nbUfnbujljt !txbwmfcb!!WJJ!lmbtTj! - Ganatleba
nbUfnbujljt !txbwmfcb!!WJJ!lmbtTj! - Ganatleba nbUfnbujljt !txbwmfcb!!WJJ!lmbtTj! - Ganatleba
- 1 - nbUfnbujljt q o r b u d a !txbwmfcb!!WJJ!lmbtTj! nfUpejlvsj!tbyfmn[Swbofmp!nbtxbwmfcmjtUwjt! avtori: zurab vaxania fsiqologiis institutis direqtoris moadgile, pedagogikis mecn. doqtori, maTematikis mecn. kandidati, ganaTlebis mecnierebaTa akademikosi. saredaqcio sabWo: vaxtang paataSvili, elza goCitaSvili sagamomcemlo samuSaoTa Semsrulebelni: zurab vaxania, naTela Sulaia, nikoloz vaxania © z.vaxania, 2007 II gamocema tel. 37-28-88 893-94-53-04 is maswavlebeli, vinc Cveneuli saxelmZRvaneloTi aswavlis, am saxelmZRvanelosac da moswavlis saxelmZRvanelosac ufasod miiRebs gamomcemlobaSi!
- Page 2 and 3: - 2 - s a r C e v i 0. saxelmZRvane
- Page 4 and 5: - 4 - garda amisa, yvelaze mniSvnel
- Page 6 and 7: - 6 - winaaRmdegobaTa gadalaxvis Cv
- Page 8 and 9: - 8 - emsaxureba ZiriTadi saxelmZRv
- Page 10 and 11: - 10 - xeebis 1/7 nawils, ese igi,
- Page 12 and 13: - 12 - fasebebSi). programa da saxe
- Page 14 and 15: _ maTematikisTvis. - 14 - moswavlez
- Page 16 and 17: - 16 - uri pasuxebi, oRond, unda mo
- Page 18 and 19: - 18 - Tuki am mcnebis namdvili gan
- Page 20 and 21: - 20 - gramatikis programaSi (dawvr
- Page 22 and 23: - 22 - naTleblo skolis moswavleTa d
- Page 24 and 25: - 24 - pirovnebis pativiscemas. mos
- Page 26 and 27: - 26 - xSirad moxdeba, rom maswavle
- Page 28 and 29: - 28 - swored maTematikis maswavleb
- Page 30 and 31: - 30 - Teoremas nairgvari (maT Sori
- Page 32 and 33: - 32 - (Tumca, amis magvari da amis
- Page 34 and 35: - 34 - sanaxevrod an TiTqmis marTeb
- Page 36 and 37: - 36 - buli (gakveTilebad da paragr
- Page 38 and 39: - 38 - qjsbebe etyvis ramdenime sit
- Page 40 and 41: - 40 - gansxvavebul amoxsnaTa urTie
- Page 42 and 43: movaleobaa: - 42 - 1) misces moswav
- Page 44 and 45: - 44 - mier aqtiurad garkveuli saki
- Page 46 and 47: - 46 - maswavlebeli. pirvel yovlisa
- Page 48 and 49: - 48 - udidesi pativiscemiTa da siy
- Page 50 and 51: - 50 - niSani, rasac imsaxurebs, yo
- 1 -<br />
<strong>nbUfnbujljt</strong><br />
q o r b u d a<br />
!<strong>txbwmfcb</strong>!!<strong>WJJ</strong>!<strong>lmbtTj</strong>!<br />
nfUpejlvsj!tbyfmn[Swbofmp!nbtxbwmfcmjtUwjt!<br />
avtori:<br />
zurab vaxania<br />
fsiqologiis institutis direqtoris moadgile,<br />
pedagogikis mecn. doqtori,<br />
maTematikis mecn. kandidati,<br />
ganaTlebis mecnierebaTa akademikosi.<br />
saredaqcio sabWo:<br />
vaxtang paataSvili, elza goCitaSvili<br />
sagamomcemlo samuSaoTa Semsrulebelni:<br />
zurab vaxania, naTela Sulaia, nikoloz vaxania<br />
© z.vaxania, 2007 II gamocema tel. 37-28-88 893-94-53-04<br />
<br />
is maswavlebeli, vinc Cveneuli saxelmZRvaneloTi<br />
aswavlis, am saxelmZRvanelosac da moswavlis<br />
saxelmZRvanelosac ufasod miiRebs gamomcemlobaSi!
- 2 -<br />
s a r C e v i<br />
0. saxelmZRvaneloebi `saymawvilo maTematika~<br />
1. `saymawvilo maTematikiT~ swavlebis miznebi<br />
2. `saymawvilo maTematikis~ Tavebi da paragrafebi<br />
3. mTavari meTodikuri principebi<br />
4. maTematikis mravalmxrivi da Tan<br />
gaerTianebuli (integrirebuli) swavleba<br />
5. wakiTxulis gaazrebis unari<br />
da `saymawvilo maTematikis~ nakli;<br />
visTvisaa upriani Cveneuli meTodikiT swavla<br />
6. maTematikis namdvili codna<br />
7. VII klasis programa saxelmwifo saswavlo gegmiT<br />
8. saxelmZRvanelos Sesabamisoba saxelmwifo<br />
saswavlo gegmasTan<br />
9. gakveTilis ageba<br />
10. maswavleblis muSaoba<br />
11. Cveneuli amocanebis tipebi<br />
12. Semamzadebeli safexurebi; erTi nimuSi<br />
13. erTi gakveTilis sanimuSo konspeqti<br />
14. Temis damuSavebis nimuSi _ usasrulobis cneba<br />
15. geometriis swavlebis safexurebi<br />
16. aramaTematikurad, gumaniT amosaxsneli amocanebi<br />
17. TvalsaCinoeba da misi farglebi<br />
18. sakontrolo werebi da SefasebaTa sistema<br />
VII klasis sakontrolo amoanebis nusxa<br />
19. amocanebis pasuxebi da miTiTebebi<br />
20. sakontrolo werebis amocanaTa pasuxebi<br />
da maTi Sefasebis kriteriumebi<br />
21. damatebiTi literatura maswavlebelTaTvis
- 3 -<br />
maTematikis maswavleblis mecxre da meaTe mcnebani:<br />
ara auwyo mowafesa Sensa saidumlo mecnierebisa mzamzareulad,<br />
aramed Tavad mowafeman gamoicnos saidumlo igi,<br />
vidre ganumartavde Sen, Tavad ipovos, rac SeiZleba meti.<br />
kurTxeul ars misaxvedri miTiTeba, iZulebiT nu<br />
moaxvev Tavs mowafes Sensa azrsa. [jorj poia]<br />
§ 0. saxelmZRvaneloebi `saymawvilo maTematika~<br />
Cveneuli saxelmZRvaneloebi agebulia axal samecniero-meTodikur<br />
da pedagogikur-fsiqologiur safuZvlebze. amasTan, isini<br />
srulad integrirebulia (maTematikis yvela dargi iswavleba er-<br />
Tianad, erTi saxelmZRvaneloTi). garda amisa, yoveli klasis<br />
programa da saxelmZRvanelo arsebiTad emyareba da aviTarebs<br />
wina klasisas. amgvarad Seqmnili gvaqvs erTiani kursi I-dan TiTqmis<br />
XI klasamde.<br />
yovelive amis gamo gaumarTlebelia romelime Cveneuli saxelmZRvanelos<br />
ganxilva zogadi safuZvlebis gareSe da sxva saxelm-<br />
ZRvaneloebisagan gamocalkevebulad. kerZod, VII klasis maswavlebelma<br />
unda naxos, rogor gvaqvs Cven damuSavebuli Semdegi,<br />
SedarebiT aratradiciuli sakiTxebi: umartivesi topologiuri<br />
cnebebi; ZiriTadi geometriuli cnebebi (nakvTi, sxeuli, mravalkuTxedi,<br />
xazi); erToblioba; logikuri mimarTeba zogadi /<br />
kerZo; proporcia, skala da masStabi.<br />
Tuki `saymawvilo maTematikiT~ swavleba VII klasidan<br />
iwyeba: maswavlebelma ar unda daiwyos VII klasis saxelmZRvanelo<br />
manam, sanam klass safuZvlianad ar gaameorebinebs<br />
wina klasebis ZiriTad sakiTxebs. Tanac, am dros klasi unda<br />
mieCvios Cveneuli meTodikiT muSaobas, rac misTvis uCveulo<br />
iqneba. amitom, sasurvelia, maswavlebelma gasameoreblad gamoiyenos<br />
Cveneuli saxelmZRvanelo VI klasisTvis.
- 4 -<br />
garda amisa, yvelaze mniSvnelovania maTematikuri teqstis<br />
wakiTxisa da gaazrebis unaris ganviTareba _ [ix. $ 5]. uamisod<br />
Cveneul meTodikas saZirkveli gamoecleba.<br />
yovelive amas dasWirdeba albaT 2-3 kvira. ZiriTadi<br />
gasameorebel-gansamtkicebel-gasaRrmavebeli sakiTxebia:<br />
1. wiladis cneba [ix. $ 3-Si]; 1-ze meti wiladebi da<br />
maTgan mTelis gamoyofa; wiladis ZiriTadi Tviseba...<br />
2. wriuli diagrama da wiladebi, romelTa jamis<br />
mniSvnelobaa 1 (procenti saWiro araa, iqneba VII klasSi).<br />
3. farTobisa da moculobis cnebebi _ da ara maTi<br />
gamosaTvleli formulebi; ori-sami aratoli<br />
marTkuTxedisgan Sedgenili nakvTis farTobis gazomva,<br />
agreTve uswormasworo da naxvretiani nakvTebis farTobis<br />
miaxloebiTi gazomva; ori-sami aguredisgan Sedgenili<br />
sxeulis moculobis gazomva.<br />
4. skala da masStabi. masStabis Caweris sxvadasxva saxeebi.<br />
5. proporciuloba. proporciis Tvisebebi.<br />
6. logikuri mimarTebani ZiriTad planimetriul nakvTebs<br />
Soris _ ix. Cveneuli geometriuli TvalsaCinoeba (plakatis<br />
saxiT, anda individualuri _ romelic dabeWdilia Cveneuli<br />
saxelmZRvaneloebis ydis Siga mxares).<br />
maswavlebeli ar unda SeuSindes am winaswar Seferxebas<br />
da mcire sirTuleebs _ maTi gadalaxvis Semdeg araTu<br />
saboloo, aramed ukve VII klasis Sedegic ki gaaxarebs!<br />
§ 1. `saymawvilo maTematikiT~ swavlebis miznebi<br />
maswavlebelma icis, rom, sazogadod, maTematikis swavlebis<br />
miznebi or ZiriTad jgufad iyofa:<br />
1) maTematikis zogadi, sayovelTaod saWiro sawyisebis daufleba<br />
_ codnisa da unarCvevaTa SeZena;<br />
2) gonebis ganviTareba da zogad saazrovno unarTa Camoyalibeba.
- 5 -<br />
gonebrivi unarebi,<br />
romlebic unda ganaviTaros saskolo maTematikis swavlebam:<br />
I. specifikuri<br />
maTematikuri<br />
unarCvevebi:<br />
Zalian<br />
mravladaa,<br />
ZiriTadad _<br />
praqtikul-<br />
gamoyenebiTi<br />
xasiaTisa.<br />
5) statistikuri _<br />
informaciis damu-<br />
Savebisa da sxvadasxva<br />
sqematuri saxiT<br />
warmodgenisa,<br />
mosalodnelobis<br />
Sefasebisa;<br />
II. zogadi<br />
maTematikuri<br />
unarebi:<br />
1) ariTmetikuli<br />
_ raodenobriv<br />
mimarTebaTa daufleba:<br />
ricxvebze<br />
moqmedebaTa,<br />
sidideTa gazomvisa<br />
an miaxloebiTi<br />
Sefasebisa;<br />
2) geometriuli<br />
_ sivrciT mimar-<br />
TebaTa da formebis<br />
daufleba;<br />
3) algebruli _<br />
maTematikuri modelirebisa;formulebisa<br />
da abstraqtuli<br />
niSnebis<br />
gamoyenebisa;<br />
4) diskretulma-<br />
6) saerTo _ maTe-<br />
Tematikuri _ almatikuri<br />
enis daufgoriTmis<br />
zustad,<br />
leba, saWiro sakiT-<br />
Tanmimdevrulad<br />
xis moZiebis da<br />
Sesrulebis, misi<br />
gaazrebis, wignze<br />
mkafiod Camoya-<br />
muSaobis unarebi.<br />
libebisa;<br />
III. saazrovno<br />
unarebi:<br />
1) sakiTxis gaazrebis,<br />
teqstis wakiTxvisa<br />
da gaazrebisa;<br />
2) arsebiTis danaxvis,<br />
misi sxva areSi<br />
gadatanisa;<br />
3) zusti azrovnebisa<br />
da metyvelebisa;<br />
4) logikuri daskvnis<br />
gamotanisa;<br />
5) analizisa da sin-<br />
Tezisa;<br />
6) argumentirebis,<br />
naTlad da Tanmimdevruladmsjelobis,<br />
dasabuTebisa<br />
Tu uaryofisa;<br />
7) gansazRvrebis gaazrebis,klasifikaciis,<br />
ganzogadebisa<br />
Tu konkretizaciisa;<br />
8) evristikuli unarebi(kanonzomierebis<br />
aRmoCenisa, raimes<br />
mixvedrisa).<br />
garda amisa, maTematikis swavlebas aRmzrdelobiTi daniSnulebac<br />
aqvs _ man unda ganaviTaros mravali zogadpirovnuli Tviseba:<br />
mowesrigebuloba, saqmis dagegmvis unarCvevebi, sibejiTe,
- 6 -<br />
winaaRmdegobaTa gadalaxvis Cveva, ganyenebul (TvaliT uxilav da<br />
aranivTier) faseulobaTa gancdis unari, kritikuli Sefasebis<br />
unari, sazogado wesebisa da kanonebis pativiscema.<br />
swored am Zvirfas pirovnul TvisebaTa da zogad unarTa ganviTarebaa<br />
maTematikis Cveneuli swavlebis mTavari mizani (da ara<br />
teqnikuri manipulaciebis triali, sqolastikur damtkicebaTa da<br />
mZime formulebis daxvaveba, rac saskolo maTematikis Cveulebrivi<br />
programis udides nawils Seadgens).<br />
§ 2. `saymawvilo maTematikis~ Tavebi da paragrafebi<br />
`saymawvilo maTematika~ iyofa Tavebad. yoveli Tavis<br />
damTavrebis Semdeg tardeba sakontrolo wera.<br />
saxelmZRvanelos bolos amocanaTa Tematikuri krebulia. igi<br />
Sedgeba ara Tavebis, aramed paragrafebisagan.<br />
Tavsa da paragrafs Soris arsebiTi gansxvavebaa. Tavi dalagebulia<br />
kalendarulad da zustadaa dayofili gakveTilebis<br />
mixedviT. amitom `saymawvilo maTematikis~ sarCevi zustad<br />
emTxveva lbmfoebsvm!hfhnbt!!da calke es gegma aRaraa saWiro.<br />
paragrafebi ki dalagebulia ara kalendarulad, aramed Tematikurad:<br />
maTSi amocanebi Temebis mixedviTaa dajgufebuli.<br />
paragrafebis gavla ar unda moxdes miyolebiT (ise, rogorc<br />
wignSia). saWiroa maTi paraleluri gavla. paragrafebis Tanmimdevrobas<br />
ara aqvs didi mniSvneloba, magram TviTeul paragrafSi<br />
ki amocanebis gavla unda moxdes im TanmimdevrobiT, rogorc<br />
wignSia. yoveli paragrafi agebulia amocanebis safexurebrivi<br />
garTulebis mixedviT! konkretulad ki Tavad maswavlebeli<br />
gaanawilebs sakuTari SexedulebiT.<br />
saxelmZRvanelos ZiriTadi nawilis anu Tavebis gavlas (sakontroloebis<br />
CaTvliT) dasWirdeba daaxloebiT 145-150 gakveTili.<br />
saswavlo wlis bolomde darCenil droSi maswavlebeli ZiriTadad<br />
iyenebs wignis meore nawils _ amocanebis Tematikur krebuls<br />
(pirveli nawili dasWirdebaT mxolod calkeuli Teoriu-
- 7 -<br />
li sakiTxebis gasameoreblad _ maswavleblis Sexedulebisamebr).<br />
sazogadod, maswavleblis TviTmoqmedebisaTvis farTo sarbieli<br />
Tematikur krebulebSia _ da ara ZiriTad gakveTilebSi.<br />
moswavleTa saxelmZRvaneloSi mocemuli amocanebi ZiriTadad<br />
saSinao davalebebisTvisaa gankuTvnili. klasSi xdeba Sesrulebuli<br />
davalebis erToblivi garCeva.<br />
Tematikur krebulSi gabneulia agreTve is amocanebi, romlebic<br />
klass sakontrolo werebze unda mieces (nusxa mocemulia<br />
danarTSi). maswavlebels TavisTvis unda hqondes moswavleTa<br />
saxelmZRvanelo, erTi cali wigni. am Tavis wignSi axlave<br />
moniSnos sakontroloebis amocanebi, raTa SemdgomSi isini<br />
SecdomiT saSinao an saklaso davalebad ar misces klass.<br />
maswavlebels ecodineba, rom mis mier wiTlad moniSnuli<br />
amocanebi _ mxolod sakontrolo Tu sagamocdo werebisaTvisaa.<br />
Cven yoveli klasis bolos vatarebT Semajamebel sakontrolo<br />
weras (mcire gamocdas) [$ 18].<br />
§ 3. mTavari meTodikuri principebi (mokled)<br />
`saymawvilo maTematika~ ganuyoflad moicavs saskolo maTema-<br />
tikis yvela dargs maTi momijnave dargebiTurT da agebulia Tanamedrove<br />
pedagogikuri fsiqologiis sayovelTaod aRiarebul<br />
principebze. amasTan, es principebi ganyenebul mowodebebad anu<br />
`cariel abrad~ ar rCeba, isini namdvilad, Tanmimdevrulad da<br />
safuZvlianad xorcieldeba yoveli calkeuli maTematikuri<br />
sakiTxis swavlebaSi.<br />
gaRrmavebuli swavleba. `saymawvilo maTematika~ gaRrmavebuli<br />
swavlebisaTvisaa, magram kargad unda iqnes gaazrebuli, Tu<br />
ras niSnavs gaRrmaveba da ras _ gaZliereba. gaRrmavebuli<br />
(meorenairad _ intensiuri) swavleba gulisxmobs programis<br />
Semofargvlas mxolod im aucilebeli sakiTxebiT, romelTa<br />
damuSaveba eswreba Rrmad, safuZvlianad, mravalmxrivad da<br />
aqtiur-SemoqmedebiTad. sakiTxi an ase iswavleba (da swored amas
- 8 -<br />
emsaxureba ZiriTadi saxelmZRvaneloebi), an sul ar iswavleba.<br />
gaZlierebuli swavleba ki sxvaa: es gulisxmobs damatebiTi<br />
rTuli sakiTxebisa da damatebiTi Zneli amocanebis Semotanas.<br />
gaZlierebuli swavlebisaTvis (zogierTi moswavlisaTvis)<br />
gankuTvnilia ufro amocanaTa Tematikuri paragrafebi da agreTve<br />
amocanaTa damatebiTi krebulebi.<br />
ese igi, gaZlierebuli swavla yvelas ar moeTxoveba, magram<br />
gaRrmavebuli ki _ TiTqmis yvelas.<br />
Cveulebrivi saskolo kursebi eqstensiuria: moswavleebs<br />
miewodeba didi odenobiT Teoriuli sakiTxebi da terminebi, ise,<br />
rom ver eswreba maTi wesierad damuSaveba, klasi `gadarbeniT~,<br />
zereled da pasurad iRebs (ufro xSirad ki _ arc iRebs!)<br />
maTematikur codnas. es iwvevs moswavleTa gulisacruebas, xolo<br />
maTi saukeTeso nawili iZulebuli xdeba, dazepirebiTa da<br />
meqanikuri gawafviT daiZvrinos Tavi.<br />
amis sapirispirod, Cveni mcnebaa: `jobia, erTi sakiTxi vaswavloT<br />
aTi sxvadasxva kuTxiT, vidre aTi sakiTxi vaswavloT TiTo<br />
kuTxiT~ [a. distervegi], vaswavloT nela, magram kargad, Rrmad<br />
da mravalmxrivad. gaRrmavebuli swavleba gulisxmobs ara<br />
saswavli Sinaarsis eqstensiur gafarToebasa da tempis aCqarebas,<br />
aramed piriqiT: saswavli Teoriuli sakiTxebis raodenobis<br />
Semcirebas, tempis Senelebas da gaRrmavebul, gaazrebul, anu<br />
intensiur swavlebas. naswavli unda Seesisxlxorcos moswavlis<br />
gonebas, unda gamoiwvios gonebis Sinagani zrda, misi mravalmxrivi<br />
ganviTareba.<br />
amgvari swavlebis Sedegad saSualo moswavleebs rCebaT aucilebeli<br />
saprogramo minimumis myari da aqtiuri codna, xolo<br />
Zlieri moswavleebi imavdroulad aswreben cota gaZlierebuli<br />
kursis gavlasac, rac damatebiT moicavs ufro rTul<br />
sakiTxebsac. amasTan, upirvelesi mniSvneloba eniWeba moswavlis<br />
rogorc saSemsruleblo, ise azrovnebis, damoukidebeli<br />
muSaobisa da kvlevis, SemoqmedebiT unarCvevaTa ganviTarebas. Cven
- 9 -<br />
veswrafviT moswavlis aramarto zusti maTematikur-logikuri<br />
azrovnebis ganviTarebas, aramed agreTve intuiciis, gumanis,<br />
mixvedrilobis ganviTarebasac.<br />
maTematikis gaRrmavebuli swavlebis arsebiTi maxasiaTebelia<br />
agreTve mTavari yuradRebis gadatana manipulaciebidan cnebebisaken.<br />
`saymawvilo maTematikaSi~ Zalian didi dro eTmoba<br />
sakvanZo cnebebis _ TviT cnebebis! _ gaazrebas. magaliTisTvis<br />
davasaxeloT: farTobis cneba; moculobis cneba; usasrulobis<br />
cneba (mis Sesaxeb ix. qvemore, § 14); cilindris cneba (ix. § 17)<br />
da wiladis cneba.<br />
xjmbejt!dofcb albaT saskolo maTematikis `nomer pirveli~<br />
cnebaa. misi ZirisZiramde gaazrebis gareSe (swored cnebisa _ da<br />
ara wiladebze moqmedebebisa!) azri ara aqvs momdevno klasebis<br />
maTematikis swavlas, iseve rogorc fizikisa, qimiisa, geografiuli<br />
masStabisa, welTaRrocxvisa, statistikisa da sxvaTa.<br />
Cven Semowmebuli gvaqvs: tradiciuli saxelmZRvaneloebiT naswavl<br />
skoladamTavrebulebsac ki (maT daaxloebiT 70-80 %-s) ar<br />
esmiT wiladis cneba. es yvelam SeiZleba Seamowmos, magaliTad,<br />
amgvari advili amocaniT:<br />
ezoSi xeebis 3/7 nawili kopitebia, amdenive _ Wadrebi.<br />
kidev ezoSi dgas 2 naZvi. sul ramdeni xe dgas am ezoSi?<br />
6 6<br />
a) 4; b) 7; g) 8; d) 14; e) 2 ; v) 2 .<br />
7 14<br />
moswavleTa didi umravlesoba irCevs pasuxs e), anda, kidev<br />
uaresi _ v) (rac imis maCvenebelia, rom wiladebis Sekrebis wesic<br />
ki ar icis). orive es pasuxi gviCvenebs, rom moswavles sruliad<br />
ar esmis, ra aris wiladi; verc imas iazrebs, rom xeebis<br />
raodenoba ar SeiZleba wiladuri iyos! moswavlem uazrod,<br />
3 3 6<br />
meqanikurad Sekriba: + + 2 = 2 . am dros ki amocanis<br />
7 7 7<br />
amoxsnas TiTqmis ar sWirdeba wiladebze moqmedebaTa wesebi,<br />
saWiroa mxolod wiladis cnebis codna: 2 naZvi Seadgens ezos
- 10 -<br />
xeebis 1/7 nawils, ese igi, ezoSi sul 14 xe mdgara. sul esaa,<br />
gantolebac ki zedmetia da am amocanisTvis gantolebis Sedgena<br />
moswavlis saazrovno unarCvevaTa ganuviTareblobas moaswavebs.<br />
maSasadame, tradiciuli saxelmZRvaneloebiT momuSave ufrosklaselTa<br />
80 % mainc, arsebiTad, fuWad dadis maTematikis, fizikisa<br />
Tu qimiis gakveTilebze, radgan maT ar esmiT wiladi.<br />
Cveneuli programiT, wiladi Semogvaqvs mxolod V klasSi (IV<br />
klasSi mas arc ki vaxsenebT!); Tanac, sul mcire 10 saaTs vuTmobT<br />
wiladis jer mxolod cnebas. V klasis ariTmetikis programa,<br />
arsebiTad, mxolod wiladebs eTmoba, aTwiladebi jer naadrevia!<br />
marTlac, aTwiladi, arsebiTad, igive wiladia (maTematikurad<br />
_ racionaluri ricxvi), oRond sxvagvarad Cawerili. Tuki<br />
moswavles bolomde ara aqvs gaazrebuli wiladis cneba,<br />
wiladebis Sedareba da maTze ariTmetikuli moqmedebebi da ukve<br />
aTwiladebs vaswavliT, es niSnavs Semdegs: CvenTvis mTavaria<br />
moqmedebaTa Sesruleba (rac aTwiladebze ufro advilia) _ da<br />
ara cnebis gaazreba da azrovneba. anu, CvenTvis mTavaria,<br />
moswavlem kalkulatoriviT imuSaos uSecdomod _ da ara is,<br />
rom wiladis arsi esmodes. marTlac, rogor SeiZleba kacma<br />
gaiazros aTwiladebze _ anu sxvagvarad Caweril da kerZo saxis<br />
wiladebze moqmedebebi _ Tuki jer wiladebze ar gauazrebia?<br />
Tuki moswavles azriT ar esmis, romelia meti _ 3/7 Tu 5/9,<br />
maSin ar dros aTwiladebze moqmedebebSi meqanikuri gawafvaa? da,<br />
sazogadod, ra saWiroa kalkulatoris saqmes esoden didi<br />
yuradReba davuTmoT? Cven ar vambobT, rom moswavlisTvis<br />
zedmetia aTwiladebze moqmedebaTa codna, es zedmeti araa, magram<br />
arc mTavaria! mTavaria wiladis cneba da wiladebze moqmedebaTa<br />
gaazreba (da aqac ara gamoTvlebSi gawafva).<br />
humanisturi swavleba, individualuri midgoma da<br />
moswavleze centrireba. igi upirvelesad gulisxmobs<br />
keTilmosurne da gulTbil damokidebulebas moswavlisadmi.<br />
magaliTad, maswavleblisa da moswavlis urTierTobaSi gamoric-
- 11 -<br />
xuli unda iyos SiSi. SiSze damyarebuli wesrigi da swavla<br />
Zalian aramyaria: gardatexis asakis Semdeg bavSvs aRar eSinia<br />
maswavleblebisa da ormagad gadauxdis maT wina wlebSi<br />
dagrovili SiSis samagieros. Tanac, SiSiTa da daZabulobiT<br />
bavSvi Znelad Tu ganviTardeba. moswavles odnavadac ar unda<br />
eSinodes imis Tqma, rom man raime ver gaigo, raRac ver gaakeTa<br />
(gamocdil maswavlebels ar gamoepareba, bavSvma marTla ver<br />
gaakeTa, Tu ar gaakeTa da cuRlutobs). moswavle ar unda<br />
iboWebodes Secdomis daSvebis SiSiT. yoveli Secdoma saqmiani da<br />
mSvidi msjelobis sagani unda gaxdes. moswavles unda vacaloT<br />
azris gamoTqma, Tundac es iyos mcdari azri. maswavlebels<br />
pirisaxis gamometyvelebaze an xmazec ki ar unda Seetyos<br />
ukmayofileba. _ SemdgomSic moindomeb da ukeTesad gaakeTeb,<br />
axla ki mousmine Sens megobars da advilad gaigeb! _ daaxloebiT<br />
amgvari ram unda iTqvas, mSvidad da gulTbilad.<br />
zogierTi maswavlebeli xazgasmiT, ganzrax gamokveTs xolme<br />
imas, rom bavSvi raRacas ver akeTebs, ver igebs, unergavs ra<br />
bavSvebs SiSsa da morCilebis grZnobas. es uxeSad arapedagogiuri<br />
saqcielia, metic, danaSaulia.<br />
moswavleze centrireba gulisxmobs kidev erT, gacilebiT<br />
ufro arsebiT da mniSvnelovan princips: mTliani swavlebis ageba<br />
ise, rogorc bunebrivia bavSvis cnobierebisaTvis; meore _<br />
SeZlebisamebr joejwjevbmvsj!!njehpnb.<br />
individualuri midgomis bolomde ganxorcieleba SeuZlebelia<br />
klasSi, miT umetes, did klasSi. magram maswavleblis didi ostatoba<br />
da xelovneba swored isaa, rom SeZlebisamebr metad moaxerxos<br />
es. man yovel moswavleSi ganumeorebeli pirovneba unda<br />
dainaxos, pativi sces am pirovnebas da, rac mTavaria, Taviseburad<br />
miudges mas. amis zogadi wesebi ar arsebobs. yovel kerZo<br />
SemTxvevaSi maswavlebelma unda moZebnos kerZo saSualebani.<br />
individualuri midgomis kidev erTi mniSvnelovani mxarea<br />
moTxovnaTa araTanabroba moswavleTa codnis mimarTac (es ki,<br />
cxadia, Sesabamisad unda aisaxos maswavleblis mier daweril Se-
- 12 -<br />
fasebebSi). programa da saxelmZRvaneloebi, maTi sruli moculobiT,<br />
gaTvlilia Zlier moswavleebze. dauSvebelia Zlier moswavleTa<br />
ganviTarebis ganzrax Seferxeba imis gamo, rom klasSi sustebic<br />
sxedan. meore mxriv, arc sustebis daCagvra egebis. kvlav<br />
gavimeorebT: maswavlebelmac da mSoblebmac unda icodnen: yvela<br />
moswavles ar moeTxoveba Cveni programis aTviseba mTeli<br />
moculobiTa da siRrmiT. saSualo moswavle yvelafers<br />
Rrmad ver gaigebs, verc yvela davalebas Seasrulebs, magram<br />
mainc kargad, myarad da aqtiurad daeufleba programis<br />
pirvel dones _ savaldebulo minimums. xolo Zlieri<br />
moswavle programis meore, maRal donesac daeufleba.<br />
Cveneuli meTodikis mixedviT, yvela moswavle swavlobs<br />
sakuTar SesaZleblobaTa Sesabamisad. klass yoveli dRisaTvis<br />
(maT Soris, dasvenebis dReebisaTvisac da ardadegebisaTvisac!)<br />
eZleva bevri davaleba, rac ukanasknel erT an or nomrad SeiZleba<br />
Seicavdes sakmaod Znel, arastandartul amocanebs.<br />
garda amisa, saswavlo wlis bolosaTvis klass ukve<br />
moTavebuli eqneba saxelmZRvanelos Tavebi da morCenili dro<br />
mTlianad meore nawils _ amocanaTa Tematikur paragrafebs<br />
daeTmoba _ yoveldRiurad 6-8 nomeri.<br />
magram saqme isaa, rom mTeli saSinao davalebis gakeTeba<br />
araa savaldebulo yvela moswavlisaTvis. magaliTad, SabaTkvirisaTvis<br />
damatebiT micemul erTi-or amocanas (Tematikuri<br />
krebulidan) mxolod mondomebuli da Zlieri moswavleebi gaake-<br />
Teben. sazogadod, vinc gaakeTebs xolme naxevarze cota mets, is<br />
iswavlis saSualod (saprogramo minimumis myari aqtiuri codniT),<br />
vinc gaakeTebs TiTqmis yvelafers _ iswavlis saukeTesod.<br />
yvelam amoxsnas imdeni, ramdensac SeZlebs. saSinao davalebaSi<br />
Sefaseba ar iwereba. saSualo moswavlisTvis sakmarisia,<br />
Tuki namdvilad damoukideblad SeZlebs davalebis daaxloebiT<br />
naxevris marTebulad Sesrulebas. es yovelive mSoblebsac unda<br />
ganvumartoT.
- 13 -<br />
ganviTarebaSi da gaZlierebaSi bavSvs TviTon amocanebi<br />
daexmareba! amocanebi dalagebulia TandaTanobiTi gaZnelebis<br />
mixedviT. Tema iwyeba Zalian advili amocanebiT, romlebsac<br />
TiTqmis yvela moswavle daZlevs. Semdeg amocanebi TandaTanobiT<br />
Zneldeba da moswavles ar gauWirdeba, mihyves maT.<br />
humanisturi swavlebis principi moiTxovs, garda individualuri<br />
midgomisa, agreTve azrovnebis ganviTarebis asakobriv<br />
da agreTve zogad kanonzomierebaTa mkacr dacvas. es kanonzomierebebi<br />
pedagogikur fsiqologiaSia dadgenili (upirvelesad,<br />
J. piaJes mier [22]). im sakiTxebs (miuxedavad maTi sirTulesimartivisa),<br />
romlebic efuZneba Zlier abstraqtul cnebebs,<br />
dawyebiTi skolis da, miT umetes, pirveli klasis moswavle<br />
azrianad ver daeufleba. aseTi sakiTxebia, magaliTad: farTobi<br />
da moculoba (maT Soris, litri); masa (TiTqos moswavles<br />
esmodes gansxvaveba masasa da wonas Soris!); wrfe _ gansxvavebiT<br />
monakveTisagan (Cven mxolod VII klasSi Semogvaqvs); kuTxe<br />
(maxvili, marTi, blagvi _ Cven mxolod VIII klasSi Semogvaqvs!).<br />
ukve am ramdenime magaliTidanac ki cxadia, rom, samwuxarod,<br />
tradiciuli meTodika naklebad iTvaliswinebs TiTqosda Tavis-<br />
Tavad cxad da ubralo WeSmaritebas: rom swavleba azrovnebis<br />
asakobrivi kanonzomierebebiT unda iyos ganpirobebuli.<br />
yovelive amas sulac ar ewinaaRmdegeba is, rom Cven pirvelive<br />
klasidan programa Sevsebuli gvaqvs Tanamedrove maTematikis<br />
sxvadasxva dargebis saymawvilo SesavlebiT, romlebsac araviTari<br />
Teoria da terminologia ar sWirdeba, moswavleebi maT daeuflebian<br />
saxaliso amocanebis sagangebo TanwyobaTa meSveobiT.<br />
CvenTvis mTavari ganmsazRvrelia moswavlis cnobiereba da misi<br />
bunebrivi interesebi, ganaTlebuli pirovnebis aRzrdis Zireuli amocana,<br />
_ da ara formaluri maTematikuri Tvalsazrisi. moswavleze<br />
centrirebis principi krZalavs saganze centrirebas, anu saswavli<br />
sagnis xedvas kerZo mecnierebis viwro egocentruli TvalTaxedviT.<br />
maTematika unda iyos moswavlisTvis da ara moswavle
_ maTematikisTvis.<br />
- 14 -<br />
moswavleze centrirebis principTan mWidrodaa dakavSirebuli<br />
jtupsj{njt! qsjodjqj/ moswavleze centrirebis gamo mTavaria<br />
is, Tu swavlis romeli gzaa bavSvis cnobierebisaTvis yvelaze<br />
bunebrivi. samwuxarod, ufro xSirad es gza ar emTxveva im gzas,<br />
romelic bunebrivi Cans Sesaswavli sagnis TvalsazrisiT. magali-<br />
Tad, mecnieruli geometriis TvalsazrisiT jer unda iswavlebodes<br />
kuTxe, Semdeg _ marTi kuTxe da Semdeg _ marTkuTxedi. magram<br />
fsiqologiurad es gza yovlad gaumarTlebeli aRmoCnda.<br />
ufro xSirad bavSvis (da sazogadod, adamianis) cnobierebisa-<br />
Tvis raime cnebis Seswavlis bunebrivi gzaa is gza, romliTac<br />
istoriulad Seimecna kacobriobam es cneba (cxadia, moswavle Zalian<br />
SemWidrovebulad da daCqarebulad gaivlis am gzas). amis<br />
kargi magaliTia VII klasSi uaryofiTi ricxvebis swavlebis Cveneuli<br />
meTodika.<br />
SemoqmedebiTi swavleba. cudi swavlebis gamo xalxSi<br />
damkvidrebulia arsebiTad mcdari warmodgena maTematikaze _<br />
TiTqos esaa raRac usaSvelod rTuli, usaSvelod mZime da<br />
usaSvelod mosawyeni, didi ricxvebi da gauTavebeli gamoTvlebi,<br />
mkvdari formulebi da wesebi...<br />
maTematikis swavleba, Tuki igi marTebuladaa agebuli da<br />
gamarTuli, ar unda iwvevdes am ulamazesi da uZlieresi<br />
mecnierebis Sejavrebas, piriqiT unda xdebodes!<br />
yovelive didi da meqanikuri _ ara adamianis SemoqmedebiTi<br />
gonebis, aramed gamoTvliTi manqanebis saqmea! adamianisTvis da<br />
gansakuTrebiT ki ymawvilisaTvis maTematikaSi mTavaria swored<br />
is, rom imoqmedos ara meqanikurad, aramed piriqiT _<br />
gaazrebulad, SemoqmedebiTad, gabedulad _ rogorc moazrovne<br />
adamians ekadreba.<br />
rasakvirvelia, yovelive es ar gamoricxavs zomieri gamoTvlebisa<br />
da sxva `Savi Sromis~ aucileblobas. iseve rogorc<br />
adamianis sulis erTerTi umSvenieresi Semoqmedeba _ musikac ki
- 15 -<br />
mxolod `Sav Sromaze~ damyarebiT ifurCqneba.<br />
Cveni meTodikis mixedviT maTematikis swavla miaxloebulia<br />
mecnierul SemoqmedebasTan, sakiTxebi isea damuSavebuli, rom<br />
axali sakiTxis arss moswavle TiTqos TviTon ikvlevs da<br />
TviTonve aRmoaCens. mTeli Cveneuli kursi _ esaa amgvar ZiebaTa<br />
da aRmoCenaTa erTiani jaWvedi.<br />
saamisod mTeli kursi daqucmacebulia mcire-mcire safexurebad,<br />
romelTa damoukideblad gavla advilad SeuZlia saSualo<br />
moswavles _ cxadia, Tuki mas gavlili aqvs wina gakveTilebi.<br />
TviTeuli am safexuris gavlas moswavle Sesabamisi patara sakiTxis<br />
arsis aRmoCenamde mihyavs da ase grZeldeba bolomde.<br />
amitom saxelmZRvaneloebis ZiriTadi nawilebi sakmaod advilebia,<br />
gaTvlilia namdvil saSualo moswavleze. magram saqme isaa, rom<br />
Zlieri moswavlisaTvisac ki am mcire-mcire kvleviTi safexurebis<br />
gavlas udidesi mniSvneloba aqvs _ Rrma swavlisaTvis.<br />
yovelive zemore Tqmulidan cxadia, rom Cven gamovricxavT<br />
saskolo maTematikisTvis damaxasiaTebel mTavar mankierebas _<br />
moswavleTa mier sityvieri debulebebis (wesebis, gansaz-<br />
Rvrebebisa Tu Teoremebis) gazuTxvas da Sesabamis moqmedebebSi<br />
meqanikur gawafvas. ufro xSirad umjobesia, moswavlem sul ar<br />
icodes raime sakiTxi, vidre amgvarad icodes. mxolod namdvilad<br />
gaazrebul, kargad gagebul codnas aqvs Rirebuleba. moswavles<br />
naTlad unda esmodes is, Tu ras ambobs da ra moqmedebas<br />
atarebs, ratom ambobs ase da ratom moqmedebs ase. es kargad<br />
mowmdeba advili arastandartuli amocanebis amoxsnisas. es<br />
amocanebi ar moicavs arc erT ucnob cnebasa Tu moqmedebas, arc<br />
Znelebia, magram amdagvari amocana moswavles jer ar amouxsnia.<br />
amitom didi mniSvneloba aqvs Tematikuri krebulebis amocanaTa<br />
uCveulo mravalferovnebas.<br />
mravali Cveni amocana sxvadasxva gzebiT amoixsneba, metic,<br />
marTebuli pasuxic ki SeiZleba ori an meti hqondes. maswavlebelma<br />
araviTar SemTxvevaSi ar unda aRkveTos moswavleTa ucna-
- 16 -<br />
uri pasuxebi, oRond, unda moiTxovos dasabuTeba. Tuki moswavle<br />
gonivrulad daasabuTebs Tavis moulodnel pasuxs, is Seqebis<br />
Rirsi iqneba, da ara gakicxvisa! aseve, yovelnairad unda<br />
waxalisdes amocanis amoxsna gansxvavebuli gzebiT.<br />
aqtiuri mzaobis meTodika. esaa `saymawvilo<br />
maTematikis~ ZiriTadi safuZveli da mTavari siaxle. igi emyareba<br />
d. uznaZisa da J. piaJes fsiqologiur Teoriebs da Cveneul gamokvlevebs<br />
azrovnebis fsiqologiaSi. es meTodika axalia ara<br />
mxolod saqarTvelosTvis, aramed, sazogadod, mecnierebisaTvisac<br />
(man ucxoelTa gamoxmaureba da mowonebac daimsaxura). Cveneuli<br />
saxelmZRvaneloebic mis safuZvelzea agebuli da maswavlebelmac<br />
gakveTilebi mis mixedviT unda warmarTos.<br />
es meTodika gulisxmobs: moswavlis did aqtiurobas, misi<br />
interesis gaRvivebas saxaliso da mcire SemoqmedebiTi ZiebebiT;<br />
yoveli sakiTxis Seswavlis win saTanado motivaciuri da agreTve<br />
inteleqtualuri mzaobis (ganwyobis) Seqmnas. da, rac mTavaria _<br />
swavlebas ara maswavleblis mier axsniT, aramed evristikuli<br />
meTodiT, aRmoCenebis gziT [dawvrilebiT _ ix. qvemore, § 12].<br />
maswavleblisTvis mTavaria, rom mieCvios or rames: acalos<br />
moswavleebs damoukidebeli fiqri, msjeloba da muSaoba da<br />
Seikavos gamzadebuli pasuxebi; yovel sakiTxs miudges<br />
SemoqmedebiTad da aseve moiTxovos moswavleebisaganac.<br />
yoveli sakiTxi isea Semzadebuli wina gakveTilebiT, rom mis<br />
arss moswavleebi TiTqmis TviTon aRmoaCenen, sakuTari<br />
aqtiurobis gziT. maswavlebeli unda aclides klass fiqrs,<br />
Secdomis daSvebasa da mis gaazrebas, mis gasworebas, unda waaxalisebdes<br />
moswavleTa msjelobasa da maT mier sakuTari azrebis<br />
gamoTqmas. maswavleblis mier sakiTxis axsnas, rogorc aseTs,<br />
iSviaTad mivmarTavT (da es sagangebodaa miTiTebuli gegmakonspeqtebSi).<br />
sakiTxi muSavdeba ZiriTadad mxolod SekiTxvebis<br />
meSveobiT, dialogurad, problemurad.<br />
maswavlebeli svams SekiTxvebs da cdilobs, sasurveli sruli
- 17 -<br />
pasuxi moswavleebs aTqmevinos. Secdomebic TviTon bavSvebma unda<br />
gaasworon, xarvezebi _ Seavson. maswavlebeli mxolod maSin<br />
unda Caerios, rodesac moswavleTa ZalebiT es veRar xerxdeba.<br />
maswavlebelma unda waaxalisos moswavleTa SekiTxvebi,<br />
uazro da mcdari SekiTxvac ki ar unda gakicxos!<br />
SekiTvis arc Cafarcxva-CaCumaTeba SeiZleba!<br />
SeiZleba, zogjer moswavlem sakmaod moulodneli SekiTxva<br />
dasvas. magaliTad, erT gonier pirvelklasels ukiTxavs, Tu<br />
ratom araa xuTkuTxedi iseTi mravalkuTxedi, romlis sami<br />
wvero erT monakveTzea (cxadia, SekiTxva ase zogadad ki ar<br />
dausvams, aramed erTi konkretuli daxazuli oTxkuTxedis<br />
Sesaxeb ikiTxa). aseTi SekiTxva, upirvelesad, maswavlebelma<br />
gansakuTrebiT unda Seaqos, imis miuxedavad, rom TviTon ara aqvs<br />
gonivruli pasuxi. Semdeg, naCqarevad naTqvam cud pasuxs meore<br />
dRes gacemuli gonivruli pasuxi sjobs. magaliTad: _ ase xom<br />
SeiZleboda, kidev erTi wveroc dagvesva (uCvenebs oTxkuTxedis<br />
naxazze, dafaze) da maSin veRar gavarkvevT, es nakvTi oTxkuTxedia,<br />
xuTkuTxedia, eqvskuTxedia Tu ramdenkuTxedia?! amitom aseTi<br />
wertilebi wveroebad ar iTvleba! wvero gverdis boloSi unda iyos,<br />
es ki sadaa? {gverdis SuaSi}. diax, wvero ar unda iyos gverdis<br />
SuaSi, unda iyos mxolod boloSi!<br />
imisaTvis, raTa SesaZlebeli iyos namdvilad aqtiuri swavleba,<br />
programa da saxelmZRvaneloebi axleburad unda iyos agebuli<br />
(mzaobis zogadi meTodikuri principica da misi konkretuli<br />
ganxorcielebac maTematikis swavlebaSi damuSavebulia Cven mier).<br />
ganmaviTarebeli swavleba. maTematika da mSobliuri ena<br />
imitomaa sayovelTaod aRiarebuli umTavres saskolo sagnebad,<br />
rom maTma swavlebam adamianis umniSvnelovanesi unarCvevebi unda<br />
Camoayalibos da ganaviTaros. es ki igivea, rac pirovnebis<br />
namdvili aRzrda-ganviTareba. swored esaa mTavari _ da ara<br />
sakuTriv maTematikis codna!<br />
zogadad es albaT yvelas moewoneba. magram saqme isaa, rom
- 18 -<br />
Tuki am mcnebis namdvili ganxorcieleba gvsurs, unda SeveguoT<br />
imas, rom moswavles bevri muSaoba mouwevs. unarCvevis<br />
ganviTarebis erTaderTi gza aris bevri damoukidebeli<br />
muSaoba da didi gamocdilebis dagroveba.! amas veraviTari<br />
meTodika Tu maswavleblis ostatoba ver Secvlis. oRond,<br />
cxadia, moswavlis es muSaoba saTanado Sinaarsisa da<br />
mimarTulebisa unda iyos.<br />
amitomaa Cveneul saxelmZRvaneloebSi Zalian mravlad sxvadasxvagvari<br />
amocanebi. Tanac, raime erTi gvaris amocanebi mizandasaxulad<br />
meordeba, TandaTanobiTi garTulebiT, Tanac, wlebis<br />
ganmavlobaSi. amocanaTa TviTeuli es Uboxzpcb romelime<br />
unarCvevas aviTarebs. amas ki, samwuxarod, didi dro da bevri<br />
muSaoba sWirdeba.<br />
Cveneul saxelmZRvaneloebSi mravladaa grZelpirobiani<br />
kompleqsuri amocanebi _ romelTa amosaxsnelad ramdenime<br />
sul sxvadasxva moqmedebis Catarebaa saWiro. magali-<br />
Tad: daTvaleT, gazomeT da SeavseT cxrili; daxazeT amaTuim<br />
saxis mravalkuTxedi, miniSneT misi umoklesi gverdi da masSi<br />
CaxazeT raime; daxazeT cxrili da daajgufeT asoebi; aRwereT<br />
sityvierad; Tuki aqvs, CawereT + niSani, Tuki ar aqvs _ CawereT<br />
_ niSani da ase Semdeg. TviTeuli es davaleba Zalian<br />
advilia, Tumca, mTlianobaSi, amocana sakmao Tanmimdevrulobas,<br />
yuradRebis mokrebasa da Zalisxmevas moiTxovs. magram saqme isaa,<br />
rom es kompleqsuroba Ubonjnefwsvmjb, anu Sedgeba ramdenime<br />
nabijisagan, romlebic cal-calke sruldeba. moswavlem jer<br />
mxolod pirvel nabijs unda miaqcios yuradReba, Seasrulos igi,<br />
Semdeg daiviwyos da momdevnoze gadavides, da ase Semdeg, ese igi,<br />
mas ar uwevs fsUespvmbe ramdenime ramis keTeba (rac Zalian<br />
Zneli iqneboda).<br />
arsebiTad, esaa amocanebi algoriTmis (instruqciis) Sesrulebaze.<br />
moswavles maTze sakmao droisa da Zalebis daxarjva<br />
mouwevs, Tanac, amiT maTematikis codnasac TiTqos bevri araferi
- 19 -<br />
emateba. samagierod, amgvari amocanebi saukeTesod aviTarebs<br />
unarCvevebs. amgvari amocanebi Zalian mniSvnelovania, radgan xels<br />
uwyobs moswavlis azrovnebis mowesrigebas da saSemsruleblo<br />
unarCvevaTa ganviTarebas. miT umetes, rom amgvari amocanebis<br />
