maTematika

guram gogiSvili, Teimuraz vefxvaZe,ia mebonia, lamara qurCiSvilim a T e m a t i k aV klasimaswavleblis wignigamomcemloba inteleqtiTbilisi 2011Guram Gogishvili, Teimuraz Vepkhvadze,Ia Mebonia, Lamara KurchishviliMaths VTeacher's BookIntelekti PublishersTbilisi 2011ISBN 978-9941-430-18-3© gamomcemloba `inteleqti~, 2011.gamomcemloba inteleqtiTbilisi, ilia WavWavaZDis gamz. 17b. tel.: 25 05 22, 91 22 83.www.intelekti.ge info@intelekti.ge inteleqti@caucasus.netintelekti Publishers17b Ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 22) 25 05 22, 91 22 83.

Sesavalimaswavleblis wignis daniSnulebaa miawodos maswavleblebs meTodikuri rekomendaciebiaxali erovnuli saswavlo gegmis mixedviT Sedgenili V klasis saxelmZRvanelosTvis (guramgogiSvili, Teimuraz vefxvaZe, ia mebonia, lamara qurCiSvili, maTematika V); daexmaros maTsaxelmZRvaneloSi warmodgenili masalis gaanalizebasa da saswavlo procesis warmarTvismeTodebis SemuSavebaSi.warmodgenilia yvela tipuri gakveTilis, aqtivobisa da savarjiSos meSveobiT swavlebisrekomendaciebi. mowodebulia damatebiTi masala, miTiTebulia literatura, rac exmarebamaswavlebels sakuTari SesaZleblobebis srulyofaSi, wigni Seicavs miTiTebebs, Tu radamatebiTi resursebis gamoyeneba SeuZlia maswavlebels saswavlo procesSi.maswavleblis wigni Seicavs erovnuli saswavlo gegmiT gaTvaliswinebuli misaRweviSedegebisa da Sesabamisi indikatorebis CamonaTvals. Sinaarsisa da miznebis rukis mixedviTki axsnilia erovnuli saswavlo gegmiT gaTvaliswinebul romel Sedegs ukavSirdebasaxelmZRvaneloSi warmodgenili TiToeuli Tema. maswavleblis wignSi SemoTavazebulimeTodika iTvaliswinebs maswavleblis profesiuli standartis moTxovnebs (maswavleblisprofesiuli standarti, maTematika, dawyebiTi safexuri, 2008 weli). magaliTad, yuradRebaeqceva ricxvebis adekvaturad gamoyenebas, poziciuri sistemebis gamoyenebis unarisganviTarebis, zomis sxvadasxva erTeulebis erTmaneTTan dakavSirebis flobas, geometriulfiguraTa swori klasifikaciis aRqmas da misi gamoyenebis unaris ganviTarebas. yoveliTemis swavlebasTan dakavSirebuli sakiTxebis gadmocemisas ganvixilavT am Temis Sesabamismecnierul safuZvlebs _ vzrunavT maswavleblis kompetenciis ganviTarebaze.vcdilobT davexmaroT maswavlebels sxvadasxva strategiis, didaqtikuri xerxisa Tuteqnologiebis gamoyenebaSi.yuradRebas vuTmobT sxvadasxva akademiuri donis moswavleebTan muSaobis Sesaxeb rekomendaciebismiwo debas; saxelmZRvaneloSi masalis gadmocemis meTodika (magaliTad, xSirimimarTeba ukve Seswavlilisadmi) exmareba maswavlebels am mimarTulebiTac. SegaxsenebT,erovnul saswavlo gegmaSi Camoyalibebul erT-erT ZiriTad princips: saganmanaTlebloprocesis centrSi dgas TiToeuli moswavle da miRweuli Sedegi.yoveli Tavisa da paragrafis mixedviT mkafiod aris Camoyalibebuli miznebi damosalodneli Sedegebi.maswavleblis sarekomendacio wignSi Tavebisa da paragrafebis numeracia da dasaxelebaemTxveva moswavlis saxelmZRvaneloSi SemoRebul numeraciasa da dasaxelebas.ZiriTadi mecnierul-meTodikuri principebi, romlebic gansazRvravs saxelmZRvanelosSinaarssa da wyobas, aris saxelmZRvanelos Sesabamisoba erovnul saswavlo gegmasTan;sistematuroba masalis dalagebaSi; Sinaarsis mecnieruloba; Teoriisa da praqtikis kavSirisprincipis ganxorcieleba; gadmocemis mosawvdomoba, enis garkveuloba, sizuste da simartive;savarjiSoebisa da maTi Sesrulebis formebis mravalferovneba. es ukanaskneli exmarebamaswavlebels sxvadasxva TamaSebis, Sejibrebebis, viqtorinebis organizebaSi. maswavlebelssaSualeba aqvs xSirad gamoiyenos sityvebi: ifiqre, moisazre, Seafase, Seadare, daasabuTe _moswavleTa saazrovno unaris ganviTareba erovnuli saswavlo gegmis erT-erTi umTavresimoTxovnaa: `moswavle unda gacdes masalis ubralod damaxsovreba-gazepirebas. kerZod, mancodnis dagrovebasTan erTad (informaciis SenarCuneba) unda SeZlos am codnis gageba,gamoyeneba, analizi, sinTezi da bolos, Sefaseba. am dros moswavle ara mxolod flobs6

informacias, aramed mas SeuZlia qmedeba, raimes swavla swored azrovnebis Sedegia~...(erovnuli saswavlo gegma, zogadsaganmanaTleblo skolebisTvis, 2009-2010 saswavlo weli).davalebebis mravalferovneba, sxvadasxva xerxiT davalebis Sesrulebis SesaZlebloba swavlisstimulirebis erT-erTi mniSvnelovani aspeqtia. sworad SerCeuli da kargad organizebuliTamaSi xels uwyobs moswavleTa yovelmriv, harmoniul ganviTarebas, exmareba moswavlesgamoimuSavos cxovrebaSi saWiro Cvevebi. kerZod, daxvewos sakuTar gancdaTa gamovleniskoreqtuli formebi. TamaSma SeiZleba xeli Seuwyos moswavleebSi mexsierebis, yuradRebis,SemoqmedebiTi warmosaxvis, operatiulobis, TviTSemowmebis unaris ganviTarebas. TamaSixels uwobs sakuTari qmedebebis dasaxuli amocanebisadmi daqvemdebarebis, dawyebuli saqmisbolomde miyvanis Cvevebis dauflebas. TamaSebSi mniSvnelovania SejibrebiToba, radganSejibrebis dros vlindeba moswavleTa aqtiuroba da gamarjvebisken swrafva.saxelmZRvaneloSi mowodebuli davalebebi saSualebas aZlevs maswavlebels SearCiossxvadasxva organizaciuli forma _ gakveTili-TamaSi, gakveTili-mogzauroba, gakveTilisazrianTaklubi, gakveTili-viqtorina da a. S. didaqtikuri TamaSebi exmareba maswavlebelssaganmanaTleblo-aRmzrdelobiTi amocanebis gadawyvetaSi.maswavlebels kvlav SevaxsenebT erovnul saswavlo gegmaSi warmodgenil ramdenimemiTiTebas _ `saxelmZRvanelo aris erT-erTi saSualeba da ara erTaderTi, romelicemsaxureba saswavlo gegmis sagnobriv programebSi mocemuli Sedegebis miRwevas.saxelmZRvanelo ar aris TavisTavad programa _ Sesabamisad, maswavlebeli ar undaiyos saxelmZRvaneloSi mocemuli `masalis gavlaze~ orientirebuli, aramed aqcentiunda gaakeTos saxelZRvaneloSi mocemuli teqstebis, savarjiSoebisa da ilustraciebisgamoyenebaze sagnobrivi standartis Sedegebis misaRwevad. sworia, roca maswavlebeliargebs saxelmZRvanelos moswavleebis saWiroebebs da ara piriqiT, rodesac maswavlebelicdilobs moergos saxelmZRvanelos. maswavlebels SeuZlia swavlebis process miudgesSemoqmedebiTad, ar gamoiyenos romelime teqsti an aqtivoba, Secvalos drois xangrZlioba,romelic romelime konkretul Temas eTmoba, gadaanacvlos paragrafebis Tanamimdevroba,daamatos sakuTari aqtivobebi da sxva~. magaliTad, maswavlebelma saswavlo wlis dasawyisSiSeiZleba meti dro dauTmos IV klasis saswavlo programiT gaTvaliswinebuli sakiTxebisgameorebas, vidre Cven mier SemoTavazebul saxelmZRvaneloSia warmodgenili.swavlebis meTodikac SeiZleba sxvadasxvanairi iyos, rogorc gamoCenili pedagogi d. poiaaRniSnavda (CvenSi farTodaa gavrcelebuli misi naSromis rusuli gamocema: `Д. Пойа.Математическoе открытие~, Москва, 1976), `swavleba xelovnebaa... yovel karg maswavlebelsswavlebis sakuTari xerxebi aqvs. amiT yoveli kargi maswavlebeli gansxvavdeba sxva kargimaswavleblisgan. yoveli efeqtiani xerxi unda Seesabamebodes Seswavlis gansazRvrul xerxs...~.Tumca, unda iTqvas, rom arsebobs zogadi meTodikuri principebi, romelTa codna daexmarebamaswavlebels muSaobaSi. es principebi Camoyalibebulia TviT d. poias cnobil samtomeulSida im masalebSi, romlebsac maswavlebelTa profesiuli ganviTarebis centri da erovnulisaswavlo gegmebisa da Sefasebis centri gamoscemen (SeiZleba mivuTiToT, `maswavlebelTaprofesiuli standartebi~, Jurnali `maswavlebeli~, meTodologiuri saxelmZRvaneloebi,magaliTad, `rogor vaswavloT azrovneba~ da sxva).maswavlebelis wigni iTvaliswinebs V klasSi swavlebis Taviseburebebs; am etapze icvlebamoswavleTa akademiuri Sefasebis sistemis gamoyeneba. cxadia, Sefasebis principebi igiverCeba _ kvlav ZalaSia maswavlebelTa komentari-Sefasebebi, maT emateba niSnebiT gamoxatuliSefasebebi _ trimestruli, sagnis wliuri, saerTo wliuri. niSnebis wera V klasidan iwyeba,Sesabamisad, davamateT davalebebi (e. w. damoukidebeli da sakontrolo werebis nimuSebi_ Sesabamisi kriteriumebiT), romlebic daexmareba maswavlebels naSromTa Sefasebisas. aqveSevaxsenebT maswavleblebs, rom aRniSnuli tipis davalebebis Semowmebis Semdeg ar undaSemoifarglos niSnis daweriT; SefasebaTa gamocxadebamde sasurvelia klasSi moswavleTaaqtiuri monawileobiT gavaanalizoT sakontrolo davalebebi. visaubroT amoxsnis gzebze7

(gamovyoT optimaluri gzebi). moswavleebs warvudginoT Sefasebis kriteriumi. amisSemdeg, vTxovoT moswavleebs maTi niSnebis gamocxadebamde Tavad daasaxelon naSromissavaraudo Sefasebebi. Tqveni Sefasebisa da moswavlis TviTSefasebis siaxlove moswavleebSipedagogis obieqturobis gancdas aZlierebs. moswavleebis namuSevrebis safuZvelze SeiZlebaganmaviTarebeli Sefasebebis gamoyenebac _ komentarebi aRniSnuli davalebis Sesaxeb,romelic aRwers namuSevris Zlier da sust mxareebs da Seicavs rekomendaciebs miRwevebisgaumjobesebis Taobaze. saxelmZRvaneloSi komentarebis savaraudo nimuSebi SesabamisdavalebebTan erTad iqneba gadmocemuli.V klasSi swavleba sxva TaviseburebebiTac gamoirCeoda. aq, rogorc wesi, icvleboda maswavlebeli, iwyeboda e. w. sagnobrivi swavleba, rac dawyebiTi safexuris damTavrebasa daaxal safexurze gadasvlas moaswavebda. es problema dResac awuxebs ruseTis pedagogiur sazogadoebas.SesaZlebelia maswavlebelma SeZlos moiZios ruseTSi gamocemuli Jurnali `Первоесен тября~, #18, 2007. am nomerSi ori statiaa gamoqveynebuli: `Проблемы преемственности из четвёртогокласса в пятый~ da `Особенности взаимодействия начальной и основной школы~. miTiTebulia, rompirvel adgilze saswavlo sagani gamodis Tavisi SinaarsiTa da meTodikiT, moswavles uCndebasiZneleebis daZlevisadmi SiSis grZnoba, igrZnoba swavlisadmi interesis dakargvis tendencia.mexuTe klasSi swavlebis Taviseburebebi moswavleTa asakis axal etaps ukavSirdeba _maTSi mkvidrdeba zrdasrulobis gancda, Tumca, igi jer kidev ar aris gamyarebuli realuripasuxismgeblobiT. izrdeba moTxovnebi damoukideblobisa da pativiscemis, seriozuli dandobis gamomxatveli midgomisadmi. moswavlesTvis TiTqmis ukve mosawyeni xdeba maTematikuriamocanis amoxsna nimuSis mixedviT, mas surs ipovos sakuTari xerxi, romelic dafuZnebuliaimprovizaciaze, ufro mniSvnelovani xdeba erTi da imave amocanis sxvadasxva xerxiTamoxsna, vidre sxvadasxva amocanis amoxsna. am mxriv, erovnuli saswavlo gegma saSualebasiZleva tradiciuli da Tanamedrove Sexedulebebis sinTezis safuZvelze gaviTvaliswinoTmemkvidreobiTi sakiTxebi me-4-dan me-5 klasSi gadasvlis procesSi. amasTanave, igi saSualebasiZleva gamoviyenoT mravali qveynis gamocdilebac, sadac SemoqmedebiTad udgebian saswavlomasalis SerCevas da misi gadacemis meTodikas. amasTanave, Tavisi agebulebiTa da SinaarsiTCveneuli V klasis saxelmZRvanelo wina IV klasis saxelmZRvanlos bunebrivi gagrZelebaa.bevri sakiTxi xelmeored, zogjer ufro gaRrmavebulad aris warmodgenili, sakiTxebisgameorebas V klasis standartic iTvaliswinebs (droisa da sigrZis erTeulebi, naturaluriricxvebis Cawera, wakiTxva). Tumca, SemoTavazebulia didi naturaluri ricxvebis wakiTxvisada Caweris xerxebis gacnobis, poziciuri sistemis warmodgenisas 10-is xarisxebis gamoyeneba.gansxvavebulia Cveni midgoma wiladebis warmodgenisas _ aq, rusuli saxelmZRvaneloebisgangansxvavebiT, TiTqmis yvela aspeqts mivaqcieT yuradReba.misaRwevi Sedegebis Sesabamisi Sinaarsis SerCevas moswavleTaTvis misawvdomobisprincipi udevs safuZvlad; es SerCeva Camoyalibebuli stabiluri birTvis bazaze xdeba. igigafiltrulia maTematikis swavlebis gamocdilebiT; tradiciuli Temebis Tanamimdevrobisaxleburi gaazrebis, maTematikis erTianobidan gamomdinare integrirebis xarisxis gazrdisada masalis axali sakiTxebiT Sevsebis aucileblobis gaTvaliswinebiT. maswavlebelsmudmivad unda axsovdes, rom maTematika erTiani mecnierebaa da ara misi sxvadasxva nawilisxelovnuri gaerTianeba, amitom gadmocemis meTodika sakiTxebis urTierTkavSirSi gadmocemasiTvaliswinebs. sistematurad mimdinareobs wina masalis gameoreba da ganmtkiceba. yovelisiaxle moswavlis aqtiur monawileobas gulisxmobs; davalebebi, maTi mravalferovneba,sxvadasxva midgomebis gamoyenebis SesaZlebloba, praqtikasTan xSiri mimarTva zrdismoswavleTa motivacias, xels uwyobs kritikuli da SemoqmedebiTi azrovnebis ganviTarebasaRmoCeniT, keTebiT, kvleviTi samuSaos ganxorcielebiT. gvxdeba savarjiSoebi, romelTaSesruleba ara mxolod werasa da kiTxvas gulisxmobs (magaliTad, sivrculi figurebisSlilebi, brtyeli figurebis `dafarvis~ amocanebi da a. S.).8

