nbUfnbujljt !txbwmfcb!!JW!lmbtTj! - Ganatleba

nbUfnbujljt 

q o r b u d a 

!txbwmfcb!!JW!lmbtTj! 

nfUpejlvsj!tbyfmn[Swbofmp!nbtxbwmfcmjtUwjt! 

avtori: 

zurab vaxania 

fsiqologiis institutis direqtoris moadgile, 

pedagogikis mecn. doqtori, 

maTematikis mecn. kandidati, 

ganaTlebis mecnierebaTa akademikosi. 

saredaqcio sabWo: 

dodo naziriSvili, Tamar bardaveliZe, elza 

goCitaSvili, manana begiaSvili, xaTuna svianaZe 

sagamomcemlo samuSaoTa Semsrulebelni: 

zurab vaxania, naTela Sulaia, nikoloz vaxania 

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

 

maswavlebelma jer dawyebiTi klasebis zogadi meTodikuri 

saxelmZRvanelo unda waikiTxos. _ es wigni, iseve rogorc 

sxva meTodikuri saxelmZRvaneloebi, internetSicaa ganTavsebuli, 

misamarTze: www.uznadzeinstitute.ge → saxelmZRvaneloebi). 

esaa me-4 klasis ZiriTadi saxelmZRvanelo, romliTac 

unda ixelmZRvanelos maswavlebelma (da ara mxolod 

moswavlis saxelmZRvaneloTi). maswavleblisTvis 

aucilebelia, rom Seasrulos aq aRwerili saklaso 

samuSaoebi - umetesad zepiri, saTanado TvalsaCinoebaze 

damyarebiT. mxolod moswavlis saxelmZRvaneloTi 

muSaoba Zalian arasakmarisia! 

- 1 -

1. me-4 klasis saxelmZRvaneloebi da TvalsaCinoeba 

ganaTlebuloba aris ara codnis didi odenoba, aramed 

naswavlis Rrma gageba da kargad gamoyeneba... codnas 

aucileblad unarCveva unda axldes. savalaloa, roca 

moswavlis goneba avsebulia codnis didi an mcire 

odenobiT, magram man ar icis misi gamoyeneba. man 

raRaca icis, magram araferi xelewifeba. a.distervegi. 

es saxelmZRvanelo agrZelebs da aviTarebs I-III klasebis 

saxelmZRvaneloTa ZiriTad mimarTulebebsa da safuZvlebs. 

mTavar safuZvlad kvlav aqtiuri mzaobis meTodika da gaRrmavebuli 

swavleba rCeba. 

zogadi meTodikuri saxelmZRvanelo saerToa yvela dawyebi- 

Ti klasisaTvis. masSi gadmocemuli zogadi Tu kerZo principebi, 

maTematikur sakiTxTa ganmartebani, meTodikuri siaxleni 

da sxva uTuod gasaTvaliswinebelia IV klasSic. amitom 

aucilebelia am saxelmZRvanelos saguldagulod wakiTxva. 

umTavres sakiTxebs mainc gavimeorebT. Cveneuli meTodika da 

saxelmZRvaneloebi ise agebuli, rom gaTvaliswinebulia 

sxvadasxva monacemebisa da midrekilebaTa mqone 

moswavleebisTvis. Cven gvsurs, rom yvela moswavles Rrmad da 

aqtiurad vaswavloT da arc Zlieri moswavlis damatebiTi 

ganviTareba SevaferxoT. amazea gaTvlili saxelmZRvaneloebis 

mTliani moculoba da sirTulis done. mTavaria, rom: 

moswavlisaTvis araa savaldebulo yvela amocanis 

amoxsna. amocanaTa mxolod naxevris amoxsnac ki 

sakmarisia. es yovelive unda ganvumartoT mSoblebsac. 

yvelam gaakeTos imdeni amocana, ramdensac SeZlebs 

zomieri ZalisxmeviT, didi daZabvisa da wvalebis 

gareSe. Semdeg ki xSirad moxdeba, rom masw.-ica da mSobelic, 

maTda gasakvirad, aRmoaCenen, rom moswavles arc ise gauWirdeba 

xolme erTi SexedviT Zneli da didi davalebis Sesruleba. esaa 

ganmaviTarebeli swavleba... 

yvela SemTxvevaSi, Tuki moswavles ara aqvs 

Sesrulebuli saSinao davalebis SedarebiT rTuli 

amocana, igi klasSi unda daweros _ am amocanis 

- 2 -

erToblivi garCevisas. 

Cveneuli meTodikiT ricxvebica da maTze moqmedebebic 

sakmaod uCveulod iswavleba. magaliTad, SedarebiT gvian 

Semogvaqvs xolme qveSmiweris wesebi. amitom masw.-ma unda 

gaafrTxilos moswavleTa mSoblebi, rom swavlebis meTodikaSi 

CareviT saqme ar gaafuWon! ufro metic, Tavad masw.-sac 

gamudmebiT unda axsovdes is, Tu ra aqvs naswavli klass da 

uneburi naadrevi gamoTqmebiT ar daarRvios programa. garda 

amisa, masw.-ma unda ixmaros mxolod is terminebi, romlebic am 

saxelmZRvaneloSia mocemulia (zedmetia, magaliTad, termini 

`saTanrigo erTeuli~ da sxva). 

IV klasis programis ZiriTadi nawilia mravalniSna 

ricxvebze moqmedebaTa wesebi (algoriTmebi). saWiroa maTi 

gaSinaarsianeba _ da ara mxolod meqanikuri damaxsovreba. 

sazogadod, masw.-i yovelnairad unda ecados, rom 

moswavleebs gaaazrebinos gamoTvlebis wesebi _ 

ratomaa saWiro swored amgvari moqmedebani. 

Cveni programa aqeTkenaa mogezili. Tumca yvela moswavlisTvis 

es gaazreba albaT miuwvdomeli iqneba; zogi moswavle gaiazrebs 

nawilobriv, zogi _ mTlianad da Tanmimdevrulad. 

magram saqme isaa, rom gegma-konspeqtebSi mocemulia mxolod 

mSrali da SekumSuli msjelobani. masw.-ma isini ki ar unda 

gaimeoros, aramed unda gaSalos. garda amisa, gaSlis Semdeg, 

isini unda `gaacocxlos~ _ unda daurTos xelis aqtiuri moZraobebi: 

dafaze daweril Canawerebze TiTiT unda aCvenos sa- 

Tanado ricxvebi Tu moqmedebaTa naSnebi. magaliTad, teqstSi 

naxsenebia sityva `gasayofi~. masw.-i am sityvas ambobs, magram Tan 

dafaze TiTs daadebs gasayofis cifrul Canawers, ramdenjerme 

daadebs, TiTiT Semoxazavs kidevac, raTa moswavleebma kargad 

dainaxon da daakavSiron ganyenebuli sityva `gasayofi~ gansaxilveli 

magaliTis konkretul gasayofTan. 

da rac mTavaria: masw.-ma ganxilvani ar unda warmarTos 

sakuTari msjelobis saxiT. yovel msjelobas 

aucileblad unda mieces kiTxva-pasuxis saxe, iseTive, 

rogoric iyo wina klasebSi! ar unda iTqvas TiTqmis 

- 3 -

arcerTi gamzadebuli TxrobiTi winadadeba manam, sanam 

klasis winaSe ar dasmula saTanado SekiTxva. 

araviTari mniSvneloba ara aqvs swrafad, sxapasxupiT 

pasuxs. zogi adamiani bunebiT nelia, zogi ki _ 

morcxvi! piriqiT, umjobesia, rom bavSvi mieCvios dafiqrebas, 

damoukideblad gaazrebas, pasuxis Camoyalibebas 

_ Tundac SecdomiT. bavSvs odnavadac ar unda eSinodes 

Secdomisa, maswavlebels ukmayofileba xmaSi an gamometyvelebazec 

ki ar unda Seetyos. 

magaliTad, teqstSi weria: `gaiyo unaSTod~. masw.-i amas ar 

ityvis. is ikiTxavs: _ unaSTod gaiyo Tu naSTiT? ratom? _ da 

mxolod moswavleTa pasuxis Semdeg daadasturebs: _ gaiyo 

unaSTod. anda, teqstSi weria: `namravli naklebia gamyofze~. 

masw.-i amas ar ambobs, is ikiTxavs: _ risi tolia namravli? igi 

metia Tu naklebia gamyofze? _ da mxolod moswavleTa pasuxis 

Semdeg daadasturebs. 

ufro metic, amocanebis garCevisas an ukve naswavli sakiTxis 

Semdgomi ganmtkicebisas dafasTan ZiriTadad mxolod moswavleebi 

unda muSaobdnen (cot-cota xniT, xSiri cvliT). masw.-i 

mxolod iSviaTad unda Caerios, Tuki vercerTma moswavlem ver 

gaarTva Tavi sakiTxs (sasurvelia, es arc xdebodes). 

Cven xSirad vwerT ase: `dafaze daiweros~, `daixazos 

dafaze~ da sxva. rogorc wesi, yvelagan vgulisxmobT, 

rom amas akeTebs moswavle (iSviaTia iseTi SemTxveva, rom 

vercerTma maTganma es ver SeZlos da masw.-s mouwios). xolo 

roca dafasTan masw.-ma unda imuSaos, amas sagangebod mivuTiTebT. 

zvsbeSfcb" roca sruldeba qveSmiweriT moqmedeba an 

advili xerxiT gamoTvla da sxva _ saWiroa moswavle 

xmamaRla, sityvierad ambobdes yvelafer imas, rasac 

akeTebs: aq davweroT es 3; 2-jer 3 tolia 6-isa, amitom aq 

vwer 6-s... aq viyenebT TanamamravlTa gadajgufebis wess, es 

Tanamamravli aqeT gadmovanacvloT, davweroT... es ar akldeba, 

amitom aqedan visesxoT erTi aseuli... ese igi, TviTeuli 

- 4 -

nabiji kidec unda aRiweros da kidec unda 

dasabuTdes. gegma-konspeqtebSi Cven amas zogan isev 

mivuTiTebT, raTa masw.-s ar daaviwydes. magram ise ki 

saWiroa yovelTvis! 

jer erTi, es aadvilebs damaxsovrebas; meorec, amcirebs Secdomis 

albaTobas; mesamec, xels uwyobs wesis gaazrebisaTvis 

gamiznul samuSaoebs. moswavleebi unda mieCvivnon gamoTvlis 

wesis (algoriTmis) TviTeuli nabijis xmamaRla aRweras. 

amrigad, gegma-konspeqtebSi mocemulia mxolod msjelobaTa 

sqemebi, romlebic masw.-ma unda gaSalos 

cocxal aqtiur ganxilvebad. 

iseve rogorc III klasSi, moswavlis saxelmZRvanelo iyofa 

Tavebad. yoveli Tavis dasrulebisas tardeba sakontrolo wera. 

saxelmZRvanelos bolos (iseve, rogorc III klasSi) 

amocanaTa Tematikuri krebulia. igi Sedgeba ara Tavebis, 

aramed paragrafebisagan. 

Tavsa da paragrafs Soris arsebiTi gansxvavebaa. 

Tavi dalagebulia kalendarulad da zustadaa dayofili 

gakveTilebis mixedviT. xolo paragrafebi Tematikuria: 

maTSi amocanebi Temebis mixedviTaa dajgufebuli. 

paragrafebis gavla unda moxdes ara miyolebiT, 

aramed erTmaneTis paralelurad. konkretulad Tavad 

masw.-i gaanawilebs sakuTari surviliT. 

moswavleTa saxelmZRvaneloSi mocemuli amocanebi ZiriTadad 

saSinao davalebebisTvisaa gankuTvnili. klasSi xdeba 

Sesrulebuli davalebis erToblivi garCeva. Tuki masw.-i 

Seatyobs, rom klass gauWirda romelime amocanis amoxsna, maSin 

damatebiT amoixsneba am amocanis magvari axali amocana, 

romelSic Secvlili iqneba mxolod ricxvebi Tu raime 

araarsebiTi monacemebi: magaliTad, kalaTaSi msxlebi Seicvleba 

buCqze vardebiT, anda erTi marTkuTxedi Seicvleba sxvagvari 

proporciebis mqone marTkuTxediT da sxva. aseT saxesxvaobebs 

masw.-i Tavadve advilad SeTxzavs, roca dasWirdeba. 

amgvarad, gakveTilze saSinao davalebis amocanebis 

- 5 -

saTiTaod Semowmeba-garCeva (da saWiroebis SemTxvevaSi 

davalebis romelime amocanis magvari axali amocanis 

amoxsnac) unda moeswros. amas zogjer naxevar 

gakveTilamde sWirdeba xolme, zogjer _ metic. es 

damokidebulia imaze, Tu rogor Seasrula klasma davaleba. amis 

winaswar dagegmva SeuZlebelia. 

Tematikur krebulSi gabneulia agreTve is amocanebi, 

romlebic klass sakontrolo werebze unda mieces. saamisod 

gaTvaliswinebul amocanaTa zusti nusxa gegma-konspeqtebSi 

saTanado adgilebSicaa mocemuli. magram am nusxas srulad aqac 

gadmovcemT. masw.-s 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. masw.-s 

ecodineba, rom mis mier wiTlad moniSnuli amocanebi _ 

mxolod sakontrolo werebisaTvisaa. 

s.w. # 1 

I r. § 1/2 § 1/3 I § 1/9 § 3/2 § 2/8 I, II 

II r. § 1/7 § 1/3 II § 1/13 § 3/5 § 2/8 III, IV 

s.w. # 2 

I r. § 1/5 § 1/17 § 1/21 I, II § 3/3 § 1/24 

II r. § 1/11 § 1/22 § 1/21 III, IV § 3/7 § 1/27 

s.w. # 3 

I r. § 1/29 I § 1/33 I § 2/11 § 1/40 I § 3/9 

II r. § 1/29 II § 1/33 II § 2/16 § 1/40 II § 3/15 

s.w. # 4 

I r. § 1/37 I-III § 1/42 § 3/10 I § 3/11 § 1/44 I 

II r. § 1/37 IV-VI § 1/46 § 3/10 II § 3/16 § 1/44 II 

s.w. # 5 

I r. § 1/48 § 1/50 I-IV § 2/18 § 1/55 I § 3/17 

II r. § 1/52 § 1/50 V-VIII § 2/21 § 1/55 II § 3/21 

s.w. # 6 

I r. § 1/56 I § 1/59 I-II § 3/18 § 2/23 § 4/4 

II r. § 1/56 II § 1/59 III-IV § 3/20 § 2/28 § 4/12 

s.w. # 7 

I r. § 1/62 I § 1/64 § 3/22 § 2/39 § 4/7 I 

II r. § 1/62 II § 1/69 § 3/26 § 2/44 § 4/7 II 

- 6 -

s.w. # 8 

I r. § 1/71 § 1/76 I § 2/46 § 1/77 § 1/81 

II r. § 1/74 § 1/76 II § 2/34 § 1/79 § 1/86 

s.w. # 9 

I r. § 1/83 § 1/90 § 1/95 I-II § 3/28 § 2/13 I-II 

II r. § 1/88 § 1/93 § 1/195 III-IV § 3/33 § 2/13 III-IV 

s.w. # 10 

I r. § 1/98 I § 1/103 I § 1/106 I-III § 2/30 § 3/30 

II r. § 1/98 II § 1/103 II § 1/106 IV-VI § 2/42 § 3/34 

s.w. # 11 

I r. § 1/108 I § 1/110 I § 2/53 § 4/38 § 4/14 

II r. § 1/108 II § 1/110 II § 2/48 § 4/44 § 4/11 

s.w. # 12 

I r. § 1/113 I § 1/116 I-III § 1/124 I § 2/50 § 3/37 

II r. § 1/113 II § 1/116 IV-VI § 1/124 II § 2/52 § 3/43 

s.w. # 13 

I r. § 1/120 I § 1/119 I § 1/126 II-III § 2/7 § 1/100 I 

II r. § 1/120 II § 1/119 II § 1/126 IV-V § 2/25 § 1/100 II 

s.w. # 14 

I r. § 1/128 I § 1/131 I § 3/94 I-IV § 3/39 § 1/122 

II r. § 1/128 II § 1/131 II § 3/94 V-VIII § 3/48 § 1/127 

s.w. # 15 

I r. § 1/133 I § 3/44 § 3/46 § 1/138 I-II § 2/35 

II r. § 1/133 II § 3/50 § 3/65 § 1/138 III-IV § 2/37 

s.w. # 16 

I r. § 1/140 I § 3/72 I-III § 1/171 I § 3/56 § 4/20 

II r. § 1/140 II § 3/72 IV-VI § 1/171 II § 3/59 § 4/32 

s.w. # 17 

I r. § 1/144 I § 3/102 § 3/60 § 2/49 § 3/69 

II r. § 1/144 II § 3/106 § 3/107 § 2/51 § 3/75 

s.w. # 18 

I r. §1/149 I § 1/152 I-III § 1/153 I-IV § 3/62 I-IV § 3/73 

II r. §1/149 II § 1/152 IV-VI § 1/153 V-VIII § 3/62 V-VIII § 3/79 

s.w. # 19 

I r. § 1/158 I § 1/159 § 3/101 § 1/173 I-III § 4/19 

II r. § 1/158 II § 1/170 § 3/108 § 1/173 IV-VI § 4/26 

- 7 -

I r. § 1/175 I § 1/161 § 3/89 § 4/33 § 4/30 

II r. § 1/175 II § 1/165 § 3/98 § 4/42 § 4/36 

I r. § 1/166 I § 1/179 § 4/39 § 4/13 § 3/87 

II r. § 1/166 II § 1/176 § 4/43 § 4/16 § 3/91 

s.w. # 20 

s.w. # 21 

sakontrolo weris Sefasebis zogadi principi gadmocemulia 

zogadi meTod-saxelmZRvanelos me-10 paragrafSi. 

sakontrolo werebis amocanaTa garCeva da maTi qulebrivi 

Sefasebis konkretuli kriteriumebi mocemulia gegma-konspeqtebis 

saTanado adgilebSi. 

sakontrolo weris dros moswavles ufleba aqvs, 

SeuzRudavad gamoiyenos saxelmZRvanelo da 

saxelmZRvanelos ydaze mocemuli TvalsaCinoebani. 

Tumca, roca TvalsaCinoeba araa saWiro, masw.-i urCevs, 

rom zedmetad ar dakargon dro. 

gegma-konspeqtebSi moswavlis saxelmZRvanelos gakveTilebi 

ormagi numereciiTaa aRniSnuli, magaliTad: 11.7. es aRniSnavs: 

`me-11 Tavis me-7 gakveTili~. 

gegma-konspeqtebSi gakveTilebi erTiani da uwyveti 

numeraciiTaa _ Tavebis gareSe. aRniSnulia agreTve sagakveTilo 

monakveTebi. magaliTad, 67.4 aRniSnavs 67-e gakveTilis me-4 

sagakveTilo monakveTs. magram 67-e gakveTilis gegma-konspeqtis 

dawyebamde, anu 66-e gakveTilis gegma-konspeqtis damTavrebis 

Semdeg, miTiTebulia agreTve moswavlis saxelmZRvanelos 

Sesabamisi gakveTilis nomeric: 11.7. ese igi, 11.7-is amocanebi 

klass saSinao davalebad eZleva 66-e gakveTilze, xolo klasSi 

gairCeva 67-e gakveTilis dasawyisSi, anu sagakveTilo 

monakveTSi 67.1. 

es yvelaferi Zalian aadvilebs saWiro gakveTilebis moZebnas. 

Tumca, yoveldRiurad, masw.-s sul ar dasWirdeba am 

nomrebisaTvis yuradRebis miqceva _ is xom uwyvetad mihyveba 

gegma-konspeqtebs da yoveldRiurad agrZelebs iqidan, sadac 

wina dRes gaCerda, nomrebisagan damoukideblad. 

IV klasSi saWiroa cota TvalsaCinoeba. isini moswavlis 

saxelmZRvanelos ydis Siga mxaresaa: erTia ariTmetikuli (gam- 

- 8 -

avleba-gayofa ricxvTa xazze), mcired Secvlili _ axlavs 

moswavlis saxelmZRvanelos (ydis Siga mxares); meorea geometriuli 

_ igive, rac iyo II-III klasebSi. cxadia, sasurvelia, 

rom amgvari TvalsaCinoeba saklaso oTaxSic ekidos. 

TvalsaCinoebani moswavlem saklaso merxze unda gamoiyenos 

xolme, da agreTve saSinao davalebis Sesrulebisas. moswavle 

maT xSir-xSirad unda iyenebdes. 

saklaso TvalsaCinoebaTagan damatebiT aucilebelia: 

1. Zveleburi koWebiani saangariSo, 10-Tanrigiani (oRond, 

unda gadakeTdes ise, rom nakluli, 4-koWiani Tanrigi bolomde 

Seevsos) _ esaa IV klasis mTavari TvalsaCinoeba! 

2. didi metriani saxazavi; 

3. didi isrebiani saaTi _ gafuWebuli, romelsac moxsnili 

aqvs wina SuSa, raTa SeiZlebodes isrebis TiTiT amoZraveba; 

4. stereometriul sxeulTa xis, muyaos an plastmasis didi 

modelebi (oRond maT wiboebi da wveroebi ar unda hqondes 

gadacveTili, mkveTrad unda etyobodes). saWiroa: kubi, aguredi 

(marTkuTxa paralelepipedi), piramida da prizma. sasurvelia, 

TiTo iyos ori-sami sxvadasxva zomebisa da formisa. 

masw.-ma unda Seaxsenos klass, rom saxelmZRvaneloSi araferi 

ar unda Caweron Tu Caxaton! wera da xazva _ mxolod rveulebSi 

iqneba. moswavleebs dasWirdebaT Cveulebrivi ujredebiani 

rveulebi. 

saukeTeso TvalsaCinoeba, Tanac aqtiuri _ kompiuterul programebSia. 

masw.-s vurCevT, rom, Tuki skolas amis saSualeba 

aqvs, xandaxan mainc gamoiyenos lpnqjvufsj: saswavlo diskebi 

an internetsaitebi, mag: http://nlvm.usu.edu/en/nav/vlibrary.html an 

http://forum.dlf.ge/saskolo-tanamshromlobiti qselebi; http://soft.dlf.ge. 

IV klasis maTematikisTvis am saitebzea, kerZod, Cveneuli 

programis erTerTi mTavari TvalsaCinoeba _ Zveleburi saangariSo, 

romlis gamoyeneba ufro nakleb saxalisoa, vidre 

kompiuteris ekranze Catareba moqmedebisa. Tanac, amas TiTqmis 

araviTari specialuri codna ar sWirdeba, Zalian advilad 

keTdeba. imave saitebzea Cveneuli programiT gaTvaliswinebuli 

sxva TvalsaCinoebac, magaliTad: jadosnuri kvadratebi, 

moqmedebebi geometrul nakvTebze, maTi xazva da sxva mravali. 

- 9 -

3. kalendaruli programa 

Sesabamisoba saxelmwifo saswavlo gegmasTan 

(mxolod ZiriTadi saprogramo xazebi _ TamaSebisa da 

amocanebis sakiTxTa gareSe) 

kalendaruli programa daaxloebiTia. zustad ver davgegmavT, 

Tu ramdeni gakveTili dasWirdeba III klasis saxelmZRvanelos 

momTavrebas. Tanac, SeiZleba ramdenjerme moxdes, rom 

gacdenebisa Tu sxva mizezTa gamo masw.-ma ramdenjerme SeaCeros 

saswavlo programaSi winsvla TiTo-TiTo gakveTiliT. Tumca, am 

dros klass saSinao davaleba mainc unda mieces _ amocanaTa 

Tematikuri krebulidan unda SevurCioT, ar unda dairRves Cveni 

wesi _ arcerTi dRe saSinao davalebis gareSe! 

aq aRwerilTa garda danarCeni xazebi (algebra, logika, 

modelireba, kombinatorika, topologia) pirdapir mihyveba 

moswavlis saxelmZRvanelos. iqve mocemulia aucilebeli 

Teoriuli ganmartebani. 

TviTeuls Tavs miwerili aqvs Sesabamisi misaRwevi Sedegebi 

saxelmwifo saswavlo gegmidan. 

Tavi I _ IV.1, IV.2, IV.5, IV.6, IV.10, IV.12, IV.14 

1. gameoreba: 050=50, 009=09=9, 023=23 da sxva. 

ricxvSi erTeulebi, aTeulebi, aseulebi, ricxvSi maTi raodenoba. 

samniSna ricxvis akrefa koWebian saangariSoze. 



samniSna ricxvis akrefa saangariSoze. 



samniSna ricxvis akrefa saangariSoze. 

4. oTxi maTematikuri (ariTmetikuli) moqmedeba da ricxviTi 

gamosaxuleba. 

5. oTxi maTemat. (ariTmetikuli) moqmedeba da ricxviTi gamosaxuleba. 

6. gameoreba _ samniSna ricxvebis qveSmiweriT Sekreba-gamokleba. 

ratomaa marTebuli misi wesebi. Sekreba-gamokleba saangariSoze. 

moqmedebaTa rigi ricxviT gamosaxulebaSi. ricxviTi gamosaxulebis 

mniSvneloba. 

7. oTxniSna ricxvebi. oTxniSna ricxvis wakiTxva. saangariSoze 

- 10 -

mocemuli raodenobis sityvieri dasaxeleba da cifruli Cawera 

dafaze. piriqiT: sityvierad an cifrulad mocemuli ricxvis àkrefa~ 

saangariSoze. 

8. oTxniSna ricxvebi. oTxniSna ricxvis wakiTxva. saangariSoze 

mocemuli raodenobis sityvieri dasaxeleba da cifruli Cawera 

dafaze. piriqiT: sityvierad an cifrulad mocemuli ricxvis àkrefa~ 

saangariSoze. 

gameoreba _ samniSna ricxvebis qveSmiweriT Sekreba-gamokleba. 

ratomaa marTebuli misi wesebi. Sekreba-gamokleba saangariSoze. 

9. sak. wera # 1. 

Tavi II _ IV.2, IV.3, IV.5, IV.6, IV.8, IV.9, IV.14 

10. Sekrebis wesebi: SesakrebTa gadanacvleba da dajgufeba 

(a+b)+c = a+(b+c) = (c+a)+b... 

oTxniSna ricxvebi. oTxniSna ricxvis wakiTxva. 

saangariSoze mocemuli raodenobis sityvieri dasaxeleba da 

cifruli Cawera dafaze. piriqiT: sityvierad an cifrulad mocemuli 

ricxvis àkrefa~ saangariSoze. 

11. Sekrebis wesebi: SesakrebTa gadanacvleba da dajgufeba 

(a+b)+c = a+(b+c) = (c+a)+b... 

11. zepiri angariSi oTxniSna ricxvebze, 10000-is farglebSi (advili 

SemTxvevebi). 

12. laTinuri asoebi. 

13. Sekreba advili xerxiT: a+b+c+d = (a+c)+(b+d) da sxva. 

oTxniSna ricxvebis zepiri Sekreba-gamokleba 10000-is farglebSi 

(advili SemTxvevebi). 

14. jamis gamokleba: a - (b+c+d) = a - b - c - d. 

saSinao davalebidan me-5 gakveTilis me-2 amocana gairCeva dawvrilebiT 

da dasknebi Camoyalibdeba am magaliTis ganzogadebiT. 

nakvTebis aRniSvna laTinuri asoebiT. 

15. jamis gamokleba: a - (b+c+d) = a - b - c - d. gamokleba advili 

xerxiT: a - b - c - d = a - (b+d) - c da sxva. 

16. nakvTebis aRniSvna laTinuri asoebiT. 

Sekreba advili xerxiT; gamokleba advili xerxiT. 

17. sak. wera # 2. 

Tavi III _ IV.2, IV.3, IV.5, IV.7, IV.9, IV.13, IV.14 

18. oTxniSna ricxvebis zepiri Sekreba-gamokleba 10000-is 

farglebSi: mrgval aseulamde an aTaseulamde Sevsebis xerxiT. 

- 11 -

19. mravalkuTxedis perimetri. 

gamravlebis wesebi: TanamamravlTa gadanacvleba da dajgufeba. 

20. gamravlebis wesebi: TanamamravlTa gadanacvleba da dajgufeba. 

oTxniSna ricxvebis zepiri Sekreba-gamokleba 10000-is 

farglebSi: mrgval aseulamde an aTaseulamde Sevsebis xerxiT. 

21. gamravlebis wesebi: TanamamravlTa gadanacvleba da dajgufeba. 

gamravleba advili xerxiT: a · b · c · d = (a · c) · (b · d) da sxva. 

22. gantoleba da misi amoxsna. 

gamravlebis wesebi: TanamamravlTa gadanacvleba da dajgufeba. 

gamravleba advili xerxiT: a · b · c · d = (a · c) · (b · d) da sxva. 

23. gamravleba advili xerxiT: a · b · c · d = (a · c) · (b · d) da sxva. 

dawvrilebiT gairCeva saSinao davalebidan me-3 amocana. 

gantoleba da misi amoxsna. 

24. sak. wera # 3. 

Tavi IV _ IV.1, IV.2, IV.3, IV.5, IV.8, IV.10, IV.14 

25. gamravlebisa da Sekrebis wesebi _ frCxilebis gaxsna: 

a · (b+c) = a · b + a · c + a · d (b+c+d) · a = b · a + c · a + d · a. 

26. gamravlebisa da Sekrebis wesebi _ frCxilebis gaxsna: 


gameoreba: samniSna ricxvebis Sedareba. 

saangariSoze mocemul raodenobaTa Sedareba aTasis farglebSi. 

27. xuTniSna da eqvsniSna ricxvebi. saTanrigo sameulebi. 

ricxvebis wakiTxva da Cawera. saangariSoze mocemuli raodenobis 

sityvieri dasaxeleba da cifruli Cawera dafaze. piriqiT: sityvierad 

an cifrulad mocemuli ricxvis àkrefa~ saangariSoze. 





gameoreba: samniSna ricxvebis Sedareba. 

saangariSoze mocemul raodenobaTa Sedareba aTasis farglebSi. 





ricxvebis Sedareba milionis farglebSi. saangariSoze mocemul 

raodenobaTa Sedareba milionis farglebSi. 

- 12 -

30. sak. wera # 4. 

Tavi V _ IV.1, IV.2, IV.3, IV.5, IV.10, IV.14 

31. mravalniSna ricxvebis qveSmiweriT Sekreba-gamokleba 

milionis farglebSi. 

32. mravalniSna ricxvebis qveSmiweriT Sekreba-gamokleba milionis 

farglebSi. 

ricxvebis Sedareba milionis farglebSi. 

saangariSoze mocemul raodenobaTa Sedareba milionis farglebSi. 

33. ricxvis gamravleba-gayofa 10-is xarisxebze. 

1000-is farglebSi advili gamravleba-gayofa _ rasac ar sWirdeba 

qveSmiwera. 

34. ramdenime ricxvis erTad Sekreba qveSmiweriT mln-is farglebSi. 

1000-is farglebSi advili gamravleba-gayofa _ rasac ar sWirdeba 

qveSmiwera. ricxvis gamravleba-gayofa 10-is xarisxebze: 1000-ze, 

10000-ze, 100000-ze. 

35. mravalniSna ricxvebis zepiri Sekreba-gamokleba milionis 

farglebSi (advili SemTxvevebi). 

ricxvis gamravleba-gayofa 10-is xarisxebze: 1000-ze, 10000-ze, 

100000-ze. 

36. sak. wera # 5. 

Tavi VI _ IV.1, IV.2, IV.3, IV.8, IV.9, IV.10, IV.14 

37. mravalniSna ricxvebis zepiri Sekreba-gamokleba 

milionis farglebSi (advili SemTxvevebi). 

mravalniSna ricxvebis qveSmiweriT Sekreba-gamokleba milionis 

farglebSi. 

38. Siga da gare areebi. 

ricxvis gamravleba-gayofa 10-is xarisxebze: 1000-ze, 10000-ze, 

100000-ze. 

39. gamravlebisa da gamoklebis wesebi _ frCxilebis gaxsna. 

a · (b-c) = a · b - a · c (b-c-d) · a = b · a - c · a - d · a. 

Siga da gare areebi. 

40. monakveTi, texili da mrudi. 

gamravlebisa da gamoklebis wesebi _ frCxilebis gaxsna. 

a · (b-c) = a · b - a · c (b-c-d) · a = b · a - c · a - d · a. 

41. Sekruli da gaxsnili xazebi. 

milionis farglebSi advili gamravleba-gayofa _ rasac ar 

sWirdeba qveSmiwera. 

- 13 -

42. erTmaneTis sapirispiro moqmedebebi; Sekrebis Semowmeba 

gamoklebiT, gamoklebisa _ SekrebiT, gamravlebisa _ gayofiT, gayofisa 

_ gamravlebiT. 

milionis farglebSi advili gamravleba-gayofa _ rasac 

ar sWirdeba qveSmiwera. 

43. sak. wera # 6. 

Tavi VII _ IV.2, IV.3, IV.6, IV.7, IV.8, IV.9, IV.10 

44. erTmaneTis sapirispiro moqmedebebi; Sekrebis 

Semowmeba gamoklebiT, gamoklebisa _ SekrebiT, gamravlebisa _ 

gayofiT, gayofisa _ gamravlebiT. 

45. wrexazi da misi xazva fargliT. 

frCxilebis gaxsnis wesis gamoyeneba gamravlebisas. 

46. SesaZleblobaTa amowera. 


47. SesaZleblobaTa amowera. 


48. nulebiT daboloebuli ricxvis advili gamravleba. 

49. wrexazi da misi xazva fargliT. 

49. sak. wera # 7. 

Tavi VIII _ IV.2, IV.3, IV.5, IV.7, IV.8, IV.11, IV.13 

50. nulebiT daboloebuli ricxvis advili gamravleba. 

51. nulebiT daboloebuli ricxvis advili gayofa. 

52. nulebiT daboloebuli ricxvis advili gayofa. 

53. gayofisa da Sekrebis wesi _ frCxilebis gaxsna: 

(a+b+c) : d = a : d + b : d + c : d. 


(a+b+c) : d = a : d + b : d + c : d. 

frCxilebis gaxsnis wesis gamoyeneba gayofisas. 


(a+b+c) : d = a : d + b : d + c : d. 


56. sak. wera # 8. 

Tavi IX _ IV.2, IV.3, IV.6, IV.7, IV.9, IV.11, IV.14 

57. frCxilebis gaxsnis wesis gamoyeneba gayofisas. 

mravalniSna ricxvis gayofa erTniSna ricxvze qveSmiweris gareSe. 

58. ormxrivi utolobani. 

frCxilebis gaxsnis wesis gamoyeneba gayofisas. mravalniSna ricxvis 

- 14 -

gayofa erTniSna ricxvze qveSmiweris gareSe. 

59. nulebiT daboloebuli ricxvis advili gayofa da frCxilebis 

gaxsnis wesis gamoyeneba gayofisas. 

60. nulebiT daboloebuli ricxvis advili gayofa da 


ormxrivi utolobani. 

61. mravalniSna ricxvis sxvagvari advili gayofa erTniSna ricxvze. 

62. mravalniSna ricxvis sxvagvari advili gayofa erTniSna ricxvze. 

63. sak. wera # 9. 

Tavi X _ IV.3, IV.6., IV.5, IV.7, IV.8, IV.11, IV.13 

64. ricxvis cifrTa jami. 

gameoreba: qveSmiweriT Sekreba-gamokleba. 

65. mravalniSna ricxvis qveSmiweriT gamravleba erTniSna ricxvze. 

66. mravalniSna ricxvis qveSmiweriT gamravleba erTniSna ricxvze. 

67. luwi da kenti ricxvebi. 

advili gamravleba orniSna ricxvze qveSmiweris gareSe. 

68. advili gamravleba orniSna ricxvze qveSmiweris gareSe. 

69. luwi da kenti ricxvebi. 

advili gamravleba orniSna ricxvze qveSmiweris gareSe. 

70. sak. wera # 10. 

Tavi XI _ IV.2, IV.3, IV.5, IV.6, IV.9, IV.10, IV.14 

71. gayofisa da gamoklebis wesi _ frCxilebis gaxsna: 

(a-b) : c = a : c - b : c. 

72. gayofisa da gamoklebis wesi _ frCxilebis gaxsna: 

(a-b) : c = a : c - b : c. 

73. mravalniSna ricxvebis qveSmiweriT gamravleba. 


gayofisa da gamoklebis wesi _ frCxilebis gaxsna: 

(a-b) : c = a : c - b : c. 


76. sak. wera # 11. 

Tavi XII _ IV.1, IV.2, IV.6, IV.7, IV.10, IV.12, IV.13 

77. milioni. ricxvebis akrefa saangariSoze. 

78. milioni. ricxvebi milionidan miliardamde. ricxvebis 

wakiTxva da Cawera. saangariSoze mocemuli raodenobis sityvieri 

dasaxeleba da cifruli Cawera dafaze. piriqiT: sityvierad an 

cifrulad mocemuli ricxvis àkrefa~ saangariSoze. 

- 15 -

79. ricxvebi milionidan miliardamde. ricxvebis wakiTxva da Cawera. 




80. ricxvebi milionidan miliardamde. ricxvebis wakiTxva da Cawera. 




mravalniSna ricxvebis qveSmiweriT gamravleba. 

81. nulebiT daboloebul TanamamravlTa gamravlebisas `nulebis 

`damaxsovreba~. 

82. nulebiT daboloebul TanamamravlTa gamravleba qveSmiweriT 

(nulebis `damaxsovreba~). 

83. nulebiT daboloebul TanamamravlTa gamravleba qveSmiweriT 

(nulebis `damaxsovreba~). 

84. sak. wera # 12. 

Tavi XIII _ IV.3, IV.4, IV.5, IV.8, IV.9, IV.10, IV.12 

85. qveSmiweriT gamravlebis gaadvilebis xerxebi. 

86. ricxvebis Sedareba miliardis farglebSi. saangariSoze 

mocemul raodenobaTa Sedareba miliardis farglebSi. 

qveSmiweriT gamravlebis gaadvilebis xerxebi. 

87. ricxvebis Cawera romauli niSnebiT. 


88. rigobrivi ricxvebis Cawera romauli niSnebiT. 

89. mravalkuTxedi. 

ricxvebis Sedareba miliardis farglebSi. saangariSoze mocemul 

raodenobaTa Sedareba miliardis farglebSi. 

90. rigobrivi ricxvebis marTlwera. 


91. sak. wera # 13. 

Tavi XIV _ IV.2, IV.3, IV.5, IV.7, IV.8, IV.12, IV.14 

92. saaTis cnoba _ jer martivad, sayofacxovrebo do- 

neze, Tan mxolod isrebiani saaTisa, amasTan erTad: 1 sT = 60 wT. 

gameoreba _ naSTiani gayofa Zneli SemTxvevebiTurT. 

93. saaTis cnoba _ jer martivad, sayofacxovrebo doneze, Tan 

mxolod isrebiani saaTisa, amasTan erTad: 1 sT = 60 wT. 


- 16 -

94. naxevari saaTi = 30 wuTi. saaTis cnoba. 


95. TamaSi: `swrafi pasuxebi~ _ ramdeni saaTisa da wuTis tolia. 

naSTiani gayofis Semowmeba. 

96. naSTiani gayofis Semowmeba. 

97. naSTiani gayofis Semowmeba. 

drois meoregvari aRricxva (saaTebi _ jer mxolod 1-dan 12-mde). 

98. sak. wera # 14. 

Tavi XV _ IV.2, IV.3, IV.5, IV.6, IV.10, IV.11, IV.14 

99. drois meoregvari aRricxva (saaTebi _ jer 

mxolod 1-dan 12-mde). 

100. drois mesamegvari aRricxva. cifruli saaTis cnoba. 

101. samniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze. 

102. dRe-Rame = 24 sT; ramdenime dReRamiT win an dReRamis Semdeg. 

samniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze. 

103. samniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze. 

drois mesamegvari aRricxva. cifruli saaTis cnoba. dRe-Rame=24 sT. 

104. dRe-Rame = 24 sT; ramdenime dReRamiT win an dReRamis 

Semdeg. drois mesamegvari aRricxva. cifruli saaTis cnoba. 

105. sak. wera # 15. 

Tavi XVI _ IV.3, IV.5, IV.6, IV.7, IV.9, IV.10, IV.11 

106. mravalniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze. 

107. drois meoTxegvari da mexuTegvari aRricxva. 

mravalniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze. 

108. mravalniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze. 

drois meoTxegvari da mexuTegvari aRricxva. 

109. 1 kvira = 7 dRe-Rame. kviris dReebis gameoreba. ramdenime kviriT 

win da kviris Semdeg. 1 Tve = 30 dRe; 1 weliwadi = 12 Tve. 


110. gayofisa da gamravlebis wesi _ frCxilebis gaxsna: 

a : (b · c) = a : b : c a : (b · c) = a : c : b. 


111. 1 kvira = 7 dRe-Rame. kviris dReebis gameoreba. ramdenime 

kviriT win da kviris Semdeg. 1 Tve = 30 dRe; 1 weliwadi = 12 Tve. 

gayofisa da gamravlebis wesi _ frCxilebis gaxsna: 

a : (b · c) = a : b : c a : (b · c) = a : c : b. 

112. sak. wera # 16. 

- 17 -

Tavi XVII _ IV.2, IV.5, IV.7, IV.9, IV.11, IV.13 

113. gayofa advili xerxiT. 

kalendari. gameoreba: weliwadis dayofa 12 Tved, weliwadis 

droebi. romeli Tve ramdendRiania, advilad dasamaxsovrebeli wesi 

_ muWebis amoburculi adgilebis mixedviT. kalendris xmareba. 

114. gayofa advili xerxiT. nulebiT daboloebuli mravalniSna 

ricxvis gayofa (nulebis `damaxsovreba~), maT Soris naSTiani 

SemTxvevebic. 



SemTxvevebic. 



SemTxvevebic. 

117. zepiri angariSi milionebze, oTxive moqmedebiT (miliardis 

farglebSi). 

118. zepiri angariSi milionebze, oTxive moqmedebiT (miliardis 

farglebSi). 

kalendari. gameoreba: weliwadis dayofa 12 Tved, weliwadis 

droebi. romeli Tve ramdendRiania, advilad dasamaxsovrebeli wesi _ 

muWebis amoburculi adgilebis mixedviT. nakiani weliwadis sakiTxi _ 

gakvriT. kalendris xmareba. 

119. sak. wera # 17. 

Tavi XVIII _ IV.1, IV.2, IV.5, IV.6, IV.9, IV.101, IV.14 

120. miliardi. akrefa saangariSoze. 

121. miliardi. ricxvebi 999 miliardamde. ricxvebis wakiTxva da 

Cawera. akrefa saangariSoze. 

122. ricxvebi 999 miliardamde. ricxvebis wakiTxva da Cawera. 

akrefa saangariSoze. 

qveSmiweriT gayofa orniSna ricxvze. 

123. qveSmiweriT gayofa orniSna ricxvze. 

124. ricxvebi 999 miliardamde. ricxvebis wakiTxva da Cawera. 

qveSmiweriT gayofa orniSna ricxvze. 

125. sak. wera # 18. 

Tavi XIX _ IV.2, IV.3, IV.5, IV.7, IV.8, IV.11, IV.14 

126. mravalniSna ricxvebis gayofis gaadvileba _ 

mxolod unaSTod: nulebiT daboloebuli ricxvebis erTmaneTze 

- 18 -

gayofa, rodesac gamyofs boloSi ara aqvs gasayofze meti nulebi. 

127. TariRebis cifruli Cawera da wakiTxva. 

mravalniSna ricxvebis gayofis gaadvileba _ mxolod unaSTod: 

nulebiT daboloebuli ricxvebis erTmaneTze gayofa, rodesac 

gamyofs boloSi ara aqvs gasayofze meti nulebi. 

128. TariRebis cifruli Cawera da wakiTxva. 


nulebiT daboloebuli ricxvebis erTmaneTze gayofa, rodesac gamyofs 

boloSi ara aqvs gasayofze meti nulebi. 

129. mravalniSna ricxvebis gayofis gaadvileba _ mxolod unaS- 

Tod: nulebiT daboloebuli ricxvebis erTmaneTze gayofa, rodesac 

gamyofs boloSi ara aqvs gasayofze meti nulebi. 

qveSmiweriT gayofa samniSna ricxvze. 

130. zepiri angariSi miliardebze, oTxive moqmedebiT. 


131. zepiri angariSi miliardebze, oTxive moqmedebiT. 


132. sak. wera # 19. 

Tavi XX _ IV.2, IV.3, IV.4, IV.6, IV.8, IV.9, IV.12 

133. stereometriis naswavli sakiTxebi _ gameoreba. 

zepiri angariSi miliardebze, oTxive moqmedebiT. 

