7 klasis matematika maswavleblis wigni

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Sesavali

saqarTveloSi amJamad mimdinare zogadsaganmanaTleblo skolis reforma arse-

biTad gansxvavebul moTxovnebs uyenebs saswavlo process. aqcenti faqtobrivi masalis aTvisebidan gadatanilia saWiro unarebis gamomuSavebaze. amasTan dakav-SirebiT Seicvala saskolo maTematikuri ganaTlebis miznebic. misi upirvelesi amo-cana gaxda moswavlis mier sasicocxlo maTematikuri unarebis aTviseba, misi momzadeba momavali profesiuli saqmianobisTvis. swored am miznebis gaTvaliswinebiT Seiqmna winamdebare kompleqti me-7 klasisaTvis: `maTematika 7~, romelic moswavlisa da maswavleblis wignebisagan Sedgeba.

saxelmZRvaneloSi aRwerili VII klasis maTematikis kursi moswavleTaTvis saWiro codnis gadacemasTan erTad miznad isaxavs praqtikaSi am codnis gamoyenebis swavlebas, maTematikuri intuiciis, msjelobis, faqtebis analizis unaris ganvi-Tarebas. am TvalsazrisiT kursis miznebi savsebiT Seesabameba zogadsaganmanaTleblo standartSi Camoyalibebul miznebs.

saxelmZRvaneloSi warmodgenili saswavlo masala moicavs Semdeg sakiTxebs:

ricxvebi da moqmedebebi ariTmetikuli moqmedebebi mTel da racionalur ricxvebze; nawilebi da procentebi, proporcia; ariTmetikuli moqmedebebis Sedegis Sefaseba; mTel ricxvTa gayofadoba, naSTi, jeradebi da gamyofebi, martiv mamravlebad daSla; xarisxi naturaluri maCvenebliT; zomis erTeulebi da kavSiri maT Soris.

kanonzomierebebi da algebra

simravleTa Teoriis elementaruli cnebebi; mimdevrobebi; cvladis Semcveli martivi gamosaxulebebi; pirdapirproporciuli da ukuproporciuli damokidebuleba; wrfivi gantolebebi da maTTan dakavSirebuli praqtikuli amocanebi; koordinatTa sistema da grafiki.

geometria

ZiriTadi geometriuli cnebebi: wertili, wrfe, sibrtye; martivi geometriuli figurebi: mravalkuTxedebi, kuTxe. maTi klasifikacia; samkuTxedebi: elementebi, Tvisebebi, tolobis niSnebi; umartivesi agebebi fargliTa da saxazaviT; geometriuli gardaqmnebi sibrtyeze.

monacemTa analizi, albaToba da statistika

monacemTa Segroveba da organizacia; monacemTa warmodgenis saSualebebi: sia, cxrili, piqtograma, sxvadasxva tipis

diagramebi.

maswavleblis wigni moicavs: saswavlo masalis Tematur ganawilebas; TiToeuli Tavis dasawyisSi mocemul miznebis rukas; TiToeuli Tavis da TiToeuli paragrafis ZiriTad miznebs; im codnisa da unarebis CamonaTvals, romelsac SeiZens moswavle TiToeuli

paragrafis Seswavlis Sedegad; TiToeuli paragrafis Sinaarsis zogad daxasiaTebas; meTodur rekomendaciebs;

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meTodur komentarebs savarjiSoebis Sesaxeb da pasuxebs; zepir savarjiSoebs; damatebiT savarjiSoebs. mocemul masalaze dayrdnobiT saxelmZRvaneloSi ganxorcielebulia standartSi

Camoyalibebuli Sedegebis realizacia. kerZod, saxelmZRvaneloSi gadmocemuli kursis aTvisebis Semdeg moswavles unda SeeZlos:

ricxvebi da moqmedebebi

mTel ricxvebze naSTiT gayofis Sesruleba, udidesi saerTo gamyofisa da umciresi saerTo jeradis povna, ricxvis daSla martiv mamravlebad;

racionaluri ricxvebis warmodgena sxvadasxva formiT (wiladi, aTwiladi, pro-centi, wertili ricxviT RerZze), maTi Sedareba da maTze moqmedebebis Sesru-leba;

racionalur ricxvebze moqmedebebis Sedegis Sefaseba; ricxvebze moqmedebisas naturalurmaCvenebliani xarisxis Tvisebebis gamoye-

neba; praqtikuli amocanebis amoxsnisas gazomvebis damoukideblad Sesruleba da

saWiro gamoTvlebis koreqtulad warmoeba. kanonzomierebebi da algebra

simravleTa Teoriis cnebebisa da aRniSvnebis gamoyeneba simravleebis Casa-werad;

mimdevrobebSi periodis an nazrdis gamoyofa; praqtikuli amocanebis amoxsnisas sidideebs Soris pirdapirproporciuli da

ukuproporciuli damokidebulebebis gamoyeneba; cvladis Semcveli martivi gamosaxulebebis Sekreba, gamokleba da gamravleba,

gamosaxulebis ricxviTi mniSvnelobis gamoTvla; cxriliT an formuliT mocemuli damokidebulebis grafikis ageba, grafikis

interpretacia; amocanebTan dakavSirebuli martivi wrfivi gantolebebis Sedgena da amoxsna.

geometria

brtyeli geometriuli figurebis (wertili, monakveTi, sxivi, wrfe, kuTxe, mravalkuTxedebi) da maTi elementebis amocnoba, daxazva da klasifikacia. geometriuli amocanebis amoxsnisas naxazis sworad Sesruleba;

koordinatTa sistemis saSualebiT wertilis miTiTeba da wertilis koor-dinatebis dadgena;

samkuTxedebis Tvisebebisa da tolobis gamoyeneba geometriuli amocanebis amoxsnisas;

izometriuli geometriuli gardaqmnebis (RerZuli simetria, paraleluri gadatana) warmoeba (koordinatTa sistemis meSveobiT an naxazis gamoyenebiT);

monacemTa analizi, albaToba da statistika monacemTa mopoveba dakvirvebiT, gazomvebiT an gamokiTxviT da warmodgena siis,

cxrilis, piqtogramis an diagramis saxiT; monacemTa miTiTebuli niSniT klasifikacia da dalageba.

momdevno gverdze moyvanili moswavlis wignis (saxelmZRvanelos) struqtura

iTvaliswinebs maTematikis Tanamedrove saskolo saxelmZRvanelos Seqmnis saerTa-Soriso gamocdilebas. saxelmZRvanelos zogierTi komponenti - proeqtebis nimuSebi, maTematikuri terminebis leqsikoni da sacnobaro masala SedarebiT axalia maTematikis saskolo saxelmZRvaneloebSi. garda uSualo funqciisa maT damatebiTi datvirTvac gaaCniaT: moswavlisTvis wignTan da informaciis sxva wyaroebTan muSao-bis kulturis Camoyalibeba.

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saxelmZRvanelos struqtura

saxelmZRvanelos sabazo erTeulia paragrafi, romelic erT an or gakveTilzea

gaTvlili. masSi saxelmwifo standartebis moTxovnebidan gamomdinare Camoya-libebulia miznebi da mosalodneli Sedegebi, romelTa miRwevasac emsaxureba mocemul paragrafSi moyvanili saswavlo Sinaarsi, savarjiSoebi da amocanebi.

paragrafis tipuri agebuleba aseTia:

Camoyalibebulia paragrafis mizani: masSi aRwerilia, ra unda aiTvisos moswavlem am paragrafSi; miznebi, rogorc wesi, emsaxureba saxelmwifo standartebSi dafiqsirebuli Sedegebis miRwevas;

ganxiluli da amoxsnilia Sesaswavl masalasTan dakavSirebuli amocana an amocanebi (rogorc wesi, praqtikuli xasiaTis);

ganxilul amocanebze dayrdnobiT axsnilia Teoriuli masala; moyvanilia axsnil masalasTan dakavSirebuli damatebiTi informacia (kavSiri

codnis sxva sferoebTan, istoriuli faqtebi da a.S.); mocemulia klasSi Sesasrulebeli savarjiSoebi da individualuri da jgufuri

aqtivobebi codnis ganmtkicebisTvis.

saxelmZRvanelo

Sesavali

S i n a a r s i

damatebiTi masala damatebiTi amocanebi proeqtebis nimuSebi sacnobaro masala terminebis leqsikoni

Tavi 1 Tavi 2

Tavis struqtura

Tavi 6

Tavfurceli miznebi proeqti `aba, scade!~

paragrafi mizani Sinaarsi kiTxvebi savarjiSoebi

Tavis mimoxilva ra viswavleT kiTxvebi da amocanebi

gavlili masalis Sesamow-meblad

jgufuri samuSao damatebiTi amocanebi

(testi)

Tavi 3 Tavi 4 Tavi 5

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paragrafis momdevno nawili gaTvaliswinebulia saSinao davalebebisa da klasSi damatebiTi muSaobisaTvis. masSi: mocemulia kiTxvebi da savarjiSoebi SeZenili codnis Sesamowmeblad; mocemulia kompleqsuri xasiaTis amocanebi, romelTa amosaxsnelad saWiroa

paragrafSi mocemuli masalis dakavSireba adre miRebul codnasTan; mocemulia kiTxvebi da amocanebi adre aTvisebuli masalis gasameoreblad; mocemulia aratipuri da kvleviTi xasiaTis amocanebi da proeqtebi.

rogorc am aRweridan Cans, ukve paragrafis doneze gaTvaliswinebulia mraval-

mxrivi aqtivobebi, maT Soris gansxvavebuli SesaZleblobebis moswavleebTan samuSaod. cxadia, konkretuli paragrafi, masSi ganxiluli saswavlo masalisa da mosa-

lodneli Sedegebis gaTvaliswinebiT, SeiZleba am komponentTagan yvelas ar Seicavdes. saxelmZRvanelos yovel Tavs aqvs Tavfurceli da Tavis mimoxilva. TavfurcelSi

Camoyalibebulia Tavis ZiriTadi miznebi. aqve mocemulia proeqti, romlis Sesruleba moiTxovs am TavSi gadmocemuli masalis codnas. Tavis mimoxilva mokled mimoixilavs am TavSi Seswavlil masalas, Seaxsenebs moswavles im Sedegebs, romelTac man miaRwia mocemul TavSi gadmocemuli masalis aTvisebisas. aq moyvanili amocanebi exmareba maswavlebelsa da moswavles, gaakontrolos am Sedegebis miRwevis done.

meTodika

saxelmZRvaneloSi gamoyenebuli meTodika iTvaliswinebs meSvide klasis moswavlis

asakobriv da fsiqologiur Taviseburebebs, agreTve saswavlo masalis specifikas. kerZod, masSi realizebulia Semdegi principebi:

masalis formaluri aTvisebidan aqcentis gadatana moswavleTaTvis rao-denobrivi da sivrculi warmodgenebis Camoyalibebaze;

maTematikuri ideebis axsnisas TvalsaCinoebis maqsimaluri gamoyeneba; yuradRebis gamaxvileba moswavlis maTematikuri da zogadi unar-Cvevebis

ganviTarebaze; saxelmZRvaneloSi gadmocemuli saswavlo masalis dakavSireba praqtikasTan.

praqtikuli amocanebis amosaxsnelad saWiro strategiebisa da meTodebis Tanamimdevruli aTviseba.

mokled SevexoT saxelmZRvaneloSi gamoyenebuli meTodikis zogierT aspeqts mimar-Tulebebis mixedviT. ricxvebi da moqmedebebi

am mimarTulebaSi mTavari yuradReba eqceva moswavlis mier ricxvis cnebis gaazrebas, gamoTvlebis strategiis sworad SemuSavebas, zepiri angariSis, Sefasebis unaris gamomuSavebas.

saxelmZRvaneloSi ricxvTa sistemebis Semotanisas da ricxvebze moqmedebebis aRwerisas Tanamimdevrulad xdeba maTi demonstrireba TvalsaCino modelebze da praqtikul amocanebze.

savarjiSoebisa da amocanebis mniSvnelovani nawili emsaxureba moswavlisTvis ricxvebze moqmedebisas Sefasebis unaris Camoyalibebas. aseTi amocanebi praqtikulad yovel paragrafSi gvxvdeba.

didi yuradReba eTmoba saswavlo procesSi kalkulatoris gamoyenebas. savarji-Soebis nawili saxelmZRvaneloSi gamiznulia kalkulatorze gamoTvlebis efeqturad Sesrulebis Sesaswavlad. amasTan, kalkulatori gamoiyeneba ara marto rogorc gamoTvlebis Sesasrulebeli instrumenti, aramed rogorc ricxviTi eqsperimentebis Casatarebeli xelsawyo.

geometria saxelmZRvaneloSi Cven Tavs vikavebT geometriis, rogorc formalur-deduqciuri

sistemis, agebisagan. vTvliT, rom swavlebis am etapze aucilebelia, moswavles

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gamoumuSavdes intuiciuri warmodgenebi geometriuli figurebisa da geometriuli gardaqmnebis Sesaxeb. amitom geometriuli faqtebis mniSvnelovani nawilis gacnoba xdeba TvalsaCinod demonstrirebiT. saxelmZRvaneloSi gaTvaliswinebulia moswavle-ebis mier Sesabamisi aqtivobebis Sesruleba geometriul modelebze, raTa dar-wmundnen gadmocemuli geometriuli faqtebis marTebulobaSi. amasTan erTad saxelm-ZRvaneloSi TandaTanobiT xdeba deduqciuri msjelobebis, rogorc geometriuli faqtebis dasabuTebis instrumentis, Semotana. kanonzomierebebi da algebra

garda tradiciuli masalisa, saxelmwifo standartiT am mimarTulebiT gaTvalis-winebulia axali moTxovnebi, kerZod obieqtTa sistemebSi kanonzomierebebis SemCnevis unaris gamomuSaveba. saxelmZRvaneloSi moyvanili aqtivobebis nawili mimarTulia swo-red am unaris ganviTarebisaken. amisaTvis gamoiyeneba TvalsaCinoebebi, praqtikidan aRebuli faqtebi, agreTve maTematikuri eqsperimentebi. aseTi aqtivobebis nawilis Sesruleba da Sedegebis ganxilva gaTvaliswinebulia moswavleTa jgufis mier, rac moswavleebSi komunikaciisa da msjelobis unaris ganviTarebis saSualebas iZleva.

