მათემატიკური წიგნიერება-მასწავლებლის სახელმძღვანელო-

ekaterine korZaZe - maTematikuri wignierebis eqsperti.

daamTavra Tbilisis ivane javaxiS-vilis saxelobis saxelmwifo univer-sitetis meqanika-maTematikis fakul- teti. wlebis ganmavlobaSi iyo skoleb-is kompiuterizaciis programa „irmis naxtomi“-s maTematikisa da bunebisme-

tyvelebis maswavlebelTa sagnobrivi qselis koordina-tori. monawileobda zogadsaganmanaTleblo skolebis maTematikis erovnuli saswavlo gegmis SemuSavebaSi. aris maTematikis swavlebis araerTi publikaciis Tanaavtori. muSaobs ilias saxelmwifo universitetSi, ganaTlebisa da mecnierebis saministroSi maTematikis eqspertad.

ekaterine korZaZe

maTematikuriwigniereba

maswavleblis saxelmZRvanelo

ekaterine korZaZe

maTematikuri wigniereba

Tbilisi 2012


ISBN 978-9941-0-5034-3

© samoqalaqo ganviTarebis instituti,2012

proeqti ,,axali SesaZleblobebi saganmanaTleblo saWiroebis mqone bavSvebs“ dafinansebulia evrokavSirisa da Ria sazogadoebis insti-tutis mier da xorcieldeba samoqalaqo ganviTarebis institutis, aso-ciacia zekarisa da marneulis axalgazrduli centris mier.

publikaciaSi gamoTqmuli mosazrebebi SeiZleba ar gamoxatavdes evro-kavSirisa da Ria sazogadoebis institutis mosazrebebs.

Project “New Opportunities for Educationally Deprived Children” was made possible by the support of European Union and Open Society Institute. Project is implementing by the Civic development Institute, Association “Zekari” and Marneuli Youth Center.

The contents of this publication do not necessarily reflect the views of European

Union and Open Society Institute

avtori: ekaterine korZaZerecenzenti: simon janaSiadizainer-damkabadonebeli: ekaterine wiqariSvili

sakonsultacio sabWoanTaZe iaboWoriSvili mananagorgoZe sofiomosiaSvili Tamar

wignis SeqmnaSi gansakuTebuli wvlili Seitanes samoqalaqo ganviT-rebis institutis TanamSromlebma, ganaTlebis eqspertebma, romleb-mac SeimuSaves ganaTlebis meore Sansis qarTuli modeli. ideis warmatebiT ganxorcielebis procesSi aqtiuradmonawileobdnen:

baqraZe maiaTofaZe qeTevankobalaZe qeTevankorZaZe ekaterinepapava paatasamsonia irinaWanturia Tamar daibeWda gamomcemloba mwignobariTbilisi, 2012

sarCevi

Sesavali ..............................................................................................5

1. maTematikuri wignierebis standarti ............................7

2. problemaze orientirebuli swavleba .........................16

3. kiTxvis strategiebis gamoyeneba ....................................22

3. Sefaseba maTematikur wignierebaSi ...............................27

5. gakveTilis gegmis nimuSebi ................................................29

6. gamoyenebuli literatura ................................................54


5

Sesavali

Tanamedrove epoqaSi maTematika Cveni cxovrebis ganuyre-li nawilia. igi gamoiyeneba adamianis saqmianobis yvela sfe-roSi: mecnierebasa da teqnologiebSi, medicinaSi, ekonomika-Si, garemos dacvasa da socialur gadawyvetilebaTa miRebaSi. Znelia moiZebnos adamianis saqmianobis sfero, romelic ga-jerebuli ar aris “maTematikuri obieqtebiT”. sabazo maTema-tikuri ganaTlebis gareSe Znelia sazogadoebis srulfasovan wevrad igrZno Tavi. Sesabamisad, maTematikuri wigniereba Se-saZlebelia ganxilul iqnas, rogorc sazogadoebis ganviTa-rebis winapiroba. ganaTlebis sistemis umniSvnelovanesi amo-canaa moswavleTa aRWurva im codniTa da unar-CvevebiT, rac gauadvilebs maT cxovrebiseuli problemebis gadaWras, miexma-reba momavlis ukeT dagegmvaSi, gadawyvetilebebis miRebaSi, gaazrebuli cxovrebiseuli arCevanis gakeTebaSi.

mniSvnelovania swori aqcentebis dasma maTematikis farTo gamoyenebaze moswavleTa yoveldRiur cxovrebaSi. saswavlo masalis Sinaarsic da swavlebis meTodologiac orientirebu-li unda iyos maTematikurad wignieri Taobis aRzrdaze, imis miuxedavad, saqmianobis romel sferos airCeven moswavleebi samomavlod.

maTematikuri wigniereba gulisxmobs elementaruli maTe-matikuri cnebebisa da procedurebis flobas, maTi gamoyene-bis unars realuri, yoveldRiuri problemebis gadaWrisas; komunikaciis im saSualebebis moxmarebis unars, romlebic saWiroa informaciis misaRebad da gadasacemad maTematikuri enis gamoyenebiT; agreTve, maTematikis adgilisa da mniSvne-; agreTve, maTematikis adgilisa da mniSvne- agreTve, maTematikis adgilisa da mniSvne-lobis codnas sxva disciplinebSi, biznesSi, xelovnebasa da adamianis moRvaweobis sxvadasxva sferoebSi.

amgvarad, maTematikuri ganaTlebis ZiriTadi miznebi da amocanebia:

• moswavleebisaTvis azrovnebis unaris ganviTareba;• msjelobis, SexedulebaTa dasabuTebis, movlenebisa da

faqtebis analizis unaris ganviTareba;

6


• maTematikis, rogorc samyaros aRwerisa da mecnierebis universaluri enis gacnobiereba;

• swavlis Semdgomi etapisaTvis an profesiuli saqmianobi-saTvis momzadeba;

• cxovrebiseuli amocanebis gadasawyvetad saWiro codnis gadacema da am codnis gamoyenebis unaris ganviTareba.

maTematikuri wignierebis kursi xels uwyobs Semdegi zogadi unar-Cvevebis Camoyalibebas:

komunikacia• informaciis gadmocemisas sakiTxis arsis warmoCena;

• informaciis gadmocemisas maTematikuri terminebsa da aR-niSvnebis sworad da efeqtianad gamoyeneba.

msjeloba-dasabuTeba• varaudis gamoTqma da kerZo SemTxvevebSi misi kvleva;

• arsebiTi Tvisebebisa da monacemebis gansazRvra;

• gamonaklisi SemTxvevebis aRniSvna da maTi ganzogadebis aramarTebulobis dasabuTeba.

problemebis gadaWra • amocanis Sinaarsis aRqma da problemis formulireba; amo-

canis monacemebisa da saZiebeli sidideebis erTmaneTisagan gamijvna; monacemebis organizeba da warmodgena.

• kompleqsuri problemis safexurebad, martiv amocanebad dayofa da etapobrivad gadaWra.

• miRebuli Sedegebis ganzogadeba.


7

1. mTematikuri wignierebis standarti

1.1. maTematikuri wignierebis standartis struqtura:standarti Sedgeba oTxi mimarTulebisagan. esenia:

1. ricxvebi da moqmedebebi: • ricxvebi, maTi gamoyeneba da ricxvis warmodgenis saSuale-

bebi;• moqmedebebi ricxvebze da ricxviTi Tanafardobebi;

• raodenobaTa Sefaseba da miaxloeba;

• sidideebi, zomis erTeulebi.

2. kanonzomierebebi da algebra:• simravleebi, asaxvebi, funqciebi da maTi gamoyeneba.

3. geometria da sivrcis aRqma: • geometriuli obieqtebi: maTi Tvisebebi, urTierTmimarTe-

ba da konstruireba;

• zoma da gazomvis saSualebebi;

• orientireba sivrceSi, koordinatebi da maTi gamoyeneba.

4. monacemTa analizi, albaToba da statistika:• monacemTa wyaroebi da monacemTa mopovebis saSualebebi;

• monacemTa mowesrigebis xerxebi da monacemTa warmodgenis saSualebebi;

• monacemTa Semajamebeli ricxviTi maxasiaTeblebi.

es mimarTulebebi mWidro urTierTkavSirSia da warmoa-Cens Sesaswavli masalis mTel speqtrs. TiToeuli mimarTu-lebisaTvis Camoyalibebulia misaRwevi Sedegebi TandarTuli indikatorebiT. indikatorebSi aRwerilia, Tu ra aqtivobebis gamovlenis SemTxvevaSia Sedegis miRweva TvalsaCino.

8


1.2.kursis dasrulebisas misaRwevi Sedegebi mimarTulebebis mixedviT:

ricxvebi da moq-medebebi

kanonzom-ierebebi da

algebra

geometria da sivrcis aRqma

monacemTa analizi,

albaToba da statistika

1) moswavle flobs da iyenebs ricxvis cnebas da ricxvis warmod-genis saSuale-bebs.

2) moswavles SeuZlia ricx-vebis Sedareba da klasificireba.

3) moswavles SeuZlia ZiriTadi ariTmetikuli operaciebis Sesruleba ricx-vebze.

4) moswavles Seu-moswavles Seu-Zlia ricxvebis damrgvaleba, moqmedebis Sede-gis miaxloebiT Sefaseba.

5) moswavles SeuZlia zomis sxvadasxva er-Teulebis erT-maneTTan dakav-Sireba.

1) moswav-moswav-les SeuZlia yoveldRiuri cxovrebidan an bunebism-etyvelebis dargebidan momdinare amocanebis amoxsna.

2) moswav-moswav-les SeuZlia simravluri cnebebisa da operaciebis gamoyeneba martivi amocanebis amoxsnisas.

1) moswavles SeuZlia fig-urebis iden-tifikacia, klasificire-ba, gamosaxva.

2) moswav-moswav-les SeuZlia saganTa da figuraTa zomebisa da obieqtTa So-ris manZilebis povna.

3) moswav-moswav-les SeuZlia orientireba sibrtyeze

1) moswavles aqvs el-ementaruli warmodgenebi informaciis mopovebis saSualebebis Sesaxeb.

2) moswavle flobs infor-maciis mowes-rigebisa da warmodgenis xerxebs (saSu-alebebs)

3) moswav-moswav-les SeuZlia Tvisebriv da raodenobriv monacemTa interpretacia da elementa-ruli analizi.


