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m a T e m a t i k a
IV klasi
maswavleblis wigni
guram gogiSvili, Teimuraz vefxvaZe,
ia mebonia, lamara qurCiSvili
gamomcemloba inteleqti
3
sarCevi
zogadi informacia saxelmZRvanelos Sesaxeb ................................................................ 5
Sesavali ........................................................................................................................................ 7
maswavleblis wignis praqtikuli nawili ........................................................................ 20
Tavi I
gameoreba. naturaluri ricxvebi aTasamde
§ 1. gavixsenoT erTidan aTasamde naturaluri ricxvebis Caweris wesi ....... 22
§ 2. ricxvebis Sedareba da dalageba ......................................................................... 30
§ 3. ricxvebis damrgvaleba ........................................................................................... 39
Tavi II
naturaluri ricxvebi milionamde
§ 1. CavweroT aTasze meti naturaluri ricxvebi ................................................ 47
§ 2. SevadaroT da davalagoT naturaluri ricxvebi ......................................... 54
§ 3. naturaluri ricxvebis damrgvalebis wesi ..................................................... 59
§ 4. viyenebT svetovan diagramas ................................................................................ 64
Tavi III
naturaluri ricxvebis Sekreba da gamokleba
§ 1. orniSna ricxvebis Sekreba .................................................................................... 70
§ 2. jamis Sefaseba damrgvalebiT .............................................................................. 75
§ 3. didi ricxvebis Sekreba .......................................................................................... 80
§ 4. SevarCioT Sekrebis wesi ........................................................................................ 87
§ 5. orniSna ricvebis gamokleba ................................................................................. 93
§ 6. sxvaobis Sefaseba damrgvalebiT ...................................................................... 100
maTematikuri viqtorina .............................................................................................. 104
§ 7. didi ricxvebis gamokleba ................................................................................... 110
§ 8. amocanebis amoxsna (SevarCioT moqmedeba); fulis odenobebis
Sekreba da gamokleba .................................................................................................... 121
§ 9. Sekrebis da gamoklebis Tvisebebi. vipovoT ricxviTi
gamosaxulebis mniSvneloba ........................................................................................ 127
§ 10. vpoulobT Sekrebisa da gamoklebis ucnob komponents ......................... 133
§ 11. gveyofa fuli sayidlebisTvis ........................................................................ 138
Semajamebeli davaleba ................................................................................................. 145
Tavi IV
dro, sigrZis gazomva, geometria
§ 1. dro (ra droa?) ....................................................................................................... 147
§ 2. mTelis nawilebi _ naxevari, meoTxedi, mesamedi ....................................... 152
§ 3. sigrZis erTeulebi ................................................................................................ 156
§ 4. texili. texilis sigrZe. manZili. marSrutis sigrZe ................................. 160
4
Tavi V
naturaluri ricxvebis gamravleba da gayofa
§ 1. gamravlebis Tvisebebi. gavixsenoT gamravlebis cxrili ......................... 165
§ 2. gamravleba da gayofa. gayofis Tvisebebi .................................................... 171
§ 3. vamravlebT nulebiT daboloebul ricxvebs ............................................... 177
§ 4. vamrgvalebT Tanamamravlebs da vpoulobT namravls miaxloebiT ...... 182
§ 5. erTniSna ricxvze gamravleba ............................................................................ 188
§ 6. orniSna ricxvze gamravleba .............................................................................. 196
§ 7. vamravlebT samniSna ricxvze ............................................................................. 201
§ 8. moqmedebaTa Tanamimdevroba .............................................................................. 207
§ 9. gayofa naSTiT ......................................................................................................... 211
§ 10. erTniSna ricxvze gayofa .................................................................................. 215
§ 11. Sesabamisoba............................................................................................................ 219
§ 12. gamravlebisa da gayofis ucnobi komponentis povna............................... 224
§ 13. amocanebis amoxsna (amocanebi zedmeti mocemulobiT) .......................... 227
Semajamebeli davaleba ................................................................................................. 231
VI Tavi
amocanebis amoxsna. geometria. monacemTa analizi
§ 1. masis erTeulebi ..................................................................................................... 233
§ 2. amocanebis amoxsna (amocanebi nakluli monacemebiT) ............................... 238
§ 3. kubi, prizma, piramida ........................................................................................... 241
§ 4. amocanebis amoxsna (viyenebT naSTiT gayofas) ............................................. 245
§ 5. cilindri. konusi. birTvi ................................................................................... 250
§ 6. SevagrovoT monacemebi ........................................................................................ 254
§ 7. monacemebis Segroveba da organizacia .......................................................... 257
§ 8. piqtograma ................................................................................................................ 260
§ 9. svetovani diagrama ................................................................................................ 263
proeqti `vakvirdebiT saojaxo xarjebs~ .............................................................. 267
Semajamebeli davaleba ................................................................................................. 267
maTematikuri simboloebi da maTi gamoyenebis magaliTebi ................................... 269
ariTmetikuli moqmedebebis Sesrulebis Tanamimdevroba ...................................... 269
ariTmetikuli moqmedebebis Tvisebebi .......................................................................... 269
zomis erTeulebi .................................................................................................................. 270
sivrculi figurebi .............................................................................................................. 270
maTematikuri terminebis leqsikoni .............................................................................. 271
Mmoswavlis ganmaviTarebeli Sefasebis Sesaxeb ............................................................ 273
beWduri da eleqtronuli resursebi ............................................................................ 278
literatura ............................................................................................................................ 280
5
zogadi informacia saxelmZRvanelos Sesaxeb
IV klasis maTematikis saxelmZRvanelo dawerilia yvela im moTxovnis gaTvaliswinebiT,
romlebsac moswavlis saxelmZRvanelo unda akmayofilebdes _ dawerilia axal (2017-
2023 wlebis) erovnul saswavlo gegmasTan sruli SesabamisobiT; daculia moswavlis
saxelmZRvanelos Sedgenis Tanamedrove, sayovelTaod aRiarebuli struqturuli da
meTodikuri principebi; gamoyenebulia moswavleze orientirebuli swavlebis dros
aqtualuri sxvadasxva strategia (yuradRebis mobilizeba, wina codnis gaaqtiureba,
masalis mTlianobaSi gaazreba, kritikuli azrovnebis, kiTxvis dasmis da weriTi unar-
Cvevebis ganviTareba, jgufuri da individualuri muSaobis unarebis ganviTareba,
motivaciis amaRleba); daculia maTematikis gadmocemis mecnieruli done da, imavdroulad,
gaTvaliswinebulia moswavleTa asakobrivi Taviseburebebi. wigni universalur
saxelmZRvanelod SeiZleba CaiTvalos _ nebismieri akademiuri donis moswavleebTan
SesaZlebelia am wigniT warmatebuli swavleba. masSi mocemul saswavlo masalas Cveni
skolis moswavle bunebrivad aRiqvams, radgan warmodgenilia misi Cveuli garemo, yofa,
maTematikis praqtikuli gamoyenebebi (magaliTad, qarTuli fulis niSnebi da misi kavSiri
ricxvebis poziciur sistemaSi warmodgenasTan). mniSvnelovania saswavlo resursebis, maT
Soris, eleqtronuli resursebis CarTulobac.
wignSi naturaluri ricxvis cneba gansxvavebuli midgomebiT aris gadmocemuli.
magaliTad, raodenobiTi aspeqtis gaTvaliswinebasTan erTad didi yuradReba eTmoba natu-
raluri ricxvis rigobiT aspeqts.
mravalferovania amocanaTa sistema _ amocanebi individualuri da jgufuri muSao-
bisTvis, `testebi~, proeqtebi, maTematikuri TamaSebi, amocanebi viqtorinebisa da
maTematikuri Sejibrebebis Casatareblad, yoveli Temis bolos _ damatebiTi amocanebi
gameorebisa da Semajamebeli mecadineobebisTvis. didi yuradReba eTmoba masalis Semzadebasa
da misawvdomi formiT gadacemas _ gavlili masalisadmi xSiri mibruneba, yvela moswavlis
SesaZleblobaTa gaTvaliswinebiT, uzrunvelyofs maT maqsimalur Semzadebas. ukve am etapze
iwyeba problemis gadawyvetaze dafuZnebuli swavlebis ganxorcieleba _ praqtikuli
amocanis dasma, misi maTematikuri modelireba da maTematikuri amocanis amoxsniT Tavad
problemis gadawyveta. maTematikis gamoyenebiTi aspeqtebis xazgasma moswavleTa motivaciis
gazrdasac uwyobs xels.
standartis moTxovnaTa dasafaravad mniSvnelovnad miviCnieT maTematikur davalebaTa
mravalferovani formebis gamoyeneba; diagramebis, cxrilebis, maTematikuri teqstebis
gacnoba; maTematikur obieqtebs Soris kavSirebisa da mimarTebebis dadgenis, msjeloba-
dasabuTebis, amoxsnis gzis realizaciis unarebis ganviTareba.
6
axal erovnul saswavlo gegmaSi calkea gamoyofili IV klasis bolos misaRwevi Sedegebi,
romelTa gaTvaliswineba IV klasis saxelmZRvanelos aucilebeli komponentia.
moswavlis saxelmZRvanelo ar Seicavs masalas, romelic moiTxovs saxelmZRvaneloSi
raimes Caweras. wigns unda gaufrTxildes moswavle, rom Semdgom wlebSic saxelmZRvanelo
sxvebisTvisac iyos gamosadegi.
dawyebiTi safexuris I-IV klasebis saxelmZRvaneloebis weris procesSi Cven vimuSaveT
eleqtronuli resursebis Seqmnazec, romlebic Cvens saxelmZRvanelozea morgebuli.
maTSi `dinamikaSi~ TvalsaCinod aris warmodgenili sxvadasxva aqtivobebi. Cven SegviZlia
mivawodoT isini maswavleblebs, romlebic arCevans Cvens saxelmZRvaneloebze gaakeTeben.
amasTanave, maswavlebelma SeiZleba gamoiyenos sxva eleqtronuli resursic; Tumca, am mxriv
saWiroa sifrTxilis gamoCena, radgan zogjer iseT masalas SeiZleba wavawydeT, romlebSic
maTematikuri Secdomebia daSvebuli.
