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AUTHOR Spikell-, Mark Ed. - TITLE ProgrammableCalculatorsz, ImplicationS for the
Mathematics _Curriculam.' , . INSTITUTION ERIC Clearinghouse,' for- S-cience, Mathematics, and EnvironmentalEdUcation, CoIumhus, Ohio.' PONS AGENCr Mational,"Inst oU Education (ED), Washington, .C. PUB DATES Dec BO CONTRACT 400-7B-4:004 NOTE, .124130-. / - AVAILABLE PRO Information--eference Center ERICIIRC), ,Ohio State 1200 Chambert Rd 3rd Floor, COlueb OH 43212 ($300).
EDRS.-PRICE mP01/PC05,Plus Postage. DESCRIPTORS *Calculators: Discovery Learning; Educational Technology: Instructional Materials; Lesson Plane: *Mathematical Enrichment: Mathematics Education: *Mathematics Instruction; *Problem Solving: *Programing: Secondary Education: *Secondary School Bathematies IDENTIFIERS *Programable Calculators STRACT This document is a collection of .repOrts presented ogmable-ra calculator symposium held in 'Seattle, Washington,in 1;1910, as ,partof the annual meeting of the National Council of eaCherS. of Mathematics. (NCTM).-. The session was designed to review hether the proTratable calculator his a place In the school thematics' program, in= light, of the current availability of the 22orcooaPnter. The pregentations at the symposium supported the view such calculators do have ,4 role to play in the:Curriculum, and e- collectd --0'peri_ of the=contributorm provide ample evidence of any _wile programable calculators can -In addition to the ed papers, bra other contributions solicited 6y the editor to &rice -the uSefulness of this work- to educators = are included.
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Mark A. $pikell Editor
[ERIC]Clearinghouse for ScienCe.,, Mathematics, and Environmentel'Ed66-A=, den. pn information- center toorganize/and disseminate inforrnifismand -materials-on: science; mathematics, and environmental -edutationto --teachers,adminiStrIptors, supervisors, researchers, and-ttie pUbhc.=A joint- project -of The Ohio :State.UnilVerSity-e.. and the Educational Resources Mfd-r-mation, 'Ginter of NIC. _ pROGRAMMABLE CALCULATORS
IMPLICATIONS FOR THE MATHEMATiCS CURRICULWA
Spike -Editor
IENRUC: Clea4ringhousd\for Science, Mah matins ancLEnvironiental 'Education The Ohio St4e University 1200 Chambers Road, Worm 310 Columbus, Ohio, 43212 TABLE OF CONTENTS
?REFACE ...... FOREWORD._
PROGRAMMABLE CALCULATOR:. R SOLUTION?
Gerald R. Rising . . 1
CAIMATOR PEDAGOGY Betty J. Krist
USING PROGRAMMABLE CALCULATORS TO ENLARGE THE PROBLEM-SOLVING WORLD OF 10-12 YEAR OLDS Gary L. Musser 14
STABILIZING ARCHIMEDES' ALGORITHM FOR PI -'--- 'John Huber = , ...... ',-ESIGNING ALGEBRA EXPERIMENTE FOR THE 2-PROGRAMMABLE CALCULATOR
Stephen L. Snover. . . .
AUSE OF THE.PROGRAMMABLE CALCULATOR -IN TRIGONOMETRY: REMOVING THE AMBIGUITY FROM THE AABIGUOUS CASE - Mary -L. Johnson
USING PROGRAMMABLE CALCULAT IONS GABS FOR MATHEMATICS ENRICHMENT Michael Battiste
DSING7THEPROGRAMMABIE CAI PROBABILITY EXPERIMENT Dave Haggerty...
&DEPENDENT STUDY WITH APROC LE CALCULATOR -Lee:Mohler
IS I1 GTHE PROGRAMABLE CAL R Tb ENHANCE SCHOOL MATHEMATICS LEARNING AND INSTUCTION IN THE 1980a: ,EUGGESTION J. F. Weaver .
SOME HIGHLIGHTS FROM WE HISTORY OF PI ON =THE .--..PROGRAMMABLE CALCULATOR Maor A ONE-CREDIT CALCUL1TOR TECIINI UES CO E- Joe Elich 85
PROGRAMMABLE CALCULATORS! A CLASSIFIED AND PARTIALLY ONOTATED BIBLIOGRAPHY Jill H, Courle a?publication was_ iprepaced with funding from ,th_ez.Satton_ n vied of _Education,U.S.Department EdfiCatitin7,--mnde onibact ric;.'4 00 781t0 004.Theopinions',---ezip_iZtiide int p rt 'd0 =nct7rie Me ita stAilyifeflect_theTiposit 11 or<
The latter part of the decade of ,the ,'70s was characterized by some important, technological and business developments with potential for significant impact on the school mathematics curriculum :at ell levels. On the technologiCal front, advanced calcurating devices, Ln the form of band-held programmable calculators-. vore developed and -Manufactured with capabilities that blurred the distinction between what is a Calculator and what is a computer.- As a result, calculations 'and Problem - solving tasks previously relegated to computers, because they were too time-consuming or complex to perforM, became petiible for students of all ability levels -fo handle. On the business front, the cost of these, programmable calculators decreased rapidly as production increased. ''By the end of the decade one could purchase a fully programmable hand-held calculator for as little as $40. .because of the low cost,- portability, And increasing availability of -these machines, seine educators began to consider the possible impact that program- mable calculators might have on the teaching of mathematics. Such considerations served as the impetus for the preparation of this monograph.
ff In the winter of 1978, a review of available li oratura on -the programmable,calculator revealed relatively few contributions discussing the role of the programmable calculator in school '-,MathematicS In the spring of 1979, proposals were being solicited for sessions by the program committee for the Aptil
-'1980:annual 'meeting of the Natienal Council of Teachers, of Mathe-_ matics to beheld in Seattle, Washington. A proposal was presented =poilitinriputLhe dearth of iniomation available on the role of --,TprogrammableAlculators ins haul mathematics, sufgestihg the potential Chese machines Might have on thjeUrrichlm, and requesting thats special session beheld at the annual meeting -where Intereateepersons could share ideas. The program'Committee,- aired by Richard Lodholz of the Parkway Schdol District in ChesterfieldMissouri, accepted the suggestion and committee ember :James :M, RuliAllo of Sucks; County Community College -in Newtown, Pennsylvania invited this writer to organize aodH lotleOte such asess4on
The session haA several:unique'features and the National Council of of:Mathema.tics should be applauded for its Willingness to- ,7_Offer-aii,experimentallsession atian annual meeting. The session', entitled ,"The Programmable CalculatorA Tool for the1980s," was specifically designedias a symp4ium for-persons interested-in sharin ,ideas on the role of programmable hand-held calculators in school +'-mathematics The forMat of the Session prcivi0-=d for opening remarks_ 4P---by the Moderator andfeleven five minute talks by- presenters', with iaSt..nne-half hour devoted to qhestions answers, and discussion
d _,the audience. _ - some possibleontrihutors -were contacted by the moderato
interest - the description of the session in rhe program booklet included an invitation to persons wishing to contribute to send copies of proposied talks to the moderator. Actual contributors were then selected by a review process and several of thoseultimately` chosen were persons who responded to the program booklet invitation.
When contributors were notified that their proposed had n selected fur presentation, they were advised that the= results symposium might be published as a collection of papers. Hence, a condition of participation was cheindividual's willingness to submit a formal paper so that the ideas shared at the symposium might be more widely disseminated at some future date. After the symposium, a proposal with first drafts of papers was sent to ERIC/SMEAC at The Ohio State Cni_versity in order that the collection might be consichired for publication, Following a review process, a favorable decision was made in the fall of 1980 to publish this monograph..-
in addition to the papers-presented at the symposium, this monograph inCludes two other contributions solicited bv-the editor to enhance the usefulness of this work to educators.Professor Gerald R. Rising, a well -known and respected mathematics educator at the State University of New York at.Buffaio, who has done extensive work with programMable calculators, was invited to prepare an introductory article for.the monograph giving a setting for the potential curriculum applications of the programmable calculator. Also,.`ts. Jill Coup, a librarian at GeOrge Mason University, was invited to help the' editor prepare a classified and partially annotated,bibliography of references on prgrammable calculators. SympoaiUmparticipants are grateful to Professor=Rising ando!'ls. for-aCcepting the invitations and providing' their respective fine'centributions.-.
" the Preparation oc-AnY manuscript for publication there are always many persons Whose assistance is invaluable.Clearly,-this monograph;-Would not have been possible without the cookratiOn and'fine,wark of the symposium contributors and the support of: others already Mentioned. But there remain three persons who deserve special mention. .1b Professor Marilyn N. SuYdam, AsSociate Director for .Mathematics Education at,the,FRIC Clearinghouse for c_tence,;Mathematics and Environmental Education, and onecifour profession'S true leaders, many thanks for her valuable'siipport and assistance in the preparation of this monograph. 'To-mY..Zoridjriend' and frerpient7collaborator, Professor Stephen L. Snover of the 4niver4ity of Hartford many thanks for his help in all Phases -- this project. Every editor needs a colleague liketeveto-call:- upon forte 2r,f--those unexpected and last-minutel.crises. And finally, my thanks appreciation, and love to my wifeLauriewhose understanding knows ojimit._At times when she deserved more of my attention, she wiliingly'sacrificed so that I night be involved in this project Dr. MArk A. Spikell Department of Education _George Madon University Fairfax, _VA- -22030 FOR'
calcula.or,sympostuM in -- 1 In opening remarks at the programmable Seattle, Ichoss to-set the stage for the contributors byproviding my:Answer to the question, "In light of the microcomputer, doeS the programmable calculator have a p'1ce in the school mathematics program?" The question needed addressing becausezbyl:980 some - .educators were suggesting that programmable calculatorswould have little or no impact on the curriculum.These educators seemed to believe that the dramatittechnological and business.developmentain the manufacture of microcomputers -- increasing machinecapabilities and rapidly decreasing costs--would.enable school's to have these dOWces in large numbqrs. Consequently, thete would be little need fetor adw..age in having-prograt table calculators available for Students to use as in the study or mathematics. The_thinking seemed to bewhy have the limited problem-solving_ capability of programmable calculators when the virtually unlimited potentialof the rocomputer was available and at reasonable. costs.
Few would dispute the potential impact 'that' uters May haVe n the school curriculum. But any suggestion that 'programmable calf ators have little to contribute is mostshort-sighted. Even as mlrocomputers proliferate and become more widelyavailable in .schools, programmable calculators have a great deal to offer ingtructionally, these calculators are effective-devices for doing mOch_of: what can be done by microcomputers. Consider the following list icfinstructional applications for which microcomputersmight be i.Bile the list is not exhaustive, it does cover some of the more bbViOms applications.
to provide drill and practice experiences rye as an independent study tool to.peorm problem-solving tasks programming skills emit simulations 6. to conduct testing to perform computer-managed instructionalactivities
dq data analysis _ Junction as -an information retrieval. device to=provide word processing dApabilities to offer;computet-literacy information to explore gaming experiences
sty of the twelve instructional applications cited ted to microcomputers Trogrammabie:taltulatOts c -1.-east eight-Of the twelve; They are:
to_provide-drill and practice experiences -.to serve as an independent study tool perform problem-solving tasks to develop programming skills permit simulatiobs
_ â. do data analysis 7. to provide coMputer literacy informrition 8. to explore gaming situations
WhAt.'s more, pr6grammable calculators offersome advantages over mieroJoMputers: The-calculatbrs are so ir.xpensive that virtually -every student could be supplied withone. !A classroom set of pr6grambables--enomgh, say, for each studentin a class of 30, cost'S no:mbei than perhaps two or three,microcomputers. Also, the programmable calculator has the superb feature ofportability, As a hand-held device it can tle easily carried wherever theuser needs or wants to use it. Flhafly, since" the calculatorsare (usually) Jiatteryoperated, they can be convenientlyused ever in places where access to eeecttiCal outlets is-not available.
In sUmmary, the view I shared in opening thesymposium was that programmable calculators do havea role to play in the school mathematics curriculum. The presentations at the symposium supported this view and the collectedpapers of the contribbtors provide ample evidence of the manyways programmable calculators' can be used in the study and reaching of mathematics.
In concluding this foreword, I provide forinterested readers a 1ef summary of the central ideas shared byeach author-ln the art es presented in this monograph.
Rising in his lead article adaresses the question ofwhethei-.± or not the programmable calculator will be,bf-onl,j passing, interest- I__.ihthe mathematics curriculum. . He seeks to place theLprograMmable t,- A culater in the historiedi,. political, and_sociological_context -w--: of Pplemencary devices and teaching techniques - Then he 77elates - Some his own experiences teach ng with programmablecalculators to sugg-t that they_are-here to s- _ but require appropriate -. ., --,_ curriculu development efforts to_ maximize_ their instructional
'
---- - Krlst pre,s'ents several pedagogical tles thatprogrammable calculator*:plain,educational.gettings. _he gives specific examples approprire for the high school cur?'..iculum1iat,2rovide____ AnsightsAnta-theH erection of students-,- calculators, and "---- mathematics Her exa-ples include an open-ended discovery activity__ di-on logarithms; an ex010 etionof a problem to find how many perfect ------' 4a',squares there areoAmoeig-_ 67 -, e numbers X X -X ...4( (where for l' -2 .--,6"---,each-n 1,2,3, ..-.-.:,Xn .:911,--1- 7), and an Interesting exercise to find whet' 7F' 'C? -.r -\ Muser discusses an instructional unit with fourth- and _ . grae on learning more about problem ribSaSeVeral activities in which children actually used p-rogrnjnable ,eeloulators'tt-Cennt:7(by'2s,-3ssquares etc.)_: to- siimSeif Various integers (consecutive iotegers, consecii, integern,- consecutive odd integera, etc.);_and, roblemyequiring partial sums of the triangular nuMbers: _ _ _
- - Huller presents by example, a philosophical issue relit The ug".ef-progr'.naliahri:7- atioris-appropriare-fcrr advanced school students, students, and teachers. He derives a .formula for the ratio of the perimeter to the diameter of a regular inscribed polygon, shows that the formula is unstable for vuting devices. and then modifies the formula to obtain a stable algorithm.
Snover's ar lo des , a standard algobra it oSsigament, to pltit the graph of v m a- pl3H- q, and shows hew the also of programmable yaleutator enables students to advance beyond the -,
.'simple graphing of a function to he able to grasp important , generalizations. He focuses on the effects of varying the'para- -s q on'the- graph and gives an interesting game called itit" to reinforce the ideas presented.
Johnson provides in her articlz. an example from trigonome
apnroach to solving a problem that is impravtdcal (bo cause t the difficulty of calculations) without the use of calculating evices.-- She considers the problem of finding the -third side and_ anAles of .a-triangle given two sides and the angle opposite one of them. Using the Law of Cosines rather than the LaW of Sines, she proceeds to show how to find the unknowns avoid
' - the ambi.-,uicv,which arises. when there are two values of arcsin,x (as passible angles of the triangle) for 041.
ga a presents for enzilchment a simulation game called 'Ghost- chip,'" which can be used with high school students. in the _s=- article hesuggests several questions, activities, and mathematical stteasions which students might explore. These include discussing g=-=ideas universal to programming such as flow charts, . loops, eoridi- -ztional tests. and .branching; asking students how they would calculate ;the'-distance between thc Ghostship and a missile shot if they knew the polar coordinates of both points; chauenging\students ie develop a program to allow a second ealculator to destroy the Ghostship;and _ exploring the game using rectangular rather than polar coordinates.
Haggerty illustrates theuse of programmable calculators to run simulated exteriMen pr&ents a probabilitveperiment app-rd= priate for high school students, included in the article are program and several in esting'questions about the experiment to use 'in the classroom.
Monier notes that programmable calculators_ can be quiteuseful_ in the: mathematico clasSrooM-as'a vehicle for stimulating independent Study.' He illustrates his point by present-ing a discussion of R3ithagorean jlew:Charta and programming suggestions are prOented for severallproblems given. 6
, Weaver suggests a specific use of the programmable calculator, _:,a'function machine) to show how 07e can direct attention to the-. drlYininature of mathematics,- -focus upon significant mathematic f :-Ideas, and exemplify an-iMportant type of learning_and latiStrates his poInfhY presenting four problem Contexts apprb-ptia0 r for students any school and teacher preparation se an end note-, eaders are invited to write for programs machines.
Maor, in the -context of discussing highlights from-the histo6r of n different ways of computing nincluding the methods of Archimedes, Viete,' Wallis, Gregory. and Euler, He includes two programs and the article should he of interest to uppergrade high school students, college students of mathematics, and tdachers.
Elich describes a oneqUartercredit programmable calculator techniques ,course with twely51class meetings given at Utah State', University. The course is-5Tlered in 0Wo sections, one for algebraic Calculators and the ocher fcr RPN calculators. The syllabus -for each section is prespnted and twelve sample problems from the exercise sets are given.
Coup presents a bibliography of 145 programmable calculator sources including books, articles, ERIC documents, and dissertal Lions. She lists the tooks and classifies the remaining references into -seven categories: Bibliographies, Elementary School,,Secondary sch-nnt,,C011-ege--andPoStgraduate,, Gatios, OtherAlsosand_Ceneral____ ny-7--of the citations are annotated for the reader!S benefit.
Mark A. Spikeli Editor -:- -SOE-
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S eway,,,;1)es pit r ess ns abou a teVe. that handql d
oOr 1'aii1,4/Lid:I.-ii-:t9.00ple to -32' , -n t o r Asp, t -'ear help u dnt-s 0 _,--0-.- -, ...,,wee,_ ma th ema a 'ideas- ...lin-on. an_ akto se-e ex '-et- ndt can :0;ra-du e matec 1 -u-.
