, this game allows illEEK--- for a choice of possible addends. Thus a WEE player must have some knowledge of which = numbers can and cannot be added to give MilIIMMM the sum he wants to cover. Another game, ROLL. THE PRODUCT, provides practice in 16MOM multiplication and r.quires the same kind Fig. I of knowledge of products and their factors. t 43 44 GAMES AND PUZZLES

To play the multiplication game, two ROLL Directions THE PRODUCT gameboards (fin2) are 1. Eachplayer in turn rolls the fourdice needed and more markers. and adds the numbers appearing on the top faces. The player with the greatest total starts the game. 2. On his turn, a player rolls the four dice. He selects two of the numbers appearing on the top faces to form a sum on the playing board. He places one of his ifiCEIDEM markers opposite that sum on his side of the playing board. IIMMMM 3. Players take turns rolling the dice. ElfZEIMM 4. Play continues in this manner until one player has placed his markers opposite LiMEMO five sums in a rowand wins the game. 54MCCES:11 Author's note.Players may test iwo or three sums before they make the selection Fig. 2 for their play. This game can be varied by making sums with three dice instead of Five sumi in a row two. Number of players:2 Kumpoz Game equipment: Two cubes with faces marked 0, 1, 2, Number of players:2 to 4 3, 4, ancI5. Game equipment: Two cubeswithfaces marked 2, 3, 6, Eighty-one cards, each 41/2" by 2", with 7, 8, and 9. nine cards of each of the following:1, 2, One playing board. (Fig. 3.) 3, 4, 5, 6, 7, 8, 9. (Fig. 4.) Markersbeans, corn, buttons, disks. 2" IllirlEM LI MW-Wi'M 11111 1 Al" MIIIIMIMMIMMENOMMEMIIIIM 5 MEE +1 I II MIA MORNIuliiii, 1$ RI itI 1111111111111113::111111 I 0

nu 111 1 !! ,1 III I II! Ana i Fig. 4 11111111111110MMINCEi,ilifillilimip 11111111111MEMMINENEMIMI Directions: IIIIIIIMMEEMP,11111MIMEIMIll 1. Cards are placed face down on play- RH 1111110111111111MEMRIIIMEM111111111 ing area. IIIIIIIIIMIIIIMMIHRMIMMEIHME1 2. To begin the game, each player turns 11111111111111111111MHINIIIMMOMMffiffill

III 11111111111111111111111.1111111111Mifirnifilffil over one card. The player turning over the

linififfHEIM i.101um!,Ho 1 ! iii highest number starts the game. (In case IMIIMIEMMIIIMPIMMOMMINffiffil of ties, players in question turn over new ;:r;11111; 1111 1111111111ffilliISM i cards.)Playthen proceeds clockwise. 11 11 +i IIIIMIifillffiifinna 1 ,,ii 1!11 Drawn cards are returned face down to PH 1 I " +," litilIMINIENIMI the playing area. 1; _____L_____1.IA1E411111111111111=11 II11111111NRUMW1.'i;INL ,,,,. '' 3. The starting player selects one num- IMM IN il lir;iiii li,,,: ber from the set (10, 11, 12, ... ,18). The selected number will be used for this ig. I ,1 ! i ,i,, sound of play. That is,the starting player Fig. 3 selects the number. WI101.E NUMBERS 45 4. Each player draws 15 cards from the playing area. 5. The first player puts down two or 5 more cards. The numbers on the played 3rd run cards must add up to the number selected 2 in direction 3. For example, suppose the Ist run selected number is 16 and the first player puts down the three cards shoWn in figure 5. This is called a run. The run must total the selected number, in. this case 16. 2nd run L -I- 10- -I-GI Fig. 5

Fig, 7 6. The second player then puts down cards to make a new run. The new run may winner, or if no player can play all his be connoted to the 1, the 6, or the 9 card cards, the player having the least number of the 'first run. The new run must also of cards when play stops is declared the total 16.''See fig. 6.) winner.

...First run total 16.

Second run total 16

Fig, 6

4th run . 7. Play continues clockwise around the table with each pl4er in turn putting down cards to make new runs. All rimc in a --10j csJ-Loa: game must total the selected number. Pos- sible third and fourth plays in the sample Fig. 8 game are shown in figures 7 and 8. 8. Play continues in this manner until Author's note.In this game players have one plays has used all of his cards, or practice in addkg two or more numbers until no player can make a play. The-player to make a given sum and on each turn who has played all his cards is declared the they must test several combinations. Mastering the basic facts withdice

HOWARD Y. GOSMAN

Mathematics coordinator and teacher at P.S. 165 in New YorkCity, Howard Gosman is.also doing additional graduate work at Teachers Collegeof Columbia University.

1 The dice arc "loaded" with activities to isthe total number of points scored. If help, master basic number facts the player rolls a double before he passes and enjoy the experience. "Double Trou- the dice to the next player, then his score ble" is a good example of an activity that for that round is zero. If the player passes can be used in any grade.This game without rolling a double, then he adds helps children master the basic facts in the score for the round to his total score. addition, subtraction, multiplication, and Once 'a player passes, he cannot lose any division. The game can be used in any of his points. classroom, and the teacher may modify it to suit his own classroom.'needs. Variations The game can be played with one;two, three, or four players. Two dice, a score Double-Double Trouble, Triple-Double sheet, and a pencil are all the materials Trouble, and other Multiplication-Double rollthe dice as needed. . Troubles. The players The players roll the dice to determine usual. Their score On each round isthe who will go first. Each player then rolls number on the dice multiplied by two,by the dice in tarn. His score is the sum of three, or by any number selected before the numbers that come up on the two the game begins. Of course, the total num- dice. In the simplest form of the game ber of points needed to win the game would (called"Trouble"), thefirstplayerto also be multiplied by the same number. score 100 points (or a lower scorein the Addition-Double Trouble. The players lower grades)isthe winner. A simple roll the dice as usual. Their score on each scoresheet should be prepared by the round isthe number rolled on the dice teacher so that the children can keep a Plus a number selected before the game running count of the score. begins. In Double Trouble each player may roll Multiplication-Addition DoubleTrouble. the dice one or more times during hic turn. The players roll the dice. The number on After each roll the player must decide to ..the diceismultiplied by a preselected. either pass or play. His score for t- he round number, and a second number is added - i .,1 , t tti 46 W1101.1: N OWE RS 47 to the resulting product. For example, the subtraction, division, and fractions. The number on the dice is multiplied by five, children canalso make up their own and eight is added to the product. (5 x games. number thrown on dice + 8 = score.) Double Trouble (or any of its varia- Triple Trouble. Three dice are used in- tions)is a fun way to learn the basic stead of two. The playerl score for the number facts. The gameiscompletely round is reduced to zero if two of the under the control of the children. They three dice thrown make a double. The must coil ,thedice, do the appropriate player's total score is reduced to zero if computation, keep track of their scores, allthree dice show the same number. and decide to pass or play after each Double-Triple Trouble would mean dou- throw of the dice. Children can learn their bling the score on the three dice. basic number facts quickly without losing Other variations can be developed using interest in numbers.

EDITORIALCoMML \T.Another interesting activity with dice was suggested by Carol Ste- phens in the April 1967 issue of the Arithmetic Teacher. The game is called "Yahoo." andthe basic information is as follows: Equipment: 2 score sheets 2 dice a multiplication fact chart a set of bonus cards from colored construthon paper with the following facts written on them: 7 x 2 through 7 x 9 8 x 2 through 8 x 9 9 x 2 through 9 x 9 Number of players: '2 Directions. Each person rolls the dice in turn. The two numbers shownon top indicate the factors. If the player knows the product, he records it on the chart. Except for the doubles facts, each product may be recorded in one of two possible places. If a player rollsa double, he gives the product and writes the answer on the chart. If correct, he selectsa bonus card from the piles, gibes its product. and records the answer on the chart. Whena playLr completes a row (vertical, horizontal. or diagonal), he earns a yahoo. A gurne consists of twenty turns for each player, and ten points are awarded for each yahoo. The objective is to earn the most points in the game. A yahoo may also be earned by completingany three consecutive bonus squares.

Turn if 12 3 4 5 6 7 8 9 tw s 3'

. ... 3 4 5 8 ' 6 '9 id. II' 9 04"*.4113n Fat Oat. t3

Scare Sheet

An obvious variation of this game would be to adapt it to drill with addition facts.

t4.1 Let's play computer

ELAINE V. BARTEL

An associate professor of education at the University of WisconsinMilwaukee, Elaine Barter---, is director of the intern Teaching Program for the School of Education on that campus.

Despite the burgeoning market of ma- possibilities of the game in lengthy detail, nipulative devices and games to enhance it might suffice to go through a few "for pupils' understanding of basic mathematics instances" at various grade levels to give concepts, classroom teachers and para- the reader a general idea of how "Com- professionalsarestillexhaustin; every puter" can be used with students. available resource in their search for ideas Imagine, if you will, that our materials and materialsthatmeetthe following include three pairs of dice; one pair each criteria: of red, green, and white. At the primary grade level, the teacher could choose two I.Relatively inexpensive youngsters, asmall group, or a group 2. Easily adaptable to various levels of divided evenly into two teams.Initially understanding the teacher would "program" the com- 3. Related to identified problem areas of puter: "R + r" or "R r" or "R + r + the mathematics curricul,(n G + g." (The letters refer to the color 4. Simple enough to be introduced and of the dice. The capital letter always de- used with a minimum of teacher guid- notes the larger of the two numbers on ance or supervision. each pair of dice, and the small letter the One idea that has passed the test of smaller of the two numbers, unless they all of these criteria and has produced very are the same.) Eventually the children positive feedback from preservice and in- will "program" the computer themselves, service teachers, tutors, and paraprofes- .writing the problem on the chalkboard sionals,is the game thatI have titled or on a large piece of paper on the table "Computer." The only materials needed where the aatre is being played. When are several pairs of dice of varying colors. the children set up a problem like "R + Since the game has been designed spe- W G," they will face the 'possibility cificallytoaidinpupils'graspand of encountering a negative number for memorization of basic addition, subtrac- their answer. This should cause no diffi- tion, and multiplication facts,it speaks culty, especially if the classroom number- directly to an identified problem area in line extends to the left beyond zero. a fresh, game approach. Since "rules" for In the- intermediate grades pupils will the game (programming the computer) tend to set up problems that match their ate established by the group using it at the ability to handle the basic addition and time, the game can be used effectively multiplication facts accurately and rapidly. with pupils at any age or level of under- This will also serve to encourage pupils standing. to work at the mastery of these facts, since Rather than attempt to explain the many the need to know them well will be clearly

48 t WHOLE NUMBERS 49 demonstrated when pupils participatein correct, the person (or his team) gets the game Such problems as that number of points added to his score. (R + r + W + w) (G + g) If the answer is under 5 or is a negative or number, the answer (to the nearest whole number) is squared to determine his score. (W + G) X (w + g) Ifthe answerisincorrect, thecorrect R r answer is deducted frem his score. will be used often at this level, but it will Needless to say, the possibilities for thi\ nett be lung before students will attempt game are endless; and ifit does happen to set up much more sophisticated prob- that children establish problems that are lems. for the moment 'beyond their ability to At the junior high level students will solve, this can be a valid learning experi- spend a great deal of time and mental ence for them, and may be just the en- energy while competing with each other couragement they need to increase their on a problem like the following: own level of expertise in mental computa- R' + ww tion. ---G X g Some of the obvious advantages of using this game approach for mastery of basic For :.;;ample, suppose six dice are rolled skills are that (a) it clearly demonstrates so that R = 4,-r = 3, W = 6, w = 2, to students the need to have the addition G = 5, and g = 4. Then, assuming that and multiplication facts "at their finger itis understood that the answer must be tips," (b) it encourages the utilization of inthe simplest form and expressed as a mental arithmetic, (c) it stresses accuracy mixed number when possible, the correct and speed in a computation situation where response would be 6%. the student must determine the balance For each game session,itis strongly between accuracy and speed, and (d)it suggested that children be permitted to places students in competition with part- establish their own "rules" for that ses- ners of similar ability. sion. For the example just given, students Rather than attempt to persuade you might conceivably establish the following to introduce this game to your students, T rules: would like to suggestthat you finda I. A referee will be selected who will willing colleague, set up a few problems, inturn choose the number and color of and play the game yourself. This will dice to be used, and select someone to serve two purposes:Itwill demonstrate "program" the computer(establishthe to you the applicability and feasibility of problem ). the game for many and varied levels, and 2. The referee will be responsible for itwillquicklydispel any preconceived tossingthe dice, checking answers, and notions that itis boring or will lose its keeping score. appeal for mathematics students. And re- 3. No participant may touch or move member, once you have introduced itto the dice while computation is taking place. your students, they might on occasion ask you to serve as referee. You would be in 4. The referee will jot down the first a much better position to serve in that answer given, wait a few moments, then capacityif you feelcomfortable about work through the problem orally to de- matching your own skills with those of termine the correct answer. your students. I can't guarantee that you 5. Participants may use scratch paper will always win, but I can guarantee some for computation. highly tense moments and the joy of see- 6. Scores he determined asfol- ing pupilsreally involvedin their own lows:If thefirstanswer called outis learning. Multi-bet

ARTHUR K. MIKI

Wayoata Elementary School, Transcona, Manitoba, Canada

Thegame "Multi-Bet" provides practice dice and calls out the product of the two with the more difficult multiplication'com- numbers turned up. The product is the binations and has proved to be one of the winning number for that round. When a most popular games in the fourth grade. Multi-Bet uses two specially marked diceeach die has the numbers 4, 5, 6, 7, MULTI BET 8, and 9; a supply of counterspoker 1 One number 15 I chips or bottle -caps are excellent; and a CU Two numbers 7 I chart like the one pictured in figure I. The EfEi RED Four nos. 4 I numbers on the chart represent all the pos- --I sible products that may result from rolling M36543024 U, tv O the two dice. The chart also shows the I 3 various combinations of numbers that can ou) 64164872a) be played and the odds to correspond with 40634520 u) the different combinations. A student may c0 bet on one number, two adjacent numbers, 2842812503 four adjacent numbers or four numbers in a row, eight numbers, or tennumbers. 35493256 Placement of the counters for the different combinations is indicated on the chart. Each player begins with ten to fifteen counters. In each round players place their counters on the chosen numbers or groups of numbers,. One player then rolls both Fig.I t,.- 50 WHOLE NUMBERS 51 player has played two numbers, if either The game can be played for a designated number is the winning product the student time period. If a player loses all his coun- gains seven counters from the house pot. ters before the game is over, then he is out Similarly, if a play as played eight num- of the game. At the end of the time period bers, he gets two coulpters from the house the person with the most counters is the pot if any of his numbers is the winning winner. product. When the number or players The game can also be used in the study played are not winners, the counters are of simple probability. The chart itself can lost to the house pot. The players then be used to introduce the subject: Which place their bets for the next round and the numbers are the most likely to occur? How next player in turn will roll the dice and were the various odds chosen? Discussions call out the product. could stimulate experimental work on the The house pot would be all the counters different combinations that are most likely left after each player had received the des- to occur when rolling two dice. ignated amount at the beginning of the For students who are just beginning to game. A student who has mastered the learn basic multiplication facts, the dice multiplication facts, or a teacher could act can be marked from 1 to 6. The game can as the house man. also be applied to addition facts. r

;

1 Arithmetic games

CLYDE. A. DILLEY and WALTER E. RUCKER Urbana, Illinois

0ften there is little variation in the set- Sum," can be played as soon as children ting in which children practice arithmetic have been introducedto theaddition skills (basic combinations and algorithms). algorithm. Ahead of time, you should In many cases the practice is unchallenging make up digit cards by writing each of the and unrewarding. An appropriate game can ten digits (0-9) on a three-by-five card. provide children with both a challenging The game is played on a table like that and rewarding experience and an opper shown in figure .1. Each player can draw tunity to discover mathematical concepts. Some games that have been used successfully with children at all levels are de- + scribed below. The firstgame, "Build the Greatest Fig. 1 ill 52 WHOLE NUMBERS 53 his own tables, or you can duplicate copies an example of a Target Number game. to distribute to the players. To play the Ask the children to copy a sentence like game you shuffle the digit cards and select the one shown in figure 2. Then, as digits one at random. Tell the children to copy are picked at random asinthe above the digit that is on the card inany of the games, children record them as they choose boxes of the table. Replace the card with in the frames of the sentence. When all the the others, reshuffle, and pick another card. frames have been filled, students are to Tell the children to copy that digit inone simplify the expression they have built. of the five remaining boxes. Repeat untila The player who has builtthe number total of six digits have been picked and nearest the target number, in this case 24, recorded in the tables. Then ask the chil- wins the game. A winning strategy for this dren to add the two three-digit 'numbers. game is not so obvious, but children who The ,player with the greatest sum wins. have "number sense" will have a distinct Since the cards are selected at random, advantage. chance-is a factor, especially at first. How- ever, after the game has been playeda few times students will begin to formulate strategies that will increase their chances of winning. The opportunity for developing such strategiesis probably the most important facet of this particular game. Fig. 2 (The other objectives are to provide reinforcement of place-valtic conceptsa digit Such a game provides an interesting in one column is worth more or less than setting for the practice of many arithmetic the same digit in another columnand to skills. Of course, the pattern sentence can provide practice using the addition algo- be deliberately designed to cultivate those rithm.) These strategies involve an intg- skills in which the students lack profic, five use of probability concepts because to ency. decide where to write a given digit the Two more pattern sentences that might child must consider the chances of getting be of interest are shown in figure 3. (Of a greater or lesser number on a later draw. course, 0 should not occur in the denom- Such a strategy does not, of course, guar- inators.) You can make up pattern sen- antee a win, but the child who uses it has tences of your own, and children should a definite advantage over the child who be encouraged to suggest others. fills in the squares at random. Our students have always enjoyed these There are many obvious variations of games, and we think that yours will too. this game. For example, one variation is not to replace each digit card before the next is drawn. In this ease there will be no repetition of digits, Another variation is to extend the table so as to provide work (C with greater numbers. Still another is to 412 have students aim for the least sum instead of the greatest sum. Other variations involve subtraction, multiplication, and division In fact, one of the most interesting of such games is the "Build the Least Dif- 1 ference" game. Another typeof game thatchildren enjoy is culled "Target Number." Here is The Witch's- Best Game

C. WINSTON SMITH, JR. College of Education, Wayne State University, Detroit, Michigan

Asstudents encounter three -digit sub- writes an addend (subtrahend) and asks traction, many who previously have "mas- the class to choose the name from the tered" two-digit subtraction falter, not only board that they think would be the most because of the complex regouping required helpful when subtracting. After surveying in some problems, but also because of the the four choices and deciding upon the wider selection of regroupings to consider. best renaming, each student indicates his While two-digit subtraction requires the choice by raising one, two, three, or four student to choose between two alternatives fingers. After the class has shown its re- no regrouping or regroupingfrom tens sponse to the first addend (subtrahend), to onesiiiree-digit subtraction can call the teacher may substitute another addend for (1) no- regrouping,(2) regrouping (subtrahend) and ask the class to indicate from tens to ones, (3) regrouping from their choices for this new problem in the hundreds to tens, or (4) regrouping from same manner.(For example, 162 may hundreds to tens and from tens to ones. be substituted for 129, and a new example, This range of choices, when linked with 245 162, presented.)After five or six the understanding and use of basic sub- problems, the teacher may wish to change traction facts and regrouping itself, creates the sum (minuend) to vary the problem. for some a task that is very perplexing. situation. An activity that encourages each child As the students raise their fingers indi- to search for the best renaming, while pro- cating their choices, the teacher will have viding a gamelike atmosphere, is one that an opportunity to give immediaterein- the students ca!! "The Witch's Best Game" forcement with a nod, a smile, or a word of (Which Is Best),To begin the game the praise. With a perplexed look the teacher teacher chooses a three-digit number. After can encourage others to reconsider their recording this on the board, she writes answers. Observation of the total class re- the four names used most often when sub- sponse will aitithe teacher in locating those tracting from this number. students who are in need of individual help and will also point out areas which re- (1)200+ 40+ 5 quire reteaching to the entire class. 245 (2)200+ 30+ 15 129 (3)100+ 140+ 5 It is important throughout for the teach- (4)100+ 130+ 15 er to stress that there are times when two In some classes students assist by suggest- or more names may beappropriate ing names and illustrating thcir choices on therefore the classis looking for a re- the twenty-bead abacus. Using this number sponse that is best or most helpful, not one as the sum (minuend), the teacherthen that is "right" or "wrong." Students who WHOLE NUMBERS 55 choose names that do not appear to be gken anopportunitytoexplaintheir helpful to the rest of the class should be choices.

EDITORIAL COMMENT. -To provide additional reinforcement with three-digit subtraction problems, create a "Pick Your Problem" game as follows: Cover three shoe boxes with deco- rative paper and mark themA. B.and C. In boxA.place cards that have only three-digit subtraction problems requiring no renaming. In boxB,place cards that have three-digit subtraction exercises requiring one renatt;ng. In box C, place cards that have three-digit subtraction examples requiring double renaming. Two to six players may participate at one time. Each student in turn spins a spinner markedA. B.and C. as shown. He then selects a card from the box indicated by the spinner and works the problem. If worked correctly, problems fromAare worth 2 points, problems fromBare worth 5 points, and problems from C are worth 10 points. The student who accumulates the most points in a specified number of turns or times is the winner. You may wish to have an answer key available for,use if the students disagree on answers.

Beanbag-toss games may also be created to provide practice. For example, the following game board may be made from posterboard or plywood:

Sorry, 326 598 141 475 Lose 5 _BittS__, Bonus too 783 249 505 94 5 Points

Bonus 184 393 609 723 670 10 Points

542 177 316 489 320 152

Sorry, 210 Lose 3 425 206 588 600 j Pcints

Have each student throw the two beanbags on the game board and then add (or subtract) the two numbers indicated by the beanbags. One point may be awarded for each correct answer. If a bag lands on a "Bonus" or a "Sorry" block, the player records the points accordingly and takes another turn. Encourage.* students to work their problems on the chalkboard or overhead projector so the results maybe verified. \

A game for reviewingbasiC facts of arithmetic

EDWARD E. ARNSDORF

Sacramento State College, Sacramento, California

Adelightful game to use in reviewing Suppose the (6, 2) card (the one with addition-subtractionandmultiplication- the 6 on top and the 2 on the bottom) is division facts for whole numbers has been the one that has been placed face up.-The developed by a Sacramento State College numbers on this card may represent 6 + education student, Mary Dunlap. 2 = 8, 6 2 = 4, 6 x 2 = 12, or 6 4- 2 Materials for the game consist of forty- = 3. If the first player has a cardcontain- five cards cut from tagboard. Three-by- ing two numbers whose sum, or difference, iwo-inch cards work well. The tagboard or product, or quotient is 8, or 4, or12, cards are numbered as shown in figure 1. or 3 respectively, he may play the card Two different colors should be used for from his hand by placing it face up on the the numerals on the cards. With all top (6, 2) card. If he does not have such a numerals one color and all bottom nu- card in his hand, he must draw the top merals another color,the cards can be card from the pack and then determine if arranged easily, right side up. it can be played. If the drawn card cannot The game may be played by two, three. be played, itis added to the player's hand or four persons. To begin play, thedeck and he draws again. The process of draw- is shuffled and five cards are dealt to each ing from the pack \continues until an 8, or player. One card is turned face up and the 4, or 12, or 3 is foimd. When a card can remaining undealt cards are placed ina be played, it is placed'on the face-up pile stack face down. The person at the im- and the next player takes his turn. For mediate left of the dealer plays first. example, assume that the first person plays

. 56 i k 1i.., WHOLE NUMBERS 57

2 R1 2 3 3 3 1 2 3 4 4 4 4 2 3 4

9 9 9 9 9 9 9 9 9 1 2 3 4 5 6 7 8 9

Fig. 1 the (8, 2) card, representing 8 ÷ 2 = 4, of the forty-five cards cannot L. used to on the opening (6, 2) card. The second name four different whole numbers. The player must now determine the new num- sum and the product may name the same bers that can be played. He may use any number; or the product and the quotient one of the four operations to name one of may name the same number; or the opera- the new numbers, in this instance either tion of division may not yield a whole 10 or 6 or 16 or 4, number. Since some cards are more likely Any time a card is played, another player to be playable than are others, there are may challenge the play by asking why that opportunities to make strategic decisions. particular card can indeed be played. The game is won by the first person to play all Players of this game are continually his cards. If all the cards have been drawn mentally performing the operations of addi- and the game has not been won, all but the tion, subtraction, multiplication, and divi- top card on the face-up, or' discard, pile sion. Hence this game can be both an effec- are reshuffled and the game continues. tive and an enjoyable way to review basic Notice that the two numbers on some facts of arithmetic.

EIMTORIAL COMMENT. Many familiar card games, such as Old Maid, War, Fish, and so on, may be adapted to serve particular needs for practice experience. For example, the Old Maid game could be renamed "Mathlete" and a set of cards constructed as follows for addition and subtraction facts. 25 matched pars of cads 1 Mattiefe Cad

1 ----1 7 15 1 L--la -J 11

1 12 I 3 _=_9_ ' 1

! 9 14 1 +51

Each game cou d be played with4, 5,or 6 players All cards are dealt to the players, and the first player chooses a card from the hand of the player on his right. The second player / '-, , 1 , chooses a card from the player on his right, and so on. Whenever a player gets a matched pair, he lays it down on the tabk. If correct, he earns a point; if not, he loses a point. The game continues until all matched pairs are on the table and a player is leftwith the "Math- lete" card. The player who is left with the "Math lete" card loses a specified number of points (or is awarded a specified bonus).

"Contig": a game to practice and sharpen skills and facts in the four fundamental operations

FRANK W. BROADBENT

Syracuse University. Syracuse, New York

"Contig"is a game that intermediate- a number on the board which is ad- grade children love to play and with which jacent vertically, horizontally, or diag- they can challenge their older brothers, onally to another covered number. One sisters, or patents. Four things are required point is scored for each adjacent cov- three.dice, a score pad, markers, and a ered number. Contig board. The dice, pad, and markers 5.When a player rolls the dice and is are rqadily available, and a supply of unable to produce a number that has not boards can be made by reproducing the already been covered, he must pass the sample provided at the end of this article dice to the next player. If he incorrectly on sheets of paper. passes the dice, believing he has no play when in fact he does have a play, any Rules of the Game of the other players may call out the 1. Two to five players may play Contig. mistake. The first player to call attest- 2. To begin play, each player in turn rci:s tion to the error may place his marker all three dice and determines the sum on the proper uncovered number. This a thethree numbers showing. The does not affect the turn of the player player with the smallest sum begins citing the error. play. Play then progresses from left to 6.A cumulative score is kept for each right. player. A playeriseliminated from 3. The first player rolls the three dice. He further play in a game when he fails in must use one or two operations on the three successive turns to produce a num- three numbers shown on the dice. He is ber that can be covered. When all playthen allowed to cover the resulting num- ers have experienced three successive ber on the board with a marker. When failures to produce a coverable number. he has finished his turn, he passes the the game ends. The player with the dice to the player on his right. A player highest cumulative score wins. may not cover a number that has been Variations of Contig previously covered. 4. To score in Contig, a player must cover 1. Use a one-minute egg timer to time the

58 CaNTIO

9 10 -11 12 13 14 15 16

17 18 19 20 21 222324

25262728 2930 31 32

333435 3637383940

41 42444548505455

60646672 75809096

100108120125144150180216 60 GAMES AND PUZZLES turn\ of each player.This will' tend to because the adjacent 10 has already been speeeLup the game. covered. Player 4 covers the 2 (4 [6 ÷ 2. Allow any player tochallengean op- 31 = 2) and scores 3 points, one for the ponent if the opponent does notchoose vertically adjacent 10 and one each for the the numbeg that will score themaximum diagonally adjacent 9 and 11. number of points. The challengershould then receive the difference between the number of points scored by the chosen Table 1 number and the greater number of Roll of Number Points points that could have been scored.. dice roLered worn' 3. For a faster game, allow onlyfive turns 0 for each player. The playerwith the Player I 2. 3. 4 9 Player 2 2. 4. S 10 1 highest score at the end oftly fifth Pla)er 3 1.5.6 II round would be the winner. Player 4 3, 4. 6 3 4. Allow students to play it as a.solitaire game and attempt to score as many points as possible before experiencing After the children hale played the game 'three successive unsuccessful, rolls of the for a while, questions like the following dide. might be explored: 5. To make the game easier, use afour- a) How were the numbers used in Contig by-five array and the numbers 1, 2, 3, selected? 4, 5, '6, 7, 8, 9, 10, 11, 12, 15, 16,18; h) What numbers would you use if you 20, 24, 25, 30, and 36. In this case, had two dice or four dice? . students use only two dice- but play by c) How many ways can you cover each the rules above. number in Contig? d) What isthe highest score (without °"- Sample play Veer players) challenges) a player can make in a game? Suppose four players rollthe dice in e) Why are some numbers between 1 and their respective turns as shown in table1. 216 left off the Contig board'? Player 1 covers the 9 on the playing board f) Would it be possible to use allthe (2 ± 3 4 = 9) but does notstdre be- numbers from I to 216 on aContig hoard if cause no other numbershave been covered. the dice went from Ito 10? Player 2 covers the 10 on the board([4 g) What is the probability of being able 21 >< 5 = 10) and scores 1 point_because to cover 216 on your first throw? the 10 is adjacent horizontally tothe al- h) Why is part of the Contig board still ready covered 9. Player 3 coversthe I1 uncovered when eseryone has passed three (1 I < 51-4- 6= I I) and scoresI point times?

EDI ro RIAI COMM!.r. --You may wish to haveeach student write a number sentence to illustrate his answig after each play.This will give him some ot the desired written practice and also make it possible for the other pliiyers tocheck his work more easily. The questions suggested at the end of the article areexcellent and should stimulate the students to do some analytical thinking. A variation of the game could be created by usingspinners instead of dice to generate the numbers. Another variation could be created byusing stacks of cards and having a player draw a card from each stack to start play.

1' . ,y

Cardematics Iusing playingcar5 as reinforcers and motivators , / in basic operations SEYMOUR METZNER and RICHARD M. SHARP California State University, Northridge, California

Materials for pupil activity-learning in 13 respectively. Other numerical values can all areas, and especially mathematics,' are be given the face cards, if desired. difficult to find even though more educators All of the described activities suggest are advocating this approach to learning. teams of students. Although any size team Some teachers haveusedconsiderable could play the game, there is greater student initiative and ingenuityto develop and participation if the teams are made up of no adapt inexpensive materials for pupil activ- more than two or three students. Small ity-learning. One of the most inexpensive, teams can better discuss possible answers easily obtained, and adaptable items that to problems. can be used is an ordinary deck of playing All games begin the same way. After the cards. / teams have been determined and the decks Six games or activities that have relevance of cards are shuffled, each team draws three tothe learning and retention of basic cards. Then the teacher draws a card (orl mathematical skills and understandings are cards) and shows it to all of the teams. The described here. The games motivate students procedures that follow then vary with the and add a certain "pizzaz" to the mathel- games, but whatever the procedure, itis matics program. Eachof theactivities repeated a predetermined number of times. requires three decks of playing cards. All In each game, the team with 'the largest nonface cards have their numerical face "bank" at the end of play is declared the value, with an ace as 1. Play can be 'limited winner. And in every game, as soon as a to the nonface cards, or if face cards are team "banks" cards, it draws to replace the used, the jack, queen, and king can be banked cards. A team always begins play assigOed the numerical values of I1, 12, and with three cards in its hand. 7061 62 GAMES AND PUZZLES These activities are self-correcting if each 5, then a team would need a 3 to add to 5 to team is paired with another team to verify get 8.) More than one card can be used by a each other's answers before being allowed team as long as each card is considered to bank cards or draw additional cards. The separately. All cardsthatare used are teacher may act as a court of last resort in added to the team's bank. questionable decisions. Divvy-do Prim -o Objective Objective Division review with remainders Identification of prime numbers, addition Procedure and subtraction practice The teacher draws two cards from the Procedure. deck; the students then divide the larger . Each team,by adding or subtracting one number by the smaller. Each team must then of their cards from that displayed by the see if they can get the same quotient and teacher, should try to reach a prime number. remainder by dividing any two of the cards Itispossible that more than one of a in their hand. A pair of cards that can be team's cards can fill this function, but the used to get the appropriate quotient and cards must be used separately. All cards remainder are added to the team's bank. that are used successfully to reach a prime number are added to the team's bank. Even-steven. Objective Round-up Divisionpractice with no remainders Objective Procedure Practice with basic number operations The teacher draws a card from the deck. Procedure Each team then tries to combine one of its Each team tries to use as many of its cards with the card displayed by the teacher cards as possible to reach the number to make a division problem with no re- drawn by the teacher. The team may use mainder. The card drawn by the teacher can any one or any combination of thebasic be used either as a divisor or a dividend. operations. (For example, if the teacher's (For example, if the teacher draws a 4, a card is a 4 and a team has a 5, an 8, and -a 10, team could use 2, since 4 ÷ 2 = 2, and an 8, then the team could use all three cards by since 8 ÷ 4 = 2.) A team may use more than showing that 5 X 8 = 40 and 40 ÷ 10 = 4.) one card as long as each card is used All cards that are used are added to the separately.All cards that are used are team's bank. added to the team's bank.

Magic number Multi-match Objective Objective Addition and subtraction practice Multiplication review Procedure Procedure The teacher announces a magic number The teacher .draws two cards from the and then draws a card. The teams add or deck. The students calculate the product subtract one of their cards from the card of the numbers on the drawn cards and drawn by the teacher to reach the magic then teams try to use any two or three of number. (For example, if the magic number their cards to get the same product. (For is 8 and the card drawn by the teacher is a example, if the teacher draws a 7 and an 8, W1101.1 Nuvtlits 63 a team could use a 2, 4, and 7 to get the Conclusion same product.) All cards that are used are All of these activities should be viewed added to the team's bank. simply as startingpoints. Teachers can This game can be varied by having teams adapt these games or activities to their own compete to see which team can get closest particular situation or to class needs and to the product of the cards drawn by the interests. The general format can be extend- teacher. The team that gets the nearest ed to areas not covered by these half-dozen product banks the cards it uses. activities.

.., Green Chimneys poker

T. RISTORCELLI Green Chimneys School, Brewster, New York

This game can be played by two, three, valued as follows: a jack is 11, a queen is 12, four, or more players using an ordinary and an ace is I. The kings and jokers are deck of cards. Six cards are dealt to each frediscards at the beginning of a player's player and the remaining cards, placed tn and both have zero point value. Jokers face down, form a drawing pile. The idea is ar ild. Three or more of a kind and a run for each player to try to rid himself of all ofree or more cards of the same suit are his cards before any of the competing also free discards at the beginning of a players by playing his cards one at a time player's turn. consistent with the rules of the game. The player who first disposes of all his Thefirstplayer (clockwise from the cards wins the hand. Points can be scored dealer) leads a card face up on the table, in one of two different ways. The winner of thus starting a playing pile. Each succeeding the hand can get credit for all the points player in his turn plays a card on the held by the other players when the hand playing pile, but when he plays the card he ends, in which case a high score is the must multiply the number on his own card goal of the game. Or, wfien a player goes by the number on the card that is face up out, the other players can be penalized the on the playing pile.If he multiplies cor- number of points they hold in their hands. rectly, the player will have gotten rid of one In the latter case a low score is the goal of card and the play goes to the next turn. the game. ! If he makes a mistake in the multiplication, he must take back the card he played and For players who are not yet up to the pick up an extra card from the drawing multiplication tables of 12, 11,10, 9, or pile. others, the corresponding cards are simply All number cards stand for the number removed from the deck before starting that is written on them and face cards are the game. ..,,-- ti 64 DIFFY HERBERT WILLS

An associate professor of mathematics education at Florida State University in Tallahassee, Herbert Wills has directed NSF summer institutes in computers, and last year was director of an NSF Academic Year Institute at Florida State University.

The game DIFFY is an academic game cally adjusting to students of diverse abil- that provides intrinsically interesting drill ities within a given classroom, DIFFY also experiences and allows for individual differ- provides productive academic recreation at ences. Moreover, teachers need not con- every grade level. This flexibility stems struct any of the exercises. The student from the fact that the game may be played himself initiates the drill, and the nature of with a variety of numbers: whole numbers, the game provides for variety in the num- nonnegative rationals, integers, all rationals, bers encountered. Thus the activity pro- or all real numbers. motes self-generated drill. This feature is To introduce a class to DIFFY, give commendable, since students won't run out each student a supply of game sheets, illus- of material and they become personally trated in figure 1, then play a sample game involved in their work. Besides automati- on the chalkboard or with the use of an

-, , Fig. 1 4 -I65 66 GAMES AND Pl'ZZI ES without writing zero in each of them, he wins. It is gratifying when a student introduces numbers in the game that produce problems that challenge him. One is also pleased with the care players take in doing their calculations or checking their work. This care assures them that a declared winis truly a success and not a fluke resulting frommis- calculation. Also, it is not uncommon to see students checking the work of others, at- Fig. 2 tempting to find an error that will refute a overhead projector.This provides each proposed win. This checking is essentially student with the information needed to additional drill, and carefully done drill at proceed independently and stimulates "Let that! Compare this with the lack of purpose me tryit myself" interest. The game is and motivation of regular drill for drill's started by placing numbers in the four sake. outermost corner cells. Players may choose Besides providing avehicleforself- any numbers they wish forthis purpose. generated drill in subtraction, DIFFY may Let us suppose that students chose the be used to introduce investigations that are numbers shown in figure 2 to start 4 game. more intellectuallystimulating.For ex- The adjacent corner cells determine the ample: number to be written between them. In this Starta sample game that requires particular game three of the numbers cen- several moves but stop short of its culmi- erated have been entered in figure 3. See nation. Instruct the pupils to continue if you can guess the remaining entry. independently. They will find by them- Once these middle cells have been filled, selves that zero is the only number that we have determined a newfour-cornered gets listed beyond some move. More- diagram (fig. 4) which, in turn, has its over, they can be encouraged to try middle cells vacant. additional games allowing them to dis- Players till these cells the same way that cover for themselves that a set of zeros the previous ones were filled. The only eventually results. This may encourage rulein D1FFY, as you probably have some to keep trying again andagain in guessed, prohibits subtracting a larger num- an effort to elude an arrival at zeros. ber from a smaller one. Younger students, Does the order in which the numbers who have no(learned to work with negative the numbers, will need only to have this rule are listed in the corner cells affect number of moves in a game? stated. More mature students can he introduced to the absolute-value function and apply it to the differences. The game being considered ends at the next move, since it yields a zero in each middle cell, as seen in figure 5. This game took only two moves before zero was written in each middle cell.If zero appears in some of the middlecells but not all of them, the game continues. The game sheet distributed to the students has room for six moves (see fig.I ). Should a player reach the innermost setof cells Fig. 3 WHOLE NUMBERS 67 DIFFY not only provides a good vehicle for drill in subtracting rational numbers but also gives practicein comparing them, since the rule does not permit a larger number to be subtracted from a smaller. Other .skills drawn on in this activityinvolvemultiplying,reducing fractions,andrecognizingequivalent fractions. Try DIFFY withany quadruple of Fig. 4 integerspositive, negative, or zero. Once a player has won the game a Try r, e, V91, 19. Before you start, few times (six moves without all zeros) consider whether you think it will ever have him try for seven moves, eight end. Observe the manipulative skills removes, nine moves, and so forth. A con- quired in this activity. tinuing contest can be held to see who can come up with the longest game. One need not stop here, as there are Others can then try to do at least one many more intriguing aspects about DIFFY. better. Thiswillalways be possible, Not the least of these is the fact that it since there is no "longest game." involves an iterative process and so lends itself very well to the use of a computer. Find a winner using only single digits Many students who arefamiliar with in the starting cells. Find another. BASIC or other programming languages How does the number of moves of may enjoy putting DIFFY on the com- one game compare with the number of puter. For this purpose, thotigh, a change moves in a second game whose starting in format proves useful. We'can show.this cells are 'entered with doubles of the by appealing once more to a sample game; entries of the first game? With triples? this one is shown in figure 6. With other multiples? Another time add 5 or some other r number to each of theinitial entries r ;1 2 9 65 and compare the number of moves. 7 3 1 3 For younger students, exhibit a completed six-move game and ask them to 4224 come up with .one that takes exactly 2020 five moves. Then four moves, and so 2222 forth. 0000 Letolderstudentstryfractions. Fig. 6

In this format the starting numbers are listed horizontally. The "distance" between neighboring numbers is listed directly below the space between them. The last entry in each row other than the first row is acquired by finding the distance between the first and last entries in the preceding row. This format also has obvious advantages in a)earch for long games, since one need ndt concern himself with expanding Fig 5 the original game chart. 68 GAMES AND PUZZLES playing DIVVY, be sure not to enter any The computer may he programmed to far, since we print out each intermediate step or just the zeros. You wouldn't get very number of steps required. One may wish can't divide by zero. DIFFY and DIVVY are but two aca- to program the game so that the player self-generated "inputs" the number of steps to heat before demic games that stimulate student. There he plays the game. The computer would drill that is purposeful to the such games having the then inform the player whether he won or are several other which is that not. The variations available provide op- same fine attributes, among differences. The portunities to apply programming skills at of providing for individual slower learners can try smallernumbers many levels. paceand win Since DIFFY proves so useful for drill and proceed at their own student may in subtraction) might we not try to get addi- in timewhile the more clever his heart tional mileage out ofit by considering choose numbers as challenging as desires and try for a wininvolving more another operation? Indeed we might. We can change, our moves than hisprevious one. operation and still use the same playing In my opinion, it is importantthat such chart. This time, instead of subtracting, we activities become more widely known,since shall divide. Furthermore, as in DIFFY we we are beginning tohear cries that "what never subtracted a largernumber from a our children needtoday is old-fashioned smaller one, in DIVVY we shall never drill." True, they need drill,but not the divide a smaller number by a larger.This barren set of exercise after exercisedevoid time instead of ending with zero in thelast of intellectual purpose. Sometimesit's bet- set of cells weshall end with something ter to give a child hismedicine with a sugar else. Can you guess what it will be? Go coating. Shouldn't we give children proper ahead and guess; then check your guess by drill and maintain their interestin mathe- trying a game of DIVVY. Of course, in matics at the same time?

EDITORIAL COMMENT.Thereare many ways to provide interesting practice in elementary school mathematics. For example, have the students look at a calendar for a given month and select any four adjacent dates in a square array, such as 9 10 Lead the students to 16,17 observe that 9 + 17 = 16 + 10. Have the students check to see if they can find similar arrangements on a calendar that do not work in this way. Test the idea on a three -bv -three ar-

rangement, such as 1.2 13 14 19 20 21 26 27 28 Does 12 +20 + 28 You can also develop coin puzzles such as the following: A box contains 3 coins. The total value of the coins is 20c. Name the coins in the box. A box contains 6 coins. There are 3 different kinds of coins in the box. The total value of the coins is 32c. Name the coins. Magic triangles practice in skills and thinking

DONALD L. ZALEWSKI

Presently an assistant professor of education at the University of NebrasAa at Omaha, Donald Zalewski teaches mathematics methods and supervises student teachers. The article was written during his investigation of mathematical problem solving while hewas at the University of Wisconsin in Madiron.

Can you place each of the numbers 1 side) containing numbers such that the through ,6 in the triangular chain of cir- sum on one side is equal to the sum on any cles ingure 1 so that each side has the other side. "Level one" magic triangles same s m? If you are successful, congratu- contain six circles in the chain. If only the lationyou have just created one example first six positive whole numbers are in- of a/ magic triangle. These simple figures serted in the circles, the configuration will may not have as many 'angles" as the well be referred to as abasic magic triangle. publicized magic squares, but theypossess Although magic squares and magic tri- a few powers that may be interesting to angles both provide prctice- in computa- both you and your students. Obviously, tionand problem solving,thetriangle. important practice in addition skills is built problem holds two distinct advantages for into the puzzle and itis possible that you the beginner: (1 t There are fewer sums to can at the same time stimulate the forma- find and check, and (2) the simple design tion of problem-solving strategies. offers children an easier opportunity touse elementary common senseandlogical thinking to solve these puzzles. Now, back to the problem. Don't feel too smug if you found one correct an- swerthere are four common sums possible using the numbers 1 through 6. Can you find the others? The four solutiOni are given in figure 2. Since each solution

Fig.I can be written in six possible forms, any forms that are related by a rotation or flip will be considered the same solution. Fig- Amagic triangle is atriangular chain of ure 3 gives the six possible representations circles (an equal number of circles on each for solution A from figure 2,

A 69 70 GANDS AND Pt1111 IS When posing this problem toelementary school children, it might bebetter to indicate which sum (9, 10,11, or 12) they should seek. For instance, youmight ask them to try to get 9 as a commonsum. If a child incidentally finds oneof the other certainly be com- possible sums, he should T mended for his efforts and youshould take the opportunity to challenge thechildren to search for othersolutions. Sometimes, hints on possiblestptegies for filling in the circles maybe necessary to help children get started orto help them organize nonproductive trials. Onehelpful suggestion is, "Pick three numbers andput them in the corner circles(vertices), then try to arrange theremaining three numbers to get equal sums."A more direct cue might have the children choose1, 2, and 3 as the vertices andthen arrange 4, 5, and 6 as interior points until common sumsare found. You might also suggestthat the children keep a record of thznumbers they have tried as vertices and interiorpoints to avoid duplication ofeffort. You may wonder if there is amethod Fig. 3 to this madness, or iftrial and error is the only procedure that can be used.Inspec- that formed tion of the trios of numbers the vertices and interiorpoints of the solu- -1- tions in figure 2 producesidentifiable patterns. If we horizontallyorder the numbers largest, we 1 through 6 from smallest to see that solution Ahas the first three numbers as vertices and the lastthree-as interior points. (Fig. 4) SolutionD has the roles of these two triosinterchanged. Solu- tions B and C share a commonpartitioning of the trios into the oddand even numbers. a (Fig. 5) Some children maynotice these regularities, but most may need tobe asked if they see any patternsthat might help them solve the puzzle. After identifying the patternsof number trios, the child has anopportunity to apply mathematicalreasoning when trying to form magic triangles.Assume that three numbers have been selectedfor the vertices. d Then the interior points canbe identified c sums. For Fig. 2 by inspecting the incomplete 4_3 W11011; NUMBERS 71

example, let1,2, and 3 be selected as vertices. (Fig. 6) Since 2 and 3 form the , largest sum thus far, that side must receive the smallest of the remaining numbers, namely, 4. Similarly, sinceI and 2 form the smallest incomplete sum, that side must get the largest of the three interior numbers, namely, 6. The third side receives the Fig. 6 remaining number, 5.

If you have children who enjoy this type ..--...Nr.---...Vertices of mathematical roblem, you need not A 1, 2, 3, zr, 5, 6 stop at the basic m4ic triangles in figure 2. For enrichment, two or three digit num- Interiors bers can be used. You may devise your own combination of numbers, or better yet, have children find some. There are an Vertices unlimited number of combinations, but any D I, 2 3, 4, 5, six consecutive whole numbers will suffice. ___../ Trifenors Fig. 4

When an incorrect triois chosen for the vertices, the same reasoning reduces the number of trials necessary to eliminate it. For example, if 1, 2, and 4 were chosen foi 'vertices (fig. 7), then 3 must be inserted between 2 and 4, and 6 must be Fig. 7 put between I and 2. The two completed Furthermore, the use of six consecutive Vertices integers permits the children to check the e....-----=..sy....------s hypotheses formed and patterns observed B I. 2 3, 4, 5, 6 in the basic magic triangles. (You might !.., _...A...... __,.... notice that new Magic triangles can be built Interiors systematically by adding or subtracting the same number to each entry of a basic Vertices magic triangle.) c'------Y------Additional challenge can be added to C I, 2, 3, 4, 5, 6 this problem by looking at a "level two" -______& ___. magic triangle consisting of nine circles. Interiors (Fig. 8) The numbers 1through 9 would Fig. 5 be used in attempting to form the equal sums of basic level-two magic triangles. sides form a common sum of 9, but putting You can decide whether to consider an- 5 between 1 and 4 gives a different sum swers with the same vertices but with dif- of 10. Since no other arrangement of 3, 5, ferent arrangements of interior points (fig. and 6 as interior points is logically pos- 9) as the same or separate solutions. The sible, we must go back and try a new trio same-questions about possible sums. pat- of numbers for the vertices. o terns, and strategies can again be raised. 2 72 GAMES AND FUZZ! ES

Fig. 8 Fig. 9

Childrenneed much practice inskill Freitag, H. T., and A. H. Freitag. "Magic of a Square."MATHEMATICS TEACHER63(Jan- development and maintenance. Building uary 1970): 5-14. magic triangles can be an interesting respite Gorts, J. "Magic Square' Patterns."ARITHMETIC from routine drill.It 'can also provide TEACHER 16 (April 1969): 314-16. opportunities for children to practice the Keepes, B. D. "Logic in the Construction of Magic Squares."ARITHMETIC TEACHER 12 searching, guessing, and thinking that is (November 1965):' 560-62. essential for problem solving. May, L. I. "Enrichment Games Get Pupils to Think."Grade Teacher83 (1966 ) : 53-54. McCombs, W. E. "Four-by-four Magic Square for theNew Year."ARITHMETICTEACHER 17 INforences (January 1970): 79. Munger, Ralph. "An Algebraic Treatment of Brown,G."Magic Squares, More than Rec- Magic Squares."MATHEMATICS TEACHER66 reations.".School Science and Mathematics (February 1973): 101-9. 66 (1966): 23-28.

COMMENT. Puzzles are excellentdevices to stimulate and interest many stu- Eotrolum. nine square dents. As a variation of the "magictriangle," you may wish to give a student beginning with pieces of blank paper and instruct him tonumber the paper squares in order, order to have the student any whole number (you maywish to specify a beginning number in nine paper practice at a specific level of difficulty).After the student has numbered the squares, instruct him to arrangethe squares in a 3 x 3 array so that I. he gets the same sum in each row(horizontally); 2. he gets the same sum in each column(vertically); 3. he gets the same sum in eachdiagonal; verti- 4. he gets the same sum simultaneouslyin each of the three directions, horizontally, cally. and diagonally. (2 x The difficulty level of this activitycould be decreased by using only four paper squares twenty-five 2 array) and the difficulty could beincreased by using sixteen (4 X 4 array) or (5 X 5 array) paper squares. Easy construction ofmagic squares for classroom use

JOHN CAPPON, SR. Willowdale, Ontario, Canada Mr. Cappon is a retired elementary school principal from Amsterdam, Holland. He also taught arithmetic at a private teachers college in Winschoten, Holland. In Canada he is an adviser to a private school.