saxelmZRvaneloSi Tanamimdevrulad xdeba algebruli meTodebis Semotana da maTi gamoyenebis demonstrireba praqtikul magaliTebze. monacemTa analizi

rogorc cnobilia, monacemebze muSaobis unaris gamomuSavebas dRemde ar eqceoda saTanado yuradReba. moswavleebma praqtikulad ar ician monacemebis Segroveba da organizacia, sustad floben Segrovili informaciis grafikuli warmodgenis me-Todebs. amitom saxelmZRvaneloSi ganxilul masalaze dayrdnobiT gaTvaliswinebulia savarjiSoebisa da amocanebis kompleqti am xarvezis aRmosafxvrelad. specialuri (maT Soris jgufuri) aqtivobebi eZRvneba monacemebis Segrovebis organizacias. isini moicaven rogorc erTjerad davalebebs, ise grZelvadian proeqtebs. zogadi maTematikuri unarebi

am terminSi vgulisxmobT im unarebs, romlebic ar aris Semofargluli erTi romelime TematikiT an mimarTulebiT. esenia msjelobis, komunikaciis, amocanebis amoxsnis, sakuTari naSromis warmoCenis unarebi. am unarebis gamomuSaveba da ganviTareba maTematikis swavlebis erT-erTi umniSvnelovanesi mizania.

saxelmZRvaneloSi am unarebis ganviTareba xdeba amocanebis amoxsnis konteqstSi. kerZod, saxelmZRvaneloSi gaTvaliswinebulia specialuri amocanebi, romlebic moicaven yvela am aspeqts: realuri situaciis maTematikur modelirebas, amocanis gadasawyvetad saWiro informaciis moZiebas, amocanis amoxsnis meTodis SemuSavebas, amoxsnis warmodgenas saWiro formiT, amoxsnis marTebulobis dasabuTebas. aseTi amocanebi saSualebas aZlevs maswavlebels, gaakontrolos moswavleSi aRniSnuli unarebis progresi. saxelmZRvaneloSi gaTvaliswinebulia agreTve zogadi organizaciuli unar-Cvevebis ganviTarebisadmi miZRvnili aqtivobebi sxvadasxva individualuri da jgufuri proeqtebisa da prezentaciebis saxiT.

saxelmZRvaneloSi realizebuli meTodikis mniSvnelovani komponentia masSi mocemuli davalebaTa sistema. igi iTvaliswinebs asakobriv fsiqologiur moTxovnebs da ganisazRvreba balansiT logikasa da intuicias, abstraqtulsa da TvalsaCinos, mixvedrilobasa da gansjas Soris. yoveli Tema sruldeba:

kiTxvebiT, romlebic moswavles aZlevs TviTSemowmebis saSualebas; `a~ jgufis savaldebulo savarjiSoebiT; `b~ jgufis garTulebuli savarjiSoebiT.

yoveli Tavis boloSi moyvanilia damatebiTi savarjiSoebi, romelTa mizania mocemul TavSi Seswavlili masalis ganmtkiceba. maTi nawili gazrdili sirTulisaa. davalebaTa aseTi struqtura saSualebas aZlevs maswavlebels, imuSaos sxvadasxva donis moswavleebTan, SeamCnios da aRmofxvras swavlis procesSi gamovlenili xarvezebi.

davalebebs Soris mniSvnelovan rols TamaSobs proeqtebi, romlebic saxel-mZRvanelos yovel TavSia mocemuli. maTematikuri codnis gaRrmavebasTan erTad maTi mizania, miaCvios moswavle samuSaos dagegmvas, saWiro informaciis damoukideblad

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mopovebasa da damuSavebas, naSromis prezentacias. moswavleTa jgufis mier Sesrulebuli proeqti ayalibebs jgufis wevrebSi Sromis organizaciis unars, xels uwyobs maTSi komunikaciis unaris ganviTarebas.

gTavazobT VII klasis maTematikis erovnul saswavlo gegmas.

mimarTuleba: ricxvebi da moqmedebebi

Sedegebi da maTi indikatorebi

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

1. aTwiladis CanawerSi uTiTebs Tanrigebs da asaxelebs TanrigebSi mdgom cifrTa mniSvnelobebs; iyenebs am codnas aTwiladebis Sedarebis an (zrdadobiT/klebadobiT) dalagebisas (mag., gaSlis sasrul aTwilads saTanrigo Sesakrebebis jamis saxiT, daalage klebiT 2.9259, 3.1, 2.93, da 2.899).

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

3. ekvivalenturi formiT wers Sereul ricxvebs, aTwiladebsa da wiladebs; adarebs da alagebs sxvadasxva saxiT mocemul ricxvebs (aTwiladi wiladi).

4. poziciuri sistemis gamoyenebiT, konkretul magaliTebze asabuTebs gayofadobis niSnebidan zogierTs (mag., 3-ze da 9-ze gayofadobis niSnebs); poulobs mocemuli ricxvebis umcires saerTo jeradsa da udides saerTo gamyofs.

VII.2. sxvadasxva xerxiT asrulebs moqmedebebs racionalur ricxvebze

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

2. iyenebs ricxvis Caweris ekvivalentur formebs, moqmedebaTa Sesrulebis Tanmimdevrobas, maT Tvisebebsa da dajgufebas gamoTvlebis gasamartiveblad (martivi magaliTebi am kompetenciis flobis Sesamowmeblad).

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

4. axdens naturalurmaCvenebiliani xarisxis Tvisebebis demonstrirebas.

5. zepiri angariSisas iyenebs procentis kavSirs ricxvis nawilebTan; poulobs mocemuli ricxvis procents da xsnis Sebrunebul amocanebs.

6. irCevs da iyenebs racionalur ricxvebze ariTmetikul moqmedebebis Sesrulebis xerxs (es xerxebia: zepiri, teqnologiebis gamoyenebiT, weriTi algoriTmebi).

VII.3. afasebs racionalur ricxvebze moqmedebaTa Sedegs.

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

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

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

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

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

1. irCevs da iyenebs Sesaferis erTeulebs sididis cvlilebis, moZraobis siCqaris, masStabisa da rukaze manZilis povnasTan dakavSirebuli amocanebis amoxsnisas.

2. xsnis praqtikul saqmianobasTan dakavSirebul da/an sxva saswavlo disciplinebidan momdinare amocanebs gamoTvlebze (mag., umartivesi xarjTaRricxva; istoriuli epoqis xangrZlivobis gansazRvra; amocanebi procentebze da proporciaze: xsnarebi, Senadnobebi da sxva).

3. mocemuli wrfivi damokidebulebis gamoyenebiT gamosaxavs erT sistemaSi mocemul erTeuls sxva sistemis erTeuliT.

4. gamosaxavs mocemul erTeuls imave sistemis sxva erTeuliT (mag., gamosaxavs did erTeulebs mciris saSualebiT: gadahyavs km/sT -ebSi mocemuli monacemi m/wm -Si).

Sinaarsi 1. mTeli ricxvebi da ariTmetikuli moqmedebebi mTel ricxvebze. 2. wiladebi da aTwiladebi, maT Soris zogierTi kavSiri. 3. procenti (100-ze naklebi an toli da 1-ze meti an toli procenti); kavSirebi:

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

mamravlebad daSla. 7. ricxvis naturalurmaCvenebliani xarisxi. 8. naSTiT gayofa, naSTi da gayofadobis niSnebidan zogierTi. 9. zomis erTeulebi, zomis erTeulebs Soris kavSirebi da zomis erTeulebis

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

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

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mimarTuleba: kanonzomierebebi da algebra


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

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

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

3. sxvadasxva xerxiT (grafikulad, cxrilis saxiT, sityvierad, algebrulad) mocemul damokidebulebebs Soris miuTiTebs erTi da imave damokidebulebis Sesabamis gamosaxvebs.

VII.6. amocanis amoxsnisas iyenebs simravleTa Teoriis cnebebsa da operaciebs.

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

2. problemis gadaWrisas iyenebs venis diagramebs simravleTa Soris mimarTebebis dasadgenad da simravluri operaciebis Sesasruleblad.

3. sworad iyenebs simravleTa Teoriis cnebebs da Sesabamis aRniSvnebs simravluri operaciebis, simravleTa Soris mimarTebis, elementsa da simravles Soris mimarTebis gamosaxvisas.

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

teqsturi amocanis amosaxsnelad adgens da xsnis erTucnobian wrfiv gan-tolebas.

iyenebs moqmedebaTa Tvisebebs, maT Tanmimdevrobas da dajgufebas algebruli (ara umetes ori cvladis Semcveli wrfivi an meore xarisxis) gamosaxulebis gasamartiveblad da misi mniSvnelobis gamosaTvlelad cvladebis mocemuli mniSvnelobebisaTvis.

algebruli gardaqmnebis an/da logikuri msjelobis gamoyenebiT asabuTebs an uaryofs ori algebruli (ara umetes ori cvladis Semcveli wrfivi an meore xarisxis) gamosaxulebis igivur tolobas.

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

1. periodul mimdevrobaSi gamoyofs mimdevrobis periods.

2. warmoadgens mimdevrobis mocemuli fragmentis gavrcobis or an met variants, ganmartavs gavrcobis variantebs da adarebs maT.

3. dasmuli amocanis konteqstidan gamomdinare irCevs mimdevrobis gavrcobis variants da asabuTebs Tavis arCevans.

4. ganavrcobs mudmivi nazrdis mqone ricxviT mimdevrobas; asaxelebs realur viTarebaSi iseTi procesis magaliTebs, romlebic aseTi mimdevrobiT aRiwereba.

Sinaarsi

13

1. pirdapirproporciuli damokidebuleba da misi gamosaxva grafikis da cxrilis saSualebiT.

2. simravleTa Teoriis cnebebi, operaciebi da Sesabamisi aRniSvnebi. 3. teqsturi amocanebis amoxsna wrfivi gantolebebis gamoyenebiT. 4. ara umetes ori cvladis Semcveli wrfivi an meore xarisxis gamosaxulebebis

gamartiveba da mniSvnelobis gamoTvla. 5. perioduli mimdevrobebi da mudmivi nazrdis mqone ricxviTi mimdevrobebi.

mimarTuleba: geometria da sivrcis aRqma


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

1. ayalibebs mimarTebebs (mag., zogadoba-kerZooba) figuris saxeobebs Soris.

2. asaxelebs figuras misi niSan-Tvisebebis mixedviT, msjelobs figuris amosacnobad maTi sakmarisobis/arasakmarisobis Sesaxeb.

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

VII.10. warmoadgens geometriul obieqtebs amocanis konteqstis Sesabamisad.

1. agebs dasmuli amocanis Sesabamis naxazs da adekvaturad iyenebs asoiT aRniSvnebs.

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

3. gamosaxavs brtyel figurebs ise, rom maTi TanakveTa/gaerTianeba iyos miTiTebuli formis an Tvisebebis mqone figura.

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

1. garemomcvel obieqtebs Soris moiZiebs simetriul obieqtebs.

2. xazavs brtyeli figuris (texili, mravalkuTxedi) simetriul figuras miTiTebuli simetriis RerZis mimarT da axdens brtyeli figuris (texili, mravalkuTxedi) paralelur gadatanas.

3. uTiTebs brtyeli figuris simetriis RerZs/RerZebs; axdens simetriulobis demonstrirebas.

VII.12. iyenebs koordinatebis meTods orientaciisaTvis.

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

2. asaxelebs mocemuli wertilis RerZulad simetriuli wertilis koordinatebs.

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

14

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

1. iyenebs samkuTxedebis tolobis niSnebs figuraTa Tvisebebis dasadgenad, figuraTa ucnobi elementebis mosaZebnad an realur viTarebaSi manZilis arapirdapiri gziT dasadgenad.

2. xsnis agebis martiv amocanebs.

3. poulobs mizezSedegobriv kavSirebs samkuTxedTan da mis elementebTan dakavSirebul debulebebs Soris.

Sinaarsi 1. wertilebi, wrfeebi da sibrtyeebi: mimarTebebi maT Soris. 2. geometriuli figurebi: klasifikacia sxvadasxva niSniT (mag., amozneqili da

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

samkuTxedebi (elementebi, klasifikacia, Tvisebebi, tolobis niSnebi). 6. koordinatTa sistema; orientireba sibrtyeze, gardaqmnebis gamosaxva. 7. agebis umartivesi amocanebi: mocemuli samkuTxedis toli samkuTxedis, kuTxis

biseqtrisis, monakveTis SuamarTobis ageba.

mimarTuleba: monacemTa analizi, albaToba da statistika


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

ganasxvavebs Tvisebriv da raodenobriv monacemebs, iyenebs monacemTa Segrovebis Sesaferis saSualebas (gazomva, dakvirveba).

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

mocemuli amocanisTvis damoukideblad gegmavs da atarebs statistikur eqsperiments da agrovebs monacemebs.

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

1. axdens Tvisebriv da raodenobriv monacemTa dalagebas/klasifikacias, warmoadgens monacemebs siis/piqtogramis saxiT, msjelobs dalagebis/klasifikaciis principze.

2. qmnis mowesrigebul monacemTa cxrilebs da asabuTebs SerCeuli dizainis mizanSewonilobas.

3. agebs sxvadasxva diagramebs erTi da imave Tvisebrivi an raodenobrivi monacemebisTvis da msjelobs, monacemTa ramdenad mniSvnelovan aspeqtebs warmoaCens TiToeuli da ra upiratesoba aqvs TiToeuls.

VII.16. akeTebs Tvisebriv da raodenobriv monacemTa interpretacias da analizs amocanis konteqstis gaTvaliswinebiT.

1. svams kiTxvebs monacemebis Sesaxeb an/da axasiaTebs monacemebs, romlebic

15

warmodgenilia siis, cxrilis, piqtogramis an diagramis saxiT, msjelobs arsebul kanonzomierebebsa da gamorCeul monacemebze.