7
Sesavali
sagnis swavlebis miznebi. standartis Sedegebis miRwevisa da Sinaarsis urTierT
kavSiris matrica
sagnis swavlebis miznebi:
• moswavleTaTvis azrovnebis, msjelobis, SexedulebaTa Camoyalilebis unaris ganv-
iTareba.
• msjeloba-dasabuTebis unaris ganviTareba.
• yoveldRiur cxovrebaSi maTematikis gamoyenebis gaazreba.
• amocanis Sinaarsis aRqma, mocemulobisa da saZiebeli sidideebis gaazreba-gamijvna.
• damxmare teqnikuri saSualebebisa da teqnologiebis SerCeva da gamoyeneba.
• yoveldRiur cxovrebaSi SeniSnuli zogierTi obieqtisa da procesis maTematikur
obieqtebTan dakavSireba.
• informaciis warmodgenis xerxebisa da meTodebis floba, maTze msjeloba.
• maTematikuri simbolikisa da terminologiis Tavad maTematikaSi da sxva disciplinebSi
gamoyenebis unaris ganviTareba.
• gansaxilveli sakiTxebis Camoyalilebisas masalis gonivruli struqturirebis unaris
ganviTareba.
standartis Sedegebis miRwevisa da Sinaarsis urTierTkavSiris matrica
Temebis CamonaTvali Mmiznebi; Sinaarsis gardaqmna miznad; erovnul saswavlo gegmaSi miTiTebuli Sesabamisi punqtebi
savaraudo
saswavlo
dro
gavixsenoT erTidan aTasam-de naturaluri ricxvebis Cawe ris wesi; ricxvebis Se-dareba da dalageba; ricx-vebis dam rgvaleba
erTidan aTasamde naturaluri ricxvebis wakiTxva, Cawera, sxvadasxva modeliT warmodgena da poziciuri sistemis demonstrireba.ricxviT CanawerSi TanrigebSi mdgomi cifrebis Sesabamisi mniSvnelobebis dasaxeleba, ricxvis warmodgena SesakrebTa jamis saxiT, ricxvebis Sedareba da damrgvaleba aTeulebamde, aseulebamde. maT. 1V.1.
14 sT
CavweroT aTasze meti na-tu raluri ricxvebi; Seva-daroT da davalagoT na-tu raluri ricxvebi; natu -raluri ric xvis damr gva-lebis wesi; viye nebT sve-tovan diagramebs.
milionamde ricxvebis wakiTxva, Cawera, sxvadasxva modeliT warmodgena, poziciuri sistemis demo-nstrireba. CanawerSi cifris mniSvnelobis miTi-Teba. mravalniSna ricxvis Cawera gaSlili formiT _ saTanrigo SesakrebTa jamis saxiT. ricxvebis Sedareba da dalageba. mravalniSna ric xvis dam-rgvaleba aTeulebamde, aseulebamde, aTa seu lebamde. damrgvalebis wesis Camoyalileba (ricxviTi sxivis gamoyeneba). raodenobebis Sedareba sxvadasxva xerxiT, maT Soris, svetovani diagramebis gamoyenebiT. aRwers svetovani diagramis saxiT warmodgenil monacemebs.maT. IV.1. maT. IV.14.
15 sT
8
orniSna ricxvebis Sekreba;jamis Sefaseba damrgva le-biT;didi ricxvebis Sekreba; SevarCioT Sekrebis wesi;orniSna ricxvebis gamok-leba;sxvaobis Sefaseba;mi lionamde naturaluri ric xvebis Sekreba kal ku-la toriT da jamis, sxva-obis Se faseba.
Sekreba sxvadasxva xerxiT: zepiri Sekreba, Sefaseba, weriTi algoriTmis gamoyeneba;Sekrebis sxadaxsva xerxis gamoyenebiT miRebuli Sedegebis Sedareba. gamoklebis sxvadasxva xerxis gamoyeneba. weriTi algoriTmi, zepiri gamokleba; sxva obis Sefaseba. sxvadasxva xerxis Sedareba. kon-kretuli magaliTisTvis yvelaze xelsayrelis arCeva.jamis, sxvaobis Sefasebis gamoyeneba kalkulatoris sa Sua lebiT miRebuli Sedegis Sesaxeb msjelobisas. we riTi algoriTmis gamoyenebiT Sesrulebuli Sek-rebis/gamoklebis nimuSSi gamotovebuli cifrebis mig neba.maT. IV.2
23 sT
Sekrebisa da gamoklebis zo gierTi Tviseba; ricx-viTi gamosaxulebis mniS-vne lobis povna; ucnobi kom po nentis povna.
drois intervalebis povna;mTelis naxevari, meoTxedi, mesamedi.
martivi maTematikuri amocanis amoxsnisas algebruli gamosaxulebis Sedgenis Cvevebis daufleba.Sekrebis da gamoklebis Tvisebebis gamoyeneba. ric-xviTi gamosaxulebis mniSvnelobis povna. Sek rebis/gamoklebis Semcveli tolobis ucnobi komponentis povna.maT. IV.7drois erTeulebs Soris Tanafardobebisa da ariTmetikuli moqmedebebiT drois intervalis gamosaxva. erTi saaTis naxevris, meoTxedis, mesamedis gamosaxva wuTebiT. mTelis nawilebis dasaxeleba da Sedareba; sxvadasxva modelze mTelis nawilebis amocnoba, dasaxeleba, Sedareba.maT. IV.4, maT. IV.5.
22 sT
sigrZis erTeulebi; obieq-t Ta Soris manZili;or obieqts Soris umoklesi gzis SerCeva;texilis sigrZe, marSruti, mar Srutis sigrZe, mraval-kuT xedis perimetri.
sigrZis romelime erTeulis gamosaxva ufro mcire erTeuliT.maT. IV.5or obieqts Soris manZilis Sefaseba, varaudis ga-moTqma.or obieqts Soris umoklesi gzis SerCeva. marSrutis aRmweri sqemis Sedgena. texilis sigrZis gazomva, mravalkuTxedis perimetris povna da gamosaxva Sesaferis standartul erTeulSimaT. IV.10, maT. IV.11
12 sT
gamravlebis Tvisebebi; gam-rav leba nulebiT dabo lo-ebul ricxvebze;erT niSna ricxvze gamrav-leba;namravlis Sefaseba;orniSna ricxvze gamrav le ba;samniS na ricxvze gamrav le ba;amo canebi zedmeti mocemu-lo biT.
ricxvis 100-ze da 1000-ze da a. S. gamravlebis da nulebiT daboloebul ricxvebis gamravlebis Se-mok lebuli wesebis daufleba. erTniSna, orniSna da samniSna ricxvebze gamravlebis sxvadasxva we sis daufleba; weriTi algoriTmebis daufleba, gamo-yenebli xerxis ganmarteba.maT. IV.3amocanebis amoxsnisas zedmeti monacemebisa da saWi-ro monacemebis gansxvavebis Cvevebis dauflebaU maT. IV.7
20 sT
9
moqmedebaTa rigi;naSTi;erTniSna ricxvze gayofa;nuliT daboloebuli ricx-ve bis gayofa;amocanebis amoxsna.Sesabamisoba;gamravlebis da gayofis uc no bi komponentis povna.
ricxviTi gamosaxulebis mniSvnelobis mosaZebnad Sek rebisa da gamravlebis Tvisebebis gamoyeneba. mar-ti vi ariTmetikuli amocanis amoxsnisas algebruli gamo saxulebis Sedgenisa da gamoyenebis Cvevebis daufleba.maT. IV.7erTniSna ricxvze gayofa zepirad, Sesabamis Sem-TxvevaSi ganayofisa da naSTis dasaxeleba. pasuxis dasabuTeba. weriTi algoriTmebis gamoyeneba gayofis Sesasruleblad, gamoyenebuli xerxis ganmarteba. amocanebis amoxsnisas naSTis interpretaciis unari.maT. IV.3Sesabamisobis gamosaxvis unaris ganviTreba; realuri viTarebis adekvaturi Sesabamisobis agebis Cvevebis dauf leba.maT. IV.6
18 sT
masis erTeulebi; amocanebi nakluli da zedmeti mona-cemebiT;kubi, prizma;konusi, cilindri, birTvi.
zomis sxvadasxva erTeulebis erTmaneTTan da-kavSireba. naSTiT gayofis gamoyeneba zomis mocemul erTeulebSi gamosaxuli sididis sxva erTeuliT gamosaxvisas.maT. IV.5sivrculi figurebis Sedarebisa da dajgufebis Cvevebis daufleba. TanamkveTi figurebis aRqma _ saerTo wer tilebis miTiTeba. sivrculi figuris elementebis dasa xaleba da miTiTeba.sivrcul figuraTa modelebisgan miTiTebuli kon fi-guraciis Seqmna;maT. IV.8, maT. IV.9
12 sT
monacemTa Segrovebisa da organizebis formebi.monacemTa warmodgenis _ saSualebebi _ cxrili, piq-tog rama, svetovani diag-rama
mocemul TemasTan an gamosakvlev obieqtTan dakavSirebiT Tvisebrivi da raodenobrivi mona-cemebis Segrovebis Cvevebis daufleba; Tvisebrivi da raodenobrivi mona ce mebis mowesrigeba; dalageba (dajgufeba) _ cxrilis, sqemis, kiTxvaris, anketis Sevsebis wesebis daufleba. monacemTa analizi, Sedareba, gansxvavebis povna.maT. IV.2, maT. IV.13, maT. V.14
10 sT
10
maswavleblis wignis mimoxilva
maswavleblis wigni Seicavs meTodikur rekomendaciebs axali erovnuli saswalo gegmis
maTematikis standartis mixedviT Sedenili me-4 klasis maTematikis saxelmZRvanelosTvis.
