maTematika - ncp.ge wlamde/mathematika.pdf2 maTematika es mimarTulebebi mWidro urTierTkavSirSia da moicavs im codnas da unar-Cvevebs, romelsac moswavle unda daeuflos zogadsaganmanaTleblo

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Sesavali

(zogadsaganmanaTlebo skolaSi maTematikis swavlebis koncefcia)

saskolo maTematikuri ganaTlebis roli da miznebi

Tanamedrove epoqaSi maTematika farTod gamoiyeneba adamianis saqmianobis yvela sferoSi: mecnierebasa da teqnologiebSi, medicinaSi, ekonomikaSi, garemos dacvasa da aRdgena-keTilmowyobaSi, socialur gadawyvetilebaTa miRebaSi. aRsaniSnavia maTematikis gansakuTrebuli roli kacobriobis ganviTarebaSi da Tanamedrove civilizaciis CamoyalibebaSi. sainformacio da gamoTvliTi teqnologiebis ganviTareba, sivrce-drois struqturis ukeT gaazreba, bunebaSi arsebuli mravali kanonzomierebis aRmoCena da aRwera naTlad warmoaCens maTematikis samecniero da kulturul Rirebulebas. rac gansakuTrebiT mniSvnelovania, maTematika xels uwyobs adamianis gonebrivi SesaZleblobebis ganviTarebas. igi iZleva efeqtiani, lakoniuri da araorazrovani komunikaciis saSualebas. maTematikis gamoyenebiT SesaZlebelia rTuli situaciis TvalsaCinod warmoCena, movlenebis axsna da maTi Sedegebis ganWvreta. maTematikaSi Seqmnili abstraqtuli sistemebi da Teoriuli modelebi gamoiyeneba kanonzomierebebis Sesaswavlad, situaciis gasaanalizeblad da problemebis gadasaWrelad.

problemis gadaWrisas aucilebelia mis arsSi wvdoma, adekvaturi maTematikuri aparatis SerCeva, xolo aseTis ar arsebobis SemTxvevaSi - misi SemuSaveba; Sesaswavli procesisa Tu obieqtis gaazrebuli modelis Seqmna, miRebuli modelis saSualebiT saWiro daskvnebis gakeTeba da Semdeg maTi gamoyenebiTi interpretacia. praqtikuli Tu samecniero problemebi, Tavis mxriv maTematikas amaragebs mniSvnelovani da saintereso amocanebiT. Sesabamisad, swavlebisas ZiriTadi yuradReba unda mieqces maTematikuri meTodebis gamoyenebas garemomcveli samyaros Semecnebisas, socialur-ekonomikuri Tu teqnikuri procesebis marTvisas, sayofacxovrebo Tu mecnieruli problemebis gadaWrisas da maTematikuri codnis, rogorc logikurad gamarTuli sistemis Camoyalibebas da gadacemas. garda amisa, maTematikis swavlebisas, ZiriTadi fokusis gadatana (rogorc praqtikuli, aseve mecnieruli xasiaTis) problemebis gadaWraze, aZlierebs moswavleTa enTuziazms da aRZravs interess maTematikisadmi.

aqedan gamomdinare, zogadsaganmanaTleblo skolaSi maTematikis swavlebis miznebia moswavleTaTvis:

•� azrovnebis unaris ganviTareba;•� deduqciuri da induqciuri msjelobis, SexedulebaTa dasabuTebis, movlenebisa

da faqtebis analizis unaris ganviTareba;•� maTematikis, rogorc samyaros aRwerisa da mecnierebis universaluri enis

aTviseba;•� maTematikis, rogorc zogadsakacobrio kulturis Semadgeneli nawilis

gacnobiereba;•� swavlis Semdgomi etapisaTvis an profesiuli saqmianobisaTvis momzadeba.•� cxovrebiseuli amocanebis gadasawyvetad saWiro codnis gadacema da am codnis

gamoyenebis unaris ganviTareba.

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ZiriTadi unar-Cvevebi, romelTa gamomuSavebasac xels uwyobs maTematikis saskolo kursi

maTematikis codna niSnavs maTematikuri cnebebisa da procedurebis flobas, maTi gamoyenebis unars realuri problemebis gadaWrisas; agreTve komunikaciis im saSu-alebebis flobas, romlebic saWiroa informaciis misaRebad da gadasacemad maTema-tikuri enisa da saSualebebis gamoyenebiT.

is ZiriTadi unar-Cvevebi, romelTa Camoyalibebasac emsaxureba problemebis gadaWraze orientirebuli maTematikuri ganaTleba, aseTia:

msjeloba-dasabuTeba• varaudis gamoTqma da misi kvleva kerZo magaliTebze• sawyisi monacemebis gadarCeva da organizeba (maT Soris aqsiomebis, ukve cno-

bili faqtebis), arsebiTi Tvisebebisa da monacemebis gamoyofa• damtkicebis, dasabuTebis meTodis SerCeva (mag. sawinaaRmdegos daSvebis meTo-

dis gamoyeneba, evristikuli meTodis gamoyeneba dasabuTebisas)•� sxvadasxva tipis gamonaTqvamis adekvaturi gamoyeneba; magaliTad: pirobiTi

(`Tu ... maSin~) da raodenobrivi xasiaTis, daSvebis, gansazRvrebis, Teoremis, hipoTezis, SemTxvevaTa CamonaTvalis

•� arCeuli strategiis vargisianobisa da misi gamoyenebis sazRvrebis ganxilva•� msjelobis xazis ganviTareba, alternatiuli gzis moZebna, miRebuli gadaw-

yvetilebis sisworisa da efeqtianobis dasabuTeba; ganzogadoebiT an deduqciT miRebuli daskvnebis axsna da dasabuTeba

•� Teoremebis-debulebebis daskvnis analizi erTi an ramdenime pirobis, SezRud-vis Sesuteba-moxniT

•� gamonaklisi SemTxvevebis aRniSvna da maTi ganzogadoebis aramarTebulobis dasabuTeba kontrmagaliTis moZebniT

komunikacia•� terminologiis, maTematikuri aRniSvnebisa da simboloebis koreqtuli gamoy-

eneba•� informaciis warmodgenis xerxebisa da meTodebis floba, gamoyeneba; sxvadasxva

gziT warmodgenili informaciis interpretacia, masze msjeloba, erTmaneTTan dakavSireba

•� sxvisi naazrevis gageba da gaanalizeba•� informaciis miRebisa da gadacemis Sesaferisi saSualebebis SerCeva audito-

riisa da sakiTxis gaTvaliswinebiT•� informaciis gadacemisas sakiTxis arsis (mag. obieqtis arsebiTi Tvisebebis)

warmoCena

gamoyeneba, modelireba•� figurebis da obieqtebis zomebis, agreTve maT Soris manZilebis, masis, temper-

aturis da drois gasazomad gzebisa da meTodebis povna da gamoyeneba; procesis an realuri viTarebis modelirebisaTvis saWiro monacemebis SerCeva da mop-oveba

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•� Cveul garemoSi (yoveldRiur cxovrebaSi) maTematikuri obieqtebisa da procesebis SemCneva da maTi Tvisebebis gamoyeneba modelis agebisas, praqtikuli (yofiTi) amocanebis gadaWrisas

•� mocemuli modelis elementebis interpretireba, im realobis konteqstSi, romelsac igi aRwers da piriqiT – realuri viTarebis dakvirvebis Sedegad miRebuli monacemebis interpretireba Sesabamisi modelis enaze

•� mocemuli modelis gaanalizeba da Sefaseba, kerZod, misi moqmedebis arealisa da modelis adekvaturobis dadgena; SesaZlo alternativebis ganxilva da Sedareba

problemebis gadaWra •� amocanis Sinaarsis aRqma, amocanis monacemebisa da saZiebeli sidideebis

gaazreba-gamijvna•� problemis gansazRvra da misi Camoyalibeba, maT Soris arastandartul

viTarebaSi (mag. rodesac problemis gadasaWrelad saWiro maTematikuri procedura calsaxad araa gansazRvruli)

•� kopleqsuri (rTuli) problemebis safexurebad, martiv amocanebad dayofa da etapobrivad gadaWra (amoxsna), maT Soris standartuli midgomebisa da procedurebis gamoyenebiT

•� problemis gadasaWrelad saWiro strategiebisa da resursebis SerCeva, maTi gamoyeneba da efeqtianobis monitoringi

•� ukve cnobili faqtebisa da strategiebis SerCeva da erTmaTeTTan dakavSireba maRali sirTulis problemebis gadasaWrelad

•� miRebuli Sedegis kritikuli Sefaseba konteqstis gaTvaliswinebiT da zRvruli SemTxvevebis kvleva

•� problemis gadaWrisas adekvaturi damxmare teqnikuri saSualebebisa da teqnologiebis SerCeva da maTi gamoyeneba

damokidebuleba•� TanamSromloba jgufuri samuSaoebis Sesrulebisas; koreqtuloba

maswavlebelTan da megobrebTan mimarTebaSi•� samuSaos organizebisa da dagegmvis xerxebisa da meTodebis floba•� maTematikis adgilisa da mniSvnelobis Sefaseba sxvadasxva disciplinebSi,

biznesSi, xelovnebaSi da adamianis moRvaweobis sxvadasxva sferoebSi•� informaciuli teqnologiebis gamoyenebisas eTikur/socialuri xasiaTis

problemebis gacnobiereba da eTikuri normebis dacva.

maTematikis erovnuli saswavlo gegmis struqtura

CamoTvlili unar-Cvevebis Camoyalibeba da ganviTareba SesaZlebelia mxolod Sesabamisi Sinaarsis (cnebebis, debulebebisa da procedurebis) gamoyenebiT.

erovnuli saswavlo gegma maTematikaSi pirobiTad dayofilia oTx ZiriTad mimarTulebad: ricxvebi da moqmedebebi; geometria da sivrcis aRqma; monacemTa analizi, statistika da albaToba; kanonzomierebebi da algebra.

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es mimarTulebebi mWidro urTierTkavSirSia da moicavs im codnas da unar-Cvevebs, romelsac moswavle unda daeuflos zogadsaganmanaTleblo skolaSi swavlis ganmavlobaSi. saswavlo gegmis dayofa mimarTulebebad ar niSnavs kursis analogiur dayofas, igi mxolod warmoaCens Sesaswavli masalis speqtrs da saSualebas iZleva mieTiTos, Tu raze unda gamaxvildes meti yuradReba swavlebis ama Tu im safexurze.

1. ricxvebi da moqmedebebi:

•� ricxvebi, maTi gamoyenebebi da ricxvis warmodgenis saSualebebi•� moqmedebebi ricxvebze da ricxviTi Tanafardobebi•� raodenobaTa Sefaseba da miaxloeba•� sidideebi, zomis erTeulebi da ricxvebis sxva gamoyenebebi

2. geometria da sivrcis aRqma:

•� geometriuli obieqtebi: maTi Tvisebebi, urTierTmimarTeba da konstruireba•� zoma da gazomvis saSualebebi•� gardaqmnebi da figuraTa simetriuloba•� koordinatebi da maTi gamoyeneba geometriaSi

3. monacemTa analizi, albaToba da statistika:

•� monacemTa wyaroebi da monacemTa mopovebis saSualebebi•� monacemTa mowesrigebis xerxebi da monacemTa warmodgenis saSualebebi•� monacemTa Semajamebeli ricxviTi maxasiaTeblebi• albaTuri modelebi•� SerCeviTi meTodi da SerCevis ricxviTi maxasiaTeblebi

4. kanonzomierebebi da algebra:

•� simravleebi, asaxvebi, funqciebi da maTi gamoyeneba•� diskretuli maTematikis elementebi da maTi gamoyeneba•� algoriTmebi da rekursia•� algebruli operaciebi da maTi Tvisebebi

mimarTulebebis miznebi swavlebis safexurebis mixedviT

saswavlo kursi zogadsaganmanaTleblo skolaSi dayofilia sam safexurad: dawyebiTi skola (I – VI klasebi), sabazo skola (VII – IX klasebi) da saSualo skola (X – XII klasebi). maTematikis saswavlo kursis agebis principi unda iTvaliswinebdes am dayofas da TiToeul safexurze maTematikis swavlebas unda hqondes mkafiod Camoyalibebuli miznebi.

ricxvebi da moqmedebebiam mimarTulebis ZiriTadi miznebia `ricxvis SegrZnebis~ ganviTareba, Tvlis

principebis aTviseba, ariTmetikuli moqmedebebisa da maTi Tvisebebis Seswavla, gamoTvlis xerxebis aTviseba da Sedegebis Sefaseba; Caweris poziciuri sistemebis Seswavla, maTi urTierTSedareba da gamoyeneba ariTmetikuli moqmedebebis

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Sesrulebisas da praqtikuli amocanebis gadaWrisas; ricxviTi sistemebis Seswavla.

dawyebiTi skola. am safexurze unda moxdes ricxvebTan dakavSirebuli problemebis gadaWrisas ariTmetikuli moqmedebebis da maTi adekvaturad gamoyenebis unaris Camoyalibeba; ariTmetikuli moqmedebebis Tvisebebisa da maT Soris kavSirebis danaxva; ariTmetikuli moqmedebebis Sedegisa da ricxviTi gamosaxulebis mniSvnelobis Sefasebis unaris ganviTareba.

garda amisa, moswavles unda Camouyalibdes aTobiTi poziciuri sistemis srulyofili gageba da mravalniSna ricxvebze moqmedebebis Sesrulebisas misi gamoyenebis unari; wiladis sxvadasxva aspeqtis (rogorc mTelis nawili, erTobliobis nawili, pozicia ricxviT RerZze da gayofis Sedegi) gageba.

sabazo skola. am safexurze moswavlem unda gaiRrmavos Tavisi codna mTel ricxvebTan, wiladebTan, aTwiladebTan da procentebTan dakavSirebiT ise, rom safexuris dasrulebis Semdeg iyenebdes wiladebis ekvivalentobas, aTwiladebs, proporcias da procentebs amocanebis amoxsnisas da realur viTarebaSi. ricxvis cnebis gageba unda gafarTovdes racionalur ricxvebamde. mas unda SeeZlos ricxviT RerZze racionaluri ricxvis miaxloebiTi poziciis miTiTeba da unda Seeqmnas sawyisi warmodgenebi iracionalur ricxvebze.

saSualo skola. ricxvebze ariTmetikuli moqmedebebis Sesrulebis unari da maTi Tvisebebis codna/gamoyeneba unda gaxdes algebruli struqturebisa da kanonzomierebebis ukeT gaazrebis safuZveli. am safexurze, moswavles unda SeeZlos rogorc ricxviTi sistemis, aseve ariTmetikuli operaciis cnebis gafarToeba, mag. veqtorebsa da matricebze. garda amisa, unda moxdes mTel ricxvTa sistemis ufro Rrmad Seswavla ricxvTa Teoriis elementebis gamoyenebiT.

kanonzomierebebi da algebra

am mimarTulebis ZiriTadi mizania moswavles Camouyalibdes kanonzomierebebis, algebruli mimarTebebisa da funqciuri damokidebulebebis amocnobis; agreTve, maTi saSualebiT movlenebis modelirebisa da problemebis gadaWris unari.

dawyebiTi skola. am safexurze mimarTulebis mizania martivi kanonzomierebebisa da sidideebs Soris damokidebulebis amocnobis unaris ganviTareba, ariTmetikuli operaciebis Tvisebebis da asoiTi aRniSvnebis gamoyenebis Seswavla.

sabazo skola. am safexurze mimarTulebis mizania sidideebs Soris damokidebulebebTan dakavSirebuli cnebebisa da procedurebis Seswavla, agreTve maTi gamosaxvis sxvadasxva xerxis erTmaneTTan dakavSirebisa da Sedarebis unaris ganviTareba; problemis gadaWrisas asoiTi gamosaxulebis gamoyenebis, maT Soris gantolebis Sedgenisa da amoxsnis unaris ganviTareba; sawyisi warmodgenebis Seqmna simravlur cnebebsa da operaciebze.

saSualo skola. am safexuris mizania funqciaTa ojaxebis, maTi Sedarebisa kvlevis meTodebis Seswavla; sxvadasxva konteqstSi arsebuli damokidebulebis gamosaxvisas iteraciuli da rekurentuli formebis gamoyenebis unaris ganviTareba; struqturis aRwerisa da Seswavlisas diskretuli maTematikis aparatis gamoyenebis unaris ganviTareba.

geometria da sivrcis aRqma

am mimarTulebis ZiriTadi mizania geometriuli obieqtebisa da maTi Tvisebebis, gazomvebis, geometriuli gardaqmnebisa da geometriaSi algebruli meTodebis gamoyenebis Seswavla.

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dawyebiTi skola. am safexurze mimarTulebis mizania geometriuli obieqtebis urTierTganlagebis aRwerisa da demonstrirebis, geometriul obieqtTa komponentebis amocnobisa da maTi urTierTmimarTebis aRweris, atributebis mixedviT figuraTa dajgufebis, sityvieri aRwerilobis mixedviT figuris amocnobisa da misi modelis Seqmnis unaris ganviTareba.

sabazo skola. am safexurze mimarTulebis mizania geometriul obieqtTa Seswavlisas, geometriul obieqtTa Soris mimarTebebis dadgenisas da geometriul obieqtTa klasifikaciisas, gazomvis, Sedarebisa da geometriuli gardaqmnebis gamoyenebis unaris ganviTareba. garemoSi orientirebisas koordinatebis gamoyenebis da arapirdapiri gziT obieqtTa zomebis dadgenis Seswavla; induqciuri/deduqciuri msjelobisa da varaudis gamoTqma-Semowmebis unaris ganviTareba.

saSualo skola. aRniSnul safexurze unda moxdes deduqciuri/induqciuri msjelobisa da geometriuli kvlevis Sedegebis ganzogadebis unaris ganmtkiceba. koordinatebis, trigonometriis da gardaqmnebis gamoyeneba praqtikuli geometriuli problemebis gadasaWrelad da am xerxebidan yvelaze efeqtianis SerCevis unaris ganviTareba.

monacemTa analizi, albaToba da statistika

zogadsaganmanaTleblo skolaSi statistikuri cnebebisa da aparatis Semotanis mizania moswavleTa intuiciuri warmodgenebis mowesrigeba, struqturebad Camoyalibeba da maTi albaTur-statistikuri intuiciisa da azrovnebis ganviTareba.

dawyebiTi skola. am safexurze mimarTulebis swavlebis mizania moswavleebi gaecnon aRweriTi statistikis elementebs – Tvisebriv da diskretul raodenobriv monacemTa Segrovebis, mowesrigebis, warmodgenisa da interpretaciis saSualebebs.

sabazo skola. am safexurze mimarTulebis swavlebis mizania moswavleebi daeuflon aRweriTi statistikis ZiriTad cnebebsa da meTodebs, raTa maTi saSualebiT gaerkvnen monacemTa TaviseburebebSi da SeZlon varaudis gamoTqma monacemebze dayrdnobiT. garda amisa, swavlebis mizania moswavleebi gaecnon albaTobis Teoriis sawyisebs da gaacnobieron gansxvaveba deterministul da SemTxveviTobis Semcvel viTarebebs Soris.

saSualo skola. am safexurze mimarTulebis swavlebis mizania moswavleebs SeeqmnaT sistematizebuli warmodgenebi albaTobis Teoriisa da statistikis Sesaxeb, raTa gaakeTon an/da Seafason daskvnebi gaurkvevlobis Semcvel viTarebaSi, amoicnon SemTxveviTobis roli ama Tu im wamowyebaSi da moaxdinon misi kvantifikacia gadawyvetilebis misaRebad.

maTematikis saskolo kursis organizacia

zogadsaganmanaTleblo skolaSi swavlis savaldebuli kursi moicavs pirvel cxra klass, xolo mecxre klasis Semdeg moswavleTa nawilma SeiZleba aRar gaagrZelos swavla zogadsaganmanaTleblo skolaSi. zogadsaganmanaTleblo skolis yovel safexurze maTematika Seiswavleba rogorc savaldebulo sagani. meaTe klasSi moswavleebi miiReben iseT ganaTlebas romelic maT xels Seuwyobs ukeT gaerkvnen TavianT midrekilebebsa da interesebSi.

