masw_wigni_5_kl

guram gogiSvili, Teimuraz vefxvaZe

ia mebonia, lamara qurCiSvili

gamomcemloba inteleqtiTbilisi 2011

m a T e m a t i k a

V klasi

maswavleblis wigni

redaqtori Teimuraz vefxvaZe

ISBN 978-9941-430-18-3

© gamomcemloba `inteleqti~, 2011.

guram gogiSvili, Teimuraz vefxvaZe,

ia mebonia, lamara qurCiSvili

m a T e m a t i k a

V klasi

maswavleblis wigni

gamomcemloba inteleqti

Tbilisi 2011

Guram Gogishvili, Teimuraz Vepkhvadze, Ia Mebonia, Lamara Kurchishvili

MaThs V Teacher's Book

InTeLeKTI PublishersTbilisi 2011

gamomcemloba inteleqtiTbilisi, ilia WavWavaZDis gamz. 17b. tel.: 25 05 22, 91 22 83.www.intelekti.ge [email protected] [email protected]

intelekti Publishers17b ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 22) 25 05 22, 91 22 83.

sarCevi

Sesavali .........................................................................................................................................................6V klasi, wlis bolos misaRwevi Sedegebi .......................................................................................10wlis bolos misaRwevi Sedegebi da maTi indikatorebi ............................................................11Sinaarsisa da miznebis ruka ................................................................................................................16saswavlo masalis wardgenis fazebi da gakveTilis dagegmvis ZiriTadi principebi ....19Sefasebis zogadi principebi ...............................................................................................................23

1. gameoreba, milionze meti ricxvebi

1. gavixsenoT naturaluri ricxvis wakiTxvisa da Caweris wesebi .....................................24 sanimuSo gakveTili #1 ..................................................................................................................272. milionze meti ricxvebi ................................................................................................................29 damoukidebeli wera #1 .................................................................................................................313. vadarebT da valagebT naturalur ricxvebs ........................................................................324. amocanebis amoxsna ...........................................................................................................................35 sakontrolo wera #1 .....................................................................................................................385. naturaluri ricxvebis Sekreba. Sekrebis Tvisebebi ..........................................................406. naturaluri ricxvebis gamokleba .............................................................................................41 damoukidebeli wera #2 .................................................................................................................447. ricxviTi sxivi ...................................................................................................................................45 sanimuSo gakveTili #2 ..................................................................................................................478. naturaluri ricxvebis gamravleba. gamravlebis Tvisebebi ............................................49 sanimuSo gakveTili #3 ..................................................................................................................52 sakontrolo wera #2 .....................................................................................................................549. gamravlebis Sedegis Sefaseba. weriTi gamravleba ..............................................................55 sanimuSo gakveTili #4 ..................................................................................................................5810. sakoordinato bade ..........................................................................................................................60 sanimuSo gakveTili #5 ..................................................................................................................62 damoukidebeli wera #3 .................................................................................................................6311. ricxviTi da asoiTi gamosaxulebebis Sedgenisa da gamartivebis meTodebi ............6512. or sidides Soris damokidebulebis gamosaxvis xerxebi ..................................................67 Ddamoukidebeli wera #4 .................................................................................................................6913. amocanebis amoxsna ricxviTi an asoiTi gamosaxulebis SedgeniT .................................7014. xarisxi. ricxvebis Caweris sxvadasxva wesi ............................................................................72 sakontrolo wera ###3 .....................................................................................................................7515. amocanebis amoxsnis sxvadasxva xerxi .......................................................................................76

2. naturaluri ricxvebi da wiladebi

1. naturaluri ricxvis jeradebi da gamyofebi .......................................................................81 Ddamoukidebeli wera #5 .................................................................................................................832. martivi da Sedgenili ricxvebi ..................................................................................................853. wrewiri. wre .......................................................................................................................................87 sakontrolo wera #4 .....................................................................................................................88

4

sanimuSo gakveTili #6 ..................................................................................................................904. wiladi _ mTelis nawili ..............................................................................................................925. wiladi, rogorc erTobliobis nawili ....................................................................................93 damoukidebeli wera #6 .................................................................................................................956. wiladi da naturaluri ricxvebis gayofa .............................................................................977. wiladebis toloba. wiladis gamartiveba ...............................................................................99 sakontrolo wera #5 ...................................................................................................................1018. Sereuli ricxvebi da wiladebi .................................................................................................1029. wiladebisa da Sereuli ricxvebis Sedareba ........................................................................105 damoukidebeli wera #7 ...............................................................................................................10810. wiladebis jamis Sefaseba ............................................................................................................10911. wiladebis Sekreba. wiladis naturalur ricxvze gamravleba ......................................112 sakontrolo wera #6 ...................................................................................................................11412. wiladebis gamokleba .....................................................................................................................11513. Sereuli ricxvebis Sekreba ........................................................................................................116 damoukidebeli wera #8 ...............................................................................................................11714. Sereuli ricxvebis gamokleba ...................................................................................................11815. vagrZelebT Sereuli ricxvebis Sekrebisa da

gamoklebis magaliTebis ganxilvas ..........................................................................................120 sakontrolo wera #7 ...................................................................................................................122 viqtorina ..........................................................................................................................................123 sanimuSo gakveTili #7 ................................................................................................................12416. amoxseniT amocanebi sinjvis xerxiT .......................................................................................125

3. sidideebis gazomva. geometriuli figurebi

1. dro. drois erTeulebi ...............................................................................................................130 sakontrolo wera #8 ...................................................................................................................1332. sigrZis erTeulebi ........................................................................................................................1343. masis erTeulebi .............................................................................................................................137 sanimuSoO gakveTili #8 ................................................................................................................140 damoukidebeli wera #9 ...............................................................................................................1424. amocanebis amoxsna (ganayofisa da naSTis gamoyeneba).....................................................1435. mravalkuTxedi ................................................................................................................................145 sakontrolo wera #9 ...................................................................................................................1476. kuTxis gazomva. marTobuli wrfeebi .....................................................................................149 sanimuSo gakveTili #9 ................................................................................................................1507. samkuTxedi ........................................................................................................................................154 amocanebi gameorebisTvis ...........................................................................................................158 damoukidebeli wera #10 .............................................................................................................1598. farTobi .............................................................................................................................................160 amocanebis nimuSebi maTematikuri SejibrebisTvis ...........................................................1649. zogierTi sivrculi figura ......................................................................................................166 sakontrolo wera #10 .................................................................................................................169 damatebiTi amocanebi ...................................................................................................................170

5

4. amovxsnaT amocanebi. movipovoT da SeviswavloT monacemebi.

1. monacemTa Segrovebis sxvadasxva xerxi .................................................................................172 2. warmovadgenT monacemebs cxrilebis gamoyenebiT ............................................................1743. viyenebT cxrils amocanis amoxsnisas ....................................................................................178 damoukidebeli wera #11 .............................................................................................................1824. piqtograma ........................................................................................................................................1845. svetovani diagrama ........................................................................................................................185 sakontrolo wera #11 .................................................................................................................187 amocanebi V klasis kursis gasameoreblad ..........................................................................189 literatura .....................................................................................................................................193

6

Sesavali

maswavleblis wignis daniSnulebaa miawodos maswavleblebs meTodikuri rekomendaciebi

axali erovnuli saswavlo gegmis mixedviT Sedgenili V klasis saxelmZRvanelosTvis (guram

gogiSvili, Teimuraz vefxvaZe, ia mebonia, lamara qurCiSvili, maTematika V); daexmaros maT

saxelmZRvaneloSi warmodgenili masalis gaanalizebasa da saswavlo procesis warmarTvis

meTodebis SemuSavebaSi.

warmodgenilia yvela tipuri gakveTilis, aqtivobisa da savarjiSos meSveobiT swavlebis

rekomendaciebi. mowodebulia damatebiTi masala, miTiTebulia literatura, rac exmareba

maswavlebels sakuTari SesaZleblobebis srulyofaSi, wigni Seicavs miTiTebebs, Tu ra

damatebiTi resursebis gamoyeneba SeuZlia maswavlebels saswavlo procesSi.

maswavleblis wigni Seicavs erovnuli saswavlo gegmiT gaTvaliswinebuli misaRwevi

Sedegebisa da Sesabamisi indikatorebis CamonaTvals. Sinaarsisa da miznebis rukis mixedviT

ki axsnilia erovnuli saswavlo gegmiT gaTvaliswinebul romel Sedegs ukavSirdeba

saxelmZRvaneloSi warmodgenili TiToeuli Tema. maswavleblis wignSi SemoTavazebuli

meTodika iTvaliswinebs maswavleblis profesiuli standartis moTxovnebs (maswavleblis

profesiuli standarti, maTematika, dawyebiTi safexuri, 2008 weli). magaliTad, yuradReba

eqceva ricxvebis adekvaturad gamoyenebas, poziciuri sistemebis gamoyenebis unaris

ganviTarebis, zomis sxvadasxva erTeulebis erTmaneTTan dakavSirebis flobas, geometriul

figuraTa swori klasifikaciis aRqmas da misi gamoyenebis unaris ganviTarebas. yoveli

Temis swavlebasTan dakavSirebuli sakiTxebis gadmocemisas ganvixilavT am Temis Sesabamis

mecnierul safuZvlebs _ vzrunavT maswavleblis kompetenciis ganviTarebaze.

vcdilobT davexmaroT maswavlebels sxvadasxva strategiis, didaqtikuri xerxisa Tu

teqnologiebis gamoyenebaSi.

yuradRebas vuTmobT sxvadasxva akademiuri donis moswavleebTan muSaobis Sesaxeb reko

mendaciebis miwo debas; saxelmZRvaneloSi masalis gadmocemis meTodika (magaliTad, xSiri

mimarTeba ukve Seswavlilisadmi) exmareba maswavlebels am mimarTulebiTac. SegaxsenebT,

erovnul saswavlo gegmaSi Camoyalibebul erTerT ZiriTad princips: saganmanaTleblo

procesis centrSi dgas TiToeuli moswavle da miRweuli Sedegi.

yoveli Tavisa da paragrafis mixedviT mkafiod aris Camoyalibebuli miznebi da

mosalodneli Sedegebi.

maswavleblis sarekomendacio wignSi Tavebisa da paragrafebis numeracia da dasaxeleba

emTxveva moswavlis saxelmZRvaneloSi SemoRebul numeraciasa da dasaxelebas.

ZiriTadi mecnierulmeTodikuri principebi, romlebic gansazRvravs saxelmZRvanelos

Sinaarssa da wyobas, aris saxelmZRvanelos Sesabamisoba erovnul saswavlo gegmasTan;

sistematuroba masalis dalagebaSi; Sinaarsis mecnieruloba; Teoriisa da praqtikis kavSiris

principis ganxorcieleba; gadmocemis mosawvdomoba, enis garkveuloba, sizuste da simartive;

savarjiSoebisa da maTi Sesrulebis formebis mravalferovneba. es ukanaskneli exmareba

maswavlebels sxvadasxva TamaSebis, Sejibrebebis, viqtorinebis organizebaSi. maswavlebels

saSualeba aqvs xSirad gamoiyenos sityvebi: ifiqre, moisazre, Seafase, Seadare, daasabuTe _

moswavleTa saazrovno unaris ganviTareba erovnuli saswavlo gegmis erTerTi umTavresi

moTxovnaa: `moswavle unda gacdes masalis ubralod damaxsovrebagazepirebas. kerZod, man

codnis dagrovebasTan erTad (informaciis SenarCuneba) unda SeZlos am codnis gageba,

gamoyeneba, analizi, sinTezi da bolos, Sefaseba. am dros moswavle ara mxolod flobs

7

informacias, aramed mas SeuZlia qmedeba, raimes swavla swored azrovnebis Sedegia~...

(erovnuli saswavlo gegma, zogadsaganmanaTleblo skolebisTvis, 20092010 saswavlo weli).

davalebebis mravalferovneba, sxvadasxva xerxiT davalebis Sesrulebis SesaZlebloba swavlis

stimulirebis erTerTi mniSvnelovani aspeqtia. sworad SerCeuli da kargad organizebuli

TamaSi xels uwyobs moswavleTa yovelmriv, harmoniul ganviTarebas, exmareba moswavles

gamoimuSavos cxovrebaSi saWiro Cvevebi. kerZod, daxvewos sakuTar gancdaTa gamovlenis

koreqtuli formebi. TamaSma SeiZleba xeli Seuwyos moswavleebSi mexsierebis, yuradRebis,

SemoqmedebiTi warmosaxvis, operatiulobis, TviTSemowmebis unaris ganviTarebas. TamaSi

xels uwobs sakuTari qmedebebis dasaxuli amocanebisadmi daqvemdebarebis, dawyebuli saqmis

bolomde miyvanis Cvevebis dauflebas. TamaSebSi mniSvnelovania SejibrebiToba, radgan

Sejibrebis dros vlindeba moswavleTa aqtiuroba da gamarjvebisken swrafva.

saxelmZRvaneloSi mowodebuli davalebebi saSualebas aZlevs maswavlebels SearCios

sxvadasxva organizaciuli forma _ gakveTiliTamaSi, gakveTilimogzauroba, gakveTili

sazrianTa klubi, gakveTiliviqtorina da a. S. didaqtikuri TamaSebi exmareba maswavlebels

saganmanaTlebloaRmzrdelobiTi amocanebis gadawyvetaSi.

maswavlebels kvlav SevaxsenebT erovnul saswavlo gegmaSi warmodgenil ramdenime

miTiTebas _ `saxelmZRvanelo aris erTerTi saSualeba da ara erTaderTi, romelic

emsaxureba saswavlo gegmis sagnobriv programebSi mocemuli Sedegebis miRwevas.

saxelmZRvanelo ar aris TavisTavad programa _ Sesabamisad, maswavlebeli ar unda

iyos saxelmZRvaneloSi mocemuli `masalis gavlaze~ orientirebuli, aramed aqcenti

unda gaakeTos saxelZRvaneloSi mocemuli teqstebis, savarjiSoebisa da ilustraciebis

gamoyenebaze sagnobrivi standartis Sedegebis misaRwevad. sworia, roca maswavlebeli

argebs saxelmZRvanelos moswavleebis saWiroebebs da ara piriqiT, rodesac maswavlebeli

cdilobs moergos saxelmZRvanelos. maswavlebels SeuZlia swavlebis process miudges

SemoqmedebiTad, ar gamoiyenos romelime teqsti an aqtivoba, Secvalos drois xangrZlioba,

romelic romelime konkretul Temas eTmoba, gadaanacvlos paragrafebis Tanamimdevroba,

daamatos sakuTari aqtivobebi da sxva~. magaliTad, maswavlebelma saswavlo wlis dasawyisSi

SeiZleba meti dro dauTmos IV klasis saswavlo programiT gaTvaliswinebuli sakiTxebis

gameorebas, vidre Cven mier SemoTavazebul saxelmZRvaneloSia warmodgenili.

swavlebis meTodikac SeiZleba sxvadasxvanairi iyos, rogorc gamoCenili pedagogi d. poia

aRniSnavda (CvenSi farTodaa gavrcelebuli misi naSromis rusuli gamocema: `Д. Пойа. Математическoе открытие~, Москва, 1976), `swavleba xelovnebaa... yovel karg maswavlebels

swavlebis sakuTari xerxebi aqvs. amiT yoveli kargi maswavlebeli gansxvavdeba sxva kargi

maswavleblisgan. yoveli efeqtiani xerxi unda Seesabamebodes Seswavlis gansazRvrul xerxs...~.

Tumca, unda iTqvas, rom arsebobs zogadi meTodikuri principebi, romelTa codna daexmareba

maswavlebels muSaobaSi. es principebi Camoyalibebulia TviT d. poias cnobil samtomeulSi

da im masalebSi, romlebsac maswavlebelTa profesiuli ganviTarebis centri da erovnuli

saswavlo gegmebisa da Sefasebis centri gamoscemen (SeiZleba mivuTiToT, `maswavlebelTa

profesiuli standartebi~, Jurnali `maswavlebeli~, meTodologiuri saxelmZRvaneloebi,

magaliTad, `rogor vaswavloT azrovneba~ da sxva).

maswavlebelis wigni iTvaliswinebs V klasSi swavlebis Taviseburebebs; am etapze icvleba

moswavleTa akademiuri Sefasebis sistemis gamoyeneba. cxadia, Sefasebis principebi igive

rCeba _ kvlav ZalaSia maswavlebelTa komentariSefasebebi, maT emateba niSnebiT gamoxatuli

Sefasebebi _ trimestruli, sagnis wliuri, saerTo wliuri. niSnebis wera V klasidan iwyeba,

Sesabamisad, davamateT davalebebi (e. w. damoukidebeli da sakontrolo werebis nimuSebi

_ Sesabamisi kriteriumebiT), romlebic daexmareba maswavlebels naSromTa Sefasebisas. aqve

SevaxsenebT maswavleblebs, rom aRniSnuli tipis davalebebis Semowmebis Semdeg ar unda

Semoifarglos niSnis daweriT; SefasebaTa gamocxadebamde sasurvelia klasSi moswavleTa

aqtiuri monawileobiT gavaanalizoT sakontrolo davalebebi. visaubroT amoxsnis gzebze

8

(gamovyoT optimaluri gzebi). moswavleebs warvudginoT Sefasebis kriteriumi. amis

Semdeg, vTxovoT moswavleebs maTi niSnebis gamocxadebamde Tavad daasaxelon naSromis

savaraudo Sefasebebi. Tqveni Sefasebisa da moswavlis TviTSefasebis siaxlove moswavleebSi

pedagogis obieqturobis gancdas aZlierebs. moswavleebis namuSevrebis safuZvelze SeiZleba

ganmaviTarebeli Sefasebebis gamoyenebac _ komentarebi aRniSnuli davalebis Sesaxeb,

romelic aRwers namuSevris Zlier da sust mxareebs da Seicavs rekomendaciebs miRwevebis

gaumjobesebis Taobaze. saxelmZRvaneloSi komentarebis savaraudo nimuSebi Sesabamis

davalebebTan erTad iqneba gadmocemuli.

V klasSi swavleba sxva TaviseburebebiTac gamoirCeoda. aq, rogorc wesi, icvleboda mas

wav lebeli, iwyeboda e. w. sagnobrivi swavleba, rac dawyebiTi safexuris damTavrebasa da

axal safexurze gadasvlas moaswavebda. es problema dResac awuxebs ruseTis pedagogiur sa

zogadoebas. SesaZlebelia maswavlebelma SeZlos moiZios ruseTSi gamocemuli Jurnali `Первое сен тября~, #18, 2007. am nomerSi ori statiaa gamoqveynebuli: `Проблемы преемственности из четвёртого класса в пятый~ da `Особенности взаимодействия начальной и основной школы~. miTiTebulia, rom

pirvel adgilze saswavlo sagani gamodis Tavisi SinaarsiTa da meTodikiT, moswavles uCndeba

siZneleebis daZlevisadmi SiSis grZnoba, igrZnoba swavlisadmi interesis dakargvis tendencia.

mexuTe klasSi swavlebis Taviseburebebi moswavleTa asakis axal etaps ukavSirdeba _

maTSi mkvidrdeba zrdasrulobis gancda, Tumca, igi jer kidev ar aris gamyarebuli realuri

pasuxismgeblobiT. izrdeba moTxovnebi damoukideblobisa da pativiscemis, seriozuli da

ndobis gamomxatveli midgomisadmi. moswavlesTvis TiTqmis ukve mosawyeni xdeba maTematikuri

amocanis amoxsna nimuSis mixedviT, mas surs ipovos sakuTari xerxi, romelic dafuZnebulia

improvizaciaze, ufro mniSvnelovani xdeba erTi da imave amocanis sxvadasxva xerxiT

amoxsna, vidre sxvadasxva amocanis amoxsna. am mxriv, erovnuli saswavlo gegma saSualebas

iZleva tradiciuli da Tanamedrove Sexedulebebis sinTezis safuZvelze gaviTvaliswinoT

memkvidreobiTi sakiTxebi me4dan me5 klasSi gadasvlis procesSi. amasTanave, igi saSualebas

iZleva gamoviyenoT mravali qveynis gamocdilebac, sadac SemoqmedebiTad udgebian saswavlo

masalis SerCevas da misi gadacemis meTodikas. amasTanave, Tavisi agebulebiTa da SinaarsiT

Cveneuli V klasis saxelmZRvanelo wina IV klasis saxelmZRvanlos bunebrivi gagrZelebaa.

bevri sakiTxi xelmeored, zogjer ufro gaRrmavebulad aris warmodgenili, sakiTxebis

gameorebas V klasis standartic iTvaliswinebs (droisa da sigrZis erTeulebi, naturaluri

ricxvebis Cawera, wakiTxva). Tumca, SemoTavazebulia didi naturaluri ricxvebis wakiTxvisa

da Caweris xerxebis gacnobis, poziciuri sistemis warmodgenisas 10is xarisxebis gamoyeneba.

gansxvavebulia Cveni midgoma wiladebis warmodgenisas _ aq, rusuli saxelmZRvaneloebisgan

gansxvavebiT, TiTqmis yvela aspeqts mivaqcieT yuradReba.

misaRwevi Sedegebis Sesabamisi Sinaarsis SerCevas moswavleTaTvis misawvdomobis

principi udevs safuZvlad; es SerCeva Camoyalibebuli stabiluri birTvis bazaze xdeba. igi

gafiltrulia maTematikis swavlebis gamocdilebiT; tradiciuli Temebis Tanamimdevrobis

axleburi gaazrebis, maTematikis erTianobidan gamomdinare integrirebis xarisxis gazrdisa

da masalis axali sakiTxebiT Sevsebis aucileblobis gaTvaliswinebiT. maswavlebels

mudmivad unda axsovdes, rom maTematika erTiani mecnierebaa da ara misi sxvadasxva nawilis

xelovnuri gaerTianeba, amitom gadmocemis meTodika sakiTxebis urTierTkavSirSi gadmocemas

iTvaliswinebs. sistematurad mimdinareobs wina masalis gameoreba da ganmtkiceba. yoveli

siaxle moswavlis aqtiur monawileobas gulisxmobs; davalebebi, maTi mravalferovneba,

sxvadasxva midgomebis gamoyenebis SesaZlebloba, praqtikasTan xSiri mimarTva zrdis

moswavleTa motivacias, xels uwyobs kritikuli da SemoqmedebiTi azrovnebis ganviTarebas

aRmoCeniT, keTebiT, kvleviTi samuSaos ganxorcielebiT. gvxdeba savarjiSoebi, romelTa

Sesruleba ara mxolod werasa da kiTxvas gulisxmobs (magaliTad, sivrculi figurebis

Slilebi, brtyeli figurebis `dafarvis~ amocanebi da a. S.).

9

davalebebi iseTi TanamimdevrobiTaa warmodgenili, rom martividan rTulisken iyos

mimarTuli. calkea gamoyofili davalebebi saklaso da saSinao muSaobisTvis, savarjiSoTa

mravalferovani sistema (`testebi~, amocanebi jgufuri muSaobisTvis, amocanebiTamaSebi,

damatebiTi amocanebi da sxva) maTematikis Segnebulad swavlasa da misi Seswavlisadmi

interesis gaRrmavebas uwyobs xels.

saxelmZRvaneloSi mocemuli zogierTi rubrika (`s~ _ sxvadasxva, `moisazre~) xels

uwyobs moswavlis cnobismoyvareobis gazrdas, ubiZgebs mas meti, SemoqmedebiTi muSaobis

ken, movlenebis Sesaxeb condis SeZenisken da TviTganviTarebisken. mowodebuli masala

maswavlebelma SeiZleba gamoiyenos viqtorinebis, warmodgenebis da sxva maTematikuri

RonisZiebebis Casatareblad.

sakiTxTa mravalferovneba, maTi ganxilvis formebis araerTgvarovneba xels uwyobs

saswavlo procesSi Sefasebis sxvadasxva formis gamoyenebas, `testebi~ kargi saSualebaa

swrafi TviTSefasebisTvis.Tumca, SesaZlebelia, zogierTi `testis~ ganxilvas Tan sdevdes msjeloba da pasuxis

dasabuTeba. amasTanave, sTxoveT moswavleebs wignSi nu miuTiTeben swor pasuxebs, CainiSnon rveulSi swori pasuxebis nomrebi, romelsac SeiZleba Tan erTvodes mokle Canawerebi, romlebic am pasuxebis miRebis gzas SeiZleba aRwerdes. yvela davaleba wignSi Canaweris gakeTebis gareSe unda Sesruldes _ mxolod rveulSi! moswavle unda movaCvioT saxelmZRvanelos sufTad xmarebas. saxelmZRvanelo regularulad Seaxsenebs xolme moswavleebs, rom gaufrTxildnen wigns. Tqvenc mimarTeT xolme moswavleebs da Tvali adevneT maT damokidebulebas wignisadmi. SesaZlebeli unda iyos wignis gamoyeneba ramdenime wlis ganmavlobaSi.

saxelmZRvanelo ori nawilisgan Sedgeba, pirvel nawilSi gadmocemulia saswavlo wlis

pirvel naxevarSi gasavleli masala. es aadvilebs saxelmZRvanelos tarebasa da moxmarebas.

saintereso da mravalferovania wignis agebuleba, igi saxelmZRvanelosadmi wayenebul

yvela moTxovnas iTvaliswinebs (ix., mag., [22]). wignis dasawyisSi gamovxatavT madlierebas

yvelas mimarT, visac Tavisi rekomendaciebiT wvlili Seaqvs wignis daxvewaSi; gansakuTrebiT

mniSvnelovania saswavlo gegmebisa da saxelmZRvanelos SemmuSavebeli jgufebis eqspertebis

Rvawli. maT mier klasebis mixedviT ganawilebuli saswavlo gegma saSualebas gvaZlevs

SevarCioT masala moswavleTa codnis gaRrmavebis, unarebisa da Cvevebis ganviTarebisa da

formirebisTvis. eqspertTa kvalificiuri rekomendaciebi da SeniSvnebi gvexmareba mTeli

`sapilote~ wlis ganmavlobaSi davxvewoT saxelmZRvanelo, yoveli axali nawilis Sedgenisas

gaviTvaliswinoT wina nawilze miRebuli daskvnebi. wlis bolos _ koreqtivebi SevitanoT

yvela nawilSi.

saxelmZRvanelo 4 Tavisgan Sedgeba, TiToeuli Tavi _ paragrafebisgan, zogierTi

paragrafi ramdenime nawiliania. sarCevi wignis dasawyisSia mocemuli, maTematikuri terminebi,

cxrilebi, maTematikuri niSnebi, sagnobrivi saZiebeli _ wignis boloSi.

Zvirfaso maswavleblebo!

warmatebebs gisurvebT Tqvens sapatio da mniSvnelovan saqmianobaSi. imedi gvaqvs, rom

Cveni TanamSromloba keTil nayofs gamoiRebs.

10

V klasi, wlis bolos misaRwevi Sedegebi:

mimarTuleba:

ricxvebi da moqmedebebi

kanonzomierebebi da algebra

geometria da sivrcis aRqma

monacemTa analizi, albaToba da statistika

1. iyenebs axal ricxviT saxelebs, poziciur sistemas da axdens naturaluri ricxvebis klasifikacias

2. kiTxulobs, gamosaxavs, afasebs, adarebs da alagebs wiladebs

3. asrulebs moqmedebebs naturalur ricxvebze da tolmniSvnelian wiladebze

4. iyenebs da erTmaneTTan akavSirebs zomis sxvadasxva erTeulebs

5. gamosaxavs da aRwers sidideebs Soris damokidebulebas

6. amocanis amoxsnisas adgens da amartivebs algebrul gamosaxulebas

7. amoicnobs, aRwers da gamosaxavs geometriul figurebs

8. adgens mimarTebebs figurebs Soris da figuris elementebs Soris

9. poulobs da adarebs brtyeli figurebis farTobebs erTmaneTs

10. orientirebs badiT dafarul areze

11. moipovebs dasmuli amocanis amosaxsnelad saWiro Tvisebriv da raodenobriv monacemebs

12. warmoadgens Tvisebriv da raodenobriv monacemebs dasmuli amocanis amosaxsnelad xelsayreli formiT

13. akeTebs Tvisebriv da raodenobriv monacemTa interpretaciasa da elementarul analizs

11

wlis bolos misaRwevi Sedegebi da maTi indikatorebi

mimarTuleba: ricxvebi da moqmedebebi

maT. V.1. iyenebs axal ricxviT saxelebs, poziciur sistemas da axdens naturaluri

ricxvebis klasifikacias

Sedegi TvalsaCinoa, Tu moswavle:

• kiTxulobs milionze did ricxvebs axali ricxviTi saxelebis gamoyenebiT (mag. trilioni

da a.S.); ganmartavs am ricxviT saxelebs.

• poulobs axali ricxviTi saxeliT mocemuli (milionze) didi ricxvis rigs (mag. ramdeni

cifrisgan Sedgeba 10-obiTi poziciuri sistemiT Sedgenili aseTi ricxvi?).

• iyenebs 10is xarisxebs didi ricxvebis Cawerisas. msjelobs aTobiTi poziciuri sistemis

upiratesobaze sxva ricxviT sistemebTan SedarebiT (mag. egvipturi an romauli

sistema).

• poulobs mocemuli erTniSna da orniSna ricxvebis jeradebsa da gamyofebs.

• ganasxvavebs kent, luw, martiv da Sedgenil ricxvebs, asabuTebs 2ze da 5ze gayofadobis

niSnebs.

• iyenebs ricxvis kvadratis cnebas, amoicnobs orniSna naturalur ricxvebs Soris

naturaluri ricxvis kvarats.

maT. V.2. kiTxulobs, gamosaxavs, afasebs, adarebs da alagebs wiladebs


• kiTxulobs da gamosaxavs Cveulebriv/Sereul wiladebs; uTiTebs maT CanawerSi wiladis

mricxvels da mniSvnels / mTel da wilad nawilebs.

• gamosaxavs erTeulis nawilebs ricxviT sxivze da aRniSnavs tol nawilebs; iTvlis

aseTi nawilebis Sesabamisi bijiT (maT Soris erTeulis gavliT).

nimuSi 1

nimuSi 3

• adarebs or wilads, maT Soris wiladis ZiriTadi Tvisebis gamoyenebiT.

• wers Sereul wilads ~arawesieri~ wiladis saxiT da piriqiT; axdens (wesieri) wiladis

cnebis sxvadasxvagvar interpretacias da msjelobs maT Soris kavSirebze (wiladi

rogorc ori naturaluri ricxvis gayofis Sedegis Canaweri, erTeulis nawili,

mTliani jgufis qvejgufi da rogorc ~ricxviT xazze~ garkveuli adgili).

maT. V.3. asrulebs moqmedebebs naturalur ricxvebze da tolmniSnvnelian

wiladebze


0 1 2

1

0 2 4

1 3 4

1 4

0 4 8

1 6 8

2 8

7 8

5 8

3 8

1 8

0 1 1 3

2 3

0 1 2 6

4 6

1 6

3 6

5 6

0 1 2 12

1 12

3 12

4 12

5 12

6 12

7 12

8 12

9 12

10 12

11 12

0 1 2 2 4

2 4

3 4

1 4

3 4

1 4

1 1 1

12

• amocanis konteqstis gaTvaliswinebiT irCevs da iyenebs naturalur ricxvebze

moqmedebaTa Sesrulebis adeqvatur xerxs; naSTiT gayofis SemTxvevaSi axdens naSTis

interpretacias amocanis konteqstis gaTvaliswinebiT.

• axdens erTnairi mniSvnelis mqone martiv wiladebze ariTmetikuli moqmedebebis

demonstrirebas da moqmedebaTa Sedegis interpretacias modelis gamoyenebiT (mag.

namcxvris naWrebi).

• msjelobs Tu rogor icvleba wiladi misi mxolod mniSvnelis an mxolod mricxvelis

~jer/iT~ gazrdiT an SemcirebiT; asabuTebs pasuxs (mag. modelis gamoyenebiT).

• iyenebs moqmedebaTa Tvisebebs da maT Soris kavSirebs Sereul ricxvebze gamoTvlebis

Sesrulebisas/maT gasamartiveblad (Sereuli ricxvebis Sekreba/gamokleba; wiladis

naturalur ricxvze gamravleba).

maT. V.4. iyenebs da erTmaneTTan akavSirebs zomis sxvadasxva erTeulebs


• erTmaneTTan akavSirebs sigrZis da farTobis erTeulebs, iyenebs ricxvis kvadratis

Canawers am konteqstSi.

• erTmaneTTan akavSirebs farTobis sxvadasxva erTeulebs; gamosaxavs farTobis did

erTeuls mcire erTeulis gamoyenebiT.

• iyenebs drois 12 da 24saaTian formatebs da ariTmetikul moqmedebebis gamoyenebiT

gansazRvravs dros da drois intervals.

• iyenebs naSTiT gayofas zomis mocemul erTeulebSi monacemis sxva erTeuliT

gamosaxvisas (mag. 50000 wami = ? saaTi).

Sinaarsi

1. naturaluri ricxvebi da maTze moqmedebebi

2. milionze meti naturaluri ricxvebi (miliardi/bilioni, trilioni da a.S.)

3. sxva ricxviTi sistemebis gacnoba (mag. anbani da asoebis ricxviTi Sesatyvisebi;

egvipturi _ aTis xarisxebis gansxvavebuli aRniSvnebi an romauli sistema).