9

ricxvebi da moqmedebebi1. moswavle flobs da iyenebs ricxvis cnebas da ricxvis war-

modgenis saSualebebsSedegi TvalsaCinoa, Tu:• ganasxvavebs ricxvis gamoyenebis sxvadasxva aspeqts (raode-

nobis aRniSvna, rigiTi nomeri, “Wde”);

• kiTxulobs da gamosaxavs ricxvebs (wiladebis SemTxvevaSi, uTiTebs maT CanawerSi wiladis mricxvels da mniSvnels, mTel da wilad nawilebs);

• axdens aTobiTi poziciuri sistemis demonstrirebas sxva-dasxva modelis gamoyenebiT; ganmartavs ricxvebis saxel-debas mSobliur enaze;

• asaxelebs ricxvis CanawerSi sxvadasxva Tanrigis cifrebis Sesabamis mniSvnelobebs, warmoadgens ricxvs saTanrigo Se-sakrebebis an sxva saxiT;

• ekvivalenturi formiT wers Sereul ricxvebs, aTwiladebsa da wiladebs.

2. moswavles SeuZlia ricxvebis Sedareba da klasificirebaSedegi TvalsaCinoa, Tu:• iyenebs poziciur sistemas naturaluri ricxvebis Sedare-

bisas;

• zrdiT an klebiT alagebs ricxvebs;

• sworad iyenebs terminebs - metia, naklebia, ar aRemateba, da Sesabamis aRniSvnebs Sedarebis Sedegis gamosaxatavad;

• ganasxvavebs naturalur da racionalur ricxvebs (luw da kent ricxvebs da sxva).

3. moswavles SeuZlia ZiriTadi ariTmetikuli operaciebis Sesruleba ricxvebze

Sedegi TvalsaCinoa, Tu:• flobs zepirad Sekrebisa da gamoklebis strategiebs (Se-

sabamisi saTanrigo erTeulebis Sekreba/gamokleba, Tanri-gamde Sevsebis xerxi, ...);

10


• moqmedebaTa Sesrulebisas iyenebs weriT algoriTms;

• angariSisas da martivi ricxviTi gamosaxulebis mniSvnelo-bis povnisas iyenebs moqmedebaTa Tanmimdevrobas da moqme-debaTa Tvisebebs.

4. moswavles SeuZlia ricxvebis damrgvaleba, moqmedebis Sede-gis miaxloebiT Sefaseba

Sedegi TvalsaCinoa, Tu:• asaxelebs mocemuli ricxvis uaxloes aTeuls, aseuls an

aTaseuls;

• mocemuli sizustiT amrgvalebs ricxvebs (maT Soris, wi-ladebsa da aTwiladebs);

• afasebs ariTmetikuli moqmedebebis Sedegs;

• amocanis konteqstis gaTvaliswinebiT irCevs, ra ufro mizanSewonilia - moqmedebaTa Sedegis Sefaseba, Sedegis miaxloebiTi Tu zusti mniSvnelobis povna.

5. moswavles SeuZlia zomis sxvadasxva erTeulebis erTmaneT-Tan dakavSireba

Sedegi TvalsaCinoa, Tu:• irCevs da iyenebs Sesaferis erTeulebs sididis cvlile-

bis, moZraobis siCqaris, masStabisa da rukaze manZilis povnasTan dakavSirebuli amocanebis amoxsnisas (igulisx-meba adekvaturi erTeulis SerCeva);

• erTmaneTTan akavSirebs sigrZis, farTobis, moculobis, drois Sesabamis erTeulebs.

•

kanonzomierebebi da algebra

1. moswavles SeuZlia yoveldRiuri cxovrebidan an bunebism-etyvelebis dargebidan momdinare amocanebis amoxsna.

Sedegi TvalsaCinoa, Tu:• martiv amocanaSi erTmaneTisagan mijnavs monacemebsa da

saZiebel sidideebs; msjelobs, sakmarisia Tu ara monaceme-


11

bi amocanis amosaxsnelad.

• amoicnobs da aanalizebs sidideebs Soris pirdapirpro-porciul damokidebulebas; iyenebs mas yidva-gayidvaze, masStabze, xsnarebze, Senadnobebze amocanebis amoxsnisas.

• poulobs ricxvis wilad nawils an procents da piriqiT, poulobs ricxvs misi wiladi nawilis (procentis) mixedviT.

• adgens da xsnis realuri viTarebis an misi sityvieri aRwe-ris Sesabamis tolobas, utolobas an gantolebas.

2. moswavles SeuZlia simravluri cnebebisa da operaciebis gamoyeneba martivi amocanebis amoxsnisas

Sedegi TvalsaCinoa, Tu:• sworad iynebs terminebs «yvela», «yoveli», «TiToeuli»,

„zogierTi», «erTi-erTi», «arc erTi», «erTaderTi».

• iyenebs zogadi-kerZo tipis mimarTebebebs gamonaTqvamis marTebulobis dasasabuTeblad (argumentirebisas).

• SeuZlia marSrutis, gegmis (sqemis) etapebis an obieqtTa Soris mimarTebebis gamomsaxavi martivi diagramebis wa-kiTxva da ageba.


1. moswavles SeuZlia figurebis identifikacia, klasificire-ba, gamosaxva.

Sedegi TvalsaCinoa, Tu:• yofiTi daniSnulebis sagnebSi an maT ilustraciebSi uTi-

Tebs geometriul figurebs.

• adarebs da ajgufebs figurebs geometriuli atributebis mixedviT.

• garemomcvel obieqtebs Soris moiZiebs simetriul obi-eqtebs; ganasxvavebs simetriul da asimetriul figurebs. saTanado SemTxvevaSi axdens figuraTa simetriulobis de-monstirebas.

12


• qmnis brtyeli da sivrculi figurebis grafikul gamosaxu-lebebs da modelebs (maT Soris, maTi sityvieri aRwerilo-bis safuZvelzec).

2. moswavles SeuZlia saganTa da figuraTa zomebisa da obieqt-Ta Soris manZilebis povna.

Sedegi TvalsaCinoa, Tu:• axdens or obieqts Soris manZilis Sefasebas Sesabamis stan-

dartul erTeulSi, zomavs mas da amowmebs Tavis varauds;

• dasmuli amocanis Sesabamisad sworad irCevs figuris zo-mis saxes (mag., Tu Robis sigrZea gamosaTvleli - es igive perimetria, Tu iatakia SesaRebi - maSin farTobis codnaa saWiro).

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

• (iTvlis mravalkuTxedis perimetrs, marTkuTxovani figu-ris farTobs, marTkuTxa paralelepipedis moculobas)

3. moswavles SeuZlia orientireba sibrtyeze.Sedegi TvalsaCinoa, Tu:• orientirebs garemoSi da aRwers obieqtTa urTierTganla-

gebas.

• aRwers obieqtis mdebareobas meore obieqtis mimarT Sesaba-misi terminebis gamoyenebiT (mag. marjvniv, marcxniv, zemoT, qvemoT).

• gascems da Tavadac asrulebs moZraobis orientaciis Se-mcvel miTiTebebs.

• aRwers rukaze ori an meti punqtis urTierTmdebareobas oTxi mimarTulebis gamoyenebiT (mag. CrdiloeTiT, dasav�leTiT).

• koordinatebis (simboloTa wyvilis) gamoyenebiT aRwers mdebareobas da iyenebs am xerxs realur viTarebaSi (mag. kinoTeatri, gemebis CaZirobana, Wadrakis dafa, rukaze obie�kaze obie�aze obie�qtis moZebna).


13

• orientirebs rukaze an sakoordinato sibrtyeze koordi-natebis gamoyenebiT (mag. asaxelebs mocemuli wertilis koordinatebis miaxloebiT an zust mniSvnelobas; pou-lobs wertils mocemuli mTelricxovani koordinatebis mixedviT).

monacemTa analizi, albaToba da statistika

1. moswavles aqvs elementaruli warmodgenebi informaciis mopovebis saSualebebis Sesaxeb.

Sedegi TvalsaCinoa, Tu:• ganasxvavebs Tvisebriv da raodenobriv monacemebs, iye-

nebs monacemTa Segrovebis Sesaferis saSualebas (gazomva, dakvirveba).

• qmnis martiv kiTxvars, gansazRvravs respodentebs, agro-vebs monacemebs.

• amokrebs saWiro monacemebs mowesrigebuli monacemebis Sesaferisi kategoriebidan (cxrili, monacemebis Semcveli teqsti).

2. moswavle flobs informaciis mowesrigebisa da warmodgenis xerxebs (saSualebebs).

Sedegi TvalsaCinoa, Tu:• sworad avsebs cxrils, sqemas, kiTxvars/anketas (magali-

Tad, Seaqvs monacemebi mza cxrilis Sesabamis ujrebSi).

• qmnis cxrils monacemis warmosadgenad (gansazRvravs Wdeebs, saTaurs, svetebisa da striqonebis raodenobas).

• agebs diagramas (svetovani, xazovani, SeiZleba wriulic). SeuZlia grafikulad warmodgenili informaciis wakiTxva.

• ajgufebs monacemebs aranakleb ori niSniT da xsnis dajgu-febis wess.

14


3. moswavles SeuZlia Tvisebriv da raodenobriv monacemTa in-terpretacia da elementaruli analizi.

Sedegi TvalsaCinoa, Tu:• iTvlis Semajamebel ricxviT maxasiaTeblebs (monacemTa sa-

Sualo, udidesi da umciresi mniSvnelobebi) diskretuli raodenobrivi monacemebisTvis da iyenebs maT monacemTa erTobliobis dasaxasiaTeblad;

• adarebs monacemTa ramdenime erTobliobas maTi winaswar mocemuli statistikuri maxasiaTeblebis saSualebiT;

• poulobs monacemTa erTobliobaSi arsebul kanonzomiere-bebs da msjelobs maTze.

1.3 maTematikuri wignierebis kursis Sinaarsiricxvebi da moqmedebebi; kanonzomierebebi da algebra1. ricxvis cneba. istoriuli cnobebi. ricxvebis gamoyeneba

(Wde, rigi, ...)2. naturaluri ricxvebi. aTobiTi poziciuri sistema. sxva ricxviTi sistemebis gacnoba (mag. anbani da asoebis

ricxviTi Sesatyvisebi; egvipturi – aTis xarisxebis gansx-vavebuli aRniSvnebi an romauli sistema).

3. ariTmetikuli moqmedebebi naturalur ricxvebze. ricx-vebis Sedareba da ariTmetikuli moqmedebebis Sedegis Se-faseba. ariTmetikuli moqmedebebis Tvisebebi: Sekrebisa da gamravlebis komutaciuroba (gadanacvlebadoba), asociaci-uroba (jufTebadoba) da Sekrebis mimarT gamravlebis dis-tribuciuloba (ganrigebadoba).

4. wiladebi, aTwiladebi da zogierTi kavSiri maT Soris. wiladebis, aTwiladebis Sedareba, dalageba da gamosaxva.

5. procentis cneba. kavSirebi erTi da imave mTelis procent-sa da nawils Soris.