instruqciaSi (algoriTmis aRweraSi) CarTulia logikuri<br />
kavSirebi da kvantorebi: Tuki, an, romelime, erTerTi,<br />
erTaderTi, yoveli...<br />
mTavaria, moswavlem gaigos, rom araa saWiro yvelaferze<br />
erTad fiqri: Seasrule jer erTi, Semdeg es daiviwye da gadadi<br />
meoreze, Semdeg esec daiviwye da gadadi mesameze... ... Tanac,<br />
rogorc yvela sxva saxis amocanebis jaWvedi, esec iwyeba jer<br />
Zalian advili, sul ornabijiani algoriTmebiT.<br />
§ 4. maTematikis mravalmxrivi da Tan<br />
gaerTianebuli (integrirebuli) swavleba<br />
Cveneuli kursi bolomde, XII klasis CaTvliT, gaerTianebulia<br />
(integrirebulia) _ maTematikis yvela sxvadasxva dargi<br />
da maTi gamoyenebani erTiani saxelmZRvaneloebiT iswavleba.<br />
amasTan, dawyebiTi klasebidanve safuZvlianad muSavdeba saymawvilo<br />
sawyisebi aramarto ariTmetikisa, algebrisa da planimetriisa,<br />
aramed agreTve: stereometriisa, mxazvelobiTi geometriisa,<br />
kombinatorikisa da simravleTa Teoriisa, miaxloebiT SefasebaTa,<br />
maTematikuri statistikisa da modelirebisa, informatikisa, topologiisa<br />
da grafTa Teoriisa, logikisa.<br />
didi yuradReba eqceva am dargTa Soris kavSirebis warmoCenas<br />
da agreTve maTematikis mravalferovan gamoyenebebs: bunebismcodneoba-biologiasa<br />
Tu geografiaSi, fizikasa Tu astronomiaSi,<br />
qimiasa Tu teqnikaSi, ekonomikasa Tu sociologiaSi, agreTve humanitarul<br />
mecnierebebSi: istoriaSi, eTnografiaSi, kulturologiaSi<br />
da xelovnebaTmcodneobaSi.<br />
yvelaze mniSvnelovania maTematikisa da logikis integracia.<br />
logikis erTi nawili Cven Caqsovili gvaqvs maTematikis<br />
programaSi, xolo meore, ufro enobrivi nawili (ritorika) _
- 20 -<br />
gramatikis programaSi (dawvrilebiT _ ix. [15]). amitom zogi<br />
sakiTxi sakuTriv maTematikuri TvalsazrisiT SeiZleba<br />
ucnauri Candes!<br />
humanisturi swavleba moiTxovs, rom didi yuradReba mieqces<br />
moswavlis Sinagan samyaros. es moicavs rogorc moswavlis kerZo<br />
pirovnul Taviseburebebs, ise zogad asakobriv kanonzomierebebs.<br />
Sinagan samyaros kidev erTi ganzomileba aqvs. esaa fspwovmj!<br />
Ubwjtfcvsfcb. eTnokulturul TaviseburebaTa gaTvaliswinebas<br />
sul ufro da ufro didi mniSvneloba eniWeba pedagogikaSi.<br />
zerele TvalsazrisiTac ki cxadia, rom maTematikisaTvis<br />
saWiro sayofacxovrebo Tu sxva magaliTebi moswavlis erovnuli<br />
kulturidan unda iyos SerCeuli. amasTan, gacilebiT ufro<br />
Rrmaa da arsebiTi sxva sakiTxi. qarTul enaSi, gansxvavebiT<br />
rusuli, evropuli da sxva mravali enisagan, ricxviTi saxelebi<br />
iwarmoeba Tvlis ocobiT-aTobiTi sistemiT. amitom pirveli<br />
klasis maTematikis is meTodika, romelic, SesaZloa, saukeTesoa<br />
evropeli bavSvisaTvis, gamousadegaria qarTvelisaTvis. Cvens mier<br />
gamoyenebuli meTodika iTvaliswinebs swored qarTulenovani<br />
cnobierebis Taviseburebas (amis Sesaxeb dawvrilebiT ix. [17:$3]).<br />
yvela am friad saWiro da saintereso sakiTxis Camateba<br />
Cveulebrivi saskolo maTematikis isedac gadatvirTul kursSi<br />
yovlad SeuZlebelia. amitom, Tuki namdvilad gvsurs kursis<br />
gamdidreba statistikis, logikis, gamoyenebiTi maTematikis sawyisebiT<br />
da misi galamazeba humanitaruli waxnagebiT _<br />
aucilebelia mkveTrad Semcirdes sxvadasxva wvrilmani teqnikurmanipulaciuri<br />
sakiTxebi, magaliTad, algebruli da trigonometriuli<br />
gardaqmnebi, formulebisa da specialuri terminebis<br />
grZel-grZeli xlarTebi _ rac Cveulebrivi saskolo maTematikis<br />
ZiriTadi nawilia.<br />
uaRresad mniSvnelovania statistika. didi xania, mTelma ganvi-<br />
Tarebulma msofliom gaacnobiera, rom saSualo aramaTematikosisTvis<br />
maTematikidan yvelaze saWiroa swored statistika
- 21 -<br />
(cxadia, ubralo ariTmetikis Semdeg). statistikis mraval<br />
sakiTxs Seicavs zogadi unarebis yvela testi da TiTqmis<br />
yoveldRiurad gazeTebSic ki qveyndeba statistikuri monacemebi.<br />
statistikis sawyisebis codnis gareSe adamiani ver iqneba<br />
demokratiuli saxelmwifos srulfasovani moqalaqe, radgan ver<br />
gaiazrebs arCevnebis procedurasa da Sedegebs. didi mniSvneloba<br />
aqvs albaTur-statistikur codnas sazogadoebrivi movlenebis<br />
marTebulad Sesafaseblad, marTebuli daskvnebis gamosatanad.<br />
adamiani, rogorc wesi, mcdarad, egocentrulad afasebs<br />
sazogadoebriv movlenasa Tu xalxis ganwyobas, radganac,<br />
misdauneburad, mxolod sakuTari garemocva aqvs mxedvelobaSi.<br />
magaliTad, esa Tu is politikuri partia darwmunebulia, rom<br />
arCevnebSi 5%-ian zRurbls gadalaxavs da varaudobs xmebis<br />
10%-is mogrovebas, Tumca, sinamdvileSi, xmebis 0,5%-sac ki ver<br />
agrovebs. statistikuri azrovnebis gareSe SeuZlebelia<br />
marTebuli daskvnis gamotana quCaSi miRebuli raime<br />
STabeWdilebidan, eqstrasensuli movlenebis garCevidan da sxva.<br />
cxadia, statistika yvelaze bunebrivad da Sinaarsianad gaer-<br />
Tianebul kursSi Caismeba. Zvelebur algebra-geometriaSi misi<br />
Casma arabunebrivia. igive iTqmis simravleTa Teoriis Sesaxebac.<br />
maTematika _ garesamyaros Semecnebis (ufro zustad ki _<br />
modelirebis) mZlavri saSualebaa. maTematika saWiroa saskolo<br />
sabunebismetyvelo Tu humanitaruli sagnebis Tanamedrove doneze<br />
Sesaswavlad; samyaros, msoflios mecnieruli xedvis Camosayalibeblad.<br />
gasaSualoebis, sixSirisa da albaTobis umartivesi<br />
cnebebis gareSe adamians gauWirdeba im cxrilebSi da diagramebSi<br />
garkveva, romlebic yoveldRiur gazeTebSic ki SeiZleba<br />
Segvxvdes. statistika uaRresad saWiroa gamoyenebiTi TvalsazrisiT:<br />
rogorc biologia-medicinaSi, aseve ekonomikaSi, sociologiaSi,<br />
demografiasa Tu fsiqologiaSi. Cveulebriv Jurnalistsac<br />
ki sakmaod sWirdeba statistikis sawyisebi.<br />
cxadia, rom Tanamedrove informaciul xanaSi zogadsaganma-
- 22 -<br />
naTleblo skolis moswavleTa didi umravlesobisaTvis informaciis<br />
daxarisxebisa da damuSavebis unarCvevebis kargi ganviTareba<br />
gacilebiT ufro mniSvnelovania, vidre mTeli algebris, trigonometriisa<br />
da maTematikuri analizis codna _ anu imisa, rac<br />
tradiciuli saskolo maTematikis albaT 80%-s Seadgens. es ukve<br />
TiTqmis mTelma msofliom gaiazra da ganaxorciela kidec.<br />
magaliTad, j. bruners [23] miaCnia, rom ganzogadebuli maTematikuri<br />
cnebebidan skolaSi saswavleblad yvelaze mniSvnelovania<br />
sami _ ricxvi, zoma da albaToba.<br />
meore aranakleb saWiro dargia logika. mas VI-VII klasamde<br />
TiTqmis ar sWirdeba specialuri terminebi da Teoria. Tumca II<br />
klasidanve unda daiwyos Zlieri mizanmimarTuli muSaoba marTebuli<br />
logikuri daskvnebis gamotanis unarebis ganviTarebisaTvis.<br />
ZiriTadi logikuri cnebebi yvelgan Tan sdevs maTematikas.<br />
amitom sakmaod gavrcelebulia azri, rom logikis sagangebo<br />
swavleba araa saWiro, radganac igi TavisTavadac iswavleba<br />
maTematikis swavlebasTan erTad. magram es azri _ Zalian<br />
mcdaria. rogorc gviCvenebs sagangebo gamokvlevebi, sazogadod<br />
maTematikis Seswavla araa sakmarisi Tundac imisaTvis, rom<br />
axalgazrdas SeeZlos umartives geometriul Tu ariTmetikul<br />
cnebaTa Soris kerZooba-zogadobis mimarTebaTa garkveva. ufro<br />
rTul logikur msjelobaze xom laparakic zedmetia. logikis<br />
sakiTxebs sworedac rom sagangebo swavleba sWirdeba, riTac<br />
arsebiTad amaRldeba moswavlis maTematikuri codnac da ufro<br />
metad ki _ misi zogadi azrovnebis done.<br />
mesame dargia kombinatorika (simravleTa TeoriasTan kavSirSi).<br />
misi sawyisebis gareSe SeuZleblia mravali sayofacxovrebo da<br />
saxaliso amocanis amoxsna, da rac mTavaria, SeuZleblia ufros<br />
klasebSi albaTobis Teoriisa da maTematikuri statistikis sawyisebis<br />
swavla.<br />
simravleTa Teoria kargad asuraTebs da aTvalsaCinoebs logikas,<br />
meores mxriv moswavlis mier maTi SeTviseba myari safuZve-
- 23 -<br />
lia sazogadod klasifikaciuri azrovnebis ganviTarebisaTvis,<br />
rac yovelgvari mecnierebis umTavresi xerxemalia (Sedarebebi,<br />
xilul siWrelesa da mravalferovnebaSi movlenaTa Tu saganTa<br />
ganTavseba jgufebSi, TviTeulisaTvis damaxasiaTebel TvisebaTa<br />
ganzogadoeba, zogadisa da kerZos dialeqtika, da sxva). logikisaTvis<br />
kargi sarbieli da sasuraTebulia agreTve geometriac.<br />
integraciis principi moiTxovs agreTve, rom sxvadasxva sagnebis<br />
programebi erToblivad iyos gaazrebuli, erTmaneTTan SeTanxmebulad<br />
da urTierTSewonilad. dauSvebelia, magaliTad, amJamad<br />
CvenSi arsebuli viTareba: dawyebiTi klasebis bunebismcodneobis<br />
programa moicavs masStabis cnebas, maSin rodesac maTematikaSi<br />
moswavleebs naswavli ara aqvT arc wiladebi da arc proporcia<br />
(amgvari magaliTebi sxvac mravladaa). amiT darRveulia araTu integraciis<br />
principi, aramed ubralod saRi azric ki.<br />
`saymawvilo maTematikaSi~ safuZvlianadaa damuSavebuli is gamoyenebiTi<br />
sakiTxebi, romlebic saskolo sagnebSia Zalian mniSvnelovani:<br />
proporcia, masStabi, geografiuli koordinatebi, welTaRricxva,<br />
ritmi, elifsi da sxva.<br />
`saymawvilo maTematika~, miuxedavad esoden didi Sinaarseuli<br />
nairgvarobisa, mainc Sinaganad erTiania. es siRrmiseuli erTianoba<br />
saerTo maTematikuri safuZvliTaa ganpirobebuli _ esaa<br />
mphjlb!eb!tjnsbwmfUb!Ufpsjb/ garda amisa, sxvadasxva dargebi<br />
erTmaneTTan kavSirdeba damuSavebisa da gadmocemis erTnairi xerxebiT,<br />
erTiani zogadi suliskveTebiTa da agreTve mravali darg-<br />
TaSorisi sakiTxiT: rogorc TeoriulebiT, aseve amocanebiT.<br />
amgvarad agebul kurss kidev erTi didi upiratesoba aqvs: mas-<br />
Si gacilebiT naklebadaa specialur-teqnikuri da sqolastikuri<br />
sakiTxebi, igi gamoyenebiTi amocanebiTaa gajerebuli. amitom moswavles<br />
ar uCndeba bunebrivi ukuqmedeba: _ raSi mWirdeba es yovelive,<br />
ratom unda vicode es? winaT, mbrZaneblur skolaSi,<br />
daTrgunul moswavles amgvari SekiTxvebi akrZaluli hqonda. magram<br />
droa, rom miveCvioT humanistur pedagogikas da moswavlis
- 24 -<br />
pirovnebis pativiscemas. moswavles aqvs sakuTari azris qonis<br />
ufleba, Secdomis uflebac aqvs. moswavle darwmunebuli unda<br />
iyos, rom mas marTlac saWiro sakiTxebs aswavlian.<br />
§ 5. wakiTxulis gaazrebis unari<br />
da `saymawvilo maTematikis~ nakli;<br />
visTvisaa upriani Cveneuli meTodikiT swavla<br />
`saymawvilo maTematika~ araviTar SemTxvevaSi araa mxolod<br />
Zlieri moswavleebisaTvis. piriqiT, Cveneul saxelmZRvaneloTa<br />
erTerT umTavares Rirsebad is migvaCnia, rom ufros klasebSic<br />
ki moswavleTa saSualod 60-70 % namdvilad Zlevs maTematikis<br />
saprogramo minimums, ganviTarebuli aqvs gonebrivi unarebi da<br />
patiosani damoukidebeli muSaobis unarCvevebi, ar ejavreba<br />
maTematika (tradiciuli meTodikiT momuSave tipuri skolebis<br />
klasebSi amgvar ufrosklaselTa wili 5-15 %-s Tu aRwevs _<br />
nacvlad Cveni 60-70 %-isa).<br />
`saymawvilo maTematika~ savsebiT gasagebia da misawvdomia<br />
Cveulebrivi saSualo moswavlisaTvis. moswavle mas damoukideblad<br />
kiTxulobs da esmis teqstis Sinaarsi.<br />
amqveynad unaklo araferia. cxadia, `saymawvilo maTematikasac~<br />
aqvs nakli. esaa is, rom swavleba arsebiTadaa damyarebuli werakiTxvaze,<br />
gansakuTrebiT _ teqstis wakiTxvisa da gaazrebis<br />
unarze. erTi mxriv, es kargia _ Cveni meTodikiT momuSave<br />
moswavleebs saukeTesod uviTardebaT es umniSvnelovanesi unari<br />
(romelic, sxvaTa Soris, erTerTi umTavresia zogadi unarebis<br />
yvela saxis testSi). magram, meore mxriv, did siZneleebs<br />
gviqmnis im bavSvebTan, romlebsac ar SeuZliaT kargad kiTxva.<br />
saqme isaa, rom kiTxvis unaris Sinaganad ganpirobebuli<br />
daqveiTeba SeiZleba aRmoaCndes bavSvebis daaxloebiT 10-15 %-s.<br />
maTgan zogierTi gonebrivadac CamorCenilia, magram zogi<br />
gonebrivad saSualo an saSualoze maRali ganviTarebisac kia.<br />
kiTxva ki mainc Zalian uWirT. garda amisa, mravalia iseTi
- 25 -<br />
moswavle, romelsac kiTxva uWirs ara Sinagan mizezTa gamo,<br />
aramed kiTxvis cudad swavlebis gamo. da kidev: ukanasknel<br />
xanebSi CvenSic SemoiWra wera-kiTxvisa da, sazogadod, enobrivgonebrivi<br />
unarebis dauZinebeli mteri _ kompiuterisa Tu<br />
televizoris ekrani (ix. [31]). yovelive amis gamo mravladaa<br />
iseTi klasi, romelSic moswavleTa 30-40 % ver kiTxulobs<br />
wesierad. wakiTxulis gaazrebaze xom laparakic zedmetia.<br />
maSasadame, imisaTvis, raTa moswavlem SeZlos `saymawvilo<br />
maTematikiT~ swavla, saWiroa Semdegi:<br />
1) normaluri gonebrivi SesaZlebloba (es Tanabrad exeba<br />
yvela sxva meTodikasac, radgan normalurTan SedarebiT<br />
gonebrivad CamorCenili moswavle Cveulebriv saskolo<br />
maTematikas vercerTi saxelmZRvaneloTi ver iswavlis);<br />
2) normaluri nebelobiTi SesaZleblobani: yuradRebis<br />
mokrebis unari, nebisyofa, sibejiTe, moTmineba (esec yvela sxva<br />
meTodikas exeba, met-naklebad);<br />
3) normalurad ganviTarebuli unarebi wera-kiTxvisa da<br />
wakiTxuli teqstis gaazrebisa (es ki sxva meTodikebs albaT<br />
naklebad exeba da CamorCena naklebad azianebs, radgan maTSi<br />
gaTvaliswinebulia maswavleblis mier axali sakiTxebis axsna).<br />
mecnierebaSi Seswavlilia, rom wakiTxvis unaris Sinagani<br />
CamorCena lbopo{pnjfsbe!!bsbb!!eblbwTjsfcvmj!!tbfsUp!!!<br />
hpofcsjw! DbnpsDfojmpcbtUbo; magram xSirad sakmaod eblbw.<br />
Tjsfcvmjb!!ofcfmpcjUj!!vobsfcjt ganviTarebis CamorCenasTan.<br />
ese igi, me-2 da me-3 xSirad erTmaneTs ukavSirdeba da erTian<br />
CamorCenilobas qmnis.<br />
nebelobiTi monacemebi ufro mniSvnelovania, vidre gonebrivi.<br />
Zlieri moswavle iqneba is, romelic orive am mxrivaa Zlieri.<br />
mxolod gonebis gamWriaxoba, mixvedriloba da kargi amTvisebloba<br />
araa sakmarisi! es yovelive unda ganvumartoT mSoblebsac.<br />
yvelam gaakeTos imdeni amocana, ramdensac SeZlebs zomieri<br />
ZalisxmeviT, didi daZabvisa da wvalebis gareSe. Semdeg ki
- 26 -<br />
xSirad moxdeba, rom maswavlebelica da mSobelic, maTda gasakvirad,<br />
aRmoaCenen, rom saSualo moswavlesac arc ise gauWirdebodes<br />
iqneba erTi SexedviT Zneli da didi davalebis Sesruleba...<br />
roca gonebrivad normalur moswavles uWirs `saymawvilo ma-<br />
TematikiT~ swavla, ver igebs mas da ver xsnis amocanebs _ amis<br />
mizezi, rogorc wesi, erTia: am moswavles Sinagan an gare<br />
mizezTa gamo daqveiTebuli aqvs teqstis wakiTxvisa da misi<br />
gaazrebis unarebi, moswavle ver kiTxulobs. amgvari SemTxvevebi<br />
gacilebiT xSiria im klasebSi, romlebsac wina wlebSi Cveneuli<br />
saxelmZRvaneloebiT ar uswavliaT da pirvelad iwyeben `saymawvilo<br />
maTematikas~.<br />
ra vqnaT? jobs gvian, vidre arasdroso. V klasi iqneba Tu<br />
IX, maswavlebelma pirvel rigSi unda Seamowmos moswavleTa kiTxvis<br />
unari: gadaaSlevinos maT saxelmZRvanelo romelime axal<br />
gakveTilze da daavalos teqstis xmamaRla wakiTxva. uceb gamoCndeba,<br />
romel moswavleebs uWirT kiTxva (umjobesia, SeirCes iseTi<br />
teqsti, romelSic mravaldaa rTuli qvewyobili winadadebebi).<br />
meore safexuria sityvis amokiTxvis sizusteze muSaoba _<br />
yvela aso zustad da mkafiod unda warmoiTqvas. moswavleTa (da<br />
mozrdilTa) umravlesobas erTmaneTSi eSleba msgavsi bgeriTi Semadgenlobis<br />
mqone sityvebi, magaliTad: erTerTi \ erTaderTi,<br />
marTebuli \ marTobuli, gamarTuli \ gamarTlebuli,<br />
gamkeTebeli \ gakeTebuli, miuwvevia \ miuRwevia...<br />
Tuki sityvebi zustad araa amokiTxuli _ romel azrovnebaze,<br />
logikaze da maTematikazea saubari?<br />
mesame safexuria sasveni niSnebis amokiTxva. maT TiTqmis sul<br />
ugulebelyofen rogorc kiTxvis, ise weris dros. gansakuTrebiT<br />
iCagreba mZime, orwertili da tire _ swored is sasveni niSnebi,<br />
romlebic gadamwyvetia azris logikisaTvis. amitom moswavle<br />
unda mieCvios teqstis wakiTxvisas sasveni niSnebis gacnobierebas,<br />
rac kiTxvisas mcire SeCerebebiT (azrobrivi pauzebiT) unda<br />
gamovlindes. sasveni niSnebi teqstSi tyuilad ar weria, maT
- 27 -<br />
jerovani yuradRebis miqceva sWirdeba!<br />
meoTxe safexuri yvelaze mniSvnelovania _ wakiTxulis<br />
gaazreba. es teqstis wakiTxvisas marTebuli azrobrivi<br />
aqcentebiT unda gamovilindes. arafrad ar varda Tundac zusti,<br />
magram meqanikur-monotonuri wakiTxva. mxatvruli teqstebis<br />
wakiTxvisas ZiriTadad mxatvrul-grZnobiTi aqcentebia saWiro,<br />
samecniero teqstebis wakiTxvisas ki _ logikur-azrobrivi<br />
aqcentebi. maxvili unda daesves im sityvas, romelic yvelaze<br />
mniSvnelovani da arsebiTia logikuri Sinaarsis TvalsazrisiT.<br />
es oTxi safexuri unda damuSavdes ara mxolod teqstis wakiTxvisaTvis,<br />
aramed agreTve zepiri metyvelebisaTvis da<br />
werisaTvis: sisrule, sizuste, sasveni niSnebi da azri. maTematikis<br />
gakveTilebze es dafasTan zepiri msjelobisas da sakontrolo<br />
weris samsjelo amocanebis amoxsnis dros unda damuSavdes.<br />
araa sakmarisi amocanis mxolod amoxsna anda Teoremis<br />
damtkiceba _ saWiroa kargad gamarTuli msjelobac. amaze<br />
maswavlebelma sagangebod unda imuSaos _ ese igi, unda<br />
Seasworos, Caasworos, Seavsos, gamarTos, daxvewos moswavlis<br />
zepiri Tu werilobiTi teqstebi. magram mxolod Sesworeba da<br />
Secdomis gacnobiereba araa sakmarisi _ moswavlem aucilebelad<br />
Tavidan unda gaimeoros (zepirad Camoayalibos an gadaweros)<br />
Sesworebul-daxvewili sruli teqsti.<br />
swored amgvari muSaobiT viTardeba gonebrivi unarebi da<br />
azrovneba. swored esaa mTavari _ da ara leqsisa Tu<br />
gramatikuli wesis gazepireba, formulis damaxsovreba,<br />
damtkicebis damaxsovreba da meore dRes moyola da gamoTvlagardaqmnebis<br />
sxapasxupiT keTeba. mTavaria ara programis gavla,<br />
aramed moswavlis gonebis gamdidreba da ganviTareba, misi<br />
SemZleobis, xelwifebis amaRleba.<br />
amgvari savarjiSoebisaTvis araa sakmarisi mxolod qarTuli<br />
literaturis teqstebi, moswavle abstraqtul-specialuri teqstebis<br />
wakiTxva-gaazrebasac da Seqmnasac unda mieCvios. amitom
- 28 -<br />
swored maTematikis maswavlebelma swored maTematikis saxelmZ-<br />
Rvanelo unda gamoiyenos. `saymawvilo maTematikaSi~ mravladaa<br />
mozrdili teqstebi mkafio Sinagani logikuri ganviTarebiT (da<br />
Tan kargi qarTuliT dawerili), mravladaa amocanebi, romlebSic<br />
msjelobis gamarTva da maTematikuri teqstis Seqmnaa saWiro, zogjer<br />
TviT axali amocanis mogonebisa da Camoyalibebis CaTvliT.<br />
wakiTxulis gaazrebis savarjiSoebis keTebisas moswavles<br />
Tvalwin hqondes saxelmZRvanelos teqsti, maswavlebelma ki dausvas<br />
SekiTxvebi am teqstidan, raTa gairkves, ramdenad gaigo<br />
moswavlem wakiTxulis Sinaarsi (SekiTxvebi unda daisves ara<br />
mxolod pirdapir, aramed arapirdapirac). cxadia, maswavlebelma<br />
unda moiTxovos zusti da gamarTuli, sruli pasuxebi.<br />
Semdeg, maswavlebelma unda Caataros amocanis pirobis gaazrebis<br />
savarjiSoebi (romlebic Cveni programiT II-III klasebSia).<br />
moswavlem waikiTxos romelime didteqstiani axali amocana<br />
(romlis amoxsna sul ar gvainteresebs!) da Semdeg Camoayalibos<br />
cal-calke: ra aris mocemuli am amocanis pirobiT da ras<br />
gvekiTxeba an ras gvavalebs amocana. momdevno safexurze ki es<br />
ukve werilobiT unda Camoyalibdes.<br />
amis Semdeg mravali maswavlebeli Tavad darwmundeba, Tu<br />
raoden didi xarvezebia mis moswavleTa unarebis ganviTarebaSi.<br />
maTematikis programaSi winsvla, cxadia, Seneldeba. magram ma-<br />
Tematikaze gacilebiT ufro mniSvnelovania wakiTxvisa da gaazrebis<br />
unarTa ganviTareba, rac mTeli wignierebis safuZvelia da<br />
ris gareSec moswavle ver Caabarebs, kerZod, zogad unarTa<br />
tests [11]. saTanado saklaso savarjiSoebi Tavidan yoveldRiurad<br />
unda Catardes. Zalian sasurvelia, rom qarTulis maswavlebelmac<br />
mohkidos am saqmes xeli. Semdeg ki maswavlebeli<br />
Tavad Seafasebs, Tu ramdenad SeiZleba am samuSaoTa gaiSviaTeba.<br />
zogierT moswavles TiTqmis ar Seetyoba gaumjobeseba. amgvar<br />
moswavles, albaT, Sinagan mizezTa gamo uWirs kiTxva. riT<br />
SeiZleba maTi Svela _ es jerjerobiT msoflio mecnierebaSic ki
- 29 -<br />
gadauWreli SesaWiria. xolo danarCen moswavleebs met-naklebi<br />
tempiT da met-naklebi xarisxiT mainc ganuviTardebaT is unari,<br />
romelic mTeli saskolo swavlebisaTvis erTerTi umniSvnelovanesia<br />
_ unari teqstis wakiTxvisa da misi gaazrebisa.<br />
amis Semdeg ki `saymawvilo maTematika~ TviTon iqneba<br />
saukeTeso saSualeba am unaris Semdgomi Seunelebeli ganviTarebisa!<br />
marTlac, Cvens mier Seqmnilia maTematikuri teqstis<br />
gaazrebis sagangebo amocanis tipi. amgvari amocana axlavs<br />
TiTqmis yvela gakveTils nulovani nomriT. es amocanebi<br />
moswavles aiZulebs, rom yuradRebiT waikiTxos gakveTilis<br />
teqsti da wakiTxva-gaazrebis unarsac aviTarebs.<br />
§ 6. maTematikis namdvili codna<br />
saWiroa imis mkafiod Camoyalibeba, Tu, fsiqologiuri Tval-<br />
sazrisiT, ra aris maTematikuri sakiTxis namdvili codna. erTi<br />
mxriv, es araviTar SemTxvevaSi ar daiyvneba imaze, rom moswavles<br />
SeuZlia am sakiTxis moyola (Cveni meTodikiT moswavlis<br />
gamoZaxeba gakveTilis mosayolad xom arc xdeba!). maTematika<br />
araa istoria an moTxroba, rom misi codna moyoliT SevamowmoT.<br />
maTematikuri sakiTxis namdvili codna imas niSnavs, rom<br />
moswavles xelewifeba am sakiTxis Tavisuflad da aqtiurad<br />
gamoyeneba saTanado amocanebis amoxsnisas. codnis<br />
kriteriumi _ esaa misi gamoyenebadoba anu is, Tu es codna<br />
ramdenad dayalibda aqtiur gonebriv unarCvevad. magram<br />
saqme isaa, rom am dros moswavles, SesaZloa, ar ZaluZdes<br />
sakuTari codnis sityvierad gamarTulad Camoyalibeba!<br />
Semdeg, mTavaria tblwbo[p!dofcfcjtb da maT Soris njnbsUf.<br />
cbUb! kargad gaazreba _ da ara manipulaciebSi (gardaqmnebSi,<br />
gamoTvlebSi da sxva) gawafva.<br />
sxvaTa Soris, Cveneul gamocdebzec ZiriTadad mowmdeba<br />
naswavli sakvanZo sakiTxebis gaazrebis xarisxi da maTi aqtiurad<br />
gamoyenebis unari. Tuki moswavle kargad iyenebs raime cnebasa Tu
- 30 -<br />
Teoremas nairgvari (maT Soris arastandartuli) amocanebis<br />
amoxsnisas _ ese igi, mas kargad scodnia es sakiTxi da am<br />
codnis calke Semowmeba aRaraa saWiro.<br />
sakontrolo werebzec da gamocdebzec moswavles vaZlevT<br />
saxelmZRvanelosa Tu cnobaris gamoyenebis uflebas.<br />
maSasadame, SeiZleba moxdes, rom moswavlem, arsebiTad, ar<br />
icodes maTematikuri sakiTxi, magram sityvierad unaklod ayalibebdes<br />
mas _ Tuki es sakiTxi wina dRiT aqvs wakiTxuli da sityvierad<br />
damaxsovrebuli anu gazepirebuli (didi xnis Semdeg es,<br />
cxadia, SeuZlebeli iqneba). meore mxriv ki, SeiZleba moxdes,<br />
rom moswavlem saukeTesod icodes sakiTxi, magram sityvierad<br />
ver ayalibebdes mas. sazogadod, Teoriuli sakiTxis gamarTuli<br />
sityvieri Camoyalibeba sakmaod Zneli amocanaa. am unaris<br />
ganviTarebas Cveni meTodika sakmao yuradRebas aqcevs, magram<br />
maswavlebels gaazrebuli unda hqondes, rom es calke unaria da<br />
mas ara aqvs uSualo kavSiri Sesabamisi maTematikuri sakiTxebis<br />
namdvil codna-arcodnasTan. Cveneuli meTodika (saSinao<br />
davalebis Semamzadebeli amocanebiT da Semdeg saklaso aqtiuri<br />
garCeviT) sakiTxs jer kargad gaaazrebinebs moswavles, da<br />
mxolod amis Semdeg dgeba sakiTxi zepiri Tu werilobiTi<br />
sityvieri Camoyalibebisa da msjelobis unaris ganviTarebisa _<br />
rac calke samuSaoa.<br />
ese igi, SesaZloa, moswavlem ukve kargad icis ganvlili gakveTilis<br />
sakiTxebi (rac imaSi vlindeba, rom am sakiTxebs kargad<br />
iyenebs saTanado amocanebis amoxsnisas!), magram sityvierad ver<br />
ayalibebdes mas gamarTulad. swored amisaTvisaa saWiro gakveTilis<br />
morCenili dro. maswavlebeli ekiTxeba moswavleebs ganvlili<br />
gakveTilis Teoriuli teqstis Sesaxeb, Tu ra azri gamoitanes<br />
maT am teqstis pirveli abzacidan, meore abzacidan, da ase<br />
Semdeg _ moswavleebi pasuxoben adgilidan, maswavlebeli iTxovs<br />
winadadebis srulad da zustad Camoyalibebas, ameorebinebs am<br />
winadadebas, sanam moswavle kargad ar gamarTavs mas. Tuki
- 31 -<br />
moswavlem daaklo raime sityva, am xarvezs sxva moswavleebs<br />
gamoavleninebs. magaliTad, moswavlem daaklo sityva `yoveli~, an<br />
`mxolod~ an `erTaderTi~ _ maswavlebeli sxva moswavleebs<br />
daavalebs, rom moigonon iseTi magaliTi, romelic daamtkicebs,<br />
rom am sityvebis dakleba ar SeiZleboda, radgan am sityvebis<br />
gareSe winadadeba mcdari xdeba, da ase Semdeg. momdevno klasebSi<br />
emateba kidev erTi saxis samuSaoc: ganvilili gakveTilis raime<br />
sakiTxis damtkiceba: moswavle gamodis dafasTan da msjelobs,<br />
maswavlebeli iTxovs unaklod gamarTul winadadebebs.<br />
amgvari samsjelo muSaobis dros moswavleebs ufleba aqvT,<br />
gadaSlili hqondeT TavianTi saxelmZRvanelo da gadaxedon<br />
xolme saTanado abzacs. samuSaos mizania ara gazepireba, aramed<br />
zusti da bolomde gamarTuli sityvieri Camoyalibebis, msjelobisa<br />
da dasabuTebis unaris ganviTareba.<br />
zepirobisas verc es unari viTardeba da verc arsebiTi maTematikis<br />
Seswavla xdeba. esaa tradiciuli meTodikis erTerTi um-<br />
Tavresi mZime naklovaneba (sxva ramdenimes Soris).<br />
`wesebis~ gazepirebaze sabolood unda vTqvaT uari!<br />
§ 9. gakveTilis ageba<br />
tradiciuli meTodikis mixedviT gakveTilis sami mTavari Semadgenelia:<br />
I. saSinao davalebis Semowmeba;<br />
II. erTi-ori moswavlis gaZaxeba dafasTan gakveTilis mosayolad;<br />
III. axali gakveTilis axsna maswavleblis mier.<br />
amasTan, miiCneven, rom mTavaria mesame Semadgeneli. magaliTad,<br />
gakveTilze myofi damswreebi yvelaze met yuradRebas swored<br />
imas aqceven, Tu rogor axsna maswavlebelma axali sakiTxebi.<br />
Cveneul gakveTilze ki araviTari axali masalis axsna ar xdeba!<br />
Cveneuli meTodika ar moicavs tradiciuli meTodikis arc<br />
meore da arc mesame Semadgenlebs: arc moswavlis dafasTan<br />
gaZaxeba misi codnis Sesamowmeblad da gakveTilis moyola xdeba
- 32 -<br />
(Tumca, amis magvari da amis Semcvleli samuSao mainc tardeba,<br />
oRond mas ufro mcire dro eTmoba); arc axali masalis axsna<br />
xdeba (es sruliad gamoricxulia, misi aranairi Semcvleli Tu<br />
saxesxvaoba ar unda iyos)! samagierod, pirvel Semadgenels<br />
gacilebiT meti yuradReba eqceva. magram amisTvis mTavaria is,<br />
rom TviTon saSinao davalebis amocanebis raoba da Sinaarsi<br />
arsebiTad gansxvavebulia tradiciulisagan, agebulia mTlianad<br />
aqtiur-SemoqmedebiT, ZiebiT-evristikul swavlebaze (es, cxadia,<br />
avtoris saqmea).<br />
Cveni zogadi meTodikis mixedviT, gakveTilze mTavaria kargad<br />
dasmul TfljUywbUb!Uboxzpcb. mxolod amgvarad, aqtiurad, Seiswavleba<br />
sakiTxi. Cven maswavlebels mTlianad vaTavisuflebT me-<br />
Todistis samuSaoebisagan da meTodikuri Semoqmedebisagan; samagierod,<br />
metad vTxovT mas SemoqmedebiT midgomas TviTeuli moswavlisadmi,<br />
individualurad. maswavlebeli karg msaxiobs unda hgavdes<br />
_ da ara dramaturgs. msaxiobs piesa gamzadebuli eZleva,<br />
misi xelovneba da Semoqmedeba piesis Seqmna ki ar aris, aramed<br />
misi gacocxleba da TviTeuli mayureblis gulamde SeRweva.<br />
akrZaluli unda iyos rogorc pasuxis wamoZaxeba, ise<br />
jgufuri, gunduri pasuxebic. vinc wamoiZaxebs an Zalian xmaurobs,<br />
mas maswavlebelma ar unda aTqmevinos. maswavleblis mier<br />
dasmul SekiTxvaze moswavleebma xeli unda awion. pasuxi ki mxolod<br />
erTma unda gasces, mere, SesaZloa, meorem, mere _ mesamemac.<br />
magram rig-rigobiT _ da ara erTad. gundur pasuxebs azri ara<br />
aqvs! Cveneuli meTodika moiTxovs, rom SekiTxva individualurad<br />
daesves moswavles (misi saxelis _ da ara gvaris dasaxelebiT!).<br />
Tuki is ver upasuxebs kargad, SekiTxva daesmis sxva moswavles.<br />
pirvelma moswavlem ki Semdeg unda gaimeoros marTebuli<br />
naTqvami. xolo roca SekiTxva daesmis mTel klass, maSin<br />
moswavleebma unda awion xeli da pasuxs ityvis mxolod is,<br />
visac maswavlebeli miscems sityvas. garda amisa, ufro xSirad,<br />
moswavle pasuxobs adgilidan, fexze audgomlad.<br />
saqme isaa, rom dasmul SekiTxvaze mosalodneli pasuxi
- 33 -<br />
moklea. am dros umjobesia, rom moswavlem upasuxos adgilidan,<br />
swrafad, fexze adgomlad. miT umetes, roca pasuxoben sakuTari<br />
rveulis an saxelmZRvanelos mixedviT da mas Tvali ar unda<br />
moswyviton. Tanac, adgoma-dajdomas didi dro miaqvs da xmaurs<br />
iwvevs, es Zalian arRvevs gakveTilis ritms. pasuxisas moswavle<br />
fexze unda adges mxolod maSin, roca grZeli msjeloba aqvs<br />
warmosaTqmeli _ rac iSviaTad xdeba!<br />
sxvaTa Soris, Tuki maswavlebeli SeZlebs, rom TviTonac<br />
ijdes xolme da fexze audgomlad SeinarCunos klasSi wesrigi<br />
_ es Zalian kargi iqneba. rac ufro metad daemsgavseba gakveTili<br />
saqmian, samuSao saubars, miT ukeTesia!<br />
davuSvaT, ramdenime moswavlem awia xeli. maswavlebelma erTerTs<br />
unda aTqmevinos. romels? aq gasaTvaliswinebelia ori ram:<br />
erTi is, rom SekiTxvebze pasuxebi daaxloebiT Tanabrad iyos<br />
xolme moswavleebze ganawilebuli, ar unda moxdes ise, rom<br />
mxolod aqtiuri moswavleebi pasuxobdnen. meorec, gasaTvaliswinebelia<br />
SekiTxvis siZnele. Tuki sakiTxi axalia, Tuki SekiTxva<br />
evristikul Ziebas emsaxureba, maSin upiratesoba ufro Zlier<br />
moswavleebs unda mivakuTvnoT. sustebi albaT isedac ar aiweven<br />
xels. magram am Zneli sakiTxis garkvevis Semdeg maswavlebelma<br />
aucileblad kidev erTxel unda hkiTxos swored sust an pasiur<br />
moswavles. maswavlebeli araviTar SemTxvevaSi ar unda dakmayofildes<br />
erTi Zlieri moswavlis pasuxiT. odnav Secvlili saxiT<br />
igive SekiTxva unda uSualod dausvas pasiur Tu sust moswavles:<br />
rati, aba axla Sen miTxari, ratomaa wiTlebis wili naklebi,<br />
vidre TeTrebisa? ara, Tqven ar gekiTxebiT, me ratis vkiTxe! acaleT,<br />
ratim icis da axla gvetyvis. aba, rati, rogor agvixsna zurikom? ...<br />
xolo Tuki sakiTxi advilia, anda mravaljer Tqmulis morigi<br />
gameorebaa, maSin pasuxi, piriqiT, ufro sust moswavleebs unda<br />
vaTqmevinoT.<br />
axla, vTqvaT, moswavlem gvipasuxa. maswavlebelma moiTxova<br />
dasabuTeba: `saidan ici?~, anda `ratom?~ moswavlem axsna, magram<br />
SeeSala. an kidev: xSirad xdeba, rom moswavlis pasuxi
- 34 -<br />
sanaxevrod an TiTqmis marTebuli kia, magram mainc raRac aklia.<br />
am SemTxvevaSic unda vaTqmevinoT sxvas, Zlier moswavles, oRond<br />
Semdeg marTebuli pasuxi aucileblad unda gavameorebinoT mas,<br />
visac SeeSala. Semdeg unda vkiTxoT kidev erTs (magaliTad,<br />
SedarebiT pasiur an sust moswavles), oRond cota sxvagvarad,<br />
magaliTad, ase: manana, axla Sen mipasuxe, amaTgan romlis wilia<br />
meti? ratom? romeli aTwiladia meti? ratom? aba, daxaze wriuli<br />
diagrama da TvalsaCinod gviCvene!<br />
jobia Tanaklaselma axsnas _ da ara maswavlebelma!<br />
Cven arsad ar vwerT, magram yvelgan vgulisxmobT, rom roca<br />
moswavles raimes pasuxi SeeSleba an dafasTan gamosuli<br />
moswavle mTlad kargad ver Seasrulebs davalebas, maswavlebeli<br />
ikiTxavs: aba daukvirdiT, rame xom ar SeSlia irines? rogor unda<br />
eTqva? (rogor unda daexaza?)<br />
Sesworebisa Tu Sevsebis Semdeg Tavad irinemac unda<br />
gaimeoros marTebulad!<br />
Tuki Tavidanve vercerTma moswavlem ver gagvca pasuxi,<br />
saWiroa misaxvedri miTiTebiT daxmareba, magaliTad: _ am aTwiladebis<br />
jami risi tolia? wriuli diagramisTvis ki ... vin ityvis?<br />
Tuki sakiTxi wina msjelobis saboloo daskvnas exeba, maSin<br />
msjeloba maswavlebelmac unda Seajamos. man SedarebiT zogadi,<br />
sruli da gamarTuli saxiT unda Camoayalibos is saboloo<br />
daskvna, romelic klasma ukve gaarkvia: _ maSasadame, aTwiladebis<br />
Sedareba ufro advilia, vidre wiladebisa, magram es im SemTxvevaSi,<br />
Tuki wiladebi aTwiladebis saxiTaa Cawerili. aTwiladebis Sedareba<br />
mTeli ricxvebis Sedarebas hgavs. wriul diagramaze ki sidideebi<br />
TvalsaCinod Cans.<br />
es yvelaferi unda gaakeTos maswavlebelma: SekiTxvis ganvrcoba,<br />
saTanado gamoTqmis SerCeva, gamosakiTx moswavleTa kargad<br />
SerCeva, SekiTxvis formis Secvla da klasSi misi `datrialeba~,<br />
pirisaxis gamometyvelebis Secvla Tu xelebis moZraoba. maswavlebels<br />
araviTar SemTxvevaSi ar unda daaviwydes, rom saWiroa<br />
SekiTxvis amgvarad `datrialeba~. mxolod erTi SekiTxva da<br />
erTi pasuxi araa sakmarisi! gansakuTrebiT didxans unda `it-
- 35 -<br />
rialos~ axal an Znel sakiTxTan dakavSirebulma SekiTxvam.<br />
SedarebiT wvrilman sakiTxebs ki arc `datrialeba~ sWirdeba<br />
da arc maswavlebliseuli Sejameba. pirdapir gadavdivarT<br />
momdevno sakiTxze.<br />
aqtiuri mzaobis meTodikis mixedviT, maswavleblis mier<br />
warmosaTqmel winadadebaTa umravlesoba _ ljUywjUj<br />
winadadebebia, romlebic klass SekiTxvas usvams da<br />
dafiqrebisaken mouwodebs; danarCenTa umravlesoba _ cs[bofcjUj<br />
winadadebebebia, romlebic klass raimes avalebs da moqmedebisaken<br />
mouwodebs (magaliTad: gadaSaleT rveuli, gamodi dafasTan, SeavseT<br />
cxrili da sxva); da mxolod Zalian mcire nawilia UyspcjUj<br />
winadadebebi, romlebic klass pasiuri mosmenisa da damaxsovrebisaken<br />
mouwodebs. rasakvirvelia, esec aucilebelia, magram<br />
Zalian mcire zomiT. isinic TiTqmis yovelTvis mas Semdeg<br />
warmoiTqmis, rac Tavad moswavleebi ityvian. SesaZloa,<br />
arazustad, arasrulad, mouxeSavad _ magram mainc ityvian ise,<br />
rom cxadi iqneba _ maT gaiges. maswavlebeli mxolod amis Semdeg<br />
ityvis TxrobiT winadadebas, romelic, arsebiTad, mxolod<br />
gaimeorebs, daadasturebs da daxvews moswavleTa mier naTqvams.<br />
da esec _ Zalian xanmoklea, sul erTi-ori mokle winadadeba.<br />
maswavleblis Tundac erTwuTiani monologi, sakiTxis `axsna~ Tu<br />
`ganvrcoba~ _ gamoricxulia!<br />
sxvaTa Soris, maTematikis swavlisaTvisac saWiroa da Zalian<br />
didi ganmaviTarebel-aRmzdelobiTi daniSnulebac aqvs imas, rom<br />
maswavlebelma aswavlos bavSvebs fsUnbofUjt! nptnfob. patarebs<br />
es uWirT, magram TandaTan unda mivaCvioT.<br />
Cveneuli maTematikis gakveTili, rogorc wesi, iwyeba saSinao<br />
davalebis erToblivi garCeviT, muSavdeba misi TviTeuli amocana.<br />
amasTan, dawyebuli V klasidan, gakveTilis agebuleba icvleba.<br />
es cvlileba imiTaa gamowveuli, rom V klasidan ZiriTadi saprogramo<br />
xazis gavla emyareba ara maswavleblis mier warmarTul<br />
sagangebo saklaso samuSaoebs (rogorc iyo I-IV klasebSi),<br />
aramed moswavlis saxelmZRvanelos. V klasamde moswavlisaTvis<br />
misi saxelmZRvanelo iyo, arsebiTad, mxolod amocanebis kre-
- 36 -<br />
buli (gakveTilebad da paragrafebad dalagebuli), xolo swavleba<br />
emyareboda maswavleblis meTodikur saxelmZRvaneloSi aRweril<br />
saklaso samuSaoebs. V klasidan moswavle ukve sakmaod<br />
momwifebulia (unda iyos sakmaod momwifebuli) saimisod, rom<br />
damoukideblad SeZlos advili Teoriuli teqstebis wakiTxva da<br />
am teqstebiT swavla (cxadia, winsvlis Zalian neli tempiT, anu<br />
Zalian mcire-mcire nabijebiT).<br />
yovelive amis gamo V klasidan maswavleblis muSaoba<br />
Zalian gaadvilebulia _ rasac ver vityviT dawyebiTi klasebis<br />
Sesaxeb. es gansxvaveba imiTaa gamowveuli, rom Cveneuli meTodika<br />
bolomde aqtiuria da moswavle sakuTari aqtiurobiT swavlobs;<br />
dawyebiT klasebSi saSualo moswavlis unari teqstis wakiTxvisa<br />
da gaazrebisa araa imdenad ganviTarebuli, rom SemecnebiTi aqtiuroba<br />
mas daemyaros. amitom saWiroa saklaso aqtiuroba, romelsac<br />
moswavlis saxelmZRvanelo mxolod nawilobriv exmareba, ZiriTadad<br />
ki maswavleblis kiserzea (Tumca dawvrilebiTaa aRwerili<br />
meTodikur saxelmZRvaneloSi). yvelaze metad es pirvel<br />
klasSia, mere da mere ki TandaTan moswavlis saxelmZRvaneloze<br />
gadadis datvirTva. xolo V klasidan ukve mTel datvirTvas<br />
moswavlis saxelmZRvanelo iRebs Tavis Tavze (rogorc iTqva zemore,<br />
vgulisxmobT, rom moswavles ukve xelewifeba teqstis damoukideblad<br />
wakiTxva da gaazreba).<br />
gakveTilis 25-35 wuTi (gaaCnia amocanebs) eTmoba saSinao davalebis<br />