2. irCevs Sesaferis Semajamebel ricxviT maxasiaTeblebs, asabuTebs Tavis arCevans, iTvlis da iyenebs maT monacemTa jgufis dasaxasiaTeblad.

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

Sinaarsi

monacemTa Segrovebis saSualebani:

gazomva da dakvirveba; gamokiTxva; statistikuri eqsperimenti.

Tvisebrivi da diskretuli raodenobrivi monacemebis organizacia: monacemebis klasifikacia (garda intervalebad dajgufebisa); monacemTa dalageba zrdadoba-klebadobiT an leqsikografiuli meTodiT.

monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi: monacemTa raodenoba, pozicia da Tanmimdevroba erTobliobaSi, monacemTa

sixSire; ganmeorebis tipis kanonzomierebani; gamorCeuli (mag., eqstremaluri, iSviaTi) monacemebi.

monacemTa warmodgenis saSualebani raodenobrivi (diskretuli) da Tvisebrivi monacemebisTvis:

sia, cxrili, piqtograma, wertilovani, meseruli, xazovani, svetovani diagramebi. monacemTa Semajamebeli ricxviTi maxasiaTeblebi Tvisebrivi da diskretuli

raodenobrivi monacemebisTvis:

maswavleblis wigni maswavleblis winamdebare wigni warmoadgens ZiriTad meTodur saSualebas saxelmZRvanelosTan samuSaod. wignSi moyvanilia saxelmZRvanelosTan dakavSirebuli meToduri xasiaTis masala. masSi saxelmZRvanelos TiToeuli paragrafisTvis mokledaa aRwerili Sesabamisi gakveTilis mizani, moyvanilia paragrafTan dakavSirebuli meToduri miTiTebebi, Sinaarsobrivi da meToduri TvalsazrisiT garCeulia paragrafSi moyvanili savarjiSoebi. paragrafebis nawilisTvis mocemulia sagakveTilo scenari an gegma. maswavleblis dasaxmareblad wignSi mocemulia sas-wavlo SinaarsTan dakavSirebuli zepiri savarjiSoebi da damatebiTi amocanebi. agreTve, sakontrolo weris nimuSebi da Sefasebis sqema.

gakveTilis elementebTan dakavSirebuli saswavlo aqtivobebi

gakveTilis mniSvnelovani elementi saswavlo aqtivoba verbaluri informacia informaciis gadacema

16

istoriebis moyola swavlebis vizualizacia

suraTebi modelebi TvalsaCino masalebi

savarjiSoebi

eqsperimentebi demonstraciebi TamaSebi

diskusia: a)maswavlebelsa da moswavleebs Soris b) moswavleebs Soris

kiTxva-pasuxi wyvilebSi muSaoba jgufebSi muSaoba proeqtebi

msjeloba, komunikacia

saklaso an jgufuri diskusiebi amocanebi problemebis gadaWris savarjiSoebi

savaraudo Tematuri gegma (kviraSi 4 saaTi, sul 140 saaTi) I Tavi arauaryofiTi ricxvebi (20 sT)

parag-rafis #

Temis dasaxeleba

gankuTvnili saaTebis raodenoba

kavSiri sax. standartTan

1.1 martivi da Sedgenili ricxvebi 2 VII.1.4; 2.2 1.2 martivi da Sedgenili ricxvebi 2 VII.1.4; 2.2 1.3 wiladi da Sereuli ricxvebi 2 VII. 1.3; 1.4; 2.1 1.4 wiladebis Sekreba-gamokleba 1 VII. 1.1; 1.3; 2.2 1.5 wiladebis gamravleba-gayofa 1 VII. 2.2; 3.3

1.6 aTwiladebis Sekreba-gamokleba 2 VII. 1.1; 1.3; 2.2 1.7 aTwiladebis gamravleba-gayofa 2 VII. 1.1; 1.3; 2.2 1.8 ricxviTi gamosaxulebis mniSvnelobis

gamoTvla 2

VII. 1.1; 1.3; 2.2; 2.6; 3.2

1.9 kalkulatori 1 VII. 2.6; 3.1; 3.2; 3.4 I Tavis mimoxilva 1 testi 1 sakontrolo wera # 1 1 jgufuri mecadineoba 1 saTadarigo dro 1

17

II Tavi geometriis sawyisi cnebebi (15 sT)

ras Seiswavlis geometria 2.1 wrfe 1 VII. 9; 10

2.2 sxivi da monakveTi 1 VII. 9; 10 2.3 monakveTis sigrZe 1 VII. 9; 10 2.4 kuTxe 1 VII. 9; 10 2.5 mosazRvre da vertikaluri kuTxeebi 2 VII. 9; 10 2.6 marTobi 1 VII. 9; 10 2.7 mravalkuTxedi 1 VII. 9; 10 2.8 samkuTxedi 1 VII. 9; 10 2.9 geometriuli amocanebi 2 VII. 9; 10 II Tavis mimoxilva 1 sakontrolo wera # 2 1 jgufuri mecadineoba 1 saTadarigo dro 1

III Tavi monacemebi da maTi warmodgenis xerxebi (29 sT)

3.1 monacemebi. monacemTa saxeebi 1 VII. 14; 15.1 3.2 rogor movipovoT monacemebi 1 VII. 15.1-2; 16. 1-2-33.3 cxrili 1 VII. 15.1-2; 16. 1-2-33.4 svetovani diagrama 1 VII. 15.1-2; 16. 1-2-33.5 nawili 2 VII. 2.3; 3.6 ricxvis procentuli Cawera 2 VII. 2.5 3.7 rogor vipovoT ricxvi misi procentis mixedviT? 2 VII. 2.5 3.8 ori sididis fardobis procentuli Cawera 2 VII. 2.3; 2.5 sakontrolo wera #3 1 3.9 proporcia 2 VII. 5.1 3.10 proporciuli damokidebuleba sidideTa Soris 2 VII. 5.1; 2.5 3.11 sididis dayofa proporciul nawilebad 2 VII.2.3; 2.5 3.12 wriuli diagrama 2 VII.15.3; 16.1-2 3.13 ricxvis damrgvaleba 1 VII. 3.3 3.14 praqtikuli gamoTvlebi 2 VII. 3.1-2; 4.1-2-3 III Tavis mimoxilva 1 testi 1 sakontrolo wera #4 1 jgufuri samuSao 1 saTadarigo dro 1

IV Tavi uaryofiTi ricxvebi (25 sT)

4.1 dadebiTi da uaryofiTi ricxvebi 1 VII. 1-2; 6 4.2 ricxviTi wrfe 1 VII. 1-2; 6 4.3 mopirdapire ricxvebi. ricxvebis Sedareba 2 VII. 1-2; 6 4.4 ricxvis moduli 2 VII. 1-2; 6 4.5 ricxvebis Sekreba 2 VII. 1-2; 6 4.6 ricxvebis gamokleba 2 VII. 1-2; 6 4.7 ricxvebis gamravleba-gayofa 2 VII. 1-2; 6 sakontrolo wera # 5 1 4.8 koordinatTa sistema 1 VII. 12.1 4.9 wertilis ageba misi koordinatebiT 1 VII. 12.1

18

4.10 simravle 1 VII. 6.1-2 4.11 simravleTa gaerTianeba da TanakveTa 2 VII. 6.2-3 4.12 xarisxi 2 VII. 2.4 IV Tavis mimoxilva 1 testi 1 sakontrolo wera #6 1 jgufuri mecadineoba 1 saTadarigo dro 1

V Tavi cvladiani gamosaxuleba (23 sT)

5.1 mimdevroba 1 VII. 8 5.2 ariTmetikuli progresia 1 VII. 8.4 5.3 cvladi sidideebi 1 VII.5 5.4 grafiki 2 VII. 5.1-2-3 5.5 pirdapirproporciuloba 2 VII. 5.1-2-3 5.6 cvladiani gamosaxuleba 1 VII. 7 5.7 toli gamosaxulebebi 1 VII. 7 5.8 gamosaxulebis gardaqmna 2 VII. 7 sakontrolo wera # 7 1 VII. 7 5.9 gantoleba 1 VII. 7 5.10 wrfivi gantolebis amoxsna 2 VII. 7 5.11 amocanis amoxsnis algebruli meTodi 3 VII. 7 V Tavis mimoxilva 1 testi 1 sakontrolo wera #8 1 jgufuri mecadineoba 1 saTadarigo dro 1

VI Tavi samkuTxedi (24 sT)

6.1 RerZuli simetria 2 VII. 7.9-10-11-12.2 6.2 simetriuli figurebi 1 VII. 7.9-10-11 6.3 samkuTxedebis klasifikacia 1 VII. 9-10-13 6.4 tolferda samkuTxedi 2 VII. 9-10-13 6.5 samkuTxedis kuTxeebis jami 2 VII. 9-10-13 sakontrolo wera #9 1 6.6 agebis amocanebi 2 VII. 13.2 6.7 samkuTxedebis ageba 2 VII. 13.2 6.8 samkuTxedebis toloba 3 VII.13 6.9 geometriuli amocanebis amoxsna algebruli

meTodiT 2

VII. 9-10-13

6.10 praqtikuli amocanebi 2 VII. 9-10-13 6.11 paraleluri gadatana 1 VII. 12.1-3 VI Tavis mimoxilva 1 testi 1 sakontrolo wera #10 1

gameoreba ------ ------------------------------------------------- 4 sT

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meToduri rekomendaciebi I Tavi arauaryofiTi ricxvebi

Tavis ZiriTadi mizani: ganumtkicos moswavleebs ricxvebze miRebuli codna da ganuviTaros maTze moqmedebebis unari.

codna da unarebi: moswavlem unda aRidginos da gaiRrmaos wina klasebSi SeZenili codna, SeZlos sxvadasxva saxiT Cawerili ricxvebiT Sedgenili gamosaxulebis mniSvnelobis gamoTvla, amocanebis amoxsna ariTmetikuli moqmedebebis gamoyenebiT.

meToduri komentarebi: TavSi mTavari yuradReba eTmoba ricxvis cnebis swor gaazrebas. imisaTvis, rom yovel moswavles ukeTesad Camouyalibdes gamoTvlebis unari, Seicvala midgoma wiladebisa da aTwiladebis swavlebisadmi. aqcenti gakeTda gamoTvlis kulturis ganviTarebaze. gamaxvilda yuradReba teqstur da praqtikuli xasiaTis amocanebze. aseT amocanebze muSaoba ayalibebs moswavleSi msjelobis, amoxsnis gzis amorCevis, situaciis gaanalizebis, sakuTari azris dasabuTebisa da dacvis unarebs.

mniSvnelovani yuradReba eTmoba zepiri gamoTvlebis unaris ganviTarebas, kalkulatoris gamoyenebas, rogorc gamoTvlebis, ise sxvadasxva eqsperimentebis Casatareblad.

1.1 martivi da Sedgenili ricxvebi mizani: vaswavloT moswavleebs gamyofebisa da jeradebis povna, ricxvis daSla martiv mamravlebad da am sakiTxebTan dakavSirebuli amocanebis amoxsna. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes ricxvis gamyofisa da jeradis, martivi da Sedgenili ricxvebis ganmarteba Sesabamisi magaliTebis moyvaniT, ricxvTa gayofadobis zogierTi niSani (2-ze, 3-ze, 5-ze, 9-ze, 10-ze). SeeZlos ricxvis daSla martiv mamravlebad, mocemuli ricxvis gamyofebisa da jeradebis povna, gayofadobis niSnebis gamoyeneba, luwi da kenti ricxvebis amocnoba. meToduri komentarebi: maswavlebelma yuradReba unda gaamaxvilos Semdeg sakiTxebze:

a) mocemul ricxvs aqvs gamyofTa sasruli da jeradTa usasrulo raodenoba; b) erTze met yovel naturalur ricxvs aqvs ori gamyofi mainc: TviT es ricxvi da

1; g) imis dasasabuTeblad, rom romelime ricxvi Sedgenilia, sakmarisia vipovoT

1-isagan da TviT am ricxvisagan gansxvavebuli gamyofi. gamyofisa da jeradis praqtikul amocanebSi gamoyenebis erT-erTi magaliTia

kalendarTan dakavSirebuli amocanebi, romlebic am paragrafSia warmodgenili. am Temaze muSaobisas saWiroa yuradReba gamaxvildes im faqtze, rom kviris dReebis dasaxeleba ar icvleba TariRisadmi 7-is jeradis damatebiT. moswavlem unda icodes weliwadis TveebSi dReebis raodenoba da nakiani wlebis amocnoba. savarjiSoebis komentarebi da pasuxebi: #10 a) ricxvi, romelic aris erTdroulad 3-isa da 8-is jeradi, agreTve 38=24-is jeradia. e. i. saZiebeli ricxvebi 24-is jeradi orniSna ricxvebia. aseTi ricxvebia: 124=24; 224=48; 324=72; 424=96. pasuxi: 24, 48, 72, 96. #11. mcdaria b) gamonaTqvami (daskvna dasabuTdeba TiToeuli gamonaTqvamis gaanalizebis Semdeg). #12. pasuxi: samSabaTi. #13. Tu romeliRac Tvis 3 ricxvi SabaTia, maSin momdevno kviris samSabaTi iqneba 6 ricxvi, danarCeni samSabaTebi ki 13, 20, 27. pasuxi: imave Tvis bolo samSabaTi iqneba 27 ricxvi. #18. pasuxi: b). miTiTeba: sasurvelia, rom savarjiSos amoxsnis dawyebamde klasSi magaliTebze naCveneb iqnes Semdegi faqtebi:

nebismieri raodenobis luwi Sesakrebebis jami luwi ricxvia; ori kenti ricxvis jami luwi ricxvia; luwi raodenobis kenti ricxvebis jami luwi ricxvia;

20

kenti raodenobis kenti ricxvebis jami kenti ricxvia. #21. amocanis garCevisas, sasurvelia, TiToeulma moswavlem moiyvanos aseTi ricxvebis magaliTebi. am magaliTebis ganxilvis Sedegad SeecadeT, moswavleebi miiyvanoT daskvnamde: TiToeuli aseTi ricxvi martivi ricxvis kvadratia. #22. radgan TiToeul darbazSi myof mayurebelTa raodenoba 15-is jeradia, amitom SesaZlebelia cxrilSi moyvanili Semdegi SemTxvevebi: #23 amocanis pirobiT nika Wadrakis wreze mxolod orSabaTobiT dadis. maSasadame, 3 maisi orSabaTi yofila. maisis Tvis Semdegi orSabaTebi iqneba 10, 17, 24, 31 ricxvebSi.

laSa wreze iyo 3 maiss. igi wreze dRegamoSvebiT dadis. es niSnavs, rom laSa maisis kent ricxvebSi ivlis Wadrakze, anu 3, 5, 9, . . . , 29, 31-Si. biWebi Wadrakis wreze erTad mivlen 17 da 31 maiss. #24. amocanis amosaxsnelad unda gaviTvaliswinoT nakiani wlebis ganawilebis wesi, romelic am paragrafSia moyvanili: 1991-dan 2222-mde 4-is jeradi 58 ricxvia. aqedan 2100 da 2200 wlebi ar aris nakiani, amitom nakiani wlebis raodenobaa 56.

