maTematika VII - Ganatleba

guram gogiSvili, Teimuraz vefxvaZe 

ia mebonia, lamara qurCiSvili 

maTematika VII 

maswavleblis wigni 

(meTodikuri rekomendaciebi, sakontrolo weris variantebis 

nimuSebi da miTiTebebi amocanebis amosaxsnelad) 

meore gadamuSavebuli gamocema 

redaqtori Teimuraz vefxvaZe 

gamomcemloba inteleqti 

Tbilisi

6 

Sesavali 

wigni Seicavs meTodikur rekomendaciebs axali erovnuli saswavlo gegmis mixedviT 

Sedgenili maTematikis VII klasis sa xelmZRvanelosTvis (guram gogiSvili, Teimuraz 

vefxvaZe, ia mebonia, lamara qurCiSvili, maTematika VII), 2005-2006 saswavlo wels am 

saxelmZRvanelom warmatebiT gaiara pilo ti reba. rekomendebulia ganaTlebis reformis 

proeqt ìlia Wav WavaZis~ erovnuli saswavlo gegmebisa da saxel mZRva ne lo ebis SemmuSavebeli 

jgufis mier. 

wignSi gadmocemulia meTodikuri rekomendaciebi yoveli Temis gadacemis, 

gakveTilebis dagegmvisa da warmarTvis, sa var jiSoebis SerCevis Sesaxeb. 

mocemulia miTiTebebi amocanaTa amosaxsnelad. 

wignSi gaecnobiT Cvens mosazrebebs axali masalis axsnis Sesaxeb; mocemuli iqneba 

Sesabamisi axsna-ganmartebebi masalis wardgenis fazebis Sesaxeb _ motivacia, sakiTxis 

dasma, amo ca nis gansazRvra, ZiriTadi saswavlo masalis gamoyofa, sawyis etapze Sesasrulebeli 

savarjiSoebi da maTi analizi, dama gvir gvinebel etapze Sesasrulebeli 

amocanebi da maTi ana li zi, damoukidebeli samuSaoebis gansazRvra da maTi Semow mebis 

formebi. mocemuli iqneba gakveTilebis dagegmvis sqemebi da ramdenime sanimuSo 

gakveTilis scenari. 

wignSi Tavebi da paragrafebi moswavlis wignis analo gi u radaa warmodgenili. 

gaecnobiT komentarebs TiToeuli Tavis, paragrafis mi xed viT, masalis gadmocemis 

Taviseburebebis motivacias, am masa lis Sesabamis mecnierul da meTodologiur safuZvlebs; 

miTi Te buli iqneba Sesabamisi literatura yvela am Temis mixedviT codnis 

gasaRrmaveblad; wigns Tan erTvis samecniero da meTo di kuri literaturis sia. 

mocemulia moswavleTa mier masalis Seswavlis aRricx vi sa da Sefasebis xerxebi; 

es rekomendaciebi xels Seuwyobs maswavlebels Sefasebis procesis dagegmvasa da 

warmarTvaSi. Se fasebis ZiriTadi mdgenelebi da am mdgenelebis mixedviT Se fasebis 

sistemis SemuSavebaSi dagexmarebaT is masala, romelic erovnuli saswavlo gegmebisa 

da Sinaarsisa da miz ne bis rukis Semdeg aris mocemuli. xsenebuli sami kom po nen ti 

mTeli saswavlo wlis ganmavlobaSi gamoiyeneba, mas wav lebels aZlevs masalis gadacemis 

orientirebsa da Se fa sebis formebs _ moswavleebis muSaobaze dakvirveba, maTi 

aq ti vobebis Sefaseba yovel gakveTilze mimdinareobs da trimes truli da wliuri 

Sefasebebis safuZvels warmoadgens. 

amasTanave, Cven mier mocemul rekomendaciebs calkeuli gakveTilebis dagegmvisa 

da warmarTvis Sesaxeb maswavlebeli SemoqmedebiTad unda miudges _ swavleba, 

cnobil maTematikoss d. poias Tu daveTanxmebiT, xelovnebaa da SesaZlebelia bevrma 

maswavlebelma gakveTilis warmarTvis, moswavleTa aqtivobis dagegmvis originaluri 

xerxebic moifiqros. Cven yovelTvis mzad varT daveswroT novatorul gakveTilebs 

da monawileoba miviRoT maT ganxilvasa da analizSi. 

swavlebis procesSi meTodikuri orientirebis garkvevis sakiTxSi dagexmarebaT 

meTodikuri literatura, romlebic wignis bolos aris mocemuli; magaliTad, dRevandel 

situacias kargad exmaureba citatebi h. froidentalis wignidan ([38], gv. 97): 

`maswavlebels unda SeeZlos swavlebis warmarTva programiT, romelic kardinalurad 

gansxvavdeba imisgan, romliTac mas aswavlidnen... 

... moTxovna, rom ganaTleba gardaqmnis Canasaxsac Sei cav des swrafad cvalebad 

sazogadoebaSi ara marto maTe ma tikis swavlebas exeba. ... feliqs klaini laparakobda 

ormag da viwyebaze: universitetSi swavlisas saskolo maTematikis daviwyebaze, Semdeg 

_ sauniversiteto maTematikis daviwyebaze, skolaSi maswavleblad dabrunebisas~. 

Cven SevecdebiT yoveli Temis warmodgenas axldes deta lu ri komentarebi da 

masalis mecnieruli da meTodologiuri safuZvlebis mimoxilva.

swavlebis procesSi meTodikuri orientirebis garkvevis sakiTxebSi daxmarebas 

gagiwevT masala, romlis gacnobas in ter netis saSualebiT SeZlebT (http://www.mccme. 

ru/edu). 

aq mocemulia ruseTisa da amerikis SeerTebuli Statebis cnobil mecnierTa, 

sazogadoebriv organizaciaTa da TviT prezident buSis mosazrebebi da rekomendaciebi 

maTematikis swavlebis mniSvnelobisa da 21-e saukuneSi misi warmarTvis axali 

orientirebis Sesaxeb. magaliTad, amerikis SeerTebuli Statebis nacionaluri komitetis 

moxsenebaSi (`jer kidev ar aris gvian~) vkiTxulobT: `miuxedavad sazogadoebis 

serio zu li cvlilebebisa, romelic gasuli saukunis meore naxe var Si ganxorcielda, 

maTematikisa da sabunebismetyvelo sag ne bis swavlebis meTodika faqtobrivad ar 

Secvlila. swav le bis praqtika srulad ar iyenebs im SesaZleblobebs, rome lic saganmanaTleblo 

procesis sxvadasxva aspeqtma mogvca...~. 

ZiriTadi aqcenti am moxsenebaSi iseve, rogorc zemoT xse nebul sxva rekomendaciebSi 

keTdeba moswavlis aqtiur Cab ma ze swavlebis procesSi, rom swavleba Ziebisa da 

dasa bu Te bebis gziT unda mimdinareobdes, moswavle TviT unda aRmoa Cendes WeSmaritebas 

motivirebuli situaciis dagegmvis Semdeg; mTavaria kvlevis procesi _ ara 

marto pasuxebis mosmena kiTxva ze _ `ra~, aramed _ `rogor~, an _ `kidev rogor~. 

amas Ta nave, mniSvnelovania imis codnac _ `ratom~ da pasuxebi kiTx vaze _ `me es 

risTvis mWirdeba~. yvelaferi es auci le belia moswavleTa momavali profesiuli 

saqmianobisTvis, demokratiul procesebSi aqtiuri monawileobisTvis _ dRevan del 

sazogadoebaSi, romelic sul ufro metad aris damo kide buli maTematikaze. 

wigni dagexmarebaT zemoTxsenebuli orientirebis ganxor cie lebis meTodikur 

xerxebis gacnobaSi _ kvlevis procesi SeiZleba individualuri iyos, SeiZleba jgufuri 

muSaobiT xorcieldebodes. iqneba rekomendaciebi sxvadasxva tipis sas wav lo 

procesis ganxorcielebisTvis. moswavlis saxel mZRva nelo am procesis erT-erTi 

damxmare saSualebaa. amitom sa Wi roa misi agebis principebi da masalis struqturirebis 

sakiTxebic gagacnoT. 

moswavlis saxelmZRvanelos Sedgenisas gaviTvaliswineT is ZiriTadi orientirebi, 

romlebic ganaTlebisa da mecni ere bis saministros proeqt ìlia WavWavaZis~ SemmuSavebuli 

jgu fis mier warmodgenil erovnul saswavlo gegmebSia miTi Te buli _ 

orientireba Sedegze da am Sedegebis demonst ri re baze, informaciis gamoyenebis 

SesaZleblobis demonst ri reba. kavSiri sagnebs Soris da TviT erTi sagnis SigniT 

mis sxvadasxva nawilebs Soris (maTematikaSi warmod ge nil mimar Tu lebebs Soris); 

saklaso da klasgareSe muSaobis saqmia no bis erTianoba; orientacia uklebliv yvela 

moswav leze (sa ku Tari SesaZleblobebidan gamomdinare, yoveli mos wavle sxvadasxva 

xarisxiT miaRwevs Sedegebs); maswavlebelTa in di vi dualuri Taviseburebebis gamovlenis 

SesaZleblobebi _ erovnuli saswavlo gegmis Sesruleba sxvadasxva gziT Se iZ leba. 

me-7 klasis saxelmZRvanelos Sedgenisas gaviTvaliswineT, rom igi sabazo safexuris 

pirvel etaps Seesabameba; amasTanave aris mosamzadebeli etapi momavali arCevanis 

(swavlis gagr Ze leba, profesiaTa daufleba) winapirobebis Sesaqmnelad; am periodSi 

iwyeba informaciis, unar-Cvevebisa da damoki debu lebebis gasaviTareblad saWiro 

pirobebis Seqmna; farTov deba Tanamimdevruli, logikurd gamarTuli msjelobis 

unaris gamo muSavebis procesis CarCoebi. 

saxelmZRvanelos Sinaarss, rogorc cnobilia, unda gan saz Rvravdes rogorc praqtikuli 

moTxovnebi, aseve maTema tikis ganviTarebis Sinagani kanonzomierebebi. im 

SemTxvevaSic ki, roca raime axali sakiTxis SemoRebas es ukanaskneli pi ro bebi ufro 

gansazRvravs, mainc vcdilobT praqtikuli sa xis sakiTxebTan maT dakavSirebas. aseT 

SemTxvevaSi saWiroa zo mierebis dacva _ praqtikasTan yoveli mimarTeba unda iyos 

ga mar Tlebuli, moswavleTaTvis gasagebi da xels uwyobdes mo tivaciis process. 

7

Sinaarsis SerCevas Sesabamisi asakis moswavleTaTvis mi sawv domobis principi udevs 

safuZvelad; es SerCeva _ Camoyalibebuli stabiluri birTvis bazaze xdeba. igi gafil 

trulia maTematikis swavlebis gamocdilebiT; tradiciuli Temebis Tanamimdevrobis 

axleburi gaazrebiT, maTematikis erTianobidan gamomdinare integraciis xarisxis 

gazrdisa da axali sakiTxebiT Sevsebis aucileblobis gaTvaliswinebiT. Sinaarsobrivi 

mimarTulebebi saswavlo gegmaSia Camoyali be bu li; maTgan maswavleblebisTvis ukve 

nacnobia wina saxel mZRva neloebis mixedviT: monacemTa analizi, aRweriTi statistikis 

elementebis cnebebi da maTi praqtikuli ilustraciebi. Tumca, midgoma da 

gadmocemis meTodika axleburia _ gazr dilia mimarTulebebis integraciis procesi; 

orientirebulia Sedegebze. maswavlebels mudmivad unda axsovdes, rom maTematika 

erTiani mecnierebaa da ara misi sxvadasxva nawilis xelovnuri gaerTianeba. amitom 

gadmocemis meTodika iTva lis wi nebs sakiTxebis urTierTkavSirSi, araizolirebulad 

gad mo cemas. sistematuria wina masalis gameorebisa da ganmtki ce bis momentebis gaTvaliswineba. 

es xels uwyobs kursis erTiani koncefciis ganxorcielebas. pirveli 

Tavic ki, rome lic TiTqos gasul wlebSi Seswavlili sakiTxebis gameorebas moicavs, 

momavali sakiTxebis gaTvaliswinebiTaa agebuli. 

yoveli siaxle isea gadmocemuli, rom ukve ganxiluli sa kiTxebis codnas emyareba. 

xSiria mimarTeba sabunebis mety ve lo sagnebTan. TiTqmis yovel paragrafSia mocemuli 

Teo riu li masalis komentarebi, saintereso mecnieruli faqtebi, terminebis warmo- 

Sobis istoria, maTi xmarebis sxvadasxva aspeqtebi, cnobil mecnierTa biografiuli 

monacemebi. 

moswavleebSi maTematikis, rogorc zogadsakacobrio kul tu ris nawilis Sesaxeb 

warmodgenis Camoyalibebas xels uw yobs SinaarsSi CarTuli epizodebi mecnierebis 

istoriidan. isi ni moswavleebs im adamianTa saxelebs acnobs, vinc qmnida ma Te matikas. 

maT Sorisaa saxelovani qarTveli maTe ma ti ko se bic. 

kacobriobis inteleqtualuri istoriis SesaniSnavi faq te bis gacnoba swavlebaSi 

kulturul-istoriul tradiciebs qmnis. vcdilobT gavufrTxildeT da ganvaviTaroT 

es tradi ciebi. 

saxelmZRvanelos Sedgenis erT-erTi principi moswavleTa asakobrivi Taviseburebebis 

gaTvaliswinebaa. vcdilobT davic vaT zomiereba, ar gadavtvirToT saxelmZRvanelo 

rTuli dam tkicebebiT; moswavle unda grZnobdes dasabuTebis auci leb lo bas. 

es gansakuTrebiT geometriuli sakiTxebis gadmocemas exeba. 

dResac aqtualuria profesor iaglomis [41] sityvebi: `Cven me TodikaSi arsebobda 

araswori warmodgena, rom mxolod geo met ria uwyobs xels logikuri momTxovnelobis 

aRzrdas da geometriaa maTematikuri disciplinebis agebis nimuSi. sinam dvi leSi 

maTematikis sxva nawilebSi debulebaTa umetesobis dasabuTebis sizuste metic ki 

aris geometriaSi mocemul damtkicebebze~. axali moTxovnebis Sesabamisadaa gadmocemuli 

geometriuli masala _ suraTebis gamoyenebiT, romlebic saSualebas aZlevs 

moswavles ifiqros, aRmoaCinos faqtebi, gamoiyenos es faqtebi sayofacxovrebo 

saqmianobaSi, gana viTaros sivrculi orientacia, intuicia. WeSmaritebisadmi pativiscema, 

dasabuTebis saWiroebis gaazreba, dasabuTebisadmi moTxovnilebis gamomuSaveba 

mniSvnelovani zneobrivi aspeqtebia da maTze muSaoba VII klasSi arc iwyeba da arc 

mTavrdeba. maswavlebels vTxovT gaixsenos samkuTxedis kuTxe ebis jamis formulis 

gamoyvanis is gza, romelic ramdenime wlis win skolaSi iswavleboda _ igi Tvalsa- 

Cino faqtebis, aqsiomebis daxmarebiT dasabuTebas emyareboda [33]. 

Tumca `maTematikuri azrovneba ar daiyvaneba mxolod deduqciur msjelobebze, 

formalur damtkicebebze ... 

cnebis gamoyofa, romelic konkretul situacias esa da geba, konkretuli SemTxvevebis 

dakvirvebis Sedegad gan z o ga debebis povna, induqciis saSualebiT msjelobebis 

8

Catareba, ana logiiT da intuiciiT daskvnis povna _ yvelaferi es _ ma Te matikuri 

azrovnebis nawilia [41]. 

am mosazrebebs Cven mTlianad veTanxmebiT; iseve, rogorc adamaris azrs: `maTematikuri 

simkacris roli SemdegSia _ dakanondes intuiciis monapovari, mas sxva mizani 

arasdros hqonia!~ 

zemoTqmuli martiv deduqciur msjelobebze uaris Tqmas ar niSnavs. viziarebT 

burbakis gamonaTqvams [13]: `berZ nebidan moyolebuli _ vityviT `maTematika~ _ 

vgulisxmobT `damtkicebas~. 

siZneleebi moswavleebs martividan rTulisken mimaval gzaze unda SexvdeT. magram 

Tu masala rTulia, zogjer _ da uZlevelic, maSin codnas moCvenebiT daufleba 

cvlis. 

saxelmZRvaneloSi zomieradaa gamoyenebuli Teoriul-sim rav luri simbolika _ 

igi damxmare masalis rols asrulebs. 

Teoriuli masalis gaazrebasa da aTvisebas xels uwyobs savarjiSoebis sistema. 

isini Teoriul masalasTan mWidrod aris dakavSirebuli, aiZulebs moswavles moZebnos 

paragrafSi Sesabamisi Teoriuli sakiTxi. raime cnebis aTvisebis mniSvne lovani niSani 

misi konkretizaciis unaria. aRniSnuli unaris gamomuSavebas emsaxureba saxelmZRvaneloSi 

warmodgenili is savarjiSoebi, romelTa saSualebiT axali cnebebi martivad, 

sakmaod damaxasiaTebel situaciebSia gamoyenebuli, CarTulia axal kavSirebSi. 

savarjiSoTa mravalferovani sistema (`testebi~, amo ca ne bi, amocanebi jgufuri 

da individualuri muSaobisTvis, da ma tebiTi amocanebi), Teoriul nawilSi mocemuli 

analizuri msje lobis magaliTebi xels uwyobs moswavleebSi analizuri az r ovnebis 

ganviTarebas. sistematurad aris warmodgenili amo canebi, romlebic Teoriis aTvisebas, 

moswavlis kritikuli azrovnebis ganviTarebasa da maTematikis Segnebulad 

Seswavlas uwyobs xels. praqtikasTan xSiri mimarTeba (axali sakiTxis motivaciisas, 

swavlebisa da codnis ganmtkicebis procesSi) xels uwyobs praqtikuli gamoyenebis 

unar-Cvevebis gamo mu Savebasa da maTematikis Seswavlisadmi interesis gaRrmavebas. 

yovel paragrafs Tan axlavs sakontrolo kiTxvebi, amo ca nebi pasuxebis variantebis 

miTiTebiT (e. w. testebi); isini xels uwyobs moswavleTa mier TviTkontrolis 

ganxorci elebas. 

`testebi~ isea warmodgenili, rom igi moswavleTa si tu aciaSi swrafad garkvevisa 

da pasuxebis sworad SerCevis unars amowmebs da ara amoxsnebis Catarebis siswores. 

`didi xania mimdinareobs dava am tipis savarjiSoTa dadebiTi da uaryofiTi mxareebis 

Sesaxeb. Cven viziarebT im Tvalsazriss, rom Tu pasuxis SerCevas amoxsnis na wilad 

CavTvliT, maSin Sesabamisi amocana gamodgeba sas wav lo miznebisTvisac da zog jer _ 

SemowmebisTvisac. maga liTad, martivi ricxvis SerCeva, roca miTiTebulia ricxvebi, 

ro mel Tagan erTi iyofa 5-ze, me ore _ 3-ze, mesame luwia da ar aris 2-is toli. 

maTematika, SesaZlebelia, erTaderTi saskolo dis cip linaa, romelic damaxsovrebisa 

da gameorebis garda zogadi ganaTlebis iseT mniSvnelovan elementebs Seicavs, 

rogoricaa amocnoba, analizi, mixvedra, arCeva. 

axla SevexoT moswavlis saxelmZRvanelos agebulebas. wignis dasawyisSi gamovxatavT 

madlierebas yvelas mimarT, vinc Tavisi yuradRebiT wvlili Seitana wignis 

daxvewaSi. gan sakuTrebiT mniSvnelovania saswavlo gegmebisa da saxel mZRvaneloebis 

SemmuSavebeli jgufis eqspertebis Rvawli; isini pilotirebis procesSi monawile 

maswavlebleTan erTad saxelmZRvanelos Seqmnis Tanamonawileni arian; miwodebuli 

masalis Sesabamisad iZleodnen mniSvnelovan miTiTebebs; Ziri Ta di iyo masalis gadmocemis 

integraciis xarisxi. Tqvenc Seam CnevT, rom es xarisxi saxelmZRvaneloSi 

TandaTanobiT izrdeba, sasurvel dones aRwevs (magaliTad, aRweriTi statis tikis 

elementebis gadmocema geometriul da algebrul mimarTulebebTan integraciiT; 

9

koordinatTa meTodis, simetri ebis gamoyenebebi) da sistematur xasiaTs atarebs. 

avtorTa madlobas misalmeba da keTili survilebi mos devs. aqvea rCevebi wigniT 

sargeblobis Sesaxeb. 

saxelmZRvanelo 5 Tavisgan Sedgeba, TiToeuli Tavi _ para gra febisgan. xSirad 

paragrafi ramdenimenawiliania. daskvniT na wilSi damxmare masalaa gadmocemuli _ 

istoriuli cno be bi, damatebiTi sakiTxebi klasgareSe da moswavleTa indivi dua luri 

muSaobisTvis. 

saxelmZRvanelo moswavlis wignisadmi wayenebuli yvela moTxovnis Sesabamisadaa 

agebuli. sarCevi wignis dasaw yis Sia mocemuli, maTematikuri terminebis leqsikoni, 

maTe ma tikuri niSnebi da maTi gamoyenebis magaliTebi, sagnobrivi saZiebeli _ wignis 

boloSi. 

moswavlis saxelmZRvanelo swavlebis procesis dagegmvis da warmarTvis erT-erTi 

saSualebaa. moswavleebTan saubari SeiZleba wignis teqstisgan gansxvavebuladac aagos 

maswav le belma. saxelmZRvanelo exmareba maswavlebels _ masSi dafiqsirebulia 

SemecnebiTi saqmianoba, romelic maswav le belma unda awarmoos. maswavleblis saqmianoba 

ki sxva saSua le bebis gamoyenebasac gulisxmobs _ maswavleblis wigni, TvalsaCino 

masala. Tumca, SevecadeT, rom ar Seqmniliyo sa var jiSoTa sxva krebulebis 

gamoyenebis saWiroeba; moswavlis wignSi SemoTavazebuli savarjiSoebi moswavleTa 

sxvadasxva SesaZleblobebis gaTvaliswinebiTaa Sedgenili; saWiroa SerCeva da konkretuli 

situaciis mixedviT maTi swori gamo yeneba. gakveTilebis warmarTvisadmi 

momzadebis sxvadasxva eta pebi (saswavlo wlisadmi momzadeba, mocemuli gakveTilisadmi 

momzadeba) maswavlebelma nayofierad unda gamoiyenos. masalis sworad Ser- 

Cevis sakiTxSi Cvenc gexmarebiT _ amox s nilia moswavlis wignSi mocemuli TiTqmis 

yvela amocana, mi Ti Tebulia meTodikuri rekomendaciebi masalis wardgenisa da 

axsnis fazebis Sesaxeb. unda gvaxsovdes, rom maTematika yve las sWirdeba da amitom 

is unda Seiswavlos yvela moswav lem (Seswavlis xarisxi SeiZleba iyos sxvadasxva _ 

igi Se fa se bebSi aisaxeba). 

Zvirfaso maswavlebelo! 

warmatebebs gisurvebT Tqven sapatio da mniSvnelovan saq mia nobaSi. 

10 

guram gogiSvili, Teimuraz vefxvaZe 

ia mebonia, lamara qurCiSvili

VII klasis maTematikis erovnuli saswavlo gegma 

wlis bolos misaRwevi Sedegebi: 

ricxvebi da 

moqmedebebi 

1. kiTxulobs, 

gamosaxavs, adarebs 

da alagebs 

racionaluri 

ricx vebs poziciuri 

sistemis 

gamoyenebiT 

2. sxvadasxva 

xer xiT asru lebs 

moqmedebebs racionalurricxvebze 

3. afasebs racio 

nalur ri cxvebze 

moq me debaTa 

Sedegs 

4. erTmaneTTan 

akavSirebs zomis 

sxvadasxva er- 

Teulebs da iyenebs 

maT amo canebis 

amoxs nisas 

kanonzomierebebi 

da algebra 

5. amoicnobs 

da gamosaxavs 

sidi de ebs Soris 

pir da pir proporci 

ul damoki debu 

le bas 

6. amocanis 

amo xs nisas iyenebs 

simravlur 

cnebebsa da opera 

ciebs 

7. amartivebs 

al gebrul gamosa 

xulebas da 

xsnis wrfiv gantolebas 

8. ganavrcobs 

da aanalizebs 

obi eqtebis pe riod 

ul mim dev robas 

an/da mudmivi 

nazrdis mqone 

ricxviT mimdevro 

bas 

mimarTuleba 

geometria da 

sivrcis aRqma 

9. amoicnobs 

ge o metriul figu 

rebs, adarebs 

maT saxeobebs da 

axdens klasifici 

rebas 

10. war mo adgens 

geo met ri ul 

obieqtebs amocanis 

kon teq s tis 

Sesabamisad 

11. axdens geo 

met riul gardaqmnebs 

da iyenebs 

maT fi gura 

Ta Tvisebebis 

da sad genad 

12. iyenebs koor 

di natebis 

meTods orientaci 

isaTvis 

13. xsnis geo met 

riul amo ca nebs 

sam kuTxe deb Tan 

dakavSi re buli 

cnebe bisa da 

faqtebis gamoyenebiT 

monacemTa ana lizi, 

alba Toba da 

sta tistika 

14. moipovebs 

das muli amo canis 

amosaxs ne lad 

saWiro Tvi sebriv 

da rao denobriv 

mo na cemebs 

15. awes rigebs 

da warmoadgens 

Tvisebriv da 

rao denobriv mon 

a cemebs das muli 

amocanis amo - 

saxsnelad xel - 

sayreli for miT 

16. akeTebs Tviseb 

riv da raodenobriv 

mo na cem 

Ta inter pre tacias 

da analizs 

amocanis kon teqstis 

gaTva liswinebiT 

11

wlis bolos misaRwevi Sedegebi da maTi indikatorebi 

mimarTuleba: ricxvebi da moqmedebebi 

maT. VII.1. kiTxulobs, gamosaxavs, adarebs da alagebs racionaluri ricxvebs 

poziciuri sistemis gamoyenebiT 

Sedegi TvalsaCinoa, Tu moswavle: 

• aTwiladis CanawerSi uTiTebs Tanrigebs da asaxelebs TanrigebSi mdgom cifrTa 

mniSvnelobebs; iyenebs am codnas aT wiladebis Sedarebis an (zrdadobiT/klebadobiT) 

da la ge bisas. (mag. gaSlis sasrul aTwilads saTanrigo Sesak re be bis jamis saxiT, 

«daalage klebiT 2.9259, 3.1, 2.93, da 2.899»). 

