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guram gogiSvili, Teimuraz vefxvaZe<br />

ia mebonia, lamara qurCiSvili<br />

m a T e m a t i k a<br />

<strong>IV</strong> <strong>klasi</strong><br />

<strong>maswavleblis</strong> <strong>wigni</strong><br />

redaqtori Teimuraz vefxvaZe<br />

gamomcemloba inteleqti<br />

Tbilisi 2011

guram gogiSvili, Teimuraz vefxvaZe,<br />

ia mebonia, lamara qurCiSvili<br />

m a T e m a t i k a<br />

<strong>IV</strong> <strong>klasi</strong><br />

<strong>maswavleblis</strong> <strong>wigni</strong><br />

gamomcemloba inteleqti<br />

Tbilisi 2011<br />

Guram Gogishvili, Teimuraz Vepkhvadze,<br />

Ia Mebonia, Lamara Kurchishvili<br />

Maths <strong>IV</strong><br />

Teacher's Book<br />

Intelekti Publishers<br />

Tbilisi 2011<br />

ISBN 978-9941-430-17-6<br />

© gamomcemloba `inteleqti~, 2011.<br />

gamomcemloba inteleqti<br />

Tbilisi, ilia WavWavaZDis gamz. 17b. tel.: 25 05 22, 91 22 83.<br />

www.intelekti.ge info@intelekti.ge inteleqti@caucasus.net<br />

intelekti Publishers<br />

17b Ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 22) 25 05 22, 91 22 83.

sarCevi<br />

Sesavali.........................................................................................................................................................5<br />

wlis bolos misaRwevi Sedegebi.........................................................................................................10<br />

Sinaarsisa da miznebis ruka.................................................................................................................14<br />

saswavlo masalis wardgenis fazebi da gakveTilis dagegmvis ZiriTadi principebi....17<br />

Sefasebis zogadi principebi...............................................................................................................20<br />

1. gameoreba; naturaluri ricxvebi erTidan aTasamde<br />

1. gavixsenoT erTidan aTasamde naturaluri ricxvebis Caweris wesi...............................21<br />

2. ricxvebis Sedareba da dalageba...................................................................................................22<br />

sanimuSo gakveTili # 1...................................................................................................................23<br />

sanimuSo gakveTili # 2...................................................................................................................25<br />

3. ricxvebis damrgvaleba......................................................................................................................26<br />

damatebiTi amocanebi........................................................................................................................28<br />

2. naturaluri ricxvebi milionamde<br />

1. CavweroT aTasze meti naturaluri ricxvebi..........................................................................29<br />

2. SevadaroT da davalagoT naturaluri ricxvebi...................................................................30<br />

3. naturalur ricxvis damrgvalebis wesi.....................................................................................31<br />

4. viyenebT svetovan diagramas..........................................................................................................33<br />

damatebiTi amocanebi........................................................................................................................34<br />

sanimuSo gakveTili # 3...................................................................................................................35<br />

3. naturaluri ricxvebis Sekreba da gamokleba<br />

1. orniSna ricxvebis Sekreba.............................................................................................................38<br />

2. jamis Sefaseba damrgvalebiT.......................................................................................................39<br />

3. didi ricxvebis Sekreba...................................................................................................................41<br />

4. SevarCioT Sekrebis wesi.................................................................................................................43<br />

sanimuSo gakveTili # 4.................................................................................................................44<br />

5. orniSna ricxvebis gamokleba.......................................................................................................46<br />

6. sxvaobis Sefaseba damrgvalebiT.................................................................................................48<br />

sanimuSo gakveTili # 5.................................................................................................................50<br />

7. didi ricxvebis gamokleba.............................................................................................................56<br />

8. amocanebis amoxsna (SevarCioT moqmedeba); fulis odenobebis<br />

Sekreba da gamokleba.......................................................................................................................57<br />

proeqti.................................................................................................................................................59<br />

9. Sekrebisa da gamoklebis Tvisebebi. vipovoT ricxviTi gamosaxulebis<br />

mniSvneloba..........................................................................................................................................60<br />

10. vpoulobT Sekrebisa da gamoklebis ucnob komponents....................................................62<br />

11. gveyofa fuli sayidlebisTvis?...................................................................................................63<br />

3

4. dro, sigrZis gazomva, geometria<br />

1. dro (ra droa)......................................................................................................................................65<br />

2. mTelis nawilebi _ naxevari, meoTxedi, mesamedi..................................................................66<br />

sanimuSo gakveTili # 6...................................................................................................................67<br />

3. sigrZis erTeulebi.............................................................................................................................70<br />

TamaSi.......................................................................................................................................................72<br />

4. texili. texilis sigrZe. manZili. marSrutis sigrZe............................................................73<br />

5. naturaluri ricxvebis gamravleba da gayofa<br />

1. gamravlebis Tvisebebi. gavixsenoT gamravlebis cxrili....................................................75<br />

2. gamravleba da gayofa. gayofis Tvisebebi.................................................................................76<br />

TamaSi.......................................................................................................................................................78<br />

3. vamravlebT nulebiT daboloebul ricxvebs...........................................................................78<br />

4. vamrgvalebT Tanamamravlebs da vpoulobT namravls miaxloebiT.................................80<br />

sanimuSo gakveTili # 7...................................................................................................................81<br />

5. erTniSna ricxvze gamravleba........................................................................................................82<br />

6. orniSna ricxvze gamravleba...........................................................................................................84<br />

7. vamravlebT samniSna ricxvze..........................................................................................................85<br />

8. moqmedebaTa Tanamimdevroba...........................................................................................................87<br />

9. gayofa naSTiT......................................................................................................................................88<br />

TamaSi.......................................................................................................................................................89<br />

sanimuSo gakveTili # 8...................................................................................................................89<br />

10. ganayofs miaxloebiT vpoulobT.................................................................................................91<br />

11. erTniSna ricxvze gayofa...............................................................................................................92<br />

jgufuri proeqti.............................................................................................................................93<br />

12. Sesabamisoba.........................................................................................................................................93<br />

13. gamravlebisa da gayofis ucnobi komponentis povna..........................................................95<br />

14. amocanebis amoxsna (amocanebi zedmeti mocemulobiT)......................................................95<br />

damatebiTi amocanebi......................................................................................................................96<br />

6. amocanebis amoxsna. geometria. monacemTa analizi<br />

1. masis erTeulebi...................................................................................................................................97<br />

2. amocanebis amoxsna (amocanebi nakluli monacemebiT)..............................................................99<br />

3. kubi. prizma. piramida.......................................................................................................................101<br />

TamaSi ....................................................................................................................................................102<br />

proeqti.................................................................................................................................................102<br />

sanimuSo gakveTili #9....................................................................................................................103<br />

4. amocanebis amoxsna (viyenebT naSTiT gayofas)..........................................................................105<br />

sanimuSo gakveTili #10..................................................................................................................107<br />

5. cilindri. konusi. birTvi...............................................................................................................109<br />

TamaSi. vis Sexvdeba ukanaskneli figura?..................................................................................109<br />

6. metobiT davamrgvaloT ricxvi Tu naklebobiT?.....................................................................110<br />

jgufuri proeqti .............................................................................................................................112<br />

7. SevagrovoT monacemebi....................................................................................................................113<br />

8. monacemebis Segroveba da organizacia.......................................................................................114<br />

9. piqtograma...........................................................................................................................................115<br />

10. svetovani diagrama...........................................................................................................................117<br />

literatura.......................................................................................................................................118<br />

4

Sesavali<br />

<strong>maswavleblis</strong> sarekomendacio <strong>wigni</strong>s daniSnulebaa miawodos maswavleblebs meTodi kuri<br />

rekomendaciebi axali erovnuli saswavlo gegmis mixedviT Sedgenili <strong>IV</strong> <strong>klasi</strong>s saxelmZRvanelosTvis<br />

(guram gogiSvili, Teimuraz vefxvaZe, ia mebonia, lamara qurCiSvili, <strong>maTematika</strong><br />

<strong>IV</strong>); daexmaros maT saswavlo procesis warmarTvis meTodikuri xerxebis SemuSavebaSi; miawodos<br />

damatebiTi masala, rekomendaciebi Sefasebis sistemis SemuSavebisTvis.<br />

<strong>maswavleblis</strong> sarekomendacio wignSi Tavebisa da paragrafebis numeracia da dasaxeleba<br />

emTxveva moswavlis saxelmZRvaneloSi SemoRebul numeraciasa da dasaxelebas.<br />

sarekomendacio <strong>wigni</strong>s struqtura erovnuli saswavlo gegmebisa da Sefasebis centris<br />

mier SemuSavebuli principebiTaa agebuli _ warmodgenilia <strong>IV</strong> <strong>klasi</strong>s saswavlo gegma, Sinaarsisa<br />

da miznebis ruka, mocemulia Sesabamisi axsna-ganmartebebi masalis wardgenis fazebis<br />

Sesaxeb _ motivacia, sakiTxis dasma, sawyis etapze Sesasrulebeli savarjiSoebi da maTi<br />

analizi, Semdgom etapebze Sesasrulebeli savarjiSoebi, amocanebi da maTi analizi, damoukidebeli<br />

samuSaoebis gansazRvra da maTi Semowmebis formebi. mocemuli iqneba gakveTilebis<br />

dagegmvis sqemebi da ramdenime sanimuSo gakveTilis scenari.<br />

gaecnobiT komentarebs TiToeuli Tavis, paragrafis mixedviT, masalis gadmocemis Taviseburebebis<br />

motivacias, am masalis Sesabamis mecnierul da meTodologiur safuZvlebs,<br />

miTiTebuli iqneba Sesabamisi literatura yvela am Temis mixedviT codnis gasaRrmaveblad;<br />

wigns Tan erTvis samecniero da meTodikuri literaturis sia.<br />

zogadi mecnierul-meTodikuri principebi, romlebic gansazRvravs saxelmZRvanelos Sinaarssa<br />

da wyobas, aris: saxelmZRvanelos Sesabamisoba erovnul saswavlo gegmasTan, sistematuroba<br />

masalis dalagebaSi, Sinaarsis mecnieruloba, Teoriisa da praqtikis kavSiris principis<br />

ganxorcieleba, gadmocemis misawvdomoba, enis garkveuloba, sizuste da simartive, savarjiSoebisa<br />

da maTi Sesrulebis formebis mravalferovneba. tradiciuli da Tanamedrove Sexedulebebis<br />

sinTezis safuZvelze mravalferovania davalebebis tipebi, romlebic saswavlo<br />

saqmianobisa (saswavlo amocanis gageba da amoxsna, algoriTmis mixedviT moqmedeba, amoxsnis<br />

gegmis Sedgena) da inteleqtualuri unarebis ganviTarebisTvis meTodikuri pirobebis Seqmnas<br />

emsaxureba (analizis, SedarebiT, <strong>klasi</strong>ficirebis, TviTSemowmebis, codnis gamoyenebis,<br />

saswavlo amocanis sxvadasxva xerxiT amoxsnis, xelsayreli xerxis SerCevis, Sefasebis).<br />

davalebebis mravalferovneba swavlis stimulirebis erT-erTi mniSvnelovani aspeqtia.<br />

meoTxeklaseli asakiT Semecnebis ganviTarebis imave safexurzea, romliTac swavlebis pirveli<br />

sami weli xasiaTdeba. piaJes [21] mixedviT 6-dan 11 wlamde asakSi wlis periodSi daZleulia<br />

SezRudvebi problemis simboluri gadaWrisa da obieqtTa <strong>klasi</strong>fikaciis unaris gamovlenis<br />

kuTxiT. am dros moswavles aqvs unari awarmoos logikuri operaciebi im codnis safuZvelze,<br />

romelsac ukve flobs. Tumca, me-4 klasSi moswavle ufro momwifebulia imisTvis, rom ganaviTaros<br />

SemecnebiTi unarebi. me-4 <strong>klasi</strong>dan skolebSi xSirad icvleba midgomebi moswavleTa<br />

saswavlo procesisadmi. kerZod, swavlebis formebis mimarT. is ufro sagnobrivi xdeba. Tumca<br />

am procesSi memkvidreobiTobis pricnipis gauTvaliswinebloba, romelic damaxasiaTebeli<br />

iyo saskolo saxelmZRvaneloebisTvis, aZnelebda axali moTxovnebis ganxorcielebas. axali<br />

erovnuli saswavlo gegmis meTodikuri orientirebi gvexmareba swavlebis axal safexurze<br />

gadasvlis umtkivneulod ganxorcielebaSi _ roca wina planze adre naswavlis ganzogadeba<br />

gamodis. ganzogadebis es procesi Zvelis gameorebis procesSi xorcieldeba.<br />

5

swavlebis axal safexurze asvla naturalur ricxvTa Seswavlis axali koncentris ganxilvasa<br />

da am ganxilvis ufro maRal doneze warmarTvas gulisxmobs. es procesi Zvel koncentrebTan<br />