wignSi warmodgenili mdidari saswavlo da meTodikuri masala maswavlebels miscems
farTo SesaZleblobebs SemoqmedebiTi da nayofieri saswavlo procesis gansaxorcieleblad.
wignSi mocemulia TiToeuli gakveTilis dagegmvis sqema. yovel paragrafs ramdenime
gakveTils vuTmobT. amasTanave, cxadia, Cveni rCevebi sarekomendacio xasiaTisaa, SeiZleba
maswavlebelma zogierT SemTxvevaSi gadauxvios maT _ Cven bevr maswavlebels vicnobT,
romlebic gamoirCevian gakveTilebis dagegmvisa da Catarebis Tavisi originaluri mignebebiT,
rac mniSvnelovnad efuZneba klasis akademiur donisa da moswavleTa individualur
Taviseburebebs. rogorc cnobili ungreli maTematikosi da meTodikis udidesi specialisti
d. poia ambobda: `ramdeni maswavlebelic aris, imdeni meTodi SeiZleba arsebobdes; Tumca,
aris meTodikaSi zogadi principebi, gakveTilis dagegmvisa da Catarebis sqemebi, romelTa
gaTvaliswineba sasurvelia~. Cven swored aseT rekomendaciebs viZleviT.
amasTanave, maswavleblis wigni Seicavs mecnierul safuZvlebs maTematikuri cnebebis
warmoSobisa da dafuZnebis Sesaxeb. am masalis codna aucilebelia dawyebiT safexurze
maTematikis swavlebisaTvis. vuTiTebT Sesabamis literaturas. magaliTad, skolaSi ricxvis
cnebis dafuZnebis sakiTxebi sainteresod da dawyebiTi klasis maswavleblebisaTvis
misawvdomi formiT aris gadmocemuli a. kolmogorovis erT-erT statiaSi (ix., mag., [15]).^
masSi meoTxe klasis ZiriTadi sakiTxis _ naturalur ricxvTa Teoriis _ mecnieruli
safuZvlebia gadmocemuli. maswavleblis wignSi es Teoria gamoyenebuli iqneba.
amasTanave, Cvens yovel rekomendacias calkeuli gakveTilis dagegmvisa da warmarTvis
Sesaxeb, maswavlebeli SemoqmedebiTad unda miudges _ swavleba xelovnebaa da,
SesaZlebelia, maswavlebelma gakveTilis warmarTvis, moswavleTa aqtivobebis dagegmvis
sakuTari originaluri xerxebic moifiqros. maswavlebeli myisierad da SemoqmedebiTad
unda exmianebodes sagakveTilo procesSi warmoqmnil (daugegmav) viTarebebsac.
swavlebis procesSi meTodikuri orientirebis sakiTxebSi maswavleblebis dasaxmareblad
warmodgenilia literatura, romelic maswavlebelis wignis boloSia dasaxelebuli.
magaliTad, axal erovnul saswavlo gegmebze gadasvlas kargad exmianeba citata h.
froidentalis @[23] wignidan: `maswavlebels unda SeeZlos swavlebis warmarTva programiT,
romelic kardinalurad gansxvavdeba imisgan, romliTac mas aswavlidnen~. es uaRresad
aqtualuria, radganac saswavlo programebis revizia iSviaTad ar xdeba.
gakveTilis warmarTvisTvis momzadebis sxvadasxva etapi (saswavlo wlisaTvis
momzadeba, mocemuli Temis SeswavlisTvis momzadeba, mocemuli gakveTilisTvis momzadeba)
maswavlebelma nayofierad unda gamoiyenos. am saqmeSi mas Cvenc vexmarebiT. sarekomendacio
wignSi TiTqmis yvela amocanaa amoxsnili, TiTqmis yvela gakveTilis Catarebis sqemaa
warmodgenili, mocemulia meTodikuri rekomendaciebi masalis wardgenisa da axsnis Sesaxeb.
11
moswavleze orientirebuli individualuri midgomebi
did mniSvnelobas vaniWebT samizne asakis Taviseburebebs, miuxedavad imisa, rom
meoTxeklaseli Semecnebis ganviTarebis imave safexurzea, romliTac swavlebis pirveli
sami weli xasiaTdeba. piaJes [21] mixedviT am asakSi daZleulia SezRudvebi problemis
simboluri gadaWrisa da obieqtTa klasifikaciis unaris gamovlenis kuTxiT, moswavles
ukve aqvs unari awarmoos logikuri operaciebi Seswavlilis safuZvelze, me-4 klasSi
moswavle ukve momwifebulia imisTvis, rom ganaviTaros SemecnebiTi unarebi. me-4 klasidan
skolebSi xSirad icvleba midgomebi moswavleTa saswavlo procesisadmi. kerZod, swavlebis
formebis mimarT _ is ufro sagnobrivi xdeba. Tumca, am procesSi memkvidreobiTobis
principis gauTvaliswinebloba, romelic axasiaTebda IV klasis zogierT saxelmZRvanelos,
aZnelebda axali moTxovnebis ganxorcielebas. axali erovnuli saswavlo gegmis
meTodikuri orientirebi gvexmareba swavlebis axal safexurze gadasvlis umtkivneulod
ganxorcielebaSi _ roca wina planze adre naswavlis ganzogadeba gamodis, ganzogadebis
es procesi Zvelis gameorebis procesSi xorcieldeba. swavlebis axal safexurze asvla,
magaliTad, naturalur ricxvTa Seswavlis axali koncentris ganxilvasa da am ganxilvis
ufro maRal doneze warmarTvas gulisxmobs. es procesi Zvel koncentrebTan (aTeulebi,
aseulebi, aTaseulebi) dakavSirebuli sakiTxebis _ numeracia, moqmedebebi _ gameorebasTan
erTad mimdinareobs. piaJes [21] mixedviT, am asakSi moswavle ukve iwyebs obieqtebis Sinagani
Tvisebebis, situaciebis gonebaSi warmodgenas _ ra moxdeboda, rom ....
Tumca, gasaTvaliswinebelia, rom jer kidev uWirT maTematikuri kvlevis iseTi
meTodebis gamoyeneba, rogoricaa abstraqcia, ganzogadeba, sinTezi; romelime konkretuli
amocanis amoxsna SeuZliaT, ZiriTadad, im SemTxvevaSi, roca es amocana ukavSirdeba
realobas. amitom Cven xSirad mivmarTavT TvalsaCino modelebiT sargeblobis xerxebs
da am xerxebis gamoyenebas amocanebis amoxsnisas. me-4 klasSi prioritetuli xdeba iseTi
saxelmZRvaneloTi muSaoba, saswavlo procesis ise warmarTva, rom ufro aqtiurad SevuwyoT
xeli moswavleebSi swavlis survilis Camoyalibebas, SemecnebiTi aqtivobis ganviTarebas,
saswavlo amocanis gagebas, algoriTmis mixedviT moqmedebas, amoxsnis gegmis Sedgenas,
zogad-mecnieruli unarebis ganviTarebas (analizi, Sedareba, klasifikacia).
maTematika sxva saskolo disciplinebze ufro metad xasiaTdeba iseTi specifikuri
kompetenciebis saWiroebiT, rogoricaa, magaliTad, amocnoba, analizi, mixvedra, arCeva,
msjeloba-dasabuTeba. tradiciuli da Tanamedrove Sexedulebebis sinTezis safuZvelze
Sinaarsi mdidrdeba logikuri xasiaTis davalebebiT, Sefasebisa da kontrolis
gansaxorcieleblad gamiznuli davalebebiT. eileris [25] Sexedulebebis gaTvaliswinebiT,
ariTmetikuli moqmedebebis algoriTmebis Camoyalibeba dasabuTebis zomierad gamoyenebebis
safuZvelze xdeba (amaSi dRes TvalsaCinoebebi da eleqtronuli resursebi gvexmareba).
eileri did yuradRebas uTmobda numeraciis sakiTxebsac _ aTobiTi sistemis, ricxvis
Caweris wesebis safuZvlian ganxilvas maTematikis safuZvlad miiCnevda.
dakvirveba da cda, Sedareba, mizez-Sedegobrivi kavSirebis aRmoCena, romlebic
TiTqos ufro bunebismetyvelebis SeswavlisTvis aris xelSemwyobi, maTematikaSic aris
mniSvnelovani. am unarebis gamoyeneba xSiria monacemTa damuSavebis, cxrilebis aRwerisa da
Sedarebisas. maTematikis Seswavlis procesSi es inteleqtualuri unarebi gamoiyeneba da,
amasTanave, viTardeba. ukve IV klasSi geometriuli figurebis ganxilvis dros, abstraqciis,
idealizaciis unarebis ganviTarebasTan gvaqvs saqme _ yoveldRiur cxovrebaSi SemCneuli
formebis idealizacia gvaZlevs im geometriul figurebs, romlebsac swavlebis procesSi
12
ganvixilavT. kanonzomierebis aRmoCenis unaris ganviTareba perioduli mimdevrobebis
Seswavlidan iwyeba (I da II klasebi) da grZeldeba IV klasSi poziciur sistemaTa TvalsaCinod
warmodgenisas.
swavlebis procesSi did mniSvnelobas vaniWebT moswavleze orientirebul mid
gomebs. moswavleebi erTmaneTisgan gansxvavdebian TavianTi SesaZleblobebiT, romlebic,
ZiriTadad, gaxsnilobiT, swavlisa da aRqmis tempis gansxvavebulobaSi gamoixateba. aqedan
gamomdinare, aSkaraa, rom aucilebelia saswavlo praqtikis mravalferovneba, gansxvavebuli
xerxebis, meTodebis, saSualebebis gamoyeneba. Tumca, aucilebeli araa TiToeuls `masze
gamoWrili~ aqtivoba mivusadagoT. mTavaria, TiToeuli moswavlis CarTva mravalferovan
aqtivobebSi da es mravalferovneba Cveni daxmarebiT maswavlebelma unda SesTavazos. Cven
mier SeTavazebuli aqtivobebi swored am mravalferovnebas sTavazobs maswavleblebs _
damoukidebeli, konkurentuli da TanamSromlobiTi swavlebis procesis Catarebas.