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erovnuli saswavlo gegmis daniSnuleba da misi struqtura

erovnuli saswavlo gegmis daniSnulebaa daexmaros saskolo ganaTlebis procesis monawileebs (maswavlebelebs, moswavleTa mSoblebs, saxelmZRvaneloebis avtorebs da ganaTlebis procesis marTvis specialistebs) am procesis dagegmvasa da warmarTvaSi.

erovnul saswavlo gegmaSi aRwerilia is savaldebulo moTxovnebi, romelsac unda akmayofilebdes yoveli moswavle TiToeuli klasis dasrulebis Semdeg. es moTxovnebi, TiToeuli mimarTulebisaTvis, Camoyalibebulia Sedegebisa da maTi indikatorebis enaze.

Sedegi aris debuleba imis Sesaxeb, Tu ra unda SeeZlos moswavles swavlis mocemuli safexuris dasrulebis Semdeg.

indikatori aris debuleba im codnisa da unar-Cvevebis demonstrirebis Sesaxeb, romelic Camoyalibebulia Sesabamis SedegSi. indikatoris ZiriTadi daniSnulebaa imis warmoCena, miRweulia Tu ara Sedegi. indikatori orientirebulia unar-Cvevebze da Camoyalibebulia aqtivobis enaze. SedegTan dakavSirebuli indikatorebis erToblioba faravs Sedegs da amave dros TiToeuli maTgani warmoaCens Sedegs romelime kuTxiT. Tu ramdenadaa esa Tu is Sedegi miRweuli, ganisazRvreba Sesabamisi indikatorebis im raodenobiT, romelsac moswavle akmayofilebs. Sedegi iTvleba miRweulad, Tu moswavle akmayofilebs am Sedegis Sesabamisi indikatorebis umetesobas.

TiToeuli safexuris Sesabamisi Sedegebisa da maTi indikatorebis erTobliobas Tan erTvis Sinaarsi, romelic aris saswavlo masalis im sakiTxTa CamonaTvali, romlis gamoyenebiTac SesaZlebelia Camoyalibebuli Sedegebis miRweva swavlebis mocemul safexurze. dokumentSi warmodgenili Sinaarsi mxolod sarekomendacio xasiaTisaa da aqedan gamomdinare igi ar ganixileba, rogorc am safexurisaTvis gankuTvnili savaldebulo saswavlo masala. igulisxmeba, rom erTi da igive Sedegis miRweva (e.i. im unarebis ganviTareba, romlebis am SedegSia aRwerili) SesaZlebelia gansxvavebul saswavlo masalaze dayrdnobiTac.

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I klasi

wlis bolos misaRwevi Sedegebi:

mimarTuleba:

ricxvebi da moqmedebebikanonzomierebebi

da algebrageometria da sivrcis

aRqma

maT. I. 1. moswavle erTmaneTs usabamebs ricxvebs, ricxviT saxelebs da raodenobebs

maT. I. 2. moswavle iyenebs ri-gobriv ricxviT saxelebs da ricxvebs rogorc Wdeebs (ìar-liyi~)

maT. I. 3. moswavle akavSirebs Tvlas, ricxvebs Soris damo-kidebulebebs da Sekreba-gamok-lebis moqmedebebs erTmaneTTan

maT. I. 4. moswavle afasebs da adarebs raodenobebs

maT. I. 5. moswavle ga-navrcobs, warmoadgens da adarebs sagnebis periodul ganlagebebs (mimdevrobebs)

maT. I. 6. moswavle amoicnobs da aRwers brtyel figuras

maT. I. 7. moswavle gamosaxavs brtyel geometriul figurebs da amoicnobs obieqtTa ur-TierTmdebareobas

wlis bolos misaRwevi Sedegebi da maTi indikatorebi

mimarTuleba: ricxvebi da moqmedebebi

maT. I.1. moswavle erTmaneTs usabamebs ricxvebs, ricxviT saxelebs da raodenobebs

Sedegi TvalsaCinoa, Tu moswavle:

• irCevs da qmnis mocemuli ricxvis Sesabamisi raodenobis grovebs da piriqiT, ricxvs usabamebs Sesabamisi raodenobis grovebs.

• qmnis toli raodenobis saganTa mowesrigebul erTobliobas maTi dawyvilebiT.

• kiTxulobs da wers ricxvebs; gamosaxavs maT sxvadasxva modelis gamoyenebiT.• gamoyofs miTiTebuli ricxvebis Sesabamisi raodenobis jgufebs grovaSi (mag.

gamoyofs aTeuls grovaSi).

maT. I.2. moswavle iyenebs rigobriv ricxviT saxelebs da ricxvebs rogorc Wdeebs (ìarliyi~)


• iTvlis win/ukan nebismieri ricxvidan, ganmartavs 11-dan 20-mde ricxvebis

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saxeldebas; asaxelebs mocemuli ricxvis wina da momdevno ricxvebs.• saganTa mowesrigebul erTobliobaSi asaxelebs miTiTebuli sagnis rigs;

mocemuli TanmimdevrobiT an/da miTiTebul pozicebze ganaTavsebs sagnebs.• iyenebs rigobriv ricxviT saxelebs movlenaTa an qmedebaTa Tanmimdevrobis

aRwerisas.• iyenebs nuls da mis aRmniSvnel simbolos Sesabamis situaciebSi.• ganasxvavebs da asaxelebs erovnuli fulis niSnebs (monetebs da banknotebs)

20-is farglebSi.

maT. I.3. moswavle akavSirebs Tvlas, ricxvebs Soris damokidebulebebs da Sekreba- gamoklebis moqmedebebs erTmaneTTan


• sityvierad aRwers Sekrebis/gamoklebis/tolobis da Sedegis cnebebebs sxvadasxva konteqstSi (mag. `davumatoT~, `movakloT~, mimateba - gazrda; gamokleba – Semcireba, gacalkeveba, gansxvaveba).

• axdens Sekreba-gamoklebis TvalsaCinod demonstrirebas, gansazRvravs gansxvavebas (mag., `ramdeniT gaizarda/Semcirda?~) da aRwers ricxvebs Soris damokidebulebebs.

• zepirad angariSisas iyenebs 1-s toli bijiT Tvlas, an sxva xerxs da axdens Sekreba-gamoklebis moqmedebaTa urTierTSebrunebulobas demonstrirebas modelis gamoyenebiT.

• mocemuli grovisaTvis asaxelebs am grovis miTiTebul raodenobamde Sesavsebad saWiro, damatebiT raodenobas; zepirad asrulebs 10-is gavliT Sekreba-gamoklebas da axdens gamoyenebuli xerxis TvalsaCinod demonstrirebas.rirebas.

maT. I.4. moswavle afasebs da adarebs raodenobebs


• dauTvlelad asaxelebs zust raodenobas erTgvarovan, mcire zomis saganTa grovaSi (saganTa raodenoba ar aRemateba 5-s) da amowmebs Tavis pasuxs.

• akavSirebs ~-iT~ metoba/naklebobas Sekreba/gamoklebis moqmedebebTan da axdens amis modelze demonstrirebas.

• saganTa dawyvilebiT adarebs raodenobebs grovebSi, iyenebs Sesabamis terminebsa da aRniSvnebsa ( , ,> < = ) da gansazRvravs gansxvavebas (`ramdeniT meti/naklebi?~).

• irCevs ori grovidan erTs, romelSic sagnebis raodenoba daaxloebiT mocemuli ricxvis tolia, amowmebs Tavis varauds.

mimarTuleba: kanonzomierebebi da algebra

maT. I.5. moswavle ganavrcobs, warmoadgens da adarebs sagnebis periodul ganlagebebs (mimdevrobebs)


• mimdevrobis mocemuli fragmentis mixedviT avsebs am mimdevrobis ramdenime

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Tanmimdevrul Ria pozicias.• adarebs erTnairi sagnebiT warmodgenil or mocemul mimdevrobas (romlebSic

saganTa raodenoba tolia) da Sesabamis SemTxvevaSi miuTiTebs im mimdevrobebs, romlebic ganlagebis erTsa da imave wess emorCileba.

• sityvierad mocemuli wesis mixedviT, mimdevrobiT ganalagebs mxolod erTi atributiT gansxvavebul sagnebs (mag. erTi zomis burTebis aseT mimdevrobas: wiTeli burTi, lurji burTi, wiTeli burTi . . .)

mimarTuleba: geometria da sivrcis aRqma

maT. I.6. moswavle amoicnobs da aRwers brtyel figuras


• yofiTi daniSnulebis sagnebSi an maT ilustraciebSi uTiTebs dasaxelebul brtyel figurebs.

• SearCevs miTiTebuli figuris models Sereuli grovidan.• aRwers miTiTebul geometriuli figuras (mag. asaxelebs mocemuli mravalku-

Txedis wveroebis raodenobas).

maT. I.7. moswavle gamosaxavs brtyel geometriul figurebs da amoicnobs obieqtTa urTierTmdebareobas


• romelime xerxiT (mag. aplikaciiT an naxatis saSualebiT) qmnis dasaxelebuli formis brtyeli figuris models an gamosaxulebas.

• uTavsebs sxvadasxva brtyeli figurebis modelebs erTmaneTs nimuSze mocemuli gamosaxulebis (naxatis) misaRebad.

• sworad pasuxobs kiTxvebze obieqtTa urTierTmdebareobis (marjvniv/marcxniv, zemoT/qvemoT, win/ukan) Sesaxeb.

• miTiTebuli wesiT aerTebs ramodenime wertils sibrtyeze an/da moniSnavs gzas miTiTebul obieqtamde martiv sqemaze.

programis Sinaarsi

Sedegebis miRweva SesaZlebelia mocemuli Sinaarsis safuZvelze:

_ naturaluri ricxvebi 20-is farglebSi da 0_ ricxvis cnebis sxvadasxva aspeqti_ ricxvebis gamoyeneba_ sagnebis saSualebiT warmodgenili perioduli mimdevrobebi._ brtyeli figurebi: samkuTxedi, oTxkuTxedi, xuTkuTxedi, eqvskuTxedi, wre._. martivi sqemebi sibrtyeze (mag. wirebiT SeerTebuli wertilebi).

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maTemat

ika

II klasi


mimarTuleba:

ricxvebi da moqmede-bebi


geometria da sivrcis

aRqma

monacemTa ana-lizi, alba-Toba da sta-

tistika

maT. II. 1. moswavle erTma-neTs usabamebs ricxvebs, ricxviT saxelebs, raode-nobebs da rigs.

maT. II. 2. moswavle akavSi-rebs Tvlas, ricxvebs, ri-cxviT saxelebs Soris da-mokidebulebebs da Sekre-ba-gamoklebis moqmedebebs erTmaneTTan.

maT. II. 3. moswavle asru-lebs ganaxevreba-gaormage-bis moqmedebebs da akavSi-rebs maT Sekreba-gamokle-basTan da erTmaneTTan.

maT. II. 4. moswavle afasebs da adarebs raodenobebs 100-is farglebSi.

maT. II. 5. moswavle iyenebs ricxvebs da maTze moqmede-bebs gamoTvlebze amocane-bis amoxsnisas

maT. II. 6. moswavle ga-navrcobs, warmoadgens da adarebs sagnebis an naxatebis/figurebis periodul ganlage-bebs (mimdevrobebs)

maT. II. 7. moswavle iyenebs Sekrebas da ga-moklebas martivi ma-Tematikuri amocanis amoxsnisas.

maT. II. 8. moswavle orientirebs garemoSi da aRwers obieqtTa urTierTganlagebas

maT. II. 9. mo-swavle iyenebs Tvisebriv da raodenobriv niSnebs figure-bis aRsawerad

maT. II. 10. mo-swavle adarebs da adgens figu-raTa zomebs

maT. II. 11. moswav-le agrovebs Tvi-sebriv monacemebs misi uSualo ga-remocvis Sesaxeb

maT. II. 12. moswa-vle awesrigebs Tvisebriv monace-mebs

maT. II. 13. moswa-vle akeTebs Tvi-sebriv monacemTa interpretacias



maT. II.1. moswavle erTmaneTs usabamebs ricxvebs, ricxviT saxelebs, raodenobebs da rigs


• kiTxulobs èrTniSna~ da òrniSna~ ricxvebs, asaxelebs maT wina da momdevno ricxvebs; nebismieri ricxvidan iTvlis bijiT win/ukan da gamosaxavs ricxvebs sxvadasxva modelis gamoyenebiT. (mag. Cawers maT poziciuri sistemis gamoyenebiT

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an gamosaxavs ricxvs saganTa Sesabamisi raodenobis groviT).• sxvadasxva xerxiT iTvlis grovaSi saganTa raodenobas da adarebs miRebul

Sedegebs erTmaneTs; axdens ricxvis aTobiTi poziciuri sistemiT Caweris de-monstrirebas saganTa grovaSi aTeulebis jgufebis gamoyofiT.

• orniSna ricxvis CanawerSi uTiTebs aTeulisa da erTeulis Tanrigebs, asaxe-lebs am TanrigebSi mdgomi cifrebis mniSvnelobas da ganmartavs erTeulis TanrigSi 0-is gamoyenebis azrs; iyenebs am codnas ricxvebis Sedarebisas.

• asaxelebs miTiTebuli elementis nomers figurebis/naxatebis mowesrigebul erTobliobaSi; asaxelebs mis Semdgom an winmswreb wevrTa rigs.

maT. II.2. moswavle akavSirebs Tvlas, ricxvebs, ricxviT saxelebs Soris damokidebulebebs da Sekreba-gamoklebis moqmedebebs erTmaneTTan


• axdens Sekreba-gamoklebis moqmedebaTa demonstrirebas modelis gamoyenebiT, daadgens moqmedebis Sedegs (`ramdeniT gaizarda, Semcirda?~).

• zepirad angariSisas iyenebs bijiT Tvlas, an sxva xerxs (mag. Tanrigebis dajgufeba, mTliani aTeuliT `gadaxtoma~); axdens moqmedebaTa urTierTSe-brunebulobis demonstrirebas.

• ganmartavs ricxvebis saxeldebas qarTul enaSi.• zepirad asrulebs aTeulis gavliT Sekreba-gamoklebas da axdens gamoyene-

buli xerxis demostrirebas (mag. ricxviT kibeze an saganTa grovaze).

maT. II.3. moswavle asrulebs ganaxevreba-gaormagebis moqmedebebs da akavSirebs maT Sekreba-gamoklebasTan da erTmaneTTan


• axdens gaormagebis moqmedebis demonstrirebas saganTa mocemuli raodenobis jgufisTvis igive raodenobis jgufis damatebiT.

• aormagebs ricxvebs 10-is farglebSi, agreTve srul 10-eulebsa da 20-eulebs; akavSirebs am moqmedebas Sesabamisi bijiT TvlasTan (mag. ganmartavs sruli aTeulis Sesabamisi ricxvebis saxeldebas qarTul enaSi).

• daadgens aris Tu ara miTiTebuli raodenoba sxva miTiTebuli raodenobis naxevari/ormagi konkretuli modelis SemTxvevaSi (mag. saganTa dawyvile-biT).

• irCevs xerxs (mag. uku Tvla an gamokleba) da anaxevrebs luw ricxvebs; axdens gaormageba-ganaxevrebis urTierTSebrunebulobis demonstrirebas.

maT. II.4. moswavle afasebs da adarebs raodenobebs 100-is farglebSi


• irCevs xerxs (mag. elementTa urTierTcalsaxa Sesabamisoba _ dawyvileba), afasebs (`daaxlebiT tolia~, `daaxloebiT naxevaria/ormagia~) da adarebs raodenobebs or grovaSi; gansazRvravs maT Soris gansxvavebas (`ramdeniT meti/naklebi?~, `toli~, òrjer meti/naklebi~).

• erTgvarovan saganTa ori/sami grovidan irCevs erTs, romlSic saganTa raode-noba daaxloebiT mocemuli ricxvis tolia da amowmebs Tavis varauds.

• asaxelebs ricxvis uaxloes oceuls, aTeuls, an xuTeuls; ganmartavs pa-suxs.

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maT. II.5. moswavle iyenebs ricxvebs da maTze moqmedebebs gamoTvlebze amocanebis amoxsnisas


• amocanis pirobis mixedviT gansazRvravs Tu ra aris mocemuli da ra aris sa-Zebni.

• irCevs Sesabamis moqmedebas, misi Sesrulebis xerxs an/da models martivi amo-canis amosaxsnelad (mag. Sekreba, gamokleba, gaormageba, an ganaxevreba; erTeu-lis bijiT win an uku Tvla; saganTa grova an ricxviTi kibe).

• iyenebs 1-is toli bijiT Tvlas da poulobs meore Sesakrebs, Tu cnobilia pir-veli Sesakrebi da jami; iyenebs erTeulis bijiT uku Tvlas ucnobi maklebis povnisTvis, mocemuli saklebiTa da sxvaobiT da axdens gamoyenebuli xerxis demonstrirebas (mag. 9 - ? = 6, ricxviT kibeze iTvlis 9-dan ukan 6-mde da axdens nabijebis raodenobis, rogorc maklebis interpretacias; axdens igive proceduris demonstrirebas ricxviT kibeze).

• ganasxvavebs, asaxelebs da realur/gaTamaSebul viTarebSi iyenebs erovnuli fulis niSnebs (monetebi da banknotebi 100-is farglebSi).bs (monetebi da banknotebi 100-is farglebSi).


maT. II.6. moswavle ganavrcobs, warmoadgens da adarebs sagnebis an naxatebis/ figurebis periodul ganlagebebs (mimdevrobebs)


•� mocemul mimdevrobaSi avsebs ramodenime gamotovebul pozicias (mag.

♠ ♦ ♣ ♠ ♦ ♣ ♠ ♣

`ra figurebi iqneba gamotovebul poziciebze ? ~).• erTmaneTs adarebs ramodenime (araumetes samisa) mimdevrobas da asaxelebs im

mimdevrobebs, romlebic ganlagebis erTsa da imave wess emorCilebian.• mocemuli wesis mixedviT warmoadgens mimdevrobas mxolod erTi atributiT

gansxvavebuli sagnebis an naxatebis/figurebis saSualebiT.bis an naxatebis/figurebis saSualebiT.

maT. II.7. iyenebs Sekrebas da gamoklebas martivi maTematikuri amocanis amoxsnisas.


• amowmebs, aris Tu ara dasaxelebuli ricxvi mocemuli tolobis (mag. ♠ + 7 = 10 ) ucnobi komponentis mniSvneloba.

• Seadgens realuri viTarebis amsaxvel, Sekrebis/gamoklebis erTi operaciis Semcvel, ekvivalentur mTelricxovan gamosaxulebebs. (mag. fuladi mone-tebis ori iseTi erTobliobisaTvis, romelic erTsa imave Tanxas Seadgens).

• iyenebs Sekrebis komutaciurobisa da asociaciurobis Tvisebebs ricxviTi gamosaxulebis mniSvnelobis gamosaTvlelad.obis gamosaTvlelad.

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maT. II.8. moswavle iyenebs Tvisebriv da raodenobriv niSnebs figurebis aRsawerad


• adarebs da ajgufebs brtyel figurebs geometriuli atributebis (mag. wveroe-bis/gverdebis raodenobis) mixedviT.

• ganasxvavebs figuris Siga da gare areebs; uTiTebs figuris SigniT, gareT da sazRvarze mdebare wertilebs.

• uTiTebs saerTo sazRvris mqone figurebis saerTo gverdebsa da wveroebs.gurebis saerTo gverdebsa da wveroebs.

maT. II.9. moswavle orientirebs garemoSi da aRwers obieqtTa urTierTganlagebas


• ganalagebs obieqtebs miTiTebuli wesis mixedviT.• aRwers obieqtis mdebareobas meore obieqtis mimarT Sesabamisi terminebis

gamoyenebiT (mag. marjvniv, marcxniv, zemoT, qvemoT).• gascems da Tavadac asrulebs moZraobis orientaciis Semcvel miTiTebebs.s orientaciis Semcvel miTiTebebs.

maT. II.10. moswavle adarebs da adgens figuraTa zomebs


• urTierTSeTavsebiT adarebs figuraTa wrfiv zomebs da gamoxatavs Sedarebis Sedegs Sesabamisi terminebiT (mag. grZeli, mokle, toli).

• moiZiebs toli figurebis nimuSebs misTvis Cveul garemoSi; axdens figuraTa tolobis demonstrirebas maTi urTierTSeTavsebiT.