(aTeulebi, aseulebi, aTaseulebi) dakavSirebuli sakiTxebis _ numeracia, moqmedebebi<br />

_ gameorebasTan erTad mimdinareobs.<br />

swavleba mimdinareobs siZneleTa ufro maRal doneze _ zomierebis dacviT, siZnelis<br />

zoma ganisazRvreba individualuri midgomebis gaTvaliswinebiT.<br />

Cveni saxelmZRvanelos ZiriTadi konceptualuri principebia:<br />

. kursis Semecnebis ZiriTadi gza induqciuria.<br />

. axali codna ixsneba problemuri situaciis saSualebiT.<br />

. maTematikuri mimarTebebis Tvisebebis Seswavla TvalsaCinod warmodgenili sqemebis,<br />

diagramebis, modelebis gamoyenebiT mimdinareobs.<br />

. mniSvneloba eniWeba saswavlo amocanis sxvadasxva xerxiT amoxsnas, maT Soris arCevanis<br />

gakeTebasa da am arCevanis dasabuTebas.<br />

. saxelmZRvanelo amkvidrebs sxvadasxva formiT mimdinare jgufur muSaobas.<br />

. tradiciuli da Tanamedrove Sexedulebebis sinTezis safuZvelze Camoyalibda Sinaarsis<br />

gamdidreba logikuri xasiaTis davalebebiTa da sakiTxebiT gacnobiTi SeswavlisaTvis,<br />

Sefasebisa da kontrolis gansaxorcieleblad gamiznuli davalebebiT; eileris [25] Sexedulebebis<br />

gaTvaliswinebiT, ariTmetikuli moqmedebebis algoriTmebis Camoyalibeba dasabuTebebis<br />

zomierad gamoyenebebis safuZvelze xdeba; eileri did yuradRebas uTmobda numeraciis<br />

sakiTxebs _ aTobiTi sistemis, ricxvis Caweris wesebis safuZvlian ganxilvas ariTmetikis<br />

safuZvlad miiCnevda. am ideebis gaTvaliswinebiT Seqmnilia bevri Tanamedrove ucxouri saxelmZRvanelo<br />

(ix., mag., [26] _ [29]). didi mniSvneloba eniWeba damrgvalebis process, moqmedebaTa<br />

Sedegebis Sefasebas, `TvaliT~ sidideebis gazomvas (codnis praqtikuli gamoyeneba),<br />

informaciis sxvadasxva formiT warmodgenas, cxovrebiseuli situaciebis gamoyenebas, realuri<br />

damokidebulebebis sqemebiT warmodgenas; metia geometriuli masalis moculobac, romelic<br />

moicavs rogorc planimetriis, aseve stereometriis elementebs (ix., mag., [6], [7], [14],<br />

[18], [20]).<br />

axali saswavlo gegmis moTxovnebi saSualebas gvaZlevs gamoviyenoT ucxo qveynebis gamocdileba<br />

am mimarTulebiT. zemoT CamoTvlil mimarTulebebs emateba moswavleTa `gamoTvliT<br />

momzadebaze~, zrunva, romliTac rusuli saxelmZRvaneloebia gamorCeuli. amasTanave,<br />

yvelgan SeiniSneba naturaluri ricxvis iseTi mniSvnelovani aspeqtis win wamowevac, romelic<br />

70-ian wlebSi saskolo <strong>maTematika</strong>Si simravleTa Teoriis globaluri gavleniT Semcirebuli<br />

iyo. Cveni maswavleblebis umetesobac 70-iani wlebis meTodikur literaturas iyenebda swavlebis<br />

procesSi. `naturaluri ricxvis safuZvels warmoadgens simravle, radgan naturaluri<br />

ricxvi aris is saerTo, is ucvleli, rac axasiaTebs erTi da imave <strong>klasi</strong>s yvela simravles. es<br />

saerTo ki aris erT klasSi Semavali eqvivalenturi simravleebis ricxovnoba (raodenobiTi<br />

rocxvi), [5]; `naturaluri ricxvi dawyebiTi swavlebis centraluri cnebaa. igi eqvivalenturi<br />

simravleebis raodenobrivi maxasiaTebelia. es cneba ganixileba simravleebisa da sidideebis<br />

gazomvisas aRiwereba [8]).<br />

naturaluri ricxvis raodenobiTi aspeqtebis win wamoweva damaxasiaTebeli iyo 70-iani<br />

wlebis saxelmZRvaneloebisTvis (maSin maT `Tanamedrove saxelmZRvaneloebs~ uwodebednen).<br />

piaJec [21] am Tvalsazrisis gamziarebeli iyo. saintereso mosazrebebi naturaluri ricxvis<br />

aranakleb mniSvnelovani aspeqtis _ rigobiTi aspeqtis Sesaxeb da piaJes Tvalsarizis kritika<br />

mocemulia g. froidentalis [23] mier.<br />

skolaSi swavlebis maRal doneze warmarTvisTvis maswavleblebs vurCevT gaecnon am mosazrebebs.<br />

skolaSi ricxvis dafuZnebis sakiTxebi sainteresod da mecnierulad maRal donezea<br />

gadmocemuli a. kolmogorovis erT-erT statiaSi (ix., mag., [15]). masSi meoTxe <strong>klasi</strong>s<br />

ZiriTadi sakiTxis _ naturalur ricxvTa Teoriis _ mecnieruli safuZvlebia gadmocemuli.<br />

es Teoria <strong>maswavleblis</strong> wignSi gamoyenebuli iqneba.<br />

6

moswavlis asakobrivi Taviseburebebis gaTvaliswineba Cveni meTodikuri orientirebis<br />

mniSvnelovani Semadgeneli nawilia. meoTxe klasSi swavlebis Taviseburebebi moswavleTa asakis<br />

axal etaps ukavSirdeba _ maTSi mkvidrdeba zrdasrulobis gancda, Tumca, igi jer kidev<br />

ar aris gamyarebuli realuri pasuxismgeblobiT. izrdeba moTxovnebi damoukideblobisa<br />

da pativiscemis, seriozuli da ndobis gamomxatveli midgomisadmi. moswavlisaTvis TiTqos<br />

ukve mosawyeni xdeba maTematikuri amocanis amoxsna nimuSis mixedviT, mas surs ipovos sakuTari<br />

xerxi, romelic dafuZnebulia improvizaciaze; ufro mniSvnelovani xdeba erTi da<br />

imave amocanis sxvadasxva xerxiT amoxsna, vidre sxvadasxva amocanis amoxsna, iseTi amocanebis<br />

amoxsna, romlebic ar xdeba `orobiTi~ logikis CarCoebSi _ sworia, ar aris swori. am<br />

asakis moswavleebisTvis ufro mizanSewonilia rTuli amocanebis amoxsnis jgufuri muSaobis<br />

formiT Catareba.<br />

jgufuri muSaobis forma xSirad gamoiyeneba maTematikuri TamaSis saxiT davalebebis Sesrulebis<br />

dros. sworad SerCeuli da kargad organizebuli TamaSi xels uwyobs moswavleTa<br />

yovelmxriv, harmoniul ganviTarebas, exmareba moswavles gamoimuSaos cxovrebaSi saWiro Cvevebi.<br />

TamaSma SeiZleba xeli Seuwyos mexsierebis, yuradRebis, SemoqmedebiTi warmosaxvis ganviTarebas,<br />

TviTSemowmebis Cvevebis ganviTarebas, sakuTari qmedebebis dasaxuli amocanebis<br />

Sesrulebisadmi daqvemdebarebas, principulobis gamomuSavebas, piradi mosazrebiT sxvaTa<br />

dainteresebis, polemikis warmoebis xelovnebas, dawyebuli saqmis bolomde miyvanis Cvevebis<br />

dauflebas. TamaSebSi mniSvnelovania Sejibrebis elementebi, radgan am dros gamovlindeba<br />

moswavleTa aqtiuroba da gamarjvebisken swrafva. saTamaSo situaciebi da siuJetebi yvelaze<br />

xSirad gakveTili-TamaSis dros gamovlindeba. TamaSis organizeba gakveTilis xvadasxva<br />

etapze SeiZleba _ Zvelis gameorebisas, Seswavlilis ganzogadebisas. Cven mier SeTavazebuli<br />

didaqtikuri TamaSebi exmareba maswavlebels sxvadasxva pedagogiuri amocanis gadawyvetaSi.<br />

Sefasebis sitemaSi dagexmarebaT is masala, romelic Sinaarsisa da miznebis rukis Semdeg<br />

aris mocemuli. igi maswavlebels masalis gadacemis orientirebsa da Sefasebis formebs<br />

aZlevs; moswavleebis muSaobaze dakvirveba yovel gakveTilze unda mimdinareobdes, igi trimestruli<br />

da wliuri Sefasebebis safuZvelia.<br />

amasTanave, Cvens yovel rekomendacias calkeuli gakveTilis dagegmvisa da warmarTvis<br />

Sesaxeb, maswavlebeli SemoqmedebiTad unda miudges _ swavleba, cnobil maTematikossa da<br />

pedagogs d. poias Tu daveTanxmebiT, xelovnebaa da, SesaZlebelia, maswavlebelma gakveTilis<br />

warmarTvis, moswavleTa aqtivobis dagegmvis sakuTari originaluri xerxebic moifiqros.<br />

Cven yovelTvis mzad varT daveswroT novatorul gakveTilebs, monawileoba miviRoT maT ganxilvasa<br />

da analizSi.<br />

swavlebis procesSi meTodikuri orientirebis garkvevis sakiTxebSi dagexmarebaT meTodikuri<br />

literatura, romelic qvemoTaa miTiTebuli; magaliTad, dRevandel situacias kargad<br />

exmaureba citatebi h. froidentalis [23] <strong>wigni</strong>dan: `maswavlebels unda SeeZlos swavlebis<br />

warmarTva programiT, romelic kardinalurad gansxvavdeba imisgan, romliTac mas aswavlidnen...~<br />

swavlebis procesSi daxmarebas gagiwevT masala, romlis gacnobas internetis saSualebiT<br />

SeiZleba (mag., http//www.mccme.ru/edu).<br />

Sinaarsis SerCevas Sesabamisi asakis moswavleTaTvis misawvdomobis principi udevs safuZvlad;<br />

es SerCeva Camoyalibebuli stabiluri birTvis bazaze xdeba. igi gafiltrulia maTematikis<br />

swavlebis gamocdilebiT; tradiciuli Temebis Tanamimdevrobis axleburi gaazrebis,<br />

maTematikis erTianobidan gamomdinare integraciis xarisxis gazrdisa da axali sakiTxebis<br />

Sevsebis aucileblobis gaTvaliswinebiT. Sinaarsobrivi mimarTulebebi saswavlo gegmaSia<br />

warmodgenili. maswavlebels mudmivad unda axsovdes, rom <strong>maTematika</strong> erTiani mecnierebaa<br />

da ara misi sxvadasxva nawilis xelovnuri gaerTianeba, amitom gadmocemis meTodika iTvaliswinebs<br />

sakiTxebis urTierTkavSirSi gadmocemas. svetovani diagramebis gamoyeneba naturaluri<br />

ricxvebisa da maTze moqmedebebis Seswavlis procesSi ukve I TavSia warmodgenili;<br />

Tumca monacemTa analizisa da statistikis elementebi me-5 TavSic ganixileba, geometriisa<br />

7

da algebris elementebis ganxilva mTeli kursis ganmavlobaSi mimdinareobs, am Temebis zogierTi<br />

sakiTxi koncentruladac aris warmodgenili. veTanxmebiT erovnuli saswavlo gegmis<br />

avtorebs am etapze algebruli simbolikis farTo gamoyenebebisgan Tavis SekavebasTan dakavSirebiT.<br />

es tendencia axal rusul saxelmZRvaneloebSic SeiniSneba (ix., mag., [6], [7], [18]).<br />

moswavlis saxelmZRvaneloSi gadmocemis meTodika, rogorc wesi, sakiTxebis urTierTkav-<br />