4. arauaryofiTi wiladebi toli mniSvneliT da maTze moqmedebebi

5. sxvadasxva mniSvneliani wiladebis Sedareba, dalageba da gamosaxva (mag. wiladebi

mniSvneliT 2,4,8; 3,6,9,12; 5,10 da 100)

6. ricxvis kvadrati farTobis konteqstSi

7. kavSiri sigrZisa da farTobis erTeulebs Soris

8. drois erTeulebi (saaTebi, wuTebi, wamebi), saaTis 12 da 24 saaTiani formati

9. wonis erTeulebi (kilogrami, grami, miligrami)

mimarTuleba: kanonzomierebebi da algebra

maT. V.5. gamosaxavs da aRwers sidideebs Soris damokidebulebas


• aRwers raime sididis Tanabar adiciur (mudmivi sididis mimatebiT/gamoklebiT miRebul)

cvlilebas (maT Soris realur viTarebaSi).

• mocemuli damokidebulebisaTvis Tvisebrivad aRwers Tu ra gavlenas axdens erTi

sididis cvlileba masze damokidebul meore sidideze da sxva atributebze. (mag.

~erTis zrda gamoiwvevs meoris zrdas~, ~zRvis donesTan SedarebiT ufro meti

simaRle rukaze ufro muqia~).

13

• erTi cvladis Semcvel mocemul asoiT gamosaxulebaSi, sxvadasxva ricxvebis CasmiT

avsebs cvladis mniSvnelobebsa da gamosaxulebis mniSvnelobebs Soris damokidebulebis

gamomsaxvel cxrils, romelSic cvladis mniSvnelobebis Sesabamisi sveti/striqoni

winaswaraa Sevsebuli.

maT. V.6. Seadgens da amartivebs algebrul gamosaxulebas maTematikuri amocanis

amoxsnisas


• adgens realuri viTarebis an misi sityvieri aRweris Sesabamis tolobas, utolobas an

gantolebas (romelSic ucnobi aris tolobis mxolod erT mxares).

• ariTmetikuli operaciebis gamoyenebiT teqsturi amocanis amoxsnisas, svams kiTxvebs

amocanis pirobaSi arasruli monacemebis Sesavsebad (mag. amocanis piroba: “moswavlem

sam fanqarSi 60 TeTri gadaixada. ra Rirs erTi fanqari?” daklebuli monacemebis

Sesavsebad SeiZleba daisvas kiTxva: “samive fanqris fasi tolia?”).

• iyenebs Sekrebisa da gamravlebis komutaciurobas, asociaciurobas da Sekrebis

mimarT gamravlebis distribuciulobis Tvisebebs (erTi cvladis Semcveli) asoiTi

gamosaxulebebis gasamartiveblad.

Sinaarsi

1. or sidides Soris damokidebuleba, romelic Sekrebis/gamoklebis Semcveli

gamosaxulebiT moicema; sidideebs Soris damokidebulebis gamosaxva cxrilis

saSualebiT.

2. Sekrebis, gamoklebisa da gamravlebis Semcveli ricxviTi da asoiTi gamosaxulebebi

da maTi gamartiveba.

3. Sekrebisa da gamoklebis Semcveli ricxviTi utolobebi da maTi Tvisebebi.

4. teqsturi amocanebi, romlebic Sekrebis, gamoklebisa da gamravlebis Semcveli

ricxviTi an erTi asoiTi aRniSvnis Semcveli algebruli gamosaxulebiT ixsneba.

mimarTuleba: geometria da sivrcis aRqma

maT. V.7. amoicnobs, aRwers da gamosaxavs geometriul figurebs


• uTiTebs wris/wrewiris elementebs; koreqtulad iyenebs wrewirTan/wresTan

dakavSirebul terminebs (centri, diametri, radiusi, qorda).

• yofs wrewirs/wres tol (naxevari, meoTxedi) rkalebad/seqtorebad; iyenebs maT

kuTxeebis Sesadareblad da dasajgufeblad (blagvi, marTi, maxvili da gaSlili).

• amzadebs marTkuTxa paralelepipedisa da kubis Slils; mocemuli Slilis mixedviT

amzadebs models da asaxelebs miRebul figuras.

maT. V.8. adgens mimarTebebs figurebs Soris da figuris elementebs Soris


• axdens samkuTxedebis klasifikacias misi kuTxeebis mixedviT (blagvkuTxa, marTkuTxa,

maxvilkuTxa).

• uTiTebs brtyeli figuris paralelur da urTierTTanamkveT gverdebs, msjelobs

gadaikveTeba Tu ara mocemuli gverdebi gagrZelebis Sedegad.

14

• sivrculi figuris modelze uTiTebs paralelur da urTierTTanamkveT waxnagebs,

msjelobs gadaikveTeba Tu ara mocemuli waxnagebi maTi gavrcobis Sedegad.

maT. V.9. poulobs da adarebs brtyeli figurebis farTobebs


• dafaravs figuras erTnairi aragadamfaravi figurebiT da asaxelebs dasafarad saWiro

figurebis mTlian raodenobas.

• figuraTa urTierTSeTavsebiT adarebs an afasebs figurebis farTobebs. (mag. rodesac

erTi Tavsdeba meoreSi, maSin misi farTobi ufro naklebia).

• iyenebs farTobis adiciurobas aragadamfaravi figurebis kombinaciiT miRebuli

figuris farTobis mosaZebnad.

maT. V.10. orientirebs badiT dafarul areze


• koordinatebis (simboloTa wyvilis) gamoyenebiT aRwers mdebareobas da iyenebs am xerxs

realur viTarebaSi (mag. kinoTeatri, gemebis CaZirobana, Wadrakis dafa, ruqaze

obieqtis moZebna).

• gadaadgildeba ujrian furcelze instruqciebis mixedviT da aRwers, rogor miaRwevs

mocemuli ujridan sxva ujramde (mag. ori ujra marcxniv, Semdeg erTi ujra

zeviT).

• aRwers ruqaze ori an meti punqtis urTierTmdebareobas oTxi mimarTulebis gamoyenebiT

(mag. CrdiloeTiT, dasavleTiT).

Sinaarsi

1. wre/wrewiri: centri, radiusi, diametri, qorda, rkali, seqtori.

2. kuTxe (araformalurad, rogorc mravalkuTxedis elementi).

3. samkuTxedis saxeobebi: blagvkuTxa, marTkuTxa, maxvilkuTxa.

4. mravalkuTxedis gverdebs Soris mimarTeba: paraleluri da TanamkveTi gverdebi;

mravalwaxnagas waxnagebs Soris mimarTeba: paraleluri da TanamkveTi waxnagebi.

5. farTobi (araformalurad, rogorc erTnaeri aragadamfaravi figurebiT dafarul

figuraSi damfaravi figurebis raodenoba).

6. koordinatebi (araformalurad, rogorc adgilmdebareobis miTiTeba simboloTa

wyviliT).

mimarTuleba: monacemTa analizi, albaToba da statistika

maT. V.11. moipovebs dasmuli amocanis amosaxsnelad saWiro Tvisebriv da raodenobriv

monacemebs


• SekiTxvebis mocemuli CamonaTvalidan SearCevs da iyenebs saWiro monacemTa

Sesagroveblad Sesaferis SekiTxvas/SekiTxvebs.

• mocemul TemasTan dakavSirebiT svams kiTxvebs Sesaferisi formiT (Ria, daxuruli,

15

ramdenime alternatiuli arCevanis momcveli) da am kiTxvebis saSualebiT moipovebs

saWiro monacemebs.

• irCevs monacemTa Segrovebis Sesaferis saSualebas (dakvirveba, gazomva, monacemTa

amokreba mocemuli erTobliobidan) da iyenebs mas, asabuTebs Tavis arCevans.

maT. V.12. warmoadgens Tvisebriv da raodenobriv monacemebs dasmuli amocanis

amosaxsnelad xelsayreli formiT


• klasificirebuli monacemebisTvis calsaxa Sesabamisobis miTiTebuli wesiT qmnis

piqtogramas, romlis erTi simbolo Seesabameba ramodenime monacems.

• qmnis martiv cxrils araumetes oci klasificirebuli da dalagebuli monacemisTvis

(mag. gansazRvravs Wdeebs, saTaurs, svetebisa da striqonebis raodenobas da adgens

monacemTa cxrils).

• klasificirebuli monacemebisTvis urTierTcalsaxa Sesabamisobis wesiT qmnis

svetovan diagramas ujrebian furcelze (mag. gansazRvravs Wdeebs, saTaurs, svetebis

raodenobas da aferadebs ujrebiani furclis Sesabamisi sigrZis zolebs).

maT. V.13. akeTebs Tvisebriv da raodenobriv monacemTa interpretaciasa da

elementarul analizs


• svams saZiebo/Semajamebel kiTxvebs monacemebis Sesaxeb, romlebic warmodgenilia

svetovani diagramis saxiT (mag. nayinis ramdeni gansxvavebuli saxeoba unda viyidoT

klasis zeimisTvis? TiToeuli saxeobis ramdeni nayini? nayinis romeli saxeoba

uyvars ufro met Cvens Tanaklasels –Sokoladis Tu marwyvis? nayinis romeli

saxeobaa yvelaze popularuli Cveni klaselebisTvis? gogonebisTvis? vaJebisTvis?

ratom?).

• adarebs monacemTa or erTobliobas da warmoaCens Tvisebriv da raodenobriv msgavsebasa

da gansxvavebas maT Soris (Tvisebrioba ukavSirdeba jgufSi monacemTa gvarobas/tips,

monacemTa ganmeorebadobas, pozicias da Tanmimdevrobas, gamorCeul monacemebs).

• gamoTqvams varauds monacemTa safuZvelze (mag. gamokiTxvis “vin ra gadaadgilebis

saSualebas iyenebs skolaSi misasvlelad” Sedegebis safuZvelze gamoTqvams

varauds, daaxloebiT ramdeni bavSvi cxovrobs skolasTan axlos).

Sinaarsi

1. Tvisebrivi da raodenobrivi monacemebis Segrovebis saSualebani: gazomva, dakvirveba,

gamokiTxva; monacemTa amokreba monacemTa umartivesi wyaroebidan (mag. cnobari,

katalogi).

2. Tvisebrivi da raodenobrivi monacemebis organizacia: monacemebis klasifikacia

(garda raodenobriv monacemTa dajgufebisa intervalebad).

3. monacemTa mowesrigebuli erTobliobebis raodenobrivi da Tvisebrivi niSnebi:

gamorCeuli (mag. eqstremaluri, iSviaTi) monacemebi.

4. monacemTa warmodgenis saSualebani raodenobrivi da Tvisebrivi monacemebisTvis:

sixSireTa cxrili, piqtograma, svetovani diagrama.

16

Sinaarsisa da miznebis ruka

Temebis CamonaTvali

miznebi: Sinaarsis gardaqmna

miznad; erovnul saswavlo gegmaSi

miTiTebuli Sesabamisi punqtebi

savaraudo

saswavlo

dro

naturaluri ricxvis wa

kiTx visa da Caweris wesebi.

milionze meti ricxvebi,

maTi Sedareba da dalageba

milionamde naturaluri ricxvebis

wakiTxvis da Caweris gameorebis

Semdeg eufleba milionze didi

ricxvebis wakiTxvis, Caweris,

klasifikaciis wesebs, poulobs axali

ricxviTi saxeliT mocemuli ricxvis

rigs. V. 1.

9 sT

naturaluri ricxvebis

Sekreba da gamokleba.

amocanebis amoxsna.

ricxviTi sxivi.

naturaluri ricxvebis

gamravleba da gayofa

amocanebis konteqstis mixedviT

naturalur ricxvebze moqmedebaTa

SerCevisa da Sesrulebis unaris

gamomuSaveba V. 3.

18 sT

sakoordinato bade,

amocanebis amoxsna.

mdebareobis aRwera simboloTa

wyvilebiT da maTi gamoyenebis unari;

ujrebian furcelze instruqciebis

mixedviT gadaadgilebaTa aRwera.

rukaze ramdenime punqtis

urTierTmdebareobis aRwera. V. 10

6 sT

asoiTi gamosaxulebis

Sedgenisa da gamartivebis

magaliTebi.

amocanebis amoxsna

realuri viTarebis aRwerisas

Sesabamisi tolobis, utolobis

an gantolebis Sedgenis unaris

gamomuSaveba V. 6.

erTi cvladis Semcveli asoiTi

gamosaxulebis mniSvnelobis povna V.5

7 sT

xarisxi. xarisxis gamoyeneba

xarisxis gamoyeneba poziciuri

sistemiT ricxvebis Cawerisas,

ricxvebis Caweris sxva wesebi da maTi

Sedareba poziciurTan V. 1

3 sT

ricxvis jeradebi da

gamyofebi.

martivi da Sedgenili

ricxvebi.

mocemuli erTniSna da orniSna

ricxvebis jeradebisa da gamyofebis

povna; kenti, luwi, martivi, Sedgenili

ricxvebis gansxvavebulobis aRwera;

2ze da 5ze gayofadobis dasabuTeba.

V. 1

6 sT

17

wrewiri, wre da maTi

nawilebi

wrewirTan dakavSirebuli terminebis

koreqtulad gamoyeneba (centri,

diametri, radiusi) wrewiris dayofa

tol rkalebad, wris dayofa tol

seqtorebad V.7

4 sT

wiladi, rogorc mTelis

nawili.

wiladi, rogorc

erTobliobis nawili.

wiladi da naturaluri

ricxvebis gayofa

wiladebis wakiTxva da Cawera, wiladis

cnebis sxvadasxvagvari interpretacia

da maT Soris kavSirebi V. 2

9 sT

wiladebis toloba, wiladis

gamartiveba.

Sereuli ricxvebi.

wiladebis Sedareba.

wiladebis Sedarebisas sxvadasxva

xerxis gamoyeneba V. 2

wiladebis cvlilebis aRwera

mricxvelis an mniSvnelis cvlilebisas

V. 3

9 sT

wiladebis jamis Sefaseba.

wiladebis Sekreba.

wiladebis gamravleba

naturalur ricxvze.

wiladebis gamokleba.

Sereuli ricxvebis Sekreba.

Sereuli ricxvebis

gamokleba.

erTnairi mniSvnelis mqone

wiladebze ariTmetikuli moqmedebis

demonstrireba, moqmedebaTa Sedegis

interpretacia modelis gamoyenebiT.

moqmedebaTa Tvisebebis gamoyeneba

Sereul ricxvebze moqmedebaTa

Sesrulebisas. V. 3

18 sT

droisa da

sigrZis erTeulebi.

masis erTeulebi.

amocanebis amoxsna (naSTis

gamoyeneba).

sigrZis erTeulebis erTmaneTTan

dakavSireba,

drois 12 da 24 saaTiani formatebis

gamoyeneba; ariTmetikul moqmedebaTa

gamoyenebiT drosa da drois

intervalis gansazRvra.

naSTiT gayofis gamoyeneba zomis

erTeulebs Soris kavSirebis

gamoyenebisas V. 4

10 sT

kuTxe. kuTxeebis Sedareba.

samkuTxedebis Sedareba.

mravalkuTxedis kuTxeebi

da gverdebi.

kuTxeeebis Sedareba da klasifikacia

(wrewiris, wris gamoyenebiT) V. 7.

samkuTxedebis klasifikaciis unari,

brtyeli figuris gverdebis Semcveli

wrfeebis urTierTganlagebis aRwera

9 sT

18

farTobi. farTobis

Sefaseba

figuraTa urTierTSeTavsebiT

farTobebis Sedareba, Sefaseba,

figuris farTobis Tvisebebis

gamoyeneba figuraTa kombinaciiT

miRebuli figuris farTobis

mosaZebnad V. 5

3 sT

marTkuTxa paralelepipedis

waxnagebi da wiboebi.

sivrculi figurebis waxnagebis

Semcveli sibrtyeebis da

wiboebis Semcveli wrfeebis

urTierTganlagebis aRwera.

kubis da marTkuTxa paralelepipedis

Slilebi V. 7. V. 8.

4 sT

viyenebT monacemTa

Segrovebis sxvadasxva xerxs

monacemTa mopoveba

SekiTxvebiT, dakvirvebiT, amokrebiT

da arCevanis dasabuTeba. V. 11.

3 sT

warmovadgenT monacemebs

cxrilis an diagramis saxiT.

vmsjelobT monacemTa

Sesaxeb.

vxsniT amocanebs cxrilis

an diagramis gamoyenebiT

monacemTa warmodgenis sxvadasxva

xerxis gamoyeneba (cxrilebi,

piqtogramebi, svetebiani diagramebi)

V. 12

monacemTa interpretaciisa

da analizis sxvadasxva xerxis

gamoyeneba (saZiebeli am Semajamebeli

kiTxvebis dasma, Sedareba; msgavsebis,

gansxvavebis aRmoCena. monacemTa

safuZvelze varaudebis gamoTqma).

V. 13

12 sT

SeniSvna:

• miTiTebul droSi naangariSevia Sesabamisi weriTi samuSaoebis CatarebagarCevac;

• maswavlebels saTadarigod rCeba miaxloebiT 14 sT.

19

saswavlo masalis wardgenis fazebi da gakveTilis dagegmvis ZiriTadi principebi

saswavlo procesis organizaciis ZiriTadi forma gakveTilia. meTodikur literatura

Si cneba _ `gakveTili~ _ ZiriTadad, ase aRiwereba: gakveTili logikurad dasrulebuli,

mTliani saswavloaRmzrdelobiTi procesis garkveuli SemosazRvruli monakveTia. masSi

rTul urTierTdamokidebulebaSia procesis yvela ZiriTadi elementi: Sinaarsi, mizani, sa

Sualebebi, meTodebi, organizacia. yovel gakveTilze gansazRvruli saganmanaTleblo da

aRmzrdelobiTi amocanebi wydeba; rac ZiriTadad konkretuli saswavlo masalis ganxilvis

procesSi mimdinareobs. dawyebiT klasebSi prioritetulia saswavloaRmzrdelobiTi funq

cia. erovnuli saswavlo gegmis mixedviT swavlebis am safexurze unda Seiqmnas safuZvlebi

sabazo safexurze efeqtiani swavlisTvis. VVI klasebi am mxriv swavlebis mniSvnelovani

etapia _ izrdeba sagnobrivi codnis SeZenis, arsebuli codnis ganzogadebis intensivoba.

yuradReba eqceva im unarCvevebis, damokidebulebebis ganviTarebas, romlebic moswavles

mTeli cxovrebis ganmavlobaSi dasWirdeba, maT Sorisaa fizikuri da sulieri Tvisebebis,

midrekilebebis gamovlena da ganviTareba.

maTematikis gakveTilisadmi wayenebuli mTavari moTxovnaa ZiriTadi didaqtikuri amo

canis arseboba _ im Temis Seswavlis miznis arseboba, romlis gadawyvetis procesi mocemul

gakveTilze mimdinareobs. yoveli gakveTilis win kargad unda gaiazroT misi Sinaarsi da miz

ani. amaSi maswavleblis es wignic dagexmarebaT _ mowodebulia miznebisa da Sedegebis ruka,

yoveli Temis axsnas win uZRvis rukaSi warmodgenili miTiTebebis sruli analizi. miznis Se

sabamisad unda iyos gaazrebuli da dagegmili saswavlo masalis wardgenis fazebi.

am miznis gaazreba moswavleebisTvis uaRresad mniSvnelovania. moswavleebTan saubari

unda daviwyoT ara imiT, Tu ras vaswavliT, aramed iseTi situaciis SeqmniT, roca moswavle

motivirebulia da poulobs pasuxebs kiTxvaze _ `risTvisaa saWiro~. motivacia SesaZlebe

lia aqtualuri praqtikuli amocanis dayenebiTa da misi amoxsnis xerxebis ZiebiT daviwyoT,

an maTematikis Sinagani kanonzomierebebis, siZneleebis gadalaxvisas warmatebis miRweviT

miRebuli STabeWdilebebis gaazrebiT. `did~ ricxvebze gadasvla orive aspeqtis warmod

geniT mimdinareobs, TandaTan xdeba im codnis gadmotanagavrceleba didi ricxvebze, rac

ukve icoda moswavlem milionamde farglebSi. gansakuTrebiT mniSvnelovania amocanis dasma

da moswavleebTan erTad misi amoxsnis Ziebis procesis ganxorcieleba. axali erovnuli sas

wavlo gegmis mixedviT saganmanaTleblo procesis centrSi dgas moswavle da misi Sesabamisi

interesebi; swavla niSnavs informaciis dagrovebas, unarCvevebisa da damokidebulebebis

ganviTarebas. swavlebaSi moiazreba ara erTi konkretuli gzis gavla, aramed maswavleblisa

da moswavlis mier erToblivad SerCeuli optimaluri variantis Zieba. amasTanave, es procesi

SeiZleba sxvadasxva formiT mimdinareobdes, es formebia _ muSaoba jgufebad (SesaZlebelia

ororadac), mTeli klasis monawileobiT, individualurad da a. S. maswavlebelma swavleba

ise unda warmarTos, rom Temis Seswavlis procesis ZiriTadi Semoqmedi Tavad moswavle aR

moCndes. maswavlebeli warmmarTvelis, `diriJoris~ funqcias unda asrulebdes; zogierTi

mecnieri maswavleblis am rols mebaRis rols adarebs _ yvavilebi mebaRis xelSewyobiT

izrdeba da alamazebs garemos.

maswavleblis sarekomendacio wigni dagexmarebaT warmatebiT daZlioT siZneleebi, gan

sakuTrebiT im davalebebis ganxilvisas, romlebic azrovnebis zeda sami donis (analizi, sin

20

Tezi, Sefaseba) amoqmedebasTan aris dakavSirebuli. nimuSis mixedviT davalebebis Sesrulebis

(gamoyenebis done) garda gvaqvs davalebebi, romlebic moswavlis SemoqmedebiTi potencialis

zrdasa da amoqmedebas iwvevs (analizisa da sinTezis done) da Sefasebisken (Sefasebis done)

aris mowodebuli. azrovnebis zeda doneebis asamoqmedeblad mowodebulia specialuri meTo

dikebi; magaliTad, dasazustebeli SekiTxvebis CamonaTvali, romelTa saSualebiT xdeba `mog

zauroba~ qveda doneebidan zeda doneebisken da Semdgom _ ukusvliT _ zeda doneebidan

qveda doneebisken.

maswavleblis wignis da moswavlis wignis struqturac dagexmarebaT TiToeul etapze

(sawyisi, damamTavrebeli) Sesasrulebeli savarjiSoebis SerCevaSi. yovel Temas ukavSirde

ba savarjiSoebi, romlebic sxvadasxva inteleqtualuri unaris (klasificirebis, Sedarebis,

gamoyenebis, TviTSefasebis Catarebis, amocanis amoxsnis Ziebis unaris) ganviTarebas emsax

ureba.

gakveTilze Seswavlili faqtebi TavisTavad aris mniSvnelovani, Tumca, kidev ufro mniS

vnelovania is, rom maTi Seswavlis dros ganxorcielebuli procesi Tavis kvals tovebs _

mimdinareobs sxvadasxva inteleqtualuri unaris ganviTareba.moswavlisa da maswavleblis wignebSi warmodgenili masaliT, swavlisadmi SemoqmedebiTi

midgomiT Tqven uTuod SeZlebT sxvadasxva akademiuri donisa da inteleqtualuri SesaZleblobebis moswavleebTan nayofieri saswavlo procesis warmarTvas. moswavleebTan deferencirebuli midgomiT (rac maTi individualuri Tvisebebis gaTvaliswinebasac gulisxmobs), mizanmimarTuli intensiuri ZalisxmeviT uTuod SeZlebTa droebiT CamorCenili

moswavleebis CarTvas saswavlo procesSi.

axla CamovayaliboT ramdenime sasargeblo rekomendacia.

• maswavlebeli cdilobs, rom yoveli axali SemecnebiTi amocana TviT moswavlem Camo

ayalibos.

• maswavleblis xelmZRvanelobiT da moswavleTa ZalisxmeviT _ dakvirvebis, cdis, konk

retuli SemTxvevebis analizis Sedegad iqmneba warmodgena arsebul kanonzomierebaze.

•maswavleblis xelmZRvanelobiT mimdinareobs amocanis gadawyvetis gzis Zieba; amocanis,

amoxsnis gegmis Sedgena; rasac xSirad mosdevs moswavleebis mier am gegmis realizacia.

axla SevexoT saswavlo procesisTvis mzadebis zogad sqemas.

• maswavleblis muSaoba saswavlo wlis win.

• gakveTilebis sistemis gaazreba _ erTi raime Temis gameorebas, ganzogadebas, axlis

Ziebas gakveTilebis erToblioba eZRvneba.

• konkretuli gakveTilisTvis momzadeba.

momzadebisTvis maswavlebelma unda gamoiyenos saxelmZRvanelo, maswavleblis wigni da,

saWiroebis SemTxvevaSi, iq miTiTebuli literatura. cxadia, pedagogi, kargad unda icnob

des erovnul saswavlo gegmas. am gegmis mixedviT Sedgenil Sinaarsisa da miznebis rukas,

SemoTavazebul sanimuSo gakveTilebis scenarebs. sarekomendacio wignis Sesavali maswavleb

els acnobs saxelmZRvanelos agebis ZiriTad principebs da meTodikur orientirebs. winaswar

ki man aqtiuri monawileoba unda miiRos Tavisi skolis saswavlo gegmis SemuSavebaSi, raTa

daakonkretos, daazustos, Seavsos erovnuli gegmiT gaTvaliswinebuli saganmanaTleblo da

aRmzrdelobiTi programebi.

amrigad, pedagogebi SemoqmedebiTad unda miudgnen Cvens rekomendaciebs, sakuTari ga

mocdilebisa da klasis moswavleTa Taviseburebebis gaTvaliswinebiT, unda daazuston,

SeiZleba Secvalon kidec zogierTi Tema da am Temis gadacema. kidev erTxel SegaxsenebT

erovnul saswavlo gegmaSi mocemul miTiTebas: `saxelmZRvanelo aris erTerTi saSualeba

da ara erTaderTi, romelic emsaxureba erovnuli saswavlo gegmis sagnobriv programaSi mo

cemuli Sedegebis miRwevas... `saxelmZRvanelo ar aris TavisTavad programa _ Sesabamisad,

maswavlebeli ar unda iyos saxelmZRvaneloSi mocemuli `masalis gavlaze~ orientirebuli,

aramed aqcenti unda gaakeTos saxelmZRvaneloSi mocemuli teqstebis, savarjiSoebisa Tu

21

ilustraciebis gamoyenebaze sagnobrivi standartebis Sedegebis misaRwevad. sworia, roca

maswavlebeli argebs saxelmZRvanelos moswavleebis saWirobebs da ara piriqiT, rodesac

maswavlebeli cdilobs moergos saxelmZRvanelos~, maswavleblis sarekomendacio wignSi Cven

SemogTavazebT konretuli Temis gadacemis, savarjiSoebis gavlis Tanamimdevrobis gansxvave

bul variantebsac.

gakveTilis dagegmvisas gasaTvaliswinebelia Casatarebeli procesis tipi:

• gameoreba;

• axali masalis gacnoba;

• masalis ganmtkiceba;

• codnis Semowmeba;

• sxva tipis gakveTilebi (gakveTili bunebaSi, gakveTiliproeqti, ...)

zogierTi tipis procesi (gansakuTrebiT V klasSi), rogorc wesi, yovel gakveTilze mim

dinareobs (gameoreba, axali masalis gacnoba, ganmtkiceba, Semowmeba) _ SesaZlebelia, sxva

dasxva moculobiT.

gakveTilze mTavaria vaswavloT da aRvzardoT: swavleba ar niSnavs mxolod codnis gad

acemas, swavla codnis SemoqmedebiTad dauflebas unda niSnavdes, codnis gamoyenebis unaris

ganviTarebas unda gulisxmobdes.

saswavlo masalis dasabuTebuli SerCeva iTvaliswinebs Semdeg moTxovnebs:

• saswavlo masalis Sesabamisoba Temis mizanTan;

• gakveTilze Sesasrulebeli samuSos moculobis swori gansazRvra;

• optimaluri Tanafardoba konkretulsa da zogads Soris;

• Teoriasa da praqtikas Soris aucilebeli urTierTkavSiris ganxorcieleba.

maswavlebelma, rogorc wesi, Tavidan bolomde detalurad unda gaiazros gakveTili,

winaswar, drois mixedviT unda iyos ganawilebuli mTeli samuSao, Tumca, rogorc yoveli

SemoqmedebiTi procesi, gakveTilic Seitans, albaT Tqven gegmebSi cvlilebebs.

kiTxvebze pasuxebis gacemis sistema ar unda iyos erTferovani _ mxolod warmatebul

moswavleebTan mimarTebiT ar unda SemovifargloT. moswavlis raime mosazrebas myisve nu

upasuxebT, swor pasuxsac ki maSinve nu daeTanxmebiT _ gaakeTeT pauza, iqneb garkveuli

eWvic ki gamoTqvaT wamoyenebuli mosazrebis sisworis mimarT. amiT miaRwevT imas, rom mos

wavleebi daubrundebian dasmuli kiTxvis analizs da male WeSmariti daskvna _ swori pasuxi

_ klasis dominantur mosazrebad gadaiqceva. klasi, erToblivi ZalisxmeviT, `gaiZulebT~

daeTanxmoT moswavlis pozicias. es axarebs, amxnevebs da aerTianebs moswavleebs. es maTi

erToblivi azris gamarjvebaa, Tqveni mizanic xom es aris _ moswavle Semoqmed, codniT

aRWurvil, iniciativian, xalisian axalgazrdad CamoayaliboT. maTematika mZlavri emociuri

muxtis matarebelia da misi amoqmedeba Tqveni ZalixmeviT miiRweva.

axali masalis ganmtkicebis procesi e. w. `testebis~ `amoxsniT~ SeiZleba daviwyoT. maTi

Sesruleba, ZiriTadad, swori pasuxis SerCeviT unda Semoifarglos _ zogjer SeiZleba ko

mentarebis gakeTebac gaxdes saWiro, moswavle TandaTan eCveva am tipis davalebebis swrafad

Sesrulebas.

swavlebis erTerTi saintereso da mniSvnelovani formaa jgufuri muSaoba. es muSao

ba SeiZleba saswavlo procesis erTerTi Semadgeneli nawili iyos _ daukavSirdes axali

masalis gaazrebas, praqtikul saqmianobas, an Semajamebeli daskvnis gamotanas; mas SeiZleba

mTeli gakveTilic daeTmos, jgufuri muSaobis warmarTvaSi saxelmZRvaneloSi warmodgenili

masala, an Sesabamisi amocanebis sistema dagexmarebaT. jgufur muSaobas SeiZleba Sejibrebis

saxec mivceT, saxelmZRvaneloSi warmodgenili TamaSebic SeiZleba jgufuri muSaobis for

miT warvmarToT.

Teoriuli masalis gadmocemis Cveneuli meTodika saSualebas gaZlevT airCioT jgufuri

muSaobis moxerxebuli forma; magaliTad, paeqroba jgufebs Soris, romlebsac kapitnebi xe

lmZRvaneloben. jgufebs SeiZleba erTnairi davalebebi mivceT. maSin Tavad jgufebi aqti

22

urad erTvebian miRebuli Sedegebis saerTo ganxilvaSi _ ukeT gauweven oponirebas sxvaTa

mosazrebebs. SeiZleba jgufebSi samuSaobi ganawildes da moswavleTa erToblivi muSaoba

raime erTi amocanis gadawyvetas gulisxmobdes; jgufis wevrebma unda gaiazron, rom maTi

warmateba erTmaneTzea damokidebuli. es zrdis wevrebis pasuxismgeblobas. TanamSromlo

bis xarisxis asamaRleblad SeiZleba cadoT saerTo Sefaseba _ jgufebi asruleben saerTo

davalebas da maTi wevrebi iReben erTnair Sefasebas. Tumca es wesi zogierTi moswavlisTvis

ar unda iqces daumsaxurebeli maRali Sefasebebis miRebis saSualebad.

sasurvelia, rom jgufuri muSaobisTvis gankuTvnili amocana sxvadasxva xerxiT ixsne

bodes. igi saintereso msjelobis saSualebas iZleva.

jgufuri muSaobis daskvniT etapze _ Sedegebis sajaro ganxilvisas Zalze mniSvnelo

vania pedagogis roli moswavleTa mier polemikis koreqtuli formebiT warmarTvaSi, sxvaTa

mosazrebis mosmenis unaris damkvidrebaSi, warmatebisa Tu warumateblobis gamoxatvis mis

aRebi formebis CamoyalibebaSi.