Inthe December, 1963, issue of THE ARITHMETIC TEACHER Bryce E. Adkins 5 gave six rules for construction of odd-cell 4 I0 inagicc.s.quares.1 His result for a five-by:five magic square is given in Figure 1. It is 3 9 15 2 8 14 20 1724i 815 7 13 19 25 23571416 6 12 8 24

46132021 11. 17 23 101219213 16 22

11 18252 2I

Figure I Figure 2

noticed that all horizontal rows, vrrtiral columns, and diagonals have the total of 3111? 9 22115 65. 20 8 2114 2 Actually, the construction of this scheme can be simplified considerably and is easier 7 25 13 I 19 to remember when certain numerals are 2412 518 6 replaced in symmetric positions outside the borders of the square as is customary, II4 17 10 23 for example, in the calculation of deter- Figure 3 minants of the third rank. A European method proceeds as fol- square along the opposite border. The re- lows: Cells are added outside the square by building a pyramid on each border line sult is shown in Figure 3. and placing the digits in the order illus- Although the integers 1-25 were used trated in Figure 2 for a five-by-five square. in the example, this is not necessary. Any Then, the extra cells of the pyramids are sequence of numbers which is part of an used to fill the empty cells within the arithmetical progression will suffice. More- over, as Frances Hewitt2 has shown for a Bryce E. Adkins. "Adapting magic squares to classroom use." Tug Artrruncrtc TeAcuss. X (December, 1963). 498- Frances Hewitt, -4X4 magic squares." Tut Aurrumsrec 500. Tsacnea. IX (November. 1962). 39r-395. 73 r

74 GANIIS AND PU711 ES four-by-four square, there can be ,gaps in 32 the progression between the parts of the eA for odd-cell 17 35 rows. This holds also all ,--7. magic squares;e.g.,,,hosequenceof 2 20 38,-. numbers for a three-by-three square can be chosen as 2, 5, 8 (11, 14), 17, 20, 23 (26, 29), 5 23 32, 35, 38, where the numbers in paren- 8 theses are omitted (Figs. 4a and 4b). Figure 5a

17 35 1 38202 3223

Fzgu,e 5b. This is the reverse of4b!

are all the same, e.g., 2, 17, 32; 5, 20, 33; Figure 4a 8, 23, 38. The gap here is 27 (Figs. 5a and Fib). 53223 All these difficult cases can be sum- 38202 iirzeFeed in one general rule: The sequence of numbers in the "pyramidally extended 17835 -..square" is formed by two arithmetical

Fzgure41) progressions, one determining the numbers from ti.e most left-placed cell to the upper Furthernioi$1, thegaps (..in be taken as cell, and another one determining the any number, positi e or negative, if they numbers from the most left.-placed cell to

15- FIRST ARITHMETICAL 13 20 PROGRESSION 11 18 25 ii, .,A 9 16 23 30

17 14 2.1 28 35 cQ_k, 4' 4 5 12 19 26 33 40 10 17 24 31 38 451 b.:, 15 22 29 36 43 ,e ,/,I. 8 co 20 27 34 41 'II e.1),18 25 32 39 ds 23 30 37 SECOND r ARITHMETIZAL 28 35 PROGRESSION q, 33

F,,i 4',6 t

s WHOI I. NUMBFRS 75 increment of the horizontal diagonal (7) 923163023 3730 equals the sum of the increments of the 3514282135287 border rows (2 and 5 respectively in the figure), while the increment of the vertical 1240193326533 diagonal (3) equals the difference ofthe 38 174524331 10 border row increments. The same applies to the oblique diagonals of the four -by- 1543221529ii36 four squares (see Fig. 7a). 41.201327203413 Furthermore, the sum of the first and last elements of either diagonalis an 18II251832 2539 importantquantity.Fortheodd-cell Figure 66. The map: square forme4 by Figure63. squares this sum equals twice the term in the middle, which in turn equals the magic the Uttom cell. Both arithmetical pro- sum divided by the rank of the square (see Fig. 6a: [3 +10 +17+24 +31 +38 +45] -:- 7 gressions can be taken arbitrarily.(See Figs. 6a and 6b.) =24). In the four-cell squares the sum of 'The rule given by Frances Hewitt, that the first and the last elements of the diagonal equals half the total of the four a new magic square can be formed from a elements. With the knowledge of this it is given square by adding or multiplying the digits with a certain number or fraction, possible to construct any magic square follows immediately from our general rule. with a given number as totals of rows, columns, or diagonals. Adkins gave an easy, rule for a four-by- four magic square formed from the num- Columbus discovered America in the bers 1-16 placed in four rows of four year 1492; let us make a four-by-four digits. The series of the numerals on each magic squarei with totals 1492. Starting diagonal must be reversed. Again, it is with the diagonal of the auxiliary figure, not necessary to restrict oneself to the the first \and the last numbers of the nunibers 1-16. In fact/ we can again build diagonal must equal 1492 +2 = 746. The up the sequence of numbers from two last number of the diagonal equals the arithmetical.progressions, one denoting the first one plus three increments, from 'rhich order in the first row and the other the it follows that the difference of the num- order in the first column. (See Fig. 7a.) Adkins' rule also applies to a square of INCREMENT S these numerals; the square of Figure 7a C R 2.`71217 is converted to a magic square by reversing E the diagonals. (See Fig. 7b.) M 5 1015 E N 131823 Practical applictians T 8 3 S. For adopting magic squares to class- 11 162126 room use it is desirable to have an easy Figure 7a method that enables us to construct a magic square for any given number. Most easily one may employ a diagonal for the 26712 II construction of the auxiliary figures of either the odd-cell squares or the four-by- 5 181520 four squares. The element's of this diagonal 815I023 are always part of an arithmetical pro- 17162I2 gression. / In the odd-cell squares (see' Fig. 6a) the Figure 7b 76 GANIFS AND PUZZI hers mint be divi,.ible by three (if we %%ant Inu,t mail our,el es of fractions. In to avoid fractions). The numbers 100 and the anNinary figure for the five-by-five 646 suffice. The increment is then (616 square, one places the number 19.1= 3.S -1001 =3= 182. The entire diagonalis in the middle. (See Fig. 9a ) therefore 100, 282, .161, anti 616. Further, e split this increment in two parts, e g , 2.8 S2 and 100, and u,e thew parts to construct the first row and the first, coluum 2.6 35 Then the rest is kno n. The example is 2 4 3.3 4.2 tlemon,trated in Figure, 8:1 and 8b. 2.2 3.1 4 I NCRSMEN1 82 2 29 38 47 I00'182264346 % 27 3.6 4 5 r 416, 34 43 5.2 200282364446 '1? 414 5 100 > 300382464546 4.8

7Iro 9 :t 1n.rtliary figure for magic number 19 400482564646 The left-cell of the lwrizontal diagonal is Fi.pire Ca unbar r less than 3.8, e.gx.1.2. The increment of '-'-`ohe diagonal is then fixed, viz. (3.8-2) ÷2= 0.9. The complete diagonal is now 64-6182'264400 known. Then the increment 0.9 is split into two numbers, e.g., 0.2 and 0.7, which 200464382446 determines the border rows. We chose these numbers so that the entire square contains all different numbers. The resulting magic 300364282546 square is Figure Ob.

100 346882564 2.4413.354 2

Fupire 11,1,11cfjti ar 7.01, drayrnell 49314.8422 2.95.63 8247 After/deteacher has made up the square tchildren check the results by 54362.84. 527 performing the fifteen possible additions: 3.42.64 33 55.2 4 rows, 4 columns, 2 diagonals, 4 two-by- two squares in the corners, and 1 two-by- Figure 9b. The magic square of 19. two square in the middle. By the choices of the first and last numbers of the diag- The magic square for 61 is made as the onal and the increment., the teacher can one for 1492. The diagonal is 4, 12, 20, and vary the difficulty of the arithmetic cal- 28. The increment (8) we split in 7 and I eulation according to the progress of the (Figs. 10a and 10b). class. If interested in the magic square for 32 The year 1961 consists c1. 19 and 64. We or 16, one can utilize the magic sqi of shall make a magic square for 19 and 61 61 by dividing all terms by 2 and 4 respec- The number 19 is a prime number. IIce, tively. (See Figs. 10c and 10d.) W1101.1: N111111RS 77 I -110. 7 N 3 4 II 18 E 2 4 MI 1219 E 5 3 5 T 6 13202.7 2 4 7142I28 3

Figure 10a1u.ritio,y..Nquare for 64. Figure Its. Auxiliary square for 9.

234 28 II 18 7 53 201326 234 6 191227

2514-21'4 Figure 11b. Magic square for 9. Figure 10b. Magic square after diagonal reterse. The magic square with totals 10 can be made with a four-by-four square. The diagonal is 1, 2, 3, and 4. The fractionis 14 9 chosen for both increments.(SeeFigs. 10'6±13 12a and 12b.) 3 6 1 1 z2 7ioi2 Ili 2 ,3 Figure 10c. Magic square for 32. 22i3 4 3 i 727-$ Figure 12a. Auxiliary square for 10. 534 443 361 643-15*

Figure 10dllogze .quare for 16.

It N not alays possible to split the increment of the diagonal in such a manner Figure 12b 7 he ',law square for 10. that no two numbers Occur twice, particularly in squares %%al' a 10 magic sum as The magic square for 11 is easily built might be used in loner grades. Suppose on a three-by-three square The middle a child in the third grade has hi'§ ninth number is birthday, for which occasion the teacher plans to make a magic square with totals 11 3-33. equal to nine. The middle term in a three- Choosefor the left cell in pyramid. by-three square is then 9 ÷3 = 3; the incre- /Magic squares for 12 and 13 are ob- ment of the diagonal is 3-1=2. This is tained from Figure 4b and Figure 3 re- divided in 1 and 1(if no fractions are to spectively. The numbers in the cells are be used). (See Figs. I la and 11b.) divided by five. 78 GAM! S AND Pt /71 US

The magic square for 1can be obtained grade up, according to the age of the by adding one to eat h cell of Figure 12h, pupils, pro% iding decimal fractions canbe by constructing a four-by-four square used in seventh- and eighth-grade arith- (diagonal 2, 3, 4, and 5; increments 1, and met ic. 1), or by constructing a sex en-by -sex en The high school teacher can use ncga- square (in the middle 2, left-cell !. tis c numerals; for example, he can sub- increments 1.216 and :290). tract 10 from each cc!! in Figure 1. He These examples aresuitableforall can also change the numerals to another elementary school grades from the third merat ion sytem.

for students to learn EDITORIAL Cessna NI Magic squares can provide many opportunities to think mathematically. However, thiswill not occur if the students' experience is limited to "plugging in" values in certain places m accordance withrules set forth by the teacher. For example, let's examine the nine-cell magic square inwhich we are to arrange the numerals 1 through 9 so that the numbers represented in each row,column, and main diagonal haze a sum of 15.

d e

g h

First,. have the students experiment with thetask on a trial -and -error basis Then focus class attention on the nine cells and ask forideas about the desired arrangement Don't tell or show the class how to arrange the numerals. Some classmenibei's slibuld begin to observe that certain cells involved will he associated in threedifferent combinations of three numbers, others in two combinations, and so on. Encourage each class member to make a list of hisideas about the arrangement of the numerals in the ntne cells Vv hen each student hascompleted and or,gamied his list of ideas on the basis of his experiments, develop a class summary. It is hoped that the students will observe thatcell a is associated with three sets of three numbers each ((a. h, (), (a. d. g)Ind (a. e, i)j and thateach corner cell is similarly associated. Other observations should include the fact that cellsh. d.i. and h are each associated with two sets of three numbers (for example, h. e. hand h, a. (). Next, encourage the students to focus on the goal of creating a sumof 15 for each row, column. and main diagonal. Again, it maybe easier to tell the students that they shouldlock -for various sets of three numbers thattotal 15. However. far more learning will take placeif you allow the students time to come to this conclusion ontheir own. Students should he encouraged to do somemathenkatical thinking for themselvcvs, and magic squares provide excellent opportunitiesYou may wish to have your students try to form nine-cell magic squares with each of thefollowing sets of nun hers Lead them to look for pi'itterns and to, make note of their observations.

'112, 4, 6, 8, 10. 12, 14. 16. 181

2x3.6, 9. 12, 15, 18, 21.24, 27) 3 11,1, 5,7, 9, II, 13, 15, 17) 4. 15, 10. 15, 20, 25, 30. 35, 40, 151 5.110, 20, 30. 40, 50, 60, 70, 80, 90) Numeration

The ideas of base and place valueare most fundamental to the understanding of our numeration system and the four fundamental operations on whole numbers Children need many concrete experiences in grouping single objects in sets of ten and then in grouping sets of ten in sets of ten tens to form a hundred. It is possible to provide much of this practice through games The first article in this section describes a game that may he used in the early grades to provide some of the desired experience with place value MacRae provides a clear set of instructions for constructing and playing her game. and many variations are possible The Placo game created by Calvo should generate many kinesthetic place-value experiences for children in a most enjoyable atmosphere. Several games and puzzles involving nondeconal numeration are suggested in the last few articles of this chaptei Shurlow suggests a ring-toss game for developing an understanding of a base-five numeration system His ideas could easily be adapted for use with other bases. including our own base-ten system. .1F0 Alfonso, Balser, ind Hartung provide ideas for several mathematical puzzles relating to base-Iwo and base-three numeration More mathematical magic relating to the binary system is suggested by Niman, and Ranucci tan- talizes the reader with several puzzles built around the base-three system

79 1/4 A Place-Value Game for First Graders' A Teacher-Made Device

IRE %E. R.NI se.R \.

MANN TEACIIERS Hsu that place %aioe Is set at 100 for second graders, and 1000 is one of the more dithLult numberfog third graders, perhaps tsso players ss ould concepts to explain to small children. Thebe more practical The game is easily and folios% ing game ss as des ised to help rein-quick!) made, so that a teacher could pro- force the understanding of place s alue aftei % ide se% eral games for a class Children could it has been introduced h the use of thees en make the game themselves. Hundreds, rensandOnesChart. This chart is familiar to primary teachers. The game Directions uses the same pocket device used in the chart. oHJE,cI' OF M,GAME:. Thus children canI einforce their initial To score a pi t deti mimed numbs r of points For first graders. 25 50 points learning by praetie e. under conditions simi- 1 or se cord graders 100 points lar to those in %% Itch the learning took place I or thud gi adt is. 1000 points The game can be used ni the finet grade, NI'MBER OFPLANI Ri but can also be adapted to the second and gust grade: 2-4 third grades merely by increasing the size Second glade: 2 of the numbers set as goalsIf the goal is Third grade: 2 set from 25 to 50 for first graders, it is recom- Spin to determine ,..ho plays first and play to the left mended that 2 to 4 children play, depending on the tune available The more players, PROCEDURE: the longer the game %sill take. If the goal 1:.1( II player places the commisIIIpiles in front of him, arranging the Colors in the same order as 1 A (Imago Tracht rs College Student Project. they are placed in the pot kt isblur at exti eine

The in Pictures STEP-I STEP -2 STEP -3 Tayboaral) Fo/a' Slaple

e 1 . 1 ,, I INOu- I NUN- I TENS 'ONES I ..-, 'SANDS I .. I MOS I I kr, / 1 SCORING POCKITS i Fad here/ I ----- Staple -----/ y -rayboard--- COIMTER i p..-1-.

sCI

i i Colored TOP DISC Construe- BOTTOM DISC I/Ot, paper

80 NU:MFIRArlo 81

right and proceeding to the left, green, red and such as, lose your next turn, tale an extra turn, %yellow. First player spins and places in the extreme add 3 ones, ;nu lose three of our ones, etc. right pocket a number of blue counters equal to the. number to which the spinner points. The How to Make the Game other players do the same On the second spin, if a player picks up enough more blue counters to The following detailedinstructions are equal tett or more when added to what he already merely suggestions. The teacher t an use any has, he "bundles up" ten blue counters, placing materials she has at hand. them with the pile of blue counters in front of him, and takes one green counter which tie 1 To make the swung pot kris, use pieces of places in the next pocket to the left. This con- tagboard, 5 i" by 9". tinues until some player has accumulated ten 2 Fold to 3I" by 9" so that there is a 1 r pocket green counters in the ten's pocket which hr then across the bottom. removes and puts bat k m dui pile in front of 3. Place a strip of brightly coloicd tinstic tape hint. Ile then may place one red counter in the along cacti edge to hold pocket in place. The hundreds pocket. The Inst. player to do this wins color helps make the game am acme, but the the game unless the goal is set at 1000. learning values would not be lost if the picket were field in place by met ely stapling. Since the game is one of chance rather 4. To divide the pockets into four sections representing the place positions, black metal rivets than skill, a fast learner may be paired with wet e used which wet e snapped together but a slogs learner without doing injustice to not spicad wall a hal11111( r.Fins tin Mod al- the fast learner. He can help the slower child lows the pockets to stand out a little so the counters can be castle insetted. If rivets are and atthe same time reinforce his own not traddy wadable, stops of nnstie tape learning as well as get fun from the gaim might be us«i, ot simple stapling would sc me There arc other %Awl to be (kris cc! Irons the purpose Ito words ...1st's/n Pocket %sem maims« iptt d across the pm km/ this game in addition to the understanding 5. lot «mine's, coma mums pap( r tai four colors of place %ableWhen used inthefirst is needed Cut sums 1" In aAW. At tract 18 Qrade, it will serge to gise practice in num- count«s art needed fatall colors ex« pt for the «shwa quest-ming the 1000 position. Only ber value as a result of spinning todetermine one of the stis net dell. (Only nine counters are who plays first. It will pros ide occasion fur 'rally nwssary fin the 10 and 100 positions.) children to reproduce alistract numbersN% uli 6For spinet rdials, cut four oh mical circles froth tagboard dMat k two into eight equal concrete counters. To those beyond the im- parts,and write the numbers hum 1to 8 in mature level of counting, there \sill lie prac- the set nuns. tice in addition by adding the number which The spinners in this particular instance were _wade sg: thegays.frouL slutts_by_ thniTay er sp-in to the number oftoufiters he punt tuts.; a now in the center with a hand has accumulated oil his pie\ ions turns N% hen punt Ii and insisting part of dine t.I In oda r this stun exceeds l)), there will he practice in half of the [net was placed at the back of the dull and the !..0 snapped together, but not subtraction by removing the group of ten, spread'1 he blank cm les of tagboard wete leaVing the remainder in the ones pocket. then glued to the backsfo make the game The ten ones are bundled and replaced more attractne in appearance, another larger etc was cut fontd poster board to match by one counter, representing a group of ten, the red on stic tape used on the pocketsl'Ins and moved one place to the left in the scor- was glued to the backs to make red borders. ing pocket. This pros ides practice in trans- ( wo spinners were made, but one would be enough if only two children were playing.) formation. The game can be gisen added interest The game can be kept together by placing by pros iding a pack of penalty and bonusin a box cost:red with gay paper or simply cards, and painting o or three numbersmarked so that the children w ill recognize its on the spinner dial in colors. When thecontents. Keep the counters, dis ided equally spinner stops on a color, the play er turns upaccording to color, in the inch\ idual pockets

a card from the pack, NShiC11is face down, to make it easier for the next players who and proceeds to do what the card directs, use the game. possible. F example, change the I EDITORAL COMMLN I. Many variations of this game are d 81 (or any others you numbers represented on the spinner face to23, 32, 12, 21, 45, 54, 18 of tens and ones wish to use). Then have each studentin turn spin the spinner and create sets with the counters to match the numberindicated by the spinner.

Placo a number-place game

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

This game emphasizes funandthe learning of positional and place-value aspects of our number system.Placo(pronounced play-so) is a number game that boys and girls can play to good advantage in the classroom. The childrenblindfold each other, one at a time, and then attempt to fit the rings on the highest place -value,pole (see diagram). One value of the game is the manipulative aspect. The children enjoy handling the rings, fitting them on the dowels, and then computing to find the mount_the_y2lwin." One player is blindfolded and given some While Placoischieflya place-value rings (9 -18) of which he may put not This ga ,in actual classroom practice it con- more than nine on any one dowel. tribut s to other operations in arithmetic blindfolded player then tries to place the as well. It provides stimulating practice in rings on the dowels. The other players computation and enables children to have (split in teams) watch in glee. The object fun while they play the game. Itcanbe of the game is to get the largest score by used with success in Grades 2 through 6 puttingtherings on thehighest-value and can be refined to challenge even the dowel. After all rings have been placed, the brightest students. It may be played by as blindfold is removed, and thP whole class few as two children or as many as the or a small group cornrows the amount. whole class. The other players then take their turn. Three colored dowels are glued on a Players can play with a goal of 10,000 or base. as shown. These dowels are station- with a time limit, the highest score winning. ary. The dowel to the leftishighest in In playing decimal-fraction Placo, the blue value, the one ter the right is least in value. dowel representsones; thereddowel, They may be painted blue, rec. and white tenths; and the white dowel, hundredths. in that order. They are labeled 100's, 10's, Placo cards may be developed to use and 1 's. Thus 100's (in a whole-number with the game. These cards are used as game) are blue, 10's are red, and l's white. follows: After each turn, a player selects ; 82 NUMERATION 83 one\of the cards and follows the directions 2. Doweling (also knownas "closet pol- given. They say, "Double your score" or ing"), 12 inches in diameter, threepieces "Cut your score in half"or "Subtract three of 7 inches each tens," etc. This serves to providevariety 3. Drapery "cafecurtain" in the game. These cardscan be developed rings,11 inches inside diameter, 12 to to reinforce specific skills needcd by the 20 4. Cards and tacks class or to provide the class withneeded 5. Sandpaper drill._ Cards might give directionsto add, / subtract, multiply, or divide. In additionto 6. Paintwhite, red, and bluehigh dramatizing and creating added interest, gloss they serve to equalizescotes of fast and 7. Carpentry tools, including drill and slow learners without penalizingthe fast saw ones. 8. For variation noted, 3-by-5 cards A word here about the use of colored 9. Large handkerchief dowels. While some experts or clean rag for say that place- blinder value instruction should not dependon 10. Oak tag for power chart color, we can't rule it out completelyat early levels of learning. In place value, it Building the game is important that the child knowsthat "this ring is in the place to the right"or "that I. Drill three holes equallydistant from peg is in the units column," but that doesn't each other in the 12-by-6-by-4 inchpiece of mean color can't be brought in to augment wood, making these holes 1 inchdeep and these instructions. It merelymeans color 1176 inches in diameter. should not be usedas a sole means of 2. Sand one end of each 7-inch pieceof identifyingnumber-place.Ifcolor can doweling smooth. Put some whiteglue on help the children gain a conceptor can the bottom inch of the dowels andthe inmake a game more attractive, thenits. side of the drilled holes. Slip thedowels in use is defensible. and allow to dry overnight. Other uses can be found for thisgame. 3. Purchase cafe curtain ringsof the One possibility is touse it with sight-sav- size specified in the materials list. ing classes by cutting niches in the base 4. Paint the game three differentcolors, to identify the COI I esp on ditsgnumber - asmentioner previously,makingthe value. Computation could be done men- dowels and the base of matchingcolors. tally. Another potentialuse would be to 5. You may want to puttwo small roll special dice and compute thescore in nails about 3 inches from thetop of each whatever number base the dice show when dowel(exact rolled. The dice could be made placement would depend so only on the thickness of the curtain rings). certain number bases would result. In this way, players could put only 9 ringson Materials list for Placo each pega 10th would have nothingto keep it from falling off. This style ofcount- 1. Piece of wood 12 inches long.4 ing uses only 9 units. When1 more is inches high, and 6 inches wide added, it becomes the numberon the left.

EDtToRtat Comm' \ ITwo other variations of Placo will provide excellent kinesthetic learning experiences for children. In the first variation, identifya whole number less than 1000 and have the blindfolded student try to represent the number correctlyon the three pegs with rings. The number may be identified by drawing a card by spinninga spinner, or simply by having someone name it orally. In the second variation, haveone student write a three- digit numeral on the chalkboard and then put rings on the threepegs to represent the number.

;*. the number. In each varia- Then have a blindfoldedstudent try to count the rings and name activity as a game. tion, it is possible to devise a scoringsystem and conduct the

The game offive

Columbus, Ohio HAROLD J. SHt7R LOWColumbus Public Schools,

Purpose Ithas been stated manytimes that the keyconceptsneededbyelementary teachers in order to teacharithmetic Eza meaningfully include thedistinction be- Figure 1 tween number andnumeral, the structure "Now in a ring-toss gamethere is of our numeration system,and the con- usually a score, so we have to getthat into cept of place value.This game is designed our game. Let's saythat if a ring is on the with an interesting to provide teachers peg to the right,it, counts I point. If a ring way Of reviewingthe important, concept of is on the next peg, it counts5 points; on place Value and of workinginto a study of the next. 25 points, and onthe next, 125 different numeration systems thathave points. (See Figure 2.)So now we know bases. what makes our score." (At this point teachers maypretend to the prospect of To a class: become visibly shaken at rings flying all over the room.) you like to learn toplaya new "Would "Now comes a change in this game.In game? Of course! Nowthere are two toss the rings before you can most ring-toss games you things you have to know and count up the score.Not so here. In learn this game. First, youmust know ahead of Well now, you all this game you know the score how to count to five. time, and it will be yourjob to put the know that. Then you mustknow how to rings on the pegs in the:fightplaces. count by 5's to 25. Let's trythat. 5-10 13. One know how to "Let's think about a score of 15- 20-25. Next you must the rings would be to put Shall we try that too? way of placing count by 25's to 125. 13 rings on the peg to theright. But there 25-50-75-190-125. This game is aring- have not won know what a ring- is more to this game. You toss game. How many this game unless you haveused the fewest toss game is?" with a score (Most children know, and atthis point member of rings possible. So of 13, there would be tworings on the peg classroom interest picks up.) marked 5 and three rings on the pegto the "Let's draw a picture of some pegsfor right." our ring-toss game." (See ,Figure 1. For lowergrades, actual pegs withstandards would be more effective than mere drawings,but for older students drawings serveadequately since most students areacquainted with peg games.) Figure 2 C 84 N l Mt RA HON' 85 To the teacher It is a simple mutter then to show your At this point itis best to go mei- a few pupils that :19 and 121 really represent the more examples, and then let. your pupils same :wore if we know Ho value of the try the game during rainy-day recess or at places. Now we need a way to show the other convenient times. The game seems N Atte of the placesHen, we introduce to work best if three or four pupils play 39,t,, = 121,r,,. So we have two pictures together. One pupil can assign a score, and (numerals) for the same score (number). then the other pupils can try to place the I must warn that in using this game the rings, using pencil and paper to record teacher must proceed slowly in order that where they have placed them. Contests of the students assimilate each idea as it speed may result. flowerer, the purpose conies. first the game and its rules, then here is for pupils to become quite familiar recordingtheresults,tliencomparing with the game before they proceed to the place values, and finally recording nu- next step. merals by means of different bases. You might next suggest that it would be Here are some good questionsfor nice to have an orderly wary of recording teachers to ask along the way: what we were doing in the gamesonic - 1 In the "game of five" w hat is the great- thing like the following est number of rings that. will ever go on a single peg? A SCORE OF 9The peg on the right has a value of 1, 13 LEADS To 2 3 . and the next peg to the left of it has a 59 .. I 2 4 53 el 2 0 3 value of 5. How do you get the value of the next peg, the next, etc. F ig u re 3 :1Suppose we wanted to play the "game (Notice if there is no ring we indicate the of " (or "game of. _"), w hat would absence with a zero.) the value On each peg (plaer) lie/ What After this orderly recording has been would be the greatest number of rings done for a while, you can draw your pupils' on a single peg' attention to the fact that the recorded In base 7, w hat i.4 the largest numeral numerals23, 124, and 203look suspi- used in any single place? Why? ciously lil.^ some old friends. liene ask the quest ion, "In our game is 121 really one Many of our elementary teachers have hundred twenty-four' No" Then what is told me that they had often wanted to try it?" some work on numeiation systems that If pupils have been playing this game, have bases other than 10, but they did they will tell you, "1 twenty-fire, 2 fi, es, not know a way to introduce welt work to and 1 ()lief," As teachers, you can see that children in an interesting manner. "The here is the chance to tell pupils that a game of five" seemed to sat isfy this need. numeral's place value is important. I hope that you, too, w ill find it useful.

E1)1101( ki COMMI N i Although the author suggests that the activity be conducted without actually playing a ring-toss game, it is possible in many situations to apply his ideas to a real ring-toss game For example, e.w.h student may he given an opportunity to toss four rings on the base-five pegs to establish some base-five numeral He may then calculate the number of points in our decimal system and score this amount if his work is done correctly. For students who need more work in our base-ten system, you may wish to try the ring-toss activity with the pegs named of es, tens, hundreds, and so on. From second base to third base

MICHAEL ALFONSO Carol City High School, Carol City, Florida RICHARD BALZER Hackettstown High School, Hackettstown, New Jersey PAUL HARTUNG Blo;omsburg State College, Bloomsburg, Pennsylvania

You may have seen the gameconsisting number. Mr. Rhee quickly adds the first of four cardscall them A, B. C, and D. number on each of those cards together; for convenienceon which sets of eight numbers are written. (See fig.1.) Using A B C t) these four cards, Mr. Rhee asks a student I 4 8 to pick a number from one to fifteen and 3 3 S 9 to tell him which cards the number ap- 5 6 6 0 7 7 7 1 pears onwhereupon Mr. Rhee quickly 9 10 12 2 announcesthe number totheamazed II II 13 3 13 14 14 4 student. 15 15 15 5 How does Mr. Rhee do it? When the student tells him which cards contain his 86 ... / NUMERATION 87 the sum is the number the student chose. constructed using base three notation. To As an example, suppose the number ap- obtain the cards, we write the numbers. peared on cards A, C, andD.Then Mr. from 1 to 26 in base three notation. (See I Rhee adds I + 4 + 8 to find the number fig. 4.) The first card, Card A, will con- is 13. sist of those numbers whose base three To see why this happens, we write the notation has a"1" in the ones place. numbers from one to fifteenin binary notation.(Seefig. 1 001 2.) Now you can -, 002 readily see how the cards are developed. 5 010 4 011

1 0001 5 012 2 0010 6 020 3 0011 7 021 4 0100 8 022 5 0101 9 100 6 0110 10 101 7 0111 11 102 8 000 9 001 10 010 II 011 25 221 12 100 26 222 13 101 14 110 Fig. 4 15 111 Fig. 2 Card B will include the numbers having "2" in ones place. Card C will include Card A contains all the numbers ha. ing the numbers with a "1" in threes place a "1" in the ones place in the base two (the second digit from the right in the notation" Card B contains all the num- base three notation). Card D will have bers having a "1" in the twos place (,the all the numbers having a "2" in threes second digit fiom the right in the base place. Card E will have all the numbers two notation). Card C contains allthe with a "I" in the nines. place. And Card numbers having a 'II" in the fours place F will have the numbers with a "2" in (the third digit from the right'in the base nines place. (See fig. c ) two notation); and Card D, all the numbers with a "1" in eights place (the fourth A B C place from the right in base two notation). , I 2 3 (Sec fig. 3.) 4 5 4 7 8 5 1 0001 A 10 II 12 2 0010 B 13 14 13 3 0011 A B 16 17 14 4 0100 C 19 20 2! 4 5 0101 A C 22 23 22 6 0110 II C 25 26 23 7 0111 A B C 8 000 1) 9 001 A 1) 4.6 D E F 10 010 B I) I., 11 011 A 11 1) 12 100 C Il 6 9 18 13 101 A C 0 7 10 19 14 110 B C 1) 8 II 20 15 111 A H C 1) 15 12 21 16 13 22 Fig. 3 17 14 23 24 V 15 24 25 16 25 It is possible to develop another set of 26 17 26 cards with which the same game can be played.. Fo example, a new set can be Fig. 5 88 GAMESAND PUZZLES E F A H C

8 3 4 1 9 13 6 7 5 5° 6 10 14 10 II 9 7 II 15 13 14 15 Fig. 6

Now to play with these cards.Suppose that each student would pick adiVerent total par- someone wits us his secretnumber ap- number, thereby encouraging pears on cards B, D, andE. Then the . the num- secret number is 2 + 6'+ 9 =17.. The easiest way to determine Base four could also be used forthis ber of cards needed isto add the digits in whatever game. A setof cards using base four of the lattgest number, written would look like those in figure 6. base yoare working with, asif the digits The 'stopping point in any base isrela- were in base 10.For example, tively unimportant. However, it.willalways 15 = 1111, = i + i + 1"1 = 4 cards, be of the-form and b" 1, 26 = 2223 = 2 2 -F. 2".= 6 cards. with where h is the base yo 're work g place values are numb.Exam You will notice that the and ti is a natural reversed when we set upthe cards. When stopping points for bpktes 2, 3, an4 re- someone picks a numberand tells you spectively, we see that what card it's on, thenumber in that base, 15 = 1, is converted to base 10.That is; for the 26 = 33 I, secret number 13(Mr. Rhee's student's number):the base two numeralis 1101 and and one 8. And r + 4 + 15 =s42 1. one 1, one 4, the game. Other stopping points could bechosen so 8 = 13, the secret of who have had considerableexperience in looking for pat- , EDITORIAL COMMENT.Students finding the sum of the first nittn- rns 'ma he answer is determined by basis for leadingthelstu- bers on the identified cards. Thiswill, of course, provide an excellent works. dents to discover why this technique fhe If you wish to create a system inwhich your secret is more difficult to discover, arrange the smallest nunyber numbers on the cards in random orderand remember that you must use For example: on each identified cardwhen you calculate the desired sum. C A B 12 9 6 15 1 .7 8 15 II 12 14 3 3 14 9 7 10 6 13 5 2 4 11 13 IS 15 .7 5 13 10 II 14

card A. you would use I, which istie smallest number on If the number is represented on smallest number card A; if the number is represented oncard B, you would use 2. which is the on card B, and so on. A game introduction to the binarynumeration system

JOHN NIMAN

An assistant professor at Hunter College of the City University o/ New York, John Niman teaches courses in mathematics and mathematics education. Hi: research interests include applied mathematics and curricula development. P ---..ave you ever encountered a teasing numeration,system. The number expressed child who would only answer you with yes by 1100010 in the binary system is ex- or no? If you ever do, tell him that you are pressed by 98 in the decimal numeration going to read his mind through his reveal- system. ing yeses and noesand all you are going To convert an expression from the binary to do is ask him the same question several to the decimal system, start with the last times! Tell him to choose a number be, digit on the right and multiply it by 2°. tween 1 and 256. Then pose the question (Note: Any number greater than zero "Is the number odd?" If he says yes, tell raised to the zero power is defined as one.) him to subtract one and divide the re- Then multiply the next digit by 21, the next mainder by two. If he says no, simply tell by 22, the next by 23, and so on. Finally, him to divide his number by two. Follow sum the results. this procedure until the question has been In our example, 1100010 becomes asked at most eight times. The game stops when the number one is reached. (0X 2)+(1 X 21)+(OX 22)+(0X12') In order to play the game, the child has -F(0X2)+(1 X 2)+(1 X24) to know how to subtract and to diVide by 2. =0+2+0+0+0+32+64 He must also know what odd and even =98. numbers are. Suppose, as an example, that your friend chose the number 98. The The game is summarized in figure 1. dialogue would go as follows:

Is it odd?(98) NO Divide by 2. Is it odd?(49) YES Subtract 1 and divide by 2. Is it odd?(24) NO Divide by 2. Is it odd?(12) NO Divide by 2. Is it odd?( 6) NO Divide by 2. Is it odd?( 3) YES Subtract 1 and divide by 2. Is it odd?(1) YES

Now you have all the information you The choice for the number does not have needthe seven yeses and noes. Starting to be restricted to numbers between 1 and with the last answer, replace each no with 256. Generally, no restrictions are neces- a "0" and each yes with a "1." Such a sary. The number of questions needed de- process transforms the result into 1100010. pends on the number to be guessed. For This is the number you are looking for! example, consider how many questions are However,itisexpressed inthe binary needed for numbers froth 1 to 1000. We 89 Cib, '90 GAMES AND PUZZI FS

[ IS IT 01)1)2Io--.1) ? lYyNNNYN Let N = 0 Y = I II 1200101 iI X 2' + I X 2' + 0 X 2' + Q X 2'1 + 0x 22+ I X21+ 0 X 2°- = 98 1

Fig. I. "Guessing" a number

V i; ican solve the problem by finding outhow of .2 such that when 2 is raised to that ex- many times it js necessary to divide suc- ponent, the resulting number is equal to or cessively by two to end up finally with one. greater than N. For numbers from 1 to 256, at most eight Tile game could be used in a variety of divisions are needed, since 28 = 256. Simi- ways to determine such characteristics as larly, at most nine divisions are needed for age, height,.and weight. The fact that one numbers from 1 to 512 (2° = 512), and does not have to ask his subject the final re- at most ten divisions are necessaryfor sult after the sequence of questions makes numbers from 1to 1024 (210 = 1024). the game quite intriguing. Furthermore, it Consequently, to guess a number between would serve as an effective tool for acquir- ing practice in the basic operations of the 1 and1000, at most ten divisions are needed. The above leads to the result that arithmetic of counting numbers, and an ex- the greatest number of questions needed to planation of how the game works could guess a particular number from 1 toN. is serve as an introduction to the-base-two 1 equal to one plus the minimum exponent numeration system. / , puzzles that appear to . DITORIALCOMMENT.This is one of many mathematical games and e "number magic." Such activities can serveto stimulate student curiosity and createinter- s" in ii.:thematics if used effectively.However, the student should ultimately investigatethe mathematics that explainf the why of the "numbermagic." '

.. rqx

..-

( Tantalizing ternary

ERNEST R. RANUCCI State University of New York at Albany.Albany: New York

Dr. Ranucci y professor of mathematics education. Lust summer he was in Colombia. South America, for threemonths on a Fulbright grant. His work there was withteacher education for both elementary and secondary levels.

Arithmetic computation in bases other Table 1 than tea is a fertile field to explore. Espe- Decimal Ternary Decimal Ternary cially interesting are certain problems that notation notation notation notation arise in the base three (ternary) system.

I 210 Place value inthe ternary .systemis I 21 2 2 22 211 based upon powers of three. The units 3 '10 23 212 digit has a .value of 1, the base digit a 4 11 24 220 value of 3, the base-two digit a value of 5 12 25 221 6 20 26 222 9, the base-three digit a value of 27, etc. 7 21 27 1000 The. number '12012 in base three has a 8 22 28 1001 value of 140 in base ten. 9 100 29 1002 10 101 30 1010 (12012,,,,.. = 140.) 11 102 31 1011 12 110 32 1012 13 111 33 1020 1 2 0 1 2 14 112 34 1021 ...=-..=.==._.-=----tr..-- 15 120 35 1022 34 33 32 31 1 Place value as a 16 121 36 1100 power of three 17 122 37 1101 ----27 9 3 1 Calculated place 18 200 38 1102 81 value 19 201 39 1110 120 2th. 20 202 40 tin =1 X81 +2 X 27+0)0 +1 X 3+2 X 1 nothing but a simple balance scale and = 81 54 0 3 2 individual weights of one, three, nine, and = 140,... twenty-seven pounds. As long as no cus- In the ternary system only three digits tomer asks for meat in a quantity requir- are needed. We shall use 0, 1,and 2. ing the use of half-pounds, the butcher Many intriguing puzzles are based on finds that he can weigh any integral num- the ternary system. For the exploration of ber of pounds of meat from one to forty. several of these, we need to convert the He can, what's more, do this in one weigh- decimal numbers from 1 to 40 into base- ing (providing the customer isnot so three numerals. (See Table 1.) Note that fastidious as to object to the placing of 40 = 1 +3-4.9 -1.27. weights on the same pan with the cut of This fact will prove to be of extreme sig- meat.) To weigh out five pounds of chopped nificance later on. beef, for example, he places a weight of Let us first consider the "butcher prob- nine pounds on one pan. He then places lem." A certain Scottish butcher finds that weights of one and three pounds on the he can get along surprisingly well with other pan and adds meat until the scales j 91 L 92 GAMES AND PUZZLES balance (Fig. 1). This leads to the simple Table 2 equation: 1 1 21 , 27- 9-1- 3 M + 1 + 3 = 9; M = 5. 2 3 - 1 22 27- 9+ 3+ 1 3 3 23 27- 3- 1 4 3 + 1 24 27-3 5 9 - 3 - 1 25 27 3+ 1 6 9 - 3 26 27-1 7 9 - 3 + 1. 27 27 - 8 9 - 1 28 27+1 9 9 29 27-1= 3- 1 10 9+ 1 30 27+3 11 9 + 3 - 1 31 27+ 3+ 1 FIGURE 1 12 9 + 3 32 27+ 9-.3- 1 13 9 + 3 + 1 33 27+ 9- 3 Weighing ten pounds of meat is easy. 14 27 - 9 - 3- 1 34 27+ 9- 3+ 1 All thii takes is the placing of weights of 15 27 - 9 - 3 35 27`.-1- 9- 1 16 27 - 9 - 3+ 1 36 27+9 one and nine pounds on one panand the 17 27 - 9 - 1 37 27+ 9+ 1 addition of meat to the other pan until 18 27 -9 38 11+ 9+ 3-1 they balance (Fig. 2). The equation here is 19 27 - 9 + 1 39 27+ 9-1- 3 20 27 - 9 + 3- 1 40 27+ 9+ 3+ 1 m = 1 + 9; M = 10. anunusual approachtothe"butcher problem." This starts with the value of the desired weight expressed in the ternary notation. To weigh out 33 pounds, for example, first consider the ternary numeral 1020,b,... Add a 1 and subtract a I in the column where a 2 occurs. It is advantageous FIGURE 2 to rewrite as follows: Curiously, each total, from one to forty, can be achieved by only onecombination 1020 1 of the 'various weights available. Table 2, The addition of a 1 and the subtraction of a based on the ternary system, indicates the unique structure behind the "butcher prob- I does not affect the value of the number. (We are really adding 1 X 3 and subtracting isbut one lem." Remember that there I X 3, but there is no real need for this weight of each size. extra step.) The addition of 2 and I means The table is used as follows: All posi- coluinn. tive weights are placed on one pan; all that we can carry a I to the adjacent The result may now be expressed as 1110, negative weights are placed on the other 3, whose value is 33. added to the pan on This means 27 + 9 pan. Meat is always Several solutions of this type are carried out: which the negative weights havebeen placed. In weighing out 20 pounds,for 20 pounds:202u,.. placed II example, weights of 27 and 3 are = 202 = 1I1I on one pan; weidtsof 9 and 1 are placed 1 1 on the other.M!at is added to the 9, 1, (This means 27 - 9 + 3 - 1 = 20.) With the side until a balance is secured. 32 pounds: 1012thr., addition of another weight of 81pounds to the set, everyintegral weight from 1 = 1012 = 1021 = 1021= 1111 1 1 to 121 may be achieved.The "butcher" (This meads 27 + 9 - 3 - 1 = 32.) approach to equation solving canbe han- grader if the 23 pou1 nds: 212th212th. dled by the average sixth 1 approach is gradual. = 212 = 1121 =1121 = 1011 Martin Gardher, in an articlein the May 1 1 1 (This means 27 - 3 - 1 = 23.) 1964 issue of the ScientificAmerican, offers .$ )./ NUMERATION 93

The first forty: numbers in the decimal Second card (Worth 9 points) system have values in the ternary system, (Front) Blonde, Brown Eyes numerals 1, 0,1, as shown in Table 3. Barbara Joy Nancy Bets) Helen Naomi Table 3 Betty Hilda Nina Bunny Ida Nora 21 1110 Carmen Ilene Ingeborg Ophelia 2 22 Carol 13 Pamela 3 10 1011 Charlotte Ingrid Patsy 4 11 24 1010 Corinne Jacqueline Paula 5 ITT 25 1011 Diane Jeannette 6 110 26. 1001 7 111 27 1000 (Back) ilItMde, Blue Eyes 8 101 28 1001 Lana Ursula 9 100 29 1011 Dinah Lena Vicki 10 101 30 1010 Dolores Dorothy Lisette Viola II tiT 31 1011 Edith Louise Wanda 12 110 32 1111 Elizabeth Mary Wendy 13 III 33 1110 Margaret Winifred 1111 34 1111 Eloise 14_ Mildred Yvette ITTO 35 1101 Ethel 15 Marilyn Yvonne 16 ITTI 36 1100 Fanny Nanette Zaza 17 1101 37 1101 Flora 18 1100 38 19 1101 39 1110 20 1111 40 tits Most puzzles of the t pe, "Find your 'Third card (Worth3points) age on each of the foll wing cards and (Front) Auburn, Brown Eyes I'll tell you what it is,"re based on the Alice Gwendolyn Nancy Naomi ternary system. So 4.... a series of puzzh Anne Harriet Audrey Hazel Nina in which names are guess t1 -mowing is Charlotte Jacqueline Petunia a description of one of the better of the Corinne Jeannette Rita Diane Joy Roberta name puzzles. Ethel Lana Ursula First prepare four cards according to Fanny Lena Vicki the directions which follow. Flora Lisette Viola

(Back) Auburn, Blue EYcs GIRL'S NAME PUZZLE Barbara Helen Pamela First card (Worth 27 points) Betsy Hilda Patsy - Betty Paula (Front) Brunette, Brown Eyes Dinah Katherine Tina Dinah Florence Helen Dolores Kay Tess Dolores Fredrika Hilda Dorothy Kyle Tracy Dorothy Gail Ida Florence Marilyn Yvette Edith Geraldine Ilene Fredrika Mildred Yvonne Elizabeth Gertrude Ingeborg Gail Nanette Zaza Voise Grace Ingrid Ethel Gwendolyn Jacqueline Fanny' Harriet -Jeannette Flora Hazel Joy Fourth card (Worth I point) (Back) Brunette. Blue Eyes '(Front) Red, Brown Eyes Nancy . Petunia Ursula Abigail Grace Nancy Naomi Rita Vicki Audrey Hazel Nora Nina Roberta Viola Betty Ida Pamela Nora Sarah Wanda Carol Ingrid Petunia Ocelie Suzanne Wendy Diane Joy Sarah Ophelia Tallulah Winifred Dorothy Katherine Tina Pamela Tina Yvette Eloise Lana Ursula Patsy Tess Yvonne Flora Louise Wanda Paula Tracy Zaza Gail Mildred Yvette 94 GAMES AND PUZZI ES

(Back) Red, BlueEyes the eyes are brown. The score isconsid- ered negative when the eyes are blue.Thus: Alice Gwendolyn Nina Barbara Helen Ophelia Rita-Bruriettc, Blue Eyes (-27); Auburn, sunny Ilene Paula Brown Eyes( 3 ). Add the scores alge- Charlotte Jacqueline Roberta braically. The, sum is -24. Look up -24 Dinah Joyce Tallulah Edith Kyle Tracy in the list of names and identify "Rita." Ethel Lisette Viola Florence Margaret Winifred Thus:Joy (27, 9, 3, 1) or 40 Geraldine Nanette Zaza Yvonne (-27, -9, -3) or -39 Harriet (27, 3) or 30 Following is a list ofpossible names, Ethel (27,-9,3, -1) or 20 with numerical value. The structure of the Girl's NamcPuzzle liosinve is identical to that of the "butcherprob- I. Abigail 21. Fanny lem." Once the desired number hasbeen 2. Alice 22. Floh .23. Florence converted to the Gardner positive-nega- .3. Anne with 4. Audrey 24. Fredrika tive number, the name associated 5. Barbara 25. Gail that number is placed on theappropriate 6. Betsy 26. Geraldine 27. Gertrude face of the proper card. 7. Betty An examination of °ther number sys- 8. Bunny 28. Grace 9. Carmen 29. Gwendolyn tems will reveal the unique natureof the 10. Carol 30. Harriet ternary system. Supposethat the weights 11. Charlotte 31. Hazel had been 12. Corinne 32. Helen used in the "butcher problem" 13. Diane 33. Hilda 1, '2', 22, 23, etc.-elementsof the binary 14. Dinah 34. Ida system. Then meat, inthe amounts indi- 15. Dolores 35. Ilene weighed as 16. Dorothy/ 36. Ingeborg cated below, could have been 17. Edith 37. "grid follows: 18. Elizabeth 38. Jacqueline 19. -Eloise 39. Jeannette 40. Joy 1.1 or 4 - 2 - 1, etc. 20. Ethel 2. 2 or 8 - 4 - 2, etc. Negarice 3.1 + 2 or 4 - 1, etc. 1. Joyce -21. Patsy 4. 4 or 8 - 4, etc. -2. Katherine -211. Paula -3. Kay -23. Petunia Such weights could, of course,be used, 4. Kyle -24. Rita unique about -25. Roberta but there would be nothing -5. Lana could be pro- -6. Lena -26. Sarah such a system. Each weight -7. Lisette 27. Suzanne duced in a variety of ways. -8. Louise 28. Tallulah system pre- -29. Tina A base-four (quaternary) -9. Mary Let us attempt to -10. Margaret -30. Tess sents its own problems. -II. Mildred 31. Tracy weigh out the :firsttwenty pounds; we -32. Ursula -12. Marilyn may use onlyindividual weights of 1, 4, 13. Nanette 33. Vicki -14. Nancy -34. Viola 16, 64, etc. -15. Naomi -35. Wanda -36. Wendy -16. Nina 1 .1 11.16- 4- I 17. Nora -37. Winifred 2. ? 12.16- 4 -18. Ocelie 38. Yvette 3. 4- 1 13.16- 4 1 -19. Ophe lia 39. Yvonne 4. 4 14.? -40. Zaza 16 -20. Pamela 5.4+ 1 15. - 1 6. ? 16.16 The puzzle is worked asfollows: Some- 7. ? 17.16+ I pick a name from thecards 8. ? 18.? one is asked to 9.? 19.16+ 4- 1 and to identify the haircolor and eyes each 10. ? 20.16+4 mentioned. The score time the name is Certain "clusters" appearwith a pre- on any cardis considered positivewhen ro k I ti NUMERATION 95 dictablc pattern, butmany "holes in the It would appear that the ternary system Swiss cheese" showup. This deficiency alone allows us to achieve allintegral becomes aggravatedwhenthe number weights in a unique manner. base increases. .1. \

/ Integers

The introduction of integers as early as third grade in recently published de- mentary school1 mathematics textbooks is an outgrowth of the natural encounter of elementary school children with many situations that require the use of integers. When primary-grade children learn to use thermometers,they note that thermometers can register temperature below zero as well as at or above zero. -title countdownsfor space launchings involve negative and posi- tivenumbers, and altitudes are recorded as above or below sea level. In football games, we make note of yardage gained and yardage lost, and we speak of budgets being in the red or in the black. The study of integers in the primary and intermediate grades should be largely informal, and many such experiences can be provided through game activities. In the opening article, Demchik suggests a car me called "In- teger Football" as one way of generating interest and enthust mfor a practice exercise involving integer addition. Milne suggests a s udent-created game that could lead to the discovery of a method for subtractionof integers as a result of various moves on a game board. In a game described by Mauthe, upward movements on a ladder correspond to positive integers, and downward movements correspond to negative integers. The composition of movementsidentified as addition. Imagine the excitement of students as they reach the tvp of the ladder or as their oppto- nents fall off the ladder. Frank incorporates much fun in hershuffleboard game while relating integers, number lines, and operations onintegers. Chil- dren will enjoy this activity and gain practice in working with integers. To provide more drill with the operations on integers, Milne has developed a spinner game; Cantlon, Homan, and Stone have devised astudent-constructed game that uses Popsicle sticks. Both activities will provide much student involvement and painless practice. In the closing pair of selections, we see how creative minds with an interest in mathematics can generate some challenging puzzles. Many upper elementary and middle school studentsill find "Grisly Grids" quite interesting.

97 Integer "football"P 4

VIRGINIA C. DEMCHIK

Bowie State College, Bowie, Maryland

Generally, students inthe upper ele- Playing the game mentary add junior.high grades are inter- Before play begins, the class is divided ested in sports. The card game "Integer into two teams. The deck is shuffled and Football" uses that enthusiasm for sports placed face down on the table. Each team as motivation for practice 'with directed starts on the "fifty yard line." A flip of a numbers. The game is designed for use coin determines which team "kicks off." with an entire class or with a small group Play begins as a member of the team of students. that won the toss turns over the top card. The team adds the integer on the card to Materials needed the fifty yards With which the team began. Integer football is played Wth a deck of For example, if the top card is 16, the 65 two- and -one- half -by -three -inchcards. play is SO + ( 16) = J4. This means that The cards are assigned numerical value% as the team now needs 34 yards to score. On follows: the other hand,. if the drawn card is +16, the play is 50 + ( +16) = 66, and the One set is numbered d through +20. team now needs 66 yards to score. Thus, Two sets arc numbered I through 20. itis to the team's advantage to draw a There are four penalty cards: +5, +15, negative number. Similarly, a penalty card 5, 15. with a negative integer ( 5 or 15) is to 98 the team's advantage; a. positive integer scoreisNhe extra point. If the next card is (1-5 or ± 15) is to the team's disadvantage. positive,Ithese i.no extra point. The "hall goes to the other team" after each- After every touchdown and the play for play. On the tcam'CS next play, the integer the extra point, play resumes again on the on the drawn card is added to the total from fifty yard line. This time the team that did the previous play. not score begin

EtwoklAt. Coxism.t.Most children gain extra enjoyment when a "football field" and "football" are used in playing this game The football field can he easily constructed from posterhoard and 'marked Wong the edges to indicate the "yard line;,' A piece of .iponge or a fbrotballrawn on posterboard may be used to mark the position of the hall.

, 9 01 02 OS Ot7 OG 017 02 02 01 9 -1- 7 7 ,

0 G 10 20304050403020 10 G las

-I

Af,% 14!v Subtraction ofintegersdiscovered through a game

ESTHER MILNE

oitv ct, 4;14 pa fi. turfy .lasortititikvw Joerta. qtr. Cin, .4.41 AscI. anui, ciV.4.,e. :ft AU.401411.4A, ru&w1.14A 434' ''?"kel."7%avt. ,u4 .%.vtdo-

This game- description was astudent's 'Two cubes are used at a time. One represents addition: it has red numerals onit. idea of a good problem requiringsubtrac- The other one is for subtraction:it has blue tion. From this suggested game, avariation numerals printed on it. It is suggested was devised to includeneg:::ive integers. each of the first cubes used be marked wt respective Thus it can be used to introducesubtrac- 0, 'I, '2, '3, '4, and '5 in its color. tion of negative integers. rolls the Materials needed are a game board, two ,.To start the game each player (buttons or small toys), cubes. The person who rollsthe higher or three markers the apt and several blank cubes to usein-place of value on the addition cube gets player with the dice. The cubes are marked with namesof (turn. In case of a tie, the higher value on the other cube goesfirst. integers. t: f.j100 INTEGERS 101 It is agreed that subtraction means to do Any combination of these or lather cubes the opposite of addition. A '3 on the addi- suggested by students should be tried. tion cube means go forward three spaces. After the students have played a few Then a '3 on.the subtfaction qube means games, make a rule that the one who goes go backward three spaces. first also chooses the pair(s) of cubes with Usually with the two cubes suggested, which to .play the game. Of course, both little progress is made by either player. (It players throw the same cubes. won't- take students long to realize that a This game, if introduced early enough '3 on both cubes gets them' exactly no- (before any models for subtraction are where.) After several turns where little offered).,cotild be the vehicle by which progress is made, a1 new cube should be - students figure out a method for subtraction introduced. A subtraction cube with 1, '2, of integers. Itis quite possible that they :3, -1, -2, and -3 on the faces could speed will intuitively arrive at the very "rule" so up the game. (If this speed-up measure often presented to first-year algebra stu- seems a bit inadequate after a fair trial, dents in theast. substitute a subtraction cube with 0, -1, -2, And at ar e chances that this game

-3, -4, and 75.) The person who reaches .could delop into a study in probability? home first is the winner.

Suggestions for some other cubes: EDITOR'SNOTE.-There is a good possibil- Addition: '2, '3, '4, '5, 1), '7 ity this might happen in a classroom in which Subtraction:0,'1,'2,'3,'4,'5 the teacher creates a learning situation where (Starter Cube) someone can ask the, question! It might be children's Additi6n: '1, -1, '4, -4, '7, -7 interestingtosurveyavailable games with reference to application of mathe- Subtraction: Similar to, or same as, matical skills and principles. It may be that addition cube many more of them than we now realize can Addition: '10, '9, -6, -5, -4, -3 be used. to introduce or to prktice mathemati- Subtraction: -10, -9, '6, '5, '4, '3 cal Understandings.-CHAm.b4E W. JUNGE. -Climb the ladder

AL-BERT H. MAUTHE

Albert Maculae is chairman of the depareent of mathematics., at 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,CohnbiaXiniversity.