6. proporcia. proporciis ucnobi elementis moZebna. ricxvis proporciul nawilebad dayofa (mag. 2 : 3 : 5).

7. teqsturi amocanebi, romlebic Sekrebis, gamoklebisa da gamravlebis Semcveli ricxviTi an erTi asoiTi aRniSvnis Semcveli algebruli gamosaxulebiT ixsneba.

8. fasdakleba/fasis gazrda: TanmimdevrobiTi da erTjeradi


15

fasdaklebebis/fasebis zrdis erTmaneTTan Sedareba da mar-tivi xarjTaRricxva.

9. simravleTa Teoriis umartivesi cnebebi da operaciebi.10. zomis erTeulebi, zomis erTeulebs Soris kavSirebi da

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


11. geometriuli figurebi (brtyeli, sivrculi). figuris el-ementebi da gamosaxva.

12. figuraTa klasifikacia sxvadasxva niSniT (mag. simetriuli da arasimetriuli, brtyeli da sivrculi).

13. figuris zomebi: sazomi xelsawyoebi da sigrZis sazomi erTeulebi. mravalkuTxedis perimetri. farTobis cneba. marTkuTxa paralelepipedis moculoba.

14. sivrculi figurebis modelebi, Slilebi.15. koordinatebi (araformalurad, rogorc adgilmdebareo-

bis miTiTeba simboloTa wyviliT). realur viTarebaSi obi-eqtTa urTierTganlagebis aRmweri sqemebi.

monacemTa analizi, albaToba da statistika16. zogadi warmodgenebi Tvisebrivi da raodenobrivi monace-

mebis saWiroebaze (daniSnulebaze) Segrovebis saSualebani: gazomva, dakvirveba, gamokiTxva; monacemTa amokreba mona-cemTa umartivesi wyaroebidan (mag. cnobaridan).

17. Tvisebrivi da raodenobrivi monacemebis organizacia: mo-nacemTa dajgufeba; raodenobriv monacemTa dalageba zr-dadoba-klebadobiT; Tvisebriv monacemTa dalageba (mag., leqsikografiuli meTodiT).

18. monacemTa warmodgenis saSualebani: cxrili, piqtograma; diagrama.

19. monacemTa Semajamebeli ricxviTi maxasiaTeblebi Tvise-brivi da raodenobrivi monacemebisTvis: centraluri ten-denciis sazomi – monacemTa saSualo; udidesi da umciresi mniSvnelobebi.

16


2. problemaze orientirebuli swavleba2.1 Sesavali

ganaTlebis specialistebis mTavari sazrunavia, rom mo-swavleebma SeZlon skolaSi SeZenili codnisa da unar-Cvevebis cxovrebiseuli problemebis gadasaWrelad gamoyeneba. swored amitomaa metad mniSvnelovani, rom skolam, sagnobriv cod-nasTan da unar-CvevebTan erTad, moswavleebs ganuviTaros problemebis gadaWris unari. es unari moswavles ara mxolod yoveldRiuri cxovrebiseuli sirTuleebis mogvarebaSi ga-moadgeba, aramed swavlis gasagrZeleblad da momaval profe-siul karieraSi warmatebis misaRwevadac.

problemis gadaWris unari arsebiTia mzrunvelobas mokle-buli bavSvebisaTvis. im mwiri faqtobrivi masalis (romelic SesaZloa maT swavlis raRac etapze aiTvises) gamoyenebis unari sasicocxlo mniSvnelobas iZens im xSiri SemTxvevebis dros, roca maT damoukideblad uwevT yofiT siZneleebTan gamklaveba, saarsebo pirobebis mopoveba.

problemaze orientirebuli swavleba aris maTematikisa da sabunebismetyvelo sagnebis swavlebis yvelaze gavrcelebuli modeli. igi moswavleze orientirebuli swavlebis strategiaa. aseTi tipis swavlebis dros moswavleebi realuri cxovrebi-seuli problemebis kvlevis gziT iZenen mniSvnelovan gamoc-dilebas, romelic maT mTeli cxovrebis manZilze gamoadge-baT.

problemaze orientirebuli swavlis dros axali codnis SeZena mimdinareobs problemebis ganxilvis konteqstSi. swav-lis amgvari forma gansxvavdeba amocanebis amoxsnis ubralo procesisagan. kerZod, problemaze orientirebuli swavlis dros axali codnis SeZena aqtualuri xdeba saWiroebis safuZ-velze, ukve dasmuli problemis mogvarebis procesSi. am proce-sis paralelurad ki viTardeba problemebis gadaWris unaric. marTlac, dasmuli problema ubiZgebs moswavleebs, moiZion da SeiZinon axali codna, romelic saWiroa problemis gadasaWre-lad. erTaderTi swori pasuxis moZebnis sanacvlod, moswavle axdens dasmuli problemis interpretirebas, agrovebs saWiro informacias, agebs models, gamoTqvams varauds, afasebs da


17

adarebs SesaZlo Sedegebs da akeTebs daskvnas. problemaze orientirebuli swavla gansakuTrebiT zrdis

moswavlis motivacias. moswavle aqtiur swavlaSia CarTuli, realuri problemebis gadaWraze muSaobs, ris Sedegadac swav-la misTvis ufro saintereso da saxaliso xdeba. amgvari swav-leba aviTarebs jgufuri muSaobis, TanamSromlobis, kriti-kuli azrovnebisa da komunikaciis unar-Cvevebs.

amgvarad, problemaze orientirebuli swavlis midgomas ori ZiriTadi niSani aqvs: • swavla eyrdnoba Riad dasmul problemur SekiTxvebs;

• maswavlebeli erTgvarad ”fasilitatoris” rolSia, ris Se-degadac moswavle iRebs met pirovnul pasuxismgeblobas sakuTar swavlaze da aqtiuri Semmecnebeli xdeba.

2.2 ra aris problemuri situacia/amocana?

gamoiyofa problemuri amocanebis ori ZiriTadi tipi: 1. amocanebi, romlebic dakavSirebulia gadawyvetilebis

miRebasTan. aseTi tipis davalebis nimuSia: moswavles surs biblioTekidan gamotanili wignebis

aslebis damzadeba. A punqtSi 1 gverdis aslis gadaReba 3 TeTri Rirs, xolo wignis ydis gakeTeba – 70 TeTri. B pun-qtSi 1 gverdis aslis gadaReba 5 TeTri Rirs, xolo wignis ydis gakeTeba – 40 TeTri. moswavlem unda gansazRvros, Tu romel punqtSia umjobesi wignebis aslis gakeTeba. ra rCevas miscemdiT moswavles?

2. amocanebi, romlebic moiTxovs sistemis analizs da mod-elirebas. aseTi tipis davalebis nimuSia:

gadawyda, rom unda aSendes saavadmyofo, romelic sam sofels moemsaxureba. Tqveni azriT, sad unda aSendes saa-vadmyofo? (ra rCevas miscemT mSeneblebs saavadmyofos mdebareobis Taobaze?)gadawyvetilebis miRebasTan dakavSirebul amocanebSi mo-

cemulia ramdenime alternativa da SezRudva. moswavles moeTxoveba iseTi gadawyvetilebis miReba, romelic am Sez-Rudvebs daakmayofilebs.

18


sistemis analizsa da modelirebasTan dakavSirebuli amocane-bi moswavlisagan moiTxovs kompleqsuri viTarebis analizs – am viTarebis logikis gasaazreblad. am tipis amocanebSi aRwerilia kompleqsuri sistemebi, romlebSic mravali urT-ierTdakavSirebuli faqtori monawileobs (parametri, Tvise-ba, maxasiaTebeli). am kavSirebis dinamikurobis gamo, pasuxi (amonaxsni), SesaZloa, ar iyos erTaderTi.

2.3 problemis gadaWris cikli

problemis gadaWris procesebs aRweren ciklis saSuale-biT, romlis erT-erT farTod gavrcelebul variants qvemoT moviyvanT. cikli aRwers TiToeul safexurs da gulisxmobs, rom Sedegis misaRwevad SesaZlebelia yovel etapze ramden-jerme dabruneba. amasTan, rogorc wesi, etapebis Tanmimdev-robas Tavad problemis tipi da problemis gadamWrelis pirovnuli Taviseburebebi ganapirobebs. problemis gadaWris cikli Semdegi etapebisgan Sedgeba: • problemis moZebna da gansazRvra;• problemis maTemartikur enaze Camoyalibebada warmodgena;• problemis Sesaxeb informaciisa da resuresebis organizeba

da codnis mobilizeba;• problemis gadaWris SesaZlo gzebis moZieba;• problemis gadaWris SesaZlo gzebis Sefaseba da maTgan

romelimes SerCeva;• SerCeuli gzis ganxorcieleba da progresze dakvirveba;• Sedegebis Semowmeba da Sefaseba realuri viTarebis konte-

qstSi.

problemis gadaWris meTodologia aris logika.• induqciuri msjeloba/argumentireba – kerZo msjelobidan

zogadis gamoyvana; • deduqciuri msjeloba/argumentireba – zogadi msjelobi-

dan kerZo msjelobis gamoyvana; • analogiiT msjeloba/argumentireba – erTi kerZo movleni-

dan analogiuri kerZo movlenis Sesaxeb daskvna.


19

2.4 konkretuli davalebebi.magaliTi 1.

rogori dizainis yuTia ufro mosaxerxebeli limonaTis Tunuqis qilebis gadasatanad?problemis gadaWris etapebi:1. problemis moZebna da gansazRvra: am etapze dadgindeba,

Tu ra niSniT ganisazRvreba “xelsayreli” yuTi (anu, ra igulisxmeba “xelsayrelSi”): mis gasakeTeblad daxarjuli qaRaldis raodenobiT, misi transportirebisaTvis mox-erxebuli formiT da a.S.

2. problemis Camoyalibeba da warmodgena: vTqvaT, ZiriTad parametrad avirCieT daxarjuli qaRaldis raodenoba. ma-Sin amocana, maTematikuri TvalsazrisiT, ase Camoyalibde-ba: ra zomebi unda hqondes yuTs (marTkuTxa paralelepi-peds), rom misi zedapiris farTobi (rac mis dasamzadeblad daxarjuli qaRaldis raodenobas ukavSirdeba) iyos umci-resi (minimaluri). amocanaSi aRwerili situaciis warmo-sadgenad, efeqturi iqneba magnituri dafis da Tanabari zomis monetebis (an magnitebis) gamoyeneba.

20


3. problemis Sesaxeb informaciisa da resursebis moZieba–organizeba, codnis mobilizeba: romeli ukve naswavli formula an debuleba SeiZleba gamoviyenoT problemis gadasaWrelad (magaliTad, marTkuTxa paralelepipedis zedapiris farTobis formula).