23

Sefasebis zogadi principebi

axal erovnul saswavlo gegmaSi saTanado adgili ukavia Sefasebis principebs. am gegmis

mixedviT TviT skolam unda daadginos zogadi Sefasebis principebi, rac yvela maswavle

blisTvis saxelmZRvanelo principebad unda iqces. kerZod, unda dadgindes is minimaluri

barieri, romlis gadalaxva aucilebelia klasidan klasSi gadasasvlelad.

erovnuli saswavlo gegma gTavazobT im aucilebel midgomebs, romlebsac Sefasebis

sistema unda iTvaliswinebdes:

`akademiuri moswrebis Sefaseba unda iyos xSiri da mravlmxrivi. unda Sefasdes ara mar

to informaciis floba, aramed SeZenili unarCvevebi, azrovnebis demonstrirebis formebi

da sxva. saswavlo procesis warmatebiT warmarTvisTvis ar aris sakmarisi moswavle mxolod

sakontroloebis da testebis Sedegebis safuZvelze Sefasdes. maswavlebeli unda afasebdes

moswavles dakvirvebis Sedegad dagrovili monacemebis, namuSevrebis krebulebis, prezen

taciebis, moswavleTa TviTSefasebis, jgufuri muSaobis Tu sxva tipis aqtivobebis mixed

viT. maswavlebelma Sefasebisas unda gaiTvaliswinos moswavlis saganmanaTleblo procesSi

CarTulobis xarisxi, saxlSi micemuli davalebis Sesrulebis xarisxi, gakveTilze aqtiuroba,

SemoqmedebiToba da sxva. aucilebelia, moswavlem winaswar icodes, Tu ra kriteriumebiT

fasdeba misi aqtivoba... moswavle arasaTanado saqcielisTvis (romelic pirdapir ar aris

dakavSirebuli saswavlo procesTan) sagnis niSniT ar unda daisajos...~ Sefasebas ara marto

ar unda dahyvebodes angariSsworebis elferi, aramed Sefaseba unda aRiqmebodes samomavlo,

pozitiuri cvlilebebisken mimarTuli qmedebad.

erovnuli saswavlo gegmiT gansazRvrulia, rom niSnebis daweris sistemis (trimestruli

niSani, wliuri niSani da a. S.) moqmedeba V klasidan iwyeba. amasTanave, oficialuri statusi

ara aqvs mimdinare yoveldRiur niSnebs, amitom maswavlebeli sakuTar wignakebSi agrovebs

Sefasebis Sesabamis monacemebs yovel moswavleze. iyenebs Sefasebis or ZiriTad tips _ ganm

sazRvrels da ganmaviTarebels. am ukanasknelis gamoyenebisas niSnis daweras Tan erTvis ko

mentarebi, romliTac vuxsniT moswavles Zlier da sust mxareebs, vexmarebiT mas Sedegebis

gaumjobesebaSi.

moswavlis niSani unda ganisazRvros (magaliTad, trimestruli niSani) mis mier sagnis

Seswavlis sxvadasxva komponentisgan _ gakveTilze CarTuloba, saSinao davaleba, saklaso

davalebebi, sakontrolo wera, proeqti da sxva.

am komponentebisgan skola kaTedrebis meSveobiT SearCevs nawils (sams mainc) romelT

ac wonebi mieniWeba. es wonebi SeiZleba sxvadasxva iyos trimestrebis mixedviTac. trimes

tris Sefaseba (umaRlesi qula 10) SeiZleba sxvadasxva komponentis Sefasebisgan davadginoT.

magaliTad, wonebi SeiZleba aseTi iyos: prezentacia _ 25%, gakveTilze aqtiuroba _ 30%,

sakontrolo wera _ 25 %, saSinao davaleba _ 20%. Tu TiToeuli komponentis saSualo Se

fasebebia 6, 9, 10, 8 maSin Sefaseba SeiZleba ase gamovTvaloT:

0,25⋅6+0,3⋅9+0,25⋅10+0,2⋅8=8,3damrgvalebis Semdeg gvaqvs Sefaseba 8.

Sefasebis sxva, Zalze mniSvnelovani detalebis gacnobis mizniT pedagogebs vurCevT xSir

ad miubrundnen xolme erovnuli saswavlo gegmis saTanado miTiTebebs, radgan es komponenti

saswavlo procesSi Zalze frTxil, obieqtur, gamWvirvale muSaobas saWiroebs. amave gegmis

moTxovniT winaswar unda gairkves: `saSemodgomo~ gamocdasTan dakavSorebuli sakiTxebi, sa

patio mizezebiT xSirad gamcden moswavleebTan damokidebulebis sakiTxebi da sxva.

24

I Tavi

gameoreba. milionze meti ricxvebi

saxelmZRvanelos pirvel Tavs gamorCeuli datvirTva aqvs _ xdeba moswavleTa CarTva

saswavlo ritmSi, swavlis axal _ ufro maRal safexurze asvla, mTeli saswavlo wlisTvis

`tonis micema~. xangrZlivi sazafxuli ardadegebis Semdeg uaRresad aqtualuria adre Seswav

lili sakiTxebis xelaxali gaazreba da SeZenili unarebis gamyarebaganviTareba. aq CamoTv

lil yvela mizans pasuxobs Cveni saxelmZRvanelos pirveli Tavi _ warmodgenili masaliTa

da misi gadacemis meTodikiT. adre ganxilul sakiTxebs moswavleebs mTeli wlis ganmavloba

Sic SevaxsenebT xolme _ axali sakiTxebis Seswavlis procesSi.

naturalur ricxvTa Tvisebebisa da maTze moqmedebebis safuZvliani codna, ricxvTa poz

iciur sistemaze naTeli warmodgenis Camoyalibeba maTematikis Seswavlis procesis erTer

Ti qvakuTxedia. am sakiTxebis codnis gaRrmavebis da gafarToebis mizniT davixmarT samo

tivacio amocanebs, TvalsaCino warmodgenebs.

pirvel sayuradRebo nabijebs gadavdgamT algebris mimarTulebiTac. SevadgenT asoiT

gamosaxulebebs, ganvixilavT maTi gamartivebis sakiTxebs. amave TavSi moswavleebi pirvel

(arasrul) warmodgenas Seiqmnian sakoordinato sibrtyeze, roca gaecnobian sakoordinato

bades da amoxsnian saTanado amocanebs. gavimeorebT naturaluri ricxvis naturalurmaCvene

blian xarisxs da misi gamoyenebis magaliTebs.

1. gavixsenoT naturaluri ricxvebis wakiTxvisa da Caweris wesebi

mizani. naturaluri ricxvebis Cawerasa da wakiTxvaSi moswavleTa gawafva; ricxvis

poziciuri sistemiT warmodgenis arsis ukeT gaazreba _ moswavle amoicnobs yoveli pozi

ciis (Tanrigis) mniSvnelobas, wers ricxvebs gaSlili formiT, iyenebs SeZenil codnas amo

canebis amoxsnisas.

es saxelmZRvanelo, rogorc Cveni sxva saxelmZRvaneloebi, ganvlili masalis gameorebiT

iwyeba. kerZod, moswavle kidev ufro unda gaiwafos naturaluri ricxvebis Cawerasa da

wakiTxvaSi (savarjiSoebi 1 6 ), man unda gaixsenos Tanrigebis klasebad gaerTianebis

principi, klasebSi Semavali Tanrigebis saxelwodeba, ricxvebis Cawera gaSlili formiT.

magaliTad,

648 850=600 000+40 000+8 000+800+50+0.

saxelmZRvanelo exmareba maswavlebels gakveTilis Catarebis gegmiT gaTvaliswinebuli

yvela komponenti gaiTvaliswinos; magaliTad moswavleTa motivacia iTvaliswinebs paragrafis

Sesavali, SeiZleba sxva praqtikuli magaliTebis ganxilvac.

ricxvis aseTi saxiT _ gaSlili formiT warmodgena abakis gamoyenebiT ufro Tval

saCino xdeba. saskolo inventarSi abakis arsebobis SemTxvevaSi, gakveTilze moswavleTa

nawilma SeiZleba dafaze daweros ricxvebi, sxvebma ki es ricxvebi abakze warmoadginos da

piriqiT, abakze warmodgenili ricxvebi moswavleebma daweron dafaze da Semdeg daasaxelon.

teqstSi suraTze udidesi eqvsniSna ricxvia warmodgenili: 999 999. suraTze gamosaxul abakze

Sesalebelia Tu ara am ricxvze 1iT meti ricxvis _ milionis warmodgena? rogori abakis

damzadebas SemogvTavazeben moswavleebi?

25

paragrafs axlavs Ria da daxurulboloebiani savarjiSoebi klasSi amosaxsnelad da sa

Sinao davalebad.

amocanebis amoxsnaSi CarTeT yvela moswavle _ TiToeuli swori pasuxi sxvaTa mierac

unda iyos gaziarebuli, xolo araswori pasuxi Tavad moswavleebma unda SeniSnon da imsjelon

mis gamosworebaze.

11 savarjiSos Sesrulebisas moswavleebi ixseneben 1ianiTa da 0ebiT Caweril ricxve

bze (10ze, 100ze da a. S.) gayofisa da gamravlebis wesebs. 13 , 14 , 15 amocanebi aris

16 amocanis amosaxsnelad Semzadeba.

13 amocanaSi erTnairi cifrebis gadanacvleba xdeba, amitom ricxvi ar icvleba.

14 amocanaSi aTaseulis cifris 3iT gazrda gamoiwvevs ricxvis 3 aTasiT (3000iT) gaz

rdas.

15 amocanaSi aTaseulis cifri icvleba 1iT naklebi cifriT, anu ricxvi Semcirdeba 1

000iT, xolo erTeulis cifri icvleba 1iT meti cifriT, anu ricxvi izrdeba 1iT. pasuxi:

ricxvi Semcirdeba 999iT.

16 amocanis amoxsnisas, isev rogorc 13 amocanaSi, moswavleebi daaskvnian, rom aTi

aTaseulisa da erTeulis cifrebi tolia. amasTanave, isini 0isgan gansxvavebulia (ricxvi

ar daiwyeba 0iT). amrigad, aTaseulis Tanrigis cifrze 8iT naklebi SeiZleba iyos mxo lod

cifri 1. moswavleebi am pirobebis gaTvaliswinebiT TanamimdevrobiT Seavseben saZiebeli

xuTniSna ricxvis Tanrigebs: 19••1.

pirobiT, aTeulis cifri aris 1+3=4, aseulis cifri aris 4+1=5. pasuxi: 19541.

klasSi SeiZleba amoixsnas amocanebi rubrikidan `moisazre~. am samuSaos SeiZleba paeqro

bis saxec mieces: moswavleebi daajgufeT ramdenime jgufad (TiToSi 45 moswavle). 15wu

Tiani muSaobis Semdeg moawyveT jgufebis naSromTa mokle prezentacia; Semdeg SeafaseT mo

swavleebi da Sedegebi gaanalizeT.

gTavazobT amocanebis amoxsnis warumatebeli mcdelobis SemTxvevaSi damxmare kiTxvebs:

1 a) • ra feris burTulebi `gvawyobs~? (lurji am mwvane)

• ra feris burTulebi àr gvawyobs~? (wiTeli)

•ras udris im burTulebis odenoba, romlebic àr gvawyobs?~ (9) e. i. 9 burTis amoRebis

SemTxvevaSi SeiZleba cxrave wiTeli aRmoCndes. am SemTxvevaSi meaTe burTula aucileblad

iqneba mwvane an lurji. pasuxi: 10.

b) • ramdenia im burTulebis odenoba, romlebic àr gvawyobs~? (9+4=13).

e. i. SeiZleba 13 burTis amoRebis SemTxvevaSi cametive aRmoCndes aramwvane. am SeTxvevaSi

me14 burTi iqneba aucileblad mwvane. pasuxi: 14.

g) yvelaze ùares~ SemTxvevaSi 13 amoRebuli burTulidan SeiZleba arc erTi ar iyos

mwva ne, maSin me14 burTi aucileblad iqneba mwvane da samive feris burTulebs SevagrovebT.

2 •wignis ramdeni gverdia gadanomrili erTniSna ricxvebiT da maT dasawerad ramdeni

cifria saWiro? (9 gverdi, 9 cifri).

• ramdeni gverdia gadanomrili orniSna ricxvebiT da maT dasawerad ramdeni cifria

saWiro? (90 gverdi, 180 cifri).

• ramdeni gverdia gadanomrili samniSna ricxvebiT da maT dasawerad ramdeni gverdia

saWiro?

(148–99=49 gverdi, 49⋅3=147 cifri)

sul gamoviyenebT 9+180+147=336 cifrs.

26

3 wina amocanis amoxsna daexmareba moswavleebs am amocanis gadawyvetaSi. damxmare

kiTxvebi:

•ramdeni gverdia am wignSi erTniSna ricxvebiT gadanomrili da ramdeni cifria amisTvis

saWiro? (9 gverdi. 9 cifri).

• sul didi ramdeni cifri SeiZleba gamoviyenoT orniSna ricxvebiT gverdebis gadasa

nomrad? (90⋅2=180).

• sul didi ramdeni cifri SeiZleba gamoviyenoT samniSna ricxvebiT gverdebis gadasa

nomrad? (900⋅3=2700).

• aris Tu ara am wignSi oTxniSna ricxvebiT gadanomrili gverdebi? (ar aris, radgan

2004<2700+189).

• ramdeni cifria saWiro orniSna da erTniSna ricxvebiT gadanomrili gverdebis gadasa

nomrad? (180+9=189).

• ramdeni cifria gamoyenebuli am wignis samniSna ricxvebiT gadanomrili gverdebis ga

dasanomrad? (2004–189=1815).

• ramdeni gverdia gadanomrili samniSna ricxvebiT? (1 815:3=605).

• ramdeni gverdia wignSi? (605+90+9=704).

4 ramdenime martiv magaliTze dakvirvebiT moswavleebma SeiZleba daaskvnan, rom mak

lebi 13 000is tolia (7–5=2, 13–6=7 _ TiToeul magaliTSi daakvirdiT, ramdeniT naklebia

sxvaoba saklebze). pirobiT, sxvaoba 5 000iT metia maklebze, e. i. sxvaobaa 13 000+5 000=18 000,

saklebia 13 000+18 000=31 000.

saSinao davalebis 12 savarjiSos a) da d) SemTxvevebSi moswavleebma SeiZleba moiy

vanon magaliTebi. SesaZloa msjelobis sagani gaxdes Semdegi faqti, rom a) SemTxveva yovelT

vis (aucileblad) sruldeba _ yoveli eqvsniSna ricxvi naklebia milionze!

d) SemTxvevac yovelTvis sruldeba, garda erTaderTi SemTxvevisa _ roca es ricxvia

100 000, anu 100 000is garda yvela eqvsniSna ricxvi metia asi aTasze. moswavleTa mier Sei

Zleba dafiqsirdes udidesi da umciresi eqvsniSna ricxvebi da amis Semdeg gaeces kiTxvas

pasuxi.

13 yovel TanrigSi vcdilobT CavweroT rac SeiZleba mcire cifri, viwyebT umaRlesi

Tanrigidan, radgan ricxvi ar SeiZleba daiwyos 0iT, amitom is iwyeba 1iT; Semdeg umciresia

0, 2 da a. S. pasuxi: 102 569.

14 aTi aTaseulis cifri Semcirda 4iT, xolo aseulis cifri gaizarda 4iT. amitom

es ricxvi Semcirda 40 000iT da gaizarda 400iT, anu Semcirda 39 600iT.

15 Semcirda 50 000iT da gaizarda 50iT. amrigad, Semcirda 49 950iT.

16 am ricxvis erTeulis cifri aTi aTaseulis cifrze 8iT nalebia. amrigad, es ricx

via 86 420 an 97 531.

17 Tu moswavleebi sinjvis meTodiT daiwyeben pasuxis Ziebas, darwmundebian, rom

ricxvi 12iT an 21iT ver daiwyeba. marTlac, 12 358? verc 40iT dawyebuli ricxvi gamodgeba

40 448? darCa: 101 123, 112 358, 202 246 da 303 369.

yoveli amocanis dawvrilebiT analizi maswavlebels daakvalianebs moswavlis saxelmZRva

nelos gamoyenebasa da saswavlo procesis organizebaSi.

27

sanimuSo gakveTili #1

aqtivoba: jgufuri muSaoba: naturaluri ricxvebis Caweris aTobiTi sistemis Sesaxeb.

aqtivobis mizani. moswavle ukeT Cawvdes ricxvis Caweris poziciuri sistemis arss, mis

upiratesobas arapoziciur sistemasTan SedarebiT; Cawvdes ricxvis Caweris aTobiTi sistemis

arss; daukavSiros zogierTi sididis sazomi erTeulebi Tvlis aTobiT sistemas. moswavlem

SeZlos miRebuli codnisa da saTanado unarebis aqtiurad gamoyeneba praqtikuli saxis

amocanebis gadawyvetaSi.

aucilebeli wina codna.

• didi ricxvebis wakiTxva da Cawera.

• ricxvis Cawera gaSlili formiT

• ricxvis Cawerisas Tanrigebis klasebad dayofa

• kavSiri sigrZis sazom erTeulebs Soris

• drois sazomi erTeulebi, kavSiri maT Soris.

aqtivobis aRwera

moswavleebs dafaze davuweroT ramdenime ricxvi, romelTa CanawerSi erTi da igive

cifrebia gamoyenebuli. magaliTad, 111, 2222.

moswavleebma TiToeuli ricxvisTvis unda daasaxelon da Caweron sxvadasxva poziciaSi

dawerili erTi da imave cifris mniSvnelobebi, Semdeg ki _ Seadaron es mniSvnelobebi.

magaliTad,

2 2 2 2 anu 2 2 2 2 anu 2 2 2 2

2000 + 200 + 20 + 2 200⋅10 + 20⋅10 + 2⋅10 + 2 2⋅10⋅10⋅10 + 2⋅10⋅10 + 2⋅10 + 2⋅1

ramdenjer metia yoveli momdevno (ufro maRali) Tanrigis cifris mniSvneloba? (10

jer).ramdennair cifrs viyenebT xolme ricxvebis Cawerisas? (10s)swored amitom, ricxvebis am wesiT Cawerisas vRebulobT naturaluri ricxvis aTobiT

Canawers da iqmneba naturaluri ricxvTa aTobiTi sistema. radgan am wesiT ricxvis Cawerisas

yoveli cifris mniSvneloba CanawerSi cifris adgilzea (poziciazea) damokidebuli, amitom

ricxvTa aTobiT sistemas uwodes poziciuri. Tu maRalia klasis akademiuri done, SeiZleba am

etapzeve arsebiTad gaaRrmavoT moswavleTa warmodgenebi poziciur sistemebze; magaliTad,

erToblivad aRweroT — Zalze mokled (aTobiTis msgavsad) xuTobiTi sistema da dasvaT

amocana:ra ricxvs warmoadgens 2222 Canaweri xuTobiT sistemaSi?

2 2 2 2

2⋅5⋅5⋅5 + 2⋅5⋅5 + 2⋅5 + 2xuTobiT sistemaSi 2222 warmoadgens 312s (2⋅5⋅5⋅5+2⋅5⋅5+2⋅5+2=312) aTobiT sistemaSi. vfiq

robT, am etapze es magaliTic ikmarebs.

amis Semdeg moswavleebs davyofT jgufebad da maT vTavazobT saxelmZRvaneloSi oTxi

cxrilis Sevsebas (I, II, III da IV). isini akvirdebian, aRweren _ masis da sigrZis sazomi

erTeulebis `gamsxvilebisas~ naklebi erTeuli mravldeba 10ze an 10is ufro maRal xarisxze.

magaliTad, 1 dm=10⋅1 sm, 1 m =10⋅1 dm, 1 km=10⋅10⋅10⋅1 m.

drois sazomi erTeulebis gamsxvilebisas es kanonzomiereba irRveva _ kavSirebi aTobiT

sistemas ar efuZneba.

amis Semdeg jgufebs vTavazobT baraTebze daweril winadadebebs:

28

5437 mm=5 m 4 dm 3 sm da 7 mm

62135 mm=6 km 2m 1dm 3 sm da 5 mm

7285 wm=7 dReRame 2 sTY 8 wT da 5 wm

627 kg=6 tona 2 cent da 7 kg.

da vTavazobT swori tolobebis amocnobas.

jgufebi pasuxebs waradgenen da Seecdebian Tavisi varaudis dasabuTebas, warmoadgenen

mcdari winadadebebis Sesworebul variantebs.

Semdeg SeiZleba kvlav davubrundeT aTobiT sistemas. magaliTad, ricxvis 111 yoveli

1iani sxvadasxva mniSvnelobisaa, Tu ar iqneba es Canaweri poziciuri, maSin yoveli 1iani,

mosalodnelia 1s aRniSnavdes da 111 simbolo 3s warmoadgendes. ricxvebis Caweris aseT wess

iyenebs `romauli sistema~, romelSic simbolo 1 aRniSnavs 1s, V _ xuTs, X _ aTs.

SevTavazoT jgufebs romauli sistemiT Caweron ricxvebi 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

warmoadginon pasuxebi cxrilis saxiT. moswavleebi Taviseburad ganaTavseben am simboloebs.

magaliTad, 9is qveS SeiZleba miaweron VIIII, ricxvebis Caweris yoveli mcdeloba unda

waxalisdes maswavleblis mxridan. Tumca, moswavleebs 12ve ricxvis Canawerebi Tqven unda

gaacnoT. iqneb dasvaT kidec SekiTxva: romauli sistemiT ricxvTa warmodgena savaraudod ra

princips efuZneba? amis Semdeg jgufebi ecnobian saxelmZRvanelos me3 punqtSi warmodgenil

cxrils. msjelobis sagani unda gaxdes 4isa da 6is Canawerebi, movisminoT moswavleTa

varaudebi _ ratom aqvs 5is Sesabamis V simbolos erT SemTxvevaSi simbolo I marcxnidan

miwerili, meore SemTxvevaSi marjvnidan? mosalodnelia, moswavleebi daaskvnian, rom

marjvnidan miwerili I emateba 5s, marcxnidan miwerili _ akldeba. am mosazrebas gaamyarebs

9isa da 11is. Canawerebi, anu romauli sistema aris ricxvebis Caweris arapoziciuri sistema.

magram, aqac gvxvdeba poziciuri sistemis elementebi _ marjvnidan miwerili naklebi

erTeulebis aRmniSvneli simbolos SemTxvevaSi gviwevs mimateba, marcxnidan miwerili _

gamoklebaze miuTiTebs.

SeiZleba amave muSaobisas (Tu dro dagvrCeba) SevTavazoT jgufebs poziciur sistemasTan

dakavSirebuli amocanebis amoxsna rubrikidan `moisazre~.

aqtivobis Sefaseba

am jgufuri muSaobis Semdeg moswavle gaiazrebs ricxvis Caweris wesebis arss _ poziciuri,

arapoziciuri (kerZod, romauli), aTobiTi; uCndeba momavalSi ricxvebis araaTobiTi

Canawerebis Seswavlis; arapoziciuri sistemebis gacnobis interesebi. SeiZleba (Tumca, es ar

aris aucilebeli) kidev gaaRrmavoT ganxilva: miuTiToT kompiuterul sistemebSi orobiTi

sistemis gamorCeuli rolis Sesaxeb, dasvaT SekiTxva: ramdeni cifri gamoiyeneba orobiT

sistemaSi, rogor Caiwereba 23 (aTobiTi Canaweri) orobiTSi? (10111). ifiqron saxlSi da Semdeg

gakveTilze gaagrZelonN saubari amis Sesaxeb.

29

2. milionze meti ricxvebi

mizani. milionze meti ricxvebis Cawerasa da wakiTxvaSi gawafva, am ricxvebis dakavSire

ba statistikur monacemebTan. ricxvTa poziciuri sistemis, kerZod, aTobiT sistemis arsis

ukeT gaazreba, misi Sedareba arapoziciur sistemebTan. romaul sistemaze pirveli, arasru

li warmodgenis Seqmna, gamoyeneba. axali ricxviTi saxeliT mocemuli ricxvis rigis povna

(anu ramdenniSnaa ricxvi, ramdeni nulia mis CanawerSi).

teqstSi didi ricxvebi dakavSirebulia statistikur monacemebTan, kerZod, sxvadasxva

qveynis mosaxleobis odenobasTan, kosmiur sxeulebs Soris manZilTan. moswavleebi ricx

vebis wakiTxvaaRweraSi gawafvasTan erTad informacias Rebuloben didi qveynebis, Tavisi

samSoblos mosaxleobis odenobis Sesaxeb. amiT iqmneba SesaZlebloba gakeTdes mravalmxrivi

analizi, romelsac calke ganxilvebi SeiZleba mieZRvnas.

CineTSi mcxovrebTa ricxvi miaxloebiT miliard 300 milionia _ 1300 000 000. aq cifri

3is mniSvnelobaa samasi milioni. dedamiwaze amJamad miaxloebiT 6500 000 000 adamini cxov

robs, am ricxvis CanawerSi rva nulia, momavalSi moswavleebi SeZleben iangariSon, planetis

mosaxleobis ra nawils Seadgens CineTis mosaxleoba.

maswavleblis momzadebis erTerTi mniSvnelovani aspeqti gakveTilis gegmis Sedgenaa.

gegmis pirveli punqti miznis Camoyalibeba da gaazrebaa. maswavleblis wigni exmareba mas am

miznis gansazRvraSi. yoveli gakveTilis garCevas viwyebT am miznis miTiTebiT. Semdegi etapebi

wina pirobebis gaazrebaa, saWiro resursebis fiqsirebaa. gakveTilis gegmis TiToeuli etapis

Sesaxeb yoveli Temis ganxilvisas gveqneba saubari maswavleblis wignSi. moswavleTa aqtivobebi,

rogorc ZiriTadi miznis daxasiaTebas, aseve maT motivacias ukavSirdeba, am mimarTulebiT

moswavlis saxelmZRvanelo dagexmarebaT _ TiTqmis yvela Temis damuSaveba garkveuli

praqtikuli situaciebisa da am situaciebSi axali maTematikuri obieqtebis gamoyenebas

ukavSirdeba. moswavleTa aqti vo bebisa da codnis gamyarebasTan dakavSirebuli davalebebi

dawvrilebiTaa garCeuli maswav leblis wignSi. maswavleblis wigni exmareba maswavlebels

saSinao da saklaso davalebebSi gamoarCios amocanebi maTematikuri SejibrebebisTvis,

amocanebi maTematikaSi maRali war matebiT gamorCeul, an SedarebiT sust moswavleebTan

muSaobisTvis; maswavleblis wignSi ki mo cemulia davalebebi damoukidebeli da Semajamebeli

sakontrolo muSaobisTvis.

davubrundeT milionze meti ricxvebis ganxilvas.

teqstSi moyvanilia yoveldRiur cxovrebaSi didi ricxvebis dasaxelebis magaliTebi. Sei

Zleba moswavleebma gaagrZelon am saxelebis gadataniTi mniSvnelobis magaliTebis moyvana.

magaliTad,

• gareT _ atlasi, SigniT _ aTasi (xalxuri);

• aTasma azrma gamielva;

• ena rom gasatexi iyos, aqamde aTas adgils gadatydeboda (ilia);

• ybad agiRo, cils gwamebs, ambobs aTas rameebs (akaki).

moswavleebma SeiZleba Tavisi sityvebiT axsnan gamonaTqvamebi:

• aTasSi erTxel gamaxsendeba;

• milioni azri mitrialebs;

• es ambavi milionma adamianma gaigo.

am magaliTebiT warmodegenilia damatebiTi sasargeblo resursebis gamoyenebis

SesaZleblobebi swavlebis procesSi.

3 da 5 savarjiSoebSi moswavleebs vTxovT ricxvis aTobiT Canawers. amaSi igulisx

meba ricxvis Canaweri Tvlis aTobiT sistemaSi am terminis mniSvnelobaSi moswavleebi para

grafis bolos gaTvaliswinebuli jgufuri muSaobisas gaerkvevian. aqvea araaTobiTi sistemis

pirveli nimuSi (arasrulad warmodgenili).

30

zogierTi didi ricxvis Canaweri da saxeli savarjiSoebSi gvxvdeba, magaliTad, 10 sa

varjiSoSi warmodgenilia trilioni, kvadrilioni da kvintilioni. am savarjiSos Sesrulebi

sas moswavleebi isargebleben saxelmZRvaneloSi warmodgenili cxriliT.

gavamaxviloT yuradReba, rom 1 kvadrilionis CanawerSi 15 nulia; 10 kvadrilionis Canaw

erSi _ 16 nuli, es ricxvi 1 kvadrilionze 10jer metia. 100 kvadrilioni 100jer metia 1

kvadrilionze, mis CanawerSi 17 nulia, es aris pasuxi 10 savarjiSos g) da d) kiTxvebze,

danarCeni klasebis TanrigebisTvisac davsvaT analogiuri kiTxvebi, gaavarjiSeT moswavleebi

erTi da imave klasis Tanrigebis erTmaneTTan dakavSirebaSi. z) miliardis CanawerSi 9 nu

lia, 1 000 miliardi miliardis 1 000ze gamravlebiT miiReba _ miliards mivawerT 3 nuls,

es ricxvi Seicavs 9+3=12 nuls.

analogiurad, T) 1000 aTasi aris 1 000 gamravlebuli 1 000ze, am ricxvis CanawerSi 6 nu

lia.

i) trilionis CanawerSi 12 nulia, Tu mas 1000ze gavamravlebT, miiReba ricxvi, romlis

CanawerSi 12+3=15 nulia.

k) 1 000 kvadrilionis CanawerSi 15+3=18 nulia.

moisazre _ am rublikiT SemoTavazebuli amocanebi SeiZleba jgufuri mu

SobisTvisac gamoviyenoT, romelsac Sejibris saxes miscemT. Sejibris dawyebamde moTel

vis mizniT gTavazobT amocanas, romelzec moswavleebi zepirad, adgilidan gascemen pa

suxs: saSinao davalebad maswavlebelma mogcaT savarjiSoebi #21dan #34is CaTvliT. ramdeni

savarjiSos Sesruleba mogviwevT? SeiZleba moswavleTa nawilisgan miiRoT mcdari pasuxi:

34–21=13. am SemTxvevaSi SeiZleba damatebiTi kiTxvebic daixmaroT: ramdeni amocanaa 1dan 34

is CaTvliT da ramdeni ar mogcaT maswavlebelma davalebad? am kiTxvis Semdeg moswavleebi

amocanis amosaxsnelad Seasruleben moqmedebas _ 34–20=14.

moTelvis Semdeg 1 amocanis I punqtis sapasuxod umciresi da udidesi samniSna ricx

vebis gaxsenebac ikmarebs: 100 da 999. samniSna ricxvebis odenobaa 999–99=900; amdeni samniSna

ricxvis Casawerad saWiroa 900⋅3=2 700 cifri,

2 miuxedavad saxelmZRvaneloSi arsebuli miTiTebisa, moswavleebma am amocanis amox

snis ramdenime gza SeiZleba SemogvTavazon. yoveli maTgani yuradRebiT unda ganvixiloT,

wavaxalisoT moswavleebi.

3 II striqoni miiReba I striqonis ricxvebis 3ze gamravlebiT, IV striqoni _ I striqo

nis ricxvebis 3ze gamravlebiT da 1is mimatebiT. III striqonisTvis aseTi martivi kanonzo

miereba ar Cans. amitom `zedmetad~ mas miviCnevT.

am tipis amocanebis amoxsnis Zieba moswavleebs enTuziazms matebs. Sedegic Zalze sayu

radReboa _ moswavleebi inviTareben kanonzomierebaTa, hipoTezebis aRmoCenis unars, kri

tikul damokidebulebas sawyisi mosazrebebis mimarT (rogorc es IIIO striqonisTvis xdeba).

sxvaTa mosazrebebis aRiarebis an oponirebis unars, iReben drois garkveul SualedSi maqsi

maluri Sedegis miRwevis gamocdilebas.

4 milioni aTasze 1 000jer metia, miliardi milionze 1 000jer metia, trilioni mil

iardze 1 000jer metia. am kanonzomierebas arRvevs `aTi miliardi~.

5 SesaZloa 4 amocanis ricxvebi gaimeoron: erTi, aTasi, milioni, miliardi, tril

ioni; an xuTi, 5 aTasi, 5 milioni, 5 miliardi, 5 trilioni.

31

am amocanis amoxsnisas SeiZleba saklaso savarjiSoebidan 10 Si warmodgenili cxrili

gamoiyenon. yuradReba gavamaxviloT im faqtze, rom yoveli momdevno klasis erTeuli (um

ciresi Tanrigis erTeuli) misi wina klasis erTeulze 1 000jer metia, iqneb mizezebzec im

sjelon. yovel klass 3 Tanrigi qnis, anu yoveli klasis erTeulis CanawerSi 3iT meti nulia

wina klasis erTeulis CanawerTan SedarebiT, 3 nulis miwera ki 1000ze gamravlebis Sedegia.

jgufuri

naturaluri ricxvebis Caweris aTobiTi sistemis Sesaxeb

• naturaluri ricxvebis cifrebiT Canawers aTobiTi Canaweri vuwodeT. rasTan aris

dakavSirebuli am terminis gamoyeneba?

• ratom vuwodebT ricxvebis Caweris Cvens mier gamoyenebul wess poziciurs?

jgufur muSaobas SeiZleba misceT paeqrobis saxe. moswavleTa warmatebuli prezentacie

bi unda Sefasdes da waxalisdes. yvela jgufi erTsa da imave davalebas asrulebs. ganxilvac

yvelas monawileobiT unda Catardes. gamorCeuli yuradReba dauTmeT moswavleTa naazrevis

gadmocemis xarisxs, formas. sayuradReboa maTi mosazrebebi romauli sistemis Sesaxeb, misi

da aTobiT sistemis SedarebiTi analizi, sayuradReboa agreTve _ aRmoaCenen Tu ara moswav

leebi, rom sigrZis erTeulTa ori erTobliobidan erTerTSi `gamsxvileba~ imave principiT

xdeba, rac aTobiT sistemaSi, meore erTobliobaSi _ ara.

maswavlebels SeiZleba damatebiTi kiTxvebiT mouwios gamarjvebuli gundis gamovlena.

moswavleebs TxoveT daasaxelon is xelsawyoebi, romelTa moqmedeba dakavSirebulia aTo

biTi principis gamoyenebasTan (gazis eleqtromricxveli, avtomobilis mier gavlili manZi

lis mricxveli da sxva). sasurvelia am xelsawyoze monacemTa Cvenebis mokle aRwera.

saswavlo procesi gulisxmobs periodul monitorings. damoukidebeli wera warmoadgens

moswavleTa mier masalis aTvisebis xarisxisa da miRebuli codnis gamoyenebis unaris

Semowmebis erTerT saSualebas. Cven periodulad SemogTavazebT weriTi samuSaoebis

savaraudo variantebs Sesabamisi Sefasebis sarekomendacio sqemiT. rekomendaciebi Sefasebis

principebis, maT Soris zogadi principebis Sesaxeb, ixileT am saxelmZRvanelos 22e gverdze.

damoukidebeli wera #1

Tema: • naturaluri ricxvebis wakiTxvisa da Caweris wesebi; • milionze meti ricxvebi.