134. mravalniSna ricxvebis gayofis gaadvileba _ mxolod 

unaSTod: gayofis advili SemTxvevebi. 

stereometriis naswavli sakiTxebi _ gameoreba. 

135. yovelgvari ricxvebis Sedareba. 


gayofis advili SemTxvevebi. 

136. yovelgvari ricxvebis Sedareba. 


gayofis advili SemTxvevebi. 

137. stereometriis sakiTxebi gonebaSi warmodgenil sxeulTaTvis. 

saerTo gameoreba. 

138. 1000 miliardi da ufro didi ricxvebi. 

saerTo gameoreba. 

139. sak. wera # 20. 

Tavi XXI _ IV.1, IV.2, IV.3, IV.4, IV.7, IV.8, IV.9 

140. gameoreba: ricxvSi aTeulTa, aseulTa, aTaseulTa raodenoba. 

- 19 -

1000 miliardi da ufro didi ricxvebi. 

141. ganayofi ramdenniSna unda miviRoT. 


stereometriis sakiTxebi gonebaSi warmodgenil sxeulTaTvis. 


stereometriis sakiTxebi gonebaSi warmodgenil sxeulTaTvis. 

144. stereometriis sakiTxebi gonebaSi warmodgenil sxeulebisTvis. 

yvela naswavli moqmedebis ganmtkiceba da ganviTareba. gansakuTrebuli 

yuradReba eqceva swrafi zepiri angariSis unarCvevebs. 

im sakiTxebis gameoreba, romlebic klass uWirs. 

145. yvela naswavli moqmedebis ganmtkiceba da ganviTareba. gansakuTrebuli 

yuradReba eqceva swrafi zepiri angariSis unarCvevebs. 


146. sak. wera # 21. 

4. me-4 klasis ZiriTadi siaxleni; 

saSinao davalebani 

IV klasis ZiriTadi siaxleebi ukavSirdeba imas, rom IV 

klasSi xdeba Semzadeba V klasisaTvis, romelSic ukve Tvisebrivad 

sxvagvari, ufro maRali donis aqtiuri swavleba iwyeba: 

swavlebis mTavari wamyvani xdeba moswavlis mier Teoriuli 

teqstebis damoukideblad wakiTxva (da ara saklaso, Tundac 

samuSaoebi, rogorc es iyo pirvel sam klasSi). 

1) gakveTilebis ageba gansxvavdeba III klasisagan [ix. $ 6]. 

2) mkveTrad momatebulia Teoriuli teqstebis raodenoba 

(Tumca, TviTeulis moculoba mcirea); maT moswavle damoukideblad 

waikiTxavs Sin (Cveneburi aqtiuri mzaobis meTodika 

ar iTvaliswinebs masw.-is mier sakiTxis axsnas). III klasSi 

amgvari teqstebi sul erTi-ori iyo, V klasidan ki TiTqmis 

yoveldRiurad iqneba. 

amasTan, IV klasSi ukve srulad xorcieldeba aqtiuri 

mzaobis meTodika: moswavle Teoriul teqsts Sin 

kiTxulobs winaswar, xolo saTanado sakiTxi klasSi 

msjelobiT damuSavdeba momdevno gakveTilze. ese igi, 

Tuki raime sakiTxi moswavlis saxelmZRvaneloSia, vTqvaT, me-5 

gakveTilze, maSin masw.-is meTod-saxelmZRvaneloSi es sakiTxi 

- 20 -

mouwevs me-6 gakveTilze (radgan meTod-saxelmZRvaneloSi 

aRwerilia saklaso samuSaoebi). 

saSinao davalebis teqstis moyola Tu moswavlis 

gaZaxeba unda gamoiricxos! am teqstis saklaso 

garCevisas merxebze msxdar moswavleebs Tvalwin 

udevT gadaSlili saxelmZRvanelo, masw.-i svams 

SekiTxvebs, moswavleebi teqstSi saWiro adgilebs 

kiTxuloben da ase pasuxoben. 

3) Semcirebulia programul sakiTxebze saklaso samuSaoTa 

wili da Sesabamisad momatebulia saSanao davalebis amocanebisa 

da sxva msgavsi amocanebis erTobliv garCevaTa wili (Tumca, 

yovel gakveTilze TiTo TamaSi mainc aucilebelia!); es masw.-is 

meTodikur saxelmZRvanelosac etyoba: igi sakmaod Semcirebulia 

winebTan SedarebiT. 

4) yovelive amis gamo IV klasSi masw.-is saklaso muSaoba 

mniSnelovnad gaadvilebulia (V klasSi kidev ufro metad 

gaadvildeba). samagierod, cxadia, ufro metia moTxovna masw.-is 

maTematikur codna-unarebze. 

5) saSinao davalebis amocanebi, rogorc yovelTvis, gakve- 

Tilis dasawyisSi gairCeva. magram IV klasSi siaxlea is, rom 

zogi amocanis garCeva uSualod CarTulia saprogramo sakiTxis 

saklaso damuSavebaSi. am amocanebis nomrebi gamotovebuli iqneba 

pirvel sagakveTilo monakveTSi (amocanebis garCevisas) da 

miTiTebuli iqneba momdevno sagakveTilo monakveTSi, saprogramo 

sakiTxTan erTad. 

iseve rogorc III klasSi, saprogramo xazi dayofilia 

safexurebad. safexuri _ esaa ramdenime (rogorc wesi, 2-dan 6mde) 

sagakveTilo monakveTisgan Semdgari Tanmimdevroba, romlis 

ganmavlobaSic am xazSi arsebiTi siaxle ar Cndeba. ese igi, 

mTeli safexuris ganmavlobaSi mis yvela sagakveTilo 

monakveTze msgavsi saklaso samuSaoebi tardeba. siaxleebi, 

cxadia, saSinao davalebaTa ganxilvisas Cndeba (rac sxva xazis 

sakiTxebsac moicavs). 

dagegmil gakveTilTa saerTo raodenobaa daaxloebiT 130. 

kidev emateba erTi ukanaskneli Tavi _ mxolod amocanaTa 20- 

- 21 -

ode gakveTili. weliwadSi sul daaxloebiT 160-165 gakveTili 

tardeba (gacdenebisa da uqmeebis gaTvaliswinebiT). ese igi, Cven 

daugegmavad vtovebT 10-15 gakveTils. isini saWiroa: Tuki masw.-i 

wlis ganmavlobaSi SeamCnevs, rom moswavleebs gauWirdaT raime 

sakiTxi, is erTi gakveTiliT SeaCerebs winsvlas dagegmil 

saswavlo programaSi da am erT gakveTils dauTmobs Zneli 

sakiTxebis gameorebasa da ganmtkicebas. am dros klass saTanado 

saSinao davalebac unda mieces _ amocanebs Tavad masw.-i SearCevs 

Tematikuri krebulidan. ar unda dairRves Cveni wesi: arcerTi 

gakveTili saSinao davalebis miucemlad! 

garda amisa, Cven gaviTvaliswineT isic, rom SeiZleba moxdes 

gauTvaliswinebeli gacdenebi, anda II klasis sakiTxebis 

gameorebasa Tu dasrulebas SeiZleba ar eyos Cven mier 

dagegmili ramdenime gakveTili. 

ukanasknel 15-ode gakveTilSi xdeba umniSvnelovanesi 

naswavli sakiTxebis saerTo gameoreba, ganmtkiceba da ganviTareba, 

erTmaneTTan dakavSirebulad (mravladaa sxvadasxva Temebis 

gamaerTianebeli amocanebi). am drosac klasi gahyveba amocanaTa 

Tematikur krebuls. amocanebs Tavad masw.-i SearCevs, 

paralelurad sxvadasxva paragrafebidan. 

Tematikuri krebulis morCenili amocanebi da sakontrolo 

weraTa yvela amocana (amoxsnili) klass sazafxulo 

davalebad eZleva. es davaleba Sedgeba 200-240 amocanisgan. 

manamde ki klass 10-20 amocana unda mieces davalebad mokle 

ardadegebze (isev Tematikuri krebulidan). ardadegebis Semdeg 

masw.-i moagrovebs moswavleTa rveulebs, gaasworebs da 

Seafasebs naSromebs. yvela bejiTi moswavle kargi niSniT unda 

waxalisdes. saardadegebo davalebaSi miRebuli Sefaseba 

sakontrolo weris Sefasebas utoldeba. 

saardadegebo davalebas udidesi mniSvneloba aqvs _ 

aranaklebi, vidre Cveulebriv saSinao davalebebs. es davaleba, 

jer erTi, moswavles karg gonebriv mdgomareobas unarCunebs da 

xels uSlis naswavlis daviwyebas. aranaklebi mniSvneloba aqvs 

zogad aRmzrdelobiT zemoqmedebas. bavSvi eCveva yoveldRiur 

rudunebas _ mcire, Zalian advil (dReSi saSualod sul 

oriode amocana), magram mainc gegmazomier da mizandasaxul 

- 22 -

yoveldRiur rudunebas. es ufro mniSvnelovania, vidre 

sazogadod maTematikis swavla. 

amocanaTa Tematikur krebuls masw.-i kidev erT SemTxvevaSi 

unda iyenebdes. SabaT-kvirisa da uqme dReebisaTvis klass 

sasurvelia mieces erTi an ramdenime (gaaCnia sidides) 

damatebiTi amocana _ arastandartuli, saazrovno, ufro bolo 

paragrafebidan, oRond mxolod ganvlili programis farglebSi 

(maTac masw.-i Tavad SearCevs). es ar iqneba aucilebeli davaleba, 

mas, iseve rogorc saardadegebo davalebas, albaT mxolod 

ramdenime mondomebuli moswavle Seasrulebs mTlianad. masw.-i 

daufaravad ityvis, rom es damatebiTi davaleba nebayoflobiTia, 

araa aucilebeli. 

amgvarad, Tuki masw.-i moiwadinebs, kargad ganxorcieldeba 

ramdenime moswavlis gaZlierebuli swavleba (mTeli klasis 

auCqarebel gaRrmavebul swavlebasTan erTad), ris karg 

saSualebasac iZleva Cveneuli saxelmZRvanelo. aseve, amocanaTa 

Tematikuri krebuli SeiZleba gamoyenebul iqnes wreobrivi 

muSaobisaTvisac. 

yvela SemTxvevaSi Tematikuri krebulis paragrafebis gavla 

unda moxdes paralelurad, oRond erTi paragrafis farglebSi 

_ daaxloebiT Tanmimdevrulad (cxadia, sakontrolo 

werebisaTvis gankuTvnil amocanaTa gamotovebiT). 

5. sakontrolo werebi 

sakontrolo werisTvis sagangebo momzadeba 

gamoricxuli unda iyos! masw.-i gakveTilebs atarebs Cveulebrivad, 

gegma-konspeqtebis mixedviT. Cven TviTon ise gvaqvs 

programa dagegmili, rom moswavles TavisTavad uwevs imdagvari 

amocanebis amoxsna, rogorebic sakontrolo weraze Sexvdeba. 

Tumca, SeiZleba mcire siaxlec Sexvdes _ raTa SemoqmedebiTi 

azrovneba aamoqmedos. 

gamoricxuli unda iyos agreTve yovelgvari siyalbe, 

gadawera, karnaxi, gamorCeul moswavleTa daxmareba 

da sxva sisaZagle. gamoricxuli unda iyos yalbi 

niSnebis werac. masw.-i Tavad darwmundeba, rom Sefasebis 

Cveneuli kriteriumebi isedac sakmaod lmobieria, 

- 23 -

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! 

TiTo siyalbe sawamlavis TiTo wveTia, romelic ryvnis 

moswavlis suls! 

yvela sakontrolo werisTvis momzadebuli gvaqvs Sefasebis 

konkretuli kriteriumebi. TiTo sakontrolo weraze xuTi 

amocanaa. rogorc wesi, pirveli, meore da mesame amocanebi _ 

oTxqulianebia, xolo meoTxe da mexuTe _ xuTqulianebi. Tuki 

es qulebi sadme Secvlilia, sagangebodaa miTiTebuli. jamSi 

udidesi SesaZlo qulaa 22. sityva `qula~ yvelgan Semoklebuli 

gvaqvs da vwerT ase: q. 

kidev erTic. masw.-ma 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 moswavles 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 weris dros masw.-i yuradRebiT unda iyos, raTa 

yvela moswavlem, jer erTi, damoukideblad imuSaos, da, 

meorec, imuSaos: bavSvs xSirad efanteba yuradReba, zogi 

- 24 -

moswavle gacilebiT ukeT niSans iRebs, roca mas aiZuleben, 

rom wesierad imuSaos! amrigad, Cven masw.-s 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 

cercetobs, cdilobs, male moamTavros wera (rac Zalian 

ezareba) da sxva rame akeTos. magram masw.-i zaris darekvamde ar 

gauSvebs moswavleebs gareT saTamaSod. maSin is moswavleebi, 

romlebmac adre moamTavres wera, cqmutaven da sxvebs uSlian 

xels. amas yvelafers didi yuradRebis miqceva sWirdeba. masw.-ma 

unda aiZulos moswavleebs, 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 qulebi miiRos, aramed 

agreTve gacilebiT ufro mniSvnelovani ramisTvis: moswavles 

mtkiced unda Camouyalibdes sakuTari naSromis Semowmebis, zogadad, 

TviTkritikisa da sakuTari namoqmedaris garedan danaxvis 

unarCveva. amave unarCvevis 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. amrigad, sakontrolo wera unda iyos sakontrolo ara 

mxolod masw.-is mier moswavlis codnis Semowmebis TvalsazrisiT, 

aramed agreTve Tavad moswavlis mier sakuTari codnisa da saku- 

Tari naSromis Semowmebisa da Semdeg gasworebis TvalsazrisiT. 

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. 

- 25 -

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 masw.-ma 

wiTlad unda moniSnos, iseve rogorc yvela sxva Secdoma da 

moswavlem unda gaasworos _ anu wesierad gadaweros _ 

Secdomebis gasworebisas. 

meorec, Tuki kaligrafiaa Zalian cudi da amasTan, Tuki 

masw.-ma icis, rom am moswavles SeuZlia ukeTesad wera da 

esoden cudi naweri celqobis an mifuCeCebis bralia. magram sa- 

Sualod cud kaligrafiaze moswavles qula ar unda daakldes, 

miT umetes, roca masw.-ma icis, rom moswavles Zalian uWirs 

lamazad wera da ukeTesad TiTqmis rom ar ZaluZs. 

moswavles ar unda daakldes qula arc gadaxazulze (Tuki 

metismetad araa gadadRabnili), Casworebulze, Canamatebze da 

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. 

6. gakveTilis ageba 

interaqtiuli meTodis gamoyeneba 

gakveTilis agebis zogadi principebi gadmocemulia dawy. kl. 

Cveneul zogad meTod-saxelmZRvaneloSi [$ 12]. xolo 

konkretulad IV klasisTvis nimuSebi ix qvemore, $ 7-Si. 

Cveneuli meTodikis mixedviT, gakveTilze mTavaria kargad 

dasmul TfljUywbUb! Uboxzpcb. mxolod amgvarad, aqtiurad, 

Seiswavleba sakiTxi. amitom gegma-konspeqtebSi swored saWiro 

SekiTxvaTa nimuSebia mocemuli. iqvea savaraudo da sasurveli 

pasuxebic, romlebic figurul { } frCxilebSia Casmuli. 

Cven zogjer vaxsenebT: `Zlieri moswavle~, an `susti, pasiu- 

- 26 -

i moswavle~. xSirad maT gansxvavebuli davalebebi eZleva 

[dawvrilebiT _ ix. dawy. klasebis masw. zogad meTodikur 

saxelmZRvanelos %!2 da %!23]. magram, cxadia, TviT moswavlem 

odnavadac ar unda igrZnos es! garda amisa, es ar unda 

iqces moswavlis mudmiv maxasiaTeblad, radgan xSirad 

xdeba, rom bavSvi TandaTan aumjobesebs Tavis miRwevebs! 

Cven maswavlebels Zalian vuadvilebT muSaobas imiT, rom 

vawvdiT dawvrilebiT ganmartebebs da, rac mTavaria, yvela gakveTilis 

gegma-konspeqts. yvelaferi gasagebadaa dawerili. magram 

gegma-konspeqti mainc moklea da mSrali, sqematuri da saSualo 

moswavlis Targze moWrili. amitom saWiroa misi morgeba 

yoveli kerZo klasis yoveli kerZo gakveTilis cvalebadi 

viTarebisadmi, moswavleTa Taviseburebebisadmi. garda amisa, rac 

mTavaria, saWiroa misi Sevseba da gacocxleba [ix. zemore, $ 1], 

SekiTxvebis gamravalferovneba da sxvadasxva moswavlisaTvis 

maTi dasma, klasis gaaqtiurebisa da gaxalisebis konkretul 

saSualebaTa gamoyeneba. erTerTi aseTi saSualebaa interaqtiuli, 

jgufuri muSaoba. 

interaqtiul samuSaoTa sixSire da formati masw.-ma Tavad 

unda SearCios, gaiTvaliswinos ra sakuTari codna-gamocdileba 

da gemovneba, skolis materialuri SesaZleblobebi, klasSi 

moswavleTa raodenoba da sxva. Cven aRwerili gvaqvs jgufuri 

muSaobis konkretuli nimuSebi [ix. qvemore, $ 7]. danarCeni ki 

masw.-ma Tavad moawyos _ gegma-konspeqtebis umravlesoba iseTia, 

rom advilad SeiZleba maTi ganxorcieleba rogorc klasis 

saerTo aqtivobis formatiT, ise interaqtiuli formatiT, 

aRwerili nimuSebis magvarad. 

es sakmaod advilia, radgan yvela Cveneuli 

gakveTili agebulia aqtiur swavlebaze. masw.-i sakiTxs 

ar xsnis _ aramed moswavleebs aZlevs davalebebs, 

raTa CarTos isini saklaso aqtivobebSi. am davalebaaqtivobaTa 

nimuSebi mzadaa _ gegma-konspeqtebSi 

TviTeuli gakveTilisTvisaa mocemuli. davalebaTa 

- 27 -

umravlesoba ki iseTia, rom advilad SeiZleba maTi 

danawileba moswavleTa calkeuli jgufebisTvis. es 

jgufebi cal-calke imuSaveben da Semdeg Sedegebs 

waradgenen dafasTan _ saerTo ganxilvaze. 

$ 7-Si mocemulia saklaso Sefasebis nimuSebic. Cven ar 

vwerT, magram yvelgan vgulisxmobT, rom Tuki moswavlem 

mTlad kargad ver Seasrula davaleba da, Sesabamisad, 

ver daimsaxura kargi Sefaseba, maSin, masw.-is davalebiT, 

Sesworebul-Sevsebuli pasuxi (moqmedeba) unda 

warmoaCinos warmatebulma moswavlem; xolo Tuki moswavleebi 

ver SeZleben, maSin _ masw.-ma, xolo moswavlem 

unda gaimeoros (mxolod mosmena araa sakmarisi!). 

IV klasSi gakveTilebis ageba gansxvavdeba III klasisagan da 

amgvaria: 

I. saSinao davalebis amocanebis dawvrilebiTi erToblivi 

aqtiuri garCeva _ rac TavisTavad moicavs erTerTi 

saprogramo xazis saTanado sakiTxis damu- 

Savebas _ 15-25 wT; 

II. TamaSi _ 5 wT; 

III. sxva saprogramo xazis aqtivoba _ 10-15 wT; 

IV. damatebiTi, ganmamtkicebeli amocanebis erToblivi 

amoxsna da garCeva (im sakiTxebze, romlebic klass 

SedarebiT uWirs) _ 5-10 wT. 

masw.-s ar unda egonos, rom TamaSi araseriozulia da amitom 

misi gamotoveba SeiZleba. aramc da aramc! TamaSi aucilebelia 

bavSvebis gansamuxtavad; garda amisa, TamaSs arsebiTi saswavlo 

datvirTva aqvs da gaTvaliswinebulia programulad. TamaSebi 

arsebiTadaa aucilebeli programuli sakiTxis Sesamzadeblad 

Tu gansamtkiceblad! 

imas, rac dafaze iwereba, moswavleebi TavianT 

rveulebSi weren. amas aRar mivuTiTebT. 

gakveTilis pirveli sagakveTilo monakveTi, rogorc wesi, 

saSinao davalebis dawvrilebiT erTobliv garCevas eTmoba. mos- 

- 28 -

wavleTa umravlesoba, bunebrivia, cdilobs, rom Tavisi azri 

gamoTqvas, Tundac sul cotaTi gansxvavebuli ram eweros, 

mainc. Cveneuli meTodika Zalze aaqtiurebs moswavleebs da es 

Zalian kargia. amitom masw.-i unda iyos yuradRebiT, raTa 

erTobliv msjelobaSi metismetad bevri dro ar gaeparos. maSin 

saklaso samuSaoebi an TamaSi veRar moeswreba. 

saSinao davalebis garCevisas yvela moswavles 

kalmistari unda ekavos xelSi da iqve avsebdes Tu 

asworebdes namuSevars, anda sul axali striqonidan 

Tavidan werdes rames (nawerSive rom ar CaaWuWynos). xolo 

masw.-ma merxebs Soris unda Camoiaros da namuSevrebs gadaxedos. 

magram amisaTvis calke drois daxarjva araa saWiro! Tan unda 

daiwyos davalebaTa ganxilva, erTi moswavle SeiZleba dafasTan 

werdes rames (raTa sxvebs auxsnas rTuli sakiTxi), masw.-ic 

saWiroebisamebr CaerTveba xolme msjelobaSi _ da Tan merxebs 

Soris ivlis da moswavleTa namuSevrebs daaTvalierebs. 

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, masw.-s evaleba, rom 

srulfasovnad amsjelos moswavleebi. mxolod am gziT miiRweva 

Rrma da kargad gaazrebuli codna da azrovnebis ganviTareba. im 

msjelobas, romelsac masw.-i 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). 

6.0. Semoklebuli aRniSvnebi amocanebisaTvis 

imisaTvis, raTa masw.-s gavuadviloT gakveTilis dagegmva, 

- 29 -

Cven vuTiTebT, Tu daaxloebiT ramdeni wuTi unda moandomos 

man saSinao davalebis TviTeuli amocanis erTobliv 

garCevas da rogori saxisaa amocana. saamisod SemoRebuli 

gvaqvs ramdenime aRniSvna, romlebic amocanis nomers aqvs 

miwerili: 

* _ amocana SedarebiT Znelia, araa gaTvaliswinebuli 

yvela moswavlisaTvis; amitom misTvis didi drois 

daTmoba araa aucilebeli, miT umetes, roca saSinao 

davalebaTa garCeva isedac Wianurdeba. 

! _ davaleba SemoqmedebiTia (Tumca, SesaZloa, arc iyos 

Zneli), sxvadasxva gzebiT keTdeba, moiTxovs SedarebiT 

met ganxilvas. 

~ _ amocana advilia, Tan iseTia, rom ar sWirdeba 

araviTari msjeloba da damatebiTi SekiTxvebi, 

sakmarisia mxolod pasuxebis swrafad Semowmeba; visac 

mcdari pasuxi aqvs, Tavis gverdiT mjdommac ki SeiZleba 

aCvenos marTebuli gza. amgvari amocanebisaTvis unda 

daixarjos saSualod naxevari wuTi, anda araumetes 

erTi wuTisa. 

# _ sakmaod advili amocana, Tan iseTi, romelsac 

SedarebiT naklebi garCeva eyofa (vTqvaT, 1-2 wuTi). 

vTqvaT, magaliTad, gegma-konspeqtSi weria: 

24.1. 3.2: 1~ 2# 3# 4! 5 6!* 7*. 

es imas niSnavs, rom: tardeba saswavlo wlis 24-e gakveTili, 

misi pirveli sagakveTilo monakveTi _ anu 23-e gakveTilze 

micemul saSinao davalebaTa erToblivi garCeva. ganixileba 

moswavleTa saxelmZRvanelodan gakveTili 3.2, anu me-3 Tavis 

me-2 gakveTili (radgan 23-e gakveTilze gairCa III Tavis 1-li 

gakveTili, xolo saSinao davalebad klass TavisTavad mieca 

momdevno, me-2 gakveTili). mis pirvel amocanas damatebiTi 

SekiTxvebi da msjeloba ar sWirdeba; meore da mesame 

amocanebs msjeloba da garCeva ki sWirdeba, magram 

SedarebiT mcire _ TviTeuls saSualod erTi-ori wuTi 

dasWirdeba; meoTxe amocana SemoqmedebiTia, meeqvse _ SemoqmedebiTi 

da Tan SedarebiT Zneli; meSvide amocana 

SedarebiT Znelia, xolo mexuTe (romelsac araviTari 

- 30 -

aRniSvna ara aqvs) _ Zneli araa, arc SemoqmedebiTia, magram 

sakmao garCeva mainc sWirdeba, radgan saSualo moswavlisaTvisac 

saWiroa misi gaazreba. 

savaraudod, am davalebebs SeiZleba daeTmos wuTebi: 

0,5 + 2 + 1,5 + 5 + 3 + 5 + 2 = 19. 

amis Semdeg, vTqvaT, meore sagakveTilo monakveTs (TamaSs) 

daeTmoba 5 wuTi, mesames _ 13 wuTi; meoTxes _ 5 wuTi. 

19 + 5 + 13 + 5 = 42. 

ukanaskneli oriode wuTi daeTmoba Jurnalidan siis amokiTxvas 

da Tan niSnebis (an àrebis~) Camoweras (es SeiZleba zaris 

darekvis Semdegac gagrZeldes). iSviaTad daemateba saSinao 

davalebis nomrebis dafaze dawera da rveulebSi gadawera _ 

roca saSinao davaleba klass Tematikuri krebulidan eZleva 

(da ara, rogorc Cveulebriv, momdevno gakveTili). 

cxadia, gegma-konspeqtebSi araa garCeuli yvela amocana. 

garCeulia SedarebiT rTuli amocanebi an iseTebi, romlebSic 

raimeze yuradRebis gamaxvilebaa saWiro. 

7. gakveTilis scenaris nimuSebi 

TviTeulis miznebs axlavs - Sesabamisoba saxelmwifo saswavlo gegmasTan 

hblwfUjmj!$!76! 

nsbwbmojTob!sjdywjt!rwfTnjxfsjU!hbnsbwmfcb!fsUojTob{f! 

! nj{ofcj; maT. IV.1. kiTxulobs ricxvebs, sxvadasxva modelis 

gamoyenebiT gamosaxavs ricxvebs da axdens poziciuri sistemis 

demonstrirebas. 

maT. IV.2. avsebs weriTi algoriTmis gamoyenebiT Sesrulebuli 

gamoTvlis nimuSSi gamotovebul cifrebs da asabuTebs pasuxs. 

maT. IV.3. iyenebs weriT algoriTms ricxvebze gamravleba-gayofis 

moqmedebaTa Sesasruleblad da ganmartavs gamoyenebul xerxs. 

maT. IV.7. ricxviTi gamosaxulebis mniSvnelobis mosaZebnad iyenebs 

Sekrebisa da gamravlebis komutaciurobas, asociaciurobas da 

Sekrebis mimarT gamravlebis distribuciulobas. 

maT. IV.8. TanamkveTi figurebis gamosaxulebaze uTiTebs rogorc 

saerTo wertilebs, ise im wertilebs, romlebic mxolod erT 

figuras ekuTvnis. 

maT. IV.9. anawevrebs brtyeli geometriuli figuris grafikul gamosaxulebas 

an models miTiTebuli figuris/figurebis misaRebad. 

- 31 -

maT. IV.13. awesrigebs raodenobriv da Tvisebriv monacemebs. 

! xjobtxbsj!dpeob!eb!vobsfcj;! asrulebs qveSmiweriT Sekrebagamoklebas 

da asabuTebs amisTvis saWiro TviTeul qmedebas. 

praqtikulad iyenebs gamravlebis ganrigebadobas Sekrebis mimarT. 

mravalniSna ricxvs daSlis saTanrigo SesakrebTa jamad. 

! tbXjsp! sftvstfcj;! moswavlis saxelmZRvanelo, Tavi X, 

gakveTili 2. dafa da feradi carcebi. koWebiani saangariSo. 

! nbtbmb!joufsbrujvmj!nvTbpcjtUwjt; baraTebi. 

SeniSvna. winaswari codna-unarebis gasaaqtiureblad iyo wina, 

64-e gakveTilis me-4 sagakveTilo monakveTi; agreTve misive sa- 

Sinao davalebis (Tavi X, gakveTili 1) amocanebi # 2 da # 3, 

mimdinare saSinao davalebis (Tavi X, gakveTili 2) amocanebi 

# 1, 2, 4. amocana # 3 aris wina gakveTilis ganmamtkicebeli. 

amocana # 6 kompleqsuria _ geometriuli aRniSvnebis ganmamtkicebeli, 

geometrul unarCvevaTa ganmaviTarebeli da Tan 

kombinatorikis unarCvevaTa ganmaviTarebeli. kompleqsuria 

agreTve amocana # 7, romelic aerTianebs ariTmetikas (ricxvTa 

Caweris poziciuri sistema) da kombinatorikas. 

! 76/2/ 10.2. 1~ 2 3~ 4# 5 6 7 

I (20 wT) _ saSinao davalebis amocanebis ganxilva _ rac 

TavisTavad moicavs gakveTilis ZiriTadi sakiTxis Semzadebasac. 

rogorc yovelTvis am dros, masw.-i merxebs Soris dadis da 

saTiTaod aTvalierebs rveulebs. Tan moswavles raimes 

gadaulaparakebs, Seaqebs an piriqiT, _ mTavaria, rom yuradRebas 

gamoiCens misi namuSevrisa da misi pirovnebis mimarT, mag.: _ 

welSi gaimarTe! (da Tan mxrebze gadausvams xels); _ guSin rom 

muceli gtkioda, xom mogirCa? (da Tan Tavze gadausvams xels)... 

1. (0,5 wT) ramdenime moswavlem adgilidanve Tqvas da 

Semowmdes mxolod pasuxebi. visac mcdari pasuxi aqvs, Tavisi 

gamoTvla Seadaros Tanaklaselisas da TviTon moZebnos Secdoma 

(Tuki saWiro gaxda _ masw.-is daxmarebiT). 

2. (3 wT) orma moswavlem dafaze daweros gamoTvla (erTma 

_ erTi svetisa, meorem _ meorisa), maT xelSi TavianTi 

rveulebi unda ekavoT da iqidan gadaweron. Tan xmamaRla 

daasabuTon. sxvebma Seadaron da gaasworon Secdomebi. 

- 32 -

sainteresoa mesame striqoni _ mofiqrebaze. dafasTan 

momuSaveTa garda ramdenime sxva moswavlemac waikiTxos Tavis 

mier mogonili magaliTebi. 

3. (0,5 wT) ramdenime moswavlem adgilidanve waikiTxos. 

sxvebma ki mosmeniT, zepirad Seumowmon, marTebulia Tu ara. 

4. (2 wT) dafasTan gamovides sami moswavle da Tavisi 

rveulidan gadaweros TiTo Semowmeba. Tan xmamaRla daasabu- 

Ton. sxvebma Seadaron da gaasworon Secdomebi. 

TviT gamoTvlebi mxolod zepirad keTdeba. 

(masw.-i Tan ganagrZobs merxebs Soris 

siaruls, rveulebis Tvalierebas da TviTeul 

moswavlesTan erTi-erTze urTierTobas). 

5. (3 wT) moswavleebma adgilidan rig-rigobiT waikiTxon 

TiTo-TiTo qveamocanis gantoleba da pasuxi, sxvebma Seadaron 

Tavisas. visac mcdari pasuxi aqvs, Tavisi gamoTvla Seadaros 

Tanaklaselisas da TviTon moZebnos Secdoma (Tuki saWiro 

gaxda _ masw.-is daxmarebiT). 

amasobaSi masw.-i dafaze amzadebs me-6 amocanis naxazs, raTa 

Semdeg xazvaze dro ar daikargos. 

6. (3 wT) dafasTan gamova is moswavle, visac jer ar 

waukiTxavs araferi da arc dafasTan yofila. gamzadebul 

naxazze jer xeliT aCvenebs oTxkuTxedebs, Semdeg ki amowers 

maT asoiT aRniSvnebs. Tuki gamorCeba romelime Camaluli 

oTxkuTxedi, sxva moswavle miexmaros. pasuxebi: 

FEKM, KMTD da ori Camaluli: AFEK, DFEK. 

MD da KT ikveTeba A wertilSi. 

AFEK oTxkuTxedis perimetria 7,5-sa da 8 sm-Soris. 

I. FE da EK; II. FD da KM (an MT da AF...). 

7. (4 wT) masw.-i ikiTxavs: _ jer gavaCeroT sityva òras~. 

vin waikiTxavs yvela ricxvs, romelic daiwyeba orasiT? Tan 

Tamrikom yvela es ricxvi dafaze daweros! sxvebi ar waikiTxoT 

_ marto isini, romlebic orasiT iwyeba! {218026, 218020, 

218826, 218820}. ricxvebis meore wyeba riT unda daiwyos _ 

orasis nacvlad? {asiT} saWiro aRaraa ricxvebis Tavidan 

Sedgena _ ukve Sedgenilebi gamoviyenoT. amisTvis ra riT unda 

SevcvaloT? {orasi _ asiT}. Tamrikom Tu dawera? {118026, 

118020, 118826, 118820}. sul ramdeni ricxvi Sedga? {rva}. 

- 33 -

7. damatebiTi davaleba (3 wT) _ axla amave ricxvebiT 

SevadginoT yvela ricxvi, aramarto eqvsniSna! gamoviyenoT ukve 

Sedgeneilebi _ dafaze rom gviweria _ da mivuweroT sxvebic. 

vin gamova dafasTan? jer 

_ modi, davweroT jer xuTniSna ricxvebi, cira, aba gadaakeTe! 

{18026, 18020, 18826, 18820, 18026, 18020, 18826, 18820}. 

_ vinme Tu mixvda, ra daaklda ciras? {sul oTxi ricxvia, 

zedmetadaa gameorebuli!} cira, waSale zedmetebi! ra darCa? 

{18026, 18020, 18826, 18820} 

_ axla gamovides cira da daweros oTxniSna ricxvebi! 

{1026, 1020, 1826, 1820}. 

_ axla gamovides luka da samniSna ricxvebi amiaweros! 

{826, 820} ra daaklda? {226, 220, 126, 120}. 

_ kidev rogori ricxvebi gvaklia? {orniSnebi} barem lukam 

amoweros! {18, 26, 20}. kidev Tu dagvrCa rame? {ara}. daTvaleT, 

sul ramdeni SesaZlebloba (am sityvis gamokveTiT) yofila! 

{23} (ar iTvleba daumTavrebeli saxelwodebebi: oras, as... ) 

rogorc yovelTvis, is, rac iwereba dafaze _ iwereba yvela 

moswavlis rveulSic! 

II. (4 wT) 76/3/!Tfkjcsj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj 

moswavleebi wreSi dgebian, anda, sadac amis saSualeba araa, 

Camwkrivdebian saklaso oTaxis gverdiTa kedelTan an, sadac 

arc amis saSualebaa, ubralod dgebian fexze. magram yvela 

SemTxvevaSi TamaSi wriulia: iwyebs romelime erTi moswavle, 

Semdeg rigi mis mezobelze gadadis da ase Semdeg; mas Semdeg, 

rac rigi mTel klass moivlis, ubrundeba kvlav pirvel 

moswavles da midis meore wreze, Semdeg mesame wreze da a.S. 

visac rame SeeSleba, maSinve gamoeTiSeba TamaSs. patieba 

gamoricxuli unda iyos, Torem, Tuki erTxel mainc 

vinmes vapatiebT, mere patiebis Txovna gauTavebeli 

iqneba da TamaSi CaiSleba. garda amisa, TamaSs eTiSeba is 

moswavlec, romelic uxeSad daarRvevs TamaSis wesebs: 

ukarnaxebs sxvas, ixmaurebs da sxva. kidev eTiSeba is moswavle, 

romelic daTqmul droze metad Seyovndeba. TamaSi swraf 

tempSi unda midiodes, ise mas azri ara aqvs. yovel moswavles, 

- 34 -

visac rigi mouwevs, eZleva dro _ taSis xuTi Semokvra 

maswavleblis mier, anu xuT Tvlaze _ magram Tvla araa kargi, 

radganac ricxvebis mosmena moswavles xels SeuSlis, gonebas 

aurevs _ amitomaa Tvlis nacvlad taSebi! Tavidan tempi 

SedarebiT nelia, meore wreze Cqardeba, mesameze kidev Cqardeba. 

magram maswavlebelma cota nela unda daukras taSi im 

moswavlesTan, romelic bunebrivad nelia da isedac odnav meti 

unda adrovos mas (cxadia, sxvebisaTvis SeumCnevlad). 

moswavleebi TandaTan eTiSebian TamaSs, vidre ar 

gamovlindebian gamarjvebulebi. gamarjvebuls yvela SemTxvevaSi 

moeTxoveba ukanasknelad kargi pasuxi _ mxolod imis gamo, rom 

sxva yvela ukve gamoeTiSa, moswavle gamarjvebulad ar 

CaiTvleba. Tuki verc is upasuxebs, arcerTi gamarjvebuli ar 

iqneba. gamarjvebuli SeiZleba iyos erTi an ramdenime moswavle. 

erTi im SemTxvevaSi, Tuki yvela danarCeni gamoeTiSa pirvelmesame 

wreebze. magram Tuki ramdenime moswavle kargad pasuxobs 

ramdenime wris ganmavlobaSi, isini yvelani gamarjvebulebad 

CaiTvlebian. maswavlebeli TandaTan kargad igrZnobs xolme, 

rodis SeiZleba darCenil moswavleTa gamarjvebulebad 

gamocxadeba (da TamaSis dasruleba). 

yvela gamarjvebuls unda daeweros raime wamaxalisebeli niSani. 

kerZod am SejibrSi Sekreba yvelanairi SeiZleba, oRond 

pirvel wreze _ SedarebiT advili SemTxvevebi, roca erTeulTa 

jami 10-s ar aRwevs, aTeulTa jami 100-s ar aRwevs, aseulTa 

jami 1000-s ar aRwevs, meore wreze _ ufro Znelebi. 

1) 34+35, 560+220, 1400+1500, 14000+71000... 

2) 34+38, 580+220, 1800+1500, 104+707, 1200 + 900... 

gamoklebis SemTxvevaSi Zneli Sesadarebeli ricxvebi araa 

saWiro: 3) roca orniSna ricxvTa Soris gansxvaveba ar 

aRemateba 20-s: 63_45; 92_78... da Sesabamisad: 380_290, 

3100_1600, 38000_29000... 

4) roca sxvaoba advili saangariSoa: 67_24; 800_150; 

9400_3000; 4000 _ 10... 

gamravleba da gayofa: 

5) cxriluri da nulmiwerili cxriluri: 9·5, 4·60, 70·8, 42:6, 

- 35 -

630:9... 

6) advili saangariSoebi: 3 · 150, 750:250, 6000:4, 6 · 11... 

III. (10 wT) 76/4/!nsbwbmojTob!sjdywjt!rwfTnjxfsjU!! 

hbnsbwmfcb!fsUojTob!sjdyw{f/!!! ! 

aucilebelia wesis Sinaarseuli gaazreba _ arsebiTad, jer 

vamravlebT erTeulebs, Semdeg _ aTeulebs, aseulebs da ase 

Semdeg, bolos ki am namravlebs vkrebT. 

_ Cven viciT qveSmiwerT Sekreba da gamokleba. aseve SeiZleba 

qveSmiweriT gamravleba. gavamravloT, magaliTad, didi ricxvi 

erTniSna ricxvze. gamovides nika (Zlieri moswavle) da 

Seasrulos dafaze 6 · 45071. mravalniSna ricxvi davSaloT 

saTanrigo SesakrebTa jamad, gamosaxulebad. dawere! 

6 · 45071 = 6 · (4 · aTiaTasi + 5 · aTasi + 0 · asi + 7 · aTi + 1) 

axla 6-ze saTiTaod unda gavamravloT ra? {aTiaTaseulebi 

(Canawerze 4 · 10000 TiTis dadebiT), aTaseulebi (5 · 1000-ze 

TiTis dadebiT), aseulebi (TiTis dadebiT), aTeulebi da 

erTeulebi}. mere ki miRebuli namravlebi isev unda SevkriboT 

da SevadginoT isev mravalniSna ricxvi. rogor gamovTvliT 

namravlebs? 6 · 4-jer aTiaTasi + 6 · 5-jer aTasi + 

6 · 5-jer aTasi + 6 · 0-jer · asi + 6 · 7 · aTi + 6 · 1 

amis gamoTvla gacilebiT ufro 

45071 

mosaxerxebelia qveSmiweriT: 

_ qveSmiweriT Sekrebas romeli mxridan viwyebT? 6 

marcxnidan Tu marjvnidan? {marjvnidan} aqac _______ 

marjvnidan daviwyoT, erTeulebidan, mere gadavalT 

aTeulebze, mere _ aseulebze da ase Semdeg. daviwyoT 

marjvnidan, jer davweroT 6 · 1 = 6. axla ra unda 

gavamravloT? {6 · 7 · aTi} ramden aTeuls miviRebT? {42 aTeuls} 

SeiZleba Tu ara, rom es 42 CavweroT aTeulebis TanrigSi 

(aTeulebis adgilas)? {ara, mxolod erTi cifri SeiZleba 

CavweroT} 42 aTeulidan ramdeni aseuli gamoiyofa da ramdeni 

aTeuli mogvrCeba naSTad? {42 aTeuli = 4 aseuli da 2 

aTeuli} hoda swored es 2 CavweroT aTeulebis TanrigSi, 

xolo 4 aseuls ra vuyoT? {davimaxsovroT} rom ar 

dagvaviwydes, aseulebs zemodan davaweroT patara 4-iani! 

kidev karga xans qveSmiweriT gamravlebisas 

- 36 -

moswavleebma ase zemodan daaweron xolme patara 

cifri, raTa ar daaviwydeT! (Caweris nimuSi mocemulia 

moswavlis saxelmZRvaneloSi, Tavi X, gakveTili 3). 

_ axla ra unda vqnaT? {6-ze unda gavamravloT aseulTa 0} 

ras miviRebT? {nuls, magram mivumatoT damaxsovrebuli 4 da 

miviRebT 4-s} CavweroT es 4 aseulTa TanrigSi. 

da ase Semdeg _ gamravleba Sesruldeba mTlianad. 

rogorc yovelTvis, is, rac iwereba dafaze _ iwereba yvela 

moswavlis rveulSic! 

esaa aqtiuri mzaobis meTodika: masw.-s araferi ar auxsnia _ 

yvelaferi moswavleebma gaakeTes; mTavari ki isaa, rom saamisod 

yvelaferi Semzadebuli iyo: saTanrigo Sesakrebebad daSla, 

qveSmiweriT Sekreba, ganrigebadoba. Tanac, es Semzadebac mxolod 

aqtiurad moxda _ da ara masw.-is mier SexsenebiT. 

amasTan, gamoTvlis wesis TviTeuli safexuri gaazrebulia _ 

da ara meqanikurad damaxsovrebuli. 

bolos, saWiroa kidev erTi gamravlebis Sesruleba. es 

SeiZleba rogorc individualurad (yvelam saTiTaod Tavis 

rveulSi iangariSos), an, umjobesia, jgufuri muSaobiT. 

IV. (5 wT) klasi daiyofa 3 jgufad, daurigdebaT baraTebi 

da miecemaT davaleba interaqtiuli muSaobisTvis: 

I jgufs: _ rvaas oTxmocdasami aTas as eqvi oTxze. 

II jgufs: _ xuTas oTxmocdaSvidi aTas eqvsas ocdaaTi samze; 

III jgufs: _ oras samocdaaTi aTas xuTas TerTmeti xuTze. 

Semdeg namuSevrebs sajarod waradgens TviTeuli jgufis 

erTerTi wevri. 

hblwfUjmj!$!248!`!hpofcbTj!xbsnpehfojm!tufsfpnfusjvm! 

tyfvmUb!xbyobhfcjt-!xjcpfcjtb!eb!xwfspfcjt!ebUwmb! 

! nj{ofcj; maT. IV.8. aRwers geometriul figurebs da axdens maT 

klasifikacias. adarebs da ajgufebs sivrcul figurebs 

geometriuli atributebis mixedviT. TanamkveTi figurebis 

gamosaxulebaze uTiTebs rogorc saerTo wertilebs, aseve im 

wertilebs, romlebic mxolod erT figuras ekuTvnis. sivrcul 

figuraSi uTiTebs mosazRvre/aramosazRvre waxnagebs, 

TanamkveT/araTanamkveT wiboebs. 

