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guram gogiSvili, Teimuraz vefxvaZe ia mebonia, lamara qurCiSvili

maTematika VI

klasi

maswavleblis wigni

grifminiWebulia saqarTvelos ganaTlebisa da mecnierebis saministros mier 2018 wels

gamomcemloba inteleqti Tbilisi


guram gogiSvili, Teimuraz vefxvaZe, ia mebonia, lamara qurCiSvili

maTematika VI klasi maswavleblis wigni gamomcemloba inteleqti Tbilisi 2018

redaqtori

Teimuraz vefxvaZe

damkabadonebeli

viola tuRuSi

teqstis amkrefi

eTer kvanWiani

ISBN 978-9941-476-61-7 Š guram gogiSvili, Teimuraz vefxvaZe, ia mebonia, lamara qurCiSvili, 2018. Š gamomcemloba `inteleqti~, 2018.

gamomcemloba inteleqti Tbilisi, ilia WavWavaZDis gamz. 5. tel.: 2-25 05 22 www.intelekti.ge

inteleqti@caucasus.net

INTELEKTI PUBLISHERS 5 Ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 32) 2-25 05 22


sarCevi zogadi informacia saxelmZRvanelos Sesaxeb................................................................ 5 Sesavali......................................................................................................................................... 7 I Tavi. gameoreba. wiladebi da aTwiladebi 1. gavixsenoT – rogor CavwerT da valagebT naturalur ricxvebs.................... 24 2. ra viciT wiladebis Sesaxeb?........................................................................................... 30 3. viwyebT aTwiladebis Seswavlas...................................................................................... 39 Semajamebeli wera #1............................................................................................................ 43 4. aTwiladebis Sedareba da dalageba.............................................................................. 46 5. aTwiladebis damrgvaleba................................................................................................. 53 6. aTwiladebis zogierTi gamoyeneba................................................................................ 56 Semajamebeli wera #2............................................................................................................ 59

II Tavi. moqmedebebi aTwiladebze 1. aTwiladebis Sekreba........................................................................................................... 61 2. aTwiladebis gamokleba...................................................................................................... 65 3. aTwiladebis gamravleba.................................................................................................... 70 4. aTwiladis gayofa naturalur ricxvze...................................................................... 74 5. aTwiladze gayofa................................................................................................................ 80 6. gayofis Sedegis damrgvaleba.......................................................................................... 85 7. ricxviTi da asoiTi gamosaxulebis Sedgena.............................................................. 87 Semajamebeli wera #3............................................................................................................ 96

III Tavi. moqmedebebi wiladebze 1. wiladis Cawera aTwiladis saxiT................................................................................... 100 2. wiladebis saerTo mniSvnelamde miyvana...................................................................... 105 3. wiladebis da Sereuli ricxvebis Sedareba............................................................... 111 4. wiladebis Sekreba da gamokleba.................................................................................... 115 5. wiladebis gamravleba......................................................................................................... 122 6. wiladebis gayofa................................................................................................................. 127 Semajamebeli wera #4............................................................................................................ 132 7. amocanebis amoxsna............................................................................................................... 134 8. ricxviTi utolobis Tvisebebi....................................................................................... 137 Semajamebeli wera #5............................................................................................................ 144

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IV Tavi. vswavlobT geometriuli figurebis Tvisebebs............................ 1. geometriuli figurebi (gameoreba).............................................................................. 146 2. wrewiri. wre........................................................................................................................... 152 3. ori wrewiris urTierTganlagebis SemTxvevebi........................................................ 156 Semajamebeli wera #6............................................................................................................ 161 4. RerZuli simetria................................................................................................................ 165 5. paraleluri gadatana......................................................................................................... 168 6. figuris farTobi. marTkuTxedis farTobis gamoTvla......................................... 171 Semajamebeli wera #7............................................................................................................ 177 7. sivrculi figurebi............................................................................................................. 179 8. moculoba................................................................................................................................ 185 9. sidideebs Soris damokidebulebebi.............................................................................. 189 Semajamebeli wera #8............................................................................................................ 195