• poulobs realuri obieqtis (mag. saklaso oTaxis, sportuli darbazis) wrfiv zomas arastandartul zomis erTeulis (mag. nabijis) gamoyenebiT.

mimarTuleba: monacemTa analizi, albaToba da statistika

maT. II.11. agrovebs Tvisebriv monacemebs misi uSualo garemocvis Sesaxeb


• agrovebs monacemebs realur obieqtebze dakvirvebiT.• amokrebs ramdenime monacems erTgvarovan monacemTa mokle siidan (araumetes

aTi monacemi).• amokrebs saWiro monacemebs umartivesi (orsvetiani an orstriqoniani) cxri-

lidan

maT. II.12. awesrigebs Tvisebriv monacemebs


• ganalagebs monacemebs mocemuli TanmimdevrobiT an mocemul poziciebze (mi-mdevrobiT gamoyofili poziciebis SemTxvevaSi).

• monacemTa erTobliobis yovel monacems miuCens adgils romelime mocemul jgufSi (monacemTa raodenoba ar aRemateba aTs, xolo jgufebis raodenoba - sams).

• erTi klasis obieqtTa (mag. geometriuli figurebi) Sesaxeb monacemebs ala-gebs/ajgufebs raime wesiT; ganmartavs dalagebis/dajgufebis wess.ajgufebs raime wesiT; ganmartavs dalagebis/dajgufebis wess.

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maT. II.13. akeTebs Tvisebriv monacemTa interpretacias


• zepirsityvierad axasiaTebs monacemTa sias (romelSic gaerTianebulia arau-metes 10 monacemisa) monacemTa saerTo raodenobis, ganmeorebis, poziciis, Tanmimdevrobis mixedviT.

• zepirsityvierad aRwers/ganmartavs piqtogramas, romelSic erTi simbolo Seesabameba erT monacems an monacemTa wyvils.

• zepirsityvierad aRwers/ganmartavs monacemTa umartives (orsvetian an ors-triqonian) cxrils.

programis Sinaarsi


_ 100-ze naklebi naturaluri ricxvebi da 0_ aTobiTi poziciuri sistema da misi demonstrireba_ ariTmetikuli moqmedebebi naturalur ricxvebze da maTi demonstrireba_ erovnuli fulis niSnebi_ sagnebis, naxatebis an figurebis saSualebiT warmodegenili perioduli

mimdevrobebi._ Sekrebis/gamoklebis (araumetes ori moqmedebis) Semcveli mTelricxovani

gamosaxulebebi da maTi ekvivalentoba._ Sekrebis komutaciuroba da asociaciuroba (araformalurad da Sesabamisi

terminebis gareSe)._ erTi ucnobi komponentisa da Sekrebis/gamoklebis erTi moqmedebis Semcveli

mTelricxovani tolobebi._ brtyeli figurebi: wertili, monakveTi, texili, mrudi wiri._ figuris Siga da gare areebi, figuris sazRvari._ saerTo sazRvris mqone figurebi, maTi saerTo gverdebi da wveroebi._ toli figurebi._ manZili: adiciuroba monakveTze, sigrZis sazomi arastandartuli

erTeulebi._ sibrtyeze orientacia da obieqtTa urTierTganlageba._ Tvisebriv monacemTa Segrovebis saSualebani: dakvirveba, monacemTa amokreba

monacemTa monacemTa siidan an/da cxrilidan._ Tvisebrivi monacemebis organizacia: monacemTa dajgufeba._ monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi

niSnebi: monacemTa saerTo raodenoba, ganmeoreba, pozicia da Tanmimdevroba erTobliobaSi.

_ monacemTa warmodgenis saSualebani Tvisebrivi monacemebisTvis: sia, cxrili, piqtograma (romelSic erTi simbolo Seesabameba erT monacems an monacemTa wyvils).

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III klasi


mimarTuleba:

ricxvebi da moq-medebebi

kanonzomierebe-bi da algebra


monacemTa ana-lizi, albaTo-ba da stati-

stika

maT. III. 1. moswavle gamosaxavs, adarebs da alagebs natu-ralur ricxvebs poziciuri sistemis gamoyenebiT

maT. III. 2. moswavle iyenebs Sekreba-gamo-klebis moqmedebaTa Sesrulebis romeli-me xerxs

maT. III. 3. moswavle asrulebs gamravle-ba-gayofis moqmede-bebs da akavSirebs maT Sekreba-gamokle-basTan da erTmaneT-Tan

maT. III. 4. moswavle wyvets gamoTvlebTan, TvlasTan da Sefase-bebTan dakavSirebul problemebs

maT. III. 5. moswavle warmoadgens, adarebs da ikvlevs sagnebis da naxatebis/fi-gurebis periodul ganlagebebs (mimde-vrobebs)

maT. III. 6. moswavle ganavrcobs, gamo-saxavs da ikvlevs sagnebs Soris an sa-gnebsa da maT atri-butebs Soris moce-mul Sesabamisobas.

maT. III. 7. moswavle problemis gada-saWrelad adgens da iyenebs ricxvi-Ti gamosaxulebis Semcvel tolobas

maT. III. 8. moswavle amoicnobs da aRwers geometriul figu-ras

maT. III. 9. moswavle qmnis brtyeli fi-gurebis grafikul gamosaxulebebs da modelebs

maT. III. 10. moswavle poulobs saganTa da figuraTa wrfiv zomebs da obieqtTa Soris manZilebs

maT. III. 11. moswavle agrovebs Tvisebriv da raodenobriv mo-nacemebs mocemul TemasTan an gamo-sakvlev obieqtTan dakavSirebiT

maT. III. 12. moswa-vle awesrigebs da warmoadgens diskretul raode-nobriv da Tvisebriv monacemebs

maT. III. 13. moswa-vle akeTebs Tvi-sebriv da raode-nobriv monacemTa interpretacias



maT. III.1. moswavle gamosaxavs, adarebs da alagebs naturalur ricxvebs poziciuri sistemis gamoyenebiT


• kiTxulobs da gamosaxavs ricxvebs, ganmartavs ricxvebis saxeldebas qarTul enaSi; axdens aTobiTi poziciuri sistemis demonstrirebas sxvadasxva mode-lis gamoyenebiT.

• asaxelebs ricxvis CanawerSi sxvadasxva TanrigebSi mdgomi cifrebis Sesabamis mniSvnelobebs, warmoadgens ricxvs saTanrigo Sesakrebebis an sxva saxiT.

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• iyenebs poziciur sistemas ricxvebis Sedarebisas, zrdiT an klebiT alagebs ricxvebs, romelTa raodenoba ar aRemateba xuTs.

• asaxelebs mocemuli ricxvis wina da momdevno ricxvebs; asaxelebs mocemuli ricxvis uaxloes aTeuls, aseuls an aTaseuls.

• Tanrigebis Sesabamisi bijiT iTvlis win/ukan mocemuli ricxvidan.isi bijiT iTvlis win/ukan mocemuli ricxvidan.

maT. III.2. moswavle iyenebs Sekreba-gamoklebis moqmedebaTa Sesrulebis romelime xerxs

Sedegi TvalsaCinoa, Tu moswavle: • konkretuli magaliTisTvis irCevs da iyenebs zepiri angariSis (Sekreba/gamo-

kleba) sxvadasxva xerxs; xsnis gamoyenebul xerxs da / an axdens mis modelze demonstrirebas. (mag. Sekreba-gamokleba Tanrigis gavliT, calkeuli Tanrigebis Sekreba/gamoklebiT, dadgenili kanonzomierebebis gamoyenebiT: radganac 7 - 3 = 4, maSin 7 aTasi _3 aTasi = 4 aTasi; gaormagebis gamoyeneba Sekrebisas, mag. 1500 + 1550 = 2-1500 + 50 = 3000; Tanrigis daSliT, mag. 8000 - 673 zepirad: radganac 8000 - 7999 = 1 da 7999 - 673 = 7326, amitom pasuxia 7326 + 1 = 7327). 159 + 22 (~gadaxtomiT~) 46 + 33 (Tanrigebis Sesabamisi bijebiT)

• irCevs da iyenebs Sekreba-gamoklebis moqmedebebis Sesrulebis adekvatur xe-rxs konkretuli magaliTis SemTxvevaSi.

• iyenebs Tanrigamde Sevsebis/Tanrigis daSlis xerxs moqmedebaTa Sesrulebi-sas; asabuTebs moqmedebaTa Sesrulebis weriT algoriTms.

• iyenebs moqmedebaTa Tanmimdevrobas zepiri angariSisas da/an martivi ricxvi-Ti gamosaxulebis mniSvnelobis povnisas. (yvela ariTmetikuli moqmedeba: mag. ras miviRebT Sedegad Tu 3 Svideuls davumatebT 7 aseuls?))

maT. III.3. moswavle asrulebs gamravleba-gayofis moqmedebebs, akavSirebs maT Sekreba-gamoklebis moqmedebebTan da erTmaneTTan


• axdens gamravlebis moqmedebis mravaljeradi SekrebiT demonstrirebas, xolo gayofis moqmedebis demonstrirebas _ grovis toli raodenobis jgufebad dayofiT.

• akavSirebs gamravleba-gayofas erTmaneTTan, rogorc urTierT-Sebrunebul moqmedebebs da axdens amis modelze demonstrirebas.

• zepirad asrulebs gamravleba-gayofas martiv SemTxvevebSi (mag. erTniSna ri-cxvebis gamravleba; erT da orniSna ricxvebis 10-ze gamravleba).

• irCevs romelime xerxs da/an models da gansazRvravs ucnob gamyofs moce-muli ganayofiTa da gasayofiT; analogiurad, erT-erT Tanamamravls _ moce-

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muli namravliTa da meore TanamamravliT; xsnis gamoyenebul xerxs (1000-is farglebSi).Si).

maT. III.4. moswavle wyvets gamoTvlebTan, TvlasTan da SefasebebTan dakavSirebul problemebs

Sedegi TvalsaCinoa, Tu moswavle: • asaxelebs Tu ramdeni wyvili, 5-euli, 10-euli da sxv. aris mocemul ricxvSi

da asabuTebs pasuxs (mag. ramdeni 10-eulia 412-Si, kidev ramdeni erTeuli rCe-ba?).

• iyenebs romelime xerxs da poulobs meore Sesakrebs, Tu cnobilia pirveli Se sakrebi da jami - poulobs ucnobi maklebis, mocemuli saklebiTa da sxvaobiT (1000-is farglebSi mainc).

• iyenebs zepiri angariSis xerxebs ricxviT gamosaxulebebis mniSvnelobaTa Sesadareblad (mag. 340+177 metia Tu 500?).

• xsnis amocanebs variantebis daTvlaze/gamoricxvaze (mag. avsebs weriTi algo-riTmis gamoyenebiT Sesrulebuli Sekrebis nimuSSi gamotovebul cifrebs da asabuTebs pasuxs).

• iyenebs ricxvebs, rogorc Wdeebs problemebis gadaWrisas; asaxelebs ricxve-bis, rogorc Wdeebis gamoyenebis magaliTebs. (mag. saxlis, telefonis, manqa-nis nomeri).


maT. III.5. moswavle warmoadgens, adarebs da ikvlevs sagnebis da naxatebis/figurebis periodul ganlagebebs (mimdevrobebs)

Sedegi TvalsaCinoa, Tu moswavle: • gamoyofs mimdevrobis periods (romlis sigrZe ar aRemateba sam pozicias).• mocemuli mimdevrobis mixedviT qmnis msgavs mimdevrobas sxva obieqtebis ga-

moyenebiT.• erTmaneTs adarebs ramdenime mimdevrobas da gamoyofs msgavs mimdevrobebs.

maT. III.6. moswavle ganavrcobs, gamosaxavs da ikvlevs sagnebs Soris an sagnebsa da maT atributebs Soris mocemul Sesabamisobas.


• analogiis an winaswar mocemuli wesis mixedviT ganavr-cobs mocemuli marti-vi Sesabamisobis fragments ( mag. mis irgvliv mdebare sagnebisaTvis mocemuli aseTi SesabamisobisaTvis: furceli →TeTri, CanTa →lurji, dafa → (?). ).

• sityvierad mocemuli Sesabamisobis mixedviT avsebs mocemul cxrils.• cxrilis saSualebiT gamosaxuli SesabamisobisaTvis poulobs miTiTebuli

elementis winasaxes (mag. mocemuli cxrilisaTvis romelic gamosaxavs Tu romelma moswavlem ra niSani miiRo, e.i. Sesabamisobas: “moswavle → niSani”, asaxelebs yvela im moswavles romelmac miiRo 6).

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maT. III.7. moswavle problemis gadasaWrelad adgens da iyenebs ricxviT gamosaxulebebis Semcvel tolobas


• qmnis realuri viTarebis gamomsaxvel mTelricxovan ekvivalentur gamosaxu-lebebs. (mag. sasworis wonasworoba, irCevs fulis Sesaferis niSnebs miTiTe-buli Tanxis warmosadgenad da dasaxurdaveblad).

• realur viTarebasTan dakavSirebuli amocanis amosaxsnelad adgens da iyenebs iseT ricxviT gamosaxulebas, romelic Sekrebis/gamoklebis erT operacias Seicavs.

• poulobs (SerCevis an raime sxva xerxiT) Sekrebis, gamoklebis Semcveli to-lobis ucnobi komponentis mniSvnelobas.


maT. III.8. moswavle amoicnobs da aRwers geometriul figuras


• amoicnobs sivrcul geometriul figurebs arqiteqturisa da xelovnebis ni-muSebSi an maT ilustraciebSi, yofiTi daniSnulebis sagnebSi an figuraTa modelebis grovaSi.

• ganasxvavebs figuris elementebs da iyenebs geometriul terminebs maTi da-saxelebisas (mag.: wvero, waxnagi, wibo).

• iyenebs geometriul figuris wveroebis asoiT aRniSvnebs figuris elementebis (wveroebi da gverdebi) dasaxelebisas.

maT. III.9. moswavle qmnis brtyeli figurebis grafikul gamosaxulebebs da modelebs


• geometriuli figuris sityvieri aRwerilobis mixedviT qmnis am figuris gra-fikul gamosaxulebas.

• irCevs brtyeli geometriuli figurebis modelebs mocemuli grovidan da qm-nis miTiTebul konfiguracias/figuras.

• anawevrebs brtyeli geometriuli figuris grafikul gamosaxulebas an mo-dels miTiTebuli figuris/figurebis misaRebad.

maT. III.10. moswavle poulobs saganTa da figuraTa wrfiv zomebs da obieqtTa Soris manZilebs


• poulobs sagnis wrfiv zomebs arastandartuli erTeulebiT (mag.oulobs sagnis wrfiv zomebs arastandartuli erTeulebiT (mag.s wrfiv zomebs arastandartuli erTeulebiT (mag.ebs arastandartuli erTeulebiT (mag. mtkaveliT) Semdeg afasebs mas standartuli erTeulebis gamoyenebiT; msjelobs standar-tuli erTeulebis gamoyenebis saWiroebis Sesaxeb.

• adarebs da afasebs obieqtTa wrfiv zomebs (maT Soris urTierTSeTavsebiT) da gamoxatavs Sedarebis Sedegs Sesabamisi terminebiT (mag. grZeli, mokle, toli).

• zomavs FfiguraTa gverdebs saxazavis gamoyenebiT da afiqsirebs gazomvis Sedegs romelime standartul erTeulebSi (mag. 3 sm an 30 mm).

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maT. III.11. moswavle agrovebs Tvisebriv da raodenobriv monacemebs mocemul TemasTan an gamosakvlev obieqtTan dakavSirebiT

Sedegi TvalsaCinoa, Tu moswavle: • kiTxulobs mokle teqsts (ori-sami martivi winadadeba) da amokrebs miTiTe-

buli obieqtis Sesaxeb teqstSi arsebul monacemebs.• svams diax/ara tipis SekiTxvebs monacemTa mosapoveblad mocemul TemasTan an

gamosakvlev obieqtTan dakavSirebiT da aRricxavs pasuxs.• irCevs monacemTa Segrovebis Sesaferis saSualebas (dakvirveba, gazomva) da

iyenebs mas.

maT. III.12. moswavle awesrigebs da warmoadgens diskretul raodenobriv da Tvisebriv monacemebs


• ajgufebs monacemebs araumetes ori niSniT da asaxelebs niSnebs, romelTa mixedviTac moaxdina dajgufeba.

• alagebs ramdenime raodenobriv monacems zrdadoba-klebadobiT.• qmnis urTierTcalsaxa Sesabamisobis wesiT piqtogramas maswavleblis mier

momzadebul badeze (mag. sqematurad gamosaxavs TiToeul obieqts badis Sesa-bamis ujraSi).

maT. III.13. moswavle akeTebs Tvisebriv da raodenobriv monacemTa interpretacias


• aRwers/ganmartavs piqtogramis da cxrilis saxiT warmodgenil monacemebs ze-pirsityvierad an/da werilobiT.

• axasiaTebs dajgufebul Tvisebriv monacemTa erTobliobas masSi monacemTa saerTo raodenobis, qvejgufebis raodenobis, TiToeul qvejgufSi monacemTa raodenobis da erTobliobaSi monacemTa ganmeorebis, poziciis, Tanmimdevro-bis mixedviT.

• svams Semajamebel kiTxvebs piqtogramis an/da umartivesi (orsvetiani an ors-triqoniani) cxrilis saxiT warmodgenili monacemebis mimarT.

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programis Sinaarsi


_ samniSna naturaluri ricxvebi_ aTobiTi poziciuri sistemis demonstrireba da gamoyeneba_ ariTmetikuli moqmedebebi naturalur ricxvebze_ ricvebis gamoyeneba_ sagnebis, naxatebis an figurebis saSualebiT warmodegenili perioduli

mimdevrobebi da maTi periodi._ Sesabamisobebi sagnebs Soris, sagnebsa da maT atributebs Soris;

Sesabamisobis gamosaxva cxrilis saSualebiT; mocemuli SesabamisobisaTvis elementis winasaxe.

_ Sekrebis/gamoklebis Semcveli mTelrixovani gamosaxulebebi da maTi ekvivalentoba.

_ erTi ucnobi komponentisa da Sekrebis/gamoklebis moqmedebis Semcveli mTelricxovani tolobebi.

_ sivrculi figurebi: kubi, marTkuTxa paralelepipedi, piramida, sfero._ sivrculi figurebis elementebi: wvero, wibo, waxnagi._ figuris wrfivi zomebi, sazomi xelsawyoebi da sigrZis sazomi erTeulebi:

metri, decimetri, santimetri._ Tvisebriv da raodenobriv monacemTa Segrovebis saSualebani: gazomva,

dakvirveba, gamokiTxva; monacemTa amokreba wakiTxuli teqstidan._ Tvisebriv da raodenobriv monacemTa organizacia: monacemTa tipebi -

Tvisebrivi da raodenobrivi monacemebi; Tvisebriv monacemTa dajgufeba; raodenobriv monacemTa dajgufeba (garda inter-valTa klasebad dayofisa); raodenobriv monacemTa dalageba zrdadoba-klebadobiT.

_ monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: monacemTa saerTo raodenoba erTobliobaSi da monacemTa raodenoba qvejgufebSi; monacemTa ganmeoreba, pozicia da Tanmimdevroba erTobliobaSi/qvejgufebSi.

_ monacemTa warmodgenis saSualebani raodenobrivi da Tvisebrivi monacemebisTvis: cxrili, piqtograma.