• gamosaxavs da adarebs uaryofiT ricxvebs poziciuri sistemis gamoyenebiT; 

axdens mopirdapire ricxvis da ricxvis absoluturi mniSvnelobis cnebebis modelze 

demonstrirebas (maT Soris ricxviT RerZze). 

• ekvivalenturi formiT wers Sereul ricxvebs, aTwiladebsa da wiladebs; adarebs 

da alagebs sxvadasxva saxiT mocemul ricxvebs (aTwiladi ↔ wiladi). 

• poziciuri sistemis gamoyenebiT, konkretul maga li Tebze asabuTebs gayofadobis 

niSnebidan zogierTs (mag. 3ze da 9ze gayofadobis niSnebs); poulobs mocemuli 

ricx vebis umciresi saerTo jeradsa da udides saerTo gamyofs. 

maT. VII.2. sxvadasxva xerxiT asrulebs moqmedebebs ra cio nalur ricxvebze 


• axdens mTel ricxvebze ariTmetikuli moqmedebebis mo delze demonstrirebas 

(mag. `dadebiTi~ da ùaryofiTi~ er Te ulovani `muxtebiT~, e.i. ori gansxvavebuli 

feris `nu lo vani wyvilebiT~). 

• iyenebs ricxvis Caweris ekvivalentur formebs, moq me debaTa Sesrulebis Tanmimdevrobas, 

maT Tvisebebsa da daj gu febas gamoTvlebis gasamartiveblad. 

• yofs ricxvs proporciul nawilebad da poulobs ricxvs misi mocemuli nawilis 

mixedviT. 

• axdens naturalur-maCvenebliani xarisxis Tvisebebis demonstrirebas. 

• zepiri angariSisas iyenebs procentis kavSirs ricxvis nawilebTan; poulobs 

mocemuli ricxvis procents da xsnis Sebrunebul amocanebs. 

• irCevs da iyenebs racionalur ricxvebze ariTmetikuli moq medebebis Sesrulebis 

xerxs; (es xerxebia: zepiri, teq no logiebis gamoyenebiT, weriTi algoriTmebi). 

maT. VII.3. afasebs racionalur ricxvebze moqmedebaTa Se degs 


• gamoTvlebze sayofacxovrebo amocanis amoxsnisas iye nebs zepiri angariSis 

xerxebs an Sesabamis SemTxvevaSi, moq me d ebaTa Sedegis Sefasebas. 

• afasebs racionalur ricxvebze ariTmetikul moq me de baTa Sedegis mniSvnelobas, 

asrulebs moqmedebebs da amow mebs Tavis varauds. 

• amrgvalebs racionlur ricxvebs miTiTebuli sizustiT; miax loebiT poulobs 

(sizustis miTiTebis gareSe) ariT me ti kuli gamosaxulebis mniSvnelobas. 

• iyenebs Sefasebas aTwiladebze (weriTi algoriTmis an kalkulatoris gamoyenebiT) 

Catarebuli gamoTvlebis Sedegis adekvaturobis Sesamowmeblad. 

maT. VII.4. erTmaneTTan akavSirebs zomis sxvadasxva erTe ulebs da iyenebs maT 

amocanebis amoxsnisas 


• irCevs da iyenebs Sesaferis erTeulebs sididis cvli l ebis, moZraobis siCqaris, 

masStabisa da ruqaze manZilis pov nasTan dakavSirebul amocanebis amoxsnisas. 

• xsnis praqtikul saqmianobasTan dakavSirebul da/an sxva saswavlo discipli- 

12

nebidan momdinare amocanebs gamoT v lebze (mag. umartivesi xarjTaRricxva; istoriuli 

epoqis xangrZlivobis gansazRvra; amocanebi procentebze da pro po rciaze: 

xsnarebi, Senadnobebi da sxva). 

• mocemuli wrfivi damokidebulebis gamoyenebiT gamo sa xavs erT sistemaSi mocemul 

erTeuls sxva sistemis erTe uliT. 

• gamosaxavs mocemul erTeuls igive sistemis sxva erTe ulis saSualebiT. (mag. 

km/sT ebSi mocemul siCqares gamo sa xavs m/wm iT). 

Sinaarsi 

1. mTeli ricxvebi da ariTmetikuli moqmedebebi mTel ricx vebze. 

2. wiladebi, aTwiladebi da zogierTi kavSiri maT Soris. 

3. procenti (100ze naklebi an toli da 1ze meti an toli procenti); kavSirebi: 

erTidaigive mTelis procenti da nawili. 

4. ricxvebis Sedareba da ariTmetikuli moqmedebebis Sedegis Sefaseba. 

5. ricxvis proporciul nawilebad dayofa (mag. 2 : 3 : 5). 

6. ricxvebis umciresi saerTo jeradi da udidesi saerTo gamyofi; ricxvis 

martiv mamravlebad daSla. 

7. ricxvis naturalurmaCvenebliani xarisxi 

8. naSTiT gayofa, naSTi da gayofadobis niSnebidan zogi er Ti. 

9. zomis erTeulebi, zomis erTeulebs Soris kavSirebi da zomis erTeulebis 

gamoyeneba: masStabi; erTi sistemis er Teulis sxva sistemis Sesabamisi erTeuliT 

gamosaxva. 

10. fasdakleba/fasis gazrda (TanmimdevrobiTi da erT je radi fasdaklebebis/ 

fasebis zrdis erTmaneTTan Sedareba) da martivi xarjTaRricxva. 

mimarTuleba: kanonzomierebebi da algebra 

maT. VII.5. amoicnobs da gamosaxavs sidideebs Soris pirdapirproporciul 

damokidebulebas 


• mocemuli damokidebulebisaTvis Tvisebrivad da raode nob rivad aRwers Tu ra 

gavlenas axdens erTi sididis cvli le ba meoris mniSvnelobaze; moyavs mudmivi da 

aramudmivi rao de nobrivi cvlilebis magaliTebi yoveldRiuri cxovrebidan. 

• sityvierad Camoyalibebul debulebas sidideebs Soris damo kidebulebisa da 

mimarTebis Sesaxeb gamosaxavs gra fi ku lad an/da cxriliT da piriqiT – grafikulad 

an/da cxri liT gamosaxul damokidebulebas aRwers sityvierad. 

• sxvadasxva xerxiT (grafikulad, cxrilis saxiT, sity vi erad, algebrulad) 

gamosaxul damokidebulebebs Soris miu TiTebs erTsa da imave damokidebulebebs. 

maT. VII.6. amocanis amoxsnisas iyenebs simravlur cnebebsa da operaciebs 


• sxvadasxva xerxiT mocemuli simravlisaTvis gansaz R vravs mocemuli elementis 

kuTvnilebas mocemuli sim rav lisadmi. 

• problemis gadaWrisas, iyenebs zogierT damxmare xerxs sim ravleTa Soris mimarTebebis 

dasadgenad da simravluri ope raciebis Sesasruleblad. 

• sworad iyenebs simravleTa Teoriis cnebebs da Sesa ba mis aRniSvnebs sasrul 

simravleebze operaciebis (ori sim rav lis TanakveTa da gaerTianeba), sasrul simravleTa 

Soris mi marTebis, elementsa da simravles Soris mimarTebis gamo sa x visas. 

maT. VII.7. amartivebs algebrul gamosaxulebas da xsnis wrfiv gantolebas 


• teqsturi amocanis amosaxsnelad adgens da xsnis erT ucnobian wrfiv gantolebas. 

13

• iyenebs moqmedebaTa Tvisebebs, maTi Tanmimdevrobas da dajgufebas algebruli 

(araumetes ori cvladis Semcveli wrfivi an meore xarisxis) gamosaxulebis 

gasamartiveblad da misi mniSvnelobis gamosaTvlelad cvladebis mocemuli mniSvnelobebisaTvis. 

• algebruli gardaqmnebisa an/da logikuri msjelobis ga moyenebiT asabuTebs an 

uaryofs ori algebruli (araumetes ori cvladis Semcveli wrfivi an meore xarisxis) 

gamosa xu lebis igivur tolobas. 

maT. VII.8. ganavrcobs da aanalizebs obieqtebis periodul mimdevrobas 

an/da mudmivi nazrdis mqone ricxviT mimdevrobas 


• periodul mimdevrobaSi gamoyofs mimdevrobis periods. 

• warmoadgens mimdevrobis mocemuli fragmentis gavr co bis or an met variants, 

ganmartavs gavrcobis variantebs da adarebs maT. 

• dasmuli amocanis konteqstidan gamomdinare irCevs mimdevrobis gavrcobis 

variants da asabuTebs Tavis arCevans. 

• ganavrcobs mudmivi nazrdis mqone ricxviT mimdevrobas; asa xelebs realur 

viTarebaSi iseTi procesis magaliTebs, rom lebic aseTi mimdevrobiT aRiwereba. 

Sinaarsi 

1. pirdapirproporciuli damokidebuleba da misi ga mo sax va grafikis da 

cxrilis saSualebiT. 

2. simravleTa Teoriis cnebebi, operaciebi da Sesabamisi aRniSvnebi sasruli 

simravleebis SemTxvevaSi. 

3. teqsturi amocanebis amoxsna wrfivi gantolebebis gamoyenebiT. 

4. araumetes ori cvladis Semcveli wrfivi an meore xa ris xis gamosaxulebebis 

gamartiveba da mniSvnelobis gamoTvla. 

5. perioduli mimdevrobebi da mudmivi nazrdis mqone ricx viTi mimdevrobebi. 

mimarTuleba: geometria da sivrcis aRqma 

maT. VII.9. amoicnobs geometriul figurebs, adarebs maT saxeobebs da axdens 

klasificirebas 


• arqiteqturisa da xelovnebis nimuSebSi an maT ilus t raciebSi, yofiTi daniSnulebis 

sagnebSi asaxelebs misTvis nac nob geometriul figurebs an maT nawilebs. 

• ayalibebs mimarTebebs (mag. zogadobakerZooba) figu ra Ta saxeobebs Soris. 

• asaxelebs figuras misi niSan-Tvisebebis mixedviT, msje lobs figuris amosacnobad 

maTi sakmarisobis/arasakmarisobis Sesaxeb. 

maT. VII.10. warmoadgens geometriul obieqtebs amocanis konteqstis Sesabamisad 


• agebs dasmuli amocanis Sesabamis naxazs da adek va tu rad iyenebs asoiT aRniSvnebs. 

• aRwers geometriul obieqtTa mocemul grafikul gamo sa xulebebs an obieqtTa 

urTierTmdebareobas Sesabamisi ter mi nologiis gamoyenebiT. (mag. marTkuTxa paralelepipedis 

ro mel waxnagebs ekuTvnis miTiTebuli wvero). 

• gamosaxavs brtyel figurebs ise, rom maTi TanakveTa/gaer Tianeba iyos miTiTebuli 

formis an Tvisebebis mqone fi gura. 

maT. VII.11. axdens geometriul gardaqmnebs da iyenebs maT figuraTa Tvisebebis 

dasadgenad 

14


• garemomcvel obieqtebs Soris moiZiebs simetriul obi eq tebs. 

• xazavs brtyeli figuris (texili, mravalkuTxedi) si met riul figuras miTiTebuli 

simetriis RerZis mimarT; ax dens brtyeli figuris (texili, mravalkuTxedi) 

paralelur gada tanas. 

• uTiTebs brtyeli figuris simetriis RerZs/RerZebs; ax dens simetriulobis 

demonstrirebas; iyenebs figuris simet riulobas figuris Tvisebis dasadgenad. 

maT. VII.12. iyenebs koordinatebis meTods orien ta ci isaT vis 


• orientirebs rukaze an sakoordinato sibrtyeze koor di natebis gamoyenebiT 

(mag. asaxelebs mocemuli wertilis koor dinatebis miaxloebiT an zust mniSvnelobas; 

poulobs wer tils mocemuli mTelricxovani koordinatebis mixedviT). 

• asaxelebs sakoordinato RerZebis mimarT mocemuli wer tilis RerZulad simetriuli 

wertilis koordinatebs. 

• poulobs paraleluri gadataniT miRebuli figuris nebis mieri wertilis koordinatebs 

misi winasaxis ko or dinatebisa da miTiTebuli paraleluri gadatanis 

meSveobiT. 

maT. VII.13. xsnis geometriul amocanebs samkuTxe deb Tan dakavSirebuli cnebebisa 

da faqtebis gamoyenebiT 


• iyenebs samkuTxedebis tolobis niSnebs figuraTa Tvi se bebis dasadgenad, figuraTa 

ucnobi elementebis mosaZebnad an realur viTarebaSi manZilis arapirdapiri 

gziT dasadgenad. 

• xsnis agebis martiv amocanebs. 

• poulobs mizez-Sedegobriv kavSirebs samkuTxedTan da mis elementebTan dakavSirebul 

debulebebs Soris. 

Sinaarsi 

1. wertilebi, wrfeebi da sibrtyeebi: mimarTebebi maT Soris. 

2. geometriuli figurebi: klasifikacia sxvadasxva niSniT (mag. amozneqili 

da araamozneqili, brtyeli da sivrculi). 

3. kuTxeebi: elementebi, zoma, klasifikacia, Tvisebebi. 

4. samkuTxedebi: elementebi, klasifikacia, Tvisebebi, tolobis niSnebi. 

5. geometriuli gardaqmnebi sibrtyeze: paraleluri gadatana, samkuTxedebi 

(elementebi, klasifikacia, Tvisebebi, tolobis niSnebi). 

6. koordinatTa sistema; orientireba sibrtyeze, gar daq mne bis gamosaxva. 

7. agebis umartivesi amocanebi: mocemuli samkuTxedis to li samkuTxedis, 

kuTxis biseqtrisa, monakveTis SuamarTobis age ba. 

mimarTuleba: monacemTa analizi, albaToba da statistika 

maT. VII.14. moipovebs dasmuli amocanis amosaxs nelad saWiro Tvisebriv da 

raodenobriv monacemebs 


• ganasxvavebs Tvisebriv da raodenobriv monacemebs, iye nebs monacemTa Segrovebis 

Sesaferis saSualebas (gazomva, dak virveba). 

• mocemul TemasTan dakavSirebiT svams kiTxvebs, gan sazRvravs respondentebs 

da moipovebs saWiro monacemebs. 

• mocemuli amocanisTvis damoukideblad gegmavs da atarebs statistikur eqsperiments 

da agrovebs monacemebs. 

15

maT. VII.15. awesrigebs da warmoadgens Tvisebriv da rao de nobriv monacemebs 

dasmuli amocanis amosaxsnelad xel say reli formiT 


• axdens Tvisebriv da raodenobriv monacemTa dalagebas/kla sifikacias, warmoadgens 

monacemebs siis/piqtogramis saxiT, msjelobs dalagebis/klasifikaciis 

principze. 

• qmnis mowesrigebul monacemTa cxrilebs da asabuTebs SerCeuli dizainis mizanSewonilobas. 

• agebs sxvadasxva diagramebs erTi da igive Tvisebrivi an raodenobrivi monacemebisTvis 

da msjelobs, Tu monacemTa ramdenad mniSvnelovan aspeqtebs warmoaCens 

TiToeuli da ra upiratesoba gaaCnia TiToeuls. 

maT. VII.16. akeTebs Tvisebriv da raodenobriv monacemTa in ter pretacias da 

analizs amocanis konteqstis gaT va lis winebiT 


• svams kiTxvebs monacemebis Sesaxeb an/da axasiaTebs mo nacemebs, romlebic 

warmodgenilia siis, cxrilis, piq tog ra mis an diagramis saxiT, msjelobs arsebul 

kanon zo mierebebsa da gamorCeul monacemebze. 

• irCevs Sesaferis Semajamebel ricxviT maxasiaTeblebs, asa buTebs Tavis arCevans, 

iTvlis da iyenebs maT monacemTa jgu fis dasaxasiaTeblad. 

• adarebs monacemTa ramdenime jgufs da warmoaCens Tvi sebriv da raodenobriv 

msgavsebasa da gansxvavebas maT Soris (Semajamebeli ricxviTi maxasiaTeblebis gare- 

Se). 

Sinaarsi 

1. monacemTa Segrovebis saSualebani: gazomva da dakvir ve ba; gamokiTxva;statistikuri 

eqsperimenti. 

2. Tvisebrivi da raodenobrivi monacemebis organizacia: mo nacemebis klasifikacia 

(garda intervalebad dajgufebisa); mo nacemTa dalageba zrdadobaklebadobiT 

an leq si ko gra fi uli meTodiT. 

3. monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: 

monacemTa raodenoba, pozicia da Tan mim devroba erTobliobaSi, monacemTa 

sixSire; ganmeorebis ti pis kanonzomierebani; gamorCeuli (mag., eqstremaluri, 

iS vi aTi) monacemebi. 

4. monacemTa warmodgenis saSualebani raodenobrivi da Tvi sebrivi monacemebisTvis: 

sia, cxrili, piqtograma, wer ti lovani, meseruli, xazovani, svetovani 

diagramebi. 

5. monacemTa Semajamebeli ricxviTi maxasiaTeblebi Tvi seb rivi da raodenobrivi 

monacemebisTvis: centraluri ten den ciis sazomebi saSualo, moda; monacemTa 

gafantulobis sa zomi gabnevis diapazoni. 

16

Sinaarsisa da miznebis ruka 

Temebis Camona Tvali miznebi; Sinaarsis gardaqmna miznad; 

erovnul saswavlo gegmaSi 

miTiTebuli Sesabamisi punqtebi 

naturalur ricxvTa mwkrivi 

naturaluri ricx vis Caweris 

sxvadasxva wesi 

jgufuri muSaoba (xa ris xis 

Tviseba) 

gayofa, gayofadobis Tvisebebi 

gayofadobis niSnebi 

jgufuri muSaoba udidesi 

saerTo gam yo fi umciresi 

saerTo je ra di 

Tematuri Semowmeba (sakontrolo 

wera # 1) 

naturaluri ricxvebisa da maTi 

Tvisebebis Sesaxeb codnis gaRr ma veba; 

poziciuri sistemis warmodgena; 

Tvlis sistemebi (arasavaldebulo) 

didi ricxvebi da maTi Caweris xer xebi. 

es masala wina wlebSi Seswav li li 

sakiTxebis gameorebasa da gan m tkicebas 

emsaxureba. VII.2(2) 

kanonzomierebebis aRmoCenis unaris 

ganviTareba axarisxebisas e. w. `bo lo 

cifris~ cxrilis saSualebiT. VII.2(4) 

poziciuri sistemis gamoyenebis unaris 

ganviTareba, naSTiani gayofis 

Tvisebebis gamoyenebiT kanonzo mie rebebis 

gansazRvra. poulobs mocemuli 

ricxvebis udides saerTo gamyofsa 

da umcires saerTo jerads, iyenebs 

maT praqtikuli amocanebis amoxs nisas. 

VII.1(4) 

umciresi saer To jeradisa da udidesi 

saerTo gamyofis cnebebis aTvisebis 

Semow meba. 

proeqti xarjTaRricxvis Sedgenis unaris 

gamomuSaveba, arCevanis dasabuTeba, 

cnobarebis gamoyeneba VII.3(1) 

mTeli ricxvebi 

mTeli ricxvis mo du li, mTeli 

ricxve bis Sedareba 

mTeli ricxvebis Sekreba 

mTeli ricxvebis ga mok leba 

mTeli ricxvebis gam rav leba 

da gayofa 

simravle, qve sim rav le, venis 

diagramebi 

naturaluri ricxvis cnebis ga far- 

Toebis algebruli gzis gacno biereba, 

geometriuli gamosaxvisa da 

misi gamoyenebis unaris gamomuSaveba; 

Sedareba, moduli. mTel ricxvebze 

moqmedebebis TvalsaCino warmodgena 

(sxvadasxva modelis gamoyenebis unari). 

VII. 2(1,2). 

Teoriul-simravluri cnebebi da maTi 

gamoyenebis unari (amocanebis amoxs na 

venis diagramebis gamo ye ne biT) VII.6 

sa vara u do 

sas wav lo 

dro 

4 sT 

1 sT 

12 sT 

1 sT 

1 sT 

7 sT 

4 sT 

jgufuri muSaoba naSTTa ariTmetikis propedevtika 1 sT 

dadebiTi racio na luri ricxvebi 

aTwiladis CanawerSi Tanrigebis miTi- 

Tebisa da cifrebis mniSvne lo be bis codnis 

gamoyeneba aTwiladebis Se darebisas. 

VII.1 

jgufuri muSaoba xarjTaRricxvis dagegmva, Sefa se ba, 

SerCeva. VII. 3(1) 

2 sT 

1 sT 

17

acionaluri ricx vebi 

moqmedebebi racio nalur 

ricxvebze. 

racionaluri ricx vis naturalurmaCve 

neb liani xarisxi 

Tematuri Semowmeba 

(sakontrolo wera # 2) 

cvladiani gamosa xu le ba. cvladiani 

gamosaxulebis mniSvnelobis 

povna. ga mo saxulebis 

gar daqmna. 


(sakontrolo wera # 3) 

monacemTa analizi da statistika 

1. monacemTa Segro veba 

2. monacemTa war mod ge na xelsayreli 

for mi T 

3. piqtograma 

geometriuli figu rebi 

1. wertili. wrfe. mo nakveTi. 

sxivi. wrfeebis urTierTganlagebis 

SemTxvevebi. 

2. wrfeebisa da sibr tye ebis 

urTierT gan la gebis SemTxvevebi 

3. kuTxe. kuTxeTa ur TierTganlageba 

4. monakveTis da kuTxis gazomva 

5. kuTxeTa klasi fi ka cia. marTobuli 

wrfe ebi 

jgufuri muSaoba (amo canebis 

amoxsna wrf eze modelirebis 

gamoyenebiT) 

Tematuri Semowmeba (sakontrolo 

wera # 4) 

gantoleba. ganto le bis fesvi 

wrfivi erTucno biani gantoleba 

wertilis koor di na tebi wrfeze 

da sib r tyeze. ganto lebaTa amoxsnis 

geometriuli xerxi 

1. wertilis koor di na tebi wrfeze 

2. wertilis koor di natebi sibrtyeze 

3. sakoordinato me oTxe debi 

18 

racionalur ricxvTa simravlis gaazreba, 

Tvisebebi, moqmedebebi, gamoyeneba 

sayofacxovrebo amocanebis amoxsnisas. 

VII.2, VII.3 

6 sT 

masalis aTvisebis Semowmeba 2 sT 

cvla diani gamosaxulebis Sedgenis 12 sT 

Cve vebi, cvladiani gamosaxulebebis 

gar daqmnis sxvadasxva wesis dauf le ba 

(moqmedebaTa Tvisebebi, Semok le buli 

gamravlebis formulebi). geometriuli 

TvalsaCinoebis gamo ye ne ba. VII. 7 


miznobrivi informaciis Segro ve bi sa 

da damuSavebis xerxebis Sed ge nis unaris 

gamomuSaveba. 