. .1 e ";4-roust _iana u c h-e41C0 70, teache a tits- maad, -binT=:Teab Tin u ,t-_c-ctilmtzir dirte-Frrit:---1---l-iigrert-h e-tril.:as-s-r- on______-_tk ft oppOrtu-niaere_ ----;=-1,-nd iy-I-Sd Eta ,.. A-- is - eaye-, c s-rabl-e-ItO, -a --6- t e_S_ 3":t.... -. 1.27...,...,.T.z tun--_ esy ts--,.So_o_n_c-%-e-- -to-realize t_ et-Ji touch C_ on -a _a wIte -end- -stir the _ cider ive=i-oiem-s 115,,,,,e, r gto,,=4.rayos,'-the s' `-test-- h:,---thi :64# imen,id7-- o=COMMUnitstb,
Ceil1eagues'4 t B e t ty Krisi-,, Carl Roes _h_ . and ,pan-Ste -..! Winced ,thatJ:JOur 'st0-dents learnedlariy,df,,ehe:., concept
tof,-.elev.e0tii and--- twelfth-grade-.. _:thernaieSPIikthkgreat tg.,,, °when,,,,,--. , : A 7 1they were taught by-..- thethe techn, qUes ,and 'Withi,°: till 14,ri,;.1.s;0fp developed for _thei laseithprog aranable ,, il,_ato erwere' able to give them, m ,cient ac vatto- rnan p egraraaing e ;function; -for exam le,l'the'yl"iJ'y:ere klige.'",eb :.-- -tvf,040 eap-Kquici--aiidtaccurately;thlisgettingrdife ,,temsth- hat,- graphing 2/exposes. ,We -were able to prow r,e,ree _ _ the.:de-elOpireUt-:i .` 4,1 cif traaltionalI ,_ dontent;,-=rt xamp,72::=:; e, un ontalefrOWIVaienverd' the mile8C,-;the,1:40 thus en_ ye 't0d0V,..,I0',thipimptittant ,gjc in,p4m0aran _ '',_=.. 1 1-::'--Zik:_i?,-
47 we _ _ ,.,alsO,ahl:02;-::341,'vf 'e'l ', F` toOrrifiii:40iiew';idie'd _ .. MoS eriancJiaof-;ttiese_%:are,,programming.,cone'Sb e' a e n'topr,431png,_,,,i c*f,,PbWeilnl'partit-iriii-`2o :ten' o i _ .intomp,iterTsii; rice te04-rseithy' -:i- p kit ltd._-s L,':-Si_''MP`iIietlifi=o7r e da-Aliti-kte-dige `s---'-unclut tared -4and, 61-9-aarg-ill` ibtf-i-er ogvmmi g --o4,,,branChingrap0 Jle'4,s,lon Ahakleg* re, klpo p, ,. ea,n_a atraightfLvart:Lthatiztherrare.,:alsafiloTt",, s .1-. tr -misunde s ta'n'an:.(00UleolitVityeYflhilli bttil und m an udents are i.fatbe'tter..;prep red,&i,fdthey.tw, "Iih _o s . _-:c tr -ay three- 6 `!_ oyes tif,f.c.o-Tifp ute E scien len -uag --a ...... _Ji..e. or_ e y seriouseatlier -Could 'Jiav i,tlfe ,,oppo - the Teague_ 1-d' I had'r,-,1.1sint',\Shetsert-ool,a0411,t 1111W-''4_pp,top a_,_ ppt mat - ' e : d r e l . ! t h e , _ , c l a s s r o o m : I x am.-- ear tain 4 th a ,Itliey(W:dri)rdttr- rd on ---o Farealablesv. -Given thisrlandrof ';pedagogl.ce_ -,--;,,s_oftwe p som,nom o w,*111i'W=Alliit -- , ep,rov __ea di: a' vtII"''' e,,3papers 1.n"'°- t ls .1.:,-, , C on-, 64144convincedilthat-00 Arearno aliii4qurn. d haven. ..,Y',fier:Tilistructio-0,-.p iy, 1 elitian0-by " othes_ ..!k,,, =, then 0, -otirA, educatlinalW _ vide tit ne e- o -"thisifribrkir( 'tool; t ",cie- sig.'s on o -,h° ,ands c°nal Firogr_- ' ti.S' b, e c-a-_culratoas $7 e oniciol ru '1-a ap _ e the s-d 1 '-ftiou_ eve bOt-ir'us ', so _h os_ , -1 n___,c rpo ate -V e , eye ns -nlos eq a win a ea_ bu_ ed cal-c . OR"TrE_ A getd-V WesSaneba a5 ,en -Sdnee
g > =yd w -7heenTworle ithitwo,te Passe en e =iced_ n rT,s 4e-e1aveoth = - a tErrOatAlliaLxs. supgarte heiNa Educative Bas ou --ant 1.0g4" 13wandventitled -Grade 'Cur Lcul m -.-- 4,,,,,,, 4,,, . -1.---`1-...,-^2 ale ,ew,Computation,, Liek4edz'tt:di'li-eao, six oms evi.a2FiriEe : final 'arm-, cu ''cubtAa'materials _nkeg- earcuIator.oisagein elev 'an 1 f th- he 4he''' ,p,rojSetts,tsff;''iPeraldi, n Ele4y, eot s chit' ndj,De Stover-hag,i6diie!d -eight; i --4- ii , , venth-g ade text and sevenchapters -Of twelfth-grade,,. e teri a l s were ; used jwita - s tudents et i' Siee t,- Home Sen i,tii Sc _co a t ,ew YoI and, 'est e eneca East 5 entlF: i1, 7P , es ndca,71.41WWIYOik. t 'The studentsWea '-e-c---iiliissued' h =.- ,ac_a d ipodel4 33E or if madell 25 ' ecientifi ,, pro grannabie-/7.' :'teethe ca cula to r.mode ls are,almoslidenticalusin sh:Nere / having Q_ pregrani-,s,teps*. to t entiwea hasdiLtit',-the1-t*y4ik State n_ El _venth Yea ut,was,s!ffiodiged tb -10.. _e teehnoI'' e eirsilab 1 e to-- eYstudents.Ter thms tudiethes3 a f une t ion re thei.Lthart" om uta,fterial t0afidk-Sii"ehet4ine=hijAirah :EOP-ialwasWirtY-iplciti on a s 411 ott unfit -ifenalo,_att4i9$116:r tunts wer y__ as Fru in f 'the- calciirdto fr.ledThrgchalyti_ h c e,r was t augh thou tr b c use' a n t._,S.Cthe' course was mot to he aird 'cliirailF-;:iiarele,fit:The'l grade a _seve don s Lander e 'Cap that benefit from "calculator availahil .t" . 4
at Di-fife=nee A drw studen steno -elm c1hesee"are randoraly.;-chosenotromxycqunt_ ewe e s tudentst who -.had tect(thectraAtleaalginth- ent-grade egants r,program e'JpreparatbrY cs -.N.Time-rit?,,rangeef rodiverygpper-;-_, ( _Yjpa dug liath 4,,
_tt grades 1Q "ands.l2t'hex''twe gadefs use _ She also:eb..-7dcreetor the tad Math _ S u lir deem 'Co tpxo-"viede.rA6 ern mathematics to we .Iva t fudents ant, Ther?%; 10=1.5 pe_c t o -o AMA a- -adt=_ t t,a'as'^e, an ..-_ n eert dr atii thy- '_n_ _ etn.1 t _eruha__a any-A. anc th...._t ..._ ,- 4rwg,A- ,-1.,--,. "-- ? O zogreftne ' ,caltculercfrai ,.- --'4-_,... ---''.,..ic...,,,,.: s u-__ n toolrittliffnal'eVairiin- 4:.--.1..--..-i_,,,-.,-:----- n r-___ genexamtnataon,..anota nsp_rep red h c a __...3.1-1.1.N.1,-=:,-.--_ni,th_estudenta-Ite ------4A-catkin-:on,1 ry ow )._,,cajatiars .-"-th.= ea_ . =-777-q-- _ -on=t _ == -^e. 5 cal curet=o y d,_,be-e or_ re. ..wdo-iir' ses,,,A4ian J y_ oflatiai-tt -sdestjlat:krerLutV:the -ijul.- hypii", ioiliesA- -slyait.--7---t141: n _c _nt dS4firene:between-air owinv a tide n i s.li t.wo4 without, thee,trVa note 'here' 'the. ,-ne&6-aOari, _
haVeftedt,cnaceptuaiconceptual differenCes in'theltast_ -, , _ , 4 ,-,-41, -494iN-ii,tv,- ,. ---4Some:_,41-r4Zue caution..must be exercised in viewing this ..analysi Th046____a as notitended ttoibaa statistic 1 sEudy,andifthe"Je,tandar-d' o s were --' sped iflcally ^mounted./ llinweve-r, the lore-CA.1n n = _^s_ 0.6'4-thtp lidgnt; I-;'4 work is -much ;like^ - the-,.ty0iCactl4t = p c research nerally' haS -4Ve,ati ''condaCted:-abolit_1101 - epriSql ts ther04C(041-wa.4ifidisti:stlidieS're:nd/Sna; o ' Sean 't9i: Ti ave.; begun .,,_ ,` ye,; can t enCf,=the t,_'.. this'Sn'afysi 4e.- Icaolle _ _. or'''!--71.1iLftsliiwiiiiaA1-6iii-:1 e ',`A`E.86'peaqase nun hypo_ t e N.. A I 4 __ 0 44tnodbi&Pence,..'11-etii-aelilVotipq. '-'1'no,zdif n '' nakric hides a--ertnt,,,TA,eted.':.inr,,7the;p::daggriicil_,,r4es:',:ttile. , p ro gram= ecalculat_R,F,.lIn-llayl---InlinredUcaki_Onal_ 4'..faetirifg-.t:'. -'441.-4---;;;(444-V7Z1-\-,7Mirr-IT;'''''-r.Ny''XIP :F:''"-40 --,-,-. it4-.-,*^,i ;,-, '0 si_ng wi_th45niot444-0 rhigh;student,s';w_1430-aale,d ptodaft a h u -o can aids4theitAstadyll'Shi.niattiesiataawwerno ®_y on _-_edethl'mater au but with the InterEtctiiin ,70,f,e udents,- ' , a e_ s Yid theireli"Cs .1 -l061c.edOcerifurilli-r044-y _Sage t_ s -t On an esite g '-`,0falar,,`6'fLth'e-kains to Iachi v s- a cu 4ato-Is -and al-s.6'4apmels ,AerV.Ttlaes.7.-?, pile aiseCove ,.-, 'v .oy, -nwatiTeM-40.10.0g'.wer ',4;parti:,(cui.:61r. ,,..aliet! a .0pada _gy e ca_ e akly scrafinxielf weIgo. . u 1 i to_n a tommu__ _ cOmpli atiMir 1'''b0t.1,.thi-Orna.rylkitaki.Y oselPmul--4c0la t 0r, u_rils7k-u,:;:fih"\-ga p 4 dreg 14 4aegue ea nde_lt'and,c-teach'afaiid beeen_situdei,lt ancrj_ _,i,:that can . 0 the me o a±irevol,:e'tkOa, ,i'101.! fiiifo 'min-11'10" 'Atip=radels/ng -and 1) enthi taboo -mat -^`=1-----'=,.-,4-- 4--v---/p;',2:,...,43___, ..__ 70,------i--,:,,,,,,,,,L- y t. x, -,sue '-'-'77,'". k. rovida. Inarghts e cur- r now yr te,,,t 0 a ,,.1:4e Cif fc;:e.'xim'pres, tha ---,.4,...--_,- ._.,..37, n tirdentS, 'calculators, an Inatheme'08.4: th rdtion,n, ,, ,1,,.. .
t
,, e_s,a o a haiibeen 0d'',133..-,,,dire dilf - n eac _ a :sr) bit th-"i"e44:lifVei-e:ft;gOtiplio uden. _.- , s a_s sth_esults hat_ been dr,amattca''^ tt-fll''y,,,,s m ' atiN an 14 -.V' n e- _ c-ve '_.-tiiv rey wi an exit_e -s 04 =-a a s_ studentw,ho were _u s _y n _rnib -es t -om on: class. AsAAS. A AK ----,...... ,---1-,.. 'A- . _...... - Awe -xisdllpl. red:tao': g° u
=---- ''---thi.3-Aty.-dn't1.tn- un _ ,-_-0cgs 6..._.,-,board= 0- tc _ --'d
=
u_
---- 184-51 ' :4771 -11* error
.6021' \ - 100 / 2 1000 joopo 4
_ t-i's,;4p_oin,two student-\made the -following/ comments:"ThiS _ ended - --3 amv.,the''real_. hingit's should be .477121255, lint,,sveK A round This ' getting, us nowhere, we should loA1..4fo't"''' -_`tt :'' 'With thisthe teacher' Uggested that students-- t el_ 're fijectn-rnsi and look for relitionghips within th0,4t6- , trv'tife it-udenti began. to.- work,independently!, making '''i;-/--1,.-- 7- ','---,,v 0 r---Af, conjectures: ' / Flog e -.-6021 10 7,4.0004 1 / 2 - 10 101 0
at iall'corde-dture =wasthe teased in each ofthe three classes-whve 'ESS'on4wa-ATUSed.-, iiav's the sameIdecimalI 1, (fn the m ,,,numberfin front.of the A . ) ----2-,alsal:viorks for 100b;-
nitwork like
ends the 5s serie 2 .,6990,so-,Ithae ortibetweep 2s and 5s. ''''t're 6 lo ive.(tha/number of) zee Ar% 1747 --241!!4.000Mg do rr een28--ati
, log before (the; log) is like the- exponent iii ticItition.Another student added, "That's OK; negativesit's one less than the number of,digi s3discussion stopped and, the keacher madeEhri7fiiil1etii the--follpwing :table: I log n
0 .3010 4,7-71 4 6031r .6990' -.7782\ 8451 1- 0- .9031 .9542 )1.0000
ong-., the [numb gain ignited:_,,
vat-, ,,.3010-+ .4 .7782 almps .3010 + .6021 = ialmost g exa4ly:-.4771 # :4771 , .9542 Tr.( tu' to ;omplete thfollowing rtable' thout a [rig their calculators , ents;IMade the fUllowlng e tes :
s i ID-
'', 11 _ 1.039116 halfway between 10 and 12::21
- -'II. 079'2 3 ana_. 4__Or and_.._6 ' '""`"' 11.1127 halfway between 1 and '1.44%-',, 14 1.14 1 2 and 7 ----,
15 - 1:17 1 : 3 and 5 '".-f'
, 16 1 .- 20),1 , ' , 2- and El,.or4 , and' 4
17 \ 11.22q7, . halfway between 1 r and,118
-18 I '1.25 '-'2 and19 or :.t't'W,i;_4,-'
, 31 and 6 --- 1 1.2 1.27 -halfway be tween _18
I 1.-30 0 , ' 10 and' 2 or
1.3011= ..- 4," and 5 1
checkthe logsbyuSing r -0-4-1',- nod discrepancies with__',,---{-. qhel,,,oldt
:p.- st:,-!,,,,----'4--rEt
th - iia1-s is.rineto,;,a:.-- close; ,so the teithe _ . ,L 41- fr homewor .'"tra'6oki3=: t ggles /and,.'conjecturesan:dr try , rs 0- why-_,Oer atements'''are e, eh nn n A;,ths. _ _ _one_ -_-ar b a-77 un-_ E - eft-eS, "'S us 'cliVith7t
t e---T-s-CuTdentworked with basesil9= Owe _-Tapr'n t4t-,ion 7.wie h :_,eadah_er 4144% arid=--studied them- _=a_ahlreation,--1frofit VignaArt
tA repisode hasomuoh' dramatic significance. ')Firat, shower's'. u tttliscoveactivities are exciting: __These students` were usi cLra ti their work.They seemed ate' be-working,bn- e.44,4 ve,--s,thry.They enjoyed theclues -givenby theircaiculat2 e sake ".Inat';!tell us ,how- it_ works.'-'-It would seem thac the dthat t.ketteachrr would have spoiled their `game ifistielhatly 4,0'ecl:f,'.e1,1 their corgbctures' as theY'arose:=401-iese
4-writ-tent pfaPerties about -roe = w h y eEhey,A ttte,yJt.temselves..-They even discovered a few,,i'athei4013eCu lugs to I+log -LI- ? , t,ntthat-,-10log -n _was genal re-d beta 3,0 ",111:e dnhsequent 'proofs and blasSW-rk
its -dTcalcul er,wag, 4 'device that,-proVidid4'4, ewo 'fielp'eli;the-stu_aens, put.the rIPOwn7,-.4eat v ity r the c_..aul,atar4.,The calculators provided a setting and itieiz'wer- 'as the- students" had --talq,beyoa:14,e ers exh bleed 8alcala tar 7-; 7;=,;71,.
'211
xemkiigis the 'Fwth":0c student4 Solving. problemswesClan, another_zeiaraPIe of the pedagogical language =aspect 04E, a ingrApg:tttl'O'iir--,clas4rooini.In .this',exaiicilg;t n o t part of the dcEi.V1.2tY"`6i even _E-3-idid:;41-d;the students in reaching theliriolAirat cotisiderVd:,hi _ 'nm- rth n -=.12-, '3, let , n Csquarea' are there among-the tturnhe'ii,-Tki, particular etudel31.14:`.'d'olut er
lip.
7
- Z6. z
'7 .2 \ ! _ ':' -\\,, CI i-' -. ,: ,,.,":N";,. v4.-.4.--' - - -, .I. . , _7 A ; - i rr,,r
J.,
= , _ ,- I--,,,,, 7.1 ,, zo, 9oz. e I cife,
77., = .
, i - - . , . t ' ' - ' -
1;,='4',*5'f'144 61e.r41.'";1'4114:1';"117' .._,- ,.., ng ihatthe-Wp"i_t tern-8 should continue, he 'announced iii-'51.,` a 1.1 , _Iiih'ii=et7ir'e"-21-4"" -When' asked if he 4AS'aiiii';`2-1-ie sita'''a` entA4lig..-t-Yes,- and-the other stUdents quit meief4=,,:tdfrig_ten- e: V --j-C., ' ,-----,r, arattd.on.:-/-"Th? teacher asked1-roa,,proo'-= f'.. a'Anife'r, _o r.studeo o uldlr,iir'' ite- a prOgrimi..",'.._ lie 'pianede a 4. to ,filiiiishu,iig4- frePgraVi. '
- o ,' ' = 4:, 11 ".g 147,0-. e7-510.1 : ' 12 G 6 b ,Rbt 1 11 a4 0 15 0 -, -_ 16-
QCi VI* t'''"' -. 09 1 u uc sisexao = Nese um -t---evaluates-=entite_eat' n esting compison-off-logic. a...fount 41 ii-7---s slideiAt.:Autckly ge ne r 1 ted,Je xampl ea_riu n -elabItidelMfi. the-.calculator_ --p_ta ternielped;;;him gaEher?his`TTideas'it,pg-echeri.'illutilitY...... -- fl.-----r.--N1--... .. - = .....a47-,--- n1which,the. choseto ablve.:the problem. Oncettpputi-1la -asquj.cler Tor him Lto_' work without the"eSloul*br. :.:.71 .... -can!balance the overwhelming- posirlve rieflthird--0., example . , -,....., or these first twoexamplesand,proclaimthat the calculato ' he toeal answer toour problems. , = L -,--_ _
erY n,--theae:yatydents were just ,beginrbing their work to at`thejAw-aite:Vresented with thefollowing axe'r-4se
Usn of the two,_ cOnversion'forthulas for gelsius,,,and'._." Fv. eraturea'C F Ci-32, answer,'-th 9 I
. ..,
f.T.Conyert 32°F'to C. CltfigikOlgf,-te,liC.