SirSi, araizolirebulad gadmocemas iTvaliswinebs; sistematurad mimdinareobs wina masalis<br />

gameoreba da ganmtkiceba.<br />

amasTanave, yoveli siaxle isea gadmocemuli, rom ukve ganxiluli sakiTxebis Sesaxeb codnas<br />

emyareba; calkea gamoyofili terminebis xmarebis sxvadasxva variantebi, maTi warmoSobis<br />

istoriaze monacemebi, cnobil mecnierTa Sesaxeb biografiuli monacemebi. mecnierebis istoriidan<br />

moxerxebulad CarTuli epizodebi xels uwyobs maTematikis, rogorc zogadsakacobrio<br />

kulturis nawilis Sesaxeb warmodgenis Camoyalibebas.<br />

davalebebi iseTi TanamimdevrobiTaa warmodgenili, rom martividan rTulisken iyos mimarTuli.<br />

calkea gamoyofili davalebebi saklaso da saSinao muSaobisTvis. savarjiSoTa mravalferovani<br />

sistema (`testebi~, amocanebi jgufuri muSaobisTvis, amocanebi-TamaSebi, damatebiTi<br />

amocanebi) xels uwyobs moswavlis kritikuli azrovnebis ganviTarebas, maTematikis<br />

Segnebulad Seswavlas. praqtikasTan xSiri mimarTeba praqtikuli gamoyenebis unar-Cvevebis<br />

gamomuSavebis da maTematikis Seswavlisadmi interesis gaRrmavebas uwyobs xels.<br />

`testebi~ isea warmodgenili, rom isini moswavleTa situaciaSi swrafad garkvevisa da<br />

pasuxebis sworad SerCevis unars amowmebs da ara amoxsnebis Catarebis siswores. didi xania<br />

mimdinareobs dava am tipis savarjiSoTa dadebiTi da uaryofiTi mxareebis Sesaxeb. Cven viziarebT<br />

Tvalsazriss: miuxedavad imisa, rom am tipis amocanebSi warmoCenilia mxolod pasuxis<br />

SerCevis etapi, is gulisxmobs amocanis amoxnis etapsac da amitom aseTi amocana saswavlo<br />

miznebisTvisac gamodgeba da zogjer _ SemowmebisTvisac.<br />

<strong>maTematika</strong>, SesaZlebelia, erTaderTi saskolo disciplinaa, romelic damaxsovrebisa da<br />

gameorebis garda zogadi ganaTlebis iseT mniSvnelovan elementebs Seicavs, rogoricaa magaliTad,<br />

amocnoba, analizi, mixvedra, arCeva, damtkiceba.<br />

axla SevexoT moswavlis saxelmZRvanelos agebulebas. <strong>wigni</strong>s dasawyisSi gamovxatavT madlierebas<br />

yvelas mimarT, vinc Tavisi rekomendaciebiT wvlils Seitans <strong>wigni</strong>s daxvewaSi; gansakuTrebiT<br />

mniSvnelovania saswavlo gegmebisa da saxelmZRvaneloebis SemmuSavebeli jgufis<br />

eqspertebis Rvawli _ maT mier klasebis mixedviT ganawilebuli saswavlo gegma saSualebas<br />

gvaZlevs SevarCioT masala moswavleTa sxvadasxva unarisa da Cvevis ganviTarebisa da formirebisTvis.<br />

eqspertTa rekomendaciebi da SeniSvnebi gvexmareba mTeli wlis ganmavlobaSi<br />

davxvewoT saxelmZRvanelo, yoveli axali nawilis Sedgenisas gaviTvaliswinoT wina nawilze<br />

miRebuli daskvnebi, wlis bolos _ koreqtivebi SevitanoT yvela nawilSi.<br />

<strong>wigni</strong>s dasawyisSi avtorTa madlobas misalmeba da keTili survilebi mosdevs. aqvea rCevebi<br />

<strong>wigni</strong>T sargeblobisTvis. saxelmZRvanelos struqturis gansazRvraSi erovnuli saswavlo<br />

gegmebisa da Sefasebis centris mier SemoTavazebuli rekomendaciebi da saskolo saxelmZRvanelos<br />

Seqmnisa da Sefasebisadmi miZRvnili literatura gvexmareba (ix., mag., [22], [19]).<br />

saxelmZRvanelo eqvsi Tavisgan Sedgeba. TiToeuli Tavi _ paragrafebisgan. zogjer paragrafi<br />

ramdenime nawiliania. xSiria sakiTxebis gameoreba _ gasul da mimdinare wels naswavlis.<br />

es aucileblad miviCnieT am asakis moswavleebisTvis. magaliTad, mravalniSna ricxvebze<br />

moqmedebebis Sesruleba, damrgvaleba, moqmedebebis Sedegebis Sefaseba, zepiri angariSi uSualod<br />

ukavSirdeba ocis, asis farglebSi Sesabamis sakiTxebTan dakavSirebuli procedurebis<br />

`avtomatizmamde~ miyvanil aTvisebas.<br />

saxelmZRvanelo moswavlis <strong>wigni</strong>sadmi wayenebuli yvela moTxovnis Sesabamisadaa agebuli.<br />

sarCevi <strong>wigni</strong>s dasawyisSia mocemuli, maTematikuri terminebi, cxrilebi, maTematikuri niSnebi,<br />

sagnobrivi saZiebeli _ <strong>wigni</strong>s boloSi.<br />

8

moswavlis saxelmZRvanelo swavlebis procesis dagegmvisa da warmarTvis erT-erTi saSualebaa.<br />

amasTanave, moswavleebTan saubari SeiZleba <strong>wigni</strong>s teqstisgan gansxvavebuladac aagos<br />

maswavlebelma. saxelmZRvanelo exmareba maswavlebels _ masSi dafiqsirebulia is Semecnebi-<br />

Ti saqmianoba, romelic moswavlem unda awarmoos. <strong>maswavleblis</strong> saqmianoba ki sxva saSualebebis<br />

gamoyenebasac gulisxmobs _ <strong>maswavleblis</strong> <strong>wigni</strong> (iq damatebiTi amocanebi, sakontrolo<br />

weris nimuSebia mocemuli), TvalsaCino masala. Tumca, SevecadeT swavlebisadmi diferencirebuli<br />

midgomis gamoyenebis saSualebac Segveqmna _ SemoTavazebuli savarjiSoebi moswavleTa<br />

sxvadasxva SesaZleblobebis gaTvaliswinebiTaa Sedgenili; saWiroa SerCeva da konkretuli<br />

situaciis mixedviT maTi swori gamoyeneba. moswavleTa maTematikuri SesaZleblobebi,<br />

rogorc wesi, aTvisebis tempebis gansxvavebasTan ufroa dakavSirebuli.<br />

gakveTilis warmarTvisTvis momzadebis sxvadasxva etapi (`saswavlo wlisTvis momzadeba,<br />

mocemuli gakveTilisTvis momzadeba) maswavlebelma nayofierad unda gamoiyenos. masalis<br />

sworad SerCevaSi maswavlebels Cvenc vexmarebiT. sarekomendacio wignSi TiTqmis yvela<br />

amocanaa amoxsnili, miTiTebulia meTodikuri rekomendaciebi masalis wardgenisa da axsnis<br />

fazebis Sesaxeb.<br />

Zvirfaso maswavleblebo!<br />

warmatebebs gisurvebT Tqvens sapatio da mniSvnelovan saqmianobaSi. imedi gvaqvs, rom<br />

Cveni TanamSromloba keTil nayofs gamoiRebs.<br />

9

wlis bolos misaRwevi Sedegebi<br />

ricxvebi da moqmedebebi<br />

kanonzomierebebi<br />

da algebra<br />

geometria da sivrcis<br />

aRqma<br />

monacemTa analizi,<br />

albaToba da statistika<br />

1. gamosaxavs, adarebs<br />

da alagebs ricxvebs<br />

poziciuri sistemis<br />

gamoyenebiT<br />

2. sxvadasxva xerxiT<br />

asrulebs naturalur<br />

ricxvebze Sekreba-gamoklebis<br />

moqmedebebs<br />

da afasebs moqmedeba-<br />

Ta Sedegs<br />

3. sxvadasxva xerxiT<br />

asrulebs naturalur<br />

ricxvebze gamravleba-gayofis<br />

moqmedebebs<br />

da afasebs moqmedebaTa<br />

Sedegs<br />

4. ganasxvavebs, asaxelebs<br />

da adarebs<br />

mTelis nawilebs<br />

(naxevari, mesamedi,<br />

meoTxedi)<br />

5. iyenebs da erTmaneTTan<br />

akavSirebs<br />

zomis sxvadasxva erTeulebs<br />

6. agebs, gamosaxavs<br />

da ikvlevs<br />

Sesabamisobas.<br />

7. martivi maTematikuri<br />

amocanis<br />

amoxsnisas adgens<br />

da iyenebs algebrul<br />

gamosaxulebas<br />

8. aRwers geometriul<br />

figurebs da axdens maT<br />

<strong>klasi</strong>fikacias<br />

9. qmnis brtyeli da<br />

sivrculi figurebis<br />

grafikul gamosaxulebebs<br />

da modelebs<br />

10. poulobs saganTa<br />

da figuraTa zomebs da<br />

obieqtTa Soris manZilebs<br />

11. orientirebs sqe maze<br />

da qmnis marSrutis aRmwer<br />

martiv sqemas<br />

12. agrovebs Tvisebriv<br />

da raodenobriv<br />

monacemebs mocemul<br />

TemasTan an gamosakvlev<br />

obieqtTan dakav-<br />

SirebiT<br />

13. awesrigebs ra o denobriv<br />

da Tvi sebriv<br />

monacemebs<br />

14. akeTebs Tvisebriv<br />

da raodenobriv monacemTa<br />

interpretacias<br />

da elementarul<br />

analizs<br />

wlis bolos misaRwevi Sedegebi da maTi indikatorebi<br />

mimarTuleba: ricxvebi da moqmedebebi<br />

maT. <strong>IV</strong>.1.<br />

gamosaxavs, adarebs da alagebs ricxvebs poziciuri sistemis gamoyenebiT<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• kiTxulobs ricxvebs, sxvadasxva modelis gamoyenebiT gamosaxavs ricxvebs da axdens<br />

poziciuri sistemis demonstrirebas (mag. struqturirebuli saganTa erToblioba,<br />

ricxviT sxivze).<br />

• asaxelebs ricxvis CanawerSi TanrigebSi mdgomi cifrebis Sesabamis mniSvnelobebs, warmoadgens<br />

ricxvs saTanrigo SesakrebTa jamis saxiT.<br />

• iyenebs poziciur sistemas ricxvebis Sedarebisas, alagebs mocemul oTx/xuT ricxvs<br />

zrdiT an klebiT.<br />

• asaxelebs mocemuli ricxvis wina da momdevno ricxvebs, agreTve uaxloes aTeuls,<br />

aseuls, aTaseuls; nebismieri oTxniSna, xuTniSna ricxvidan iTvlis Tanrigebis Sesabamisi<br />

bijiT win/ukan.<br />

10

maT. <strong>IV</strong>.2.<br />

sxvadasxva xerxiT asrulebs naturalur ricxvebze Sekreba-gamoklebis<br />

moqmedebebs da afasebs moqmedebaTa Sedegs<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• zepirad asrulebs Sekreba-gamoklebis moqmedebebs romelime xerxis gamoyenebiT da<br />

xsnis gamoyenebul xerxs.<br />

• asrulebs Sekreba-gamoklebas sxvadasxva xerxis (Sefaseba, zepiri angariSi, weriTi<br />

algoriTmebi) gamoyenebiT; konkretuli magaliTisaTvis irCevs maTgan ufro<br />

xelsayrels.<br />

• adarebs gamoTvlebis Sedegs mis mierve winaswari SefasebiT miRebul pasuxs da msjelobs<br />

gamoTvlebis Sedegis marTebulobis Sesaxeb.<br />

• avsebs weriTi algoriTmis gamoyenebiT Sesrulebuli Sekrebis/gamoklebis nimuSSi gamotovebul<br />

cifrebs da asabuTebs pasuxs.<br />

maT. <strong>IV</strong>.3.<br />

iyenebs gamravleba-gayofis moqmedebebis Sesrulebis romelime xerxs<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• zepirad yofs orniSna ricxvs erTniSnaze, Sesabamis SemTxvevaSi asaxelebs ganayofsa da<br />

naSTs; asabuTebs pasuxs.<br />

• xsnis ricxvis 100-ze da 1000-ze da a.S. gamravlebis da nulebiT daboloebul ricxvebis<br />

gamravlebis Semoklebul wesebs; iyenebs maT gamoTvlebis Sesrulebisas.<br />

• iyenebs weriT algoriTms ricxvebze gamravleba-gayofis moqmedebaTa Sesasruleblad<br />

da ganmartavs gamoyenebul xerxs (erTniSna ricxvze gayofisas); Sesabamis SemTxvevaSi<br />

uTiTebs naSTs.<br />

• gamoTvlebze amocanebis amoxsnisas, naSTiT gayofis SemTxvevaSi, axdens naSTis interpretacias<br />

amocanis konteqstis gaTvaliswinebiT.<br />

maT. <strong>IV</strong>.4.<br />

ganasxvavebs, asaxelebs da adarebs mTelis nawilebs<br />

(naxevari, mesamedi, meoTxedi)<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• amoicnobs da asaxelebs mTelis naxevar/mesamed/meoTxed nawilebs sxvadasxva modelze<br />