garCevas. swored am dros swavlobs moswavle sakiTxs _<br />
ara maswavleblis mier axsniT, aramed aqtiurad. TviTon saxelmZRvaneloa<br />
ise agebuli da iseTi meTodikiT gamarTuli (rac<br />
avtoris saqmea), rom misi gaazrebuli damuSaveba sruliad sakmarisia<br />
saswavlo programis aqtiurad, cocxlad, Sinaarsianad da<br />
Rrmad SesaTviseblad. maSasadame, maswavleblis amocanaa, rom moswavleebs<br />
daamuSavebinos da gaaazrebinos saxelmZRvanelo, misi<br />
Teoriuli teqstebiTa da ZiriTadi amocanebiT (rogorc yvela<br />
Cveneul saxelmZRvaneloSi, yvela amocanis amoxsna Tu gaazreba
- 37 -<br />
yvela moswavles ar moeTxoveba).<br />
maswavlebeli rig-rigobiT ekiTxeba sxvadasxva moswavleebs,<br />
Tu rogor amoxsnes esa Tu is amocana, ra Sedegi miiRes, gzadagza<br />
_ Tu rogor gaiges esa Tu is sakiTxi Teoriuli teqstidan.<br />
moswavleebi erTmaneTs adareben namuSevrebs, msjeloben, gamoTqvamen<br />
azrebs. maswavleblis Careva mxolod mcire miniSnebiTa Tu<br />
miTiTebiT an mokle SeniSvnebiT unda Semoifarglos.<br />
rogorc yovelTvis saSinao davalebis garCevisas, amocanebs<br />
rig-rigobiT TviTon moswavleebi arCeven. erTerTi maTgani<br />
dafasTanaa da wers, sxvebi aRniSnaven, Tuki raime sxvagvarad<br />
aqvT. yvela cdilobs sakuTaris dasabuTebas da irkveva<br />
marTebuli pasuxi. maswavlebeli mxolod maSin Caereva, Tuki<br />
vercerTi moswavle (maT Soris Zlierebic) Tavs ver gaarTmeven<br />
sakiTxs. magram es Carevac mxolod mimniSnebeli, sanaxevrod<br />
Tqmuli an misaxvedri SekiTxvis formisa unda iyos.<br />
dafasTan momuSave moswavle Cumad ar unda muSaobdes _ Tvi-<br />
Teul Sesrulebul moqmedebas is xmamaRal sityvier ganmartebebs<br />
unda urTavdes, raTa sxvebs auxsnas sakiTxi (magaliTad: am<br />
wiladis mricxvelSi gavxsnaT frCxilebi, miviRebT amas, axla<br />
SevkvecoT... axla gavavloT am wrexazis sxva radiusi, ai es.<br />
igi ar gadakveTs am rkals...). maswavlebelic saWiroebisamebr<br />
CaerTveba xolme msjelobaSi, magram mxolod mokle-mokle<br />
CanamatebiT.<br />
maswavleblis mTavari saqme am dros sul sxvaa. is saTiTaod<br />
Camoivlis yvela merxs da Tvalis gadavlebiT Seamowmebs, Tu<br />
rogor aqvT Sesrulebuli davaleba moswavleebs. cxadia, yvela<br />
moswavlis saSinao davalebis yvela amocanis Semowmeba SeuZlebelia,<br />
magram maswavlebelma mainc unda gadaxedos namuSevars<br />
da raime Tqvas. amis gakeTeba aucilebelia, raTa TviTeulma<br />
moswavlem icodes, rom mis namuSevars maswavlebeli aucileblad<br />
naxavs. maswavlebeli wiTeli melniT erTi-or Sesworebasac<br />
gaakeTebs (rac TvalSi moxvdeba), yovel moswavles aucileblad
- 38 -<br />
qjsbebe etyvis ramdenime sityvas, naxazs Seuqebs an formulis<br />
Canawers dauwunebs, mxrebSi gaasworebs an Tavze xels<br />
gadausvams, Tuki saWiroa, sayvedursac etyvis, an, piriqiT,<br />
Seaqebs. cxadia, am dros SeuZlebelia saSinao namuSevarTa<br />
yuradRebiT naxva (yuradRebiT da safuZvlianad sakontrolo<br />
nawerebi isinjeba).<br />
maSasadame, yovel gakveTilze maswavlebeli unda mivides<br />
TviTeul moswavlesTan, fsUjfsU{f, cota xniT mainc<br />
gaesaubros mas piradad, Seaqos an muTiTos xarvezze.<br />
sxva moswavleebi ki am dros TavianT saqmes akeTeben: amocanebis<br />
garCeva gakveTilis pirvelive wuTidan iwyeba. yvelas gadaSlili<br />
aqvs wigni da rveuli da avsebs da asworebs Tavis namuSevars.<br />
yvela moswavles kalmistari unda ekavos xelSi da iqve<br />
avsebdes Tu asworebdes namuSevars, anda sul axali striqonidan<br />
Tavidan werdes rames (nawerSive rom ar CaaWuWynos). yvela<br />
SemTxvevaSi, Tuki moswavles ara aqvs Sesrulebuli saSinao<br />
davalebis SedarebiT rTuli amocana, igi klasSi unda<br />
daweros _ am amocanis erToblivi garCevisas.<br />
am dros dafasTan ukve sxva moswavle SeiZleba muSaobdes _<br />
visac daevaleba. maswavlebeli ki ganagrZobs merxebis Camovlas,<br />
rveulebis Tvalierebasa da TviTeul moswavlesTan piradad<br />
urTierTobas _ saamisod mas sakmao dro aqvs.<br />
Tuki adamians ar SeuZlia ori ramis erTad keTeba, mas maswavlebloba<br />
gauWirdeba. magram ori ramis erTad keTebis unari<br />
gavarjiSebadia, amitom, Tuki maswavlebels es uWirs, sagangebod<br />
unda cdilobdes, rom gaivarjiSos da ganiviTaros es unari.<br />
moswavleTa umravlesoba, bunebrivia, cdilobs, rom Tavisi<br />
azri gamoTqvas, Tundac sul cotaTi gansxvavebuli ram eweros,<br />
mainc. Cveneuli meTodika Zalze aaqtiurebs moswavleebs da es<br />
Zalian kargia. amitom maswavlebeli unda iyos yuradRebiT, raTa<br />
erTobliv msjelobaSi metismetad bevri dro ar gaeparos _<br />
Zneli amocanebis dawvrilebiT garCevas im donemde, rom sustma
- 39 -<br />
moswavlemac gaiazros isini, azri ara aqvs. sustma moswavlem<br />
kargad unda gaiazros gakveTilis ZiriTadi sakiTxebi da<br />
amocanebis naxevari mainc. saamisod zogjer saWiroa xolme<br />
saSinao davalebis romelime amocanis saxesxvaobis damatebiT<br />
amoxsna. es saxesxvaoba maswavlebelma Tematikuri krebulidan<br />
unda aarCios (zogjer, SesaZloa, TviTonac mouwios Sedgena _<br />
arsebuli amocanis pirobaSi ricxvebisa da araarsebiTi<br />
monacemebis SecvliT).<br />
cxadia, gamosaTvlel an standartuli pasuxis mqone<br />
amocanebis garCevas SedarebiT ufro naklebi dro unda daeTmos.<br />
umjobesia, maswavlebelma TviTon daawerinos romelime moswavles<br />
dafaze axali amgvari amocana (Secvlili ricxvebiT) da yvelas<br />
mosTxovos misi Sesruleba (mcire saklaso wera).<br />
Tuki saSinao davalebis garCevas Zalian bevri dro ar<br />
dasWirda, misi damTavrebis Semdeg sasurvelia, Catardes 2-4wuTiani<br />
maTematikuri TamaSi (TamaSebis nimuSebi mocemulia II-IV<br />
klasebis meTodikur saxelmZRvaneloebSi).<br />
axla mokled aRvweroT, Tu rogor xdeba amocanaTa<br />
erToblivi garCeva. pirveli amocana xSirad gamosaTvlelia,<br />
amasTan, igi ramdenime qveamocanisgan Sedgeba (es qveamocanebi<br />
zogjer gadanomrilia romauli cifrebiT, zogjer _ ara).<br />
maswavlebeli erT moswavles adgilidanve aTqmevinebs pirveli<br />
qveamocanis (gamosaTvleli magaliTis) pasuxs, sxva moswavles _<br />
meore qveamocanis pasuxs da ase Semdeg. TviTeul SemTxvevaSi<br />
gairkveva, vinmes sxvagvari pasuxi xom ara aqvs. visac Secdoma<br />
aRmouCndeba, iqve Tavidan Seasrulebs gamoTvlas. am dros yvela<br />
moswavle zis da ise muSaobs Tu pasuxobs.<br />
sazogadod, yovelTvis, roca amocana ramdenime qveamocanisagan<br />
Sedgeba, maswavlebeli maT amoxsnebs sxvadasxva moswavleebs<br />
Camoayalibebinebs. cxadia, TviTeul SemTxvevaSi imarTeba<br />
erToblivi msjeloba (amocanis Sesabamisi xangrZliobisa).<br />
SemoqmedebiTi amocanis garCevisas sasurvelia, sxvadasxva<br />
amoxsna mravalma moswavlem warmoadginos da moxdes
- 40 -<br />
gansxvavebul amoxsnaTa urTierTSedareba. sazogadod, samsjelo<br />
amocanis amoxsnisas erTi moswavle dafasTan muSaobs (xazavs<br />
geometriul nakvTsa Tu logikur sqemas, wers gamosaxulebas da<br />
sxva) an adgilidan msjelobs, da Tavis pasuxs asabuTebs, Semdeg<br />
sxvebic erTvebian msjelobaSi.<br />
Cveneuli meTodikis mkacri moTxovnaa: Tuki amocana samsjeloa<br />
(rasac Cven yovelTvis vuTiTebT), maSin mTavaria ara amocanis<br />
marTebuli pasuxis migneba da Tqma, aramed swored msjeloba.<br />
bavSvs msjeloba ezareba, msjelobas sakmao Zalisxmeva sWirdeba,<br />
miT umetes, rom man pasuxi ukve icis. miuxedavad amisa,<br />
maswavlebels evaleba, rom srulfasovnad amsjelos moswavleebi.<br />
mxolod am gziT miiRweva Rrma da kargad gaazrebuli codna da<br />
azrovnebis ganviTareba. im msjelobas, romelsac maswavlebeli<br />
moswavles klasSi xmamaRla warmoaTqmevinebs, SemdgomSi moswavle<br />
ukve gonebaSi Caatarebs (anda, Caatarebs Cumi burtyunis<br />
TanxlebiT) _ da swored amgvarad yalibdeba namdvili azrovneba,<br />
romelic ganuyofelia sityvierebisagan (es exeba sakuTriv<br />
azrovnebas; aranaklebi mniSvneloba aqvs gumans anu intuicias,<br />
romelic ar aris sityvieri da ar aris metyvelebaze<br />
damokidebuli _ mis gansaviTareblad ki sul sxvagvari, faruli<br />
muSaoba gvaqvs Caqsovili Cveneul programaSi, amocanaTa<br />
Tanwyobis meSveobiT [$ 16]).<br />
morCenili 10-15 wuTis ganawileba damokidebulia imaze, ganvlil<br />
gakveTilSi iyo Tu ara Teoriuli teqsti. Tuki Teoriuli<br />
teqsti ar yofila, maSin morCenili dro eTmoba damatebiTi<br />
erTi-ori amocanis garCevas Tematikuri krebulidan _ isini maswavlebels<br />
winaswar unda hqondes SerCeuli gakveTilis<br />
momzadebisas, unda SearCios iseTebi, romlebic Seexeba SedarebiT<br />
Znel sakiTxs ganvlili gakveTilebidan. amitom maswavlebels,<br />
cxadia, yovelTvis CaniSnuli unda hqondes, Tu ra sakiTxi<br />
gauWirdaT moswavleebs da am sakiTxze damatebiTi muSaoba unda<br />
dagegmos momdevno gakveTilebze. sazogadod, Tuki moswavleTa<br />
codnaSi raime xarvezi aRmoCndeba, maswavlebeli atarebs
- 41 -<br />
ramdenimewuTian saklaso weras saTanado Temaze.<br />
garda amisa, maswavlebels CaniSnuli unda hqondes is<br />
amocanebic, romelTa garCeva mizezTa gamo ver moeswro wina<br />
gakveTilebze. isinic morCenil droSi unda gairCes.<br />
Tuki ganvlili gakveTili moicavda Teoriul teqsts, maSin<br />
morCenili dro unda daeTmos am teqstis gaazrebas da mis<br />
irgvliv msjelobas. saSinao davalebis aqtiuri garCevis Sedegad<br />
am teqstis ZiriTadi sakiTxebi moswavleebma ukve gaiazres,<br />
magram saWiroa am codnis ganmtkiceba da, rac mTavaria, zepiri<br />
msjelobisa da dasabuTebis unarebis ganviTareba. am unarebis<br />
ganviTareba xom maTematikis swavlebis calke mizania [§ 6].<br />
sasurvelia, maswavlebelma moitovos gakveTilis bolo orisami<br />
wuTi imisaTvis, raTa ukve TviTon Seajamos ganvlili<br />
sakiTxi. moswavleebma es sakiTxi ukve waRma-ukuRma atriales da<br />
amuSaves, aqtiuradac gamoiyenes da sityvieradac Camoayalibes,<br />
ukve ician igi. magram, sakiTxis arsi saerTo msjelobaSi rom ar<br />
gafermkrTaldes, maswavlebeli bolos TviTon mkafod Seajamebs<br />
da ganazogadebs naswavl sakiTxs: _ maSasadame, Cven viswavleT<br />
Semdegi: ... ... (cxadia, saxelmZRvanelos mixedviT). magram saqme<br />
isaa, rom es araa axsna. maswavlebeli mxolod Seajamebs ukve<br />
naswavl _ da ara axal _ sakiTxs, Tanac, mxolod mas Semdeg,<br />
rac sakiTxi ukve sajarod garCeulia da namsjelia.<br />
klass saSinao davalebad yovelTvis Semdegi gakveTili eZleva.<br />
Cveulebriv arcaa saWiro saSinao davalebis micema: moswavleebma<br />
unda icodnen, rom yovelTvis, ugamonaklisod, davalebad momdevno<br />
gakveTili eZlevaT (Tuki maswavlebels sxva araferi uTqvams).<br />
moswavleebi Sin damoukideblad Seiswavlian am axal<br />
gakveTils, amoxsnian mis amocanebs. rasac ver daZleven, meore<br />
dRes klasSi gaigeben _ sajaro ganxilvis dros.<br />
bavSvma amocanaze damoukideblad unda imuSaos!<br />
`repetitoroba~ Zalian uaryofiTad moqmedebs moswavlis<br />
azrovnebis ganviTarebaze. Sin mSoblisa da klasSi maswavleblis
movaleobaa:<br />
- 42 -<br />
1) misces moswavles saTanado davalebebi da yuradReba<br />
miaqcios bavSvs _ gulisyuriT mecadineobs Tu ara;<br />
2) Seamowmos da Seasworos moswavlis namuSevari;<br />
3) sTxovos moswavles, Tavad moZebnos, anda uCvenos mas<br />
amocanis sxvagvari amoxsnis gzebi (roca moiZebneba);<br />
4) saWiroebis SemTxvevaSi ganumartos ucnobi sityva;<br />
5) mxolod aucileblobis SemTxvevaSi daexmaros moswavles<br />
amocanis pirobis gaazrebaSi;<br />
6) ukidures SemTxvevaSi (magaliTad, Tuki moswavle mizezTa<br />
gamo CamorCenilia saswavlo programas) daexmaros moswavles<br />
amocanis amoxsnaSi, calkeul miTiTebaTa an miniSnebaTa meSveobiT.<br />
maswavleblisTvis mTavaria, rom mieCvios or rames: acalos<br />
moswavleebs damoukidebeli fiqri, msjeloba da muSaoba da<br />
Seikavos gamzadebuli pasuxebi; yovel sakiTxs miudges SemoqmedebiTad<br />
da aseve moiTxovos moswavleebisaganac.<br />
amrigad, sabolood, Cveneuli meTodikiT agebul gakveTilze<br />
ar xdeba arc moswavlis mier gakveTilis moyola, arc maswavleblis<br />
mier axali masalis axsna da, Cveulebriv, arc saSinao davalebis<br />
micema (Tumca, Cveni wesia: arcerTi dRe saSinao davalebis<br />
gareSe).<br />
§ 10. maswavleblis muSaoba<br />
Cveneuli meTodikis mixedviT, maswavleblis mTavari amocanebia:<br />
1) amuSaos moswavle ise, raTa man safuZvlianad da<br />
gaazrebulad daamuSaos Tavisi saxelmZRvanelo;<br />
2) sagudagulod gaasworos sakontrolo nawerebi da moswavleebsac<br />
bolomde Seasworebinos;<br />
3) guldasmiT Seasrulebinos moswavleebs gamarTuli, zusti<br />
da Tanmimdevruli msjelobisa da dasabuTebis gansaviTarebeli<br />
savarjiSoebi (rogorc aRvwereT § 6-Si).<br />
danarCens TviTon saxelmZRvanelo gaakeTebs.<br />
roca klass miecema saweri, dasaxazi, dasaxati,
- 43 -<br />
dasaTvleli Tu sxva saklaso samuSao, maswavlebelma<br />
merxebs Soris unda iaros da Tvalyuri adevnos muSaobas:<br />
ramdenad kargad asruleben davalebas, rogor ukaviaT<br />
kalmistari Tu rogor xazaven, vinmes rame xom ar eSleba,<br />
welSi xom ar ixrebian da sxva.<br />
amgvarad, Cveneuli sagakveTilo muSaobis sqemaa:<br />
Tanaklaselisa da agreTve maswavlebelis mosmena →<br />
dafiqreba, azrovneba, Zieba → azris gaCena →<br />
azris sityvierad Camoyalibeba → naTqvamis daxvewa,<br />
Sesworeba, Sevseba, azrobrivad da enobrivad gamarTva.<br />
sabolood ki unarCvevis ganviTareba, mcired damaxsovrebac.<br />
yvelaze Zvirfasia unarCvevis ganviTareba. mis misaRwevad ki<br />
mTavaria meore rgoli, romelsac Cveulebrivi swavlebisas<br />
Caenacvleba gaxseneba. Cveulebrivi sagakveTilo muSaobis sqemaa:<br />
maswavleblisa da agreTve (Zalian mcired) Tanaklaselis mosmena →<br />
mosmenilis damaxsovreba →<br />
damaxsovrebulis aRdgena anu gaxseneba...<br />
rogorc vxedavT, Cveneuli meTodikiT erT sakiTxs sakmao ganxilva<br />
da msjeloba SeiZleba dasWirdes. amitom maswavlebeli<br />
Zalian unda gaufrTxildes sagakveTilo dros da ar daxarjos<br />
igi iseT rameebze, rac ar gvaqvs miTiTebuli gegma-konspeqtebSi!<br />
kidev erTxel: maswavlebels evaleba, rom gaacocxlos da<br />
`xorci Seasxas~ imas, rac mokled da mSraladaa mocemuli gegmakonspeqtSi.<br />
xolo sakuTar monologebze (`gakveTilis axsnaze~<br />
da sxva) drois daxarjva fuWia. bavSvisaTvis _ da ara mxolod<br />
bavSvisaTvis, aramed, sazogadod, adamianisaTvis, bavSvisaTvis ki<br />
miT umetes _ gacilebiT ufro nayofieria sakuTari<br />
dafiqrebisa Tu grZnobis Sedegad warmoTqmuli ori sityva,<br />
vidre sxvisi oci sityvis mosmena.<br />
maswavleblis mier sakiTxis mokled axsna Cven gadatanili<br />
gvaqvs sul boloSi. arsebiTad, es arcaa axsna. esaa moswavleTa
- 44 -<br />
mier aqtiurad garkveuli sakiTxis mxolod Sejameba da<br />
gamarTulad Camoyalibeba _ raTa sakiTxis arsi ar Caikargos<br />
saerTo msjelobaSi. xolo winaswar maswavlebeli arafers ar<br />
xsnis. Cveneuli meTodika gamoricxavs `gakveTilis axsnas~.<br />
maswavlebelma aqtiurad unda amuSaos klasi da Tavad<br />
moswavleebs aRmoaCeninos sakiTxis arsi. amisaTvis sakmarisia,<br />
rom man kargad Seasrulos Cveneuli zogadi principebi da<br />
mxolod gaacocxlos saxelmZRvanelo.<br />
amgvarad, maswavlebeli xan merxebs Soris dadis, moswavleebs<br />
asworebs (an ise Seexeba maT) da Tan muSaobs, xan dafasTan muSaobs,<br />
xan ki Tavis skamze zis da saubrobs. maswavlebeli unda<br />
moeridos leqtoris qcevebs (garda, SesaZloa, namsjelis mokle<br />
Sejameba-ganzogadebis wuTebSi), unda moeridos agreTve mqadageblis<br />
qcevebs. es Zalian iSviaTad unda xdebodes, magaliTad, rame<br />
gansakuTrebul SemTxvevasTan dakavSirebiT (Tuki amgvari qceva<br />
gaxSirdeba, igi gaufasurdeba).<br />
xSirad xdeba, rom maswavlebeli klass TiTqos kargad amuSavebs,<br />
moswavleebi aqtiuroben, pasuxoben da sxva. magram es moCvenebiTi,<br />
zerele aqtiurobaa xolme. saqme isaa, rom xSirad<br />
moswavle ambobs, arsebiTad, sxvis nafiqrsa Tu nagrZnobs.<br />
moswavle mxolod ixsenebs da imeorebs imas, rac sxvisgan<br />
(kerZod, maswavleblisgan, ufrosebisgan, televizoridan)<br />
mousmenia. es ki sul sxvaa. nayofieria ara ubralod sakuTari<br />
`ori sityva~, aramed, rac mTavaria, sakuTari dafiqrebisa Tu<br />
grZnobis Sedegad warmoTqmuli `ori sityva~! sxvisi (umjobesia,<br />
Tanaklaselis) nafiqr-nagrZnobis gameorebas ki mxolod maSin<br />
unda mivmarToT, roca Tavad moswavle ver gaumklavdeba sakiTxs<br />
_ ifiqrebs, Seecdeba, magram ver SeZlebs. am dros is<br />
qvecnobierad mainc Semzadebulia sxvisi naTqvamis gasaTaviseblad<br />
_ swored imis wyalobiT, rom winaswar ukve Tavad aqvs nafiqri.<br />
swored nafiqri _ da ara ganaxsenebi.<br />
moswavleTa namdvili aqtiuroba _ esaa maTi damoukidebeli<br />
azrovneba, msjeloba, Zieba, kvleva, Semoqmedeba, warmosaxva Tu<br />
grZnoba. maTi sakuTari, maTi nebelobisa da Zalisxmevis nayofi.
- 45 -<br />
Cveneuli meTodikiT momuSave maswavlebeli mxolod krebis<br />
karg TavmjdomaresaviTaa _ kargad da ostaturad anawilebs<br />
SekiTxvebs da bolos gamarTulad da srulad Seajamebs daskvnas.<br />
xolo gakveTilis ZiriTadi dro eTmoba Tavad moswavleTa<br />
msjelobas, moqmedebas, muSaobasa da sxva. oRond, mxolod<br />
`krebis Tavmjdomareoba~ araa sakmarisi. maswavlebeli cotaTi<br />
mainc `fsiqologi~ unda iyos, raTa SeZlebisamebr (ramdenadac<br />
saSualebas aZlevs klasSi moswavleTa raodenoba) ganaxorcielos<br />
individualuri midgomis principi. sul mcire, rac mas<br />
moieTxoveba am mxriv _ esaa moswavleebis unar-SesaZleblobaTa<br />
mudmivi gaTvaliswineba. da kidev _ maswavlebeli cotaTi mainc<br />
`msaxiobi~ unda iyos: qmediTad unda iyenebdes xelebis<br />
gamaTvalsaCinoebel moZraobas, pirisaxis gamometyvelebasa da<br />
xmis mimoxras.<br />
amrigad, saswavli sagnis Cveneuli kargi maswavlebeli (aq<br />
ganvixilavT swored sagnis maswavleblobas _ da ara aRmzrdeldamrigeblobas)<br />
_ esaa ZiriTadad `krebis kargi Tavmjdomare~,<br />
romelic, amasTan, cotaTi `fsiqologia~ da cotaTi _<br />
`msaxiobi~. swored esaa misi SemoqmedebiTi sarbieli. xolo sxva<br />
mxriv is saukeTeso Semsrulebeli unda iyos _ zedmiwevniT da<br />
gulmodgined asrulebdes yvela meTodikur miTiTebas. TviT<br />
sametyvelo sityva-terminebiTac ki ar unda gascdes gegmakonspeqtebs,<br />
ar unda ixmaros is sityvebi, romlebic<br />
arabunebrivia bavSvisaTvis da namdvili qarTulisaTvis<br />
(magaliTad: konfliqti, konfliqturi, funqcionirebs,<br />
problema, stabiluri, winaaRmdegobaSi moxvedi, pozicia,<br />
situacia da sxva mravali `sarevela~). maswavlebels<br />
gauWirdeba uceb, sakuTari metyvelebis kvaldakval, gaarkvios,<br />
sityva Tu gamoTqma ramdenad kargia da bunebrivi mocemuli<br />
wlovanebis bavSvisaTvis. amitom Cven mas advil gzas vTavazobT:<br />
ganuxrelad ganaxorcielos gegma-konspeqtebSi mocemuli ConCxi.<br />
rogorc iTqva zemore, swored rom ganaxorcielos, anu `xorci<br />
Seasxas~, `gaacocxlos~.<br />
axla CamovayaliboT isic, Tu ra ar unda iyos Cveneuli kargi
- 46 -<br />
maswavlebeli. pirvel yovlisa, esaa `mecnier-leqtori~: grZeli<br />
monologebiT, `mecnieruli~ terminebiT, maRalfardovanebiT. da<br />
agreTve _ `novator-meTodisti~. Cven yvelanairad mxars vuWerT<br />
maswavleblis novatorul meTodistur Ziebebsa da Semoqmedebas,<br />
magram es unda xdebodes ara maswavleblobisas, aramed calke,<br />
skolidan Sin dabrunebisas _ Tuki eqneba amis dro, Zal-Rone da<br />
SesaZlebloba, Tanac, ara momdevno gakveTilisaTvis saWiro<br />
Cveulebrivi momzadebis xarjze! maSin es iqneba maswavleblis<br />
meore saqmianoba, `SeTavsebiT meore profesia~ _ mkvlevarmeTodistisa.<br />
amgvarma maswavlebelma Tavisi namuSevari Cven unda<br />
mogvawodos. Cven amas madlierebiTa da gulisyuriT SevxvdebiT,<br />
erTad ganvixilavT, avwon-davwoniT da davsaxavT samoqmedo<br />
gegmasac. magram uamisod, naCqarevi, zerele da TviTneburi<br />
`novator-meTodistoba~ Zalian sarisko ramea da ufro xSirad<br />
bavSvebisaTvis savalalod mTavrdeba.<br />
xSirad Segvxvedria maswavlebeli, romelic `novatorobs~ da<br />
garegnulad gakveTilebs fuWi zizilpipiloebiT amSvenebs. sinamdvileSi<br />
ki gaurbis an zereled asrulebs mis mTavar movaleobas<br />
_ yoveldRiur `Sav samuSaos~: saguldagulod moemzados gakve-<br />
TilisaTvis, mowesrigebuli hqondes TvalsaCinoeba, kargad gaasworos<br />
sakontrolo namuSevrebi da moswavleebsac safuZvlianad<br />
gaasworebinos Secdomebi, izrunos mravali moswavlis codnasa<br />
Tu unarCvevebSi arsebul xarvezTa gamosasworeblad da sxva.<br />
erTi sityviT, svevofcb icvleba zerele garegnuli zizilpipiloebiT,<br />
da Sedegic saTanadoa: moswavleTa codna da unar-<br />
Cvevebic zerelea da zizilpipiloebiviT umaqnisi.<br />
maswavlebels ar unda gaeparos pedagogikuri Secdomebi:<br />
1) moswavlis Secdomaze ukmayofilebis gamoxatva (uxeS sityvebze<br />
xom laparakic zedmetia) _ moswavles ar unda eSinodes Secdomisa;<br />
2) advili SekiTxvis micema Tu advili amocanis garCevis davaleba<br />
Zlieri moswavlisaTvis;<br />
3) pasiuri moswavlisTvis yuradRebis dakleba, bolomde<br />
argarkveva imisa, sustma moswavlem gaigo Tu vera programuli
- 47 -<br />
(ara damatebiTi Zneli) sakiTxi;<br />
4) maTematikuri teqstebis zepirad swavleba, anda, imis<br />
ugulebelyofa, rom moswavles sakiTxi zepirad aqvs naswavli da<br />
wesierad ar esmis;<br />
5) roca msjelobisas moswavle winadadebas kargad ver gamar-<br />
Tavs, Semofargvla mxolod SesworebiT: moswavlem SeiZleba<br />
kargad gaigo Tavisi xarvezi, maswavleblis Sesworebac gaigo,<br />
magram es araa sakmarisi _ moswavlem sakuTari piriT unda<br />
gaimeoros (anda, sakontrolo naweris gasworebisas _ sakuTari<br />
xeliT unda gadaweros) Sesworebul-Sevsebuli winadadeba.<br />
Cveneuli mrwamsiT, maswavleblis ostatobis umTavresi maCveneblebia:<br />
1) `lsfcjt!Ubwnkepnbsf~ _ ramdenad aaqtiurebs, aazrovnebs,<br />
damoukideblad amuSavebs moswavleebs, ramdenad metad Canan<br />
moswavleebi da naklebad _ TviTon; ramdenad Tanabrad aZlevs<br />
sityvas sxvadasxva moswavleebs. _ Tanac, ara mxolod erTi<br />
gakveTilis farglebSi. ese igi, maswavlebels isic unda<br />
axsovdes, Tu romel moswavleebs alaparakebda Tu amuSavebda<br />
wina gakveTilze _ raTa axla sxvebs dauTmos sarbieli. garda<br />
amisa, SekiTxvebis ganawilebisas unda gaiTvaliswinos TviTeuli<br />
moswavlis Taviseburebani.<br />
2) `Tfntsvmfcfmj~ _ ramdenad srulad da zustad<br />
asrulebs meTodikur saxelmZRvaneloTa miTiTebebsa da gegmas;<br />
`ConCxs~ ramdenad kargad `asxmas xorcs~; ramdenad bolomde<br />
CaeZieba xarvezsa Tu naklovanebas da ar miafuCeCebs mas.<br />
3) `lsfcjt! Ubwnkepnbsf~ _ ramdenad kargad iyenebs<br />
sagakveTilo dros, magaliTad: winaswar aqvs Tu ara momzadebuli<br />
dafaze warwerebi da saklaso TvalsaCinoeba; roca erTi<br />
moswavle dafasTan raimes akeTebs, sxvebi uqmad xom ar hyavs<br />
mocdenili da sxva; ramdenad inarCunebs gakveTilisTvis saWiro<br />
ritms.<br />
4) `gtjrpmphj~ _ ramdenad axerxebs individualuri<br />
midgomis ganxrocielebas, ramdenad axerxebs imas, rom hqondes<br />
urTierToba calkeul moswavlesTan, romlis pirovnebasac
- 48 -<br />
udidesi pativiscemiTa da siyvaruliT emsWvalvis, romlis<br />
Taviseburebebsac iTvaliswinebs da romelsac ar Cakargavs<br />
saerTo usaxur `klasSi~.<br />
5) `gtjrpmphj~ _ ramdenad axerxebs bavSvis<br />
cnobierebisaTvis bunebrivi samyaros Seqmnas da ramdenad<br />
naklebad arRvevs mas mxolod mozrdili adamianisaTvis<br />
naSandoblivi sxvadasxva sityviT, azriT, qceviTa da sxva;<br />
ramdenad qmnis imis pirobebs, rom bavSvis cnobiereba gaRrmavdes,<br />
gamdidrdes; Sinaganad, bunebrivad ganviTardes da gaifurCqnos,<br />
rogorc lamazi yvavilis kokori, sakuTari bunebrivi niadagidan<br />
rom ikvebeba da izrdeba da jansaR nayofs gamoiRebs (romelic<br />
rogors SeZlebs!) _ da ara ufrosTagan mosmenilis dagrovebiT<br />
gaberili da damZimebuli, rogorc xelovnuri sasuqebiT uxvad<br />
napativebi da gafuyuli fuWyvavila, romlis nayofi garegnulad<br />
viTom lamazia, magram sinamdvileSi futuro da ugemuria.<br />
6) `ntbyjpcj.rbsUwfmj~ _ rogoria misi metyveleba,<br />
gamoTqma, ramdenad gamarTuli, mdidari da wminda qarTuliT<br />
metyvelebs.<br />
7) `ntbyjpcj~ _ ramdenad kargad iyenebs xelebis<br />
gamomxatvel moZraobas, miTiTebebs, pirisaxis gamometyvelebas,<br />
xmis mimoxrasa da sxva.<br />
8) `Tfntsvmfcfmj~ _ ramdenad saguldagulod da<br />
xarisxianad asworebs da afasebs sakontrolo Tu sagamocdo<br />
namuSevrebs;<br />
9) `nfdojfs.npb{spwof~ _ ramdenad kargad esmis saswavli<br />
sagnis mecnieruli, logikuri da SemoqmedebiTi mxareebi.<br />
SeiZleba vinmes moeCvenos, rom Cven Zalian maRal moTxovnebs<br />
vuyenebT maswavlebels. es ase araa. Cveneul moTxovnaTa `sanacvlod~<br />
Cven maswavlebels vaTavisuflebT sxva, gacilebiT ufro<br />
sapasuxismgeblo an rTul movaleobaTagan: meTodikuri Ziebani da<br />
sakiTxis damuSavebis formaTa da saSualebaTa mogoneba;<br />
damatebiTi amocanebis, TamaSebis, literaturisa Tu<br />
TvalsaCinoebis moZieba; gakveTilis winaswari gegma-konspeqtis<br />
Sedgena da sxva. ukanasknel sakiTxTan dakavSirebiT: Cven ar
- 49 -<br />
miviCnevT maswavleblis naklad, Tuki is gakveTilis ganmavlobaSi<br />
meTodikur saxelmZRvanelos, gegma-konspeqtebs gamoiyenebs xolme.<br />
mravali wvrilmanis zepirad damaxsovreba Znelia da araa saWiro.<br />
sazogadod, maswavleblis muSaobaSi (sxvaTa Soris, asevea<br />
mecnierebaSic!) cxra wilia ruduneba da Savi Sroma da mxolod<br />
erTi wilia Semoqmedeba-STagoneba, sixaruli da xelovneba _ rac,<br />
Tanac, im cxra wilis gareSe _ unayofo iqneba. magram, cxadia, es<br />
erTi wili aranakleb Zvirfasia, vidre danarCeni cxra wili!<br />
sxvaTa Soris, amasve ukavSirdeba erTi Zalian mniSvnelovani<br />
zogadpedagogikuri sakiTxi. saqarTveloSi metismetad mravladaa<br />
`Zalian niWieri~, magram zarmaci~ adamianebi. winaT ase ar<br />
yofila, qarTveli kaci, piriqiT, gamorCeulad Sromismoyvare iyo<br />
(amas ueWvelad amtkicebs Tundac qvaSi nakveTi CuqurTmebi da<br />
umaRles doneze ganviTarebuli meRvineoba). Cveni erovnuli<br />
ubedureba isaa, rom, maxinjur garemoebaTa gamo, davSordiT Cvens<br />
namdvil znes (sxva mxrivac, da ara mxolod Sromismoyvareobis<br />
mxriv). Cveneuli meTodika isea agebuli, rom `Zalian niWieri,<br />
magram zarmaci~ moswavle karg Sedegebs verasdros miaRwevs.<br />
ramdeni mniSvnelobac aqvs gonebriv monacemebs, imdenive<br />
mniSvneloba aqvs sibejiTesac. rogorc adamianis marjvena da<br />
marcxena xelebia. `Zalian niWieri, magram zarmaci~ calxela<br />
adamians hgavs.<br />
maswavlebelma niSani unda daweros mxolod Sedegis mixedviT<br />
_ da ara imis mixedviT, Tu ra `SeuZlia~ moswavles! zarmacs<br />
ufro metad `ar SeuZlia~, vidre saSualo gonebriv monacemTa<br />
mqones! swored ufro sizarmace, vidre gonebriv monacemTa<br />
nakleboba, iwvevs siZneleebs swavlaSi, saqmeSi da ufro metic,<br />
cxovrebaSi (da xSirad am siZneleTa gamo adamianis zneobac ki<br />
fuWdeba, is Suriani da RvarZliani xdeba).<br />
kidev erTxel: maswavlebelma niSani unda daweros<br />
mxolod Sedegis mixedviT _ da ara imis mixedviT, Tu ra<br />
`SeuZlia~ moswavles. Semfasebelma SeiZleba arc icodes, Tu<br />
romel moswavles ekuTvnis Sedegi! garda amisa, yovelad<br />
dauSvebelia niSnebis mikerZoebiT wera. yvelas unda eweros is
- 50 -<br />
niSani, rasac imsaxurebs, yovelgvari siyalbis gareSe! uazro da<br />
fuWi lmobierebiT bavSvs daTvur samsaxurs gavuwevT!<br />
maswavlebels yovelTvis unda axsovdes, rom is _ moswavlis<br />
aRmzrdelicaa. xolo yoveli nayalbevi niSani _ sawamlavis TiTo<br />
wveTia, romelic ryvnis moswavlis suls. axalgazrda unda mie-<br />
Cvios sakuTari gonebiT, sibejiTiTa da patiosani SromiT warmatebis<br />
mopovebas, sakuTari Zalebis rwmena unda gamoumuSavdes da<br />
gergilianoba, Taosnoba ganuviTardes. es yovelive gacilebiT ufro<br />
mniSvnelovania, vidre maTematikis codna.<br />
§ 11. Cveneuli amocanebis tipebi<br />
Cveneuli amocanebi saguldagulod gaazrebul Tanwyobas qmnis.<br />
a m o c a n e b i<br />
programulebi damakavSirebelni damatebiTebi am samis naerTebi<br />
(romlebic<br />
uSualod<br />
mimdinare<br />
programul<br />
sakiTxzea)<br />
(ori naswavli<br />
sakiTxisa<br />
erTmaneTTan _<br />
gvaqvs sakiTxebis<br />
TiTqmis yvela<br />
SesaZlo wyvilze)<br />
(zogad saazrovno da<br />
SromiT-saSemsruleblo<br />
unarCvevaTa<br />
ganmaviTarebelni<br />
_ Zalian mravali<br />
sxvadasxva saxisaa)<br />
Semamzadebelni (saswavli sakiTxis win) ganmamtkicebelni<br />
I tipi II tipi III tipi I tipi II tipi III tipi<br />
Semamzadebeli amocanebis sami ZiriTadi tipia:<br />
I. axali sakiTxisaTvis xSirad saWiroa xolme romelime adre<br />
naswavli sakiTxi. misi gameoreba swored Semamzadebeli amocanis<br />
meSveobiT xdeba, aqtiurad. Tuki moswavles daviwyebuli aqvs es<br />
Zveli sakiTxi an kargad veRar iyenebs mas, am amocanas ver<br />
amoxsnis. magram am saSinao davalebis klasSi garCevisas mainc<br />
moxdeba Sexseneba da naswavlis gaaqtiureba, moswavle am amocanas
- 51 -<br />
klasSi mainc amoxsnis. amitom momdevno gakveTilisaTvis<br />
(romelSic is axali sakiTxia) moswavle mainc mzad iqneba.<br />
II. axal Teoriul teqstSi xSirad aris xolme erTi an ramdenime<br />
Zneli adgili (magaliTad, damtkicebaSi raime gardaqmna, an<br />
SedarebiT rTuli msjeloba). am nawyvetis gaadvilebuli<br />
varianti Semamzadebel amocanaSi muSavdeba. Tuki moswavleebs igi<br />
gauWirdebaT da Sesabamis amocanas ver amoxsnian, am saSinao<br />
davalebis klasSi garCevisas sxva amocanebs Soris es Zneli<br />
adgilic gairCeva. amitom momdevno gakveTilis Teoriuli<br />
teqstisaTvis (romelSic is Zneli adgilia) moswavleebi mainc<br />
mzad iqnebian.<br />
III. esaa namdvili, sakuTriv Semamzadebeli amocanis tipi. moswavle<br />
xsnis erT an ramdenime advil amocanas axali cnebisa Tu<br />
axali wesis konkretul saxeebze ise, rom TviT es cneba Tu wesi<br />
naxsenebic ki araa. am tipis Semamazadebeli amocanebi, rogorc<br />
wesi, advilebia da maT xsnis moswavleTa didi umravlesoba<br />
(Tuki Sin ara, klasSi mainc).<br />
aseve aqtiur swavlebazea agebuli Cveneuli ganmamtkicebeli<br />
amocanebic. maTi sami ZiriTadi tipia:<br />
I. naswavli sakiTxis Cveulebrivi, pirdapiri gamoyenebis amocanebi.<br />
amgvari amocanebi uSualod mosdevs axladnaswavl sakiTxs<br />
(Tanac, pirveli amocana Zalian advilia xolme, raTa misi amoxsna<br />
TiTqmis yvela moswavles SeeZlos). garda amisa, amgvari ganmamtkicebeli<br />
amocanebi drodadro gvxvdeba xolme SemdgomSic, maT<br />
Soris momdevno klasebSic _ raTa es naswavli sakiTxi aqtiurad<br />
gameordes xolme.<br />
II. ganmamtkicebeli amocanis es tipi hgavs II tipis<br />
Semamzadebel amocanas: Tuki axladnaswavl Teoriul teqstSi<br />
gamotovebulia raime sakiTxi, an damtkicebis nawyveti, an<br />
romelime msgavsi SemTxvevis ganxilva, an damtkicebis dayvana<br />
advil saxeze da sxva _ es amocanaSi muSavdeba. Tuki moswavle<br />
mas Tavs ver gaarTmevs, am saSinao davalebis klasSi garCevisas
- 52 -<br />
mainc gairkveva.<br />
III. naswavli sakiTxis ganzogadebis, an raimegvari ganviTarebis<br />
amocanebi.<br />
amrigad, aqtiuri swavlebis mTavari sayrdeni aris mravali<br />
tipis amocanebis sagangebo Tanwyoba. amitom vaniWebT Cven udides<br />
mniSvnelobas amocanebis saguldagulod garCevas saklaso<br />
muSaobisas. magram imisaTvis, raTa amocanis saklaso garCeva<br />
nayofieri iyos, saWiroa, rom moswavles winaswar nafiqri<br />
hqondes masze damoukideblad (saSinao davalebaSi).<br />
§ 12. Semamzadebeli safexurebi<br />
yoveli saxelmZRvanelo iwyeba wina klasSi naswavlis gameorebiT,<br />
rasac mcire istoriuli cnobebi axlavs (istoriuli<br />
cnobebi axlavs sxva sakiTxebsac, oRond, cxadia, ara<br />
gasazepireblad). Semdeg TandaTan iwyeba axali sakiTxebi. magram,<br />
rogorc iTqva, TiTqmis yoveli axali sakiTxis gadmocemas win<br />
uZRvis, erTi dRiT uswrebs saTanado Semamzadebeli safexuri. es<br />
safexuri sagangebod Sedgenili, Tftbtxbwm!tbljUy{f!nphf{jmj<br />
Semamzadebeli amocanebisagan Sedgeba, Teoriuli teqstis gareSe.<br />
swored Semamzadebeli amocanebiT iqmneba hbovxzwfufmj!<br />
brujvsj! n{bpcb. esaa Cveni kursis yvelaze mTavari siaxle da<br />
Tavisebureba.<br />
Semamzadebeli safexuri moicavs an erT mTlian gakveTils _<br />
da maSin am gakveTilis saTauria `amocanebi~, an gakveTilis<br />
nawils _ mxolod ramdenime amocanas. ese igi, Semamzadebel<br />
safexurs an calke gakveTili eTmoba, an igi SeerTebulia wina<br />
gakveTilis ganmamtkicebel da sxva amocanebTan.<br />
cxadia, maswavlebelma unda icodes, Tu risi Semzadeba xdeba<br />
ama Tu im Sesamzadebeli amocanebiT. magram man araviTar<br />
SemTxvevaSi ar unda Tqvas es, ar unda ixmaros is cnebaterminebi<br />
Tu moqmedebebi, romelTa Semzadebac xdeba. unda<br />
amoixsnas mxolod mocemuli gakveTilis konkretuli amocanebi,
- 53 -<br />
im terminebis farglebSi, romlebic an maTive pirobaSia<br />
mocemuli, an adrea naswavli.<br />
amrigad, maswavlebeli mTlianad saxelmZRvanelos unda mihyves,<br />
mas win ar unda gauswros. saWiroebis SemTxvevaSi unda<br />
daexmaros moswavleebs sakiTxis arsis aRmoCenaSi, da Tan Seavsos<br />
saTanado amocanebiT Tematikuri krebulidan. Tuki klasi,<br />
mizezTa gamo, sakmaod kargad ver iswavlis ama Tu im Znel<br />
sakiTxs, maswavlebeli erTi dRiT SeaCerebs winsvlas ZiriTad<br />
saxelmZRvaneloSi da dauTmobs erT damatebiT gakveTils Znel<br />
sakiTxs, SeurCevs ra mas saTanado amocanebs Tematikuri<br />
paragrafebidan _ rogorc saklaso, ise saSinao samuSaoebisaTvis.<br />
ase rom, maswavlebels damoukideblad mouwevs mxolod Semavsebeli<br />
amocanebis SerCeva Tematikuri krebulidan.<br />
maswavleblis TviTmoqmedebisaTvis Tematikur krebulebSi<br />
gacilebiT ufro didi sarbielia, vidre ZiriTad gakveTilebSi.<br />
Semamzadebel safexurTa nimuSebi<br />
magaliTisTvis ganvixiloT XIV Tavis me-3 da me-4<br />
gakveTilebi:<br />
upmgbtj!hboupmfcboj da hboupmfcjt!psj![jsjUbej!Uwjtfcb.<br />
me-3 gakveTilSi 9 amocanaa. amocana nulovani nomriT, rogorc<br />
yovelTvis, gakveTilis teqstis gasaazrebelia. 1-li, me-2 da me-3<br />
amocanebi naswavli sakiTxis _ tolfasobis _ gansamtkicebladaa.<br />
Tanac, 1-li da me-2 _ advili amocanebia, romlebsac sWirdeba<br />
naswavlis pirdapiri, standartuli gamoyeneba _ maT naxevars<br />
mainc albaT amoxsnis TiTqmis yvela moswavle, Tan gaimeoreben<br />
ricxvis modulis cnebas; me-3 amocanac advilia, magram mas<br />
logikuri gaazreba sWirdeba. aseve, ramdenime gakveTilis win<br />
naswavlis gansamtkicebelia me-7, geometriuli amocana, romelic<br />
agreTve Zalian advilia. adre naswavlis gansamtkicebelia agreTve<br />
me-5 amocanac, Tanac, igi ukve naxevrad Semamzadebelia, radgan<br />
msgavs SesakrebTa SeerTebis gaxseneba aucilebelia momdevno<br />
gakveTilebis Sesaswavlad. sazogadod, Cven naswavls yovelTvis
- 54 -<br />
swored ase _ aqtiurad, amocanis saSualebiT vaxsenebT klass.<br />
mTavari ki danarCeni amocanebia: me-4, me-6 da me-8. samive es<br />
amocana _ momdevno gakveTilis Semamzadebelia. amasTan, me-4<br />
amocanis amoxsnas TiTqmis ar sWirdeba maTematikis codna, mcire<br />
mosazrebac eyofa da mas advilad amoxsnis is moswavlec,<br />
romelsac maTematika naklebad exerxeba, magram mosazrebulia.<br />
asea Tu ise, es amocana klasSi gairCeva da, momdevno<br />
gakveTilisTvis mainc, moswavleebs ecodinebaT, rom 2x + 15 = 25<br />
+ x gantolebis tolfasia x + 15 = 25 gantoleba. ese igi,<br />
SeiZleba ucnobis gadatana gantolebis meore mxares, oRond<br />
sapirispiro niSniT. amave daskvnas ganamtkicebs me-6 amocanis I<br />
da II qveamocanebi. arsebiTad, moswavleebi gamoiyeneben<br />
gantolebis ZiriTad Tvisebas ise, rom Camoyalibebulic ki ar<br />
eqnebaT igi _ da wesis zogadi Camoyalibeba am gakveTilze arcaa<br />
saWiro! mas momdevno gakveTilisTvis moswavle TviTon<br />
`aRmoaCens~, Semdeg sityvieradac gaiazrebs, roca waikiTxavs me-4<br />
gakveTilis teqsts.<br />
aseve aqtiurad Seamzadebs gantilebis meore ZiriTad Tvisebas<br />
me-6 amocanis III da IV qveamocanebi da me-8 amocana.<br />
amrigad, me-3 gakveTilidan sami amocana aris momdevno, me-4<br />
gakveTilis Semamzadebeli. maswavlebelma es amocanebi unda<br />