VII.14, VII.15 

geometriuli figurebis warmod ge nis, 

urTierTganlagebis SemTxve ve bis aTviseba, 

gazomvis operaciis arsis Sesaxeb 

warmodgenis Camoya li be ba (monakveTis 

sigrZe, kuTxis zo ma); kuTxeebis zomebis 

mixedviT kla si fikaciis warmodgena. 

nacnobi geo met riuli figurebis amocnobis 

una ris gamomuSaveba. VII.9; VII.10 

3 sT 

5 sT 


gantolebis, rogorc bunebasa da sazo 

gadoebaSi mimdinare mraval fe ro van 

movlenaTa aRweris ZiriTadi mo delis 

Sesaxeb, modelirebis meTo de bis Sesaxeb 

warmodgenebis gaazreba; gan tolebis, 

rogorc maTematikis erT-erTi ZiriTadi 

enis dauflebis unar ze muSaoba. VII.7 

maTematikis mniSvnelovani aparatis 

_ koordinatTa meTodis gacnoba; mis 

gamoyenebis unarze muSaoba; ko or di natebis 

gamoyenebiT amocanebis amox snis 

xerxebis Seswavla.VII. 7 

3 sT 

4 sT

amocanebis amoxsna gantolebis 

saSua le biT 

amocanebis amoxsnis sxvadasxva 

xerxi 

naturalurmaCvenebli ani 

xarisxis Tvisebebi 

jgufuri muSaoba 

(naturalurmaCve neb li ani 

xarisxis Tvise be bis gamoyeneba) 

teqsturi amocanebis amosaxsnelad gantolebis 

Sedgenisa da amoxsnis unaris 

gamomuSaveba 

amocanebis amoxsnis sxvadasxva xerxebis 

warmodgena VII.7, VII.6 

naturalurmaCvenebliani xarisxis Tvisebebis 

warmodgena da gamoyeneba VII.2(4) 

texili. mra val kuTxe di asaxelebs da amoicnobs geomet riul 

figurebs: texils, mraval ku Txeds. 

gamosaxavs brtyel figurebs ise, rom 

maTi TanakveTa (gaerTianeba miTiTebuli 

formis an Tvisebebis mqone figura iyos). 

VII.10; VII.11 

jgufuri muSaoba (kombinatorikis 

amocanis amoxsna 

xisebri diagramis saSualebiT) 

RerZuli simetria 

paraleluri gada ta na 

amocanebis amoxsna 

individualuri mu Sa obis 

proeqti (ùmok lesi marSruti~ 

_ te xilis sigrZe) 


(amocanebis amoxsnis sxvadasxva 

xerxi, na tu ra lurmaCvenebliani 

xarisxi 

sakontrolo wera # 5) 

Sefardeba. pro por cia. 

proporciis gamo ye ne ba. procenti 

amo ca ne bis amoxsna 

procen tis gamoyenebaze. 

jgu fu ri muSaobis proeq ti 

(martivi xarjTaR ricx vis 

Sedgena) 


sakontrolo wera # 6 

iyenebs paralelur gadatanas, Rer Zul 

simetrias figuraTa Tvisebebis dasadgenad; 

garemomcvel obieqtebs Soris 

moiZiebs si metriul obieq tebs; xazavs 

texilis, mraval kuTxe dis simetriul 

figuras mi Ti Te bu li RerZis mimarT da 

axdens maT pa ra lelur gadatanas. VII.12 

iyenebs proporcias, procents praq tikuli 

amocanebis amoxsnisas (xsna re bi, 

Senadnobebi, ....); xsnis praq ti kul amocanebs 

umartives xarjT aRricxvasTan 

dakavSirebiT; iyenebs ricxvis proporciuli 

gayofis wess amocanebis amoxsnisas. 

iyenebs proporcias masStabTan da rukaze 

manZilis povnasTan dakavSi re buli 

amocanebis amoxsnisas. akav Sirebs zomis 

sxvadasxva erTeulebs 

VII.2(3,5), VII.4 

8 sT 

2 sT 

1 sT 

3 sT 

1 sT 

6 sT 

1 sT 

2 sT 

9 sT 

2 sT 

2 sT 

19

statistikis elemen te bi. monacemTa 

Sema ja mebeli ricxviTi 

ma xa siaTeblebi. 

samkuTxedi. samku Txe dis elementebi 

samkuTxedebis klasi fi kacia 

20 

avlens Tvisebriv da raodenobriv monacemTa 

interpretaciisa da analizis 

unars, Semajamebeli ricxviTi ma xasiaTeblebis 

warmodgenis da gamo ye nebis 

unars VII. 16 

samkuTxedisa da misi elementebis warmodgena; 

klasifikacia sxvadasxva niSnis 

mixedviT. asaxelebs misTvis nacnob geometriul 

figuras yofiTi daniSnulebis 

sagnebSi VII. 9(3) 

mimdevrobebi mimdevrobis mocemuli fragmen tebis 

mixedviT gavrcobaTa warmodgena. 

obieqtebis perioduli mimdevrobebi. 

mudmivi nazrdis mqone ricxviTi mimdevrobebis 

warmodgena; periodul mimdevrobaSi 

periodis dasaxeleba. VII.8 

jgufuri muSaoba (gantolebis 

Sed gena, perioduli mimdev 

robebis gamoye ne ba) 

tolferda samkuT xedi 

samkuTxedebis to lo bis niSnebi 

s a m k u T x e d i s k u T x e e b i s 

jami 

marTkuTxa samkuT xedi 

monakveTis SuamarTo bis 

Tviseba 

jgufuri muSaoba ( samkuTxedebis 

tolo bis niSnebis 

gamoyeneba) 

individualuri mu Sa o bis 

proeqti (xarj TaR ricx vis 

Sedgena) 

samkuTxedebis tolobis niSnebis warmodgena 

da maTi gamoyeneba praqtikuli 

amocanebis amoxsnisas 

tolferda samkuTxedis Tvisebebis 

dasabuTeba 

samkuTxedis kuTxeebis jamis for mu lis 

gamoyvana, TvalsaCino war mod gena. marTkuTxa 

samkuTxedis Tvi se be bis warmodgena 

da gamoyeneba praq ti kuli amocanis amoxsnisas. 

mona kve Tis SuamarTobis Tvisebis 

gamo ye neba. 

2 sT 

3 sT 

2 sT 

2 sT 

1 sT 

8 sT 

2 sT 

1 sT 

1 sT 

kuTxis biseqtrisis Tviseba kuTxis biseqtrisis Tviseba da misi 

gamoyeneba. agebis amocanebi. VII.13(1) 2 sT 

Tematuri Semowmeba, sakontrolo 

wera #7 (samkuTxedi, 

Tvisebebi) 

amocanebi agebaze samkuTxedis toli samkuTxedis age ba 

monakveTis SuamarTobis, biseqt risis 

agebis amocanebi VII.13(2) 

2 sT 

4 sT 

gameoreba 6 sT

saswavlo masalis wardgenis fazebi da gakveTilis 

dagegmvis zogadi principebi 

saswavlo procesis organizaciis ZiriTadi forma gakve Ti lia; saganmanaTleblo, 

aRmzrdelobiTi da praqtikuli miz ne bis ganxorcieleba gakveTilze xdeba. amitom 

maTematikis swav lebis ZiriTadi sakiTxi gakveTilis kargi momzadeba da Cata rebaa. 

meTodikur literaturaSi didaqtikuri cneba _ `gakve Ti li~ ZiriTadad ase aRiwereba: 

gakveTili logikurad das ru lebuli, mTliani saswavlo-aRmzrdelobiTi procesis 

gar kve uli SemosazRvruli monakveTia. masSi rTul urTierT da mo kidebulebaSia 

procesis yvela ZiriTadi elementi: Sinaarsi, mizani, saSualebebi, meTodebi, organizacia; 

yovel gakveTilze gansazRvruli saganmanaTleblo da aRmzrdelobiTi 

amocanebi wydeba; am amocanebis gadawyveta konkretuli saswavlo ma sa lis ganxilvis 

procesSi mimdinareobs. 

maTematikis gakveTilisadmi wayenebuli mTavari moTxov naa ZiriTadi didaqtikuri 

amocanis arseboba _ im Temis Ses wav lis miznis arseboba, romlis gadawyvetis procesi 

mo ce mul gakveTilze mimdinareobs. yoveli gakveTilis win kargad unda gaiazroT misi 

Sinaarsi da mizani, amaSi Cven mier mo wo debuli ruka dagexmarebaT. miznis Sesabamisi 

unda iyos kar gad gaazrebuli da dagegmili saswavlo masalis wardgenis fazebi: 

• motivacia; ar aris sakmarisi, rom maswavlebels gaaz re buli hqondes mizani; 

saWiroa, rom igi moswavleebisTvisac gax des ZiriTadi mizani. moswavleebTan saubari 

unda daviwyoT ara imiT, Tu ras vaswavliT, aramed mniSvnelovania Tavidanve interesis 

aRZvra da iseTi situaciis Seqmna, roca moswavle motivirebulia da poulobs 

pasuxebs kiTxvaze _ `risTvisaa saWiro~. motivacia SesaZlebelia praqtikuli 

amocanis dayenebiTa da misi amoxsnis xerxebis ZiebiT daviwyoT, an maTematikis Siga 

kanonzomierebis gaazrebiT, problemuri situaciis Seqmnis xelSewyobiT; mxolod 

amis Semdeg xdeba Sesabamisi amocanis dasma da misi amoxsnis Zieba. 

• amocanis gansazRvris Semdeg mimdinareobs misi amoxsnis gzis Ziebis procesi. 

amoxsnis Ziebis procesis warmarTvis sxvadasxva meTodikuri saSualeba arsebobs. yovel 

amocanas, rogorc wesi, amoxsnis Ziebis garkveuli forma miesadageba. es formebia _ 

muSaoba jgufebad (SesaZlebelia or-oradac), mTeli klasis erToblivi monawileobiT, 

individualuri muSaoba da a. S. im SemTxvevaSic ki, roca amocanis amoxsnis ZiebaSi 

mTeli klasi erTdrouladaa Cabmuli, maswavlebelma swavleba ise unda warmarTos 

(moxerxubuli kiTxvis dasmis saSualebiT), rom Temis Seswavlis procesis ZiriTadi 

Semoqmedebi Tavad moswavleebi aRmoCndnen; maswavlebeli am SemTxvevaSi warmarTvelis, 

diriJoris funqcias unda asru lebdes. 

moswavlis saxelmZRvanelos teqsti mogcemT aseTi gakve Ti lebis Catarebis sa- 

Sualebas. igi ZiriTadi saswavlo saSua le baa da amitom gakveTilisTvis mzadebis 

procesSi teqs tebis, ilustraciebis, savarjiSoebis gacnobas, daxarisxebas didi dro 

unda dauTmoT. Tumca, maswavleblis wignic dagex ma rebaT TiToeul etapze (sawyisi, 

damagvirgvinebeli) Sesas rulebeli savarjiSoebis SerCevis sakiTxSi. 

Sinarsisa da miznebis rukaSi ver mivuTiTeT Temis Seswav lis yvela mizani, romelic 

saganmanaTleblo da aRmzrde lo biTi amocanebis gadawyvetis komponentebisgan Sedgeba. 

ar unda gavaigivoT mizani SinaarsTan. magaliTad, samkuTxedis kuTxeebis jamis formulis 

gamoyvana saxelmZRvaneloSi isea warmodgenili, rom igi am Temis Sinaarsis miznad 

gardaqmnis process kargad warmogvidgens; dasmuli amocanis amoxsna kritikuli 

azrovnebis, msjelobisa da dasabuTebis unaris gamomuSaveba, geometriul figuraTa 

gamosaxvis da am figu raTa Tvisebebis SeswavlaSi eqsperimentis gamoyenebis unaris 

gamomuSavebaa. am miznebis gansaxorcieleblad qaRaldisgan gamoWril samkuTxedze 

(modelze) Catarebuli eqsperimentis (gamoiyeneba ramdenime RerZuli simetria) an 

21

dasabuTebis xer xia mowodebuli. meore maTgani dasmul kiTxvebze pasuxis SerCevis, 

msjelobisa da dasabuTebebis fonze mimdinareobs. aq maswavlebelma SeiZleba sxva 

eqsperimentic daamatos (tran s portiris gamoyenebiT). ufro dawvrilebiT am gakveTilis 

Sesaxeb qvemoT mogaxsenebT. 

am konkretuli magaliTidan Cans, rom aqtivobis mizani saganmanaTleblo da aRmzrdelobiTi 

miznebis erTobliobaa. 

gakveTilze Seswavlili faqtebi TavisTavad aris mniSv ne lovani; Tumca, kidev 

ufro mniSvnelovania is, rom maTi Ses wav lis dros ganxorcielebuli procesi Tavis 

kvals tovebs _ kritikuli azrovnebis ganviTareba, msjeloba da dasabuTebis unaris 

gamomuSaveba. amasTanave, moswavleTa gonebaSi gaiazreba mniSvnelovani sakiTxebi: 

adamianis mier faqtebis aRmoCena, aRmocenebuli amocanebis amoxsna da Sedegebis 

dafiqsireba azrovnebaSi. konkretuli deduqciuri msjeloba, dasabuTeba im SemTxvevaSi 

atarebs aRmzrdelobiT funqcias, roca mos wav les gavagebinebT misi Catarebis 

mniSvnelobas. 

axla davasaxelebT ramodenime sasargeblo zogad reko men dacias, romelic meTodikis 

sakiTxebs ganekuTvneba: 

• maswavlebeli cdilobs, rom yoveli axali SemecnebiTi amo cana TviT moswavlem 

Camoayalibos; 

• maswavleblis xelmZRvanelobiT da moswavleTa Za lis xme viT _ dakvirvebis, cdis, 

konkretuli SemTxvevebis ana li zis Sedegad iqmneba warmodgena, hipoTeza arsebul 

kanon zomierebaze. 

• maswavleblis xelmZRvanelobiT mimdinareobs dasa bu Te biT gzis Zieba, amocanis 

amoxsnis gegmis Sedgena; rasac xSi rad mosdevs TviT moswavleebis mier am gegmis 

realizacia. 

axla SevexoT saswavlo procesisTvis mzadebis zogad sqemas: 

• maswavleblis muSaoba saswavlo wlis win. 

• gakveTilTa sistemis gaazreba saswavlo wlis win; 

• konkretuli gakveTilisTvis momzadeba. 

momzadebisas maswavlebelma unda gamoiyenos moswavlis wigni, maswavlebelis 

sarekomendacio wigni da, saWiroebis SemTxvevaSi, iq miTiTebuli literatura. 

sarekomendacio wig nis mixedviT, sasurvelia, maswavlebeli Tavdapirvelad sas wav - 

lo gegmas, Sefasebis sistemas da gakveTilebis Semo Ta va zebul nimuSebs gaecnos. 

sarekomendacio wignis Sesavali mas gaacnobs avtorTa mier saxelmZRvanelos agebis 

ZiriTad prin cipebs. 

maswavlebeli SemoqmedebiTad unda miudges Cven reko men da ciebs; igi safuZvlad 

iRebs Cven mier SemoTavazebul gegmas da azustebs mas sakuTari gamocdilebiTa 

da klasis Tavise bu rebebis gaTvaliswinebiT. es dazustebebi gansakuTrebiT mniSvnelovania 

sawyis etapze Sesasrulebeli savarjiSoebis da damamTavrebel etapze 

Sesasrulebeli savarjiSoebisa da sakontrolo weris variantebis SerCevisas. 

gakveTilis dagegmva iTvaliswinebs Casatarebeli procesis tips: 

• axali masalis gacnoba 

• masalis ganmtkiceba 

• codnis Semowmeba 

• sxva tipis gakveTilebi (gakveTili bunebaSi, gakveTili-proeqti ...) 

Tumca, zogierTi tipis procesi (axali masalis gacnoba, gan mtkiceba, Semowmeba), 

rogorc wesi, yovel gakveTilze mim di nareobs, SesaZlebelia _ sxvadasxva moculobiT. 

moswavleTa codnis Semowmeba, moswavleTa muSaobaze dak vir veba yovel gakveTilze 

mimdinareobs. gakveTilze mTavaria vaswavloT da aRvzardoT. swavleba ar niSnavs 

22

mxolod codnis gadacemas _ swavleba codnis SemoqmedebiTad dauf lebas unda 

niSnavdes, misi gamoyenebis unaris ganviTarebaze unda iyos mogebuli. moswavleTa 

Sefaseba swavlis procesze maswavleblis dakvirvebebiT, moswavleTa mier sakontrolo 

da damoukidebeli samuSaoebis Sesrulebis xarisxiT gani saz Rv reba. yoveli 

gakveTilis Semdeg sakuTar wignakSi Cai niS neT moswavleebze dakvirvebebis Sedegebi, 

gaiTvaliswineT mos wavleTa SemoqmedebiTi aqtiuroba (masalis aTvisebis done kargad 

Cans savarjiSoebis amoxsnis drosac _ ganmtkicebis pro cesSi). 

saswavlo masalis dasabuTebuli SerCeva iTvaliswinebs Sem deg moTxovnebs: 

• saswavlo masalis Sesabamisoba Temis mizanTan 

• gakveTilze Sesasrulebeli samuSaos moculobis swori gansazRvra 

• optimaluri Tanafardoba konkretulsa da zogads Soris 

• Teoriasa da praqtikas Soris aucilebeli urTierT kav Siris ganxorcieleba 

maswavlebelma, rogorc wesi, Tavidan bolomde deta lu rad unda gaiazros gakveTili, 

winaswar, drois mixedviT unda iyos ganawilebuli mTeli samuSao. 

magaliTad, Tu gakveTilze axal Temaze gadasvlac aris gaT valiswinebuli, maSin 

SeiZleba im sakiTxebis Sesaxeb msje loba, romelTa bunebriv da kanonzomier gagr- 

Zelebis axali sa kiTxebi Seicavs, SeiZleba gakveTili pirdapir im praqtikuli amocanis 

ganxilviT daviwyoT, romlis maTematikuri modelis Seswavla axali maTematikuri 

faqtebis aRmoCenas, hipoTezis Camoyalibebasa da dasabuTebebs moiTxovs. am procesis 

buneb rivi gagrZeleba Sesabamisi savarjiSoTa sistemis ganxilvaa. 

kiTxvebze pasuxebis gacemis sistema ar unda iyos erT fe rovani _ mxolod warmatebuli 

moswavleebTan mimarTebiT ar unda SemoifargloT; moswavlis raime mosazrebas 

myisve nu upasuxebT. swor pasuxsac ki maSinve nu daeTanxmebiT xol me _ gaakeTeT 

pauza, iqneb garkveuli eWvic ki gamoTqvaT misi mosazrebis sisworis mimarT. amiT 

miaRwevT imas, rom bavSvebi daubrundebian dasmuli kiTxvis analizs da male WeSmariti 

daskvna _ swori pasuxi klasis dominantur mosazrebad gadaiq ceva. klasi, 

erToblivi ZalisxmeviT, `gaiZulebT~ daeTanxmoT mis pozicias. es axarebs, amxnevebs 

da aerTianebs axal gazr debs. es maTi erToblivi azris gamarjvebaa. Tqvens mizansac 

xom es warmoadgens _ moswavle CamoayaliboT Semoqmed, codniT aRWurvil, iniciativian, 

xalisian axalgazrdad. maTematika mZlavri emociuri muxtis matarebelia da 

misi amoqmedeba Tqveni ZalisxmeviT miiRweva~. 

yuradRebiT unda movisminoT yvela pasuxi, uxeSi Sec do mis SemTxvevaSic ki dauSvebelia 

mkacri uaryofiTi Sefa se bebis gamoTqma. 

yuradReba miaqcieT, rom terminebi, cnebebi da movlenebi sworad iyos ganmartebuli. 

gakveTilebis dagegmvasa da warmarTvaSi xels SegiwyobT sanimuSo gakveTilebis 

scenarebi. 

axali masalis ganmtkicebis procesi SeiZleba e. w. `tes turi~ amocanebis àmoxsniT~ 

daviwyoT, maTi Sesruleba swori pasuxis SerCeviT unda Semoifarglos _ zogjer 

SeiZleba komentarebis gakeTebac gaxdes saWiro. yovel paragrafSi mo ce muli masala, 

rogorc wesi, 2 gakveTilzea gaTvalis wi ne buli; meore gakveTilze codnis ganmtkicebaze 

zrunviT Semo vifarglebiT. 

swavlebis erT-erTi saintereso da mniSvnelovani forma jgu furi muSaobaa. 

es muSaoba SeiZleba gakveTilis procesis erT-erTi Semadgeneli nawili iyos _ 

daukavSirdes axali ma salis gaazrebas, praqtikuli saqmianobis (eqsperimentis) an 

Semajamebeli daskvnebis gamotanas, an, SesaZlebelia, mas mTe li gakveTilic davuTmoT. 

mis warmarTvaSi moswavlis sa xel mZRvaneloSi warmodgenili amocanebis sistema 

dagex ma rebaT (magaliTad, amocanebi jgufuri muSaobisTvis). am Sem TxvevaSi jgufur 

muSaobas SeiZleba Sejibris saxec ki miv ceT. 

23

organizaciulad jgufuri muSaobis es varianti _ paeq ro ba _ jgufuri muSaoba 

SeiZleba ase movawyoT: 

winaswar vacxadebT Catarebis dRes; moswavleebs vavalebT samuSao rveulis ormagi 

furceli iqonion. paeqroba mos wav leTagan kapitnebisa da maTi TanaSemweebis 

dasaxelebiT iwyeba. optimaluria 4-5 moswavlisgan Semdgari jgufebi _ gundebi. 

gun debis dakompleqteba SeiZleba kapitnebsac miandoT. mTa varia, àrCevnebma~ didi 

dro ar wagarTvaT. 

mas Semdeg, rac gundebi dakompleqtdeba, winaswar gamrav le buli amocanebi 

daurigeT gundebs (an dafaze amowereT pirobebi). am SemTxvevaSi yvela gunds erTnairi 

davaleba miecema. 

gakveTilis dasrulebamde 10-12 wuTiT adre kapitnebs eva lebaT warmoadginon 

maTi gundebis Sedegebi _ maTi amocanebis amoxsnebi. CaibareT es amoxsnebi winaswari 

komentarebis gareSe da saukeTeso amoxsnebis avtorebi rigrigobiT miipatiJeT 

dafasTan naSromTa mokle prezentaciisTvis. cxadia, kritika da polemika, Tu amis 

safuZveli arsebobs, unda iyos uSeRa vaTo, magram koreqtuli. am bWobaSi Tqvenc 

mogiwevT xandaxan Cabma; zogjer mediatoris rolis Sesrulebac. es procedura aRniSnul 

droze mets ar moiTxovs, radgan amocanebi yvelas kar gad aqvs gaazrebuli da 

mxolod sakvanZo punqtebia xaz gasasmeli. 

am gansjis dasrulebisTanave unda aRdges merxebis Tavda pirveli ganlageba, Semdeg 

acxadebT gundebis mier mopovebul qulebs (TiToeuli amocana SeiZleba 2-quliani 

skaliT Sefasdes) da dakavebul adgilebs am paeqrobaSi. SeiZleba daawesoT damatebiTi 

qulebi prezentaciis Sesafaseblad. 

jgufuri muSaobisTvis amocanebi SeiZleba SeirCes saxel mZRva neloSi Sesabamisi 

niSnakiT gamoyofili adgilidan. Tumca, winaswar yvela amocana rom ar iyos `gaSifruli~, 

iqneb maTi pirobebi odnav SecvaloT, an, zogierTi amocana pa rag rafis 

damatebiTi savarjiSoebis krebulidan airCioT. 

moswavlis saxelmZRvaneloSi SemoTavazebuli jgufuri mu Saobis zogierTi proeqti 

raime erTi axali Temis Sesabamisi struq turirebuli kiTxvebisgan Sedgenili amocanaa 

_ yoveli kiTxva winas ukavSirdeba _ kiTxvebze pasuxebis sistema raime erTi axali 

Temis Sesabamisi problemis dasmas da amoxsnas gulisxmobs. am SemTxvevaSi pasuxebis 

sistema, romelic mos wav leTa jgufis erToblivi Zalisxmevis Sedegia, maTi ko leqtiuri 

SemoqmedebiTi naSromia da misi Sefaseba prezen ta ciisas ganxorcieldeba. 

zogjer jgufebs sxvadasxva saxis davalebebi SeiZleba mivceT _ gansxvavebuli 

eqsperimentis Catareba (magaliTad, geometriuli obieqtis Tvisebebis dasadgenad 

an dasabuTebis gziT geometriuli faqtis ganxilva-dasabuTeba, Sedegis war modgena. 

am SemTxvevaSi moswavleebi msjeloben warmodgenili varaudebis koreqtulobaze da 

adareben maT. aseTi tipis jgufuri muSaobebi, rogorc zemoT aRvniSneT, gakveTilis 

fragmenti SeiZleba iyos da xSirad unda gamoviyenoT. 

Teoriuli masalis gadmocemis Cveneuli meTodika saSua lebas gaZlevT airCioT, 

yvelaze ufro mosaxerxebeli forma Sinaarsisa da miznebis rukaSi miTiTebuli moTxovnebis 

Sesas ruleblad. 

amrigad, gakveTilis dagegmva SesaZlebelia Semdegi sqemiT mimdinareobdes. 

• saswavlo miznebi _ gakveTilis ideuri Sinaarsi; is, rasac unda miaRwioT, ra 

codna da unar-Cvevebi unda SeiZinon moswavleebma am gakveTilze; 

• Sefaseba _ ra kriteriumebiT SevafasebT dagegmilis Sesrulebis xarisxs da ra 

SemTxvevaSi vityviT, rom gakveTilma warmatebulad Caiara; 

• miRwevis saSualebebi _ ra meTodebiT, moswavleTaA ra organizebiTa da ra 

damatebiTi resursebis saSualebiT apirebT dasaxuli amocanebis gadaWras; sasurvelia 

winaswar gaTvaloT drois savaraudo xarji; aucileblad winaswar gansazRvreT 

saSinao davalebis xasiaTi da moculoba. 