. 071-a '6?F:to' _- i,4(-1,-..-,!-;.- ..... , a . w n_ ixperimen5ing when F- and rare the's
4Ite ., r..1-.4-05,4:,4.-.4,,,, ,_...;,--f 4"" u`e.' ,3, i tudent an. no...trouoie,-oolngz-i _tnrougn. 4bit few OW '''e-i-pon.hel was-,="I-Yd_en-4,:rkno-w- - 11.::--,gw^.,tildehtS,did tryra few- specific values for F,__acnd,C;tbut,..1.- _ o"._cesi",,C4-&-e,:iiraticl:' nd their results wereVirot nrge-rala, n .,-,,,r. 1 _ , 4---42. a__ ; .- one tu`deiitlsai'd° '"l'i T -.4 32'then-TLI.s-1-n-egatiiie',11- when he _ -..'&-dthet, ,a-C,F,:,,de-br ased '1' ,----C-.decrease-dutliheo:ie e'd
d , '',',F-AnelCq.:dou d-,16e identleil and negtire _ly_ thtee, _ti eori'eelY1 so 1-Ve' ,.. tb e .,,,f. rob len' en_ their own''..., SS 4' h hree=---,..laas-roonvexamO_lei -clearinilatis =mown tliktt..:°a,l,ealeulatorliceitvbe ,an...rapet,taiit---a t, oumaartlidtle1c°`andirpedegogicalVlanguageT-.....criWica: rhe serianc_el'ci:gii-.7.1;..,Whet rae,i,ineed17te-coneire-i' are 40-pr agecb",_i171aut'tlits' t'a.liappati4afieli:0 rettiOci-_- that -' t net happea; -1-iliat.,,eaoo ,4,tese cis'as requi: a_ .._ .-1.,_._ -,,i4 studen__ :calculator was h u
.-..: v . ..=-9 udshea --de to g a _ n_ y what' _a- our student, do: at ema their:0_7-es ...-L,-...,=,
. -7=A , - -13 aa- i ror mpn cero-;pze adin ng at a _', glrdhit e'dldli v.'an _, Ad --,,r------0,--..- a o-p wd:'. pRylide-,Cis-;-wi naratto -s c dren,s_tudieg,, hill`alahOwn the;?,coil Ell -o to poat_the earnettg'_Ua tradit-arithmetic-x-arithm_ as w q _ ppo _uni fomany=,:mora propm bl-Vag,Vcpa e n- a . m 2estga:uitVitfaEfeiti VI ,gel aeZare;e6aib e a en_ u aid ced.ndiVtdualizingTrinstruction. -.iin.4aiiditiOn;frito- = - -.., _r,-,k * ,Tc,;, zr , e. -a be,_used 'to=.teaehzchidren7progr.,amrrp,.y_kg-,,ski'klsto_n
sV_ cpipPlicated,iPrirlblenia:-4 ='' =-=
i 1 ;44-4 . _,-,- -.: 6u hicrocompUtera for _ab1 w
IS es y fpr=ekch-dlill4akdrayene,Secomin,More_a _ c_ p -erb' a p og able77dalcolntors.icUrteay i_ h t.w_ en c *torsand ciocoMputera:both.010xide,wik-1.4,, and ._4141....°:-.14-ENieY,A, .-Ift,1 o- 'es thaaeIOU,oneAcan.r. porcuasea_ Irian qp t , ab' r 'orate, 'Whi;a-'ca:rikVused,t--- niTroidtie :_,Mdien efts vi ,program'ag.- .4Mai7edyar'awn_` entirec 'a oo #0,110 /,mil,,mt- puck onsch_o'bui .ding=,at) -cancan pu =chased 0 th t_a _omputedye-term, us, a bri--o ou go b em, ap,Arin-g t rough -grog amzias _,the pc, n a b'eisk'sbre!:t---',Vietk-WjE -his"-drerown o uforin s hc5o1s Andr-rnous. Sincethe tr ghno gi- 0: -able oat 'a tors 'arid CompifferS lataa - ow'n t valp.--_,. ---1,,- .777 . cam S. OnyMOUS W -t n _3,94yearre (hopefully much eon aa_ 'a w ',, hiveI clifiltiiii2ff1Milier. - , _ ,- = teMpe I asked _ %alb tan_ do_ th t--,:a. stu en howere, nvoltre a. a math, matics -__- in -4,-_---ty-. ,.. . two --minute iods%per,,wea d'ichlis*e had a-- e- o oram arSails s 't4-.- a--ws_e e q wit. ' aril -a" capab it as 4--.. A es an__ -heir enthus o w iipTo g -atm ./sokiwi. -' . uspected, the dp`et Tr5nadiate e a -h t a la aF inad,dung even w_e s w m , was i cu t to hp pro onged dcuss -0 a, ecaus_ eA. w_n to be rk e'c -u t--
us _ -- ep ea,-v' a:nser-vei ern=a _T- _ac as. asgiven Many kaho- n
v -_ s ar c- h--_ b -n pub zsh d e su 4C ma cs a a_ b hm h .
_ um t"se a orm , p t m The main go,.rl of our instructional unit was to learn more abou roblem solving, particularly how to apply,technique's of programming to -s'i:ilve-problerrisi which would not be within, reach of atypical 4= -mathematics program at this level. A second-goal, was to help the :children see that certain problems had various solution types and thatsometimesa programed solution might be "best," whereas an ..-irtsightful mathematicalsolutionmight, be "best " " - other yimes. The follouiirrg discussion provides a glimpseinto the -students' accofn- plishments during our seven-weeksession.-
(.; After beroming. familiar withmanyof the keys of, the calculator constEuctad our first program. (Our calculators Used ReVerse 1,1-t Natation,which isa.great convenience and posed no difficulty- the children.) We programmed the calculator to count by Is as- ollows:
a. The number 1 was entered in Storage Register
b.lhe following program wasentered:
Step Operation'
01 RCL 0 egister,_ to whatever we have.-
This makes the calculator pause for_ second sowe can see thenumber-.--
04 GM 01 This sends us back to Step 01.
--+,1-. en _the- children ,pressed their Run/Stop keys, they-werethrillq-d_tb , . -,:,_ ,--# a- Vtliefr__. calculator count. _ It vas easy forthem , to-get the _--... -1-CUlator to -count by 2s or 3s or-any other digit by simply changing 011--t.in--Reglster-0 to 2 or 3, etc,- ._ _--_
next step was totryto, count using the odd numbers. -=Once._ aT,'-stddents observed thateach odd r:Umber wasone less than arievent";- ---:, =they_ wrote the following program.°
Store the number 2 in Register O. - 01- RCL 0 02 This counts by 2s and store 03 -STO 1_ in Register 1. I This prodUces - nrierdit__ the.cairnato o the
r s47-0u-r1-_-=1a StreVerffnamb 77- °10
After the students had n thiZpr'ogram success-fully,I listed the following. two programs on the board and asked them 'to determine the outputfor each program.
Store the nwther 1 Store the,number 2 Register 0=. Register
01 Ra 0 01 02 02 [PAUSE 03 03 RCL 0 04 04 + 05 -- 05 GTO 02., 06-- (PAUSE
07 1 - -08 RCL 0 09 10 ST O 0 11- GTO01
To obtain the, outputs, the students pretended that they were the calculator. Each in turn =executed one step of program just= as the `calculator would. For example,-forthe pcng-ram,on the left above, theywould say 2, 1 x 2, 1, 1 x2- 1, -Pause, 1, 1,; 2,...5tore.in 0, tor,ete.-,.-Id their amazement,Abe outputsof-allth-reke programs- re-identical. This gaveus anopportunity to observe that programs --- and problems) can have different solutiens and that many 'times it is 1 Orthwhile to try to'writepIogramsmore simply and, perhaps, with__ werf.-steps. :'After 'nountingbythe odd numbers, they wrote-aprOgrath
count =bythe-squares-, i.e.7 1 47 9 16 etc.
our. calculators had an easy_ method-to cumulate nuMbers as counted them; namely,thecommand ."STO + 1'' would storethe numb,er:_-_-- entlydisplayed by aldingli to register 1." Iaskedthestude-ntS- oW---ii'rea program which would c:eaunt by-odd numbers 'a35 I7I9-I add-them as-it went alon4 (1,. 143, 1+3+5....). After a considerable amount of work, we arrived at_xprogram that produced the desired .sums After :several of the sums were displayed,onestudent exclaimed, "tiny, this. program is giving us all the square numbers," Uponfurther '-_, _exariinatton the student .could actually see *hy the sums of nonsecutiv odd'numbers bekinnidgtatbI always produced square numbers by considering the felioiqing arrays:
---.1told the famoux-story--aboutr;Cauis-,haVing° toe-add"- Whenasked.how they'oul.cl_rrk--' separated-Intotwngroups;!,db-quOihn: d,,-
_ ;co---'Use pencil and paper,andthe rest wanted to write a prograM.. Before the end' of out period, all of the calculatot students had written and run their program and .none of the paper-and-pencil oup had ,come close to the Correct !answer.
At our,n meeting ked the students howrho,would add heno-abet-a-1 to 1000.- They all said, "By using a nrogrammabje calcdlator." Then mused,- 'How -fongwill it take if we have-one ifPAUSE in oUrzprogram?" Sine an fl'AUSElasts about one:se,cor41,- (itheir. replywas, "About L000 seconds."'Since we did not nave time- -to run such a program, I showed themhow,Gauss was supposed to have done the 1 to 100 problem:
2 99 + 100 100+ 99+ +799 101 + 101 + + 101 + 101
Therefore, two + 2 + + 99 + 100 equals 100 x101, 100 x 101 so 1 2 k 99 100a 5 050
,akfter they had calculated 1 + 1000 in a similarfashion titSingtheir calculators,forthe multiplication and division, we cussed__' how problems,,involving calculations could b'e dons in k_ several ways:, mentally, using paper and pencil, using a calculator, 1:- usLing.aprograrmmable calculator, or any combination of these with :-'mathematical reasonifig" as Gauss did.
Since Christmas was nearing, T presented the following problems- to solve-:
Chris rues' trees,below arecomposed of ornaments and are to *,-appealon greeting.cards 24=limensional). c.
199 drnamentsits the last row
Imianrornaments are 'in each
. e ts--recegnized tree ii--asca.-_auss, yr a1 d=_much: _&/T-Gailasdid. The-zoiiiy--infutinnTiohtairieri--foi whic-summed the eagniz_ a.pteblieffit-,Cilasritife7,611d=t-VEZ-fidds=-Wir eh diarely led finding the apptopriate square (1002, in this case) we had 'tudone _earlier;After problem C had heefi I asked the students ifthey could see-sny connection among the three problems.After a few hints they obselyed that thenumber of ornaments on tree B together with the ntunher of ornaments on. tree C total the number on tree A.Thus, if problems A and C were solved first, the solution to problem B could be obtained using subtraction.
II. Now imagine the two3-dimensionaltrees made from sph meats.Tree D has 1 ornament on the top layer, 3 on the second, 6 on the third, 10 on the fourth, etc. (these are the- riarigylar. numbers),, down to the bottom layer whichis equilateral triangle' with 100 ornaments on each side.Tree E has the same conf igurarion, eN..lept the layers are represented by the square nurnbeds 11,4,9,1'6.. .1002).How many ornaments are required to make these trees'
One studentrecognized the numbers in each layer asthe Partial sums leading up to Gauss' problem, 'Having successfully written a program tb find "Gauss" sums, it was an easy step to rite aprograM for tree D:
01 02 1 This counts by 1- 03 +- andstores 'the numbers in 104 STO 0 I Register 0, and - -05 fPAUSE , = pauses roshow the number of the layer.-
. I - 06 STO+ I : This adds corsech`_ive numberd- to Reginte 07 RCL 1 formingZahss" sums and recalls these
STO RCL 2 This adds successive 110 fPAUSE- to display them.
.' Theanswer this- problem is 171,700.The solution to CheTre E problem can -be found in a similar manner. °--3 ---
% These final problems inirplving'. Christmas trees were extremely use-----.. -11 v . . ,-. -...- ailin--.helping the_childfen to seethe role of a programmable calculato_ is; a: tool in ' the-- problem7so lying process. Where the problems involving Cr ---_--es -fil,n,_ and. C were most...easily solved -using mathematichl_thiniting '" Gauss' --odijand patterns (the sums '.f conseehtive_orld_niirnberk- ,,a..-T Sitar:di-ft -ard--s4hare numbers),_ the- only techniqugs,;evsilable-:tqL=----- 71-041=-4'L-----e ohrth---==hfid'fifthProblemsVereNprograia '-fi-3-i'th-e caitUlato_ --__ l-----obite-ms-U4.s 4k-the-Se- which-formerly were beyondthe--rehlmof: oietv
_ 'Oh1-1-ataderlii; can now,-be:---so ved- fairly---ronrinelY,,h3r__S-Onie--Ifif th--g . +j , :.This unit I taught fifth- and sixth-grade Students would also. 11e;_ap-propriate for students in grades 8 and up.- (I agi going to integrateit into a class for elementary teachers I teach.) -Judging from the excitement` which came out of this work with ,programmabe calculators, teachers whoiCan incorporate teaching problem SolVkng via programmable calculators will have an appre dative audience: students, parents, and =administrators, alike. DES ALL
John Huber Department of Mathematics Pan American University Edinburg-, Texas 78539
Using_the fact that the circumference of a Circle lies between_ the Perimeter of any egular inscribed polygon and -that of any - regular circumscribed` Polygon, Archimedes was able to show that n 46a between ,223/71 and 22/7. The purpose of this .paper is to Orly the usual recursive formula for,,the ratio, of the perimeter to they diameter of regular inscribed Polygon; show that this fordula is unstable_ for-computing devices, and modify the formula to a stable tOrithm-.
let en- 'demote the length of .a side of a regular polygon of n dea:{inseribed in a circle of -radius ri and let s2n.denote the =length _cif the sideof th'e tegulae-polygon of 2n sides formed by- _-,Aiffeciirig the arc containing consecutive vertices of the original regular. incribed.polygon of rCsides " .
Huber t e a ch e Imathematics education -courses' for - He has-_ n---.Widel range of interests including: resear_ tit-tide-an!dAnxiety;`coritivepncesssesa ling culato_r_s an 4....computers gain usidvFigure 1 and the Pythagorean Theorem we hav
2\ 2' _en considering a re [Liar hexagon inscribed in a circle of tad becomed:-.
where s6 and perimeter /diameter = Using4 programme.. calculator,we have the results in Table I.- "(See AppendOe.for
programs.) Clearly the :retie does not converge to '71'.
(nowing thatit% = 7,why does the algorithm not converge on the calculatbe The' lack of cenvergenee is caused
by the large relative erfrorin the difference-;i2 t.fi 4 s Since ,.
n . s6 is cibete'2 QseeTahle,II)s the rounding error along with he closeness of 2 and 4 causes a large relative error = sgs:resulting inan unstable-algorithm (C4teland de-Beet :1972S pp_:-13714). ;
TO stabilize ' algorithm; we must remove; the diffdrence-:- - - s2 k can be accomplished by .rationalizing the 'nursers a.. -der the radical id (11) giving LIS: /
result .
e :1 'iriating-thp difference results in a stable algorithm that verges-to--Tr. Using a-programmable calcuietor,we:have the esars in Tablfe III
,and. -de Boors Carl- Elements Numetical :-Analysis.(2n MEGra_ ,- __, NewYork: 'w-Hill 1972, - -- --/- 23
TABLE
NuMber Sides Length of Side Diameter
1,0000000000 3.000000000 10.5176380902 3.105828541 24 0.2610523844 3.132638613 48 "0.1308062585 3.139350203 96 0.0654381654 3.141031951 -192 0.0327234633 3.141452473 384 0.6163622792 3.141557615 768 0.0081812081 3.141583911 1536 0.0040906127 3.141590529 3072 0.0020453076 3.141592407 6144 0.0010226544 3.141595284 12288 -- 0.0005113277 3.141597288 24576 0.0002556658 3.141621319 49152 0.0001278348 -.3.141693413 98304 -` .0:0000639218 3.141885657 0.0000319687 3.142654499 393216 0.6000160000 3.145728000 =786432 0.0000080623 3.170208743 =1572864 7 0.0h00041232 3_242542203 0.0 00022361 ...3.517030823 62914.6; 0.0b0014142 4.448731201
TAB!
= ofides: 2
6 1.732050809 0.2679491924 12 1931851653 0.0681483474- 24 1.982889723 0.0171102772 1.995717846 0.0042821535_`- 1.998929175 0.0010708250 192 1.999732276 0.0002677243 1.999933068 0,0000669322' 1.999983267 0,0000167331
=pH erof Sides `Length of Side Perimeter /Diameter
6 ' 1.0000000000 3.000000000 12 0.5176380902 3.105828541 24 0-2610523855 3.132628613 48 0.1308062585 3.139350203 '96 0-0654381656 3.141031951 192 0.0327324633 3.141452472 384 0.016362792 3.141557608 768 0.0081812081 3v141583892 1536 0.0040906126 3.141590463 3072 0.0020433074 3.141592106 6144 0.0010226538 3.141592517 12288 0.0005113269 3 141592619 24576 0.0002556635 3.141592645 49152 0.0001278317 3.141592651
98304' 0`.0000639159 3.141592653 I - 196608 0.0000319579 3.141592653 393216 0.0000159790 3.141592654 786432 0.0000079895 141592654 z,-1572864 0.0000039947 3.141592654 ry=3145728 0.00000199 4 3.141592654 6291456 0.0000009987 3.141592654 APPENDIX
PROGRAMS FOR. GENERATING
and 59
PRGM oer-2nd CP 00f Clear Prgm 01 STO 01 if FIX 9 02 01 02STO 1 03 -R/S 03R/S 04 , STO 04STO 2 1 05 02 05 R/5 -06R/S 06g 07 -x2 - 07RCL 2 _ 08 RCL 08 CRS 09 02 09 ENTER '10 10 4 11 11 + 12 4 2 f 1-7t 13 ENTER 14 CRS 15 2 16 17STO 2 8 2 S 19 STO x 1 °02" 20 RCL 1 2 21R/S 222nd 22RCL 2 =23Prd 23f irrf A'24 01 24'R/S RCL 25 ENTER 26RCL 1 =R/S 27 x RCL = 28ENTER 29, 02 29 2 .30-= 30 R/S 32. GTO 07 ROR- g RTN ,Appendix (confirmed
T1 58 and 59
PR GM
00 f Clear Prgm = 01 9 02STO 1 03 R/S 04 STO 2 05 R/S 06 g Xi 07 RCL 2 08 CRS 09 ENTER. 10 4 11 12 El-Z. 13ENTER ' i4 2 15 16' 16 g 17= 17 ENTER `18 RCL 2 19 19 20 RCL 20STO 2 21 02 21 2 -22_, 22 STOxl 23 STO 23RCL 1 24 02 24 R/S -5 / 2 25- RCL 2 26 2nd Prd '26 brr -27 01 2. R/s 2S- RCL 28- ENTER- 29:r 29 RCL 1 30 R/S 30, x 31 RCL 31ENTER 2 't _ 02 32 2 33 R/S 34 R/S 35 GTO 07
RUN-
'0UTPU DESIGNING ALGEBRA EXPERIMENTS FOR THE' PRDGRA2-131ABLE CALCULATOR
Stephen L. Snover mathematics Department' University of,Hartford West Hartford, Connecticut 06117
As the programmable calculator (and computer) are so adept-at --programming routine calculations accurately and quickly, teachers Should be taking advantage of their worthwhile features in the high #010Ot classroom. Not only do these calculators extend the ability to solve mathematical problems, they also bring studentsmore quickly to he'frontierof discoverable mathemntica. This article describes an 741gebre experiment which will demonstrate the powerof the program mable calculator as a function-evaluating machine.