- 37 -

maT. IV.14. sworad avsebs cxrils, sqemas, kiTxvars/anketas (mag. 

Seaqvs monacemebi mza cxrilis Sesabamis ujrebSi). 

! xjobtxbsj!dpeob!eb!vobsfcj;! TvalsaCinod mocemul 

stereometriul mravalwaxnaga sxeulze iTvlis waxnagebs, wiboebsa 

da wveroebs; Zebnis ori mosazRvre waxnagis saerTo wibos da ori 

mosazRvre wibos saerTo wveros. xazavs da avsebs cxrils. 

! tbXjsp! sftvstfcj;! ! moswavlis saxelmZRvanelo, Tavi XX, 

gakveTili 5. moswavlis saxelmZRvanelos Siga ydis TvalsaCinoeba 

`gamravleba-gayofa ricxvTa xazze~ (anda, Tuki skolas aqvs, amave 

TvalsaCinoebis plakati).!! 

kubis, aguredis, ori sxvadasxvagvari piramidis, samkuTxa prizmis, 

ori sxvadasxvagvari prizmis modelebi. dafa da feradi carcebi. 

! nbtbmb!joufsbrujvmj!nvTbpcjtUwjt; igive. 

248/2/ 20.5. 1# 2# 3# 4# 5# 6# 7 

I (13 wT) _ saSinao davalebis amocanebis ganxilva 

1. (2 wT) moswavleebi adgilidan pasuxoben SekiTxvebs: _ 

romelia namravlia yvelaze didi? {720·20} ratom? {720-ic 

yvelaze didia I Tanamamravlebs Soris da 20-ic II 

Tanamamravlebs Soris} Semdeg? {586·16} ratom? (axsna igivea) ... 

da ase Semdeg: 468·15, 457·8, 457·6, 342·6, 342·4, 425. 

2. (1,5 wT) moswavleebi adgilidan pasuxoben SekiTxvebs: _ 

romelia meti da ratom? (dasabuTeba advilia). 

3. (1,5 wT) es amocana momdevno gakveTilis Semamzadebelia. 

garda amisa, umniSvnelovanesia gonebrivi Tvalsawieris 

gasafarToveblad, radgan miemarTeba usasrulobis cnebas. 

albaT, moswavleTa umravlesoba ityvis, rom ar arsebobs, 

radgan ricxvebi uTvalavia. magram ratom? 

_ vTqvaT, xatiam daiCema, rom arsebobs udidesi ricxvi. rogor 

davumtkicoT, rom eSleba? {am ricxvs daumatos 1 da miiRebs 

kidev ufro did ricxvs} diax, ricxvisTvis yovelTvis 

SegviZlia kidev erTis an metis damateba da es arasdros 

damTavrdeba! ricxvebi uTvalavia! 

4. (2 wT) dafasTan 

gava moswavle, daxazavs iqiT fexiT: 30 wT 

sqemas da imsjelebs. saxli __________________ tba 

aqeT velosip: 10 wT 

- 38 -

30 wT + 30 wT = 1 sT. 

5. (2 wT) dafasTan 

gava moswavle da dawers simcdaris dasabuTebas: 

I. marTebulia; II. marTebulia; III. mcdaria: 3·3 = 9. 

IV. mcadria iseve, rogorc III, radgan roca erTerTi 

Tananamravli kentia, meorec SeiZleba kenti iyos. 

6. (2 wT) moswavleebi adgilidan pasuxoben: 

I. gamotovebulia 40; II. orjeraa 170; 

III. araTanabaria danayofebi; 

IV. jer matebaa 10-iT, Semdeg ki 100-iT; 

V. jer matebaa 100-iT, Semdeg ki 1000-iT. 

7. (4 wT) erTi moswavle adgilidan waikiTxavs amocanis 

pirobis dasrulebas: `ramdeni kg Tafli darCeba mefutkres 5 

wlis ganmavlobaSi?~ _ sxvagvari dasrulebac SeiZleboda! aba 

vis aqvs sxvagvarad? ise, rom 1 wlis ganmavlobaSi iyos! 

{ramdeni kg Tafli darCeba mefutkres erT weliwadSi 5 

skidan?} _ diax, erT skaSi erTi ojaxi cxovrobs. 

dafasTan gadis moswavle da wers amocanis erT SesaZlo 

amoxsnas: (35 _ 16)·5 = 19·5 = 95 (kg). 

sxva moswavle dawers meore gzas: 

(35 _ 16)·5 = 35·5 _ 16·5 = 175 _ 80 = 95 (kg). 

romeli gza jobda da ratom? {pirveli, radgan meoreSi 

orjer dagvWirda did ricxvebze gamravleba}. 

II. (8 wT) 248/3/!tufsfpnfusjjt!tbljUyfcj!! 

hpofcbTj!xbsnpehfojm!tyfvmUbUwjt/!! 

moswavlem gonebaSi warmodgeniT unda daTvalos (gaarkvios) 

zogierTi mravalwaxnaga sxeulis wveroebis, waxnagebisa da 

wiboebis raodenoba (uadvilesi SemTxvevebi). 

SekiTxvaTa nimuSebi: _ ramdeni wvero, ramdeni waxnagi da 

ramdeni wibo aqvs am piramidas? [prizmas?] {oTxi} ramdenkuTxa 

mravalkuTxedia am piramidis [prizmis] es fuZe? (Tan fuZes 

vuTiTebT) {oTxkuTxedi}. 

_ axla warmoidgine amdagvari sxeuli, oRond romelsac fuZeSi 

aqvs SvidkuTxedi [eqvskuTxedi, aTkuTxedi, cxrakuTxedi, 

TormetkuTxedi]. ramdeni wvero eqneba mas? ramdeni waxnagi? 

ramdeni wibo eqneba mas? 

- 39 -

am dros moswavles xelSi ukavia mocemuli oTxkuTxa 

piramida [prizma]. 

saWiroa bevri varjiSi, ori-sami savarjiSo araa sakmarisi. 

sxeulTa saxelwodebebze (paralelepipedi, piramida da sxva) 

yuradRebis gamaxvileba araa saWiro. masw.-i moswavles awvdis 

miTiTebuli geometriuli sxeulis models da ambobs mis 

saxelwodebas. kargia, Tuki moswavleebi daimaxsovreben maT. 

magram es Zaldautaneblad unda moxdes. saxelwodebaTa codna 


III. (10 wT) klasi daiyofa 4 jgufad, daurigdebaT furclebi, 

flomasterebi, sxvadasxvagvari mravalwaxnaga sxeulebi 

da miecemaT davaleba interaqtiuli muSaobisTvis: 

warmoidgineT, rom amas viTom ockuTxedi (TxuTmetkuTxedi) 

aqvs aq. moisazreT, ramden ramdeni eqneba mas wveroebi, 

waxnagebi, wiboebi. pasuxebi CawereT cxrilSi. 

sanam moswavleebi fiqroben, masw.-i dafaze daxazavs cxrils: 

20-kuTxa piramida 

20-kuTxa prizma 

15-kuTxa prizma 

15-kuTxa piramida 

wvero wibo waxnagi 

Semdeg namuSevrebs sajarod waradgens TviTeuli jgufis 

erTerTi wevri da Seavsebs cxrilis Sesabamis striqons. 

IV. (6 wT) 248/4/!! ! UbnbTj.Tfkjcsj;!!txsbgj!qbtvyfcj!`!! 

hbnsbwmfcb.hbzpgb!sjdywUb!yb{{f•/ 

SekiTxvebi moicavs gamravleba-gayofis naswavl magaliTebs, 

moswavleebi pasuxoben saxelmZRvanelos ydis TvalsaCinoebis 

gamoyenebiT (an, ukeTesia, plakatis gamoyenebiT). 

es TamaSi xSirad Catardeba, rac saukeTeso saSualebaa, 

kerZod, gamravlebis cxrilis TavisTavad dasazepirebladac 

(rac TviTmiznad ar unda iqces). 

V. (7 wT) ! 248/5/! tbfsUp! hbnfpsfcb/! gameordeba algebraariTmetikis 

is sakiTxebi, romlebic moswavleebs uWirT. 

danarCeni gakveTilebis scenarebi mocemulia gegma-konspeqtebSi. 

- 40 -

8. gakveTilebis gegma-konspeqtebi 

mizanSewonilia, rom amocanis siZnele da sirTule cotaTi 

uswrebdes saSualo moswavlis SesaZleblobebs. Zalian advilia 

trivialuri, arafrismTqmeli, metismetad martivi SekiTxvebis 

dasma. agreTve sakmaod advilia metismetad Zneli 

SekiTxvebis dasmac. yvelaze saWiro da yvelaze Zneli _ 

esaa im saSualo sirTulis SekiTxvebis dasma, romlebze pasuxic 

bavSvs xelewifeba da Tan romlebic mas win waiyvans, 

gaafarTovebs ra mis SesaZleblobebs. j. bruneri 

saprogramo xazebis dawyebamde: sazafxulo namuSevrebis ganxilva; 

da, SesaZloa, III klasis programis dasruleba. 

sul: savaraudod 1 kvireuli anu 5 gakveTili 

IV klasis programa isea Semofargluli, rom gaTvaliswinebulia 

saswavlo wlis dasawyisSi erTi kvireulis daTmoba 

III klasis sakiTxebis ara mxolod gameoreba-ganmticebisaTvis, 

aramed agreTve III klasis saxelmZRvanelos momTavrebisaTvisac 

_ Tuki es Tavis droze ver moeswro. 

pirveli ori-sami dRe daeTmoba saardadegebo davalebidan 

sayuradRebo amocanebis klasSi garCevas da saklaso TamaSebs. es 

iqneba naswavlis mokle gameoreba. amocanebs masw.-i SearCevs 

sakuTari SexedulebiT: arastandartulebs, romlebic SedarebiT 

gauWirdaT moswavleebs, romlebsac masw.-i sainteresod miiCnevs 

da sxva. SedarebiT swrafad gairCeva imdeni, ramdenic moeswreba 

or-sam gakveTilSi. 

cxadia, am dros gamomJRavndeba moswavleTa codnis xarvezebic, 

romlebic SeZlebisamebr gamoswordeba (an damatebiTi 

mecadineoba dainiSneba). 

saSinao davalebad miecemaT amocanebi Tematikuri krebulidan 

_ paragrafebis dasawyisebidan, dReSi saSualod 6 amocana. 

momdevno gakveTilebis numeracias pirobiTad viwyebT pirveli 

nomridan (Tumca, sinamdvileSi, es iqneba, albaT, meoTxe, 

mexuTe Tu sulac meSvide gakveTili). 

Tavi I 

h!2/!sjdywTj!fsUfvmfcj-!bUfvmfcj-!btfvmfcj-!nbUj!sbpefopcb/! 

2/2/ rogorc yovelTvis, wina gakveTilze micemuli saSinao 

davalebis amocanebis garCeva. am gakveTilze ki ukve viwyebT IV 

- 41 -

klasis programas: klass saSinao davalebad miecema ukve Tavi 1is 

pirveli gakveTili 1.1 (romelic momdevno gakveTilis dasawyisSi 

gairCeva). Sin moswavlem damoukideblad unda waikiTxos 

gakveTilis teqsti da Semdeg amoxsnas amocanebi. 

2/3/! UbnbTj! `! txsbgj! qbtvyfcj•/ moswavleebi rig-rigobiT 

asaxeleben: orniSna ricxvs, samniSna ricxvs. Tan yoveli moswavle 

ambobs imasac, Tu ramdeni aseulia mis mier dasaxelebul 

ricxvSi. 

_________________________________________________ 

iwyeba pirveli xazi, ariTmetikisa: mravalniSna ricxvebis 

Tvla da Sedareba. am xazSi ZiriTadi TvalsaCinoebaa Zveleburi 

saangariSo. masze aucileblad unda gaTvalsaCinovdes 

ricxvebi, msjelobebi da gamoTvlebi mravalniSna 

ricxvebze! 

_________________________________________________ 

2/4/! hbnfpsfcb; 16161-! 11:1::-! 13434! eb! tywb/! sjdywTj! 

fsUfvmfcj-! bUfvmfcj-! btfvmfcj-! sjdywTj! nbUj! sbpefopcb/! 

tbnojTob!sjdywjt!blsfgb!tbbohbsjTp{f/! 

_ gamovides dafasTan Sorena da daweros ricxvebi `cxra~ da 

òrmoci~, oRond sami cifriT, viTom samniSna ricxviviT 

(daiwereba: 009, 040). sinamdvileSi ki es ricxvebi 

ramdenniSnaa? {pirvelia erTniSna, meorea orniSna} marcxena 

nulebi xom ar iTvleba! diax, eseni samniSnas hgavs, magram... 

gaimeoros beqam (susti moswavle). 

amdagvaradve gakeTdeba kidev erTi-orjer: 005, 063, 000, 

025. Semdeg gairkveva, dafaze daweril TviTeul ricxvSi 

ramdeni calia aTeuli. 

_ axla vaJam daweros marTlac samniSna ricxvi! (mag.: 324). 

gairkveva, ramdeni calia aTeulia am samniSna ricxvSi. 

daiwereba: 324 = 3 · 100 + 2 · 10 + 4. sul ramdeni aTeuli 

iqneba? 100-Sia 10 cali aTeuli, kidev 10, kidev 10, da kidev 

ori. qveviT miewereba: 10 + 10 + 10 + 2 = 32. 

_ maSasadame, samniSna ricxvSi aTeulTa raodenobas gviCvenebs... 

ra? misi Canaweris... romeli cifrebi? {misi Canaweris pirveli 

da meore cifrebi}. 

TviTeuli dasaxelebuli ricxvi Tan aikrifeba saangariSoze. 

- 42 -

zvsbeSfcb" masw.-ma ar unda gaimeoros gavrcelebuli 

Secdoma: 324-Si aris ara 2, aramed 32 aTeuli! 

h!3/!sjdywTj!fsUfvmfcj-!bUfvmfcj-!btfvmfcj-!nbUj!sbpefopcb/! 

3/2/! 1.1. 1~ 2 3~ 4~ 5~ 6 

3/3/!UbnbTj!`!txsbgj!qbtvyfcj•/! 

masw.-i TviTeul moswavles rig-rigobiT usaxelebs samniSna 

ricxvs, moswavle swrafad pasuxobs, Tu ramdeni aTeulia 

dasaxelebul ricxvSi. zogi samniSna ricxvi unda iyos aTeulTa 

TanrigSi nuliani: 304, 707, 109, 600, 900... 

3/4/! hbnfpsfcb;! 16161-! 11:1::-! 13434! eb! tywb/! sjdywTj! 

fsUfvmfcj-!bUfvmfcj-!btfvmfcj-!sjdywTj!nbUj!sbpefopcb/! 

tbnojTob! sjdywjt! blsfgb! tbbohbsjTp{f/! swrafad gameordeba 

1.3, oRond bolos daemateba naSTiani gayofis zepirad 

Sesrulebac, magaliTad: 508 : 10 = 50 (naSTi 8). 

508 : 100 = 5 (naSTi 8). 

933 : 10 = 93 (naSTi 3). 

933 : 100 = 9 (naSTi 33). 

TviTeuli dasaxelebuli ricxvi Tan aikrifeba saangariSoze. 

h! 4/! sjdywTj! fsUfvmfcj-! bUfvmfcj-! btfvmfcj-! nbUj! 

sbpefopcb/ 

4/2/ 1.2. 1~ 2# 3 4# 5 6 

4/3/!UbnbTj!`!txsbgj!qbtvyfcj•/! 

masw.-i TviTeul moswavles rig-rigobiT usaxelebs samniSna 

ricxvis 10-ze an 100-ze naSTiani gayofis magaliTs, moswavle 

swrafad angariSobs ganayofsa da naSTs. 

4/4/! hbnfpsfcb;! 16161-! 11:1::-! 13434! eb! tywb/! sjdywTj! 

fsUfvmfcj-! bUfvmfcj-! btfvmfcj-! sjdywTj! nbUj! sbpefopcb/! 

tbnojTob! sjdywjt! blsfgb! tbbohbsjTp{f/! ! swrafad gameordeba 

2.3. TviTeuli dasaxelebuli ricxvi Tan aikrifeba saangariSoze. 

h!5/!pUyj!nbUfnbujlvsj!nprnfefcb!eb!sjdywjUj!hbnptbyvmfcb/! 

5/2/ 1.3. 1# 2# 3# 4 5 6! 

_______________-----------___________________________ 

iwyeba mesame xazi, ariTmetika-algebrisa: angariSi _ 

oTxi ariTmetikuli moqmedeba. es xazi mihyveba pirvel xazs, 

ese igi: moqmedebebia jer 100-isa da 1000-is farglebSi 

- 43 -

(gameoreba), Semdeg _ 10000-is farglebSi, Semdeg _ milionis 

farglebSi, Semdeg cotaTi miliardis farglebSic da 

bolos sul cotaTi _ miliardis zeviTac. 

mTavaria ara qveSmiweriTi, aramed gaazrebuli zepiri 

angariSis unarCvevaTa ganviTareba. advili zepiri gamoTvlebi 

mravalniSna ricxvebze grZeldeba mTeli saswavlo 

wlis ganmavlobaSi. aucilebelia am moqmedebaTa gaTvalsaCinoveba 

saangariSoze! 

dafaze (da Tan rveulebSi) Sesrulebuli gamoTvlebis 

TviTeuli moqmedeba xmamaRali msjelobiT unda aRiweros 

(ras vakeTebT) da unda aixnas _ yovelTvis! 

________________________________________________ 

5/3/!UbnbTj;!txsbgj!qbtvyfcj!`!Tflsfcb.hbnplmfcb•/! 

SekiTxvebi moicavs zepir Sekreba-gamoklebas 100-is farglebSi. 

pirvel wreze mTeli aTeulebisa Tu oceulebisa: 56 + 20, 

13 + 40... 56 _ 30, 93 _ 40... 

meore wreze _ erTeulebi rom advilad iangariSeba: 34 + 34, 

51 + 38... 84 _ 34, 51 _ 8, 89 _ 83... 

mesame wreze _ Znelebic: 45 + 38, da romlebic `SevsebiT~ 

akldeba (sxvaoba ar aRemateba ocs): 78 _ 59, 36 _ 17... 

5/4/! pUyj! nbUfnbujlvsj! )bsjUnfujlvmj*! nprnfefcb! eb! 

sjdywjUj! hbnptbyvmfcb/! ! moswavleebs win unda edoT 

gadaSlili saxelmZRvanelo, me-3 gakveTilis teqstze (romelic 

Sin unda hqondeT wakiTxuli). masw.-is SekiTxvebiT gameordeba 

am teqstis TviTeuli nawili. 

SeniSvna. termini àriTmetikuli~ jer araa saWiro, iqneba 

V klasSi. 

h!6/!nprnfefcbUb!sjhj!sjdywjU!hbnptbyvmfcbTj/!njtj!nojTwofmpcb/ 

6/2/ 1.4. 1# 2~ 3~ 4* 5 6! 

6/3/!UbnbTj;!txsbgj!qbtvyfcj!`!sjdywjt!eblblvofcb•/! 

dafaze daxazulia ori Setyupebuli CarCo (ujredebi). erTSi 

weria: òceulebi~ meoreSi weria: `naSTi~. masw.-i maCvenebeli 

joxis wveriT akakunebs am ujredebSi, moswavlem ki unda 

gamoicnos Sesabamisi ricxvi. magaliTad: 

masw.-ma daakakuna erTxel òceulebSi~ da samjer _ `naSTSi~ 

{ocdasami}; arcerTxel òceulebSi~ da rvajer _ `naSTSi~ 

{rva}; samjer _ oceulebSi da arcerTxel _ `naSTSi~ {samoci} 

- 44 -

da ase Semdeg. 

6/4/! pUyj! nbUfnbujlvsj! )bsjUnfujlvmj*! nprnfefcb! eb! 

sjdywjUj! hbnptbyvmfcb/! nprnfefcbUb! sjhj! sjdywjU! 

hbnptbyvmfcbTj/! sjdywjUj! hbnptbyvmfcjt! nojTwofmpcb/! 

moswavleebs win unda edoT gadaSlili saxelmZRvanelo, me-4 

gakveTilis teqstze. masw.-is SekiTxvebiT gameordeba am teqstis 

TviTeuli nawili. 

TfojTwob/ termini àriTmetikuli~ jer araa saWiro, iqneba 

V klasSi. 

h!7/!sjdywjUj!hbnptbyvmfcjt!nojTwofmpcb/!! 

rwfTnjxfsjU!Tflsfcb.hbnplmfcb/! 

7/2/ 1.5. 1~ 2# 3~ 4# 5 6 

me-5 amocanisTvis moswavlem dafaze daxazos borbali. 

7/3/! hbnfpsfcb! `! tbnojTob! sjdywfcjt! rwfTnjxfsjU! Tflsfcb. 

hbnplmfcb/! sbupnbb! nbsUfcvmj! njtj! xftfcj/! Tflsfcb. 

hbnplmfcb! tbbohbsjTp{f/! ! moswavle dafaze dawers samniSna 

ricxvebis qveSmiweriT Sekrebis magaliTs, gverdiT _ imave 

Sekrebas qveSmiwerili gamosaxulebebiT. sruli msjelobiT 

Sesruldeba Sekreba. Semdeg aseve Sesruldeba gamokleba. 

rodesac erTi moswavle dafasTan muSaobs da wers, imavdroulad 

meore moswavle imave moqmedebas masw.-is daxmarebiT 

asrulebs saangariSoze. sxvebi ki msjeloben, romelimes rame 

xom ar SeSlia da ratom. 

masw.-ma mravaljer unda moiTxovos dasabuTeba, Tu ratomaa 

marTebuli moqmedebaTa TviTeuli nabiji. 

5· 

100 + 5· 

10 + 6 

+ 

7· 

10 + 3 

556 

+ 

73 

7 · 100 

2 · 100 

4 · 10 

5 · 10 

+ 1 

+ 8 

am dros mTavaria, rom klasma gaiazros qveSmiweris wesi da ar 

moxdes misi mxolod meqanikuri damaxsovreba: ratom vwerT 

Sesakrebebs erT svetSi, ratom ar SeiZleba maTi wanacvleba 

(aTeuli zustad aTeulis qveS rom unda iyos), ratom vkrebT 

erTmaneTis qveS miweril ricxvebs. amisaTvis damatebiT 

maxvildeba yuradReba qveSmiweriT Canaweris sisworeze: _ 

SekrebiT qveSmiweriT 464 da 89. romeli Canaweria swori? 

464 464 

464 

+ Tu 

+ 

− Tu 

89 

89 

89 

− 

+ 

+ 

464 

− 

89 

- 45 -

_ ratom? {raki qveSmiweriT Sekrebisas erTeulebs erTad 

vkrebT, amitom erTeuli erTeulis qveS unda eweros, aseve 

aTeuli (roca gvaqvs) aTeulis qveS, xolo aseuli (roca 

gvaqvs) aseuli qveS} (aseve gamoklebis SemTxvevaSic). 

7/4/! UbnbTj;! ebwehfU! hbnptbyvmfcjt! njyfewjU•/! ! masw.-i 

dafaze dawers erT gamosaxulebas: 5 · 3 + 2; 5 · 3 + 0; 

5 · 4 + 1, 1 · 9 + 7 da sxva. moswavleebi unda gamovidnen 

TiTo-TiTod, Tanac zustad im TanmimdevrobiT, rogorc sxedan: 

jer marcxena rigis pirveli merxis marcxena, Semdeg misi 

mezobeli, Semdeg meore merxi da ase Semdeg. SeiZleba am Tanmimdevrobis 

Secvla, mTavari isaa, rom klasma kargad icodes, ra 

TanmimdevrobiT unda gamovidnen da es Tanmimdevroba aRar unda 

Seicvalos arc am gakveTilze da arc SemdgomSi. 

moswavleebi gamodian da dgebian ise, rom miviRoT moswavleTa 

5 · 3 + 2 (Tu sxva). Semdeg masw.-i sxva Canawers dawers, gamodian 

is moswavleebi, visac pirvel jerze ar mouwia gamosvla (oRond 

kvlav imave TanmimdevrobiT) da ase Semdeg. 

7/5/! nprnfefcbUb! sjhj! sjdywjU! hbnptbyvmfcbTj/! sjdywjUj! 

hbnptbyvmfcjt!nojTwofmpcb/ swrafad gameordeba moswavlis 

saxelmZRvanelos me-4 gakveTilis teqstis Sinaarsi _ 

sasurvelia, wignSi Cauxedavad, magram Tuki moswavles 

SekiTxvaze pasuxi gauWirdeba, Caixedos. 

h!8/!pUyojTob!sjdywfcjt!xbljUywb!eb!Dbxfsb! 

8/2/! 1.6. 1 2# 3~ 4# 5* 6 

8/3/!UbnbTj;!txsbgj!qbtvyfcj!`!sjdywfcjt!ubTfcj/•! 

am TamaSSi rig-rigobiT erTi TiTo moswavle ki ar pasuxobs, 

aramed _ ori-ori (erT merxze msxdarni). masw.-i daasaxelebs 

100-ze nakleb raime ricxvs. moswavleTa wyvilidan marcxenam 

unda Semokras taSi imdenjer, ramdeni oceulicaa am ricxvSi, 

xolo marjvenam _ am oceulTa garda naSTi risi tolicaa. 

magaliTebi: _ ocdaori {taSi erTxel, taSi orjer}; 

ormcodaTerTmeti {taSi orjer, taSi 11-jer}; aTi {siCume, 

taSi aTjer}; nuli {siCume, siCume}; xuTi {siCume, taSi 

xuTjer}; samoci {taSi samjer, siCume}; da ase Semdeg. 

- 46 -

8/4/! pUyojTob! sjdywfcj/! ! pUyojTob! sjdywjt! xbljUywb/! 

tbbohbsjTp{f! npdfnvmj! sbpefopcjt! tjuzwjfsj! ebtbyfmfcb! 

eb! djgsvmj! Dbxfsb! ebgb{f/! qjsjrjU;! tjuzwjfsbe! bo! 

djgsvmbe!npdfnvmj!sjdywjt!blsfgb•!tbbohbsjTp{f/! 

_ ra aris aTaseuli? {esaa 10 aseuli} rogor iwereba samniSna 

ricxvi, romelSic 9 aseulia da naSTi 87? {9 · 100 + 87 = 

987 (cxriani iwereba marcxniv!)} ese igi, aseve rogor Caiwereba 

samniSna ricxvi, romelSic 10 aseulia da naSTi 87? {10 · 100 

+ 87 = 1087 (aTiani iwereba marcxniv!) _ aTas 

oTxmocdaSvidi}. avkrifoT saangarSoze! 

aseve: 20 · 100 + 87 = 2087 35 · 100 + 87 = 3587 

_ axla am raodenobebSi gamovyoT aTaseulebi da naSTi _ 

romelic samniSna ricxvia (da romlis Cawerac ukve gvaqvs 

naswavli): 2087 = 2 · 1000 + 87 3587 = 3 · 1000 + 587 

_ avkrifoT saangarSoze! 

oTxniSna ricxvis wakiTxva: gairCeva moswavlis saxelmZRvanelos 

me-6 gakveTilSi mocemuli TvalsaCinoeba. Tan es 

ricxvi aikrifeba saangarSoze. 

h!9/!pUyojTob!sjdywfcjt!xbljUywb!eb!Dbxfsb! 

9/2/ 1.7. (1 _ ara) 2~ 3~ 4# 5 6# 

9/3/! pUyojTob! sjdywfcj/! ! pUyojTob! sjdywjt! xbljUywb/! 

tbbohbsjTp{f! npdfnvmj! sbpefopcjt! tjuzwjfsj! ebtbyfmfcb! 

eb! djgsvmj! Dbxfsb! ebgb{f/! qjsjrjU;! tjuzwjfsbe! bo! 

djgsvmbe!npdfnvmj!sjdywjt!blsfgb•!tbbohbsjTp{f/! 

gameordeba 7.3, oRond, cxadia, sxva ricxvebiT. Tan gairCeva 

moswavlis saxelmZRvanelos me-7 gakveTilSi mocemuli 

TvalsaCinoeba. Semdeg gairCeva amocana 1. 

9/4/!UbnbTj;!txsbgj!qbtvyfcj!`!sjdywjt!eblblvofcb•/! 

Catardeba iseve, rogorc 5.2, oRond 999-is farglebSi. 

amisaTvis, cxadia, dafaze saWiro iqneba sami ujredi: 

aseulebisa, aTeulebisa da erTeulebisa. axali magaliTebi: 

dakakuneba samjer, dakakuneba eqvsjer, dakakuneba erTxel {samas 

samocdaerTi}; dakakuneba orjer, dakakuneba arcerTxel, 

dakakuneba rvajer {oras rva}; da ase Semdeg. 

9/5/! hbnfpsfcb! `! tbnojTob! sjdywfcjt! rwfTnjxfsjU! Tflsfcb. 

hbnplmfcb/! sbupnbb! nbsUfcvmj! njtj! xftfcj/! Tflsfcb. 

- 47 -

hbnplmfcb! tbbohbsjTp{f/ gameordeba 6.2, oRond, cxadia, sxva 

ricxvebiT. 

h!:/!tblpouspmp!xfsb!$!2 

1. udides 4 qulas TiTo SecdomisTvis akldeba TiTo q. 

SeiZleba erTi meqanikuri Secdomis patieba. 

2. udides 4 qulas or-ori SecdomisTvis akldeba TiTo q. 

3. advilia, amitom udides 4 qulas TiTo SecdomisTvis 

akldeba TiTo q. 

4. Tuki dawyebulia gamravleba, ukve iwereba 1 q. aseve, Tuki 

dawyebulia namravlidan gamokleba, iwereba 2 q. xolo 

gamoTvlebis or-ori SecdomisTvis udides 5 qulas TiTo 

SecdomisTvis akldeba TiTo q. 

Tuki Tavidanve mcdari moqmedebebiTaa dawyebuli, magram 

gamoTvlebi marTebuladaa, maSin iwereba 1 q. 

5. Tuki daxazulia erTmaneTze nebismierad (Tundac mcdarad) 

midgmuli oTxkuTxedi da xuTkuTxedi, ukve iwereba 1 q. Tuki 

marTebuladaa Sesrulebuli mxolod erTi davaleba _ 3 q. 

orive miRebuli mravalkuTxedis gverdebis raodenobis TiTo 

SecdomisTvis udides 5 qulas akldeba TiTo q. 

Tavi II 

iwyeba meore xazi, ariTmetika-algebrisa: ariTmetikul 

moqmedebaTa wesebi. aq formulebi weria mxolod masw.isaTvis, 

da ara moswavleTaTvis! 

wesebis gazepireba araa saWiro. moswavle gaazrebulad, 

sakuTari sityvebiT unda ayalibebdes maT, Tundac arc 

mTlad zustad _ masw.-i Seusworebs. mTavaria, moswavles 

arsebiTad esmodes es wesebi da gaazrebulad iyenebdes maT 

saWiroebisamebr. 

am xazis safexurebi ZiriTadad moswavlis saxelmZRvanelos 

amocanebSia dawvrilebiT damuSavebuli. sagakveTilo 

muSaoba Tavidan bevri ar sWirdeba, magram sakiTxis 

gavlis Semdeg aucilebelia misi xangrZlivi praqtikuli 

ganmtkiceba. amitom TviTeul safexurs eTmoba saSualod 3 

sagakveTilo monakveTi. xolo ganmtkiceba SemdgomSi 

TavisTavad moxdeba saSinao davalebis amocanebis amoxsnisas 

da garCevisas. 

- 48 -

am xazis bolo mesamedis safexurebi Znelia da yvela 

moswavle ver daeufleba. maTze muSaoba gagrZeldeba V klasSi. 

Cven Zalian did yuradRebas vuTmobT angariSis advil 

xerxebs. masw.-s ar unda egonos, rom maTi mTavari mizania 

swored angariSi da gamoTvlebi. rasakvirvelia, esecaa 

(gansakuTrebiT, zepiri angariSisa), magram mTavari mainc 

sxvaa: swored angariSis advili xerxebia bavSvisTvis 

misawvdomi, bunebrivi da misaRebi saSualeba algebris 

swavlebisa. mTavari mizania is, rom bavSvi daeuflos algebrul 

moqmedebaTa sawyisebs, an moqmedebebs gamosaxulebebze. 

magaliTad: 

800 · 7 = 8 · 100 · 7 = 8 · 7 · 100 = 56 · 100 = 5600; 

24000 · 3 = 24 · 1000 · 3 = 24 · 3 · 1000 = 72000; 

30015 · 4 = (30000 + 15) · 4 = 30000 · 4 + 15 · 4 = 120000 + 60 = 120060. 

dafaze (da Tan rveulebSi) Sesrulebuli gamoTvlebis 

TviTeuli moqmedeba xmamaRali msjelobiT unda aRiweros 

(ras vakeTebT) da unda aixnas (ratomaa ase) _ yovelTvis! 

________________________________________________ 

h!21/!pUyojTob!sjdywfcj/!TftblsfcUb!hbebobdwmfcb!eb!ebkhvgfcb/! 

21/2/ 1.8. 1~ 2~ 3# 4# 5# 6 

2.1. 1~ 2~ 3~ 4~ (5 _ ara) 6# 

21/3/! Tflsfcjt! xftfcj;!TftblsfcUb! hbebobdwmfcb! eb! ebkhv. 

gfcb (a+b)+c = a+(b+c) = (c+a)+b... wesebi zogadad jer ar 

Camoyalibdeba _ ubralod, saSinao davalebidan 1-li gakveTilis 

me-5 amocana gairCeva dawvrilebiT da daskvnebi Camoyalibdeba am 

kerZo SemTxvevisaTvis. 

21/4/!UbnbTj;!txsbgj!qbtvyfcj!`!sjdywfcjt!ubTfcj/•! 

ix. 7.2, oRond oceuli Seicvalos aTeuliT. magaliTebi: 

_ ormocdasami {taSi oTxjer, taSi samjer}; samoci {taSi 

eqvsjer, siCume}; rva {siCume, taSi rvajer} ocdacameti {taSi 

samjer, taSi samjer}; oci {taSi orjer, siCume}; nuli {siCume, 

siCume} da ase Semdeg. 

21/5/!pUyojTob!sjdywfcj/!!pUyojTob!sjdywjt!xbljUywb/! 

! tbbohbsjTp{f! npdfnvmj! sbpefopcjt! tjuzwjfsj! 

ebtbyfmfcb!eb!djgsvmj!Dbxfsb!ebgb{f/!qjsjrjU;!tjuzwjfsbe! 

bo! djgsvmbe! npdfnvmj! sjdywjt! blsfgb•! tbbohbsjTp{f/!! 

- 49 -

gameordeba 7.3, oRond, cxadia, sxva ricxvebiT. 

h!22/!{fqjsj!bohbsjTj!pUyojTob!sjdywfc{f/!!Tflsfcjt!xftfcj! 

22/2/ 2.2. 1# 2~ 3~ 4 5~ 6~ 

22/3/! Tflsfcjt! xftfcj;! TftblsfcUb! hbebobdwmfcb! eb! ebkhv. 

gfcb (a+b)+c = a+(b+c) = (c+a)+b... moswavleebs win unda 

edoT gadaSlili saxelmZRvanelo, me-2 gakveTilis teqstze 

(romelic Sin unda hqondeT wakiTxuli). masw.-is SekiTxvebiT 

gameordeba am teqstis TviTeuli nawili. 

22/4/!UbnbTj;!yfmfcjt!tjhs[jt!njyfewjU!ebehpnb•/! 

gamova ori moswavle, sworad gaSlian xelebs da daetolebian 

_ vis ufro grZeli aqvs. visac ufro grZeli aqvs, dadgeba 

ufro marcxniv. gamova Semdegi moswavle da daetoleba _ Tavis 

adgils moZebnis. gamova meoTxe da _ isic. ase dalagdeba mTeli 

klasi xelebis sigrZis mixedviT. 

Tuki zog or moswavles toli sigrZis xelebi aRmoaCndeba, 

erTmaneTis win dgebian, ese igi, rigSi erTsadaimave adgilas. 

Zalian bevri datoleba rom ar gaxdes saWiro, axalgamosul 

moswavles masw.-i miuTiTebs, Tu romel moswavleebs daetolos 

_ daetolos daaxloebiT misi simaRlis moswavleebs, ufro 

maRlebsa da ufro dablebs nu daetoleba. 

bolos, yuradRebas mivaqcevT imas, xelebis sigrZis mixedviT 

dadgoma igivea Tu ara, rac simaRlis mixedviT dadgoma. 

22/5/! {fqjsj! bohbsjTj! pUyojTob! sjdywfc{f-! 21111.jt! 

gbshmfcTj! )bewjmj! TfnUywfwfcj*/! sruldeba saangariSoze 

gaTvalsaCinovebis TanxlebiT: 630 + 670 1500 + 3500 

4122 + 4003 6305 + 900 1811 _ 805 1411 _ 800 

7800 _ 4300 1500 · 3 8400:2 

h!23/!mbUjovsj!btpfcj/!!Tflsfcb!bewjmj!yfsyjU! 

23/2/ 2.3. 1~ 2 3~ 4 5 6# 7# 

23/3/! mbUjovsj! btpfcj/! ! swrafad gaimeoreben laTinuri 

asoebis wakiTxvas: masw.-i dawers dafaze, moswavle pasuxobs. 

SeiZleba, moswavleebma inglisuri an sxvaenovani gamoTqmiT 

waikiTxon (rogorc aqvT naswavli) _ Sesworeba araa saWiro, 

waikiTxon ise, rogorc eadvilebaT. 

sxva asoebi araa saWiro, radgan bavSvs SeiZleba gauZneldes 

- 50 -

_ saqme isaa, rom sxva asoebi rusul da laTinur enebSi sxvadasxvagvarad 

ikiTxeba, rac bavSvs damatebiT siZneles uqmnis. 

23/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj!pUyojTob! 

sjdywfc{f-!21111.jt!gbshmfcTj!)bewjmj!TfnUywfwfcj*/! ! ! 

! ! nimuSebi ix. 11.4-Si. 

Sekreba advili xerxiT: a+b+c+d = (a+c)+(b+d) da sxva. 

Sekrebis naswavli wesis gamoyenebiT: daiwereba: 250 + 105 + 

650 + 195 _ rogor jobia Sekreba, rogori TanmimdevrobiT? 

Sesakrebebi advilad rom Seikribos? Cven xom viciT, rom 

SesakrebTa gadanacvleba da dajgufeba SegviZlia, amiT jamis 

mniSvneloba xom ar Seicvleba! 

Sekrebis naswavli wesis gamoyenebiT: (250+650) + (105+195) = 

900 + 300 = 1200. 

aseve Sesruldeba: 128 + 3010 + 22; 5070 + 251 + 49 + 

30. ukanasknelSi sxva Tanmimdevrobis SerCevaa saWiro. 

Sekrebis win yovelTvis unda davukvirdeT da yvelaze advili 

Tanmimdevroba SevarCioT! 

h!24/!bewjmj!Tflsfcb/!! 

pUyojTob!sjdywfcjt!{fqjsj!Tflsfcb.hbnplmfcb! 

24/2/ 2.4. 1# 2# 3 4# 5# 6 7 

24/3/! Tflsfcb! bewjmj! yfsyjU; a+b+c+d = (a+c)+(b+d) eb! 

tywb/!Catardeba iseve, rogorc 12.3, oRond sxva magaliTebiT: 

26 + 275 + 225 + 64; 1011 + 1243 + 889 + 7; 

2100 + 1800 + 2900 + 2200. 

TviTeul SemTxvevaSi xmamaRla vityviT, Tu ratom 

davawyvileT Sesakrebebi swored ase _ me-4 gakveTilis teqstSi 

mocemuli wesis Sesabamisad. am wess kidev erTxel wavikiTxavT 

da gaviazrebT. rogorc yovelTvis, TviTeuli moqmedeba 

xmamaRali msjelobiT unda aRiweros (ras vakeTebT) da unda 

aixnas (ratomaa ase). 

24/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!txsbgj!qbtvyfcj!`!sjdywjt! 

eblblvofcb•/!Catardeba iseve, rogorc 8.3, oRond ukve 9999-is 

farglebSi. amisaTvis, cxadia, dafaze saWiro iqneba oTxi 

ujredi: aTaseulebisa, aseulebisa, aTeulebisa da erTeulebisa. 

axali magaliTebi: dakakuneba xuTjer, dakakuneba samjer, dakakuneba 

eqvsjer, dakakuneba erTxel {xuTi aTas samas samoc- 

- 51 -

daerTi}; dakakuneba arcerTxel, dakakuneba orjer, dakakuneba 

arcerTxel, dakakuneba rvajer {oras rva}; da ase Semdeg. 

cxadia, am TamaSSi kakunis mosmenisa da mosmenilis 

damaxsovrebis siZnelecaa. 

24/5/!pUyojTob!sjdywfcjt!{fqjsj!Tflsfcb.hbnplmfcb!21111.jt! 

gbshmfcTj! )bewjmj! TfnUywfwfcj*/! sruldeba saangariSoze 

gaTvalsaCinovebis TanxlebiT: 360 + 740 2500 + 4500 

6005 + 2103 800 + 5012 8612 _ 505 2333 _ 600 

8700 _ 3400 3400 · 2 9300:3 

h!25/!kbnjt!hbnplmfcb/!!oblwUfcjt!bSojTwob!mbUjovsj!btpfcjU! 

25/2/ 2.5. 1# (2 _ ara) 3# 4# 5# 6~ 

25/3/! kbnjt! hbnplmfcb; a - (b+c+d) = a - b - c - d. saSinao 

davalebidan me-5 gakveTilis me-2 amocana gairCeva dawvrilebiT 

da dasknebi Camoyalibdeba am magaliTis ganzogadebiT. 

25/4/!UbnbTj;!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj• _ 110 +_ `20~.! 

aRwerilia meTod-saxelmZRvanelos zogad nawilSi. masw.-s taSis 

SemokvriT TamaSi gadahyavs ricxvTa ritmuli matebidan maT 

ritmul klebaze, rac TamaSs ufro sainteresos xdis. 

amjerad daviwyoT 110-dan, vumatoT oc-oci (masw.-is 

taSamde), Semdeg vakloT oc-oci (masw.-is taSamde), isev 

vumatoT or-ori da ase Semdeg. 

es yovelive ase aRiniSneba: 110 +_ `20~. 

25/5/!oblwUfcjt!bSojTwob!mbUjovsj!btpfcjU/! 

gairkveva, me-4 gakveTilis teqstSi mocemuli TviTeuli mcdari 

Canaweri ratomaa mcdari. 

h!26/!kbnjt!hbnplmfcb/!!hbnplmfcb!bewjmj!yfsyjU! 

26/2/ 2.6. 1~ 2~ (3 _ ara) 4# 5 6# 7 

me-7 amocanaSi gavixsenoT zooparkebi. 

26/3/! UbnbTj;! txsbgj! qbtvyfcj! `! sbnefojUbb! nfuj•/! 

SekiTxvebi moicavs ricxvTa zepir Sedarebas 100-is farglebSi. 

Zneli Sesadarebeli ricxvebi araa saWiro. misacemi magaliTebi: 

1) 40-is farglebi: 38 da 29; 16 da 31...; 2) roca ricxvTa Soris 

gansxvaveba ar aRemateba 20-s: 45 da 63; 78 da 92...; 3) 

roca sxvaoba advili saangariSoa: 67 da 24; 80 da 15; 30 da 

94... 

- 52 -

26/4/!kbnjt!hbnplmfcb; a - (b+c+d) = a - b - c - d. hbnplmfcb! 

bewjmj!yfsyjU; a - b - c - d = a - (b+d) - c eb!tywb/ 

saSinao davalebidan me-6 gakveTilis me-3 amocana gairCeva 

dawvrilebiT. rogorc yovelTvis, yvelaferi Sesruldeba 

xmamaRali msjelobiT _ TviTeuli nabijis aRweriTa da axsniT. 

Semdeg saklaso samuSaoebi: 9000 _ 5125 = 

9000 _ (5000 + 100 + 25) = 9000 _ 5000 _ 100 _ 25 = 

4000 _ 100 _ 25 = 3900 _ 25 = 3875. 

axla gadajgufebiT: 8500 _ 4531 = 8500 _ (4000 + 500 + 

31) = 8500 _ 500 _ 4000 _ 31 = 8000 _ 4000 _ 31 = 3969. 

_ gamoklebisas SesakrebTa gadajgufeba ratom SeiZleba? {Cven 

ukve naswavli gvaqvs, rom SesakrebTa gadajgufebiT jamis mniSvneloba 

ar icvleba}. winaswar (4000 + 500 + 31) jamSi movaxdenT 

saTanado gadanacvlebas _ Sekrebis naswavli wesis ZaliT! 

h!27/!oblwUfcjt!bSojTwob!btpfcjU/!! 