(monakveTis, marTkuTxedis da wris modelebze, mag. torti/saaTi, Sokoladis fila).<br />

• axdens nawilis, rogorc mTelis tol nawilebad dayofis Sedegis da saganTa struqturirebuli<br />

grovis toli raodenobis jgufebad dayofis Sedegis demonstrirebas.<br />

• iyenebs gaormagebas da erTmaneTTan akavSirebs mTelis meoTxeds da naxevars.<br />

• adarebs mTelis nawils mTelis naxevars modelze (naxevarze metia, naklebia, tolia).<br />

maT. <strong>IV</strong>.5.<br />

iyenebs da erTmaneTTan akavSirebs zomis sxvadasxva erTeulebs<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• gamosaxavs sigrZis/wonis romelime did erTeuls (agreTve didi erTeulis naxevars)<br />

mcire erTeuliT. (mag. 2m = 2dm, 2m = 200sm; 4kg = 4000g).<br />

• iyenebs drois erTeulebs (saaTebi da wuTebi) Soris cnobil Tanafardobas da<br />

ariTmetikuli moqmedebebis gamoyenebiT poulobs drois (erT saaTamde) intervals.<br />

• erTi saaTis naxevars/meoTxeds gamosaxavs wuTebiT.<br />

• iyenebs naSTiT gayofas zomis mocemul erTeuleSi monacemis sxva erTeuliT gamosaxvisas<br />

(mag. 320sm = ?m, ?sm, 100wuTi = ? saaTi).<br />

11

Sinaarsi<br />

1. naturaluri ricxvebi milionis farglebSi<br />

2. moqmedebebi naturalur ricxvebze<br />

3. naSTiT gayofa<br />

4. mTelis naxevari, mesamedi da meoTxedi nawilebi mxolod gacnobis wesiT (nawilis<br />

wiladad Cawera da wiladebis Sesaxeb codna ar igulisxmeba)<br />

5. sigrZis erTeulebi<br />

6. drois erTeulebi: saaTebi da wuTebi, sawyisi warmodgenebi saaTis 12 saaTiani formatis<br />

Sesaxeb<br />

7. wonis erTeulebi: kilogrami, grami.<br />

mimarTuleba: kanonzomierebebi da algebra<br />

maT. <strong>IV</strong>.6.<br />

agebs, gamosaxavs da ikvlevs Sesabamisobas.<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• asaxelebs erTsa da imaves Sesabamisobas misi gamosaxvis xerxisagan damoukideblad.<br />

• raime xerxiT (mag. sityvierad, cxrilis an sqemis saSualebiT) mocemuli Sesabamisobisa-<br />

Tvis poulobs miTiTebuli elementis winasaxes.<br />

• agebs realuri viTarebis adekvatur Sesabamisobas obieqtTa mocemul or jgufs Soris<br />

(mag. moswavleebi da merxebi saklaso oTaxSi) da cxrilis an sqemis saSualebiT gamosaxavs<br />

mas.<br />

maT. <strong>IV</strong>.7. martivi maTematikuri amocanis amoxsnisas adgens da iyenebs algebrul<br />

gamosaxulebas.<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• xsnis martiv proporciul damokidebulebasTan dakavSirebul amocanebs (romlebSic<br />

erTeulis Sesabamisi ricxvis mixedviT saWiroa ramodenime erTeulis Sesabamisi ricxvis<br />

gamoTvla, mag. erTeulis Rirebulebis mixedviT ramdenime erTeulis Rirebulebis<br />

gamoTvla).<br />

• ricxviTi gamosaxulebis mniSvnelobis mosaZebnad iyenebs Sekrebisa da gamravlebis komutaciurobas,<br />

asociaciurobas da Sekrebis mimarT gamravlebis distribuciulobas.<br />

• poulobs Sekrebis, gamoklebis, gamravlebis, gayofis Semcveli tolobis ucnobi komponentis<br />

mniSvnelobas.<br />

• amocanis amoxsnisas ganasxvavebs saWiro da zedmet monacemebs.<br />

Sinaarsi<br />

1. Sesabamisobebi sagnebs Soris, sagnebsa da maT atributebs Soris; Sesabamisobis gamosaxva<br />

cxrilis da sqemis saSualebiT; mocemuli SesabamisobisaTvis elementis<br />

winasaxe.<br />

2. Sekrebis, gamoklebisa da gamravlebis Semcveli mTelricxovani gamosaxulebebi da<br />

maTi ekvivalentoba.<br />

3. Sekrebisa da gamravlebis komutaciuroba, asociaciuroba da Sekrebis mimarT gamravlebis<br />

distribuciuloba.<br />

12

4. teqsturi amocanebi, romlebic Sekrebis, gamoklebisa da gamravlebis Semcveli algebruli<br />

gamosaxulebebis saSualebiT ixsneba.<br />

mimarTuleba: geometria da sivrcis aRqma<br />

maT. <strong>IV</strong>.8.<br />

aRwers geometriul figurebs da axdens maT <strong>klasi</strong>fikacias<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• adarebs da ajgufebs sivrcul figurebs geometriuli atributebis mixedviT.<br />

• TanamkveTi figurebis gamosaxulebaze uTiTebs rogorc saerTo wertilebs, aseve im<br />

wertilebs, romlebic mxolod erT figuras ekuTvnis.<br />

• sivrcul figuraSi uTiTebs mosazRvre/aramosazRvre waxnagebs, TanamkveT/araTanamkveT<br />

wiboebs.<br />

maT. <strong>IV</strong>.9.<br />

qmnis brtyeli da sivrculi figurebis grafikul gamosaxulebebs da<br />

modelebs<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• nimuSis mixedviT qmnis miTiTebuli sivrculi figuris models an karkass sxvadasxva<br />

masalis gamoyenebiT.<br />

• qmnis brtyeli figuris an figuraTa jgufis grafikul gamosaxulebas misi sityvieri<br />

aRwerilobis safuZvelze (mag. daxaze erTi da igive perimetris mqone kvadrati da<br />

marTkuTxedi).<br />

• sivrculi geometriuli figurebis modelebisagan qmnis miTiTebul konfiguracias/<br />

figuras; anawevrebs brtyeli geometriuli figuris grafikul gamosaxulebas an models<br />

miTiTebuli figuris/figurebis misaRebad.<br />

maT. <strong>IV</strong>.10. poulobs saganTa da figuraTa zomebs da obieqtTa Soris manZilebs<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• axdens or obieqts Soris manZilis Sefasebas Sesabamis standartul erTeulSi, zomavs<br />

mas da amowmebs Tavis varauds.<br />

• zomavs da iTvlis texilis sigrZes, mravalkuTxedis perimetrs da afiqsirebs Sedegs<br />

Sesaferis standartul erTeulSi.<br />

• realuri viTarebis Sesabamisi sqematuri gamosaxulebis (romelzec manZilebia aRniSnuli)<br />

mixedviT poulobs or obieqts Soris umokles manZils (mag. saxlidan skolamde<br />

marSrutis sigrZe).<br />

maT. <strong>IV</strong>.11. orientirebs sqemaze da qmnis marSrutis aRmwer martiv sqemas<br />

Sedegi TvalsaCinoa, Tu moswavle:<br />

• gamoarCevs simboloebis gamoyenebiT miTiTebul marSruts sqemaze.<br />

• iyenebs simboloebs (mag. asoiT aRniSvnebs) sqemaze miTiTebul or wrtils Soris mar-<br />

Srutis aRsawerad.<br />

• sqematurad gamosaxavs realuri viTarebis Sesabamis marSruts (mag. marSruti saxlidan<br />

skolamde).<br />

Sinaarsi<br />

1. sivrculi figurebi: prizma, konusi, cilindri.<br />

2. sivrculi figuris elementTa urTierT-ganlageba: mosazRvre da aramosazRvre waxnagebi,<br />