davalebebi iseTi TanamimdevrobiTaa warmodgenili, rom martividan rTulisken iyosmimarTuli. calkea gamoyofili davalebebi saklaso da saSinao muSaobisTvis, savarjiSoTamravalferovani sistema (`testebi~, amocanebi jgufuri muSaobisTvis, amocanebi-TamaSebi,damatebiTi amocanebi da sxva) maTematikis Segnebulad swavlasa da misi Seswavlisadmiinteresis gaRrmavebas uwyobs xels.saxelmZRvaneloSi mocemuli zogierTi rubrika (`s~ _ sxvadasxva, `moisazre~) xelsuwyobs moswavlis cnobismoyvareobis gazrdas, ubiZgebs mas meti, SemoqmedebiTi muSaobisken,movlenebis Sesaxeb condis SeZenisken da TviTganviTarebisken. mowodebuli masalamaswavlebelma SeiZleba gamoiyenos viqtorinebis, warmodgenebis da sxva maTematikuriRonisZiebebis Casatareblad.sakiTxTa mravalferovneba, maTi ganxilvis formebis araerTgvarovneba xels uwyobssaswavlo procesSi Sefasebis sxvadasxva formis gamoyenebas, `testebi~ kargi saSualebaaswrafi TviTSefasebisTvis.Tumca, SesaZlebelia, zogierTi `testis~ ganxilvas Tan sdevdes msjeloba da pasuxisdasabuTeba. amasTanave, sTxoveT moswavleebs wignSi nu miuTiTeben swor pasuxebs, CainiSnonrveulSi swori pasuxebis nomrebi, romelsac SeiZleba Tan erTvodes mokle Canawerebi,romlebic am pasuxebis miRebis gzas SeiZleba aRwerdes. yvela davaleba wignSi CanawerisgakeTebis gareSe unda Sesruldes _ mxolod rveulSi! moswavle unda movaCvioTsaxelmZRvanelos sufTad xmarebas. saxelmZRvanelo regularulad Seaxsenebs xolmemoswavleebs, rom gaufrTxildnen wigns. Tqvenc mimarTeT xolme moswavleebs da TvaliadevneT maT damokidebulebas wignisadmi. SesaZlebeli unda iyos wignis gamoyeneba ramdenimewlis ganmavlobaSi.saxelmZRvanelo ori nawilisgan Sedgeba, pirvel nawilSi gadmocemulia saswavlo wlispirvel naxevarSi gasavleli masala. es aadvilebs saxelmZRvanelos tarebasa da moxmarebas.saintereso da mravalferovania wignis agebuleba, igi saxelmZRvanelosadmi wayenebulyvela moTxovnas iTvaliswinebs (ix., mag., [22]). wignis dasawyisSi gamovxatavT madlierebasyvelas mimarT, visac Tavisi rekomendaciebiT wvlili Seaqvs wignis daxvewaSi; gansakuTrebiTmniSvnelovania saswavlo gegmebisa da saxelmZRvanelos SemmuSavebeli jgufebis eqspertebisRvawli. maT mier klasebis mixedviT ganawilebuli saswavlo gegma saSualebas gvaZlevsSevarCioT masala moswavleTa codnis gaRrmavebis, unarebisa da Cvevebis ganviTarebisa daformirebisTvis. eqspertTa kvalificiuri rekomendaciebi da SeniSvnebi gvexmareba mTeli`sapilote~ wlis ganmavlobaSi davxvewoT saxelmZRvanelo, yoveli axali nawilis SedgenisasgaviTvaliswinoT wina nawilze miRebuli daskvnebi. wlis bolos _ koreqtivebi SevitanoTyvela nawilSi.saxelmZRvanelo 4 Tavisgan Sedgeba, TiToeuli Tavi _ paragrafebisgan, zogierTiparagrafi ramdenime nawiliania. sarCevi wignis dasawyisSia mocemuli, maTematikuri terminebi,cxrilebi, maTematikuri niSnebi, sagnobrivi saZiebeli _ wignis boloSi.Zvirfaso maswavleblebo!warmatebebs gisurvebT Tqvens sapatio da mniSvnelovan saqmianobaSi. imedi gvaqvs, romCveni TanamSromloba keTil nayofs gamoiRebs.9

V klasi, wlis bolos misaRwevi Sedegebi:mimarTuleba:ricxvebi damoqmedebebikanonzomierebebida algebrageometria dasivrcis aRqmamonacemTa analizi,albaToba dastatistika1. iyenebs axalricxviT saxelebs,poziciursistemas da axdensnaturaluriricxvebisklasifikacias2. kiTxulobs,gamosaxavs, afasebs,adarebs da alagebswiladebs3. asrulebsmoqmedebebsnaturalurricxvebze datolmniSvnelianwiladebze5. gamosaxavsda aRwerssidideebs Sorisdamokidebulebas6. amocanisamoxsnisas adgensda amartivebsalgebrulgamosaxulebas7. amoicnobs, aRwersda gamosaxavsgeometriulfigurebs8. adgens mimarTebebsfigurebs Soris dafiguris elementebsSoris9. poulobs daadarebs brtyelifigurebisfarTobebserTmaneTs10. orientirebsbadiT dafarulareze11. moipovebs dasmuliamocanis amosaxsneladsaWiro Tvisebrivda raodenobrivmonacemebs12. warmoadgensTvisebriv daraodenobrivmonacemebs dasmuliamocanis amosaxsneladxelsayreli formiT13. akeTebs Tvisebrivda raodenobrivmonacemTainterpretaciasa daelementarul analizs4. iyenebs daerTmaneTTanakavSirebszomis sxvadasxvaerTeulebs10

wlis bolos misaRwevi Sedegebi da maTi indikatorebimimarTuleba: ricxvebi da moqmedebebimaT. V.1. iyenebs axal ricxviT saxelebs, poziciur sistemas da axdens naturaluriricxvebis klasifikaciasSedegi TvalsaCinoa, Tu moswavle:• kiTxulobs milionze did ricxvebs axali ricxviTi saxelebis gamoyenebiT (mag. trilionida a.S.); ganmartavs am ricxviT saxelebs.• poulobs axali ricxviTi saxeliT mocemuli (milionze) didi ricxvis rigs (mag. ramdenicifrisgan Sedgeba 10-obiTi poziciuri sistemiT Sedgenili aseTi ricxvi?).• iyenebs 10-is xarisxebs didi ricxvebis Cawerisas. msjelobs aTobiTi poziciuri sistemisupiratesobaze sxva ricxviT sistemebTan SedarebiT (mag. egvipturi an romaulisistema).• poulobs mocemuli erTniSna da orniSna ricxvebis jeradebsa da gamyofebs.• ganasxvavebs kent, luw, martiv da Sedgenil ricxvebs, asabuTebs 2-ze da 5-ze gayofadobisniSnebs.• iyenebs ricxvis kvadratis cnebas, amoicnobs orniSna naturalur ricxvebs Sorisnaturaluri ricxvis kvarats.maT. V.2. kiTxulobs, gamosaxavs, afasebs, adarebs da alagebs wiladebsSedegi TvalsaCinoa, Tu moswavle:• kiTxulobs da gamosaxavs Cveulebriv/Sereul wiladebs; uTiTebs maT CanawerSi wiladismricxvels da mniSvnels / mTel da wilad nawilebs.• gamosaxavs erTeulis nawilebs ricxviT sxivze da aRniSnavs tol nawilebs; iTvlisaseTi nawilebis Sesabamisi bijiT (maT Soris erTeulis gavliT).nimuSi 10 1210 121330 12443410 1 2 3 4 5 1666660 1 2 3 48 8 8 858687810 1 2 3 4 5 6 7 8 9 10 11 11212121212121212121212nimuSi 30 1 2 3 1 111213 2444444• adarebs or wilads, maT Soris wiladis ZiriTadi Tvisebis gamoyenebiT.• wers Sereul wilads ~arawesieri~ wiladis saxiT da piriqiT; axdens (wesieri) wiladiscnebis sxvadasxvagvar interpretacias da msjelobs maT Soris kavSirebze (wiladirogorc ori naturaluri ricxvis gayofis Sedegis Canaweri, erTeulis nawili,mTliani jgufis qvejgufi da rogorc ~ricxviT xazze~ garkveuli adgili).maT. V.3. asrulebs moqmedebebs naturalur ricxvebze da tolmniSnvnelianwiladebzeSedegi TvalsaCinoa, Tu moswavle:11

• amocanis konteqstis gaTvaliswinebiT irCevs da iyenebs naturalur ricxvebzemoqmedebaTa Sesrulebis adeqvatur xerxs; naSTiT gayofis SemTxvevaSi axdens naSTisinterpretacias amocanis konteqstis gaTvaliswinebiT.• axdens erTnairi mniSvnelis mqone martiv wiladebze ariTmetikuli moqmedebebisdemonstrirebas da moqmedebaTa Sedegis interpretacias modelis gamoyenebiT (mag.namcxvris naWrebi).• msjelobs Tu rogor icvleba wiladi misi mxolod mniSvnelis an mxolod mricxvelis~-jer/-iT~ gazrdiT an SemcirebiT; asabuTebs pasuxs (mag. modelis gamoyenebiT).• iyenebs moqmedebaTa Tvisebebs da maT Soris kavSirebs Sereul ricxvebze gamoTvlebisSesrulebisas/maT gasamartiveblad (Sereuli ricxvebis Sekreba/gamokleba; wiladisnaturalur ricxvze gamravleba).maT. V.4. iyenebs da erTmaneTTan akavSirebs zomis sxvadasxva erTeulebsSedegi TvalsaCinoa, Tu moswavle:• erTmaneTTan akavSirebs sigrZis da farTobis erTeulebs, iyenebs ricxvis kvadratisCanawers am konteqstSi.• erTmaneTTan akavSirebs farTobis sxvadasxva erTeulebs; gamosaxavs farTobis diderTeuls mcire erTeulis gamoyenebiT.• iyenebs drois 12 da 24-saaTian formatebs da ariTmetikul moqmedebebis gamoyenebiTgansazRvravs dros da drois intervals.• iyenebs naSTiT gayofas zomis mocemul erTeulebSi monacemis sxva erTeuliTgamosaxvisas (mag. 50000 wami = ? saaTi).Sinaarsi1. naturaluri ricxvebi da maTze moqmedebebi2. milionze meti naturaluri ricxvebi (miliardi/bilioni, trilioni da a.S.)3. sxva ricxviTi sistemebis gacnoba (mag. anbani da asoebis ricxviTi Sesatyvisebi;egvipturi _ aTis xarisxebis gansxvavebuli aRniSvnebi an romauli sistema).4. arauaryofiTi wiladebi toli mniSvneliT da maTze moqmedebebi5. sxvadasxva mniSvneliani wiladebis Sedareba, dalageba da gamosaxva (mag. wiladebimniSvneliT 2,4,8; 3,6,9,12; 5,10 da 100)6. ricxvis kvadrati farTobis konteqstSi7. kavSiri sigrZisa da farTobis erTeulebs Soris8. drois erTeulebi (saaTebi, wuTebi, wamebi), saaTis 12 da 24 saaTiani formati9. wonis erTeulebi (kilogrami, grami, miligrami)mimarTuleba: kanonzomierebebi da algebramaT. V.5. gamosaxavs da aRwers sidideebs Soris damokidebulebasSedegi TvalsaCinoa, Tu moswavle:• aRwers raime sididis Tanabar adiciur (mudmivi sididis mimatebiT/gamoklebiT miRebul)cvlilebas (maT Soris realur viTarebaSi).• mocemuli damokidebulebisaTvis Tvisebrivad aRwers Tu ra gavlenas axdens erTisididis cvlileba masze damokidebul meore sidideze da sxva atributebze. (mag.~erTis zrda gamoiwvevs meoris zrdas~, ~zRvis donesTan SedarebiT ufro metisimaRle rukaze ufro muqia~).12