Here is a new game.lt is called "Climb the ladder the numberOf stepi indicated the Ladder." The rules of the game are (1, 2, 3, or 4). We will call the numbers 'very easy to learn, and you will have lots 1, -2, -3, -4 on the spinner-dial "negative of fun playing Climb the Ladder. numbers." To play the game, you will have to make If the spinner stops on the 0 space, the a ladder and a dial-spinner. (Thesewill be player does not move up or down the' shown later.) ladder. We will call this number "zero." As many players, who wish to play the Each player takes one turn at a time game together may do so. Theobject of until he has fallen off the ladder or some the game is to go above the '12 step on player has been declared the winner. the ladder. Anyone who falls be1oW the -12 Let us watch Betty, Bob, and Dave play step on the ladder has fallen off the ladder, the game. We will watch each player's and he is out of the game. Thus a player progress up the ladder on the demonstra- can win thegame in either of two ways: tion-laddei. At the same time, we will keep a tecord of each player's steps at theright 1. if he is the first person to go above side of this page. the 12 step on theladder,''he wins the game; or Betty begins. She starts BF1TY at Zero, 0. Start 0 2. if he is the last person remaining on She spins a +3. Thus, she the ladder after all the other players have goes 3 steps up the ladder. Add +3 fallen off the ladder (by falling below the She is now on step +3. Step +3 -12 step), he wins the game. Bob and Dave each take their first turns. Everyone starts by putting his marker at Bob-lands on -2, Dave on -4. The second "Zero", the "0" step. We will call the turns for Betty, Bob, and Dave 1k like numbers above 0 "positive numbers," and this: we will call the numbers below0 "negative BETTY numbers." We will not call 0 either "posi- Betty was at +3. tart ^41 tive" or "negative"; 0 will be,called She spins a +1. Thus:she Add +1 The first person spins the spinner: If the goes 1 step up the ladder. spinnq stops on a 1, '2, '3, or '4 space, She is now on step +4. Step +4 the player moves his marker iffl_the ladder the number of steps indicated (1,2,---; BOB 4). We will call the numbers '1, '3, and Bob war at -2. Start -2 He spins a +3. Thus, he '4 on the spinner-dial "positive numbers." goes 3 steps up the ladder. Add +3 If the spinner stops on a -1, -2, '3, or -4 space, the player moves his markerdown He is now on step +1. Step +1 A a 102 INTEGERS 103 DAVE 1. Note that ,there are nine different sec- Dave was at -4. Start -4 tors, four for the numbers '1, '2, '3', and '4, He spins a -2. Thus, he goes 2 steps down the ladder. Add -2 four for the numbers 1, -2, -3, and -4, and one for 0. He is now on step -6. Step -6 Dial BOB Bob spins a -2 on his third turn. He was at +1. Start +1 Thus, he goes 2 steps down the ladder. Add -2

He is now at -1. Step -1

BOB Bob spins a 0 on his fourth turn. He was at -1. Start 71 Thus, he does not move either IQor down the ladder, Add 0

He is still at -1. Step The game continues, until one player passes '12 or all but one player passes -12. FIGURE 1

Make your own game Do you remember that there are 360 degrees in one complete turn around the Materials needed.We'll want to play center of a circle? How many degrees must this game in class. But before we begin there be in each of the angles around the playing, we'll want to make our own game circle if we use 9 sectors that have equal to keep. To make the game, each person angles? in the room will make the game from these Draw a radius from the center of the materials: circle you have drawn to any point on the 1. A square piece of heavy cardboard, circle.Measuring with, your protractor, about 15 centimeters on a -side (15 draw another radius so that the two radii centimeters is about 53/4 inches )_ make an angle of 40degrees at the center 2. A piece of heavy paper or cardboard, of the circle. approximately' 8 centimeters by 28 cen- Continue making 40-degree angles until timeters {thatis.about 3 inches by you have nine equally spaced radii, and 11 inches). nine sectors of the same size around the center of the circle. Label these sectors as 3. A paper clip. shown in Figure 1., 4. A thumbtack. Take a paper clip and bend the last 5. Several small pieces 9f cardbohrd or loop out so that it looks, like the one shown other objects to serve 'as markers. in Figure 2. 6. A ruler, preferably one with centimeter Spn ne r markings.' 7. A pfotractor. 8. Compasses ( tO draw a circle). FIGURE 2

The dial-spinner.On the heavy card- , I board, draw a circle, pith center in the Place a thumbtack through the oval middle of the cardboard, with a radius of still remaining in the paper clip, and place 5 centimeters (about 2 inches). the thumbtack through the center of the Study the spinner-dial shown in Figure circle you have drawn. Your spinner-dial ti ki .1, t 104 GAMES AND PUZZLES should look like the one shown in Figure 3. If your game finishes before the others, p1ay another game. Keep score for this game also. If time runs outbefore anyone has won the game, the player highest on the ladder will be the winner. Have fun!

Ladder

+12 +12 FIG. 3. Thumbtack through oust of paper +II +11 clip. *10 +10 Make-sure_the_paper_ clipspins freely +9 +9 around the dial. ft +g The laddei:On a heavy piece of A +7 +7 P.: -paper (or cardboard), draw a ,straight line CI +6 +6 8 segment 24 centimeters long (or 9inches, +5 +5 D. if you do not have a ruler market in Q +4 +4 centimeters). Make a mark at the bottom J +3 +3 of the line segment. Then mark off each +2 + centimeter (use % inch, if yob used a 9- 141 +1 +1 inch line segment), until you get to the 0 -ZERO- 0 top of the line segment. H-1 Label each step from -12 to '12, as was -2 done with the ladder in Figure 4. m -2 -3 tot's play! -4 Although any number of players can C.) -s- -5id play togethei; let's divide into groupsof -6 -6 about three players each and try playing -7 r. our new game. Each groupwill use one -8' ladder; each player will use his own -9 marker. -10 -101 With each turn, be sure to move your marker up or down the ladder. And be -11 11 sure to keep score, as wedid for Betty, - /2 Bob, and Dave. FIGURE 4

write a number'sentence to de- EDITORIAL COMMENT.Youmay wish to have the student the ma- scribe each c f his moves on the ladder.This will provide the connecting link between nipulations and movements on the ladderand the symbolic representation of these movements. Many times thestudent fails to realize that these gameexperiences have any relationship to the symbolic languageof m.0 hematics.

d4 I /-..0 Play shuffleboard with negative numbers CHARLOTTE FRANK Albert Einstein Intermediate School 131, Bronx, New York

Shuffleboard,anyone?" Can you imagine gram. Each player tries to push his own how asttklent-filled math classroom would disk into one of the scoring, areas or to respond to that invitation? Compare that knock his opponent's disk out of a scoring, with the/reaction to a teacher announcing, area and'hopefully into a penalty area. "Students, today we are going to learn the By adding positive and negative signs, four fundamental operations of the set of the playing field (Fig. 1) can easily be integers" By the time the teacher had transformed into Negative Shuffleboard finished! saying the word "fundamental" (Fig. 2). his pupilt would have tuned out both him Other adjustments necessary to fit our and his lesson. Negative Shuffleboard can new game to the facilities of a classsroom are the following: be used to introduce-the four basic opera- 4 tions of negative integers to the elementary 1. Target diagrams are drawn on paper school'youngster in a fun setting. sheeting large enough to cover the stu- For the few who have never played, regular Shuffleboard is a game in which dent's table. players. use long-handled cues to shove 2. Red and black checker-size disks w disks into scoring beds of a dia- replace the larger disks normally used.

CO ti

CD O

FIGURE 1

ti O 0

CO O

FIGURE 2 -4 %. ;,105 106 GAMES AND PUZZLES 3. A pencil, pen, or ruler is used as a Some of the computations at another propelling instrument. table, where Matthew and Sidney were 4. The ratio of the playing field to the seated, were as follows: disk is increased in order to allow for Situation 1. Matthewhad 1, '8, '10 more scoring activity. and one disk out of limits, giving him a 5. A number lineisused to keep a total of '25. When Sidney's turn came he running score. successfully knocked Matthew's diskout .1 When Anne and Harley played the game, of the '8 box and into the -10 box. Mat- their mathematical computations developed thew's '8 was taken away; written math- this way: ematically, it was '8. This number sentence describes the last play: ROUND I '8 --I = 0. Situation2.Sidney successfully knocked I0 his own disk out of the -10 penalty box. +8 Translated into our language of numbers, +8 \ Sidney had a -10. When he took away the +7+7 -10,it gave him a 0 for the box. This number sentence is written: -10 -10 = 0. /0-100-1° Situation 3. AfterRound 4 Matthew Anne(0 disks) -10 + 10 =-20. or 2 x 10 = -20. had 30 points more than Sidney had. With Harley ( disks) 13 + 7 + -10 = 5. three disks still to shoot, Sidney told the group watching that he could win if Mat- Round 2 thew were unlucky enough to end the round with a -30. That would be written by the following number sentence: -30 4- 3 = 10. The target diagrams vary from table to table to' meet the needs of the individual

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

. Round 3

Anne0 disks) 3 X +7) + '10 = 1 I. Harley ( disks) 4 x -10 = "40. This game continues tit a previously agreed upon winning score Ireached. FIGURE3

A ,-- 1 . i-i `. INTEGERS 107 children.Some suggestions for a few possible changes to match the learning/styles of your students are indicated in Figure 3.

EDITORIAL COMMENT. Many variations of this activity are possible. A beanbag-game format could be used in a manner similar to that suggested for the shuffleboard activities. The game arrangement will actively involve the children andprovicie an interesting format for skill development. If the students can be encouraged to recordtheir scores on a number line and write equations to represent theoperations, they will gain additional insight into the operations with directed numbers:_. a

Disguised practice for multiplication and addition of directed numbers

ESTHER MILNE

Esther Milne is a Mathematics Helping Teacher in the Tucson Public Schools, Tucson, Arizona.

Practicing computation can be drudgery. spinner, label a couple of the numet But practicing computation with integers "power" and the rest positive andnegative. is necessary. Games aren't drudgery. Let's (There might be a zero on a dial.) combine games with practice. Then students can have fun as they improve the accuracy of their computation. Try this game.

INTEGER GAME How many can play?Two, three, or a whole class divided into twoteams., What etiuipment is needed?Two spin- How do you play?Spin the first spinner. ners and paper on which to write\ scores. is 6 two The number on which the pointer stops (When a large group is playing the first factor. Spin the secondspinner. teams,it is best to have replicasif the Where the pointer stops gives the second spinner dials on an overhead projector factor, or tells us the power to whichthe transfcncy.) first factor is to be raised. (If alarge group Ma k some numerals + (positive) on is playing, place a marker oneach dial the first spinner. Mark the rest of ,the show where (negative). On the sccimd on the overhead projector to numerals the pointer stopped. A coin could beused as a marker.) Record theproduct. Keep If you don't have your own favorite source for a running score. purchasing spinners. these are available from Vro- Decide before the game man's. 367 S. Pasadena Ave.. Pasadena, Calif. 91105. Who wins? .t".u108 INTEGERS 109 starts how many turns will constitute a Each one keeps his own score and that game. After the last turn the player or of his opponent(s). team whose score is farthest from zero is If IO turns is to be a game, the score the winner. Score is kept by all players. sheet looks something like 'this:

ME THEE , Running Running Factors Product Score Factors ,. Product, Score

+30 I. -5 +4 -20 -20 1. -5 -6 10 2. +4. +4 +16 -4 2. +3 +4 +12 +42 3. (+3)' +27 +23 3. (-5)2 +25 +67 4. 0 -6 0 +23 4. 0 -6 0 +67 +I 5. 0 .. +1 0 +23 5. (IP +6g 6. -8 -6 +48 +71 6. -6 +I -6 +62 7. (+I)' +1 +72 7. +2 -5 -10 +52 8. (t3)+ +9 +81 8. (-6)3 -216 -164 9. 0 -5 0 +8 9. -8 +I -8 -172 10. (-7)' +49 +130 10. (+2)3 +8 .-I64

Thee is the winner!

EDITORIAL COMMENT.Manyvariations of this game are possible. For example, a basic addition game may be played by having students add the two integersidentified by spinning the two spinners. A subtraction game may be played byhaving a student spin the first spinner to identify the minuend and then spin the second spinner to identifythe subtrahend. The student then finds the difference between the two integers. If you are looking for a way to design spinners that work extremelywell, borrow a pair of plastic lazy Susan turntables from your kitchen cabinets and cutposterboard disks to fit in the turntables. The disks can be marked and numberedaccording to your needs and a pointer can be placed on the desk top beside theturntable.

Pasteboard risk cut to fit in the turntable. Plasticlazy Susan turntable - witty ball beariv base.' I.--- I Pointer card

-4f A. i v

A student- constructed game for drill with integers

MERLE MAE CANTLON Jefferson Junior High School, Caldwell, Idaho DORIS HOMAN Vallivue Junior-Senior High School, Caldwell, Idaho BARBARA STONE Central Junior High School, Nampa, Idaho

In teaching a unit on integers,teachers For this activity, the students should be are invariably faced with theage-old prob- divided into groups of three or four. With- lem of how to provide interesting and outfurther instructions, each group is meaningful drill. Popsicle sticks can be given its assignment sheet for the day (fig. used to provide a motivating, "hands on" 1). laboratory exercise in which the students The second activity involves the actual build their own game. The anticipation of playing of the Popsicle-Stick Game with using what they are studying in a fun situa- the numbered Popsicle sticks that the chil- tion .seems to be an added motivational dren have made. Instructions for playing factor for students. The exercise shown in the game follow: figure 1 is a suggestion for the first activity, in which the 'children make the materials 1. The class is divided into teams of and experiment with the "Popsicle-Stick three or four students, and each team is Game." given a set of numbered Popsicle sticks. INTEGERS 1 1 1

Building a Popsicle Stick Game

Names of team members' Cberig the WI at the right when the:team has completed each step, please. I.Get 15 Popsicle sticks and one felt marker. 2. Mark 1 on the first stick, k4eping the sticks in vertical position. Mark =a on the second stick, and so on thrqugh _=. 3. Mark 0 on both sides of one stick. 0 4. Mark 1 on one side of a stick and 1 on the opposite side. Mark 2 on one side of a stick and 2 on the opposite side, continuing on in this manner through 9 and 9. 5. Build with Popsicle sticks the following math phrases (expressions) that are equal to 4. a. 3 'i- 1 = b. 8+4= 4' c. (-4 x 2) + 4 = 6.With your sticks, write three math phrases that are equal to 48. a. - 48 b. -48 c = 48 7. Using at least one negative integer in each, write three math phrases equal to 3, a. -3 o b =3 0 c. - 3 o

Fig. 1

Nen 2. Students on each team take turns sponsibility for writing the correct opera- acting as captain, the honor rotating around tions as well as the correct integers, and for the table with each new problem. A per- indicating the order in which the opera- manent scorekeeper should be chosen for tions are to be performed. each team. 6. If the problem prepared by the first 3. The object of the game is to see team whose hands were raised is correct, which team, using one or more of the four the team receives four points. Two points operations on integers, can be the. first to are awarcizd to each of the other teams that construct a mathematics problem with an has a correct problem. Teams with incor- answer equal to the integer that the teacher rect problems receive a score of 1. Each has written on the board. Numbers from (cam keeps its own score. 100 through 100 can be used for the in- 7. For variations, the teacher may place teger answers, and for variation, these can restrictions on the problems to be con- be written on slips of paper and then drawn structed. For example, some possible re- from a box by the captain of the last win- strictions are: ning team. a. There must be at least one negative 4. When a team has constructed a prob- integer in the problem. lem with an answer equal to the integer on b. At least two operations must be used the board, all members of the team must in the problem. raise their hands. The teacher notes the c. The problem must be worked with first team whose hands areallraised. a minimum of two sticks, or three When it is apparent that most groups have sticks, and so on. / completed a problem, time is called. The Popsicle-Stick Game has several 5. The completed problems are written advantages. It promotes total student par- on the board (or on the overhead projec- ticipation--groups are small. The honor of tor) by the respective team captains, and acting as captain isrotated within the the accuracy of all problems is checked by groups. All membeis of the team must the class. The team captains have the re- raise their hands in order for the team to 4 l'ij . 4- 112 GAMES AM) PUZZLES

be recognized asfirst.All students are In summary, this experience provides involved in checking the problems that are students with useful and enjoyable practice put on the board, since numerous correct for operations with integers through the responses arc possible. The scpring pro- use of manipulative materials in a labora- cedure provides additional opportunities tory setting. In the words of a Chinese for practice. "Hands on" materials are philosopher: I hear and I forget; I see and ci used, and little teacher direction is needed. I remember; I do and 1 understand.,

.....;

EIMORIAL COMMENT. The first activity shown in figurel' requires the students,to "write math phrases" in steps 5, 6, and 7, using Popsicle sticks; however, there is no provisionfor the operation signs. You may wish to provide each team of students with a number of papa squares on which they may write the desired operational symbols inordcr to compete their

math phrases with the Popsicle sticks. . Game activities such as these can provide very interesting and stimulating practice experiences, but these experiences should be preceded by meaningfulinstruction to develop an understanding of the operations with integers. 1

r .. em,

Grisly grids

WILLIAM G. MEHL and DAVID W. MEHL

William Mehl is a teacher of mathematics at Wilson Junior High School in Pasadena, California.

David Mhl, his son, was a student in Mr. Mehrs summer class in exploratory algebra and assisted in the writing of the ankle.

The multiplication lattice grid is fulfil:1r 2. A negative product consisting of two tomost teachersand many students digits is always recorded such that the throughout the country. A more cttalleng- tens digit is negative and the ones digit ing version of this grid type might include is not negative. directed numbers. The writers have modi- 3. A positive product consisting of two fied the traditional presentation and intro- digitsisalways recorded such that duced a grid incorporating a new set of neither digit may be negative. "rules." The grid can be used to stimulate 4. The integers in the margins opposite and to entertain the most ambitious pre- any cell may have one digit only. algebra or algebri student. 5. Marginal sums are found b' adding Let's look at the grid's anatomy. Each obliquely from right to left, starting in small square shall be called a cell. Each the lower right corner of the rectangle. cell consists of two congruent triangular 6. The solution consists of any necessary regions. The upper region is designated marginal or tabular entries, and in some UR and the lower region LR (Fig.I ). cases requires that the marginal sums The total grid may be in the form of a be totaled and entered in a frame as square or other rectangle, thus enabling directed by an arrow (Fig. 4). the teacher to regulate the number of factois and cells for any given presentation. All marginal entries at the top and right of the grid shall represent factors. ...- while the entries at the left and bottom FIGURE 2 shall represent sums. Each cell contains a product. All entries consist of integers, The teacher might begin simply by pre- and all marginal entries are integers x senting a grid such as Figure 3: A varia- such that -10 < x < '10.

C.41

FIGURE .6 0 The following' rules regulate. the opera- FIGURE 3 tion of the grids: 1. All tens are recorded in UR, while all ones are entered.in LR. tion of the first presentation might appear 113 114 GAMES AND PUZZLES as shown in Figure 4. Note the required answer recorded in the frame at theh6ad arrow. 4-2.8

I--I 44 48 Present to students -lz/ry -2A -I7+2 *.6443 -7 +8 +6 -3 Solution 7 +2/1/ +4 +4 -4 -3/ 4, +4 44 /6 /0ZO/0 / '4.8 .3 /44/./+3/8+5 -4 0 o/.5 FIGURE 4 +6. 1/ 1 / -1 -2 /14/6 A2 +6 It is suggested that the teacher formulate -2/ 22/t. several grids of the type previously shown +3 j. v .A"2 I to provide sufficient orientation for students +4 1 +9 .+2 and as a prerequisite to attemptingthe Solution grid types that follow. Thestudent must :+251 be thoroughly 'familiar with the multiplica- FIGURE 5 tion, division, and addition of directed numbers, some minor rules for divisibility, and have a talent for observation and deductive reasoning. Of course, the teacher may elect to have students attempt some simpler solutions than those shown here, ill or confine the entitd presentation tomerely determining products. 44OPP . Complete the grid (Fig. 5) and place your final answer in the frame at thehead 'OMA of the arrow. As the students attempt the solution of this and,similar grids, many observations Present to students. may be noted, and the writershave found '2 7 that successful students are most eager to 1/ 44/ 11/1,6 share them. These observations include `? the following: o y +8I/.2:1* 1. No product can exceed 81 or beless 1.6A-2/ 'than +91431 .7'4.9 2. No factor can consist of two digits.

3. If UR is positive or negative, than +I 1 +9 'LR is positive. '6'3/' 1/1.5 both 0 y+5_2_+5] 4. If UR is positive, the signs of .2 factors are alike. +2 +3 +5 5. If UR is negative, the signs of both factors are unlike: 6. If UR is 0, then LR is 0, negative FIGURE 6 re') INTEGERS 115 or positive. An alternate method of presenting and 7. If LR is negative, then-UR is 0. solving grids of our type follows. 8. If LR is even, then either the upper Complete the grid in accordance with or right -hand marginal entry ,for that the mathematical instructions below: cell is even. 9.If LR is 0 or 5 and UR 7i-- 0, then either the upper or right-hand mar&inal 1. A is not positive. entry for that cell is 5 or "5. 2. d = 5. 10. If "carrying" is necessary while 'adding 3.-3 < g < -1. obliquely; a positive sum "carries" a 4.k is not negative. ,positive tens digit, while a negOve . 5.(A)(F) > O. sum "carries" a negative tens digit. 6. /3 7. The reader may enjoy attempting the '7.7 < p < 9. next grid (Fig. 6) while paying particular 8.i > 0. attention to the observatidn'Spreviously 9. IA I + = (C)(D). noted. 10. D A = j. Perhaps some explanation or advice may 11. q4 0. be useful to the teacher who desires to formulate grids of his own. For hest results, the writers recommend that a completed grid be formulated at the beginning. The teacher would then contribute to the difficulty of the solution by removing appropriate factors, products, sums, and par- tial products. Here it is advisable. for the teacher to attempt his own solutionin order to decide whether or not he may have abbreviated the original grid beyond the point of effecting a lomplele solution. FIGURE7- Editorial feedback

MARY HELEN BEAN Bel Air JuniorHigh School, Bel Air, Maryland

Myeighth graders thoroughly enjoyed p. 358). Some of the studentsexamined it working with "Grisly Grids" (May 1969) and found that the inclusion of one addi- after their study of integers. The grids tional number would have resulted in a proved to be a highly motivating type of unique solution. (Note fig. 1.) drill. The circle can be replaced with 1, '8, The students found that one of the grids -7, or '7, in order for the original grid to does not have a unique solution (fig.5, have but one solution. The four possible

+2 + -5 -3EOM6 AIM +3MEM-4 MOM 9 WWII +2BEng MIPAM ISO-6Arl VAME+4 NEMO -4 +3 0 +2

+17

FIGURE 2 FIGURE

.4../.,."±116 -o-

INTEGERS 117 solutions obtained when theadditional Our thanks to the authors William G. number cloe is provided are shown in fig- Mehl and David W. Mehl for making drill ures 2-5. so enjoyable!

-5 +2 -.5 f2 Lv "3 5 +2 -3 CE) -3 + -4. + -I 6 v-3 ,+i, +4 .6 +3 -6 0 ' 2 ,+2, +2 +8 --.', 0 A prririr r +2 p +2 0/+2 41 +1 +I '4 +31+2 0 i -2 -4 -4 0 8 8 2 /+8 *2 i 01/13, 41 6 -2 9 421-4 +i .,,+6 -2 , +9 +8 3 +5 +8 43, +7 i K n X A 'OP ORM !+I 0 -2 0 3 +1, -3 +2 -3 +5 -6."'+I +9 1 4.5 + -4 5 +41-2'+1 /43; +5 +5 -di Effier v+5/ 0- MAI°80" +2 +2 +4 42 +5 + +2 +2 +3 +5 I

1 4-34 +19

! 1

FIGUR), 4 FIGURL 5 Rational Numbers

Aconsiderable portion of the elementary school mathematics curriculum is devoted to the teaching of fractional numbers and their operations. Although whole numbers are extremely important in everyday living, itis the use of rational numbers with their fractional and decimal representations that has made progress possible in any field requiring mevist:nment. Many of the games selected for this section deal with the equival '.nce of names for rational numbers. These include equivalences ofdecimals and percents as well as the usual equivalences with fractions. To createinstructional games for use with operations on fractions, many of the gamessuggested for whole-number operations may be adapted. The first article, by Junge, gives directions for a most useful fraction activity: fraction strips. She suggests several games with these materials that are either based on equivalences or designed to show how fractional parts relate to make a whole. If Ce.riiSie'S "Crazy Fractions" reminds you of Crazy Eights, it should. Students ,get much practice recognizing like denominators and fraction equivalences with this game. "Fracto," by Molinoski, is a game of fraction, decimal, and percent equivalences. It is strongly reminiscent of the universal game called poker. Lazerkk's "Conversion Game" is a variation of Bingo using percent, decimal. and fraction equivalences. Be sure to let the winner of one game be the caller for the next game. Armstrong's "Fradecent" is a card game related to the same equivalences..

4 120 GAMES AND PUZZLES The activity discussed in "An Adventure in Division" helpschildren to understand the concept of repeating decimals and to recognizethat rational numbers expressed in fraction form can also be expressed inrepeating decimal natation. Lots of practice in long division as well as ingraphing is provided in this activity. The article "Introduction to Ratio and Proportion" willgive you some ideas on presenting rate pairs to children and for showing how ratepairs differ from ordinary fractions. The function machine can beused effectively with this-activity. Rode's "Make-a' Whole" and 7.ytkowski's "Game with FractionNum- bers" are useful when working with operations on rationalnumbers. Make -a- Whole gives students an intuitive feeling for many ways to generate aunit region from fractional parts. "A Game with Fraction Numbers"provides much practi6e with computations of the four basic operations.'

.4 $ .4.A... I A 1igame of fractions

CHRLOTTE W. JUNGE

This game may be played by two, three, These partsmaybe used for several or fog children. Each one makes his own games. For example: set of fractional parts as follows: I. Each player empties all the fractional 1. Cut from lightweight tagboard five parts from his envelope into a common rectangular regions 12" x 4".. pile. Each player in turn draws a part until 2. Very carefully mark four of them into all have been drawn. Then each player. fractional parts, and label them as shown begins assembling the parts he has drawn below.Leave one region unmarked as a into rectangular regions, the same shape measure. and size as the uncut card. The first child toassemblethree"wholes" winsthe game.

1 II. Start as before. Set a limit of three to four minutes. The player who in this time has the greatest number of equivalent expressions for V2.. wins. III. This game is something like "Au- , 1 1 1 Mors." Each player in turn quickly draws 2 4 4 two fractional parts from the common pile until he has taken ten. The remaining parts are put aside in a box and not used in the 1 1 1 1 game. The player who first fitsallof the 3 3 6 6 parts he has drawn into a region or regions equivalent to one, wins. Besides the parts he has drawn, each player in turn may ask any other player for the parts he needs, 1 I 1 1 1 1 1 1 1 giving in return an equivalent, thus:3/1 2 6 6 6 6 12121212 for %; 1/ for %; % for1,4, etc. A variation of this game may be played without reference to objective materials. Children often refer to this game as "Make 1 1 1 1 1 1 1 One." One player starts the game, saying 4 8 8 8 8 8 8 "1/2,i/6, make one." The next player adds another fraction which will "make 3. Cut along the lines drawn.Write one," thus: "1/2, 1/4, 1/2," ."1/2, 1/4, %," or your name or initial on the back of each -1/2, Ms, %." This game is played orally, of the fractional parts and place them in an without reference to written solutions, ex- envelope for ease of storing. cept where they are needed for clarification.

EDITORIAL COMMENT. This game should provide some very interesting practice with a variety of fractional-r.umber concepts. Additional variety could be provided by the construction of five circular regions, which are marked and cut into the same sets of fractional parts suggested for the rectangular regions. This would enable the students to view the fractional parts in another spectrum.

AS ( 121 Crazy fractions: An equivalence game

EARNEST CARLISLE

Columbus College, Columbus, Georgia

The game "Crazy Fractions" is useful in in the center of the table facedown.Play helping fifth and .sixth graders learn equiv- starts with the player to the left ofthe alent fractions and the renaming of whole dealer discarding a card of the same suit numbers as fractions. (same denominator) or of equivalent value The game isplayed with a deck of to the top card of the discardpile, or a seventy-two cards. There are cardsfor the whole number card. Whole number cards following sets of fractions: halves, thirds, act as wild cards because player may suit he fourths,fifths,sixths, sevenths, eighths, rename a whole number card to any ninths, tenths and fourteenths. There are chooses. During his turn, a player may discard or also cards forthewhole numbers one, two, does three, and four. draw a card from the deck. A player The game works ideally with four play- not both draw anddiscard unless the ers; however, any reasonablenumber of drawn card is discardable. The obj.ct of of his players may play. The gamebegins with the game is for a player to get rid the dealer giving each player sevencards, cards. follows turning one card faceup (starting thedis- A typical hand may begin as 0/4faceup; card pile), and placing the remain)figcards (refer to fig. 1) : Dealer turns

4 DECK

r 2 I 7 4

CRAZY FRACTIONS

Fig. I Al 122 /

-- player one discards 2/4; player two has of a blocked game, no one receives any no fourths but discards the equivalent frac- points. Otherwise, a player receives twenty- tion1/6;player three discards 5/6; dealer five points for getting rid\of his cards plus, has no sixths or equivalent fraction but the total of the cards remaining in all other draws 2 froin the deck, which he plays and handsone point for each fraction less names sevenths as the suit; player one than 1/2; two points for each fraction follows by discarding 1/7; player two dis- _equal to or greater than 1/2; 'and three cards 4/7; and the play continues until one points for each whole number. Before play player is rid of all of his cards. begins, players should agree on the number' When one player gets rid of all of his of points that will constitute the game. cards, or whet, no one can play, the hand Fifty-andhundred-point games have is over and the points are totaled. In case worked well in the past.

EDITORIAL COMMENT.--The cards for homemade card games such as "Crazy Fractions" can be made more attractive and more durable if you cover the back side of the card with a color- ful pattern ofcontactpaper and then cover the front, or playing, side with clear contact paper. The cards-may be cut from index cards, posterboard, or other lightweight cardboard.

Fracto

MARIE MOLINOSKI

Worth Junior High School, Worth, Illinois

Upperelementary and junior high math- fractions with a denominator of 100. ematics teachers are always searching for 20 25 371/2 40 SO 60 ways of teaching or reviewing the relation- 100 '100 ' 100 ' 100 ' 100 ' 100 ' ships between decimals, percents, and frac- 62 1/2 80 66 2/3 75 83 1/3 tions.Fractois a card game designed to 100 ' 100 ' 100 ' 100 ' 100 ' make the review a little more fun. 87 1/2 88 8/9 ,andand 100 One set consists of thirteen equivalent Materials mosdsd decimals..20, .25, .37 1/2, .40, .50, .60, Fracto isplayed with a deck of fifty-two .62 1/2, .66 2/3, .75, .80, .83 1/3, .87 1/2, cards. The cards can be of any convenient and .88 8/9. size; index cards maybeused. The make- One set consists of thirteen equivalent up of a deck ofFractocards can vary de- percents.20%,25%, 371/2%,40%, pending on the type of practice desired. 50%,60%, 621/2%, 662/3%,75%, Following are some suggetted decimals, 80%,83 1/3%,87 1/2%, and 888/9%. percents, and equivalent fractions: One set consists of thirteen fractions. Numbir of plows 1/5, 1/4, 3/8, 2/5, 1/2, 3/5, 5/8, 2/3, The number of players can be from two 3/4, 4/5, 5/6, 7/8, and 8/9. to six. A class could be divided into groups One set consists of thirteen equivalent of varied size. A ' ki 1231- . Rules of tho game have been recorded, a new hand is dealt Shuffle the fifty-two cards and deal five and the game continues until a player cards to each player one at a time. Players reaches 300 points. sort their cards, trying to keep equivalent cards together. Starting with the player to Point values of cards in hands. the left of the dealer, each player is given Two of a kind or one pair 20 points achance todiscard from one to three cards Two pairs 40 points and ask for new ones. A player may choose 50 points to keep his hand as it was dealt. After Three of a kind every player has had an opportunity to Straight 60 points draw new cards, the player to the left of Example: 5/8, 2/3, 3/4, 4/5, 5/6 the dealer lays down his hand with the Four of a kind 70 points other players following in order. The player Full house 80 points with the best hand wins. When the points Example: 1/2, .50, 50%, 1/4, .25

The conversiongame

BETH ELLEN iLAZERICK The Dalton School, New York, New York

Just how do you convincea sixth or used cards measuring five inches by five seventh grader that he should sou able inches, but three-by-three or four-by-four quickly to recall fraction-decimal-percent cards could be used with slight alterations. equivalents? After students have used the My key list contained 28 sets of basic ....,,,pt of (for example) facts and their equivalents. This number can vary according to the level -of the = .50 = 50% class. Thelistincludedhalves,thirds, fourths, fifths, sixths, eighths, tenths, elev- in word problems and computations, they enths,twentieths,andone-hundredths. frequently forget their facts; and to help Fractions in lowest terms, nonreduced frac- them retain mastery of the. various equiv: tions, decimals (repeating and nonrcpeat- ing), and percentages comprised the list. alencies 1 fashionedthe"Conversion Game." This game also reinforces the con- A portion of it is reproduced in figure 1. cept that a "number" is represented by The students had learned that.6and .6 many "numerals." ("point 6 repeating") are not the same

Playing cards fashioei-z.: :ike bingo cards and that .6, 2, ,1;;, and 662percent are were made from brightly colored oaktag. I all numerals representing the same numb7t.

.4c 124 .$ I.. // RATIONAL NUMBERS 125

KEY LIST 1

+ 100% I 5 50 50% i 33; .3 33i% 333%

20 A 05 5V. - ,_--- _-_-.------L------

Fig. 1

These and other facts were placed ran- and call out "Conversion!" The caller then domly on the cards so that each card had checks the winning line by looking for an equivalent of 24 of the28 facts and equivalents of all the covered numerals as each card had about eight fractions, eight listed on the chalkboard. decimals, and eight percentages. Samples The game is over if the student is cor- of the cards are shown in figures 2 and3. rect. Boards can then be cleared and the game repeated two or three times without, To play the game, the caller (teacher or in most cases, any loss 'of interest. -student) simply chooses randomly from The advantage of putting an equivalent the key list any equivalentor he can of nearly every key number on each card select from duplicates of the list that are is that students then know they have to placed in a box, as in bingo. He might look carefully after each call. Better stu- select 1/3 or 10%00. He then calls the num- dents simply find more answers on their ber and writes it on a chalkboard while the cards, while slower students can always players search for any equivalent (i.e.,.3 identify halves, thirds, and other simple or 331,4 percent) on their cards.When a equivalents. player locates. an equivalent on his card, Since most students enjoy games, the he covers it with a simple marker. After Conversion Game proves to be a fun way many pulls from the key-numbersbox or: to review important facts long after the selections from the list, someone will com- "real" mathematics of these concepts have plete a horizontal, vertical, or diagonal line -been stored away!

The Conversion Game The Conversion Game

8i% 90% .07 375 i 46 60% 15% 33i% 1

I 4 100% 5 83i 66i% i TIE H If A FREE 70% 1% 375% FREE 5% 20% i 4-

tg if 87 2% 3 mti. ,A 05 4 1.0

7 90 * 1 75 30% w ::: & 80% k

Fig. 2 Fig. 3 1

"FradeCent"a game using equivalent fractions, decimals, and percents

CHARLESARMSTRONG

Claypit Hill School Wayland, Massachusetts

Fradecent is a card game that is useful in begins by drawing the top card from the helping sixth graders learn equivalent frac- deck. Each player in turn tries to make a tions, decimals, and percents as well as aid- scoring set. There are two possible scoring ing them in adding and subtracting fractions. sets: (1) a set of three cards consisting of The word fradecent contains parts of the an equivalent fraction, a decimal, and a words fraction, decimal, and percent. percent (one of each ), and (2) a set of The game is played with a deck of eighty- three cards consisting of three equivalent one cards. The following fractions are used: fractions. When a player is able to make 1 '2, 2,4, 4 8, 5 10,1 /4, 28, 4/16, 3 4.6 '8, a scoring set, he places the cards face up 12/16,115. 2/10,3/ 15,2/5, 4/10,6'15. in front of himself. Whether or not he is 4/5, 8/10, 12/15, 18, 38, 5/8, 7'8,1110, able to make a scoring set, each player 3,10, 7, 10, and 9,10. Cards with the cor- must discard (place a card face up beside responding equivalent decimals and per- the deck of cards in the center of the table) cents are used with these fraction cards, to complete his turn. A player may not go, Play begins with the dealer giving each outwithoutdiscarding.Playcontinues player seven cards and placing the remain- clockwise. ing cards in the center of the table face Additional scoring is possible by adding down, The player to the left of the dealer on to existing scoring sets. This may be c, ) 126 RATIONAL NUMBERS 127

.375 1. .6. D .1 E __I _.. _ _I Discardsr, 56%- C i 1 §H K -nio I I -I- i , .25 8; ./ 1

IF-R _.....I A \ i _J '1 D EC Li

E , ., iitt N ,1- Top 3 cards 1-4__ T1 i Fig.I. Drawing from the discards

done by any player during his turn. He may three cards, he must use the .25 card in a build on his own sets or on the sets of his scoring set and add the 3/15 and 4, 8 cards opponents. A player builds on his oppo- to his hand. nents' sets by placing the scoring card in 4 front of himself and declaring the set on When a player gets r1/41 of all his cards, which he is building. the hand is over and the points are added. During his turn, a player may draw a The face values of the cards are counted card from the deck in the center of the for points. The total of the cards remaining table.or any number of cards from the pile in a player's hand is deducted from the of discards. If he draws from the discards. total of his scoring cards. The winner of he must use the last card he takes in a the game is the player who gets five or ten scoring set during that turn and keep all pointsfirst.Beforeplaystarts,players the other discards in his hand. In figure 1, should agree on whether they will play a for example, if the player takes the top five-point game or a ten-point game.

EDITORIAL. COMMENT. This game should provide some meaningful practice for students, and once the rules are learned, they should be able to play independently. As a possible variation of the game, you might suggest that the goal is to build a set of four cards equal to one (for example: 1/4.1' Is 30%..375),

Af-, 4, .1io' C. An adventure indivision 4

L O I S STEPHENS Camarillo, California

Lois Stephens is principal of McKinna School in Oxnard, California. She recently completed her doctoral dissertationand is now making a study of mathematics in the primary grades.

6 The adventure upon which you are about 0.0909... it =0.5454... to embark requires some computation, so - before reading any further, get a pencil «- 0.1818... 7= 0.4363... and some scratch paper 11 11 Ready? Here we go! 3- 0.2727... 8= 0.7272... iT 11 Until a few years ago I had viewed di- 9 vision as a process to be used only when 4= 0.3636... = 0.8181... 11 ii needed to solve a practicalproblem. What 5= 0.4545... 10= 0.9090... changed my attitude was an investigation it ri of the patterns to be found in the repetends' of the decimal equivalents of common- Again, at a glance there appear tobe fractions having prinic numbers as denom- ten patterns, but closerinspection reveals inators. Try these operations with me: that one eleventh and ten eleventhsfollow the same pattern, two and nineelevenths , For a start, find the equivalents of the the same, three and eight, and so on,for a sevenths, carrying each division problem total of five different patterns of twodigits to the . umber of places necessary to re- each. peat the cattern twice. Let's try one more easy seriesthe thir- Finished? teenthi. You know now what Co do, so You should have these answers: perform the operations and check your re- sulti with the answers given here: t 4 = 0.142857142857... = 0.571428571428... 7 7 7 ij = 0.076923076923... = 0.538461538461... 2 5 13 = 0.285714285714... = 0.714285714285... 7 7 8 2= 0.153846153846...rj = 0.615384615384... 6 3= 0.428571428571.... ;0.857142857142... 7 7 9 = 0.230769230769... = 0.692307692307... 13 13 Although at a glance these appear to 4 10 be differedt patterns,itis easily shown ri = 0.307692307692... 5 =0.769230769230... that the same sequence of six digits ap- 11 equivalents, 5= 0.384615384615... 5 =0.846153846153... pears in each of the decimal 13 and there is only one pattern for the sev- 6 L2= 0.923076923076... enths, this one made up of six digits. 13 = 0.461538461538...13 Moving on to the next prime number, eleven, find the decimal values of the ten This time, did you find the two patterns proper fractions, then check your answers of six digits each? have with those given below. Now try putting all the results you

.1..: :ti--- 128 RATI9NAL NUMBERS 129 obtained so far into a chart with four And using the saw: kind of desig:ation, columns. Head the first column "N" for the thirteenths produced the graph shown the prime number which is the denomi- in Figure 3. nator. The second column will be "D" for Numerators the number of digits in pattern. Head- orsprosomusan nowarnionama 181MUNB/c1111rawn mrawmpla...hief ing the third column will be "P" for the prisamminum imminaepumumm 4,YIP 111B/2=11111111 number of different patterns. You can de- nmmismoon1110111111rIm&iimonwasZ11111111711 termine the heading for the fourth column Baur= -rammicam a ouseiraimamoss by discovering the relationship among the entries made in each row. Your chart FIGURE 3 ,should look like the one in Figure 1. Going back to the chart you made, did you try) the product of D and P and find N D P ? it to be N 1? Test this as a pOssible heading for the fourth column by'calculat- 7 6 1 , ing The decimal equivalents of the seventy- 11 2 -5 thirds: Before you begin calculations, pre-

13 6 2 dict the number of digits and the number of patterns by making a factor tree for FIGURE 1 72, N 1 in this case. (Sec Fig. 4.) If you are having difficulty in heading the last column, go on to 17, 19, and other prime numbers to obtain more entries for your chart. (More about this later in the 2 36 article.)

Now let's go down another path on this 2/ 2 18 adventurous journey. Looking back to the decimal values of the elevenths, designate the different patterns by letters, thus: 2 2 2 s 9

2 2 2 3 3

2 6 10 FIGURE 4 if A This gives many possibilities: 7 3 c N = 2, D = 36; D N = 4, D = 18; 8, D = 9; 4 N= 12, D= 6; D-8 C N = 24, D= 3; These patterns can then be graphed in the and the opposites of all these! manner shown in Figure 2. Write your prediction at the top of your paper, and start dividing. Numerators Finished? Now wasn't that fun? And was your prediction correct? You may ask, "What is the value of this so-called adventure?" Not only have I had many pleasant surprises as I have explored these repetends myself, but I have found many junior high school students FIGURE 2 who would spend hours on problems such

02, !30 GAMES AND PUZZLES as 1/97 =1, even though theyhad pre- Cliss became self-appointed "checkers," viously avoidecVall practict. in division. and ali were participating in camouflaged (This problem is not rsimmodefl'unless drill as a.sideline while the regular class- you are ready for a laKf practice indivi- room assignment was being pursued. sion, which of course also means practice If you have followed me down this path in multiplication and sulliaction!) so far, do not think you haveeached the One successful method of using' this end, for I predict you will find yourself computational game was to allow one stu- wondering such things as, "How do you dent to work at the chalkboard while the suppose 997 behaves as adenominator?" other students looked on. In this way,.with- ....And do try 101 next! out any teacher direction, the rest ofthe Have fun! AprioniAL CommENT. If you are loOking for something a little different after the work in division suggested by Stephens, have your students do the following puzzle bycoloring in each region that is labeled with a symbol for one-half. This puzzle would beespecially good just before Halloween. Introductionto ratio and proportion

ROLAND L. BROUSSEAU North Attleboro High School, North Attleboro, Massachusetts T..110MAS A. BROWN Merrill Junior High School, Des Moines, Iowa PETER J. JOHNSON Jim Bridger Junior High School, North Las Vegas, Nevada

The objective of this,lesson is to introduce The girl in the example cited drew three fifth- and sixth-grade students to ratio and objects, while the boy drew four. There- proportion through a discovery situation. after, and with each successive drawing, This approach would also be appli,:able the total number of objects was recorded for junior and senior high general mathe- in the table. (If by chance the same num- matics students. This technique should al- ber of objects are drawn by both children, , low students to find patterns that will en- these numbers should be recorded, but able them to verbalize algorithms for the another drawing is made to get different solution of proportions. The following is numbers. After three or .four drawings, from an actual demonstration with fifth children should be able to supply new graders. values without actually drawing objects As a starting point, in developing a feel- from the container. If the children have ing for ratio, the children took objects from trouble supplying the values, they should a container. In order to keep the number return, tothecontainerforadditional property small, large objects were used. drawings.) A comparison of the number of objects After the first table was completed, a drawn by each of two children was made. second table was constructed without hav- A table was constructed to keep a rec- ing the children draw objects from the con- ord of the number of objects drawn. tainer. At this point the word "rate" was (See Example 1.) Each time the children introduced. The children were asked to came to the container, they were told to think of the table as a record of the rate draw the same number of objects they at which these objects were selected from had drawn on their first turn. The objects the container (Example 2). The girl drew drawn were displayed where they cowl be three to the boy's fiveexpressed as a rate seen by the entire group. of 3 to 5.

EXAMPLE I EAAMPLE 2 Girl Boy Girl Boy (G, B) 3 4 3 5 (3, 5) 6 8 6 10 (6, 10) 9 12 12 20 (12, 20) 15 25 (15,'25)

15 -zti

131 132 GAMES AND PUZZLES

Children were then asked to fill in the EXAMPLE4 table. Once again it should be stressed that (2, ) = (6, 9) (12, 16) (, 4) the table can be verified by having the childrenactuallydraw the number of The symbol of replacement should be objects specified:: used in all positions. At this stage, the notation of ordered A game based on equivalent pairs was pits was introduced. The tables were con- devised to help the children see the re- structed vertically to make this transition lationship between the product of the easy for the children. The valueslisted in means and extremes. The gamerevolves the second table were written using ordered around an expression like this: (3; P ) = pairs which were referred to as "rate pairs." (0, 4). The rules of the game are asfol- Once the concept of rate pairs had been lows: The girls pick a pair of numbers established, the next step was to show that would yield equivalent rate pairs. They equivalence between rate pairs. At this then give one value and let the boys find point, a child was asked to explain to his the other. In a similar fashion the boys classmates how the objects were drawn could pick a pair of replacement values. from the container and how the L.-4es were A table like the one below couldbe con- constructed. Children were asked to volun- structed to record replacement values. teer other rate pairs thatshowed the same

1 6 2 rate of drawing. (See Example3.) L ) , 121 1 3 4 1 12 4 3 2 6 EXAMPLE 3 E this game (3,5) =(9,15) Other good examples for 13,5) =(6,10) would be (0, 6) (5, ,n,) and (8, ) = 2). 'Hopefully, the children will soon realize that the product of the replacementslisted It is necessary or the children to see in the table is the same as the productof that equivalence e fists between other rate the numbers in the examples. Thismight pairs .from the se table. The children set the stage for a verbalization ofthe rule. were asked td fiqd a ratepair that was Itis possible to see that replacement equivalent to (21ji 35), but they were told could also include fractions. A last example not to use (3, 5): A secondrate-pair was might be one such as this:(4, 1/i ) = introduced and children supplied rate pairs. (A, 2). If the children have discovered The term "equivalent rate pairs" was the rule of the game, examples such asthe used to express the same rate. Thepair one above bilould not cause anyfrustration, (3, 5) expresses the same rate as (6,10) but merely make the game more interesting. even though different numbers areused. In conclusion, the brief development of As further exercises on findingequiva- proportion as outlined above would, with lent rate pairs, examples can be devised extension, lerd itself to problems involv- which call for replacements suppliedby ing percent, scale drawing, conversions, the students. similar figures, area, and volume. Makeawhole=pagame using simple fractions

JOANN RODE University of West Florida. Pensacola, Florida

The game called "Make a Whole" helps star and the /following fractions:1/2,1/2, develop the concept of fractional num- IA, IA , a n d ye. bers by using concrete examples; it gives reinforcing experience with the use of frac-. Rules of play tions and ttrfr equivalents. It is suggested The object of the game is to use the for use at third-grade level and above, de- fractional sections to construct a multi- pending on the achievement of the stu- colored disk, the "Whole," in the black dents. mat frame. The Whole may be constructed The material required for tin game is with any of the differently colored sections pictured in figure 1 and described more so long as they fit together to be equivalent completely in a later section. The sections to a whole. are made in four different colors, equally Two to eight children may play the distributed; that is, of 8 sections represent- game in individual competition (in which ing V2, 2 could be red, 2 blue, 2 green, case the color of the area for the Whole and 2 yellow. The area for the "Whole" makes no difference), or they may play is also prepared in four different colors. on color teams as determined by the color The faces of the die are marked with a of that area. --0,,.

EDITORIAL COMMENT.Thegame 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 ofpaper with fraction names. A student draws a piece from the pile andmarks on his number line a segment representing that fraction.

l_aninated number line

Grease pend

Draw pile

133 134 GAMES AND PUZZLES Nfulticolorrd Sections the (Black name)

(Colored Area for "Whole")

Fig. I

identifica- The fractional sections areplaced in a advanced students, the numeral box, from which they are to beselected tion may be omitted from thesections. the by the children as they rollthe die, in The materials may also be used by in clockwise order. teacher to show equivalent fractions classroom demonstrations and by theindi- To determine who goes first, eachchild rolls the die seen in figure1. The child vidual student in working with equivalences rolling the star or the fraction withthe at his desk. - greatest value goes first. At each turn the child mustdecide Construction of game materials whether he can use one of the sections A complete list oftiieces follows, the represented by the fraction showing on color of the mat disks andsections being top of the die. equally divided amongwhatever four col- If the player cannot use the section,he ors are used. passes. Mats, 8 If he decides to take the sectionand can 1!,sections,8 his mat. 'use it, he places the section in 'sections, 12 Whole (If the section taken completes the 1 , 4/sections, 16 and for that player, then the game is over sections, 24 the that player has won. If it does not, I/sections, 32 roll of the die passes to the textplayer.) Die,I If he decides to take the sectionand , black mat cannot use it, he loses his nextturn. The material used for the sections is inex- If he passes up a section he couldhave frame and the fractional in differ- used, he loses his next turn. pensive tagboard (poster board) Wholes constructed If the starisrolled, the player may ent colors. Because the by the students are to be multicoloredand choose any section he can use. in The winner is the first to construct a because the fractional sections come colors, the multicolored Whole. equal numbers of the different in various colors give no clues to value. The game may be played cutting 8 other ways, according to rulesdesigned by The mats were made first by tagboard. the teacher. For example, achild ,might eight-inch squares from black with no Disks six inchesindiameter were cut be asked to construct five Wholes and the more than one of akind of fractional sec- from the center of these squares, frames were then glued toother eight- tion(1'2,1 a, etc.) to go into each.For inch squares, 2 of each of the four colors of the children, the sections may or may chosen. not be marked with the fractions to which Five disks in each of the four colors they correspond, as seen in figure 1. were then constructed, with a diameterof The die was made from a cube cut at a six inches. A compass 'was used for ac- lumber yard so that each edge had a curacy. The disks were thendivided, with measurement of three-fourths of an inch. the aid of a protractor, into sections cor- Thefivefractions were painted, one to a responding to IA, 1/2,1/4,1,;, and 1/8. The face on each of five faces of the cube. central-angle measurements for thefive A star was placed on the sixth face. The sections are 180", 120', 90", 60' ,and cube was sprayed with a clear varnish to 45°, respectively. Depending on the level prevent smudging.

A game with fraction numbers RICHARD THOMAS ZYTKOWSK 1 University Heights, Ohio

The game described below is a unique r-11F471 way to provide practice in the addition, LI subtraction, multiplication, and division of fractions. The, game can also be used to IX1 teach ratios as well as the sequence of L 5 numbers. FIGURE 2 Use a 2-by-2 ft. piece of oaktag or card- +'/6;x1/7; x v.); 11/2;x23.6; board to make a playing board. Divide it +3;%; +4%; x5%; 6%; +7%; and into spaces as shown below, then write the x 8%. fractions as figure 1 shows. In figure 2, examples are shown of how the front side of the small pieces of tag- 1 2-,..1"5IL- A[Burt2 46 10 1,12 board would be labeled. 2 4 6 8 10412'1 On the back of each small piece of tag- 3 6 9 12 15 18 board wriie the answers for the problem 1,1 1 C4,11. -2 8 12 16 20 142la4 that is on the front. There will only be six 4 8 12 16 20' 24 possible answers, since there are only six D5 10 15 20 251 30 rows: A, B, C, D, E, F; the rest of the E 10 15 2025 12 18 2430 ; 36 ,fractions in-therow-are-equivalent to the * 6 24 1130 F Win- 12 18 first fraction in that row. Figure 3 shows ner 14 21 287 35 how the back side of 1/2, +1/2, xy4, +1/4 FIGURE 1 would look respectively. On some small pieces of tagboard 31/2 Two to six students can easily play this in. by 8 in. write the following symbols game, or the class may be divided into and numerals:1/2; +1/3; x IA; + groups.

135 136 GAMES AND PUZZLES

square on the large oaktagboard (1/2 ), the piayer must multiply 1/2 x 1/4 . IA 2i A 0 A 1- A* Player A's computations: 13* B1 Bi- B3} ix1=1 C 1 i C i C 3t Player B checks the back of the card being used to see if the answer is correct. D I i, D,4 D5 D The answer is correct, so player A moves E El-k EA E 4i to the next space, It is now player B's Fa F 1 4 r a F44 turn. Player B misses his problem. There- fore, he Must remain at the same place FIGURE 3 until it is his turn again. The rules of the game are as follows: If there are no more than two players, it will be player A's turn again. Only this Each player will begin with the first numeral time instead of being at1/29' he has now fraction 1/2. on the large tagboard. which is the moved to He draws another card; the He will. without looking, pick a small piece of tagboard with a symbol and fraction on it. card states 1/2. Player A must subtract He wilt then perform the mathematical opera- 1/2 from %. tion in order to arrive at the correct answer. Player A's computations: If the player gets the answer correct, he moves one space. If he is wrong, he remains where he : I I - 0. is. (The answer can be verified by checking the Player B checks his' answer by looking at back of the card used by the player in performing the mathematical operation.) the back side of 1/2 to find the answer. The firststudent to reach "winner"is the He secs that the answer is correct, soplayer "mathematical champ" or "whiz kid." Amoves to the next space, :M. Itis now All answers must be reduced to their lowest player B's turn again. Player B should terms. Improper fractions must be changed to mixed numbers. always get his turn last, since player A Each playeris given only one chance per was first. card, then itis the next player's turn. If the game is used by the teacher in The fraction on the small piece of tagboard row competition a few times,the pupils or cardboard should always he considered to be- represent the second numeral in the problem: should be led to see the relationship thus, itwill either be the second addend, the tween numerals. The studentshould be minuend, the multiplier, or the divisor. asked what numeral would followthe this can be Example: sequence in each row; then To begin the game, player A, without expanded to lead the pupils to see the looking. picks a card. The card states x relationship between the numerator and the denominator of each fraction. .Since all players begin with the first Number Theory and Patterns

Most contemporary elementary and middle ool programs contain topics from number theoryfactors, primes. multi les,divisibility, patterns, sequences, and so on. There are two major reaso sfor including these topics in the curriculum: 1. Number theory topics are convenient /tools usedin conjunction with fractional number computations. 2. There is an intrinsic fascination in therelatedness of number proper- tiesthis natural intrigue can be used to provide manyopportunities for practice with whole-number computations. Most of the articles in this section reflect the sentimentsexpressed by Kapp and Hamada in the lead article, "Fun Can BeMathematics""Every mathematics classroom can be a laboratory where studentsexperiment with numerical ideas.... The games :...ggested ... can stimulatethe students' mathematical thinking by use of number sequences andpatterns." The first article gives several suggestions for turning sequenceinvestiga- tions into games. There is an especially good game fordeveloping the concept of greatest common factor. This game is easilyadapted to the concept of lowest common multiple. Allen's description of "Bang, Buzz, Buzz-Bang, andPrime" is guaranteed to give a vigorous presentation ofmultiples and primes. Be prepared for a high level of excitement when you play this game. The ever-popular Bingo game is the basis forHoldan's game of "Prime." This is a game of disguised practice in the recognition-ofprime and composite- numbers. Most students enjoy playing Bingo games,especially when the reward for winning a game is being the next caller. If you are interested in a game requiring the developmentof a strategy to win rather than just chance, then try Harkin andMartin's "Factor Game." The game rules are presented in the form of a computerpriiitouthowever, you don't need a computer to playthe game! If you do have programming ca- pabilities, then by all means make the computetste opponent.