V Tavi. monacemTa analizi da statistika 1. monacemTa Segroveba........................................................................................................... 199 2. monacemTa moxerxebuli warmodgena............................................................................. 202 3. svetovani da wriuli diagramebi................................................................................... 205 4. monacemTa Semajamebeli ricxviTi maxasiaTeblebi................................................. 207 5. amocanebis amoxsnis sxvadasxva xerxi........................................................................... 211 Semajamebeli wera #9............................................................................................................ 217 amocanebi gameorebisaTvis (Semajamebeli)................................................................. 219 Semajamebeli wera #10.......................................................................................................... 221

Semajamebeli amocanebi...................................................................................................... 224 beWduri da eleqtronuli resursebi.......................................................................... 231

literatura........................................................................................................................... 233

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zogadi informacia saxelmZRvanelos Sesaxeb maTematikis VI klasis saxelmZRvanelo amave avtorTa V klasis saxelmZRvanelos Sedgenis principebs eyrdnoba da misi uSualo gagrZelebaa – saswavlo kursi agebulia 2017-2021 wlebis erovnuli saswavlo gegmis moTxovnebTan sruli SesabamisobiT, moswavleTa asakobrivi Taviseburebebis gaTvaliswinebiT da xels uwyobs moswavleTa aRWurvas qmediTi codniT. saxelmZRvaneloSi warmodgenili saswavlo masala xels uwyobs moswavleTa motivaciis gaZlierebas, masSi kvlav mravladaa sxvadasxva sirTulis testebi da amocanebi sxvadasxva akademiuri donis moswavleebTan diferencirebuli midgomis gansaxorcieleblad, masala siRrmiseuli swavlebisTvis. teqsti xels uwyobs moswavleTa codnis etapobriv konstruirebas winare codnaze dafuZnebiT. maswavlebels eqmneba SesaZlebloba, gaaZlieros moswavleTa CarTuloba saswavlo procesSi. moyvanilia saintereso istoriuli faqtebi, warmodgenilia masala sxvadasxva TamaSis, Sejibrebis, viqtorinis Casatareblad. VI klasis saxelmZRvanelo xuTi Tavisgan Sedgeba, yovel Tavs axlavs damatebiTi amocanebi ganvlili masalis gasameoreblad. yoveli Tavi paragrafebisgan Sedgeba. calke gamoyofilia amocanebi klasSi Sesasruleblad da amocanebi, romelTa Sesruleba moswavleebs Sin evalebaT. aseTi dayofa pirobiTia da mxolod rekomendaciis xasiaTs atarebs. cxadia, maswavlebels SeuZlia koreqtivebi Seitanos am dayofaSi. ZiriTadi mecnierul-meTodikuri principebi, romlebic gansazRvravs saxelmZRvanelos Sinaarss da wyobas, aris saxelmZRvanelos Sesabamisoba erovnuli saswavlo gegmis maTematikis programasTan, sistematuroba masalis dalagebaSi, Sinaarsis mecnieruloba, Teoriisa da praqtikis kavSiris principis ganxorcieleba, samotivacio amocanebis mravalferovneba, gadmocemis misawvdomoba, enis garkveuloba, sizuste da simartive, savarjiSoebisa da maTi Sesrulebis formebis mravalferovneba.

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Sesavali maTematikis swavlebis miznebis, amocanebis, standartis Sedegebis miRwevisa da Sinaarsis urTierTkavSiris matrica sagnis swavlebis miznebi SeiZleba mokled ase CamovayaliboT: • moswavleTaTvis azrovnebis, msjelobis, SexedulebaTa dasabuTebis unarisa da sxvadasxva kompetenciis ganviTareba; • codnis integraciaSi xelSewyoba; • amocanis Sinaarsis aRqma, mocemulobisa da saZiebeli sidideebis gaazreba-gamijvna; • yoveldRiur cxovrebaSi maTematikis gamoyenebis gaazreba, SromiTi da sasicocxlo unarebis ganviTarebis xelSewyoba; • yoveldRiur cxovrebaSi maTematikuri obieqtebis modelebisa da procesebis SemCneva, aRqma da kreatiuli unarebis ganviTareba; • informaciis warmodgenis xerxebisa da meTodebis floba da maTze msjeloba; • socialur da kulturul aRzrdaSi xelSewyoba.

standartis Sedegebis miRwevisa da Sinaarsis urTierTkavSiris matrica Temebis CamonaTvali

miznebi: Sinaarsis gardaqmna miznad; erovnul saswavlo gegmaSi miTiTebuli Sesabamisi punqtebi

savaraudo saswav­lo dro

naturaluri ricxvebis Cawera da Sedareba

naturaluri ricxvebis Cawerisa da Sedarebis gamoyenebis unaris Semdgomi ganviTareba. maT. VI.1.