gaarCevinos moswavleebs klasSi Cveulebriv, saSinao davalebis<br />
garCevisas, oRond ise, rom ar Camoayalibos, arc ki axsenos<br />
is axali wesebi, romlebsac es amocanebi Seamzadebs.<br />
es wesi moswavleebs momdevno gakveTilis Teoriul teqstSi<br />
SexvdebaT (da mas Sin damoukideblad waikiTxaven). am gakveTilze ki<br />
amocanebi amoixsneba mxoloddamxolod konkretulad, yovelgvari<br />
ganzogadebis gareSe, anu maTi konkretuli Sinaarsidan gamomdinare!<br />
zogadi wesis Camoyalibeba maswavlebelma ar unda moiTxovos. am<br />
wesis gasaazreblad moswavleebi ukve mzad iqnebian da TviTon unda<br />
waikiTxon da gaiazron igi Sin momdevno gakveTilis momzadebisas.<br />
sxvaTa Soris, ganxiluli gakveTilidan TvalsaCinod gamoCnda
- 55 -<br />
`saymawvilo maTematikis~ kidev erTi arsebiTi maxasiaTebeli:<br />
neli gaRrmavebuli swavleba, Zalian mcire-mcire nabijebiT<br />
winsvliT _ raTa saSualo moswavlesac xelewifebodes axali<br />
sakiTxis damoukideblad aRmoCena da gaazreba (arc Zlieri<br />
moswavle moiwyens, radgan misTvisac xom gvaqvs sakmao saazrovno<br />
masala _ bolo amocanebi). rogorc vnaxeT, Cven TiTo gakveTils<br />
vuTmobT iseT sakiTxebs, romlebic sxva kursebSi, rogorc wesi,<br />
erTdroulad iswavleba.<br />
§ 13. erTi gakveTilis sanimuSo konspeqti:<br />
Tavi I, gakveTili 6<br />
gakveTilis dasawyisSi maswavlebeli Camoivlis da Tvalis<br />
erTi movlebiT Seamowmebs, Tu rogor aqvT Sesrulebuli<br />
davaleba moswavleebs. amis gakeTeba kargia imisaTvis, raTa<br />
TviTeulma moswavlem icodes, rom mis namuSevars maswavlebeli<br />
aucileblad naxavs. yvela moswavles kalmistari ukavia da avsebs<br />
da xvews Tavis saSinao namuSevars [dawvrilebiT _ ix. $ 9].<br />
1. maswavlebeli rig-rigobiT akiTxebs moswavleebs am<br />
amocanis I, II, III da IV qveamocanebs da sTxovs maT axsnan, ratomaa<br />
marTebuli an mcdari Sesabamisi ormxrivi utoloba. magaliTad,<br />
IV utoloba araa marTebuli, radganac 1/125 = 0,008, es araa<br />
naklebi 0,008-ze.<br />
2. maswavlebeli isev rig-rigobiT gamoiyvans moswavleebs<br />
dafasTan da dawerinebs Sesabamisi qveamocanebis pasuxebs.<br />
araswori pasuxis SemTxvevaSi Secdomas Seasworebs sxva<br />
moswavle, da ara maswavlebeli.<br />
3. maswavlebeli romelime moswavles SeekiTxeba, Tu romel<br />
ricxvebs SeiZleba aRniSnavdes w da ratom. ramdenime aseTi<br />
ricxvis CamoTvlis Semdeg pasuxs gaagrZelebinebs sxva moswavles.<br />
4. am amocanis garCevisas maswavlebelma moswavleebis<br />
yuradReba unda gaamaxvilos imaze, rom winadadebaTa simcdaris
- 56 -<br />
dasamtkiceblad sakmarisia erTi magaliTic ki, xolo<br />
marTebulobis dasamtkiceblad magaliTebis moyvana araa sakmarisi.<br />
magaliTad, I winadadebis marTebuloba unda daasabuTon<br />
amgvari msjelobiT:<br />
raki a < g < b, amitom a < g da g < b. maSin, cxadia,<br />
marTebuli iqneba aramkacri utolobebic: a ≤ g da g ≤ b. ese igi<br />
marTebuli iqneba utoloba a ≤ g ≤ b.<br />
II winadadebis dasamtkiceblad ki moviyvanoT amgvari<br />
magaliTi: vTqvaT, a = g = b = 2. am SemTxvevaSi miviRebT<br />
utolobebs: 2 ≤ 2 ≤ 2 da 2 < 2 < 2 . cxadia, am<br />
utolobebidan pirveli marTebulia, meore _ mcdari. amitom II<br />
winadadeba mcdaria.<br />
5. es amocana, iseve rogorc wina amocana, logikur<br />
msjelobebs moiTxovs da amitom moswavleTaTvis sakmaod<br />
Znelebia. maswavlebeli ar unda moelodos, rom moswavleebi<br />
unaklod Caatareben saWiro msjelobebs. magram swored amgvari<br />
amocanebis amoxsna TandaTanobiT ganuviTarebs moswavleebs<br />
logikuri azrovnebis metad saWiro unar-Cvevebs.<br />
maswavlebeli, iseve rogorc sxva amocanebis garCevisas,<br />
TviTeul qveamocanas sxvadasxva moswavleebs akiTxebs da pasuxs<br />
asabuTebinebs. magaliTad: I ar SeiZleba moxdes, imitom, rom<br />
Tu a > b -ze, maSin a ver iqneba b-ze naklebi.<br />
II SeiZleba moxdes im SemTxevaSi, roca a = b.<br />
IV SeiZleba moxdes, radganac utolobebs Soris weria sityva<br />
an. maswavlebeli ikiTxavs: _ rodis ar SeiZleba moxdes IV ? {im<br />
SemTxvevaSi, roca a = b, radganac maSin arc a>b da arc a
- 57 -<br />
farTobi {fuZis perimetrisa da simaRlis namravlis tolia}.<br />
pirvel rigSi unda gamovTvaloT fuZis perimetri, Semdeg ki<br />
gamovTvliT ucnobi gverdis sigrZes.<br />
8. maswavlebeli akiTxebs moswavleebs maT mogonil amocanebs<br />
da ambobs, ramdenad kargad aris Sedgenili isini. Seaqebs iseT<br />
moswavleebs, romlebac Seadgines moulodneli, ucnauri, saintereso,<br />
Tanac marTebulad dasabuTebuli amocanebi.<br />
amis Semdeg maswavlebeli gaarkvevs, Tu rogor gaiges moswavleebma<br />
gakveTilis Teoriuli nawili, anu gakveTilis teqsti.<br />
amisaTvis maswavlebeli romelime moswavles gaiyvans dafasTan da<br />
daawerinebs mas raime ormxriv, magram mkacr utolobas,<br />
waakiTxebs da gaarkvevinebs, marTebulia dawerili utoloba Tu<br />
ara da ratom. magaliTad: 3,2 < 4,5 < 4,5 . moswavlem unda<br />
upasuxos, rom es utoloba mcdaria, radganac 4,5 araa naklebi<br />
4,5-ze. Tuki moswavle arasworad an arasrulad upasuxebs, maSin<br />
SekiTxvas upasuxebs sxva moswavle. amiT moxdeba wina, me-5<br />
gakveTilis mokle gameoreba.<br />
amasobaSi maswavlebeli dafaze<br />
swrafad xeliT daxazavs wriul<br />
rgols (nax.1) da ityvis:<br />
_ CawereT ormxrivi utoloba,<br />
romelic gviCvenebs, rom A<br />
wertili am rgolis wertilia.<br />
maswavlebeli gamoiyvans romelime moswavles da daawerinebs<br />
pasuxs dafaze. moswavleebs swori pasuxis dawera ar unda gau-<br />
WirdeT, radganac Sesabamisi Semamzadebeli amocana gairCeoda wina<br />
gakveTilze (me-4 amocana). daweren: 12 m ≤ |AO| ≤ 16 m.<br />
_ ratomaa saWiro aq aramkacri utolobebis niSnebi?<br />
{imitom, rom A wertili SeiZleba iyos sazRvris wertilic.<br />
am SemTxvevaSi ki gveqneba tolobebi}. _ ra moxdeboda, erT
- 58 -<br />
mxares mkacri utoloba, rom dagvewera, magaliTad:<br />
12 m < |AO| ≤ 16 m ? {es utoloba mcdari iqneba, roca A<br />
wertili rgolis Siga wrexazis wertilia}.<br />
maswavlebeli dafaze dawers raime ormxriv utolobas,<br />
magaliTad, aseTs: 1,2 ≤ a < 1,6 da romelime moswavles sTxovs<br />
waikiTxos is ornairad: {1,2 naklebia an tolia a-ze da a<br />
naklebia 1,6-ze; meorenairad: a metia an tolia 1,2-ze da<br />
naklebia 1,6-ze}. sxva moswavles ki SeekiTxeba, marTebulia Tu<br />
ara es utoloba da ratom, roca a = 1,2 da a = 1,6 .<br />
gakveTilis bolos maswavlebeli amoikiTxavs sias da Tan<br />
dawers moswavleTa Sefasebebs.<br />
sazogadod, gamosaTvlel an standartuli pasuxis mqone sxva<br />
amocanebis garCevas SedarebiT ufro naklebi dro unda daeTmos.<br />
aseT SemTxvevaSi umjobesia maswavlebelma TviTon daawerinos<br />
romelime moswavles dafaze axali savarjiSo da yvelas<br />
mosTxovos misi Sesruleba. magaliTad, garCeuli amocanebidan<br />
pirvelis wakiTxvisas maswavlebelma SeiZleba moswavles dafaze<br />
daawerinos utoloba 0,06 ≤ 3/50 ≤ 2 da mosTxovos yvelas, rom<br />
Tav-TavianT rveulebSi CaweriT gaarkvion, marTebulia es<br />
utoloba Tu ara.<br />
TfojTwob; ganxiluli gakveTili sakmaod datvirTulia. amitom<br />
SedarebiT susti klasis SemTxvevaSi, SesaZlebelia, yvela amocanis<br />
safuZvliani garCeva verc moeswros. es araa sagangaSo. maswavlebelma<br />
davalebad mainc unda misces momdevno gakveTili. zogierT<br />
gakveTilze SeiZleba piriqiT moxdes, morCes dro. aseT<br />
SemTxvevaSi maswavlebelma SeiZleba gaarCevinos adre gaurCeveli,<br />
vermoswrebuli amocana an gaakeTebinos axali amocana saxelmZRvanelos<br />
Tematikuri paragrafebidan, Tavisi Sexedulebisamebr.<br />
§ 14. Temis damuSavebis nimuSi _ usasrulobis cneba<br />
VII klasis programis mTavari Taviseburebaa usasrulobis cnebis<br />
Rrma da safuZvliani damuSaveba. igi maTematikis, fizikis, as-
- 59 -<br />
tronomiis, logikis, filosofiisa da Teologiis erTerTi umniSvnelovanesi<br />
da uaRresad saintereso cnebaa [12]. am cnebis<br />
gareSe SeuZlebelia wrfis, sibrtyis, kuTxis, ricxviT simravle-<br />
Ta, iracionaluri ricxvisa da mravali sxva umniSvnelovanesi ma-<br />
Tematikuri cnebis namdvili swavla. SeuZlebelia maTematikuri<br />
analizis umartivesi sawyisebis gaazrebac (`analizis sawyisebi~,<br />
tradiciuli programiT viTom `iswavleboda~, Tumca mas TiTqmis<br />
veravin swavlobda. arsebiTad, es arcaa maTematika, aramed uSinaarso<br />
da maTematikuri TvalsazrisiT gaumarTav cnebebze da formulebze<br />
meqanikuri manipulaciebis sistemaa; mas kompiuteruli<br />
programa gacilebiT ukeTesad gaakeTebda).<br />
yovelive amis gamo usasrulobis cneba uTuod imsaxurebs imas,<br />
rom safuZvlianad iqnes damuSavebuli. cxadia, amas sakmao raodenobis<br />
saswavlo saaTebi sWirdeba. amitom gviwevs zogi sakiTxis<br />
gadatana Semdgom klasebSi (magaliTad, aqsioma da Teorema geometriaSi;<br />
algebruli gardaqmnebi). magram es mainc aucilebelia.<br />
rogorc wesi, saskolo kursebSi usasrulobis cneba Zalian<br />
zereled muSavdeba an TiTqmis ar muSavdeba. amis gamo, rogorc<br />
gviCvenebs Cveni sagangebo gamokvlevebi, es cneba ar esmis ara<br />
mxolod skoladamTavrebulTa 95 %-ze mets, aramed maTematikursabunebismetyvelo<br />
ganxris specialistebsac ki.<br />
usasrulobis cnebis swavlebis Cveneuli meTodika emyareba,<br />
garda zemore aRwerili zogadi safuZvlebisa, Cven mier Catarebul<br />
sagangebo fsiqologiur eqsperimentsac [12]. cnebis swavlebaSi<br />
TvalsaCinod vlindeba da xorcieldeba Cveni yvela mTavari safuZveli:<br />
mzaobis principi, aqtiur-problemuri swavleba, humanisturi<br />
swavleba, gaRrmavebuli swavleba, istorizmis principi da sxva.<br />
gansakuTrebiT mkveTrad vlindeba gaerTianebuli (integrirebuli)<br />
swavleba. usasrulobis cnebis swavlebis Cveneuli meTodika<br />
bunebrivad aerTianebs maTematikis TiTqmis yvela saskolo dargs<br />
_ Tema Semdegi safexurebiT muSavdeba:<br />
I. ariTmetika: Zalian didi naturaluri ricxvebi da 10-is
- 60 -<br />
xarisxebi;<br />
II. algebra: ricxvis Cawera standartuli saxiT, moqmedebebi<br />
Zalian did ricxvebze;<br />
III. simravleTa Teoria: usasrulo simravleni;<br />
IV. gamoyenebiTi maTematika: usasrulobis cneba bunebaSi da moCvenebiTi<br />
usasrulobani;<br />
V. geometria: ricxvTa sxivi, geometriuli sxivi da wrfe;<br />
VI. maTematikis istoria.<br />
pirvelad swored am dros Semogvaqvs SemousazRvreli geometriuli<br />
nakvTebi _ sxivi da wrfe.<br />
saWiroa, kargad iqnes gaazrebuli Semdegi ZiriTadi cnebebi:<br />
1. SemousazRvreli geometriuli nakvTi _ ewodeba iseT<br />
nakvTs, romelic ver moTavsdeba vercerT wreSi (ragind didic<br />
unda iyos igi).<br />
2. usasrulo simravle _ ewodeba simravles, Tuki ar<br />
arsebobs iseTi naturaluri ricxvi, romelic metia, vidre am<br />
simravlis wevrebis raodenoba.<br />
3. or simravles ewodeba tolZalovani, Tuki maTi `wevrebis<br />
raodenoba erTidaigivea~, ufro zustad ki: Tuki arsebobs urTierTcalsaxa<br />
asaxva erTi simravlisa meoreze; ufro martivad: Tuki<br />
SesaZlebelia maTi wevrebis `dawyvileba~. sxvaTa Soris, Cveneuli<br />
meTodikiT, I klasis pirvelsave TveSi, swored dawyvilebas<br />
emyareba raodenobaTa metnaklebobisa da tolobis cnebebi.<br />
4. Tvladi simravle _ ewodeba iseT usasrulo simravles,<br />
romelic naturalur ricxvTa simravlis tolZalovania. magali-<br />
Tad, Tvladi simravleebia: mTel ricxvTa simravle, racionalur<br />
ricxvTa simravle, 45-is jerad ricxvTa simravle, 45-is<br />
naturalur xarisxTa simravle. ese igi, racionaluri ricxvi<br />
`imdenivea~, ramdenicaa naturaluri ricxvi, anda, ramdenicaa<br />
mxolod 45-is naturaluri xarisxebis simravle. swored esaa<br />
usasrulobis pirveli paradoqsi: yovel usasrulo simravles<br />
moeZebneba misive sakuTrivi qvesimravle (misi nawili), romelic
- 61 -<br />
mTliani simravlis tolZalovania!<br />
Tvladi simravleebi _ esaa yvelaze `mcirericxovani~ usasrulo<br />
simravleebi. Semdegi usasruloba masze mZlavria:<br />
5. kontinualuri (uwyveti) simravle _ ewodeba iseT<br />
usasrulo simravles, romelic namdvil ricxvTa simravlis<br />
tolZalovania. magaliTad, kontinualuri simravleebia: dadebiT<br />
ricxvTa simravle; 0-sa da 1-s Soris moTavsebul namdvil<br />
ricxvTa simravle; raime or ricxvs Soris moTavsebul anda<br />
yvela iracionalur ricxvTa simravle; monakveTis wertilTa<br />
simravle; sibrtyeze kvadratebis simravle da sxva.<br />
kontinualuri simravlis simZlavre metia, vidre _ Tvladisa.<br />
ese igi, monakveTze `imdenive~ wertilia, ramdenic _ wrfeze!<br />
magram `gacilebiT meti~ wertilia, vidre racionalurkoordinatebiani<br />
wertilebia mTel wrfeze!<br />
sxvaTa Soris, Zalian advilia imis<br />
damtkiceba, rom monakveTi tolZalovania<br />
masze grZeli nebismieri monakveTisa.<br />
amisaTvis sakmarisia, ganisazRvros<br />
amgvari urTierTcalsaxa<br />
asaxva (`dawyvileba~):<br />
Cveneuli programiT, me-3, me-4 da me-5 sakiTxebi mxolod X-<br />
XI klasebSi iswavleba. xolo 1-li da me-2 _ VII klasSi.<br />
aqve ganvixiloT ramdenime sakiTxi, raTa dazRveuli viyoT<br />
gavrcelebuli mcdari warmodgenebisagan.<br />
o!b!l!w!U!f!c!j!<br />
!<br />
tbtsvmj;!!!!!)vtbtsvmp*!Tfnptb{Swsvmj;!!!!!)vtbtsvmp*!Tfnpvtb{Swsfmj;!<br />
wertili<br />
monakveTi, Sekruli mrudi,<br />
Sekruli texili, mravalkuTxedi,<br />
wre, rgoli, elifsi...<br />
wrfe, sxivi,<br />
naxevarsibrtye,<br />
sibrtye, kuTxe...<br />
1) monakveTi, wre Tu samkuTxedi _ SemosazRvruli nakvTebia,
- 62 -<br />
magram, viTarc wertilTa simravleebi _ usasruloebia. sazogadod,<br />
yvela geometriuli nakvTi, wertilis garda, usasrulo<br />
simravlea; xolo SemousazRvreli nakvTia kuTxe, rac gansakuTrebiT<br />
arsebiTia! amgvarad, nakvTebi SeiZleba ise davajgufoT ↑<br />
2) raimes daTvla rom SeuZlebelia, es imas sulac ar niSnavs,<br />
rom raodenobaa usasrulo. magaliTad, bunebaSi, dedamiwaze Tu<br />
mTel mzis sistemaSi arcerTi nivTieri erToblioba araa usasrulo.<br />
sasrulia yvela mdinareSi, zRvasa da okeaneSi wylis wveTebisa<br />
Tu qviSis marcvalTa raodenoba, odesme mcxovreb adamianTa<br />
Tmis Rerebis raodenoba, anda maT mier odesme warmoTqmuli sityvebis<br />
raodenoba, mTel dedamiwaze molekulebisa Tu atomebis<br />
raodenobac ki! amis damtkiceba araa Zneli _ igi mocemulia VII<br />
klasis saxelmZRvaneloSive.<br />
mTavaria ara daTvlis SesaZlebloba, aramed raodenobis zemodan<br />
SemosazRvris, zemodan Sefasebis SesaZlebloba. ese igi, iseTi<br />
ricxvis moZebna, romelic namdvilad metia, vidre es raodenoba.<br />
3) maTematikaSi Tvladi simravle, rogorc iTqva zemore,<br />
usasruloa da ara sasruli!<br />
4) racionalur ricxvTa simravle araa uwyveti, Tumca igi<br />
yvelgan mkvrivia namdvil ricxvTa simravleSi. ese igi, ricxvTa<br />
wrfeze racionalurkoordinatebiani wertilebi arsad ar qmnis<br />
uwyvet monakveTs, magram yovel monakveTSi (ragind mcireSi)<br />
mainc usasrulod mravali racionaluri ricxvia.<br />
yovelive amis gamoa, rom usasrulobis cnebis gaazreba arsebiTad<br />
afarTovebs adamianis gonebis Tvalsawiers da Tvisebrivad,<br />
naxtomiseburad aviTarebs azrovnebas.<br />
§ 15. geometriis swavlebis safexurebi<br />
maTematikis gaerTianebuli swavlebas erT rames uwuneben:<br />
geometria `daiCagrebao~. geometria aris erTaderTi saskolo<br />
sagani, romelic moswavles mkacri deduqciuri Teoriis agebas<br />
uCvenebs. amitom misi daknineba dauSvebelia.
- 63 -<br />
es asea, magram saamisod ramdenad saWiroa mTeli Teoriis ageba?<br />
Cveneul gaerTianebul kursSi geometriis wili gacilebiT metia,<br />
vidre amJamad arsebul sxva kursebSi. Tanac, es iwyeba pirvelive<br />
klasidan, romlis programis ramdenime siaxleTagan erTerTi<br />
swored geometriis mkveTri gaZlierebaa (logikasTan erTad).<br />
saxelganTqmuli maTematikosi, akademikosi v. arnoldi abstraqtul-formalistur<br />
midgomas saskolo geometriisadmi moixseniebs<br />
`sqolastikuri tvinisWyletis~ saxeliT, romelic namdvili<br />
`kiseliovuri~ maTematikuri codnis mospobas cdilobs! (ruseTSi<br />
ukve arsebiTadaa Secvlili geometriis kursi, xolo<br />
inglisSi _ da dasavleTis TiTqmis yvela saxelmwifoSi asea _<br />
60-iani wlebis Semdeg aRar yofila geometriis aqsiomatikuri<br />
swavleba, mravali Teoremis grZel-grZeli damtkicebebiT).<br />
TavisTavad, geometriac logikasa da simravleTa Teoriaze<br />
(topologiasTan erTad) gvaqvs dafuZnebuli. geometriis swavlebaSi<br />
ramdenime safexuri gamoikvTeba:<br />
I-IV klasebSi vaswavliT mraval geometriul cnebas: kubi,<br />
kvadrati, marTkuTxedi, wre, wertili, monakveTi, texili, mrudi,<br />
samkuTxedi, oTxkuTxedi, xuTkuTxedi, ... , mravalkuTxedi, misi<br />
gverdi, waxnagi, wibo, wvero da sxva. magram TiTqmis yovelTvis<br />
gansazRvrebaTa gareSe vaswavliT. ganisazRvreba mxolod is cnebebi,<br />
romlebic eqvemdebareba advil, da rac mTavaria, TvalsaCino<br />
gansazRvrebas (magaliTad, texili, Siga are, sazRvari da sxva).<br />
swavlebis sayrdenia uSualo TvalsaCino Cveneba, xatovani warmodgena<br />
da moswavlis mier mravali geometriuli amocanis amoxsna.<br />
maT Soris mTavaria mravalferovani amocanebi xazvaze, romel-<br />
Ta Sesrulebisas moswavle sakuTari xeliT xazavs nakvTebs.<br />
V-VI klasebSi iwyeba mcire Teoria, ganisazRvreba ramdenime<br />
mniSvnelovani geometriuli cneba (magaliTad, wre da wrexazi).<br />
safuZvlianad iswavleba farTobisa da moculobis cnebebi da<br />
gazomva, simetria. planimetriis paralelurad stereometriac<br />
muSavdeba: sfero, birTvi, cilindri, kubi, aguredi (marTkuTxa
- 64 -<br />
paralelepipedi)... es xazi dagvirgvindeba eileris ulamazesi<br />
TeoremiT mravalwaxnagebisaTvis.<br />
sazogadod, stereometriis swavleba planimetriis<br />
paralelurad _ Cveneuli meTodikis erTerTi Taviseburebaa. V<br />
klasSi erTad iswavleba birTvi, sfero, wre da wrexazi; Semdeg,<br />
kvlav wresTan dakavSirebulad _ cilindric. aseve, VI klasSi<br />
erTad iswavleba marTkuTxedi da aguredi (marTkuTxa<br />
paralelepipedi), kubi da kvardati; xolo momdevno klasebSi _<br />
samkuTxedi da samkuTxa prizma, samkuTxedi da piramida da sxva.<br />
VII klasSi pirvelad Semogvaqvs SemousazRvreli nakvTebi<br />
(sxivi, wrfe, kuTxe). viwyebT damtkicebebs. viTardeba stereometriis<br />
Temebic (Slilebi da sxva).<br />
VIII-IX klasebSi logikur da istoriul konteqstSi vaswavliT,<br />
ra aris gansazRvreba, aqsioma da Teorema. mxedvelobiT<br />
iluziebze da logikis xaziT naswavl kerZo/zogadis mimarTebaze<br />
damyarebiT vasabuTebT zogadi da zusti damtkicebis saWiroebas.<br />
aRvwerT aqsiomaTa nimuSebs da maTgan ramdenime Teoremis gamoyvanas<br />
(mxolod nimuSisaTvis). mkacrad ganvsazRvravT wina<br />
wlebSi TvalsaCino doneze naswavl geometriul cnebebs, da zogierT<br />
axalsac. vaswavliT umTavres planimetriul Teoremebs<br />
damtkicebebiT (gavdivarT daaxloebiT piTagorasa da Talesis Teoremebamde),<br />
oRond ara formalur-aqsiomatikuri midgomiT.<br />
vaswavliT mxazvelobiTi geometriis sawyisebsac _ sxeulTa<br />
xedebsa da izometrias. danarCeni sakiTxebi maRal skolaSi, X-<br />
XII klasebSi iqneba.<br />
geometriis namdvil dafuZnebas mxolod maTematikuri fakultetis<br />
studentebi (da Tan maTi mxolod nawili) Tu iswavlian.<br />
skolis moswavlisaTvis sakmarisia mxolod imis zogadi codna,<br />
rom maTematikaSi saWiroa yvela Teoriis aqsiomebze dafuZneba da<br />
cnebaTa mkacri gansazRvreba, Semdeg ki Teoremebis mkacrad<br />
damtkiceba (rac mdgomareobs yoveli axali Teoremis logikur<br />
dayvanaSi ukve damtkicebul Teoremebze an aqsiomebze). sakmarisia
- 65 -<br />
amis ramdenime konkretuli nimuSis swavleba. xolo mTeli<br />
geometriis amgvari ageba saskolo kursSi, jer erTi, ubralod<br />
SeuZlebelia da mraval logikur siyalbesa da winaaRmdegobas<br />
iwvevs. amgvari geometria araTu ar emsaxureba logikas (rac<br />
geometriis mTavari mowodeba unda iyos), aramed azianebs da<br />
aferxebs kidec mas. da meorec, amgvari geometria arc aravis<br />
sWirdeba, moswavleTa im 0,1%-is garda, romlebmac es umaRles<br />
saswavlebelSi mainc wesierad unda iswavlon. Tumca,<br />
samwuxarod, mizezTa gamo, verc iq swavloben. amitom<br />
maswavlebelTa udidesma nawilmac ki ar icis es!<br />
Cven veTanxmebiT im azrs, rom moswavlem unda gaiazros, Tu<br />
ras niSnavs Teoriis mkacri deduqciuri ageba, radganac esaa nam-<br />
dvili maTematikis erTerTi mTavari maxasiaTebeli. magram saqme<br />
isaa, rom amisaTvis srulebiT araa aucilebeli, nUfmj! hfp.<br />
nfusjb deduqciurad aigos. skolis farglebSi es ubralod SeuZlebelia<br />
da Tanac, amgvari mcdeloba sapirispiro Sedegsac<br />
iZleva. rogorc stereometria iswavleba arasrul deduqciurad<br />
(yvela saskolo kursSi asea), aseve SeiZleba iswavlebodes<br />
planimetriac. sakmarisia, rom moswavlem gaiazros, Tu ra aris:<br />
sazogadod cneba da pirveladi cneba; aqsioma da Teorema; mkacri<br />
damtkiceba da sapirispiro magaliTi; aucilebeli, sakmarisi,<br />
aucilebeli da sakmarisi piroba; da ras niSnavs mwyobri Teoriis<br />
mkacrad (deduqciurad) ageba. amisaTvis Cveneul kursSi<br />
gaTvaliswinebuli gvaqvs sagangebo logikuri sqemebi, romlebic<br />
TvalsaCinod aCvenebs, Tu rogor daiyvaneba erTi Teorema<br />
meoreze, meore _ mesameze, da ase Semdeg _ aqsiomebamde. gamoCndeba<br />
isic, rom SeiZleboda Teoriis sxvagvarad ageba: sxva aqsiomebiT,<br />
anda jer im Teoremis damtkicebiT, romelic sxvagvari<br />
agebisas mogvianebiT mtkicdeboda. mTavaria, rom ar dairRves<br />
TviT agebis wesi (principi). TvalsaCinod vaCvenebT imis magaliTsac,<br />
rom dauSvebelia warmoiqmnas logikuri `Sekruli wre~.<br />
Semdeg sruli gansazRvrebebiTa da bolomde mkacri damtkicebe-
- 66 -<br />
biT avagebT _ ara mTliani geometriis, Tundac mxolod planimetriis<br />
Teorias (rac SeuZlebelicaa) _ aramed mxolod mis<br />
ramdenime xazs, nimuSisaTvis.<br />
kursis gaerTianeba am mxrivac did upiratesobas mogvcems. saqme<br />
isaa, rom moswavles ar unda egonos, rom simkacre<br />
(deduqciuroba) mxolod geometriisTvisaa niSneuli. arada, swored<br />
amgvari mcdari warmodgena Seeqmneba mas, radganac yvela<br />
danarCeni saskolo maTematikuri dargi Zalian Sorsaa<br />
maTematikuri simkacrisa da deduqciurobisagan. amis Secvla ki<br />
yovlad SeuZlebelia: skolaSi naturalur ricxvTa metaTeoriisa<br />
Tu dedekindiseuli gankveTebis swavlebas xom ver daviwyebT!<br />
magram gaerTianebul kursSi kvlav TvalsaCino sqemebisa da<br />
ramdenime nimuSis saxiT kargad gamoCndeba, rom geometriis<br />
magvarad SeiZleboda sxva maTematikuri Teoriebis agebac,<br />
magaliTad, ariTmetikisa. moswavleebs SeeqmnebaT warmodgena<br />
imaze, Tu ra aris sazogadod maTematikuri Teoriis arsi _ da<br />
ara mxolod planimetriisa.<br />
§ 16. aramaTematikurad, gumaniT amosaxsneli amocanebi<br />
gumans (intuicias) udidesi mniSvneloba aqvs aramarto<br />
cxovrebasa da xelovnebaSi, aramed agreTve yovelgvar<br />
saqmianobaSi, mkacr mecnierebaSic ki. cxadia, sul sxvaa eqimis<br />
gumani da sul sxva _ xelosnisa; sul sxvaa istorikosis gumani<br />
da sul sxva _ maTematikosisa; ufro metic, gumani geometriaSi<br />
gansxvavdeba gumanisgan ariTmetikaSi. gumanis TviTeul am ganStoebas<br />
sakuTrivi ganviTareba sWirdeba.<br />
saskolo maTematikaSi sayovelTaodaa cnobili, rom moswavle<br />
mraval cnebasa Tu moqmedebas gumaniT Seimecnebs _ da ara mkacri<br />
logikiT. gansakuTrebiT xSirad es dawyebiT klasebSi xdeba. magali-<br />
Tad, meoreklaseli mxolod gumaniT swvdeba cnebebs `marTkuTxedi~<br />
Tu `Sekreba~. xolo cnebebi `wrfe~, `sibrtye~ `raodenoba~<br />
da sxva _ ufros klasebSic ki mxolod gumanis amaraa.
- 67 -<br />
magram es yovelive _ gumanis udabalesi donea, roca gumani,<br />
arsebiTad, mxolod xatovan warmodgenebsa da calkeul kerZo<br />
magaliTebs emyareba. am donis gumans ganviTarebac ki TiTqmis ar<br />
sWirdeba _ igi normalur gonebas bunebrivad aqvs.<br />
maTematikosis gumanis maRali done _ esaa Zneli, misaxvedri<br />
(mosasazrebeli) amocanis amoxsnis gzis mixvedris unari. amgvari<br />
amocanebi gvxvdeba rogorc saskolo maTematikaSi (gansakuTrebiT,<br />
olimpiadebsa da konkursebSi), ise namdvil mecnierul kvlevebSi.<br />
maT winaSe logikuri msjeloba umweoa, mxolod gumani Tu gaikvlevs<br />
gzas.<br />
magram es maRali donis gumani mxolod gamorCeuli maTematikuri<br />
niWis mqone adamians SeiZleba ganuviTardes. amitom misi<br />
ganviTareba mxolod calkeul moswavleebs exeba.<br />
gumanis yvelaze mniSvnelovani donea saSualo, romelic Cveulebriv<br />
moswavles unda ganuviTardes. ganvixiloT ramdenime magaliTi.<br />
Semdegi amocana VI-VII klasebisTvisaa:<br />
mocemulia marTkuTxedi: SemdegTagan aarCieT is<br />
marTkuTxedi, romlis gverdebis sigrZeebic am marTkuTxedis<br />
gverdebis sigrZeTa proporciulia:<br />
a) b) g) d) e)<br />
amocana testuri formisaa _ mocemulia savaraudo pasuxebi.<br />
pasuxebi yovelTvis gadanomrilia qarTuli asoebiT:<br />
a) ; b) ; g) ; d) ; e) ; v) ; z) ...<br />
moswavlem unda icodes, rom amgvar amocanebSi (sadac pasuxi<br />
unda avirCioT da SesaZlo pasuxebi qarTuli asoebiTaa gadanomrili),<br />
yovelTvis mxolod erTi pasuxia xolme marTebuli.<br />
mixvedriT unda gamoiricxos yvela pasuxi, garda erTisa _ da<br />
maS swored es erTi iqneba marTebuli pasuxi.<br />
moswavleebs am droisaTvis ukve naswavli aqvT proporcia da
- 68 -<br />
misi sami Tviseba. am amocanis amoxsna SeiZleboda gazomvebiTa da<br />
gamoTvlebiT, magram amas Zalian didi dro dasWirdeboda. vinc<br />
aqtiurad icis proporciuloba (Tundac cnebaTa gareSe _ mxolod<br />
praqtikulad _ magaliTad, xelovnebaTmcodnem), is pirdapir,<br />
gumaniT mixvdeba Tu `dainaxavs~, rom marTebuli pasuxia (g).<br />
gumani saTanado mimarTulebiT dagrovebul gamocdilebas<br />
emyareba (dawvrilebiT _ ix. [10]). magaliTad, zemore amocanaSi<br />
marTebul pasuxs pirdapir ver `dainaxavs~ is adamiani, romelsac<br />
proprociuloba axali naswavli aqvs da jer kidev Rrmad ver<br />
`grZnobs~ mas _ Tundac rom proporciulobis Teoriul-cnebiTi<br />
codna sruliad unaklo hqondes!<br />
gumanis dabali done ufro `dablaa~, vidre Teoriul-cnebiTi<br />
codnis done, jer kidev araa amaRlebuli am gaazrebuli cnebis<br />
donemde. magaliTad, proporciulobis SemTxvevaSi es iqneboda is<br />
codna, romelic eqneboda mesameklasels, romelsac TvalsaCino<br />
magaliTebiT auxsnidnen, Tu ra aris proporciuloba. saSualo<br />
mesameklaselisTvis proporciulobis cnebiTi gaazreba ubralod<br />
miuwvdomelia. xolo gumanis saSualo done, piriqiT, ufro `maRlaa~,<br />
vidre Teoriul-cnebiTi codnis done, radgan misi momdevno<br />
donea, mas moicavs da damatebiT aqtiur gamocdilebasac moicavs!<br />
gumanis umaRlesi done _ esaa SemoqmedebiTi gumani, romelic<br />
mxolod saTanado niWierebis SemTxvevaSi Tu ganviTardeba [10].<br />
wmindad gumaniT amoixsneba agreTve amocanebi njbympfcjU!Tf.<br />
gbtfcb{f/ isinic testuri formisaa, mxolod erTi pasuxia mar-<br />
Tebuli. gamoTvlebis gareSe, mixvedriT unda gamoiricxos yvela<br />
pasuxi, garda erTisa _ da maS swored es erTi iqneba marTebuli<br />
pasuxi. ganvixiloT amgvari amocanis magaliTi (III klasidan):<br />
risi tolia saSualo cxrasarTuliani saxlis simaRle?<br />
a) 8 m; b) 244 m; g) 20 dm; d) 110 m; e) 30 m; v) 1 km.<br />
moswavle am droisaTvis ukve sakmaod gawafulia gazomvasa da<br />
sigrZis erTeulebSi. es Cveni zogadi wesia: amocana raimes miaxloebiT<br />
Sefasebaze Semodis mxolod mas Semdeg, rac saTanado sa-
- 69 -<br />
kiTxi ukve karga xnis ganmavlobaSi muSavdeboda da moswavles<br />
tblnbp! hbnpdejmfcb unda hqondes dagrovebuli. amitom moswavle<br />
gumaniT unda mixvdes, rom cxrasarTuliani saxlis simaRle<br />
ver iqneba 8 m Tu 20 dm (Zalian mcirea!) da verc 244 m, 110<br />
m Tu 1 km (Zalian didia!). maS, rCeba erTi SesaZlebloba _ 30 m<br />
(raki viciT, rom erTi pasuxi namdvilad marTebulia!).<br />
kidev erTi saxis amocanebi amoixsneba gumaniT, oRond saWiroa<br />
Semdgomi logikuri dasabuTeba da msjeloba. Cven xSirad gvaqvs<br />
amocanebi lbopo{pnjfsfcb{f: an kanonzomierebis damrRvevi wevris<br />
moZebnaze, an kanonzomierebis gagrZelebaze (Sesabamisi carieli<br />
adgilebis SevsebiT). amgvar amocanebSi mTavaria: kanonzomierebas<br />
mxolod erTi wevri (anda, ori _ Tuki asea miTiTebuli<br />
amocanis pirobaSi) unda arRvevdes, igi unda iyos `gamonaklisiviT~,<br />
`zedmetiviT~ danarCenTa Soris; Tanac _ yvela danarCeni<br />
wevri raimeTi unda erTiandebodes, unda ukavSirdebodes erTmaneTs.<br />
ganvixiloT, magaliTad, amocana (IV klasidan):<br />
romeli sityva arRvevs kanonzomierebas?<br />
ru; arxi; Rele; wyaro; mdinare; gube.<br />
miTiTeba: unda daukvirdeT sityvebis azrsac (mniSvnelobasac)<br />
da maT asoebsac.<br />
ganvixiloT jer asoebis mixedviT. aSkarad mcdari pasuxia,<br />
magaliTad, es: `ru radgan am sityvaSia ori aso~, es ki marTalia,<br />
magram maSin yvela danarCeni sityva riTiRa ukavSirdeba<br />
erTmaneTs? (yvela danarCen sityvaSi rom yofiliyo, magaliTad,<br />
oTx-oTxi aso, maSin es pasuxi marTali iqneboda). asoebis mxriv<br />
kanonzomierebas arRvevs arxi _ radgan mxolod igi iwyeba<br />
xmovniT, xolo yvela danarCeni iwyeba TanxmovniT. arxi _ esaa<br />
erTi marTebuli pasuxi.<br />
$ 3-Si iTqva, rom maswavlebelma unda waaxalisos moswavleTa<br />
uCveulo pasuxebi _ Tuki isini dasabuTebulia. magaliTad, Cvens<br />
amocanaSi moswavles SeiZleboda aseTi pasuxi aRmoeCina:<br />
kanonzomierebas arRvevs wyaro, radgan mxolod am sityvaSia
- 70 -<br />
xmovani o.<br />
maswavlebeli dafiqrdeba da xmamaRla daiwyebs msjelobas: _<br />
modi, SevamowmoT, danarCeni xmovnebi rogoraa? (da SeekiTxeba sxvadasxva<br />
moswavleebs): sul ramdeni xmovania qarTul enaSi? {xuTi} a<br />
xmovani ramden sityvaSia? {samSi}; i xmovani? {or sityvaSi}; e<br />
xmovani? {or sityvaSi}; u xmovani? {esec orSi}; romeli xmovani<br />
dagvrCa? {o} es o marTlac erTaderT sityvaSia! ese igi, zurikos<br />
pasuxSic marTebuli yofila, yoCaR, zuriko, Zalian kargia!<br />
sxvaTa Soris, aseTi kanonzomierebis danaxvac SeiZleba: a, i, e<br />
da u xmovanebi or-or sityvaSia, mxolod mexuTe xmovani o aris<br />
iseTi, romelic mxolod erT sityvaSia, amitom es erTi, meoTxe<br />
sityva wyaro arRvevs am kanonzomierebas. Cven xom viciT, rom<br />
kanonzomiereba zogjer ori an ramdenimec ki SeiZleba iyos!<br />
magram, moswavles rom eTqva, kanonzomierebas arRvevs ru,<br />
radgan mxolod am sityvaSiao xmovani u _ es mcdari iqneboda,<br />
radgan u xmovani aris agreTve sityvaSi gube! aseve, moswavles<br />
rom eTqva, kanonzomierebas arRvevs wyaro, radgan mxolod am<br />
sityvaSiao Tanxmovani w _ esec mcdari iqneboda, radgan<br />
gaugebari darCeboda danarCenTa gamaerianebeli kanonzomiereba _<br />
maSin Tanxmovani x aris mxolod meore sityvaSi, d da n _<br />
mxolod mexuTeSi da ase Semdeg. maSin, romeli arRvevs<br />
kanonzomierebas? araa bunebrivi, amitom pasuxi naZaladevia, ar<br />
varga. sxva yvela Tanxmovani rom marTlac or an ramdenime<br />
sityvaSi yofiliyo (rogorcaa xmovnebis SemTxvevaSi), maSin<br />
marTebuli iqneboda es pasuxi!<br />
axla aRmovaCinoT kanonzomiereba sityvebis azris, mniSvnelobis<br />
mixedviT: yvela sityvis mniSvneloba ukavSirdeba wyals,<br />
magram rogor wyals? romelia danarCeTagan gansxvavebuli? iqneb,<br />
arxi _ imiT, rom igi xelovnuria? ara _ ruc xelovnuria (mas<br />
saxeldaxelod gaTxrian xolme sarwyavad, ru aris mcire droebiTi<br />
arxi). iqneb wyaro _ radgan daileva? ara, zogjer Relis<br />
wyalsac svamen, zogjer _ arxisa! maSasadame, es arcerTi pasuxi
- 71 -<br />
ar varga. kanonzomierebas ki arRvevs gube, radgan igia erTaderTi<br />
damdgari wyali, yvela danarCeni ki moZravi wyalia.<br />
rogorc vxedavT, amgvar amocanebSi SeiZleba arsebobdes gansxvavebuli<br />
Tvalsazrisebi da gansxvavebuli pasuxebi, rac maTematikaSi<br />
uCveuloa. sazogadod, zogi sakiTxis uCveuloba logikasTan<br />
integraciiTaa gamowveuli [$ 4].<br />
§ 17. saWiro TvalsaCinoeba<br />
upirvelesad, gasarkvevia kalkulatoris sakiTxi. misi xmareba<br />
sakmaod SezRuduli unda iyos.<br />
1) yovlad dauSvebelia kalkulatoris xmareba VII-VIII klasebamde,<br />
radgan es daangrevs ariTmetikis codnas _ bavSvi ver<br />
SeZlebs zepirad Seasrulos amgvari gamoTvlebi (rogorc esaa<br />
aSS-Si da sxvagan): 126:3; 1 m 40 sm : 5; 320•4; 1 kg _ 360 g; 1 sT 45 wT + 2 sT 35 wT;<br />
3,5 miliardis naxevari; 400 aTasi laris 15 %<br />
da, rac mTavaria, naSTiani gayofa (romelic dawyebiTi<br />
klasebis mTeli ariTmetikis gvirgvinia):<br />
90:21, 1 mln : 150 aTasi, 2 t : 6, 105/45 = 2 1/3 .<br />
amgvari gamoTvlebi, jer erTi, gacilebiT advilad, uSecdomod<br />
da Tan swrafad sruldeba zepirad, gonebaSi, vidre kalkulatoriT;<br />
meorec, aviTarebs azrovnebasa da ganamtkicebs maTematikis<br />
codnas; mesamec, aucilebelia praqtikulad, radgan yvelgan kalkulators<br />
ver ixmar. amasTan, uaRresad mniSvnelovania miaxloebiTi<br />
angariSisa da miaxloebiTi Sefasebis unarCvevebi, romlebic<br />
efuZneba raodenobis gumaniT wvdomas. es unari ki ver ganviTardeba<br />
gonebaSi mravalwliani angariSis gareSe!<br />
2) VII-VIII klasebidan moswavles kalkulatoris xmarebis ufleba<br />
unda mieces mxolod im amocanebis amoxsnisas, romlebSic<br />
marTlac didi gamoTvlebia Casatarebeli (magaliTad, statistikuri<br />
Sinaarsis amocanebSi) _ saxelmZRvaneloSi sagangebod miTiTebuli<br />
iqneba, rom saWiroa kalkulatoris gamoyeneba. xolo sadac<br />
es ar iqneba miTiTebuli, iq kalkulatoris xmareba unda aikrZa-
- 72 -<br />
los. maswavlebelma es wesi unda daicvas saklaso, sakontrolo<br />
da sagamocdo werebis dros, xolo saSinao davalebis Sesrulebisas,<br />
sasurvelia, daicvas mSobelma. cxadia, amas ver daveyrdnobiT,<br />
amitom TviTon moswavlesac unda avuxsnaT, rom Tuki is<br />
borotad gamoiyenebs kalkulators, mas dauClungdeba zepiri<br />
angariSis unari da amitom sakontrolo werebze da gamocdebze<br />
dabal niSnebs daimsaxurebs.<br />
moswavlisTvis saWiro TvalsaCinoebas TviTon moswavlis saxelmZRvanelo<br />
Seicavs. aucilebelia damatebiTi stereometriuli<br />
saklaso TvalsaCinoeba: stereometriuli sxeulebis modelebi<br />
(umjobesia, mTliani xisa an plastmasisa, anda maTi muyaos modelebi);<br />
saklaso saxazavi da fargali. moswavleebi sakuTari xeliT<br />
unda zomavdnen stereometriuli sxeulebis saTanado ganzomilebebs.<br />
Zalian kargia Slilis daxazva, gamoWra da modelad<br />
Sewebeba (`saymawvilo maTematikaSi~ sakmaodaa amgvari davalebebi).<br />
CvenTvis mTavari da namdvili TvalsaCinoeba _ esaa xazva, anu<br />
moswavleebis mier sakuTari xeliT daxazuli cxrilebi, diagramebi,<br />
naxazebi da sqemebi, geometriulebic, logikurebic, ariTmetikulebic<br />
da agreTve yvela sxva saxisa. amitom `saymawvilo<br />
maTematikaSi~ Zalian mravladaa amocanebi xazvaze, zomvaze,<br />
cxrilebze, diagramebze, sqemebze, mravalgvar praqtikul<br />
samuSaoze da sxva. sxvaTa Soris, swored amgvari amocanebi aer-<br />
Tianebs organulad maTematikis sxvadasxva dargebs erTmaneTTanac<br />
da gamoyenebiT mimarTulebebTanac; swored amgvari amocanebi<br />
aviTarebs yvelaze kargad zogad gonebriv unarebs.<br />
amrigad, maTematikis gakveTilebze moswavleebs dasWirdebaT<br />
fanqari, saxazavi da zogjer _ fargali.<br />
TvalsaCinoebis farglebi<br />
TvalsaCinoebis principi klasikuri humanisturi pedagogikis<br />
ZiriTadi mcnebaa. cneba ver ganzogaddeba, ver gaiazreba da<br />
mxolod yalb, futuro, uazrod gazepirebul sityvier garsad<br />
darCeba, Tuki mas TvalsaCino sayrdeni ar eqna. TvalsaCinoebis
- 73 -<br />
mniSvneloba gansakuTrebiT didia aqtiuri swavlebis pirobebSi.<br />
es yvelaferi asea, magram TvalsaCinoebis gamoyenebis farglebi,<br />
zoma da safexurebi saguldagulod unda iyos damuSavebuli,<br />
raTa igi azrovnebisTvis _ nacvlad sayrdenisa _ Semaferxeblad<br />
ar iqces. j. bruneri [22] sagangebod aRniSnavs, rom TvalsaCinoeba<br />
mxolod zomierebis farglebSia kargi. d. uznaZec [1] aRniSnavs,<br />
rom Warbi TvalsaCinoeba Semaferxebelia cnebiTi azrovnebis<br />
ganviTarebisaTvis, romlis gareSec adamiani Tavs ver daaRwevda<br />
cxovelur dones: imwamierad mocemul, SemTxveviT, erTeul<br />
STabeWdilebaTa da aqtualuri aRqmis tyveobas, qcevis impulsur<br />
dones. adamiani cnebaTa sistemaSi sinamdvilis mTlianobas swvdeba<br />
