nbUfnbujljt !txbwmfcb!!WJJ!lmbtTj! - Ganatleba

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nbUfnbujljt !txbwmfcb!!WJJ!lmbtTj! - Ganatleba

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nbUfnbujljt

q o r b u d a

!txbwmfcb!!WJJ!lmbtTj!

nfUpejlvsj!tbyfmn[Swbofmp!nbtxbwmfcmjtUwjt!

avtori:

zurab vaxania

fsiqologiis institutis direqtoris moadgile,

pedagogikis mecn. doqtori,

maTematikis mecn. kandidati,

ganaTlebis mecnierebaTa akademikosi.

saredaqcio sabWo:

vaxtang paataSvili, elza goCitaSvili

sagamomcemlo samuSaoTa Semsrulebelni:

zurab vaxania, naTela Sulaia, nikoloz vaxania

© z.vaxania, 2007 II gamocema tel. 37-28-88 893-94-53-04


is maswavlebeli, vinc Cveneuli saxelmZRvaneloTi

aswavlis, am saxelmZRvanelosac da moswavlis

saxelmZRvanelosac ufasod miiRebs gamomcemlobaSi!


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s a r C e v i

0. saxelmZRvaneloebi `saymawvilo maTematika~

1. `saymawvilo maTematikiT~ swavlebis miznebi

2. `saymawvilo maTematikis~ Tavebi da paragrafebi

3. mTavari meTodikuri principebi

4. maTematikis mravalmxrivi da Tan

gaerTianebuli (integrirebuli) swavleba

5. wakiTxulis gaazrebis unari

da `saymawvilo maTematikis~ nakli;

visTvisaa upriani Cveneuli meTodikiT swavla

6. maTematikis namdvili codna

7. VII klasis programa saxelmwifo saswavlo gegmiT

8. saxelmZRvanelos Sesabamisoba saxelmwifo

saswavlo gegmasTan

9. gakveTilis ageba

10. maswavleblis muSaoba

11. Cveneuli amocanebis tipebi

12. Semamzadebeli safexurebi; erTi nimuSi

13. erTi gakveTilis sanimuSo konspeqti

14. Temis damuSavebis nimuSi _ usasrulobis cneba

15. geometriis swavlebis safexurebi

16. aramaTematikurad, gumaniT amosaxsneli amocanebi

17. TvalsaCinoeba da misi farglebi

18. sakontrolo werebi da SefasebaTa sistema

VII klasis sakontrolo amoanebis nusxa

19. amocanebis pasuxebi da miTiTebebi

20. sakontrolo werebis amocanaTa pasuxebi

da maTi Sefasebis kriteriumebi

21. damatebiTi literatura maswavlebelTaTvis


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maTematikis maswavleblis mecxre da meaTe mcnebani:

ara auwyo mowafesa Sensa saidumlo mecnierebisa mzamzareulad,

aramed Tavad mowafeman gamoicnos saidumlo igi,

vidre ganumartavde Sen, Tavad ipovos, rac SeiZleba meti.

kurTxeul ars misaxvedri miTiTeba, iZulebiT nu

moaxvev Tavs mowafes Sensa azrsa. [jorj poia]

§ 0. saxelmZRvaneloebi `saymawvilo maTematika~

Cveneuli saxelmZRvaneloebi agebulia axal samecniero-meTodikur

da pedagogikur-fsiqologiur safuZvlebze. amasTan, isini

srulad integrirebulia (maTematikis yvela dargi iswavleba er-

Tianad, erTi saxelmZRvaneloTi). garda amisa, yoveli klasis

programa da saxelmZRvanelo arsebiTad emyareba da aviTarebs

wina klasisas. amgvarad Seqmnili gvaqvs erTiani kursi I-dan TiTqmis

XI klasamde.

yovelive amis gamo gaumarTlebelia romelime Cveneuli saxelmZRvanelos

ganxilva zogadi safuZvlebis gareSe da sxva saxelm-

ZRvaneloebisagan gamocalkevebulad. kerZod, VII klasis maswavlebelma

unda naxos, rogor gvaqvs Cven damuSavebuli Semdegi,

SedarebiT aratradiciuli sakiTxebi: umartivesi topologiuri

cnebebi; ZiriTadi geometriuli cnebebi (nakvTi, sxeuli, mravalkuTxedi,

xazi); erToblioba; logikuri mimarTeba zogadi /

kerZo; proporcia, skala da masStabi.

Tuki `saymawvilo maTematikiT~ swavleba VII klasidan

iwyeba: maswavlebelma ar unda daiwyos VII klasis saxelmZRvanelo

manam, sanam klass safuZvlianad ar gaameorebinebs

wina klasebis ZiriTad sakiTxebs. Tanac, am dros klasi unda

mieCvios Cveneuli meTodikiT muSaobas, rac misTvis uCveulo

iqneba. amitom, sasurvelia, maswavlebelma gasameoreblad gamoiyenos

Cveneuli saxelmZRvanelo VI klasisTvis.


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garda amisa, yvelaze mniSvnelovania maTematikuri teqstis

wakiTxisa da gaazrebis unaris ganviTareba _ [ix. $ 5]. uamisod

Cveneul meTodikas saZirkveli gamoecleba.

yovelive amas dasWirdeba albaT 2-3 kvira. ZiriTadi

gasameorebel-gansamtkicebel-gasaRrmavebeli sakiTxebia:

1. wiladis cneba [ix. $ 3-Si]; 1-ze meti wiladebi da

maTgan mTelis gamoyofa; wiladis ZiriTadi Tviseba...

2. wriuli diagrama da wiladebi, romelTa jamis

mniSvnelobaa 1 (procenti saWiro araa, iqneba VII klasSi).

3. farTobisa da moculobis cnebebi _ da ara maTi

gamosaTvleli formulebi; ori-sami aratoli

marTkuTxedisgan Sedgenili nakvTis farTobis gazomva,

agreTve uswormasworo da naxvretiani nakvTebis farTobis

miaxloebiTi gazomva; ori-sami aguredisgan Sedgenili

sxeulis moculobis gazomva.

4. skala da masStabi. masStabis Caweris sxvadasxva saxeebi.

5. proporciuloba. proporciis Tvisebebi.

6. logikuri mimarTebani ZiriTad planimetriul nakvTebs

Soris _ ix. Cveneuli geometriuli TvalsaCinoeba (plakatis

saxiT, anda individualuri _ romelic dabeWdilia Cveneuli

saxelmZRvaneloebis ydis Siga mxares).

maswavlebeli ar unda SeuSindes am winaswar Seferxebas

da mcire sirTuleebs _ maTi gadalaxvis Semdeg araTu

saboloo, aramed ukve VII klasis Sedegic ki gaaxarebs!

§ 1. `saymawvilo maTematikiT~ swavlebis miznebi

maswavlebelma icis, rom, sazogadod, maTematikis swavlebis

miznebi or ZiriTad jgufad iyofa:

1) maTematikis zogadi, sayovelTaod saWiro sawyisebis daufleba

_ codnisa da unarCvevaTa SeZena;

2) gonebis ganviTareba da zogad saazrovno unarTa Camoyalibeba.


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gonebrivi unarebi,

romlebic unda ganaviTaros saskolo maTematikis swavlebam:

I. specifikuri

maTematikuri

unarCvevebi:

Zalian

mravladaa,

ZiriTadad _

praqtikul-

gamoyenebiTi

xasiaTisa.

5) statistikuri _

informaciis damu-

Savebisa da sxvadasxva

sqematuri saxiT

warmodgenisa,

mosalodnelobis

Sefasebisa;

II. zogadi

maTematikuri

unarebi:

1) ariTmetikuli

_ raodenobriv

mimarTebaTa daufleba:

ricxvebze

moqmedebaTa,

sidideTa gazomvisa

an miaxloebiTi

Sefasebisa;

2) geometriuli

_ sivrciT mimar-

TebaTa da formebis

daufleba;

3) algebruli _

maTematikuri modelirebisa;formulebisa

da abstraqtuli

niSnebis

gamoyenebisa;

4) diskretulma-

6) saerTo _ maTe-

Tematikuri _ almatikuri

enis daufgoriTmis

zustad,

leba, saWiro sakiT-

Tanmimdevrulad

xis moZiebis da

Sesrulebis, misi

gaazrebis, wignze

mkafiod Camoya-

muSaobis unarebi.

libebisa;

III. saazrovno

unarebi:

1) sakiTxis gaazrebis,

teqstis wakiTxvisa

da gaazrebisa;

2) arsebiTis danaxvis,

misi sxva areSi

gadatanisa;

3) zusti azrovnebisa

da metyvelebisa;

4) logikuri daskvnis

gamotanisa;

5) analizisa da sin-

Tezisa;

6) argumentirebis,

naTlad da Tanmimdevruladmsjelobis,

dasabuTebisa

Tu uaryofisa;

7) gansazRvrebis gaazrebis,klasifikaciis,

ganzogadebisa

Tu konkretizaciisa;

8) evristikuli unarebi(kanonzomierebis

aRmoCenisa, raimes

mixvedrisa).

garda amisa, maTematikis swavlebas aRmzrdelobiTi daniSnulebac

aqvs _ man unda ganaviTaros mravali zogadpirovnuli Tviseba:

mowesrigebuloba, saqmis dagegmvis unarCvevebi, sibejiTe,


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winaaRmdegobaTa gadalaxvis Cveva, ganyenebul (TvaliT uxilav da

aranivTier) faseulobaTa gancdis unari, kritikuli Sefasebis

unari, sazogado wesebisa da kanonebis pativiscema.

swored am Zvirfas pirovnul TvisebaTa da zogad unarTa ganviTarebaa

maTematikis Cveneuli swavlebis mTavari mizani (da ara

teqnikuri manipulaciebis triali, sqolastikur damtkicebaTa da

mZime formulebis daxvaveba, rac saskolo maTematikis Cveulebrivi

programis udides nawils Seadgens).

§ 2. `saymawvilo maTematikis~ Tavebi da paragrafebi

`saymawvilo maTematika~ iyofa Tavebad. yoveli Tavis

damTavrebis Semdeg tardeba sakontrolo wera.

saxelmZRvanelos bolos amocanaTa Tematikuri krebulia. igi

Sedgeba ara Tavebis, aramed paragrafebisagan.

Tavsa da paragrafs Soris arsebiTi gansxvavebaa. Tavi dalagebulia

kalendarulad da zustadaa dayofili gakveTilebis

mixedviT. amitom `saymawvilo maTematikis~ sarCevi zustad

emTxveva lbmfoebsvm!hfhnbt!!da calke es gegma aRaraa saWiro.

paragrafebi ki dalagebulia ara kalendarulad, aramed Tematikurad:

maTSi amocanebi Temebis mixedviTaa dajgufebuli.

paragrafebis gavla ar unda moxdes miyolebiT (ise, rogorc

wignSia). saWiroa maTi paraleluri gavla. paragrafebis Tanmimdevrobas

ara aqvs didi mniSvneloba, magram TviTeul paragrafSi

ki amocanebis gavla unda moxdes im TanmimdevrobiT, rogorc

wignSia. yoveli paragrafi agebulia amocanebis safexurebrivi

garTulebis mixedviT! konkretulad ki Tavad maswavlebeli

gaanawilebs sakuTari SexedulebiT.

saxelmZRvanelos ZiriTadi nawilis anu Tavebis gavlas (sakontroloebis

CaTvliT) dasWirdeba daaxloebiT 145-150 gakveTili.

saswavlo wlis bolomde darCenil droSi maswavlebeli ZiriTadad

iyenebs wignis meore nawils _ amocanebis Tematikur krebuls

(pirveli nawili dasWirdebaT mxolod calkeuli Teoriu-


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li sakiTxebis gasameoreblad _ maswavleblis Sexedulebisamebr).

sazogadod, maswavleblis TviTmoqmedebisaTvis farTo sarbieli

Tematikur krebulebSia _ da ara ZiriTad gakveTilebSi.

moswavleTa saxelmZRvaneloSi mocemuli amocanebi ZiriTadad

saSinao davalebebisTvisaa gankuTvnili. klasSi xdeba Sesrulebuli

davalebis erToblivi garCeva.

Tematikur krebulSi gabneulia agreTve is amocanebi, romlebic

klass sakontrolo werebze unda mieces (nusxa mocemulia

danarTSi). maswavlebels TavisTvis unda hqondes moswavleTa

saxelmZRvanelo, erTi cali wigni. am Tavis wignSi axlave

moniSnos sakontroloebis amocanebi, raTa SemdgomSi isini

SecdomiT saSinao an saklaso davalebad ar misces klass.

maswavlebels ecodineba, rom mis mier wiTlad moniSnuli

amocanebi _ mxolod sakontrolo Tu sagamocdo werebisaTvisaa.

Cven yoveli klasis bolos vatarebT Semajamebel sakontrolo

weras (mcire gamocdas) [$ 18].

§ 3. mTavari meTodikuri principebi (mokled)

`saymawvilo maTematika~ ganuyoflad moicavs saskolo maTema-

tikis yvela dargs maTi momijnave dargebiTurT da agebulia Tanamedrove

pedagogikuri fsiqologiis sayovelTaod aRiarebul

principebze. amasTan, es principebi ganyenebul mowodebebad anu

`cariel abrad~ ar rCeba, isini namdvilad, Tanmimdevrulad da

safuZvlianad xorcieldeba yoveli calkeuli maTematikuri

sakiTxis swavlebaSi.

gaRrmavebuli swavleba. `saymawvilo maTematika~ gaRrmavebuli

swavlebisaTvisaa, magram kargad unda iqnes gaazrebuli, Tu

ras niSnavs gaRrmaveba da ras _ gaZliereba. gaRrmavebuli

(meorenairad _ intensiuri) swavleba gulisxmobs programis

Semofargvlas mxolod im aucilebeli sakiTxebiT, romelTa

damuSaveba eswreba Rrmad, safuZvlianad, mravalmxrivad da

aqtiur-SemoqmedebiTad. sakiTxi an ase iswavleba (da swored amas


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emsaxureba ZiriTadi saxelmZRvaneloebi), an sul ar iswavleba.

gaZlierebuli swavleba ki sxvaa: es gulisxmobs damatebiTi

rTuli sakiTxebisa da damatebiTi Zneli amocanebis Semotanas.

gaZlierebuli swavlebisaTvis (zogierTi moswavlisaTvis)

gankuTvnilia ufro amocanaTa Tematikuri paragrafebi da agreTve

amocanaTa damatebiTi krebulebi.

ese igi, gaZlierebuli swavla yvelas ar moeTxoveba, magram

gaRrmavebuli ki _ TiTqmis yvelas.

Cveulebrivi saskolo kursebi eqstensiuria: moswavleebs

miewodeba didi odenobiT Teoriuli sakiTxebi da terminebi, ise,

rom ver eswreba maTi wesierad damuSaveba, klasi `gadarbeniT~,

zereled da pasurad iRebs (ufro xSirad ki _ arc iRebs!)

maTematikur codnas. es iwvevs moswavleTa gulisacruebas, xolo

maTi saukeTeso nawili iZulebuli xdeba, dazepirebiTa da

meqanikuri gawafviT daiZvrinos Tavi.

amis sapirispirod, Cveni mcnebaa: `jobia, erTi sakiTxi vaswavloT

aTi sxvadasxva kuTxiT, vidre aTi sakiTxi vaswavloT TiTo

kuTxiT~ [a. distervegi], vaswavloT nela, magram kargad, Rrmad

da mravalmxrivad. gaRrmavebuli swavleba gulisxmobs ara

saswavli Sinaarsis eqstensiur gafarToebasa da tempis aCqarebas,

aramed piriqiT: saswavli Teoriuli sakiTxebis raodenobis

Semcirebas, tempis Senelebas da gaRrmavebul, gaazrebul, anu

intensiur swavlebas. naswavli unda Seesisxlxorcos moswavlis

gonebas, unda gamoiwvios gonebis Sinagani zrda, misi mravalmxrivi

ganviTareba.

amgvari swavlebis Sedegad saSualo moswavleebs rCebaT aucilebeli

saprogramo minimumis myari da aqtiuri codna, xolo

Zlieri moswavleebi imavdroulad aswreben cota gaZlierebuli

kursis gavlasac, rac damatebiT moicavs ufro rTul

sakiTxebsac. amasTan, upirvelesi mniSvneloba eniWeba moswavlis

rogorc saSemsruleblo, ise azrovnebis, damoukidebeli

muSaobisa da kvlevis, SemoqmedebiT unarCvevaTa ganviTarebas. Cven


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veswrafviT moswavlis aramarto zusti maTematikur-logikuri

azrovnebis ganviTarebas, aramed agreTve intuiciis, gumanis,

mixvedrilobis ganviTarebasac.

maTematikis gaRrmavebuli swavlebis arsebiTi maxasiaTebelia

agreTve mTavari yuradRebis gadatana manipulaciebidan cnebebisaken.

`saymawvilo maTematikaSi~ Zalian didi dro eTmoba

sakvanZo cnebebis _ TviT cnebebis! _ gaazrebas. magaliTisTvis

davasaxeloT: farTobis cneba; moculobis cneba; usasrulobis

cneba (mis Sesaxeb ix. qvemore, § 14); cilindris cneba (ix. § 17)

da wiladis cneba.

xjmbejt!dofcb albaT saskolo maTematikis `nomer pirveli~

cnebaa. misi ZirisZiramde gaazrebis gareSe (swored cnebisa _ da

ara wiladebze moqmedebebisa!) azri ara aqvs momdevno klasebis

maTematikis swavlas, iseve rogorc fizikisa, qimiisa, geografiuli

masStabisa, welTaRrocxvisa, statistikisa da sxvaTa.

Cven Semowmebuli gvaqvs: tradiciuli saxelmZRvaneloebiT naswavl

skoladamTavrebulebsac ki (maT daaxloebiT 70-80 %-s) ar

esmiT wiladis cneba. es yvelam SeiZleba Seamowmos, magaliTad,

amgvari advili amocaniT:

ezoSi xeebis 3/7 nawili kopitebia, amdenive _ Wadrebi.

kidev ezoSi dgas 2 naZvi. sul ramdeni xe dgas am ezoSi?

6 6

a) 4; b) 7; g) 8; d) 14; e) 2 ; v) 2 .

7 14

moswavleTa didi umravlesoba irCevs pasuxs e), anda, kidev

uaresi _ v) (rac imis maCvenebelia, rom wiladebis Sekrebis wesic

ki ar icis). orive es pasuxi gviCvenebs, rom moswavles sruliad

ar esmis, ra aris wiladi; verc imas iazrebs, rom xeebis

raodenoba ar SeiZleba wiladuri iyos! moswavlem uazrod,

3 3 6

meqanikurad Sekriba: + + 2 = 2 . am dros ki amocanis

7 7 7

amoxsnas TiTqmis ar sWirdeba wiladebze moqmedebaTa wesebi,

saWiroa mxolod wiladis cnebis codna: 2 naZvi Seadgens ezos


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xeebis 1/7 nawils, ese igi, ezoSi sul 14 xe mdgara. sul esaa,

gantolebac ki zedmetia da am amocanisTvis gantolebis Sedgena

moswavlis saazrovno unarCvevaTa ganuviTareblobas moaswavebs.

maSasadame, tradiciuli saxelmZRvaneloebiT momuSave ufrosklaselTa

80 % mainc, arsebiTad, fuWad dadis maTematikis, fizikisa

Tu qimiis gakveTilebze, radgan maT ar esmiT wiladi.

Cveneuli programiT, wiladi Semogvaqvs mxolod V klasSi (IV

klasSi mas arc ki vaxsenebT!); Tanac, sul mcire 10 saaTs vuTmobT

wiladis jer mxolod cnebas. V klasis ariTmetikis programa,

arsebiTad, mxolod wiladebs eTmoba, aTwiladebi jer naadrevia!

marTlac, aTwiladi, arsebiTad, igive wiladia (maTematikurad

_ racionaluri ricxvi), oRond sxvagvarad Cawerili. Tuki

moswavles bolomde ara aqvs gaazrebuli wiladis cneba,

wiladebis Sedareba da maTze ariTmetikuli moqmedebebi da ukve

aTwiladebs vaswavliT, es niSnavs Semdegs: CvenTvis mTavaria

moqmedebaTa Sesruleba (rac aTwiladebze ufro advilia) _ da

ara cnebis gaazreba da azrovneba. anu, CvenTvis mTavaria,

moswavlem kalkulatoriviT imuSaos uSecdomod _ da ara is,

rom wiladis arsi esmodes. marTlac, rogor SeiZleba kacma

gaiazros aTwiladebze _ anu sxvagvarad Caweril da kerZo saxis

wiladebze moqmedebebi _ Tuki jer wiladebze ar gauazrebia?

Tuki moswavles azriT ar esmis, romelia meti _ 3/7 Tu 5/9,

maSin ar dros aTwiladebze moqmedebebSi meqanikuri gawafvaa? da,

sazogadod, ra saWiroa kalkulatoris saqmes esoden didi

yuradReba davuTmoT? Cven ar vambobT, rom moswavlisTvis

zedmetia aTwiladebze moqmedebaTa codna, es zedmeti araa, magram

arc mTavaria! mTavaria wiladis cneba da wiladebze moqmedebaTa

gaazreba (da aqac ara gamoTvlebSi gawafva).

humanisturi swavleba, individualuri midgoma da

moswavleze centrireba. igi upirvelesad gulisxmobs

keTilmosurne da gulTbil damokidebulebas moswavlisadmi.

magaliTad, maswavleblisa da moswavlis urTierTobaSi gamoric-


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xuli unda iyos SiSi. SiSze damyarebuli wesrigi da swavla

Zalian aramyaria: gardatexis asakis Semdeg bavSvs aRar eSinia

maswavleblebisa da ormagad gadauxdis maT wina wlebSi

dagrovili SiSis samagieros. Tanac, SiSiTa da daZabulobiT

bavSvi Znelad Tu ganviTardeba. moswavles odnavadac ar unda

eSinodes imis Tqma, rom man raime ver gaigo, raRac ver gaakeTa

(gamocdil maswavlebels ar gamoepareba, bavSvma marTla ver

gaakeTa, Tu ar gaakeTa da cuRlutobs). moswavle ar unda

iboWebodes Secdomis daSvebis SiSiT. yoveli Secdoma saqmiani da

mSvidi msjelobis sagani unda gaxdes. moswavles unda vacaloT

azris gamoTqma, Tundac es iyos mcdari azri. maswavlebels

pirisaxis gamometyvelebaze an xmazec ki ar unda Seetyos

ukmayofileba. _ SemdgomSic moindomeb da ukeTesad gaakeTeb,

axla ki mousmine Sens megobars da advilad gaigeb! _ daaxloebiT

amgvari ram unda iTqvas, mSvidad da gulTbilad.

zogierTi maswavlebeli xazgasmiT, ganzrax gamokveTs xolme

imas, rom bavSvi raRacas ver akeTebs, ver igebs, unergavs ra

bavSvebs SiSsa da morCilebis grZnobas. es uxeSad arapedagogiuri

saqcielia, metic, danaSaulia.

moswavleze centrireba gulisxmobs kidev erT, gacilebiT

ufro arsebiT da mniSvnelovan princips: mTliani swavlebis ageba

ise, rogorc bunebrivia bavSvis cnobierebisaTvis; meore _

SeZlebisamebr joejwjevbmvsj!!njehpnb.

individualuri midgomis bolomde ganxorcieleba SeuZlebelia

klasSi, miT umetes, did klasSi. magram maswavleblis didi ostatoba

da xelovneba swored isaa, rom SeZlebisamebr metad moaxerxos

es. man yovel moswavleSi ganumeorebeli pirovneba unda

dainaxos, pativi sces am pirovnebas da, rac mTavaria, Taviseburad

miudges mas. amis zogadi wesebi ar arsebobs. yovel kerZo

SemTxvevaSi maswavlebelma unda moZebnos kerZo saSualebani.

individualuri midgomis kidev erTi mniSvnelovani mxarea

moTxovnaTa araTanabroba moswavleTa codnis mimarTac (es ki,

cxadia, Sesabamisad unda aisaxos maswavleblis mier daweril Se-


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fasebebSi). programa da saxelmZRvaneloebi, maTi sruli moculobiT,

gaTvlilia Zlier moswavleebze. dauSvebelia Zlier moswavleTa

ganviTarebis ganzrax Seferxeba imis gamo, rom klasSi sustebic

sxedan. meore mxriv, arc sustebis daCagvra egebis. kvlav

gavimeorebT: maswavlebelmac da mSoblebmac unda icodnen: yvela

moswavles ar moeTxoveba Cveni programis aTviseba mTeli

moculobiTa da siRrmiT. saSualo moswavle yvelafers

Rrmad ver gaigebs, verc yvela davalebas Seasrulebs, magram

mainc kargad, myarad da aqtiurad daeufleba programis

pirvel dones _ savaldebulo minimums. xolo Zlieri

moswavle programis meore, maRal donesac daeufleba.

Cveneuli meTodikis mixedviT, yvela moswavle swavlobs

sakuTar SesaZleblobaTa Sesabamisad. klass yoveli dRisaTvis

(maT Soris, dasvenebis dReebisaTvisac da ardadegebisaTvisac!)

eZleva bevri davaleba, rac ukanasknel erT an or nomrad SeiZleba

Seicavdes sakmaod Znel, arastandartul amocanebs.

garda amisa, saswavlo wlis bolosaTvis klass ukve

moTavebuli eqneba saxelmZRvanelos Tavebi da morCenili dro

mTlianad meore nawils _ amocanaTa Tematikur paragrafebs

daeTmoba _ yoveldRiurad 6-8 nomeri.

magram saqme isaa, rom mTeli saSinao davalebis gakeTeba

araa savaldebulo yvela moswavlisaTvis. magaliTad, SabaTkvirisaTvis

damatebiT micemul erTi-or amocanas (Tematikuri

krebulidan) mxolod mondomebuli da Zlieri moswavleebi gaake-

Teben. sazogadod, vinc gaakeTebs xolme naxevarze cota mets, is

iswavlis saSualod (saprogramo minimumis myari aqtiuri codniT),

vinc gaakeTebs TiTqmis yvelafers _ iswavlis saukeTesod.

yvelam amoxsnas imdeni, ramdensac SeZlebs. saSinao davalebaSi

Sefaseba ar iwereba. saSualo moswavlisTvis sakmarisia,

Tuki namdvilad damoukideblad SeZlebs davalebis daaxloebiT

naxevris marTebulad Sesrulebas. es yovelive mSoblebsac unda

ganvumartoT.


- 13 -

ganviTarebaSi da gaZlierebaSi bavSvs TviTon amocanebi

daexmareba! amocanebi dalagebulia TandaTanobiTi gaZnelebis

mixedviT. Tema iwyeba Zalian advili amocanebiT, romlebsac

TiTqmis yvela moswavle daZlevs. Semdeg amocanebi TandaTanobiT

Zneldeba da moswavles ar gauWirdeba, mihyves maT.

humanisturi swavlebis principi moiTxovs, garda individualuri

midgomisa, agreTve azrovnebis ganviTarebis asakobriv

da agreTve zogad kanonzomierebaTa mkacr dacvas. es kanonzomierebebi

pedagogikur fsiqologiaSia dadgenili (upirvelesad,

J. piaJes mier [22]). im sakiTxebs (miuxedavad maTi sirTulesimartivisa),

romlebic efuZneba Zlier abstraqtul cnebebs,

dawyebiTi skolis da, miT umetes, pirveli klasis moswavle

azrianad ver daeufleba. aseTi sakiTxebia, magaliTad: farTobi

da moculoba (maT Soris, litri); masa (TiTqos moswavles

esmodes gansxvaveba masasa da wonas Soris!); wrfe _ gansxvavebiT

monakveTisagan (Cven mxolod VII klasSi Semogvaqvs); kuTxe

(maxvili, marTi, blagvi _ Cven mxolod VIII klasSi Semogvaqvs!).

ukve am ramdenime magaliTidanac ki cxadia, rom, samwuxarod,

tradiciuli meTodika naklebad iTvaliswinebs TiTqosda Tavis-

Tavad cxad da ubralo WeSmaritebas: rom swavleba azrovnebis

asakobrivi kanonzomierebebiT unda iyos ganpirobebuli.

yovelive amas sulac ar ewinaaRmdegeba is, rom Cven pirvelive

klasidan programa Sevsebuli gvaqvs Tanamedrove maTematikis

sxvadasxva dargebis saymawvilo SesavlebiT, romlebsac araviTari

Teoria da terminologia ar sWirdeba, moswavleebi maT daeuflebian

saxaliso amocanebis sagangebo TanwyobaTa meSveobiT.

CvenTvis mTavari ganmsazRvrelia moswavlis cnobiereba da misi

bunebrivi interesebi, ganaTlebuli pirovnebis aRzrdis Zireuli amocana,

_ da ara formaluri maTematikuri Tvalsazrisi. moswavleze

centrirebis principi krZalavs saganze centrirebas, anu saswavli

sagnis xedvas kerZo mecnierebis viwro egocentruli TvalTaxedviT.

maTematika unda iyos moswavlisTvis da ara moswavle


_ maTematikisTvis.

- 14 -

moswavleze centrirebis principTan mWidrodaa dakavSirebuli

jtupsj{njt! qsjodjqj/ moswavleze centrirebis gamo mTavaria

is, Tu swavlis romeli gzaa bavSvis cnobierebisaTvis yvelaze

bunebrivi. samwuxarod, ufro xSirad es gza ar emTxveva im gzas,

romelic bunebrivi Cans Sesaswavli sagnis TvalsazrisiT. magali-

Tad, mecnieruli geometriis TvalsazrisiT jer unda iswavlebodes

kuTxe, Semdeg _ marTi kuTxe da Semdeg _ marTkuTxedi. magram

fsiqologiurad es gza yovlad gaumarTlebeli aRmoCnda.

ufro xSirad bavSvis (da sazogadod, adamianis) cnobierebisa-

Tvis raime cnebis Seswavlis bunebrivi gzaa is gza, romliTac

istoriulad Seimecna kacobriobam es cneba (cxadia, moswavle Zalian

SemWidrovebulad da daCqarebulad gaivlis am gzas). amis

kargi magaliTia VII klasSi uaryofiTi ricxvebis swavlebis Cveneuli

meTodika.

SemoqmedebiTi swavleba. cudi swavlebis gamo xalxSi

damkvidrebulia arsebiTad mcdari warmodgena maTematikaze _

TiTqos esaa raRac usaSvelod rTuli, usaSvelod mZime da

usaSvelod mosawyeni, didi ricxvebi da gauTavebeli gamoTvlebi,

mkvdari formulebi da wesebi...

maTematikis swavleba, Tuki igi marTebuladaa agebuli da

gamarTuli, ar unda iwvevdes am ulamazesi da uZlieresi

mecnierebis Sejavrebas, piriqiT unda xdebodes!

yovelive didi da meqanikuri _ ara adamianis SemoqmedebiTi

gonebis, aramed gamoTvliTi manqanebis saqmea! adamianisTvis da

gansakuTrebiT ki ymawvilisaTvis maTematikaSi mTavaria swored

is, rom imoqmedos ara meqanikurad, aramed piriqiT _

gaazrebulad, SemoqmedebiTad, gabedulad _ rogorc moazrovne

adamians ekadreba.

rasakvirvelia, yovelive es ar gamoricxavs zomieri gamoTvlebisa

da sxva `Savi Sromis~ aucileblobas. iseve rogorc

adamianis sulis erTerTi umSvenieresi Semoqmedeba _ musikac ki


- 15 -

mxolod `Sav Sromaze~ damyarebiT ifurCqneba.

Cveni meTodikis mixedviT maTematikis swavla miaxloebulia

mecnierul SemoqmedebasTan, sakiTxebi isea damuSavebuli, rom

axali sakiTxis arss moswavle TiTqos TviTon ikvlevs da

TviTonve aRmoaCens. mTeli Cveneuli kursi _ esaa amgvar ZiebaTa

da aRmoCenaTa erTiani jaWvedi.

saamisod mTeli kursi daqucmacebulia mcire-mcire safexurebad,

romelTa damoukideblad gavla advilad SeuZlia saSualo

moswavles _ cxadia, Tuki mas gavlili aqvs wina gakveTilebi.

TviTeuli am safexuris gavlas moswavle Sesabamisi patara sakiTxis

arsis aRmoCenamde mihyavs da ase grZeldeba bolomde.

amitom saxelmZRvaneloebis ZiriTadi nawilebi sakmaod advilebia,

gaTvlilia namdvil saSualo moswavleze. magram saqme isaa, rom

Zlieri moswavlisaTvisac ki am mcire-mcire kvleviTi safexurebis

gavlas udidesi mniSvneloba aqvs _ Rrma swavlisaTvis.

yovelive zemore Tqmulidan cxadia, rom Cven gamovricxavT

saskolo maTematikisTvis damaxasiaTebel mTavar mankierebas _

moswavleTa mier sityvieri debulebebis (wesebis, gansaz-

Rvrebebisa Tu Teoremebis) gazuTxvas da Sesabamis moqmedebebSi

meqanikur gawafvas. ufro xSirad umjobesia, moswavlem sul ar

icodes raime sakiTxi, vidre amgvarad icodes. mxolod namdvilad

gaazrebul, kargad gagebul codnas aqvs Rirebuleba. moswavles

naTlad unda esmodes is, Tu ras ambobs da ra moqmedebas

atarebs, ratom ambobs ase da ratom moqmedebs ase. es kargad

mowmdeba advili arastandartuli amocanebis amoxsnisas. es

amocanebi ar moicavs arc erT ucnob cnebasa Tu moqmedebas, arc

Znelebia, magram amdagvari amocana moswavles jer ar amouxsnia.

amitom didi mniSvneloba aqvs Tematikuri krebulebis amocanaTa

uCveulo mravalferovnebas.

mravali Cveni amocana sxvadasxva gzebiT amoixsneba, metic,

marTebuli pasuxic ki SeiZleba ori an meti hqondes. maswavlebelma

araviTar SemTxvevaSi ar unda aRkveTos moswavleTa ucna-


- 16 -

uri pasuxebi, oRond, unda moiTxovos dasabuTeba. Tuki moswavle

gonivrulad daasabuTebs Tavis moulodnel pasuxs, is Seqebis

Rirsi iqneba, da ara gakicxvisa! aseve, yovelnairad unda

waxalisdes amocanis amoxsna gansxvavebuli gzebiT.

aqtiuri mzaobis meTodika. esaa `saymawvilo

maTematikis~ ZiriTadi safuZveli da mTavari siaxle. igi emyareba

d. uznaZisa da J. piaJes fsiqologiur Teoriebs da Cveneul gamokvlevebs

azrovnebis fsiqologiaSi. es meTodika axalia ara

mxolod saqarTvelosTvis, aramed, sazogadod, mecnierebisaTvisac

(man ucxoelTa gamoxmaureba da mowonebac daimsaxura). Cveneuli

saxelmZRvaneloebic mis safuZvelzea agebuli da maswavlebelmac

gakveTilebi mis mixedviT unda warmarTos.

es meTodika gulisxmobs: moswavlis did aqtiurobas, misi

interesis gaRvivebas saxaliso da mcire SemoqmedebiTi ZiebebiT;

yoveli sakiTxis Seswavlis win saTanado motivaciuri da agreTve

inteleqtualuri mzaobis (ganwyobis) Seqmnas. da, rac mTavaria _

swavlebas ara maswavleblis mier axsniT, aramed evristikuli

meTodiT, aRmoCenebis gziT [dawvrilebiT _ ix. qvemore, § 12].

maswavleblisTvis mTavaria, rom mieCvios or rames: acalos

moswavleebs damoukidebeli fiqri, msjeloba da muSaoba da

Seikavos gamzadebuli pasuxebi; yovel sakiTxs miudges

SemoqmedebiTad da aseve moiTxovos moswavleebisaganac.

yoveli sakiTxi isea Semzadebuli wina gakveTilebiT, rom mis

arss moswavleebi TiTqmis TviTon aRmoaCenen, sakuTari

aqtiurobis gziT. maswavlebeli unda aclides klass fiqrs,

Secdomis daSvebasa da mis gaazrebas, mis gasworebas, unda waaxalisebdes

moswavleTa msjelobasa da maT mier sakuTari azrebis

gamoTqmas. maswavleblis mier sakiTxis axsnas, rogorc aseTs,

iSviaTad mivmarTavT (da es sagangebodaa miTiTebuli gegmakonspeqtebSi).

sakiTxi muSavdeba ZiriTadad mxolod SekiTxvebis

meSveobiT, dialogurad, problemurad.

maswavlebeli svams SekiTxvebs da cdilobs, sasurveli sruli


- 17 -

pasuxi moswavleebs aTqmevinos. Secdomebic TviTon bavSvebma unda

gaasworon, xarvezebi _ Seavson. maswavlebeli mxolod maSin

unda Caerios, rodesac moswavleTa ZalebiT es veRar xerxdeba.

maswavlebelma unda waaxalisos moswavleTa SekiTxvebi,

uazro da mcdari SekiTxvac ki ar unda gakicxos!

SekiTvis arc Cafarcxva-CaCumaTeba SeiZleba!

SeiZleba, zogjer moswavlem sakmaod moulodneli SekiTxva

dasvas. magaliTad, erT gonier pirvelklasels ukiTxavs, Tu

ratom araa xuTkuTxedi iseTi mravalkuTxedi, romlis sami

wvero erT monakveTzea (cxadia, SekiTxva ase zogadad ki ar

dausvams, aramed erTi konkretuli daxazuli oTxkuTxedis

Sesaxeb ikiTxa). aseTi SekiTxva, upirvelesad, maswavlebelma

gansakuTrebiT unda Seaqos, imis miuxedavad, rom TviTon ara aqvs

gonivruli pasuxi. Semdeg, naCqarevad naTqvam cud pasuxs meore

dRes gacemuli gonivruli pasuxi sjobs. magaliTad: _ ase xom

SeiZleboda, kidev erTi wveroc dagvesva (uCvenebs oTxkuTxedis

naxazze, dafaze) da maSin veRar gavarkvevT, es nakvTi oTxkuTxedia,

xuTkuTxedia, eqvskuTxedia Tu ramdenkuTxedia?! amitom aseTi

wertilebi wveroebad ar iTvleba! wvero gverdis boloSi unda iyos,

es ki sadaa? {gverdis SuaSi}. diax, wvero ar unda iyos gverdis

SuaSi, unda iyos mxolod boloSi!

imisaTvis, raTa SesaZlebeli iyos namdvilad aqtiuri swavleba,

programa da saxelmZRvaneloebi axleburad unda iyos agebuli

(mzaobis zogadi meTodikuri principica da misi konkretuli

ganxorcielebac maTematikis swavlebaSi damuSavebulia Cven mier).

ganmaviTarebeli swavleba. maTematika da mSobliuri ena

imitomaa sayovelTaod aRiarebuli umTavres saskolo sagnebad,

rom maTma swavlebam adamianis umniSvnelovanesi unarCvevebi unda

Camoayalibos da ganaviTaros. es ki igivea, rac pirovnebis

namdvili aRzrda-ganviTareba. swored esaa mTavari _ da ara

sakuTriv maTematikis codna!

zogadad es albaT yvelas moewoneba. magram saqme isaa, rom


- 18 -

Tuki am mcnebis namdvili ganxorcieleba gvsurs, unda SeveguoT

imas, rom moswavles bevri muSaoba mouwevs. unarCvevis

ganviTarebis erTaderTi gza aris bevri damoukidebeli

muSaoba da didi gamocdilebis dagroveba.! amas veraviTari

meTodika Tu maswavleblis ostatoba ver Secvlis. oRond,

cxadia, moswavlis es muSaoba saTanado Sinaarsisa da

mimarTulebisa unda iyos.

amitomaa Cveneul saxelmZRvaneloebSi Zalian mravlad sxvadasxvagvari

amocanebi. Tanac, raime erTi gvaris amocanebi mizandasaxulad

meordeba, TandaTanobiTi garTulebiT, Tanac, wlebis

ganmavlobaSi. amocanaTa TviTeuli es Uboxzpcb romelime

unarCvevas aviTarebs. amas ki, samwuxarod, didi dro da bevri

muSaoba sWirdeba.

Cveneul saxelmZRvaneloebSi mravladaa grZelpirobiani

kompleqsuri amocanebi _ romelTa amosaxsnelad ramdenime

sul sxvadasxva moqmedebis Catarebaa saWiro. magali-

Tad: daTvaleT, gazomeT da SeavseT cxrili; daxazeT amaTuim

saxis mravalkuTxedi, miniSneT misi umoklesi gverdi da masSi

CaxazeT raime; daxazeT cxrili da daajgufeT asoebi; aRwereT

sityvierad; Tuki aqvs, CawereT + niSani, Tuki ar aqvs _ CawereT

_ niSani da ase Semdeg. TviTeuli es davaleba Zalian

advilia, Tumca, mTlianobaSi, amocana sakmao Tanmimdevrulobas,

yuradRebis mokrebasa da Zalisxmevas moiTxovs. magram saqme isaa,

rom es kompleqsuroba Ubonjnefwsvmjb, anu Sedgeba ramdenime

nabijisagan, romlebic cal-calke sruldeba. moswavlem jer

mxolod pirvel nabijs unda miaqcios yuradReba, Seasrulos igi,

Semdeg daiviwyos da momdevnoze gadavides, da ase Semdeg, ese igi,

mas ar uwevs fsUespvmbe ramdenime ramis keTeba (rac Zalian

Zneli iqneboda).

arsebiTad, esaa amocanebi algoriTmis (instruqciis) Sesrulebaze.

moswavles maTze sakmao droisa da Zalebis daxarjva

mouwevs, Tanac, amiT maTematikis codnasac TiTqos bevri araferi


- 19 -

emateba. samagierod, amgvari amocanebi saukeTesod aviTarebs

unarCvevebs. amgvari amocanebi Zalian mniSvnelovania, radgan xels

uwyobs moswavlis azrovnebis mowesrigebas da saSemsruleblo

unarCvevaTa ganviTarebas. miT umetes, rom amgvari amocanebis

instruqciaSi (algoriTmis aRweraSi) CarTulia logikuri

kavSirebi da kvantorebi: Tuki, an, romelime, erTerTi,

erTaderTi, yoveli...

mTavaria, moswavlem gaigos, rom araa saWiro yvelaferze

erTad fiqri: Seasrule jer erTi, Semdeg es daiviwye da gadadi

meoreze, Semdeg esec daiviwye da gadadi mesameze... ... Tanac,

rogorc yvela sxva saxis amocanebis jaWvedi, esec iwyeba jer

Zalian advili, sul ornabijiani algoriTmebiT.

§ 4. maTematikis mravalmxrivi da Tan

gaerTianebuli (integrirebuli) swavleba

Cveneuli kursi bolomde, XII klasis CaTvliT, gaerTianebulia

(integrirebulia) _ maTematikis yvela sxvadasxva dargi

da maTi gamoyenebani erTiani saxelmZRvaneloebiT iswavleba.

amasTan, dawyebiTi klasebidanve safuZvlianad muSavdeba saymawvilo

sawyisebi aramarto ariTmetikisa, algebrisa da planimetriisa,

aramed agreTve: stereometriisa, mxazvelobiTi geometriisa,

kombinatorikisa da simravleTa Teoriisa, miaxloebiT SefasebaTa,

maTematikuri statistikisa da modelirebisa, informatikisa, topologiisa

da grafTa Teoriisa, logikisa.

didi yuradReba eqceva am dargTa Soris kavSirebis warmoCenas

da agreTve maTematikis mravalferovan gamoyenebebs: bunebismcodneoba-biologiasa

Tu geografiaSi, fizikasa Tu astronomiaSi,

qimiasa Tu teqnikaSi, ekonomikasa Tu sociologiaSi, agreTve humanitarul

mecnierebebSi: istoriaSi, eTnografiaSi, kulturologiaSi

da xelovnebaTmcodneobaSi.

yvelaze mniSvnelovania maTematikisa da logikis integracia.

logikis erTi nawili Cven Caqsovili gvaqvs maTematikis

programaSi, xolo meore, ufro enobrivi nawili (ritorika) _


- 20 -

gramatikis programaSi (dawvrilebiT _ ix. [15]). amitom zogi

sakiTxi sakuTriv maTematikuri TvalsazrisiT SeiZleba

ucnauri Candes!

humanisturi swavleba moiTxovs, rom didi yuradReba mieqces

moswavlis Sinagan samyaros. es moicavs rogorc moswavlis kerZo

pirovnul Taviseburebebs, ise zogad asakobriv kanonzomierebebs.

Sinagan samyaros kidev erTi ganzomileba aqvs. esaa fspwovmj!

