nbUfnbujljt !txbwmfcb!!JW!lmbtTj! - Ganatleba

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nbUfnbujljt !txbwmfcb!!JW!lmbtTj! - Ganatleba

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!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.

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

ricxvSi erTeulebi, aTeulebi, aseulebi, ricxvSi maTi raodenoba.

samniSna ricxvis akrefa saangariSoze.

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

ricxvSi erTeulebi, aTeulebi, aseulebi, ricxvSi maTi raodenoba.

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 `akrefa~

saangariSoze.

8. oTxniSna ricxvebi. oTxniSna ricxvis wakiTxva. saangariSoze

mocemuli raodenobis sityvieri dasaxeleba da cifruli Cawera

dafaze. piriqiT: sityvierad an cifrulad mocemuli ricxvis `akrefa~

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 `akrefa~ 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:

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

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 `akrefa~ saangariSoze.

28. 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 `akrefa~ saangariSoze.

gameoreba: samniSna ricxvebis Sedareba.

saangariSoze mocemul raodenobaTa Sedareba aTasis farglebSi.

29. 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 `akrefa~ saangariSoze.

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.

frCxilebis gaxsnis wesis gamoyeneba gamravlebisas.

47. SesaZleblobaTa amowera.

frCxilebis gaxsnis wesis gamoyeneba gamravlebisas.

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.

54. gayofisa da Sekrebis wesi _ frCxilebis gaxsna:

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

frCxilebis gaxsnis wesis gamoyeneba gayofisas.

55. gayofisa da Sekrebis wesi _ frCxilebis gaxsna:

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

frCxilebis gaxsnis wesis gamoyeneba gayofisas.

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

frCxilebis gaxsnis wesis gamoyeneba gayofisas.

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.

74. mravalniSna ricxvebis qveSmiweriT gamravleba.

gayofisa da gamoklebis wesi _ frCxilebis gaxsna:

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

75. mravalniSna ricxvebis qveSmiweriT gamravleba.

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 `akrefa~ saangariSoze.

- 15 -


79. ricxvebi milionidan miliardamde. ricxvebis wakiTxva da Cawera.

saangariSoze mocemuli raodenobis sityvieri dasaxeleba da

cifruli Cawera dafaze. piriqiT: sityvierad an cifrulad mocemuli

ricxvis `akrefa~ saangariSoze.

80. ricxvebi milionidan miliardamde. ricxvebis wakiTxva da Cawera.

saangariSoze mocemuli raodenobis sityvieri dasaxeleba da

cifruli Cawera dafaze. piriqiT: sityvierad an cifrulad mocemuli

ricxvis `akrefa~ saangariSoze.

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.

qveSmiweriT gamravlebis gaadvilebis xerxebi.

88. rigobrivi ricxvebis Cawera romauli niSnebiT.

89. mravalkuTxedi.

ricxvebis Sedareba miliardis farglebSi. saangariSoze mocemul

raodenobaTa Sedareba miliardis farglebSi.

90. rigobrivi ricxvebis marTlwera.

qveSmiweriT gamravlebis gaadvilebis xerxebi.

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.

gameoreba _ naSTiani gayofa Zneli SemTxvevebiTurT.

- 16 -


94. naxevari saaTi = 30 wuTi. saaTis cnoba.

gameoreba _ naSTiani gayofa Zneli SemTxvevebiTurT.

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.

mravalniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze.

110. gayofisa da gamravlebis wesi _ frCxilebis gaxsna:

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

mravalniSna ricxvis qveSmiweriT gayofa erTniSna ricxvze.

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.

115. gayofa advili xerxiT. nulebiT daboloebuli mravalniSna

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

SemTxvevebic.

116. gayofa advili xerxiT. nulebiT daboloebuli mravalniSna

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

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.

mravalniSna ricxvebis gayofis gaadvileba _ mxolod unaSTod:

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.

qveSmiweriT gayofa samniSna ricxvze.

131. zepiri angariSi miliardebze, oTxive moqmedebiT.

qveSmiweriT gayofa samniSna ricxvze.

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.

mravalniSna ricxvebis gayofis gaadvileba _ mxolod unaSTod:

gayofis advili SemTxvevebi.

136. yovelgvari ricxvebis Sedareba.

mravalniSna ricxvebis gayofis gaadvileba _ mxolod unaSTod:

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.

142. ganayofi ramdenniSna unda miviRoT.

stereometriis sakiTxebi gonebaSi warmodgenil sxeulTaTvis.