137 138 GAMES AND PUZZLES

You will find additional strategy-based games Fn Trotter's "Five 'Non- trivial' Number Games." The secrcts behind severzl very clever and useful games are exposed. One of the secrets involves the Fibonacci sequence. At first glance, Sawada's article on "Magic Squares" may seem somewhat complex for elementary or middle school students. However, the author's purpose is not to encourage the teaching of abstract conceptsbut rather to "suggest the strategy of taking interesting ideas, such as magic squares, and extending them in ways that lead to the discovery of other mathematical structures,'-all the while providing interesting practice of fundamentals as riginally intended." In addition, the article gives some very good illustra- tio s of activity cards, using magic squares as the subject. Th final two articles provide a potpourri of tricks and games designed as "Intert Getters" and "Just for Fun." Such diversions provide the perfect fillers for those in-between times.

-I( Fun can be mathematics

AUDREY KOPPand ROBERT HAMADA Edison Model Mathematics Demonstration Center, Los Angeles City Sc.':e-ds, California

Every mathematics classroom can be a time, usually everyone knows the secret laboratory where students experiment with and you may have a perfect response on numerical ideas. Two-way communication the fourth try. between teacher and class by means of The game can be used at the end of a games can foster ap atmosphere of eager class period in order,to set the groundwork participation in mathematical activities. The for the lessons to come on patterns and games suggested below have characteristics sequence. If only a few students discover that can stimulate the student's mathemat- the secret one day, try again the next. ical thinking by the use of number ideas 2. Number sequences may be introduced and number sequences.and patterns. Some by the teacher presenting the first numbers of the exercises call for use of paper and on the chalkboard: pencil' by students and either the chalk- 1, 4, 7, --, board or the overhead projector for the The class discovers more numbers in the teacher to show collection of data. Often sequence each child may be asked for an oral re- i, 4, 7, 10, 13, 16, 19, sponse, thus allowing all to participate, as well as permitting the teacher to check if Then the teacher asks the class to verbalize each student understands the rules of the the rule governing the sequence. If prac- game. tical, the class 'should be asked to write the rule in some form depending on the 1. An easy way to establish the notion ability of the class: Add 3 to the previous of patterns and sequences is to first ask number:I + (3.0), 1 + (3.1),1 + the class to count by five. Tell the students (3.2), 1 + [3 (n 1)) where n is the to listen for a secret as each child in order nth term in the sequence. Here is another gives an answer. Then again count by fic example: but use 1, 2, 3, or 4 as the initial number. For--example; count: 37 8, --13;--I8;--23. Again suggest .that students listen for the After the class suggests the correct num- secret. Try a third time. The teacher can bers in sequence: judge if students know the secret by the 1, 4, 9, 16, 25, 36, promptness of their responses. If they have itis time to formulate rules again. Some not caught on by the third round, the classes may come up with the rule of teacher may repeat each answer, enabling squares; others may suggest adding the odd students to hear the terminal digit. By this numbers in sequence. Teachers may find

139 I /.1:;(6 H

140 GAMES AND PUZZLES

sequences or make up some that are com- number." When Ernest was1, he an- mensurate with 'class levels. The patterns nounced that he would stand for every should not be difficult at first so that stu- question that had to do with factors. He dents can easily be acquainted with the learnedthathe was wrong when the idea of formulating a rule. After a few teacher asked all the prime factors to stand. sequences the students can participate by Questions are easily found for each direc- offering sequences themselves. One stu- tion. After the even numbers are stand- dent gives a series of numbers which the ing, for instance, one may ask 8 why she teacher records on the board. The student is standing. If she doesn't know, somebody calls on other members of the class to will probably state that it's because 8 con- complete the sequence and can either sup- tains a factor of 2. The question of why ply the rule himself or ask another student 13 should stand in the prime numbers was to do so. Rules and sequences maybe answered by Mary as "13 has only 2 dif- recorded by all students. ferent factors." Jerome, number 17, was Occasionally not enough data is supplied tired of sitting so he suggested "the factors to predict just one pattern. of 34 stand." Here is another example of a series of statements. 1, 2, 4, , , . - As many possibilities as can be found Factors of 12: should be examined along with their rules. Students assigned to numbers 1, 2, 3, 4, 6, and 12 should stand. I, 2, 4, 8, 16, 32. - 1, 2, 4, 7, 11, 16, - Factors of 18: 3. Student Participation Game Students assigned to numbers 1, 2, 3, 6, Each child in the room is assigned a 9, and 18 should stand. number in sequence. Students may count Common factors of 12 and 18: off in order to make sure each knows his Only those students who stood up both own number and those of the others around times previously should stand. them. The teacher then asks students to stand according to a particular rule. Greatest common factor of 12 and 18: The relationship between ordinal and The student with the largest number cardinal numbers may be illustrated by remains standing. having every third student stand, or asking Another drill on fundamentals would have each studentafter the tenth student to the teacher make a statement such as: stand. The game extends to practice with 6. factors and primes. After the teacher makes 4 plus 3, multiplied by 2, minus 2, divided by a statement, the classmembers respond by Obviously, the student assigned to the num- standing, making it easy to check answers ber 2 should stand up. and locate missing ones. Sample directions: Instead of having pupils stand, each may be given a 3 x 5 card withhis num- The even numbers Multiples of three ber written on both sides in bold color. Factors of twelve Cards may be held up for answers. After Numbers with only two factors ______14 given out Theprime numbers a -while-the numbers may randomly, which may increaset\he difficulty Eventually, students will be able to make of the game. up appropriate directions themselves. Stu- dents often express ideas as they play this 4. Ordered pairslend themselvesto game. After a series of statements regard- games wherein students select andverbalize ing factors, Carl asked, "Why do I have to rules. Visual reproduction of a function standallthe time?" Dave replied im- machine is often helpful. Basic parts are mediately, "Because 1is a factor of every Input, Output, and Function Rule.' Many 4 I A) rules can be used, depending again on the Students enjoy working out their own rules level and needs of the class. Here are two and trying them on the other members of sample sets of number pairs: the -class. Thus the notion of function can (2, A) (3, 6) (8, 16) (12, -) (-, 100) be introduced at even a very elementary (4, 9) (5. 11) (6, 13) (9. 19) (10, --) (-, 51) level. .

EDITORIAL COMMENT. The student-participation game described above may be adapted for use with lowest common multiples (LCM). To find the LCM of 6 and4, give directions such as the following: "Multiples of 6 stand:" 6, 12, 18, 24, 30, .. . "Now sit down." "Multiples of 4 stand:" 4, 8, 12, 16, 20, 24, 28, ... "Only those of you who stood up both times remain standing:" The student with the lowest number remains standing. This will be the LCM of 6 and 4.

Bang, buzz, buzz-bang,and prime

ERNEST E. ALLEN Southern Colorado State College, Pueblo, Colorado

Games often provide an excellent tech- "Bang" is by far the easiest version. nique to stimulate motivation and main- Here are the rules: taininterest. This article discusses four versions of a simple counting game. 1. Seat a sr-all number of students, say less than ten, in a circle. Though the game is easily played by primary youngsters, itis also applicable to 2. Have a student begin the counting in a older children and adults. clockwise manner.

141 3. When it is a student's turn to say "5" diagram indicates the correct moves up to or any multiple of five, such as 10, 15, thirty. _.....- 20,... ,he says "bang"instead. -- 4. At this point the 'counting direction re- (I) (2) verses, beginningina counter-clock- Buzz Buzz (16) wise manner, so. that the person who (18) said "4" now says "6," etc. 5, The game continues inthis manner. When a student (3) a) utters the wrong number name Buzz b)forgets to say "bang" (13) Bang c)doesn't say anythinglas when count- Bang Students . (19) ing reverses direction), Buzz the game starts over with this peison saying "1." Bang Bang (4) "Buzz" is played much like Bang but (24) (9) (6) with the following exception: (26) (11) (8) The student says "buzz', on multiples of Buzz (23) (12) Buzz Bang 7 and numbers which have names that in- (29) (22) clude the numeral 7, such as17, 27, Bang 37, .,.. "Buzz-Bang" isa combination of the "Prime" is played exactly like Buzz ex- above two games.Itrequires a better cept that the student says "prime" instead knowledge of our number/numeral system pf the particular name for the prime num- and more concentration. The following ber.

Prime:adrill in the recognition of and composite numbers

GREGORY H O L D A N Indiana University of Pennsylvania

Drill bas always been of fundamental tributions become functional in later learn- importance in mathematics for mastering ing experiences. Whenever drill is disguised and reinforcing the basic concepts. To dis- in some other not-so-obvious form, such courage the development of unfavorable as a game, it tends to be more fruitful. attitudes towards mathematics, a drill ex- "Prime," as I have devised it, is basically ercise should not becomeaboring, rote a disguised drill in the recognition of prime exercise.Instead,drillshould be made and composite numbers lessthan100. meaningful for the student so that its con- The game sheets, rule slips, and rules for NUMBER THEORY AND PATTERNS 143

playing "Prime" are very similar to those (In items 21 and 22. you may select the for "Bingo." As you read the rules for number to take the place of the blank.) constructing and playing "Prime," you will 1.A prime number probably find that the game is quite versa- The largest prime number tile; it can be readily adapted, in terms of subject matter and the methods used in 3.The smallest prime number constructing and playing it, to practically 4.A composite 'number any situation. 5.The largest composite number 4 6.The smallest composite number t.. Coristruction of the game sheets 7.A twin prime On a ditto master, construct a five-inch 8.A composite number between a pair square; subdivide it into twenty-five oneof twin primes inch ,squares. Above the five columns of 9.A single-digit prime number squares you have constructed, print, the 10.A two-digit prime number letters P-R-I-M-E, one letter per column. II.A two-digit composite number From this master ditto, duplicate the num- P.A two-digit number ifeitherof its need. ber of game sheets you will digits is a prime number Seleet\sets of twenty-five numbers be- 13.A two-digit number, the' sum of whose tween1 and 99 from a table of random numbers. Place one number in each square digits is a compolite number on each game sheet. (When I constructed 14.A two-digit number, the sum of whose the game sheets, I used a black ditto master digits is a prime.number and, on the cooks, wrote in the numbers 15.A two-digit numb- r with a prime number in the ones place ....with a red fine-tipped felt pen.) Your game sheetS should look sonic - A two-digit number with a composite thing like that shown Figure 1. number in the ones place A two-digit number with a prime number in the tens piaci: . A two-digit number with a composite PR I M E number in thz tens place 47 13 547738 A two-digit composite number with both of its digits prime numbers 7238 16 II 5 20.A prime number less than 'I1 A prime number greater than 28469 7623 , 12.An even prime number 13,An odd prime number 2762 61 9941 Your rule slips should look something 3097542039 like this: r- M:atwo-digit prime number J I

Playing the game f1;UKI. I Pass out the game sheets to the students. Spread out the rule slipsface downin Constructing the rules frontof you. Randomly choose arule On slips of paper. write or type an ap- and read italoud to the students. If a propriate selection of the following state- student has a number on his game sheet, ments foreachof the letters P-R-I-M-E: tinder the appropriate letter P, R, I. M, or 144 GAMES AND PUZYI FS

E, which has the property defined by the rule read, he is to put an X in that square. (If you require that only a pencil can ,be used for marking the game sheets, and that the Xs are to be marked lightly, the game sheets' can easily be erased and reused.) For checking purposes, have the students write the number of the rule in the lower right-hand corner of the square in which he places an X. For instanc4 the first rule read would be considered number one; the second rul:. read %you'd be--cilsidered number two, and so on. As each rule is read, place it-to the side. Keep these rules in the order that they are read! Make sure that the students understand that only one square can be crossed out per rule, and that after a square is crossed FIGURE 2 out, the number in it cannot be considered for any follOwing rules. Checking Continue selectingrulesand reading Have the winner read aloud thenumbers' them aloud to the students until someonein the row, column, or diagonal, along has crossed out all the blocks in a single with the rule that permitted him to cross column, row,, or diagonal. .' out each of those numbers. Recordthese Possible winning game sheets might'look rules and numbers on the chalkboard and like one of those pictured in Figure 2. then have the class help you to check the The student signifies that he feels he is validity of the winner's card. a winner by saying aloud theword "Prime." Anyone fora gaine of "Prime"?

EDITORIAl. COMM! N I. To help students disCover which whole numbers are prime and which are omposite, you may develop a game using the pegboard. Show' ;:l7.:.!-..1.ren that price numbers have peg arrays that are single rows or single columns. Composite numbers have other kinds of arrays. 7

L. 5 6 Prime Composite

studer.u.talboard and a supply of pegs. Call out a number.The first team (or ,,,,,------dent) to deterininFgether the number isprime or composite using an array ,cores a point. Play the game to a fixed number of points.' The factor game

J.B. HARKIN and D. S. MARTIN

Both authors are on the faculty of New York State University College at Brockport. I. B., Harkin is a professor of mathematics with responsibility for preservice and in-service mathematics programs for elementary school teachers. D. S. Martin is an instructor of mathematics with an interest in computer-assisted instruction in the mathematics curriculum of the elementary school.

In a systems approach to teaching, par- cepts embodied in the game. It involes the ticularly in the individualization of activi- student with concepts of prime and com- ties, manipulatives and other media hard- posite as well as applications of the funda- ware play an essential role. An activity mental theorem of arithmetic. These ex- package in the delivery system is a sequence periences are basic preparation for addition of multigroup activities developed for a of rationals. content unit. A variety of media hardware In the framework of an activity package (a term that embraces everything from a designed to treat algorithms for the addi- geoboard, a projector, or a video set to tion of rationals, we included the factor a piece of chalk) can be exploited in the game as an individualized computer in- activity package. This includes a computer, struction activity. The following computer when it is available to a school. print out displays the student and machine The "factor game" has appealed to pre- dialogue that reveals how the factor game service and in-service elementary teachers is played as well as the strategies of the because of the diversity of skills and con- game.

WELCOME TO THE GAME OF FACTORING. THI RULES ARE EASY. WE START WITH THE FOLLOWING TABLE: 2 3 4 5 6 7 8 9 10 , 11 12 13 14 15 16 17 18 19 20 21/ 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 THE IDEA IS SIMPLE. WE TAKE TURNS PICKING NUMBERS FROM THE BOARD. AFTER ONE OF US MKS A NUMBER, THE OTHER ,MAY THEN CLAIM ALL THE DIVISORS OF THE NUMBER THAT ARE NOT ALREADY USED. EACH NUMBER IN THE TABLE MAY BE USED ONLY ONCE, WHETHER IT IS PICKED OR CLAIMED. THE SCORE WILL BE THE SUM OF ALL THE NUMBERS PICKED OR CLAIMED BY A PLAYER. HIGHER SCORE WINS. IF YOU WANT TOsFFAN UP-TO-DATE CHART AT ANY TIME, JUST PUSH THE RETURN BUTTON. YOU WILL START THE FIRST GAME AND WE WILL ALTERNATE AFTER THAT. YOU MAY WANT TO RIP OFF THE PAPER CONTAINING THE BOARD, SO THAT YOU CAN X OUT THE NUMBERS ALREADY USED. START BY TYPING ANY NUMBER ON THE CHART. REMEMBER, YOU MUST PUSH THE RETURN BUTTON AFTER YOU TYPE.

145 ;, 146 GANTES AND PUZZI I-S

?37 YOUR SCQRE IS NOW 37 MY CHOICE IS 31 AND MY SCORE IS-NOW 31 IT IS NOW YOUR TURN. YOU CAN EITHER CLAIMkDIVISOR OF MY NUMBER (IF THERE IS ONE NOT YET USED).OR YOU MAY PICK A NUMBER OF YOUR OWN, IF YOU WANT TO CLAIM. TYPE 1. IF YOU WANT TO PICK, TYI'E THE NUMBER. IF YOU WANT TO SEE THE CHART, JUST PUSH THERETURN. YOUR CHOICE IS ?29 YOUR SCORE IS NOW 66 MY CHOICE IS 23 AND MY SCORE IS NOW 54 IT IS YOUR TURN. TYPE EITHER 1 (TO CLAIM), A NUMBER, OR JUST PUSH THE RETURN BUTTON. YOURCHOICE IS ?35 YOUR SCORE IS NOW 101 I CLAIM THE NUMBER (S) 5 7 MY SCORE IS NOW 66 MY CHOICE IS 25 AND MY SCORE IS NOW 91 IT IS YOUR TURN. TYPE EITHER I (TOCLAIM), A NUMBER, OR JUST I'USH THE RETURN BUTTON. YOURCHOICE IS ?I WHAT IS A NUMBER YOU WISH TOCLAIM ?5 1 AM SORRY BUT IWE HAVE ALREADYUSED THAT NUMBER. IF YOU WANT TO!CLAIM ANY OTHERNUMBERS, TYPE ONE OF THEM IN. OTHERWISE. TYPE I. YOUR CHOICEIS 11 YOUR NEXT CHOICE IS?I9 YOUR SCORE IS NOW 120 MY CHOICE IS 33 AND MY SCORE IS NOW 124 IT IS YOUk TURN. TYPE EITHER A 1 (TOCLAIM). A NUMBER, OR JUST I'USH THE RETURN BUTTON, YOURCHOICE IS?I WHAT IS A NUMBER YOU WISH TO CLAIM ?3 YOUR SCORE IS NOW 123 IF YOU WANT TO CLAIM ANY OTHERNUMBERS:, TYPE ONE OF THEM IN. OTHERWISE. TYPE I. YOURCHOICE IS?II YOUR SCORE IS NOW 134 IF YOU WANT TO CLAIM ANY OTHERNUMBERS, TYPE ONE OF THEM EN. OTHERWISE, TYPE 1. YOURCHOICE IS 7 I AM SORRY BUT WE HAVE ALREADYUSED THAT NUMBER. IF YOU WANT TO CLAIM ANY OTHERNUMBERS, TYI'E ONE OF THEM IN. OTHERWISE, TYPE I. YOUR CHOKE:ES '0 2 x 4 x 6 8 9 10 x 12 -13 !4 15 16 17 18 20 gi 22 x 24 26 27 28 x 30 32 x 34 x 36 YOUR NEXT CHOICE IS99 NO,9IS NOT A FACTOR OF 3333;9=3.66667 IF YOU WANT TO CLAIM ANY OTHERNUMBERS. TYPE ONE OF THEM IN. OTHERWISE. TYPE I. YOUR CHOICEIS 1 YOUR NEXT CHOICE IS?34 YOUR SCORE IS NOW 168 I CLAIM THE NUMBER(S) 2 17 MY SCORE IS NOW 143

Program strategies The numhers 2 through 37 wcrc chosen nice spread of both primes and composi- (1) to avoid 1, (2) to fit into a convenient ties that are low enough for easy calcula- array (here, six by six), and (3) togive a tion by the student. NUMBER THEORY AND PATTERNS 147

The first strategy is to choose (in order) tive of the net gain rather than maximizing 37, 31, 29, and 23. The choice of 19 next the net gain). is not optimal as, for example, the choice If the machine were to choose the opti- of 35 will result in a forfeit of 5 and 7 and,mal choice every time, the benefit to the the net gain is thus 35 12 = 23, which student would be limited and, perhaps, is/greater than 19. psychologically destructive.Inthis pro- This quantitychoice minus forfeitgram, the machine will pick the optimal can be calculated very easily by the ma- choice one-half of the time. The rest of the chine. This net 'gain is a measure of the time, the machine will pick randomly (oc- value sof the choice. Without attempting to casionally picking an optimal choice). justify our decision by game theoretic con- After two losses, the machine will pick siderations, we will say that a choice is the random choice only one-third of the optimal if (1) no other choice has a larger time. Experience has shown that on the net gain and (2) among all choices with first try, most students can play two or the same net gain, itis the largest in nu- three games in thirty minutes, More experi- merical value (for reasons of convenience, enced players will probably encounter the the program actually minimizes the nega- better machine strategy.

EDITORIAL. COMNIFNr.( I) Adapt the card game Fish for use with prime factorizations. In this game, a "run" consists of a card for a whole number and cards for its prime factorization (12, 2, 2, 3).When you prepare the playing deck,,include cards for the numbers 2 through 24 and a liberal number of cards fcr the prime factors 2, 3, and 5. Includea few cards for the prime factors 7 and I I. See the description of Fish in Hunt's article in chapter 9. (2) You can develop dice games to work the "multiple of and "factor of relationships. Start with two blank cubes of different colors. On one cube write the numerals 2,3, 4, 5, 6, and 7. On the other cube write the numerals 10, 12, 14, 16, 18, and 20. Players rollthe two cubes. Points are scored if the number named on one cube is "a multiple of'or "a factor of the number named on the other cube.

No Score Score 14 is not a multiple of 4 12 is a multiple of 4 4 is not a factor of 14 4 ts a factor of 12

A. i . . ;, Five "nontrivial" number games TERREL TROTTER, JR.

Formerly a teacher of high school mathematics at the NorwichHigh Sc lunfi in Norwich. Kansas, Terre! Trotter is a partici.,ant inthe I.971-72 Academic Year Institute for mathematics teachers atthe University of Illinois.

ATA. Number games can provide elementary ent to make the total 100 or more. students with a good opportunity to test Game 3,The name of this game is their observation skills for problem solving, "Aliquot." It was devised by David L. and at the same time they will be prac- Silverman and appeared in the problem ticing such drill skills as _adding and sub- section ofthe Journal of Recreational tracting. Here are five trivial number games Mathematics.Here is Mr. Silverman's own guaranteed to provide hours of fascination description from that journal (1970[a]): and lots of bidden drill. The wordtrivial Two players start with a positive integer and is used here with two meanings in mind. alternatelysubtract any aliquot part (factor) The games are trivial in the usual mathe- with the exception of the number itself from matical sensenamely, once the secret is the number left by the opponent. Winner is the last player able to perform such asubtraction. known to both players, the game's winner By way of example, if the original number is ispredetermined, based on the starting 12, first player may subtract either 1, 2, 3, 4, or number and who begins play. The games 6 (but not 12). If he subtracts 2, leaving 10, are nontrivial from the point ofview that second player may subtract 1, 2, or 5. they are valuable teaching aids for moti- The objective is to leave your opponent without a move. This can only be doneby leaving vating student thinking. him a 1, since 1is the only positive integer with The rules for the games will be pre- no aliquot part other than itself. sented first, followed by discussions of the Game4.This game, called "Proper respectivewinning strategies. You are en- Aliquot." v,as also devised by Mr. Si!"er- couraged to try the games before reading theJournal of the solutions section. This way you may man. In the same issue of he gives the have the pleasure of discovering the secrets Recreational Mathematics, for yourself before challenging your stu- rules thus: dents to attempt the same. The rules are the same as those of Aliquot with the exception that only proper divisors may Rules be subtracted. Consider 1 an improper divisor. Game 1.The first player selects any Game 5.The first player selects any integer from 1 to 10. Then the two players reasonable large number. Then the two alternately add any integer from 1to 10 players alternately subtract any number to the sum left by the opponent. Playcon- they choose from the number left by the tintv:s until one player can make an addi- opponent, provided the chosen number tion giving a grand total of 100. That mccts one requirement: the numbersub- player is thereby declared the winner. tracted must not exceed twice the value Game2. The rulesarethe same as of the number subtracted by the opponent those for game 1,exceptthat now the on the previous play.. Forexample, if winner is the player who forces his oppon- player A subtracts 4, then player 13 can r:: ...14 .1. ..) 148 NUMBER THEORY AND PATTERNS 149 select any integer from Ito 8. The game If a winning number is defined to be one is won when one player can "take it all," from which a player can force a win, even that is, leave 0. with the best play from his opponent, a loser will be a number from which a Solutions player can only leave a winning number to Game 1.After playing this game a the opponent, no matter what factor is sub- time or two, most students realize that tracted. Thus, 2 is a winnerthe ultimate whoever makes the total 89 can force the winner, of course, since it leads directly to win. A player does this by adding the dif- the game's objective. On the other hand, 3 ference between II and whatever his op- is a loser because it leaves 2 to the oppo- ponent adds next. That is,if A adds 3 to nent. Though 4 has two possible subtrac- 89, B adds 11 3, or 8, making 100 tions, it still can be classified as a winner and winning. because the best play (the factor I) leaves Similar reasoning leads to the conclusion a loser. that 78 is the next desirable total to ob- Furtherinvestigationalongthisline tain, since 78 = 89 11. A continuation yields the conclusion that all even num- of the same reasoning yields the winning bers are winners and all odd numbers are sequence 67, 56, 45, 34, 23, 12, and 1. losers.This suggests a simple rule for (An easy way to remember these numbers winning: If player A faces an even number, is to recognize that the tens digit is one he subtracts one of its odd factors so that less than the units digit.) Therefore, by he leaves an odd number for his opponent starting with 1, the first player can force to play from. The logic behind this is that the win, regardless of what the other player the only factors of an odd number are also does. At that point the game becomes odd, and odd odd = even. Therefore, trivial. (Of course, if a player doesn't know the opponent must return some other even the secret,his opponent can concealit number, andthecycleisestablished. from him longer by simply "entering" the Eventually he must leave the 2, and then sequence at some later point.) player A wins. Game 2. Inasmuch as 99 is the largest Game 4. The exclusion of 1 from the total a player can make without leaving or setoffactorsmakes some interesting exceeding 100, it and the sequence derived changes in the strategy. Now all the primes by subtracting1 Is are the secret totals. are the objectives of the game and there- (Naturally, they are the multiples of I 1.) fore are immediate losers. Testing the other Therefore, by allowing his opponent to se- numbers as was done for game 3 leads to, lect the first number, a player can once the conclusion that all odds are still losers, again force the win. but not all evens are winners. For example, Game 3. The proper way to analyze this 8 is the first even loserafter 2, that is, game is also to begin at the end. Only now which is also prime. Identification of the the end is the integer1.First, examine losing even numbers becomes less difficult allthepossiblesubtractionsfor some when the next even loser (32) is found. integers near the end of the game, say 2 Factoring these three even losers yields through 10: 2', 23, and 2. These are odd powers of 2, that is, 2^ where n is odd. 2 3 4 4 5 6 6 6 In summary then, the winning procedure -1-1-2 -1-1 3- 2-1 is to subtract such factors that leave either ______1 2 2 3 4 3 4 5 an odd number or an odd power of 2.

8 8 8 9 9 10 10 10 Game 5. The secret behind this game is

1 a great deal more sophisticated and elusive ______-4-2-1 -3______-1-_ 5 __2 4 6 7 6 8 5 8 9 than the secrets behind the other four. It A 150 GAMES AND PUZZLES

relies on some high - powered concepts from take the I. Now B can only subtract 1 or higher mathematics. But, the secret is rela- 2. neither of which will take all the 3 in tively easy to learn and use. the representation. Whatever B subtracts, A A preliminary observation about the will then take whatever isleft of the 3. game should be noted. A player should This presents B with the Fibonacci number never subtract a value that is one-third or 13. The greatest possible number B can more of the number. If he does, the op- now subtract is 4, depending on howthe ponent can, always take all the remainder. two previous subtractions were made. In For example,if a player faced 17 and any case, A will have a new non-Fibonacci subtracted 6,leaving11,his opponent number to play from, and he can return could "take it all" because the opponent to the winning procedure described above. can go as high as 12. As mentioned in the beginning, these The'Fibonacci numbers are an essential games are indeed trivial from a pure,math- part of the secret. Simply stated, the Fibo- ematicalstandpoint.Obviously,ifboth nacci numbers are integers from the se- players know the secret, there would be no quence 1 , 2, 3, 5, 8, 13, 21,. . .,where point in playing a game. The first to play succeeding members are found by adding would hand the other a losing number, and the two preceding members A theorem for all practical purposes the game would from number theory, the Zeckendorf the- he over right away. If only one player orem, says, in effect, that all non-Fibonacci knew a game's secret, he could always win numbers can he represented uniquely as too, much to the consternation of his patsy. the sum of two or more nonconsecutive But, if both players are unaware of the Fibonacci numbers. (For a proof of this 'ecrets, the game's provide a lot of practice theorem, see Noma 11969. p 74) Here in basic skills. The importance is that the are a few examples to illustrate thins con- drill is only a means to an end, not an cept: end to itself. In order to enjoy mathematics, students a) 17 13-1- 3-1-I, need to experience some success along the f..) 29 r=21 way. These games often allow the poorer

c) 40 33 5 2. studentto achieve some wins over the better student, especially at first, before the It turns out that when playing from a strategies are either discovered by the stu- non-Fibonacci number, a player can force dents themselves (it is hoped) or revealed the win by subtracting the smallest Fibo- to them. The more talented student should nacci number appearing in the representa- be encouraged to seek the strategies as tion. This succeeds as a result of the easily exercises intrue problem solving.Itis proven fact that for ally two ,-.-,ncouseett- recommended that the secrets not be re- tive Fibonacci numbers, the smaller one is vealedaled too soon, ifiiit all. Telling the secrets always less than one-half the larger. The of the games lest oys the fun and denies significance of this is that it always allows an individual thc satisfactionof finding the player to give his opponent a Fibonacci them for himself. Either way, the games number at some future stage of the game. are far from trivial as learningexperiencd. From there. the opponent must then return a non-Fibonacci number to thc player. It References will haVe a new representationwith a 1-logitatt. Verner F ,Jr.Fibonacci and Lucas smallest Fibonacci addend, andthccycle Nionbets.Boston: Houghton Mifflin Co , 196g, Silverman, David L. (a)"Problems and Con- is repeated. jectures"JournalofRecreationalMathe- An illustration will serve to clear up the matics2 (October 1970) : 227. preceding discussion.Playing from17, ---- (h).YourMore. New York: McGraw- which equals 13 + 3 + 1, player A will Hill Book Co.. 1971.

e Magic squares: extensions into mathematics

DAIYO SAWADA

An assistant professor in the Department of Elementary Education at the University of Alberta in Edmonton, Daiyo Sawada's main responsibility is the preparatiqn of teachers for the teaching of mathematics.

Many a potentially dull practice session magic square for addition, and the exis- on addition and subtraction has been trans- tence of additive inverses. formed into a lively encounter with mathe- Any consideration of these properties matics through the introduction of magic with operations with magic squares as- squares. Figure 1 shows three of the more sumes that the students have had prior ex- familiar ways in which magic squares are perience with the system of whole numbers. used. Normally work with magic squares Some students may think of comparing in the elementary school does not go be- magic squares with integers as a system. yond the types of examples shown there. The emphasis would not be on proof but Further exploration of magic squaresis rather on the use of basic propertie!. to usually limited to finding simple procedures characterize some new mathematical en- for constructing 3 x 3 squares, 4 x 4 tities. In short, the students would be par- squares, and n x n squares where n is ticipating in the construction and explora- odd. The procedures are frequently pre- tion of mathematical systemsexperiences sented in a rather rote fashion since the that they seldom get in the usual elemen- emphasis is on practice for addition, not tary mathematics program. Furthermore, on the properties of magic squares per se. the strategy of asking questions concerning As an introduction to more novel ways closure,associativity, commutativity, ex- of using magic squares, consider the se- istence of inverses, and so on whenever quence of activity cards shown in figures 2, interesting and new ideas are encountered 3, and 4. After reading the cards in order. is a highly mathematical behavior. A spirit the reader can probably sense the direction of inquiry can be encouraged in students of the activities. When students have com- without insisting on formalization. pleted the three activities, the discussion All of the suggestions so far have been that follows might consider other ways of rather general. The rest of this article adding magic squares, including ways that focuses on details that teachers should find the students themselves suggest. Different helpful if they are interested in exploring ways of adding magic squares could he the idea of magic squares as a system. compared and, indeed, a fourth activity card could investigate the closure property Magic squares as a mathematical of the operation of- addition with magic system squares. Subsequent cards could ask about Forteachers who believe*thatitis the associative and commutative properties worthwhiletoprovide someinformal of addition with magic squares. the identity experiences with mathematical systems for 151 -4 r- At ,( 152 GAMES AND PUZZLES

Testing Magic Squares Magic squares have row, column, and diagonal sums that are all equal. Test to see if the array below is a magic square.

4 3 8 0 9 5 1 0 2 7 6 0 0r 0001 1 1 \0

Completing Magic Squares Complete the following magic square.

12 89 45

Constructing Magic Squares

Construct a magic square that has a sum of 27. s

Fig.

Card 1 Complete the following magic squares Write the row,column, and diagonal sums in the circles.

(---), 0,\ ./ r----r ; ------1 .--- 8 15 I ! / ) 6 1 I .) r- I ______2, , 18 10 51 9 I 1 ( j 2 H 0 ii

, 1 1 _l____I _ i

1 I I I 1 00C) NUMBER THEORY AND PATTERNS 153

Card 2

Using the squares on card 1, multiply each number in each of the nine cells by 2. Do you still have a magic square' Use the squares below to carry out your work. ON 0 ON 0 30 o o --0 _o _o-o

I I I I I I

, 000 000_

Fig. 3

Card 3 Suppose we wanted to add the squares given on card 1. One way of adding is started below. Can you finish the adding? o 23 -o I9 -o 0 000 Did you get another magic square? Test it

Fig. 4 their students, it is useful to know what properties on some magic squares.) These kinds of mathematicalsystems canbe four properties (closure, associativity, iden- easily constructed using magic squares as tity, inverse) are the axioms of a group. In the elements and addition as the operation. (i) Closure. X C) Y is a magic square. As the reader has probably already guessed, (ii) Associativity. (X C) Y) C) Z = X C) (Y C) Z) the set of 3 X 3 magic squares can be used (Hi) Identity. The identity square is to form a group. That is,if we let M 0 0 0 designate the set of 3 X 3 magic squares with integer entries in the cells and let 0 0 0 0 designate magic square addition as suggested 0 0 0 on card 3 (see fig. 4), then for any magic (iv) Inverse. The inverse of a given magic square is squares X, Y, Z E M the properties shown generated by multiplying each cell entry by 1. in figure 5 hold. (Students can test these Fig. 5 154 GAMES AND PUZILFS other words, the set M for the operation of readers who have had experiences with addition 0 is a group. Since it is also true matrices, itis probably evident that the that X(-1-) Y = Y0 X -that is, the property proofs correspond quite closely to proofs of commutativity holds the set M for the related to the properties of matrix addition.) operation of additionisa commutative Perhaps the property that is most inter- group. esting is the property of closure. Many stu- As indicated earlier, it would not be a dents have some difficulty grasping what is good idea to expect all students to construct meant by closure as related to the set of proofs of these properties of addition of natural numbers under addition or multi- 3 X 3 magic squares. The proofs, however, plication. To them it is too obvious thdt are relatively simple, if you begin with the "when you add two natural numbers you assumption thatthe properties hold for get another natural number"why prove integers, and they are rather typical of what is so obviously true? When students elementaryproofsinvolvedinnumber are confronted with the analogous problem systemslike provingthatadditionof with magic squares, however, itis not so rationalnumbersiscommutative,for obvious that adding two magic squares will example. Interested students may findit produce another magic square. Hence the very worthwhile to engage in some "prov- property of closure begins to take on meaning" experiences. ing; it begins to make a difference. Indeed The proof for X Y = Y (1)- X. namely. many students will want to test it out on that the operatiob of addition for 3 X 3 several magic squares before accepting the magic squares is commutative, is shown in hypothesis that magic square addition is figure 6. The other proofs are similar. (For closed.

To be proved: X0Y= YOt Proof:

h Let X = e I and Y g It t

where a. h. e. . r are integers

a ti Then XEDY = e It g It r

+ b k e

4- In e -5- n f-1-tt By definition of C). g P It 4 ti t4-r

+a k / 01-1- (I e I By commutativityof addition ith integers I)-f- g q It t

/ h e n 0 r By definition of C) p r g Ir t

f' (D X Fig6 ' NUMBER THEORY AND PATTERNS 155

As suggested on activity card 2, a mul- 2.Incolumn (a) tiplication operation can also be defined. the units digits run from 1 through 9. For example, Similar patterns hold for the tens, hundreds, and thousands columns.

6 1 8 2x 62x 12 x 8 1089-410+ 89=99 2 75 3 = 2X 72 x 52 x 3 2178-421+ 78=99 294 2'22 x 92 x 4 and so on. 1089-4 1+ 0+ 8+ 9=18 12 2 16 2178-4 2+ 1+ 7+ 8=18 14 10 6 and so on. 418 8 3. Patterns observed for column (a) could be tested for column (b). Other where, the multiplication of the number 2 relationships may be di'-overed. But what and the magic square is indicated merely has this to do with magic squares? by writing them adjacent to each other. 1 1089 9801 With this new operation we can get an 2 2178 8712 additional property, a(X C) Y) = aX C) aY, 3 3267 7623 4 4356 6534 where a is an inteber. For example,: 5 X 1089 5445 5445 6 6534 4356 7 7623 3267 6 72 14 3 8 8 8712 2178 1 59 9 9801 1089 8 34 Cs 137 16 ( ) (a) (b)

67 2 4 3 8 Fig. 7

= 2 1 59 C) 2 9'5 I Two squares are shown infigure8. 3 276 8 4 Square A is the standard 3 x 3 magic square. Square B can be generated from It is easy to show that the property holds the numbers in column (a) of figure 7 for this example, and other examples can by looking atthe thousands digitand be tried. Some students may even try a matching it with the numbers in square A. proof. Students may also recognize this (Is squareBa magic square? Students as an example of the distributive property. could find out by adding.)

"1089 Overture" I8 8712 1089 6534 As a closing example, consider an en- 3267 5445 7623 richment activity based on the interesting 4 9 2 4356 9801 2178 number pattern in figure 7. As a catalyst for discussion, column (b)is generated Fig. 8 after the products are written in column At some point, however, it should be (a). The teacher could say, "Let's pre- noted that the solution to the open sen- tend we have a mirror beside our column tence in figure 9 is relevant. That is, what of products. What would the mirror images look like?" The images would then be is the result if square A is multiplied by written as shown in column (b). Students could then be encouraged tolookfor i 6 interesting relationships like the following: 1089 3 5 7

4 9 2 1. Column (b) is column (a) written "upside down." F g. 9 156 GAMES AND PUZZLES 1089? If the students have already con- tion of the well known structure of a vinced themselves that multiplication of vector space. a 3 x 3 magic squareby a whole number Although modules, vector spaces, mul- or integer is a closedoperation, then the tiples, and groups are important math- answer to this question must be }es,with- matical concepts in and of themselves, the out calculating. Or the students can con- purpose in wri.ing this article was notto firm the result by actual calculation. imploreteachersofupper elementary Several other questions could be asked grades, or even junior high, to teach such about square B: concepts. Rather, the purpose was to suggest the strategy of takinginteresting ideas, 1. Considering only the units digits in such as magic squares, and extendingthem each cell, do you get a magic square? in ways that lead to the discoveryof other 2. Similarly, do you get a magic square mathematical.structures, all the while pro- if you consider just the tens digits? the viding interesting-practice of fundamentals hundreds digits? the thousands digits? as originally intended. Itis this process 3. Knowing that the row sum of square of extension that is important, not the A is15, can you figure out the row sum particular outcome. Such extensions re- of square B wiihout adding? (Hint: Use quire the asking of the right questions. multiplication. ) As students discover what sort of questions lead to interesting payoffs, theywill Summary be well on the road to converting inter- Itis hoped that enough detail has been estingpracticeintointerestingmathe- given so that many teachers will be able matics. to use magic squares in new ways.Ac- tivities such as those described can lead to some profound mathematics (groups. modules, theory of multiples) at the same References time they provide needed computational Heath, K. V. Mathemagic. New York: Dover practice with whole numbers and integers. Publications. Inc., 1933. Or if students are working with fractions. Keedy, M. L. A Modern Introduction to Basic magic squares could be multiplied byfrac- Afathentatics. Reading, Mass. Addison-Wes- tions. The latter could lead to the explora- ley, 1963.

Arithmetic Teacher. Anne Mae Cox EDITORIA1COMNIFNr. In the March 1974 issue of the suggests using magic spiderwebs in "Magic While They Are Young."

2W 2 1 9 \ ,4), 4 6

1. Place pairs of numbers atthreelocationsin the outer ring. 2. Write the sum of thesenumbers in the adjacent triangle. in the two adjacent triangles. 3. In the remaining triangles,write the sum of the numbers 4. Now find the sums of thenumbers in opposite triangles. Surprise!

/ "Intel getters"

KARL G. Z A H N t7'.P. Roseman Campus Elementary School Wisconsin State University, Whitewater, Wisconsin

Itis generally recognized that it is im- EXAMPLE 3 portant for a teacher to maintain -pupil = ? 992 = ? interest in arithmetic. The use of a variety 9992 = 1 of learning activities is sir excellent-tech- 9,9992 = ? interest. One nique for retaining pupil EXAMPLE 4 technique is to use one class period per 222,222,222 x9 = 1.999,999,998 weekforpresenting"interestgetters." 333,333.333 x9 = 2,999,999,997 Friday is usually a good day to present 444,444,444 x9 = these,for the teacher can reviewthe 355,555,555 x9 = 666,666.666 x9= past week's activities and then prdent 777.777,777 x9 = some stimulating items to create and main- 888,888.888 x9 = tain pupil interest. 999.999,999 x9 = Continuedinterestin arithmetic is also One purpose for using Examples 1, 2, maintained by patient, sympathetic guid- 3, and 4 is to develop interest in arithmetic ance given to the student by the teacher. for the student. Another purpose might The following are examples of some be to practice computational skills. Third, activities that I have had success in using but not least, it is hoped that these exer- with elementary and junior high school cises will help students develop insight and students: sec relationships in numberi. In each ex- EXAMPLE I ercisc, I would ask students to work out a few of the problems and then make some "educated guesses" concerning the pattern that is evolving. Dividing 10 by 81 has an interesting result. Ask your students to do this and -carry it out for three or four places. Then ask themif they can predict what the EXAMPLE 2 eighth, ninth, and tenth digits will be.

1 2 3 4 5 6 7 9 Students always enjoy stories that are

x 198 == 111.111.111? humorous or challenging. Here are two x 27 = ? favorites that I have told to children (and xx45 36 adults) for a number of years: X 54 = ? Pat and Mike X7X 623 ; x 81 = Pat and Mike went into, the coal busi- X 90 + 1 = ?? ness. Pat said, "Seven tons of coal at $16

s ;s 158 GAMES AND PUZZLES giving $49." Mike said, "1had better his old mare to the sons, thereby per ton is them 18 horses to divide. The sonsjoy- multiply that andcheck to sec if you are fully made the division; the eldest son took correctfz one-half of 18 or 9 horses, the second 7 times 6 is 42 16 , ti sononeAhird of ,18 or 6 horses, and the Put down the 42. third son one-ninth of 18 or 2 horses for a is 7. 7 times ! 42 17horses. So the wise man 7 totalof Put it tinder the 42. mounted the remaining horse and rode 49 away, leaving the sonshappy and colt- Result is 49. tented. Pat said, "I had bettercheck your mte- An interesting game that can be played, "tiplying by dividing 7 into,49 and see if in any classroom is Nim. I have usually we'get 16. Seven doesn't gointo 4 but it introduced this game by puttingmarks on goes into 9 once. the chalkboard.

1 Itples 7 is7. Row Subtract 7 from 9 and 16 Row2 III bring dolt n the 4. 7 7,5 Row3 11111 7 Row4 1111111 7 divides into 42 six 42 times. Any number of marks or rows -may be les correct' used; however,Ihatie always used the Mike said, "Let's check one moretime by 1, 3, 5, 7 combination with children at adding." the beginning for the sake of simplicity. Two crs take turns in making their 6- 6= 12 6= 1 S 6= 24 ' 6 =30 6 = 36 -+6 = 42 move . Each player, during his turn, may ( then coming down theI'S at the top) cross off all or only some of themarks 43, 44, 45, 46, 47, 48, 49. from any one row, but he must cross off until 16 at least one mark. The play continues 16 the player wllo succeeds in crossing off 16 the last mark wins the game. 16 To illustrate,let us follow the moves 16 16 in a game: 16

49 Row 1 Player No. 1 It checks' (Now try this with 7 and13.) Row 2 crosses out all of Row 3 Row 4. Row 4 The Arab and His Three Sons Step 2 There is a story related about anArab be divided 1 Player No. 2 crosses who died, leaving 17 horses to III off 2 in Row 3. among his three sons.According to the III++ Arab's will, the oldest son was toreceive Step 3 one-half the horses, the second son was Playe? No. 1 crosses toreceiveone-third, and the youngest 11 l off 1 in Row I . .son's, share was to be one- ninth.The sons ill could not agree upon the divisionof the Step 4 horses and quarreled violently. Theirdis- 111 Player No. 2 crosses oft 2 in Row 3. pute woverheard one day by art ancient 11+ wise man who was riding by on anold Step 5 gray mare. The wise manconsidered the 144 Player No. I crosses problem at length and at last presented 1 oft 2 in Row 2; / NUMBER THEORY AND PATTERNS 159

Now player No. 2 can only crossoff game, but a player who knows the secret 1, so No. Iis the winner. can almost always win. After some experience, children can usually tell what the This game can also be playedwith game-ending winning distributions should matches, pebbles, or toothpicks.Itis a be. Can youIf someone becomes sus- good "rainy day" game children Canplay picious of your winning frequently, change al-their desks or at the chalkboard. the game so that the last mark crossed Nim is frequently used as a gambling Out is the loser rather than the winner.

EDITORIAL. CONINIIN I. -Here are some other curiosities to consider: A. "number pyramid." Place I's at each end. Find the numbers inthe pyramid blocks by taking the sum of the two numbers in the blocks above.

1I 4 I 6L4 I 1

I 5 10 10 5 1 1

Do you recognize this as Pascal'striangle? Predictable answers!

1 x 8+ 1=9 0 x 9 + 1= 1 12 x 8+ 2=98 I x 9 + 2=II 123 x 8+3=987 12 x 9 + 3 III 1234 x 8+4=9876 123 x 9+ 4= 1111 12345x8+5.98765 1234 x 9 + 5=11111 123456 x 8+ 6=? 12345 x 9 + 6=? Just for fun

J.D. CALDWELL Public Schools, Windsor, Ontario

Some of our readers might like to try the numbers which he is supplying, he says following diversion with their classes. As them aloud and encourages the children to many times asI have seenitused,it say them with him, as soon as they have never fails to stithulate pupil interest and discovered how itis done. You will be further exploration. We might also note surprisedat how quickly some of the chil- that this particular lesson is a good ex- dren can join the chorus and how pleased ample of the inductive-deductive approach. they will be with their own perception. The The teacher asks a pupil to give' the last examples can go on until at least half the five digits of his telephone number. These class has seen through the"trick,','-at are recorded on the chalkboard. Let us which time a pupil might lasked to ex- suppose the recorded number is 34,487. plain the procedure to everyone. The teacher suggests that they will add Follow-up on "Why does this work?" four more 5-digit numbers to this one, and produces much worthwhile thought. he is prepared to predict the answer. He says the sum will be 234,485 and places Examples this in the appropriate position. He then A few examples of the chalkboard work asks a pupil to supply the second 5-digit are given below* and you areleft to figure number, which is recorded. The teacher out the trick for yourself. The numbers are supplies the third number (if he wants it given in the following order (but remember to work), a pupil supplies the fourth num- that the prediction is the second step): ber, and the teacher the fifth. At this point Starting number the addition is carried out orally, and the Number supplied by pupil predicted answer is seen to be correct. Number supplied by teacher Number supplied by pupil Successive examples are done. As the Number supplied by teacher teacher writes down his prediction and the Prediction

34,487 99,896 66,831 48,320 26,356 18,723 73,491 61,133 73,643 81,276 26,508 38,866 48,915 43,711 91,735 29,114 51,084 56,288 8,264 70,885 234,485 299,894 266,829 248,318 with an apology from the teacherthat in this case he can use only a 1 -digit number .!*.160 Geometry and Measurement

IInterestin geometry and related measurement activities has increased in the elementary school mathematics program during the past several years. On the one hand, geometric concepts are seen as important in helping the child give structure to his environment. On the other hand, there is a movement in the United States to adopt the metric system of measurement. Both forces have combined to make us more aware of the need for sound instruction in geometrical ideas in the elementary school years. One measurement concept that is taught early in the mathematics program is telling time. Some children come to school already possessing this skill. In the first article, by Porlier, we find not only a game that helps children learn this skill but also a technique for maintaining the interest of children who already know how to tell time. The second article, by Trueblood And Szabo, gives hints on games to use for introducing the metric system. Me checklist they provide for helping teachers develop games is especially good,,ind it can also be applied to many other games. The next pair of articles is related to the concept of area measurement. In addition to relationships develornd in Berman's "Geo-Gin," we find the germ of an idea for many different card games. Games such as Old Maid,1.ish, and rummy can be converted into activittus for review and practice with geometric terms and symbols. The tangrani square puzzle has already become familiar to many teachers. The puzzle pieces take the form of many of the basic geometric shapes included in the elementary curriculum Dickoff's presentation uses many intuitive geometric constructions to create the .,even pieces through paper folding. The paper-folding activity reminds us that he oriental art of origami is also a very appealing activity for children. Grogan's article, "A Game with Shapes," and the article on mirror cards, by Walter, describe games designed to help students see the relationships involved in the geometric transformations known as reflections, rotations, and translations. Both games are easy and exciting for children to play; both are loaded with significant geometric ideas. A r., ...-, ,) 161 162 GAMES AND PUZZLES Aliso& Popgicle sticks and tongue depressors have found many uses in themathe-, matics'program. They make excellent counters and bundle easily into groups of ten for work on place-value concepts. The childrenin Lund's class probably have found the way to have the most fun withPopsicle sticksmaking polygons out of them! And in addition to the fun ofmaking them, they fly, too! This sounds like a good outdoor activity. Reminiscent of the tangram acti *ty is Hall's "PythagoreanPuzzle." The puzzle activity confirms the basic form lation of thePythagorean theorem and is also used to explore the relationip between areas of various figures triangles, squares, and parallelograms. The last games in this section provide a means todevelop and practiCe skills associated with graphing 'with Cartesian coordinates.Timmon's tic-tac- toe game is a very good example ofusing a game to develop a significant mathematical idea. One might even expect some students to perceivethe numerical relationships existing between coordinates of pointsfalling along a line. The article by Overholser shows how to incorporate area conceptsinto work with coordinates. On the lighter side, and especiallysuitable for the earlier elementary grades, are Deatsman's "HolidayPlot Dots" and Bell's "Car- tesian Coordinates and Battleship," a game that mostchildren (and adults) attack with fervor.