A standard-Algebra:II ahsignment is to plot the graph of ' ) + 47a soniewhat time7cdnsumioli process. With a rogrammable- calculator though, the formula (x73)2 +.4'. is easily emorized" and its values are quickly and accuraiely calculated,_
OmrebY: allowing students to .°-efficiently plot the graph of y'= ---,--,,i- '------(X73)2- + 4.,., SpecificallY,teachers Might have students.group in -_.--1" ..---, r r-s-.1.-C6 do this graphing One student might input--the- alue_a ---,r.t1 0.1 ;-2.. -.-1P'. into -..'the calculator and read .of f the corresponding -_, ___ ltputValues of y --- 7-5,0,3:4,0,..., -45, while a= second taidenti--jfm-Err
lo,ta-- the- indicaled points (0,75), (1 0) '(2,3)' etc. on graph ------__= ,--,--- gpit-ifFlith the increased speed'and 'AceOracy of such graphing,- -- ,.-s-tudents will be able to advince more easily 'beyond the simple 7-7-.74,,--- giihiniCef function and grasp important generalize' iii5hs 1--, &-resc_of this article illustrates going beyond simple graph'7-,,
rig: by showing how students can develop an understanding, of the _ ffeets-of the-parameters a, p, and q on the graph of y a- (x p) e, 'Elie programmable calculator can memorize a- (x p)2 *cijust as -71Ylas (x- - 3)2. + 4, students can choose their own values -fpr p-ad q, plot the corresponding graph, and discover' for themselvesfl e effects of their choices of parameters. The appendix atthe-'end' hiarticle- shows in detail, how to program the Texas Instruments
Hewlettjackard_ HP 33E to evaluate a-(x p)2 q.
7_27eachers, can direct= students' experimentation a, bit by asking 2.0 find a, p, and q so that the graph (i) goes through the 10EiTi_(0,-p), and (ii) -goes through both the origin andsome other °
___ - , 0 _en-,41-2 .0r-re-Fitly teaches_ computer 0 clened:_ 4d.-maehema flog-7 ...... ,- -0 d "d uate79Cydents- , _lie- haS' -a =ke-enA,nt resr-J-11-n=math" -±,---,- ---,RI13.risiaialid4`s-7:8a*ew-iii*-- ii ','C'elaraes -.-. -iiii--mitiff-fkuiaatEa5170-eigTrwaLwar_ '-rn-g1=,_, rA__,_, ompttwe,=-.== .46-z4a-lia aver---i:SittLktap#:!!bnp_ ath_- -=7=,-- --n- t : T gaidla egad t,_CSTI-;..-----
_ -,,,----;-:----ri- --:-= ------,--- 28
-= This:parameter-choosing arid graphing can be built toT an interesting game called "Catapult."Imagine a catapult located at theoriitand an enemy located at the point (10,5) ,- _ Theobjakt"-of":_, the game to launch a stene from the catapult at (0,0) ariChheiitY= it the enemy at (1G,-5). Assume that the stone will follow the rphofy: y a.(x +.q.
TO play the game, sturThnes will need to guess a, p, ?.d graptil the function y a- x P)2 q for x 0, 1,-2, ..., 10 and thereby,, "see" the "flight" of the stone. Does the stone get launched properly? Does it hit the enemy? See Figure 1-
, Stone launched from catapult ,-_--,=---, and following y=a(x+p)2÷q
Enemy_loc_c
, = - , --,-- _ _ _-,, -,-:__-_-.1-_7- - _--- _- _- _ _ -. _,..------____
.,-. _ -3-i'''''; -,E,±".' ii
Catapult -at- (0, 0)
Guess a,, p, and q so_ that the stone --..--
(i) is launched from the catapult (-ii) follows the trajectory y _ hits the enemylocated at ,(10;----15)-i7;_:_
_ - , The., Game _of Catapult:
rg7-41.1- n UM q a am t n sud it n Ei° n- Elmssitdnew ea h s h- - s =ts aua h n edo a. 6 a au c __ a hgame.
egrl ain h a th Ott . nth asaanyb rus_ n a u emen ke th ra gav" o1r0.44ke th'-at h- g-roCint.ns edtby14 mu?' helghCa5-61,-1,q-417st p _ t stan- w nay each he enegy. m the e emlip-fote Tvg,--AtiVma mum Er it o aunched.botrpuutheraglaireek,;param: -cis that a,A Eha 4.12e- I
meta pgr 4apto 4'canbe - ' -ea e -amen e,,;thein unc 147:ia- nowroens quad.,bediie di: ph nd-ong, _to u _andleg4-'" 01? a h -2,, to 'n; ants w h ca dv n a u tail -a one
.4`tr,' on t _*e Tasns a ,ac_ -rd HP 33Erogrammab, Ca at n ,
A
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or ut _ng Ht(ci 2,A.d Rgt, p a Iry a' STO, lace .pfAn iplay;'tSTO p-aee, :4E14 ,dtriaiialLua g o t_verue: rPadte:.x4n:d e,yrin
pE,pgralx' g x2' 0-1L- aqpapse npy4iling8yalties.of a-, p, and e:v-,,,_f,.,_ __ _, tvalue of a in 'd-iii77,-"-§T .1W1.--;.-' :id Oa eix,Xiilie1-pf,,p-,11 ;display,,4 dy ,, SIn2' -,,-)41.71-'27C ad'vaiiik ,-',00'cliWgiiY't, -T04 --'' sVenSP or eachz-va ne in' the 0 NO 1 Ttl S Mary -.ohneen vtlertiog Soa ,Nerth bah-piette0;360-_,.--_
OrLom t_ C tedi- _nt -_ n =-7 cul_a c o _ ormer m -. - di y nvolved. , uous,Ipase : hew, trian !one a rb,re -o lee On- ira'tiaASA:64-461-erinthe -Wife _71),Tth e A VAeat, adADelii'3acbiattaVdercaiiiallimaCare 4h,a h- no;;damn xw 4e,VP:iteed t tails tie?? mesh qn a dp d keit.rai.i6n''IM11;idirig-!,'. ea_ no - Ii4IY,g44 fteat'Oifk.a'65--t050MpOVA.'bOt 11:061 Dr the e olution no -t.,!easent4ial 04 W , this, is a ,nicepr9blerni- - to )am you,aye t_Ari elegant solution reqiiir-es'.'edatallla tor o
p hat - triangle -ABC weknow- angle,A, - s si de "co', (it, itexists) .6 By the. Le.'w df,,ICosines- -ww 2b c ,dos-,
e-u on *quadra and scan be arranged 0 OW 11 )`. -t,-
, t 2 2b coe A) c t (1) a2 cp2
qua
4 .K. tea -- w range _ courses dva p a.sement. c13 74. a -oedu na hs en a 0 ada emi ear on s b a Atomeppi du_ a tot mp u_ rpos-
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3,,, . ,'..b 7 -it.010'- 9'.... nit r-,,--butAi- 'not-rany.moreefficie o>ipyrpose u a on ,on 11'61'3 , a,,c-arculetbr'wrt h two_ 'ilea!, . finethe s v aluesiko ar_et ValfUsolutions -L'Negatiye, taro, and n l'o'utionn:muir e rejected:_'Negative,;d4 ltieS';`indicat _ con ' 116.11_SUMAtcent,lbf:EITO '),tig 71111Mao n_ ang e.-:-Nnrsaillituea-r-indicate-h be-aide ogosice -rt -3 too Short ,C_ ?iirco-tirrei,line,.'coniailingtre--othe aid _ ,. -4.1...... ,-,.,.. -, ..,- ..% -h ng e. Agurn sh i -some , Of ,thigpb'S's cases. f41 av found-the ehi dola:lbf,\*EltaCeaVnire9one may find th es bytuffiiirl'hli,traiid f:-_,ZitaineV:riNttha o us _ neg. aOr ilkts-kbe;iir.dantfC,,astowho aeu_orr obtus
. p a__ cal ap,,-Zicat.d.offi one.usualg;y31anek eithe unknown e ence berweent..(WeT.tWerpeSSibl'e_valueso the nnoun cfd,{ t aer,oaso, the solutinn"qa_y tol: g#estairmilw,-,
va0556---S7-; those o p e_ hersS a gem prat"which reques on eor -_-,and a bdof ri 9., Ar-r b b 1?,k,
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,afp. 6'i -7 647
'171
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ga Values ,", ne_selibel valVues -.7 - Flgure'
*tYt3crt- - ,::,-.7,_,r4 ______-- . - - t,, - ....7.4.14,if erea a good AmbignOym,,C.ase-problem: artier'Brown, a h ad man e_ th- goat to the,'-co-k-ner,tpf,,,,;,the)harn,,viith1.-.1- ,-_ 'F., . , ,,,, ,.-,1 . i'15_,f0o1' - -..11r,f3:.iri! .chain. he ot -e = carteOttt6S4'740400e.,,Iepg 12art-ii`_,T To indge eaten_ o 00ea_ an angit 0,1!_wrtht th'eMilitt t. muc_ of the h_g_ t_- a go emolVili4?r -- t, ,,,---- r on (see:gore c '1767ndT.n* 15 iPeltig.rp 1 11' -1204 154-
2ndscat 31
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- ons,are:_nocites t.a 'negative- number ing' zeros in memories 4,, 5, a . , , ppear; ;and 'atpred4i.nimatiories , o solvenother -triangle, tp CS _ N.
. thael Departmen_ as_a ay6tte.ndian47907 '
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-_ mathema.,ca':..4 ''''' _ ' d ,--_ ' ' ::'''''4 ;_,_ .',._.:-.1.5 -:0 ...,40,trt.---j---''' a .,,ate, willflb'rief-tly:fiescribea jai i -_aac adapted liOrartheV-Telcas, triderthien _ p pxfa ...,7:1A -&..'4,vilm ye, .,,:o- __II , 1 t.",,,zZ Wit. n.._ n , .Tr-nta.-to -a-theop T, n shbiY "Illattleatar ,Galactr C",',thel,gemelusa_ ,_ -,aipragrassnab, _ 1 _, _e,eVdeep----,-,, pate--.- bat tle .betr.irk re l',?' Wipe, " a to - _,, P ,,-,_ - e .,_ ,,-_-0. ,,,,d,..4: . _.*.,- ,t.:'9,i,iq. ,,Gtistt...111i*-_,Thepyiper,carr es tour
a thite,'muairtdiSp-ans eYwft hMth 6, Gkeis t a hipIttreth. a tvis__
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a4des_,nc_ ed.,shove entedras .ng n ,n a pp led mathemati.mat hem . , e game. -can us _d st.-.,0-,4--v-,-.Aco,-.....T..- _ ants to studyk.apecifticl.topicsauchstudy ,- as_- pa-- ,a-and 01e4Z-31-,.- lr -. .%Z.--` .r, o d m t_ geomet-ty ,ot ',later-sec& g cir as os , nd, cgipute_v,pr,,,ogramming-,..a as the -apo n man_ un " ,umeroOpiptaofs4fixi or th-e t,e4chei _- . P t__-1 udents w' vestigat they ay an- am am . _a obje-- AO_ the parp.Cular tre tm n_ -. e we _63, bav_ si:use4raathamab-ics_ tn, sov -ii-,2--ml,,,L, r,-,.E.....F.0,',-: a_ MS __a t-oatroduc- 4:,) them (grog rammi raT:'eompu ting, ma chine . 7-, ----Nnk.-... - .--.. rk L n 4:-.4 t---0r,te-' t ifer '.. agraa 13,, ca cuto -I? ii !CT theu ese _ ut tithe gamed th , yes ort an amusement- d4,f_not1,4-- be eduaat Tna.
ns ;13qty amathema ,-,addeacors 'and teachs ., _ta , ,,,3,k, -...,m, 'educes_ _anal v ...,erliitQesigruractivi.e= t e , av-- ..,_ , m- _a, wa s tnatigw1.11 motatuden...te ,r itaznino -4- -u teasoen a a n on e- s_ h n nan_ m um aan Ang aons 42- t_ un a v e h m u a r have ons un a. t nvo ve v op 'P'onitilve ud_ Ards no o u
- AI_ e Onleulato '3,6r, ocotiter's ?' -10h -1-5 ;better for -use in mathemst-i s instruction, grog Gana 0 or midracormputers db_v-itrusly..nt--th44anSwer TO; thcfneso a ove"'ambvSjective--,a -;,_-the ,teaeher'.-dIllav_ingboth.-,, ,, _ ',91.41..._ a a- ' b ,:ti must chbase- ,, consider [the foe p -amine, ulators,areirrsts,,tery,Aimes,,Icheaper than camp er:n. YEitica-.can _Or tkn_,77,171;c11.171a or h " 094:4, iTilS4CIZMI:=CincOniPlite?:. Far /,'n.,,, sMa11.Z5E,,,s!nr;, n S wou lebffer eacht-s`tudentI significantly noprerid Fence 4hanfe'=9 ngleratcroelomputer.So, in these,timeseo ' n budgets seems,t at, asaru'instructional.aid for 'tenc - at -.'''^-131-'`calculatorsa _e''"'"Iir- ":---oer ia 'ST"'v ae -a1terns e MpUt
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-,tt ..,4,--:,-; ,t'-, t-lein ,`"'l -4 __ -=....a., ._-,,,,- ..._. os_ a -e. yi,haVe;_baen-"atteeking,y0 t4 rp , ur iiai7iio, i 4 inraris t he rfuarine" way .4 k Gh6s t- ocir , afO-a,Vkp rV ini'ing' thtcaamd'' dit- i a f- eL a y -et ativer position. ;LnoatshipI shoo tw-so, or w r g iteyiNat the Vi.per2whielp eventually ,-,render." the Vipe vex. -eshiple,therigi '4 iiiii49. leiil thel'IlliPerLi.itast in
o_ n d _eris _.!Viper'has'againet;,theis e siciz.Zialle-dteGh-o-stsh_-s,._. bli, te o,:.ouzltttattc;misal,iless:- Armissisleiva-fe Shrei a ehh: Ghos ttih 'oUght';tdIbirikeatea. :if tha h _ o b__nit_ri-tit, ;fiat, :gliontailip:.d.`ti liitnina ed. u -p- --,41-_..%21,_,. ....c, as, the demise "rheAyipe:4 truiritel. ,.: t- hue' .nese. Avi ty-ocialeu atom- ty on0 __. Was_-e 14 son a _ SIO 0 _nd be the ania 0 And0-a y u a a __. To hfa®_ an Vac. niaazler, -chopsa , -ne- hai to locate t' G poin__ o__ yo_oca e... a_ ne_ on c_a ato-r and ptesS_,R S. en. an asES.- h- Mt ifftShV- th -um --- ....? a ou diased by ash on the eil.feu-__ to .-'i onlislae,1 distieha, Atm miss_edmissed by _.9 Youship ts , 1:1A- '''C'''iPa;9:MOilcgiV-Ae this io_ a on o next -epee-"the abovp- ie . have on y_ ou -Ahots. IA ...th hunt spa_ y u_ eve Ghoets..,-._p,, th.i!fantintSritisplay a_ 9 ' n ou havA' Stttie-leaite ,. S _s
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Progra TL-58 Calculator
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CATION CODE ICY COMMENTS,.
-000 36 2nd, Pgin
:001 15 , 15 : 002 71 SBH ob3 .88. 2nd;. O. MN 004 65- x . Enters locationorGhostship 005' 01. '- I (d, a) using,random number 006 -00_ , 0 - generator from library.- 007 -06- 0 5-- 008 95 ,.. ,--...-009 42 ' STOST 010 s- 00 00 46'11 36 2nd, Pgm 12,- 15 15 ... 013 = 71 ..c' SBR. 7,1_-_01.4.1_- .88 2nd, --D.M i - -5- .65. x 0 -=---' 09- =-9 IP 00 =1 0 95 = i------'----=":019,----?_ =42 STO =gin 01 01 ,_ 00= 0= 1-r2022.7 9i R/S ... ._-;-.--_=-I-f02-3-.- _ .42 STO 2woi4---_,, ,,_0177------__Qz1 Acceptsguess- 91 -- R/S ------___ distance first, then _42 STO 03- 03
04 4 j _ _32 c x qL t ff _Puts "4" into the- t- regikt,0 01 1 Register 4fa,used as -,a- 44 ' SUM "counter." 04- 04
43 , RCL,f ....,ctlii-ir, 03 03/. -'-`035--,- 1 .25- .' m036 -,-- 43--- RCL 037: 1 '- ''' 01
6--8 ._95 . 2ne, -_ 1 x Galen -'_es',diit-eric-e- , RCL be ftigen.---gueei an-ar stair =,00 using...thseaiaioQ .%' a 1'''.-- e 7.---:S fare-s d --a ce
-registar,---...... --.-_,-- 74 4
LOCATION CO E KEY COMMENTS
65 047 02 2 ,048 94 +1- -1049 85 050 43 Rat 051 00 00 ;_,052 33 x2 051:: 85 054 '43 R. 055 02 056 13 /- ±=.- 057 95 - 050 r 059 42 ISTO 060_ 05. '05 061 77 2n 11 Test; If DZ4, go re label_ 063 -85 A.If n &t, go to,,"+ +." 064 85 Since the double plus is _{165 z_ 91 --R/S illegal, calculator display- 066, 76 -2nd; Lhl -flashes. 11 A -06V --_RCL 9 = _04' 04.: Recall number ° times s f070 77 2nd ,xe- t through program. If-it greater than. nr_ equal to 4 -- .1..R-6;;071,.. 12 B 4)72-- REL. go to end at label B.- Otberm, 05. 05 wise, display neW 91 guess, and Start--over at 61 GTO location 023. 00 0 23 23 76 2nd, Lbl - 12 -B Label B, end. -Dividing by 00 0 zero is illegal and results- 35 1/x in flashing 9s. 91 RiS
qz-
USING TH E C SIMULATION--A PROBABrLITY EXPERIMENT Dave Haggerty Monroe High School Monroe, Oregon 47456_
The programmable calculator can be used in many different ways in the classroom.One way is for simulation.The calculator can be prograrmned to runsimulated experimentsthat wcruld be cumbersome or timeconsurring.For example, It-can be programmed to roll 1, 2, 3,----- or moredice, total them, count the number of times a particular outcome fs roiled, and stop after a specified number of rolls. We used the calculator to simulate the following probability qxperinterit:We considered a setup of three, stacks of blocks; four f-'-blocks in the first stack, six in the second;andeight in the third.---- :We decided to haveone of the stacks chosen at random and from that,: one block removed.This process was to be repeated until-one, the stacks was depleted, thus ending. the experiment.
We pragrammedthe HP 33E calculator with the programpreseritiAL, at- -the end of this article and found that each experiment took ab-out= 15=-Seconds.--By pressing RCL 1, RCL 2arid RCL 3, we could see and== ecard how many 61ockewere..taken from each stack.Also by presar ---- ,Wet obtained the total number of blocks taken from all the =stacks_. Dace the data were recorded, the experiment could be oil ,agairi by pressing R/S . ° ., en We did±niis in the cla oomeach of the 10 student the experiment 20 times', recorded their data,andanswered the allowingquesticins.-(These are but some of the many related liestions that could be asked.) 1.What was the mean (avere blocks taken on each run?
2. What was themean number, ofblockstaken from stack #1? ...from stack 112?...stack#3? =How many_ times was ick #1depleted be e the other- stacks?...stack #2 -ck #3?
--- Dave-Haggerty is the mathematicsacuity at Monroe High, .a.. -s EhVollirCe="farinifig.:and lumberarea 6f Oregon.- '-He -tea&Hig i -general-- Midi ttiiough pre-calculiii,iifiCusis-progiinie-- , ___ _ _._ 1-iLalgebreaad- caMputer- science courses.---. con .-- ,,==,.. ..- --o Minables-for-sectinda-teachers at eWide" = ence. -7
. . -A774.7,7=7 4. Using your data, what is your experimental probabilitythat a total of 10 blocks will be taken before one of the stacks depleted?