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VII klasi


mimarTuleba:

ricxvebi da mo-qmedebebi



monacemTa ana-lizi, alba-Toba da sta-

tistika

maT. VII. 1. moswa-vle kiTxulobs, gamosaxavs, adarebs da alagebs racio-naluri ricxvebs poziciuri sistemis gamoyenebiT

maT. VII. 2. moswavle sxvadasxva xerxiT asrulebs moqmede-bebs racionalur ricxvebze

maT. VII. 3. moswavle afasebs racionalur ricxvebze moqmede-baTa Sedegs

maT. VII. 4. moswavle erTmaneTTan akav-Sirebs zomis sxva-dasxva erTeulebs da iyenebs maT amo-canebis amoxsnisas

maT. VII. 5. moswavle amoicnobs da gamosa-xavs sidideebs Soris pirdapirproporciul damokidebulebas

maT. VII. 6. moswavle amocanis amoxsnisas iyenebs simravlur cnebebsa da opera-ciebs

maT. VII. 7. moswavle amartivebs algebrul gamosaxulebas da xsnis wrfiv gantole-bas

maT. VII. 8. moswavle ganavrcobs da aanali-zebs obieqtebis pe-riodul mimdevrobas an/da mudmivi nazrdis mqone ricxviT mimde-vrobas

maT. VII. 9. moswav-le amoicnobs geometriul figu-rebs, adarebs maT saxeobebs da axdens klasificirebas

maT. VII. 10. moswavle warmoadgens geome-triul obieqtebs amocanis konteqstis Sesabamisad

maT. VII. 11. moswavle axdens geometriul gardaqmnebs da iye-nebs maT figuraTa Tvisebebis dasadge-nad

maT. VII. 12. moswavle iyenebs koordinate-bis meTods orienta-ciisaTvis

maT. VII. 13. moswavle xsnis geometriul amocanebs samkuTxe-debTan dakavSirebuli cnebebisa da faqtebis gamoyenebiT

maT. VII. 14. mo-swavle moipovebs dasmuli amoca-nis amosaxsnelad saWiro Tvisebriv da raodenobriv monacemebs

maT. VII. 15. mo-swavle awesrigebs da warmoadgens Tvisebriv da raodenobriv mo-nacemebs dasmuli amocanis amosax-snelad xelsayre-li formiT

maT. VII. 16. mo-swavle akeTebs Tvisebriv da raodenobriv monacemTa in-terpretacias da analizs amocanis konteqstis gaTva-liswinebiT

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maT. VII.1. moswavle kiTxulobs, gamosaxavs, adarebs da alagebs racionaluri ricx vebs poziciuri sistemis gamoyenebiT


• aTwiladis CanawerSi uTiTebs Tanrigebs da asaxelebs TanrigebSi mdgom cifrTa mniSvnelobebs; iyenebs am codnas aTwiladebis Sedarebis an (zrdadobiT/klebadobiT) dalagebisas. (mag. gaSlis sasrul aTwilads saTanrigo Sesakrebebis 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 magaliTebze asabuTebs gayofadobis niSnebidan zogierTs (mag. 3-ze da 9-ze gayofadobis niSnebs); poulobs mocemuli ricxvebis umciresi saerTo jeradsa da udides saerTo gamyofs.

maT. VII.2. moswavle sxvadasxva xerxiT asrulebs moqmedebebs racionalur ricxvebze


• axdens mTel ricxvebze ariTmetikuli moqmedebebis modelze demonstrirebas (mag. `dadebiTi~ da ùaryofiTi~ erTeulovani `muxtebiT~, e.i. ori gansxvavebuli feris `nulovani wyvilebiT~).

• iyenebs ricxvis Caweris ekvivalentur formebs, moqmedebaTa Sesrulebis Tanmimdevrobas, maT Tvisebebsa da dajgufebas 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 moqmedebebis Sesrulebis

xerxs; (es xerxebia(es xerxebia: zepiri, teqnologiebis gamoyenebiT, weriTi algoriTmiebi).

maT. VII.3. moswavle afasebs racionalur ricxvebze moqmedebaTa Sedegs


• gamoTvlebze sayofacxovrebo amocanis amoxsnisas iyenebs zepiri angariSis xerxebs an Sesabamis SemTxvevaSi, moqmedebaTa Sedegis Sefasebas.

• afasebs racionalur ricxvebze ariTmetikul moqmedebaTa Sedegis mniSvnelobas, asrulebs moqmedebebs da amowmebs Tavis varauds.

• amrgvalebs racionlur ricxvebs miTiTebuli sizustiT; miaxloebiT poulobs (sizustis miTiTebis gareSe) ariTmetikuli gamosaxulebis mniSvnelobas.

• iyenebs Sefasebas aTwiladebze (weriTi algoriTmis an kalkulatoris gamoyenebiT) Catarebuli gamoTvlebis Sedegis adekvaturobis Sesamowmeblad.

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maT. VII.4. moswavle erTmaneTTan akavSirebs zomis sxvadasxva erTeulebs da iyenebs maT amocanebis amoxsnisas


• irCevs da iyenebs Sesaferis erTeulebs sididis cvlilebis, moZraobis siCqa-ris, masStabisa da ruqaze manZilis povnasTan dakavSirebul amocanebis amoxs-nisas.

• xsnis praqtikul saqmianobasTan dakavSirebul da/an sxva saswavlo disci-plinebidan momdinare amocanebs gamoTvlebze (mag. umartivesi xarjTaRricxva; istoriuli epoqis xangrZlivobis gansazRvra; amocanebi procentebze da pro-porciaze: xsnarebi, Senadnobebi da sxva).

• mocemuli wrfivi damokidebulebis gamoyenebiT gamosaxavs erT sistemaSi moce-mul erTeuls sxva sistemis erTeuliT.

• gamosaxavs mocemul erTeuls igive sistemis sxva erTeulis saSualebiT. (mag. km/sT -ebSi mocemul siCqares gamosaxavs m/wm -iT).


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


• mocemuli damokidebulebisaTvis Tvisebrivad da raodenobrivad aRwers Tu ra gavlenas axdens erTi sididis cvlileba meoris mniSvnelobaze; moyavs mudmivi da aramudmivi raodenobrivi cvlilebis magaliTebi yoveldRiuri cxovrebidan.

• sityvierad Camoyalibebul debulebas sidideebs Soris damokidebulebisa da mimarTebis Sesaxeb gamosaxavs grafikulad an/da cxriliT da piriqiT – grafikulad an/da cxriliT gamosaxul damokidebulebas aRwers sityvierad.

• sxvadasxva xerxiT (grafikulad, cxrilis saxiT, sityvierad, algebrulad) gamosaxul damokidebulebebs Soris miuTiTebs erTsa da imave damokidebulebebs.

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


• sxvadasxva xerxiT mocemuli simravlisaTvis gansazRvravs mocemuli elementis kuTvnilebas mocemuli simravlisadmi.

• problemis gadaWrisas, iyenebs zogierT damxmare xerxs simravleTa Soris mimarTebebis dasadgenad da simravluri operaciebis Sesasruleblad.

• sworad iyenebs simravleTa Teoriis cnebebs da Sesabamis aRniSvnebs sasrul simravleebze operaciebis (ori simravlis TanakveTa da gaerTianeba), sasrul simravleTa Soris mimarTebis, elementsa da simravles Soris mimarTebis gamosaxvisas.

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


• teqsturi amocanis amosaxsnelad adgens da xsnis erTucnobian wrfiv

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gantolebas.• 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 gamoyenebiT asabuTebs an uaryofs ori algebruli (araumetes ori cvladis Semcveli wrfivi an meore xarisxis) gamosaxulebis igivur tolobas.

maT. VII.8. moswavle ganavrcobs da aanalizebs obieqtebis periodul mimdevrobas an/ da mudmivi nazrdis mqone ricxviT mimdevrobas


• periodul mimdevrobaSi gamoyofs mimdevrobis periods.• warmoadgens mimdevrobis mocemuli fragmentis gavrcobis 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; asaxelebs realur

viTarebaSi iseTi procesis magaliTebs, romlebic aseTi mimdevrobiT aRiwere-ba.

SmimarTuleba: geometria da sivrcis aRqma

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


• arqiteqturisa da xelovnebis nimuSebSi an maT ilustraciebSi, yofiTi daniSnulebis sagnebSi asaxelebs misTvis nacnob geometriul figurebs an maT nawilebs.

• ayalibebs mimarTebebs (mag. zogadoba-kerZooba) figuraTa saxeobebs Soris.• asaxelebs figuras misi niSan-Tvisebebis mixedviT, msjelobs figuris

amosacnobad maTi sakmarisobis/arasakmarisobis Sesaxeb.

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


• agebs dasmuli amocanis Sesabamis naxazs da adekvaturad iyenebs asoiT aRniS-vnebs.

• aRwers geometriul obieqtTa mocemul grafikul gamosaxulebebs an obieqtTa urTierTmdebareobas Sesabamisi terminologiis gamoyenebiT. (mag. marTkuTxa paralelepipedis romel waxnagebs ekuTvnis miTiTebuli wvero).

• gamosaxavs brtyel figurebs ise, rom maTi TanakveTa/gaerTianeba iyos miTiTe-buli formis an Tvisebebis mqone figura.

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


• garemomcvel obieqtebs Soris moiZiebs simetriul obieqtebs.• xazavs brtyeli figuris (texili, mravalkuTxedi) simetriul figuras

miTiTebuli simetriis RerZis mimarT; axdens brtyeli figuris (texili, mravalkuTxedi) paralelur gadatanas.

• uTiTebs brtyeli figuris simetriis RerZs/RerZebs; axdens simetriulobis demonstrirebas; iyenebs figuris simetriulobas figuris Tvisebis dasadgenad.

maT. VII.12. moswavle iyenebs koordinatebis meTods orientaciisaTvis


• orientirebs ruqaze an sakoordinato sibrtyeze koordinatebis gamoyenebiT (mag. asaxelebs mocemuli wertilis koordinatebis miaxloebiT an zust mniSvnelobas; poulobs wertils mocemuli mTelricxovani koordinatebis mixedviT).

• asaxelebs sakoordinato RerZebis mimarT mocemuli wertilis RerZulad simetriuli wertilis koordinatebs.

• poulobs paraleluri gadataniT miRebuli figuris nebismieri wertilis koordinatebs misi winasaxis koordinatebisa da miTiTebuli paraleluri gadatanis meSveobiT.

maT. VII.13. moswavle xsnis geometriul amocanebs samkuTxedebTan dakavSirebuli cnebebisa da faqtebis gamoyenebiT


• iyenebs samkuTxedebis tolobis niSnebs figuraTa Tvisebebis 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.


maT. VII.14. moswavle moipovebs dasmuli amocanis amosaxsnelad saWiro Tvisebriv da raodenobriv monacemebs


• ganasxvavebs Tvisebriv da raodenobriv monacemebs, iyenebs monacemTa Seg-rovebis Sesaferis saSualebas (gazomva, dakvirveba).

• mocemul TemasTan dakavSirebiT svams kiTxvebs, gansazRvravs respondentebs da moipovebs saWiro monacemebs.

• mocemuli amocanisTvis damoukideblad gegmavs da atarebs statistikur eqs-periments da agrovebs monacemebs.

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maT. VII.15. moswavle awesrigebs da warmoadgens Tvisebriv da raodenobriv monacemebs dasmuli amocanis amosaxsnelad xelsayreli formiT


• axdens Tvisebriv da raodenobriv monacemTa dalagebas/klasifikacias, 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. moswavle akeTebs Tvisebriv da raodenobriv monacemTa interpretacias da analizs amocanis konteqstis gaTvaliswinebiT


• svams kiTxvebs monacemebis Sesaxeb an/da axasiaTebs monacemebs, romlebic war-modgenilia siis, cxrilis, piqtogramis an diagramis saxiT, msjelobs arse-bul kanonzomierebebsa da gamorCeul monacemebze.

• irCevs Sesaferis Semajamebel ricxviT maxasiaTeblebs, asabuTebs Tavis arCev-ans, iTvlis da iyenebs maT monacemTa jgufis dasaxasiaTeblad.

• adarebs monacemTa ramdenime jgufs da warmoaCens Tvisebriv da raodenobriv msgavsebasa da gansxvavebas maT Soris (Semajamebeli ricxviTi maxasiaTeblebis gareSe).

programis Sinaarsi


_ mTeli ricxvebi da ariTmetikuli moqmedebebi mTel ricxvebze._ wiladebi, aTwiladebi da zogierTi kavSiri maT Soris._ procenti (100-ze naklebi an toli da 1-ze meti an toli procenti);

kavSirebi: erTidaigive mTelis procenti da nawili._ ricxvebis Sedareba da ariTmetikuli moqmedebebis Sedegis Sefaseba. _ ricxvis proporciul nawilebad dayofa. _ ricxvebis umciresi saerTo jeradi da udidesi saerTo gamyofi; ricxvis

martiv mamravlebad daSla._ ricxvis naturalurmaCvenebliani xarisxi_ naSTiT gayofa, naSTi da gayofadobis niSnebidan zogierTi._ zomis erTeulebi, zomis erTeulebs Soris kavSirebi da zomis erTeulebis

gamoyeneba: masStabi; erTi sistemis erTeulis sxva sistemis Sesabamisi erTeuliT gamosaxva.

_ fasdakleba/fasis gazrda (TanmimdevrobiTi da erTjeradi fasdaklebebis/fasebis zrdis erTmaneTTan Sedareba) da martivi xarjTaRricxva.

_ pirdapirproporciuli damokidebuleba da misi gamosaxva grafikis da cxrilis saSualebiT.

_ simravleTa Teoriis cnebebi, operaciebi da Sesabamisi aRniSvnebi sasruli simravleebis SemTxvevaSi.

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_ teqsturi amocanebis amoxsna wrfivi gantolebebis gamoyenebiT._ araumetes ori cvladis Semcveli wrfivi an meore xarisxis gamosaxulebebis

gamartiveba da mniSvnelobis gamoTvla._ perioduli mimdevrobebi da mudmivi nazrdis mqone ricxviTi mimdevrobebi._ wertilebi, wrfeebi da sibrtyeebi: mimarTebebi maT Soris._ geometriuli figurebi: klasifikacia sxvadasxva niSniT (mag. amozneqili da

araamozneqili, brtyeli da sivrculi)._ kuTxeebi: elementebi, zoma, klasifikacia, Tvisebebi._ samkuTxedebi: elementebi, klasifikacia, Tvisebebi, tolobis niSnebi._ geometriuli gardaqmnebi sibrtyeze: paraleluri gadatana, RerZuli

simetria, samkuTxedebi (elementebi, klasifikacia, Tvisebebi, tolobis niSnebi).

_ koordinatTa sistema; orientireba sibrtyeze, gardaqmnebis gamosaxva._ agebis umartivesi amocanebi: mocemuli samkuTxedis toli samkuTxedis,

kuTxis biseqtrisi, monakveTis SuamarTobis ageba._ monacemTa Segrovebis saSualebani: gazomva da dakvirveba; gamokiTxva;

statistikuri eqsperimenti._ Tvisebrivi da raodenobrivi monacemebis organizacia: monacemebis

klasifikacia (garda intervalebad dajgufebisa); monacemTa dalageba zrdadoba-klebadobiT an leqsikografiuli meTodiT.

_ monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: monacemTa raodenoba, pozicia da Tanmimdevroba erTobliobaSi, monacemTa sixSire; ganmeorebis tipis kanonzomierebani; gamorCeuli (mag., eqstremaluri, iSviaTi) monacemebi.

_ monacemTa warmodgenis saSualebani raodenobrivi da Tvisebrivi monacemebisTvis: sia, cxrili, piqtograma, wertilovani, meseruli, xazovani, svetovani diagramebi.

_ monacemTa Semajamebeli ricxviTi maxasiaTeblebi Tvisebrivi da raodenobrivi monacemebisTvis: centraluri tendenciis sazomebi - saSualo, moda; monacemTa gafantulobis sazomi - gabnevis diapazoni.

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VIII klasi

wlis bolos misaRwevi Sedegebi

mimarTuleba:


kanonzomiere-bebi da alge-

bra


monacemTa anali-zi, albaToba da

statistika

maT. VIII. 1. moswavle iyenebs poziciur sistemas da ricxvis Caweris standartul formas

maT. VIII. 2. moswavle asrulebs moqmede-bebs racionalur ricxvebze da afasebs moqmedebaTa Sedegs

maT. VIII. 3. moswavle iyenebs msjeloba-dasabuTebis zogierT xerxs

maT. VIII. 4. moswavle xsnis gamoTvlebTan dakavSirebul amoca-nebs

maT. VIII. 5. moswa-vle amoicnobs, aanalizebs da ga-mosaxavs sidideebs Soris wrfiv damo-kidebulebas

maT. VIII. 6. mo-swavle agebs, gamo-saxavs da ikvlevs Sesabamisobas or simravles Soris

maT. VIII. 7. moswa-vle iyenebs ganto-lebaTa sistemebs da utolobebs problemis ga-daWrisas

maT. VIII. 8. moswav-leiyenebs figuraTa Tvisebebs figuraTa saxeobebis Sedarebi-sa da klasificirebi-sTvis

maT. VIII. 9. moswavle poulobs figuris an/da misi elementebis zomebs

maT. VIII. 10. moswavle asabuTebs geometriu-li debulebebis mar-Tebulobas

maT. VIII. 11. moswav-le moipovebs monace-mebs da warmoadgens maT dasmuli amo-canis amosaxsnelad xelsayreli formiT

maT. VIII. 12. moswavle amoicnobs SemTxveviT movlenebs da iTvlis xdomilobaTa alba-Tobebs

maT. VIII. 13. moswavle fardobiT sixSiresa da albaTobas Soris kavSiris gamoyenebiT afasebs xdomilobaTa albaTobebs da msje-lobs xdomilobaTa mosalodnelobis Se-saxeb



maT. VIII.1. moswavle iyenebs poziciur sistemas da ricxvis Caweris standartul formas


• mocemuli sizustiT amrgvalebs mTel ricxvebsa da aTwiladebs, ganasxvavebs perioduli aTwiladis SemoklebiT Caweras damrgvalebisgan. (mag. `daamrgvale measedis sizustiT da Seadare 0.7(6) da 0.767~).

• ganmartavs mTelmaCveneblian xarisxs da axdens misi Tvisebebis demonstri-rebas.

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• poziciuri sistemis gamoyenebiT asabuTebs gayofadobis niSnebs; (erTniSna) ricxvis Tanmimdevruli xarisxebis ganxilvisas msjelobs erTeulebis TanrigebSi mdgom cifrTa perioduli ganmeorebis Sesaxeb (mag. “romeli cifri iqneba erTeulebis TanrigSi, Tu poziciuri sistemiT CavwerT 2 xarisxad 11-s?”).

• wers ricxvebs standartuli formiT da piriqiT, standartuli formiT mocemul ricxvs wers poziciuri sistemis gamoyenebiT; adarebs ricxvis Caweris sxvadasxva formebs (mag. ra upiratesoba aqvs standartul formas ricxvebze moqmedebebis Sesrulebisas).

maT. VIII.2. moswavle asrulebs moqmedebebs racionalur ricxvebze da afasebs moqmedebaTa Sedegs


• iyenebs Sefasebas racionalur ricxvebze Sesrulebuli gamoTvlebis (maT So-ris xarisxi da fesvi) Sedegis adekvaturobis Sesamowmeblad.

• iyenebs ricxvis Caweris ekvivalentur formebs (mag. standartuli forma) ga-moTvlebis Sesrulebis da/an gamoTvlebis Sedegis Sefasebisas.

• amocanis konteqstis gaTvaliswinebiT irCevs ra ufro mizanSewonilia - moq-medebaTa Sedegis Sefaseba Tu misi zusti mniSvnelobis povna.

• axdens ricxvidan kvadratuli/kuburi fesvis amoRebisa da ricxvis kvadratSi/kubSi ayvanis operaciebis Tvisebebis (maT Soris, am operaciebis urTierTSe-brunebulobis) demonstrirebas.

maT. VIII.3. moswavle iyenebs msjeloba-dasabuTebis zogierT xerxs


• ganasxvavebs debulebis wanamZRvars/wanamZRvrebs da daskvnas; cvlis debulebis wanamZRvars da msjelobs daskvnis marTebulobis Sesaxeb.

• ayalibebs da asabuTebs martiv debulebas mTeli ricxvebis Tvisebebis an maTze moqmedebebis Sedegis Sesaxeb. (mag. ~Tu kent ricxvs davumatbT kent ricxvs SedegSi miviRebT...~).

• Sesabamis SemTxvevaSi axdens ricxvebis Tvisebebis Sesaxeb gamonaTqvamis aramarTebulobis dasabuTebas (mag. kontrmagaliTis gamoyenebiT); ayalibebs mocemuli debulebis sawinaaRmdego debulebas.

• asabuTebs an xsnis amocanis amoxsnisas gamoyenebul xerxs.

maT. VIII.4. moswavle xsnis gamoTvlebTan dakavSirebul amocanebs


• ori (wrfivi modeliT mocemuli) samomaxmareblo kontraqtidan an momsax-urebis gegmidan ukeTesis SesarCevad asrulebs gamoTvlebs da iRebs gadaw-yvetilebas.

• xsnis bunebismetyvelebis dargebidan momdinare amocanebs gamoTvlebze.• iyenebs gamoricxvis an amowurvis meTods ricxvebze amocanebis amoxnisas da

ganmartavs gamoyenebul xerxs (mag. avsebs ariTmetikuli moqmedebis weriTi algoriTmis nimuSs, sadac zogierTi ricxvi simboloebiT aris Secvlili).

• irCevs da iyenebs sididis cvlilebis Sesaferis erTeulebs; gamosaxavs mcire erTeuls didi erTeulis gamoyenebiT.