wignSi warmodgenili gakveTilebis scenarebSi mraval meTodikur miTiTebasTan erTad
sanimuSod SeTavazebulia xerxebi, rogor CavrToT yvela moswavle saswavlo procesSi _
magaliTad, amocanebis amoxsnis dros damxmare, damatebiTi kiTxvebis sistemiT, wamoWril
sakiTxTa sajaro ganxilviT, jgufuri mecadineobebiT, didaqtikuri TamaSebiT, romlebic
xSirad Sejibrebis xasiaTs atarebs.
xangrZlivi kvlevebi adasturebs, rom moswavle yvelaze ukeT maSin swavlobs, rodesac
igi aqtiuri da qmediTia, da ara maSin, rodesac mxolod aqtiur da qmediT maswavlebels
uyurebs. Cven mier warmodgenili sqemebidan kargad Cans, rom gakveTilebi ar aris (da arc
unda iyos) saleqcio xasiaTis, yoveli gakveTili interaqtiuli xasiaTisaa. amasTanave,
moswavleebi swavloben aramxolod maswavleblisgan, aramed sxva saSualebiTac _
TanaklaselebTan urTierTobiT (amas vaRwevT sakiTxebis sajaro ganxilviT, gundebs Soris
dagegmili SejibrebebiT), davalebebiT (xSirad davalebebi klasSi Sesrulebuli davalebebis
analogiuria) _ SedarebiT sust moswavleebsac ar unda gauWirdeT Sin maTi Sesruleba. arc
niWier da gamorCeul moswavleebs viviwyebT _ xSirad maT davalebebad vaZlevT `moisazres~
rubrikiT warmodgenil amocanebs. damoukidebeli swavlebis ganxorcielebisas moswavleebi
klasSi, an Sin damoukideblad asruleben davalebebs. magaliTad, maswavlebeli mimarTavs
moswavleebs, aZlevs maT garkveul dros amocanebis amosaxsnelad da TiToeuli maTgani
damoukideblad fasdeba. ar viviwyebT konkurentul swavlebasac. amisTvis, magaliTad,
specialuri rubrikac gvaqvs _ `vip~ (vin iqneba pirveli), aq warmatebis misaRwevad saWiroa
sxvebze ukeT da sxvebze adre Sesruldes samuSao. aseTi konkurentuli garemo zrdis
moswavleTa motivacias da eqstremalur pirobebSi maqsimaluri mobilizebis unarsac
uviTarebs.
TanamSromlbiTi swavlebis mravali nimuSic gvaqvs warmodgenili da garCeuli _
moswavleebi iyofa gundebad, gundebi ejibrebian erTmaneTs davalebebis SesrulebaSi,
prezentaciaSi. dasaxuli miznis miRwevis erTaderTi gza TanaklaselebTan TanamSromlobaa.
am dros yvela unda iyos CarTuli davalebebis Sesrulebis procesSi, gundis yovel wevrs
unda SeeZlos Sesrulebuli samuSaos prezentacia, miuxedavad misi monawileobis xarisxisa
davalebis Sesrulebis saqmeSi (SeiZleba amocanebi gundis gamorCeulma wevrebma amoxsnan,
magram es amoxsnebi gundis yvela wevrisTvis unda iyos gasagebi, SeiZleba romelime maTgans
mouwios prezentacia).
kargi Sedegis misaRwevad, saswavlo procesSi sxvadasxva SesaZleblobis mqone moswavleTa
CasarTavad samive forma SeiZleba gamoiyenos maswavlebelma.
moswavleze orientirebuli swavlebis xelSemwyobi garemos uzrunvelyofa saswavlo
procesis moswavleTa saWiroebebze morgebis saSualebiT warimarTeba. am formas meTodikur
13
literaturaSi diferencirebuli swavleba ewodeba. am dros erTi da imave klasSi
gansxvavebuli miRwevis mqone moswavleebs gansxvavebulad vaswavliT, moswavleTa sxvadasxva
jgufTan gansxvavebuli meTodebiT, zogjer, gansxvavebuli Sinaarsis masaliT vaswavliT.
diferencirebuli swavlebis mizania TiToeulma moswavlem miaRwios saukeTeso Sedegs
Tavisi SesaZleblobebis farglebSi. es ar niSnavs, cxadia, Tanabari Sedegebis miRwevas.
diferencirebuli, individualuri midgomebiT moswavleebs eZlevaT saSualeba daZlion
swavlis procesSi arsebuli barierebi da mkveTrad gaiumjobeson Sedegebi. maswavlebeli
amas uwyveti Sefasebisa da moswavleTa moqnili dajgufebis gziT aRwevs. saswavlo procesSi
SeiZleba iseTi elementebis diferencireba, rooricaa Sinaarsi, davalebebi, garemo.
swavlebis efeqtianoba damokidebulia imaze, ramdenad kargad vicnobT Cven moswavleebs,
maT interesebs, saWiroebebs. amasTanave, maswavlebeli kargad unda icnobdes standartis
Sinaarss, moTxovnebs da fokusirebuli unda iyos standartiT moTxovnili miznebis
auCqareblad da xarisxianad Sesrulebaze.
diferencirebuli swavlebis dros moswavleTa nawili damoukideblad muSaobs, zogierTi
jgufi _ Cveni meTvalyureobiT.
diferencirebuli swavleba ganmaviTarebeli Sefasebis momdevno aucilebeli nabijia.
cxadia, moswavleze orientirebuli individualuri midgomebis ganxorcieleba
did Zalisxmevas moiTxovs; sxvadasxva variantebis Sedgena (e. w. `xelmisawvdomi~ da
`garTulebuli), miznobrivi daxmareba akademiuri sirTuleebis mqone moswavleebisadmi
_ nabij-nabij instruqciebis, miniSnebebis micema, TanaklaselebTan erTad muSaobis
saSualeba; ganmeorebiT, gansxvavebuli meTodiT swavleba; axali terminebis, cnebebis
miwodeba, damatebiTi drois gamoyofa.
gansxvavebulia midgomebi maRali mzaobis moswavleebTan muSaobis dros; ufro rTuli
davalebebis micema, romlebsac moswavleebi damoukideblad asruleben. iseTi davalebebis
micema, romlebic saWiroebs kargad gaazrebas, SedarebiT abstraqtuli sakiTxebis
dafiqsirebas, axali informaciis gamoyenebas; zogjer SeiZleba arasworad amoxsnili
amocanebi mivceT da vTxovoT _ gaasworos. ar viyenebT TvalsaCino miniSnebebs; vTxovT
moswavleebs Tavad Seqmnan diagramebi, sqemebi, sxvadasxva gziT gamoxaton TavianTi ideebi
da mosazrebebi.
diferencirebuli swavleba ukavSirdeba swavlebis teqnikas, romelic meTodikur
literaturaSi cnobilia skalfoldingis saxelwodebiT. skalfoldingi inglisuri terminia
da xaraCos agebas, am SemTxvevaSi _ saswavlo xaraCos agebas niSnavs. es aris wabiZgeba,
xaraCos Seqmna, romelic Sinaarsis gagebas uwyobs xels. skalfoldingi ganamtkicebs im
winare unar-Cvevebs, romlebsac moswavleebi axali masalis gacnobamde unda flobdnen.
skalfoldingi niSnavs im moswavleebis daxmarebas, romlebic mzad ar arian
damoukideblad Seasrulon davaleba. rogorc xaraCoebi exmareba mSenebels da droebiTia,
aseve droebiTi unda iyos maswavleblis mier saswavlo `xaraCos~ saSualebiT gaweuli
daxmareba.
moswavleze orientirebuli swavlebis strategiebidan unda gamovyoT problemaze
orientirebuli swavleba da proeqtebi. problemaze orientirebuli swavlebiT moswavleebs
uviTardebaT is unar-Cvevebi, romlebsac mTeli cxovrebis ganmavlobaSi gamoiyeneben. es
problemis gadaWris, drois racionalurad ganawilebis, saqmis dagegmvisa da sakiTxis
SemoqmedebiTad gaazrebis unarebia. Cven mier SemoTavazebuli proeqtebi saintereso da
daZlevadia moswavleebisTvis. maTematikis praqtikuli gamoyenebebis SemTxvevebis Seswavla
(magaliTad, ricxvebis poziciur sistemaSi warmodgenisas avtomanqanebSi spidometrebze
14
gavlili manZilis aRricxva, aZlierebs maTematikis Seswavlisadmi interess. aseve aZlierebs
maTematikis Seswavlisadmi interess maTematikuri TamaSebi, romelTa Catarebis wesebi
aRwerilia moswavlis saxelmZRvaneloSi.
maswavlebels SeiZleba klasSi hyavdes specialuri saganmanaTleblo saWiroebis mqone
moswavleebi (SezRuduli da Seferxebuli ganviTarebis mqone moswavleebi, socialur-
emociuri problemebis mqone moswavleebi, gamorCeulad niWieri bavSvebi). amitom kargad
unda vicodeT maTTan muSaobis wesebi _ rogor CavrToT isini saswavlo procesSi ise, rom
ar vagrZnobinoT, rom sWirdebaT gansakuTrebuli yuradReba, mzrunveloba da TanagrZnoba...
saWiroa daigegmos moswavlis akademiuri winsvla individualuri saswavlo gegmebis
SedgeniT (SezRuduli SesaZleblobis mqone moswavleebisTvis) da damatebiTi informaciis
moZiebaSi daxmarebiT (maRali akademiuri donis bavSvebisTvis): maTTvis maswavlebeli
advilad moiZiebs masalas Cvens saxelmZRvaneloSi da Sesabamis saswavlo resursebSi.