What is your experlmen probability that stack 1Je depleted
Once,the students completed this exercise all of the data were and:the class answered similar ;questions for the entire200 runs. The students then compared their results against the totals. sing the programmable calculator in this way, the_claesTuas-Able. TO compile the Composite resultSpf-200-eSPeriMentsin one:class: periode task that would not have been poseible to do. without Of:pre-graMmable-ealeulOtor:Or,similarmachine. Furthermore, the,., .1,-200 experiments were sufficient for us to .obtainexperimental_ rOfiabilities that 660A:be trusted for predictionpurposes.
With rhe_prn- rote for the .HP -33E, a whole range of ar probabilitY experiments could be-simulated. By inserting il7differentnnmbers_in eteps-26, 31, and-Wof the' ccompanying V-Mgrani (see Appendix)``; the numberof blocks in each of the three --gfackSiran be changtd. 'You can experiment ifyou like and see what effect thesal-changes have on the outcome.
he-kinds-Of simul&pion, experiments p -g ammable calculators at.bn_pregrdramed to_performnre endless, Moreover, because `of -Teir-'4Fficiency,:.siniulations:privionsly, impossibleto do in a nserpomnectinbecinsesetting of time` considerations become . very 02able-: PRQ4 Mode-- KEYSTROk DISPLAY- KEYSTROKE DISPLAY
1 28- STO + o-2-?3 51 4 STO + 2 29 -23.51 -2- 1101 03- 24 0' RCL 2 .24.2- 7 - 04-'-24' 7 6 6' 61 . GTO 17 32- 13 37 06- 15 33 1 33-- 1 07-- -230 STO + 1 34-23 51 08 1 RCL. 1 35- 24 1 '09& 4 36-
10 _ 61 f x 37- 14`61 11- 15 32 GTO 01 "- 38- 13 01- 12-j 15.71 RCL 4 39- 244', .13d- 13 03 R/S 40- 74 14- 0 41r_ 0 15-- 3 , STO 1 42--23 1 167 61- STO 2 43- 23 17- 15 32, STO 3 44- 23- 18 15 71 STO 4 45- 23 4,_ 0 -197 13 33 GTO 01 . 1-3:01' '20- 1 21-, _41 22- 15 71 _0_;28 -23- 13 28 STO 4 3 =24-23 51 3 257: 24- 3
7 27- 13 37
-Mode =Enter a 6-digit decimarFfraction, last digit being odd but net snare it in the 0 register.STO 0 er 114,7_ and store it in the 7 register.147 STO 7 turn tlie,pxogram to the beginning.g rtn - Data-areneir-reatr to be generated. R/S r` ran, iush-the R/S button and new data fur 3. he-d ItEP DENT STUDY WITH 'A PROGRAALAVLE CALCULATOR Lee Millet
Decrement of Mathematics . University. of Alabamh. irmingh Birmingham, Alabama 35294
E lec.rronic cgmpuring d Vlces have not yet made significant mathematics' curriculum, ads- into. the highcnool. aloughai there -a beginningtobe widespread sgreemehr thdt they should.It 'is :clear that computers ,dre-revolutionizing scientific practite (along ifh -just about everything else); thus, if the schools,ace to -_-ListicallyPrepare,studentS for the technological w&rid of the s. they must _ come to gr,ips with_electronic .comPutot ion.This per prisenta-:a sample-- forMa=ti for using prograismable calcujators in dependent,-study. . following, material- is an excerpt., from tfathermatical =c ens/ for thl Programmablefdaltulater (Re, and Hoffman,, 9 -a collection of programming prial3lerm designed to teach th andard techniques of programming' The/ probOms preainted nvoive theuse.of nested loops, a technique iptroduced to studedts reviouschapter of the book..-!- The student is to read-the the ;problems and attempt/co solve them before lo?king the =-soletitins.The solutions-theinselves are quite complete, but not_inailde:program listings The/ idea is to encourage dent to de as much as possible befOre looking at the sol student who :Works. through. this:material- t411, learn all cite 'Weir about nested 'loops.'Exercikee are, after all, the heart o efaearning-process in mathematics!The goal of the book /end. the bieMs--giVen hire is to=presenC-Vrogratruiling exercises withina ficiently interesting- setting that"students will be stimulated , owant Co solve themn their own is worth noting that this particular excerpt cdritains a good eilof matheniaticalistory. --Aside from IA fntrinsiic'interest, material,pcpmo the idea that mathematics is a creation of _ nibeings'-and is even now developing and changing/if Ambitious "7- udifiti-iney' one d hope to make theiown mark on /that history.
7- -scripted'Ns.: appearing in thelowchartsfgthi solutions re-blame r pre-ent memories. e arrow notation means roughly<<= x mple, m1 m0+ 1 means add l/ to the contents o u t in mi (the con e of m0 ,remains unchariged)=.--
u Pyrk-raortIIMl.es
call the -famous Pythagorean theorem: if a, b,-And carethe legs and hypotenuse of a right triangle -(see Figufe 1), then a2 b2 e2. If a, b, and d are (positive) whole numbers satisfying this equation 'then a, 6, c is called a Pythagorean triple. The simples example is the triple' 3, 4, 5 You are probably already familiar with-the,,fact that 32+42 5-. Pythagorean triples have fascinated mathematicians for millenia (see the Notes).' There are lotsof them -infinitely -many _ to be precise-but.'they are rather thinly scattered. The next interesting one after 3 4, 5 is 5, 12,'-13 (52 +',122 = 25 +
144 169 I 132)41-
1- --- -rablem ;Write a -program -whrch searches out and findsall in Pyrhagor_ean'triples.. The program shouLd generate triples some - _ systematic way,- screening forand-outputtingonly the_Pythagorean .. , ones: ,--- = and run your program, you will, have if-yoU have solved P-roblere i the i 11-._,-noticed how slow itis,and that the larger the numbersin -o. triples Aet-i the longer it -takes the calculator to find them. You . in,imagine, then,-how;.hard it would be for humans to findPy,thagotea- -lpies;:by- themethod of reearchini,! So a long time \ago certain etiii-lookingbegan -looking for rormulas which would generate.- egorean triples automatically. liere is a schemeattributed-toL -dgora.all-Limself9 .1 . thagotas nottdedthatthe -sum of all th-ei-odd numbets s added upto a perfect square:I-1 +-5 -= 16 etc.,:-NU-Wiaupp.oe4tt istself, a perfect ,s-qulareladlik-the -who] e sumala-a-p-erlfeet=sqUa'raz: thqr:tie-ileleE i-1-1aVei 9 the sum up to but not Including the last term, and the itself), thereby. creeting a Pythagorean --25 2 + 3 + 5 +-7 +T'(1 + 3 + 5 + 7) ±(9) =16+9=4- + - next. sum for which this holds true works 4_ 5 + 21 + 25: = 169 23 +25- = (1 + 3 + (25) 144 + 25 elding the Pythagorean t 4the It becomea clear that any odd'perfect'squarem.,its,at "trtd- m of-consecitLVa.'odd'aombsro;- + + ar- - 2) + 1-- sPate you the :de ;A1101' shOw that + - 5 +_ (M2 - ds Op tol(.i--(ra2 L.1))2 and that the tihole sub 1 +3+5 + - adds to ( -1=-1)2',;'-priidue trig the Pythagorean_ triple --1),4(02+ 1)"1.11itia':-.We hhvelPythagoras!--fermUla: m in y -odd number,. thou mr 1(m2 : 1) ..1.2(m."--1- 1) is triple. in a.prableMweleave you to':.Work-on'yaur own'own (no .(no solution in d)
Write.:a. program which genera Pythagorean triples, -Ing--_pythegOras,formula; t kagoras'cheme. does not 'generate all Pythagorean 'triples -haVe solvd'-Problem 2, you/have probably nottbdd, chef -the am- producesonly triplee a, 1,, where c b + , (1232 t 1 4-1) Thus will npt Igenerate the ,triple 8,15, 17 irt,yen car rectilyverify tob6 Pythagorean.What .1* needed Jai a 74#1,11a- taking o numbers as inpdt instead of oneThis will
enough exibility to produce all Pythagorean eiiples. II ere it is:-Let z and w be/anytwo positive whole numberswilth Then a 2_ w=2, -b = 2zwi and c = z2 + w2-is al Pythagorean _Oroof: 2 + b2 . (z2-w2)2 + (2,02= 21-1.- 2z2w2 + w4+4z_ zwFw4= (z2 + w2)2 = c2).For example, let z = 7 and -1- a.-=49 - 16 = 33, b= 2.7-4 = 56, and c = 49 +_16 =165. 'eck--on your own that 3 56, 65 is a Pythagorean triple. _S-Cheme generates 1-but a few.uninterestinvaiples'.-- r-litstance, gene ate the;-triple-9-, 5,,',1blown -up" by a factor-of 3.)4 (i.e), ones which_are not' datiples
11ti i 1.
-1 .Problem 3: Write a prOgram for genera ng all Pythagorean triples, .i.tsitt& thp :preceding formulas.
Solutions
/ 2 Notice that if a- b = then c b2. Thus the thagorean triple a, b, c .can be rewritten a, b,a2 The idea
I p he solution i-s to generate all triples a, -b, a2 b2, cheating a ime. to see if 142 b2 is a whole niithber. If it is then a, 112.-Is a Pythagorean triple, and the program stops to output 011 only generate thane triples for which a 4A b, since . edundant ( the triple 4, 3, 5 -is no different from_
'5
So- first- we need a-part of the program-whichgenerates all ,= ::passible pairs of whole. numbers A, b with a :E This we accomplis with nested-loops.' Here is-the flOwChart:
aF
No all -the rest at die program has to do is compute it in a memory (incasett turns out to be a whole numbs to see if it is a whoLe'number(fractional part output the triple`a,:._h, c. We will leave it Lo---iNu to arrange the output..if .42 177,2 is not a whole 6umier, it is time to generate a new pair-a b" - which, takes____Fau--17:rto the part of theprogram we haVe already desc:ab-d------. Hefe is the flowchart for the rest of the
--f--- ,roblem_3: 'In this program we want to generate all pairs_ z, w witif;=,--f =an ,to generate _from each pair z, w the b,- e-,,-usin before the- statement of the problem. Thg:part_o th-eiformulas given , , _-,=---- - togram generating the pairs z, w_will be just like,:thefs-e,herrie.,__ --e-fiertitingllttle pairs a, -b in the previous solution,ekapptfftfie" .,- ---_-7,.-Imr?gets_replaced by the testmo=-_mi-__?- _The-flowe a 6f-the:solutionlooks_ like this:_
lc 4 40. I I.
I
`.."1"" 71' I I 4- , , 1,1;,, I,."
j1"---), 1 ,.. r'l '''''',';',1, 1,,;:,1',I,- .""[""' l' ''"Iv 1/2
,
lIli ill 1,1.1k, ,
I TIEN I rii , As noted earlier, there is much histo associated with this anal. Students might find the following of interest.Pythagoras_ got his llama attached to the Pythagorean theorem and Pythagorean triplesmore or less by historicalaccident. Until the 20th century,, it had been thought that there was verylittlemathematics -worth mentioning before the time of Pythagoras,' However we now know that .the Egy-ptians and Babylonians were producing respectable' ma_thematics asfar` back as 2Q00 B.C. (1400 years bafore-Pythagords1)------Indeed; the more we learnabout their work(especially thatof the-74;-- - the more impressive itbecomes. Clay tablets dating , -int about-1800B.C.-,showthatBabylonians of that period knew how- generate Pythagorean triples.
As e' notedearLier,-__the scheme-for=generating Pythagoreans,- ' Ett_-. i Pi o e- n3-produces all the ,,, .e n in "p tiM i " triples ries ,QtefjOhcCh all others_canbe ib tained-by- Mni tipLiCatirn. : Fns_'------...... pte -3i-12r13A is.primitive.-_ From it wecaw:generate-0-e- -,10_2C.- 26-(milt iplY by 2); 1 36, 39-20;'-48-,52; -etc, afnottree -a- m -_-Afry: pit y _agbreall .aearly,:ifwe-,ate.:,: te:tested:in,generating=111-_BY,t_ago p,e, tb73eferate - a h ---i ii i '. 1ial"-giVen_iffProb1eri'3-rdelsnOt"46 thia7:InaddieiCnto-Fte----,-,F ------iiira-trvgtripleait "gen'atafer-s- ovine'(thbaghnet=A.11)----- rinn-p-ri=mi.tilVe_.-.----_ - . _.,-.:,-,,..- a . The, s-rib-riprinfitive triples can be ayoided_'-bicimp o sing:, ---L--- ietions-oni-the-snumbers- the;, .numbers,- ,_z and- w: '-( i) , theil:alideld-Ila_Vi- cOnattin: divisdrother-then 1:0- :e :-.,they should- be-relativelY-z-ljrtSie)..- ----7?-ii_they ;libtild- not brith be add- (they cannot 'both_beeverrJ-1------_het_,,, since- then 2 .Would_ be a__ common rdivisOr) .It -is-trickyl e -a program _embodying these restrictions; we leave it- to-,_ at cs among our readers, . -,-,, .._ ---,_,, Pythagorean triples can be gerierali-zeA in various ways-._-/
.Ef,_, for example, "Pythagorean_quadruples;" ,three 2erfect.4s-u rjes.-_- -idding:_--up ,to a perfect square, such-As-67,-1102, 15'4,- and 19 --in, Alto .fiad-;rinadruples of_ perfect cubes (33 + 43 + 5 ,?,.;4= citiiittli3les_-_of.perfeetfouttfipowers (304 --i--1204__f 2724 ,1+-115L__ eiit-iipLeTerTo-f-' perfect fifth powers, etc. However,=there-farejip 'painsi=- 6.asibeefriddirig;lup-,_ta: a -perfect cube; i.e.-,t there -Are7no ''triP7fes° _ ,-,,, --_ tOsla_Vg)jw-holeijrumbe_ra a; -b,--and c such that- -'1,3- nertiiiifira'il-a-ia_WrOf-P f_l't his _fact'--Wns-,-,grv-nn ,by _Euler in= tb l8.th$ g-nts, -I9theiiitY Ganis',- perhapt- the- greenest-illliVaiaeiteiWP--- -_--_ _ 7,- -4--....' ti.r_e, t - , en t i rg 1 y !-correct,-proof-.-:--
-wl w hrrou_this tin ter _41y,Lthey,cii1L-a themeticszan_ cave meno Cu urn and _,ins_ uc_
n _ t. _ on: 175
sue p- Sul use _o p-0- amnia a, a ! aocepla.,,c.prat o_o t -au aa ' ne, -a a gn , an i__ c n a as uc-_ 0 at emat 4nd (c)in _mm tan sp and _natru on. -
thematida -arDisOpline. w - cs no, pang 767 a .ariety tit waya e-ems to be -more, than -a- sat _least -,in rela
eatises,Sawxe4955
umpo, eo this Vero. -e rieY. (a ttia.1 c cation and studof 1. pasibr a pa to a to be understood in a, very,w de sena "f Ma a 7g, ar,i4y ''`tha-I4t can z : -: 2. mate_ yearlater, level 9 -o ten t a_
0etter thaalooil w th_ourbaki gup e ans'uteri,taon amat tics :ah_ a tes o to- stu y (38 a to . 6p. 2*tal, os added' n Ginpb (f1977)leanten ed th t
-us_ a n: only to ca-eu a_ _u must_ea1 ga_ _ a _s
s aat erfra4ca ediat tow o has dnum _us salon oug aeseah, wos na. auant WO Aka W th ese dvane d g adun at den s e, "au d 'a
n nt tiatb me Itea _Conte
ns .g. ,toir 745 h v d_ o on_ ep t 41:mi n c ,nne w- _n_ t ems: te. 01._ thin, ma a S -9 gest-a gnel t 'us_ It -7d a ye,etyo wa mb e,C-n etrenwSehool Mathemati=cs -,0 ;W. :z _ `cli,4inClude--gictUred "fun n 4,-1.-C 4-ft It.--.711i,"- " _a nc Mtd- lthin ac 00m t e -a 1041 caw' Ptr a c dda -_- ins, . . 7'"B -tV-the --devepmept., C de-a '--Co-ficagnarriong,,ich401 studerits5haalbeencth'efau Van Wait' 've tor1.2 ( 6114 z -r F. ;
Mimpor tai] ee t, Mathematical Leatning:and,:in atea-419_77,) ident ions&f dif fitalic ; type .-Jeatn_ n n_ t8iiiy-4edifr4,-IniriOny g elle t a 1,,';.{,13!-Iti,',P,.4.1 fear? Wrilkin u-t. ,ir>TAv-i. arlrand 'tS h ettl eV° r &A e.. eirma*, arivimX;_can, -athematIcalrkl_eur _ng,:,andAnsirUctiork-linv41,yes- _he proes a...ch . --- is' %akin, t 6ck-' 197417hypothes- 4 . '.; e,:netily, ,, Y7's stn ted .-iliat,r.", puma butlist ,,, ",.._ e ,nw ,,th under 9-0,-,generatlytigAgcess 1, -i ,,-..-, , rig t e co uc (a), 0_-genizationa, _a st g andetely.,ingarzjn..I1,,tP .42t.rmat, a-n, aAdji a attng nk, ,inakina4Plon.to.ehe a '
K., !thor y'ialvearning',tia-it:nvoe a na discover_.-- -- ,earn,i-g--., ,.....1-1 i
-, , , :, -k'hyp_otheals nut.uiii.dReSt..,,,,.,,,,,,,,-.-0,,,,r, rridut_ kft,-, , , --, u es " --emgha-,--...... ,,au,,-, isnigtin;trdianvety.,",....., -Y a 5 -a neat. --payeb' _a era e n
MTT,,_67) ungrtgez__ _ ..