24

Sefasebis zogadi principebi 

saskolo Sefasebis axali sistema, romelic erovnuli saswavlo gegmiT aris gansaz- 

Rvruli, iTvaliswinebs Semdeg aucilebel midgomebs: akademiuri moswrebis Sefaseba 

unda iyos xSiri da mravalmxrivi. unda Sefasdes ara marto informaciis floba, 

aramed SeZenili unar-Cvevebi. ar aris sakmarisi moswavle mxolod sakontrolo werebis 

safuZvelze Sefasdes. maswavlebeli unda afasebdes prezentaciebis, mos wav lisave 

TviTSefasebis, jgufuri muSaobis, Tu sxva tipis aqti vobebis mixedviT. maswavlebelma 

Sefasebisas unda gaiT valiswinos saganmanaTleblo procesSi moswavlis CarTulobis 

xarisxi (saxlSi micemuli davalebis Sesrulebis xarisxi, gak ve Tilze aqtiuroba, 

SemoqmedebiToba da sxva), amasTanave, mi zan Sewonilia moswavlesac gavacnoT winaswar 

Sefasebis kri te riumebi. am kriteriumebis SedgenaSi SeiZleba moswavleTa Car Tvac. 

sakontrolo werebis Sefasebis sqemebs am wignSi gaec no biT. dauSvebelia moswavleTa 

qcevis gaTvaliswineba akade miuri moswrebis Sefasebisas _ gakveTilze arasa- 

Tanadod moqceva, rogorc wesi, TavisTavad aisaxeba akademiur moswrebaze. 

moswavlis niSani unda gamomdinareobdes mis mier sagnis Seswavlis sxvadasxva 

komponentisgan _ sakontrolo weris Sesruleba, gakveTilze msjeloba, jgufuri 

muSaoba, prezen tacia da sxva. 

am komponentebs maswavlebeli wonebs aniWebs. wonebis miniWebis sistema SeiZleba 

maswavlebelma sxvadasxvanairad da gegmos. es wonebi SeiZleba sxvadasxva iyos sxvadasxva 

tri mes trisTvis, magram unda gvaxsovdes, rom trimestris niSa ni (umaRlesi 

qulaa 10) aucileblad sxvadasxva komponen tis Sefasebebisgan unda gamomdinareobdes. 

magaliTad, maT SeiZleba Semdegi wonebi mivakuTvnoT: prezentacia _ 25%, gakveTilze 

aqtivoba, msjeloba _ 30%, sakontrolo wera _ 25% da sa Si nao davalebis Sesruleba 

_ 20%. magaliTad, Tu saSualo Sef asebebi TiToeuli komponentis mixedviT aris 6, 

9, 10, 8, maSin Sefaseba ase gamoiTvleba: 

0,25·6+0,3·9+0,25·10+0,2·8=8,6 ... 

aq saboloo Sefaseba moswavlis sasargeblod unda moxdes da daiweros `9~. 

wliuri Sefaseba trimestruli Sefasebebis saSualo SeiZ le ba iyos; Tumca, aqac 

SesaZlebelia koreqtivebis Setana _ trimestrebis mixedviT progresis gaTvaliswineba. 

axla gTavazobT proeqt ìlia WavWavaZis~ erovnuli sas wav lo gegmebisa da saxelmZ- 

Rvaneloebis SemmuSavebeli jgufis mier momzadebul sqemas Sefasebis ZiriTadi 

mdgenelebisa (saSinao davaleba, Semajamebeli samuSao, klasSi CarTuloba, proeqti) 

da Sesabamisi unarebis Sesaxeb, zogierTi maTganis mixedviT niSnis gamoyvanis variants. 

25

26 

Sefasebis ZiriTadi mdgenelebi 

saSinao davaleba 

unarebi 

1. maTematikuri cnebebisa da debulebebis gamoyeneba. 

2. kavSirebisa da mimarTebebis dadgena. 

3. maTematikuri obieqtebis warmodgena da maTematikuri enis floba. 

4. damxmare teqnikuri saSualebebisa da sainformacio teqnologiebis gamoyeneba. 

unarebi fasdeba Semdegi aqtivobebiT: 

moswavle 

1. aRiqvams amocanis Sinaarss, gaiazrebs da gamijnavs amo ca nis monacemebsa da 

saZiebel sidideebs. axdens monacemebis (maT Soris problemis gadasaWrelad saWiro 

monacemebis) or ga nizebas da maT warmodgenas. 

2. gadmocemisas sworad da efeqtianad iyenebs maTematikur terminebsa da aRniSvnebs. 

adekvaturad irCevs simkacris dones da rodesac saWiroa, dasabuTebisas iyenebs 

mkacr maTematikur msjelobas (maT Soris induqciur da deduqciur msjelobas). 

3. poulobs, irCevs da iyenebs gzebsa da meTodebs (maT Soris teqnologiebs) figurebis 

da obieqtebis zomebis, agreTve maT Soris manZilebis, masis, temperaturis 

gasazomad. ir Cevs da moipovebs procesis an realuri viTarebis mode li rebisTvis 

saWiro monacemebs. 

4. kompleqsur (rTul) problemas yofs safexurebad, martiv amocanebad da Wris 

etapobrivad (amoxsna), maT Soris standartuli midgomebisa da procedurebis gamoyenebiT. 

5. keTilsindisierad asrulebs davalebebs (vadebisa da raodenobis TvalsazrisiT). 

Semajamebeli samuSao 

unarebi 

1. maTematikuri obieqtebis warmodgena da enis floba. 

2. maTematikuri cnebebisa da debulebebis gamoyeneba. 

3. kavSirebi da mimarTebebi. 

4. amocanis amoxsnis gza da misi realizeba. 

5. gamoTvlebi. 


moswavle 

1. axdens monacemebis (maT Soris problemis gadasaWrelad saWiro monacemebis) 

organizebas da maT warmodgenas. 

2. gadmocemisas sworad da efeqtianad iyenebs maTe ma ti kur terminebsa da aRniSvnebs. 

3. amocanebis gadaWrisas (maT Soris yofiTi), iyenebs ma Te matikur obieqtebs, 

procesebs da maT Tvisebebs. 

4. irCevs efeqtian strategias da mokled aRwers prob le mis gadaWris safexurebs. 

miyveba arCeul strategias. 

5. irCevs gamoTvlebis adekvatur/optimalur xerxs da axdens mis realizebas. 

6. amyarebs kavSirebs (mag. sxva maTematikur struq tu rebTan, obieqtebTan an sxva 

disciplinebTan) da iyenebs am kavSirebs rogorc problemis gadaWrisas aseve miRebuli 

Sedegebis gaanalizebisas.

klasSi CarTuloba 

unarebi 

1. cnebebisa da debulebebis gamoyeneba 

2. maTematikuri obieqtebis warmodgena 

3. maTematikuri enis floba 

4. msjeloba _ dasabuTeba 


moswavle 

1. moswavle irCevs dasabuTebis meTods (mag. sawinaaRm de gos daSvebis gamoyeneba 

damtkicebisas, evristikuli meTodis ga moyeneba dasabuTebisas). 

2. koreqtulad iyenebs terminologias, maTematikur aR niSv nebsa da simboloebs. 

3. informaciis gadacemisas warmoaCens sakiTxis arss (mag. obieqtis arsebiT 

Tvisebebs). 

4. koreqtulia maswavlebelTan da megobrebTan mimar Te baSi. igebs da aanalizebs 

sxvis naazrevs. 

5. TanamSromlobs TanaklaselebTan jgufuri samuSaoebis Sesrulebisas. 

6. auditoriisa da saprezentacio masalis mixedviT irCevs prezentaciis formas 

da damxmare saSualebebs (maT Soris sa in formacio teqnologiebs). 

7. axdens problemis formulirebas auditoriisTvis ga sa gebi formiT. asabuTebs 

problemis aqtualurobas da mniSv ne lobas (igulisxmeba problemis praqtikuli an/ 

da wminda mec nieruli aqtualuroba). 

8. msjelobisas, sxvadasxva safexurze, adekvaturad ir Cevs simkacris dones da 

rodesac saWiroa, dasabuTebisas iye nebs mkacr maTematikur msjelobas (maT Soris 

induqciur da deduqciur msjelobas). 

proeqti 

fasdeba Semdegi unarebi 

1. cnebebisa da debulebebis gamoyeneba 

2. kavSirebisa da mimarTebebis dadgena 

3. amocanis Camoyalibeba da axsna 

4. maTematikuri obieqtebis warmodgena 

5. maTematikuri enis floba 

6. damxmare teqnikuri saSualebebisa da sainformacio teqnologiebis gamoyeneba 

7. modelireba 


moswavle 

1. poulobs, irCevs da iyenebs gzebsa da meTodebs (maT So ris teqnologiebs) figurebis 

da obieqtebis zomebis, ag reT ve maT Soris manZilebis, masis, temperaturis 

da drois ga sazomad. irCevs da moipovebs procesis an realuri viTarebis modelirebisTvis 

saWiro monacemebs. 

2. axdens monacemebis (maT Soris problemis gadasaWrelad saWiro monacemebis) 

organizebas da maT warmodgenas. 

3. Cveul garemoSi (maT Soris yoveldRiur cxovrebaSi) amCnevs maTematikur 

obieqtebsa da procesebs. iyenebs maT Tvi sebebs praqtikul (maT Soris yofiTi) amocanebis 

gadaWrisas. 

4. gansazRvravs da ayalibebs problemas, maT Soris arastandartul viTarebaSi 

27

(mag. rodesac problemis gada saWrelad saWiro maTematikuri procedura calsaxad 

araa gansazRvruli). 

5. irCevs efeqtian strategias da mokled aRwers prob le mis gadaWris safexurebs. 

miyveba arCeul strategias. aana lizebs arCeul strategias da asabuTebs arCeuli 

strategiis efeqtianobas, mimoixilavs SesaZlo alternatiul stra te gi ebs da msjelobs 

maT upiratesobebsa da naklze. 

6. amyarebs kavSirebs (sxva maTematikur struqturebTan, obieqtebTan an sxva disciplinebTan) 

da iyenebs am kavSirebs rogorc problemis gadaWrisas aseve miRebuli 

Sedegebis gaanalizebisas. 

7. axdens mocemuli modelis elementebis interpretirebas im realobis konteqstSi, 

romelsac modeli aRwers da piriqiT _ realuri viTarebis dakvirvebis 

Sedegad miRebuli mo na ce mebis interpretirebas Sesabamisi modelis enaze. gan saz Rvravs 

modelis vargisianobas da afasebs misi gamo yenebis saz Rvrebs. 

8. auditoriisa da saprezentacio masalis mixedviT irCevs prezentaciis formas 

da damxmare saSualebebs (maT Soris sain formacio teqnologiebs). efeqtianad iyenebs 

prezen ta ci isT vis gankuTvnil dros. 

9. axdens problemis formulirebas auditoriisTvis ga sa gebi formiT. asabuTebs 

problemis aqtualurobas da mniSv ne lobas (igulisxmeba problemis praqtikuli an/ 

da wminda mec ni eruli aqtualuroba). 

10. zogadi Sedegebis sademonstraciod iyenebs magaliTebs, rogorc realuri 

viTarebidan, aseve maTematikidan. 

11. axdens miRebuli Sedegebis ganzogadebas, amyarebs kavSirebs (mag. sxva maTematikur 

struqturebTan, obieqtebTan an sxva disciplinebTan) da iyenebs am kavSirebs 

rogorc prob le mis gadaWrisas, aseve miRebuli Sedegebis gaanalizebisas. 

12. afasebs maTematikis adgilsa da mniSvnelobas sxva da sxva disciplinebSi, biznesSi, 

xelovnebaSi da adamianis moR va weobis sxvadasxva sferoebSi. 

13. informaciuli teqnologiebis gamoyenebisas acno bi erebs eTikur/socialuri 

xasiaTis problemebs da icavs eTikur normebs. 

28 

semestruli niSnis Semadgeneli komponentebis 

ganawilebis magaliTi 

saSinao 

davaleba, 

15% 

klasSi 

CarTuloba, 

15% 

Sefasebis ZiriTadi mdgenelebi 

Semajamebeli 

samuSao, 40% 

proeqti, 

30% 

sakontrolo 

samuSao 

proeqti 

klasSi CarTuloba 

saSinao davaleba

maTematikuri obieqtebisa 

da enis gamoyeneba 

amocanis amoxsnis 

gza 

kavSirebi da mimarTebebi 

gamoTvlebi 

Semajamebeli samuSao (VII-X klasebi) 

(wera) 

1-3 4-5 6-7 8-10 

araadekvaturad 

iye nebs cnebebs, 

ter minebs, aRniSvnebs 

/ arasruly 

o filad war moadgens 

monacemebs. 

ver poulobs amoxs 

nis romelime 

gzas. 

amoicnobs or, odnav 

gansxvavebuli 

formiT warmod genil, 

identur ma- 

Te ma tikur obieqts 

(mag. gansxvavebuli 

asoiTi SeniSvnebiT 

Caweril erTsa da 

imave algebrul 

ga mo saxulebas). 

g a m o T v l e b i a r 

met yvelebs amocanis 

gagebaze, 

Sedegi araadekvaturia 

iyenebs amocanis 

amosaxsnelad saWiro 

zogierT cnebas, 

terminsa da aR niSvnas 

/ nawilobriv, 

Tumca araorganizebuladwarmoadgens 

monacemebs. 

poulobs amoxsnis 

romelime gzas, 

iwyebs magram ver 

asrulebs amoxsnas. 

amoicnobs or, gansxvavebuli 

for miT 

war modgenil, identur 

maTe ma tikur 

obieqts (mag. sxvada 

sxva formiT warmod 

genil erTsa da 

imave asaxvas). 

Seicavs gamoT vle 

bis ZiriTad nawils, 

realizeba 

na wilobriv araswo 

ria. 


amosaxsnelad sa- 

Wi ro cnebebis, 

debu le bebisa da 

aRniS vnebis umetesobas 

/ axdens 

monacemTa organizebas 

da warmodgenas. 

poulobs amocanis 

amoxsnis gzas, miyveba 

mas, Tumca 

xarvezebiT. 

aRwers kavSirs 

sxva dasxva standar 

tul maTemati 

kur obieqtsa 

Tu proceduras 

Soris (mag. gantolebisamoxsna 

da mo ce mu li 

asaxvisaTvis moce 

muli mniSv nelo 

bis winasazis 

povna). 

gamoTvlebi Ziri 

Tadad sworia, 

T u m c a S e i c a v s 

teq nikuri xasiaTis 

xarvezebs. 


amosaxsnelad sa- 

Wi ro yvela cnebas, 

debulebas 

da aR niS vnas / 

irCevs mo na cemTa 

orga ni ze bisa da 

war mod ge nis optimalur 

for mas 

poulobs amocanis 

amoxsnis 

(opti ma lur) gzas, 

bo lo mde axdens 

mis rea li zebas 

aRwers da iyenebs 

kavSirs sxvadasxvamaTematikur 

obieq teb sa 

Tu pro ce durebs 

Soris (mag. iyene 

b s a l g e b r u l 

gar daqmnebs da 

to lobis / utolo 

bis Tvisebebs 

geo met riul 

d e b u l e b a T a 

dasabu Te bi sas) 

gamoTvlebi sworia, 

arCeuli da 

g a m o y e n e b u l i 

aqvs gamoTvlebis 

opti ma luri 

xerxi. 

29

gazomvebi, modelireba, gamoyeneba 

msjeloba-dasabuTeba 

kavSirebi da mimarTebebi 

komunikacia 

30 

proeqti (VII,X klasebi) 

1-3 4-5 6-7 8-10 

Caketili tipisumartivesi 

teqsturi 

a m o c a n i s 

amoxsna stand 

a r t u l i 

pro cedurebis 

gamoyenebiT. 

u m a r t i v e s i 

stan dartuli 

gamoT vlebis 

Catareba. 

ori, odnav 

gan s xva vebuli 

for miT warm 

o d g e n i l i , 

i d e n t u r i 

maTe ma tikuri 

o b i e q t i s 

amoc no ba. 

martivi maTema 

tikuri obieq 

tis saxelwo 

de bisa da 

Z i r i T a d i 

T v i s e b e b i s 

g a m e o r e b a 

(maT Soris saku 

Tari sityvebi). 

nacnobi mode lis 

gamoyeneba nacnobi 

viTa rebis Sesas wavlad. 

Caketili tipis 

amocanis amoxsna 

stan dar tuli 

pro ce durebis ga moyenebiT. 

nac nob damxma 

re saSuale be bisa 

da teqno lo giebis 

gamo ye neba. 

martivi ma Te ma tik 

u r i o b i e q t e b i s 

Tvise bebis gamoye 

nebiT, stan dartuli 

ga moTv lebisa 

da msjelobis Ca tareba. 

o r s t a n d a r t u l 

maTe ma ti kur obieqtsa 

Tu proce duras 

Soris kav Si ris 

amoc no ba. 

martivi maTe matikuri 

obieq tis 

saxelwo debisa da 

Ziri T adi Tvi se bebis 

gadmo ce ma. 

im viTarebisa amoc 

noba, rom lis Sesaswavlad 

Se saZlebelia 

nac no bi 

modelis ga moyeneba. 

amo ca nis 

amoxsnisas misT vis 

nacnob maTema tikur 

pro ce du rebs 

Soris sa Wi ro pro - 

ceduris Ser Ceva 

da gamo ye neba. 

nacnobi damx ma re 

saSua lebe bi sa da 

teqno lo gi ebis ga - 

mo yeneba. 

maTematikuri obi 

e q tis Tvise be bis 

gamoyenebiT, standartuligamoTvlebisa 

da msjelobis 

Cata reba 

sxvadasxva standar 

tul maTe mati 

kur obieq teb sa 

Tu pro ce du rebs 

Soris kav Si ris 

amocnoba. 

maTematikuri obieq 

tis saxel wo debisa 

da Zi ri Tadi 

Tvi se bebis gadmoce 

ma. komu ni kacii 

isas sim kacris 

ade k va tu ri donis 

Ser Ceva. infor maci 

is ga da cemisas 

sa ki Txis arsis war - 

mo Cena. 

problemis gadasaWrelad 

rea luri viTarebis modeli 

reba, maSinac ki, rodesac 

es viTareba sagrZnoblad 

gansx vav deba misTvis 

nacnobi Sem Txve vebisagan. 

problemis da mo uki deblad 

dasma da misi for mulireba 

maTematikur ena ze. sxvadasxva 

damxmare teqnikuri 

saSualebebisa da teqnologiebis 

arsebobis, maTi maxasiaTeblebis, 

gamo ye nebis 

arealisa da Sez Rud ve bis 

codna. problemis gadaWrisas 

adekvaturi damxmare 

teqnikuri saSualebebisa 

da teqnologiebis SerCeva 

da ma Ti gamoyeneba. 

varaudis gamoTqma da misi 

kvle va kerZo magaliTebze. 

gamoTvlebisas Tu damtki - 

cebisas arCeuli strategiis 

var gisianobisa da misi 

gamo yenebis sazRvrebis 

ganxilva. msje lobis xazis 

ganvi Ta re ba, alternatiuli 

gzis mo Zeb na, miRebuli 

gadawyve ti le bis sisworisa 

da efeq tia no bis dasabuTeba. 

miRebuli Sedegebis ganzoga 

deba. kavSirebis damyareba 

sxva maTematikur struq tureb 

Tan, an sxva dis cip lineb 

Tan. am kavSirebis gamoyeneba 

prob lemis gadaWrisas 

da mi Re buli Sedegebis gaanalizebisas. 

sxvadasxva gziT warmod geni 

li informaciis interpre 

ta cia, masze msjeloba, 

erT maneT Tan dakavSireba. 

infor m aciis miRebisa da 

gadacemis Sesaferisi sa- 

Sualebebis Ser Ce va auditoriisa 

da sa kiTxis gaTvaliswinebiT. 

in for maciis 

gadacemisas sa kiT xis arsis 

warmoCena.

sanimuSo gakveTilebi 

I 

a q i t i v o b a: 

naturaluri ricxvebis gayofadobis Tvisebebis gamokvleva da gamoyeneba praqtikuli 

amocanebis amoxsnisas. 

reziume: 

moswavleebi praqtikuli amocanebis amoxsnisas, eqspe ri mentis Catarebisas aRmoaCenen 

naturaluri ricxvebis gayofis Tvisebas: erTi naturaluri ricxvis meore 

naturalur ricxv ze gayofisas miiReba ganayofi da naSTi, romelic zogjer nu lis 

tolia _ aseT SemTxvevaSi erTi naturaluri ricxvi iyo fa meoreze; ayalibeben sakuTar 

pozicias am operaciis ga mo saxvisa da dasabuTebis Sesaxeb. 

aqtivobis mizani: 

• moswavleebi gaiwafon praqtikuli amocanebis amoxsnisas varaudebis gamoTqmasa 

da maT SemowmebaSi. 

• CamouyalibdeT kritikuli azrovneba, msjelobis, dasa bu Tebis unari. 

• eCvevian ricxvebis geometriuli warmodgenebis gamo ye nebas. 

savaraudo dro: 

1 gakveTili 

masala: 

• saskolo saxelmZRvanelo 

• saweri saSualebebi 

• Cxirebi (an kenWebi) 

aucilebeli wina codna: 

• gayofadobis niSnebi 

• gayofadoba. gayofadobis Tvisebebi 

aqtivobis aRwera 

sTxoveT moswavleebs moewyon 5-5-ad, 6-6-ad. dasviT SekiTxva: 

_ ra operaciis Sesrulebaa saWiro imisTvis, rom davad gi noT, ramdeni xuTeuli 

SeiZleba Seiqmnas klasis mos wav le ebisgan, ramdeni eqvseuli SeiZleba Seiqmnas klasis 

mos wav leebisgan. 

Tu SesaZlebelia ganixileT is SemTxvevac, roca klasis mos wavleebis ricxvi iyofa 

dasaxelebul ricxvze da is Sem Txve vac, roca ar iyofa. am ukanasknel SemTxvevaSi 

SeiZleba dav svaT SekiTxva: 

• ramdeni 5-kaciani gundi SeiZleba Sedges am moswav le ebis gan? 

• ramdeni 6-kilogramiani amanaTi SeiZleba gaamzadon mzru n velobas moklebul 

bavSvTa saxlisTvis, Tu TiToeuli mos wavle TiTo kilogram xils miitans? 

• sul mcire ramdeni yuTi iqneba saWiro bavSvTa sax lis Tvis wignebis gadasacemad, 

Tu TiToeuli moswavle TiTo wigns mo itans da TiToeul yuTSi 4 wigni eteva? 

sTxoveT moswavleebs gayofis procesi ricxviTi tolobiT Caweron. 

CaatareT fragmentuli muSaoba jgufebad; erT jgufs gar kveuli raodenobis 

kenWebi daurigeT, meore jgufs _ sxva ra odenobis kenWebi, mesame jgufs daavaleT 

saskolo saxel mZRva nelos teqstis mixedviT gaarCios raime magaliTze (roca ga sayofi 

metia gamyofze) ricxviTi wrfis gamoyenebis SesaZ leb loba. sTxoveT am ukanaskneli 

31

jgufis wevrebs ricxviTi wrfis saSualebiT aRweron gayofis procesi nebismieri a 

naturaluri ricxvisa da b naturaluri ricxvis SemTxvevaSi (a>b). sTxoveT jgufebs 

Seadginon ricxviTi tolobebi da Sesa ba misi tolobebi asoebis gamoyenebiT. SeiZleba 

Tqveni Carevac gaxdes saWiro moswavleTa Sedegebis reviziis mizniT an am Sedegebis 

prezentaciisas. 

ganixileT is magaliTebic, romlebic saxelmZRvaneloSia da praqtikuli Sinaarsis 

amocanebs ukavSirdeba, (magaliTi 1 da magaliTi 2). 

aqtivobis gaRrmaveba-gafarToeba 

daavaleT jgufebs ganixilon is SemTxvevebi, roca gasa yo fi naklebia gamyofze, 

gasayofi udris gamyofs. SeiZleba daviw yoT ricxviTi magaliTebiT, Sesabamisi ricxviTi 

tolo be bis CaweriT. Semdeg _ gayofis procesis aRmweri formula Caiweros 

asoebis gamoyenebiT. 

SeecadeT kiTxva-pasuxis saSualebiT SeajamoT yvela Sem Txve va. SeamowmoT aTvisebis 

xarisxi kiTxvebis dasmis saSua lebiT; 

_ ra naSTi SeiZleba miviRoT 5-ze gayofisas? 

_ vTqvaT, ricxvi iyofa 5-ze, risi toolia am SemTxvevaSi naSTi? _ rogor CavwerT 

formuliT im ricxvebs, romlebic 5-ze iyofa? 

_ ra formuliT SeiZleba Caiweros is ricxvebi, romlebic 5-ze gayofisas naSTSi 

1-s iZleva? 

_ daakvirdiT tolobis 43=4·8+11. rogor CawerdiT mas, rom gamoikveTos 43-is 

8-ze gayofisas miRebuli ganayofi da naSTi. 