da amis Sedegad eZleva mis qcevas cnobieri, nebismieri da mizandasaxuli<br />
xasiaTi. d. uznaZis azriT, saskolo swavleba, arsebi-<br />
Tad, mecnieruli cnebiTi azrovnebis ganviTarebas emsaxureba.<br />
aRmqmeli adamianis cnobiereba SezRudulia, mas axasiaTebs `situaciuroba~,<br />
ese igi, `aq da axla~ mocemuliT SezRuduloba,<br />
konkretul-materialuroba. azrovnebis upiratesobas aRqmasTan<br />
SedarebiT J. piaJe [21] amgvarad aRwers: `mokle manZilebisa da<br />
realuri gzebidan Tavdaxsna mxolod azrovnebas SeuZlia, Tavisi<br />
im miswrafebiT, rom mTeli garemomcveli samyaro moicvas, TviT<br />
uxilavis CaTvliT, zogjer imis CaTvliTac ki, risi warmodgenac<br />
SeuZlebelia. swored subieqtsa da obieqts Soris arsebuli<br />
manZilis es usasrulo mateba aris mTavari siaxle, warmomqneli<br />
cnebiTi inteleqtisa da im Zalmosilebisa, romelic inteleqts<br />
operaciebis warmoSobas SeaZlebinebs~.<br />
ekonomikur-teqnologiurad ganviTarebul qveynebSi mimdinare<br />
procesebi gviCvenebs, rom Tanamedrove cxovrebis wesi xels uwyobs<br />
ekranul sainformacio saSualebaTa sayovelTao gavrcelebas,<br />
rogoricaa: televizia, kompiuteri, video, kalkulatori, kino,<br />
Sou, sareklamo dafebi, moda, podiumi, mdidrulad dasuraTebuli<br />
bukletebi da Jurnalebi. Sesabamisad kiTxvisa da weris<br />
mniSvneloba knindeba. sityva-cnebas mxedvelobiTi xatebi aZevebs,
- 74 -<br />
mkiTxvels _ mayurebeli, sityvier-wignier adamians _ mxedvelobiT-ekranieri<br />
adamiani, mwignobrul kulturas _ komfort-materializmi.<br />
es seriozul uaryofiT Sedegebs iwvevs, rogorc sulieri,<br />
pirovnul-fsiqologiuri mxriv, ise kulturul-inteleqtualuri<br />
da socialuri mxriv. ekranuli civilizacia Zlier<br />
amuxruWebs agreTve adamianis nebelobis ganviTarebas, mis mier<br />
infantiluri egocentrizmis daZlevas, pasuxismgeblobisa da zneobrivi<br />
Segnebis ganviTarebas, pirovnebis Camoyalibebas.<br />
televizoris an kompiuteris ekrani mxolod mexsierebas avsebs<br />
nawilobriv, Tanac mouwesrigebeli da daqsaqsuli informaciis<br />
groviT. Sedegebic kanonzomieria. specialuri gamokvlevebi<br />
gviCvenebs dasavleTis qveynebSi maTematikuri codnis mkveTr daqveiTebas,<br />
agreTve zogadi wignierebis, saerTo codnis dakninebas,<br />
enobriv unarTa dasustebas, zepiri da weriTi metyvelebis ukiduresad<br />
gaRaribebas, gramatikuli struqturebis gamartivebas,<br />
enis metaforul SreTa standartizacias (Jargonad gadagvarebas),<br />
leqsikis uaRres simwires. maSin, roca yvela es Tviseba pirovnuli<br />
cnobierebis simdidrea da pirovnebis ganviTarebas ganapirobebs.<br />
ekrani gansakuTrebiT damaClungeblad dawyebiTi klasebis asakis<br />
bavSvze moqmedebs. $ 5-Si Cven ganvixileT bavSvis orgvari<br />
CamorCena _ gonebrivi da nebelobiTi. ekrani orives gamomwvevi<br />
SeiZleba iyos.<br />
gonebaWvretiTi wignierebis burjia ori mTavari saskolo sagani:<br />
mSobliuri ena-literatura da maTematika _ xatobriv-niv-<br />
Tieri konkretulobis sapirispirod. am sagnebis swavlebisas gansakuTrebiT<br />
saWiroa TvalsaCinoebisa da suraTovnebis mozRudva<br />
rogorc saxelmZRvaneloTa gaformebis, ise arsebiTi mxriv.<br />
zedmeti da araarsebiTi (sakiTxTan mxolod zereled dakavSirebuli)<br />
suraTovneba gansakuTrebiT amerikuli stilis saxelmZRvaneloebs<br />
axasiaTebs.<br />
maTematikisTvis suraTi araa kargi TvalsaCinoeba dawyebiT<br />
klasebSic ki (dawvrilebiT amis Sesaxeb ix. [16:$1,2,3,4,6]). ukve I
- 75 -<br />
klasSi Cven nivTieri TvalsaCinoebis paralelurad Semogvaqvs<br />
sqematuri TvalsaCinoeba. TandaTan nivTieri TvalsaCinoebis wili<br />
mcirdeba, sqematurisa _ imatebs, II klasidan ki nivTier<br />
TvalsaCinoebas TiTqmis aRarc viyenebT (garda stereometriuli<br />
modelebisa). yvela klasisaTvis maTematikis ZiriTadi Tvalsa-<br />
Cinoeba _ esaa sxvadasxvagvari sqemebi da naxazebi.<br />
sayovelTaod gavrcelebulia mosazreba, rom TvalsaCinoeba<br />
swavlis motivaciis gamaZlierebeli uebari saSualebaa. sinamdvileSi<br />
ki bavSvisTvis mimzidveli Wreli suraTebiT swavlisa da<br />
azrovnebis motivacia ki ar Zlierdeba, aramed mxolod _<br />
saxelmZRvanelos Tvalierebisa da zerele gadakiTxvis, TamaSis<br />
motivacia. TvalsaCinoeba mxolod im formiTa da zomiT unda<br />
gamoiyenebodes, rac uSualodaa saWiro ama Tu im programuli<br />
sakiTxis gaazrebulad, Rrmad da aqtiurad saswavleblad.<br />
amas ar iTvaliswinebs ara mxolod saqarTveloSi, aramed<br />
msoflioSi yvelaze popularuli, interaqtiuli meTodikebi da<br />
Tanamedrove stilis saxelmZRvaneloebi.<br />
namdvili swavlisa da azrovnebis motivaciis gamaZlierebeli<br />
mTavari saSualeba unda iyos ara xelovnurad gazviadebuli<br />
Wreli suraTebi, aramed kvleviT-ZiebiTi swavleba, evristikuli<br />
meTodi, romelic swavlis motivaciasac aZlierebs da, rac<br />
mTavaria, Rrma da aqtiuri codnis miRebis mTavari saSualebaa.<br />
amasTan, daufaravad unda iTqvas simarTle: namdvili swavla<br />
verasdros gaxdeba mTlianad saxaliso _ swavlis aucilebeli<br />
Tanmxlebi mainc nebelobis daZabvaa (cxadia, asakis Sesabamisad!).<br />
uamisod mxolod zerele codna Tu SeiZineba. vinc swavlis Ziris<br />
simwares gaeqceva, is ver igemebs kenweros gatkbilebas.<br />
pedagogikis mizania, SeZlebisamebr Seamciros `Ziris simware~<br />
da gaaZlieros `kenweros gatkbileba~.<br />
meTodikaSi ori urTierTsapirispiro mimarTuleba arsebobs:<br />
I. sqolastikuri (aseTia, kerZod, sabWoTa meTodika), romelic<br />
mTlianad an TiTqmis mTlianad ugulebelyofs saxaliso-sa-
- 76 -<br />
yofacxovrebo sakiTxebs, maTematikuri codnis praqtikul mxareebs;<br />
swavleba mZimea da mTlianad maTematikazea centrirebuli, moswavlis<br />
pirovnebis, misi asakobrivi fsiqologiis ugulebelyofiT;<br />
II. pedocentruli (moswavleze centrirebuli), romelic<br />
ugulebelyofs maTematikur safuZvlianobasa da siRrmes (aseTia,<br />
kerZod, amerikuli meTodika). gadaWarbebuli mniSvneloba eniWeba<br />
saxalisobas, praqtikul sakiTxebsa da gazviadebul TvalsaCinoebas.<br />
ganvixiloT erTi konkretuli magaliTi. vTqvaT, magaliTad,<br />
moswavleebma iswavles cilindri. rogor damuSavdes es sakiTxi?<br />
sqolastikuri meTodika TiTqmis mTlianad geometriuli<br />
TeoriiTa da mkacri formaluri gansazRvrebebiTaa SezRuduli,<br />
praqtikul TvalsaCinoebas wvrilmanad miiCnevs.<br />
pedocentruli meTodika, piriqiT, upirveles mniSvnelobas<br />
aniWebs cilindris codnis praqtikul gamoyenebasa da mis dakav-<br />
Sirebas yofacxovrebasTan (aq ganvixilavT cilindris swored<br />
rom cnebas _ da ara zedapiris farTobisa Tu moculobis gamoTvlas,<br />
rac calke sakiTxebia!). am mizniT SeiZleba gamoviyenoT sami<br />
donis TvalsaCinoeba (klasikur pedagogikaSic ase iyo):<br />
1) nivTieri. magaliTad, maswavlebels klasSi Seaqvs nairnairi<br />
feradi fanqrebis 3 sxvadasxva kolofi (maTgan zogi<br />
mrgvalia, zogi _ waxnagovani, zogi wverwaTlilia, zogi _ ara).<br />
interaqtiuli meTodikis mixedviT, klass dayofs 3 jgufad,<br />
TiTos miscems TiTo kolofs da daavalebs: amoarCieT cilindris<br />
formis fanqrebi. Semdeg, roca jgufebi warmoadgenen TavianT<br />
namuSevars, kargi maswavlebeli msjelobasac wamoiwyebs: ratom<br />
ara aqvs cilindris forma am wiTel fanqars? {zedapiri aqvs<br />
waxnagovani da fuZeSi eqvskuTxedia} am Sav fanqars? {waTlilia da<br />
amitom am fuZeSi wre ara aqvs}. da ase saTiTaod ganixileba yvela<br />
tipis SemTxveva.<br />
amgvari midgoma kargia, saxalisoa, nayofieria, magram erTi<br />
arsebiTi nakli aqvs: Zalian bevr dros STanTqams. Tuki<br />
maswavlebelma amgvari gakveTili xSirad Caatara, mas saswavlo
- 77 -<br />
programa gverdiT darCeba. xolo Tuki zerele msjelobiT dasrulda,<br />
maSin amgvari muSaoba maTematikisTvis TiTqmis fuWicaa.<br />
2) suraTovani. magaliTad, saxelmZRvaneloSi mTeli gverdi<br />
ukavia did ferad suraTebs: nair-nairi feradi fanqrebi,<br />
konservis qilebi da sxva. iqvea foto: industriuli qarxana<br />
didi cilindruli avziT.<br />
amgvari midgoma yvelaze uaresia, radgan zerele, araarsebiT<br />
mxareze gadaaqvs moswavlis yuradReba. saxelmZRvanelos<br />
moculoba da Rirebuleba maTematikisaTvis fuW rameebze cdeba.<br />
siRrmisaTvis arc saxelmZRvaneloSia adgili darCenili da arc<br />
moswavles Seqmnia saTanado ganwyoba.<br />
Tumca, xSirad es midgoma pirvelTanaa Serwymuli.<br />
gavrcelebuli Secdomaa, rom TiTqos saswavlo TvalsaCinoeba<br />
am ori tipiT (doniT) amoiwureba. sinamdvileSi ki klasikurma<br />
pedagogikamac ki icoda, rom arsebobs mesame donis TvalsaConoebac:<br />
3) warmodgeniTi. Cveneuli meTodika swored amgvar Tvalsa-<br />
Cinoebas emyareba. ucnaurad JRers, magram ueWvelia: Cveneul saxelmZRvaneloebSi,<br />
romlebic Sav-TeTria da Zalian cota suraTs<br />
Seicavs, TvalsaCinoeba metia, vidre sxvebSi. oRond esaa ara suraTovani,<br />
aramed warmodgeniTi da agreTve sqematuri TvalsaCinoeba,<br />
Tanac romelic uSualod ukavSirdeba maTematikur amocanebs.<br />
magaliTisTvis ganvixiloT Cveneuli programis saTanado nawyveti,<br />
kvlav cilindris Sesaxeb (V klasidan). gasaTvaliswinebelia,<br />
rom Cveneuli meTodika, jer erTi, stereometrias planimetriis<br />
paralelurad amuSavebs, magaliTad, cilindri wris paraleluradaa<br />
[ix. $ 15]. meorec, sakiTxis swavleba Semamzadebeli amocanebiT<br />
iwyeba [ix. zemore, $ 11, 12], kerZod, cilindrisaTvis,<br />
esaa amocanebi wris Tvisebebis Sesaxeb (brunviTi sxeulebi da<br />
sxeulis miReba nakvTis brunviT mxolod IX klasSi Semogvaqvs).<br />
saxelmZRvaneloSi cilindris mxolod sqematuri naxazia. samagierod,<br />
gakveTilis teqsts saSinao davalebis aseTi amocanebi<br />
axlavs (maTgan zogi momdevno gakveTilebzecaa gadanawilebuli,
- 78 -<br />
ariTmetika-algebris sakiTxebs Soris, maT gasaxaliseblad):<br />
1. warmoidgineT 4 fanqari: mrgvali wverwaTlili, mrgvali uxmari<br />
(wverwauTleli), kuTxovani wverwaTlili da kuTxovani uxmari.<br />
romel maTgans ar aqvs cilindris magvari forma? ratom?<br />
maSasadame, TvalsaCinoebis 1-l doneze namdvili nivTiebi<br />
fanqrebi iyo, me-2 doneze _ fanqrebis naxatebi (TiTqos V-VI<br />
klasel bavSvs uWirdes fanqris gonebaSi warmodgena!), xolo me-3<br />
doneze _ fanqrebis warmodgeniTi xatebi. asevea sxva SemTxvevebSic.<br />
2. warmoidgineT nivTebi: gaberili sacurao rgoli,<br />
Sededebuli rZis qila, Cveulebrivi boTli, manqanis<br />
borbali, swori mili, xis kasri, xurda fuli (moneta),<br />
sanTeli, muTaqa, vedro. romel maTgans aqvs cilindris<br />
magvari forma? ratom?<br />
3. warmoidgineT, rom birTvi gakveTes organ,<br />
erTmaneTis gaswvriv ise, rom erTerTi<br />
miRebuli sxeuli aseTia: mas fuZeebad aqvs ori<br />
erTmaneTis toli wre. moifiqreT, iqneba Tu<br />
ara es sxeuli cilindri. ratom? dasabuTeba CawereT.<br />
4. warmoidgineT dauWreli Zexvi. rodesac Zexvs yiduloben,<br />
gamyidveli mas cerad CamoaWris xolme naWers da wonis. aqvs<br />
Tu ara aseTnairad CamoWril Zexvis naWers cilindris magvari<br />
forma? ratom? SeiZleba Tu ara Zexvis ise CamoWra, rom<br />
daaxloebiT cilindruli formis naWeri miiRebodes?<br />
5. nagebobebisa da ezoebis dasamSveneblad xSirad iyeneben<br />
cilindrisa da birTvis formebs. ra SeiZleba iTqvas<br />
naxazze gamosaxuli cilindris fuZis diametrisa da<br />
birTvis diametris Sesaxeb? daxateT amis magvari ori<br />
iseTi naxati, rom erTze cilindris fuZis diametri<br />
meti iyos birTvis diametrze, xolo meoreze _ piriqiT.<br />
mocemulia ramdenime cnobili xuroTmoZRvruli Zeglis<br />
suraTi, oRond ara Wrel-Wreli, aramed sqematuri. amocanaa:<br />
6. am nagebobebSi moZebneT Semdegi geometriuli formebi:
- 79 -<br />
naxevarsfero, cilindri, naxevarcilindri. amowereT<br />
TviTeuli am formis saxelwodeba da yovel maTgans gverdiT<br />
miuwereT is ricxvi, romelic gviCvenebs, Tu ramdenjer Segvxvdeba<br />
am saxelwodebis forma naxatze gamosaxul nagebobebSi<br />
(yvelaSi erTad).<br />
gvaqvs amocanebi aqtiobazec, magram amaze dro klasSi ar ixarjeba<br />
_ klasSi mxolod namuSevrebi ganixileba erToblivi<br />
msjelobiT:<br />
7. moZebneT Sin nivTebi, romlebsac daaxloebiT cilindris<br />
magvari forma aqvs da CawereT maTi saxelebi.<br />
8. aiReT qaRaldis furceli da sworad dagragneT igi<br />
cilindris gverdiTi zedapiris formis magvarad. ra aklia<br />
mas cilindruli zedapiris magvar formamde?<br />
rogorc vxedavT, Cven udides mniSvnelobas vaniWebT sakiTxis<br />
intuiciur wvdomas [$ 16]. magram, zomierad, gvaqvs amocanebi<br />
Zalian faqiz, wmindad maTematikur, Rrma sakiTxzec, magaliTad:<br />
9. warmoidgineT, rom fuZeebis gaswvriv gakveTeT ara cilindri,<br />
aramed misi zedapiri. ra nakvTs miiRebT gakveTis<br />
adgilas? {ara wres, aramed wrexazs!}<br />
garda amisa, mravlad gvaqvs cilindris cnebis sxva naswavl<br />
geometriul cnebebTan damakavSirebeli amocanebic, magaliTad:<br />
10. warmoidgineT, rom sfero mTlianadaa cilindrSi<br />
moTavsebuli da aris udidesi amgvari sfero. SemdegTagan romeli<br />
daskvnis gamotana SeiZleba aqedan?<br />
a) sfero exeba cilindris erTaderT fuZes;<br />
b) sfero exeba cilindris orive fuZes;<br />
g) sferosi da cilindris fuZis radiusebi tolia;<br />
d) sferos radiusi cilindris simaRlis tolia;<br />
e) sferos da cilindrs erTnairi moculoba aqvs.<br />
dakavSireba ariTmetikasTan:<br />
11. warmoidgineT, rom cilindrs daadges meore cilindri,<br />
romelsac pirvelis toli fuZe aqvs, xolo simaRle _<br />
nax. 9
- 80 -<br />
pirvelis simaRlis naxevari. cilindrebis fuZeebi erTmaneTs<br />
SeuTavses. ra sxeuls miiRebdnen?<br />
yuradReba mivaqcioT, rom amdeni, 11 amocana cilindris<br />
mxolod cnebas eZRvneba! cilindris radiusi, diametri da<br />
simaRle _ Semdegaa. momdevno klasebSi ki es sakiTxebi<br />
bunebrivad erwymis algebras _ saTanado formulebis gamoyeneba.<br />
Semdeg kavSirdeba fizikasTan (kargi amocana _ nacvlad<br />
zemorexsenebuli fotosi industriuli qarxnis xediT):<br />
mocemulia navTis kuTri wona _<br />
12. gamoTvaleT, ras iwonis is navTi, rac eteva 2 m simaRlisa<br />
da 70 sm radiusis sigrZis mqone cilindrul avzSi.<br />
cotaTi rTuli, saazrovno, arastandartuli amocana:<br />
13. gaarkvieT, rogor Seicvleboda wina amocanis pasuxi, avzs<br />
sqeli, 5-santimetriani fskeri, kedlebi da sarqveli rom hqonoda.<br />
sarqveli rom ar hqonoda (anu, avzi TavRia rom yofiliyo)?<br />
dabolos, TvalsaCinoebis umaRlesi gamoyeneba _ logikuri<br />
sqemebia. kerZod, Zalian kargia geometriuli sxeulebis Tvalsa-<br />
Cino saklasifikacio sqemebi (xisebri diagrama da agreTve venis<br />
diagramebi), romlebzec TvalsaCinod gamoCndeba, ra adgili ukavia<br />
cilindris cnebas sxva stereometriuli sxeulebis cnebebs<br />
Soris da ra logikuri mimarTebebia am cnebebs Soris (IX kl).<br />
Cveneuli meTodikisaTvis kidev erTi sakiTxia Zalian<br />
mniSvnelovani _ magaliTebis saxeobani. pirveli, Cveulebrivi<br />
saxea cnebis dadebiTi magaliTebi, cilindris SemTxvevaSi,<br />
magaliTad: mrgvali kasri, konservis qila, morgvi da sxva. meore<br />
saxea uaryofiTi magaliTebi: burTi, borbali, konusi, 8waxnaga<br />
gumbaTi (rogoric aqvs, magaliTad, mcxeTis jvarsa Tu<br />
atenis taZars). ese igi, saWiroa Cveneba: eseni cilindrebia, eseni<br />
ki araa cilindrebi. magram arc esaa sakmarisi. saWiroa agreTve<br />
kidura, ukiduresi (marginaluri), anu aratipuri magaliTebis<br />
garCeva, raTa kargad moixazos cnebis sazRvari (dawvrilebiT _<br />
ix. [12]). cilindris SemTxvevaSi, magaliTad:
- 81 -<br />
14. CamowereT, romel nivTebs aqvs badros magvari forma:<br />
rogorc vxedavT, badros simaRle gacilebiT<br />
naklebia, vidre misi diametri. miaxloebiT<br />
gamoTvaleT, Tqvens mier dasaxelebul erTerT<br />
nivTs ramdenjer naklebi aqvs simaRle, vidre diametri. axla<br />
warmoidgineT amis sapirispiro Tvisebis mqone cilindri:<br />
romlis simaRle daaxloebiT 100-jer metia, vidre diametri.<br />
romel nivTs aqvs daaxloebiT am cilindris magvari forma?<br />
ese igi, kidura dadebiTi magaliTia iseTi cilindri, romelic<br />
ar hgavs cilindrs, romlis cilindrad aRqma mraval adamians<br />
gauWirdeba. Tanac, raki cilindri fuZeze ar dgas, Zneli<br />
gasaazrebelia, romelia cilindris simaRle da romeli _<br />
diametri. aseve, kidura uaryofiTi magaliTia iseTi sxeuli,<br />
romelic araa cilindri, magram hgavs cilindrs, magaliTad, 12-<br />
14-16-waxnaga gumbaTebi (rogoric aqvs, magaliTad, nikorwmindas,<br />
gelaTs, metexsa Tu martvilis taZars).<br />
cilindris cnebis amgvari aqtiuri da mravalmxrivi damuSaveba<br />
kargad gviCvenebs imasac, Tu ras niSnavs gaRrmavebuli swavleba.<br />
amrigad, aqtiuri mzaobis meTodikaSi, warmodgeniT TvalsaCinoebaze<br />
da Semamzadebel amocanebze dayrdnobiT, moZebnilia is<br />
oqros Sualedi, romelic zomierad Seuwonis erTmaneTs pedagogikis<br />
or sapirispiro mimarTulebas; TviTon igi arc sqolastikuria<br />
(Tumca, inarCunebs mecnierul siRrmesa da cnebiT azrovnebas)<br />
da arc calmxrivad pedocentrulia (Tumca, aZlierebs humanistur<br />
midgomas, praqtikulobasa da gamoyenebiT mxareebs).<br />
sabolood, mSobliuri ena-literaturisa da maTematikis swavlebisas,<br />
agreTve logikuri azrovnebis ganviTarebisaTvis Tvalsa-<br />
Cinoeba aucilebelia, magram igi Zalian mozomili da TiTqmis<br />
mxolod warmodgeniTi an sqematuri unda iyos.<br />
maTematikis saxelmZRvanelo moswavles unda uqmnides ara<br />
zerele, sanaxaobiT, komiqsur-multfilmur-kompiuterul, anu<br />
saekrano ganwyobas, aramed piriqiT: azrovnebis, gonebaWvretis,
- 82 -<br />
dinji CaRrmavebis, maTematikis Sinagani simwyobrisa da silamazis<br />
ganWvretis ganwyobas _ im silamazisa, romelic mxolod<br />
daxvewili gonebis TvaliT dainaxeba. udidesi moazrovnisa da<br />
poetis _ platonis akademiis karibWis Tavze ewera:<br />
`nu Semova aq nuravin, vinc ar icis maTematika!~<br />
rogorc Zvel berZnul, aseve indur Tu Cinur kulturaSi<br />
maTematika `samyaros musikas~ aRwers, im musikas, romelsac<br />
zecaze mnaTobTa harmoniuli moZraoba warmoqmnis da romelic<br />
mxolod maRalganviTarebuli gonebis yuriT moismineba.<br />
§ 17. sakontrolo werebi da SefasebaTa sistema<br />
tblpouspmp!xfsfct!wjxzfcU!JJ!lmbtjt!nfpsf!obyfwsjebo/!<br />
kfs!jTwjbUbe!ubsefcb!`!tvm!nypmpe!tbnj!tblpouspmp/!<br />
ypmp! JJJ! lmbtjebo! tblpouspmp! xfsfcj! yTjsefcb! eb!<br />
tbTvbmpe! frwt.Twje! hblwfUjmTj! fsUyfm! ubsefcb/!<br />
! tblpouspmp!xfsfct![bmjbo!ejej!nojTwofmpcb!brwt/!jtjoj!<br />
hwjDwfofct! nptxbwmfUb! obnewjm! dpeobt-! Ubobd-! gsjbe!<br />
obzpgjfsjb!txbwmfcjtbUwjtbd/!bnjtbUwjt!tblpouspmp!xfsjt!<br />
obTspnfcj! lbshbe! voeb! hbbtxpspt! nbtxbwmfcfmnbd! eb!<br />
Tfnefh!nptxbwmfnbd/!bnjt!njgvDfDfcb!bs!Tfj[mfcb-!sbehbobd!<br />
nptxbwmf! zwfmb{f! lbshbe! tblvUbsj! Tfdepnfcjtb! eb!<br />
ybswf{fcjt! hbtxpsfcjU! txbwmpct/! bnjupn-! [bmjbo!<br />
tbtvswfmjb-! spn! <strong>nbUfnbujljt</strong>! 5! lwjsfvm! tbbUt! ebfnbupt! 1!<br />
damatebiTi mecadineoba-!spnfmjd!hblwfUjmfcjt!Tfnefh!<br />
ebjojTofcb/! bn! ebnbufcjU! hblwfUjm{f! lmbtj! hvmebtnjU!<br />
hbbtxpsfct! tblpouspmp! obnvTfwsfct-! nUbwbsj! Tfdepnfcj!<br />
ebgb{fd! hbjsDfwb/! wjtbd! sb! blmeb-! jnbtbd! hbjhfct! eb!<br />
Tfbwtfct! tblvUbs! obxfst/! bn! hblwfUjmjebo! nptxbwmfUb!<br />
vnsbwmftpcb! 6.21! xvUjt! Tfnefh! xbwb-! spdb! npsDfcjbo!<br />
tblvUbsj! Tfdepnfcjt! hbtxpsfcbt/! ebsDfcjbo! nypmpe! jt!<br />
nptxbwmffcj-! spnmfctbd! nsbwbmj! Tfdepnb! irpoebU! eb! ftf!<br />
jhj-!tb{phbepebd!DbnpsDfcjbo!)Uvoebd!espfcjU-!hbdefofcjt!<br />
hbnp*/! bnhwbs! nptxbwmffct! lj! jtfebd! tXjsefcbU! ebnbufcjUj!
- 83 -<br />
nfdbejofpcb/!<br />
Tuki damatebiTi gakveTilis Catareba ar xerxdeba, maSin<br />
Secdomebi unda gaswordes erTerTi gakveTilis bolos mas<br />
Semdeg, rac gairCeva saSinao davalebis amocanebi.<br />
sakontrolo werisTvis araviTari winaswari<br />
momzadeba araa saWiro! nbtxbwmfcfmj!hblwfUjmfct!bub.<br />
sfct! Dwfvmfcsjwbe-! qsphsbnjt! njyfewjU/! Dwfo! UwjUpo! jtf!<br />
hwbrwt! qsphsbnb! ebhfhnjmj-! spn! nptxbwmft! UbwjtUbwbe!<br />
vxfwt! jnebhwbsj! bnpdbofcjt! bnpytob-! sphpsfcjd! tblpou.<br />
spmp! xfsb{f! Tfywefcb/! Uvndb-! Tfj[mfcb! ndjsf! tjbymfd!<br />
Tfyweft!`!sbUb!TfnprnfefcjUj!b{spwofcb!bbnprnfept/!<br />
nptxbwmffcj! voeb! njfDwjpo! xjobbSnefhpcjt! hbebmbywbt-!<br />
tblvUbsj! dpeojt! xbsnpDfobtb! eb! ebnuljdfcbt-! eb[bcvm!<br />
wjUbsfcbTj!Ubwjt!hbubobt!eb!qbujptbo-!tbnbsUmjbo!Tfkjcst/!<br />
sakontrolo werisas gamoricxuli unda iyo daxmareba; gamoricxuli<br />
unda iyos agreTve yovelgvari siyalbe, gadawera,<br />
karnaxi da sxva sisaZagle. gamoricxuli unda iyos yalbi<br />
niSnebis werac. maswavlebeli Tavad darwmundeba, rom Sefasebis<br />
Cveneuli kriteriumebi isedac sakmaod lmobieria, isedac<br />
vcdilobT, rom moswavles ar daekargos Tundac mcire codna,<br />
mondomeba Tu mosazreba. nel moswavles cotaTi met xansac<br />
vadrovebT, yvelanairad xels vuwyobT da veferebiT moswavleebs<br />
_ magram siyalbe mainc sastikad unda gamoiricxos! yvela<br />
moswavle ver miiRebs umaRles niSnebs da arc unda miiRos (amis<br />
Sesaxeb ix. $ 1-Si).<br />
TiTo siyalbe sawamlavis TiTo wveTia, romelic ryvnis<br />
moswavlis suls!<br />
sakontrolo wera tardeba saxelmZRvanelos TviTeuli Tavis<br />
bolos. sakontrolo weris win bavSvebs davalebad eZlevaT<br />
Sesabamisi Tavis damatebiTi amocanebi. sakontrolo weris<br />
momdevno dRisaTvis ki bavSvebma unda moamzadon momdevno Tavis<br />
pirveli gakveTili. swored am davalebis garCeva unda moxdes<br />
sakontrolo weris momdevno gakveTilze. sasurvelia wina Tavis<br />
damatebiTi amocanebidan zogierTis, arastandartulis ganxilvac.
- 84 -<br />
sakontrolo werisaTvis SerCeuli yvela amocana aRebulia<br />
moswavlis saxelmZRvanelodan _ amocanaTa Tematikuri krebulidan.<br />
maswavlebelma dafaze unda Camoweros mxolod amocanebis<br />
nomrebi, ori rigisTvis. moswavleebs ar unda movTxovoT<br />
amocanebis pirobaTa gadawera. rveulSi moswavleebma<br />
unda Caweron amocanis nomeri da misi sruli amoxsna.<br />
sakontrolo amocanebi isea SerCeuli, rom maT amosaxsnelad<br />
erTi gakveTilisaTvis gankuTvnili dro _ 45 wuTi sakmarisi<br />
unda iyos. Tumca zogierTi moswavle mainc ver aswrebs am<br />
droSi. Tuki SesaZlebelia, kargi iqneba, mivceT maT damatebiTi<br />
dro sakontrolo samuSaos dasamTavreblad. magram es unda<br />
moxdes maSinve, da ara sxva gakveTilebis Semdeg!<br />
gasarkvevia erTi sakiTxic _ naweris xarisxisa. Cveni azriT,<br />
cudi nawerisa Tu naxazebis gamo moswavles SeiZleba daakldes<br />
qula: qula _ da ara niSani, ukidures SemTxvevaSi _ ori qula,<br />
anda, ufro kargi iqneba, aRar epatios is wvrilmani Secdoma,<br />
romlis patiebac Cveni kriteriumebis mixedviT SeiZleboda.<br />
magram: mainc rogor cud nawerze unda moxdes es?<br />
upirvelesad, Tuki naweri cudia gaformebis TvalsazrisiT:<br />
mindvrebi, sityvebsa Tu maTematikur gamosaxulebebs Soris Sualedebis<br />
dacva, amocanis pirobis mokle Canaweris wesierad Sedgena,<br />
CamonaTvalis garkveviT Camowera, naxazis ise daxazva, rom<br />
nakvTi nakvTs hgavdes da sxva. amgvari xarvezebi maswavlebelma<br />
wiTlad unda moniSnos, iseve rogorc yvela sxva Secdoma da<br />
moswavlem unda gaasworos _ anu wesierad gadaweros Tu gadaxazos<br />
_ Secdomebis gasworebisas.<br />
meorec, Tuki kaligrafiaa Zalian cudi da amasTan, Tuki<br />
maswavlebelma icis, rom am moswavles SeuZlia ukeTesad wera da<br />
esoden cudi naweri maimunobis an mifuCeCebis bralia. magram<br />
saSualod cud kaligrafiaze moswavles qula ar unda daakldes,<br />
miT umetes, roca maswavlebelma icis, rom moswavles Zalian<br />
uWirs lamazad wera da ukeTesad TiTqmis rom ar ZaluZs.<br />
moswavles ar unda daakldes qula arc gadaxazulebze (Tuki<br />
metismetad araa gadadRabnili), Casworebulebze, Canamatebze da
- 85 -<br />
sxva. sakontrolo weris rveuli _ esaa ara saCvenebeli, aramed<br />
samuSao rveuli!<br />
iseve rogorc, zogadad, Cveneuli meTodikiT warmarTuli gakveTili<br />
_ esaa ara saCvenebeli, lamazi da saintereso gakveTili,<br />
aramed _ gakveTili yoveldRiuri muSaobisa, rudunebisa, msjelobisa,<br />
daxvewisa, Ziebisa, azrovnebisa da, rac mTavaria, gonebrivi<br />
varjiSisa da isev da isev varjiSisa.<br />
kidev erTic. maswavlebelma xSirad unda Seaxsenos<br />
moswavleebs, rom sakontrolos rveulSive _ da ara calke<br />
furcelze! _ gaakeTon xolme samuSao Canawerebi. es Canawerebi<br />
SeiZleba ar iyos lamazad Sesrulebuli, SeiZleba iyos<br />
gadaxazuli, Canamatebiani da sxva _ magram amaSi moswavle qula<br />
ar daakldeba! piriqiT, qula daakldeba im moswavles, romelsac<br />
ar eqneba amgvari Canawerebi da pirdapir eqneba dawerili pasuxi.<br />
amocanis piroba romc ar moiTxovdes dasabuTebas _ samuSao<br />
Canawerebi mainc unda Candes. sakontrolo naweris gasworebisas<br />
yovelTvis unda mieqces yuradReba: aris Tu ara rveulSi<br />
moswavlis mier gakeTebuli raime Canawerebi, romlebic<br />
adasturebs, rom man namdvilad amoxsna amocana _ da ara<br />
gadaiwera mzamzareuli pasuxi!<br />
es gansakuTrebiT mniSvnelovania maSin, roca amocanis an misi<br />
qveamocanebis pasuxebi Sedgeba erTi-ori asosgan, erTi-ori<br />
ricxvisa an erTi-ori sityvisagan.<br />
sakontrolo werisas maswavlebeli yurdRebiT unda iyos,<br />
raTa yvela moswavlem, jer erTi, damoukideblad imuSaos, da,<br />
meorec, imuSaos: bavSvs xSirad efanteba yuradReba, zogi<br />
moswavle gacilebiT ukeT niSans iRebs, roca mas aiZuleben, rom<br />
wesierad imuSaos! amrigad, Cven maswavlebels vukrZalavT<br />
moswavlis mixmarebas maTematikis mxriv, magram vTxovT mixmarebas<br />
nebelobis mxriv.<br />
amas ukavSirdeba kidev erTi sakiTxi. zogi moswavle bejiTia,<br />
Tanmimdevruli da Seupovari, bolomde zis da muSaobs. zogi ki<br />
cercetaa, cdilobs, male moamTavros wera (rac Zalian ezareba)<br />
da sxva rame akeTos. magram maswavlebeli zaris darekvamde ar
- 86 -<br />
gauSvebs moswavleebs gareT saTamaSod. maSin is moswavleebi,<br />
romlebmac adre moamTavres wera, cqmutaven da sxvebs uSlian<br />
xels. amas yvelafers didi yuradRebis miqceva sWirdeba. maswavlebelma<br />
unda aiZulos moswavleebi, rom kidev erTxel yuradRebiT<br />
waikiTxon TaviaTi naweri, gadaamowmon, xelaxla iangariSon<br />
(amocanaSi romc ar iyos moTxovnili Semowmeba, mainc!). moswavle<br />
unda mieCvios sakuTari codnis warmoCenasa da damtkicebas, nebisyofis<br />
mokrebiT maRali da ufro maRali Sedegebis mopovebas.<br />
yovelive amas udidesi mniSvneloba aqvs. da ara mxolod<br />
imisaTvis, rom moswavlem ukeTesi niSani miiRos, aramed agreTve<br />
gacilebiT ufro mniSvnelovani ramisTvis: moswavles mtkiced<br />
unda Camouyalibdes sakuTari naSromis Semowmebis, zogadad,<br />
TviTkritikisa da sakuTari namoqmedaris garedan danaxvis<br />
unari da Cveva. amave unaris gansaviTareblad, iseve rogorc sakuTriv<br />
maTematikuri codnis Sesavsebad da gansamtkiceblad<br />
yovlad aucilebelia sakontrolo nawerebis saguldagulo gasworeba<br />
moswavleTa mier!<br />
TviTkritikisa da sakuTari namoqmedaris garedan danaxvis<br />
unarCveva _ adamianis erTerTi umniSvnelovanesi da uZvirfasesi<br />
Tvisebaa (amis Sesaxeb ix. paragrafSi 9). amrigad, sakontrolo<br />
wera unda iyos sakontrolo ara mxolod maswavleblis mier<br />
moswavlis codnis Semowmebis TvalsazrisiT, aramed agreTve<br />
Tavad moswavlis mier sakuTari codnisa da sakuTari naSromis<br />
Semowmebisa da Semdeg gasworebis TvalsazrisiT...<br />
sakontrolo weraze xuTi amocanaa. rogorc wesi, pirveli,<br />
meore da mesame amocanebi _ oTxqulianebia, xolo meoTxe da<br />
mexuTe _ xuTqulianebi. Tuki es qulebi sadme Secvlilia,<br />
sagangebodaa miTiTebuli.<br />
sakontrolo davalebaTa SerCevisas gaTvaliswinebulia, rom<br />
pirveli ori-sami amocanis amoxsna unda SeZlon SedarebiT<br />
sustma moswavleebmac. meoTxe, da gansakuTrebiT mexuTe amocana<br />
ufro Znelebia. xSirad mexuTe amocanaSi aris xolme damatebiTi<br />
davalebac, romlis kargad Sesrulebisas moswavle (Tuki man sxva<br />
davalebebic kargad Seasrula), miiRebs umaRles Sefasebas (`9~-s,
- 87 -<br />
`10~-s an `5+ ~-s Cveulebrivi sistemiT).<br />
yvela sakontrolo weris yoveli amocanisTvis momzadebuli<br />
gvaqvs Sefasebis konkretuli kriteriumebi. isini gadmocemulia<br />
gegma-konspeqtebSi, saTanado adgilas. iq zogierTi amocana<br />
SedarebiT dawvrilebiTaa amoxsnili, amoxsnis gzis etapebis<br />
aRweriT, zogierTisa ki mxolod pasuxia mocemuli. TviTeuli<br />
etapis (an saboloo pasuxis) bolos frCxilebSi miTiTebulia<br />
qulebis maqsimaluri raodenoba, riTac SeiZleba Sefasdes moswavle<br />
am etapis (an mTlianad davalebis) SesrulebisTvis. Tuki moswavles<br />
kargad aqvs Sesrulebuli amoxsnis aRwerili etapi (an mTlianad<br />
davaleba), maSin igi unda Sefasdes miTiTebuli maqsimaluri quliT;<br />
srulad da marTebulad Sesrulebul davalebaSi maswavlebelma<br />
unda daweros 4 qula (pirvel, meore an mesame amocanaSi) an 5<br />
qula (meoTxe an mexuTe amocanaSi). xolo Tuki amocana amoxsnilia<br />
nawilobriv an sul araa amoxsnili, maSin moswavle iRebs ufro<br />
nakleb qulas: 3-s, 2-s, 1-s an 0-s _ imisdamixedviT, Tu amocanis<br />
amosaxsnelad ra nabijebia gadadgmuli. kerZod:<br />
Tuki Sesrulebulia davalebis daaxloebiT naxevari, mesamedi,<br />
meoTxedi (maswavleblis azriT), maSin aRwerili etapi (an<br />
mTlianad davaleba) unda Sefasdes Sesabamisad miTiTebuli maqsimaluri<br />
qulis naxevriT, mesamediT, meoTxediT. Tuki moswavles<br />
dawerili aqvs amosanis amosaxsnelad saWiro erTi mainc marTebuli<br />
gardaqmna, gamoTvla Tu msjeloba, mas am amocanaSi 0,5 an 1<br />
qula mainc unda daeweros. Tuki moswavles amoxsnisas warmatebiT<br />
aqvs gamoyenebuli mis mier mignebuli, uCveulo, skolaSi arnaswavli<br />
xerxi, mas 0,5 an 1 qula unda daematos. miRebuli<br />
qulebi unda Seikribos (rogorc wesi, jamis maqsimaluri mniSvnelobaa<br />
22) da moswavlis sakontrolo namuSevari Sefasdes 10doniani<br />
sistemiT qvemore marcxena cxrilis mixedviT:<br />
sagamocdo davaleba, rogorc wesi, 10 amocanisgan Sedgeba da<br />
maqsimaluri jamuri qula, romelic SeiZleba moswavlem daagrovos,<br />
42-ia (rogorc wesi). amitom moswavlis sagamocdo namuSevari<br />
unda Sefasdes 10-doniani sistemiT marjvena cxrilis mixedviT.
- 88 -<br />
yvela SemTxvevaSi maswavlebelma moswavlis sakontrolo<br />
namuSevarze garkveviT unda aRniSnos, Tu konkretulad<br />
raSi daaklo Tu moumata qula da, ese igi, ramdeni<br />
quliT Seafasa TviTeuli amocana.<br />
qulaTa jami Sefaseba<br />
0-1 0<br />
2-4 1<br />
5-7 2<br />
8-10 3<br />
11-12 4<br />
13-14 5<br />
15-16 6<br />
17-18 7<br />
19-20 8<br />
21-22 9<br />
qulaTa jami Sefaseba<br />
0-2 0<br />
3-8 1<br />
9-14 2<br />
15-19 3<br />
20-23 4<br />
24-27 5<br />
28-31 6<br />
32-35 7<br />
36-39 8<br />
40-42 9<br />
sakontrolo naSromebisTvis Sefasebis 10-doniani sistema gacilebiT<br />
ukeTesia, ramdenime TvalsazrisiT. xolo yoveldRiuri,<br />
mimdinare SefasebebisTvis sjobs 3-4-doniani. saqme isaa, rom aqtiuri<br />
meTodikiT agebul gakveTilze ver xerxdeba moswavlis<br />
codnis safuZvliani Semowmeba (raki ar gvaqvs moswavlis gaZaxeba<br />
gakveTilis mosayolad da sxva). samagierod, Cans TiTqmis yvela<br />
moswavlis (Zalian did klasSi _ moswavleTa umravlesobis) aqtiuroba,<br />
maTi pasuxebi da msjeloba, calkeul sakiTxTa codna.<br />
Cveneuli aqtiuri gakveTilis Semdeg maswavlebels SeuZlia 20<br />
moswavlesac ki Camouweros niSnebi. xolo moswavleTa codnis<br />
safuZvliani Semowmeba sakontrolo werebze iqneba.<br />
amitom, maswavlebels SeuZlia, gakveTilis bolos Seafasos<br />
TiTqmis yvela moswavle (anda, maTi umravlesoba) _ magram es<br />
ver iqneba iseTi dazustebuli da myari niSnebi, rogorebic<br />
iwereba sakontrolo werebze. amitom mimdinare Sefasebebad<br />
ukeTesia amgvari: + (CaTvla), _ (arCaTvla), \ (saSualo). Tanac,<br />
+ araa igive, rac 5-iani, igi aRniSnavs mxolod imas, rom
- 89 -<br />
moswavlem mimdinare gakveTili `CaTvala~: Sesrulebuli hqonda<br />
saSinao davalebis ZiriTadi nawili, erT-or SekiTxvasac kargad<br />
upasuxa. Tuki moswavlem damatebiT kidev TamaSSic gaimarjva, an<br />
raimeTi Zalian gamoiCina Tavi, mas daewereba ori +. xolo \<br />
iwereba, Tuki moswavlem mxolod sanaxevrod CaTvala. semestris<br />
bolos es niSnebi Sejamdeba, TiTo _ niSani gaabaTilebs TiTo +<br />
niSans, xolo sami cali \ CaiTvleba erT + niSnad. saboloo<br />
niSani gamoyvaneba miRebuli ricxvisa da sakontrolo werebis<br />
saSualo niSnis SejerebiT.<br />
SefasebaTa amgvari sistema rTulia, magram gacilebiT ukeT<br />
asaxavs moswavlis codnas aqtiuri swavlebisas. skolas, Tavisi<br />
SexedulebiT, SeuZlia miiRos an ar miiRos amgvari sistema.<br />
VII klasis sakontrolo amoanebis nusxa<br />
zvsbeSfcb" maswavlebelma swavlebis dawyebamde Tavis<br />
saxelmZRvaneloSi wiTlad unda Semoxazos amocanaTa<br />
Tematikuri krebulidan sakontrolo werebisaTvis<br />
gankuTvnil amocanaTa nomrebi, raTa romelime maTgani ar<br />
misces klass damatebiT saSinao an saklaso weris davalebad!<br />
sakontrolo werebisa da gasworebis Sesaxeb ix. $ 10.<br />
aRniSvnebi qvemore siaSi: sw _ sakontrolo wera;<br />
3/9 I, II _ amocanebis Tematikuri krebulis me-3 paragrafis<br />
me-9 amocanis I da II qveamocanebi;<br />
siis marcxena sveti _ I rigi (I varianti);<br />
marjvena sveti _ II rigi (II varianti).<br />
sw # 1<br />
1. 3/4 3/12<br />
2. 3/9 I, II 3/9 III, IV<br />
3. 2/8 2/12<br />
4. 4/11 4/20<br />
5. 5/10 5/19<br />
sw # 2<br />
1. 3/16 I, III, V 3/16 II, IV, VI<br />
2. 3/17 I-III 3/17 IV-VI<br />
3. 4/27 4/34<br />
4. 3/13 3/18<br />
5. 4/28 4/35
sw # 3<br />
1. 3/20 I-III 3/20 IV-VI<br />
2. 3/21 I-II 3/21 III-IV<br />
3. 4/39 I, II 4/39 III, IV<br />
4. 2/17 2/20<br />
5. 4/12 4/21<br />
sw # 5<br />
1. 3/29 I-IV 3/29 V-VIII<br />
2. 3/30 I, III, V 3/30 II, IV, VI<br />
3. 4/52 I, II 4/52 III-IV<br />
4. 2/23 2/27<br />
5. 4/53 4/56<br />
sw # 7<br />
1. 3/31 3/38<br />
2. 2/33 I 2/33 II<br />
3. 2/41 2/49<br />
4. 2/46 2/50<br />
5. 4/60 4/66<br />
sw # 9<br />
1. 3/39 I 3/39 II<br />
2. 3/44 I 3/44 II<br />
3. 2/56 2/60<br />
4. 4/81 4/85<br />
5. 4/75 4/83<br />
sw # 11<br />
1. 3/52 I-IV 3/52 V-VIII<br />
2. 3/53 I-IV 3/53 V-VIII<br />
3. 3/54 I-IV 3/54 V-VIII<br />
4. 2/63 I 2/63 II<br />
- 90 -<br />
sw # 4<br />
1. 3/24 I-III 3/24 IV-VI<br />
2. 3/25 I 3/25 II<br />
3. 3/22 3/27<br />
4. 5/25 I-VIII 5/25 IX-XVI<br />
5. 4/43 4/50<br />
sw # 6<br />
1. 3/36 3/36<br />
I, III, V II, IV, VI<br />
2. 5/28 5/34<br />
3. 2/28 2/29<br />
4. 4/57 4/64<br />
5. 5/32 5/38<br />
sw # 8<br />
1. 3/46 3/51<br />
2. 2/47 2/54<br />
3. 4/58 I 4/58 II<br />
4. 2/40 2/48<br />
5. 2/43 2/45<br />
sw # 10<br />
1. 3/45 I-IV 3/45 V-VIII<br />
2. 3/47 I-IV 3/47 V-VIII<br />
3. 3/48 I-IV 3/48 V-VIII<br />
4. 2/57 2/58<br />
5. 4/90 4/82<br />
sw # 12<br />
1. 3/57 I-III 3/57 IV-VI<br />
2. 3/58 I-II 3/58 III-IV<br />
3. 3/55 3/59<br />
4. 2/71 I 2/71 II<br />
5. 5/12 5/13
5. 4/16 4/30<br />
- 91 -<br />
sw # 13<br />
sw # 14<br />
1. 3/61 I 3/61 II 1. 3/66 I 3/66 II<br />
2. 3/62 I-IV 3/62 V-VIII 2. 3/67 I-II 3/67 III-IV<br />
3. 3/63 I 3/63 II 3. 3/68 I 3/68 II<br />
4. 2/76 2/79<br />
4. 2/77 I 2/77 II<br />
5. 4/40 4/47<br />
5. 4/80 4/92<br />
sagamocdo (Semajamebeli) wera<br />
1. 3/35 3/43<br />
2. 4/15 I-IV 4/15 V-VIII<br />
3. 4/26 I-II 4/26 III- IV<br />
4. 3/64 I, III 3/64 II, IV<br />
5. 5/41 I-IV 5/41 V-VIII<br />
6. 3/69 I, II 3/69 III, IV<br />
7. 3/71 I 3/71 II<br />
8. 2/34 2/44<br />
9. 2/78 I 2/78 II<br />
10. 4/62 4/69<br />
§ 18. amocanebis pasuxebi da miTiTebebi<br />
Tavi I. g. (= gakveTili) 1. 1. I. 2,0036 _ 0,8 = 1,2036;<br />
II. 202,55 + 319,066 + 0,23 = 511,846;<br />
III. 75,05 : 0,8 = 93,8125; IV. 86,508 _ 68,8941 = 17,6139;<br />
V. 0,625 · 60,3 · 15 = 565,3125.<br />
2. I. 7,5 milioni; II. 240 milioni; III. 140 miliardi; IV. 1,5<br />
aTasi. 3. 130-135 kg-iT. 4. mcdaria II. mag., winadadeba `8 iyofa<br />
2-ze an 3-ze~ marTebulia, xolo `8 iyofa 6-ze.~ _ mcdaria.<br />
5. 80 mm : 200 = 0,8 mm. (80 : 0,2) · 3 = 1200 (xvia).<br />
6. 0,051 = an , xolo 3,4442 =<br />
gulwrfelad<br />
gisurvebT<br />
warmatebas<br />
Cvens saerTo<br />
saqmeSi!<br />
an .