Ubwjtfcvsfcb. eTnokulturul TaviseburebaTa gaTvaliswinebas

sul ufro da ufro didi mniSvneloba eniWeba pedagogikaSi.

zerele TvalsazrisiTac ki cxadia, rom maTematikisaTvis

saWiro sayofacxovrebo Tu sxva magaliTebi moswavlis erovnuli

kulturidan unda iyos SerCeuli. amasTan, gacilebiT ufro

Rrmaa da arsebiTi sxva sakiTxi. qarTul enaSi, gansxvavebiT

rusuli, evropuli da sxva mravali enisagan, ricxviTi saxelebi

iwarmoeba Tvlis ocobiT-aTobiTi sistemiT. amitom pirveli

klasis maTematikis is meTodika, romelic, SesaZloa, saukeTesoa

evropeli bavSvisaTvis, gamousadegaria qarTvelisaTvis. Cvens mier

gamoyenebuli meTodika iTvaliswinebs swored qarTulenovani

cnobierebis Taviseburebas (amis Sesaxeb dawvrilebiT ix. [17:$3]).

yvela am friad saWiro da saintereso sakiTxis Camateba

Cveulebrivi saskolo maTematikis isedac gadatvirTul kursSi

yovlad SeuZlebelia. amitom, Tuki namdvilad gvsurs kursis

gamdidreba statistikis, logikis, gamoyenebiTi maTematikis sawyisebiT

da misi galamazeba humanitaruli waxnagebiT _

aucilebelia mkveTrad Semcirdes sxvadasxva wvrilmani teqnikurmanipulaciuri

sakiTxebi, magaliTad, algebruli da trigonometriuli

gardaqmnebi, formulebisa da specialuri terminebis

grZel-grZeli xlarTebi _ rac Cveulebrivi saskolo maTematikis

ZiriTadi nawilia.

uaRresad mniSvnelovania statistika. didi xania, mTelma ganvi-

Tarebulma msofliom gaacnobiera, rom saSualo aramaTematikosisTvis

maTematikidan yvelaze saWiroa swored statistika


- 21 -

(cxadia, ubralo ariTmetikis Semdeg). statistikis mraval

sakiTxs Seicavs zogadi unarebis yvela testi da TiTqmis

yoveldRiurad gazeTebSic ki qveyndeba statistikuri monacemebi.

statistikis sawyisebis codnis gareSe adamiani ver iqneba

demokratiuli saxelmwifos srulfasovani moqalaqe, radgan ver

gaiazrebs arCevnebis procedurasa da Sedegebs. didi mniSvneloba

aqvs albaTur-statistikur codnas sazogadoebrivi movlenebis

marTebulad Sesafaseblad, marTebuli daskvnebis gamosatanad.

adamiani, rogorc wesi, mcdarad, egocentrulad afasebs

sazogadoebriv movlenasa Tu xalxis ganwyobas, radganac,

misdauneburad, mxolod sakuTari garemocva aqvs mxedvelobaSi.

magaliTad, esa Tu is politikuri partia darwmunebulia, rom

arCevnebSi 5%-ian zRurbls gadalaxavs da varaudobs xmebis

10%-is mogrovebas, Tumca, sinamdvileSi, xmebis 0,5%-sac ki ver

agrovebs. statistikuri azrovnebis gareSe SeuZlebelia

marTebuli daskvnis gamotana quCaSi miRebuli raime

STabeWdilebidan, eqstrasensuli movlenebis garCevidan da sxva.

cxadia, statistika yvelaze bunebrivad da Sinaarsianad gaer-

Tianebul kursSi Caismeba. Zvelebur algebra-geometriaSi misi

Casma arabunebrivia. igive iTqmis simravleTa Teoriis Sesaxebac.

maTematika _ garesamyaros Semecnebis (ufro zustad ki _

modelirebis) mZlavri saSualebaa. maTematika saWiroa saskolo

sabunebismetyvelo Tu humanitaruli sagnebis Tanamedrove doneze

Sesaswavlad; samyaros, msoflios mecnieruli xedvis Camosayalibeblad.

gasaSualoebis, sixSirisa da albaTobis umartivesi

cnebebis gareSe adamians gauWirdeba im cxrilebSi da diagramebSi

garkveva, romlebic yoveldRiur gazeTebSic ki SeiZleba

Segvxvdes. statistika uaRresad saWiroa gamoyenebiTi TvalsazrisiT:

rogorc biologia-medicinaSi, aseve ekonomikaSi, sociologiaSi,

demografiasa Tu fsiqologiaSi. Cveulebriv Jurnalistsac

ki sakmaod sWirdeba statistikis sawyisebi.

cxadia, rom Tanamedrove informaciul xanaSi zogadsaganma-


- 22 -

naTleblo skolis moswavleTa didi umravlesobisaTvis informaciis

daxarisxebisa da damuSavebis unarCvevebis kargi ganviTareba

gacilebiT ufro mniSvnelovania, vidre mTeli algebris, trigonometriisa

da maTematikuri analizis codna _ anu imisa, rac

tradiciuli saskolo maTematikis albaT 80%-s Seadgens. es ukve

TiTqmis mTelma msofliom gaiazra da ganaxorciela kidec.

magaliTad, j. bruners [23] miaCnia, rom ganzogadebuli maTematikuri

cnebebidan skolaSi saswavleblad yvelaze mniSvnelovania

sami _ ricxvi, zoma da albaToba.

meore aranakleb saWiro dargia logika. mas VI-VII klasamde

TiTqmis ar sWirdeba specialuri terminebi da Teoria. Tumca II

klasidanve unda daiwyos Zlieri mizanmimarTuli muSaoba marTebuli

logikuri daskvnebis gamotanis unarebis ganviTarebisaTvis.

ZiriTadi logikuri cnebebi yvelgan Tan sdevs maTematikas.

amitom sakmaod gavrcelebulia azri, rom logikis sagangebo

swavleba araa saWiro, radganac igi TavisTavadac iswavleba

maTematikis swavlebasTan erTad. magram es azri _ Zalian

mcdaria. rogorc gviCvenebs sagangebo gamokvlevebi, sazogadod

maTematikis Seswavla araa sakmarisi Tundac imisaTvis, rom

axalgazrdas SeeZlos umartives geometriul Tu ariTmetikul

cnebaTa Soris kerZooba-zogadobis mimarTebaTa garkveva. ufro

rTul logikur msjelobaze xom laparakic zedmetia. logikis

sakiTxebs sworedac rom sagangebo swavleba sWirdeba, riTac

arsebiTad amaRldeba moswavlis maTematikuri codnac da ufro

metad ki _ misi zogadi azrovnebis done.

mesame dargia kombinatorika (simravleTa TeoriasTan kavSirSi).

misi sawyisebis gareSe SeuZleblia mravali sayofacxovrebo da

saxaliso amocanis amoxsna, da rac mTavaria, SeuZleblia ufros

klasebSi albaTobis Teoriisa da maTematikuri statistikis sawyisebis

swavla.

simravleTa Teoria kargad asuraTebs da aTvalsaCinoebs logikas,

meores mxriv moswavlis mier maTi SeTviseba myari safuZve-


- 23 -

lia sazogadod klasifikaciuri azrovnebis ganviTarebisaTvis,

rac yovelgvari mecnierebis umTavresi xerxemalia (Sedarebebi,

xilul siWrelesa da mravalferovnebaSi movlenaTa Tu saganTa

ganTavseba jgufebSi, TviTeulisaTvis damaxasiaTebel TvisebaTa

ganzogadoeba, zogadisa da kerZos dialeqtika, da sxva). logikisaTvis

kargi sarbieli da sasuraTebulia agreTve geometriac.

integraciis principi moiTxovs agreTve, rom sxvadasxva sagnebis

programebi erToblivad iyos gaazrebuli, erTmaneTTan SeTanxmebulad

da urTierTSewonilad. dauSvebelia, magaliTad, amJamad

CvenSi arsebuli viTareba: dawyebiTi klasebis bunebismcodneobis

programa moicavs masStabis cnebas, maSin rodesac maTematikaSi

moswavleebs naswavli ara aqvT arc wiladebi da arc proporcia

(amgvari magaliTebi sxvac mravladaa). amiT darRveulia araTu integraciis

principi, aramed ubralod saRi azric ki.

`saymawvilo maTematikaSi~ safuZvlianadaa damuSavebuli is gamoyenebiTi

sakiTxebi, romlebic saskolo sagnebSia Zalian mniSvnelovani:

proporcia, masStabi, geografiuli koordinatebi, welTaRricxva,

ritmi, elifsi da sxva.

`saymawvilo maTematika~, miuxedavad esoden didi Sinaarseuli

nairgvarobisa, mainc Sinaganad erTiania. es siRrmiseuli erTianoba

saerTo maTematikuri safuZvliTaa ganpirobebuli _ esaa

mphjlb!eb!tjnsbwmfUb!Ufpsjb/ garda amisa, sxvadasxva dargebi

erTmaneTTan kavSirdeba damuSavebisa da gadmocemis erTnairi xerxebiT,

erTiani zogadi suliskveTebiTa da agreTve mravali darg-

TaSorisi sakiTxiT: rogorc TeoriulebiT, aseve amocanebiT.

amgvarad agebul kurss kidev erTi didi upiratesoba aqvs: mas-

Si gacilebiT naklebadaa specialur-teqnikuri da sqolastikuri

sakiTxebi, igi gamoyenebiTi amocanebiTaa gajerebuli. amitom moswavles

ar uCndeba bunebrivi ukuqmedeba: _ raSi mWirdeba es yovelive,

ratom unda vicode es? winaT, mbrZaneblur skolaSi,

daTrgunul moswavles amgvari SekiTxvebi akrZaluli hqonda. magram

droa, rom miveCvioT humanistur pedagogikas da moswavlis


- 24 -

pirovnebis pativiscemas. moswavles aqvs sakuTari azris qonis

ufleba, Secdomis uflebac aqvs. moswavle darwmunebuli unda

iyos, rom mas marTlac saWiro sakiTxebs aswavlian.

§ 5. wakiTxulis gaazrebis unari

da `saymawvilo maTematikis~ nakli;

visTvisaa upriani Cveneuli meTodikiT swavla

`saymawvilo maTematika~ araviTar SemTxvevaSi araa mxolod

Zlieri moswavleebisaTvis. piriqiT, Cveneul saxelmZRvaneloTa

erTerT umTavares Rirsebad is migvaCnia, rom ufros klasebSic

ki moswavleTa saSualod 60-70 % namdvilad Zlevs maTematikis

saprogramo minimums, ganviTarebuli aqvs gonebrivi unarebi da

patiosani damoukidebeli muSaobis unarCvevebi, ar ejavreba

maTematika (tradiciuli meTodikiT momuSave tipuri skolebis

klasebSi amgvar ufrosklaselTa wili 5-15 %-s Tu aRwevs _

nacvlad Cveni 60-70 %-isa).

`saymawvilo maTematika~ savsebiT gasagebia da misawvdomia

Cveulebrivi saSualo moswavlisaTvis. moswavle mas damoukideblad

kiTxulobs da esmis teqstis Sinaarsi.

amqveynad unaklo araferia. cxadia, `saymawvilo maTematikasac~

aqvs nakli. esaa is, rom swavleba arsebiTadaa damyarebuli werakiTxvaze,

gansakuTrebiT _ teqstis wakiTxvisa da gaazrebis

unarze. erTi mxriv, es kargia _ Cveni meTodikiT momuSave

moswavleebs saukeTesod uviTardebaT es umniSvnelovanesi unari

(romelic, sxvaTa Soris, erTerTi umTavresia zogadi unarebis

yvela saxis testSi). magram, meore mxriv, did siZneleebs

gviqmnis im bavSvebTan, romlebsac ar SeuZliaT kargad kiTxva.

saqme isaa, rom kiTxvis unaris Sinaganad ganpirobebuli

daqveiTeba SeiZleba aRmoaCndes bavSvebis daaxloebiT 10-15 %-s.

maTgan zogierTi gonebrivadac CamorCenilia, magram zogi

gonebrivad saSualo an saSualoze maRali ganviTarebisac kia.

kiTxva ki mainc Zalian uWirT. garda amisa, mravalia iseTi


- 25 -

moswavle, romelsac kiTxva uWirs ara Sinagan mizezTa gamo,

aramed kiTxvis cudad swavlebis gamo. da kidev: ukanasknel

xanebSi CvenSic SemoiWra wera-kiTxvisa da, sazogadod, enobrivgonebrivi

unarebis dauZinebeli mteri _ kompiuterisa Tu

televizoris ekrani (ix. [31]). yovelive amis gamo mravladaa

iseTi klasi, romelSic moswavleTa 30-40 % ver kiTxulobs

wesierad. wakiTxulis gaazrebaze xom laparakic zedmetia.

maSasadame, imisaTvis, raTa moswavlem SeZlos `saymawvilo

maTematikiT~ swavla, saWiroa Semdegi:

1) normaluri gonebrivi SesaZlebloba (es Tanabrad exeba

yvela sxva meTodikasac, radgan normalurTan SedarebiT

gonebrivad CamorCenili moswavle Cveulebriv saskolo

maTematikas vercerTi saxelmZRvaneloTi ver iswavlis);

2) normaluri nebelobiTi SesaZleblobani: yuradRebis

mokrebis unari, nebisyofa, sibejiTe, moTmineba (esec yvela sxva

meTodikas exeba, met-naklebad);

3) normalurad ganviTarebuli unarebi wera-kiTxvisa da

wakiTxuli teqstis gaazrebisa (es ki sxva meTodikebs albaT

naklebad exeba da CamorCena naklebad azianebs, radgan maTSi

gaTvaliswinebulia maswavleblis mier axali sakiTxebis axsna).

mecnierebaSi Seswavlilia, rom wakiTxvis unaris Sinagani

CamorCena lbopo{pnjfsbe!!bsbb!!eblbwTjsfcvmj!!tbfsUp!!!

hpofcsjw! DbnpsDfojmpcbtUbo; magram xSirad sakmaod eblbw.

Tjsfcvmjb!!ofcfmpcjUj!!vobsfcjt ganviTarebis CamorCenasTan.

ese igi, me-2 da me-3 xSirad erTmaneTs ukavSirdeba da erTian

CamorCenilobas qmnis.

nebelobiTi monacemebi ufro mniSvnelovania, vidre gonebrivi.

Zlieri moswavle iqneba is, romelic orive am mxrivaa Zlieri.

mxolod gonebis gamWriaxoba, mixvedriloba da kargi amTvisebloba

araa sakmarisi! es yovelive unda ganvumartoT mSoblebsac.

yvelam gaakeTos imdeni amocana, ramdensac SeZlebs zomieri

ZalisxmeviT, didi daZabvisa da wvalebis gareSe. Semdeg ki


- 26 -

xSirad moxdeba, rom maswavlebelica da mSobelic, maTda gasakvirad,

aRmoaCenen, rom saSualo moswavlesac arc ise gauWirdebodes

iqneba erTi SexedviT Zneli da didi davalebis Sesruleba...

roca gonebrivad normalur moswavles uWirs `saymawvilo ma-

TematikiT~ swavla, ver igebs mas da ver xsnis amocanebs _ amis

mizezi, rogorc wesi, erTia: am moswavles Sinagan an gare

mizezTa gamo daqveiTebuli aqvs teqstis wakiTxvisa da misi

gaazrebis unarebi, moswavle ver kiTxulobs. amgvari SemTxvevebi

gacilebiT xSiria im klasebSi, romlebsac wina wlebSi Cveneuli

saxelmZRvaneloebiT ar uswavliaT da pirvelad iwyeben `saymawvilo

maTematikas~.

ra vqnaT? jobs gvian, vidre arasdroso. V klasi iqneba Tu

IX, maswavlebelma pirvel rigSi unda Seamowmos moswavleTa kiTxvis

unari: gadaaSlevinos maT saxelmZRvanelo romelime axal

gakveTilze da daavalos teqstis xmamaRla wakiTxva. uceb gamoCndeba,

romel moswavleebs uWirT kiTxva (umjobesia, SeirCes iseTi

teqsti, romelSic mravaldaa rTuli qvewyobili winadadebebi).

meore safexuria sityvis amokiTxvis sizusteze muSaoba _

yvela aso zustad da mkafiod unda warmoiTqvas. moswavleTa (da

mozrdilTa) umravlesobas erTmaneTSi eSleba msgavsi bgeriTi Semadgenlobis

mqone sityvebi, magaliTad: erTerTi \ erTaderTi,

marTebuli \ marTobuli, gamarTuli \ gamarTlebuli,

gamkeTebeli \ gakeTebuli, miuwvevia \ miuRwevia...

Tuki sityvebi zustad araa amokiTxuli _ romel azrovnebaze,

logikaze da maTematikazea saubari?

mesame safexuria sasveni niSnebis amokiTxva. maT TiTqmis sul

ugulebelyofen rogorc kiTxvis, ise weris dros. gansakuTrebiT

iCagreba mZime, orwertili da tire _ swored is sasveni niSnebi,

romlebic gadamwyvetia azris logikisaTvis. amitom moswavle

unda mieCvios teqstis wakiTxvisas sasveni niSnebis gacnobierebas,

rac kiTxvisas mcire SeCerebebiT (azrobrivi pauzebiT) unda

gamovlindes. sasveni niSnebi teqstSi tyuilad ar weria, maT


- 27 -

jerovani yuradRebis miqceva sWirdeba!

meoTxe safexuri yvelaze mniSvnelovania _ wakiTxulis

gaazreba. es teqstis wakiTxvisas marTebuli azrobrivi

aqcentebiT unda gamovilindes. arafrad ar varda Tundac zusti,

magram meqanikur-monotonuri wakiTxva. mxatvruli teqstebis

wakiTxvisas ZiriTadad mxatvrul-grZnobiTi aqcentebia saWiro,

samecniero teqstebis wakiTxvisas ki _ logikur-azrobrivi

aqcentebi. maxvili unda daesves im sityvas, romelic yvelaze

mniSvnelovani da arsebiTia logikuri Sinaarsis TvalsazrisiT.

es oTxi safexuri unda damuSavdes ara mxolod teqstis wakiTxvisaTvis,

aramed agreTve zepiri metyvelebisaTvis da

werisaTvis: sisrule, sizuste, sasveni niSnebi da azri. maTematikis

gakveTilebze es dafasTan zepiri msjelobisas da sakontrolo

weris samsjelo amocanebis amoxsnis dros unda damuSavdes.

araa sakmarisi amocanis mxolod amoxsna anda Teoremis

damtkiceba _ saWiroa kargad gamarTuli msjelobac. amaze

maswavlebelma sagangebod unda imuSaos _ ese igi, unda

Seasworos, Caasworos, Seavsos, gamarTos, daxvewos moswavlis

zepiri Tu werilobiTi teqstebi. magram mxolod Sesworeba da

Secdomis gacnobiereba araa sakmarisi _ moswavlem aucilebelad

Tavidan unda gaimeoros (zepirad Camoayalibos an gadaweros)

Sesworebul-daxvewili sruli teqsti.

swored amgvari muSaobiT viTardeba gonebrivi unarebi da

azrovneba. swored esaa mTavari _ da ara leqsisa Tu

gramatikuli wesis gazepireba, formulis damaxsovreba,

damtkicebis damaxsovreba da meore dRes moyola da gamoTvlagardaqmnebis

sxapasxupiT keTeba. mTavaria ara programis gavla,

aramed moswavlis gonebis gamdidreba da ganviTareba, misi

SemZleobis, xelwifebis amaRleba.

amgvari savarjiSoebisaTvis araa sakmarisi mxolod qarTuli

literaturis teqstebi, moswavle abstraqtul-specialuri teqstebis

wakiTxva-gaazrebasac da Seqmnasac unda mieCvios. amitom


- 28 -

swored maTematikis maswavlebelma swored maTematikis saxelmZ-

Rvanelo unda gamoiyenos. `saymawvilo maTematikaSi~ mravladaa

mozrdili teqstebi mkafio Sinagani logikuri ganviTarebiT (da

Tan kargi qarTuliT dawerili), mravladaa amocanebi, romlebSic

msjelobis gamarTva da maTematikuri teqstis Seqmnaa saWiro, zogjer

TviT axali amocanis mogonebisa da Camoyalibebis CaTvliT.

wakiTxulis gaazrebis savarjiSoebis keTebisas moswavles

Tvalwin hqondes saxelmZRvanelos teqsti, maswavlebelma ki dausvas

SekiTxvebi am teqstidan, raTa gairkves, ramdenad gaigo

moswavlem wakiTxulis Sinaarsi (SekiTxvebi unda daisves ara

mxolod pirdapir, aramed arapirdapirac). cxadia, maswavlebelma

unda moiTxovos zusti da gamarTuli, sruli pasuxebi.

Semdeg, maswavlebelma unda Caataros amocanis pirobis gaazrebis

savarjiSoebi (romlebic Cveni programiT II-III klasebSia).

moswavlem waikiTxos romelime didteqstiani axali amocana

(romlis amoxsna sul ar gvainteresebs!) da Semdeg Camoayalibos

cal-calke: ra aris mocemuli am amocanis pirobiT da ras

gvekiTxeba an ras gvavalebs amocana. momdevno safexurze ki es

ukve werilobiT unda Camoyalibdes.

amis Semdeg mravali maswavlebeli Tavad darwmundeba, Tu

raoden didi xarvezebia mis moswavleTa unarebis ganviTarebaSi.

maTematikis programaSi winsvla, cxadia, Seneldeba. magram ma-

Tematikaze gacilebiT ufro mniSvnelovania wakiTxvisa da gaazrebis

unarTa ganviTareba, rac mTeli wignierebis safuZvelia da

ris gareSec moswavle ver Caabarebs, kerZod, zogad unarTa

tests [11]. saTanado saklaso savarjiSoebi Tavidan yoveldRiurad

unda Catardes. Zalian sasurvelia, rom qarTulis maswavlebelmac

mohkidos am saqmes xeli. Semdeg ki maswavlebeli

Tavad Seafasebs, Tu ramdenad SeiZleba am samuSaoTa gaiSviaTeba.

zogierT moswavles TiTqmis ar Seetyoba gaumjobeseba. amgvar

moswavles, albaT, Sinagan mizezTa gamo uWirs kiTxva. riT

SeiZleba maTi Svela _ es jerjerobiT msoflio mecnierebaSic ki


- 29 -

gadauWreli SesaWiria. xolo danarCen moswavleebs met-naklebi

tempiT da met-naklebi xarisxiT mainc ganuviTardebaT is unari,

romelic mTeli saskolo swavlebisaTvis erTerTi umniSvnelovanesia

_ unari teqstis wakiTxvisa da misi gaazrebisa.

amis Semdeg ki `saymawvilo maTematika~ TviTon iqneba

saukeTeso saSualeba am unaris Semdgomi Seunelebeli ganviTarebisa!

marTlac, Cvens mier Seqmnilia maTematikuri teqstis

gaazrebis sagangebo amocanis tipi. amgvari amocana axlavs

TiTqmis yvela gakveTils nulovani nomriT. es amocanebi

moswavles aiZulebs, rom yuradRebiT waikiTxos gakveTilis

teqsti da wakiTxva-gaazrebis unarsac aviTarebs.

§ 6. maTematikis namdvili codna

saWiroa imis mkafiod Camoyalibeba, Tu, fsiqologiuri Tval-

sazrisiT, ra aris maTematikuri sakiTxis namdvili codna. erTi

mxriv, es araviTar SemTxvevaSi ar daiyvneba imaze, rom moswavles

SeuZlia am sakiTxis moyola (Cveni meTodikiT moswavlis

gamoZaxeba gakveTilis mosayolad xom arc xdeba!). maTematika

araa istoria an moTxroba, rom misi codna moyoliT SevamowmoT.

maTematikuri sakiTxis namdvili codna imas niSnavs, rom

moswavles xelewifeba am sakiTxis Tavisuflad da aqtiurad

gamoyeneba saTanado amocanebis amoxsnisas. codnis

kriteriumi _ esaa misi gamoyenebadoba anu is, Tu es codna

ramdenad dayalibda aqtiur gonebriv unarCvevad. magram

saqme isaa, rom am dros moswavles, SesaZloa, ar ZaluZdes

sakuTari codnis sityvierad gamarTulad Camoyalibeba!

Semdeg, mTavaria tblwbo[p!dofcfcjtb da maT Soris njnbsUf.

cbUb! kargad gaazreba _ da ara manipulaciebSi (gardaqmnebSi,

gamoTvlebSi da sxva) gawafva.

sxvaTa Soris, Cveneul gamocdebzec ZiriTadad mowmdeba

naswavli sakvanZo sakiTxebis gaazrebis xarisxi da maTi aqtiurad

gamoyenebis unari. Tuki moswavle kargad iyenebs raime cnebasa Tu


- 30 -

Teoremas nairgvari (maT Soris arastandartuli) amocanebis

amoxsnisas _ ese igi, mas kargad scodnia es sakiTxi da am

codnis calke Semowmeba aRaraa saWiro.

sakontrolo werebzec da gamocdebzec moswavles vaZlevT

saxelmZRvanelosa Tu cnobaris gamoyenebis uflebas.

maSasadame, SeiZleba moxdes, rom moswavlem, arsebiTad, ar

icodes maTematikuri sakiTxi, magram sityvierad unaklod ayalibebdes

mas _ Tuki es sakiTxi wina dRiT aqvs wakiTxuli da sityvierad

damaxsovrebuli anu gazepirebuli (didi xnis Semdeg es,

cxadia, SeuZlebeli iqneba). meore mxriv ki, SeiZleba moxdes,

rom moswavlem saukeTesod icodes sakiTxi, magram sityvierad

ver ayalibebdes mas. sazogadod, Teoriuli sakiTxis gamarTuli

sityvieri Camoyalibeba sakmaod Zneli amocanaa. am unaris

ganviTarebas Cveni meTodika sakmao yuradRebas aqcevs, magram

maswavlebels gaazrebuli unda hqondes, rom es calke unaria da

mas ara aqvs uSualo kavSiri Sesabamisi maTematikuri sakiTxebis

namdvil codna-arcodnasTan. Cveneuli meTodika (saSinao

davalebis Semamzadebeli amocanebiT da Semdeg saklaso aqtiuri

garCeviT) sakiTxs jer kargad gaaazrebinebs moswavles, da

mxolod amis Semdeg dgeba sakiTxi zepiri Tu werilobiTi

sityvieri Camoyalibebisa da msjelobis unaris ganviTarebisa _

rac calke samuSaoa.

ese igi, SesaZloa, moswavlem ukve kargad icis ganvlili gakveTilis

sakiTxebi (rac imaSi vlindeba, rom am sakiTxebs kargad

iyenebs saTanado amocanebis amoxsnisas!), magram sityvierad ver

ayalibebdes mas gamarTulad. swored amisaTvisaa saWiro gakveTilis

morCenili dro. maswavlebeli ekiTxeba moswavleebs ganvlili

gakveTilis Teoriuli teqstis Sesaxeb, Tu ra azri gamoitanes

maT am teqstis pirveli abzacidan, meore abzacidan, da ase

Semdeg _ moswavleebi pasuxoben adgilidan, maswavlebeli iTxovs

winadadebis srulad da zustad Camoyalibebas, ameorebinebs am

winadadebas, sanam moswavle kargad ar gamarTavs mas. Tuki


- 31 -

moswavlem daaklo raime sityva, am xarvezs sxva moswavleebs

gamoavleninebs. magaliTad, moswavlem daaklo sityva `yoveli~, an

`mxolod~ an `erTaderTi~ _ maswavlebeli sxva moswavleebs

daavalebs, rom moigonon iseTi magaliTi, romelic daamtkicebs,

rom am sityvebis dakleba ar SeiZleboda, radgan am sityvebis

gareSe winadadeba mcdari xdeba, da ase Semdeg. momdevno klasebSi

emateba kidev erTi saxis samuSaoc: ganvilili gakveTilis raime

sakiTxis damtkiceba: moswavle gamodis dafasTan da msjelobs,

maswavlebeli iTxovs unaklod gamarTul winadadebebs.

amgvari samsjelo muSaobis dros moswavleebs ufleba aqvT,

gadaSlili hqondeT TavianTi saxelmZRvanelo da gadaxedon

xolme saTanado abzacs. samuSaos mizania ara gazepireba, aramed

zusti da bolomde gamarTuli sityvieri Camoyalibebis, msjelobisa

da dasabuTebis unaris ganviTareba.

zepirobisas verc es unari viTardeba da verc arsebiTi maTematikis

Seswavla xdeba. esaa tradiciuli meTodikis erTerTi um-

Tavresi mZime naklovaneba (sxva ramdenimes Soris).

`wesebis~ gazepirebaze sabolood unda vTqvaT uari!

§ 9. gakveTilis ageba

tradiciuli meTodikis mixedviT gakveTilis sami mTavari Semadgenelia:

I. saSinao davalebis Semowmeba;

II. erTi-ori moswavlis gaZaxeba dafasTan gakveTilis mosayolad;

III. axali gakveTilis axsna maswavleblis mier.

amasTan, miiCneven, rom mTavaria mesame Semadgeneli. magaliTad,

gakveTilze myofi damswreebi yvelaze met yuradRebas swored

imas aqceven, Tu rogor axsna maswavlebelma axali sakiTxebi.

Cveneul gakveTilze ki araviTari axali masalis axsna ar xdeba!

Cveneuli meTodika ar moicavs tradiciuli meTodikis arc

meore da arc mesame Semadgenlebs: arc moswavlis dafasTan

gaZaxeba misi codnis Sesamowmeblad da gakveTilis moyola xdeba


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(Tumca, amis magvari da amis Semcvleli samuSao mainc tardeba,

oRond mas ufro mcire dro eTmoba); arc axali masalis axsna

xdeba (es sruliad gamoricxulia, misi aranairi Semcvleli Tu

saxesxvaoba ar unda iyos)! samagierod, pirvel Semadgenels

gacilebiT meti yuradReba eqceva. magram amisTvis mTavaria is,

rom TviTon saSinao davalebis amocanebis raoba da Sinaarsi

arsebiTad gansxvavebulia tradiciulisagan, agebulia mTlianad

aqtiur-SemoqmedebiT, ZiebiT-evristikul swavlebaze (es, cxadia,

avtoris saqmea).

Cveni zogadi meTodikis mixedviT, gakveTilze mTavaria kargad

dasmul TfljUywbUb!Uboxzpcb. mxolod amgvarad, aqtiurad, Seiswavleba

sakiTxi. Cven maswavlebels mTlianad vaTavisuflebT me-

Todistis samuSaoebisagan da meTodikuri Semoqmedebisagan; samagierod,

metad vTxovT mas SemoqmedebiT midgomas TviTeuli moswavlisadmi,

individualurad. maswavlebeli karg msaxiobs unda hgavdes

_ da ara dramaturgs. msaxiobs piesa gamzadebuli eZleva,

misi xelovneba da Semoqmedeba piesis Seqmna ki ar aris, aramed

misi gacocxleba da TviTeuli mayureblis gulamde SeRweva.

akrZaluli unda iyos rogorc pasuxis wamoZaxeba, ise

jgufuri, gunduri pasuxebic. vinc wamoiZaxebs an Zalian xmaurobs,

mas maswavlebelma ar unda aTqmevinos. maswavleblis mier

dasmul SekiTxvaze moswavleebma xeli unda awion. pasuxi ki mxolod

erTma unda gasces, mere, SesaZloa, meorem, mere _ mesamemac.

magram rig-rigobiT _ da ara erTad. gundur pasuxebs azri ara

aqvs! Cveneuli meTodika moiTxovs, rom SekiTxva individualurad

daesves moswavles (misi saxelis _ da ara gvaris dasaxelebiT!).

Tuki is ver upasuxebs kargad, SekiTxva daesmis sxva moswavles.

pirvelma moswavlem ki Semdeg unda gaimeoros marTebuli

naTqvami. xolo roca SekiTxva daesmis mTel klass, maSin

moswavleebma unda awion xeli da pasuxs ityvis mxolod is,

visac maswavlebeli miscems sityvas. garda amisa, ufro xSirad,

moswavle pasuxobs adgilidan, fexze audgomlad.

saqme isaa, rom dasmul SekiTxvaze mosalodneli pasuxi


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moklea. am dros umjobesia, rom moswavlem upasuxos adgilidan,

swrafad, fexze adgomlad. miT umetes, roca pasuxoben sakuTari

rveulis an saxelmZRvanelos mixedviT da mas Tvali ar unda

moswyviton. Tanac, adgoma-dajdomas didi dro miaqvs da xmaurs

iwvevs, es Zalian arRvevs gakveTilis ritms. pasuxisas moswavle

fexze unda adges mxolod maSin, roca grZeli msjeloba aqvs

warmosaTqmeli _ rac iSviaTad xdeba!

sxvaTa Soris, Tuki maswavlebeli SeZlebs, rom TviTonac

ijdes xolme da fexze audgomlad SeinarCunos klasSi wesrigi

_ es Zalian kargi iqneba. rac ufro metad daemsgavseba gakveTili

saqmian, samuSao saubars, miT ukeTesia!

davuSvaT, ramdenime moswavlem awia xeli. maswavlebelma erTerTs

unda aTqmevinos. romels? aq gasaTvaliswinebelia ori ram:

erTi is, rom SekiTxvebze pasuxebi daaxloebiT Tanabrad iyos

xolme moswavleebze ganawilebuli, ar unda moxdes ise, rom

mxolod aqtiuri moswavleebi pasuxobdnen. meorec, gasaTvaliswinebelia

SekiTxvis siZnele. Tuki sakiTxi axalia, Tuki SekiTxva

evristikul Ziebas emsaxureba, maSin upiratesoba ufro Zlier

moswavleebs unda mivakuTvnoT. sustebi albaT isedac ar aiweven

xels. magram am Zneli sakiTxis garkvevis Semdeg maswavlebelma

aucileblad kidev erTxel unda hkiTxos swored sust an pasiur

moswavles. maswavlebeli araviTar SemTxvevaSi ar unda dakmayofildes

erTi Zlieri moswavlis pasuxiT. odnav Secvlili saxiT

igive SekiTxva unda uSualod dausvas pasiur Tu sust moswavles:

rati, aba axla Sen miTxari, ratomaa wiTlebis wili naklebi,

vidre TeTrebisa? ara, Tqven ar gekiTxebiT, me ratis vkiTxe! acaleT,

ratim icis da axla gvetyvis. aba, rati, rogor agvixsna zurikom? ...

xolo Tuki sakiTxi advilia, anda mravaljer Tqmulis morigi

gameorebaa, maSin pasuxi, piriqiT, ufro sust moswavleebs unda

vaTqmevinoT.

axla, vTqvaT, moswavlem gvipasuxa. maswavlebelma moiTxova

dasabuTeba: `saidan ici?~, anda `ratom?~ moswavlem axsna, magram

SeeSala. an kidev: xSirad xdeba, rom moswavlis pasuxi


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sanaxevrod an TiTqmis marTebuli kia, magram mainc raRac aklia.

am SemTxvevaSic unda vaTqmevinoT sxvas, Zlier moswavles, oRond

Semdeg marTebuli pasuxi aucileblad unda gavameorebinoT mas,

visac SeeSala. Semdeg unda vkiTxoT kidev erTs (magaliTad,

SedarebiT pasiur an sust moswavles), oRond cota sxvagvarad,

magaliTad, ase: manana, axla Sen mipasuxe, amaTgan romlis wilia

meti? ratom? romeli aTwiladia meti? ratom? aba, daxaze wriuli

diagrama da TvalsaCinod gviCvene!

jobia Tanaklaselma axsnas _ da ara maswavlebelma!

Cven arsad ar vwerT, magram yvelgan vgulisxmobT, rom roca

moswavles raimes pasuxi SeeSleba an dafasTan gamosuli

moswavle mTlad kargad ver Seasrulebs davalebas, maswavlebeli

ikiTxavs: aba daukvirdiT, rame xom ar SeSlia irines? rogor unda

eTqva? (rogor unda daexaza?)

Sesworebisa Tu Sevsebis Semdeg Tavad irinemac unda

gaimeoros marTebulad!

Tuki Tavidanve vercerTma moswavlem ver gagvca pasuxi,

saWiroa misaxvedri miTiTebiT daxmareba, magaliTad: _ am aTwiladebis

jami risi tolia? wriuli diagramisTvis ki ... vin ityvis?

Tuki sakiTxi wina msjelobis saboloo daskvnas exeba, maSin

msjeloba maswavlebelmac unda Seajamos. man SedarebiT zogadi,

sruli da gamarTuli saxiT unda Camoayalibos is saboloo

daskvna, romelic klasma ukve gaarkvia: _ maSasadame, aTwiladebis

Sedareba ufro advilia, vidre wiladebisa, magram es im SemTxvevaSi,

Tuki wiladebi aTwiladebis saxiTaa Cawerili. aTwiladebis Sedareba

mTeli ricxvebis Sedarebas hgavs. wriul diagramaze ki sidideebi

TvalsaCinod Cans.

es yvelaferi unda gaakeTos maswavlebelma: SekiTxvis ganvrcoba,

saTanado gamoTqmis SerCeva, gamosakiTx moswavleTa kargad

SerCeva, SekiTxvis formis Secvla da klasSi misi `datrialeba~,

pirisaxis gamometyvelebis Secvla Tu xelebis moZraoba. maswavlebels

araviTar SemTxvevaSi ar unda daaviwydes, rom saWiroa

SekiTxvis amgvarad `datrialeba~. mxolod erTi SekiTxva da

erTi pasuxi araa sakmarisi! gansakuTrebiT didxans unda `it-


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rialos~ axal an Znel sakiTxTan dakavSirebulma SekiTxvam.

SedarebiT wvrilman sakiTxebs ki arc `datrialeba~ sWirdeba

da arc maswavlebliseuli Sejameba. pirdapir gadavdivarT

momdevno sakiTxze.

aqtiuri mzaobis meTodikis mixedviT, maswavleblis mier

warmosaTqmel winadadebaTa umravlesoba _ ljUywjUj

winadadebebia, romlebic klass SekiTxvas usvams da

dafiqrebisaken mouwodebs; danarCenTa umravlesoba _ cs[bofcjUj

winadadebebebia, romlebic klass raimes avalebs da moqmedebisaken

mouwodebs (magaliTad: gadaSaleT rveuli, gamodi dafasTan, SeavseT

cxrili da sxva); da mxolod Zalian mcire nawilia UyspcjUj

winadadebebi, romlebic klass pasiuri mosmenisa da damaxsovrebisaken

mouwodebs. rasakvirvelia, esec aucilebelia, magram

Zalian mcire zomiT. isinic TiTqmis yovelTvis mas Semdeg

warmoiTqmis, rac Tavad moswavleebi ityvian. SesaZloa,

arazustad, arasrulad, mouxeSavad _ magram mainc ityvian ise,

rom cxadi iqneba _ maT gaiges. maswavlebeli mxolod amis Semdeg

ityvis TxrobiT winadadebas, romelic, arsebiTad, mxolod

gaimeorebs, daadasturebs da daxvews moswavleTa mier naTqvams.

da esec _ Zalian xanmoklea, sul erTi-ori mokle winadadeba.

maswavleblis Tundac erTwuTiani monologi, sakiTxis `axsna~ Tu

`ganvrcoba~ _ gamoricxulia!

sxvaTa Soris, maTematikis swavlisaTvisac saWiroa da Zalian

didi ganmaviTarebel-aRmzdelobiTi daniSnulebac aqvs imas, rom

maswavlebelma aswavlos bavSvebs fsUnbofUjt! nptnfob. patarebs

es uWirT, magram TandaTan unda mivaCvioT.

Cveneuli maTematikis gakveTili, rogorc wesi, iwyeba saSinao

davalebis erToblivi garCeviT, muSavdeba misi TviTeuli amocana.

amasTan, dawyebuli V klasidan, gakveTilis agebuleba icvleba.

es cvlileba imiTaa gamowveuli, rom V klasidan ZiriTadi saprogramo

xazis gavla emyareba ara maswavleblis mier warmarTul

sagangebo saklaso samuSaoebs (rogorc iyo I-IV klasebSi),

aramed moswavlis saxelmZRvanelos. V klasamde moswavlisaTvis

misi saxelmZRvanelo iyo, arsebiTad, mxolod amocanebis kre-


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buli (gakveTilebad da paragrafebad dalagebuli), xolo swavleba

emyareboda maswavleblis meTodikur saxelmZRvaneloSi aRweril

saklaso samuSaoebs. V klasidan moswavle ukve sakmaod

momwifebulia (unda iyos sakmaod momwifebuli) saimisod, rom

damoukideblad SeZlos advili Teoriuli teqstebis wakiTxva da

am teqstebiT swavla (cxadia, winsvlis Zalian neli tempiT, anu

Zalian mcire-mcire nabijebiT).

yovelive amis gamo V klasidan maswavleblis muSaoba

Zalian gaadvilebulia _ rasac ver vityviT dawyebiTi klasebis

Sesaxeb. es gansxvaveba imiTaa gamowveuli, rom Cveneuli meTodika

bolomde aqtiuria da moswavle sakuTari aqtiurobiT swavlobs;

dawyebiT klasebSi saSualo moswavlis unari teqstis wakiTxvisa

da gaazrebisa araa imdenad ganviTarebuli, rom SemecnebiTi aqtiuroba

mas daemyaros. amitom saWiroa saklaso aqtiuroba, romelsac

moswavlis saxelmZRvanelo mxolod nawilobriv exmareba, ZiriTadad

ki maswavleblis kiserzea (Tumca dawvrilebiTaa aRwerili

meTodikur saxelmZRvaneloSi). yvelaze metad es pirvel

klasSia, mere da mere ki TandaTan moswavlis saxelmZRvaneloze

gadadis datvirTva. xolo V klasidan ukve mTel datvirTvas

moswavlis saxelmZRvanelo iRebs Tavis Tavze (rogorc iTqva zemore,

vgulisxmobT, rom moswavles ukve xelewifeba teqstis damoukideblad

wakiTxva da gaazreba).

gakveTilis 25-35 wuTi (gaaCnia amocanebs) eTmoba saSinao davalebis

garCevas. swored am dros swavlobs moswavle sakiTxs _

ara maswavleblis mier axsniT, aramed aqtiurad. TviTon saxelmZRvaneloa

ise agebuli da iseTi meTodikiT gamarTuli (rac

avtoris saqmea), rom misi gaazrebuli damuSaveba sruliad sakmarisia

saswavlo programis aqtiurad, cocxlad, Sinaarsianad da

Rrmad SesaTviseblad. maSasadame, maswavleblis amocanaa, rom moswavleebs

daamuSavebinos da gaaazrebinos saxelmZRvanelo, misi

Teoriuli teqstebiTa da ZiriTadi amocanebiT (rogorc yvela

Cveneul saxelmZRvaneloSi, yvela amocanis amoxsna Tu gaazreba


- 37 -

yvela moswavles ar moeTxoveba).

maswavlebeli rig-rigobiT ekiTxeba sxvadasxva moswavleebs,

Tu rogor amoxsnes esa Tu is amocana, ra Sedegi miiRes, gzadagza

_ Tu rogor gaiges esa Tu is sakiTxi Teoriuli teqstidan.

moswavleebi erTmaneTs adareben namuSevrebs, msjeloben, gamoTqvamen

azrebs. maswavleblis Careva mxolod mcire miniSnebiTa Tu

miTiTebiT an mokle SeniSvnebiT unda Semoifarglos.

rogorc yovelTvis saSinao davalebis garCevisas, amocanebs

rig-rigobiT TviTon moswavleebi arCeven. erTerTi maTgani

dafasTanaa da wers, sxvebi aRniSnaven, Tuki raime sxvagvarad

aqvT. yvela cdilobs sakuTaris dasabuTebas da irkveva

marTebuli pasuxi. maswavlebeli mxolod maSin Caereva, Tuki

vercerTi moswavle (maT Soris Zlierebic) Tavs ver gaarTmeven

sakiTxs. magram es Carevac mxolod mimniSnebeli, sanaxevrod

Tqmuli an misaxvedri SekiTxvis formisa unda iyos.

dafasTan momuSave moswavle Cumad ar unda muSaobdes _ Tvi-

Teul Sesrulebul moqmedebas is xmamaRal sityvier ganmartebebs

unda urTavdes, raTa sxvebs auxsnas sakiTxi (magaliTad: am

wiladis mricxvelSi gavxsnaT frCxilebi, miviRebT amas, axla

SevkvecoT... axla gavavloT am wrexazis sxva radiusi, ai es.

igi ar gadakveTs am rkals...). maswavlebelic saWiroebisamebr

CaerTveba xolme msjelobaSi, magram mxolod mokle-mokle

CanamatebiT.

maswavleblis mTavari saqme am dros sul sxvaa. is saTiTaod

Camoivlis yvela merxs da Tvalis gadavlebiT Seamowmebs, Tu

rogor aqvT Sesrulebuli davaleba moswavleebs. cxadia, yvela

moswavlis saSinao davalebis yvela amocanis Semowmeba SeuZlebelia,

magram maswavlebelma mainc unda gadaxedos namuSevars

da raime Tqvas. amis gakeTeba aucilebelia, raTa TviTeulma

moswavlem icodes, rom mis namuSevars maswavlebeli aucileblad

naxavs. maswavlebeli wiTeli melniT erTi-or Sesworebasac

gaakeTebs (rac TvalSi moxvdeba), yovel moswavles aucileblad


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qjsbebe etyvis ramdenime sityvas, naxazs Seuqebs an formulis

Canawers dauwunebs, mxrebSi gaasworebs an Tavze xels

gadausvams, Tuki saWiroa, sayvedursac etyvis, an, piriqiT,

Seaqebs. cxadia, am dros SeuZlebelia saSinao namuSevarTa

yuradRebiT naxva (yuradRebiT da safuZvlianad sakontrolo

nawerebi isinjeba).

maSasadame, yovel gakveTilze maswavlebeli unda mivides

TviTeul moswavlesTan, fsUjfsU{f, cota xniT mainc

gaesaubros mas piradad, Seaqos an muTiTos xarvezze.

sxva moswavleebi ki am dros TavianT saqmes akeTeben: amocanebis

garCeva gakveTilis pirvelive wuTidan iwyeba. yvelas gadaSlili

aqvs wigni da rveuli da avsebs da asworebs Tavis namuSevars.

yvela moswavles kalmistari unda ekavos xelSi da iqve

avsebdes Tu asworebdes namuSevars, anda sul axali striqonidan

Tavidan werdes rames (nawerSive rom ar CaaWuWynos). yvela

SemTxvevaSi, Tuki moswavles ara aqvs Sesrulebuli saSinao

davalebis SedarebiT rTuli amocana, igi klasSi unda

daweros _ am amocanis erToblivi garCevisas.

am dros dafasTan ukve sxva moswavle SeiZleba muSaobdes _

visac daevaleba. maswavlebeli ki ganagrZobs merxebis Camovlas,

rveulebis Tvalierebasa da TviTeul moswavlesTan piradad

urTierTobas _ saamisod mas sakmao dro aqvs.