ZiriTadi midgoma, romelsac eyrdnoba saxelmZRvanelo martividan rTulisken
svla, Teoriisa da praqtikis kavSiris principis ganxorcielebaa, dakvirvebis, Sedareba-
Sepirispirebis, sinjvis meTodis gamoyenebaa. im codnis aTvisebas, romelsac aqvs zogadi da
abstraqtuli xasiaTi, win uswrebs ufro kerZo da konkretuli codnis gacnoba. Semecnebis
ZiriTadi gza induqciuria. axali codnis SeZena problemuri situaciis ganxilviT iwyeba.
unda gamovyoT muSaoba iseTi inteleqtualuri unarebis ganviTarebaze, rogoricaa
maTematikuri obieqtebis Sedareba (riT gansxvavdeba da ra aris saerTo?), klasifikaciis
unari (dayavi or jgufad raime niSnis mixedviT), sxvadasxva xerxiT saswavlo amocanis
amoxsna (magaliTad, gamoiyene qveSmiweriT Sekreba, an zepiri Sekrebis romelime xerxi
da ipove gamosaxulebis mniSvneloba); xelsayreli xerxis SerCeva (Sekrebis romel xerxs
gamoiyenebdi?).
dawyebiT skolaSi efeqturi saswavlo garemos uzrunvelyofas damxmare saswavlo
resursebis gamoyenebiTac SeiZleba mivaRwioT. magaliTad, ricxvebis Caweris poziciuri
sistemisa da am sistemis gamoyenebebis aTvisebaSi SeiZleba CavrToT damxmare resursebic
_ kuburebi, qarTuli fulis niSnebis modelebi, baraTebi, romlebzec gamosaxulia
calkeuli cifrebi, moqmedebaTa niSnebi, naturaluri ricxvebi (maTi gamoyeneba, magaliTad,
didaqtikuri TamaSebis dros xdeba); eleqtronuli resursebi.
ZiriTadi principebi, romlebsac efuZneba Cven mier SemoTavazebuli programa,
aris sruli Sesabamisoba erovnuli saswavlo gegmiT gansazRvrul saswavlo programasTan,
sistemuroba masalis dalagebaSi, gadmocemis misawvdomoba, enis garkveuloba, sizuste da
simartive, savarjiSoebisa da maTi Sesrulebis formebis mravalferovneba, Tavad saswavlo
procesis SemoqmedebiTad warmarTvis mravalferovani nimuSebis SeTavazeba.
moswavlis Sefasebis meqanizmebi eyrdnoba Sefasebis miznebsa da ZiriTad kriteriumebs,
romlebic kargad aris cnobili literaturidan _ magaliTad, moswavlis Zlieri da susti
mxareebis gamovlena da moswavlis swavlis xelSewyoba.
aqve unda aRiniSnos, rom erovnuli saswavlo gegmebis Zvel da axal redaqciebs Soris
gansxvaveba swored SefasebasTan dakavSirebul Tavebsa da muxlebSia. axal dokumentSi,
romlis safuZvelze Seiqmna Cveni saxelmZRvaneloebi, vkiTxulobT: `Sefaseba unda iZleodes
informacias moswavlis individualuri progresis Sesaxeb, igi unda daefuZnos swavlebis
konstruqtivistul principebs~. konstruqtivizmi ki niSnavs imas, rom moswavle axal codnas
gaazrebulad agebs, mis konstruirebas axdens. saswavlo-maTematikuri konstruqtivistuli
aqtivoba gulisxmobs interacias moswavlesa da maswavlebels Soris; an TviT moswavleebs
Soris maswavleblis davalebiT. konstruqtivistuli swavleba samfaziania:
15
I. `winaswar~, anu Semzadeba _ axali sakiTxisaTvis winare codnis gaaqtiureba, moswavleTa
daintereseba axali sakiTxiT, moswavleTa gonebis mobilizeba saTanado aqtiurobisaTvis
(gonebrivi mzaoba).
II. `ganmavlobaSi~ _ axali sakiTxis ageba an aRmoCena; misi magaliTebisa da kont-
rmagaliTebis ganxilva; problemis gadaWrisaTvis saWiro moqmedebaTa SerCeva da ganxilva;
moswavleebi iazreben Sesaswavl sakiTxs, euflebian codnasa da unar-Cvevebs.
III. `Semdgom~ _ naswavli sakiTxis ganmtkiceba, ganviTareba; misi gamoyeneba, sirTuleTa
gaazreba; misi sxvadasxva waxnagis warmoCena. moswavleebi ganamtkiceben naswavls da
gaivarjiSeben mis gamoyenebaSi.
ganaTlebisa da mecnierebis saministrodan miRebuli rekomendaciiT (`mas wavlebelis
wignis sarekomendacio struqtura~) moTxovnili iyo calkeuli gakveTi lebis scenarebis
warmodgena (Cven yvela gakveTilis scenari warmovadgineT) da ara calkeuli sakiTxis
swavlebis scenari, amitom samfaziani modelis ganxorcieleba ramdenime gak veTilzea
ganawilebuli da TiToeuli es faza calkeuli (SeiZleba oric) gakveTilis sce nariT aris
warmodgenili.
erovnuli saswavlo gegmis axal dokumentSi gaCnda axali Canaweri Sefasebis amocanebis
Sesaxeb, romelic radikalurad cvlis ganmaviTarebeli Sefasebis arssa da teqnologias.
kerZod, Sefasebis ZiriTad amocanas warmoadgens: a) aCvenos, rogor mimdinareobs moswavlis
codnis konstruirebis procesi da mexsierebaSi codnaTa urTierTdakavSireba. b) axali
sakiTxis dawyebamde daadginos moswavlis winare codna da warmodgenebi. g) gamoavlinos,
ramdenad axerxebs moswavle sakuTari Zlieri da susti mxareebis damoukideblad Sefasebas,
ramdenad gaazrebul da efeqtian nabijebs dgams sakuTari winsvlis xelSesawyobad.
rogorc cnobilia, arsebobs Sefasebis ori forma _ ganmaviTarebeli da ganmsazRvruli
(maT Soris – diagnostikuri). I-IV klasebSi pirveli formiT SemovifarglebiT _ vexmarebiT
moswavleebs ganviTarebaSi sxvadasxva rCevis, rekomendaciis, Tu problemis gadaWris gzebis
SemuSavebis saSualebiT. magaliTad, saklaso samuSaoebi, rogorc wesi, interaqtiuli formiT,
kiTxva-pasuxis `reJimSi~ mimdinareobs _ am dros moswavleTa pasuxebis analizi, Sesworeba da
rCeva metad mniSvnelovan rols asrulebs maTi ganviTarebis saqmeSi. mniSvnelovnad gvesaxeba
Cveni rCevebi maswavleblis mimarT, saswavlo procesSi wamoWrili sakiTxebis ganxilvisas
Tavad moswavleTa iniciativis waxaliseba, maT mier sakuTari poziciis dafiqsireba,
SeniSnuli xarvezebis sajarod aRniSvna, gansxvavebuli mosazrebebis SemoTavazeba,
bundovani sakiTxebis xelmeored ganxilvis moTxovna. moswavleTa Sefasebis efeqturad
ganxorcielebis saqmeSi metad mniSvnelovania saxelmZRvaneloSi warmodgenil davalebaTa
mravalferovneba _ diagramebisa da cxrilebis wakiTxva, monacemebis analiziT daskvnebis
gadmocema; ricxviTi gamosaxulebebis mniSvnelobebis povna sxvadasxva xerxiT; geometriuli
figurebis amocnoba, dasaxeleba, gamoyeneba yoveldRiur cxovrebaSi. `moisazres~ da `vip~-
is rubrikiT warmodgenili amocanebis amoxsnisas msjeloba-dasabuTebis unaris gamovlena;
jgufuri muSaobis proeqtebi, didaqtikuri TamaSebi _ romelTa Catarebisas fokusSi eqceva
iseTi mniSvnelovani unar-Cvevebi, rogoricaa SemoqmedebiToba, TanamSromloba (jgufis
wevrebTan). gvaqvs davalebebi figurebis konstruirebaze, amocnobaze, figuraTa dayofaze,
sivrculi sxeulebis aRqmaze, sazomi erTeulebis gamoyenebaze, miTiTebuli kriteriumebiT
monacemTa dalagebasa, dajgufebasa da klasifikaciaze. moswavleTa Sefasebisas ZiriTadad,
is meqanizmebi gamoiyeneba, romlebic saSinao davalebebis SesrulebiT, klasSi kiTxva-
pasuxSi aqtiuri da SemoqmedebiTi CarTviT, jgufur proeqtebSi da TamaSebSi nayofieri
monawileobiT ganisazRvreba.