• erTi cvladis Semcvel mocemul asoiT gamosaxulebaSi, sxvadasxva ricxvebis CasmiTavsebs cvladis mniSvnelobebsa da gamosaxulebis mniSvnelobebs Soris damokidebulebisgamomsaxvel cxrils, romelSic cvladis mniSvnelobebis Sesabamisi sveti/striqoniwinaswaraa Sevsebuli.maT. V.6. Seadgens da amartivebs algebrul gamosaxulebas maTematikuri amocanisamoxsnisasSedegi TvalsaCinoa, Tu moswavle:• adgens realuri viTarebis an misi sityvieri aRweris Sesabamis tolobas, utolobas angantolebas (romelSic ucnobi aris tolobis mxolod erT mxares).• ariTmetikuli operaciebis gamoyenebiT teqsturi amocanis amoxsnisas, svams kiTxvebsamocanis pirobaSi arasruli monacemebis Sesavsebad (mag. amocanis piroba: “moswavlemsam fanqarSi 60 TeTri gadaixada. ra Rirs erTi fanqari?” daklebuli monacemebisSesavsebad SeiZleba daisvas kiTxva: “samive fanqris fasi tolia?”).• iyenebs Sekrebisa da gamravlebis komutaciurobas, asociaciurobas da SekrebismimarT gamravlebis distribuciulobis Tvisebebs (erTi cvladis Semcveli) asoiTigamosaxulebebis gasamartiveblad.Sinaarsi1. or sidides Soris damokidebuleba, romelic Sekrebis/gamoklebis SemcveligamosaxulebiT moicema; sidideebs Soris damokidebulebis gamosaxva cxrilissaSualebiT.2. Sekrebis, gamoklebisa da gamravlebis Semcveli ricxviTi da asoiTi gamosaxulebebida maTi gamartiveba.3. Sekrebisa da gamoklebis Semcveli ricxviTi utolobebi da maTi Tvisebebi.4. teqsturi amocanebi, romlebic Sekrebis, gamoklebisa da gamravlebis SemcveliricxviTi an erTi asoiTi aRniSvnis Semcveli algebruli gamosaxulebiT ixsneba.mimarTuleba: geometria da sivrcis aRqmamaT. V.7. amoicnobs, aRwers da gamosaxavs geometriul figurebsSedegi TvalsaCinoa, Tu moswavle:• uTiTebs wris/wrewiris elementebs; koreqtulad iyenebs wrewirTan/wresTandakavSirebul terminebs (centri, diametri, radiusi, qorda).• yofs wrewirs/wres tol (naxevari, meoTxedi) rkalebad/seqtorebad; iyenebs maTkuTxeebis Sesadareblad da dasajgufeblad (blagvi, marTi, maxvili da gaSlili).• amzadebs marTkuTxa paralelepipedisa da kubis Slils; mocemuli Slilis mixedviTamzadebs models da asaxelebs miRebul figuras.maT. V.8. adgens mimarTebebs figurebs Soris da figuris elementebs SorisSedegi TvalsaCinoa, Tu moswavle:• axdens samkuTxedebis klasifikacias misi kuTxeebis mixedviT (blagvkuTxa, marTkuTxa,maxvilkuTxa).• uTiTebs brtyeli figuris paralelur da urTierTTanamkveT gverdebs, msjelobsgadaikveTeba Tu ara mocemuli gverdebi gagrZelebis Sedegad.13

• sivrculi figuris modelze uTiTebs paralelur da urTierT-TanamkveT waxnagebs,msjelobs gadaikveTeba Tu ara mocemuli waxnagebi maTi gavrcobis Sedegad.maT. V.9. poulobs da adarebs brtyeli figurebis farTobebsSedegi TvalsaCinoa, Tu moswavle:• dafaravs figuras erTnairi aragadamfaravi figurebiT da asaxelebs dasafarad saWirofigurebis mTlian raodenobas.• figuraTa urTierTSeTavsebiT adarebs an afasebs figurebis farTobebs. (mag. rodesacerTi Tavsdeba meoreSi, maSin misi farTobi ufro naklebia).• iyenebs farTobis adiciurobas aragadamfaravi figurebis kombinaciiT miRebulifiguris farTobis mosaZebnad.maT. V.10. orientirebs badiT dafarul arezeSedegi TvalsaCinoa, Tu moswavle:• koordinatebis (simboloTa wyvilis) gamoyenebiT aRwers mdebareobas da iyenebs am xerxsrealur viTarebaSi (mag. kinoTeatri, gemebis CaZirobana, Wadrakis dafa, ruqazeobieqtis moZebna).• gadaadgildeba ujrian furcelze instruqciebis mixedviT da aRwers, rogor miaRwevsmocemuli ujridan sxva ujramde (mag. ori ujra marcxniv, Semdeg erTi ujrazeviT).• aRwers ruqaze ori an meti punqtis urTierTmdebareobas oTxi mimarTulebis gamoyenebiT(mag. CrdiloeTiT, dasavleTiT).Sinaarsi1. wre/wrewiri: centri, radiusi, diametri, qorda, rkali, seqtori.2. kuTxe (araformalurad, rogorc mravalkuTxedis elementi).3. samkuTxedis saxeobebi: blagvkuTxa, marTkuTxa, maxvilkuTxa.4. mravalkuTxedis gverdebs Soris mimarTeba: paraleluri da TanamkveTi gverdebi;mravalwaxnagas waxnagebs Soris mimarTeba: paraleluri da TanamkveTi waxnagebi.5. farTobi (araformalurad, rogorc erTnaeri aragadamfaravi figurebiT dafarulfiguraSi damfaravi figurebis raodenoba).6. koordinatebi (araformalurad, rogorc adgilmdebareobis miTiTeba simboloTawyviliT).mimarTuleba: monacemTa analizi, albaToba da statistikamaT. V.11.moipovebs dasmuli amocanis amosaxsnelad saWiro Tvisebriv da raodenobrivmonacemebsSedegi TvalsaCinoa, Tu moswavle:• SekiTxvebis mocemuli CamonaTvalidan SearCevs da iyenebs saWiro monacemTaSesagroveblad Sesaferis SekiTxvas/SekiTxvebs.• mocemul TemasTan dakavSirebiT svams kiTxvebs Sesaferisi formiT (Ria, daxuruli,14

amdenime alternatiuli arCevanis momcveli) da am kiTxvebis saSualebiT moipovebssaWiro monacemebs.• irCevs monacemTa Segrovebis Sesaferis saSualebas (dakvirveba, gazomva, monacemTaamokreba mocemuli erTobliobidan) da iyenebs mas, asabuTebs Tavis arCevans.maT. V.12. warmoadgens Tvisebriv da raodenobriv monacemebs dasmuli amocanisamosaxsnelad xelsayreli formiTSedegi TvalsaCinoa, Tu moswavle:• klasificirebuli monacemebisTvis calsaxa Sesabamisobis miTiTebuli wesiT qmnispiqtogramas, romlis erTi simbolo Seesabameba ramodenime monacems.• qmnis martiv cxrils araumetes oci klasificirebuli da dalagebuli monacemisTvis(mag. gansazRvravs Wdeebs, saTaurs, svetebisa da striqonebis raodenobas da adgensmonacemTa cxrils).• klasificirebuli monacemebisTvis urTierTcalsaxa Sesabamisobis wesiT qmnissvetovan diagramas ujrebian furcelze (mag. gansazRvravs Wdeebs, saTaurs, svetebisraodenobas da aferadebs ujrebiani furclis Sesabamisi sigrZis zolebs).maT. V.13. akeTebs Tvisebriv da raodenobriv monacemTa interpretaciasa daelementarul analizsSedegi TvalsaCinoa, Tu moswavle:• svams saZiebo/Semajamebel kiTxvebs monacemebis Sesaxeb, romlebic warmodgeniliasvetovani diagramis saxiT (mag. nayinis ramdeni gansxvavebuli saxeoba unda viyidoTklasis zeimisTvis? TiToeuli saxeobis ramdeni nayini? nayinis romeli saxeobauyvars ufro met Cvens Tanaklasels –Sokoladis Tu marwyvis? nayinis romelisaxeobaa yvelaze popularuli Cveni klaselebisTvis? gogonebisTvis? vaJebisTvis?ratom?).• adarebs monacemTa or erTobliobas da warmoaCens Tvisebriv da raodenobriv msgavsebasada gansxvavebas maT Soris (Tvisebrioba ukavSirdeba jgufSi monacemTa gvarobas/tips,monacemTa ganmeorebadobas, pozicias da Tanmimdevrobas, gamorCeul monacemebs).• gamoTqvams varauds monacemTa safuZvelze (mag. gamokiTxvis “vin ra gadaadgilebissaSualebas iyenebs skolaSi misasvlelad” Sedegebis safuZvelze gamoTqvamsvarauds, daaxloebiT ramdeni bavSvi cxovrobs skolasTan axlos).Sinaarsi1. Tvisebrivi da raodenobrivi monacemebis Segrovebis saSualebani: gazomva, dakvirveba,gamokiTxva; monacemTa amokreba monacemTa umartivesi wyaroebidan (mag. cnobari,katalogi).2. Tvisebrivi da raodenobrivi monacemebis organizacia: monacemebis klasifikacia(garda raodenobriv monacemTa dajgufebisa intervalebad).3. monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi:gamorCeuli (mag. eqstremaluri, iSviaTi) monacemebi.4. monacemTa warmodgenis saSualebani raodenobrivi da Tvisebrivi monacemebisTvis:sixSireTa cxrili, piqtograma, svetovani diagrama.15

Sinaarsisa da miznebis rukaTemebis CamonaTvalinaturaluri ricxvis wakiTxvisa da Caweris wesebi.milionze meti ricxvebi,maTi Sedareba da dalagebanaturaluri ricxvebisSekreba da gamokleba.amocanebis amoxsna.ricxviTi sxivi.naturaluri ricxvebisgamravleba da gayofasakoordinato bade,amocanebis amoxsna.asoiTi gamosaxulebisSedgenisa da gamartivebismagaliTebi.amocanebis amoxsnaxarisxi. xarisxis gamoyenebaricxvis jeradebi dagamyofebi.martivi da Sedgeniliricxvebi.miznebi: Sinaarsis gardaqmnamiznad; erovnul saswavlo gegmaSimiTiTebuli Sesabamisi punqtebimilionamde naturaluri ricxvebiswakiTxvis da Caweris gameorebisSemdeg eufleba milionze didiricxvebis wakiTxvis, Caweris,klasifikaciis wesebs, poulobs axaliricxviTi saxeliT mocemuli ricxvisrigs. V. 1.amocanebis konteqstis mixedviTnaturalur ricxvebze moqmedebaTaSerCevisa da Sesrulebis unarisgamomuSaveba V. 3.mdebareobis aRwera simboloTawyvilebiT da maTi gamoyenebis unari;ujrebian furcelze instruqciebismixedviT gadaadgilebaTa aRwera.rukaze ramdenime punqtisurTierTmdebareobis aRwera. V. 10realuri viTarebis aRwerisasSesabamisi tolobis, utolobisan gantolebis Sedgenis unarisgamomuSaveba V. 6.erTi cvladis Semcveli asoiTigamosaxulebis mniSvnelobis povna V.5xarisxis gamoyeneba poziciurisistemiT ricxvebis Cawerisas,ricxvebis Caweris sxva wesebi da maTiSedareba poziciurTan V. 1mocemuli erTniSna da orniSnaricxvebis jeradebisa da gamyofebispovna; kenti, luwi, martivi, Sedgeniliricxvebis gansxvavebulobis aRwera;2-ze da 5-ze gayofadobis dasabuTeba.V. 1savaraudosaswavlodro9 sT18 sT6 sT7 sT3 sT6 sT16

wrewiri, wre da maTinawilebiwiladi, rogorc mTelisnawili.wiladi, rogorcerTobliobis nawili.wiladi da naturaluriricxvebis gayofawrewirTan dakavSirebuli terminebiskoreqtulad gamoyeneba (centri,diametri, radiusi) wrewiris dayofatol rkalebad, wris dayofa tolseqtorebad V.7wiladebis wakiTxva da Cawera, wiladiscnebis sxvadasxvagvari interpretaciada maT Soris kavSirebi V. 24 sT9 sTwiladebis toloba, wiladisgamartiveba.Sereuli ricxvebi.wiladebis Sedareba.wiladebis jamis Sefaseba.wiladebis Sekreba.wiladebis gamravlebanaturalur ricxvze.wiladebis gamokleba.Sereuli ricxvebis Sekreba.Sereuli ricxvebisgamokleba.droisa dasigrZis erTeulebi.masis erTeulebi.amocanebis amoxsna (naSTisgamoyeneba).kuTxe. kuTxeebis Sedareba.samkuTxedebis Sedareba.mravalkuTxedis kuTxeebida gverdebi.wiladebis Sedarebisas sxvadasxvaxerxis gamoyeneba V. 2wiladebis cvlilebis aRweramricxvelis an mniSvnelis cvlilebisasV. 3erTnairi mniSvnelis mqonewiladebze ariTmetikuli moqmedebisdemonstrireba, moqmedebaTa Sedegisinterpretacia modelis gamoyenebiT.moqmedebaTa Tvisebebis gamoyenebaSereul ricxvebze moqmedebaTaSesrulebisas. V. 3sigrZis erTeulebis erTmaneTTandakavSireba,drois 12 da 24 saaTiani formatebisgamoyeneba; ariTmetikul moqmedebaTagamoyenebiT drosa da droisintervalis gansazRvra.naSTiT gayofis gamoyeneba zomiserTeulebs Soris kavSirebisgamoyenebisas V. 4kuTxeeebis Sedareba da klasifikacia(wrewiris, wris gamoyenebiT) V. 7.samkuTxedebis klasifikaciis unari,brtyeli figuris gverdebis Semcveliwrfeebis urTierTganlagebis aRwera9 sT18 sT10 sT9 sT17