1. dawereT mocemul ricxvSi xazgasmuli cifris mniSvneloba: 587 641.

1) 7641 2) 7 3) 7 000 4) 700 000.

2. CawereT mocemuli ricxvi gaSlili formiT: 406080331.

1) 40+60+80+331 2) 400000000+6000000+80000+300+30+1 3) 400000000+60000000+30000000+300000+1 4) 400000+60000+8000+300+30+1.

3. warmoadgineT as xuTi miliard 7 milion asi aTas ocdaerTis aTobiTi Canaweri.

1) 105700100021 2) 1050710021 3) 105007100021 4) 105710021.

4. warmoadgineT ricxvis mokle sityvieri Canaweri: 260425310.

1) 260 miliard 425 aTas 310 2) 2 miliard 60 milion 425 aTas 310 3) 26 miliard 42 milion 5 aTas 310 4) 260 milion 425 aTas 310.

32

5. Tu eqvsniSna ricxvSi wavSliT asi aTaseulis Tanrigis cifrs _ 3s, maSin es ricxvi

Semcirdeba

1) 300000iT 2) 3iT 3) 3jer 4) 30000iT.

6. rvaniSna ricxvSi Secvales erTi cifri da es ricxvi 40 aTasiT gaizarda. Secvlili

cifri

1) aTeulis Tanrigisaa 2) aTaseulis Tanrigisaa

3) aTi aTaseulis Tanrigisaa 4) asi aTaseulis Tanrigisaa.

7. mocemulia ricxvi 9068347. misi romeli cifrebi unda wavSaloT, rom miviRoT: a) umciresi SesaZlebeli oTxniSna ricxvi; b) udidesi SesaZlebeli xuTniSna ricxvi?

8. mocemulia cifrebi: 0, 1, 2, 4, 9. SeiZleba TuU ara am xuTive cifris gamoyenebiT CaweroT

xuTniSna ricxvi, romlis aTaseulebis klasis cifrTa jami 2iT aRemateba erTeulebis klasis

cifrebis jams? dadebiTi pasuxis SemTxvevaSi warmoadgineT yvela SesaZlebeli ricxvi.

Sefasebis sarekomendacio sqema.

testuri tipis (daxurulboloiani) amocanebidan TiToeulSi swori pasuxis miTiTeba

Sefasdes 1 quliT, arasworis _ 0 quliT.

me7 amocana _ 2; 1 an 0 quliT _ apunqti 1 quliT da bpunqti 1 quliT.

me8 amocanis amoxsnac 2; 1 an 0 quliT Sefasdes _ erTi mainc SesaZlebeli ricxvis

warmodgenisTvis 1 qula, eqvsive SesaZlebeli ricxvis warmodgenisTvis kidev 1 qula.

pasuxebi da miTiTebebi amocanebis amosaxsnelad.

1 2 3 4 5 6

3 2 3 4 1 3

7. a) mocemuli ricxvi SvidniSnaa; amrigad, moswavlem unda waSalos sami cifri: 9; 0 da

8: 9068347 , miiReba: 6347.

b) unda waiSalos ori cifri: 0 da 6, anu 9068347 , miiReba: 98347.

8. SeiZleba, es ricxvebia: 90421, 90412, 90241, 90214, 90142, 90124.

damoukidebeli wera SeiZleba maswavlebelma CamorCenil moswavleebTan damatebiTi

muSaobis Casatareblad gamoiyenos, maT SeTavazos gamotovebuli varianti, an im dros, roca

sxvebi asruleben davalebas, maTTan damatebiTi mecadineoba Caataros.

maswavleblis yuradRebis miRma arcerTi moswavle ar unda darCes.

3. vadarebT da valagebT naturalur ricxvebs

mizani. ricxvebis Sedarebaze moswavleTa warmodgenebis gaRrmaveba; raodenobrivi maCve

neblis mixedviT monacemebis elementaruli analizis unaris ganviTareba. kerZod, monacemTa

erTobliobaSi monacemebis poziciis gansazRvra (magaliTad, qveynebis dalageba mosaxleobis

odenobis zrdis an klebis mixedviT).

paragrafSi moyvanil cxrilSi qveynebi ar aris mosaxleobis odenobis mixedviT dalagebu

li. cxrili anbanis mixedviTaa Sedgenili. moswavles am gakveTilze mouwevs Seadaros cxril

Si moyvanili ricxvebi Sedarebis misTvis ukve cnobili wesis mixedviT. Sedarebas viwyebT

33

Tanrigebis odenobaTa SedarebiT. maTi tolobis SemTxvevaSi vadarebT ricxvebis udidesi

Tanrigebis erTeulebs. metia is ricxvi, romlis Sesabamisi Tanrigis pirvelive gansxvavebuli

cifri aris meti. ricxvebis klebis mixedviT dalagebis Semdeg moswavleebi amave Tanamim

devrobiT daalageben Sesabamis qveynebs. es statistikuri monacemebis damuSavebis pirveli

nabijebia.

moswavleebs proeqtis saxiT vTavazobT or amocanas, moswavleTa erT nawils mouwevs

cnobarebSi moiZios monacemebi sxvadasxva qveynis mosaxleobis odenobis Sesaxeb. maT SeuZli

aT qveynebi Tavisi interesis mixedviT amoirCion, warmoadginon monacemebis cxrilis saxiT.

momdevno gakveTilze SesaZloa mopovebuli informaciebiT mTelma klasma erTi mozrdili

cxrili gaakeTos. magaliTad, viRebT romelime moswavlis cxrils, yoveli momdevno moswavle

masSi ganalagebs Tavis monacemebs (qveynebs, mosaxleobis odenobas) saTanado adgilze. am sa

muSaos Sesruleba kompiuteris daxmarebiT umjobesia: viyenebT programa words, mis SesaniS

nav Tvisebas _ svetebad CamobeWdil qveynebs Soris momdevno moswavlis mier qveynis Casmi

sas svetSi qveynebis dasaxelebebi Tavad wainacvlebs; SeiZleba baraTebze dawerili qveynebis

saxelebi dafaze miamagroT da saWiroebis SemTxvevaSi waanacvloT.

am samuSaos Sesrulebisas moswavle iwafeba ricxvebis SedarebaSi, kompiuteris gamoy

enebaSi. SesaZloa es gakveTili bunebis maswavlebelTan erTad Catardes da integrirebuli

saxe mieces. danarCeni moswavleebi analogiuri gzebiT moiZieben informacias planetebis Ses

axeb. am SemTxvevaSi moswavleebs mouwevT erTi da imave erTobliobis orgvarad warmodgena

(masebis mixedviT, manZilebis mixedviT). aRsaniSnavia orive saproeqto amocanis SemecnebiTi

mniSvnelobac

zemoT warmodgenili rekomendaciebi exmareba maswavlebels sxvadasxva strategiis,

teqnikis, didaqtikuri xerxisa Tu informaciuli da sakomunikacio teqnologiis gamoyenebaSi.

7 savarjiSos v) z) magaliTebSi marcxena ricxvi eqvsniSnaa, marjvena _ xuTniSna. mo

swavleebi kidev erTxel gaixseneben: Tu Sesadarebeli ori ricxvidan erTerTi `metniSnaa~

_ misi Tanrigebis odenoba metia, maSin am ricxvTa cifrebis Sedareba saWiro aRar aris _

`metniSna~ ricxvi metia.

11 savarjiSo moswavleebs uviTarebs martivi cxriliT warmodgenili monacemebis el

ementaruli analizis unars. kerZod, cxrilis saxiT warmodgenili monacemebis `wakiTxvis~/

gagebis unars.

13 amocanaSi a) da b) savarjiSoebSi mocemuli ricxvebidan metia `metniSna~ ricxvebi, g)

savarjiSoSi, Tu II ricxvi iwyeba 9ze naklebi cifriT, maSin cxadia, I metia IIze, Tu meore

ricxvi iwyeba 9iT, maSin momdevno _ aTaseulis Tanrigis SedarebiT kvlav vRebulobT: I metia IIze, e. i. nebismier SemTxvevaSi I ricxvi metia IIze. analogiurad, d) savarjiSoSi II ricxvi daiwyeba 1iT, Tu 1ze meti ricxviT, mainc II ricxvi metia Ize.

14 amocanaSi meore ricxvis erTeulis cifria 0, aTeulis cifri nulze metia. amitom

es ricxvi metia pirvel ricxvze.

15 amocanaSi imsjeleT ninikos Secdomebze. mogiwevT sigrZis, masis, dros erTeuleb

is da maT Soris kavSirebis gaxseneba. miaqcieT yuradReba: b) punqtSi zogierTi moswavle

Seecdeba 1000 wm gadaiyvanos saaTebSi, es ufro rTuli gzaa, magram sworia da waxalisebas

imsaxurebs.

20 amocanis amoxsnisas SeiZleba moswavlem gamoTvalos 3⋅55+50 gamosaxulebis mniS

vneloba da Seadaros 2 lars, anu 200 TeTrs. SeiZleba ase imsjelos: 4 cali 50TeTriani

34

1

2 3 4 5 6 7 8 9

2

1 3 4 5 6 7 8 9

3 4 5 6 7 8 9

1 2 4 5 6 7 8 9 . . .

Seadgens 2 lars. 1 cali 50TeTriani mWirdeba nayinisTvis, 3 cal 55 TeTrian purs 3 cali

50 TeTriani ar eyofa.

21 or avtobusSi 72 adgilidan 23+18=41 adgili ukve dakavebulia da darCenilia

72–41=31 adgili, es adgilebi yofnis ninikos klasis moswavleebs, mSobels da damrigebels,

maTi odenoba zustad 31ia, (29+1+1). msjeloba SeiZleba asec warimarTos: I avtobusSi aris

36–23=13 Tavisufali adgili, IISi _ 36–18=18. sul _ 13+18=31 (adgili).

moisazre

1 saZiebeli ricxvi samniSnaa, misi aseulis Tanrigis cifria 1, erTeulis Tanrigis ci

fria _ 3. aTeulis Tanrigi moswavlem SeiZleba sinjvis meTodiT daadginos. pasuxi: 123.

2 999 981ze meti eqvsniSna ricxvis aseulis an ufro maRali Tanrigis cifri ar Sei

Zleba meti iyos mocemuli ricxvis Sesabamisi Tanrigis cifrze. erTeulis Tanrigis cifri,

pirobiT, 1ia. darCa aTeulis Tanrigi, romelic unda iyos 9is toli. pasuxi: 999 991.

3 xelis CamorTmevaTa odenobis dasaTvlelad ninikos kargi gza aurCevia, Tumca yoveli

xelis CamorTmeva orjeraa daTvlili _ mis mier CamoTvlil wyvilebSi erTnairi cifrebiT

Cawerili ricxvebi erTi da imave wyvilis xelis CamorTmevas aRniSnavs (magaliTad, 36 da 63

_ orive warmoadgens, me3 da me6 megobris xelis CamorTmevas). amitom cxrilis elementebis

odenoba _ 72 unda gaiyos 2ze. pasuxi: 36.

SeiZleba moswavleTa msjeloba warvmarToT damxmare kiTxvebiT.

• ramden megobars CamoarTva xeli TiToeulma? (8s).

8⋅9=72•ramdenjer davTvaleT TiTo xelis CamorTmeva? (2jer. magaliTad me6 da me3 megobars

Soris xelis CamorTmeva orjer davTvaleT).

• ramdeni xelis CamorTmeva Sedga? (72:2=36).

orive SemTxvevaSi gadaTvlis erTsa da imave princips viyenebT, Tumca II gzam cxrilis

Sedgenas agvarida.

am amocanis amoxsna xisebri diagramiTac SeiZleba warmovadginoT:

cifrebis yoveli wyvili (megobrebis nomrebi) orjeraa SeerTebuli wiboTi. sul 72 wyvi

lia, yoveli maTgani orjer dasaxelda.

me3 amocanis ramdenime gziT amoxsnis Semdeg me4 da me5 amocanebis yovel SemTxvevas

moswavleebi swrafad upasuxeben. Tu megobrebis odenoba 6ia, xelis CamorTmevaTa odenoba

iqneba (6⋅5):2=15;

roca megobrebis odenoba 7ia _ (7⋅6):2=21;

roca megobrebis odenoba aris 8 _ (8⋅7):2=28.

5 (10⋅9):2=45.

am amocanebis seria daexmareba moswavleebs SemdgomSi ufro rTuli kombinatoruli amo

canebis amoxsnisas gamravlebis wesis gamoyenebaSi.

wignSi warmodgenili esa Tu is amocana klass SeTavazeT xolme mxolod mas Semdeg, roca

miiCnevT, rom is Tqven mier Sedgenili gakveTilis gegmis organuli nawilia, gaqvT misi

ganxilvisTvis saWiro dro da es ganxilva iqneba warmatebuli. erovnuli saswavlo gegmis

35

moTxovnebis warmatebuli gadawyveta ar gulisxmobs uklebliv yvela amocanis amoxsnas.

Tumca, Tu klasis akademiuri done maRalia, saxelmZRvaneloSi warmodgenili yvela amocanis

ganxilva savsebiT miRwevadia.

4. amocanebis amoxsna

mizani. amocanebis amoxsnisas amocanis Sinaarsis gagebis, amoxsnis strategiis gansazRvr

is da ganxorcielebis, Sesasrulebeli samuSaos Sedegis Semowmebisa da gaanalizebis unaris

ganviTareba.

paragrafSi aRwerilia amocanis amoxsnis etapebi _ 4 etapi. yoveli maTgani gasagebi

gaxdeba moswavlisTvis konkretuli amocanebis amoxsnisas. amocanebis amoxsnis dawyebamde

SeiZleba gavixsenoT:

• ra moqmedebas vasrulebT saerTo odenobis sapovnelad? (Sekrebas).

sasurvelia moswavleebma Tavad moifiqron Sesabamisi situaciebi. magaliTad, diliT am

ovxseni 6 amocana, saRamos _ 5. sul ramdeni amocana amovxseni?

dedam momca 15 lari, mamam _ 28 lari. ramdeni lari momces mSoblebma?

• ra moqmedebiT SeiZleba SevcvaloT Sekreba, roca toli Sesakrebebi gvaqvs? (gamrav

lebiT).

magaliTad, maswavlebelma klasis 20 moswavlidan TiToeuls 3 rveuli moatanina. ramdeni

rveuli Segrovda klasSi?

3 + 3 + . . . + 3 = 2 0 ⋅ 3 = 6 0 . 20jer

• ra moqmedebas viyenebT nivTebis tolad gadanawilebisas? (gayofas).

magaliTad, 48 wigns 3 Taroze tolad vanawilebT. TiTo Taroze unda davdoT 48:3=16

wigni.

20 kanfets 4 megobari Tanabrad inawilebs. TiTos Sexvdeba 20:4=5 kanfeti.

• ra moqmedebas vasrulebT ricxvebis Sedarebisas? (gamoklebas)

magaliTad, ramdeni wliT ufrosia 51 wlis marika 23 wlis rusudanze?

am kiTxvebiT gavixsenebT im SemTxvevebs, roca oTxive moqmedebidan ama TuU im moqmedebas

virCevT. moswavleTa mier SemoTavazebuli yoveli amocana unda iqnes ganxiluli, maT Soris

sainteresoTa avtorebi _ waxalisebuli.

# 7 amocanaSi marTkuTxedis sigrZea 160⋅3=480 m, perimetris gamosaTvleli gamosaxule

baa 160⋅2+480⋅2 an (160+480)⋅2. am gamosaxulebis miReba mniSvnelovnad gagviadvilebs momavalSi

marTkuTxedis perimetris gamosaTvleli formulebis miRebas.

9 moswavle 70⋅6 gamosaxulebas Seadarebs 420s da daaskvnis _ am manZilis gavlas zus

tad 6 sT sWirdeba. SeiZleba zogierTma maTganma 420:6 ganayofi Seadaros 70s, anu upasuxos

kiTxvebs: ra siCqariT unda imoZraos avtomobilma, rom 6 sTSi 420 km gaiaros?

10 570is 90ze gayofisas ganayofia 6, naSTia 30. am 30 adamianisTvisac saWiroa kidev

erTi _ me7 reisi, romlis drosac TviTmfrinavi mTlianad ar Seivseba.

36

13 cxadia, sqemaze (romelic rveulSi unda gakeTdes) unda miuTiTon qalaqebs Soris

manZili: 70⋅4=280 km.

280 km

? 3 15 km.

14 yuTebSi 350 wignia. TiToeul yuTSi wignebis odenobis dasadgenad saWiroa vicodeT

yuTebis odenoba (magaliTad, 10), amasTanave yuTebSi wignebis Tanabari odenoba unda iyos.

msgavsi amocanebis mofiqrebiT moswavleebs unviTardebaT SemoqmedebiTi azrovnebis un

ari, uCndebaT motivacia amoxsnan megobris mier mofiqrebuli amocana; Seadaron, gaanali

zon es amocanebi; unviTardebaT oponirebis unari. magaliTad, kritikulad unda ganixilon

amocana, romelSic avtomobilis siCqare aRmoCnda 2 km/sT, an 500 km/sT, an qalaqebs Soris

manili 20 000 km miiRes da a. S.

16 moswavleebSi xSirad `2jer meti~ ukavSirdeba gamravlebis operacias, xolo `nak

lebi~ _ gamoklebas. yuradRebiT gaacaniT amocana moswavleebs da isini I SemTxvevaSi Seas

ruleben gayofas, II SemTxvevaSi _ mimatebas.

saSinao saSinao davalebis amocanebi

1 2 sTSi — 35 km; 105 aris 35ze 3jer meti, e. i. droc 3jer meti daixarjeba _ 6 sT.

2 nucas zedmeti 8 lari Tanabrad unda ganawildes: 4 lari ubrundeba nucas, 4 lari

_ paatas.

3 Tu davubrunebT giorgis Tavis 12 lars, miviRebT mis Tavdapirvel Tanxas: 30+12=42.

4 daxarja 9, darCa _ 18, hqonda 18+9=27.

5 nikam daxarja 10, darCa _ 7, hqonda _ 17. Tornikesac hqonda 17. daixarja 17+10=27.

6 2 sTSi Camoedineba 1 litri wyali, 10 sTSi _ 5 litri.

7 3674 km aris amocanis zusti pasuxi. miaxloebiT gaiara 2 000+900+1 000=3 900 km.

17 amocana momdevno gakveTilze gansjis sagani unda gaxdes: orive sacavSi darCa

12+24=36 kg Warxali.

• orive sacavidan erTad 100–(12+24)=64 kg Warxali gautaniaT.

•ramdeni gautaniaT TiToeulidan _ amis gageba SeuZlebelia damatebiTi pirobis gareSe.

gavaanalizoT TiToeuli moswavlis mier damatebuli piroba. magaliTad, SeiZleba daematos,

rom

• Tavdapirvelad orive sacavSi toli odenobis Warxali iyo (TiToSi 50 kg, Idan gauta

niaT 50–12=38 kg, meoredan _ 50–24=26 kg).

18 natam gacura 8⋅50=400 m, gasacuri darCa 1000–400=600 metri. 600 metris gasacurad

natam auzi 600:50=12jer unda gadacuros.

37

moisazre

1 vagrZelebT kombinatoruli xasiaTis amocanebze muSaobas. saxelmZRvaneloSi moswav

les vTavazobT variantebis amoweras Semdegi kanonzomierebiT:

kodi, romelic Seicavs zustad erT 1ians _ 0001, 0010, 0100, 1000;

kodi romelic Seicavs or 1ians _ 1100, 1010, 1001, 0110, 0101, 0011;

kodi, romeli Seicavs zustad sam 1ians _ 1110, 1101, 1011, 0111. aq SeiZleba moswavle

mixvdes, rom tolia im kodebis odenobebi, romlebic Seicavs zustad erT 1ians an zustad

sam 1ians (igivea, rac erT 0ians). miviReT 14 varianti.

SesaZloa moswavleTa nawilma ase imsjelos: kodis yovel TanrigSi aris ori SesaZle

bloba 0 an 1. wina TanrigSi yovel arCevans Seesabameba momdevno Tanrigis ori SesaZlebloba,

sul 2⋅2⋅2⋅2=16 varianti, romlidanac pirobis Tanaxmad unda gamoiricxos ori varianti: 0000

da 1111. SesaZloa xisebri diagramis gamoyenebac, Tumca yoveli me5 klaselisTvis varian

tebis aRweris aseTi forma SesaZloa ar iyos martivad aRsaqmeli.

I Tanrigi 0 an 1

II Tanrigi 0 1 0 1

III Tanrigi 0 1 0 1 0 1 0 1

IV Tanrigi 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

16 varianti

zemoT miTiTebuli ori variantis gamoricxviT miiReba 14 varianti.

38

sakontrolo wera #1

Tema: • naturaluri ricxvebis Sedareba da dalageba

• amocanebis amoxsna

standartis moTxovnis gadawyvetilebiT, Semajamebeli sakontrolo werisTvis davalebebis

SerCeva maswavleblis sagnobrivi kompetenciebis Semadgeneli nawilia; gansxvavebuli

sirTulis savarjiSoebis SerCevisa da Sedgenis mimarTulebiT kompetenciebis ganviTarebaSi

maswavlebels Cven mier SemoTavazebuli nimuSebi da maTi Sefasebis kriteriumebi daexmareba.

cxadia, maswavlebels SeuZlia gansxvavebuli variantebi SesTavazos moswavleebs, SesaZlebelia

saxelmZRvaneloSi warmodgenili amocanebi Seicvalos analogiuri amocanebiTac. Cven swored

aseT rekomendacias vuwevT pedagogebs, raTa Tavidan aviciloT sakontrolo amocanaTa

winaswari gacnoba moswavleebis mier.

1. daalageT klebis mixedviT ricxvebi: 12 miliard 416 aTasi, 21 milion 641 aTasi, 12

milion 164 aTasi.

1) 21 milion 641 aTasi, 12 miliard 416 aTasi, 12 milion 164 aTasi

2) 21 milion 641 aTasi, 12 milion 164 aTasi, 12 miliard 416 aTasi

3) 12 miliard 416 aTasi, 21 milion 641 aTasi, 12 milion 164 aTasi

4) 12 milion 164 aTasi, 21 milion 641 aTasi, 12 miliard 416 aTasi.

2. ra cifri unda CavsvaT gamotovebul adgilze, rom swori utolobebi miviRoT: 810535<81 435<819535?

1 ) mxolod 5 2) mxolod 9

3) nebismieri cifri 4) nulisgan gansxvavebuli nebismieri cifri.

3. gamoiyeneT samive mocemuli cifri 5, 8, 9 da CawereT ricxvi, romelic metia 895ze

da naklebia 985ze.

1) 958 2) 589 3) 859 4) 598.

4. daamrgvaleT 2590867 aseulamde, aTaseulamde, aTi aTaseulamde, asi aTaseulamde da

milionamde. daalageT, miRebuli ricxvebi zrdis an klebis mixedviT. romelia am dalagebaSi

mesame ricxvi?

1) 2600000 2) 2590000 3) 2590900 4) 3000000.

5. vTqvaT, ◊ niSnakebiT dafarulia cifrebi. SeiZleba Tu ara SevadaroT xuTniSna

ricxvebi: 13◊5◊ da ◊9◊7◊?

1) ar SeiZleba

2) SeiZleba; meore ricxvia meti

3) SeiZleba; pirveli ricxvia meti

4) SeiZleba, es ricxvebi xuTniSnaa, amitom tolia.

6. marTkuTxedis gverdebia 12 sm da 10 sm, samkuTxedisa ki _ 19 sm, 11 sm da 10 sm. romeli

figuris perimetria meti?

1) pasuxis gasacemad monacemebi sakmarisi ar aris

2) samkuTxedis

3) am figurebis perimetrebi tolia

4) marTkuTxedis.

7. quTaisidan sofelSi mimavalma giorgim avtobusiT 2 saaTi imgzavra. yovel saaTSi is

60 kms gadioda. amis Semdeg mas gasavleli darCa 35 km. sul ra manZili hqonia gasavleli

giorgis?

39

8. puris qarxanam sam maRazias Tanabari odenobis puri miawoda. pirvelma maRaziam gayida

56 cali puri, meorem _ mxolod 14 _ zustad imdeni, rac darCa gasayidi pirvel maRazias; mesame maRaziam gayida imdenive puri, rac pirvelma da meorem erTad. ramdeni puri darCa

gasayidi samive maRazias?

SeniSvna: warmodgenilia sakontrolo weris savaraudo varianti. sasurvelia misi

gamoyenebiT Sedges odnav gansxvavebuli ramdenime bileTi, gamravldes da daurigdes

moswavleebs Sesasruleblad.

pasuxebi da miTiTebebi amocanebis amosaxsnelad:

1 2 3 4 5 6

3 4 1 2 2 4

7. 60⋅2+35=155 (km)

8. amocanis piroba SeiZleba sqemiT warmovadginoT:

I I I I I Igaiyida gaiyida gaiyidadarCa darCa darCa

56 14 56+1414 56 0

amrigad, gasayidi darCa 14+56=70 (puri).

Sefasebis sarekomendacio sqema.

testuri tipis amocanebidan TiToeulSi swori pasuxis miTiTeba Sefasdes 1 quliT, araswori _ 0 quliT. me7 amocana _ 2; 1 an 0 quliT (mTeli manZilis gamosaTvleli

gamosaxulebis Sedgena, an avtobusiT gavlili manZilis povna _ 1 qula, amocanis pasuxis

miReba _ kidev 1 qula); me8 amocana _ 2; 1; an 0 quliT (sqemis Sedgena, an TiToeul maRaziaSi

mitanili puris odenobis, an meore maRaziaSi darCenili odenobis povna _ 1 qula, pasuxi

_ kidev 1 qula).pedagogebs kidev erTxel SevaxsenebT, rom skolaSi moqmedi Sefasebis zogadi principebi

skolis saswavlo gegmis Semadgenel nawils warmoadgens _ maT Tavad skola amtkicebs. am

principebs unda efuZnebodes yvela saxis Sefaseba, kerZod, sakontrolo naSromis gasworebis

kriteriumic. pedagogma Tavad unda SeimuSavos gasworebis kriteriumebi. Cvens mosazrebebs

mxolod sarekomendacio xasiaTi aqvs. pedagogi unda iTvaliswinebdes, rom misi kriteriumi

garkveulwilad gansazRvravs mis damokidebulebas swavlebisa da moswavleebis mimarT.

sakontrolo wera Semajamebeli davalebaa da misi Sefasebis tipis arCevisas pedagogma

upiratesoba ganmaviTarebel Sefasebas unda mianiWos (miuTiTos, auxsnas moswavles rogor

SeiZleboda Secdomis Tavidan acileba, amoxsnis ra gzebs mianiWos upiratesoba, rogor

gavaumjobesoT Sedegi...). moswavleTa CarTva miRebuli Sefasebebis, Sedegebis analizSi, met

gamWvirvalobas SesZenda pedagogis gadawyvetilebebs.

Zalze mniSvnelovania sakontrolo weris procesic. pedagogma unda uzrunvelyos

samuSao atmosferos Seqmna _ mSvidi, saqmiani garemo. TviT yvelaze `liberali~ pedagogic

ki Seurigebeli unda iyos gadaweris, sxvisi naSromis miTvisebis faqtebisadmi. dauSvebelia,

rom aseT saqciels, zogierTi pedagogi `uwyinar ancobad~ miiCnevs. gadawera, miTviseba

erovnul saswavlo gegmaSi senad, qurdobad aris miCneuli da is unda aRmoifxvras saskolo

cxovrebidan.

maswavlebeli piradi magaliTiT unda aCvenebdes moswavles koreqtuli, guliTadi,

gulisxmieri, samarTliani, urTierTpativiscemaze damyarebuli qcevis nimuSs.

40

5. naturaluri ricxvebis Sekreba. Sekrebis Tvisebebi

mizani. naturaluri ricxvebis Sekrebis Tvisebebis gameoreba, maTi warmodgena zogadi _

algebruli saxiT. am TvisebaTa gamoyenebiT gamoTvlebis `teqnikis~ daxvewa. jamis Sefasebis

unaris ganviTareba. asoiTi gamosaxulebis mniSvnelobis gamoTvlaSi gawafva da am codnis

praqtikuli gamoyenebis unaris gaRrmaveba. terminebis `nebismieri~, `zogierTi~ aTviseba da

adeqvaturi gamoyeneba.

kvlav did yuradRebas vuTmobT moswavleTa motivacias praqtikuli magaliTebis aRwerisas

ricxvebis Sekrebisa da Sevsebis Tvisebebis gamoyenebas.

qveSmiweriT Sekrebis wess sailustracio magaliTebiT vixsenebT. Sekrebis moqmedebis

Sesrulebamde Sesakrebebs vamrgvalebT da zepiri SekrebiT vafasebT/vamowmebT jams, anu

Sekrebis siswores.

gavixsenoT, rom ramdenime ricxvis Sekrebisas SegviZlia Sesakrebebi nebismierad dava

jgufoT. teqstSi moyvanilia xelsayreli dajgufebis magaliTi. SemoTavazebuli didi ricx

vebis Sekrebisas Sesakrebebi SeiZleba ase davajgufoT:

(13+27) miliardi + (6+24) miliardi = 70 miliardi.

konkretul magaliTebze dakvirvebiT moswavleebs vaxsenebT da asoebis gamoyenebiT

zogadad _ algebruli saxiT vwerT Sekrebis Tvisebebs. moswavleebi `patara~ ricxvebze

moqmedebis Catarebisas am wesebis gamoyenebaSi ukve gawafulni arian.

3 savarjiSoSi mosalodnelia, rom moswavleebi Sesakrebebs ase daajgufeben:

a+b+c=(a+c)+b da Sekrebas zepirad Seasruleben. sasurvelia, aqve davafiqsiroT Sekrebis

TvisebaTa gamoyenebis faqti.

16 da 17 savarjiSoebi asoiTi gamosaxulebis gamartivebaSi gawafvisTvisaa, nimuSi mo

swavleTaTvis me16 savarjiSoSia warmodgenili, gamartivebis procesSi moswavleebi Sekrebis

gadanacvlebisa da jufTebadobis Tvisebebs iyeneben, magaliTad, 28+m+62=(28+62)+m=90+m.

19 a) 3 6 milionisa da 19 milionis jami 496 milionze naklebi ar aris (≥496 mil

ionze), 4 5 milioni ki 495 milionze meti ar aris. e. i.

3 6 milioni + 19 milioni >4 5 milionze.

b) 4 7 milionisa da 29 milionis jami 697 milionze naklebi ar aris, 6 6 milioni

ki 696 milions ar aRemateba.

jgufuri

jgufuri muSaobis proeqti

gaanawileT klasi ramdenime jgufad (sasurvelia Tanabari Zalis jgufebad) da TiToeul

jgufs daavaleT samive SemoTavazebuli amocanis amoxsna. amoxsnisTvis winaswar gansaz

RvreT dro, magaliTad, 15 wuTi, drois amowurvis Semdeg moawyveT naSromTa prezentacia

_ erTerTi jgufis wevri warmoadgens amoxsnis SeTanxmebul versias; danarCeni jgufebi mas

oponirebas gauweven. maT naSromebs Tqven winaswar unda gaecnoT.

am mecadineobas SeiZleba mieces Sejibrebis saxec _ yoveli swori amoxsna, yoveli kargi

SekiTxva oponirebisas SesaZloa Sefasdes raime qulebiT. sasurvelia, Sefaseba Tavad moswav

leebma moaxdinon _ es xels Seuwyobs maTSi samarTlianobisa da koreqtulobis TvisebaTa

ganmtkicebas.

ganvixiloT jgufuri muSobisTvis gankuTvnili amocanebi.

41

1 I SesakrebSi aseulis Tanrigis cifri Seamcires 2iT, anu jami Semcirda 200iT; II Sesakrebis SecvliT jami Semcirda 3iT; III Sesakrebis SecvliT jami gadidda 100iT, sab

olood miiRes mocemuli ricxvebis jamze 103iT naklebi ricxvi, e. i. mocemuli ricxvebis

jamia 12 083+103=12 186.

2 I Sesakrebi gaizarda 3 aseuliT, II Sesakrebi _ 5 aTaseuliT, anu maTi jami gaizarda

5 300iT; III Sesakrebi Semcirda 7iT, IV Semcirda 30iT, e. i. am ori Sesakrebis jami Semci

rebula 37iT.

amrigad, mocemuli jamia

28 975–5 300+37=23 712.