Semdeg gadavdivarT `testebSi~ pasuxebis SerCevis pro ces ze (moswavlis saxelmZ- 

Rvanelo _ savarjiSoebi #1-#8) 

mizanSewonilad migvaCnia klasSi amovxsnaT 23-e, 24-e, 26-e da 29-e amocanebi, romlebic 

codnis ganmtkicebis, da sa buTebis, msjelobis unaris ganviTarebisTvisaa mocemuli. 

yve la es amocana iseve, rogorc am paragrafis sxva amocanebi dawv rilebiTaa 

amoxsnili $1.2-is warmodgenisas. 

a q i t i v o b a: 

simravlis klasifikacia mTel ricxvTa simravlis klasebad dayofis magaliTebis 

mixedviT 

forma: 

jgufuri muSaoba 

reziume: 

ganixileba mTel ricxvTa Z simravlis sxvadasxva kla si fikaciis magaliTebi: 

• mTeli dadebiTi ricxvebi, mTeli uaryofiTi ricxvebi, nuli. 

• mTel ricxvTa simravlis dayofa raime konkretul 1-is gan gansxvavebul naturalur 

ricxvze gayofisas miRebuli naSTebis mixedviT. 

• ganzogadeba _ nebismieri naturaluri ricxvis (mo du lis) ganxilva. 

aqtivobis mizani. 

propedevqtikuli muSaoba swori kla si fikaciis (raime erTi niSnis mixedviT da 

a. S.) Sedgenis Cve vebis misaRebad. msjelobis, dasabuTebis, kritikuli az rov nebis 

unaris gan vi Tareba; ganzogadebis procesis Catarebis una ris ganviTareba. 

32 

II


1 gakveTili 

samuSao masala: 

moswavlis saxelmZRvanelo, 

$ 1.9. jgufuri muSaobis proeqti 

aucilebeli wina codna 

• mTeli ricxvebi 

• moqmedebebi mTel ricxvebze 

• mTeli ricxvis naturalur ricxvze gayofisas naSTis miTiTeba 

• simravle. moqmedebebi simravleebze 

aqtivobis aRwera 

gakveTils viwyebT saWiro wina masalis gaxsenebiT, codnis ganmtkicebisa da gafar- 

Toebis procesiT, romelic mTeli ricxvis naturalur ricxvze gayofasa da naSTis 

miTiTebas Seexeba. sasargebloa aseTi saxis savarjiSoebis winaswari ganxilva: 

_ miuTiTeT ganayofi da naSTi 37-is 5-ze gayofisas: 

37=5·7+2 

_ miuTiTeT ganayofi da naSTi -37-is 5-ze gayofisas: 

–37=5·(-8)+3 

_ miuTiTeT ganayofi da naSTi -40-is 6-ze gayofisas: 

–40=6·(–7)+2 

_ vTqvaT, a mTeli ricxvis b naturalur ricxvze gayo fi sas miiReba ganayofi q 

da naSTi r, CawereT es faqti for mulis saxiT. SeiZleba Tu ara r nulis toli iyos? 

SeiZleba Tu ara r ricxvi b-s toli an b-ze meti iyos? 

winaswari `moTelvis~ Semdeg klass vyofT jgufebad da vavalebT saxelmZRvaneloSi 

mocemuli proeqtis Sesrulebas. erTi jgufi asrulebs davalebas, roca 

xdeba 5-ze gayofa, meore _ 4-ze gayofis SemTxvevas ixilavs, mesame _ 6-ze gayofis 

SemTxvevas. 

TiToeuli jgufi warmoadgens pasuxebis variantebisgan Sedgenil moxsenebas. am 

moxsenebebis prezentacias axdenen jgufebis warmomadgenlebi. 

Semdeg erToblivad mimdinareobs msjeloba miRebuli Sedegebis Sesaxeb; 

_ ramdeni qvesimravlis gaerTianebad warmovadgineT TiToeul SemTxvevaSi mTel 

ricxvTa simravle? 

_ amowuravs Tu ara am qvesimravleebis gaerTianeba mTlianad mTel ricxvTa 

simravles? 

_ aqvs Tu ara saerTo elementebi nebismier or qve sim ravles? 

_ SeiZleba Tu ara raime sxva niSnis mixedviT analogiuri we siT warmovadginoT 

mTel ricxvTa simravle? 

_ SeiZleba Tu ara moviyvanoT raime sxva magaliTi, roca sim ravle warmodgenilia 

aRniSnuli wesiT. 

aq SeiZleba moswavleebs daxmareba dasWirdeT (SeiZleba miuTiToT raime cxovrebiseuli 

magaliTi, magaliTad, cxo ve lebis dayofa raime niSniT da a. S.). 

SeiZleba vimsjeloT ganzogadebaze, ramdeni qvesimravle miiReba nebismieri n 

naturaluri ricxvis SemTxvevaSi? 

33

aqtivoba: 

kvadratebis sxvaobis formula da marTkuTxa miwis nakveTebis farTobebis Sedareba. 

reziume: 

marTkuTxedis farTobisa da kvadratebis sxvaobis formulis erTmaneTTan dakav- 

Sireba. sidideebs Soris damokidebulebebis kvleva marTkuTxedis farTobis magali- 

Tze. iyeneben dakvirvebis Sedegebs algebruli Tanafardobis misaRebad, realur 

viTarebasTan dakavSirebuli amocanis amoxsnisas. 


1 gakveTili. 

aqtivobis mizani. 

algebruli gamosaxulebebis igivuri tolobis miReba wminda algebruli gziT, 

romelic dakavSirebulia ope ra tiuli unarebis ganviTarebasTan da am tolobis geometriuli 

Sinaarsis warmodgena. sxvadasxva (algebrul da geometriul) obieqtebsa da 

operaciebs Soris kavSirebis gacnobiereba. geometriuli optimizaciis amocanasTan 

algeb ru li igiveobis dakavSireba. 

aqtivobis aRwera. 

gakveTils viwyebT im kvadratisa da marTkuTxedis farTobebis 

Seda re biT, romelTac toli perimetrebi aqvs. 

_ am figurebs toli perimetrebi aqvs. SeecadeT SeadaroT 

maTi far To bebi. 

_ ramdeni kvadratuli erTeuliTaa meti kvadratis far To bi 

marTkuTxedis farTobze? 

_ SeadareT marTkuTxedis gverdebi kvadratis gverdebs. 

am faqtis ganzogadebis mizniT SevTavazoT moswavlveebs (Se saZlebelia maT wyvilebSi 

imuSaon), Seavson cxrili imave kanonzomierebiT (marTkuTxedis sigrZe kvadratis 

gverdze 2-iT metia, sigane _ 2-iT naklebi). 

gverdis 

sigrZe 

4 

5 

6 

7 

8 

9 

34 

kvadrati marTkuTxedi farTobebis 

sxvaoba 

farTobi sigrZe sigane farTobi 

III 

16 6 2 12 4 

_ ra kanonzomierebas amCnevT Sevsebuli cxrilis mi xed viT? 

diskusiis Semdeg moswavleebi midian daskvnamde, rom Ziri Ta di kanonzomiereba 

aris is, rom farTobebis sxvaoba mudmivia da 4-is tolia. 

SeiZleba sxva magaliTis ganxilvac, roca kvadratis gver di sxva ricxviT icvleba

(magaliTad, ganvixilavT marT kuTxe debs, romelTa sigrZe 3-iT metia kvadratis gverdis 

sigrZeze, sigane _ 3-iT naklebia). 

amis Semdeg klass vTavazobT miRebuli kanonzomiereba for mulis saxiT Caweros 

_ magaliTad, 

(a+2)(a–2)=a 2 –4, (a+3)(a–3)=a 2 –9. 

am tolobebis Cawera kvlav kiTxva-pasuxis procesiT SeiZ le ba ganxorcieldes. 

_ vTqvaT, kvadratis gverdis sigrZe a-s tolia, risi to lia, misi farTobi? 

_ Tu marTkuTxedis sigrZe 2-iT metia kvadratis gverdze, si gane _ 2-iT naklebi, 

risi tolia misi farTobi? 

_ risi tolia miRebuli marTkuTxedis farTobi? 

moswavleebi advilad SeZleben am igiveobebisa da misi ganzogadoebis miRebas 

algebruli xerxiTac (frCxilebis gaxsnis wesis gamo ye nebiT). 

axla SeiZleba gamoviyenoT formulis geometriuli dasa bu Tebis xerxi. 

SevTavazoT moswavleebs daxazon kvadrati, romlis gverdi a-s tolia. 

Semdeg SeiZleba davsvaT kiTxvebi: 

_ ra figuris farTobi SeiZleba iyos warmodgenili (a+b)(a-b) 

formuliT. 

_ risi tolia daStrixuli figuris farTobi? 

_ (a 2 -b 2 )-is. 

_ tolia Tu ara daStrixuli fi gu ris farTobi im marTkuTxedis 

farTobis, romlis gverdebia a+b da a-b? 

_ ra tolobis dawera SeiZleba? 

am kiTxvebisa da pasuxebis sistema warmoadgens (a+b)(a-b)=a 2 -b 2 tolobis miRebis 

process. Semdeg SeiZleba igive to lo ba moswavleTaTvis cnobili algebruli xerxebiTac 

miviRoT (frCxi lebis gaxsnis wesi). 

am procesis Catarebis Semdeg moswavleebs vekiTxebiT: 

_ risi tolia mocemuli kvadratis perimetri? 

_ risi tolia misi erTi gverdis b-Ti SemcirebiTa da meore gver dis b-Ti gazrdis 

Semdeg miRebuli marTkuTxedis peri met ri? 

_ kvadratsa da marTkuTxeds toli perimetrebi aqvs; rom lis farTobia metia 

da ramdeniT? 

am kiTxva-pasuxis procesSi moswavleebi aRmoaCenen mniS vne lovan faqts: mocemuli 

perimetris mqone marTkuTxedebs So ris udidesi farTobi aqvs kvadrats. aq 

SeiZleba yuradReba mi vaqcioT kvadratis `met~ simetriulobazec marTkuTxedTan 

Se darebiT. 

saxelmZRvaneloSi mocemuli ($2.2.) amocanebis amoxsna xels uwyobs codnis gan mtkicebas, 

formulis gamoyenebis unaris gamomuSavebas. 76-e amo cana kvlav geometriul 

interpretacias Seexeba da sasur ve lia misi amoxsna klasSi. 

aqtivoba: 

proporcia. proporciis gamoyenebebi. 

reziume: 

moswavleebi euflebian proporciis gamoyenebas sxvadasxva sayofacxovrebo an 

maTematikis momijnave disciplinebidan amocanebis amosaxsnelad. ecnobian proporciul 

sidideebs da maT geometriul gamosaxvas. 

IV 

35

savaraudi dro: 

1 gakveTili. 


• moswavleebma ganiviTaron Sefardebisa da proporciis gamoyenebis unar-Cvevebi 

amocanaTa amosaxsnelad. 

• moswavleebi daeuflon proporciuli sidideebis cnebis, proporciuli damokidebulebis 

cnebasa da mis geometriul ga mosaxvas. 

aucilebeli wina codna: 

• Sefardeba 

• proporcia. proporciis ZiriTadi Tviseba 

• sakoordinato sibrtye 

ativobis aRwera 

muSaoba SeiZleba daviwyoT im ori praqtikuli amocanis gan xilviT, romelic moswavlis 

saxelmZRvanelos $3.4-is me-3 punqtSi 1 da 2 magaliTiT aris warmodgenili. 

isini rukis mixedviT manZilis gansazRvras da sayofacxovrebo amocanis amox snas 

ukavSirdeba. Tu saskolo oTaxSi kedelze saqarT ve los rukaa Camokiduli, maSin 

SeiZleba aRniSnuli amocanis mi xedviT maswavlebelma am rukis mixedviT saqarTvelos 

raime or qalaqs Soris manZilis povnis amocana dasvas. 

Semdeg moswavleebs davyofT ramdenime jgufad da Sev Ta vazebT amocanebs proporciul 

sidideebze. 

• magaliTad, erT-erT jgufs vTxovT Seavsos cxrili 

36 

vaSlis masa 

m (kg) 

m kg vaSlis 

Rirebudeba p 

(lari) 

0,5 1 1,5 2 2,5 3 3,5 4 4,5 

0,8 

• ipovos m 

p sididis ricxviTi mniSvnelobebi. 

• ipovos sakoordinato sibrtyeze (m; p) wyvilebis Sesabamisi ricxvTa wyvilebis 

gamomsaxveli wertilebi. 

sxva jgufs SeiZleba mivceT Sesavsebad cxrili, romelic Tanabari moZraobisas 

gavlil manZilsa (S km) da dros (t sT) Soris damokidebulebas Seesabameba, SeiZleba 

aseTi amocanac Sev TavazoT: vixilavT marTkuTxedebs, romelTa erT-erTi gver dia a. 

SeavseT cxrili, romelic meore gverdis sididesa da saTanado marTkuTxedis far- 

Tobs Soris damokidebulebas See sabameba. 

erT-erT jgufs SeiZleba davusvaT iatakis Tanabrad Se Reb visas S farTobsa da 

daxarjuli saRebavis odenobis Sesa ba misi cxrilis Sevsebisa da sakoordinato sibrtyeze 

am damo kid ebulebis mixedviT wertilebis agebis amocana. 

jgufebis mier warmodgenili cxrilebisa da sako or di na to sibrtyeze Sesabamisi 

damokidebulebis Sesabamisad age bu li wertilebis mixedviT mimdinareobs msjeloba 

da vara udis gamoTqma aRniSnuli damokidebulebebis Tvisebebisa da geometriuli 

gamosaxvis Sesaxeb.

proporciis gamoyenebiT mimdinareobs Semzadeba Senad no beb ze, narevebze amocanebis 

amosaxsnelad. 

am SemTxvevaSi gamoiyeneT saxelmZRvaneloSi mocemuli me-3 magaliTi, roca cnobilia 

ori biznesmenis saerTo Tanxa (10 000 lari) da am Tanxebis Sefardeba: 

a 

= 2 

b 3 

proporciul nawilebad gayofis procesi SeiZleba kiTxva-pasuxis reJimSi gavagrZeloT: 

_ kidev ra proporciis dawera SeiZleba mocemuli pro por ciis mixedviT ( a b 

2 

= )? 

3 

_ ra SeiZleba vTqvaT a da b ricxvebis Sesaxeb (a da b aris 2 da 3-is proporciuli)? 

_ vTqvaT, toli Sefardebebi k-s tolia (proporciulobis koeficienti). ris 

tolia a? b? (a=2k, b=3k). 

_ 10 000 larSi ra wilia erTi biznesmenisa da ra wilia meore biznesmenis? 

_ sul ramden wils warmoadgens 10000? 

_ ra Tanxa modis 1 wilze? 

_ ra Tanxa Seesabameba 2 wils? 3 wils? 

aqtivobis Semdgomi gafarToeba-gaRrmaveba 2 an 2-ze meti Sesakrebis povnis zogadi 

wesis Camoyalibebaa, roca cnobilia maTi jami da ricxvebi, romelTa proporciulia 

mocemuli ricx vebi. arasavaldebulo nawilad SeiZleba CaiTvalos sa xel mZRvaneloSi 

mocemuli is nawili, romelic oqros kve Tas _ harmoniul proporcias exeba. 

sakontrolo kiTxvebi da `testebi~ (mesame punqtis #1-#9) moswavleTa TviT Semowmebis 

kargi saSualebaa. 

sasargebloa klasSi gavarCioT da amovxsnaT amocanebi Se nad no bebze, xsnarebze, 

masStabis gamoyenebaze (#12, #14, #26, #28). 

29-e amocana proporciuli sidideebis geometriuli ga mosaxvis gamoyenebis kargi 

magaliTia; 

33-e amocana, romelic struqturirebuli kiTxvebisgan Sed geba, statistikur 

monacemTa diagramis aRwerisas pro por ciulobis gamoyenebas Seicavs. 

aqtivoba: 

samkuTxedis kuTxeebis jamis formulis gamoyvana sxva da sxva gziT. 

reziume: 

moswavleebi aRmoaCenen samkuTxedis Tvisebas (samkuTxedis kuTxeebis jamis Sesaxeb) 

modelebze, asabuTeben am Tvisebas sxvadasxva xerxiT, msjeloben dasabuTebis 

xerxebis sarwmunoobis Sesaxeb. 


1 gakveTili. 


• miiRos eqsperimentis Catarebis Cvevebi 

gaiwafos amocanis konteqstis Sesabamisad geometriul obieqtTa gamosaxvaSi 

• Camouyalibdes kritikuli azrovneba, msjelobisa da dasabuTebis unari 

V 

37

ZiriTadi SekiTxvebi: 

1. SeiZleba Tu ara figuris Tvisebis aRmoCena misi mo de lis mixedviT? 

2. SesaZlebelia Tu ara sazomi xelsawyoebis gamoyenebiT fi guris raime Tvisebis 

Sesaxeb varaudis gamoTqma? 

3. debulebis samarTlianobis dadgenis romeli xerxia uf ro sarwmuno _ dasabuTebis 

gziT miRebuli, Tu sazomi xel saw yoebis gamoyenebiT? 

masala: 

furclebi, transportiri, saxelmZRvanelo. 

aqtivobis aRwera: 

muSaoba iwyeba gavlili masalis gameorebiT: paraleluri wrfeebis Tvisebebi, ori 

wrfis paralelurobis niSani, samkuTxedebis klasifikacia kuTxeebis mixedviT (am 

masalis didi nawili $ 4.5-is pirvel nawilSia gadmocemuli da wina ori gakveTilis 

Sinaarss Seesebameba). 

klasis moswavleebs vyofT jgufebad: erT-erTi jgufis mos wavleebs vTxovT gamoiyenon 

transportiri da sxva da sxva gva ri samkuTxedebis kuTxeebis zomis mixedviT 

gamoTqvan va ra udi kuTxeebis jamis Sesaxeb. 

38 

meore jgufi iye nebs samkuTxedis mo de lebs, 

gamoWris sam tol samkuTxeds da mi ad gams maT ise, 

ro gorc suraTzea gamo saxu li. 

mesame jgufi iyenebs saxelmZRvaneloSi war modgenil su raTs da iq 

dasmul kiTxvaze pa suxebis Ziebis procesSi cdi lobs gamo Tqvas varaudi 

samkuTxedebis kuTxeebis jamis Se sa xeb. 

meoTxe jgufi iyenebs saxel mZRvanelos mixedviT ABC 

sam kuTxedis models, aagebs mis si maRles _ BD-s; poulobs 

AD da CD monakveTebis Sua mar To bebs da gadakecvebis sa SualebiT 

(ise, ro gorc saxel mZRva neloSia miTi Te buli) cdi lobs 

gamoTqvas vara udi samkuTxedebis kuTxeebis ja mis Se saxeb. 

oTxive jgufi axdens Tavisi namuSevris prezentacias. 

imar Teba diskusia _ romeli xerxia ufro ioli, romeli uf ro sarwmuno? 

saxelmZRvaneloSi mocemuli sxva Teoriuli sakiTxebi (marT kuTxa samkuTxedis 

maxvili kuTxeebis jami, tolgverda sam kuTxedis TiToeuli kuTxis sidide, gare 

kuTxis Tviseba) Se iZ leba davalebad mivceT moswavleebs. 

samkuTxedis aRniSnuli Tvisebis aTvisebis TviTkont rolisTvis sasargebloa $ 4.5-is 

me-2 nawilis #1-#8 `testebSi~ pasuxebis SerCeva. dasabuTebisa da msjelobis unaris 

ganviTarebas emsaxureba #10, #12, #14, #15, #20 da #28 amocanebis amoxsnis proce sebi. 

sasurvelia isini maswavleblis xelmZRvanelobiT Sesruldes.

I Tavi. racionaluri ricxvebi 

I Tavi, ZiriTadad, ricxvis cnebis gafarToebis, racionaluri ricxvebisa da maTze 

moqmedebebis aRweras eZRvneba. poziciuri sistemis gamoyenebiT racionaluri ricxvebis 

Sedarebisa da dalagebis Cvevebis aRdgenisa da ganmtkicebis safuZvelze mimdinareobs 

uaryofiTi ricxvebis gamosaxvis, Sedarebisa da dalagebis Sesaxeb masalis 

gadacema. 

pirveli Tavis sawyisi paragrafebi naturaluri ricxvebis mwkrivis ganxilviT 

iwyeba, naturaluri ricxvebis Caweris xerxebisa da poziciuri sistemis gamoyenebis 

magaliTebis ganxilvas eZRvneba. 

maswavleblebs SevaxsenebT, rom naturalur ricxvTa dafuZnebis Teoriis gacnobisas, 

maT, ZiriTadad, or midgomaze miuTiTebdnen _ aqsiomuri midgoma, romelic 

peanos saxelTan aris dakavSirebuli da simravluri koncefcia (kantoris mier dafuZnebuli). 

pirveli midgomisas naturalur ricxvTa mwkrivi esaa {N, I, F} struqtura, romelic 

1. Sedgeba N simravlisgan (am simravlis elementebs naturaluri ricxvebi hqvia); 

2. gamoyofilia elementi, romelsac erTi (1) hqvia; 

3. gvaqvs mimarTeba F(x; y), y aris x-is momdevno. 

garda amisa mocemuli struqturis Tvisebebia (aqsiomebi); 

1) 1 ar aris raime naturaluri ricxvis momdevno; 

2) nebismieri naturaluri ricxvisTvis arsebobs erTi da mxolod erTi misi momdevno; 

3) yoveli naturaluri ricxvi araumetes erTi naturaluri ricxvis momdevnoa; 

4) (induqciis aqsioma) N-is yoveli qvesimravle, romelSic Sedis 1 da yovel x-Tan 

erTad Sedis misi momdevno naturaluri ricxvi, N-s emTxveva. 

aqsiomuri meTodis Tanmimdevruli gatarebisas asec amboben: elementebis buneba, 

romlebisganac N Sedgeba, araa arsebiTi. 

amis Semdeg SeiZleba SemoviRoT naturalur ricxvTa monakveTis cneba, Semdeg 

masze dafuZnebiT sasruli simravlisa da sasrul simravleSi elementTa odenobis 

(simZlavris) cnebebi. 

naturaluri ricxvis cnebis dafuZnebis meore midgoma kantoris saxelTan aris 

dakavSirebuli. aq elementebis ricxvi gaigeba, rogorc simravlis simZlavris kerZo 

SemTxveva. safuZvlad erTi simravlis meore simravleze urTierTcalsaxa Sesabamisobis 

cneba aiReba. am mimarTebiT xdeba simravleTa faqtorizacia, klasebad dayofa. 

am koncefciis Sesabamisad TiToeuli klasi xasiaTdeba misi simZlavriT. aracarieli 

sasruli simravlis simZlavre, gansazRvrebis Tanaxmad, naturaluri ricxvia. 

aseTi Semotanisas sasruli simravle ganusazRvrelia. kidev erTi daSveba am dros 

aris usasrulo simravlis arseboba (magaliTad, mimdevrobis usasrulo gagrZelebadobis 

daSveba). 

peanos Tvalsazrisis Sesabamisad ki saWiroa imis daSveba, rom aqsiomuri sistemis 

modeli arsebobs. es daSveba bevr maTematikoss (ix. mag., [21]) ufro bunebrivad miaCnia. 

cxadia, rom skolaSi ariTmetikis sawyisi Seswavla TvalsaCino unda iyos. igi ar 

unda misdevdes arc aqsiomur meTods da arc sxva logikuri agebis sqemas. es ar niSnavs 

imas, rom ricxvTa sistemis swavlebas ar unda hqondes Tavisi logikuri agebuleba. 