143. ganayofi ramdenniSna unda miviRoT.

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.

im sakiTxebis gameoreba, romlebic klass uWirs.

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 `arebis~) 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 `oras~.

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

araa aucilebeli.

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

`ormoci~, 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 `ariTmetikuli~ 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: `oceulebi~ meoreSi weria: `naSTi~. masw.-i maCvenebeli

joxis wveriT akakunebs am ujredebSi, moswavlem ki unda

gamoicnos Sesabamisi ricxvi. magaliTad:

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

{ocdasami}; arcerTxel `oceulebSi~ 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 `ariTmetikuli~ 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.


100 + 5·

10 + 6

+


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

akldeba TiTo q.

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 `aveji~). 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

TviTeuli nawili.

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.

32/5/! hbnsbwmfcjt! xftfcj;! UbobnbnsbwmUb! hbebobdwmfcb! eb!

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

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

swrafad gameordeba am teqstis ZiriTadi sakiTxebi.

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

33/5/! hbnsbwmfcjt! xftfcj;! UbobnbnsbwmUb! hbebobdwmfcb! eb!

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.

35/!tblpouspmp!xfsb!$!4/!

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;!

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

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;!

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

- 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

gadaSlili saxelmZRvanelo, me-3 gakveTilis teqstze (romelic

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

am teqstis TviTeuli nawili.

teqstSi mocemuli yvela ricxvi unda aikrifos saangariSoze!

Semdeg gairCeva amocanebi 1 da 2.

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!

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

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

gadaSlili saxelmZRvanelo, me-4 gakveTilis teqstze (romelic

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

am teqstis TviTeuli nawili.

teqstSi mocemuli yvela ricxvi unda aikrifos saangariSoze!

Semdeg gairCeva amocanebi 1 da 2.

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!

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

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

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/!

moswavleebs win unda edoT gadaSlili saxelmZRvanelo, me-5

gakveTilis teqstze (romelic Sin unda hqondeT wakiTxuli).

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.

41/!tblpouspmp!xfsb!$!5/!

2/ udides 4 qulas TiTo SecdomisTvis akldeba TiTo q.

3/ udides 4 qulas TiTo SecdomisTvis akldeba TiTo q.

4/ 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.

+


100

+


100


100

+

+


10

7 · 10

+ 6

+ 3

5456

+

873

7 · 1000


1000


100


100


10


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~

43/3/! nsbwbmojTob! sjdywfcjt! rwfTnjxfsjU! Tflsfcb.

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.

46/4/!UbnbTj;!txsbgj!qbtvyfcj!`!hbnsbwmfcb.hbzpgb!sjdywUb!

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.

47/!tblpouspmp!xfsb!$!6/!

1. or-ori meqanikuri SecdomisTvis udides 4 qulas akldeba

TiTo q.

2. udides 4 qulas TiTo SecdomisTvis 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.

48/5/! nsbwbmojTob! sjdywfcjt! rwfTnjxfsjU! Tflsfcb.

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

gadaSlili saxelmZRvanelo, me-2 gakveTilis teqstze (romelic

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

am teqstis TviTeuli nawili.

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

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

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/!

moswavleebs win unda edoT gadaSlili saxelmZRvanelo, me-5

gakveTilis teqstze (romelic Sin unda hqondeT wakiTxuli).

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.

rogorc yovelTvis, TviTeuli moqmedeba xmamaRali

msjelobiT unda aRiweros (ras vakeTebT) da unda aixnas.

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.

rogorc yovelTvis, TviTeuli moqmedeba xmamaRali

msjelobiT unda aRiweros (ras vakeTebT) da unda aixnas.

54/!tblpouspmp!xfsb!$!7/!

yuradReba! am gakveTilze masw.-ma gaafrTxilos moswavleebi,

rom momdevno gakveTilisaTvis moitanon farglebi.

1. udides 4 qulas TiTo SecdomisTvis akldeba TiTo q.

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

dasrulebis Semdeg is jdeba da sxva moswavle gamodis

dafasTan), sxvebi _ rveulebSi:

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).

rogorc yovelTvis, TviTeuli moqmedeba xmamaRali

msjelobiT unda aRiweros (ras vakeTebT) da unda aixnas.