3 am amocanis amoxsnisas sasurvelia, rom moswavleebma axsnan:

Caatares Tu ara maT winaswari analizi nawilebis SerCevis mizniT _ es

gaaiolebs amoxsnas (3 cali 4ujriani da 1 cali 3ujriani nawili erTad

gvaZlevs 15 ujras _ ramdenicaa mocemul figuraSi). iqneb maT sinjvis gza

airCies da ase miagnes sasurvel dayofas?

sainteresoa rublika `sxvadasxvaSi~ mowodebuli masala magiuri kvadratebis Sesaxeb; mo

swavleebi amowmeben suraTze gamosaxuli kvadratebis magiurobas, iyeneben Sekrebis wesebs.

am kvadratis striqonebSi, svetebSi da did diagonalebze ricxvebis jami

(samsami ricxvis) aris 15.

am suraTze magiuri kvadrati araa warmodgenili _ erTerT diago

nalze ganTavsebuli ricxvebis jamia 15, meorze _ 11.

Sesavsebad SemoTavazebul kvadratSi es jami 33ia _ 12+7+14=33. mis Sesavsebad moswav

leebi pouloben ucnob Sesakrebs jamisa da cnobili ori Sesakrebis mixedviT. faqtobrivad,

moswavle xsnis gantolebebs: ra unda mivumatoT 14isa da 11is jams, rom miviRoT 33? pasuxia

8 _ qveda marcxena ujraSi vwerT 8s da a. S.

magiuri kvadratis Sedgenis zogierT wess moswavleebi momavalSi gaecnobian.

6. naturaluri ricxvebis gamokleba

mizani. didi ricxvebis gamoklebaSi gawafva. SekrebiT gamoklebis Semowmebis unaris

ganviTareba; gamoklebis komponentebis damrgvalebiTa da zepiri angariSiT Sesrulebuli

moqmedebis Semowmebis unaris ganviTareba. amocanis konteqstis mixedviT moqmedebebis

SerCevisa da Sesrulebis unaris gamomuSaveba. jgufuri muSaobiT moswavleTa codnis gaaqti

ureba da amiT moswavleTa mzaobis miRweva momdevno sakiTxebze dinamiuri gadasvlisTvis.

paragrafis e. w. Teoriuli nawilis gacnoba moswavleebs daavaleT, Semdeg sakiTxi

erToblivad ganixileT.

teqstSi moyvanili praqtikuli magaliTiT moswavleebi ixseneben qveSmiweriT gamoklebis

wess, gamoklebis sisworis Semowmebas jer Sekrebis saSualebiT Maxdenen, Semdeg _

damrgvalebuli komponentebis gamoklebiT. msjeloben, ratom aris miaxloebiTi pasuxi zustze

2 9 4

7 5 3

6 1 8 3 6 2

4 5 2

4 0 7

42

meti _ komponentebis damrgvalebisas saklebica da maklebic Semcirda, magram maklebi metad

Semcirda, vidre saklebi, amitom miaxloebiTi sxvaobac zustze meti aRmoCnda. aq gavixsenoT,

rom saklebis zrda (kleba) iwvevs sxvaobis zrdas (klebas), xolo maklebis zrda, sxvaobas

amcirebs, kleba _ adidebs. es faqti saSinao davalebis Sesrulebisas gamoadgebaT.

es ganxilva moswavleebis aqtiuri monawileobiT warmarTeT. mxolod aseTi midgomiT

SeZlebT maqsimalurad gaaqtiuroT moswavleTa codna, axali impulsi SesZinoT mimdinare

sakiTxebis Seswavlis process.

paragrafis me2 punqtSi gamoklebis zogierT Tvisebas da SekrebasTan kavSirs asoebis

saSualebiT CavwerT.

2 savarjiSoSi ar igulisxmeba asoiTi gamosaxulebebis gamartiveba (a+b–a=b), aramed

igulisxmeba kavSiri Sekrebasa da gamoklebas Soris, romelic paragrafis me2 punqtSia: TuU

a+b=c, maSin b=c–a da a=c–b.

12 da 13 amocanebis ganxilvisas moswavleebis mier gamoTqmuli yoveli mosazreba

klasma uyuradRebod ar datovos. SesaZloa zogierT kiTxvaze pasuxi gamoklebis garda

sxva moqmedebis Sesrulebasac moiTxovdes. aseTi kiTxvebic unda waxalisdes. am monacemebiT

moswavleebi daadgenen CamoTvlilTagan romel qveynebSi gaizarda mosaxleoba, romelSi

Semcirda, ramdeniT, SesaZloa imsjeloT kidec savaraudo mizezebze.

15 savarjiSoSi 4 moswavle dafaze erTdroulad asrulebs qveSmiweriT gamoklebas,

danarCenebi amowmeben maT Sedegebs, samuSaos aseTi ganawilebiT izrdeba moswavleTa

CarTuloba saswavlo procesSi. procesic ufro dinamiuri xdeba.

16 amocanis amoxsnisas SeiZleba erTis dawyebuli gamoTvlebi sxvas gaagrZelebinoT _

yvela unda iyos mzad saamisod.

a) b)

saSinao davalebis 1 amocanis amoxsna moswavlisgan ara marto damrgvalebis wesebis

codnas, aramed analizis unarsac moiTxovs; aTeulamde damrgvalebisas ricxvi 5ze meti

ricxviT ar gaizrdeba da 4ze meti ricxviT ar Semcirdeba. aseulamde damrgvalebisas ricxvi

SeiZleba gaizardos nebismieri ricxviT 1dan 50mde (CaTvliT) an Semcirdes nebismieri

ricxviT 1dan 49mde (CaTvliT), e. i. samniSna ricxvis damrgvalebisas 13iT Semcireba

SeiZleba gamoiwvios mxolod aseulamde damrgvalebam.

11 a) 192 300–82 100=110 200, zusti mniSvnelobaa 110 119, miaxloebiTi zustze 81iT metia;

b) miaxloebiTia _ 1 210 000–249 300=960 700, rac naklebia zust mniSvnelobaze _ 960 720ze

20iT.

g) 200 000–32 300=167 700 maixloebiTi mniSvneloba naklebia zustze (167 741ze) 41iT;

d) miaxloebiTi mniSvneloba 200 500–4500=196 000, rac metia zustze (195 944ze) 56iT.

13 16 amocanebis amoxsnis Semdeg sasurvelia kidev erTxel Camoayalibon daskvna:

nebismieri Sesakrebis gazrda (Semcireba) iwvevs jamis gazrdas (Semcirebas); saklebis gazrda

an maklebis Semcireba iwvevs sxvaobis gazrdas.

`sxvadasxvaSi~ SemoTavazebulia zepiri da swrafi angariSis erTerTi xerxi, aTeulamde

Sevsebis xerxi. magaliTad, Tu gvWirdeba ricxvs gamovakloT 999, ricxvs jer vaklebT 1000s

da Semdeg vamatebT 1s (radgan gamovakeliT mocemulze erTiT meti ricxvi).

SesTavazeT moswavleebs 999is zepirad da swrafad mimateba (unda daumaton 1000 da

gamoaklon 1).

101200- 29372

71828

100000- 31957

68043

43

723-303

523+200

643-120

420+320

564+79

740-352

388+401

789-225

i

amocana 1 2 3 4 5

pasuxi 4 1 2 3 2

2 amocanis amoxsnisas jobs suraTis gamoyeneba, 60 smiani naWrebidan sams

akldeba 55 sm, sul daakldeba 15 sm.

pasuxi: 4⋅60–15=225 sm.

3 swori pasuxi SeiZleba savaraudo pasuxebidan sinjvis meTodiT SevarCioT.

5 da 19 ar gamodgeba, radgan 19>a>5. a=9 utolobebs ar akmayofilebs. pasuxi:L a=10.

# 4 am amocanis pasuxic SeiZleba savaraudo pasuxebidan sinjvis meTodiT SevarCioT.

2 165 189–115 189=2 050 000 _ ar akmayofilebs,

2 175 189–115 189=2 060 000 _ ar akmayofilebs,

2 185 189–115 189=2 070 000 _ akmayofilebs,

2 195 189–115 189=2 080 000 _ akmayofilebs.

swori pasuxia 8.

5 savaraudo pasuxebis SemowmebiT:

carieli yuTi O2 kgia, forToxali _ 20 kg. 20 ar aris 2ze 20iT meti;

carieli yuTi 1 kgia, forToxali _ 21 kg. 21 metia 1ze 20iT _ akmayofilebs pirobas.

swori pasuxia 2) 1 kg.

II gza. Tu `zedmet~ 20 kg forToxals `gadavyriT~, darCeba 2 kg — forToxali yuTiT.

forToxali _ 1 kg, yuTi _ 1 kg.

TamaSi

`jaWvi~

TamaSis dawyebamde, instruqciis gacnobis Semdeg, jgufebi ecnobian saxelmZRvaneloSi

SemoTavazebul `jaWvs~. rwmundebian, rom yovel baraTze gamosaxuli moqmedebis Sedegi

(gamosaxulebis mniSvneloba — magaliTis pasuxi) aris mezobeli baraTis I komponenti.

akvirdebian jaWvis agebis princips. am jaWvis Semowmebisas, axali jaWvis agebisas da

mowinaaRmdegis jaWvis aRdgenisas moswavleebi iwafebian SekrebagamoklebaSi, eCvevian

jgufur muSaobas.

moswavleebi, albaT, pirveli 7 baraTis Sedgenisas izruneben mxolod imaze, rom

TiToeulSi moqmedebis Sedegi iyos naturaluri ricxvi da am Svideulis yovel Semdeg

baraTSi I komponenti iyos wina baraTSi miRebuli Sedegis toli, merve baraTiT ki Seikras

es jaWvi. magaliTad, jaWvi SeiZleba ase gamoiyurebodes.

44

damoukidebeli wera #2

Tema.: • naturaluri ricxvebis Sekreba. Sekrebis Tvisebebi.

• naturaluri ricxvebis gamokleba

1. elenes aqvs 136 lari, kaxas ki _ 122 lari. Tanxebis gasaTanabreblad elenem kaxas

unda gadasces

a) 6 lari 2) 7 lari 3) 14 lari 4) 8 lari.

2. or konodarbazSi adgilebis erTi da igive odenobaa. pirvelSi kinoseansze 170 bileTi

gaiyida, darCa gasayidi _ 65. meoreSi gaiyida 140. ramdeni bileTi darCa gasayidi meore

kinodarbazSi?

1) 135 2) 60 3) 75 4) 95.

3. Tu a, b da c nebismieri ricxvebia, maSin (a+b)+c=a+(b+c) _

1) Sekrebis ganrigebadobis Tvisebaa

2) Sekrebis jufTebadobis Tvisebaa

3) toloba SeiZleba ar Sesruldes

4) toloba mcdaria.

4. gemi 3 saaTSi 20 kms gadis. es gemi 60 kms gaivlis

1) 6 saaTSi 2) 7 saaTSi 3) 9 saaTSi 4) 8 saaTSi.

5. Tu saklebi maklebze 15432iT metia, maSin sxvaoba

1) aris 15432 2) 15432ze metia 3) 15432ze naklebia

4) nebismieri naturaluri ricxvi SeiZleba iyos.

6. or gemze 895 turistia; amasTanave pirvel gemze 15iT meti turistia, vidre meoreze.

ramdeni turistia pirvel gemze?

1) 880 2) 440 3) 455 4) 425.

7. AB monakveTis sigrZe CD monakveTis sigrZeze 6jer metia; amasTanave, CD monakveTi

AB monakveTze 30 smiT moklea. ipoveT am monakveTebis sigrZeTa jami.

8. aRadgineT swori toloba: 3*1*2*+ *8*6*9**81042

pasuxebi da miTiTebebi amocanebis amosaxsnelad

1 2 3 4 5 6

2 4 2 3 1 3

7. SeiZleba isargebloT sqemiT:

CD=30:5=6 (sm) AB=6+30=36 (sm) AB+CD=42 (sm)moswavleTa codnis done am etapze SesaZleblobas aZlevs moswavleebs airCion amoxsnis

`algebruli gzac~. magaliTad, CDs sigrZe _ x sm, ABs sigrZe _ 6x. CD aris ABze 30 smiT

mokle, anu 6x–x=30, 5x=30, x=6, 6x=36, AB+CD=42 sm.

ar aris gamoricxuli sxva gzebic SemogvTavazon moswavleebma. maTi Canafiqris

realizaciisas miRweuli raime sayuradRebo Sualeduri Sedegi SeiZleba 1 quliT Sefasdes.

amoxsnis dasruleba kidev 1 qulas SesZens maT.

A

C D 30 sm

B

45

8. 391423+ 6896191081042

Tu am amocanaSi 8 saZiebeli cifridan mxolod oria arasworad dadgenili, vfiqrobT,

dasaSvebia 1 qulis dawera, uares SemTxvevaSi Sefaseba _ 0 qulaa, ukeTes SemTxvevaSi _ 2

qula.

yuradRebiT aris gansaxilveli amoxsnis is variantebi, romlebSiac erTi Secdomis daSveba

iwvevs danarCen cdomilebebs _ faqtobrivad daSvebulia erTi Secdoma. am SemTxvevaSic,

vfiqrobT, SesaZlebelia amoxsna erTi quliT Sefasdes.

amrigad, pirveli eqvsi amocana TiTo quliania, bolo ori amocana oror quliani, sul

naSromis umaRlesi Sefaseba SeiZleba iyos _ 10.

sasurvelia, rom Sedges damoukidebeli weris amocanebis ramdenime varianti, SedarebiT

rTuli amocanebi SeiZleba yvelas ar SevTavazoT, didi sifrTxilea saWiro aseTi araTanabari

sirTulis davalebaTa Sesrulebis Sefasebisas, gvaxsovdes rom moswavlis codnis obieqtur

Sefasebas aqvs mastimulirebeli efeqti. Tu Sefasebas araobieqturobis, dasjis an

angariSsworebis `elferi dahkravs~, maSin Sefaseba iqceva saswavlo procesis Semakavebel,

arahumanur aqtad.

sasurvelia, rom damoukidebel weras mosdevdes naSromTa ganxilva _ tipuri Secdomebis

analizi, saTanado komentarebi, romlebSic dasaxelebuli iqneba Sedegebis gaumjobesebis

gzebi; Zalze mniSvnelovania moswavleTa mosazrebebi am amocanaTa Sinaarsisa da Sefasebis

kriteriumebis Sesaxeb.

Akomentarebi sasurvelia weriTi formiT axldes TiToeul naSroms. zogjer, roca amas

pedagogi saWirod miiCnevs, dasaSvebia zepiri ganxilvac. yvelaze nayofieri mainc komentarebis

orive formis gamoyenebaa.

es saswavlo weli garkveul siaxleebs gvTavazobs Sefasebis sistemaSi. amitom sasurvelia

kidev erTxel gaecnoT saTanado masalas am wignSi (ix. gverdi 22) es sakiTxebi warmodgenilia

erovnuli saswavlo gegmis mixedviT, masSi Cveni gamocdilebac aris gaTvaliswinebuli.

gaiTvaliswineT Tqvens skolaSi SemuSavebuli zogadi Sefasebis saxelmZRvanelo principebic

da amis safuZvelze miiReT konkretuli gadawyvetilebebi.

7. ricxviTi sxivi

mizani. moswavlem SeZlos ricxviT sxivze naturaluri ricxvebis ganlageba, gaiwafos

ricxviT sxivze naturalurkoordinatiani wertilis koordinatis povnaSi da piriqiT,

mocemuli koordinatiT sxivze wertilis mdebareobis gansazRvraSi. gaixsenos gasul wels

aTvisebuli masala, gaaRrmavos is, ganiviTaros saTanado unarebi. amiT moswavle eufleba

analizuri geometriis elementebs, axdens codnis integrirebas, rac mis saganmanaTleblo

procesSi erTerT umniSvnelovanes process warmoadgens.

mniSvnelovani praqtikuli amocanebas amoxsnisas. ricxviTi sxivis da geometriuli

TvalsaCinoebis gamoyeneba.

paragrafis teqstSi moswavle suraTis mixedviT poulobs wertilebis koordinatebs.

magaliTad, I suraTze A da B wertilebis koordinatebia 6 da 8, ABs Sua wertilis koordinatia 7. II suraTze 5is bijiT svlis Sedegi _ C wertili gamosaxavs 20s. III suraTze

pirveli monakveTi 1 milionia, Es koordinatia 3 milioni, Gs koordinatia 5 milioni, Bs

koordinatia 8 milioni.

savarjiSoebiT gviwevs sxvadasxva bijiT svlisas mocemuli ricxvebis Sesabamisi wertilebis

sxivebze ganlagebis dadgena, monakveTis Sua wertilis Sesabamisi ricxvis povna.

46

A(a) B(b)

A

3

O

0

B

7

10 a) sami danayofiT gadaadgileba koordinatis 6 erTeuliT cvlilebas axdens (24–

18=6), erTi danayofi Seesabameba 2s. vpoulobT koordinatebs: 20 da 22;

b) erTi danayofis `fasia~ (1032–1022):2=5. wertilebs vawerT: 1027, 1037, 1042, 1047;

g) erTi danayofia 20. wertilebs vawerT: 20, 60, 80, 100, 120, 140, 160.

11 da 12 savarjiSoebis Sesrulebis Semdeg SeiZleba zogadi daskvna gavakeToT: A(a) da

B(b) wertilebidan is aris marjvniv, romlis koordinatic metia. es faqti SeiZleba suraTzec

gamovsaxoT: Tu b>a, maSin

B marjvnivaa.

14 savarjiSoSi moswavle agebs naxazs amocanis pirobis mixedviT; mniSvnelovania, rom

man es samuSao damoukideblad, maswavleblis Carevis gareSe Seasrulos da misi naSromi

pedagogs uyuradRebod ar darCes. monakveTis oTx tol nawilad gasayofad zogierTi

moswavle 8 sms gayofs 4ad da saxazaviT gadazomavs 2 smian monakveTebs, TiTo danayofi

4 erTeulia. zogierTi moswavle miRebul monakveTs jer or tol nawilad gayofs, Semdeg

miRebuli monakveTebidan TiToeuls _ kvlav 2 tol nawilad.

saSinao davalebis 2 savarjiSoSiO SeiZleba moswavlem iolad daiTvalos 101ze meti

da 115ze naklebi ricxvebis odenoba. 3 savarjiSoSi ki mouwevs ufro xelsayreli gzis

Zieba, raTa daiTvalos ricxvebi 97dan 256mde (CaTvliT). kerZod, man unda Seasrulos

gamokleba, magram 2 savarjiSos gamocdilebiTac SeamCnevs, rom swori pasuxis misaRebad

mas mouwevs 256–97+1 gamosaxulebis mniSvnelobis gamoTvla (1is mimatebis gareSe pasuxidan

gamoiricxeboda erTi ricxvi _ 97).

amrigad, vRebulobT saZiebel odenobas _ 160.

8 am monakveTis bolo wertilebi aris 4 da 9, Sesabamisi jamia 13.

9 OA=3 sm, OB=7 sm, AB=4 sm. maTi jamia 14 sm.

10 A wertilis koordinats ase miviRebT: 120+4⋅20=200.

B wertilis _ 120–3⋅20=60.

proeqti: movemzadoT maTematikuri viqtorinisTvis

es SemoqmedebiTi xasiaTis samuSaoa. moswavleebs uviTardebaT sacnobaro saswavlo

literaturis, internetis gamoyenebis unari. SesaZloa da misasalmebelia, Tu isini moZiebul

amocanebs Secvlian/gadaakeTeben Tavisi gemovnebiT, Seadgenen axals. moswavleebi amocanebis

Sedgenisas iwafebian amocanebis amoxsnaSi, pasuxis gaanalizebaSi _ isini unda cdilobden,

rom maTi amocana iyos aqtualuri, saintereso, originaluri, Seesabamebodes maTi codnis

dones, iyos daZlevadi da iqneb _ saxalisoc ki. naSromTa Sefasebisas swored es parametrebi

unda iyos ganmsazRvreli.

maswavleblis mier SerCeuli amocanebiT viqtorina Catardeba an amave klasis jgufebs

Soris, an is sxvadasxva klasis moswavleTa konkursad gadaiqceva. gundebis warmateba fasdeba

ara marto mowinaaRmdegis amocanebis amoxsniT, aramed Tavisi wardgenili amocanebiTac da

maTi avtorebis mier SemoTavazebuli amoxsnis meTodebiTac.

ajobebs, Tu am viqtorinis Jiuris mxolod maswavleblebi Seadgenen, radgan moswavlee

bisTvis erTmaneTis amocanebis gacnoba azrs daukargavda viqtorinas.

47

amocanebis nimuSebis amoxsnis meTodebi

1. Cafiqrebulia umciresi 7niSna ricxvi _ milioni.

2. miliardi aris 1000 milioni, anu milioni aTaseuli. pasuxi: miliardSi aris milioni

aTaseuli.

3. am tipis amocanebi ukve amoxsnili aqvT. am wignSi aris erTniSna ricxvebiT gadanomrili

9 gverdi; orniSna ricxvebiT gadanomrilia 90 gverdi, romelTa gadasanomrad gamoyenebulia

90⋅2=180 cifri; danarCeni 1989 _ (180+9)=1800 cifri gamoyenebulia samniSna ricxvebiT

gadanomril 1800:L3=600 gverdze, miviRebT: 600+90+9=699.

gadanomrili gverdebis odenoba aris 699.

4. sxvaobis misaRebad saklebisTvis 12 milioni gamoukliaT, e. i. maklebi aris 12 milioni.

sxvaoba maklebze, anu 12 milionze 5 milioniT metia, udris 17 milions. saklebia 12 milioni

+ 17 milioni=29 milioni.

5. samniSna ricxvebis odenoba 900ia. maT Casawerad saWiroa 2 700 cifri.

6. ewodeba aTobiTi, radgan viyenebT 10 cifrs da yoveli momdevno Tanrigis erTeuli

winaze 10jer metia.

poziciuria, radgan yoveli cifris mniSvneloba ricxvis CanawerSi misi poziciiT

_ dakavebuli adgiliT ganisazRvreba. mimarTeT moswavleebs moiyvanon sailustracio

magaliTebi da aqve miuTiTon raime arapoziciuri sistemis magaliTi.

7. informacia (laTinuri) _ Setyobineba, cnobebi visime moRvaweobis, raime mdgomareobis,

momxdari ambis da sxvaTa Sesaxeb.

8. abaki _ sangariSo mowyobilobaa.

9. ciferblati (germanuli) _ sazom xelsawyoebSi (magaliTad, saaTi) danayofebiani

firfita, romelzedac moZraobs isari da mianiSnebs dros, an raime sxva sidides.

misasalmebelia, Tu me7 da me9 amocanebSi moswavleebi axsnian warmodgenili sityvebis

warmomavlobasac;

10. (4⋅3):2=6.

11. TiTo xelze 5 TiTia, 10⋅5=50 (TiTi).

12. yvela Tve, garda Tebervlisa _ 11 Tve.

13. 3⋅15=45.

sanimuSo gakveTili #2

aqtivoba. ricxviTi sxivi ($7)

aqtivobis mizani:• moswavlem aRmoaCinos da aiTvisos naturalur ricxvTa gamosaxvis axali forma. pirveli

warmodgenebi Seiqmnas ricxviT simravlesa (amjerad naturalur ricxvTa simravlesa) da

sxivis garkveul wertilTa simravles Soris urTierTcalsaxa Sesabamisobaze _ moswavlem

SeZlos mocemuli ricxvis mixedviT ricxviT sxivze Sesabamisi wertilis mdebareobis dadgena,

wertilis gadaadgilebis aRwera, SeZlos ricxviT sxivze aRebuli wertilis mixedviT misi

Sesabamisi ricxvis (koordinatis) dadgena;• SeZlos am axali warmodgenebis gamoyenebiT praqtikuli amocanebis amoxsna.

• kvlevis arsenalSi CarTos axali meTodis SesaZleblobebi.

aucilebeli wina codna

• naturaluri ricxvebi, moqmedebebi naturalur ricxvebze

• ricxvebis damrgvaleba

• ricxvebis Sedareba

• geometriuli figurebis (wrfe, sxivi, monakveTi) amocnoba, aRwera.

48

aqtivobis aRwera. moswavleebTan erTad vixsenebT, aRvwerT, dafaze gamovsaxavT wrfes,

monakveTs, sxivs. SevCerdebiT sxivze. Semdeg davsvaT amocana: ra wesiT gamovsaxoT ricxvebi

am sxivze? am ganxilvaSi SeiZleba Tqvenc mogiwioT CarTva: SevusabamoT nuls sxivis saTave.

davsvaT SekiTxva: ra wesiT SevusabamoT nulisgan gansxvavebul ricxvebs sxivis wertilebi?

aq SeiZleba gamoiTqvas mosazreba, rom 10s SevusabamoT am sxivis esa Tu is wertili. Tqven

am mosazrebas daupirispireT aseTi mosazreba: rogor (ra wesiT) movZebnoT maSin sxivze,

magaliTad, 13is Sesabamisi wertili? warmoqmnili sirTuleebi moswavleebs daarwmunebs,

rom ajobebs ricxviT sxivze nulis Semdgomi raime wertili SevusabamoT ricxv 1s. nulisa

da 1is Sesabamisi O da U wertilebiT SerCeulia erTeuli monakveTi. aseTi sxivi ricxviTi

sxivia. amis Semdeg daavaleT moswavleebs 2is, 5is Sesabamisi wertilebis miTiTeba am sxivze.

dasviT SekiTxva: ra sirTuleebi SeiZleba wamoiWras ricxvebis wertilebiT gamosaxvisas,

Tu 1s SevusabamebT wina SemTxvevaSi SerCeulisgan gansxvavebul wertils? moswavleebTan

erToblivi ganxilva moswavleebs daarwmunebs, rom aseTi arCevani araviTar principul

gansxvavebas ar Semoitans. amrigad, erTeulovani monakveTi SeiZleba iyos, magaliTad, 1 sm,

7dm, 4 km da a.S.

vTqvaT, erTeuli monakveTi OU 4 smia, maSin 3is Sesabamisi M wertilisTvis OM iqneba

12 sm.

Semdgomi ganxilvisTvis SeiZleba moswavleebs SevTavazoT paragrafis teqstSi war

modgenili savarjiSoebi, saklaso savarjiSoebi 1 15 , axali savarjiSoebi, magaliTad,

1) 1

BUO A

vipovoT mocemul sxivze A–sa da B –s koordinatebi, AB monakveTis Sua wertilis

koordinatebi.

2) MO ricxviT sxivze M wertilis koordinatia 500. rogor ipoviT

2000isa da 2500is koordinatebs?

vfiqrobT, am amocanebis ganxilva moswavleTa aqtiurobiT warimarTeba da arazusti

pasuxebis gasworebas Tavad moswavleebi SeZleben.

3) ricxviT sxivze moniSnulia F(20) wertili. amave sxivze gamosaxeT is wertilebi,

romelTa aTeulamde damrgvalebiT miiReba 20.

FO

1

4) planeta marsis masa miaxloebiT 9jer naklebia dedamiwis masaze. Tu marsis masa

(tonebSi) K wertiliT gamoisaxeba, maSin am ricxviT sxivze sad moniSnavT (TvalzomiT) im D

wertils, romelic dedamiwis masas miaxloebiT warmogvidgens?

moswavleebs mouwevT Sesabamisobis wesis gaxseneba da TvalzomiT swori arCevnis unaris

demonstrireba. mimarTeT mTel klass gamoTqvan sakuTari mosazreba sxvaTa arCevnis sizustis

Sesaxeb. Tqvenc mogiwevT moswavleTa polemikaSi CarTva.

Tu SeniSnavT, rom moswavleebisTvis aseTi amocana (mravali siaxlis gamo) bolomde naTeli

ar iyo, miawodeT raime msgavsi amocana da misi ganxila bolomde miandeT maT. vfiqrobT aseTi

intensiuri ganxilva uTuod nayofieri aRmoCndeba.

aqtivobis gafarToeba. vTavazobT amocanas _ ricxviT sxivze moswavleebma moniSnon A(8) wertili, Semdeg moZebnon AAdan marjvniv 3 erTeuliT daSorebuli B wertili (anu A wertili

gadaadgilon 3 erTeuliT marjvniv), ipovon Bs koordinati; moniSnon Adan 6 erTeuliT

marjvniv daSorebuli C wertili. ipovon misi koordinati.

K (k )O

49

Adan 50 erTeuliT daSorebuli D wertilis moniSvnas ver moaxerxeben, magram koordinatis

sapovnelad ivaraudeben, rom unda Seasrulon mimateba: 8+50, D(58).ra moqmedebas SeasrulebT, Tu gadaadgileba moxdeba Adan marcxniv _ sxivis saTavisken?

moswavleebi ivaraudeben, rom es gamoklebaa da Semdeg praqtikulad Seasruleben am samuSaos.

magaliTad, ipovon Adan marcxniv 5 erTeuliT daSorebuli E wertilis koordinati (8–5=3,

E(3)). amis Semdeg vTavazobT klasSi 16 savarjiSos.

SeiZleba es savarjiSoebi did ricxvebze SeasrulebinoT da ricxviTi sxivis dauflebasTan

erTad gaagrZelon gawafva ricxvebze moqmedebis SesrulebaSi.

magaliTad,

• vTqvaT, ricxviT sxivze mocemulia wertili A(257). ipoveT saTavidan Aze 3jer ufro

Sors mdebare B wertilis koordinatebi.

• vTqvaT, ricxviT sxivze mocemulia M(12125) wertili. ipoveT Misgan 355 erTeuliT

daSorebuli A da B wertilebis koordinatebi.

• ipoveT A(317) da B(1024) wertilebs Soris manZili.

• vTqvaT, ricxviT sxivze mocemulia wertilebi: A(607) da B(3217). ramdeni erTeuliTaa

daSorebuli Adan AB monakveTis P Sua wertili? ipoveT P wertilis koordinati.

sayuradRebo arCevnis winaSe aRmoCndebian moswavleebi, Tu maT SeTavazebT aseT amocanas: ricxviT sxivze gamosaxeT saqarTvelos, azerbaijanis da somxeTis mosaxleobaTa odenobis

Sesabamisi wertilebi: G(4500 000), A(800 000), R(3 500 000). moswavleebs mouwevT saWiro moqmedebaTa

dagegmva. Semdgom SeiZleba aseTi gadawyvetileba miiRon: ricxviT sxivze gamosaxon 500000is

Sesabamisi wertili _ P da OP monakveTis, Sesabamisad, 9jer, 16jer da 7jer gadazomviT

miiRon miTiTebuli wertilebi. SesaZlebelia, rom jer moiniSnos wertili K(1000000) da OK

bijiT moiniSnos sxivze wertilebi `mTeli~ milionebis koordinatebiT _ `viaroT~ OK bijiT.

am gziT aRmovaCenT azerbaijanis Sesabamis wertils, danarCeni ori wertilidan TiToeuli

or `mTel~ wertils Soris iqneba _ Sua wertili.

aqtivobis Sefaseba. am aqtivobis warmatebiT ganxorcielebis SemTxvevaSi moswavle

akavSirebs Sekrebisa da gamoklebis operaciebs ricxviT sxivze wertilis gadaadgilebasTan; gaiwafeba ricxviT sxivze wertilis mdebareobis gansazRvraSi; emzadeba sakoordinato

badis, sakoordinato sistemis gacnobisTvis; iTvisebs analizuri geometriis elementebs,

axdens kvlevis sxvadasxva meTodis integrirebas. dekartisa da fermas epoqaSi daiwyo ori

maTematikuri disciplinis algebrisa da geometriis kvlevis meTodebis Serwyma, ramac

udidesi impulsi da SesaZleblobebi misca maTematikis ganviTarebas.

8. naturaluri ricxvebis gamravleba. gamravlebis Tvisebebi

mizani. moswavlem kidev erTxel gaiazros gamravlebis operaciis arsi, kavSiri Sekrebisa

da gamravlebis ariTmetikuli moqmedebebis Sesrulebisas, amocanebis amoxsnisas. ganaviTaros

konkretuli magaliTebiT gaazrebuli raime kanonzomierebis, Tvisebis, debulebis zogadi

_ algebruli saxiT Cawerisa da Camoyalibebis unari.

paragrafis dasawyisSi SemoTavazebuli amocanis amosaxsnelad saWiroa toli Sesakrebebis

jamis povna, es jami SeiZleba gamravlebiT SevcvaloT. aqve ganvixiloT aseTi amocana:

maTematikur SejibrSi 6 gundi monawileobs da TiToeul gundSi 5 moswavlea. ramdeni

moswavle monawileobs SejibrSi? am amocanis amosaxsnelad unda SevasruloT moqmedebebi

5+5+5+5+5+5, an 6⋅5, pasuxi kvlav 30ia.

gavamaxviloT yuradReba, rom erTi da imave odenobis moswavleebi I SemTxvevaSi 5

(6wevrian) gundad, meore SemTxvevaSi 6 (5wevrian) gundad gaanawiles _ 5⋅6=6⋅5=30. es

gamravlebis gadanacvlebis Tvisebaa: a⋅b=b⋅a.

50

am da sxva Tvisebebs magaliTebis ganxilviT paragrafis teqstSi cxrilis saxiT vawvdiT.

teqstSi erTerT Tvisebas _ gamravlebis ganrigebadobis Tvisebas jamis mimarT _

formulis saxiT vuwerT: a(b+c)=ab+ac da vTavazobT danarCeni Tvisebebi damoukideblad

Caweron asoebis daxmarebiT. maswavlebelTan erTad klasi yoveli Tvisebis sailustraciod

ganixilavs magaliTebs da Semdeg Tavad moswavleebi am Tvisebas asoebiT warmoadgenen.

magaliTad, 1ze gamravlebis Tviseba: 9⋅1=9, 10⋅1=10, 102⋅1=102, ..., a⋅1=a.

gadanacvlebis Tvisebis asoiTi Canaweri amocanebis amoxsnisTanave SeiZleba Sesrul des.