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maT. VIII.5. moswavle amoicnobs, aanalizebs da gamosaxavs sidideebs Soris wrfiv damokidebulebas


• misTvis nacnobi sidideebisaTvis asaxelebs sidideebs Soris wrfiv damokide-bulebebs (mag. Tanabari moZraobisas ganvlili manZilis damokidebuleba droze).

• ganasxvavebs wrfiv da arawrfiv damokidebulebebs miuxedavad damokideb-ulebis gamosaxvis xerxisa; msjelobs wrfiv da arawrfiv damokidebulebebs Soris gansxvavebaze.

• sityvierad Camoyalibebul debulebas sidideebs Soris damokidebulebisa da mimarTebis Sesaxeb gamosaxavs algebrulad; algebrulad mocemul damokideb-ulebas gamosaxavs grafikulad, cxriliT an ayalibebs sityvierad.

maT. VIII.6. moswavle agebs, gamosaxavs da ikvlevs Sesabamisobas or simravles Soris


• agebs realuri viTarebis adekvatur Sesabamisobas or mocemul simravles So-ris (mag. moswavleebi da merxebi saklaso oTaxSi) da cxrilis an sqemis saSu-alebiT gamosaxavs mas.

• asaxelebs erTsa da imave Sesabamisobas Sesabamisobis gamosaxvis xerxisagan damoukideblad.

• raime xerxiT (sityvierad, cxrilis an sqemis saSualebiT) mocemuli Sesabam-isobisaTvis poulobs miTiTebuli simravlis anasaxs/winasaxes.

maT. VIII.7. moswavle iyenebs gantolebaTa sistemebs da utolobebs problemis gadaWrisas


• teqsturi amocanis amosaxsnelad adgens da xsnis orucnobian wrfiv gantolebaTa sistemas; axdens amonaxsnis interpretacias amocanis konteqstis gaTvaliswinebiT.

• irCevs xerxs da xsnis orucnobian wrfiv gantolebaTa sistemas; axdens amonaxsnis simravlur da geometriul interpretacias.

• teqsturi amocanebis amoxsnisas an/da realuri viTarebis modelirebisas adgens da xsnis erTucnobian wrfiv utolobebs; axdens amonaxsnis simravlur interpretacias.


maT. VIII.8. moswavle iyenebs figuraTa Tvisebebs figuraTa saxeobebis Sedarebisa da klasificirebisTvis


• ayalibebs mimarTebebs (mag. zogadoba-kerZooba) figuraTa saxeobebs an Tvise-

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bebs Soris, sqematurad gamosaxavs am mimarTebebs (mag. cxrilis an diagramis saSualebiT).

• figuris mocemul Tvisebebs (maT Soris simetriuloba) Soris irCevs Tviseba-Ta im minimalur erTobliobas, romelic calsaxad gansazRvravs am figuras.

• mocemuli xedebis mixedviT asaxelebs sivrculi figuris SesaZlo saxeobas.

maT. VIII.9. moswavle poulobs figuris an/da misi elementebis zomebs


• iyenebs figuraTa Tvisebebs da toli figurebis Sesabamisi elementebis Sedar-ebis meTods figuris elementis ucnobi zomis mosaZebnad.

• iyenebs dekartes koordinatebs figuris an misi elementis ucnobi zomis mosa-Zebnad.

• poulobs figuris farTobs martiv figurebad dayofis an/da martiv figu-ramde Sevsebis xerxiT.

maT. VIII.10. moswavle asabuTebs geometriuli debulebebis marTebulobas


• deduqciuri da induqciuri msjelobis nimuSSi aRadgens gamotovebul safex-urs/safexurebs.

• iyenebs algebrul gardaqmnebs, tolobisa da utolobebis Tvisebebs geome-triul debulebaTa dasabuTebisas.

• iyenebs dekartes koordinatebs geometriuli obieqtis Tvisebebis dasadgenad da dasabuTebisTvis (mag. marTkuTxedis diagonalebis tolobis saCveneblad).

• iyenebs geometriul gardaqmnebs da maT kompoziciebs sibrtyeze figuraTa So-ris mimarTebis (mag. tolobis) dasabuTebisTvis.


maT. VIII.11. moswavle moipovebs monacemebs da warmoadgens maT dasmuli amocanis amosaxsnelad xelsayreli formiT


• atarebs SemTxveviT eqsperiments SemTxveviTobis warmomqmneli romelime mow-yobilobiT, agrovebs monacemebs da wamoadgens maT sixSiruli cxrilis sax-iT.

• qmnis martiv kiTxvars, gansazRvravs respondentebs, agrovebs monacemebs da warmoadgens maT grafikuli formiT.

• erTi grafikuli formiT warmodgenil monacemebs warmoadgens gansxvavebuli grafikuli formiT da warmoaCens TiToeuli formis xelsayrel da araxel-sayrel mxareebs

maT. VIII.12. moswavle amoicnobs SemTxveviT movlenebs da iTvlis xdomilobaTa albaTobebs


• asaxelebs aucilebel da SeuZlebel xdomilobebs, mocemuli xdomilobis

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sawinaaRmdego xdomilobas, Tanabrad mosalodnel xdomilobebs, mocemul xdomilobaze metad/naklebad mosalodnel xdomilobebs.

• aRwers SemTxveviTi eqsperimentis xdomilobebis erTobliobas, iyenebs vari-antebis daTvlis xerxebs xdomilobaTa albaTobebis gamosaTvlelad.

• iyenebs albaTobis Tvisebebs xdomilobaTa albaTobebis gamosaTvlelad, ga-mosaxavs xdomilobaTa albaTobebs wiladebis, aTwiladebis da procentebis saSualebiT

maT. VIII.13. moswavle fardobiT sixSiresa da albaTobas Soris kavSiris gamoyenebiT afasebs xdomilobaTa albaTobebs da msjelobs xdomilobaTa

mosalodnelobis Sesaxeb


• akeTebs monacemTa pirvelad damuSavebas da mis safuZvelze gamoTqvams varauds xdomilobis Sesaxeb – aris Tu ara ori an ramdenime xdomiloba Tanabrad mo-salodneli, erTi romelime xdomiloba ufro mosalodneli, vidre meore da ramdenjer.

• atarebs SemTxveviT eqsperiments SemTxveviTobis warmomqmneli mowyobilobiT da afasebs xdomilobis albaTobas fardobiTi sixSiris saSualebiT, msjelobs gansxvavebaze Teoriul (mosalodnel) Sedegebsa da empiriul (eqsperimentul) Sedegebs Soris.

• qmnis SemTxveviTobis warmomqmnel mowyobilobas fardobiTi sixSiris kerZo mniSvnelobis misaRebad.

programis Sinaarsi


_ racionaluri ricxvebi da maTi ekvivalenturi formebiT Cawera._ 1-ze naklebi / 100-ze meti procenti._ ricxvis Caweris standartuli forma da misi kavSiri poziciur sistemasTan._ mTelmaCvenebliani xarisxi_ ariTmetikuli fesvi ricxvidan; kuburi fesvi ricxvidan._ ricxvebisa da ricxviTi gamosaxulebebis (maT Soris xarisxebis an

ariTmetikuli fesvebis Semcveli gamosaxulebebis) Sedareba._ ariTmetikuli moqmedebebi ricxvebze; moqmedebebis Sedegis Sefaseba._ naSTiT gayofa; naSTi da gayofadobis niSnebi._ zomis erTeulebi, maT Soris kavSirebi da gamoyeneba: mimarTeba sigrZisa

da farTobis erTeulebs Soris; erTi sistemis erTeulis sxva sistemis Sesabamisi erTeuliT gamosaxva

_ ~samomxmareblo ariTmetika~: martivad daricxuli saprocento ganakveTi; sxvadasxvagvari fasdakleba; martivi xarjTaRricxva

_ wrfivi damokidebuleba da misi gamosaxva grafikis, cxrilis da gantolebis saSualebiT.

_ Sesabamisobebi sasrul simravleebs Soris da maTi gamosaxvis xerxebi; simravlis anasaxi da winasaxe.

_ orucnobian wrfiv gantolebaTa sistemebi da maTi gamoyeneba teqsturi amocanebis amoxsnisas.

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_ erTucnobiani wrfivi utolobebi._ oTxkuTxedebi: elementebi, klasifikacia, Tvisebebi._ marTkuTxedis, paralelogramis, trapeciis, wesieri mravalkuTxedis

farTobi, marTi prizmisa da wesieri piramidis zedapiris farTobi._ piTagoras Teorema._ koordinatTa sistema: or wertils Soris manZili sibrtyeze, gamoyeneba

figuraTa Tvisebebis kvlevaSi._ geometriuli gardaqmnebi sibrtyeze: mobruneba, gardaqmnaTa kompoziciebi,

maTi gamoyeneba figuraTa tolobis dasadgenad._ monacemTa Segrovebis saSualebani: kiTxvaris/anketis Sedgena da

respondentTa gamokiTxva (warmomadgenlobiTi jgufis SerCevis gareSe); SemTxveviTi eqsperimenti, SemTxveviTobis warmomqmneli mowyobilobebi - moneta, urna, kamaTeli, ruleti.

_ monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: monacemTa fardobiTi sixSire monacemTa warmodgenis saSualebani: wriuli diagrama fardobiTi sixSiris diagrama

_ albaToba: aucilebeli da SeuZlebeli xdomilobani, mocemuli xdomilobis sawinaaRmdego xdomiloba; variantebis daTvlis xerxebis gamoyeneba SemTxveviTi eqsperimentis aRsawerad (magaliTad, xisebri diagrama an sxva sqemebi); xdomilobis albaToba, albaTobis Tvisebebi; fardobiT sixSiresa da albaTobas Soris kavSiri da gansxvaveba

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stika

maT. IX. 1. moswavle adarebs racionalur ricxvebs da axdens maT klasifikacias

maT. IX. 2. moswavle sxvadasxva xerxiT asrulebs moqmede-bebs racionalur ri-cxvebze da afasebs am moqmedebebis Sedegs

maT. IX. 3. moswavle iyenebs msjeloba-dasabuTebis zogierT xerxs

maT. IX. 4. moswavle xsnis gamoTvlebTan da raodenobis Sefa-sebasTan dakavSire-bul amocanebs

maT. IX. 5. moswa-vle problemebis gadaWrisas iyenebs diskretuli maTema-tikis elementebs

maT. IX. 6. moswavle iyenebs funqciebs da maT Tvisebebs sidi-deebs Soris damoki-debulebis aRsawerad da gamosakvlevad

maT. IX. 7. moswavle iyenebs gantolebaTa sistemebs da uto-lobebs problemis gadaWrisas

maT. IX. 8. moswavle poulobs/afasebs figurebis an maTi elementebis zo-mebs da iyenebs maT praqtikuli proble-mebis gadaWrisas

maT. IX. 9. moswavle ikvlevs da iyenebs geometriul gar-daqmnebs da maT kom-poziciebs

maT. IX. 10. moswavle iyenebs wertilTa geometriuli adgi-lis cnebas obieqtTa gamosaxvisa da maTi Tvisebebis aRsawerad

maT. IX. 11. moswav-le awesrigebs da warmoadgens mo-nacemebs dasmuli amocanis amosaxsne-lad xelsayreli formiT

maT. IX. 12. moswa-vle iTvlis/afa-sebs damoukidebel xdomilobaTa al-baTobebs SemTxvevi-Ti eqsperimentebi-saTvis dabrunebiT da dabrunebis gareSe

maT. IX. 13. moswav-le akeTebs mo-nacemTa analizs da ayalibebs daskv-nebs



maT. IX.1. moswavle adarebs racionalur ricxvebs da axdens maT klasifikacias


• amrgvalebs, adarebs da alagebs sxvadasxva saxiT mocemul racionalur ricxvebs.

• ganasxvavebs racionalur da iracionalur ricxvebs, rogorc periodul da ara-periodul aTwiladebs da moyavs iracionaluri ricxvis magaliTebi (mag.

2 ).• aRniSnavs naSTis periodulobas erTniSna ricxvze naturaluri ricxvebis

Tanmimdevrulad gayofisas; ganmartavs SemCneul kanonzomierebas.• wers racionalur ricxvebs ekvivalenturi (maT Soris standartuli) formiT;

adarebs da alagebs sxvadasxva saxiT mocemul racionalur ricxvebs (xarisxi, standartuli forma da a.S.).

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maT. IX.2. moswavle sxvadasxva xerxiT asrulebs moqmedebebs racionalur ricxvebze da afasebs am moqmedebebis Sedegs

Sedegi TvalsaCinoa, Tu moswavle:: • iyenebs gayofadobis niSnebs da naSTis Tvisebebs ricxvebisa da ariTmetikuli

moqmedebebis Sedegis Tvisebebze msjelobisas (mag. “ras miviRebT naSTSi Tu 2345 gavyofT 3-ze?”).

• irCevs da iyenebs racionaluri ricxvebze ariTmetikul moqmedebaTa, agreTve xarisxSi ayvanisa da fesvis amoRebis operaciebis Sesrulebis optimalur xe-rxs. (mag. martiv mamravlebad Slis ricxvs da poulobs am ricxvidan fesvis mniSvnelobas).

• amocanis konteqstis gaTvaliswinebiT irCevs, ra ufro mizanSewonilia - moq-medebaTa Sedegis Sefaseba, Sedegis miaxloebiTi Tu zusti mniSvnelobis povna. (mag. ~sayofacxovrebo~ amocana, romelic dakavSirebulia ramdenime sagnis SesaZenad saWiro Tanxis qona/arqonasTan).

• iyenebs moqmedebaTa Tvisebebs, Tanmimdevrobas da maT Soris kavSirs raciona-luri ricxvebze moqmedebebis (maT Soris mTelmaCvenebliani xarisis da ariT-metikuli fesvis) Semcveli gamosaxulebis gasamartiveblad.

• amrgvalebs ricxviT wevrebs (mag. Sekrebis dros - Sesakrebebs) da poulobs racionalur ricxvebze ariTmetikul moqmedebaTa Sedegis miaxloebiT mniSvn-elobas.

maT. IX.3. moswavle iyenebs msjeloba-dasabuTebis zogierT xerxs


• ayalibebs da asabuTebs (ricxvebs Soris damokidebulebebze, maT Tvisebebze an maTze moqmedebebis Sedegis Sesaxeb) martiv debulebas; Sesabamis SemTxvevaSi axdens gamonaTqvamis uaryofas (moyavs konatrmagaliTi); ayalibebs sawinaaRmdego debulebas.

• amocanebis amoxsnisas iyenebs ricxviT simravleebs Soris damokidebulebis gamosaxvis zogierT xerxs (mag. diagramebs an sxva grafikul gamosaxulebebs).

3 axdens ricxvebis saSualo ariTmetikulisa da saSualo geometriulis interpretaciasa da erTmaneTTan Sedarebas; iyenebs maT Tvisebebs amocanebis amoxsnisas.

• iyenebs naSTTa ariTmetikis elementebs (ricxvebis Sekreba/gamokleba moduliT 12, 60 an 360. mag. iseTi amocanebis amoxsnisas, romlebic dakavSirebulia saaTTan an kuTxiT mobrunebasTan).

maT. IX.4. moswavle xsnis gamoTvlebTan da raodenobis SefasebasTan dakavSirebul amocanebs


• asrulebs gamoTvlebs da adarebs or martivad daricxul saprocento ganakveTs, sxvadasxvagvar fasdaklebas, dabegvras; msjelobs maT Soris Soris gansxvavebaze.

• amyarebs kavSirs mTlian Semosavalsa/mogebasa da sacalo fass Soris, moTxovnasa da cnobili xarjebiT miwodebas Soris mocemuli wrfivi damokidebulebis mixedviT. (mag. Tu wignis fasia 20 lari, maSin gaiyidena 20000

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cali. gamocdilebiT cnobilia, rom sacalo fasis yoveli 3 lariani mateba gayidvebis raodenobis 500 caliT Semcirebas iwvevs. ra unda iyos minimaluri sacalo fasi, rom Semosavali iyos 576000 lari?).

• asrulebs pirad xarjTaRricxvasTan, SemosavlTan dakavSirebul gamoTvlebs an/da Sefasebebs Semdgomi moqmedebis dagegmvis mizniT.

• xsnis sxva saswavlo disciplinebidan momdinare gamoTvlebTan dakavSirebul amocanebs.


maT. IX.5. moswavle problemebis gadaWrisas iyenebs diskretuli maTematikis elementebs


• gegmis an ganrigis Sedgenisas iyenebs xisebr diagramebs; erTi niSniT optimal-uri marSrutis (mag. umoklesi gzis) mosaZebnad (algoriTmebis gareSe) iyenebs grafebs.

• irCevs da iyenebs romelime xerxs (CamonaTvali, xisebri diagrama) variantebis daTvlisas (mag. sxvadasxva simravleebis elementebis kombinaciebis yvela Ses-aZlo variantis, simravlis elementTa amorCevis, dalagebisa da gadanacvle-bebis raodenobis mosaZebnad).

• realuri procesebis diskretuli modelebiT aRwerisas iyenebs rekurentul wess (mag. mosaxleobis raodenobis yovelwliuri mudmivi procentuli zrda); ganavrcobs rekurentuli wesiT mocemul mimdevrobas (n-uri wevris formu-lis gareSe).

• akavSirebs simravlur operaciebs (gaerTianeba, TanakveTa, damateba) Sesabamis logikur operaciebTan (an, da, ara).

maT. IX.6. moswavle iyenebs funqciebs da maT Tvisebebs sidideebs Soris damokidebulebis aRsawerad da gamosakvlevad


•� mocemuli funqciisaTvis, romelic aRwers realur viTarebas, poulobs fun-qciis mniSvnelobas, nulebs, maqsimums/minimums, zrdadoba/klebadobisa da niSanmudmivobis Sualedebs da axdens maT interpretacias am viTarebis konte-qstSi.

•� axdens funqciis grafikis Tvisebebis (daxris koeficienti da sakoordinato RerZebTan gadakveTa) interpretirebas sidideebs Soris damokidebulebis gas-aanalizeblad.

•� cvlis funqciis parametrebs da aRwers am cvlilebis Sedegis interpretire-bas im procesis konteqstSi romelic am funqciiT a R i w e r e b a (mag. man-Zilis droze damokidebulebis aRmwer funqciaSi ra gavlenas axdens siCqaris cvlileba ganvlil manZilze?).

maT. IX.7. moswavle iyenebs gantolebaTa sistemebs da utolobebs problemis gadaWrisas

Sedegi TvalsaCinoa, Tu moswavle::

• teqsturi amocanis amosaxsnelad adgens da xsnis orucnobian wrfiv gan-

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tolebaTa sistemas; axdens sistemis amonaxsnis interpretacias amocanis konteqstis gaTvaliswinebiT.

• teqsturi amocanebis amoxsnisas da/an realuri viTarebis modelirebisas ad-gens da xsnis erTucnobian utolobaTa sistemas; axdens amonaxsnis simravlur in-terpretacias.

• adarebs or funqcias, romlebic realur process gamosaxavs (poulobs im sim-ravles sadac erTi funqcia metia/naklebia meore funqciaze, tolia meore funqciis) da axdens Sedarebis Sedegis interpretirebas konteqstTan mimar-TebaSi.


maT. IX.8. moswavle poulobs/afasebs figurebis an maTi elementebis zomebs da iyenebs maT praq-tikuli problemebis gadaWrisas


• axdens sibrtyeze mocemuli wiris miaxloebas texilis saSualebiT da iyenebs am meTods wiris sigrZis Sefasebisas an miaxloebiT gamoTvlisas. (mag. mrud wirze moZraobis marSrutis sigrZis miaxloebiTi gamoTvla; wrewiris sigrZis miaxloebiTi gamoTvla).

• daadgens figuris zomebs Soris damokidebulebis tips da iyenebs am damokidebulebas amocanebis amosaxsnelad (mag. kvadratis farTobis damokidebuleba gverdze; wris farTobis damokidebuleba mis radiusze).

• iyenebs marTkuTxa samkuTxedis gverdebsa da kuTxeebs Soris trigonometriul Tanafardobebs realur viTarebaSi obieqtTa zomebis an obieqtebs Soris manZilebis dasadgenad (mag. im sagnis simaRlis gazomva, romlis fuZe miudgomelia, miudgomel wertilamde manZilis gamoTvla).

• figurebis Tvisebebis mixedviT msjelobs mocemuli figurebis gamoyenebiT sibrtyis nawilis dafarvis Sesaxeb (maT Soris realur viTarebaSi).

maT. IX.9. moswavle ikvlevs da iyenebs geometriul gardaqmnebs da maT kompoziciebs


• msjelobs Tu ra geometriuli gardaqmna SeiZleba iyos mocemuli ori geomet-riuli gardaqmnis kompozicia; asabuTebs Tavis mosazrebas.