Tuki adamians ar SeuZlia ori ramis erTad keTeba, mas maswavlebloba

gauWirdeba. magram ori ramis erTad keTebis unari

gavarjiSebadia, amitom, Tuki maswavlebels es uWirs, sagangebod

unda cdilobdes, rom gaivarjiSos da ganiviTaros es unari.

moswavleTa umravlesoba, bunebrivia, cdilobs, rom Tavisi

azri gamoTqvas, Tundac sul cotaTi gansxvavebuli ram eweros,

mainc. Cveneuli meTodika Zalze aaqtiurebs moswavleebs da es

Zalian kargia. amitom maswavlebeli unda iyos yuradRebiT, raTa

erTobliv msjelobaSi metismetad bevri dro ar gaeparos _

Zneli amocanebis dawvrilebiT garCevas im donemde, rom sustma


- 39 -

moswavlemac gaiazros isini, azri ara aqvs. sustma moswavlem

kargad unda gaiazros gakveTilis ZiriTadi sakiTxebi da

amocanebis naxevari mainc. saamisod zogjer saWiroa xolme

saSinao davalebis romelime amocanis saxesxvaobis damatebiT

amoxsna. es saxesxvaoba maswavlebelma Tematikuri krebulidan

unda aarCios (zogjer, SesaZloa, TviTonac mouwios Sedgena _

arsebuli amocanis pirobaSi ricxvebisa da araarsebiTi

monacemebis SecvliT).

cxadia, gamosaTvlel an standartuli pasuxis mqone

amocanebis garCevas SedarebiT ufro naklebi dro unda daeTmos.

umjobesia, maswavlebelma TviTon daawerinos romelime moswavles

dafaze axali amgvari amocana (Secvlili ricxvebiT) da yvelas

mosTxovos misi Sesruleba (mcire saklaso wera).

Tuki saSinao davalebis garCevas Zalian bevri dro ar

dasWirda, misi damTavrebis Semdeg sasurvelia, Catardes 2-4wuTiani

maTematikuri TamaSi (TamaSebis nimuSebi mocemulia II-IV

klasebis meTodikur saxelmZRvaneloebSi).

axla mokled aRvweroT, Tu rogor xdeba amocanaTa

erToblivi garCeva. pirveli amocana xSirad gamosaTvlelia,

amasTan, igi ramdenime qveamocanisgan Sedgeba (es qveamocanebi

zogjer gadanomrilia romauli cifrebiT, zogjer _ ara).

maswavlebeli erT moswavles adgilidanve aTqmevinebs pirveli

qveamocanis (gamosaTvleli magaliTis) pasuxs, sxva moswavles _

meore qveamocanis pasuxs da ase Semdeg. TviTeul SemTxvevaSi

gairkveva, vinmes sxvagvari pasuxi xom ara aqvs. visac Secdoma

aRmouCndeba, iqve Tavidan Seasrulebs gamoTvlas. am dros yvela

moswavle zis da ise muSaobs Tu pasuxobs.

sazogadod, yovelTvis, roca amocana ramdenime qveamocanisagan

Sedgeba, maswavlebeli maT amoxsnebs sxvadasxva moswavleebs

Camoayalibebinebs. cxadia, TviTeul SemTxvevaSi imarTeba

erToblivi msjeloba (amocanis Sesabamisi xangrZliobisa).

SemoqmedebiTi amocanis garCevisas sasurvelia, sxvadasxva

amoxsna mravalma moswavlem warmoadginos da moxdes


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gansxvavebul amoxsnaTa urTierTSedareba. sazogadod, samsjelo

amocanis amoxsnisas erTi moswavle dafasTan muSaobs (xazavs

geometriul nakvTsa Tu logikur sqemas, wers gamosaxulebas da

sxva) an adgilidan msjelobs, da Tavis pasuxs asabuTebs, Semdeg

sxvebic erTvebian msjelobaSi.

Cveneuli meTodikis mkacri moTxovnaa: Tuki amocana samsjeloa

(rasac Cven yovelTvis vuTiTebT), maSin mTavaria ara amocanis

marTebuli pasuxis migneba da Tqma, aramed swored msjeloba.

bavSvs msjeloba ezareba, msjelobas sakmao Zalisxmeva sWirdeba,

miT umetes, rom man pasuxi ukve icis. miuxedavad amisa,

maswavlebels evaleba, rom srulfasovnad amsjelos moswavleebi.

mxolod am gziT miiRweva Rrma da kargad gaazrebuli codna da

azrovnebis ganviTareba. im msjelobas, romelsac maswavlebeli

moswavles klasSi xmamaRla warmoaTqmevinebs, SemdgomSi moswavle

ukve gonebaSi Caatarebs (anda, Caatarebs Cumi burtyunis

TanxlebiT) _ da swored amgvarad yalibdeba namdvili azrovneba,

romelic ganuyofelia sityvierebisagan (es exeba sakuTriv

azrovnebas; aranaklebi mniSvneloba aqvs gumans anu intuicias,

romelic ar aris sityvieri da ar aris metyvelebaze

damokidebuli _ mis gansaviTareblad ki sul sxvagvari, faruli

muSaoba gvaqvs Caqsovili Cveneul programaSi, amocanaTa

Tanwyobis meSveobiT [$ 16]).

morCenili 10-15 wuTis ganawileba damokidebulia imaze, ganvlil

gakveTilSi iyo Tu ara Teoriuli teqsti. Tuki Teoriuli

teqsti ar yofila, maSin morCenili dro eTmoba damatebiTi

erTi-ori amocanis garCevas Tematikuri krebulidan _ isini maswavlebels

winaswar unda hqondes SerCeuli gakveTilis

momzadebisas, unda SearCios iseTebi, romlebic Seexeba SedarebiT

Znel sakiTxs ganvlili gakveTilebidan. amitom maswavlebels,

cxadia, yovelTvis CaniSnuli unda hqondes, Tu ra sakiTxi

gauWirdaT moswavleebs da am sakiTxze damatebiTi muSaoba unda

dagegmos momdevno gakveTilebze. sazogadod, Tuki moswavleTa

codnaSi raime xarvezi aRmoCndeba, maswavlebeli atarebs


- 41 -

ramdenimewuTian saklaso weras saTanado Temaze.

garda amisa, maswavlebels CaniSnuli unda hqondes is

amocanebic, romelTa garCeva mizezTa gamo ver moeswro wina

gakveTilebze. isinic morCenil droSi unda gairCes.

Tuki ganvlili gakveTili moicavda Teoriul teqsts, maSin

morCenili dro unda daeTmos am teqstis gaazrebas da mis

irgvliv msjelobas. saSinao davalebis aqtiuri garCevis Sedegad

am teqstis ZiriTadi sakiTxebi moswavleebma ukve gaiazres,

magram saWiroa am codnis ganmtkiceba da, rac mTavaria, zepiri

msjelobisa da dasabuTebis unarebis ganviTareba. am unarebis

ganviTareba xom maTematikis swavlebis calke mizania [§ 6].

sasurvelia, maswavlebelma moitovos gakveTilis bolo orisami

wuTi imisaTvis, raTa ukve TviTon Seajamos ganvlili

sakiTxi. moswavleebma es sakiTxi ukve waRma-ukuRma atriales da

amuSaves, aqtiuradac gamoiyenes da sityvieradac Camoayalibes,

ukve ician igi. magram, sakiTxis arsi saerTo msjelobaSi rom ar

gafermkrTaldes, maswavlebeli bolos TviTon mkafod Seajamebs

da ganazogadebs naswavl sakiTxs: _ maSasadame, Cven viswavleT

Semdegi: ... ... (cxadia, saxelmZRvanelos mixedviT). magram saqme

isaa, rom es araa axsna. maswavlebeli mxolod Seajamebs ukve

naswavl _ da ara axal _ sakiTxs, Tanac, mxolod mas Semdeg,

rac sakiTxi ukve sajarod garCeulia da namsjelia.

klass saSinao davalebad yovelTvis Semdegi gakveTili eZleva.

Cveulebriv arcaa saWiro saSinao davalebis micema: moswavleebma

unda icodnen, rom yovelTvis, ugamonaklisod, davalebad momdevno

gakveTili eZlevaT (Tuki maswavlebels sxva araferi uTqvams).

moswavleebi Sin damoukideblad Seiswavlian am axal

gakveTils, amoxsnian mis amocanebs. rasac ver daZleven, meore

dRes klasSi gaigeben _ sajaro ganxilvis dros.

bavSvma amocanaze damoukideblad unda imuSaos!

`repetitoroba~ Zalian uaryofiTad moqmedebs moswavlis

azrovnebis ganviTarebaze. Sin mSoblisa da klasSi maswavleblis


movaleobaa:

- 42 -

1) misces moswavles saTanado davalebebi da yuradReba

miaqcios bavSvs _ gulisyuriT mecadineobs Tu ara;

2) Seamowmos da Seasworos moswavlis namuSevari;

3) sTxovos moswavles, Tavad moZebnos, anda uCvenos mas

amocanis sxvagvari amoxsnis gzebi (roca moiZebneba);

4) saWiroebis SemTxvevaSi ganumartos ucnobi sityva;

5) mxolod aucileblobis SemTxvevaSi daexmaros moswavles

amocanis pirobis gaazrebaSi;

6) ukidures SemTxvevaSi (magaliTad, Tuki moswavle mizezTa

gamo CamorCenilia saswavlo programas) daexmaros moswavles

amocanis amoxsnaSi, calkeul miTiTebaTa an miniSnebaTa meSveobiT.

maswavleblisTvis mTavaria, rom mieCvios or rames: acalos

moswavleebs damoukidebeli fiqri, msjeloba da muSaoba da

Seikavos gamzadebuli pasuxebi; yovel sakiTxs miudges SemoqmedebiTad

da aseve moiTxovos moswavleebisaganac.

amrigad, sabolood, Cveneuli meTodikiT agebul gakveTilze

ar xdeba arc moswavlis mier gakveTilis moyola, arc maswavleblis

mier axali masalis axsna da, Cveulebriv, arc saSinao davalebis

micema (Tumca, Cveni wesia: arcerTi dRe saSinao davalebis

gareSe).

§ 10. maswavleblis muSaoba

Cveneuli meTodikis mixedviT, maswavleblis mTavari amocanebia:

1) amuSaos moswavle ise, raTa man safuZvlianad da

gaazrebulad daamuSaos Tavisi saxelmZRvanelo;

2) sagudagulod gaasworos sakontrolo nawerebi da moswavleebsac

bolomde Seasworebinos;

3) guldasmiT Seasrulebinos moswavleebs gamarTuli, zusti

da Tanmimdevruli msjelobisa da dasabuTebis gansaviTarebeli

savarjiSoebi (rogorc aRvwereT § 6-Si).

danarCens TviTon saxelmZRvanelo gaakeTebs.

roca klass miecema saweri, dasaxazi, dasaxati,


- 43 -

dasaTvleli Tu sxva saklaso samuSao, maswavlebelma

merxebs Soris unda iaros da Tvalyuri adevnos muSaobas:

ramdenad kargad asruleben davalebas, rogor ukaviaT

kalmistari Tu rogor xazaven, vinmes rame xom ar eSleba,

welSi xom ar ixrebian da sxva.

amgvarad, Cveneuli sagakveTilo muSaobis sqemaa:

Tanaklaselisa da agreTve maswavlebelis mosmena →

dafiqreba, azrovneba, Zieba → azris gaCena →

azris sityvierad Camoyalibeba → naTqvamis daxvewa,

Sesworeba, Sevseba, azrobrivad da enobrivad gamarTva.

sabolood ki unarCvevis ganviTareba, mcired damaxsovrebac.

yvelaze Zvirfasia unarCvevis ganviTareba. mis misaRwevad ki

mTavaria meore rgoli, romelsac Cveulebrivi swavlebisas

Caenacvleba gaxseneba. Cveulebrivi sagakveTilo muSaobis sqemaa:

maswavleblisa da agreTve (Zalian mcired) Tanaklaselis mosmena →

mosmenilis damaxsovreba →

damaxsovrebulis aRdgena anu gaxseneba...

rogorc vxedavT, Cveneuli meTodikiT erT sakiTxs sakmao ganxilva

da msjeloba SeiZleba dasWirdes. amitom maswavlebeli

Zalian unda gaufrTxildes sagakveTilo dros da ar daxarjos

igi iseT rameebze, rac ar gvaqvs miTiTebuli gegma-konspeqtebSi!

kidev erTxel: maswavlebels evaleba, rom gaacocxlos da

`xorci Seasxas~ imas, rac mokled da mSraladaa mocemuli gegmakonspeqtSi.

xolo sakuTar monologebze (`gakveTilis axsnaze~

da sxva) drois daxarjva fuWia. bavSvisaTvis _ da ara mxolod

bavSvisaTvis, aramed, sazogadod, adamianisaTvis, bavSvisaTvis ki

miT umetes _ gacilebiT ufro nayofieria sakuTari

dafiqrebisa Tu grZnobis Sedegad warmoTqmuli ori sityva,

vidre sxvisi oci sityvis mosmena.

maswavleblis mier sakiTxis mokled axsna Cven gadatanili

gvaqvs sul boloSi. arsebiTad, es arcaa axsna. esaa moswavleTa


- 44 -

mier aqtiurad garkveuli sakiTxis mxolod Sejameba da

gamarTulad Camoyalibeba _ raTa sakiTxis arsi ar Caikargos

saerTo msjelobaSi. xolo winaswar maswavlebeli arafers ar

xsnis. Cveneuli meTodika gamoricxavs `gakveTilis axsnas~.

maswavlebelma aqtiurad unda amuSaos klasi da Tavad

moswavleebs aRmoaCeninos sakiTxis arsi. amisaTvis sakmarisia,

rom man kargad Seasrulos Cveneuli zogadi principebi da

mxolod gaacocxlos saxelmZRvanelo.

amgvarad, maswavlebeli xan merxebs Soris dadis, moswavleebs

asworebs (an ise Seexeba maT) da Tan muSaobs, xan dafasTan muSaobs,

xan ki Tavis skamze zis da saubrobs. maswavlebeli unda

moeridos leqtoris qcevebs (garda, SesaZloa, namsjelis mokle

Sejameba-ganzogadebis wuTebSi), unda moeridos agreTve mqadageblis

qcevebs. es Zalian iSviaTad unda xdebodes, magaliTad, rame

gansakuTrebul SemTxvevasTan dakavSirebiT (Tuki amgvari qceva

gaxSirdeba, igi gaufasurdeba).

xSirad xdeba, rom maswavlebeli klass TiTqos kargad amuSavebs,

moswavleebi aqtiuroben, pasuxoben da sxva. magram es moCvenebiTi,

zerele aqtiurobaa xolme. saqme isaa, rom xSirad

moswavle ambobs, arsebiTad, sxvis nafiqrsa Tu nagrZnobs.

moswavle mxolod ixsenebs da imeorebs imas, rac sxvisgan

(kerZod, maswavleblisgan, ufrosebisgan, televizoridan)

mousmenia. es ki sul sxvaa. nayofieria ara ubralod sakuTari

`ori sityva~, aramed, rac mTavaria, sakuTari dafiqrebisa Tu

grZnobis Sedegad warmoTqmuli `ori sityva~! sxvisi (umjobesia,

Tanaklaselis) nafiqr-nagrZnobis gameorebas ki mxolod maSin

unda mivmarToT, roca Tavad moswavle ver gaumklavdeba sakiTxs

_ ifiqrebs, Seecdeba, magram ver SeZlebs. am dros is

qvecnobierad mainc Semzadebulia sxvisi naTqvamis gasaTaviseblad

_ swored imis wyalobiT, rom winaswar ukve Tavad aqvs nafiqri.

swored nafiqri _ da ara ganaxsenebi.

moswavleTa namdvili aqtiuroba _ esaa maTi damoukidebeli

azrovneba, msjeloba, Zieba, kvleva, Semoqmedeba, warmosaxva Tu

grZnoba. maTi sakuTari, maTi nebelobisa da Zalisxmevis nayofi.


- 45 -

Cveneuli meTodikiT momuSave maswavlebeli mxolod krebis

karg TavmjdomaresaviTaa _ kargad da ostaturad anawilebs

SekiTxvebs da bolos gamarTulad da srulad Seajamebs daskvnas.

xolo gakveTilis ZiriTadi dro eTmoba Tavad moswavleTa

msjelobas, moqmedebas, muSaobasa da sxva. oRond, mxolod

`krebis Tavmjdomareoba~ araa sakmarisi. maswavlebeli cotaTi

mainc `fsiqologi~ unda iyos, raTa SeZlebisamebr (ramdenadac

saSualebas aZlevs klasSi moswavleTa raodenoba) ganaxorcielos

individualuri midgomis principi. sul mcire, rac mas

moieTxoveba am mxriv _ esaa moswavleebis unar-SesaZleblobaTa

mudmivi gaTvaliswineba. da kidev _ maswavlebeli cotaTi mainc

`msaxiobi~ unda iyos: qmediTad unda iyenebdes xelebis

gamaTvalsaCinoebel moZraobas, pirisaxis gamometyvelebasa da

xmis mimoxras.

amrigad, saswavli sagnis Cveneuli kargi maswavlebeli (aq

ganvixilavT swored sagnis maswavleblobas _ da ara aRmzrdeldamrigeblobas)

_ esaa ZiriTadad `krebis kargi Tavmjdomare~,

romelic, amasTan, cotaTi `fsiqologia~ da cotaTi _

`msaxiobi~. swored esaa misi SemoqmedebiTi sarbieli. xolo sxva

mxriv is saukeTeso Semsrulebeli unda iyos _ zedmiwevniT da

gulmodgined asrulebdes yvela meTodikur miTiTebas. TviT

sametyvelo sityva-terminebiTac ki ar unda gascdes gegmakonspeqtebs,

ar unda ixmaros is sityvebi, romlebic

arabunebrivia bavSvisaTvis da namdvili qarTulisaTvis

(magaliTad: konfliqti, konfliqturi, funqcionirebs,

problema, stabiluri, winaaRmdegobaSi moxvedi, pozicia,

situacia da sxva mravali `sarevela~). maswavlebels

gauWirdeba uceb, sakuTari metyvelebis kvaldakval, gaarkvios,

sityva Tu gamoTqma ramdenad kargia da bunebrivi mocemuli

wlovanebis bavSvisaTvis. amitom Cven mas advil gzas vTavazobT:

ganuxrelad ganaxorcielos gegma-konspeqtebSi mocemuli ConCxi.

rogorc iTqva zemore, swored rom ganaxorcielos, anu `xorci

Seasxas~, `gaacocxlos~.

axla CamovayaliboT isic, Tu ra ar unda iyos Cveneuli kargi


- 46 -

maswavlebeli. pirvel yovlisa, esaa `mecnier-leqtori~: grZeli

monologebiT, `mecnieruli~ terminebiT, maRalfardovanebiT. da

agreTve _ `novator-meTodisti~. Cven yvelanairad mxars vuWerT

maswavleblis novatorul meTodistur Ziebebsa da Semoqmedebas,

magram es unda xdebodes ara maswavleblobisas, aramed calke,

skolidan Sin dabrunebisas _ Tuki eqneba amis dro, Zal-Rone da

SesaZlebloba, Tanac, ara momdevno gakveTilisaTvis saWiro

Cveulebrivi momzadebis xarjze! maSin es iqneba maswavleblis

meore saqmianoba, `SeTavsebiT meore profesia~ _ mkvlevarmeTodistisa.

amgvarma maswavlebelma Tavisi namuSevari Cven unda

mogvawodos. Cven amas madlierebiTa da gulisyuriT SevxvdebiT,

erTad ganvixilavT, avwon-davwoniT da davsaxavT samoqmedo

gegmasac. magram uamisod, naCqarevi, zerele da TviTneburi

`novator-meTodistoba~ Zalian sarisko ramea da ufro xSirad

bavSvebisaTvis savalalod mTavrdeba.

xSirad Segvxvedria maswavlebeli, romelic `novatorobs~ da

garegnulad gakveTilebs fuWi zizilpipiloebiT amSvenebs. sinamdvileSi

ki gaurbis an zereled asrulebs mis mTavar movaleobas

_ yoveldRiur `Sav samuSaos~: saguldagulod moemzados gakve-

TilisaTvis, mowesrigebuli hqondes TvalsaCinoeba, kargad gaasworos

sakontrolo namuSevrebi da moswavleebsac safuZvlianad

gaasworebinos Secdomebi, izrunos mravali moswavlis codnasa

Tu unarCvevebSi arsebul xarvezTa gamosasworeblad da sxva.

erTi sityviT, svevofcb icvleba zerele garegnuli zizilpipiloebiT,

da Sedegic saTanadoa: moswavleTa codna da unar-

Cvevebic zerelea da zizilpipiloebiviT umaqnisi.

maswavlebels ar unda gaeparos pedagogikuri Secdomebi:

1) moswavlis Secdomaze ukmayofilebis gamoxatva (uxeS sityvebze

xom laparakic zedmetia) _ moswavles ar unda eSinodes Secdomisa;

2) advili SekiTxvis micema Tu advili amocanis garCevis davaleba

Zlieri moswavlisaTvis;

3) pasiuri moswavlisTvis yuradRebis dakleba, bolomde

argarkveva imisa, sustma moswavlem gaigo Tu vera programuli


- 47 -

(ara damatebiTi Zneli) sakiTxi;

4) maTematikuri teqstebis zepirad swavleba, anda, imis

ugulebelyofa, rom moswavles sakiTxi zepirad aqvs naswavli da

wesierad ar esmis;

5) roca msjelobisas moswavle winadadebas kargad ver gamar-

Tavs, Semofargvla mxolod SesworebiT: moswavlem SeiZleba

kargad gaigo Tavisi xarvezi, maswavleblis Sesworebac gaigo,

magram es araa sakmarisi _ moswavlem sakuTari piriT unda

gaimeoros (anda, sakontrolo naweris gasworebisas _ sakuTari

xeliT unda gadaweros) Sesworebul-Sevsebuli winadadeba.

Cveneuli mrwamsiT, maswavleblis ostatobis umTavresi maCveneblebia:

1) `lsfcjt!Ubwnkepnbsf~ _ ramdenad aaqtiurebs, aazrovnebs,

damoukideblad amuSavebs moswavleebs, ramdenad metad Canan

moswavleebi da naklebad _ TviTon; ramdenad Tanabrad aZlevs

sityvas sxvadasxva moswavleebs. _ Tanac, ara mxolod erTi

gakveTilis farglebSi. ese igi, maswavlebels isic unda

axsovdes, Tu romel moswavleebs alaparakebda Tu amuSavebda

wina gakveTilze _ raTa axla sxvebs dauTmos sarbieli. garda

amisa, SekiTxvebis ganawilebisas unda gaiTvaliswinos TviTeuli

moswavlis Taviseburebani.

2) `Tfntsvmfcfmj~ _ ramdenad srulad da zustad

asrulebs meTodikur saxelmZRvaneloTa miTiTebebsa da gegmas;

`ConCxs~ ramdenad kargad `asxmas xorcs~; ramdenad bolomde

CaeZieba xarvezsa Tu naklovanebas da ar miafuCeCebs mas.

3) `lsfcjt! Ubwnkepnbsf~ _ ramdenad kargad iyenebs

sagakveTilo dros, magaliTad: winaswar aqvs Tu ara momzadebuli

dafaze warwerebi da saklaso TvalsaCinoeba; roca erTi

moswavle dafasTan raimes akeTebs, sxvebi uqmad xom ar hyavs

mocdenili da sxva; ramdenad inarCunebs gakveTilisTvis saWiro

ritms.

4) `gtjrpmphj~ _ ramdenad axerxebs individualuri

midgomis ganxrocielebas, ramdenad axerxebs imas, rom hqondes

urTierToba calkeul moswavlesTan, romlis pirovnebasac


- 48 -

udidesi pativiscemiTa da siyvaruliT emsWvalvis, romlis

Taviseburebebsac iTvaliswinebs da romelsac ar Cakargavs

saerTo usaxur `klasSi~.

5) `gtjrpmphj~ _ ramdenad axerxebs bavSvis

cnobierebisaTvis bunebrivi samyaros Seqmnas da ramdenad

naklebad arRvevs mas mxolod mozrdili adamianisaTvis

naSandoblivi sxvadasxva sityviT, azriT, qceviTa da sxva;

ramdenad qmnis imis pirobebs, rom bavSvis cnobiereba gaRrmavdes,

gamdidrdes; Sinaganad, bunebrivad ganviTardes da gaifurCqnos,

rogorc lamazi yvavilis kokori, sakuTari bunebrivi niadagidan

rom ikvebeba da izrdeba da jansaR nayofs gamoiRebs (romelic

rogors SeZlebs!) _ da ara ufrosTagan mosmenilis dagrovebiT

gaberili da damZimebuli, rogorc xelovnuri sasuqebiT uxvad

napativebi da gafuyuli fuWyvavila, romlis nayofi garegnulad

viTom lamazia, magram sinamdvileSi futuro da ugemuria.

6) `ntbyjpcj.rbsUwfmj~ _ rogoria misi metyveleba,

gamoTqma, ramdenad gamarTuli, mdidari da wminda qarTuliT

metyvelebs.

7) `ntbyjpcj~ _ ramdenad kargad iyenebs xelebis

gamomxatvel moZraobas, miTiTebebs, pirisaxis gamometyvelebas,

xmis mimoxrasa da sxva.

8) `Tfntsvmfcfmj~ _ ramdenad saguldagulod da

xarisxianad asworebs da afasebs sakontrolo Tu sagamocdo

namuSevrebs;

9) `nfdojfs.npb{spwof~ _ ramdenad kargad esmis saswavli

sagnis mecnieruli, logikuri da SemoqmedebiTi mxareebi.

SeiZleba vinmes moeCvenos, rom Cven Zalian maRal moTxovnebs

vuyenebT maswavlebels. es ase araa. Cveneul moTxovnaTa `sanacvlod~

Cven maswavlebels vaTavisuflebT sxva, gacilebiT ufro

sapasuxismgeblo an rTul movaleobaTagan: meTodikuri Ziebani da

sakiTxis damuSavebis formaTa da saSualebaTa mogoneba;

damatebiTi amocanebis, TamaSebis, literaturisa Tu

TvalsaCinoebis moZieba; gakveTilis winaswari gegma-konspeqtis

Sedgena da sxva. ukanasknel sakiTxTan dakavSirebiT: Cven ar


- 49 -

miviCnevT maswavleblis naklad, Tuki is gakveTilis ganmavlobaSi

meTodikur saxelmZRvanelos, gegma-konspeqtebs gamoiyenebs xolme.

mravali wvrilmanis zepirad damaxsovreba Znelia da araa saWiro.

sazogadod, maswavleblis muSaobaSi (sxvaTa Soris, asevea

mecnierebaSic!) cxra wilia ruduneba da Savi Sroma da mxolod

erTi wilia Semoqmedeba-STagoneba, sixaruli da xelovneba _ rac,

Tanac, im cxra wilis gareSe _ unayofo iqneba. magram, cxadia, es

erTi wili aranakleb Zvirfasia, vidre danarCeni cxra wili!

sxvaTa Soris, amasve ukavSirdeba erTi Zalian mniSvnelovani

zogadpedagogikuri sakiTxi. saqarTveloSi metismetad mravladaa

`Zalian niWieri~, magram zarmaci~ adamianebi. winaT ase ar

yofila, qarTveli kaci, piriqiT, gamorCeulad Sromismoyvare iyo

(amas ueWvelad amtkicebs Tundac qvaSi nakveTi CuqurTmebi da

umaRles doneze ganviTarebuli meRvineoba). Cveni erovnuli

ubedureba isaa, rom, maxinjur garemoebaTa gamo, davSordiT Cvens

namdvil znes (sxva mxrivac, da ara mxolod Sromismoyvareobis

mxriv). Cveneuli meTodika isea agebuli, rom `Zalian niWieri,

magram zarmaci~ moswavle karg Sedegebs verasdros miaRwevs.

ramdeni mniSvnelobac aqvs gonebriv monacemebs, imdenive

mniSvneloba aqvs sibejiTesac. rogorc adamianis marjvena da

marcxena xelebia. `Zalian niWieri, magram zarmaci~ calxela

adamians hgavs.

maswavlebelma niSani unda daweros mxolod Sedegis mixedviT

_ da ara imis mixedviT, Tu ra `SeuZlia~ moswavles! zarmacs

ufro metad `ar SeuZlia~, vidre saSualo gonebriv monacemTa

mqones! swored ufro sizarmace, vidre gonebriv monacemTa

nakleboba, iwvevs siZneleebs swavlaSi, saqmeSi da ufro metic,

cxovrebaSi (da xSirad am siZneleTa gamo adamianis zneobac ki

fuWdeba, is Suriani da RvarZliani xdeba).

kidev erTxel: maswavlebelma niSani unda daweros

mxolod Sedegis mixedviT _ da ara imis mixedviT, Tu ra

`SeuZlia~ moswavles. Semfasebelma SeiZleba arc icodes, Tu

romel moswavles ekuTvnis Sedegi! garda amisa, yovelad

dauSvebelia niSnebis mikerZoebiT wera. yvelas unda eweros is


- 50 -

niSani, rasac imsaxurebs, yovelgvari siyalbis gareSe! uazro da

fuWi lmobierebiT bavSvs daTvur samsaxurs gavuwevT!

maswavlebels yovelTvis unda axsovdes, rom is _ moswavlis

aRmzrdelicaa. xolo yoveli nayalbevi niSani _ sawamlavis TiTo

wveTia, romelic ryvnis moswavlis suls. axalgazrda unda mie-

Cvios sakuTari gonebiT, sibejiTiTa da patiosani SromiT warmatebis

mopovebas, sakuTari Zalebis rwmena unda gamoumuSavdes da

gergilianoba, Taosnoba ganuviTardes. es yovelive gacilebiT ufro

mniSvnelovania, vidre maTematikis codna.

§ 11. Cveneuli amocanebis tipebi

Cveneuli amocanebi saguldagulod gaazrebul Tanwyobas qmnis.

a m o c a n e b i

programulebi damakavSirebelni damatebiTebi am samis naerTebi

(romlebic

uSualod

mimdinare

programul

sakiTxzea)

(ori naswavli

sakiTxisa

erTmaneTTan _

gvaqvs sakiTxebis

TiTqmis yvela

SesaZlo wyvilze)

(zogad saazrovno da

SromiT-saSemsruleblo

unarCvevaTa

ganmaviTarebelni

_ Zalian mravali

sxvadasxva saxisaa)

Semamzadebelni (saswavli sakiTxis win) ganmamtkicebelni

I tipi II tipi III tipi I tipi II tipi III tipi

Semamzadebeli amocanebis sami ZiriTadi tipia:

I. axali sakiTxisaTvis xSirad saWiroa xolme romelime adre

naswavli sakiTxi. misi gameoreba swored Semamzadebeli amocanis

meSveobiT xdeba, aqtiurad. Tuki moswavles daviwyebuli aqvs es

Zveli sakiTxi an kargad veRar iyenebs mas, am amocanas ver

amoxsnis. magram am saSinao davalebis klasSi garCevisas mainc

moxdeba Sexseneba da naswavlis gaaqtiureba, moswavle am amocanas


- 51 -

klasSi mainc amoxsnis. amitom momdevno gakveTilisaTvis

(romelSic is axali sakiTxia) moswavle mainc mzad iqneba.

II. axal Teoriul teqstSi xSirad aris xolme erTi an ramdenime

Zneli adgili (magaliTad, damtkicebaSi raime gardaqmna, an

SedarebiT rTuli msjeloba). am nawyvetis gaadvilebuli

varianti Semamzadebel amocanaSi muSavdeba. Tuki moswavleebs igi

gauWirdebaT da Sesabamis amocanas ver amoxsnian, am saSinao

davalebis klasSi garCevisas sxva amocanebs Soris es Zneli

adgilic gairCeva. amitom momdevno gakveTilis Teoriuli

teqstisaTvis (romelSic is Zneli adgilia) moswavleebi mainc

mzad iqnebian.

III. esaa namdvili, sakuTriv Semamzadebeli amocanis tipi. moswavle

xsnis erT an ramdenime advil amocanas axali cnebisa Tu

axali wesis konkretul saxeebze ise, rom TviT es cneba Tu wesi

naxsenebic ki araa. am tipis Semamazadebeli amocanebi, rogorc

wesi, advilebia da maT xsnis moswavleTa didi umravlesoba

(Tuki Sin ara, klasSi mainc).

aseve aqtiur swavlebazea agebuli Cveneuli ganmamtkicebeli

amocanebic. maTi sami ZiriTadi tipia:

I. naswavli sakiTxis Cveulebrivi, pirdapiri gamoyenebis amocanebi.

amgvari amocanebi uSualod mosdevs axladnaswavl sakiTxs

(Tanac, pirveli amocana Zalian advilia xolme, raTa misi amoxsna

TiTqmis yvela moswavles SeeZlos). garda amisa, amgvari ganmamtkicebeli

amocanebi drodadro gvxvdeba xolme SemdgomSic, maT

Soris momdevno klasebSic _ raTa es naswavli sakiTxi aqtiurad

gameordes xolme.

II. ganmamtkicebeli amocanis es tipi hgavs II tipis

Semamzadebel amocanas: Tuki axladnaswavl Teoriul teqstSi

gamotovebulia raime sakiTxi, an damtkicebis nawyveti, an

romelime msgavsi SemTxvevis ganxilva, an damtkicebis dayvana

advil saxeze da sxva _ es amocanaSi muSavdeba. Tuki moswavle

mas Tavs ver gaarTmevs, am saSinao davalebis klasSi garCevisas


- 52 -

mainc gairkveva.

III. naswavli sakiTxis ganzogadebis, an raimegvari ganviTarebis

amocanebi.

amrigad, aqtiuri swavlebis mTavari sayrdeni aris mravali

tipis amocanebis sagangebo Tanwyoba. amitom vaniWebT Cven udides

mniSvnelobas amocanebis saguldagulod garCevas saklaso

muSaobisas. magram imisaTvis, raTa amocanis saklaso garCeva

nayofieri iyos, saWiroa, rom moswavles winaswar nafiqri

hqondes masze damoukideblad (saSinao davalebaSi).

§ 12. Semamzadebeli safexurebi

yoveli saxelmZRvanelo iwyeba wina klasSi naswavlis gameorebiT,

rasac mcire istoriuli cnobebi axlavs (istoriuli

cnobebi axlavs sxva sakiTxebsac, oRond, cxadia, ara

gasazepireblad). Semdeg TandaTan iwyeba axali sakiTxebi. magram,

rogorc iTqva, TiTqmis yoveli axali sakiTxis gadmocemas win

uZRvis, erTi dRiT uswrebs saTanado Semamzadebeli safexuri. es

safexuri sagangebod Sedgenili, Tftbtxbwm!tbljUy{f!nphf{jmj

Semamzadebeli amocanebisagan Sedgeba, Teoriuli teqstis gareSe.

swored Semamzadebeli amocanebiT iqmneba hbovxzwfufmj!

brujvsj! n{bpcb. esaa Cveni kursis yvelaze mTavari siaxle da

Tavisebureba.

Semamzadebeli safexuri moicavs an erT mTlian gakveTils _

da maSin am gakveTilis saTauria `amocanebi~, an gakveTilis

nawils _ mxolod ramdenime amocanas. ese igi, Semamzadebel

safexurs an calke gakveTili eTmoba, an igi SeerTebulia wina

gakveTilis ganmamtkicebel da sxva amocanebTan.

cxadia, maswavlebelma unda icodes, Tu risi Semzadeba xdeba

ama Tu im Sesamzadebeli amocanebiT. magram man araviTar

SemTxvevaSi ar unda Tqvas es, ar unda ixmaros is cnebaterminebi

Tu moqmedebebi, romelTa Semzadebac xdeba. unda

amoixsnas mxolod mocemuli gakveTilis konkretuli amocanebi,


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im terminebis farglebSi, romlebic an maTive pirobaSia

mocemuli, an adrea naswavli.

amrigad, maswavlebeli mTlianad saxelmZRvanelos unda mihyves,

mas win ar unda gauswros. saWiroebis SemTxvevaSi unda

daexmaros moswavleebs sakiTxis arsis aRmoCenaSi, da Tan Seavsos

saTanado amocanebiT Tematikuri krebulidan. Tuki klasi,

mizezTa gamo, sakmaod kargad ver iswavlis ama Tu im Znel

sakiTxs, maswavlebeli erTi dRiT SeaCerebs winsvlas ZiriTad

saxelmZRvaneloSi da dauTmobs erT damatebiT gakveTils Znel

sakiTxs, SeurCevs ra mas saTanado amocanebs Tematikuri

paragrafebidan _ rogorc saklaso, ise saSinao samuSaoebisaTvis.

ase rom, maswavlebels damoukideblad mouwevs mxolod Semavsebeli

amocanebis SerCeva Tematikuri krebulidan.

maswavleblis TviTmoqmedebisaTvis Tematikur krebulebSi

gacilebiT ufro didi sarbielia, vidre ZiriTad gakveTilebSi.

Semamzadebel safexurTa nimuSebi

magaliTisTvis ganvixiloT XIV Tavis me-3 da me-4

gakveTilebi:

upmgbtj!hboupmfcboj da hboupmfcjt!psj![jsjUbej!Uwjtfcb.

me-3 gakveTilSi 9 amocanaa. amocana nulovani nomriT, rogorc

yovelTvis, gakveTilis teqstis gasaazrebelia. 1-li, me-2 da me-3

amocanebi naswavli sakiTxis _ tolfasobis _ gansamtkicebladaa.

Tanac, 1-li da me-2 _ advili amocanebia, romlebsac sWirdeba

naswavlis pirdapiri, standartuli gamoyeneba _ maT naxevars

mainc albaT amoxsnis TiTqmis yvela moswavle, Tan gaimeoreben

ricxvis modulis cnebas; me-3 amocanac advilia, magram mas

logikuri gaazreba sWirdeba. aseve, ramdenime gakveTilis win

naswavlis gansamtkicebelia me-7, geometriuli amocana, romelic

agreTve Zalian advilia. adre naswavlis gansamtkicebelia agreTve

me-5 amocanac, Tanac, igi ukve naxevrad Semamzadebelia, radgan

msgavs SesakrebTa SeerTebis gaxseneba aucilebelia momdevno

gakveTilebis Sesaswavlad. sazogadod, Cven naswavls yovelTvis


- 54 -

swored ase _ aqtiurad, amocanis saSualebiT vaxsenebT klass.

mTavari ki danarCeni amocanebia: me-4, me-6 da me-8. samive es

amocana _ momdevno gakveTilis Semamzadebelia. amasTan, me-4

amocanis amoxsnas TiTqmis ar sWirdeba maTematikis codna, mcire

mosazrebac eyofa da mas advilad amoxsnis is moswavlec,

romelsac maTematika naklebad exerxeba, magram mosazrebulia.

asea Tu ise, es amocana klasSi gairCeva da, momdevno

gakveTilisTvis mainc, moswavleebs ecodinebaT, rom 2x + 15 = 25

+ x gantolebis tolfasia x + 15 = 25 gantoleba. ese igi,

SeiZleba ucnobis gadatana gantolebis meore mxares, oRond

sapirispiro niSniT. amave daskvnas ganamtkicebs me-6 amocanis I

da II qveamocanebi. arsebiTad, moswavleebi gamoiyeneben

gantolebis ZiriTad Tvisebas ise, rom Camoyalibebulic ki ar

eqnebaT igi _ da wesis zogadi Camoyalibeba am gakveTilze arcaa

saWiro! mas momdevno gakveTilisTvis moswavle TviTon

`aRmoaCens~, Semdeg sityvieradac gaiazrebs, roca waikiTxavs me-4

gakveTilis teqsts.

aseve aqtiurad Seamzadebs gantilebis meore ZiriTad Tvisebas

me-6 amocanis III da IV qveamocanebi da me-8 amocana.

amrigad, me-3 gakveTilidan sami amocana aris momdevno, me-4

gakveTilis Semamzadebeli. maswavlebelma es amocanebi unda

gaarCevinos moswavleebs klasSi Cveulebriv, saSinao davalebis

garCevisas, oRond ise, rom ar Camoayalibos, arc ki axsenos

is axali wesebi, romlebsac es amocanebi Seamzadebs.

es wesi moswavleebs momdevno gakveTilis Teoriul teqstSi

SexvdebaT (da mas Sin damoukideblad waikiTxaven). am gakveTilze ki

amocanebi amoixsneba mxoloddamxolod konkretulad, yovelgvari

ganzogadebis gareSe, anu maTi konkretuli Sinaarsidan gamomdinare!

zogadi wesis Camoyalibeba maswavlebelma ar unda moiTxovos. am

wesis gasaazreblad moswavleebi ukve mzad iqnebian da TviTon unda

waikiTxon da gaiazron igi Sin momdevno gakveTilis momzadebisas.

sxvaTa Soris, ganxiluli gakveTilidan TvalsaCinod gamoCnda


- 55 -

`saymawvilo maTematikis~ kidev erTi arsebiTi maxasiaTebeli:

neli gaRrmavebuli swavleba, Zalian mcire-mcire nabijebiT

winsvliT _ raTa saSualo moswavlesac xelewifebodes axali

sakiTxis damoukideblad aRmoCena da gaazreba (arc Zlieri

moswavle moiwyens, radgan misTvisac xom gvaqvs sakmao saazrovno

masala _ bolo amocanebi). rogorc vnaxeT, Cven TiTo gakveTils

vuTmobT iseT sakiTxebs, romlebic sxva kursebSi, rogorc wesi,

erTdroulad iswavleba.

§ 13. erTi gakveTilis sanimuSo konspeqti:

Tavi I, gakveTili 6

gakveTilis dasawyisSi maswavlebeli Camoivlis da Tvalis

erTi movlebiT Seamowmebs, Tu rogor aqvT Sesrulebuli

davaleba moswavleebs. amis gakeTeba kargia imisaTvis, raTa

TviTeulma moswavlem icodes, rom mis namuSevars maswavlebeli

aucileblad naxavs. yvela moswavles kalmistari ukavia da avsebs

da xvews Tavis saSinao namuSevars [dawvrilebiT _ ix. $ 9].

1. maswavlebeli rig-rigobiT akiTxebs moswavleebs am

amocanis I, II, III da IV qveamocanebs da sTxovs maT axsnan, ratomaa

marTebuli an mcdari Sesabamisi ormxrivi utoloba. magaliTad,

IV utoloba araa marTebuli, radganac 1/125 = 0,008, es araa

naklebi 0,008-ze.

2. maswavlebeli isev rig-rigobiT gamoiyvans moswavleebs

dafasTan da dawerinebs Sesabamisi qveamocanebis pasuxebs.

araswori pasuxis SemTxvevaSi Secdomas Seasworebs sxva

moswavle, da ara maswavlebeli.

3. maswavlebeli romelime moswavles SeekiTxeba, Tu romel

ricxvebs SeiZleba aRniSnavdes w da ratom. ramdenime aseTi

ricxvis CamoTvlis Semdeg pasuxs gaagrZelebinebs sxva moswavles.

