maswavleblis wigni 7 klasi

gamomcemloba inteleqtiTbilisi

guram gogiSvili, Teimuraz vefxvaZe

ia mebonia, lamara qurCiSvili

maTematika VII

maswavleblis wigni

(meTodikuri rekomendaciebi, sakontrolo weris variantebis nimuSebi da miTiTebebi amocanebis amosaxsnelad)

meore gadamuSavebuli gamocema

redaqtori Teimuraz vefxvaZe

6

Sesavali

wigni Seicavs meTodikur rekomendaciebs axali erovnuli saswavlo gegmis mixedviT Sedgenili maTematikis VII klasis sa xelmZRvanelosTvis (guram gogiSvili, Teimuraz vefxvaZe, ia mebonia, lamara qurCiSvili, maTematika VII), 2005-2006 saswavlo wels am saxelmZRvanelom warmatebiT gaiara pilo ti reba. rekomendebulia ganaTlebis reformis proeqt `ilia Wav WavaZis~ erovnuli saswavlo gegmebisa da saxel mZRva ne lo ebis Sem-muSavebeli jgufis mier.

wignSi gadmocemulia meTodikuri rekomendaciebi yoveli Temis gadacemis, gakveTilebis dagegmvisa da warmarTvis, sa var jiSoebis SerCevis Sesaxeb.

mocemulia miTiTebebi amocanaTa amosaxsnelad.wignSi gaecnobiT Cvens mosazrebebs axali masalis axsnis Sesaxeb; mocemuli iqneba

Sesabamisi axsna-ganmartebebi masalis wardgenis fazebis Sesaxeb _ motivacia, sakiTxis dasma, amo ca nis gansazRvra, ZiriTadi saswavlo masalis gamoyofa, sawyis etapze Ses-asrulebeli savarjiSoebi da maTi analizi, dama gvir gvinebel etapze Sesasrulebeli amocanebi da maTi ana li zi, damoukidebeli samuSaoebis gansazRvra da maTi Semow me-bis formebi. mocemuli iqneba gakveTilebis dagegmvis sqemebi da ramdenime sanimuSo gakveTilis scenari.

wignSi Tavebi da paragrafebi moswavlis wignis analo gi u radaa warmodgenili.gaecnobiT komentarebs TiToeuli Tavis, paragrafis mi xed viT, masalis gadmocemis

Taviseburebebis motivacias, am masa lis Sesabamis mecnierul da meTodologiur sa-fuZvlebs; miTi Te buli iqneba Sesabamisi literatura yvela am Temis mixedviT codnis gasaRrmaveblad; wigns Tan erTvis samecniero da meTo di kuri literaturis sia.

mocemulia moswavleTa mier masalis Seswavlis aRricx vi sa da Sefasebis xerxebi; es rekomendaciebi xels Seuwyobs maswavlebels Sefasebis procesis dagegmvasa da warmarTvaSi. Se fasebis ZiriTadi mdgenelebi da am mdgenelebis mixedviT Se fasebis sistemis SemuSavebaSi dagexmarebaT is masala, romelic erovnuli saswavlo gegmebisa da Sinaarsisa da miz ne bis rukis Semdeg aris mocemuli. xsenebuli sami kom po nen ti mTeli saswavlo wlis ganmavlobaSi gamoiyeneba, mas wav lebels aZlevs masalis gadace-mis orientirebsa da Se fa sebis formebs _ moswavleebis muSaobaze dakvirveba, maTi aq ti vobebis Sefaseba yovel gakveTilze mimdinareobs da trimes truli da wliuri Sefasebebis safuZvels warmoadgens.

amasTanave, Cven mier mocemul rekomendaciebs calkeuli gakveTilebis dagegm-visa da warmarTvis Sesaxeb maswavlebeli SemoqmedebiTad unda miudges _ swavleba, cnobil maTematikoss d. poias Tu daveTanxmebiT, xelovnebaa da SesaZlebelia bevrma maswavlebelma gakveTilis warmarTvis, moswavleTa aqtivobis dagegmvis originaluri xerxebic moifiqros. Cven yovelTvis mzad varT daveswroT novatorul gakveTilebs da monawileoba miviRoT maT ganxilvasa da analizSi.

swavlebis procesSi meTodikuri orientirebis garkvevis sakiTxSi dagexmarebaT meTodikuri literatura, romlebic wignis bolos aris mocemuli; magaliTad, dRevan-del situacias kargad exmaureba citatebi h. froidentalis wignidan ([38], gv. 97): `maswavlebels unda SeeZlos swavlebis warmarTva programiT, romelic kardinalurad gansxvavdeba imisgan, romliTac mas aswavlidnen...

... moTxovna, rom ganaTleba gardaqmnis Canasaxsac Sei cav des swrafad cvalebad sazogadoebaSi ara marto maTe ma tikis swavlebas exeba. ... feliqs klaini laparakobda ormag da viwyebaze: universitetSi swavlisas saskolo maTematikis daviwyebaze, Semdeg _ sauniversiteto maTematikis daviwyebaze, skolaSi maswavleblad dabrunebisas~.

Cven SevecdebiT yoveli Temis warmodgenas axldes deta lu ri komentarebi da masalis mecnieruli da meTodologiuri safuZvlebis mimoxilva.

7

swavlebis procesSi meTodikuri orientirebis garkvevis sakiTxebSi daxmarebas gagiwevT masala, romlis gacnobas in ter netis saSualebiT SeZlebT (http://www.mccme.ru/edu).

aq mocemulia ruseTisa da amerikis SeerTebuli Statebis cnobil mecnierTa, sazogadoebriv organizaciaTa da TviT prezident buSis mosazrebebi da rekomenda-ciebi maTematikis swavlebis mniSvnelobisa da 21-e saukuneSi misi warmarTvis axali orientirebis Sesaxeb. magaliTad, amerikis SeerTebuli Statebis nacionaluri komite-tis moxsenebaSi (`jer kidev ar aris gvian~) vkiTxulobT: `miuxedavad sazogadoebis serio zu li cvlilebebisa, romelic gasuli saukunis meore naxe var Si ganxorcielda, maTematikisa da sabunebismetyvelo sag ne bis swavlebis meTodika faqtobrivad ar Secvlila. swav le bis praqtika srulad ar iyenebs im SesaZleblobebs, rome lic sa-ganmanaTleblo procesis sxvadasxva aspeqtma mogvca...~.

ZiriTadi aqcenti am moxsenebaSi iseve, rogorc zemoT xse nebul sxva rekomendacie-bSi keTdeba moswavlis aqtiur Cab ma ze swavlebis procesSi, rom swavleba Ziebisa da dasa bu Te bebis gziT unda mimdinareobdes, moswavle TviT unda aRmoa Cendes WeSmar-itebas motivirebuli situaciis dagegmvis Semdeg; mTavaria kvlevis procesi _ ara marto pasuxebis mosmena kiTxva ze _ `ra~, aramed _ `rogor~, an _ `kidev rogor~. amas Ta nave, mniSvnelovania imis codnac _ `ratom~ da pasuxebi kiTx vaze _ `me es risTvis mWirdeba~. yvelaferi es auci le belia moswavleTa momavali profesiuli saqmianobisTvis, demokratiul procesebSi aqtiuri monawileobisTvis _ dRevan del sazogadoebaSi, romelic sul ufro metad aris damo kide buli maTematikaze.

wigni dagexmarebaT zemoTxsenebuli orientirebis ganxor cie lebis meTodikur xerxebis gacnobaSi _ kvlevis procesi SeiZleba individualuri iyos, SeiZleba jgu-furi muSaobiT xorcieldebodes. iqneba rekomendaciebi sxvadasxva tipis sas wav lo procesis ganxorcielebisTvis. moswavlis saxel mZRva nelo am procesis erT-erTi damxmare saSualebaa. amitom sa Wi roa misi agebis principebi da masalis struqturi-rebis sakiTxebic gagacnoT.

moswavlis saxelmZRvanelos Sedgenisas gaviTvaliswineT is ZiriTadi orientirebi, romlebic ganaTlebisa da mecni ere bis saministros proeqt `ilia WavWavaZis~ Sem-muSavebuli jgu fis mier warmodgenil erovnul saswavlo gegmebSia miTi Te buli _ orientireba Sedegze da am Sedegebis demonst ri re baze, informaciis gamoyenebis SesaZleblobis demonst ri reba. kavSiri sagnebs Soris da TviT erTi sagnis SigniT mis sxvadasxva nawilebs Soris (maTematikaSi warmod ge nil mimar Tu lebebs Soris); saklaso da klasgareSe muSaobis saqmia no bis erTianoba; orientacia uklebliv yvela moswav leze (sa ku Tari SesaZleblobebidan gamomdinare, yoveli mos wavle sxvadasxva xarisxiT miaRwevs Sedegebs); maswavlebelTa in di vi dualuri Taviseburebebis gamovlenis SesaZleblobebi _ erovnuli saswavlo gegmis Sesruleba sxvadasxva gziT Se iZ leba.

me-7 klasis saxelmZRvanelos Sedgenisas gaviTvaliswineT, rom igi sabazo safexuris pirvel etaps Seesabameba; amasTanave aris mosamzadebeli etapi momavali arCevanis (swavlis gagr Ze leba, profesiaTa daufleba) winapirobebis Sesaqmnelad; am periodSi iwyeba informaciis, unar-Cvevebisa da damoki debu lebebis gasaviTareblad saWiro pirobebis Seqmna; farTov deba Tanamimdevruli, logikurd gamarTuli msjelobis unaris gamo muSavebis procesis CarCoebi.

saxelmZRvanelos Sinaarss, rogorc cnobilia, unda gan saz Rvravdes rogorc praq-tikuli moTxovnebi, aseve maTema tikis ganviTarebis Sinagani kanonzomierebebi. im SemTxvevaSic ki, roca raime axali sakiTxis SemoRebas es ukanaskneli pi ro bebi ufro gansazRvravs, mainc vcdilobT praqtikuli sa xis sakiTxebTan maT dakavSirebas. aseT SemTxvevaSi saWiroa zo mierebis dacva _ praqtikasTan yoveli mimarTeba unda iyos ga mar Tlebuli, moswavleTaTvis gasagebi da xels uwyobdes mo tivaciis process.

8

Sinaarsis SerCevas Sesabamisi asakis moswavleTaTvis mi sawv domobis principi udevs safuZvelad; es SerCeva _ Camoyalibebuli stabiluri birTvis bazaze xdeba. igi ga-fil trulia maTematikis swavlebis gamocdilebiT; tradiciuli Temebis Tanamimdevro-bis axleburi gaazrebiT, maTematikis erTianobidan gamomdinare integraciis xarisxis gazrdisa da axali sakiTxebiT Sevsebis aucileblobis gaTvaliswinebiT. Sinaarsobrivi mimarTulebebi saswavlo gegmaSia Camoyali be bu li; maTgan maswavleblebisTvis ukve nacnobia wina saxel mZRva neloebis mixedviT: monacemTa analizi, aRweriTi statis-tikis elementebis cnebebi da maTi praqtikuli ilustraciebi. Tumca, midgoma da gadmocemis meTodika axleburia _ gazr dilia mimarTulebebis integraciis procesi; orientirebulia Sedegebze. maswavlebels mudmivad unda axsovdes, rom maTematika erTiani mecnierebaa da ara misi sxvadasxva nawilis xelovnuri gaerTianeba. amitom gadmocemis meTodika iTva lis wi nebs sakiTxebis urTierTkavSirSi, araizolirebulad gad mo cemas. sistematuria wina masalis gameorebisa da ganmtki ce bis momentebis gaT-valiswineba. es xels uwyobs kursis erTiani koncefciis ganxorcielebas. pirveli Tavic ki, rome lic TiTqos gasul wlebSi Seswavlili sakiTxebis gameorebas moicavs, momavali sakiTxebis gaTvaliswinebiTaa agebuli.

yoveli siaxle isea gadmocemuli, rom ukve ganxiluli sa kiTxebis codnas emyareba. xSiria mimarTeba sabunebis mety ve lo sagnebTan. TiTqmis yovel paragrafSia mocemuli Teo riu li masalis komentarebi, saintereso mecnieruli faqtebi, terminebis warmo-Sobis istoria, maTi xmarebis sxvadasxva aspeqtebi, cnobil mecnierTa biografiuli monacemebi.

moswavleebSi maTematikis, rogorc zogadsakacobrio kul tu ris nawilis Sesaxeb warmodgenis Camoyalibebas xels uw yobs SinaarsSi CarTuli epizodebi mecnierebis istoriidan. isi ni moswavleebs im adamianTa saxelebs acnobs, vinc qmnida ma Te matikas. maT Sorisaa saxelovani qarTveli maTe ma ti ko se bic.

kacobriobis inteleqtualuri istoriis SesaniSnavi faq te bis gacnoba swavlebaSi kulturul-istoriul tradiciebs qmnis. vcdilobT gavufrTxildeT da ganvaviTaroT es tradi ciebi.

saxelmZRvanelos Sedgenis erT-erTi principi moswavleTa asakobrivi Tavisebure-bebis gaTvaliswinebaa. vcdilobT davic vaT zomiereba, ar gadavtvirToT saxelmZR-vanelo rTuli dam tkicebebiT; moswavle unda grZnobdes dasabuTebis auci leb lo bas. es gansakuTrebiT geometriuli sakiTxebis gadmocemas exeba.

dResac aqtualuria profesor iaglomis [41] sityvebi: `Cven me TodikaSi arsebobda araswori warmodgena, rom mxolod geo met ria uwyobs xels logikuri momTxovnelo-bis aRzrdas da geometriaa maTematikuri disciplinebis agebis nimuSi. sinam dvi leSi maTematikis sxva nawilebSi debulebaTa umetesobis dasabuTebis sizuste metic ki aris geometriaSi mocemul damtkicebebze~. axali moTxovnebis Sesabamisadaa gadmo-cemuli geometriuli masala _ suraTebis gamoyenebiT, romlebic saSualebas aZlevs moswavles ifiqros, aRmoaCinos faqtebi, gamoiyenos es faqtebi sayofacxovrebo saqmianobaSi, gana viTaros sivrculi orientacia, intuicia. WeSmaritebisadmi pativis-cema, dasabuTebis saWiroebis gaazreba, dasabuTebisadmi moTxovnilebis gamomuSaveba mniSvnelovani zneobrivi aspeqtebia da maTze muSaoba VII klasSi arc iwyeba da arc mTavrdeba. maswavlebels vTxovT gaixsenos samkuTxedis kuTxe ebis jamis formulis gamoyvanis is gza, romelic ramdenime wlis win skolaSi iswavleboda _ igi Tvalsa-Cino faqtebis, aqsiomebis daxmarebiT dasabuTebas emyareboda [33].

Tumca `maTematikuri azrovneba ar daiyvaneba mxolod deduqciur msjelobebze, formalur damtkicebebze ...

cnebis gamoyofa, romelic konkretul situacias esa da geba, konkretuli SemTx-vevebis dakvirvebis Sedegad gan z o ga debebis povna, induqciis saSualebiT msjelobebis

9

Catareba, ana logiiT da intuiciiT daskvnis povna _ yvelaferi es _ ma Te matikuri azrovnebis nawilia [41].

am mosazrebebs Cven mTlianad veTanxmebiT; iseve, rogorc adamaris azrs: `maTema-tikuri simkacris roli SemdegSia _ dakanondes intuiciis monapovari, mas sxva mizani arasdros hqonia!~

zemoTqmuli martiv deduqciur msjelobebze uaris Tqmas ar niSnavs. viziarebT burbakis gamonaTqvams [13]: `berZ nebidan moyolebuli _ vityviT `maTematika~ _ vgulisxmobT `damtkicebas~.

siZneleebi moswavleebs martividan rTulisken mimaval gzaze unda SexvdeT. ma-gram Tu masala rTulia, zogjer _ da uZlevelic, maSin codnas moCvenebiT daufleba cvlis.

saxelmZRvaneloSi zomieradaa gamoyenebuli Teoriul-sim rav luri simbolika _ igi damxmare masalis rols asrulebs.

Teoriuli masalis gaazrebasa da aTvisebas xels uwyobs savarjiSoebis sistema. isini Teoriul masalasTan mWidrod aris dakavSirebuli, aiZulebs moswavles moZebnos paragrafSi Sesabamisi Teoriuli sakiTxi. raime cnebis aTvisebis mniSvne lovani niSani misi konkretizaciis unaria. aRniSnuli unaris gamomuSavebas emsaxureba saxelmZR-vaneloSi warmodgenili is savarjiSoebi, romelTa saSualebiT axali cnebebi martivad, sakmaod damaxasiaTebel situaciebSia gamoyenebuli, CarTulia axal kavSirebSi.

savarjiSoTa mravalferovani sistema (`testebi~, amo ca ne bi, amocanebi jgufuri da individualuri muSaobisTvis, da ma tebiTi amocanebi), Teoriul nawilSi mocemuli analizuri msje lobis magaliTebi xels uwyobs moswavleebSi analizuri az r ovnebis ganviTarebas. sistematurad aris warmodgenili amo canebi, romlebic Teoriis aT-visebas, moswavlis kritikuli azrovnebis ganviTarebasa da maTematikis Segnebulad Seswavlas uwyobs xels. praqtikasTan xSiri mimarTeba (axali sakiTxis motivaciisas, swavlebisa da codnis ganmtkicebis procesSi) xels uwyobs praqtikuli gamoyenebis unar-Cvevebis gamo mu Savebasa da maTematikis Seswavlisadmi interesis gaRrmavebas.

yovel paragrafs Tan axlavs sakontrolo kiTxvebi, amo ca nebi pasuxebis varian-tebis miTiTebiT (e. w. testebi); isini xels uwyobs moswavleTa mier TviTkontrolis ganxorci elebas.

`testebi~ isea warmodgenili, rom igi moswavleTa si tu aciaSi swrafad garkvevisa da pasuxebis sworad SerCevis unars amowmebs da ara amoxsnebis Catarebis siswores. `didi xania mimdinareobs dava am tipis savarjiSoTa dadebiTi da uaryofiTi mxareebis Sesaxeb. Cven viziarebT im Tvalsazriss, rom Tu pasuxis SerCevas amoxsnis na wilad CavTvliT, maSin Sesabamisi amocana gamodgeba sas wav lo miznebisTvisac da zog jer _ SemowmebisTvisac. maga liTad, martivi ricxvis SerCeva, roca miTiTebulia ricxvebi, ro mel Tagan erTi iyofa 5-ze, me ore _ 3-ze, mesame luwia da ar aris 2-is toli.

maTematika, SesaZlebelia, erTaderTi saskolo dis cip linaa, romelic damaxsovre-bisa da gameorebis garda zogadi ganaTlebis iseT mniSvnelovan elementebs Seicavs, rogoricaa amocnoba, analizi, mixvedra, arCeva.

axla SevexoT moswavlis saxelmZRvanelos agebulebas. wignis dasawyisSi gamovx-atavT madlierebas yvelas mimarT, vinc Tavisi yuradRebiT wvlili Seitana wignis daxvewaSi. gan sakuTrebiT mniSvnelovania saswavlo gegmebisa da saxel mZRvaneloebis SemmuSavebeli jgufis eqspertebis Rvawli; isini pilotirebis procesSi monawile maswavlebleTan erTad saxelmZRvanelos Seqmnis Tanamonawileni arian; miwodebuli masalis Sesabamisad iZleodnen mniSvnelovan miTiTebebs; Ziri Ta di iyo masalis gad-mocemis integraciis xarisxi. Tqvenc Seam CnevT, rom es xarisxi saxelmZRvaneloSi TandaTanobiT izrdeba, sasurvel dones aRwevs (magaliTad, aRweriTi statis tikis elementebis gadmocema geometriul da algebrul mimarTulebebTan integraciiT;

10

koordinatTa meTodis, simetri ebis gamoyenebebi) da sistematur xasiaTs atarebs.avtorTa madlobas misalmeba da keTili survilebi mos devs. aqvea rCevebi wigniT

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

paragrafi ramdenimenawiliania. daskvniT na wilSi damxmare masalaa gadmocemuli _ istoriuli cno be bi, damatebiTi sakiTxebi klasgareSe da moswavleTa indivi dua luri muSaobisTvis.

saxelmZRvanelo moswavlis wignisadmi wayenebuli yvela moTxovnis Sesabamisadaa agebuli. sarCevi wignis dasaw yis Sia mocemuli, maTematikuri terminebis leqsikoni, maTe ma tikuri niSnebi da maTi gamoyenebis magaliTebi, sagnobrivi saZiebeli _ wignis boloSi.

moswavlis saxelmZRvanelo swavlebis procesis dagegmvis da warmarTvis erT-erTi saSualebaa. moswavleebTan saubari SeiZleba wignis teqstisgan gansxvavebuladac aa-gos maswav le belma. saxelmZRvanelo exmareba maswavlebels _ masSi dafiqsirebulia SemecnebiTi saqmianoba, romelic maswav le belma unda awarmoos. maswavleblis saqmi-anoba ki sxva saSua le bebis gamoyenebasac gulisxmobs _ maswavleblis wigni, Tval-saCino masala. Tumca, SevecadeT, rom ar Seqmniliyo sa var jiSoTa sxva krebulebis gamoyenebis saWiroeba; moswavlis wignSi SemoTavazebuli savarjiSoebi moswavleTa sxvadasxva SesaZleblobebis gaTvaliswinebiTaa Sedgenili; saWiroa SerCeva da konk-retuli situaciis mixedviT maTi swori gamo yeneba. gakveTilebis warmarTvisadmi momzadebis sxvadasxva eta pebi (saswavlo wlisadmi momzadeba, mocemuli gakveTi-lisadmi momzadeba) maswavlebelma nayofierad unda gamoiyenos. masalis sworad Ser-Cevis sakiTxSi Cvenc gexmarebiT _ amox s nilia moswavlis wignSi mocemuli TiTqmis yvela amocana, mi Ti Tebulia meTodikuri rekomendaciebi masalis wardgenisa da axsnis fazebis Sesaxeb. unda gvaxsovdes, rom maTematika yve las sWirdeba da amitom is unda Seiswavlos yvela moswav lem (Seswavlis xarisxi SeiZleba iyos sxvadasxva _ igi Se fa se bebSi aisaxeba).

Zvirfaso maswavlebelo!warmatebebs gisurvebT Tqven sapatio da mniSvnelovan saq mia nobaSi.

guram gogiSvili, Teimuraz vefxvaZe

ia mebonia, lamara qurCiSvili

11

VII klasis maTematikis erovnuli saswavlo gegma

wlis bolos misaRwevi Sedegebi:

mimarTuleba

ricxvebi da moqmedebebi

kanonzomierebebi da algebra

geometria da sivrcis aRqma

monacemTa ana lizi, alba Toba da sta tistika

1. kiTxulobs, gamosaxavs, ada-rebs da alagebs racionaluri ricx vebs pozi-ciuri sistemis gamoyenebiT

2. sxvadasxva xer xiT asru lebs moqmedebebs ra-cionalur ricx-vebze

3. afasebs ra-cio nalur ri cx-vebze moq me de-baTa Sedegs

4. erTmaneTTan akavSirebs zomis sxvadasxva er-Teulebs da iye-nebs maT amo ca-nebis amoxs nisas

5. amoicnobs da gamosaxavs sidi de ebs Soris pir da pir propor-ci ul damoki de-bu le bas

6. amocanis amo xs nisas iye-nebs simravlur cnebebsa da ope-ra ciebs

7. amartivebs al gebrul gamo-sa xulebas da xsnis wrfiv gan-tolebas

8. ganavrcobs da aanalizebs obi eqtebis pe ri-od ul mim dev ro-bas an/da mudmivi nazrdis mqone ricxviT mimdev-ro bas

9. amoicnobs ge o metriul fi-gu rebs, adarebs maT saxeobebs da axdens klasifi-ci rebas

10. war mo ad-gens geo met ri ul obieqtebs amo-canis kon teq s tis Sesabamisad

11. axdens ge-o met riul gar-daqmnebs da iye-nebs maT fi gu-ra Ta Tvisebebis da sad genad

12. iyenebs ko-or di natebis meTods orienta-ci isaTvis

13. xsnis geo me-t riul amo ca nebs sam kuTxe deb Tan dakavSi re buli cnebe bisa da faqtebis gamoyen-ebiT

14. moipovebs das muli amo ca-nis amosaxs ne lad saWiro Tvi sebriv da rao denobriv mo na cemebs

15. awes rigebs da warmoadgens Tvisebriv da rao denobriv mo-n a cemebs das muli amocanis amo -saxsnelad xel -sayreli for miT

16. akeTebs Tvi-seb riv da rao-denobriv mo na ce-m Ta inter pre ta-cias da analizs amocanis kon teqs-tis gaTva lis-winebiT

12

wlis bolos misaRwevi Sedegebi da maTi indikatorebi

mimarTuleba: ricxvebi da moqmedebebi

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

Sedegi TvalsaCinoa, Tu moswavle: • aTwiladis CanawerSi uTiTebs Tanrigebs da asaxelebs TanrigebSi mdgom cifrTa

mniSvnelobebs; iyenebs am codnas aT wiladebis Sedarebis an (zrdadobiT/klebadobiT) da la ge bisas. (mag. gaSlis sasrul aTwilads saTanrigo Sesak re be bis jamis saxiT, «daalage klebiT 2.9259, 3.1, 2.93, da 2.899»).

• gamosaxavs da adarebs uaryofiT ricxvebs poziciuri sistemis gamoyenebiT; axdens mopirdapire ricxvis da ricxvis absoluturi mniSvnelobis cnebebis modelze demonstrirebas (maT Soris ricxviT RerZze).