- 92 -<br />
g. 2. 1. x + 1 x = 15 , saidanac x = 12 . 2. Crd. yin. okeane _<br />
4<br />
1/20 naw. I. 2-jer; II. 2,5-jer; III. 5-jer; IV. 4-jer.<br />
3. 40-jer. 4. raki samniSna ricxvis Canaweri unda Seicavdes<br />
8-iansac da 5-iansac, amitom pasuxebia: I. 885, 855, 585;<br />
II. 885, 855, 585, 858, 588, 558.<br />
3 2<br />
3 2<br />
5. x + 5x<br />
− 2x<br />
+ 1 = 0,<br />
5 + 5 · 0,<br />
5 − 2 · 0,<br />
5 + 1 = 1,<br />
375.<br />
6. I. daaxloebiT 17 saukunis win; II. daaxloebiT 370 wlis win.<br />
g. 3. 1. a ⋅ 1 = 1.<br />
2. { 100 , 101,<br />
102,<br />
103,<br />
104,<br />
105,<br />
106}<br />
.<br />
a<br />
3. I. 0,93 > 0,83. II. 0,10 < 0.12 an 0,11 < 0,12.<br />
III. 5,24 > 5.08 an 5,24 > 5,18. IV. 3,72 < 3,81 an 3,72 < 3,91.<br />
V. 59,09 > 59,009. 4. { 1300 , 6000,<br />
0,<br />
100500600,<br />
200}.<br />
5. I. 19 ivlisi, kvira, dilis 4-is naxevari. II. 12 ivlisi, kvira,<br />
dilis 12-is naxevari. III. 9 ivlisi, xuTSabaTi, dilis 4-is<br />
naxevari. 6. erTsa da imaves niSnavs.<br />
g. 4. 1. I. { 96 , 26}.<br />
II. { 96 , 100,<br />
26}.<br />
2. {0,3; 0,35; 10,25; 8;<br />
0,8; 5,05} 3. x > 18 da 18 < x; x ≥ 18 da 18 ≤ x;<br />
4. s ≤ 15.<br />
5. a ≥ b.<br />
6. 6 ≤ x.<br />
| |<br />
g. 5. 1. I. marTebulia. II. marTebulia. 2. I. 0,<br />
3<<br />
< 3,<br />
5.<br />
2<br />
QP<br />
II. k<br />
3<br />
< ( k + 2)<br />
< m .<br />
a<br />
3. { 0 , 1;<br />
0,<br />
5;<br />
0,<br />
6;<br />
0,<br />
065;<br />
5,<br />
06;<br />
6,<br />
005}.<br />
4. 12650 km. 5. I. |AC| > 2 m; II. |KC| ≥ 2 m; III. |KC| ≤ 5 m; IV.<br />
|FC| = 2 m; V. |GC| ≥ 5 m. 6. { I, II, IV, V, VI}.<br />
. 7. {I, II, VI, VII}.<br />
(IV-s simcdares adasturebs SemTxveva: c = d = 0).<br />
g. 6. 1. I. marTebulia. II. mcdaria. III. mcdaria. IV. mcdaria.<br />
2. I. b ≤ EF < 7,<br />
5 m; II. 0 < r ≤ 1,<br />
2;<br />
III. k ≤ c<br />
5<br />
≤ ( 2g<br />
+ 1).<br />
3. {1, 7, 0,9, 1,9, 11, 0,5} 4. I. marTebulia. II. mcdaria.<br />
III. marTebulia. 5. 5,6 sm. 6. I. ar SeiZleba. II. SeiZleba. III.<br />
ar SeiZleba. IV. SeiZleba. V. SeiZleba. VI. SeiZleba.<br />
7. I. ar SeiZleba. II. SeiZleba. III. ar SeiZleba. IV. ar SeiZleba.<br />
g. 7. 1. maTematika saZirkvelia codnisa. 2. {A(0), B(0,5),
- 93 -<br />
C(1,5), D(1,75), K(3), E(3,5), T(4), F(5,5)}. erTeul. monakv. sigrZea 2<br />
sm. I. { 4 ), 5),<br />
6),<br />
9),<br />
10)<br />
}. II. { 2 ), 7)<br />
}. III. { 1 ), 3),<br />
8)<br />
}. 3. I. ar<br />
SeiZleba. II. ar SeiZleba. 5. I. metia. II. naklebia. III. marjvnivaa.<br />
IV. Tuki A wertili B-s marcxnivaa, maSin a naklebia b-ze. 6.<br />
( b − a)<br />
-si 7. g) 1/100; es moxdeba, Tuki 10 xmovnidan 5 aris a.<br />
g. 8. 1. I. 5,08. II. 10,32. III. 0. 2. I. ar SeiZleba. II.<br />
SeiZleba Tu d=7. 3. b) 827; 4. AB = b.<br />
wertilis koordinati<br />
tolia am wertilis daSorebisa ricxvTa sxivis saTavidan.<br />
5. AB = 3( b − a)<br />
sm, AB = 0, 3(<br />
b − a)<br />
sm.<br />
6. Tuki samives wili naklebia 0,3-ze, maSin maTi jamuri wili<br />
naklebi iqneba 0,9-ze. amitom danarCen erovnebaTa wili gamova<br />
0,1-ze meti. es ki ewinaaRmdegeba pirobas.<br />
g. 9. 1. 1, 2 ≤ x ≤ 3,<br />
6 2. I. c ≤ z < d.<br />
II. 0 < y ≤ e.<br />
3. I. gare<br />
wrexazis gareT. II. Siga wrexazis SigniT. III. Siga wrexazze. IV.<br />
gare wrexazze. V. gare wrexazis gareT. VI. gare wrexazis gareT.<br />
VII. gare wrexazis SigniT. VIII. gare wrexazis SigniT da Siga<br />
wrexazis gareT. IX. wertili ekuTvnis rgols oRond ara Siga<br />
wrexazs. X. Siga wrexazis SigniT oRond ara Tavad C wertili.<br />
4. I. x = 0 , 6.<br />
II. y = 3,<br />
8.<br />
5. 2,568 6. I. 0, 005 ≤ 0,<br />
005;<br />
II. 10, 905 ≥ 10,<br />
905;<br />
III. 21,911>21,9109 an 21,911>21,9009;<br />
IV. 55
- 94 -<br />
dasWirdeba 10 litri Saqris fxvnili.<br />
4 4<br />
g. 2. 1. I. 1. II. 1. III. = ( 13 · 21)<br />
= ( 3)<br />
. 2. I. x<br />
4<br />
⋅ y<br />
4<br />
;<br />
7 26 2<br />
II. 3<br />
5<br />
⋅ a<br />
5<br />
⋅b<br />
5<br />
⋅c<br />
5<br />
; III. 3<br />
n<br />
⋅ 4<br />
n<br />
. 4. I.<br />
3 b<br />
; II.<br />
3ac<br />
.<br />
4k<br />
5b<br />
5. I. 40735332; II. 25043766, 9306402, 71702202, 40735332.<br />
6. es toloba mcdaria, magaliTad, im SemTxvevaSi, roca<br />
a = b = 1 , n = 2 . amitom mTlianobaSi is mcdaria, magram SeiZleba<br />
marTebuli iyos kerZo SemTxvevebSi. magaliTad, marTebulia I da<br />
II SemTxvevebSi, III SemTxvevaSi ki _ mcdaria.<br />
g. 3. 1. I. 2; II. ( 208 · 125<br />
3 ) = 2<br />
3<br />
= 8 ; III. ( 0 , 01:<br />
0,<br />
001)<br />
6<br />
= 10<br />
6<br />
=<br />
25 104<br />
100000; IV.<br />
625<br />
; V. 32. 4. 9,6 erTeuliT; e = 12,<br />
8 .<br />
81<br />
10,<br />
9−8,<br />
6<br />
5. m = 8,<br />
6 + = 9,<br />
75 . 6. es toloba mcdaria,<br />
2<br />
magaliTad, im SemTxvevaSi, roca a = 2 , b = 1,<br />
n = 2 . amitom<br />
mTlianobaSi is mcdaria, magram SeiZleba marTebuli iyos kerZo<br />
SemTxvevebSi. magaliTad, marTebulia I, II da III SemTxvevebSi, IV<br />
SemTxvevaSi ki _ mcdaria.<br />
g. 4. 1. I. 5,5; II. 47; III. 8. 2. 8 wlis. 3. 150.<br />
4. I. ( 1<br />
6 ) = 1<br />
; II. 1; III. 5 625<br />
2 64<br />
4 = . 5. saSualod 2 kovzi<br />
gamodis. 6. Tbilissa da nalCiks Soris manZili iqneba<br />
220 + 440 −220<br />
= 330 km. Zalian kargi iqneba, Tuki vinme<br />
2<br />
dainaxavs, rom am manZilis gamoTvla SeiZleba daukavSirdes<br />
Suawertilis koordinatis gamoTvlas:<br />
220 + 440 = 330 km.<br />
2<br />
manZili Tbilisidan joxaryalas gadasaxvevamde tolia 342 _ 12<br />
= 330 km-is, xolo vladikavkazisa da maxaWyalas Suagzamde _<br />
220 + 500 = 360 (km). amitom pasuxia 360 _ 330 = 30 (km).<br />
2<br />
7. 72· 7 = 21 weli.<br />
24<br />
g. 5. 1. magaliTad, mama - 70 kg, asuli - 50 kg, vaJi - 60
- 95 -<br />
kg. 2. I. 53 sm; II. 950 g; III. 365 l; IV. 2,65 sT; V. 30,5<br />
wm; 3. I. raime ricxvisa da 0-is saSualo tolia am ricxvis<br />
naxevrisa; II. ori toli ricxvis saSualo tolia mocemuli<br />
ricxvisa. 4. I. 14,6; II. 198; III. 0,4; 5. I. ar ewinaaRmdegeba,<br />
radgan 1 = 25<br />
, 7 = 35 , 2 = 40<br />
; II. giorgis pasuxi, radgan<br />
4 100 20 100 5 100<br />
tolmniSvneliani wiladebis Sedareba ufro advilia; III. me-4<br />
klasSi _ 40 ⋅ 1 = 10 , me-5 klasSi _ 40 ⋅ 7 = 14 , me-6 klasSi _<br />
4<br />
20<br />
40 ⋅ 2 = 16 . 6. I. 28<br />
n<br />
; II. ( )<br />
5<br />
3<br />
xyz ; III. ( ) 5<br />
a<br />
; IV.<br />
9<br />
m<br />
2 ;<br />
g. 6. 4. ufro meti sididis raime procenti metia ufro<br />
naklebi sididis imave procentze! 6. d) mxolod I da III.<br />
g. 7. 1. I. a pirveli c monakveTis sigrZe udris orjer<br />
a nulovani a pirveli monakveTis sigrZes; II. b meeqvse udris<br />
2 xarisxad a meeqvse; III. y mexuTe naklebia an tolia y<br />
meeqvsesa da erTnaxevari p nulovanis jamze.<br />
a1<br />
b3<br />
2. I. a 1 + a2<br />
+ a3<br />
+ a 4 + a5<br />
+ a6<br />
; II.<br />
k<br />
y3<br />
−<br />
c0<br />
c1<br />
; III. ⋅ ⋅ c<br />
2 3 2<br />
3. I. 1 sm ≤ |AC| ≤ 2 sm; II. 3 sm ≤ |AC| ≤ 4 sm; III. 40 sm ≤<br />
|AC| ≤ 41 sm; IV. n sm ≤ |AC| ≤ n+1 sm. 5. I. 6<br />
3<br />
⋅ d<br />
3<br />
;<br />
II. 2<br />
n<br />
⋅ 7<br />
n<br />
; III. 4<br />
3<br />
⋅ x<br />
3<br />
⋅ y<br />
3<br />
⋅ z<br />
3<br />
5<br />
k<br />
; IV.<br />
c<br />
; V.<br />
7<br />
.<br />
5<br />
k<br />
9 5<br />
g. 8. 2. I. a 1 + a2<br />
+ ... + a60<br />
; II. c<br />
3 3 3<br />
1<br />
+ c<br />
2<br />
+ ... + c<br />
100<br />
.<br />
1 6·<br />
0,<br />
05 1 0,<br />
3<br />
3. I. marTebulia, radgan = = 0,<br />
07 ≥ 0,<br />
01<br />
10 10<br />
−<br />
−<br />
;<br />
II. mcdaria, radgan<br />
0<br />
≤ 0,<br />
01;<br />
III. marTebulia, radgan<br />
10<br />
10+<br />
6<br />
= 1,<br />
6 ≥ 0,<br />
01.<br />
4. 20% aris 8 spilo. amitom sul iqneba<br />
10<br />
8 ··5 = 40 spilo. 5. I. SeiZleba; II. SeiZleba; III. ar<br />
SeiZleba; IV. SeiZleba; V. ar SeiZleba; VI. ar SeiZleba.<br />
7. magaliTad: I. a<br />
2 2 2<br />
1<br />
+ a<br />
2<br />
+ ... + a<br />
10<br />
;
- 96 -<br />
II. b 71 + b73<br />
+ b75<br />
+ ... + b99<br />
; III. b71 − b70<br />
− b69<br />
− ... − b1<br />
.<br />
g. 9. 2.<br />
6<br />
:<br />
36<br />
=<br />
360<br />
:<br />
100<br />
. es toloba mcdaria, amitom<br />
2 5 10 1<br />
proporcia ar aris. 3. (3) Tviseba proporciis ZiriTadi<br />
Tvisebaa: `proporciis jvaredina wevrebis namravli erTmaneTis<br />
tolia~; (2) Tviseba ki proporciulobis mesame Tvisebaa:<br />
`proporciis Sua wevrebis adgilebis gacvliT isev proporcia<br />
miiReba~. 4. II. ( 4·<br />
0,<br />
25)<br />
26<br />
= 1<br />
26<br />
= 1;<br />
III. ( 0,<br />
2·<br />
5)<br />
8<br />
· 5 = 5 ;<br />
IV. 3 81<br />
4 = ; V. 1. 5. ( b b b ) n<br />
b<br />
n<br />
b<br />
n<br />
b<br />
n<br />
1 · 2 · ... · 100 =<br />
1<br />
·<br />
2<br />
· ... ·<br />
100<br />
.<br />
6. I. 960 kg; II. 3 t. 7. I marTebulia, radgan<br />
2<br />
4<br />
+ 2<br />
3<br />
+ 2<br />
2<br />
+ 2<br />
1<br />
= 16+8+4+2=30≤30; II. mcdaria, radgan<br />
2<br />
4<br />
+ 4<br />
3<br />
+ 5<br />
2<br />
+ 9, 7 30<br />
1 > ; III. marTebulia, radgan<br />
0<br />
4<br />
+ 1<br />
3<br />
+ 0,<br />
7<br />
2<br />
+ 0,<br />
7<br />
1<br />
= 1 + 0,<br />
49 + 0,<br />
7 < 30.<br />
8. proporciuloba gamoiyeneba:<br />
aTwiladebis gayofis, rukis masStabis, ricxvebis SefardebaTa<br />
tolobis, skalis, svetovani diagramis SemTxvevebSi.<br />
Tavi III. g. 1. 7. I. 10 . . . 0 ; II. 10 . . . 0.<br />
2n-jer 3n-jer<br />
g. 2. 1. I. 1 , 2<br />
14<br />
; II. 29<br />
4 ; III. 220<br />
g ; IV. 38<br />
h ;<br />
V. 21<br />
3 ; VI. 13<br />
5 .<br />
2. 6<br />
10<br />
= 6<br />
2<br />
· 6<br />
3<br />
· 6<br />
5<br />
= 6<br />
1<br />
· 6<br />
4<br />
· 6<br />
5<br />
= 6<br />
2<br />
· 6<br />
4<br />
· 6<br />
4<br />
. 3. 0.<br />
4. tolebia. 5. I. 4<br />
7 ; II. 10<br />
a . 6. nayinSi _ 20%, anu 6<br />
lari; wignebSi _ 12 lari.<br />
g. 3. 1. 3 81<br />
4 = ; 2 124<br />
7 = ; 3 , 45 . 2. I. 10 , 2<br />
68<br />
;<br />
II. 4<br />
b ; III. ( ) 10 7 ; IV. 0,44; V. ( )<br />
23<br />
9<br />
a + b . 4. I. SeiZleba;<br />
II. SeiZleba; III. SeiZleba; IV. SeiZleba; V. SeiZleba; VI. ar<br />
SeiZleba. 5. I. 0; II. 0; III. 1 : 1,<br />
6 =<br />
10<br />
=<br />
5<br />
;<br />
16 8<br />
5<br />
7 6<br />
IV.<br />
5<br />
=<br />
1<br />
=<br />
1<br />
; V. 64; VI.<br />
1 5 · 2<br />
; VII. =<br />
1<br />
.<br />
8 3<br />
5 5 125<br />
81<br />
7 7<br />
5 · 2 2
- 97 -<br />
6. ara, radgan 12m-ic da 8n-ic luwia, amitom maTi jamic<br />
luwia! 7. ganayofi _ sxvaoba; Tanamamravli _ Sesakrebi;<br />
gamyofi _ maklebi; gamravleba _ Sekreba; gayofa _ gamokleba.<br />
g. 4. 1. 1; 2. 2. tolebia. 4. = 6· = 6·<br />
0,<br />
04 =<br />
2<br />
9 9<br />
6 · c<br />
8 7 c<br />
6 · c<br />
= 0,<br />
24 . 5. zedapiris farTobi metia 4-jer (9-jer); moculoba<br />
_ 8-jer (27-jer). 6. azoti _ 48 kg; fosfori _ 7 kg;<br />
kaliumi _ 4 kg; kalciumi _ 6 kg. 7. yoveli ricxvi (mesamedan<br />
dawyebuli) wina ori ricxvis saSualo ariTmetikulis tolia.<br />
amitom ? niSnebis nacvlad unda eweros 9 da 3,75.<br />
g. 5. 1. I. 3 , 5<br />
12<br />
; II. d<br />
n 5 ; III. 33<br />
y . 2. I. ( ) 3<br />
13<br />
4<br />
;<br />
4<br />
⎛ 9<br />
II. ( 2 ⎞<br />
⎜ 23 ) ⎟ ; III. ( )<br />
⎝ 19 ⎠<br />
2<br />
2<br />
2<br />
2 ⎛ 2<br />
; IV. ( 1 ⎞<br />
⎜ ) ⎟ ; V. ( )<br />
⎝ 3 ⎠<br />
3<br />
b<br />
5<br />
; VI.<br />
( ) 3<br />
d<br />
m<br />
. 3. g) 4. cxrilSi %-ebis maCveneblebi 100-jer metia<br />
Sesabamisi nawilebis maCveneblebze. 5. I. gadiddeba 2-jer; II.<br />
Semcirdeba 5-jer; III. ar Seicvleba. 6. I maRaziaSi fasdaklebis<br />
Semdeg televizori eRireba 368,6 lari, meoreSi _ 365,8.<br />
g. 6. 2. daaxloebiT 694 kg. 3. 17% metia 17/12-jer.<br />
4.<br />
720<br />
· 100%<br />
12%<br />
6000<br />
II. ( ) ( ) 2 2 3 2<br />
3<br />
2<br />
2<br />
2<br />
2<br />
3<br />
= . 5. I. ( ) · ( ) · 11 121<br />
3 3<br />
3<br />
2<br />
3<br />
· · =<br />
3<br />
; III. 0,<br />
25<br />
4<br />
· 4<br />
4<br />
· 4<br />
3<br />
= 64 ;<br />
4<br />
2 2<br />
4 4 3<br />
2<br />
·<br />
4<br />
:<br />
8<br />
= · · =<br />
4<br />
·<br />
1<br />
=<br />
1<br />
.<br />
IV. ( ) ( ) ( ) 9 9 3 9 9 8 9 36 81<br />
2<br />
= ;<br />
6. I.<br />
1<br />
3<br />
d ; II. 1<br />
; III. b<br />
m−3<br />
.<br />
k<br />
g. 7. 1. I.<br />
125<br />
; II.<br />
10<br />
. 2. I. 27%; II. 160%. 3. 72 kg.<br />
6 9<br />
4. I. 1,3; II. 1,7. 6. I. 532; II. 2,4. 7. II. raki a < b,<br />
amitom m =<br />
a+<br />
b<br />
><br />
a+<br />
a<br />
= a . aseve damtkicdeba, rom m < b.<br />
2 2
- 98 -<br />
g. 8. 1. I. 1 m; II. 1,5 t. III. 1,5 weli; IV. 390 lari;<br />
V. 1 sT 5 wT; VI. 1 kg. 3. I.<br />
a<br />
; II. TviT am ricxvisa!<br />
n+<br />
1<br />
4. saSualo 8-is tolia. 3 < 8 < 16. 5. saSualo 9-is tolia.<br />
9 ≤ 9 ≤ 9. 6. 2 kg-iani sazamTro yofila 600 cali, 3 kg-iani<br />
_ 150 cali. maTi mTliani wona iqneba 1200 + 450 = 1650 kg. 2<br />
kg-iani sazamTroebi sul gayidula, radgan 750-is 80% tolia<br />
600-is, xolo 3 kg-ianebi ki _ ara, radgan 1650 kg-is 20%<br />
tolia 330 kg-isa da ara 450-isa!<br />
g. 9. 2. roca yvela ricxvi tolia! 4. ara, radganac<br />
SesakrebTa gadanacvlebiT jamis mniSvneloba ar icvleba.<br />
5. I. meore 15/8-jer; II. meore 81/5 = 16,2-jer. 6. I. 45,5;<br />
II. 19; III. 8,8. 7. 0,07a + 0,07b + 0,07c = 0,07(a + b + c).<br />
2<br />
amitom uvargisia mTliani wonis 7%. 8. II. ( 2<br />
3 ) = 2<br />
6<br />
;<br />
3<br />
III. ( 10<br />
2 ) = 10<br />
6<br />
3<br />
; IV. ( 2<br />
2 ) = 2<br />
6<br />
2<br />
; V. ( 10<br />
3 ) = 10<br />
6<br />
.<br />
g. 10. 1. I. 26<br />
26<br />
; II. 1/2 + 8. 2. mcdaria II.<br />
3. I. 24<br />
5 ; II. 5<br />
c ; III.<br />
2<br />
11 . 4. siraqlemas kvercxebis wona<br />
1,<br />
4·<br />
15<br />
0,<br />
050·<br />
16<br />
misi wonis · 100 = 25 %-ia, qaTmisa _ · 100 = 40 %.<br />
84<br />
2<br />
amitom Tavis wonasTan SedarebiT qaTami ufro meti wonis<br />
kvercxs debs,<br />
40<br />
vidre siraqlema. 5. 3<br />
200<br />
= ( 3<br />
5 ) = 243<br />
40<br />
> 200<br />
40<br />
> 200<br />
3<br />
.<br />
6. keplerma _ 21 wliT. 7. I. gadiddeba 3-jer;<br />
II. Semcirdeba 4-jer; III. ar Seicvleba. 8. ar SeiZleba,<br />
radgan Tuki TviTeul TveSi mzis naTebis xangrZlioba ≤ 200 sTze,<br />
maSin 1 weliwadSi ≤ 12 · 200 = 2400 sT-ze, rac<br />
ewinaaRmdegeba mocemulobas.<br />
Tavi IV. g. 1. 1. I. 10000. II. 99999. III. 81253. IV. 7002.<br />
2. I. 4 ⋅ 1000 + 0 ⋅100<br />
+ 7 ⋅10<br />
+ 2 ⋅1=<br />
4 ⋅<br />
10<br />
3<br />
+ 0 ⋅10<br />
2<br />
+ 7 ⋅10<br />
1<br />
+ 2 ⋅10<br />
0<br />
.
- 99 -<br />
II. 8 ⋅100000<br />
+ 9 ⋅10000<br />
+ 0 ⋅1000<br />
+ 7 ⋅100<br />
+ 0 ⋅10<br />
+ 1⋅1<br />
= 8⋅10<br />
5<br />
+<br />
+ 9 ⋅10<br />
4<br />
+ 0 ⋅10<br />
3<br />
+ 7 ⋅10<br />
2<br />
+ 0 ⋅10<br />
1<br />
+ 1⋅10<br />
0<br />
. 3. m > n 30-iT.<br />
4. n > m 9-iT. 5. I. 51. II. 15. 6. I. 75%. II. 52%. III. 190%.<br />
7. 3003 aguri. 8. I. 1 saaTSi. II. naxevar saaTSi. III. 1 saaTSi.<br />
4 3 2 1<br />
g. 2. 1. a 4a3a<br />
2a1a0<br />
= a4<br />
⋅10<br />
+ a3<br />
⋅10<br />
+ a2<br />
⋅10<br />
+ a1<br />
⋅10<br />
+<br />
0<br />
5 4 3 2<br />
+ a 0 ⋅10<br />
; a 5a4a3a2a<br />
1a0<br />
= a5<br />
⋅10<br />
+ a 4 ⋅10<br />
+ a3<br />
⋅10<br />
+ a2<br />
⋅10<br />
+<br />
1 0<br />
59<br />
58<br />
0<br />
a 1 ⋅10 + a0<br />
⋅10<br />
; a 59a58...<br />
a0<br />
= a59<br />
⋅10<br />
+ a58<br />
⋅10<br />
+ ... + a0<br />
⋅10<br />
.<br />
2. 18-is. 3. abcd ≥ abcd (tolia mxolod maSin, roca yvela aso<br />
0-is tolia!) 4. 30001000. 5. 5-is. 6. I. 540000.<br />
II. 54 miliardi. III. 400 miliardi. IV. 600000.<br />
7. I. 2 dRe. II. 4 dRe. III. 2 dRe. IV. 1 dRe. V. 2 dRe.<br />
8. ab = a ⋅10<br />
+ b ⋅1;<br />
ba = b ⋅10<br />
+ a ⋅1.<br />
340 . II. 7900000;<br />
9700000; 11; 0,71; 1,4. III. ar SeiZleba. 3. I. mcdaria. mag: 547<br />
da 386. II. mcdaria. mag: 386 da 547. III. marTebulia. IV. mcdaria.<br />
6<br />
7 4<br />
5<br />
mag: 386 da 364. 4. 10 ≤ 2507500<br />
< 10 ; 10 ≤ b 4b3b2b1b<br />
0 < 10 .<br />
21<br />
5. I. m ; II. 1<br />
p ; III. 25<br />
5 . 6. xuTkuTxa. 7. 90.<br />
27 28 0 1 0 1<br />
g. 4. 1. 10 ≤ h < 10 ; 10 ≤ 1 < 10 ; 10 ≤ 6 < 10 ;<br />
3<br />
4 1<br />
10 ≤ 3560 < 10 ; 10 20 4 2 5<br />
6 4<br />
≤ < 10 ; 10 ≤ 100000 < 10 ; 10 ≤<br />
7<br />
5<br />
82000 , 8 < 10 . 2. I. b 3 . II. b 6 . 3. 123456789. 4. 9050.<br />
5. I. ara. II. ki. III. ki. IV. ara. V. ki. 6. qaTmis<br />
sicocxlis xangrZlioba gamodis bus sicocxlis xangrZliobas<br />
mimatebuli misi 0,8 nawili. ese igi, bus sicocxlis xangrZliobis<br />
0,8 nawili yofila 10 weli. aqedan miviRebT, rom bu<br />
cocxlobs 10· 10 = 12,5 wels, qaTami _ 22,5 wels.<br />
8<br />
g. 5. 1. I. gadiddeba; II. Semcirdeba; 2. I. 11,25 sm; II. 22,5<br />
sm; III. ar SeiZleba, radgan SesaZloa moTova, magram maSinve<br />
g. 3. 1. I. { ; 34;<br />
0;<br />
3;<br />
55000;<br />
100;<br />
0,<br />
1;<br />
2000}
- 100 -<br />
gadna; 3. I. {mtkvari, Tergi, Woroxi, alazani, iori, debeda};<br />
II. udidesebi: {mtkvari, alazani, iori, rioni, enguri, xrami};<br />
saSualo sigrZe aRmosavleTSi 240+ 385+<br />
300 ≈ 308 km; samxreTSi _<br />
3<br />
150 + 220 = 185 km; dasavleTSi _ 327 + 221 = 274 km; III. udides mdi-<br />
2<br />
2<br />
nareTa saSualo sigrZeebi; 4. I. 4 saaTSi erTxel; II. 13<br />
0<br />
; 5. I.<br />
0,0; 0,5; 0,6; 1,4; 5,5; 10,0; II. 3; 6. 900. 7. I. 450; II. 15%.<br />
6+<br />
7,<br />
5+<br />
6+<br />
5,<br />
5+<br />
7<br />
6 + 6,<br />
5+<br />
7,<br />
5+<br />
7,<br />
5+<br />
6+<br />
5+<br />
5,<br />
5+<br />
6,<br />
5+<br />
7+<br />
5<br />
g. 6.1.I. = 6,<br />
4 ; II.<br />
5<br />
10<br />
= 6,25. 2. I. 14-jer; II. 1910-1920 wlebSi mateba iyo<br />
daaxloebiT 500 ha, 1920-1930-Si _ 500 ha, 1930-1940-Si _<br />
2000 ha, 1940-1950-Si _ 500 ha, 1950-1960-Si _ 1500 ha, 1960-<br />
1970-Si _ 3000 ha, 1970-1980-Si _ 5000 ha. amitom saSualo<br />
aTwliani mateba iqneba:<br />
500+ 500+<br />
2000+<br />
500+<br />
1500+<br />
3000+<br />
5000 13000 = ≈ 1860 ha = 18 600 000 kv.m<br />
7<br />
7<br />
3.I. mcdaria (ix. XI Tve). II. saSualo wliuri simaRlea baxmaroSi:<br />
125 + 175+<br />
200+<br />
150+<br />
50+<br />
0+<br />
0+<br />
0+<br />
0+<br />
15+<br />
15+<br />
50 = 65 sm, gudaurSi _ ≈ 38 sm.<br />
12<br />
Tovlis saSualo wliuri simaRle metia baxmaroSi ≈ 1,7-jer.<br />
III. marTebulia. 4. ufro zustia cxrilis xerxi, ufro TvalsaCino<br />
_ diagramisa. 5. I. 176 kg; II. raki 1 kub. m. nedli SeSa<br />
176 kg wyals Seicavs, amitom danarCeni nivTierebis wona (wylis<br />
garda) iqneba 440_176=264 kg. es wona Seesabameba 100_24=76%s.<br />
amitom 1 kub. m xmeli SeSis wonis gamosaTvlelad<br />
264 kg − − − 76%<br />
Sedgeba proporcia:<br />
. aqedan x ≈ 347 kg.<br />
x kg − − − 100%<br />
pasuxi miaxloebiTia, radgan monacemebic miaxloebiTia.<br />
g. 7. 1. I. rema; II. fara; III. jogi; IV. gundi; V. xrova;<br />
VI. kona; VII. Zna; VIII. anbani. 2. I. 11, 13 da 15; II. 9, 10<br />
da 14; III. 9 da 10; IV. 11, 14 da 15. gogonebi izrdebian<br />
ufro swrafad 11-12 wlis asakSi, vaJebi _ 14-15 wlis asakSi.
- 101 -<br />
3. 2), 3), 2) da 4). saSualod igulisxmeba 1), 3) da 4)<br />
winadadebebSi. 2) winadadebis marTebulobidan ar gamomdinareobs<br />
arcerTi sxva winadadebis marTebuloba, Tumca SedarebiT sandod<br />
SeiZleba miviCnioT 3), xolo yvelaze naklebad sandod _ 4)<br />
winadadeba. 4. magaliTad, 2 sm × 8 sm × 4 sm da 8 sm × 8 sm × 4 sm.<br />
5. I. 70 lari; II. 17,5 lari; III. 367,5 lari.<br />
g. 8. 1. I. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}; II. {iisferi, muqi<br />
lurji, cisferi, mwvane, yviTeli, narinjisferi, wiTeli};<br />
III. {qarTli, samcxe, javaxeTi, tao-klarjeTi, lazeTi da a.S};<br />
IV. {qarTuli, maTematika, ucxo ena, istoria, geografia, religiis<br />
istoria, biologia}; V. {2, 3, 5, 7, 11, 13, 17, 19};<br />
VI. {a, b, g, d, e, v, z, da a.S}; 2. marTebulia, marTebulia,<br />
mcdaria, marTebulia, marTebulia, marTebulia, mcdaria, mcdaria,<br />
mcdaria, mcdaria; 3. UI. erTi; II. oTxi; III. sami; 4. I. Seicavs;<br />
II. ar Seicavs; III. Seicavs; IV. ar Seicavs; 5. I. ana∈ D ;<br />
II. zuriko∉ A ; III. curva∉ E ; IV. e ∈ E ; V. f ∉ F ;<br />
VI. fanqari∉ D ; VII. gia∉ D ; VIII. {cira, Tamriko, naTela}<br />
⊂ D ; IX. CogburTi∉ C ; X. B ⊃ {Cikori, bzriala, burTi};<br />
XI. {Cikori} ⊂ B ; XII. Tojina∈ B ; XIII. k ∈ E ;<br />
6. I. 3 9<br />
2<br />
6 4 2<br />
3 ⋅(<br />
3 ) 6 8 14<br />
3 ⋅3<br />
3 = = = = ;<br />
12 12 12<br />
3 3 3<br />
9 7 3<br />
2 ⋅(<br />
2 ) ⋅8<br />
9 21 3 33<br />
II. = 2 ⋅2<br />
⋅2<br />
= 2 = 1 = 1<br />
;<br />
35<br />
35 35 2<br />
2<br />
2 2 2 4<br />
7. 1 ≤ a 2 + a1<br />
+ a0<br />
≤ 27 .<br />
Tavi V. g. 1. 1. I. V, VI, VII, IV, III, IX, II, VIII, X, XI,<br />
I, XII; II. {III, IX}, {II, VIII, X}, {I, XII};<br />
IV. 20+30+40+45+70+60+50+30+40+30+25+20=460 mm;<br />
3500 12700<br />
V. ≈ 7,<br />
6 -jer; VI. ≈ 27,<br />
6 -jer;<br />
460<br />
460<br />
VII. 460-jer; 2. gv. 38, 41, 42; 3. g); 4. S ≈ 330 km,<br />
330 = 33 sT-Si; 5. 5+15+25+35=80 m; me-4 wamis ganmavlobaSi.<br />
10<br />
6. rezo _ 18 ⋅ 5 = 90 , xviCa _ 12 · 5 + 24 = 84 . amitom daeweva.
- 102 -<br />
7.<br />
1<br />
nawils; Zmac igives; mama _<br />
1<br />
nawils; I. 10 sT; II. 5 sT.<br />
20<br />
10<br />
85 10<br />
g. 2. 1. = 42,<br />
5 km/sT; 2. 0 , 3⋅<br />
60 = 18 cida; ≈ 33,<br />
3 wm;<br />
2<br />
0.<br />
3<br />
ag hb ad<br />
3. I. 1 km/wT; II. 60 km/sT; 4. g) 1,8; 5. I. ; II. ; III. ;<br />
b a bc<br />
150000000<br />
6. 300000 ⋅ 3600 = 1080000000 km; I. = 500 wm;<br />
300000<br />
288000000<br />
II. = 960 wm; 7. damtkiceba advilia, Tuki gaviTvalis-<br />
300000<br />
winebT, rom S _ gasayofia, t _ gamyofi, v ki _ ganayofi.<br />
5 30<br />
g. 3. 1. ⋅ 3,<br />
6 = 3 km/sT; t = = 10 sT. 2. 2,3 sm; a) D ,<br />
6<br />
3<br />
b) B , g) E , d) A ;A 3. I wT-10 km/sT, II wT-10 km/sT, III wT-0<br />
km/sT, IV wT-8 km/sT, V wT-0 km/sT, BVI wT-9 km/sT; saSualo _<br />
10+ 10+<br />
0+<br />
8+<br />
0+<br />
9 ≈ 6,<br />
1 km/sT; 4. es amocana momdevno gakveTilis<br />
6<br />
Semamazadebelia. amitom unda dafaze unda gairCes ise, rogorc<br />
aqaa mocemuli: cxadia, v1<br />
= s t<br />
1 :<br />
3<br />
= s 3<br />
1 ⋅ t<br />
=<br />
3 s1<br />
. amgvaradve<br />
t<br />
3s2<br />
miviRebT, rom: v2<br />
= ,<br />
t<br />
v3<br />
=<br />
3s3<br />
. am siCqareTa saSualo v<br />
t<br />
saS. =<br />
.<br />
=<br />
=<br />
=<br />
1<br />
3<br />
1<br />
3<br />
v +<br />
v + v<br />
3 3 1<br />
3 3 3<br />
( s s<br />
1 2 +<br />
s<br />
+ 3 )<br />
t t t<br />
·<br />
1<br />
( s + s + s )<br />
3<br />
1<br />
2<br />
t<br />
2<br />
3<br />
3<br />
=<br />
1<br />
=<br />
( v + v + v )<br />
1<br />
3<br />
·<br />
=<br />
3s<br />
t<br />
2<br />
1<br />
3<br />
=<br />
5<br />
5. 10 km/sT; 50 km/sT; 50 · = 14 1<br />
m/wm. 7. kubSi Casmuli<br />
18 9<br />
ricxvi unda aviyvanoT mesame xarisxSi, xolo kvadratSi Casmuli<br />
_ meore xarisxSi. amitom ? = 98.<br />
3<br />
g. 4. 1. Tanabari moZraobis siCqarisa! 2. 45 wT= sT. amitom<br />
4<br />
3 v 2<br />
saS. = 14 : = 18 km/sT. 3. v) 84. 4. 250 m/wT = 15 km/sT. 1<br />
4 3<br />
dRe-RameSi frenen 10 sT-is ganmavlobaSi, amitom gaifrenen 150<br />
km-s. 2500 km-is gavlas moandomeben 2500 : 150 ≈ 17 dRe-Rames.<br />
xanmokle frenisas siCqare metia 55 : 15 ≈ 3,7-jer.<br />
g. 5. 1. saerTo wesiT: u ≈ 440 da v ≈ 440 000 000 000 000.<br />
·<br />
3<br />
3s1+<br />
3s2<br />
+ 3s<br />
t<br />
s<br />
t<br />
.<br />
=<br />
3<br />
=
- 103 -<br />
meore SemTxvevaSi: u ≈ 400 da v ≈ 400 000 000 000 000. damrgvaleba<br />
ufro uxeSia meore SemTxvevaSi. 2. milioni = 10 6 ; miliardi<br />
anu bilioni = 10 9 ; trilioni = 10 12 ; kvadrilioni = 10 15 ; kvintilioni<br />
= 10 18 ; seqstilioni = 10 21 ; septilioni = 10 24 ; oqtalioni<br />
= 10 27 ; nonalioni = 10 30 ; dekalioni = 10 33 . xarisxis<br />
maCvenebeli 3-iT imatebs. 3. vidre 10n–1 da naklebia, vidre<br />
10n . ese igi, yoveli n-niSna ricxvi, garda 10...0-isa (sadac 0<br />
aRebulia (n-1)-jer), moTavsebulia 10n–1-sa da 10n-s Soris.<br />
g) 10 n – 2 da 10 n–1 + 1. 4. e) 5. v saS. =<br />
16 + 48+<br />
128+<br />
240 = 108 ft/wm = 32,4 m/wm. cxadia, v<br />
4<br />
1 = 16 ft/wm;<br />
v 2 = 48 ft/wm; v 3 = 128 ft/wm; v 4 = 240 ft/wm. am siCqareebis<br />
saSualo daemTxveva vsaS. -s, radgan vardnis dro Tanabar<br />
15 + 5<br />
1200<br />
Sualedebadaa dayofili. 6. I. = 10 m/wm; II. t 80<br />
2<br />
1 = =<br />
15<br />
1200<br />
1200 + 1200 2400<br />
wm; t 2 = = 240 wm; amitom v<br />
5<br />
saS. = = = 7,<br />
5 m/wm.<br />
80+<br />
240 320<br />
ar daemTxva imitom, rom irmis moZraobis dro araa or tol<br />
Sualedad dayofili (tborisken _ 80 wm, tboridan _ 240 wm).<br />
7. 6 · 2 · 2 = 24 varianti. 8. g) 35-jer.<br />
g. 6. 1. I. 15 · 10 7 km; II. 4 · 10 13 km; III. 10 31 ; IV. 2 · 10 16 ;<br />
V. 10 15 . 2. I. 1000; II. 10 000 000; III. 15000; IV. 2,3;<br />
V. 34. 3. I.<br />
55 −50<br />
=<br />
12<br />
5<br />
12<br />
sT=25 wT; II.<br />
77−50 27 = = 2<br />
12 12<br />
1<br />
4<br />
sT=2sT15wT.<br />
4. I. mcdaria, radgan mZimis 2 TanrigiT marcxniv gadmotanisas<br />
aTwiladi mcirdeba 10 2 -jer da ara 10 3 -jer! II da III mcdaria,<br />
orivegan unda iyos 10 3 . 5. I. 2-jer; II. 5,5-jer; III. 10-jer;<br />
IV. 10 2 -jer; V. 10 4 -jer; VI. 10 m -jer. 6. I. riwa, yelis tba,<br />
bazaleTi; II. paliastomi, tabawyuri, xanjali, Wandara,<br />
madataba, saRamos tba; III. faravani, karwaxi;<br />
IV. xanjali, madataba, saRamos tba, bazaleTi;
- 104 -<br />
V. faravani, karwaxi, paliastomi, Wandara, riwa, yelis tba;<br />
VI. tabawyuri. saSualo udidesi siRrmea 22,7 m.<br />
g. 7. 1. 3 = 3 · 10 0 ; 10 = 1 · 10 1 ; 27 = 2,7 · 10 1 ; 25,6 = 2,56 · 10 1 ;<br />
1 = 1 · 10 0 ; 523 = 5,23 · 10 2 ; 52300000 = 5,23 · 10 7 ; 1 800 000 000<br />
000 = 1,8 · 10 12 ; 1,0040 = 1,004 · 10 0 ; 23 · 10 8 = 2,3 · 10 9 ; 0,9 · 10 6<br />
= 9 · 10 5 ; sami milioni = 3 · 10 6 ; ocdaori miliardi = 2,2 · 10 9 .<br />
3<br />
2. {7; 4,5; 1 ; 2,007}, {10; 89,63} {45000; 50001}.<br />
4<br />
3. I. erTeulebamde; II. meaTedebamde. 4. I. 1,5 kv.km = 1,5 · 10 2<br />
ha; II. 5,5 · 10 9 litri. 5. I. p = 1; II. n = 0; III. n = 3.<br />
g. 8. 1. 1,45093 ≈ 1,5 · 10 0 ; 7,8 = 7,8 · 10 0 ; 7,08 ≈ 7,1 · 10 0 ; 10<br />
= 1 · 10 1 ; 349093 ≈ 3,5 · 10 5 ; 3,625 · 10 9 ≈ 3,7 · 10 9 ; 825 · 10 15 ≈ 8,3<br />
· 10 17 ; cxranaxevari milioni = 9,5 · 10 6 ; 148 miliardi ≈ 1,5 · 10 11 .<br />
21<br />
2. 81 · 10 18 6·<br />
10 600<br />
kv.m. 3. I. = ≈ 86 jer;<br />
19<br />
7·<br />
10 7<br />
27<br />
5<br />
2·<br />
10 20·<br />
10<br />
5<br />
II. = ≈ 3,<br />
3·<br />
10 -jer. 4. dedamiwa _ 60 astrotona;<br />
21<br />
6·<br />
10 6<br />
mzis _ 2 · 10 7 astrotona; mTvare _ 0,7 astrotona. 5. aklia z)<br />
damrgvalebiT; araa qristiani (b) 0,44 nawili. meoTxeds Seadgenen<br />
kaTolikeebi da isini metni arian iudaistebze (g) 12,5-jer.<br />
6. mravalkuTxedi. 7. (d) 510066 mln. kv.km ≈ 5,1 · 10 11 kv.km =<br />
5,1 · 10 17 kv.m = 5,1 · 10 23 kv.mm. amitom gugolpleqsis cifruli<br />
Canaweri dedamiwis zedapirze ar daeteva!<br />
g. 9. 1. 3,6 · 3,6 ≈ 13 (kv.mm). 1,5 · 10 17 · 13 kv.mm = 19,5 · 10 17<br />
kv.mm ≈ 2 · 10 18 kv.mm = 2 · 10 12 kv.m = 2 · 10 8 ha.<br />
2. gamravlebisTvis, gayofisTvis da axarisxebisTvis.<br />
3. I. 500 wm; II. 37500 sT ≈ 1563 dRe-Rame ≈ 4,3 weli.<br />
4. I. a ∈ S , a ∈ S0 ; II. b ∉ S, b ∉ S0 ; III. c ∉ S , c<br />
∉ S 0 ; IV. d ∈ S , d ∉ S 0 ; V. e ∉ S , e ∉ S 0 ; VI. S ⊃ S 0 ,
- 105 -<br />
S0 ⊂ S . 5. aguredi (marTkuTxa paralelepipedi).<br />
6. 1 dReSi _ 20 · 6 = 120 mg nikotini, 1 weliwadSi _ 365 · 120<br />
≈ 44000 mg = 44 g nikotini ≈ 200 abi. amocanis SekiTxvebze<br />
pasuxis gasacemad pirveli abzacidan saWiroa mxolod is, rom 1<br />
Rer sigaretSi nikotini 6 mg-ia.<br />
Tavi VI. g. 1. 1. I. ki (kvadratisgan gansxvavebuli<br />
marTkuTxedi); II. ara; III-IV. ara; V. ki; VI. ara; VII. ki;<br />
VIII. ki. M da K sxvadasxvaa, T da K _ erTidaigive. K 2. z) 21.<br />
3. I. ara; II. ara; III. ara; IV. ki; V. ki; VI. ki. 4. erTeulis<br />
sigrZe = 2,5 mm. D0 (8), D3 (23), D4 (26), D5 (35), D6 (48), D7 (53),<br />
D8 (57), D9 (62). 5. I. v) measeds (klasSi 100 moswavle ver<br />
iqneba!); II. z) 10%-90% (arafrismTqmeli monacemia!). 7. 3·10 14.<br />
g. 2. 1. I. {1, 3, 5, 7, 9}; II. {19, 28, 37, 46, 55, 64, 73, 82,<br />
91}; III. {1, 2, 3, 4, 6, 12}; IV. {1000}; V. {avTandili,<br />
tarieli, TinaTini, ...}. 2. I. mag.: luw martiv ricxvTa<br />
erToblioba; II. mag.: 21-ze nakleb 5-is jerad ricxvTa<br />
erToblioba. 3. I. 2; II. 6; III. 0; IV. 4; V. 1; VI. 0; VII. 0;<br />
VIII. 0. SeniSvna: am amocanis garCevisas aqcenti unda gakeTdes<br />
imaze, rom erTobliobaSi SeiZleba arcerTi wevri ar iyos!<br />
4. I. ≡ II. ≡/ III. ≡/ IV. ≡ V. ≡/ VI. ≡/ VII. ≡ .<br />
III SemTxvevaSi ≡/ niSani SeiZleba Seicvalos ⊂ niSniT.<br />
5. I. ara; II. ara; III. ki; IV. ki.<br />
g. 3. 1. I. mcdaria; II. mcdaria; III. marTebulia; IV. marTebulia;<br />
V. marTebulia. 2. I. ∅ ≡/ {5}; II. 5 ∉ ∅; III. {a, b} ≡<br />
{b, a}; IV. {a, b} ≡/ {a, b, c}. ≡/ niSnis Secvla ⊂ niSniT SeiZleba<br />
I da IV SemTxvevebSi. 3. I. ara (2); II. ki; III. ara (tolferda<br />
samkuTxedi); IV. ki; V. ki; VI. ki; VII. ara (aguredi);<br />
VIII. ara (9). 4. pirveli boZidan meoTxemde 3 Sualedia, xolo<br />
meSvidemde 2-jer meti _ 6 Sualedi. amitom pasuxia 24 wuTi.<br />
5. I . marjvena; II. marcxena; III. marjvena; IV. marjvena.
- 106 -<br />
g. 4. 1. ≈ 38. sul iqneba ≈ 1590 · 4 · 38 ≈ 240 000 = 2,4 · 10 5<br />
aso. 2. ≈ 2,4 · 10 5 · 10 12 = 2,4 · 10 17 . 3. d) 4. 1 km 3 =10 9<br />
m 3 =10 15 sm 3 =5·10 14 kovzi. amitom Sav zRvaSi iqneba 80000 · 5 · 10 14 =<br />
4 · 10 19 kovzi, baltiis zRvaSi _ 10000·5·10 14 =5·10 18 kovzi, xolo<br />
kaspiis zRvaSi _ 15000 · 5 · 10 14 = 7,5 · 10 18 kovzi. 5. I. ≈1,3 · 10 151 ;<br />
II. 2,8 · 10 71 ; III. 3,5 · 10 43 . 6. I. prizmas minimum 3 gverdiTi<br />
waxnagi aqvs da ori fuZe aqvs! II. minimum 9 wibo aqvs!<br />
III. wveroebis raodenoba luwi unda iyos! prizmas SeiZleba<br />
hqondes 1348 wvero, magram ar SeiZleba hqondes amdeni wibo,<br />
radgan wiboebis raodenoba 3-is jeradi unda iyos!<br />
7. Tvlis dros didi ricxvis dasaxelebas meti dro sWirdeba.<br />
amitom CavTvaloT, rom saSualod erTi ricxvis dasaxelebas<br />
sWirdeba 10 wami. maSin wuTSi dasaxeldeba 6 ricxvi, 1 sT-Si _<br />
360, 1 dRe-RameSi _ 16· 360 ≈ 5800 (davuSvaT, rom adamians 8 sT<br />
sZinavs), 1 weliwadSi _ ≈ 2 000 000; 80 weliwadSi _ 160 000<br />
000 = 160 milioni. cxadia, es pasuxi Zalian pirobiTia. amitom<br />
dadebiTad unda CaiTvalos moswavleTa mier miRebuli sxva<br />
pasuxebic, oRond maT unda SeeZloT TavianTi pasuxebis metnaklebad<br />
dasabuTeba.<br />
g. 5. 1. bundovania: II, IV, VI, VII, IX. 2. I. 5; II. 1; III.<br />
5; IV. 8; V. 2; VI. 15; VII. 4. 3. I. {1, 29}; II. {16, 32, 48,<br />
64, 80, 96}. 4. mcdaria III (radgan `ori~ sityvaa da ara aso!)<br />
da V (unda iyos ∈ niSani). 5. erTeulovani monakveTis sigrZea<br />
4 sm : 10 = 0,4 sm. e) 6. 1,5 km = 1500 m sigrZeze saWiroa<br />
3000 fila, 3,5 m siganeze _ 7 fila. sul _ 21 000 fila. 180<br />
sm × 50 sm zomis fila 180:50=3,6-jer naklebi iqneba saWiro, sul<br />
21000:3,6 ≈ 5834 cali. 7. {a}, {b}, {g}, {a, b}, {a, g}, {b, g}, {a, b, g}.<br />
bolo igivuria mocemuli igiveobisa. arcerTi maTganis wevrTa rao-
- 107 -<br />
denoba ar aRemateba mocemuli erTobliobis wevrTa raodenobas.<br />
g. 6. 1. `milionis~ Sesatyvisia `uSqari~; uStisuSti 100<br />
milionis tolia; miliardi tolia 1000 uSqarisa. 2. qviSis<br />
marcvalTa raodenoba naklebi iqneba 6·10 21 t : 1 /1 0 6 g = 6·10 27 ·10 6<br />
= 6 · 10 33 -ze. 3. me-5 nabiji: 10 10 kvadr.; me-10 nabiji: 10 20 kvadr.<br />
4. ara, radgan rogori ricxvic ar unda daiweros, yovelTvis<br />
SeiZleba masze metis dawera! sami Zmis SemTxvevaSi araferi<br />
icvleba! 5. ara! 6. ar SeiZleba! 7. weliwadSi daewveTeba 365<br />
mg=0,365 g kiri, saukuneSi _ 36,5 g. stalaqtidis xnovaneba<br />
iqneba 100 000 g : 36,5 ≈ 2740 saukune.<br />
g. 7. 1. sasrulebia: I, II, III, IV, VI, VIII. 2. ara, imitom<br />
rom sxivi erT mxares usasrulod grZeldeba! 3. SeiZleba. samive<br />
SemTxvevaSi (I, II, III) sxivze usasrulod mravali monakveTi<br />
moizomeba. 4. erTeul. monakv. sigrZea 4 sm : 0,8 5 sm.<br />
(d) 0,2 < x < 0,8; y > 0,2. 5. magaliTad: I. C erTniSna ricxvTa<br />
simravle, D _ naturalur ricxvTa simravle; II. C kent<br />
ricxvTa simravle, D _ naturalur ricxvTa simravle;<br />
6. I. raki naturalur ricxvTa simravle usasruloa, amitom<br />
usasrulo iqneba mTel ricxvTa simravlec; II. raki monakveTebis<br />
simravle usasruloa da TviTeuli monakveTi romeliRac kubis<br />
wiboa, amitom kubebis simaravlec usasruloa! III. kent ricxvTa<br />
simravle sasruli rom iyos, maSin iarsebebda udidesi kenti<br />
ricxvi! IV. koncentruli wrexazebis radiusebis sigrZeebi, magali-<br />
Tad, metrebSi nebismieri dadebiTi ricxvis toli SeiZleba iyos.<br />
amitom koncentrul wrexazTa simravle ver iqneba sasruli.<br />
g. 8. 1. 7. 4. I. 1-isa; II. 2-isa; III. 1-isa; IV. 1-isa.<br />
5. {(A, C), (E, D)}. 6. I. B rom sasruli yofiliyo, maSin A-c<br />
sasruli gamovidoda (marTlac, Tuki iarsebebs ricxvi, romelic<br />
metia B-s wevrebis raodenobaze, maSin, cxadia, es ricxvi meti<br />
iqneba A-s wevrebis raodenobazec!); II. Tuki arsebebs ricxvi, ro-
- 108 -<br />
melic metia A-s wevrebis raodenobaze, maSin, cxadia, es ricxvi<br />
meti iqneba A-s nebismieri qvesimravlis wevrebis raodenobazec!<br />
g. 9. 1. mcdaria I (monakveTi moTavsdeba wreSi, romlis diametri<br />
am monakveTis tolia!) da IV (aseTi xazi SeiZleba iyos<br />
agreTve usasrulo texili an mrudi). 2. 2 wrfe da 10 sxivi<br />
(aq Sedis is sxivebi, romelTa saTavec moniSnulia!). 4. nakvTi<br />
dagalkevebuli ar unda iyos. amitom Tuki ori wertili<br />
ekuTvnis nakvTs, maSin am nakvTs unda ekuTvnodes am wertilebis<br />
SemaerTebeli raime xazic, romelic, cxadia, usasrulod bevri<br />
wertilebisgan Sedgeba! 6. aseTi wertili oria!<br />
g. 10. 1. I. SeiZleba; II. SeiZleba; III. ara; IV. ara; V.<br />
ara; VI. ara. 2. magaliTad, cilindri imitomaa<br />
SemosazRvruli, rom igi Caeteva iseT birTvSi, romlis radiusi<br />
cilindris simaRleze 2-jer metia. 3. I. ara; II. ki; III. ki;<br />
IV. ara. 5. 1,2 : 0,8 = 1,5 (lari). gaZvirebis Semdeg _<br />
1,5+1,5·1,2 = 3,3 (lari). 7. I. {6, 4, 3, 9} ⊂ {7, 3, 6, 5, 4, 9};<br />
II. {10, 20, 30} ≡/ {30, 40, 10}; III. {a, i, o, u, e} ⊃ {e, o, i, a};<br />
IV. {maia, mzia, lia, naTia, ia} ≡/<br />
{sandro, irakli, biZina, vaxtangi}; V. {a , a , a , a } ⊃ {a , a , a }.<br />
1 2 3 4 2 3 4<br />
8. davalebis Sesruleba advilia moswavlis saxelmZRvanelos<br />
ydis Siga mxares mocemuli TvalsaCinoebis gamoyenebiT.<br />
Tavi VII. g. 1. 1. 75 : 23 ≈ 3,3-jer. 2. 675-450=225 g-iT meti.<br />
3. xazis sisqea daaxloebiT 0,5 mm = 500 mikr. wertilis<br />
diametris sigrZe iqneba daaxloebiT 1 mm = 1000 mikr.<br />
4. iseTi TvalsaCino magaliTi, rogoric sxivis SemTxvevaSia,<br />
wrfisTvis Zneli mosaZebnia. amitom mosalodneli pasuxebic<br />
nairgvari SeiZleba iyos: eleqtrogadacemis gaWimuli sadeni,<br />
swori gzatkecilis kide, rkinigzis liandagi da sxva.<br />
5. erT. monakv. sigrZea 4 sm : (3,07-3,06) = 400 sm = 4 m.<br />
e) 3,065. 6. (40_4) : 4 = 9 (km/sT).