farTobi. farTobisSefasebamarTkuTxa paralelepipediswaxnagebi da wiboebi.viyenebT monacemTaSegrovebis sxvadasxva xerxswarmovadgenT monacemebscxrilis an diagramis saxiT.vmsjelobT monacemTaSesaxeb.vxsniT amocanebs cxrilisan diagramis gamoyenebiTfiguraTa urTierTSeTavsebiTfarTobebis Sedareba, Sefaseba,figuris farTobis Tvisebebisgamoyeneba figuraTa kombinaciiTmiRebuli figuris farTobismosaZebnad V. 5sivrculi figurebis waxnagebisSemcveli sibrtyeebis dawiboebis Semcveli wrfeebisurTierTganlagebis aRwera.kubis da marTkuTxa paralelepipedisSlilebi V. 7. V. 8.monacemTa mopovebaSekiTxvebiT, dakvirvebiT, amokrebiTda arCevanis dasabuTeba. V. 11.monacemTa warmodgenis sxvadasxvaxerxis gamoyeneba (cxrilebi,piqtogramebi, svetebiani diagramebi)V. 12monacemTa interpretaciisada analizis sxvadasxva xerxisgamoyeneba (saZiebeli am SemajamebelikiTxvebis dasma, Sedareba; msgavsebis,gansxvavebis aRmoCena. monacemTasafuZvelze varaudebis gamoTqma).V. 133 sT4 sT3 sT12 sTSeniSvna:• miTiTebul droSi naangariSevia Sesabamisi weriTi samuSaoebis Catareba-garCevac;• maswavlebels saTadarigod rCeba miaxloebiT 14 sT.18

saswavlo masalis wardgenis fazebi dagakveTilis dagegmvis ZiriTadi principebisaswavlo procesis organizaciis ZiriTadi forma gakveTilia. meTodikur literatura­Si cneba _ `gakveTili~ _ ZiriTadad, ase aRiwereba: gakveTili logikurad dasrulebuli,mTliani saswavlo-aRmzrdelobiTi procesis garkveuli SemosazRvruli monakveTia. masSirTul urTierTdamokidebulebaSia procesis yvela ZiriTadi elementi: Sinaarsi, mizani, sa­Sualebebi, meTodebi, organizacia. yovel gakveTilze gansazRvruli saganmanaTleblo daaRmzrdelobiTi amocanebi wydeba; rac ZiriTadad konkretuli saswavlo masalis ganxilvisprocesSi mimdinareobs. dawyebiT klasebSi prioritetulia saswavlo-aRmzrdelobiTi funqcia.erovnuli saswavlo gegmis mixedviT swavlebis am safexurze unda Seiqmnas safuZvlebisabazo safexurze efeqtiani swavlisTvis. V-VI klasebi am mxriv swavlebis mniSvnelovanietapia _ izrdeba sagnobrivi codnis SeZenis, arsebuli codnis ganzogadebis intensivoba.yuradReba eqceva im unar-Cvevebis, damokidebulebebis ganviTarebas, romlebic moswavlesmTeli cxovrebis ganmavlobaSi dasWirdeba, maT Sorisaa fizikuri da sulieri Tvisebebis,midrekilebebis gamovlena da ganviTareba.maTematikis gakveTilisadmi wayenebuli mTavari moTxovnaa ZiriTadi didaqtikuri amocanisarseboba _ im Temis Seswavlis miznis arseboba, romlis gadawyvetis procesi mocemulgakveTilze mimdinareobs. yoveli gakveTilis win kargad unda gaiazroT misi Sinaarsi da mizani.amaSi maswavleblis es wignic dagexmarebaT _ mowodebulia miznebisa da Sedegebis ruka,yoveli Temis axsnas win uZRvis rukaSi warmodgenili miTiTebebis sruli analizi. miznis Sesabamisadunda iyos gaazrebuli da dagegmili saswavlo masalis wardgenis fazebi.am miznis gaazreba moswavleebisTvis uaRresad mniSvnelovania. moswavleebTan saubariunda daviwyoT ara imiT, Tu ras vaswavliT, aramed iseTi situaciis SeqmniT, roca moswavlemotivirebulia da poulobs pasuxebs kiTxvaze _ `risTvisaa saWiro~. motivacia SesaZlebeliaaqtualuri praqtikuli amocanis dayenebiTa da misi amoxsnis xerxebis ZiebiT daviwyoT,an maTematikis Sinagani kanonzomierebebis, siZneleebis gadalaxvisas warmatebis miRweviTmiRebuli STabeWdilebebis gaazrebiT. `did~ ricxvebze gadasvla orive aspeqtis warmodgeniTmimdinareobs, TandaTan xdeba im codnis gadmotana-gavrceleba didi ricxvebze, racukve icoda moswavlem milionamde farglebSi. gansakuTrebiT mniSvnelovania amocanis dasmada moswavleebTan erTad misi amoxsnis Ziebis procesis ganxorcieleba. axali erovnuli saswavlogegmis mixedviT saganmanaTleblo procesis centrSi dgas moswavle da misi Sesabamisiinteresebi; swavla niSnavs informaciis dagrovebas, unar-Cvevebisa da damokidebulebebisganviTarebas. swavlebaSi moiazreba ara erTi konkretuli gzis gavla, aramed maswavleblisada moswavlis mier erToblivad SerCeuli optimaluri variantis Zieba. amasTanave, es procesiSeiZleba sxvadasxva formiT mimdinareobdes, es formebia _ muSaoba jgufebad (SesaZlebeliaor-oradac), mTeli klasis monawileobiT, individualurad da a. S. maswavlebelma swavlebaise unda warmarTos, rom Temis Seswavlis procesis ZiriTadi Semoqmedi Tavad moswavle aRmoCndes.maswavlebeli warmmarTvelis, `diriJoris~ funqcias unda asrulebdes; zogierTimecnieri maswavleblis am rols mebaRis rols adarebs _ yvavilebi mebaRis xelSewyobiTizrdeba da alamazebs garemos.maswavleblis sarekomendacio wigni dagexmarebaT warmatebiT daZlioT siZneleebi, gansakuTrebiTim davalebebis ganxilvisas, romlebic azrovnebis zeda sami donis (analizi, sin­19

Tezi, Sefaseba) amoqmedebasTan aris dakavSirebuli. nimuSis mixedviT davalebebis Sesrulebis(gamoyenebis done) garda gvaqvs davalebebi, romlebic moswavlis SemoqmedebiTi potencialiszrdasa da amoqmedebas iwvevs (analizisa da sinTezis done) da Sefasebisken (Sefasebis done)aris mowodebuli. azrovnebis zeda doneebis asamoqmedeblad mowodebulia specialuri meTodikebi;magaliTad, dasazustebeli SekiTxvebis CamonaTvali, romelTa saSualebiT xdeba `mogzauroba~qveda doneebidan zeda doneebisken da Semdgom _ ukusvliT _ zeda doneebidanqveda doneebisken.maswavleblis wignis da moswavlis wignis struqturac dagexmarebaT TiToeul etapze(sawyisi, damamTavrebeli) Sesasrulebeli savarjiSoebis SerCevaSi. yovel Temas ukavSirdebasavarjiSoebi, romlebic sxvadasxva inteleqtualuri unaris (klasificirebis, Sedarebis,gamoyenebis, TviTSefasebis Catarebis, amocanis amoxsnis Ziebis unaris) ganviTarebas emsaxureba.gakveTilze Seswavlili faqtebi TavisTavad aris mniSvnelovani, Tumca, kidev ufro mniSvnelovaniais, rom maTi Seswavlis dros ganxorcielebuli procesi Tavis kvals tovebs _mimdinareobs sxvadasxva inteleqtualuri unaris ganviTareba.moswavlisa da maswavleblis wignebSi warmodgenili masaliT, swavlisadmi SemoqmedebiTimidgomiT Tqven uTuod SeZlebT sxvadasxva akademiuri donisa da inteleqtualuriSesaZleblobebis moswavleebTan nayofieri saswavlo procesis warmarTvas. moswavleebTandeferencirebuli midgomiT (rac maTi individualuri Tvisebebis gaTvaliswinebasacgulisxmobs), mizanmimarTuli intensiuri ZalisxmeviT uTuod SeZlebTa droebiT CamorCenilimoswavleebis CarTvas saswavlo procesSi.axla CamovayaliboT ramdenime sasargeblo rekomendacia.• maswavlebeli cdilobs, rom yoveli axali SemecnebiTi amocana TviT moswavlem Camoayalibos.• maswavleblis xelmZRvanelobiT da moswavleTa ZalisxmeviT _ dakvirvebis, cdis, konkretuliSemTxvevebis analizis Sedegad iqmneba warmodgena arsebul kanonzomierebaze.• maswavleblis xelmZRvanelobiT mimdinareobs amocanis gadawyvetis gzis Zieba; amocanis,amoxsnis gegmis Sedgena; rasac xSirad mosdevs moswavleebis mier am gegmis realizacia.axla SevexoT saswavlo procesisTvis mzadebis zogad sqemas.• maswavleblis muSaoba saswavlo wlis win.• gakveTilebis sistemis gaazreba _ erTi raime Temis gameorebas, ganzogadebas, axlisZiebas gakveTilebis erToblioba eZRvneba.• konkretuli gakveTilisTvis momzadeba.momzadebisTvis maswavlebelma unda gamoiyenos saxelmZRvanelo, maswavleblis wigni da,saWiroebis SemTxvevaSi, iq miTiTebuli literatura. cxadia, pedagogi, kargad unda icnobdeserovnul saswavlo gegmas. am gegmis mixedviT Sedgenil Sinaarsisa da miznebis rukas,SemoTavazebul sanimuSo gakveTilebis scenarebs. sarekomendacio wignis Sesavali maswavlebelsacnobs saxelmZRvanelos agebis ZiriTad principebs da meTodikur orientirebs. winaswarki man aqtiuri monawileoba unda miiRos Tavisi skolis saswavlo gegmis SemuSavebaSi, raTadaakonkretos, daazustos, Seavsos erovnuli gegmiT gaTvaliswinebuli saganmanaTleblo daaRmzrdelobiTi programebi.amrigad, pedagogebi SemoqmedebiTad unda miudgnen Cvens rekomendaciebs, sakuTari gamocdilebisada klasis moswavleTa Taviseburebebis gaTvaliswinebiT, unda daazuston,SeiZleba Secvalon kidec zogierTi Tema da am Temis gadacema. kidev erTxel SegaxsenebTerovnul saswavlo gegmaSi mocemul miTiTebas: `saxelmZRvanelo aris erT-erTi saSualebada ara erTaderTi, romelic emsaxureba erovnuli saswavlo gegmis sagnobriv programaSi mocemuliSedegebis miRwevas... `saxelmZRvanelo ar aris TavisTavad programa _ Sesabamisad,maswavlebeli ar unda iyos saxelmZRvaneloSi mocemuli `masalis gavlaze~ orientirebuli,aramed aqcenti unda gaakeTos saxelmZRvaneloSi mocemuli teqstebis, savarjiSoebisa Tu20

ilustraciebis gamoyenebaze sagnobrivi standartebis Sedegebis misaRwevad. sworia, rocamaswavlebeli argebs saxelmZRvanelos moswavleebis saWirobebs da ara piriqiT, rodesacmaswavlebeli cdilobs moergos saxelmZRvanelos~, maswavleblis sarekomendacio wignSi CvenSemogTavazebT konretuli Temis gadacemis, savarjiSoebis gavlis Tanamimdevrobis gansxvavebulvariantebsac.gakveTilis dagegmvisas gasaTvaliswinebelia Casatarebeli procesis tipi:• gameoreba;• axali masalis gacnoba;• masalis ganmtkiceba;• codnis Semowmeba;• sxva tipis gakveTilebi (gakveTili bunebaSi, gakveTili-proeqti, ...)zogierTi tipis procesi (gansakuTrebiT V klasSi), rogorc wesi, yovel gakveTilze mimdinareobs(gameoreba, axali masalis gacnoba, ganmtkiceba, Semowmeba) _ SesaZlebelia, sxvadasxvamoculobiT.gakveTilze mTavaria vaswavloT da aRvzardoT: swavleba ar niSnavs mxolod codnis gadacemas,swavla codnis SemoqmedebiTad dauflebas unda niSnavdes, codnis gamoyenebis unarisganviTarebas unda gulisxmobdes.saswavlo masalis dasabuTebuli SerCeva iTvaliswinebs Semdeg moTxovnebs:• saswavlo masalis Sesabamisoba Temis mizanTan;• gakveTilze Sesasrulebeli samuSos moculobis swori gansazRvra;• optimaluri Tanafardoba konkretulsa da zogads Soris;• Teoriasa da praqtikas Soris aucilebeli urTierTkavSiris ganxorcieleba.maswavlebelma, rogorc wesi, Tavidan bolomde detalurad unda gaiazros gakveTili,winaswar, drois mixedviT unda iyos ganawilebuli mTeli samuSao, Tumca, rogorc yoveliSemoqmedebiTi procesi, gakveTilic Seitans, albaT Tqven gegmebSi cvlilebebs.kiTxvebze pasuxebis gacemis sistema ar unda iyos erTferovani _ mxolod warmatebulmoswavleebTan mimarTebiT ar unda SemovifargloT. moswavlis raime mosazrebas myisve nuupasuxebT, swor pasuxsac ki maSinve nu daeTanxmebiT _ gaakeTeT pauza, iqneb garkveulieWvic ki gamoTqvaT wamoyenebuli mosazrebis sisworis mimarT. amiT miaRwevT imas, rom moswavleebidaubrundebian dasmuli kiTxvis analizs da male WeSmariti daskvna _ swori pasuxi_ klasis dominantur mosazrebad gadaiqceva. klasi, erToblivi ZalisxmeviT, `gaiZulebT~daeTanxmoT moswavlis pozicias. es axarebs, amxnevebs da aerTianebs moswavleebs. es maTierToblivi azris gamarjvebaa, Tqveni mizanic xom es aris _ moswavle Semoqmed, codniTaRWurvil, iniciativian, xalisian axalgazrdad CamoayaliboT. maTematika mZlavri emociurimuxtis matarebelia da misi amoqmedeba Tqveni ZalixmeviT miiRweva.axali masalis ganmtkicebis procesi e. w. `testebis~ `amoxsniT~ SeiZleba daviwyoT. maTiSesruleba, ZiriTadad, swori pasuxis SerCeviT unda Semoifarglos _ zogjer SeiZleba komentarebisgakeTebac gaxdes saWiro, moswavle TandaTan eCveva am tipis davalebebis swrafadSesrulebas.swavlebis erT-erTi saintereso da mniSvnelovani formaa jgufuri muSaoba. es muSaobaSeiZleba saswavlo procesis erT-erTi Semadgeneli nawili iyos _ daukavSirdes axalimasalis gaazrebas, praqtikul saqmianobas, an Semajamebeli daskvnis gamotanas; mas SeiZlebamTeli gakveTilic daeTmos, jgufuri muSaobis warmarTvaSi saxelmZRvaneloSi warmodgenilimasala, an Sesabamisi amocanebis sistema dagexmarebaT. jgufur muSaobas SeiZleba Sejibrebissaxec mivceT, saxelmZRvaneloSi warmodgenili TamaSebic SeiZleba jgufuri muSaobis formiTwarvmarToT.Teoriuli masalis gadmocemis Cveneuli meTodika saSualebas gaZlevT airCioT jgufurimuSaobis moxerxebuli forma; magaliTad, paeqroba jgufebs Soris, romlebsac kapitnebi xelmZRvaneloben.jgufebs SeiZleba erTnairi davalebebi mivceT. maSin Tavad jgufebi aqti­21