• ekvivalenturi formiT wers Sereul ricxvebs, aTwiladebsa da wiladebs; ada-rebs da alagebs sxvadasxva saxiT mocemul ricxvebs (aTwiladi ↔ wiladi).

• poziciuri sistemis gamoyenebiT, konkretul maga li Tebze asabuTebs gayofado-bis niSnebidan zogierTs (mag. 3ze da 9ze gayofadobis niSnebs); poulobs mocemuli ricx vebis umciresi saerTo jeradsa da udides saerTo gamyofs.

maT. VII.2. sxvadasxva xerxiT asrulebs moqmedebebs ra cio nalur ricxvebzeSedegi TvalsaCinoa, Tu moswavle: • axdens mTel ricxvebze ariTmetikuli moqmedebebis mo delze demonstrirebas

(mag. `dadebiTi~ da `uaryofiTi~ er Te ulovani `muxtebiT~, e.i. ori gansxvavebuli feris `nu lo vani wyvilebiT~).

• iyenebs ricxvis Caweris ekvivalentur formebs, moq me debaTa Sesrulebis Tan-mimdevrobas, maT Tvisebebsa da daj gu febas gamoTvlebis gasamartiveblad.

• yofs ricxvs proporciul nawilebad da poulobs ricxvs misi mocemuli nawilis mixedviT.

• axdens naturalur-maCvenebliani xarisxis Tvisebebis demonstrirebas.• zepiri angariSisas iyenebs procentis kavSirs ricxvis nawilebTan; poulobs

mocemuli ricxvis procents da xsnis Sebrunebul amocanebs.• irCevs da iyenebs racionalur ricxvebze ariTmetikuli moq medebebis Sesrulebis

xerxs; (es xerxebia: zepiri, teq no logiebis gamoyenebiT, weriTi algoriTmebi).maT. VII.3. afasebs racionalur ricxvebze moqmedebaTa Se degsSedegi TvalsaCinoa, Tu moswavle: • gamoTvlebze sayofacxovrebo amocanis amoxsnisas iye nebs zepiri angariSis

xerxebs an Sesabamis SemTxvevaSi, moq me d ebaTa Sedegis Sefasebas.• afasebs racionalur ricxvebze ariTmetikul moq me de baTa Sedegis mniSvnelobas,

asrulebs moqmedebebs da amow mebs Tavis varauds.• amrgvalebs racionlur ricxvebs miTiTebuli sizustiT; miax loebiT poulobs

(sizustis miTiTebis gareSe) ariT me ti kuli gamosaxulebis mniSvnelobas.• iyenebs Sefasebas aTwiladebze (weriTi algoriTmis an kalkulatoris gamoye-

nebiT) Catarebuli gamoTvlebis Sedegis adekvaturobis Sesamowmeblad.maT. VII.4. erTmaneTTan akavSirebs zomis sxvadasxva erTe ulebs da iyenebs maT

amocanebis amoxsnisasSedegi TvalsaCinoa, Tu moswavle: • irCevs da iyenebs Sesaferis erTeulebs sididis cvli l ebis, moZraobis siCqaris,

masStabisa da ruqaze manZilis pov nasTan dakavSirebul amocanebis amoxsnisas.• xsnis praqtikul saqmianobasTan dakavSirebul da/an sxva saswavlo discipli-

13

nebidan momdinare amocanebs gamoT v lebze (mag. umartivesi xarjTaRricxva; istoriuli epoqis xangrZlivobis gansazRvra; amocanebi procentebze da pro po rciaze: xsnarebi, Senadnobebi da sxva).

• mocemuli wrfivi damokidebulebis gamoyenebiT gamo sa xavs erT sistemaSi mo-cemul erTeuls sxva sistemis erTe uliT.

• gamosaxavs mocemul erTeuls igive sistemis sxva erTe ulis saSualebiT. (mag. km/sT ebSi mocemul siCqares gamo sa xavs m/wm iT).

Sinaarsi1. mTeli ricxvebi da ariTmetikuli moqmedebebi mTel ricx vebze.2. wiladebi, aTwiladebi da zogierTi kavSiri maT Soris.3. procenti (100ze naklebi an toli da 1ze meti an toli procenti); kavSi

rebi: erTidaigive mTelis procenti da nawili.4. ricxvebis Sedareba da ariTmetikuli moqmedebebis Sedegis Sefaseba. 5. ricxvis proporciul nawilebad dayofa (mag. 2 : 3 : 5).6. ricxvebis umciresi saerTo jeradi da udidesi saerTo gamyofi; ricxvis

martiv mamravlebad daSla.7. ricxvis naturalurmaCvenebliani xarisxi8. naSTiT gayofa, naSTi da gayofadobis niSnebidan zogi er Ti.9. zomis erTeulebi, zomis erTeulebs Soris kavSirebi da zomis erTeulebis

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

10. fasdakleba/fasis gazrda (TanmimdevrobiTi da erT je radi fasdaklebebis/fasebis zrdis erTmaneTTan Sedareba) da martivi xarjTaRricxva.

mimarTuleba: kanonzomierebebi da algebramaT. VII.5. amoicnobs da gamosaxavs sidideebs Soris pirdapirproporciul

damokidebulebasSedegi TvalsaCinoa, Tu moswavle: • mocemuli damokidebulebisaTvis Tvisebrivad da raode nob rivad aRwers Tu ra

gavlenas axdens erTi sididis cvli le ba meoris mniSvnelobaze; moyavs mudmivi da aramudmivi rao de nobrivi cvlilebis magaliTebi yoveldRiuri cxovrebidan.

• sityvierad Camoyalibebul debulebas sidideebs Soris damo kidebulebisa da mimarTebis Sesaxeb gamosaxavs gra fi ku lad an/da cxriliT da piriqiT – grafikulad an/da cxri liT gamosaxul damokidebulebas aRwers sityvierad.

• sxvadasxva xerxiT (grafikulad, cxrilis saxiT, sity vi erad, algebrulad) gamosaxul damokidebulebebs Soris miu TiTebs erTsa da imave damokidebulebebs.

maT. VII.6. amocanis amoxsnisas iyenebs simravlur cnebebsa da operaciebsSedegi TvalsaCinoa, Tu moswavle: • sxvadasxva xerxiT mocemuli simravlisaTvis gansaz R vravs mocemuli elementis

kuTvnilebas mocemuli sim rav lisadmi.• problemis gadaWrisas, iyenebs zogierT damxmare xerxs sim ravleTa Soris mi-

marTebebis dasadgenad da simravluri ope raciebis Sesasruleblad.• sworad iyenebs simravleTa Teoriis cnebebs da Sesa ba mis aRniSvnebs sasrul

simravleebze operaciebis (ori sim rav lis TanakveTa da gaerTianeba), sasrul simra-vleTa Soris mi marTebis, elementsa da simravles Soris mimarTebis gamo sa x visas.

maT. VII.7. amartivebs algebrul gamosaxulebas da xsnis wrfiv gantolebasSedegi TvalsaCinoa, Tu moswavle: • teqsturi amocanis amosaxsnelad adgens da xsnis erT ucnobian wrfiv gantole-

bas.

14

• iyenebs moqmedebaTa Tvisebebs, maTi Tanmimdevrobas da dajgufebas alge-bruli (araumetes ori cvladis Semcveli wrfivi an meore xarisxis) gamosaxulebis gasamartiveblad da misi mniSvnelobis gamosaTvlelad cvladebis mocemuli mniS-vnelobebisaTvis.

• algebruli gardaqmnebisa an/da logikuri msjelobis ga moyenebiT asabuTebs an uaryofs ori algebruli (araumetes ori cvladis Semcveli wrfivi an meore xarisxis) gamosa xu lebis igivur tolobas.

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

Sedegi TvalsaCinoa, Tu moswavle: • periodul mimdevrobaSi gamoyofs mimdevrobis periods.• warmoadgens mimdevrobis mocemuli fragmentis gavr co bis or an met variants,

ganmartavs gavrcobis variantebs da adarebs maT.• dasmuli amocanis konteqstidan gamomdinare irCevs mimdevrobis gavrcobis

variants da asabuTebs Tavis arCevans.• ganavrcobs mudmivi nazrdis mqone ricxviT mimdevrobas; asa xelebs realur

viTarebaSi iseTi procesis magaliTebs, rom lebic aseTi mimdevrobiT aRiwereba.Sinaarsi1. pirdapirproporciuli damokidebuleba da misi ga mo sax va grafikis da

cxrilis saSualebiT.2. simravleTa Teoriis cnebebi, operaciebi da Sesabamisi aRniSvnebi sasruli

simravleebis SemTxvevaSi.3. teqsturi amocanebis amoxsna wrfivi gantolebebis gamoyenebiT.4. araumetes ori cvladis Semcveli wrfivi an meore xa ris xis gamosaxulebebis

gamartiveba da mniSvnelobis gamoTvla.5. perioduli mimdevrobebi da mudmivi nazrdis mqone ricx viTi mimdevrobe

bi.

mimarTuleba: geometria da sivrcis aRqmamaT. VII.9. amoicnobs geometriul figurebs, adarebs maT saxeobebs da axdens

klasificirebasSedegi TvalsaCinoa, Tu moswavle: • arqiteqturisa da xelovnebis nimuSebSi an maT ilus t raciebSi, yofiTi daniS-

nulebis sagnebSi asaxelebs misTvis nac nob geometriul figurebs an maT nawilebs.• ayalibebs mimarTebebs (mag. zogadobakerZooba) figu ra Ta saxeobebs Soris.• asaxelebs figuras misi niSan-Tvisebebis mixedviT, msje lobs figuris amosac-

nobad maTi sakmarisobis/arasakmarisobis Sesaxeb.maT. VII.10. warmoadgens geometriul obieqtebs amocanis konteqstis Sesabami

sadSedegi TvalsaCinoa, Tu moswavle: • agebs dasmuli amocanis Sesabamis naxazs da adek va tu rad iyenebs asoiT aRniS-

vnebs.• aRwers geometriul obieqtTa mocemul grafikul gamo sa xulebebs an obieqtTa

urTierTmdebareobas Sesabamisi ter mi nologiis gamoyenebiT. (mag. marTkuTxa paralelepipedis ro mel waxnagebs ekuTvnis miTiTebuli wvero).

• gamosaxavs brtyel figurebs ise, rom maTi TanakveTa/gaer Tianeba iyos miTiTe-buli formis an Tvisebebis mqone fi gura.

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

15

Sedegi TvalsaCinoa, Tu moswavle: • garemomcvel obieqtebs Soris moiZiebs simetriul obi eq tebs.• xazavs brtyeli figuris (texili, mravalkuTxedi) si met riul figuras miTiTe-

buli simetriis RerZis mimarT; ax dens brtyeli figuris (texili, mravalkuTxedi) paralelur gada tanas.

• uTiTebs brtyeli figuris simetriis RerZs/RerZebs; ax dens simetriulobis demonstrirebas; iyenebs figuris simet riulobas figuris Tvisebis dasadgenad.

maT. VII.12. iyenebs koordinatebis meTods orien ta ci isaT visSedegi TvalsaCinoa, Tu moswavle:• orientirebs rukaze an sakoordinato sibrtyeze koor di natebis gamoyenebiT

(mag. asaxelebs mocemuli wertilis koor dinatebis miaxloebiT an zust mniSvnelobas; poulobs wer tils mocemuli mTelricxovani koordinatebis mixedviT).

• asaxelebs sakoordinato RerZebis mimarT mocemuli wer tilis RerZulad si-metriuli wertilis koordinatebs.

• poulobs paraleluri gadataniT miRebuli figuris nebis mieri wertilis ko-ordinatebs misi winasaxis ko or dinatebisa da miTiTebuli paraleluri gadatanis meSveobiT.

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

Sedegi TvalsaCinoa, Tu moswavle: • iyenebs samkuTxedebis tolobis niSnebs figuraTa Tvi se bebis dasadgenad, fi-

guraTa ucnobi elementebis mosaZebnad an realur viTarebaSi manZilis arapirdapiri gziT dasadgenad.

• xsnis agebis martiv amocanebs.• poulobs mizez-Sedegobriv kavSirebs samkuTxedTan da mis elementebTan da-

kavSirebul debulebebs Soris.Sinaarsi1. wertilebi, wrfeebi da sibrtyeebi: mimarTebebi maT Soris.2. geometriuli figurebi: klasifikacia sxvadasxva niSniT (mag. amozneqili

da araamozneqili, brtyeli da sivrculi).3. kuTxeebi: elementebi, zoma, klasifikacia, Tvisebebi.4. samkuTxedebi: elementebi, klasifikacia, Tvisebebi, tolobis niSnebi.5. geometriuli gardaqmnebi sibrtyeze: paraleluri gadatana, samkuTxedebi

(elementebi, klasifikacia, Tvisebebi, tolobis niSnebi).6. koordinatTa sistema; orientireba sibrtyeze, gar daq mne bis gamosaxva.7. agebis umartivesi amocanebi: mocemuli samkuTxedis to li samkuTxedis,

kuTxis biseqtrisa, monakveTis SuamarTobis age ba.

mimarTuleba: monacemTa analizi, albaToba da statistikamaT. VII.14. moipovebs dasmuli amocanis amosaxs nelad saWiro Tvisebriv da

raodenobriv monacemebsSedegi TvalsaCinoa, Tu moswavle: • ganasxvavebs Tvisebriv da raodenobriv monacemebs, iye nebs monacemTa Segro-

vebis Sesaferis saSualebas (gazomva, dak virveba).• mocemul TemasTan dakavSirebiT svams kiTxvebs, gan sazRvravs respondentebs

da moipovebs saWiro monacemebs.• mocemuli amocanisTvis damoukideblad gegmavs da atarebs statistikur eqs-

periments da agrovebs monacemebs.

16

maT. VII.15. awesrigebs da warmoadgens Tvisebriv da rao de nobriv monacemebs dasmuli amocanis amosaxsnelad xel say reli formiT

Sedegi TvalsaCinoa, Tu moswavle: • axdens Tvisebriv da raodenobriv monacemTa dalagebas/kla sifikacias, war-

moadgens monacemebs siis/piqtogramis saxiT, msjelobs dalagebis/klasifikaciis principze.

• qmnis mowesrigebul monacemTa cxrilebs da asabuTebs SerCeuli dizainis mi-zanSewonilobas.

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

maT. VII.16. akeTebs Tvisebriv da raodenobriv monacemTa in ter pretacias da analizs amocanis konteqstis gaT va lis winebiT

Sedegi TvalsaCinoa, Tu moswavle: • svams kiTxvebs monacemebis Sesaxeb an/da axasiaTebs mo nacemebs, romlebic

warmodgenilia siis, cxrilis, piq tog ra mis an diagramis saxiT, msjelobs arsebul kanon zo mierebebsa da gamorCeul monacemebze.

• irCevs Sesaferis Semajamebel ricxviT maxasiaTeblebs, asa buTebs Tavis arCe-vans, iTvlis da iyenebs maT monacemTa jgu fis dasaxasiaTeblad.

• adarebs monacemTa ramdenime jgufs da warmoaCens Tvi sebriv da raodenobriv msgavsebasa da gansxvavebas maT Soris (Semajamebeli ricxviTi maxasiaTeblebis gare-Se).

Sinaarsi1. monacemTa Segrovebis saSualebani: gazomva da dakvir ve ba; gamo

kiTxva;statistikuri eqsperimenti.2. Tvisebrivi da raodenobrivi monacemebis organizacia: mo nacemebis klasifika

cia (garda intervalebad dajgufebisa); mo nacemTa dalageba zrdadobaklebadobiT an leq si ko gra fi uli meTodiT.

3. monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: monacemTa raodenoba, pozicia da Tan mim devroba erTobliobaSi, monacemTa sixSire; ganmeorebis ti pis kanonzomierebani; gamorCeuli (mag., eqstremaluri, iS vi aTi) monacemebi.

4. monacemTa warmodgenis saSualebani raodenobrivi da Tvi sebrivi monacemebisTvis: sia, cxrili, piqtograma, wer ti lovani, meseruli, xazovani, svetovani diagramebi.

5. monacemTa Semajamebeli ricxviTi maxasiaTeblebi Tvi seb rivi da raodenobrivi monacemebisTvis: centraluri ten den ciis sazomebi saSualo, moda; monacemTa gafantulobis sa zomi gabnevis diapazoni.

17

Sinaarsisa da miznebis ruka

Temebis Camona Tvali miznebi; Sinaarsis gardaqmna miznad; erovnul saswavlo gegmaSi miTiTebuli Sesabamisi punqtebi

sa vara u do sas wav lo

dro

naturalur ricxvTa mwkrivi naturaluri ricx vis Caweris sxvadasxva wesi

naturaluri ricxvebisa da maTi Tvisebebis Sesaxeb codnis gaRr ma ve-ba; poziciuri sistemis warmodgena; Tvlis sistemebi (arasavaldebulo) didi ricxvebi da maTi Caweris xer xe-bi. es masala wina wlebSi Seswav li li sakiTxebis gameorebasa da gan m tkicebas emsaxureba. VII.2(2)

4 sT

jgufuri muSaoba (xa ris xis Tviseba)

kanonzomierebebis aRmoCenis unaris ganviTareba axarisxebisas e. w. `bo lo cifris~ cxrilis saSualebiT. VII.2(4)

1 sT

gayofa, gayofadobis Tvise-bebi gayofadobis niSnebi

jgufuri muSaoba udide-si saerTo gam yo fi umciresi saerTo je ra di

poziciuri sistemis gamoyenebis un-aris ganviTareba, naSTiani gayofis Tvisebebis gamoyenebiT kanonzo mie re-bebis gansazRvra. poulobs mocemuli ricxvebis udides saerTo gamyofsa da umcires saerTo jerads, iyenebs maT praqtikuli amocanebis amoxs ni-sas. VII.1(4)

12 sT

Tematuri Semowmeba (sakon-trolo wera # 1)

umciresi saer To jeradisa da udide-si saerTo gamyofis cnebebis aTvisebis Semow meba.

1 sT

proeqti xarjTaRricxvis Sedgenis unaris gamomuSaveba, arCevanis dasabuTeba, cnobarebis gamoyeneba VII.3(1)

1 sT

mTeli ricxvebi mTeli ricxvis mo du li, mTeli

ricxve bis SedarebamTeli ricxvebis SekrebamTeli ricxvebis ga mok lebamTeli ricxvebis gam rav leba

da gayofa

naturaluri ricxvis cnebis ga far-Toebis algebruli gzis gacno bi-ereba, geometriuli gamosaxvisa da misi gamoyenebis unaris gamomuSaveba; Sedareba, moduli. mTel ricxvebze moqmedebebis TvalsaCino warmodgena (sxvadasxva modelis gamoyenebis un-ari). VII. 2(1,2).

7 sT

simravle, qve sim rav le, venis diagramebi

Teoriul-simravluri cnebebi da maTi gamoyenebis unari (amocanebis amoxs na venis diagramebis gamo ye ne biT) VII.6

4 sT

jgufuri muSaoba naSTTa ariTmetikis propedevtika 1 sT

dadebiTi racio na luri ricx-vebi

aTwiladis CanawerSi Tanrigebis miTi-Tebisa da cifrebis mniSvne lo be bis cod-nis gamoyeneba aTwiladebis Se darebisas. VII.1

2 sT

jgufuri muSaoba xarjTaRricxvis dagegmva, Sefa se ba, SerCeva. VII. 3(1)

1 sT

18

racionaluri ricx vebimoqmedebebi racio nalur

ricxvebze.racionaluri ricx vis natura-

lurmaCve neb liani xarisxi

racionalur ricxvTa simravlis gaaz-reba, Tvisebebi, moqmedebebi, gamoyeneba sayofacxovrebo amocanebis amoxsnisas. VII.2, VII.3

6 sT

Tematuri Semowmeba(sakontrolo wera # 2)

masalis aTvisebis Semowmeba 2 sT

cvladiani gamosa xu le ba. cv-ladiani gamosaxulebis mniS-vnelobis povna. ga mo saxulebis gar daqmna.

cvla diani gamosaxulebis Sedgenis Cve vebi , cvladiani gamosaxulebebis gar daqmnis sxvadasxva wesis dauf le ba (moqmedebaTa Tvisebebi, Semok le buli gamravlebis formulebi). geometriuli TvalsaCinoebis gamo ye ne ba. VII. 7

12 sT

Tematuri Semowmeba(sakontrolo wera # 3)


monacemTa analizi da statis-tika1. monacemTa Segro veba2. monacemTa war mod ge na xelsay-reli for mi T3. piqtograma

miznobrivi informaciis Segro ve bi sa da damuSavebis xerxebis Sed ge nis unaris gamomuSaveba.

VII.14, VII.15

3 sT

geometriuli figu rebi1. wertili. wrfe. mo nakveTi. sxivi. wrfeebis urTierTgan-lagebis SemTxvevebi.2. wrfeebisa da sibr tye ebis urTierT gan la gebis SemTxveve-bi3. kuTxe. kuTxeTa ur TierT-ganlageba 4. monakveTis da kuTxis gazom-va5. kuTxeTa klasi fi ka cia. marTo-buli wrfe ebijgufuri muSaoba (amo canebis amoxsna wrf eze modelirebis gamoyenebiT)

geometriuli figurebis warmod ge nis, urTierTganlagebis SemTxve ve bis aT-viseba, gazomvis operaciis arsis Sesaxeb warmodgenis Camoya li be ba (monakveTis sigrZe, kuTxis zo ma); kuTxeebis zome-bis mixedviT kla si fikaciis warmodgena. nacnobi geo met riuli figurebis amocno-bis una ris gamomuSaveba. VII.9; VII.10

5 sT

Tematuri Semowmeba (sakontrolo wera # 4)


gantoleba. ganto le bis fesvi wrfivi erTucno biani gantole-ba

gantolebis, rogorc bunebasa da sa-zo gadoebaSi mimdinare mraval fe ro van movlenaTa aRweris ZiriTadi mo delis Sesaxeb, modelirebis meTo de bis Sesaxeb warmodgenebis gaazreba; gan tolebis, rogorc maTematikis erT-erTi ZiriTadi enis dauflebis unar ze muSaoba. VII.7

3 sT

wertilis koor di na tebi wrfeze da sib r tyeze. ganto lebaTa amox-snis geometriuli xerxi 1. wertilis koor di na tebi wr-feze2. wertilis koor di natebi si-brtyeze3. sakoordinato me oTxe debi

maTematikis mniSvnelovani aparatis _ koordinatTa meTodis gacnoba; mis gamoyenebis unarze muSaoba; ko or di na-tebis gamoyenebiT amocanebis amox snis xerxebis Seswavla.VII. 7

4 sT

19

amocanebis amoxsna gantoleb-is saSua le biTamocanebis amoxsnis sxva-dasxva xerxi

teqsturi amocanebis amosaxsnelad gan-tolebis Sedgenisa da amoxsnis unaris gamomuSaveba

amocanebis amoxsnis sxvadasxva xerxebis warmodgena VII.7, VII.6

8 sT

naturalurmaCvenebli ani xarisxis Tvisebebi

naturalurmaCvenebliani xarisxis Tvise-bebis warmodgena da gamoyeneba VII.2(4)

2 sT

jgufuri muSaoba(naturalurmaCve neb li ani xarisxis Tvise be bis gamoy-eneba)

1 sT

texili. mra val kuTxe di asaxelebs da amoicnobs geomet ri-ul figurebs: texils, mraval ku Txeds. gamosaxavs brtyel figurebs ise, rom maTi TanakveTa (gaerTianeba miTiTebuli formis an Tvisebebis mqone figura iyos). VII.10; VII.11

3 sT

jgufuri muSaoba (kombi-natorikis amocanis amox-sna xisebri diagramis saSu-alebiT)

1 sT

RerZuli simetriaparaleluri gada ta naamocanebis amoxsna

iyenebs paralelur gadatanas, Rer Zul simetrias figuraTa Tvisebebis dasa-dgenad; garemomcvel obieqtebs Soris moiZiebs si metriul obieq tebs; xazavs texilis, mraval kuTxe dis simetriul figuras mi Ti Te bu li RerZis mimarT da axdens maT pa ra lelur gadatanas. VII.12

6 sT

individualuri mu Sa obis proeqti (`umok lesi marSru-ti~ _ te xilis sigrZe)

1 sT

Tematuri Semowmeba(amocanebis amoxsnis sxva-dasxva xerxi, na tu ra lur-maCvenebliani xarisxisakontrolo wera # 5)

2 sT

S e f a r d e b a . p r o p o r c i a . proporciis gamo ye ne ba. pro-centi amo ca ne bis amoxsna procen tis gamoyenebaze.

iyenebs proporcias, procents praq ti-kuli amocanebis amoxsnisas (xsna re bi, Senadnobebi, ....); xsnis praq ti kul amo-canebs umartives xarjT aRricxvasTan dakavSirebiT; iyenebs ricxvis propor-ciuli gayofis wess amocanebis amoxs-nisas.

iyenebs proporcias masStabTan da ru-kaze manZilis povnasTan dakavSi re buli amocanebis amoxsnisas. akav Sirebs zomis sxvadasxva erTeulebs

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

9 sT

jgu fu ri muSaobis proeq ti (martivi xarjTaR ricx vis Sedgena)

2 sT

Tematuri Semowmebasakontrolo wera # 6

2 sT

20

statistikis elemen te bi. mo-nacemTa Sema ja mebeli ricx-viTi ma xa siaTeblebi.

avlens Tvisebriv da raodenobriv mo-nacemTa interpretaciisa da anali-zis unars, Semajamebeli ricxviTi ma xa-siaTeblebis warmodgenis da gamo ye nebis unars VII. 16

2 sT

samkuTxedi. samku Txe dis el-ementebi

3 sT

samkuTxedebis klasi fi ka-cia

samkuTxedisa da misi elementebis war-modgena; klasifikacia sxvadasxva niSnis mixedviT. asaxelebs misTvis nacnob geome-triul figuras yofiTi daniSnulebis sagnebSi VII. 9(3)

2 sT

mimdevrobebi mimdevrobis mocemuli fragmen te-bis mixedviT gavrcobaTa warmodgena. obieqtebis perioduli mimdevrobebi. mudmivi nazrdis mqone ricxviTi mimdev-robebis warmodgena; periodul mimdev-robaSi periodis dasaxeleba. VII.8

2 sT

jgufuri muSaoba (gantoleb-is Sed gena, perioduli mim-dev robebis gamoye ne ba)

1 sT

tolferda samkuT xedi samkuTxedebis to lo bis niS-nebis a m k u T x e d i s k u T x e e b i s jamimarTkuTxa samkuT xedi monakveTis SuamarTo bis Tviseba

samkuTxedebis tolobis niSnebis war-modgena da maTi gamoyeneba praqtikuli amocanebis amoxsnisas

tolferda samkuTxedis Tvisebebis dasabuTeba

samkuTxedis kuTxeebis jamis for mu lis gamoyvana, TvalsaCino war mod gena. marT-kuTxa samkuTxedis Tvi se be bis warmodgena da gamoyeneba praq ti kuli amocanis amox-snisas. mona kve Tis SuamarTobis Tvisebis gamo ye neba.