Temis bolos moswavleebi ecnobian `Cvens leqsikonSi~ mocemul termins _ `erTi mainc

ricxvi~. moviyvanoT magaliTebi da `gavSifroT~.

• deda, erTi lari mainc momeci _ anu dedas sTxovs 1 lars an 1 larze met Tanxas.

• `erT moswavles mainc mouwevs azris gamoTqma~ _ erT moswavles an mets (Tundac

yvelas) mouwevs azris gamoTqma.

Semdeg moswavleebs vTavazobT: iqneb moifiqron iseTi magaliTi Sekrebaze, romelSic:

Sesakrebebi nulisgan gansxvavebuli toli ricxvebia, xolo maTi jami _ 0is toli. xanmokle

gansjis Semdeg moswavleebi ivaraudeben, rom es SeuZlebelia. amis Semdeg daasruleben

paragrafis bolo winadadebas: Tu ramdenime ricxvis namravli nulis tolia, maSin am

ricxvTagan erTi mainc nulis tolia.

am ukanasknel debulebas iyeneben moswavleebi saklaso savarjiSoebidan 13 , 14 da 15is ganxilvisas. 13 Si x–2 aris 0is toli, e. i. x=2;

14 Si x–5 aris 0is toli, e. i. x=5;

15 Si x=0 an x–3=0, saidanac x=0 an x=3.

6 savarjiSoSi swrafi angariSisTvis viyenebT gamravlebis gadanacvlebadobisa da

jufTebadobis Tvisebebs, agreTve, 100ze gamravlebis wess:

4⋅7⋅25=4⋅(7⋅25)=4⋅(25⋅7)=(4⋅25)⋅7=700.

7 Si viyenebT gamravlebis gadanacvlebadobisa da ganrigebadobis Tvisebas Sekrebis

mimarT, saidanac vRebulobT bc+ac=(b+a)c. amrigad,

7⋅n+9⋅n=n⋅7+n⋅9=n(7+9)=n⋅16=16⋅n,

rasac mokled ase CavwerT: 7n+9n=(7+9)n=16n.

18 , 20 da 21 amocanebi gaTvaliswinebulia amocanis pirobis mixedviT maTematikuri

modelis SeqmnaSi gawafvisTvis. moswavleebi adgenen asoiT gamosaxulebas da iTvlian mis

mniSvnelobas asoebis mocemuli mniSvnelobebisTvis.

22 amocanaSi jer namravlebs Caweren jamis saxiT, Semdeg jams _ kvlav namravlis

saxiT, magaliTad,

g) 3b+5b+b=b+b+b+b+b+b+b+b+b=9b 3b 5b SesaZlebelia, ganrigebadobis Tvisebebze dayrdnobiT, asec vimsjeloT:

3b+5b+b=(3+5+1)⋅b=9b.

msgavsi savarjiSoebi saSinao davalebaSic SexvdebiT ( 15 ).

saSinao davalebis 3 savarjiSos pasuxi moswavlem SeiZleba kerZo magaliTis safuZvel

ze daadginos. magaliTad, ganixilos namravli 2⋅4=8. Secvalos Tanamamravlebi amocanis pirobis

mixedviT, miiRebs: 4⋅12=48. namravli gaizarda 6jer. SeiZleba imsjelos ase: I Tanamamravlis

2jer gazrdiT namravli gaizarda 2jer (gamravlda 2ze), II Tanamamravlis 3jer gazrdiT

es gaorkecebuli namravli gamravlda 3ze, e. i. namravli gamravlda 6ze _ gaizarda 6jer.

51

11 savarjiSoSi 10a+10b+10c jamis gamosaTvlelad moswavleebs saklaso ganxilvebi

gamoadgebaT:

10b+10c=10(b+c)=10⋅48=480

14 Si vsargeblobT gamravlebis gadanacvlebadobisa da jufTebadobis TvisebebiT:

a) 125⋅256⋅8=125(256⋅8)=125⋅(8⋅256)=(125⋅8)⋅256=1000⋅256=256000,

b) 102⋅500=51000.

21 savarjiSos msgavsi klasSi ar amouxsniaT, amitom savarjiSos axlavs nimuSi,

romliTac isargeblebs moswavle. es amocana axali samuSaos an mocemuli instruqciis

damoukideblad gagebis unaris gamomuSavebasac emsaxureba.

25 amocanis amoxsnisas umjobesia sqema gamoiyenon

C

A

D30 sm

30 sm 30 sm

B

31 am amocanis ori xerxiT amoxsna gamoklebis mimarT ganrigebadobis Tvisebis

ilustrirebaa. amoxsna: 5(16–8) an 5⋅16–5⋅8.

32 es amocanac ornairad SeiZleba amoxsnan (wina amocanis msgavsad):

8⋅(42+53+55) an 8⋅42+8⋅53+8⋅55.

TamaSi

`romeli Tvisebaa gamoyenebuli?~

xSirad moswavleebi asruleben ricxvebze moqmedebebs, swrafi da zepiri angariSis dros

anacvleben Sesakrebebs, Tanamamravlebs, amartiveben asoiT gamosaxulebebs, magram ar ician

moqmedebaTa romel Tvisebebs iyeneben am SemTxvevaSi (magaliTad, 4a–4b=4(a–b) gardaqmnisas

gamoyenebulia gamravlebis ganrigebadobis Tviseba gamoklebis mimarT).

amocanebis Sesadgenad SeiZleba kvlav gadavxedoT paragrafs da movifiqroT msgavsi

amocanebi. magaliTad,

1. 57⋅4–4⋅57=57⋅4–57⋅4=0. gamoyenebulia Tviseba

a) namravlis gadanacvlebadobis b) jamis gadanacvlebadobis

g) namravlis jufTebadobis d) 1is Tviseba.

2. 25⋅4⋅57=(25⋅4)⋅57=5700. gamoyenebulia Tviseba

a) namravlis gadanacvlebadobis b) namravlis jufTebadobis

g) Sekrebis gadanacvlebadobis d) namravlis ganrigebadobis.

3. roca 5(x–2)=0dan vRebulobT x–2, viyenebT

a) namravlis gadanacvlebadobis Tvisebas b) erTis Tvisebas

g) nulis Tvisebas

d) namravlis ganrigebadobis Tvisebas.

4. 4⋅52+4⋅48=4⋅(52+48)=4⋅100=400. gamoyenebulia

a) namravlis gadanacvlebadobis Tviseba

b) jamis gadanacvlebadobis Tviseba

g) namravlis ganrigebadobis Tviseba Sekrebis mimarT

d) nulis Tviseba.

52

5. 8025⋅(2⋅8–2⋅8)=0 gamoyenebulia

a) sxvaobis gadanacvlebadobis Tviseba

b) nulis Tviseba g) erTis Tviseba

d) namravlis jufTebadobis Tviseba.

cxrilis saxiT warmodgenili pasuxebi:

am TamaSis dros moswavleebi kidev erTxel ixseneben moqmedebebis Tvisebebs, iwafebian

maT gamoyenebaSi, amocanebis pasuxebs cxrilis saxiT warmoadgenen _ es erTgvari wvrTnaa

momavali muSaobis warmatebiT Sesasruleblad.

SeiZleba TamaSi gavarTuloT iseTi amocanebiT, romlebSic ori an meti Tvisebis gamoyeneba

mouwevT, magaliTad,

4⋅57⋅25=4⋅(57⋅25)=4⋅(25⋅57)=(4⋅25)⋅57=5700gamoyenebulia gamravlebis jufTebadobisa da gadanacvlebadobis Tvisebebi, amasTan

jufTebadobis Tviseba _ orjer;

4⋅57+43⋅4=4⋅57+4⋅43=4⋅(57+43)=4⋅100=400.

gamoyenebulia gamravlebis gadanacvlebadobisa da gamravlebis ganrigebadobis Tviseba

Sekrebis mimarT.

sanimuSoO gakveTili #3

aqtivoba. naturaluri ricxvebis gamravleba. gamravlebis Tvisebebi ($8)

aqtivobis mizani:• moswavleebma aRidginon da gaifarTovon codna naturaluri ricxvebis gamravlebis

Sesaxeb, gamravlebis operaciis Sekrebis operaciasTan kavSiris Sesaxeb.

• magaliTebis saSualebiT gaixsenon gamravlebis Tvisebebi; moaxdinon maTi zogadi saxiT

formulireba; • gamoimuSavon ariTmetikuli moqmedebebis Sesrulebis dros am Tvisebebis gamoyenebis

unari;• gaiwafon amocanis konteqstis mixedviT mis amosaxsnelad moqmedebaTa mimdevrobis

SerCevisa da am moqmedebebis SesrulebaSi.

• gaiRrmavon ganzogadebis unari, zogadi saxis debulebebis Camoyalibebis unari;• ganiviTaron algebrul gamosaxulebebTan operirebis unari.


• naturaluri ricxvebis Sekreba

• Sekrebis Tvisebebi

• erTniSna ricxvebze gamravleba.

aqtivobis aRwera. moswavleebs vTavazobT paragrafis dasawyisSi Omocemul amocanas (5

gundia, TiToSi 6 moswavlea, sul ramdeni moswavlea?). misi amoxsnisas toli Sesakrebebis

Sekrebaa saWiro. am moqmedebas (toli Sesakrebebis jams) vcvliT gamravlebiT.

1 2 3 4 5

a *

b * *

g * *

d

53

saklaso savarjiSoebidan 5 , 16 , 17 is da an maTi msgavsi savarjiSoebis Sesrulebis

Semdeg vTavazobT dasawyisSi SemoTavazebuli amocanis msgavs amocanas: SejibrebaSi 6 gundi

monawileobs, TiToSi 5 moswavlea, ramdeni moswavle monawileobs SejibrSi?

SevadaroT am ori amocanis amoxsnebi.

I amocana _ 6+6+6+6+6=5⋅6=30,

II amocana _ 5+5+5+5+5+5=6⋅5=30.

mivaqcioT yuradReba im faqts, rom Tanamamravlebis gadanacvlebiT namravli ar Seicvleba.

es gamravlebis gadanacvlebadobis Tvisebaa. kidev rogor SeiZleba es 30 moswavle erTnair

gundebad davajgufoT? 3 gundi _ TiToSi 10 moswavle, 10 gundi _ TiToSi 3 moswavle: 3⋅10=10⋅3; an, 2 gundi _ TiToSi 15 moswavle, 15 gundi _ TiToSi 2 moswavle: 2⋅15=15⋅2.

ramdenime savarjiSos Sesrulebis Semdeg vTavazobT gadanacvlebadobis Tvisebis

formuliT (asoebiT) Caweras: a⋅b=b⋅a.magaliTebis mixedviT moswavleebi ixseneben erTis Tvisebas, nulis Tvisebas, weren am

Tvisebebs formulebis saxiT: 7⋅1= 7⋅0=

8⋅1= 8⋅0=

1017⋅1= 1017⋅0=

a⋅1= a⋅0=

am Tvisebebis gamoyenebaSi gawafvis mizniT vasrulebT saklaso savarjiSoebidan 1 ,

4 , 12 , 13 , 14 , 15 , 19 s. bolo oTxi savarjiSos amoxsnamde aucileblad yuradReba

unda gamaxvildes im faqtze, rom ori ricxvis namravli 0ia maSin da mxolod maSin, roca

am Tanamamravlebidan erTi mainc nulia, am Tvisebis ilustracia magaliTebiT movaxdinoT: 0⋅7=0, 0⋅0=0, 7⋅0=0, 7⋅5≠0.

jufTebadobisa da ganrigebadobis Tvisebebis gasaxseneblad SeiZleba dafa gavyoT orad

da paralelurad ori moswavlis dafasTan muSaobiT, mTeli klasis aqtiurobiT, amovxsnaT

magaliTebi, CavweroT formulebi: (5⋅3)⋅7= 5⋅(3⋅7)= (2⋅12)⋅5= 2⋅(12⋅5)= formula (a⋅b)c=a(b⋅c)

5(3+7)= 5⋅3+5⋅7=

10(2+9)= 10⋅2+10⋅9=

formula a(b+c)=a⋅b+a⋅c

gamravlebis Tvisebebis gamoyenebaSi gawafvis mizniT sasurvelia amoixsnas saklaso

savarjiSoebidan yvela darCenili savarjiSo da amocana.

aqtivobis gafarToeba. am aqtivobis gafarToeba gamravlebis Tvisebebis gamoyenebis

kidev ufro Rrmad gacnobierebas gulisxmobs.

magaliTad, vTavazobT moswavleebs Seadaron 3(a⋅5) namravli 15⋅as. moswavleTa

umravlesoba miuTiTebs, rom isini tolia, magram SeiZleba ver CamoTvalon gamravlebis

Tvisebebi, romlebsac efuZneba maTi varaudi. amovarCioT ramdenime moswavle, romlebic

Seafaseben sxva TanaklaselTa mier gadadgmul yovel nabijs:3⋅(a⋅5) gadanacvlebadobis TvisebiT =3⋅(5⋅a), jufTebadobis TvisebiT=(3⋅5)⋅a=15⋅a.

amrigad, 3⋅a⋅5 CanaweriT mocemuli namravli SeiZleba warmovadginoT, rogorc (3⋅a)⋅5, an

3⋅(a⋅5), an 15a, an a⋅15.

am magaliTebiT vemzadebiT momdevno gakveTilze maTematikuri TamaSisTvis: `romeli

Tvisebaa gamoyenebuli?~

54

gamravlebis TvisebaTa gamoyenebis erTi sayuradRebo nimuSia 17m+13m jamis gamartiveba.

saxeldobr,

17m+13m=(17+13)m=30m; an a, b da c ricxvebisTvis, ab+cb=(a+c)b.

aqtivobis Sefaseba. moswavleebi daakavSireben Sekrebisa da gamravlebis operaciebs; gaecnobian, gaixseneben gamravlebis Tvisebebs da gaiwafebian gamoTvlebisas maT gamoyenebaSi,

asoiTi gamosaxulebebis martivi gardaqmnebisas, amocanebis amoxsnisas, riTac gaiRrmaveben

am sakiTxebis codnas da ganiviTareben misi gamoyenebis unarebs; sayuradRebo magaliTs

miiReben konkretuli faqtebis safuZvelze zogadi saxis varaudebis, debulebebis gaazrebasa

da formulirebaSi.

sakontrolo wera #2

Tema: • ricxviTi sxivi

• naturaluri ricxvebis gamravleba. gamravlebis Tvisebebi

1. A(12)O B(k)

suraTis mixedviT ipoveT k.

1) 24 2) 30 3) 5 4) 36.

2. M(45)O N P

suraTis mixedviT N da P wertilebs Soris Sua wertilis koordinatia.

1) 90 2) 60 3) 120 4) 6.

3. ricxviTi sxivis M wertili Seesabameba 250⋅18⋅40 namravlis mniSvnelobas. romel

wertilebs Soris mdebareobs M wertili?

A(50000) B(100000) C(150000) D(200000)O

1) O da A 2) A da B 3) B da C 4) C da D.

4. ipoveT G wertilis koordinati.

O

+90

G

1) 90 2) 120 3) 180 4) 150.

5. 96⋅12⋅25–25⋅12⋅96 gamosaxulebis MmniSvnelobaa

1) 28800 2) 57600 3) 14400 4) 0.

6. vTqvaT, a, b da c nebismieri naturaluri ricxvebia, maSin gamravlebis ganrigebadobis

kanoni gamoklebis mimarT SeiZleba ase CavweroT:1) (ab)c–a(bc)=0 2) (a–b)–c=(a–c)–b3) a(b–c)=ab–ac 4) (a–b)c=ab–bc.

7. suraTis mixedviT ipoveT m, n, p da t ricxvebi.

D( )t

B( )n

-48 -36

A( )m C( )p E( )215

+21

+53

55

8. ipoveT x, y, z da k ricxvebi:a) 7(x–5)=0; b) y⋅(9–1)=8;

g) z⋅4530=4530⋅16; d) 81652⋅(3+k)=816520.


1 2 3 4 5 6

2 1 4 3 4 3

7. • t+21=215, t=215–21=194; • n+53=t, n+53=194, n=194–53=141; • p=t–36, p=194–36, p=158; • m=p–48, m=158–48=110.

8. a) 7(x–5)=0, x–5=0, x=5 _ nulis Tviseba; b) y⋅(9–1)=8, y⋅8=8, y=1 _ erTis Tviseba; g) z⋅4530=4530⋅16, z=16;

d) 81652⋅(3+k)=816520, 81652⋅(3+k)=81652⋅10, 3+k=10, k=7.

Sefasebis sqema.

testuri tipis amocanebi _ TiToeuli swori amoxsna Sefasdes 1 quliT.

me7 da me8 amocanebidan TiToeuli Sefasdes maqsimum 2 quliT _ TiToeuli cvladis

povna 0,5 quliT.

naSromTa gasworebisas gamoiyeneT 22e gverdze warmodgenili komentarebi.

9. gamravlebis Sedegis Sefaseba. weriTi gamravleba

mizani. praqtikuli amocanis gadawyvetisas namravlis winaswari Sefasebis unaris ganvi

Ta reba dawvrilebiTi gamoTvlebis Tavidan acilebisa da viTarebis swrafi Sefasebis

mizniT; Tanamamravlebis damrgvalebiT miRebuli namravlis zust namravlTan Sedarebisas

swori daskvnebis gamotanis unaris ganviTareba; qveSmiweriT gamrav lebaSi gawafva; nulebiT

daboloebuli ricxvebis gamravlebisas gamravlebis Tvisebebis da 10is xarisxebze gamravlebis

wesebis gamoyenebis unaris ganviTareba.

xSirad, roca moswavleebi ar axdenen Sedegebis winaswar Sefasebas, mcire e. w. `meqa

niku ri~ Secdomis gamo, Rebuloben amocanis SinaarsisTvis sruliad miuRebel pasuxs.

paragrafSi mocemuli masala xels uwyobs aseTi damokidebulebis Secvlas, miznis ufro

naTlad warmodgenas.

paragrafis I punqtSi moyvanili amocanis amoxsnisas orive Tanamamravli metobiT

daamrgvales, amitom miRebuli namravli zustze metia _ zusti namravli naklebia

miaxloebiT mniSvnelobaze. momdevno magaliTis damrgvalebebi naklebobiTaa Sesrulebuli

_ miaxloebiTi namravli naklebia zustze. sasurvelia moswavleebma am magaliTis Sesabamisi

amocana TviTon moifiqron; saswavlo procesi am SemTxvevaSi SemoqmedebiT xasiaTs miiRebs,

aamaRlebs moswavleTa CarTulobas saswavlo procesSi.

moswavleebma unda moifiqron iseTi amocana, sadac monacemebis miaxloebiT mniSvnelobaTa

namravli Sesadarebelia zust mniSvnelobasTan. maT amocanebSi unda iyos kiTxva: `eyofa Tu

ara...~ magaliTad, eyofa Tu ara organizacias 7 950 lari, 41 cali 238 lariani kondicioneris

sayidlad?

56

238⋅41≈200⋅40=8000 _ amisTvis 8000ze meti lari yofila saWiro da pasuxia _ ar eyofa.

amave punqtis mesame magaliTSi ganxilulia SemTxveva, roca Tanamamravlebidan erTi

metobiTaa damrgvalebuli, meore _ naklebobiT.

SevajamoT pirveli punqtis Sedegebi, ganvixiloT sami SemTxveva:

I _ orive Tanamamravli metobiT davamrgvaloT:

238⋅48≈240⋅50=12000. vambobT namravlis zusti mniSvnelona (zusti namravli) naklebia 12000

ze. magaliTad, am saqmes moxmarda 12 000 laramde, an damzadda 12000mde tona...

II orive Tanamamravli davamrgvaloT naklebobiT:

238⋅52≈200⋅50=10000. vambobT _ zusti namravli metia 10 000ze. magaliTad, TamaSs daeswro

10 000ze meti mayurebeli.

III _ Tanamamravlebidan erTi metobiT damrgvalda, meore _ naklebobiT:

238⋅48≈200⋅50=10000. vambobT _ movida miaxloebiT 10 000 mayurebeli, daixarja miaxloebiT 10

000 lari. am SemTxvevaSi ar vazustebT _ miaxloebiTi mniSvneloba metia Tu naklebi zust ze.

Tu SeniSnavT, rom aq warmodgenili masala zogierTi moswavlisTvis rTulad aRsaqmeli

_ moswavle ver axerxebs gamravlebis Sedegis Sefasebas, daubrundiT umartives SemTxvevebs.

SeexeT moswavlisTvis kargad cnobil yofiT situaciebs, CarTeT is aqtiur ganxilvaSi,

uTuod miaRwevT warmatebas.

meore punqti eTmoba weriTi gamravlebis wesebis gaxsenebas, moswavleTa gawafvas

qveSmiweriT gamravlebaSi, 10is xarisxze gamravlebis wesebis gaxsenebasa da nulebiT

daboloebuli ricxvebis gamravlebisas gamravlebis Tvisebebis gamoyenebas. magaliTad,

50⋅3=(10⋅5)⋅3=10⋅(5⋅3)=10⋅15=150. an mokled,

50⋅300=(5⋅3)⋅(10⋅100)=15⋅1000=15000.

gamoviyeneT gamravlebis gadanacvlebadobis, jufTebadobis Tvisebebi da 10ze

gamravlebis wesi.

SeaxseneT moswavleebs, rom am punqtis teqstSi da, saerTod, mTel saxelmZRvaneloSi

dauSvebelia Canawerebis gakeTeba, saxelmZRvanelom unda SeinarCunos Tavdapirveli saxe

4-5 wlis ganmavlobaSi mainc.

klasSi

13 g) 36⋅11=36⋅(10+1)=360+36=396.

16 a) 41⋅23≈40⋅20, an 41⋅23≈40⋅23, an 41⋅23≈41⋅20. samive es miaxloeba naklebobiTaa. amocanis

pirobis gaTvaliswinebiT SeiZleba gamravleba Sesruldes aseTi miaxloebiT

41⋅23≈40⋅25=1000.

17 19⋅28≈20⋅30=600, metobiT.

18 Tanamamravlebi: 185 da 14 davamrgvaloT umaRles Tanrigebamde, miviRebT:

6⋅185⋅14≈6⋅200⋅10=12000, e. i. yuTebSi miaxloebiT 12000 fanqaria.

19 38⋅29 unda SevadaroT 1220s, radgan 38⋅29≈40⋅30=1200 metobiT. amitom 25 dReSi

damzaddeba 1200ze naklebi detali — 1220 detals ver daamzadeben.

20 SevafasoT 128⋅12 namravli.

128⋅12≈128⋅10 ≈1280 naklebobiT.

1280>1200, e. i. Tonis sruli datvirTviT amuSaveba ar aris saWiro, sruli datvirTviT

muSaobisas ukve 10 sTSi SekveTa Sesrulebuli iqneboda.

24 386⋅28 unda SevadaroT 12140kms.

386⋅28≈400⋅30=12 000 metobiT. e. i. 4 kviraSi/28 dReSi navi gaivlis 12 000 kmze naklebs.

57

25 a) ganxilvis pirvel etapze moswavleebi amoicnoben II Tanamamravls, is 21ia. (ra

erTniSna ricxvze unda gavamravloT 3, rom miviRoT 3 an 3iT daboloebuli ricxvi? _ 1ze).

meore etapze amoicnoben I Tanamamravls, is 483ia. Semdeg aRadgenen danarCen cifrebs.

b) ra ricxvze unda gavamravloT 8, rom, miviRoT 0 an 0iT daboloebuli ricxvi? _ 0ze

an 5ze. 0 ar gamodgeba. masze gamravlebiT oTxniSna ricxvi ar miiReba, darCa _ 5. es cifri

aris II Tanamamravlis erTeulis cifri.

ra ricxvze unda gavamravloT 8, rom miviRoT 4iT daboloebuli ricxvi? _ 3ze an 8ze.

8 ar gamodgeba, radgan 318⋅8 ar aris samniSna ricxvi, e. i. II Tanamamravlis aTeulis cifria 3.

danarCeni cifrebis aRdgena qveSmiweriT gamravlebis wesiT moxdeba, miviRebT: 318⋅35=11 130.

26 maklebia a, saklebi _ 5a, sxvaoba _ 5a–a=4a. 4a metia aze 4jer. pirobiT, a=720;

saklebia 5⋅720=3600, sxvaoba _ 4⋅720=2880.

27 maklebia a, saklebi _ 6a, sxvaoba _ 6a–a=5a. sxvaoba 5jer metia maklebze. sxvaoba

tolia 5⋅324=1620is.

saSinao davalebis 23 savarjiSoSi:

a) oTxniSna ricxvis 9ze gamrvlebiT miiRes kvlav oTxniSna ricxvi. amitom, I

Tanamamravlis aTaseulis Tanrigis cifria 1. ra ricxvze unda gavamravloT 9, rom miviRoT

7ze daboloebuli ricxvi? _ 3ze. amrigad, I Tanamamravlis erTeulis Tanrigis cifri 3ia,

aTeulis cifri 0 unda iyos. miviReT: 1 1 0 3× 9

9 9 2 7b) mocemuli samniSna ricxvi 1ze unda gamravldes, rom kvlav samniSna ricxvi miviRoT.

e. i. II Tanamamravlis erTeulis Tanrigis cifri 1ia, I Tanamamravlis erTeulis cifri _ 5.

II Tanamamravlis aTeulis cifri luwia da am luwi ricxvis 6ze gamravlebiT miiReba 2iT

dawyebuli orniSna ricxvi. es ricxvia 4. miviReT, rom II Tanamamravli 41ia, I Tanamamravli

_ 625. pasuxi: 625⋅41=25625.

24 amocanis amoxsna SeiZleba mxolod pirobis damatebiT. saxelmZRvanelos pasuxebSi

SemoTavazebulia dasamatebeli piroba. SeiZleba daematos sxva pirobac. magaliTad, meoreSi

Tavdapirvelad iyo 5 000 litri benzini; an _ meoreSi Tavdapirvelad iyo 2jer meti benzini,

vidre pirvelSi; an meoreSi darCa imdeni, ramdenic gaitanes da sxva.

moisazre

1 as mniSvnelobebidan moswavleebs SeiZleba gamorCeT TviT 8 an 1. mivaqcioT amas

yuradReba! pasuxi: 1; 2; 4; 8.

2 a⋅c=c tolobidan daadgenen, rom a=1. amitom b da c gansxvavebulebia 1isgan. darCa

b=4, c=2 an b=2, c=4. miviReT:

1 4×

22 8

1 2×

42 8

an

SeiZleba gamoiyenoT sinjvis meTodi, romlis drosac unda ganixiloT cs mxolod oTxi

mniSvneloba 1; 2; 4; 8 (ixileT wina amocana).

3 maklebi iyos x, saklebi iqneba 3x, sxvaoba 3x–x=2x. sxvaoba maklebze 2jer metia,

sxvaoba 1400ia, maklebi 2jer naklebia _ 700, saklebi 3jer metia maklebze _ 2 100.

58

4 z unda iyos 5 an 0. 0 gamoiricxeba (pirobis Tanaxmad), e. i. z=5; a ar SeiZleba iyos

1ze meti ricxvi, winaaRmdeg SemTxvevaSi namravli ar iqneba samniSna, e. i. a=1.

c kentia, radgan misi 5ze gamravlebis Sedegi 5iT bolovdeba, e. i. c=9.

x metia, an tolia 5is, e. i. x=6.

b da yisTvis darCa luwi ricxvebi 2 da 4. 5⋅b bolovdeba 0iT, e. i. y=4, b=2.

5 II ricxvi iyos x, I ricxvi _ 5x, II ricxvi 3jer naklebia x+5x=6xze. e. i. III ricxvia 2x.

II ricxvi 2jer naklebia IIIze.

6 I _ 3x II _ x 4x

III _ 2x III metia IIze 2jer.

7 warwera Sav CanTaze _ `CanTaSi qarTulis da maTematikis wignebia~ niSnavs, rom

CanTaSi aris qarTulis wignic da maTematikis wignic. es warwera arasworia mxolod maSin,

roca am CanTaSi aris mxolod qarTulis wigni, an mxolod maTematikis wigni. radgan mcdaria

warwera lurj CanTaze, amitom masSi iqneba maTematikis wigni, SavSi aRmoCndeba qarTulis

wigni.

8 4ze da 5ze namravlebis jami iqneba 643is 9ze namravlis toli:

643⋅9=3215+2572=5787.

gamravleba ase dasruldeba: 643× 945

3215 25725787607635

9 yuradRebas vaqcevT termins èrTi mainc wiTeli vaSli~ (adre gvqonda èrTi mainc

ricxvi~), anu 1 an 1ze meti wiTeli vaSli unda amoviRoT.

aseTi amocanebis amoxsnisas vixilavT yvelaze ùiRblo~ (uares) variants, rodesac sul

mwvane vaSlebi amodis. aseTi SemTxveva ar SeiZleba iyos 10ze meti. 10 uiRblo amoRebis

Semdeg me11 vaSli aucileblad wiTeli aRmoCndeba. pasuxi: 11.


aqtivoba. gamravlebis Sedegis Sefaseba ($9, I punqti).

aqtivobis mizani. mocemuli ricxviTi gamosaxulebis daxasiaTebisas mniSvnelobis

Sromatevadi, araaucilebeli gamoTvlebis Tavidan acilebis mizniT moswavlem SeZlos

moqmedebis komponentebis damrgvaleba da Semdeg maTze moqmedebebis swrafad Sesruleba;

amasTanave, man unda SeZlos, amocanis pirobidan gamomdinare, dagegmos ganxilvis gza, SearCios

damrgvalebis wesi _ metobiT daamrgvalos, Tu — naklebobiT; gaiwafos gamravlebis Sedegis

SefasebaSi.


• 0ebze daboloebuli ricxvebis gamravleba

• erTniSna ricxvze gamravleba

59

• ricxvebis damrgvaleba

• amocanis konteqstis mixedviT moqmedebebis Tanamimdevrobis SerCeva.

aqtivobis aRwera. Temis Seswavlas viwyebT amocanis ($9is I punqti, I amocana) dasmiT,

sadac moswavles mouwevs amocanis amoxsnisTvis saWiro moqmedebis Sedegis 626 000Tan

Sedareba. es Sedareba SesaZlebelia dawvrilebiTi gamoTvlebis gareSe _ swrafi angariSiT

moxerxdes. moswavleebi amocanis gadasawyvetad saWiro moqmedebas weren dafaze 26 700⋅18 da

Semdeg TiToeuli Tanamamravlis umaRles Tanrigamde metobiT damrgvalebiT miRebul ricxvs

(600 000s) adareben 626 000s. cxadia, kompania am davalebas ver Seasrulebs.

dafaze davtovoT Sefaseba: 26 700⋅18<600 000.

axla SeiZleba SevTavazoT igive amocana Secvlili monacemebiT: kompanias dReSi SeuZlia

32 300 litris gamoSveba; SeZlebs Tu ara es kompania 22 dReSi 600 000 litris gamoSvebas?

moswavleebi imave meTodiT Tu isargebleben, miiReben, rom 32 300⋅22≈30 000⋅20=600 000.

imsjelon, ra igulisxmeba miRebul miaxloebiT pasuxSi _ kompanias zustad 600 000 lit

ris damzadeba SeuZlia, metis, Tu naklebis? erToblivi gansjiT miviRebT 32 300⋅22>600 000.

pasuxia _ kompania am davalebis Sesrulebas SeZlebs.

am ori amocanidan unda gamoitanon daskvnebi: I SemTxvevaSi Tanamamravlebi metobiT

damrgvalda Tu naklebobiT?

sasurvelia moswavleebma orive SemTxveva sityvierad aRweron. magaliTad, I SemTxvevaSi

kompania amzadebs 600 000 litrze naklebs, II SemTxvevaSi 600 000 litrze mets.

am ori amocanis Semdeg SeiZleba gavagrZeloT I punqtSi SemoTavazebuli amocanebidan

meore _ fexburTis matCze mayurebelTa odenobebze.

am amocanebis ganxilvis Semdeg moswavlem unda SeZlos 3 SemTxvevis gamoyofa:

a) erTerTi Tanamamravli damrgvalebulia metobiT _ miaxloebiTi Sedegi miRebulia

metobiT, anu miaxloebiTi Sedegi metia zustze.

b) erTerTi Tanamamravli damrgvalebulia naklebobiT _ miaxloebiTi Sedegi miRebulia

naklebobiT, anu miaxloebiTi Sedegi naklebia zustze.

g) Tanamamravlebidan zogi metobiTaa damrgvalebuli, zogi naklebobiT, am dros

gamravlebis Sedegis Sefasebas Semdgomi dazusteba esaWiroeba.

gamravlebis Sedegis Sefasebasa da mis mravalmxriv gamoyenebaSi gawafvisTvis SeiZleba

amoixsnas saklaso savarjiSoebidan 1 12 , 14 , 17 , 18 , 19 , 20 , 24 savarjiSoebi.

saSinao davaleba SevarCioT savarjiSoebidan: 1 9 , 12 , 14 , 15 , 16 ,

17 , 18 .

aqtivobis gafarToeba. ganvixiloT saklaso savarjiSo 16 .

41⋅23 gavamravloT miaxloebiT sxvadasxva xerxiT. magaliTad, maswavlebeli sTavazobs

moswavles daamrgvalos I Tanamamravli: 41⋅23≈40⋅23, a=40⋅23;

daamrgvalos II Tanamamravli: 41⋅23≈41⋅20, b=41⋅20,

daamrgvalos orive Tanamamravli: 41⋅23≈40⋅20, c=40⋅20.