urad erTvebian miRebuli Sedegebis saerTo ganxilvaSi _ ukeT gauweven oponirebas sxvaTamosazrebebs. SeiZleba jgufebSi samuSaobi ganawildes da moswavleTa erToblivi muSaobaraime erTi amocanis gadawyvetas gulisxmobdes; jgufis wevrebma unda gaiazron, rom maTiwarmateba erTmaneTzea damokidebuli. es zrdis wevrebis pasuxismgeblobas. TanamSromlobisxarisxis asamaRleblad SeiZleba cadoT saerTo Sefaseba _ jgufebi asruleben saerTodavalebas da maTi wevrebi iReben erTnair Sefasebas. Tumca es wesi zogierTi moswavlisTvisar unda iqces daumsaxurebeli maRali Sefasebebis miRebis saSualebad.sasurvelia, rom jgufuri muSaobisTvis gankuTvnili amocana sxvadasxva xerxiT ixsnebodes.igi saintereso msjelobis saSualebas iZleva.jgufuri muSaobis daskvniT etapze _ Sedegebis sajaro ganxilvisas Zalze mniSvnelovaniapedagogis roli moswavleTa mier polemikis koreqtuli formebiT warmarTvaSi, sxvaTamosazrebis mosmenis unaris damkvidrebaSi, warmatebisa Tu warumateblobis gamoxatvis misaRebiformebis CamoyalibebaSi.22

Sefasebis zogadi principebiaxal erovnul saswavlo gegmaSi saTanado adgili ukavia Sefasebis principebs. am gegmismixedviT TviT skolam unda daadginos zogadi Sefasebis principebi, rac yvela maswavleblisTvissaxelmZRvanelo principebad unda iqces. kerZod, unda dadgindes is minimaluribarieri, romlis gadalaxva aucilebelia klasidan klasSi gadasasvlelad.erovnuli saswavlo gegma gTavazobT im aucilebel midgomebs, romlebsac Sefasebissistema unda iTvaliswinebdes:`akademiuri moswrebis Sefaseba unda iyos xSiri da mravlmxrivi. unda Sefasdes ara martoinformaciis floba, aramed SeZenili unar-Cvevebi, azrovnebis demonstrirebis formebida sxva. saswavlo procesis warmatebiT warmarTvisTvis ar aris sakmarisi moswavle mxolodsakontroloebis da testebis Sedegebis safuZvelze Sefasdes. maswavlebeli unda afasebdesmoswavles dakvirvebis Sedegad dagrovili monacemebis, namuSevrebis krebulebis, prezentaciebis,moswavleTa TviTSefasebis, jgufuri muSaobis Tu sxva tipis aqtivobebis mixedviT.maswavlebelma Sefasebisas unda gaiTvaliswinos moswavlis saganmanaTleblo procesSiCarTulobis xarisxi, saxlSi micemuli davalebis Sesrulebis xarisxi, gakveTilze aqtiuroba,SemoqmedebiToba da sxva. aucilebelia, moswavlem winaswar icodes, Tu ra kriteriumebiTfasdeba misi aqtivoba... moswavle arasaTanado saqcielisTvis (romelic pirdapir ar arisdakavSirebuli saswavlo procesTan) sagnis niSniT ar unda daisajos...~ Sefasebas ara martoar unda dahyvebodes angariSsworebis elferi, aramed Sefaseba unda aRiqmebodes samomavlo,pozitiuri cvlilebebisken mimarTuli qmedebad.erovnuli saswavlo gegmiT gansazRvrulia, rom niSnebis daweris sistemis (trimestruliniSani, wliuri niSani da a. S.) moqmedeba V klasidan iwyeba. amasTanave, oficialuri statusiara aqvs mimdinare yoveldRiur niSnebs, amitom maswavlebeli sakuTar wignakebSi agrovebsSefasebis Sesabamis monacemebs yovel moswavleze. iyenebs Sefasebis or ZiriTad tips _ ganmsazRvrelsda ganmaviTarebels. am ukanasknelis gamoyenebisas niSnis daweras Tan erTvis komentarebi,romliTac vuxsniT moswavles Zlier da sust mxareebs, vexmarebiT mas SedegebisgaumjobesebaSi.moswavlis niSani unda ganisazRvros (magaliTad, trimestruli niSani) mis mier sagnisSeswavlis sxvadasxva komponentisgan _ gakveTilze CarTuloba, saSinao davaleba, saklasodavalebebi, sakontrolo wera, proeqti da sxva.am komponentebisgan skola kaTedrebis meSveobiT SearCevs nawils (sams mainc) romelTacwonebi mieniWeba. es wonebi SeiZleba sxvadasxva iyos trimestrebis mixedviTac. trimestrisSefaseba (umaRlesi qula 10) SeiZleba sxvadasxva komponentis Sefasebisgan davadginoT.magaliTad, wonebi SeiZleba aseTi iyos: prezentacia _ 25%, gakveTilze aqtiuroba _ 30%,sakontrolo wera _ 25 %, saSinao davaleba _ 20%. Tu TiToeuli komponentis saSualo Sefasebebia6, 9, 10, 8 maSin Sefaseba SeiZleba ase gamovTvaloT:0,25⋅6+0,3⋅9+0,25⋅10+0,2⋅8=8,3damrgvalebis Semdeg gvaqvs Sefaseba 8.Sefasebis sxva, Zalze mniSvnelovani detalebis gacnobis mizniT pedagogebs vurCevT xSiradmiubrundnen xolme erovnuli saswavlo gegmis saTanado miTiTebebs, radgan es komponentisaswavlo procesSi Zalze frTxil, obieqtur, gamWvirvale muSaobas saWiroebs. amave gegmismoTxovniT winaswar unda gairkves: `saSemodgomo~ gamocdasTan dakavSorebuli sakiTxebi, sapatiomizezebiT xSirad gamcden moswavleebTan damokidebulebis sakiTxebi da sxva.23

I Tavigameoreba. milionze meti ricxvebisaxelmZRvanelos pirvel Tavs gamorCeuli datvirTva aqvs _ xdeba moswavleTa CarTvasaswavlo ritmSi, swavlis axal _ ufro maRal safexurze asvla, mTeli saswavlo wlisTvis`tonis micema~. xangrZlivi sazafxuli ardadegebis Semdeg uaRresad aqtualuria adre SeswavlilisakiTxebis xelaxali gaazreba da SeZenili unarebis gamyareba-ganviTareba. aq CamoTvlilyvela mizans pasuxobs Cveni saxelmZRvanelos pirveli Tavi _ warmodgenili masaliTada misi gadacemis meTodikiT. adre ganxilul sakiTxebs moswavleebs mTeli wlis ganmavloba­Sic SevaxsenebT xolme _ axali sakiTxebis Seswavlis procesSi.naturalur ricxvTa Tvisebebisa da maTze moqmedebebis safuZvliani codna, ricxvTa poziciursistemaze naTeli warmodgenis Camoyalibeba maTematikis Seswavlis procesis erT-er­Ti qvakuTxedia. am sakiTxebis codnis gaRrmavebis da gafarToebis mizniT davixmarT samotivacioamocanebs, TvalsaCino warmodgenebs.pirvel sayuradRebo nabijebs gadavdgamT algebris mimarTulebiTac. SevadgenT asoiTgamosaxulebebs, ganvixilavT maTi gamartivebis sakiTxebs. amave TavSi moswavleebi pirvel(arasrul) warmodgenas Seiqmnian sakoordinato sibrtyeze, roca gaecnobian sakoordinatobades da amoxsnian saTanado amocanebs. gavimeorebT naturaluri ricxvis naturalurmaCveneblianxarisxs da misi gamoyenebis magaliTebs.1. gavixsenoT naturaluri ricxvebis wakiTxvisa da Caweris wesebimizani. naturaluri ricxvebis Cawerasa da wakiTxvaSi moswavleTa gawafva; ricxvispoziciuri sistemiT warmodgenis arsis ukeT gaazreba _ moswavle amoicnobs yoveli poziciis(Tanrigis) mniSvnelobas, wers ricxvebs gaSlili formiT, iyenebs SeZenil codnas amocanebisamoxsnisas.es saxelmZRvanelo, rogorc Cveni sxva saxelmZRvaneloebi, ganvlili masalis gameorebiTiwyeba. kerZod, moswavle kidev ufro unda gaiwafos naturaluri ricxvebis Cawerasa dawakiTxvaSi (savarjiSoebi 1 - 6 ), man unda gaixsenos Tanrigebis klasebad gaerTianebisprincipi, klasebSi Semavali Tanrigebis saxelwodeba, ricxvebis Cawera gaSlili formiT.magaliTad,648 850=600 000+40 000+8 000+800+50+0.saxelmZRvanelo exmareba maswavlebels gakveTilis Catarebis gegmiT gaTvaliswinebuliyvela komponenti gaiTvaliswinos; magaliTad moswavleTa motivacia iTvaliswinebs paragrafisSesavali, SeiZleba sxva praqtikuli magaliTebis ganxilvac.ricxvis aseTi saxiT _ gaSlili formiT warmodgena abakis gamoyenebiT ufro TvalsaCinoxdeba. saskolo inventarSi abakis arsebobis SemTxvevaSi, gakveTilze moswavleTanawilma SeiZleba dafaze daweros ricxvebi, sxvebma ki es ricxvebi abakze warmoadginos dapiriqiT, abakze warmodgenili ricxvebi moswavleebma daweron dafaze da Semdeg daasaxelon.teqstSi suraTze udidesi eqvsniSna ricxvia warmodgenili: 999 999. suraTze gamosaxul abakzeSesalebelia Tu ara am ricxvze 1-iT meti ricxvis _ milionis warmodgena? rogori abakisdamzadebas SemogvTavazeben moswavleebi?24

paragrafs axlavs Ria da daxurulboloebiani savarjiSoebi klasSi amosaxsnelad da sa­Sinao davalebad.amocanebis amoxsnaSi CarTeT yvela moswavle _ TiToeuli swori pasuxi sxvaTa mieracunda iyos gaziarebuli, xolo araswori pasuxi Tavad moswavleebma unda SeniSnon da imsjelonmis gamosworebaze.11 savarjiSos Sesrulebisas moswavleebi ixseneben 1-ianiTa da 0-ebiT Caweril ricxvebze(10-ze, 100-ze da a. S.) gayofisa da gamravlebis wesebs. 13 , 14 , 15 amocanebi aris16 amocanis amosaxsnelad Semzadeba.13 amocanaSi erTnairi cifrebis gadanacvleba xdeba, amitom ricxvi ar icvleba.14 amocanaSi aTaseulis cifris 3-iT gazrda gamoiwvevs ricxvis 3 aTasiT (3000-iT) gazrdas.15 amocanaSi aTaseulis cifri icvleba 1-iT naklebi cifriT, anu ricxvi Semcirdeba 1000-iT, xolo erTeulis cifri icvleba 1-iT meti cifriT, anu ricxvi izrdeba 1-iT. pasuxi:ricxvi Semcirdeba 999-iT.16 amocanis amoxsnisas, isev rogorc 13 amocanaSi, moswavleebi daaskvnian, rom aTiaTaseulisa da erTeulis cifrebi tolia. amasTanave, isini 0-isgan gansxvavebulia (ricxviar daiwyeba 0-iT). amrigad, aTaseulis Tanrigis cifrze 8-iT naklebi SeiZleba iyos mxo lodcifri 1. moswavleebi am pirobebis gaTvaliswinebiT TanamimdevrobiT Seavseben saZiebelixuTniSna ricxvis Tanrigebs: 19••1.pirobiT, aTeulis cifri aris 1+3=4, aseulis cifri aris 4+1=5. pasuxi: 19541.klasSi SeiZleba amoixsnas amocanebi rubrikidan `moisazre~. am samuSaos SeiZleba paeqrobissaxec mieces: moswavleebi daajgufeT ramdenime jgufad (TiToSi 4-5 moswavle). 15-wu­Tiani muSaobis Semdeg moawyveT jgufebis naSromTa mokle prezentacia; Semdeg SeafaseT moswavleebida Sedegebi gaanalizeT.gTavazobT amocanebis amoxsnis warumatebeli mcdelobis SemTxvevaSi damxmare kiTxvebs:1 a) • ra feris burTulebi `gvawyobs~? (lurji am mwvane)• ra feris burTulebi `ar gvawyobs~? (wiTeli)• ras udris im burTulebis odenoba, romlebic `ar gvawyobs?~ (9) e. i. 9 burTis amoRebisSemTxvevaSi SeiZleba cxrave wiTeli aRmoCndes. am SemTxvevaSi meaTe burTula aucilebladiqneba mwvane an lurji. pasuxi: 10.b) • ramdenia im burTulebis odenoba, romlebic `ar gvawyobs~? (9+4=13).e. i. SeiZleba 13 burTis amoRebis SemTxvevaSi cametive aRmoCndes aramwvane. am SeTxvevaSime-14 burTi iqneba aucileblad mwvane. pasuxi: 14.g) yvelaze `uares~ SemTxvevaSi 13 amoRebuli burTulidan SeiZleba arc erTi ar iyosmwva ne, maSin me-14 burTi aucileblad iqneba mwvane da samive feris burTulebs SevagrovebT.2 • wignis ramdeni gverdia gadanomrili erTniSna ricxvebiT da maT dasawerad ramdenicifria saWiro? (9 gverdi, 9 cifri).• ramdeni gverdia gadanomrili orniSna ricxvebiT da maT dasawerad ramdeni cifriasaWiro? (90 gverdi, 180 cifri).• ramdeni gverdia gadanomrili samniSna ricxvebiT da maT dasawerad ramdeni gverdiasaWiro?(148–99=49 gverdi, 49⋅3=147 cifri)sul gamoviyenebT 9+180+147=336 cifrs.25