TanamkveTi da araTanamkveTi wiboebi.<br />

3. mravalkuTxedis perimetri.<br />

4. realur viTarebaSi obieqtTa urTierTganlagebis aRmweri sqemebi.<br />

13

Sinaarsisa da miznebis ruka<br />

Temebis CamonaTvali<br />

miznebi; Sinaarsis gardaqmna miznad;<br />

erovnul saswavlo gegmaSi miTiTebuli<br />

Sesabamisi punqtebi<br />

savaraudo<br />

saswavlo<br />

dro<br />

gavixsenoT erTidan aTa<br />

sam de naturaluri ricxve<br />

bis Ca we ris wesi;<br />

ricxvebis Sedareba da<br />

da lageba;<br />

ricxvebis damrgvaleba<br />

erTidan aTasamde naturaluri ricxvebis<br />

wakiTxva, Cawera, sxvadasxva modeliT<br />

war modgena da poziciuri sistemis demonstrireba.<br />

ricxviT CanawerSi TanrigebSi mdgomi cifrebis<br />

Sesabamisi mniSvnelobebis dasaxeleba,<br />

ricxvis warmodgena SesakrebTa jamis<br />

saxiT, ricxvebis Sedareba da damrgvaleba<br />

aTeulebamde, aseulebamde.<br />

<strong>IV</strong>.1<br />

8 sT<br />

CavweroT aTasze meti<br />

natu raluri ricxvebi;<br />

SevadaroT da davalagoT<br />

naturaluri ricxvebi;<br />

naturaluri ricxvis damrgva<br />

lebis wesi;<br />

viyenebT svetovan diagra<br />

mebs.<br />

milionamde ricxvebis wakiTxva, Cawera,<br />

sxvadasxva modeliT warmodgena, poziciuri<br />

sistemis demonstrireba. CanawerSi cifris<br />

mniSvnelobis miTiTeba. mravalniSna ricxvis<br />

Cawera gaSlili formiT _ saTanado<br />

SesakrebTa jamis saxiT. ricxvebis Sedareba<br />

da dalageba. mravalniSna ricxvis<br />

damrgvaleba aTeulebamde, aseulebamde,<br />

aTaseulebamde. damrgvalebis wesis Camoyalibeba<br />

(ricxviTi sxivis gamoyeneba).<br />

rao denobebis Sedareba xvadasxva xerxiT,<br />

maT Soris, svetovani diagramebis gamoyenebiT.<br />

aRwers svetovani diagramis saxiT<br />

warmodgenil monacemebs.<br />

<strong>IV</strong>.1 <strong>IV</strong>. 14<br />

12 sT<br />

orniSna ricxvebis Sekreba;<br />

jamis Sefaseba damrgvale<br />

biT;<br />

didi ricxvebis Sekreba;<br />

SevarCioT Sekrebis wesi;<br />

orniSna ricxvebis gamokleba;<br />

sxvaobis Sefaseba;<br />

milionamde naturaluri<br />

ricxvebis Sekreba kalkulatoriT<br />

da jamis, sxvaobis<br />

Sefaseba.<br />

Sekreba sxvadasxva xerxiT zepiri Sekreba,<br />

Sefaseba, weriTi algoriTmis gamoyeneba;<br />

Sekrebis sxvadasxva xerxis gamoyenebiT<br />

miRebuli Sedegebis Sedareba. gamoklebis<br />

sxvadasxva xerxis gamoyeneba; weriTi<br />

algoriTmi, zepiri gamokleba; sxvaobis Sefaseba.<br />

sxvadasxva xerxis Sedareba. konkretuli<br />

magaliTisTvis yvelaze xelsayrelis<br />

arCeva.<br />

jamis, sxvaobis Sefasebis gamoyeneba<br />

kalkulatoris saSualebiT miRebuli Sede<br />

gis Sesaxeb msjelobisas. weriTi algoriTmis<br />

gamoyenebiT Sesrulebuli Sekrebis/gamoklebis<br />

nimuSSi gamotovebuli<br />

cifrebis migneba.<br />

<strong>IV</strong>.2<br />

13 sT<br />

14

Sekrebisa da gamoklebis<br />

zogierTi Tviseba; ricxvi<br />

Ti gamosaxulebis mni-<br />

S vne lobis povna; ucnobi<br />

kom ponentis povna.<br />

drois intervalebis povna;<br />

mTelis naxevari, meoTxedi,<br />

mesamedi.<br />

martivi maTematikuri amocanis amoxnisas<br />

algebruli gamosaxulebis Sedgenis Cvevebis<br />

daufleba.<br />

Sekrebis da gamoklebis Tvisebebis gamo<br />

yeneba. ricxviTi gamosaxulebis mni-<br />

Svnelobis povna. Sekrebis/gamoklebis<br />

Se mcveli tolobis ucnobi komponentis<br />

povna<br />

<strong>IV</strong>.7<br />

drois erTeulebs Soris Tanafardobebisa<br />

da ariTmetikuli moqmedebebiT drois<br />

intervalis gamosaxva. erTi saaTis naxevris,<br />

meoTxedis, mesamedis gamosaxva wuTebiT.<br />

mTeli nawilebis dasaxeleba da Sedareba;<br />

sxvadasxva modelze mTelis nawilebis<br />

amocnoba, dasaxeleba, Sedareba.<br />

<strong>IV</strong>.4, <strong>IV</strong>.5<br />

14 sT<br />

sigrZis erTeulebi; obieqt<br />

Ta Soris manZili;<br />

or obieqts Soris umoklesi<br />

gzis SerCeva;<br />

texilis sigrZe, mar Sruti,<br />

marSrutis sigr Ze,<br />

mra val kuTxedis perimet<br />

ri.<br />

sigrZis romelime erTeulis gamosaxva<br />

mcire erTeuliT.<br />

<strong>IV</strong>.5<br />

or obieqts Soris manZilis Sefaseba, varaudis<br />

gamoTqma.<br />

or obieqts Soris umoklesi manZilis<br />

SerCeva. marSrutis aRmweri sqemis wedgena.<br />

texilis sigrZis gazomva, mravalkuTxedis<br />

perimetris povna da gamosaxva Sesaferis<br />

standartul erTeulSi<br />

<strong>IV</strong>.10, <strong>IV</strong>.11<br />

17 sT<br />

gamravlebis Tvisebebi;<br />

gam ravleba nulebiT dabo<br />

loebul ricxvebze;<br />

erTniSna ricxvze gamravleba;<br />

namravlis Sefaseba;<br />

orniSna ricxvze gamravleba;<br />

samniSna ricxvze gamravleba;<br />

amocanebi zedmeti mocemu<br />

lobiT.<br />

ricxvis 100-ze da 1000-ze da a. S. gamravlebis<br />

da nulebiT daboloebul ricxvebis<br />

gamravlebis Semoklebuli wesebis daufleba.<br />

erTniSna, orniSna da samniSna<br />

ricxvebze gamravlebis sxvadasxva wesis<br />

daufleba; weriTi algoriTmebis daufleba,<br />

gamoyenebuli xerxis ganmarteba.<br />

UV.3<br />

amocanebis amoxsnisas zedmeti monacemebisa<br />

da saWiro monacemebis gansxvavebis Cvevebis<br />

daufleba<br />

<strong>IV</strong>.7<br />

18 sT<br />

15

moqmedebaTa rigi;<br />

naSTi;<br />

erTniSna ricxvze<br />

gayofa;<br />

nuliT daboloebuli<br />

ricxvebis gayofa;<br />

amocanebis amoxsna.<br />

Sesabamisoba;<br />

gamravlebis da gayofis<br />

ucnobi komponentis<br />

povna.<br />

ricxviTi gamosaxulebis mniSvnelobis<br />

mo saZebnad Sekrebisa da gamravlebis<br />

Tvi sebebis gamoyeneba. martivi ariTmetikuli<br />

amocanis amoxsnisas algebruli<br />

gamosaxulebis Sedgenisa da gamoye nebis<br />

Cvevebis daufleba.<br />

<strong>IV</strong>.7<br />

erTniSna ricxvze gayofa zepirad, Sesabamis<br />

SemTxvevaSi ganayofisa da naSTis<br />

dasaxeleba. pasuxis dasabuTeba. weriTi<br />

algoriTmebis gamo yeneba gayofis Sesa -<br />

sruleblad, gamoyenebuli xerxis ganmar<br />

teba. amocanebis amoxsnisas naSTis<br />

interpretaciis unari.<br />

<strong>IV</strong>.3<br />

Sesabamisobis gamosaxvis unaris ganvi-<br />

Tareba; realuri viTarebis adekvaturi<br />

Sesabamisobis agebis Cvevebis daufleba.<br />

<strong>IV</strong>.6<br />

16 sT<br />

masis erTeulebi; amocanebi<br />

nakluli da zedmeti<br />

monacemebiT;<br />

kubi, prizma;<br />

konusi, cilindri, birTvi.<br />

zomis sxvadasxva erTeulebis erTmaneTTan<br />

dakavSireba. naSTiT gayofis gamoyeneba<br />

zomis mocemul erTeulebSi gamosaxuli<br />

sididis sxva erTeuliT gamosaxvisas.<br />

<strong>IV</strong>.5<br />

sivrculi figurebis Sedarebisa da<br />

dajgufebis Cvevebis daufleba. TanamkveTi<br />

figurebis aRqma _ saerTo wertilebis<br />

miTiTeba. sivrculi figuris elementebis<br />

dasaxeleba da miTiTeba.<br />

sivrcul figuraTa modelebisgan miTi Tebuli<br />

konfiguraciis Seqmna;<br />

<strong>IV</strong>.8, <strong>IV</strong>.9<br />

8 sT<br />

monacemTa Segrovebisa<br />

da organizaciis for mebi.<br />

monacemTa warmodgenis<br />

_ saSualebebi _<br />

cxrili, piqtograma, svetovani<br />

diagrama<br />

mocemul TemasTan an gamosakvlev obieqtTan<br />

dakavSirebiT Tvisebrivi da raodenobrivi<br />

monacemebis Segrovebis Cvevebis daufleba;<br />

Tvisebrivi da raodenobrivi monacemebis<br />

mowesrigeba; dalageba (dajgufeba) _<br />

cxrilis, sqemis, kiTxvaris, anketis Sevsebis<br />

wesebis daufleba. monacemTa analizi,<br />

Sedareba, gansxvavebis povna.<br />

<strong>IV</strong>.12, <strong>IV</strong>.13, UV.14<br />

10 sT<br />

miaxloebiT 10 sT gameorebisTvis da miaxloebiT 8 sT saTadarigo dro.<br />

16

saswavlo masalis wardgenis fazebi da gakveTilis dagegmvis<br />

ZiriTadi principebi<br />

saswavlo procesis organizaciis ZiriTadi forma gakveTilia; saganmanaTleblo,<br />

aRmzrdelobiTi da praqtikuli miznebis ganxorcieleba gakveTilze xdeba, Sesabamisad<br />

gakveTilis kargi momzadeba da Catareba uzrunvelyofs swavlebis maRal xarisxs.<br />

meTodikur literaturaSi cneba _ `gakveTili~ _ ZiriTadad, ase aRiwereba: gakveTili<br />

logikurad dasrulebuli, mTliani saswavlo-aRmzrdelobiTi procesis garkveuli<br />

SemosazRvruli monakveTia. masSi rTul urTierTdamokidebulebaSia procesis yvela<br />

ZiriTadi elementi: Sinaarsi, mizani, saSualebebi, meTodebi, organizacia. yovel gakveTilze<br />

gansazRvruli saganmanaTleblo da aRmzrdelobiTi amocanebi wydeba; am amocanebis gadawyveta<br />

konkretuli saswavlo masalis ganxilvis procesSi mimdinareobs. dawyebiT klasebSi<br />

prioritetulia saswavlo-aRmzrdelobiTi funqcia; Tumca, TandaTan, me-4 <strong>klasi</strong>dan izrdeba<br />

sagnobrivi codnis SeZenis, arsebuli codnis ganzogadebis intensivoba.<br />

maTematikis gakveTilisadmi wayenebuli mTavari moTxovnaa ZiriTadi didaqtikuri amocanis<br />

arseboba _ im Temis Seswavlis miznis arseboba, romlis gadawyvetis procesi mocemul<br />

gakveTilze mimdinareobs. yoveli gakveTilis win kargad unda gaiazroT misi Sinaarsi da<br />

mizani. amaSi Cven mier mowodebuli ruka dagexmarebaT. miznis Sesabamisad unda iyos kargad<br />

gaazrebuli da dagegmili saswavlo masalis wardgenis fazebi.<br />

am miznis gaazreba mxolod <strong>maswavleblis</strong> mier ar aris sakmarisi; saWiroa is<br />

moswavleTaTvisac gaxdes ZiriTadi mizani; moswavleebTan saubari unda daviwyoT ara imiT,<br />

Tu ras vaswavliT, aramed mniSvnelovania Tavidanve interesis aRZvra da iseTi situaciis<br />

Seqmna, roca moswavle motivirebulia da poulobs pasuxebs kiTxvaze _ `risTvisaa saWiro~.<br />

motivacia SesaZlebelia praqtikuli amocanis dayenebiTa da misi amoxsnis xerxebis ZiebiT<br />

daviwyoT, an _ maTematikis Siga kanonzomierebebis gaazrebiT. saxelmZRvaneloSi `did~<br />

ricxvebze gadasvla orive aspeqtis warmodgeniT mimdinareobs. TandaTan xdeba im codnis<br />

gadmotana-gavrceleba did ricxvebze, rac ukve icoda moswavlem aTaseulebis farglebSi.<br />

problemuri situaciis Seqmnis Semdeg xdeba Sesabamisi amocanis dasma da amoxsnis Zieba.<br />

amocanis gansazRvris Semdeg mimdinareobs misi amoxsnis Ziebis procesi. amoxsnis Ziebis<br />

warmarTvis sxvadasxva meTodikuri saSualeba arsebobs. yovel amocanas, rogorc wesi, amoxsnis<br />

Ziebis garkveuli forma Seesabameba. es formebia _ muSaoba jgufebad (SesaZlebelia ororadac),<br />

mTeli <strong>klasi</strong>s erToblivi monawileobiT, individualurad da a. S. im SemTxvevaSic ki,<br />

roca amocanis amoxsnis ZiebaSi mTeli <strong>klasi</strong> erTdrouladaa Cabmuli, maswavlebelma swavleba<br />

ise unda warmarTos, rom Temis Seswavlis procesis ZiriTadi Semoqmedi Tavad moswavle<br />

aRmoCndes. maswavlebeli warmmarTvelis, `diriJoris~ funqcias unda asrulebdes; zogierTi<br />

mecnieri <strong>maswavleblis</strong> am rols mebaRis rols adarebs _ yvavilebi mebaRis xelSewyobiT<br />

izrdeba da alamazebs garemos.<br />

moswavlis saxelmZRvanelos teqsti, rogorc zemoTmoyvanili magaliTidanac Cans,<br />

mogcemT aseTi gakveTilis Catarebis saSualebas. igi ZiriTadi saswavlo saSualebaa da amitom<br />

gakveTilisTvis mzadebis procesSi teqstebis, ilustraciebis, savarjiSoebis gacnobas,<br />

daxarisxebas didi dro unda dauTmoT. <strong>maswavleblis</strong> <strong>wigni</strong>c da moswavlis <strong>wigni</strong>s struqturac<br />

dagexmarebT TiToeul etapze (sawyisi, damamTavrebeli). naxatis Sesasrulebeli savarjiSoebis<br />

SerCevaSi Sinaarsisa da miznebis rukaSi SeuZlebeli iyo yvela miznis miTiTeba. yovel Temas<br />

17

ukavSirdeba savarjiSoebi, romlebic sxvadasxva inteleqtualuri unaris (<strong>klasi</strong>ficirebis,<br />

Sedarebis, cxovrebaSi gamoyenebis, TviTSefasebis Catarebis, amocanis gadawyvetis xerxis<br />

SerCevis) ganviTarebas emsaxureba. ar unda gavaigivoT mizani SinaarsTan. saxelmZRvaneloSi<br />

maTematikuri amocanis warmodgenisa da axsnis procesi isea mocemuli, rom igi Sesabamisi Temis<br />

Sinaarsis miznad gardaqmnis process kargad warmogvidgens. magaliTad, dasmuli amocanis<br />

amoxsna SeiZleba emsaxurebodes kritikuli azrovnebis, msjelobis, varaudis gamoTqmis,<br />

zogjer _ dasabuTebis unaris gamomuSavebas. Tumca, me-4 klasSi masalis gadmocemis forma,<br />

ZiriTadad, induqciuria.<br />

gakveTilze Seswavlili faqtebi, sxvadasxva algoriTmebis gacnoba, TavisTavad aris<br />

mniSvnelovani. maTi gamoyenebis unari `avtomatizmamde~ unda iyos miyvanili. Tumca, kidev<br />

ufro mniSvnelovania is, rom maTi Seswavlis dros ganxorcielebuli procesi Tavis kvals<br />

tovebs _ mimdinareobs sxvadasxva inteleqtualuri unaris ganviTareba.<br />

axla davasaxelebT ramdenime sasargeblo zogad rekomendacias, romelic meTodikas<br />

ganekuTvneba.<br />

• maswavlebeli cdilobs, rom yoveli axali SemecnebiTi amocana TviT moswavlem<br />