• figurebis Sesaxeb sxvadasxva monacemebis safuZvelze gamoTqvams varauds imis Sesaxeb SeiZleba Tu ara, mocemuli gardaqmnis gamoyenebiT, mocemuli figurisagan meore mocemuli figuris miReba.

• iyenebs geometriuli figuris Tvisebebs da geometriul gardaqmnebs imis da-sabuTebisaTvis, SesaZlebelia Tu ara sibrtyis dafarva; axdens dafarvis demon-strirebas sibrtyis nawilze.

maT. IX.10. moswavle iyenebs “wertilTa geometriuli adgilis” cnebas obieqtTa gamosaxvisa da maTi Tvisebebis aRsawerad


• wertilTa geometriuli adgilis sityvieri aRweris mixedviT asaxelebs an gamosaxavs im geometriul figuras an figuris elements romelic am aRweras

S(t)=v t+S

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Seesabameba (mag. “im wertilTa simravle romelic Tanabradaa daSorebuli mocemuli kuTxis gverdebidan aris am kuTxis biseqtrisa”).

• iyenebs “wertilTa geometriuli adgilis meTods” agebis amocanebis amoxsnisas (mag. “kuTxis biseqtrisa aris am kuTxis gverdebidan Tanabrad daSorebul wertilTa simravle, e.i. imisaTvis rom avagoT biseqtrisa saWiroa . . . ”).

• wertilTa geometriuli adgilebis sxvadasxva aRwerebis mixedviT daadgens mimarTebas Sesabamis figurebs Soris (mag. erTi da igivea Tu ara es figurebi? erTi figura meore figuris nawilia Tu ara?).


maT. IX.11. moswavle awesrigebs da warmoadgens monacemebs dasmuli amocanis amosaxsnelad xelsayreli formiT


• ajgufebs raodenobriv monacemebs intervalTa klasebSi da agebs Sesabamis cxrils/histogramas (maT Soris teqnologiebis gamoyenebiT).

• arCevs daujgufebel raodenobriv monacemTa warmodgenis Sesaferis grafi-kul formas, asabuTebs arCevans da qmnis cxrils/diagramas (teqnologiebis gamoyenebiT an mis gareSe).

• erTi grafikuli formiT warmodgenil monacemebs warmoadgens gansxvavebuli grafikuli formiT da warmoaCens TiToeuli formis xelsayrel da araxel-sayrel mxareebs

maT. IX.12. moswavle iTvlis/afasebs damoukidebel xdomilobaTa albaTobebs SemTxveviTi eqsperimentebisaTvis dabrunebiT da dabrunebis gareSe


• iyenebs albaTobis Tvisebebsa da formulebs (jamisa da namravlis) xdomilo-baTa albaTobis gamosaTvlelad.

• gegmavs SemTxveviT eqsperiments, SemTxveviTi eqsperimentis Casatareblad erT mowyobilobas Caanacvlebs sxva mowyobilobiT da asabuTebs arCevans.

• asaxelebs rTuli xdomilobis xelSemwyob elementarul xdomilobebs da iy-enebs albaTobis klasikur gansazRvras rTuli xdomilobis albaTobis gamo-saTvlelad.

maT. IX.13. moswavle akeTebs monacemTa analizs da ayalibebs daskvnebs


• amocanis konteqstis gaTvaliswinebiT irCevs Sesaferis Semajamebel ricxviT maxasiaTeblebs, asabuTebs Tavis arCevans, iTvlis da iyenebs maT monacemTa erTobliobebis dasaxasiaTeblad/Sesadareblad.

• iyenebs grafikuli formiT warmodgenil monacemebs statistikuri Sinaarsis mosazrebaTa/argumentebis Camosayalibeblad an Sesafaseblad.

• gamoTqvams varauds xdomilobis mosalodnelobis Sesaxeb monacemTa safuZvelze (mag. fardobiTi sixSiris mixedviT) da asabuTebs varaudis marTlzomierebas.

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programis Sinaarsi

Sedegebis miRweva SesaZlebelia mocemuli Sinaarsis safuZvelze: _ racionaluri ricxvTa simravle da misi qvesimravleebi (naturalur da

mTel ricxveTa simravleebi)._ iracionalur ricxvebTan gacnoba (mag. 2 )._ ariTmetikuli moqmedebebi da maTi Sedegis Sefaseba._ fesvis amoReba; fesvis Semcveli martivi ricxviTi gamosaxulebis

mniSvnelobis Sefaseba._ sxvadasxva saxiT mocemuli ricxvebis Sedareba._ proporcia da ukuproporcia._ naSTTa ariTmetikis elementebi (gacnobis wesiT: ~bolo cifris ariTmetika~,

naSTiT gayofa)_ zomis erTeulebi, maT Soris kavSirebi da gamoyeneba: farTobisa da

moculobis erTeulebs Soris mimarTebebi._ ~samomxmareblo ariTmetika~: martivad da rTulad daricxuli saprocento

ganakveTi; xarjTaRricxva; sxvadasxva gadasaxadi, fasdakleba, amortizacia._ wrfivi funqcia, kvadratuli funqcia maTi grafikebi da Tvisebebi:

zrdadoba/klebadoba, niSanmudmivobis Sualedebi, nulebi, mocemul intervalze maqsimumis/minimumis wertilebi da Sesabamisi mniSvnelobebi, gansazRvris are da mniSvnelobaTa simravle.

_ erTucnobian utolobaTa sistemebi._ orucnobian gantolebaTa sistemebi (erTi gantoleba mainc wrfivia, xolo

meoris xarisxi ar aRemateba ors)._ optimizaciis amocanebi grafebis gamoyenebiT (algoriTmebis gareSe)._ ariTmetikuli/geometriuli progresia da zogierTi sxva rekurentuli

wesiT mocemuli mimdevroba._ msgavsi mravalkuTxedebi._ trigonometriuli Tanafardobebi marTkuTxa samkuTxedSi._ geometriuli gardaqmnebi da maTi kompoziciebi: msgavsebis gardaqmna,

mimarTebebi gardaqmnaTa kompoziciebs Soris._ wrewiri da wre: maTTan dakavSirebuli monakveTebi da maTi Tvisebebi,

centraluri da Caxazuli kuTxeebi._ wrewiris sigrZe da wris farTobi (damtkicebis gareSe)._ geometriuli adgilis cneba da misi gamoyeneba agebis amocanebSi._ veqtorebi sibrtyeze: Sekreba, skalarze gamravleba._ monacemTa organizacia: raodenobriv monacemTa dajgufeba intervalTa

klasebad_ monacemTa warmodgenis saSualebani raodenobrivi da dajgufebuli

monacemebisTvis: foTlebiani Reroebis msgavsi diagrama; sixSiruli poligoni, histograma

_ Semajamebeli ricxviTi maxasiaTeblebi raodenobrivi monacemebisTvis: centraluri tendenciis sazomi – mediana; monacemTa gafantulobis sazomi - saSualo kvadratuli gadaxra

_ albaToba: elementaruli da rTuli xdomilobani; albaTobaTa jamisa da namravlis formulebis gamoyeneba damoukidebel xdomilobaTa albaTobebis gamosaTvlelad

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maT. X. 1. moswavle ganasxvavebs namdvil ricxvTa qvesisiste-mebs

maT. X. 2. moswavle akavSirebs sxvadasxva poziciur sistemebs / namdvil ricxvTa qvesimravleebs erT-maneTTan

maT. X. 3. moswavle asrulebs moqmede-bebs namdvil ri-cxvebze da afasebs maT Sedegs

maT. X. 4. moswavle iyenebs msjeloba -dasabuTebis sxva-dasxva xerxs

maT. X. 5. moswavle wyvets praqtikuli saqmianobidan momdi-nare problemebs

maT. X. 6. moswavle ikvlevs funqciis Tvisebebs da iyenebs maT sidideebs Soris damokidebulebis Se-saswavlad

maT. X. 7. moswav-le iyenebs gantole-baTa da utolobaTa sistemebs modelire-bis saSualebiT pro-blemis gadaWrisas

maT. X. 8. moswa-vle iyenebs diskre-tuli maTematikis elementebs proble-mis modelirebisa da analizisTvis.

maT. X. 9. moswavle flobs da iyenebs geometriul figu-raTa warmodgenisa da debulebaTa for-mulirebis xerxebs

maT. X. 10. moswavle poulobs obieqtTa zomebsa da obieqtTa Soris manZilebs

maT. X. 11. moswavle asabuTebs geome-triuli debulebe-bis marTebulobas

maT. X. 12. moswavle ikvlevs figuraTa geometriul gar-daqmnebs sibrtyeze da iyenebs maT geo-metriuli proble-mebis gadaWrisas

maT. X. 13. moswavle moipovebs dasmuli amocanis amosaxsne-lad saWiro Tvise-briv da raodeno-briv monacemebs

maT. X. 14. moswavle awesrigebs da war-moadgens Tvisebriv da raodenobriv monacemebs dasmuli amocanis amosaxsne-lad xelsayreli formiT

maT. X. 15. moswavle aRwers SemTxvevi-Tobas albaTuri mo-delebis gamoyenebiT

maT. X. 16. moswav-le iyenebs stati-stikur da albaTur cnebebs / Tval-sazrisebs yovel-dRiur viTarebebSi



maT. X.1. moswavle ganasxvavebs namdvil ricxvTa qvesisistemebs


• ganasxvavebs racionalur da iracionalur ricxvebs, rogorc periodul da araperiodul aTwiladebs; asabuTebs ricxvis iracionalurobas / raciona-lurobas an/da axdens iracionalurobis / racionalurobis demonstrirebas modelis gamoyenebiT; axdens iracionaluri ricxvis racionaluri ricxvebis mimdevrobiT miaxloebis demonstrirebas modelis gamoyenebiT.

• mocemuli sizustiT amrgvalebs namdvil ricxvebs; ganasxvavebs usasrulo perioduli aTwiladis SemoklebiT Caweras damrgvalebisagan.

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• ori mocemuli namdvili ricxvisaTvis asaxelebs maT Soris moTavsebul ra-cionalur ricxvs. (mag. asaxelebs 0.6(5) –sa da 0.66 –s Soris moTavsebul ra-cionalur ricxvs).

• axdens namdvili ricxvis aTobiTi poziciuri sistemiT Caweris interpreta-cias da/an mis demonstrirebas modelis gamoyenebiT (mag. axdens 1-ze naklebi dadebiTi namdvili ricxvis miaxloebas [0, 1] monakveTis Tanmimdevruli dan-awilebiT).

maT. X.2. moswavle akavSirebs sxvadasxva poziciur sistemebs / namdvil ricxvTa qvesimravleebs erTmaneTTan


• adarebs sxvadasxva poziciur sistemebs erTmaneTs; msjelobs TiToeulis upiratesobaze ricxvebis Cawerisas. (mag. aTobiTi poziciuri sistema, romauli, egvipturi – sadac aTis xarisxebisTvis Sesabamisi ricxviTi saxelebi/ieroglifebi arsebobda).

• akavSirebs namdvil ricxvTa qvesimravleebs erTmaneTTan simravleTa Teoriis enis gamoyenebiT (qvesimravle, simravleTa TanakveTa, gaerTianeba, sxvaoba, damateba; am mimarTebebis gamosaxva sxvadasxva xerxiT, maT Soris diagramebis saSualebiT).

• sxvadasxva formiT gamosaxavs namdvil ricxvebs (mag. periodul aTwilads Cawers wiladis saxiT); adarebs da alagebs sxvadasxva formiT Caweril namdvil ricxvebs (aTwiladi, wiladi; erTi da igive mTelis nawili da procenti; ricxvis standartuli forma, aTobiTi da orobiTi poziciuri sistema; ricxvis xarisxi da iracionaluri gamosaxuleba).

maT. X.3. moswavle asrulebs namdvil ricxvebze moqmedebebs da afasebs maT Sedegs


• amartivebs namdvil ricxvebze moqmedebebis (agreTve modulis) Semcvel gamosaxulebas moqmedebaTa Tvisebebis, Tanmimdevrobisa da maT Soris kavSiris gamoyenebiT.

• axdens wiladi maCveneblis mqone xarisxis interpretacias da misi Tvisebebis demonstrirebas; adarebs da alagebs erTi da igive fuZis mqone xarisxebs.

• amocanis konteqstis gaTvaliswinebiT irCevs ra ufro mizanSewonilia – moqmedebaTa Sedegis Sefaseba, Tu misi zusti mniSvnelobis povna; iyenebs Sefasebas namdvili ricxvebze Sesrulebuli gamoTvlebis Sedegis adeqvaturobis Sesamowmeblad.

• erTi ariTmetikuli moqmedebis Semcvel gamosaxulebaSi amrgvalebs wevrebs (namdvili ricxvebs) da poulobs moqmedebebaTa Sedegis miaxloebiT mniSvnelobas; msjelobs damrgvalebiT gamowveul gansxvavebaze.

• moyavs fardobiTi azriT ̀ Zalian didi~ da ̀ Zalian mcire~ sidideTa magaliTebi (mag. sinaTlis weli, eleqtronis masa);

maT. X.4. moswavle iyenebs msjeloba-dasabuTebis sxvadasxva xerxs


• asabuTebs martiv debulebas ricxvebis Tvisebebis an ricxviTi kanonzomierebebis

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Sesaxeb; Sesabamis SemTxvevaSi axdens hipoTezis uaryofas kontrmagaliTiT (mag. WeSmaritia Tu mcdari: nebismieri ori kenti ricxvis namravli kentia; nebismieri luwi da kenti ricxvebis sxvaoba luwia da a.S.).

• msjelobis nimuSebSi amoicnobs deduqcias, ganzogadebas da analogias; iyenebs maT mTel ricxvebs Soris damokidebulebebis dasadgenad (mag. romeli cifri dgas ricxvis 23455 erTeulebis TanrigSi?).

• amocanebis amoxsnisas iyenebs ricxviT simravleebs Soris damokidebulebis gamosaxvis zogierT xerxs (mag. venis diagramebs).

• iyenebs “winaaRmdegis daSvebis” meTods ricxvebis Sesaxeb martivi debulebebis damtkicebisas.

maT. X.5. moswavle wyvets praqtikuli saqmianobidan momdinare problemebs


• asrulebs gamoTvlebs da adarebs or martivad/rTulad daricxul saprocento ganakveTs, sxvadasxvagvar fasdaklebas, dabegvras; msjelobs maT Soris Soris gansxvavebaze.

• msjelobs teqnologiebis gamoyenebasTan dakavSirebiT wamoWrili eTikuri/socialuri xasiaTis problemebis Sesaxeb (prezentaciis magaliTi: sxvadasx-vagvari informacia internetSi; sainformacio teqnologiebis/programuli uzrunvelyofis momxmareblis ufleba/movaleobebi, momsaxure mxaris ufleba/movaleobebi).

• msjelobsmsjelobs informaciis Teoriisa da ricxvTa Teoriis praqtikul mxareze, maT rolze/gavlenaze Zvel/Tanamedrove sazogadoebaSi (jgufuri samuSaos nimuSi: teqsturi informaciis kodireba / dekodireba romelime xerxiT; prezentaciis magaliTi: oqros kveTa arqiteqturasa da xelvnebaSi; an fibonaCis mimdevroba da bunebrivi procesebis modelireba/simulireba; anbanis wanacvlebiT daSifvris magaliTebi istoriidan - iulius keisaris Sifri: 5-asoTi wanacvlebuli anbani; mag. meore msoflio omis droindeli germanuli daSifvris manqana ~enigma~).

• iyenebs kuTxis zomis erTeulebs Soris kavSirebs wrewirze mobrunebasTan da/an brunvis Sedegad gadaadgilebasTan dakavSirebuli amocanebis amoxsnisas (mag. lilvTan dakavSirebuli amocanebi).

SeniSvna: me-2 da me-3 indikatorebidan erTi mainc savaldebuloa.


maT. X.6. moswavle ikvlevs funqciis Tvisebebs da iyenebs maT sidideebs Soris damokidebulebis Sesaswavlad


• sidideebs Soris damokidebulebis aRmweri funqciisaTvis (maT Soris realur viTarebaSi) asaxelebs funqciis tips (wrfivi, modulis Semcveli, kvadrat-

uli, xk)x(f = ) am funqciis gamosaxvis xerxisagan damoukideblad.

• sidideebs Soris damokidebulebis aRmweri funqciisaTvis, maT Soris realur viTarebaSi, poulobs funqciis nulebs, funqciis maqsimums/minimums, zrdado-ba/klebadobisa da niSanmudmivobis Sualedebs; axdens am monacemebis interpre-

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tacias realuri viTarebis konteqstSi.• cvlis funqciis parametrebs da axdens am cvlilebis Sedegis interpretirebas

im procesis konteqstSi, romelic am funqciiT aRiwereba (mag. manZilis droze

damokidebulebis aRmwer funqciaSi - 0

( ) = ⋅ +S t v t S , ra gavlenas axdens siCqa-ris cvlileba ganvlil manZilze?).

• adarebs or funqcias, romlebic realur process gamosaxavs (poulobs im simravles sadac erTi funqcia metia/naklebia meore funqciaze, tolia meore funqciis) da axdens Sedarebis Sedegis interpretacias konteqstTan mimarTe-baSi.

maT. X.7. moswavle iyenebs gantolebaTa da utolobaTa sistemebs modelirebis saSualebiT problemis gadaWrisas


• teqsturi amocanis amosaxsnelad adgens da xsnis orucnobian gantolebaTa sistemas; axdens amonaxsnis interpretacias amocanis konteqstis gaTvalis-winebiT.

• irCevs da iyenebs gantolebaTa/utolobaTa sistemis amoxsnis xerxs (mag. Cas-mis, Sekrebis); grafikulad gamosaxavs amonaxsns da axdens amonaxsnis simrav-lur interpretacias.

• wrfivi utolobis an/da ori wrfivi utolobis Semcveli sistemis saSuale-biT gamosaxavs amocanis pirobaSi mocemul SezRudvebs (mag. firmam sarekla-mo kompaniaze unda daxarjos araumetes 2000 larisa. maT dagegmili aqvT ga-moaqveynon aranakleb 10 sareklamo gancxadebisa. dasvenebis dReebSi sareklamo gancxadebis Rirebulebaa 20 lari, xolo kviris danarCen dReebSi 10 lari.).

maT. X.8. moswavle iyenebs diskretuli maTematikis elementebs problemis modelirebisa da analizisTvis.


• iyenebs xisebr diagramebs an/da grafebs, variantebis dasaTvlelad, gegmis/ganrigis Sesadgenad, optimizaciis diskretuli amocanebis amosaxsnelad (al-goriTmebis gareSe) (mag. or obieqts Soris umciresi manZilis moZebna).

• mimdevrobis gamosaxvisas iyenebs rekurentul wess (maT Soris realuri proce-sebis diskretuli modelebiT aRwerisas. mag., mosaxleobis raodenobis yovel-wliuri mudmivi procentuli zrda); ganavrcobs rekurentuli wesiT mocemul mimdevrobas.

• adekvaturad iyenebs simravlur terminebs da cnebebs (mag., funqciis gansaz-Rvris are da mniSvnelobaTa simravle) da operaciebs sasrul simravleebze (TanakveTa, gaerTianeba, sxvaoba, damateba), maT Soris realuri viTarebis mod-elirebisas an/da aRwerisas.


maT. X.9. moswavle flobs da iyenebs geometriul figuraTa warmodgenisa da debulebaTa formulirebis xerxebs


• aRwers geometriul obieqtebs da maT grafikul gamosaxulebebs Sesabamisi

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terminologiis gamoyenebiT.• iyenebs maTematikur simboloebs geometriuli debulebebisa da faqtebis gad-

mocemisas; sworad iyenebs terminebs: `yvela~, àrcerTi~, `zogierTi~, `yove-li~, `nebismieri~, àrsebobs~ da `TiToeuli~.

• msjeloba-dasabuTebisas iyenebs mocemuli pirobiTi winadadebis/debulebis Sebrunebul, mopirdapire da Sebrunebulis mopirdapire winadadebas/debule-bebs.

maT. X.10. moswavle poulobs obieqtTa zomebsa da obieqtTa Soris manZilebs


• obieqtTa zomebisa da obieqtTa Soris manZilebis dasadgenad (maT Soris re-alur viTarebaSi) iyenebs figuraTa (mravalkuTxedebis, wreebis/wrewirebis) msgavsebas an/da damokidebulebebs figuris elementebis zomebs Soris (mag-aliTad im sagnis simaRlis gazomva, romlis fuZe miudgomelia, miudgomel wertilamde manZilis gamoTvla).