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 -


57/5/!gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbnsbwmfcjtbt/!

erTi moswavle Seasrulebs dafaze (TviTeuli gamoTvlis

dasrulebis Semdeg is jdeba da sxva moswavle gamodis

dafasTan), sxvebi _ rveulebSi:

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

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

_ vin moigonebs advili xerxiT gamravlebis kidev erT amgvar

magaliTs? (moswavlis mier Camoyalibebuli gamravlebac aseve

Sesruldeba).

rogorc yovelTvis, TviTeuli moqmedeba xmamaRali

msjelobiT unda aRiweros (ras vakeTebT) da unda aixnas.

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/!

masw.-is SekiTxvebiT damuSavdeba me-4 gakveTilis teqstis

ZiriTadi sakiTxebi. Semdeg dawvrilebiT gairCeva amocana 1.

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

58/5/!gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbnsbwmfcjtbt/!

erTi moswavle Seasrulebs dafaze (TviTeuli gamoTvlis

dasrulebis Semdeg is jdeba da sxva moswavle gamodis

dafasTan), sxvebi _ rveulebSi:

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

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

_ vin moigonebs advili xerxiT gamravlebis kidev erT amgvar

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/!!

moswavleebs win unda edoT gadaSlili saxelmZRvanelo, me-5

- 72 -


gakveTilis teqstze (romelic Sin unda hqondeT wakiTxuli).

masw.-is SekiTxvebiT damuSavdeba me-5 gakveTilis teqstis

TviTeuli nawili.

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.

2. udides 4 qulas dalagebis TiTo SecdomisTvis akldeba

TiTo q.

3. udides 4 qulas dalagebis TiTo SecdomisTvis akldeba

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/!

moswavleebs win unda edoT gadaSlili saxelmZRvanelo, me-2

gakveTilis teqstze (romelic Sin unda hqondeT wakiTxuli).

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 -


gadaSlili saxelmZRvanelo, me-4 gakveTilis teqstze (romelic

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

am teqstis TviTeuli nawili.

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.

65/4/!hbzpgjtb!eb!Tflsfcjt!xftj!`!gsDyjmfcjt!hbytob;!

(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!66/!gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbzpgjtbt/!

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.

66/4/!hbzpgjtb!eb!Tflsfcjt!xftj!`!gsDyjmfcjt!hbytob;!

(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

!

h!68/!gsDyjmfcjt!hbytojt!xftjt!hbnpzfofcb!hbzpgjtbt/!

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#

72/3/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj@•!

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 `erTis 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.

h!81/!tblpouspmp!xfsb!$!21/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

3. 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.

! 83/4/!hbzpgjtb!eb!hbnplmfcjt!xftj!`!gsDyjmfcjt!hbytob;!

(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!

h!85/!nsbwbmojTob!sjdywfcjt!rwfTnjxfsjU!hbnsbwmfcb/!

85/2/ 11.4. 1 (2 _ ara) 3 4 5 6#

! 85/3/!nsbwbmojTob!sjdywfcjt!rwfTnjxfsjU!hbnsbwmfcb/!

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

! 85/4/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj@•!

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...

! 85/5/!hbzpgjtb!eb!hbnplmfcjt!xftj!`!gsDyjmfcjt!hbytob;!

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

masw.-is SekiTxvebiT mokled gameordeba me-2 gakveTilis

ZiriTadi sakiTxebi, Semdeg dawvrilebiT gairCeva amocana 2.

h!86/!nsbwbmojTob!sjdywfcjt!rwfTnjxfsjU!hbnsbwmfcb/!

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.

! 86/4/!nsbwbmojTob!sjdywfcjt!rwfTnjxfsjU!hbnsbwmfcb/!

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

h!87/!tblpouspmp!xfsb!$!22/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

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!

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

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

aikrifeba saangariSoze.

Semdeg masw.-is SekiTxvebiT damuSavdeba me-3 gakveTilis

teqstis TviTeuli nawili. teqstSi mocemuli yvela ricxvi

aikrifeba saangariSoze.

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

93/3/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj@•!

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.

Semdeg gairCeva amocana 1.

! 94/4/!UbnbTj;!wjo!bnpyb{bwt!zwfmb{f!nfut@•!

ix. 55.2, oRond axla amoxazon sxvadasxvanairi samkuTxedebi.

h!95/!tblpouspmp!xfsb!$!23/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

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#

96/3/!UbnbTj;!txsbgj!qbtvyfcj!`!hbnsbwmfcb.hbzpgb!sjdywUb!

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

TviTeuli nawili.

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 -


Sesruldeba qveSmiweriT gamravleba 51 · 38 da 38 · 51.

_ 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/!

masw.-is SekiTxvebiT damuSavdeba me-3 gakveTilis teqstis

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

Sesruldeba qveSmiweriT gamravleba 38 · 55 da 55 · 38.