4. am amocanis garCevisas maswavlebelma moswavleebis

yuradReba unda gaamaxvilos imaze, rom winadadebaTa simcdaris


- 56 -

dasamtkiceblad sakmarisia erTi magaliTic ki, xolo

marTebulobis dasamtkiceblad magaliTebis moyvana araa sakmarisi.

magaliTad, I winadadebis marTebuloba unda daasabuTon

amgvari msjelobiT:

raki a < g < b, amitom a < g da g < b. maSin, cxadia,

marTebuli iqneba aramkacri utolobebic: a ≤ g da g ≤ b. ese igi

marTebuli iqneba utoloba a ≤ g ≤ b.

II winadadebis dasamtkiceblad ki moviyvanoT amgvari

magaliTi: vTqvaT, a = g = b = 2. am SemTxvevaSi miviRebT

utolobebs: 2 ≤ 2 ≤ 2 da 2 < 2 < 2 . cxadia, am

utolobebidan pirveli marTebulia, meore _ mcdari. amitom II

winadadeba mcdaria.

5. es amocana, iseve rogorc wina amocana, logikur

msjelobebs moiTxovs da amitom moswavleTaTvis sakmaod

Znelebia. maswavlebeli ar unda moelodos, rom moswavleebi

unaklod Caatareben saWiro msjelobebs. magram swored amgvari

amocanebis amoxsna TandaTanobiT ganuviTarebs moswavleebs

logikuri azrovnebis metad saWiro unar-Cvevebs.

maswavlebeli, iseve rogorc sxva amocanebis garCevisas,

TviTeul qveamocanas sxvadasxva moswavleebs akiTxebs da pasuxs

asabuTebinebs. magaliTad: I ar SeiZleba moxdes, imitom, rom

Tu a > b -ze, maSin a ver iqneba b-ze naklebi.

II SeiZleba moxdes im SemTxevaSi, roca a = b.

IV SeiZleba moxdes, radganac utolobebs Soris weria sityva

an. maswavlebeli ikiTxavs: _ rodis ar SeiZleba moxdes IV ? {im

SemTxvevaSi, roca a = b, radganac maSin arc a>b da arc a


- 57 -

farTobi {fuZis perimetrisa da simaRlis namravlis tolia}.

pirvel rigSi unda gamovTvaloT fuZis perimetri, Semdeg ki

gamovTvliT ucnobi gverdis sigrZes.

8. maswavlebeli akiTxebs moswavleebs maT mogonil amocanebs

da ambobs, ramdenad kargad aris Sedgenili isini. Seaqebs iseT

moswavleebs, romlebac Seadgines moulodneli, ucnauri, saintereso,

Tanac marTebulad dasabuTebuli amocanebi.

amis Semdeg maswavlebeli gaarkvevs, Tu rogor gaiges moswavleebma

gakveTilis Teoriuli nawili, anu gakveTilis teqsti.

amisaTvis maswavlebeli romelime moswavles gaiyvans dafasTan da

daawerinebs mas raime ormxriv, magram mkacr utolobas,

waakiTxebs da gaarkvevinebs, marTebulia dawerili utoloba Tu

ara da ratom. magaliTad: 3,2 < 4,5 < 4,5 . moswavlem unda

upasuxos, rom es utoloba mcdaria, radganac 4,5 araa naklebi

4,5-ze. Tuki moswavle arasworad an arasrulad upasuxebs, maSin

SekiTxvas upasuxebs sxva moswavle. amiT moxdeba wina, me-5

gakveTilis mokle gameoreba.

amasobaSi maswavlebeli dafaze

swrafad xeliT daxazavs wriul

rgols (nax.1) da ityvis:

_ CawereT ormxrivi utoloba,

romelic gviCvenebs, rom A

wertili am rgolis wertilia.

maswavlebeli gamoiyvans romelime moswavles da daawerinebs

pasuxs dafaze. moswavleebs swori pasuxis dawera ar unda gau-

WirdeT, radganac Sesabamisi Semamzadebeli amocana gairCeoda wina

gakveTilze (me-4 amocana). daweren: 12 m ≤ |AO| ≤ 16 m.

_ ratomaa saWiro aq aramkacri utolobebis niSnebi?

{imitom, rom A wertili SeiZleba iyos sazRvris wertilic.

am SemTxvevaSi ki gveqneba tolobebi}. _ ra moxdeboda, erT


- 58 -

mxares mkacri utoloba, rom dagvewera, magaliTad:

12 m < |AO| ≤ 16 m ? {es utoloba mcdari iqneba, roca A

wertili rgolis Siga wrexazis wertilia}.

maswavlebeli dafaze dawers raime ormxriv utolobas,

magaliTad, aseTs: 1,2 ≤ a < 1,6 da romelime moswavles sTxovs

waikiTxos is ornairad: {1,2 naklebia an tolia a-ze da a

naklebia 1,6-ze; meorenairad: a metia an tolia 1,2-ze da

naklebia 1,6-ze}. sxva moswavles ki SeekiTxeba, marTebulia Tu

ara es utoloba da ratom, roca a = 1,2 da a = 1,6 .

gakveTilis bolos maswavlebeli amoikiTxavs sias da Tan

dawers moswavleTa Sefasebebs.

sazogadod, gamosaTvlel an standartuli pasuxis mqone sxva

amocanebis garCevas SedarebiT ufro naklebi dro unda daeTmos.

aseT SemTxvevaSi umjobesia maswavlebelma TviTon daawerinos

romelime moswavles dafaze axali savarjiSo da yvelas

mosTxovos misi Sesruleba. magaliTad, garCeuli amocanebidan

pirvelis wakiTxvisas maswavlebelma SeiZleba moswavles dafaze

daawerinos utoloba 0,06 ≤ 3/50 ≤ 2 da mosTxovos yvelas, rom

Tav-TavianT rveulebSi CaweriT gaarkvion, marTebulia es

utoloba Tu ara.

TfojTwob; ganxiluli gakveTili sakmaod datvirTulia. amitom

SedarebiT susti klasis SemTxvevaSi, SesaZlebelia, yvela amocanis

safuZvliani garCeva verc moeswros. es araa sagangaSo. maswavlebelma

davalebad mainc unda misces momdevno gakveTili. zogierT

gakveTilze SeiZleba piriqiT moxdes, morCes dro. aseT

SemTxvevaSi maswavlebelma SeiZleba gaarCevinos adre gaurCeveli,

vermoswrebuli amocana an gaakeTebinos axali amocana saxelmZRvanelos

Tematikuri paragrafebidan, Tavisi Sexedulebisamebr.

§ 14. Temis damuSavebis nimuSi _ usasrulobis cneba

VII klasis programis mTavari Taviseburebaa usasrulobis cnebis

Rrma da safuZvliani damuSaveba. igi maTematikis, fizikis, as-


- 59 -

tronomiis, logikis, filosofiisa da Teologiis erTerTi umniSvnelovanesi

da uaRresad saintereso cnebaa [12]. am cnebis

gareSe SeuZlebelia wrfis, sibrtyis, kuTxis, ricxviT simravle-

Ta, iracionaluri ricxvisa da mravali sxva umniSvnelovanesi ma-

Tematikuri cnebis namdvili swavla. SeuZlebelia maTematikuri

analizis umartivesi sawyisebis gaazrebac (`analizis sawyisebi~,

tradiciuli programiT viTom `iswavleboda~, Tumca mas TiTqmis

veravin swavlobda. arsebiTad, es arcaa maTematika, aramed uSinaarso

da maTematikuri TvalsazrisiT gaumarTav cnebebze da formulebze

meqanikuri manipulaciebis sistemaa; mas kompiuteruli

programa gacilebiT ukeTesad gaakeTebda).

yovelive amis gamo usasrulobis cneba uTuod imsaxurebs imas,

rom safuZvlianad iqnes damuSavebuli. cxadia, amas sakmao raodenobis

saswavlo saaTebi sWirdeba. amitom gviwevs zogi sakiTxis

gadatana Semdgom klasebSi (magaliTad, aqsioma da Teorema geometriaSi;

algebruli gardaqmnebi). magram es mainc aucilebelia.

rogorc wesi, saskolo kursebSi usasrulobis cneba Zalian

zereled muSavdeba an TiTqmis ar muSavdeba. amis gamo, rogorc

gviCvenebs Cveni sagangebo gamokvlevebi, es cneba ar esmis ara

mxolod skoladamTavrebulTa 95 %-ze mets, aramed maTematikursabunebismetyvelo

ganxris specialistebsac ki.

usasrulobis cnebis swavlebis Cveneuli meTodika emyareba,

garda zemore aRwerili zogadi safuZvlebisa, Cven mier Catarebul

sagangebo fsiqologiur eqsperimentsac [12]. cnebis swavlebaSi

TvalsaCinod vlindeba da xorcieldeba Cveni yvela mTavari safuZveli:

mzaobis principi, aqtiur-problemuri swavleba, humanisturi

swavleba, gaRrmavebuli swavleba, istorizmis principi da sxva.

gansakuTrebiT mkveTrad vlindeba gaerTianebuli (integrirebuli)

swavleba. usasrulobis cnebis swavlebis Cveneuli meTodika

bunebrivad aerTianebs maTematikis TiTqmis yvela saskolo dargs

_ Tema Semdegi safexurebiT muSavdeba:

I. ariTmetika: Zalian didi naturaluri ricxvebi da 10-is


- 60 -

xarisxebi;

II. algebra: ricxvis Cawera standartuli saxiT, moqmedebebi

Zalian did ricxvebze;

III. simravleTa Teoria: usasrulo simravleni;

IV. gamoyenebiTi maTematika: usasrulobis cneba bunebaSi da moCvenebiTi

usasrulobani;

V. geometria: ricxvTa sxivi, geometriuli sxivi da wrfe;

VI. maTematikis istoria.

pirvelad swored am dros Semogvaqvs SemousazRvreli geometriuli

nakvTebi _ sxivi da wrfe.

saWiroa, kargad iqnes gaazrebuli Semdegi ZiriTadi cnebebi:

1. SemousazRvreli geometriuli nakvTi _ ewodeba iseT

nakvTs, romelic ver moTavsdeba vercerT wreSi (ragind didic

unda iyos igi).

2. usasrulo simravle _ ewodeba simravles, Tuki ar

arsebobs iseTi naturaluri ricxvi, romelic metia, vidre am

simravlis wevrebis raodenoba.

3. or simravles ewodeba tolZalovani, Tuki maTi `wevrebis

raodenoba erTidaigivea~, ufro zustad ki: Tuki arsebobs urTierTcalsaxa

asaxva erTi simravlisa meoreze; ufro martivad: Tuki

SesaZlebelia maTi wevrebis `dawyvileba~. sxvaTa Soris, Cveneuli

meTodikiT, I klasis pirvelsave TveSi, swored dawyvilebas

emyareba raodenobaTa metnaklebobisa da tolobis cnebebi.

4. Tvladi simravle _ ewodeba iseT usasrulo simravles,

romelic naturalur ricxvTa simravlis tolZalovania. magali-

Tad, Tvladi simravleebia: mTel ricxvTa simravle, racionalur

ricxvTa simravle, 45-is jerad ricxvTa simravle, 45-is

naturalur xarisxTa simravle. ese igi, racionaluri ricxvi

`imdenivea~, ramdenicaa naturaluri ricxvi, anda, ramdenicaa

mxolod 45-is naturaluri xarisxebis simravle. swored esaa

usasrulobis pirveli paradoqsi: yovel usasrulo simravles

moeZebneba misive sakuTrivi qvesimravle (misi nawili), romelic


- 61 -

mTliani simravlis tolZalovania!

Tvladi simravleebi _ esaa yvelaze `mcirericxovani~ usasrulo

simravleebi. Semdegi usasruloba masze mZlavria:

5. kontinualuri (uwyveti) simravle _ ewodeba iseT

usasrulo simravles, romelic namdvil ricxvTa simravlis

tolZalovania. magaliTad, kontinualuri simravleebia: dadebiT

ricxvTa simravle; 0-sa da 1-s Soris moTavsebul namdvil

ricxvTa simravle; raime or ricxvs Soris moTavsebul anda

yvela iracionalur ricxvTa simravle; monakveTis wertilTa

simravle; sibrtyeze kvadratebis simravle da sxva.

kontinualuri simravlis simZlavre metia, vidre _ Tvladisa.

ese igi, monakveTze `imdenive~ wertilia, ramdenic _ wrfeze!

magram `gacilebiT meti~ wertilia, vidre racionalurkoordinatebiani

wertilebia mTel wrfeze!

sxvaTa Soris, Zalian advilia imis

damtkiceba, rom monakveTi tolZalovania

masze grZeli nebismieri monakveTisa.

amisaTvis sakmarisia, ganisazRvros

amgvari urTierTcalsaxa

asaxva (`dawyvileba~):

Cveneuli programiT, me-3, me-4 da me-5 sakiTxebi mxolod X-

XI klasebSi iswavleba. xolo 1-li da me-2 _ VII klasSi.

aqve ganvixiloT ramdenime sakiTxi, raTa dazRveuli viyoT

gavrcelebuli mcdari warmodgenebisagan.

o!b!l!w!U!f!c!j!

!

tbtsvmj;!!!!!)vtbtsvmp*!Tfnptb{Swsvmj;!!!!!)vtbtsvmp*!Tfnpvtb{Swsfmj;!

wertili

monakveTi, Sekruli mrudi,

Sekruli texili, mravalkuTxedi,

wre, rgoli, elifsi...

wrfe, sxivi,

naxevarsibrtye,

sibrtye, kuTxe...

1) monakveTi, wre Tu samkuTxedi _ SemosazRvruli nakvTebia,


- 62 -

magram, viTarc wertilTa simravleebi _ usasruloebia. sazogadod,

yvela geometriuli nakvTi, wertilis garda, usasrulo

simravlea; xolo SemousazRvreli nakvTia kuTxe, rac gansakuTrebiT

arsebiTia! amgvarad, nakvTebi SeiZleba ise davajgufoT ↑

2) raimes daTvla rom SeuZlebelia, es imas sulac ar niSnavs,

rom raodenobaa usasrulo. magaliTad, bunebaSi, dedamiwaze Tu

mTel mzis sistemaSi arcerTi nivTieri erToblioba araa usasrulo.

sasrulia yvela mdinareSi, zRvasa da okeaneSi wylis wveTebisa

Tu qviSis marcvalTa raodenoba, odesme mcxovreb adamianTa

Tmis Rerebis raodenoba, anda maT mier odesme warmoTqmuli sityvebis

raodenoba, mTel dedamiwaze molekulebisa Tu atomebis

raodenobac ki! amis damtkiceba araa Zneli _ igi mocemulia VII

klasis saxelmZRvaneloSive.

mTavaria ara daTvlis SesaZlebloba, aramed raodenobis zemodan

SemosazRvris, zemodan Sefasebis SesaZlebloba. ese igi, iseTi

ricxvis moZebna, romelic namdvilad metia, vidre es raodenoba.

3) maTematikaSi Tvladi simravle, rogorc iTqva zemore,

usasruloa da ara sasruli!

4) racionalur ricxvTa simravle araa uwyveti, Tumca igi

yvelgan mkvrivia namdvil ricxvTa simravleSi. ese igi, ricxvTa

wrfeze racionalurkoordinatebiani wertilebi arsad ar qmnis

uwyvet monakveTs, magram yovel monakveTSi (ragind mcireSi)

mainc usasrulod mravali racionaluri ricxvia.

yovelive amis gamoa, rom usasrulobis cnebis gaazreba arsebiTad

afarTovebs adamianis gonebis Tvalsawiers da Tvisebrivad,

naxtomiseburad aviTarebs azrovnebas.

§ 15. geometriis swavlebis safexurebi

maTematikis gaerTianebuli swavlebas erT rames uwuneben:

geometria `daiCagrebao~. geometria aris erTaderTi saskolo

sagani, romelic moswavles mkacri deduqciuri Teoriis agebas

uCvenebs. amitom misi daknineba dauSvebelia.


- 63 -

es asea, magram saamisod ramdenad saWiroa mTeli Teoriis ageba?

Cveneul gaerTianebul kursSi geometriis wili gacilebiT metia,

vidre amJamad arsebul sxva kursebSi. Tanac, es iwyeba pirvelive

klasidan, romlis programis ramdenime siaxleTagan erTerTi

swored geometriis mkveTri gaZlierebaa (logikasTan erTad).

saxelganTqmuli maTematikosi, akademikosi v. arnoldi abstraqtul-formalistur

midgomas saskolo geometriisadmi moixseniebs

`sqolastikuri tvinisWyletis~ saxeliT, romelic namdvili

`kiseliovuri~ maTematikuri codnis mospobas cdilobs! (ruseTSi

ukve arsebiTadaa Secvlili geometriis kursi, xolo

inglisSi _ da dasavleTis TiTqmis yvela saxelmwifoSi asea _

60-iani wlebis Semdeg aRar yofila geometriis aqsiomatikuri

swavleba, mravali Teoremis grZel-grZeli damtkicebebiT).

TavisTavad, geometriac logikasa da simravleTa Teoriaze

(topologiasTan erTad) gvaqvs dafuZnebuli. geometriis swavlebaSi

ramdenime safexuri gamoikvTeba:

I-IV klasebSi vaswavliT mraval geometriul cnebas: kubi,

kvadrati, marTkuTxedi, wre, wertili, monakveTi, texili, mrudi,

samkuTxedi, oTxkuTxedi, xuTkuTxedi, ... , mravalkuTxedi, misi

gverdi, waxnagi, wibo, wvero da sxva. magram TiTqmis yovelTvis

gansazRvrebaTa gareSe vaswavliT. ganisazRvreba mxolod is cnebebi,

romlebic eqvemdebareba advil, da rac mTavaria, TvalsaCino

gansazRvrebas (magaliTad, texili, Siga are, sazRvari da sxva).

swavlebis sayrdenia uSualo TvalsaCino Cveneba, xatovani warmodgena

da moswavlis mier mravali geometriuli amocanis amoxsna.

maT Soris mTavaria mravalferovani amocanebi xazvaze, romel-

Ta Sesrulebisas moswavle sakuTari xeliT xazavs nakvTebs.

V-VI klasebSi iwyeba mcire Teoria, ganisazRvreba ramdenime

mniSvnelovani geometriuli cneba (magaliTad, wre da wrexazi).

safuZvlianad iswavleba farTobisa da moculobis cnebebi da

gazomva, simetria. planimetriis paralelurad stereometriac

muSavdeba: sfero, birTvi, cilindri, kubi, aguredi (marTkuTxa


- 64 -

paralelepipedi)... es xazi dagvirgvindeba eileris ulamazesi

TeoremiT mravalwaxnagebisaTvis.

sazogadod, stereometriis swavleba planimetriis

paralelurad _ Cveneuli meTodikis erTerTi Taviseburebaa. V

klasSi erTad iswavleba birTvi, sfero, wre da wrexazi; Semdeg,

kvlav wresTan dakavSirebulad _ cilindric. aseve, VI klasSi

erTad iswavleba marTkuTxedi da aguredi (marTkuTxa

paralelepipedi), kubi da kvardati; xolo momdevno klasebSi _

samkuTxedi da samkuTxa prizma, samkuTxedi da piramida da sxva.

VII klasSi pirvelad Semogvaqvs SemousazRvreli nakvTebi

(sxivi, wrfe, kuTxe). viwyebT damtkicebebs. viTardeba stereometriis

Temebic (Slilebi da sxva).

VIII-IX klasebSi logikur da istoriul konteqstSi vaswavliT,

ra aris gansazRvreba, aqsioma da Teorema. mxedvelobiT

iluziebze da logikis xaziT naswavl kerZo/zogadis mimarTebaze

damyarebiT vasabuTebT zogadi da zusti damtkicebis saWiroebas.

aRvwerT aqsiomaTa nimuSebs da maTgan ramdenime Teoremis gamoyvanas

(mxolod nimuSisaTvis). mkacrad ganvsazRvravT wina

wlebSi TvalsaCino doneze naswavl geometriul cnebebs, da zogierT

axalsac. vaswavliT umTavres planimetriul Teoremebs

damtkicebebiT (gavdivarT daaxloebiT piTagorasa da Talesis Teoremebamde),

oRond ara formalur-aqsiomatikuri midgomiT.

vaswavliT mxazvelobiTi geometriis sawyisebsac _ sxeulTa

xedebsa da izometrias. danarCeni sakiTxebi maRal skolaSi, X-

XII klasebSi iqneba.

geometriis namdvil dafuZnebas mxolod maTematikuri fakultetis

studentebi (da Tan maTi mxolod nawili) Tu iswavlian.

skolis moswavlisaTvis sakmarisia mxolod imis zogadi codna,

rom maTematikaSi saWiroa yvela Teoriis aqsiomebze dafuZneba da

cnebaTa mkacri gansazRvreba, Semdeg ki Teoremebis mkacrad

damtkiceba (rac mdgomareobs yoveli axali Teoremis logikur

dayvanaSi ukve damtkicebul Teoremebze an aqsiomebze). sakmarisia


- 65 -

amis ramdenime konkretuli nimuSis swavleba. xolo mTeli

geometriis amgvari ageba saskolo kursSi, jer erTi, ubralod

SeuZlebelia da mraval logikur siyalbesa da winaaRmdegobas

iwvevs. amgvari geometria araTu ar emsaxureba logikas (rac

geometriis mTavari mowodeba unda iyos), aramed azianebs da

aferxebs kidec mas. da meorec, amgvari geometria arc aravis

sWirdeba, moswavleTa im 0,1%-is garda, romlebmac es umaRles

saswavlebelSi mainc wesierad unda iswavlon. Tumca,

samwuxarod, mizezTa gamo, verc iq swavloben. amitom

maswavlebelTa udidesma nawilmac ki ar icis es!

Cven veTanxmebiT im azrs, rom moswavlem unda gaiazros, Tu

ras niSnavs Teoriis mkacri deduqciuri ageba, radganac esaa nam-

dvili maTematikis erTerTi mTavari maxasiaTebeli. magram saqme

isaa, rom amisaTvis srulebiT araa aucilebeli, nUfmj! hfp.

nfusjb deduqciurad aigos. skolis farglebSi es ubralod SeuZlebelia

da Tanac, amgvari mcdeloba sapirispiro Sedegsac

iZleva. rogorc stereometria iswavleba arasrul deduqciurad

(yvela saskolo kursSi asea), aseve SeiZleba iswavlebodes

planimetriac. sakmarisia, rom moswavlem gaiazros, Tu ra aris:

sazogadod cneba da pirveladi cneba; aqsioma da Teorema; mkacri

damtkiceba da sapirispiro magaliTi; aucilebeli, sakmarisi,

aucilebeli da sakmarisi piroba; da ras niSnavs mwyobri Teoriis

mkacrad (deduqciurad) ageba. amisaTvis Cveneul kursSi

gaTvaliswinebuli gvaqvs sagangebo logikuri sqemebi, romlebic

TvalsaCinod aCvenebs, Tu rogor daiyvaneba erTi Teorema

meoreze, meore _ mesameze, da ase Semdeg _ aqsiomebamde. gamoCndeba

isic, rom SeiZleboda Teoriis sxvagvarad ageba: sxva aqsiomebiT,

anda jer im Teoremis damtkicebiT, romelic sxvagvari

agebisas mogvianebiT mtkicdeboda. mTavaria, rom ar dairRves

TviT agebis wesi (principi). TvalsaCinod vaCvenebT imis magaliTsac,

rom dauSvebelia warmoiqmnas logikuri `Sekruli wre~.

Semdeg sruli gansazRvrebebiTa da bolomde mkacri damtkicebe-


- 66 -

biT avagebT _ ara mTliani geometriis, Tundac mxolod planimetriis

Teorias (rac SeuZlebelicaa) _ aramed mxolod mis

ramdenime xazs, nimuSisaTvis.

kursis gaerTianeba am mxrivac did upiratesobas mogvcems. saqme

isaa, rom moswavles ar unda egonos, rom simkacre

(deduqciuroba) mxolod geometriisTvisaa niSneuli. arada, swored

amgvari mcdari warmodgena Seeqmneba mas, radganac yvela

danarCeni saskolo maTematikuri dargi Zalian Sorsaa

maTematikuri simkacrisa da deduqciurobisagan. amis Secvla ki

yovlad SeuZlebelia: skolaSi naturalur ricxvTa metaTeoriisa

Tu dedekindiseuli gankveTebis swavlebas xom ver daviwyebT!

magram gaerTianebul kursSi kvlav TvalsaCino sqemebisa da

ramdenime nimuSis saxiT kargad gamoCndeba, rom geometriis

magvarad SeiZleboda sxva maTematikuri Teoriebis agebac,

magaliTad, ariTmetikisa. moswavleebs SeeqmnebaT warmodgena

imaze, Tu ra aris sazogadod maTematikuri Teoriis arsi _ da

ara mxolod planimetriisa.

§ 16. aramaTematikurad, gumaniT amosaxsneli amocanebi

gumans (intuicias) udidesi mniSvneloba aqvs aramarto

cxovrebasa da xelovnebaSi, aramed agreTve yovelgvar

saqmianobaSi, mkacr mecnierebaSic ki. cxadia, sul sxvaa eqimis

gumani da sul sxva _ xelosnisa; sul sxvaa istorikosis gumani

da sul sxva _ maTematikosisa; ufro metic, gumani geometriaSi

gansxvavdeba gumanisgan ariTmetikaSi. gumanis TviTeul am ganStoebas

sakuTrivi ganviTareba sWirdeba.

saskolo maTematikaSi sayovelTaodaa cnobili, rom moswavle

mraval cnebasa Tu moqmedebas gumaniT Seimecnebs _ da ara mkacri

logikiT. gansakuTrebiT xSirad es dawyebiT klasebSi xdeba. magali-

Tad, meoreklaseli mxolod gumaniT swvdeba cnebebs `marTkuTxedi~

Tu `Sekreba~. xolo cnebebi `wrfe~, `sibrtye~ `raodenoba~

da sxva _ ufros klasebSic ki mxolod gumanis amaraa.


- 67 -

magram es yovelive _ gumanis udabalesi donea, roca gumani,

arsebiTad, mxolod xatovan warmodgenebsa da calkeul kerZo

magaliTebs emyareba. am donis gumans ganviTarebac ki TiTqmis ar

sWirdeba _ igi normalur gonebas bunebrivad aqvs.

maTematikosis gumanis maRali done _ esaa Zneli, misaxvedri

(mosasazrebeli) amocanis amoxsnis gzis mixvedris unari. amgvari

amocanebi gvxvdeba rogorc saskolo maTematikaSi (gansakuTrebiT,

olimpiadebsa da konkursebSi), ise namdvil mecnierul kvlevebSi.

maT winaSe logikuri msjeloba umweoa, mxolod gumani Tu gaikvlevs

gzas.

magram es maRali donis gumani mxolod gamorCeuli maTematikuri

niWis mqone adamians SeiZleba ganuviTardes. amitom misi

ganviTareba mxolod calkeul moswavleebs exeba.

gumanis yvelaze mniSvnelovani donea saSualo, romelic Cveulebriv

moswavles unda ganuviTardes. ganvixiloT ramdenime magaliTi.

Semdegi amocana VI-VII klasebisTvisaa:

mocemulia marTkuTxedi: SemdegTagan aarCieT is

marTkuTxedi, romlis gverdebis sigrZeebic am marTkuTxedis

gverdebis sigrZeTa proporciulia:

a) b) g) d) e)

amocana testuri formisaa _ mocemulia savaraudo pasuxebi.

pasuxebi yovelTvis gadanomrilia qarTuli asoebiT:

a) ; b) ; g) ; d) ; e) ; v) ; z) ...

moswavlem unda icodes, rom amgvar amocanebSi (sadac pasuxi

unda avirCioT da SesaZlo pasuxebi qarTuli asoebiTaa gadanomrili),

yovelTvis mxolod erTi pasuxia xolme marTebuli.

mixvedriT unda gamoiricxos yvela pasuxi, garda erTisa _ da

maS swored es erTi iqneba marTebuli pasuxi.

moswavleebs am droisaTvis ukve naswavli aqvT proporcia da


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misi sami Tviseba. am amocanis amoxsna SeiZleboda gazomvebiTa da

gamoTvlebiT, magram amas Zalian didi dro dasWirdeboda. vinc

aqtiurad icis proporciuloba (Tundac cnebaTa gareSe _ mxolod

praqtikulad _ magaliTad, xelovnebaTmcodnem), is pirdapir,

gumaniT mixvdeba Tu `dainaxavs~, rom marTebuli pasuxia (g).

gumani saTanado mimarTulebiT dagrovebul gamocdilebas

emyareba (dawvrilebiT _ ix. [10]). magaliTad, zemore amocanaSi

marTebul pasuxs pirdapir ver `dainaxavs~ is adamiani, romelsac

proprociuloba axali naswavli aqvs da jer kidev Rrmad ver

`grZnobs~ mas _ Tundac rom proporciulobis Teoriul-cnebiTi

codna sruliad unaklo hqondes!

gumanis dabali done ufro `dablaa~, vidre Teoriul-cnebiTi

codnis done, jer kidev araa amaRlebuli am gaazrebuli cnebis

donemde. magaliTad, proporciulobis SemTxvevaSi es iqneboda is

codna, romelic eqneboda mesameklasels, romelsac TvalsaCino

magaliTebiT auxsnidnen, Tu ra aris proporciuloba. saSualo

mesameklaselisTvis proporciulobis cnebiTi gaazreba ubralod

miuwvdomelia. xolo gumanis saSualo done, piriqiT, ufro `maRlaa~,

vidre Teoriul-cnebiTi codnis done, radgan misi momdevno

donea, mas moicavs da damatebiT aqtiur gamocdilebasac moicavs!

gumanis umaRlesi done _ esaa SemoqmedebiTi gumani, romelic

mxolod saTanado niWierebis SemTxvevaSi Tu ganviTardeba [10].

wmindad gumaniT amoixsneba agreTve amocanebi njbympfcjU!Tf.

gbtfcb{f/ isinic testuri formisaa, mxolod erTi pasuxia mar-

Tebuli. gamoTvlebis gareSe, mixvedriT unda gamoiricxos yvela

pasuxi, garda erTisa _ da maS swored es erTi iqneba marTebuli

pasuxi. ganvixiloT amgvari amocanis magaliTi (III klasidan):

risi tolia saSualo cxrasarTuliani saxlis simaRle?

a) 8 m; b) 244 m; g) 20 dm; d) 110 m; e) 30 m; v) 1 km.

moswavle am droisaTvis ukve sakmaod gawafulia gazomvasa da

sigrZis erTeulebSi. es Cveni zogadi wesia: amocana raimes miaxloebiT

Sefasebaze Semodis mxolod mas Semdeg, rac saTanado sa-


- 69 -

kiTxi ukve karga xnis ganmavlobaSi muSavdeboda da moswavles

tblnbp! hbnpdejmfcb unda hqondes dagrovebuli. amitom moswavle

gumaniT unda mixvdes, rom cxrasarTuliani saxlis simaRle

ver iqneba 8 m Tu 20 dm (Zalian mcirea!) da verc 244 m, 110

m Tu 1 km (Zalian didia!). maS, rCeba erTi SesaZlebloba _ 30 m

(raki viciT, rom erTi pasuxi namdvilad marTebulia!).

kidev erTi saxis amocanebi amoixsneba gumaniT, oRond saWiroa

Semdgomi logikuri dasabuTeba da msjeloba. Cven xSirad gvaqvs

amocanebi lbopo{pnjfsfcb{f: an kanonzomierebis damrRvevi wevris

moZebnaze, an kanonzomierebis gagrZelebaze (Sesabamisi carieli

adgilebis SevsebiT). amgvar amocanebSi mTavaria: kanonzomierebas

mxolod erTi wevri (anda, ori _ Tuki asea miTiTebuli

amocanis pirobaSi) unda arRvevdes, igi unda iyos `gamonaklisiviT~,

`zedmetiviT~ danarCenTa Soris; Tanac _ yvela danarCeni

wevri raimeTi unda erTiandebodes, unda ukavSirdebodes erTmaneTs.

ganvixiloT, magaliTad, amocana (IV klasidan):

romeli sityva arRvevs kanonzomierebas?

ru; arxi; Rele; wyaro; mdinare; gube.

miTiTeba: unda daukvirdeT sityvebis azrsac (mniSvnelobasac)

da maT asoebsac.

ganvixiloT jer asoebis mixedviT. aSkarad mcdari pasuxia,

magaliTad, es: `ru radgan am sityvaSia ori aso~, es ki marTalia,

magram maSin yvela danarCeni sityva riTiRa ukavSirdeba

erTmaneTs? (yvela danarCen sityvaSi rom yofiliyo, magaliTad,

oTx-oTxi aso, maSin es pasuxi marTali iqneboda). asoebis mxriv

kanonzomierebas arRvevs arxi _ radgan mxolod igi iwyeba

xmovniT, xolo yvela danarCeni iwyeba TanxmovniT. arxi _ esaa

erTi marTebuli pasuxi.

$ 3-Si iTqva, rom maswavlebelma unda waaxalisos moswavleTa

uCveulo pasuxebi _ Tuki isini dasabuTebulia. magaliTad, Cvens

amocanaSi moswavles SeiZleboda aseTi pasuxi aRmoeCina:

kanonzomierebas arRvevs wyaro, radgan mxolod am sityvaSia


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xmovani o.

maswavlebeli dafiqrdeba da xmamaRla daiwyebs msjelobas: _

modi, SevamowmoT, danarCeni xmovnebi rogoraa? (da SeekiTxeba sxvadasxva

moswavleebs): sul ramdeni xmovania qarTul enaSi? {xuTi} a

xmovani ramden sityvaSia? {samSi}; i xmovani? {or sityvaSi}; e

xmovani? {or sityvaSi}; u xmovani? {esec orSi}; romeli xmovani

dagvrCa? {o} es o marTlac erTaderT sityvaSia! ese igi, zurikos

pasuxSic marTebuli yofila, yoCaR, zuriko, Zalian kargia!

sxvaTa Soris, aseTi kanonzomierebis danaxvac SeiZleba: a, i, e

da u xmovanebi or-or sityvaSia, mxolod mexuTe xmovani o aris

iseTi, romelic mxolod erT sityvaSia, amitom es erTi, meoTxe

sityva wyaro arRvevs am kanonzomierebas. Cven xom viciT, rom

kanonzomiereba zogjer ori an ramdenimec ki SeiZleba iyos!

magram, moswavles rom eTqva, kanonzomierebas arRvevs ru,

radgan mxolod am sityvaSiao xmovani u _ es mcdari iqneboda,

radgan u xmovani aris agreTve sityvaSi gube! aseve, moswavles

rom eTqva, kanonzomierebas arRvevs wyaro, radgan mxolod am

sityvaSiao Tanxmovani w _ esec mcdari iqneboda, radgan

gaugebari darCeboda danarCenTa gamaerianebeli kanonzomiereba _

maSin Tanxmovani x aris mxolod meore sityvaSi, d da n _

mxolod mexuTeSi da ase Semdeg. maSin, romeli arRvevs

kanonzomierebas? araa bunebrivi, amitom pasuxi naZaladevia, ar

varga. sxva yvela Tanxmovani rom marTlac or an ramdenime

sityvaSi yofiliyo (rogorcaa xmovnebis SemTxvevaSi), maSin

marTebuli iqneboda es pasuxi!

axla aRmovaCinoT kanonzomiereba sityvebis azris, mniSvnelobis

mixedviT: yvela sityvis mniSvneloba ukavSirdeba wyals,

magram rogor wyals? romelia danarCeTagan gansxvavebuli? iqneb,

arxi _ imiT, rom igi xelovnuria? ara _ ruc xelovnuria (mas

saxeldaxelod gaTxrian xolme sarwyavad, ru aris mcire droebiTi

arxi). iqneb wyaro _ radgan daileva? ara, zogjer Relis

wyalsac svamen, zogjer _ arxisa! maSasadame, es arcerTi pasuxi


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ar varga. kanonzomierebas ki arRvevs gube, radgan igia erTaderTi

damdgari wyali, yvela danarCeni ki moZravi wyalia.

rogorc vxedavT, amgvar amocanebSi SeiZleba arsebobdes gansxvavebuli

Tvalsazrisebi da gansxvavebuli pasuxebi, rac maTematikaSi

uCveuloa. sazogadod, zogi sakiTxis uCveuloba logikasTan

integraciiTaa gamowveuli [$ 4].

§ 17. saWiro TvalsaCinoeba

upirvelesad, gasarkvevia kalkulatoris sakiTxi. misi xmareba

sakmaod SezRuduli unda iyos.

1) yovlad dauSvebelia kalkulatoris xmareba VII-VIII klasebamde,

radgan es daangrevs ariTmetikis codnas _ bavSvi ver

SeZlebs zepirad Seasrulos amgvari gamoTvlebi (rogorc esaa

aSS-Si da sxvagan): 126:3; 1 m 40 sm : 5; 320•4; 1 kg _ 360 g; 1 sT 45 wT + 2 sT 35 wT;

3,5 miliardis naxevari; 400 aTasi laris 15 %

da, rac mTavaria, naSTiani gayofa (romelic dawyebiTi

klasebis mTeli ariTmetikis gvirgvinia):

90:21, 1 mln : 150 aTasi, 2 t : 6, 105/45 = 2 1/3 .

amgvari gamoTvlebi, jer erTi, gacilebiT advilad, uSecdomod

da Tan swrafad sruldeba zepirad, gonebaSi, vidre kalkulatoriT;

meorec, aviTarebs azrovnebasa da ganamtkicebs maTematikis

codnas; mesamec, aucilebelia praqtikulad, radgan yvelgan kalkulators

ver ixmar. amasTan, uaRresad mniSvnelovania miaxloebiTi

angariSisa da miaxloebiTi Sefasebis unarCvevebi, romlebic

efuZneba raodenobis gumaniT wvdomas. es unari ki ver ganviTardeba

gonebaSi mravalwliani angariSis gareSe!

2) VII-VIII klasebidan moswavles kalkulatoris xmarebis ufleba

unda mieces mxolod im amocanebis amoxsnisas, romlebSic

marTlac didi gamoTvlebia Casatarebeli (magaliTad, statistikuri

Sinaarsis amocanebSi) _ saxelmZRvaneloSi sagangebod miTiTebuli

iqneba, rom saWiroa kalkulatoris gamoyeneba. xolo sadac

es ar iqneba miTiTebuli, iq kalkulatoris xmareba unda aikrZa-


- 72 -

los. maswavlebelma es wesi unda daicvas saklaso, sakontrolo

da sagamocdo werebis dros, xolo saSinao davalebis Sesrulebisas,

sasurvelia, daicvas mSobelma. cxadia, amas ver daveyrdnobiT,

amitom TviTon moswavlesac unda avuxsnaT, rom Tuki is

borotad gamoiyenebs kalkulators, mas dauClungdeba zepiri

angariSis unari da amitom sakontrolo werebze da gamocdebze

dabal niSnebs daimsaxurebs.

moswavlisTvis saWiro TvalsaCinoebas TviTon moswavlis saxelmZRvanelo

Seicavs. aucilebelia damatebiTi stereometriuli

saklaso TvalsaCinoeba: stereometriuli sxeulebis modelebi

(umjobesia, mTliani xisa an plastmasisa, anda maTi muyaos modelebi);

saklaso saxazavi da fargali. moswavleebi sakuTari xeliT

unda zomavdnen stereometriuli sxeulebis saTanado ganzomilebebs.

Zalian kargia Slilis daxazva, gamoWra da modelad

Sewebeba (`saymawvilo maTematikaSi~ sakmaodaa amgvari davalebebi).

CvenTvis mTavari da namdvili TvalsaCinoeba _ esaa xazva, anu

moswavleebis mier sakuTari xeliT daxazuli cxrilebi, diagramebi,

naxazebi da sqemebi, geometriulebic, logikurebic, ariTmetikulebic

da agreTve yvela sxva saxisa. amitom `saymawvilo

maTematikaSi~ Zalian mravladaa amocanebi xazvaze, zomvaze,

cxrilebze, diagramebze, sqemebze, mravalgvar praqtikul

samuSaoze da sxva. sxvaTa Soris, swored amgvari amocanebi aer-

Tianebs organulad maTematikis sxvadasxva dargebs erTmaneTTanac

da gamoyenebiT mimarTulebebTanac; swored amgvari amocanebi

aviTarebs yvelaze kargad zogad gonebriv unarebs.

amrigad, maTematikis gakveTilebze moswavleebs dasWirdebaT

fanqari, saxazavi da zogjer _ fargali.