3 wina amocanis amoxsna daexmareba moswavleebs am amocanis gadawyvetaSi. damxmarekiTxvebi:• ramdeni gverdia am wignSi erTniSna ricxvebiT gadanomrili da ramdeni cifria amisTvissaWiro? (9 gverdi. 9 cifri).• sul didi ramdeni cifri SeiZleba gamoviyenoT orniSna ricxvebiT gverdebis gadasanomrad?(90⋅2=180).• sul didi ramdeni cifri SeiZleba gamoviyenoTsamniSna ricxvebiT gverdebis gadasanomrad?(900⋅3=2700).• aris Tu ara am wignSi oTxniSna ricxvebiT2004

sanimuSo gakveTili #1aqtivoba: jgufuri muSaoba: naturaluri ricxvebis Caweris aTobiTi sistemis Sesaxeb.aqtivobis mizani. moswavle ukeT Cawvdes ricxvis Caweris poziciuri sistemis arss, misupiratesobas arapoziciur sistemasTan SedarebiT; Cawvdes ricxvis Caweris aTobiTi sistemisarss; daukavSiros zogierTi sididis sazomi erTeulebi Tvlis aTobiT sistemas. moswavlemSeZlos miRebuli codnisa da saTanado unarebis aqtiurad gamoyeneba praqtikuli saxisamocanebis gadawyvetaSi.aucilebeli wina codna.• didi ricxvebis wakiTxva da Cawera.• ricxvis Cawera gaSlili formiT• ricxvis Cawerisas Tanrigebis klasebad dayofa• kavSiri sigrZis sazom erTeulebs Soris• drois sazomi erTeulebi, kavSiri maT Soris.aqtivobis aRweramoswavleebs dafaze davuweroT ramdenime ricxvi, romelTa CanawerSi erTi da igivecifrebia gamoyenebuli. magaliTad, 111, 2222.moswavleebma TiToeuli ricxvisTvis unda daasaxelon da Caweron sxvadasxva poziciaSidawerili erTi da imave cifris mniSvnelobebi, Semdeg ki _ Seadaron es mniSvnelobebi.magaliTad,2 2 2 2 anu 2 2 2 2 anu 2 2 2 22000 + 200 + 20 + 2 200⋅10 + 20⋅10 + 2⋅10 + 2 2⋅10⋅10⋅10 + 2⋅10⋅10 + 2⋅10 + 2⋅1ramdenjer metia yoveli momdevno (ufro maRali) Tanrigis cifris mniSvneloba? (10-jer).ramdennair cifrs viyenebT xolme ricxvebis Cawerisas? (10-s)swored amitom, ricxvebis am wesiT Cawerisas vRebulobT naturaluri ricxvis aTobiTCanawers da iqmneba naturaluri ricxvTa aTobiTi sistema. radgan am wesiT ricxvis Cawerisasyoveli cifris mniSvneloba CanawerSi cifris adgilzea (poziciazea) damokidebuli, amitomricxvTa aTobiT sistemas uwodes poziciuri. Tu maRalia klasis akademiuri done, SeiZleba ametapzeve arsebiTad gaaRrmavoT moswavleTa warmodgenebi poziciur sistemebze; magaliTad,erToblivad aRweroT — Zalze mokled (aTobiTis msgavsad) xuTobiTi sistema da dasvaTamocana:ra ricxvs warmoadgens 2222 Canaweri xuTobiT sistemaSi?2 2 2 22⋅5⋅5⋅5 + 2⋅5⋅5 + 2⋅5 + 2xuTobiT sistemaSi 2222 warmoadgens 312-s (2⋅5⋅5⋅5+2⋅5⋅5+2⋅5+2=312) aTobiT sistemaSi. vfiqrobT,am etapze es magaliTic ikmarebs.amis Semdeg moswavleebs davyofT jgufebad da maT vTavazobT saxelmZRvaneloSi oTxicxrilis Sevsebas (I, II, III da IV). isini akvirdebian, aRweren _ masis da sigrZis sazomierTeulebis `gamsxvilebisas~ naklebi erTeuli mravldeba 10-ze an 10-is ufro maRal xarisxze.magaliTad, 1 dm=10⋅1 sm, 1 m =10⋅1 dm, 1 km=10⋅10⋅10⋅1 m.drois sazomi erTeulebis gamsxvilebisas es kanonzomiereba irRveva _ kavSirebi aTobiTsistemas ar efuZneba.amis Semdeg jgufebs vTavazobT baraTebze daweril winadadebebs:27

2. milionze meti ricxvebimizani. milionze meti ricxvebis Cawerasa da wakiTxvaSi gawafva, am ricxvebis dakavSirebastatistikur monacemebTan. ricxvTa poziciuri sistemis, kerZod, aTobiT sistemis arsisukeT gaazreba, misi Sedareba arapoziciur sistemebTan. romaul sistemaze pirveli, arasruliwarmodgenis Seqmna, gamoyeneba. axali ricxviTi saxeliT mocemuli ricxvis rigis povna(anu ramdenniSnaa ricxvi, ramdeni nulia mis CanawerSi).teqstSi didi ricxvebi dakavSirebulia statistikur monacemebTan, kerZod, sxvadasxvaqveynis mosaxleobis odenobasTan, kosmiur sxeulebs Soris manZilTan. moswavleebi ricxvebiswakiTxva-aRweraSi gawafvasTan erTad informacias Rebuloben didi qveynebis, TavisisamSoblos mosaxleobis odenobis Sesaxeb. amiT iqmneba SesaZlebloba gakeTdes mravalmxrivianalizi, romelsac calke ganxilvebi SeiZleba mieZRvnas.CineTSi mcxovrebTa ricxvi miaxloebiT miliard 300 milionia _ 1300 000 000. aq cifri3-is mniSvnelobaa samasi milioni. dedamiwaze amJamad miaxloebiT 6500 000 000 adamini cxovrobs,am ricxvis CanawerSi rva nulia, momavalSi moswavleebi SeZleben iangariSon, planetismosaxleobis ra nawils Seadgens CineTis mosaxleoba.maswavleblis momzadebis erT-erTi mniSvnelovani aspeqti gakveTilis gegmis Sedgenaa.gegmis pirveli punqti miznis Camoyalibeba da gaazrebaa. maswavleblis wigni exmareba mas ammiznis gansazRvraSi. yoveli gakveTilis garCevas viwyebT am miznis miTiTebiT. Semdegi etapebiwina pirobebis gaazrebaa, saWiro resursebis fiqsirebaa. gakveTilis gegmis TiToeuli etapisSesaxeb yoveli Temis ganxilvisas gveqneba saubari maswavleblis wignSi. moswavleTa aqtivobebi,rogorc ZiriTadi miznis daxasiaTebas, aseve maT motivacias ukavSirdeba, am mimarTulebiTmoswavlis saxelmZRvanelo dagexmarebaT _ TiTqmis yvela Temis damuSaveba garkveulipraqtikuli situaciebisa da am situaciebSi axali maTematikuri obieqtebis gamoyenebasukavSirdeba. moswavleTa aqti vo bebisa da codnis gamyarebasTan dakavSirebuli davalebebidawvrilebiTaa garCeuli maswav leblis wignSi. maswavleblis wigni exmareba maswavlebelssaSinao da saklaso davalebebSi gamoarCios amocanebi maTematikuri SejibrebebisTvis,amocanebi maTematikaSi maRali war matebiT gamorCeul, an SedarebiT sust moswavleebTanmuSaobisTvis; maswavleblis wignSi ki mo cemulia davalebebi damoukidebeli da Semajamebelisakontrolo muSaobisTvis.davubrundeT milionze meti ricxvebis ganxilvas.teqstSi moyvanilia yoveldRiur cxovrebaSi didi ricxvebis dasaxelebis magaliTebi. Sei­Zleba moswavleebma gaagrZelon am saxelebis gadataniTi mniSvnelobis magaliTebis moyvana.magaliTad,• gareT _ atlasi, SigniT _ aTasi (xalxuri);• aTasma azrma gamielva;• ena rom gasatexi iyos, aqamde aTas adgils gadatydeboda (ilia);• ybad agiRo, cils gwamebs, ambobs aTas rameebs (akaki).moswavleebma SeiZleba Tavisi sityvebiT axsnan gamonaTqvamebi:• aTasSi erTxel gamaxsendeba;• milioni azri mitrialebs;• es ambavi milionma adamianma gaigo.am magaliTebiT warmodegenilia damatebiTi sasargeblo resursebis gamoyenebisSesaZleblobebi swavlebis procesSi.3 da 5 savarjiSoebSi moswavleebs vTxovT ricxvis aTobiT Canawers. amaSi igulisxmebaricxvis Canaweri Tvlis aTobiT sistemaSi am terminis mniSvnelobaSi moswavleebi paragrafisbolos gaTvaliswinebuli jgufuri muSaobisas gaerkvevian. aqvea araaTobiTi sistemispirveli nimuSi (arasrulad warmodgenili).29

zogierTi didi ricxvis Canaweri da saxeli savarjiSoebSi gvxvdeba, magaliTad, 10 savarjiSoSiwarmodgenilia trilioni, kvadrilioni da kvintilioni. am savarjiSos Sesrulebisasmoswavleebi isargebleben saxelmZRvaneloSi warmodgenili cxriliT.gavamaxviloT yuradReba, rom 1 kvadrilionis CanawerSi 15 nulia; 10 kvadrilionis CanawerSi_ 16 nuli, es ricxvi 1 kvadrilionze 10-jer metia. 100 kvadrilioni 100-jer metia 1kvadrilionze, mis CanawerSi 17 nulia, es aris pasuxi 10 savarjiSos g) da d) kiTxvebze,danarCeni klasebis TanrigebisTvisac davsvaT analogiuri kiTxvebi, gaavarjiSeT moswavleebierTi da imave klasis Tanrigebis erTmaneTTan dakavSirebaSi. z) miliardis CanawerSi 9 nulia,1 000 miliardi miliardis 1 000-ze gamravlebiT miiReba _ miliards mivawerT 3 nuls,es ricxvi Seicavs 9+3=12 nuls.analogiurad, T) 1000 aTasi aris 1 000 gamravlebuli 1 000-ze, am ricxvis CanawerSi 6 nulia.i) trilionis CanawerSi 12 nulia, Tu mas 1000-ze gavamravlebT, miiReba ricxvi, romlisCanawerSi 12+3=15 nulia.k) 1 000 kvadrilionis CanawerSi 15+3=18 nulia.moisazre_ am rublikiT SemoTavazebuli amocanebi SeiZleba jgufuri mu­SobisTvisac gamoviyenoT, romelsac Sejibris saxes miscemT. Sejibris dawyebamde moTelvismizniT gTavazobT amocanas, romelzec moswavleebi zepirad, adgilidan gascemen pasuxs:saSinao davalebad maswavlebelma mogcaT savarjiSoebi #21-dan #34-is CaTvliT. ramdenisavarjiSos Sesruleba mogviwevT? SeiZleba moswavleTa nawilisgan miiRoT mcdari pasuxi:34–21=13. am SemTxvevaSi SeiZleba damatebiTi kiTxvebic daixmaroT: ramdeni amocanaa 1-dan 34-is CaTvliT da ramdeni ar mogcaT maswavlebelma davalebad? am kiTxvis Semdeg moswavleebiamocanis amosaxsnelad Seasruleben moqmedebas _ 34–20=14.moTelvis Semdeg 1 amocanis I punqtis sapasuxod umciresi da udidesi samniSna ricxvebisgaxsenebac ikmarebs: 100 da 999. samniSna ricxvebis odenobaa 999–99=900; amdeni samniSnaricxvis Casawerad saWiroa 900⋅3=2 700 cifri,2 miuxedavad saxelmZRvaneloSi arsebuli miTiTebisa, moswavleebma am amocanis amoxsnisramdenime gza SeiZleba SemogvTavazon. yoveli maTgani yuradRebiT unda ganvixiloT,wavaxalisoT moswavleebi.3 II striqoni miiReba I striqonis ricxvebis 3-ze gamravlebiT, IV striqoni _ I striqonisricxvebis 3-ze gamravlebiT da 1-is mimatebiT. III striqonisTvis aseTi martivi kanonzomierebaar Cans. amitom `zedmetad~ mas miviCnevT.am tipis amocanebis amoxsnis Zieba moswavleebs enTuziazms matebs. Sedegic Zalze sayuradReboa_ moswavleebi inviTareben kanonzomierebaTa, hipoTezebis aRmoCenis unars, kritikuldamokidebulebas sawyisi mosazrebebis mimarT (rogorc es IIIO striqonisTvis xdeba).sxvaTa mosazrebebis aRiarebis an oponirebis unars, iReben drois garkveul SualedSi maqsimaluriSedegis miRwevis gamocdilebas.4 milioni aTasze 1 000-jer metia, miliardi milionze 1 000-jer metia, trilioni miliardze1 000-jer metia. am kanonzomierebas arRvevs `aTi miliardi~.5 SesaZloa 4 amocanis ricxvebi gaimeoron: erTi, aTasi, milioni, miliardi, trilioni;an xuTi, 5 aTasi, 5 milioni, 5 miliardi, 5 trilioni.30