4 ---

Don't miss the train

CORINNA PORLIER

Go ldie King School, East Gary, Indiana

In our fast-moving society of schedules, just avoid having to deal with exact times. everyone finds a need for telling time, but In the -first grade, where the textbook whose responsibilityisitto teach this deals only with the hour and half-hour, critical topic? Is it the teacher's? Is it the the child who already knows that is bored. parent's? Or, is the child left to learn on On the other hand, the child who does not his own? know this much is not given enough class- Evidently, the textbook authors feel that room time or experiences to learn to read the child already knows how to read a these times on the clock. The same thing clock before coming to school. A check happens in the second and third grades. of several arithmetic textbooks will show By the time a child is in the fifth or sixth that an average of only two to four pages grade, he may be thoroughly confused and per grade level (for the first,second, and too embarrassed to ask for help from the thirdgrades)aredevotedtoreading teacher or his parents. clocks. There is usually no space atall This problem has bothered me for many given to telling time after the third grade. years. After much thought, I have come Many parents do teach their children to up with a solution. tell time at a very young age and some- I have developed a game'that not only times reward them with a watch or clock provides a fun way to teach time to chil- of their very own when the children can dren, but also holds the interest of children successfully read them. But what about who already know how to tell time. the child who is left to learn on his own? In playing this game, the older child He may struggle through for a few years can learn more about telling timewithout telling time by the locations of the "big having to be classified by his classmates hand" and the "little hand," or he may as a dummy. ,; 163 I .. ADVANCE CLOCK BONUS 25 MINUTES

Bonus cards (20)

Space Marker

(1 red I green I blueI yellow

TIME

Time cards (20)

Players' clocks

(1 red 1 green 12 11 1 blue Iyellow) 0' 55clock 10 2 50 10

9 45

ao 20 Dice 8 4 35 25 30 7 5 EXAMPLE 6 ,Time GAME BOARD Bonus _1 f GLOMI: 1 RY AND MI.ASUREMI.NT 165

Don't Miss The Train space markers that match the colors of the players' clocks. The markers are used (Rules for playing the clock game.) to pace off the spaces on the round track. How the game is played: BONUScards. The noNys cards give The game is played by 2, 3, or 4 players. players extra chances toadvancetheir Each player moves his marker around the clocks. A player must land on a space railroad track according to the throw of marked BONUS in order to draw a card the dice. A player may advance his clock from the top of theBONUS ,pile.Bonuses when he lands on a space that tells him range from five minutes to one hour. to do so or when he draws aBONUScard. There are a few occasions when a player TIMEcards. The TIME cards state the must set his clock back. time that the train leaves and the present, The master Clock on the playing board or starting, time. The top card of the is set at the time the train is to leave. The stack of TIME cards is turned up at the train's departure timeis determined by beginning of the game; only one TIME card theTIMEcard. The player whose clock is used in each game. The first player sets first shows the departure time of the train the master clock with the time shown on is the one who "catches the train" and is the TIME card that he turns up, and each the winner. All other players miss the player sets his own clock at the present train. time shown on the card. The object of the game: Don't miss the train. Playing the game: I. The players throw th, dice to see Equipment: who starts first: The player with the largest 1 playing board total plays first, and play thereafter passes 4 clocks of different colors to the left. 4 colored space markers to match the 2. At the beginning -of the game, the clocks markers are placed on the spur tracks in, 2 dice the station. The cards are shuffled and 1 pack ofBONUScards (20) placed intheir proper places. The first 1 pack ofTIMEcards (20) player turns up the topTIMEcard and sets 1.et of directions the master clock at the time indicated on Playing board. The playing board shows the TIME card. Each player sets his own a station with four spur lines, a circular clock at the "time now" indicated on the track that is marked off as a master 'clock, TIME card. When the game begins, the and spaces at the bottom of the board for clocks of all of the players are set at the theBONUSandTIMEcards. The master same time. clock has its own set of movable hands. 3. The beginning player must roll either In some of the spaces between the railroad a one or a six on at least one of the dice ties,therearespecialinstructionsfor in order to bring his marker out of the players; other spaces are blank. station. If he does so, he places bis marker on the space marked start. He then totals Clocks. There are four clocks, one yel- his throw of the dice and advances his low, one blue, one green, and one red. mark7 clockwise around the tracks for Each player has his own clock on which that many spaces. If he lands on a space he keeps his time as he advances. These that tas special instructions, he follows the clocks have special faces and hands that directions, that are given. If a player does make telling time easier for the players. .. - not throw a one or a six on his first throw, Space markers. There are four colored he has to remain in the station and /the

A , 1 166 GAMES AND PUZILFS play moves to the player on his left. his clock back, only his marker is moved 4. If a player throws doubles, he gets back. fifteen another turn. If lie throws doubles a second 7. A player advances his clock time, he gets an added turn. If he throws minutes every timehis marker moves doubles a third time, he does not take the around the entire circle and passes the count, and his turn of play is ended, space with the star. '5. A player is entitled to only the top 8. When a player is sent back to start, he does not get the additional advancement BONUScard when his marker lands on a of 15 minutes on his clock. The advance- space markedBONUS.After he reads the ment is given only when a playerhas BONUScard and advances his clock as directed, the player places the card face completed the circle. down on the bottom of theBONUSstack. 9. When a player's clock reaches or 6.If a player's marker lands on a space passesthe time shown onthe master where there is already a marker, the first clock, that player wins the gamehe has marker must be moved back to start. The caught the train. All others have missed penalized player does not have to move the train.

EDITORIAL. COMMENT. -To provide practice inreading a clock,try CLOCK bingo. Two versions are suitable. In one version, the CLOCK card containsthe names of times. The caller shows a large cloekf.iee. Players read the time fromthe dock and find the notation on the card.

r T 1- 1 1 C L 0 t C I K

19 30 Bet', 1115 12 Oa; 145

specific times. in the second verson., the TIMEcard contains pictures of clockfaces set to The caller names a time Theplayers look for a clockface showing thattime.

With some groups it is wise to use cards with fewer cellsperhaps a 3 x 3 arrangement. This makes the game go faster. Procedures for designingyour own

, metric games for pupil involvement

CECIL' R. TRUEBLOOD and MICHAEL SZABO

Currently an associate professor in mathematics education at Pennsylvania State University, Cecil Trueblood is particularly interested in the teaching of mathematics in the elementary school. Michael Szabo, also at Pennsylvania State, is an associate professor of science education. His educational interests center on instructional development, individualized instruction, and complex problem-solving.

Although much has been written on the their own games? The available profes- valueof mathematical games in the ele- sionalopinion supportedthe following me ary grades and many game books have conclusions: bepublished, little has been written that 1. Games can be used with modest w uld help classroom teachers design, pro- success with verbally unskilled and emo- uce, and evaluate games for use in their tionally disturbed students, and students classroom. The focus of this article is to for whom English is a second language. present a set of seven criteria that were de- 2. Games have helped some teachers veloped in a summer workshop for inser- deal with student who present discipline vice elementary teachers who decided that problems because they are bored with the they wanted to be able to produce metric regular classroom routine. games and related activities that would fit into their "metrication" program. 3. Games seem to fitwell into class- The teachers in the workshop began by rooms where the laboratory or learning- asking a practical question: Why should I center approach isused. This seems re- be interested in producing my own metric lated to the feature that games can be games? They concluded that the game operated indeperdent of direct teacher con- format provided them with specific activi- trol thus freeing the teacher to observe and ties for pupils who did not respond to the provide individual pupils with assistance more typical patterns of instruction. They on the same or related content. felt that in the game format they could provide activities of a higher cognitive Plan for development level for pupils who had difficulty respond- If for any of the reasons just cited you ing to material requiring advanced reading are interested in designing and evaluating skills. several of your own metric games, how The teachers then asked a second ques- should you begin? Simply use the following tion: Does the literature on the use of checklist as a step-by-step guide to help mathematics games contain any evidence you generate the materials needed to create thatwould encouragebusyclassroom your game. Use the exemplar that follows teachers to use planning time to develop the checklist as a source for more detailed , 4 t167 168 GAMES AND PUZZLES objects, the -suggestions. Each item in the checklisthas 1. Given a set of common been keyed to the exemplar tofacilitate students estimate the objects' weight cor- kilogram. (Observa- cross referencing. rect to the nearest tion and estimation) CHECKLIST GUIDE 2. Given an equal-armbalance,the Write down what you want your students weigh and record the weightsof _ the nearest students to learn from playing your common objects correct to game. (Establish specificoutcomes) centigram. (Measurement) __Develop the materials required to 3. Given an object's estimatedand ob- play the game. (Make simple 'ma- served weight correct to the nearestcenti- terials) gram, the student computesthe amount Develop the rules and procedures over or under hisestimate. (Computation needed to tell each player how to and number relationships) participateinthegame.(Write simple rules and procedures) Make simple materials __Decide how you want students to The following materials wereconstructed obtain knowledge of results. (Pro- or assembled tohelp students attain the vide immediate feedback) objectives previously stated in aninterest- _Create some way for chance to en- ing and challenging manner. terinto the playing of the game. 1. Sets of 3-by-5 cards with tasksgiven (Build in some suspense) on the front and correct answersand points __Pick out the features that canbe to be scored on theback. (See fig. 1.) easily changed to vary the focus or 2. A cardboard track (seefig. 2) made rules of the game. (Create the ma- from oak tag. Shuffle the E's(estimate terials to allow variation) cards), O's (observed cards), andthe D's _Find out what the students thinkof (difference cards) and place them onthe the game and decide whetherthey gameboard in the places indicated. learned what you intended them to 3. An equal-arm balance that canweigh learn. (Evaluate the game) objects up to 7 kilograms. 4. A pair of dice and onedifferent The exemplar colored button per player. Establish specific outcomes 5. A set of common objects thatweigh By carefully choosing objectivesthat in- less than 7 kilograms and morethan 1 volve both mathematics and science pro- kilogram. cessessuch as observing, measuring,and 6. Student record card. (See fig. 3.) classifyingthe teachers created a game integrated ac- that involves players in the Write .simple rules andprocedures tivities. This approach reinforces thephi- The rules and procedures are crucialto losophy that science and mathematics can In the be taught together when theactivities are making a game self-instructional. mutually beneficial. That is, in manyin- following set of directions notice how a stu- be used to dent leader and an answer carddeck serve stances integrated activities can processing needed to conserve instructional timeand to promote to ease the answer from one keep the game moving smoothlyfrom one the transfer of process skills keep the exemplar's player to another. It is essential to subject area to the other. The that hjectives are labeled to show their rela- rules simple and straightforward so student to inship to science and mathematical pro- play moves quickly from one cesses. the other. GEOMETRY AND MEASUREMENT 169

ECard

2 points

The estimated weignt of thebrick is kg kg

Front of card Back of card QCard

T2points

The observed brick's weight is kg kg

L Front of card Back of card DCard

4 points Fillin the blanks below --] r Then compute the difference between the brick's estimated and observed weight Estimated Difference kg. weightis _kg Observed weight of brick is kg J Front of card Back of card Fig. 1

1. Number of players, two to six. The player then records the card number, his answer, and the points awarded by the 2. The student leader or teacher aide student leader on his record card. The stu- begins by rolling the dice. dent leader checks each player's answer The highest roll goes first. All players start and awards theappropriate number of with their buttons in the "Start Here" block. points by reading the back side on the The first player rolls one die and moves his task card. He then places that card on the button the number of spaces indicated on bottom of the appropriate deck and play the die. If he lands on a space containing moves to the right of the first player. The an E, 0, or D he must choose the top card player who reaches "Home" square with in the appropriate deck located in the cen- the highest number of points is the winner. ter of the playing board or track and per- At the end of the play each player turns form the task indicated. (In the example in his score card to the student leader who shown in figure 1this would be Card E3.) gives them to the teacher. AJ

:,...4* ?

9 170 GAMES AND PUZZLES

Skip Forward 2 spaces

Go back

I Space

D D E _ Add Subtract 1 1 n You lose 2 points 1 UP ID i Iturn 10(DI a 2 points

Fig. 2

..,

Provide immediate feedback This evaluative feature can .bebuilt By placing the answer on the back of into most games by using an individual the task card and appointing a student record card, by having the student leader leader, the teacher who developed this pile cards yielding right answers in one game built into the game an important pile and cards with wrong answers in an- characteristic, immediate knowledge of the other pile, or by having the student leader record the results of each play on a class results of each player's performance. In most cases this feedback feature can be record sheet. built into a gameby using the back of task cards, by creating an answer deck, or Build in some suspense by using a student leader whose level of Experience has shown that games en- performance would permit him to judge joyed by si udents contain some element of the adequacy of other students' perform- risk or chance. In this particular game a ance in a reliable manner. Feedback is one player gets a task card based upon the of the key features of an instructional game roll of the die. He also has the possibility because it has motivational as well as in- of being skipped forward or skipped back structional impact. spaces, or of losing his turn. Skipping Have students record diagnostic infor- back builds in the possibility of getting ad- mation.The student record card is an im- ditional opportunities to score points; this portant feature of the game. The cards feature helps low-scoring students catch up. help the teacher to judge when the diffi- Skipping forward cuts the number of op- culty of the task card should be altered portunities a 'high-scoring player has to and which players should play together in accumulate points. The possibility of add- a game, and to designate student leaders ing or subtracting points also helps create for succeeding games. The card also pro- some suspense. These suspense-creating vides the player with a record that shows features help make the game what the stu- his scores and motivates him to improve. dents call "a fun game."

5 4 K\ AND MEASUREMI.NT 171

Student's name Date

Card number I Anwergiven I Number of points ? t- E3 2 kg. _ _ 2_ E2 Tkg. J

Fig. 3

Create the materials to allow variation 2. Which face indicates how you felt A game that has the potential for varia- when you were playing the game? tion with minor modifications of the .rules or materials has at least two advantages. 0 0 ® First, it allows a new game to be created 3. What part of the game did you like without a large time investment on the part best? of the teacher. Second, it keeps the game from becoming stale because the students 4. How would you improve the game? know all the answers. For ingunce"the Concluding remarks exemplar game can be quickly changed by making new task cards that require that The procedure just illustrated can be students estimate and measure the area generalized to other topics in science and of common surfaces found in the classroom mathematics. The following list provides such as a desk or table tops. By combin- some suggested topics. ing the two decks mixed practice could I. Classifying objects measured in metric be provided. units by weight and shape 2. Measuring volume and weightwith Evaluate the game metric instruments Try the game and variations with a small 3. Measuring length and area with metric group of students and observe their ac- instruments tions. Use the first-round record cards as 4. Classifying objects measured in metric a pretest. Kelp the succeeding record cards units by size and shape foi each student in correct order. By com- 5. Comparing the weight of a liquid to its paring the last-round record cards with the first-round record cards for a specific stu- volume dent, you can keep track of the progress 6. Comparing the weight of a liquid with a particular student is making. Filing the the weight of an equal volume of water cards by student names will provide a 7. Predicting what will happen to a block longitudinal record of a student's progress on an inclined plane for a given skill as well as diagnostic infor- 8. Comparing - theweights ofdifferent mation for future instruction. metals of equal volume Finally, decide whether the students en- Why don't you try and create some joy the game. The best way is to use a games for each of these topics? Then share self-report form containing sevcral single the results with your colleagues. Additional questions like the following, which can be examples developed by the authors are answered in an interview or in writing: available in "Metric Games and Bulletin I. Would you recommend the game to boards" in The Instructor Handbag. k Series someone else in the class? _Yes _No No. 319 (Dansville, New York, 1973).

4 Irt

..' Geo-gin

.C1 JANIS A. BERMAN

Student, University of South Florida, Tampa, Florida /

41Cost students in the upper elementary On one set of 16 cards, draw a circle gradeshaveplayedand enjoyed card (two inches in diameter.) on the white side games.Geo-ginis a card game that makes of each card. There should be four circles use of some important geometrid concepts in each of the four colors. On another set such as spatial percepti identification, of -16 cards, draw a two-inch square on the anddiscrimination.Itis edfor white side of each card. There should be groups of two to four students. four squares in each of the four colors. Ott the last set of 16 cards, draw a parallelo- Materials needed gram with two-inch sides on the white side of each card. There should be four paral- Twb sheets of poste-ard(pr1111 lelograms in each of the four colors. bly, one sick white and the other si In the upper left-hand corner of each bright color) card, write the letter "T" in the color of Four sheets of construction paper, one the figure. This marks the top of the card. each of four different colors On each color of construction paper, Magic markers in four different colors draw the three different figures. Cut each (preferably, the same four colors as the of the construction-paper figures into four construction paper) k..". parts as in figure 1. \ Compass, scissors, ruler, and glue.

Constructing the game Cut out 48 three-by-four-inch rectangular cards from the poster board. Separate these cards into three sets of 16 each. Fig. I ,..;

%.. 1172 GEON11: FRY AND MEASURLNILNT 173 Now glue one fourth of each color of Playing the game the construction-paper figuresinitsre- spective position and in the corresponding color on each- card. Only one pieceis The deck is shuffled and one person glued on any one card. Figure 2 shows the starts the game by dealing eight cards to placement of the four parts of a parallelo- each player. The dealer then turns over the gram on each of four cards. On each card, top card in the deck and places the re- draw a dotted line to show the other three maining cards face down on the table. One fourths of each figure. person should be selected to keep score with a pencil and paper. The object of the game is to obtain two complete figures. A complete figure consists of four cards of the same figure, with ft a different portion of each figure colored all four parts would fit together to make a complete figure. The cards must all be turned so that the "T" is in the upper left hand corner. In making a complete figurer itis not necessary to use only one color. Sine:, this is very hard to do, extra points are given if a figure is completed in only one color. --1 Play begins as the player left of the fr dealer decides whether to use the card turned face up or to draw thz top card on the face-down pile. A player must decide which two figures he will attempt to complete, but he may change his mind at any point of the game, since he is the only one who sees his hand. For every card he adds Fig 2 to his hand, a player must discard one; therefore, he should have eight cards in his hand at the completion of his turn. After a player has put down his discard, (You can also =decorate the colored side the player on his left then decides whether of each card with the name of the game to use the last card placed face up or to and an appropriate design to make: it Took pick up the top card in the face-down deck. `,`professional" as in figure 3.) Play continues in a clockwise manner until one player Wins by getting two complete figures in his hand. The winner calls out "Geo-gin" and lays down his hand. He scores ten points for winning and an additional five points for each figure he completed using only one color. Any other player that has a completed figure in his hand in one color only, scores five points. The player on the left of the dealer then shuffles the cards again and the same pro- \ cedures are resumed until one player ob- tains 50 points. Paper folding and cutting a set of tangrampieces

STEVEN S. DICKOFF Montgomery County Public Schools, Rockville, Maryland

Stetrn Dukoll is an elementary mathematics teacher specialist in the Department of Supervision and Curriculum Development in Montgomery County. Alarylan.l. Ile has led many in-service workshops for dementia) teachers in the states of Maryland and New York.

Inarecentteacher-training workshop conducted by the author. a question arose concerning a \method for duplicating the seven pieces Of the ancient Chinese tan - gram puzzle without having to trace the pieces of anotherpuzzle.After some thought abouttherelationshipsoftie pieces in the puzzle to one another, the following paper-folding and cutting method, illustrateddiagrammatically,wascon- ceived. Start with a rectangular shsst:(1_pap' A as in figure I (usually 8' x II). Fold F C A B 1-ig2

AL F!)remains( tie.3). Cut along fold El) Congruent triangles, AEI) andDEP

A E

D C / 11g 1

edge AD to. coincide with edge1)Cas in figure 2. Cut off the excel', figure E/3CF, and discarditUnfold the shape. Square Fig

174 GI 0%11 I Rl AN.I) ASL, RUM' N I 175 result(see fig. 4 . Fold each of the tri- Place pieces numbered "I" and "2" in figure 6 asideThey arethefirsttwo E tangrani pieces Fold triangle EFD No that point r coincides %%ith point G see lie. 7).

D F Fig4 angles inhal! as in figure 5. Unfold the D II Fw. 7 Unfold tuangle D and cut along fold

F q: S

tri,int2les and L.ut ,Ilong fold ,ll; only II A F!g g -7 otd,Set piece numbered "3- m figure

F Fig 6 176 (;Amt's rst) I.S

S aside Fold figure /II /// so that point L The Chinese tangram purile has always coincides with point G as in figure 9 Un- been popular tt ith children. who, once they fold figure DEM Cut along folds IK and hat e suited it, are delighted to "fool their (;JSet pieces nwnhcred "4- and "5" in friends l'his same putzle may also he figure10 asideFoldfigure DGE/ so sawed from a piece ofI cinch pit wood or

II 'gasoline and the pieces kept ina shoe- Fig10 hot. along with different purrle shapes on that point G coincides with point Il( see aetitit cards, for the children's useAre. war. it,it's sure to capture the fig.11)Unfold figure DG./// Cut Aunt! ton- use interest of the children as it has for ages

fold J 1):: c..6. nninbercd6,ind "7in figure 12are now formed. hus,the tangram puirle is now complete withall seven piecesPIC pacts C.in now be plied hack together, as in figure 13, to foim the

square in figure 3 Fic 13

olt mum C otitti N i langramlike purrle for basic area formulas arc easy to make For example, to show that the area of a parallelogram and the areaof a rectangle are related. start with a parallelogram of two pieces

Rearrange the pieces to make a rectangle A game with shapes

DAISY GOGAN

Daisy GOSUIII is stocking on her din loyal program at Teachers College. Columbia University She has had etperience in teaching high cc Iwo!, serving as chairman of the mathematics department at Northern Highlands Regional High School in Allendale, Net, Jersey, and as an cissistant in the nu:the:mugs department at Teachers College,Columbia Unarsity.

....._, Duringtheir lunch period, Toni and his his turn again for the fifth square. He made friend Greg invented a new game they sure that he filled it in so as to make the called "Shapes." On a sheet of graph paper final shape a new one. He won that point. they took turns filling in one square at a The boys continued The game went time with a big X to see whether they could very quickly for several shapes, since they form various shapes. They made these had little trouble making them different. rules: Tom had just made a point by finishing this shape. I. They would toss a coin to-see who w....ild start the first shape. After that they would,take turns starting. 2. The "starter" could fill in any square he wished. 3. Each person, as his turn came, could then fill in any square next to one already and the boys were working on the next filled in. round. 4. They would stop after filling in five Greg placed the fifth square so that he squares. had this shape:

5. When the five squares were finished. 1 -,--1 they would examine the resultIf the shape were a new one, the boy who had finished it would win that point. If the shape were just like one already made, the boy who "My point," he said finished it would lose and the point would "Oh, no!" replied Tom "That shape is go to the other person just like the one I just finished " Tom won the toss to he the first starter, "How can you say that. Tom? Why, and they filled in the squares quickly this one uses three columns of squares, and the other one takes only two columns." Tom replied, "If you turn your paper clockwise, you will see that the one you It was Rim's point, since he :hidfin- justfinished takes only two columns. It ished the shape. and of course since it was is exactly the same as the other one. Let's the first shape it was a new one. use scissors and cut yours out so that we Greg started the second shape When can see that itfits on mine exactlyIf it four squares had been completed. it was fits exactly, they are the same'"

177 178 GANIFS AND PL Z/1 IS

Greg reluctantly agreed that Tom was right, and the point went to Tom. The boys continued more slowly now. , Tom was trying to keep his lead and Greg was trying to catch up. Tom made a point by finishing this ,/\; T shape: , p

Several turns later, Greg finishedthis one:

I -t

"You lose another point, Greg," cried Tom: "How come?If you try turningthis one, you'll get

The gameeame proceeded more slowly, as it became harder to find new shapes. But it was more fun; too, trying totrip each other up. and that's different." The game with shapes that Tom and "If I use scissors and cut out the shape Greg invented can be extended by having and flip it over, I can show you that yours the players try to establish a relationship willfit exactly on mineIt really is the between successive reflections and a rota- same shape. Or better yet. if Jane willlend tion or to find a method ofdetecting re- me a mirror, I can show you easily See flection without using a mirror. yours is a reflection of mine." "Yes,itisthe samer can seethat How many different shapes can you find? now," said Greg. Here is how the boys How would the game change if we used used the mirror to show the reflection six squares?

with other geomet- EDITOKIAI \ t Instead of using squares as the base shape, start plastic pieces ric figures, such as triangles orhexagons. This can be done with small paper or Give students a fixed number of the shapesand ask them to make as many regions as pos- units arnd sible, using all the given regionsI his ent.ourages thinking about nonstandard area the recognition that different shapes havethe same area

Four triangles Some configurations 5

An example of informal geometry: Mirror Cards*

MARION WALTER Educational Services Incorporated, Watertown, Massachusetts

Marion Walter is a part-time mathematics instructor at the Harvard University Graduate School of Education. She is on the staffs of Educational Services Incorporated in the Elementary Science Study and the Cambridge Conference on School Mathematics, She teaches mathematics to the students in elementary school education at the Harvard Graduate School of Education.

The need forinformal geometry, es- The problems range from the simplest, such peciallyinthe earlier grades,isbeing as the one shown above, to moredifficult' recognized by educators, psychologists, and ones, such as the one shown in Figures2 mathematicians. The Mirror Cards were and 2a.2 Some patterns are possible to created by the author to provide a means match and others are not.3 of obtaining, on an informal level, some geometric experience that combines the possibility of genuine spatial insight with a strong element of play. The basic problem posed by the Mirror Cards is one of matching, by means of a

mirror,' a pattern on one card with a FIGURE 2 FIGURE 2a pattern shown on another card.For example, can one, by using a mirror on the card shown in Figure 1, see the pattern shown on the card in Figure 1 a9

FIGURE 2h FIGURE 2c

Using the mirror on rho card shown in Figura 2, which of this patterns shown in Figures 2., 2b, and 2c can you mak.? FIGURI.I FIGURE I a

f Its work was begun while the author was work "Farb box of AI irror ( arils contains. in addition to mg during the summer of 1961 with the Fiementars mirrors. 171) cards ztrange.1 in fourteen different sets 'rcience StmlY, a protect supported by grants from the Although the instructions for the sets sary, the basic Nattonal Science Foundation and administered by Edu problem is the same for all the sets and is the one cattonal Services Incosporated a nonprofit organization described abuse A trial editiGn of !Mirror Cards was engaged in educationalresearch She would liketo produced and copyrighted by the Elementary Science thank the numbers of the group she worked %%WI that StudsIII Jone 1965!he% are being used on atrial summer and the group an optics of the protons Will basis in user 250 classrooms around the country The met for the:: help and encouragement, she is espr clans author %Judd liketo acknowledge the help received gratefultoProfessorPhiltpAforroson,Mrsphshs from Mrs A Nainian, Sirs F Myer. and Mrs J Wtl Singer, and sirs Lore Rasmussen Lams in editing the guide and producing the cards I he reader should have a small rectangular pocket he posttion of the pattern Watts e to the edge of mirror handy before readtlig on the card is to be ignored

.117) t80 GAMES AND PUZZITS

.10/.

o'?

We have noticed that the children usually and learn through experience the natureA find the colors and shapes pleasing and of the scientific method: enjoy the challenge presented by the cards. While moving the mirror around on the They do not think of this work as "mathe- cards, the children notice and experiment matics," and they often findthe cards with the position of object and image in stimulatingoverand above theactual relation to the edge of the mirror. The geometr!, involved. The cards may be a player can decide where to place the mir- means of reclaiming the children who al- ror; and he soon learns that he can control ready dislike mathematics or are bored its position, but that for any given position or frightened by it. The cards do not call of the mirror he cannot control the position thr verbal response from the children, and of the image! no mathematical notation is needed. Closer The students also learn that a mirror connection with science and mathematics does not carry out a translation. (See Figs. classes will be explored by the author in 3, 3a, 3b.) the future, since the cards can give insight intosomemathematicalandphysical principles. CI)]

One advantage that the cards have is I that the children can see for themselves FIGURE 3a FIGURE 3b whether or not they have made a pattern. FIGURE 3 They don't need to resort to authority to Can ono by using, a mirror on Figure 3 make dm check whdther they have solved the prob.- patterns showninFigures 30 and 3b? alas, the lem correctly.In addition, while playing mirror doss not carry out a translation! with the cards they'ne, in effect, constantly making predictions and are immediately They learn by experience that congruence able to test these predictions and amend of two parts is a necessary but not a suf- them, if necessary; and itis fun to do so! ficient condition for a pattern to be made Thus, while working with the cards they by use of a mirror. Most children do not should gain confidence in their own powers know the expression "symmetric with re- a

GI.ON41.1 R1 AND 11l'ASURI.N41 `r 181 spect to a line" or "reflection in a line " example, the circle. By putting the mirror They may, nevertheless, by using the cards, on a diameter they can see the whole circle. gain experience that will enable them to More than that, any diameter will do and understand the concepts that these expres- any chord not a diameter will not do. This sions describe.This does not imply that may gke young children their first fc'eling they could give, or should be expected to for a diameter of a circle, long before they give, a formal or verbal definition of these know the word "diameter expressions. Eventually they do notice that With the diamond, (see Fig6) they for a pattern to be reproducible by use of notice that there are two places where the a mirror, it must have two parts that lie mirror may he placed to enable them to on either side of some line and that these see the whole diamond On the other hand, must "match exactly." They soon learn, for example, that the pattern shown in Figure 4a cannot be made from the pattern in Figure 4, and they probably have a good feeling for why this is so.

Fintikr 6 FIGUR! 7

L the pattern shown inFigure 7 does not of FI6UPIr 4 FIGI RI 4a FIGUR! 4b havethisproperty--tothesurprise many! Or, again, take the triangle(seeFig Pattern 4a cannot be obtained from 4Whar-about the pattern in 4121 8). the children may notice that the effect of putting the mirror along AB is in some way "the same" as that of putting it along The cards provide opportunity to practice recognizing congruent figures and selecting parts of figures congruent to another.

FIGURI 8

BC, but that itis quite different from that Fi6ukt 5 FR,L.R1 5a of putting it along AC What about BD'' Other patterns on the cards, such as the ladybugs, arrows, etc , can he explored in Where must you place the mirror in Figure 5 to see the pattern shown in figure 5,1 similar ways For a few cards the children can obtain patterns that look somewhat like the one The children must be observant, not only required but are not congruent nor actually about a shape and the position of that 'amilar in the mathematical senseI intend shape, but also about its whirs.Ststne of to devise cards where congruent and similar the patterns match in shape but notin patterns are obtainable, and similar but not color. congruent ones They may also notice a variety of ge- Unfortunately, none of the present cards ometric properties of figures. Consider, for hat e Lades with arrows on them to show 182 GAmt s A \ D //I'S perhaps more clearly; that the orientation gets reversed under a mirror mappina or reflectionThus Figure 9 becomes Figure 9a. I 12 FIGURE. 12a The imagined placement of points "A" and "B" illus. trees the fact that the mirror does not "rotate" the figure. Actually the mirror "flips" the image. (Points "A" and "8" are not marked on the actual cards.) ru.t.ai 9 116l RI 9a

The concept of orientationis.of course. brought outb!,the cards, although the arrows arc not used for this purposeOften patterns with orientation reversed and not reversed are included to make the idea Examples taken from the more obvious bi(xlit 13 I.JouRi 13a ladybug and the circle set are n below

The cards may he used at any age level. they have been used by children as young as five and by sophisticatedprofessional -01 scientists or mathematicians.It is interesting to note that some adults who "know

HOUR! 10 FIGURI 10:1 Fag RI 10b

Can one by using the mirror on Figure 10 obtain the patterns shown in Figures 10a and 106 respectively?

FIGURE1 lb

Can one by using the mirror on Figure 11 obtainthe patterns shown in Figures I la and llbrespectively?

The fact that a mirror does not carry out a rotation in the plane is often masked by the symmetry of the figure.For ex-, ample, one can make Figure12a from Figure 12, but not Figure I 3a from Figure 13 GI Mil I R\ AND All.ASURINII.N1 183 all the rules" verbally (such as "There must at all whether a pattern can or cannot be be a line of symmetry" or "Image distance made, they will have a clear demonstration = object distance") often have more dif- of the power of 'reasoning based on ex- ficulty in working through the sets than periencei.e., that it is possible to predict children who have not yet memorized such results with confidence by thinking rather phrases. The one barrier to the effective than doing! (And they are able to check use of the cards by adults appears to be their thinking if they wish.) In this way an ingrained habit of respect for authority. they are savoring an essential part of the Adults often do not want to rely on their nature of rational thought. own C.Jility to see whether they have made There are many questions that still need a pattern correctly. to be answered. I mention just a few. Will When the children find the problems use of the cards make children moreob- becoming too easy, they may want to add servant about other geometric patterns? the rule, "You may put the mirror down Will it enable them to see figures within only once for each pattern," so that all the figures more easily? Does it improve their trial and error must go on in their heads. ability to visualize? Will they be able to They may wish to make some of their own describe patterns more clearly?Willit cards. When, as happens often, children help or hinder children with reading dif- are able to predict without using a mirror ficulties?

LIJI I. )R1.11 C (NMI i In ideation to the measurement concepts discussed in the v arious articles of this section. temperature. weight. and volume are measurement systems commonly taught in the elementary mathematics program. To encourage estimation of these and other measures, or to develop the connection betvkcen a characteristic and its measure. youmight tonstruct a measurement game board

Question piles Home

Spinet- Start Pioying board

for Players spin to identify the question stackfrom which to draw. Question stacks exist each of the following time measure. volume measure.weight measure. temperature measure. linear measure. and area measure I-ach questioncard contains a question and an indication of the number of spz .es to move if the question isanswered correctly You might use questions such as these "What time will it be two hours fromnow')" "How long is this room''" Popsicle sticks and flyingpolygons

CHARLES LUND The American School of the International School,of the Hague, The Netherlands

In the teaching of a unit on basicideas SAS11(11Sin''11(iNS from geometry to children in grades 5 -.S. Popsicle sticks can be used to create a mo- tit tivating. "hands-on," laboratory exercise. The following, exercises have been used with children in both the United States and the Netherlands. and the message of identification of polygons and patterns in mathematics has been put across each time The flying polygon pu/i.les and a few sample solutions are illustrated in figure 1

IM YING Ptinis I.. ryto construct a "flying mangle" using fist Popsicle sticks (No glue' 2 yto construct a "flying squat c' using sot Popsiele sticks

3 it-)toconstruct a "flying pentagon" using F-161. RI I eight PolsiLle sticks .1 ry to L instruLt.t"fly mg hexagon" using perform each nine Popsicle sticks ber of sticks necessary to What is the minimum number of Popsicle construction Here arc two interesting con- sticks neLessary Cu construct fly mg jectures that have beenformulated by my heptagon" students in the past: 6.0I there a pattern to thefly mg poly gon constructions If so, shat is /.//en "If a flying triangle can be constructLd using fivePopsicle stuks and aflying flow are these puizles used in the class- square canbe constructedusingsix room.' A dittoed copy of the directions in Popsicle sticks,thena flying pentagon I and a small container filled with and a 11:.ing, hexagon can he constructed figure usingseven and eightPopsiclesticks Popsicle sticks are placed in what I call a respectively (She vvasn't able to con- "thinkers' corner" near the pencil sharp- strus t them but feels certain she is correct ) ener of the classroom at thebeginning of Although no formal classroom IrmAat "A flying polygon skull any number of the unit requiredsidescanbeconstructedby discussion of the problems takes place. ,impi> adding mu) sticks to the previous interest m the puizles usually builds quite shapeI or example. a flying pentagon can be constructed by simplyadding to rapidly after the first day sticks to the model of a flying square" Students are encouraged to formulate (Six of the ten shapes he %vas able to conjectures regarding the minimum num- construct are pictured in fig, 2 )

184 Bons constructed by students, the only special equipment necessaryis a100-foot A 1 measuring tape. You may wish to have the C: .7--..... \-....) class make that,too! By breaking the 'class up into teams, each member can be kl assigned a different task to perform. Re- cording, measuring distances with hand- made and official measuring tapes, flying polygons, making repairs, and keeping records are a few possible jobs. All of my classes have found that the triangh con- structionwillsailthefarthest. My students say it is because "the flying triangle is the smallest and the strongest Try this series of exercises featuring simple materials. You'll enjoy them.

The Try-Angie Puzzle

4 (;11)11il.L .1 \ \ Irkl OMSchotrl, Elmwood Pad III

pIll alio% e pattern ut dot., '.t1g,g(--t- tilt 1 idea thattin a e to, he connected The :iii%er r i, kit do not take I/III ---1.1 IWIICII(Or 111:11kOil111(' pat leill.:11111do not r7 ) .1, ...`..' go Igo h or tetra' c a, .. hoc .., A - ..... - - _ ___,._ _ ...... IVIwil you compete th pattern-.m.- ,._ ..% ressfully.on w in :old von ',cow 16 pollit.

If you fall.;Indtallint , a trapthe ...,'- '.._). gany. euil:-. Your rearing i. Ipoint tom a emiiiilvi 'y A-.-- _, ,._.-- triangle: 2 3 point for 2 emnpleted 1(1.... 1-16t RI2 rind 1/3 point for oily mule ot any tilanalc In lower grade.. you may %%ant to .cure Fhe Popsicle-stick models of polygons i II term. of %%hole monk I. all '. poem. per can also be utilizedas a bridge to an triangleconipleteil. point, andopt exercise in direct measurement For exam- pond for }than.% Iilillpil 11IIiamtit - ple. conduct a contest to see which flying 1 totoid the :thot, pal tcl o to iii.:,t !mu- polygon will sail the greatest distance. 1 his Int mg thinking excursion can take place on the school It look ea.y- Nall 1 a Iliallengt to i !It- grounds or, if the weather is melem mt. m o althinking tillall.tialent.ItI, fun the gymnasium. Aside from the flying poly- ;old if 1 n11111111411(' -111 I 85 related to shapes, try gettingopinions on EDITORIAL.CONIMI,Ni For something provocative these two questions. do you see? How many squares do yousee? How many triangles

A P,ythag puzzle

G A Ry D..FIALLNorth Judson San-Pierre Schools, North Judson, Indiana

des'cribed above). Each This puzzle was constructed as aproject of the triangle of these figures has on areaof 36 square for a graduate course. IthaS been used in inches. They should be paintedthe same, N\ teaching sixth-grade mathematics. color. ' ose 1 square (side 8 inches)and Iparallelo- and 10 incheS; in The purpose of thisprOject is to teach gram (sides 8 inches angled of the Pythagorean theorem tochildren who angle equal to the larger acute Eacti of have no background inplane geometry. the triangle described above). 64 square The ideas of squarednumbers and the these figures has an area of the same concept of a 3-4-5 righttriangle are also inches. They should be painted introduced. color. The child is taught theserelated concepts through the use of abrightly colored manipulative puzzle that guideshim to form relationships involving area.

Materials inches. 8 1 right triangle with sides .6 inches, and 10 inches. e 1 square (side 6 inches) and Iparallelo- gram (sides 6 inchesand 10 inches: an -4 Fig. 1 angle equal to the smaller acuteangle

186 .r.. e GEOMETRY AND MEASUREMENT 187

1 square (side 10 inches) with an area of Empty the puzzle again and give the 100 square inches. child the pieces, substituting the smaller 50 squares (side 2 inches), each with art parallelogram for the smallest of the three area of 4 square inches; 9 should be squares.(Note:These two piecesare painted to match the 6-inch square; 16 painted the same color to facilitate the con- should be painted to match the 8-inch clusion that they are the same area.) The square; and 25 should be painted to child should then conclude that the small match the 10-inch square. parallelogram and the small square are the 1-frame like that shown in figure 1. same size by constructing figure 3.* After emptying the puzzle the third time, givethechildthelargestsquare,the smallest square, and the triangle and sub-

1 Procedure Fig. 4 El. -Ay the puzzle frame and give the stitute the larger parallelogram for the child the basic puzzle pieces (the triangle medium square.(Thesetwoarealso and the three large squares). He should painted the same- color and are the same construct a figure such as that shown in size.) The child should construct figure 4. figure 2. The fourthconstruction(fig.5)is

Fig. 5 ,.; e '....; \ `-,

188 GAMES AND PUZZLES in the fifty frame and Empty the puzzle and put formed by emptying the puzzle squared painted parallelograms, the small squares. The nine giving the child the two go in the small smallest square and the medium square, to match.the six-inch square compartment, the sixteen squarespainted and the triangle. Aftermaking figure 4, that the two to match the eight-inchsquare go in the the child should conclude middle-sized compartment, and the twenty- parallelograms are the samesize as the five squares painted to matchthe ten-inch largest squa hence the two smaller square go in the largestcompartment, ilso, squares are atthe same size as the largest put the triangle inthe middle. Have the square. the large coin- The next series ofconstructions is de- child take the squares from . .partment and fill up the twosmaller com- signed to reinforce the idea ofthe Pythago- introduce the idea of partments. Then have himtake the squares rean theorem and to compartments and squared numbers and the conceptof a from the two smaller- construct various designs,such as those 3-4-5 right triangle.

t. shown in figure 6, in the large compart- Enroa'sNOTE.Puzzlesgood onesare ment. helpfulindevelopinganalyticalthinking. Use of the puzzle would not necessari:, They should be readily accessible in all class- require a rigid lecture type of presentation rooms. They are particularly apptopriate in mathematics-laboratory settings where chil- such as has been outlined. If students are dren make originaldiscoveries and solve merely allowed access to the puzzle, many quantitativeproblems individually.---CHAR- of them will make interesting discoveries LO:TEW. JUNGE. in their free time.

...

I Tic-tac-toea mathematical game for Grades 4 through9 ROBERT A. TIMMONS Commack, New York

played with The knowledge of negative andpositive Here is a game that can be It equal enthusiasm in Grades 4 through9. numbers would not have to be very great. of The game changes only in the amountof is sufficient for them to simply be aware strategy used by the students in the upper their existence. grades. The only prerequisite for the students x 0 0 is that they have had the conceptof negative and positive numbersintroduced to x 0 them before the game and arc familiarwith the ordinary game of tic-tac-toe. x. x

189 190 GAMES AND PUZZLES

The game is played with the entire class. 3. Once you say a number you may not An overhead projector with aprepared change your mind. Think before you tell us your numbers, but if you take too long you will lose your turn. 4.You,are not allowed to help your team I members. (This rule can be altered at 1111111111111111111116111111111MIMON the teacher's discretion.) 1111111111111111111111111111RWM The game follows the ruleS of coordinate 1111111111111111111111111m1111111 111111111111111111111111 axes with the studentssupplying the two variables. The teacher should be careful that he or she does not count by pointing asiminaminamin to the lines but rather simplyplacing the marks in the correct place., co

11111111111111MIMI The first couple of games played will most likeiy be played and won allin the

ION 1111111111111111 111111111111111111011111 first quadrant. I have been surprised each time I have MiliaMillrnmainumns played this game it the speed at which grid the students discover how to locate the grid makes things easy. However, a is the student who well. points. Frequently it drawn on the board will do just as is having difficulty with his regular pro- The instructions for the game are pur- gram who is the first to discover it. posely short and simple. It willbe up to the participants to fill in the gaps. A 0 The classisdivided into two teams (e.g.. boys versus girls). The teamsdo X 7 5 number or ability. O 7 3 not have to be of equal X 3 3 One pupil is assigned the job as arecorder. O 9 7 He or she may or may notparticipate in X 4 2 the game. His job is torecord, in two O 9 6 that X 51 columns on the board, .the numbers O 0 0 are given to them bythe students. E.g.. X9 8 0 O 0 2 x 2 3 X4 9 0 4 O 10 3 x 3 0 X 2 4 the following in- O 1 5 The teacher then gives X 3 9 structions to the students: ,. 0,,9 5 X 9 4 1.This is a game of tic-tac-toe,but in O 3 6 this game in order to win you must get X 6 0 five "X" or "0" in a row. If a student should happen togive a 2.In order to tell me where toplace your "X" or "0" youmust give me coordinate that has been already given or number must be is not in the limits previouslyset, he is two numbers. Each informed that he cannot go there and that equal to or less than 10. Therecorder will write these numbers on theboard he loses his turn. "0" in In order to force the game outof the and I will place your "X" or reduce the the correct place on thegrid. Watch first quadrant, the teacher may if you can under- limits set in rule #2 to"numbers equal me closely and see The game will then stand how I place them. to or less than five." ,v f GEOMETRYAND'MEASUREMENT 191

0 020 1111111111111111BM X'24 0 1.1 OINR11111111113 11111 1 1111111111 X3 1111111111RMIN 04.1 11111111111MIIIMEPIPIIIIM X22 mommrammmum-mom 02 1 11111111M1Ill 11111111111Fi141141641111111M111 X 1 1 1111111M11\egr%Ira 010 1111111111l9llMidi&MINIM 1111111111MIIIMMINIME X24 1111111111Md 0-3 1 miamosm X-32 1111111111111111M1111 1111 R 00-5 IEM11E X00 0-3-3 X53 quickly come to a stalemate with the first 021 alreadygiven qydrant completely filled in. X114 05 At this point the students will urge the X52 teacher to tell them how they can get out 014 of the first quadrant. The teacher should X -1 not give in but rather keep on promoting 033 2 them on with questions like this: X-2 "Give me the right pair of numbers and it will gel you out of that corner. Try a matics, A Text for Teachers, by Robert B. different type of number. Think back. Davis. (Reading, Mass.: Addison-Wesley Didn't we learn aboutny other types of Publishing Company, 1964). numbers?" The value of this activity and others Sooner or later soone will come out likeitis the enthusiasm it generates. It with a negative numbc or a pair of .nega- allows all to participate no matter what tive numbers. The game canitreir continue their degree of competency, and lends with the previous limits or any limits set itself to the discovery method of teaching., by the teactier. with little effort on the teacher's part. If One of the difficulties with this game a student is unable to comprehend how to is .that the children would like to play it locate the points, there is no loss in the all the time. I have yet to find a class that mathematics sequence.If ,on the other tires of it. hand. he becomes proficient in locating I believe the activity originated in the points in all four quadrants, he has been Madison Project. A complete description brought to. thethresholdof analytical of it can be found in Discovery in Mathe- geometry.

: : Hide-a-regionN?-2 can play

JEAN S. OVERHOLSER Oregon State University, Corvallis, Oregon

Hide -a- Region is a game that can be by calling out ordered pairs, while the played by two or more persons, from the group that has hidden the region calls out first to the twelfth grade. Its purpose is to "Inside!" or "Outside!" in response to each give practice in locating points on a grid, trial. The region is located when the op- and in the concept of the area .of a region. posing team has named all four vertices. In its simplest form, all players are given A tally is kept of the number of guesses. graph paper or A grid. One group decides It is now the turn of the second team on the locatioa of a squareregion on the to hide a region. After; games, the grid, with the vertices at ordered pairs of winning team will have the fewest num- whole numbers. In figure1, a region of ber of guesses. area 16 is shown on aten-by-ten grid. For ayariation, a rectangular region of The other group tries to locate the region (1,10) (3.10) 10

8 7 6 5 4 (3,4) 3 2

1 0 0 I 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Fro. 1.A "hidden" square region of area 16. Flo. 2.A "hidden" rectangular region of area 12. ( tj 192 GEOMETRY AND MEASUREMENT . 193

5I 4

Flo. 3.A"hidden" rectangular regionofarea12.

a given area can be hidden as shown in that has hidden the rectangular regionsaysl figure 2. The group who hides the region "Rectangular region, area 12, hidden says, "Rectangular region, area 12." The the region where the x-coordinates go from solution is (1,4), (3,4), (3,10), (1,10), negative 6 to positive 6, and the y-coordi- in any order. nates go from negative 5 to positive 5." For another variation, the team that A further variation would involve put- hides the region could specify that the ting the vertices at points determined by boundaries are excluded. Then, if the op- ordered pairs such as (11/2, 23/4), (91/2, posing team calls out, a point on the boun- 23/4), (9%, 13/4), (11/2, 13/4) to .deter- dary, the team that hid the region can mine a rectangle of area 8. say, "Outside!" Here the vertices determine the region, although they are outside the region. Eirroa's No-re.Is it possible that some After negative numbers and three more regions could have the same area but have quadrants are introduced, the area avail- different dimensions? It occurs to me that the able for hiding a region is expanded. Figure region described as "rectangular, area 12," 3 shows a hidden region of area 12 with might bearegion 2 by 6 units as well as 3 vertices in all four quadrants. The group by 4!CHARLOTTEW. JUNGE.

EDITORIAL COMMENT.An appealing variation on Hide-a-Region is Hide-a-Name. And if you make the name hidden that of one of the children, it is even more exciting! Cartesian coordinatesand battleship WILLIAM R. BELL Boca Raton Middle School, BocaRaton, Florida

mathe(natical mes are a way of y relievinge d dgerr of practicefor some students, nds adaptation of an oldand familiar awhichisoftenplayed surreptitioly by students) can also be justified as peing educational. In the version of Battleship described here, thebasic difference ithat the intersections of grid lines are named instead of the squares between the grid lines. The game is best introduced after a study of integers and as a prelude to graphingequations of lines. The game is played on thestandard two-dimensional grid. (See fig. 1.) Students must undeistand that allpoipts on the grid areidentified by a pair of 'numbers (x, y), where x is counted horizontally andis -Y always named and ,located first, and yis Fig. 1 counted vertically and is alwaysnamed I), and located second. A few examples(lr his si. 1 at (2, 2), (-2, 2), (I, 1)may be appropriate. and on the other he records is allowed Limiting the size of the grid to six unitsin his opponent's forces. Each player consists of each direction from zero is best if the game three battleships. 'A battleship vertically, di: is to be completed within the normalclass three adjacent (horizontally, agonally, or on a corner) points. period of forty or fifty minutes. three Each player should have a sing,I.;sheet Players take turns firing volleys of it. The shots. (The decision rcgdi dingwho fires of paper with two grids drawn on When player spots his own battleships on onegrid, first can be made by any method.)

194 feammommi hitsare made, they are acknowledged his sheet. Notebooks placed on endcan immediately. A game ends when a player serve as improvised walls. has lost all three of his batkhips. Be prepared to be challenged by your During a game, players mast be placed students. You will be pleasantly surprised in such a way that they cannot see their at their ability and determination to bea opponent's sheet or the opponent marking the teacher at his own game.

EDITORIAL COMMENT.Rather than using battleships on the grid, you might consider placing fish on the oceah and have a "Fishing Rodeo." Whales might be five connected locations, mackerel could be four points, and so on to minnows, which are single points.

Holiday plot -dots

GARY A,. DEATSMAN

Moorhead State College, Moorhead, Minnesota

Here is an activity that will interest and of natural numbers to let points on a co- challenge third graders and advanced sec- ordinate system. Each point is labeled with ond graders. By carefully following written a number and when the plotting is done directions, each child plots ordered pairs the child has constructed a follow-the-dots

195 196 ---dAMES AND PUZZI.ES if the pic- puzzle which he can then complete.Direc- Accuracy is very important tures arc to look right.Giving each child tkins for three of these "plot-dots" are essentill;grying to given here. The first turns out tobe a an instruction sheet is class will Halloween pumpkin and thesecond a read the instructions aloud to the Christmas tree. The third, which isshown result ichaos. I would encourage the chil- rk in pairs or small groups and in figure 1, is a Valentifie heart. dren to child to check eacother's work. It seems to be . To do one of the plot-dots, each lightly so should be supplied\ with aninstruction futile to try to get them to write sheet and a piece of special graph paper. they can erase easily if they make an error, children Half-inch squarer must be used ifthe fig- but maybe it's wprili d,try. Some ure is to fitproperly on 81/2-by-11-inch confuse- "over" with "up," so sometpre- should be on the liminary practice in this area mayhelp. paper. The coordinates of be teacher- Sometimes demonstrating the plotting paper. The graph paper can blackboard helps. If made. I made mine very easily bydrawing some points on the children I worked the coordinates on a pieceof ordinary your pupils are like the their work graph paper and then making aspirit dupli- with, many of them will finish with complete accuracy and get verynice cating master.

19 ..

17 14 13 16 15' 14 13 12

II 10 9 18 7 UP 6 5 4 3 2

12 13 14 15 16 - 1 2 3 4 5 6 7 8 9 10 11 )OVER A. , ...? results. A few may get discouraged and 18. Go over 6 and un 1 for dot number 23. quit, so I would try to keep competition at 19. Go over 10 and up 9 for dot number 6. . 20. Go over 9 and up 12 for dot number 8. a low key. 21. Go over 6 and up 15 for dot number 14. When a child-come plot-dots he 22. Go over 4 and up 12 for dot number 15. should then be allowed to a d to his pic- 23. Go over 2 and up 6 for dnumber 19. ture-draw a face on the pumpkin, draw Directions for Valentine Heart Plot-Dots decorations on the Christmas tree, or write something appropriate ofthe heart. He I. Go over 13 and up 7 for dot number t3. 2. Go over 8 and up 13 for dot number 11. then can color his creation. 3. Go over 8 and up 3 for dot number 1. Although I can't prove it, it seems rea- 4. Go over 1 and up 12 for dot number 17. sonable to me that experience with plot- S. Go over 15 and up 12 for dot number S. 6. Go over 15 and up 13'for dot number 6. dots may help prepare children for later 7. Go over 12 isnd up 16 for dot number 8. work with linear measurement and graph- 8. Go over 2 and up 9 for dot number 18. ing. In any case, it's fun. 9. Go over S and up 16 for dot number 13. 10. Go over 2 abd up 1S for dot number 15. 1 I. Go over 14 and up 15 for dot number 7. 12. Go over II and up 16 for dot number 9. Directions for Halloween Pumpkin Plot- 13. Go over 11 and up S for dot number 2. Dots 14. Go over S and up S for dot number 20. IS. Go over 3 and up 7 for dot number 19. I. Go over 14 and up 10 for dot number 5. 16. Gp over 9 and up 15 for dot number 10. 2. Go over 12 and up 13 for dot number 7. 17. Go over I and up 13 for dot number 16. 3. Go over 4 and up 4 for dot number 19. 8. GO-over 4 and up 16 for dot number 14. 4. On over 6 and up 16 for dot number 1 I. 19. Go over 14 and up 9 for dot number 4. 5. Go over 9 and up 14 for dot number 9. 20. Go over 1 and up IS for dot number 12. .6. Go over 4 and up 13 for dot number 14. 21.. Go over d and up 3 for dot number 21. 7. Go over 14 and up 7 for dot number 4. 8. Go over 10 and up 3 for dot number I. 9. Go over 2 and up 7 for dot number 17. 1 10. Go over 3 and up 12 for dot number IS. 11. Go over 8 and up 16 for dot number 10. 12. Go over 7 and up 14 for dot number 12. 13. Go over 2 and up 10 for dot number 16. 14. Go over 3 and up 5 for dot number 18. IS. Go over 6 and up 3 for dot number 20. 16. Go over 10 and up 14 for dot number 8. 17. Go over 12 and up 4 for dot number 2. 18. Go over 6 and up 14 for dot number 13. 19. Go over 13 and up S for dot number 3. Stick Puzzle 20. Go over 13 and up 12 for dot number 6. 21. do over 10 and up 3 for dot number 21. Ifyou have 12 sticks of equal length arranged as in the diagram below, show how Directions for Christmas Tree Plot-Dots ou can make the following rearrangements. 1. Remove 4 stick and leave 2 squares. I. Go over 8 and up I for dot number I. 2. Go over 8 and up 3 for dot numter 2. 2. Remove 4 sticks and leave 1 square. 3. Go over 10 and up 12 for dot number 9. 3. Change 3 sticks and have 3 squares. 4. Go over S and up 12 for dot number 16. 4.Remove 2 sticks and leave 2 squares. S. Go over 7 and up 18 for dot number 12. 6. Go over 9 and up 15 for dot number 1 I. 7. Go over II and up 6 for dot number 4. \ 8. Go over 3 and up 6 for dot number 20. 4., 9. Go over 4 and up 9 for dot number 18. 10: Go over 13 and up 3 for dot number 3. H I 1 I. Go over 6 and up 3 for dot number 22. 12. Go over 8 and up IS for dot number 10. 13. Go over 5and up IS for dot number 13. r 14. Go over 11- and up 9 for dot number 7. [ IS. Go over 12 and up 6 for dot number 5. 16. Go over I and up 3 for dot number 21. 170 Go over 3 and up 9 for dot number 17. A Game of Squares Cross Figure Puzzle tleasures GEonQE JAxICKI GEORGE JANWKI .Ebn School, Millwood Park, III. --Elmiwood Park, Illinois

mill. past tern ninny-suggests a,game of I dots to any bright student. In my 14 S 7 Is classes, I used this puzzle asfollows: First set a time limit of oneminute. 19 31 (This makes it very exciting. The idea is to start at any 'ace and to 23, connect each (10, and to try tuL,,mplete at many whole squares as possible. ACROSS DEFINITIONS You do not permit any diagonal con- OFF 1-1 of a mile in feet nections. NEVER TAKE PENCIL 4-1/10 of a mile in feet THE PAPER!. 7A certain type of fire alarm You cannot retrace or cross any;previ- 8-144 minutes in seconds 9 t dozen ous line. 10.Number of ounces in 1 pound When you end up in a blind alley,the 11Baker's dozen game is finished. 14Ounces in pound (Troy system) 16A full day in hours The scoring is: 1 point for Acompleted 17-6 feet 5 inches expressed in inches square; 3/4 point for3 sides of a square 19-466 X1 +0 = completed; 1/2 point for two sides com- 21Largest 3 place whole number 22--,A 1/12 of a mile in feet pleted; and 1 point for one side since you 23-19 X3 X4 X4= might possibly have such an arrangement. The average score is: 7 points; thereal DOWN DEFINITIONS bright students reach 8 3/4. This game 1Product of 23 and 27 2-38% pounds changed-To0 nces is really fascinating. 3A penr.y in decimal form I recommend it highly to anyarith- 4-5 feet less 2 inches in inches metic teacher: hi grades 4, 5, 6, 7,8. 5=100 X2.71 = pointsior 6-3212 X I = (You can change the scoring 12One hour in minutes lower grade r.lw: silts to allwhole num- 13-3 feet in inches bers: 4 points, cS points, 2 points,and one 14-7,-44i gross unfamiliar with 15--2k dozen is how many units? point since they may be 17An odi number adding unlike fractions.' insuch eases, 18An e:en Lumber 48 is top score; see howhigh they can 20-5 feet tall is how many inches? make using these scoring rules.) 21Largest 2 place whole number

<198 Reasoningand Logic

Reasoning and logicare pervasive throughout all mathematics. The canons of logic form the bases of the deductive nature of abstract mathematics. Stu- dents naturally acquire logical proweis through reasoning in everyday affairs. This occurs throughout the elementary school years. Formal instruction in logical ideas frequently occurs in contemporary mathematicts programs in the late elementary and junior high school years. Most mathematical games require some degree of reasoning or logic (although some rely more on chance factors). Some games are designed specifi- cally to encourage the development of reasoning, and others depend heavily "on the development of strategy. This sectiOn contains illustrations of games of both types. The lead article, "Rainy-Day Games." illustrates four games designed to.. help students "learn an essential process of mathematics, namely, asking good questions and piecing information together to draw a conclusion." Be sure to follow the authors' suggestions about supplying props tothe stu-, dents--it'will help the students put structure on the situations. Ruderman's "Nu-Tic-Tac-Toe" is a variant on checkers, requiring strategies for timing moves, blockinvand positioning. The tic-tac-toe array becomes the playing board for the hexapawn ga medescribed in Ackerman's article, "Computers Teach Math." (You don't really need a computer for this game.) The article provides a good example of a procedure for analyzin games having a finite number of possible moves. Such gamesand their an lytical procedure's-demonstrate clew y that there is really much more th n motivation availableh the use classroom games. The game Kalah has been around for thousands of years. It has been used \in different versions in many cultures, particularly for thedevelopment' of quantitative judgment-ft is a game-in which victory depends solely on reasoning rather than on chance. ljaggerty's description of the game provides both a clear exposition of the rules and some interesting historical notes. For those with limited resources, be sure to note the inexpensive way to construct the Kalah board in the follow-up article by Brill.