Tflsfcb.hbnplmfcb!bewjmj!yfsyjU! 

27/2/ 2.7. 1~ (2 _ ara) 3# 4# 5# 6 

27/3/!oblwUfcjt!bSojTwob!mbUjovsj!btpfcjU/! 

dawvrilebiT gairCeva saSinao davalebidan me-7 gakveTilis me-2 

amocana. Semdeg gairkveva, Tu ratomaa mcdari Semdegi aRniSvnebi: 

marTkuTxedi ADME, gaxsnili texili MATKDE, gaxsnili 

texili AMKTDE. 

27/4/! UbnbTj;! txsbgj! qbtvyfcj! `! sjdywfcjt! ubTfcj/•! 

Catardeba iseve, rogorc 10.3, oRond ukve 999-is farglebSi. 

amitom, cxadia, ukve sam-sami moswavle iqneba saWiro da 

erTmaneTs moswavleTa sameulebi Seejibrebian. axali magaliTebi: 

_ oras samocdaeqvsi {taSi orjer, taSi eqvsjer, taSi 

eqvsjer}; as oTxmoci {taSi erTxel, taSi rvajer, siCume}; 

ormocdaori {siCume, taSi oTxjer, taSi orjer}; xuTasi {taSi 

xuTjer, siCume, siCume}; oTxas cxra {taSi oTxjer, siCume, 

taSi cxrajer} da ase Semdeg. 

27/5/!Tflsfcb!bewjmj!yfsyjU

(3000 + 4000) + (400 + 600) + (25 + 75) = 

7000 + 1000 + 100 = 8100. 

3425 _ 1475 = 3425 _ (1000 + 400 + 75) = 3425 _ 1000 

_ 400 _ 75 = 2425 _ 400 _ 75 = 2000 _ 75 = 1925. 

3425 _ (95 + 25 + 600) = 3425 _ 95 _ 25 _ 300 = (aq umjobesia, 

rom jer 25 gamovakloT!) = 3400 _ 95 _ 300 = (aq 

umjobesia, rom jer 300 gamovakloT!) = 3100 _ 95 = 3005. 

28/!tblpouspmp!xfsb!$!3/! 

2/ Tuki erTi isari mainc marTebuli mimarTulebiTaa, ukve 

iwereba 1 q. Tuki meorec marTebulia _ 2 q da ase Semdeg _ 4 q. 

3/ advilia, amitom udides 4 qulas TiTo SecdomisTvis 


4/ Tuki erTerTi gamoTvlilia marTebulad, magram ara 

advili xerxiT, maSin iwereba 3 q. gamoTvlis TiTo SecdomisTvis 

akldeba TiTo q. Tuki SerCeulia advili xerxebi, maSin 

gamoTvlis or-ori SecdomisTvis akldeba TiTo q. 

5/ Tuki dawyebulia gamravleba da Sekreba, ukve iwereba 1 q. 

aseve, Tuki dawyebulia jamidan gamokleba, iwereba 2 q. xolo 

gamoTvlebis or-ori SecdomisTvis udides 5 qulas TiTo 

SecdomisTvis akldeba TiTo q. 

Tuki Tavidanve mcdari moqmedebebiTaa dawyebuli, magram 

gamoTvlebi marTebuladaa, maSin iwereba 1 q. 

6/ TiTo samniSna ricxvis marTebulad Sedgenaze iwereba 

TiTo q. erTaderTobis TiTo marTebul pasuxzec _ TiTo q. 

Tuki dasabuTebac sanaxevrod mainc marTebulia _ 5 q. 

orive SemTxvevaSi pasuxia: araa erTaderTi. dasabuTeba 

amgvaria: 

I _ magaliTad: 480 da 370; II _ magaliTad: 106 da 116. 

Tavi III 

h!29/!Tfls/.hbnplm/!nshwbm!btfvmbnef!bo!bUbtfvmbnef!TfwtfcjU/! 

29/2/ 2.8. 1~ 2# 3# 4# 5! 6# 

3.1. (1 _ ara) 2~ 3~ 4# 5# 6# 7 

29/3/!UbnbTj;!Uwmb!eb!vlvUwmb•!)2111.jt!gbshmfcTj*/! 

wesi aseTia: pirveli moswavle iwyebs Tvlas ise, rom meoremesame 

moswavles mouwios aseulze gadasvla (magaliTad, iwyebs 

- 54 -

487-iT). masw.-is taSis Semokvraze is wyvets Tvlas, agrZelebs 

momdevno moswavle _ masw.-is Semdegi taSis Semokvramde, da ase 

Semdeg. uceb masw.-i daiZaxebs: _ axla ukuRma! da momdevno 

moswavlem ukuTvla unda ganagrZos zustad im ricxvidan, 

romelzedac wina moswavles Seawyvetina masw.-ma. da ase Semdeg... 

29/4/!pUyojTob!sjdywfcjt!{fqjsj!Tflsfcb.hbnplmfcb!21111.jt! 

gbshmfcTj;! nshwbm! btfvmbnef! bo! bUbtfvmbnef! Tfwtfcjt! 

yfsyjU/! jer dawvrilebiT gairCes saSinao davalebidan 1-li 

gakveTilis 1-li amocana. Semdeg aseve viangariSebT: 

5071 + 2098 = 5071 + 2100 _ 2 =7111 _ 2 = 7109. 

3536 + 4005 = 3536 + 4000 + 5 = 7536 + 5 = 7541. 

4999 + 3999 = 5000 – 1 + 4000 – 1 = 9000 – 1 – 1 = 8998. 

h!2:/!qfsjnfusj/!!UbobnbnsbwmUb!hbebobdwmfcb!eb!ebkhvgfcb! 

2:/2/ 3.2. 1~ 2# 3~ 4# (5 _ ara) 6 

2:/3/! nsbwbmlvUyfejt! qfsjnfusj/!gavimeorebT, ra aris mravalkuTxedis 

perimetri da Semdeg gamovTvliT dafis perimetrs. 

gamoTvlebi dafaze (da, ese igi, rveulebSic) daiwereba. 

2:/4/!UbnbTj;!v[sbwUb!ebxzwjmfcb•/!Catardeba TamaSis saxiT, 

saxalisod. masw.-i dafaze swrafad daxatavs sxvadasxva naswavl 

nakvTebs: wreebs, rgolebs, mravalkuTxedebs, marTkuTxedebs, 

kvadratebs, xazebsa da sxva. maTi raodenoba iyos daaxloebiT 

imdeni, ramdeni erTeuli avejic dgas saklaso oTaxSi, moswavleTa 

skamebis garda (Tuki oTaxSi moswavleebi cal-calke 

skamebze sxedan; xolo sizustes ara aqvs mniSvneloba). _ axla 

gavarkvioT, Cvens saklaso oTaxSi avejis raodenobaa meti Tu am 

nakvTebis raodenoba. (oRond, moswavleTa skamebs nu CavTvliT, 

Torem aveji Zalian bevri gamogviva; maswavlebis skami ki 

CavTvaloT!). rogor davawyviloT? {moZravi saTvleli nivTebis 

meSveobiT}. modi, saTvleli nivTebis nacvlad moswavleebi 

iyaviT! aba, TiTo avejTan TiTo moswavle dawyvildes! (gzadagza 

gairkveva, Tu kerZod romeli nivTebia àveji~). axla ra vqnaT? 

{dawyvilebuli moswavleebi nakvTebTan dawyvildnen}. modiT, 

dawyvildiT, TiTo moswavlem TiTo nakvTs daados TiTi! Zalian 

kargi. ras gvetyvis cara nakvTebisa da da avejis raodenobaTa 

Sesaxeb? {nakvTebis raodenoba samiTaa meti, vidre avejisa}. nika, 

saidan icis es ciram? {radgan sami nakvTi mogvrCa mewyviles 

- 55 -

gareSe}. kargi, axla jer nu dasxdebiT, isev daadeT TiTebi Tqvens 

nakvTebs! daviwyoT aqedan da rig-rigobiT: TviTeulma moswavlem 

daasaxelos is nakvTi, romelic mas Sexvda! 

2:/5/! hbnsbwmfcjt! xftfcj;! UbobnbnsbwmUb! hbebobdwmfcb! eb! 

ebkhvgfcb/! wesebi zogadad jer ar Camoyalibdeba _ ubralod, 

saSinao davalebidan me-2 gakveTilis me-5 amocana gairCeva 

dawvrilebiT da dasknebi Camoyalibdeba am kerZo SemTxvevisaTvis. 

h!31/!UbobnbnsbwmUb!hbebobdwm/!eb!ebkhvg/!!Tfwtfcjt!yfsyj! 

31/2/ 3.3. (1 da 2 _ ara) 3~ 4~ 5 6 

31/3/! hbnsbwmfcjt! xftfcj;! UbobnbnsbwmUb! hbebobdwmfcb! eb! 

ebkhvgfcb/!dawvrilebiT gairCeva saSinao davalebidan 1-li da 

me-2 amocanebi. Semdeg moswavleebma win unda daidon gadaSlili 

saxelmZRvanelo, me-3 gakveTilis teqstze (romelic Sin unda 

hqondeT wakiTxuli). masw.-is SekiTxvebiT gameordeba am teqstis 


31/4/!UbnbTj;!txsbgj!qbtvyfcj!`!tbqjsjtqjspoj•/! 

masw.-i ambobs romelime sityvas naswavl raodenobriv, sivrciT 

Tu droiT mimarTebaTagan Tu moqmedebaTagan _ moswavlem unda 

daasaxelos sapirispiro sityva. sityvebi: 

bevri, mravali, cota, meti, naklebi, didi, patara, maRali, 

dabali, grZeli, mokle, zemo, zemoT, zeda, qvemo, qvemoT, qveda, 

maRla, dabla, wina, win, ukana, ukan, SigniT, Siga, gareT, gare, 

marjvena, marjvniv, marcxena, marcxniv, adre, gvian, ufrosi, 

umcrosi, mimateba, gamokleba. 

meore wreze ufro Znelebi: msxvili, wvrili, sqeli, 

Txeli, farTo, ganieri, viwro, xangrZlivi, xanmokle, Semdeg; 

dagrZeleba, damokleba, daviwroeba, gafarToeba, gamsxvileba, 

dawvrileba, gasqeleba, gaTxeleba. 

31/5/! pUyojTob! sjdywfcjt! {fqjsj! Tflsfcb.hbnplmfcb! 21111. 

jt! gbshmfcTj;! nshwbm! btfvmbnef! bo! bUbtfvmbnef! Tfwtfcjt! 

yfsyjU/!Catardeba iseve, rogorc 18.3: 

5071 + 2098 = 5071 + 2100 _ 2 =7111 _ 2 = 7109. 

3536 + 4005 = 3536 + 4000 + 5 = 7536 + 5 = 7541. 

4999 + 3999 = 5000 _ 1 + 4000 _ 1 = 9000 _ 1 _ 1 = 8998. 

h!32/!hbnsbwmfcjt!xftfcj/!!hbnsbwmfcb!bewjmj!yfsyjU/! 

- 56 -

32/2/ 3.4. (1 da 2 _ ara) 3# 4~ 5 6 7! 

32/4/! UbnbTj;! txsbgj! qbtvyfcj! `! Dbwjgjrsf! sjdywj•/! ucnobi 

saklebis, maklebis, Sesakrebis, Tanamamravlis, gasayofisa an gamyofis 

moZebnaze. oRond, Zneli saangariSoebi ar unda gamoerios. 

magaliTad: 1) Cavifiqre ricxvi, gamovakeli 60, miviRe 1620. 

ra ricxvi Camifiqrebia? 

2) Cavifiqre ricxvi, mivumate 1300, miviRe 8000. 

3) Cavifiqre ricxvi, es ricxvi gamovakeli 2801-s, miviRe 2799. 

4) Cavifiqre ricxvi, gavamravle 8-ze, miviRe 48. 

5) Cavifiqre ricxvi,gavyavi 7-ze, miviRe 3. 

6) Cavifiqre ricxvi, am ricxvze gavyavi 45, miviRe 9. 


ebkhvgfcb/!! hbnsbwmfcb!bewjmj!yfsyjU; a · b · c · d = (a · c) · 

(b · d) eb!tywb. dawvrilebiT gairCeva saSinao davalebidan 1-li 

da me-2 amocanebi. Semdeg aseve Sesruldeba: 7 · 40 da 300 · 6. 

bolos, swrafad gameordeba me-3 gakveTilis wesebi. 

h!33/!hboupmfcb!eb!njtj!bnpytob/!!hbnsbwmfcb!bewjmj!yfsyjU! 

33/2/ 3.5. (1 da 2 _ ara) 3~ 4# 5! 6 

33/3/! hboupmfcb! eb! njtj! bnpytob/! dawvrilebiT gairCeva 

saSinao davalebidan 1-li amocana. Semdeg moswavleebma win unda 

daidon gadaSlili saxelmZRvanelo, me-5 gakveTilis teqstze 


swrafad gameordeba am teqstis ZiriTadi sakiTxebi. 

33/4/!UbnbTj;!txsbgj!qbtvyfcj!`!Dbwjgjrsf!sjdywj•/!ix. 21.3. 


ebkhvgfcb/!!hbnsbwmfcb!bewjmj!yfsyjU; 

a · b · c · d = (a · c) · (b · d) eb!tywb/ 

dawvrilebiT gairCeva saSinao davalebidan me-2 amocana. Semdeg 

advili xerxiT gamovTvliT: 5 · 37 · 2 = 5 · 2 · 37 = 10 · 37 = 370. 

h!34/!hboupmfcb!eb!njtj!bnpytob/!!hbnsbwmfcb!bewjmj!yfsyjU! 

34/2/ 3.6. (1, 2 da 3 _ ara) 4~ 5# 6# 7 

34/3/!hbnsbwmfcb!bewjmj!yfsyjU; a · b · c · d = (a · c) · (b · d) eb! 

tywb/!dawvrilebiT gairCeva saS. davalebidan me-3 amocana. 

34/4/! UbnbTj;! txsbgj! qbtvyfcj! `! Tfnefhjt! Tfnefhj-! xjojt! 

xjob•/! masw.-i sityvierad asaxelebs ricxvs (magaliTad, 5398), 

- 57 -

moswavlem unda daasaxelos am ricxvis Semdegis Semdegi ricxvi 

{5400}. meore wreze _ am ricxvis winis wina ricxvi {5396}, 

mesame wreze _ misi Semdegis Semdegis Semdegi {5401}, meoTxeze 

_ winis winis wina {5395}. masw.-ma ricxvebi ise unda 

daasaxelos, rom moswavles mouwios aseulze an aTaseulze 

gadasvla: 7101, 12999, 16001... 

34/5/!hboupmfcb!eb!njtj!bnpytob/!! 

dawvrilebiT gairCeva saSinao davalebidan 1-li da me-2 

amocanebi. me-2 amocanisTvis SevadgenT Sesabamis gantolebebs: 

x + 2080 = 3000 y _ 117 = 117 2050 _ z = 1510. 


2/ TiTo marTebul gamravlebaSi iwereba TiTo q. 

3/ TiTo SecdomaSi an or-ori SesaZleblobis gamotovebaSi 

udides 4 qulas TiTo SecdomisTvis akldeba TiTo q. 

4/ Tuki dawyebulia gamravleba an sm-ebSi gadayvana da 

gamravleba, ukve iwereba 2 q. Tuki es araa, magram erTi mainc 

Sekrebaa Catarebuli marTebulad (sigrZis gaormageba mainc) _ 1 

q. gamravlebis SecdomisTvis udides 4 qulas akldeba 1 q. 

5/ sul xuTi ricxvia gasarkvevi, TiToSi iwereba TiTo q. 

6/ Tuki dawyebulia gamravleba da gamokleba, ukve iwereba 1 q. 

Semdeg TiTo marTebul gamoTvlaze iwereba TiTo q, sul _ 5 q. 

Tavi IV 

h!36/!hbnsbwmfcjtb!eb!Tflsfcjt!xftfcj/! 

36/2/ 3.7. 1~ 2# 3~ 4~ 5# 6# 

4.1. (1 da 2 _ ara) 3~ 4~ 5# 6# 

36/3/!UbnbTj;!Uwmb!eb!vlvUwmb•!)2111.jt!gbshmfcTj*/!ix. 18.2. 

36/4/!hbnsbwmfcjtb!eb!Tflsfcjt!xftfcj!`!gsDyjmfcjt!hbytob;! 


moswavleebs win unda edoT gadaSlili saxelmZRvanelo, 1-li 

gakveTilis teqstze (romelic Sin unda hqondeT wakiTxuli). 

masw.-is SekiTxvebiT gameordeba am teqstis TviTeuli nawili. 

Semdeg gairCeva amocanebi 1 da 2. 

h!37/!hbnsbwm/!eb!Tflsfcjt!xftfcj/!tbnojTob!sjdywfcjt!Tfebsfcb 

37/2/ 4.2. (1 da 2 _ ara) 3# 4 5! 6# 7 

37/3/!hbnsbwmfcjtb!eb!Tflsfcjt!xftfcj!`!gsDyjmfcjt!hbytob;! 


- 58 -

dawvrilebiT gairCeva amocanebi 1 da 2. 

Semdeg Sesruldeba erTi savarjiSo, romelSic gamoyenebulia 

frCxilebis gaxsnaca da TanamamravlTa gadanacvlebac: 

5407 · 3 = (5 · 1000 + 4 · 100 + 7) · 3 = 5 · 1000 · 3 + 4 · 100 · 3 + 7 · 3 

= 5 · 3 · 1000 + 4 · 3 · 100 + 7 · 3 = 15 · 1000 + 12 · 100 + 21 = 

15000 + 1200 + 21 = 16221. 

rogorc yovelTvis, TviTeuli moqmedeba xmamaRali 

msjelobiT unda aRiweros (ras vakeTebT) da unda aixnas. 

37/4/!UbnbTj;!hb{pnwb•/!swrafad da saxalisod, dawvrilebiT 

SekiTxvaTa gareSe, gaizomeba: nabijebiTa da terfebiT _ 

saklaso oTaxis sigrZe; mtkavelebiTa da TiTebiT _ fanjris 

rafis sigrZe. orive gazomvas erTdroulad Seasrulebs orori 

moswavle (romlebic yvelaze kargad iqceodnen da kargad 

pasuxobdnen), sxvadasxva adgilas (erTmaneTis paralelurad). 

masw.-i winaswar gaafrTxilebs, rom xmamaRla ar Tqvan gazomvis 

Sedegi. 

oTaxi rafa 

sanam moswavleebi zomaven, 

cira _ 

masw.-i dafaze daxazavs cxrils 

nika _ 

(moswavleebi mas ar gadaxazaven, 

mediko _ 

radgan TamaSia). 

amirani _ 

Tan masw.-i merxebze 

mjdom moswavleebs ekiTxeba: _ levan, rogor ggonia, ciras 

ramdeni nabiji iqneba oTaxis sigrZe? cira, kargad gamoicno 

levanma? sinamdvileSi ramdeni nabiji iyo? beso, rogorc 

ggonia, ciras ramdeni terfi iyo naSTi?.. merab, rogor ggonia, 

amiranis ramdeni mtkaveli iqneba rafis sigrZe?.. salome, 

rogorc ggonia, ramdeni TiTi iqneboda naSTi? (da aseve 

danarCen gazomvebzec). 

bolos, gazomvis yvela Sedegi cxrilSi Caiwereba. 

37/5/! hbnfpsfcb;! tbnojTob! sjdywfcjt! Tfebsfcb/! 

tbbohbsjTp{f! npdfnvm! sbpefopcbUb! Tfebsfcb! bUbtjt! 

gbshmfcTj/!swrafad Sedardeba 430 da 416; 201 da 353. 

oRond, ricxvebi arc daiwereba da arc dasaxeldeba _ mxolod 

saangariSoze aikrifeba erTi ricxvi _ cal mxares da meore _ 

meore mxares. 

Sedareba Zalian advilia, siaxlea mxolod saangariSoze 

akrefil raodenobaTa Sedareba. 

- 59 -

h!38/!yvUojTob!eb!frwtojTob!sjdywfcj/!!tbUbosjhp!tbnfvmfcj/! 

38/2/ 4.3. (1 da 2 _ ara) 3# 4~ 5 6 7 

38/3/!UbnbTj;!txsbgj!qbtvyfcj!`!hbnsbwmfcb.hbzpgb!sjdywUb! 

yb{{f•/! SekiTxvebi moicavs gamravleba-gayofis naswavl magali- 

Tebs, moswavleebi pasuxoben saxelmZRvanelos ydis Tvalsa- 

Cinoebis gamoyenebiT. 

es TamaSi xSirad Catardeba, rac saukeTeso saSualebaa, 

kerZod, gamravlebis cxrilis dasazepirebladac. 

38/4/! yvUojTob! eb! frwtojTob! sjdywfcj/! ! tbUbosjhp! tbnfv. 

mfcj/! ! sjdywfcjt! xbljUywb! eb! Dbxfsb/! tbbohbsjTp{f! npdfnv. 

mj! sbpefopcjt! tjuzwjfsj! ebtbyfmfcb! eb! djgsvmj! Dbxfsb! 

ebgb{f/! qjsjrjU;! tjuzwjfsbe! bo! djgsvmbe! npdfnvmj! sjd. 

ywjt! blsfgb•! tbbohbsjTp{f/! moswavleebs win unda edoT 




teqstSi mocemuli yvela ricxvi unda aikrifos saangariSoze! 


h!39/!yvUojTob!eb!frwtojTob!sjdywfcj/!!tbUbosjhp!tbnfvmfcj/! 

39/2/ 4.4. (1 da 2 _ ara) 3 4# 5~ 6 7! 

39/3/!yvUojTob!eb!frwtojTob!sjdywfcj/!!tbUbosjhp!tbnfvmf. 

cj/! sjdywfcjt! xbljUywb! eb! Dbxfsb/! tbbohbsjTp{f! npdfnvmj! 

sbpefopcjt! tjuzwjfsj! ebtbyfmfcb! eb! djgsvmj! Dbxfsb! 


ywjt! blsfgb•! tbbohbsjTp{f/! moswavleebs win unda edoT 




teqstSi mocemuli yvela ricxvi unda aikrifos saangariSoze! 


39/4/!UbnbTj;!!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj;!! 8500! ± 

`250~/!ix. 14.3. 

39/5/!hbnfpsfcb;!tbnojTob!sjdywfcjt!Tfebsfcb/! 

! tbbohbsjTp{f! npdfnvm! sbpefopcbUb! Tfebsfcb! bUbtjt! 

gbshmfcTj/!!Catardeba iseve, rogorc 26.4, swrafad. 

h!3:/!yvUojTob!eb!frwtojTob!sjdywfcj/!!tbUbosjhp!tbnfvmfcj! 

- 60 -

3:/2/ 4.5. (1, 2, 3 da 4 _ ara) 5~ 6 7 

3:/!yvUojTob!eb!frwtojTob!sjdywfcj/!!tbUbosjhp!tbnfvmfcj/!! 

sjdywfcjt! xbljUywb! eb! Dbxfsb/! tbbohbsjTp{f! npdfnvmj! 



ywjt! blsfgb•! tbbohbsjTp{f/! swrafad gameordeba moswavlis 

saxelmZRvanelos me-4 gakveTilis ZiriTadi sakiTxebi. Semdeg 

dawvrilebiT gairCeva saSinao davalebidan me-5 gakveTilis 1-li 

amocana. 

yvela mocemuli Tu naxsenebi ricxvi aikrifeba saangariSoze! 

3:/4/! UbnbTj;! Uwbmjt! ebybnybnfcb•/! ! magnitur dafaze 

mikrulia 2 kvadrati marjvniv da 4 rgoli marcxniv (areuli 

ganlagebiT), oRond dafas farda aqvs Camofarebuli. masw.-i erTi 

wamze naklebi droiT gadawevs fardas. moswavlem unda Tqvas, 

ra iyo dafaze {sami kvadrati da oTxi rgoli}. dafaze icvleba 

rogorc nakvTebi, ise maTi raodenoba. momdevno wreebze 

raodenobani imatebs, oRond ar unda iyos 5-ze meti, radgan 

meti raodenobis ase erTbaSad aRqma Zalian Znelia. 

3:/5/! sjdywfcjt! Tfebsfcb! njmjpojt! gbshmfcTj/! ! tbbohbsj. 

Tp{f!npdfnvm!sbpefopcbUb!Tfebsfcb!njmjpojt!gbshmfcTj/! 



masw.-is SekiTxvebiT gameordeba am teqstis TviTeuli nawili. 

teqstSi mocemuli yvela ricxvi unda aikrifos saangariSoze: 

erTi ricxvi _ cal mxares da meore _ meore mxares. 

Semdeg gairCeva amocanebi 2, 3 da 4. 


2/ udides 4 qulas TiTo SecdomisTvis akldeba TiTo q. 



5/ Tuki dawyebulia gayofa, ukve iwereba 1 q. Semdeg TiTo 

marTebul xolo gamoTvlaze iwereba TiTo q, sul _ 5 q. 

6/ Tuki TiTo gamoTvlilia marTebulad, magram araa naCvenebi 

advili xerxi, maSin iwereba TiTo q. gamoTvlis TiTo SecdomisTvis 

akldeba TiTo q. Tuki kargadaa SerCeuli gamoTvlis 

advili xerxebi, maSin gamoTvlis or-ori SecdomisTvis udides 

5 qulas akldeba TiTo q. 

- 61 -

Tavi V 

h!42/!rwfTnjxfsjU!Tflsfcb.hbnplmfcb!njmjpojt!gbshmfcTj/! 

42/2/ 4.6. 1~ 2~ 3# 4 5~ 6~ 7# 

5.1. 1~ 2~ 3~ 4! 5~ 6# 

42/3/!UbnbTj;!txsbgj!qbtvyfcj!`!hbnsbwmfcb/! 

gamosaTvlel namravlTa nimuSebi: _ erTjer TxuTmeti; ocjer 

nuli; cxrajer erTi; nuljer asTerTmeti; erTisa da 

ormocis namravli; erTisa da nulis namravli; pirveli 

Tanamamravlia rva, meore _ ori aTasi, risi tolia namravli? 

pirveli Tanamamravlia nuli, meore _ erTi, risi tolia 

namravli? orjer sami aTasi; gaormagebuli xuTi aTasi; 

gaorkecebuli aTi aTasi... 

42/4/! nsbwbmojTob! sjdywfcjt! rwfTnjxfsjU! Tflsfcb. 

hbnplmfcb!njmjpojt!gbshmfcTj/!Catardeba iseve, rogorc 6.2. 

_ Cven viciT samniSna ricxvebis qveSmiweriT Sekreba-gamokleba. 

oTxniSna da xuTniSna ricxvebis SemTxvevaSic asevea. 

moswavle dafaze dawers samniSna ricxvebis qveSmiweriT 

Sekrebis magaliTs, gverdiT _ imave Sekrebas qveSmiwerili 

gamosaxulebebiT. sruli msjelobiT Sesruldeba Sekreba. Semdeg 

aseve Sesruldeba gamokleba. 

rodesac erTi moswavle dafasTan muSaobs da wers, imavdroulad 

meore moswavle imave moqmedebas masw.-is daxmarebiT 

asrulebs saangariSoze. sxvebi ki msjeloben, romelimes rame 

xom ar SeSlia da ratom. 

masw.-ma mravaljer unda moiTxovos dasabuTeba, Tu ratomaa 

marTebuli moqmedebaTa TviTeuli nabiji. 

+ 

5· 

100 

+ 

4· 

100 

8· 

100 

+ 

+ 

5· 

10 

7 · 10 

+ 6 

+ 3 

5456 

+ 

873 

7 · 1000 

2· 

1000 

1· 

100 

2· 

100 

4· 

10 

5· 

10 

+ 1 

+ 8 

am dros mTavaria, rom klasma gaiazros qveSmiweris wesi da ar 

moxdes misi mxolod meqanikuri damaxsovreba: ratom vwerT 

Sesakrebebs erT svetSi, ratom ar SeiZleba maTi wanacvleba 

(aTeuli zustad aTeulis qveS rom unda iyos), ratom vkrebT 

erTmaneTis qveS miweril ricxvebs. amisaTvis damatebiT 

maxvildeba yuradReba qveSmiweriT Canaweris sisworeze: _ 

SekrebiT qveSmiweriT 464 da 89. romeli Canaweria swori? 

− 

+ 

+ 

+ 

+ 

- 62 -

4064 4064 

4064 4064 

+ Tu 

+ 

− Tu − 

859 

859 

859 

859 

_ ratom? {raki qveSmiweriT Sekrebisas erTeulebs erTad 

vkrebT, amitom erTeuli erTeulis qveS unda eweros, aseve 

aTeuli (roca gvaqvs) aTeulis qveS, aseuli (roca gvaqvs) aseulis 

qveS, xolo aTaseuli (roca gvaqvs) aTaseulis qveS} 

(aseve gamoklebis SemTxvevaSic). 

h!43/!rwfTnjxfsjU!Tflsfcb.hbnplmfcb!eb!Tfebsfcb! 

njmjpojt!gbshmfcTj/! 

43/2/ 5.2. 1~ (2 _ ara) 3# 4# 5 6~ 7~ 


hbnplmfcb! njmjpojt! gbshmfcTj/! Catardeba iseve, rogorc 

31.3, oRond ufro swrafad da, cxadia, sxva ricxvebiT. 

43/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj/! 

Sekreba yvelanairi SeiZleba, oRond pirvel wreze _ SedarebiT 

advili SemTxvevebi, roca erTeulTa jami 10-s ar aRwevs, 

aTeulTa jami 100-s ar aRwevs, aseulTa jami 1000-s ar aRwevs, 

meore wreze _ ufro Znelebi. 

1) 34+35, 560+220, 1400+1500, 14000+71000... 

2) 34+38, 580+220, 1800+1500, 104+707, 1200 + 900... 

gamoklebis SemTxvevaSi Zneli Sesadarebeli ricxvebi araa 

saWiro: 3) roca orniSna ricxvTa Soris gansxvaveba ar 

aRemateba 20-s: 63_45; 92_78... da Sesabamisad: 380_290, 

3100_1600, 38000_29000... 

4) roca sxvaoba advili saangariSoa: 67_24; 800_150; 

9400_3000; 4000 _ 10... 

gamravleba da gayofa: 

5) cxriluri da nulmiwerili cxriluri: 9·5, 4·60, 70·8, 42:6, 

630:9... 

6) advili saangariSoebi: 3 · 150, 750:250, 6000:4, 6 · 11... 

43/5/! sjdywfcjt! Tfebsfcb! njmjpojt! gbshmfcTj/! tbbohbsj. 

Tp{f!npdfnvm!sbpefopcbUb!Tfebsfcb!njmjpojt!gbshmfcTj/ 

swrafad gameordeba moswavlis saxelmZRvanelos me-5 gakve- 

Tilis ZiriTadi sakiTxebi. Semdeg dawvrilebiT gairCeva saSinao 

davalebis me-2 amocana. masSi mocemuli yvela ricxvi aikrifeba 

saangariSoze. 

- 63 -

h!44/!hbnsbwm/.hbzpgb!21.jt!ybsjtyfc{f/!bewjmj!hbnsbwm/.hbzpgb! 

44/2/ 5.3. (1, 2 da 3 _ ara) 4~ 5~ 6 7 

44/3/! sjdywjt! hbnsbwmfcb.hbzpgb! 21.jt! ybsjtyfc{f/! moswavleebs 

win unda edoT gadaSlili saxelmZRvanelo, me-3 gakveTilis 

teqstze (romelic Sin unda hqondeT wakiTxuli). masw.-is 

SekiTxvebiT gameordeba am teqstis TviTeuli nawili. 

Semdeg gairCeva amocanebi 1, 2 da 3. 

44/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!hbnsbwm/.hbzpgb!21.{f!bo!211.{f/! 

44/5/!2111.jt!gbshmfcTj!bewjmj!hbnsbwmfcb.hbzpgb!`!sbtbd! 

bs! tXjsefcb! rwfTnjxfsb/! erTi moswavle Seasrulebs dafaze 

(TviTeuli gamoTvlis dasrulebis Semdeg is jdeba da sxva 

moswavle gamodis dafasTan), sxvebi _ rveulebSi: 

4 · 210 = 4 · (200 + 10) = 800 + 40 = 840. 

30 · 20 = 30 · 2 · 10 = 60 · 10 = 600. 

115 · 6 = (100 + 15) · 6 = 600 + 3 · 5 · 6 = 600 + 3 · 30 = 690. 

800 : 2 = 400 amitom 800 : 20 = ? { = 40}. 

250 : 2 = 125. 680 : 2 = 340. 900 : 2 = 450. 

rogorc yovelTvis, TviTeuli moqmedeba xmamaRali msjelobiT 

unda aRiweros (ras vakeTebT) da unda aixsnas. 

h!45/!sjdyw/!fsUbe!Tflsfcb/!hbnsbwm/.hbzpgb!21.jt!ybsjtyfc{f! 

45/2/ 5.4. 1~ 2 3~ 4 5~ 6 

45/3/! sbnefojnf! sjdywjt! fsUbe! Tflsfcb! rwfTnjxfsjU! njmj. 

pojt!gbshmfcTj/!Sevruldeba Sekreba 42088 + 386 + 375092 + 

29 ori gziT: I _ rigrigobiT vkrebT; II _ erTbaSad vkrebT. 

romeli gza jobia? 

45/4/!UbnbTj;!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj;! 5000 +_ 

`1500~. ix. 14.3. 

45/5/!2111.jt!gbshmfcTj!bewjmj!hbnsbwmfcb.hbzpgb!`!sbtbd! 

bs! tXjsefcb! rwfTnjxfsb/! ! sjdywjt! hbnsbwmfcb.hbzpgb! 21.jt! 

ybsjtyfc{f;!!2111.{f-!!21111.{f-!!211111.{f/! 

iseve, rogorc 33.4-Si, Sesruldeba weriT: 306 · 3, 40 · 20, 

700 : 2, 1000 : 4, 64 · 10000, 345000 : 1000, 7 · 100000. 

h!46/!{fqjsj!Tflsfcb.hbnplm/!hbnsbwm/.hbzpgb!21.jt!ybsjtyfc{f! 

46/2/ 5.5. 1~ 2~ 3~ 4# 5# 6~ 7 

46/3/! nsbwbmojTob! sjdywfcjt! {fqjsj! Tflsfcb.hbnplmfcb! 

- 64 -

njmjpojt!gbshmfcTj!)bewjmj!TfnUywfwfcj*/! 

gamoTvlebi sruldeba, arsebiTad, zepirad, magram ricxvebi mainc 

iwereba, radgan mravalniSna ricxvebis smeniT aRqma da maTze 

moqmedebani Znelia. masw.-i karnaxobs ricxvebis saxelwodebebs, 

moswavleebi iweren, zepirad asruleben moqmedebas da pasuxs 

weren. erTi moswavle Seasrulebs dafaze (TviTeuli gamoTvlis 

dasrulebis Semdeg is jdeba da sxva moswavle gamodis 

dafasTan), sxvebi _ rveulebSi: 

120000 + 4509; 340000 + 110000; 47000 _ 39000; 

780190 _ 31; 800095 _ 200034. 


yb{{f•/!!ix. 27.2. 

46/5/! sjdywjt! hbnsbwmfcb.hbzpgb! 21.jt! ybsjtyfc{f;! ! 2111.{f-!! 

21111.{f-! ! 211111.{f/! masw.-is SekiTxvebiT damuSavdeba me-5 

gakveTilis teqstis TviTeuli nawili. Semdeg, iseve rogorc 

35.2-Si, Sesruldeba weriT-zepirad: 

1000 · 85, 560000:10000, 402 · 1000, 10000 · 3, 10000 · 23. 


1. or-ori meqanikuri SecdomisTvis udides 4 qulas akldeba 

TiTo q. 


3. Tuki dawyebulia sm-ebSi gadayvana da gayofa, ukve iwereba 

2 q. gamoTvlebis TiTo SecdomisTvis udides 4 qulas akldeba 

1 q. 

4. pirveli gamoTvla _ 2 q, meore _ 1 q, mesame _ 2 q. 

5. Tuki dawyebulia simaRleTa Sekreba, ukve iwereba 1 q. 

Semdeg TiTo marTebuli moqmedebis SerCevaze iwereba TiTo q; 

aseve, saTanado gamoTvlis marTebulobaSic iwereba TiTo q, 

sul _ 5 q. 

Tuki dawyebulia mcdari gziT, magram gamoTvlebi 

marTebulia, iwereba 1 q. 

Tavi VI 

h!48/!Tfls/.hbnplm/!njmjpojt!gbshm/!{fqjsbe!eb!rwfTnjxfsjU! 

48/2/ 5.6. 1~ 2~ 3~ 4 5# 6# 7 

6.1. 1~ 2~ 3~ 4# 5 6~ 

48/3/! nsbwbmojTob! sjdywfcjt! {fqjsj! Tflsfcb.hbnplmfcb! 

- 65 -

njmjpojt!gbshmfcTj!)bewjmj!TfnUywfwfcj*/! 

iseve rogorc 35.2-Si, Sesrudeba weriT-zepirad: 

678555 + 45; 600025 + 100026; 568969 _ 2202; 

900000 _ 230000; 47000 _ 17000. 

48/4/!UbnbTj;!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj;!!!84111!`! 

22111•/!!ix. 14.3. 


hbnplmfcb! njmjpojt! gbshmfcTj/! ! Catardeba iseve, rogorc 

31.3, oRond ufro swrafad da, cxadia, sxva ricxvebiT. 

h!49/!Tjhb!eb!hbsf!bsffcj/!!hbnsbwmfcb.hbzpgb!21.jt!ybsjtyfc{f! 

49/2. 6.2. (1 _ ara) 2~ 3 4# 5 6 

49/3/! Tjhb! eb! hbsf! bsffcj/! moswavleebs win unda edoT 




Semdeg gairCeva 1-li amocana. sul 15 Siga area, maTgan 

mxolod Sua 3 ar esazRvreba gare ares arcerTi wertiliT. 

49/4/!UbnbTj;!txsbgj!qbtvyfcj!`!TjhojUbb!Uv!hbsfU•/! 

dafaze daxazulia Zlier daklaknili, uswormasworo da 

SeWyletili Sekruli xazi. masw.-i maCvenebeli joxis wvers 

adebs raime wertils, moswavlem ki unda upasuxos, es wertili 

xazis SigniTaa Tu gareT. 

49/5/! sjdywjt! hbnsbwmfcb.hbzpgb! 21.jt! ybsjtyfc{f;! ! 2111.{f-!! 

21111.{f-!!211111.{f/! 

h!4:/!hbnsbwmfcjtb!eb!hbnplmfcjt!xftfcj/!!Tjhb!eb!hbsf!bsffcj! 

4:/2/ 6.3. (1 da 2_ ara) 3~ 4~ 5 6 

4:/3/!hbnsbwmfcjt!eb!hbnplmfcjt!xftfcj!`!gsDyjmfcjt!hbytob/!!! 

a · (b-c) = a · b - a · c (b-c-d) · a = b · a - c · a - d · a. moswavleebs 

win unda edoT gadaSlili saxelmZRvanelo, me-2 gakveTilis 

teqstze (romelic Sin unda hqondeT wakiTxuli). masw.-is 

SekiTxvebiT gameordeba am teqstis TviTeuli nawili. 

Semdeg gairCeva 1-li amocana. 

4:/4/! UbnbTj;! txsbgj! qbtvyfcj! `! sbnefojUbb! nfuj! 21111.jt! 

gbshmfcTj•/! klass Sesadareblad eZleva iseTi ricxvebi, 

romelTa zepirad gamokleba advilia: 1) mrgvali ricxvebi: 

- 66 -

9000 da 3400; 4100 da 3800; 780 da 310... 2) toli 

aTaseulebisa da aseulebis mqone ricxvebi: 508 da 511; 4164 da 

4148; 920 da 992... 3) toli naSTis mqone ricxvebi: 413 da 

813; 4) iseTi ricxvebi, romelTa Soris gansxvaveba ar 

aRemateba 40-s: 689 da 912, 1515 da 1534; anda, zogadad, iseTi 

ricxvebi, romelTa gansxvaveba advilad gamoiTvleba Sevsebis anu 

Sekrebis xerxiT: 9905 da 9320, 800 da 157... 

4:/5/! Tjhb! eb! hbsf! bsffcj/! swrafad gameordeba moswavlis 

saxelmZRvanelos me-2 gakveTilis ZiriTadi sakiTxebi. Semdeg 

dawvrilebiT gairCeva saSinao davalebis me-2 amocana. 

h!51/!npoblwfUj-!ufyjmj!eb!nsvej/!hbnsbwm/.hbnplmfcjt!xftfcj! 

51/2/ 6.4. (1, 2 da 3 _ ara) 4~ 5 6 7# 

51/3/! npoblwfUj-! ufyjmj! eb! nsvej/! moswavleebs win unda 

edoT gadaSlili saxelmZRvanelo, me-4 gakveTilis teqstze 


gameordeba am teqstis TviTeuli nawili. 

Semdeg gairCeva 1-li amocana. misi amoxsnaa: 

I. 2, 10; II. 4, 6, 7, 8, 11; III. 1, 3, 5, 9; IV. 4, 7, 9, 11; 

V. 1, 2, 3, 8, 10. 

51/4/! UbnbTj;! tbjebobb! ynb@•! moswavle, romelic im gakve- 

Tilze yvelaze kargad iqceoda da kargad pasuxobda, gamodis da 

dgeba Sua oTaxSi, merxebs Soris. masw.-i mas Tvalebs auxvevs. 

Semdeg masw.-i aniSnebs romelime sxva moswavles, romelic 

adgilidan asaxelebs romelime maTematikur sityvas, rac 

maTematikis gakveTilebze aqvT naswavli. Tvalebaxveulma 

moswavlem unda gaimeoros es sityva da Semdeg Tqvas: visi xma 

moesma; da rac mTavaria _ saidan moesma xma: ukana marcxena 

mxridan, wina marjvena mxridan da ase Semdeg. 

51/5/! hbnsbwmfcjtb! eb! hbnplmfcjt! xftfcj! `! gsDyjmfcjt! 

hbytob/!!!a · (b-c) = a · b - a · c (b-c-d) · a = b · a - c · a - d · a. 

swrafad gameordeba moswavlis saxelmZRvanelos me-3 

gakveTilis ZiriTadi sakiTxebi. Semdeg dawvrilebiT gairCeva 

saSinao davalebis me-2 da me-3 amocanebi. 

h!52/!Tflsvmj!eb!hbytojmj!yb{fcj/!bewjmj!hbnsbwmfcb.hbzpgb! 

52/2/ 6.5. (1 da 2_ ara) 3~ 4# 5~ 6 7 

- 67 -

52/3/!Tflsvmj!eb!hbytojmj!yb{fcj/! 



masw.-is SekiTxvebiT gameordeba es teqsti. gairkveva, iqve 

naxazze mocemuli xazebidan TviTeuli ratomaa Sekruli 

texili Tu gaxsnili mrudi. 

Semdeg gairCeva amocanebi 1 da 2. maTi amoxsnaa: 

1. oTxi xazi: Sekruli araa imitom, rom aqvs bolo wertilebi 

(xolo Sekrul xazs ar unda hqondes bolo wertilebi); 

gaxsnili ki araa imitom, rom SemosazRvravs Siga ares 

(gaxsnili xazi ki ar unda SemosazRvravdes arcerT Siga ares). 

2. 1-iT gansxvaveba iman gamoiwvia, rom gaxsnil texilTan 

SedarebiT Sekrul texilis boloebi erTmaneTs unda daemTxves. 

amitom A da T wertilebi SeerTebuli unda iyos monakveTiT da 

swored es erTi monakveTi emateba. 

52/4/!UbnbTj;!txsbgj!qbtvyfcj!`!tbqjsjtqjspoj•/!ix. 15.3. 

52/5/! njmjpojt! gbshmfcTj! bewjmj! hbnsbwmfcb.hbzpgb! `! 

sbtbd! bs! tXjsefcb! rwfTnjxfsb/! erTi moswavle Seasrulebs 

dafaze (TviTeuli gamoTvlis dasrulebis Semdeg is jdeba da 

sxva moswavle gamodis dafasTan), sxvebi _ rveulebSi: 

4 · 2100 = 4 · (2000 + 100) = 8000 + 400 = 8400. 

300 · 20 = 300 · 2 · 10 = 600 · 10 = 6000. 

96000 : 3 = ? romeli ricxvi unda gavamravloT 3-ze, raTa 

miviRoT 960? {30000+2000 = 32000}. 