am amocanis amoxsnisas SeiZleba saklaso savarjiSoebidan 10 -Si warmodgenili cxriligamoiyenon. yuradReba gavamaxviloT im faqtze, rom yoveli momdevno klasis erTeuli (umciresiTanrigis erTeuli) misi wina klasis erTeulze 1 000-jer metia, iqneb mizezebzec imsjelon.yovel klass 3 Tanrigi qnis, anu yoveli klasis erTeulis CanawerSi 3-iT meti nuliawina klasis erTeulis CanawerTan SedarebiT, 3 nulis miwera ki 1000-ze gamravlebis Sedegia.jgufurinaturaluri ricxvebis Caweris aTobiTi sistemis Sesaxeb• naturaluri ricxvebis cifrebiT Canawers aTobiTi Canaweri vuwodeT. rasTan arisdakavSirebuli am terminis gamoyeneba?• ratom vuwodebT ricxvebis Caweris Cvens mier gamoyenebul wess poziciurs?jgufur muSaobas SeiZleba misceT paeqrobis saxe. moswavleTa warmatebuli prezentaciebiunda Sefasdes da waxalisdes. yvela jgufi erTsa da imave davalebas asrulebs. ganxilvacyvelas monawileobiT unda Catardes. gamorCeuli yuradReba dauTmeT moswavleTa naazrevisgadmocemis xarisxs, formas. sayuradReboa maTi mosazrebebi romauli sistemis Sesaxeb, misida aTobiT sistemis SedarebiTi analizi, sayuradReboa agreTve _ aRmoaCenen Tu ara moswavleebi,rom sigrZis erTeulTa ori erTobliobidan erT-erTSi `gamsxvileba~ imave principiTxdeba, rac aTobiT sistemaSi, meore erTobliobaSi _ ara.maswavlebels SeiZleba damatebiTi kiTxvebiT mouwios gamarjvebuli gundis gamovlena.moswavleebs TxoveT daasaxelon is xelsawyoebi, romelTa moqmedeba dakavSirebulia aTobiTiprincipis gamoyenebasTan (gazis eleqtromricxveli, avtomobilis mier gavlili manZilismricxveli da sxva). sasurvelia am xelsawyoze monacemTa Cvenebis mokle aRwera.saswavlo procesi gulisxmobs periodul monitorings. damoukidebeli wera warmoadgensmoswavleTa mier masalis aTvisebis xarisxisa da miRebuli codnis gamoyenebis unarisSemowmebis erT-erT saSualebas. Cven periodulad SemogTavazebT weriTi samuSaoebissavaraudo variantebs Sesabamisi Sefasebis sarekomendacio sqemiT. rekomendaciebi Sefasebisprincipebis, maT Soris zogadi principebis Sesaxeb, ixileT am saxelmZRvanelos 22-e gverdze.damoukidebeli wera #1Tema: • naturaluri ricxvebis wakiTxvisa da Caweris wesebi;• milionze meti ricxvebi.1. dawereT mocemul ricxvSi xazgasmuli cifris mniSvneloba: 587 641.1) 7641 2) 7 3) 7 000 4) 700 000.2. CawereT mocemuli ricxvi gaSlili formiT: 406080331.1) 40+60+80+331 2) 400000000+6000000+80000+300+30+13) 400000000+60000000+30000000+300000+1 4) 400000+60000+8000+300+30+1.3. warmoadgineT as xuTi miliard 7 milion asi aTas ocdaerTis aTobiTi Canaweri.1) 105700100021 2) 1050710021 3) 105007100021 4) 105710021.4. warmoadgineT ricxvis mokle sityvieri Canaweri: 260425310.1) 260 miliard 425 aTas 310 2) 2 miliard 60 milion 425 aTas 3103) 26 miliard 42 milion 5 aTas 310 4) 260 milion 425 aTas 310.31

5. Tu eqvsniSna ricxvSi wavSliT asi aTaseulis Tanrigis cifrs _ 3-s, maSin es ricxviSemcirdeba1) 300000-iT 2) 3-iT 3) 3-jer 4) 30000-iT.6. rvaniSna ricxvSi Secvales erTi cifri da es ricxvi 40 aTasiT gaizarda. Secvlilicifri1) aTeulis Tanrigisaa 2) aTaseulis Tanrigisaa3) aTi aTaseulis Tanrigisaa 4) asi aTaseulis Tanrigisaa.7. mocemulia ricxvi 9068347. misi romeli cifrebi unda wavSaloT, rom miviRoT:a) umciresi SesaZlebeli oTxniSna ricxvi;b) udidesi SesaZlebeli xuTniSna ricxvi?8. mocemulia cifrebi: 0, 1, 2, 4, 9. SeiZleba TuU ara am xuTive cifris gamoyenebiT CaweroTxuTniSna ricxvi, romlis aTaseulebis klasis cifrTa jami 2-iT aRemateba erTeulebis klasiscifrebis jams? dadebiTi pasuxis SemTxvevaSi warmoadgineT yvela SesaZlebeli ricxvi.Sefasebis sarekomendacio sqema.testuri tipis (daxurulboloiani) amocanebidan TiToeulSi swori pasuxis miTiTebaSefasdes 1 quliT, arasworis _ 0 quliT.me-7 amocana _ 2; 1 an 0 quliT _ a-punqti 1 quliT da b-punqti 1 quliT.me-8 amocanis amoxsnac 2; 1 an 0 quliT Sefasdes _ erTi mainc SesaZlebeli ricxviswarmodgenisTvis 1 qula, eqvsive SesaZlebeli ricxvis warmodgenisTvis kidev 1 qula.pasuxebi da miTiTebebi amocanebis amosaxsnelad.1 2 3 4 5 63 2 3 4 1 37. a) mocemuli ricxvi SvidniSnaa; amrigad, moswavlem unda waSalos sami cifri: 9; 0 da8: 9068347 , miiReba: 6347.b) unda waiSalos ori cifri: 0 da 6, anu 9068347 , miiReba: 98347.8. SeiZleba, es ricxvebia: 90421, 90412, 90241, 90214, 90142, 90124.damoukidebeli wera SeiZleba maswavlebelma CamorCenil moswavleebTan damatebiTimuSaobis Casatareblad gamoiyenos, maT SeTavazos gamotovebuli varianti, an im dros, rocasxvebi asruleben davalebas, maTTan damatebiTi mecadineoba Caataros.maswavleblis yuradRebis miRma arcerTi moswavle ar unda darCes.3. vadarebT da valagebT naturalur ricxvebsmizani. ricxvebis Sedarebaze moswavleTa warmodgenebis gaRrmaveba; raodenobrivi maCveneblismixedviT monacemebis elementaruli analizis unaris ganviTareba. kerZod, monacemTaerTobliobaSi monacemebis poziciis gansazRvra (magaliTad, qveynebis dalageba mosaxleobisodenobis zrdis an klebis mixedviT).paragrafSi moyvanil cxrilSi qveynebi ar aris mosaxleobis odenobis mixedviT dalagebuli.cxrili anbanis mixedviTaa Sedgenili. moswavles am gakveTilze mouwevs Seadaros cxril­Si moyvanili ricxvebi Sedarebis misTvis ukve cnobili wesis mixedviT. Sedarebas viwyebT32

Tanrigebis odenobaTa SedarebiT. maTi tolobis SemTxvevaSi vadarebT ricxvebis udidesiTanrigebis erTeulebs. metia is ricxvi, romlis Sesabamisi Tanrigis pirvelive gansxvavebulicifri aris meti. ricxvebis klebis mixedviT dalagebis Semdeg moswavleebi amave TanamimdevrobiTdaalageben Sesabamis qveynebs. es statistikuri monacemebis damuSavebis pirvelinabijebia.moswavleebs proeqtis saxiT vTavazobT or amocanas, moswavleTa erT nawils mouwevscnobarebSi moiZios monacemebi sxvadasxva qveynis mosaxleobis odenobis Sesaxeb. maT SeuZliaTqveynebi Tavisi interesis mixedviT amoirCion, warmoadginon monacemebis cxrilis saxiT.momdevno gakveTilze SesaZloa mopovebuli informaciebiT mTelma klasma erTi mozrdilicxrili gaakeTos. magaliTad, viRebT romelime moswavlis cxrils, yoveli momdevno moswavlemasSi ganalagebs Tavis monacemebs (qveynebs, mosaxleobis odenobas) saTanado adgilze. am samuSaosSesruleba kompiuteris daxmarebiT umjobesia: viyenebT programa word-s, mis SesaniSnavTvisebas _ svetebad CamobeWdil qveynebs Soris momdevno moswavlis mier qveynis CasmisassvetSi qveynebis dasaxelebebi Tavad wainacvlebs; SeiZleba baraTebze dawerili qveynebissaxelebi dafaze miamagroT da saWiroebis SemTxvevaSi waanacvloT.am samuSaos Sesrulebisas moswavle iwafeba ricxvebis SedarebaSi, kompiuteris gamoyenebaSi.SesaZloa es gakveTili bunebis maswavlebelTan erTad Catardes da integrirebulisaxe mieces. danarCeni moswavleebi analogiuri gzebiT moiZieben informacias planetebis Sesaxeb.am SemTxvevaSi moswavleebs mouwevT erTi da imave erTobliobis orgvarad warmodgena(masebis mixedviT, manZilebis mixedviT). aRsaniSnavia orive saproeqto amocanis SemecnebiTimniSvnelobaczemoT warmodgenili rekomendaciebi exmareba maswavlebels sxvadasxva strategiis,teqnikis, didaqtikuri xerxisa Tu informaciuli da sakomunikacio teqnologiis gamoyenebaSi.7 savarjiSos v) z) magaliTebSi marcxena ricxvi eqvsniSnaa, marjvena _ xuTniSna. moswavleebikidev erTxel gaixseneben: Tu Sesadarebeli ori ricxvidan erT-erTi `metniSnaa~_ misi Tanrigebis odenoba metia, maSin am ricxvTa cifrebis Sedareba saWiro aRar aris _`metniSna~ ricxvi metia.11 savarjiSo moswavleebs uviTarebs martivi cxriliT warmodgenili monacemebis elementarulianalizis unars. kerZod, cxrilis saxiT warmodgenili monacemebis `wakiTxvis~/gagebis unars.13 amocanaSi a) da b) savarjiSoebSi mocemuli ricxvebidan metia `metniSna~ ricxvebi, g)savarjiSoSi, Tu II ricxvi iwyeba 9-ze naklebi cifriT, maSin cxadia, I metia II-ze, Tu meorericxvi iwyeba 9-iT, maSin momdevno _ aTaseulis Tanrigis SedarebiT kvlav vRebulobT: Imetia II-ze, e. i. nebismier SemTxvevaSi I ricxvi metia II-ze. analogiurad, d) savarjiSoSi IIricxvi daiwyeba 1-iT, Tu 1-ze meti ricxviT, mainc II ricxvi metia I-ze.14 amocanaSi meore ricxvis erTeulis cifria 0, aTeulis cifri nulze metia. amitomes ricxvi metia pirvel ricxvze.15 amocanaSi imsjeleT ninikos Secdomebze. mogiwevT sigrZis, masis, dros erTeulebisda maT Soris kavSirebis gaxseneba. miaqcieT yuradReba: b) punqtSi zogierTi moswavleSeecdeba 1000 wm gadaiyvanos saaTebSi, es ufro rTuli gzaa, magram sworia da waxalisebasimsaxurebs.20 amocanis amoxsnisas SeiZleba moswavlem gamoTvalos 3⋅55+50 gamosaxulebis mniSvnelobada Seadaros 2 lars, anu 200 TeTrs. SeiZleba ase imsjelos: 4 cali 50-TeTriani33

Seadgens 2 lars. 1 cali 50-TeTriani mWirdeba nayinisTvis, 3 cal 55 TeTrian purs 3 cali50 TeTriani ar eyofa.21 or avtobusSi 72 adgilidan 23+18=41 adgili ukve dakavebulia da darCenilia72–41=31 adgili, es adgilebi yofnis ninikos klasis moswavleebs, mSobels da damrigebels,maTi odenoba zustad 31-ia, (29+1+1). msjeloba SeiZleba asec warimarTos: I avtobusSi aris36–23=13 Tavisufali adgili, II-Si _ 36–18=18. sul _ 13+18=31 (adgili).moisazre1 saZiebeli ricxvi samniSnaa, misi aseulis Tanrigis cifria 1, erTeulis Tanrigis cifria_ 3. aTeulis Tanrigi moswavlem SeiZleba sinjvis meTodiT daadginos. pasuxi: 123.2 999 981-ze meti eqvsniSna ricxvis aseulis an ufro maRali Tanrigis cifri ar Sei­Zleba meti iyos mocemuli ricxvis Sesabamisi Tanrigis cifrze. erTeulis Tanrigis cifri,pirobiT, 1-ia. darCa aTeulis Tanrigi, romelic unda iyos 9-is toli. pasuxi: 999 991.3 xelis CamorTmevaTa odenobis dasaTvlelad ninikos kargi gza aurCevia, Tumca yovelixelis CamorTmeva orjeraa daTvlili _ mis mier CamoTvlil wyvilebSi erTnairi cifrebiTCawerili ricxvebi erTi da imave wyvilis xelis CamorTmevas aRniSnavs (magaliTad, 36 da 63_ orive warmoadgens, me-3 da me-6 megobris xelis CamorTmevas). amitom cxrilis elementebisodenoba _ 72 unda gaiyos 2-ze. pasuxi: 36.SeiZleba moswavleTa msjeloba warvmarToT damxmare kiTxvebiT.• ramden megobars CamoarTva xeli TiToeulma? (8-s).8⋅9=72• ramdenjer davTvaleT TiTo xelis CamorTmeva? (2-jer. magaliTad me-6 da me-3 megobarsSoris xelis CamorTmeva orjer davTvaleT).• ramdeni xelis CamorTmeva Sedga? (72:2=36).orive SemTxvevaSi gadaTvlis erTsa da imave princips viyenebT, Tumca II gzam cxrilisSedgenas agvarida.am amocanis amoxsna xisebri diagramiTac SeiZleba warmovadginoT:123 4 5 6 7 8 92 3 4 5 6 7 8 91 3 4 5 6 7 8 91 2 4 5 6 7 8 9 . . .cifrebis yoveli wyvili (megobrebis nomrebi) orjeraa SeerTebuli wiboTi. sul 72 wyvilia,yoveli maTgani orjer dasaxelda.me-3 amocanis ramdenime gziT amoxsnis Semdeg me-4 da me-5 amocanebis yovel SemTxvevasmoswavleebi swrafad upasuxeben. Tu megobrebis odenoba 6-ia, xelis CamorTmevaTa odenobaiqneba (6⋅5):2=15;roca megobrebis odenoba 7-ia _ (7⋅6):2=21;roca megobrebis odenoba aris 8 _ (8⋅7):2=28.5 (10⋅9):2=45.am amocanebis seria daexmareba moswavleebs SemdgomSi ufro rTuli kombinatoruli amocanebisamoxsnisas gamravlebis wesis gamoyenebaSi.wignSi warmodgenili esa Tu is amocana klass SeTavazeT xolme mxolod mas Semdeg, rocamiiCnevT, rom is Tqven mier Sedgenili gakveTilis gegmis organuli nawilia, gaqvT misiganxilvisTvis saWiro dro da es ganxilva iqneba warmatebuli. erovnuli saswavlo gegmis34