39

icxvis cnebis gafarToebas meTodikur literaturaSi [21] xSirad sqematurad ase 

warmoadgenen: 

(2) mTeli ricxvebi 

(3) racionaluri ricxvebi 

(4) algebruli ricxvebi 

algebris TvalsazrisiT ganzogadebis bunebrivi gza aseTia: (1) → (2) → (3) → (4) 

skolaSi ki Tanmimdevroba aseTia: 

(1) → (5) → (3) → (7) 

axali erovnuli saswavlo gegmac ricxvis cnebis gafarToebis am Tanamimdevrobas 

gvTavazobs: Tumca, uaryofiTi ricxvebis Seswavla me-7 klasidan aris gaTvaliswinebuli. 

uaryofiTi ricxvebis Seswavla erT-erTi rTuli sakiTxia. misi Camoyalibebis istoriac 

amas adasturebs (didi xnis ganmavlobaSi mecnierebis mier ar iyo miRebuli). 

es sakiTxebi kargad aris gadmocemuli kolmogorovis statiaSi (ix; mag; [21] da 

maswavleblebs vurCevT gaecnon mas. Cven mier SemoTavazebuli gadmocemis meTodika 

TvalsaCinoebis farTo gamoyenebas gulisxmobs. 

miuxedavad imisa, rom moswavleebi am droisTvis TiTqos ukve safuZvlianad unda 

flobdnen arauaryofiT racionalur ricxvebs, Cven mainc Sesabamisi sakiTxebis gadmocemas 

didi dro davuTmeT. erovnuli saswavlo gegmac iTvaliswinebs zogierTi adre 

naswavli sakiTxis gameorebasa da formaluri TvalsazrisiT ufro maRal doneze 

gadmocemas _ magaliTad, gayofadobis zogierTi niSnis swavlebisas deduqciuri 

msjelobis gamoyenebas. 

maswavleblebs kidev erTxel SegaxsenebT, rom Sefasebis zogad principebs wignis 

Sesaval nawilSi calke paragrafi daeTmo, es pedagogebisTvis saxelmZRvanelo 

principebia – yovel konretul SemTxvevaSi pedagogma am principebis safuZvelze 

unda miiRos gadawvetileba. moswavlis Sefaseba, TviT dabali Sefasebac ki, unda 

iyos dafiqsireba arsebuli realobis, romelic unda aRiqmebodes samomavlo 

pozitiuri cvlilebebisken mimarTul qmedebad. am Sefasebas ar unda dahyvebodes 

niSnis mogebis, angariSsworebis elferi. saerTod, yofaqceva da akademiuri donis 

Sefaseba unda gaimijnos erTmaneTisgan. warumateblobis yoveli gamovlineba ar unda 

iqces moswavlisTvis gamousworebel Cavardnad. akademiuri Sefaseba unda iyos xSiri 

40 

(1) arauaryofiTi mTeli ricxvebi 

(8) kompleqsuri ricxvebi 

(5) arauaryofiTi racionaluri 

ricxvebi 

(6) arauaryofiTi namdvili 

ricxvebi 

(7) namdvili ricxvebi

da mravalmxrivi, pedagogiur praqtikaSi unda gamoiricxos adre damkvidrebuli 

ualternativo procesi – gamoZaxeba, moswavlis xangrZlivi monologi, informaciis 

flobis donis warmoCena, maswavleblis ramdenime replika da Sefaseba. 

dRes SeiZleba SevafasoT erTi sagulisxmo CarTvis, sayuradRebo varaudis 

gamoTqmis, ganxilvisas konstruqciuli rolis, TanaklaselTa naSromebis prezentaciis 

aqtiuri oponireba, gakveTilis procesSi gamovlenili SemoqmedebiTi aqtiuroba. am 

midgomiT SeiZleba moswavleTa akademiuri aqtivobis amaRleba, motivaciis gaZliereba, 

SemoqmedebiTi unarebis gaRrmaveba. Sefasebis procesi unda iyos gamWvirvale. 

mis gareSe SeuZlebelia pedagogma maRali avtoriteti daimkvidros moswavleTa 

Soris. ufro metic, TviT moswavleebic SeiZleba CavabaT SefasebaTa kriteriumebis 

SemuSavebaSi. 

sakontrolo naSromebis ganxilvas unda eswrebodes Sefasebis kriteriumis sajaro 

wardgena klasSi _ es sayuradRebo meTodikuri xerxia, romlis saSualebiTac 

moswavleebi, sakmaod maRali sizustiT, Tavad ivaraudeben sakuTar Sefasebebs, 

zogierTi amocanis Sefasebisas gamoyenebuli nawilobrivi Sefasebebic mimarTuli 

unda iyos moswavlis naSromSi `dadebiTi nabijebis~ aRmoCenisa da dafiqsirebisken. 

moswavlem unda igrZnos pedagogis TanagrZnoba, keTili ganwyoba da imavdroulad, 

maRali principuloba da profesiis erTguleba moswavlis akademiuri moswrebis 

Sefasebisas. 

gavixsenoT, rom Sefasebis ZiriTadi miznebia: moswavlis swavlis xelSewyoba, 

moswavlis Zlieri da susti mxareebis gamovlena (saTanado analiziTa da 

daskvnebiT), swavlebis konkretuli meTodis efeqtianobis Semowmeba, saswavlo 

programebis efeqturobis dadgena da srulyofa da sxva. amrigad, Sefasebis 

formebi ganmaviTarebeli, ganmsazRvreli da diagnostikuria da am formebs vaniWebT 

aRmzrdelobiT funqciasac. gamovyoT Sefasebis kidev ori funqcia: swavlis 

procesis Sefasebis funqcia (mimdinare Sefaseba) da Sedegebis Sefaseba (Semajamebeli 

Sefaseba). 

SefasebaSi unda aisaxos prezentaciisas gamovlenili unarebi, moswavleTa mier 

erTmaneTis Sefasebebi. 

sagangebod unda SevCerdeT CvenSi farTod gavrcelebul erT-erT mankier Cvevaze, 

romlis Sedegebi xandaxan did gavlenas axdens Sefasebis obieqturobaze. migvaCnia, 

rom zogjer ùwyinar ancobad~ miCneuli sxvaTa naSromebis gadawera unda CaiTvalos 

moswavlisTvis miuRebel, uRirs saqcielad, moswavlem unda icodes, rom is iTvisebs 

sxvaTa inteleqtualur sakuTrebas, raTa daumsaxureblad miiRos maRali niSani, 

miiRos `jildo~ sicrueze agebuli qmedebisTvis. aseT SemTxvevaSi maswavleblis mier 

gamoCenili principuloba klasSi yovelTvis moipovebs gagebas, garkveuli drois 

Semdeg ki – aRiarebas. 

sayuradRebo rekomendaciebi gvaqvs jgufuri muSaobis warmarTvisa da Sefasebis 

mimarTac (ixileT wignis Sesaval nawilSi `gakveTilis dagegmvis zogadi principebi~)O 

jgufuri muSaobis warmarTvas, axlavs garkveuli sirTuleebi, magram am formis didi 

aRmzrdelobiTi, ganmaviTarebeli, emociuri mxare migvaniSnebs, rom is aucileblad 

da jerovnad unda tardebodes. 

wignis Teoriul nawilSia agreTve ganxiluli moswavleTa mier proeqtis 

Sesrulebis da am naSromis SefasebasTan dakavSirebuli sakiTxebi, dro da dro 

daubrundiT xolme SefasebasTan dakavSirebuli Cvens am rekomendaciebs, gaamdidreT 

es masala Tqveni gamocdilebis kvalobaze aRmoCenili detalebiT. Cven ki gzadagza 

kvlav SegaxsenebT xolme SefasebasTan dakavSirebul zogierT mosazrebas, konkretuli 

amocanebis Sefasebis kriteriumis dadgenasTan dakavSirebul sakiTxebs. 

unda gvaxsovdes, rom moswavleTa SefasebaSi daSvebulma Secdomebma SeiZleba 

arsebiTad daakninos sxva mimarTulebiT mopovebuli warmatebebi. 

41

42 

§ 1.1. naturaluri ricxvebi. naturaluri ricxvis 

Caweris sxvadasxva forma 

mizani: 

• naturaluri ricxvebis Caweris sxvadasxva formis gaxsenebiT, moswavleebma ganiviTaron 

Caweris formebis, poziciuri sistemis gamoyenebis unar-Cvevebi. 

• xarisxis cnebis gamoyenebiT moswavleebi gaiwafon ricxvebis, kerZod ki `giganturi~ 

ricxvebis Caweris xerxebis gamoyenebebSi. 

• naturalur ricxvTa mwkrivis Tvisebebis gamoyenebiT moswavleebs eqmnebaT 

pirveli warmodgenebi usasrulobis (usasrulod gagrZelebadobis) Sesaxeb. 

erovnuli saswavlo gegmis im moTxovnebis Sesasruleblad, romlebic VII.I da VII.II 

indikatorebiTaa warmodgenili, saWiroa CavataroT Semamzadebeli muSaoba, rac wina 

wlebSi Seswavlili sakiTxebis Sesaxeb codnis gafarToebas gulisxmobs. 

pirvel paragrafSi warmodgenili masala moswavleebisTvis SeiZleba ar iyos axali. 

aseTi dasawyisi bunebrivia, radgan saswavlo wlis pirveli gakveTilebi mizanSewonilia 

da sasurvelia ucnobi maTematikuri faqtebis gadmocemiT ar daviwyoT. 

calkeuli gakveTilebis dagegmvisa da warmarTvisTvis konkretuli miTiTebebis 

garda, mizanSewonilad vTvliT zogadi saxis zogierTi rekomendaciis Sesaxeb saubarsac, 

romlebic momavalSi SeiZleba aRar gavimeoroT. 

saswavlo masalis aTvisebisa da ganmtkicebisTvis saWiro muSaobis ZiriTadi nawili 

gakveTilze unda modiodes; gakveTili saswavlo procesis organizaciis ZiriTadi 

formaa; gakveTilze xdeba saganmanaTleblo miznebis ganxorcieleba. xSirad sauke- 

Teso Sedegamde mivyavarT aseT zogad rekomendaciebs swavlebis meTodikidan: 

1) maswavlebeli unda cdilobdes, rom yoveli axali SemecnebiTi amocana TviT moswavleebma 

Camoayalibon, 

2) maswavleblis da moswavleebis erToblivi ZalisxmeviT, dakvirvebis, konkretuli 

SemTxvevebis analizis Sedegad iqmneba garkveuli warmodgena arsebul maTematikur 

kanonzomierebebze, 

3) maswavleblis xelmZRvanelobiT moswavleebi eZeben amocanis amoxsnis gegmas da 

xSirad TviT axdenen gegmis realizacias. 

maswavleblebs kargad moexsenebaT, rom saswavlo procesisTvis mzadebis zogadi 

sqema gulisxmobs:^ 

1) saswavlo wlis dawyebamde Catarebul muSaobas, 2) gakveTilebis sistemis gaazrebas, 

3) konkretuli gakveTilisTvis momzadebas. 

axali saxelmZRvaneloebis SemoRebis procesSi saswavlo wlis dawyebamde muSaobaSi 

igulisxmeba erovnuli saswavlo gegmis moTxovnebis kargad gaazreba. ZiriTadi 

masala, romelsac unda daeyrdnos maswavlebeli saswavlo procesSi, erovnuli saswavlo 

gegmaa. saxelmZRvanelo mxolod damxmare saSualebaa erovnuli saswavlo gegmis 

moTxovnebis Sesasruleblad. saxelmZRvanelo erovnuli saswavlo gegmis mixedviT 

iqmneba, Tumca misi Sinaarsi, sakiTxebis Tanamimdevroba xSirad avtoris SexedulebebiTaa 

Sedgenili da rekomendaciebis doneze unda aRiqvas maswavleblebma. erovnuli 

saswavlo gegmis Sesabamisad, maswavlebels SeuZlia damoukideblad SearCios sakiTxebi 

da maTi gadacemis Tanamimdevroba. moswavlis saxelmZRvanelo, maswavleblis 

sarekomendacio wigni, romlebmac gaiares eqspertiza, pilotireba da meore gamocemisas 

eqspertTa da maswavlebelTa mosazrebebisa da rekomendaciebis Sesabamisad gadamuSavda, 

exmareba maswavleblebs moswavleze da Sedegze orientirebuli saswavlo 

procesis dagegmvasa da warmarTvaSi.

saswavlo wlis pirveli gakveTilebis specifikidan gamomdinare, pirvel paragrafs 

SeiZleba 2 saaTic ki davuTmoT. aq mocemuli masala moswavleTaTvis SeiZleba nacnobic 

iyos, amitom maswavlebels meti saSualeba aqvs gakveTilebi moswavleTa aqtiuri 

monawileobiT, interaqtiuri meTodiT warmarTos. aqcenti gakeTdes swavlebis iseT 

formebze, roca yvela moswavle aqtiurad erTveba swavlis procesSi. gakveTilebi 

ise SeiZleba daigegmos, rom yovelma moswavlem ipovos Tavisi adgili, ar darCes pasiuri 

an gariyuli. maswavlebeli da moswavleebi aqtiurad urTierToben erTmaneT- 

Tan. moxerxebulad SerCeuli kiTxvebis dasmis saSualebiT moswavleebi Cabmuli arian 

saswavlo procesSi. Tumca, ar igulisxmeba mxolod kiTxva-pasuxis reJimis damkvidreba 

_ moswavle TviTon monawileobs saganmanaTleblo procesis mimdinareobaSi; 

monawileobs diskusiaSi _ icavs sakuTar mosazrebebs da oponirebas uwevs sxvebs. 

erT-erTi sadiskusio Tema pirvel paragrafSi SeiZleba iyos naturalur ricxvTa 

mwkrivis usasrulod gagrZelebadobis SesaZlebloba, potenciuri usasrulobis Sesaxeb 

diskusia; SeiZleba gamoviyenoT hilbertis magaliTi rubrikidan `sxvadasxva~. 

SeiZleba gaimarTos diskusia poziciuri da arapoziciuri sistemebisa da maTi Taviseburebebis 

Sesaxeb. 

maswavlebeli warmarTavs gakveTils, SearCevs sadiskusio Temas, Tumca diskusia 

SeiZleba daugegmavadac _ gakveTilze wamoWrili problemis irgvliv gaimarTos. 

moswavleebi monawileoben diskusiaSi, avseben erTmaneTis pasuxebs; magaliTad, naturaluri 

ricxvebis Tanrigebis saSualebiT Cawerisas, sxvadasxva Tvisebis CamoTvlis 

dros, damrgvalebis wesebis warmodgenisas. maTematikuri faqtebis codnis Sesamowmeblad 

SeiZleba gamoviyenoT amocanebi savaraudo pasuxebiT. maswavleblebi am tipis 

davalebebis Sesrulebis sxvadasxva meTods mimarTaven xolme. rac mTavaria, am 

davalebebis Sesrulebas SeiZleba drois mcire monakveTi davuTmoT da SeiZleba 

garkveuli TanamimdevrobiT yvela moswavle CavrToT am procesSi. araswori pasuxis 

miTiTebis SemTxvevaSi SeiZleba gaimarTos msjeloba Secdomis mizezebis Sesaxeb. 

pirvel gakveTilze SeiZleba amoixsnas 9, 11, 13, 15, 17, 19, 21 amocanebic. maTi 

momdevno amocanebi moswavleebs saSinao davalebad miecemaT. 

meore gakveTilze vagrZelebT amocanebis amoxsnas. sruldeba 22, 24, 26, 28, 30, 32, 

34, 36, 38, 40, 42, 44 amocanebi. maTi `mezobeli~ _ kentnomriani amocanebi moswavleebs 

saSinao davalebad SeiZleba mivceT. 

pasuxebi da miTiTebebi 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 

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

19 20 21 22 

3 4 1 2 

27. a) 27148 ≈ 27150; b) 27148 ≈ 27100; g) 27148 ≈ 27000; 

28. a) 9753; b) 86540; g) 97542. 

29. a) 13456; b) 1057; g) 30689. 

37. mefe daviT IV (aRmaSenebeli) daibada 1073 wels. is 16 wlisa gamefda. XII saukuneSi 

saqarTvelos aRorZineba daviTis saxels ukavSirdeba. 1121 wels didgorSi didi 

gamarjvebis Semoqmedi mefe daviTia. 

43

42. 32a - 23a = 320 - 230 = 90. 

43. 1ab0 saxis ricxvebis odenoba ase daviTvaloT: cifri a SeiZleba SeirCes nebismierad 

aTi cifridan _ 0; 1; 2; 3; 4; 5; 6; 7; 8; 9 _ sul 10 SesaZlebloba. aseve aTi SesaZlebloba 

aqvs b cifris arCevasac. amitom sul asi ricxvi gveqneba _ 10 . 10=100. 

44. amowerilia pirveli eqvsi naturaluri ricxvis kvadrati. 

52. 9 cifriT gadainomreba pirveli gverdi, 180 cifriT _ Semdegi 90 gverdi, 2103 

cifriT _ Semdegi 701 gverdi; amrigad, gvaqvs 800 gverdi, anu 400 furceli. 

jgufuri muSaobis proeqti 

mizani: 

• kanonzomierebis aRmoCena da gavrceleba; 

• kritikuli azrovnebis, msjelobisa da dasabuTebis unaris gamomuSaveba. 

jgufuri muSaoba aris saswavlo procesis organizaciis iseTi forma, roca xdeba 

moswavleTa jgufebad dayofa da jgufebs eZleva saziaro kompleqsuri davaleba. 

davaleba miznad isaxavs moswavleebis CarTvas TanamSromlobaSi. jgufuri davaleba 

moswavleebisgan amocanis erToblivad gadaWras, an saerTo produqtis Seqmnas unda 

moiTxovdes. jgufSi moswavleTa odenoba davalebis specifikiT unda ganisazRvros. 

moswavleebi SeiZleba daiyos sam jgufad. jgufebs erTnairi davaleba eZleva _ 

Seadginon ricxvebis xarisxebis `bolo cifris~ cxrili. TiToeul jgufSi davalebebis 

ganawileba jgufis moswavleebzea damokidebuli. SeiZleba erTma moswavlem 

cxrilis romelime erTi striqonis Sedgenasa da Sesabamisi kanonzomierebis aRmo- 

Cenaze ifiqros; Semdeg moxdes am striqonebis `gaerTianeba~ erT cxrilSi da kanonzomierebebis 

erToblivi prezentacia. amrigad, am jgufuri muSaobis Sesrulebisas 

TiTToeul jgufSi moswavleTa ricxvi SeiZleba 6-s ar aRematebodes. maswavlebeli 

afasebs mTlianad jgufis muSaobas. Tumca, is valdebulia Tvali adevnos am muSaobis 

process da gamorCeviT aRniSnos yvelaze produqtiuli monawileebi. prezentacias 

jgufis erT-erTi wevri warmoadgens. sxvadasxva jgufis Sedegebi Sedardeba 

erTmaneTs da imarTeba diskusia mTeli klasis monawileobiT. mxedvelobaSi SeiZleba 

miviRoT warmodgenili magaliTebis odenobac, romlebic kanonzomierebebis gamoyenebiT 

xarisxebis bolo cifrebis povnas exeba. gamarTuli diskusiisas deduqciuri 

dasabuTebis elementebic iqneba gamoyenebuli, romlebsac maswavlebelma unda yuradReba 

miaqcios da Sefasebisas gaiTvaliswinos. 

44

mizani. 

§ 1.2. naturaluri ricxvebis gayofa. gayofadobis niSnebi 

• poziciuri sistemis gamoyenebiT naturaluri ricxvis gayofis procedurebis 

Catarebis unarebis ganviTareba; gamoyeneba praqtikul amocanebSi. 

• deduqciuri msjelobis Catarebis unaris ganviTareba gayofadobis niSnebis ganxilviT. 

meore gadamuSavebul variantSi, eqspertTa mosazrebebis gaTvaliswinebiT ramdenime 

paragrafi gavaerTianeT. gayofis algoriTmi calke ar aris warmodgenili. viziarebT 

maT mosazrebas, rom am algoriTmis warmodgena a=bq+r, 0≤r

18. TiToeuls imdeni moneta aqvs, ramdenjerac 150 TeTri metia (10+20) TeTrze: 

150:30=5 — xuT-xuTi moneta hqoniaT. 

19. wina amocanis analogiuria, 200:(20+5)=8 — rva-rva mo neta. 

21. am ricxvTagan umciresi aRvniSnoT x-iT, maSin 

x+(x+2)+(x+4)+(x+6)+(x+8)+(x+10)=1326 

6x+30=1326 

x=216. 

22. a) 13·4=52; b) 13·4+1=53. 

23. mocemulobis Tanaxmad, yuTebis odenobaze erTiT naklebi ricxvi 6-is jeradicaa 

da 7-isac, Tanac 50-ze naklebia. aseTi ricxvia 42. maSasadame, 43 yuTi yofila. 

24. 25-is 7-ze gayofisas naSTi 4-ia, e. i. nika meoTxe yofila. 

25. 366=7·52+2. 

27. gaiTvaliswineT, rom 2009 wlis 1 ianvari iyo xuTSabaTi. aTvla SeiZleba sxva 

wlidanac daiwyoT isargebleT Semdegi informaciiT _ 2008 weli nakiani weli iyo; 

nakian weliwadSi 366 dRea, aranakianSi _ 365 dRe. 2007 wlis 1 ianvari iyo orSabaTi. 

28. gamoiyeneT jgufuri muSaobisas miRebuli cxrili da SeamowmeT 4-iT dabo- 

loebuli ricxvis axarisxebisas bolo cifrisTvis SemCneuli kanonzomiereba — 

kentmaCvenebliani xarisxis bolo cifria 4, luwmaCveneblianisa ki — 6. amitom 

a) 10-ze gayofisas naSTi 4-ia, Tu xarisxis maCvenebeli kentia — aseTi xarisxebis 

odenoba 15-ia; 

b) luwmaCvenebliani xarisxebis odenoba 14-ia; 

g) naSTi 8 arc erT SemTxvevaSi ar gvxvdeba — rviT daboloebuli oTxis xarisxi 

ar arsebobs. 

29. wina amocanis analogiurad gamoiyeneT `bolo cifris cxrili~. 

a) 1-iT bolovdeba 7-is is xarisxebi, romelTa maCve neb lebi 4-is jeradia, aseTi 

(mocemul SemTxvevaSi) 11-ia; 

11-ia; 

11-ia; 

12-ia; 

b) 9-iT bolovdeba 7-is is xarisxebi, romelTa maCve neb lebi 4k+2 saxisaa — aseTi 

g) 3-iT bolovdeba 7-is is xarisxebi, romelTa maC ve neb lebi 4k+3 saxisaa — aseTic 

d) 7-iT bolovdeba 7-is is xarisxebi, romelTa maCve neb lebi 4k+1 saxisaa — aseTi 

e) 7-is arc erTi xarisxi 6-iT ar bolovdeba. 

30. Tu a=5n+r, b=5q+r, maSin a–b=5(n–q) — iyofa. 

46

31. vTqvaT, samive biWs erTad n burTi ergo, maSin n iyofa 3-ze; cxrave gogonas 

ki — m burTi, maSin m iyofa 9-ze da, amitom, m iyofa 3-zec. Tu n+m=20, maSin 20ic 

unda iyofodes 3-ze, rac ar sruldeba. amrigad, SeuZlebelia maT hqonodaT 20 

burTi. 

32. fanqris fasic da kalmis fasic luwi ricxvebia, lu wia nayidi rveulebisa da 

albomebis odenobebic. amrigad, na vaW ris Rirebulebac, rogorc luwi ricxvebis jami 

— luwia; 557 ki kentia. amrigad, gamyidveli Secda. 

33. a) TiToeuli ricxvi iyofa 3-ze — cifrebis jami iyofa 3-ze; 

b) 6-ze gaiyofa mxolod 132 da 312 — orive luwicaa da 3-is jeradic; 

g) 9-ze arc erTi ar gaiyofa — cifrTa jami (6) ar iyo fa 9-ze. 

34. a) 999998; b) 999999; g) 999995; d) 999999; e) 999990. 

35*. jer vipovoT im ricxvebis odenoba, romlebic 2-ze an 3-ze (erT-erT ricxvze 

mainc) iyofa; 2-is jeradia yoveli meore naturaluri ricxvi — aseTTa odenoba 500ia; 

3-is jeradia yoveli mesame naturaluri ricxvi — maTi odenobaa 333; 500+333 

jamSi 6-is jeradi ricxvebi orjeraa daTvlili, maTi odenobaa 166. 

miviReT, rom 1-dan 1000-mde im ricxvebis odenoba, romelic arc 2-ze da arc 3-ze 

ar iyofa, aris 1000-(500+333-166)=333. 

36. 1·2·3·4·5· ... ·10·11·12·13 — am TanamamravlTa Soris aris 2, 5 da 10, romelTa 

namravlia 100. amitom namravlis bolo ori cifri nulebia. 

37. a) dasaxelebul Tanamamravlebs Soris gamovyoT iseTi wyvilebi, romlebic 

nuliT daboloebul ricxvebs mogvcemen: 2 da 5; 4 da 15; 1 da 10 — sul sami. isini 

mxolod sami nuliT daboloebul ricxvs warmogvidgenen. 

38. ricxvi rom iyofodes 45-ze, is 9-zec unda iyofodes da 5-zec, amitom cifrTa 

jami 9-is jeradi unda iyos da bolo cifri ki — 0 an 5. saZiebeli ricxvebia: 

123840, 123345. 

39. gamoviyenoT 3-ze gayofadobis niSani da miviRebT, rom 3-is jeradia 8·10 7 +1; 

3·10 4 +2·10 2 +1; 2·10 n +1. 

47

mizani 

48 

§ 1.3. martivi da Sedgenili ricxvebi. 

udidesi saerTo gamyofi da umciresi saerTo jeradi 

• naturalur ricxvTa klasifikaciis warmodgena da misi gamoyeneba. 

• ariTmetikis ZiriTadi Teoremis propedevtika. misi gamoyeneba umciresi saerTo 

jeradisa da udidesi saerTo gamyofis povnisas. 

• praqtikuli amocanebis amoxsnisas udidesi saerTo gamyofisa da umciresi saer- 

To jeradis gamoyenebis unaris ganviTareba. 

naturalur ricxvTa klasebad dayofa _ martiv ricxvTa klasi, ricxvi 1, Sedgenil 

ricxvTa klasi _ mniSvnelovan rols TamaSobs ricxvTa Teoriasa da mis gamoyenebebSi. 

amitom maT Sesaxeb warmodgenebis gaRrmavebasa da gafarToebas me-7 klasSic 

vagrZelebT. 

erT-erTi mniSvnelovani sakiTxi, romelic martiv ricxvTa klasTan aris dakav- 

Sirebuli, aris imis dadgena, martiv ricxvTa klasi usasruloa, Tu ara. 

maswavleblebi sasurvelia icnobdnen evklides Teoremis damtkicebas ([5], gv. 49), 

romelic sawinaaRmdegos daSvebis meTodiT tardeba. 

paragrafi sami nawilisgan Sedgeba; TiToeul nawils Sesabamis amocanebTan erTad 

SeiZleba TiTo gakveTili davuTmoT. 

pasuxebi da miTiTebebi: 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 

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

17 18 19 20 21 22 23 24 25 26 27 28 20 30 

2 4 2 2 4 2 2 3 2 4 2 3 4 2 

14. n=2 3 ·3 2 — am ricxvis gamyofebia 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 — sul 12 

gamyofi. 