800000 : 2 = 400000, amitom 800000 : 20 = 40000. 

2500 : 2 = 1250 68000 : 2 = 34000. 



h!53/!bewjmj!hbnsbwm/.hbzpgb/!!nprnfe/!Tfnpxn/!tbqjsjtqjspUj/! 

53/2/ 6.6. (1 _ ara) 2~ 3~ 4 5 6! 7# 

53/3/!fsUnbofUjt!tbqjsjtqjsp!nprnfefcfcj

sbtbd! bs! tXjsefcb! rwfTnjxfsb/! ! erTi moswavle Seasrulebs 

dafaze (TviTeuli gamoTvlis dasrulebis Semdeg is jdeba da 

sxva moswavle gamodis dafasTan), sxvebi _ rveulebSi: 

25 · 30 = 25 · 3 · 10 = 75 · 10 = 750. 

115 · 6 = (100 + 15) · 6 = 600 + 15 · 6 = 3 · 5 · 6 = 

= 600 + 3 · 30 = 690. 

6000 : 2 = 3000 amitom 6000 : 4 = 1500. 

900000 : 3 = 300000. 

120000 : 2 = 60000. 120000 : 4 = 30000. 120000 : 3 = 40000. 




yuradReba! am gakveTilze masw.-ma gaafrTxilos moswavleebi, 

rom momdevno gakveTilisaTvis moitanon farglebi. 


2. sul oTxi nabijia: ori gamoTvla da ori Semowmeba. 

TiToSi iwereba TiTo q. 

3. udides 4 qulas dalagebis TiTo SecdomisTvis akldeba 

TiTo q. 

4. esaa 4-quliani amocana: pirveli xazis daxazva _ 1 q; 

meorisa _ 1 q; mesamisa _ 2 q. Tuki area mcdari an mesame 

xazis gadakveTebi _ akldeba 1 q. 

5. pasuxebi: I rigi: I. monakveTebi (41_11)-iT; II. monakveTebi 

(41_11_3)-iT; III. xazebi (11+3)-iT; IV. texilebi 11-iT. 

II rigi: I. monakveTebi (25_4)-iT; II. monakveTebi (25_4_2)iT; 

III. xazebi 25-iT; IV. xazebi (4+2)-iT. 

esaa 6-quliani amocana: I srulad _ 1 q, II srulad _ 1 q, 

III srulad _ 2 q da IV srulad _ 2 q; III an IV nawilobriv 

(mxolod marTebuli pasuxi metobaze ramdeniT metobis gareSe) 

_ 1 q. gamoTvlis erTi meqanikuri Secdoma qulas ar aklebs. 

Tavi VII 

h!55/!nprnfefcjt!Tfnpxnfcb!tbqjsjtqjspUj/!gbshmjt!ynbsfcb/ 

55/2/ 6.7. (1 _ ara) 2~ 3# 4# 5~ 6~ 7~ 

7.1. 1 2 3 4 5~ 6# 

55/3/! UbnbTj;! txsbgj! qbtvyfcj! `! Tfnefhjt! Tfnefhj-! xjojt! 

xjob•/!ix. 23.3. 

- 69 -

55/4/! fsUnbofUjt! tbqjsjtqjsp! nprnfefcfcj

oTxmonakveTiani Sekruli texili; askuTxedi. 

_ ra kanonzomierebiTaa Sedgenili es cxrili? {cxrilis zeda 

striqonSi weria nakvTi, qveda striqonSi _ misi sazRvari}. 

Camoayalibos Teonamac (susti moswavle). 

56/4/!UbnbTj;!txsbgj!qbtvyfcj!`!Dbwjgjrsf!sjdywfcj•/! 

Catardeba iseve, rogorc 21.3, oRond gansxvaveba isaa, rom 

gamosacnobia ori ricxvi _ sakleb-maklebi, an ori Sesakrebi, an 

ori Tanamamravli, an gasayof-gamyofi. Tanac, gansxvavebiT wina 

TamaSebisagan, pasuxTa sxvadasxva SesaZleblobaa. masw.-i ambobs: _ 

Cavifiqre ricxvi, gamovakeli meore ricxvi, sxvaoba miviRe 

7250. romeli saklebi da maklebi SeiZleboda Camefiqrebina? 

{magaliTad, 7300 da 50; anda 7260 da 10} Cavifiqre ricxvi, 

mivumate meore ricxvi, jami miviRe 69000. romeli Sesakrebebi 

SeiZleboda Camefiqrebina? {60000 da 9000, anda 65000 da 

4000}. oRond, 0 ar iTvleba! 

es pirveli wre Zalian advilia. momdevno wreze cota 

rTuldeba: Cavifiqre ori Tanamamravli, gadavamravle, miviRe 90 

(anda: 360, 6400, 30000, 44 da sxva). romeli Tanamamravlebi 

SeiZleboda Camefiqrebina? oRond, 1 ar iTvleba! 

56/5/!gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbnsbwmfcjtbt/! 

erTi moswavle Seasrulebs dafaze (TviTeuli gamoTvlis 



97 · 5 = (100 _ 3) · 5 = 100 · 5 _ 3 · 5 = 500 _ 15 = 485. 

3 · 994 = 3 · (1000 _ 6) = 3000 _ 18 = 2982. 

_ vin moigonebs advili xerxiT gamravlebis kidev erT amgvar 

magaliTs? (moswavlis mier Camoyalibebuli gamravlebac aseve 

Sesruldeba). 



h!57/!Tftb[mfcm/!bnpxfsb/!gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb/! 

57/2/ 7.3. (1 _ ara) 2 3# 4 5# 6~ 

57/3/!Tftb[mfcmpcbUb!bnpxfsb/! 

masw.-is SekiTxvebiT damuSavdeba me-3 gakveTilis teqstis 

ZiriTadi sakiTxebi. Semdeg dawvrilebiT gairCeva amocana 1. 

57/4/!UbnbTj;!hb{pnwb•/! 

ix. 26.3. axla, cxadia, sxva rame gaizomeba. 

- 71 -





2 · 9960 = 2 · (10000 _ 40) = 20000 _ 80 = 19920. 

1997 · 4 = (2000 _ 3) · 4 = 8000 _ 12 = 7988. 


magaliTs? (moswavlis mier Camoyalibebuli gamravlebac aseve 

Sesruldeba). 



h!58/!Tftb[mfcmpcbUb!bnpxfsb/!gsDyjm/!hbytojt!xftjt!hbnpzfo/! 

! 58/2/ 7.4. (1 _ ara) 2# 3 4~ 5# 6 

58/3/!Tftb[mfcmpcbUb!bnpxfsb/! 


ZiriTadi sakiTxebi. Semdeg dawvrilebiT gairCeva amocana 1. 

58/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj/!!ix. 32.3. 





3 · 149000 = 3 · (150000 _ 1000) = 450000 _ 3000 = 447000. 

195000 · 4 = (200000 _ 5000) · 4 = 800000 _ 20000 = 780000. 


magaliTs, oRond SekrebisaTvis? (moswavlis mier Camoyalibebuli 

gamravlebac aseve Sesruldeba). 

rogorc yovelTvis, TviTeuli moqmedeba xmamaRali msjelobiT 

unda aRiweros (ras vakeTebT) da unda aixnas. 

amgvari gamoTvlebisas umjobesia mravalniSna ricxvis 

zepirad dasaxeleba. magaliTad, 800000 _ 20000 zepirad Zneli 

Sesasrulebelia, magram advilia ase: 

800 aTasi _ 20 aTasi = 780 aTasi. 

h!59/!ovmfcjU!ebcpmpfcvmj!sjdywjt!bewjmj!hbnsbwmfcb/! 

! 59/2/ 7.5. (1 _ ara) 2 3 4# 5 6 7~ 

59/3/! ovmfcjU! ebcpmpfcvmj! sjdywjt! bewjmj! hbnsbwmfcb/!! 


- 72 -




Semdeg gavimeoroT ukve naswavli, advilad saangariSo SemTxveva: 

30015 · 4 = (30000 + 15) · 4 = 120000 + 60 = 120060. 

bolos dawvrilebiT gairCeva amocana 1. 

59/4/!UbnbTj;!txsbgj!qbtvyfcj!`!sbnefoj!tboujnfusjb•/! 

masw.-s ukavia saklaso metriani saxazavi. masze sxvadasxva 

adgilas adebs TiTs. moswavlem unda upasuxos, Tu ramdeni 

santimetria dasawyisidan aqamde. 

59/5/! xsfyb{j! eb! njtj! yb{wb! gbshmjU/! ! _ gavixsenoT, ras 

ewodeba wrexazi {wrexazi aris wris sazRvari} sWirdeba Tu 

ara mas gaferadeba? {ara} ratom? {imitom, rom mxolod 

sazRvaria, mxolod xazia, Signidan carielia}. 

bolos, moswavleebi sakuTar rveulebSi fargliT daxazaven 

erTi-or wrexazs, masw.-i Seamowmebs, Tu ramdenad kargad xmaroben 

fargals. 

h!5:/!tblpouspmp!xfsb!$!8/! 

1. udides 4 qulas moqmedebaTa rigis arevisaTvis akldeba 2 

q; or-ori meqanikuri SecdomisTvis akldeba TiTo q. 


TiTo q. 


TiTo q. 

4. udides 5 qulas wveroebis ganlagebis TiTo SecdomisTvis 

akldeba TiTo q. aseve akldeba wrexazis cudad daxazvis gamoc. 

Tuki texilis sigrZe saWiro sigrZisagan gansxvavdeba 5 mm-iT an 

metiT 1 sm-mde _ akldeba 1 q, Tuki gansxvavdeba 1 sm-ze metiT 

_ akldeba 2 q. 

5. pasuxebi: I rigi: 24810, 24806, 16810, 16806, 24010, 

24006, 16010, 16006, 24800, 16800, 24000, 16000. 

II rigi: 12607, 12610, 33607, 33610, 12007, 12010, 33007, 

33010, 12600, 33600, 12000, 33000. 

Secdoma SeiZleba iyos saWiro ricxvis gamotoveba an zedmeti 

ricxvis Camateba. or-ori amgvar SecdomisTvis udides 5 

qulas akldeba TiTo q. 

Tavi VIII 

- 73 -

h!61/!ovmfcjU!ebcpmpfcvmj!sjdywjt!bewjmj!hbnsbwmfcb/! 

61/2/ 7.6. 1~ 2~ 3# 4~ 5~ 6~ 

8.1. 1~ 2~ 3 4~ 5~ 6~ 7 

61/3/! UbnbTj;! txsbgj! qbtvyfcj! `! sbnefojUbb! nfuj! 21111.jt! 

gbshmfcTj•/!!ix. 39.3. 

61/4/!ovmfcjU!ebcpmpfcvmj!sjdywjt!bewjmj!hbnsbwmfcb/! 

masw.-is SekiTxvebiT gameordeba me-5 gakveTilis teqsti. 

Semdeg swrafad Sesruldeba advili gamravleba. masw.-i 

erToblivi msjelobisas gamokveTs, rom amisTvis gvWirdeba 

TanamamravlTa gadajgufebis naswavli wesi. 

6 · 8000 (anu 8 aTasi) = 6 · 8 · 1000 = 48 · 1000 = 48000. 

250000 (anu 250 aTasi) · 3 = 250 · 1000 · 3 = 250 · 3 · 1000 = 

750 · 1000 = 750000. 

8 · 11011 = ? jer gamovTvaloT 8 · 11 = 8 · (10 + 1) = 

8 · 10 + 8 = 88. 

8 · 11011 = 8 · (11000 + 11) = 8 · 11000 + 8 · 11 = 

8 · 11 · 1000 + 88 = 88000 + 88 = 88088 (88 aTas 88). 

h!62/!ovmfcjU!ebcpmpfcvmj!sjdywjt!bewjmj!hbzpgb/! 

62/2/!! 8.2. (1 _ ara) 2~ 3# 4 5 6 

62/3/!UbnbTj;!`hvebTj!yfmjt!Dbzpgb!eb!hbnpdopcb~. 

masw.-i naWris gudaSi Cayris kubebs, aguredebs, piramidebs, 

burTebsa da sxva sxeulebis modelebs; agreTve nakvTebis 

modelebs: plastmasis an magari muyaos kvadratebs, wreebs, 

rgolebs, sxvadasxvagvar samkuTxedebs, mkveTrad wagrZelebul 

marTkuTxedebs, sxvagvar oTxkuTxedebs, tolgverdebian an 

eqvskuTxedebs an rvakuTxedebs (an erTs, an meores _ radganac 

isini erTmaneTisagan Zneli gansarCevia!) da ramdenime 

uswormasworo nakvTs. moswavleebma rig-rigobiT unda Cayon 

gudaSi xeli, aiRon erTi modeli da gudaSi Cauxedavad, xeliT 

mosinjviT gamoicnon, ra aris. uswormasworo nakvTze unda 

Tqvan `sxva nakvTi~; kubisa da burTis garda sxva sxeulze _ 

`sxva sxeuli~. pasuxis Semdeg moswavle amoiRebs models 

gudidan da yvela naxavs, SeeSala Tu ara. Semdeg models isev 

Caabruneben gudaSi, masw.-i aurevs da momdevno moswavles 

Caayofinebs xels. 

TamaSis dawyebamde masw.-i klass kargad uCvenebs, Tu radara 

- 74 -

modelebs Cayris gudaSi da urCevs, daimaxsovron. _ aqaa 

kvadratebi, wreebi, rgolebi, sxvadasxvagvari samkuTxedebi da 

kidev sxva nakvTebi _ xuTkuTxedebi, eqvskuTxedebi da sxva. 

Tqven rig-rigobiT unda CayoT gudaSi xeli, aiRoT erTi nakvTi 

da gudaSi Cauxedavad, xeliT mosinjviT swrafad gamoicnoT, es 

ra nakvTia: kvadrati, wre, rgoli, sxvadasxvagvari samkuTxedi 

Tu sxva rame. uswormasworo eqvskuTxedebisa da oTxkuTxedebis 

gamocnoba Znelia. amitom maT nu gamoicnobT, TqviT mxolod: 

sxvagvari nakvTi. 

62/4/!ovmfcjU!ebcpmpfcvmj!sjdywjt!bewjmj!hbzpgb/! 



masw.-is SekiTxvebiT damuSavdeba am teqstis TviTeuli nawili. 

Semdeg dawvrilebiT gairCeva amocana 1. 

h!63/!ovmfcjU!ebcpmpfcvmj!sjdywjt!bewjmj!hbzpgb/! 

63/2/ 8.3. (1 _ ara) 2 3 4 5~ 6 7 

63/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!txsbgj!qbtvyfcj!`!sjdywjt! 

eblblvofcb•/!!ix. 13.3. 

63/4/!ovmfcjU!ebcpmpfcvmj!sjdywjt!bewjmj!hbzpgb/! 

masw.-is SekiTxvebiT gameordeba me-2 gakveTilis teqsti. Semdeg 

gairCeva amocana 1. 

! 

h!64/!hbzpgjtb!eb!Tflsfcjt!xftj!`!gsDyjmfcjt!hbytob/! 

64/2/ 8.4. (1 _ ara) 2~ 3 4# 5# 6 7 

64/3/!UbnbTj;!txsbgj!qbtvyfcj!`!oblwUfcjt!dopcb•/! 

am TamaSisaTvis masw.-s winaswar unda hqondes momzadebuli 

muyaosgan gamoWrili sxvadasxvagvari da sxvadasxva feris 

wreebi, rgolebi, kvadratebi, samkuTxedebi, oTxkuTxedebi, 

xuTkuTxedebi, eqvskuTxedebi, SvidkuTxedebi da ase Semdeg _ 

yvela naswavli nakvTi. masw.-i moswavles wamierad uCvenebs 

nakvTs, Semdeg xeliT faravs an parkSi malavs. moswavlem unda 

Tqvas, romeli nakvTi iyo. 

rac ufro Zneli sacnobia nakvTi (sxvadasxvagvari xuTkuTxedebi, 

eqvskuTxedebi da ase Semdeg), miT ufro met xans vuCvenebT 

nakvTs. advilad sacnobebs (wre, rgoli, samkuTxedebi) ki 

vuCvenebT myisierad da isic mxolod pirvel wreze. 

64/4/!hbzpgjtb!eb!Tflsfcjt!xftj!`!gsDyjmfcjt!hbytob;! 

(a+b+c) : d = a : d + b : d + c : d. moswavleebs win unda edoT 

- 75 -




Semdeg gairCeva amocana 1. 

h!65/!gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbzpgjtbt/! 

65/2/ 8.5. (1 _ ara) 2# 3 4~ 5~ 6~ 7! 

65/3/! UbnbTj;! txsbgj! qbtvyfcj! `! Dbwjgjrsf! sjdywfcj•/! ix. 

45.3. 


(a+b+c) : d = a : d + b : d + c : d. gsDyjmfcjt! hbytojt! xftjt! 

hbnpzfofcb!hbzpgjtbt/!jer swrafad gameordeba me-4 gakveTilis 

teqstis ZiriTadi sakiTxebi. gameordeba amdagvarive naswavli 

wesi gamravlebisaTvis. Semdeg gairCeva amocana 1. 

! 


66/2/ 8.6. (1 _ ara) 2~ 3 4~ 5 6# 7! 

66/3/!UbnbTj;!wjo!bnpyb{bwt!zwfmb{f!nfut@•!moswavleebi win 

daideben saxelmZRvanelos gare ydas. daakvirdebian varskvlavebian 

naWrelas. _ gaimarjvebs is, vinc Tavis rveulSi 

swrafad amoxazavs yvelaze met sxvadasxvanair oTxkuTxeds! 

masw.-i dainiSnavs dros (2-3 wuTi, meti araa saWiro), Semdeg 

brZanebs: _ yvelam dados fanqari! _ da momdevno erT wuTSi 

gamoavlens ramdenime gamarjvebuls. 


(a+b+c) : d = a : d + b : d + c : d. gsDyjmfcjt! hbytojt! xftjt! 

hbnpzfofcb!hbzpgjtbt/!jer swrafad gameordeba me-4 gakveTilis 

teqstis ZiriTadi sakiTxebi. gameordeba amdagvarive naswavli 

wesi gamravlebisaTvis. Semdeg gairCeva amocana 1. 

h!67/!tblpouspmp!xfsb!$!9/! 

1. udides 4 qulas or-ori SecdomisTvis akldeba TiTo q. 

2. TiTo gayofaSi or-ori nabijia: gasayofis warmodgena 

saWiro jamad da Semdeg gayofis Sesruleba. TiTo am nabijSi 

iwereba TiTo q. or-ori meqanikuri Secdoma mxolod TiTo 

qulas aklebs. 

3. Tuki dawyebulia sigrZeTa sm-ebSi marTebulad gadayvana _ 

ukve iwereba 1 q. Tuki es araa, magram sigrZeTa erTi mainc 

Sekrebaa Catarebuli marTebulad _ 1 q. Semdeg Sekrebis TiTo 

- 76 -

SecdomisTvis udides 4 qulas akldeba TiTo q. 

4. pasuxebi: I rigi: 81351 > 80861 (erTaderTia); 917560 < 

920200 an 907560 < 920200 (aq SeiZleba Caiweros ori 

cifri); 6889 < 6892 (erTaderTia). 

II rigi: 3909 > 3895 (erTaderTia); 137508 < 137514 

(erTaderTia); 86106 > 78000 an 96106 > 78000 (aq SeiZleba 

Caiweros ori cifri). 

sul dasaweria 4 utoloba, TiTo utolobis daweraSi _ 

TiTo q. erTaderTobis orjerve marTebul miTiTebaSi _ 1 q. 

sul _ 5 q. 

5. pasuxebi: I rigi: I. 8274; II. 2074; III. 6074. 

II rigi: I. 3185; II. 9385; III. 4985. 

pirveli ricxvi _ 1 q, meore _ 1 q, mesame _ 3 q. udides 5 

qulas or-ori mcire SecdomisTvis (saWiro cifris nacvlad 

misi momdevno cifris SerCeva uyuradRebobis gamo) akldeba 

TiTo q; magram TiTo uxeSi SecdomisaTvis (sul sxva cifris 

SerCeva) akldeba TiTo q. 

Tavi IX 

! 


68/2/ 8.7. (1 _ ara) 2# 3~ 4# 5~ 6~ 7~ 

9.1. 1~ 2~ 3 4~ 5~ 6# 7! 

68/3/!UbnbTj!`!txsbgj!qbtvyfcj;!211.jtb!eb!2111.jt!hbzpgb/! 

TamaSis saxiT gameordeba gakveTili 1-is teqstSi mocemuli 

ganayofebi. SekiTxvebis nimuSebi: 1000:25; 100:5; 1000:20... 

masw.-i moswavleebs daavalebs, rom kargad iswavlon zepirad 

es ganayofebi da gakveTili 1-is teqstSi mocemuli namravlebi! 

68/4/! gsDyjmfcjt! hbytojt! xftjt! hbnpzfofcb! hbzpgjtbt/! 

nsbwbmojTob! sjdywjt! hbzpgb! fsUojTob! sjdyw{f! rwfTnjxfsjt! 

hbsfTf/! 

gairCeva me-7 gakveTilis amocana 1. Semdeg: 

63735:7 = (63 aTasi + 700 + 35):7 = 63 aTasi :7 + 700:7 + 

35:7 = 9 aTasi + 100 + 5 = 9105. 

Semdeg Sesruldeba amdagvarive gayofa, magram naSTiani 

SemTxveva: 6907:3 = (6000 + 900 + 7):3 = 

= 2000 + 300 + 2 (n. 1) = 2302 (n. 1). 

h!69/!psnysjwj!vupmpcboj/!!gsDyjmfcjt!hbytob!hbzpgjtbt/! 

- 77 -

69/2/ 9.2. 1, 2 da 3 _ ara) 4~ 5 6 7 

69/3/! psnysjwj! vupmpcboj/!moswavleebs win unda edoT gadaSlili 

saxelmZRvanelo, me-2 gakveTilis teqstze (romelic 

Sin unda hqondeT wakiTxuli). masw.-is SekiTxvebiT damuSavdeba 

es teqsti, Semdeg dawvrilebiT gairCeva amocanebi 1 da 2. 

69/4/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj@•! 

winadadebaTa nimuSebi: 

1) kvadratis oTxive gverdi tolia. 

2) kvadratis xuTive gverdi tolia. 

3) eqvskuTxeds yovelTvis eqvsi gverdi aqvs. 

4) xuTkuTxeds yovelTvis xuTi wvero aqvs. 

5) oTxkuTxedis oTxive wvero wertilebia. 

6) samkuTxedis oTxive wvero wertilebia. 

7) marTkuTxedis yvela gverdi monakveTia. 

8) yvela kvadrati oTxkuTxedia. 

9) yvela marTkuTxedi oTxkuTxedia. 

10) yvela marTkuTxedi samkuTxedia. 

11) yvela marTkuTxedi rvakuTxedia. 

12) arcerTi marTkuTxedi araa xuTkuTxedi. 

13) arcerTi kvadrati araa eqvskuTxedi. 

14) yvela monakveTs ori bolo aqvs. 

15) monakveTis orive bolo wertilebia. 

16) monakveTis orive bolo wreebia. 

17) monakveTis samive bolo wertilebia. 

18) monakveTis oTxive bolo wertilebia. 

69/5/! gsDyjmfcjt! hbytojt! xftjt! hbnpzfofcb! hbzpgjtbt/! 

nsbwbmojTob! sjdywjt! hbzpgb! fsUojTob! sjdyw{f! rwfTnjxfsjt! 

hbsfTf/!!jer gairCeva amocana 3. Semdeg iseve, rogorc 57.3-Si, 

Sesruldeba (naSTiani) gayofa: 32009:5 990050:9 

75024:3 = magaliTad: 75024:3 = (75000 + 24):3 = 75 

aTasi :3 + 24:3 = 25 aTasi + 8 = 25008. 

820030:4 = (800 aTasi + 20 aTasi + 30):4 = 200 aTasi + 

5 aTasi + 7 (n. 2) = 205 aTas 7 (n. 2) _ 205007 (n.2). 

h!6:/!ovmfcjU!ebcpmpfcvmjt!eb!bewjmj!hbzpgjt!TfnUywfwfcj/! 

6:/2/ 9.3. (1 da 2 _ ara) 3! 4# 5# 6~ 7 

6:/3/!UbnbTj!`!txsbgj!qbtvyfcj;!211.jtb!eb!2111.jt!hbzpgb/! 

- 78 -

ix. 57.2. SigadaSig daemateba `swrafi pasuxebi _ gamravlebagayofa 

ricxvTa xazze~ _ ix. 27.2. 

masw.-i moswavleebs daavalebs, rom kargad iswavlon zepirad 

gakveTili 1-is teqstSi mocemuli namravlebi da Sesabamisi 

ganayofebi! 

6:/4/! ovmfcjU! ebcpmpfcvmj! sjdywjt! bewjmj! hbzpgb! eb! 

gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbzpgjtbt/! 

jer gairCeva amocanebi 1 da 2. Semdeg Sesruldeba gayofa: 

140000:7 = 140 aTasi :7 = 20 aTasi = 20000. 

36031:6 = (36000 + 31):6 = 6000 + 5 (n. 1) = 6005 (n. 1). 

600000:12 = 5000; 5000:25 = 200. 

yuradReba! yuradReba unda mivaqcioT mesame gayofisas 

nulebis damaxsovrebas: erTi nuli xom 6-ians mivakuTvneT, 

davitoveT 60, amitom damaxsovrebuli gvqonda ara xuTi cali, 

aramed oTxi cali nuliani! 

aseve, meoTxe gayofisas erTi cali nuliani mivakuTvneT 5ians, 

davitoveT 50, amitom damaxsovrebuli gvqonda ara sami 

cali, aramed ori cali nuliani! 

h!71/!bewjmj!hbzpgjt!TfnUywfwfcj/!!psnysjwj!vupmpcboj/! 

! 71/2/ 9.4. (1, 2 da 3 _ ara) 4~ 5 6 7 

71/3/! ovmfcjU! ebcpmpfcvmj! sjdywjt! bewjmj! hbzpgb! eb! 

gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbzpgjtbt/! 

jer gairCeva amocana 1. Semdeg Sesruldeba gayofa: 

40060:5 = (40000 + 60):5 = 8000 + 12 = 8012. 

300071:6 = (300000 + 60 + 11):6 = 50000 + 10 + 1 (n. 5) = 

50011 (n. 5). 

400024:8 = (400000 + 24):8 = 50000 + 3 = 50003. 

20050:4 = (20000 + 50):4 = 5000 + (40 + 10):4 = 

5000 + 10 + 2 (n.2) = 5012 (n. 2). 

yuradReba! yuradReba unda mivaqcioT pirveli gayofisas 

nulebis damaxsovrebas: erTi nuli xom 4-ians mivakuTvneT, 

davitoveT 40, amitom damaxsovrebuli gvqonda ara oTxi cali, 

aramed sami cali nuliani! 

aseve, meore gayofisas erTi nuli xom 3-ians mivakuTvneT, 

davitoveT 30, amitom damaxsovrebuli gvqonda ara xuTi cali, 

aramed oTxi cali nuliani! 

aseve iyo mesame da meoTxe gayofisasac da saSinao davalebis 

- 79 -

1-li amocanis pirveli da meore gayofisasac! 

71/4/!UbnbTj;!tjuzwfcjt!Tfehfob!)btpfcjt!hbnfpsfcjU*•/! 

esaa kombinatorikuli TamaSi. masw.-i dafaze msxvilad dawers 

ramdenime asos, xmovnebsac da Tanxmovnebsac, magaliTad: a, i, g, 

T, w, n. gaimarjveben isini, vinc am asoebiT yvelaze met 

sityvas Seadggens. asoebis gameoreba SeiZleba (magaliTad: TiTi, 

wigni, TaTi, gia, gani, ia, ai, win... SeiZleba zmnebica da 

sazogadod, yovelgvari sityva!). masw.-i dainiSnavs dros (3-4 

wuTi), moswavleebi Cumad daweren sityvebs TavianT rveulebSi. 

drois gasvlis Semdeg masw.-i brZanebs: _ aba, yvelam dados 

kalmistari! vis ramdeni sityva aqvs Sedgenili? sityvebis 

wakiTxva jer ar gvinda, jer daTvaleT da miTxariT, vis 

ramdeni sityva aqvs! (ityvian) aha, yvelaze meti hqoniaT natos, 

ias da tariels. waikiTxeT Tqveni sityvebi! (waikiTxaven) ara, 

ias SeSlia! aba, ratom ar varga sityvebi `ginda~ da `winda`? 

{imitom, rom gamoyenebulia aso d _ romelic araa Cvens siaSi} 

amitom ias oriT naklebi sityva hqonia. gaimarjves natom da 

tarielma. axla sxvebma miTxariT, raime sxva sityvac xom ar 

SegidgeniaT? 

71/5/! psnysjwj! vupmpcboj/! masw.-is SekiTxvebiT swrafad 

gameordeba me-2 gakveTilis teqsti, Semdeg dawvrilebiT 

gairCeva amocanebi 2 da 3. 

h!72/!tywbhwbsj!bewjmj!hbzpgb!fsUojTob!sjdyw{f/! 

72/2/ 9.5. (1 _ ara) 2~ 3 4~ 5 6! 7# 


winadadebaTa nimuSebi: 1) wris yvela wvero wertilebia. 

2) zog monakveTs mxolod erTi bolo aqvs. 

3) rgolis yvela wvero wertilebia. 

4) wris yvela gverdi monakveTebia. 

5) rgolis yvela gverdi monakveTebia. 

6) kvadratis oTxive gverdi wertilebia. 

7) SvidkuTxedis Svidive wvero monakveTebia. 

8) zogi cxrakuTxedis Svidive gverdi tolia. 

9) yovel monakveTs ori gverdi aqvs. 

10) yovel texils xuTi wvero aqvs. 

11) zog texils xuTi wvero aqvs. 

- 80 -

12) yoveli texili mxolod monakveTebisagan Sedgeba. 

13) texili swori xazia. 

14) monakveTi swori xazia. 

15) mrudi xazi ar SeiZleba iyos texili. 

72/4/! nsbwbmojTob! sjdywjt! tywbhwbsj! bewjmj! hbzpgb! 

fsUojTob!sjdyw{f/!!jer gairCeva 1-li amocana. 

Semdeg Sesruldeba cota sxvagvari gayofani _ roca gasayofis 

daSla SesakrebTa jamad sxvagvaradaa saWiro _ ara 100-isa Tu 

1000-is gamoyofiT: 

100:3 = ? umjobesia, ase: 100 = 90 + 10, amitom: 100:3 = 

(90 + 10):3 = 30 + 3 (n. 1) = 33 (n. 1). 

111:3 = ? _ rogor davSaloT 111? bunebrivia 100 + 11, 

magram es ar ivargebs, radgan 100... iyofa Tu ara unaSTod 3ze? 

{100 ar iyofa unaSTod 3-ze} magram amgvari daSlisas 

pirveli Sesakrebi unaSTod unda iyofodes! maS, rogor 

davSaloT 111? {90 + 21}. amitom: 

97:4 = (80 + 17):4 = 20 + 17:4 = 20 + 4 (n. 1) = 24 (n. 1). 

h!73/!tywbhwbsj!bewjmj!hbzpgb!fsUojTob!sjdyw{f/! 

! 73/2/ 9.6. (1 _ ara) 2~ 3# 4# 5~ 6! 7 

73/3/!UbnbTj;!UbnbTj;!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj;!!!! 

961!±!26•/!!ix. 14.3. 

73/4/! nsbwbmojTob! sjdywjt! tywbhwbsj! bewjmj! hbzpgb! fsUoj. 

Tob!sjdyw{f/!jer gairCeva amocana 1. Semdeg Sesruldeba gayofa: 

1000:3 = (900 + 90 + 10):3 = 300 + 30 + 3 (n. 1) = 333 (n. 1). 

111:3 = (90 + 21):3 = 30 + 7 = 37. 

7000:6 = (6000 + 600 + 360 + 40):6 = 1000 + 100 + 60 + 6 (n.4) = 

1166 (n.4). 

h!74/!tblpouspmp!xfsb!$!:/! 

1. or-ori meqanikuri SecdomisTvis udides 4 qulas akldeba 

TiTo q. 

2. TiTo SecdomisTvis udides 4 qulas akldeba TiTo q. 

3. mcdaria meore utoloba, radgan 400 araa 400-ze meti da 

300 araa 300-ze meti. gamoTvlis TiTo SecdomisTvis udides 4 

qulas akldeba TiTo q. 

4. gasarkvevia 8 wona, or-orSi iwereba TiTo q. metnaklebobis 

gamoTvalSic iwereba 1 q, sul _ 5 q. 

- 81 -

5. udides 4 qulas akldeba TiTo q saWiro mraval- 

kuTxedis gverdebis raodenobis TiTo SecdomisTvis. 

Tavi X 

h!75/!sjdywjt!djgsUb!kbnj/!hbnfpsfcb;!rwfTnjx/!Tfls/.hbnplm/! 

75/2/ 9.7. 1~ 2# 3 4 5# 6~ 7~ 

10.1. (1 da 2 _ ara) 3# 4# 5# 6! 

! 75/3/! sjdywjt! djgsUb! kbnj/!_ ras ewodeba ricxvis cifrTa 

jami? Semdeg swrafad gairCeva amocana 1. 

! 75/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!djgsUb!kbnj/! 

masw.-i moswavles usaxelebs ricxvs (magaliTad: samocdaCvidmeti, 

aTas oci, orasi aTasi, xuTas ormocdaaTi...), moswavle 

angariSobs cifrTa jams. 

! 75/5/! hbnfpsfcb;! rwfTnjxfsjU! Tflsfcb.hbnplmfcb/! Sesruldeba 

igive Sekreba da igive gamokleba, rac saSinao davalebis 

me-2 amocanaSi, oRond gverdiT miewereba igive qveSmiweriT 

moqmedeba saTanrigo Sesakrebebze _ iseve, rogorc iyo 6.2-Si. 

Sesruldeba qveSmiweriT Sekreba, rogorc yovelTvis, TviTeuli 

nabijis xmamaRali aRweriTa da dasabuTebiT: 

45071 4 · aTiaTasi + 5 · aTasi + 0 · asi + 7 · aTi + 1 

+ 

9099 9 · aTasi + 0 · asi + 9 · aTi + 9 

________ _____________________________________ 

mTavaria, moswavleebma kargad daasabuTon, Tu ratom unda 

davweroT erTeulTa qveS erTeulebi, aTeulTa qveS _ aTeulebi 

da ase Semdeg, ras niSnavs èrTis damaxsovreba~ da sxva. mTeli 

algoriTmi kargad unda gaiazron da daasabuTon. aseve, 

qveSmiweriT gamoklebisas ras niSnavs `sesxeba~ da sxva. 

45071 4 · aTiaTasi + 5 · aTasi + 0 · asi + 7 · aTi + 1 

_ 

9099 9 · aTasi + 0 · asi + 9 · aTi + 9 

________ _____________________________________ 

h!76/!nsbwbmojTob!sjdywjt!rwfTnjx/!hbnsbwmfcb!fsUojTob{f/! 

dawvrilebiTi aRwera _ ix. $ 7 

qveSmiweriT gamravlebisas 

gamravlebis niSnad jobia -is dawera · -is nacvlad. 

h!77/!nsbwbmojTob!sjdywjt!rwfTnjx/!hbnsbwmfcb!fsUojTob{f/! 

- 82 -

! 77/2/ 10.3. 1# 2 3 4 5~ 6 7# 

! 77/3/!UbnbTj;!Uwmb!eb!vlvUwmb!fsUjt!hbnpupwfcjU•/! 

iseve, rogorc Tvlisa da ukuTvlisas (18.2), oRond unda 

gamotovon TiTo ricxvi. daiwyeben, magaliTad, 191-dan, unda 

daasaxelon mxolod kenti ricxvebi (cxadia, am sityvis gareSe!) 

_ masw.-is taSamde. 

momdevno wreze unda daiwyon, magaliTad, 880-dan da 

daasaxelon mxolod luwi ricxvebi _ masw.-is taSamde. 

! 77/4/! nsbwbmojTob! sjdywjt! rwfTnjxfsjU! hbnsbwmfcb! 

fsUojTob! sjdyw{f/! Catardeba iseve, rogorc 65.3, srulad, 

cota garTulebuli gamoTvla: roca damaxsovrebulTa Sekrebis 

Semdeg kidev unda davimaxsovroT: 47990 · 8. 

h!78/!mvxj!eb!lfouj!sjdyw/!bewjmj!hbnsbwm/!psojTob!sjdyw{f/! 

78/2/ 10.4. (1 _ ara) 2~ 3# 4 5 6~ 7! 

! 78/3/! mvxj! eb! lfouj! sjdywfcj/! masw.-is SekiTxvebiT 

damuSavdeba me-4 gakveTilis teqsti, Semdeg gairCeva amocana 1. 

! 78/4/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj@•! 

winadadebaTa nimuSebi: 1) kvadratis oTxive gverdi tolia. 

2) kvadratis xuTive gverdi tolia. 

3) eqvskuTxeds yovelTvis eqvsi gverdi aqvs. 

4) xuTkuTxeds yovelTvis xuTi wvero aqvs. 

5) oTxkuTxedis oTxive wvero wertilebia. 

6) samkuTxedis oTxive wvero wertilebia. 

7) marTkuTxedis yvela gverdi monakveTia. 

8) yvela kvadrati oTxkuTxedia. 

9) yvela marTkuTxedi oTxkuTxedia. 

10) yvela marTkuTxedi samkuTxedia. 

11) yvela marTkuTxedi rvakuTxedia. 

12) arcerTi marTkuTxedi araa xuTkuTxedi. 

13) arcerTi kvadrati araa eqvskuTxedi. 

14) yvela monakveTs ori bolo aqvs. 

15) monakveTis orive bolo wertilebia. 

16) monakveTis orive bolo wreebia. 

17) monakveTis samive bolo wertilebia. 

18) monakveTis oTxive bolo wertilebia. 

! 78/5/! bewjmj! hbnsbwmfcb! psojTob! sjdyw{f! rwfTnjxfsjt! 

- 83 -

hbsfTf/! arsebiTad, es ukve naswavli gvaqvs. viyenebT naswavl 

wesebs: gamravlebisa da Sekrebisa a · (b+c) = a · b + a · c da 

TanamamravlTa gadajgufebisa a · (b · c) = a · b · c. 

12 · 15 = 12 · (10+5) = 120 + 60 = 180. 

250 · 12 = 250 · (4 · 3) = 250 · 4 · 3 = 1000 · 3 = 3000. 

40 · 35 = 40 · (5 · 7) = 40 · 5 · 7 = 200 · 7 = 1400. 

h!79/!bewjmj!hbnsbwmfcb!psojTob!sjdyw{f!rwfTnjxfsjt!hbsfTf/! 

79/2/ 10.5. 1# 2~ 3# 4~ 5# 6 7 

! 79/3/!UbnbTj;!txsbgj!qbtvyfcj!`!hbnsbwmfcb.hbzpgb!sjdywUb! 

yb{{f•/!ix. 27.2. 

! 79/4/! bewjmj! hbnsbwmfcb! psojTob! sjdyw{f! rwfTnjxfsjt! 

hbsfTf/!Catardeba iseve, rogorc 67.4: 

16 · 2500 = 4 · 4 · 2500 = 4 · 10000 = 40000. 

360 · 11 = 360 · (10+1) = 3600 + 360 = 3960. 

35 · 12 = 35 · (10+2) = 350 + 70 = 420. 

25 · 800 = 25 · 8 · 100 = 200 · 100 = 20000. 

h!7:/!mvxj!eb!lfouj!sjdyw/!!bew/!hbnsbwm/!psojTob!sjdyw{f/! 

7:/2/ 10.6. (1 da 2 _ ara) 3# 4 5 6~ 7! 

! 7:/3/!mvxj!eb!lfouj!sjdywfcj/! 

jer gameordeba, Tu ras ewodeba luwi ricxvi da ras _ kenti 

ricxvi. Semdeg dawvrilebiT gairCeva amocana 2. 

! 7:/4/!UbnbTj;!Uwmb!eb!vlvUwmb!fsUjt!hbnpupwfcjU•/!! 

ix. 66.2. 

! 7:/5/! bewjmj! hbnsbwmfcb! psojTob! sjdyw{f! rwfTnjxfsjt! 

hbsfTf/! kargad gairCeva amocana 1. Sesruldeba erTi-ori 

amdagvarive gamoTvla. 


1. or-ori meqanikuri SecdomisaTvis udides 4 qulas 


2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q. 


4. xuTkuTxedis or-ori mocemuli sigrZis gverdis 

daxazvisaTvis iwereba TiTo q. perimetris gamoTvlis marTebuli 

dawyebisaTvis (sigrZeTa Sekrebis dawyebisaTvis) _ 1 q. mexuTe 

gverdis gazomvisaTvis _ 1 q. perimetris marTebulad 

- 84 -

gamoTvlisaTvis _ kidev 1 q. sul _ 5 q. 

5. gamoklebisTvis 7 lari _ 3 lari (an, II rigSi: 5 lari _ 

2 lari) iwereba 1 q. Tuki mosinjulia (gaangariSebulia) 

Tevzebis raodenobis erTi mainc SesaZlebloba, Tund mcdari _ 

1 qula. dasabuTebis magvari msjelobisaTvis _ 1 q. sul, 

gamarTuli dasabuTebianad _ 5 q. 

amoxsna SeiZleba Tanxebis gayofiT (28:4 an 18:3), an 

sxvadasxva SesaZleblobaTa mosinjviT. orive gziT SeiZleba. 

Tavi XI 

h!82/!hbzpgjtb!eb!hbnplmfcjt!xftj!`!gsDyjmfcjt!hbytob/! 

82/2/ 10.7. 1# 2~ 3# 4# 5~ 6~ 7# 

11.1. (1 _ ara) 2 3# 4# 5 6~ 

! 82/3/!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj; 8500 +_ `1200~. ix. 14.3. 

! 82/4/!hbzpgjtb!eb!hbnplmfcjt!xftj!`!gsDyjmfcjt!hbytob;! 

(a-b):c = a:c - b:c. wesebi zogadad jer ar Camoyalibdeba _ 

ubralod, saSinao davalebidan 1-li gakveTilis me-5 amocana 

gairCeva dawvrilebiT da daskvnebi Camoyalibdeba am kerZo 

SemTxvevisaTvis. 

h!83/!hbzpgjtb!eb!hbnplmfcjt!xftj!`!gsDyjmfcjt!hbytob/! 

83/2/ 11.2. (1 _ ara) 2# 3 4~ 5# 6 

! 83/3/!UbnbTj;!txsbgj!qbtvyfcj!`!Dbwjgjrsf!sjdywfcj•/!ix. 45.3. 


(a-b):c = a:c - b:c. masw.-is SekiTxvebiT damuSavdeba me-2 

gakveTilis teqsti, Semdeg dawvrilebiT gairCeva amocana 1. 

h!84/!nsbwbmojTob!sjdywfcjt!rwfTnjxfsjU!hbnsbwmfcb/! 

84/2/ 11.3. 1 2 3 4# 5# 6~ 

! 84/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!txsbgj!qbtvyfcj!`!sjdywjt! 

eblblvofcb•/!ix. 13.3. 

! 84/4/!nsbwbmojTob!sjdywfcjt!rwfTnjxfsjU!hbnsbwmfcb/! 

vasrulebT moqmedebaTa naswavli wesebis gamoyenebiT. meore 

Tanamamravli iyos, magaliTad, samniSna ricxvi: 741 · 128 = 

=741 · (8 + 2 · 10 + 1 · 100)= 741 · 8 + 741 · 2 · 10 + 741 · 1 · 100. 

CavweroT TviTeuli gamravleba jer cal-calke, qveSmiweriT: 

741 741 741 

× 8 × 2 × 1 

5928 1482 741 

- 85 -

axla gamoviyenoT nulebis damaxsovrebis wesi, namravlebs 

mivuweroT saTanado `damaxsovrebuli~ nulebi da Semdeg 

qveSmiweriT SevkriboT: 

5928 + 14820 + 72100. miviRebT saboloo pasuxs 94848. 

magram yvelaferi amis gakeTeba SeiZleboda ufro 

mosaxerxeblad da naklebi weriT. amisaTvis pirvelad 

miRebuli namravlebi Tavidanve erTmaneTis qveS 

unda davweroT, raTa isini pirdapir gam- 

zadebuli iyos qveSmiweriT Sesakrebad! qveSmiweriT 

gamravleba davweroT srulad: 

_ aTeulebze gamravlebisas namravli yovelTvis erTi an meti nuliT 

bolovdeba, aseulebze gamravlebisas _ ori an meti nuliT. 

moswavleebma es nulebi jer-jerobiT weron! 