_ 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].

masw.-is SekiTxvebiT damuSavdeba me-5 gakveTilis teqstis

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/!

1. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

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/!

masw.-is SekiTxvebiT damuSavdeba me-4 gakveTilis teqstis

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/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

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.

Semdeg gairCeva amocana 1.

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!

sjdywUb!yb{{f•/!ix. 27.2.

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.

h!216/!tblpouspmp!xfsb!$!26/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

3. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

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/!

masw.-is SekiTxvebiT damuSavdeba me-2 gakveTilis teqstis

TviTeuli nawili. Semdeg gairCeva amocana 1.

_ 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.

gairCeva amocana 1.

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

TviTeuli nawili. Semdeg gairCeva amocana 1.

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/!

masw.-is SekiTxvebiT gameordeba me-4 gakveTilis teqstis

TviTeuli nawili. Semdeg gairCeva amocana 2.

! 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.

h!223/!tblpouspmp!xfsb!$!27/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

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

TviTeuli nawili. Semdeg gairCeva amocana 1.

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.

!

h!226/!ovmfcjt!ebnbytpwsfcb!rwfTnjxfsjU!hbzpgjtbt/!

226/2/ 17.3. 1# 2# 3 4 5# 6 7!

226/3/! 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).

! 226/4/! hbzpgb! bewjmj! yfsyjU/! ovmfcjU! ebcpmpfcvmj!

nsbwbmojTob! sjdywjt! hbzpgb! )ovmfcjt! ebnbytpwsfcb*-! nbU!

Tpsjt! obTUjboj! TfnUywfwfcjd/! gameoodeba 114.3, oRond sxva

ricxvebiT: 678000:6 da 679000:6.

h!227/!ovmfcjt!ebnbytpwsfcb!rwfTnjxfsjU!hbzpgjtbt/!

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!

nsbwbmojTob! sjdywjt! hbzpgb! )ovmfcjt! ebnbytpwsfcb*-! nbU!

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.

h!22:/!tblpouspmp!xfsb!$!28/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

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

ekidos mimidinare wlis didi kalendari. masw.-i moswavleebs

rig-rigobiT ekiTxeba: _ ra dRea 5 maisi? ra ricxvia

Tebervlis pirveli orSabaTi? (da misTanani).

! 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/!

masw.-is SekiTxvebiT damuSavdeba me-3 gakveTilis teqstis

TviTeuli nawili. teqstSi mocemuli ricxvi aikrifeba

saangariSoze (121.3-Si mocemuli xerxiT).

Semdeg gairCeva amocana 1. masSi mocemuli TviTeuli ricxvic

aikrifeba saangariSoze.

! 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.

! 234/4/!rwfTnjxfsjU!hbzpgb!psojTob!sjdyw{f/!

gairCeva amocana 1.

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

aikrifeba saangariSoze.

! 235/4/! UbnbTj;! txsbgj! qbtvyfcj•! `! sb! nbo[jmjb! UjUfct!

Tpsjt•/!ix. 108.3.

! 235/5/!rwfTnjxfsjU!hbzpgb!psojTob!sjdyw{f/!

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.

h!236/!tblpouspmp!xfsb!$!29/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

3. sul oTxi gamoTvlaa, TiToSi iwereba TiTo q. erTi

arsebiTi an or-ori meqanikuri (namdvilad uneblie)

SecdomisaTvis udides 4 qulas akldeba TiTo q.

4. or-ori meqanikuri SecdomisaTvis udides 5 qulas

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

gairCeva amocana 1.

! 238/4/! UbnbTj;! txsbgj! qbtvyfcj! `! hbnsbwmfcb.hbzpgb!

sjdywUb!yb{{f•/!ix. 27.2.

238/5/! nsbwbmojTob! sjdywfcjt! hbzpgjt! hbbewjmfcb! .!

nypmpe! vobTUpe;! ! ovmfcjU! ebcpmpfcvmj! sjdywfcjt!

fsUnbofU{f! hbzpgb-! speftbd! hbnzpgt! cpmpTj! bsb! brwt!

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

gairCeva amocana 1.

! 239/4/!UbnbTj;!wjo!bnpyb{bwt!zwfmb{f!nfut@•!ix. 55.2, oRond

axla amoxazon sxvadasxvanairi mravalkuTxedebi.