1.2 udidesi saerTo gamyofi da umciresi saerTo jeradi mizani: ramdenime ricxvis udidesi saerTo gamyofisa da umciresi saerTo jeradis povnis wesisa da urTierTmartivi ricxvebis cnebis Seswavla. amocanebis amoxsna ricxvebis udidesi saerTo gamyofisa da umciresi saerTo jeradis gamoyenebiT. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes urTierTmartivi ricxvebis ganmarteba; SeeZlos ramdenime ricxvis udidesi saerTo gamyofisa da umciresi saerTo jeradis gamoTvla, amocanebis amoxsna ricxvebis saerTo jeradebisa da gamyofebis gamoyenebiT. meToduri komentarebi: masala ori gakveTilisTvisaa navaraudevi. maswavlebels SeuZlia pirvel gakveTilze imuSaos ramdenime ricxvis udides saerTo gamyofze, xolo meore gakveTilze umcires saerTo jeradze da am cnebebTan dakavSirebul amocanebze. moswavleebma unda SeZlon udidesi saerTo gamyofisa da umciresi saerTo jeradis povna rogorc SerCevis, ise martiv mamravlebad daSlis xerxiT. savarjiSoebis komentarebi da pasuxebi: #4. 50-is jeradi ricxvis erTeulebis TanrigSi iqneba cifri 0, aTeulebis TanrigSi ki 5 an 0. pasuxi: 100. #5. b), g) da d) savarjiSoebSi moTxovnilia, rom miRebuli ricxvebi iyofodes Sesabami-sad 5-ze, 10-ze da 2-ze. samive SemTxvevaSi pasuxis SerCeva moxdeba bolo cifris mixed-viT. #6. unda gairkves, aris Tu ara 9-is jeradi a) 126, b) 113. #12. pasuxi: 4 . #14. pasuxi: 99. #15. jamis 2 3 5 13 16 + pirveli Sesakrebi 15-is jeradia. maSasadame, naSTi iqneba is ricxvi, rac iqneba 16-is 15-ze gayofisas, anu 1. #16. pasuxi:12. #17. eqskursiis monawileTa raodenoba jeradia rogorc 15-is, ise 9-isa. unda vipovoT usj(15, 9). usj(15, 9) = 45. pasuxi: damsvenebelTa umciresi raodenobaa 45. #18. usj(40; 50)=200. 200wT=3sT 20wT. pasuxi: 3 saaTisa da 20 wuTis Semdeg. #19. gamravldeba am naturalur ricxvze. #20. WeSmaritia. #21. sami 14-adgiliani da ori 15-adgiliani. 14 3 15 2 42 30 72 + = + = . #22. 2 3 17 19 8 2 3 17 19 17 9 = + . pasuxi:9. #23. jer gamoviTvaloT, ramdeni 7sm-iani da ramdeni 12sm-iani naWrebi miiRes gadaWris Sedegad. aseTi SesaZlebloba mxolod erTia: eqvsi 12-metriani da 4 Svidmetriani naWeri, sul 10. aT nawilad gadaWras dasWirdeba 9 gadaxerxva, TiToeulze ixarjeba 3 wuTi, sul dasWirdeba 27 wuTi. zepiri savarjiSoebi 1. ris tolia urTierTmartivi ricxvebis udidesi saerTo gamyofi? 2. rogor ricxvs ewodeba Sedgenili ricxvi? 3. martivia Tu Sedgenili ricxvi 12; 54; 23; 33; 41; 27?

I darbazi II darbazi1 15 6 15 2 15 5 15 3 15 4 15 4 15 3 15 5 15 2 15 6 15 1 15

21

4. martivia Tu Sedgenili ricxvi 1? 5. ricxvebidan: 2, 3, 4, 5, 6 romelia 35-is gamyofi? 6. ricxvebidan: 102, 36, 891, 256, 148, 801 romelia 9-is jeradi ricxvi? 7. gamoiyene sityvebi: `gamyofi~, `iyofa~ da Camoayalibe daskvna mocemuli tolo-

bidan: a) 324 = 369; b) 120= 1210. 8. zrdis mixedviT CamoTvale gamyofebi ricxvisa: a) 20; b) 36. 9. ramdeni gamyofi aqvs a) 4-s? b) 8-s? g) 13-s? 10. CamoTvale 15-is jeradi sami ricxvi. 11. ipove sami ricxvi, romelic jeradia erTdroulad a) 3-is da 2-isa; b) 4-is da 6-isa. 12. aris Tu ara ricxvi 51 martivi? 13. ra naSTi miiReba 80-is 9-ze gayofiT? damatebiTi masala sasurvelia, maswavlebelma dainteresebuli moswavleebis yuradReba gaamaxvilos ramdenime ricxvis udidesi saerTo gamyofisa da umciresi saerTo jeradis Semdeg Tvisebebze:

Tu erTi ricxvi iyofa meore ricxvze, maSin meore ricxvi maTi udidesi saerTo gamyofia. magaliTad, usg( 60, 3060)=60;

mocemuli ricxvebis ganayofebi maT udides saerTo gamyofze urTierTmartivi ricxvebia. magaliTad, usg(540, 1512)=108; 540:108=5; 1512:108=14. ricxvebi 5 da 14 urTierTmartivi ricxvebia, radgan usg(5, 14)=1;

ramdenime ricxvis saerTo gamyofi aris gamyofi maTi udidesi saerTo gamyofisa da piriqiT, mocemuli ricxvebis udidesi saerTo gamyofis yoveli gamyofi warmoadgens TiToeuli mocemuli ricxvis gamyofs;

Tu mocemul ricxvebs erTsa da imave ricxvze gavamravlebT, maSin am ricxvze gamravldeba maTi umciresi saerTo jeradi da udidesi saerTo gamyofic;

mocemuli ori ricxvis usg-s da usj-s namravli am ricxvebis namravlis tolia; usj-s ganayofebi mocemul ricxvebze urTierTmartivi ricxvebia; Tu mocemuli ricxvebi urTierTmartivia, maSin maTi umciresi saerTo jeradi am ricxvebis namravlia. SesaZlebelia am masalis kidev ufro gamdidreba rogorc Teoriulad, ise sayofacxovrebo amocanebis moZiebiT. am sakiTxis damuSaveba SeiZleba individualur, jgufur Tu sxva saxis mecadineobebze. damatebiTi amocanebi #1. rveuli 15 TeTri Rirs. Teklas aTi cali ocTeTriani moneta aqvs. gamyidvels ki xurda ara aqvs. ramdeni rveulis yidva SeiZleba xurdis gareSe? rveulebis ra umciresi da ra udidesi raodenoba SeiZleba SeiZinos Teklam xurdis gareSe? amoxsna:

Teklas aqvs 2010=200 TeTri vipovoT usj(15, 20).=60 vipovoT sj(15, 20)=60; 120; 180. SesaZlebelia am TanxiT unaSTod iyidos 60:15=4; 120:15=8; 180:15=12 rveuli. maT

Soris umciresi raodenobaa 4, udidesi ki 12. #2. ipove ricxvTa wyvili, romelTa udidesi saerTo gamyofi tolia 12-is, xolo umciresi saerTo jeradi 720-is. amocanas erTze meti amonaxsni aqvs. ipove yvela maTgani. miTiTeba: amocanis amosaxsnelad unda gamoviyenoT udidesi saerTo gamyofisa da umciresi saerTo jeradis is Tviseba, romlis mixedviTac maTi namravli mocemuli ricxvebis namravlis tolia. #3. erT klasSi 24 moswavlea, meoreSi 30. orive klasis moswavleebi unda moewyvnen erT kolonad (erTi klasi meoris gverdiT) ise, rom yovel rigSi moswavleTa raodenoba rogorc erTi, ise meore klasidan Tanabari iyos. aseTi mowyobis ramdeni SesaZlo SemTxveva arsebobs?

22

amoxsna: amocanis amosaxsnelad unda vipovoT 24-isa da 30-is saerTo gamyofebi. esenia: 1, 2, 3, 6. cal-calke ganvixiloT 2, 3 da 6 rigad mowyobis variantebi. SemoviRoT aRniSvna: ocdaoTxmoswavliani klasi aRvniSnoT I kl `-, xolo ocdaaTmoswavliani klasi IIkl `+~ niSnebiT.

1. Tu moswavleebi or rigad moewyobian, maSin TiTo rigSi I klasidan moewyoba 12, xolo II klasidan 15 moswavle.

moswavleTa ganawilebis es varianti SegviZlia sqematurad ase warmovidginoT: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

2. Tu moswavleebi sam rigad moewyobian, maSin TiTo rigSi I klasidan moewyoba 8, xolo II klasidan 10 moswavle.

moswavleTa ganawilebis es varianti SegviZlia sqematurad ase warmovidginoT: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3. Tu moswavleebi eqvs rigad moewyobian, maSin TiTo rigSi I klasidan moewyoba 4, xolo II klasidan 5 moswavle.

ganawilebis es varianti SegviZlia sqematurad ase warmovidginoT: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

#4. mgzavrebi, romelTa Soris 145 qali da 87 mamakacia, ise unda ganawildnen mikroav-tobusebSi, rom yovel avtobusSi qalebisa da kacebis raodenoba erTnairi iyos. a) avtobusebis ra umciresi raodenobaa saWiro? b) ramdeni kaci iqneba TiToeul avtobusSi? g) ramdeni qali iqneba TiToeul avtobusSi? amoxsna: usg(145, 87)=29. pasuxi: saWiro avtobusebis umciresi raodenobaa 29, TiToeul avtobusSi iqneba 3 mamakaci da 5 qali. #5. patara yuTSi 24 Wiqa eteva, didSi ki 30. a) gansazRvreT Wiqebis is umciresi raodenoba, romelic SeiZleba ganTavsdes rogorc did, ise patara yuTebSi ise, rom yuTebi mTlianad gaivsos; b) ramdeni patara yuTi iqneba saWiro Wiqebis am raodenobisaTvis? g) ramdeni didi yuTi iqneba saWiro? amoxsna: a) imisaTvis, rom ganvsazRroT Wiqebis umciresi raodenoba, romelic SeiZleba ganTavsdes rogorc did, ise patara yuTebSi, unda gamoviTvaloT usj(24, 30), romelic 120-is tolia; b) saWiro iqneba 120:24=5 patara yuTi; g) saWiro iqneba 120:30=4 didi yuTi. 1.3 wiladi da Sereuli ricxvebi mizani: gavaxsenoT moswavleebs wesieri da arawesieri wiladebi da Sereuli ricxvebi, maTi wakiTxva da Cawera, wiladis ZiriTadi Tviseba. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes wiladisa da Sereuli ricxvis Cawera da wakiTxva. unda SeeZlos wiladis dakavSireba gayofis procesTan, xolo Sereuli ricxvisa naSTian gayofasTan da demonstrireba TvalsaCinoebiT; wesieri da arawesieri wiladebis garCeva; ganayofis warmodgena wiladis saxiT; arawesieri wiladis Cawera Sereuli ricxvis saxiT; wiladebis Sedareba; wiladis ZiriTadi Tvisebis gamoyeneba. meToduri komentarebi: wiladis da misi Tvisebebis (kerZod, wiladis ZiriTadi Tvisebis, wiladis sxvadasxva formiT Caweris, wiladebis Sedarebis) demonstrireba maswavlebelma sasurvelia TvalsaCinoebebze dayrdnobiT moaxdinos.

23

savarjiSoebis komentarebi da pasuxebi: #12. wiladis mricxvelis 15-ze gamravleba gamoiwvevs wiladis sididis 15-jer gazrdas (gasayofi gaizarda 15-jer, gamyofis Seucvlelad). a) Tu 15-jer gazrdili wiladis mniSvnels 3-ze gavamravlebT, miviRebT mocemul

wiladze 5-jer did wilads. b) imisaTvis, rom miviRoT mocemul wiladze 3-jer patara ricxvi, 15-jer gazrdili

wiladis mniSvneli unda gamravldes 15-ze 3-jer did ricxvze, anu 45-ze. #13. moswavleebma unda Seadaron wiladebi sxvadasxva xerxiT, kerZod, erTnair mniSvnelze an mricxvelze dayvaniT an gayofis SesrulebiT.