- 109 -<br />
7. uTvalavi yofiTi sityvaa (aramaTematikuri!) da gulisxmobs,<br />
rom daTvla SeuZlebelia. amitom uTvalavi SeiZleba iyos<br />
sasruli simravlec. magaliTad, dedamiwaze qviSis marcvlebis<br />
simravle. am azriT ufro zogadia usasrulo simravle, radgan<br />
misi daTvla SeuZlebelia.<br />
g. 2. 1. orma monakveTma SeiZleba gadakveTos erTmaneTi erT<br />
wertilSi. or monakveTs SeiZleba hqondes uamravi saerTo<br />
wertili. 2. 4 wrfe. 3. 65 km/sT ≈ 18 m/wm. Tevzis frenis<br />
saUalo siCqarea 4 m/wm. 18 m/wm metia 4 m/wm-ze 4,5-jer, anu<br />
350%-iT. 4. uamravi. erTi. arcerTi. I. uamravi. erTi. uamravi.<br />
II. arcerTi. wrfes bolo ar aqvs. erTi. 6. I. {A, E, B}; II. {E,<br />
C, D}; III. {E}; IV. {F, G}; V. {A, E, B, C, D}. 7. sxivi 1<br />
welSi gairbens 365 · 24 · 3600 · 300000 ≈ 9,5 · 10 11 km-s, rac<br />
9,5 · 10 5 -jer metia 10 6 km-ze. amitom masStabi iqneba 1 : 9,5 · 10 16 .<br />
g. 3. 1. ar iqneba marTebuli: or wertilze uamravi swori<br />
xazi gaivleba! 2. sam an oTx wertilze, sazogadod, swori xazi<br />
ar gaivleba. magram Tu wertilebi isea ganlagebuli, rom maTze<br />
swori xazis gavleba SesaZlebelia, maSin gaivleba uamravi aseTi<br />
swori xazi. wrfis SemTxvevaSi ki Tu gaivlo, mxolod erTi<br />
wrfe gaivleba. 3. I. 1,3 · 105-jer; II. 3 m : 12,5 mm ≈ 2,4 · 102- jer; III. 2 · 108-jer; 4. ar varga imitom, rom uamravia iseTi<br />
sxivi, romelic am or wertilze gadis! magram Tu moniSnulia<br />
sxivis saTave da am sxivis nebismieri sxva erTi wertili, maSin<br />
am ori wertiliT es sxivi ukve calsaxad ganisazRvreba.<br />
5. I. SeiZleba; II. ara (maSin es wertilebi aRar iqneba<br />
gansxvavebuli!); III. SeiZleba. 6. I. ki; II. ara; III. ki; IV. ki;<br />
V. ara (magaliTad, monakveTi arcerT sxivs ar Seicavs). 7.<br />
g. 4. 1. uamrav wertilSi. 2. 3. 4. 0,8 litrian qilaSi<br />
Caeteva 1,5 · 0,8 = 1,2 kg Tafli. amitom 12 kg TaflisTvis saWiroa<br />
10 0,8 litriani qila. 5. 1,5-jer, anu 50%-iT. 6. I. ki; II.<br />
ara; III. ki. 7. I. mcdaria (magaliTad, texilic SeiZleba iyos
- 110 -<br />
SemousazRvreli); II. marTebulia; III. marTebulia; IV. mcdaria<br />
(SeiZleba moicavdes!); V. marTebulia; VI. marTebulia; VII.<br />
marTebulia (radgan nakvTi ar SeiZleba iyos dacalkevebuli).<br />
8. I. yovelgvari swori xazebisTvis (aq gadakveTa ufro yofiTi<br />
azriTaa da ara ise, rogorc maTematikuri termini `TanakveTa~!);<br />
II. mxolod wrfeebisTvis.<br />
g. 5. 2. PQ da AB (wrfe), PQ da AB (sxivi), PQ da BA<br />
(sxivi), PQ da AB (monakveTi), m da NK (wrfe), m da NK<br />
(sxivi), m da KN (sxivi), m da NK (monakveTi), CD monakveTi da<br />
FE monakveTi, DC sxivi da FE monakveTi, CD monakveTi da FE<br />
sxivi, DC sxivi da FE sxivi. 3. orisa. 4. ara. 6. I da IV.<br />
7. ar SeiZleba.<br />
g. 6. 2. I. NP, PQ, ND, CK; II. BP PC, BC, HN; III. AB,<br />
BD, DE, AD, BE, AE. 4. ar SeiZleba. 5. manZili kaspiis zRvidan<br />
mtkvris gavliT aragvis saTaveebamde aris daaxloebiT 600 km.<br />
amitom oraguls dasWirdeba daaxloebiT 12 dRe.<br />
6. I. mcdaria; II. marTebulia; III. mcdaria, mxolod sami ar<br />
SeiZleba hqondes! IV. mcdaria.<br />
g. 7. 1. d || g, d || b, b || g. 4. I. monakveTia; II. sxivia;<br />
III. sxivia. 5. I. iqneba; II. iqneba.<br />
7. maSin 1/x axlosaa 0-Tan, amitom swori pasuxia (e) 6.<br />
8. erT. mon. sigrZea 1,5 : 0,1 = 15 sm. swori pasuxia (e).<br />
g. 8. 2. A3B3 da A4B4 , A1B1 da A2B2 , A6B6 da A5B5 .<br />
4. erT. monakv. sigrZea 6 sm : (18,75 _ 18,25) = 12 sm. e) 18,375<br />
da 18,72. 6. mcdaria I, radgan 3-is jeradi ricxvebis simravle<br />
moicavs 9-is jeradi ricxvebis simravles. amitom is Tviseba, rac<br />
aqvT 9-is jerad ricxvebs, SeiZleba ar hqondes 3-is jeradebs!<br />
g. 9. 1. 2. AB ⊥ AE ; CD || GH ; DH || AE ; EH ⊥ HG ;<br />
AD || BC. 3. 4. 5; 19,5; 4. 5. mcdaria: I, III, IV, V.<br />
6. 3 araqarTuli wigni.<br />
g. 10. 1. d) 3,3 . 2. BC||AD, BC||EK, AD||EK, AB||DC, CF||DK,<br />
CF||AE, DK||AE. 3. aris. 4. mcdaria: I, II, III, IV. 5. 20 t.
- 111 -<br />
7. pasuxi unda SevarCioT miaxloebiTi gamoTvlebis Sedegad. 10dan<br />
80-mde sul 70 ricxvia, maTi saSualo (10+80) : 2 = 45-is<br />
tolia. amitom yvela am ricxvis jami daaxloebiT 45 · 70-is<br />
toli unda iyos. axla gaviTvaliswinoT, rom 3-is jeradia<br />
yoveli mesame ricxvi, amitom vivaraudoT, rom saZebni jamis<br />
mniSvnelobaa daaxloebiT 45 · 70 : 3 = 70 · 15 = 1050. amitom<br />
swori pasuxia (d) 10035.<br />
g. 11. 1. araa marTebuli. magaliTad, pirvel oTxkuTxeds<br />
aqvs es Tviseba, magram araa marTkuTxedi. 2. I. paraleluria;<br />
II. erTmaneTs gadakveTs; III. erTmaneTs emTxveva. 4. ese igi,<br />
saSualod 45 kg rZisgan miiReba 4 kg yveli. amitom 1 t rZisgan<br />
miiReba 4 · 1000 ≈ 90 kg yveli. 5. g) 0,4. 6. usasrulod<br />
45<br />
mravali. 7. marTebulia: II, III, VI.<br />
Tavi VIII. g. 1. 2. z-RerZis - 3,4 sm : 0,001 = 3400 sm = 34 m;<br />
x-RerZis - 3 sm : 800 ≈ 0,0036 sm; y-RerZis - 3 sm : 2700 ≈ 0,001 sm.<br />
7. I. ara; II. ki. eqvskuTxedis SemTxvevaSi diagonalebi SeiZleba<br />
iyos erTmaneTis paraleluri, SeiZleba _ marTobulic.<br />
g. 2. 1. 1 sm-iT marjvniv wanacvlebiT koordinati<br />
matulobs 0,01 erTeuliT. amitom, magaliTad, T1 (123,45)-dan<br />
marjvniv 1 sm-is daSorebiT SeiZleba moiniSnos wertili<br />
A1 (123,46), xolo marcxniv _ B1 (123,44). 2. 1 sm-iT marjvniv<br />
wanacvlebiT koordinati matulobs 100 erTeuliT. amitom,<br />
magaliTad, T2 (123,45)-dan marjvniv 1 sm-is daSorebiT SeiZleba<br />
moiniSnos wertili A2 (223,45), xolo marcxniv _ B2 (23,45).<br />
3. 120 kg. 4. I. BH-is paraleluria NP, PD, ND, CK; BH-is<br />
marTobulia NH, BP, PC, BC, DK. II. EK-s paraleluria AH, HM,<br />
AM; EK-s marTobulia ED, DB, BA, EB, DA, EA, CT, NC, NM,<br />
NT, NC, TM. III. NC-s paraleluria AB, BD, DE, AD, BE, AE;<br />
NC-s marTobulia AH, HM, AM, EK, KT, ET. 7. d) 9 · 8 · 7 · 5.<br />
g. 3. 1. tolia, radgan urTierTmarTobuli wrfeebi
- 112 -<br />
erTmaneTis simetriis RerZebia. meoTxed nawils. 2. I. e) 1 sm :<br />
2 m ; II. 10 rigi, rigSi – 12 skami; III. scenis miaxloebiTi<br />
zomebia naxazze 0,2 sm × 3,8 sm. sinamdvileSi iqneba 0,4 m × 7,6<br />
m. amitom farTobi iqneba daaxloebiT 3 kv.m. IV. darbazis<br />
miaxloebiTi zomebia naxazze 5,8 sm × 4,5 sm. sinamdvileSi iqneba<br />
11,6 m × 9 m. amitom darbazis moculoba iqneba 11,6 · 9 · 5,5 ≈ 574<br />
(kub.m). V. (1; 7), (2; 4), (2; 12), (4; 9), (4; 11), (6; 12), (7; 4),<br />
(7; 9), (8; 1), (8; 2), (8; 10), (9; 2), (9; 9), (10; 1), (10; 5), (10; 10),<br />
(10; 11), (10; 12).<br />
3. C1-is koordinatia 4, C2-is _ 3. O-dan C-Si misasvleli<br />
kodia, magaliTad, O ~<br />
⎛ 4<br />
3<br />
⎞<br />
⎜ ⎯ ⎯→ ↑ ⎟ ~ C . mis Casawerad sakmarisia<br />
⎝ ⎠<br />
ori ricxvi. 4. 1 sm-iT marjvniv wanacvlebiT koordinati<br />
matulobs 0,001 erTeuliT. amitom, magaliTad, M(263,6)-dan<br />
marjvniv 1 sm-is daSorebiT SeiZleba moiniSnos wertili<br />
E(263,601), xolo marcxniv _ F(263,599). 5. 1 sm-iT marjvniv<br />
wanacvlebiT koordinati matulobs 1000 erTeuliT. amitom,<br />
magaliTad, M(263,6)-dan marjvniv 1 sm-is daSorebiT SeiZleba<br />
moiniSnos wertili E(1263,6), xolo marcxniv _ F(–736,4).<br />
g. 4. 1. B(2; 0), D(2,5; 1), O(0; 0), F(0; 1,5), K(9; 2), M(9; 3),<br />
L(6; 2). 3. A da G, B da F, C da K, H da P, L da M.<br />
5. axlos: N(0,08 ; 1), Sors: K(10 ; 112,1).<br />
g. 5. 1. P(3,5; 2) I, Q(1,5; 4) II, J(1,5; 0) II (an III), R(4; 3)<br />
III, T(4,5; 2) IV. 4. kvadratisa. 5. mravali wertilis<br />
dasaxeleba SesaZlebelia I da II SemTxvevebSi.<br />
7. mcdaria: V. pasuxi Seicvleba V da VI SemTxvevebSi.<br />
g. 6. 1. I. E(4,6; 2,3) IV, F(2; 2,4) I, G(3,4; 0,6) I, H(2,2; 2) II,<br />
K(2,6; 1) II, L(2,2; 1) III, M(2,1; 2,7) III, N(1; 2,3) IV. II. P(2,7; 0)<br />
I (an IV), Q(0; 0,5) I (an II), R(0,5; 0,5) II, S(1,5; 0,8) III, T(0,6; 2,5)<br />
I, U(0; 1,7) III (an IV), V(2; 1) IV, W(1,8; 1,5) IV. 3. ≈7,4%.<br />
4. 19,7%. 5. {(0; 7)I, (2; 4)I, (1; 4)I, (3; 1)I, (1; 1)I, (4; 2)IV,<br />
(1; 2)IV, (1; 4,5)IV, (1; 4,5)III, (1; 2)III, (4; 2)III, (1; 1)II, (3; 1)II,
- 113 -<br />
(1; 4)II, (2; 4)II }. {(7; 1)I, (8; 1)I, (8; 2)I, (9; 3)I, (10; 4)I}.<br />
g. 7. 2. A(0,5; 3,6)I, B(1,6; 0,8)II, D(2,7; 2,7)III, F(2,6;<br />
2,7)IV, C(2,4; 1,5)I. 3. I. 4 l; II. 1,7-2 kg. 5. 54 lari.<br />
6. I. pirveli; II. meore.<br />
g. 8. 1. A(30; 50)I, B(20; 60)II, C(23; 90)III, D(25; 50)IV, E(2;<br />
2)I, F(1; 2,5)II, G(1,6; 2)III, H(2,1; 1,3)IV, K(2; 2,5)II, L(1,5; 2,7)III,<br />
M(1,1; 0,5)II, T(1,7; 3,2)I. 2. I. ≈ 4/7 mm; II. ≈ 1/6 mm. 3. 178,2 l.<br />
5. cxadia, 4y = 50, amitom swori pasuxi iqneba: (d) 55,5.<br />
6. I. saydris _ (120; 70)II; II. Zveli eklesiis _ (165; 70)I; III.<br />
mwyemsebisa _ (50; 30)III. gareubani _ (180-240; 20-40)IV.<br />
soflidan Zvel eklesiamde: (120; 25)II, (45; 15)II, (120; 20)IV,<br />
(195; 20)I. am gzis sigrZea rukaze daaxloebiT 12 sm. amitom<br />
sinamdvileSi iqneba _ 24 km. avtobusi gaivlis am manZils<br />
daaxloebiT 50 wT-Si.<br />
g. 9. 1. E - 180° ag da 60° Cg, F - 30° ag da 70° Cg, G - 0° ag (an<br />
dg) da 45° sg, H - 50° dg da 0° Cg (an sg), J - 42° ag da 0° Cg (an<br />
sg), K - 30° dg da 45° sg, L - 15° dg da 60° Cg, M - 70° ag da 40°<br />
sg, N - 110° ag da 0° Cg (an sg). 2. I. sankt-peterburgi;<br />
II. axali orleani; III. kito. 3. I. q. Tbilisi - 45° ag da 42° Cg;<br />
II. q. moskovi - 38° ag da 57° Cg; III. q. londoni - 0° dg da 52°<br />
Cg; IV. q. brazilia - 42° dg da 15° sg; V. q. sidnei - 151° ag da 33°<br />
sg; VI. q. los-anJelesi - 118° dg da 34° Cg. 4. 1 wlis Semdeg<br />
_ 103 lari; 2 wlis Semdeg _ 106,09 lari.<br />
g. 10. 2. I. soxumi; II. ozurgeTi; III. Sxara;<br />
IV. dedofliswyaro. 3. I. 43,6° da 41,6°; II. 44,5° da 42,8°;<br />
III. 40,7° da 43,2°; IV. 45,9° da 41,9°; V. 44° da 42,2°;<br />
VI. 41,8° da 42,5°. 5. I. 8%; II. marili - 500 g, niori<br />
- 300 g, Zmari - 40 g, mwvanili - 160 g.<br />
Tavi IX. g. 1. 3. cila - 2 kg, cximi - 1 kg, saxamebeli - 1,45<br />
kg. 4. zRarbi - 32°-iT, Tria - 29°-iT, zazuna - 34°-iT, Ramura -<br />
32°-iT. 5. dedamiwa sicocxlis gareSe yofila ≈ 4,5 _ 3,55 =<br />
0,95 miliardi weli. amitom pasuxia 4,45 : 0,95 ≈ 3,7-jer.
- 114 -<br />
6. oTaxSi myofTa asakTa jamia Tavidan 8 · 24 = 192 weli, Semdeg<br />
_ 7 · 22 = 154 weli. amitom pasuxia 192 _ 154 = 38 weli.<br />
g. 2. 1. I. ianvarSi - 10 aTasiT, aprilSi - 10 aTasiT,<br />
ivnisSi - 30 aTasiT; II. TebervalSi - 20 aTasiT, martSi - 15<br />
aTasiT, maisSi - 20 aTasiT.<br />
2. I. momatebula 105 aTasi ha-Ti; II. yvelaze didi - 1940 wels<br />
(1910 welTan SedarebiT 160 aTasi ha-Ti meti iyo), yvelaze<br />
mcire - 1980 wels (1910 welTan SedarebiT 280 aTasi ha-Ti<br />
naklebi iyo). III. 27 aTasi ha; IV. ar SeiZleba, radgan ar<br />
viciT ramdeni iyo TviTeulis farTobi 1910 wels!<br />
3. 1° siTbo (pasuxi SeiZleba miviRoT orgvarad: 1) TandaTan<br />
viangariSoT Sesabamisi temperaturebi: 1° siTbo, 0°, 1° siTbo, 2°<br />
siTbo, 1° siTbo, 0°, 2° yinva, 1° siTbo; 2) jer gamovTvliT,<br />
rom dRis 12 sT-ze temperatura wina Ramis 12 sT-Tan SedarebiT<br />
Seicvala -2°-iT, anu Semcirda 2°-iT. amitom saboloo<br />
temperatura iqneba 3°_2° = 1° siTbo).<br />
4. (7; 4), (5; 6), (6; 9), (8; 12), (9; 15), (11; 18), (14; 18), (17;<br />
16), (20; 13), (23; 11), (22; 10). 5. I. gogona - 550; vaJi - 537.<br />
II. 1976, 1978, 1979 wlebSi; III. 1971, 1973, 1975, 1977 wlebSi.<br />
6. 150-200 kg. 15-20 kg-is saSualoa 17,5 kg. ese igi, 100 kg<br />
Warxlisgan saSualod miiReba 17,5 kg Saqari. amitom 60 kg<br />
Saqris misaRebad saWiro iqneba ≈340 kg Warxali.<br />
g. 3. 1. yvelaze bevri fuli eqneba noembris bolos,<br />
yvelaze didi vali _ ivnisis bolos. 2. I. (d) 5,5; II. (a) 1,2;<br />
III. (e) 1,6. 3. I. II-Si 10 aTasi, III-Si 5 aTasi; II. IV-Si 15 aTasi;<br />
III. ixeira, radgan winasTan SedarebiT qoneba gaizarda; IV.<br />
TebervalSi 15 aTasi lariT. 4. I. 0 lari; II. 10 lari vali;<br />
III. 20 lari vali. 5. cvlilebaa -2°. diliT - 2° yinva. 6.<br />
sokos wona mcirdeba saSualod 80 %-iT. amitom: (I gza) 20-30<br />
kg axali sokos wona Semcirdeba 16-24 kg-iT. ese igi miiReba 4-6<br />
kg (20_16=4, 30_24=6). (amoxsnis II gza) darCeba wonis 20%.<br />
Sesabamisad, 20-30 kg axali sokosgan miiReba 4-6 kg soko.
- 115 -<br />
g. 4. 1. I. +7, +3,1, 3,1, 1/5, 7; II. –7, –3,1, –1/5; III. +0,<br />
–0, 0. tolebia: +7 da 7, +3,1 da 3,1, +0, –0 da 0.<br />
2. I. 550 l – 400 l = 150 l; II. 550 l – 550 l = 0 l;<br />
III. 550 l – 650 l = –100 l; IV. 50 l – 200 l = –150 l.<br />
3. an 5,5 m simaRleze, an 0,5 m-ze. 4. I. 180,5; II. 19,5; III. 0;<br />
mdebareobs A0-s marjvniv. 5. +4°. +22,5°; _8° 6. _24 sm.<br />
g. 5. 1. D(–0,3), E(+0,7), F(+2,2), K(–2,3), G(+1,6), J(+0,7),<br />
H(–0,5), T(–1,7) . 3. 1962 wels. 4. 1871 wels. 5. I. marjvniv;<br />
1 sm; C wertils; II. marcxniv; 3/4 sm; F wertils;<br />
III. marjvniv; 75 sm; D wertils; IV. marjvniv; 3/800 sm; E<br />
wertils. 6. I. 13; II. –16; III. –900; IV. –50.<br />
g. 6. 1. daviT aRmaSenebeli - 52 weli; oqtaviane avgustusi<br />
- 14+63–1=76 weli (1 weli imitom gamoaklda, rom `nulovani~<br />
weli ar arsebobs!). I msoflio omi grZeldeboda 4 weli,<br />
spartakis ajanyeba - 3 weli. 2. unda eTqva: _ es moxda Zveli<br />
welTaRricxvis me-4 saukuneSi. 3. im droisTvis, cxadia, Zveli<br />
welTaRricxvis cneba ar arsebobda! 4. B1 (–9,5), B2 (6,3), B3 (0,1),<br />
B4 (–256,03). 5. I. –1. 7. mag.: –12 da –10, –13 da –9 da a.S.<br />
g. 7. 3. dabadebula Zv.w. 5 w., gardacvlila ax.w. 66 w.<br />
4. {–0,5; 1/2}, {0,02; –1/50}, {1,3 ; –13 /1 0 }.<br />
6. I. 3 milion 800 aTasi; II. 700 aTasiT; III. 5-jer.<br />
7. I. C-Si; II. D-Si; III. E-Si.<br />
g. 8. 2. raki 1600 weli XVII saukunesaa mikuTvnebuli,<br />
amitom dekarti dabadebula 1596 wlis martSi, xolo<br />
gardacvlila _ 1650 wlis TebervalSi. 3. unda eTqva: `Zv.w.<br />
VIII-VI saukuneebSi~. 4. I. 400 l + 150 l = 550 l; II. –1000<br />
l + 1500 l = 500 l; III. –50600 l + 40000 l = –10600 l;<br />
IV. –15000 l + 10000 l = –5000 l. 5. A(1), B(–5), C(–1,5).<br />
6. mcdaria III. marTlac: 0-is mopirdapire araa uaryofiTi,<br />
magram 0 araa uaryofiTi. 7. amocanis pirobaSi mocemuli Tvalsazrisis<br />
gaziarebis SemTxvevaSi I saukuneebi (Zv.w. da ax.w.)<br />
`nakiani~ ar iqneboda; Tu nulovani weli iarsebebda, maSin ax.w.
- 116 -<br />
I saukune aRar iqneboda `nakiani~ (gagrZeldeboda 100 wels da<br />
damTavrdeboda 99 wels), xolo Zv.w. I saukune ki mainc `nakiani~<br />
darCeboda (daiwyeboda -99 wels da damTavrdeboda -1 wels).<br />
g. 9. 2. I. 3,7; II. 7,8; III. 0; IV. 200. 3. I. 12 da –12;<br />
II. 41,09 da –41,09. 4. waTe I mefobda ax.w. 510-530 wlebSi,<br />
iuliusi cxovrobda Zv.w. 100-44 wlebSi, ovidiusi _ Zv.w. 43 -<br />
ax.w. 17 wlebSi, filoni _ Zv.w. 25 - ax.w. 45 wlebSi. 5. I.<br />
waTe I mefobda ax.w. VI sauk. I naxevarSi; II. iuliusi _ Zv.w. I<br />
sauk. I da II meoTxedebSi; III. ovidiusi daibada Zv.w. I sauk.<br />
III meoTx. da gardaicvala ax.w. I sauk. I meoTx.; IV. filoni _<br />
Zv.w. I sauk. IV meoT. da ax.w. I sauk. I da II meoTxedebSi.<br />
6. I. A; II. C; III. F; IV. M. 7. I. pirvels: 1700 l > 1100 l;<br />
II. meores: –1700 l < 0 l; III. pirvels: –200 l > –1000 l.<br />
g. 10. 3. I. 25,5 > 3,4; II. 4,5 > –800; III.–36,7 > –84,3;<br />
IV. –7,5 < –7,2; V.–1025 < 0. 4. I. –4,3 < 0; II. 27,1 >0;<br />
III. a < 0; IV. c ≤ 0; V. b > 0; VI. d ≥ 0. 5. I. 8;<br />
II. 150; III. 7,6. 6. 7. 7. I. x = 9 an x = –9; II. y =<br />
0 ; III. aseTi z ar arsebobs! 8. 500 t madnisgan miiRebdnen ≈<br />
32,5 kg spilenZs, 600 t madnisgan _ ≈ 25,5 kg-s. sxvaobaa 7 kg.<br />
g. 11. 1. I. ki; (mag.: 3); II. ara; III. ki (mag.: 3,7); IV. ki<br />
(aseTia 0); V. ki (mag.: 3); VI. ki (mag.: –3,7). 2. rkina, nikeli,<br />
spilenZi, vercxli, alumini, tyvia, kala, naftalini, gayinuli<br />
glicerini, gayinuli vercxliswyali, gayinuli acetoni.<br />
acetoni. 3. I. 11,3; II. –46,9. 4. I. –5083 > –5183;<br />
II. –15,44 > –25,44; III. –999,0 > –999,1; IV. –90,701 <<br />
–90,602. erTaderTi cifris Caweraa SesaZlebelia I SemTxvevaSi.<br />
6. II. mcdaria (mag., roca m = 7, n = –7); IV. mcdaria<br />
(magaliTad, roca m = –7, n = –6); VI. mcdaria (mag., roca m = –<br />
7, n = –8); VII. mcdaria (magaliTad, roca m = –7, n = 6);<br />
7. I. 0; II. a; III. –a. 8. I. 0 < x; II. y> 0; III. m < 0;<br />
IV. 0 > n; V. x > m; VI. n < x; VII. y > n; VIII. m < y.
- 117 -<br />
Tavi X. g. 1. 1. I. {1; 3,0; 50000; 43}; II. {1; 3,0; 50000;<br />
43; 0; –19; – 30<br />
; –345,00}; III. { 3<br />
1<br />
; –2,5;<br />
6<br />
; –0,05 ; – 12<br />
;<br />
6<br />
5 5<br />
5<br />
0,7}; IV. {–19; – 30<br />
; –345,00; –2,5; –0,05 ; – 12<br />
}. 2. I. –4;<br />
6<br />
5<br />
II. –5, –4, –3, –2, –1, 0, 1, 2. 3. marTebulebia: I, II, III, VI,<br />
VIII, IX, XII. 4. I. –607; II. 30,04. 5. mag.: {Savi, lurji,<br />
TeTri, yviTeli, yavisferi, nacrisferi, iasamnisferi, narinjisferi}.<br />
6. araa igive! SeiZleba ar iyo boroti, magram arc keTili iyo!<br />
7. mcdaria: I (ekuTvnis niSani simravleebs Soris ar daismis!),<br />
III (dadebiTi mTeli ricxvebic xom arsebobs!), IV (yvela<br />
dadebiTi ricxvi araa naturaluri!). 8. I. uaryofiTi;<br />
II. dadebiTi; III. nuli; IV. an dadebiTia, an nulia.<br />
g. 2. 2. I. {0, 1, 2, 3, 4, 5}; II. {–2, –1, 0}. 3. I. 2100<br />
wlis 1 ianvars; II. 1900 wlis 1 ianvars; III. 900 wlis 1<br />
ianvars; IV. 1599 wlis 31 dekembers; V. Zv.w. 99 wlis 1<br />
ianvars; VI. Zv.w. 700 wlis 31 dekembers. 4. I. daibada Zv.w.<br />
VII sauk. IV meoTxedSi, gardaicvala Zv.w. VI sauk. III meoTx.;<br />
II. daibada ax.w. VI sauk. III meoTx., gardaicvala ax.w. VII sauk.<br />
II meoTx. 5. mcdaria: II (wrexazis centri ar ekuTvnis<br />
wrexazs!), III (sferos centric ar ekuTvnis sferos!). 6.<br />
7. I. aseTia 0; II. ar arsebobs! III. ar arsebobs! IV. aseTia 0.<br />
g. 3. 1. mtkvar-araqsis kulturis `asaki~ gamodis daaxloebiT<br />
3,5+2=5,5 aTaswleuli, xolo kreta-mikenisa - 2+2=4 aTaswleuli.<br />
amitom pirveli Zvelia 5,5 : 4 ≈ 1,4-jer. 2. I. N(2); II. A(–<br />
6); III. toladaa daSorebuli. 3. I. 2000 wlis 1<br />
ianvars; II. 1000 wlis 1 ianvars; III. 999 wlis 31 dekembers;<br />
IV. Zv.w. 1 wlis 31 dekembers; V. Zv.w. 1000 wlis 31 dekembers;<br />
VI. Zv.w. 2999 wlis 1 ianvars. 4. {4} ⊂ {3} ⊂ {1} ⊂ {2} ⊂ {5}.<br />
5. N ⊂ Z ⊂ Q, N ⊂ P ⊂ V ⊂ Q , U ⊂ T ⊂ Q.<br />
6. sul 13-isa (1.1.11; 11.1.11; 1.11.11; 11.11.11; 2.2.22; 22.2.22;<br />
3.3.33; 4.4.44; 5.5.55; 6.6.66; 7.7.77; 8.8..88, 9.9.99).<br />
g. 4. 1. I. 1 miliardi weli; II. 4 aTaswleuli; III. 4,5
- 118 -<br />
aTaswleuli; IV. Zv.w. XV saukuneSi; V. 35 saukune; VI. 65<br />
saukunis win; VII. 9 aTaswleulis win; VIII. 900 000 wlis<br />
win. 2. qristes dabadebidan Cvens xanamde daaxloebiT 2000<br />
welia gasuli. amitom Sesabamisi wertili daSorebuli unda iyos<br />
Cveni xanis Sesabamisi wertilidan (2) skalaze<br />
2000<br />
≈ 0,<br />
013 sm-<br />
150000<br />
iT, xolo (1) skalaze _<br />
2000<br />
= 1 sm. aseTi mcire<br />
300000000 150000<br />
manZilebis mozomva skalaze, cxadia, SeuZlebelia!<br />
3. mcdaria: I, IV, VI, VII, VIII, IX, XII. 4. Tanamedroveni iyvnen.<br />
ufrosi iyo platoni, ufro adre gardaicvala evdoqse.<br />
7. 1 lomi ~ 200 kg, 1 Cliqosani ~ 200 kg, 1 foToli ~ 20 t.<br />
g. 5. 1. 195 lari vali anu –195 lari. 2. dabadebula<br />
savaraudod Zv.w. 64 wels, Zv.w. I saukuneSi, gardacvlila ax.w. I<br />
saukuneSi. aTaswleulTa cvlas moeswro. 3. daRupula –212<br />
wels, Zv.w. III saukunis IV meoTxedSi. 4. I. 1; II. 0 an 2.<br />
5. Tbilisi +4,5°, manglisi +0,5°, TianeTi –2°, quTaisi - 6°.<br />
4,<br />
5+<br />
0,<br />
5−2<br />
+ 6<br />
saSualo temperaturaa = 2,<br />
25°<br />
.<br />
4<br />
6. I. 2200 l; II. 1200 l; III. 0 l. 7. I. 250 l; II. –50 l;<br />
III. 200 l. 8. I. 30 l + (–20) l = 10 l; II. 3000 l + (–2000)<br />
l = 1000 l; III. 3000 l + (–4000) l = –1000 l.<br />
g. 6. 1. I. _20; II. _96,6; III. _3/13; IV. 6,5; V. –3,5;<br />
VI. –7,5; VII. 2,5. 2. I. –10; II. –0,1; III. –2; IV. –1,1.<br />
3. I. = –3,75 + 12,5 = 8,75; II. = 16,7 – 34,18 = –17,48.<br />
4. I. mcdaria; II. marTebulia; III. marTebulia.<br />
6. I. _ 6a; II. –1,2m; III. 0,9v; IV. _a; V. 2k.<br />
g. 7. 2. dabadebula Zv.w. 276 wels, Zv.w. III saukuneSi,<br />
gardacvlila Zv.w. II saukuneSi. aTaswleulTa cvlas ver<br />
moeswro. 3. –5,5°. 4. I. –80 l + (–20) l = –100 l;<br />
II. –50 dol. + (–30) dol. = –80 dol.<br />
5. = 2 · 67,24 – 20,5 + 25,8 – 10 = 134,48 – 4,7 = 129,78. 6. a.<br />
7. =<br />
108000000<br />
=<br />
1080<br />
= 6 m. 8. mcdaria: II, IV.<br />
2400000·<br />
7,<br />
5 180
g. 8. 1. I. –3; II. ( 4<br />
2)<br />
- 119 -<br />
− ; III. –22; IV. –2,2; V. –5,6;<br />
3<br />
VI. –12,8. 2. I. = –56 + 23 + 6 = –27; II. = 64–8,4–2,4–3,2 = 50.<br />
4. gaaCnia a-s: SeiZleba metic, tolic da naklebic! 5. mefobda<br />
65 wels, dabadebula Zv.w. 394 wels, gardacvlila Zv.w. 302<br />
wels, Zv.w. I aTaswl. II naxevarSi. 6. I. –3,1; II. –15,5; III. 0;<br />
IV. –1,5; V. –5. 7. gaxmobis Semdeg darCeboda 4,8 kg Ciri. 1 kg Ciri<br />
daujdeboda 12 : 4,8 = 2,5 (lari). dauzogavs 4,8 · 0,5 = 2,4 l.<br />
g. 9. 2. mag.: I. –5,8 = –2,8 + (–3); II. –5,8 = 1,2 + (–7).<br />
3. yoveli racionaluri ricxvi SeiZleba iyos an uaryofiTi,<br />
an nuli, an dadebiTi. 4. dabadebis dRemde: +6 lari, dabadebis<br />
dRis Semdeg +10 lari. 4 lari valis daklebiT misma qonebam<br />
moimata 4 lariT. 5. I. yvelas meore koordinati 0-is tolia;<br />
II. yvelas pirveli koordinati 0-is tolia. 6. Raribi valisgan<br />
gaTavisufldeboda orive SemTxvevaSi. mdidris qoneba mogebis<br />
Sedegad Semcirdeboda 10 oqroTi! 8. I. 8; II. 20; III. 10.<br />
g. 10. 1. I. 13; II. –12; III. 4<br />
3<br />
; IV. –2,5; V. –10,36;<br />
17<br />
VI. – 1,3; VII. − 7<br />
3<br />
. 2. I. = 1,64–2,1–5,73 = 1,64–7,83 =<br />
7<br />
–6,19; II. = 22+10 = 32. 3. I. 5; II. 0; III. 0,5; IV. –2.<br />
4. I. diddeba; II. mcirdeba; III. mcirdeba; IV. diddeba.<br />
7. I. 120 aTasiT; II. 5/4-jer; III. sul qarTveli mosaxleoba<br />
yofila 16+1,5+20+7,5+0,5+11+0,5 = 57 `kacuna~, amitom<br />
megrelebi iqnebodnen 11 · 100 ≈ 19 %.<br />
57<br />
g. 11. 1. I. = –8,81+6,42–6,4 = –8,81+ 0,02 = –8,71; II. = –2,8 +<br />
6,38 – 7,9 = 6,38 – 10,7 = –4,32. 2. –7,1. 3. I. 0° – 4° = –4°;<br />
II. –6,5° – 5° = –11,5°; III. –6° – 8,5° = –14,5°; IV. –3° – 0° = –3°.<br />
4. mcdaria: II, IV, VI. 5. marTebulia: I, VII, VIII. 7. mcdaria II, V.<br />
Tavi XI. g. 1. 1. |EB|=7 sm, |BC|=4 sm, |AF|=5 sm. 2. mesame<br />
gverdis sigrZe a unda akmayofilebdes samkuTxedis utolobas:<br />
5,5 m < a < 6,9 m. amitom es sigrZea 6 m. 3. I da II sawarmoebis
- 120 -<br />
SemaerTebeli monakveTis SuamarTobis mdinaresTan gadakveTis<br />
wertilSi. 4. I. –18; II. – 16,6; III. –5; IV. –3,5; V. 25;<br />
VI. –<br />
5<br />
. 5. I. z = –12 ; II. x = 0; III. y = 2; IV. z = –8.<br />
7<br />
g. 2. 1. I. E, K, F; II. M, T, N. 5. I. = – 4,4 – 12 = –16,4;<br />
II. = – 2,3 – 24,8 + 20 = –7,1. 6. raki |BC| = 14 sm, amitom |BE| +<br />
|EC| = 40 – 14 = 26 (sm). magram |BE| = |AE|, amitom |AE| + |EC| = 26<br />
sm, anu |AC| = 26 sm. 7. mcdaria: I, III, IV, VI.<br />
g. 3. 4. = 3 · 16 + 22,5 – 5 = 48 + 17,5 = 65,5. 5. am monakveTis<br />
SuamarTobs, garda am monakveTis Suawertilisa (monakveTis<br />
boloebiT da SuawertiliT samkuTxedi ar Seiqmneba!). 7. 120 kg.<br />
g. 4. 2. |EK| = 0,75 sm. 3. ara (samkuTxedis utoloba ar<br />
sruldeba). 4. I. –20; II. 2,4; III. –2,6; IV. –12,2; V. 18,6;<br />
VI. 56,8. 5. I. –14a; II. –0,4 k; III. –4x; IV. – 8b; V. 8y.<br />
6. gavavloT ori qorda. wrexazis centri iqneba maT Suamar-<br />
TobTa gadakveTis wertili.<br />
g. 5. 1. 1 sm. 3. a. 4. LB monakveTi. 7. 1 m.<br />
g. 6. 3. aseTi wrfe mravali gaivleba. 4. I. ara; II. ki;<br />
III. ki; IV. ki. 5. 1. Tbilisi; 2. basra; 3. bombei; 4. kolombo;<br />
5. kalkuta; 6. deli; 7. novosibirski; 8. moskovi. 6. I. AB<br />
marTobia, AD - daxrili; II. AD marTobia, AB – daxrili.<br />
g. 7. 1. Q(3; 3), S(–4; 2), D(–1,5; –2,5), K(3,5; –2).<br />
3. I. {A, B, E}; II. {A, F, D}; III. {D, G, C}; IV. {B, H, C};<br />
V. {B, E, A, P}; VI. {A, B, E, P, T, K, M, F, H, L}; VII. yvela, garda<br />
A, F da D wertilebisa; VIII. ∅. 4. 1) A, C, D;<br />
2) A, C, D, P, K, O; 3) B, E, L, O; 4) F; 5) H, P, O; 6) A, B,<br />
C, D, E, P; 7) A, C, D, G, K. 5. I. meore koordinati metia an<br />
toli 0-ze; II. pirveli koordinati naklebia an toli 0-ze.<br />
g. 8. 2. I. x-RerZis wertilebs; II. y-RerZis wertilebs;<br />
III. koordinatTa saTaves. 3. I. (4; 0); II. (0; 4); III. (–3; 1);<br />
IV. (1; –2). 4. I. = 1 + 2,6u = 14; II. = 3 · 125 – 30 – 9,3 – 1 – 4<br />
= 375 – 44,3 = 330,7. 5. nakvTis sazRvari Sedgeba 4 naxevarwrexa-
- 121 -<br />
zisgan. aqvs simetriis 4 RerZi.<br />
6. dReRameSi: 7000 · 60 · 24 = 10080000 kub.sm ≈ 10000 litri.<br />
weliwadSi: 3650000 litri = 3650 kub.m.<br />
g. 9. 1. |DB| = |BC| = |AC| = |AD| = 5 sm. 3. I. {C, E, F, M, K};<br />
II. {A, B, D, N, H, O}; III. {C, D, M, H}; IV. {A, L, O}; V. {C, D,<br />
M, H}. 4. I. 18; II. 62; III. 9,4; IV. –8,6; V. –11,4; VII. –<br />
15,6. 5. I. = 48,2 – (–45) = 93,2; II. = –1,6 + 6,9 + 3,8 – 9 = 0,1.<br />
6. I. ki; II. ki; III. ara. 7. araa toli.<br />
Tavi XII. g. 1. 1. A(0,2), B(–0,8), C(–0,1). 2. I. 0; II. 0; III.<br />
k 3 . 3. I. A1 (2102,5); II. A2 (–102,5); III. A3 (97,5); IV. A4 (–302,5).<br />
4. I. A1 (–500,5); II. A1 (0,5). 6. b) 25. 8. mcdaria II da V.<br />
g. 2. 1. = –9,3. 2. 100. 3. I. –3+y; II. a + 47;<br />
III. –100 + x; IV. –14; V. –33 – a. 4. I. x = –8,3; II. x = 4.<br />
5. p – q = p + (–p). 6. I. uaryofiTisTvis; II. dadebiTisTvis;<br />
III. dadebiTisTvis. 7. e) 569 000. 8. g).<br />
g. 3. 1. I. 49; II. –38; III. –6; IV. 7,8. 2. I. = –67 – a =<br />
–16,4; II. = x – y – 3 = –34,8. 3. I. 9,8; II. –3. 4. mtkvararaqsis<br />
_ 50-55, TrialeTis _ 35-40, kolxeTis _ 25-35<br />
saukunis win. 5. sanam xuro urmiT naxevar gzas gaivlis,<br />
mWedels SeeZleba mTeli gzis gavla. magram mas ukve hqonda<br />
raRac manZili gavlili. amitom ufro adre Cava mWedeli.<br />
7. mcdaria II da III.<br />
g. 4. 3. I. –3; II. 6; III. 38. 4. I. 2,4; II. –24,9; III. a+k;<br />
IV. 0. 5. I. x = 1,25; II. x = –4. 6. mecnierma. 770 km.<br />
g. 5. 2. I. –4; II. –5; III. 8,1; IV. –32,2.<br />
3. II. –8,7; III. 0; IV. b. 4. I. x = 42; II. y = 29/9.<br />
5. perimetri – 28 erT.; farTobi – 48 kv. erT. 6. I. 0-s, 1-s<br />
an 2-s; II. 0-s, 1-s, 2-s; III. 0-s, 1-s an 2-s; IV. 0-s, 1-s, 2-s,<br />
3-s an 4-s; V. 0-s an 1-s. 7. I. 0; II. nebismieri dadebiTi ricxvi.<br />
g. 6. 1. I. somexi; II. azerbaijaneli; III. –Ze; IV. imerelraWvel-leCxumel-guruli.<br />