TvalsaCinoebis farglebi

TvalsaCinoebis principi klasikuri humanisturi pedagogikis

ZiriTadi mcnebaa. cneba ver ganzogaddeba, ver gaiazreba da

mxolod yalb, futuro, uazrod gazepirebul sityvier garsad

darCeba, Tuki mas TvalsaCino sayrdeni ar eqna. TvalsaCinoebis


- 73 -

mniSvneloba gansakuTrebiT didia aqtiuri swavlebis pirobebSi.

es yvelaferi asea, magram TvalsaCinoebis gamoyenebis farglebi,

zoma da safexurebi saguldagulod unda iyos damuSavebuli,

raTa igi azrovnebisTvis _ nacvlad sayrdenisa _ Semaferxeblad

ar iqces. j. bruneri [22] sagangebod aRniSnavs, rom TvalsaCinoeba

mxolod zomierebis farglebSia kargi. d. uznaZec [1] aRniSnavs,

rom Warbi TvalsaCinoeba Semaferxebelia cnebiTi azrovnebis

ganviTarebisaTvis, romlis gareSec adamiani Tavs ver daaRwevda

cxovelur dones: imwamierad mocemul, SemTxveviT, erTeul

STabeWdilebaTa da aqtualuri aRqmis tyveobas, qcevis impulsur

dones. adamiani cnebaTa sistemaSi sinamdvilis mTlianobas swvdeba

da amis Sedegad eZleva mis qcevas cnobieri, nebismieri da mizandasaxuli

xasiaTi. d. uznaZis azriT, saskolo swavleba, arsebi-

Tad, mecnieruli cnebiTi azrovnebis ganviTarebas emsaxureba.

aRmqmeli adamianis cnobiereba SezRudulia, mas axasiaTebs `situaciuroba~,

ese igi, `aq da axla~ mocemuliT SezRuduloba,

konkretul-materialuroba. azrovnebis upiratesobas aRqmasTan

SedarebiT J. piaJe [21] amgvarad aRwers: `mokle manZilebisa da

realuri gzebidan Tavdaxsna mxolod azrovnebas SeuZlia, Tavisi

im miswrafebiT, rom mTeli garemomcveli samyaro moicvas, TviT

uxilavis CaTvliT, zogjer imis CaTvliTac ki, risi warmodgenac

SeuZlebelia. swored subieqtsa da obieqts Soris arsebuli

manZilis es usasrulo mateba aris mTavari siaxle, warmomqneli

cnebiTi inteleqtisa da im Zalmosilebisa, romelic inteleqts

operaciebis warmoSobas SeaZlebinebs~.

ekonomikur-teqnologiurad ganviTarebul qveynebSi mimdinare

procesebi gviCvenebs, rom Tanamedrove cxovrebis wesi xels uwyobs

ekranul sainformacio saSualebaTa sayovelTao gavrcelebas,

rogoricaa: televizia, kompiuteri, video, kalkulatori, kino,

Sou, sareklamo dafebi, moda, podiumi, mdidrulad dasuraTebuli

bukletebi da Jurnalebi. Sesabamisad kiTxvisa da weris

mniSvneloba knindeba. sityva-cnebas mxedvelobiTi xatebi aZevebs,


- 74 -

mkiTxvels _ mayurebeli, sityvier-wignier adamians _ mxedvelobiT-ekranieri

adamiani, mwignobrul kulturas _ komfort-materializmi.

es seriozul uaryofiT Sedegebs iwvevs, rogorc sulieri,

pirovnul-fsiqologiuri mxriv, ise kulturul-inteleqtualuri

da socialuri mxriv. ekranuli civilizacia Zlier

amuxruWebs agreTve adamianis nebelobis ganviTarebas, mis mier

infantiluri egocentrizmis daZlevas, pasuxismgeblobisa da zneobrivi

Segnebis ganviTarebas, pirovnebis Camoyalibebas.

televizoris an kompiuteris ekrani mxolod mexsierebas avsebs

nawilobriv, Tanac mouwesrigebeli da daqsaqsuli informaciis

groviT. Sedegebic kanonzomieria. specialuri gamokvlevebi

gviCvenebs dasavleTis qveynebSi maTematikuri codnis mkveTr daqveiTebas,

agreTve zogadi wignierebis, saerTo codnis dakninebas,

enobriv unarTa dasustebas, zepiri da weriTi metyvelebis ukiduresad

gaRaribebas, gramatikuli struqturebis gamartivebas,

enis metaforul SreTa standartizacias (Jargonad gadagvarebas),

leqsikis uaRres simwires. maSin, roca yvela es Tviseba pirovnuli

cnobierebis simdidrea da pirovnebis ganviTarebas ganapirobebs.

ekrani gansakuTrebiT damaClungeblad dawyebiTi klasebis asakis

bavSvze moqmedebs. $ 5-Si Cven ganvixileT bavSvis orgvari

CamorCena _ gonebrivi da nebelobiTi. ekrani orives gamomwvevi

SeiZleba iyos.

gonebaWvretiTi wignierebis burjia ori mTavari saskolo sagani:

mSobliuri ena-literatura da maTematika _ xatobriv-niv-

Tieri konkretulobis sapirispirod. am sagnebis swavlebisas gansakuTrebiT

saWiroa TvalsaCinoebisa da suraTovnebis mozRudva

rogorc saxelmZRvaneloTa gaformebis, ise arsebiTi mxriv.

zedmeti da araarsebiTi (sakiTxTan mxolod zereled dakavSirebuli)

suraTovneba gansakuTrebiT amerikuli stilis saxelmZRvaneloebs

axasiaTebs.

maTematikisTvis suraTi araa kargi TvalsaCinoeba dawyebiT

klasebSic ki (dawvrilebiT amis Sesaxeb ix. [16:$1,2,3,4,6]). ukve I


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klasSi Cven nivTieri TvalsaCinoebis paralelurad Semogvaqvs

sqematuri TvalsaCinoeba. TandaTan nivTieri TvalsaCinoebis wili

mcirdeba, sqematurisa _ imatebs, II klasidan ki nivTier

TvalsaCinoebas TiTqmis aRarc viyenebT (garda stereometriuli

modelebisa). yvela klasisaTvis maTematikis ZiriTadi Tvalsa-

Cinoeba _ esaa sxvadasxvagvari sqemebi da naxazebi.

sayovelTaod gavrcelebulia mosazreba, rom TvalsaCinoeba

swavlis motivaciis gamaZlierebeli uebari saSualebaa. sinamdvileSi

ki bavSvisTvis mimzidveli Wreli suraTebiT swavlisa da

azrovnebis motivacia ki ar Zlierdeba, aramed mxolod _

saxelmZRvanelos Tvalierebisa da zerele gadakiTxvis, TamaSis

motivacia. TvalsaCinoeba mxolod im formiTa da zomiT unda

gamoiyenebodes, rac uSualodaa saWiro ama Tu im programuli

sakiTxis gaazrebulad, Rrmad da aqtiurad saswavleblad.

amas ar iTvaliswinebs ara mxolod saqarTveloSi, aramed

msoflioSi yvelaze popularuli, interaqtiuli meTodikebi da

Tanamedrove stilis saxelmZRvaneloebi.

namdvili swavlisa da azrovnebis motivaciis gamaZlierebeli

mTavari saSualeba unda iyos ara xelovnurad gazviadebuli

Wreli suraTebi, aramed kvleviT-ZiebiTi swavleba, evristikuli

meTodi, romelic swavlis motivaciasac aZlierebs da, rac

mTavaria, Rrma da aqtiuri codnis miRebis mTavari saSualebaa.

amasTan, daufaravad unda iTqvas simarTle: namdvili swavla

verasdros gaxdeba mTlianad saxaliso _ swavlis aucilebeli

Tanmxlebi mainc nebelobis daZabvaa (cxadia, asakis Sesabamisad!).

uamisod mxolod zerele codna Tu SeiZineba. vinc swavlis Ziris

simwares gaeqceva, is ver igemebs kenweros gatkbilebas.

pedagogikis mizania, SeZlebisamebr Seamciros `Ziris simware~

da gaaZlieros `kenweros gatkbileba~.

meTodikaSi ori urTierTsapirispiro mimarTuleba arsebobs:

I. sqolastikuri (aseTia, kerZod, sabWoTa meTodika), romelic

mTlianad an TiTqmis mTlianad ugulebelyofs saxaliso-sa-


- 76 -

yofacxovrebo sakiTxebs, maTematikuri codnis praqtikul mxareebs;

swavleba mZimea da mTlianad maTematikazea centrirebuli, moswavlis

pirovnebis, misi asakobrivi fsiqologiis ugulebelyofiT;

II. pedocentruli (moswavleze centrirebuli), romelic

ugulebelyofs maTematikur safuZvlianobasa da siRrmes (aseTia,

kerZod, amerikuli meTodika). gadaWarbebuli mniSvneloba eniWeba

saxalisobas, praqtikul sakiTxebsa da gazviadebul TvalsaCinoebas.

ganvixiloT erTi konkretuli magaliTi. vTqvaT, magaliTad,

moswavleebma iswavles cilindri. rogor damuSavdes es sakiTxi?

sqolastikuri meTodika TiTqmis mTlianad geometriuli

TeoriiTa da mkacri formaluri gansazRvrebebiTaa SezRuduli,

praqtikul TvalsaCinoebas wvrilmanad miiCnevs.

pedocentruli meTodika, piriqiT, upirveles mniSvnelobas

aniWebs cilindris codnis praqtikul gamoyenebasa da mis dakav-

Sirebas yofacxovrebasTan (aq ganvixilavT cilindris swored

rom cnebas _ da ara zedapiris farTobisa Tu moculobis gamoTvlas,

rac calke sakiTxebia!). am mizniT SeiZleba gamoviyenoT sami

donis TvalsaCinoeba (klasikur pedagogikaSic ase iyo):

1) nivTieri. magaliTad, maswavlebels klasSi Seaqvs nairnairi

feradi fanqrebis 3 sxvadasxva kolofi (maTgan zogi

mrgvalia, zogi _ waxnagovani, zogi wverwaTlilia, zogi _ ara).

interaqtiuli meTodikis mixedviT, klass dayofs 3 jgufad,

TiTos miscems TiTo kolofs da daavalebs: amoarCieT cilindris

formis fanqrebi. Semdeg, roca jgufebi warmoadgenen TavianT

namuSevars, kargi maswavlebeli msjelobasac wamoiwyebs: ratom

ara aqvs cilindris forma am wiTel fanqars? {zedapiri aqvs

waxnagovani da fuZeSi eqvskuTxedia} am Sav fanqars? {waTlilia da

amitom am fuZeSi wre ara aqvs}. da ase saTiTaod ganixileba yvela

tipis SemTxveva.

amgvari midgoma kargia, saxalisoa, nayofieria, magram erTi

arsebiTi nakli aqvs: Zalian bevr dros STanTqams. Tuki

maswavlebelma amgvari gakveTili xSirad Caatara, mas saswavlo


- 77 -

programa gverdiT darCeba. xolo Tuki zerele msjelobiT dasrulda,

maSin amgvari muSaoba maTematikisTvis TiTqmis fuWicaa.

2) suraTovani. magaliTad, saxelmZRvaneloSi mTeli gverdi

ukavia did ferad suraTebs: nair-nairi feradi fanqrebi,

konservis qilebi da sxva. iqvea foto: industriuli qarxana

didi cilindruli avziT.

amgvari midgoma yvelaze uaresia, radgan zerele, araarsebiT

mxareze gadaaqvs moswavlis yuradReba. saxelmZRvanelos

moculoba da Rirebuleba maTematikisaTvis fuW rameebze cdeba.

siRrmisaTvis arc saxelmZRvaneloSia adgili darCenili da arc

moswavles Seqmnia saTanado ganwyoba.

Tumca, xSirad es midgoma pirvelTanaa Serwymuli.

gavrcelebuli Secdomaa, rom TiTqos saswavlo TvalsaCinoeba

am ori tipiT (doniT) amoiwureba. sinamdvileSi ki klasikurma

pedagogikamac ki icoda, rom arsebobs mesame donis TvalsaConoebac:

3) warmodgeniTi. Cveneuli meTodika swored amgvar Tvalsa-

Cinoebas emyareba. ucnaurad JRers, magram ueWvelia: Cveneul saxelmZRvaneloebSi,

romlebic Sav-TeTria da Zalian cota suraTs

Seicavs, TvalsaCinoeba metia, vidre sxvebSi. oRond esaa ara suraTovani,

aramed warmodgeniTi da agreTve sqematuri TvalsaCinoeba,

Tanac romelic uSualod ukavSirdeba maTematikur amocanebs.

magaliTisTvis ganvixiloT Cveneuli programis saTanado nawyveti,

kvlav cilindris Sesaxeb (V klasidan). gasaTvaliswinebelia,

rom Cveneuli meTodika, jer erTi, stereometrias planimetriis

paralelurad amuSavebs, magaliTad, cilindri wris paraleluradaa

[ix. $ 15]. meorec, sakiTxis swavleba Semamzadebeli amocanebiT

iwyeba [ix. zemore, $ 11, 12], kerZod, cilindrisaTvis,

esaa amocanebi wris Tvisebebis Sesaxeb (brunviTi sxeulebi da

sxeulis miReba nakvTis brunviT mxolod IX klasSi Semogvaqvs).

saxelmZRvaneloSi cilindris mxolod sqematuri naxazia. samagierod,

gakveTilis teqsts saSinao davalebis aseTi amocanebi

axlavs (maTgan zogi momdevno gakveTilebzecaa gadanawilebuli,


- 78 -

ariTmetika-algebris sakiTxebs Soris, maT gasaxaliseblad):

1. warmoidgineT 4 fanqari: mrgvali wverwaTlili, mrgvali uxmari

(wverwauTleli), kuTxovani wverwaTlili da kuTxovani uxmari.

romel maTgans ar aqvs cilindris magvari forma? ratom?

maSasadame, TvalsaCinoebis 1-l doneze namdvili nivTiebi

fanqrebi iyo, me-2 doneze _ fanqrebis naxatebi (TiTqos V-VI

klasel bavSvs uWirdes fanqris gonebaSi warmodgena!), xolo me-3

doneze _ fanqrebis warmodgeniTi xatebi. asevea sxva SemTxvevebSic.

2. warmoidgineT nivTebi: gaberili sacurao rgoli,

Sededebuli rZis qila, Cveulebrivi boTli, manqanis

borbali, swori mili, xis kasri, xurda fuli (moneta),

sanTeli, muTaqa, vedro. romel maTgans aqvs cilindris

magvari forma? ratom?

3. warmoidgineT, rom birTvi gakveTes organ,

erTmaneTis gaswvriv ise, rom erTerTi

miRebuli sxeuli aseTia: mas fuZeebad aqvs ori

erTmaneTis toli wre. moifiqreT, iqneba Tu

ara es sxeuli cilindri. ratom? dasabuTeba CawereT.

4. warmoidgineT dauWreli Zexvi. rodesac Zexvs yiduloben,

gamyidveli mas cerad CamoaWris xolme naWers da wonis. aqvs

Tu ara aseTnairad CamoWril Zexvis naWers cilindris magvari

forma? ratom? SeiZleba Tu ara Zexvis ise CamoWra, rom

daaxloebiT cilindruli formis naWeri miiRebodes?

5. nagebobebisa da ezoebis dasamSveneblad xSirad iyeneben

cilindrisa da birTvis formebs. ra SeiZleba iTqvas

naxazze gamosaxuli cilindris fuZis diametrisa da

birTvis diametris Sesaxeb? daxateT amis magvari ori

iseTi naxati, rom erTze cilindris fuZis diametri

meti iyos birTvis diametrze, xolo meoreze _ piriqiT.

mocemulia ramdenime cnobili xuroTmoZRvruli Zeglis

suraTi, oRond ara Wrel-Wreli, aramed sqematuri. amocanaa:

6. am nagebobebSi moZebneT Semdegi geometriuli formebi:


- 79 -

naxevarsfero, cilindri, naxevarcilindri. amowereT

TviTeuli am formis saxelwodeba da yovel maTgans gverdiT

miuwereT is ricxvi, romelic gviCvenebs, Tu ramdenjer Segvxvdeba

am saxelwodebis forma naxatze gamosaxul nagebobebSi

(yvelaSi erTad).

gvaqvs amocanebi aqtiobazec, magram amaze dro klasSi ar ixarjeba

_ klasSi mxolod namuSevrebi ganixileba erToblivi

msjelobiT:

7. moZebneT Sin nivTebi, romlebsac daaxloebiT cilindris

magvari forma aqvs da CawereT maTi saxelebi.

8. aiReT qaRaldis furceli da sworad dagragneT igi

cilindris gverdiTi zedapiris formis magvarad. ra aklia

mas cilindruli zedapiris magvar formamde?

rogorc vxedavT, Cven udides mniSvnelobas vaniWebT sakiTxis

intuiciur wvdomas [$ 16]. magram, zomierad, gvaqvs amocanebi

Zalian faqiz, wmindad maTematikur, Rrma sakiTxzec, magaliTad:

9. warmoidgineT, rom fuZeebis gaswvriv gakveTeT ara cilindri,

aramed misi zedapiri. ra nakvTs miiRebT gakveTis

adgilas? {ara wres, aramed wrexazs!}

garda amisa, mravlad gvaqvs cilindris cnebis sxva naswavl

geometriul cnebebTan damakavSirebeli amocanebic, magaliTad:

10. warmoidgineT, rom sfero mTlianadaa cilindrSi

moTavsebuli da aris udidesi amgvari sfero. SemdegTagan romeli

daskvnis gamotana SeiZleba aqedan?

a) sfero exeba cilindris erTaderT fuZes;

b) sfero exeba cilindris orive fuZes;

g) sferosi da cilindris fuZis radiusebi tolia;

d) sferos radiusi cilindris simaRlis tolia;

e) sferos da cilindrs erTnairi moculoba aqvs.

dakavSireba ariTmetikasTan:

11. warmoidgineT, rom cilindrs daadges meore cilindri,

romelsac pirvelis toli fuZe aqvs, xolo simaRle _

nax. 9


- 80 -

pirvelis simaRlis naxevari. cilindrebis fuZeebi erTmaneTs

SeuTavses. ra sxeuls miiRebdnen?

yuradReba mivaqcioT, rom amdeni, 11 amocana cilindris

mxolod cnebas eZRvneba! cilindris radiusi, diametri da

simaRle _ Semdegaa. momdevno klasebSi ki es sakiTxebi

bunebrivad erwymis algebras _ saTanado formulebis gamoyeneba.

Semdeg kavSirdeba fizikasTan (kargi amocana _ nacvlad

zemorexsenebuli fotosi industriuli qarxnis xediT):

mocemulia navTis kuTri wona _

12. gamoTvaleT, ras iwonis is navTi, rac eteva 2 m simaRlisa

da 70 sm radiusis sigrZis mqone cilindrul avzSi.

cotaTi rTuli, saazrovno, arastandartuli amocana:

13. gaarkvieT, rogor Seicvleboda wina amocanis pasuxi, avzs

sqeli, 5-santimetriani fskeri, kedlebi da sarqveli rom hqonoda.

sarqveli rom ar hqonoda (anu, avzi TavRia rom yofiliyo)?

dabolos, TvalsaCinoebis umaRlesi gamoyeneba _ logikuri

sqemebia. kerZod, Zalian kargia geometriuli sxeulebis Tvalsa-

Cino saklasifikacio sqemebi (xisebri diagrama da agreTve venis

diagramebi), romlebzec TvalsaCinod gamoCndeba, ra adgili ukavia

cilindris cnebas sxva stereometriuli sxeulebis cnebebs

Soris da ra logikuri mimarTebebia am cnebebs Soris (IX kl).

Cveneuli meTodikisaTvis kidev erTi sakiTxia Zalian

mniSvnelovani _ magaliTebis saxeobani. pirveli, Cveulebrivi

saxea cnebis dadebiTi magaliTebi, cilindris SemTxvevaSi,

magaliTad: mrgvali kasri, konservis qila, morgvi da sxva. meore

saxea uaryofiTi magaliTebi: burTi, borbali, konusi, 8waxnaga

gumbaTi (rogoric aqvs, magaliTad, mcxeTis jvarsa Tu

atenis taZars). ese igi, saWiroa Cveneba: eseni cilindrebia, eseni

ki araa cilindrebi. magram arc esaa sakmarisi. saWiroa agreTve

kidura, ukiduresi (marginaluri), anu aratipuri magaliTebis

garCeva, raTa kargad moixazos cnebis sazRvari (dawvrilebiT _

ix. [12]). cilindris SemTxvevaSi, magaliTad:


- 81 -

14. CamowereT, romel nivTebs aqvs badros magvari forma:

rogorc vxedavT, badros simaRle gacilebiT

naklebia, vidre misi diametri. miaxloebiT

gamoTvaleT, Tqvens mier dasaxelebul erTerT

nivTs ramdenjer naklebi aqvs simaRle, vidre diametri. axla

warmoidgineT amis sapirispiro Tvisebis mqone cilindri:

romlis simaRle daaxloebiT 100-jer metia, vidre diametri.

romel nivTs aqvs daaxloebiT am cilindris magvari forma?

ese igi, kidura dadebiTi magaliTia iseTi cilindri, romelic

ar hgavs cilindrs, romlis cilindrad aRqma mraval adamians

gauWirdeba. Tanac, raki cilindri fuZeze ar dgas, Zneli

gasaazrebelia, romelia cilindris simaRle da romeli _

diametri. aseve, kidura uaryofiTi magaliTia iseTi sxeuli,

romelic araa cilindri, magram hgavs cilindrs, magaliTad, 12-

14-16-waxnaga gumbaTebi (rogoric aqvs, magaliTad, nikorwmindas,

gelaTs, metexsa Tu martvilis taZars).

cilindris cnebis amgvari aqtiuri da mravalmxrivi damuSaveba

kargad gviCvenebs imasac, Tu ras niSnavs gaRrmavebuli swavleba.

amrigad, aqtiuri mzaobis meTodikaSi, warmodgeniT TvalsaCinoebaze

da Semamzadebel amocanebze dayrdnobiT, moZebnilia is

oqros Sualedi, romelic zomierad Seuwonis erTmaneTs pedagogikis

or sapirispiro mimarTulebas; TviTon igi arc sqolastikuria

(Tumca, inarCunebs mecnierul siRrmesa da cnebiT azrovnebas)

da arc calmxrivad pedocentrulia (Tumca, aZlierebs humanistur

midgomas, praqtikulobasa da gamoyenebiT mxareebs).

sabolood, mSobliuri ena-literaturisa da maTematikis swavlebisas,

agreTve logikuri azrovnebis ganviTarebisaTvis Tvalsa-

Cinoeba aucilebelia, magram igi Zalian mozomili da TiTqmis

mxolod warmodgeniTi an sqematuri unda iyos.

maTematikis saxelmZRvanelo moswavles unda uqmnides ara

zerele, sanaxaobiT, komiqsur-multfilmur-kompiuterul, anu

saekrano ganwyobas, aramed piriqiT: azrovnebis, gonebaWvretis,


- 82 -

dinji CaRrmavebis, maTematikis Sinagani simwyobrisa da silamazis

ganWvretis ganwyobas _ im silamazisa, romelic mxolod

daxvewili gonebis TvaliT dainaxeba. udidesi moazrovnisa da

poetis _ platonis akademiis karibWis Tavze ewera:

`nu Semova aq nuravin, vinc ar icis maTematika!~

rogorc Zvel berZnul, aseve indur Tu Cinur kulturaSi

maTematika `samyaros musikas~ aRwers, im musikas, romelsac

zecaze mnaTobTa harmoniuli moZraoba warmoqmnis da romelic

mxolod maRalganviTarebuli gonebis yuriT moismineba.

§ 17. sakontrolo werebi da SefasebaTa sistema

tblpouspmp!xfsfct!wjxzfcU!JJ!lmbtjt!nfpsf!obyfwsjebo/!

kfs!jTwjbUbe!ubsefcb!`!tvm!nypmpe!tbnj!tblpouspmp/!

ypmp! JJJ! lmbtjebo! tblpouspmp! xfsfcj! yTjsefcb! eb!

tbTvbmpe! frwt.Twje! hblwfUjmTj! fsUyfm! ubsefcb/!

! tblpouspmp!xfsfct![bmjbo!ejej!nojTwofmpcb!brwt/!jtjoj!

hwjDwfofct! nptxbwmfUb! obnewjm! dpeobt-! Ubobd-! gsjbe!

obzpgjfsjb!txbwmfcjtbUwjtbd/!bnjtbUwjt!tblpouspmp!xfsjt!

obTspnfcj! lbshbe! voeb! hbbtxpspt! nbtxbwmfcfmnbd! eb!

Tfnefh!nptxbwmfnbd/!bnjt!njgvDfDfcb!bs!Tfj[mfcb-!sbehbobd!

nptxbwmf! zwfmb{f! lbshbe! tblvUbsj! Tfdepnfcjtb! eb!

ybswf{fcjt! hbtxpsfcjU! txbwmpct/! bnjupn-! [bmjbo!

tbtvswfmjb-! spn! nbUfnbujljt! 5! lwjsfvm! tbbUt! ebfnbupt! 1!

damatebiTi mecadineoba-!spnfmjd!hblwfUjmfcjt!Tfnefh!

ebjojTofcb/! bn! ebnbufcjU! hblwfUjm{f! lmbtj! hvmebtnjU!

hbbtxpsfct! tblpouspmp! obnvTfwsfct-! nUbwbsj! Tfdepnfcj!

ebgb{fd! hbjsDfwb/! wjtbd! sb! blmeb-! jnbtbd! hbjhfct! eb!

Tfbwtfct! tblvUbs! obxfst/! bn! hblwfUjmjebo! nptxbwmfUb!

vnsbwmftpcb! 6.21! xvUjt! Tfnefh! xbwb-! spdb! npsDfcjbo!

tblvUbsj! Tfdepnfcjt! hbtxpsfcbt/! ebsDfcjbo! nypmpe! jt!

nptxbwmffcj-! spnmfctbd! nsbwbmj! Tfdepnb! irpoebU! eb! ftf!

jhj-!tb{phbepebd!DbnpsDfcjbo!)Uvoebd!espfcjU-!hbdefofcjt!

hbnp*/! bnhwbs! nptxbwmffct! lj! jtfebd! tXjsefcbU! ebnbufcjUj!


- 83 -

nfdbejofpcb/!

Tuki damatebiTi gakveTilis Catareba ar xerxdeba, maSin

Secdomebi unda gaswordes erTerTi gakveTilis bolos mas

Semdeg, rac gairCeva saSinao davalebis amocanebi.

sakontrolo werisTvis araviTari winaswari

momzadeba araa saWiro! nbtxbwmfcfmj!hblwfUjmfct!bub.

sfct! Dwfvmfcsjwbe-! qsphsbnjt! njyfewjU/! Dwfo! UwjUpo! jtf!

hwbrwt! qsphsbnb! ebhfhnjmj-! spn! nptxbwmft! UbwjtUbwbe!

vxfwt! jnebhwbsj! bnpdbofcjt! bnpytob-! sphpsfcjd! tblpou.

spmp! xfsb{f! Tfywefcb/! Uvndb-! Tfj[mfcb! ndjsf! tjbymfd!

Tfyweft!`!sbUb!TfnprnfefcjUj!b{spwofcb!bbnprnfept/!

nptxbwmffcj! voeb! njfDwjpo! xjobbSnefhpcjt! hbebmbywbt-!

tblvUbsj! dpeojt! xbsnpDfobtb! eb! ebnuljdfcbt-! eb[bcvm!

wjUbsfcbTj!Ubwjt!hbubobt!eb!qbujptbo-!tbnbsUmjbo!Tfkjcst/!

sakontrolo werisas gamoricxuli unda iyo daxmareba; gamoricxuli

unda iyos agreTve yovelgvari siyalbe, gadawera,

karnaxi da sxva sisaZagle. gamoricxuli unda iyos yalbi

niSnebis werac. maswavlebeli Tavad darwmundeba, rom Sefasebis

Cveneuli kriteriumebi isedac sakmaod lmobieria, isedac

vcdilobT, rom moswavles ar daekargos Tundac mcire codna,

mondomeba Tu mosazreba. nel moswavles cotaTi met xansac

vadrovebT, yvelanairad xels vuwyobT da veferebiT moswavleebs

_ magram siyalbe mainc sastikad unda gamoiricxos! yvela

moswavle ver miiRebs umaRles niSnebs da arc unda miiRos (amis

Sesaxeb ix. $ 1-Si).

TiTo siyalbe sawamlavis TiTo wveTia, romelic ryvnis

moswavlis suls!

sakontrolo wera tardeba saxelmZRvanelos TviTeuli Tavis

bolos. sakontrolo weris win bavSvebs davalebad eZlevaT

Sesabamisi Tavis damatebiTi amocanebi. sakontrolo weris

momdevno dRisaTvis ki bavSvebma unda moamzadon momdevno Tavis

pirveli gakveTili. swored am davalebis garCeva unda moxdes

sakontrolo weris momdevno gakveTilze. sasurvelia wina Tavis

damatebiTi amocanebidan zogierTis, arastandartulis ganxilvac.


- 84 -

sakontrolo werisaTvis SerCeuli yvela amocana aRebulia

moswavlis saxelmZRvanelodan _ amocanaTa Tematikuri krebulidan.

maswavlebelma dafaze unda Camoweros mxolod amocanebis

nomrebi, ori rigisTvis. moswavleebs ar unda movTxovoT

amocanebis pirobaTa gadawera. rveulSi moswavleebma

unda Caweron amocanis nomeri da misi sruli amoxsna.

sakontrolo amocanebi isea SerCeuli, rom maT amosaxsnelad

erTi gakveTilisaTvis gankuTvnili dro _ 45 wuTi sakmarisi

unda iyos. Tumca zogierTi moswavle mainc ver aswrebs am

droSi. Tuki SesaZlebelia, kargi iqneba, mivceT maT damatebiTi

dro sakontrolo samuSaos dasamTavreblad. magram es unda

moxdes maSinve, da ara sxva gakveTilebis Semdeg!

gasarkvevia erTi sakiTxic _ naweris xarisxisa. Cveni azriT,

cudi nawerisa Tu naxazebis gamo moswavles SeiZleba daakldes

qula: qula _ da ara niSani, ukidures SemTxvevaSi _ ori qula,

anda, ufro kargi iqneba, aRar epatios is wvrilmani Secdoma,

romlis patiebac Cveni kriteriumebis mixedviT SeiZleboda.

magram: mainc rogor cud nawerze unda moxdes es?

upirvelesad, Tuki naweri cudia gaformebis TvalsazrisiT:

mindvrebi, sityvebsa Tu maTematikur gamosaxulebebs Soris Sualedebis

dacva, amocanis pirobis mokle Canaweris wesierad Sedgena,

CamonaTvalis garkveviT Camowera, naxazis ise daxazva, rom

nakvTi nakvTs hgavdes da sxva. amgvari xarvezebi maswavlebelma

wiTlad unda moniSnos, iseve rogorc yvela sxva Secdoma da

moswavlem unda gaasworos _ anu wesierad gadaweros Tu gadaxazos

_ Secdomebis gasworebisas.

meorec, Tuki kaligrafiaa Zalian cudi da amasTan, Tuki

maswavlebelma icis, rom am moswavles SeuZlia ukeTesad wera da

esoden cudi naweri maimunobis an mifuCeCebis bralia. magram

saSualod cud kaligrafiaze moswavles qula ar unda daakldes,

miT umetes, roca maswavlebelma icis, rom moswavles Zalian

uWirs lamazad wera da ukeTesad TiTqmis rom ar ZaluZs.

moswavles ar unda daakldes qula arc gadaxazulebze (Tuki

metismetad araa gadadRabnili), Casworebulebze, Canamatebze da


- 85 -

sxva. sakontrolo weris rveuli _ esaa ara saCvenebeli, aramed

samuSao rveuli!

iseve rogorc, zogadad, Cveneuli meTodikiT warmarTuli gakveTili

_ esaa ara saCvenebeli, lamazi da saintereso gakveTili,

aramed _ gakveTili yoveldRiuri muSaobisa, rudunebisa, msjelobisa,

daxvewisa, Ziebisa, azrovnebisa da, rac mTavaria, gonebrivi

varjiSisa da isev da isev varjiSisa.

kidev erTic. maswavlebelma xSirad unda Seaxsenos

moswavleebs, rom sakontrolos rveulSive _ da ara calke

furcelze! _ gaakeTon xolme samuSao Canawerebi. es Canawerebi

SeiZleba ar iyos lamazad Sesrulebuli, SeiZleba iyos

gadaxazuli, Canamatebiani da sxva _ magram amaSi moswavle qula

ar daakldeba! piriqiT, qula daakldeba im moswavles, romelsac

ar eqneba amgvari Canawerebi da pirdapir eqneba dawerili pasuxi.

amocanis piroba romc ar moiTxovdes dasabuTebas _ samuSao

Canawerebi mainc unda Candes. sakontrolo naweris gasworebisas

yovelTvis unda mieqces yuradReba: aris Tu ara rveulSi

moswavlis mier gakeTebuli raime Canawerebi, romlebic

adasturebs, rom man namdvilad amoxsna amocana _ da ara

gadaiwera mzamzareuli pasuxi!

es gansakuTrebiT mniSvnelovania maSin, roca amocanis an misi

qveamocanebis pasuxebi Sedgeba erTi-ori asosgan, erTi-ori

ricxvisa an erTi-ori sityvisagan.

sakontrolo werisas maswavlebeli yurdRebiT unda iyos,

raTa yvela moswavlem, jer erTi, damoukideblad imuSaos, da,

meorec, imuSaos: bavSvs xSirad efanteba yuradReba, zogi

moswavle gacilebiT ukeT niSans iRebs, roca mas aiZuleben, rom

wesierad imuSaos! amrigad, Cven maswavlebels vukrZalavT

moswavlis mixmarebas maTematikis mxriv, magram vTxovT mixmarebas

nebelobis mxriv.

amas ukavSirdeba kidev erTi sakiTxi. zogi moswavle bejiTia,

Tanmimdevruli da Seupovari, bolomde zis da muSaobs. zogi ki

cercetaa, cdilobs, male moamTavros wera (rac Zalian ezareba)

da sxva rame akeTos. magram maswavlebeli zaris darekvamde ar


- 86 -

gauSvebs moswavleebs gareT saTamaSod. maSin is moswavleebi,

romlebmac adre moamTavres wera, cqmutaven da sxvebs uSlian

xels. amas yvelafers didi yuradRebis miqceva sWirdeba. maswavlebelma

unda aiZulos moswavleebi, rom kidev erTxel yuradRebiT

waikiTxon TaviaTi naweri, gadaamowmon, xelaxla iangariSon

(amocanaSi romc ar iyos moTxovnili Semowmeba, mainc!). moswavle

unda mieCvios sakuTari codnis warmoCenasa da damtkicebas, nebisyofis

mokrebiT maRali da ufro maRali Sedegebis mopovebas.

yovelive amas udidesi mniSvneloba aqvs. da ara mxolod

imisaTvis, rom moswavlem ukeTesi niSani miiRos, aramed agreTve

gacilebiT ufro mniSvnelovani ramisTvis: moswavles mtkiced

unda Camouyalibdes sakuTari naSromis Semowmebis, zogadad,

TviTkritikisa da sakuTari namoqmedaris garedan danaxvis

unari da Cveva. amave unaris gansaviTareblad, iseve rogorc sakuTriv

maTematikuri codnis Sesavsebad da gansamtkiceblad

yovlad aucilebelia sakontrolo nawerebis saguldagulo gasworeba

moswavleTa mier!

TviTkritikisa da sakuTari namoqmedaris garedan danaxvis

unarCveva _ adamianis erTerTi umniSvnelovanesi da uZvirfasesi

Tvisebaa (amis Sesaxeb ix. paragrafSi 9). amrigad, sakontrolo

wera unda iyos sakontrolo ara mxolod maswavleblis mier

moswavlis codnis Semowmebis TvalsazrisiT, aramed agreTve

Tavad moswavlis mier sakuTari codnisa da sakuTari naSromis

Semowmebisa da Semdeg gasworebis TvalsazrisiT...

sakontrolo weraze xuTi amocanaa. rogorc wesi, pirveli,

meore da mesame amocanebi _ oTxqulianebia, xolo meoTxe da

mexuTe _ xuTqulianebi. Tuki es qulebi sadme Secvlilia,

sagangebodaa miTiTebuli.

sakontrolo davalebaTa SerCevisas gaTvaliswinebulia, rom

pirveli ori-sami amocanis amoxsna unda SeZlon SedarebiT

sustma moswavleebmac. meoTxe, da gansakuTrebiT mexuTe amocana

ufro Znelebia. xSirad mexuTe amocanaSi aris xolme damatebiTi

davalebac, romlis kargad Sesrulebisas moswavle (Tuki man sxva

davalebebic kargad Seasrula), miiRebs umaRles Sefasebas (`9~-s,


- 87 -

`10~-s an `5+ ~-s Cveulebrivi sistemiT).

yvela sakontrolo weris yoveli amocanisTvis momzadebuli

gvaqvs Sefasebis konkretuli kriteriumebi. isini gadmocemulia

gegma-konspeqtebSi, saTanado adgilas. iq zogierTi amocana

SedarebiT dawvrilebiTaa amoxsnili, amoxsnis gzis etapebis

aRweriT, zogierTisa ki mxolod pasuxia mocemuli. TviTeuli

etapis (an saboloo pasuxis) bolos frCxilebSi miTiTebulia

qulebis maqsimaluri raodenoba, riTac SeiZleba Sefasdes moswavle

am etapis (an mTlianad davalebis) SesrulebisTvis. Tuki moswavles

kargad aqvs Sesrulebuli amoxsnis aRwerili etapi (an mTlianad

davaleba), maSin igi unda Sefasdes miTiTebuli maqsimaluri quliT;

srulad da marTebulad Sesrulebul davalebaSi maswavlebelma

unda daweros 4 qula (pirvel, meore an mesame amocanaSi) an 5

qula (meoTxe an mexuTe amocanaSi). xolo Tuki amocana amoxsnilia

nawilobriv an sul araa amoxsnili, maSin moswavle iRebs ufro

nakleb qulas: 3-s, 2-s, 1-s an 0-s _ imisdamixedviT, Tu amocanis

amosaxsnelad ra nabijebia gadadgmuli. kerZod:

Tuki Sesrulebulia davalebis daaxloebiT naxevari, mesamedi,

meoTxedi (maswavleblis azriT), maSin aRwerili etapi (an

mTlianad davaleba) unda Sefasdes Sesabamisad miTiTebuli maqsimaluri

qulis naxevriT, mesamediT, meoTxediT. Tuki moswavles

dawerili aqvs amosanis amosaxsnelad saWiro erTi mainc marTebuli

gardaqmna, gamoTvla Tu msjeloba, mas am amocanaSi 0,5 an 1

qula mainc unda daeweros. Tuki moswavles amoxsnisas warmatebiT

aqvs gamoyenebuli mis mier mignebuli, uCveulo, skolaSi arnaswavli

xerxi, mas 0,5 an 1 qula unda daematos. miRebuli

qulebi unda Seikribos (rogorc wesi, jamis maqsimaluri mniSvnelobaa

22) da moswavlis sakontrolo namuSevari Sefasdes 10doniani

sistemiT qvemore marcxena cxrilis mixedviT:

sagamocdo davaleba, rogorc wesi, 10 amocanisgan Sedgeba da

maqsimaluri jamuri qula, romelic SeiZleba moswavlem daagrovos,

42-ia (rogorc wesi). amitom moswavlis sagamocdo namuSevari

unda Sefasdes 10-doniani sistemiT marjvena cxrilis mixedviT.


- 88 -

yvela SemTxvevaSi maswavlebelma moswavlis sakontrolo

namuSevarze garkveviT unda aRniSnos, Tu konkretulad

raSi daaklo Tu moumata qula da, ese igi, ramdeni

quliT Seafasa TviTeuli amocana.

qulaTa jami Sefaseba

0-1 0

2-4 1

5-7 2

8-10 3

11-12 4

13-14 5

15-16 6

17-18 7

19-20 8

21-22 9

qulaTa jami Sefaseba

0-2 0

3-8 1

9-14 2

15-19 3

20-23 4

24-27 5

28-31 6

32-35 7

36-39 8

40-42 9

sakontrolo naSromebisTvis Sefasebis 10-doniani sistema gacilebiT

ukeTesia, ramdenime TvalsazrisiT. xolo yoveldRiuri,

mimdinare SefasebebisTvis sjobs 3-4-doniani. saqme isaa, rom aqtiuri

meTodikiT agebul gakveTilze ver xerxdeba moswavlis

codnis safuZvliani Semowmeba (raki ar gvaqvs moswavlis gaZaxeba

gakveTilis mosayolad da sxva). samagierod, Cans TiTqmis yvela

moswavlis (Zalian did klasSi _ moswavleTa umravlesobis) aqtiuroba,

maTi pasuxebi da msjeloba, calkeul sakiTxTa codna.

Cveneuli aqtiuri gakveTilis Semdeg maswavlebels SeuZlia 20

moswavlesac ki Camouweros niSnebi. xolo moswavleTa codnis

safuZvliani Semowmeba sakontrolo werebze iqneba.

amitom, maswavlebels SeuZlia, gakveTilis bolos Seafasos

TiTqmis yvela moswavle (anda, maTi umravlesoba) _ magram es

ver iqneba iseTi dazustebuli da myari niSnebi, rogorebic

iwereba sakontrolo werebze. amitom mimdinare Sefasebebad

ukeTesia amgvari: + (CaTvla), _ (arCaTvla), \ (saSualo). Tanac,

+ araa igive, rac 5-iani, igi aRniSnavs mxolod imas, rom


- 89 -

moswavlem mimdinare gakveTili `CaTvala~: Sesrulebuli hqonda

saSinao davalebis ZiriTadi nawili, erT-or SekiTxvasac kargad

upasuxa. Tuki moswavlem damatebiT kidev TamaSSic gaimarjva, an

raimeTi Zalian gamoiCina Tavi, mas daewereba ori +. xolo \

iwereba, Tuki moswavlem mxolod sanaxevrod CaTvala. semestris

bolos es niSnebi Sejamdeba, TiTo _ niSani gaabaTilebs TiTo +

niSans, xolo sami cali \ CaiTvleba erT + niSnad. saboloo

niSani gamoyvaneba miRebuli ricxvisa da sakontrolo werebis

saSualo niSnis SejerebiT.

SefasebaTa amgvari sistema rTulia, magram gacilebiT ukeT

asaxavs moswavlis codnas aqtiuri swavlebisas. skolas, Tavisi

SexedulebiT, SeuZlia miiRos an ar miiRos amgvari sistema.

VII klasis sakontrolo amoanebis nusxa

zvsbeSfcb" maswavlebelma swavlebis dawyebamde Tavis

saxelmZRvaneloSi wiTlad unda Semoxazos amocanaTa

Tematikuri krebulidan sakontrolo werebisaTvis

gankuTvnil amocanaTa nomrebi, raTa romelime maTgani ar

misces klass damatebiT saSinao an saklaso weris davalebad!

sakontrolo werebisa da gasworebis Sesaxeb ix. $ 10.

aRniSvnebi qvemore siaSi: sw _ sakontrolo wera;

3/9 I, II _ amocanebis Tematikuri krebulis me-3 paragrafis

me-9 amocanis I da II qveamocanebi;

siis marcxena sveti _ I rigi (I varianti);

marjvena sveti _ II rigi (II varianti).

sw # 1

1. 3/4 3/12

2. 3/9 I, II 3/9 III, IV

3. 2/8 2/12

4. 4/11 4/20

5. 5/10 5/19

sw # 2

1. 3/16 I, III, V 3/16 II, IV, VI

2. 3/17 I-III 3/17 IV-VI

3. 4/27 4/34

4. 3/13 3/18

5. 4/28 4/35


sw # 3

1. 3/20 I-III 3/20 IV-VI

2. 3/21 I-II 3/21 III-IV

3. 4/39 I, II 4/39 III, IV

4. 2/17 2/20

5. 4/12 4/21

sw # 5

1. 3/29 I-IV 3/29 V-VIII

2. 3/30 I, III, V 3/30 II, IV, VI

3. 4/52 I, II 4/52 III-IV

4. 2/23 2/27

5. 4/53 4/56

sw # 7

1. 3/31 3/38

2. 2/33 I 2/33 II

3. 2/41 2/49

4. 2/46 2/50

5. 4/60 4/66

sw # 9

1. 3/39 I 3/39 II

2. 3/44 I 3/44 II

3. 2/56 2/60

4. 4/81 4/85

5. 4/75 4/83

sw # 11

1. 3/52 I-IV 3/52 V-VIII

2. 3/53 I-IV 3/53 V-VIII

3. 3/54 I-IV 3/54 V-VIII

4. 2/63 I 2/63 II

- 90 -

sw # 4

1. 3/24 I-III 3/24 IV-VI

2. 3/25 I 3/25 II

3. 3/22 3/27

4. 5/25 I-VIII 5/25 IX-XVI

5. 4/43 4/50

sw # 6

1. 3/36 3/36

I, III, V II, IV, VI

2. 5/28 5/34

3. 2/28 2/29

4. 4/57 4/64

5. 5/32 5/38

sw # 8

1. 3/46 3/51

2. 2/47 2/54

3. 4/58 I 4/58 II

4. 2/40 2/48

5. 2/43 2/45

sw # 10

1. 3/45 I-IV 3/45 V-VIII

2. 3/47 I-IV 3/47 V-VIII

3. 3/48 I-IV 3/48 V-VIII

4. 2/57 2/58

5. 4/90 4/82

sw # 12

1. 3/57 I-III 3/57 IV-VI

2. 3/58 I-II 3/58 III-IV

3. 3/55 3/59

4. 2/71 I 2/71 II

5. 5/12 5/13


5. 4/16 4/30

- 91 -

sw # 13

sw # 14

1. 3/61 I 3/61 II 1. 3/66 I 3/66 II

2. 3/62 I-IV 3/62 V-VIII 2. 3/67 I-II 3/67 III-IV

3. 3/63 I 3/63 II 3. 3/68 I 3/68 II

4. 2/76 2/79

4. 2/77 I 2/77 II

5. 4/40 4/47

5. 4/80 4/92

sagamocdo (Semajamebeli) wera

1. 3/35 3/43

2. 4/15 I-IV 4/15 V-VIII

3. 4/26 I-II 4/26 III- IV

4. 3/64 I, III 3/64 II, IV

5. 5/41 I-IV 5/41 V-VIII

6. 3/69 I, II 3/69 III, IV

7. 3/71 I 3/71 II

8. 2/34 2/44

9. 2/78 I 2/78 II

10. 4/62 4/69

§ 18. amocanebis pasuxebi da miTiTebebi

Tavi I. g. (= gakveTili) 1. 1. I. 2,0036 _ 0,8 = 1,2036;

II. 202,55 + 319,066 + 0,23 = 511,846;

III. 75,05 : 0,8 = 93,8125; IV. 86,508 _ 68,8941 = 17,6139;

V. 0,625 · 60,3 · 15 = 565,3125.

2. I. 7,5 milioni; II. 240 milioni; III. 140 miliardi; IV. 1,5

aTasi. 3. 130-135 kg-iT. 4. mcdaria II. mag., winadadeba `8 iyofa

2-ze an 3-ze~ marTebulia, xolo `8 iyofa 6-ze.~ _ mcdaria.

5. 80 mm : 200 = 0,8 mm. (80 : 0,2) · 3 = 1200 (xvia).

6. 0,051 = an , xolo 3,4442 =

gulwrfelad

gisurvebT

warmatebas

Cvens saerTo

saqmeSi!

an .


- 92 -

g. 2. 1. x + 1 x = 15 , saidanac x = 12 . 2. Crd. yin. okeane _

4

1/20 naw. I. 2-jer; II. 2,5-jer; III. 5-jer; IV. 4-jer.

3. 40-jer. 4. raki samniSna ricxvis Canaweri unda Seicavdes

8-iansac da 5-iansac, amitom pasuxebia: I. 885, 855, 585;

II. 885, 855, 585, 858, 588, 558.

3 2

3 2

5. x + 5x

− 2x

+ 1 = 0,

5 + 5 · 0,

5 − 2 · 0,

5 + 1 = 1,

375.

6. I. daaxloebiT 17 saukunis win; II. daaxloebiT 370 wlis win.

g. 3. 1. a ⋅ 1 = 1.

2. { 100 , 101,

102,

103,

104,

105,

106}

.

a

3. I. 0,93 > 0,83. II. 0,10 < 0.12 an 0,11 < 0,12.

III. 5,24 > 5.08 an 5,24 > 5,18. IV. 3,72 < 3,81 an 3,72 < 3,91.

V. 59,09 > 59,009. 4. { 1300 , 6000,

0,

100500600,

200}.

5. I. 19 ivlisi, kvira, dilis 4-is naxevari. II. 12 ivlisi, kvira,

dilis 12-is naxevari. III. 9 ivlisi, xuTSabaTi, dilis 4-is

naxevari. 6. erTsa da imaves niSnavs.

g. 4. 1. I. { 96 , 26}.

II. { 96 , 100,

26}.

2. {0,3; 0,35; 10,25; 8;

0,8; 5,05} 3. x > 18 da 18 < x; x ≥ 18 da 18 ≤ x;

4. s ≤ 15.

5. a ≥ b.