moTxovnebis warmatebuli gadawyveta ar gulisxmobs uklebliv yvela amocanis amoxsnas.Tumca, Tu klasis akademiuri done maRalia, saxelmZRvaneloSi warmodgenili yvela amocanisganxilva savsebiT miRwevadia.4. amocanebis amoxsnamizani. amocanebis amoxsnisas amocanis Sinaarsis gagebis, amoxsnis strategiis gansazRvrisda ganxorcielebis, Sesasrulebeli samuSaos Sedegis Semowmebisa da gaanalizebis unarisganviTareba.paragrafSi aRwerilia amocanis amoxsnis etapebi _ 4 etapi. yoveli maTgani gasagebigaxdeba moswavlisTvis konkretuli amocanebis amoxsnisas. amocanebis amoxsnis dawyebamdeSeiZleba gavixsenoT:• ra moqmedebas vasrulebT saerTo odenobis sapovnelad? (Sekrebas).sasurvelia moswavleebma Tavad moifiqron Sesabamisi situaciebi. magaliTad, diliT amovxseni6 amocana, saRamos _ 5. sul ramdeni amocana amovxseni?dedam momca 15 lari, mamam _ 28 lari. ramdeni lari momces mSoblebma?• ra moqmedebiT SeiZleba SevcvaloT Sekreba, roca toli Sesakrebebi gvaqvs? (gamravlebiT).magaliTad, maswavlebelma klasis 20 moswavlidan TiToeuls 3 rveuli moatanina. ramdenirveuli Segrovda klasSi?3 + 3 + . . . + 3 = 2 0 ⋅ 3 = 6 0 .20-jer• ra moqmedebas viyenebT nivTebis tolad gadanawilebisas? (gayofas).magaliTad, 48 wigns 3 Taroze tolad vanawilebT. TiTo Taroze unda davdoT 48:3=16wigni.20 kanfets 4 megobari Tanabrad inawilebs. TiTos Sexvdeba 20:4=5 kanfeti.• ra moqmedebas vasrulebT ricxvebis Sedarebisas? (gamoklebas)magaliTad, ramdeni wliT ufrosia 51 wlis marika 23 wlis rusudanze?am kiTxvebiT gavixsenebT im SemTxvevebs, roca oTxive moqmedebidan ama TuU im moqmedebasvirCevT. moswavleTa mier SemoTavazebuli yoveli amocana unda iqnes ganxiluli, maT SorissainteresoTa avtorebi _ waxalisebuli.# 7 amocanaSi marTkuTxedis sigrZea 160⋅3=480 m, perimetris gamosaTvleli gamosaxulebaa160⋅2+480⋅2 an (160+480)⋅2. am gamosaxulebis miReba mniSvnelovnad gagviadvilebs momavalSimarTkuTxedis perimetris gamosaTvleli formulebis miRebas.9 moswavle 70⋅6 gamosaxulebas Seadarebs 420-s da daaskvnis _ am manZilis gavlas zustad6 sT sWirdeba. SeiZleba zogierTma maTganma 420:6 ganayofi Seadaros 70-s, anu upasuxoskiTxvebs: ra siCqariT unda imoZraos avtomobilma, rom 6 sT-Si 420 km gaiaros?10 570-is 90-ze gayofisas ganayofia 6, naSTia 30. am 30 adamianisTvisac saWiroa kideverTi _ me-7 reisi, romlis drosac TviTmfrinavi mTlianad ar Seivseba.35

13 cxadia, sqemaze (romelic rveulSi unda gakeTdes) unda miuTiTon qalaqebs SorismanZili: 70⋅4=280 km.280 km? 3 . 15 km14 yuTebSi 350 wignia. TiToeul yuTSi wignebis odenobis dasadgenad saWiroa vicodeTyuTebis odenoba (magaliTad, 10), amasTanave yuTebSi wignebis Tanabari odenoba unda iyos.msgavsi amocanebis mofiqrebiT moswavleebs unviTardebaT SemoqmedebiTi azrovnebis unari,uCndebaT motivacia amoxsnan megobris mier mofiqrebuli amocana; Seadaron, gaanalizones amocanebi; unviTardebaT oponirebis unari. magaliTad, kritikulad unda ganixilonamocana, romelSic avtomobilis siCqare aRmoCnda 2 km/sT, an 500 km/sT, an qalaqebs Sorismanili 20 000 km miiRes da a. S.16 moswavleebSi xSirad `2-jer meti~ ukavSirdeba gamravlebis operacias, xolo `naklebi~_ gamoklebas. yuradRebiT gaacaniT amocana moswavleebs da isini I SemTxvevaSi Seasrulebengayofas, II SemTxvevaSi _ mimatebas.saSinaosaSinao davalebis amocanebi1 2 sT-Si — 35 km; 105 aris 35-ze 3-jer meti, e. i. droc 3-jer meti daixarjeba _ 6 sT.2 nucas zedmeti 8 lari Tanabrad unda ganawildes: 4 lari ubrundeba nucas, 4 lari_ paatas.3 Tu davubrunebT giorgis Tavis 12 lars, miviRebT mis Tavdapirvel Tanxas: 30+12=42.4 daxarja - 9, darCa _ 18, hqonda 18+9=27.5 nikam daxarja 10, darCa _ 7, hqonda _ 17. Tornikesac hqonda 17. daixarja 17+10=27.6 2 sT-Si Camoedineba 1 litri wyali, 10 sT-Si _ 5 litri.7 3674 km aris amocanis zusti pasuxi. miaxloebiT gaiara 2 000+900+1 000=3 900 km.17 amocana momdevno gakveTilze gansjis sagani unda gaxdes: orive sacavSi darCa12+24=36 kg Warxali.• orive sacavidan erTad 100–(12+24)=64 kg Warxali gautaniaT.• ramdeni gautaniaT TiToeulidan _ amis gageba SeuZlebelia damatebiTi pirobis gareSe.gavaanalizoT TiToeuli moswavlis mier damatebuli piroba. magaliTad, SeiZleba daematos,rom• Tavdapirvelad orive sacavSi toli odenobis Warxali iyo (TiToSi 50 kg, I-dan gautaniaT50–12=38 kg, meoredan _ 50–24=26 kg).18 natam gacura 8⋅50=400 m, gasacuri darCa 1000–400=600 metri. 600 metris gasacuradnatam auzi 600:50=12-jer unda gadacuros.36

moisazre1 vagrZelebT kombinatoruli xasiaTis amocanebze muSaobas. saxelmZRvaneloSi moswavlesvTavazobT variantebis amoweras Semdegi kanonzomierebiT:kodi, romelic Seicavs zustad erT 1-ians _ 0001, 0010, 0100, 1000;kodi romelic Seicavs or 1-ians _ 1100, 1010, 1001, 0110, 0101, 0011;kodi, romeli Seicavs zustad sam 1-ians _ 1110, 1101, 1011, 0111. aq SeiZleba moswavlemixvdes, rom tolia im kodebis odenobebi, romlebic Seicavs zustad erT 1-ians an zustadsam 1-ians (igivea, rac erT 0-ians). miviReT 14 varianti.SesaZloa moswavleTa nawilma ase imsjelos: kodis yovel TanrigSi arisori SesaZlebloba0 an 1. wina TanrigSi yovel arCevans Seesabameba momdevno Tanrigis ori SesaZlebloba,sul 2⋅2⋅2⋅2=16 varianti, romlidanac pirobis Tanaxmad unda gamoiricxos ori varianti: 0000da 1111.SesaZloa xisebri diagramis gamoyenebac, Tumca yoveli me-5 klaselisTvis variantebisaRweris aseTi forma SesaZloa ar iyos martivad aRsaqmeli.I Tanrigi 0 an 1II Tanrigi 0 1 0 1III Tanrigi 0 1 0 1 0 1 0 1IV Tanrigi 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 116 variantizemoT miTiTebuli ori variantis gamoricxviT miiReba 14 varianti.37

sakontrolo wera #1Tema:• naturaluri ricxvebis Sedareba da dalageba• amocanebis amoxsnastandartis moTxovnis gadawyvetilebiT, Semajamebeli sakontrolo werisTvis davalebebisSerCeva maswavleblis sagnobrivi kompetenciebis Semadgeneli nawilia; gansxvavebulisirTulis savarjiSoebis SerCevisa da Sedgenis mimarTulebiT kompetenciebis ganviTarebaSimaswavlebels Cven mier SemoTavazebuli nimuSebi da maTi Sefasebis kriteriumebi daexmareba.cxadia, maswavlebels SeuZlia gansxvavebuli variantebi SesTavazos moswavleebs, SesaZlebeliasaxelmZRvaneloSi warmodgenili amocanebi Seicvalos analogiuri amocanebiTac. Cven sworedaseT rekomendacias vuwevT pedagogebs, raTa Tavidan aviciloT sakontrolo amocanaTawinaswari gacnoba moswavleebis mier.1. daalageT klebis mixedviT ricxvebi: 12 miliard 416 aTasi, 21 milion 641 aTasi, 12milion 164 aTasi.1) 21 milion 641 aTasi, 12 miliard 416 aTasi, 12 milion 164 aTasi2) 21 milion 641 aTasi, 12 milion 164 aTasi, 12 miliard 416 aTasi3) 12 miliard 416 aTasi, 21 milion 641 aTasi, 12 milion 164 aTasi4) 12 milion 164 aTasi, 21 milion 641 aTasi, 12 miliard 416 aTasi.2. ra cifri unda CavsvaT gamotovebul adgilze, rom swori utolobebi miviRoT:810535

8. puris qarxanam sam maRazias Tanabari odenobis puri miawoda. pirvelma maRaziam gayida56 cali puri, meorem _ mxolod 14 _ zustad imdeni, rac darCa gasayidi pirvel maRazias;mesame maRaziam gayida imdenive puri, rac pirvelma da meorem erTad. ramdeni puri darCagasayidi samive maRazias?SeniSvna: warmodgenilia sakontrolo weris savaraudo varianti. sasurvelia misigamoyenebiT Sedges odnav gansxvavebuli ramdenime bileTi, gamravldes da daurigdesmoswavleebs Sesasruleblad.pasuxebi da miTiTebebi amocanebis amosaxsnelad:1 2 3 4 5 63 4 1 2 2 47. 60⋅2+35=155 (km)8. amocanis piroba SeiZleba sqemiT warmovadginoT:I II IIIgaiyida darCa gaiyida darCa gaiyida darCa56 14 14 56 56+14 0amrigad, gasayidi darCa 14+56=70 (puri).Sefasebis sarekomendacio sqema.testuri tipis amocanebidan TiToeulSi swori pasuxis miTiTeba Sefasdes 1 quliT,araswori _ 0 quliT. me-7 amocana _ 2; 1 an 0 quliT (mTeli manZilis gamosaTvleligamosaxulebis Sedgena, an avtobusiT gavlili manZilis povna _ 1 qula, amocanis pasuxismiReba _ kidev 1 qula); me-8 amocana _ 2; 1; an 0 quliT (sqemis Sedgena, an TiToeul maRaziaSimitanili puris odenobis, an meore maRaziaSi darCenili odenobis povna _ 1 qula, pasuxi_ kidev 1 qula).pedagogebs kidev erTxel SevaxsenebT, rom skolaSi moqmedi Sefasebis zogadi principebiskolis saswavlo gegmis Semadgenel nawils warmoadgens _ maT Tavad skola amtkicebs. amprincipebs unda efuZnebodes yvela saxis Sefaseba, kerZod, sakontrolo naSromis gasworebiskriteriumic. pedagogma Tavad unda SeimuSavos gasworebis kriteriumebi. Cvens mosazrebebsmxolod sarekomendacio xasiaTi aqvs. pedagogi unda iTvaliswinebdes, rom misi kriteriumigarkveulwilad gansazRvravs mis damokidebulebas swavlebisa da moswavleebis mimarT.sakontrolo wera Semajamebeli davalebaa da misi Sefasebis tipis arCevisas pedagogmaupiratesoba ganmaviTarebel Sefasebas unda mianiWos (miuTiTos, auxsnas moswavles rogorSeiZleboda Secdomis Tavidan acileba, amoxsnis ra gzebs mianiWos upiratesoba, rogorgavaumjobesoT Sedegi...). moswavleTa CarTva miRebuli Sefasebebis, Sedegebis analizSi, metgamWvirvalobas SesZenda pedagogis gadawyvetilebebs.Zalze mniSvnelovania sakontrolo weris procesic. pedagogma unda uzrunvelyossamuSao atmosferos Seqmna _ mSvidi, saqmiani garemo. TviT yvelaze `liberali~ pedagogicki Seurigebeli unda iyos gadaweris, sxvisi naSromis miTvisebis faqtebisadmi. dauSvebelia,rom aseT saqciels, zogierTi pedagogi `uwyinar ancobad~ miiCnevs. gadawera, miTvisebaerovnul saswavlo gegmaSi senad, qurdobad aris miCneuli da is unda aRmoifxvras saskolocxovrebidan.maswavlebeli piradi magaliTiT unda aCvenebdes moswavles koreqtuli, guliTadi,gulisxmieri, samarTliani, urTierTpativiscemaze damyarebuli qcevis nimuSs.39