8 sT

2 sT

j g u f u r i m u S a o b a ( s a m -kuTxedebis tolo bis niSne-bis gamoyeneba)

1 sT

individualuri mu Sa o bis proeqti (xarj TaR ricx vis Sedgena)

1 sT

kuTxis biseqtrisis Tviseba kuTxis biseqtrisis Tviseba da misi gamoyeneba. agebis amocanebi. VII.13(1) 2 sT

Tematuri Semowmeba, sakon-trolo wera #7 (samkuTxe-di, Tvisebebi)

2 sT

amocanebi agebaze samkuTxedis toli samkuTxedis age ba monakveTis SuamarTobis, biseqt risis agebis amocanebi VII.13(2)

4 sT

gameoreba 6 sT

21

saswavlo masalis wardgenis fazebi da gakveTilis dagegmvis zogadi principebi

saswavlo procesis organizaciis ZiriTadi forma gakve Ti lia; saganmanaTleblo, aRmzrdelobiTi da praqtikuli miz ne bis ganxorcieleba gakveTilze xdeba. amitom maTematikis swav lebis ZiriTadi sakiTxi gakveTilis kargi momzadeba da Cata rebaa.

meTodikur literaturaSi didaqtikuri cneba _ `gakve Ti li~ ZiriTadad ase aRi-wereba: gakveTili logikurad das ru lebuli, mTliani saswavlo-aRmzrdelobiTi pro-cesis gar kve uli SemosazRvruli monakveTia. masSi rTul urTierT da mo kidebulebaSia procesis yvela ZiriTadi elementi: Sinaarsi, mizani, saSualebebi, meTodebi, or-ganizacia; yovel gakveTilze gansazRvruli saganmanaTleblo da aRmzrdelobiTi amocanebi wydeba; am amocanebis gadawyveta konkretuli saswavlo ma sa lis ganxilvis procesSi mimdinareobs.

maTematikis gakveTilisadmi wayenebuli mTavari moTxov naa ZiriTadi didaqtikuri amocanis arseboba _ im Temis Ses wav lis miznis arseboba, romlis gadawyvetis procesi mo ce mul gakveTilze mimdinareobs. yoveli gakveTilis win kargad unda gaiazroT misi Sinaarsi da mizani, amaSi Cven mier mo wo debuli ruka dagexmarebaT. miznis Sesabamisi unda iyos kar gad gaazrebuli da dagegmili saswavlo masalis wardgenis fazebi:

• motivacia; ar aris sakmarisi, rom maswavlebels gaaz re buli hqondes mizani; saWiroa, rom igi moswavleebisTvisac gax des ZiriTadi mizani. moswavleebTan saubari unda daviwyoT ara imiT, Tu ras vaswavliT, aramed mniSvnelovania Tavidanve in-teresis aRZvra da iseTi situaciis Seqmna, roca moswavle motivirebulia da pou-lobs pasuxebs kiTxvaze _ `risTvisaa saWiro~. motivacia SesaZlebelia praqtikuli amocanis dayenebiTa da misi amoxsnis xerxebis ZiebiT daviwyoT, an maTematikis Siga kanonzomierebis gaazrebiT, problemuri situaciis Seqmnis xelSewyobiT; mxolod amis Semdeg xdeba Sesabamisi amocanis dasma da misi amoxsnis Zieba.

• amocanis gansazRvris Semdeg mimdinareobs misi amoxsnis gzis Ziebis procesi. amoxsnis Ziebis procesis warmarTvis sxvadasxva meTodikuri saSualeba arsebobs. yovel amocanas, rogorc wesi, amoxsnis Ziebis garkveuli forma miesadageba. es formebia _ muSaoba jgufebad (SesaZlebelia or-oradac), mTeli klasis erToblivi monawileobiT, individualuri muSaoba da a. S. im SemTxvevaSic ki, roca amocanis amoxsnis ZiebaSi mTeli klasi erTdrouladaa Cabmuli, maswavlebelma swavleba ise unda warmarTos (moxerxubuli kiTxvis dasmis saSualebiT), rom Temis Seswavlis procesis ZiriTadi Semoqmedebi Tavad moswavleebi aRmoCndnen; maswavlebeli am SemTxvevaSi warmarTve-lis, diriJoris funqcias unda asru lebdes.

moswavlis saxelmZRvanelos teqsti mogcemT aseTi gakve Ti lebis Catarebis sa-Sualebas. igi ZiriTadi saswavlo saSua le baa da amitom gakveTilisTvis mzadebis procesSi teqs tebis, ilustraciebis, savarjiSoebis gacnobas, daxarisxebas didi dro unda dauTmoT. Tumca, maswavleblis wignic dagex ma rebaT TiToeul etapze (sawyisi, damagvirgvinebeli) Sesas rulebeli savarjiSoebis SerCevis sakiTxSi.

Sinarsisa da miznebis rukaSi ver mivuTiTeT Temis Seswav lis yvela mizani, romelic saganmanaTleblo da aRmzrde lo biTi amocanebis gadawyvetis komponentebisgan Sedgeba. ar unda gavaigivoT mizani SinaarsTan. magaliTad, samkuTxedis kuTxeebis jamis for-mulis gamoyvana saxelmZRvaneloSi isea warmodgenili, rom igi am Temis Sinaarsis miz-nad gardaqmnis process kargad warmogvidgens; dasmuli amocanis amoxsna kritikuli azrovnebis, msjelobisa da dasabuTebis unaris gamomuSaveba, geometriul figuraTa gamosaxvis da am figu raTa Tvisebebis SeswavlaSi eqsperimentis gamoyenebis unaris gamomuSavebaa. am miznebis gansaxorcieleblad qaRaldisgan gamoWril samkuTxedze (modelze) Catarebuli eqsperimentis (gamoiyeneba ramdenime RerZuli simetria) an

22

dasabuTebis xer xia mowodebuli. meore maTgani dasmul kiTxvebze pasuxis SerCevis, msjelobisa da dasabuTebebis fonze mimdinareobs. aq maswavlebelma SeiZleba sxva eqsperimentic daamatos (tran s portiris gamoyenebiT). ufro dawvrilebiT am gakveTi-lis Sesaxeb qvemoT mogaxsenebT.

am konkretuli magaliTidan Cans, rom aqtivobis mizani saganmanaTleblo da aRmz-rdelobiTi miznebis erTobliobaa.

gakveTilze Seswavlili faqtebi TavisTavad aris mniSv ne lovani; Tumca, kidev ufro mniSvnelovania is, rom maTi Ses wav lis dros ganxorcielebuli procesi Tavis kvals tovebs _ kritikuli azrovnebis ganviTareba, msjeloba da dasabuTebis unaris gamomuSaveba. amasTanave, moswavleTa gonebaSi gaiazreba mniSvnelovani sakiTxebi: adamianis mier faqtebis aRmoCena, aRmocenebuli amocanebis amoxsna da Sedegebis dafiqsireba azrovnebaSi. konkretuli deduqciuri msjeloba, dasabuTeba im SemTx-vevaSi atarebs aRmzrdelobiT funqcias, roca mos wav les gavagebinebT misi Catarebis mniSvnelobas.

axla davasaxelebT ramodenime sasargeblo zogad reko men dacias, romelic meTo-dikis sakiTxebs ganekuTvneba:

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

• maswavleblis xelmZRvanelobiT da moswavleTa Za lis xme viT _ dakvirvebis, cdis, konkretuli SemTxvevebis ana li zis Sedegad iqmneba warmodgena, hipoTeza arsebul kanon zomierebaze.

• maswavleblis xelmZRvanelobiT mimdinareobs dasa bu Te biT gzis Zieba, amocanis amoxsnis gegmis Sedgena; rasac xSi rad mosdevs TviT moswavleebis mier am gegmis realizacia.

axla SevexoT saswavlo procesisTvis mzadebis zogad sqemas:• maswavleblis muSaoba saswavlo wlis win.• gakveTilTa sistemis gaazreba saswavlo wlis win;• konkretuli gakveTilisTvis momzadeba.momzadebisas maswavlebelma unda gamoiyenos moswavlis wigni, maswavlebelis

sarekomendacio wigni da, saWiroebis SemTxvevaSi, iq miTiTebuli literatura. sarekomendacio wig nis mixedviT, sasurvelia, maswavlebeli Tavdapirvelad sas wav -lo gegmas, Sefasebis sistemas da gakveTilebis Semo Ta va zebul nimuSebs gaecnos. sarekomendacio wignis Sesavali mas gaacnobs avtorTa mier saxelmZRvanelos agebis ZiriTad prin cipebs.

maswavlebeli SemoqmedebiTad unda miudges Cven reko men da ciebs; igi safuZvlad iRebs Cven mier SemoTavazebul gegmas da azustebs mas sakuTari gamocdilebiTa da klasis Tavise bu rebebis gaTvaliswinebiT. es dazustebebi gansakuTrebiT mniSvnelovania sawyis etapze Sesasrulebeli savarjiSoebis da damamTavrebel etapze Sesasrulebeli savarjiSoebisa da sakontrolo weris variantebis SerCevisas.

gakveTilis dagegmva iTvaliswinebs Casatarebeli procesis tips:• axali masalis gacnoba• masalis ganmtkiceba• codnis Semowmeba• sxva tipis gakveTilebi (gakveTili bunebaSi, gakveTili-proeqti ...)Tumca, zogierTi tipis procesi (axali masalis gacnoba, gan mtkiceba, Semowmeba),

rogorc wesi, yovel gakveTilze mim di nareobs, SesaZlebelia _ sxvadasxva moculo-biT.

moswavleTa codnis Semowmeba, moswavleTa muSaobaze dak vir veba yovel gakveTilze mimdinareobs. gakveTilze mTavaria vaswavloT da aRvzardoT. swavleba ar niSnavs

23

mxolod codnis gadacemas _ swavleba codnis SemoqmedebiTad dauf lebas unda niSnavdes, misi gamoyenebis unaris ganviTarebaze unda iyos mogebuli. moswavleTa Sefaseba swavlis procesze maswavleblis dakvirvebebiT, moswavleTa mier sakon-trolo da damoukidebeli samuSaoebis Sesrulebis xarisxiT gani saz Rv reba. yoveli gakveTilis Semdeg sakuTar wignakSi Cai niS neT moswavleebze dakvirvebebis Sedegebi, gaiTvaliswineT mos wavleTa SemoqmedebiTi aqtiuroba (masalis aTvisebis done kargad Cans savarjiSoebis amoxsnis drosac _ ganmtkicebis pro cesSi).

saswavlo masalis dasabuTebuli SerCeva iTvaliswinebs Sem deg moTxovnebs:• saswavlo masalis Sesabamisoba Temis mizanTan• gakveTilze Sesasrulebeli samuSaos moculobis swori gansazRvra• optimaluri Tanafardoba konkretulsa da zogads Soris• Teoriasa da praqtikas Soris aucilebeli urTierT kav Siris ganxorcielebamaswavlebelma, rogorc wesi, Tavidan bolomde deta lu rad unda gaiazros gakveTili,

winaswar, drois mixedviT unda iyos ganawilebuli mTeli samuSao.magaliTad, Tu gakveTilze axal Temaze gadasvlac aris gaT valiswinebuli, maSin

SeiZleba im sakiTxebis Sesaxeb msje loba, romelTa bunebriv da kanonzomier gagr-Zelebis axali sa kiTxebi Seicavs, SeiZleba gakveTili pirdapir im praqtikuli amocanis ganxilviT daviwyoT, romlis maTematikuri modelis Seswavla axali maTematikuri faqtebis aRmoCenas, hipoTezis Camoyalibebasa da dasabuTebebs moiTxovs. am procesis buneb rivi gagrZeleba Sesabamisi savarjiSoTa sistemis ganxilvaa.

kiTxvebze pasuxebis gacemis sistema ar unda iyos erT fe rovani _ mxolod warmate-buli moswavleebTan mimarTebiT ar unda SemoifargloT; moswavlis raime mosazrebas myisve nu upasuxebT. swor pasuxsac ki maSinve nu daeTanxmebiT xol me _ gaakeTeT pauza, iqneb garkveuli eWvic ki gamoTqvaT misi mosazrebis sisworis mimarT. amiT miaRwevT imas, rom bavSvebi daubrundebian dasmuli kiTxvis analizs da male WeS-mariti daskvna _ swori pasuxi klasis dominantur mosazrebad gadaiq ceva. klasi, erToblivi ZalisxmeviT, `gaiZulebT~ daeTanxmoT mis pozicias. es axarebs, amxnevebs da aerTianebs axal gazr debs. es maTi erToblivi azris gamarjvebaa. Tqvens mizansac xom es warmoadgens _ moswavle CamoayaliboT Semoqmed, codniT aRWurvil, inicia-tivian, xalisian axalgazrdad. maTematika mZlavri emociuri muxtis matarebelia da misi amoqmedeba Tqveni ZalisxmeviT miiRweva~.

yuradRebiT unda movisminoT yvela pasuxi, uxeSi Sec do mis SemTxvevaSic ki dauS-vebelia mkacri uaryofiTi Sefa se bebis gamoTqma.

yuradReba miaqcieT, rom terminebi, cnebebi da movlenebi sworad iyos ganmarte-buli.

gakveTilebis dagegmvasa da warmarTvaSi xels SegiwyobT sanimuSo gakveTilebis scenarebi.

axali masalis ganmtkicebis procesi SeiZleba e. w. `tes turi~ amocanebis `amoxsniT~ daviwyoT, maTi Sesruleba swori pasuxis SerCeviT unda Semoifarglos _ zogjer SeiZleba komentarebis gakeTebac gaxdes saWiro. yovel paragrafSi mo ce muli masala, rogorc wesi, 2 gakveTilzea gaTvalis wi ne buli; meore gakveTilze codnis ganmtkice-baze zrunviT Semo vifarglebiT.

swavlebis erT-erTi saintereso da mniSvnelovani forma jgu furi muSaobaa. es muSaoba SeiZleba gakveTilis procesis erT-erTi Semadgeneli nawili iyos _ daukavSirdes axali ma salis gaazrebas, praqtikuli saqmianobis (eqsperimentis) an Semajamebeli daskvnebis gamotanas, an, SesaZlebelia, mas mTe li gakveTilic davuT-moT. mis warmarTvaSi moswavlis sa xel mZRvaneloSi warmodgenili amocanebis sistema dagex ma rebaT (magaliTad, amocanebi jgufuri muSaobisTvis). am Sem TxvevaSi jgufur muSaobas SeiZleba Sejibris saxec ki miv ceT.

24

organizaciulad jgufuri muSaobis es varianti _ paeq ro ba _ jgufuri muSaoba SeiZleba ase movawyoT:

winaswar vacxadebT Catarebis dRes; moswavleebs vavalebT samuSao rveulis or-magi furceli iqonion. paeqroba mos wav leTagan kapitnebisa da maTi TanaSemweebis dasaxelebiT iwyeba. optimaluria 4-5 moswavlisgan Semdgari jgufebi _ gundebi. gun debis dakompleqteba SeiZleba kapitnebsac miandoT. mTa varia, `arCevnebma~ didi dro ar wagarTvaT.

mas Semdeg, rac gundebi dakompleqtdeba, winaswar gamrav le buli amocanebi daurigeT gundebs (an dafaze amowereT pirobebi). am SemTxvevaSi yvela gunds erT-nairi davaleba miecema.

gakveTilis dasrulebamde 10-12 wuTiT adre kapitnebs eva lebaT warmoadginon maTi gundebis Sedegebi _ maTi amocanebis amoxsnebi. CaibareT es amoxsnebi winas-wari komentarebis gareSe da saukeTeso amoxsnebis avtorebi rigrigobiT miipatiJeT dafasTan naSromTa mokle prezentaciisTvis. cxadia, kritika da polemika, Tu amis safuZveli arsebobs, unda iyos uSeRa vaTo, magram koreqtuli. am bWobaSi Tqvenc mogiwevT xandaxan Cabma; zogjer mediatoris rolis Sesrulebac. es procedura aRniS-nul droze mets ar moiTxovs, radgan amocanebi yvelas kar gad aqvs gaazrebuli da mxolod sakvanZo punqtebia xaz gasasmeli.

am gansjis dasrulebisTanave unda aRdges merxebis Tavda pirveli ganlageba, Semdeg acxadebT gundebis mier mopovebul qulebs (TiToeuli amocana SeiZleba 2-quliani skaliT Sefasdes) da dakavebul adgilebs am paeqrobaSi. SeiZleba daawesoT damate-biTi qulebi prezentaciis Sesafaseblad.

jgufuri muSaobisTvis amocanebi SeiZleba SeirCes saxel mZRva neloSi Sesabamisi niSnakiT gamoyofili adgilidan. Tumca, winaswar yvela amocana rom ar iyos `gaSi-fruli~, iqneb maTi pirobebi odnav SecvaloT, an, zogierTi amocana pa rag rafis damatebiTi savarjiSoebis krebulidan airCioT.

moswavlis saxelmZRvaneloSi SemoTavazebuli jgufuri mu Saobis zogierTi proeqti raime erTi axali Temis Sesabamisi struq turirebuli kiTxvebisgan Sedgenili amocanaa _ yoveli kiTxva winas ukavSirdeba _ kiTxvebze pasuxebis sistema raime erTi axali Temis Sesabamisi problemis dasmas da amoxsnas gulisxmobs. am SemTxvevaSi pasuxebis sistema, romelic mos wav leTa jgufis erToblivi Zalisxmevis Sedegia, maTi ko leq-tiuri SemoqmedebiTi naSromia da misi Sefaseba prezen ta ciisas ganxorcieldeba.

zogjer jgufebs sxvadasxva saxis davalebebi SeiZleba mivceT _ gansxvavebuli eqsperimentis Catareba (magaliTad, geometriuli obieqtis Tvisebebis dasadgenad an dasabuTebis gziT geometriuli faqtis ganxilva-dasabuTeba, Sedegis war modgena. am SemTxvevaSi moswavleebi msjeloben warmodgenili varaudebis koreqtulobaze da adareben maT. aseTi tipis jgufuri muSaobebi, rogorc zemoT aRvniSneT, gakveTilis fragmenti SeiZleba iyos da xSirad unda gamoviyenoT.

Teoriuli masalis gadmocemis Cveneuli meTodika saSua lebas gaZlevT airCioT, yvelaze ufro mosaxerxebeli forma Sinaarsisa da miznebis rukaSi miTiTebuli moTx-ovnebis Sesas ruleblad.

amrigad, gakveTilis dagegmva SesaZlebelia Semdegi sqemiT mimdinareobdes.• saswavlo miznebi _ gakveTilis ideuri Sinaarsi; is, rasac unda miaRwioT, ra

codna da unar-Cvevebi unda SeiZinon moswavleebma am gakveTilze;• Sefaseba _ ra kriteriumebiT SevafasebT dagegmilis Sesrulebis xarisxs da ra

SemTxvevaSi vityviT, rom gakveTilma warmatebulad Caiara;• miRwevis saSualebebi _ ra meTodebiT, moswavleTaA ra organizebiTa da ra

damatebiTi resursebis saSualebiT apirebT dasaxuli amocanebis gadaWras; sasurvelia winaswar gaTvaloT drois savaraudo xarji; aucileblad winaswar gansazRvreT saSinao davalebis xasiaTi da moculoba.

25

Sefasebis zogadi principebi

saskolo Sefasebis axali sistema, romelic erovnuli saswavlo gegmiT aris gansaz-Rvruli, iTvaliswinebs Semdeg aucilebel midgomebs: akademiuri moswrebis Sefaseba unda iyos xSiri da mravalmxrivi. unda Sefasdes ara marto informaciis floba, aramed SeZenili unar-Cvevebi. ar aris sakmarisi moswavle mxolod sakontrolo were-bis safuZvelze Sefasdes. maswavlebeli unda afasebdes prezentaciebis, mos wav lisave TviTSefasebis, jgufuri muSaobis, Tu sxva tipis aqti vobebis mixedviT. maswavlebelma Sefasebisas unda gaiT valiswinos saganmanaTleblo procesSi moswavlis CarTulobis xarisxi (saxlSi micemuli davalebis Sesrulebis xarisxi, gak ve Tilze aqtiuroba, SemoqmedebiToba da sxva), amasTanave, mi zan Sewonilia moswavlesac gavacnoT winaswar Sefasebis kri te riumebi. am kriteriumebis SedgenaSi SeiZleba moswavleTa Car Tvac.

sakontrolo werebis Sefasebis sqemebs am wignSi gaec no biT. dauSvebelia moswav-leTa qcevis gaTvaliswineba akade miuri moswrebis Sefasebisas _ gakveTilze arasa-Tanadod moqceva, rogorc wesi, TavisTavad aisaxeba akademiur moswrebaze.

moswavlis niSani unda gamomdinareobdes mis mier sagnis Seswavlis sxvadasxva komponentisgan _ sakontrolo weris Sesruleba, gakveTilze msjeloba, jgufuri muSaoba, prezen tacia da sxva.

am komponentebs maswavlebeli wonebs aniWebs. wonebis miniWebis sistema SeiZleba maswavlebelma sxvadasxvanairad da gegmos. es wonebi SeiZleba sxvadasxva iyos sxva-dasxva tri mes trisTvis, magram unda gvaxsovdes, rom trimestris niSa ni (umaRlesi qulaa 10) aucileblad sxvadasxva komponen tis Sefasebebisgan unda gamomdinareobdes. magaliTad, maT SeiZleba Semdegi wonebi mivakuTvnoT: prezentacia _ 25%, gakveTilze aqtivoba, msjeloba _ 30%, sakontrolo wera _ 25% da sa Si nao davalebis Sesruleba _ 20%. magaliTad, Tu saSualo Sef asebebi TiToeuli komponentis mixedviT aris 6, 9, 10, 8, maSin Sefaseba ase gamoiTvleba:

0,25·6+0,3·9+0,25·10+0,2·8=8,6 ... aq saboloo Sefaseba moswavlis sasargeblod unda moxdes da daiweros `9~.wliuri Sefaseba trimestruli Sefasebebis saSualo SeiZ le ba iyos; Tumca, aqac

SesaZlebelia koreqtivebis Setana _ trimestrebis mixedviT progresis gaTval-iswineba.

axla gTavazobT proeqt `ilia WavWavaZis~ erovnuli sas wav lo gegmebisa da saxelmZ-Rvaneloebis SemmuSavebeli jgufis mier momzadebul sqemas Sefasebis ZiriTadi mdgenelebisa (saSinao davaleba, Semajamebeli samuSao, klasSi CarTuloba, proeqti) da Sesabamisi unarebis Sesaxeb, zogierTi maTganis mixedviT niSnis gamoyvanis vari-ants.

26

Sefasebis ZiriTadi mdgenelebi

saSinao davaleba

unarebi1. maTematikuri cnebebisa da debulebebis gamoyeneba.2. kavSirebisa da mimarTebebis dadgena.3. maTematikuri obieqtebis warmodgena da maTematikuri enis floba.4. damxmare teqnikuri saSualebebisa da sainformacio teqnologiebis gamoyen-

eba.

unarebi fasdeba Semdegi aqtivobebiT:moswavle1. aRiqvams amocanis Sinaarss, gaiazrebs da gamijnavs amo ca nis monacemebsa da

saZiebel sidideebs. axdens monacemebis (maT Soris problemis gadasaWrelad saWiro monacemebis) or ga nizebas da maT warmodgenas.

2. gadmocemisas sworad da efeqtianad iyenebs maTematikur terminebsa da aRniS-vnebs. adekvaturad irCevs simkacris dones da rodesac saWiroa, dasabuTebisas iyenebs mkacr maTematikur msjelobas (maT Soris induqciur da deduqciur msjelobas).

3. poulobs, irCevs da iyenebs gzebsa da meTodebs (maT Soris teqnologiebs) fig-urebis da obieqtebis zomebis, agreTve maT Soris manZilebis, masis, temperaturis gasazomad. ir Cevs da moipovebs procesis an realuri viTarebis mode li rebisTvis saWiro monacemebs.

4. kompleqsur (rTul) problemas yofs safexurebad, martiv amocanebad da Wris etapobrivad (amoxsna), maT Soris standartuli midgomebisa da procedurebis gamoy-enebiT.