239/5/! nsbwbmojTob! sjdywfcjt! hbzpgjt! hbbewjmfcb! .!

nypmpe! vobTUpe;! ! ovmfcjU! ebcpmpfcvmj! sjdywfcjt!

fsUnbofU{f! hbzpgb-! speftbd! hbnzpgt! cpmpTj! bsb! brwt!

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:/!

1. or-ori meqanikuri SecdomisaTvis udides 4 qulas

akldeba TiTo q.

2. erTi arsebiTi an or-ori meqanikuri (namdvilad uneblie)

SecdomisaTvis udides 4 qulas akldeba TiTo q.

3. erTi arsebiTi an or-ori meqanikuri (namdvilad uneblie)

- 119 -


SecdomisaTvis udides 4 qulas akldeba TiTo q.

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

araa aucilebeli.

! 244/4/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj@•!

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.

Semdeg gairCeva amocana 3.

! 246/4/! UbnbTj;! txsbgj! qbtvyfcj! `! hbnsbwmfcb.hbzpgb!

sjdywUb!yb{{f•/!ix. 27.2.

! 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.

! 248/4/! UbnbTj;! txsbgj! qbtvyfcj! `! hbnsbwmfcb.hbzpgb!

sjdywUb!yb{{f•/!ix. 27.2.

! 248/5/! tbfsUp! hbnfpsfcb/! gameordeba is algera-ariTmetikis

is sakiTxebi, romlebic moswavleebs uWirT.

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):

masw.-is SekiTxvebiT damuSavdeba me-6 gakveTilis teqstis

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

is sakiTxebi, romlebic moswavleebs uWirT.

h!24:/!tblpouspmp!xfsb!$!31/!

1. TiTo meqanikuri SecdomisaTvis udides 4 qulas akldeba

TiTo q. xolo moqmedebaTa rigis darRvevis TiTo SecdomisaTvis

- 2 q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo 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.

Semdeg gairCeva amocana 2.

! 251/4/!UbnbTj;!txsbgj!qbtvyfcj!`!uzvjmjb!Uv!nbsUbmj@•!

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

wiboebi aqvs mas;

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

wiboebi aqvs mas;

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/!

masw.-is SekiTxvebiT damuSavdeba me-2 gakveTilis teqstis

TviTeuli nawili. Semdeg gairCeva amocana 1.

h!253/!hbobzpgj!sbnefoojTob!voeb!njwjSpU/!tufsfpnfusjb/!

253/2/ 21.3. (1 - ara) 2~ 3# 4# 5# 6# 7

! 253/3/!hbobzpgj!sbnefoojTob!voeb!njwjSpU/!

masw.-is SekiTxvebiT gameordeba me-2 gakveTilis teqstis

TviTeuli nawili. Semdeg gairCeva amocana 1.

! 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~

! 254/3/!hbobzpgj!sbnefoojTob!voeb!njwjSpU/!

- 125 -


masw.-is SekiTxvebiT gameordeba me-2 gakveTilis teqstis

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.

! 254/5/! tufsfpnfusjjt! tbljUyfcj! hpofcbTj! xbsnpehfojm!

tyfvmUbUwjt/!Catardeba iseve, rogorc 137.2.

h!255/!tufsfpnfusjb/!tbfsUp!hbnfpsfcb/!

255/2/ 21.5. 1 2# 3~ 4# 5 6 7#

! 255/3/! tufsfpnfusjjt! tbljUyfcj! hpofcbTj! xbsnpehfojm!

tyfvmUbUwjt/!Catardeba iseve, rogorc 137.2.

! 255/4/! 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).

! 255/5/! zwfmb! obtxbwmj! nprnfefcjt! hbonuljdfcb! eb!

hbowjUbsfcb/! gansakuTrebuli yuradReba eqceva swrafi zepiri

angariSis unarCvevebs.

im sakiTxebis gameoreba, romlebic klass uWirs.

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.

im sakiTxebis gameoreba, romlebic klass uWirs.

h!257/!tblpouspmp!xfsb!$!32/!

1. TiTo meqanikuri SecdomisaTvis udides 4 qulas akldeba

TiTo q. xolo moqmedebaTa rigis darRvevis TiTo SecdomisaTvis

- 2 q.

2. TiTo SecdomisaTvis udides 4 qulas akldeba TiTo q.

3. pasuxebi: I rigi: ricxvs rig-rigobiT emateba an 40000,

an 30000.

II rigi: ricxvs rig-rigobiT akldeba an 5000, an 2000.

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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

akldeba TiTo q.

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 (`an~-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 _ `iangariSeT

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.

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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.

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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? }.

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