#24. miTiTeba: unda moxdes wiladebis 15

20 da

7

10 Sedareba.

#25. amoxsna: unda moxdes wiladebis 81

100,

29

50,

20

25 Sedareba.

pasuxi: giorgim ukeTesad icis qarTuli ena. #29. 13-ze naSTiani gayofisas SesaZlo naSTebs Soris udidesia 12. saZiebeli ricxvia 142. #31. miTiTeba: amocanis amoxsnisas gaiTvaliswineT, rom saZiebelze erTiT meti ricxvi 4-is, 5-is da 6-is, e. i. 60-is jeradia. pasuxi: 959. #32. miTiTeba: 18 2 3 3= , 630 2 3 3 5 7= , amitom saZiebeli ricxvebia: 2 3 3 5 90 = , 2 3 3 7 126 = . zepiri savarjiSoebi 1. daasaxele ricxvi, romelic aRniSnavs naxevris naxevars. 2. daasaxele ricxvi, romelic aRniSnavs naxevarze orjer mets. 3. ramdeni santimetria naxevari metri? 4. ramdeni metria naxevari kilometri? 5. ramdeni kilogramia erTi meoTxedi tona? 6. ramdeni wuTia naxevari saaTi? 7. ramdeni wuTia meoTxedi saaTi?

8. romelia meti: a) 35

Tu 37

? b) 1 41 2 3

Tu 47123

?

9. romelia naklebi: 1 34

Tu 4 1 57 8 9

?

10. Seadare ricxvebi: 7458

da 2 67

.

damatebiTi savarjiSoebi 1. daasaxele mricxveli wiladisa: a) 1 2

2 0; b) 3

5.

2. daasaxele mniSvneli wiladisa: a) 34

; b) 7 45 8

.

3. erTnaxevari Cawere sami sxvadasxva saxiT.

4. wiladebidan: 1220

, 6

10,

910

, 1220

amoiwere wiladebi, romelic tolia 35

-is.

5. Sekvece wiladebi: 9 21 108 144, , ,12 28 441 63

.

6. Sekvece wiladebi: 3 5 1 12 2 1 5

, 3 1 2 1 8

7 6 2 3 2

.

7. Seadare 1-s ricxvi: a) 56

; b) 65

; g) 99

.

8. ricxvi 1 14

warmoadgine: a) sxva wiladis saxiT; b) Sereuli ricxvis saxiT;

9. warmoadgine wiladis saxiT: a) 1; b) 5; g) 0.

24

1.4 wiladebis Sekreba da gamokleba

mizani: vaswavloT moswavles wiladebisa da Sereuli ricxvebis Sekreba da gamokleba, Sereuli ricxvis Cawera arawesieri wiladis saxiT. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes wiladebisa da Sereuli ricxvebis Sekrebisa da gamoklebis wesebi, SeeZlos wiladebis Sekreba-gamokleba da maTi gamoyeneba gamoTvlebis sawarmoeblad da amocanebis amosaxsnelad. meToduri komentarebi: masala gaTvlilia or gakveTilze. gTavazobT II gakveTilis scenars: aqtivoba `wiladebis Sekreba da gamokleba~. mizani:

moswavlis gawafva wiladebis Sekreba-gamoklebaSi; daakavSiros ricxvebi realur cxovrebasTan.

resursebi: baraTebi davalebebiT. aqtivobis aRwera: maswavlebeli klass yofs jgufebad da TiToeul jgufs aZlevs winaswar gamzadebul baraTs, romlebzec Camowerilia davalebebi:

1. sami an oTxi kiTxva; 2. savarjiSos nomeri saxelmZRvanelodan; 3. amocana-naxati (magaliTad, Tavis dasawyisSi mocemuli `aba, scade~); 4. cxovrebiseuli amocana, romelic amoixsneba wiladebis Sekreba-gamoklebis

gamoyenebiT. baraTebisaTvis SeiZleba Semdegi kiTxvebis gamoyeneba: rogori wiladebis Sekreba da gamokleba SeiZleba gaerTmniSvnelianebis gareSe? (erTnairmniSvnelianis). rogori wiladebis Sekreba da gamokleba moiTxovs gaerTmniSvnelianebas? (sxvadasxvamniSvnelianis). ra gardaqmnebi unda movaxdinoT sxvadasxvamniSvneliani wiladebis Seskrebad an gamosaklebad? ras niSnavs wiladebis gaerTmniSvnelianeba? ra icvleba da ra ar icvleba wiladebis gaerTmniSvnelianebis dros? ras udris ramdenime wiladis saerTo mniSvneli, roca maTi mniSvnelebi urTierTmartivi ricxvebia? ras udris ramdenime wiladis umciresi saerTo mniSvneli, roca erT-erTi maTgani danarCenebze unaSTod iyofa? raSi mdgomareobs wiladis ZiriTadi Tviseba? rogor SevkriboT Sereuli ricxvebi? rogor unda gamovakloT mTel ricxvs wesieri wiladi? rogor unda gamovakloT Sereul ricxvs Sereuli ricxvi? rogor unda SevasruloT Sereuli ricxvebis gamokleba im SemTxvevaSi, roca saklebis wiladi nawili naklebia maklebis wilad nawilze? aRvweroT aqtivoba: sakiTxi 1. kiTxvebze pasuxi moeTxovebaT zepirad. moswavleTa yvela pasuxi unda iyos sruli, logikurad gamarTuli. yvela wess unda axldes magaliTebis demonstrireba. erT kiTxvaze pasuxs iZleva erTi moswavle, meoreze meore da a. S. saWiroebis SemTxvevaSi komentars an Sesworebas akeTebs sxva jgufis wevri (Tu ar iqneba sworad Camoyalibebuli pasuxi an warmodgenili magaliTi arasworad iqneba SerCeuli). swori pasuxi fasdeba 2 quliT. sakixTi 2. SeiZleba saxelmZRvanelodan mieces #11(ganawildes a), b), g) sxvadasxva bileTebSi), #7, #8 da #10 savarjiSoebi. swori pasuxi fasdeba 2 quliT. sakiTxi 3. Tavfurcelze mocemulia amocana-naxati. SeiZleba misi msgavsis mofiqreba rogorc Sekrebaze, ise gamoklebaze an orive moqmedebaze erTad. es amocanebi amo-saxsnelad gadaecema sxva gundebs. SesaZlebelia ramdenimes warmodgena. gundis namu-Sevari fasdeba imdeni quliT, ramdeni sworad Sedgenili amocanac warmoadgina am gundma. sakiTxi 4. Seadginon amocana teqstis saxiT, Tan daurTon amoxsna. swori pasuxi fasdeba 2 quliT.

25

gamarjvebulad CaiTvleba is gundi, romelsac meti qula daugrovdeba. savarjiSoebis komentarebi da pasuxebi:

#10. pirvel bidonSi (20 34

10 )l = 9 14

l rZea, meoreSi 20l1 1

(6 9 )2 4

+ l= 4 14

l, mesameSi

(201

132

)l = 162

l.

#13. wiladebi: 1

3 da

2

3 unda daviyvanoT mniSvnel 15-mde.

1

3=

5

15;

2

3=

10

15; saZiebeli

wiladi metia 5

15-ze da naklebia

10

15-ze. misi mniSvneli 15-is tolia. aseTi wiladebs

Soris udidesia 9

15. #15. pasuxi:

173

. #16. pasuxi:449

#17. pasuxi:1220

.

#18. miTiTeba: TiToeuli Sesakrebi warmovadginoT ori wiladis sxvaobis saxiT,

magaliTad, 1 1 1

2 3 2 3= ,

1 1 16 7 6 7

= . pasuxi: 9

10.

1.5 wiladebis gamravleba da gayofa mizani: wiladebis gamravlebisa da gayofis wesebis aTviseba. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes wiladebis gamravlebisa da gayofis wesebi da SeeZlos maTi gamoyeneba wiladebis, Sereuli ricxvebisa da naturaluri ricxvebis gamravleba-gayofisas. meToduri komentarebi: vinaidan ricxvis nawili jer ar Semogvitania, am punqtSi wiladze gamravleba ganixileba rogorc moqmedeba, romlis daxmarebiTac amoixsneba iseTive Sinaarsis amocanebi, rogoric ixsneba naturalur ricxvze gamravlebiT (gamravlebas, rogorc ricxvis wiladis sapovnel moqmedebas, mogvianebiT ganvi-xilavT).

yuradReba unda gamaxvildes imaze, rom umravles SemTxvevaSi ukeTesia jer Caiweros mricxvelebis da mniSvnelebis namravlebi, Seikvecos, rac Sesakvecia da Sekvecis

Semdeg Sesruldes gamravleba. magaliTad, 16 45 16 45 4 5 20

27 28 27 28 3 7 21

= = =

.

savarjiSoebis komentarebi da pasuxebi:

#4. 1 6 3 6 3 18 3

1 3 35 5 1 5 1 5 5

= = = =

(lari) = 3lari 60TeTri.

pasuxi: sami kilogrami brinji Rirs 3 lari da 60 TeTri. #11. pasuxi: 96km/sT. #12. Tbilisidan TelavSi Casasvlelad saWiroa 180

8100

l= 180 8100

l=

18 4

5

l=

72

5l=

214

5l benzini.

#14. sasurvelia, amocana moswavleebma jgufurad amoxsnan, raTa meti saSualeba hqondeT msjelobis, Tavisi azris gamoTqmisa da dacvisaTvis. maswavlebelma SeiZleba saamisod gakveTilis bolos ramdenime wuTi gamoyos, klasi dayos jgufebad da misces es amocana amosaxsnelad. pasuxs warmoadgens da daicavs jgufidan erTi moswavle. pasuxis xarisxi ganisazRvreba imiT, Tu vin ufro swrafad da sworad amoxsna amocana, vin ufro damajereblad asabuTebs Sesrulebul moqmedebebs. amoxsna. amocanis pirobiT laSam unda iyidos 6l `civi Cai~. mas aqvs SesaZlebloba: iyidos 4 didi boTliT 6l `civi Cai~, an 12 patara boTliT, amdenive `civi Cai~.

4 didi boTli `civi CaisaTvis~ saWiroa 41

15

lari, anu 4 45

lari, xolo 12 patara

boTlisaTvis 6 lari. fasebis Sedareba cxadyofs, rom laSas vurCioT, `civi Cai~ didi

boTlebiT iyidos. Cveni rCevis mixedviT laSa gaakeTebs 6lari4 45

lari=1

15

laris

26

ekonomias. #15. pasuxi: d) #17. miwis nakveTis erTi kvadratuli metris dasamuSaveb-

lad saWiroa 1

: 102

g= 1

20g xsnari. gamoviTvaloT ramdeni kilogrami xsnari iqneba

saWiro mocemuli miwis nakveTisaTvis: 1

14

1000 25

1000 120

= 5 1000 2 1000 14 5 20

=1000 100

1000 25 250004

= = g=25kg pasuxi: dasWirdebaT 25 kilogrami xsnari. 1.6 aTwiladebis Sekreba da gamokleba mizani: vaswavloT aTwiladebi, maTi Sekreba-gamokleba da Sedareba. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes aTwiladebis wakiTxva da Cawera, SeeZlos aTwiladis Tanrigis dasaxeleba, aTwiladis warmodgena Tanrigobrivi Sesakrebebis jamis saxiT, sxvadasxva saxiT Cawerili ricxvebis Sedareba, aTwiladebis Sekreba da gamokleba. meToduri komentarebi: aTwiladebis sworad Cawerisa da wakiTxvis dauflebaSi maswavlebels karg daxmarebas gauwevs TamaSi, romelSic mTeli klasi monawileobs. maswavlebels dafasTan gamohyavs 5-6 bavSvi, maTgan erTs aZlevs qaRaldze daweril mZimes, xolo danarCenebs ricxvebs. TamaSis wesi aseTia: dafasTan gasul moswavleebs ukaviaT ricxvebi. maT Soris dgeba mZimis niSani. moswavleebi xmamaRla kiTxuloben miRebul ricxvs. Semdeg mZimis niSani icvlis adgils, miiReba axali ricxvi, axla am ricxvs kiTxuloben. icvlis adgils romelime cifri, axla am ricxvs kiTxuloben da a.S.

Tu romelime moswavles uWirs iseTi wiladis Cawera aTwiladis saxiT, romlis mniSvneli 10-is xarisxia, maSin SegviZlia mivmarToT Semdeg xerxs: wiladis mricxvels marcxnidan imdeni nuli mivuweroT, rom cifrebis raodenoba mricxvelSi da nulebis raodenoba mniSvnelSi Tanabari iyos. amis Semdeg davwerT mTel nawils, davsvamT mZimes

da marjvniv mivuwerT nulebian mricxvels. magaliTad, 7 007

5 5 5, 0071000 1000

= = . aTwiladis saxiT Cawerili ricxvebis Sedarebisas moswavleebi xSirad uSveben Secdomas. naturaluri ricxvebis Sedarebisas isini miCveulni arian, metad CaTvalon is ricxvi, romlis CaweraSic meti cifria. magaliTad, radgan 452891>96857, amitom Tvlian, rom 4,52891>9,6857. am Secdomas Tavidan avicilebT, Tu moswavleebs Sedarebas davawyebinebT erTnairi mTelis mqone aTwiladebiT, Semdeg erTnairi meaTedebiT da a.S. kargi iqneba am dros ricxviTi sxivis gamoyenebac. gansakuTrebuli yuradReba unda mieqces imas, rom sidideebis Sekreba da gamokleba SeiZleba Sesruldes mxolod maSin, roca isini erTsa da imave erTeulebSi iqnebian mocemuli. savarjiSoebis komentarebi da pasuxebi: #6. mwvervalebi simRleebis klebis mixedviT rom davalagoT, miviRebT aseT mimdevrobas: ialbuzi: 5,642km, Sxara: 5,642km0,574km=5,068km, myinvarwveri: 5,068km0,035km=5,033km. #8. amocanis ganxilvamde maswavlebelma sasurvelia ganixilos wylis zedapirze moZraobis sxvadasxva SemTxvevebi. #9. am SemTxvevaSi umjobesia Cveulebrivi wiladebis aTwiladis saxiT Cawera da Semdeg Sedareba. #11. pasuxi: 2kg; #12. amocanis pirobiT erT gadaxerxvas 2,5 wuTi sWirdeba. ficris oTx nawilad gayofas sami gadaxerxva sWirdeba, razec (2,5+2,5+2,5)wT, anu 7,5 wuTi daixarjeba. pasuxi: ficris oTx nawilad gadaxerxvas 7,5 wuTi dasWirdeba. #13. Tu naxevari sazamTro 5,4 kg-s iwonis, mTeli 10,8kg-s aiwonis. #14. wiwilis wonaa (100-5):2gr=47,5gr, kvercxisa ki 52,5gr. 1.7. aTwiladebis gamravleba da gayofa mizani: vaswavloT aTwiladebis gamravleba da gayofa. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes aTwiladis aTwiladze da naturalur ricxvze gamravleba da gayofa, aTwiladis aTwiladze