85/2/ 11.4. 1 (2 _ ara) 3 4 5 6# 


Catardeba iseve, rogorc 74.3, gamravldeba igive ricxvebi, 

oRond boloSi daemateba msjeloba: 

_ simoklisaTvis SeiZleba aRar vweroT 

xolme nulebi, magram ar unda dagvaviwydes 

wanacvleba! Semoklebisas mesame striqonSi 

ewereboda 741, magram sinamdvileSi 

igulisxmeba 74100: 

masw.-i dafaze gakeTebul CanawerebSi 

waSlis marjvena nulebs: _ es nulebi xom zedmetia, 

tvirTavs Canawerebs! oRond unda dagvaviwydes wanacvleba! 

741 

× 

128 

5928 

+ 14820 

74100 

94848 

741 

× 

128 

5928 

+ 1482 

741 

94848 


1) raime ricxvis erTze gamravlebiT viRebT nuls; 

2) raime ricxvis nulze gamravlebiT viRebT nuls; 

3) raime ricxvis erTze gamravlebiT viRebT TviTon am ricxvs; 

4) raime ricxvis nulze gamravlebiT viRebT TviTon am ricxvs; 

5) erTis gamravlebiT raime ricxvze viRebT erTs; 

6) nulis gamravlebiT raime ricxvze viRebT nuls; 

7) erTis gamravlebiT raime ricxvze viRebT TviTon am ricxvs; 

8) nulis gamravlebiT raime ricxvze viRebT TviTon am ricxvs; 

9) raime ricxvis erTze gayofiT viRebT nuls; 

- 86 -

10) raime ricxvis nulze gayofiT viRebT nuls; 

11) raime ricxvis erTze gayofiT viRebT TviTon am ricxvs; 

12) raime ricxvis nulze gayofiT viRebT TviTon am ricxvs; 

13) raime ricxvis TavisTavze gayofiT viRebT nuls; 

14) raime ricxvis TavisTavze gayofiT viRebT am ricxvs; 

15) raime ricxvis TavisTavze gayofiT viRebT erTs... 


(a-b):c = a:c - b:c. 

masw.-is SekiTxvebiT mokled gameordeba me-2 gakveTilis 

ZiriTadi sakiTxebi, Semdeg dawvrilebiT gairCeva amocana 2. 


86/2/ 11.5. 1~ 2 3 (4 _ ara) 5 6 

! 86/3/!UbnbTj;!wjo!bnpyb{bwt!zwfmb{f!nfut@•! 

ix. 55.2, oRond axla amoxazon sxvadasxvanairi Sekruli 

texilebi _ mravalkuTxedebis sazRvrebi. 


jer gairCeva me-4 amocana. 

Semdeg, iseve rogorc 74.2-Si, srulad iwereba qveSmiweriT 

gamravleba, oRond rodesac meore Tanamamravls SuaSi nuli 

aqvs: 

312 

× 

201 

312 

+ 0000 

62400 

62712 

rogorc viciT, SeiZleba ar vweroT 

marjvena nulebi. aseve, SeiZleba Sua 

nulebis striqonic aRar vweroT, 

magram samagierod ar unda dagvaviw- 

ydes oriT wanacvleba! 

312 

× 

201 

+ 312 

624 

62712 





3. xuTkuTxedis daxazva da Sig saWiro samkuTxedisa _ 2 q 

(Tuki samkuTxedi SecdomiT gareTaa, an Sigaa magram mcdarad _ 

1 q). saerTo gverdis amowera (mxolod erTia) _ 1 q, samive 

saerTo wverosi _ 1 q. sul _ 4 q. Tuki yvelaferi kargadaa, 

erTi wveros gamorCenisaTvis SeiZleba ar daakldes q. 

4. pasuxebi: I rigi: mcdarebia II (magaliTad: 1+2=3) da III 

(magaliTad: 5_3=2). 

- 87 -

II rigi: mcdarebia I (mag.: 1+3=4) da IV (mag.: 5_2=3). 

TiTo mcdari winadadebis amorCevaSi _ TiTo q. TiTo dasa- 

sabuTebeli magaliTis dasaxelebaSi _ TiTo q. marTebuli wina- 

dadebebis ar amorCevaSi _ 1 q. sul _ 5 q. ese igi, amorCevis 

TiTo SecdomisaTvis an dasasabuTebeli magaliTis gamotovebisa 

an SecdomisaTvis akldeba TiTo q. 

5. pasuxebi: I rigi: ruxi satvirTo manqanaa 2. sul ki ruxi 

SeiZleba iyos an 2 manqana (Tuki danarCenebSi arcerTi araa 

ruxi), an 3 manqana (Tuki danarCenebSi erTi manqanaa ruxi), an 4 

manqana (Tuki danarCenebSi ori manqanaa ruxi). 

II rigi: kedelze kidia 3 Wreli fanduri. sul ki kedelze 

SeiZleba ekidos an 3 fanduri (Tuki danarCenebSi arcerTi araa 

kedelze ), an 4 fanduri (Tuki danarCenebSi erTia kedelze), an 

4 fanduri (Tuki danarCenebSi oria kedelze). 

pirvel marTebul pasuxSi yovelgvari dasabuTebisa da SesaZleblobaTa 

gareSe iwereba 1 q. dasabuTebis magvar msjelobaSi 

iwereba 1 q. damatebiT TiTo SesaZleblobis ganxilvaSic _ 

TiTo q. sul _ 5 q. 

Tavi XII 

h!88/!njmjpoj/! 

88/2/ 11.6. 1~ 2# 3 4~ 5~ 6~ 

12.1. 1~ (2 da 3 _ ara) 4 5# 6~ 7 

88/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj/!!ix. 32.3. 

88/4/!njmjpoj/!sjdywfcjt!blsfgb!tbbohbsjTp{f/! 

dawvrilebiT gairCeva amocanebi 2 da 3 _ am gakveTilis 

Semamzadeblebia. 

_ maS, ra udidesi raodenobis aTaseulebs SeiZleba Seicavdes 

eqvsniSna ricxvi? {999}. vTqvaT, mocemulia aseTi raodenoba: 

2000 · 1000 + 502. rogor CavweroT ricxvis saxiT? Tuki 

aTaseulTa raodenoba 1000-ze metia, eqvsniSna ricxvad xom 

veRar CavwerT! amitom gvWirdeba axali Tanrigebi! 

1000 · 1000 _ ewodeba milioni. cxadia, ase Caiwereba: 

1000000. 

h!89/!sjdyw/!njmjpojebo!njmjbsebnef/!nbUj!xbljUywb!eb!Dbxfsb/! 

89/2/ 12.2. (1 da 2 _ ara) 3 4# 5# 6 7! 

89/3/!UbnbTj;!txsbg/!qbtvy/!`!sbnefoj!tboujnfusjb•/!ix. 48.3. 

- 88 -

89/4/! njmjpoj/! sjdywfcj! njmjpojebo! njmjbsebnef/!! 

sjdywfcjt! xbljUywb! eb! Dbxfsb/! tbbohbsjTp{f! npdfnvmj! 


ebgb{f/! qjsjrjU;! tjuzwjfsbe! bo! djgsvmbe! npdfnvmj! 

sjdywjt! blsfgb•! tbbohbsjTp{f/! ! jer swrafad gameordeba, ra 

aris milioni {1000 aTaseuli, 1000-jer 1000}. gairCeva 

amocanebi 2 da 3. 

Semdeg masw.-is SekiTxvebiT damuSavdeba me-2 gakveTilis 

teqstis TviTeuli nawili. teqstSi mocemuli yvela ricxvi 

aikrifeba saangariSoze. 

h!8:/!sjdyw/!njmjpojebo!njmjbsebnef/!nbUj!xbljUywb!eb!Dbxfsb/! 

8:/2/ 12.3. (1 _ ara) 2 3 4# 5 6 

8:/3/! UbnbTj;! tjuzwfcjt! Tfehfob! )btpfcjt! hbnfpsfcjt! 

hbsfTf*•/! igivea, rac 60.3-Si aRwerili TamaSi, oRond ar 

SeiZleba asoebis gameoreba. amitom asoebis raodenoba cota 

ufro meti unda iyos, raTa ramdenime sityva mainc Sedges. 

8:/4/! sjdywfcj! njmjpojebo! njmjbsebnef/! ! sjdywfcjt! 

xbljUywb! eb! Dbxfsb/! tbbohbsjTp{f! npdfnvmj! sbpefopcjt! 

tjuzwjfsj! ebtbyfmfcb! eb! djgsvmj! Dbxfsb! ebgb{f/! qjsjrjU;! 

tjuzwjfsbe! bo! djgsvmbe! npdfnvmj! sjdywjt! blsfgb•! 

tbbohbsjTp{f/! ! jer gairCeva amocana 1, Tan yvela ricxvi 


Semdeg masw.-is SekiTxvebiT damuSavdeba me-3 gakveTilis 

teqstis TviTeuli nawili. teqstSi mocemuli yvela ricxvi 


h!91/!sjdyw/!njmjpojebo!njmjbsebnef/!!rwfTnjx/!hbnsbwmfcb/! 

91/2/ 12.4. (1 _ ara) 2# 3 4 5 6 

91/3/! sjdywfcj! njmjpojebo! njmjbsebnef/! ! sjdywfcjt! 

xbljUywb! eb! Dbxfsb/! tbbohbsjTp{f! npdfnvmj! sbpefopcjt! 

tjuzwjfsj! ebtbyfmfcb! eb! djgsvmj! Dbxfsb! ebgb{f/! qjsjrjU;! 

tjuzwjfsbe! bo! djgsvmbe! npdfnvmj! sjdywjt! blsfgb•! 

tbbohbsjTp{f/! ! jer masw.-is SekiTxvebiT gameordeba me-3 

gakveTilis teqstis TviTeuli nawili. teqstSi mocemuli yvela 

ricxvi aikrifeba saangariSoze. 

Semdeg dawvrilebiT gairCeva amocana 1, Tan yvela ricxvi 

(maT Soris Sevsebuli striqonebisac) aikrifeba saangariSoze. 

! 91/4/!UbnbTj;!hvebTj!yfmjt!Dbzpgb!eb!hbnpdopcb•/!ix. 51.2. 

- 89 -

! 91/5/!nsbwbmojTob!sjdywfcjt!rwfTnjxfsjU!hbnsbwmfcb/!! 

iseve, rogorc 75.3-Si, Sesruldeba erTi qveSmiweriT gamravleba. 

dafasTan Seasrulebs is moswavle, romelsac es 

SedarebiT uWirs. 

h!92/!ovmfcjt!ebnbytpwsfcb•!hbnsbwmfcjtbt/! 

92/2/ 12.5. 1 2 (3 da 4 _ ara) 5# 6# 7 

92/3/!UbnbTj;!UbnbTj;!Uwbmjt!ebybnybnfcb•/!ix. 29.3. 

! 92/4/! ovmfcjU! ebcpmpfcvm! UbobnbnsbwmUb! hbnsbwmfcjtbt! 

ovmfcjt! ebnbytpwsfcb•/! gairCeva amocana 4 da amocana 3-is 

bolo ori gamravleba. _ roca raime ricxvs vamravlebT 900-ze, 

rogor SegviZlia moviqceT? {am nulebs ar mivaqcioT yuradReba 

da namravls bolos mivuweroT} diax, davimaxsovroT es ori 

nuli da maT gareSe gavamravloT. oRond, ar unda dagvaviwydes 

am nulebis isev miwera namravlisaTvis. 

Semdeg `nulebis damaxsovrebiT~ Sesruldeba gamravleba 

qveSmiweris gareSe: 70000 · 60 = 42 00000 = 4200000 

500 · 8000 = 40 00000 = 4000000. 

h!93/!!ovmfcjt!ebnbytpwsfcb•!rwfTnjxfsjU!hbnsbwmfcjtbt/! 

93/2/ 12.6. (1 _ ara) 2~ 3# 4 5# 6# 7 


SekiTxvebis nimuSebi: 1) wris yvela wvero wertilebia. 

2) zog monakveTs mxolod erTi bolo aqvs. 

3) rgolis yvela wvero wertilebia. 

4) wris yvela gverdi monakveTebia. 

5) rgolis yvela gverdi monakveTebia. 

6) kvadratis oTxive gverdi wertilebia. 

7) SvidkuTxedis Svidive gverdi monakveTebia. 

8) zogi ockuTxedis ocive gverdi tolia. 

9) yovel monakveTs ori waxnagi aqvs. 

10) yovel texils xuTi wibo aqvs. 

11) zog texils xuTi wvero aqvs. 

12) yoveli texili mxolod monakveTebisagan Sedgeba. 

13) texili mrude xazia. 

14) monakveTi swori xazia. 

15) yvelanairi samkuTxedi mravalkuTxedia. 

16) yvelanairi mravalkuTxedi samkuTxedia. 

17) wre mravalkuTxedia. 

- 90 -

! 93/4/! ovmfcjU! ebcpmpfcvm! UbobnbnsbwmUb! hbnsbwmfcb! rwfT. 

njxfsjU! )ovmfcjt! ebnbytpwsfcb*/! jer gairCeva amocana 1. Semdeg 

masw.-is SekiTxvebiT damuSavdeba me-6 gakveTilis teqsti. 

Semdeg ganixileba, magaliTad: 1700 · 230 = (17 · 100) · (23 · 

10) = (gavixsenoT TanamamravlTa gadajgufebis wesi!) = (17 

· 23) · 100 · 10. ese igi, 17-isa da 23-is namravls unda 

mivuweroT 2+1 cali nuli _ anu ramdenic hqonda orive 

Tanamamravls erTad. 

amrigad, gamravlebisas SegviZlia, rom nulebs ar mivaqcioT 

yuradReba, oRond isini unda `davimaxsovroT~ da namravls unda 

mivuweroT imdeni, ramdenic davimaxsovreT. 

h!94/!ovmfcjt!ebnbytpwsfcb•!rwfTnjxfsjU!hbnsbwmfcjtbt/! 

94/2/ 12.7. (1 _ ara) 2 3 4 5 6 7 

94/3/! ovmfcjU! ebcpmpfcvm! UbobnbnsbwmUb! hbnsbwmfcb! 

rwfTnjxfsjU!)ovmfcjt!ebnbytpwsfcb*/! 

jer masw.-is SekiTxvebiT gameordeba me-6 gakveTilis teqsti. 


! 94/4/!UbnbTj;!wjo!bnpyb{bwt!zwfmb{f!nfut@•! 

ix. 55.2, oRond axla amoxazon sxvadasxvanairi samkuTxedebi. 





3. gasarkvevia 4 ricxvi, TiToSi iwereba TiTo q. 

4. oTxkuTxedis daxazva da Sig saWiro samkuTxedisa _ 2 q 

(Tuki samkuTxedi SecdomiT gareTaa, an Sigaa magram mcdarad _ 

1 q). saerTo gverdis amowera (mxolod erTia, magram Tuki 

moswavle saerTo gverdad miiCnevs sanaxevrod saerTos, ara 

uSavs) _ 1 q, orive saerTo wverosi _ 1 q; perimetrebis 

metobis dasabuTebis magvari msjeloba _ 1 q. sul _ 5 q. 

5. orive amocanaSi erTi kramiti faravs 20 sm siganes. amaSi 

ukve iwereba 1 q. pasuxebi: I rigi: 6 m : 20 sm = 30 cali. 

II rigi: 4 m : 20 sm = 20 cali. ganayofis daweraSi iwereba 1 

q, misi mniSvnelobis marTebulad gamoTvlaSi _ 1 q. 

zedmeti monacemia saxuravis sigrZe _ 1 q. sul miviReT 4 q, 

romelic im SemTxvevaSic iwereba, Tuki moswavles aeria 

saxuravis sigrZe da sigane. xolo Tuki ar arevia, maSin emateba 

- 91 -

kidev 1 q, sul _ 5 q. 

Tavi XIII 

h!96/!rwfTnjxfsjU!hbnsbwmfcjt!hbbewjmfcjt!yfsyfcj/! 

96/2/ 12.8. 1~ 2~ 3# 4# 5 6# 7# 

13.1. 1~ 2# (3 _ ara) 4~ 5~ 6 7# 


yb{{f•/!ix. 27.2. 

96/4/! rwfTnjxfsjU! hbnsbwmfcjt! hbbewjmfcjt! yfsyfcj/! jer 

gairCeva amocana 3. Semdeg dafaze erTmaneTis gverdi-gverd 

Sesruldeba qveSmiweriT gamravleba 678 · 65 da 65 · 678. 

_ romelia ufro advili? {678 · 65} ratomaa ufro advili 678 

· 65, vidre 65 · 678? {imitom, rom samniSna ricxvze 

gamravlebisas qveSmiweraSi gveqneba sami striqoni, xolo 

orniSna ricxvze gamravlebisas _ ori striqoni}. 

h!97/!sjdywfcjt!Tfebsfcb/!rwfTnjx/!hbnsbwmfcjt!hbbewjmfcb/! 

97/2/ 13.2. (1 da 2 _ ara) 3# 4 5! 6 

97/3/!sjdywfcjt!Tfebsfcb!njmjbsejt!gbshmfcTj/!!tbbohbsj. 

Tp{f! npdfnvm! sbpefopcbUb! Tfebsfcb! njmjbsejt! 

gbshmfcTj/! ! ! moswavleebs win unda edoT gadaSlili 

saxelmZRvanelo, me-2 gakveTilis teqstze (romelic Sin unda 

hqondeT wakiTxuli). masw.-is SekiTxvebiT gameordeba am teqstis 


Semdeg gairCeva amocana 1. masSi mocemuli yvela ricxvi unda 

aikrifos saangariSoze: erTi ricxvi _ cal mxares da meore _ 

meore mxares. 

! 97/3/! UbnbTj;! tjuzwfcjt! Tfehfob! )btpfcjt! hbnfpsfcjt! 

hbsfTf*•/! igivea, rac 60.3-Si aRwerili TamaSi, oRond ar 

SeiZleba asoebis gameoreba. amitom asoebis raodenoba cota 

ufro meti unda iyos, raTa ramdenime sityva mainc Sedges. 

! 97/4/!rwfTnjxfsjU!hbnsbwmfcjt!hbbewjmfcjt!yfsyfcj/! 

jer gavimeoroT wina xerxi: _ romeli ufro advilia 

qveSmiweriT _ samniSna ricxvis gamravleba xuTniSnaze Tu 

xuTniSnas gamravleba samniSnaze? ratom? {imitom, rom 

qveSmiweraSi jobia sami striqoni gvqondes, vidre xuTi}. 

Semdeg gairCeva amocana 2. _ ratomaa ufro mosaxerxebeli 

94 · 31, vidre 31 · 94? _ dafaze erTmaneTis gverdi-gverd 

- 92 -


_ ra xdeba, roca meore TanamamravlSi Sedis 1-iani? {pirveli 

Tanamamravlis 1-ze gamravleba advilia, pirdapir gadavwerT mas} 

ratomaa ufro advili gamravleba: 14 · 38-s sjobia 

38 · 14, 51 · 38-s sjobia 38 · 51. 

h!98/!spnbvmj!ojTofcj/!!rwfTnjxfsjU!hbnsbwmfcjt!hbbewjmfcb/! 

98/2/ 13.3. (1 da 2 _ ara) 3 4 5# 6# 7 

98/3/!sjdywfcjt!Dbxfsb!spnbvmj!ojTofcjU/! 


TviTeuli nawili. Semdeg gairCeva amocana 1. 

romauli niSnebi me-4 klasSi iswavleba mxolod 30-mde. 

zogadi wesis axsna araa saWiro. aixsneba is, Tu ra SemTxvevaSi 

miewereba romaul niSans meore niSani marjvnidan da ra 

SemTxvevaSi _ marcxnidan, oRond mxolod 30-mde konkretuli 

ricxvebisaTvis, yvelasaTvis saTiTaod, rig-rigobiT CamovuyvebiT 

yvelas da Tan avxsniT kanonzomierebas. moswavleebma TviTon 

scadon gamocnoba. 

! 98/4/!UbnbTj;!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj;!!!61111!`! 

:611•/!ix. 14.3. 

! 98/5/! rwfTnjxfsjU! hbnsbwmfcjt! hbbewjmfcjt! yfsyfcj/! jer 

gavimeoroT wina xerxebi: _ romeli ufro advilia qveSmiweriT 

_ samniSna ricxvis gamravleba xuTniSnaze Tu xuTniSnas 

gamravleba samniSnaze? ratom? {imitom, rom qveSmiweraSi jobia 

sami striqoni gvqondes, vidre xuTi}. magram: 567 · 1811 Tu 

1811 · 567? {567 · 1811} ratom? {pirveli Tanamamravlis 1-ze 

gamravleba advilia, pirdapir gadavwerT mas}. 

Semdeg gairCeva amocana 2. _ ratomaa ufro mosaxerxebeli 

38 · 55, vidre 55 · 38? _ dafaze erTmaneTis gverdi-gverd 


_ ratomaa ufro advili gamravleba 22-ze, 333-ze, 555-ze? 

{qveSmiweris samive striqoni erTidaigive iqneba da xelaxla 

aRar viangariSebT}. 

h!99/!sjhpcsjwj!sjdywfcjt!Dbxfsb!spnbvmj!ojTofcjU/! 

99/2/ 13.4. (1 da 2 _ ara) 3~ 4# 5 6 

99/3/!UbnbTj;!txsbgj!qbtvyfcj!`!sjhpcsjwj!sjdywfcj•/!masw.-i 

asaxelebs ricxvs 1000-is farglebSi, moswavle _ Sesabamis 

rigobriv ricxvs. magaliTad: samas ormoci _ samas meormoce, as 

- 93 -

samocdaSvidi _ as samocdameSvide, xuTasi _ mexuTase... 

99/4/!sjhpcsjwj!sjdywfcjt!Dbxfsb!spnbvmj!ojTofcjU/! 

masw.-is SekiTxvebiT jer swrafad gameordeba me-3 gakveTilis 

teqsti; Semdeg saguldagulod damuSavdeba me-4 gakveTilis 

teqstis TviTeuli nawili. bolos gairCeva amocana 1. 

rigobrivi ricxvebis marTlwera masw.-ma xSirad unda 

gaameorebinos xolme klass _ es qarTuli enis, gramatikis 

sakiTxicaa! 

h!9:/!nsbwbmlvUyfej/!!sjdyw/!Tfebsfcb!njmjbsejt!gbshm/! 

9:/2/ 13.5. (1 da 2 _ ara) 3 4 5~ 6# 7 

9:/3/! nsbwbmlvUyfej/! winaswar masw.-ma xelaxla unda 

waikiTxos mravalkuTxedis Sesaxeb zogad meTodikur saxelmZRvaneloSi 

[paragrafSi 5]. 


TviTeuli nawili. Semdeg gairCeva amocana 1. misi pasuxebia _ 

araa mravalkuTxedi imitom, rom: 

I. aqvs ori Siga are, mravalkuTxeds ki unda hqondes erTi; II. 

sazRvars aqvs ori zedmeti monakveTi, sazRvari araa Sekruli 

xazi (Tumca, arc gaxsnilia!); III. nakvTi Signidan carielia, 

ese igi, nakvTs ar ekuTvnis Siga are, mravalkuTxeds ki unda 

ekuTvnodes; IV. sazRvari Sedgeba ara erTi, aramed ori 

Sekruli texilisagan. 

II-Tan dakavSirebiT masw.-i moswavleebs saxelmZRvanelos 

gadaaSlevinebs VI Tavis me-5 gakveTilze. _ daasabuTeT, rom am 

nakvTis sazRvari ar aris Sekruli texili {Sekrul texils ar 

unda hqondes arcerTi bolo, amas ki aqvs ori bolo 

(wertili)}. maS, ratom ar yofila mravalkuTxedi? 

! 9:/4/! UbnbTj;! txsbgj! qbtvyfcj! `! hbnsbwmfcb.hbzpgb! 

sjdywUb!yb{{f•/!ix. 27.2. 

! 9:/5/! sjdywfcjt! Tfebsfcb! njmjbsejt! gbshmfcTj/! ! tbbohb. 

sjTp{f! npdfnvm! sbpefopcbUb! Tfebsfcb! njmjbsejt! gbsh. 

mfcTj/! gameordeba me-2 gakveTilis teqstis TviTeuli nawili. 

Semdeg Sedardeba saangariSoze akrefili ori ricxvi. 

Semdeg dawvrilebiTgairCeva amocana 2. 

h!:1/!sjhpcsjwj!sjdyw/!nsbwbmlvUyfej/!rwfTnjx/!hbbewjmfcb/! 

:1/2/ 13.6. (1 _ ara) 2 3 4 5! 6 7! 

:1/3/!sjhpcsjwj!sjdywfcjt!nbsUmxfsb/! 

- 94 -

masw.-is SekiTxvebiT jer gameordeba me-4 gakveTilis teqsti; 

Semdeg gairCeva amocana 1. misi amoxsnaa: 

1-li, me-6, XX, 21-e, XXI, XXIV, 90-e, me-800, me-1000. 

kanonzomierebas (matebis mixedviT) arRvevs XXIX. 

! :1/4/!UbnbTj;!nsbwbmlvUyfej!`!uzvjmjb!Uv!nbsUbmj•/! 

SekiTxvaTa nimuSebi: 1) zogi xazi aris mravalkuTxedi. 

2) yoveli oTxkuTxedi aris mravalkuTxedi. 

3) yoveli mravalkuTxedi aris oTxkuTxedi. 

4) zogierTi mravalkuTxedi aris Sekruli oTxkuTxedi. 

5) yoveli aTkuTxedi aris mravalkuTxedi. 

6) zogi mravalkuTxedi aris marTkuTxedi. 

7) zogi rvakuTxedi aris marTkuTxedi. 

8) zogi mravavalkuTxedi aris kvadrati. 

9) yoveli kvadrati aris mravalkuTxedi. 

10) zogi mravalkuTxedi aris xazi. 

11) arcerTi xazi ar aris mravalkuTxedi. 

12) arcerTi mravalkuTxedi ar aris xazi... 

! :1/5/!rwfTnjxfsjU!hbnsbwmfcjt!hbbewjmfcjt!yfsyfcj/! 

gavimeorebT samive naswavl xerxs: umjobesia 892 · 45; magram 

umjobesia 892 · 1151; aseve umjobesia 67 · 444, vidre 444 · 67. 

_ axla Tqven TviTon moigoneT samive am xerxis TiTo 

axali magaliTi! 

h!:2/!tblpouspmp!xfsb!$!24/! 



3. sul oTxi nabijia: ori gantolebis Sedgena da ucnobis 

mniSvnelobis ori gamoTvla. TiToSi iwereba TiTo q, sul _ 4 

q. gamoTvlis erTi meqanikuri (ara arsebiTi!) Secdoma qulas ar 

aklebs, Tuki am gamoTvlis moqmedeba marTebuladaa SerCeuli. 

4. mocemuli erTi sigrZis gverdis mqone marTkuTxedis daxazvaSi 

iwereba 1 q (Tuki meore gverdis sigrZe mcdaria). Tuki 

kargadaa gamoTvlili meore gverdis sigrZe _ 2 q. sul _ 5 q. 

5. Tuki gamoTvlilia jamis mniSvneloba 670+570+710 (an, 

II rigSi, 1180+1140+1100), ukve iwereba 1 q. Tuki am mniSvnelobas 

gamoklebulia meoTxe mocemuli ricxvi da, ese igi, 

Sevsebulia erTi ujredi _ 2 q. momdevno TiTo ujredis 

- 95 -

marTebulad SevsebisTvis iwereba TiTo q. angariSis or-ori 

meqanikuri SecdomisTvis akldeba TiTo q. sul _ 5 q. 

Tavi XIV 

h!:3/!tbbUjt!dopcb/!!obTUjboj!hbzpgb!hbobzpgjU!ovmj/! 

:3/2/ 13.7. 1~ 2# 3# 4 5# 6 7 

14.1. 1~ 2~ (3 _ ara) 4# 5# 6# 7 

________________________________________________ 

drois safexurebis mizania, rom moswavleebma kargad gaiazron 

is yvelaferi, rac mokledaa mocemuli amocanaTa krebulis 

Sesabamisi paragrafis SesavalSi. amitom am Sesavlis 

erTbaSad Seswavla ar unda movTxovoT klass. mas Tanda- 

TanobiT aiTviseben. yoveli am safexuris swavleba aucileblad 

unda emyarebodes yvelafris TvalsaCinod Cvenebas _ 

isrebian saaTze. amitom sasaurvelia klasSi iyos gafu- 

Webuli kedlis saaTi, romelsac moxsnili eqneba wina SuSa, 

raTa SeiZlebodes TiTiT isrebis Tavisuflad moZraoba. 

masw.-ma araviTar SemTxvevaSi ar unda ifiqros, rom 

bavSvma saaTis cnoba isedac iciso. dro gacilebiT ufro 

rTuli cnebaa, vidre sigrZe Tu wona! drois erTeulebic 

gacilebiT ufro rTulia, vidre sigrZisa Tu wonis 

erTeulebi. amitom am yvelafris safuZlianad swavla 

sakmaod Znelia (imdenad, rom drois gazomvisa da misi erTeulebis 

Tema me-5 klasSic grZeldeba). amitom moswavleebma 

Tavidanve yvelaferi kargad unda gaiazron, myarad da 

Tavisuflad unda daeuflon pirvelsave SedarebiT advil 

safexurebs, raTa mere da mere aRar gauZneldeT. 

sayofacxovrebo doneze saaTis cnoba da zerele warmodgenebi 

araa sakmarisi! 

________________________________________________ 

! :3/3/! tbbUjt! dopcb! `! kfs! nbsujwbe-! tbzpgbdypwsfcp! 

epof{f-!Ubo!nypmpe!jtsfcjboj!tbbUjtb-!bnbtUbo!fsUbe;!!2!tU!! 

71!xU/!! _ risi gasazomi erTeulebia wuTi da saaTi? {drois 

erTeulebia} diax, wuTsa da saaTs drois gasazomad viyenebT 

(me-4 klasSi wami araa saWiro). ramdeni wuTis tolia 1 

saaTi? (msxvilad daiwereba: 1 sT = 60 wT). 

! :3/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!xvUjb/! 

- 96 -

masw.-i moswavleebs rig-rigobiT usaxelebs: _ 3 saaTi; 10 

saaTi; 20 saaTi; 5 saaTi... (advilad saangariSoebi). 

moswavlem unda gamoTvalos, ramdeni wuTis tolia. 

! :3/5/!hbnfpsfcb!`!obTUjboj!hbzpgb![ofmj!TfnUywfwfcjUvsU/ 

! gairCeva amocana 3. Semdeg gairCeva Semdegi Zneli SemTxvevebi: 

10:9 = 1 (n. 1); 6:9 = 0 (n. 6); 12:20 = 0 (n. 12). 

TviTeul SemTxvevaSi saWiroa dasabuTeba, magaliTad: 12-Si 

ramdeni oceuli moTavsdeba? {arcerTi, nuli} maS, risi tolia 

ganayofi? {nulisa} naSTi ra dagvrCeba? {12 mTlianad naSTia}. 

h!:4/!tbbUjt!dopcb/!!obTUjboj!hbzpgb!hbobzpgjU!1/! 

:4/2/ 14.2. (1 da 2 _ ara) 3# 4~ 5 6! 7 

! :4/3/! tbbUjt! dopcb! `! kfs! nbsujwbe-! tbzpgbdypwsfcp! 

epof{f-!Ubo!nypmpe!jtsfcjboj!tbbUjtb-!bnbtUbo!fsUbe;!!2!tU!! 

71! xU/! ! ! ! ! !moswavleTa mier axsna-ganmarteba-dasabuTebiT 

gairCeva amocanebi 1 da 2. 

! :4/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!xvUjt!upmjb/! 

masw.-i moswavleebs rig-rigobiT usaxelebs: _ sami saaTi; ori 

saaTi da aTi wuTi; saaTi da ormocdaori wuTi; xuTi saaTi... 

moswavlem unda iangariSos, ramdeni wuTis tolia. 

! :4/5/!hbnfpsfcb!`!obTUjboj!hbzpgb![ofmj!TfnUywfwfcjUvsU/!!!! 

Catardeba iseve, rogorc 92.4, axsna-dasabuTebiT: 

1 mln. : 750 aTasi = 1 (n. 250 aTasi); 1000:300 = 3 (n. 100); 

750 aTasi : 1 mln. = 0 (n. 750 aTasi); 300:1000 = 0 (n. 300). 

h!:5/!obyfwbsj!tbbUj!!41!xvUj/!!obTUjboj!hbzpgb/! 

:5/2/ 14.3. 1~ (2 _ ara) 3# 4 5~ 6 7 

:5/3/!obyfwbsj!tbbUj!!41!xvUj/!!tbbUjt!dopcb/! 

`naxevari saaTi~ iswavleba naxevris zogadi cnebis 

gamoukveTavad (es araa saWiro!), iswavleba saaTze TvalsaCinod 

CvenebiT: _ vTqvaT, axla 12-is naxevaria. mamuka, daayene saaTis 

isrebi 12-is naxevarze! cira, ramdeni wuTi uklia 12-s? 

{ocdaaTi} ese igi, naxevari saaTi ramdeni wuTi yofila? {30} 

davweroT msxvilad: naxevari saaTi = 30 wuTi. 

! :5/4/!UbnbTj;!txsbg/!qbtvy/!`!sjhpcsjwj!sjdywfcj•/!ix. 88.2. 

! :5/5/!hbnfpsfcb!`!obTUjboj!hbzpgb![ofmj!TfnUywfwfcjUvsU/!!!!! 

Catardeba iseve, rogorc 92.4, axsna-dasabuTebiT: 

1 mln. : 300 aTasi = 3 (n. 100 aTasi); 

- 97 -

300 aTasi : 1 mln = 0 (n. 300 aTasi); 

2 mln. : 700 aTasi = 2 (n. 400 aTasi). 

h!:6/!obTUjboj!hbzpgjt!Tfnpxnfcb/! 

:6/2/ 14.4. (1 da 2 _ ara) 3~ 4 5 6 7! 

! :6/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!tbbUjtb!eb!xvUjt! 

upmjb/! masw.-i moswavleebs rig-rigobiT usaxelebs: _ 

oTxmocdaCvidmeti wuTi; asi wuTi; orasi wuTi; 

asormocdaaTi wuTi; oTxmocdaaTi wuTi; asoTxmoci wuTi 

(advilad saangariSoebi)... 

moswavlem iangariSos, ramdeni saaTis da ramdeni wuTis tolia. 

! :6/4/!obTUjboj!hbzpgjt!Tfnpxnfcb/! 


TviTeuli nawili. Semdeg gairCeva amocanebi 1 da 2. 

h!:7/!obTUjboj!hbzpgjt!Tfnpxnfcb/!!obyfwbsj!tbbUj!!41!xvUj/! 

:7/2/ 14.5. 1# 2# 3 (4 _ ara) 5 6 7 

! :7/3/!obTUjboj!hbzpgjt!Tfnpxnfcb/! 

mswavleblis SekiTxvebiT swrafad gameordeba me-4 gakveTilis 

teqsti. Semdeg gairCeva amocana 4. 

! :7/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!obyfwbsj!tbbUjb/! 

masw.-i moswavleebs rig-rigobiT usaxelebs: _ xuTi saaTi; rva 

saaTi da aTi wuTi; oTxmocdaCvidmeti wuTi; asi wuTi; 

orasi wuTi; asormocdaaTi wuTi; oTxmocdaaTi wuTi; 

asoTxmoci wuTi; sami saaTi da ormoci wuTi; ori saaTi da 

ocdaaTi wuTi... 

moswavlem unda iangariSos, es dro ramden naxevar saaTs 

moicavs (naSTis dasaxelebis gareSe). moswavleebs mosafiqreblad 

cota meti xani vadrovoT. 

h!:8/!obTUjboj!hbzpgjt!Tfnpxnfcb/!espjt!nfpsfhwbsj!bSsjdywb/! 

:8/2/ 14.6. (1 da 2 _ ara) 3 4! 5 6 7 

:8/3/!obTUjboj!hbzpgjt!Tfnpxnfcb/!! 

gairCeva amocana 2. Tan TviTeul SemTxvevaSi naSTiani gayofa 

Semowmdeba naswavli wesiT. 

! :8/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj/!ix. 32.3. 

! :8/5/! espjt! nfpsfhwbsj! bSsjdywb! )tbbUfcj! `! kfs! nypmpe! 

2.ebo! 23.nef*/! masw.-is SekiTxvebiT damuSavdeba me-6 gakveTilis 

teqstis TviTeuli nawili. Semdeg gairCeva amocana 1. Semdeg 

- 98 -

moxdeba Sedareba: _ dRis 12 saaTi da Ramis 12 saaTi; 

saRamos 7 saaTi da dilis 7 saaTi. am droebs Soris ramdeni 

xania gasuli? {12 saaTi} vaCvenoT saaTze! 

_ Ramis 3 saaTi da 30 wuTi anu... ? {Ramis 4-is naxevari}. 

h!:9/!tblpouspmp!xfsb!$!25/! 




Secdomad iTvleba isic, Tuki naSTiani gayofa araa marTebuli 

wesiT Semowmebuli. 

3. TiTo drois garkvevaSi iwereba TiTo q. 

4. amoxsna Sedgeba Semdegi moqmedebebisagan: 1) sarTulebisa 

da sadarbazoebis raodenobaTa gadamravleba; 2) gamravleba 

binebis raodenobaze; 3) sami Sesakrebis jamis mniSvnelobis 

gamoTvla; 4) me-2 ricxvisTvis me-3 ricxvis gamokleba. 

SeiZleba 3) da 4)-is nacvlad sxvagvarad: 3) erTis 

gamokleba; 4) meoris gamokleba; 5) mesamis gamokleba. 

udidesia 5 q. Tuki moqmedebani marTebuladaa awyobili, 

gamoTvlis erTi SecdomisTvis qula ar akldeba. gamoTvlis orori 

SecdomisaTvis udides 5 qulas akldeba TiTo q. 

5. am amocanaSi bevri davaleba kia, magram saqme isaa, rom 

pirveli qveamocana xels uwyobs meoresa da mesames, xolo 

meoTxe _ advilia. Tanac, saweri cotaa _ mxolod pasuxebi. 

amocanis pirobis mixedviT moswavles msjeloba ar moeTxoveba! 

magram mainc, davalebaTa simravlis gamo, qulebi didi 

SeRavaTebiT daiwereba! 

I rigi. I. 987; 

II. raki ricxvi udidesia, pirveli cifrebi unda iyos 9ianebi. 

amitom: 99 · · ·. meti 9-iani ar SeiZleba, radgan piroba 

moiTxovs, rom arcerTi cifri ar unda iyos or calze meti. 

amitom Semdeg unda iyos 8-ianebi: 99887; 

III. raki ricxvi umciresia, pirveli cifrebi unda iyos 1ianebi 

(0-iani ar SeiZleba, radgan maSin xuTniSna ricxvs ver 

miviRebT!). amitom: 1 · · · · . amitom Semdeg unda iyos 0-ianebi: 

100 · · ; Semdeg SeiZleba mxolod erTi cali 1-iani, meti ar 

SeiZleba, radgan piroba moiTxovs, rom arcerTi cifri ar unda 

- 99 -

iyos or calze meti. amitom pasuxia: 10012. 

IV. 98. 

II rigSic amdagvarive msjelobaa. pasuxebi: 

I. 102; II. 100122; III. 9988; IV. 11. 

pasuxebSi or-ori cifris Secdoma aklebs TiTo qulas. ese 

igi, erTi cifris Secdomaze qula ar akldeba. 

Tavi XV 

h!::/!espjt!nfpsfhwbsj!bSsjdywb/! 

::/2/ 14.7. 1 2 3# 4# 5~ 6# 

15.1. (1 _ ara) 2# 3# 4 5 6! 7# 

! ::/3/!UbnbTj;!tbjebobb!ynb@•!ix. 40.3. 

! ::/4/! espjt!nfpsfhwbsj! bSsjdywb! )tbbUfcj! `! kfs! nypmpe! 2. 

ebo!23.nef*/!gairCeva amocana 1, Tan misi pasuxebi Caiwereba drois 

meoregvari aRricxviTac _ Catardeba iseve, rogorc 97.4. 

h!211/!espjt!nftbnfhwbsj!bSsjdywb/!!djgsvmj!tbbUjt!dopcb/! 

211/2/ 15.2. (1 _ ara) 2# 3# 4! 5 6 

! 211/3/!UbnbTj;!txsbg/!qbtvy/!`!Dbwjgjrsf!sjdywfcj•/!ix. 45.3. 

! 211/4/! espjt! nftbnfhwbsj! bSsjdywb/! ! djgsvmj! tbbUjt! 

dopcb/! ! !masw.-is SekiTxvebiT damuSavdeba me-2 gakveTilis 

teqstis TviTeuli nawili da cxrili Tavisi naxazebiTurT. 


h!212/!tbnojTob!sjdywjt!rwfTnjxfsjU!hbzpgb!fsUojTob{f/! 

212/2/ 15.3. 1# 2# 3# 4 5 6! 7 

212/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!tbbUjtb!eb!xvUjt! 

upmjb/!masw.-i moswavleebs rig-rigobiT usaxelebs: _ asi wuTi; 

orasi wuTi; asormocdaaTi wuTi; asoTxmoci wuTi... 

moswavlem unda iangariSos, ramdeni saaTisa da ramdeni 

wuTis tolia. 

! 212/4/! tbnojTob! sjdywjt! rwfTnjxfsjU! hbzpgb! fsUojTob! 

sjdyw{f/!!!!!dawvrilebiT (da, rogorc yovelTvis, xmamaRali 

msjelobis TanxlebiT) gairCeva erTi magaliTi, Tan 

TvalsaCinoebisaTvis gamoiyeneba dafaze gverdiT mizazuli 

`kedlebi, svetebi da agurebi~. 

magaliTi qveSmiweris wesis gaSinaarsianebisaTvis: 

745 : 3 = (7 · 100 + 4 · 10 + 5) : 3 = 

_ gamoviyenoT jamis ricxvze gayofis naswavli wesi! 

- 100 -

(7 · 100 + 4 · 10 + 5) : 3 = 700 : 3 + 40 : 3 + 5 : 3 

_ ramdeni sameulia 700-Si? 100? 200? 300? {ramdeni 

sameulic moTavsdeba 7-Si, imdeni aseuli!}. amitom vwerT 

ganayofSi 2 · 100 da vagrZelebT, magram xom unda gamovakloT 2 

· 100-jer 3! es ki risi tolia? 2 · 100 · 3 = 600. davweroT es 

600 mocemuli 745-is qveS da gamovakloT. ratom vaklebT? 

imitom, rom es 600 xom ukve gavyaviT 3-ze da miviReT kidevac 

2 · 100. amitom gamovakloT da sxvaobac gavyoT 3-ze: 

745 : 3 = (7 · 100 + 4 · 10 + 5) : 3 = 2 · 100 + 

600 

1 

dagvrCa 1 aseuli, mivumatoT amas 4 aTeuli, anu `CamovitanoT~ 

4, iqneba 140. ramdeni sameulia 140-Si? ramdeni sameulic 

moTavsdeba 14-Si, imdeni aTeuli! ese igi, ganayofSi iqneba: 

745 : 3 = (7 · 100 + 4 · 10 + 5) : 3 = 2 · 100 + 4 · 10 + 

600 

140 

120 

2 

bolos, am 2 aTeuls, anu 20-s mivumatebT 5-sac: 

20 + 5 = 25 da gavyofT amas 3-ze: 

745 : 3 = (7 · 100 + 4 · 10 + 5) : 3 = 2 · 100 + 4 · 10 + 8 (naSTi 1) = 

248 (naSTi 1). 

TviTeuli nabijis gadadgmisas masw.-i Tan xeliT aCvenebs 

gverdiT daxazul TvalsaCinoebaze saTanado moqmedebebs: 

Semoxazavs, anawilebs da sxva. 

h!213/!eSf.Sbnf = 35!tU/!!rwfTnjx/!hbzpgb!fsUojTob!sjdyw{f/! 

213/2/ 15.4. (1 da 2 _ ara) 3 4! 5 6~ 7 

! 213/3/! eSf.Sbnf! ! 35! tU

_ dRe-Rame _ esaa drois erTeuli, romelic 24 saaTis tolia. 

Tuki axlaa, magaliTad, saRamos 9-is naxevari, erTi dRe- Ramis 

Semdeg ra dro iqneba? {xvalindeli dRis saRamos 9-is naxevari} 

(dawvrilebiT gairCes). ori dRe-Ramis Semdeg? ratom? 

sami? erTi dRe-Ramis win? xuTi dRe-Ramis win? ratom? 