4. problemis gadaWris SesaZlo xerxebi: marTkuTxa paralel-epipedis formis yuTis garda, SesaZlebelia samkuTxa priz-mis formis yuTis gamoyenebac (unda ganisazRvros, qilebis ra raodenobisTvis aris SesaZlebeli am formis yuTis gam-oyeneba).

5. problemis gadaWris strategiis SerCeva da ganxorcieleba:

magaliTad, rogor SeiZleba daaxloebiT gamovTvaloT im samkuTxa prizmis fuZis farTobi, romelSic zustad 6 qila eteva.6. Sedegebis Semowmeba da Sefaseba: problemisa da miRebuli

amonaxsnis kvleva realuri viTarebis konteqstSi – dadge-na, Tu saSualod ra raodenobis qilas yidulobs myidveli erT jerze da am informaciaze dayrdnobiT yuTis formis SerCeva.

magaliTi 2.SearCieT moZraobis optimaluri marSruti skolis avtobu-

sisaTvis, romelsac qalaqis sxvadasxva ubnidan skolaSi mo-swavleebi da maswavleblebi mihyavs, problemis gadaWris etapebi:1. problemis moZebna da gansazRvra: am etapze dadgindeba,

Tu ra niSniT ganisazRvreba “optimaluri” marSruti (anu, ra igulisxmeba “optimalurSi”): umoklesi manZili, nakleb datvirTuli da usafrTxo gza, umciresi dro am marSrutis


21

dasafarad da a.S.2. problemis Camoyalibeba da warmodgena: vTqvaT, ZiriTad

parametrad am marSrutis dasafarad saWiro minimaluri dro avirCieT. maSin amocana, maTematikuri TvalsazrisiT, amgvarad Camoyalibdeba: SeirCes iseTi marSruti, rom misi sigrZisa da mgzavrobis saSualo siCqaris Sefardeba iyos umciresi.

3. problemis Sesaxeb informaciisa da resursebis moZieba–or-ganizeba, codnis mobilizeba: moswavleTa da maswavlebelTa gamokiTxvis saSualebiT unda dadgindes, vin ra transpor-tiT sargeblobs skolaSi mosvlisas. unda Seiqmnas martivi kiTxvari, SeirCnen respodentebi, Catardes gamokiTxva da monacemebi warmodgenili unda iqnas cxrilis an grafikuli saxiT.

4. problemis gadaWris SesaZlo xerxebi: am monacemebis sa-am monacemebis sa-fuZvelze moswavleebma unda daadginon is ubnebi, sadac moswavleebisa da maswavleblebis mniSvnelovani raodenoba cxovrobs. uTuod SeamCnevT, rom xelaxla dgeba sakiTxi, ras ewodeba “mniSvnelovani raodenoba”. anu, problema Txoulobs ciklis pirveli etapisken mibrunebas da cne-bebis xelaxla dazustebas.

5. problemis gadaWris strategiis SerCeva da ganxorcieleba: unda moxdes problemis vizualizacia – amocanis sqematu-rad gamosaxva.

A

B

C

D

E

skola - O

22


rukaze unda aRiniSnos skolis avtobusis gaCerebis sa-kaze unda aRiniSnos skolis avtobusis gaCerebis sa-aze unda aRiniSnos skolis avtobusis gaCerebis sa-unda aRiniSnos skolis avtobusis gaCerebis sa- skolis avtobusis gaCerebis sa-varaudo punqtebi, am punqtebis erTmaneTTan damakavSirebeli gzebi.

rukaze miTiTebuli masStabis gamoyenebiT unda dadgindes punqtebs Soris manZilebi (daaxloebiT) da am manZilebis dasa- manZilebi (daaxloebiT) da am manZilebis dasa-manZilebi (daaxloebiT) da am manZilebis dasa-i (daaxloebiT) da am manZilebis dasa- am manZilebis dasa-farad saWiro dro. moswavleebma yvela marSrutis savaraudo sigZe unda daTvalon da es monacemebi cxrilis saxiT war-unda daTvalon da es monacemebi cxrilis saxiT war-da es monacemebi cxrilis saxiT war-es monacemebi cxrilis saxiT war-cxrilis saxiT war- war-moadginon.

marSruti1 OA AB BC CD DE EOmarSruti2 OA CO OD DE EOmarSruti3

Sedegebis Semowmeba da Sefaseba: problemisa da miRebuli amonaxsnis kvleva realuri viTarebis konteqstSi – moswav-leebma unda Seajeron moZiebuli informacia da saukeTeso marSruti airCion. Semdeg unda imsjelon arCeuli gzis opti- arCeuli gzis opti- opti-opti-malurobis Sesaxeb – maTematikuri da eTikuri TvalsazrisiT (mizanSewonlia Tu ara im punqtis gavliT mgzavroba, sadac erTi maswavlebeli, an moswavleTa/maswavlebleTa mcireric-xovani jgufi cxovrobs?). SeiZleba, SeTavazebul iqnas aseTi tipis balansis dacvis gzebi (mag. maswavleblis xelfasSi mg- mg-mg-zavrobis xarjebis gaTvaliswineba J).

3. kiTxvis strategiebis gamoyeneba

maTematikis daufleba SeuZlebelia mSobliuri enis Rr-mad da gaazrebulad flobis gareSe. mniSvnelovan sirTules maTematikis swavlebis procesSi warmoadgens is faqti, rom moswavleTa did nawils uWirs wakiTxuli masalis _ teqstis Sinaarsis gageba-gaazreba. Sesaswavli teqstebi xom infor-maciuli xasiaTisaa, xSirad gadatvirTuli cnebebiTa da ter-minebiT, romelic Seicavs sakmaod rTul argumentirebul msjelobas. araerTi kvleva adasturebs, rom maTematikaSi moswavleTa warumeteblobas mniSvnelovanwilad ganapirobebs


23

ara imdenad amocanis amosaxsnelad saWiro procedurebis ar-codna, aramed is, rom moswavle ver igebs, ras ekiTxebian, ra davaleba aqvs mas.

Tanamedrove saganmanaTleblo meTodologia gulisxmobs wignierebis (Rrmad da gaazrebulad kiTxvis) strategiebis aq-tiur swavleba-gamoyenebas ara mxolod mSobliuri enis, ar-amed sxva sagnebis gakveTilebzec. wignierebis ZiriTad star-egiebad aris miCneuli:

• prognozireba, hipoTezebisa da varaudebis gamoTqma. es strategia gulisxmobs varaudebisa da mosazrebebis gamoTqmas imis Taobaze, Tu ras SeiZleba exebodes teqsti da rogor SeiZleba ganviTardes teqstis calkeul nawil-Si aRwerili viTareba. maTematikis gakveTilze es SeiZle-ba iqnas gamoyenebuli, magaliTad, axali masalis axsnisas. maswavlebelma SeiZleba gamoaTqmevinos mosazrebebi mo-swavleebs, Tu ras SeiZleba exebodes maTi varaudiT para-grafi, ra mniSvnelovani sakiTxis Seswavlas moelodebian isini saTauris Sesabamisi Temis irgvliv.

• kavSirebis damyareba _ es aris gansaxilavi masalis dakav-Sireba moswavlis pirad gamocdilebasTan an adre Seswav-lil masalasTan.

• SekiTxvebis dasma: teqstis kiTxvis yovel etapze SekiTx-vebis dasma _ ra aris mocemuli, ras veZebT, ras gvekiTxeba amocana, raTa ukeT iqnas gagebuli teqstis Sinaarsi.

• Sejameba, daskvnebis gakeTeba, msjeloba: yoveli abzacis Semdeg mokle Semajamebeli daskvnis gakeTeba.

• vizualizacia: damxmare sqemebisa da naxazebis ageba.• mniSvnelovani da meorexarisxovani faqtebis gansazRvra _

risi codnaa aucilebeli amocanis amosaxsnelad, ra aris nakleb mniSvnelovani am etapze, ra ar gamogvadgeba.

• sinTezi gulisxmobs teqstis moswavliseuli interpreta-ciiT moTxrobas, amoxsnis gansxvavebuli xerxebis moZiebas, miRebuli daskvenebis ganzogadebas da maTi sxva sferoebSi gamoyenebis SesaZleblobis SemCnevas.

24


maswavleblis funqciaa sworad dasmuli SekiTxvebis an damx-mare sqemebis meSveobiT miexmaros moswavles teqstis gaazre-baSi. amasTan, es ar unda iyos “erTjeradi” daxmareba. moswav-les unda vaswavloT teqstis damuSavebis xerxebi.

ganvixiloT ramdenime davaleba:

• amocana satelefono kompaniis momsaxurebis Sesaxeb.

yovelTviurad, mobiluri telefoniT sargeblobis pirve-li 200 wuTi ninos eZleva ufasod. yoveli momdevno wuTisT-vis igi ixdis 5 TeTrs. ianvarSi ninos satelefono saubrebis xangrZlivoba iyo 230 wuTi, TebervalSi ki – 300 wuTi. sul ra Tanxis gadaxda mouwevs ninos satelefono kompaniisaTvis?

davalebis teqsti informaciulia da SesaZloa moswavle-Ta nawils gauWirdes davalebaSi aRweril viTarebaSi bolomde garkveva. aseT dros SesaZloa dagvexmaros maorientirebeli (mimgezavi) SekiTxvebi:6. razea amocana? 7. ra pirobebi aqvs satelefono kompanias? 8. ramdeni wuTi saubaria ufaso? 9. rogor gesmiT “yoveli momdevni wuTi”? 10. 230 wuTidan ramdeni iyo ufaso da ramdeni wuTisaTvis

mouwevda gadaxda? 11. raze mogvaniSnebs sityva “sul”?

Zalian mniSvnelovania cxrilebisa da sqemebis gamoyeneba, romlebic iZleva monacemebis struqturirebisa da TvalsaCi-nod warmodgenis saSualebas. mag., zemoT ganxiluli amocani-saTvis cxrils SeiZleba hqondes aseTi saxe:

Tve wuTebis raodenoba ufasofa-siani

ianvari 230 wuTi 200 wuTi 30 wuTi

Tebervali 300 wuTi 200 wuTi 100 wuTi


25

• sagarantio momsaxureoba

kompania myidvels, nivTis SeZenisas sTavazobs sagarantio momsaxurebas, romelic gulisxmobs SeZenili nivTis axliT Secvlas, Tu nivTi dazianda 1 wlis ganmavlobaSi. rom 1 wlis ganmavlobaSi kompaniam gayida 300 erTeuli TiToeuli 180 la-rad. cnobilia, rom aqedan dazianda 6%. risi tolia kompaniis Semosavali?

amocanis teqsti sakmaod rTuli da informaciulia. Sinaar-sis ukeT gasaazreblad maswavlebelma unda ikiTxos, smeniaT Tu ara moswavleebs rame sagarantio momsaxurebis Sesaxeb, Zi-riTadad ra saxis sagnebze moqmedebs xolme amgvari wesebi, xom ar SeuZeniaT ojaxSi rame nivTi, romelsac hqonda sagarantio vada da a.S.Semdeg unda ganixilon, Tu rogor unda moxdes kompaniis Se-mosavlis gamoTvla. aq kargi iqneba sqemis gamoyeneba:

gayiduli nivTebis saerTo raodenoba

maTgan xarisx-iani

uxarisxo

300 mTliani raodenobis 6%.

kompaniis Semosavali zarali

saWiro gamoTvlebi SeiZleba Sesruldes cxrilis Sesabamis svetSi.