2. mag.: I. ia, vardi, ia, ia, ia, ia,
- 122 -<br />
SroSani, vardi, yayaCo, gvirila; II. ku, ku, ku, ku, ku, ku, ku,<br />
ku, ku, ku. 3. CamonaTvalis `moda~ gviCvenebs, romelia am<br />
CamonaTvalis yvelaze xSiri wevri. 4. I. 1; II. –0,3.<br />
5. I. 3; II. – 2a – 7; III. – 7; IV. y. 6. I. ara; II. ki.<br />
frCxilebis gaxsnis wesi modulebisTvis araa marTebuli. 7.<br />
CamonaTvalSi wevrebis gameoreba dasaSvebia, erTobliobaSi _ ara.<br />
g. 7. 1. Tuki ori ricxvis jami 0-ia, maSin isini sxvadasxvaniSnianebia,<br />
mastan maTi modulebi toil unda iyos. amitom isini<br />
mopirdapire ricxvebi iqnebian! 2. TviTeul SemTxvevaSi<br />
vaCvenoT, rom mocemuli ricxvebis jami 0-is tolia!<br />
3. I. x = –2; II. x = 4; III. x = –11; IV. x = –15. 4. I. varskvlavi;<br />
II. kata; III. qarTuli ena. 6. marTebulia: II, IV, V.<br />
7. azoti _ 3,9 kg, Jangbadi _ 1,05 kg, danarCeni _ 0,05 kg.<br />
g. 8. 1. I. x = –9; Z-s. II. x = 8,45; Q-s. III. x = –4,37; Q-s.<br />
IV. x = –6,99. Q-s. 2. –15. 6. I. ar aqvs; II. aqvs; III. ar aqvs.<br />
7. I. perimetri = 12 erT., farTobi = 9 kv.erT.; II. perimetri<br />
= 20 erT., farTobi = 25 kv.erT.; 8. I dRes - 20%, meore<br />
dRes – 80 · 30% = 24%. darCa – 56%. ese igi, misatani<br />
wignebis raodenobis 56% Aaris 42. amitom pasuxia: 75.<br />
g. 9. 2. A2B2C2 , A5B5C5 . 4. Tanabradaa daSorebuli.<br />
6. mcdaria: II, VI, VII. 9. marTebulia: I, II, IV, V.<br />
g. 10. 1. Q Tanabradaa daSorebuli A da D wertilebidan.<br />
amitom is AD-s SuamarTobze mdebareobs. 2. paralelobasa da<br />
marTobulobas Soris kavSiris Tanaxmad, erTi wrfis marTobuli<br />
ori wrfe erTmaneTis paraleluria. 3. ar SeiZleba, radgan ma-<br />
Sin gamovidoda, rom a da b wrfeebi erTmaneTis paraleluria<br />
(paraleluri wrfeebis marTobuli wrfeebi xom erTmaneTis paraleluria!).<br />
5. kveTs erT wertilSi. 6. I. 8,3; II. –0,3;<br />
III. –57. 7. I. – 2y – 7; II. – 2a – 2b + 8; III. –7; IV. 0,6 – y.<br />
8. wertilebi. 9. mag.: I. x +2 = 8; II. x +2 = 1; III. 3x = 8.<br />
g. 11. 2. marTkuTxedis diagonalebis gadakveTis wertili
- 123 -<br />
iqneba Semoxazuli wrexazis centri. 3. ara, radgan FG da EH<br />
gverdebis SuamarTobebi saerTod ar gadakveTs erTmaneTs.<br />
4. perimetri = 20 erT., farTobi = 21 kv. erT.<br />
5. I. M; II. B; III. B. 7. paraleluria!<br />
g. 12. 1. I. d; II. 2; III. – 2a. 4. aqvs. I. 0,5 sm; II. 0,25 sm;<br />
III. 0,5 mm; IV. 0,00005 mm. 5. 2 mm, 1 mm, 0,5 mm da a.S.<br />
monakveTi wertilTa usasrulo simravlea. 6. mcdaria: II, III,<br />
V, VI, VIII, X, XII. 7. 5227 wels. aRmoaCines XIX saukunis<br />
III meoTxedSi, daumarxavT Zv.w. XXXIV saukunis II meoTxedSi,<br />
IV aTaswleulis II naxevarSi. 8. mcdaria: I, III.<br />
Tavi XIII. g. 1. 1. I. 2; II. 9. 2. |MN| = 8 erT. 3. I. 14,2<br />
erT.; II. 4,76 erT. 4. I. –6; II. –6; III. –15; IV. –26,73.<br />
5. pirvel SemTxvevaSi cvlilebaa 7 aTasi lari, meoreSi – 15<br />
aTasi. mewarmis qonebis cvlileba Tvis ganmavlobaSi orive<br />
SemTxvevaSi tolia misi saboloo (Tvis bolos) qonebasa da<br />
Tavdapirvel (Tvis dasawyisis) qonebas Soris sxvaobis. 6. I. x =<br />
11; II. y = –18,06; III. z = –0,9. 7. qarTuladac da rusuladac<br />
laparakobs mcxovrebTa (65 + 55) – 100 = 20 %. amitom<br />
mxolod qarTulad ilaparakebs 65 – 20 = 45 %.<br />
g. 2. 1. I. –6,1; II. 6,1; III. 2,1. 2. I. 0,9; II. –1,2; III.<br />
10,3; IV. –3,7. 4. –90. 7. –1,7 da –9,7.<br />
8. I. a–7; II. –x–9,2; III. –6+c–k; IV. 4 – n + m.<br />
g. 3. 1. –5°c. 2. 76 m. 3. I. –0,5 mln. lari; II. 2 mln., 1<br />
mln.; III. 0,5 mln., 1,5 mln.; IV. 1) 1,5 mln.; 2) 1 mln.; 3) 0,5<br />
mln. 4. ufro zogadia tolferda samkuTxedi. 5. I. –4 da –<br />
3; II. 5 da 6; III. –1 da 0. 6. I. x = 1,7; II. x = –18,99.<br />
7. cvlilebaTa jamia 40 m. raki mTliani cvlileba 0-is toli<br />
unda iyos, amitom bolo ujredSi unda eweros –40.<br />
g. 4. 1. –11,4. 2. hipotenuzis sigrZe gaizomeba saxazaviT.<br />
5. I. 2a; II. 9a; III. ta.<br />
g. 5. 1. I. –27; II. –3; III. 56; IV. 0; V. –2,5; VI.<br />
0,209. 2. (d) 30. 3. I. 4 aTasi; II. –8 aTasi; III. Semcirdeboda
- 124 -<br />
6 aTasiT; IV. gadiddeboda 16 aTasiT. 4. ricxvis (–1)-ze<br />
gamravlebiT miiReba am ricxvis mopirdapire ricxvi. 6. I.<br />
12; II. –60. 7. 7,8 mm + 8,75 mm < 16,9 mm. amitom ar SeiZleba!<br />
2 g. 6. 1. I. –1981; II. 103; III. –3 3 ; IV. –57. 2. I. 3 + (–4c);<br />
II. –7,5 + 6a + (–9a). 3. I. –7,9b, 4c da –8; II. 1/k, –2/b da –<br />
1,5. 4. a(b – c) = a(b +(–c)) = ab+a(–c) = ab – ac. 5. I. 3(3b+3);<br />
II. 4(c–d); III. a(3–5) = –2a; IV. 3b. 6. I. x = –9; II. x = 10.<br />
7. I. perimetri _ 16 erT., farTobi – 15 kv.erT. II. perimetri<br />
_ 15 erT., farTobi – 12,5 kv.erT. zedmeti monacemia me-4<br />
wveros koordinatebi! 8. I. dadebiTi; II. uaryofiTi.<br />
g. 7. 1. I. 15x; II. –5y; III. –4a; IV. 2x – 12; V. 0; VI. –<br />
2; VII. x; VIII. –5+2c. 2. I. –a – 2; II. 1 + 3x; III. t + z; IV.<br />
2,8 – 8,5y. 3. I. 20 km-iT marjvniv; II. 20 km-iT marcxniv;<br />
III. 20 km-iT marcxniv; IV. 20 km-iT marjvniv. l = vt formula<br />
marTebulia masSi Semavalma asoebis nebismieri<br />
mniSvnelobebisTvis. 4. I. kaTeti ar SeiZleba hipotenuzis toli<br />
iyos! II. fuZe unda iyos hipotenuza. 5. (a) 13. 6. raki<br />
mercxali isev budeSi dabrunda, amitom cvlilebaTa jami 0-is<br />
toli unda iyos. amitom swori pasuxia –11,26.<br />
g. 8. 3. I. AFEDC; II. B. 4. gamravleba / gayofa.<br />
5. I. 21,5; II. 3; III. 0. 6. racionaluri ricxvebis gadamravlebis<br />
wesis mixedviT namravlis moduli TanamamravlTa<br />
modulebis namravlis tolia, modulebi arauaryofiTi ricxvebia. aseTi<br />
ricxvebis SemTxvevaSi ki Tanamamravli tolia namravli gayofili<br />
meore Tanamamravlze.<br />
7. damtkiceba eyrdnoba racionaluri ricxvebis gadamravlebis<br />
wess. 8. I. x = –2; II. x = 5 an x = –5; III. x = –3.<br />
9. ufro zogadia Tanamedrove mniSvneloba, radgan is gamoiyeneba<br />
ara mxolod im SemTxvevaSi, roca fuZeSi oTxkuTxedia, aramed<br />
sxva mravalkuTxedebis SemTxvevaSic.<br />
g. 9. 1. I. –2; II. –3; III. 6; IV. 500; V. 0,31; VI. –100.<br />
2. I. ki; II. ara (= –40); III. ara (= –2,7); IV. ki.
- 125 -<br />
3. I. 25; II. –11. 4. orive tolia 19-is. 5. I. –10,7; II. 36.<br />
6. I. x = 2; II. z = 27. 7. g) 98% .<br />
g. 10. 1. I. 5; II. –1/3; III. a = – 2,5; IV. a = – 7/4.<br />
2. Tuki a < 0, maSin a = –|a|, amitom misi Sebrunebuli iqneba −<br />
1<br />
.<br />
| a|<br />
3. I. –4/5; II. –6. 4. I. 3600; II. –42. 6. SeiZleba (amis dasasabuTeblad<br />
warmovidginoT ori piramida, romelebic miRebulia<br />
erTi piramidis gakveTiT ise, rom kveTa wveroze da fuZeze<br />
gadiodes!). 7. I. (ab) : c = (ab) · 1/c =(a · 1/c) · b = (a : c) · b.<br />
8. I. 160; II. 65. 9. I. x = –4; II. y = –3; III. z = –5/8.<br />
g. 11. 2. ufro zogadia piramida. `klebis mixedviT~: piramida;<br />
samkuTxa piramida; iseTi piramida, romlis fuZe _<br />
tolferda samkuTxedia; tetraedri. 4. I. = (–12,3 – 6) · (–0,1)<br />
= 1,83; II. = –2,5 + 6,4 – 1,9 = 2; III. 1. 5. x = –1,8.<br />
6. I. 2d+7c; II. 2(5m+4k+n+3k) = 10m+14k+2n.<br />
7. metad _ indielTa piramidebi, naklebad _ zikurati.<br />
8. ufro zogadia mniSvneloba Zveli berZnuli sityvisa `tetraedri~,<br />
radgan is ar gulisxmobs, rom waxnagebi aucileblad<br />
tolia!<br />
Tavi XIV. g. 1. 4. (d) 12 l 25 T. 5. 1 sT =<br />
20+1+20+1+18. amitom swori pasuxia 58 · 30 = 17,4 lari.<br />
`mudmivi nazrdi~ iqneboda, mag., aseT SemTxvevaSi: pirveli<br />
nasaubrevi 20 wuTis Semdeg 1 wuTis saubaria ufaso, kidev 20<br />
wT-is Semdeg _ 2 wuTia ufaso, kidev 20 wT-is Semdeg _ 3<br />
wuTia ufaso da a.S. 6. I. xuTkuTxeds; II. samkuTxeds.<br />
g. 2. 1. I. 1; II. 3, 5; III. 1, 3, 5. 2. I. 10; II. 0,3.<br />
3. I. 2; II. a + 8; III. –10b – 2; IV. – 3x – 4.<br />
4. 1) 13 weli gamodis toli daaxloebiT gedis sicocxlis xangrZliobis<br />
1 – 0,4 = 0,6 nawilisa. amitom gedi cocxlobs daaxloebiT<br />
22 wels, mercxali _ 8-10 wels. 2) yvavi da Woti _<br />
60 w = 0,6 sauk.; qori, arwivi, TuTiyuSi _ 4 w = 0,4 sauk.;<br />
wero _ 37,5 w = 0,375 sauk.; qaTami _ 25 w = 0,25 sauk.; toro-
- 126 -<br />
la, Zera, Sevardeni _ 20 w = 0,2 sauk.; ixvi, bati, Tolia, SaSvi,<br />
bu, yanCa _ 17,5 w = 0,175 sauk.; SoSia _ 12,5 w. = 0,125 sauk.<br />
3) siraqlema _ 30-40 w, gareuli qaTami _ 12-16 w, mtredi _ 15-20 w.<br />
4) ozurgeTis mosaxleobis raodenobis 4/7 nawili yofila<br />
12000. amitom ozurgeTSi iqneba 21000 mosaxle, lagodexSi _<br />
9000. sayofacxovrebo SinaarsiT erTmaneTs hgavs 1), 2) da<br />
3), maTematikuri SinaarsiT _ 1) da 4).<br />
5. I. x = 30,4; II. x = 1 an x = –2.<br />
6. ar aqvs, radgan marcxena mxare yovelTvis 9-iTaa meti<br />
marjvenaze. 7. gamartivebiT miviRebT: 3x – 1 = 3x – 1. es<br />
toloba ki x-is nebismieri mniSvnelobisTvisaa marTebuli.<br />
g. 3. 1. I. tolfasia; II. tolfasia; III. tolfasia;<br />
IV. araa tolfasi. 3. I. marTebulia; II. mcdaria. 4. x + 15 =<br />
25. x = 10. es gantolebebi tolfasi iqneba. 5. I. marTebulia;<br />
II. marTebulia; III. marTebulia; IV. mcdaria. 7. I. = –3,5x –<br />
0,5; II. –1,8y + 2,5. 8. I. ki; II. ara; III. ki; IV. ara.<br />
g. 4. 1. I. ki; II. ki; III. ara; IV. ki. 2. I. x = –60; II. y =<br />
37,9; III. x = 6; IV. y = 36. 6. I. 16x + 5x = 25 – 7; II. 9x 2 – 6,5x<br />
+ x2 = 1 + 2,3. 7. e) 97,5. 8. Tuki aravis ar eqneboda<br />
gatanili 4 an ufro meti goli, maSin 8 moTamaSes SeeZlo<br />
gaetana maqsimum 3 · 8 = 24 goli. es ki ewinaaRmdegeba pirobas!<br />
g. 5. 1. I. x = 5; II. y = 2; III. z = 3; IV. x = 10. 2. I. 4;<br />
II. 2; III. 1. 3. me-5 wevria 17, me-9 wevri - 29. me-100 wevri<br />
metia 97-e wevrze 3 · 3 = 9-iT. 4. 1 000 000 : 6 = 166 666 (naSTi<br />
4). amitom memilione iqneba me-4 aso I . 6. I. 0,01; 0,02;<br />
0,03; 4; II. a, d, e, a, r; III. .<br />
7. II. 1 kvira; III. 32 dRe; IV. 5 weli; V. 1 Tve; VI. 1 weli.<br />
yvelaze zustia II, radgan Tvisa Tu wlis xangrZlioba araa<br />
ucvleli, kvirisa ki ucvlelia.<br />
g. 6. 1. 6). 2. me-3 wevria 18, me-5 wevri - 162. me-10 wevri<br />
metia me-8-ze 9-jer. 6. I. (3; –2); II. (–3; 2). 7. AB monakveTi.<br />
g. 7. 1. {AM, CO, PF, MO, PK}. 7. I. 16-jer; II. me-18 -
- 127 -<br />
jixuri, 31-e - Zelskami. III. raki 3904-is 5-ze gayofisas naSTia<br />
4, amitom Sexvdeba Zelskami, xolo 5000-is 5-ze gayofisas<br />
naSTia 0, amitom Sexvdeba yvavilnari. periodi iqneba:<br />
yvavilnari, Zelskami, anZa, jixuri, Zelskami. 8. simetria.<br />
g. 8. 1. I. x = –7000; IV. x = 0. 2. I. x = –1/3; II. ∅;<br />
III. x = –3; IV. usasrulod mravali amonaxsni aqvs. 4. I. 193;<br />
II. 2. 8. I. 10+9+8+1 = 28; II. 4 · 3 + 1 = 13; III.<br />
10+9+8+1 = 28; IV. 10+1 = 11; V. 9+8+8+2 = 27.<br />
g. 9. 1. I. x = 2; II. y = 5; III. x = –2; IV. y = –7.<br />
7. 1 danayofis `wonaa~ 2 : 5 = 0,4. amitom ? niSnebis nacvlad<br />
unda eweros (marjvnidan marcxniv): 2,6, 1,8, 0,2 da –2,6.<br />
yvelaze Sesaferisi saTauria: d) mudmivi uaryofiTi nazrdi.<br />
8. zedmetia `mezobeli~.<br />
g. 10. 3. I. x = 2/19; II. t = 1; III. y = 0; IV. x = 30 an x = –<br />
30; V. x = 2 an x = –2. 4. a) 24,75 km. 5. I. d); II. b).<br />
7. nebismieri sami ricxvidan 2 mainc iqneba kenti an 2 mainc<br />
iqneba luwi. maTi jami ki unaSTod gaiyofa 2-ze.<br />
g. 11. 1. I. A1 (2; 1) da C1 (7; 5); II. C2 (–1; –3); x-RerZis<br />
mimarT simetriuli arekvliT miRebuli samkuTxedis wveroebis<br />
koordinatebi iqneba: (-6; -1), (-1; -1) da (-1; -5). 2. I. b) E(-4; -1),<br />
F(1; -1), G(1; 3); II. d) D(-6; 1), T(-6; -4), S(-2; -4);<br />
III. g) P(6; 1), Q(1; 1), R(1; 5). 3. l = 6a, sadac a erTi wibos<br />
sigrZea. 5. Tu TiTo TveSi 4 moswavles eqneba dabadebis dRe,<br />
maSin klasSi 48 moswavlis SemTxvevaSic ki 5 moswavles<br />
dabadebis dRe erT TveSi ar eqneba. magram Tu TiTo TveSi 3<br />
moswavles eqneba dabadebis dRe, maSin maqsimum 36 moswavle unda<br />
iyos klasSi, es ki ewinaaRmdegeba pirobas. amitom aucileblad<br />
moiZebneba iseTi Tve, roca dabadebis dRe eqneba am klasis 4<br />
moswavles mainc. 6. yvelaze xelsayrelia: (v) 1 wuTis saubris<br />
fasia 28 TeTri; yvelaze araxelsayreli: (e) 1 wuTis saubris<br />
fasia 30 TeTri, oRond yoveli nasaubrevi 1 saaTis Semdeg 1 wuTis<br />
saubari ufasod. wesis `1 wuTis saubris fasia 30 TeTri, oRond
- 128 -<br />
yoveli nasaubrevi 20 wuTis Semdeg 1 wuTis saubari ufasoa~<br />
tolfasia (d).<br />
§ 19. sakontrolo werebis amocanaTa pasuxebi<br />
da maTi Sefasebis kriteriumebi<br />
sakontrolo wera # 1 rigi I<br />
1. 33,6 (1 q); 32,5 (1 q); 35,5 (1 q); 33 (1 q).<br />
(TviTeuli arasworad amowerili ricxvisTvis unda daakldes 1<br />
qula, oRond, cxadia, Tuki sxvaoba 0-ze naklebi gamodis, maSin<br />
am amocanis saboloo qula unda iyos 0).<br />
2. I. 6,3 m ≤ |AK| < 42,8 m (2 q); II. b3 < b+ 9,4 ≤ a · 35 (2 q).<br />
(Tuki romelime ormxrivi utolobis mxolod erTi mxarea<br />
swori, maSin am utolobis CawerisTvis daeweros 1 qula).<br />
3. A(3) da C(4) wertilebis sworad moniSvna - 1 q; E da F<br />
wertilebisa - 1 q; |AC| = 2 sm (1 q); |EF| = 5 sm (1 q).<br />
4. I. 15 kg (1 q); II. 15/40 an 3/8 (1 q); III. 120 lari (2 q).<br />
5. I. musikis Semswavleli moswavleebidan sportze dadis<br />
moswavleTa saerTo raodenobis 1 · 1 = 1 nawili (1 q), xolo<br />
4 10 40<br />
danarCenebidan _ 3 · 1 = 1 nawili (1 q). amitom sul sportze<br />
4 3 4<br />
dadis moswavleTa saerTo raodenobis 1 + 1 = 11 nawili (1 q);<br />
40 4 40<br />
II. swavlobs musikas an dadis sportze moswavleTa saerTo<br />
raodenobis 1 + 11 − 1 = 1 naw.<br />
4 40 40 2<br />
(SeiZleboda asec gamogveTvala: 1 + 1 = 1 ) (3 q).<br />
4 4 2<br />
rigi II<br />
1. 23,7 (1 q), 27,4 (1 q), 24,3 (1 q), 24 (1 q).<br />
2. III. 2,7 km ≤ |DC| < 12,3 km (2 q); IV. d 2 < d+ 0,7 ≤ c–23 (2 q).<br />
3. M(4) da N(5) wertilebis sworad moniSvna _ 1 q; A da B<br />
wertilebisa _ 1 q; |MN| = 2 sm (1 q); |AB| = 7 sm (1 q).<br />
4. I. 25 m (1 q); II. 24/60 an 2/5 (1 q); III. 45 TeTri (2 q).
- 129 -<br />
5. I. katis myol mobinadreTagan ZaRli hyavs mobinadreTa saerTo<br />
raodenobis 1 · 1 = 1 nawils (1 q), xolo danarCenebidan -<br />
5 10 50<br />
4 · 1 = 1 nawils (1 q). amitom sul ZaRli hyavs mobinadreTa<br />
5 4 5<br />
saerTo raodenobis 1 + 1 = 11 nawils. (1 q); II. ZaRli an<br />
50 5 50<br />
kata hyavs mobinadreTa saerTo raodenobis 1 + 11 − 1 = 2<br />
5 50 50 5<br />
nawils. (SeiZleboda asec gamogveTvala: 1 + 1 = 2 naw.) (3 q).<br />
5 5 5<br />
sakontrolo wera # 2 rigi I<br />
1. I. 1 (1 q); III. 27 (1 q); V. = ( 0,<br />
25⋅<br />
8)<br />
5<br />
· 8 = 32·<br />
8 = 256 (2 q).<br />
2. I. 2 4 m 4 n 4 (1 q); II. k 6 : 3 6 (1 q); III. mag.: 3 n · 8 n (2 q).<br />
3. I. 5,7 (2 q); II. 5,6 (2 q).<br />
4. I. 0,8 · 40 = 32 ≤ 32. utoloba marTebulia (2 q);<br />
II. 0,6 · 38 + 1 + 6 = 22,8 + 7 = 29,8 ≤ 32. utoloba marTebulia (2 q).<br />
TfojTwob/ yoveli swori, magram dausabuTebeli pasuxisTvis - 1 q.<br />
5. I. 44,8 kg (3 q); II. 87,5 kg (3 q).<br />
rigi II<br />
1. II. 1 (1 q); IV. 81 (1 q); VI. = ( 0,<br />
4⋅<br />
5)<br />
6<br />
· 5 = 64·<br />
5 = 320 (2 q).<br />
2. IV. 3 5 k 5 d 5 (1 q); V. b 8 : 7 8 (1 q); VI. mag.: 4 m · 8 m (2 q).<br />
3. I. 8,8 (2 q); II. 8,4 (2 q).<br />
4. I. 0,7 · 50 = 35 ≤ 30. utoloba mcdaria (2 q);<br />
II. 5 + 1 + 0,5 · 48 = 6 + 24 = 30 ≤ 30. utoloba marTebulia (2 q).<br />
TfojTwob/ yoveli swori, magram dausabuTebeli pasuxisTvis - 1 q.<br />
5. I. 17,6 kg (3 q); II. 12,5 kg (3 q).<br />
sakontrolo wera # 3 rigi I<br />
1. I. a 35 (1 q); II. c 14 (1 q) ; III. 9<br />
d (2 q).<br />
2. I. 2<br />
3 an 9 (2 q); II. 1/62 an 1/36 (2 q).<br />
3. I. 16 (2 q); II. 735 kg (2 q).<br />
4. e) 257,5 (4 q). TfojTwob/ g) 259 an d) 256 - 2 q.<br />
5. Svlebia 70% (1 q). irmebis raodenoba naklebia 70%–<br />
30%=40%-iT (1 q). ese igi, irmebisa da Svlebis mTliani
- 130 -<br />
raodenobis 40% tolia 144-is (1 q). amitom mTliani raodenoba<br />
iqneba 144 : 0,4 = 360 (2 q), Svlebis raodenoba ki iqneba<br />
360 · 0,7 = 252 (1 q).<br />
! TfojTwob/ SeiZleba vinmem amoxsna ase daamTavros: raki<br />
irmebisa da Svlebis mTliani raodenobis 40% tolia 144-isa,<br />
amitom Svlebis raodenoba iqneba 144 · 70 = 252 (3 q).<br />
40<br />
rigi II<br />
1. IV. t 27 (1 q); V. b 30 (1 q); VI. 24<br />
k (2 q).<br />
2. I. 2<br />
7 an 49 (2 q); II. 1/52 an 1/25 (2 q).<br />
3. III. 33 (2 q); IV. 1069 g (2 q).<br />
4. d) 317,5 (4 q). TfojTwob/ g) 316 an e) 319 - 2 q.<br />
5. cxvrebia 65% (1 q). cxvrebis raodenoba metia<br />
65% – 35% = 30%-iT (1 q). ese igi, cxvrebisa da Zroxebis<br />
mTliani raodenobis 30% tolia 234-isa (1 q). amitom mTliani<br />
raodenoba iqneba 234 : 0,3 = 780 (2 q), cxvrebis raodenoba ki<br />
iqneba 780 · 0,65 = 507 (1 q).<br />
! TfojTwob/ SeiZleba vinmem amoxsna ase daamTavros: raki<br />
cxvrebisa da Zroxebis mTliani raodenobis 30% tolia 234-isa,<br />
amitom cxvrebis raodenoba iqneba 234 · 65 = 507 (3 q).<br />
30<br />
sakontrolo wera # 4 rigi I<br />
1. I. a 32 (1 q); II. 1 (1 q); III. b 13 (2 q).<br />
2. I. a4a3a2a1a0 = a4 · 104 + a3 · 103 + a2 · 102 + a1 · 101 + a0 · 100 (2 q);<br />
a79a78 ... a0 = a79 · 1079 + a78 · 1078 + ... + a0 · 100 (2 q).<br />
3. 10 24 ≤ d < 10 25 (2 q); 10 3 ≤ 4960 < 10 4 (1 q); 10 5 ≤ 100060 < 10 6 (1 q);<br />
4. I. ana ∉ D (0,5 q); II. zurabi ∈ F (0,5 q);<br />
III. e ∈ E (0,5 q); IV. qaTami ∈ B (0,5 q);<br />
V. B ⊃ D (0,5 q); VI. C ⊂ A (0,5 q);<br />
VII. {gia, lia} ⊂ F (0,5 q); VIII. A ⊃ {muxa, ia, cacxvi} (0,5 q).<br />
5. mesame kviraSi SeakeTes mTeli gzis 1/5 nawili (1 q), meore<br />
kviraSi ki - 1 − 1 − 1 = 7 nawili (2 q). ese igi, mTeli gzis 7/15<br />
3 5 15
- 131 -<br />
nawili tolia 21 km-is (1 q). amitom mTeli gzis sigrZe iqneba<br />
21: 7 = 45 km (2 q).<br />
15<br />
rigi II<br />
1. I. d 7 (1 q); II. 1 (1 q); III. m 49 (2 q).<br />
2. I. a3a2a1a0 = a3 · 103 + a2 · 102 + a1 · 101 + a0 · 100 (2 q);<br />
a89a88 ... a0 = a89 · 1089 + a88 · 1088 + ... + a0 · 100 (2 q).<br />
3. 10 8 ≤ r < 10 9 (2 q); 10 3 ≤ 8190 < 10 4 (1 q); 10 6 ≤ 1000700 < 10 7 (1 q).<br />
4. IX. Tina ∈ F (0,5 q); X. lia ∉ A (0,5 q);<br />
XI. g ∈ E (0,5 q); XII. indauri ∈ B (0,5 q);<br />
XIII. D ⊂ B (0,5 q); XIV. C ⊂ A (0,5 q);<br />
XV. {vaJa, Tea} ⊂ F (0,5 q); XVI. A ⊃ {Tela, enZela, cacxvi} (0,5 q).<br />
5. mesame dRes gaTibes mTeli saTibis 3/10 nawili (1 q), meore<br />
dRes ki - 1 − 1 − 3 = 8 nawili (2 q). ese igi, mTeli saTibis<br />
6 10 15<br />
8/15 nawilis farTobi tolia 48 aris (1 q). amitom mTeli<br />
saTibis farTobi iqneba 48 : 8 = 90 ari (2 q).<br />
15<br />
sakontrolo wera # 5 rigi I<br />
1. I. 7,5 · 10 2 (1 q); II. 4,7 · 10 4 (1 q);<br />
III. 5 · 10 7 (1 q); IV. 5,2 · 10 13 (1 q);<br />
2. I. 2,1 · 10 22 (1 q); III. 8 · 10 12 (1 q); V. 8,1 · 10 57 (2 q).<br />
3. I. 1 m/wT = 5/3 sm/wm (2 q); II. 1 dm/wm = 6 m/wT (2 q).<br />
4. e) 207,5 (4 q). TfojTwob/ g) 206,5 an d) 209 - 2 q.<br />
5. patara mwvervalamde manZilia 3,6 · 2,5 = 9 (km) (2 q). ukan<br />
dabrunebas moundeboda 9 : 4,5 = 2 (sT) (2 q). mTamsvlelis saS.<br />
siCqare iqneba (9+9) km : (2,5+0,5+2) sT = 3,6 km/sT (2 q).<br />
rigi II<br />
1. V. 4,9 · 10 2 (1 q); VI. 6,3 · 10 4 (1 q);<br />
VII. 4 · 10 7 (1 q); VIII. 7,4 · 10 17 (1 q);<br />
2. II. 3,6 · 10 22 (1 q); IV. 8 · 10 5 (1 q); VI. 6,4 · 10 52 (2 q).<br />
3. III. 1 m/sT = 1/6 dm/wT (2 q); IV. 1 sm/wm = 3/5 m/wT (2 q).<br />
4. e) 318,5 (4 q). TfojTwob/ g) 317 an d) 319,5 - 2 q.
- 132 -<br />
5. tyemde manZilia 7,5 · 1,2 = 9 (km) (2 q). ukan dabrunebas<br />
moundeboda 9 : 15 = 0,6 (sT) (2 q). bavSvis saSualo siCqare iqneba<br />
(9+9) km : (1,2+0,2+0,6) sT = 9 km/sT (2 q).<br />
sakontrolo wera # 6 rigi I<br />
1. I. 1,4 · 10 29 (1 q); III. 1,3 · 10 32 (1 q); V. 1,4 · 10 68 (2 q).<br />
2. I. bundovania (0,5 q); II. moicema erToblioba (0,5 q);<br />
III. bundovania (0,5 q); IV. moicema erToblioba (0,5 q);<br />
V. moicema erToblioba (0,5 q); VI. bundovania (0,5 q);<br />
VII. moicema erToblioba (0,5 q); VIII. moicema erToblioba (0,5 q).<br />
3. swori naxazi - 4 q.<br />
4. Semcirebis Semdeg siCqare iqneba 48 km/sT (2 q). isev 60<br />
km/sT siCqariT rom imoZraos, memanqanem siCqare unda gaadidos<br />
60 – 48 = 12 km/sT-iT, anu 12 · 100 =25 %-iT (2 q).<br />
48<br />
5. I. sasrulia (0,5 q); II. usasruloa (0,5 q); III. sasrulia (0,5 q);<br />
IV. usasruloa (0,5 q); V. sasrulia (1 q); VI. sasrulia (1 q);<br />
carielia VI, radgan Tuki ricxvis gamyofia 4, maSin misi<br />
gamyofi iqneba 2-ic (2 q).<br />
rigi II<br />
1. II. 3,7 · 10 56 (1 q); IV. 1,5 · 10 14 (1 q); VI. 1,2 · 10 114 (2 q).<br />
2. I. moicema erToblioba (0,5 q); II. bundovania (0,5 q);<br />
III. bundovania (0,5 q); IV. moicema erToblioba (0,5 q);<br />
V. moicema erToblioba (0,5 q); VI. bundovania (0,5 q);<br />
VII. moicema erToblioba (0,5 q); VIII. moicema erToblioba (0,5 q).<br />
3. swori naxazi - 4 q.<br />
4. momatebis Semdeg siCqare iqneba 75 km/sT (2 q). isev 60 km/sT<br />
siCqariT rom imoZraos, mZRolma siCqare unda Seamciros<br />
75 – 60 = 15 km/sT-iT, anu 15 · 100 =20 %-iT (2 q).<br />
75<br />
5. I. sasrulia (0,5 q); II. usasruloa (0,5 q); III. sasr. (0,5 q);<br />
IV. usasruloa (0,5 q); V. sasrulia (1 q); VI. sasrulia (1 q);<br />
carielia VI, radgan Tu ricxvis gamyofia 6-iani, maSin misi<br />
gamyofi iqneba 2-ianic da 3-ianic (2 q).
- 133 -<br />
sakontrolo wera # 7 rigi I<br />
1. 105,8 (4 q).<br />
2. I. meore gverdis sigrZea 3 · 108 m (2 q). marTkuTxedis<br />
farTobia 4,2 · 1024 · 3 · 108 ≈ 1,3 · 1033 m2 (2 q).<br />
3. B wveroze CD-s marTobuli wrfis gavleba - 1 q, CD-s<br />
paraleluri wrfis gavleba - 1 q. D wveroze BC-s marTobuli<br />
wrfis gavleba - 1 q. saTanado aRniSvnebis gamoyenebiT marTobul<br />
an paralelur wrfeTa yvela wyvilis amowera - 1 q.<br />
4. 20,1-isa da 20,5-is Sesabamis wertilebs Soris manZilia 6<br />
sm. amitom erTeulovani monakveTis sigrZea 6 sm : (20,5–20,1) =<br />
15 sm (2 q). E da F wertilebis koordinatebia: d) 20,13 da<br />
20,47 (2 q).<br />
TfojTwob/ Tu SearCia g) 20,15 da 20,42 - 1 q.<br />
5. 3 wuTis Semdeg maT Soris manZili iqneba 1800 m (3 q).<br />
2 km 100 m manZili iqneba 2100 : 600 = 3,5 wuTis Semdeg (3 q).<br />
rigi II<br />
1. 29 (4 q).<br />
2. II. meore gverdis sigrZea 1,5 · 109 m (2 q). marTkuTxedis<br />
farTobia 4,2 · 1024 · 1,5 · 109 = 6,3 · 1033 m2 (2 q).<br />
3. N wveroze MQ-s marTobuli wrfis gavleba - 1 q, AD-s<br />
paraleluri wrfis gavleba - 1 q. M wveroze NP-s marTobuli<br />
wrfis gavleba - 1 q. saTanado aRniSvnebis gamoyenebiT marTobul<br />
an paralelur wrfeTa yvela wyvilis amowera - 1 q.<br />
4. 14,5-isa da 14,7-is Sesabamis wertilebs Soris manZilia 6 sm.<br />
amitom erTeulovani monakveTis sigrZea 6 sm : (14,7–14,5) = 30<br />
sm (2 q). P da T wertilebis koordinatebia: e) 14,55 da 14,67 (2 q).<br />
Tuki SearCia g) 14,52 da 14,65 - 1 q.<br />
5. 3 wuTis Semdeg maT Soris manZili iqneba 2400 m (3 q).<br />
3 km 600 m manZili iqneba 3600 : 800 = 4,5 wuTis Semdeg (3 q).<br />
1. = 64–8+18,6 = 74,6 (4 q).<br />
sakontrolo wera # 8 rigi I
- 134 -<br />
2. wertilebi sworadaa moniSnuli - 3 q, TanmimdevrobiTaa<br />
SeerTebuli - 1 q.<br />
3. I. praRa ≈ 13° ag da 50° Cg (1 q); dublini ≈ 14° dg da 55° Cg (1 q);<br />
soxumi ≈ 41° ag da 43° Cg (1 q); gori ≈ 44,2° ag da 41,9° Cg (1 q).<br />
4. aguredis simaRlea 3,2 · 1014 : 4 · 109 = 8 · 104 m (2 q),<br />
moculoba - 3,2 · 1014 · 5 · 104 · 8 · 104 = 128 · 1022 ≈ 1,3 · 1024 m3 (2 q).<br />
5. raki erTeulovani monakveTis sigrZea 1 m, amitom 1 sm iqneba<br />
0,01 erTeulovani monakveTis sigrZis toli. Sesabamisad, A<br />
wertilis koordinati iqneba 51,99-is toli (2 q), xolo B<br />
wertilisa - 52,03 (2 q).<br />
rigi II<br />
1. = 27+81–1,8 = 106,2 (4 q).<br />
2. wertilebi sworadaa moniSnuli - 3 q, TanmimdevrobiTaa<br />
SeerTebuli - 1 q.<br />
3.II. oslo ≈ 12° ag da 60° Cg (1 q); reikiaviki ≈ 22° dg da 61° Cg (1 q);<br />
maikopi ≈ 40° ag da 44,2° Cg (1 q); baTumi ≈ 41,6° ag da 41,6° Cg (1 q).<br />
4. aguredis simaRlea 6,4 · 1016 : 8 · 107 = 8 · 108 m (2 q),<br />
moculoba - 4 · 1018 · 6,4 · 1016 · 8 · 108 = 204,8 · 1042 ≈ 2 · 1044 m3 (2 q).<br />
5. raki erTeulovani monakveTis sigrZea 0,1 mm, amitom 1 sm<br />
iqneba 100 erTeulovani monakveTis sigrZis toli. Sesabamisad, M<br />
wertilis koordinati iqneba 720-is toli (2 q), xolo N<br />
wertilisa - 1220 (2 q).<br />
sakontrolo wera # 9 rigi I<br />
1. I. –341 /6 ; –7,3 ; –7 ; –1; 0 ; 0,74 ; 21 (4 q).<br />
2. I. = 24 – 9,3 + 2,3 = 17 (4 q).<br />
3. sxivze A(–6), B(+4,5), D(–2,5), E(–2) wertilebi sworadaa<br />
moniSnuli - 2 q, M(–5,5) - 1 q, N(+1,5) - 1 q.<br />
4. aleqsandre gardacvlila ax.w. 38 wlis 18 aprils (1 q),<br />
mariami - ax.w. 131 wlis 18 aprils (1 q). aleqsandre<br />
dabadebula Zv.w I sauk. III meoTxedSi (1 q), mariami - ax.w. I sauk.
- 135 -<br />
II meoTxedSi (1 q).<br />
5. xaliCebi TebervalSi eRireboda 276 lari (2 q), martSi -<br />
345 lari (2 q). aprilSi xaliCebi gaiafebula 345–240 =<br />
105 lariT (1 q), anu 105 · 100 ≈ 30,4 %-iT (1 q).<br />
345<br />
rigi II<br />
1. II. 14 ; 43 /4 ; +1,8 ; –0,14 ; –3; –14; –21 (4 q).<br />
2. II. = 40+6,7–12,7 = 34 (4 q) .<br />
3. sxivze M(3), N(–1,5), P(+6,5), Q(–3). wertilebi sworadaa<br />
moniSnuli - 2 q, A(–6,5) - 1 q, B(+0,5) - 1 q.<br />
4. nino gardacvlila ax.w. 19 wlis 23 maiss (1 q), amirani -<br />
ax.w. 134 wlis 23 maiss (1 q). nino dabadebula Zv.w I sauk. II<br />
meoTxedSi (1 q), amirani - ax.w. I sauk. III meoTxedSi (1 q).<br />
5. macivari gazafxulze eRireboda 504 lari (2 q), zafxulSi<br />
- 630 lari (2 q). Semodgomaze macivari gaiafebula 630–420<br />
= 210 lariT (1 q), anu 210 · 100 ≈ 33,3 %-iT (1 q).<br />
630<br />
sakontrolo wera # 10 rigi I<br />
1. I. –14 (1 q); II. –28,8 (1 q); III. 20,5 (1 q); IV. 27 (1 q).<br />
2. I. _3a (1 q); II. –2k (1 q); III. –4,6a (1 q); IV. _3b (1 q).<br />
3. I. x = 20 (1 q); II. y = 3,5 (1 q);<br />
III. x = 4,5 (1 q); IV. y = –8 (1 q).<br />
4. swori naxazi - 4 q.<br />
5. vTqvaT, manqanis siCqarea x km/sT, maSin matareblis siCqare<br />
iqneba x+5 km/sT (1 q). miviRebT gantolebas: 4x + 7(x+5) = 640<br />
(2 q). am gantolebis amonaxsnia x = 55 (2 q). matareblis siCqare<br />
yofila 60 km/sT (1 q).<br />
rigi II<br />
1. V. _22 (1 q); VI. 13,6 (1 q); VII. –13,6 (1 q); VIII. 22 (1 q).<br />
2. V. _2b (1 q); VI. –2,6m (1 q); VII. –3,2k (1 q); VIII. _8c (1 q).<br />
3. V. y = 23 (1 q); VI. y = 5,6 (1 q);<br />
VII. x = 3,5 (1 q); VIII. x = –4 (1 q).<br />
4. swori naxazi - 4 q.<br />
5. vTqvaT, matareblis siCqarea x km/sT, maSin manqanis siCqare
- 136 -<br />
iqneba x+6 km/sT (1 q). miviRebT gantolebas: 4x + 2(x+6) = 312<br />
(2 q). am gantolebis amonaxsnia x = 50 (2 q). manqanis siCqare<br />
yofila 56 km/sT (1 q).<br />
sakontrolo wera # 11 rigi I<br />
1. I. _16 (1 q); II. –17,7 (1 q); III. 40,5 (1 q); IV. 32 (1 q).<br />
2. I. _2a (1 q); II. 2,9k (1 q); III. 4,7a (1 q); IV. _8b (1 q).<br />
3. I. x = 144 (1 q); II. y = 4 (1 q); III. x = 9 (1 q); IV. y = –8 (1 q).<br />
4. I. sworad daxaza ABCD marTkuTxedi (1 q). marTkuTxedis<br />
perimetria 32 erTeulovani monakveTis sigrZe (2 q). C wvero<br />
marTkuTxedis BD diagonalidan daSorebulia daaxloebiT 5<br />
erTeulovani monakveTis sigrZiT (1 q).<br />
5. vTqvaT, nakveTis sigrZea 5x m, sigane - 3x m (1 q). miviRebT<br />
gantolebas: 5x – 3x = 300 (1 q). aqedan x = 150 (1 q). nakveTis<br />
perimetri iqneba 2400 m (1 q). mevele am nakveTs Semouvlis<br />
2,4 km : 3 km/sT = 0,8 sT = 48 wT-Si (2 q).<br />
rigi II<br />
1. V. _28 (1 q); VI. 21,8 (1 q); VII. –29,6 (1 q); VIII. 36 (1 q).<br />
2. V. _7b (1 q); VI. –31,2m (1 q);<br />
VII. –1,8k (1 q); VIII. _23c (1 q).<br />
3. V. y = 115 (1 q); VI. y = 20 (1 q);<br />
VII. x = 4 (1 q); VIII. x = –7 (1 q).<br />
4. I. sworad daxaza ABCD marTkuTxedi (1 q). marTkuTxedis<br />
perimetria 28 erTeulovani monakveTis sigrZe (2 q). C wvero<br />
marTkuTxedis BD diagonalidan daSorebulia daaxloebiT 4,8<br />
erTeulovani monakveTis sigrZiT (1 q).<br />
5. vTqvaT, nakveTis sigrZea 7x m, sigane - 5x m (1 q). miviRebT<br />
gantolebas: 7x – 5x = 600 (1 q). aqedan x = 300 (1 q). nakveTis<br />
perimetri iqneba 7200 m (1 q). mevele am nakveTs Semouvlis<br />
7,2 km : 4 km/sT = 1,8 sT = 1 sT 48 wT-Si (2 q).<br />
sakontrolo wera # 12 rigi I<br />
1. I. y = –7 (1 q); II. x = 25 (1 q);<br />
III. z = –23,6 (1 q); IV. t = –32 (1 q).
- 137 -<br />
2. I. 2a – 5 (2 q); II. y + 1 (2 q).<br />
3. vTqvaT, Caifiqres x. miviRebT gantolebas:<br />
x – 14 – (–28) + 18 + (–27) = –21 (2 q). aqedan x = –26 (2 q).<br />
TfojTwob/ Cafiqrebuli ricxvi SeiZleba gantolebis gareSec<br />
gamovTvaloT: –21 – (–27) – 18 + (–28) + 14 = –26 (4 q).<br />
4. MNP samkuTxedis sworad daxazva - 2 q, masze wrexazis Semoxazva<br />
- 2 q.<br />
5. I. fleitaze ukravs 2,5-jer meti moswavle, vidre violinoze (2 q).<br />
II. mxolod erT sakravze dakvras swavlobs moswavleTa<br />
100% – (20%+50%) = 30%, anu 3/5 nawili (2 q).<br />
Tuki I SekiTxvaSi sityvas `ramdenjer~ SevcvlidiT sityviT<br />
`ramdeniT~, maSin pasuxs ver gavcemdiT, radgan saWiro iqneboda,<br />
damatebiT gvcodnoda, sul ramdeni moswavle swavlobs samusiko<br />
skolaSi (an ramdeni ukravs fleitaze, an violinoze) (2 q).<br />
1. V. t = –8 (1 q); VI. y = –2 (1 q);<br />
VII. x = 4,7 (1 q); VIII. z = –1 (1 q).<br />
2. III. –15 + m (2 q); IV. – 2 – y (2 q).<br />
3. vTqvaT, Caifiqres x. miviRebT gantolebas:<br />
rigi II<br />
x + 12 + (–17) – 28 – (–13) = –30 (2 q). aqedan x = –10 (2 q).<br />
TfojTwob/ Cafiqrebuli ricxvi SeiZleba gantolebis gareSec<br />
gamovTvaloT: –30 – 12 – (–17) + 28 + (–13) = –10 (4 q).<br />
4. ABD samkuTxedis sworad daxazva - 2 q, masze wrexazis Semoxazva<br />
- 2 q.<br />
5. I. fleitaze ukravs 4-jer meti moswavle, vidre violinoze (2 q).<br />
II. mxolod erT sakravze dakvras swavlobs moswavleTa<br />
100% – (15%+60%) = 25%, anu 1/4 nawili (2 q).<br />
Tuki I SekiTxvaSi sityvas `ramdenjer~ SevcvlidiT sityviT<br />
`ramdeniT~, maSin pasuxs ver gavcemdiT, radgan saWiro iqneboda<br />
damatebiT gvcodnoda sul ramdeni moswavle swavlobs samusiko<br />
skolaSi (an ramdeni ukravs fleitaze, an violinoze) (2 q).<br />
sakontrolo wera # 13 rigi I
- 138 -<br />
1. I. = (–7,2 – 0,6) · (–0,2) = –7,8 · (–0,2) = 15,6 (4 q).<br />
2. I. 12x (1 q); II. –4a (1 q); III. –10b (1 q); IV. –12x + 30 (1 q).<br />
3. I. –0,5x + 0,3 = 1,2 (2 q); x = –1,8 (2 q).<br />
4. ABC, MNK da PQR samkuTxedebis sworad daxazva - 2 q.<br />
marTkuTxaa MNK samkuTxedi (1 q).<br />
misi kaTetebia MN da NK, hipotenuza - MK (1 q).<br />
5. I. patara goWi 1 dReSi Wams 1/4 kg saWmels (1 q), didi ixvi -<br />
1/7 kg-s (1 q). amitom ufro met saWmels Wams patara goWi (1 q).<br />
II. 2 patara goWi da 5 didi ixvi 1 dReSi Wams 2 + 5 = 34 = 17<br />
4 7 28 14<br />
kg saWmels (2 q), amitom 17 kg saWmels SeWamen 14 dReSi (1 q).<br />
rigi II<br />
1. II. = (–14,4 + 0,7) · (–0,4) = (–13,7) · (–0,4) = 5,48 (4 q).<br />
2. V. –13y (1 q); VI. 5a (1 q); VII. –11x (1 q); VIII. –4 + 8c (1 q).<br />
3. II. –0,4y + 0,4 = 3,2 (2 q); y = –7 (2 q).<br />
4. EFK, BCD da ATP samkuTxedebis sworad daxazva - 2 q.<br />
marTkuTxaa ATP samkuTxedi (1 q).<br />
misi kaTetebia AT da TP, hipotenuza - AP (1 q).<br />
5. I. 1 TuSi mqsoveli 1 kviraSi qsovs 1/5 fardags (1 q), 1 xevsuri<br />
- 1/6 fardags (1 q). amitom ufro swrafad qsovs TuSi (1 q).<br />
II. 3 TuSi da 4 xevsuri mqsoveli erTad muSaobiT 1 kviraSi<br />
moqsovs 3 + 4 = 38 = 19 fardags (2 q), amitom 19 aseTive<br />
5 6 30 15<br />
fardags moqsoven 15 kviraSi (1 q).<br />
sakontrolo wera # 14 rigi I<br />
1. I. = (–64 – 81)·(–2)–28 = (–145)·(–2)–28 = –290 –28 = –318 (4 q).<br />
2. I. –19x +18 (2 q); II. 22a – 32 (2 q).<br />
3. I. 5y – 48 = 18 + 4y – 1 (2 q); y = 65 (2 q).<br />
4. swori naxazi - 4 q.<br />
5. a) 1 x + 1 x = 1 (3 q).<br />
4 5<br />
dasabuTeba: xelosani 1 sT-Si Seasrulebs mTeli samuSaos 1/4<br />
nawils, Segirdi _ 1/5 nawils, orive erTad x sT-Si Seasruleben
- 139 -<br />
1 x + 1 x nawils, rac 1 mTelis tolia (3 q).<br />
4 5<br />
rigi II<br />
1. II. = (9 – 125)·(–3)–12 = (–116)·(–3)–12 = 348 –12 = 336 (4 q).<br />
2. III. –5b – 35 (2 q); IV. 9x – 42 (2 q).<br />
3. I. 6x + 40 = –32 + 5x + 2 (2 q); x = –70 (2 q).<br />
4. swori naxazi - 4 q.<br />
5. d) 1 + 1 = 1 (3 q).<br />
6 7 y<br />
dasabuTeba: glexi 1 sT-Si Toxnis yanis 1/6 nawils, misi vaJi _<br />
1/7 nawils, orive erTad - 1 + 1 nawils, rac 1 -is toli unda<br />
6 7<br />
y<br />
iyos, radgan orive erTad mTel yanas y sT-Si Toxnian (3 q).<br />
sagamocdo wera rigi I<br />
1. = 25 · (–8) + 30 + 2 = 232 (4 q).<br />
2. I. 8 (1 q); II. 3 (1 q); III. 1,5 sm (1 q); IV. 24 kg (1 q).<br />
3. I. 1700 (2 q); II. 30% (2 q).<br />
4. I. 6 · 10 41 (2 q); III. 2 · 10 10 (2 q).<br />
5. I. sasrulia (1 q); II. usasruloa (1 q);<br />
III. usasruloa (1 q); IV. sasrulia (1 q).<br />
6. I. 20z –19 (2 q); II. 15m + 16 (2 q).<br />
7. I. –2x – 18 = 26 – 12x (2 q); x = 4,4 (2 q).<br />
8. A wveroze CD gverdis paraleluri wrfis gavleba - 2 q;<br />
D wveroze BC gverdis marTobuli wrfis gavleba - 2 q.<br />
9. I. ABC samkuTxedis daxazva - 2 q; wrexazis Semoxazva - 2 q.<br />
10. wlis bolos fiWvis simaRle iqneba 1,8 m (3 q).<br />
meore wlis bolos fiWvi gaizarda 2,34 m – 1,8 m = 0,54 m-iT,<br />
0,<br />
54<br />
anu · 100 = 30%-iT (3 q).<br />
1,<br />
8<br />
rigi II<br />
1. = 27 · 4 + 12 – 2 = 118 (4 q).<br />
2. V. 14 (1 q); VI. 7,5 (1 q); VII. 5 m (1 q); VIII. 40 t (1 q).<br />
3. III. 1300 (2 q); IV. 40% (2 q).<br />
4. II. 9 · 10 26 (2 q); IV. 9 · 10 15 (2 q).
- 140 -<br />
5. V. sasrulia (1 q); VI. usasruloa (1 q);<br />
VII. sasrulia (1 q). VIII. usasruloa (1 q);<br />
6. III. –78x +30 (2 q); IV. 12n – 12 (2 q).<br />
7. II. –3y + 15 = 5 – 5y (2 q); y = –5 (2 q).<br />
8. Q wveroze MN gverdis paraleluri wrfis gavleba - 2 q;<br />
N wveroze MQ gverdis marTobuli wrfis gavleba - 2 q.<br />
9. I. MNP samkuTxedis daxazva - 2 q; wrexazis Semoxazva - 2 q.<br />
10. wlis bolos naZvis simaRle iqneba 1,5 m (3 q).<br />
meore wlis bolos fiWvi gaizarda 1,8 m – 1,5 m = 0,3 m-iT, anu<br />
0,<br />
3<br />
· 100 = 20%-iT (3 q).<br />
1,<br />
5<br />
§ 20. damatebiTi literatura maswavlebelTaTvis<br />
1. qarTuli maTematikuri terminebis leqsikoni, Tb.-1998.<br />
2. d. uznaZe. zogadi fsiqologia. _ Sromebi, t. 3-4. Tb.-1964.<br />
3. d. uznaZe. bavSvis fsiqologia. _ Sromebi, t. 5. Tb.-1966.<br />
4. S. CxartiSvili. nebisyofa da misi aRzrda. Tb.-1972.<br />
5. S. nadiraSvili. ganzogadebis ganviTareba saskolo asakis bavSvebSi. Tb.-1963.<br />
6. m. yolbaia. bavSvis fsiqologia. Tb.-1998.<br />
7. e. imerliSvili. saskolo maTematikis ganviTarebis istoriisaTvis. Tb.-1984.<br />
8. l. mWedliSvili, n. ivaniZe. logika. Tb.-1995.<br />
9. z. vaxania. saymawvilo logika. Tb.-2006.<br />
10. z. vaxania. raodenobis intuiciuri wvdoma. _ saqarTvelos mecnier.<br />
akad. d. uznaZis sax. fsiqologiis inst. `macne~, # 2, 2005 w.<br />
11. z. vaxania, T. maxaraZe, S. xelaZe. zogadi unarebi _ testebi da<br />
garCeva. _ t. 1, t. 2, Tb. "testirebis centri"_2005.<br />
12. z. vaxania. cneba `usasrulobis~ fsiqologiuri maxasiaTeblebi. _<br />
`mecniereba da teqnika~, 1999, # 7-9.<br />
13. z. vaxania. gabindul cnebaTa raobis Sesaxeb _ krebuli<br />
`fsiqologia~, t. 18, 1998, 26-34.<br />
14. z. vaxania. saskolo saganmanaTleblo standartebis fsiqologiuri<br />
safuZvlebi. _ `fizika da maTematika skolaSi~, 1997, # 112, 10-17.<br />
15. z. vaxania. saskolo da sauniversiteto logikis swavlebis mokle<br />
koncefcia. _ gaz. `axali ganaTleba~, 31\03-07\04, 2005.<br />
16. z. vaxania. saskolo maTematikis swavlebis aqtiuri mzaobis meTodika.<br />
_ avtoreferati pedagogikis mecn. doqtoris sam. xar. mosap. Tb.-1999.<br />
17. vaxania. maTematikis swavleba dawyebiT klasebSi _ meTodikuri<br />
saxelmZRvanelo maswavlebelTaTvis. meore gamocema. Tb.-2003.<br />
18. История математики, т. 1. М.-1970.<br />
19. А. Пуанкаре. О Науке. М.-1983.<br />
20. Ж. Адамар. Исследование психологии процесса изобретения в области
- 141 -<br />
математики. М.-1970.<br />
21. М. Клайн. Математика - утрата определенности. М.-1984.<br />
22. Ж. Пиаже. Избранные психологические труды. М.-1969 (1994).<br />
23. Дж. Брунер. Процесс обучения. М.-1962.<br />
24. А. Дистервег. Избранные педагогические сочинения. М.-1956.<br />
25. Л.И.Божович. Психологический анализ формализма в усвоении школьных<br />
знаний. - в сб. "Хрестоматия по возрастной и педагогич. психологии", т. 1. М.-1980.<br />
26. В.А.Крутецкий. Общие вопросы структуры математических способностей.<br />
- в сб. "Хрестоматия по возрастной и педагогич. психологии", т. 2. М.-1981.<br />
27. В.А. Крутецкий. Основы педагогической психологии. М.-1972.<br />
28. Fellows M., Koblitz A.H., Koblitz N. Сultural Aspeсts of Mathematiсal<br />
Eduсation Reform. - “Notiсes of the Mathematiсal Soсiety”, 1994, v.41, # 1.<br />
29. Madison B.L. Eduсation for Numeraсy: A Сhallenging Responsibility. -<br />
“Notiсes of the Mathematiсal Soсiety”, 2000, # 2, 181.<br />
30. З. Ваханиа. Начала математики или система манипуляций? -<br />
"Математика в школе", Москва-1999, # 2.<br />
31. З. Ваханиа. Современное образование: Книга и Экран. -<br />
"The Journal of Eurasian Research", Москва-2002, # 3.<br />
is maswavlebeli, vinc Cveneuli saxelmZRvaneloTi<br />
aswavlis, am saxelmZRvanelosac da moswavlis<br />
saxelmZRvanelosac ufasod miiRebs gamomcemlobaSi!