6. 6 ≤ x.

| |

g. 5. 1. I. marTebulia. II. marTebulia. 2. I. 0,

3<

< 3,

5.

2

QP

II. k

3

< ( k + 2)

< m .

a

3. { 0 , 1;

0,

5;

0,

6;

0,

065;

5,

06;

6,

005}.

4. 12650 km. 5. I. |AC| > 2 m; II. |KC| ≥ 2 m; III. |KC| ≤ 5 m; IV.

|FC| = 2 m; V. |GC| ≥ 5 m. 6. { I, II, IV, V, VI}.

. 7. {I, II, VI, VII}.

(IV-s simcdares adasturebs SemTxveva: c = d = 0).

g. 6. 1. I. marTebulia. II. mcdaria. III. mcdaria. IV. mcdaria.

2. I. b ≤ EF < 7,

5 m; II. 0 < r ≤ 1,

2;

III. k ≤ c

5

≤ ( 2g

+ 1).

3. {1, 7, 0,9, 1,9, 11, 0,5} 4. I. marTebulia. II. mcdaria.

III. marTebulia. 5. 5,6 sm. 6. I. ar SeiZleba. II. SeiZleba. III.

ar SeiZleba. IV. SeiZleba. V. SeiZleba. VI. SeiZleba.

7. I. ar SeiZleba. II. SeiZleba. III. ar SeiZleba. IV. ar SeiZleba.

g. 7. 1. maTematika saZirkvelia codnisa. 2. {A(0), B(0,5),


- 93 -

C(1,5), D(1,75), K(3), E(3,5), T(4), F(5,5)}. erTeul. monakv. sigrZea 2

sm. I. { 4 ), 5),

6),

9),

10)

}. II. { 2 ), 7)

}. III. { 1 ), 3),

8)

}. 3. I. ar

SeiZleba. II. ar SeiZleba. 5. I. metia. II. naklebia. III. marjvnivaa.

IV. Tuki A wertili B-s marcxnivaa, maSin a naklebia b-ze. 6.

( b − a)

-si 7. g) 1/100; es moxdeba, Tuki 10 xmovnidan 5 aris a.

g. 8. 1. I. 5,08. II. 10,32. III. 0. 2. I. ar SeiZleba. II.

SeiZleba Tu d=7. 3. b) 827; 4. AB = b.

wertilis koordinati

tolia am wertilis daSorebisa ricxvTa sxivis saTavidan.

5. AB = 3( b − a)

sm, AB = 0, 3(

b − a)

sm.

6. Tuki samives wili naklebia 0,3-ze, maSin maTi jamuri wili

naklebi iqneba 0,9-ze. amitom danarCen erovnebaTa wili gamova

0,1-ze meti. es ki ewinaaRmdegeba pirobas.

g. 9. 1. 1, 2 ≤ x ≤ 3,

6 2. I. c ≤ z < d.

II. 0 < y ≤ e.

3. I. gare

wrexazis gareT. II. Siga wrexazis SigniT. III. Siga wrexazze. IV.

gare wrexazze. V. gare wrexazis gareT. VI. gare wrexazis gareT.

VII. gare wrexazis SigniT. VIII. gare wrexazis SigniT da Siga

wrexazis gareT. IX. wertili ekuTvnis rgols oRond ara Siga

wrexazs. X. Siga wrexazis SigniT oRond ara Tavad C wertili.

4. I. x = 0 , 6.

II. y = 3,

8.

5. 2,568 6. I. 0, 005 ≤ 0,

005;

II. 10, 905 ≥ 10,

905;

III. 21,911>21,9109 an 21,911>21,9009;

IV. 55


- 94 -

dasWirdeba 10 litri Saqris fxvnili.

4 4

g. 2. 1. I. 1. II. 1. III. = ( 13 · 21)

= ( 3)

. 2. I. x

4

⋅ y

4

;

7 26 2

II. 3

5

⋅ a

5

⋅b

5

⋅c

5

; III. 3

n

⋅ 4

n

. 4. I.

3 b

; II.

3ac

.

4k

5b

5. I. 40735332; II. 25043766, 9306402, 71702202, 40735332.

6. es toloba mcdaria, magaliTad, im SemTxvevaSi, roca

a = b = 1 , n = 2 . amitom mTlianobaSi is mcdaria, magram SeiZleba

marTebuli iyos kerZo SemTxvevebSi. magaliTad, marTebulia I da

II SemTxvevebSi, III SemTxvevaSi ki _ mcdaria.

g. 3. 1. I. 2; II. ( 208 · 125

3 ) = 2

3

= 8 ; III. ( 0 , 01:

0,

001)

6

= 10

6

=

25 104

100000; IV.

625

; V. 32. 4. 9,6 erTeuliT; e = 12,

8 .

81

10,

9−8,

6

5. m = 8,

6 + = 9,

75 . 6. es toloba mcdaria,

2

magaliTad, im SemTxvevaSi, roca a = 2 , b = 1,

n = 2 . amitom

mTlianobaSi is mcdaria, magram SeiZleba marTebuli iyos kerZo

SemTxvevebSi. magaliTad, marTebulia I, II da III SemTxvevebSi, IV

SemTxvevaSi ki _ mcdaria.

g. 4. 1. I. 5,5; II. 47; III. 8. 2. 8 wlis. 3. 150.

4. I. ( 1

6 ) = 1

; II. 1; III. 5 625

2 64

4 = . 5. saSualod 2 kovzi

gamodis. 6. Tbilissa da nalCiks Soris manZili iqneba

220 + 440 −220

= 330 km. Zalian kargi iqneba, Tuki vinme

2

dainaxavs, rom am manZilis gamoTvla SeiZleba daukavSirdes

Suawertilis koordinatis gamoTvlas:

220 + 440 = 330 km.

2

manZili Tbilisidan joxaryalas gadasaxvevamde tolia 342 _ 12

= 330 km-is, xolo vladikavkazisa da maxaWyalas Suagzamde _

220 + 500 = 360 (km). amitom pasuxia 360 _ 330 = 30 (km).

2

7. 72· 7 = 21 weli.

24

g. 5. 1. magaliTad, mama - 70 kg, asuli - 50 kg, vaJi - 60


- 95 -

kg. 2. I. 53 sm; II. 950 g; III. 365 l; IV. 2,65 sT; V. 30,5

wm; 3. I. raime ricxvisa da 0-is saSualo tolia am ricxvis

naxevrisa; II. ori toli ricxvis saSualo tolia mocemuli

ricxvisa. 4. I. 14,6; II. 198; III. 0,4; 5. I. ar ewinaaRmdegeba,

radgan 1 = 25

, 7 = 35 , 2 = 40

; II. giorgis pasuxi, radgan

4 100 20 100 5 100

tolmniSvneliani wiladebis Sedareba ufro advilia; III. me-4

klasSi _ 40 ⋅ 1 = 10 , me-5 klasSi _ 40 ⋅ 7 = 14 , me-6 klasSi _

4

20

40 ⋅ 2 = 16 . 6. I. 28

n

; II. ( )

5

3

xyz ; III. ( ) 5

a

; IV.

9

m

2 ;

g. 6. 4. ufro meti sididis raime procenti metia ufro

naklebi sididis imave procentze! 6. d) mxolod I da III.

g. 7. 1. I. a pirveli c monakveTis sigrZe udris orjer

a nulovani a pirveli monakveTis sigrZes; II. b meeqvse udris

2 xarisxad a meeqvse; III. y mexuTe naklebia an tolia y

meeqvsesa da erTnaxevari p nulovanis jamze.

a1

b3

2. I. a 1 + a2

+ a3

+ a 4 + a5

+ a6

; II.

k

y3


c0

c1

; III. ⋅ ⋅ c

2 3 2

3. I. 1 sm ≤ |AC| ≤ 2 sm; II. 3 sm ≤ |AC| ≤ 4 sm; III. 40 sm ≤

|AC| ≤ 41 sm; IV. n sm ≤ |AC| ≤ n+1 sm. 5. I. 6

3

⋅ d

3

;

II. 2

n

⋅ 7

n

; III. 4

3

⋅ x

3

⋅ y

3

⋅ z

3

5

k

; IV.

c

; V.

7

.

5

k

9 5

g. 8. 2. I. a 1 + a2

+ ... + a60

; II. c

3 3 3

1

+ c

2

+ ... + c

100

.

1 6·

0,

05 1 0,

3

3. I. marTebulia, radgan = = 0,

07 ≥ 0,

01

10 10



;

II. mcdaria, radgan

0

≤ 0,

01;

III. marTebulia, radgan

10

10+

6

= 1,

6 ≥ 0,

01.

4. 20% aris 8 spilo. amitom sul iqneba

10

8 ··5 = 40 spilo. 5. I. SeiZleba; II. SeiZleba; III. ar

SeiZleba; IV. SeiZleba; V. ar SeiZleba; VI. ar SeiZleba.

7. magaliTad: I. a

2 2 2

1

+ a

2

+ ... + a

10

;


- 96 -

II. b 71 + b73

+ b75

+ ... + b99

; III. b71 − b70

− b69

− ... − b1

.

g. 9. 2.

6

:

36

=

360

:

100

. es toloba mcdaria, amitom

2 5 10 1

proporcia ar aris. 3. (3) Tviseba proporciis ZiriTadi

Tvisebaa: `proporciis jvaredina wevrebis namravli erTmaneTis

tolia~; (2) Tviseba ki proporciulobis mesame Tvisebaa:

`proporciis Sua wevrebis adgilebis gacvliT isev proporcia

miiReba~. 4. II. ( 4·

0,

25)

26

= 1

26

= 1;

III. ( 0,


5)

8

· 5 = 5 ;

IV. 3 81

4 = ; V. 1. 5. ( b b b ) n

b

n

b

n

b

n

1 · 2 · ... · 100 =

1

·

2

· ... ·

100

.

6. I. 960 kg; II. 3 t. 7. I marTebulia, radgan

2

4

+ 2

3

+ 2

2

+ 2

1

= 16+8+4+2=30≤30; II. mcdaria, radgan

2

4

+ 4

3

+ 5

2

+ 9, 7 30

1 > ; III. marTebulia, radgan

0

4

+ 1

3

+ 0,

7

2

+ 0,

7

1

= 1 + 0,

49 + 0,

7 < 30.

8. proporciuloba gamoiyeneba:

aTwiladebis gayofis, rukis masStabis, ricxvebis SefardebaTa

tolobis, skalis, svetovani diagramis SemTxvevebSi.

Tavi III. g. 1. 7. I. 10 . . . 0 ; II. 10 . . . 0.

2n-jer 3n-jer

g. 2. 1. I. 1 , 2

14

; II. 29

4 ; III. 220

g ; IV. 38

h ;

V. 21

3 ; VI. 13

5 .

2. 6

10

= 6

2

· 6

3

· 6

5

= 6

1

· 6

4

· 6

5

= 6

2

· 6

4

· 6

4

. 3. 0.

4. tolebia. 5. I. 4

7 ; II. 10

a . 6. nayinSi _ 20%, anu 6

lari; wignebSi _ 12 lari.

g. 3. 1. 3 81

4 = ; 2 124

7 = ; 3 , 45 . 2. I. 10 , 2

68

;

II. 4

b ; III. ( ) 10 7 ; IV. 0,44; V. ( )

23

9

a + b . 4. I. SeiZleba;

II. SeiZleba; III. SeiZleba; IV. SeiZleba; V. SeiZleba; VI. ar

SeiZleba. 5. I. 0; II. 0; III. 1 : 1,

6 =

10

=

5

;

16 8

5

7 6

IV.

5

=

1

=

1

; V. 64; VI.

1 5 · 2

; VII. =

1

.

8 3

5 5 125

81

7 7

5 · 2 2


- 97 -

6. ara, radgan 12m-ic da 8n-ic luwia, amitom maTi jamic

luwia! 7. ganayofi _ sxvaoba; Tanamamravli _ Sesakrebi;

gamyofi _ maklebi; gamravleba _ Sekreba; gayofa _ gamokleba.

g. 4. 1. 1; 2. 2. tolebia. 4. = 6· = 6·

0,

04 =

2

9 9

6 · c

8 7 c

6 · c

= 0,

24 . 5. zedapiris farTobi metia 4-jer (9-jer); moculoba

_ 8-jer (27-jer). 6. azoti _ 48 kg; fosfori _ 7 kg;

kaliumi _ 4 kg; kalciumi _ 6 kg. 7. yoveli ricxvi (mesamedan

dawyebuli) wina ori ricxvis saSualo ariTmetikulis tolia.

amitom ? niSnebis nacvlad unda eweros 9 da 3,75.

g. 5. 1. I. 3 , 5

12

; II. d

n 5 ; III. 33

y . 2. I. ( ) 3

13

4

;

4

⎛ 9

II. ( 2 ⎞

⎜ 23 ) ⎟ ; III. ( )

⎝ 19 ⎠

2

2

2

2 ⎛ 2

; IV. ( 1 ⎞

⎜ ) ⎟ ; V. ( )

⎝ 3 ⎠

3

b

5

; VI.

( ) 3

d

m

. 3. g) 4. cxrilSi %-ebis maCveneblebi 100-jer metia

Sesabamisi nawilebis maCveneblebze. 5. I. gadiddeba 2-jer; II.

Semcirdeba 5-jer; III. ar Seicvleba. 6. I maRaziaSi fasdaklebis

Semdeg televizori eRireba 368,6 lari, meoreSi _ 365,8.

g. 6. 2. daaxloebiT 694 kg. 3. 17% metia 17/12-jer.

4.

720

· 100%

12%

6000

II. ( ) ( ) 2 2 3 2

3

2

2

2

2

3

= . 5. I. ( ) · ( ) · 11 121

3 3

3

2

3

· · =

3

; III. 0,

25

4

· 4

4

· 4

3

= 64 ;

4

2 2

4 4 3

2

·

4

:

8

= · · =

4

·

1

=

1

.

IV. ( ) ( ) ( ) 9 9 3 9 9 8 9 36 81

2

= ;

6. I.

1

3

d ; II. 1

; III. b

m−3

.

k

g. 7. 1. I.

125

; II.

10

. 2. I. 27%; II. 160%. 3. 72 kg.

6 9

4. I. 1,3; II. 1,7. 6. I. 532; II. 2,4. 7. II. raki a < b,

amitom m =

a+

b

>

a+

a

= a . aseve damtkicdeba, rom m < b.

2 2


- 98 -

g. 8. 1. I. 1 m; II. 1,5 t. III. 1,5 weli; IV. 390 lari;

V. 1 sT 5 wT; VI. 1 kg. 3. I.

a

; II. TviT am ricxvisa!

n+

1

4. saSualo 8-is tolia. 3 < 8 < 16. 5. saSualo 9-is tolia.

9 ≤ 9 ≤ 9. 6. 2 kg-iani sazamTro yofila 600 cali, 3 kg-iani

_ 150 cali. maTi mTliani wona iqneba 1200 + 450 = 1650 kg. 2

kg-iani sazamTroebi sul gayidula, radgan 750-is 80% tolia

600-is, xolo 3 kg-ianebi ki _ ara, radgan 1650 kg-is 20%

tolia 330 kg-isa da ara 450-isa!

g. 9. 2. roca yvela ricxvi tolia! 4. ara, radganac

SesakrebTa gadanacvlebiT jamis mniSvneloba ar icvleba.

5. I. meore 15/8-jer; II. meore 81/5 = 16,2-jer. 6. I. 45,5;

II. 19; III. 8,8. 7. 0,07a + 0,07b + 0,07c = 0,07(a + b + c).

2

amitom uvargisia mTliani wonis 7%. 8. II. ( 2

3 ) = 2

6

;

3

III. ( 10

2 ) = 10

6

3

; IV. ( 2

2 ) = 2

6

2

; V. ( 10

3 ) = 10

6

.

g. 10. 1. I. 26

26

; II. 1/2 + 8. 2. mcdaria II.

3. I. 24

5 ; II. 5

c ; III.

2

11 . 4. siraqlemas kvercxebis wona

1,


15

0,

050·

16

misi wonis · 100 = 25 %-ia, qaTmisa _ · 100 = 40 %.

84

2

amitom Tavis wonasTan SedarebiT qaTami ufro meti wonis

kvercxs debs,

40

vidre siraqlema. 5. 3

200

= ( 3

5 ) = 243

40

> 200

40

> 200

3

.

6. keplerma _ 21 wliT. 7. I. gadiddeba 3-jer;

II. Semcirdeba 4-jer; III. ar Seicvleba. 8. ar SeiZleba,

radgan Tuki TviTeul TveSi mzis naTebis xangrZlioba ≤ 200 sTze,

maSin 1 weliwadSi ≤ 12 · 200 = 2400 sT-ze, rac

ewinaaRmdegeba mocemulobas.

Tavi IV. g. 1. 1. I. 10000. II. 99999. III. 81253. IV. 7002.

2. I. 4 ⋅ 1000 + 0 ⋅100

+ 7 ⋅10

+ 2 ⋅1=

4 ⋅

10

3

+ 0 ⋅10

2

+ 7 ⋅10

1

+ 2 ⋅10

0

.


- 99 -

II. 8 ⋅100000

+ 9 ⋅10000

+ 0 ⋅1000

+ 7 ⋅100

+ 0 ⋅10

+ 1⋅1

= 8⋅10

5

+

+ 9 ⋅10

4

+ 0 ⋅10

3

+ 7 ⋅10

2

+ 0 ⋅10

1

+ 1⋅10

0

. 3. m > n 30-iT.

4. n > m 9-iT. 5. I. 51. II. 15. 6. I. 75%. II. 52%. III. 190%.

7. 3003 aguri. 8. I. 1 saaTSi. II. naxevar saaTSi. III. 1 saaTSi.

4 3 2 1

g. 2. 1. a 4a3a

2a1a0

= a4

⋅10

+ a3

⋅10

+ a2

⋅10

+ a1

⋅10

+

0

5 4 3 2

+ a 0 ⋅10

; a 5a4a3a2a

1a0

= a5

⋅10

+ a 4 ⋅10

+ a3

⋅10

+ a2

⋅10

+

1 0

59

58

0

a 1 ⋅10 + a0

⋅10

; a 59a58...

a0

= a59

⋅10

+ a58

⋅10

+ ... + a0

⋅10

.

2. 18-is. 3. abcd ≥ abcd (tolia mxolod maSin, roca yvela aso

0-is tolia!) 4. 30001000. 5. 5-is. 6. I. 540000.

II. 54 miliardi. III. 400 miliardi. IV. 600000.

7. I. 2 dRe. II. 4 dRe. III. 2 dRe. IV. 1 dRe. V. 2 dRe.

8. ab = a ⋅10

+ b ⋅1;

ba = b ⋅10

+ a ⋅1.

340 . II. 7900000;

9700000; 11; 0,71; 1,4. III. ar SeiZleba. 3. I. mcdaria. mag: 547

da 386. II. mcdaria. mag: 386 da 547. III. marTebulia. IV. mcdaria.

6

7 4

5

mag: 386 da 364. 4. 10 ≤ 2507500

< 10 ; 10 ≤ b 4b3b2b1b

0 < 10 .

21

5. I. m ; II. 1

p ; III. 25

5 . 6. xuTkuTxa. 7. 90.

27 28 0 1 0 1

g. 4. 1. 10 ≤ h < 10 ; 10 ≤ 1 < 10 ; 10 ≤ 6 < 10 ;

3

4 1

10 ≤ 3560 < 10 ; 10 20 4 2 5

6 4

≤ < 10 ; 10 ≤ 100000 < 10 ; 10 ≤

7

5

82000 , 8 < 10 . 2. I. b 3 . II. b 6 . 3. 123456789. 4. 9050.

5. I. ara. II. ki. III. ki. IV. ara. V. ki. 6. qaTmis

sicocxlis xangrZlioba gamodis bus sicocxlis xangrZliobas

mimatebuli misi 0,8 nawili. ese igi, bus sicocxlis xangrZliobis

0,8 nawili yofila 10 weli. aqedan miviRebT, rom bu

cocxlobs 10· 10 = 12,5 wels, qaTami _ 22,5 wels.

8

g. 5. 1. I. gadiddeba; II. Semcirdeba; 2. I. 11,25 sm; II. 22,5

sm; III. ar SeiZleba, radgan SesaZloa moTova, magram maSinve

g. 3. 1. I. { ; 34;

0;

3;

55000;

100;

0,

1;

2000}


- 100 -

gadna; 3. I. {mtkvari, Tergi, Woroxi, alazani, iori, debeda};

II. udidesebi: {mtkvari, alazani, iori, rioni, enguri, xrami};

saSualo sigrZe aRmosavleTSi 240+ 385+

300 ≈ 308 km; samxreTSi _

3

150 + 220 = 185 km; dasavleTSi _ 327 + 221 = 274 km; III. udides mdi-

2

2

nareTa saSualo sigrZeebi; 4. I. 4 saaTSi erTxel; II. 13

0

; 5. I.

0,0; 0,5; 0,6; 1,4; 5,5; 10,0; II. 3; 6. 900. 7. I. 450; II. 15%.

6+

7,

5+

6+

5,

5+

7

6 + 6,

5+

7,

5+

7,

5+

6+

5+

5,

5+

6,

5+

7+

5

g. 6.1.I. = 6,

4 ; II.

5

10

= 6,25. 2. I. 14-jer; II. 1910-1920 wlebSi mateba iyo

daaxloebiT 500 ha, 1920-1930-Si _ 500 ha, 1930-1940-Si _

2000 ha, 1940-1950-Si _ 500 ha, 1950-1960-Si _ 1500 ha, 1960-

1970-Si _ 3000 ha, 1970-1980-Si _ 5000 ha. amitom saSualo

aTwliani mateba iqneba:

500+ 500+

2000+

500+

1500+

3000+

5000 13000 = ≈ 1860 ha = 18 600 000 kv.m

7

7

3.I. mcdaria (ix. XI Tve). II. saSualo wliuri simaRlea baxmaroSi:

125 + 175+

200+

150+

50+

0+

0+

0+

0+

15+

15+

50 = 65 sm, gudaurSi _ ≈ 38 sm.

12

Tovlis saSualo wliuri simaRle metia baxmaroSi ≈ 1,7-jer.

III. marTebulia. 4. ufro zustia cxrilis xerxi, ufro TvalsaCino

_ diagramisa. 5. I. 176 kg; II. raki 1 kub. m. nedli SeSa

176 kg wyals Seicavs, amitom danarCeni nivTierebis wona (wylis

garda) iqneba 440_176=264 kg. es wona Seesabameba 100_24=76%s.

amitom 1 kub. m xmeli SeSis wonis gamosaTvlelad

264 kg − − − 76%

Sedgeba proporcia:

. aqedan x ≈ 347 kg.

x kg − − − 100%

pasuxi miaxloebiTia, radgan monacemebic miaxloebiTia.

g. 7. 1. I. rema; II. fara; III. jogi; IV. gundi; V. xrova;

VI. kona; VII. Zna; VIII. anbani. 2. I. 11, 13 da 15; II. 9, 10

da 14; III. 9 da 10; IV. 11, 14 da 15. gogonebi izrdebian

ufro swrafad 11-12 wlis asakSi, vaJebi _ 14-15 wlis asakSi.


- 101 -

3. 2), 3), 2) da 4). saSualod igulisxmeba 1), 3) da 4)

winadadebebSi. 2) winadadebis marTebulobidan ar gamomdinareobs

arcerTi sxva winadadebis marTebuloba, Tumca SedarebiT sandod

SeiZleba miviCnioT 3), xolo yvelaze naklebad sandod _ 4)

winadadeba. 4. magaliTad, 2 sm × 8 sm × 4 sm da 8 sm × 8 sm × 4 sm.

5. I. 70 lari; II. 17,5 lari; III. 367,5 lari.

g. 8. 1. I. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}; II. {iisferi, muqi

lurji, cisferi, mwvane, yviTeli, narinjisferi, wiTeli};

III. {qarTli, samcxe, javaxeTi, tao-klarjeTi, lazeTi da a.S};

IV. {qarTuli, maTematika, ucxo ena, istoria, geografia, religiis

istoria, biologia}; V. {2, 3, 5, 7, 11, 13, 17, 19};

VI. {a, b, g, d, e, v, z, da a.S}; 2. marTebulia, marTebulia,

mcdaria, marTebulia, marTebulia, marTebulia, mcdaria, mcdaria,

mcdaria, mcdaria; 3. UI. erTi; II. oTxi; III. sami; 4. I. Seicavs;

II. ar Seicavs; III. Seicavs; IV. ar Seicavs; 5. I. ana∈ D ;

II. zuriko∉ A ; III. curva∉ E ; IV. e ∈ E ; V. f ∉ F ;

VI. fanqari∉ D ; VII. gia∉ D ; VIII. {cira, Tamriko, naTela}

⊂ D ; IX. CogburTi∉ C ; X. B ⊃ {Cikori, bzriala, burTi};

XI. {Cikori} ⊂ B ; XII. Tojina∈ B ; XIII. k ∈ E ;

6. I. 3 9

2

6 4 2

3 ⋅(

3 ) 6 8 14

3 ⋅3

3 = = = = ;

12 12 12

3 3 3

9 7 3

2 ⋅(

2 ) ⋅8

9 21 3 33

II. = 2 ⋅2

⋅2

= 2 = 1 = 1

;

35

35 35 2

2

2 2 2 4

7. 1 ≤ a 2 + a1

+ a0

≤ 27 .

Tavi V. g. 1. 1. I. V, VI, VII, IV, III, IX, II, VIII, X, XI,

I, XII; II. {III, IX}, {II, VIII, X}, {I, XII};

IV. 20+30+40+45+70+60+50+30+40+30+25+20=460 mm;

3500 12700

V. ≈ 7,

6 -jer; VI. ≈ 27,

6 -jer;

460

460

VII. 460-jer; 2. gv. 38, 41, 42; 3. g); 4. S ≈ 330 km,

330 = 33 sT-Si; 5. 5+15+25+35=80 m; me-4 wamis ganmavlobaSi.

10

6. rezo _ 18 ⋅ 5 = 90 , xviCa _ 12 · 5 + 24 = 84 . amitom daeweva.


- 102 -

7.

1

nawils; Zmac igives; mama _

1

nawils; I. 10 sT; II. 5 sT.

20

10

85 10

g. 2. 1. = 42,

5 km/sT; 2. 0 , 3⋅

60 = 18 cida; ≈ 33,

3 wm;

2

0.

3

ag hb ad

3. I. 1 km/wT; II. 60 km/sT; 4. g) 1,8; 5. I. ; II. ; III. ;

b a bc

150000000

6. 300000 ⋅ 3600 = 1080000000 km; I. = 500 wm;

300000

288000000

II. = 960 wm; 7. damtkiceba advilia, Tuki gaviTvalis-

300000

winebT, rom S _ gasayofia, t _ gamyofi, v ki _ ganayofi.

5 30

g. 3. 1. ⋅ 3,

6 = 3 km/sT; t = = 10 sT. 2. 2,3 sm; a) D ,

6

3

b) B , g) E , d) A ;A 3. I wT-10 km/sT, II wT-10 km/sT, III wT-0

km/sT, IV wT-8 km/sT, V wT-0 km/sT, BVI wT-9 km/sT; saSualo _

10+ 10+

0+

8+

0+

9 ≈ 6,

1 km/sT; 4. es amocana momdevno gakveTilis

6

Semamazadebelia. amitom unda dafaze unda gairCes ise, rogorc

aqaa mocemuli: cxadia, v1

= s t

1 :

3

= s 3

1 ⋅ t

=

3 s1

. amgvaradve

t

3s2

miviRebT, rom: v2

= ,

t

v3

=

3s3

. am siCqareTa saSualo v

t

saS. =

.

=

=

=

1

3

1

3

v +

v + v

3 3 1

3 3 3

( s s

1 2 +

s

+ 3 )

t t t

·

1

( s + s + s )

3

1

2

t

2

3

3

=

1

=

( v + v + v )

1

3

·

=

3s

t

2

1

3

=

5

5. 10 km/sT; 50 km/sT; 50 · = 14 1

m/wm. 7. kubSi Casmuli

18 9

ricxvi unda aviyvanoT mesame xarisxSi, xolo kvadratSi Casmuli

_ meore xarisxSi. amitom ? = 98.

3

g. 4. 1. Tanabari moZraobis siCqarisa! 2. 45 wT= sT. amitom

4

3 v 2

saS. = 14 : = 18 km/sT. 3. v) 84. 4. 250 m/wT = 15 km/sT. 1

4 3

dRe-RameSi frenen 10 sT-is ganmavlobaSi, amitom gaifrenen 150

km-s. 2500 km-is gavlas moandomeben 2500 : 150 ≈ 17 dRe-Rames.

xanmokle frenisas siCqare metia 55 : 15 ≈ 3,7-jer.

g. 5. 1. saerTo wesiT: u ≈ 440 da v ≈ 440 000 000 000 000.

·

3

3s1+

3s2

+ 3s

t

s

t

.

=

3

=


- 103 -

meore SemTxvevaSi: u ≈ 400 da v ≈ 400 000 000 000 000. damrgvaleba

ufro uxeSia meore SemTxvevaSi. 2. milioni = 10 6 ; miliardi

anu bilioni = 10 9 ; trilioni = 10 12 ; kvadrilioni = 10 15 ; kvintilioni

= 10 18 ; seqstilioni = 10 21 ; septilioni = 10 24 ; oqtalioni

= 10 27 ; nonalioni = 10 30 ; dekalioni = 10 33 . xarisxis

maCvenebeli 3-iT imatebs. 3. vidre 10n–1 da naklebia, vidre

10n . ese igi, yoveli n-niSna ricxvi, garda 10...0-isa (sadac 0

aRebulia (n-1)-jer), moTavsebulia 10n–1-sa da 10n-s Soris.

g) 10 n – 2 da 10 n–1 + 1. 4. e) 5. v saS. =

16 + 48+

128+

240 = 108 ft/wm = 32,4 m/wm. cxadia, v

4

1 = 16 ft/wm;

v 2 = 48 ft/wm; v 3 = 128 ft/wm; v 4 = 240 ft/wm. am siCqareebis

saSualo daemTxveva vsaS. -s, radgan vardnis dro Tanabar

15 + 5

1200

Sualedebadaa dayofili. 6. I. = 10 m/wm; II. t 80

2

1 = =

15

1200

1200 + 1200 2400

wm; t 2 = = 240 wm; amitom v

5

saS. = = = 7,

5 m/wm.

80+

240 320

ar daemTxva imitom, rom irmis moZraobis dro araa or tol

Sualedad dayofili (tborisken _ 80 wm, tboridan _ 240 wm).

7. 6 · 2 · 2 = 24 varianti. 8. g) 35-jer.

g. 6. 1. I. 15 · 10 7 km; II. 4 · 10 13 km; III. 10 31 ; IV. 2 · 10 16 ;

V. 10 15 . 2. I. 1000; II. 10 000 000; III. 15000; IV. 2,3;

V. 34. 3. I.

55 −50

=

12

5

12

sT=25 wT; II.

77−50 27 = = 2

12 12

1

4

sT=2sT15wT.

4. I. mcdaria, radgan mZimis 2 TanrigiT marcxniv gadmotanisas

aTwiladi mcirdeba 10 2 -jer da ara 10 3 -jer! II da III mcdaria,

orivegan unda iyos 10 3 . 5. I. 2-jer; II. 5,5-jer; III. 10-jer;

IV. 10 2 -jer; V. 10 4 -jer; VI. 10 m -jer. 6. I. riwa, yelis tba,

bazaleTi; II. paliastomi, tabawyuri, xanjali, Wandara,

madataba, saRamos tba; III. faravani, karwaxi;

IV. xanjali, madataba, saRamos tba, bazaleTi;


- 104 -

V. faravani, karwaxi, paliastomi, Wandara, riwa, yelis tba;

VI. tabawyuri. saSualo udidesi siRrmea 22,7 m.

g. 7. 1. 3 = 3 · 10 0 ; 10 = 1 · 10 1 ; 27 = 2,7 · 10 1 ; 25,6 = 2,56 · 10 1 ;

1 = 1 · 10 0 ; 523 = 5,23 · 10 2 ; 52300000 = 5,23 · 10 7 ; 1 800 000 000

000 = 1,8 · 10 12 ; 1,0040 = 1,004 · 10 0 ; 23 · 10 8 = 2,3 · 10 9 ; 0,9 · 10 6

= 9 · 10 5 ; sami milioni = 3 · 10 6 ; ocdaori miliardi = 2,2 · 10 9 .

3

2. {7; 4,5; 1 ; 2,007}, {10; 89,63} {45000; 50001}.

4

3. I. erTeulebamde; II. meaTedebamde. 4. I. 1,5 kv.km = 1,5 · 10 2

ha; II. 5,5 · 10 9 litri. 5. I. p = 1; II. n = 0; III. n = 3.

g. 8. 1. 1,45093 ≈ 1,5 · 10 0 ; 7,8 = 7,8 · 10 0 ; 7,08 ≈ 7,1 · 10 0 ; 10

= 1 · 10 1 ; 349093 ≈ 3,5 · 10 5 ; 3,625 · 10 9 ≈ 3,7 · 10 9 ; 825 · 10 15 ≈ 8,3

· 10 17 ; cxranaxevari milioni = 9,5 · 10 6 ; 148 miliardi ≈ 1,5 · 10 11 .

21

2. 81 · 10 18 6·

10 600

kv.m. 3. I. = ≈ 86 jer;

19


10 7

27

5


10 20·

10

5

II. = ≈ 3,


10 -jer. 4. dedamiwa _ 60 astrotona;

21


10 6

mzis _ 2 · 10 7 astrotona; mTvare _ 0,7 astrotona. 5. aklia z)

damrgvalebiT; araa qristiani (b) 0,44 nawili. meoTxeds Seadgenen

kaTolikeebi da isini metni arian iudaistebze (g) 12,5-jer.

6. mravalkuTxedi. 7. (d) 510066 mln. kv.km ≈ 5,1 · 10 11 kv.km =

5,1 · 10 17 kv.m = 5,1 · 10 23 kv.mm. amitom gugolpleqsis cifruli

Canaweri dedamiwis zedapirze ar daeteva!

g. 9. 1. 3,6 · 3,6 ≈ 13 (kv.mm). 1,5 · 10 17 · 13 kv.mm = 19,5 · 10 17

kv.mm ≈ 2 · 10 18 kv.mm = 2 · 10 12 kv.m = 2 · 10 8 ha.

2. gamravlebisTvis, gayofisTvis da axarisxebisTvis.

3. I. 500 wm; II. 37500 sT ≈ 1563 dRe-Rame ≈ 4,3 weli.

4. I. a ∈ S , a ∈ S0 ; II. b ∉ S, b ∉ S0 ; III. c ∉ S , c

∉ S 0 ; IV. d ∈ S , d ∉ S 0 ; V. e ∉ S , e ∉ S 0 ; VI. S ⊃ S 0 ,


- 105 -

S0 ⊂ S . 5. aguredi (marTkuTxa paralelepipedi).

6. 1 dReSi _ 20 · 6 = 120 mg nikotini, 1 weliwadSi _ 365 · 120

≈ 44000 mg = 44 g nikotini ≈ 200 abi. amocanis SekiTxvebze

pasuxis gasacemad pirveli abzacidan saWiroa mxolod is, rom 1

Rer sigaretSi nikotini 6 mg-ia.

Tavi VI. g. 1. 1. I. ki (kvadratisgan gansxvavebuli

marTkuTxedi); II. ara; III-IV. ara; V. ki; VI. ara; VII. ki;

VIII. ki. M da K sxvadasxvaa, T da K _ erTidaigive. K 2. z) 21.

3. I. ara; II. ara; III. ara; IV. ki; V. ki; VI. ki. 4. erTeulis

sigrZe = 2,5 mm. D0 (8), D3 (23), D4 (26), D5 (35), D6 (48), D7 (53),

D8 (57), D9 (62). 5. I. v) measeds (klasSi 100 moswavle ver

iqneba!); II. z) 10%-90% (arafrismTqmeli monacemia!). 7. 3·10 14.

g. 2. 1. I. {1, 3, 5, 7, 9}; II. {19, 28, 37, 46, 55, 64, 73, 82,

91}; III. {1, 2, 3, 4, 6, 12}; IV. {1000}; V. {avTandili,

tarieli, TinaTini, ...}. 2. I. mag.: luw martiv ricxvTa

erToblioba; II. mag.: 21-ze nakleb 5-is jerad ricxvTa

erToblioba. 3. I. 2; II. 6; III. 0; IV. 4; V. 1; VI. 0; VII. 0;

VIII. 0. SeniSvna: am amocanis garCevisas aqcenti unda gakeTdes

imaze, rom erTobliobaSi SeiZleba arcerTi wevri ar iyos!

4. I. ≡ II. ≡/ III. ≡/ IV. ≡ V. ≡/ VI. ≡/ VII. ≡ .

III SemTxvevaSi ≡/ niSani SeiZleba Seicvalos ⊂ niSniT.

5. I. ara; II. ara; III. ki; IV. ki.

g. 3. 1. I. mcdaria; II. mcdaria; III. marTebulia; IV. marTebulia;

V. marTebulia. 2. I. ∅ ≡/ {5}; II. 5 ∉ ∅; III. {a, b} ≡

{b, a}; IV. {a, b} ≡/ {a, b, c}. ≡/ niSnis Secvla ⊂ niSniT SeiZleba

I da IV SemTxvevebSi. 3. I. ara (2); II. ki; III. ara (tolferda

samkuTxedi); IV. ki; V. ki; VI. ki; VII. ara (aguredi);

VIII. ara (9). 4. pirveli boZidan meoTxemde 3 Sualedia, xolo

meSvidemde 2-jer meti _ 6 Sualedi. amitom pasuxia 24 wuTi.

5. I . marjvena; II. marcxena; III. marjvena; IV. marjvena.


- 106 -

g. 4. 1. ≈ 38. sul iqneba ≈ 1590 · 4 · 38 ≈ 240 000 = 2,4 · 10 5

aso. 2. ≈ 2,4 · 10 5 · 10 12 = 2,4 · 10 17 . 3. d) 4. 1 km 3 =10 9

m 3 =10 15 sm 3 =5·10 14 kovzi. amitom Sav zRvaSi iqneba 80000 · 5 · 10 14 =

4 · 10 19 kovzi, baltiis zRvaSi _ 10000·5·10 14 =5·10 18 kovzi, xolo

kaspiis zRvaSi _ 15000 · 5 · 10 14 = 7,5 · 10 18 kovzi. 5. I. ≈1,3 · 10 151 ;

II. 2,8 · 10 71 ; III. 3,5 · 10 43 . 6. I. prizmas minimum 3 gverdiTi

waxnagi aqvs da ori fuZe aqvs! II. minimum 9 wibo aqvs!

III. wveroebis raodenoba luwi unda iyos! prizmas SeiZleba

hqondes 1348 wvero, magram ar SeiZleba hqondes amdeni wibo,

radgan wiboebis raodenoba 3-is jeradi unda iyos!

7. Tvlis dros didi ricxvis dasaxelebas meti dro sWirdeba.

amitom CavTvaloT, rom saSualod erTi ricxvis dasaxelebas

sWirdeba 10 wami. maSin wuTSi dasaxeldeba 6 ricxvi, 1 sT-Si _

360, 1 dRe-RameSi _ 16· 360 ≈ 5800 (davuSvaT, rom adamians 8 sT

sZinavs), 1 weliwadSi _ ≈ 2 000 000; 80 weliwadSi _ 160 000

000 = 160 milioni. cxadia, es pasuxi Zalian pirobiTia. amitom

dadebiTad unda CaiTvalos moswavleTa mier miRebuli sxva

pasuxebic, oRond maT unda SeeZloT TavianTi pasuxebis metnaklebad

dasabuTeba.

g. 5. 1. bundovania: II, IV, VI, VII, IX. 2. I. 5; II. 1; III.

5; IV. 8; V. 2; VI. 15; VII. 4. 3. I. {1, 29}; II. {16, 32, 48,

64, 80, 96}. 4. mcdaria III (radgan `ori~ sityvaa da ara aso!)

da V (unda iyos ∈ niSani). 5. erTeulovani monakveTis sigrZea

4 sm : 10 = 0,4 sm. e) 6. 1,5 km = 1500 m sigrZeze saWiroa

3000 fila, 3,5 m siganeze _ 7 fila. sul _ 21 000 fila. 180

sm × 50 sm zomis fila 180:50=3,6-jer naklebi iqneba saWiro, sul

21000:3,6 ≈ 5834 cali. 7. {a}, {b}, {g}, {a, b}, {a, g}, {b, g}, {a, b, g}.

bolo igivuria mocemuli igiveobisa. arcerTi maTganis wevrTa rao-


- 107 -

denoba ar aRemateba mocemuli erTobliobis wevrTa raodenobas.

g. 6. 1. `milionis~ Sesatyvisia `uSqari~; uStisuSti 100

milionis tolia; miliardi tolia 1000 uSqarisa. 2. qviSis

marcvalTa raodenoba naklebi iqneba 6·10 21 t : 1 /1 0 6 g = 6·10 27 ·10 6

= 6 · 10 33 -ze. 3. me-5 nabiji: 10 10 kvadr.; me-10 nabiji: 10 20 kvadr.

4. ara, radgan rogori ricxvic ar unda daiweros, yovelTvis

SeiZleba masze metis dawera! sami Zmis SemTxvevaSi araferi

icvleba! 5. ara! 6. ar SeiZleba! 7. weliwadSi daewveTeba 365

mg=0,365 g kiri, saukuneSi _ 36,5 g. stalaqtidis xnovaneba

iqneba 100 000 g : 36,5 ≈ 2740 saukune.

g. 7. 1. sasrulebia: I, II, III, IV, VI, VIII. 2. ara, imitom

rom sxivi erT mxares usasrulod grZeldeba! 3. SeiZleba. samive

SemTxvevaSi (I, II, III) sxivze usasrulod mravali monakveTi

moizomeba. 4. erTeul. monakv. sigrZea 4 sm : 0,8 5 sm.

(d) 0,2 < x < 0,8; y > 0,2. 5. magaliTad: I. C erTniSna ricxvTa

simravle, D _ naturalur ricxvTa simravle; II. C kent

ricxvTa simravle, D _ naturalur ricxvTa simravle;

6. I. raki naturalur ricxvTa simravle usasruloa, amitom

usasrulo iqneba mTel ricxvTa simravlec; II. raki monakveTebis

simravle usasruloa da TviTeuli monakveTi romeliRac kubis

wiboa, amitom kubebis simaravlec usasruloa! III. kent ricxvTa

simravle sasruli rom iyos, maSin iarsebebda udidesi kenti

ricxvi! IV. koncentruli wrexazebis radiusebis sigrZeebi, magali-

Tad, metrebSi nebismieri dadebiTi ricxvis toli SeiZleba iyos.

amitom koncentrul wrexazTa simravle ver iqneba sasruli.

g. 8. 1. 7. 4. I. 1-isa; II. 2-isa; III. 1-isa; IV. 1-isa.

5. {(A, C), (E, D)}. 6. I. B rom sasruli yofiliyo, maSin A-c

sasruli gamovidoda (marTlac, Tuki iarsebebs ricxvi, romelic

metia B-s wevrebis raodenobaze, maSin, cxadia, es ricxvi meti

iqneba A-s wevrebis raodenobazec!); II. Tuki arsebebs ricxvi, ro-


- 108 -

melic metia A-s wevrebis raodenobaze, maSin, cxadia, es ricxvi

meti iqneba A-s nebismieri qvesimravlis wevrebis raodenobazec!

g. 9. 1. mcdaria I (monakveTi moTavsdeba wreSi, romlis diametri

am monakveTis tolia!) da IV (aseTi xazi SeiZleba iyos

agreTve usasrulo texili an mrudi). 2. 2 wrfe da 10 sxivi

(aq Sedis is sxivebi, romelTa saTavec moniSnulia!). 4. nakvTi

dagalkevebuli ar unda iyos. amitom Tuki ori wertili

ekuTvnis nakvTs, maSin am nakvTs unda ekuTvnodes am wertilebis

SemaerTebeli raime xazic, romelic, cxadia, usasrulod bevri

wertilebisgan Sedgeba! 6. aseTi wertili oria!

g. 10. 1. I. SeiZleba; II. SeiZleba; III. ara; IV. ara; V.

ara; VI. ara. 2. magaliTad, cilindri imitomaa

SemosazRvruli, rom igi Caeteva iseT birTvSi, romlis radiusi

cilindris simaRleze 2-jer metia. 3. I. ara; II. ki; III. ki;

IV. ara. 5. 1,2 : 0,8 = 1,5 (lari). gaZvirebis Semdeg _

1,5+1,5·1,2 = 3,3 (lari). 7. I. {6, 4, 3, 9} ⊂ {7, 3, 6, 5, 4, 9};

II. {10, 20, 30} ≡/ {30, 40, 10}; III. {a, i, o, u, e} ⊃ {e, o, i, a};

IV. {maia, mzia, lia, naTia, ia} ≡/

{sandro, irakli, biZina, vaxtangi}; V. {a , a , a , a } ⊃ {a , a , a }.