• poulobs brtyeli figuris farTobs da iyenebs mas optimizaciis zogierTi problemis gadasaWrelad (maT Soris realur viTarebaSi).

• iyenebs koordinatebs sibrtyeze geometriuli figuris zomebis dasadgenad.

maT. X.11. moswavle asabuTebs geometriuli debulebebis marTebulobas


• deduqciuri da induqciuri msjelobis nimuSSi aRadgens gamotovebul safexurs/safexurebs.

• iyenebs algebrul gardaqmnebs, tolobisa da utolobebis Tvisebebs geometriul debulebaTa dasabuTebisas.

• iyenebs koordinatebs geometriuli obieqtis Tvisebebis dasadgenad da dasabuTebisTvis.

maT. X.12. moswavle ikvlevs figuraTa geometriul gardaqmnebs sibrtyeze da iyenebs maT geometriuli problemebis gadaWrisas


• axdens geometriul gardaqmnebs sibrtyeze da martiv SemTxvevebSi iyenebs maT figuraTa tolobis dasadgenad.

• iyenebs koordinatebs geometriuli gardaqmnebis (paraleluri gadatana, Rer-Zuli/centruli simetria) Sesrulebisa da gamosaxvisaTvis.

• msjelobs da akeTebs daskvnas erTi da igive tipis geometriuli gardaqmnebis (paraleluri gadatana, mobrunebebi erTi da igive centris garSemo, RerZuli simetriebi paraleluri RerZebis mimarT, saerTo centris mqone homoTetiebi) kompoziciebis Sesaxeb.

• figuris da/an geometriuli gardaqmnebis Tvisebebis mixedviT msjelobs moce-muli figurebiT sibrtyis dafarvis SesaZleblobis Sesaxeb; Sesabamis SemTx-vevaSi axdens sibrtyis (lokalurad) dafarvis demonstrirebas.

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maT. X.13. moswavle moipovebs dasmuli amocanis amosaxsnelad saWiro Tvisebriv da raodenobriv monacemebs


• iyenebs monacemTa Segrovebis xerxebs (dakvirveba, gazomva, miTiTebul respon-dentTa jgufis gamokiTxva mza anketiT/kiTxvariT).

• atarebs statistikur (maT Soris, SemTxveviT) eqsperiments da agrovebs mona-cemebs.

• ikvlevs da iyenebs monacemTa sxvadasxva istoriul da Tanamedrove wyaroebs (mag., sainformacio cnobari, interneti, katalogi da sxva).

maT. X.14. moswavle awesrigebs da warmoadgens Tvisebriv da raodenobriv monacemebs dasmuli amocanis amosaxsnelad xelsayreli formiT


• irCevs Tvisebriv da raodenobriv (daujgufebel) monacemTa warmodgenis Ses-aferis grafikul formas, asabuTebs Tavis arCevans da qmnis cxrils/diagram-as.

• agebs sxvadasxva diagramebs erTi da igive Tvisebrivi an raodenobrivi monace-mebisTvis da msjelobs, Tu monacemTa ramdenad mniSvnelovan aspeqtebs warmo-aCens TiToeuli da ra upiratesoba gaaCnia TiToeuls.

• axdens monacemTa dajgufebas/dalagebas, msjelobs dajgufebis/dalagebis principze.

maT. X.15. moswavle aRwers SemTxveviTobas albaTuri modelebis saSualebiT


• aRwers SemTxveviTi eqsperimentis elementarul xdomilobaTa sivrces, iTv-lis xdomilobaTa albaTobebs variantebis daTvlis xerxebis gamoyenebiT (mag., xisebri diagramis saSualebiT).

• atarebs eqsperiments SemTxveviTobis warmomqmneli romelime mowyobilobiT da afasebs xdomilobis albaTobas eqsperimentuli monacemebis safuZvelze (fardobiTi sixSiris saSualebiT), msjelobs gansxvavebaze Teoriul (mosa-lodnel) Sedegsa da empiriul (eqsperimentul) Sedegs Soris.

• mocemuli sasruli albaTuri sivrcisaTvis aRwers SemTxveviTobis warmomqm-nel mowyobilobas, romlis albaTur modelsac warmoadgens es sivrce, asab-uTebs mowyobilobis dizains.

maT. X.16. moswavle iyenebs statistikur da albaTur cnebebs/Tvalsazrisebs yoveldRiur viTarebebSi


• ganixilavs im statistikur viTarebebs, romelTa gamocdilebac gaaCnia (mag. mosaxleobis aRwera, arCevnebi, sazogadoebrivi azris gamokiTxva), iyenebs gamoqveynebul faqtebs/monacemebs da msjelobs mocemuli problemis Sesaxeb (mag. ekologiuri sakiTxebis Sesaxeb).

• msjelobs albaTuri modelebis gamoyenebis Sesaxeb dazRvevaSi, sociologiur

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kvlevebSi, demografiaSi.• mohyavs albaTur-statistikuri modelebis gamoyenebis magaliTebi bunebism-

etyvelebaSi da medicinaSi (mag., mikro da makro nawilakebis fizika, genetika), xsnis movlenebs SemTxveviTobis meqanizmis moqmedebis saSualebiT.

programis Sinaarsi


_ namdvil ricxvTa qvesimravleebi (racionalur da iracionalur ricxvTa simravleebi): iracionaluri ricxvis miaxloeba racionaluri ricxvebis mimdevrobiT

_ aTobiTisgan gansxvavebuli ricxviTi sistemebi: aTobiTisagan gansxvavebul sistemaSi ricxvebis Caweris praqtikuli magaliTebi (mag. orobiT sistemaSi); kavSirebi sxvadasxva poziciur sistemebs Soris (mag. aTobiTi poziciuri sistemaSi mocemuli ricxvis warmodgena orobiT sistemaSi da piriqiT)

_ aTobiT sistemaSi mocemuli ricxvis Cawera standartuli formiT; standartuli formiT mocemuli ricxvis Cawera aTobiT poziciur sistemaSi

_ sxvadasxva saxiT mocemuli ricxvebis Sedareba/dalageba._ ariTmetikuli moqmedebebi namdvil ricxvebze_ namdvili ricxvebis damrgvaleba da ariTmetikuli moqmedebebis Sedegis

Sefaseba_ racionalur-maCvenebliani xarisxi da misi Tvisebebi_. naSTebis ariTmetikis elementebi (gacnobis wesiT, ~bolo cifris

ariTmetika~)_ zomis erTeulebi: kuTxis radianuli zoma.1. wrfivi, modulis Semcveli,

kvadratuli da xk)x(f = funqciebi.

_ simravlis cneba; operaciebi sasrul simravleebze: TanakveTa, gaerTianeba, simravlis damateba; venis diagramebi.

_ funqciis gansazRvris are da mniSvnelobaTa simravle._ funqciis zrdadoba/klebadobisa da niSanmudmivobis Sualedebi._ funqciis nulebi da maqsimumis/minimumis wertilebi da Sesabamisi

mniSvnelobebi._ orucnobian gantolebaTa iseTi sistemebi, romelSic erTi gantoleba

wrfivia xolo meoris xarisxi ar aRemateba ors._ orucnobian wrfiv utolobaTa sistema._ grafebi (aramkacrad – rogorc wirebiT SeerTebuli wertilebi sibrtyeze),

maTi zogierTi saxeoba da Tviseba gacnobis wesiT: orientirebuli/araorientirebuli, ciklebi, grafis ori wveros SemaerTebeli gzebi.

_ ricxviTi mimdevrobis mocemis rekurentuli xerxi._ figuraTa msgavseba da msgavsebis niSnebi._ or wertils Soris manZilis formula koordinatebSi._ geometriuli gardaqmnebi sibrtyeze: RerZuli simetria, mobruneba,

homoTetia, paraleluri gadatana; geometriuli gardaqmnebis kompoziciebi._ mravalwaxnagebi da maTi niSan-Tvisebebi.

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_ monacemTa wyaroebi da monacemTa mopovebis xerxebi mecnierebaSi (sabunebismetyvelo, humanitaruli, socialuri, teqnikuri mecnierebebi), warmoebaSi, marTvaSi, ekonomikaSi, ganaTlebaSi, sportSi, medicinaSi, momsaxurebasa da soflis meurneobaSi: dakvirveba, eqsperimenti, mza kiTxvariT gamokiTxva

_ monacemTa klasifikacia da organizacia: Tvisebrivi da raodenobrivi monacemebi; monacemTa dalageba zrdadoba-klebadobiT an leqsikografiuli meTodiT

_ monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: monacemTa raodenoba, pozicia da Tanmimdevroba erTobliobaSi; monacemTa sixSire da fardobiTi sixSire

_ monacemTa warmodgenis saSualebani Tvisebrivi da raodenobrivi (maT Soris dajgufebuli monacemebisTvis): sia, cxrili, piqtograma; diagramis nairsaxeobani (wertilovani, meseruli, xazovani, svetovani, wriuli)

_ Semajamebeli ricxviTi maxasiaTeblebi Tvisebrivi da daujgufebeli raodenobrivi monacemebisTvis: centraluri tendenciis sazomebi (saSualo, moda, mediana); monacemTa gafantulobis sazomebi (gabnevis diapazoni, saSualo kvadratuli gadaxra)

_ albaToba: SemTxveviTi eqsperimenti, elementarul xdomilobaTa sivrce (sasruli sivrcis SemTxveva); SemTxveviTobis warmomqmneli mowyobilobebi (moneta, kamaTeli, ruleti, urna); xdomilobis albaToba, albaTobebis gamoTvla variantebis daTvlis xerxebis gamoyenebiT

_ fardobiT sixSiresa da albaTobas Soris kavSiri

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XI klasi


mimarTuleba:

ricxvebi da moqmedebebi



monacemTa ana-lizi, albaToba da statistika

maT. XI. 1. moswavle akavSirebs ricxvTa poziciur siste-mebs / namdvil ri-cxvTa simravleebs erTmaneTTan

maT. XI. 2. moswavle sxvadasxva xerxiT asrulebs moqmede-bebs namdvil ri-cxvebze da afasebs am moqmedebebis Sedegs

maT. XI. 3. moswavle iyenebs msjeloba-dasabuTebis sxva-dasxva xerxs

maT. XI. 4. moswavle wyvets praqtikuli saqmianobidan mom-dinare problemebs

maT. XI. 5. moswavle iyenebs funqcieb-sa da maT Tvisebebs realuri viTarebis modelirebisas da mis Sesaswavlad

maT. XI. 6. moswav-le iyenebs grafikul, algebrul meTodebs an/da teqnologiebs funqciis / funqciaTa ojaxis Tvisebebis Sesaswavlad

maT. XI.7. moswa-vle iyenebs diskre-tuli maTematikis cnebebsa da aparats modelirebisas da problemebis ga-daWrisas

maT. XI. 8. moswavle asrulebs operaciebs veqtorebze da iyenebs veqtorebs geome-triuli da sabunebi-smetyvelo probleme-bis gadaWrisas

maT. XI. 9. moswavle iyenebs deduqciur / induqciur msjelo-bas da/an algebrul teqnikas geometriul debulebaTa dasamtki-ceblad

maT. XI.10. moswavle axasiaTebs geome-triul gardaqmnebs da iyenebs maT geome-triuli problemebis gadaWrisas

maT. XI. 11. moswavle iyenebs sivrculi figuris kveTebsa da gegmilebs sivrculi figuris Sesaswavlad

maT. XI.12. moswavle moipovebs dasmuli amocanis amosaxsne-lad saWiro monace-mebs

maT. XI.13. moswavle warmoadgens monace-mebs dasmuli amo-canis amosaxsnelad xelsayreli formiT da axdens maT inter-pretacias

maT. XI. 14.moswavle aRwers SemTxveviTo-bas albaTuri mode-lebis saSualebiT

maT. XI. 15. moswavle akeTebs monacemTa analizs da ayali-bebs daskvnebs



maT. XI.1. moswavle akavSirebs ricxvTa poziciur sistemebs / namdvil ricxvTa simravleebs erTmaneTTan


• moyavs informaciis cifruli kodirebis/teqnologiebis magaliTebi; akav-Sirebs ricxvis sxvadasxva poziciur sistemaSi Caweras erTmaneTTan (mag. oro-biT poziciur sistemaSi Caweril ricxvs wers aTobiT poziciur sistemaSi).

• axdens iracionaluri ricxvis racionaluri ricxvebis mimdevrobiT miaxloe-bis demonstrirebas praqtikul amocanebTan dakavSirebuli gamoTvlebis kon-

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teqstSi (mag. neperis ricxvi e).• axdens usasrulod didi da usasrulod mcire sidideebis, maTze moqmedebebisa

da moqmedebaTa Sedegis interpretacias.• msjelobs racionalur da iracionalur ricxvebs Soris gansxvavebaze maTi

poziciuri sistemiT Cawerisas.

maT. XI.2. moswavle sxvadasxva xerxiT asrulebs moqmedebebs namdvil ricxvebze da afasebs am moqmedebebis Sedegs


•amartivebs namdvil ricxvebze moqmedebebis (maT Soris xarisxisa da logariT-mis) Semcvel gamosaxulebas an poulobs mis mniSvnelobas moqmedebaTa Tvise-bebis, Tanmimdevrobisa da maT Soris kavSiris gamoyenebiT.

• poulobs ariTmetikuli moqmedebis Sedegs dasaxelebuli sizustiT; msjelobspoulobs ariTmetikuli moqmedebis Sedegs dasaxelebuli sizustiT; msjelobs Sedegis cvlilebaze an cdomilebis sizusteze, romelic gamowveulia gamosax-ulebis wevrebis damrgvalebiT.

•amocanis konteqstis gaTvaliswinebiT irCevs ra ufro mizanSewonilia moqmede-baTa Sedegis Sefaseba, misi miaxlobiTi, Tu zusti mniSvnelobis povna.

•iyenebs Sefasebis sxvadasxva xerxs namdvil ricxvebze Sesrulebuli gamoTv-lebis (maT Soris fesvi da logariTmi martiv SemTxvevebSi) Sedegis adeqva-turobis Sesamowmeblad.

maT. XI.3. moswavle iyenebs msjeloba-dasabuTebis sxvadasxva xerxs


•� iyenebs sawinaaRmdegos daSvebis meTods amocanebis amoxsnisas an ricxvebis Sesaxeb martivi debulebebis damtkicebisas (mag. sawinaaRmdegos daSvebiT amt-

kicebs rom 2 iracionaluri ricxvia).•� ayalibebs da gamosaxavs ricxvebis Tvisebebis an ricxviTi kanonzomierebebis

Sesaxeb gamonaTqvamebs Soris kerZo/zogadi tipis mimarTebebs, iyenebs gamosax-vis xerxs gamoTqmuli mosazrebis marTebulobis Semowmebisas/dasabuTebisas.

•� raodenobrivi msjelobis nimuSze axdens msjelobis xazis da daskvniTi nawilis analizs, aRniSnavs mis sust da Zlier mxareebs (mag., mocemuli sabuTebidan romeli Sematebda msjelobas met damajereblobas / an yvelaze ufro romeli daayenebda mas eWvqvS?).

maT. XI.4. moswavle wyvets praqtikuli saqmianobidan momdinare problemebs


• iyenebs ricxvis xarisxsa da logariTms, xarisxisa da logariTmis Tvisebebs praqtikuli saqmianobidan an mecnierebis sxvadasxva dargebidan momdire amocanebis amoxsnisas (mag. entropia biologiasa da fizikaSi, radioaqtiuli daSla da daTariRebis meTodebi).

• gansazRvravs da iyenebs Sesaferis erTeulebs sididis cvlilebis siCqaris aRsawerad; adgens sxvadasxva erTeulebs Soris Tanafardobas.

• asrulebs informaciis daSifvrasTan dakavSirebul gamoTvlebs da axdens informaciis gaSifvra-wakiTxvas romelime misTvis cnobili algoriTmis gamoyenebiT (mag. ( ) (m od )f x a x b n= ⋅ + (modn) gardaqmnis Sebrunebuli gardaqmnis, anu gaSifvris ~gasaRebis~ mosaZebnad iyenebs evklides algoriTms; axdens am proceduris demonstrirebas kalkulatoris an kompiuteris gamoyenebiT).

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maT. XI.5. moswavle iyenebs funqciebsa da maT Tvisebebs realuri viTarebis modelirebisas da mis Sesaswavlad


• iyenebs (trigonometriul, uban-uban wrfiv, safexurebriv, maCveneblian, log-ariTmul) funqciebs da maT Tvisebebs realuri procesebis modelirebisas.

• axdens funqciis nulebis, funqciis maqsimumis/minimumis interpretirebas im realuri procesis/viTarebis konteqstSi, romelic am funqciiT aRiwereba.

• iyenebs sibrtyeze wrfivi optimizaciis meTodebs realur viTarebasTan dakav-Sirebul amocanebSi (mag. SezRuduli resursebis efeqtianad gamoyenebis amo-canebSi) wrfivis funqciis maqsimumis/minimumis moZebnisas.

maT. XI.6. moswavle iyenebs grafikul, algebrul meTodebs an/da teqnologiebs funqciis/ funqciaTa ojaxis Tvisebebis Sesaswavlad


• iyenebs funqciis grafikis geometriul niSnebs (sakoordinato RerZis parale-luri wrfis mimarT simetriuloba, koordinatTa saTavis mimarT centrulad simetriuloba, paraleluri gadatanis mimarT simetriuloba) funqciis Tvise-bebis dasadgenad.

• iyenebs Sesaferis grafikul, algebrul meTodebs an teqnologiebs (trigo-nometriuli, uban-uban wrfivi, safexurebrivi, maCvenebliani, logariTmuli) funqciis iseTi Tvisebebis dasadgenad, rogoricaa: zrdadoba/klebadoba, niSan-mudmivoba, perioduloba/periodi, fesvebi, eqstremumebi.

• aRwers Tu ra gavlenas axdens funqciis parametrebis cvlileba funqciis grafikze.

maT. XI.7 moswavle iyenebs diskretuli maTematikis cnebebsa da aparats modelirebisas da problemebis gadaWrisas


• asaxelebs iseT struqturebs (mag. mimdevrobebs, asaxvebs; maT Soris realur viTarebaSi), romelTa aRwerisas SesaZlebelia rekurentuli wesis gamoyeneba; iyenebs rekurentul wess aseTi struqturis aRsawerad.

• debulebebis damtkicebisas, Sesabamis SemTxvevebSi, iyenebs maTematikur induq-cias (maT Soris ariTmetikul/geometriul progresiasTan dakavSirebuli zo-gierTi formulis misaRebad).

• iyenebs xisebr diagramebs an/da grafebs variantebis dasaTvlelad, gegmis/gan-rigis Sesadgenad, optimizaciis diskretuli amocanebis amosaxsnelad.


maT. XI.8 moswavle asrulebs operaciebs veqtorebze da iyenebs veqtorebs geometriuli da sabunebismetyvelo problemebis gadaWrisas


• axdens veqtoris sigrZisa da mimarTulebis, veqtorebze operaciebis (Sekreba,

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skalarze gamravleba, skalaruli/veqtoruli namravli) da maTi Tvisebebis geometriul da fizikur interpretacias.

• iyenebs veqtorebs geometriuli debulebebis dasamtkiceblad da zomebis dasa-dgenad sibrtyeze.

• iyenebs koordinatebs veqtorebisa da veqtorebze operaciebis gamosaxvisas.

maT. XI.9 moswavle iyenebs deduqciur/induqciur msjelobas da/an algebrul teqnikas geometriul debulebaTa dasamtkiceblad


• poulobs logikur kavSirebs (mag. ̀ gamomdinareobs~) mocemul geometriul de-bulebebs Soris; iyenebs deduqciur da induqciur msjelobas.

• ganazogadebs calkeul geometriul debulebebs; ayalibebs hipoTezas da asab-uTebs/uaryofs mas (maT Soris maTematikuri induqciis gamoyenebiT; mag. eile-ris formula sibrtyeze da sivrceSi).

• iyenebs algebrul gardaqmnebs geometriul debulebaTa dasamtkiceblad.

maT. XI.10 moswavle axasiaTebs geometriul gardaqmnebs da iyenebs maT geometriuli problemebis gadaWrisas


• asaxelebs geometriuli figuris im maxasiaTeblebs, romlebic ar icvleba mo-cemuli geometriuli gardaqmnisas (gardaqmnis invariantebs).