5. keTilsindisierad asrulebs davalebebs (vadebisa da raodenobis Tvalsaz-risiT).

Semajamebeli samuSao

unarebi1. maTematikuri obieqtebis warmodgena da enis floba.2. maTematikuri cnebebisa da debulebebis gamoyeneba.3. kavSirebi da mimarTebebi.4. amocanis amoxsnis gza da misi realizeba.5. gamoTvlebi.

unarebi fasdeba Semdegi aqtivobebiT:moswavle1. axdens monacemebis (maT Soris problemis gadasaWrelad saWiro monacemebis)

organizebas da maT warmodgenas.2. gadmocemisas sworad da efeqtianad iyenebs maTe ma ti kur terminebsa da aRniS-

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

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

miyveba arCeul strategias.5. irCevs gamoTvlebis adekvatur/optimalur xerxs da axdens mis realizebas.6. amyarebs kavSirebs (mag. sxva maTematikur struq tu rebTan, obieqtebTan an sxva

disciplinebTan) da iyenebs am kavSirebs rogorc problemis gadaWrisas aseve miRe-buli Sedegebis gaanalizebisas.

27

klasSi CarTuloba

unarebi1. cnebebisa da debulebebis gamoyeneba2. maTematikuri obieqtebis warmodgena3. maTematikuri enis floba4. msjeloba _ dasabuTeba

unarebi fasdeba Semdegi aqtivobebiT:moswavle1. moswavle irCevs dasabuTebis meTods (mag. sawinaaRm de gos daSvebis gamoyeneba

damtkicebisas, evristikuli meTodis ga moyeneba dasabuTebisas).2. koreqtulad iyenebs terminologias, maTematikur aR niSv nebsa da simboloebs.3. informaciis gadacemisas warmoaCens sakiTxis arss (mag. obieqtis arsebiT

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

sxvis naazrevs.5. TanamSromlobs TanaklaselebTan jgufuri samuSaoebis Sesrulebisas.6. auditoriisa da saprezentacio masalis mixedviT irCevs prezentaciis formas

da damxmare saSualebebs (maT Soris sa in formacio teqnologiebs).7. axdens problemis formulirebas auditoriisTvis ga sa gebi formiT. asabuTebs

problemis aqtualurobas da mniSv ne lobas (igulisxmeba problemis praqtikuli an/da wminda mec nieruli aqtualuroba).

8. msjelobisas, sxvadasxva safexurze, adekvaturad ir Cevs simkacris dones da rodesac saWiroa, dasabuTebisas iye nebs mkacr maTematikur msjelobas (maT Soris induqciur da deduqciur msjelobas).

proeqti

fasdeba Semdegi unarebi1. cnebebisa da debulebebis gamoyeneba2. kavSirebisa da mimarTebebis dadgena3. amocanis Camoyalibeba da axsna4. maTematikuri obieqtebis warmodgena5. maTematikuri enis floba6. damxmare teqnikuri saSualebebisa da sainformacio teqnologiebis gamoyen-

eba7. modelireba

unarebi fasdeba Semdegi aqtivobebiT:moswavle1. poulobs, irCevs da iyenebs gzebsa da meTodebs (maT So ris teqnologiebs) fig-

urebis da obieqtebis zomebis, ag reT ve maT Soris manZilebis, masis, temperaturis da drois ga sazomad. irCevs da moipovebs procesis an realuri viTarebis modeli-rebisTvis saWiro monacemebs.

2. axdens monacemebis (maT Soris problemis gadasaWrelad saWiro monacemebis) organizebas da maT warmodgenas.

3. Cveul garemoSi (maT Soris yoveldRiur cxovrebaSi) amCnevs maTematikur obieqtebsa da procesebs. iyenebs maT Tvi sebebs praqtikul (maT Soris yofiTi) amo-canebis gadaWrisas.

4. gansazRvravs da ayalibebs problemas, maT Soris arastandartul viTarebaSi

28

(mag. rodesac problemis gada saWrelad saWiro maTematikuri procedura calsaxad araa gansazRvruli).

5. irCevs efeqtian strategias da mokled aRwers prob le mis gadaWris safexurebs. miyveba arCeul strategias. aana lizebs arCeul strategias da asabuTebs arCeuli strategiis efeqtianobas, mimoixilavs SesaZlo alternatiul stra te gi ebs da msjelobs maT upiratesobebsa da naklze.

6. amyarebs kavSirebs (sxva maTematikur struqturebTan, obieqtebTan an sxva dis-ciplinebTan) da iyenebs am kavSirebs rogorc problemis gadaWrisas aseve miRebuli Sedegebis gaanalizebisas.

7. axdens mocemuli modelis elementebis interpretirebas im realobis kon-teqstSi, romelsac modeli aRwers da piriqiT _ realuri viTarebis dakvirvebis Sedegad miRebuli mo na ce mebis interpretirebas Sesabamisi modelis enaze. gan saz R-vravs modelis vargisianobas da afasebs misi gamo yenebis saz Rvrebs.

8. auditoriisa da saprezentacio masalis mixedviT irCevs prezentaciis formas da damxmare saSualebebs (maT Soris sain formacio teqnologiebs). efeqtianad iyenebs prezen ta ci isT vis gankuTvnil dros.

9. axdens problemis formulirebas auditoriisTvis ga sa gebi formiT. asabuTebs problemis aqtualurobas da mniSv ne lobas (igulisxmeba problemis praqtikuli an/da wminda mec ni eruli aqtualuroba).

10. zogadi Sedegebis sademonstraciod iyenebs magaliTebs, rogorc realuri viTarebidan, aseve maTematikidan.

11. axdens miRebuli Sedegebis ganzogadebas, amyarebs kavSirebs (mag. sxva maTema-tikur struqturebTan, obieqtebTan an sxva disciplinebTan) da iyenebs am kavSirebs rogorc prob le mis gadaWrisas, aseve miRebuli Sedegebis gaanalizebisas.

12. afasebs maTematikis adgilsa da mniSvnelobas sxva da sxva disciplinebSi, biz-nesSi, xelovnebaSi da adamianis moR va weobis sxvadasxva sferoebSi.

13. informaciuli teqnologiebis gamoyenebisas acno bi erebs eTikur/socialuri xasiaTis problemebs da icavs eTikur normebs.

semestruli niSnis Semadgeneli komponentebis ganawilebis magaliTi

Sefasebis ZiriTadi mdgenelebi

saSinao davaleba,

15%

sakontrolo

samuSao

proeqti

klasSi CarTuloba

saSinao davaleba

klasSi CarTuloba,

15%

Semajamebeli samuSao, 40%

proeqti, 30%

29

Semajamebeli samuSao (VII-X klasebi)

(wera)

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

maT

emat

iku

ri

obi

eqt

e-bi

sa d

a en

is g

amo

yene

ba

araadekvaturad iye nebs cnebebs, ter minebs, aRniS-vnebs / arasrul-y o filad war moa-dgens monacemebs.

iyenebs amocanis amosaxsnelad saWi-ro zogierT cnebas, terminsa da aR niS-vnas / nawilobriv, Tumca araorgani-zebulad warmoad-gens monacemebs.

iyenebs amocanis amosaxsnelad sa-Wi ro cnebebis, debu le bebisa da aRniS vnebis um-etesobas / axdens monacemTa orga-nizebas da war-modgenas.

iyenebs amocanis amosaxsnelad sa-Wi ro yvela cne-bas, debulebas da aR niS vnas / irCevs mo na cemTa orga ni ze bisa da war mod ge nis op-timalur for mas

amo

can

is a

mox-

snis

gz

a

ver poulobs amo-xs nis romelime gzas.

poulobs amoxs-nis romelime gzas, iwyebs magram ver asrulebs amoxsnas.

poulobs amocanis amoxsnis gzas, mi-yveba mas, Tumca xarvezebiT.

p o u l o b s a m o -canis amoxsnis (opti ma lur) gzas, bo lo mde axdens mis rea li zebas

kavS

ireb

i d

a mi

mar

Teb

ebi

amoicnobs or, od-nav gansxvavebuli formiT warmod ge-nil, identur ma-Te ma tikur obieqts (mag. gansxvavebuli asoiTi SeniSvnebiT Caweril erTsa da imave algebrul ga mo saxulebas).

amoicnobs or, gans-xvavebuli for miT war modgenil, iden-tur maTe ma tikur obieqts (mag. sxva-da sxva formiT war-mod genil erTsa da imave asaxvas).

aRwers kavSirs sxva dasxva stan-dar tul maTema-ti kur obieqtsa Tu proceduras Soris (mag. gan-tolebis amoxs-na da mo ce mu li asaxvisaTvis mo-ce muli mniSv ne-lo bis winasazis povna).

aRwers da iyen-ebs kavSirs sxva-dasxva maTemati-kur obieq teb sa Tu pro ce durebs Soris (mag. iyen-e b s a l g e b r u l gar daqmnebs da to lobis / uto-l o b i s T v i s e -bebs geo met riul d e b u l e b a T a dasabu Te bi sas)

gamo

Tvl

ebi

g a m o T v l e b i a r met yvelebs amo-canis gagebaze, Sedegi araadekva-turia

Seicavs gamoT v-le bis ZiriTad na-wils, realizeba na wilobriv aras-wo ria.

gamoTvlebi Zi-ri Tadad sworia, T u m c a S e i c a v s teq nikuri xasi-aTis xarvezebs.

gamoTvlebi swo-ria, arCeuli da g a m o y e n e b u l i aqvs gamoTvleb-is opti ma luri xerxi.

30

proeqti (VII,X klasebi)

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

gaz

omv

ebi,

mo

del

ireb

a, g

amo

yene

ba

Caketili ti-pis umartive-si teqsturi a m o c a n i s amoxsna stan-d a r t u l i pro cedurebis gamoyenebiT.

nacnobi mode lis gamoyeneba nacnobi viTa rebis Sesas wav-lad. Caketili ti-pis amocanis amox-sna stan dar tuli pro ce durebis ga mo-yenebiT. nac nob dam-xma re saSuale be bisa da teqno lo giebis gamo ye neba.

im viTarebisa amo-c noba, rom lis Se-saswavlad Se saZ-lebelia nac no bi m o d e l i s g a m o -yeneba. amo ca nis amoxsnisas misT vis nacnob maTema ti-kur pro ce du rebs Soris sa Wi ro pro -ceduris Ser Ce-va da gamo ye neba. nacnobi damx ma re saSua lebe bi sa da teqno lo gi ebis ga -mo yeneba.

problemis gadasaWrelad rea luri viTarebis mode-li reba, maSinac ki, rode-sac es viTareba sagrZno-blad gansx vav deba misTvis nacnobi Sem Txve vebisagan. problemis da mo uki deblad dasma da misi for mulireba maTematikur ena ze. sxva-dasxva damxmare teqnikuri saSualebebisa da teqnolo-giebis arsebobis, maTi max-asiaTeblebis, gamo ye nebis arealisa da Sez Rud ve bis codna. problemis gadaW-risas adekvaturi damxmare teqnikuri saSualebebisa da teqnologiebis SerCeva da ma Ti gamoyeneba.

msj

elo

ba-d

asab

uT

eba

u m a r t i v e s i stan dartuli gamoT vlebis Catareba.

martivi ma Te ma ti-k u r i o b i e q t e b i s Tvise bebis gamo-ye nebiT, stan dar-tuli ga moTv lebisa da msjelobis Ca ta-reba.

maTematikuri ob-i e q tis Tvise be bis gamoyenebiT, stan-dartuli gamoT-vlebisa da msje-lobis Cata reba

varaudis gamoTqma da misi kvle va kerZo magaliTebze. gamoTvlebisas Tu damtki -cebisas arCeuli strate-giis var gisianobisa da misi gamo yenebis sazRvrebis ganxilva. msje lobis xazis ganvi Ta re ba, alternatiu-li gzis mo Zeb na, miRebuli gadawyve ti le bis sisworisa da efeq tia no bis dasabuTe-ba.

kavS

ireb

i d

a mi

mar

Teb

ebi o r i , o d n a v

gan s xva vebuli for miT war-m o d g e n i l i , i d e n t u r i maTe ma tikuri o b i e q t i s amoc no ba.

o r s t a n d a r t u l maTe ma ti kur obieqt-sa Tu proce du-ras Soris kav Si ris amoc no ba.

sxvadasxva stan-dar tul maTe ma-ti kur obieq teb sa Tu pro ce du rebs Soris kav Si ris amocnoba.

miRebuli Sedegebis ganzo-ga deba. kavSirebis damyareba sxva maTematikur struq tu-reb Tan, an sxva dis cip li-neb Tan. am kavSirebis gamoy-eneba prob lemis gadaWrisas da mi Re buli Sedegebis gaan-alizebisas.

komu

nika

cia

martivi maTe-ma tikuri obi-eq tis saxel-wo de bisa da Z i r i T a d i T v i s e b e b i s g a m e o r e b a (maT Soris sa-ku Tari sit-yvebi).

m a r t i v i m a T e m a -tikuri obieq tis saxelwo debisa da Ziri T adi Tvi se bebis gadmo ce ma.

maTematikuri obi-eq tis saxel wo de-bisa da Zi ri Tadi Tvi se bebis gadmo-ce ma. komu ni kaci-i isas sim kacris ade k va tu ri donis Ser Ceva. infor ma-ci is ga da cemisas sa ki Txis arsis war -mo Cena.

sxvadasxva gziT warmod g-eni li informaciis inter-pre ta cia, masze msjeloba, erT maneT Tan dakavSireba. infor m aciis miRebisa da gadacemis Sesaferisi sa-Sualebebis Ser Ce va audi-toriisa da sa kiTxis gaT-valiswinebiT. in for maciis gadacemisas sa kiT xis arsis warmoCena.

31

sanimuSo gakveTilebi

Ia q i t i v o b a:

naturaluri ricxvebis gayofadobis Tvisebebis gamokvleva da gamoyeneba praq-tikuli amocanebis amoxsnisas.

reziume:

moswavleebi praqtikuli amocanebis amoxsnisas, eqspe ri mentis Catarebisas aR-moaCenen naturaluri ricxvebis gayofis Tvisebas: erTi naturaluri ricxvis meore naturalur ricxv ze gayofisas miiReba ganayofi da naSTi, romelic zogjer nu lis tolia _ aseT SemTxvevaSi erTi naturaluri ricxvi iyo fa meoreze; ayalibeben sa-kuTar pozicias am operaciis ga mo saxvisa da dasabuTebis Sesaxeb.

aqtivobis mizani:

• moswavleebi gaiwafon praqtikuli amocanebis amoxsnisas varaudebis gamoTqmasa da maT SemowmebaSi.

• CamouyalibdeT kritikuli azrovneba, msjelobis, dasa bu Tebis unari.• eCvevian ricxvebis geometriuli warmodgenebis gamo ye nebas.

savaraudo dro:

1 gakveTili

masala:

• saskolo saxelmZRvanelo• saweri saSualebebi• Cxirebi (an kenWebi)

aucilebeli wina codna:

• gayofadobis niSnebi• gayofadoba. gayofadobis Tvisebebi

aqtivobis aRwera

sTxoveT moswavleebs moewyon 5-5-ad, 6-6-ad. dasviT SekiTxva:_ ra operaciis Sesrulebaa saWiro imisTvis, rom davad gi noT, ramdeni xuTeuli

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

Tu SesaZlebelia ganixileT is SemTxvevac, roca klasis mos wavleebis ricxvi iyofa dasaxelebul ricxvze da is Sem Txve vac, roca ar iyofa. am ukanasknel SemTxvevaSi SeiZleba dav svaT SekiTxva:

• ramdeni 5-kaciani gundi SeiZleba Sedges am moswav le ebis gan?• ramdeni 6-kilogramiani amanaTi SeiZleba gaamzadon mzru n velobas moklebul

bavSvTa saxlisTvis, Tu TiToeuli mos wavle TiTo kilogram xils miitans?• sul mcire ramdeni yuTi iqneba saWiro bavSvTa sax lis Tvis wignebis gadasacemad,

Tu TiToeuli moswavle TiTo wigns mo itans da TiToeul yuTSi 4 wigni eteva?sTxoveT moswavleebs gayofis procesi ricxviTi tolobiT Caweron.CaatareT fragmentuli muSaoba jgufebad; erT jgufs gar kveuli raodenobis

kenWebi daurigeT, meore jgufs _ sxva ra odenobis kenWebi, mesame jgufs daavaleT saskolo saxel mZRva nelos teqstis mixedviT gaarCios raime magaliTze (roca ga sayofi metia gamyofze) ricxviTi wrfis gamoyenebis SesaZ leb loba. sTxoveT am ukanaskneli

32

jgufis wevrebs ricxviTi wrfis saSualebiT aRweron gayofis procesi nebismieri a naturaluri ricxvisa da b naturaluri ricxvis SemTxvevaSi (a>b). sTxoveT jgufebs Seadginon ricxviTi tolobebi da Sesa ba misi tolobebi asoebis gamoyenebiT. SeiZleba Tqveni Carevac gaxdes saWiro moswavleTa Sedegebis reviziis mizniT an am Sedegebis prezentaciisas.

ganixileT is magaliTebic, romlebic saxelmZRvaneloSia da praqtikuli Sinaarsis amocanebs ukavSirdeba, (magaliTi 1 da magaliTi 2).

aqtivobis gaRrmaveba-gafarToeba

daavaleT jgufebs ganixilon is SemTxvevebi, roca gasa yo fi naklebia gamyofze, gasayofi udris gamyofs. SeiZleba daviw yoT ricxviTi magaliTebiT, Sesabamisi ricx-viTi tolo be bis CaweriT. Semdeg _ gayofis procesis aRmweri formula Caiweros asoebis gamoyenebiT.

SeecadeT kiTxva-pasuxis saSualebiT SeajamoT yvela Sem Txve va. SeamowmoT aT-visebis xarisxi kiTxvebis dasmis saSua lebiT;

_ ra naSTi SeiZleba miviRoT 5-ze gayofisas?_ vTqvaT, ricxvi iyofa 5-ze, risi toolia am SemTxvevaSi naSTi? _ rogor Cavw-

erT formuliT im ricxvebs, romlebic 5-ze iyofa?_ ra formuliT SeiZleba Caiweros is ricxvebi, romlebic 5-ze gayofisas naSTSi

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

8-ze gayofisas miRebuli ganayofi da naSTi.Semdeg gadavdivarT `testebSi~ pasuxebis SerCevis pro ces ze (moswavlis saxelmZ-

Rvanelo _ savarjiSoebi #1-#8)mizanSewonilad migvaCnia klasSi amovxsnaT 23-e, 24-e, 26-e da 29-e amocanebi, rom-

lebic codnis ganmtkicebis, da sa buTebis, msjelobis unaris ganviTarebisTvisaa mo-cemuli. yve la es amocana iseve, rogorc am paragrafis sxva amocanebi dawv rilebiTaa amoxsnili $1.2-is warmodgenisas.

II

a q i t i v o b a:

simravlis klasifikacia mTel ricxvTa simravlis klasebad dayofis magaliTebis mixedviT

forma:

jgufuri muSaoba

reziume:

ganixileba mTel ricxvTa Z simravlis sxvadasxva kla si fikaciis magaliTebi:• mTeli dadebiTi ricxvebi, mTeli uaryofiTi ricxvebi, nuli.• mTel ricxvTa simravlis dayofa raime konkretul 1-is gan gansxvavebul natu-

ralur ricxvze gayofisas miRebuli naSTebis mixedviT.• ganzogadeba _ nebismieri naturaluri ricxvis (mo du lis) ganxilva.

aqtivobis mizani.

propedevqtikuli muSaoba swori kla si fikaciis (raime erTi niSnis mixedviT da a. S.) Sedgenis Cve vebis misaRebad. msjelobis, dasabuTebis, kritikuli az rov nebis unaris gan vi Tareba; ganzogadebis procesis Catarebis una ris ganviTareba.

33

savaraudo dro:

1 gakveTili

samuSao masala:

moswavlis saxelmZRvanelo,$ 1.9. jgufuri muSaobis proeqti

aucilebeli wina codna

• mTeli ricxvebi• moqmedebebi mTel ricxvebze• mTeli ricxvis naturalur ricxvze gayofisas naSTis miTiTeba• simravle. moqmedebebi simravleebze

aqtivobis aRwera

gakveTils viwyebT saWiro wina masalis gaxsenebiT, codnis ganmtkicebisa da gafar-Toebis procesiT, romelic mTeli ricxvis naturalur ricxvze gayofasa da naSTis miTiTebas Seexeba. sasargebloa aseTi saxis savarjiSoebis winaswari ganxilva:

_ miuTiTeT ganayofi da naSTi 37-is 5-ze gayofisas:37=5·7+2_ miuTiTeT ganayofi da naSTi -37-is 5-ze gayofisas:–37=5·(-8)+3_ miuTiTeT ganayofi da naSTi -40-is 6-ze gayofisas:–40=6·(–7)+2_ vTqvaT, a mTeli ricxvis b naturalur ricxvze gayo fi sas miiReba ganayofi q

da naSTi r, CawereT es faqti for mulis saxiT. SeiZleba Tu ara r nulis toli iyos? SeiZleba Tu ara r ricxvi b-s toli an b-ze meti iyos?

winaswari `moTelvis~ Semdeg klass vyofT jgufebad da vavalebT saxelmZR-vaneloSi mocemuli proeqtis Sesrulebas. erTi jgufi asrulebs davalebas, roca xdeba 5-ze gayofa, meore _ 4-ze gayofis SemTxvevas ixilavs, mesame _ 6-ze gayofis SemTxvevas.

TiToeuli jgufi warmoadgens pasuxebis variantebisgan Sedgenil moxsenebas. am moxsenebebis prezentacias axdenen jgufebis warmomadgenlebi.

Semdeg erToblivad mimdinareobs msjeloba miRebuli Sedegebis Sesaxeb; _ ramdeni qvesimravlis gaerTianebad warmovadgineT TiToeul SemTxvevaSi mTel

ricxvTa simravle?_ amowuravs Tu ara am qvesimravleebis gaerTianeba mTlianad mTel ricxvTa

simravles?_ aqvs Tu ara saerTo elementebi nebismier or qve sim ravles?_ SeiZleba Tu ara raime sxva niSnis mixedviT analogiuri we siT warmovadginoT

mTel ricxvTa simravle?_ SeiZleba Tu ara moviyvanoT raime sxva magaliTi, roca sim ravle warmodge-

nilia aRniSnuli wesiT.aq SeiZleba moswavleebs daxmareba dasWirdeT (SeiZleba miuTiToT raime cxovre-

biseuli magaliTi, magaliTad, cxo ve lebis dayofa raime niSniT da a. S.).SeiZleba vimsjeloT ganzogadebaze, ramdeni qvesimravle miiReba nebismieri n

naturaluri ricxvis SemTxvevaSi?

34

III

aqtivoba:

kvadratebis sxvaobis formula da marTkuTxa miwis nakveTebis farTobebis Se-dareba.

reziume:

marTkuTxedis farTobisa da kvadratebis sxvaobis formulis erTmaneTTan dakav-Sireba. sidideebs Soris damokidebulebebis kvleva marTkuTxedis farTobis magali-Tze. iyeneben dakvirvebis Sedegebs algebruli Tanafardobis misaRebad, realur viTarebasTan dakavSirebuli amocanis amoxsnisas.

savaraudo dro:

1 gakveTili.

aqtivobis mizani.

algebruli gamosaxulebebis igivuri tolobis miReba wminda algebruli gziT, romelic dakavSirebulia ope ra tiuli unarebis ganviTarebasTan da am tolobis geome-triuli Sinaarsis warmodgena. sxvadasxva (algebrul da geometriul) obieqtebsa da operaciebs Soris kavSirebis gacnobiereba. geometriuli optimizaciis amocanasTan algeb ru li igiveobis dakavSireba.

aqtivobis aRwera.

gakveTils viwyebT im kvadratisa da marTkuTxedis farTo-bebis Seda re biT, romelTac toli perimetrebi aqvs.

_ am figurebs toli perimetrebi aqvs. SeecadeT SeadaroT maTi far To bebi.

_ ramdeni kvadratuli erTeuliTaa meti kvadratis far To bi marTkuTxedis farTobze?