1 2 3 4 2 3 4

8. davalebis Sesruleba advilia moswavlis saxelmZRvanelos

ydis Siga mxares mocemuli TvalsaCinoebis gamoyenebiT.

Tavi VII. g. 1. 1. 75 : 23 ≈ 3,3-jer. 2. 675-450=225 g-iT meti.

3. xazis sisqea daaxloebiT 0,5 mm = 500 mikr. wertilis

diametris sigrZe iqneba daaxloebiT 1 mm = 1000 mikr.

4. iseTi TvalsaCino magaliTi, rogoric sxivis SemTxvevaSia,

wrfisTvis Zneli mosaZebnia. amitom mosalodneli pasuxebic

nairgvari SeiZleba iyos: eleqtrogadacemis gaWimuli sadeni,

swori gzatkecilis kide, rkinigzis liandagi da sxva.

5. erT. monakv. sigrZea 4 sm : (3,07-3,06) = 400 sm = 4 m.

e) 3,065. 6. (40_4) : 4 = 9 (km/sT).


- 109 -

7. uTvalavi yofiTi sityvaa (aramaTematikuri!) da gulisxmobs,

rom daTvla SeuZlebelia. amitom uTvalavi SeiZleba iyos

sasruli simravlec. magaliTad, dedamiwaze qviSis marcvlebis

simravle. am azriT ufro zogadia usasrulo simravle, radgan

misi daTvla SeuZlebelia.

g. 2. 1. orma monakveTma SeiZleba gadakveTos erTmaneTi erT

wertilSi. or monakveTs SeiZleba hqondes uamravi saerTo

wertili. 2. 4 wrfe. 3. 65 km/sT ≈ 18 m/wm. Tevzis frenis

saUalo siCqarea 4 m/wm. 18 m/wm metia 4 m/wm-ze 4,5-jer, anu

350%-iT. 4. uamravi. erTi. arcerTi. I. uamravi. erTi. uamravi.

II. arcerTi. wrfes bolo ar aqvs. erTi. 6. I. {A, E, B}; II. {E,

C, D}; III. {E}; IV. {F, G}; V. {A, E, B, C, D}. 7. sxivi 1

welSi gairbens 365 · 24 · 3600 · 300000 ≈ 9,5 · 10 11 km-s, rac

9,5 · 10 5 -jer metia 10 6 km-ze. amitom masStabi iqneba 1 : 9,5 · 10 16 .

g. 3. 1. ar iqneba marTebuli: or wertilze uamravi swori

xazi gaivleba! 2. sam an oTx wertilze, sazogadod, swori xazi

ar gaivleba. magram Tu wertilebi isea ganlagebuli, rom maTze

swori xazis gavleba SesaZlebelia, maSin gaivleba uamravi aseTi

swori xazi. wrfis SemTxvevaSi ki Tu gaivlo, mxolod erTi

wrfe gaivleba. 3. I. 1,3 · 105-jer; II. 3 m : 12,5 mm ≈ 2,4 · 102- jer; III. 2 · 108-jer; 4. ar varga imitom, rom uamravia iseTi

sxivi, romelic am or wertilze gadis! magram Tu moniSnulia

sxivis saTave da am sxivis nebismieri sxva erTi wertili, maSin

am ori wertiliT es sxivi ukve calsaxad ganisazRvreba.

5. I. SeiZleba; II. ara (maSin es wertilebi aRar iqneba

gansxvavebuli!); III. SeiZleba. 6. I. ki; II. ara; III. ki; IV. ki;

V. ara (magaliTad, monakveTi arcerT sxivs ar Seicavs). 7.

g. 4. 1. uamrav wertilSi. 2. 3. 4. 0,8 litrian qilaSi

Caeteva 1,5 · 0,8 = 1,2 kg Tafli. amitom 12 kg TaflisTvis saWiroa

10 0,8 litriani qila. 5. 1,5-jer, anu 50%-iT. 6. I. ki; II.

ara; III. ki. 7. I. mcdaria (magaliTad, texilic SeiZleba iyos


- 110 -

SemousazRvreli); II. marTebulia; III. marTebulia; IV. mcdaria

(SeiZleba moicavdes!); V. marTebulia; VI. marTebulia; VII.

marTebulia (radgan nakvTi ar SeiZleba iyos dacalkevebuli).

8. I. yovelgvari swori xazebisTvis (aq gadakveTa ufro yofiTi

azriTaa da ara ise, rogorc maTematikuri termini `TanakveTa~!);

II. mxolod wrfeebisTvis.

g. 5. 2. PQ da AB (wrfe), PQ da AB (sxivi), PQ da BA

(sxivi), PQ da AB (monakveTi), m da NK (wrfe), m da NK

(sxivi), m da KN (sxivi), m da NK (monakveTi), CD monakveTi da

FE monakveTi, DC sxivi da FE monakveTi, CD monakveTi da FE

sxivi, DC sxivi da FE sxivi. 3. orisa. 4. ara. 6. I da IV.

7. ar SeiZleba.

g. 6. 2. I. NP, PQ, ND, CK; II. BP PC, BC, HN; III. AB,

BD, DE, AD, BE, AE. 4. ar SeiZleba. 5. manZili kaspiis zRvidan

mtkvris gavliT aragvis saTaveebamde aris daaxloebiT 600 km.

amitom oraguls dasWirdeba daaxloebiT 12 dRe.

6. I. mcdaria; II. marTebulia; III. mcdaria, mxolod sami ar

SeiZleba hqondes! IV. mcdaria.

g. 7. 1. d || g, d || b, b || g. 4. I. monakveTia; II. sxivia;

III. sxivia. 5. I. iqneba; II. iqneba.

7. maSin 1/x axlosaa 0-Tan, amitom swori pasuxia (e) 6.

8. erT. mon. sigrZea 1,5 : 0,1 = 15 sm. swori pasuxia (e).

g. 8. 2. A3B3 da A4B4 , A1B1 da A2B2 , A6B6 da A5B5 .

4. erT. monakv. sigrZea 6 sm : (18,75 _ 18,25) = 12 sm. e) 18,375

da 18,72. 6. mcdaria I, radgan 3-is jeradi ricxvebis simravle

moicavs 9-is jeradi ricxvebis simravles. amitom is Tviseba, rac

aqvT 9-is jerad ricxvebs, SeiZleba ar hqondes 3-is jeradebs!

g. 9. 1. 2. AB ⊥ AE ; CD || GH ; DH || AE ; EH ⊥ HG ;

AD || BC. 3. 4. 5; 19,5; 4. 5. mcdaria: I, III, IV, V.

6. 3 araqarTuli wigni.

g. 10. 1. d) 3,3 . 2. BC||AD, BC||EK, AD||EK, AB||DC, CF||DK,

CF||AE, DK||AE. 3. aris. 4. mcdaria: I, II, III, IV. 5. 20 t.


- 111 -

7. pasuxi unda SevarCioT miaxloebiTi gamoTvlebis Sedegad. 10dan

80-mde sul 70 ricxvia, maTi saSualo (10+80) : 2 = 45-is

tolia. amitom yvela am ricxvis jami daaxloebiT 45 · 70-is

toli unda iyos. axla gaviTvaliswinoT, rom 3-is jeradia

yoveli mesame ricxvi, amitom vivaraudoT, rom saZebni jamis

mniSvnelobaa daaxloebiT 45 · 70 : 3 = 70 · 15 = 1050. amitom

swori pasuxia (d) 10035.

g. 11. 1. araa marTebuli. magaliTad, pirvel oTxkuTxeds

aqvs es Tviseba, magram araa marTkuTxedi. 2. I. paraleluria;

II. erTmaneTs gadakveTs; III. erTmaneTs emTxveva. 4. ese igi,

saSualod 45 kg rZisgan miiReba 4 kg yveli. amitom 1 t rZisgan

miiReba 4 · 1000 ≈ 90 kg yveli. 5. g) 0,4. 6. usasrulod

45

mravali. 7. marTebulia: II, III, VI.

Tavi VIII. g. 1. 2. z-RerZis - 3,4 sm : 0,001 = 3400 sm = 34 m;

x-RerZis - 3 sm : 800 ≈ 0,0036 sm; y-RerZis - 3 sm : 2700 ≈ 0,001 sm.

7. I. ara; II. ki. eqvskuTxedis SemTxvevaSi diagonalebi SeiZleba

iyos erTmaneTis paraleluri, SeiZleba _ marTobulic.

g. 2. 1. 1 sm-iT marjvniv wanacvlebiT koordinati

matulobs 0,01 erTeuliT. amitom, magaliTad, T1 (123,45)-dan

marjvniv 1 sm-is daSorebiT SeiZleba moiniSnos wertili

A1 (123,46), xolo marcxniv _ B1 (123,44). 2. 1 sm-iT marjvniv

wanacvlebiT koordinati matulobs 100 erTeuliT. amitom,

magaliTad, T2 (123,45)-dan marjvniv 1 sm-is daSorebiT SeiZleba

moiniSnos wertili A2 (223,45), xolo marcxniv _ B2 (23,45).

3. 120 kg. 4. I. BH-is paraleluria NP, PD, ND, CK; BH-is

marTobulia NH, BP, PC, BC, DK. II. EK-s paraleluria AH, HM,

AM; EK-s marTobulia ED, DB, BA, EB, DA, EA, CT, NC, NM,

NT, NC, TM. III. NC-s paraleluria AB, BD, DE, AD, BE, AE;

NC-s marTobulia AH, HM, AM, EK, KT, ET. 7. d) 9 · 8 · 7 · 5.

g. 3. 1. tolia, radgan urTierTmarTobuli wrfeebi


- 112 -

erTmaneTis simetriis RerZebia. meoTxed nawils. 2. I. e) 1 sm :

2 m ; II. 10 rigi, rigSi – 12 skami; III. scenis miaxloebiTi

zomebia naxazze 0,2 sm × 3,8 sm. sinamdvileSi iqneba 0,4 m × 7,6

m. amitom farTobi iqneba daaxloebiT 3 kv.m. IV. darbazis

miaxloebiTi zomebia naxazze 5,8 sm × 4,5 sm. sinamdvileSi iqneba

11,6 m × 9 m. amitom darbazis moculoba iqneba 11,6 · 9 · 5,5 ≈ 574

(kub.m). V. (1; 7), (2; 4), (2; 12), (4; 9), (4; 11), (6; 12), (7; 4),

(7; 9), (8; 1), (8; 2), (8; 10), (9; 2), (9; 9), (10; 1), (10; 5), (10; 10),

(10; 11), (10; 12).

3. C1-is koordinatia 4, C2-is _ 3. O-dan C-Si misasvleli

kodia, magaliTad, O ~

⎛ 4

3


⎜ ⎯ ⎯→ ↑ ⎟ ~ C . mis Casawerad sakmarisia

⎝ ⎠

ori ricxvi. 4. 1 sm-iT marjvniv wanacvlebiT koordinati

matulobs 0,001 erTeuliT. amitom, magaliTad, M(263,6)-dan

marjvniv 1 sm-is daSorebiT SeiZleba moiniSnos wertili

E(263,601), xolo marcxniv _ F(263,599). 5. 1 sm-iT marjvniv

wanacvlebiT koordinati matulobs 1000 erTeuliT. amitom,

magaliTad, M(263,6)-dan marjvniv 1 sm-is daSorebiT SeiZleba

moiniSnos wertili E(1263,6), xolo marcxniv _ F(–736,4).

g. 4. 1. B(2; 0), D(2,5; 1), O(0; 0), F(0; 1,5), K(9; 2), M(9; 3),

L(6; 2). 3. A da G, B da F, C da K, H da P, L da M.

5. axlos: N(0,08 ; 1), Sors: K(10 ; 112,1).

g. 5. 1. P(3,5; 2) I, Q(1,5; 4) II, J(1,5; 0) II (an III), R(4; 3)

III, T(4,5; 2) IV. 4. kvadratisa. 5. mravali wertilis

dasaxeleba SesaZlebelia I da II SemTxvevebSi.

7. mcdaria: V. pasuxi Seicvleba V da VI SemTxvevebSi.

g. 6. 1. I. E(4,6; 2,3) IV, F(2; 2,4) I, G(3,4; 0,6) I, H(2,2; 2) II,

K(2,6; 1) II, L(2,2; 1) III, M(2,1; 2,7) III, N(1; 2,3) IV. II. P(2,7; 0)

I (an IV), Q(0; 0,5) I (an II), R(0,5; 0,5) II, S(1,5; 0,8) III, T(0,6; 2,5)

I, U(0; 1,7) III (an IV), V(2; 1) IV, W(1,8; 1,5) IV. 3. ≈7,4%.

4. 19,7%. 5. {(0; 7)I, (2; 4)I, (1; 4)I, (3; 1)I, (1; 1)I, (4; 2)IV,

(1; 2)IV, (1; 4,5)IV, (1; 4,5)III, (1; 2)III, (4; 2)III, (1; 1)II, (3; 1)II,


- 113 -

(1; 4)II, (2; 4)II }. {(7; 1)I, (8; 1)I, (8; 2)I, (9; 3)I, (10; 4)I}.

g. 7. 2. A(0,5; 3,6)I, B(1,6; 0,8)II, D(2,7; 2,7)III, F(2,6;

2,7)IV, C(2,4; 1,5)I. 3. I. 4 l; II. 1,7-2 kg. 5. 54 lari.

6. I. pirveli; II. meore.

g. 8. 1. A(30; 50)I, B(20; 60)II, C(23; 90)III, D(25; 50)IV, E(2;

2)I, F(1; 2,5)II, G(1,6; 2)III, H(2,1; 1,3)IV, K(2; 2,5)II, L(1,5; 2,7)III,

M(1,1; 0,5)II, T(1,7; 3,2)I. 2. I. ≈ 4/7 mm; II. ≈ 1/6 mm. 3. 178,2 l.

5. cxadia, 4y = 50, amitom swori pasuxi iqneba: (d) 55,5.

6. I. saydris _ (120; 70)II; II. Zveli eklesiis _ (165; 70)I; III.

mwyemsebisa _ (50; 30)III. gareubani _ (180-240; 20-40)IV.

soflidan Zvel eklesiamde: (120; 25)II, (45; 15)II, (120; 20)IV,

(195; 20)I. am gzis sigrZea rukaze daaxloebiT 12 sm. amitom

sinamdvileSi iqneba _ 24 km. avtobusi gaivlis am manZils

daaxloebiT 50 wT-Si.

g. 9. 1. E - 180° ag da 60° Cg, F - 30° ag da 70° Cg, G - 0° ag (an

dg) da 45° sg, H - 50° dg da 0° Cg (an sg), J - 42° ag da 0° Cg (an

sg), K - 30° dg da 45° sg, L - 15° dg da 60° Cg, M - 70° ag da 40°

sg, N - 110° ag da 0° Cg (an sg). 2. I. sankt-peterburgi;

II. axali orleani; III. kito. 3. I. q. Tbilisi - 45° ag da 42° Cg;

II. q. moskovi - 38° ag da 57° Cg; III. q. londoni - 0° dg da 52°

Cg; IV. q. brazilia - 42° dg da 15° sg; V. q. sidnei - 151° ag da 33°

sg; VI. q. los-anJelesi - 118° dg da 34° Cg. 4. 1 wlis Semdeg

_ 103 lari; 2 wlis Semdeg _ 106,09 lari.

g. 10. 2. I. soxumi; II. ozurgeTi; III. Sxara;

IV. dedofliswyaro. 3. I. 43,6° da 41,6°; II. 44,5° da 42,8°;

III. 40,7° da 43,2°; IV. 45,9° da 41,9°; V. 44° da 42,2°;

VI. 41,8° da 42,5°. 5. I. 8%; II. marili - 500 g, niori

- 300 g, Zmari - 40 g, mwvanili - 160 g.

Tavi IX. g. 1. 3. cila - 2 kg, cximi - 1 kg, saxamebeli - 1,45

kg. 4. zRarbi - 32°-iT, Tria - 29°-iT, zazuna - 34°-iT, Ramura -

32°-iT. 5. dedamiwa sicocxlis gareSe yofila ≈ 4,5 _ 3,55 =

0,95 miliardi weli. amitom pasuxia 4,45 : 0,95 ≈ 3,7-jer.


- 114 -

6. oTaxSi myofTa asakTa jamia Tavidan 8 · 24 = 192 weli, Semdeg

_ 7 · 22 = 154 weli. amitom pasuxia 192 _ 154 = 38 weli.

g. 2. 1. I. ianvarSi - 10 aTasiT, aprilSi - 10 aTasiT,

ivnisSi - 30 aTasiT; II. TebervalSi - 20 aTasiT, martSi - 15

aTasiT, maisSi - 20 aTasiT.

2. I. momatebula 105 aTasi ha-Ti; II. yvelaze didi - 1940 wels

(1910 welTan SedarebiT 160 aTasi ha-Ti meti iyo), yvelaze

mcire - 1980 wels (1910 welTan SedarebiT 280 aTasi ha-Ti

naklebi iyo). III. 27 aTasi ha; IV. ar SeiZleba, radgan ar

viciT ramdeni iyo TviTeulis farTobi 1910 wels!

3. 1° siTbo (pasuxi SeiZleba miviRoT orgvarad: 1) TandaTan

viangariSoT Sesabamisi temperaturebi: 1° siTbo, 0°, 1° siTbo, 2°

siTbo, 1° siTbo, 0°, 2° yinva, 1° siTbo; 2) jer gamovTvliT,

rom dRis 12 sT-ze temperatura wina Ramis 12 sT-Tan SedarebiT

Seicvala -2°-iT, anu Semcirda 2°-iT. amitom saboloo

temperatura iqneba 3°_2° = 1° siTbo).

4. (7; 4), (5; 6), (6; 9), (8; 12), (9; 15), (11; 18), (14; 18), (17;

16), (20; 13), (23; 11), (22; 10). 5. I. gogona - 550; vaJi - 537.

II. 1976, 1978, 1979 wlebSi; III. 1971, 1973, 1975, 1977 wlebSi.

6. 150-200 kg. 15-20 kg-is saSualoa 17,5 kg. ese igi, 100 kg

Warxlisgan saSualod miiReba 17,5 kg Saqari. amitom 60 kg

Saqris misaRebad saWiro iqneba ≈340 kg Warxali.

g. 3. 1. yvelaze bevri fuli eqneba noembris bolos,

yvelaze didi vali _ ivnisis bolos. 2. I. (d) 5,5; II. (a) 1,2;

III. (e) 1,6. 3. I. II-Si 10 aTasi, III-Si 5 aTasi; II. IV-Si 15 aTasi;

III. ixeira, radgan winasTan SedarebiT qoneba gaizarda; IV.

TebervalSi 15 aTasi lariT. 4. I. 0 lari; II. 10 lari vali;

III. 20 lari vali. 5. cvlilebaa -2°. diliT - 2° yinva. 6.

sokos wona mcirdeba saSualod 80 %-iT. amitom: (I gza) 20-30

kg axali sokos wona Semcirdeba 16-24 kg-iT. ese igi miiReba 4-6

kg (20_16=4, 30_24=6). (amoxsnis II gza) darCeba wonis 20%.

Sesabamisad, 20-30 kg axali sokosgan miiReba 4-6 kg soko.


- 115 -

g. 4. 1. I. +7, +3,1, 3,1, 1/5, 7; II. –7, –3,1, –1/5; III. +0,

–0, 0. tolebia: +7 da 7, +3,1 da 3,1, +0, –0 da 0.

2. I. 550 l – 400 l = 150 l; II. 550 l – 550 l = 0 l;

III. 550 l – 650 l = –100 l; IV. 50 l – 200 l = –150 l.

3. an 5,5 m simaRleze, an 0,5 m-ze. 4. I. 180,5; II. 19,5; III. 0;

mdebareobs A0-s marjvniv. 5. +4°. +22,5°; _8° 6. _24 sm.

g. 5. 1. D(–0,3), E(+0,7), F(+2,2), K(–2,3), G(+1,6), J(+0,7),

H(–0,5), T(–1,7) . 3. 1962 wels. 4. 1871 wels. 5. I. marjvniv;

1 sm; C wertils; II. marcxniv; 3/4 sm; F wertils;

III. marjvniv; 75 sm; D wertils; IV. marjvniv; 3/800 sm; E

wertils. 6. I. 13; II. –16; III. –900; IV. –50.

g. 6. 1. daviT aRmaSenebeli - 52 weli; oqtaviane avgustusi

- 14+63–1=76 weli (1 weli imitom gamoaklda, rom `nulovani~

weli ar arsebobs!). I msoflio omi grZeldeboda 4 weli,

spartakis ajanyeba - 3 weli. 2. unda eTqva: _ es moxda Zveli

welTaRricxvis me-4 saukuneSi. 3. im droisTvis, cxadia, Zveli

welTaRricxvis cneba ar arsebobda! 4. B1 (–9,5), B2 (6,3), B3 (0,1),

B4 (–256,03). 5. I. –1. 7. mag.: –12 da –10, –13 da –9 da a.S.

g. 7. 3. dabadebula Zv.w. 5 w., gardacvlila ax.w. 66 w.

4. {–0,5; 1/2}, {0,02; –1/50}, {1,3 ; –13 /1 0 }.

6. I. 3 milion 800 aTasi; II. 700 aTasiT; III. 5-jer.

7. I. C-Si; II. D-Si; III. E-Si.

g. 8. 2. raki 1600 weli XVII saukunesaa mikuTvnebuli,

amitom dekarti dabadebula 1596 wlis martSi, xolo

gardacvlila _ 1650 wlis TebervalSi. 3. unda eTqva: `Zv.w.

VIII-VI saukuneebSi~. 4. I. 400 l + 150 l = 550 l; II. –1000

l + 1500 l = 500 l; III. –50600 l + 40000 l = –10600 l;

IV. –15000 l + 10000 l = –5000 l. 5. A(1), B(–5), C(–1,5).

6. mcdaria III. marTlac: 0-is mopirdapire araa uaryofiTi,

magram 0 araa uaryofiTi. 7. amocanis pirobaSi mocemuli Tvalsazrisis

gaziarebis SemTxvevaSi I saukuneebi (Zv.w. da ax.w.)

`nakiani~ ar iqneboda; Tu nulovani weli iarsebebda, maSin ax.w.


- 116 -

I saukune aRar iqneboda `nakiani~ (gagrZeldeboda 100 wels da

damTavrdeboda 99 wels), xolo Zv.w. I saukune ki mainc `nakiani~

darCeboda (daiwyeboda -99 wels da damTavrdeboda -1 wels).

g. 9. 2. I. 3,7; II. 7,8; III. 0; IV. 200. 3. I. 12 da –12;

II. 41,09 da –41,09. 4. waTe I mefobda ax.w. 510-530 wlebSi,

iuliusi cxovrobda Zv.w. 100-44 wlebSi, ovidiusi _ Zv.w. 43 -

ax.w. 17 wlebSi, filoni _ Zv.w. 25 - ax.w. 45 wlebSi. 5. I.

waTe I mefobda ax.w. VI sauk. I naxevarSi; II. iuliusi _ Zv.w. I

sauk. I da II meoTxedebSi; III. ovidiusi daibada Zv.w. I sauk.

III meoTx. da gardaicvala ax.w. I sauk. I meoTx.; IV. filoni _

Zv.w. I sauk. IV meoT. da ax.w. I sauk. I da II meoTxedebSi.

6. I. A; II. C; III. F; IV. M. 7. I. pirvels: 1700 l > 1100 l;

II. meores: –1700 l < 0 l; III. pirvels: –200 l > –1000 l.

g. 10. 3. I. 25,5 > 3,4; II. 4,5 > –800; III.–36,7 > –84,3;

IV. –7,5 < –7,2; V.–1025 < 0. 4. I. –4,3 < 0; II. 27,1 >0;

III. a < 0; IV. c ≤ 0; V. b > 0; VI. d ≥ 0. 5. I. 8;

II. 150; III. 7,6. 6. 7. 7. I. x = 9 an x = –9; II. y =

0 ; III. aseTi z ar arsebobs! 8. 500 t madnisgan miiRebdnen ≈

32,5 kg spilenZs, 600 t madnisgan _ ≈ 25,5 kg-s. sxvaobaa 7 kg.

g. 11. 1. I. ki; (mag.: 3); II. ara; III. ki (mag.: 3,7); IV. ki

(aseTia 0); V. ki (mag.: 3); VI. ki (mag.: –3,7). 2. rkina, nikeli,

spilenZi, vercxli, alumini, tyvia, kala, naftalini, gayinuli

glicerini, gayinuli vercxliswyali, gayinuli acetoni.

acetoni. 3. I. 11,3; II. –46,9. 4. I. –5083 > –5183;

II. –15,44 > –25,44; III. –999,0 > –999,1; IV. –90,701 <

–90,602. erTaderTi cifris Caweraa SesaZlebelia I SemTxvevaSi.

6. II. mcdaria (mag., roca m = 7, n = –7); IV. mcdaria

(magaliTad, roca m = –7, n = –6); VI. mcdaria (mag., roca m = –

7, n = –8); VII. mcdaria (magaliTad, roca m = –7, n = 6);

7. I. 0; II. a; III. –a. 8. I. 0 < x; II. y> 0; III. m < 0;

IV. 0 > n; V. x > m; VI. n < x; VII. y > n; VIII. m < y.


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Tavi X. g. 1. 1. I. {1; 3,0; 50000; 43}; II. {1; 3,0; 50000;

43; 0; –19; – 30

; –345,00}; III. { 3

1

; –2,5;

6

; –0,05 ; – 12

;

6

5 5

5

0,7}; IV. {–19; – 30

; –345,00; –2,5; –0,05 ; – 12

}. 2. I. –4;

6

5

II. –5, –4, –3, –2, –1, 0, 1, 2. 3. marTebulebia: I, II, III, VI,

VIII, IX, XII. 4. I. –607; II. 30,04. 5. mag.: {Savi, lurji,

TeTri, yviTeli, yavisferi, nacrisferi, iasamnisferi, narinjisferi}.

6. araa igive! SeiZleba ar iyo boroti, magram arc keTili iyo!

7. mcdaria: I (ekuTvnis niSani simravleebs Soris ar daismis!),

III (dadebiTi mTeli ricxvebic xom arsebobs!), IV (yvela

dadebiTi ricxvi araa naturaluri!). 8. I. uaryofiTi;

II. dadebiTi; III. nuli; IV. an dadebiTia, an nulia.

g. 2. 2. I. {0, 1, 2, 3, 4, 5}; II. {–2, –1, 0}. 3. I. 2100

wlis 1 ianvars; II. 1900 wlis 1 ianvars; III. 900 wlis 1

ianvars; IV. 1599 wlis 31 dekembers; V. Zv.w. 99 wlis 1

ianvars; VI. Zv.w. 700 wlis 31 dekembers. 4. I. daibada Zv.w.

VII sauk. IV meoTxedSi, gardaicvala Zv.w. VI sauk. III meoTx.;

II. daibada ax.w. VI sauk. III meoTx., gardaicvala ax.w. VII sauk.

II meoTx. 5. mcdaria: II (wrexazis centri ar ekuTvnis

wrexazs!), III (sferos centric ar ekuTvnis sferos!). 6.

7. I. aseTia 0; II. ar arsebobs! III. ar arsebobs! IV. aseTia 0.

g. 3. 1. mtkvar-araqsis kulturis `asaki~ gamodis daaxloebiT

3,5+2=5,5 aTaswleuli, xolo kreta-mikenisa - 2+2=4 aTaswleuli.

amitom pirveli Zvelia 5,5 : 4 ≈ 1,4-jer. 2. I. N(2); II. A(–

6); III. toladaa daSorebuli. 3. I. 2000 wlis 1

ianvars; II. 1000 wlis 1 ianvars; III. 999 wlis 31 dekembers;

IV. Zv.w. 1 wlis 31 dekembers; V. Zv.w. 1000 wlis 31 dekembers;

VI. Zv.w. 2999 wlis 1 ianvars. 4. {4} ⊂ {3} ⊂ {1} ⊂ {2} ⊂ {5}.

5. N ⊂ Z ⊂ Q, N ⊂ P ⊂ V ⊂ Q , U ⊂ T ⊂ Q.

6. sul 13-isa (1.1.11; 11.1.11; 1.11.11; 11.11.11; 2.2.22; 22.2.22;

3.3.33; 4.4.44; 5.5.55; 6.6.66; 7.7.77; 8.8..88, 9.9.99).

g. 4. 1. I. 1 miliardi weli; II. 4 aTaswleuli; III. 4,5


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aTaswleuli; IV. Zv.w. XV saukuneSi; V. 35 saukune; VI. 65

saukunis win; VII. 9 aTaswleulis win; VIII. 900 000 wlis

win. 2. qristes dabadebidan Cvens xanamde daaxloebiT 2000

welia gasuli. amitom Sesabamisi wertili daSorebuli unda iyos

Cveni xanis Sesabamisi wertilidan (2) skalaze

2000

≈ 0,

013 sm-

150000

iT, xolo (1) skalaze _

2000

= 1 sm. aseTi mcire

300000000 150000

manZilebis mozomva skalaze, cxadia, SeuZlebelia!

3. mcdaria: I, IV, VI, VII, VIII, IX, XII. 4. Tanamedroveni iyvnen.

ufrosi iyo platoni, ufro adre gardaicvala evdoqse.

7. 1 lomi ~ 200 kg, 1 Cliqosani ~ 200 kg, 1 foToli ~ 20 t.

g. 5. 1. 195 lari vali anu –195 lari. 2. dabadebula

savaraudod Zv.w. 64 wels, Zv.w. I saukuneSi, gardacvlila ax.w. I

saukuneSi. aTaswleulTa cvlas moeswro. 3. daRupula –212

wels, Zv.w. III saukunis IV meoTxedSi. 4. I. 1; II. 0 an 2.

5. Tbilisi +4,5°, manglisi +0,5°, TianeTi –2°, quTaisi - 6°.

4,

5+

0,

5−2

+ 6

saSualo temperaturaa = 2,

25°

.

4

6. I. 2200 l; II. 1200 l; III. 0 l. 7. I. 250 l; II. –50 l;

III. 200 l. 8. I. 30 l + (–20) l = 10 l; II. 3000 l + (–2000)

l = 1000 l; III. 3000 l + (–4000) l = –1000 l.

g. 6. 1. I. _20; II. _96,6; III. _3/13; IV. 6,5; V. –3,5;

VI. –7,5; VII. 2,5. 2. I. –10; II. –0,1; III. –2; IV. –1,1.

3. I. = –3,75 + 12,5 = 8,75; II. = 16,7 – 34,18 = –17,48.

4. I. mcdaria; II. marTebulia; III. marTebulia.

6. I. _ 6a; II. –1,2m; III. 0,9v; IV. _a; V. 2k.

g. 7. 2. dabadebula Zv.w. 276 wels, Zv.w. III saukuneSi,

gardacvlila Zv.w. II saukuneSi. aTaswleulTa cvlas ver

moeswro. 3. –5,5°. 4. I. –80 l + (–20) l = –100 l;

II. –50 dol. + (–30) dol. = –80 dol.

5. = 2 · 67,24 – 20,5 + 25,8 – 10 = 134,48 – 4,7 = 129,78. 6. a.

7. =

108000000

=

1080

= 6 m. 8. mcdaria: II, IV.

2400000·

7,

5 180


g. 8. 1. I. –3; II. ( 4

2)

- 119 -

− ; III. –22; IV. –2,2; V. –5,6;

3

VI. –12,8. 2. I. = –56 + 23 + 6 = –27; II. = 64–8,4–2,4–3,2 = 50.

4. gaaCnia a-s: SeiZleba metic, tolic da naklebic! 5. mefobda

65 wels, dabadebula Zv.w. 394 wels, gardacvlila Zv.w. 302

wels, Zv.w. I aTaswl. II naxevarSi. 6. I. –3,1; II. –15,5; III. 0;

IV. –1,5; V. –5. 7. gaxmobis Semdeg darCeboda 4,8 kg Ciri. 1 kg Ciri

daujdeboda 12 : 4,8 = 2,5 (lari). dauzogavs 4,8 · 0,5 = 2,4 l.

g. 9. 2. mag.: I. –5,8 = –2,8 + (–3); II. –5,8 = 1,2 + (–7).

3. yoveli racionaluri ricxvi SeiZleba iyos an uaryofiTi,

an nuli, an dadebiTi. 4. dabadebis dRemde: +6 lari, dabadebis

dRis Semdeg +10 lari. 4 lari valis daklebiT misma qonebam

moimata 4 lariT. 5. I. yvelas meore koordinati 0-is tolia;

II. yvelas pirveli koordinati 0-is tolia. 6. Raribi valisgan

gaTavisufldeboda orive SemTxvevaSi. mdidris qoneba mogebis

Sedegad Semcirdeboda 10 oqroTi! 8. I. 8; II. 20; III. 10.

g. 10. 1. I. 13; II. –12; III. 4

3

; IV. –2,5; V. –10,36;

17

VI. – 1,3; VII. − 7

3

. 2. I. = 1,64–2,1–5,73 = 1,64–7,83 =

7

–6,19; II. = 22+10 = 32. 3. I. 5; II. 0; III. 0,5; IV. –2.

4. I. diddeba; II. mcirdeba; III. mcirdeba; IV. diddeba.

7. I. 120 aTasiT; II. 5/4-jer; III. sul qarTveli mosaxleoba

yofila 16+1,5+20+7,5+0,5+11+0,5 = 57 `kacuna~, amitom

megrelebi iqnebodnen 11 · 100 ≈ 19 %.

57

g. 11. 1. I. = –8,81+6,42–6,4 = –8,81+ 0,02 = –8,71; II. = –2,8 +

6,38 – 7,9 = 6,38 – 10,7 = –4,32. 2. –7,1. 3. I. 0° – 4° = –4°;

II. –6,5° – 5° = –11,5°; III. –6° – 8,5° = –14,5°; IV. –3° – 0° = –3°.

4. mcdaria: II, IV, VI. 5. marTebulia: I, VII, VIII. 7. mcdaria II, V.

Tavi XI. g. 1. 1. |EB|=7 sm, |BC|=4 sm, |AF|=5 sm. 2. mesame

gverdis sigrZe a unda akmayofilebdes samkuTxedis utolobas:

5,5 m < a < 6,9 m. amitom es sigrZea 6 m. 3. I da II sawarmoebis


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SemaerTebeli monakveTis SuamarTobis mdinaresTan gadakveTis

wertilSi. 4. I. –18; II. – 16,6; III. –5; IV. –3,5; V. 25;

VI. –

5

. 5. I. z = –12 ; II. x = 0; III. y = 2; IV. z = –8.

7

g. 2. 1. I. E, K, F; II. M, T, N. 5. I. = – 4,4 – 12 = –16,4;

II. = – 2,3 – 24,8 + 20 = –7,1. 6. raki |BC| = 14 sm, amitom |BE| +

|EC| = 40 – 14 = 26 (sm). magram |BE| = |AE|, amitom |AE| + |EC| = 26

sm, anu |AC| = 26 sm. 7. mcdaria: I, III, IV, VI.

g. 3. 4. = 3 · 16 + 22,5 – 5 = 48 + 17,5 = 65,5. 5. am monakveTis

SuamarTobs, garda am monakveTis Suawertilisa (monakveTis

boloebiT da SuawertiliT samkuTxedi ar Seiqmneba!). 7. 120 kg.

g. 4. 2. |EK| = 0,75 sm. 3. ara (samkuTxedis utoloba ar

sruldeba). 4. I. –20; II. 2,4; III. –2,6; IV. –12,2; V. 18,6;

VI. 56,8. 5. I. –14a; II. –0,4 k; III. –4x; IV. – 8b; V. 8y.

6. gavavloT ori qorda. wrexazis centri iqneba maT Suamar-

TobTa gadakveTis wertili.

g. 5. 1. 1 sm. 3. a. 4. LB monakveTi. 7. 1 m.

g. 6. 3. aseTi wrfe mravali gaivleba. 4. I. ara; II. ki;

III. ki; IV. ki. 5. 1. Tbilisi; 2. basra; 3. bombei; 4. kolombo;

5. kalkuta; 6. deli; 7. novosibirski; 8. moskovi. 6. I. AB

marTobia, AD - daxrili; II. AD marTobia, AB – daxrili.

g. 7. 1. Q(3; 3), S(–4; 2), D(–1,5; –2,5), K(3,5; –2).

3. I. {A, B, E}; II. {A, F, D}; III. {D, G, C}; IV. {B, H, C};

V. {B, E, A, P}; VI. {A, B, E, P, T, K, M, F, H, L}; VII. yvela, garda

A, F da D wertilebisa; VIII. ∅. 4. 1) A, C, D;

2) A, C, D, P, K, O; 3) B, E, L, O; 4) F; 5) H, P, O; 6) A, B,

C, D, E, P; 7) A, C, D, G, K. 5. I. meore koordinati metia an

toli 0-ze; II. pirveli koordinati naklebia an toli 0-ze.

g. 8. 2. I. x-RerZis wertilebs; II. y-RerZis wertilebs;

III. koordinatTa saTaves. 3. I. (4; 0); II. (0; 4); III. (–3; 1);

IV. (1; –2). 4. I. = 1 + 2,6u = 14; II. = 3 · 125 – 30 – 9,3 – 1 – 4

= 375 – 44,3 = 330,7. 5. nakvTis sazRvari Sedgeba 4 naxevarwrexa-


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zisgan. aqvs simetriis 4 RerZi.

6. dReRameSi: 7000 · 60 · 24 = 10080000 kub.sm ≈ 10000 litri.

weliwadSi: 3650000 litri = 3650 kub.m.

g. 9. 1. |DB| = |BC| = |AC| = |AD| = 5 sm. 3. I. {C, E, F, M, K};

II. {A, B, D, N, H, O}; III. {C, D, M, H}; IV. {A, L, O}; V. {C, D,

M, H}. 4. I. 18; II. 62; III. 9,4; IV. –8,6; V. –11,4; VII. –

15,6. 5. I. = 48,2 – (–45) = 93,2; II. = –1,6 + 6,9 + 3,8 – 9 = 0,1.

6. I. ki; II. ki; III. ara. 7. araa toli.

Tavi XII. g. 1. 1. A(0,2), B(–0,8), C(–0,1). 2. I. 0; II. 0; III.

k 3 . 3. I. A1 (2102,5); II. A2 (–102,5); III. A3 (97,5); IV. A4 (–302,5).

4. I. A1 (–500,5); II. A1 (0,5). 6. b) 25. 8. mcdaria II da V.

g. 2. 1. = –9,3. 2. 100. 3. I. –3+y; II. a + 47;

III. –100 + x; IV. –14; V. –33 – a. 4. I. x = –8,3; II. x = 4.

5. p – q = p + (–p). 6. I. uaryofiTisTvis; II. dadebiTisTvis;

III. dadebiTisTvis. 7. e) 569 000. 8. g).

g. 3. 1. I. 49; II. –38; III. –6; IV. 7,8. 2. I. = –67 – a =

–16,4; II. = x – y – 3 = –34,8. 3. I. 9,8; II. –3. 4. mtkvararaqsis

_ 50-55, TrialeTis _ 35-40, kolxeTis _ 25-35

saukunis win. 5. sanam xuro urmiT naxevar gzas gaivlis,

mWedels SeeZleba mTeli gzis gavla. magram mas ukve hqonda

raRac manZili gavlili. amitom ufro adre Cava mWedeli.

7. mcdaria II da III.

g. 4. 3. I. –3; II. 6; III. 38. 4. I. 2,4; II. –24,9; III. a+k;

IV. 0. 5. I. x = 1,25; II. x = –4. 6. mecnierma. 770 km.

g. 5. 2. I. –4; II. –5; III. 8,1; IV. –32,2.

3. II. –8,7; III. 0; IV. b. 4. I. x = 42; II. y = 29/9.

5. perimetri – 28 erT.; farTobi – 48 kv. erT. 6. I. 0-s, 1-s

an 2-s; II. 0-s, 1-s, 2-s; III. 0-s, 1-s an 2-s; IV. 0-s, 1-s, 2-s,

3-s an 4-s; V. 0-s an 1-s. 7. I. 0; II. nebismieri dadebiTi ricxvi.

g. 6. 1. I. somexi; II. azerbaijaneli; III. –Ze; IV. imerelraWvel-leCxumel-guruli.

2. mag.: I. ia, vardi, ia, ia, ia, ia,


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SroSani, vardi, yayaCo, gvirila; II. ku, ku, ku, ku, ku, ku, ku,

ku, ku, ku. 3. CamonaTvalis `moda~ gviCvenebs, romelia am

CamonaTvalis yvelaze xSiri wevri. 4. I. 1; II. –0,3.

5. I. 3; II. – 2a – 7; III. – 7; IV. y. 6. I. ara; II. ki.

frCxilebis gaxsnis wesi modulebisTvis araa marTebuli. 7.

CamonaTvalSi wevrebis gameoreba dasaSvebia, erTobliobaSi _ ara.

g. 7. 1. Tuki ori ricxvis jami 0-ia, maSin isini sxvadasxvaniSnianebia,

mastan maTi modulebi toil unda iyos. amitom isini

mopirdapire ricxvebi iqnebian! 2. TviTeul SemTxvevaSi

vaCvenoT, rom mocemuli ricxvebis jami 0-is tolia!

3. I. x = –2; II. x = 4; III. x = –11; IV. x = –15. 4. I. varskvlavi;

II. kata; III. qarTuli ena. 6. marTebulia: II, IV, V.

7. azoti _ 3,9 kg, Jangbadi _ 1,05 kg, danarCeni _ 0,05 kg.

g. 8. 1. I. x = –9; Z-s. II. x = 8,45; Q-s. III. x = –4,37; Q-s.

IV. x = –6,99. Q-s. 2. –15. 6. I. ar aqvs; II. aqvs; III. ar aqvs.

7. I. perimetri = 12 erT., farTobi = 9 kv.erT.; II. perimetri

= 20 erT., farTobi = 25 kv.erT.; 8. I dRes - 20%, meore

dRes – 80 · 30% = 24%. darCa – 56%. ese igi, misatani

wignebis raodenobis 56% Aaris 42. amitom pasuxia: 75.

g. 9. 2. A2B2C2 , A5B5C5 . 4. Tanabradaa daSorebuli.

6. mcdaria: II, VI, VII. 9. marTebulia: I, II, IV, V.

g. 10. 1. Q Tanabradaa daSorebuli A da D wertilebidan.

amitom is AD-s SuamarTobze mdebareobs. 2. paralelobasa da

marTobulobas Soris kavSiris Tanaxmad, erTi wrfis marTobuli

ori wrfe erTmaneTis paraleluria. 3. ar SeiZleba, radgan ma-

Sin gamovidoda, rom a da b wrfeebi erTmaneTis paraleluria

(paraleluri wrfeebis marTobuli wrfeebi xom erTmaneTis paraleluria!).

5. kveTs erT wertilSi. 6. I. 8,3; II. –0,3;

III. –57. 7. I. – 2y – 7; II. – 2a – 2b + 8; III. –7; IV. 0,6 – y.

8. wertilebi. 9. mag.: I. x +2 = 8; II. x +2 = 1; III. 3x = 8.

g. 11. 2. marTkuTxedis diagonalebis gadakveTis wertili


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iqneba Semoxazuli wrexazis centri. 3. ara, radgan FG da EH

gverdebis SuamarTobebi saerTod ar gadakveTs erTmaneTs.

4. perimetri = 20 erT., farTobi = 21 kv. erT.

5. I. M; II. B; III. B. 7. paraleluria!

g. 12. 1. I. d; II. 2; III. – 2a. 4. aqvs. I. 0,5 sm; II. 0,25 sm;

III. 0,5 mm; IV. 0,00005 mm. 5. 2 mm, 1 mm, 0,5 mm da a.S.

monakveTi wertilTa usasrulo simravlea. 6. mcdaria: II, III,

V, VI, VIII, X, XII. 7. 5227 wels. aRmoaCines XIX saukunis

III meoTxedSi, daumarxavT Zv.w. XXXIV saukunis II meoTxedSi,

IV aTaswleulis II naxevarSi. 8. mcdaria: I, III.

Tavi XIII. g. 1. 1. I. 2; II. 9. 2. |MN| = 8 erT. 3. I. 14,2

erT.; II. 4,76 erT. 4. I. –6; II. –6; III. –15; IV. –26,73.

5. pirvel SemTxvevaSi cvlilebaa 7 aTasi lari, meoreSi – 15

aTasi. mewarmis qonebis cvlileba Tvis ganmavlobaSi orive

SemTxvevaSi tolia misi saboloo (Tvis bolos) qonebasa da

Tavdapirvel (Tvis dasawyisis) qonebas Soris sxvaobis. 6. I. x =

11; II. y = –18,06; III. z = –0,9. 7. qarTuladac da rusuladac

laparakobs mcxovrebTa (65 + 55) – 100 = 20 %. amitom

mxolod qarTulad ilaparakebs 65 – 20 = 45 %.

g. 2. 1. I. –6,1; II. 6,1; III. 2,1. 2. I. 0,9; II. –1,2; III.