! 213/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!tbbUjb/! 

masw.-i moswavleebs rig-rigobiT usaxelebs: _ sami dRe-Rame; 

oTxi dRe-Rame da 5 saaTi; ori dRe-Rame da 10 saaTi; erTi 

dRe-Rame da 18 saaTi; xuTi dRe-Rame... 

moswavlem unda gamoTvalos, ramdeni saaTis tolia. 

! 213/5/! tbnojTob! sjdywjt! rwfTnjxfsjU! hbzpgb! fsUojTob! 

sjdyw{f/! gairCeva me-2 amocana. misi is gayofa, romelic 

yvelaze met moswavles SeeSala, qveSmiweriT Sesruldeba 

dafasTan imdagvarive dawvrilebiTi xmamaRali msjelobebiT, 

rogorebic iyo 101.3-Si. 

h!214/!rwfTnjxfsjU!hbzpgb/!!espjt!nftbnfhwbsj!bSsjdywb/! 

214/2/ 15.5. (1 da 2 _ ara) 3 4 5 6~ 7 

214/3/! tbnojTob! sjdywjt! rwfTnjxfsjU! hbzpgb! fsUojTob! 

sjdyw{f/! gairCeva me-2 amocana. misi is gayofa, romelic 

yvelaze met moswavles SeeSala, qveSmiweriT Sesruldeba 

dafasTan imdagvarive dawvrilebiTi xmamaRali msjelobebiT, 

rogorebic iyo 101.3-Si. 

214/4/! UbnbTj;! txsbgj! qbtvyfcj! `! hbnsbwmfcb.hbzpgb! 


214/5/!espjt!nftbnfhwbsj!bSsjdywb/!djgsvmj!tbbUjt!dopcb/!! 

eSf.Sbnf!!35!tU/!gairCeva amocana 1 da Tan gameordeba me-4 

gakveTilis teqsti. 

drois maCvenebeli saaTebi jer-jerobiT kvlav mxolod 0-dan 

12-mde icvleba. 

h!215/!rwfTnjx/!hbzpgb/!!sbnefojnf!eSfSbnjU!xjo!eb!Tfnefh/! 

215/2/ 15.6. (1, 2 da 3 _ ara) 4# 5# 6! 7 

215/3/! eSf.Sbnf! ! 35! tU

! 215/4/! UbnbTj;! txsbgj! qbtvyfcj•! `! sbnefoj! eSf.Sbnf! eb! 

tbbUjb/!masw.-i moswavleebs rig-rigobiT usaxelebs: _ 30 saaTi; 

50 saaTi; 60 saaTi; 72 saaTi... (advilad saangariSoebi). 

moswavlem unda gamoTvalos, ramdeni dRe-Ramisa da ramdeni 

saaTis tolia. 

215/5/ gairCeva me-3 amocana. misi is gayofa, romelic yvelaze 

met moswavles SeeSala, qveSmiweriT Sesruldeba dafasTan 

imdagvarive dawvrilebiTi xmamaRali msjelobebiT, rogorebic 

iyo 101.3-Si. 






4. moqmedebis SerCevis TiTo SecdomisaTvis udides 5 qulas 

akldeba TiTo q. aseve TiTo q. akldeba gamoTvlis TiTo 

SecdomisaTvis. 

moqmedebaTa dagegmva SeiZleba orgvarad: an rig-rigobiT 

gamokleba (mimateba), an jer Sekreba da Semdeg erTianad 

gamokleba (Sekreba). Tuki pirveli moqmedeba mainc dawyebulia, 

ukve 1 q iwereba. 

5. areTa raodenobis an maTi mosazRvreobis TiTo SecdomisaTvis 

udides 5 qulas akldeba TiTo q. II rigis davaleba 

cota ufro Znelia naxazis sirTuliT, magram, samagierod, 

ufro advilia imis gamo, rom araa aucilebeli texilebis xazva 

_ mrudebis daxazva ufro advilia da maTi saxesxvaobac ufro 

advilad miiReba. 

Tavi XVI 

h!217/!nsbwbmojTob!sjdywjt!rwfTnjx/!hbzpgb!fsUojTob{f/! 

217/2/ 15.7. 1# 2~ 3~ 4 5# 6~ 7# 

16.1. 1~ 2~ 3~ 4~ 5 6 7# 

217/3/!UbnbTj;!nbUfnbujlvsj!tjuzwfcjt!ebtbyfmfcb•/! 

moswavleebma rig-rigobiT unda daasaxelon is sityvebi, romlebic 

iswavles maTematikis gakveTilebze, garda ricxvebisa, 

ricxviTi saxelebisa. masw.-i winaswar waukiTxavs amgvari 

sityvebis sias: Sekreba, mimateba, gamokleba, moqmedeba, 

Sesakrebi, jami, saklebi, maklebi, sxvaoba, namravli, 

- 103 -

gamravleba, Tanamamravli, gayofa, gasayofi, ganayofi, gamyofi, 

Tanrigi, gamosaxuleba, frCxilebi, SesaZlebloba, Sesabamisi, 

dajgufeba, ricxvTa xazi; metia, naklebia, metoba, nakleboba, 

tolia, toloba, igive, imdenive da ase Semdeg; sazRvari, 

nakvTi, are, perimetri da ase Semdeg; wre, rgoli, wertili, 

xazi, texili, mrudi, monakveTi, marTkuTxedi, mravalkuTxedi, 

piramida, prizma, aguredi, kubi; waxnagi, wibo, wvero da ase 

Semdeg (samkuTxedi, oTxkuTxedi _ ar SeiZleba, radgan es 

sityvebi Seicavs ricxviT saxelebs)... roca yvela moswavlis 

rigi Caivlis, daiwyeba meore wre. 

ar iTvleba: damaxinjebulad warmoTqmuli sityva, sxvis mier 

ukve dasaxelebuli sityva da xuTi taSis Semdeg naTqvami. 

yovel momdevno wreze tempi Cqardeba. moswavleebi eTiSebian 

Sejibrs, sanam ar gamovlindebian gamarjvebulni. 

! 217/4/! nsbwbmojTob! sjdywjt! rwfTnjxfsjU! hbzpgb! fsUojTob! 

sjdyw{f/! gayofa qveSmiweriT Sesruldeba dafasTan imdagvarive 

dawvrilebiTi xmamaRali msjelobebiT, rogorebic iyo 101.3-Si. 

19506:4 = ? 

srulad iwereba qveSmiweriT gayofa. Tan TviTeul nabijs 

axlavs axsna-ganmartebebi, raTa wesi gaSinaarsiandes da 

gaazrebul iqnes. 

19506 = 19 · 1000 + 5 · 100 + 6. jer gavyoT aTaseulebi. 

ramdenia gasayofSi aTaseuli? {cxrameti}. 19 gayofili 4-ze 

tolia 4-isa. amdeni aTaseuli iqneba ganayofSi. ganayofis 

aTaseulebis TanrigSi vwerT 4-s. naSTi ramdeni rCeba? amisaTvis 

19-s unda gamovakloT 4 · 4=16. rCeba 3 aTaseuli _ es jer ar 

gagviyvia {Sesamowmeblad SevadaroT naSTi 3 gamyofs 4 _ marTlac 

naklebia}. magram 3 aTasi 4-ze rom gavyoT, ganayofi ukve aTasze 

naklebi iqneba. amitom gadavdivarT aseulebze. 

gvaqvs 3 aTasi da kidev 5 aseuli, sul _ 35 aseuli. es 35 

gavyoT 4-ze _ miviRebT 8-s. amdeni aseuli iqneba ganayofSi. 

ganayofis aseulebis TanrigSi vwerT 8-s. naSTia 35 _ 8 · 4 = 3. 

amdeni aseuli jer ar gagviyvia. magram 3 aseuli 4-ze rom 

gavyoT, ganayofi ukve asze naklebi iqneba. amitom gadavdivarT 

aTeulebze. 

gvaqvs 3 aseuli da kidev 0 aTeuli, sul _ 30 aTeuli. es 

30 gavyoT 4-ze _ miviRebT 7-s. amdeni aTeuli iqneba 

- 104 -

ganayofSi. ganayofis aTeulebis TanrigSi vwerT 7-s. naSTia 30 

_ 7 · 4 = 2. amdeni aTeuli jer ar gagviyvia. magram 2 aTeuli 

4-ze rom gavyoT, ganayofi ukve aTze naklebi iqneba. amitom 

gadavdivarT erTeulebze. 

gvaqvs 2 aTeuli da kidev 6 erTeuli, sul _ 26 erTeuli. 

es 26 gavyoT 4-ze _ miviRebT 6-s. amdeni erTeuli iqneba 

ganayofSi, xolo naSTSi gvrCeba 26 _ 6 · 4 = 2. ganayofis 

erTeulebis TanrigSi vwerT 6-s. naSTi ki 2 erTeulis, anu 2-is 

tolia. ganayofSi vwerT: (n. 2). 

amrigad, ganayofia 4 · 1000 + 8 · 100 + 7 · 10 + 6 = 4876, 

naSTia 2. 

h!218/!espjt!nfpUyf!.!nfyvUfhwbsj!bSsjdywb/!rwfTnjx/!hbzpgb/! 

218/2/ 16.2. (1 da 2 _ ara) 3# 4~ 5# 6 

! 218/3/!espjt!nfpUyfhwbsj!eb!nfyvUfhwbsj!bSsjdywb/! 



_ dRe-Rame davyoT 24 saaTad, daviwyoT 0 saaTiT, anu Ramis 12 

saaTiT. miviRebT: dRis 12 saaTis Semdeg iqneba 13 saaTi, 14, 15 

da ase Semdeg 23 saaTi. bolos _ 24 saaTi, romelic igivea, 

rac Semdegi dRe-Ramis... romeli saaTi? {0 saaTi}. 

es yvelafris TvalsaCinod unda vaCvenoT saaTze. 

_ dRis 3 saaTs uklia 15 wuTi anu dRis 2 saaTi da 45 wuTi 

igivea, rac 14 saaTi da 45 wuTi _ mokled: 14.45. saRamos 9 

saaTi da 10 wuTi anu saRamos 10-is 10 wuTi igivea, rac 

21.10. 

! 218/4/!UbnbTj;!kbXwvsbe!.jU!nfuj \ .jU!oblmfcj;!!623!`!32•/!ix. 14.3. 

! 218/5/! nsbwbmojTob! sjdywjt! rwfTnjxfsjU! hbzpgb! fsUojTob! 

sjdyw{f/! iseve, rogorc 106.3-Si, dawvrilebiT Sesruldeba: 

8055:9 = ? srulad iwereba qveSmiweriT gayofa. Tan 

TviTeul nabijs kvlav axlavs xmamaRali axsna-ganmartebani. 

8055 = 8 · 1000 + 5 · 10 + 5. jer gavyoT aTaseulebi. 

ramdenia gasayofSi aTaseuli? (rva). 8 gayofili 9-ze tolia 0isa. 

amdeni aTaseuli iqneba ganayofSi. magram ricxvis dasawyisSi 

0 SeiZleba ar davweroT. ganayofis aTaseulebis TanrigSi 

arafers ar vwerT. naSTi ramdeni rCeba? TviTon 8 aris naSTi! 

magram 8 aTasi 9-ze rom gavyoT, ganayofi ukve aTasze naklebi 

iqneba. amitom gadavdivarT aseulebze. 

- 105 -

gvaqvs 8 aTasi da kidev 0 aseuli, sul _ 80 aseuli. es 80 

gavyoT 9-ze _ miviRebT 8-s. amdeni aseuli iqneba ganayofSi. ganayofis 

aseulebis TanrigSi vwerT 8-s. naSTia 80 – 8 · 9 = 8. amdeni 

aseuli jer ar gagviyvia. magram 8 aseuli 9-ze rom gavyoT, 

ganayofi ukve asze naklebi iqneba. amitom gadavdivarT aTeulebze. 

gvaqvs 8 aseuli da kidev 5 aTeuli, sul _ 85 aTeuli. es 

85 gavyoT 9-ze _ miviRebT 9-s. amdeni aTeuli iqneba ganayofSi. 

ganayofis aTeulebis TanrigSi vwerT 9-s. naSTia 85 _ 9 · 9 = 

4. amdeni aTeuli jer ar gagviyvia. magram 4 aTeuli 9-ze rom 

gavyoT, ganayofi ukve aTze naklebi iqneba. amitom gadavdivarT 

erTeulebze. 

gvaqvs 4 aTeuli da kidev 5 erTeuli, sul _ 45 erTeuli. 

es 45 gavyoT 9-ze _ miviRebT 5-s. amdeni erTeuli iqneba 

ganayofSi, xolo naSTi ar gvrCeba, radganac 9 · 5 = 45. 

ganayofis erTeulebis TanrigSi vwerT 5-s. unaSTod gaiyo. 


h!219/!rwfTnjx/!hbzpgb/!espjt!nfpUyf!eb!nfyvUfhwbsj!bSsjdywb/! 

219/2/ 16.3. (1 _ ara) 2~ 3# 4~ 5# 6 7 

! 219/3/!nsbwbmojTob!sjdywjt!rwfTnjxfsjU!hbzpgb!fsUojTob{f/! 

7660:7 = ? yvelaferi keTdeba iseve, rogorc 107.4-Si, 

magram erTi gansxvavebaa. ganayofis aTaseulTa TanrigSi vwerT 

1-s, naSTi ar gvrCeba. gauyofavi dagvrCa 6 aseuli, magram 6 

aseuli rom 7-ze gavyoT, ganayofi ukve asze naklebi iqneba! 6:7 

= 0 (n. 6). amitom ganayofis aseulebis TanrigSi vwerT 0-s da 

gadavdivarT aTeulebze. darCenili gvaqvs 6 aseuli da kidev 6 

aTeuli, sul _ 66 aTeuli. 66:7 = 9 (n. 1). ganayofis 

aTeulebis TanrigSi vwerT 9-s da gadavdivarT erTeulebze. 

dagvrCa 1 aTeuli da kidev 0 erTeuli, anu 10. es 10:7 = 1 (n. 

3). ganayofis erTeulebis TanrigSi vwerT 1-s da naSTia 3. 

saboloo pasuxia 1091 _ da ara 191! saWiroa moswavleTa 

yuradRebis miqceva: amgvar SemTxvevebSi ar unda dagvaviwydes 

ganayofis saTanado TanrigSi 0-is dawera! 

! 219/4/! UbnbTj;! txsbgj! qbtvyfcj•! `! sb! nbo[jmjb! UjUfct! 

Tpsjt•/!masw.-s xelSi ukavia saklaso metriani saxazavi da masze 

TiTebiT aCvenebs monakveTebs. SekiTxvebi: pirvel wreze _ 

advilebi (did danayofebs Soris), momdevno wreebze _ Znelebi 

- 106 -

(saSualo danayofebs Soris). 

! 219/5/!espjt!nfpUyfhwbsj!eb!nfyvUfhwbsj!bSsjdywb/! 

masw.-is SekiTxvebiT gameordeba me-2 gakveTilis teqstis 


h!21:/!2!lwjsb!!8!eSf.Sbnf/!2!Uwf!!41!eSf/!2!xfmjxbej!!23!Uwf/! 

21:/2/ 16.4. (1, 2 da 3 _ ara) 4 5 6 7# 

! 21:/3/! 2! lwjsb! ! 8! eSf.Sbnf/! ! lwjsjt! eSffcjt! hbnfpsfcb/!! 

sbnefojnf!lwjsjU!xjo!eb!lwjsjt!Tfnefh/!!2!Uwf!!41!eSf

xfmjxbej!!23!Uwf/! 



! 222/4/!UbnbTj;!txsbgj!qbtvyfcj!`!oblwUfcjt!dopcb•/!ix. 53.2. 

! 222/5/!hbzpgjtb!eb!hbnsbwmfcjt!xftj!`!gsDyjmfcjt!hbytob;! 

a:(b · c) = a:b:c a:(b · c) = a:c:b. 

dawvrilebiT gairCeva amocana 1. 





3. unda daiweros gamosaxuleba: 14$30 _ 8. Tuki 

moswavles sigrZe-siganis ujredebis gadaTvlisas erTiT 

SeeSala, qula ar daakldeba, Tuki 2-iT _ daakldeba. 

Tuki dawerilia namravlis gamosaxuleba _ 1 q. Tuki 

gamosaxulebaSi dawerilia gamoklebac _ 2 q. Tuki 

marTebuladaa gamoTvlili misi mniSvneloba _ 4 q. gamoTvlis 

TiTo Secdoma TiTo qulas aklebs. 

4. moqmedebis SerCevis TiTo SecdomisaTvis udides 5 qulas 

akldeba TiTo q. aseve TiTo q. akldeba gamoTvlebis or-ori 

SecdomisaTvis. 

moqmedebaTa dagegmva SeiZleba orgvarad: an rig-rigobiT 

gamokleba an jer Sekreba da Semdeg erTianad gamokleba. Tuki 

pirveli moqmedeba mainc dawyebulia, ukve 1 q iwereba. 

5. I rigi. unda ganvixiloT ori SemTxveva: marTebulia 

pirveli Setyobineba meore _ mcdaria, an _ piriqiT. pirveli 

SesaZlebloba gamoiricxeba, radgan gamodis, rom urCxuli 

daamarcxa usupma da Tan mcdaria is, rom urCxuli ar mouklavs 

badris. ese igi, urCxuli mouklavs badris, magram maSin mas ver 

moklavda usupi! amitom gvrCeba meore SesaZlebloba: pirveli 

Setyobineba mcdaria da meorea marTebuli, ese igi, usups ar 

mouklavs urCxuli, arc badris mouklavs. maSasadame, mouklavs 

amirans. 

Tuki pasuxi marTebulia da msjelobac Catarebulia (Tundac 

cotaTi xarvezovani, magram, ZiriTadad, marTebuli) _ iwereba 

udidesi 5 q. Tuki msjeloba marTebuli kia, magram Zlier 

- 108 -

naklulia _ 4 q. Tuki mxolod marTebuli pasuxia _ 1 q. 

Tuki pasuxi sul araa, magram marTebuli msjelobaa dawyebuli 

_ 1 q. Tuki marTebuli pasuxicaa da msjelobis magvaricaa _ 2 

q. Tuki msjeloba sanaxevrod Catarebulia, magram pasuxi araa 

_ 3 q. 

II rigi. TaRliTaZe pirdapir gamoiricxeba, radgan 

danaSaulis Cadenisas sxva qalaqSi yofila. Tuki qurdava 

damnaSavea, maSin bacacaSvilic iqneba damnaSave, radgan qurdava 

marto ar dadis saZarcvavad, xolo TaRliTaZe sxva qalaqSi 

yofila. Tuki qurdava araa damnaSave, maSin bacacaSvili mainc 

damnaSavea, radgan qurdava da TaRliTaZe gamoiricxnen, mesame ki 

bacacSvilia da ese igi, is damnaSave yofila. maSasadame, pasuxia: 

bacacaSvili namdvilad damnaSavea, xolo qurdavaze verafers 

vityviT _ is an damnaSavea, an ara. 

Tuki pasuxebi marTebulia da msjelobac Catarebulia 

(Tundac cotaTi xarvezovani, magram, ZiriTadad, marTebuli) _ 

iwereba udidesi 5 q. Tuki msjeloba marTebuli kia, magram 

Zlier naklulia _ 4 q. Tuki mxolod marTebuli pasuxebia _ 

2 q. Tuki mxolod erTi marTebuli pasuxia _ 1 q. Tuki 

pasuxebi sul araa, magram marTebuli msjelobaa dawyebuli _ 1 

q. Tuki erTi marTebuli pasuxicaa da msjelobis magvaricaa _ 

2 q. Tuki ori marTebuli pasuxicaa da msjelobis magvaricaa 

_ 3 q. Tuki msjeloba sanaxevrod Catarebulia, magram pasuxebi 

araa _ 3 q. 

Tavi XVII 

! 

h!224/!hbzpgb!bewjmj!yfsyjU/!!lbmfoebsj!eb!njtj!ynbsfcb/! 

224/2/ 16.7. 1~ 2~ (3 _ ara) 4~ 5# 6~ 

17.1. (1 da 2 _ ara) 3~ 4~ 5# 6 7 

224/3/!hbzpgb!bewjmj!yfsyjU/! 

gairCeva me-7 gakveTilis me-3 da 1-li gakveTilis me-2 amocanebi. 

Semdeg amdagvaradve Sesruldeba swrafad: 

36048:24 = 36048:6:4 = 6008:4 = 1502. 

! 224/4/! UbnbTj;! txsbgj! qbtvyfcj•! `! lbmfoebsj/! klasSi unda 

ekidos mimidinare wlis didi kalendari. masw.-i moswavleebs 

rig-rigobiT ekiTxeba: _ ra dRea 5 maisi? ra ricxvia 

Tebervlis pirveli orSabaTi? (da misTanani). 

- 109 -

Tuki aRmoCndeba, rom romelime moswavlem ar icis 

kalendris xmareba, mas Tanaklaselma auxsnas. 

! 224/5/! lbmfoebsj/! ! hbnfpsfcb;! ! xfmjxbejt! ebzpgb! 23! Uwfe-! 

xfmjxbejt! espfcj/! ! spnfmj! Uwf! sbnefoeSjbojb-! bewjmbe! 

ebtbnbytpwsfcfmj! xftj! `! nvXfcjt! bnpcvsdvmj! behjmfcjt! 

njyfewjU/!!lbmfoesjt!ynbsfcb/! 

masw.-is SekiTxvebiT gameordeba 1-li gakveTilis teqstis 


nakiani weliwadis sakiTxi _ mxolod gakvriTaa, 

dawvrilebiT iqneba V klasSi. 

h!225/!ovmfcjt!ebnbytpwsfcb!rwfTnjxfsjU!hbzpgjtbt/! 

225/2/ 17.2. 1# 2~ 3~ 4 5 6~ 7# 

! 225/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!spnfm!UwfTj!sbnefoj! 

eSfb/!masw.-i moswavleebs rig-rigobiT ekiTxeba: _ ramdeni dRea 

agvistoSi? ramdendRiania noemberi? romelia pirveli 30dRiani 

Tve? romelia ukanaskneli 30-dRiani Tve? romelia 

pirveli 31-dRiani Tve? romelia ukanaskneli 31-dRiani Tve? (da 

misTanani). 

cxadia, moswavles aqvs ufleba, rom sakuTar muWebze 

gadaTvalos, amitom sapasuxod metxans unda vadrovoT. 

! 225/4/! hbzpgb! bewjmj! yfsyjU/! ! ovmfcjU! ebcpmpfcvmj! 

nsbwbmojTob! sjdywjt! hbzpgb! )ovmfcjt! ebnbytpwsfcb*-! nbU! 

Tpsjt!obTUjboj!TfnUywfwfcjd/! 

ganvixilavT jer jer unaSTo gayofas: 49500:9 = ? 

_ ese igi, 495 aseuli gayofili 9-ze, es ki igivea, rac 495:9 

aseuli, ese igi: 49500:9 = (495 · 100):9 = (495:9) · 100. 

_ amrigad, gayofisas SegviZlia, rom nulebs ar mivaqcioT 

yuradReba, oRond isini unda `davimaxsovroT~ da mere ganayofs 

unda mivuweroT imdeni nuli, ramdenic davimaxsovreT. 

_ vnaxoT, ra xdeba naSTiani gayofisas. 4600:9 = (46 · 100):9. 

gayofa Sesruldeba ori gziT: jer pirdapir, qveSmiweriT; 

Semdeg ki `nulebis damaxsovrebiT~. davrwmundebiT, rom naSTiani 

gayofis SemTxvevaSi `nulebis damaxsovrebis~ wesi mcdar pasuxs 

mogvcems. ese igi, naSTiani gayofisas nulebis damaxsovreba ar 

SeiZleba, gayofa Cveulebrivad unda ganvagrZoT bolomde! 

_ axla kargad wavikiTxoT da davimaxsovroT me-2 gakveTilis 

- 110 -

teqsti. 

! 


226/2/ 17.3. 1# 2# 3 4 5# 6 7! 

226/3/! UbnbTj;! txsbgj! qbtvyfcj•! `! lbmfoebsj/! klasSi unda 




! 226/4/! hbzpgb! bewjmj! yfsyjU/! ovmfcjU! ebcpmpfcvmj! 


Tpsjt! obTUjboj! TfnUywfwfcjd/! gameoodeba 114.3, oRond sxva 

ricxvebiT: 678000:6 da 679000:6. 


227/2/ 17.4. (1 _ ara) 2~ 3~ 4 5! 6~ 7# 

! 227/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!eSfb/! 

masw.-i moswavleebs rig-rigobiT usaxelebs: _ 6 kvira; 10 Tve; 

20 kvira; 5 Tve; 11 kvira; 100 Tve; 50 kvira; 4 Tve... 

(advilad saangariSoebi). 

moswavlem unda gamoTvalos, ramdeni dRis tolia. 

1 Tve = 30 dRe. 

! 227/4/! hbzpgb! bewjmj! yfsyjU/! ovmfcjU! ebcpmpfcvmj! 


Tpsjt! obTUjboj! TfnUywfwfcjd/! gairCeva amocana 1. Tan 

Catardeba iseTive msjeloba, rogorc 114.3-Si. 

h!228/!{fqjsj!bohbsjTj!njmjpofc{f-!pUyjwf!nprnfefcjU/! 

228/2/ 17.5. (1 da 2 _ ara) 3~ 4 5 6 7~ 

228/3/! UbnbTj;! txsbgj! qbtvyfcj•! `! sbnefoj! eSf.Sbnf! eb! 

tbbUjb/!ix. 104.3. 

! 228/4/! {fqjsj! bohbsjTj! njmjpofc{f-! pUyjwf! nprnfefcjU! 

)njmjbsejt!gbshmfcTj*/! 

gairCeva amocanebi 1 da 2. Sesruldeba advili zepiri 

gamravleba-gayofac: 1 mln 500 aTasi gamravlebuli 4-ze; 60 

mln gayofili 3-ze; 800 aTasi gamravlebuli 10-ze an 100-ze; 

40 mln. gayofili 10-ze an 100-ze da misTanani. 

h!229/!{fqjsj!bohbsjTj!njmjpo/!!lbmfoebsj-!xfmj!eb!Uwffcj/! 

229/2/ 17.6. 1~ 2~ (3 da 4 _ ara) 5# 6! 7# 

229/3/! {fqjsj! bohbsjTj! njmjpofc{f-! pUyjwf! nprnfefcjU! 

- 111 -

)njmjbsejt!gbshmfcTj*/! 

gairCeva amocana 3. Sesruldeba advili zepiri gamravlebagayofac: 

1 mln 200 aTasi gamravlebuli 5-ze; 150 mln 

gayofili 3-ze; 350 aTasi gamravlebuli 10-ze an 100-ze; 25 

mln. gayofili 10-ze an 100-ze da misTanani. 

! 229/4/! UbnbTj;! txsbgj! qbtvyfcj•! `! sbnefoj! xfmjxbej! eb! 

Uwfb/!!masw.-i moswavleebs rig-rigobiT usaxelebs: _ 30 Tve; 50 

Tve; 60 Tve; 72 Tve... (advilad saangariSoebi). 

moswavlem unda gamoTvalos, ramdeni weliwadisa da ramdeni 

Tvis tolia. 

! 229/5/! lbmfoebsj/! hbnfpsfcb;! xfmjxbejt! ebzpgb! 23! Uwfe-! 

xfmjxbejt!espfcj/!spnfmj!Uwf!sbnefoeSjbojb-!bewjmbe!eb. 

tbnbytpwsfcfmj! xftj! `! nvXfcjt! bnpcvsdvmj! behjmfcjt! nj. 

yfewjU/! obljboj! xfmjxbejt! tbljUyj! `! hblwsjU/! ! lbmfoesjt! 

ynbsfcb/!dawvrilebiT gairCeva amocana 4 da misi wriuli sqema. 





3. Tuki moqmedeba marTebuladaa dagegmili, or-ori 

meqanikuri SecdomisaTvis udides 4 qulas akldeba TiTo q. 

4. TiTo q. iwereba TiTo nabijSi: 1) naxazis marTebulad 

Sedgena; 2) didi marTkuTxedis sigrZis gamoTvla; 3) didi 

marTkuTxedis siganis gamoTvla; 4) didi marTkuTxedis 

sigrZisa da siganis Sekreba; 5) me-4 sididis gaormageba anu 

didi marTkuTxedis perimetris gamoTvla. 

5. pasuxebi: I rigi: 7+20 (ramdeni wuTis ganmavlobaSi hyavda 

daWerili mwerebi gogonas); 4+12 (sul ramdeni mweri hyavda da- 

Werili gogonas); 12:4 (ramdenjer meti iyo pepela, vidre Wiamaia); 

20_7 (ramdeniT met xans hyavda peplebi, vidre Wiamaiebi). 

II rigi: 5+15 (ramdeni wuTis ganmavlobaSi hyavda daWerili 

mwerebi gogonas); 15:5 (ramdenjer met xans hyavda xoWoebi, 

vidre cicinaTelebi); 18_6 (ramdeniT meti iyo pepela, vidre 

Wiamaia); 6+18 (sul ramdeni mweri hyavda daWerili gogonas). 

udides 5 qulas akldeba TiTo q. Semdegi SecdomebisaTvis: 

azriani gamosaxulebis gamotoveba; uazro gamosaxulebis 

- 112 -

amowera; ar miwera imisa, Tu ras Seesabameba amowerili 

gamosaxuleba; gamoTvlis ori meqanikuri Secdoma. 

Tavi XVIII 

h!231/!njmjbsej/!!blsfgb!tbbohbsjTp{f/! 

231/2/ 17.7. 1~ 2~ 3 4 5 6~ 7# 

18.1. (1 da 2 _ ara) 3# 4~ 5 6~ 7# 

! 231/3/! UbnbTj;! txsbgj! qbtvyfcj•! `! lbmfoebsj/! klasSi unda 




! 231/4/! njmjbsej/! ! blsfgb! tbbohbsjTp{f/! jer gairCeva 

amocanebi 2 da 3 _ am gakveTilis Semamzadeblebia. 

_ maS, ra udidesi raodenobis milionebs SeiZleba Seicavdes 

cxraniSna ricxvi? {999}. vTqvaT, mocemulia aseTi raodenoba: 

2000 · mln + 502. rogor CavweroT ricxvis saxiT? Tuki 

milionTa raodenoba 1000-ze metia, cxraniSna ricxvad xom 

veRar CavwerT! amitom gvWirdeba axali Tanrigebi! 

_ 1000 milioni _ ewodeba miliardi. cxadia, ase Caiwereba: 

1000000000. 

_ is Cveni 2000 · mln + 502 ra ricxvi iqneba? {2 miliard 

502} rogor Caiwereba? {2 000 000 502} ramdenniSnaa? ratom? 

avkrifoT saangariSoze! 

_ axla me avkref saangariSoze da Tqven miTxariT, ra ricxvi 

iqneba! {xuTi miliard samasi milion ocdaxuTi aTas oTxasi}. 

SeniSvna. Tuki saangariSos aklia meaTe Tanrigi, maSin aqve 

gamoviyenoT is xerxi, romelic 121.3-Sia aRwerili (kubebis an 

sxva nivTebis moSvelieba). 

h!232/!sjdywfcjt!xbljUywb!eb!Dbxfsb!:::!njmjbsebnef/! 

232/2/ 18.2. (1 da 2 _ ara) 3 4 5 6 7~ 

3 da 5 _ momdevno gakveTilisTvis Semamzadebeli amocanebia da 

kargad unda gairCes! 

! 232/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!sbnefoj!Uwf!eb!eSfb/! 

masw.-i moswavleebs rig-rigobiT usaxelebs: _ 70 dRe; 150 dRe; 

305 dRe; 600 dRe... (advilad saangariSoebi). 

moswavlem gamoTvalos, ramdeni Tvisa da ramdeni dRis tolia. 

! 232/4/! njmjbsej/! sjdywfcj! :::! njmjbsebnef/! sjdywfcjt! 

- 113 -

xbljUywb!eb!Dbxfsb/!blsfgb!tbbohbsjTp{f/!masw.-is SekiTxvebiT 

damuSavdeba me-2 gakveTilis teqstis TviTeuli nawili. _ 

saangariSoze Tu avkrefT am raodenobebs? {vera, radgan 

saangariSoze Tanrigebi aRar gveyofa} modi, ase moviqceT: 

saangariSos davaxmaroT kubebi (SeiZleba raime sxva nivTebic!). 

davdgaT saangariSo da gverdiT mivuwyoT kubebi, romlebic 

aRniSnavs... ras? {aTeul miliardebs, aseul miliardebs} axla 

teqstSi mocemuli ricxvebi ase avkrifoT saangariSoze. 

Semdeg gairCeva amocanebi 1 da 2. me-2 amocanis TviTeuli 

ricxvic aikrifeba saangariSoze. 

h!233/!sjdyw/!:::!njmjbsebnef/!rwfTnjx/!hbzpgb!psojT/!sjdyw{f/! 

233/2/ 18.3. (1 da 2_ ara) 3~ 4~ 5 6~ 7 

! 233/3/! sjdywfcj! :::! njmjbsebnef/! sjdywfcjt! xbljUywb! eb! 

Dbxfsb/!blsfgb!tbbohbsjTp{f/! 


TviTeuli nawili. teqstSi mocemuli ricxvi aikrifeba 

saangariSoze (121.3-Si mocemuli xerxiT). 

Semdeg gairCeva amocana 1. masSi mocemuli TviTeuli ricxvic 


! 233/4/!UbnbTj;!txsbgj!qbtvyfcj•!`!lbmfoebsj/! 

klasSi unda ekidos mimidinare wlis didi kalendari. masw.-i 

moswavleebs rig-rigobiT ekiTxeba: _ ra dRea 5 maisi? ra 

ricxvia Tebervlis pirveli orSabaTi? (da misTanani). 

! 233/5/!rwfTnjxfsjU!hbzpgb!psojTob!sjdyw{f/! 

jer gairCeva amocana 2. 

qveSmiweriT gayofa orniSna ricxvze iseve xdeba, rogorc 

erTniSnaze gayofa, magram erTi damatebiTi siZnele SeiZleba 

wamoiWres. magaliTad: 4938:82 = ? 

_ pirdapir gayofa Znelia. amitom gavyoT ara 82-ze, aramed 80ze. 

ganayofis pirveli cifris sapovnelad 49 gavyoT 8-ze! 

es nabiji emyareba naSTiani gayofis Semowmebis naswavl wess 

(romelic gavixseneT kidevac wina gakveTilis Semamzadebeli 

amocanebiT 3 da 5!). gaviazroT: 49:8 = 6 (n. 1). magram es 6 

sacdeli ricxvia, misi Cawera ganayofSi jer ar SeiZleba, 

radganac iqneb arc gamogvadges. es imitom, rom gavyaviT 80-ze, 

da ara 82-ze, razec unda gagveyo! magram mainc movsinjoT es 6: 

- 114 -

amisTvis gavamravloT 6 · 82 = 492. es naklebia 493-ze, amitom 

6 cali 82-euli gasayofSi namdvilad eteva. CavweroT 6 

ganayofSi. gasayofSi ra dagvrCeba? {4938 _ 492}. gamovakloT 

da ganvagrZoT gayofa ise, rogorc naswavli gvaqvs. 

magram napovni sacdeli ricxvi zogjer SeiZleba ar 

gamogvadges. amgvar SemTxvevas momdevno gakveTilze gavarCevT. 

h!234/!rwfTnjxfsjU!hbzpgb!psojTob!sjdyw{f/! 

234/2/ 18.4. (1 _ ara) 2~ 3 4 5~ 6# 7# 

! 234/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj/!ix. 32.3. 



moZebnili sacdeli ricxvi zogjer SeiZleba ar gamogvadges, 

magaliTad: 28456:47 = ? jer 284 gavyoT 40-ze. amisTvis 28 

gavyoT 4-ze, miviRebT 7-s. esaa sacdeli ricxvi. movsinjoT, 

varga Tu ara: 7 · 47 = 329. es metia, vidre 284, ese igi, amdeni 

47-euli gasayofSi ar moTavsdeba! amitom 7 ar varga. maSin 

vcadoT 6. movsinjoT: 6 · 47 = 282. es ukve naklebia, vidre 

284, ese igi, 6 varga. CavweroT 6 ganayofSi da Semdeg 

ganvagrZoT gayofa ise, rogorc naswavli gvaqvs. 

nulebis damaxsovrebac iseve xdeba, rogorc erTniSna 

gamyofis SemTxvevevaSi _ ese igi, rogor? (gavimeoroT XVII 

Tavis me-2 gakveTilis teqsti). 

h!235/!sjdyw!:::!njmjbsebnef/!rwfTnjx/!hbzpgb!psojTob!sjdyw{f/! 

235/2/ 18.5. (1 da 2 _ ara) 3~ 4~ 5 6# 7! 

! 235/3/! sjdywfcj! :::! njmjbsebnef/! sjdywfcjt! xbljUywb! eb! 

Dbxfsb/! masw.-is SekiTxvebiT gameordeba me-2 da me-3 

gakveTilebis teqstis TviTeuli nawili. teqstSi mocemuli 

ricxvebi aikrifeba saangariSoze. 

Semdeg gairCeva amocana 1. masSi mocemuli TviTeuli ricxvic 


! 235/4/! UbnbTj;! txsbgj! qbtvyfcj•! `! sb! nbo[jmjb! UjUfct! 

Tpsjt•/!ix. 108.3. 


gairCeva amocana 2. is gayofa, romelic moswavleebs yvelaze 

metad gauWirdaT, Sesruldeba klasSi, rogor yovelTvis, 

xmamaRali aRwera-msjelobis TanxlebiT. 

- 115 -

nulebis damaxsovrebac iseve xdeba, rogorc erTniSna 

gamyofis SemTxvevevaSi _ ese igi, rogor? (gavimeoroT XVII 

Tavis me-2 gakveTilis teqsti). 

orniSna ricxvze gayofaSi metismeti gawafva araa aucilebeli. 





3. sul oTxi gamoTvlaa, TiToSi iwereba TiTo q. erTi 

arsebiTi an or-ori meqanikuri (namdvilad uneblie) 



akldeba TiTo q. magram Tuki Secdoma arsebiTia (Tanrigebis 

areva, sigrZis erTeulTa areva da sxva), maSin TiTo 

SecdomisaTvis akldeba TiTo q. 

5. 1) didi wignis (rgolis) wonis gamravleba maT 

raodenobaze: moqmedebis marTebulad dawyeba _ 1q, marTebulad 

gamoTvla _ 1 q; 2) mTliani wonisTvis gamokleba miRebuli 

winisa: moqmedebis marTebulad dawyeba _ 1q, marTebulad 

gamoTvla _ 1 q; 3) miRebuli wonis gayofa patara wignis 

(qvis) wonaze _ 1 q. sul _ 5 q. 

Tuki moqmedebani mcdaradaa dagegmili, magram marTebuladaa 

Sesrulebuli erTi mainc saWiro moqmedeba _ iwereba TiTo q. 

Tavi XIX 

h!237/!ovmfcjU!ebcpmpfcvmj!sjdywfcjt!fsUnbofU{f!hbzpgb/! 

237/2/ 18.6. 1~ 2~ 3# 4# 5 6 

19.1. (1 - ara) 2# 3~ 4# 5# 6~ 7 

237/3/!UbnbTj;!txsbgj!qbtvyfcj•!.!sbnefoj!lwjsb!eb!eSfb/! 

masw.-i moswavleebs rig-rigobiT usaxelebs: - 700 dRe; 140 

dRe; 80 dRe; 710 dRe... (advilad saangariSoebi). 

moswavlem unda gamoTvalos, ramdeni kvirisa da ramdeni 

dRis tolia. 

237/4/! nsbwbmojTob! sjdywfcjt! hbzpgjt! hbbewjmfcb! .! 

nypmpe! vobTUpe;! ovmfcjU! ebcpmpfcvmj! sjdywfcjt! 

fsUnbofU{f! hbzpgb-! speftbd! hbnzpgt! cpmpTj! bsb! brwt! 

- 116 -

hbtbzpg{f! nfuj! ovmfcj/!masw.-is SekiTxvebiT damuSavdeba 1-li 

gakveTilis teqstis TviTeuli nawili. Semdeg gairCeva amocana 

1. 

h!238/!UbsjS/!djgs/!Dbxfsb!eb!xbljUywb/!ovmfcjt!ebnbytpws/! 

238/2/ 19.2. (1 da 2 - ara) 3~ 4~ 5~ 6# 7 

238/3/!UbsjSfcjt!djgsvmj!Dbxfsb!eb!xbljUywb/! 

masw.-is SekiTxvebiT damuSavdeba me-2 gakveTilis teqsti. Semdeg 


! 238/4/! UbnbTj;! txsbgj! qbtvyfcj! `! hbnsbwmfcb.hbzpgb! 



nypmpe! vobTUpe;! ! ovmfcjU! ebcpmpfcvmj! sjdywfcjt! 


hbtbzpg{f! nfuj! ovmfcj/!Tavidan bolomde gameordeba 126.3: 1li 

gakveTilis teqstis SekiTxvebiT damuSaveba. Semdeg gairCeva 

amocana 2-is pirveli gayofa. 

h!239/!UbsjS/!djgs/!Dbxfsb!eb!xbljUywb/!ovmfcjt!ebnbytpws/! 

239/2/ 19.3. (1 da 2 - ara) 3~ 4 5~ 6 7# 

239/3/!UbsjSfcjt!djgsvmj!Dbxfsb!eb!xbljUywb/! 

masw.-is SekiTxvebiT gameordeba me-2 gakveTilis teqsti. Semdeg 


! 239/4/!UbnbTj;!wjo!bnpyb{bwt!zwfmb{f!nfut@•!ix. 55.2, oRond 

axla amoxazon sxvadasxvanairi mravalkuTxedebi. 


nypmpe! vobTUpe;! ! ovmfcjU! ebcpmpfcvmj! sjdywfcjt! 


hbtbzpg{f! nfuj! ovmfcj/! swrafad gameordeba 1-li gakveTilis 

teqsti. Semdeg gairCeva amocana 2. 

h!23:/!ovmfcjt!ebnbytpws/!rwfTnjx/!hbzpgb!tbnojTob!sjdyw{f/! 

23:/2/ 19.4. (1 - ara) 2~ 3~ 4 5# 6 7 

23:3/! nsbwbmojTob! sjdywfcjt! hbzpgjt! hbbewjmfcb! .! nyp. 

mpe!vobTUpe;!!ovmfcjU!ebcpmpfcvmj!sjdywfcjt!fsUnbofU. 

{f! hbzpgb-! speftbd! hbnzpgt! cpmpTj! bsb! brwt! hbtbzpg{f! 

nfuj! ovmfcj/! swrafad gameordeba 1-li gakveTilis teqsti. 

Semdeg swrafad gairCeva amocana 1-is mxolod pirveli gayofa. 

! 23:/4/!UbnbTj;!tjuzwfcjt!Tfehfob!)btpfcjt!hbnfpsfcjU*•/! 

- 117 -

ix. 60.3, oRond, cxadia, axla sxva asoebi daiwereba. 

! 23:/5/!rwfTnjxfsjU!hbzpgb!tbnojTob!sjdyw{f/! 

qveSmiweriT gayofa samniSna ricxvze iseve xdeba, rogorc 

orniSnaze gayofa, magram erTi damatebiTi siZnele SeiZleba 

wamoiWres. magaliTi: 10185:127 = ? jer 1016 gavyoT 100-ze, 

miviRebT 10-s. magram 10 ar SeiZleba iyos TanrigSi Casaweri 

ricxvi, radganac igi orniSnaa! amitom 10 ar gamogvadgeba. 

amitom movsinjoT 9. gavamravloT: 

9 · 127 = 1143. es gasayofis 1018-ze metia, amitom arc 9 varga! 

maSin movsinjoT 8. gavamravloT: 8 · 127 = 1016. ese igi, 8 varga. 

CavwerT mas ganayofSi da Semdeg ganvagrZobT gayofas ise, 

rogorc naswavli gvaqvs. 

ese igi, samniSna gamyofis SemTxvevaSi ufro meti mosinjva 

SeiZleba dagvWirdes. 

nulebis damaxsovreba iseve xdeba, rogorc naswavli gvaqvs. 

samniSna ricxvze gayofaSi gawafva araa saWiro. 

bolos gairCeva amocana 1-is mxolod meore gayofa. 

h!241/!{fqjsj!bohbsjTj!njmjbsefc{f/!!hbzpgb!tbnojTob!sjdyw{f/! 