27

gayofa, zepirad asrulebdes aTwiladis 10-ze, 100-ze da a. S., agreTve 0,1-ze, 0,01-ze da a. S. gamravlebasa da gayofas. unda SeeZlos Cveulebrivi wiladis aTwiladad gadaqceva. meToduri komentarebi: teqstSi gamravlebis ori gansxvavebuli tipis magaliTia ganxiluli: aTwiladis gamravleba naturalur ricxvze da aTwiladis gamravleba aTwiladze. gayofis magaliTebic ori tipisaa: aTwiladis gayofa naturalur ricxvze da aTwiladis gayofa aTwiladze. aqac warmatebiT SeiZleba iqnes gamoyenebuli wina paragrafSi ganxiluli TamaSi: Tu mZimis niSani ori TanrigiT marjvniv gadava, maSin ricxvi 100-jer gaizrdeba. aTwiladis 0.01-ze gamravlebisas mZimis niSani (moswavle, romelsac mZime ukavia) ori TanrigiT (moswavliT) marcxniv gadava da a. S. savarjiSoebis komentarebi da pasuxebi: #15. turists Suagzamde gasavleli aqvs kidev 9,6km. mTeli gzis sigrZe iqneba (3,2+9,6)2km =12,8km 2 =25,6km. SeiZleba gzis sigrZe gamoiTvalos gamosaxulebiT: 43,22km=25,6km. #16. amocanis ganxilvisas urCieT moswavleebs, amocanis pirobaSi mocemuli dro wuTebSi gadaiyvanon. #17. erTi rveuli Rirs: 1,8 : (74)=0,6 (lari)=60 TeTri. #19. 4-jer naklebi gzis gavlas avtomobili moandomebs 4-jer nakleb dros, anu 3,2:4sT=0,8sT-s. 2-jer meti siCqariT 2-jer nakleb dros daxarjavs. #20. pasuxi: 3 (12,5 5 12,5) 225 + = km. #21. saZiebel dros gamoviTvliT gamosaxulebiT: 4.5 : (6,34,5). pasuxi: daeweva 2,5 saaTSi. 1.8 ricxviTi gamosaxulebis mniSvnelobis gamoTvla mizani: ganvumtkicoT moswavles moqmedebaTa Tanmimdevrobis wesebis codna. gamovumu-SaoT mas moqmedebaTa Tvisebebis gamoyenebis unari ricxviTi gamosaxulebis mniSvnelobis gamosaTvlelad. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes moqmedebaTa Tanmimdevrobis wesebi da iyenebdes maT ricxviTi gamosaxulebis mniSvnelobis gamosaTvlelad. unda SeeZlos gamoTvlis optimaluri gzis SerCeva da ariTmetikul moqmedebaTa Tvisebebis gamoyeneba ricxviTi gamosaxulebis mniSvnelobis gamoTvlisas, amocanis Sesabamisi ricxviTi gamosaxulebis Sedgena da amocanis Sedgena ricxviTi gamosaxulebis mixedviT. meToduri komentarebi: es paragrafi or gakveTilzea gaTvaliswinebuli. umjobesia, pirvel gakveTilze gavimeoroT moqmedebaTa Tanmimdevrobis wesebi da mivceT saTanado magaliTebi. meore gakveTilze SevaxsenoT ariTmetikul moqmedebaTa Tvisebebi da SevasrulebinoT saTanado savarjiSoebi. moswavleTa yuradReba unda gamaxvildes imaze, rom moqmedebaTa Tvisebebis gamoyeneba xSirad amartivebs gamoTvlebs. gansakuTrebuli yuradReba unda mieqces gamoTvlebis Sesrulebisas swori gzis arCevisa da zepiri angariSis unarebis gamomuSavebas. amis samagaliTod gTavazobT gakveTilis nimuSs.

gamovTvaloT 23 31 3 5 3 748 : :36 63 4 21 5 8

+ + ariTmetikuli gamosaxulebis mniSvne-loba oTxi xerxiT:

a) moqmedebaTa Cveulebrivi TanmimdevrobiT:

1) 23 31 161 124 285 95 111 ;36 63 252 252 84 84

++ = = = = 2) 3 5 63 20 834 21 84 84

++ = = ;

3) 11 83 95 83 12 1 1 481 ; 4) 4884 84 84 84 7 7 7

= = = = ;

5) 3 7 3 8 24:5 8 5 7 35

= = ; 48 24 48 356) : 2 5 107 35 7 24

= = = .

b) SevTavazoT moswavleebs gamoTvalon es gamosaxuleba 4 moqmedebiT, magaliTad:

28

23 31 3 5 23 31 3 5 161 124 189 60 36 136 63 4 21 36 63 4 21 252 252 7

+ + + = + = = = , xolo Semdeg Seasrulon a) xerxiT gamoTvlaSi miTiTebuli 4), 5) da 6) punqtebi.

g) moswavleTa yuradReba mivaqcioT im faqts, rom 4), 5) da 6) punqtebSi aRwerili sami moqmedebis nacvlad SegviZlia SevasruloT ori moqmedeba:

3 7 24:5 8 35

= da 1 2448 2 5 107 35

= = , e. i gamosaxuleba gamovTvaloT 3 moqmedebiT.

d) SesTavazeT klass amoxsnis kidev ufro mokle gzis moZebna. magaliTad:

1)23 31 3 5 161 124 189 60 36 136 63 4 21 252 252 7

+ + = = = 2) 1 7 3 1 48 7 548 : 2 5 107 8 5 7 8 3

= = = . erTi da igive gamosaxuleba 4 gansxvavebuli gziT gamovTvaleT, amasTan yoveli momdevno gza ufro racionaluri iyo, vidre wina. savarjiSoebis komentarebi da pasuxebi: #3. ori rveuli eRireba 0,6 2 lari, 3 fanqari 1,4 3 lari, yvelaferi erTad ki ( 0,6 2 1,4 3 + ) lari. pasuxi: ekam gadaixada 5,4 lari. #4. marTkuTxedis farTobi misi sigrZisa da siganis namravlis tolia. namravli gaizrdeba imdenjer, ramdenjerac gavzrdiT TiToeul Tanamamravls. amitom im

nakveTis farTobi, romlis sigrZe 2,5-jer metia, xolo sigane 213

-jer naklebia, iqneba

21200 2,5 :1 18003

= kv.m. kv.m.

#5. pasuxi: orive qvevrSi erTad aris 3 32 2 :1,35 5

+ t= 4,6t Rvino.

#6. a) 45,6788,789

36,889

b) 2,4418,2353,322

13,998

+

#7. mocemuli ricxviTi gamosaxulebebi 3, 4 da 5 amocanebSi miRebuli gamosaxulebebis analogiuria. 1.9 kalkulatori mizani: SevaswavloT moswavles kalkulatorze gamoTvlebis efeqturad Sesruleba. codna da unarebi: paragrafis Seswavlis Sedegad moswavlem unda icodes kalkulato-ris klaviSebis daniSnuleba, unda SeeZlos kalkulatorTan muSaobisas moqmedebaTa Tanmimdevrobis wesebis dacva, mexsierebis efeqturi gamoyeneba, kalkulatoriT ricxviTi eqsperimentebis Catareba, ricxviTi gamosaxulebis aRdgena, romlis gamoTvlac sqemiTaa mocemuli. resursebi: kalkulatori. meToduri komentarebi: moswavleebi xalisiT asruleben gamoTvlebs kalkulatoris daxmarebiT. magram yvela maTgani ver iyenebs mas efeqturad. magaliTad, 17,2 13 14,5 18 + mniSvnelobis gamosaTvlelad umravlesoba Semdegnairad iqceva: gamoTvlis pirvel namravls, amowers mas furcelze, Semdeg daiTvlis meore namravls da miumatebs furcelze amoweril ricxvs. mivceT moswavleebs msgavsi gamosaxuleba da SevTavazoT, gamoTvalon igi Sualeduri pasuxis amoweris gareSe, risTvisac gamoiyenon mexsierebasTan samuSao M+, M- da MR klaviSebi.

sanam uSualo gamoTvlebs Seasruleben, moswavleebs SevTavazoT, jer Sesabamisi sqema Seadginon. mniSvnelovania rogorc mocemuli gamosaxulebis gamosaTvleli sqemis Sedgenaze (savarjiSo 3), ise sqemis mixedviT gamosaTvleli gamosaxulebis

29

aRdgenaze (savarjiSo 8) varjiSi. am savarjiSoebis Sesasruleblad kalkulatori ar dagvWirdeba. savarjiSoebis komentarebi: # 7. amocanaSi mocemul sqemaSi araa daculi moqmedebaTa Tanmimdevroba, # 8. a) pasuxi: 0,5 - 3,21,5 # 9. d) kalkulatorze SevasruloT moqmedebebi (9999 23) : 29 . miviRebT 344-s, e.i. winadadeba samarTliania. I Tavis mimoxilva

#9. amoxsna: matarebels dasWirdeba 160 160 2 1,6 3,680 100

+ = + = saaTi. #12. quTaisis mimarTulebiT mimavalma avtobusma Sexvedramde erTi saaTiT meti imoZrava, amitom mis mier 60 km-is gavlis Semdeg orive avtobusis mier gavlili manZili iqneba 240km-60km=180km. isini erTmaneTs uaxlovdebodnen 60km/sT+75km/sT=135km/sT siCqariT, erTmaneTs Sexvdebodnen quTaisidan gamosuli avtobusis gamosvlidan

180km:135km/sT=113

sT-Si, anu 1saaTsa da 20 wuTSi, rac imas niSnavs, rom avtobusebi

erTmaneTs 11saaTsa da 20 wuTze Sexvdnen. #13. 70 2 5 7, 60 2 2 3 5.= = usj(70;60)=420. mamisa da Svilis nabijebi yoveli 420sm-is Semdeg 5-jer daemTxva, amave dros, siaruli erTidaimave xidan daiwyes, amitom xeebs Soris manZili yofila 420 4 1680 = sm, anu 16,8 metri. #14. Tu saZiebel ricxvs davumatebT 1-s, maSin miRebuli ricxvi unaSTod gaiyofa 2-ze, 3-ze, 4-ze, 5-ze, 6-ze, 7-ze, 8-ze, 9-ze, 10-ze. aseTi umciresi ricxvi aris 4 7 9 10 2520 = , xolo masze 1-iT naklebi ricxvi 2519. sakontrolo weris nimuSi I varianti savaldebulo nawili

1. Sekvece wiladebi: 1 24 2

, 355 6

, 6 07 2

. romeli maTgania udidesi?

2. aTwiladis saxiT Cawere ricxvebi: 3610

, 310000

, 35

. maT Soris romelia umciresi?

3. gamoTvale 1 514 7,2 (1 ) :1,23 6

+ gamosaxulebis mniSvneloba. 4. nanam SeiZina 2,5 kilogrami kexura vaSli, 1 3

4kg gulabi msxali da erTi sazamTro,

romelic orjer ufro mZimea, vidre vaSli da msxali erTad. ramden kilograms iwonis sazamTro? damatebiTi amocana eqskursiaze wasasvlelad Sekrebilia 63 gogona da 72 biWi. ramden 18-adgilian mikroavtobusSi unda gavanawiloT mgzavrebi ise, rom TiToeulSi rogorc gogonebis, ise biWebis raodenoba erTnairi iyos? ramdeni gogona da ramdeni biWi iqneba TiToeul avtobusSi?

. II varianti savaldebulo nawili

1. Sekvece wiladebi: 2135

, 1872

, 6193

. romeli maTgania udidesi?

30

2. aTwiladis saxiT Cawere ricxvebi: 13

1000,

25

, 56

100. maT Soris romelia umciresi?

3. gamoTvale 1 523 8,4 ( 1 ) :1,43 6

+ gamosaxulebis mniSvneloba.

4. nodaris deda 50,3 kilograms iwonis, mama ki 7634

kilograms. ramden kilograms

iwonis nodari, Tu igi 4-jer ufro msubuqia, vidre misi ded-mama erTad? damatebiTi amocana saswavlo wlis dawyebasTan dakavSirebiT raionis xelmZRvanelobam Tavisi raionis skolebisaTvis saCuqrebis gadacema gadawyvita. SeiZines 90 TeTri da 75 yavisferi burTi. saCuqari yuTebiT gadaeca raionis yvela skolas, amasTanave, yvela yuTis SigTavsi erTnairi iyo. daadgine, ramdenma skolam miiRo saCuqari, Tu maTi raodenoba orniSna ricxviT gamoisaxeba? ramdeni TeTri da ramdeni yavisferi burTi ergo TiToeul skolas?

Sefasebis sqema savaldebulo nawili 1. 0,5 qula TiTo wiladis SekvecisaTvis.