_ SeadareT marTkuTxedis gverdebi kvadratis gverdebs.am faqtis ganzogadebis mizniT SevTavazoT moswavlveebs (Se saZlebelia maT wyvi-

lebSi imuSaon), Seavson cxrili imave kanonzomierebiT (marTkuTxedis sigrZe kvadra-tis gverdze 2-iT metia, sigane _ 2-iT naklebi).

kvadrati marTkuTxedi farTobebis sxvaoba

gverdis sigrZe

farTobi sigrZe sigane farTobi

456789

16 6 2 12 4

_ ra kanonzomierebas amCnevT Sevsebuli cxrilis mi xed viT?diskusiis Semdeg moswavleebi midian daskvnamde, rom Ziri Ta di kanonzomiereba

aris is, rom farTobebis sxvaoba mudmivia da 4-is tolia.SeiZleba sxva magaliTis ganxilvac, roca kvadratis gver di sxva ricxviT icvleba

35

(magaliTad, ganvixilavT marT kuTxe debs, romelTa sigrZe 3-iT metia kvadratis gver-dis sigrZeze, sigane _ 3-iT naklebia).

amis Semdeg klass vTavazobT miRebuli kanonzomiereba for mulis saxiT Caweros _ magaliTad,

(a+2)(a–2)=a2–4, (a+3)(a–3)=a2–9.am tolobebis Cawera kvlav kiTxva-pasuxis procesiT SeiZ le ba ganxorcieldes._ vTqvaT, kvadratis gverdis sigrZe a-s tolia, risi to lia, misi farTobi?_ Tu marTkuTxedis sigrZe 2-iT metia kvadratis gverdze, si gane _ 2-iT naklebi,

risi tolia misi farTobi?_ risi tolia miRebuli marTkuTxedis farTobi?moswavleebi advilad SeZleben am igiveobebisa da misi ganzogadoebis miRebas

algebruli xerxiTac (frCxilebis gaxsnis wesis gamo ye nebiT).axla SeiZleba gamoviyenoT formulis geometriuli dasa bu Tebis xerxi.SevTavazoT moswavleebs daxazon kvadrati, romlis gverdi a-s tolia.Semdeg SeiZleba davsvaT kiTxvebi:_ ra figuris farTobi SeiZleba iyos warmodgenili (a+b)(a-b)

formuliT._ risi tolia daStrixuli figuris farTobi?_ (a2-b2)-is._ tolia Tu ara daStrixuli fi gu ris farTobi im marTkuTxe-

dis farTobis, romlis gverdebia a+b da a-b?_ ra tolobis dawera SeiZleba?am kiTxvebisa da pasuxebis sistema warmoadgens (a+b)(a-b)=a2-b2 tolobis miRebis

process. Semdeg SeiZleba igive to lo ba moswavleTaTvis cnobili algebruli xerx-ebiTac miviRoT (frCxi lebis gaxsnis wesi).

am procesis Catarebis Semdeg moswavleebs vekiTxebiT:_ risi tolia mocemuli kvadratis perimetri?_ risi tolia misi erTi gverdis b-Ti SemcirebiTa da meore gver dis b-Ti gaz-

rdis Semdeg miRebuli marTkuTxedis peri met ri?_ kvadratsa da marTkuTxeds toli perimetrebi aqvs; rom lis farTobia metia

da ramdeniT?am kiTxva-pasuxis procesSi moswavleebi aRmoaCenen mniS vne lovan faqts: moce-

muli perimetris mqone marTkuTxedebs So ris udidesi farTobi aqvs kvadrats. aq SeiZleba yuradReba mi vaqcioT kvadratis `met~ simetriulobazec marTkuTxedTan Se darebiT.

saxelmZRvaneloSi mocemuli ($2.2.) amocanebis amoxsna xels uwyobs codnis gan mtki-cebas, formulis gamoyenebis unaris gamomuSavebas. 76-e amo cana kvlav geometriul interpretacias Seexeba da sasur ve lia misi amoxsna klasSi.

IV

aqtivoba:

proporcia. proporciis gamoyenebebi.

reziume:

moswavleebi euflebian proporciis gamoyenebas sxvadasxva sayofacxovrebo an maTematikis momijnave disciplinebidan amocanebis amosaxsnelad. ecnobian propor-ciul sidideebs da maT geometriul gamosaxvas.

36

savaraudi dro:

1 gakveTili.

aqtivobis mizani:

• moswavleebma ganiviTaron Sefardebisa da proporciis gamoyenebis unar-Cvevebi amocanaTa amosaxsnelad.

• moswavleebi daeuflon proporciuli sidideebis cnebis, proporciuli damok-idebulebis cnebasa da mis geometriul ga mosaxvas.

aucilebeli wina codna:

• Sefardeba• proporcia. proporciis ZiriTadi Tviseba• sakoordinato sibrtye

ativobis aRwera

muSaoba SeiZleba daviwyoT im ori praqtikuli amocanis gan xilviT, romelic mo-swavlis saxelmZRvanelos $3.4-is me-3 punqtSi 1 da 2 magaliTiT aris warmodgenili. isini rukis mixedviT manZilis gansazRvras da sayofacxovrebo amocanis amox snas ukavSirdeba. Tu saskolo oTaxSi kedelze saqarT ve los rukaa Camokiduli, maSin SeiZleba aRniSnuli amocanis mi xedviT maswavlebelma am rukis mixedviT saqarTvelos raime or qalaqs Soris manZilis povnis amocana dasvas.

Semdeg moswavleebs davyofT ramdenime jgufad da Sev Ta vazebT amocanebs propor-ciul sidideebze.

• magaliTad, erT-erT jgufs vTxovT Seavsos cxrili

vaSlis masam (kg)

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

m kg vaSlisRirebudeba p(lari)

0,8

• ipovos mp sididis ricxviTi mniSvnelobebi.

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

sxva jgufs SeiZleba mivceT Sesavsebad cxrili, romelic Tanabari moZraobisas gavlil manZilsa (S km) da dros (t sT) Soris damokidebulebas Seesabameba, SeiZleba aseTi amocanac Sev TavazoT: vixilavT marTkuTxedebs, romelTa erT-erTi gver dia a. SeavseT cxrili, romelic meore gverdis sididesa da saTanado marTkuTxedis far-Tobs Soris damokidebulebas See sabameba.

erT-erT jgufs SeiZleba davusvaT iatakis Tanabrad Se Reb visas S farTobsa da daxarjuli saRebavis odenobis Sesa ba misi cxrilis Sevsebisa da sakoordinato sibr-tyeze am damo kid ebulebis mixedviT wertilebis agebis amocana.

jgufebis mier warmodgenili cxrilebisa da sako or di na to sibrtyeze Sesabamisi damokidebulebis Sesabamisad age bu li wertilebis mixedviT mimdinareobs msjeloba da vara udis gamoTqma aRniSnuli damokidebulebebis Tvisebebisa da geometriuli gamosaxvis Sesaxeb.

37

proporciis gamoyenebiT mimdinareobs Semzadeba Senad no beb ze, narevebze amo-canebis amosaxsnelad.

am SemTxvevaSi gamoiyeneT saxelmZRvaneloSi mocemuli me-3 magaliTi, roca cno-bilia ori biznesmenis saerTo Tanxa (10 000 lari) da am Tanxebis Sefardeba:

ba =

32

proporciul nawilebad gayofis procesi SeiZleba kiTxva-pasuxis reJimSi gav-agrZeloT:

_ kidev ra proporciis dawera SeiZleba mocemuli pro por ciis mixedviT ( a b2 3

= )?

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

_ vTqvaT, toli Sefardebebi k-s tolia (proporciulobis koeficienti). ris tolia a? b? (a=2k, b=3k).

_ 10 000 larSi ra wilia erTi biznesmenisa da ra wilia meore biznesmenis?_ sul ramden wils warmoadgens 10000?_ ra Tanxa modis 1 wilze?_ ra Tanxa Seesabameba 2 wils? 3 wils?aqtivobis Semdgomi gafarToeba-gaRrmaveba 2 an 2-ze meti Sesakrebis povnis zogadi

wesis Camoyalibebaa, roca cnobilia maTi jami da ricxvebi, romelTa proporciulia mocemuli ricx vebi. arasavaldebulo nawilad SeiZleba CaiTvalos sa xel mZRvaneloSi mocemuli is nawili, romelic oqros kve Tas _ harmoniul proporcias exeba.

sakontrolo kiTxvebi da `testebi~ (mesame punqtis #1-#9) moswavleTa TviT Se-mowmebis kargi saSualebaa.

sasargebloa klasSi gavarCioT da amovxsnaT amocanebi Se nad no bebze, xsnarebze, masStabis gamoyenebaze (#12, #14, #26, #28).

29-e amocana proporciuli sidideebis geometriuli ga mosaxvis gamoyenebis kargi magaliTia;

33-e amocana, romelic struqturirebuli kiTxvebisgan Sed geba, statistikur monacemTa diagramis aRwerisas pro por ciulobis gamoyenebas Seicavs.

V

aqtivoba:

samkuTxedis kuTxeebis jamis formulis gamoyvana sxva da sxva gziT.

reziume:

moswavleebi aRmoaCenen samkuTxedis Tvisebas (samkuTxedis kuTxeebis jamis Ses-axeb) modelebze, asabuTeben am Tvisebas sxvadasxva xerxiT, msjeloben dasabuTebis xerxebis sarwmunoobis Sesaxeb.

savaraudo dro:

1 gakveTili.

aqtivobis mizani:

• miiRos eqsperimentis Catarebis Cvevebigaiwafos amocanis konteqstis Sesabamisad geometriul obieqtTa gamosaxvaSi• Camouyalibdes kritikuli azrovneba, msjelobisa da dasabuTebis unari

38

ZiriTadi SekiTxvebi:

1. SeiZleba Tu ara figuris Tvisebis aRmoCena misi mo de lis mixedviT?2. SesaZlebelia Tu ara sazomi xelsawyoebis gamoyenebiT fi guris raime Tvisebis

Sesaxeb varaudis gamoTqma?3. debulebis samarTlianobis dadgenis romeli xerxia uf ro sarwmuno _ das-

abuTebis gziT miRebuli, Tu sazomi xel saw yoebis gamoyenebiT?

masala:

furclebi, transportiri, saxelmZRvanelo.

aqtivobis aRwera:

muSaoba iwyeba gavlili masalis gameorebiT: paraleluri wrfeebis Tvisebebi, ori wrfis paralelurobis niSani, samkuTxedebis klasifikacia kuTxeebis mixedviT (am masalis didi nawili $ 4.5-is pirvel nawilSia gadmocemuli da wina ori gakveTilis Sinaarss Seesebameba).

klasis moswavleebs vyofT jgufebad: erT-erTi jgufis mos wavleebs vTxovT gam-oiyenon transportiri da sxva da sxva gva ri samkuTxedebis kuTxeebis zomis mixedviT gamoTqvan va ra udi kuTxeebis jamis Sesaxeb.

meore jgufi iye nebs samkuTxedis mo de lebs, gamoWris sam tol samkuTxeds da mi ad gams maT ise, ro gorc suraTzea gamo saxu li.

mesame jgufi iyenebs saxelmZRvaneloSi war modgenil su raTs da iq dasmul kiTxvaze pa suxebis Ziebis procesSi cdi lobs gamo Tqvas vara-udi samkuTxedebis kuTxeebis jamis Se sa xeb.

meoTxe jgufi iyenebs saxel mZRvanelos mixedviT ABC sam kuTxedis models, aagebs mis si maRles _ BD-s; poulobs AD da CD monakveTebis Sua mar To bebs da gadakecvebis sa Sua-lebiT (ise, ro gorc saxel mZRva neloSia miTi Te buli) cdi lobs gamoTqvas vara udi samkuTxedebis kuTxeebis ja mis Se saxeb.

oTxive jgufi axdens Tavisi namuSevris prezentacias. imar Teba diskusia _ romeli xerxia ufro ioli, romeli uf ro sarwmuno?

saxelmZRvaneloSi mocemuli sxva Teoriuli sakiTxebi (marT kuTxa samkuTxedis maxvili kuTxeebis jami, tolgverda sam kuTxedis TiToeuli kuTxis sidide, gare kuTxis Tviseba) Se iZ leba davalebad mivceT moswavleebs.

samkuTxedis aRniSnuli Tvisebis aTvisebis TviTkont rolisTvis sasargebloa $ 4.5-is me-2 nawilis #1-#8 `testebSi~ pasuxebis SerCeva. dasabuTebisa da msjelobis unaris ganviTarebas emsaxureba #10, #12, #14, #15, #20 da #28 amocanebis amoxsnis proce se-bi. sasurvelia isini maswavleblis xelmZRvanelobiT Sesruldes.

39

I Tavi. racionaluri ricxvebi

I Tavi, ZiriTadad, ricxvis cnebis gafarToebis, racionaluri ricxvebisa da maTze moqmedebebis aRweras eZRvneba. poziciuri sistemis gamoyenebiT racionaluri ricx-vebis Sedarebisa da dalagebis Cvevebis aRdgenisa da ganmtkicebis safuZvelze mim-dinareobs uaryofiTi ricxvebis gamosaxvis, Sedarebisa da dalagebis Sesaxeb masalis gadacema.

pirveli Tavis sawyisi paragrafebi naturaluri ricxvebis mwkrivis ganxilviT iwyeba, naturaluri ricxvebis Caweris xerxebisa da poziciuri sistemis gamoyenebis magaliTebis ganxilvas eZRvneba.

maswavleblebs SevaxsenebT, rom naturalur ricxvTa dafuZnebis Teoriis gacno-bisas, maT, ZiriTadad, or midgomaze miuTiTebdnen _ aqsiomuri midgoma, romelic peanos saxelTan aris dakavSirebuli da simravluri koncefcia (kantoris mier da-fuZnebuli).

pirveli midgomisas naturalur ricxvTa mwkrivi esaa {N, I, F} struqtura, romel-ic

1. Sedgeba N simravlisgan (am simravlis elementebs naturaluri ricxvebi hqvia);2. gamoyofilia elementi, romelsac erTi (1) hqvia;3. gvaqvs mimarTeba F(x; y), y aris x-is momdevno. garda amisa mocemuli struqturis Tvisebebia (aqsiomebi);1) 1 ar aris raime naturaluri ricxvis momdevno;2) nebismieri naturaluri ricxvisTvis arsebobs erTi da mxolod erTi misi mom-

devno;3) yoveli naturaluri ricxvi araumetes erTi naturaluri ricxvis momdevnoa;4) (induqciis aqsioma) N-is yoveli qvesimravle, romelSic Sedis 1 da yovel x-Tan

erTad Sedis misi momdevno naturaluri ricxvi, N-s emTxveva.aqsiomuri meTodis Tanmimdevruli gatarebisas asec amboben: elementebis buneba,

romlebisganac N Sedgeba, araa arsebiTi.amis Semdeg SeiZleba SemoviRoT naturalur ricxvTa monakveTis cneba, Semdeg

masze dafuZnebiT sasruli simravlisa da sasrul simravleSi elementTa odenobis (simZlavris) cnebebi.

naturaluri ricxvis cnebis dafuZnebis meore midgoma kantoris saxelTan aris dakavSirebuli. aq elementebis ricxvi gaigeba, rogorc simravlis simZlavris kerZo SemTxveva. safuZvlad erTi simravlis meore simravleze urTierTcalsaxa Sesabamiso-bis cneba aiReba. am mimarTebiT xdeba simravleTa faqtorizacia, klasebad dayofa. am koncefciis Sesabamisad TiToeuli klasi xasiaTdeba misi simZlavriT. aracarieli sasruli simravlis simZlavre, gansazRvrebis Tanaxmad, naturaluri ricxvia.

aseTi Semotanisas sasruli simravle ganusazRvrelia. kidev erTi daSveba am dros aris usasrulo simravlis arseboba (magaliTad, mimdevrobis usasrulo gagrZeleba-dobis daSveba).

peanos Tvalsazrisis Sesabamisad ki saWiroa imis daSveba, rom aqsiomuri sistemis modeli arsebobs. es daSveba bevr maTematikoss (ix. mag., [21]) ufro bunebrivad mi-aCnia.

cxadia, rom skolaSi ariTmetikis sawyisi Seswavla TvalsaCino unda iyos. igi ar unda misdevdes arc aqsiomur meTods da arc sxva logikuri agebis sqemas. es ar niS-navs imas, rom ricxvTa sistemis swavlebas ar unda hqondes Tavisi logikuri ageb-uleba.

40

ricxvis cnebis gafarToebas meTodikur literaturaSi [21] xSirad sqematurad ase warmoadgenen:

algebris TvalsazrisiT ganzogadebis bunebrivi gza aseTia: (1) → (2) → (3) → (4)skolaSi ki Tanmimdevroba aseTia:(1) → (5) → (3) → (7)

axali erovnuli saswavlo gegmac ricxvis cnebis gafarToebis am Tanamimdevrobas gvTavazobs: Tumca, uaryofiTi ricxvebis Seswavla me-7 klasidan aris gaTvaliswineb-uli.

uaryofiTi ricxvebis Seswavla erT-erTi rTuli sakiTxia. misi Camoyalibebis is-toriac amas adasturebs (didi xnis ganmavlobaSi mecnierebis mier ar iyo miRebu-li). es sakiTxebi kargad aris gadmocemuli kolmogorovis statiaSi (ix; mag; [21] da maswavleblebs vurCevT gaecnon mas. Cven mier SemoTavazebuli gadmocemis meTodika TvalsaCinoebis farTo gamoyenebas gulisxmobs.

miuxedavad imisa, rom moswavleebi am droisTvis TiTqos ukve safuZvlianad unda flobdnen arauaryofiT racionalur ricxvebs, Cven mainc Sesabamisi sakiTxebis gadmo-cemas didi dro davuTmeT. erovnuli saswavlo gegmac iTvaliswinebs zogierTi adre naswavli sakiTxis gameorebasa da formaluri TvalsazrisiT ufro maRal doneze gadmocemas _ magaliTad, gayofadobis zogierTi niSnis swavlebisas deduqciuri msjelobis gamoyenebas.

maswavleblebs kidev erTxel SegaxsenebT, rom Sefasebis zogad principebs wignis Sesaval nawilSi calke paragrafi daeTmo, es pedagogebisTvis saxelmZRvanelo principebia – yovel konretul SemTxvevaSi pedagogma am principebis safuZvelze unda miiRos gadawvetileba. moswavlis Sefaseba, TviT dabali Sefasebac ki, unda iyos dafiqsireba arsebuli realobis, romelic unda aRiqmebodes samomavlo pozitiuri cvlilebebisken mimarTul qmedebad. am Sefasebas ar unda dahyvebodes niSnis mogebis, angariSsworebis elferi. saerTod, yofaqceva da akademiuri donis Sefaseba unda gaimijnos erTmaneTisgan. warumateblobis yoveli gamovlineba ar unda iqces moswavlisTvis gamousworebel Cavardnad. akademiuri Sefaseba unda iyos xSiri

(1) arauaryofiTi mTeli ricxvebi

(2) mTeli ricxvebi

(3) racionaluri ricxvebi

(4) algebruli ricxvebi

(8) kompleqsuri ricxvebi

(5) arauaryofiTi racionaluri ricxvebi

(6) arauaryofiTi namdvili ricxvebi

(7) namdvili ricxvebi

41

da mravalmxrivi, pedagogiur praqtikaSi unda gamoiricxos adre damkvidrebuli ualternativo procesi – gamoZaxeba, moswavlis xangrZlivi monologi, informaciis flobis donis warmoCena, maswavleblis ramdenime replika da Sefaseba.

dRes SeiZleba SevafasoT erTi sagulisxmo CarTvis, sayuradRebo varaudis gamoTqmis, ganxilvisas konstruqciuli rolis, TanaklaselTa naSromebis prezentaciis aqtiuri oponireba, gakveTilis procesSi gamovlenili SemoqmedebiTi aqtiuroba. am midgomiT SeiZleba moswavleTa akademiuri aqtivobis amaRleba, motivaciis gaZliereba, SemoqmedebiTi unarebis gaRrmaveba. Sefasebis procesi unda iyos gamWvirvale. mis gareSe SeuZlebelia pedagogma maRali avtoriteti daimkvidros moswavleTa Soris. ufro metic, TviT moswavleebic SeiZleba CavabaT SefasebaTa kriteriumebis SemuSavebaSi.

sakontrolo naSromebis ganxilvas unda eswrebodes Sefasebis kriteriumis sajaro wardgena klasSi _ es sayuradRebo meTodikuri xerxia, romlis saSualebiTac moswavleebi, sakmaod maRali sizustiT, Tavad ivaraudeben sakuTar Sefasebebs, zogierTi amocanis Sefasebisas gamoyenebuli nawilobrivi Sefasebebic mimarTuli unda iyos moswavlis naSromSi `dadebiTi nabijebis~ aRmoCenisa da dafiqsirebisken.

moswavlem unda igrZnos pedagogis TanagrZnoba, keTili ganwyoba da imavdroulad, maRali principuloba da profesiis erTguleba moswavlis akademiuri moswrebis Sefasebisas.

gavixsenoT, rom Sefasebis ZiriTadi miznebia: moswavlis swavlis xelSewyoba, moswavlis Zlieri da susti mxareebis gamovlena (saTanado analiziTa da daskvnebiT), swavlebis konkretuli meTodis efeqtianobis Semowmeba, saswavlo programebis efeqturobis dadgena da srulyofa da sxva. amrigad, Sefasebis formebi ganmaviTarebeli, ganmsazRvreli da diagnostikuria da am formebs vaniWebT aRmzrdelobiT funqciasac. gamovyoT Sefasebis kidev ori funqcia: swavlis procesis Sefasebis funqcia (mimdinare Sefaseba) da Sedegebis Sefaseba (Semajamebeli Sefaseba).

SefasebaSi unda aisaxos prezentaciisas gamovlenili unarebi, moswavleTa mier erTmaneTis Sefasebebi.

sagangebod unda SevCerdeT CvenSi farTod gavrcelebul erT-erT mankier Cvevaze, romlis Sedegebi xandaxan did gavlenas axdens Sefasebis obieqturobaze. migvaCnia, rom zogjer `uwyinar ancobad~ miCneuli sxvaTa naSromebis gadawera unda CaiTvalos moswavlisTvis miuRebel, uRirs saqcielad, moswavlem unda icodes, rom is iTvisebs sxvaTa inteleqtualur sakuTrebas, raTa daumsaxureblad miiRos maRali niSani, miiRos `jildo~ sicrueze agebuli qmedebisTvis. aseT SemTxvevaSi maswavleblis mier gamoCenili principuloba klasSi yovelTvis moipovebs gagebas, garkveuli drois Semdeg ki – aRiarebas.

sayuradRebo rekomendaciebi gvaqvs jgufuri muSaobis warmarTvisa da Sefasebis mimarTac (ixileT wignis Sesaval nawilSi `gakveTilis dagegmvis zogadi principebi~)O jgufuri muSaobis warmarTvas, axlavs garkveuli sirTuleebi, magram am formis didi aRmzrdelobiTi, ganmaviTarebeli, emociuri mxare migvaniSnebs, rom is aucileblad da jerovnad unda tardebodes.

wignis Teoriul nawilSia agreTve ganxiluli moswavleTa mier proeqtis Sesrulebis da am naSromis SefasebasTan dakavSirebuli sakiTxebi, dro da dro daubrundiT xolme SefasebasTan dakavSirebuli Cvens am rekomendaciebs, gaamdidreT es masala Tqveni gamocdilebis kvalobaze aRmoCenili detalebiT. Cven ki gzadagza kvlav SegaxsenebT xolme SefasebasTan dakavSirebul zogierT mosazrebas, konkretuli amocanebis Sefasebis kriteriumis dadgenasTan dakavSirebul sakiTxebs.

unda gvaxsovdes, rom moswavleTa SefasebaSi daSvebulma Secdomebma SeiZleba

arsebiTad daakninos sxva mimarTulebiT mopovebuli warmatebebi.

42

§ 1.1. naturaluri ricxvebi. naturaluri ricxvis Caweris sxvadasxva forma

mizani:• naturaluri ricxvebis Caweris sxvadasxva formis gaxsenebiT, moswavleebma ganiv-

iTaron Caweris formebis, poziciuri sistemis gamoyenebis unar-Cvevebi.• xarisxis cnebis gamoyenebiT moswavleebi gaiwafon ricxvebis, kerZod ki `gigan-

turi~ ricxvebis Caweris xerxebis gamoyenebebSi.• naturalur ricxvTa mwkrivis Tvisebebis gamoyenebiT moswavleebs eqmnebaT

pirveli warmodgenebi usasrulobis (usasrulod gagrZelebadobis) Sesaxeb.erovnuli saswavlo gegmis im moTxovnebis Sesasruleblad, romlebic VII.I da VII.II

indikatorebiTaa warmodgenili, saWiroa CavataroT Semamzadebeli muSaoba, rac wina wlebSi Seswavlili sakiTxebis Sesaxeb codnis gafarToebas gulisxmobs.

pirvel paragrafSi warmodgenili masala moswavleebisTvis SeiZleba ar iyos axa-li. aseTi dasawyisi bunebrivia, radgan saswavlo wlis pirveli gakveTilebi mizanSe-wonilia da sasurvelia ucnobi maTematikuri faqtebis gadmocemiT ar daviwyoT.

calkeuli gakveTilebis dagegmvisa da warmarTvisTvis konkretuli miTiTebebis garda, mizanSewonilad vTvliT zogadi saxis zogierTi rekomendaciis Sesaxeb saubar-sac, romlebic momavalSi SeiZleba aRar gavimeoroT.

saswavlo masalis aTvisebisa da ganmtkicebisTvis saWiro muSaobis ZiriTadi nawili gakveTilze unda modiodes; gakveTili saswavlo procesis organizaciis ZiriTadi formaa; gakveTilze xdeba saganmanaTleblo miznebis ganxorcieleba. xSirad sauke-Teso Sedegamde mivyavarT aseT zogad rekomendaciebs swavlebis meTodikidan:

1) maswavlebeli unda cdilobdes, rom yoveli axali SemecnebiTi amocana TviT mo-swavleebma Camoayalibon,

2) maswavleblis da moswavleebis erToblivi ZalisxmeviT, dakvirvebis, konkretuli SemTxvevebis analizis Sedegad iqmneba garkveuli warmodgena arsebul maTematikur kanonzomierebebze,

3) maswavleblis xelmZRvanelobiT moswavleebi eZeben amocanis amoxsnis gegmas da xSirad TviT axdenen gegmis realizacias.

maswavleblebs kargad moexsenebaT, rom saswavlo procesisTvis mzadebis zogadi sqema gulisxmobs:^

1) saswavlo wlis dawyebamde Catarebul muSaobas, 2) gakveTilebis sistemis gaaz-rebas, 3) konkretuli gakveTilisTvis momzadebas.

axali saxelmZRvaneloebis SemoRebis procesSi saswavlo wlis dawyebamde muSao-baSi igulisxmeba erovnuli saswavlo gegmis moTxovnebis kargad gaazreba. ZiriTadi masala, romelsac unda daeyrdnos maswavlebeli saswavlo procesSi, erovnuli sas-wavlo gegmaa. saxelmZRvanelo mxolod damxmare saSualebaa erovnuli saswavlo geg-mis moTxovnebis Sesasruleblad. saxelmZRvanelo erovnuli saswavlo gegmis mixedviT iqmneba, Tumca misi Sinaarsi, sakiTxebis Tanamimdevroba xSirad avtoris Sexedulebe-biTaa Sedgenili da rekomendaciebis doneze unda aRiqvas maswavleblebma. erovnuli saswavlo gegmis Sesabamisad, maswavlebels SeuZlia damoukideblad SearCios sakiTx-ebi da maTi gadacemis Tanamimdevroba. moswavlis saxelmZRvanelo, maswavleblis sarekomendacio wigni, romlebmac gaiares eqspertiza, pilotireba da meore gamoce-misas eqspertTa da maswavlebelTa mosazrebebisa da rekomendaciebis Sesabamisad ga-damuSavda, exmareba maswavleblebs moswavleze da Sedegze orientirebuli saswavlo procesis dagegmvasa da warmarTvaSi.