26


• mogzaurobis dagegmva

vTqvaT Tqven gegmavT 12 dRian mogzaurobas avtomobiliT. dReSi apirebT saSualod 60 kilometris gavlas. informaciis Segrovebisa da damuSavebis Sedegad SearCieT 3 kompania, rom-lebic avtomobilebs aqiraveben Semdegi pirobebiT: pirveli kompania iTxovs dRiur gadasaxads 33 lars, romelsac emateba 10 TeTri yovel gavlil kilometrze; meore kompaniaSi dRiu-ri gadasaxadia 50 lari, xolo yovel gavlil kilometrze ga-dasaxadi Seadgens 4 TeTrs; mesame kompanias aqvs fiqsirebuli yoveldRiuri gadasaxadi 60 lari gavlili manZilis miuxeda-vad.

romel kompanias airCevT samogzaurod? am SemTxvevaSic sasurvelia amoxsnis procesi daviwyoT

saubriT – smeniaT Ta ara rame kompaniebis (turistuli sa-agentoebis) Sesaxeb, romlebic aqiraveben manqanebs? ramdenad mniSvnelovnad miaCniaT mogzaurobis dagegmva da ras gulisx-moben “dagegmvaSi”?

Semdeg etapze unda moxdes teqstSi mocemuli informaciis vizualizacia. am mizniT SeiZleba Sedges Semdegi cxrili:

kompania gadasaxadi

dRiuri yovel gavlil km-ze

I 33 lari 10 TeTri

II 50 lari 4 TeTri

III 60 lari 0

Semdegi etapia ariTmetikuli gamoTvlebis ganxorcieleba da imis dadgena, Tu ra Tanxis gadaxda mouwevT TiToeuli kompaniis aRcevis SemTxvevaSi.

Semdegi sadiskusio Tema SeiZleba iyos sakiTxi – mxolod fasi asrulebs mniSvnelovan rols kompaniis aRcevaSi, Tu sxva parametrebic mniSvnelovania swori gadawyvetilebis misaRe-bad? (igulisxmeba kompaniis avtoriteti, gamarTuli avtomo-bilebi da a.S.).


27

4. Sefaseba maTematikur wignierebaSi

Sefaseba swavlebis procesis ganuyofeli nawilia. is Tanabrad mniSvnelovania swavlebis procesSi monawile yvela mxarisaTvis. moswavle yovelTvis unda flobdes informacias, ramdenad warmatebulia saganSi; ra unda Secvalos, rom uke-Tesi Sedegi miiRos. maswavlebeli Sefasebis Sedegebis gaTval-iswinebiT unda gegmavdes yovel Semdeg nabijs.

maTematikaSi Sefasebis komponentebi (anu, dakvirvebis obi-eqti) aris:1. maTematikuri cnebebisa da debulebebis codna da gamoye-codna da gamoye-gamoye-

neba; 2. maTematikuri obieqtebis warmodgena (mag., daxazva, kon- (mag., daxazva, kon-

struireba);3. maTematikuri enis floba (mag., terminologiis da aRniS- (mag., terminologiis da aRniS-

vnebis koreqtulad gamoyeneba, maTematikuri Sinaarsis wi-nadadebebis gageba da Camoyalibeba);

4. msjeloba - dasabuTeba;5. gamoTvla, gazomva da sxva standartuli procedurebi;6. damxmare teqnikuri saSualebebisa da sainformacio teqno-

logiebis gamoyeneba.socialuri unar-Cvevebidan yuradReba unda gamaxvildes

SemoqmedebiTobaze da TanamSromlobaze (mewyvilesTan, jgu- (mewyvilesTan, jgu-fis wevrebTan).

Sefasebis komponentebis, rogorc unar–Cvevebisa da codnis flobis, xarisxis dadgena SeiZleba Semdegi saxis aqtivobebiT:

moswavleze yoveldRiuri dakvirvebis Sedegad unda dadg-indes: • aRiqvams Tu ara moswavle amocanis Sinaarss, SeuZlia Tu

ara amocanis monacemebisa da saZiebeli sidideebis gamijvna, axerxebs Tu ara monacemebis organizebas da warmodgenas.

• sworad iyenebs Tu ara maTematikur terminebsa da aRniS-vnebs.

• SeuZlia Tu ara kompleqsuri davalebis dayofa safex-urebad – martiv amocanebad.

• SeuZlia Tu ara ZiriTadi maTematikuri procedurebis gam-oyeneba (mag., ariTmetikuli operaciebis Sesruleba).

28


Sefasebisas SeiZleba gamoyenebul iqnas sxvadasxva saxis dava-lebebi, magaliTad:

• amocanebi, sadac moswavles evaleba ramdenime SesaZlo pa-suxs Soris swori pasuxis SerCeva, Sesabamisobis damyareba, swori TanmimdevrobiT dalageba.

• teqstis wakiTxva da monacemTa analiziT (gamoTvlebis an logikuri msjelobis safuZvelze) miRebuli daskvnis gad-mocema da dasabuTeba (maT Soris, iseTi teqstis, romelic diagramebs da cxrilebs Seicavs).

• geometriuli amocana, romelSic moswavles moeTxoveba figuris Tvisebebis dadgena, zomebis gansazRvra, figuris ageba.

Sefasebis rubrikis nimuSi:

flobis saSualoze dabali done

flobis saSualo done

flobis maRali done

xSirad araadek-araadek-vaturad iyenebs cnebebs, terminebs, aRniSvnebs / aras-rulyofilad, araorganizebulad warmoadgens mona-cemebs.

iyenebs amocanis amosaxsnelad saWiro cnebebis, debulebe-bisa da aRniSvnebis umetesobas / axdens monacemTa organize-bas da warmoadgens maT.

iyenebs amocanis amosaxsnelad saWiro yvela cnebas, debu-lebas da aRniSvnas / irCevs monacemTa organizebisa da warmodgenis opti-malur formas.

ver poulobs amosx-nis romelime gzas. an poulobs amoxs-poulobs amoxs-nis romelime gzas, iwyebs, magram ver asrulebs amoxs-nas; ver poulobs logikur kavSirebs, uWirs zogadidan kerZos gamoyofa.

poulobs amoca-nis amoxsnis gzas, mihyveba mas, Tumca, xarvezebiT.

poulobs amocanis amoxsnis (optim-alur) gzas, bo-lomde axdens mis realizebas.

gamoTvlebi ar adasturebs amocanis gagebas, Sedegi ara-adekvaturia; an ume-tesi nawili xarveze-biTaa Sesrulebuli.

gamoTvlebi ZiriTa-dad sworia, Tumca, teqnikuri xasiaTis xarvezebs Seicavs.

gamoTvlebi sworia, arCeuli da gamoy-enebuli aqvs gamoT-vlebis optimaluri xerxi.


29

5. gakveTilis gegmis nimuSebi1) gakveTilis gegma – procenti

gakveTilis mTavari Tema procenti

gakveTilis mniSvneloba, aqtualuroba

yoveldRiur realobaSi farTod gamoi-yeneba procentis cneba: sakvebisa Tu wamlis yuTebze, maRaziebis vitrinebze yoveli Cvengani xvdeba procentis aRniSvnas. sazogadoebaSi srulfa-sovnad funqcionireba SeuZlebelia procentTan dakavSirebuli umartivesi procedurebis flobis gareSe.

gakveTilis miznebi da Sedegebi

Sedegi: moswavles SeuZlia yoveldRi-uri cxovrebidan momdinare amocanebis amoxsnamizani: moswavle gaerkves procentis cnebaSi, gaigos misi saWiroeba, ricxvis gamosaxvis sxvadasxva formas Soris mocemuli viTarebisaTvis adekvaturis SerCeva, sxvadasxva formiT gamosaxuli ricxvebis Sedareba da Sefaseba.

winaswari codna ricxvis wiladi nawilis povna, wiladis cnebis codna, sidideebs Soris pirda-pirproporciuli damokidebulebis amocnoba.

gakveTilis msvleloba 1. moTelva, klasis mobilizeba:moswavleebs magaliTebis saSualebiT gavaxsenoT ricxvis nawilis arsi da misi povnis xerxi (mag. risi tolia 25 – is 3/5 nawili, 120 – is 7/10 da a.S). SesaZlebelia gakeTdes erTi–ori umartivesi amocana Temaze: Rirda m lari, gaZvirda Rirebulebis p/q nawil-iT, ra Rirs axla? iwonida m kilo-grams, moaklda wonis p/q nawili, ras iwonis axla? igulisxmeba ara amgvari zogadi saxiT miwodebuli amocana, aramed amocana p, q da m-is konkre-tuli ricxviTi mniSvnelobebisaTvis (5 wuTi).

30


2. samotivacio amocana, problemis formulireba:

ganvixiloT amocana: supermarket “meg-alainis” (an, “pasaJis”) samma maRaziam saSobaod saqonlis gaiafeba gadawyvi-ta. warmovidginoT, rom vitrinebze gamotanili informacia gaiafebis Sesaxeb am saxisaa: “saqoneli iafdeba Rirebulebis 1/9 nawiliT” (I maRa-zia), “saqoneli iafdeba Rirebulebis 2/17 nawiliT” (II maRazia), “saqoneli iafdeba Rirebulebis 9/100 nawiliT” (III maRazia). romel maRaziaSi ufro momgebiania saqonlis SeZena?savaraudod, moswavleebi daiwyeben msjelobas imis Sesaxeb, rom am kiTx-vaze pasuxis gasacemad saWiroa ama Tu im saqonlis sawyisi fasis codna. Tumca, sabolood SeTanxmdebian, rom saWiroa wiladebis Sedareba (10 wuTi).