-0 gtaMme, oea40_414/47a- di c_., 1 e -ft exceagerimeans (c dlir_ &f rig n -_a e 4 maths abics w "th a_cusU in * -): ,d Al t thtatt rC -S ed, 4 *tarp o.mat ems-ca, _ n an n at'Thy sail"' -__ -.. o s ad_ sa: -_ t _. c r_9xt a n -emsoWng ott7e Re-se -ed -o_
° an, _ Wr. calf lo 1 a m in..b - o: Ott -L, P- °feast, -'Wee _a ecitu__:.gt grams' fo'r-va d a 'a-V- been. "Flo ---g with th'a
n -a -ha' y or p-cint,sut, _ Tte'd P w a our,,p.robrecontexEs
sp ethentAry. ,e graphic_S1.! reps included.)
yl odewhat different from and m !'popeis4Pclaqk thanAha' preceding contexts), .r , _ xne tJ.f,Ort,an_..eXample`o-r two to get started aparticular"..15t -- a the =suggested by -the , inimuctor ;qs tudents should yo tun `own' rairie's ''anci"-tdlirdTfy
u ;_ Eh 'the ulatti:r,4..chriaTithi0iP1416
4Intout' nf erre-inn-es sl_se s It;the; case'0_, :122_
*dostOdent-i4-0011614-141640, c -onn onewl 'h SIXEW '6:0 t 41Atd.ppi:-_, L. instari do
th . OWn. vok t' on -mov"&lititthel,:if -act10-n o. -- ,. 5,2*. ;at ..7 . ,,.,,. a " int __a ach (Se -our. andil:Shodd,,,,1793 "`';What 'kinkand nc an/shot.' .,4 '. zfuct®rs',g IT t vgajtJdents .
n,,s1 -e bp' 0 othefl.gmoTaTnathemat do -stu eata Ham IP 4 cisti-g*--,th'45604eifre-d..
ng. ":011WOW,..tedl! '_1.11pyi-e _a :,,,p :0 apu_.,m_ e 0 4 hthingso b y and ea doraba as
0 = _ b te eha am a a dist--bUt there Athat stud nts n u __m t m stu_ en tan_ en- g ._ -_a .e_ s niuh -
a th thappend d ea the t on (SP 'MU E, OUR E,
A s mkt e,onget_ _n
haw P a _er .to tc,--OfintQ e a va ues and the d ±-cr messages es o_- a a n ut Or output.,
__u have made, program-fmoa a for use with- ca c company ng ,!13f-ater c Ma-to -/ om nA '666-_-parlp,c6Tr7f a thus as 6 _ jp ese, f4ndL4cYcadvAnt ageause,tw use culati=Wialt:nonvointire-program memories or with magnati.c-'4,4`., A VJ iff 7
-,End,,,Notes
. ,,, -11' o erid;a,,,.._,, 're-00y of '',the POUHULE'f.rift4TR.1111._TL r th. Ed' 'q (v t_tif'821'41t ,p4itter)_cS1.culatar-,tp shji- an _, 44. P, yeaver, ,;.,,papartmenti; f,i Cur ric- iitilf4,P ugfi;..,Y,rit,e M And 2ns t-p.k.ct ion i.,Univei-s 1 t3,16 f '.. ki Saari iin.:aka-iv-4272'S - Mactieorr,''wr '3705'.'' "'''I''''''''-'"---"-=-"'211'1'±-#.- --- C iSZS tr e' t ----;,:4----- +"+. , same applieay, gKRULE,,whichlinawt ,-,--y- ..,1,--- ,,,,s.1_, :hdyein.alrdrer t - Qand',prfht-ax):_lli=97;;54Higq7, H -15 HP e on for 1 il4I'd --,.....Lci.,-...--.,.---_,..--.-4- , H .. C, 1HP.m,lpg, nstrumentai'SR7.52z(wi,th p_raitt -. .,,_ 9 (withrunt 6fAMULTIRifila so has bee --par d _n VAS' _ y uter.
e '-August P KEY ,NO,TES° No. a red use afithe ord !Vane-aon w, ,th u- ean`34gert th- uno coull have been bz A _ew s ad be ttp_a nd TRfRuttE frog am-
varalian a_ *AO prep S`ICr myP.E m.1.0 mU eir -0, d o,tas .e she _ts a1-1,:i-o,,calcu?.at,,itiirVjproCidllgs,,useof da regisi tn.,a, and Colisiderat_a-nt. , ase portppu _ a
tr AIS t_TU , tyou, 4 b
co-nclu a Oa v 9-
1-
0
,4
5_ -Trials 10-18,lhoweve_ _u ''seMe modi,ficsatibffi inTzt
p eel.,43pss/bLy
*Frmaybe
La Tistanc
a' reject AP -o_g re4 t ecture, or'tsysubsequem, d__ ,ca- , .
Amul,E.pr_s n
An . . _rin _ _ r Craw- PY-
s '01181d:et,- th- 'Uric-061P nth e'
S as fvMath_ domeo_me ch-e-r7(Billlablii2N321V0)'. o s tit Vidies1- "FliirlEdu-eatralitgitfOrg.
uric.-_ on C^.- Baia (Ed'.,);.Mathemab_fcs 'E it a at-lanai SRciety forlth ;tu ]y Uiversityvf-"Chicago Pry (f ,_ he
Can roeSaboal Mathematics..,Coali-Tfar-SChg-allMa she-- t on ir-11oa'ghto-n' Mifflin (for Educational,'STarkiga
emetic Palo Ca 1964:- XplarielonsinMathematica. P Wtson-Wee1evrA9.07-
Inc_ oar A hradtl&Me0cteri;:14:; 65740,
aaandi ris-co,f,,earning-
Ch dram:a_ Ari,thmet aces,. N Yo OS .and
Al A. Aciltilme 27-_ 3_ 2 ;
4P, Sam "p ttie heiConcep-of a o. Ste_ AiTatrckV Pding_a ti er, eveomenRes -:arch nd ematical, duca 13:'Na- iota°Until of-Iea
MS t,uily*i , dZs-'AbiliityittilitUnderktanth u'caii., 003-61AishedMIdetoralrdistete& t_db, -ii101§1ahoma;*. , 9-.:. ..7 4701_:,--I., ; 44 re: ,! : --, a See_ ii eaStUdprofittesDeveraOmen.-- .wolf th th -aaatib-e-abaaraSe-c-olfdaSOrtYcilitChardre_ a Welttitata'"WfAblUty. UtiOSIilifiihT 51f; ''s UIMill." BelfaedP- 70,
n ons , a un amen_ IL_opici 4mo cs- *0 ea'
C tic s r 6 el ;cutcle 6
Concept Jo t,TEun lcop,f (Ed.; ITarAitien-ra 3v1hia-anzstrid VaiesE, 1. 7 ctVerbalnstructign.Arithmetic fescher10 72 -77, `t4 . M. Math Wa d en. "Chieego ,tier%die4Pieee,
and Nueie , C.Discovery re:,Elimeft'a "tics'Chicago:, :Encyelopedia BritannicaPiesss
-nere iVe-/MC-del. of' Mathematics L''aitiing tioe se_ a' ea ,u_, a hay_ 6) among n
as o _e O&' okic u ' asho e__ t.4,,
704.47- -elat onsnv _entwoset-:
-- ea a 4 defined b that -= -int-ague element
,t-a.110-iol- s d aylb be e compactly 0 '"1 .,, a -ion-ion et' of or&red,paits!,,,L ilaf'0 (I) 1 le -erit`:`9 ',.1,.604Y,''(2)..'-,i'S''::airefament,,i_o,-, '''; an pairs, inhave the `Sniii'firillalein -:-1-
t ecea gtA,, ef* in i tiO n?(or theies E fe newt .th..,,e,,,, reb,..7f)isi onmo _j accep and-- used, Buck. (1970) ties conteeaed that
Elcpe-rdfen cr-ahotuthat theY-'a:FfunCti6nkissan____ crrd dT-pai-Fskapproac r onewhich ev r _stionsvynOorCthe'A,tndentiafid',pro-vges a poo =Pd e a_a-_ on 0_ 'atije 4.rts;:t'eltIat s _hooo
d1 cl.ten p
on- s on. no longer,,Preach,-,thl a, 41.a -osat ofrordered' parrs d by Buc , 0 7r
urp se_ o_ this P4Art. , .eulivtabl tO.a_op ,characearizat'innqo land, o_n:
A 13; a rstlii_ which we ,is a.chember (*lig defij
rf it416-jr-Alt.A'A aa e ill IZ:4a13i papersuggest oaab2 IgraGNiete f 61 4uriori e-k4han-on a_ts de_ A R
and tit den
. - a fci -' sny on Jesn ica-d ,,,,,
, ---1=----*--X .4WAVV° 4 rbt pheral!Priritar,output b. azt
--Pb _
ci)b . (M- - a) b.
..;)b. -*(a
rOtTRULt
. nuid6els:--t :.stidey,,-, ealptiliifel h esndica ed tre-, ,w' (A -hr ' -,. W,.evk-.,,-,= I je;%1-A-11 o 0 _ tt_C' 821434..6_iphera1F Pr in_ I i,..iy + s se_as:. -,,, _pL,,x-e,-. 4-1,4 tt
Ur. A. f a /is thelliCF of z and Ty. . .,,,,,,l- a -rof -- ..anandd. st a nemb-ers"s4Ein_ d"-y, 1 cn-let ter
e es -1797-ndisated belowg,C;',ori o 4 C' g '8214347Pe 9 se _
C. w tY @4HCF to, w is th 00
eynothe thrrule
,=04'cir ft 't en linye,"roue EweY eilotien =whamAs-dikidg-__et, -.-e-. , `y (x/r1];'..a-whei_ 1 num_ e
. .,------,,,t,;r -0- , , ... -- 1-4_, 4, s .._ '' i S 7 the',re remaznder'_whenma xIscdivided:!-,by S dai t ,clq? +:_y (q a whole-'eumber,7y----<.d). .y is a` whole ftumber
14*, ma,, ebYfatirdants,,tp',"input_pai o, posit Ve e n -a _e vcprrespendirtuttitresef-apOertika e. t a organ b`uch-tbat,a2 c_t
p e up -11 van, inpu rcji!mab t-r. - - / HCF :tive-4.1. generat_, enalLer the quo on,'svia _ user' inPU_ a d ca en lito know edarit'ait,',1_rom ± -pu:
.I0315(9),;14rUSTp ,i1p,S0_,,,TILAT.1`, 14A_ BE `A 0 MC, A 'P,R6eiTITZ' fit-4 a'b ,,- A. ..,--,-ff =, 49,1 9- besed7LPoil' ,p am wr e ---t.....t ..,..,dInt ah 4 ea: c ndl'accompa ying182141-A.Eeriphere. ete ' ,t.,-,ir.-4-,-,..., _,.:.r._ ti_#;-- :e T.,pie e?spi , ro Iles ages ta t,tA.e e:',. 9 m ifg , o_Ei'rnialia =sof' b e _npu V *.tct " ;1;7.4,,,$,.ttti4T4 A..s
-fQa, U -2-
mss- t _, _
I 1
I _---.
i ..
1_ 1
1 1 .. .A. . ,
,.. . i
,,. .. _ , _ L _ , .. .,
I... a , I
1. « 1 ,t « i .
114 _ _ .._ .
«.... ' -=-.=-1-
. «
' 1
..- .. , . ., - V'IL4,4 . 4 _or (wit hasean,,prog amm_ a a _gnmen the knp:oti41.ngarm
nPti Ou t_Rut
NTE uch t a- , ple-ome triplesare& at
a_ resat ';,Ont.the attached sheet (with __ratitvetSr sRal_ a vaaeagera than 100, for instance) to a ma- p ngs sre,fass,i ants that will _enable youoraii.swer:_the ng,tquest4Ori
a,b 'a)t-does (n_;m) generate -4Saine
Undat what-condltion(s) will, annput ed -and-- nagenerst-e'anyl_triple? ' adis tin ive;,triple;p,',fyorn, ong;,
6'11 zed-_ hat -O-V7-LditTiiri(Irro,a" erate) wthar v (pa_ n9_
72
Us ing PYTR: 2
Step I. !pi ENTER t
*Step 2. at2;= bt2 SUM et 2
.. PRIMITIVEor NOT PRIMITIVE
PR1InTI go to 1. for new input if desired_, f NOTRIMITI VE continue: Seep 4 CFCT a
=; *Step 5. at 2 bt 2 = = SUM t 2=
Return -to Step 1 for .newinput if desired.
OBAL st0.- .y At, step 4: -What -does the value of CPC/ represent? How are a, b, c generated? , Will the Step 6- printed outpdt al a
If so,1191; If not,__Whi-h6-T -- a =r1--- P M TIVE, :dad- dorlaqued- r =With--- START
Input:
PRIMITIVE
0E' . -NOT _PRIM..
Primi tive?
#'R?
-CFCT Press r 74
INPUT/OUTPUTSamplas
4 -rENTER': « iENTER,, ,. =
[BEEP] [BEtP) Idv4,11, . -PY 4E6 :NP,-.
1-47 6 TrifER.
9 A =
2=-0 b = 45_ es:- ITIVE
'12 =St 18 Dj
MEN
.;4 NOTES
STARWLE: The HP 41C program has been written. to accommodate any one of the ollowing options(with prestored a and b restricted accordingly)
1. Input /Output are restricted to nc'n-negatiz,. int egers, an explicit error message printed if thisrestriction i violated.
Input/Output are restricted to integers (negative as well as non-negative);with an explicit error message printed if this restriction is violated.
Input/Output are restricted to none naga rational monbers, with an explicit'. error message printed this restriction is isolated.
jnput/Outputrmay be any rational number range), negative ae well as non-negative
FOURLUEE:
, The HP 41C program has been written toprint n e licitrror rsage-ifinput,x ory, is nota countingnumber.
TUBULE: The HP 41C program has been written to p Message if .input a- is not a counting number. _ - (The instructor-prestored d shOuld also be a eijtinting number; ever, NO-provislon has been bade in the program as written for an -err-or message if this condition is violated.)
*Since these values are instructor-prestored, it is assumed that OPplic-able restrictions have been met and no error-message provi- kalonhas-1,een included in the program to cover "violations."
tratiue err pmessages, each of which is preceded by an audible BEEP
el IT 4 a0Ri TPYFiEk, IXRULE
ENTp
rENTER.tj t CF. I
F: TENTER SFj
Prevtore/ Ini a ze R LE
A: [ENTER1
B: -ENTER-:j 1 CH5
CHSj FENTERt
- g¥ @1-
Recall !, iLl C4*Ept EN7EFt F. 47 94= E 4E 127 TNT sF ee 4Z xt2 128 RE 07 81 V. 4i :29 STO 06 XE9 5 A6 130 09 Fs, al '70 ST 04 10STGP 71Da 07 132 Xce' ao 72 R. 9: :33 GT0 3 XE12 C 73 X?? 134 STO 07. :3 XE9 74 RCL d2 ;25 GTO 02 14x=Y" 75 X.Y2 136.181 r- 1..5 Rim 16 1;7 1 i6 XE 3 77 ST1 4! 138 EC 1?Fs, ec, O6 = 78.LEL 0: 139 XcY) XE,C, 79 -6. 140 410-,04 tel XEP 80 ARCL 14f SF 12 29n. 81 AVID 14? CF.13 82 '8 143 'NOT ' 22 AIN 83 ARCL Y 144 FICA 23 5F 1: ##U 145 CI IL 24 CF :3 85 'C l6'NRRIVE 25 -CFCT 86 ARCL A 147, ACP 26 APCL'i. 37 01E11 148 PEBUF AVIEW 88 RTA 149 PIN 28 D? C 1541.81. W4 - ADV: :51 12 _34ST; 0Z 152 CF I: , --- 31 ST/ 04- 9912 153 -PRIMITIVE- sif 05 93 PEI, 05 154 ACP . 33 ecxt2. -3.43T/ e? 155 PPBUr 95#E84 156 P111 15--eck 03 % Xt2 157.18L 86 36 Re, 97 RC', B3 158 SF 91 --77-RCL Q 9C, Xt2 159 (F-12 LTC 6: 99 -At2 2' 18 CF 12 _ HO ARC .4 161 'IMOD 49 CF /IA 162 PRA 41 SF _ 102 ACV , 163"TRY-NEN INPuT,- 42 iF 12 103 "Bt2 a :44 PPA -- 43 PIN 104 CV Y 1E5 'USING --4-4 ROY 135 ACP 166 PCP J5&m 02 106 ADV - 4 krre 167 SF 1Z 468 SF 13= A7 STO .19813 48 70 t - <. 6¥ =. 169 BEEP 170 M $ APC1.--X 118 PPCL y 171 PCP '1E11 111 ACP '172 PPBUF '11E2 KW 173 RTN 113 SF 13 1744181 7, 14 'CU = 175 5F 91 _ 115 APCL 116 RAI_ 117 igAr
1I P:h Additional PYTR Notes
1. The program is to be executed with the calculato USER mode,anct the printer in MAN mode. No-provision has been made _ executingthe program without the printer.
2 The pregraM assumes that the user is "intellectually honest" and restricts inp0t to positive integers, although an error met.sage is
printed if (:),is mistakenly (or-otherwise!) used for m or'n. In its 7, Present forM, however, the pfogram does not reject a nonintegral input fir m or n, nor does it reject an integral input for Which m < 0 or n c 0. (The program could be Modified, of course, to reject such inpnta'also:.)
3. d,B,c is computed from n as follows: a = 'horun. b = lm2 n21. c = m2
B using the absolute value of the difference between 4 and n2
-.= to compute 5, it is unnecessary to invoke the input condition that m > n*It is left-For the student to. "discover" that input order -(m,n vs n,m) has effect upon the il,b,c triple computed (which is pr tive iff in and n are relatively prime and of opposite'parT__
4. In Step 3 (FYTR-1=or PYTR 2)the Eee1ideart,eigorithm is used-to calculate the HC.F of a and Yfprthe aji,c -triple geneKeted by Step- andthevalue-Of:the HCF (1 if aprimitive triple) isdisinOYed brit not printed identified as such asfinal calculator output When:Step3-- ruminates.
For-J'YTR 2, if` HCF # 1 its value -.is =printed=at the outs St p-,,Aand'idontif1ed as -CFCT but its meaningreMains to be isOovered":by_USers, The HCF is used aa'the"diVisor of a and o Of the Step 1 nonprimitive triple to generate theiPrimitive _iple printed in Step 4.
_Two &hoetcuts-have been programmed-in connection With PYTR 2:
FENTER t
Executes rEl aj as subroutines, and ifnOces -= executes rrq M On to generate a primitive trip
m ENTER F I
Same as (1), except [r] is not executed.
;.: and R 4 R -05 ,Flag and 13 are used to control printout format.
nag 00 is used to determine whether 11:1 is included as a ::subroutine of TM or excluded as asubroutine of (2].nag 01 is e and to this also-. 0
Here :noes
Archambs1au Siste 'l: Pyt agorean Tripled Grouped into'Fam_ 61e atics -Tear 6 q251-252; 1968.
Barret_ 0 de as A. ut ,Number Theory, Washington: a 0P411- CO: of Tear herhatIcs. 1961. pp. 48-49. ,
ardner. h hematiralGames S enti c Ncw'York: Scr bn 7 pp.. 158-160.
Glen: V H. and J-hnson Fl ovan. The Pythagorean Theorem 'Explorin Mathematics oYour Manchester, Missouri: Webst 960, Opt: 9-12.
0gi1v}rx G. S.i and A_ nders ,_T Excursions n Numbe New :York: Oxfor. Univecity Press, 1966,-pp.
Ore-_ Lnvitatn to umber Theory (Vol_. New Mat} e cp '.Liba Yor .11 ndom House, 1967 49-53
ch.,: S. Fi ding.,Pythagroan Trip os. Wisecan T e o Ma he tca 28(1):'. 177.
RdtHbart dPauls 11, 8 Pythaocean Trip e AN w. Easy -to- - 'Perive F Ula with S me Germs is Applications. a heMatits Teacher 67: -215-21 1974.
.11 thagorrean 1Tri les: 'What Kind?, .MathematicsTeach 69: 108-110 19762
offing ,, F. R P a dartNuncbers on:runnc A:Finite Arithmetic - 'oMe y in the mber Plane exin o ssachusetts: nn 1961, pp. -5.
f
e SOMEtitcHuGHTSFROM THE HISTORY OF PI ON niEPROGRAMMABLE CALCULATOR-
Eli Maur Department of Mathematics University of Wisconsin-Eau Cl;?ire Diu Claire, Wisconsin 54701
.The history of. mathematics offers an almost endless source of enrichment material with which to enlighten a classroom discussion. :NOWadays,,with the programmable,hand-held calculator having become Ean-InexPensive, one can illustrate many problemson one's:own linstrament and run thefeat home, in the classroom, or even ona vacation. Let me discuss here some highlights from the history of the number n. Of course, for a mare detailed discussion, one :Shoald consult any of the many sources on the history of mathematics, Csome--of.:ylliCh are mentioned in the bibliographical list.
The number it-tia1 intrigued scholars and laymen alike since the lawn of recorded history. The famous Mhind papyrus (ca. 1650 B.C.)