16
moswavlis wignis mimoxilva
moswavlis saxelmZRvaneloSi Sinaarsis SerCevasa da gadmocemas Sesabamisi asakis
moswavleTaTvis misawvdomobis pripcipi udevs safuZvlad. es SerCeva Camoyalibebuli,
stabiluri birTvis bazaze xdeba. igi gafiltrulia maTematikis swavlebis gamocdilebiT,
tradiciuli Temebis Tanamimdevrobis axleburi gaazrebis, maTematikis erTianobidan
gamomdinare integraciis xarisxis gazrdisa da axali sakiTxebiT Sevsebis aucileblobis
gaTvaliswinebiT. Sinaarsobrivi mimarTulebebi saswavlo gegmis wliur programaSia war-
modgenili.
gaTvaliswinebulia moswavlis saxelmZRvanelos yvela funqcia, romelic literaturaSia
miTiTebuli (ix. mag., [2]); codnis gadacemis funqcia, unarebisa da kompetenciebis
ganviTarebis funqcia, codnis ganmtkicebis funqcia, codnis Sefasebis funqcia; codnis
integrirebis funqcia _ vertikaluri integracia (erTi da imave disciplinis farglebSi),
horizontaluri integracia (sxvadasxva disciplinis farglebSi), socialuri da
kulturuli aRzrdis funqcia. am mxriv mniSvnelovania mdgradi ganviTarebis principebis
ganxorcieleba _ genderuli Tanasworoba, usafrTxo garemoze zrunva, jansaRi cxovrebis
wesze zrunva, qalaqebisa da sxva dasaxlebuli punqtebis mdgradi ganviTareba, energiis
axali wyaroebis gamoyeneba. saxelmZRvaneloSi warmodgenili masala iTvaliswinebs am
moTxovnebs _ genderuli Tanasworobaa daculi Cvens `personaJebSi~, saqarTvelos sazRvao
da mTis kurortebze monacemebis Segroveba, saubari saqvelmoqmedo fondebis mier gaweuli
daxmarebebis Sesaxeb; saubari gafuWebuli onkanidan wylis danakargebis raodenobis
dadgenaze (saxelmZRvaneloSi ganxiluli amocana), saubrebi sportze, xelovnebis nimuSebis
gamofenebze, sasoflo-sameurneo samuSaoebze. specialuri rubrikac ki gvaqvs: `sxvadasxva~,
romliTac moswavleebi imaRleben TavianT ganaTlebas sxvadasxva kulturuli movlenebis
Sesaxeb. gvaqvs specialuri proeqti saklaso oTaxSi wesrigis, sisufTavis SenarCunebis
mizniT morigeobis Sesaxeb, wesrigis, sisufTavis uzrunvelyofis aucileblobis Sesaxeb.
amocanebSi xSiria miTiTeba xeebis dargvaze, usafrTxo moZraobis uzrunvelyofaze, sakuTari
istoriis codnis aucileblobaze; sxva qveynebis socialur-kulturul daxasiaTebaze.
msoflio mniSvnelobis kulturuli Zeglebis Sesaxeb (am ZeglebSi geometriuli figurebis
amocnoba).
vcdilobT saxelmZRvaneloSi ar iyos gadaumowmebeli informacia; yvela informacia
iyos swori, gansazRvrebebi gasagebi.
Cveni wigni novatorulia _ igi ar hgavs arc erT adre sxva avtorebis mier (CvenTan, an
sazRvargareT) Seqmnil saxelmZRvanelos.
did yuradRebas vaqcevT teqstis aRqmadobis sakiTxs.
teqsts aqvs mkafio struqtura; advili gasamijnia dasawyisi, Sua nawili da dasasruli.
yoveli saTauri Seesabameba Sinaarss da e. w. `cocxali~ formulirebiT gamoirCeva;
magaliTad, `gavixsenoT erTidan aTasamde naturaluri ricxvebis Caweris wesi~, `vpoulobT
Sekrebisa da gamoklebis ucnob komponents~, `gveyofa fuli sayidlebisTvis?~ `SevarCioT
Sekrebis wesi~, `gamravlebis Tvisebebi, gavixsenoT gamravlebis cxrili~. Tumca, aris mokle
dasaxelebebic, magaliTad, `masis erTeulebi~, `piqtograma~, `svetovani diagrama~. TamaSebic
`cocxali~ dasaTaurebiT gamoirCeva, `vis Sexvdeba ukanaskneli figura?~
yuradRebas vaqcevT, rom savarjiSoebis tipebi `dabalansebulad~ iyos warmodgenili
_ codnis, aq erTi mxriv, SemecnebiTi (kognituri), motoruli da qceviTi unarebis
17
ganviTarebaze mimarTuli savarjiSoebis dabalansebazea saubari, meore mxriv, sirTulis
doneebis mixedviT dabalansebaze.
maswavlebels mudam unda asxovdes, rom maTematika erTiani mecnierebaa da ara misi
sxvadasxva nawilis xelovnuri gaerTianeba, amitom gadmocemis meTodika iTvaliswinebs
sakiTxebis urTierTkavSirs. svetovani diagramebis gamoyeneba naturaluri ricxvebisa da
maTze moqmedebebis Seswavlis procesSi ukve I TavSia warmodgenili, Tumca, monacemTa
analizisa da statistikis elementebi me-5 TavSic ganixileba, geometriisa da algebris
elementebis ganxilva mTeli kursis ganmavlobaSi mimdinareobs. am Temebis zogierTi nawili
koncentrulad aris warmodgenili. veTanxmebiT erovnuli saswavlo gegmis avtorebs am
etapze algebruli simbolikis farTo gamoyenebisgan Tavis SekavebasTan dakavSirebiT. es
problema araerT rusul saxelmZRvaneloSi SeiniSneba (ix., mag., [6], [7], [18]).
davalebebi iseTi TanamimdevrobiTaa warmodgenili, rom martividan rTulisken iyos
mimarTuli. calkea gamoyofili davalebebi saklaso da saSinao muSaobisTvis, savarjiSoTa
mravalferovani sistema (`testebi~, amocanebi jgufuri muSaobisTvis, amocanebi-TamaSebi,
damatebiTi amocanebi gameorebisTvis) xels uwyobs moswavleTa kritikuli azrovnebis
ganviTarebas, maTematikis Segnebulad Seswavlas. praqtikasTan xSiri mimarTeba praqtikuli
gamoyenebis unar-Cvevebis gamomuSavebisa da maTematikis Seswavlisadmi interesis gaR-
rmavebas uwyobs xels. `testebi~ isea SerCeuli, rom isini upirvelesad moswavleTa
mier situaciaSi swrafad garkvevisa da pasuxebis sworad SerCevis unars amowmebs da
ara amoxsnebis detaluri Catarebis proceduras. didi xania mimdinareobs dava am tipis
savarjiSoTa dadebiTi da uaryofiTi mxareebis Sesaxeb. Tumca, miuxedavad imisa, rom am
tipis amocanebSi warmoCenilia mxolod pasuxis SerCevis etapi, maswavleblis Sexedulebis
mixedviT SeiZleba amocanis amoxsnis etapzec iyos msjeloba, amitom aseTi tipis amocanebi
SeiZleba saswavlo miznebisTvisac gamodges da TviTSemowmebisTvisac.
axla SevexoT moswavlis wignis agebulebas: wignis dasawyisSi avtorTa madlobas
eqspertebisa da maswavleblebisadmi, trening-seminarebis organizatorebisadmi meoT xek -
laselTa mimarT misalmeba da keTili survilebi mosdevs. aqvea rCevebi wigniT sargeb-
lobisTvis.
saxelmZRvanelo eqvsi Tavisgan Sedgeba. TiToeuli Tavi _ paragrafebisgan. zogjer
paragrafi ornawiliania. yoveli Tema mTavrdeba Semajamebeli amocanebiT _ gameorebisa
da codnis ganmtkicebisTvis.
saxelmZRvanelo moswavlis wignisadmi wayenebuli yvela moTxovnis Sesabamisadaa
agebuli. sarCevi wignis dasawyisSia, maTematikuri simboloebis cxrili, maTematikuri
terminebis leqsikoni, zomis erTeulebi da sxva sacnobaro masala wignis boloSi.
moswavlis saxelmZRvanelo swavlebis procesis dagegmvisa da warmarTvis erT-erTi
saSualebaa. amasTanave, moswavleebTan saubari SeiZleba wignis teqstisgan gansxvavebuladac
aigos. saxelmZRvanelo exmareba maswavlebels _ masSi dafiqsirebulia is SemecnebiTi
saqmianoba, romelic moswavlem unda awarmoos.
moiTxoveba, rom moswavlis wigni ar Seicavdes masalas, romelic moiTxovs saxel-
mZRvaneloSi raimes Caweras, Caxazvas.
18
moswavlis saxelmZRvanelos ZiriTadi konceptualuri principebia:
• kursis Semecnebis ZiriTadi gza induqciuria;
• axali codnis daufleba problemuri situaciis saSualebiT iwyeba;
• maTematikuri mimarTebebis Tvisebebis Seswavla TvalsaCinod warmodgenili sqemiT
mimdinareobs;
• mniSvneloba eniWeba saswavlo amocanis sxvadasxva xerxiT amoxsnas, am xerxebis
SedarebiT analizs, arCevanis gakeTebas da am arCevanis dasabuTebas. es mniSvnelovani
aqtivobebia kritikuli azrovnebis unaris gasaviTareblad.
• saxelmZRvanelo amkvidrebs sxvadasxva formiT mimdinare jgufur da individualur
muSaobas.
moswavlis wignSi Temebi 6 TavSi integrirebuladaa warmodgenili. magaliTad, me-6 TavSi
monacemTa analizi (maT. IV. 11 _ maT. IV. 13), masis erTeulebi (maT. IV. 5) da geometriis
sakiTxebia warmodgenili (maT. IV. 8., maT. IV. 9). erT-erTi mniSvnelovani Tema _ Sesabamisoba
(maT. IV. 6) III TavSia mocemuli, aqvea naturaluri ricxvebis gamravlebisa da gayofis
sakiTebi (maT. IV. 3). naturaluri ricxvebis gamosaxvis, Sedarebisa da dalagebis sakiTxebi
(maT. IV. 1., maT. IV.2) I da II TavebSia gadanawilebuli. Tumca, I Tavi, ZiriTadad, wina wels
Seswavlilis gameorebas da aTasamde ricxvebis Sesaxeb sakiTxebs eZRvneba. meoTxe TavSi
integrirebuladaa warmodgenili mTelis nawilebi da drois erTeulebi, marSruti da
marSrutis sigrZe (maT. IV.4., maT. IV.5., maT. IV.10., maT. IV.11).
amrigad, saxelmZRvanelo srulad faravs IV klasis standartis yvela moTxovnas.