Camoayalibos.<br />

• <strong>maswavleblis</strong> xelmZRvanelobiT da moswavleTa ZalisxmeviT _ dakvirvebis, cdis,<br />

konkretuli SemTxvevebis analizis Sedegad iqmneba warmodgena arsebul kanonzomierebaze.<br />

• <strong>maswavleblis</strong> xelmZRvanelobiT mimdinareobs amocanis gadawyvetis gzis Zieba;<br />

amocanis amoxsnis gegmis Sedgena; rasac xSirad mosdevs TviT moswavleebis mier am gegmis<br />

realizacia.<br />

axla SevexoT saswavlo procesisTvis mzadebis zogad sqemas;<br />

• <strong>maswavleblis</strong> muSaoba saswavlo wlis win,<br />

• gakveTilebis sistemis gaazreba saswavlo wlis win. es gansakuTrebiT mniSvnelovania <strong>IV</strong><br />

klasSi; erTi raime Temis gameorebas, ganzogadebas, axlis Ziebasa da gadawyvetas gakveTilebis<br />

erToblioba eZRvneba.<br />

• konkretuli gakveTilisTvis momzadeba.<br />

momzadebisTvis maswavlebelma unda gamoiyenos moswavlis <strong>wigni</strong>, <strong>maswavleblis</strong> sarekomendacio<br />

<strong>wigni</strong> da, saWiroebis SemTxvevaSi, iq miTiTebuli literatura. sasurvelia, sarekomendacio<br />

<strong>wigni</strong>s mixedviT Tavdapirvelad saswavlo gegmisa da SemoTavazebuli sanimuSo gakveTilebis<br />

gacnoba. sarekomendacio <strong>wigni</strong>s Sesavali maswavlebels avtorTa mier saxelmZRvanelos agebis<br />

ZiriTad principebsa da meTodikur orientirebs gaacnobs.<br />

maswavlebeli SemoqmedebiTad unda miudges Cvens rekomendaciebs; man safuZvlad unda<br />

aiRos SemoTavazebuli rekomendaciebi da, sakuTari gamocdilebisa da <strong>klasi</strong>s moswavleTa<br />

Taviseburebebis gaTvaliswinebiT, daazustos igi. es dazustebebi gansakuTrebiT sawyis da<br />

damamTavrebel etapebze Sesasrulebeli savarjiSoebisa da sakontrolo weris variantebis<br />

SerCevisas aris mniSvnelovani.<br />

gakveTilis dagegmvisas gasaTvaliswinebelia Casatarebeli procesis tipi:<br />

• gameoreba;<br />

• axali masalis gacnoba;<br />

• masalis ganmtkiceba;<br />

• codnis Semowmeba;<br />

• sxva tipis gakveTilebi (gakveTili bunebaSi, gakveTili _ proeqti).<br />

zogierTi tipis procesi (gansakuTrebiT <strong>IV</strong> klasSi), rogorc wesi, yovel gakveTilze<br />

mimdinareobs (gameoreba, axali masalis gacnoba, ganmtkiceba, Semowmeba, Tumca, SesaZlebelia<br />

_ sxvadasxva moculobiT).<br />

gakveTilze mTavaria vaswavloT da aRvzardoT. swavleba ar niSnavs mxolod codnis<br />

gadacemas _ swavleba codnis SemoqmedebiTad dauflebas unda niSnavdes, codnis gamoyenebis<br />

unaris ganviTarebas unda gulisxmobdes.<br />

18

moswavleTa Sefaseba swavlis procesze <strong>maswavleblis</strong> dakvirvebiT, sakontrolo samuSaoebs<br />

Sesrulebis xarisxiT ganisazRvreba. yoveli gakveTilis Semdeg sakuTar wignakSi unda<br />

CainiSnoT moswavleTa aqtiuroba.<br />

saswavlo masalis dasabuTebuli SerCeva iTvaliswinebs Semdeg moTxovnebs:<br />

• saswavlo masalis Sesabamisoba Temis mizanTan;<br />

• gakveTilze Sesasrulebeli samuSaos moculobis swori gansazRvra;<br />

• optimaluri Tanafardoba konkretulsa da zogads Soris.<br />

• Teoriasa da praqtikas Soris aucilebeli urTierTkavSiris ganxorcieleba.<br />

maswavlebelma, rogorc wesi, Tavidan bolomde detalurad unda gaiazros gakveTili,<br />

winaswar, drois mixedviT unda iyos ganawilebuli mTeli samuSao; Tumca, rogorc yoveli<br />

SemoqmedebiTi procesi, gakveTilic Seitans, albaT, Tqvens gegmebSi cvlilebebs (zogjer<br />

mniSvnelovansac).<br />

kiTxvebze pasuxebis gacemis sistema ar unda iyos erTferovani _ mxolod warmatebul<br />

moswavleebTan mimarTebiT ar unda SemovifargloT; moswavlis raime mosazrebas myisve nu<br />

upasuxebT, swor pasuxsac ki maSinve nu daeTanxmebiT _ gaakeTeT pauza, iqneb garkveuli eWvic<br />

ki gamoTqvaT wamoyenebuli mosazrebis sisworis mimarT. amiT miaRwevT imas, rom bavSvebi<br />

daubrundebian dasmuli kiTxvis analizs da male WeSmariti daskvna _ swori pasuxi _ <strong>klasi</strong>s<br />

dominantur mosazrebad gadaiqceva. <strong>klasi</strong>, erToblivi ZalisxmeviT, `gaiZulebT~ daeTanxmoT<br />

mis pozicias. es axarebs, amxnevebs da aerTianebs moswavleebs. es maTi erToblivi azris<br />

gamarjvebaa. Tqven mizansac xom es warmoadgens _ moswavle Semoqmed, codniT aRWurvil,<br />

iniciativian, xalisian axalgazrdad CamoayaliboT. <strong>maTematika</strong> mZlavri emociuri muxtis<br />

matarebelia da misi amoqmedeba Tqveni ZalisxmeviT miiRweva.<br />

axali masalis ganmtkicebis procesi SeiZleba e. w. `testuri~ amocanebis `amoxsniT~<br />

daviwyoT. maTi Sesruleba, ZiriTadad, swori pasuxis SerCeviT unda Semoifarglos _ zogjer<br />

SeiZleba komentarebis gakeTebac gaxdes saWiro. aseTi tipis davalebebis odenoba Semdgom<br />

wlebSi gaizrdeba; moswavle TandaTan eCveva pasuxebis analizs, maT Soris swori pasuxis<br />

Sesaxeb varaudis gamoTqmas, SerCevas.<br />

swavlebis erT-erTi saintereso da mniSvnelovani formaa jgufuri muSaoba. es muSaoba<br />

SeiZleba gakveTilis procesis erT-erTi Semadgeneli nawili iyos _ daukavSirdes axali<br />

masalis gaazrebas, praqtikul saqmianobas (eqsperiments), an Semajamebeli daskvnebis<br />

gamotanas; an, SesaZlebelia, mas mTeli gakveTilic daeTmos. jgufuri muSaobis warmarTvaSi<br />

moswavlis saxelmZRvaneloSi warmodgenili Teoriuli masala, an Sesabamisi amocanebis<br />

sistema dagexmarebaT. jgufur muSaobas SeiZleba Sejibris saxec mivceT. saxelmZRvaneloSi<br />

warmodgenili maTematikuri TamaSebic, jgufuri muSaobis formiT unda warimarTos.<br />

Teoriuli masalis gadmocemis Cveneuli meTodika saSualebas gaZlevT airCioT jgufuri<br />

muSaobis moxerxebuli forma Sinaarsisa da miznebis rukaSi miTiTebuli moTxovnebis Sesabamisad;<br />

magaliTad, paeqroba jgufebs Soris, romlebsac kapitnebi (SesaZloa TanaSemweebTan<br />

erTad) xelmZRvaneloben. jgufebs SeiZleba erTnairi davalebebi mivceT. maSin Tavad jgufebic<br />

aqtiurad CaerTvebian miRebuli Sedegebis analizSi da ukeT gauweven oponirebas sxvaTa<br />

mosazrebebs. SeiZleba jgufebSi samuSaoebSi ganawildes da maTi erToblivi muSaoba raime<br />

erTi amocanis gadawyvetas gulisxmobdes; jgufis wevrebma unda aRiqvan, rom maTi warmateba<br />

erTmaneTzea damokidebuli. es zrdis jgufis wevrebis pasuxismgeblobas. TanamSromlobis<br />

xarisxis asamaRleblad rekomendebulia saerTo Sefaseba _ jgufebi asruleben saerTo<br />

davalebas da misi wevrebi iReben erTnair Sefasebas (Tumca, `qolgis qveS gancxromis~<br />

sindromis asacileblad, jgufis wevrTa diferencirebuli Sefasebac gamoiyeneT xolme).<br />

sasurvelia, rom jgufuri muSaobisTvis gankuTvnili amocana sxvadasxva xerxiT ixsnebodes.<br />

igi saintereso msjelobis saSualebas iZleva.<br />

19

Sefasebis zogadi principebi 1<br />

saskolo Sefasebis axali sistema, romelic erovnuli saswavlo gegmiT aris gansazRvruli,<br />

iTvaliswinebs Semdeg aucilebel midgomebs: akademiuri moswrebis Sefaseba unda iyos xSiri<br />

da mravalmxrivi. unda Sefasdes ara marto informaciis floba, aramed SeZenili unar-Cvevebi.<br />

ar aris sakmarisi moswavle mxolod sakontrolo werebis safuZvelze Sefasdes. maswavlebeli<br />

unda afasebdes prezentaciebis, moswavlisve TviTSefasebis, jgufuri muSaobis, Tu sxva tipis<br />

aqtivobebis mixedviT.<br />

maswavlebelma Sefasebisas unda gaiTvaliswinos saganmanaTleblo procesSi moswavlis<br />

CarTulobis xarisxi (saxlSi micemuli davalebebis Sesrulebis xarisxi, gakveTilze<br />

aqtiuroba, SemoqmedebiToba da sxva), amasTanave, mizanSewonilia moswavleebsac gavacnoT<br />

winaswar Sefasebis kriteriumebi. am kriteriumebis SedgenaSi SeiZleba moswavleTa CarTvac.<br />

sakontrolo werebis Sefasebis sqemebs am wignSi gaecnobiT. dauSvebelia moswavleTa<br />

qcevis gaTvaliswineba akademiuri moswrebis Sefasebisas _ gakveTilze arasaTanadod moqceva,<br />

rogorc wesi, TavisTavad aisaxeba akademiur moswrebaze.<br />

moswavlis niSani unda gamomdinareobdes mis mier sagnis Seswavlis sxvadasxva<br />

komponentisgan _ sakontrolo weris Sesruleba, gakveTilze msjeloba, jgufuri muSaoba,<br />

prezentacia da sxva. am komponentebs maswavlebeli wonebs aniWebs. wonebis miniWebis sistema<br />

SeiZleba maswavlebelma sxvadasxvanairad dagegmos. es wonebi SeiZleba sxvadasxva iyos<br />

sxvadasxva trimestrisTvis. trimestris Sefaseba (umaRlesi qulaa 10) SeiZleba sxvadasxva<br />

komponentebis Sefasebebisgan gamoviyvanoT. magaliTad, wonebi SeiZleba aseTi iyos _<br />

prezentacia 25%, gakveTilze aqtivoba _ 30%, sakontrolo wera _ 25%, saSinao davaleba _<br />

20%, Tu TiToeuli komponentisTvis saSualo Sefasebebi, Sesabamisad, 6, 9, 10, 8, maSin Sefaseba<br />

ase SeiZleba gamovTvaloT.<br />

.......................................................................................................................................................<br />

0,25⋅6+0,3⋅9+0,25⋅10+0,2⋅8=8,3.<br />

Tumca, zogierT skolaSi SeiZleba romelime komponentze (magaliTad, sakontrolo weraze)<br />

uaric ki Tqvan. maSin Seicvleba sxva komponentebis wonebi. Sefasebis ZiriTadi mdgenelebi<br />

SeiZleba aseTi iyos:<br />

saSinao davaleba, Semajamebeli samuSao, klasSi CarTuloba, proeqti. TiToeul mdgenels<br />

unarebis garkveuli sistema Seesabameba da es unarebi Sesabamisi aqtivobebiT fasdeba.<br />

saSinao davaleba,<br />

15%<br />

Semajamebeli<br />

samuSao, 40%<br />

Semajamebeli samuSao<br />

klasSi CarTuloba<br />

proeqti,<br />

15%<br />

klasSi CarTuloba,<br />

30%<br />

proeqti<br />

saSinao davaleba<br />

20<br />

1<br />

gamoyenebulia proeqt `ilia WavWavaZis~ erovnuli saswavlo gegmebisa da saxelmZRvanelobis SemmuSavebeli<br />

jgufis mier momzadebuli masala.