199 200 GAMES AND PUZZLES

Other inexpensive strategy games are describedin Masset "Drill Some Fun into Your Mathematics Class." Massegives directions for playing as well as for constructing the game boards. Inaddition to the strategy features, addition-subtraction relationships .these games are useful for practice with and number-theory concepts. The last two articles are v rsions of the"Whodunnit" puzzles or riddles; "Jupiter Horse Race" is audent-created puzzlea reminder thatourstti- dents can be a rich sourcef legitimate mathematical activity. "Paper,Pen - cil,and Book," like Kalah,. ame of ancient origins. This onedates from medieval times. Games survive not only because theyprovide diversions for man but because the best oneswhet and test his intellectual appetite.

0 Rainy-day games \ ROBERT C.GESSE,k-7-4.11ROLYN JOHNSON, M.ARTY BOREN, and CHARLESSMITH

At-the time this article was written, allfour of the authoti were Miller Mathematics specialistswith the Fullerton,, ft ealifornia, Elementary Schools. Since then,both Robert Gessel and Marty Boren have left Fullerton.Robert Gessel is currently teaching mathematics in theAudubon !ardor High School in Cleveland, Ohio.

is There are _many occasions when "rainy; Pico means that one of the digits day- games", have their special value and right, but itis in the wrong place. usefulness, and, of course, rainy-day games Fumi means that one of the digits is may even be played on sunny days too. right, and it is in the right place. The games described here give children To illustrate how the terms are used, a chance to ask questionsthat enable you might pick the number43 and say, them to obtain information; the informa- "I have selected a number between 9 and tion is then used to reach a conclusion. 100; guessthenumber."Ifsomeone In playing the games. children frequently guesses that the number is 63, your answer ask redundant or useless questions. They would be, "That is one fumi," since the 3 realize their questions are useless when is a correct digit in the right place." You no'information is gained. Consequently. would then record 63 on the board with the children listen to classmates' questions "one fumi" next to it. Suppose the next to learn how to ask questions that will guess is 74. You would answer,"That is elicit useful information. This type of learn- one pico," since the 4is a correct digit but ing experience provides an opportunity for in the wrong position." Under tilt 63 you the ,children to develop an intuitive feeling would record the 74 with "onepico" for deductive reasoning. written beside it.If someone guessed 98, One cannot say that because of playing you would call it nothing,since it is neither these games students will improvetheir pico nor fumi. The 98 would be recorded arithmetic skills. However, they will learn with "nothing" besideit under the 74. anessential'processofmathematics. More examples may be given, if it seems namely. asking good questions and piecing necessary, to further establishthe answers information togethef to draw a conclusion. that may be given. The children ,will be actively invoked in an To actually play the game, the class is experience that encourages deductive rea- divided into two teams, and a member soning. Finally. the games are fun! .from each team, say Thad and Wendy, is picked to write a"seEret" number be-;' Pico-Fumi tween 9 and 100 on a slip of paper.They When the game of "Pico-Fumi" is pre- come to the front of the room.Since the sented to a class for the first time. itis teacher should record the answers for the best to begin with. the definitions of the first few games, you look at their, '1m- terms written on the chalkboard for easy bers; suppose Thad's is 25 and Wendy's reference as follows: . is 61, Thad picks a member, of his team 201 ,', , ii10/ G

202 . GAMESAND PUZZLES to guess Wendy's number. Ifthe person see a red hat. The game is then to seeif guesses 56, then you record."one pico" each student can figure out what the color next to the56 under Wendy's name.Wendy of the hat he is wearing is. then chodses a member of her team to Name Game guess Thad's number. If the person guesses 46, a 46 with "nothing" beside it would The "Name Game" is a fun game that be recorded under Thad's name. The game allows children more practice in logic and continues until one team gets two fumis, good question asking. It discourages wild which will be the correct number. The guessing and ragged thinking. progesslof a game might be recorded on The Name Game begins when the the board in a table like that shown in teacher selects a child. No one in the class, including the child himself, knows who figureI has been chosen. The teacher gives a clue. The children use the clue to ask a question Thad'sNumber Wendy'sNumber that will yield more information. If the nothing4,4677 56 --.1 pico question uses the clues the teacher has 1 Ulm: .--35 63 41 fume given, the yielding of information con- 2 picos--52 67 41 fumi tinues until the chosen child has been determined. The following is an example of 2 fumis4,25 69 --1 fumi how the game proceeds.

Fig. 1 TEACHER:The person I have chosen is not a girl. -dAt t4e end of a game, another person BOBBY:Is it Charles? from each team is picked to write down - No, but you used my clue in a secret number. Thus, a new gameis TEACHER: another clue. started. After one or two games, the stu- your question; so you get The person sits on the right side of the dents can record the guesses themselves. A few suggestions may be helpful.'Play room. T T 0, the game with a friend first in older that LAURA:.Is it Sherry? you clearly see some of thestrategies of TEACHER:No. Sherry sits on the right the game. Do not expect the students to side of the classroom, but is Sherry *a boy? develop good strategies at the start. Did you use both the clues? Sorry, no new clues. hats MARGARET: ISit Danny? frhe game of "Hats" requires six, hats TEACHER: No. although Danny is aboy artd a paper bag big enough to hold them. and sits on the right side of theclassroom. of the hats ate of one color and Good guess. The person sits in row one. threeThree are of another. NI. the purposes of JOSE:It's Bill! this explanation, let us assume that three Yes, how did you know? ,Of the hats are red and three are black. TEACHER: To play the game, select three students JOSE:In row one on the right side of Rind arrange them in a circle so that each the room, Bill is the only boy. may see the other. With thesix hats all TEACHER:Very good thinking, Jose. hidden in the bag, ask the students to close Number on the Shoe their eyes. Then remove three of the hats and place one on each head. Tell the stu- The entire class can play this game.The sturdy i: dents to look carefully atthe others but teacher chooses a student with a caution them not to give away what they shoe sole and asks the student tothink of and one hundred. see. a number between one Ask them to raise their hands if they The teacher records thenumber on the student's sole with a piece of chalk. The opportunity to listen, to verbalize, and to class then tries to guess the number by use and build on informationgotten from asking questions that can be answered yes-, students' previous questions. The teacher from the chit. or no., may want to record data The students usually begin by asking dren's questions on a number line on the questions like "Is it 73?" But eventually chalkboard. someone asks, "Is it in thethirties?" The latter type of question should be encour- These games give children an excellent aged by the teacher because it takesin a opportunity to develop gaine stategy and \greater'range of numbers. game planning. Overall, they Offer a non- The object of the game is, for the stu- threatening kind of total-class involvement dents to ask the minimum numberof by which deductive reasoning is exemplified questions to deteroline a classmte's num- through aquestion,--answerinformation ber,Iiring the game students.11,ave an further-deduction type of sequence.

is Nu-Tac oe

HARRY D. RUDERMAN Hunter CollegeHigh School, New York City

Games like checkersand chess offer the learn this game in less than one minute, child an opportunity to think aheadby get to grips with the challenge very consideritg alternate moves for himself quickly; and seem to &rive considerable as welLas for his opponent.Unfortunately, enjoyment in playing it. It is a game that in this fast-moving period many are re- I .invented recently and would like to luctant to sit down and ponder over such share with others. games, mainly because ittakes too much Directions for playing the game time\ ThO.game described here is a type of tic- 1 Start with four pieces marked with tac-toeame that has ingredientsof crosses and four pieces 7rarked with strategymmon to checkers. and chess: circles, and arrange them as shown on timing, blomg, position, anticipation. the twenty-square board in Figure 1. Moreover, ch. dren, as well as adults, 2 Two players play. One moves the pieces

203- 204 GAMES AND PUZZLES

0 0 0 000 0 0

0 0

a b Figure 1 Figure 2 with crosses, the other moves the pieces with circles,taking turns,first one player then the other. 0 3 A move consists'. of pushing your own 0 0 piece into an adjacent vacant square up or down, to the right or to the left, but 0 NOT diagonally. There is no "jump- O 'ing" or "taking" in this game, and if a Cr. square is occupied, no other piece may 0 be moved into this occupied square. a 4 The object, of the game is to place three Figure 3 of your pieces in the same line: vertically, horizontally, or diagonally without -any intervening vacant squares. A variation Examples of wins .for the circles are One player places all eight pieces on the shown in Figures 2a and b. Examples of board into squares in any artamgement he no wins for the circles are shown in chooses, one to a square. The other player Figures 3a and makes the choice of moving first or seleond.

EDITORIAL COMMENT. A quick but intriguing game is the game of NMI :There are many variations on this game. One of the simplest involves three containers of beads. Container A has five beads, container B has f6ur beads, and container C has three beads. Two persons play, alternately taking beads from the containers. When it is a player's turn, he may take as many beads as he wants from a single conOner as long as he takes at least one bead. The person to take the last bead is the winner. (The strategyfor winning uses con- ! ceOts from binary numeration.) Computers teachmath

JUDY ACKERMAN West Boylston, Massachusetts, SchoolSystem, Worcester, Massachusetts

order to maintain a high degreeof diagonally to the left or right to capture In captured marker interestin my sixth-grade mathematics an enemy marker. The classes, I .decided to introduce ayear-long is taken out of play. The game is won in class project. Each child belongs to a com- three ways: mittee and is. able to help with thework. 1. By moving a marker to the oppo- Our.. project for the yearis to learn nent's side. about, and, build game-learningcomputers. 2. By capturing all of the opponent's The classes are workingwiththe simplest markers. kind of game-learning- computersones 3. By planning moves so that a point that play games that can becompletely is reached in the game where the oppo- design analyzed., The ultimate goal is to nent is unable, to move. and buildagame-learning tick tack toe computer. The first phase ofthis project in- As the class learned to play hexapawn, volved studying and Wilding asmall game- they observed that a draw was impossible learning computer for hexapawn,2 a game and that there was a decided advantage of the players. (See if you can designed by Martin Gardner. for on discover which player has the advantage.) When the class became familiar with the game, they charted all the possible combinations of moves that could face the computer during any game if the computer went second. (There are only twenty-four possible Combinations.) For each combination, arrows were drawn in differ- FIGURE 1 ent colors to illustrate the choicesthat the The game of hexapawn isplayed on a Move 2 3-by-3 board with threemarkers for each player. (See Fig. 1.) The markers are movedin the same way as the pawnsin chess: \ 0t4;4 1. A marker may bemoved forward ,:x45', one squaretoan empty square. 2. A marker may bemoved forward A combination diagram FIGURE 2 Journal C. E. Shannon. "Game PlayingMachines." computer faces on each move(Fig. 2). of theFranklinInstitute,CCLX (De:Amber1955). of 447-53: The "brain" of the computer consists IMartin Gardner. "MathematicalGamest"Scientific twenty-four cups which weresuspended American, CCV1(March 1962). 140-43. 205 206 GAMES AND PUZZLES

last move of the game. If the computer lost,it was punished by permanently removing from its memory the marble that corresponded to its last move of the game. The other marbles removed during that game were replaCed. As the members of the class took turns playing the computer, the rest of the class was busy keeping records of the results of The hexapawn computer each game. In this way there was an ex- FIGURE 3 cellent opportunity to introduce a unit on in the side of a cardboard carton (Fig. 3). graphing. The class found that the line The combination diagrams were stapled tograph was one of the more effective ways the cups. All the possible second moves,of illustrating the results. (See Fig. 4.) fourth moves, andsixthmoves were grouped together. HER'S RECORD nomerriguillaminnntens4MMA The "memory" was made up of marbles 1:11mmtimma imillim 1.4 SI" placed in the cups. The number and colors Eirigilihigliennig'l iprirIPU'Irr SR OMR of the marbles in each cup were determined Sus O..'aam V EXIMMIIMUNM MIMEO dbilbtkOW NEUVEllilin by the number and color of the arrows nugsw.BOISEMrs SOMMOMEMIUM r.dr :MUM ONORIMITNIMMVI on each combination diagram. %AS SIN ilinemUM ONOMMIM fall 'NOV MOIMMERIMMOUNIM The learning took place as the class *Op ossums sippomossmosaiss sussussimummmisussommussso played the computer. The human player always startedfirst. The operator then Games Played, located the cup that contained the present board position. He randomly removed one FIGURE 4 marble from that cup and made a move The experience thit my classes have for the computer that corresponded to thegained from building the hexapawn game- color of the marble. The game continuedlearning machine has given them many until there was a winner. When the com-ideas for the design and construction of a puter won, it was rewarded by replacingticktacktoe-learning computer. They are all the marbles removed during the gamemost enthusiastic and willing math stu- and by adding an extra marble to cor-dents who look forward to coming to respond to the marble removed on theclass.

EDITORIAL COMMENT.-- Be sure to make a file of all thebrainteasers" you run across. Put each one on a separate file card and keep in yqur puzzle corner. Students can pull out cards to work on wh have a spare moment. For example: Ca circular pie into eight pieces by making only three cuts. KALAHan ancient game of mathematical skill

JOHN B. HAGGERTY PublicSchools, Al elrose, Massachusetts Mr. liaggerty is an instructor of mathematics in the Calvin Coolidge Schoolin Melrose.

A. large number of elementary and of probability. To the best of my know- secondary teachers of mathematics ,have'ledge there are but a handful of games found it advantageous to use games as part whickdo not, and most of these are listed of their instruction. The use of these games in Recreations Al athematique, Volume III, is much more than a mere whim upon the pike 105 ...written by Lucas."t teacher's part. It has been shown that In the October, 1963, issue of Scientific games, when properly used withinthe American, page 124, two of these games classroom environment, do actually pro- are described, together with two gamesof duce an improvement in the ability)rnown more modern origin. The two older games as quantitative judgment. are The French Military WarGame and We realize that this term, quantitative Tafl, a game of Scandinavian origin. judgment, is inclusive; there are many These games combine the characteristics facets involved. The important point is, of extreme simplicity and unusual strategy however, that the pupil is led by involve- based upon immediate decision depending ment in the game to make a decision based solely upon the last move by the opponent. upon the relative position of his opponent. We have used all these games through- The past few years have seen a sudden out the system in Grades 1 through 8 with upsurge in the use ofmathematical games some limited success. The gameslacked in -in the classroom. More teachers than ever most cases an essential element which we before are becoming interested in the use felt was highly desirable: cultural signifi- of games in teaching. More people than cance and a relationship to thehistorical ever before areplaying these games. development of the number systems, Games such as chess and checkers are in- bases, and systems of numeration. teresting, and do involve to some extent Dr. Kelley did not know back in 1947 the quality which this articledescribes, that there was, indeed, a game which pos- namely, quantitative judgment. There are sessed all of the essential elements to pro- games which do even betterin this regard.. mote the development of pure reasonin For severatyears we experimented with the form of quantitative judgment with- various games in the classroom. What we out a trace of chance or probability as itis were looking for were gamesin which the knowninmost modern mathematics element of chance was minimal or absent books and courses. Further, and equally altogether, These are extremely hard to important, was the fact that the game was find and, in\many cases, are not altogether as old as civilization itself andhad been practical frem an educational pointof continuously played throughout the Near view. and Far East for seven thousand years. Quoting from notes which the author The name of this fascinating game was made in a class in advanced statistics at Kalah. While we were busy taking notes Harvard University in 1947, "Most games I Truman L Kelley. "Advanced Statistics," HarvardUnl- now known contain atleast some elements venFili. 1947. e- 207 " 208 GAMES AM) PUZZLES in Dr. Kelley's_ classes,less than fifty Troy; he found African chieftains playing miles away from us, in Holbrook, Mas- for stakes of female slaves, and mahara- sachusetts, 67-year-old William Champion jahs using rubies and star sapphires as was developing a modern version of this counters. He finally traced it back some ancient game after a lifetime of personal 7,000 years to the ancient Sumerians, who historical, as well as archaeological, re- evolved the six-twelve-sixty system of search throughout the cotintrics of the keeping numerical records.* eastern half of the world. These people inhabited the fertile val- It is a fact of recorded history that leys between the Tigris and Euphrates many games which were popular at one rivers in that part of the world presently time or another became lost in antiquity. known as Iraq. This has long since been Many of these games never appeared accepted in archaeological circles as the again. We only know about them through birthplace of civilization. archaeological research. So far as we After having searched for several years know, the game of Kalah is the only game for the game which was most adaptable to continuously played in widely separated the modern math approach, I was natur- parts of the world on at least three conti- ally intrigued by the Time article. A letter nents from the time of the first civilized to both the periodical staff and the author country, Surneria, down to the present. drew unexpected responses. Time granted The following quotation from the June unqualified authority to quote at length 14, 1963, issue of Time gives us a vivid from the article entitled "Pits and Peb- account of the background and ancient bles," but most pleasant of all was the origin of this game. surprise of receiving a call from the sub- Carved on a vast block of rock in the ject in person from his home in Holbrook, ancient Syrian city of Aleppo are two-fac- Massachusetts. He thanked me for my ing ranks of six shallow pits with larger letter and invited me to meet him at the hollows scooped out at each end. The same Boston Public Library where the game was design is carved on colurrins of the temple being granted prime exhibit space during at Karnak in Egypt, and it appears in the month of September in the Sargent early tomb paintings in the valley of the Gallery which is devoted exclusively to Nile. It is carved in the Theseum in subjects of documented archaeological and Athens, and in rock ledges along caravan cultural significance., routes of the ancient world. Today the Naturally, I went to meet this remark- same pits and hollows are to be foundall able man, who at the age of 83 can still over Asia and Africa, scratched in the bare run at full speed up three flights of stairs. earth, carved in rare woods or ivory inlaid He spelled out for me the fascinating with gold. story of his travels over much of the earth's In 1905, the year he graduated from surface in his quest of the ancient origin of Yale, William Champion read an article this game which is still widely played in about an exhibit of African game boards Semitic countries of the Near East and at the Chicago Exposition of 1893 in which Africa. the author noted that Kalah "has served He told me he first became interested in for ages to divert the inhabitants of the game that his porters on an African nearly half the inhabited area of the trip were playing in the sand. pe place globe." Fascinated by the failure of such a was the Makahlai Desert of Africa. These pandemic pastime to catch on in the U.S. Arabs called it Mancalah,Vr "The Game and Europe, Champion began tracing its of Intelligence." migrations and permutations. "Two players sit behind the two ranks He found an urn painting of Ajax and of six pits on the board between them. Achilles playing it during the siege of Courtesy Tian.Copyright. Tune, Toe.. 1963.

^ REASONING AND LOGIC 209

Each pit contains three (for beginners) or pebbles can be used in playing, much as in six "pebbles" (which may be anything the game of hopscotch. The mathematical from matches to diamonds). Purpose of the implications are, however, much more im- game is to accumulate as many pebbles as portant than those of hopscotch. possible in the larger bin (kalah) to his The modern version of the game is right. Each 'player in turn picks up all the played upon a wooden game board which pebbles in any one of his own six pits and is made in the loft of a factory in Hol- sows them, one by one, in each pitaround brooks, Massachusetts. Some 24 styles of the board to the right, including, if there boards in all arc turned k,ut. The simplest are enough, his ownkalah, and on into his board is routed out of ponderosa pine by opponent's PITS (but not his kaiak). If automated machining. The counters are a the player's last counter lands in his own species of Italian bead which are extremely kalah, he g9ts another turn, and if it lands durable and adaptable to vegetable dyeing in an empty pit on his own side, [only) he procesies. The most elaborate of these captures all his opponent's counters in the boards is curved in the forth of an original opposite pit and puts' them in his kalah Nile skiff, a craft which is so common to together with the capturing pebble. The that area of the world. game is over when all six pits on oneside or The city of Boston's 160 playgrounds another are empty. It is not always an have 1600 of these game boards. Los advantage for a player to go "out,"since Angeles, Cleveland, Chicago, and New all the pebbles in the pits on theopposite York playgrounds are also equipped with side go into the ,opponent's kalah.The this highly interesting educational-diver- score is determined by whohas the most sionary aid. pebbles. Each player cleans out his own The most frequent question asked by kalah at the end of the game and replaces parents is, "Will this game increase my three pebbles in each pit on his ownside. child's mathematical ability?" Our most MI pebbles left over represent the margin frequent answer is not lightly stated: of victory."' "Kalah is the best all round teaching aid in This method of tallying enables even the country." We are not alone in this be- very young children toplaythe game and, lief. The Harvard Graduate School of Edu- although they are unable to conatfor- cation also thinks highly of the game and mally, they know whether they won or is publishing a teaching manual to cover lost by this tally. li they endwith excess not only the rules for play but alsothe counters aft,cripadiog each pit,they know ancient cultural origins of the game. Also intuitively tliitt they have won. It is pre- discussed in the Harvard manual are the cisely this intuitive outcome ofthe. game implications of the game for the teaching which we feel is one of theimportant'outof "modern mathematics." comes of playing the game. One must again recognize that the A sophisticated player learns not to term "Quantitative judgment" is a gen- accept all short-termadvantages, how- eral term which has several facets. Among ever tempting.Sometimes the early gain these is one which, for want of a better is wiped out in the later stages.It is this term, might be defined as a "quality of the fact which adds to the fascinationof the mind to make a specific decision baso:1 game. We would invitethe reader to play upon the array of theopposing position. one full game to checkthis fact for himself. This decision is entirely intuitive and is Figure 1 on page 329 is a sketch of the based upon a grasp of special relationships layout of the board. These pits can,be within the scope of that which is loosely scooped out of the ground, andordinary defined as' 'human intelligence.' "2 Jerome S. Bruner. TA. Protein of Education (Cambridge: Courtesy Time. Copyright. Tim.. Inc.. INS. ffarsard University Prow. 196!) P. et 210 GAMES AND PUZZLES

This ability seems to be best fostered in against there being two identical games in children by providing them with problem- a lifetime. *. solving situations whichrequire them to The following demonstration game has otp use whatever natural intuitiveskill they been found effective for teaching even possess. In this sense the game ofKalah is very young pupils how the gameis actu- purely mathematical, no element of chance ally played. An interesting projection of enters in, and the basic rules are so simple this game, which the classroom, teacher that even a young pre-school child can will find useful in introducing the game to play the game after a short .demonstra- large numbers of pupils in Grades 1 tion. through 12, will be described later in the We have found that this ancient game of article. Kalah not only induces very young child- In the game illustrated Player A began ren to think quantitativelybut, also by moving the.three pebbles in his pit A4, developsintuitivedecision-making so ending in his Kalah and thus earning necessary in problem-solving. Italso helps another move, which he used to play from pupils think computatively. The method pit Al, ending on empty pit A4 and there- of playdistributing counters one by one by capturing B's ;men. By similar moves to the rightconfirms and structuresthe andl captures, A, by the foUrth turn,has habit of moving from left to right as in beconi pebble-proud with eleven ;ti hist reading and writing. bash a pathetic one in B's (seeFig. Small children learn to count without 1). The sequence of moves in the sample using their fingers; to distinguish units from multiples; to assess special array in a physical order to objects as well as to acquire other mathematical concepts in a natural way while playing an interesting and absorbing game. The four fundamental processesin math are all used in determining the number of counters in the twelve pits at any Figure I. The board after Al given time. This attribute isalso used in computing the score. Pure reason isvital to victory. The strategic move must.bd game is A4, A1-B6-A3, A2, Al-114, 136, immediately evident to the player in- B1-133A6-B2, 131. By the fourth turn A volved. is. dangerously concentrated inthe two Thereisabsolutely no element of pits, A5 and A6. II, seeding six pebbles on chance in this game. It is purely a gameof his own side, forces A to startdistributing skill, as it has always been. The opponent his hoard around the board. By theeighth moves, and the playerresponds intuitively turn (see Fig. 2) A still hastwelVe in his to the situation at his ownlevel of sophis- Relish to five in B's, but. B moves the five tication. The number of counters in eachpit can be increased to six or reduced to onefor pupils of widely varying mental ages.The use of increasednumbers of counters creates so many variablesthat no mathe- -matician could compute thecombina- tions. No doubt some Oftoday's larger computers could be programmedto do this, but the probabilities are sogreat Figure2. B'swinning move REASONING AND LOGIC 2 1 1 We feel that the pupils' inability toread pebbles in and then has only to move B1 to capture is no justification foromitting problems the single pebble in his pit The teacher A's seven remainingpebbles, ending the from these early programs. it by a score of 21-12. ran present thesesituations orally to the game and winning of the We have found that the use of ademon- pupils. The game of Kalah is one of accomplishing stration game such as thatjust dmcribed best nonverbal means is.the most practical way tointroduce tbf this end. pupils. In our own case In addition to its value as adiversion game to a group of the intuitive two applications ofthis method were and as a means of developing adopted at different levels ofinstruction. abilities so importanttoolem-solving, equally valuable. For groups of pupils inGrades 4 and up there is another4mteiime This outcome is the recognitionof the we used an overheadprojector with the lialahs and the pits drawn on theplaten close identification of the gamethrough- overlay. We used corn seeds as counters out the histor y ofeiviNation with the which appeared on the screen asirregular development of systemits of montmtion number. This opaque pebbles. Asthese objects were hud the concept and idea:. manipulated upon the platen, each move outcome alone a mildmake it a valuable lab- was clearly visible onthe overhead ;111-Ce 11_ acti% it y for pupils in the mathematics to the watching group. oratory. With younger pupils who ha% e amuch At the %cry lea:t it makes theclassroom shorter attention span we foundthat an a woreinteresting place. Pupils are given actual demonstration game on ateaching an opportunity toidentify themselves board which used separate colorsfor each closely with a game whichcertainly has a side of the board was more interesting. rich 'cultural heritage. The game was taught developmentally Ve do nut cc ish to imply, however, that while the children observed, asking and mathematics can be completely taughtby answering many questions. This demon- mans of a game or anyseries of games. stration was followed by several practice We merely think that this game ofKalah games between pupilsselected from the is one way of producing somedesirable class. The two basic rules of the game were outcomes in the teaching ofmathematics structured by discussion and additional since pupils usually play it with great demonstration: We then paired' off the interest and enthusiasm. whole class at boards previously set up. The u fderstanding, enthusiasm.and all Considerableinterestresulted, andIt interest that Kalah engenders is not at seemed to be entirely spontaneous. remarkable a1101 Omit' reliAtiers that the it has long been felt in professional cir- game has persisted for seventhousand cles that experiences in problem-solving years in a idely separated areasall over and the development of intuitive pom yrs in the globe. Ve tend to agree with the solving problems have been sparingly used ancienthistorianIlerodotus. who was in pre-school and Grades 1 and 2.One of quote! as having said, "A cultUral heri- the difficulties has been that many young tage may be lost once and yet beretrieved pupils do not read a ell at all. some teach- by someone of persistence and dedication, ers have attempted to surmountthese but to ice lo-,t It may prove to be irretriev- difficulties through extensive use of dia- able.-' As a result of our own ,brief grammatic and pictorial representation. perience a ith this fascinating game we This procedure, a bile useful in sonic situa- certainly hope th:'t other school pupils tions, is generally not the aay to start. We throughout the country a ill soon be given think now that a mathematical game such a chance to play it. as Ka lah has more value. irdwur. Gmat avg.Sena, I V, M. A project for the low-budget mathematics laboratory: the game of kalah

RANDALL L.- BRILL

A teacher' 01 3ixth-grade mathematics at St. John Vianny School in Northlake, Randy Brill has studied with Dr. fames Lockwood at Northeastern Illinois University.

eachers often see a kit or game that they, materials to fit your/Situation. For instance, feel would be ideal for use in the mathe- egg cartons are a possible substitute for the matics-laboritory activities of their class- cardboard and paper cups, and marbles or rooms but find it financially out of reach. beads might be used as counters.) By using d little imagination, they very CIA all the paper cups so that they are often can make similar kits or games and 1/2 inch to 3/4 inch deep as shown in fig- at a much lower cost. ure 1. Kalah, a fascinating count-and-capture game, is easily made and offers endless opportunities for the student to develop his basic arithmetic skills while having fun. It involves the players ir. a strategic and logical pattern of thinking. We are told that this game, which was firs: played in ancient Africa, was found to be so simple and fascinating that it became Fig. 1 a popular form of gambling. It is saidthat men even, determined their empiresand harems over a game of Kalah. Although Two large containers, called kalahs, are itstill needed. To make a kalah, prepare three of no longer used for such purposes, the cups as shown in figure 2. One cup holds a great fascination for young and old 'alike and his spread to many parts of the should be trimmed on both sides and the world. Following is a list of materials needed to make this game: 1: One piece of stiff cardboard approximately 8" x 24" 2, Eighteen paper cups (the 4 oz. Si7C is best) 3. Glue and cellophane tape 4. Dried beans to be used as counters (You may want to vary the sizes or the Fig. 2

6 .212 REASONING AND LOGIC 213 other two cups should be trimmed on one of the board, he captures all his'opponent's them side only. Taping these three cups together counters in the cup opposite and puts provides the large container that is used to in his own kalah, along with his capturing gather the playing pieces during the`game. piece. In figure 4, player A has emptied His last (Other aVailable materialsfor example, cup A-3 for his opening move. and so milk shake or cottage cheese containers counter landed in his own kalah, can be used to form thekalahs.) he is entitled to another move. He now Afrange the remaining twelve cups in empties cup A-6 to capture the opponent's two rows of six on the cardboard sothat counters in cup B-4 because his own cup the members of each pair aredirectly A-3 was empty when hislast counter across from eachother as shown in figure 3. landed init. All the counters from B-4 and the single counter- that landed in A-3 go into his kalah. Once a counteris placed in either kalah, it remains there until the n0000CDOn round is ended. A capture ends the move, and play goes to the opponent.

(-5000000U Player B Fig. 3 8-I 8-2 8-3 8-4 8-5 8-6

LP The two rows should be about twoinches K apart. Glue the cups into place.At each a end Of the cardboard glue one of thekalahs. B Aimed with your beans, which will serve A-6 A-5 A-4 A-3 A-2 A-1 as the counters, you are nowready to play a game of Kalah. Player A The game is played by two playerswho Fig. 4 sit on opposite sides of the playingboard. Each player deposits three counters in each of the six cups on his side of theboard. A round of play is over when all six cups The object of the game is to collect as many on one side are empty.The other player adds the remaining counters in his cups to counters as possible in thekalah, the large container, at each player's right. the ones that are in his kalah. The.score is The method of determining who moves determined by who finishes with the most first can be decided by the players. Each counters. If the winning player has col- player, in turn, takes all the counters that lected 23 counters, his score is S because are in any one of the six cups onhis side each player begins with 18 counters. The of the ,board and distributes them oneby winning score for the game can be deter- one in each cup going todills right. If a mined according to the situation or playing player has enough counters to gobeyond time allowed, but a score of 40 is usually his kalah, he distributes them in his oppo- the goal. nent's cup, skipping the opponent'skalah. As you can see, the rules of the game are Those counters now belong to the oppo- quite simple, but play can become very complex depending on the abilities of the nent.. There' are two important elements that opponents. Lower grades or slow learners give the game its challenging strategy.If can begin learning to playby simply putting More experienced the player's last counter lands inhis own one counter in each cup. kalah, he gets another turn; if hislast players or faster students can play with as each cup. counter lands in an empty cup onhis side many as six counters in

J SJ -.Pt: 214 CiAMES AND PUZZLES

Three general objectives in having pupils creases or decreases according tothe moves play the game of Kalah are ( I ) to offer that are made, and increasing the number a recreational learningexperience, (2) to of counters in the cups at the beginning sharpen basic computational skills and con- of the game encourages new strategic pat- cepts, and (3) to involvestudents in a terns. situation in which they use a logical pattern The board and counters can be used as other concepts, of strategy. a model to demonstrate In addition to being played as an inde- even without playing the game.The count- pendent activity, Kalah is very effective ers in a cup can become a setand the when used as a part ;-,f a specific lesson or board a universe, or using simple modifi- guided activity. 4, may be used to present cations on the board, such as markings or concepts as wellas to reinforceskills. coloring, can be helpful in discussing place Counting can be demonstrated in playing value and grouping. Creative individuals the game and tallying the score,addition may find many othervariations of Kalah and subtraction are used during play as that will prove effective and interesting in the number of counters in the cupsin- their classrooms. ,

classroom games can be created wising setsof objects variously EDITORIAL COMMhNT.--M any of these called "logical bfocks" or "attributeblocks." You can makean inexpensive version colors,, materials with file cards and the threeattributescolor, number, and shape. Use two red and blue; four numbers, 1, 2,3, 4; and three shapes, * Using these characteristics, combinations of the three construct twenty-four attributecards representing all the possible attributes. For example, you will have cardslooking like these: 40/

3 2 0000 blue red 1 0 0 0disks stars

Place one card face up on atable top. Distribute the 24 cards to asmall group ofchildren. their hands following the game Students then take turnsdiscarding (one card per turn) from involves discarding cards if they aredifferent in one and only rules. For exampie, one game the cards is the card on top of the discardpile. The first person to discard all one way from the is like the top card in one and winner. Another game is based onthis rule: Discard if the card only one attribute. the rule "different in two Students can make up their ownrules. The student who made up way." ways" soon realized that this wasequivalent to the rule "alike in one Drill some fun into your mathematicsclass

MARIE MASSE Central Elementary School West Newbury, Massachusetts

Would you like to drill some fun into 19, any of which leaves the next movethe is your mathematics class?Put aside the winning play. After discovering that 16 usual mathematics drill and get out an a crucial point, aplayer must then figure electric drill, some scrap wood, some golf out how he can control theplay so that it tees, and some dice. From theseinexpensive he can land there every time. Gradually materials you can develop a varietyof becomes *al- that the multiples of four are mathematics games, even some that ac- the key to winning every time.The sly complish the same results as the usual player learns that it's not only polite to let an unusually your opponentplay first,it's also good mathematics drill,butin be varied by enjoyable way. strategy.' The game can chancing the number of holes on theboard Project 1 Materials needed A piece of wood measuring approximately 1011 1" by 3" by 8" in which two parallel rows 9 12 of 10 holes each are drilled (see fig. 1), and 8 13 two golf tees. 7 14 Object of the game 6 15 To land exactly on 20. 5 16 Player A moves his tee one, two, or three 4 17 spaces; player B moves his tee one,two, where A landed. Play 318 Ior three spaces from continues alternately until someone !ands 219 on 20. l20 The strategy, which some children see almost immediately, is to land on !6thereby Fig.I forcing your opponent to land on 17, 18, or 3'

216 GAMESAND PUZZLES I or by taking moves of one throughfour, 20 or more, spaces. 20

Frolct 2 Materials needed A piece pf wood measuring approximately 1" by 4" by 4" inihich a square array of nine holes is drilled (see fig. 2), three white tees, three red

0 0

Fig. 3

- Project 4

Fig. 2 Materials needed A piece of wood measuring I" by 6" by 6" with lines marked and intersections drilled Object of the game as in figure 4, thirteen white tees,and one To get your three tees in row. other colored tee. Each player gets thre6 lees. Players take turns placing tees, one at a time; on the 'Board. When all six tees have been placed, play continues by players alternately moving tees one at a time and one space at a time in any direction until someone is able to ge'his three tees in a row.

Project 3 Materials needed A piece of wood measuring approximately 1" by 3" by 10" in which two parallel rows of twenty holes each have been drilled (see fig. 3) and numbered fromIto 20; two white dice, one red die, two tees.

Object of the game Fig. 4 To reach 20, or beyond. The players take turns throwing the dice. On each throw, the player moves his tee Object of the game forward the total of the white dice and To pen the fox (colored tee). backward the number on the red die. The The name of the game is "Fox and player who first finishes his play on 20 or Geese"the white tees are the geese and beyond wins the game. A player does not the odd color, the fox. The fox is placed in need to end up on 20 to win. in the center (see star in fig. 4). The geese

. c, .)4 are lined up on one side as shown in figure 4. should be able .to trap the fox. This is a strategy game in which the geese With these as a starter, children can try to hem in the fox. The problem is that design their own games. Children can mark the fox can jump any lone goose and remove out the boards to be drilled even though, him from the board as in checkers. Double generally, they would not be allowed to and triple jumps are allowed. Clever geese use the power drill.

,.,

Jupiter horse race

ELINOR J. WRITT

Brunswick Junior High School, Brunswick, Maine

Eighthgraders at our school recently is presented here as it was submitted by had a great deal of fun trying to solve a the student. problem in logical reasoning that was in- Jupiter hers* race by Oliver Batson vented by one of their classmates. Perhaps your students will enjoy solving such a (Do not be put off by the weird things puzzle, and they might possibly be moti- you see in this puzzle.) vated to create similar ones. The problem Fill out the chart below.

Stall Name of Color Color Sex Place Owner number horse of ribbon Age of hair

First

Second ,

Third

Fourth 1 .:.

Fifth

.# 4217 218 GAMES AND PUZZLES wasn't the same color ashis/her coat. Clues: 23. Runner lives next tothe 3rd-place foal at the age of 1. Jupiter horses all winner. three, with one exception. 24. Purple horses alwayswin hor e,,shows. 2. Prepon has themiddle stall. 25. When Preponflies, you 1e hh/her 3. First place is green. against the clouds. 4. Curly cane secondin his/her heat, 26. Mr. Peters bet $50t1t Curly would but did better in thefinals. come in second. 5. Speedball lost toRunning Thing and 27. 4th place brought ablue ribbon. Prepon by an inch. 28. All Jupiter horses raceat the age of his, horse's ribbon, 6. Mr. Smith washed one. and his white socksturned pink. 29. Curly won -a prizefor one-year-olds 7. Mr. Peteis collected5th-place money. last year. 8. Curly is a celebratedstallion. 30. The electronicpictureput Prepon 9. If Running Thinghad come in fifth, ahead of Running Thing atiny bit. Runner wouldn't haveplacled. 31. Runner has thecolor of a tree trunk. 10. Prepon hatedRunning Thing, and 32. Mr. Evans jeeringlycalls Speedball a kicked his/her wall. flower. 11. Prepon loves Curlybecause of his/her 33. Mr. Evans got $5Q on abet he made polka dots. on his horse. 12. Speedball was born twoyears before 34. Running Thing is inheat. the latest leap year. 35. Speedball was thehead of a wild horse mixture of black 13. Prepon's ribbon is a band. and blackish-black'. 36. Running Thing wonfirst prize for 14. Mr. Jones' favoritecolor is violet. beauty two months ago. 15. Mr. Peters andMr. Evans were seen 37. Mr. Davis owns afemale horse wlIr betting together. is one of the fivewinners. 16. This was Prepon'sfirst race. 38. Runner is a year youngerthan his/her 17. Running Thing isthe same age as mother, a Jupiter horse,when she his/her stall number,less one. foaled. 18. Runner foaled lastmonth. Note 1: The stalls arein this order: 1, 19. Runner is his/hermoth9r's first foal. You must find out which shoWs are put in 2, 3, 4, and 5. 20. Winners of horse horse goes in whichstall. stall two. Note 2: Placing meanscoming in first, 21. Speedball lives next toPrepon. second, third, fourth, orfifth place. /2. Runner wishes thathis/her ribbon

Solution. Color Color Stall Name of Sex of hair horse of ribbon Age ______Place' Owner number - polka Curly green 2 First Evans 5 dot _ _ ----- white l'repon black Second Dais 3 purple red F Third Smith 2 Running Thing 3 violet Speed- blue Jones 1 Fourth ball brown brown 2 F Fifth Peters gunner A Coin Trick Suppose you have nine coins identical in size and appearance but one is counterfeit and heavier than the other eight. How can you with a simple balance scale locate the counterfeit coin in just two balance tests? For your first step place any three on one side of the balance and any other three on the other side. Now complete the test and find the counterfeit coin. 5.

Paper, pencil, and book

HENRY LULLI

Henry Clay Junior HigySchool, Los Angeles, California

This is a mathematical game that has makes it easier for the students to mem- been used successfully in the classroom. orize. The game uses a box with twenty- It was tried with various levels of seventh four counterspaper clips can be used as and eighth grade classes at an inner-city the countersand three items: a sheet of school. paper, a pencil, and a book. (Any items The origin of the game dates back to could be used. Cards with the names or medieval times. The earliest reference is in pictures of the items could also be used the eighth century. Many versions of the if desired.). ,game have been published and sold since To play the game, choose three students, 1900. A,B, and C. GiveAone counter, B two In the present version of the game, the counters, and C three counters out of the code key has been alphabetized, which box. A fourth student turns his back to 219 220 GAMES AND PUZZLES in the the first three stu- Suppose three counters remain the class while each of is CANOE. This dents chooses one of thethree selected box. The code key for 3 the paper means that thefirst student, A, took the articles. The student who chose student, B, took the then takes from the box ofcounters as paper; the second book; and the third student,C, took the many counters ashe was given initially; pencil takes twice pencil. the one who chose the vowels. Each he was given; and the Each code key has three as many counters as of the three student who chose thebook takes four vowel corresponds to one itemsin this case, A correspondsto the times as many counters ashe was given. each taken paper, E to the pencil,and Q to the book. After the three students have code key of counters, the The order of the vowels in the the appropriate number three items. fourth student turns aroundand counts the indicates the distribution of the Notice that there is no codeword for 4. counters remainingin the box. With the It is impossible forfour...counters to remain analysis of the Number of in the box. Why? A careful Articles counters Code AO' situation provides the answer. Paper, pencil, book Take any .of the six possiblecombina- Pencil, paper, book 2 BEACON tions of the numbers of counterstaken by Paper, book, pencil 3 CANOE A, B, and C. (See fig. 2.)The total num- 5 ENCOMPASS Pencil, book. paper 6 FLOATER Book, paper. pencil ber of counters remaining inthe box after 7 GOLDLEAF Book, pencil. paper the first 6 are distributed (1 toA, 2 to B, Fig.I and 3 to C) is 18. If the sumof the numbers of counters taken byA, B, and C is aid of the code key infigure 1 he instantly subtracted from 18, the differenceis from tells which student tookwhich article. 1 to 7, but never 4.

Number of counters Counters left over taken by in box References A B C Mathe- 4 12 Ball, W. W. R. and H. S. M.Coxeter. London: Mac- 6 8 3 matical Recreations and Essays. 2 2 2 12 millan, 1942. 6 Mathematics Magic and Mys- 2 6 4 Gardner, Martin. 7 1956. 3 4 4 tery.New York: Dover Publications, 5 New York: 3 2 8 Gibson, Walter.Magic with Science. Grosset lc Dunlap, 1968. Fig. 2

- C Multipurpose Games and Puzzles

By the timeyou have reached this point, you have probably said to yourself many times over, "You know, if we just changedthita little, we could use that game for ..." To encourage this kind of thinking, we included in this last section a collection of games whose utility seems multifaceted. Articles appearing in this section were selected because I. they describe several games; 2. they show how one game has many uses; 3. they describe a game with a multipurpose potential. By this time, you have also found that games wisely used in the mathematics classroom provide- 1. both diversion and motivation; 2. a technique for accomplishing the necessary repetition for sound drill and practice; : 3. an intellectually honest means for developing new mathematical ideas. As you read through the articles of this last section, keep in mind the more se- rious purposes of instructional gamesinitial teaching, drill, and eval- uationas well as the obvious and much-desired char icteristic of fun. To create multipurpose games for mathematics instruction, we borrow heavily -from games and diversions in our ordinary lives. We use action games, especially for the early grades. We adapt all kinds of games that use playing Cads. Athletic gamesbaseball, football, the Olympicsare a rich source of ideas. Television has more games than you can count on yourfin- gers. Children are fond of creating their own games patternedafter popular TV shows. Familiar games such as dominoes, checkers, and Monopoly are highly versatile. And the ever-popular crossword puzzles and crostics find their matches in cross-number puzzles and "mathematicalosterms." Where would we be if we couldn't find some use for dice (which you may prefer to call "number cubes"). Familiar .eaching aids, such as the geoboard and geometric shape cutouts, provide the raw material for many activities. For those who enjoy brainteasers, there is a multitude of number oddities, codes, and riddles to ponder. Even the bulletin board can do a turn as a mathematical game board! We hope that you have found some of these games useful and that you have been able to create some of your own. Rememberbe sure to share your ideas with other teachers. After all, this collection was created from the ideas of classroom teachers. 221 ' #, 4 1'.. l'-'....) Active games: anapproach to teaching mathematicalskills tothe educablementally retarded

GEORGE R. TAYLOR and SUSAN T. WATKINS

As associate professor and chairmanof the Department of Special Education at Coppin StateCollege in Baltimore, George Taylor teaches methods courses inspecial education to both undeigraduate and graduatestudents. He has taught both educable and trainable retarded childrenin the District of Columbia Public Schools. Susan Watkins is a teacherof the educable mentally retarded at Shady Side ElementarySchool in Prince Georges County, Maryland.She has used the active-game idea for a number of years in herclasses.

Pragmatic experience has shown that maximal level in skill and conceptdevelop- curriculum adjustment must be madeif ment in school subject areastraditionally .(Humphrey schools are to meet the everydayneeds of oriented to verbal learning educable mentally retarded children.Many 1970). indicates retarded individuals have been unable to Research (Cruickshank 1946) differences in the cope 'successfully with the'curriculum of that there are significant retarded children the regular grades; they are toodifferent ways by which mentally from the average child to adjust tothe and normal children learnmathematics. children tend usual academic demands. For this reason Educable mentally retarded mathematical skills and various forms of instructional procedures to learn certain have evolved. "Active games" are an ex- concepts better throughactive games than through traditional approaches andmedia. ample. iie illustrative The active-game approach to learning is The following activities major concerned with how children candevelop examples of active games related to skills and concepts in various school sub- areas in elementaryschool mathematics. ject areas while actively engaged in game Number system situations. Although all children differin Objective one or more characteristics,the fact remains that they are more alike than they The student will demonstratehis ability are different. One commonlikeness of all to match numerals and setsof objects with children is that they move and live in a eighty percent accuracy. mobile world. The active-game approach to Activity: Hot spot learning is based essentially on the theory numerals 1 that children will learn better when learn- Pieces of paper with the phys- through 10 are placed invarious spots ing takes place through pleasurable There should ical activity; that is, when the motor com- around the floor or play area. with the same ponent of an individual operatesat a be several pieces of paper

4 -a:222 MULTIPURPOSE GAMES AND PUZZLES 223

numeral. The teacher has a collection of Addition large posters picturing various numbers of Objective different objects. He shows a poster to the class. An overhead projector can also be Given numbers from 1 through 9, the used to show different quantities of objects. student will correctly add them during a The children must identify the number of specified period of time. objects on the poster and then run to that Activity: Addition tag numeral on the floor. A child gets a point every time he finds the right numeral. Any Each child is given a card bearing a child who is left without a spot gets no numeral from 1 through 9, which he keeps points for that round. Any child who has throughout the game. One child is then five points at the entl of the period of play chosen to be the tagger. He may go to any is considered a winner. child and tag him. The tagger then adds his number to that of the child he tagged, Expected outcomes and so on. Children are helped to count from one through ten objects and then to identify Expected outcomes numerals that represent the number of ob- Such a game provides practice in adding jects. After a game, the posters can be put for those who need it most. There are no on display around the room along with the penalties for children who make mistakes correct numerals. Procedures may be re- and have to drop out of the game. For peated for children who do not master the children who score perfectly for several skill. days, some other enrichment work or t_l R . i"arN --,,,

224 GAMES AND PUZZLES

games should bepiadned while the other teacher may elect to-have a child be the children are playing addition tag. one in the center callingthi problerps and bouncing the ball. In such case, it should st. Subtraction, be emphasized thatall children in the circle should be given a chance to catch ,Objective the ball and answer problems. When presented with simpleproblems in subtraction, the child will demonstratehis Expected outcomes understanding by giving the correct answer This game is a little easier for children eighty percent of the time. to play than the regular game ofcall ball, where they have to be very quick to re- Activity:Musical chairs member the answer to a giyen problem. Chairs, one less than there are children This activity helps to provide the repeti- playing the game, are arrargedin a circle, tion necessary to develop quick recall of facing rut. As the music plays,children the multiplication facts. Addition, subtrac- walk, run, or hop around thechairs. When tion, and division facts might also be used the music stops, all children tryto find a for this game. seat. The child leftwithout a chair drops 1 out of the game-andtakes a chair out of. Fractions also the circle of chairs. The teacher may Objective time. take out two or more chairs at a his knowl- The activity may be repeated asmany The student will demonstrate the teacher. edge of simple fractions by being able to times as seems appropriate to verbalize values, to know that fractions are part of a whole, and to expresswholes Expected outcomes in simple fractional parts..., Children see, concrete examplesof subtraction; . chairsaretaken away. The Activity:Hit or.miss , teacher may have the childrendetermine each time the Children are divided into teams. Each the number of chairs left team is given an eraser, which is set on Results may be recorded game is repeated. a table, a chalk tray, orthe floor. The on the chalkboard,and later on chart specified distance review at their seats. teams stand in rows a paper, for children to from the erasers. Each child is given three or fou: erasers. He tossesthe erasers, Multiplication trying to knock down the one set up on Objective the table. As each child plays, hecalls out Given a pair of one-digit numbers,the his score, expressing it as a- fraction.One student will demonstratehis knowledge of hit in four tries equalsone-fourth; three simple multiplication facts. hitsinfour attempts would be three- fourths, and so on. The team withthe Activity:Call ball highest score wins. The The childrin stand in a circle. Expected outcomes teacherstands in the center of the circle with a ball. The teacher calls out a com- Children can learn to use the relation- bination (for example, 2 x 3) andbounces ship between successful and unsuccessful develop an the ball to a child in the circle.The child attempts at making points to children must try to catch theball and give the understanding of fractions. The those situations in correct answer before theteacher counts to can use this concept in shooting ten. One point is givenfor knowing the which they are practicing skills; fact and another for catching theball. The baskets, for example. ,

MULTIPURPOSE GAMES ANI) PUZZLES 225

Measurement gets to the chosen hour, the chase begins. Hour chases Minute clockwise around the Objective outside of the circle. If Hour can catch The student will demonstrate his knowl- Minute before the children in the circle edge of linear measurement (centimeters once again count up to the chosen number, and meters) by being able to use a meter he chooses another child to become Hour. stick to measure distances from various The game can also be played by counting points. by half-hours.