19
rCevebi wigniT sargeblobisTvis
waikiTxeT es rCevebi, gaecaniT specialuri simboloebis daniSnulebas.
es simbolo klasSi Sesasrulebeli samuSaoebis dawyeba-
ze migviTiTebs. am amocanebis nomrebi wreebSia Casmu-
li, feradicaa, magaliTad, 4 .
es simbolo Sin muSaobisTvis gankuTvnil amocanebs
warmogvidgens. am amocanebis nomrebi samkuTxedebSia
Casmuli, magaliTad, 5 .
`sxvadasxva~ _ saintereso faqtebsa da istoriebs
warmogvidgens.
am simbolos mosdevs amocanebi, romelTa amosaxsnelad
xSi rad sakmao gamWriaxobis gamovlenac mogiwevT. am amo-
canebis nomrebi kvadratebSia Casmuli, magaliTad, 1 .
am simboloTi jgufuri muSaobis proeqtebia gam-
oyofili _ aseTi davalebebi TanaklaselebTan er-
Toblivad muSaobis gamocdilebas SegmatebT.
am simboloTi TamaSebia aRniSnuli – maTi saSualebiT
maTematikuri codnis gamoyenebis gamocdilebas SeiZ-
enT.
vin iqneba pirveli. am niSnakiT warmodgenili amo-
canebis amoxsnisas ecadeT, gamWriaxobasTan erTad
sxa rti azrovnebac gamoavlinoT.
teqstis ukeT gagebis mizniT saWirod CavTvaleT zo-
gierTi sityvis warmomavlobis, misi msgavsi mniSvne-
lobebis miTiTeba. aseTi Canawerebi teqstSi moniSnu-
lia aq gamosaxuli niSnakiT.
am simboloTi gamoyofili yvela amocana unda gadaita-
noT rveulSi da DSemdeg SeasruloT.
Cveni
leqsikoni
jgufuri
saSinao
moisazre
sxvadasxva
TamaSi
i
klasSi
rv
eul
Si!
gaufrTxildiT wigns!
masSi Tqvenma Canawerebma (amoxsnebma, miTiTebebma) wigni SeiZleba sxvebisTvis
gamousadegari gaxados!
avtorTa mier moswavlis wignSi gamoyenebuli
pirobiTi niSnebis ganmartebebi
20
maswavleblis wignis praqtikuli nawili
maswavleblis wignis Sedgenis rekomendaciebSi, romelic Cven ganaTlebisa da
mecnierebis saministrodan miviReT (`maswavleblis wignis sarekomendacio struqtura~),
miTiTebuli iyo, rom gvevaleboda yoveli Tavis aranakleb 2 gakveTilis scenaris war-
modgena. Cven gaviTvaliswineT maswavlebelTa Txovna, romliTac isini wina wlebSi mog-
vmarTavdnen da maswavleblis wignSi yvela gakveTilis Catarebis sqema warmovadgineT.
meTodikur literaturaSi, rogorc wesi, sakiTxis swavlebis savaraudo sqemas warmoadgenen
xolme. sakiTxis Seswavlas ki, magaliTad, samfaziani konstruqtivistuli strategiis
ganxorcielebis SemTxvevaSi, rogorc wesi, ramdenime gakveTili eTmoba. am SemTxvevaSi
sagnis swavlebis sqema SeiZleba ase gamoiyurebodes.
maswavlebeli
sagani
maTematika
klasi IV
Tema: vuTiTebT Temis saTaurs dro: vuTiTebT gakveTilebis raodenobas
gakveTilebis mizani/Sedegebi (codna, unar-Cvevebi, damokidebulebebi)
gakveTilebis mizani/Sedegi: moswavleebi SeZleben gaiziaron...
erovnuli saswavlo gegmis standarti
maT. IV. 10. amoviwerT indikators da misaRwev Sedegebs
moswavleTa organizaciis formebi: individualuri, wyvilebi, jgufuri, mTeli klasi
gakveTilze gamoyenebuli ZiriTadi meTodebi da aqtivobebi: konstruqtivistuli meTodi
(interaqcia, keTebiT swavleba, skalfoldingi)
saswavlo masala da resursebi:
vuTiTebT, magaliTad, fanqari, sigrZis sazomi lenti, kuburebi
diferencirebis SemTxvevaSi SeiZleba mivuTiToT, rogor vanawilebT sxvadasxva tipis da
sirTulis masalas
saWiro winare codna da unar-Cvevebi:
sakiTxis swavlebis msvleloba, magaliTad, samfaziani modelis mixedviT:
1. `winaswar~ (mivuTiToT dro). aqtivobis mizania moswavleTa winare codnis gaaqtiureba,
motivaciis amaRleba da jgufuri muSaobis unar-Cvevebis ganviTareba, radgan moxdes axali
codnis konstruirebisaTvis xelSewyoba.
21
aqtivoba 1.maswavlebeli amowmebs saSinao davalebis Sesrulebas. moswavleebi amowmeben Sesru-
lebuli samuSaos Sesrulebis xarisxs maswavleblisa da Tanaklaselebis daxmarebiT.
SerCevis raime wesis gamoyenebiT maswavlebeli iZaxebs dafasTan moswavles da igi wers
im amocanis amoxsnas saSinao davalebidan, romelic maswavlebelma SearCia. moswavleebi
adareben TavianT pasuxebs da raime xerxiT afiqsireben TavianT pozicias. maswavlebeli
im moswavleebs, romlebsac uSecdomod hqondaT Sesrulebuli davaleba, aZlevs damatebiT
samuSaos saxelmZRvanelodan. xolo is moswavleebi, romlebsac Secdomebi hqondaT,
jguf debian erT magidasTan, maswavlebeli skalfoldingis daxmarebiT asworebinebs maT
Secdomebs. Semdeg maswavlebeli ubrundeba damoukideblad momuSave moswavleebs da
amowmebs maT mier Sesrulebul samuSaos (kiTxvebis dasmis saSualebiT).
II. ganmavlobaSi (vuTiTebT savaraudo dros). aqtivobis mizania moswavleebma moaxdinon
codnis konstruireba.
axali Temis garCeva kvlav kiTxva-pasuxis reJimSi xdeba da SeiZleba ramdenime aqtivobad
daiSalos. SeiZleba pirveli aqtivoba iyos cxovrebiseuli magaliTebi, romlebsac
maTematikuri procedurebis ganxilvis Semdeg (II aqtivoba) SeiZleba mivubrundeT. amasTanave,
SeiZleba gamoviyenoT mecadineobis sxvadasxva forma, romelic miTiTebuli unda iyos.
III. `Semdgom~ (vuTiTebT dros). am fazaze xdeba miRebuli codnis ganmtkiceba.
Tu maswavlebels aqvs monacemebi moswavleebis sxvadasxva profilis mixedviT
ganawilebis Sesaxeb, maSin am monacemebis mixedviT xdeba jgufebis Sedgena da davalebebis
micema. Sesrulebis prezentacia SeiZleba Seicavdes amocanis amoxsnas, cxrilebis Sedgenas,
teqstis saxiT pasuxebis gadmocemis (magaliTad, axali cnebebis gansazRvrebebi). drois
amowurvis Semdeg moswavleebi akeTeben prezentacias. maswavlebeli ajamebs gakveTils,
aZlevs saSinao davalebas.
es sqema SeiZleba gansxvavdebodes Cven mier mowodebuli gakveTilebis scenarebidan.
Tumca, am scenarebSi advilad ikiTxeba fazebi da sxva niuansebi, romlebic zogad sqemaSi
iyo warmodgenili. gansxvaveba gamowveulia im garemoebis gamo, rom Cven gadavwyviteT
yoveli gakveTilis Catarebis sarekomendacio sqema warmovgvedgina: ase iyo miTiTebuli
ganaTlebisa da mecnierebis saministrodan miRebul dokumentSi: `maswavleblis wignis
praqtikuli nawili, gakveTilis scenarebi, ...~.
22
Tavi I
gameoreba. naturaluri ricxvebi aTasamde
am TavSi warmodgenili masalis swavlebas gamorCeuli mniSvneloba eniWeba saswavlo
procesSi. migvaCnia, rom sazafxulo ardadegebis Semdeg moswavleTa CarTva saswavlo
procesSi unda ganxorcieldes III klasSi Seswavlili ZiriTadi sakiTxebis gameorebiT.
umartivesi samotivacio amocanebis ganxilviT xdeba moswavleTa Semzadeba II Tavis
masalis, kerZod, milionamde naturaluri ricxvebis gasacnobad. yuradRebas vamaxvilebT
ricxvTa poziciur (aTobiT) sistemaSi Tanrigebis arsze. ganvixilavT naturalur ricxvTa
miaxloebiT mniSvnelobebs, damrgvalebas da maT gamoyenebebs. mniSvnelovania agreTve
naturalur ricxvTa Sedarebisa da dalagebis Sesaxeb warmodgenebis gameoreba. Tavis
Seswavlas davasrulebT damatebiTi amocanebis ganxilviT.
martivi amocanebis siWarbe, vfiqrobT, aamaRlebs saswavlo procesSi moswavleTa
CarTulobis xarisxs. aqvea aratrivialuri amocanebic, romelTa amoxsna xSirad sakmao
gamWriaxobas moiTxovs.
§ 1. gavixsenoT erTidan aTasamde naturaluri ricxvebis Caweris wesi
am Temas SeiZleba 4 gakveTili davuTmoT
1-li gakveTili
Tema: aTasamde naturaluri ricxvebis weriTi numeracia.
mizani: ricxvebis gamosaxvisa da poziciuri sistemis demonstrirebis unaris ganviTareba.
cifris mniSvnelobis dasaxeleba (esg, wlis bolos misaRwevi Sedegebi da indikatorebi, maT. IV.