Semdeg vTavazobTB a, b da c namravlis zust d mniSvnelobas (misi gamoTvlis gareSe) Seada

ron:

• c (cxadia, c<d) radgan orive Tanamamravli naklebobiTaa damrgvalebuli;

• a (a<d);• b (b<d);• a da bs Sedareba SeiZleba msjelobis sagani gaxdes.

a zust namravlze 1jer 23iT anu 23iT naklebia,

b zust namravlze 3jer 41iT naklebia, e. i. b<a.

60

gamravlebis Sedegis Sefasebis aq warmodgenili magaliTebis ganxilvam moswavleebs ar

unda Seuqmnas iluzia, rom msgavsi miaxloebiTi gamoTvlebi yovelTvis SevZlebT iolad

SevafasoT namravlebi. sailustraciod asa da bs Sedareba SeiZleba moviyvanoT.

moswavleebSi analizis unaris ganviTarebas xels Seuwyobs 25 saklaso amocanis sajaro

ganxilvac. am tipis amocanebi, rogorc wesi, moswavleTa interess iwvevs da amitom maTi

ganxilva maRali aqtiurobiT warimarTeba.

aqtivobis Sefaseba. am aqtivobis warmatebiT ganxorcielebis SemTxvevaSi moswavleebi

gaiwafebian gamravlebis Sedegis swrafad SefasebaSi Tanamamravlebis damrgvalebis

saSualebiT. amocanis konteqstis mixedviT isini gamoimuSaveben damrgvalebis saWiro xerxis

(metobiT, naklebobiT) SerCevis unars. es unari maT gamoadgebaT momavalSi praqtikuli

(ekonomikuri, samSeneblo da sxva) amocanebis gadawyvetaSi.

10. sakoordinato bade

mizani. sakoordinato badis gamoyenebiT moswavle araformalurad gaecnos koordina

tebs da SeZlos raime obieqtis adgilmdebareobis aRwera simboloTa wyvilis gamoyenebiT, da

piriqiTac, obieqtis koordinatebis dadgena mocemuli badis mixedviT; gamoiyenos es xerxi

realur viTarebaSi; moswavlem SeZlos sakoordinato badeze marSrutis gamosaxva da piriqiT,

gamosaxuli marSrutiT aRweros obieqtis gadaadgileba, kerZod — oTxi mimarTulebiT

gadaadgileba.

paragrafis dasawyisSive xdeba koordinatebis araformaluri gacnoba. suraTze Tbilisis

erTerTi ubnis ramdenime obieqtia warmodgenili. maTi mdebareoba koordinatebiT (amjerad

simboloTa wyviliT _ asoTi da ricxviT) moswavleTa erToblivi ZalisxmeviT unda

ganisazRvros. zogierTi obieqtis mdebareobis gansazRvrisTvis ori an meti wyvilia saWiro.

magaliTad, ipodromi d4 da g4 kvadratebSia.

umravlesobisTvis nacnobia Wadrakis dafa da saWadrako figurebi. am dafaze yoveli

figuris mdebareoba ganisazRvreba ori koordinatiT (aso, ricxvi).

moswavleebi adgilidan asaxeleben yoveli figuris koordinatebs. SeiZleba koordinatebis

saSualebiT figurebis ramdenime svlac SemogvTavazon. magaliTad, upasuxon kiTxvebs:

• dasaxeleT ujrebi, romlebzec erTi svliT SeiZleba gadaadgildes TeTri mefe (b1, c1, d1, b2, d2, b3, c3, d3).

• romelia am ujrebidan usafrTxo mefisTvis? da a. S.

4 savarjiSoSi suraTze warmodgenilia turistis moZraobis marSruti. am savarjiSos

Sesrulebis Semdeg 5 savarjiSoSi moswavles Tavad mouwevs aseTive marSrutis Sedgena

sawyisi, saboloo da Sualeduri punqtebis miTiTebiT, anu is iwafeba instruqciis mixedviT

sakoordinato badeze gadaadgilebaSi.

9 amocana wyvilebSi samuSaodaa gankuTvnili. am savarjiSos Sesrulebisas moswavleebi

iwafebian koordinatebiT obieqtis mdebareobis aRweraSi, xalisdebian, eCvevian jgufurad

muSaobas.

saWadrako dafa kargi magaliTia sakoordinato badisa. saSinao davalebis 3 savar

jiSoSi moswavlem saWadrako figuris _ mxedris svlac unda icodes. amitom Cven vacnobT

moswavles mxedris SesaZlo svlebs. am amocaniT moswavle sayuradRebo da sasargeblo

informacias gaecnoba. gegmaze muSaobisas is ukeT gaecnoba metropolitenis sadgurTa

umravlesobas, maT Tanamimdevrobas da urTierTganlagebas.

61

5 da 6 savarjiSoSi moswavleebs vTavazobT

Tavad moifiqron bade da miuTiTon obieqtebis koordinatebi.

am savarjiSoebis Sesrulebisas isini ukve gawafulni arian

koordinatebiT obieqtis mdebareobis gansazRvraSi. saklaso

oTaxi SeiZleba Teatris darbaziviT dafaron badiT: anu

jer miuTiTon rigi, Semdeg adgili. magaliTad, Teonas

koordinatebia rigi 3, adgili 4.

7 savarjiSo moswavleTaTvis saxaliso aRmoCndeba. isini iwafebian garemos ganaSenia

nebis gegmis SedgenaSi.

rubrika `sxvadasxvaSi~ SemoTavazebuli TamaSi SeiZleba mezobel moswavleebs an

moswavleTa jgufebs Soris Catardes. yoveli `moxvedris~ Semdeg moswavleebi cdiloben

ipovon SesaZlo didi gemis danarCeni nawilebi _ eZeben maT horizontalurad an

vertikalurad mezobel ujrebSi. yoveli gasrolisas unda moniSnon dasaxelebuli

koordinatebi TavianT badezec. TamaSis dasrulebis Semdeg msjeloben TamaSis strategiaze

_ rogor jobda gemebis ganlageba, romel ujrebze jobda Seteva. magaliTad, teqstSi

SemoTavazebulia didi gemebis erTmaneTTan axlos ganlagebis idea. ra upiratesoba aqvs am

ganlagebas?

suraTze mocemulia didi gemebis ganlagebis erTi maga

liTi, roca didi `sivrce~ rCeba katarRebisTvis da maTi

`pov na~ Zneldeba.

SeiZleba Seicvalos wesebi ase: ramdenime erTdrouli

gasrola xdeba erTi moTamaSis mier, Semdeg amave odenobis

gasrolebs axdens meore mowinaaRmdege.

SeiZleba metoqes Seatyobinos mxolod moxvedraTa,

CaZirvebis da acdenebis saerTo odenoba.

TamaSis warmatebiT Catarebis Semdeg SeiZleba CavTvaloT,

rom Tqveni moswavleebi floben sakoordinato bades.

Tu sakoordinato badeebSi ujrebis odenoba didia, horizontalur RerZze asoebis

nacvlad ricxvebis miTiTeba iqneboda mosaxerxebeli. maSin gemis koordinatebi ricxvTa

wyviliT gamoisaxeba.

proeqtSi SemoTavazebulia internetis SesaZleblobaTa CarTva sakoordinato badis

ukeT aTvisebis procesSi. miTiTebuli gzebiT moswavle dainaxavs (kompiuterebisa da

internetis misawvdomobis SemTxvevaSi) sakuTari sacxovrebeli ubnis, qalaqis, qveynis

saocar suraTebs. Tanamgzavrebidan gadaRebul am suraTebs man unda miusadagos mis mierve

damzadebuli sakoordinato badeebi. SeiZleba geografiuli rukebis, atlasebis gamoyenebac,

samuSaos dasrulebisas moswavleebs gaaxalisebda Sejibri punqtebisa da maTi koordinatebis

amocnobaSi.

adgili

rig

i

62


aqtivoba. sakoordinato bade ($10)

aqtivobis mizani. moswavle daeuflos sakoordinato badeze obieqtis mdebareobis

gansazRvras koordinatebiT; daeuflos badeze gadaadgilebis aRweras koordinatebiT;

gamoiyenos miRebuli codna momijnave disciplinebSi (magaliTad, geografia). am aqtivobiT

safuZveli eyreba koordinatTa sistemis gaazrebasa da mis aqtiur gamoyenebas.


• sakoordinato sxivze wertilis mdebareobis da misi gadaadgilebis aRwera koordinatebis

saSualebiT.

• ariTmetikuli moqmedebebi naturalur ricxvebze.

• sakoordinato sxivze wertilis gadaadgilebis dakavSireba Sekrebisa da gamoklebis

operaciebTan.

masala:

• saxelmZRvanelos $10is suraTebi.

• ori sakoordinato bade yoveli moswavlisTvis.

aqtivobis aRwera.

Temis axsnas paragrafis dasawyisSi mocemuli suraTis ganxilviT viwyebT. erTerTi

raionis gegma sakoordinato badiTaa dafaruli, koordinatebiT obieqtis mdebareobis

miTiTebaSi moswavleebs daexmarebaT maswavleblis kiTxvebi:

• ra obieqtia g svetSi (lokomotivis stadioni, WavWavaZis gamziris nawili, TamaraSvilis

quCis nawi li, ipodromis da vakis parkis nawili), • ra obieqtia me2 striqonSi?

• ra obieqtia g svetsa da me2 striqonSi? _ lokomotivis stadioni, vakis parkis

nawili,

lokomotivis stadionis koordinatebia (g; 2), vakis parkis nawilic (g; 2) kvadratSia

gamosaxuli.

am suraTis saSualebiT moswavleebi gaiwafebian obieqtis koordinatebis dasaxelebaSi da

piriqiT, koordinatebiT obieqtis mignebaSi.

amis Semdeg moswavleebs davavaloT maTTvis darigebul cariel badeebze ganaTavson maT

mier `daarsebuli~ `qalaqebi~, `regionebi~, `saxelmwifoebi~, ganalagon obieqtebi da aRweron

isini koordinatebiT.

isargebleT romelime moswavlis gegmiT, saxelmZRvaneloSi $10is teqstis bolo suraTiT

da dasviT kiTxvebi:

• ra koordinatebis mqone ujrebi unda gaiaros mgzavrma klubidan saavadmyofoSi

misasvlelad?

• miuTiTeT sxva traeqtoria imave gadaadgilebisTvis.

• mocemuli koordinatebiT aRwereT mgzavris gadaadgileba:

(6; 2) → (6; 3) → (6; 4) → (7; 4) → (8; 4). (saavadmyofo _ tba ).• miuTiTeT umoklesi traeqtoria imave gadaadgilebisTvis da sxva.

es savarjiSo moswavleebs saSinao davalebis 8 amocanis SesrulebaSi daexmareba.

am da sxva misi msgavsi kiTxvebiT moswavleebi gaiwafebian koordinatebiT sakoordi na

to badeze gadadgilebis aRweraSi. paragrafis yoveli savarjiSo am unarCvevebis ganviTa re

bisTvisaa gankuTvnili. yuradReba gaamaxvileT (a; b) da (b; a) wertilTa ganlagebaze.

63

9 savarjiSos Sesasruleblad moswavleebs kvlav davurigebT sakoordinato badeebs.

am savarjiSoTi maT mouwevT Tanaklaselis mier DdaSifruli cifris amocnoba Tanaklaselis

mier miTiTebuli koordinatebis mqone ujrebis gaferadebiT.

SeiZleba `daSifron ~ saqarTvelos droSa.

Wadrakis mcodne moswavleebi koordinatebis saSualebiT aswavlian Tanaklaselebs

figurebis svlebs.

magaliTad, Wadrakis partia ufro xSirad iwyeba paikis svliT e2-e4. romel ujraze

SeiZleba gadaadgildes mxedari b2? da sxva.

aqtivobis gafarToeba SeiZleba momdevno gakveTilze `sazRvao brZolebiT~

ganxorcieldes, TamaSi saxelmZRvaneloSia aRwerili, misi ganxorcieleba sakoordinato badis

flobis garda anviTarebs sabrZolo strategiebis SemuSavebis, mowinaaRmdegis strategiis

amocnobis unars.

aqtivobis Sefaseba. aqtivobis ganxorcielebis Sedegad moswavleebi gaiwafebian

sakoordinato badeze obieqtis mdebareobisa da gadaadgilebis aRweraSi koordinatebis

meSveobiT. es aqtivobebi amzadebs moswavleebs geografiis, informatikis gakveTilebisTvisac.

isini ukve SeZleben obieqtis grZedis da ganedis saSualebiT mignebas; figuris koordinatebiT

gamosaxva daexmareba kompiuteruli suraTebis SeqmnaSi.

Ddamoukidebeli wera #3

Tema: • gamravlebis Sedegis Sefaseba. weriTi gamravleba.

• sakoordinato bade

1. Tanamamravlebis umaRles Tanrigamde damrgvalebiT ipoveT miaxloebiT 71693⋅296

namravli.

1) 21600000 2) 14000000 3) 21000000 4) 24000000.

2. sasadilosTvis sayidelia 73 tomara kartofili, TiToeuli — 43 larad. am

SenaZenisTvis sakmarisia

1) 3 aTas 200 lari

2) 3 aTas 100 lari

3) 2 aTas 300 lari

4) 2 aTas 100 lari.

3. daamrgvaleT Tanamamravlebi naklebobiT aseulamde da ipoveT namravli: 2569⋅731.

1) 1820000 2) 1750000 3) 2100000 4) 1400000.

4. amJamad mefrinveleobis fermaSi 19 aTas 384 frinvelia; maTi ricxvis 2jer gazrdis

SemTxvevaSi fermaSi iqneba

1) 38700ze naklebi frinveli

2) 38000ze naklebi frinveli

3) 40 aTasze naklebi frinveli

4) 40 aTasze meti frinveli.

64

4 skola•

zooparki•

3 cirki•

2 stadioni•

baRi•

1 auzi•

a b g d e

6. mocemul sakoordinato badeze (e; 4) kvadratSi ganlagebulia

1) zooparki 2) auzi 3) skola 4) cirki.

7. gaarkvieT, rogor Seicvleba ori ricxvis namravli, Tu

a) erTerTi Tanamamravli Semcirdeba 9jer, meore ki ar Seicvleba;b) erTerTi Tanamamravli gaizrdeba 3jer, meore ki 5jer.

8. marikam peplis amoqargva gadawyvita

da ujrian furcelze Sesafarisi naxatic

Seasrula, Tumca frTebze `xalebis~

moniSvna bolomde ver moaswro _

mosaniSni darCa erTi erTujriani da

erTic orujriani `xalebi~. frTebis

mokecvisas `xalebi~ erTmaneTs unda

emTxveodes. daasaxeleT mosaniSni ujrebis

koordinatebi da moniSneT es wertilebi.

pasuxebi da miTiTebebi amocanebis amoxsnisTvis

1 2 3 4 5 6

3 1 2 3 4 1

7. a) erTerTi Tanamamravlis 9jer Semcireba gamoiwvevs namravlis 9jer Semcirebas; b) namravli gaizrdeba 3⋅5=15jer.

8. gasaferadebeli ujrebia (7; 14), (15, 9) da (16; 8).

5. mocemul sakoor

di nato badeze `baRis~

Se sabamisi kvadratebis

koor dinatebia:a) (a; 2)2) (b; 4) 3) (d; 3) 4) (g; 2).

12

1920

16 191

1

5

9

16

3

7

14

11

18

2

6

13

10

17

4

8

15

73 9 145 112 8 134 10 15 186 12 17 20

65

Sefasebis sqema.

yoveli testidan TiToeulis amoxsna _ 1 qula.

me7 amocana _ a) _ 0,5 q.

b) _ 0,5 q.

me8 amocana _ sworad moniSnuli yoveli wertilisTvis da misTvis sworad amowerili

koordinatebisTvis _ 1 qula.

srulyofili pasuxebisTvis _ 3 qula.

naSromTa gasworebisas gamoiyeneba 22e gverdze warmodgenili rekomendaciebi.

11. ricxviTi da asoiTi gamosaxulebebis Sedgenisa da gamartivebis meTodebi

mizani. moswavlem SesZlos realuri vaTarebis, teqsturi amocanis sityvieri aRweris

mixedviT Sesabamisi ricxviTi gamosaxulebis an erTi cvladis Semcveli gamosaxulebis Sedgena;

ricxviTi gamosaxulebis mniSvnelobis povna ricxvebze moqmedebaTa Tvisebebis gamoyenebiT;

cvladiani gamosaxulebis mniSvnelobebis povna cvladis sxvadasxva mniSvnelobisTvis.

gamoTvlebSi gawafvis kvalobaze gamoTvlebis racionaluri gzebis SerCevis unaris

ganviTareba; ricxvTa Caweris poziciur sistemaze warmodgenebis gaRrmaveba.

paragrafis teqstSi moyvanilia teqsturi amocana da aRwerilia misi Sesabamisi ricxviTi

gamosaxulebis Sedgenis procesi. Semdeg mocemulia meore amocana, romelic miiReba

pirvelisgan erTerTi ricxviTi monacemis SecvliT. Sesabamisi ricxviTi gamosaxulebebis

SedarebiT xdeba amocanis ganzogadeba _ miTiTebuli ricxviTi mniSvnelobis cvladiT

Canacvleba. ganxilulia is kerZo SemTxvevebi, roca cvladi sawyis etapze mocemul

mniSvnelobebs iRebs.

savarjiSoebi isea SerCeuli, rom moswavleebma Teoriuli codna ganimtkicon praqtikuli

gamoyenebebiT. daxurulboloiani (testuri) tipis amocanebis amoxsnisas ( 1 20 ) zogierT

SemTxvevaSi ar SemoifargloT mxolod swori pasuxis miTiTebiT _ aucileblad gaarCieT Tu

ra mosazrebas, an moqmedebaTa ra Tvisebebs efuZneba moswavleTa arCevani. magaliTad, 1 _

ricxviTi gamosaxulebis mniSvnelobas vpoulobT moqmedebaTa Tanamimdevrobis wesis dacvis

gziT, 6 , 7 , 8 _ viyenebT ganrigebadobis kanons, 18 , 19 , 20 _ marTkuTxedis

perimetris gamosaTvleli formulis gamoyeneba.

gansakuTrebuli yuradRebiT moekideT cvladiani gamosaxulebebis mniSvnelobis povnis

process. magaliTad, 21 amocanaSi SesaZlebelia an cvladis mniSvnelobis pirdapiri Casma,

an gamosaxulebis winaswar gamartiveba:

I xerxi. Tu n=75, maSin (135+n)–23=(135+75)–23=210–23=187;

II xerxi. Tu n=75, maSin (135+n)–23=135–23+n=112+n=112+75=187.

ricxviTi an cvladiani gamosaxulebis mniSvnelobis povnis racionaluri gzis Ziebisas

moswavleebi uTuod darwmundebian, rom winaswari gamartiveba xSirad arsebiTad amcirebs

Sromatevad gamoTvlebs. aseTive analizi sasurvelia Catardes 29 amocanis amoxsnisas.

kanonzomierebis aRmoCenis, hipoTezis SemoTavazebis, misi kritikuli analizisa da Caweris

unarebis ganviTarebas emsaxureba 33 amocana _ a) meore svetis ricxvebi 3iT aRemateba

pirveli svetis Sesabamis ricxvebs; b) meore svetis ricxvebi 3iT naklebia pirveli svetis

Sesabamis ricxvebze; g) meore svetis ricxvebi erTiT aRemateba pirveli svetis Sesabamisi

ricxvebis gaorkecebuls. 35 amocanaSi SesaZloa zogierTma moswavlem orjer dasaxelebuli

`nakleboba~, orivejer gayofas daukavSiros. Semdgom aseT Secdomebs mxolod klasSi sajaro

ganxilva agvaSorebs.

66

pedagogebs SevaxsenebT, rom sxvadasxva akademiuri donis moswavleebTan warmodgenili

saklaso da klasgareSi muSaoba mxolod diferencirebuli _ moswavleTa individualuri

Taviseburebebis gaTvaliswinebiTaa SesaZlebeli.

saSinao davalebis amocanebi saklaso amocanebis analogiuria. gamonakliss warmoadgens

24 _ suraTis mixedviT moswavleebi iReben gantolebas 10x=50, saidanac x=5 (sm). BC

monakveTis sigrZe 3xia, anu 3⋅5=15 smia.

36 mniSvnelovania sidideTa (Tanxis) erTsa da imave zomis erTeulebSi (TeTrebSi)

gamosaxva.

1 amocana sinjvis meTodiT amovxsnaT; amasTanave, erTniSna ricxvis `dasawuneblad~

sakmarisia gamravlebisas ori mainc gansxvavebuli cifris miReba. magaliTad,

cxadia, rom 2 ar gamodgeba. amocanis pirobas

akmayofilebs erTaderTi erTniSna ricxvi _ 7:

2 pirobiT, 1abcd7

× 5

71abcd

gamravlebis TanamimdevrobiT SesrulebiTa da

miRebuli Sedegis TanamamravlSi gaTvaliswinebiT vipoviT ucnobi cifrebis

mni Sv nelobebs. magaliTad, 7⋅5=35, e. i. d=5; d⋅5+3=25+3=28, e. i. c=8 da a. S. b=2, a=4:

3 gaviTvaliswinoT, rom pirobiT, 5abcde:4=abcde5, amitom, abcde5⋅4=5abcde. gamoviyenoT

wina amocanaSi aRwerili meTodi da miviRebT: a=1, b=2, c=8, d=2, e=0.

4 es amocana SeiZleba, magaliTad, ase amovxsnaT:

vato, zura, _ , _ 11

vato, _, irakli _ 12

_ , zura, irakli _ 13.

amrigad, 11+12+13=36 aris biWebis kaklebis odenobaTa gaorkecebulebis jami. Sesabamisad,

saerTo odenoba iqneba 36is naxevari, anu 18.

b) calcalke odenobebis povna SeiZleba aseTi sqemebiT:

vato, zura, irakli vato, irakli, zura an irakli, zura

11 12 7

18 18 13

iraklis: 18–11=7 zuras: 18–12=6 zuras: 13–7=6.

cxadia, vatos aqvs 18–7–6=5 kakali.

5 amocanis pirobis mixedviT, daWerili Tevzebis umciresi odenoba aranakleb erTia.

sinjvis meTodiT SeiZleba davrwmundeT, rom arsebobs SesaZlo odenobaTa erTaderTi

oTxeuli: 1; 2; 3; 4. radgan 1+4=2+3, amitom biWebma da gogonebma Tanabari odenobis Tevzebi

daiWires?

6 amocana variantebis SerCevis saintereso magaliTs warmogvidgens. SemoTavazebulia

SesaZlo Secvlis mxolod ori nimuSis moZieba. erTeulebi calsaxad ganisazRvra. mniSvnelovan

SezRudvas cifrebis gameorebis dauSvebloba qmnis. klasSi erToblivma ganxilvam, Ziebam

SeiZleba or aseT nimuSamde migviyvanos:

142857

× 2

... 14

142857

× 7

999999

142857

× 5

714285

6450+

3120

9570

3450+

6120

9570

67

12. or sidides Soris damokidebulebis gamosaxvis xerxebi

mizani. moswavlem SeZlos sityvieri aRweriT, cxriliT, an diagramiT mocemuli ori si di

dis konkretuli mniSvnelobebis analizis gziT aRmoaCinos am sidideTa Soris damoki de bu

lebis zogadi wesi, Semdeg ki am wesis mixedviT mocemuli sidideebisTvis miuTiTos sa Tanado

mniSvnelobaTa wyvilebi; mocemuli damokidebulebisTvis Tvisebrivad aRweros erTi sididis

cvlilebisas masze damokidebuli meore sididis cvlileba (magaliTad, erTis zrda iwvevs meoris

zrdas/klebas); gadadgas mniSvnelovani nabiji Sesabamisobis, funqciis arsis gaazrebisken.

monacemTa analiziT kanonzomierebis aRmoCenis da asoiTi gamosaxulebis saxiT am

kanonzomierebis warmodgenis nimuSebi wina paragrafis amocanebis amoxsnisasac Segvxvda _

ixileT §11is 33 , 27 da sxva. zogierTma maswavlebelma axal masalaze gadasvlisTvis

saWiro Semamzadebeli samuSao SeiZleba am amocanebis ganxilviT Caataros. gakveTilis

gegmis Sedgenisas gasaTvaliswinebelia sxva aspeqtebic: resursebi, aqtivobebi (magaliTad,

motivacia, moswavleTa Cabmuloba, muSaobis Sefaseba, davalebis SerCeva, diferencirebuli

muSaoba). am mxriv saxelmZRvanelo TiTqmis yvela saWiro masalasa da miTiTebas Seicavs,

igi erovnul saswavlo gegmasTan srul SesabamisobaSia (miaqcieT yuradReba rukaze ferebis

cvlilebis magaliTis ganxilvas). warmodgenili paragrafiT Cven miznad visaxavT sidideebs

Soris damokidebulebebze moswavleTa yuradRebis fokusirebas, realur cxovrebaSi aseTi

damokidebulebebis aRmoCenisa da formalizebis, maTematikurad warmodgenis unarCvevebis

Camoyalibeba/ganmtkicebas.

savarjiSoebis umravlesoba warmogvidgens sidideTa Soris praqtikul damokidebulebebs:

siCqare _ gavlili manZili, tvirTis odenoba _ masa, saqonlis odenoba _ Rirebuleba. didi

yuradReba eqceva formuliT mocemuli damokidebulebisTvis Sesabamisobis cxrilis Sevsebis

problemebs _ ganxilulia pirdapiri da Sebrunebuli amocanebi.

sigrZis sxvadasxva zomis erTeulebs Soris damokidebulebas exeba 9 amocana: AB=2

dm=20 sm, amitom erekles saxazavze B wertili SeuTavsdeba 23is Sesabamis wertils.

10 vixsenebT drois erTeulebs Soris damokidebulebebs: 1 wT=60 wm, 1 sT=60 wT=3600

wm. saSinao davalebebis 6 , 7 da 12 amocanebis ganxilvisas sasurvelia aRwerili

damokidebulebis analizuri (formuliT) warmodgena. 15 amocanaSi pirveli cxrilis

mixedviT aRmovaCenT savaraudo Sesabamisobebs: x → 500 000x. meore cxrilis mixedviT x →

600000x. I firmas gza gayavs ufro iafad. daizoga 6(600000–500000)=600000 (lari).

6 Tu 1 ivnisis Semdeg gavida x dRe, gayiduli dRiuri odenoba iqneba 240+24x;

Sesabamisad, 5 dRis Semdeg gaiyideba 240+24⋅5=360 cali nayini.

7 x Tvis Semdeg simaRle iqneba 40+3x sm; Sesabamisad, Tu x=4, simaRlea 40+3⋅4=52 sm.

12 xwuTiani saubris Rirebulebaa 10+24x TeTri. amrigad, vRebulobT:

10+24x=58, x=2.

14 am amocanis amoxsnisas moswavleTa mier SemoTavazebuli mosazrebebis ganxilvis

Semdeg SeTavazeT klass geometriuli amocanebi:

• tolferda samkuTxedis ferdi x smia, xolo fuZe ferdze 1 smiT moklea. gamosaxeT am

samkuTxedis perimetri xiT.

• marTkuTxedis erTi gverdis sigrZea x sm, meore gverdi pirvelze 1 smiT grZelia.

gamosaxeT am marTkuTxedis perimetri.

• marTkuTxedis erTi gverdis sigrZea x sm, es gverdi meore gverdze 12 smiT grZelia.

gamosaxeT am marTkuTxedis perimetri xiT.

imsjeleT TiToeul gamosaxulebaSi xis SesaZlo mniSvnelobebis Sesaxeb; ipoveT

gamosaxuli sidideebis mniSvneloba (miRebuli asoiTi gamosaxulebis mniSvneloba) xis

romelime mniSvnelobisTvis.

68

jgufuri muSaobis proeqti

es proeqti xels uwyobs moswavleebs monacemTa daxarisxeba/warmodgenis unarCvevebis

gamomuSavebasa da ganmtkicebaSi. es proeqti statistikis propedevtikadac CaiTvleba.

sacalo da sabiTumo fasebs Soris damokidebulebis sityvieri aRweris mixedviT

moswavleebi miiReben Sesabamisobas x → x–1.

• pirveli davalebis Sesruleba SeiZleba gadanawildes jgufis wevrebs Soris. magaliTad,

erTi da imave klasebis saxelmZRvaneloebis mixedviT. davalebis Sesrulebis Semdeg

moswavleebs eqnebaT daxarisxebuli da mowesrigebuli monacemebi.

• am davalebis Sesasruleblad moswavleebs dasWirdebaT wina etapze Sedgenili erTerTi

cxrili _ IX klasis Sesabamisi saxelmZRvaneloebis sacalo fasebi.

• mesame davalebis Sesasruleblad moswavleebs daWirdebaT V klasis saxelmZRvaneloebis

sacalo da sabiTumo fasebi; aucilebelia klasis moswavleTa odenobis codnac.

• fasdaklebis axali wesis Sesabamisad Seicvleba monacemTa cxrilebic. maT SeiZleba

aseTi saxec mieceT, Tumca misaRebia sxva tipis cxrilis Sedgenac.

III klasis saxelmZRvaneloebis fasebi

sagani sacalosabiTumo

x–1 x–2

germanuli 5 4 _

inglisuri 8 _ 6

1 111=37⋅3; pirobis Sesabamisad, klasSi 37 moswavlea.

2 I xerxi. SegviZlia sinjvis meTodi gamoviyenoT. erTaderTi SesaZlo cifri (ricxvi),

romelic 7ze gamravlebisas erTeulebSi isev am cifrs mogvcems, aris 5. Sesabamisad, orniSna

ricxvia 35.

II xerxi (pedagogebisTvis). vTqvaT, es orniSna ricxvia 10a+b. maSin, pirobis Tanaxmad,

10a+b=7b, 10a=6b, 5a=3b, b=5, a=3, saZiebeli ricxvia 35.

3 gaviTvaliswinoT, rom ukanaskneli gverdis nomeri luwi ricxviT gamoisaxeba, anu 312

iT, amovardnili gverdebis odenobaa 312–212=100, rac 50 furcels Seadgens.

4 TanamimdevrobiT xdeba kidura wevrebis gadaadgileba. miviRebT ⋅

69

Ddamoukidebeli wera #4

Tema: • ricxviTi da asoiTi gamosaxulebebis Sedgena da gamartiveba;

• or sidides Soris damokidebulebis gamosaxvis xerxebi.

1. ipoveT (52–12):4+90 gamosaxulebis mniSvneloba.

a) 10 b) 40 g) 100 d) 94.

2. ipoveT (32+a)–51 gamosaxulebis mniSvneloba, roca a=28.

a) 28 b) 60 g) 51 d) 9.

3. yoveli x ricxvisTvis 52+12x–9x+3x=

a) 6x+52 b) 52x+3 g) 52+3x d) 52+15x.

4. CawereT gamosaxulebis saxiT: gaorkecebuli nisa da 17is jamis 8ze namravli.

a) 2n+17⋅8 b) (2n+17)⋅8 g) 2(n+17⋅8) d) 2(n+17)⋅8.

5. SearCieT sworad Sevsebuli cxrili.

a) b) g) d)

6. daCis xelfasi ori Tvis ganmavlobaSi TveSi 540 lari iyo, Semdegi 3 Tvis ganmav lobaSi

ki _ TveSi x lari. gamosaxeT daCis mier am 5 TveSi miRebuli Tanxa cvladiani gamosaxulebiT.

a) 540+x b) 2⋅540+x g) 2⋅540+5x d) 2⋅540+3x.

7. samkuTxedis erTi gverdis sigrZea x sm, meore gverdi pirvelis gaorkecebulze 3 sm

iT grZelia, xolo mesame 1 smiT moklea meoreze.

a) warmoadgineT am samkuTxedis perimetri cvladiani gamosaxulebis saxiT;

b) ipoveT am samkuTxedis perimetri, roca x=5.

8. telefonis gadasaxadi Sedgeba saabonento da wuTobrivi gadasaxadebisgan. vTqvaT,

saabonentoa _ TveSi 7 lari, wuTobrivi _ 1 wuTian saubarSi 2 TeTri.

ra Tanxa unda gadaixados ojaxma zafxulis ganmavlobaSi telefoniT sargeblobaSi, Tu

saubrebis saerTo xangrZlivoba 472 wuTia?


1 2 3 4 5 6

g d a b g d

7. a) x+(2x+3)+(2x+2), anu 5x+5;

b) 5x+5=25+5=30 (sm).

8. 3⋅700+472⋅2=2100+944=3044 (TeTri).

pasuxi: 3044 TeTri, anu 30 lari da 44 TeTri.

x x⋅5+2

10 52

7 73

6 23

x x⋅5+2

10 52

7 73

6 32

x x⋅5+2

10 52

7 37

6 32

x x⋅5+2

10 52

7 37

6 23

70

Sefasebis sqema

testuri tipis amocanebi _ TiToeuli swori pasuxi Sefasdes 1 quliT.

me7 da me8 amocanebidan TiToeuli SeiZleba Sefasdes maqsimum 2 quliT. es amocanebi

SeiZleba Sefasdes TiTo quliTac (maTSi xarvezebis arsebobisas). magaliTad, me7 amocanis

mxolod pirveli davalebis Sesrulebisas daiwereba 1 qula.