3. codnis konstruireba:amocanis amoxsnis Semdeg davsvaT SekiTxva: ra sirTuleebs waawydeba myidveli, Tu aseTi formiT gamoaqvey-nebs gaiafebis pirobebs supermarketis yvela maRazia? vTxovoT moswavleebs, moifiqron raime xerxi, rac myidvels msgavs situaciebSi gadawyvetilebis miRebas Seumsubuqebs.wiladebis Sedarebis dros garkveuli sirTuleebi warmoiqmneba imasTan dakavSirebiT, rom es wiladebi sx-vadasxvamniSvneliania. amitom, myidv-els rom saqme gavuadviloT, sasurve-lia gaiafebis maCvenebeli wiladebi erTnairmniSvneliani iyos. msjeloba SeiZleba gagrZeldes manam, vidre mo-swavleebi ar mivlen erT daskvnamde: sasurvelia, arCeul iqnas iseTi ricxvi, romelic sayovelTao standarti iqneba amgvari saxis raodenobrivi infor-maciis gamoqveynebisas. amis Semdeg, bunebrivad SeiZleba procentis cnebis Semotana.


31

mokled gavacnoT moswavleebs pro-centis da misi gamomsaxveli simbolos warmoSobis istoria, ganvumartoT sityva “procentis” etimologia.vTxovoT moswavleebs, gaixsenon, sad SexvedriaT es simbolo. vaCvenoT raime, razec Semadgenel nawilebs Soris Tanafardoba procentuli saxiT aris aRniSnuli (mag. wamlis yuTi an Sokoladis fila) da procentTan dakavSirebuli warwerebis ganmarteba vTxovoT (20 wuTi).

4. gamTlianeba, refleqsia: vavarjiSoT moswavleebi amgvari dava-lebebis SesrulebaSi: wiladebisa da aTwiladebis gamosaxva procentis saxiT (martivi magaliTebi), procentis gamosaxva wiladis saxiT, kvadratis gaferadeba miTiTebuli procentiT (mag., gaaferadeT kvadratis 75%), miTiTebuli procentiT gaferadebuli figurebis amocnoba (mag., romeli figuraa gaferadebuli 20%–iT?) (10 wuTi).

Sefasebis sagani da pro-cedurebi

yuradReba unda mieqces, gaigo Tu ara moswavlem procentis cneba, esmis Tu ara procentis daniSnuleba, SeuZlia Tu ara erTmaneTs daukavSiros wiladi, aTwiladi da procenti. Sefasebis forma – ganmaviTarebeli.

saWiro resursebi saxelmZRvanelo, wamlis an raime sakvebis yuTi (romelzedac procen-tuladaa gamosaxuli ingredientebis xvedriTi wili).

32


2) gakveTilis gegma – koordinatebi

gakveTilis mTavari Tema

koordinatebi


sivrceSi da sibrtyeze ori-entireba adamianisaTvis mniS-vnelovani unaria. metros sadgurebSi, qalaqis quCebSi Tavisuflad samoZraod saWiroa martivi sqemebis gaSifvra, rac gulisxmobs obieqtis mdebareobis povnas koordinatebis mixedviT, an obieqtis mdebareobis aRweras koordinatebis meSveobiT.


Sedegi: moswavles SeuZlia orien-tireba sibrtyeze.

mizani: moswavlem unda SeZlos garemoSi orientireba da obi-eqtTa urTierTganlagebis aRwera, orientireba ruqaze an sakoor-dinato sibrtyeze koordinatebis gamoyenebiT.

winaswari codna moswavlem unda icodes mimar-Tulebis gamomxatveli sityvebi: marjvniv, marcxniv, zemoT, qve-moT, ....

gakveTilis msvleloba 1. moTelva, klasis mobilizeba:

maswavlebelma moswavleebs unda sTxovos, aRweron saklaso oTaxi mimarTulebis gamomxatveli sityvebis gamoyenebiT: mag., kare-bidan marjvniv kedelze dafaa, dafidan yvelaze daSorebuli fanjris rafaze qoTniT dgas yvavili da a.S. SeiZleba aRweron skolis ezo da Semogareni (5 wuTi).


33

2. samotivacio amocana, proble�mis formulireba:

maswavlebelma unda warmarTos amgvari Sinaarsis diskusia: sakma-risi iqneboda Tu ara TiToeuli Cvenganis identifikaciisaTvis mx-olod saxeli (anu, ra moxdeboda, gvari rom ar gvqonoda)? yovelT-vis erT adamians Seesabameba Tu ara saxelisa da gvaris konkretu-li kombinacia? kidev riTi mianiS-neben xolme romelime konkretul adamianze? (mag. dabadebis adgi-lis miTiTebiT – Tales mileTe-li, maia wyneTeliJ).

Semdeg maswavlebelma unda sTxo-vos moswavleebs, obieqtebis mde-bareobis aRweris Sesaxeb gamoTq-van mosazreba: ramdenad xSirad aris amis saWiroeba? TviTon Tu dasWirvebiaT raime adgilis mi-gneba, an Tavad Tu miuswavlebiaT vinmesTvis raime adgili?

diskusia imgvarad unda warimar-Tos, rom aSkarad gamoikveTos koordinatebis aucilebloba (10 wuTi).

3. codnis konstruireba:

maswavlebeli auxsnis moswavlee-bs, rom, rogorc wesi, sibrtyeze wertilis mdebareobis gansasaz-Rvrad ori ricxvia sakmarisi (cxadia, igulisxmeba, rom sibrtye badiTaa dafaruli).

34


eubneba, rom, zogjer, ricxvebis nacvlad laTinur asoebsac iy-eneben. moswavleebma unda daasax-elon koordinatebis gamoyenebis nimuSebi: adgili kinoTeatrSi, Wadraki. maswavlebelma moswav-leebs unda aCvenos ruqa rome-lime atlasidan da moaxdinos imis demonstrireba, rom cnobili koordinatebis pirobebSi nebi-smieri, Tundac, Zalian patara dasaxlebuli punqtis zusti mde-bareobis gansazRvraa SesaZle-beli (15 wuTi).

4. gamTlianeba, refleqsia:

unda Sesruldes saxelmZR-vanelos Sesabamis paragrafSi mocemuli amgvari davalebebi: moswavlem unda daasaxelos, romeli obieti mdebareobs koor-dinatebiT miTiTebul wertilSi, aRniSnos badeze miTiTebuli koordinatebis mqone wertili, SesaZlebelia, rom koordi-natebis mixedviT badeze gamosax-on cxovelebi – spilo, kurdReli (15 wuTi).

klasgareSe aqtivobisaTvis Ses-aZlebelia “gemebis CaZirobanas” TamaSis swavlebac (am TamaSis eleqtronuli versiebi internet-Sia ganTavsebuli).


35


yuradReba unda mieqces, gaigo Tu ara moswavlem koordinatis cneba, esmis Tu ara koordi-natebis daniSnuleba, SeuZlia Tu ara megobars aswavlos obieqtis moZebna koordinatebis mixedviT, SeuZlia Tu ara vinmes mianiSnos obieqtis mdebareobaze koordi-natebis meSveobiT.Sefasebis forma – ganmaviTare-beli.

saWiro resursebi saxelmZRvanelo, kinoTeatris bileTi, Wadrakis dafa, geografi-uli atlasi, ruka, ujrebiani furclebi.

36


3)gakveTilis gegma – simetria


simetria


Cvens garemomcvel samyaroSi uamravi obieqtis Tvisebaa sime-triuloba. obieqtis Tvisebebis Seswavla SeuZlebelia simetriaze garkveuli warmodgenebis gareSe.


Sedegi: moswavle garemomcvel obieqtebs Soris simetriul obieqtebs moiZiebs; ganasxvavebs simetriul da arasimetriul figurebs; saTanado SemTxvevaSi axdens figuraTa simetriulobis demonstirebas.

mizani: moswavlem SeZlos figu-raTa simetriulobis gamoyeneba realur viTarebaSi (mag., parketis dizainis Seqmna).

winaswari codna elementaruli brtyeli fig-urebis saxeldeba da umartivesi Tvisebebis codna.


37

gakveTilis msvleloba 1. moTelva, klasis mobilizeba:

maswavlebelma unda sTxovos moswavleebs, brtyeli geometri-uli figurebi daasaxelon. dafaze dasaxelebuli figurebi unda daiweros da TiToeuli figuris eskizi gakeTdes. maswavlebeli unda Seecados, rom dasaxelebuli figurebis nusxa, ramdenadac SesaZlebelia, mravalferovani iyos (mag., mravalkuTxedebi, wrewiri, Ria da gaxsnili tex-ilebi da mrudebi, amozneqili da araamozneqili figurebi). Semdeg moswavleebi unda Seecadon, rom es figurebi raime niSniT daax-arisxon – mravalwaxnagebi da tex-ilebi erTad, wrewiri da mrudi wirebi calke da a.S. (5 wuTi).

2. samotivacio amocana, problemis formulireba:

maswavlebelma unda Seajamos wina aqtivoba da xazi gausvas, rom figurebi dajgufda ZiriTadad erTi niSniT – mravalkuTxedebi da “aramravalkuTxedebi”. Semdeg unda daisvas SekiTxva, kidev rame niSani xom ar ganasxvavebs erT-maneTisagan sxvadasxva figuras? maswavlebelma SeiZleba moswav-leebs iatakis dafarvis ori nimuSi aCvenos (kafeli, metlaxi, xaliCa):

38


maswavlebelma moswavleebs unda sTxovos, es nimuSebi erTmaneTs Seadaron ara feriT, aramed geo-metriuli atributebiT (ar aris aucilebeli dakonkretdes, Tu ra igulisxmeba geometriul atrib-utebSi). maswavlebelma gezis mim-cemi SekiTxvebis meSveobiT unda miiyvanos moswavleebi simetri-ulobis cnebamde (10 wuTi).


39


saintereso iqneba moswavleTaT-vis slaidSous meSveobiT sime-triulobis nimuSebis Cveneba yoveldRiuri realobidan: pep-lebi, foTlebi, arqiteqturuli nagebobebi, parketis mozaika da sxva. Semdeg yuradReba unda gamaxvildes simetriis yvelaze gavrcelebul formaze – RerZul simetriaze. unda moxdes figuris simetriulobis demonstrireba – figuris nawilebi rom erTma-neTs SeuTavsdeba, Tuki mas Rer-Zis gaswvriv gadavkecavT. metad efeqturi iqneba kompiuteruli TamaSebis Cveneba. saintereso TamaSia ganTavsebuli saitze http://buki.ge/ (TamaSis saxelwode-baa “xelovneba”). moswavlem unda aagos RerZulad simetriuli kon-struqcia, winaaRmdeg SemTxvevaSi, konstruqciis mdgradoba irRveva. es TamaSi moswavleebs RerZuli simetriis arsis ukeT garkvevaSi miexmareba.