Tavi I<br />

gameoreba. naturaluri ricxvebi aTasamde<br />

am TavSi warmodgenili masalis swavlebas gamorCeuli mniSvneloba eniWeba saswavlo<br />

procesSi. migvaCnia, rom sazafxulo ardadegebis Semdeg moswavleTa CarTva saswavlo<br />

procesSi unda ganxorcieldes III klasSi Seswavlili ZiriTadi sakiTxebis gameorebiT.<br />

umartivesi samotivacio amocanebis ganxilviT xdeba moswavleTa Semzadeba II Tavis<br />

masalis, kerZod, milionamde naturaluri ricxvebis gasacnobad. yuradRebas vamaxvilebT<br />

ricxvTa poziciur (aTobiT) sistemaSi Tanrigebis arsze. ganvixilavT naturalur ricxvTa<br />

miaxloebiT mniSvnelobebs, damrgvalebas da maT gamoyenebebs. mniSvnelovania agreTve<br />

naturalur ricxvTa Sedarebisa da dalagebis Sesaxeb warmodgenebis gameoreba. Tavis<br />

Seswavlas davasrulebT damatebiTi amocanebis ganxilviT.<br />

martivi amocanebis siWarbe, vfiqrobT, aamaRlebs saswavlo procesSi moswavleTa<br />

CarTulobis xarisxs. aqvea aratrivialuri amocanebic, romelTa amoxsna xSirad sakmao<br />

gamWriaxobas moiTxovs.<br />

1. gavixsenoT aTasamde naturaluri ricxvebis Caweris wesi<br />

am paragrafSi yuradRebas vamaxvilebT ricxvebis Caweris poziciur sistemaze, Ti-<br />

Toeuli poziciis Tanrigis mniSvnelobis gaazrebaze. es sakiTxebi, garkveulwilad wina<br />

klasebSic iyo ganxiluli, Tumca, amjeradac _ sazafxulo ardadegebis Semdeg, uTuod<br />

gamarTlebulia maTi xelaxeli ganxilva.<br />

paragrafis teqstSi I da II skolis mier nayidi fanqrebis odenobis gamomsaxveli ricxvebis<br />

aTobiT CanawerebSi cifri 2 fanqrebis sxvadasxva odenobas aRwers. I SemTxvevaSi _ 200<br />

fanqars, II SemTxvevaSi _ 2 fanqars. es faqti, suraTebze warmodgenisas, TvalsaCinoa.<br />

diagramis gamoyenebiT TvalsaCino warmodgena xels uwyobs pozitiuri sistemis ukeT<br />

gaazrebas. savarjiSoebSi mocemuli ricxvebi sasurvelia moswavleebma xmamaRla warmoTqvan<br />

_ gaiwafon ricxvis wakiTxvaSi. ramdenime savarjiSoSi mouwevT mocemulis wina da<br />

momdevno ricxvebis dasaxeleba da gaSlili formiT _ saTanrigo SesakrebTa jamis saxiT<br />

warmodgena.<br />

15 amocanaSi moswavle iwafeba kombinatorikis gamoyenebaSi amocanebis amoxsnisas. am<br />

ricxvebis Camowerisas moswavlem sasurvelia raime wesiT isargeblos. magaliTad, Camoweros<br />

jer 2-iT dawyebuli ricxvebi (237, 273), Semdeg 3-iT, Semdeg 7-iT dawyebuli ricxvebi.<br />

16 arsebobs oci aseTi ricxvi: 800, 811, 822, ... , 899, 900, 911, ... , 999.<br />

17 arsebobs oTxi aseTi ricxvi: 607, 707, 807, 907.<br />

18 arsebobs ori aseTi ricxvi: 802, 903.<br />

19 amocanis amoxsnamde SeiZleba <strong>klasi</strong> SevamzadoT kiTxva-pasuxis reJimiT:<br />

• dawereT ramdenime samniSna ricxvi, romlis aTeulebis cifri erTeulebis da aseulebis<br />

cifrebis jamis tolia (magaliTad, 781, 132, ...). akmayofilebs es ricxvebi amocanis pirobas?<br />

pirobis ra nawilia darRveuli?<br />

21

• SeiZleba aseulebis cifri iyos 4, 5 an 6 cifrebidan romelime?<br />

moisazre<br />

rubrikiT warmodgenili amocanebis nawili SeiZleba klasSi amoixsnas,<br />

nawili, magaliTad, 1 da 4 amocana, saSinao davalebas daematos. aseTi tipis amocanebi<br />

xels uwyobs moswavleTa inteleqtualuri unarebis ganviTarebas.<br />

1 amocanis amoxsnisas, Tu garkveuli sirTuleebi SeeqmnaT moswavleebs, SeiZleba romelime<br />

maTgans TxovoT daasaxelos Tavisi asaki da upasuxos kiTxvebs: ramdeni wlis iyo is ori<br />

wlis win; ramdeni wlis iqneba 10 wlis Semdeg da a. S.<br />

2 amocanis amoxsnis procesSi SeiZleba kerZo SemTxvevebis ganxilvac moviSvelioT<br />

(magaliTad, biWebs hqondaT 20-20 lari da a. S.). SeiZleba vimsjeloT ase: nikam jer 2 lari<br />

gadado calke. ramdeniT meti aqvs axla Tornikes? _ 2 lariT. Semdeg gadadebuli 2 lari<br />

aiRo Tornikem, ramdeniT meti gauxda mas? _ 4 lariT.<br />

3 amocana: lukas darCenil 7 lars rom 6 lari daematos, miRebuli Tanxa burTis fasze<br />

1 lariT metia. burTi 12 lari Rirebula. Sesabamisad, Tavdapirvelad lukas 19 lari hqonda.<br />

4 dayofis Sedegi ufro TvalsaCino rom iyos, sasurvelia mocemul<br />

figuraSi mcire formis nawilebi gamovaCinoT. magaliTad, ase _<br />

• moswavleTa aqtivoba ufro nayofieri gaxdeba, Tu isini am figuraTa<br />

modelebs gamoiyeneben.<br />

TamaSi<br />

ipove Seni adgili<br />

es TamaSi ganamtkicebs moswavleTa mier samniSna ricxvebis wakiTxvisa da cifrebiT<br />

gamosaxvis unars. Secdomis daSvebis SemTxvevaSi SeTavazeT gunds gadaxedos Tavis gadawyvetilebas<br />

_ waaxaliseT erTmaneTis mimarT maTi Tanadgoma.<br />

TamaSi<br />

airCie swori strategia<br />

TamaSi xels uwyobs strategiebTan dakavSirebuli kompetenciebis Camoyalibebas da ricxvebis<br />

ukeT gagebas. garda amisa, saxelmZRvanelos am adgilze TamaSis ganTavsebiT Cven vaxsenebT<br />

moswavleebs ricxviT sxivs da mis Tvisebebs, rasac aqtiurad viyenebT momdevno paragrafSi.<br />

22

2. ricxvebis Sedareba da dalageba<br />

am etapze SemovifargleT 1000-ze naklebi ricxvebis SedarebiT (II TavSi Sesadarebel ricxvebs<br />

milionis farglebSi SevarCevT). SeiZleba gamoiyenoT ricxvebis warmodgena diagramebiT,<br />

maTi saSualebiT advilad aRiqmeba Sedarebis procesi. ricxvTa Sedarebisas vadarebT<br />

saTanado Tanrigebis cifrebs. viwyebT udidesi TanrigiT. paragrafSi moyvanil magaliTSi<br />

yuradReba unda gavamaxviloT: Tu ricxvSi Tanrigi araa miTiTebuli, is Tanrigi 0-is tolia<br />

_ 79-Si aseulebis Tanrigi igulisxmeba 0-is tolad.<br />

sainteresoa 3 savarjiSos amoxsna: SeiZleba moswavlem pasuxebidan SearCios udidesi,<br />

SesaZloa man Tavad Seadginos sami cifrisgan udidesi SesaZlo ricxvi aseTi msjelobiT:<br />

udidesi ricxvi udidesi cifriT iwyeba _ am SemTxvveaSi 6-iT, Semdegi Semdegi darCenilTagan<br />

udidesia 4 da bolos 3.<br />

14 amocanaSi moswavle kvlav iyenebs ricxvebis Sedarebas TanrigebiT. is msjelobs: *-is<br />

nacvlad udidesi cifris _ 9-is Caweris SemTxvevaSic ki levanis Sedegi giorgis Sedegze<br />

uaresia _ 194

aqtivoba 1.<br />

reziume.<br />

ricxvebis damrgvaleba<br />

moswavleebi asaxeleben mocemuli ricxvebis uaxloes aTeuls, uaxloes aseuls, gamosaxaven<br />

maT ricxviT sxivze sailustracio magaliTebis safuZvelze adgenen damrgvalebis wess,<br />

aTasze naklebi ricxvebisTvis ayalibeben am wess, iyeneben amocanebis amoxsnisas, sidideebis<br />

Sefasebisas. moswavleebs esmiT damrgvalebis mizanSewoniloba saTanado SemTxvevebSi da Cans<br />

maTi mzaoba am wesis gamoyenebis mimarT.<br />

aqtivobis mizani<br />

• moswavleebma SeZlon nebismieri erTniSna, orniSna da samniSna ricxvis uaxloesi aTeulis,<br />

uaxloesi aseulis dasaxeleba;<br />

• moswavleebma magaliTebis safuZvelze SeimuSaon damrgvalebis wesi;<br />

• moswavleebi daeuflon ricxvebis (1000-ze naklebi) damrgvalebis wesis gamoyenebas amocanebis<br />

amoxsnisas, sidideebis Sefasebisas.<br />

• moswavleebma aRmoaCinoT damrgvalebuli ricxvebis gamoyenebis mizanSewoniloba (upiratesobac<br />

ki) saTanado SemTxvevebSi.<br />

• damrgvalebis dadgenili wesi SemdegSi bunebrivad ganavrcon im SemTxvevebzec, roca<br />

dasamrgvalebeli ricxvi metia samniSna ricxvze.<br />

aucilebeli wina codna:<br />

• 1000-mde ricxvebis Cawera/wakiTxva;<br />

• mocemuli ricxvis momdevno da wina ricxvis dasaxeleba;<br />

• ricxvebis Sedareba;<br />

• dasaxelebuli Tanrigis amocnoba.<br />

saWiro masalebi:<br />

• ujrebiani furclebi;<br />

• saxazavi da fanqari;<br />

• saweri saSualeba.<br />

aqtivobis aRwera<br />

moswavleebs vusaxelebT orniSna ricxvebs, isini asaxeleben uaxloes aTeuls. magaliTad,<br />

rasTan ufro axlosaa 48 _ 40-Tan, Tu 50-Tan? (50-Tan). 48-is uaxloesi aTeulia 50.<br />

skolis <strong>IV</strong> klaselTa odenobaa 87. rasTan ufro axlosaa es ricxvi _ 80-Tan Tu 90-Tan?<br />

am ori ricxvidan romlis SerCeva sjobs 87-is miaxloebad? (90).<br />

drois ekonomiis mizniT SeiZleba winawar gavumzadoT moswavleebs ujrebian furcelze<br />

daxazuli ricxviTi sxivi, masze mivuTiToT zogierTi danayofis Sesabamisi ricxvi; danarCen<br />

danayofebze moswavleebi Tavad aweren ricxvebs. moswavleebs vTavazobT ricxviT wrfeze monaxon,<br />

magaliTad, 27-is Sesabamisi wertili, misi mezobeli aTeulebi da fanqriT moniSnon<br />

isini. 27-is Sesabamisi wertilidan 20-isa da 30-is Sesabamis wertilebamde manZilebis SedarebiT<br />

moswavleebi daadgenen, rom 27 aris 30-Tan ufro axlos, vidre 20-Tan.<br />

kvleva-Zieba.<br />

davukvirdeT, miTiTebul Tanrigamde damrgvalebisas ra SemTxvevaSi xdeba am Tanrigis cifris<br />

gazrda 1-iT, ra SemTxvevaSi ar xdeba. anu, rodis xdeba mocemul ricxvTan miaxloeba<br />

metobiT, rodis _ naklebobiT _ Tu damrgvalebisas miTiTebuli Tanrigis marjvniv aris 5,<br />

an 5-ze meti cifri (5; 6; 7; 8; 9). damrgvaleba xdeba metobiT, 5-ze naklebi cifris SemTxvevaSi<br />

(0; 1; 2; 3; 4) _ naklebobiT.<br />

24

SeZenili codnis gamoyenebaSi gawafvis mizniT moswavleebs dasamrgvaleblad vTavazobT<br />

konkretul ricxvebs.<br />

sawyis etapze aqtiurad viyenebT ricxviT sxivs. masze moswavleebi moniSnaven manZilebs<br />

mezobel aTeulebamde (aseulebamde) da daasaxeleben uaxloes aTeuls (aseuls). garda amisa,<br />

klasSi viyenebT 1 - 5 savarjiSoebs.<br />

miRebuli codnis gaRrmavebis mizniT SeiZleba gamoviyenoT saklaso savarjiSoebi 6 -<br />