43

saswavlo wlis pirveli gakveTilebis specifikidan gamomdinare, pirvel paragrafs SeiZleba 2 saaTic ki davuTmoT. aq mocemuli masala moswavleTaTvis SeiZleba nacno-bic iyos, amitom maswavlebels meti saSualeba aqvs gakveTilebi moswavleTa aqtiuri monawileobiT, interaqtiuri meTodiT warmarTos. aqcenti gakeTdes swavlebis iseT formebze, roca yvela moswavle aqtiurad erTveba swavlis procesSi. gakveTilebi ise SeiZleba daigegmos, rom yovelma moswavlem ipovos Tavisi adgili, ar darCes pa-siuri an gariyuli. maswavlebeli da moswavleebi aqtiurad urTierToben erTmaneT-Tan. moxerxebulad SerCeuli kiTxvebis dasmis saSualebiT moswavleebi Cabmuli arian saswavlo procesSi. Tumca, ar igulisxmeba mxolod kiTxva-pasuxis reJimis damkvi-dreba _ moswavle TviTon monawileobs saganmanaTleblo procesis mimdinareobaSi; monawileobs diskusiaSi _ icavs sakuTar mosazrebebs da oponirebas uwevs sxvebs. erT-erTi sadiskusio Tema pirvel paragrafSi SeiZleba iyos naturalur ricxvTa mwkrivis usasrulod gagrZelebadobis SesaZlebloba, potenciuri usasrulobis Ses-axeb diskusia; SeiZleba gamoviyenoT hilbertis magaliTi rubrikidan `sxvadasxva~.

SeiZleba gaimarTos diskusia poziciuri da arapoziciuri sistemebisa da maTi Tav-iseburebebis Sesaxeb.

maswavlebeli warmarTavs gakveTils, SearCevs sadiskusio Temas, Tumca diskusia SeiZleba daugegmavadac _ gakveTilze wamoWrili problemis irgvliv gaimarTos. moswavleebi monawileoben diskusiaSi, avseben erTmaneTis pasuxebs; magaliTad, natu-raluri ricxvebis Tanrigebis saSualebiT Cawerisas, sxvadasxva Tvisebis CamoTvlis dros, damrgvalebis wesebis warmodgenisas. maTematikuri faqtebis codnis Sesamow-meblad SeiZleba gamoviyenoT amocanebi savaraudo pasuxebiT. maswavleblebi am ti-pis davalebebis Sesrulebis sxvadasxva meTods mimarTaven xolme. rac mTavaria, am davalebebis Sesrulebas SeiZleba drois mcire monakveTi davuTmoT da SeiZleba garkveuli TanamimdevrobiT yvela moswavle CavrToT am procesSi. araswori pasuxis miTiTebis SemTxvevaSi SeiZleba gaimarTos msjeloba Secdomis mizezebis Sesaxeb.

pirvel gakveTilze SeiZleba amoixsnas 9, 11, 13, 15, 17, 19, 21 amocanebic. maTi momdevno amocanebi moswavleebs saSinao davalebad miecemaT.

meore gakveTilze vagrZelebT amocanebis amoxsnas. sruldeba 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44 amocanebi. maTi `mezobeli~ _ kentnomriani amocanebi moswav-leebs saSinao davalebad SeiZleba mivceT.

pasuxebi da miTiTebebi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 182 3 3 1 2 1 3 4 2 3 2 2 2 1 2 4 4 3

19 20 21 223 4 1 2

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

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

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

37. mefe daviT IV (aRmaSenebeli) daibada 1073 wels. is 16 wlisa gamefda. XII sauku-neSi saqarTvelos aRorZineba daviTis saxels ukavSirdeba. 1121 wels didgorSi didi gamarjvebis Semoqmedi mefe daviTia.

44

42. a a32 23 320 230 90- = - = .

43. ab1 0 saxis ricxvebis odenoba ase daviTvaloT: cifri a SeiZleba SeirCes nebis-mierad aTi cifridan _ 0; 1; 2; 3; 4; 5; 6; 7; 8; 9 _ sul 10 SesaZlebloba. aseve aTi Ses-aZlebloba aqvs b cifris arCevasac. amitom sul asi ricxvi gveqneba _ 10.10=100.

44. amowerilia pirveli eqvsi naturaluri ricxvis kvadrati.

52. 9 cifriT gadainomreba pirveli gverdi, 180 cifriT _ Semdegi 90 gverdi, 2103 cifriT _ Semdegi 701 gverdi; amrigad, gvaqvs 800 gverdi, anu 400 furceli.

jgufuri muSaobis proeqti

mizani:

• kanonzomierebis aRmoCena da gavrceleba;• kritikuli azrovnebis, msjelobisa da dasabuTebis unaris gamomuSaveba.

jgufuri muSaoba aris saswavlo procesis organizaciis iseTi forma, roca xdeba moswavleTa jgufebad dayofa da jgufebs eZleva saziaro kompleqsuri davaleba. davaleba miznad isaxavs moswavleebis CarTvas TanamSromlobaSi. jgufuri davaleba moswavleebisgan amocanis erToblivad gadaWras, an saerTo produqtis Seqmnas unda moiTxovdes. jgufSi moswavleTa odenoba davalebis specifikiT unda ganisazRvros.

moswavleebi SeiZleba daiyos sam jgufad. jgufebs erTnairi davaleba eZleva _ Seadginon ricxvebis xarisxebis `bolo cifris~ cxrili. TiToeul jgufSi davale-bebis ganawileba jgufis moswavleebzea damokidebuli. SeiZleba erTma moswavlem cxrilis romelime erTi striqonis Sedgenasa da Sesabamisi kanonzomierebis aRmo-Cenaze ifiqros; Semdeg moxdes am striqonebis `gaerTianeba~ erT cxrilSi da kanon-zomierebebis erToblivi prezentacia. amrigad, am jgufuri muSaobis Sesrulebisas TiTToeul jgufSi moswavleTa ricxvi SeiZleba 6-s ar aRematebodes. maswavlebeli afasebs mTlianad jgufis muSaobas. Tumca, is valdebulia Tvali adevnos am muSao-bis process da gamorCeviT aRniSnos yvelaze produqtiuli monawileebi. prezenta-cias jgufis erT-erTi wevri warmoadgens. sxvadasxva jgufis Sedegebi Sedardeba erTmaneTs da imarTeba diskusia mTeli klasis monawileobiT. mxedvelobaSi SeiZleba miviRoT warmodgenili magaliTebis odenobac, romlebic kanonzomierebebis gamoy-enebiT xarisxebis bolo cifrebis povnas exeba. gamarTuli diskusiisas deduqciuri dasabuTebis elementebic iqneba gamoyenebuli, romlebsac maswavlebelma unda yura-dReba miaqcios da Sefasebisas gaiTvaliswinos.

45

§ 1.2. naturaluri ricxvebis gayofa. gayofadobis niSnebi

mizani.

• poziciuri sistemis gamoyenebiT naturaluri ricxvis gayofis procedurebis Catarebis unarebis ganviTareba; gamoyeneba praqtikul amocanebSi.

• deduqciuri msjelobis Catarebis unaris ganviTareba gayofadobis niSnebis ganx-ilviT.

meore gadamuSavebul variantSi, eqspertTa mosazrebebis gaTvaliswinebiT ramden-ime paragrafi gavaerTianeT. gayofis algoriTmi calke ar aris warmodgenili. vi-ziarebT maT mosazrebas, rom am algoriTmis warmodgena a=bq+r, 0≤r<b saxiT gan-sazRvravs operacias (e. w. evklides operacias), romelic naturalur ricxva yovel (a; b) wyvils Seusabamebs (q; r) wyvils; sadac a aris gasayofi, b aris gamyofi, q _ ganayofi, r _ naSTi. am warmodgenas, cxadia, Tan unda sdevdes Sesabamisi Tvisebebis ganxilvac, rac me-7 klasSi ar aris mizanSewonili. amasTanave, gadmocemis gamartive-bis mizniT, wina gamocemisgan gansxvavebiT, amjerad ganzogadeba (cvladebiT warmod-gena) II TavSi gadavitaneT.

SemovifargleT mxolod a naturaluri ricxvis b naturalur ricxvze Cveulebrivi, `grZeli~ (ix., mag. [8], gv. 75) gayofis proceduris gaxsenebiT. moswavleebma ician, rom procedura manamde gagrZeldeba, vidre naSTi gamyofze naklebi ar gaxdeba. magali-Tad, Tu a=648 da b=7, vRebulobT ganayofs q=92 da naSTs r=4; 648=7.92+4.

648 7 63 92 18 14 4erovnuli saswavlo gegmis Sesabamisad, warmodgenilia gayofadobis niSnebis das-

abuTebis elementebi _ konkretul ricxvebze (`specializaciis~ meTodi) Catarebis saSualebiT. momavalSi cvladebze gadasvliT (`ganzogadebis~ procesi) es Tvisebebi SeiZleba nebismier naturalur ricxvebze gadavitanoT.

paragrafs SeiZleba sami gakveTili davuTmoT pirvel nawils da Sesabamis amo-canebs _ 1 gakveTili, meore da mesame nawilebs (Sesabamisi amocanebiT) _ TiTo-TiTo gakveTili.

klasSi SeiZleba Sesruldes savarjiSoebi: #1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 21, 23, 25, 27, 28, 33, 35*, 37. danarCeni savarjiSoebi sasurvelia moswavleebs saSinao davalebad mieces.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 173 2 2 1 4 2 3 4 2 1 4 2 3 1 4 4 1

7. x=7q+2, x+3=7q+5 — naSTia 5. 8. x=3·5·7+1=106.

10. I xerxi: ricxvis luwobis moTxovna niSnavs, rom bolo cif ri luwi unda iyos — Cvens SemTxvevaSi, 2 — erTerTi cif ri oTxi mocemulidan. amrigad, ricxvebis meoTxedi luwia — 24:4=6.

II xerxi: ricxvis luwobis moTxovna niSnavs, rom bolo cifri luwi unda iyos — Cvens SemTxvevaSi, 2. danarCeni sami cifriT samniSna ricxvis Sedgenis 3·2·1=6 SesaZlebloba arsebobs.

46

18. TiToeuls imdeni moneta aqvs, ramdenjerac 150 TeTri metia (10+20) TeTrze: 150:30=5 — xuT-xuTi moneta hqoniaT.

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

21. am ricxvTagan umciresi aRvniSnoT x-iT, maSin x+(x+2)+(x+4)+(x+6)+(x+8)+(x+10)=1326 6x+30=1326 x=216.

22. a) 13·4=52; b) 13·4+1=53.

23. mocemulobis Tanaxmad, yuTebis odenobaze erTiT naklebi ricxvi 6-is jeradi-caa da 7-isac, Tanac 50-ze naklebia. aseTi ricxvia 42. maSasadame, 43 yuTi yofila.

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

25. 366=7·52+2.

27. gaiTvaliswineT, rom 2009 wlis 1 ianvari iyo xuTSabaTi. aTvla SeiZleba sxva wlidanac daiwyoT isargebleT Semdegi informaciiT _ 2008 weli nakiani weli iyo; nakian weliwadSi 366 dRea, aranakianSi _ 365 dRe. 2007 wlis 1 ianvari iyo orSa-baTi.

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

loebuli ricxvis axarisxebisas bolo cifrisTvis SemCneuli kanonzomiereba —

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

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

ebis odenoba 15-ia;b) luwmaCvenebliani xarisxebis odenoba 14-ia;g) naSTi 8 arc erT SemTxvevaSi ar gvxvdeba — rviT daboloebuli oTxis xarisxi

ar arsebobs.

29. wina amocanis analogiurad gamoiyeneT `bolo cifris cxrili~.a) 1-iT bolovdeba 7-is is xarisxebi, romelTa maCve neb lebi 4-is jeradia, aseTi

(mocemul SemTxvevaSi) 11-ia;b) 9-iT bolovdeba 7-is is xarisxebi, romelTa maCve neb lebi 4k+2 saxisaa — aseTi

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

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

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

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

47

31. vTqvaT, samive biWs erTad n burTi ergo, maSin n iyofa 3-ze; cxrave gogonas ki — m burTi, maSin m iyofa 9-ze da, amitom, m iyofa 3-zec. Tu n+m=20, maSin 20-ic unda iyofodes 3-ze, rac ar sruldeba. amrigad, SeuZlebelia maT hqonodaT 20 burTi.

32. fanqris fasic da kalmis fasic luwi ricxvebia, lu wia nayidi rveulebisa da albomebis odenobebic. amrigad, na vaW ris Rirebulebac, rogorc luwi ricxvebis jami — luwia; 557 ki kentia. amrigad, gamyidveli Secda.

33. a) TiToeuli ricxvi iyofa 3-ze — cifrebis jami iyofa 3-ze;b) 6-ze gaiyofa mxolod 132 da 312 — orive luwicaa da 3-is jeradic;g) 9-ze arc erTi ar gaiyofa — cifrTa jami (6) ar iyo fa 9-ze.

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

35*. jer vipovoT im ricxvebis odenoba, romlebic 2-ze an 3-ze (erT-erT ricxvze mainc) iyofa; 2-is jeradia yoveli meore naturaluri ricxvi — aseTTa odenoba 500-ia; 3-is jeradia yoveli mesame naturaluri ricxvi — maTi odenobaa 333; 500+333 jamSi 6-is jeradi ricxvebi orjeraa daTvlili, maTi odenobaa 166.

miviReT, rom 1-dan 1000-mde im ricxvebis odenoba, romelic arc 2-ze da arc 3-ze ar iyofa, aris 1000-(500+333-166)=333.

36. 1·2·3·4·5· ... ·10·11·12·13 — am TanamamravlTa Soris aris 2, 5 da 10, romelTa namravlia 100. amitom namravlis bolo ori cifri nulebia.

37. a) dasaxelebul Tanamamravlebs Soris gamovyoT iseTi wyvilebi, romlebic nuliT daboloebul ricxvebs mogvcemen: 2 da 5; 4 da 15; 1 da 10 — sul sami. isini mxolod sami nuliT daboloebul ricxvs warmogvidgenen.

38. ricxvi rom iyofodes 45-ze, is 9-zec unda iyofodes da 5-zec, amitom ci-frTa jami 9-is jeradi unda iyos da bolo cifri ki — 0 an 5. saZiebeli ricxvebia: 123840, 123345.

39. gamoviyenoT 3-ze gayofadobis niSani da miviRebT, rom 3-is jeradia 8·107+1; 3·104+2·102+1; 2·10n+1.

48

§ 1.3. martivi da Sedgenili ricxvebi. udidesi saerTo gamyofi da umciresi saerTo jeradi

mizani

• naturalur ricxvTa klasifikaciis warmodgena da misi gamoyeneba.• ariTmetikis ZiriTadi Teoremis propedevtika. misi gamoyeneba umciresi saerTo

jeradisa da udidesi saerTo gamyofis povnisas.• praqtikuli amocanebis amoxsnisas udidesi saerTo gamyofisa da umciresi saer-

To jeradis gamoyenebis unaris ganviTareba.

naturalur ricxvTa klasebad dayofa _ martiv ricxvTa klasi, ricxvi 1, Sedge-nil ricxvTa klasi _ mniSvnelovan rols TamaSobs ricxvTa Teoriasa da mis gamoyen-ebebSi. amitom maT Sesaxeb warmodgenebis gaRrmavebasa da gafarToebas me-7 klasSic vagrZelebT.

erT-erTi mniSvnelovani sakiTxi, romelic martiv ricxvTa klasTan aris dakav-Sirebuli, aris imis dadgena, martiv ricxvTa klasi usasruloa, Tu ara.

maswavleblebi sasurvelia icnobdnen evklides Teoremis damtkicebas ([5], gv. 49), romelic sawinaaRmdegos daSvebis meTodiT tardeba.

paragrafi sami nawilisgan Sedgeba; TiToeul nawils Sesabamis amocanebTan erTad SeiZleba TiTo gakveTili davuTmoT.

pasuxebi da miTiTebebi:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 164 3 1 2 4 1 3 3 4 1 2 4 3 2 4 2

17 18 19 20 21 22 23 24 25 26 27 28 20 302 4 2 2 4 2 2 3 2 4 2 3 4 2

14. n=23·32 — am ricxvis gamyofebia 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 — sul 12 gamyofi.

36. pirvel rva aTeulSi cxriani rvajer gvxvdeba 9; 19; ...; 79; mecxreSi — orjer: 89; 90; meaTeSi — aTjer; 91, 92, ..., 99, e. i. sul ocjer.

37. am ricxvs 5-is garda sxva martivi gamyofic Tu aqvs, maSin mas oTxi gamyofi eqneba (ix. amocana 7). amrigad, es ricxvia mxolod 52, anu 25 — gamyofebia 1; 5; 25.

38. ricxvi, romelic 21-ze iyofa, gaiyofa 3-zec da 7-zec. amrigad, misi gamyofebia 1; 3; 7; 21. am ricxvebisgan gansxvavebuli p martivi gamyofi rom hqonoda, kidev sul cota oTxi gamyofic eqneboda — p, 3p, 7p, 21p. amrigad, saZiebeli ricxvis martiv mamravlebad daSlaSi 3 an 7 orjer mainc monawileobs. 3 unda gamoiricxos — maSin ricxvi 9-ze gaiyofoda. amrigad, gameordeba 7. Sesabamisad, gamyofebi iqneba: 1; 3; 7; 21; 49; 147, TviT ricxvia 147.

39. radgan x ricxvi orze metia, x–1 da x+1 ricxvebi erTze metia da (x–1)(x+1) namravls sami gamyofi mainc aqvs — 1, x–1, x+1 — amitom Sedgenili ricxvia.

49

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

A B A-sa da B-sSeniSvnebi

Sedareba

m–10 n A>B m–10=n+14–10=n+4m n+20 A<B n+14<n+20

m–10 n+10 A<B m–10=n+4<n+10m–7 n+7 A=B m–7=n+14–7=n+7

(m+n)-is 2-ze gayo fiT miRebuli

naSTi

0 A=B m+n=2m+14=2(m+7)

(3m+5n)-is 2-zegayofiT miRebuli

naSTi

1 A<B 3m+5n=8n+42=2(4n+21)

m-is 7-ze gayofiTmiRebuli naSTi

n-is 7-ze gayofiTmiRebuli naSTi

A=B Tu n=7k+r, maSin m=n+14

=7k+r+14=7(k+2)+r(m–n)-is 7-ze gayo-

fiT miRebuli naSTi

0 A=B m–n=14=7·2

42. ar SeiZleba, radgan ricxvi ar SeiZleba Tavis gam yof ze naklebi iyos.

43. gaviTvaliswinoT, rom 322=46·7 da 230=46·5. Tu Ti To eul amanaTSi movaTavsebT 7 mandarinsa da 5 vaSls, 46 ama na Tis Sedgenas SevZlebT. 46-ze meti amanaTis Sedgena SeuZ le belia — 46 aris udidesi saerTo gamyofi 322-isa da 230-is.

44. a) SesaZlebelia. magaliTad, usg (2; 12)=2;b) ar SeiZleba — Tu b kentia, maSin 3b-c kentia da 2 misi gamyofi ver iqneba;g) SeiZleba. magaliTad, usg (4; 6)=2;d) SeiZleba. magaliTad, usg (2; 6)=2;e) SeiZleba. magaliTad, usg (14; 12)=2.

47. sami gamyofi aqvs nebismieri martivi ricxvis kvadrats: Tu p martivia, p2-is gamyofebia mxolod 1, p da p2 (Tu raime ricxvs ori gansxvavebuli martivi gamyofi aqvs, maSin mis gam yofTa odenoba ≥4. Tu gamyofia martivi ricxvis kubi, gam yo fTa ricxvi kvlav ≥4). martiv ricxvTa kvadratebi uam ra via, radgan uamravia TviT mar-tivi ricxvebic.

48. a) usg (p; q)=1; b) usg (p2; pq)=p; g) usg (p–q, 2)=2 – p da q kenti ricxvebia, maTi sxva oba — luwi; d) usg (q+1; 2p)=2 — radgan q martivia da 2-ze me tia, amitom q kentia, maSin

q+1 — luwia; 2p-c luwia. am ri gad, 2 aris (q+1)-isa da 2p-s saerTo gamyofi; is udi-desic iq ne ba saerTo gamyofebs Soris, radgan 2p-s gamyofebia mxo lod 1, 2, p da 2p, q+1 ki arc p-ze da arc 2p-ze ar gaiyofa — q<p, orive kentia, e. i. q+1<p.

49. usj (60; 28)=420, e. i. pirveli disko unda mobrundes 420:20=7-jer, meore ki — 420:28=15-jer.

50

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

51. 374 -sa da

415 -ze gayofa

437 -sa da

541-ze gamrav le bas niSnavs, amitom saZiebeli

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

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

53. mocemuli ricxvebi unda iyos 9m da 9n saxis, sadac m da n Tanamartivebia. vTqvaT, 9m<9n, maSin m=5m1 da vRebulobT 9·5m1·9n=3645; m1·n=9; m1=1 da n=9.

amrigad, mocemuli ricxvebia 45 da 81.

54. movZebnoT usj (400; 600; 900)=3600, radgan 3600 metrs TiToeuli morbenali 5 m/wm siCqariT faravs, amitom Sexvedra Sedgeba 3600:5=720 wamSi, anu 12 wuTSi.

55*. pirobis Tanaxmad, Tu mocemul ricxvs x-iT aRv niS navT, maSin x+3 iyofa 5-zec, 8-zec, 10-zec da 15-zec. umciresi aseTi ricxvi 120-ia, e. i. saZiebeli ricxvi x=117.

56. Tu ricxvi iyofa 36-ze, is iyofa 9-zec da 4-zec. aTive SesaZlebeli cifriT Sedgenili aTniSna ricxvis cifrTa jami 45-ia, amitom aseTi ricxvi 9-is jeradia. 4-ze gayo fa do bisTvis ricxvis Canaweris bolo ori cifriT (aTeulebisa da erTeulebis) Sedgenili orniSna ricxvi unda iyofodes oTx ze. aRniSnuli Tvisebis aTniSna ricx-vebs Soris umciresia 1023457896.

57*. radgan 0 da 6 raime naturaluri ricxvis kvadrati ar aris, amitom, vTqvaT, gansaxilveli ricxvis aTobiT CanawerSi ori cifri mainc aris. ganvixiloT bolo ori cifri. pirobis gaTvaliswinebiT, vixilavT oTx SemTxvevas: 66, 06, 60, 00 — sinjvis meTodiT mivagnebT, rom naturaluri ricxvis kvadrati ver dabolovdeba 66-iT, 06-iTa da 60-iT. amrigad, is unda bolovdebodes 00. axla gansaxilveli ricxvi ase warmovadginoT 100·a, an 102·a da a-s mimarT igive msjeloba CavataroT. daskvna — CanawerSi SesaZlebelia mxolod 0-ebis arseboba — es ricxvi naturaluri ricxvis kvadrati ar aris.

pasuxi. ar SeiZleba.

58*. Tu oTxniSna ricxvi iyofa 12-ze, maSin is unda iyofodes 3-zec da 4-zec. pi-robiT, ricxvis cifrebis jami 3-ia _ is iyofa 3-ze. 4-ze gayofadoba ki niSnavs, rom am ricxvis bolo ori cifriT Sedgenili orniSna ricxvi iyofa oTxze _ Tumca es niSani ar aris savaldebulo da Tqven SegiZliaT sinjvis meTodi gamoiyenoT: 1020, 1200, 2100, 3000.

sul oTxi ricxvia.

jgufuri muSaoba _ proeqti

mizani• cnobarebis gamoyenebis Cvevebis daufleba.• maTematikis gamoyenebiT optimaluri (Rirebulebisa da sxva garemoebebis gaT-

valiswinebiT) variantis SerCeva.proeqtis Sesruleba SeiZleba jgufuri muSaobis formiT CavataroT. jgufebs

evalebaT Seiswavlon SemoTavazebuli variantebi, warmoadginon sakuTari varianti, daasabuTon arCevani.

prezentaciis dros mimdinareobs diskusia, msjeloba arCevanis dasabuTebis Sesax-eb.

51

sakontrolo wera #1

(sakontrolo weris variantebis nimuSebi)I varianti

1. ipoveT 76-is 12-ze gayofisas miRebuli ganayofisa da naSTis jami. 1) 6 2) 8 3) 12 4) 10.