40


arsebobs sxva kompiuteruli pro-gramac, romelic moswavles gardaqm-nebis gamoyenebiT sibrtyis dasafari ornamentebis (e.w. parketis dizainis) SeqmnaSi daexmareba. es programa gan-Tavsebulia eleqtronul misamarTze:

http://illuminations.nctm.org/ActivityDetail.aspx?ID=205.

programiT sargebloba martivia: moswavlem unda airCios mraval-kuTxedis saxe, romliTac surs parketis dizainis Seqmna. sain-tereso iqneba Seqmnili dafarvebis jgufurad ganxilva. kargia msj-eloba, Tu simetriis ramdeni RerZi aqvs wresa da sferos, kvadratsa da kubs, farniT gaSuqebul ko-nuss, egviptur oTxkuTxa pirami-das, futkris fiWas an krazanis budes, abreSumis parks, frinvelis kvercxs, pepelas da sxvas mravals. simetriulia Tu ara adamianis saxe da sxeuli, mcenaris foToli, skolis Senoba. da, saerTod, arse-bobs Tu ara bunebaSi xelTuqmneli idealuri simetria. imazec kargia saubari, Tu ratom „uyvars“ bunebas simetria, ratom aSenebs futkari fiWas eqvskuTxa da ara samkuTxa prizmebiT an wesieri marTkuTxa paralelepipedebiT (25 wuTi).


41

4. gamTlianeba, refleqsia:

moswavleebi unda vavarjiSoT simetriuli da arasimetriuli figurebis Tu konstruqciebis amocnobaSi. aseve sainteresoa da ganmaviTarebeli funqciis matare-beli aseTi davalebebi: maswavle-beli warmoadgens (aCvenebs)arasi-metriuli figuris, ujredebian furcelze Sesrulebul naxazs da moswavleebs sTxovs, Seavson is simetriulamde umciresi raode-nobis ujrebis gaferadebiT. mas-wavlebelma Tavidan unda miuTi-Tos simetriis RerZi, Semdeg ki moswavleebs arCevanis gakeTebis Tavisufleba misces (5 wuTi).


yuradRebis centrSi unda iyos, gaigo Tu ara moswavlem simetriis cneba, SeuZlia Tu ara mas ganasx-vavos simetriuli da arasimetri-uli obieqtebi, SeuZlia Tu ara simetriuli konstruqciis ageba.

Sefasebis forma – ganmaviTarebe-li.

saWiro resursebi saxelmZRvanelo, internetSi Car-Tuli kompiuteri, proeqtori.

42


4) gakveTilis gegma-cxrili


cxrili

gakveTilis mniSvn-eloba, aqtualuroba

cxrili informaciis struqturize-bis saukeTeso saSualebaa. infor-maciis didi nawili - matareblebis moZraobis ganrigi, romelime eqimis samuSao grafiki Tu sxva -swored cxrilis saxiT mogvewodeba. Sesabam-isad, cxrilidan saWiro informaciis amokreba mniSvnelovani kompetenciaa, romelsac TiToeuli moswavle unda flobdes.


Sedegi: moswavle flobs informaciis mowesrigebisa da warmodgenis xerxebs (saSualebebs).

mizani: moswavlem SeZlos cxrilidan saWiro monacemebis amokreba.

sworad Seavsos cxrili (sworad Sei-tanos monacemebi mza cxrilis Sesa-bamis ujrebSi).

SeZlos cxrilis Seqmna monacemis warmosadgenad Wdeebis, saTauris, svetebisa da striqonebis raodenobis gansazRvra.

moswavles hqondes elementaruli warmodgenebi eleqtronul cxrile-bze da SeeZlos umartivesi opera-ciebis Sesruleba maTi meSveobiT.

winaswari codna moswavlem unda icodes mimarTulebis gamomxatveli sityvebi: marjvniv, marcxniv, zemoT, qvemoT... unda See-Zlos Sesabamisi striqonisa da svetis TanakveTaze mdebare ujris miTiTeba.


43

gakveTilis msvl-eloba

1. moTelva, klasis mobilizeba:

maswavlebelma moswavleebi or jgu-fad unda dayos. orive jgufs erTi da igive davaleba eqneba - maT xelT arsebuli teqstidan matareblebis moZraobis ganrigis Sesaxeb amoi-kiTxon da daadginon, romel saaTze gadis Tbilisidan matarebeli, magal-iTad, ozurgeTis mimarTulebiT. erT jgufs masala unda miewodos cxri-lis saxiT, meores ki - teqstis saxiT (narativis saxiT). savaraudod, meore jgufi ufro swrafad gaarTmevs Tavs davalebas (10 wuTi).


maswavlebelma amgvari Sinaarsis dis-kusia unda warmarTos: ram ganapiroba meore jgufis warmateba? aqvs Tu ara raime saxis upiratesoba infor-maciis cxrilis saxiT warmodgenas? Tu unaxavT sadme cxrilebi da ra saxis informacia iyo maTi meSveobiT gadmocemuli? diskusiis mizania, rom moswavleebi mixvdnen informaciis struqturizebis aucileblobas. Sem-deg maswavlebelma moswavleebs unda aCvenos Tbilisis rkinigzis sadguris ganrigi (realuri,internetSi ganTavse-buli), romelime wamlis instruqcia (sadac cxrilis saxiTaa xSirad war-modgenili, ra asakis adamianma ra doziT unda miiRos aRniSnuli wamali), cxrilebis gamoyenebis sxva realuri nimuSebi. pirvel etapze moswavlee-bma cxrilidan saWiro informaciis amokrebaze unda ivarjiSon. yuradReba unda gamaxvildes imaze, rom moswavlem zustad miuTiTos saWiro striqoni, sveti, ujra (15 wuTi).

44


eleqtronuli cxrilebis gamoyeneba:

maswavlebeli gauSvebs programas Microsoft Office Excel. aCvenebs, ra martivad SeiZleba cxrilisaTvis sasurveli formis micema, magali-Tad, svetis siganis daregulireba. Semdeg yuradReba unda gamaxvildes imaze, Tu ramdenad mniSvnelovania svetis da striqonis saxeldeba. mo-swavleebma unda Seadginon sakuTari monacemebis cxrili - gvari, sqesi, dabadebis TariRi da adgili da a.S. moswavleebs mxolod informaciis saxiT (cnobad) unda miewodoT in-formacia eleqtronuli cxrilebis upiratesobis Sesaxeb - rom SesaZle-belia filtrebis dayeneba, monace-mebis daxarisxeba zrdiT an klebiT, an leqsikografiuli meTodiT da a.S. (20 wuTi).

Sefasebis sagani da procedurebi

yuradReba unda mieqces, gaigo Tu ara moswavlem, ra saxis informaciis warmosadgenad aris ukeTesi cxrilis gamoyeneba; SeuZlia Tu ara moswav-les saWiro informaciis amokreba cxrilidan; axerxebs Tu ara cxrilis sworad Sevsebas.

Sefasebis forma – ganmaviTarebeli.

saWiro resursebi saxelmZRvanelo, sxvadasxva saxis cxrilebi, internetSi CarTuli kompiuteri da proeqtori, eleqtro-nuli cxrilebis programa Microsoft Office Excel.


45

5. gakveTilis gegma - monacemTa saSualo


monacemTa saSualo

gakveTilis mniSvnelo-ba, aqtualuroba

raime movlenis dasaxasiaTeblad, zogadi tendenciis dasanaxad, monacemTa gasaanalizeblad mniS-vnelovania monacemTa saSualos cnebis da misi gamoyenebis far-glebis codna.


Sedegi: moswavles SeuZlia Tvise-briv da raodenobriv monacemTa interpretacia da elementaruli analizi

mizani: moswavlem SeZlos Semaj-amebeli ricxviTi maxasiaTeblis - saSualo ariTmetikulis gamoTvla da gamoyeneba monacemTa erTo-bliobis dasaxasiaTeblad.

winaswari codna moswavlem unda icodes ricxvebis Sekreba da gayofa.

gakveTilis msvleloba 1. samotivacio aqtivoba, proble�mis formulireba: maswavlebelma unda waukiTxos moswavleebs Sem-degi Sinaarsis teqsti sagazeTo an internet gamocemidan: magali-Tad, bolo wlebSi temperaturam dedamiwaze moimata saSualod 3 gradusiT, an, saSualo mervek-laselis aqtiuri leqsika sul 800 sityvas moicavs da a.S. – da sTxovos moswavleebs, TavianTi mosazrebebi gamoTqvan teqstSi gadmocemuli Sinaarsis Sesaxeb. aqcenti unda gakeTdes sityvaze

46


“saSualod”. moswavleebi maswav-leblis daxmarebiT unda mividnen im daskvnamde, rom temperatura dedamiwis yvela regionSi Tanabrad ver moimatebda, rom SesaZloa zo-gan namati ufro metia da zogan ki naklebi. maswavlebelma unda sTxovos moswavleebs, TviTon Ca-moayalibon msgavsi debulebebi - magaliTad, TbilisSi puris fasi saSualod 70 TeTria, an saskolo saxelmZRvanelos fasi saSualod 10 laria (10 wuTi).

2. codnis konstruireba: maswavle-beli ganmartavs, rom arsebobs situaciebi, roca mniSvnelovania procesis Sesaxeb zogadi war-modgenis Seqmna, situaciisTvis damaxasiaTebeli tendenciis da-naxva, da ar arsebobs faqtebis gansakuTrebuli detalizaciis aucilebloba. zogjer CvenT-vis saintereso procesis Sesaxeb iseTi miaxloebiTi informaci-is flobaa sakmarisi, romelic zogad warmodgenas Segviqmnis procesze masSi metismetad CaR-rmavebis gareSe. aseTi saxis “mi-axloebiTi mniSvnelobaa” saSua-lo ariTmetikuli. maswavlebeli moswavleebs ganumartavs, rogor iTvleba ramdenime ricxvis saSu-alo ariTmetikuli (sasurvelia, rom ricxvebis raodenoba ar aRe-matebodes 4-s).


47

3. Semdeg, sasurvelia, gairCes ram-denime amocana, magaliTad, dadg-indes xaWapuris saSualo fasi Cvens qalaqSi, an iqve klasSi dadgindes moswavleTa saSualo simaRle. Semdeg aucileblad unda gamaxvildes yuradReba, sak-marisia Tu ara saSualo ariTme-tikuli ufro zogadi daskvnebis gasakeTeblad; anu, Tu mxolod erT ubanSi mdebare saxaWapuree-bis fasebis saSualos vipovniT, es monacemi SesaZloa mniSvnelovnad gansxvavdebodes qalaqis sxva ub-nebis monacemebisagan (20 wuTi).