Acti Vity: Ring toss Expected outcomes A regular ring toss game can be used Children are exposed to and given prac- for this activity. A meter stick is needed. tice in telling time by the hour, as well The class is divided into two teams. Each as, developing and understanding the term team has a ring, and both teams try to clockwise. Children may play this game ring the same post. The first child on each independently to review or reinforce pre- team takes a turn tossing his ring. Each viously learned skills. child then takes the meter stick and measures the distance between the postand his Monitorysystem ring. The ring closest to the post scores Objective one point for the playci's team. Children should use centimeters or meters, and Given symbols that represent coins (pen- tenths of hundredths of meters. The team nies, nickels, dimes, quarters, and fifty-cent with the highest score wins. pieces) the student will demonstrate that he recognizes them and that he can cor- Expected outcomes rectly use their values in different com- Each child has the opportunity to de- binations. velop his measuring skills when he deter- Activity: Banker and coins mines the closeness of his ring to the post. To get additional practice, a child could The teacher has a set of signs denoting check his opponent's measurement. the basic coins or their values: 50, five cents, nickel, penny, 10, one cent, and so Mina tie:s on. One child serves as the Banker. The Banker calls out a coin. The children then Objective run and group themselves with otherchil- The student will demonstrate that he can dren until their group amounts to the value correctly and with little, hesitation tell time of the coin called. Every child who is a by the -hour and that he understands the part of a correct group gets a point. term clockwise. Expected outcomes Activity:. Tick -cock The children become aware of the terms The class forms a circ!e that represents for and value of different coins. The chil- a clock. Two children, called Hourand dren get practice in combining different Minute, are runners. The children in the amounts of money to arrive at a specific circle chant "What timeis it?" Minute amount. To check that a group is correct, then chooses the hour and callsit out the teacher can helpThe-whole class count (for example, sixo'clock). Hour and itout. In this manner children can be Minute must stand still while the children helped to see how it takes two nickels to in the circle call "one o'clock, two o'clock, make one dime, or five pennies and a three o'clock ..." until they reach the nickel. And they learn that "five cents" is time that Minute chose. When the count the same as "one nickel." .

... study of mathematics than themajority of Implications children, itis suggested that the useof Inplanning for learningthrough active purposeful active. games for thedevelop- games, anattempt is made to arrange an ment of mathematical conceptsis a very active learning situation sothat a funda- effective instructional process. mental skill or idea isbeing practiced in the course ofparticipating in the active game ,situation.Actiftlies are selected on References degree of involvement the basis of their Brueckner, Leo J. How to MakeAlithnictic of (1) a skill or conceptin a given school Meaningful? Philadelphia: John C.Winston physicalactivity Company, 1947. . subject area, and (2) appropriate to thephysical and ,social Cruickshank. William M. "AComparative Study given group of of Psychological Factors Involvedin the Re- developmental level of a sponses of Mentally.Retarded and Normal 7 children. Boys to Problems inArithmetic." Doctoral Carefully, selected active gamesprovide Dissertation, University of Michigan,1946. experiences. Hood, H. "An FxperimentalStudy of Piaget's purposeful and pleasurable Numbers in Chil- actively and give Theory of Development of They involve the learner dren."BritishJournalofPsychology53 the child the opportunityto dramatize the (1962): 273-86. idea. Active games reinforce aconcept and Huripkwey, James H. ChildLearning Through the repetitive Elementary School PhysicalEducation. Du- develop skills :by providing buque, Iowa: William C. BrownCompany, retarded chil- drill that educable mentally 1966. dren need. ind Dorothy D Sullivan.Teaching Slow Since educable mentallyretarded chil- Learners Throufg, /lime Games.Springfield, Charles C. Thomas Publisher,1970. dren are more severelyhandicapped in the

More games forthe 'earlygrades

EDWINA DEANS

the weight and recordhis guess. Then the Guessing and checking object is weighed to seewhich children came closest to thecorrect weight. Chil- Playa game ofguessing and checking the of differ-, dren are encouraged to usegood judgment weights of different children and example, if a child Let several chil- in making guesses. For ent objects in your room. weighed stands beside dren lift an object in turn.Each will guess who has just been 4 i 6 226 MULTIPURPOSE GAMES AND PUZZI ES 227 the child who will he weighed next, there is Hauling lumber some basis for comparison. Individual children can benefit greatly If children are to improve in the ability from stimulating game-like experiences to make approximate estimates, they must which are designed to help them overcome discover and keep in mind that large ob- some difficulty in number understanding. jects are not necessarily heavy nor are Anita Riess gives a report of a retarded small ones' always light. They must learn child of eleven who was helped to build up to, estimate weight by actually lifting ob- an understanding of our number system jects or by some visual comparison of and of money values by means of a game similar objects. necessitating the use of money and of tens. "Guessing and checking" can also be A miniature lumber yard was set up and played with measuring. Mark off nine feet the lumber (wooden sticks) was bundled along the floor to serve as a model measure into tens. This enabled the lumber man to just as a ruler or yardstick. The door, a ship his lumber more easily. A toy truck window, the chalkboard may be measured was used for making deliveries. The boy, and marked as other model measures. pretending that he was the truck driver, Provide rulers, yardsticks, and tape for did the bundling, the loading and unload- measuring stationary objects. Guess the ing, and kept account of his deliveries. At width, length, and height of 1, arious ob- the beginning he drew heavy black lines to jects, record the guesses, and measure to stand for each bundle of ten and thin lines determine correct measurements. to stand for the single sticks. Payment was Many of our decisions w ith regard to made in pennies only. or in dimes for the number experiences are reached, not by bundles and pennies for the single sticks. the pencil and paper method, but by rapid, Finally, figures were used to sum up the rough calculation which gives a working number of bundles and single sticks which estimate of the amount. Any number were checked with the number of dimes learning which leads the child to improve and pennies. his ability to )cason sensibly and arrive at a fairly aeourate estimation will proveof I'm thinking of a number infinite north to him. One child says, "I am thinking of two numbers that make six. What are they?" Button game The leader answers "No" until the To play this game, assemble a paper correct guess has been made. The child cup, a paper plate, and six to tenbuttons who guesses correctly then Lakes the place (two oe three different colors will facilitate of the leader. the recognition of groups). Glue the paper Some concrete means of working out or cup exactly in the middle of the paper demonstrating wrong ausw ers should be plate and allow it, to dry thoroughly. provided for the children For example, if Two to four children play. The players a child should say, "Are you thinking of take turns throwing the buttons one at a four and three?" the leader will challenge time into the paper eup. The exact dis- him to prove or demonstrate by a request, tance to stand from the plate is deter- ouch as, -Show us four and three. See if it mined, measured accurately, and marked. could be six." Scoring for this game is done according to The child will use sticks, block?, or other the desired scoring device. Each child available materials to demonstrate and to playing will determine these facts: "How check his thinking. many did I throw into the cup?" "How This game can be played by any number many did I miss?" "Do the ones in the cup of children. After it is well learned by the and the ones outside equal the total num- children, itimay he played in small groups ber I had to begin with?" of six to ten to give all children an oppo- 228 GAMES AND PUZZLES the circles tunity for active participation. It canalso and matching the pieces. When children name the sets be adapted for use with largeraddition are all together, the combinations and with columnaddition on each of the twopieces of the puzzle and the total number represented onthe disks. and multiplication. other one in a "I am thinking of twonumbers that If one player can catch the mistake, he can take the circle.When all make thirteen." "I am thinking of three numbersthat the disks have been named, the person who has the most disks is the winner. make nine." other "I am thinking of twonumbers that Similar puzzles can be made from multiplied together will maketwenty- shapes, such as squares and oblongs. four." children be Shooting rubber bands It is recommended that Games invented by teacher andchildren encouraged to use thelanguage form often great "six to serve a spikific purpose are "eight three's are twenty-four," or kindergarten immedi- fun for all. A precocious four's are twenty-four," which morning method of youngster arrived 'at school one ately suggests to the child a with a broken broom handle and asupply checking his answer by the useof draw- of rubber bands. lie wasanxious to dem- ings, a counting frame, or asimilar device. rubber into play as onstrate his skill in shooting the The guessing element comes handle. The which of the bands off the end of the broom the childr:In attempt to guess within given num- teacher, wary f his ability to keep combinations that make up a bounds, drew a target on tagboard.The ber the leader has inmind. The teacher that outside area oiletirclewas painted red, helps the child analyze the guesses the next circle blue, and theinside circle show lack of understanding sothat the profit white (Fig.2). The children and the individual child who needs help will teacher decided that the littlecircle should from the game. count the most because it washardest to hit. Values decided upon for thewhite, Number puzzles blue, and red areas were five,four, and Makeas01, of disks the same size.Draw sets on the isksfourdots, five crosses, three points, respectively. six squaresfive ears, etc. Divide each ,,, RED circle into wo parts. Be carefulthat no ..-- BLUE 0003 two circlesre divided inexactlythe same RED way (Fig. II). BLUE co' g 4 WHITE goo° 5 --WHITE

Figure2 The target was placed on thefloor. A chart on the blackboard served as a reminder of the number values. Thechildren took turns shooting the rubber bands.The idea was to hit the bull's eye and get a score of five ifpossible. Interest in this Figure 1 game lasted for days. turned a Two children work together aspartners. An ingenious teacher had potentially troublesome situation into a They mix the pieces well andplace them face down. Then they take turnsdrawing worthwhile learning activity. A deck of cards, a bunch of kids, and thou

PETER K. GURAU Springfield College, Springfield, Massachusetts

Peter Gurau is an assistant professor of education at Springfield College.

One of the more frustrating problems deck of cards from which all the jacks, faced by the kindergarten and first-grade queens, and kings have been removed. In classroom teacher is the problem of pro- each initance, divide the deck into the viding for individual differences. four distinct suits and give each of three All too often the range of initial abili- children suit, keeping one suit aside. ties of a kindergarten class is just large Now arrange the cards in the extra suit enough to invalidate the best-planned les- in order of size, from ace to 10, and place son. In arithmetic, for example, we may them face up on the table. Explain to the have one child who cannot even count children that the object of the game is to when he enters the kindergarten or first- match all the cards in the players' decks grade classroom, while another may actu- with the cards already on the table. Each ally be able ,to do simple addition and child in turn places a card from his deck subtractida, and more. The problemis on top of the appropriate card on the complicated by the fact that youngsters of table. The first few times, the children this age group have not as yet learned should receive their suits in order of size how to sitstillfor any great length of so that they will be able to associate their time, and thus the more we can individual- rote counting with the game. Each child ize their instruction, the better the chances will then in turn place his first card (the that we will be able to hold their attention. ace) on the ace on the board and call Finally, if a lesson can be presented in out, "One!"' In the same manner, they pure game form, the lesson is much more will next place their 2's on the 2 on the likely to go down smoothly and without board, and call out, "Two!" emotional or academic indigestion. After having the children play this game The following series of card games is once, twice, or three times, you can add a offered almost as a curriculum in begin- variation by having the cards preshuffled. ning mathematics.It presupposes only Now each child will come up with a dif- that the youngsters have all learned to ferent card)and should try to call out the recite the counting numbers up through right name for the card as he places it on 10an event which is usually quite easy the right card on the board. However, no to achieve through a series of counting great stress should be placed on calling songs. out the correct name of the numeral, since ,The stage that follows rote counting is usually considered to be symbol recogni- I It should be noted thatlittletrouble is en- tion. To implement the first stage, seat countered in the ambiguous labeling of the ace as a "1." However. if you feel that it is preferable to the youngsters around tables, with three have a "1" symbol,itshould be separately con- players at each. Every table will have a structed. Sf 229 230 i GAMES AND PUZZLES likely to at this stage the primaryobjective will kind of game immensely and are want to play it forlong periods of time, still be to have the youngsters recognize such as the visual symbol subverbally. You, as a especially for visiting "firemen" other teachers. teacher, can go around acting as referee. parents, the principal, or which also In the process, you will have an automatic An alternate gameone coding device telling you which youngster stresses symbolrecognitionis "Concen- played with is not placing the right numeral onthe tration." This game can be appropriate pile (because of the suit des- two to five youngstersworking on one shuffled and, ignation). Thus, if Johnny has the hearts deck of cards. The deck is 5-by-8 array. at Table 3 and is placing a 6 onthe 9's laid out in a face-down, the game off by pile, you cati make a mental note ofthat One youngster starts If both cards fact. The most common errors at this stage turning over any two cards. he takes them as are reversals of 6 and 9,reversals of the are numerically equal, again. As long ace ("A") and 4.(or of the aceand 7),-his winnings and may go and an occasional reversal of 3 and 8. as he continues topick up equals, he may he Within a very short time the children continue to go on. However, once misses, the pair of cards are againturned can ' not only recognizethe symbols but also name them. Thosechildren who, at face down, and the next youngster goes. - be Memory plays a major part inthis game, this stage, still need more practice may better regrouped for more practice with this game, and you will find that children are while the remainder of the class go on to at it than adults. I knowthat I have been beaten repeatedly by my youngsters,de- the next game. The next game can still be played in a spite my most sincere efforts towin. have mastered group, but it requires eachchild to work By this time, youngsters with his own suit in front of him.Receiv- symbol, recognition and the appropriate In addition, they suitin preshuffled form, each name for each \symbol.. ing the numerals in correct youngster is instructed tolay the cards are able to plai.* the out before him in order of size (i.e., A, order of size. 71* stressing ofthe serial leads to 2, 3, 4, 5, 6, 7, 8, 9, 10). Ifcompetition order of the fi si ten numbers is deemed appropriate, then a speed con-: the concept of line nifty andsize. mastered the pretest can be instituted, withthe winners Children who' ha from each of the tables competingwith viously mentioned games -usually have no each other in an elimination contest. trouble whatsoever with the gamecalled Once the children have gained profi- "War." While War can beplayed with ciency with this game, a game of"magic" more than twoparticipants, for the sake exactly of simplicity it wit! beexplained for two is in order. This game starts off the deck in like the previous one, with eachchild players. Each child 'gets half laying the cards 'out in front ofhimself a preshuffledform. Both players hold in order of size. Next, all tencards are their decks face down andsimultaneously player whose turned face down on the table,stillin reveal one card apiece. The value wins order, however. Some youngster startsoff card has the higher numerical the bot- the game by pointing to one ofthe cards both cards, which he places on controversy in front of the child to his left.The child tom of his deck. To prevent "predict" what nu- over which of twonumbers is the greater, to his left must then provided with a meral is under the card, with points or each youngster could be picture of the number line. Ties areset- .prizes of some kind being awardedfor correct 'responses, if theteacher desires tledinthe following manner. If both example, each it. In effect, each child is amagician^ who players display a three, for given card by player counts out four morecards, which can tell what is under a of himself, secretly counting. Youngstersenjoy this he places face down in front and then places a fifth card face up. These traction, or .?t least in thinking in terms fifth cards then determine the outcome of of addition and subtraction. Actually, the the 'tie. Of course, -should another tie re- rules of the game are identical to those sult, the procedure is repeated. Children of "War" except that the player with the tend to enjoy this game immensely and card of higher value gets as many cards are likely to wish to play the gamelong from the loser as are represented by the after the apparent pedagogic values seem numericaldifferencebetweenthetwo to have disappeared. Again, it is possible cards. Thus, if the players show respec- to make this sort of situation into a com- tively a 3 and a 7, the player .holding the petition by having the winners from one 7 gets four cards from the player holding group play the winners from another group the 3, plus the originally played cards. until all but one individual are eliminated. Chips can, of course, be substituted; and A natural extension of this game, as well variations on the rules can Jengthen or as of all the previous games, would be to shorten the game. create more cards by adding the numbers The presentation of these materials can from 11 through 15, or even from 11 be so designed as to allow advanced learn- through 20. ers and youngsters new to numerical con- A follow-up of the game "War" actu- cepts to work together in harmony, thus ally involves the child in a form of sub- providing for productive peer learning.

I i

/ Arithmetic card games MARTIN H. HUNT Boston Public Schools, Boston, Massachusetts c

We all realize how important itis to child who has not memorized a particular provide youngsters with a variety of rein- fact can compute it by counting. Any in- forcing opportunities ,that promotethe ac- volvement in games, however, soon makes quisition of a knowledge of number facts. it clear that "knowing the number facts Playing arithmetic card games that have well" makes the game more enjoyable been adapted -from games familiar to chil- and winning easier. dren and young adults not only is an en- The five games described below were joyable social experience for pupils, but initially introduced at a recent summer serves as an interesting school orhome program and should not beconsidered, a activitythat -enhancesfacility with the complete list by any means. Both students number facts. and teachers should feel free to designnew Because each cardisimprinted with games or modify any of the given ones. both numerals (the ace represents a one) While we encourage children to play the and the equivalent number of elements games, we would never compel anypupil (spades, hearts, clubs, and diamonds), the to participate, as this would detract from

231 I .',..,, .

232 GAMES AND PUZZLES before the value of the activity as a game tobe previously thrown by the player player, enjoyed. Generally, the children willbe him. The "trick" is made when a succeeds in employing the aces through the 9's.How- through the card he plays, number, ever, if the goal is toachieve a Fombina- getting the cards to total a given thaii 10, the such as 10 or 21. Analternate version in- tion equaling a number less the highest valued card used wouldbe that volves having the cards totaled until 14 and 20, then sub- which is one less than the desiredgoal. sum falls between to,combine tracting the value of eachsuccessive dis- For example, if the purpose is reached. If cards equaling 5, the 4 would bethe high- card until the value of 13 is on subtraction est card used. one wishes to concentrate skills, the players might begin with25 and War then subtract the value Ofeach card until given each The deck is divided equally amongthe 3 is reached. Chips may be players. Each player insuccession turns time a "trick" is completed. over a card from hispack, -which is held the highest Number Scrabble face down. The ,player with similar valued card takes in the set,and either the This game is played in a manner of "Scrabble" in player who ends up with allthe cards or, to the commercial game player with the that a player attempts tocomplete com- if a time limit is set, the of letters) most cards at the endof the time, is the binations of numbed (instead over equaling a designated sum suchas 5 or winner. If two or more players turn players may results. Two 10. For more able students, cards of equal value, "war" combination of cards packs belonging attempt to make the cards from the tops of the combination of the to these latter players arediscarded face.; equal 85, using any arithmetical processes. Eachtime a player down, and the third cardis played faceup. successfully combines thecards, he re- The player engaged in"war" whose third takes in all the ceives a chip and drawsthe appropriate card is greatest in value number of cards from thedeck to replace cards played. This gameenables the chil- those he has just used.After all the cards dren to learn torecognize numerals and the deck, the It promotes an have been removed from their ordered relationship. student with the most chipsis declared the understanding of the conceptof "equals," winker. "greater than," and"less than." SAMPLE COMBINATIONS Fish Depending upon the needsof the chil- (a) (9 X 9) + 4 =85 be to obtain 'Tuns" (b) (3 X 4 X 7) + 1 =85 dren, the goal might 6 = 85 of three cards (A, 2,3; 2, 3, 4; 8, 9, 10) (c)(7 +6) X 7 to find two cardswhose difference equals (b) a specific number,or to obtaincombina- tions of cards whosevalues equal a given in turn asks another number. Each player (a) 9 9 player for the neededcard. If the card is 6 not available, therequesting player selects (c) the closed deck. The the top card from E3 player with the 'mist"books" at the end of the game is thewinner. Number Rummy Each player is dealt sevencards. The Number Trick first card from theremaining closed deck Each player discards onecard at a time, is turned over. Thechildren in turn choose adding the value ofhis card to the one MULTIPURPOSE GAMES AND PUZZLES 233

a card from the open pileof discards or that calculate, when using any combina- froth the closed deck in an attempt to ob- tion of the arithmetical processes, to 100 tain combinations of two or more cards (a more complicated version). Again, the --;--,that equal 5 (in the simplest version) or player with the most "tricks" at the end of the game is the winner.

EDITORIAL CommENT.--,-Sporting ev,ents often prot;ide a successful way to motivate students to do practice exercis. In the spring, baseball is, appropriate. A playingfield is neededa square piece of to°ard with the bases and a pitcher's mound marked out will do. Place a spinner on t cher's mound.

3 Question Saxes 3

Place questions in boxes marked "1," "2," "3,"-and"H." The hardest problems, correspond- -ing to homeruns, should be placed in the box marked"H." Theleasiest problems, singles, should be, placed in the box marked "1." To play thegame, two teams are needed (the game iseasier to manage if there are no more than four students on a team). Each player needs amarkt°layers spin to see froni which question box to .choose. If the question isanswered corret..y, the student gets a single, double, and, so on, according to the number marked on thebox. Play changes hands when a team misses three questions. - Supply answers to an "umpire," give him a hat markedUMPIRE, and you're oft'! Arithetic Football EA k A. K. itAU Sagin* 4fichigan

o A0 YO SO .JO /O-- U WIRE . l0 JO X.° -..- -

... 6

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_ARITHMETIC F00141ALL is a gaMe of men-.mark off the ten _yards to a first "down:" \ _Mal arithmetie.-It is played without theThis hangs frdm the top -wire and- must aid- of pencil or paper. A football-fieldmoveable. It is placed and used as inan made of roll paper or oilcloth -is Iaid outactual gaMe. _It _ shows how many yards as-slvii in the drawing; -It is 120 inchesare needed for-a -first"down." long anck-36_ inchesn wide. The-line markers You -will need enough problem play areten inches apart-for easy layoutof thecards to carryon the game. The cards-are ofiplay field; -put in piles according to the-type Over the top edge of the field a Wire isthereon. The following types of plays strung with-marks one- inch -apart. Put 'awere found suitable: small dent at every inch mark so it is easy Gain 1 to-3 -yds. to set- the ball Wire inplace. One_ football Off-guard(Easiest problems) on a moveable -wireis used. The wire Off- tackle! 1 to 8 j'ds. should -be at least twelve inches long and "Reverse} (Easy peoblvms) 1-to 25 yds. 1 to 15 yds. the size Of the football should beabout twoEnd run J inch-es- by three-quarters of an inch.This ,Pass (Ranier problems) 1 to 60 yds: ball moves up and down the 'field aseach team plays the _part. ofthe offensive team. Punts -FoUPdoWn" cards are used in ,place of Field goals hang tip the 'number Kick =off a "down" box. You Runbacks , of- the "down" so thequarterback. will know -what_play to call; A wire teninchesnue are some examples of plays andtheir long -is used like the chain measuretoproblems: PuNTS OFF- GuARD END , Ru id\ 34- Aacir'eulvcr 50% 8 4 = 5°4* L. .t.r; 3 Ia.. 211._ .44, aoali N Psss Rum sitek 3 9 )13/ = 79f a. X T = 32.

G. L.

1:;+;214N MULTIPURPOSE-GAMES AND PUZZLES 235

Each card carries a gain or a loss. If the play of the game. Quarterback kpieks-a problem is answered correctly within the-team player and- the -type\ of play he 30Atacond time limit you use the gain sec- wants, him to run. The quart() back can't tion. If the time runs- out or the wrongcall on one team member again till every answer is given you use. the loss section.one on his team has had a chance to carry The loss may. be: no yards gained, yardsthe ball The quarterback has fivdiffer- loss, fumble,_or in case of a pass, it mayent types of plays he may-call: off-:ard, be incomplete or intercepted. The cardsOff-tackle, end -run,reverse, and arc turned over,.one at a time,_se even theThe easier-problems give less yards 01 teacher doesn't know what problem isplay. The-pass plays-are-the hardest bu coming up next in any of the play stacks.the greatest gains are made on this type A -stopwatch is fine for timing but, anyof.play. On each card is written the num- sweep-secend-hand watch will-do. A fairber of yards to gain if- answered correctly knowledge of football is needed by theand the card also carries the loss if the teachertas he-acts as referee-of the game. wrong answeris given. The teacher makes :IV° teams are picked and -a quarter-up his own problems-and adds new,prob- back.from-each-team is-elected to-run thelems as new work-iS learned=in class: You team. The quarterback_ Should know acan fit your cards-to any_grAdelevel-you little- about-football ashe calls all- thewish._ For seventh gradeit is best to have plays including the person to carry themore than a hundred play eards_and use ball. The-teacher flips a coin to see whichthem over again.-The students work-ont team shall kick-off.' If team :1kicks-offside of class to_ learn what types of prob- the -moVeable football is placed on thelems arc _most difficult for them. They :1's- forty-yard line. A kick-off card isWork, on their weak points so as, not to picked up-b the teacher and quarterbacklet their team doWn. Field-goal cards are used for the extra :1picks, a- team member to answer -the problem on the card. The problem ispoint %%hen a touchdown is made. You field goal- if given, and if answered by the player, themay also let a team try a for these ball is moved down the field the number ofwithin the twenty-yard line earths the point is made if the problem is yards shown on the card. Now team 13 has the chance to run the kick-off hack soanswered correctly and the point-is missed the B quarterback picks a team memberif the problem is missed. to answ&the problem the teacherhas Penalties can also be applied, for example, if one team member helps-angther picked the top of the rim-hack stack., The player B has 30 seconds toteam member it is fifteen-yards and the answer the problem. If he answersthe"down" remains the same. -If tearn problem correctly the ball is moved thebothers team B by talking, team A_ is number of yards shown On the card, butpenalized fie yards and the "down" -remains the same. Not many penalties-are if he misses the problem the loss may be: Marked off as each team really wants to -ball down at that point or fumble, de- win.* The teacher may set any time limit -pending on the card.' he wishes for the duration of the ganie. Team-B now has the ball and the move- The use of this game in seventh and able ten yard marker is placed so that oneeighth grade classes has been highly suc- end -is -at the point of the ball and thecessful in stimulating rapid and accurate -other end _is ten yards away. This is thecomputation without pencil and paper. same principle as the chain measureused Both the boys and girls enjoyed it_and at a real -game. "Down" 1 is put up on aworked hard to improve their skillin hook and team B is ready- to try its firstmental arithmetic. Television games adapted for use in junior highmathematics classes

DORIS HOMAN

Moue. Junior and Senior High School, Caldwell, Idaho

- a small card and putsthe card into his TOday, as,in the past, the use of =the- matical- gaMes is an effective way to pro- - team's box. Players take their turns at play vide Meaningful, interesting drill for,jun- as their numbers are drawnfroni the boxes. ior high students. Highly, competitive by Mathword nature, students -at the juniorhigh level respond with enthusiasm to activities of "Mathword," a game based on the tele- this kind. When these games can be asso- vision show "Password," providespractice ciated with -familiar, pleasant experiences, with mathematical terms. there is an added motivational factor. Why Materials: not, then, relate-mathematics tothe most familiar activity of -all viewing television. A list of mathematical terms ofeither Three games that have been adapted frdm, one or two words. (Ex' mples:equation; television shows are described here. product, numeral, assoc ative, commuta- For each of _the games, the class is di- tive, ratio, inverse, pri, set,diagonal, vided into two teams. Each team numbers polygon, integer, square, rectangle, prism, off, and each member writes his number on triangle, obtuse triangle, right triangle, re- 7,4236 1, MULTIPURPOSE GAMES AND PUZZLES 237

ciprocal, terminating decimal, decimal, ver- the game is over. Ned, contestants are tex, altitude, disjoint, volume, area, par- drawn, and another term is chosen.* circle, perpendicular, quadrilateral,- allel, Eye Guns Math cylinder, circumference, diameter, radius, factor.) "Eye Guess Math" is a game based on the television* shows "Eye Guess" and "Concentration" that provides drill with Instructions for play: integers, 'fractions, decimals, whole num- 1. Place a chair at the head Of each team bers, and mathematical terms. so that the occupant will facethe class and his back will be -turned to the overhead Materials:' screen or the chalkboard. Eye Guess Math board on an overhead 2. Choose the -first two contestants by projector transparency,_ as in figure 1(a drawing a' number from each team box. large piece of posterboard can be used), The two contestants come forward and sit nine problems, overlay transparency for the in the- team-chairs. Eye Guess Math board (on the overlay, as 3. Write a mathematical term on the shown in figure 2, an answer to one of the overhead -projector (or chalkboard) so selected problems is written in each of the that the class can see the term, but the eighteutside squares. The ninth answer, the contestants cannot. Then turnoff- the answer for the Eye Guess square, is not put overhead projector (or erase the term). on the overlay until later), and Nine card- Tell the contestants whether the term -is board squares to cover the answers on the one word or two words. Eye Guess Math board. 4. Draw another number from each team box. The two students whose numbers are drawn are the -fast to give -a clue; I 2 3 the student -with the-lower number =goes first. The- fiat player can give -the first clue, or he can pass and allow the other 4 5 6 team- to give the first- clue. 5. The clues may be only one word or one number. The opportunity togive clues alternates between the teams. After 7 8 9 one play,er giveS a clue, another member of his team is selected, by drawing, to give the next-due for the team. Fig. 1 6. The contestants stay the same until the term has been identified, or untilthe five-point clue has been given. The game Instructions for play: is over- when -the list of terms has been L Show the blank Eye Guess Math exhausted. board to the students. (See fig. J.).Explain to the students that answers toproblems Scoring: will be put in the squares on the board. When a. contestant gets the term ont,The answers in the eight outside squares will then be revealed for fifteen seconds. the first clue, his team receives ten points. the The team gets nine points if its contestant The students are to try to remember gets the term on the second clue; eight positions of as many answers as possible. pOints,-Ionithe third clue; and so on. When 2. After the eight answers have been the fiVe-point clue is given andmissed, revealed, the ninth answer is put in the Eye 238 PAWS AND PUZZLES unmarked on Phis list and goes on to the 2 3 next problem. After going through the list 0 '1 1 of problems once, the teacher then goes 1/ back and uses the probleins missed in the 4/ 6 first round. This continueis until all prob- 13 2 lems have been answered correctly. The / team with the larger score is declared the winner. 7 /8/-18/9 Celebrity Math

Fig. 2 "Celebrity Math," which is a game based on the television show "Celebritydame," Guess square, and all squares are covered provides drillin ?factions, decimals, or with small cards as- shown in figure 3. whole numbers. It also encouragesmental calculation.and approximations. Materials: 1 A list of thirty problems that cannot be 1 easily worked without pencil and paper. (Examples: 53 x 79 = 3798 4---- 5 6 - 59 = 5800 x 500 = 19 ÷ 78 + 53 + 98 = Instructions for play: 7..._ 8 9 1. Each team numbers off 1, 2, 1, 2, and so on. Each one is a "celebrity"for his team, and each two is a contestant. Fig. 3 2. Ea'ch student who is selected to he a celebrity must think of some celebrity that 3. The teacher then reads the first prob- he would like to be. He writes the name lem on his list. After sufficient time to of that person on the small card provided. work the problem has elapsed, the teacher The cards are collected, and the celebrities draws -a. number from the box o( the team for each team are listed on the chalkboard. chosen to go first. The team member whose number .is drawn will then call out the 3. The contestants (the twos) for each number of the square in which he thinks team number.off 1, 2, 3, 4, and so on, and the answer is located. The teacher lifts place their numbers. in their team box. off the cover and reveals the number in 4. One team isselected to go first. The that- square.If the number in the un- game begins as the teacher draws a num- covered square is the right answer, then ber from the starting team's box of con- the team receives one point. Right or testant numbers. The contestant whose wrong, the square is quicklycovered again. number is called selects one of the opposite The teacher then proceeds to givethe team's celebrities to answer his problem. next problem to the other team, following 5. The teacher then reads the problem the same procedure as before. or writes it on the overhead projector.The 4. When a problem is answered cor- celebrities can use pencil and paper, but rectly, the teacher- crosses off that problem the contestants cannot. The selected celeb- on his list. If aproblem is not answered rity works the problem as quickly as pos- vorrectfy, the teacher leaves the .problem sible and gives his answer. The contestant A l kJ MULTIPURPOSE- GAMES AND-PUZZLES 239 must immediately agree or disagree, If the been used, or until the time allotted for the contestant is correct, his team receives the game has elapsed. point. If the contestant is not correct, the 7. Since the .object of the game is to celebrity's team receives the point. The secure points for the team, acelebrity may next problem is given to a contestant on deliberately give a wrong answer in an the 'other team. A number is drawn -from attempt to trick the contestant fromthe that team's box of numbers, and the con- other team. The contestant cannot use pen- testant chooses a celebrity -from the ()filer cil and paper; but he can approximate the team's list to answer the problem. answer or work the probleth mentally,if 6. The game continues in this manner he can do so in the time it take..1s the celeb- until -all- the problems from the list have rity to give an answer.

EDITORIAL COMMENT. In addition to the n.anS,television_games _of the contest type that may be adapted for use inmathematics classrooms, you should consider incorporatingchil- dren's favorite TV- characters into boardtames. Spending afew hours watching the Saturday Morning cartoons will give you many ideas. If you -feelthat you cannot draw the cartoon fig- urtis, you can usually find a coloring book from which to tracethe characters.

t. TheconcetitratiorLgame

HOMER F. HAMPTON CentralMisspiIri State College, Warrensburg, Missouri

The use of drill has fallen into disfavor in are not shown when the cardshang down. recent years. In spite of the shortcomings but appear when they are turned up. 'The of drill in and of itself,it does seem to space arrangements are such that a turned- make a .worthy confributidn to a student's up card does not cover another card.The learning experience. Perhaps we can im- rows and columns are numbered, so that prove the setting in which drill experiences a pupil selects a card to be turned up by are conducted and benefit from drill.with- choosing an ordered pair of numbers. (See out...incurring its Iaults. The current image fig. I.) With a little practice you can ad- of drat is vague, but its theme could well just the opening of the eyelet: so that the be "reoccurring but varied contacts with a cards can be easily and quickly replaced concert, skill, Or procedure." if a need should arise. Extensive use of physical models and The concepts or skills that can be pur- gamelike activities will assist a teacher in sued with this game are limited only by carrying out This theme. Ishould like to your imagination. I shall describe only one describe a game that provides fordrill in some detail to serve as a prototype. activities in a setting that can be both in- Suppose you have a fourth-grade class viting and satisfying to pupils from grades and wish to improve their facility with /-:;two to six. multiplication facts of the sixes, sevens, This game is patterned after a television and eights. Choose forty-nine whole num- 'program called "Concentration."The prin- bers so that you can find a large number cipal, piece of equipment is a board that of multiplication facts utilizing three of can be constructed from a pieceof ''/i3-inch them at a time.I suggest the inclusion of plywood 18 inches wide and24 inches a symbol for a placeholder (variable), but .long. Eyelets are used to attach small cards it should be used as a factot Only=not as in a seven-by-seven array. Whole numbers a product. are written on one side of thesecards, and Locate thek cards, with the numbers the-cards-OnIttached-so-that the numbers selected as described above, at random on

240 4

MULTIPURPOSE GAMES AND PUZZLES 241

CONCENTRATION

o o o o 7 fJ 12 6 L6J

6 6 6 4

2 o a

6 6 2

I 2 3 4 5 6 7

Fig.I

the board. Divide yolk. class into two teams numbers on those three cards. If they are and each team. into subgroups of two or successful, their team scores a point and three players each. Alternate- plays from all cards are turned down. If it is not pos- team to team and rotate among the sub- sibleto make a product, the cards are groups. , turned down and the play moves to the We are now ready for the procedure and Other team. rules. Each subgroup selects three cards to If a product is made, then that particular be turned up by choosing, an ordered pair product and its commutative counterpart for each card. They are to "make a prod- are not to be used again. They should be uct" (state &multiplication fact) using the written on the chalkboard for all to see. N 242 GAMES AND I'VZ:ZI.IES

But one or two of those numbers canbe suceest that a successful thirdchoice is used to form a different product in another very unlikely, You might permit achal- situation. Fol. example, suppose a subgroup lenge from the opposing team to an attempt :inmake a product ora. failure to .chose 6, 2, and .12. The product 6 x 2 = 12 (or 2 x 6= 12) could not be used recognize one, and award or take away again later, but if .a subgroup found a 3 a point depending on the outcdmeof such and could' recall the locations of the 6 and an effort. the 2, then they could score with 2 x 3 If a class is not strong, a subarray, say = 6. a five.-by-five array issuggested. You can The excitement really starts when the build up several "decks" of cards andbe placeholder appears. It plays the role of a prepared to "reload" the'board for different "wild card," and the pupil's ingenuity is situations. now-unleashed. If used as a product, how- This game can also serve as areadiness ever, it would permit scoring almost atwill activity for a unit on graphs. The notiohs Of course, there are many possible vari- of ordered pairs of numbers representing ations, and you are encouraged to make use points and perpendicular lines as references of them. A subgroup may wish to !lass on are the -basic essentialsof giaphs. This their third choice of a card during their alone will almost justify the use ofthis play if the two. nunibers already showing game in your classroom.Have fun.

fbr EDITORIAL. COMMI:Ni.Concentration games can be constructed using two boards, one questions and the other for answers.

Students call for one card from board I (example.C,4) and one from boariF2example' D,8). If the answer matches the problem, a point is scored. r

The Match Game

LARRY HOLTK A,MP

Itforimont School Cincinnati, Ohio

After the concept is, understood,,children Explanaticin need Much drillin changing percents to Before the game begins, divide the class decimals and fractions or changing frac- into two teams, A and B, and turn over tions to decimals and percents. However, any four squares on the MatchBoard. needless to say. -drill can be very boring To start the game. a member of team A unless there is some means of motivation picks a card from Potluck and hands it to To provide such motivation, I developed the teacher. The player then goes to the a game that I call the "Match Game." blackboard and the teacher dictates the problem to him and the rest of the class. Preparation (I have all the children' do the problems You will need fifty three-by-three-inch for practice.)If the player answers the squares (I cut them from yellow construc- problem correctly,his team scores one tion paper) and between forty and fifty point and he advances to the Match Board. three-by-five-inchplainfilecards. Take If the problem isincorrect, there is no twenty-five of the squares and write a frac- score and the other team takes, its turn. tion or a decimal on each. On the remain- At ,the Match Board the player alms ing squares write aft equivalent percent, over any one of the squares and sayseither one for each of the fractions anddecimals, "Match" and makes his claim or "No. thus making a total of twenty-five matches. match." Mix the squares and, using double-stick Several possibilities now exist: Scotch tape, attach them face down on the 1, The player claims a match correctly and blackboard in a five-by-ten arrangement. I scores one additional point. call this the Match Board. On the file cards write various problems Thplayer claims a match incorrectly that the children have had in the past and and I s one point. put them in some type of container.' Icall 3. The player claims no match and scores this container Potluck. no points.

243 Anytime a player says "No match," the Itis possible that you may uncover a teacher asks the ,other team for a challenge match before the game when -you expose there is a possibility that a match is over- the four squares. This is fine. In fact, there looked. If the opposing team challenges is a slim poisibility that in exposing the and makes a correct match, it scores one faur sqUares two matches occur. This is point. they challenge and make an in- also fine, but it is a rule that a player may correct match they lose one point. make only one match a turn, no matter - The/teams alternate turns- for as many how many matches there may happen ,round's as are needed to give each child to be. a chance at the Match Board. Atthe com- Remember, the only time a challenge plelion of the play, the team with.the most may occur is when a.player at the Match .points is the winner. Board states, "No match."

A Cross Number Puzzlefor St. Patrick's Day

7. The number of prime numbers between and 20. 8. The least common denominator of I. and 1. 12 .9. The altitude of a triangle whose base is units and whose area is 42 square units. 10. The number of days from St. Patrick'sDay to Christmas. (Do not count St..Patrick's 6 7 8 Day or Christmas Day.) - 11. Thenumber of letters in the name of the color associated with St. Patrick's Day. O I I 12. The square root' of 1936. 13. The number of degrees in a right angle. la 13 14 DOWN 1. The day of the month on which wecelebrate St. Patrick's Day. Now 2. The number of sides an octagon has. 3. The average of 97, 98, 94, 80 and 86. 6. The number of sides of a triangle. 7. Two angles of a triangle are 42° and 50°. ACROSS How many degrees ai in the third angle of the triangle? 1. Product of 3X9 X7. 10. Two dozen. 4. The unit's digit in the dateof St. Pairicle., 11. A pentagon has - sides. Day. date of St,.Patrick's 5. The ten's digit in the Contributed by MARGARET WILLERDINGof Day. San Diego State College, California 6. March is the month of the year. 4- ": 244 $ -A. Dominoes in the mathematicsclassroom

TOM E. MASSEY P. K. Laboratory School, Univel-sity of Florida,Gainesville, Florida

During a summer mathematics program for ninth-grade students at the P. K. Yonge .. Laboratory School of the University of Florida, the mathematics staff found a .traditional game to be a very popular and . effective learning aid. The game of domi- ' , noes was enjoyedand played frequently by students of all levels of ability. The game was an effective vehicle for practicein addition and recognition of multiples. The game as played tradition-

ally (by two to four players) provides Fig.1 practicein addition and recognition of multiples of five. This practice results from duce practice in a particularelementary he rule that,a player scores by producing mathematics skill. In one particularlyeffec- n addition total, on theexposed halves of tive variation the basic ideas ofdominoes he dominoes, which is a multiple offive. were transferred toplaying tiles with frac- An example of play resulting in a scoreof tions rather than spots. The fractionvaria- ten is shown in figure 1. tion of dominoes, also, was wellreceived Practice in recognition of multiples of by the students partidipating inthe pro- numbers other than five may be obtained gram. The play of the gameis like domi- by requiring_that the total be a multiple of noes in that tiles areplayed by matching a number other than five to score. the adjacent halves of the tiles.Scoring is There seems to be no limit to the vari- accomplished by a player when he pro- ations of rules of the game that will produces an additional total that is anatural 245 number. tigUre 2 shows pray thatresulted experiments with the game. Another set of twenty-eight playing tiles consisted of the in a score of two. numbers 0, %, 1/2; %, %, and I. If the

-IN numbers of both of the above sets are used in constructing a set, it will contain forty- five playing tiles. Again, there seems to.be win no limit to the number.ofuseful variations. Any teacher can manufacture a crude set of plywood tiles for use inthe class.; room. The experimental tiles were constructed out of 34i-inch plywood measuring light coat, of Fig. 2 one inch by two inches. A- varnish was- applied to preserve the mark- Theievel of,difficulty can be altered by ings on the tiles. The students in the pro- the choice of -fractions to be used in con- gram were- eager to assist in thedetermi- structing the tiles and the number of dif- nation of which fractions were utiliied in ferent fractions used. A set consistingof the construction of the sets. Each set con- twenty - eight -tiles with thenumbers 0, %, structed is expected to provide many hours was used in initial of enjoyable practice with-fractions. 1/3,1/2, %, and I

A Cross Number Puzzlefor Valentine's Day

r 5. There me generally __days intile month of February. 11311111141111 1 2 6, There are -- months in a year. 7. St. Valentim!, lived in the ___ century. 8. Seven months of the 3 ear have ___days., 5 6 10. Number of letters in the month inwhich Valentine's Day is. I0 I I. The number of dais in a not mill }ear multiplied by 31. 13. Four months of the year have __ days. DOWN I. T%%o less than the prodnet of 5X5 X5, 14 2. 8 X6 = 3. 216 divided by 6. 4. The product of 22 and 24. 8, In a normal year there are _ daysbe- tween Valentine's Day and Christmas. (Do not count Valentine's Day orChristmas Day.) 0. The number Of quarters in $40. I I. The number of days in the year wesend ACROSS Valentines. 1. Date of Valentine's Da,, relo v _ 12. The number of days in a week plus one. 3. Valentine's Day is the ---- _thda3 of the Contributed MAROARET Wli.I.ERDING of year. San Diego State College, California 246 A CROSS-NUMBER PUZZLEFOR JUNIOR HIGH SCHOOL MARGARET F. W1LLERDING San Diego State College

-10 II

12 -13 14

Is 1

Il 19

20 21

22 23

ACROSS. DOWN 1. How many yards are in 4,890 feet? 2. What is the perimeter of aigarden 180 feet by 4. 152 168 feet? 6. 8 dozen. 3. Write the number of days in a regular year 7. Write five minutes past eight in figures from January 1 through December 30 8. How many acres are in one square mile? 4. Express 1/5 as a per cent 10. XLIV represents what Hindu-Arabic numeral? 5. Con.pute the interest for one year on $127.00 13. What is the total cost of 4 andyards of cambric at 2% at 8.40 a yard, 1 package of pins at 5.10, and 9. Reverse the digits in f miles (express in feet) 3 spools of thread at 5.05 each? 11. What is the area in square.feet of a rectangle 15. What is the average of 728, 964, 247, 425, and 1 yard 2 feet by 9 feet? 316? 12. What is the total cost, including 2% state sales 17. What is 90% of 100? tax, for 1 dozen cream puffs at 5.06 each, 2 18. What is the perimeter of an equilateral triangle loaves of bread at $.17 a loaf, and 1 coffee cake with a bran of 104 feet? at $.40? 21. What is the cost of 1 yard of bunting if 27 14. How many ounces are in one pound? yardi east $11.61? 16. How many dozen cookies are needed to feed a 22. How many quarters are there in $33.75? troop of 33 Boy Scouts if each boy gets 4 cookies? 23.,Write 4i% as a decimal 18. How many days remain in a normal year after January has passed? 19. What is the age in 1957 of a man born in 1930? Omit demal points and percent signs in the20. Write I as a decimal uzile. Juswrite the figures. 21. Write 2/5 as a decimal 1 247 t. g T.

A CROSS-NUMBERPUZZLE FOR INTERMEDIATEGRADE MARGARET F.WILLERDING San Diego State College 4' 2 3

. . 8

12 10 I I ; .

. 14 'IS 13 1

18 19 20 16 17-

. 21

21. From 10,272 subtract he yearColumbus dis- ACROSS covered America 1. 100-15 3. 3,240+36 DOWN 4. 7+8+9+6+5+3 1. 9X9 5. 2,650+25 2. 884-379 7. 11,820 +60 4. 2.765 +7 8. 8 X7 6. 3,356+3,288 9. 315+9 7. 2,408-1,059 11. 52X92 10. 63+9 12. 12 4 13. 780-711 13. 8X8 15. 616+7 14. 14X7 16. 6+7+8+9+10 16. 7X7 17. 2X4+9 18. Reverse the digits in the productof 9X9 17. 121+11 20. 100-80 19. of 164

r 248 A CROSS-NUMBER PUZZLE FORPRIMARY GRADES

MI't,RGARET F. WILLERDINC San Diego State College

5. What time does this dock say? 1

1 4

4 I- 7. When counting by 5's what number comes after 18 9 20?

10 11 zr- I. :

'ACROSS This is a number pictu're of what number? 1. What - number comes after 110? 10. When counting by 2's what number comes after 3. What time does this clock say? 150?

DOWN 2. How many eggs in a dozen? 1 What number comes after 117? 4. What three numbers come after 3? 7. 1+1 =? 9. What number comes after 74? 10. What is 6 take-away 5? 4. Dow many tens arc there in 40? 11. What is 7 take away 5?

EDITORIA1. COMMENT.Hereis another word activity. Unscramble these mathematical words: NOTEQUAI (equation) EST (set) DADNOITI (addition) CRAFTINO (fraction) MADELIC (decimal) MEETGORY (geometry) RASOUE (square) RIMPE (prime)

249 Malhem'aticakisterms

_S_A.LLY MATHISON Northuiior High School, Portage,Michigan

the title 1967 issue ofTHE MATHENIATICS TEACHER If you had trouble deciphering this puzzle. I q.as of this article, perhaps you.shouldread no gave me the .idea for high students would further. Robert E., .Reys'"Mathematical sure that my junior Reys Word Search" that appeared inthe April enjoy a puzile of this kind, but Mr.

E R M P E R I ME T 'U B T R ACT A

S R'N'l N-EVBCD UND1"EDEiE A_REASEFUPGJI-IHDR AT.ER A GAT IVE,EI S I CI-OS1RP ANE MCP E'RKARR L'I R M CZ GGNAU AGONAL T COTEl P--NR T G E T R AIQR'ESAT.'Z' R 0 L W I t r - E N N V E A C S P 0 W.E 0 F )(CET A D I VI DEF IX YT ZCEA R I ME R B C T T DE OGOWP P Y V A I, DM OF ANAGAHHIJUHB N.----X N NLCIRNUABERf ACODD,PQ E OT'Al R S T M PT YSET E D U E L N LEL EMENTUOUVEQUAL H E'

T U N .I t IV W X I Is YR VUUZDEN W W X V I Y Z A CT I HAE ACAMDG T N Y N T X U R DT B CO/F E I 1 L BEH ENN U/ M E R A L _F P G' HI

S P HER ES1.11. KL\SOMN'OEP 1 T T R. S P O I N G R A T E R E Q Y tE MULTIPURPOSE GAMES AND PUZZLES 251 made use of words such as ellipse, con-, Can you find all of the numbers from zero .gruence, calculus, polynomial, contraposi- throughten? Watch for words within tive, etc. Junior high students would find words, for example, the word "exponent" these words as nonsensical as the whole contains the word "one." Both of these- array of letters. So I decid A to create my are mathematical, so you can count this down puzzle using mathematically related as two words. Watch for words that over- words tht a junior high student would lap, such as those in_ the title of this arti- recognize. It is a challenge to hide as many cle. It.might be fun to challenge your stu- of. these words as. possible in a 20 -by-20 dents to make a puzzle of this kind that square. Try it and see. contains- all of the numbers from one to As Ianticipated,the studentsloved twenty, using as few letters as possible: In working with this- (and so did some of the the meantime, check your own ability to teachers!) I gave both my seventh- and locate mathematical words in this. puzzle. eighth-grade classes half of one class pe- If you get stuck, look for help in my list riod (about twenty -five minutes) to work of hidden terms. on the puzzle, and the better students found This is a list -of the words that can be as many as. fifty math-related words and found in the word puzzlein no partic- several others that are not mathematical ular order. The word that appears three and therefore did not - count. At the end different times in the puzzle is the word of the class period the students were so "odd." captivated that they asked me to postpone

giving them the complete solution until ::- rate divide they h_ad moretime to search on their zero own. The next day I posted a solution. one ratio add two meter term The students -were delighted with them- three total ate selves- overthe. many words they had four fraction length' found, and surprised to have revealed to five width prime them some words-that were hidden so well six area integer! that they missed them. I'm quite sure you seveh multiply inverse eight set paralle,k- will findthis an interesting class activity nine empty set diagonal if you try sit. ten average .. distance The' wordsarelocatedhorizontally, triangle cube clement vertically, and diagonally. All of the words angle place value factor are spelled= forwards. How many words unit base graph digit negative numeral can you find? That depends onhow4hard height positive sphere you look! .But, if you look very hard you measure ray math can find seventy-five .words that I've placed estimate decimal error there, and maybe even more that'I haven't number lino greater discovered myself! See if you can find the hundred evert less arithmetic odd perimeter .math Word that appears three different volume t4pi power of ten timesinthe puzzleonce horizontally, subtract point circle once vertically, and once diagonally.(In- equation percent square cidentally, this only counts as one word!) equal exponent accuracy

4-4r b 1

Editorial 'feedback VERNE G., JEFFERS 'Mansfield Stale College, Mansfield Pennsylvania .

Inoted with interest the articleeniitled but had never attached a name to it. How- "Mathernaticalostermsr by Sally Mathison ever, seeing the idea in print didgive rise in the January 1969 issue of THE ARITH- to another idea which, like "Mathematica- METIC- TEACHER. I have beenusing a ver- losterms," may not be original, but has not sion of this puzzle for a number ofyear? had popular" usage. It involves using nu-

60 7 12 19 57 76 23

(6+ 8 = 1 i)25 33 58

42 30 56 3 44 66

28 11 17 35 16 II (36 + 6 = 6) 4 27 31 5 1 + 48 8 80 7 4 9 36 12

5 15 45 40 3 10

8 3 6 18

4 19 21 5

(58+ 1840 = 36) 21 28 36 4

'57 20 25 45

3 72 17 (4 X (7 + 6)= 52) 2 21?50 72 252 A- r;j MULTIPURPOSE GAMES AND PUZZLES 253 morals rather than words in a matrix. My 12 = 32), simple subtraction (28 11 = graduate students in "Mathematics for the 17), and compound-subtraction (44 17 Elementary School Teacher" thought = 27) may also befound. puzzle of the following type had possibili- Variations may be used in marking com- ties and encouraged me to develop it.I binations, 'possibly depending on grade am sure that nonc of usrealized its poten- level,and purpose. These may range from tial at- that time. merely encirclingthe combinations for Number combinations appear in the grid identification purposes to inserting the, ap- vertically, horizontally, and diagonally. propriate signs to complete .a tnumber sen- you examine the matrix .closely youwill tence. A few combinatiOng'arc -encircled find many of the basic facts for addition here for illustratiot (6 + -8 = 14), subtraction (8 3 = 5), I first tried this o, _..y own'children, who Multiplication (9 x 2 = 18), and division are third- and fifth-grade students.Both (27 y = 3). Also. inverse operations were intrigued and found much enjoyment (28 = 4 x 7 and 28 4 = 7) and com- in seeing how many, familiar combinations mutative, assoeiative, and distributive prop- thcy could find. My wife's' fourth-grade erties can be shown. Decade addition (19 class found the puzzle equally interesting + 9 = 28), column.addition (7 + 13 + and challenging.

EDITORIAL COMMENT.-Trya mathematicalosterm using the namesof famous mathemati- cianS. Can you find these names: S SAUGAUSG 0 Gauss P YTH AGO.R A-S Newton D.1 LCUE EUtLN Einstein CATASUUSSO Pythagoras ASWIISACST T Euclid N I ETSN I EOW Russell TONOWENLR E Cantor NTORPYTL N a

Two mathematical gameswith dice

RONALD G. GOODrT"----

Florida State University, Tallahassee, Florida

Each of the games described ,here uses Game 1 a pair of dice. The twodice must be dis- tinguishable by color, one die of one color A number line (see\fig. 1), a marker for and /theother of a different color.The each player, and a pair of colored dice are variations in color can be, in the dice or needed to play the game. The winner's in the spots on the dice. Such dice canbe spot,, marked by x in figure 1, couldbe purchased commercially :or they can be either to The left or tq the "right,of 0. The produced by making wooden cubes and number that turns up on the black- die then using magic markers to make-the means a move to the right.The number required spots on the faces of thecube. that turns up on the, red die means a move Only two colors. are needed and itmight to the left. Each player begins at0 and be a valuable activity to allowstudents to on each turn rolls a reddie and a black make their own dice before learninghow die. A few sample plays will illustrate the to play the. games. game. In the descriptions of the gamesthat Suppose player A rolls first and rolls a follow, one dieis red and the-'other is red 2 and a black 5. He ends up on black 3. Player B then rolls and follows the same black.