36. pirvel rva aTeulSi cxriani rvajer gvxvdeba 9; 19; ...; 79; mecxreSi — orjer: 

89; 90; meaTeSi — aTjer; 91, 92, ..., 99, e. i. sul ocjer. 

37. am ricxvs 5-is garda sxva martivi gamyofic Tu aqvs, maSin mas oTxi gamyofi 

eqneba (ix. amocana 7). amrigad, es ricxvia mxolod 5 2 , anu 25 — gamyofebia 1; 5; 

25. 

38. ricxvi, romelic 21-ze iyofa, gaiyofa 3-zec da 7-zec. amrigad, misi gamyofebia 

1; 3; 7; 21. am ricxvebisgan gansxvavebuli p martivi gamyofi rom hqonoda, kidev sul 

cota oTxi gamyofic eqneboda — p, 3p, 7p, 21p. amrigad, saZiebeli ricxvis martiv 

mamravlebad daSlaSi 3 an 7 orjer mainc monawileobs. 3 unda gamoiricxos — maSin 

ricxvi 9-ze gaiyofoda. amrigad, gameordeba 7. Sesabamisad, gamyofebi iqneba: 1; 3; 

7; 21; 49; 147, TviT ricxvia 147. 

39. radgan x ricxvi orze metia, x–1 da x+1 ricxvebi erTze metia da (x–1)(x+1) 

namravls sami gamyofi mainc aqvs — 1, x–1, x+1 — amitom Sedgenili ricxvia.

40. m=n+14 — amis gaTvaliswinebiT gveqneba. 

A B A-sa da B-s 

SeniSvnebi 

Sedareba 

m–10 n A>B m–10=n+14–10=n+4 

m n+20 A

50. es ricxvi iqneba usj(4;5;7)+3=143. 

51. 4 -sa da 

5 

-ze gayofa 37 -sa da 

41 

-ze gamrav le bas niSnavs, amitom saZiebeli 

37 41 

4 5 

ricxvi 4-zec unda iyofodes da 5-zec — aseT ricxvebSi umciresia 20. 

52. Tu a ar aris p-s jeradi, maSin usg(a; p)=1, usj(a; p)=ap; Tu a aris p-s jeradi, 

maSin usg(a; p)=p, usj(a; p)=a. 

53. mocemuli ricxvebi unda iyos 9m da 9n saxis, sadac m da n Tanamartivebia. 

vTqvaT, 9m

sakontrolo wera #1 

(sakontrolo weris variantebis nimuSebi) 

I varianti 

1. ipoveT 76-is 12-ze gayofisas miRebuli ganayofisa da naSTis jami. 

1) 6 2) 8 3) 12 4) 10. 

2. x ricxvis 5-ze gayofisas naSTSi miiReba 2. ipoveT (x+6)-is 5-ze gayofisas 

miRebuli naSTi. 

1) 2 2) 4 3) 1 4) 3. 

3. x martivi ricxvia, x≠2 da x≠3. ipoveT 6x-is gamyofebis ricxvi. 

1) 4 2) 8 3) 12 4) 16. 

4. a da b gansxvavebuli martivi ricxvebia. mocemulia tolobebi: 

I. usg (a; b)=1, II. usj (a; b)=ab, 

III. usg (a; b)=a, IV. usj (a; b)=1. 

maTgan WeSmaritia 

1) mxolod I 2) III 3) IV 4) I da II. 

5. ipoveT umciresi naturaluri ricxvi, romelic 3-ze, 5-ze da 7-ze gayofisas 

naSTSi 2-s iZleva. 

6. saskolo Sejibrebaze 36 gogona da 96 vaJi gundebadaa dayofili. gundebSi 

mxolod gogonebi an mxolod vaJebia. 

gundebSi moTamaSeTa odenobebi erTi da igivea. moTamaSeTa ra udidesi o de no ba 

SeiZleba iyos TiToeul gundSi? ramdeni gundi Camoyalibdeba am Sem Txve vaSi? 

II varianti 

1. ipoveT is a ricxvi, romlis 17-ze gayofisas miiReba ganayofi 3 da naSTi 2. 

1) 51 2) 50 3) 53 4) 54. 

2. x ricxvis 7-ze gayofisas miiReba naSTi 6. ra naSTi miiReba (x+2)-is 7-ze gayofisas? 

1) 1 2) 8 3) 2 4) 3. 

3. x martivi ricxvia, x≠3 da x≠7. ipoveT 21x-is gamyofebis ricxvi. 

1) 2 2) 8 3) 4 4) 16. 

4. x da y gansxvavebuli martivi ricxvebia. mocemulia tolobebi: 

I. usg (x; y)=y II. usg (x; y)=xy, III. usj (x; y)=xy, 

IV. usg (x; y)=1, V. usj (x; y)=1. 

maTgan WeSmaritia: 

1) II da III 2) III da IV 3) I 4) V. 

5. ipoveT umciresi naturaluri ricxvi, romelic 5-ze, 7-ze da 11-ze gayofisas 

naSTSi 2-s iZleva. 

6. or punqts Soris boZebi yovel 45 m-is manZilze iyo gan la gebuli. es boZebi 

axliT Secvales da isini erTmane Tis gan yovel 60 m-Si ganalages. ipoveT manZili 

pirveli punq ti dan uaxloes boZamde, romelic Zveli boZis adgilasaa dar Wo bili. 

51

Sefasebis sqema 

pirveli oTxi amocanidan TiToeulSi swori pasuxis Ser Ceva Sefasdeba 1 quliT. 

me-5 da me-6 amocanebidan Ti To euli – 3 quliT. am ori amocanis Sefasebisas Se iZ le ba 

gamoviyenoT 1 an 2 qulac im SemTxvevaSi, roca amocanis na wilia amox s nili – amoxsna 

bolomde ar aris miyvanili. 

magaliTad, Tu me-5 amocanaSi napovnia dasaxelebuli Tvisebis raime naturaluri 

ricxvi (romelic umciresi ar aris), SeiZleba es amoxsna 2 quliT Sefasdes. 

sakontrolo weris variantebis nimuSebSi SemoTavazebuli zogierTi amocanis 

amoxsnebi: 

36 2 

18 2 

9 3 

3 3 

1 

5. aseTi ricxvia 5·7·11+2=387. 

52 

96 2 

48 2 

24 2 

12 2 

6 2 

3 3 

1 

I varianti 

5. aseTi ricxvia 3·5·7+2=107. 

6. movZebnoT 36-isa da 96-is saerTo gamyofebs Soris udidesi: 

usg (36; 96)=2·2·3=12. 

maSasadame, gundebSi SeiZleba iyos sul didi 12-12 moTamaSe. 

aseTi odenobebis moTamaSeTa gundebis odenoba iqneba 11. maTgan 

3 — gogonebisa da 8 — vaJebis. 

II varianti 

6. unda movZebnoT 45-isa da 60-is umciresi saerTo jeradi. 

45 $ 60 

usj (45; 60)= 

= 

45 $ 60 

180 

usg ( 45; 60) 

15 

= . 

(SeiZleba es usj movZebnoT 45-isa da 60-is martiv mamravlebad daSlis gzi- 

Tac). 

saZiebeli manZili 180 metria.

§ 1.4. mTeli ricxvebi 

mizani 

• mTeli ricxvis Sesaxeb pirveli warmodgenebis Seqmna 

• mTeli ricxvebis praqtikuli interpretaciebis gaazreba urTierTmopirdapire 

situaciebis aRwerisas. mTeli ricxvebis gamoyenebis unaris ganviTareba 

• mTeli ricxvebis TvalsaCino warmodgenis gaazreba. 

erovnuli saswavlo gegmis mixedviT paragrafSi mocemuli masala axali unda iyos 

moswavleTaTvis, Tumca, yoveldRiur cxovrebaSi maT aucileblad SexvedriaT situaciebi, 

roca uaryofiTi mTeli ricxvebis gamoyeneba xdeboda, am gamoyenebebze saubriT 

iwyeba paragrafis Teoriuli nawili da igi moswavleebTan saubrisTvis SeiZleba 

gamoviyenoT. saubaric moswavleTa monawileobiT SeiZleba warimarTos. magaliTad, 

visaubroT moswavleebTan zamTarSi haeris temperaturis Sesaxeb, xom ar icnoben 

isini Termometrs, romliTac haeris temperaturas vzomavT, ra SemTxvevaSi vityviT, 

rom haeris temperatura –5 0 -ia _ Termometri am dros 0-s qvemoT 5 0 -s uCvenebs. gadacemis 

meTodika konkretul-induqciuri unda iyos; maSasadame, axlis gagebis safuZveli 

moswavleTa cxovrebiseuli gamocdileba da sakoordinato wrfeze Tvalsa- 

Cino warmodgenebi unda iyos. sidideTa cvlilebis dasaxasiaTeblad axali ricxvebis 

aucilebloba moswavleTa azrovnebaSi ganzogadebas, sayofacxovrebo enaSi (zogjer, 

lakonurobis mizniT) `moqnili~ maTematikuri terminologiis damkvidrebas ukavSirdeba. 

magaliTad, Cveulebriv enaze _ temperatura nuls qvemoT 3 0 , maTematikurad _ 

temperatura: –3 0 . 

axali ricxvebis Semotanis motivaciis procesis gaazreba ar mTavrdeba am paragrafSi. 

igi Semdeg paragrafebSi moqmedebebis (mag. gamoklebis) operaciis warmodgenisas 

grZeldeba. 

cxadia, maswavlebelma ar unda Caataros leqcia mTeli ricxvebis Sesaxeb, moyves 

Teoriuli nawilis Sinaarsi; saWiroa saubari moswavleebTan _ Sesabamisi kiTxvebis 

dasma, pasuxebis mosmena. mniSvnelovania Semajamebeli sja-baasi mTel ricxvTa zogierTi 

Tvisebis Sesaxeb. 

mniSvnelovania mopirdapire ricxvebis gaazreba. 


1 2 3 4 5 6 7 8 9 24 

4 4 2 4 2 2 3 4 3 3 

7. 

8. 

9. 

19. a) 12; b) 6; g) 18; d) 0. 

20. koordinatebi moimatebs or-or erTeuls. 

25. 2a=12, a=6. 

53

mizani 

54 

§ 1.5. mTeli ricxvis moduli (absoluturi mniSvneloba). 

mTeli ricxvebis Sedareba 

• mTeli ricxvebis Tvisebebis Sesaxeb codnis ganmtkiceba. 

• mTeli ricxvebis geometriuli warmodgenebis gamoyeneba dalagebis mimarTebis 

SemoRebisas. 

vagrZelebT mTeli ricxvebis Tvisebebis Seswavlas, geometriuli warmodgenis 

gamoyenebebs. ricxviTi wrfis saSualebiT Semogvaqvs dalagebis mimarTeba mTel ricxvTa 

simravleSi. 

aqac SeiZleba praqtikuli interpretaciebis gamoyeneba; magaliTad, haeris temperaturis 

gamzomi Termometri SeiZleba ricxviTi wrfis analogiurad ganvixiloT. 

moswavleebs SeiZleba SevTavazoT kiTxvebi, romlebic maT daafiqrebs uaryofiTi 

ricxvebis Sedarebaze; 

_ ra temperaturaa Tqveni azriT meti 3 0 yinva, Tu 5 0 yinva, –3 0 , Tu –5 0 ? 

#1 _ #6, #10 amocanebis amosaxsnelad sasurvelia SedarebiT susti moswavleebi 

moiwvioT _ es saSualebas mogcemT CarToT isini saswavlo procesSi, miTumetes, rom 

es amocanebi maTTvis sruliad misawvdomia da iolad dasaZlevi; #7 _ #8 amocanebi 

iTxovs meti informaciis damuSavebas, xolo #9, #12, #13 amocanebis amosaxsnelad 

saWiroa kanonzomierebis SemCneva, ganzogadeba, debulebis saxiT Camoyalibeba. 

paragrafs SeiZleba mxolod 1 saaTi davuTmoT. 


1 2 3 4 5 6 7 8 9 10 11 16 

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

17. a) 

b) 

22. a) a>b; b) |a|>|b|. 

23. a) a>b; b) |a|

mizani 

§ 1.6. mTeli ricxvebis Sekreba da gamokleba 

• mTeli ricxvebis Sekrebisa da gamoklebis TvalsaCinod warmodgenis saSualebiT 

am operaciaTa aTviseba, mTel ricxvTa simravleze moswavleTa warmodgenebis gafar- 

Toeba. 

• mTeli ricxvebis Sekrebisa da gamoklebis praqtikuli gamoyenebebis unarebis 

ganviTareba. 

SesavalSi aRvniSneT, rom ricxvis cnebis gafarToebis algebruli gza ar emTxveva 

skolaSi ricxviTi sistemebis swavlebis Tanamimdevrobas. es sakiTxi kargad aris axsnili 

kolmogorovisa [21] da froidentalis [38] wignebSi. 

mTel ricxvebze moqmedebebis wesebis `receptulobisa~ da praqtikasTan naklebi 

kavSirebis Sesaxeb kolmogorovTan ([21], gv. 21) vkiTxulobT: `dabal klasSi kiTxva 

_ SeiZleba Tu ara ricxvs, romelic naklebia, gamovakloT meti ricxvi; SeiZleba Tu 

ara, magaliTad, 3-s gamovakloT 5 _ iwvevs gamococxlebas moswavleebSi da maswavleblis 

kargi xelmZRvanelobiT SeiZleba gaxdes uaryofiT ricxvebze moqmedebebis 

swavlebis dasawyisi. magram konkretul gamoyenebebTan kavSiris gareSe es yvelaferi 

gasarTob TamaSad aRiqmeba. 

sakiTxi _ SeiZleba Tu ara 3 orze gavyoT, ufro bunebrivad moeCveneba moswavleebs. 

maswavleblis daxmarebis gareSec am sakiTxis aRqma ufro advilia dawyebiT 

klasebSi; rom gavyoT 3 vaSli, erT-erTi unda Suaze gavWraT da miviRebT erTnaxevar 

vaSls~. amitom kolmogorovs ufro bunebrivad miaCnia ricxvebis swavlebis is Tanamimdevroba, 

romelic istoriulad Camoyalibda (swavlebis Tanamimdevroba emTxveva 

istoriuli ganviTarebis Tanamimdevrobas). 

SevecadeT e. w. dadebiTi da uaryofiTi muxtebis, valisa da qonebis, ricxviTi 

wrfis gamoyenebiT, TvalsaCinod warmogvedgina mTeli ricxvebis Sekrebisa da gamoklebis 

operaciebi. 

nu movTxovT moswavleebs Sekrebisa da gamoklebis wesebis zepirad gameorebas _ 

mTavaria moswavles hqondes wesebis swori gamoyenebis unari, romelic TvalsaCino 

warmodgenebiT uyalibdebaT. 

SeiZleba maswavlebelma sxva praqtikuli interpretaciebic moiSvelios. magali- 

Tad, dRis ganmavlobaSi temperaturis cvlileba zamTris romelime dRes _ diliT 

SeiZleba temperatura nuls qvemoT iyos, Semdeg gaizardos, saRamos isev Semcirdes. 

es procesi SeiZleba mTeli ricxvebis SekrebiTa da gamoklebiT warmoadginos maswavlebelma. 

SeiZleba Sesabamisi plakatic ki moitanos skolaSi (sadac miTiTebuli 

iqneba dRis ganmavlobaSi temperaturis cvlileba) da moswavleebTan erTad imsjelos 

am cvlilebis Sesaxeb, es cvlilebebi mTel ricxvebze moqmedebebiT aRweros. 

paragrafs SeiZleba daeTmos sami gakveTili. amocanebi ise gaanawileT, rom erTi 

tipis savarjiSoebi gadabmulad ar sruldebodes. magaliTad, 22-e savarjiSos a) Sei- 

Zleba gakeTdes klasSi, b) da g) _ mieces davalebad, 23-is a) da b) _ klasSi, g) da 

d) _ davalebad da a. S. 


1 2 3 4 5 6 7 8 9 10 11 12 

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

13 14 15 16 17 18 19 20 

3 3 1 3 2 1 2 1 

55

7. 3) –5+7. 

8. gavixsenoT, rom • aRniSnavs –1-s, 0 ki — 1-s. amis gaTvaliswinebiT, swori pasuxia 

2) . 

9. gaiTvaliswineT, rom TiTo dadebiTi da uaryofiTi `muxti~, anu — nulia. 

amis gaTvaliswinebiT, swori pasuxia 2) –3. 

19. (a+(–4))-is Sesabamisi wertili mdebareobs A(a) wer ti lidan 4 erTeuliT marcxniv. 

20. b=a+3. 

32. a) (+)–(–)=(–) — arasodes, radgan (+)–(–)=(+)+(+)=(+); 

b) (+)–(+)=(+) — zogjer; magaliTad, 3–2=1, 2–3= –1; 

g) (–)–(+)=(–) — yovelTvis, radgan (–)–(+)=(–)+(–)=(–); 

d) (–)–(–)=(+) — zogjer; magaliTad, (–2)–(–5)=3, (–5)–(–2)= –3; 

e) (–)–(+)=(+) — arasodes, radgan (–)–(+)=(–)+(–)=(–); 

v) (+)–(+)=(–) — zogjer; magaliTad, 5–7=–2, 7–5=2; 

z) (+)–(–)=(+) — yovelTvis, radgan (+)–(–)=(+)+(+)=(+). 

42. a) 99–(–99)=99+99=198; 

b) 999–(–999)=999+999=1998. 

43. a) yoveli momdevno ricxvi winaze 3-iT naklebia; 

4; 1; –2; –5; –8; –11; –14; –17; 

b) yoveli momdevno ricxvi winaze 4-iT metia; 

–15; –11; –7; –3; 1; 5. 

44. moxvedraTa odenoba aRvniSnoT n-iT, acdenaTa odenoba m-iT, maSin qulebis 

odenoba iqneba — 7n+(–5)m. 

a) n=5, m=2, 35–10=25; 

b) n=2, m=5, 14–25=–11. 

46. WeSmaritia: b); g); v). 

56

mizani. 

§ 1.7. mTeli ricxvebis gamravleba da gayofa 

• mTeli ricxvebis Tvisebebis warmodgeniTa da gamoyenebiT gamravlebisa da gayofis 

operaciebis aRwera. 

wina paragrafSi naxseneb `receptulobaze~ ufro metad `receptuli~ xasiaTisaa 

mTeli ricxvebis gamravlebisa da gayofis wesebi. am operaciebis SemoReba Sekrebisa 

da gamoklebis (magaliTad, jeradi Sekreba _ gamravleba) operaciebze dayrdnobiT, 

naturalur ricxvebze moqmedebaTa Tvisebebis (magaliTad, gadanacvlebadobis, ganrigebadobis 

gavrcobis saSualebiT mimdinareobs. 

gamravlebisas `niSnebis wesis~ aTviseba gacilebiT swrafad mimdinareobs, vidre, 

magaliTad, mTeli ricxvebis gamoklebis operaciis Catareba. nu movTxovT moswavleebs 

am wesebis raime saxiT dasabuTebis warmodgenas. 

amocanebis amoxsnisas ecadeT zogierTi moqmedeba moswavleebs zepirad SeasrulebinoT 

_ amiT gaaxsenebT moqmedebaTa Tvisebebsac da cota gaaxalisebT gamoTvlebTan 

dakavSirebul mosawyen procedurebs, SesaZlebloba mogecemaT mTeli klasi 

CarToT ganxilvebSi, SeajibroT moswavleebi swraf gamoTvlebSi. 

Zalian saintereso savarjiSoa 11 _ aq ar arsebobs SekiTxvaze calsaxa pasuxi, 

radgan ar aris miTiTebuli Sefasebis raime SezRudva, amitom yoveli swori Sefaseba 

SeiZleba misaReb pasuxad CaTvaloT; Tumca saintereso iqneba, Tu moswavleebs SemoTavazebuli 

Sefasebebidan yvelaze `zusts~ SearCevinebT. magaliTad, 

a) –24⋅35≈-20⋅40=-800 

-24⋅35>24⋅40=-960 

-24⋅35

mizani 

58 

§ 1.8. simravle. qvesimravle 

• simravlis, qvesimravlis, carieli simravlis cnebebis aRwera da maTi sxvadasxva 

gamoyenebis unaris ganviTareba 

• ricxviT simravleebs Soris mimarTebebis warmodgenisas simravluri enis gamoyenebis 

daufleba 

paragrafs SeiZleba ori gakveTili davuTmoT. pirveli gakveTili SeiZleba Sereuli 

tipis iyos; mimdinareobs wina gakveTilze axsnili masalis aTvisebis Semowmeba, ganmtkiceba. 

SeiZleba amoixsnas damatebiTi amocanebidan ramdenime Sesabamisi amocana. amis 

Semdeg gadavdivarT simravlis cnebis aRweris procesze. es procesic SeiZleba moswavleTa 

monawileobiT daigegmos. Tavidanve miaqcieT yuradReba imas, rom simravle 

SeiZleba erTi, ori, an meti elementisgan Sedgebodes, an ar hqondes elementi, rom 

simravle ganisazRvreba raRac TvisebiT, romelic axasiaTebs gansaxilvel obieqtebs 

da mxolod maT (maxasiaTebeli Tviseba). garkveuli Tvisebebis miTiTebiT moswavleebi 

asaxeleben simravles, romelsac am Tvisebis mqone yvela obieqti qmnis. magaliTad, 

SeiZleba moswavleebTan ganvixiloT naturaluri ricxvebi da mivuTiToT Tviseba _ 

ìyos martivi~. moswavleebi TviTon adgenen simravles, romelic Sedgeba yvela martivi 

_ 2; 3; 5; 7; ... ricxvisgan. 

simravleTa Teoriis sawyisi elementebis swavleba xels uwyobs maTematikis sxvadasxva 

nawilis erTianobis gaazrebas. zogierTi simravlisTvis specialuri aRniSvnebi 

Semogvaqvs. am aRniSvnebis gamoyeneba momavalSi moswavleebs xSirad mouwevT. 

maswavlebelma ar unda ifiqros, rom aRniSnuli masaliT moswavleebs simravleTa 

Teorias vaswavliT skolaSi _ am Teoriis sawyisi cnebebis gadmocema damxmare saSualebaa. 

simravleebTan dakavSirebiT asoebis gamoyeneba maTematikuri niSnebis gamoyenebis 

erT-erTi mniSvnelovani magaliTia _ aso aRniSnavs, gansazRvravs `sagans~ _ 

konkretul simravles an am simravlis elements. momavalSi ufro xSirad mogviwevs 

im SemTxvevebis ganxilva, roca aso simravlis nebismieri elementebis aRsaniSnavad 

gamoiyeneba (cvladi; ix., mag. [11]); magaliTad, Canaweri `x nebismieri naturaluri 

ricxvia~, niSnavs, rom ar SeiZleba iyos 1, 2, 3 an romelime sxva naturaluri ricxvi, 

erT-erTi yvelaze gavrcelebuli Secdoma, rac simravleTa swavlebisas gvxvdeba es 

erTelementiani simravlis Tavis elementTan gaigivebaa _ xSirad moswavle fiqrobs, 

rom {a} da a erTsa da imave obieqts warmogvidgens. saTanado amocanebis ganxilviT 

mniSvnelovania am sakiTxze moswavleTa yuradReba gansakuTrebiT gaamaxviloT _ 

{a}∈a. am momentis aTvisebis xarisxs me-6 testis amoxsniT SeamowmebT. 

meore gavrcelebuli Secdomaa simravluri CarTvisa da mikuTvnebis niSnebis araswori 

gamoyenebebi _ moswavleebs erevaT erTmaneTSi ⊃ da ∈ niSnebi. am Secdomis 

prevencias movaxdenT 1-5, agreTve, 16 savarjiSoebis SesrulebiT. 


1 2 3 4 5 6 8 20 

2 1 4 3 4 1 4 2 

7. {ianvari, marti, maisi, ivlisi, agvisto, oqtomberi, dekemberi}. 

13. a) {2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47}; 

b) {2; 4; 6; 8; 10; 12; 14; ... ; 48; 50}; 

am simravleebis saerTo elementia mxolod 2.

15. sasruli simravleebia: 

g) N 3 — masSi sami elementia; 

d) N 6 — masSi eqvsi elementia; 

e) 5-ze meti da 9-ze naklebi mTeli ricxvebis simravlea — {6; 7, 8}, masSi sami 

elementia. 

16. a) N⊃N n ; b) N n ⊂N m ; g) N m ⊂N; d) Z⊃N; e) N n ⊂Z. 

17. N m–1 simravleSi m–1 elementia; pirobiT, m–1=9, m=10. 

18. N k ⊂N 7 ⇒ k≤7, anu k=1; 2; 3; 4; 5; 6; 7. 

19. N m ⊂N n ⊂N 6 , e. i. 1≤m≤n≤6; gveqneba (m; n) SesaZlo wyvilebi: 

(1; 1), (1; 2), (1; 3), (1; 4), (1; 5), (1; 6); 

(2; 2), (2; 3), (2; 4), (2; 5), (2; 6); 

(3; 3), (3; 4), (3; 5), (3; 6); 

(4; 4), (4; 5), (4; 6); 

(5; 5), (5; 6); 

(6; 6). 

20. Tu ricxvi iyofa 6-ze, is gaiyofa 3-zec, amitom B aris A-s qvesimravle — 

B⊂A. 