3. gamTlianeba: maswavlebelma unda auxsnas moswavleebs, rom didi raodenobis monacemebis sa-Sualos eleqtronuli meTodebiT iTvlian. SesaZlebelia am faqtis demonstrirebac Microsoft Office Excel-is meSveobiT (15 wuTi).

Sefasebis sagani da procedurebi

yuradReba unda mieqces, gaigo Tu ara moswavlem, ra saxis informa-ciis warmosadgenad aris ukeTesi saSualo ariTmetikulis gamoyene-ba. axerxebs Tu ara moswavle 3 an 4 ricxvis saSualos gamoTvlas.

Sefasebis forma – ganmaviTarebe-li.

saWiro resursebi saxelmZRvanelo, sxvadasxva saxis realuri monacemebi - produqtebis fasebi da sxva., internetSi Car-Tuli kompiuteri da proeqtori, eleqtronuli cxrilebis programa Microsoft Office Excel.

48


6. gakveTilis gegma – geometriuli figurebi

gakveTilis mTavari Tema geometriuli figurebi


cxovrebaSi, yofiTi daniS-nulebis sagnebSi an maT il-ustraciebSi xSirad gvxvdeba geometriuli figurebi. maTi formebisa da Tvisebebis Seswav-la ganapoirobebs infrastruq-turis sworad dagegmvas da gan-viTarebas.


mizani: moswavles unda SeeZlos amoicnos, miuTiTos da gamosax-os geometriuli figurebi misi garemomcveli samyarodan.

Sedegi: moswavle SeZlebs amoic-nos da miuTiTos yofiTi daniS-nulebis sagnebsa da ilustra-ciebSi geometriuli figurebi; Seadaros da daajgufos brtyeli da sivrciTi figurebi; Seqmnas maTi grafikuli gamosaxulebis modelebi (plastelinisa da asan-Tis RerebiT – karkasebi).

winaswari codna elementaruli brtyeli fig-urebis amocnoba da dasaxeleba.

gakveTilis msvleloba moTelva, klasis mobilizeba: maswavlebeli warmoudgens mo-swavleebs sxvadasxva arqiteqtu-ruli nagebobebis suraTebs da sTxovs, amoicnon maTTvis cno-bili geometriuli figurebi. mag., samkuTxedi, oTxkuTxedi, wre. maswavlebelma unda Seadgi-nos maT mier dasaxelebuli fig-urebis nusxa.


49

samotivacio amocana: maswav-lebels winaswar momzadebuli aqvs sivrciTi figurebis suraTebi da modelebi. klass yuofs jgufebad, urigebs sivrciTi figurebis mod-elebs da sTxovs moswavleebs, aRniSnul suraTebze aRmoaCinon modelebis msgavsi figurebi da daajgufon isini msgavsi niSnebvis mixedviT.

codnis konstruireba: savarau-dod moswavleebi warmodgenil TvalsaCinoebebSi daasaxeleben maTTvis cnobil figurebs. Semdeg, SeZlebisdagvarad, ganasxvaveben sivrcul da brtyel figurebs, ris Semdegac maswavleblis daxmarebiT moxdeba saxeldeba sivrciTi fig-urebis. Semdeg, maswavlebeli war-moadgens qaRaldisagan damzade-bul marTkuTxa paralelepipedis models (asanTis kolofi, wamlis kolofi), gaWris mas wiboebis gas-wvriv da moswavleTa daxmarebiT gaSlis sibrtyeze (Slili umjobe-sia iyos damzadebuli feradi qa-Raldisagan).

refleqsia: moswavleebi unda vavarjiSoTb rtyeli da sivrciTi figurebis amocnobasa da klasi-ficirebaSi. amisaTvis, maswavle-beli jgufebs urigebs winaswar gamzadebul or nimuSs (erTi– brtyeli, meore – sivrciTi figu-ra) – marTkuTxedi da marTkuTxa-paralelepipedi. maswavlebeli

50


sTxovs daawebon furcelze mo-cemuli figurebi, rac TvalnaTliv daanaxebs bavSvebs, rom sivrciTi figuris Cveulebrivi saxiT dawe-beba SeuZlebelia. maswavlebeli daavalebT maT, rom gaakeTon sivr-ciTi figuris Slilebi da is daawe-bon qaRaldze.

davubrundeT Temis dasawy-isSi Semotanil sailustracio suraTebs. maswavlebelma unda daavalos moswavleebs amoicnon maTSi sivrciTi figurebi, daamza-don erTerTi maTganis Slili da modeli. proeqtorisa da kom-piuteris arsebobis SemTxvevaSi SesaZlebelia gamoyenebul iqnas prezentacia brtyeli da sivrciTi figurebis gamosaxulebebiT. mo-swavleebma unda amoicnon figurebi da moaxdinon maTi klasificireba.


yuradRebis centrSi unda iyos, gaigo Tu ara moswavlem gansxvave-ba brtyel da sivrciT figurebs Soris, axerxebs Tu ara figuraTa klasificirebas, SeuZlia Tu ara figurebis gamosaxva da Slilebis Seqmna.

Sefasebis forma - ganmaviTarebeli

saWiro resursebi foto- masala, geometriuli figurebis naxatebi da mod-elebi, arqiteqturuli Zeglebis suraTebi, feradi qaRaldi, makrateli, webo, plastelini, asanTis Rerebi, saxelmZRvane-lo, kompiuteri da proeqtori.

http://www.youtube.com/watch?v=15likNwNMjk


51

7. gakveTilis gegma _ marTkuTxedi. marTkuTxedis perimetri.

gakveTilis mTavari Tema marTkuTxedi. marTkuTxedis perimetri


marTkuTxedis modeli xSirad gvxvdeba yofiTi daniSnulebis sagnebSi. amitom, misi Tvise-bebis Seswavla Zalian mniS-vnelovania.


moswavle SeZlebs yofiTi dan-iSnulebis sagnebSi an maT il-ustraciebSi marTkuTxedis amocnobas; SeZlebs miTiTebuli zomebis marTkuTxedis daxa-zvas; SeZlebs marTkuTxedis gverdebis sigrZeebis povnas gazomviT da gamoTvlis marT-kuTxedis perimetrs.

winaswari codna moswavlem unda icodes monakve-Tis cneba, unda SeeZlos misi sigrZis gazomva, aTwiladebis Sekreba-gamokleba.

gakveTilis msvleloba 1. Temisa da miznis gacnoba da klasis organizeba (maswavle-beli klass dayofs 3-4 moswav-lisagan Semdgar jgufebad).

2. TiToeul jgufs maswavle-beli daurigebs feradi fur-clebisagan gamoWril mraval-kuTxedebs – marTkuTxedi, trapecia, samkuTxedi, xuT-kuTxedi,

52


3. kvadrati _ da Txovs maT amoicnon, romeli figuraa marTkuTxedi da daasabu-Ton TavianTi arCevani. sax-elwodebidan gamomdinare, moswavleebi albaT airCeven marTkuTxedsa da kvadrats da imsjeleben am figurebis maxasiaTeblebze.

4. maswavlebeli Seajamebs jgufebis namuSevars da gansazRvravs marTkuTxedis da kvadratis cnebebs.

5. maswavlebeli dasvams SekiTx-vas – xom ar gsmeniaT sityva perimetri? ras SeiZleba niS-navdes? ra asociacia giCnde-baT am sityvis mosmenisas? moswavleTa pasuxebs Soris SesaZloa gaiJReros perimet-ric erTgvarma gansazRvrebam. maswavlebeli Seajamebs mo-swavleTa mosazrebebs da aux-snis maT figuris perimetris cnebas.

6. maswavlebeli TiToeul jgufs aZlevs praqtikul davalebas. pirvelma jgufma unda gai-gos merxis perimetri, meore jgufma – maswavleblis magi-dis, mesame jgufma – wignis, meoTxe jgufma ki – dafis perimetri. maswavlebeli ak-virdeba jgufebs da aZlevs miTiTebebs. davalebis Ses-rulebis Semdeg, TiToeuli jgufi aRwers, rogor gazomes Sesabamisi figurebi da ra Sedegi miiRes.


53

7. maswavlebeli ajamebs moswav-leTa namuSevrebs da Semoaqvs perimetris gamosaTvleli formula.

8. Semdeg, moswavleebs eZlevaT individualuri davalebebi: miTiTeli zomis mqone marT-kuTxedis daxazva da perime-tris gamoTvla.

Sefasebis sagani da proce-durebi

maswavleblis dakvirvebis fokusSi unda iyos SeuZlia Tu ara moswavles marTkuTxedis daxazva, gvardebis sigrZeebis gazomva da perimetris gamoT-vla

saWiro resursebi saxelmZRvanelo, saxazavi, san-timetri (xveula), feradi fur-clebisagan gamoWrili brtyeli figurebis modelebi.

54


gamoyenebuli literatura:

1. Teaching and Learning Primary Numeracy: Policy, Practice and Effectiveness (A review of British research for the British Edu-cational Research Association in conjunction with the British Society for Research in the Learning of Mathematics. Editors: Mike Askew and Margaret Brown);

2. Steen, LA (ed.) 2001, Mathematics and democracy: the case for quantitative literacy,

3. The National Council on Education and the Disciplines, US. Available at: http://www.maa.org/ql/mathanddemocracy.html;

4. Thomson, S 2004, ‘Numeracy in the early years: Project Good Start’, Australian Primary Mathematics Classroom, vol. 9, no. 4, pp. 14–17;

5. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathemat-ics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan;

6. Reynolds, D., & Muijs, D. (2000). School effectiveness and teacher effectiveness in mathematics: Findings from the evalu-ation of the mathematics Enhancement Program (Primary). School Effectiveness & School Improvement.

7. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

8. Polya, G. (1957). How to solve it? (2nd Edition). New York: Penguin Books.

9. problemaze dafuZnebuli swavla. erovnuli saswav-lo gegmebisa da Sefasebis centri. 2008 weli.

ekaterine korZaZe - maTematikuri wignierebis eqsperti.

daamTavra Tbilisis ivane javaxiS-vilis saxelobis saxelmwifo univer-sitetis meqanika-maTematikis fakul- teti. wlebis ganmavlobaSi iyo skoleb-is kompiuterizaciis programa „irmis naxtomi“-s maTematikisa da bunebisme-

tyvelebis maswavlebelTa sagnobrivi qselis koordina-tori. monawileobda zogadsaganmanaTleblo skolebis maTematikis erovnuli saswavlo gegmis SemuSavebaSi. aris maTematikis swavlebis araerTi publikaciis Tanaavtori. muSaobs ilias saxelmwifo universitetSi, ganaTlebisa da mecnierebis saministroSi maTematikis eqspertad.

ekaterine korZaZe

maTematikuriwigniereba


Documents

მათემატიკური წიგნიერება-მასწავლებლის სახელმძღვანელო-