1.).
winapirobebi: codna da gamocdileba, romelsac unda flobdes moswavle masalis
Sesaswavlad, maT Soris kavSirebi sxva sagnebSi naswavl masalasTan:
aTobiTi poziciuri sistemis sxvadasxva modeliT demonstrirebis, aTasamde ricxvis
saTanrigo Sesakrebebis jamis saxiT warmodgenis unarebi.
moswavle unda flobdes kiTxvis teqnikas, xmamaRla kiTxulobdes mcire zomis teqstebs,
icavdes pauzas sasven niSnebTan (esg, qarT. III.4.).
resursebi: kuburebi, Cxirebi.
poziciuri sistemis gaazrebasTan dakavSirebuli aqtivobebi III klasis saxelmZRvaneloSi
iyo mocemuli. SeiZleba im aqtivobebis gameoreba. SemdgomSi moxdeba am aqtivobebis ganvrcoba
1000-ze meti ricxvebis SemTxvevaSi. 1000-mde ricxvebis demonstrireba SeiZleba damxmare
saswavlo resursebis gamoyenebiT (kuburebi, Cxirebi). yoveli kubura _ _ erTi erTeulia.
aTi aseTi kubura _ aTeulia , asi aseTi kubura _ aseulia.
23
SeiZleba dafaze gamovsaxoT erTeulebi, aTeulebi da aseulebi.
sxvadasxva konstruqciis warmodgenis, an dafaze, an kompiuterSi
gamosaxvis Semdeg moswavleebi asaxeleben am konstruqciiT
warmodgenil ricxvs, Caweren am ricxvs da uTiTeben mis CanawerSi
TiToeuli cifris mniSvnelobas.
es aqtivobebi ramdenjerme unda gavimeoroT. ori ricxvis
warmodgena suraTzea gamosaxuli.
aseulebi aTeulebi erTeulebi warmodgenili ricxvi
111
234
erTi aqtivobidan meoreze gadasvla umtkivneulod warimarTeba, roca aqtivobebi
logikurad ebmis erTmaneTs. am SemTxvevaSi moswavleebs ar uWirT erTi aqtivobidan meoreze
gadarTva, maswavleblis instruqciebic SedarebiT martivia da nakleb dros moiTxovs.
moswavlis saxelmZRvaneloSi warmodgenili magaliTi logikurad ebmis im TvalsaCino
warmodgenebs, romlebiTac daviwyeT gakveTili. winapirobebis gaTvaliswinebiT, moswavleebs
unda SeeZloT teqstis wakiTxva da gaazreba. Tumca, teqstis wakiTxvis Semdeg misi gaazreba
Cven mier dasmuli kiTxvebis gamoyenebiTac mimdinareobs:
_ kalmebis ra raodenobaa warmodgenili 2 kolofiT, 4 paketiTa da kidev 3 kalmiT?
aq SeiZleba damatebiTi kiTxvebic dagvWirdes:
_ ra raodenobaa warmodgenili 2 kolofiT? (200)
24
_ ra raodenobaa warmodgenili 4 paketiT? (40)
_ kidev ramdeni kalami gvaqvs? sul ramdeni gvaqvs? (243).
aseulebis, aTeulebis, erTeulebis cifrebis mniSvnelobebsac moswavleebi asaxeleben.
SeiZleba moswavlis dafasTan gamoZaxebac da aRniSnuli ricxvis saTanrigo Sesakrebebis jamis
saxiT Cawera. aqve SeiZleba vTxovoT sxva moswavles moZebnos saxelmZRvaneloSi pasuxebi
dasmul kiTxvebze: (cifrebis mniSvnelobebis dasaxeleba, saTanrigo Sesakrebebis jamad
warmodgena).
miuxedavad imisa, rom mimdinareobs III klasSi Seswavlili masalis gameoreba da ganmtkiceba,
bevri ram SeiZleba moswavleebs daviwyebuli hqondeT. saswavlo procesi unda mimdinareobdes
moswavleTa aqtiuri monawileobiT. romelime moswavlesTan xangrZlivi dialogis dros
dundeba sxvaTa yuradReba da masalis aRqmac uxarisxoa. yoveli aqtivobisas moswavleTa Cabmis
mzaoba maqsimaluri unda iyos.
yovel gakveTilze mimdinareobs codnis ganmtkicebis procesi. damatebiTi aqtivobebis
saxiT SeiZleba ganvixiloT samniSna ricxvis warmodgenis da Caweris sxva SemTxvevebi.
magaliTad, ganvixilavT samniSna ricxvs _ 432-s. SevarCioT kiTxvebi da moswavleTa pasuxebis
mixedviT warmovadginoT, CavweroT, wavikiTxoT es ricxvi.
_ vTqvaT, gvaqvs kalmebis 4 kolofi, 3 paketi da kidev 2 kalami. kalmebis ra raodenoba
gveqneba; (432)
_ ra aris am ricxvSi TiToeuli cifris mniSvneloba?
_ CawereT es ricxvi saTanrigo Sesakrebebis jamis saxiT.
_ ramdeni cifrisgan Sedgeba es ricxvi?
_ ras vuwodebT sami cifriT Caweril ricxvs? ori cifriT Caweril ricxvs? (samniSna
ricxvs, orniSna ricxvs).
individualuri midgomis gaTvaliswinebiT, moswavleebs vavalebT I Tavis damatebiTi
amocanebidan 1 - 3 amocanebis amoxsnas – samniSna ricxvSi cifrebis mniSvnelobebis
miTiTeba, saTanrigo Sesakrebebis jamad warmodgena, raime raodenobis ricxviT warmodgena
da am ricxvis poziciur sistemaSi Caweris analizi. klasis interess gamoiwvevs 4 davaleba
amave amocanebidan. is klasis erToblivi ZalisxmeviT da Tqveni komentarebiT daiZleva.
mniSvnelovania saswavlo Sedegebis Semowmeba, romlis mixedviTac moxdeba dasaxuli
miznis miRwevis donis Sefaseba. am mizniT gakveTilis bolos vamowmebT moswavleebis mier
Sesrulebul samuSaos; vuTiTebT Secdomebs, vakeTebT komentarebs, zogjer vaxalisebT da
aRvniSnavT Sesrulebuli samuSaos Zlier mxareebs. moswavleebs qulebiT ar vafasebT.
saSinao davalebad vaZlevT I Tavis damatebiTi amocanebidan 5 - 9 davalebebs.
25
me-2 gakveTili
Tema: aTasamde ricxvebis Cawera, wakiTxva.
mizani: 1000-mde ricxvis poziciuri warmodgenis gameoreba, cifris mniSvnelobis
miTiTebis unaris ganviTareba (maT. IV. 1.)
winapirobebi: sxvadasxva modeliT aTasamde ricxvis poziciuri sistemis gamoyenebiT
gamosaxvis Cvevebi.
resursebi:^ kuburebi, saswavlo Cxirebi, saxelmZRvanelo.
am gakveTilze kvlav vubrundebiT saxelmZRvaneloSi warmodgenili magaliTis ganxilvas
da kiTxva-pasuxis reJimSi warmovadgenT kalmebis sxvadasxva raodenobis samniSna ricxviT
Caweris SemTxvevebs poziciuri sistemis gamoyenebiT, saTanrigo Sesakrebebis jamis saxiT
(gaSlili formiT) ricxvis warmodgenas, cifrebis mniSvnelobaTa povnas.
am Semzadebis Semdeg moswavleebi mzad arian 1-11 amocanebSi swori pasuxebis SesarCevad.
Cven SesavalSi ukve visaubreT am tipis (e. w. testuri) davalebebis saswavlo procesSi CarTvis
mniSvnelobaze; aseTi davalebebis Sesrulebas dabali klasebidanve unda mieCvios moswavle.
amasTanave, maTi Sesruleba codnis gamovlenasTan erTad dakavSirebulia sxvadasxva
mniSvnelovani unaris ganviTarebasTanac (pasuxebis swrafi analizi, alRo, mixvedra).
magaliTad, 1 amocanis amoxsnisas 648 migviTiTebs, rom 4 savaraudo pasuxidan unda
SevarCioT 3), radgan 648 aris 6 aseuli, kidev 4 aTeuli da kidev 8 erTeuli. 2 davalebis
swori pasuxia 4), radgan 208 Seicavs 2 aseuls da kidev 8 erTeuls.
analogiuria 3 da 4 amocanebi, amitom 1 da 2 -is Sesaxeb erToblivi muSaobis
Semdeg es ori amocana gamodgeba dasaxuli miznis miRwevis SefasebisTvisac _ sasurvelia am
or amocanaSi moswavleebma damoukideblad SearCion pasuxebi.
5 amocana e. w. orsafexuriania _ moswavlem unda SeZlos 803-is wina ricxvis (802-is)
Cawerac da misi saTanrigo Sesakrebebis jamis saxiT warmodgenac _ masSi aseulebis, aTeule-
bisa da erTeulebis odenobaTa miTiTeba.
6 amocanaSi ricxvi sityvieradaa warmodgenili; 7 amocanac analogiuria. am or
davalebasac moswavleebi damoukideblad asruleben; iseve, rogorc, 8 da 11 davalebebs,
radgan wina gakveTilze, III klasis masalis gameorebis dros, analogiuri davalebebi ukve
Sesrulebulia.
9 da 10 amocanebSi cifris mniSvnelobis miTiTeba moiTxoveba. maTi Sesrulebis
Semdeg SeiZleba vTxovoT moswavleebs dasaxelebul ricxvebSi sxva cifrebis mniSvnelobebis
miTiTebac.
am davalebebis dawvrilebiTi ganxilvis Semdeg SeiZleba SevudgeT 12 da 13 amocanebis
amoxsnas. am amocanebSi warmodgenili davalebebidan nawils erToblivi ZalisxmeviT, mTeli
klasis CarTulobiT vxsniT, danarCens moswavleebs vaZlevT damoukidebeli muSaobisTvis.
TxoveT moswavleebs Tavad daasaxelon ricxvebi da klasma damoukideblad ganixilos
analogiuri amocanebi.
saSinao davalebad vaZlevT analogiur amocanebs: 1 - 13 .