Sefasebis gamocxadebamde SeecadeT sajarod ganixiloT yvela amocana da daavaloT

moswavleebs Tavad daasaxelon maTi savaraudo Sefasebebi. moswavleTa aseTi aqtivobis

mravalmxriv mniSvnelobaze Cven araerTxel visaubreT analogiuri wesebis mimoxilvisas.

yuradRebiT ganixileT moswavleTa kritikuli mosazrebebic, maTi ganxilvisas moswavlem

unda miiRos obieqturi argumentebi, mowodebuli uaRresad koreqtuli formiT.

13. amocanebis amoxsna ricxviTi an asoiTi gamosaxulebis SedgeniT

mizani. moswavlem arsebuli informaciisa da situaciis adeqvaturad Seadginos ricxviTi

da cvladiani gamosaxulebebi; teqsturi amocanebis amoxsnisas mosaxerxeblad SemoiRos

cvladi, Seadginos da amoxsnas gantoleba, moaxerxos gantolebis amonaxsnis interpretacia.

am aqtivobiT mniSvnelovnad gaiRrmavos codna da unarebi maTematikuri modelirebis

sawyisebSi.

am paragrafis damuSavebisas sasurvelia gaiTvaliswinoT ramdenime mniSvnelovani momenti:

• vidre moswavle daiwyebdes amocanis amoxsnas, darwmundiT, rom man sworad gaigo piroba

_ amis gakeTeba miznobrivi kiTxvebiT SeiZleba; • ricxviTi an cvladiani gamosaxulebis

Sedgenis procesSi moiTxoveT sityvieri komentarebi _ moswavlem aucileblad unda axsnas

ariTmetikuli gamosaxulebebis praqtikuli Sinaarsi; magaliTad, Tu erTi puris fasi 70

TeTria da nayidia 8 puri, maSin gamosaxuleba 8⋅70 warmogvidgens 8 puris saerTo Rirebulebas;

• mravalgvari amocanebis amoxsnisa da ganxilvis kvalobaze moswavleebi darwmundebian, rom

erTi da imave amocanis amoxsna sxvadasxva gamosaxulebis saSualebiT SeiZleba; am faqtis

demonstrirebisas SeiZleba gamoiyenoT Sekrebisa da gamravlebis gadanacvlebadobis,

jufTebadobisa da Sekrebis mimarT gamravlebis ganrigebadobis kanonebi; • ricxviTi

gamosaxulebis mniSvnelobis an gantolebis fesvis povnis Semdeg aucilebelia miRebuli

pasuxis kritikuli analizi. magaliTad, Tu veZebT adamianTa odenobas, maSin dauSvebelia,

rom pasuxi ar iyos mTeli.

savarjiSoebi isea SerCeuli, rom gagiadgvildebaT zemoT CamoTvlil momentebze

moswavleTa yuradRebis gamaxvileba da Sesabamisi Cvevebis gamomuSavebis xelSewyoba.

zogierTi amocanis amoxsnisas sasurvelia Sesaferisi sqemis gamoyeneba; magaliTad:

12 240–(120+40)=80 (km)

13 1150–137=1013 (km).

71

14 1135+(1135–645)=1625 (km).

gantolebis gamoyenebiT amocanebis amoxsnisas mniSvnelovania xelsayreli SerCeva im

sididisa, romlis mimarTac SeadgenT gantolebas.

17 xiT aRvniSnoT alublis wvenis odenoba (kovzebiT). wylis odenoba 4jer metia, anu

4xia, Taflia x. pirobiT, sruli odenoba 18ia.; amitom x+4x+x=18, 6x=18, x=3 _ saWiroa 3

kovzi wveni.

22 da 23 amocanebis amoxsnisas SeiZleba moswavleebi wyvilebad, an ufro mravalricxovan

jgufebad dayoT da TiToeul jgufSi gaanawiloT (an TviTon gainawilon) amocanebis amoxsnis

sxvadasxva xerxi. Semdeg imsjeleT ama Tu im xerxis upiratesobaze.

22 I xerxi: 1 m 50 sm + 50 sm + 25 sm=2m 25 sm.

II xerxi: (x–50)–25=150, x–50=175, x=225 (sm)

an x–(50+25)=150, x–75=150, x=225 (sm).

23 I xerxi: gamoviyenoT `ukusvlis~ meTodi

35–16+9–5+8=31.

II xerxi: x–8+5–9+16=35, x=31.

amocanebi jgufuri muSaobisTvis

am amocanebze samuSaod moswavleebi SeiZleba dayoT wyvilebad _ es gazrdis maTi

saqmianobis efeqtianobas, Tqven ki kidev erTxel mogecemaT SesaZlebloba daakvirdeT

TiToeuli moswavlis msjelobas, sakuTari poziciis dacvasa, Tu mis mimarT kritikulad

ganwyobil oponentTan dialogis warmarTvis unarCvevebis ganviTarebis process, moaxdinoT

reagireba SeniSnul xarvezebze, daxvewoT jgufuri muSaobis formebi.

1 10, 20, 30, 40, ...

• 10n, n Rebulobs mniSvnelobebs: 1; 2; 3; 4; ...

pirveli ricxvia 10, yoveli Semdegi wina ricxvze 10iT metia

10 ---→ 20 ---→ 30 ---→ 40 ---→

• 50, 60, 70.

2 a) 100, 200, 300, 400, ...



100 ------→ 200 ------→ 300 ------→ 400 ------→ • 500, 600, 700.

b) 1000, 2000, 3000, 4000, ...



1000 ------→ 2000 ------→ 3000 ------→ 4000 ------→ • 5000, 6000, 7000.

+10

+100

+1000 +1000 +1000 +1000

+100 +100 +100

+10 +10

72

3 pedagogebi iolad aRmoaCenT, rom samive SemTxvevaSi mocemulia ariTmetikuli

progresia. TqvenTvis (mxolod TqvenTvis da ara moswavleebisTvis) warmovadgenT:

• a1=10, a

n+1=a

n+10, n∈N, amrigad, a

n=a

1+10(n-1), n∈N, anu a

n=10n, n∈N;

• a1=100, a

n+1=a

n+100, n∈N, a

n=100n, n∈N.

• a1=1000, a

n+1=a

n+1000, n∈N, a

n=1000n, n∈N.

moswavleebs es davaleba (saWiroebis SemTxvevaSi) Tqven SeiZleba ase daukonkretoT:

mimdevrobis wevri, romlis rigiTi nomeria x, rogor CavweroT xisa da pirveli ricxvis

saSualebiT? mosalodneli pasuxebia:

• 10x, • 100x, • 1000x. klasma unda Seamowmos es savaraudo pasuxebi.

me5 klaselTa maTematikuri momzadebis done savsebiT sakmarisia am saxaliso fokusis

asaxsnelad.

vTqvaT, Cafiqrebuli ricxvia m. maSin davwerT

m, m, m

Semdegi moqmedebiT vRebulobT:

m–6, m+12, m–6

m–6, m+12–(m–6), m–6

anu m–6, 18, m–6

amrigad, nebismieri (6ze meti) Cafiqrebuli ricxvis SemTxvevaSi Sua ricxvi aRmoCndeba

18is toli (anu Cafiqrebul ricxvze 3jer meti). Tu TqvenTvis cnobili gaxdeba (18+m)is

mniSvneloba, maSin Tqven msac martivad miagnebT.

Tu 6s SecvliT 9iT, maSin Sua ricxvi 27is toli gaxdeba da, cxadia, am SemTxvevaSic

martivad miagnebT Cafiqrebul ricxvs.

14. xarisxi. ricxvebis Caweris sxvadasxva wesi

mizani. ricxvis naturaluri xarisxis gacnoba da misi damkvidreba moswavleTa CanawerebSi.

ricxvTa aTobiTi poziciuri sistemis ganxilvis gafarToeba, romauli da egvipturi sistemis

ganxilva. yvela am sistemis Sedareba da aTobiTi sistemis upiratesobaTa dadgena.

naturaluri ricxvis gaSlili formiT Cawerisas moswavlem aiTvisos 10is xarisxis

gamoyeneba. jgufuri muSaobiT gaecnos naSTTa ariTmetikis elementebs.

gakveTils viwyebT ricxvis Tavis Tavze ramdenjerme gamravlebisas xarisxis gamoyenebis

magaliTebiT. ricxvis meore da mesame xarisxebs `ricxvTa kvadratsa~ da `ricxvis kubs~

ecnobian kvadratis farTobisa da kubis moculobis faqtobrivi dasaxelebis gareSe.

ixseneben ricxvis Caweras gaSlili formiT da am CanawerebSi iyeneben 10is xarisxebs.

paragrafis teqstis me2 punqtSi yuradRebas vamaxvilebT ricxvis Caweris poziciur

aTobiT sistemaze. moswavleebs SeiZleba SevTavazoT iseTi ricxvebis gaSlili formiT

_ 10is xarisxebis daxmarebiT Cawera, romlebSic cifri meordeba; magaliTad,

55555=5⋅104+5⋅103+5⋅102+5⋅10+5. maTze dakvirvebiT moswavleebi Seecdebian kidev erTxel axsnan

ricxvis Caweris am wesis saxelwodeba: poziciuri _ radgan yoveli cifris mniSvneloba

CanawerebSi am cifris adgilze (poziciaze) aris damokidebuli; aTobiTi _ viyenebT 10

cifrs, ricxvis mniSvnelobas gamovsaxavT 10is xarisxebiT; yoveli Semdegi (ufro maRali)

Tanrigis erTeuli 10-jer metia wina Tanrigis erTeulze.

ricxvebis Caweris am wesis gacnobis paralelurad moswavle interesiT ganixilavs

ricxvebis Caweris sxva wesebsac. saxelmZRvaneloSi SemoTavazebulia ricxvebis Cawera

73

romauli cifrebiT _ romauli wesi, moswavleebi iwafebian romaulad Cawerili ricxvebis

wakiTxvaSi da aTobiT sistemaSi Cawerili ricxvebis romaulad gadaweraSi.

magaliTad, XXXV=35, CD=500-100=400, LXV=50+10+5=65, 86=LXXXVI,

=10000+10000=20000 da sxva.

ricxvebis Caweris am or wess Soris gansxvaveba da romelimes upiratesobaTa moZieba

msjelobis sagani unda gaxdes. magaliTad, aTobiTi sistemis upiratesobaa, rom ricxvi 86

is CanawerSi aTobiT sistemaSi ori simbolo gamoiyeneba, romaulSi _ 6, egvipturSi _ 14;

aTobiTi Canaweri gviCvenebs ricxvis erTeulebis odenobas, romaulSi ki ricxviT gamosaxuli

erTeulebis odenobis dasadgenad gamoTvlebia saWiro. magaliTad, LXXXVI=50+30+6=86;

poziciur sistemaSi mniSvnelovania cifri 0is gamoyeneba; aTobiT sistemaSi nebismieri

ricxvi Caiwereba mxolod aTi cifris gamoyenebiT. romauli sistemis sirTuleebi matulobs

didi ricxvebis Cawerisas.

sasurvelia moswavleebma am da sxva gansxvavebebze Tavad imsjelon da maswavlebelma

mxolod mniSvnelovan momentebze gaamaxvilos maTi yuradReba.

analogiurad SeiZleba SeadaroT ricxvis Caweris aTobiTi poziciuri sistema egvipturs,

romelic paragrafis bolosaa warmodgenili.

rubrikaSi `sxvadasxva~ warmodgenilia cifrebis nacvlad asoebis gamoyenebis zogierTi

magaliTi.

am paragrafis Seswavlis Semdeg SeiZleba moswavles gauCndes saqarTveloSi an sxva

romelime qveyanaSi Zvelad ricxvebis Caweris wesebis ufro dawvrilebiT Seswavlis survili,

SeiZleba gamoiyenoT internetis SesaZleblobebi. magaliTad, http://ka.wikipedia.org/wiki/ qarTuli

_ damwerloba.

sxvadasxva sistemaSi ricxvebis Cawerasa da wakiTxvaSi gawafvas emsaxureba savarjiSoebi 11 22 (klasSi). me16 savarjiSoSi moswavle jer egvipturi sistemiT daweril ricxvs gadawers

aTobiTU sistemaSi gaSlili formiT, Semdeg 10is xarisxis gamoyenebiT: 4⋅100+2⋅10+4=4⋅102+2⋅10+4.

(swori pasuxia d)

25 amocanis amoxsnisas moswavle TiToeul Tanamamravls Tu Cawers namravlis saxiT,

miiRebs: 5⋅5⋅5⋅5⋅5⋅5⋅5=57. aq ar igulisxmeba xarisxis Tvisebis (an⋅am=an+m) gamoyeneba, romelsac

moswavleebi maRal klasebSi gaecnobian.

analogiurad amoixsneba 33 amocana; masze muSaobisas SeiZleba zogierTma moswavlem

Tanamamravlebis Camoweris gareSe SeZlos maTi odenobis dadgena. magaliTad, 1015⋅108 _

msjelobs: 10 aviRoT Tanamamravlad 15jer, Semdeg 8jer, miiReba: 10 Tanamamravlad

aRebulia 23jer _ 1023. SeiZleba romelime moswavlem ganazogados miRebuli Sedegi da

is sityvieradac Camoayalibos. am aqtivobis ganxorcielebis SemTxvevaSi moswavle xdeba

erTnairfuZiani xarisxebis Tvisebis `aRmomCeni~ _ rac SeiZleba Sefasdes, rogorc am asakis

moswavlis azrovnebis umaRlesi donis dadgenis erTerTi kriteriumis dakmayofileba.

jgufuri muSaoba `bolo cifris cxrili~

mizani. moswavle gaiwafos ricxvis xarisxis gamoTvlaSi, kanonzomierebis aRmoCenaSi,

samuSaos Sesrulebis strategiis dadgenasa da swrafad ganxorcielebaSi; Sesrulebuli

samuSaos SefasebaSi, anu SeZlos aRmoCenili kanonzomierebis ganzogadeba, gamoyeneba

amocanebis amoxsnisas.

moswavleebs cxrilis Sevsebamde vTavazobT daasaxelon ramdenime magaliTi: 34is aTobiTi

Canaweris bolo cifria 1 (34=81), 83is bolo cifria 2 (83=512). rodesac darwmundebiT, rom

klasi gaerkva davalebis SinaarsSi, jgufebad gaerTianebul moswavleebs vTavazobT cxrilis

74

Sevsebas. muSaobas SeiZleba mivceT Sejibris saxe. jgufis muSaobis Sefasebisas gaiTvaliswineT

samuSaos Sesrulebis dro da xarisxi. magaliTad, yoveli Secdoma TiTo sajarimo quliT

SeafaseT. daakvirdiT jgufebis muSaobis process. savaraudoa, rom garkveuli drois

Semdeg — ramdenime striqonis Sevsebis Semdeg, moswavleebi aRar Seasruleben gamravlebas

srulad da yoveli momdevno striqonis Sesavsebad gamoiyeneben wina striqonis monacemebs _

gamoimuSaveben cxrilis Sedgenis strategias.

xarisxis maCve nebeli

bolo cifri

1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1

2 2 4 8 6 2 4 8 6 2

3 3 9 7 1 3 9 7 1 3

4 4 6 4 6 4 6 4 6 4

5 5 5 5 5 5 5 5 5 5

6 6 6 6 6 6 6 6 6 6

7 7 9 3 1 7 9 3 1 7

8 8 4 2 6 8 4 2 6 8

9 9 1 9 1 9 1 9 1 9

Sevsebuli cxrilis sisworis Semowmebisa da Sedegebis sajaro gamocxadebis Semdeg

vTxovT jgufebs, an calkeul moswavleebs daasaxelon SemCneuli kanonzomierebebi yoveli

ricxvis xarisxis bolo cifrebis monacvleobis Sesaxeb. isini metad mniSvnelovan Sedegebs

miiReben: 0, 1, 5 da 6is xarisxebis bolo cifrebi ucvlelia, danarCeni ricxvebis xarisxebis

aTobiTi Canaweris bolo cifrebi ki periodulad meordeba. magaliTad, 4 da 9is _ yoveli

oris Semdeg, 2, 3, 7 da 8is _ yoveli oTxis Semdeg. aRsaniSnavia, rom 4 da 9 aris 2isa da

3is kvadratebi, sayuradReboa 2is da 4is, 3isa da 9is striqonebis Sedareba.

am kanonzomierebis ganxilvis Semdeg jgufebs vTavazobT aqtivobis gafarToebas _ ipoveT

magaliTad, 411, 910, 325, 731, 9131is bolo cifrebi.

aqtivobis gafarToeba SeiZleba ganvaxorcieloT mocemul striqonSi warmodgenili

cifrebiT daboloebuli ricxvebis xarisxebze dakvirvebiTac. magaliTad, 23is, 73is, 253

is xarisxebi bolovdeba imave cifrebiT, rac 3is xarisxebi. jgufebs vTavazobT, daadginon,

magaliTad, 235is, 7311is, 12586is bolo cifrebi.

moisazre

1 a) 4=22, b) 1=12, g) 27=33, d) 256=44.

2 a) 62=36, 26=64, 26>62.

b) ar aris aucilebeli orive xarisxis gamoTvla:

49=16⋅16⋅16⋅16⋅9. es namravli ki metia 94ze.

miviReT: 49>94.

am amocanis amoxsnisas moswavleebma SeiZleba kalkulatori gamoiyenon. zogierT

kalkulatorze xarisxis gamoTvlis wesi rubrika `sxvadasxvaSia~ aRwerili. amjerad

sasurvelia Sedareba kalkulatoris gamoyenebis gareSe Sesruldes.

75

3 orniSna ricxvebSi cifri 0 gvxvdeba mxolod erTeulebis TanrigSi 9jer (10, 20, ...,

90) da ricxv 100Si 2jer. sul 11jer.

cifri 1 erTeulebis TanrigSi gvxvdeba 10jer, aTeulebis TanrigSi 10jer, aseulebis

TanrigSi erTxel, sul 21jer.

4 analogiuri moqmedebebi SeiZleba jer aTniSna ricxvebze SevasrulebinoT.

magaliTad,

32:3=3, 52:5=5, ...

pasuxi: 4 820 063 6172:4 820 063 617=4 820 063 617.

5 da 6 amocanebSi viyenebT jgufuri muSaobisas Sevsebul cxrils: 1iT bolovdeba

mxolod 1iT daboloebuli ricxvis kubi; 4iT bolovdeba 2iT an 8iT daboloebuli

ricxvebis kvadratebi, 2iT daboloebuli ricxvebis kvadratebi. 2iT daboloebuli ricxvis

kubi bolovdeba 8iT. 8iT daboloebuli ricxvis kubi bolovdeba 2iT, 6 is pasuxia: d)

2iT an 8iT.

sakontrolo wera ## 3

• amocanebis amoxsna ricxviTi an asoiTi gamosaxulebebis SedgeniT.

• xarisxi, ricxvebis Caweris sxvadasxva wesi.

1. Tu sami Sesakrebis jamia 682 000, amasTanave, pirveli da meore Sesakrebebis jamia

540 900, maSin mesame Sesakrebis povna SegviZlia Semdegi gamosaxulebiT:

a) 682000+540900 b) 682000–2⋅540900

g) 682000–540900 d) 2⋅(682000–540900).

2. SearCieT (2x+4):3=12 gantolebis fesvi.

a) 16 b) 36 g) 32 d) 20.

3. sawyobidan pirveli 5 dRis ganmavlobaSi dReSi 8 tona fqvili gahqondaT, Semdegi

5 dRis ganmavlobaSi ki _ dReSi 7 tona fqvili. ramdeni tona fqvili gautaniaT am sawyobidan

aTive dRis ganmavlobaSi? SeadgineT ricxviTi gamosaxuleba.

a) 10(8+7) b) 5⋅8+5⋅7 g) 10(5+8+7) d) 10⋅5+8⋅7.

4. vTqvaT, mocemuli kubis gverdis sigrZea 5 erTeuli. ramdeni mcire kubisgan (gverdis

sigrZe 1 erTeulia) Sedgeba mocemuli kubi?

a) 25 b) 10 g) 125 d) 20.

5. saklebi 52iT aRemateba sxvaobas da 15iT metia maklebze. saklebis sapovnelad

SeiZleba SevadginoT ricxviTi gamosaxuleba:

a) 52–15 b) 52⋅15 g) 52+15 d) 52–2⋅15.

6. 62⋅63=

a) (2⋅3)6 b) 66 g) 2⋅2⋅2⋅2⋅2⋅2⋅6⋅6⋅6 d) 65.

7. marTkuTxedis AB gverdi 18 smiT moklea BC gverdze. ipoveT es

gverdebi, Tu markuT xedis perimetria 96 sm.A D

B C

76

8. ipoveT gamosaxulebis mniSvneloba:

a) 92:3+(10–6)3⋅5=

b) (82–72)⋅10:(52+25)=

SegaxsenebT, rom sasurvelia gqondeT sakontrolo weris ramdenime varianti, maT SedgenaSi

Cvens mier SemoTavazebuli amocanebi dagexmarebaT.


1 2 3 4 5 6

g a b g g d

7. 2(x+18+x)=96

x=15 (sm),

x+18=33 (sm)

8. a) 92:3+(10–6)3⋅5=81:3+43⋅5=27+320=347

b) (82–72)⋅10:(52+25)=(64–49)⋅10:(25+25)=15⋅10:50=150:50=3.

Sefasebis sqema

me7 amocanis Sefasebisas, Tu sworadaa Sedgenili gantoleba, vwerT 1 qulas, gantolebis

sworad amoxsnis SemTxvevaSi kidev erT qulas.

me8 amocanis TiToeuli davalebis umaRles Sefasebad SeiZleba miiCnioT 1 qula.

SeiZleba Sefasebis ufro moqnili sistemac airCioT. Tu moswavle avlens xarisxis codnas,

magram daudevrobis gamo ver iRebs swor pasuxebs, SeiZleba es amoxsna SeafasoT 1 quliT.

Tumca aqve saTanado komentari daurToT Tqvens gadawyvetilebas da mouwodoT moswavles

izrunos xarvezebis gamosasworeblad. moismineT Tavad moswavlis komentarebic _ daSvebuli

Secdomebis mizezebis axsna.

15. amocanebis amoxsnis sxvadasxva xerxi

Temis Seswavlas viwyebT amocanebis amoxsnis ramdenime xerxis gaxsenebiT. magaliTad,

ricxviTi gamosaxulebis Sedgenis xerxiT rogor amovxsnaT amocana: nikas xelfasi 360 laria,

rac 25 lariT metia sandros xelfasze. ramden lars Seadgens orives xelfasi?

Semdeg SevTavazoT paragrafis pirvel punqtSi ganxiluli amocanis amoxsnis ori xerxi:

suraTebis SesaZlo wyvilebis Camoweris (1,2; 1,3; 1,4; 1,5; 1,6; 1,7; 2,3; 2,4; 2,5; 2,6; 3,4; 3,5; 3,6;

4,5; 4,6 da 5,6) da gamravlebis (5⋅6:2). me2 punqtSi SemoTavazebuli amocanis amoxsnisas xisebri

diagramaa gamoyenebuli. saxelmZRvaneloSi warmodgenil suraTze Cans, rom ori totidan

TiToeuli oTxi totiT bolovdeba, e. i. sul aris SerCevis 2⋅4=8 varianti. me3 punqtSi

SemoTavazebulia amocanis amoxsnis sinjvis xerxi, SeiZleba cxriliTac visargebloT.

swori pasuxebis odenoba

araswori pasuxebis odenoba

qulebis odenoba

15 0 150 _ metia

12 3 120–9=111 _ metia

10 5 100–15=85 _ metia

9 6 90–18=72 _ tolia

x

x+18

77

SeiZleba moswavle mixvdes, rom sworad amoxsnilTa svetidan arasworad amoxsnilTa

svetSi yoveli amocanis gadmotanisas mopovebul qulebs akldeba 13 qula. 72 qulis misaRebad

150 qulas unda gamoakldes 78=6⋅13 qula, e. i. sworad amoxsna 15–6=9 amocana.

paragrafSi SemoTavazebuli amocanebis amoxsnisas, saklaso da saSinao savarjiSoebis

Sesrulebis Semdeg moswavle gaiwafeba ara marto TiToeuli xerxiT amocanebis amoxsnaSi,

aramed amoxsnis xerxis/strategiis dadgenasa da mis ganxorcielebaSi.

klasSi

1 I gza. TuU monawileebs gadavnomravT, miviRebT wyvilebs (1,2; 1,3; 1,4; 1,5; 1,6; 1,7;

2,3; 2,4; 2,5; 2,6; 3,4; 3,5; 3,6; 4,5; 4,6; 5,6) _ 15 partia.

II gza. TiToeuli monawile TamaSobs 5 partias _ 5⋅6.

magram aseTi daTvlisas yoveli partia orjer viangariSeT, amitom partiaTa odenobaa

5⋅6:2=15.

2 Tu am amocanis amoxsnisas isev gadavnomravT monawileebs, rogorc wina amocanaSi,

maSin yoveli wyvili orjer unda davasaxeloT (radgan gundebi ororjer xvdebian erTmaneTs)

_ 7⋅6. pasuxi: 42.

3 gundebis (10⋅9):2=45 wyvilidan yoveli wyvili 4 Sexvedras atarebs. pasuxia 45⋅4=180.

4 saxelmZRvaneloSi SemoTavazebulia ori gza: I _ sameulebis daTvla: mds, msd, dms,

dsm, smd da sdm _ aseTi sameuli 6ia. momavalSi moswavleebi iswavlian, rom es aris 3

elementis gadanacvlebaTa odenoba, romelic gamoiTvleba P3=3! formuliT (P

n=1⋅2⋅⋅⋅⋅⋅⋅n=n! _

n elementis gadanacvlebaTa odenobaa).

II _ xisebri diagrama, romlis saSualebiT moswavle variantebis daTvlis gamravlebis

wesSi gaiwafeba _ 3 totidan TiToeuls gamoeba 2, miRebuli 3⋅2=6 totidan TiToeuls

gamoeba 1. pasuxi: 3⋅2⋅1=6.

5 da 6 amocanebi me 4 s analogiuria, sadac s, m da d asoebi Secvlilia 3, 4 da 5

cifrebiT an ferebiT. pasuxi: 3⋅2⋅1=6.

7 3⋅4=12.

8 sami sportsmenidan TiToeuli farikaobs 3jer, sul _ 3⋅3=9.

SeiZleba sqemac gamoiyenon. magaliTad, mofarikaveebi wertilebiT gamo

vsaxoT da daviTvaloT wertilTa damakavSirebeli wirebi. aseTi wiria 9.

9 I dRes morigis SerCevis 3 variantia, yoveli SerCeuli morigisTvis II dRes 2

variantia, III dRes yoveli wyvilisTvis III dRes aris darCenili 1 varianti, sul 3⋅2⋅1=6.

10 sinjvis meTodiT

mwvane cisferi wiTeli

1 6 20–7 _ ar akmayofilebs pirobas

2 12 20–14 _ akmayofilebs pirobas

3 18 SeuZlebelia

pasuxi: 6.

11 aTaseulebis TanrigSi mxolod 1 SeiZleba eweros; aseulebis, aTeulebis da erTeulebis

TanrigebSi cifrebis SerCevis orori variantia.

pasuxi: 1⋅2⋅2⋅2=8.

78

12 erT jibeSi SeiZleba CavidoT sami monetidan erTi romelime (3 varianti), sami

monetidan ori romelime (3 varianti), samive moneta (1 varianti), an arc erTi (1 varianti).

sul 3+3+1+1=8 varianti.

pedagogebs SevaxsenebT amoxsnis kidev erT xerxs: erTerT (SerCeul) jibeSi 5TeTrians

an CavidebT, an ara (gvaqvs 2 SemTxveva). TiToeuli SemTxvevaSi 10TeTrians an CavidebT amave

jibeSi an ara (sul — 4 SemTxveva). am oTxi SemTxvevisas mesame monetasac an CavidebT an ara

(sul 8 SemTxveva).

13 SeiZleba amoixsnas ricxviTi gamosaxulebis SedgeniT. es ricxvebia 90:3⋅1=30 da

90:3⋅2=60.

SeiZleba saZiebeli ricxvebi x–iTa da 2xiT aRvniSnoT, miviRebT x+2x=90, saidanac x=30.

pasuxi: 30 da 60.

15 ricxviTi gamosaxulebis SedgeniT: erT saaTSi zeinklebi erTad amzadeben 19+17=36

detals, samuSaos daasruleben 144:36=4 sTSi.

gantolebis SedgeniT: vTqvaT, samuSaos dasrulebas x sT sWirdeba, maSin 17x+19x=144,

saidanac x=4 (sT).

16 SeiZleba suraTi gamoviyenoT:

35:7=5 (sm), BC=10 sm.

gantolebis SedgeniT: ABs sigrZe aRvniSnoT xiT, maSin BC=2x, CD=4x. miviRebT gan

tolebas

x+2x+4x=35, saidanac x=5.

pasuxi: BC=10 sm.

saSinao

4 aTeulebis Tanrigis 5 variantidan (1, 3, 5, 7 an 9) yovelisTvis erTeulebis TanrigSi

5 variantia. pasuxi: 5⋅5=25.

5 aTeulebis Tanrigis 4 variantidan yovelisTvis erTeulebis TanrigSi 3 varianti

arsebobs, pasuxi: 4⋅3=12.

6 aseulebis Tanrigis 3 variantidan yovelisTvis aTeulebis TanrigSi 3 variantia

(3⋅3=9), miRebuli 9 variantidan yovelisTvis erTeulebis TanrigSic 3 variantia. pasuxi:

33=27.

7 Camoayalibon 5 is analogiuri amocana: ramdenia im samniSna ricxvebis odenoba,

romlebic Caiwereba 1, 3, 5, 7 cifrebiT da cifrebi ar meordeba. pasuxi: 4⋅3⋅2=24.

8 bebia SviliSvilze 44 wliT ufrosia.

I gza. SviliSvilis wlovaneba im momentisTvis, roca bebiis asakze 5jer naklebia xiT

aRvniSnoT, bebiis _ 5xiT. pirobiT, 5x–x=44, saidanac x=11. e. i. dasaxelebuli momentisTvis

SviliSvili 11 wlis iyo _ 4 wlis win. SeiZleba sqemac gamoviyenoT:

SviliSvili

bebia

suraTidan Cans, rom am dros SviliSvili 11 wlisaa. es sqema gantolebis Sedgenis tolfasia.

79

SevniSnoT, rom gantolebis Sedgenis gziT am amocanis amoxsnisas ufro bunebrivi iqneboda

xiT aRgveniSna saZiebeli ricxvi. maSin miviRebdiT gantolebas: 5(15–x)=59–x, romlis amoxsna

problemebs Seuqmnida me5 klaselebs. am mosazrebaTa gaTvaliswinebiT avirCieT zemoT

warmodgenili gza. cxadia, moswavleTaTvis ufro martivi iqneba _

II gza. sinjvis meTodi. SevadginoT wlovanebis cxrili,

bebia SviliSvili

59 15 ar akmayofilebs

58 14 `~

57 13 `~

56 12 `~

55 11 akmayofilebs

SeiZleba moswavlem pirdapir 55is mosinjviT daiwyos, radgan bebiis asaki 5ze unda

iyofodes.

9 I gza. sinjvis xerxi.

indauri cxvari fexebi

10 10 60

9 11 62

8 12 64

pasuxi: 8 indauri, 12 cxvari.

II gza. oci Tavidan TiToeuls `CamovurigoT~ orori fexi. darCa 24 fexi. `davumatoT~

Tavebs orori fexi _ 12 Tavi `gaxdeba oTxfexa~. miviRebT _ 12 cxvaria da 8 indauri.

III gza. gantolebis SedgeniT: cxvari _ x, indauri _ (20–x).

4x+2(20–x)=64

4x+40–2x=64,

2x+40=64

2x=64–40

2x=24

x=12.

pasuxi: cxvari 12, indauri 8.

10

I varianti II varianti

I gakv. maTematika istoria

II gakv. qarTuli maTematika

III gakv. istoria qarTuli

cxrilis Sevseba SeiZleba maTematikiT daviwyoT. Tu mas I gakveTils davuTmobT, maSin

danarCeni sagnebisTvis erTaderTi arCevani rCeba. analogiuri Sedegi miiReba maSinac, roca

maTematikas II gakveTils davuTmobT.

cotaa, e. i. cxvrebis

raodenoba unda

gavzardoT.

80

11 (8⋅7):2=28. 12 variantebis odenobaa 3⋅2⋅1=6. 13 3⋅2⋅1.

17 es amocana saSinao davalebisTvis sagangebodaa gankuTvnili, raTa moswavlem

auCqareblad sinjvis meTodiT, sxvadasxva variantis ganxilvisas miRebuli gamocdilebiT

SeZlos sasurveli sameulebis povna. is aRmoaCens, rom aseTi sameuli erTaderTia _ 1; 2; 3.

18 gantolebis SedgeniT: I ricxvi _ x, II _ 2x, III _ 4x. pirobiT, x+2x+4x=777, x=111.

pasuxi: 111, 222 da 444.

19 umjobesia xiT aRvniSnoT yvelaze mcire _ I ricxvi, maSin II masze orjer metia _

2x, III metia 2xze orjer _ 4x. miviRebT: x+2x+4x=111 111, saidanac x=15873.

pasuxi: 15873, 31746, 63492.

20 gantolebis SedgeniT: es ricxvebi aRvniSnoT x–iTa da x+20iT. pirobiT, x+x+20=140,

2x=120, x=60.

pasuxi: 60 da 80.

ricxviTi gamosaxulebis SedgeniT: am ricxvebidan umciresia (140–20):2.

masw_wigni_5_kl

Documents