10,3; IV. –3,7. 4. –90. 7. –1,7 da –9,7.

8. I. a–7; II. –x–9,2; III. –6+c–k; IV. 4 – n + m.

g. 3. 1. –5°c. 2. 76 m. 3. I. –0,5 mln. lari; II. 2 mln., 1

mln.; III. 0,5 mln., 1,5 mln.; IV. 1) 1,5 mln.; 2) 1 mln.; 3) 0,5

mln. 4. ufro zogadia tolferda samkuTxedi. 5. I. –4 da –

3; II. 5 da 6; III. –1 da 0. 6. I. x = 1,7; II. x = –18,99.

7. cvlilebaTa jamia 40 m. raki mTliani cvlileba 0-is toli

unda iyos, amitom bolo ujredSi unda eweros –40.

g. 4. 1. –11,4. 2. hipotenuzis sigrZe gaizomeba saxazaviT.

5. I. 2a; II. 9a; III. ta.

g. 5. 1. I. –27; II. –3; III. 56; IV. 0; V. –2,5; VI.

0,209. 2. (d) 30. 3. I. 4 aTasi; II. –8 aTasi; III. Semcirdeboda


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6 aTasiT; IV. gadiddeboda 16 aTasiT. 4. ricxvis (–1)-ze

gamravlebiT miiReba am ricxvis mopirdapire ricxvi. 6. I.

12; II. –60. 7. 7,8 mm + 8,75 mm < 16,9 mm. amitom ar SeiZleba!

2 g. 6. 1. I. –1981; II. 103; III. –3 3 ; IV. –57. 2. I. 3 + (–4c);

II. –7,5 + 6a + (–9a). 3. I. –7,9b, 4c da –8; II. 1/k, –2/b da –

1,5. 4. a(b – c) = a(b +(–c)) = ab+a(–c) = ab – ac. 5. I. 3(3b+3);

II. 4(c–d); III. a(3–5) = –2a; IV. 3b. 6. I. x = –9; II. x = 10.

7. I. perimetri _ 16 erT., farTobi – 15 kv.erT. II. perimetri

_ 15 erT., farTobi – 12,5 kv.erT. zedmeti monacemia me-4

wveros koordinatebi! 8. I. dadebiTi; II. uaryofiTi.

g. 7. 1. I. 15x; II. –5y; III. –4a; IV. 2x – 12; V. 0; VI. –

2; VII. x; VIII. –5+2c. 2. I. –a – 2; II. 1 + 3x; III. t + z; IV.

2,8 – 8,5y. 3. I. 20 km-iT marjvniv; II. 20 km-iT marcxniv;

III. 20 km-iT marcxniv; IV. 20 km-iT marjvniv. l = vt formula

marTebulia masSi Semavalma asoebis nebismieri

mniSvnelobebisTvis. 4. I. kaTeti ar SeiZleba hipotenuzis toli

iyos! II. fuZe unda iyos hipotenuza. 5. (a) 13. 6. raki

mercxali isev budeSi dabrunda, amitom cvlilebaTa jami 0-is

toli unda iyos. amitom swori pasuxia –11,26.

g. 8. 3. I. AFEDC; II. B. 4. gamravleba / gayofa.

5. I. 21,5; II. 3; III. 0. 6. racionaluri ricxvebis gadamravlebis

wesis mixedviT namravlis moduli TanamamravlTa

modulebis namravlis tolia, modulebi arauaryofiTi ricxvebia. aseTi

ricxvebis SemTxvevaSi ki Tanamamravli tolia namravli gayofili

meore Tanamamravlze.

7. damtkiceba eyrdnoba racionaluri ricxvebis gadamravlebis

wess. 8. I. x = –2; II. x = 5 an x = –5; III. x = –3.

9. ufro zogadia Tanamedrove mniSvneloba, radgan is gamoiyeneba

ara mxolod im SemTxvevaSi, roca fuZeSi oTxkuTxedia, aramed

sxva mravalkuTxedebis SemTxvevaSic.

g. 9. 1. I. –2; II. –3; III. 6; IV. 500; V. 0,31; VI. –100.

2. I. ki; II. ara (= –40); III. ara (= –2,7); IV. ki.


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3. I. 25; II. –11. 4. orive tolia 19-is. 5. I. –10,7; II. 36.

6. I. x = 2; II. z = 27. 7. g) 98% .

g. 10. 1. I. 5; II. –1/3; III. a = – 2,5; IV. a = – 7/4.

2. Tuki a < 0, maSin a = –|a|, amitom misi Sebrunebuli iqneba −

1

.

| a|

3. I. –4/5; II. –6. 4. I. 3600; II. –42. 6. SeiZleba (amis dasasabuTeblad

warmovidginoT ori piramida, romelebic miRebulia

erTi piramidis gakveTiT ise, rom kveTa wveroze da fuZeze

gadiodes!). 7. I. (ab) : c = (ab) · 1/c =(a · 1/c) · b = (a : c) · b.

8. I. 160; II. 65. 9. I. x = –4; II. y = –3; III. z = –5/8.

g. 11. 2. ufro zogadia piramida. `klebis mixedviT~: piramida;

samkuTxa piramida; iseTi piramida, romlis fuZe _

tolferda samkuTxedia; tetraedri. 4. I. = (–12,3 – 6) · (–0,1)

= 1,83; II. = –2,5 + 6,4 – 1,9 = 2; III. 1. 5. x = –1,8.

6. I. 2d+7c; II. 2(5m+4k+n+3k) = 10m+14k+2n.

7. metad _ indielTa piramidebi, naklebad _ zikurati.

8. ufro zogadia mniSvneloba Zveli berZnuli sityvisa `tetraedri~,

radgan is ar gulisxmobs, rom waxnagebi aucileblad

tolia!

Tavi XIV. g. 1. 4. (d) 12 l 25 T. 5. 1 sT =

20+1+20+1+18. amitom swori pasuxia 58 · 30 = 17,4 lari.

`mudmivi nazrdi~ iqneboda, mag., aseT SemTxvevaSi: pirveli

nasaubrevi 20 wuTis Semdeg 1 wuTis saubaria ufaso, kidev 20

wT-is Semdeg _ 2 wuTia ufaso, kidev 20 wT-is Semdeg _ 3

wuTia ufaso da a.S. 6. I. xuTkuTxeds; II. samkuTxeds.

g. 2. 1. I. 1; II. 3, 5; III. 1, 3, 5. 2. I. 10; II. 0,3.

3. I. 2; II. a + 8; III. –10b – 2; IV. – 3x – 4.

4. 1) 13 weli gamodis toli daaxloebiT gedis sicocxlis xangrZliobis

1 – 0,4 = 0,6 nawilisa. amitom gedi cocxlobs daaxloebiT

22 wels, mercxali _ 8-10 wels. 2) yvavi da Woti _

60 w = 0,6 sauk.; qori, arwivi, TuTiyuSi _ 4 w = 0,4 sauk.;

wero _ 37,5 w = 0,375 sauk.; qaTami _ 25 w = 0,25 sauk.; toro-


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la, Zera, Sevardeni _ 20 w = 0,2 sauk.; ixvi, bati, Tolia, SaSvi,

bu, yanCa _ 17,5 w = 0,175 sauk.; SoSia _ 12,5 w. = 0,125 sauk.

3) siraqlema _ 30-40 w, gareuli qaTami _ 12-16 w, mtredi _ 15-20 w.

4) ozurgeTis mosaxleobis raodenobis 4/7 nawili yofila

12000. amitom ozurgeTSi iqneba 21000 mosaxle, lagodexSi _

9000. sayofacxovrebo SinaarsiT erTmaneTs hgavs 1), 2) da

3), maTematikuri SinaarsiT _ 1) da 4).

5. I. x = 30,4; II. x = 1 an x = –2.

6. ar aqvs, radgan marcxena mxare yovelTvis 9-iTaa meti

marjvenaze. 7. gamartivebiT miviRebT: 3x – 1 = 3x – 1. es

toloba ki x-is nebismieri mniSvnelobisTvisaa marTebuli.

g. 3. 1. I. tolfasia; II. tolfasia; III. tolfasia;

IV. araa tolfasi. 3. I. marTebulia; II. mcdaria. 4. x + 15 =

25. x = 10. es gantolebebi tolfasi iqneba. 5. I. marTebulia;

II. marTebulia; III. marTebulia; IV. mcdaria. 7. I. = –3,5x –

0,5; II. –1,8y + 2,5. 8. I. ki; II. ara; III. ki; IV. ara.

g. 4. 1. I. ki; II. ki; III. ara; IV. ki. 2. I. x = –60; II. y =

37,9; III. x = 6; IV. y = 36. 6. I. 16x + 5x = 25 – 7; II. 9x 2 – 6,5x

+ x2 = 1 + 2,3. 7. e) 97,5. 8. Tuki aravis ar eqneboda

gatanili 4 an ufro meti goli, maSin 8 moTamaSes SeeZlo

gaetana maqsimum 3 · 8 = 24 goli. es ki ewinaaRmdegeba pirobas!

g. 5. 1. I. x = 5; II. y = 2; III. z = 3; IV. x = 10. 2. I. 4;

II. 2; III. 1. 3. me-5 wevria 17, me-9 wevri - 29. me-100 wevri

metia 97-e wevrze 3 · 3 = 9-iT. 4. 1 000 000 : 6 = 166 666 (naSTi

4). amitom memilione iqneba me-4 aso I . 6. I. 0,01; 0,02;

0,03; 4; II. a, d, e, a, r; III. .

7. II. 1 kvira; III. 32 dRe; IV. 5 weli; V. 1 Tve; VI. 1 weli.

yvelaze zustia II, radgan Tvisa Tu wlis xangrZlioba araa

ucvleli, kvirisa ki ucvlelia.

g. 6. 1. 6). 2. me-3 wevria 18, me-5 wevri - 162. me-10 wevri

metia me-8-ze 9-jer. 6. I. (3; –2); II. (–3; 2). 7. AB monakveTi.

g. 7. 1. {AM, CO, PF, MO, PK}. 7. I. 16-jer; II. me-18 -


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jixuri, 31-e - Zelskami. III. raki 3904-is 5-ze gayofisas naSTia

4, amitom Sexvdeba Zelskami, xolo 5000-is 5-ze gayofisas

naSTia 0, amitom Sexvdeba yvavilnari. periodi iqneba:

yvavilnari, Zelskami, anZa, jixuri, Zelskami. 8. simetria.

g. 8. 1. I. x = –7000; IV. x = 0. 2. I. x = –1/3; II. ∅;

III. x = –3; IV. usasrulod mravali amonaxsni aqvs. 4. I. 193;

II. 2. 8. I. 10+9+8+1 = 28; II. 4 · 3 + 1 = 13; III.

10+9+8+1 = 28; IV. 10+1 = 11; V. 9+8+8+2 = 27.

g. 9. 1. I. x = 2; II. y = 5; III. x = –2; IV. y = –7.

7. 1 danayofis `wonaa~ 2 : 5 = 0,4. amitom ? niSnebis nacvlad

unda eweros (marjvnidan marcxniv): 2,6, 1,8, 0,2 da –2,6.

yvelaze Sesaferisi saTauria: d) mudmivi uaryofiTi nazrdi.

8. zedmetia `mezobeli~.

g. 10. 3. I. x = 2/19; II. t = 1; III. y = 0; IV. x = 30 an x = –

30; V. x = 2 an x = –2. 4. a) 24,75 km. 5. I. d); II. b).

7. nebismieri sami ricxvidan 2 mainc iqneba kenti an 2 mainc

iqneba luwi. maTi jami ki unaSTod gaiyofa 2-ze.

g. 11. 1. I. A1 (2; 1) da C1 (7; 5); II. C2 (–1; –3); x-RerZis

mimarT simetriuli arekvliT miRebuli samkuTxedis wveroebis

koordinatebi iqneba: (-6; -1), (-1; -1) da (-1; -5). 2. I. b) E(-4; -1),

F(1; -1), G(1; 3); II. d) D(-6; 1), T(-6; -4), S(-2; -4);

III. g) P(6; 1), Q(1; 1), R(1; 5). 3. l = 6a, sadac a erTi wibos

sigrZea. 5. Tu TiTo TveSi 4 moswavles eqneba dabadebis dRe,

maSin klasSi 48 moswavlis SemTxvevaSic ki 5 moswavles

dabadebis dRe erT TveSi ar eqneba. magram Tu TiTo TveSi 3

moswavles eqneba dabadebis dRe, maSin maqsimum 36 moswavle unda

iyos klasSi, es ki ewinaaRmdegeba pirobas. amitom aucileblad

moiZebneba iseTi Tve, roca dabadebis dRe eqneba am klasis 4

moswavles mainc. 6. yvelaze xelsayrelia: (v) 1 wuTis saubris

fasia 28 TeTri; yvelaze araxelsayreli: (e) 1 wuTis saubris

fasia 30 TeTri, oRond yoveli nasaubrevi 1 saaTis Semdeg 1 wuTis

saubari ufasod. wesis `1 wuTis saubris fasia 30 TeTri, oRond


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yoveli nasaubrevi 20 wuTis Semdeg 1 wuTis saubari ufasoa~

tolfasia (d).

§ 19. sakontrolo werebis amocanaTa pasuxebi

da maTi Sefasebis kriteriumebi

sakontrolo wera # 1 rigi I

1. 33,6 (1 q); 32,5 (1 q); 35,5 (1 q); 33 (1 q).

(TviTeuli arasworad amowerili ricxvisTvis unda daakldes 1

qula, oRond, cxadia, Tuki sxvaoba 0-ze naklebi gamodis, maSin

am amocanis saboloo qula unda iyos 0).

2. I. 6,3 m ≤ |AK| < 42,8 m (2 q); II. b3 < b+ 9,4 ≤ a · 35 (2 q).

(Tuki romelime ormxrivi utolobis mxolod erTi mxarea

swori, maSin am utolobis CawerisTvis daeweros 1 qula).

3. A(3) da C(4) wertilebis sworad moniSvna - 1 q; E da F

wertilebisa - 1 q; |AC| = 2 sm (1 q); |EF| = 5 sm (1 q).

4. I. 15 kg (1 q); II. 15/40 an 3/8 (1 q); III. 120 lari (2 q).

5. I. musikis Semswavleli moswavleebidan sportze dadis

moswavleTa saerTo raodenobis 1 · 1 = 1 nawili (1 q), xolo

4 10 40

danarCenebidan _ 3 · 1 = 1 nawili (1 q). amitom sul sportze

4 3 4

dadis moswavleTa saerTo raodenobis 1 + 1 = 11 nawili (1 q);

40 4 40

II. swavlobs musikas an dadis sportze moswavleTa saerTo

raodenobis 1 + 11 − 1 = 1 naw.

4 40 40 2

(SeiZleboda asec gamogveTvala: 1 + 1 = 1 ) (3 q).

4 4 2

rigi II

1. 23,7 (1 q), 27,4 (1 q), 24,3 (1 q), 24 (1 q).

2. III. 2,7 km ≤ |DC| < 12,3 km (2 q); IV. d 2 < d+ 0,7 ≤ c–23 (2 q).

3. M(4) da N(5) wertilebis sworad moniSvna _ 1 q; A da B

wertilebisa _ 1 q; |MN| = 2 sm (1 q); |AB| = 7 sm (1 q).

4. I. 25 m (1 q); II. 24/60 an 2/5 (1 q); III. 45 TeTri (2 q).


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5. I. katis myol mobinadreTagan ZaRli hyavs mobinadreTa saerTo

raodenobis 1 · 1 = 1 nawils (1 q), xolo danarCenebidan -

5 10 50

4 · 1 = 1 nawils (1 q). amitom sul ZaRli hyavs mobinadreTa

5 4 5

saerTo raodenobis 1 + 1 = 11 nawils. (1 q); II. ZaRli an

50 5 50

kata hyavs mobinadreTa saerTo raodenobis 1 + 11 − 1 = 2

5 50 50 5

nawils. (SeiZleboda asec gamogveTvala: 1 + 1 = 2 naw.) (3 q).

5 5 5

sakontrolo wera # 2 rigi I

1. I. 1 (1 q); III. 27 (1 q); V. = ( 0,

25⋅

8)

5

· 8 = 32·

8 = 256 (2 q).

2. I. 2 4 m 4 n 4 (1 q); II. k 6 : 3 6 (1 q); III. mag.: 3 n · 8 n (2 q).

3. I. 5,7 (2 q); II. 5,6 (2 q).

4. I. 0,8 · 40 = 32 ≤ 32. utoloba marTebulia (2 q);

II. 0,6 · 38 + 1 + 6 = 22,8 + 7 = 29,8 ≤ 32. utoloba marTebulia (2 q).

TfojTwob/ yoveli swori, magram dausabuTebeli pasuxisTvis - 1 q.

5. I. 44,8 kg (3 q); II. 87,5 kg (3 q).

rigi II

1. II. 1 (1 q); IV. 81 (1 q); VI. = ( 0,

4⋅

5)

6

· 5 = 64·

5 = 320 (2 q).

2. IV. 3 5 k 5 d 5 (1 q); V. b 8 : 7 8 (1 q); VI. mag.: 4 m · 8 m (2 q).

3. I. 8,8 (2 q); II. 8,4 (2 q).

4. I. 0,7 · 50 = 35 ≤ 30. utoloba mcdaria (2 q);

II. 5 + 1 + 0,5 · 48 = 6 + 24 = 30 ≤ 30. utoloba marTebulia (2 q).

TfojTwob/ yoveli swori, magram dausabuTebeli pasuxisTvis - 1 q.

5. I. 17,6 kg (3 q); II. 12,5 kg (3 q).

sakontrolo wera # 3 rigi I

1. I. a 35 (1 q); II. c 14 (1 q) ; III. 9

d (2 q).

2. I. 2

3 an 9 (2 q); II. 1/62 an 1/36 (2 q).

3. I. 16 (2 q); II. 735 kg (2 q).

4. e) 257,5 (4 q). TfojTwob/ g) 259 an d) 256 - 2 q.

5. Svlebia 70% (1 q). irmebis raodenoba naklebia 70%–

30%=40%-iT (1 q). ese igi, irmebisa da Svlebis mTliani


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raodenobis 40% tolia 144-is (1 q). amitom mTliani raodenoba

iqneba 144 : 0,4 = 360 (2 q), Svlebis raodenoba ki iqneba

360 · 0,7 = 252 (1 q).

! TfojTwob/ SeiZleba vinmem amoxsna ase daamTavros: raki

irmebisa da Svlebis mTliani raodenobis 40% tolia 144-isa,

amitom Svlebis raodenoba iqneba 144 · 70 = 252 (3 q).

40

rigi II

1. IV. t 27 (1 q); V. b 30 (1 q); VI. 24

k (2 q).

2. I. 2

7 an 49 (2 q); II. 1/52 an 1/25 (2 q).

3. III. 33 (2 q); IV. 1069 g (2 q).

4. d) 317,5 (4 q). TfojTwob/ g) 316 an e) 319 - 2 q.

5. cxvrebia 65% (1 q). cxvrebis raodenoba metia

65% – 35% = 30%-iT (1 q). ese igi, cxvrebisa da Zroxebis

mTliani raodenobis 30% tolia 234-isa (1 q). amitom mTliani

raodenoba iqneba 234 : 0,3 = 780 (2 q), cxvrebis raodenoba ki

iqneba 780 · 0,65 = 507 (1 q).

! TfojTwob/ SeiZleba vinmem amoxsna ase daamTavros: raki

cxvrebisa da Zroxebis mTliani raodenobis 30% tolia 234-isa,

amitom cxvrebis raodenoba iqneba 234 · 65 = 507 (3 q).

30

sakontrolo wera # 4 rigi I

1. I. a 32 (1 q); II. 1 (1 q); III. b 13 (2 q).

2. I. a4a3a2a1a0 = a4 · 104 + a3 · 103 + a2 · 102 + a1 · 101 + a0 · 100 (2 q);

a79a78 ... a0 = a79 · 1079 + a78 · 1078 + ... + a0 · 100 (2 q).

3. 10 24 ≤ d < 10 25 (2 q); 10 3 ≤ 4960 < 10 4 (1 q); 10 5 ≤ 100060 < 10 6 (1 q);

4. I. ana ∉ D (0,5 q); II. zurabi ∈ F (0,5 q);

III. e ∈ E (0,5 q); IV. qaTami ∈ B (0,5 q);

V. B ⊃ D (0,5 q); VI. C ⊂ A (0,5 q);

VII. {gia, lia} ⊂ F (0,5 q); VIII. A ⊃ {muxa, ia, cacxvi} (0,5 q).

5. mesame kviraSi SeakeTes mTeli gzis 1/5 nawili (1 q), meore

kviraSi ki - 1 − 1 − 1 = 7 nawili (2 q). ese igi, mTeli gzis 7/15

3 5 15


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nawili tolia 21 km-is (1 q). amitom mTeli gzis sigrZe iqneba

21: 7 = 45 km (2 q).

15

rigi II

1. I. d 7 (1 q); II. 1 (1 q); III. m 49 (2 q).

2. I. a3a2a1a0 = a3 · 103 + a2 · 102 + a1 · 101 + a0 · 100 (2 q);

a89a88 ... a0 = a89 · 1089 + a88 · 1088 + ... + a0 · 100 (2 q).

3. 10 8 ≤ r < 10 9 (2 q); 10 3 ≤ 8190 < 10 4 (1 q); 10 6 ≤ 1000700 < 10 7 (1 q).

4. IX. Tina ∈ F (0,5 q); X. lia ∉ A (0,5 q);

XI. g ∈ E (0,5 q); XII. indauri ∈ B (0,5 q);

XIII. D ⊂ B (0,5 q); XIV. C ⊂ A (0,5 q);

XV. {vaJa, Tea} ⊂ F (0,5 q); XVI. A ⊃ {Tela, enZela, cacxvi} (0,5 q).

5. mesame dRes gaTibes mTeli saTibis 3/10 nawili (1 q), meore

dRes ki - 1 − 1 − 3 = 8 nawili (2 q). ese igi, mTeli saTibis

6 10 15

8/15 nawilis farTobi tolia 48 aris (1 q). amitom mTeli

saTibis farTobi iqneba 48 : 8 = 90 ari (2 q).

15

sakontrolo wera # 5 rigi I

1. I. 7,5 · 10 2 (1 q); II. 4,7 · 10 4 (1 q);

III. 5 · 10 7 (1 q); IV. 5,2 · 10 13 (1 q);

2. I. 2,1 · 10 22 (1 q); III. 8 · 10 12 (1 q); V. 8,1 · 10 57 (2 q).

3. I. 1 m/wT = 5/3 sm/wm (2 q); II. 1 dm/wm = 6 m/wT (2 q).

4. e) 207,5 (4 q). TfojTwob/ g) 206,5 an d) 209 - 2 q.

5. patara mwvervalamde manZilia 3,6 · 2,5 = 9 (km) (2 q). ukan

dabrunebas moundeboda 9 : 4,5 = 2 (sT) (2 q). mTamsvlelis saS.

siCqare iqneba (9+9) km : (2,5+0,5+2) sT = 3,6 km/sT (2 q).

rigi II

1. V. 4,9 · 10 2 (1 q); VI. 6,3 · 10 4 (1 q);

VII. 4 · 10 7 (1 q); VIII. 7,4 · 10 17 (1 q);

2. II. 3,6 · 10 22 (1 q); IV. 8 · 10 5 (1 q); VI. 6,4 · 10 52 (2 q).

3. III. 1 m/sT = 1/6 dm/wT (2 q); IV. 1 sm/wm = 3/5 m/wT (2 q).

4. e) 318,5 (4 q). TfojTwob/ g) 317 an d) 319,5 - 2 q.


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5. tyemde manZilia 7,5 · 1,2 = 9 (km) (2 q). ukan dabrunebas

moundeboda 9 : 15 = 0,6 (sT) (2 q). bavSvis saSualo siCqare iqneba

(9+9) km : (1,2+0,2+0,6) sT = 9 km/sT (2 q).

sakontrolo wera # 6 rigi I

1. I. 1,4 · 10 29 (1 q); III. 1,3 · 10 32 (1 q); V. 1,4 · 10 68 (2 q).

2. I. bundovania (0,5 q); II. moicema erToblioba (0,5 q);

III. bundovania (0,5 q); IV. moicema erToblioba (0,5 q);

V. moicema erToblioba (0,5 q); VI. bundovania (0,5 q);

VII. moicema erToblioba (0,5 q); VIII. moicema erToblioba (0,5 q).

3. swori naxazi - 4 q.

4. Semcirebis Semdeg siCqare iqneba 48 km/sT (2 q). isev 60

km/sT siCqariT rom imoZraos, memanqanem siCqare unda gaadidos

60 – 48 = 12 km/sT-iT, anu 12 · 100 =25 %-iT (2 q).

48

5. I. sasrulia (0,5 q); II. usasruloa (0,5 q); III. sasrulia (0,5 q);

IV. usasruloa (0,5 q); V. sasrulia (1 q); VI. sasrulia (1 q);

carielia VI, radgan Tuki ricxvis gamyofia 4, maSin misi

gamyofi iqneba 2-ic (2 q).

rigi II

1. II. 3,7 · 10 56 (1 q); IV. 1,5 · 10 14 (1 q); VI. 1,2 · 10 114 (2 q).

2. I. moicema erToblioba (0,5 q); II. bundovania (0,5 q);

III. bundovania (0,5 q); IV. moicema erToblioba (0,5 q);

V. moicema erToblioba (0,5 q); VI. bundovania (0,5 q);

VII. moicema erToblioba (0,5 q); VIII. moicema erToblioba (0,5 q).

3. swori naxazi - 4 q.

4. momatebis Semdeg siCqare iqneba 75 km/sT (2 q). isev 60 km/sT

siCqariT rom imoZraos, mZRolma siCqare unda Seamciros

75 – 60 = 15 km/sT-iT, anu 15 · 100 =20 %-iT (2 q).

75

5. I. sasrulia (0,5 q); II. usasruloa (0,5 q); III. sasr. (0,5 q);

IV. usasruloa (0,5 q); V. sasrulia (1 q); VI. sasrulia (1 q);

carielia VI, radgan Tu ricxvis gamyofia 6-iani, maSin misi

gamyofi iqneba 2-ianic da 3-ianic (2 q).


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sakontrolo wera # 7 rigi I

1. 105,8 (4 q).

2. I. meore gverdis sigrZea 3 · 108 m (2 q). marTkuTxedis

farTobia 4,2 · 1024 · 3 · 108 ≈ 1,3 · 1033 m2 (2 q).

3. B wveroze CD-s marTobuli wrfis gavleba - 1 q, CD-s

paraleluri wrfis gavleba - 1 q. D wveroze BC-s marTobuli

wrfis gavleba - 1 q. saTanado aRniSvnebis gamoyenebiT marTobul

an paralelur wrfeTa yvela wyvilis amowera - 1 q.

4. 20,1-isa da 20,5-is Sesabamis wertilebs Soris manZilia 6

sm. amitom erTeulovani monakveTis sigrZea 6 sm : (20,5–20,1) =

15 sm (2 q). E da F wertilebis koordinatebia: d) 20,13 da

20,47 (2 q).

TfojTwob/ Tu SearCia g) 20,15 da 20,42 - 1 q.

5. 3 wuTis Semdeg maT Soris manZili iqneba 1800 m (3 q).

2 km 100 m manZili iqneba 2100 : 600 = 3,5 wuTis Semdeg (3 q).

rigi II

1. 29 (4 q).

2. II. meore gverdis sigrZea 1,5 · 109 m (2 q). marTkuTxedis

farTobia 4,2 · 1024 · 1,5 · 109 = 6,3 · 1033 m2 (2 q).

3. N wveroze MQ-s marTobuli wrfis gavleba - 1 q, AD-s

paraleluri wrfis gavleba - 1 q. M wveroze NP-s marTobuli

wrfis gavleba - 1 q. saTanado aRniSvnebis gamoyenebiT marTobul

an paralelur wrfeTa yvela wyvilis amowera - 1 q.

4. 14,5-isa da 14,7-is Sesabamis wertilebs Soris manZilia 6 sm.

amitom erTeulovani monakveTis sigrZea 6 sm : (14,7–14,5) = 30

sm (2 q). P da T wertilebis koordinatebia: e) 14,55 da 14,67 (2 q).

Tuki SearCia g) 14,52 da 14,65 - 1 q.

5. 3 wuTis Semdeg maT Soris manZili iqneba 2400 m (3 q).

3 km 600 m manZili iqneba 3600 : 800 = 4,5 wuTis Semdeg (3 q).

1. = 64–8+18,6 = 74,6 (4 q).

sakontrolo wera # 8 rigi I


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2. wertilebi sworadaa moniSnuli - 3 q, TanmimdevrobiTaa

SeerTebuli - 1 q.

3. I. praRa ≈ 13° ag da 50° Cg (1 q); dublini ≈ 14° dg da 55° Cg (1 q);

soxumi ≈ 41° ag da 43° Cg (1 q); gori ≈ 44,2° ag da 41,9° Cg (1 q).

4. aguredis simaRlea 3,2 · 1014 : 4 · 109 = 8 · 104 m (2 q),

moculoba - 3,2 · 1014 · 5 · 104 · 8 · 104 = 128 · 1022 ≈ 1,3 · 1024 m3 (2 q).

5. raki erTeulovani monakveTis sigrZea 1 m, amitom 1 sm iqneba

0,01 erTeulovani monakveTis sigrZis toli. Sesabamisad, A

wertilis koordinati iqneba 51,99-is toli (2 q), xolo B

wertilisa - 52,03 (2 q).

rigi II

1. = 27+81–1,8 = 106,2 (4 q).

2. wertilebi sworadaa moniSnuli - 3 q, TanmimdevrobiTaa

SeerTebuli - 1 q.

3.II. oslo ≈ 12° ag da 60° Cg (1 q); reikiaviki ≈ 22° dg da 61° Cg (1 q);

maikopi ≈ 40° ag da 44,2° Cg (1 q); baTumi ≈ 41,6° ag da 41,6° Cg (1 q).

4. aguredis simaRlea 6,4 · 1016 : 8 · 107 = 8 · 108 m (2 q),

moculoba - 4 · 1018 · 6,4 · 1016 · 8 · 108 = 204,8 · 1042 ≈ 2 · 1044 m3 (2 q).

5. raki erTeulovani monakveTis sigrZea 0,1 mm, amitom 1 sm

iqneba 100 erTeulovani monakveTis sigrZis toli. Sesabamisad, M

wertilis koordinati iqneba 720-is toli (2 q), xolo N

wertilisa - 1220 (2 q).

sakontrolo wera # 9 rigi I

1. I. –341 /6 ; –7,3 ; –7 ; –1; 0 ; 0,74 ; 21 (4 q).

2. I. = 24 – 9,3 + 2,3 = 17 (4 q).

3. sxivze A(–6), B(+4,5), D(–2,5), E(–2) wertilebi sworadaa

moniSnuli - 2 q, M(–5,5) - 1 q, N(+1,5) - 1 q.

4. aleqsandre gardacvlila ax.w. 38 wlis 18 aprils (1 q),

mariami - ax.w. 131 wlis 18 aprils (1 q). aleqsandre

dabadebula Zv.w I sauk. III meoTxedSi (1 q), mariami - ax.w. I sauk.


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II meoTxedSi (1 q).

5. xaliCebi TebervalSi eRireboda 276 lari (2 q), martSi -

345 lari (2 q). aprilSi xaliCebi gaiafebula 345–240 =

105 lariT (1 q), anu 105 · 100 ≈ 30,4 %-iT (1 q).

345

rigi II

1. II. 14 ; 43 /4 ; +1,8 ; –0,14 ; –3; –14; –21 (4 q).

2. II. = 40+6,7–12,7 = 34 (4 q) .

3. sxivze M(3), N(–1,5), P(+6,5), Q(–3). wertilebi sworadaa

moniSnuli - 2 q, A(–6,5) - 1 q, B(+0,5) - 1 q.

4. nino gardacvlila ax.w. 19 wlis 23 maiss (1 q), amirani -

ax.w. 134 wlis 23 maiss (1 q). nino dabadebula Zv.w I sauk. II

meoTxedSi (1 q), amirani - ax.w. I sauk. III meoTxedSi (1 q).

5. macivari gazafxulze eRireboda 504 lari (2 q), zafxulSi

- 630 lari (2 q). Semodgomaze macivari gaiafebula 630–420

= 210 lariT (1 q), anu 210 · 100 ≈ 33,3 %-iT (1 q).

630

sakontrolo wera # 10 rigi I

1. I. –14 (1 q); II. –28,8 (1 q); III. 20,5 (1 q); IV. 27 (1 q).

2. I. _3a (1 q); II. –2k (1 q); III. –4,6a (1 q); IV. _3b (1 q).

3. I. x = 20 (1 q); II. y = 3,5 (1 q);

III. x = 4,5 (1 q); IV. y = –8 (1 q).

4. swori naxazi - 4 q.

5. vTqvaT, manqanis siCqarea x km/sT, maSin matareblis siCqare

iqneba x+5 km/sT (1 q). miviRebT gantolebas: 4x + 7(x+5) = 640

(2 q). am gantolebis amonaxsnia x = 55 (2 q). matareblis siCqare

yofila 60 km/sT (1 q).

rigi II

1. V. _22 (1 q); VI. 13,6 (1 q); VII. –13,6 (1 q); VIII. 22 (1 q).

2. V. _2b (1 q); VI. –2,6m (1 q); VII. –3,2k (1 q); VIII. _8c (1 q).

3. V. y = 23 (1 q); VI. y = 5,6 (1 q);

VII. x = 3,5 (1 q); VIII. x = –4 (1 q).

4. swori naxazi - 4 q.

5. vTqvaT, matareblis siCqarea x km/sT, maSin manqanis siCqare


- 136 -

iqneba x+6 km/sT (1 q). miviRebT gantolebas: 4x + 2(x+6) = 312

(2 q). am gantolebis amonaxsnia x = 50 (2 q). manqanis siCqare

yofila 56 km/sT (1 q).

sakontrolo wera # 11 rigi I

1. I. _16 (1 q); II. –17,7 (1 q); III. 40,5 (1 q); IV. 32 (1 q).

2. I. _2a (1 q); II. 2,9k (1 q); III. 4,7a (1 q); IV. _8b (1 q).

3. I. x = 144 (1 q); II. y = 4 (1 q); III. x = 9 (1 q); IV. y = –8 (1 q).

4. I. sworad daxaza ABCD marTkuTxedi (1 q). marTkuTxedis

perimetria 32 erTeulovani monakveTis sigrZe (2 q). C wvero

marTkuTxedis BD diagonalidan daSorebulia daaxloebiT 5

erTeulovani monakveTis sigrZiT (1 q).

5. vTqvaT, nakveTis sigrZea 5x m, sigane - 3x m (1 q). miviRebT

gantolebas: 5x – 3x = 300 (1 q). aqedan x = 150 (1 q). nakveTis

perimetri iqneba 2400 m (1 q). mevele am nakveTs Semouvlis

2,4 km : 3 km/sT = 0,8 sT = 48 wT-Si (2 q).

rigi II

1. V. _28 (1 q); VI. 21,8 (1 q); VII. –29,6 (1 q); VIII. 36 (1 q).

2. V. _7b (1 q); VI. –31,2m (1 q);

VII. –1,8k (1 q); VIII. _23c (1 q).

3. V. y = 115 (1 q); VI. y = 20 (1 q);

VII. x = 4 (1 q); VIII. x = –7 (1 q).

4. I. sworad daxaza ABCD marTkuTxedi (1 q). marTkuTxedis

perimetria 28 erTeulovani monakveTis sigrZe (2 q). C wvero

marTkuTxedis BD diagonalidan daSorebulia daaxloebiT 4,8

erTeulovani monakveTis sigrZiT (1 q).

5. vTqvaT, nakveTis sigrZea 7x m, sigane - 5x m (1 q). miviRebT

gantolebas: 7x – 5x = 600 (1 q). aqedan x = 300 (1 q). nakveTis

perimetri iqneba 7200 m (1 q). mevele am nakveTs Semouvlis

7,2 km : 4 km/sT = 1,8 sT = 1 sT 48 wT-Si (2 q).

sakontrolo wera # 12 rigi I

1. I. y = –7 (1 q); II. x = 25 (1 q);

III. z = –23,6 (1 q); IV. t = –32 (1 q).


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2. I. 2a – 5 (2 q); II. y + 1 (2 q).

3. vTqvaT, Caifiqres x. miviRebT gantolebas:

x – 14 – (–28) + 18 + (–27) = –21 (2 q). aqedan x = –26 (2 q).

TfojTwob/ Cafiqrebuli ricxvi SeiZleba gantolebis gareSec

gamovTvaloT: –21 – (–27) – 18 + (–28) + 14 = –26 (4 q).

4. MNP samkuTxedis sworad daxazva - 2 q, masze wrexazis Semoxazva

- 2 q.

5. I. fleitaze ukravs 2,5-jer meti moswavle, vidre violinoze (2 q).

II. mxolod erT sakravze dakvras swavlobs moswavleTa

100% – (20%+50%) = 30%, anu 3/5 nawili (2 q).

Tuki I SekiTxvaSi sityvas `ramdenjer~ SevcvlidiT sityviT

`ramdeniT~, maSin pasuxs ver gavcemdiT, radgan saWiro iqneboda,

damatebiT gvcodnoda, sul ramdeni moswavle swavlobs samusiko

skolaSi (an ramdeni ukravs fleitaze, an violinoze) (2 q).

1. V. t = –8 (1 q); VI. y = –2 (1 q);

VII. x = 4,7 (1 q); VIII. z = –1 (1 q).

2. III. –15 + m (2 q); IV. – 2 – y (2 q).

3. vTqvaT, Caifiqres x. miviRebT gantolebas:

rigi II

x + 12 + (–17) – 28 – (–13) = –30 (2 q). aqedan x = –10 (2 q).

TfojTwob/ Cafiqrebuli ricxvi SeiZleba gantolebis gareSec

gamovTvaloT: –30 – 12 – (–17) + 28 + (–13) = –10 (4 q).

4. ABD samkuTxedis sworad daxazva - 2 q, masze wrexazis Semoxazva

- 2 q.

5. I. fleitaze ukravs 4-jer meti moswavle, vidre violinoze (2 q).

II. mxolod erT sakravze dakvras swavlobs moswavleTa

100% – (15%+60%) = 25%, anu 1/4 nawili (2 q).

Tuki I SekiTxvaSi sityvas `ramdenjer~ SevcvlidiT sityviT

`ramdeniT~, maSin pasuxs ver gavcemdiT, radgan saWiro iqneboda

damatebiT gvcodnoda sul ramdeni moswavle swavlobs samusiko

skolaSi (an ramdeni ukravs fleitaze, an violinoze) (2 q).

sakontrolo wera # 13 rigi I


- 138 -

1. I. = (–7,2 – 0,6) · (–0,2) = –7,8 · (–0,2) = 15,6 (4 q).

2. I. 12x (1 q); II. –4a (1 q); III. –10b (1 q); IV. –12x + 30 (1 q).

3. I. –0,5x + 0,3 = 1,2 (2 q); x = –1,8 (2 q).

4. ABC, MNK da PQR samkuTxedebis sworad daxazva - 2 q.

marTkuTxaa MNK samkuTxedi (1 q).

misi kaTetebia MN da NK, hipotenuza - MK (1 q).

5. I. patara goWi 1 dReSi Wams 1/4 kg saWmels (1 q), didi ixvi -

1/7 kg-s (1 q). amitom ufro met saWmels Wams patara goWi (1 q).

II. 2 patara goWi da 5 didi ixvi 1 dReSi Wams 2 + 5 = 34 = 17

4 7 28 14

kg saWmels (2 q), amitom 17 kg saWmels SeWamen 14 dReSi (1 q).

rigi II

1. II. = (–14,4 + 0,7) · (–0,4) = (–13,7) · (–0,4) = 5,48 (4 q).

2. V. –13y (1 q); VI. 5a (1 q); VII. –11x (1 q); VIII. –4 + 8c (1 q).

3. II. –0,4y + 0,4 = 3,2 (2 q); y = –7 (2 q).

4. EFK, BCD da ATP samkuTxedebis sworad daxazva - 2 q.

marTkuTxaa ATP samkuTxedi (1 q).

misi kaTetebia AT da TP, hipotenuza - AP (1 q).

5. I. 1 TuSi mqsoveli 1 kviraSi qsovs 1/5 fardags (1 q), 1 xevsuri

- 1/6 fardags (1 q). amitom ufro swrafad qsovs TuSi (1 q).

II. 3 TuSi da 4 xevsuri mqsoveli erTad muSaobiT 1 kviraSi

moqsovs 3 + 4 = 38 = 19 fardags (2 q), amitom 19 aseTive

5 6 30 15

fardags moqsoven 15 kviraSi (1 q).

sakontrolo wera # 14 rigi I

1. I. = (–64 – 81)·(–2)–28 = (–145)·(–2)–28 = –290 –28 = –318 (4 q).

2. I. –19x +18 (2 q); II. 22a – 32 (2 q).

3. I. 5y – 48 = 18 + 4y – 1 (2 q); y = 65 (2 q).

4. swori naxazi - 4 q.

5. a) 1 x + 1 x = 1 (3 q).

4 5

dasabuTeba: xelosani 1 sT-Si Seasrulebs mTeli samuSaos 1/4

nawils, Segirdi _ 1/5 nawils, orive erTad x sT-Si Seasruleben


- 139 -

1 x + 1 x nawils, rac 1 mTelis tolia (3 q).

4 5

rigi II

1. II. = (9 – 125)·(–3)–12 = (–116)·(–3)–12 = 348 –12 = 336 (4 q).

2. III. –5b – 35 (2 q); IV. 9x – 42 (2 q).

3. I. 6x + 40 = –32 + 5x + 2 (2 q); x = –70 (2 q).

4. swori naxazi - 4 q.

5. d) 1 + 1 = 1 (3 q).

6 7 y

dasabuTeba: glexi 1 sT-Si Toxnis yanis 1/6 nawils, misi vaJi _

1/7 nawils, orive erTad - 1 + 1 nawils, rac 1 -is toli unda

6 7

y

iyos, radgan orive erTad mTel yanas y sT-Si Toxnian (3 q).

sagamocdo wera rigi I

1. = 25 · (–8) + 30 + 2 = 232 (4 q).

2. I. 8 (1 q); II. 3 (1 q); III. 1,5 sm (1 q); IV. 24 kg (1 q).

3. I. 1700 (2 q); II. 30% (2 q).

4. I. 6 · 10 41 (2 q); III. 2 · 10 10 (2 q).

5. I. sasrulia (1 q); II. usasruloa (1 q);

III. usasruloa (1 q); IV. sasrulia (1 q).

6. I. 20z –19 (2 q); II. 15m + 16 (2 q).

7. I. –2x – 18 = 26 – 12x (2 q); x = 4,4 (2 q).

8. A wveroze CD gverdis paraleluri wrfis gavleba - 2 q;

D wveroze BC gverdis marTobuli wrfis gavleba - 2 q.

9. I. ABC samkuTxedis daxazva - 2 q; wrexazis Semoxazva - 2 q.

10. wlis bolos fiWvis simaRle iqneba 1,8 m (3 q).

meore wlis bolos fiWvi gaizarda 2,34 m – 1,8 m = 0,54 m-iT,

0,

54

anu · 100 = 30%-iT (3 q).

1,

8

rigi II

1. = 27 · 4 + 12 – 2 = 118 (4 q).

2. V. 14 (1 q); VI. 7,5 (1 q); VII. 5 m (1 q); VIII. 40 t (1 q).

3. III. 1300 (2 q); IV. 40% (2 q).

4. II. 9 · 10 26 (2 q); IV. 9 · 10 15 (2 q).


- 140 -

5. V. sasrulia (1 q); VI. usasruloa (1 q);

VII. sasrulia (1 q). VIII. usasruloa (1 q);

6. III. –78x +30 (2 q); IV. 12n – 12 (2 q).

7. II. –3y + 15 = 5 – 5y (2 q); y = –5 (2 q).

8. Q wveroze MN gverdis paraleluri wrfis gavleba - 2 q;

N wveroze MQ gverdis marTobuli wrfis gavleba - 2 q.

9. I. MNP samkuTxedis daxazva - 2 q; wrexazis Semoxazva - 2 q.

10. wlis bolos naZvis simaRle iqneba 1,5 m (3 q).

meore wlis bolos fiWvi gaizarda 1,8 m – 1,5 m = 0,3 m-iT, anu

0,

3

· 100 = 20%-iT (3 q).

1,

5

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"Математика в школе", Москва-1999, # 2.

31. З. Ваханиа. Современное образование: Книга и Экран. -

"The Journal of Eurasian Research", Москва-2002, # 3.

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