.m.ties the approximation ir '.(4/3)4 , which is within _6 percent':of:rhe exact valuM.- Another interesting apprOxiMation di-wris:%the easilr-remembered fraction 355/113 '3.1415929... g-ScoVeredby'rhe-Chinese Tsu Ch'ung-chih around 480 A.D. It aUrpris:ing that the.Biblical value for Tris simply-3, as is clear_ a ftatement in i'Kings-vii 21: And he made molten sea, ten.ciiits from the one brim to the Other: it was round -all about, g=and-hia-lheight:vas five cubits: and aline of thirty cubits did compass it rouneaboUt." Thus ate ancient Egyptians had already. _ rbeen usink,a much better approximationsome fifteen centuries hdfore.- _
-But while all-the ancient vtilues mre based on ah actual Measure- -ant of the 'circumference-to-diameter ratio for given circles,-,itwas he.=.-Greeka who'firat,proposeCan algorithm-that.ia, aiaystematic raeedure--to find Itto any destredi--acturacI.This was the 'famous method of exhaustion, invented by-Archimedes-of Syracuse (ca, 287 2122,13J.C.)'. By inscribing and circumscribing regular polygons of an dreasing-number of sides n around a circle of radius R, he showed that]. _the value of Tris squeezed" between the values n-sin(180 a /n) n.tan(-180'/n)-for any given it (Of course, he did not use the dern-trigonometricnotation. but the formulas are essentially his'.) -- Megan with an equilateral triangle (n m 3) and then doubled n five ='--
Mr' teaches undergraduate and graduate mathematics courses pecial interests in =applied mathematics, mathematics educe=_ -thethistory of mathematics. In 1978 he developed cout-ses for gifted elementary-and janieii-tr-- -61-ehildre- times up to n w-9_ which 1 is squeezed between 3.1410320..
and 1.1427146... . It is an easy task to write a program ,that will these values for n 3, 6, 12, (Do not forget to put your calculator in the "degree" mode1)
All subsequent methods of approximating x _ essentially variations of,the exhaustion method. It was n`et until 1579 that the French mathematician Francois' Viate gave a new method based on an infinite product:
+ 2 + 2 2
Thi6 remarkable formula shows that a can be calculated;' olely from _ the number 2 by. ,successlon ofjadditions, multtplications0divisions, --and square:reat extractions...Once again, it is interesting to write .--- -__ - , - 5 a program which will approximate from thiSiormula, us in ',1 -,-_, partial products. A program for the Texas instruments TI 5,fallows. -1 -.,
. 2 ROL 1 : ROL 1 - 1/x: IT 4.. STO Ou 6TO1 2 Lbl 0 ROL 0 =. 9 1/x Prd 0 (18 program SUM-1 Pause GTO 0
It becomes a-fascinating experience to watch,.the nunbers in. the as-they gradually approach 11-. The convergance is very fast, and after-AM only 15 partial products, the displayed value is correct to seven decimal places.
Anothg remarkable- product leading to IT was discovered by thel- English mathematician John Wallis in'1650 and is named after him:-
6
Againit is easy and Instructive rite a program_wh will approx-s--.-- _Limate x from the partial product this formula. The details 'are ft to the reader.
Let me-now mention some infinite series which involvex,:many,lof - , - .. . _-- --- chrlark' milestones in the history of mrifhematics. The first-Wileh'------
.es-:was discovered in 1671 by the Scotch James:Gregory-- frOMIL-the 2--s. --esforran-lxf
+ -- (The series is also known the Gregory-Leibniz series.) It was one o the first applications-of the newly invented different 1 and integral p'atz:ulus, even though it is quite'alseless as a p ctical means to calculate r, due to its -slow convergence. In writing a :program tio approximate r-,from the partial sums of this series,one ha% to take into account the alternating signs .of the terms.,This .can be done by storing (-1) in some memory and then instruct the instrument to Multiply the content of this memory by (-1) at every execution of the loop.
It is well known that the harmonic seriesthe sum of the re iprecals of the natural numbersdiverges. However, for many it,Vas not known if the corresponding series with the sates the natural numbers divergef or converges, and if it converges, o what-limit. Intrigningquestion was solved in 1736 by,' Eisler, who showed that the Series, converges to r2/6:
1/12 1/22 ±1/33+ 1/42 +... arr2 /6.
tiways fas inating--mystifying, indeed--to disc ver such ».-tremarkable relation between the natural numbers and r, which- -is =--.transcendencal. A-TI 57 program to approximate ITfrom thii series Hy
1. Pause -SUN 0' SUM 1 RST -RC', 0 act 1 LRN EST R/S
One cannot onlyuse the programmable calculator to approximate from these series,but also to compare .their rates of convergence: ,only has 4o haltthe program at any desired stage by Pressing t denRCL 0, andthe number of partial sums will be displayed. sing R/S-agairi-U111 resume-execution of the:program.
It turns out -that both the Gregory series an the Euler series T, ,etge.ivery slowly, taking 628 and 600 rerris re'spectively to find. -1Cwn decimal places (i.e., r = 3.14). For tie Gregory series,,_ se,_.the convergence will be oscillaring,/approaching trriatingly from above and below. 1 /
_-6ther "ggneralized harmonic series/ (i.e., series inVolvIng- ala= Of powerS-of_the natural numbers) alsO.-_involve i-fasC.-denverging series
-4 -1; 1 Many moire aeries of this kind and known and can be derived. either,fro the Taylor or Fourier expansiori of various elementary -funct,ions.Fof derallssee Courant (1956).
References
BeckmannPetr, Histo y of Boulde Colorado:Golan Press 1971 Courant, Richard= 'Differe and -In eaCalculus, VoluMes I aria II,; London:: Blackie Sens,195 Y . Hord An Introduct to the ory of hematics Ath York:- Ho nehart-and Winston, 1976.,
Dwmoor ,-uGtr, army r ,rthe "'ho we he, in dud wo enceur,n4e students nrc:.ents 2zri:layyinpmhI involving not omlid i t Iv k ,ritin p rob 1 er:
t red ri-71 m© p2yni.11even t the I y2-- el the fiP-',33E :::""hers'."11,.-211 tin., ex »n ban inte,:er. H 11s, re 1.-ite hum r.m-,L.Sin rh', r quo T. n sr compositl,)n
I t'.
L I
11P (I ry roins
ctilt.,Ars r Ii 19;`)
i 1 ii. u ii r.tct Crt,. (:ltn tt11S inLiitl{jttlt Jntrnt - 111111 wr inFlort-_s tor Lostrt nvls
hecntI tptilrrcrm inilnil
92 -I 4't:1 It'IL i,srs Assr0.17.1
Ir
r;1:T11_ I C., , ) I i)
-70m)tc1-1 cul r _LL1. _ _ _ Prot,1,-;
;=-,;= I;
; W. lolul .1. tor in [ _ ptcrIher- 1q7' .
Articl i the first in A sorios on the HP 25 pr- tam ()lc calculator. Sasic operations are presented.
Wavr r...m.ihltCaloti I t or:; IorI' ha !it arY Tmelt tots Calt' it.1t 11.1ga-21(Th.t N, .tuhe .197
A lesson on simple programming for students in grades 4-6 is Liven, Several prohl.ems and ex..umptes are g ivon, includlng
temporatur conversion, estimation, And num r 01-1,2 tons.
Wavrik, ,John J. A . htirti 'resentatiou in "Compu iteracy" using mmable Calculators. Calculators/Compute_s Naga:duo 9 -1l; November/December 1978.
rho pro4rammable calculator is stisted a IT . ducing elementary school_ students to certain leature6oi'IllrO- coNpulters. A ono -hour lesson for grades 5 and h using (he HP 2) p0d4rammahle calculator is presented, in which students arw g_. -n experience entering and running programs,
hn 1. Programmable Calculators for Elementary School Stu -Its. lug ToachAr 6: 39-41; May 1979. 1W -o units of study .Ire given for teaching elementary sch -1( students the use ,it programmable calculators StAra re4Lsters the program memory ArC consider
C, h L
ndary 1 Agriculture (Voational)
Tr e, Larry D. Usin? Programmable Calculator in Vo-Ag. Acrichltural l '! V 52: 17-1S: April 1980, lie capAbilit the programmable calcula(or And its po4aihle uses by A vnca tonal Agriculture teacher aro discussed,
School BAo10,v
Pro4rammahlt! Calcut.ittir,in siol(gy Classes.- Teactier 16: 4964498: November 1974. several uses r` calculators in blot classes are mentinned, including charting e:-Tponenrial population curves, evoiution by natural selection, and random genetic drift. Secondary School tnemistry
Ehrlich, Amos. Programmahlo (II,mititer and Kinetics of Chemical Reactions. hateynational Journal of Mathematjyad Ittdueat ion in Saience and Technology 11: 38)-389: July/September 1980. 8imulation techniques using a proera-'"Ible calculator in the 41 df ahomiallfl'ACt ions are presenttAL
I
. Woldert, K.W. Programmable Pocket Electronic C4ulators in the
C,lassroem. Aournnl of Chemical Education I October 1977. I IThe uses of programmable calcu-.ators in high school chomistrY alasses are discussed. in mrading, lahoratory exercises, aamputing T-scores, and a quantitative approich to chemical equilibrium.
8eaondary School Mathematics
Andersen. Lyle et al. Making Comparisons: Ratios. Topical Module for Use in a Mathematics Laborafory Setting.,'1973. ERIC: (31 p1 ge,0 --- .. UD Is::11 Tn-, .,n1,,ti,,, of this module on making comparison and ratios no 1 ude using ratios to compare sets of obje k ts and expressing
r It !,os as decimals or tractions in lowest tems.(t Six experiments
I pints directions for urilizine a programmable c:ilculator or computer.
,
Bawtree Micha,-'1. Numerical Solutions it ilalLais. Mathematics in School 8: 19-20; March 1979. The prograrnahle calculator-can he used to obtain numerical So lutions to equations. The program is given and the method
illustrated. .
Dennis, J. Richard and Thomas, David. Low-budget Comp3iter Program- ming in Your School (An Alternative to the Cost ot\ Lat Computers). fllinoits Series or Educational Applfrlftions of Computers. No IA. 1976. ERIC: ED 139 291. (6 Ttages) The programmable calculator can be used in teaching the concepts and the rudiments of computer programming and in computer problem solving. fwenty-five programming activities related to high
school mathematics are 1 isted. '
Dur,ipau. V.J. and Bernard. John. From Games to Mathematical Concepts eta the Hand-held Programmable Calculator. International Journal of Mathematical rducation in Science and Technology10: 417-424; JulviSePtemher 1979. -A few games ire suggested for programmable calculators which can create an environment in which mathematical concepts are more easily formed. 99
Holtman, iui c.t Itu Mathematics Irim an View with the F:se or_ Pocket Cilculator. Vnpublished M.S. thesis, Tel Aviv University, 1977. A unit for teach in. algorithms and the use of the SR 56 program- mable picket calculitor was developed for teachers of science or mathemati,-s in hi0h ;al s or caprolionsive
Kastner, Sheldon B. Remedial Mathematics Skills Program for Optional Assignment Pupils; School Year 1074-75. (New York City Board of Education Function Ni. 09-59678). t975. ERIC. ED 137 477. (20 pages) This is in evaluation of a New York City School District educa- tional project, the major objective of which was to increase student competency in math computational skills. Math labs equipped with calculators, printing calculators, and programmable calculators were available for student use. Program participants, on the average, made one-year gains in actual achievement.
Krist, Betty A. The Programmable Calculator in Senior High School: A Didactical Analysis. Unpublishfld Doctoral dissertation, State lrniversity of New York nt Buffalo, 1980.
Krist Betty Uses of Calculators in Secondary Mathematics. Columbus, 011: Calculator Information Center, Information Bulletin No. 8, September 1980.
LaBor, Martin; Wilcox, Floyd; and Richman, Claude M. Programmable Calculators as Teaching Aids and Alternatives to Computers. School Science and Mathematics 74: 647-650; Decembe7 1974. The author's provide tolist of calculators which have a eawitv for handling programs, and,a list of programs for such calcula- tors which are available at cost. They argue that the use of these materials at many levels of mathematics instruction enhances both motivation and understanding.
Mattel, K.C. Courses about Computers--for Secondary School Students. Australian MathematicsTeacher 30: 118-121; June 1974. A method of teaching introductory ideas about computer operations by using a programmable calculator is suggested.
Quinn, Doald Ray, The Effect ofthe Usage of a ProgrammableCalcu- __laTor upu Achievement and Attitude of Eiglath and Ninth Grade Algebra Students Unpublished Doctoral dissertation', St. Louis University, 1975. The programmable calculator was used in eighth- and ninth-grade algebra classes. When compared with students in noncalculator algebra classes, students using calculators showed less "anxiety toward mathematics" and had better "self-concept in mathematics," but no difference in achievement. lull
Sigurdsou. orviHe cr al. Area. Topical Module tnt Ii- in a Mathe- matics 1.aborator.- 3etrillg. 1973. FRIC: 111 181 '405. (iii pages) rilk area package emphas12..cs throe tacets: the concept At area oovering, the square tin it. And formula development. Two enrichment it are included, the Tirst of which requires the ii A: a oco.frammaTilc al,.ulater Ar computer.
Snoyer. Stephen L. Ana Stkell, Mark A. Flt= Rile cif Programmable Calcular7-1 And Computers\ in Mathematical Proms. Mathriaiies Taf,yhr 1: 1-.739; December 197S. This articlo illustrates the role of programmable calculators and computers in the creation cite mathematical prof s by exploring A simple prAblem trom number theory.
Snoyer. Stephen L.aria Spikell, Nark A. Soneraliv, How Do YOH Solve Valuations? Mathematics Tooaner 72 26-336: MaY 1070. Iteratiyo reehniques are presented for solving difficult equa- tians with numerical methods that can he used easily on oregrimmoble calculators. Flowcharts and programs are given CAr El57, HP 23, and BASIC.
Snover, Stfihon I..and Spikell, Mark A. x + 1, A Programmable Calculator Activitv. New Teisey MathomAcje_rLJc,iehps 37: 6-8; Fall 1979. Prograias are given for the Ti57 and the HP 33E. programmable calculators far Anononstandard problem.
Snovor, Stephen L.and Spikell, Mark A. A Programmable Calculator Activity, x t. 1979. ERIC: ED 170 117. (7 pages) nonstandard activity which could not he oasily explored without the use of a programmable calculator is presented, and flowcharts and programs for different programmable (xelculators are given.
Snover, Stephen L. and Spikell, Mark A. Programmable Calculators Facilitate Simple Solutions Co Mathematical Problems. 1979
ERIC: ED 170 115. (8 pages) . Many types,T problems ordinarily requiring advanced techniquos or special insight to solve can now be done as simple program-- Tine Axor-l-ies Of inipenC leO nrogrammnble calcutarors. The rAllowing examples are given: evaluating polynomials, finding limits, evaluating finite and infinite series, computing variable length products, searching for data, and developing proof.
Sneer. Stephen L. and Spikell, Mark A. Using Programmable Calcu- lators to Evaluate Complicated Formulas. 1979, ERIC: ED 170 116. (9 pages). Also in Virginia Mathematics Teacher 6: 23-21); February 1980. The use of the programmable calculator in evaluating complicated formulas is illustrated by considering the formula for finding 101
the a Y! triangle when only the lengths the t1 stiles are mowcharts and programs are )41 tctr TI 57 and h le calculators.
tr -alt 1 Physics geare, ichar Ind New. Peter immablo C, lulators menrar= Physics Teching. Phy'si 12: -426; ember 1977.
_ ring charao erist Le and features programmable hand- held calculators arc compared.
E o.tre, Richard. .1,1rammable Calculators, fart II: Their Use in ppIcingSimple Laws in Physics to Some Complex Problems. Sollool_Science Review 59: 269-284; December 1977. (Part I of this article is listed below under Secondary School Science. use of pregraimmible calculators to solve complex phYslcs problems i describ d.
Reiland. Robert J. A Realistic Model Rocket Program for a Small Programmable Calculator. Calculators/Computers Magazine 2: 72-74; September/October 108- A program for a programmable calculator is given which predicts the altitude achievable by a model rocket.
Smith, Clifton L. Computing in Secondary Physics at Amndale, W.A. Australian Science Teachers Journal 22: 33-40; May 1976 An Australian secondary school physics course utilizing an electronic programmable calculator and computer is described. Calculation techniques and functions, prOgramming techniques, and simulation of physical systems are detailed. A summary of student responses to the program is included.
Summers, . K. Use of a Programmable Pocket Calculator in A-level! Physics Courses. School Science Review 60: 316-325; December 1975
Scare, Richard, Programmable Calculators, Part T: Their Use in Teaching : "-moo. School Science Review 59: 36-48; September 1977. Advanced secondary schnoi science exercises which are amenable to the use of a prograammable calculator are given, and use of e calculator is compared to use of a computer terminal. lUd
Craig, James C. Simuliting Air Quality Investigations with the i'rogrammahl Calculator. nce Teacher 41: 18-42; April
ammable later to obtain . :iir !ThI tut L
1r . tutt t t r(ITIOnt iht1.
Cott g_ and Postgradu.te
ilogv
Blumenberg, tRennett and Spikell, Nick .\. Calculator Programs in General Go. ics: IL. Nei's indices of Genetic Distance, Protein Llent it", Migration Rate, and Divergence Time. Journal of Heredity 69: 278-280; July/August 1978. --- Two programs of general interest to the population geneticist trc presented, written in algebraic notation for Texas Instruments calculators, These programs are written for a genetic °ystem composd of five diatiolic loci, but may he modified to accommo- date locf'comprised of more than two alleles.
Biumenberg, Bennett and Spikell, Mark A. Calculator Programs in General Genetics: III. Tatter's Indices of Iletc TositY. Population Differentiation, and Genetic Distance, Journal of Heredity 71: 293-294 July/August 1980. A program written in algebraic notation for Texas instrummts calculators, and intended a genetic system embracing an infinite number of di,allelic loci,is presented. This program may al be modified to accommodate loci some or all of which consist of more than two Alleles.
FOrbes,!,'-',on L. Simulation of Natural Selection on the Program= thi ,alculator.-Journal of College Bcien o Teaching 95-96; November 1978. A model or game is described which enables students' to experiment with equilibria ; ind to trace rapidly gene frequency changes through rime under postulated conditions of selection.
0 is ::.D. Tc!dhing -5431c Tlioto tqucs with the Programmable Calculator. 1972. 7,D 079 990. pages) An introductory course on digital-computer simulation in Biology, taught at Michigan Technological University using the Olivetti programmable 101 calculator, is discussed.'
Spikell, Mark A, and Blumenberg,. Bennett. Calculator Programs in General Genetics: I. Computing Genetic Distance. Journal of Heredity. 63:.187-190; May/June 1977. A program written for Texas Instruments calculators is Presented which can he used for computing genetic difference and genetic distance from data from five diallelic loci. The prOgram can easily be expari'ded,to include any number of loci. 1
'Coll Gh
Attard, Alfred E. and Lee, Henry C. X-ray Crystaltographic.Computa- tions Using a Programmable Calculator. Journal of Chemical Education 56: 650: October 1979.
Six crystallographic programs developed toit lustr; of usefulness of programmable calculators in ihemical ana are described. The programs are suitable for the laborato analysis of X-ray diffractiondatai.