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

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

3. x martivi ricxvia, x≠2 da x≠3. ipoveT 6x-is gamyofebis ricxvi. 1) 4 2) 8 3) 12 4) 16.

4. a da b gansxvavebuli martivi ricxvebia. mocemulia tolobebi: I. usg (a; b)=1, II. usj (a; b)=ab, III. usg (a; b)=a, IV. usj (a; b)=1. maTgan WeSmaritia

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

5. ipoveT umciresi naturaluri ricxvi, romelic 3-ze, 5-ze da 7-ze gayofisas naSTSi 2-s iZleva.

6. saskolo Sejibrebaze 36 gogona da 96 vaJi gundebadaa dayofili. gundebSi mxolod gogonebi an mxolod vaJebia.

gundebSi moTamaSeTa odenobebi erTi da igivea. moTamaSeTa ra udidesi o de no ba SeiZleba iyos TiToeul gundSi? ramdeni gundi Camoyalibdeba am Sem Txve vaSi?

II varianti1. ipoveT is a ricxvi, romlis 17-ze gayofisas miiReba ganayofi 3 da naSTi 2. 1) 51 2) 50 3) 53 4) 54.

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

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

3. x martivi ricxvia, x≠3 da x≠7. ipoveT 21x-is gamyofebis ricxvi. 1) 2 2) 8 3) 4 4) 16.

4. x da y gansxvavebuli martivi ricxvebia. mocemulia tolobebi:I. usg (x; y)=y II. usg (x; y)=xy, III. usj (x; y)=xy, IV. usg (x; y)=1, V. usj (x; y)=1. maTgan WeSmaritia: 1) II da III 2) III da IV 3) I 4) V.

5. ipoveT umciresi naturaluri ricxvi, romelic 5-ze, 7-ze da 11-ze gayofisas naSTSi 2-s iZleva.

6. or punqts Soris boZebi yovel 45 m-is manZilze iyo gan la gebuli. es boZebi axliT Secvales da isini erTmane Tis gan yovel 60 m-Si ganalages. ipoveT manZili pirveli punq ti dan uaxloes boZamde, romelic Zveli boZis adgilasaa dar Wo bili.

52

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

me-5 da me-6 amocanebidan Ti To euli – 3 quliT. am ori amocanis Sefasebisas Se iZ le ba gamoviyenoT 1 an 2 qulac im SemTxvevaSi, roca amocanis na wilia amox s nili – amoxsna bolomde ar aris miyvanili.

magaliTad, Tu me-5 amocanaSi napovnia dasaxelebuli Tvisebis raime naturaluri ricxvi (romelic umciresi ar aris), SeiZleba es amoxsna 2 quliT Sefasdes.

sakontrolo weris variantebis nimuSebSi SemoTavazebuli zogierTi amocanis amoxsnebi:

I varianti

5. aseTi ricxvia 3·5·7+2=107.

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

usg (36; 96)=2·2·3=12.maSasadame, gundebSi SeiZleba iyos sul didi 12-12 moTamaSe.

aseTi odenobebis moTamaSeTa gundebis odenoba iqneba 11. maTgan 3 — gogonebisa da 8 — vaJebis.

II varianti

5. aseTi ricxvia 5·7·11+2=387.

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

usj (45; 60)=( ; )45 60

45 6015

45 60 180usg

$ $= = .

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

saZiebeli manZili 180 metria.

36 2 18 2 9 3 3 3 1

96 2 48 2 24 2 12 2 6 2 3 3 1

53

§ 1.4. mTeli ricxvebi

mizani• mTeli ricxvis Sesaxeb pirveli warmodgenebis Seqmna• mTeli ricxvebis praqtikuli interpretaciebis gaazreba urTierTmopirdapire

situaciebis aRwerisas. mTeli ricxvebis gamoyenebis unaris ganviTareba• mTeli ricxvebis TvalsaCino warmodgenis gaazreba.

erovnuli saswavlo gegmis mixedviT paragrafSi mocemuli masala axali unda iyos moswavleTaTvis, Tumca, yoveldRiur cxovrebaSi maT aucileblad SexvedriaT situa-ciebi, roca uaryofiTi mTeli ricxvebis gamoyeneba xdeboda, am gamoyenebebze sau-briT iwyeba paragrafis Teoriuli nawili da igi moswavleebTan saubrisTvis SeiZle-ba gamoviyenoT. saubaric moswavleTa monawileobiT SeiZleba warimarTos. magaliTad, visaubroT moswavleebTan zamTarSi haeris temperaturis Sesaxeb, xom ar icnoben isini Termometrs, romliTac haeris temperaturas vzomavT, ra SemTxvevaSi vityviT, rom haeris temperatura –50-ia _ Termometri am dros 0-s qvemoT 50-s uCvenebs. ga-dacemis meTodika konkretul-induqciuri unda iyos; maSasadame, axlis gagebis sa-fuZveli moswavleTa cxovrebiseuli gamocdileba da sakoordinato wrfeze Tvalsa-Cino warmodgenebi unda iyos. sidideTa cvlilebis dasaxasiaTeblad axali ricxvebis aucilebloba moswavleTa azrovnebaSi ganzogadebas, sayofacxovrebo enaSi (zogjer, lakonurobis mizniT) `moqnili~ maTematikuri terminologiis damkvidrebas ukavSird-eba.

magaliTad, Cveulebriv enaze _ temperatura nuls qvemoT 30, maTematikurad _ temperatura: –30.

axali ricxvebis Semotanis motivaciis procesis gaazreba ar mTavrdeba am para-grafSi. igi Semdeg paragrafebSi moqmedebebis (mag. gamoklebis) operaciis warmod-genisas grZeldeba.

cxadia, maswavlebelma ar unda Caataros leqcia mTeli ricxvebis Sesaxeb, moyves Teoriuli nawilis Sinaarsi; saWiroa saubari moswavleebTan _ Sesabamisi kiTxvebis dasma, pasuxebis mosmena. mniSvnelovania Semajamebeli sja-baasi mTel ricxvTa zogi-erTi Tvisebis Sesaxeb.

mniSvnelovania mopirdapire ricxvebis gaazreba.


1 2 3 4 5 6 7 8 9 244 4 2 4 2 2 3 4 3 3

7.

8.

9.

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

20. koordinatebi moimatebs or-or erTeuls.

25. 2a=12, a=6.

54

§ 1.5. mTeli ricxvis moduli (absoluturi mniSvneloba). mTeli ricxvebis Sedareba

mizani

• mTeli ricxvebis Tvisebebis Sesaxeb codnis ganmtkiceba.• mTeli ricxvebis geometriuli warmodgenebis gamoyeneba dalagebis mimarTebis

SemoRebisas.

vagrZelebT mTeli ricxvebis Tvisebebis Seswavlas, geometriuli warmodgenis gamoyenebebs. ricxviTi wrfis saSualebiT Semogvaqvs dalagebis mimarTeba mTel ricx-vTa simravleSi.

aqac SeiZleba praqtikuli interpretaciebis gamoyeneba; magaliTad, haeris tem-peraturis gamzomi Termometri SeiZleba ricxviTi wrfis analogiurad ganvixiloT. moswavleebs SeiZleba SevTavazoT kiTxvebi, romlebic maT daafiqrebs uaryofiTi ricxvebis Sedarebaze;

_ ra temperaturaa Tqveni azriT meti 30 yinva, Tu 50 yinva, –30, Tu –50?#1 _ #6, #10 amocanebis amosaxsnelad sasurvelia SedarebiT susti moswavleebi

moiwvioT _ es saSualebas mogcemT CarToT isini saswavlo procesSi, miTumetes, rom es amocanebi maTTvis sruliad misawvdomia da iolad dasaZlevi; #7 _ #8 amocanebi iTxovs meti informaciis damuSavebas, xolo #9, #12, #13 amocanebis amosaxsnelad saWiroa kanonzomierebis SemCneva, ganzogadeba, debulebis saxiT Camoyalibeba.

paragrafs SeiZleba mxolod 1 saaTi davuTmoT.


1 2 3 4 5 6 7 8 9 10 11 163 4 3 4 3 1 4 3 2 3 1 2

17. a)

b)

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

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

24. SesaZlebelia — magaliTad, 23-e amocanaSi garCeul SemTxvevaSi.

27. a)

b)

g)

d)

28. a) mocemuli monakveTis Sua wertilis marjvniv;b) mocemuli monakveTis Sua wertilSi;g) mocemuli monakveTis Sua wertilis marcxniv.

55

§ 1.6. mTeli ricxvebis Sekreba da gamokleba

mizani

• mTeli ricxvebis Sekrebisa da gamoklebis TvalsaCinod warmodgenis saSualebiT am operaciaTa aTviseba, mTel ricxvTa simravleze moswavleTa warmodgenebis gafar-Toeba.

• mTeli ricxvebis Sekrebisa da gamoklebis praqtikuli gamoyenebebis unarebis ganviTareba.

SesavalSi aRvniSneT, rom ricxvis cnebis gafarToebis algebruli gza ar emTxveva skolaSi ricxviTi sistemebis swavlebis Tanamimdevrobas. es sakiTxi kargad aris ax-snili kolmogorovisa [21] da froidentalis [38] wignebSi.

mTel ricxvebze moqmedebebis wesebis `receptulobisa~ da praqtikasTan naklebi kavSirebis Sesaxeb kolmogorovTan ([21], gv. 21) vkiTxulobT: `dabal klasSi kiTxva _ SeiZleba Tu ara ricxvs, romelic naklebia, gamovakloT meti ricxvi; SeiZleba Tu ara, magaliTad, 3-s gamovakloT 5 _ iwvevs gamococxlebas moswavleebSi da maswav-leblis kargi xelmZRvanelobiT SeiZleba gaxdes uaryofiT ricxvebze moqmedebebis swavlebis dasawyisi. magram konkretul gamoyenebebTan kavSiris gareSe es yvelaferi gasarTob TamaSad aRiqmeba.

sakiTxi _ SeiZleba Tu ara 3 orze gavyoT, ufro bunebrivad moeCveneba moswav-leebs. maswavleblis daxmarebis gareSec am sakiTxis aRqma ufro advilia dawyebiT klasebSi; rom gavyoT 3 vaSli, erT-erTi unda Suaze gavWraT da miviRebT erTnaxevar vaSls~. amitom kolmogorovs ufro bunebrivad miaCnia ricxvebis swavlebis is Tana-mimdevroba, romelic istoriulad Camoyalibda (swavlebis Tanamimdevroba emTxveva istoriuli ganviTarebis Tanamimdevrobas).

SevecadeT e. w. dadebiTi da uaryofiTi muxtebis, valisa da qonebis, ricxviTi wrfis gamoyenebiT, TvalsaCinod warmogvedgina mTeli ricxvebis Sekrebisa da gamok-lebis operaciebi.

nu movTxovT moswavleebs Sekrebisa da gamoklebis wesebis zepirad gameorebas _ mTavaria moswavles hqondes wesebis swori gamoyenebis unari, romelic TvalsaCino warmodgenebiT uyalibdebaT.

SeiZleba maswavlebelma sxva praqtikuli interpretaciebic moiSvelios. magali-Tad, dRis ganmavlobaSi temperaturis cvlileba zamTris romelime dRes _ diliT SeiZleba temperatura nuls qvemoT iyos, Semdeg gaizardos, saRamos isev Semcirdes. es procesi SeiZleba mTeli ricxvebis SekrebiTa da gamoklebiT warmoadginos mas-wavlebelma. SeiZleba Sesabamisi plakatic ki moitanos skolaSi (sadac miTiTebuli iqneba dRis ganmavlobaSi temperaturis cvlileba) da moswavleebTan erTad imsje-los am cvlilebis Sesaxeb, es cvlilebebi mTel ricxvebze moqmedebebiT aRweros.

paragrafs SeiZleba daeTmos sami gakveTili. amocanebi ise gaanawileT, rom erTi tipis savarjiSoebi gadabmulad ar sruldebodes. magaliTad, 22-e savarjiSos a) Sei-Zleba gakeTdes klasSi, b) da g) _ mieces davalebad, 23-is a) da b) _ klasSi, g) da d) _ davalebad da a. S.


1 2 3 4 5 6 7 8 9 10 11 123 3 3 1 3 3 3 2 2 3 3 4

13 14 15 16 17 18 19 203 3 1 3 2 1 2 1

56

7. 3) –5+7.

8. gavixsenoT, rom • aRniSnavs –1-s, 0 ki — 1-s. amis gaTvaliswinebiT, swori pa-suxia 2) .

9. gaiTvaliswineT, rom TiTo dadebiTi da uaryofiTi `muxti~, anu — nulia. amis gaTvaliswinebiT, swori pasuxia 2) –3.

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

20. b=a+3.

32. a) (+)–(–)=(–) — arasodes, radgan (+)–(–)=(+)+(+)=(+);b) (+)–(+)=(+) — zogjer; magaliTad, 3–2=1, 2–3= –1;g) (–)–(+)=(–) — yovelTvis, radgan (–)–(+)=(–)+(–)=(–);d) (–)–(–)=(+) — zogjer; magaliTad, (–2)–(–5)=3, (–5)–(–2)= –3;e) (–)–(+)=(+) — arasodes, radgan (–)–(+)=(–)+(–)=(–);v) (+)–(+)=(–) — zogjer; magaliTad, 5–7=–2, 7–5=2;z) (+)–(–)=(+) — yovelTvis, radgan (+)–(–)=(+)+(+)=(+).

42. a) 99–(–99)=99+99=198;b) 999–(–999)=999+999=1998.

43. a) yoveli momdevno ricxvi winaze 3-iT naklebia;4; 1; –2; –5; –8; –11; –14; –17;b) yoveli momdevno ricxvi winaze 4-iT metia;–15; –11; –7; –3; 1; 5.

44. moxvedraTa odenoba aRvniSnoT n-iT, acdenaTa odenoba m-iT, maSin qulebis odenoba iqneba — 7n+(–5)m.

a) n=5, m=2, 35–10=25;b) n=2, m=5, 14–25=–11.

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

57

§ 1.7. mTeli ricxvebis gamravleba da gayofa

mizani.

• mTeli ricxvebis Tvisebebis warmodgeniTa da gamoyenebiT gamravlebisa da gay-ofis operaciebis aRwera.

wina paragrafSi naxseneb `receptulobaze~ ufro metad `receptuli~ xasiaTisaa mTeli ricxvebis gamravlebisa da gayofis wesebi. am operaciebis SemoReba Sekrebisa da gamoklebis (magaliTad, jeradi Sekreba _ gamravleba) operaciebze dayrdnobiT, naturalur ricxvebze moqmedebaTa Tvisebebis (magaliTad, gadanacvlebadobis, gan-rigebadobis gavrcobis saSualebiT mimdinareobs.

gamravlebisas `niSnebis wesis~ aTviseba gacilebiT swrafad mimdinareobs, vidre, magaliTad, mTeli ricxvebis gamoklebis operaciis Catareba. nu movTxovT moswav-leebs am wesebis raime saxiT dasabuTebis warmodgenas.

amocanebis amoxsnisas ecadeT zogierTi moqmedeba moswavleebs zepirad Seas-rulebinoT _ amiT gaaxsenebT moqmedebaTa Tvisebebsac da cota gaaxalisebT gamoT-vlebTan dakavSirebul mosawyen procedurebs, SesaZlebloba mogecemaT mTeli klasi CarToT ganxilvebSi, SeajibroT moswavleebi swraf gamoTvlebSi.

Zalian saintereso savarjiSoa 11 _ aq ar arsebobs SekiTxvaze calsaxa pasuxi, radgan ar aris miTiTebuli Sefasebis raime SezRudva, amitom yoveli swori Sefaseba SeiZleba misaReb pasuxad CaTvaloT; Tumca saintereso iqneba, Tu moswavleebs SemoT-avazebuli Sefasebebidan yvelaze `zusts~ SearCevinebT. magaliTad,

a) –24⋅35≈-20⋅40=-800 -24⋅35>24⋅40=-960 -24⋅35<20⋅35=-700 samive Sefaseba misaRebia _ aq uaryofiTi ricxvebis Sedarebis wesebic gamovi-

yeneT.amocanebi dalagebulia mzardi sirTulis mixedviT, rac moswavleTAa dasaqmebis

dros maTi momzadebis donis gaTvaliswinebis saSualebas iZleva.


1 2 3 4 5 6 7 8 93 2 2 3 2 1 2 3 1

20. a) –30=1·(–30)=2·(–15)=3·(–10)=5·(–6)=6·(–5)=10·(–3)=15·(–2)=30·(–1).b) sailustraciod mogvyavs ramdenime warmodgena nam rav lis saxiT: –30=(–2)·5·3=10·(–3)·1=(–6)·(–5)·(–1).

21. 4=2·2, 4= –2·(–2).

22. g) 16·(–1)·...·(–1)=16·1=16. 10-jer

24. cxrilSi Casmulia mxolod niSnebi — Tavad ricxvebs (ufro zustad, maT modulebs) mniSvneloba am amocanaSi ara aqvs.

+ – + – – – + – +

58

§ 1.8. simravle. qvesimravle

mizani

• simravlis, qvesimravlis, carieli simravlis cnebebis aRwera da maTi sxvadasxva gamoyenebis unaris ganviTareba

• ricxviT simravleebs Soris mimarTebebis warmodgenisas simravluri enis gamoy-enebis daufleba

paragrafs SeiZleba ori gakveTili davuTmoT. pirveli gakveTili SeiZleba Sereuli tipis iyos; mimdinareobs wina gakveTilze axsnili masalis aTvisebis Semowmeba, ganmt-kiceba. SeiZleba amoixsnas damatebiTi amocanebidan ramdenime Sesabamisi amocana. amis Semdeg gadavdivarT simravlis cnebis aRweris procesze. es procesic SeiZleba mos-wavleTa monawileobiT daigegmos. Tavidanve miaqcieT yuradReba imas, rom simravle SeiZleba erTi, ori, an meti elementisgan Sedgebodes, an ar hqondes elementi, rom simravle ganisazRvreba raRac TvisebiT, romelic axasiaTebs gansaxilvel obieqtebs da mxolod maT (maxasiaTebeli Tviseba). garkveuli Tvisebebis miTiTebiT moswavleebi asaxeleben simravles, romelsac am Tvisebis mqone yvela obieqti qmnis. magaliTad, SeiZleba moswavleebTan ganvixiloT naturaluri ricxvebi da mivuTiToT Tviseba _ `iyos martivi~. moswavleebi TviTon adgenen simravles, romelic Sedgeba yvela mar-tivi _ 2; 3; 5; 7; ... ricxvisgan.

simravleTa Teoriis sawyisi elementebis swavleba xels uwyobs maTematikis sxva-dasxva nawilis erTianobis gaazrebas. zogierTi simravlisTvis specialuri aRniS-vnebi Semogvaqvs. am aRniSvnebis gamoyeneba momavalSi moswavleebs xSirad mouwevT. maswavlebelma ar unda ifiqros, rom aRniSnuli masaliT moswavleebs simravleTa Teorias vaswavliT skolaSi _ am Teoriis sawyisi cnebebis gadmocema damxmare saSu-alebaa. simravleebTan dakavSirebiT asoebis gamoyeneba maTematikuri niSnebis gamoy-enebis erT-erTi mniSvnelovani magaliTia _ aso aRniSnavs, gansazRvravs `sagans~ _ konkretul simravles an am simravlis elements. momavalSi ufro xSirad mogviwevs im SemTxvevebis ganxilva, roca aso simravlis nebismieri elementebis aRsaniSnavad gamoiyeneba (cvladi; ix., mag. [11]); magaliTad, Canaweri `x nebismieri naturaluri ricxvia~, niSnavs, rom ar SeiZleba iyos 1, 2, 3 an romelime sxva naturaluri ricxvi, erT-erTi yvelaze gavrcelebuli Secdoma, rac simravleTa swavlebisas gvxvdeba es erTelementiani simravlis Tavis elementTan gaigivebaa _ xSirad moswavle fiqrobs, rom {a} da a erTsa da imave obieqts warmogvidgens. saTanado amocanebis ganxil-viT mniSvnelovania am sakiTxze moswavleTa yuradReba gansakuTrebiT gaamaxviloT _ {a}∈a. am momentis aTvisebis xarisxs me-6 testis amoxsniT SeamowmebT.

meore gavrcelebuli Secdomaa simravluri CarTvisa da mikuTvnebis niSnebis aras-wori gamoyenebebi _ moswavleebs erevaT erTmaneTSi ⊃ da ∈ niSnebi. am Secdomis prevencias movaxdenT 1-5, agreTve, 16 savarjiSoebis SesrulebiT.


1 2 3 4 5 6 8 202 1 4 3 4 1 4 2

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

13. a) {2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47};b) {2; 4; 6; 8; 10; 12; 14; ... ; 48; 50};am simravleebis saerTo elementia mxolod 2.

59

15. sasruli simravleebia: g) N3 — masSi sami elementia; d) N6 — masSi eqvsi elementia; e) 5-ze meti da 9-ze naklebi mTeli ricxvebis simravlea — {6; 7, 8}, masSi sami

elementia.

16. a) N⊃Nn; b) Nn⊂Nm; g) Nm⊂N; d) Z⊃N; e) Nn⊂Z.

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

18. Nk⊂N7 ⇒ k≤7, anu k=1; 2; 3; 4; 5; 6; 7.

19. Nm⊂Nn⊂N6, e. i. 1≤m≤n≤6; gveqneba (m; n) SesaZlo wyvilebi: (1; 1), (1; 2), (1; 3), (1; 4), (1; 5), (1; 6); (2; 2), (2; 3), (2; 4), (2; 5), (2; 6); (3; 3), (3; 4), (3; 5), (3; 6); (4; 4), (4; 5), (4; 6); (5; 5), (5; 6); (6; 6).

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

21. a) aseTi qvesimravle sami arsebobs. isini SeiZleba ase CavweroT: A0={3k\ k∈N, 4≤k≤33}; A1={3k+1\ k∈N, 3≤k≤32}; A2={3k+2\ k∈N, 4≤k≤32}. b) A0, A1, A2 simravleTagan TiToeulis nebismieri ori elementi 3-ze gayofi-

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

22. A={1; 2; 3; 5; 6; 10; 15; 30}; B={1; 2; 3; 6; 7; 14; 21; 42}; a) saerTo elementebis simravlea {1; 2; 3; 6}; b) {1; 2; 3; 5; 6; 7; 10; 14; 15; 21; 30; 42}; g) {5; 10; 15; 30}; d) {7; 14; 21; 42}.

60

§ 1.9. venis diagramebi

mizani

• simravleebis, simravleebze operaciebis (gaerTianeba, TanakveTa) TvalsaCinod warmodgenis unaris ganviTareba

• amocanebis amoxsnis TvalsaCino xerxebis gamoyenebis unarebis ganviTareba.

vecnobiT simravleebis TvalsaCino gamosaxvis xerxs _ venis daigramebs. venis di-agramebiT warmovadgenT simravleTa gaerTianebasa da TanakveTas. venis diagramebs didi gamoyeneba aqvs _ maTi saSualebiT mravali amocana TvalsaCinod da martivad ixsneba.

simravleebze operaciebis venis diagramebiT warmodgenisas yuradReba miaqcieT, rom yvela logikuri SesaZlebloba iyos ganxiluli; magaliTad, roca saubaria ori simravlis saerTo elementebis arsebobaze, an erT simravleSi iseTi elementebis ar-sebobaze, romlebic meore simravleSi ar Sedis. magaliTad, ori simravlis TanakveTis ama Tu im SemTxvevis gamosaxvasTan erTad moswavleebi ganixilaven sxva SemTxvvebsac da TanakveTis sxva SesaZleblobebsac Tavad warmoadgenen venis diagramebiT.

simravleebis gaerTianebis SesaZlo SemTxvevebis ganxilvisas sayuradReboa saer-To elementebis arseboba _ isini axal simravleSi TiTo. TiTojer moiazreba. am SemTxvevaze yuradRebis miqceva momavalSi _ logikuri amocanebis amoxsnisas swori daskvnebis gamotanis sawindaria.

paragrafis amocanebSi, garda ZiriTadi Temisa, mniSvnelovani yuradReba eqceva gayofadobasa da naSTebs _ magaliTad, amocanebi 10, 12, 13, 14, 15, 29, 30; amave Tema-tikas ganekuTvneba paragrafis boloSi SemoTavazebuli jgufuri muSaobis proeqtic. gTavazobT ramdenime amocanas uaryofiT ricxvebze moqmedebebis Tvisebebis gasaxsen-eblad _ amocanebi 20, 21, 22, 23, 24, 25, 27. Tematikis mravalferovneba saSualebas mogcemT gakveTili ise warmarToT, rom is ar iyos erTferovani, mosawyeni, amasTa-nave, SeZlebT gaiTvaliswinoT Tqveni klasis momzadebis specifika da meti yuradReba miaqcioT im masalas, romelic moswavleebs naklebad aqvT aTvisebuli. moswavleebTan mudmivi kontaqtiT, sakiTxebis erToblivi ganxilvebiT kidev ufro daazustebT arse-bul xarvezebs moswavleTa codnaSi, codnis gamovlenis unarSi. es informacia Tqvens samomavlo muSaobas ufro mizanmimarTuls gaxdis.


1 2 3 4 5 6 7 8 10 18 19 20 21 22 26 27 29 302 3 1 2 2 4 4 3 4 2 3 3 2 4 2 4 2 3

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

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

g) C∩(A∩B)=C.