• figurebis Sesaxeb sxvadasxva monacemebis (mag. figuraTa zomebi, figuraTa wveroebis koordinatebi, figuraTa elementebs Soris algebruli Tanafardo-bebi) gamoyenebiT asabuTebs an uaryofs ori geometriuli figuris ekvivalen-tobas mocemuli gardaqmnis an gardaqmnis tipis mimarT.

• figuris geometriul gardaqmnas (mobrunebis SemTxvevaSi - mxolod π/2-is jeradi kuTxiT) sibrtyeze gamosaxavs dekartes koordinatebis saSualebiT.

• asaxelebs koordinatebSi mocemuli geometriuli gardaqmnis SesaZlo tips (paraleluri gadatana, saTavis mimarT centruli simetria, sakoordinato Re-rZebis mimarT RerZuli simetria).

maT. XI.11 moswavle iyenebs sivrculi figuris kveTebsa da gegmilebs sivrculi figuris Sesaswavlad


• msjelobs sivrculi figuris kveTis SesaZlo formaze da agebs sivrculi fig-uris miTiTebul kveTas.

• poulobs figuris gegmils miTiTebuli paraleluri dagegmilebisas.• msjelobs sivrculi figuris SesaZlo formaze misi kveTis/kveTebis mixed-

viT.• msjelobs figuris SesaZlo formaze misi anasaxis mixedviT paraleluri

dagegmilebisas.

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maT. XI.12 moswavle moipovebs dasmuli amocanis amosaxsnelad saWiro monacemebs


• irCevs da iyenebs monacemTa Segrovebis Sesaferis saSualebas (dakvirveba, gazomva, miTiTebul respondentTa jgufis gamokiTxva mza anketiT/kiTxvariT, monacemTa mopoveba monacemTa sxvadasxva wyaroebidan), asabuTebs Tavis arCev-ans.

• gansazRvravs respondentebs, irCevs kiTxvebis dasmis Sesaferis formas (Ria kiTxvebi, daxuruli kiTxvebi, ujredis moniSvna, Skalaze moniSvna), qmnis mar-tiv kiTxvars da iyenebs mas monacemTa Sesagroveblad.

• warmoadgens sakiTxis Sesaswavlad Sesaferisi eqsperimentis gegmas, atarebs eqsperiments da agrovebs monacemebs.

maT. XI.13 moswavle warmoadgens monacemebs dasmuli amocanis amosaxsnelad xelsayreli formiT da axdens maT interpretacias


• irCevs monacemTa warmodgenis Sesaferis grafikul formebs, asabuTebs Tavis arCevans, agebs da ganmartavs cxrilebs/diagramebs (maT Soris intervalTa klasebad dajgufebuli monacemebisaTvis).

• adgens sixSireTa ganawilebas, warmoadgens mas grafikuli formiT da aRwers mas simetriulobis, modebis raodenobis, gaSlilobis an sxva niSnebis saSu-alebiT.

• erTi grafikuli formiT warmodgenil monacemebs warmoadgens gansxvavebuli grafikuli formiT da warmoaCens TiToeuli formis xelsayrel da araxel-sayrel mxareebs.

• amoicnobs diagramis mcdar interpretaciebs an arakoreqtulad agebul/ga-formebul diagramebs, ganmartavs da asworebs nakls.

maT. XI.14 moswavle aRwers SemTxveviTobas albaTuri modelebis saSualebiT


• aRwers SemTxveviTi eqsperimentis elementarul xdomilobaTa sivrces, iT-vlis damoukidebel xdomilobaTa albaTobebs (maT Soris jamis albaTobis formulebis gamoyenebiT).

• iTvlis rTul xdomilobaTa albaTobebs kombinatoruli analizis gamoy-enebiT.

• SemTxveviTi eqsperimentis Casatareblad erT mowyobilobas cvlis misi ekviva-lenturi sxva mowyobilobiT da asabuTebs arCevans.

maT. XI.15 moswavle akeTebs monacemTa analizs da ayalibebs daskvnebs


• iTvlis da iyenebs Semajamebel ricxviT maxasiaTeblebs daujgufebel mona-cemTa erTobliobebis dasaxasiaTeblad/Sesadareblad da mosazrebaTa/argu-mentebis Sesafaseblad.

• gansazRvravs modalur klass da afasebs saSualos, medianas da diapazons

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dajgufebul monacemTa simravlisTvis, iTvaliswinebs maT realur viTarebaSi gadawyvetilebis miRebisas.

• gamoTqvams varauds xdomilobis mosalodnelobis Sesaxeb monacemTa safuZ-velze (mag. fardobiTi sixSiris mixedviT) da asabuTebs varaudis marTlzom-ierebas.

programis Sinaarsi


_ namdvil ricxvTa qvesistemebi (racionalur da iracionalur ricxvTa simra-vleebi)

_ sxvadasxva poziciuri sistemebi da maT Soris kavSirebi._ sxvadasxva saxiT mocemuli ricxvebis Sedareba/dalageba._ algebruli moqmedebebi namdvil ricxvebze_ namdvili ricxvis damrgvaleba da ariTmetikuli moqmedebebis Sedegis Sefa-

seba, ariTmetikuli moqmedebebis Sedegis miaxloebiTi mniSvnelobis povna_ ricxvis xarisxi da logariTmi (nebismieri fuZiT)_ naSTebis ariTmetikis elementebi._ usasrulod didi da usasrulod mcire sidideebi da maTze moqmedebebi (aram-

kacrad)_ mimdevrobis zRvari (aramkacrad)_ trigonometriuli, uban-uban wrfivi, safexurebrivi, maCvenebliani, logariT-

muli funqciebi: gansazRvris are da mniSvnelobaTa simravle; nulebi, maqsimu-mebi da minimumebi; zrdadobis/klebadobis da niSanmudmivobis Sualedebi.

_ funqciis perioduloba da periodi._ funqciis grafikis geometriuli Tvisebebi._ wrfivi programirebis amocanebi sibrtyeze._ maTematikuri induqcia da misi gamoyeneba rekurentuli wesiT mocemuli ri-

cxviTi mimdevrobis zogadi wevris formulis misaRebad (mag. ariTmetikuli/geometriuli progresia).1. wrfeebs Soris, wrfesa da sibrtyes Soris, sibrtyeebs Soris mimarTebebi sivrceSi.

_ veqtorebi da maTze operaciebi (Sekreba, skalarze gamravleba, skalaruli/ve-qtoruli namravli).

_ veqtorebisa da veqtoruli operaciebis gamosaxva koordinatebSi._ geometriuli gardaqmnebi sibrtyeze: gadaadgilebebi da msgavsebis gardaq-

mnebi._ figuris (mravalkuTxedis, wris) invariantebi geometriuli gardaqmnis mi-

marT._ sivrculi figuris kveTebi da gegmilebi._ monacemTa Segrovebis saSualebani: kiTxvaris/anketis Sedgena da respondentTa

gamokiTxva (warmomadgenlobiTi jgufis SerCevis gareSe)_ monacemTa klasifikacia da organizacia: raodenobriv monacemTa dajgufeba

sasruli raodenobis intervalTa klasebad_ monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSne-

bi: tipuri da gamorCeuli (mag., eqstremaluri, iSviaTi) monacemebi; sixSireTa

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ganawileba; dagrovili sixSire, dagrovili fardobiTi sixSire; monacemTa po-ziciis maxasiaTebeli - rangi.

_ monacemTa warmodgenis saSualebani Tvisebrivi da raodenobrivi monacemebis-Tvis: diagramis nairsaxeobani (foTlebiani Reroebis msgavsi diagramebi, his-tograma, sixSiruli poligoni, ogiva, dagrovil fardobiT sixSireTa dia-grama)

_ Semajamebeli ricxviTi maxasiaTeblebi Tvisebrivi da daujgufebeli raode-nobrivi monacemebisTvis: monacemTa gafantulobis sazomebi (standartuli ga-daxra)

_ albaToba: operaciebi xdomilobebze (xdomilobaTa gaerTianeba, TanakveTa); damoukidebel xdomilebaTa albaTobebis gamoTvla jamis albaTobisa da kom-binatoruli analizis gamoyenebiT; geometriuli albaToba monakveTze da br-tyel figuraze

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XII klasi


mimarTuleba:

ricxvebi da moqmedebebi



monacemTa ana-lizi, albaToba da statistika

maT. XII. 1. mo-swavle wyvets praqtikuli saqmia-nobidan momdinare problemebs

maT. XII. 2. moswa-vle axdens msje-loba-damtkicebis procesisa da misi Sedegis analizs

maT. XII. 3. moswavle ikvlevs da adgens funqciis an funqcia-Ta ojaxis Tvisebebs da axdens am Tvisebe-bis interpretirebas konteqstTan mimarTe-baSi

maT. XII. 4. moswavle iyenebs diskretuli maTematikis meTo-debs modelirebisas da problemebis ga-daWrisas

maT. XII. 5. moswavle poulobs/afasebs figurebis an maTi elementebis zo-mebs da iyenebs maT praqtikuli proble-mebis gadaWrisas

maT. XII. 6. moswavle ikvlevs da iyenebs zogierT faqts are-vkliduri geome-triidan

maT. XII. 7. moswavle ganasxvavebs SerCevis meTods populaciis sruli aRwerisa-gan da msjelobs SerCevis mixedviT populaciis Sesaxeb daskvnebis gamotanis SesaZleblobaze

maT. XII. 8. moswavle warmoadgens monace-mebs dasmuli amo-canis amosaxsnelad xelsayreli formiT da axdens maT inter-pretacias

maT. XII. 9. moswavle aRwers SemTxveviTo-bas albaTuri mode-lebis saSualebiT

maT. XII. 10. moswavle akeTebs monacemTa analizs da ayali-bebs daskvnebs



maT. XII.1. moswavle wyvets praqtikuli saqmianobidan momdinare problemebs


• iyenebs maCvenebliani da logariTmuli funqciebis Tvisebebs praqtikuli

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saqmianobidan an mecnierebis sxvadasxva dargebidan momdinare gamoTvlebTan dakavSirebuli amocanebis amoxsnisas (mag. uwyvetad daricxuli saprocento ganakveTi, entropia biologiasa da fizikaSi, informaciis raodenoba, radio-aqtiuli daSla da daTariRebis meTodebi).

• sididis cvlilebis grafikuli gamosaxvisas irCevs da iyenebs Sesaferis ska-las (magaliTad, logariTmul skalas).

• axdens mocemuli algoriTmiT (magaliTad, RSA) monacemTa daSifvra-wakiTx-vis demonstrirebas; msjelobs informaciisa da ricxvTa Teoriebis praqti-kul

demonstrirebas; msjelobs informaciisa da ricxvTa Teoriebis praqtikul mxareze/maT rolze Tanamedrove samyaroSi. (mag. informacis dacva; informa-

ciis Rirebuleba da daSifvris gaxsnisas saWiro gamoTvlebis xarjebi; ~Ria tipis gasaRebiT~ daSifrvis sistemis socialur aspeqtebi - misi usafrTxoe-bis dacvis meqanizmebi - ~gamWvirvalobis principi moqmedebaSi~).

maT. XII.2. moswavle axdens msjeloba-damtkicebis procesisa da misi Sedegis analizs


• axdens ricxvebis Sesaxeb debulebis an raodenobrivi msjelobis nimuSis da misi Sedegis analizs erTi an ramdenime pirobis, SezRudvis an daSvebis Sesus-teba-moxsniT.

• asabuTebs ricxvebis Tvisebebis an ricxviT kanonzomierebebis Sesaxeb ganzoga-doebiT, analogiiT miRebul daskvnebs an debulebebs (maT Soris maTematikuri induqciis gamoyenebiT).

• raodenobrivi msjelobis nimuSze axdens msjelobis xazis da daskvniTi nawilis kritikul analizs. (mag. mocemuli sabuTebidan romeli Sematebda msjelobas met damajereblobas; yvelaze ufro romeli daayenebda mas eWvqvS; moyavs Sesa-Zlo argumentebi romlebic metad/naklebad sarwmunos gaxdida Sedegs).


maT. XII.3. moswavle ikvlevs da adgens funqciis an funqciaTa ojaxis Tvisebebs da axdens am Tvisebebis interpretirebas konteqstTan mimarTebaSi


• aRwers da adarebs Seswavlil funqciaTa ojaxebs iseTi Tvisebebis mixedviT, rogoricaa: gansazRvris are da mniSvnelobaTa simravle, fesvebisa da eqstre-mumis wertilTa SesaZlo raodenoba, niSanmudmivobisa da zrdadoba/kleba-dobis Sualedebi, perioduloba, asimptoturi qceva, grafikis geometriuli Tvisebebi; axdens am Tvisebebis interpretirebas konteqstTan mimarTebaSi.

• iyenebs Sesaferis grafikul, algebrul meTodebs an/da teqnologiebs funq-ciis iseTi Tvisebebis (gansazRvris are da mniSvnelobaTa simravle, fesvebi da eqstremumis wertilebi, niSanmudmivobisa da zrdadoba/klebadobis Sualede-bi, luwoba/kentoba, perioduloba, asimptoturi qceva, grafikis geometriu-li Tvisebebi) dasadgenad. axdens am Tvisebebis interpretirebas konteqstTan mimarTebaSi.

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• aRwers Tu ra gavlenas axdens funqciis parametrebis cvlileba funqciis Tvisebebze; axdens am gavlenis interpretirebas konteqstTan mimarTebaSi.

• iyenebs Seswavlil funqciebs da maT Tvisebebs modelirebisas da problemis gadaWrisas.

maT. XII.4. moswavle iyenebs diskretuli maTematikis meTodebs modelirebisas da problemebis gadaWrisas


• iyenebs iteracias, rekursias da maTematikur induqcias modelirebisas, debulebebis dasabuTebisas, formulebis gamoyvanisas, kombinatoruli amocanebis amoxsnisas.

• iyenebs grafebs, xisebr diagramebs da maT Tvisebebs modelirebisas da amocanebis amoxsnisas.

• diskretuli optimizaciis zogierTi problemis gadaWrisas iyenebs algoriTmebs an/da teqnologiebs.


maT. XII.5. moswavle poulobs/afasebs figurebis an maTi elementebis zomebs da iyenebs maT praqtikuli problemebis gadaWrisas


• iyenebs sivrculi figuris zomebs Soris funqciur damokidebulebas opti-mizaciis zogierTi problemis gadasaWrelad (maT Soris realuri viTarebis Sesabamis amocanebSi; mag. cilindruli formis Ria konservis yuTis damzade-baze ixarjeba S sm2 masala. rogori unda iyos yuTis wrfivi zomebi, rom misi moculoba udidesi iyos?)

• iyenebs veqtorebs geometriuli debulebebis dasamtkiceblad da zomebis dasa-dgenad.

• iyenebs figuris zomebs da maT Soris kavSirebs geometriuli albaTobis dasa-dgenad.

maT. XII.6. moswavle ikvlevs da iyenebs zogierT faqts arevkliduri geometriidan


• msjelibs, evklides geometriis romeli debulebebi sruldeba an ar sruldeba romelime araevklidur geometriaSi (mag.: cnobilia, rom erT wrfeze mdebare sami wertilidan mxolod erTi mdebareobs danarCen ors Soris. samarTliania Tu es debuleba sferuli geometriis SemTxvevaSi?)

• asabuTebs martiv debulebebs romelime araevklidur geometriaSi (mag.: loba-Cevskis geometriaSi samkuTxedis Suaxazi naklebia fuZis naxevarze).

• poulobs obieqtTa zomebs an/da obieqtTa Soris manZilebs romelime arae-vklidur geometriaSi (maT Soris realuri viTarebis Sesabamis amocanebSi; mag.: manZili or wertils Soris sferoze).

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maT. XII.7. moswavle ganasxvavebs SerCevis meTods populaciis sruli aRwerisagan da msjelobs SerCevis mixedviT populaciis Sesaxeb daskvnebis gamotanis SesaZleblobaze


• ganasxvavebs SerCevasa da populacias.

• amoicnobs jgufs, romelic warmomadgenlobiTia populaciisTvis.

• mocemuli SerCevis SemTxvevaSi asaxelebs faqtorebs, romlebsac zegav-lena SeuZlia moaxdinos SerCevis mixedviT populaciis Sesaxeb gamotanili daskvnebis saimedoobaze (mag. gazomvis sizuste, SerCevis warmomadgenlobiTo-ba).

maT. XII.8. moswavle warmoadgens monacemebs dasmuli amocanis amosaxsnelad xel sayreli for miT da axdens maT interpretacias


• arCevs monacemTa warmodgenis Sesaferis grafikul formebs, asabuTebs Tavis arCevans, agebs da ganmartavs cxrilebs/diagramebs.

• dawyvilebuli monacemebisTvis qmnis gafantulobis diagramas, Tvisebrivad aRwers mis formas (romelime wiris mag. wrfis, parabolis, midamoSi koncen-tracia), agebs saukeTeso misadagebis wrfes.

• adgens sixSireTa ganawilebas, warmoadgens mas grafikulad da aRwers mis formas (mag. simetriuloba/asimetriuloba, maqsimumis/minimumis wertilebi).

maT. XII.9. moswavle aRwers SemTxveviTobas albaTuri modelebis saSualebiT


• ganasxvavebs damoukidebel da damokidebul xdomilobebs, asaxelebs maT mag-aliTebs da iTvlis xdomilobaTa pirobiT albaTobebs.

• iTvlis rTul xdomilobis albaTobas jamisa da namravlis formulebis gamo-yenebiT.

• atarebs eqsperiments mravaljeradi dabrunebiT da am eqsperimentis saSuale-biT adgens urnis Sedgenilobas _ afasebs gansxvavebuli feris burTulebis raodenobaTa Sefardebas.

• iyenebs simulaciebs SerCevis statistikebis (mediana, saSualo mniSvneloba, saSualo kvadratuli gadaxra) variabelurobis gamosakvlevad da SerCevis ganawilebaTa asagebad.

maT. XII.10. moswavle akeTebs monacemTa analizs da ayalibebs daskvnebs


• irCevs mocemuli SerCevisTvis iseT ricxviT maxasiaTeblebs, romlebic xel-sayrelia dasmuli amocanis gadasawyvetad da asabuTebs Tavis arCevans, iTv-

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lis da iTvaliswinebs arCeul maxasiaTeblebs gadawyvetilebis miRebisas.• axdens monacemTa interpolacias/eqstrapolacias saukeTeso misadagebis

wrfis saSualebiT.• amoicnobs Canacvlebas SerCevisa da gamokiTxvis nimuSSi, msjelobs Tu ro-

gor zegavlenas axdens SerCeviTi meTodi da SerCevis moculoba daskvnaTa san-doobaze.

• iTvlis korelaciis koeficients da msjelobs dawyvilebul monacemebs Soris wrfivi kavSiris Sesaxeb.

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programis Sinaarsi


_ monacemTa daSifvris romelime algoriTmi._ informaciisa da ricxvTa Teoriebis aqtualuroba da gamoyenebebi Tanmedrove

sazogadoebaSi_ logariTmuli skala_ polinomialuri, wilad-wrfivi, kvadratuli/kuburi fesvis Semcveli funq-

ciebi._ variantebis daTvlis xerxebi da formulebi, kombinatoruli formulebi._ ori simravlis dekartuli namravli; or simravles Soris asaxva, Sebrunebuli

asaxva, simravlis winasaxe._ grafebi da xisebri diagramebi: grafis gansazRvreba simravlur enaze; grafis

gamosaxvis algebruli da geometriuli xerxebi._ zomebs Soris funqciuri damokidebuleba._ veqtorebi sivrceSi._ araevkliduri geometriis (magaliTad sferuli) elementebi._ monacemTa Segrovebis saSualebani: SerCeviTi meTodi, SerCeva da variaciuli

mwkrivi; SerCevis ricxviTi maxasiaTeblebi (mediana, saSualo mniSvneloba, sa-Sualo kvadratuli gadaxra)

_ monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: dawyvilebuli monacemebi, korelacia

_ monacemTa warmodgenis saSualebani Tvisebrivi da raodenobrivi monacemebis-Tvis. gafantulobis diagrama, misadagebis wiri

_ albaToba: pirobiTi albaToba, xdomilobaTa damoukidebloba.; albaTobaTa jamisa da namravlis formulebi; did ricxvTa kanoni (gacnobis wesiT).

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