9 . moswavleebs saintereso gamokvlevas SavTavazebT 5 amocanis ganxilviT.<br />

aqtivobis ganxilva/Sefaseba.<br />

aqtivobis warmatebiT ganxorcielebis SemTxvevaSi moswavle Tavad mouyris Tavs damrgvalebisas<br />

SeniSnul faqtebs da Camoayalibebs ricxvebis damrgvalebis wess (1000-ze naklebi<br />

ricxvebisTvis), daeufleba am wesis gamoyenebas amocanebis amoxsnisas, sidideebis Sefasebisas.<br />

es aqtivoba Soreul miznebsac isaxavs _ momavalSi am wess gavavrcelebT jer nebismier<br />

naturalur, Semdeg _ aTwilad ricxvebze; ekonomikuri da sxva gamoyenebiTi xasiaTis<br />

amocanebis gadawyvetisas damrgvalebasa da Sefasebas xSirad gamoviyenebT.<br />

aqtivoba 2.<br />

TamaSi _ vasaxelebT uaxloes aseuls.<br />

reziume.<br />

moswavleebi iyofian jgufebad/gundebad; asaxeleben orniSna da samniSna ricxvebs, amrgvaleben<br />

maT. `arbitrebi~ ki qulebiT afaseben jgufebis mier Sesrulebul samuSaos (siswores,<br />

siswrafes).<br />

aqtivobis mizani<br />

• moswavle gaiwafos ricxvis damrgvalebis wesis gamoyenebaSi;<br />

• moswavle mieCvios jgufSi muSaobas _ sakuTari azris gamoTqmas, dacvas, sxvisi azris<br />

gaTvaliswinebas (saWiroebis SemTxvevaSi _ kompromiss), prezentacias;<br />

• moswavlem SeZlos sxvisi namuSevris analizi da Sefaseba.<br />

aucilebeli wina codna<br />

• 1000-mde ricxvebis Cawera/wakiTxva;<br />

• dasaxelebuli Tanrigis amocnoba;<br />

• ricxvebis damrgvalebis wesi.<br />

aqtivobis aRwera<br />

maswavlebeli gamoikiTxavs ricxvebis damrgvalebis wess, moswavleebi zepirad amrgvaleben<br />

ramdenime ricxvs miTiTebul Tanrigamde.<br />

saSinao davalebis Semowmebisas yuradRebas vamaxvilebT davalebis 4 amocanaze. moswavleebi<br />

Tbilisidan goramde manZils _ 89 km-s amrgvaleben jer aTeulebamde, Semdeg aseulebamde.<br />

SeiZleba ganvixiloT moswavleTa mosazrebebi _ romel Tanrigamde damrgvalebas miiCneven<br />

isini ufro mizanSewonilad da ratom, upasuxon kiTxvas: romeli damrgvalebis Sedegia<br />

ufro axlos realobasTan?<br />

amis Semdeg <strong>klasi</strong> 3 jgufad/gundad davyoT. erTi jgufi dausaxelebs meores jer<br />

aTeulebamde dasamrgvalebel ricxvebs. Cvenc CaverToT xolme am paeqrobaSi da davusaxeloT<br />

`Znelad dasamrgvalebeli~ ricxvebi. magaliTad, 299 (≈ 300), 8 (≈ 10), 3 (≈ 0), 30 (≈ 30).<br />

meore jgufi amrgvalebs I-is dasaxelebul ricxvebs. mesame jgufi afasebs qulebiT Sesrulebul<br />

davalebas (siswores, siswrafes).<br />

25

periodulad gundebi funqciebs cvlian. SeiZleba gundebis movaleobebi sawyis etapze ken-<br />

WisyriTac ganawildes (mizani _ kenWisyris wesebis gacnoba).<br />

Semdeg igive samuSao Sesruldes ricxvebis aseulebamde damrgvalebis Casatareblad.<br />

saintereso etapia Sedegebis Sejameba, gundebis waxaliseba.<br />

• Zalze mniSvnelovania SeniSnuli xarvezebis aRnusxva da zrunva maT gamosasworeblad.<br />

aqtivobis gaRrmaveba<br />

aqtivobis gaRrmavebis mizniT maswavlebelma gundebs SeiZleba dausvas damatebiTi amocanebi:<br />

daasaxelon im ricxvTa odenoba, romlebic akmayofilebs saSinao davalebis me-7 amocanis<br />

pirobebs (I gundi), me-8 amocanis pirobebs (II gundi), me-9 amocanis pirobebs (III gundi),<br />

me-6 amocanis pirobebs (samive gunds). am kiTxvebze swori pasuxis SemTxvevaSi moswavleebma<br />

SeiZleba damatebiTi qulebi SesZinon Tavis gundebs.<br />

aqtivobis ganxilva/Sefaseba<br />

aqtivobis mizani, garda ricxvis damrgvalebis wesis gamoyenebaSi gawafvisa, aris is, rom<br />

moswavle SeeCvios jgufur muSaobas, jgufSi sakuTari azris gamoTqmas, dacvas, sxvisi azris<br />

mosmenas, gaTvaliswinebas, sxvis mier Sesrulebuli samuSaos sajaro ganxilvas da obieqtur<br />

Sefasebas.<br />

gundis interesebis dacvis survili momavalSi usaTuod gadaizrdeba ojaxis, koleqtivis,<br />

sazogadoebis interesebis gaTvaliswinebisa da dacvis Tvisebad.<br />

3. ricxvebis damrgvaleba<br />

am paragrafSi moswavleebi ecnobian 1000-mde ricxvebis raime Tanrigamde damrgvalebis<br />

wess magaliTebis saSualebiT. isini <strong>maswavleblis</strong> dauxmareblad asaxeleben ricxvis uaxloes<br />

srul aTeuls an aseuls (magaliTad, 87-Tan uaxloesi aTeulia 90). ramdenime magaliTis ganxilvis<br />

Semdeg moswavleebi Tavs mouyrian calkeul faqtebs da Camoayalibeben damrgvalebis<br />

wess. sasurvelia aRiniSnos damrgvalebisas ra SemTxvevaSi xdeba miaxloeba metobiT (5-isa da<br />

5-ze meti cifrebis SemTxveva) da rodis _ naklebobiT (5-ze naklebi cifrebis SemTxveva).<br />

1 savarjiSoSi miaqcieT yuradReba im SemTxvevas, roca miTiTebuli Tanrigis cifri<br />

9-iania. Tu mas 5-iani mosdevs, 9-ianis 1-iT gadidebis Semdeg miiReba 10, magaliTad, 95-is<br />

aTeulebamde damrgvalebisas miiReba 100, 970-is aseulebamde damrgvalebisas _ 1000. sasurvelia<br />

sawyis etapze aRniSnuli SemTxvevebis ricxviT sxivze ganxilva.<br />

4 manZilebis aseulebamde damrgvalebis Semdeg cxrili miiRebs saxes:<br />

qalaqebi<br />

manZili miaxloebiT (km)<br />

Tbilisi-soxumi 400<br />

Tbilisi-baTumi 400<br />

Tbilisi-ozurgeTi 300<br />

Tbilisi-quTaisi 300<br />

Tbilisi-ambrolauri 200<br />

Tbilisi-zugdidi 300<br />

Tbilisi-Telavi 100<br />

Tbilisi-axalcixe 200<br />

kiTxvebze pasuxebi: a) Telavi, b) ambrolauri da axalcixe, g) ozurgeTi, quTaisi da zugdidi,<br />

d) baTumi da soxumi, e) yvelaze Sorsaa soxumi, yvelaze axlos _ Telavi.<br />

26

5 pasuxi: 187.<br />

6 amocanis pirobas akmayofilebs naturaluri ricxvebi 45-dan 54-is CaTvliT. aseTi<br />

ricxvebis odenobaa 10.<br />

7 es ricxvebia 100, 200, ..., 900, maTi odenobaa 9.<br />

8 moswavleTa ricxvi metia 296-ze. SeuZlebelia es ricxvi 302-ze naklebi iyos.<br />

9 umciresi ricxvi, romlis damrgvalebiT miiReba 180, aris 175, udidesi _ 184.<br />

moisazre<br />

1 amocanis amoxsnisas moswavleebma SeiZleba sinjvis meTodi gamoiyenon,<br />

SeiZleba suraTi davixmaroT:<br />

?<br />

SeWama<br />

5<br />

darCa<br />

pasuxi: Tavidan ido 10 orcxobila.<br />

SeiZleba monakveTic gamoviyenoT _<br />

?<br />

SeWama<br />

7<br />

darCa<br />

2 Saqarlamebi langarze Tanabrad gavyoT:<br />

?<br />

SeWama<br />

SeWama – 1<br />

darCa – 3<br />

amocanis pirobidan vRebulobT: Tavidan arsebuli odenobis<br />

naxevari aris 4, e. i. Tavidan ido 8 Saqarlama.<br />

3 orive klasSi yvelaze maRali moswavlis simaRlea 140 sm, yvelaze dablis _ 111 sm.<br />

4 12 sm SeuZlebelia, radgan 13 sm-iani sxvaoba ukve arsebobs (I gundSi). 16 santimetriani<br />

sxvaoba SesaZlebelia. magaliTad, es moxdeba, Tu I gundSi aris 180 sm simaRlis moTamaSe, II<br />

gundSi 164 sm simaRlis.<br />

5 mocemuli ricxvi SeiZleba iyos erT-erTi ricxvi Semdegi ricxvebidan: 51, 52, ..., 59,<br />

61, ..., 69; damrgvalebisas SeiZleba miviRoT: 50, 60 an 70.<br />

6 mocemuli ricxvi SeiZleba iyos 43, 44, 45, 46 an 47. damrgvalebisas SeiZleba miviRoT<br />

40 an 50.<br />

saSinao<br />

4 amocanaSi erTsa da imave ricxvs amrgvaleben jer aTeulebamde (89 ≈ 90), Semdeg<br />

aseulebamde (89 ≈ 100) da Semdeg afaseben Sedegebs _ romeli miaxloebaa ufro mizanSewonili.<br />

SeiZleba davalebis Semowmebisas davazustoT: ramden km-s Seadgens sxvaoba realur<br />

27

da miaxloebiT manZilebs Soris I SemTxvevaSi, II SemTxvevaSi (anu SevadaroT damrgvalebisas<br />

cdomilobebi).<br />

5 270 miiReba 265, 266, ..., 274 ricxvebis damrgvalebiT.<br />

6 aseTia 10 ricxvi: 185, 186, 187, 188, 189, 190, 191, 192, 193 da 194.<br />

7 es ricxvebia: 95, 96, 97, 98 da 99.<br />

8 es ricxvebia: 5, 6, 7, 8 da 9.<br />

9 es ricxvebia: 1, 2, 3 da 4.<br />

jgufuri<br />

_ vasaxelebT uaxloes aseuls. am muSaobaSi sami jgufi monawileobs.<br />

erTi jgufis mier dasaxelebul amocanas samive jgufi xsnis, Tumca isini gansxvavebulad<br />

motivirebulni arian am samuSaos SesrulebisTvis.<br />

I jgufi xsnis Tavis dasmul amocanas, raTa darwmundes `Jiuris~ daskvnis samarTlianoba-<br />

Si, II xsnis qulebis mosapoveblad, III jgufi _ SefasebisTvis. jgufebis rolebi Sejibris<br />

procesSi icvleba. maswavlebels miecema samive jgufis waxalisebis safuZveli.<br />

moswavleebs am muSaobisas uviTardebaT jgufSi muSaobis, sxvisi azris gaziarebis, sajaro<br />

diskusiis warmoebis, obieqturi Sefasebis, prezentaciis unarebi.<br />

damatebiTi amocanebis pasuxebi da miTiTebebi:<br />

SearCie swori pasuxi:<br />

testi 1 2 3 4 5 6 7 8 9 10 11 12<br />

pasuxi 2 2 1 2 1 4 2 1 2 4 4 4<br />

amoxseniT amocanebi:<br />

15. a) 60; 70, b) 65; 70, g) 56; 66, d) 124, 4.<br />

16. a) . b) .<br />

17. a) 30 aTeuli aris 3 aseuli: .<br />

d) 44 aTeuli aris 4 aseuli da 4 aTeuli: .<br />

e) 23 aTeuli da 4 erTeuli aris 2 aseuli, 3 aTeuli da 4 erTeuli: .<br />

19. a) 300, b) 30, g) 3.<br />

20. 300 da 310, 300

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