4*2`:54) MULTIPURPOSE GAMES AND PUZZLES 255 directional moves. Suppose I) rolls a red paper Markedtby an x (see fig. 2). A 6 and a black 2. B then ends up on the player rolls one red die and one black die. red 4. On his next turn A rolls a red 4 and The black die means move either right or a black 2, which puts him at the black1. left and the red die means move either up, B then rolls a. black 5 and a red I, which or down. A player cannot move both up takes -him back to 0. The, play continues and down in the same' move, nor can he until a player gets a roll that places him move both left and right in the same move. on or beyond the win slot. That player is The following series of moves illustrates then the winner and the game is over. the game. Player A rolls a-red 4 and a black 3. illiELTETELKO He proceeds from 0 to the right 3 spaces 9 8 7 6-5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 and then up 4 spaces. He puts his marker at (3, 4). Player B rolls a red 6 and a Red Black black 5 and puts his marker at (5, 6). On his next turn, A rolls a red 4 and a black Fig. 1 7, which puts hjm at(10, 8). Then B rolls a red 6 and al black 3, which puts him The main intent of the game is to intro- at (8, 1-2). A then rolls a red 3 and a black duce subtraction in a concrete way. Al- 2. Since A is already beyond the x, he thotigh negative numbers as such do not

appear on the game board, the fact that 12 there are numbers to the left of 0 is one way of looking-Tit negative numbers on a 1 1 number line. 10 ,, The game can be varied in several ways. 9 Number lines of different lengths can be 8 used. A "lose" spot could be substituted 7 for the "win" ipot=children might want '6 alI IFA to see, if this changed their luck." Because the "laws of chance" are op- 5 .. erating When the dice are being rolled, the 4h___ e children will likely gain an intuitive idea I II 3 iii na of such laws. because of the tendency for 2 the play to remain near 0. In fact, if number lines of-different length's are used, chil- g dren will probably notice thatitis more 0 . n- A =C-----7. 01III difficult to win with a longer linc than with Fig. 2 a shorter line. chooses to move to the left 2 spaces and up Game 2 3. This puts him a '(8, 4-1). (The direction Coordinate paper with axes drawn; pins, of a player's move may be dictated by a map tacks, or other small markers;and a boundary. Thus, if A had rolled a black 3 pair of colored dice are needed to play the he would have been forced to move to the game. The coordinate paper (see fig. 2) left A move of 3 to the right would have should be prepared by the teacher in ad- p t him "out of bounds.") Onhis next vance so that children can take the dice t rn B rolls a red 2 and a black 1. This 'enables B to move I space to the right and and begin the game after minimal initial / instructions. 2 spaces down, which puts him on x. B The object of this game is to move the thereby wins the game. 4 playing pieces from 0 to the point on the Number pairs are obviously involved in 256 GAMES AND PUZZLES quadrants of the graph this game, as well as rudimentary ideasof to include all four The x the whole process of graphing. Addingand thereby involving negative numbers: could be put in any of the quadrants. subtracting, as useful processes, are em- are re- phasized. Perhaps most importantly,the Minimal computational skills in planning quired to participate with understanding game ,requires some strategy, in ahead and considering the consequencesof both dice games and yet the outcomes terms of other kinds ofmathematical think- different moves. their own As with any game withpredetermined ing are potentially great. If left to devise rules, the children should beallowed, and devices, it is likely that children will which perhaps even encouraged, to "seewhat other variations of the two 'games valuable to happens" if they change the rules in some might prove to be even more abilities. way. The "game board" canbe extended their mathematical thinking

practice all the basic facts EDITORIAL. COMMINT.Ordinary dice do not have enough faces to for two dodecahedra with one pair of dice. Dodecahedrondice will do the trick. Make models and write the numerals 1 through 12, one to aface, on each die. Roll the dice and add,subtract, multiply, or divide withthe two numbers named on the tofaces.

I Three games BRUCE F'. GODSA-E

Bruce Godsave is associate professor of mathematics at Gallaudet College, Washington, D.C. All hisundergrlduate students are deaf. The ideas for these games weredeveloped while he was teaching a graduate course inmathematics education to future teachers of the deaf.

when Rene Descartes sat up in bed Treasure Hunt one day and created coordinate geometry, This is like the game "Battleship," he did a fantastic thing for matheinatics. cept it is nonviolent! Use 4 x 4 or51 5 Many students and teachers believethat geoboards, balls of clay, and two teams. much of mathematics came to usin a Each team will put five balls of'gold" manner similar to that of theTen Com- (clay) on the pins or nails ofthie geo- mandmentsIn...fact, something likethe board without letting the other tea now coordinate system was made up by a where the gold is (see fig. 1). The teams mortal man, in much the same waythat man created Monopoly. Itwould be dif- o 4 0. 0 N O 0 o ficult, if not impossible, to devise anin- O 0 0 tuitive way of learning the rules of Monop- o es, 0 oly. We are told the rules, and after we $0 00 0 0 0 intelli- learn them, we use our skill and O 0 0 0 gence to send our friends tothe "poor- O 0 0 0 house." The same is true with coordinate Team I Team II geometry and most other mathematical Fig. I concepts:. we need to learn the,rules. The idea of a point, howl to name a point, whereto begin numbering, and the take turns naming points. When oneteam direction the numbers go are all part of names a point that has gold onit, this team its five the rules of a coordinate system. Assume gets the gold. The first team to lose now that we have a class thathas been pieces of clay loses the game. given the rules; the children recognize the The rules of the game arc simple, but axis and have been given a method for the skills required to win efficiently involve finding a point. We need to add "experi- practice with using ordered pairs to name ence." points and organizing data so that stu- The first two games described below are dents know whether or not a point has al- geoboard activities that provide such ex- ready been named. While one team gets perience in naming points, checking points. practice by naming points, the other team writing the names of points, and organizing gets practice by checking eachpoint to data.1 This will help children become, fa- see if it has any gold on it. nuliiJ with coordinate systems before we For the first game or so, after naming students will for- complicate theidea Withlines,circles, ten or more points the graphs, and things like that. The activities get whether they have alreadycalled a will also be fun for the class. particular point or not. This should lead pairs as they The third game described disguises gen- to writing down the ordered eral review. as a game of tag. are named so that theplayers don't have 257

4. ; 258 GAMES AND PUZZLES

to waste time checkingthe same point To play, someone on team 1 names a over and .oWicook.childrendon't do this point on the maze. If there is a straight-line on their own, suggest it.Here again is the path between where he is and where he wants to go, he is told he can moire. For lpractice they ,need. Later, their. listwill be. rather long and they may spend a example, on Maze II he could make one of long, time checking the list. This should two possible first moves, (2,1) or (3,1 )., lead to organizing the list as shownin All other choices'are blocked. figure 2. Again, if children don't do this on This game requires a different recording their own, suggest it. method from that used in-Treasure Hunt. Points that are blocked while a player is in one position will not be-bTocked if he is in First List another position. Here are some suggestions (0,0) (2,2) (3.4) (4,4) for this game: (2,1) (0,1) 1. Use a marker to show the position of (1,3) (1,1) (4,2) the team in the maze. (1,2) 2. The first time the game is played, (0.3) make up a maze and play while the class (4.1) (2.3) looks at the maze. Explain why various moves can or can't be made. Be sure there Organized List is a path through the maze. (0,0) (1.3) (2.1) (3,4) (4,2) (0,3) (1,2) (2.3) (4,1) 3. For the next game, make up a maze (0,1) (1,1) (2,2) (4,4) and show it only briefly to the class. Then Fig. 2 have the class work together to choose points. Once they understand the method of playing, they can divide into groups or Through the Maze pairs to play. This is also a team game involving group averyinteresting be played Children findthis versus group, although it may game, although it can't be played fast. It by one student against another. Eap)team can be played during the time before or Winds constructs a maze, using rubber; It'sa ordered after school, or during lunchtime. and a geoboard. By calling off quiet, game that provides needed practice pairs, one team is to get through theother in finding points on a plane. team's maze without seeing the maze.Fig- ure 3 shows two mazeswith their solutions. Another game that could be used to In this game, no rubber bands are tobe provide experiencein naming pointsis crossed, nor may anyone land on a peg "Four -in -a- Row," a game used in the Mad- with a rubber band on it. The beginningis ison Project. It is similar to tic-tac-toe, ex- at (0,0), and the goal is to get to(n,n). cept that there is an extra column and row.

Maze 11 Out

(0.0)-0,3)(2,0.4-(3,3) (3,3) 4(3,1)-.40,21

Fig. 3 NIULTIPORPOSI: GAMES AND l'Ui.ZLES 259

The teams take turns naming points where he must tag a person whose number is a an X or an C) is.to go. member of the solution set. The person tagged is then "it," and the new "it" draws "Matheraatical Tag a card. This can continueuntil the desire Son-Mimes in the teaching of mathe- to hold class outside is eliminatedfrom matics we feel compelled to do a little everyone, or until the kids aretired enough review. I'm sure most of you have tried to to do seatwork. find different ways to review that don't What do you do when "it" tags someone seem like review. I was faced with areview whose number is not a member of the and-also needed to keep a promise that we solution set? You could impose a penalty, could have mathematics outside someday but that is not necessary. The truth (set) My solution-to both problems was a varia- will out. tion on the game of tag. I called it "Mathe- Just a few more comments. The levelof matical:Tag," since that described very well difficulty-of the cards depends on the abili- what-was going-on. ties of the students. For example, compare .Mathematical Tag is based on the rules (0 is not equal to 5) with (x 5) and of-the_tag you played (or still play). Here (0 is between 3 and 3) with(ly( < 3). is how it works: After the game is played once, the class Everyone inclass choosesa number can be asked;ito make up apack of cards fromI -through the number of studentsin for the next time. They will see howhard the class. No two pupils can have the same itis to get the number of occurrencesof number. Each child will wear his number so each number the same. it can be seen from the front or the back. Try to work out a way of usiregative Wide tape does a good job. You will have numbers. already prepared a pack of 3 x 5 cards on Try to keep the playing area restricted which are writtenexpressionslikethe to a small space. following: This game can be played indoors; but o is not equal to 5. itis a noisy game, so choose a place ac- 3 is greater than 0. cordingly. + 3 is an odd number. O is a member of {5. 10. 15, 20}. I have played this with my classes,and + 4 = + 4. it is a lot of fun. If your class sizeis about You will need to keep a tally of your solu- twenty-five, you could use thefollowing tion sets to check that each number is sentences for your deck of cards.Copy each used about the same number of times.If sentence on a card by itself.IfI figured after checking you find you need another right, each number will appear in seven I,16, and 19, you can make up a card different solution sets. with the expression is even. o is a member ofJ. 16, 19} is odd. speak. Try to 0is not even. and yokro all set, so to is not odd. avoid cards whose solution set willhave 0is a multiple of 2. only one or two members. It'shard on 0is a multiple of 3. whoever is "it" to go after only twopeople. 4- 1 is a multiple of 2. is a prime. Now comes the hard partdeciding is not a prime. vho will be "it." Let me suggest that you 0is less than II. be "it." This shows that you're a real sport, is greater than 17. is between 10 and 18. and it will show the children how the game is a multiple of 5. is played.. is a member of {I, 2. 4. 7, 8,II, 13). is a member of ;14. 16, 17, 19, 22, 23, 25). Now "it" dAws a card, readsit.to him- is rtctIs. self, and figures out the solution set.With- out telling anyone what the solution setis, Have fun with mathematics! 04

Takeamathematical holiday

-,"Mr., NE McCALLA ,Christian Academy in Japan, Tokyo, Japan

The day before vacation usually brings vacation plans of your own. Happy holi- an air of-;carefree excitement into the class- day! room as the children come,bubbling over with thoughts of "No school tomorrow!" To the teacher, educationally speaking, I'm afraid it often means a wastedday. I have fouhd a way to/put some of this restlessness to work, and so I invite you to join me as we "take amathematical holiday"--the day before Christmas vacation. The ideas presented herec/bemade as simple or complex asdorred, so they can be adapted to mostgrade levels. They can also be alteredto/fit the particular mathematical skill or concept you wish to incorporate; i.e., from basic operations of whole numbers, fractions and decimals, tp geometry and probability. Oneholiday idea can be used for almost any otherholiday, simply by adapOng the figure. Good old St. Nick (Fig. 1) can beused to 'construct and to namedifferent geo- FIGURF: metric figures. Halloween could use afew GEOAN1A and witches made from squares, circles 1. Draw the followinggeometric figures:. cut triangles;Thanksgivingvacationcould them out; put them together tomake a easily fatten up a geo-turkey; andspring Santa. AMU. TRI- vacationcouldbring geometricflowers 2. For Santa's hat, make an r-Out! ANGI E measuring 1 inch on each side. Die into 1_,....21oLuu.-----" RI CI AN(.I 1that meas- Santa Claus (or any otherholiday fig- brim is made from ;t "joining the ures J,4 by I inch. ure) can easily be drawn by 3. Santa's face is madefrom a sou %g r.Iinch dots" (Fig. 2), employingmathematical on each side. Cose him molittle no mt I s problems of varying difficulty. I hope that these ideas willgive birth ----- to many more mathematicalholidays ps t Younger children can morecacti). draw these ypu use your imaginationto create some shapes on graph paper. ; 260 1I C MULTIPURPOSE GAMES ANDPUZZLES 261 for eyes, a little SQUARE for a nose,and a SANTA CLAUS DOTS little CIRCLE for a mouth. Join the dots from 1 to 29 in the correct order. .23+0 4. Make Santa's beard with an ISOSCELESTRI- 25 -I ANGLE, measuring 1inch On one side, 11/2 ft, inches on each of the other twosides. 10+10+5/*" 25+1 20+123-1 5. His body is made from anISOSCELES TRAPE- 27+1- the two equal s 5+5+5+5 ZOID measuring 3 inches on 0+27+1 btil-nonpaiallel sides, 3 inches on one parallel base and 1 inch on the otherbase. 9+1+9 30-1 6. Santa's armsareoutstretchedwithtwo RECTANGLES tha't are 1 inch long and 1/2 .20-2 3+10+4 inch wide. 1+0* .1 +9 +6 1+3-2. 7. His'hands are made fromtwo4aRct.Es that .20-5 are 1/2 inch in diameter. 1+1+1 6-2' . 7+7 8. Each of Santa's legi is madefrom a RHOM- BUS, inch on a side.

3+2 9+4 9. His feet are made trom twoRECTANGLES, 14-2 2+2+2. 3+3 +3 11/2 by 1/2 inches. 3+4. 8+3 10-2 14-4 "- 10. Give him three CIRCLEbuttons, and you've FIGURE2 finished your task!

can* turned into personalizedpuzzles. Do the prob- EDITORIAL CONIMENT.Worksheets the answers. Can lems at the bottom of this sheet.Then color in the letters corresponding to you make your children oneof these using each child's name? Classroom Experiences with Recreational Arithmetic

RUTI1 H. NIES* Sixth Grade, Wright School, St. Louis County, Mo.

MUCH -OF THE SO-CALLED DISLIKE for ten on the chalk board will fill the pupil's arithmeticintheintermediatetime pleasantly and to good advantage. grades -is -onlypretense.Attitudes of HERE ARE A FEW EXAMPLES: apathy and hatred can be changed to con- tagious enthusiasm when methods of I. Selecta number less than 10 recreationalarithmeticareemployed. Multiply it by 9 When-pupils participate in the healthy Use the result aS a multiplier of enjoyment of number games, they are not 123456789 ashamed to 'admit, their enjoyment to the (The final product will be a succes- world An general. Original puzzles, tricks, sion of the digit originally chosen) and recreational units devised by pupils (Expect for a zero in the 10's place.) prOve that working with numbers is not 7 123456789 really distasteful. X9 X63 Recreatioaal mathematics can serve as 63 370370367 a "pepper=upper" to start a school day or 740740734 to begin an arithmetic period. A short 777rurrUr impromptu game will sharpen wits on a Select a three-digit dull day Amd willrelieve tensions and number boredom. Kometimes betweenperiods. reverse the order of 53 just before lunch, or before dismissal time digits 351 there are five, or ten minutes %vide!' seem subtract the smaller to "dangle" and. whieh.are opportune for 99+9= II number from the a game which can buildti-worthwitile interest in working with numbers. An oc- larger casional interruption of the usual routine divide the result by (The answer can be with a Puzzle. joke, story. riddle, or game not only provides a pleasurable break read backward or forward) but may also be stimulatingfor interest inlearning. Thereare- many stories, 3. Add: 123456789 games, and puzzles %vial a mathematieal 987651321 content. 123456.789 Iii our area some children must come to 987654321 school early and have free time before 9 schooland muststayindoors when 222222222 weather does not permit outdoor play. A "number Oddity" or a number pro- (Don't show the answer, let pupils gression or some pattern of numbers writ- find it.) Mrs. Nies developed an interest inand a Boys and girls like to aee for themselves basis for leereational arithmetic in acourse how these exercises "work out." They with Dr. Margaret F.Willerding at.Harris Teachers College.St.Louis, Mo, will test one repeatedly, try it with other

4 -y 362 MULTIPURPOSE GAMES AND PUZZLES 263 numbers, 'and in-the meantime they areof mathematicsinasimplified version developing speed and accuracy withoutfascinates the elementary school child. sensing any drill. They think it is"kindaMy pupils particularlylive such things of counting, ancient 'fun." Some daysI will put a number odd-as the "stick method" ity,or riddle on the chalk board orbulletinways of telling time, earlycalendars, etc. board< and make no comment aboutit.These topics also help them to appreciate modern arithmetic, The formula for find- _Many childrenwill"surprise" meby solving the oddity in their sparetime. ing area seems more useful to achild Like Ratty teachers, I have a stock ofwhen he sees how troublesome area prob- seasonal items.Oeeasionally codes arelems were to the Egyptian rope-stretchers. used and both the boys and girlshave aThe story of our modern standardized developed great fondness for them. measures and how these were can be both dramaticand worthwhile. 1. CopE Pitom,Km: The humor of impractical, methodsof Add: Ill-the sum each let ter representsmeasuring yards and feet in, the days of a distinct digit. Findthe numbers. King Henry II helps emphasize the importance of accuracy. 0. S E N 1) The interest of almostall children can E M 0 R be captured by stories, but ci3OITICwill M 0 N E Y prefer action.When pupils experiment with an abacus, geometricpaper folding, I focus Pocus I 1.1 t.r.owEEN: 2. or a set of Napier's"bones" which they Add. In the sum each letter representsthemselves construct.,I find their con- a distinct digit. Findthe numbers. cepts of numbers and processesstrength- games and puzzles afford 1 0 C S ened. Certain P 0 C U S action, but not every childwill enjoy arithmetic games immediately.- He may P R,E S '1' 0 snot "catch on" to the tricksand short 3. Aram Foot.'s PuzzLE: computations or he may not "see" a Write anunilTr of t hree digits on yourpattern or relationship in number pro- paper. lie sure that thedifference between gressions, Or he may not grasp the essen- the first and last digit exceeds one.Re- tialpoint8c a riddle. On- the other hand, verse the digits.Find the difference be-there are times when recreational mate- rela- t ween the two numbers.Reverse therials will help a child to sense number digits of this difference. Addthese twotiOnships which he cannot understand numbers. Multiply by a million.Subtract throughbasictechniques.In- certain 966,685,33. Substitute theseletters forremedial cases, when all other methods figures: under every figure1, write thehave failed, "light has dawned" during letter L, under every figure 2,%%rite the useof recreational devices. But it would letter 0, under every figure 3,write thebe a mistake to forces child to participate letter F, under every figure 4,write thein number games. Let him obSerie pas- letter I, under every figure 5,write thesively, he may soon take an active interest letter 11. under every figure 6,write thealong with his' classmates. letter P, under every figure 7,write the Likewise, it is a mistake to "explain" which may letter A. Read the resultbackwards. some Short computations of the children. A trick I have observed that frequentlypupilsonlyconfuse some will take a greater interest innumbersor a puzzle which callsfor processes be- and measures ifthey know somethingyond the current learning level of the about how these originated.The historyclass is not useful. Conversely, one which 264 GAMES AND PUZZLES makes application of a child's )current Ihave no wishto over-emphasize stock of number facts provides him with aarithmetic in our course of study. Any thrill when he sees it work. The fourthsixth grade teacher knows that she must grader who sees for the first time the inter-cover much ground in teaching many estinF patterns made by multiples of 9 isother subjects. Occasionally correlation actually gleeful when he reaches gradeswith arithmetic proves valuable. In science ,five and six and more advanced patternsthe study of the solar system has provided he can work out with "Magic 9," "Trickyinformation and entertainment for pupils :3," and "Lucky 7." who have devised recreational type units. They prepare. these in committees and TRICKY 3 small study groups beyond the minimum 37X3 =111 assignment. Our language arts activities 37X ,6 =222 include dramatizations. We have written 37X 9=333 andpresentedseveraldramatizations 37X12=-144 based upon arithmetic.In , thesocial 37 X15 555 studies our units the Latin American 37X1,13.=666"' countries, the British Isles, Canada, and :37X21 =777 Australia. Xossibilitiesforrecreational 37X24 =888 units iniolving arithmetic are obvious. 37 X27 = 999 Comparisons and contrasts relative to sizeinarea,population,etc.always LUCKY 7 -strengthen knowledge in both fields.I 15,873X 7=111,111 have had many such units worked out by 15,873 X14 =222,222 pupils.Subjectmatterofthesixth, 15:873 X21 = 333,333 seventh, and eighth grades iS.especially 15,873 X28 =444,-144 good for incidental correlation' of other 15,873 X35 = 555,555 subjects. with arithmetic. 15,873 X42 = 666,666 Drills and numbei games for fun can 15,873 X 48 = 777,777 be used in lower grades, but recreational 15,873 X56 = 888,888 arithmetic begins to come into its own in 15,873 X63 = 999,999 grades four through eight, after basic fundamentals have beeiimastered. Of MAGIC 9 course, these same fundamentals need 123456789 X 9 = 111111111 constant reviei: 1.2345()789 X I8 = 222222222 Because sixth-gradersloveguessing 113456789 X 27 = 333333133 games, I may start one out by saying, 3 123456789 X36 = 444444444 "Think of a number between 1 and 10, 123456789 X45 = 555555555 and I'llbetI can guess your number! 123456789 X54 = 666666666 Write it down, but don't let anyone see it. 123456789 X 63 = 777777777 Xo multiply your number by 3. Add I. 1234156789 X 72 = 888888888 Multiply the result by 3. Add the number 123456789 X 81 =999999999 which you selected in the beginning and 222222222 X9 =1999999998 be ready to tell me your final answer. I'll 331331333 X 9 =2909999997 promise to guess your original number 'if 444444444 X9=3999999996 you have made no careless mistake!" 55oo5aaoa X 9 = 4999999995 By striking cut the units digit in the 66666666(1 X 9 = 5999999994 final answer, the child's number will re- ututmX9= 6990099993 main, but of courseIdon'ttellthat 888888888 X9 = 7999'J99992 "secret" to the class. Until they discover 990099999 X 9 = 8999999991 it, they are amazed at my ability, and MULTIPURPOSE GAMES AND PUZZLES 265 they all clamor to(have me guess their MAGIC SQUARE chosen- numbers. I may add afew more difficult guessing tricks with two orthree- ? 2 -digits. When the secrets areexposed, the 4 pupils all try the tricks onMom, Dad, Big Brother, or Sister whenthey go home. 3 5 It gives my sixth-graders a.chance to -"Show-off" at home, perhaps togain added prestige. Often, as Dr.Willerding 1 4 -has pointed out, children canestablish, through recreational arithmetic, a new type-of comradeship ivith theelder members of their families. Thisleads to an whichI give them for review,they make puizles exchange of tricks and puzzles, with classmates. stimulates _interest in mathematics.Eyesof their,own to exchange tricks to try.None of these has beenrequired, but I and cars are alerted for new the number of I am always sielight0when my pupilshave been surprised at home orpuzzles turned in on avoluntary basis. bring a trick worked out at recreational magazine. My files arc full of really good spotted in a newspaptr or creations Guessing ages, birth dates, smallchange,materia s whichiare the original is sufficient fun to.of puriil. pages in a book, etc. leisure old that working When ,oungsters choose to use prove to an 'eleven year and when with numbers is not all drearybusiness.time. to iplay" with numbers Ithey, beg to remain indoors at recessto On the basis of personal experience, (which plea I feel impelled to reiteratethat, going be-work on number puzzles fun, workingdo not grant in good weather!),I am sure yond the point. of having arithmetic. with magic squares and crossnumberthat sixth-graders do like mathe-But of course the true testcomes in puizles can actually strengthen 'where matical ability. When I present amagicevaluating the results of a program explain what one isrecreational methods are used.Standard squiare to my class, I per(oynance to be demonstratetests show the pupils' and how it is constructed. I Each year the im- with one simple 9-celled squareand tellwell above the norm. seven other pos-provement in arithmeticachievement. is the class that there are "gratifying. During* this year the progres- sible combinations. Within afew minutes with the othersive growth in skills has beenespecially many .pupils come up noteworthy. I must in no wayimply-that squares filled in:they beg for more pat- difficult squares. recreational, arithmetic is solely respon- terns. larger and more record. There CrOss-number puzzles serve asgoodsible for a good achievement I teach area.are various contributingfactors. review of. basic processes. recognize that many teaching pro- Tierimeter, percents anddecimals as ap- I cedures are necessary for athoroughly 'pliedtopractical,every-daylife, of processes androunded arithmetic program.Recreational course. But the necessary but cross-numberdevices are only among the many, formulae can be reviewed in overlooked. puzzles. These are not usuallyincluded intheir value should not be Some teachers will say thatthey have no text - books, and, whilethe text -hook is time for fun in arithmetic.I think that important in an arithmetic program, ma- for it. I terials not found betveenits pages seemthey should allocate some time believe improvement in pupils'skills and to have more magnetismfor the minds of sixth- graders. If the pupilslike the puzzlesattitudes will result.

4 v.,

_ EDITORIAL COMMENT. Can:you solve this code problem? FIVE -FOUR . ONE

"Arithmecode" puzzle DAVID F. WINICKMinneapolis, MinnesolT

The dashes below, labeled with Roman identify the number described. Once the numerals, represent the words of a quota- number is identified it must be decoded by tion or phrase. Each dash stands for a selecting the appropriate code letter for single letter. For example, five dashes rep- each digit of the number. Each digit cor- - resent a five-letterword. To determine each responds to exactly one letter of the word word, the reader must examine the clues in the same order, but there may be two by the corresponding Roman numerals and or three code letters for any given digit.

I II III IV V VI

VII VIII IX X XI XII XIII N, -FRANCOIS, DUC DE LA ROCHEFOUCAULD

CODE_, CLUES A0 I.04,011,272 D 1 II.(59 x 103) + (47 x 10') + 10° E2 III. 5 plus the number of feet in 148 fathoms F3 IV.(14046)2-' 266 4- MULTIPURPOSE GAMES)/0) PUZZLES 267

of/degrees in.all the interior 4, V.64 plus the total number angles of a 46 ,sided polygon H5 I6 VI.V3,969- L,7 VII.(26 + 2'5 + 24 .. + ++ 1) + 1 N8 VIII.(5! x 254) 7 09 IX. 6 is subtracted from qs number.the result will Q0 be what Euclid called a Verfectnumber." R X.79 plus the product of the greatesttwo-digit prime number and the least three - digitprime number T3 3434,,e expressed as a base-ten numeral U4- If this number is X, then log (X -IN42)=c W-5 XII. Y6 MMDCXII

SOLUTION ..QUARRELS I.04,011,272 .. WOULD II.(59 x 101) 4- (47 x 10') +10° = 59,471 893 .NOT III.5 plus the number of feet in148 fathoms LAST IV.(14,046)2-' 7023 . , V.64 plus the total number ofdegiees of all the interior angles of a 46-sided` polygon/7,984 ...... : . IF .63 . VI.1/375TT 352 .... THE VII.(26 + 25 + 24' -I-..,1-/ 3 + 2 4: 1) + 1 FAULT VIII.(5! x 254) 714,,, 30,473/ IX.If 6 is subtractedfrom/this number-the result will be what Euclidcalled 4/"perfect number." :502 WAS itgreatesttwo-digit prime num- X.79 Oils the product of the bcr and the least Three-digit primenumber .9,876 .. ONLY . ON XI.343 - 98,,,, ? - five 9820NE XII.If this number is X, then logr.(X + 42) = 5 , , SIDE XIII.MMDCXII 2,612 . . .. ,, , I

EDITORIAL. CONIMF.NT.Here is an easier one:

I II III IV V VI VII VIII IX (Everyman)

Clues: 1.2 x 3 IV. 3 x 3 'II. 2 x 2 V. 3 x I 111.2x I VI. 4 x 2

Code: A4 //I 3 S 8 F M6 T 2 . H9 N 7 Arithmetical brain- teasersfor the young

HENRY,WINTHOP, University of South Florida

Dr. Winthrop is professor and chairman of theDepartment of Interdisciplinary Soda! Sciences at the University of South Florida, Tampa,Florida.

.A.n interestin numbers %is sometimes digits from 1 through9,thS( following stimulated inthe elementary grades by brain-teaser will be in order. arithmetical brain - teasers. Provided below 4.Q. Notice that 43 = 42 + 33, that are a number of arithmeticalbrain-teasers is to say, the digits that make up 43 are that range indifficulty from items that so relatea_that the square of thefirst-digit may be employed in thelower grades to plus the cube of the second digit will yield items that can be fused in those schools the 'number 43 itself. How many other than have introduced a littleelemvithry numbers can you find exhibiting the follow- algebra in the seventh and eighth years. ing property? A two- or three-cr.:it number Without more ado let me present these in is to be found such that if its digits from an order thatreflects, in my judgment, left to right are raised to the first, second, their increasing difficulty. and third poweis, respectivelyand in that orderthe sum of these digits raised 1.Q.%Perform any operations you wish with the digit 9, which is to be used four to the first, second, and third powerswill times, so that the result of all the oper- give the number itself. ations employed will give the number 100. A. Sample answers:11 + 32 + 53 = 135; 5' + 12 + 8' = 518; 62 '+ 3' = 63; A. 9'9 + 9/9. 11 + 72 + 53 = 175; and 5' + 92 + 83 2. Q. ,Name two coins, that add up to = 598. 11 but one of them mustn't be a penny; and give the reason for your answer. 5.a) Q. A ,number has three digits. One of these digits multiplied by the square A. One dime and. one penny. root of the other two digits is equal tothe Many, children and adults fail to dis- square root of the number itself.What is tinguish between. the condition which demands that neither of them must he a the number? penny and the mildercondition which A. \MS = 3 AfT57 demands only that ottel.of them mustn't be b) Q. A number, has three digits. The a' penny. square root of this number is equal tothe 3. Q. If a bottle and corktogether product of one of these three digits multiplied by the square root of the other two cost $1.05 -butthebottlecosts$1.00 more thark the cork, howmuch does the digits. What is the number? cork cost? A. r783 = 3 V87.. ° A. 2't. If the notion of factorial is explainedto' Many individuals develop a mental block youngsters, then the followingbrain-teaser tothis problem, seeking for a solution can be given to them. that shall be an integer. 6.O. Notice that some factorials are Where youngsters have been taught how the product of two or more other factorials. to obtain the square and cubeof the Thus 6! 7! = 10! Using nonconsecutive '268 ' Nttit,TtPURNAti,GANttis AND PUZZLEs 269

/how many more examples of thought of. Thus, at the end of Step5, the numbers, Divid- this type:of relationship can yousupply? selector will give you his ;result, 48. ing 48 by 8 will immediately give youthe 23!.=24!; A,ample answers:4! number he originally selected, 6. 2! 4!47! = 48! 2! 3!4!287! -= 288!; ,Obviously this procedure is such that and !5! 7! = 10! the numb&Y, of operations towhich the /7. "Take a number." selector can be submittedisunlimited, / Ask a personio take a number but not while the nature of the operations canbe to let you know what itis. Then ask him more varied than the fewoperations shown to pe:form thefollowing operationsin in our example. 1 mystify sequence. The whole procedure tends to a) Double the numberthought of. those wno select a number. Onceshowh l b) Square the result of doubling. what has been done, however,the- effort c) Multiply the resultobtained in Step to duplicate this type ofperformance with 2 by.the number 3. others tends to sharpen theindividual's d) Ask the person involved todivide sense of the meaningof arithmetical oper- the result tof Step 3 by the numberhe or ations. This is all to the good,This last of course, most sheriginiilly thought of. , type of brain-teaser is, e) Now askthisperson to subtract, appropriate for individuals in the seventh from the result of Step 4, an amountequal and eighth years, provided theyhave been to four times the productof the original exposedtoa modicum ofelementary number. algebra. question to f) Nay ask the person in think of give you' his last result fromStep 5. As The toughest brain-teaser I can tell him the giving' youngsters, one that willreally keep soon as he/d5es so you can business for number he originally thoughtof. As an them going on unfinished \ orig- quite a while, is to interestthem in auto- example,' let us assume the person if any of inally thought of 6. Then theresults of morphic numbers and then see above them can find automorphicnumbers for the firstfive operations described number is would be the following: (1) 12;(2) 144; -themselves. An automorphic (3) 432; (4) 72; (5) 48.If you now one whose squareends with the number number itself. Two large examples;ofautomorphic divide 48 by 8, you will obtain the and originally thought of, 6. numbers are 43,740,081,787,109,376 56,259,918,212,890,625. In asking young- What is the rationale of thisprocedure? For a person whose task is to guessthe sters to find otherautomorphic numbers, be surprisedto number that was originally thoughtof by many individualswill is a slight discover that a fina! few or moredigits the other, all that is required numbers already acquaintance withthe most elementary of the two automorphic themselves, be automorphic aspects of beginningalgebra. Suppose you gi.enwill, task is to guess the numbers, Thus 762 ends with76; 906252 are the person whose 87,109,3762 ends number. In your own mind,think of the ends with 90625; and rember selected by the other party as x. with 87,109,376. present Then the five results that youtOtild get Itis an interesting exercise to duffined can the first two largeautomorphic numbers from performing the steps then to tell be represented as follows:(1) 2x; (2) given- above to youngsters; 4x2; (3) 12,r; (4) 12x; (5)8x. When the them that there are sequencesof successive number gives you digits in these two numbersthat are them- person who selected a and to ask the answer corresponding to8x, you quickly selves automorphic numbers, divide mentally by 8. Thisimmediately them to find these new automorphic num>. will give you the number thatthis person bers. "Parallelograms ": a simple answer to drillmotivation and individualized instruction

BENNY F. TUCKER

Benny Tucker is mathematics. consultant for grades KI2 for the Parks?ay School District, Chesterfield. ,Vissouri. fie is the author of a handbook of activities for elementary school mathenuuics Jot use in the Parkway schools.

Tdrill or not to drillthat is the ques- isa game similartotic tac-toecalled tion that "bugs" many teachers'of elemen- "Parallelograms." tary inematics. Some who would have us Nearly every child knov.s how to play teach "n\odern" mathematics have said tic-tac-toe. By requiring the plays to answer that ...drill iTd memorization have no place questions col reedyatthe risk of loSing in contemmary mathematics programs. turnsi teacher can effLetively use ticic- Experience and common sense tell us that toe for drill or review. Out of a search for these spokesmen of "modem math" are a game that is similar to tic-tac-toe but thlat either wrong or misunderstood. Indeed, few would seem quite different to the student, have actually said that memorization and the game Parallelograms evolved. The ob- drill do not have their place in mathematics ject of the game is to be the first player to education. Most experts agree that both mark the four vertices of one of the many must be an integral part of our programs parallelograms contained infigure1. Of but that they must be kept in proper p/er- course one's opponent will be trying to do spective. the same thing. so each player must try to' Drill, and the memorization and ski II anticipate his opponent's strategy and block development that can result, should come him. Although itwill be obvious to the only after, the student adequately under- teacher thatsquares are also parallelo- stands underlying concepts.Drill should grams, it may be necessary to point out to never be used as a substitute for teaching the students that the vertices of any of the and should not be a club to be used on four squares in the figure could also be a children who. have trouble understanding winning combination. . mathematics teachers. In the game at the. left in figure 1, "X" Teachers sometimes forget that if a child needs one more move to win. In the sec- is to maintain a receptive attitude toward ond game, "0" needs one more move to drill over any significant period oftime, win. every effort must be made to keep the drill In addition to the obvious benefits from procedures varied and interesting. There the drill, t;ie use of tills figure gives students are many drill games on the market which practice in the visualization of geometric can be of valuable help to the teacher w ho shapes and in the use of game strategies. wants to maintain interest. Parallelograms is simple enough for third- A game that has been particularly sue- grade students but challenging enough for ces-sfu: in the Parkway elementary schools the most talented sixth-grade students.

270 4 r Fg 'MULTIPURPOSE GAMES ANDPUZZLES 271

Fig.

Since the figure for Parallelograms is pared by the teacher to meet the specific rather complicated and not easy to draw, it needcan be used. is suggested that dittoed copiesbe provided A question that elementary teachers per- for the students. Or an overhead trans- petually face is what to do with the stu- parency car be used as a game sheetand dent who is having trouble with topics nor- mlirked with a crayon or wax pencil. When mally learned in an earlier grade. Certainly the game is finished the acetate can be thee students should be taught what they wiped clean with a tissue to make ready need to learn, but many teachers do not for the/next game. find the time that they feel is necessary to Perhaps the greatest value of the game satisfy the needs of such students. Another common problem is the student whoalways isits adaptaUffirci-sto many situations. A ahead of thei' sequence of questions can be used, withthe seems to finish his assignment players aikswering them alternately.If the rest of the class. The pat answer to this player ans)vers correctly, he may mark a problem is that such a studentshould be vertex. if hemisses, he loses his turn., The allowed to continue at his own ratelie questions used in the game could just as should not wait for the other studentsbut / well come from an academic areaother shout(o on with the next"lesson." Parallelograms can sui)ply than mathematics .(for example.spelling. The gi -ne problem students. science, or geography). Questions couldbe one solutipn for the needing\ used from several academic areas allin the Workingin pairs oth students d accelerated students can same game. The teacher can,by using care- remedial tasks oft ,,. fully selected questions, teach newmaterial enjoy playing' the game, and a deck in the context of the game. The best ap- cards containing a progiammed sequence plication of the game may be in theindi- of questions on a needed or interestingtopic vidualization of mathematics. If a teacher will allow those students to effectively teach who is assigned thirty students is to beef- themselves while requiring only indirect fective in dealing with individual needs,he supervision from the teacher. Work with must have at his disposal aWealth of ma- programmed materials is far more interest- terials that are worthwhile but thatrequire ing to the student when it is plated in the only indirect teacher supervision;Paral- context of a game like Parallelograms. In the following game between Billand lelogramsfitsthis description. A/ set of multiplication flash cards can be the source ferry, the questions are drill questions on of questions for a pairof students whoneed decimals. 13111is marking with an X. and drill on their multiplication flict. the Jerry is marking with an 0. teacher sees a need for work inn ar or the which there are no commercially prepared 01/TioN: Where would you place I 2 3 4 5 so cards, a deck of cards with questions on imal point in the numeral hundreds place? one side and answers onthe otherpre- at the 3 woulfl be in the

2.1 272 GAMES AND PUZZLES

Btu.: After the 5. (Correct; Bill marks a QUESTION: What is 5.62 .13? vertex as shown in fig. 2.) BILL: 5.49. (Correct; see fig. 5.)

Fig. 2 Fig. 5

QuEsTioN: What is .023 .009? - 'QUESTION: What is..984 + 6? JERRY: .0194. (Wrong; Jerry loses his JERRY : .1.64. (Correct; see fig 6.) turn.) QUESTION: Which is greater, .0098 or .012? ,,BILL: .012. (Correct; see fig. 3.)

Fig. 6 .

r QUESTION: Answer the 'following: 7.96' = 24.3. Fig. 3 BILL :32.26.(Wrong; 13i11 loseshis .ttrn. ) QUESTION: Where would you place the QUESTION: What is .32 .4-1.5? decimal point in the numeral 1 2 3 4 5 so JERRY: 1.82.(Correct;11,1C-C-fig.--7-:)--- that the 4 would be inthe hundredths place? JERRY: Between the 2 and the 3. (Cor- rect; see fig. 4.)

z/ Fig. 7 QUESTION: What is 5 + .2? Bit I,: 5.2. (Correct; see fig. 8.)/ MIIITIPURPOSE GAMES AND PUZZLES 273

I. 0000 0 0 0 0 0000 0 0 0 0 0000 0 0 0 0 0000 + + + + 0000 + + + + { is the 1.3 5 fours fours + 0 fours same as 2.000000 0 0 0 0 0 0 7 sixes 0 0 0 0 0 0 Fig. 8 0 0 0 0 0 0 is the same as QUESTION: What is .32 x 1.5? ++ + + ++ JERRY: .48. (Correct; see fig. 9.) + + + + + + 0 sixes -f 3 sixes ++ + + ++

3.++ + ++ ++ + ++ 6 x 5 (2 X.5) 0 0 0 0 0 . - 0 0 0 0 0 +(4 X 0) 0 0 0 0 0 0 0 0 0 0

4.0 0 0 0 0 0 0 0:' 0 0 0 0 Fig. 9 0 X 4=(4 X 4) 0 0 0 0 + (3 X 4) QUESTION: What is 6.5 x 12.4? 0 0 0 0 BILL; 80.60. (Correct; see fig. 10.) 0 0 0 0 0 0 0 0

5. 9 threes = 5 threes + 0 threes 6.9 X 5 = (4 X 5) + (0 X 5)

7.6 X 3 (1 + 5) X 3 = (I X 3) + (0 X3) 8.-6 X 9 = (3 + 3) X 9 = (3 X 9) + (3 X 9) = 27 + 0 9.7 X 8 = (5 + 2) X 8 Fig. 10 = (5 X 8) + (2 X 8) 0 X A Bill wins the game. 10. 12 x 7 = (10 + 2) X 7 The following is an example of a pro- = 70 + 0 grammed sequence of questions, for third- grade students or for fourth-grade students.I L 14 x 6 = (10 + 4) X 6 needing remedialwork,developingthe = 0+24 distributive property as a tool to be used 12. 13 X 8 = (10 + 3) X 8 in multiplying one-digit numbers by two- = 80 + 24 digit numbers. = o 4:11,

1 ',.. Like most games, Parallelograms will 13.15 X 5=(10 -t- 5) X 5 =(10 X 5) + (5 X 5) only be as effective as the ingenuity ,and = creativity of the classroom teacher alloys. It is hoped that the reader will adaptthe 14.32''X 6=(30 + 2) X 6 game to his, own needs and feel freeto', =(30 X 6) -I- (2 X 6) use it. =0 + A 15.27 X 4,=(20 + 7) X 4 NOTE. The.apthor would appreciate hear- = (20 X 4) + (7 X 4) ing from readers about any success or failvrc = experienced with Thegame.

Take a chance withthe wheel of fortune

BARBARA ROSSER - SpartaElementary School, Sparta, New Jersey

Through 'the years I have noticed that rated into a suspenseful game that takes no any of my friends who havechosen to more than five minutes out of theperiod make mathematics their life's work have and turns drudgery into a sport. I use a thought of numbers in terms of a game. "wheel of fortune" for fact-study motiva- Working out a mathematical problem is as tion. Yes, we're gambling in theclassroom exciting and challenging to them as decid- and the children love it! ing on a move in a chess tournament. Here's how it works. A spinner from converted into a num- Why not, then, injectalittle of the any game is simply intrigue of the game into the math class? ber wheel by cutting a mask to fit around Undoubtedly there are distasteful aspects the outer edge of the board so as not to of any subject, and obviously wecan't interfere with the movement of the spin- thefamily's make itallfun and games. ner(minecomesfrom Take multiplication facts, for instance. "Twister" game). I don't know of a single child,gifted or Enough divisions are made on the mask otherwise, who has enjoyed learning them. to accommodate a number for everychild is assigned the What's more,inthe day of "concept in the class. Each child teaching" the teacher is hesitant to take number that corresponds to his name in my class time for old-fashioneddrill. But in grade book. the few years that I h?te used the "game When the spinner lands on a number, of chance," the chore has beenincorpo- the child who has that number gets the

274 MULTIPURPOSE GAMES AND PUZZLES 275

"oral test of the day," consisting of fire then. To cut down on "test tension"I randomly-chosen multiplication facts, with usuallytell my pupils thattheir lowest about three or four seconds allowed for mark w in be discarded. They all want to an answer. Iusually give three of these "try for high," so niglitly review becomes tests a day. Whichever facts are missed a must. ---- arc practiced ten times for homework in A gimmick? Perhaps, but no more so addition to the regular assignment. than many forms of motivation.It'sa Going one step further,it's an easy game. It's exciting, and everyone wants to matter to make up index cards with facts win. It's suspenseful, it adds excitement to on them along "theleft edge, with space w hat could easily be another dull math at the top for puriils' numbers. An or \,'routinethelearningofmultiplication .. in the appropriate box on your ruled grid facts. provides an at-a-glance record on Johnny It's a challenge. No one knows when it or Mary for quick checks on their prog- will be his turn to be tested, so everyone ress. studies every day. Sometimes a few min- Iusua..iy limit my tests to three per utes' study time is given in class for those marking period so that everyone has at few children who let the home study slip. least two or three grades to average in Seeing almost everyone else cramming for with the others. the "big maybe" is usually enough to prod At first the idea of being tested "outthe one or two lards into action. of the blue" produces groans of horror, but it's stfrprising how, after a week or two Immoral? Not at/all. Life's a gamble. the children beg for more: "Let's do fourThe math teacher/could call on anyone for an answer. She could also be accused today!" "Oh, please!Ireally studied last nightget me today." "We should do sixof "picking on" someone. The wheel of fortune is impartial. todayinsteadof .three:remember we skipped a day last week.- One final advantage is that conscientious Sureit's an easy 100ifthey know youngsters learn the lesson of cause and their facts, and the oral quiz Me facts effect by watching the upward trend of to zicustomer) carries the weight of a their graphs of weekly speed tests, as well regular quiz. One thing to keep in mind as being able to reap the immediate re- is that eeryone has a bad da\ nowand wards of winning at a game of chance.

A Using functionalbulletin boards in elementarymathematics WILLIAM E. SCHALL State University College, Fredonia,Nework

Visual aidsfilms, still pictures, models, A good bulletin board should alsobe bulletin boards, and so-onare among the supportive of and adaptable to classroom most useful tools ,in education, butthey activities in a particular subject area,ele- Since do not teach without intelligent planning mentary mathematics in this case. and use (Glenn 0. Blough and AlbertJ. children like activities or iames in Which Hugget,Elementary School Science and they participate or are actively involved, a for use here. How to TeachIt New York: Dryden game approach is suggested Press, 1957], pp. 33-34). Bulletinboards The rest of this paper desdribes several can play an importantroleintoday's bulletin boards in various areas ofelemen- mathematics program. However, abulletin tary school mathematics. Eachsuggested board, if it is to be successful inachieving bulletin board includes a shortdiscussion its purpose, must gain and beworthyof of the bulletin board's purpose, the appro- objectives the class's attention. priate grade level, concepts and that the bulletin board is intended todevelop orreinforce,suggestedquestions The writer wishes to extend hisapprecia- connection Deborah that the teacher might use in tion to Mary Jane Koepfle and addi- Lewis, students in his mathematicsmethods with the bulletin board to stimulate Cincinnati, for tional thought and discovery, and ashort course at the University of could their held with this article. description of a class activity that

276 A'fr.,; ; 1..1,0 ,9 MULTIPURPOSE GAMES AND PUZZLES 277 involve pupils with the bulletin board.' different mathematical problems. Be- The first bulletin board is shown infig- sides reviewing renaming, this will ure 1. alsoevie the basic mathematical operations. A.PURPOSE: To reviewandreinforce the basic mathematical operations as Another bulletin board is shown in figure 2. wellastheconceptofrenaming. It is used as follows: numbers. The code or the message can A.PURPOSE: be changed frequently, depending on I. To use the written nunierals as a the class, activities, season, and so on. daily practice in counting 2. To serve as a daily practice to match B.GRADELEVEL: Fourth or higher depending on the code used. For higher a set with the appropriate cardinal grades a rational-number code can be number used. 3: To use the displayed objects as a basis of comparing sets C. , BEHAVIORALOBJECTIVES:. 1. The children will be able to workthe B.,GRADE LEVEL: Thisbulletinboard problems and read the message. could be used in the kindergarten and 2. The children will be able torecognize first grade. Counting, the introduction that a number can be renamedin of sets, arid the comparison. of sets are many ways. usually introduced in the kindergarten 3. The children will be able to rename and could be used fort review purposes numbers in different ways. jn the first grade. (However, the specific time it is used depends on thechildren's D.DISCUSSION QUESTIONS: 1. What is a code? progress.) 2. Why do people sometimes writein C.BEHAVIORAL OBJECTIVES:, codes? 1. The child will be able to count to 10. 3. How do you read a code? 2. The child will be able to identify E. DESCRIPTION OF THEACTIVITY: each numeral; this means knowing I . The 'children work the problems to a "5" or a "7" when he seesit. 3. The child will be able to name the . read the message. 2. For additional practice in renaming, geometricfiguresthatarethe each child can write his own message elements of the sets. in code and rename the letters as 4. The child will be able' to match the

CAN YOU READ THIS MESSAGE??

3 + (4 + 3)(5 + 5) 3 (8 + 2) + 0 (5 + 8) + 13

3+20 i5 + 5 + 5) 10(7 + 7) 13 (25 + 4) 26

(1 + 1) + 0 2 + 2 + 2 + 2(6 + 6) 7(4 + 4) + 0

12 ( 1 7 + 10) 6(24 8) + 5(1 + I) 1 (5 + 6 + 6) (1 + 2) + 0 3 + {28 27) 79 53 (18 7) 2

(3 + 1) + (1+ 3) (8 8) + I a = 8 = 25k = 20p = 21u =2 b = 14g = 12I=4 = 6 24 c = 22h 3 5 m = 19 = 15w=,10 d = 9 i= 26n = 17 S = 23 16 e = 7 j= 13o = 3 = Ii z= 18 Fig.I X278278 GAM AND PUZZLES

How Many ? 0 Li ,LA

.L_LLAA 0000000'DO AA AL LAA LI ® a a) 000000000b

Fig. 2

sets with the correct cardinal number. 2. Another child can match the correct Rope mounted on the bulletin board set with the cardinal number. can be usedtoindicate' the cor- 3. If the second child gets the correct respondence. answe'r. he gets to select a number. 5. He will be able to compare the sets, 4. The activity can bevaried. The thus using the ideas of more or less, teacher can point to a numeral and ask children to identify the correct D. QUESTIONS TO STIMULATE THOUGHT: I. Can you name the numerals written set. shown in on the bulletin board? The nextbulletinboardis 2. Can you show me where the "5" is figure 3, and its use is outlined below. (similarly for other numerals up to A. PURPOSE: ten)? This bulletin boardis designed to 3. Can you show me the set of four stimulatethinkingaboutgeometric objects? Or, which group has the shapes- how they are made and, what four circles? they are called. There is to be transfer of 4. Which set: of objects has the largest learningfromthegeometric shapes cardinaynumber? illustrated onthebulletin board to 5. Which set of objects has the smallest geometric shapes in the everyday en- cardinal number? vironment. 6. Are there more circles (0) than B. GRADE LEVEL: squares (0)? How can yotitell? Thisparticularbulletinboardis E. DESCRIPTION OF THE GAME: designed for the primary grades; how- I. A child can select a number and ever, the basic idea of thebulletin point to the .numeral that represents board (the geo-board) can be used for the number. all gr:ide levels in the elementary school. MULTIPURPOSE GAMES AND PUZZLES 279 t

0 0 0 0 0 0 (Geo Whizz o 0 0 0 0 0 Can you make and identify o o the shapes on the geo-board ? o o

O 0 0 0 0 0

O 0 0 0 0 a I

Fig. 3

C. BEHAVIORAL OBJECTIVES: 4. How many different geometric shapes I. The child will be able to identify the did you see on the way to school six geometric shapes pictured on the this morning? bulletin board. The last bulletin board to be discussed 2. The child will be able to copy the here is shown in figure 4; its use is outlined indicated shapes on the geo-board. 3. The child will be able to' recognize below. the' number of points connected in A. PURPOSE: To motivate children as they work each shape. 4. The child will be able to distinguish withthe fundamental operations of between the geometric shape and its arithmetic; also, to increase the learners' region. skill in computation in the fundamental 5. The child wilt be able to recognize a operatiolis wide variety of geometric shapes in B. GRADE LEVEL: the classroom.. Mostintermediategradelevels depending on the difficulty of the com- D. MATERIALS: Ii I. Individual geo-boards putation and skills involved 2. Rubber bands C. BEHAVIORAL OBJECTIVES: 3. Yarn (for use on the bulletin board) I. The learnerswill work examples E. QUESTIONS TO STIMULATE THINKING: using the four fundamental opera- I. The same geometric shapes that you tions of arithmetic with increased made on the geo-board can be found accuracy and speed. within the classroom. Can you find 2. The learners will demonstratein- some examples? creased interest in arithmetic through 2. Can you make a given shape on the participation in the "Grand Prix" geo-board? activity and other mathematical ac- 3. If given the number of points or tivities. sides in a geometric form, can you 3. The ' earners will demonstrate co- create the corresponding form? opative learning and working skills

I ...,t . 1,,/ 280 GA,MF.S AND PUZZLES

rand Prix

... De eloped by Mrs. Beth McCracken. Westwoo,lElementary School, Cinonnati, Ohio. Fig. 4 I.

through -participation in the Grand .operation; the emphasis is onrefinement Prix activities. of the skill, that is, accuracy, speed, retention,' and operational ease. Tob1egin the D. DESCRIPTION OP THE ACTIvITY: activities, the addition facts are given for Grand Prix is designed to motivate the qualification day. The learnersare then ahigher levelof children toattain eiven 'a chance to have a "trial run. ' They proficiency with the four fundamental have a week to practice with their p rtners operations of arithmeticaddition, sub- before "race day," when their combined traction, multiplication, and division. scores (these scores canbe predetermined Children within each class are divided values) determine where their racerinoves into teams of two. These teams on the track. The same procedure\s re. 1. Are composed of one studentfrom peated for substruction, multiplication and more skilled or moreable groups division. Trophies, certificates, and st on, and one from the less skilled or presentedto teams that reach the slower-moving groups to maintain a "checkpoint." The checkpoints are .t am balance; scores of 200 Or whatevercategories are each 2. Are in twos go they can help chosen. / otl*r with flash cards and other team One corner of the room has letters activities; suspended overitspelling "THE PIT," 3. Provide good "workingtogether" where children may go when otherwork experience. is finished to get "tuned up." Thereisla These activitiesare done, of course, box labeled "Mechanics' Tools" that 'con- devicbs after good, meaningful experienceshave tains flash cards and various other been provided for the leainerswith the for practice with the basic facts.

1 ' C