0,5 qula swori pasuxisaTvis. maqsimalur qulaTa raodenoba 2

2. 0,5 qula TiTo wiladis gardaqmnisaTvis. 0,5 qula swori pasuxisaTvis. maqsimalur qulaTa raodenoba 2

3. 1 qula moqmedebaTa rigis dacvisaTvis. 4. 0,5 qula sworad Sesrulebuli TiToeuli moqmedebisaTvis. maqsimalur qulaTa raodenoba 3 5. 1,5 qula Sesabamisi ricxviTi gamosaxulebis CawerisaTvis

1 qula ricxviTi gamosaxulebis mniSvnelobis gamoTvlisaTvis 0,5 qula sruli pasuxis dawerisaTvis an , safexurebad amoxsnis SemTxvevaSi: 0,5 qula gamarTuli msjelobisaTvis 0,5 qula I moqmedebis SerCevisaTvis 0,5 qula I moqmedebis SesrulebisaTvis 0,5 qula II moqmedebis SerCevisaTvis 0,5 qula II moqmedebis SesrulebisaTvis 0,5 qula sruli pasuxis dawerisaTvis maqsimalur qulaTa raodenoba 3

testi

II Tavi geometriis sawyisi cnebebi (14 saaTi) Tavis ZiriTadi mizani: gaacnos moswavleebs umartivesi geometriuli figurebi. Seaswavlos maTi zogierTi Tviseba. codna da unarebi: mocemuli Tavis Seswavlis Sedegad moswavles SeuZlia umartivesi geometriuli figurebis: wertilis, wrfis, sibrtyis, sxivis, monakveTis, kuTxis, texilis, mravalkuTxedis amocnoba, dasaxeleba, daxazva, maTi elementebis aRniSvna, miTiTeba da gazomva, figurebis klasifikacia. SeuZlia Seswavlil figuraTa martivi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25g a e d g g b d a g b e d a a b d g a g e a b b a

31

Tvisebebis Camoyalibeba, maT Sesaxeb msjeloba da saWiro SemTxvevaSi dasabuTeba. meToduri komentarebi: am TavSi moswavle pirvelad ecnoba geometriul obieqtebs,

rogorc abstraqtul cnebebs, acnobierebs mimarTebas realur sagnebsa da maT geometriul modelebs Soris (burTi birTvi). aseTi mimarTebis kerZo saxea naxazi, rogorc geometriuli figuris gamosaxuleba. Tavis erT-erTi ZiriTadi daniSnulebaa, moswavles Seaswavlos geometriuli naxazebis sworad aRqma da Sesruleba. amasTan moswavlem unda SeZlos saWiro SemTxvevaSi (amocanis konteqstidan gamomdinare) Seasrulos rogorc sqematuri naxazi, aseve naxazi zomebis zusti dacviT. sqematuri naxazis Sesrulebisas yuradReba unda gamaxvildes naxazis elementebis zomebs Soris swori Tanafardobis dacvaze (magaliTad, samkuTxedis udidesi gverdi an kuTxe naxazzec udidesi unda iyos).

Tavis mizania agreTve, ganumtkicos moswavles geometriuli xelsawyoebiT saxaza-viT, fargliT da transportiriT sargeblobis Cvevebi. Seaxsenos, xolo saWiro SemTxvevaSi Seaswavlos gazomvebis warmoeba am xelsawyoebiT. im SemTxvevaSi, rodesac gazomvebi kompleqsur xasiaTs atarebs, magaliTad texilis sigrZis an mravalkuTxedis perimetris povnisas, sasurvelia, moswavlem winaswar Camoayalibos sakuTari moqmedebebis gegma.

saxelmZRvaneloSi Cven Tavs vikavebT geometriis, rogorc formalur- deduqciuri sistemis agebis mcdelobisgan, radgan igi naadrevad migvaCnia. vTvliT, rom swavlebis am etapze pirvel rigSi aucilebelia moswavles gamoumuSavdes intuiciuri warmodgenebi geometriuli figurebisa da geometriuli gardaqmnebis Tvisebebis Sesaxeb. amitom geometriuli cnebebisa da faqtebis gacnoba xdeba TvalsaCinoebebze, kerZod naxazebze demonstrirebiT. amasTan erTad saxelmZRva-neloSi TandaTanobiT xdeba msjelobis, rogorc geometriuli faqtebis dasabuTebis instrumentis, Semotana. geometriuli faqtebis damtkiceba arsebiTad eyrdnoba geometriuli gardaqmnebis, kerZod paraleluri gadatanisa da RerZuli simetriis gamoyenebas. ukve mocemul TavSi xdeba monakveTebisa da kuTxeebis tolobis Semotana moZraobebze dafuZnebiT. aseTi tipis geometriuli msjelobebis Catarebisas sasurvelia TvalsaCinoebis gamoyeneba rogorc maswavleblis, ise moswavlis mier.

yoveli Temis damTavrebis Semdeg sasurvelia CaTvlebis Catareba. magaliTad, meore Tavis Seswavlisas SeiZleba Catardes 3 CaTvla:

2.4 paragrafis Semdeg 2.8 paragrafis Semdeg Tavis dasrulebisa da sakontrolo weris Semdeg.

ras Seiswavlis geometria

es paragrafi Tavis Sesavals warmoadgens. mizani: gavacnoT moswavleebs geometriis Seswavlis ZiriTadi obieqtis geomet-

riuli figuris cneba; davaxasiaToT umartivesi geometriuli figurebi wertili da sibrtye; mivuTiToT brtyel da sivrcul figurebs Soris gansxvavebaze; avuxsnaT geometriis mniSvneloba da misi gamoyenebis sferoebi.

codna da unarebi: paragrafis Seswavlis Sedegad moswavles SeuZlia, realur saganTa geometriuli Tvisebebi gamoarCios maTi sxva Tvisebebisagan, gaacnobieros mimarTeba realur sagnebsa da maT geometriul modelebs Soris (msgavseba gansxvaveba), miuTiTos wertilisa da sibrtyis damaxasiaTebeli Tvisebebi, gaarCios erTmaneTisgan brtyeli da sivrculi geometriuli figurebi, daasaxelos misTvis nacnobi geometriuli figurebi da realur situaciaSi miuTiTos Sesabamisi formis sagnebi.

meToduri komentarebi: miuxedavad imisa, rom mocemuli paragrafi saxelmZRva-nelos geometriuli nawilis garkveul Sesavals warmoadgens, masSi ganxilulia sxvadasxva mniSvnelovani sakiTxebi, romlebzec aucilebelia yuradRebis gamaxvileba:

geometriuli figurebi abstraqtuli, warmosaxvaSi arsebuli obieqtebia, romelTac realuri sagnebis mxolod zogierTi Tviseba aqvT.

umartivesi geometriuli figuraa wertili, sxva figurebi wertilebisagan Sedgeba.

geometriuli figurebi iyofa brtyel da sivrciT geometriul figurebad. paragrafisadmi miZRvnili gakveTilis dawyeba sasurvelia saubriT, sadac

32

moswavleebi imsjeleben yoveldRiur saqmianobaSi saganTa formebis, zomebisa da maTi urTierTganlagebis mniSvnelobaze. gaixseneben maTTvis nacnob geometriul formebs, isaubreben maT kavSirze realurad arsebul sagnebTan. savarjiSoebis komentarebi da pasuxebi: #1 am savarjiSos Sesaxeb msjelobisas SeiZleba daisvas damatebiTi kiTxva: aqvT Tu ara saerTo geometriuli Tviseba suraTze gamosaxul sagnebs. igive kiTxvebi SeiZleba davsvaT paragrafSi moyvanili im suraTis mimarT, sadac sxvadasxva zomis burTebia gamosaxuli. #4 aseTi nagebobebis yvelaze cnobili magaliTi, cxadia, piramidebia, Tumca moswavlebma SeiZleba sxva magaliTebic moiyvanon, magaliTad, romis kolizeumi, romelsac cilindris forma aqvs. moswavleebs SeiZleba damatebiT davalebad mieceT pasuxSi moiyvanon romelime aseTi nagebobis zomebi, fotosuraTi an naxati. davalebis garCevisas, sasurvelia, moswavleebs uCvenoT plakati romelime aseTi nagebobis gamosaxulebiT. saerTod am davalebis irgvliv msjeloba SeiZleba Catardes, rogorc saubari geometriis rolze arqiteqturaSi. #5 am davalebis garCevisas SeiZleba moswavleebs mouyvanoT geometriis gamoyenebis magaliTi romelime sferoSi. magaliTad, esaubroT leonardo da vinCize da geometriis rolze mis SemoqmedebaSi. agreTve xatvisas perspeqtivis kanonebis dacvis mniSvnelobaze. aCvenoT da erTmaneTs SeadaroT suraTebi, romlebic Seqmnilia perspeqtivis kanonebis dacvis gareSe (magaliTad, Zvelegvipturi gamosaxulebebi) da maTi dacviT. Tu klasma daintereseba gamoxata, SeiZleba mouyveT filipo di ser bruneleskos (Filippo Di Ser Brunellesco, 1377-1446) dakvirvebebis Sesaxeb, romelTac perspeqtivis kanonebis aRmoCena mohyva. 2.1 wrfe

mizani: gavacnoT moswavleebs axali geometriuli figura wrfe, SevaswavloT misi aRniSva da gamosaxva geometriul naxazze. aRvweroT gadamkveTi wrfeebis ZiriTadi Tviseba.

codna da unarebi: paragrafis Seswavlis Sedegad moswavles SeuZlia aagos wrfis naxazi, ganasxvaos wrfeze mdebare da aramdebare wertilebi da Caweros es mimarTebebi simravleTa Teoriis aRniSvnebis gamoyenebiT. aRniSnos da daasaxelos wrfe misi ori wertilis mixedviT. man icis da iyenebs or wertilze gamavali wrfis erTaderTobis Tvisebas.

resursebi: saxazavi. meToduri komentarebi: am paragrafSi pirvelad Semodis naxazi, rogorc geomet-

riuli masalis gadmocemis, amocanis Camoyalibebisa da amoxsnis erT-erTi saSualeba. naxazis sworad agebisa da interpretaciis unars mniSvnelovani roli eniWeba moswavlis geometriuli warmodgenebis CamoyalibebaSi.

aqedan gamomdinare, gaZlierebuli yuradReba unda daeTmos moswavleebSi am unaris Camoyalibebas. rogorc maswavleblis, aseve moswavlis mier amoxsnili amocana, rogorc wesi, ilustrirebuli unda iyos faqizad Sesrulebuli naxaziT. yuradReba unda mieqces, raTa moswavleebma naxazis Sesrulebisas maqsimalurad gamoiyenon saxazavi xelsawyoebi: saxazavi, fargali da transportiri.

amasTan erTad, saWiroa yuradReba gamaxvildes im faqtze, rom naxazi geometriuli obieqtis, kerZod wrfis, miaxloebiTi gamosaxulebaa, amitom figuris Tvisebebis dadgenisas naxazze dayrdnobas zogjer Secdomebamde mivyavarT. magaliTad, Tu naxazze wrfeebi ar ikveTeba, es jer kidev ar niSnavs, rom isini paraleluria (maTi gagrZelebebi SeiZleba gadaikveTon!).

amave paragrafSi pirvelad Semodis simravleTa Teoriis simboloebi da (`ekuTvnis~ da `ar ekuTvnis~). amjerad is mxolod geometriul konteqstSi ixmareba. momavalSi Cven am simbolos mniSvenelobas gavafarToebT, magram saWiroa moswavleebi Tavidanve mieCvion mis sworad dasaxelebas: Tu gamoyenebulia aRniSvna P a , es ase ikiTxeba: ` P wertili ekuTvnis a wrfes~ da ara ` P wertili mdebareobs a wrfeze~ an ` a wrfe gadis P wertilze~. savarjiSoebis komentarebi da pasuxebi: #1 am amocanaSi moswavles ar moeTxoveba sakuTari naxazis gakeTebisas zustad daicvas me-5 naxazze mocemuli manZilebi wertilebs Soris, sakmarisia, naxazi

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sqematurad Sesruldes ise, rom am wertilebze gamaval wrfeTa yoveli wyvili ikveTebodes. #2 aseTi wertili SeiZleba ar arsebobdes, an arsebobdes erTi, ori an sami wertili (ixile nax. 2.1, sadac moyvanilia yvela SemTxveva). TiToeuli napovni SemTxvevisTvis moswavlem unda warmoadginos Sesabamisi naxazi. #3 arsebobs sami aseTi wertili (ixile nax. 2.2). #4 nax. 2.2 - ze moyvanilia wrfeebisa da wertilebis amocanaSi moTxovnili ganlageba. #6 pasuxis (AC da BC wrfeebi erTmaneTs emTxveva) dasasabuTeblad moswavlem unda gaixsenos paragrafSi moyvanili wrfis Tviseba, agreTve aagos Sesabamisi naxazi. #7 oTx wertilze SegviZlia gavataroT 1, 4 an 6 wrfe. Sesabamisi SemTxvevebi ix. nax. 2.3-ze.

imis dasabuTeba, rom oTx wertilze 2, 3 , 5 an 6-ze meti wrfis gatareba ar SeiZleba, garkveuli msjelobis Catarebas moiTxovs. Tu klasis momzadebis done amis saSualebas iZleva, maswavlebelma amocanis garCevisas SeiZleba romelime an yvela SemTxvevisTvis es msjeloba Caataros.

radgan wertilTa wyvilebis raodenoba sul eqvsia, amitom wertilTa am wyvilebze 6-ze met wrfes ver gavatarebT.

Tu yvela wertili erT wrfeze mdebareobs, maSin gvaqvs nax. 2.3 - is a) SemTxveva da am wertilebze gamavali erTaderTi wrfe.

Tu romelime sami wertili erT wrfeze mdebareobs, xolo meoTxe am wrfes ar ekuTvnis, maSin gvaqvs nax. 2.3-is b) SemTxveva. am wertilebze oTxi wrfe gadis.

dabolos, Tu arc erTi sami wertili erT wrfeze ar mdebareobs, maSin wertilTa yovel wyvils gansxvavebuli wrfe Seesabameba. aseTi wyvilebi ki, rogorc viciT, sul eqvsia, amitom am SemTxvevaSi mocemul wertilebze eqvsi wrfe gaivlis (nax. 2.3-is g) SemTxveva).

SeiZleba agreTve daisvas am amocanis gamartivebuli varianti roca wertilebis raodenoba samia.