Brabson, G, Dana and Seegmiller, David W. Programmable Calculi01 s Add a Now Dimension to Laboratories. Journal of Chemical Education 47: 117-119; February 1970 Uses of programmable calculators in college chemistry slat are disciassed, and as specific example is given: studycat he homogeneous N104 - equilibrium.
Clark.C. J.; ' Haymin, H.J.G. Stereoscopic Diagrams Prepared by a DeskCalculaator and Plotter. Journal_ of Chemical Education 54: 31-34: January 1977. The use of a Hewlett - Packard 9810A prorammable calculator with plotter for drawing ball-and-line stereopairs as well as three- dimnsional structural formulas which are useful for teaching stereochemicat principleS and molecular structure is discussed, Holdsworth, David. Applications of Programmable Calculators in Chemistry Classes. Australian Science Teachers Journal '3: 74-76;. May 1977. Two 0-Tcriments in calulators are used ,arc described. In the first, the relative atomic mass of magnesium is deter- mined. In the second, a constant for gaseous concentrations of two reactants and the product at equilibrium are determined Ho s, -rth, David K. High Resolution Mass Spectra Analysis w Programmable Calculator. Journal of Chemical Education 57: 99-100; February 1950- 'H14, es, B. G. and Bundschuh, J.E. The Use of a Hand-held Program- mable Calculator in Evaluating Freshman Experiments. Journal of Chemical Education 55: 136-337: May 1978. The use of a programmable calculator for evaluating a student's performance on a rwantitntive iaboratory experiment is reviewed. McwfIliam, I. 1i'i.iI:Ii.iluCalculators. Journal of Chemical! Education 31: AA2-:,8,..; July 1974, the use of programmable calculators for the simulation ol experiments is discussed, And iLve exaMples of specific applica R11119ni,L. 01a1 et al. Programmable Calculators: simulated ments. Journal of Clm,mioal Education 49: 265-266: April 1977. Simulated chemistry experiments with a Wang 360 programmable calculator aro described. and data on a sample titration simulation are provided. Sevmour, M. D. and Fernando. Quintus. Effect or Ionic Strength on Equilibrium Constants. Journal cifChomical Educatiep 54: 225- 777; April 1.977. An pxporiment examining the effect of ionic strength on equilib- rium!.'-instantsis described. lie experiment involves the use of A programmable calculator and the concepts of activity and ictivity ooefficionts. Shearer. i1dmund C. Applications of a programmable Calculator in a Freshman Laboratoryrournal of College Science Teaching 5: 2--245: March 1976. Luc of a programmable calculator for student experiments, in grad Log laboratory reports .and in assigning accuracy and precision scores is discussed. Snadden, R.P. and Sfinquist, O. Simulated Experiments. Education in Chemisti-v 12: 73. 77; May 1975. A progriummable calculator is used as a data-generating system for. a ,;:mulated experiment involving conductimetric titration of an aqueous solution of OCT with an aqueous solution of NAOH. College Demography andery, P. The Programmable Calculator as an Aid. South Australian 7,!, Tach,--rs Triurnal 71,7: 49-'31:Jul':1n7!,, A program is described which can be used to explore the effects of various value-s of R and C on population growth. -41 105 College t. Fl. Addi? The I., 1tf Progra m t C1lnllla:Clrs in tlltTeaching or Economics, Part 1. ies 3-8; :;l fir. )78. culator Iriputer-based economic simulations 1. ar., described, each or whi gives objectives, operating instroc- \ tints, notes t the tea he and detailed instructions for students. Addis, G, H. rll , Use raimnahle Calculators in the Teaching tai Economics, Part Economics 0-58; Summer 1978. The complete program for exploring the dynamists of the Harrod- Demur equation is given, and some statistical uses nro mentioned. College ologv Shea, dames H. Treatment of Earthquake Hvpoconter Data with a Programmable Calculator. Journal of Geological Educati 29-14; January 1973. Throe investigations a-- developed using earthquake data and a calculator system with a d reader, X-Y plotter, and-rintout dovice. The investigations involve determining the spatial distribution of earthquake hypocenters with the goal of having students work with realistic data. e Mathes: Eisberg, Robert Programmable Pocket Calculators in College Science Teaching. ournal of College Science Teaching 7: 305: May 1978. The use of programmable calculators to solve second-order differential equations is di cussed in an article which con is of examples selected by the author from his 1976 book. Candor, Abdus Sattar, A Short' Program for SimOsonfs or Ga-,(1 r!s Pul--intoraffon -,n Handhei'l Pro Calculators, Two `'ear College Mathematic--Jonrnal 9 132-185; June 1978 lowchart and program for numerical integration are presented Gerald, Curtis F. Interactive Computing with a. Programmable Cal u later: Student Experimentations in Numerical Methods. 1 .73. ERIC: ED 032 470. (8 pages) In this paper presented at the June 1973 Conference on Computers in the Undergraduate Curricula in Claremont, California, the advantageous use of the Compucorp Model 025 programmable calculator in courses at California Polytechnic State University at San Luis Obispo is discussed. Students learned to solve non- linear equations and differential equations and were able to perform mathematical experiments analogous to the laboratory experiments of the physical sciences. Hallden-Abberton, PAtti And Waits, Bert K. The PrograrmiLdge cAlculator--An Inexpensive Teachini., Machine. MATYC_Journal 12: 215-219: Fall 1978. rtY.t-tm T.-o-7 prarimabla -11olit,t 1-4 permits evaluation ot students' Arithmetic computational skills. Wri Lae, Harr., Rudolph and Iturkett, iluvh MALI. Invosti gation g rrir urn ib I iirI i:hip ProqrammAhle Pocket Calculators and cAlculator Systems for lIse at Naval Postcraduate Schil Lnd the Nay ii t.11-) I npuh It Ii LI I i L r ' thes Pa-s-t,tradu:Ite School Monterey, Cal i fornia, March 1977. The usefulness of card-programmable hand-held calculators in the mAnagement curricula of the Naval Postgraduate Suhool Ar.d in thJ fleet were investigated. It was concluded that calculators nrovide significant adyantaces in teauhing er learning mathematical concepts and that they arc potentially importAnt management and tactical support tools Navy -wide. in addition. theuser's overall analytic capacity isimproved. -Mohrman, Kathryn (Ed.). EnnOvation:2 in Science Teaching. TheForum for Liberal Education-, Volume IT Number 4, February 1980. ERIC: ED 181 8:4-4, (II pagos) Curri:(1Lar development in undergraduate program- ,in the biologicfl, physical, and mathematical sciences at n number of colleges and universities is described. Included is The Ohio State UnIlferAity's program for teaching calculus with programmable calculators. O'Loughlin, Thomas. Using Electronic Programmable Calculators (Mini- Computers) in Calculus tristrUction. American Mathematical Monthly Si: 281-283; April 1976. An experiment L3 describQd in which a minicomputer was used as an instructional aid In a calculus classroom and as a labora- tory device. Papers Presented at the Association for Educational Data Systems Annual Convention Phoenix, Arizona. May1-976----ERIC: ED l_a yip. (93 PageS) Included among papers on the use of computers and electronic equipment in instruction is one paper Oti the use of program- mable calculators for calculus Lnstriction. Peckham, Norbert D. and Weir, Maurice D. Introduction to the T1-59 Programmable Calculator. Calculators/Computers Magazine 7: 52-57; Septctmberiectober 1978. Activities designed to familiarize- calculus students with the use -4, of the TI 59 are.presentd, and instructions for creating a library of programs on magnetic cards are given. 107 Schlaphoft, Carl W. CAI on a Programmable Calculator. MATYC Journal 9: 42-46; Winter 1973. A procedure is described for presenting routine practice problems ou a programmable calculator witi. atiachei teletype. rho program uses a random-number generator to write problems, 4Lves feedback, ant assigns grades ae,-ording to the procedures outlined and Llow-charted by the author. Stover, Clifford W., Jr. and Tingeylienry, B. Binomial Probabilities and the Pooket Calculator. Math Sciences Roundtable I: 27-31; 1979. Use of the calculator in computing probabilities for the binomial is discussed, with specific examples and problems. Saover, Stephen L. and Spikeil, Mark A. Because of Programmable Calculators, Why Avoid These Problems Any Longer? 1979. ERIC: ED 170 11-'4. (Li pages) Also appears AS: Problems Now Solvable in Calculus and Other Beginning Undergraduate Mathematics Courses with the Use Programmable Calculators. In Looking at Calculus: Perspectives for Techers (edited by A. David Burdoin). Milon, MA: AssoTiation of Advanced Placement Mathematics Teachers, 1980. pp.31-40. Several examples are given of the types of nonstandard problems that students can solve by using programmable calculators, Finding limits, evaluating and calculating infinite series, mpuftni; variable length products, searching for data, and developing proofs are among the examples. Wolfe D.B: Natural Frequencies and !'kule Shapes of MultiDegrees of Freedom Systems on a Programmable Calculator. R.C.A. Reviews 39: 604; 1978, Zimmerman, Mark. Random Numbers and Pocket Calculators. Calculators/ Computers Magazine' 2: 42-43; November/December 1978. A procedure for grnerating random numbers with an HP 55 program tlor p re,-;ent College PhystcadEducation Miller, Doris I. Simulation of Sports Techniques by Digital Computer and Programmable Calculator. Journa.1 of Hcalth, Physical Edura- tion and Recreation 45: 65-67; 'larch 1974. Use of_the.--programmable calculator in blomechanics . research and -Ln pityical education classes is discussed. College Physics Albergotti, J. C. instructional Lses.of the Computer: Satellite r-tnahlo ca ou (1I1Iit tTn :41: 1 1,:i-116;January 1973. M. N. ..tn.1 cower, col in N. (1..1s.). Reseancla in Sc lence Educ.it ion_p_ Vol Iii e 2. Proceedings o Ole rt2tIce th0 Austril i nrichaticeEdu,:at ion Research Assoc i at i on (7th. Ctl,2Uni, rs f New ym f 14cw South 1 il May1 7719 1976) 1976. ERR:: 14) 'AL (142 pages) Among these papers is one on clic effects of the proguannable .:alculator on attitudes LOWACW-1physics. Phil1ips, R. F. Simple Gravitation UsingAProgrammable Pocket Catcolator. Physics Education 1.2: 160-161; September 1977. Cal culat in us lug a programmable calculator., of the potent ill of the eart:114 gravitational iold strength and the energies of sate' 1 ttesin orb it Around the earth is described. l'itairn, Cameron C. nod Baker, Crevory L. The Rooket Game. its Teacher 12: 427-429; October 1974. A program for a orogrammable calculator is provided which simulats the problems of .Rot propuluion, hovering, and soft landing. Reiland, Robert J. A Realistic Model Rocket Program for a Small rograilmtable Calculator. Calcultors/Computers Magazine 2: 72-74; SeptemberiOctobe A program for the prediction of the alt tilde achievable by a :o(le 1 rocket is given. Schmidt. Stan1ey A. Fourier Analysis and Synthesis with n Pocket Calculator. American iiourna 1of Physics 45: 79=82; January 1977. -Two programs for performing Fourier analysis and synthesis with Loy arc Fiummer, X. K. Programmable Gal-oulators as an Aid in Physics Te1 -C6in4. Nvs 1 Educat ion 13: 246-250; 'I 1978. L:qc f a programmable calculator t solve two kinds of ditter- ential equat ions( those de f ining s imp lc harmonic and quantum harmon tr. mot ton)is de6rLhed. P. Games Calculator Programs. Sky and Telescope 54: 292; October 1977; 55: 102; February 1978; 5.5: 101; April 1978. Selected programs of astronomical interest that have been written for calculators are noted. Topic and source are indicated. 1)1111 1du, ildvld J. 1-11tdiro Nostermind with Progrdmmahle Hand CdIcilldtora C.11.:n! ttors/:multers Mdgadin,i 2: 11-3h; Miv 147S. :or tuie with other progrdmmable cdlculdtors. Using deductive !);LC, ORO LWO plavors Attempt to determine the code. Two proArims, witn AlrLIL tin.ui; A record:n Moot Arc provided. How td. Pr. mr1111 Cdlouldtors :or run And admca. FIRCritRIIkV-; II: 1M hiRC1977. A 1 1 tinit o; six g.imos for the p re g ramm.11) lev.) 1 cu 1 d to r .1 re the Dive Bomber, Footha 1 1 Blackjack. 1-;Paci' niorhvthm Forecast,Ind Test Yoar ESP, Goals dnd rules Tre dosoribed, with prodrdm,s ;or tInHP 2n, Johnston, iJnti. d Redetion,3 (to tho Tin Can Vrohled). Corinuters 1daaitino 2: A5-47: Fehrnary 1974 . tTit loiter, 1 programmdble'II uldtor program to salve the ofpiltidu I It Tin CR Problem (Clyde, 197S)is presented. Johndton, Pd-1 ./. Letters. Cadeulators/Computors Magdgine 2: 81: Aprll 1978. Ehree DrO4rAMN fir oxerolans in Scott (1978) are given in this letier. Juhnston, 'A. Letters. C11211torsiComputers 1,1dgdmtne 12: 12-11; t:.6amhcritohr 147. A ..--tIcaldtor program to -;olve for mean, standdrd devidtion :nut "normdi duric eTiition is given, to supplement Crothdriel 19:8) ogleshy, Hdo. "Hilo," "Hurkle." Calctilators _ Mag;izinc : Al2-ji7; Xdy 1977. hirct'tlon:3 :or rw6 cdi(oildtor manes are given. One cdn he pldved en either d four-tuwtion rm- d progrdmmable calculator, the-other only on T calculAcer. Flowcharts and Art. dri)V10,'j. 0-41e--;bv, 1.1.1 Froz..3 rdlculdtrs/Computers Ndgdzine 1: 5-S; 1dther 197:, ii n totin v the _inc "Fros" an dn SR 52 progrdmmnble calculdtor jr c o iintl. rlowch pr-6.i:rt:: 1 L t irig, 1 r 1 , 1 1 '4'1E11' are in-luded. WAV Findin4 the Fl inn in Your Cdlculdtor. Crilculators/ ':ompaters MA!",AHR 29-33; January 1974. -d1ealitar progrdm ;1or finding a Klingon spdcoship is given. ivriz, Teha J. -r:: r-;in! Telcher1s Notes and Answers. Calculators/ Computers H1 ,6mine 2: 7)-76; Fu1)rnary 197g. Answers far Wdvriks January 1174 article are given with comments. Id k. kOni h Klingon in Your Calculator. Calculators/ Computers Magline 02: ep embc r Mc tobe r1978. A ALI t. (11,110r t, CXLVP.s.1tiit it ivitv preaented by 'oLls k in January 1978. Otner Cses liradshaw, M .Eugene. 1 1 Pro.2,rammahl e r A8 A TC-; r0 I r moch Lae pe,ligo on: 64-6 );Apr t I 97a. Carat. John F. Crade Analysia with Progralmnable Po ke tElectronic CAI 1_11 it r . Journal of Chemical Euc at it 541 :14; February 1977. Advantages 01 using a proyyalmnahle calculator in computing atudent grades (e.4_,in fiuuring weighted averages And in the LineAr adiustment otraw acores) aro pointed out. eters, F. Fit e HP 25 aa a Digital Clock and Timer. EolottLir RI erot.itoa 117 57-58: August 1977. 0,1w mo program An SP 75 calculator to serve as a, lock/ t imer with 1.ap1 ivin hours, minutes, and seconds is shown. Reuhens, Arthur. Track in Down Equation Roota.. Maclanc_Dosign 50: 110; Aotgust in,1978. A program for an HP 55 calculator is given. Rowe, A.J. Machine-Markin.7 of MultApLe-Choice Tests: A Simple And Lnexpensive System Cs ii a Desk-top Calculalor. Journal of Biolou,ioal Ednoation 6: 13-16, February 1972. Schado, Herbert C. A Comparison of Student Characteristics Between lDwu Academic Years, 1971-72 and 1974-75. Mistitutional Research Report 3-75. 1975. ERIC: ED 130 722. (31 Pages) Statiattcal cnmir isns were made botwoen 23 characteristics otl. atudenta enroll.ed at Crowder Collet, (Neosho, MO) during two 000r;, o i)ro,4r Mt r.)r- the JiLt wwawr t lcn for a Howlett-Packard prog-ammablo calculator. The program LA 10pened. Tiny rl6mputcrs Specd Hut- it-us Doc is ions. Collage Store Journal 4461, soc 1 o f 2) : 11:6- 117: Apr il/Moy 1977. The use it programmAB 1 e cal cu ta tors Ii inyostmt_nt analysis, product ion achedu 1 ins inventory control and figuring compound interest rates is discussed. Twoddato, R.RrIA:e. Difficult Budgetary Decisions: A Desk-top 61culator Model to Facilitate Executive Decisions. 1916. ERIC: ED 126 846. (16 pages) o .1 p 1 i 1 v cA , Min Lmum. t:t=, Jmn. Compur 2 iues L m, -AL it 1 eve 1o: comp 1 yLry A p et =m ird HP's are dcscrihed Fro- Jmkn. t;pec 1i 'l d Cdl ou! Ato rsm-P rep ro...,,rmmcd to :Lot ye Prmc, r 1ms \lic ii propm:4ramm:! .mn -1Ls,..s1ssoT, . mor.11 delsIre describd. FeAtures ot business or financiAL calcu- : At m- tre ment Loned .onn, 1 hm,-4e ....'ork-Sny Proh cm-Sol y Prom,rAmmAble 11U1 rs iurIce uo , 70: L-NruAry Th ree 4roups of pro..4rammah l t1uI t ors are idoy: ILod ro r rnmnh le w ,h vn lati :e memory , o a rd p rulrAmmall 1 e ,and ke: 2 ro r amma h l o w i t n non yo la, i le memo ry El even c:11 oilAt,) r!--4 A ro ..:0m2Ared ,s1Cie t L tow Inc 1ca L arcs : p rog cam stej), ii rAnch ing, lTdre,m4h1 mon: r f0,4, o s tack registerst rove rs e Po 1 ish u.,t p ren -s Ls 5)11._.1-1LInc (1c,c-brcL r IL ii iys tem) and pricc. C:o t rheimer, Don rA Moru Power to th,.2 Cal cut :Icor Adm t rat i ye 'iii 1rioiit 1) 14-.. July cccimrmi! e 11 m il a t o r s ino 1 Iin ud-ht.:1( As wel Icc desk-top mode 1 s iro J, s Lbed. Two charts are incl udvd, p1tirivc rend Karp Luwa rt LII oil ltOrc Icr the Chemi st . Journal of Chemical uly 197.m In the r t or two art Le les a I ell 1 a to rs of interest tto the hind icc :Thrv..Iyud and their eapahilLties discuSSed- ! 111= . H: rind L._;t1i; i t',111L 1,1pr.lonU, kl A Sour, flo 1- ist c.: !:11 108 4- -; ri It 1 1.1 r-`) d il , 7in=iiirin n i 1. It r r I Ill (711,711.1r !7: April pr dt,,r d prinr,r) ;Ind Ind[Itor 7 t!1 1.7 ,7 (7.711 :ti I rn ; cul 7: April I c) 12. '.11 r Lho cort:un tftio:=; Ii '''' irS 'Sr ''' , ,/1"'3;'il'. 4, ''' 'ZI-4121 , rj 0`71- ,b. '' '4,7gt ' V '11711,i. , 4 1Z' ,404 ro: r `r A