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

61

11. a) b) g)

d) z) v) z)

12. A={3k| k∈N, 4≤k≤33}; B={7n| n∈N, 2≤n≤14}. A∩B={21; 42; 63; 84} — sul oTxi elementi.

13. orive mocemuli ricxvi warmovadginoT martivi mam rav le bis namravlis sax-iT:

210 231

2 105 3 77

2 3 35

2 3 5 7 3 7 11

210=2·3·5·7 231=3·7·11 A={2; 3; 5; 7} B={3; 7; 11} A∩B={3; 7}.

14. warmovadginoT mocemuli ricxvebis gamyofTa simrav le ebi: 36—A={1; 2; 3; 4; 6; 9; 12; 18; 36}; 40—B={1; 2; 4; 5; 8; 10; 20; 40}. A∩B={1; 2; 4}.

15. A={3k+1\ k∈N, 3≤k≤32}; B={3k+2\ k∈N, 3≤k≤32}; • A∪B aris 3k+1 an 3k+2 saxis orniSna ricxvTa simravle, anu im orniSna ricxvebis simravle, romlebic ar aris 3-is jeradi.

17. a) A∩B=B b) A∪B=A.

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

24. a) 6+(–1); ; b) –3+2 .

A B

C

A B

C

A B

C

A B

C

A B

C

A B

C

A B

C

62

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


es proeqti naSTTa ariTmetikis mniSvnelovani propedevtikaa, amitom mis Sesas-ruleblad maswavlebelma dro ar unda dainanos.

1. • am simravleebs saerTo elementi ara aqvs — raime naturaluri ricxvis 5-ze gayofisas ori gansxvavebuli naSTi ar miiReba.

• arsebobs — 5-is jeradi ricxvebi arc erT mocemul simravleSi ar Sedis.• magaliTad, ase: `simravle im naturaluri ricxvebisa, romlebic ar iyofa

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

SeiZleba — A0.

•

N

2. wina amocanis analogiurad, n ricxvisTvis Sedgeba sim rav leebi A1={nk+1| k∈Z, 0≤k}, ..., An–1={n(k+1)–1| k∈Z, 0≤k}.

A0 A1 A2 A3 A4

63

§ 1.10. dadebiTi racionaluri ricxvebi

mizani

• dadebiTi racionaluri ricxvebis warmosadgenad da Tvisebebis Sesaswavlad poziciuri sistemis gamoyenebis unaris ganviTareba

• dadebiTi racionaluri ricxvebis gamoyenebebis Sesaxeb codnis gaRrmaveba da gafarToeba

• sxvadasxva formiT (procentiT, naturaluri ricxvebis SefardebiT, aTwiladeb-iT) dadebiTi racionaluri ricxvebis Cawerisa da gamoyenebis unaris ganviTareba.

warmodgenili masala nawilobriv nacnobia moswavleebisTvis. amitom gadmocemisas moswavleTa aqtiuri monawileoba advilad miiRweva.

erovnuli saswavlo gegmis mixedviT arauaryofiTi racionaluri ricxvebis Ses-wavlis Semdeg aTwiladebis swavleba mimdinareobas. Tumca, amas win uswrebs (ukve me-4 klaSi) propedevtikuli muSaoba wiladis Sesaxeb (Semodis naxevari, mesamedi, meoTxedi).

Semdeg wlebSi konkretuli modelebis (wris, kvadratis marTkuTxedis) gamoyen-ebiT Semodis `wiladi~, `mniSvneli~, `mricxveli~, iswavleba wiladebis Sedareba, ricx-vis nawilis povna, erTobliobis nawilis povna.

swavlebis forma induqciuria _ xSiria TvalsaCino warmodgenebis, moswavleTa praqtikuli gamocdilebis gamoyeneba. amasTanave, wiladi ricxvebis swavlebis parale-lurad mimdinareobs aTwiladebis swavlebis procesi, romelia poziciuri warmod-genis gaazreba naturalur ricxvebze Sesabamisi warmodgenebis gamoyenebiT advil-deba. maTi praqtikuli gamoyenebebis sferoc gacilebiT farToa, ariTmetikuli moq-medebebis Sesruleba poziciuri sistemis gamoyenebiT ufro advilad Seiswavleba.

aTwiladebze moqmedebebis Tvisebas eZleva induqciuri dafuZneba.paragrafSi warmodgenili masalis gadmocemisas mimdinareobs im codnis

sistematizacia, romelic ukve aqvs me-7 klasels. upirveles yovlisa, es aris zomis metruli sistemis gameoreba, aTwiladebi da Cveulebrivi wiladebi. gansakuTrebiT mniSvnelovania am codnis praqtikuli amocanebis amoxsnebTan dakavSireba. amitom maswavleblebs vurCevT saswavlo procesSi gamoiyenon is amocanebi, magaliTad, #26, #27, romlebic dakavSirebulia sidideebs Soris kavSirebze _ metrebidan santime-trebze gadasvla, wamebidan _ saaTebze, kilogramebidan _ centnerebze da tonebze da a. S. amocanebi, romlebSic xdeba gadasvla m/wm-dan km/sT-ze da a. S.

ukve SeiZleba codnis gaRrmavebis mimarTulebiT mivaRwioT saTanrigo erTeuleb-is gamoyenebiT ricxvebis Caweris unaris ganviTarebas.

aTwiladebis gamoyenebebis istoriuli aspeqtebis win wamowevis mizniT, SesaZle-belia maswavlebelma Caataros saubari:

`SevadaroT erTi da imave ricxvis ori Canaweri:

210043 da 2,43.

pirveli Seicavs 7 niSans, meore _ 4-s. meore Canaweri ufro `ekonomiuria~. XV saukuneSi aTwiladebis warmoSoba maTematikaSi aRmoCenad aris miCneuli.

SemdgomSi zomis aTobiTi sistemebis SemoRebisas am wiladebma didi gamoyeneba hpova~.

paragrafSi gTavazobT mravalferovani Tematikis amocanebs: proporcia _ # #9, # #20, #21, wiladebze moqmedebebi _ @#10, #11, #12, #13, #17, #18, #19, poziciuri

64

sistemaA _ #5, #6, #7, #8, Pprocentebi _ #1, #22, #23, sidideebs ASoris kavSirebi _ #26, #27. es saSualebas mogcemT zogierTi moswavlis SesaZlo CamorCenis SemTxvevaSi masalis gameorebas meti praqtikuli datvirTva misceT. gansakuTrebuli yuradRebiT moekideT #16-s _ es kombinirebuli tipis amocanaa, saWiroa terminebis gaxseneba, SesaZlo mniSvnelobebis simravlis dadgena da analizi.


1 2 3 4 5 6 7 82 2 3 1 2 1 3 3

14. gaviTvaliswinoT, rom ori naturaluri ricxvis Se far debis saxiT Cawerili ricxvi Caiwereba sasruli aT wi ladis saxiT mxolod im SemTxvevaSi, Tu es wiladi

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

martivi ricxvi. amis gaTvaliswinebiT, usas ru lo aTwiladiT gamoisaxeba: 9

20 ; 121 ;

sasruli aTwiladiT — 207 ;

1245 ;

3259 ;

1258 .

16. radgan n5

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

a) n5

<1; b) 1<n6 ; g) n–5<0;

d) n2+4>n; e) n·100<500; v) n5

–1<0.

17. vTqvaT, wiladi x-ze Seikveca, anu misi sawyisi saxe iyo xx

114 , maSin, pirobiT,

4x+11x=375, x=25. maSasadame, moce mu li wiladia 275100 .

18. magaliTad,

31

93

91

92= = + ,

21

63

61

62= = + ,

71

284

281

283= = + .

20. 18 wuTSi — 15 detali, 3-jer nakleb droSi (6 wuTSi) — 3-jer naklebi ode-nobis detali, anu 5 detali.

21. pirobiT, erT Zroxas 1 kviraSi 31 zvini sWirdeba, Txas ki —

61 . orives erTad

1 kviraSi — 31 +

61 =

21 , 4 kvi ra Si — 4·

21 =2 zvini.

25. Zafis siganis sapovnelad zolis sigane xviaTa ode no baze unda gavyoT.

27. a) gaiTvaliswineT, rom a m/wm = a·10003600 km/sT. amitom 325 m/wm=1170 km/sT; 25

m/wm=90 km/sT; 100 m/wm = 360 km/sT.

b) gaiTvaliswineT, rom a km/sT = a·36001000 m/wm.

amitom 120 km/sT = 33,(3) m/wm; 250 km/sT = 69,(4) m/wm;80 km/sT = 22,(2) m/wm.

65


mizani

• maTematikis saskolo kursis cnebebis gamoyenebis unaris ganviTareba• maTematikis gamoyenebiT optimaluri variantis SerCevis unaris ganviTareba.

1. sanimuSod warmogidgenT dagegmvis erT-erT variants: vTqvaT, avtobusis saSualo siCqarea 50 km/sT, maSin gzaSi yof nis droa sT, Sesabamisi gadasaxdeli Tanxa — ·25=37 (lari), darCenili Tanxa — 123,2–37=85,2 lari rom gavyoT 25-ze, miviRebT Segrovili TanxiT iyalToSi yofnisTvis gan kuTvnil udides dasaSveb dros (saaTebSi).

2. • 850-iT. • 500-iT. • niniko — 350:4=87,5; mariami — –300:4= –75; kote — –500:4= –125; giorgi — 200:4=50.

• magaliTad, niniko — 1000+250-750-150mariami — 1000-400-750-150kote — 250-250+250-750giorgi — 250-150+250-150

• 700-quliani an 1000-quliani. ajobebs 700-qulianis SekveTa, radgan mosal-odnelia, rom is ufro martivi iqneba, vidre 1000 quliani. Tumca, Tavad ninikoc, SesaZloa, kidev moipovebs qulebs, amitom 1000 quliani SekiTxvis SekveTis riskic gamarTlebulia.

66

§ 1.11. racionaluri ricxvebi

mizani

• dadebiTi da uaryofiTi racionaluri ricxvebis simravleebis warmodgena; maTze moqmedebebis aRwera

• dadebiTi da uaryofiTi racionaluri ricxvebis praqtikuli gamoyenebebis un-aris ganviTareba.

es paragrafi ajamebs ricxvis cnebis gafarToebis or sqemas, romelic saxelmZR-vaneloSia warmodgenili. faqtobrivad, ricxvis cnebis gafarToebis algebruli Tana-mimdevrobaa warmodgenili.

racionalur ricxvTa Q simravle Sedgeba dadebiTi racionaluri ricxvebisgan, uaryofiTi racionaluri ricxvebisgan da nulisgan. N⊂Z⊂Q mimarTebis venis dia-gramebiT gamosaxva aZlierebs am simravleTa aRqmas. ganvsazRvreT racionaluri ricxvis modulic:

|a|=a, Tu a>0; |a|=–a, Tu a<0; |0|=0.

racionaluri ricxvebis Sekreba da gamravleba mTel ricxvebze am moqmedebebis analogiuria. racionaluri ricxvebis gamravlebisas namravlis modulisa da niS-nis dadgenas mTel ricxvebze imave moqmedebis msgavsad ganisazRvreba. gamravlebis Sebrunebuli moqmedeba _ gayofa _ erTi gamonaklisis garda Q simravleSi yovelT-vis sruldeba (nulze gayofa ar SeiZleba) _ ganayofi kvlav racionaluri ricxvia. klasSi, moswavleebTan erTad, SeadareT am Tvisebis mixedviT Q da Z simravleebi.

mTeli uaryofiTi ricxvebis Seswavlis analogiurad mimdinareobs uaryofiTi racionaluri ricxvebis praqtikuli interpretaciebis warmodgena.

moswavleebi ukve icnoben mTel uaryofiT ricxvebs, amitom ar gauWirdebaT uaryofiTi racionaluri ricxvis, misi modulis gaazreba. Sedareba isev geometriu-li TvalsaCinoebis gamoyenebiT xorcieldeba. cxadia, axali masalis axsna kvlav mo-swavleTa aqtiuri monawileobiT unda mimdinareobdes. SeiZleba gamoviyenoT garemos temperaturis gamzomi Termometri, romelic uCvenebs temperaturas nuls zemoT da nuls qvemoT, magaliTad, nuls qvemoT 2,50 temperatura aris –2,50.

teqstSi uxvad aris sanimuSo magaliTebi, rac mniSvnelovnad dagexmarebaT sailus-tracio masalis moZiebasa da axsnaSi. cadeT Tavad moswavleebs miandoT gakveTilzeve maTi operatiuli gacnoba da prezentacia. gamocxadebuli ndoba, swrafad miRwevadi warmatebis molodini moswavleebs ubiZgebs aqtiurobisken, matebs sakuTari Zalebis rwmenas.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 23 24 253 1 2 4 1 3 3 1 1 2 3 4 1 1 3 3 4

67

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

ricxvia a1 .

23. gaiTvaliswineT, rom uaryofiTi ricxvis mopirdapire ricxvi dadebiTia.

24. nebismieri racionaluri ricxvis moduli dadebiTia an nulia.

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

39. vTqvaT, pirvel yuTSi kanfetis masaa x kg, meoreSi — x–72 (kg).a) unda gadavdoT 7,2 kg-is naxevari — 3,6 (kg). marTlac, maSin I-Si darCeba x–3,6,

II-Si — (x–7,2)+3,6=x–3,6 (kg);b) unda gadavdoT (10–7,2)-is naxevari — 1,4 (kg). marT lac, maSin II-Si darCeba:

x–8,6 (kg), I-Si aRmoCndeba — x+1,4=(x–8,6)+10;g) unda gadavdoT (7,2-4,8)-is naxevari — 1,2 (kg); marT lac, maSin II-Si aRmoCndeba

x–6, I-Si ki darCeba — x–1,2=(x–6)+4,8 (kg).

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

niu

a) 8848–(–400)=9248; b) 8848–(–1,5)=8849,5;g) 8848–(–27,9)=8875,9.

68

§ 1.12. racionaluri ricxvis naturalurmaCvenebliani xarisxi

mizani

• naturalurmaCvenebliani xarisxis Tvisebebis gamoyenebis unaris ganviTareba• ricxvis Cawerisas xarisxis praqtikuli gamoyenebis Cvevebis daufleba.

gavixseneT ricxvis xarisxis saxiT Caweris wesi: a ricxvis n-uri xarisxi (n∈N, n>1) aris n-cali Tanamamravlis namravli, romelTagan TiToeuli aris a _

a⋅a⋅...⋅a=an, n>1; a1=a. n-jer

xarisxis cnebis Seswavlas konkretuli ricxviTi magaliTebis ganxilviT viwyebT. am masalis propedevtikas adrec sakmao dro daeTmo _ ganixileboda ricxvis kvadra-ti, naturaluri ricxvis xarisxi, `giganturi~ ricxvebis xarisxis saxiT warmodgenebi. amitom, axal masalas, romelic nebismieri racionaluri ricxvis naturalurmaCven-ebliani xarisxis aRweras eZRvneba, moswavleebi advilad euflebian.

paragrafs SeiZleba 2 saaTi davuTmoT.#1 – #9 amocanebiT SeiZleba moswavleTaA faqtobrivi codnis Semowmeba, #11

da # #12 _ erTgvari ganmazogadebeli amocanebia; saintereso amocanaa # 15 _ moswavleebs _ kidev erTxel gaamaxvilebineT yuradReba faqtze: Tu a>b>0 da n∈N, maSin an>bn; amitom 26>62, radgan 26=82.


1 2 3 4 5 6 7 8 9 11 12 13 144 2 3 1 4 2 4 2 2 3 3 1 2

15. 26=2·2·2·2·2·2=82>62.

22. 2·2·2·2·2=25=32. Sesabamisad, sisqe iqneba 32·0,1=3,2 (mm). 2·2·...·2=25·25·25=323. Sesabamisad, sisqe iqneba

15-jer

323·0,1=32768·0,1=3276,8 (mm).

23. mexuTe _ 1–(21 )5

, meaTe _ 1–(21 )10

, mease _ 1–(21 )100

.

sakontrolo wera #2

(sakontrolo weris variantebis nimuSebi)

I varianti

1. ricxviT RerZze A(5) wertilidan 7 erTeuliT daSo re buli wertilebis Se sa-bamisi koordinatebia:

1) 2 da 12 2) –2 da –9 3) 12 da 19 4) –2 da 12.

69

2. muxtebis modeliT –3 SeiZleba ase gamoisaxos: 1) 2) 3) 4)

3. uaryofiTi ricxvebis jamis uaryofiT ricxvze nam rav lis moduli 1) uaryofiTia 2) dadebiTia 3) nulia 4) dadebiTi ar aris.

4. or faraSi 100 cxvaria. amasTanave, meoreSi 12 cxv riT metia, vidre pirvelSi. maSin pirvel faraSi cxvrebis odenoba Semdegi gamosaxulebis mniSvnelobis tolia:

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

5. Tu a aris –56:8+(–14) gamosaxulebis mniSvneloba, maSin 1) a∈N 2) a∉Z 3) a=Z 4) a∈Z.

6. umciresi elementi, romelic (N5∩Z)-simravles ekuTv nis, aris 1) 1 2) 5 3) 0 4) –1.

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

8. cnobilia, rom n∈N, n4 ar aris wesieri, n

62+ we si eri wiladia. ipoveT n-is

yvela SesaZlo mniSvnelobaTa sim rav le.

II varianti

1. ricxviT RerZze A(–5) wertilidan 6 erTeuliT da So rebuli wertilebis Se sa-bamisi koordinatebia:

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

2. muxtebis modeliT –4 SeiZleba ase gamoisaxos: 1) 2) 3) 4)

3. uaryofiTi ricxvebis jamis dadebiT ricxvze namravlis moduli 1) uaryofiTia 2) dadebiTia 3) nulia 4) dadebiTi ar aris.

4. or oTaxSi 59 skamia. amasTanave, pirvel oTaxSi 13 ska miT metia, vidre meoreSi. maSin meore oTaxSi skamebis odenoba Semdegi gamosaxulebis mniSvnelobis tolia:

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

5. Tu b aris 49:(–7)–(–12) gamosaxulebis mniSvneloba, maSin 1) b∉Z 2) b∉N 3) b=N 4) b∈N.

6. udidesi elementi, romelic (N7∩Z)-simravles ekuTv nis, aris 1) 1 2) 7 3) –1 4) –7.

7. SeasruleT moqmedebebi:

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

8. cnobilia, rom p∈N, p6 ar aris wesieri wiladi, p

83+ wesieri wiladia. ipoveT

p-s yvela SesaZlo mniSv ne lo baTa simravle.

70

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

Sefasdes 1 quliT; me-7 da me-8 amocanebidan TiToeuli – 2 quliT, Tumca am or amo-canaSi SeiZleba gamoviyenoT nawilobrivi Sefasebac _ davweroT 0,5; 1; 1,5 qulebi. magaliTad, I variantis me-8 amocanaSi, Tu moswavlem sworad gaaanaliza is faqti, rom erTi wiladi aris wesieri (an meore ar aris wesieri) da miiRo n-is saTanado Sefaseba, weiZleba dauweros 1 qla. sazogadod, Tu moswavle poulobs amoxsnis gzas, iwyebs amoxsnas, magram ver asrulebs, maSin mas SeiZleba mTeli Sefasebis 50% davu-

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

am SefasebaSi). Tu gaiTvaliswina mxolod is faqti, rom n4 ar aris wesieri wiladi

da miiRo n-isTvis arazusti Sefaseba n<4, am etapis Sefaseba iqneba 0,5 qula). meore wiladis SeswavlaSi kidev 0,5 qula daiwereba. miRebuli Sedegebis mixedviT n-is yvela SesaZlo mniSvnelobis dadgenaSi kidev 0,5 qula daiwereba. sul _ maqsimaluri Sefasebaa 2 qula.

uxeSi Secdomis daSvebisas (magaliTad, gamoTvlebSi uxeSi Secdomebi, cnebebis araadekvaturi gamoyeneba, an Sedegis araadekvaturi warmodgena), Tu amocana 2-quli-ania, sazogadod _ maqsimaluri Sefaseba ar unda aRematebodes 0,5 qulas, miaxloe-biT 30%-s (maqsimaluri Sefasebis).

sakontrolo weris niSnebis gamocxadebamde, ganixileT sakontrolo amocanebi, maTi amoxsnis gzebi da Sefasebis kriteriumi. mimarTeT, moswavleebs, am informaciis gaTvaliswinebiT, Tavadve (winaswar) gamoacxadon sakuTari savaraudo niSnebi. SeaqeT isini, vinc SeZlebs obieqturad Seafasos Tavisi naSromi.

sakontrolo #2 ajamebs I Tavs, romelic sakmaod didia (12 paragrafi) da, Sesabamisad, sakiTxebis farTo Ospeqtrs moicavs. maswavlebels vurCevT am sakontrolo weris Sedegebis Tematuri analizi Caataros, risTvisac SeiZleba, magaliTad, aseTi formis gamoyeneba:

sakontrolo wera #2

#moswavlis gvari,

saxeli

Tema

sabo

lo

o S

efas

eba.

koo

rd

inat

ebi

wrf

eze

(#1

) ma

qs.

Sef

aseb

a 1

q.

muxt

ebis

mo

del

i

(#2

) ma

qs.

Sef

aseb

a

1 w

.

ric

xviT

i ga

mo-

saxu

le b

e bi

(#3

,

#4

, #

5, #

7)

maqs

.

Sef

aseb

a 5

q.

ric

xviT

i si

mrav

le-

ebi

(#6

, #

8)

maqs

.

Sef

aseb

a 3

q.

1 anTiZe TaTuli 1 1 5 3 10

2 baramiZe elene 1 0 4 2 7

3 giorgaZe aleqsandre 0 0 2,5 3 5,5≈6

4 daraxveliZe levani 1 1 3 1 6

5 vawaZe Tika 1 1 4 1 7

Sedegi %-Si 80% 60% 74% ≈67% ≈70%

procentebSi gamosaxuli Sedegebi iZleva saSualebas SevafasoT Catarebuli samuSao da davgegmoT Semdgomi procesi _ magaliTad, moyvanil nimuSSi kargi Sedegia TemaSi `koordinatebi wrfeze~ da SedarebiT dabali Sedegi ricxviTi simravleebisa da maTi modelebis sakiTxebSi.

71

gTavazobT sakontrolo weris variantebis SemoTavazebuli nimuSebis zogierTi amocanis amoxsnas:

varianti I

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

8. radgan n4 ar aris wesieri, amitom n≤4; n

62+ — wesi e ria, amitom n+2<6, n<4.

maSasadame, n<4, n∈N, e. i. n∈{1; 2; 3}.

varianti II

7. (–768):((–4)·27+(–24)·(–5))–(4,3·(–2))2=(–768):(–108+120)– –(–8,6)2=(–768):12–73,96= –64––73,96= –137,96.

8. radgan p6 ar aris wesieri, amitom p≤6; p

83+ — wesi eria, amitom p+3<8, p<5.

amrigad, p<5, p∈N, e. i. n∈{1; 2; 3; 4}.

damatebiTi amocanebi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 212 4 2 2 3 4 4 2 1 3 3 4 2 2 3 2 2 3 4 3 4

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 421 2 2 3 3 2 3 2 2 3 1 3 3 1 3 3 2 4 1 3 1

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 682 1 1 3 3 3 1 3 2 4 2 3 1 3 3 4 3

59. ricxvi Tu iyofa 3-ze, am ricxvis cifrTa jamic unda iyofodes 3-ze. Sesabam-isad, * unda CavanacvloT 3-iT^, 6-iT an 9-iT.

60*. Tina cxovrobs mexuTe sarTulze _ e. i. Tavis sadarbazoSi misi bina me-19-ea; amrigad, danarCeni 82-18=64 bina 16 sarTuls Seadgens _ es odenoba unda iyos saxlSi sarTulebis odenobis jeradi. nino cxovrobs mesame sarTulze _ e. i. Tavis sadarbazoSi misi bina me-11-ea; danarCeni 160 bina qmnis 40 sarTuls _ es odenobac unda iyofodes saxlSi sarTulebis odenobaze, romelic, sxvaTa Soris, ar aris 5-ze naklebi. sarTulebis odenobaa 8.

61. 20:4.5=25; 25-20=5.

62. I dRes _ waikiTxa 73 nawili, darCa

74 nawili;

II dRes _ 74 -is naxevari, anu

72 waikiTxa, darCa

72 nawili;

III dRes _ darCenili 32 gverdi.

amrigad, mTeli wignis 72 nawili 32 gverdia _ wignSi 32:2.7=112 gverdia.

63. ekuTvnis, gamoiyeneT venis diagrama: aM P

72

64. 1 0 2 9 3 5 5 1 4 5 3 0 8 7 3 6 0 1 5

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

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

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

69. 3.4=12 _ 12 sxvadasxva variantia.

70.yv. mwv. cisf. wT.

cisf. wT. mwv. yv.

mwv. yv. wT. cisf.

wT. cisf. yv. mwv.

71. 12 saaTi da 40 wuTi.

73. a=3–8=–5 b=3+8=11

74. a) orjer;

b) lurjebi _ 31 nawils, TeTrebi _

32 nawils.

g) lurjebi _ 24, TeTrebi _ 48.

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

76. SeiZleba ekuTvnodes, SeiZleba arc ekuTvnides:

77. asoebi periodulad meordeba, yoveli xuTi asos Semdeg. 116-e adgilze iqneba `a~. adgilis rigiTi nomris 5-ze gayofisas miRebuli naSTis mixedviT vipoviT asos pirvel xuTeulSi.

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

weTSi gadis, xolo fexiT _ 141 saaTSi. Sesabamisad, samsaxurSi misvlasa da ukan dab-

runebas fexiT is 2,5 saaTSi moaxerxebs.

b bM MP P

maswavleblis wigni 7 klasi

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