#8 iseve, rogorc #6 amocanis SemTxvevaSi, am amocanaSi moyvanili meTodis dasabuTebac emyareba or wertilze gamavali wrfis erTaderTobas. #9 polaruli varskvlavi fotosuraTze gamosaxuli kaSkaSa varskvlavebidan yvelaze

a b c

a b

c

a c b

a

b c

a)

g )

b)

d) nax. 2.1

a

c b

P R

Q

nax. 2.2

A B C D A B C

D A

D

B

C

a) b) g)

nax. 2.3

34

marcxenaa. saerTod am amocanasTan dakavSirebiT maswavlebelma SeiZleba isaubros maTematikisa da, kerZod, geometriis rolze astronomiaSi (magaliTad, geometriuli meTodebiT ciur sxeulebamde manZilis dadgenaze). 2.2 sxivi da monakveTi

mizani: gavacnoT moswavleebs sxivi, monakveTi da texili, maTi aRniSvna da gamosax-va naxazze.

codna da unarebi: paragrafis Seswavlis Sedegad moswavles unda SeeZlos sxivisa da monakveTis aRniSvna da gamosaxva naxazze, mocemuli monakveTis an sxivis Semcveli wrfis daxazva, texilis daxazva, aRniSvna da miTiTeba.

resursebi: saxazavi. meToduri komentarebi: cnebas `sxivi~ garda geometriuli mniSvnelobisa, fizikuri

da sayofacxovrebo mniSvnelobac gaaCnia: `sinaTlis sxivi~. sasurvelia, gakveTilis msvlelobaSi es kavSiri miTiTebul iqnes: sxivi warmosaxvaSi arsebobs, magram realurad arsebuli sagnis sinaTlis sxivis Tvisebebi aqvs (ix. saxelmZRvaneloSi moyvanili fotosuraTi, agreTve me-7 amocana).

arsebiTia, moswavlem Tavidanve gaacnobieros is faqti, rom monakveTi da sxivi wrfis nawilebia, amisaTvis SeiZleba naxazze monakveTis an sxivis dasaxelebisas movTxovoT maTi Semcveli wrfis daxazva saxazaviT (monakveTis an sxivis `gagrZeleba~ wrfemde). saerTod aq, iseve rogorc wina paragrafSi, win wamoweulia simravluri Tanafardobani (wrfe-sxivi-monakveTi-wertili) da savarjiSoebis didi nawilic am sakiTxisadmi aris miZRvnili, Tumca am etapze simravleTa Teoriis terminologiisagan Tavs vikavebT. saWiroa moswavlem TiToeul SemTxvevaSi TvalnaTliv warmoidginos da Sesabamisi naxaziT daasuraTos figurebis urTierTganlageba (magaliTad, ra SemTxvevaSia ori sxivis saerTo nawili sxivi? monakveTi? wertili?).

savarjiSoebis nawilSi xdeba wrfis wina paragrafSi naswavli Tvisebis gameoreba mis axal cnebebTan dakavSirebis gziT (kiTxvebi 5, 6, amocanebi 5, 9). moswavleebma es kavSiri unda gaacnobieron, rac SeiZleba ganxorcieldes am amocanebis garCevisas Sesabamisi kiTxvebis dasmiT.

texilTan dakavSirebuli sakiTxebis gadmocemisas SegiZliaT moswavleebs dauxazoT monakveTebisagan Semdgari figurebi, romlebic ar warmoadgenen texils (magaliTad, orze met monakveTs saerTo wvero aqvs, an mezobeli monakveTebi erT wrfeze mdebareoben) da SesTavazoT, TiToeul SemTxvevaSi axsnan, ratom ar arian isini texilebi.

grZeldeba moswavleTaTvis geometriul naxazebSi aRniSvnebis sistemis gacnoba, kerZod Seiswavleba sxivis, monakveTisa da texilis aRniSvnebi. geometriuli obieqtebis sworad miTiTebis aTviseba am etapze erT-erTi mTavari mizania da savarjiSoebis nawilic (1, 2, 4 amocanebi) am sakiTxs eTmoba.

savarjiSoebis komentarebi da pasuxebi: am etapze ar moviTxovT amocanebis gaformebisas garkveuli wesebis dacvas (magaliTad, mocemulobis daweras), magram yoveli savarjiSos amoxsnas sasurvelia Tan axldes saxazavi xelsawyoebis saSualebiT Sesrulebuli naxazi. miT umetes, rom aseTi naxazis Sesruleba moiTxoveba amocanebis umravlesobis pirobiT. #5 amocanis garCevisas dasviT kiTxvebi: ramdeni saerTo wertili aqvT am monakveTebis Semcvel wrfeebs? SeiZleba Tu ara es wrfeebi gansxvavebuli iyos? #6 amocanis garCevisas maswavlebelma SeiZleba moiyvanos texilis araswori aRniSvna da dasvas kiTxva: ratomaa aRniSvna araswori? #7 amocana saSualebas aZlevs maswavlebels, kidev erTxel gausvas xazi kavSirs sinaTlis sxivsa da geometriul sxivs Soris. #8 amocanaze msjelobisas SeiZleba moswavleebs SesTavazoT, daTvalon wveroebi da gverdebi maT mier daxazul Sekrul da gaxsnil texilebSi. Seadaron erTmaneTs miRebuli Sedegebi, raTa TviTonve aRmoaCinon kanonzomiereba: wveroebis raodenoba Sekrul texilSi emTxveva gverdebis raodenobas, xolo gaxsnil texilSi erTiT metia. #9 amocanis amosaxsnelad sakmarisia msjeloba Catardes naxazze. sityva `damtkiceba~ aq ixmareba rogorc sinonimi frazisa `maCvene, ratom~. saWiro SemTxvevaSi maswavlebeli SeiZleba daexmaros moswavles kiTxvebiT: SeiZleba Tu ara, am monakveTebs Soris ar mdebareobdes texilis arc erTi monakveTi? mdebareobdes

35

texilis erTi monakveTi? ori monakveTi? rogoria naxazze monakveTebis ganlageba ukanasknel SemTxvevaSi? 2.3 monakveTis sigrZe

mizani: SevaswavloT moswavles monakveTis sigrZisa da or wertils Soris manZilis gazomva, texilis sigrZis povna.

codna da unarebi: paragrafis Seswavlis Sedegad moswavles unda SeeZlos monakveTis sigrZisa da or wertils Soris manZilis gazomva, mocemuli sigrZis monakveTis daxazva, farglis gamoyenebiT mocemuli monakveTis toli monakveTis ageba, texilis sigrZis gazomva. man unda icodes monakveTisa da texilis sigrZis Tviseba da iyenebdes maT amocanebis amoxsnisas.

resursebi: danayofebiani saxazavi, fargali. meToduri komentarebi: paragrafSi Semodis umniSvnelovanesi cnebebi: monakveTis

sigrZe da manZili or wertils Soris, monakveTebis toloba. TiToeuli am cnebis aTviseba, mizanSewonilia, moxdes naxazebze praqtikuli moqmedebebis safuZvelze, romelic yvela moswavlem unda Seasrulos. am moqmedebebs Soris pirvel rigSi unda Sesruldes monakveTis sigrZis gazomva saxazaviT.

im SemTxvevaSi, roca maswavlebeli xedavs, rom moswavleTa garkveulma nawilma ar icis, rogor Caataros gazomvebi saxazavis saSualebiT, man pirvel rigSi yuradReba unda miaqcios am unaris gamomuSavebas an aRdgenas. am SemTxvevaSi sasurvelia, sademonstracio gazomva Caataros maswavlebelma, xolo Semdeg moswavleebma maswavleblis xelmZRvanelobiT. maswavlebeli aRniSnavs gazomvisas daSvebul damaxasiaTebel Secdomebs. gazomvebis Sedegebis ganxilvisas maswavlebeli Seaxsenebs moswavleebs, rom gazomvis Sedegad miRebuli ricxvi damokidebulia ara marto gasazomi monakveTis sigrZeze, aramed zomis erTeulzec da sTxovs, gazomvis Sedegebi warmoadginon sxvadasxva erTeulebSi (santimetrebSi da milimetrebSi). aseve maswavleblis xelmZRvanelobiT SeiZleba Caataron moswavleebma saxazaviT mocemuli sigrZis monakveTis daxazva da farglis daxmarebiT mocemulis toli monakveTis ageba. farglis gamoyenebis unaris gamomuSaveba mniSvnelovania, radgan igi aucilebelia momavalSi fargliTa da saxazaviT agebebis sworad SesrulebisTvis.

toli monakveTebis cneba Semodis monakveTebis sigrZeebis Sedarebis safuZvelze: tolia monakveTebi, romelTac toli sigrZe aqvT. am faqtis demonstrireba SeiZleba farglis saSualebiT. monakveTTa tolobis es ganmarteba erTgvarad scdeba figuraTa tolobis zogad koncefcias, romelic am saxelmZRvaneloSia ganviTarebuli (Tumca mas ar ewinaaRmdegeba): tolia figurebi, romlebic SeiZleba moZraobiT erTmaneTs SeuTavsdes. amitom sasurvelia maswavlebelma moswavleTa yuradReba miaqcios im faqts, rom toli sigrZis monakveTebi `erTmaneTze dadebisas erTmaneTs emTxvevian~.

or wertils Soris manZilis cnebis axsnisas maswavlebeli SeiZleba daeyrdnos moswavleTa intuicias: magaliTad, miuTiTos ori wertili dafaze, sTxovos moswavleebs, uCvenon, romeli gziT SeiZleba erTi wertilidan meoremde miRweva umokles droSi da am sakiTxze gamarTuli diskusiidan bunebrivad gadavides ori wertilis SemaerTebeli umoklesi wiris miTiTebaze.

savarjiSoebis komentarebi da pasuxebi: #1. moswavleTa yuradReba miaqcieT im faqts, rom fotosuraTze gamosaxul saxazavze 7 duimis aRmniSvneli xazi daaxloebiT 17,5sm da 18sm aRmniSvneli xazebis SuaSia. Tu miviRebT, rom miaxloebiT 7duimi 17,7sm, maSin 1 duimi 17,7sm/7 2,53sm (zusti mniSvnelobaa 2,54sm). analogiuri tipisaa me-4 amocana II Tavis mimoxilvaSi. #2-#3. Tu moswavleTa momzadebis done amis saSualebas iZleva, SegiZliaT maT SesTavazoT, amoxsnan amocana im pirobis gareSe, rom AB, BC da AC monakveTebidan AC yvelaze grZelia. maSin gansaxilveli ori SemTxveva (C wertili mdebareobs AB monakveTze an mis gareT) mogvcems #2 da #3 amocanebs Sesabamisi pasuxebiT. amocanis garCevisas xazi gausviT im faqts, rom amocanis pirobas SeiZleba ramdenime sxvadasxva naxazi Seesabamebodes da Sesabamisad amocanis pasuxic gansxvavebuli iqneba. #5. es amocana #2-#3 amocanebis erTgvarad garTulebuli versiaa. zustad Sesrulebuli naxazis SemTxvevaSi (monakveTebis zomebis Tanafardobis dacviT) naxazidan pirdapir Cans pasuxi: 2/3.

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#6. 2.4 naxazze moyvanilia A, B, C da D wertilebis wrfeze ganlagebis ZiriTadi SesaZ-lebeli SemTxvevebi (yvela sxva SemTxveva am naxazebidan simetriiT miiReba) da TiToeul SemTxvevaSi miTiTebulia AD monakveTis sigrZe (a -Ti aRniSnulia saerTo mo-nakveTis sigrZe):

amocanis erT-erTi mizania, miaCvios moswavle amoxsnisas naxazis agebasa da mis gamoyenebas. amocana SeiZleba amoixsnas klasSi maswavleblis xelmZRvanelobiT. moswavleebs SesTavazeT, monaxon wertilebis SesaZlo ganlagebis rac SeiZleba meti varianti (sul 12 gansxvavebuli ganlageba arsebobs). TiToeul SemTxvevaSi dasviT kiTxva: Seesabameba Tu ara es ganlageba amocanis pirobebs da Tu ar Seesabameba, ra mizeziT? (magaliTad, ganlageba A, C, D, B SeuZlebelia, radgan am SemTxvevaSi monakveTebis saerTo nawilis sigrZe 5,8sm iqneba). #8 amocanaSi moiTxoveba mxolod sqematuri naxazis ageba, e. i. naxazze zomebi SeiZleba daculi ar iyos, saWiroa mxolod zomaTa Tanafardobis dacva. naxazis ujrian qaRaldze Sesrulebisas SeiZleba davicvaT Tanafardoba: ujris gverdi = 0,5dm. TiToeul monakveTs sasurvelia miuweros misi sigrZe. #9 moswavleebs ganumarteT, rom laparakia ara konkretuli, naxazze agebuli texilis boloebs Soris manZilze, aramed iseTi texilis agebis SesaZleblobaze, romlis perimetri, magaliTad, 1,5m iqneba (rac SeuZlebelia texilis Tvisebis gamo). #10 amocanis amoxsnisTvis saWiroa yvela SesaZlebeli texilis daxazva, romelic amocanis pirobas akmayofilebs (aseTi texili sul eqvsia) da TiToeuli maTganis sigrZis gazomva. amocana masze jgufuri muSaobis saSualebasac iZleva. jgufSi TiToeulma moswavlem SeiZleba gazomvebi Caataros texilis gansxvavebul variantze. moswavleebs winaswar sTxoveT, gamoTqvan varaudi, Tu romeli texilis sigrZe iqneba umciresi. #11 amocana geometriul konteqstSi Seaxsenebs moswavleebs procentebze moqmedebas. am amocanaSi igulisxmeba, rom ABCD texili ar aris Sekruli. radgan amocanis pirobidan gamomdinare BC monakveTis sigrZe (1,4m) texilis sigrZis 28%-s Seadgens, amitom texilis sigrZea 5m. paragrafis bolos moyvanili savarjiSo `TvalzomiT ipove monakveTis sigrZe~ moswavleebSi monakveTis sigrZis TvalzomiT Sefasebis unaris gamomuSavebas emsaxureba da amitom sasargebloa misi periodulad Sesruleba. sasurvelia, am SejibrebaSi maswavlebelmac miiRos monawileoba. 2. 4 kuTxe

mizani: gavacnoT moswavleebs kuTxe, SevaswavloT kuTxis gazomva. codna da unarebi: paragrafis Seswavlis Sedegad moswavles unda SeeZlos kuTxis

7 klasis matematika maswavleblis wigni

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