21. a) aseTi qvesimravle sami arsebobs. isini SeiZleba ase CavweroT: 

A 0 ={3k\ k∈N, 4≤k≤33}; 

A 1 ={3k+1\ k∈N, 3≤k≤32}; 

A 2 ={3k+2\ k∈N, 4≤k≤32}. 

b) A 0 , A 1 , A 2 simravleTagan TiToeulis nebismieri ori elementi 3-ze gayofisas 

erTsa da imave naSTs iZleva, amitom am elementebis sxvaoba iyofa 3-ze — naSTi 

nulia. 

22. A={1; 2; 3; 5; 6; 10; 15; 30}; 

B={1; 2; 3; 6; 7; 14; 21; 42}; 

a) saerTo elementebis simravlea {1; 2; 3; 6}; 

b) {1; 2; 3; 5; 6; 7; 10; 14; 15; 21; 30; 42}; 

g) {5; 10; 15; 30}; 

d) {7; 14; 21; 42}. 

59

mizani 

60 

§ 1.9. venis diagramebi 

• simravleebis, simravleebze operaciebis (gaerTianeba, TanakveTa) TvalsaCinod 

warmodgenis unaris ganviTareba 

• amocanebis amoxsnis TvalsaCino xerxebis gamoyenebis unarebis ganviTareba. 

vecnobiT simravleebis TvalsaCino gamosaxvis xerxs _ venis daigramebs. venis diagramebiT 

warmovadgenT simravleTa gaerTianebasa da TanakveTas. venis diagramebs 

didi gamoyeneba aqvs _ maTi saSualebiT mravali amocana TvalsaCinod da martivad 

ixsneba. 

simravleebze operaciebis venis diagramebiT warmodgenisas yuradReba miaqcieT, 

rom yvela logikuri SesaZlebloba iyos ganxiluli; magaliTad, roca saubaria ori 

simravlis saerTo elementebis arsebobaze, an erT simravleSi iseTi elementebis arsebobaze, 

romlebic meore simravleSi ar Sedis. magaliTad, ori simravlis TanakveTis 

ama Tu im SemTxvevis gamosaxvasTan erTad moswavleebi ganixilaven sxva SemTxvvebsac 

da TanakveTis sxva SesaZleblobebsac Tavad warmoadgenen venis diagramebiT. 

simravleebis gaerTianebis SesaZlo SemTxvevebis ganxilvisas sayuradReboa saer- 

To elementebis arseboba _ isini axal simravleSi TiTo. TiTojer moiazreba. am 

SemTxvevaze yuradRebis miqceva momavalSi _ logikuri amocanebis amoxsnisas swori 

daskvnebis gamotanis sawindaria. 

paragrafis amocanebSi, garda ZiriTadi Temisa, mniSvnelovani yuradReba eqceva 

gayofadobasa da naSTebs _ magaliTad, amocanebi 10, 12, 13, 14, 15, 29, 30; amave Tematikas 

ganekuTvneba paragrafis boloSi SemoTavazebuli jgufuri muSaobis proeqtic. 

gTavazobT ramdenime amocanas uaryofiT ricxvebze moqmedebebis Tvisebebis gasaxseneblad 

_ amocanebi 20, 21, 22, 23, 24, 25, 27. Tematikis mravalferovneba saSualebas 

mogcemT gakveTili ise warmarToT, rom is ar iyos erTferovani, mosawyeni, amasTanave, 

SeZlebT gaiTvaliswinoT Tqveni klasis momzadebis specifika da meti yuradReba 

miaqcioT im masalas, romelic moswavleebs naklebad aqvT aTvisebuli. moswavleebTan 

mudmivi kontaqtiT, sakiTxebis erToblivi ganxilvebiT kidev ufro daazustebT arsebul 

xarvezebs moswavleTa codnaSi, codnis gamovlenis unarSi. es informacia Tqvens 

samomavlo muSaobas ufro mizanmimarTuls gaxdis. 


1 2 3 4 5 6 7 8 10 18 19 20 21 22 26 27 29 30 

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

9. A={1; 2; 7; 9; 10; 12; 15}, B={0; 3; 5; 6; 7; 8; 9; 11}. 

a) A∪B={0; 1; 2; 3; 5; 6; 7; 8; 9; 10; 11; 12; 15}; A∩B={7, 9}; 

b) C SeiZleba SeirCes Semdegi simravleebidan: ∅, {7}, {9}, {7; 9}; 

g) C∩(A∩B)=C. 

10. A={0; 1; 2; 3; 4}, B={0; 1; 2; 3; 4; 5; 6}, A∩B={0; 1; 2; 3; 4}.

11. a) A B 

b) A B g) 

C 

d) z) v) z) 

A B 

C 

12. A={3k| k∈N, 4≤k≤33}; 

B={7n| n∈N, 2≤n≤14}. 

A∩B={21; 42; 63; 84} — sul oTxi elementi. 

13. orive mocemuli ricxvi warmovadginoT martivi mam rav le bis namravlis saxiT: 

210 231 

2 105 3 77 

2 3 35 

2 3 5 7 3 7 11 

210=2·3·5·7 231=3·7·11 

A={2; 3; 5; 7} B={3; 7; 11} 

A∩B={3; 7}. 

14. warmovadginoT mocemuli ricxvebis gamyofTa simrav le ebi: 

36—A={1; 2; 3; 4; 6; 9; 12; 18; 36}; 

40—B={1; 2; 4; 5; 8; 10; 20; 40}. 

A∩B={1; 2; 4}. 

15. A={3k+1\ k∈N, 3≤k≤32}; B={3k+2\ k∈N, 3≤k≤32}; • A∪B aris 3k+1 an 3k+2 saxis 

orniSna ricxvTa simravle, anu im orniSna ricxvebis simravle, romlebic ar 

aris 3-is jeradi. 

17. a) A∩B=B b) A∪B=A. 

19. A={–6; –5; ...; 4}, B={–2; –1; ...; 9}, 

A∩B={–2; –1; ...; 4}. 

24. a) 6+(–1); ; b) –3+2 . 

C 

A B 

C 

A B 

C 

A B 

C 

A B 

C 

61

29. m=9n=3(3n). 30. m=9n+6=3(3n+2). 

62 


es proeqti naSTTa ariTmetikis mniSvnelovani propedevtikaa, amitom mis Sesasruleblad 

maswavlebelma dro ar unda dainanos. 

1. • am simravleebs saerTo elementi ara aqvs — raime naturaluri ricxvis 5-ze 

gayofisas ori gansxvavebuli naSTi ar miiReba. 

• arsebobs — 5-is jeradi ricxvebi arc erT mocemul simravleSi ar Sedis. 

• magaliTad, ase: `simravle im naturaluri ricxvebisa, romlebic ar iyofa 

5-ze~. 

• magaliTad, ase: `5-is jeradi ricxvebi~. aRniSvnis SerCeva naSTis (0-is) mixedviT 

SeiZleba — A 0 . 

• 

A0 A1 A2 A3 A4 N 

2. wina amocanis analogiurad, n ricxvisTvis Sedgeba sim rav leebi 

A 1 ={nk+1| k∈Z, 0≤k}, ..., A n–1 ={n(k+1)–1| k∈Z, 0≤k}.

mizani 

§ 1.10. dadebiTi racionaluri ricxvebi 

• dadebiTi racionaluri ricxvebis warmosadgenad da Tvisebebis Sesaswavlad 

poziciuri sistemis gamoyenebis unaris ganviTareba 

• dadebiTi racionaluri ricxvebis gamoyenebebis Sesaxeb codnis gaRrmaveba da 

gafarToeba 

• sxvadasxva formiT (procentiT, naturaluri ricxvebis SefardebiT, aTwiladebiT) 

dadebiTi racionaluri ricxvebis Cawerisa da gamoyenebis unaris ganviTareba. 

warmodgenili masala nawilobriv nacnobia moswavleebisTvis. amitom gadmocemisas 

moswavleTa aqtiuri monawileoba advilad miiRweva. 

erovnuli saswavlo gegmis mixedviT arauaryofiTi racionaluri ricxvebis Seswavlis 

Semdeg aTwiladebis swavleba mimdinareobas. Tumca, amas win uswrebs (ukve 

me-4 klaSi) propedevtikuli muSaoba wiladis Sesaxeb (Semodis naxevari, mesamedi, 

meoTxedi). 

Semdeg wlebSi konkretuli modelebis (wris, kvadratis marTkuTxedis) gamoyenebiT 

Semodis `wiladi~, `mniSvneli~, `mricxveli~, iswavleba wiladebis Sedareba, ricxvis 

nawilis povna, erTobliobis nawilis povna. 

swavlebis forma induqciuria _ xSiria TvalsaCino warmodgenebis, moswavleTa 

praqtikuli gamocdilebis gamoyeneba. amasTanave, wiladi ricxvebis swavlebis paralelurad 

mimdinareobs aTwiladebis swavlebis procesi, romelia poziciuri warmodgenis 

gaazreba naturalur ricxvebze Sesabamisi warmodgenebis gamoyenebiT advildeba. 

maTi praqtikuli gamoyenebebis sferoc gacilebiT farToa, ariTmetikuli moqmedebebis 

Sesruleba poziciuri sistemis gamoyenebiT ufro advilad Seiswavleba. 

aTwiladebze moqmedebebis Tvisebas eZleva induqciuri dafuZneba. 

paragrafSi warmodgenili masalis gadmocemisas mimdinareobs im codnis 

sistematizacia, romelic ukve aqvs me-7 klasels. upirveles yovlisa, es aris zomis 

metruli sistemis gameoreba, aTwiladebi da Cveulebrivi wiladebi. gansakuTrebiT 

mniSvnelovania am codnis praqtikuli amocanebis amoxsnebTan dakavSireba. amitom 

maswavleblebs vurCevT saswavlo procesSi gamoiyenon is amocanebi, magaliTad, #26, 

#27, romlebic dakavSirebulia sidideebs Soris kavSirebze _ metrebidan santimetrebze 

gadasvla, wamebidan _ saaTebze, kilogramebidan _ centnerebze da tonebze 

da a. S. amocanebi, romlebSic xdeba gadasvla m/wm-dan km/sT-ze da a. S. 

ukve SeiZleba codnis gaRrmavebis mimarTulebiT mivaRwioT saTanrigo erTeulebis 

gamoyenebiT ricxvebis Caweris unaris ganviTarebas. 

aTwiladebis gamoyenebebis istoriuli aspeqtebis win wamowevis mizniT, SesaZlebelia 

maswavlebelma Caataros saubari: 

`SevadaroT erTi da imave ricxvis ori Canaweri: 

2 

43 

da 2,43. 

100 

pirveli Seicavs 7 niSans, meore _ 4-s. meore Canaweri ufro èkonomiuria~. XV 

saukuneSi aTwiladebis warmoSoba maTematikaSi aRmoCenad aris miCneuli. 

SemdgomSi zomis aTobiTi sistemebis SemoRebisas am wiladebma didi gamoyeneba 

hpova~. 

paragrafSi gTavazobT mravalferovani Tematikis amocanebs: proporcia _ # #9, 

# #20, #21, wiladebze moqmedebebi _ @#10, #11, #12, #13, #17, #18, #19, poziciuri 

63

sistemaA _ #5, #6, #7, #8, Pprocentebi _ #1, #22, #23, sidideebs ASoris kavSirebi _ 

#26, #27. es saSualebas mogcemT zogierTi moswavlis SesaZlo CamorCenis SemTxvevaSi 

masalis gameorebas meti praqtikuli datvirTva misceT. gansakuTrebuli yuradRebiT 

moekideT #16-s _ es kombinirebuli tipis amocanaa, saWiroa terminebis gaxseneba, 

SesaZlo mniSvnelobebis simravlis dadgena da analizi. 


64 

1 2 3 4 5 6 7 8 

2 2 3 1 2 1 3 3 

14. gaviTvaliswinoT, rom ori naturaluri ricxvis Se far debis saxiT Cawerili 

ricxvi Caiwereba sasruli aT wi ladis saxiT mxolod im SemTxvevaSi, Tu es wiladi 

ukvecia da misi mniSvnelis gamyofi ar aris 2-isa da 5-isgan gansx va vebuli raime 

martivi ricxvi. amis gaTvaliswinebiT, usas ru lo aTwiladiT gamoisaxeba: 20 ; 

1 

; 

9 12 

sasruli aTwiladiT — 7 ; 

45 

; 59 ; 

8 

. 

20 12 32 125 

16. radgan n wesieri wiladia, n∈{1; 2; 3; 4}. amitom 

5 

a) n 5

mizani 


• maTematikis saskolo kursis cnebebis gamoyenebis unaris ganviTareba 

• maTematikis gamoyenebiT optimaluri variantis SerCevis unaris ganviTareba. 

1. sanimuSod warmogidgenT dagegmvis erT-erT variants: vTqvaT, avtobusis saSualo 

siCqarea 50 km/sT, maSin gzaSi yof nis droa sT, Sesabamisi gadasaxdeli Tanxa — ·25=37 

(lari), darCenili Tanxa — 123,2–37=85,2 lari rom gavyoT 25-ze, miviRebT Segrovili 

TanxiT iyalToSi yofnisTvis gan kuTvnil udides dasaSveb dros (saaTebSi). 

2. • 850-iT. 

• 500-iT. 

• niniko — 350:4=87,5; 

mariami — –300:4= –75; 

kote — –500:4= –125; 

giorgi — 200:4=50. 

• magaliTad, niniko — 1000+250-750-150 

mariami — 1000-400-750-150 

kote — 250-250+250-750 

giorgi — 250-150+250-150 

• 700-quliani an 1000-quliani. ajobebs 700-qulianis SekveTa, radgan mosalodnelia, 

rom is ufro martivi iqneba, vidre 1000 quliani. Tumca, Tavad ninikoc, 

SesaZloa, kidev moipovebs qulebs, amitom 1000 quliani SekiTxvis SekveTis riskic 

gamarTlebulia. 

65

mizani 

66 

§ 1.11. racionaluri ricxvebi 

• dadebiTi da uaryofiTi racionaluri ricxvebis simravleebis warmodgena; maTze 

moqmedebebis aRwera 

• dadebiTi da uaryofiTi racionaluri ricxvebis praqtikuli gamoyenebebis unaris 

ganviTareba. 

es paragrafi ajamebs ricxvis cnebis gafarToebis or sqemas, romelic saxelmZRvaneloSia 

warmodgenili. faqtobrivad, ricxvis cnebis gafarToebis algebruli Tanamimdevrobaa 

warmodgenili. 

racionalur ricxvTa Q simravle Sedgeba dadebiTi racionaluri ricxvebisgan, 

uaryofiTi racionaluri ricxvebisgan da nulisgan. N⊂Z⊂Q mimarTebis venis diagramebiT 

gamosaxva aZlierebs am simravleTa aRqmas. ganvsazRvreT racionaluri 

ricxvis modulic: 

|a|=a, Tu a>0; 

|a|=–a, Tu a

14. gavixsenoT: nulisgan gansxvavebuli nebismieri a ricx vis Tvis a-s Sebrunebuli 

ricxvia a 1 . 

23. gaiTvaliswineT, rom uaryofiTi ricxvis mopirdapire ricxvi dadebiTia. 

24. nebismieri racionaluri ricxvis moduli dadebiTia an nulia. 

29. a) N∩A=∅; b) Z∩A={0; –1}; g) Q∩A=A. 

39. vTqvaT, pirvel yuTSi kanfetis masaa x kg, meoreSi — x–72 (kg). 

a) unda gadavdoT 7,2 kg-is naxevari — 3,6 (kg). marTlac, maSin I-Si darCeba x–3,6, 

II-Si — (x–7,2)+3,6=x–3,6 (kg); 

b) unda gadavdoT (10–7,2)-is naxevari — 1,4 (kg). marT lac, maSin II-Si darCeba: 

x–8,6 (kg), I-Si aRmoCndeba — x+1,4=(x–8,6)+10; 

g) unda gadavdoT (7,2-4,8)-is naxevari — 1,2 (kg); marT lac, maSin II-Si aRmoCndeba 

x–6, I-Si ki darCeba — 

x–1,2=(x–6)+4,8 (kg). 

43. ajobebs, Tu monacemebs ricxviT wrfeze moniSnavT (sqe ma turad) da dafaze am 

wrfes vertikalurad ga mosaxavT. 

niu 

a) 8848–(–400)=9248; b) 8848–(–1,5)=8849,5; 

g) 8848–(–27,9)=8875,9. 

67

§ 1.12. racionaluri ricxvis naturalurmaCvenebliani xarisxi 

mizani 

• naturalurmaCvenebliani xarisxis Tvisebebis gamoyenebis unaris ganviTareba 

• ricxvis Cawerisas xarisxis praqtikuli gamoyenebis Cvevebis daufleba. 

gavixseneT ricxvis xarisxis saxiT Caweris wesi: a ricxvis n-uri xarisxi (n∈N, n>1) 

aris n-cali Tanamamravlis namravli, romelTagan TiToeuli aris a _ 

a⋅a⋅...⋅a=a n , n>1; a 1 =a. 

68 

n-jer 

xarisxis cnebis Seswavlas konkretuli ricxviTi magaliTebis ganxilviT viwyebT. 

am masalis propedevtikas adrec sakmao dro daeTmo _ ganixileboda ricxvis kvadrati, 

naturaluri ricxvis xarisxi, `giganturi~ ricxvebis xarisxis saxiT warmodgenebi. 

amitom, axal masalas, romelic nebismieri racionaluri ricxvis naturalurmaCvenebliani 

xarisxis aRweras eZRvneba, moswavleebi advilad euflebian. 

paragrafs SeiZleba 2 saaTi davuTmoT. 

#1 – #9 amocanebiT SeiZleba moswavleTaA faqtobrivi codnis Semowmeba, #11 

da # #12 _ erTgvari ganmazogadebeli amocanebia; saintereso amocanaa # 15 _ 

moswavleebs _ kidev erTxel gaamaxvilebineT yuradReba faqtze: Tu a>b>0 da n∈N, 

maSin a n >b n ; amitom 2 6 >6 2 , radgan 2 6 =8 2 . 


1 2 3 4 5 6 7 8 9 11 12 13 14 

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

15. 2 6 =2·2·2·2·2·2=8 2 >6 2 . 

22. 2·2·2·2·2=2 5 =32. Sesabamisad, sisqe iqneba 32·0,1=3,2 (mm). 

2·2·...·2=2 5 ·2 5 ·2 5 =32 3 . Sesabamisad, sisqe iqneba 

15-jer 

32 3 ·0,1=32768·0,1=3276,8 (mm). 

23. mexuTe _ 1–( 2 1 ) 5 

, meaTe _ 1–( 2 1 ) 10 

, mease _ 1–( 2 1 ) 100 

. 

sakontrolo wera #2 

(sakontrolo weris variantebis nimuSebi) 

I varianti 

1. ricxviT RerZze A(5) wertilidan 7 erTeuliT daSo re buli wertilebis Se sabamisi 

koordinatebia: 

1) 2 da 12 2) –2 da –9 

3) 12 da 19 4) –2 da 12.

2. muxtebis modeliT –3 SeiZleba ase gamoisaxos: 

1) 2) 3) 4) 

3. uaryofiTi ricxvebis jamis uaryofiT ricxvze nam rav lis moduli 

1) uaryofiTia 2) dadebiTia 3) nulia 4) dadebiTi ar aris. 

4. or faraSi 100 cxvaria. amasTanave, meoreSi 12 cxv riT metia, vidre pirvelSi. 

maSin pirvel faraSi cxvrebis odenoba Semdegi gamosaxulebis mniSvnelobis tolia: 

1) 100:2–12 2) (100–12):2 3) (100+12):2 4) 100:2+12. 

5. Tu a aris –56:8+(–14) gamosaxulebis mniSvneloba, maSin 

1) a∈N 2) a∉Z 3) a=Z 4) a∈Z. 

6. umciresi elementi, romelic (N 5 ∩Z)-simravles ekuTv nis, aris 

1) 1 2) 5 3) 0 4) –1. 

7. SeasruleT moqmedebebi: 

(–480):((–2)·15)+(–2)·(–7)+(1,3·(–3)) 2 . 

8. cnobilia, rom n∈N, 4 ar aris wesieri, n + 2 we si eri wiladia. ipoveT n-is 

n 6 

yvela SesaZlo mniSvnelobaTa sim rav le. 

II varianti 

1. ricxviT RerZze A(–5) wertilidan 6 erTeuliT da So rebuli wertilebis Se sabamisi 

koordinatebia: 

1) 1 da 7 2) –17 da –11 3) –11 da 1 4) –1 da 11. 

2. muxtebis modeliT –4 SeiZleba ase gamoisaxos: 

1) 2) 3) 4) 

3. uaryofiTi ricxvebis jamis dadebiT ricxvze namravlis moduli 

1) uaryofiTia 2) dadebiTia 3) nulia 4) dadebiTi ar aris. 

4. or oTaxSi 59 skamia. amasTanave, pirvel oTaxSi 13 ska miT metia, vidre meoreSi. 

maSin meore oTaxSi skamebis odenoba Semdegi gamosaxulebis mniSvnelobis tolia: 

1) (59–13)·2 2) (59–13):2 3) 59–13·2 4) 59·2–13. 

5. Tu b aris 49:(–7)–(–12) gamosaxulebis mniSvneloba, maSin 

1) b∉Z 2) b∉N 3) b=N 4) b∈N. 

6. udidesi elementi, romelic (N 7 ∩Z)-simravles ekuTv nis, aris 

1) 1 2) 7 3) –1 4) –7. 

7. SeasruleT moqmedebebi: 

(–768):((–4)·27+(–24)·(–5))–(4,3·(–2)) 2 . 

8. cnobilia, rom p∈N, 6 p 

ar aris wesieri wiladi, 

+ 3 

wesieri wiladia. ipoveT 

p 8 

p-s yvela SesaZlo mniSv ne lo baTa simravle. 

69

* * * 

Sefasebis sqema: yoveli variantis pirveli eqvsi amoca ni dan TiToeuli SeiZleba 

Sefasdes 1 quliT; me-7 da me-8 amocanebidan TiToeuli – 2 quliT, Tumca am or amocanaSi 

SeiZleba gamoviyenoT nawilobrivi Sefasebac _ davweroT 0,5; 1; 1,5 qulebi. 

magaliTad, I variantis me-8 amocanaSi, Tu moswavlem sworad gaaanaliza is faqti, 

rom erTi wiladi aris wesieri (an meore ar aris wesieri) da miiRo n-is saTanado 

Sefaseba, weiZleba dauweros 1 qla. sazogadod, Tu moswavle poulobs amoxsnis gzas, 

iwyebs amoxsnas, magram ver asrulebs, maSin mas SeiZleba mTeli Sefasebis 50% davu- 

weroT (aqve gasaTvaliswinebelia daSvebuli Secdomebis `simZime~, rac unda aisaxos 

am SefasebaSi). Tu gaiTvaliswina mxolod is faqti, rom n 4 ar aris wesieri wiladi 

da miiRo n-isTvis arazusti Sefaseba n

gTavazobT sakontrolo weris variantebis SemoTavazebuli nimuSebis zogierTi 

amocanis amoxsnas: 

varianti I 

7. (–480):((–2)·15)+(–2)·(–7)+(1,3·(–3)) 2 =-480:(–30)+14+(–3,9) 2 =16+14+15,21=45,21. 

8. radgan 4 ar aris wesieri, amitom n≤4; n + 2 — wesi e ria, amitom n+2

64. 

72 

1 0 2 9 

3 5 

5 1 4 5 

3 0 8 7 

3 6 0 1 5 

65. 125:25=5; 375:75=5; 250:50=5 _ sul sami aseTi wyvilia. 

66. manJili –4-isa da 4-is Sesabamis wertilebs Soris 8 erTeulia. 

68. visac SenobaSi ar uSveben, mas ara aqvs saSvi _ winaaRmdeg SemTxvevaSi, anu 

saSvi rom hqonoda, mas SenobaSi SeuSvebdnen. 

69. 3 . 4=12 _ 12 sxvadasxva variantia. 

70. 

yv. mwv. cisf. wT. 

cisf. wT. mwv. yv. 

mwv. yv. wT. cisf. 

wT. cisf. yv. mwv. 

71. 12 saaTi da 40 wuTi. 

73. a=3–8=–5 

b=3+8=11 

74. a) orjer; 

b) lurjebi _ 3 1 nawils, TeTrebi _ 3 2 nawils. 

g) lurjebi _ 24, TeTrebi _ 48. 

75. A _ martiv ricxvTa simravlea, B _ Sedgenil ricxvTa simravle; 1∈N da 1∉A, 

1∉B. 

76. SeiZleba ekuTvnodes, SeiZleba arc ekuTvnides: 

M 

P 

M P 

b b 

77. asoebi periodulad meordeba, yoveli xuTi asos Semdeg. 116-e adgilze iqneba 

à~. adgilis rigiTi nomris 5-ze gayofisas miRebuli naSTis mixedviT vipoviT asos 

pirvel xuTeulSi. 

78. saxlsa da samsaxurs Soris manZils daTo avtobusiT saaTis meoTxedSi, anu 15 

weTSi gadis, xolo fexiT _ 1 4 1 saaTSi. Sesabamisad, samsaxurSi misvlasa da ukan dab- 

runebas fexiT is 2,5 saaTSi moaxerxebs.

maTematika VII - Ganatleba

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