241/2/ 19.5. (1 - ara) 2~ 3~ 4~ 5 6 7~ 

241/3/!{fqjsj!bohbsjTj!njmjbsefc{f-!pUyjwf!nprnfefcjU/! 

gariCeva amocana 1. damatebiT zepirad Sesruldeba ramdenime 

amdagvari, ufro advili gamoTvla: 5 miliard aTasi 

gamravlebuli 5-ze; 2 miliard 4 milioni gayofili 2-ze; 2 

miliard 4 milioni gamravlebuli 3-ze; 7 miliard 600 

milions mimatebuli 5 miliard 600 milioni; 15 miliards 

gamoklebuli 8 miliard 100 milioni. 

! 241/4/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj•/! 

SekiTxvebis nimuSebi: 1) namravli zogjer tolia cxrisa. 

2) pirveli Tanamamravli yovelTvis tolia meore Tanamamravlisa. 

3) meore Tanamamravli zogjer tolia pirveli Tanamamravlisa. 

4) pirveli Tanamamravli yovelTvis tolia namravlisa. 

5) meore Tanamamravli zogjer tolia namravlisa. 

6) pirveli Tanamamravli yovelTvis tolia nulisa. 

7) meore Tanamamravli zogjer tolia nulisa. 

8) pirveli Tanamamravli yovelTvis tolia aTisa. 

9) meore Tanamamravli zogjer tolia aTisa. 

- 118 -

10) namravli yovelTvis tolia ocisa. 

11) namravli arasdros araa Tvrametis toli. 

12) pirveli Tanamamravli arasdros araa TerTmetis toli. 

13) meore Tanamamravli arasdros araa ocdaxuTis toli. 

14) pirveli Tanamamravli yovelTvis metia, vidre meore. 

15) pirveli Tanamamravli yovelTvis naklebia, vidre meore. 

16) pirveli Tanamamravli zogjer metia, vidre meore. 

17) ganayofi arasdros araa TerTmetis toli. 

18) gasayofi arasdros araa as Tormetis toli. 

19) gamyofi arasdros araa samocdaoris toli. 

20) gasayofi zogjer ormocis tolia. 

21) gamyofi zogjer cxrametis tolia. 

22) ganayofi zogjer ocdacxris tolia. 

23) gasayofi zogjer gamyofis tolia. 

24) gamyofi yovelTvis oris tolia. 

! 241/5/! rwfTnjxfsjU! hbzpgb! tbnojTob! sjdyw{f/! gameordeba 

129.4. Semdeg gairCeva me-2 amocanis mxolod pirveli gayofa. 

h!242/!{fqjsj!bohbsjTj!njmjbsefc{f/!!hbzpgb!tbnojTob!sjdyw{f/! 

242/2/ 19.6. 1 (2 da 3 - ara) 4 5 6# 7 

242/3/!{fqjsj!bohbsjTj!njmjbsefc{f-!pUyjwf!nprnfefcjU/! 

gairCeva me-3 amocana. Semdeg damatebiT zepirad Sesruldeba 

ramdenime advili gamoTvla: 4 miliard 2 aTasi gamravlebuli 

4-ze; 12 miliard 40 milioni gayofili 2-ze; 1 miliard 3 

milioni gamravlebuli 3-ze; 7 miliard 600 milions 

gamoklebuli 5 miliard 600 milioni; 15 miliard 200 

milions gamoklebuli 8 miliard 100 milioni. 

! 242/4/! UbnbTj;! txsbgj! qbtvyfcj! `! sbnefojUbb! nfuj! 21111.jt! 

gbshmfcTj•/!ix. 39.3. 

! 242/5/!rwfTnjxfsjU!hbzpgb!tbnojTob!sjdyw{f/! 

swrafad gameordeba 129.4. Semdeg gairCeva me-2 amocana. 

h!243/!tblpouspmp!xfsb!$!2:/! 



2. erTi arsebiTi an or-ori meqanikuri (namdvilad uneblie) 


3. erTi arsebiTi an or-ori meqanikuri (namdvilad uneblie) 

- 119 -


4. I - 1 q (radgan advilia); II - 2 q (radgan gadideba- 

Semcireba unda Sebrundes); III 2 q (ar Seicvleba - cota 

Znelia, radgan wina orive unda Sejerdes). sul - 5 q. 

Tuki II an III SemTxvevaSi moswavles SeeSala, magram 

marTebuli msjelobis magvari aqvs (ara mxolod mcdari 

pasuxi), maSin daewereba 1 q. msjeloba-dasabuTeba iqneba, albaT, 

gamoklebis konkretuli magaliTebis ganxilva. 

5. pasuxebi: I rigi: I. m s; II. f r T; III. f i t a r e T m s; 

IV. i t a e; V. m s f r T. 

II rigi: I. s v c o l; II. m; III. e t x i; IV. m e t x i s v c 

o l; V. s v c o l m. 

V jgufi unda iyos I da II jgufebis gaerTianeba. 

TiTo qveamocanaSi iwereba TiTo q. or-ori SecdomisaTvis udides 

5 qulas akldeba TiTo q. Secdomaa: asos gamotoveba, zedmeti asos 

amowera, erT jgufSi erTidaimave asos orjer Cawera. 

Tavi XX 

h!244/!tufsfpnfu/!obtxbwmj!tbljUy/!{fq/!bohbsjTj!njmjbsefc{f/! 

244/2/ 19.7. 1~ 2 3# 4 5~ 6 7 

20.1. 1 (2 - ara) 3~ 4# 5# 6# 7 

! 244/3/!tufsfpnfusjjt!obtxbwmj!tbljUyfcj!`!hbnfpsfcb/! 

swrafad, mxolod TvalsaCino miTiTebebiT unda gameordes: 

wveroebi, waxnagebi da wiboebi: kubisa; paralelepipedisa; 

prizmebisa; piramidebisa. 

waxnagebis saerTo wibo da wiboebis saerTo wvero. 

SekiTxvaTa nimuSebi: 

- daakvirdiT am samkuTxa prizmas. ramdeni wvero, ramdeni 

waxnagi da ramdeni wibo aqvs mas? am xuTkuTxa prizmas? 

- daakvirdiT am oTxkuTxa piramidas. ramdeni wvero, ramdeni 

waxnagi da ramdeni wibo aqvs mas? am eqvskuTxa piramidas? 

- ramdeniT metia wiboebi, vidre waxnagebi? vidre wveroebi? 

- romelia am waxnagebis saerTo wibo? 

- romelia am wiboebis saerTo wvero? 

sxeulTa saxelwodebebze (paralelepipedi, piramida da sxva) 

yuradRebis gamaxvileba araa saWiro. masw.-i moswavles awvdis 

- 120 -

miTiTebuli geometriuli sxeulis models da ambobs mis 

saxelwodebas. kargia, Tuki moswavleebi daimaxsovreben maT. 

magram es Zaldautaneblad unda moxdes. saxelwodebaTa codna 



am TamaSisas moswavleebs Tvalwin unda hqondeT stereometriuli 

sxeulebi. winadadebaTa nimuSebi: 

1) kubis yvela wibo tolia. 

2) kubis yvela waxnagi tolia. 

3) agureds yovelTvis eqvsi waxnagi aqvs. 

4) piramidas yovelTvis xuTi wvero aqvs. 

5) prizmis yoveli wvero wertilia. 

6) kubis ocive wvero wertilebia. 

7) aguredis yvela wibo monakveTia. 

8) yvela kubi piramidaa. 

9) arcerTi aguredi araa piramida. 

10) kubis zogierTi wibo erTmaneTis mezobelia. 

11) piramidis yvela wibo erTmaneTis mezobelia. 

12) prizmis zogierTi waxnagi erTmaneTis mezobelia. 

13) aguredis yvela waxnagi erTmaneTis mezobelia. 

14) piramidis zogierTi wibo erTmaneTis gadamkveTia. 

15) kubis yvela wibo erTmaneTis gadamkveTia. 

16) kubis yvela waxnagi kvadratebia. 

17) kubis yvela waxnagi marTkuTxedebia. 

18) aguredis yvela waxnagi kvadratebia. 

19) prizmis yvela waxnagi samkuTxedebia. 

20) piramidis yvela waxnagi samkuTxedebia. 

21) piramidis yvela waxnagi marTkuTxedebia. 

22) piramidis erTerTi waxnagi SeiZleba iyos kvadrati. 

22) prizmis erTerTi waxnagi SeiZleba iyos kvadrati. 

23) prizmis erTerTi waxnagi SeiZleba iyos xuTkuTxedi 

! 244/5/!{fqjsj!bohbsjTj!njmjbsefc{f-!pUyjwf!nprnfefcjU/! 

gairCeva amocana 2. Tuki morCeba dro, zepirad gamoiTvleba: 

igive ricxvebi, gayofili asze; TviTeul im ricxvs mimatebuli 

miliard samasi milioni. 

h!245/!nsbwbmo/!sjdywfcjt!hbzpgjt!hbbewjmfcb/!tufsfpnfusjb/! 

- 121 -

245/2/ 20.2. 1 2~ 3~ 4 5 6 7# 

2453/! nsbwbmojTob! sjdywfcjt! hbzpgjt! hbbewjmfcb! –! nyp. 

mpe! vobTUpe;! ! hbzpgjt! bewjmj! TfnUywfwfcj/! magaliTad: 

25600:256; 240000:120 da misT. amgvar SemTxvevebSi sagangebo 

wesebi araa saWiro, ganayofs mivxvdebiT azriT. magaliTad: 

25600:256 = 256 · 100:256; 240000:120 = 240 · 1000:120. 

! 245/4/!UbnbTj;!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj;!!!34211!±! 

2211•/!!ix. 14.3. 

! 245/5/!tufsfpnfusjjt!obtxbwmj!tbljUyfcj!`!hbnfpsfcb/! 

Catardeba iseve, rogorc 133.2. 

h!246/!zpwfmhwbsj!sjdywfcjt!Tfebsfcb/!!bewjmj!hbzpgb/! 

246/2/ 20.3. 1~ 2~ (3 - ara) 4# 5 6 7~ 

246/3/! zpwfmhwbsj! sjdywfcjt! Tfebsfcb/! masw.-is SekiTxvebiT 

gameordeba IV Tavis me-5 gakveTilis teqsti. 




! 246/5/! nsbwbmojTob! sjdywfcjt! hbzpgjt! hbbewjmfcb! .! 

nypmpe! vobTUpe;! ! hbzpgjt! bewjmj! TfnUywfwfcj/! TviTon 

moswavleebma moigonon sazepiro gayofis iseTi magaliTebi, 

romlebSic ganayofis mniSvnelobaa: 2, 3, 10, 100, 1000, 30, 20, 

200, 500. 

h!247/!zpwfmhwbsj!sjdywfcjt!Tfebsfcb/!!bewjmj!hbzpgb/! 

247/2/ 20.4. (1 - ara) 2 3~ 4# 5# 6# 7 

247/3/!zpwfmhwbsj!sjdywfcjt!Tfebsfcb/! 

dawvrilebiT gairCeva amocana 1. msjelobiT dasabuTdes, rom 

amgvar SemTxvevaSi araa saWiro ricxvebis cifrulad Cawera, es 

ufro gaarTulebda Sedarebas. 

! 247/4/!UbnbTj;!kbXwvsbe!.jU!nfuj!!.jU!oblmfcj;!!!:111!`! 

3311•/!ix. 14.3. 

! 247/5/! nsbwbmojTob! sjdywfcjt! hbzpgjt! hbbewjmfcb! .! 

nypmpe! vobTUpe;! ! hbzpgjt! bewjmj! TfnUywfwfcj/! TviTon 

moswavleebma moigonon sazepiro gayofis iseTi magaliTebi, 

romlebSic ganayofis mniSvnelobaa: 2, 3, 10, 100, 1000, 30, 20, 

200, 500. 

h!248!tufsfpnfu/!hpofcbTj!xbsnpeh/!tyfvmUbUwjt/!hbnfpsfcb/! 

- 122 -

248/2/ 20.5. 1# 2# 3# 4 5# 6# 7 

248/3/! tufsfpnfusjjt! tbljUyfcj! hpofcbTj! xbsnpehfojm! 

tyfvmUbUwjt/!! ! dawvrilebiTaa aRwerili _ ix. $ 7. 



! 248/5/! tbfsUp! hbnfpsfcb/! gameordeba is algera-ariTmetikis 


h!249/!2111!njmjbsej!eb!vgsp!ejej!sjdywfcj/!hbnfpsfcb/! 

249/2/ 20.6. (1, 3 da 3 - ara) 4~ 5! 6~ 7~ 

! 249/3/!2111!njmjbsej!eb!vgsp!ejej!sjdywfcj/! 

ra vqnaT 1000 miliardze meti raodenobebisaTvis? Zalian 

mokled (Semdgomi ganmticeba-ganviTarebis gareSe): 


TviTeuli nawili. Semdeg gairCeva amocanebi 1, 2 da 3. 

249/4/!UbnbTj;!hb{pnwb•/! 

ix. 26.3. axla, cxadia, sxva rame gaizomeba. 

! 249/5/! tbfsUp! hbnfpsfcb/! gameordeba is algera-ariTmetikis 



1. TiTo meqanikuri SecdomisaTvis udides 4 qulas akldeba 

TiTo q. xolo moqmedebaTa rigis darRvevis TiTo SecdomisaTvis 

- 2 q. 


3. Tuki drois sxvagvari Canaweri mxolod erTia da ara 

ori - iwereba 2 q. Tuki garkveulia mxolod dila-saRamo - 1 q. 

drois Caweris TiTo SecdomisaTvis udides 4 qulas akldeba 

TiTo q. 

4. pasuxebi: I rigi: I. yoveli; II. zogierTi; III. zogierTi 

(an: yoveli); IV. arcerTi. 

II rigi: I. zogierTi (an: yoveli); II. zogierTi; III. 

yoveli; IV. arcerTi. TiTo am pasuxSi iwereba TiTo q, sul 

- 4 q. 

5. amocana Znelia, amitom 6-quliania. pasuxebi: I rigi: 6jer; 

15-jer; II rigi: 5-jer; 15-jer. 

marTebulisagan moswavlis pasuxebis gansxvavebaTa mixedviT 

udides 6 qulas akldeba: 

- 123 -

1 - ar akldeba; 2-3 - akldeba 1 q; 4-5 - akldeba 2 q; 6-9 - 

akldeba 3 q; 10-14 - akldeba 4 q; 15-19 - akldeba 5 q; 20 

an meti - akldeba 6 q. 

cxadia, Tuki orive pasuxi mcdaria, qulebis saTanado daklebac 

orjer moxdeba. amitom, Tuki, magaliTad, orive pasuxi 1iTaa 

gansxvavebuli marTebulisagan, maSin mainc daakldeba 1 q. 

ese igi, orive pasuxis gansxvavebani unda Seikribos! 

magram Tuki moswavles ar amouweria saTanado ricxvebi 

dasasabuTeblad, maSin mas pirdapir ewereba 0 q. 

Tavi XXI 

h!251/!bUfvm.btfvm.bUbt/!sbpefopcb/![bmjbo!ejej!sjdywfcj/! 

251/2/ 20.6. 1~ 2# 3! 4~ 5# 6# 7# 

21.1. (1 da 2 - ara) 3# 4~ 5# 6~ 7 

! 251/3/! hbnfpsfcb;! sjdywTj! bUfvmUb-! btfvmUb-! bUbtfvmUb! 

sbpefopcb/! saguldagulod gairCeva amocana 1 (Semdegi 

gakveTilis Semamzadebeli). masw.-is mier dasaxelebul ramdenime 

ricxvSi gairkveva aTeulTa, aseulTa, aTaseulTa raodenoba. 



masw.-s ar unda gamouvides orazrovani winadadebani, magaliTad: 

`rac ufro didia prizma, miT ufro grZeli wiboebi aqvs mas~. 

es winadadeba, sazogadod, mcdaria, Tumca, Tuki vigulisxmebiT 

msgavs prizmebs, maSin marTebuli iqneba. amitom masw.-ma zustad 

unda Camoayalibos winadadebebi: 

1. rac ufro didia kubi, miT ufro didi wveroebi aqvs mas; 

2. rac ufro didia kubi, miT ufro meti raodenobis wiboebi aqvs mas; 

3. rac ufro mcirea kubi, miT ufro mokle wiboebi aqvs mas; 

4. rac ufro mcirea aguredi, miT ufro naklebi raodenobis 

wiboebi aqvs mas; 

5. rac ufro didia kubi, miT ufro mokle wiboebi aqvs mas; 

6. rac ufro didia kubi, miT ufro didi waxnagebi aqvs mas; 

7. rac ufro mcirea kubi, miT ufro naklebi raodenobis wveroebi 

aqvs mas; 

8. rac ufro mcirea kubi, miT ufro naklebi raodenobis wiboebi 

aqvs mas; 

9. rac ufro didia aguredi, miT ufro mcire waxnagebi aqvs mas; 

- 124 -

10. rac ufro didia kubi, miT ufro grZeli wiboebi aqvs mas; 

11. rac ufro didia aguredi, miT ufro meti raodenobis wveroebi 

aqvs mas; 

12. rac ufro mcirea aguredi, miT ufro naklebi raodenobis 


13. rac ufro didia aguredi, miT ufro mcire waxnagebi aqvs mas; - 

14. rac ufro didia piramida, miT ufro meti raodenobis wveroebi 

aqvs mas; 

15. rac ufro mcirea piramida, miT ufro naklebi raodenobis 


16. rac ufro didia piramida, miT ufro meti raodenobis waxnagebi 

aqvs mas; 

17. rac ufro didia prizma, miT ufro meti raodenobis wveroebi 

aqvs mas; 

18. rac ufro mcirea prizma, miT ufro naklebi raodenobis 

waxnagebi aqvs mas; 

19. rac ufro didia prizma, miT ufro meti raodenobis wiboebi 

aqvs mas... 

!!!251/5/!2111!njmjbsej!eb!vgsp!ejej!sjdywfcj/!gameordeba 138.2. 

h!252/!hbobzpgj!sbnefoojTob!voeb!njwjSpU/! 

252/2/ 21.2. (1 - ara) 2# 3~ 4# 5~ 6 7 

! 252/3/!UbnbTj;!txsbgj!qbtvyfcj•!`!{fqjsj!bohbsjTj/!ix. 32.3. 

! 252/4/!hbobzpgj!sbnefoojTob!voeb!njwjSpU/! 



h!253/!hbobzpgj!sbnefoojTob!voeb!njwjSpU/!tufsfpnfusjb/! 

253/2/ 21.3. (1 - ara) 2~ 3# 4# 5# 6# 7 




! 253/4/!UbnbTj;!nbUfnbujlvsj!tjuzwfcjt!ebtbyfmfcb•/!ix. 106.3. 

! 253/5/! tufsfpnfusjjt! tbljUyfcj! hpofcbTj! xbsnpehfojm! 

tyfvmUbUwjt/!Catardeba iseve, rogorc 137.2. 

h!254/!hbobzpgj!sbnefoojTob!voeb!njwjSpU/!tufsfpnfusjb/! 

254/2/ 21.4. (1 da 2 - ara) 3 4~ 5# 6 7~ 


- 125 -


TviTeuli nawili. Semdeg gairCeva amocanebi 1 da 2. 

! 254/4/!UbnbTj;!txsbgj!qbtvyfcj•!.! hbobzpgj!sbnefoojTob!vo. 

eb!njwjSpU/!moswavleebs rig-rigobiT eZleva advili gasarkvevi 

magaliTebi: 1312:5, 367:9, 8672:4, 6700:6 da ase Semdeg. 



h!255/!tufsfpnfusjb/!tbfsUp!hbnfpsfcb/! 

255/2/ 21.5. 1 2# 3~ 4# 5 6 7# 



! 255/4/! UbnbTj;! txsbgj! qbtvyfcj•! .! lbmfoebsj/! klasSi unda 


rig-rigobiT ekiTxeba: - ra dRea 5 maisi? ra ricxvia 


! 255/5/! zwfmb! obtxbwmj! nprnfefcjt! hbonuljdfcb! eb! 

hbowjUbsfcb/! gansakuTrebuli yuradReba eqceva swrafi zepiri 

angariSis unarCvevebs. 


h!256/!tbfsUp!hbnfpsfcb/! 

256/2/ 21.6. 1~ 2 3! 4 5# 6# 7 

! 256/3/! UbnbTj;! txsbgj! qbtvyfcj•! .! hbobzpgj! sbnefoojTob! 

voeb! njwjSpU/! moswavleebs rig-rigobiT eZleva advili gasarkvevi 

magaliTebi: 1312:5, 367:9, 8672:4, 6700:6 da a.S. 

! 256/4/! zwfmb! obtxbwmj! nprnfefcjt! hbonuljdfcb! eb! hbowj. 

Ubsfcb/! gansakuTrebuli yuradReba eqceva swrafi zepiri 

angariSis unarCvevebs. 



1. TiTo meqanikuri SecdomisaTvis udides 4 qulas akldeba 

TiTo q. xolo moqmedebaTa rigis darRvevis TiTo SecdomisaTvis 

- 2 q. 


3. pasuxebi: I rigi: ricxvs rig-rigobiT emateba an 40000, 

an 30000. 

II rigi: ricxvs rig-rigobiT akldeba an 5000, an 2000. 

- 126 -

Tuki mwkrivi kanonzomieradaa gagrZelebuli, magram aklia 

kanonzomierebis sityvieri Camoyalibeba - iwereba 3 q; Tuki 

ZiriTadad Camoyalibebulia (Tundac cota naklovanad) - 4 q; 

xolo gamoTvlebis TiTo SecdomisaTvis udides 4 qulas 


4. ganvixiloT I rigis amocana. gasayofSi moTavsdeba 

ramdeniRac 9-euli da naSTi rCeba 7. orjer 9 aris 18. amitom 

es ramdeniRac 9-euli SeiZleba zustad daemTxves 18-eulebs da 

maSin naSTi iqneba 0. SeiZleba, ar daemTxves da maSin naSTi 

iqneba 7+9 = 16. amitom pasuxia 0 an 16. 

aseve II rigSi pasuxia: 0 an 11 (radgan 3+8 = 11). 

Tuki pasuxi marTebulia da Tan dasabuTebis magvari 

msjelobaa Catarebuli (Tundac nakluli) - iwereba udidesi 5 

q. Tuki pasuxi marTebulia, magram msjelobaa araa - 4 q. Tuki 

pasuxSi mxolod erTi Secdomaa da dasabuTebis magvari 

msjelobaa Catarebuli (Tundac nakluli) - 4 q. Tuki pasuxSi 

mxolod erTi Secdomaa da msjeloba ki araa - 3 q. Tuki 

dawyebulia marTebuli msjelobis magvari ram - 1 q. Tuki weria 

erTerTi marTebuli pasuxi (àn~-is gareSe) - 2 q. 

5. saWiro moqmedebani: I rigi: 1) 150:15 = 10, 150:10 = 15; 

ricxvebi isea SerCeuli, rom gayofis gareSec, zepiradac 

SeiZleba daskvna, rom pirveli mRebavi erT dReSi SeRebavs 10 

CarCos, meore - 15-s; 2) 10 + 15 = 25; 3) 150:25 = 6. 

asevea II rigis amocanac. 

amocanis amoxsna Sedgeba msjelobisa da 4 moqmedebisagan, 

TiToSi iwereba TiTo q, sul - 5 q. 

258/2/ 21.7. 1# 2~ 3~ 4# 5 6~ 7 

7. I-IV klasebis Semajamebeli programa 

(programul moTxovnaTa dakonkreteba codnisa Tu unarCvevaTa 

mixedviT mocemulia sagamocdo amocanebis nimuSebis saxiT) 

1. bsjUnfujlb 

1.1. jufTebi da jufTebiT Tvla. raodenobis warmodgena 

jufTebad da naSTad. 

1.2. raodenobaTa Sedareba dawyvilebiT. 

1.3. raodenobis miaxloebiTi Sefaseba TvaliT. 

1.4. Tvla da ukuTvla. Tanrigebi. ricxvebis saxelwodebani 

- 127 -

da cifruli Canawerebi _ asobiT miliardis farglebSi. 

1.5. ricxvTa Sedareba, utolobani. 

10 ricxvis dalageba matebiT an klebiT. 

1.6. ormxrivi utolobebi, ormxrivi an calmxrivi utolobis 

saWiro niSnebis dasma miaxloebiTi Sefasebis safuZvelze. 

1.7. rigobiTi ricxviTi saxelebi 1000-is farglebSi. Tanmimdevroba. 

Soris. amocanebi Tanmimdevrobaze (rigobiTi ricxviTi 

saxelebiT), roca mniSvneloba aqvs pirvelisa Tu ukanasknelis 

CaTvla-arCaTvlas. 

1.8. I-dan XXX-mde romauli ricxvebis Cawera da wakiTxva. 

1.9. ricxvebis Tu sidideTa gadideba-Semcireba `-iT~ an `-jer~. 

1.10. kenti da luwi ricxvebi (cnebebis CaTvliT). 

1.11. ricxvis cifrTa jami. 

ariTmetikuli moqmedebani zepiri gamoangariSebiT: 

1.12. Sekreba-gamokleba 100-is farglebSi (gamoklebis Zneli 

SemTxvevebis garda). 

1.13. gamravlebisa da gayofis cxrili (9×9). 

1.14. 1000-is daSla Sesakrebebad. 

1.15. zepiri Sekreba, gamokleba, gamravleba da gayofa 1000is 

farglebSi (garda Zneli SemTxvevebisa). 

1.16. rva aTass, orasi aTass, sam milions da misT. gamoklebuli 

erTniSna ricxvi an ramdenime aTeuli. 

1.17. gamravleba-gayofa 10-ze; 100-ze da 1000-ze. 

1.18. naSTiani gayofa _ 100-is farglebSi. 

1.19. iseTi SemTxvevebi, romlebic erTi martivi moqmedebiT 

daiyvaneba CamoTvlilebze (1.2-1.8). 

1.20. martivi (ormoqmedebiani), magram didricxvebiani ariTmetikuli 

gamosaxulebis mniSvnelobis miaxloebiTi Sefaseba. 

1.21. ricxvis warmodgena mis saTanrigo SesakrebTa jamis 

saxiT da piriqiT, ricxvis aRdgena am jamis mixedviT (aTi milionis 

farglebSi). 

1.22. ricxvebSi aTeulebis, aseulebisa da aTaseulebis saerTo 

raodenobis garkveva. 

ariTmetikuli moqmedebani weriTi gamoangariSebiT: 

1.23. miliardis farglebSi: qveSmiweriT Sekreba, gamokleba 

da gamravleba. qveSmiweriT gayofa erTniSna ricxvze da, advil 

SemTxvevebSi, orniSna ricxvze. 

- 128 -

1.24. jufTebad Caweril raodenobaTa Sedareba, Sekreba da 

gamokleba. 

1.25. frCxilebian gamosaxulebaTa mniSvnelobis gamoTvla, 

oTxive ariTmetikuli moqmedebiT, mxolod erTmagi frCxilebiT, 

1000-is farglebSi. 

2. bmhfcsb!eb!npefmjsfcb 

2.1. terminebi: Sesakrebi, Tanamamravli, saklebi, maklebi, ganayofi, 

gamyofi, toloba, utoloba, metoba, nakleboba. 

2.2. oTxive ariTmetikuli moqmedebis Sedegebis Semowmeba sapirispiro 

moqmedebiT. 

2.3. ucnobi Sesakrebis, Tanamamravlis, saklebis, maklebis, ganayofis 

an gamyofis povna. Sekreba-gamoklebisa da gamravlebagayofis 

komponentTa Soris mimarTebani (mxolod praqtikulad). 

2.4. umartives (2.3-is Sesabamis) gantolebaTa amoxsna. 

2.5. moqmedebaTa Tanmimdevroba ariTmetikul gamosaxulebaSi. 

frCxilebis xmareba. sityvierad mocemuli ori moqmedebis 

Sesabamisi gamosaxulebis Cawera frCxilebiT an ufrCxilebod; 

da piruku: dawerili gamosaxulebis zusti sityvieri aRwera. 

2.6. ariTmetikul moqmedebaTa umartivesi algebruli Tvisebani: 

SesakrebTa gadanacvlebadoba da dajgufebadoba; TanamamravlTa 

gadanacvlebadoba; Sekrebisa da gamoklebis sapirispiroba; 

gamravlebisa da gayofis sapirispiroba; ganrigebadoba: 

am wesebis mxolod praqtikuli gamoyeneba _ ìangariSeT 

advili xerxiT~: 

a) a+b = b+a b) (a + b) + c = a + (b + c) = a + b + c 

g) a · b = b · a d) (a ± b) · c = a · c ± b · c 

e) a – (b + c + d) = a – b – c – d 

v) a · (b · c)=(a · b) · c = a · b · c 

z) a + (b–c) = a+b–c T) a : (b · c) = a : b : c 

i) (a+b) : c = a : c + b : c k) frCxilebs gareT gatana. 

2.7. sxvadasxva SesaZleblobiT: ricxvis mimateba an gamokleba 

ori moqmedebiT. maTgan yvelaze advilis arCeva: `nawilnawil~ 

Sekreba-gamoklebis xerxebi. zepiri angariSis gaadvilebis 

sxva umartivesi xerxebi. 

2.8. mravalmoqmedebiani gamoTvlebi (ricxvTa kibeze an zepirad). 

ormagi da sammagi tolobani. Sekreba-gamoklebisa da 

- 129 -

gamravleba-gayofis mravalmoqmedebiani cxrilebi. 

2.9. rogor icvleba jami, namravli, sxvaoba Tu ganayofi 

erTerTi komponentis gadidebiT an SemcirebiT. 

2.10. sayofacxovrebo amocanebi: yvela tipis ormoqmedebianni 

da saSualo siZnelis sammoqmedebianni. 

2.11. sayofacxovrebo amocanis pirobis, amoxsnisa da pasuxis 

mokle Cawera. 

2.12. sayofacxovrebo amocanis pirobis dakvirvebiT gaazreba. 

amocanis pirobaSi xarvezis aRmoCena. nakluli pirobis Sevseba. 

amocanisaTvis SekiTxvebis dasma. ormoqmedebiani amocanis Sedgena 

mocemuli naxatis, naxazis, sqemis, cxrilisa Tu ariTmetikul 

moqmedebaTa mixedviT. 

sayofacxovrebo amocanebis tipebi: 

I. sidideTa ori sxvadasxva wyviliT, mag.: wonis ori 

erTeuliT da fulis ori erTeuliT, sammoqmedebiani; 

II. raodenobebze (-iT/-jer meti/naklebi), romelSic ramdenime 

nawilis da mTlianis mixedviT mosaZebnia darCenili nawili; 

nimuSi: erTi _ x, meore _ x – 3, mesame _ 2x, sul _ a. 

ramdenia meoTxe? (aq asoebi mxolod masw.-isTvisaa!) 

III. tol nawilebad dayofaze, sammoqmedebiani; 

IV. sxva tipisa, am sami tipis amocanebis tolfasi siZnelisa. 

3. hfpnfusjb-!upqpmphjb-!!yb{wb! 

3.1. nakvTebi da maTi klasifikacia (dajgufebani): wertili, 

xazebi, monakveTi, Sekruli Tu gaxsnili mrudi, texili, 

samkuTxedi, oTxkuTxedi, da a.S. mravalkuTxedi, kvadrati, 

marTkuTxedi, wre, wrexazi. 

3.2. ori wertilis SemaerTebeli xazi; xazebis gadakveTis 

wertili. 

3.3. mravalkuTxedTa saerTo gverdebi Tu wveroebi, mezobeli 

da mopirdapire wveroebi Tu gverdebi. 

3.4. mravalkuTxedis (10-12-kuTxedi, maT Soris Cazneqili, 

varskvlaviseburi) wveroebisa da gverdebis raodenoba; 

wveroebis raodenobisa da gverdebis raodenobis toloba. 

3.5. kubi; sxva mravalwaxnaga sxeulebi (aguredi (anu 

marTkuTxa paralelepipedi), piramida, prizma). 

3.6. mravalwaxnaga sxeulis waxnagebi, wveroebi da wiboebi. 

- 130 -

wiboTa saerTo wvero da waxnagTa saerTo wibo. 

3.7. mravalwaxnaga sxeulis wveroebis, wiboebisa da 

waxnagebis raodenoba (maT Soris martivi warmosaxuli sxeulebisTvisac). 

3.8. SigniT, gareT, sazRvarze. xazebi, maTi gadakveTebi, Siga da 

gare areebi, nakvTis sazRvari. nakvTebisa Tu areebis mosazRvreoba. 

3.9. nakvTebis toloba, toli da aratoli nakvTebi. 

3.10. monakveTis, wertilis, texilis, mravalkuTxedis asoiTi 

aRniSvnebi. 

3.11. mocemuli sigrZis gverdis (gverdebis) mqone nakvTebis 

xazva _ 3.1-Si CamoTvlilebisa; wrexazis daxazva fargliT. 

4/!hbnpzfofcjUj!nbUfnbujlb-!tjejeffcj!eb!hb{pnwb! 

4.1. sigrZe da gazomva. sigrZis erTeulebi da maT Soris 

Tanafardoba, maTi urTierTgadayvana: nn, tn, en, n da ln. 

4.2. texilis sigrZe. mrude xazis miaxloebiTi sigrZe. 

4.3. sigrZis miaxloebiTi Sefaseba TvalzomiT. 

4.4. marTkuTxedis sigrZe da sigane. 

4.5. mravalkuTxedis perimetri. 

4.6. manZilebis gazomva ori erTeuliT da ori erTeuliT 

mocemuli sigrZis mqone monakveTebis xazva. 

4.7. fulis erTeulebi da maT Soris Tanafardoba: `TeTri~ 

da `lari~. advili amocanebi fasebze. 

4.8. awona. wonis erTeulebi da maT Soris Tanafardoba, 

maTi urTierTgadayvana: 1 u = 1000 lh, 1 lh = 1000 h. 

4.9. drois erTeulebi da maT Soris Tanafardoba, maTi ur- 

TierTgadayvana: 1 weliwadi = 12 Tve, 

1 kvira = 7 dRe-Rame, 1 dRe-Rame = 24 sT, 1 sT = 60 wT. 

4.10. saaTis cnoba (maT Soris, eleqtruli saaTisac) da 

Sesabamisi gamoTvlebi. 

4.11. kalendris xmareba. 

4.12. sivrciT-droiTi mimarTebani; Tanmimdevroba sivrcesa da 

droSi. gadanomrva. 

4.13. sxvadasxvagvari cxrilebi. 

4.14. martivi sqemebis daxazva, maT mixedviT moqmedeba. 

4.15. monacemebis sistematizacia da warmodgena siis an 

cxrilis saxiT; siidan an cxrilidan monacemebis amokiTxva. 

- 131 -

4.16. dasaxelebuli gverdisa Tu amocanis moZebna saxelmZRvaneloSi: 

paragrafis nomrisa da amocanis nomris, an romauli 

nomris mixedviT. 

5. tjnsbwmfUb!Ufpsjb-!lpncjobupsjlb-! 

mphjlb!eb!lbopo{pnjfsfcfcj 

5.1. praqtikulad _ logikuri terminebi: 

Uvlj-!spdb-!nbTjo-!eb-!bo-!zpwfmj-!fsUfsUj-!fsUbefsUj-!bsdfsUj-! 

{phjfsUj-!nypmpe-!UwjUfvmj-!bhsfUwf-!sbehbobd-!sblj-!jnjupn-! 

bnjupn-!ftf!jhj-!jnefojwf-!sbnefojd-!spnfmjnf-!spnfmjd-!tbfsUp-! 

jhjwf-! fsUjebjnbwf-! Tftb[mfcmpcb-! Tftbcbnjtj-! hbombhfcb-! 

Ubonjnefwspcb, lbopo{pnjfsfcb... 

5.2. simravleTa Teoriis sawyisebi: ori TvalsaCino erTobliobis 

TanakveTis, gaerTianebisa Tu sxvaobis moZebna; erTobliobis 

qvesimravlis moZebna (mxolod kargad nacnobi da konkretuli 

masaliT). 

5.3. kombinatorika: kargad nacnob da konkretul viTarebebSi _ 

yvela SesaZlo SesaZleblobis (kombinaciebis) Camowera, roca maTi 

raodenoba ar aRemateba 16-s (gameorebebiT an gameorebaTa gareSe). 

yvela SesaZleblobis amowurva _ ariTmetikul, geometriul, 

grafikul, enobriv Tu sayofacxovrebo masalaze. 

5.4. dajgufeba (klasifikacia). sagnebis erTobliobaSi saer- 

To niSanTvisebis aRmoCena da umartivesi ganzogadeba. 

5.5. praqtikulad: ormimarTebiani konkretuli kanonzomierebebi. 

logikuri kanonzomierebis aRmoCena: geometriul, ariTmetikul, 

grafikul Tu sayofacxovrebo (maT Soris sityvierad 

mocemul) masalaze. 

5.6. logikuri sqemebi da logikuri cxrilebi. 

5.7. umartivesi logikuri mimarTebani da logikuri msjeloba. 

8. zepiri Semajamebeli gamokiTxva 

IV klasis bolos sasurvelia, Catardes oTxi klasis Semajamebeli 

gamokiTxva (V klasidan xom ukve sxvanair maTematikas viwyebT). 

erTi-ori wina kvira ki daeTmoba ZiriTadi sakiTxebis 

gameorebas. 

Zalian mniSvnelovania gamokiTxvis zepiri nawili, radgan 

pirveli oTxi klasis maTematikaSi mTavaria zepiri 

angariSis, zepiri Sefasebis, sivrciTi warmodgenisa da martivi 

- 132 -

zepiri msjelobis unarCvevebi. yvela amgvar sakiTxs amowuravs 

gamokiTxvis programa. igi yvelaze kargad da konkretulad 

bileTebis nimuSebSi gamoCndeba. 

sasurvelia, es zepiri gamokiTxva calke dRes, gamocdasaviT 

Catardes _ moswavleebic mieCvevian sagamocdo viTarebas da 

gamokiTxvis xarisxic ukeTesi iqneba. am SemTxvevaSi moswavlis 

pasuxi Seafasos erTma-orma sxva masw.-mac. TviTeulma maTganma 

erTmaneTisagan damoukideblad dauweros moswavles qulebi 

(bileTis TiTo sakiTxSi 0-dan 4-mde), Semdeg es qulebi 

Sejamdes _ erTad yvela. am qulebs unda daematos weriTi 

naSromis (ukanaskneli, Semajamebeli sakontrolo weris) 

qulebi; saboloo niSani gamoiyvaneba am mTlian-jamuri qulisa 

da wliuri niSnis SejerebiT. saqme isaa, rom, mTlianobaSi, 

Semajamebeli gamokiTxvisa da Semajamebeli sakontrolo weris 

davalebebi sakmaod advilia. Cveulebriv sakontrolo werebze 

klass xSirad ufro Zneli amocanebi eZleoda. swored amitomaa, 

rom moswavlis saboloo niSani unda gamovTvaloT wliur 

niSanTan SejerebiT _ romelSic mTavari wili sakontrolo 

werebis niSnebs unda hqondes! 

magram mTavari araa Sefaseba da niSani. mTavaria oTx klasSi 

naswavlis Semajamebeli gameoreba-ganmtkiceba-gaaqtiureba da 

moswavlis codnaSi arsebuli xarvezebis gamovlena. 

ukanaskneli, Semajamebeli sakontrolo weris amocanebi Cveulebrivadaa 

miTiTebuli sakontrolo werebis amocanebis nusxis 

bolos. amocanebis raodenoba gacilebiT metia, vidre Cveulebriv 

(Tumca isini gacilebiT advilebia). moswavleebs samuSaod ori 

gakveTili unda mieceT, nel moswavleebs dro kidev davumatoT. 

zepiri Semajamebeli gamokiTxvis bileTebis saerTo raodenoba 

unda iyos 14-15 an meti (roca moswavleebis erTi wyeba 

daamTavrebs, isev igive bileTebi SeiZleba datrialdes). moswavles 

winaswari mofiqrebisaTvis sul ramdenime wuTi eyofa, amitom 

klasSi erTdroulad mxolod ori-sami moswavle ijdes; garda 

amisa, moswavleebi davyoT xuTeulebad da skolaSi davibaroT 

sxvadasxva dros, winaswar Sedgenili ganrigis mixedviT. 

bileTebis sxvadasxva variantebi (maT Soris weriTi 

gamocdisac) am meTodikuri saxelmZRvanelos CanarTzea. 

aq ki sanimuSod mocemulia mxolod erTi varianti. 

- 133 -

ileTi # ... 

{ figurul frCxilebSi mocemuli SekiTxvebi moswavlis 

bileTSi ar Sedis, magram maT zepirad dasvams gamomcdeli; sxva 

damatebiTi SekiTxvebi aRaraa saWiro! } 

1. 6890347; 378208; 9605893022; XVIII; XXI. 

Tuki moswavle romaul cifrebs waikiTxavs raodenobriv 

ricxviT saxelebad, maSin gamomcdeli mosTxovs, rom waikiTxos 

rigobrivi ricxviTi saxelebis magvarad! 

2. I. ramdeni 200-eulia 1000-Si? 

II. ramdeni aTeulia 3203-Si? aseuli? 

III. ramdeni rvaeulia 60-Si da risi tolia naSTi? 

3. upasuxeT swrafad _ risi tolia: 7 · 8 ; 63 : 9 ; 

17 aTass gamoklebuli 8 aTasi; 

Svid milions mimatebuli eqvsi milioni. 

aqaa cxriluri gamravleba-gayofa da 20-is farglebSi Sekrebagamokleba, 

oRond aTaseulebze da milionebze. gamomcdeli 

damatebiT zepirad dasvams erT SekiTxvas gamravlebaze da erTs 

_ gayofaze: kvlav 9×9 cxrilis farglebSi. 

4. iangariSeT, ramdeni aklia 1000-mde Semdeg ricxvebs: 

400, 220, 982, 366 . 

5. iangariSeT: I. 25 + 35 II. 3001 + 157 III. 60 – 36 

IV. 7800 – 3000 V. rva aTass gamoklebuli sami. 

VI. ormoci aTasi xuTjer. 

6. dauTvlelad gaarkvieT, romelia meti da ramdeniT: 

{ _ ratom? saidan ici? } _ moswavlem unda gaiazros 

dawyvileba; { _ rogor ggonia, daaxloebiT ramdeni 

patara nakvTia aq sul daxazuli? swrafad gvipasuxe! } _ 

yvelgan daaxloebiT 50-90 nakvTia) 

7. iangariSeT: I. 150 · 3 II. 60 : 15 III. 36 · 100 

IV. 56000 : 100 V. xuTasi aTasi xuTjer. 

8. I. 17345 + 1887 = ? a) 204232; b) 19232; 

g) 11342; d) 8355; e) 30000. 

- 134 -

II. angariSis gareSe moifiqreT, romelia meti: 

325 + 1751 Tu 2745 : 9. 

9. iangariSeT advili xerxiT: I. 377 + 299 

II. 4 · 17 · 25 III. 230 + (95 + 70) IV. 7 · 112 _ 7 · 12 

10. vTqvaT, eleqtruli saaTi aCvenebs dros 21.05. 

dilaa Tu saRamo? meorenairad rogor vityviT, 

romeli saaTia? 

{ _ ra dros aCvenebs es saaTi? _ vaCvenebT saaTs } 

11. kalendris mixedviT gaarkvieT, kviris ra dRea ama wlis 

3.04. { _ ra ricxvi iqneba Semdegi [wina] `SabaTi~? 

(romeli dRec moswavlem daasaxela); ra ricxvia `seqtembris~ 

(imave Tvis) pirveli [ukanaskneli] SabaTi? } 

12. daakvirdiT am samkuTxa prizmas. ramdeni wvero, ramdeni 

waxnagi da ramdeni wibo aqvs mas? 

xuTkuTxa prizmas? 

moswavles vawvdiT bileTSi miTiTebuli pirveli 

geometriuli sxeulis models. xolo meore man gonebaSi unda 

warmoidginos da ase upasuxos SekiTxvas. 

{ romelia am waxnagebis saerTo wibo? } 

{ romelia am wiboebis saerTo wvero? } 

{ daakvirdi aq daxazul 

marjvena [variantSi _ 

marcxena] mravalkuTxeds. 

daTvale, ramdeni gverdi 

aqvs mas. ramdeni wvero aqvs mas? } (meore SekiTxvaze 

unda upasuxos swrafad, xelmeored daTvlis gareSe) 

13. giorgim wignSi waikiTxa moTxroba, romelic iwyeboda 

me-5 gverdis dasawyisidan da mTavrdeboda 128-e 

gverdis boloSi. 

ramdengverdiani moTxroba waukiTxavs giorgis? 

moswavles SeuZlia, Caixedos romelime wignSi, raTa 

daakvirdes gverdebis numeracias. { _ saidan ici? }. 

- 135 -

nbUfnbujljt !txbwmfcb!!JW!lmbtTj! - Ganatleba

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