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aTwiladebi, Sedareba, dalageba, damrgvaleba, aTwiladebis zogierTi gamoyeneba

aTwiladis gamosaxva sxvadasxva saxiT; aTwiladebis wakiTxvis, Caweris, Sedarebisa da damrgvalebis unarebis ganviTareba. maT.VI.1, VI.2, VI.3.

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aTwiladebze ariTmetikuli moqmedebebis Sesrulebis una­ris ganviTareba maT.VI.2

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ricxvebze moqmedebebis Tvisebebis gamoyeneba, ricxviTi gamosaxulebebis mniSvnelobebis povna. gamoTvlebze amocanis amoxsnisas ariTmetikul moqmedebaTa Sedegebis Sefasebis unaris ganviTareba maT.VI.2, VI.4.

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moqmedebebi aTwiladeb­ze ricxviTi gamosaxulebis mniSvnelobis povna

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asoiTi gamosaxuleba. gamartiveba, gantoleba da utoloba, maTi gamoyeneba. wiladebis sxvadasxva aspeqtebi, erTnairmniSvneliani wiladebis Sedareba, Sekreba. wiladis ZiriTadi Tviseba.

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wiladis gamosaxulebaSi mTeli da wiladi nawilebis mi­ TiTeba. wiladis ukveci formiT gamosaxva, wiladebze ariTmetikuli moqmedebebis Seswavla. maT.VI.1, VI.2.

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wiladebis Sekveca, wiladebis Cawera aTwiladis saxiT, wiladebis Sekreba da gamokleba.

wiladebis ZiriTadi Tvisebis gamoyeneba wiladebis Sekrebisa da gamoklebis operaciebis Catarebisas. racionaluri ricxvebis ekvivalenturi formebiT Caweris Cvevebis daufleba. maT. VI.2.

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wiladebis gamravleba da gayofa. ricxvis nawilis povna.

mocemuli ricxvis nawilis povnisa da misi Sebrunebuli amocanis amoxsnis Cvevebis daufleba. wiladebze moqmedebaTa Sedegebis Sefaseba. maT.VI.2.

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wiladebisa da aTwiladebis zogierTi gamoyeneba. ricxviTi da asoiTi gamosaxulebebis Sedgena da gamartiveba.

zomis sxvadasxva erTeulis erTmaneTTan dakavSirebis amocanebis amoxsnisas wiladebisa da aTwiladebis gamoyenebis Cvevebis daufleba. maT.VI.3. komutaciurobis, asociaciurobisa da distribuciulobis Tvisebebis gamoyeneba maT. VI.6.

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geometriuli figurebi: wrewiri, wre. wrewirebis urTierTganlageba.

sibrtyeze wrewirebis urTierTganlagebis SemTxvevebis ganxilva maT.VI.9.

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geometriuli gardaqmnebi; RerZuli simetria, paraleluri gadatana.

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asoiTi gamosaxulebis gamartivebisa da misi mniSvnelobis povnis Cvevebis daufleba. aTwiladebis Semcveli gantolebebisa da utolobebis gamoyenebiT amocanebis amoxsna. maT.VI.6

mocemuli brtyeli figuris paraleluri gadatanis gan­ xorcielebis unari; mocemuli brtyeli figuris simetriuli figuris ageba; figuris simetriis RerZebis povna. maT.VI.8. geometriuli gardaqmnebis gamoyeneba maT.VI.9

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figuris farTobi. marTkuTxedis farTobis gamoTvla

brtyeli figuris farTobis Sefaseba. farTobis adiciurobis gaazreba da gamoyeneba. maT.VI.10

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sivrculi figurebi

sivrculi figurebis erTmaneTisgan sxvadasxva atributebis mixedviT garCevis unari, sivrcul figuraTa Slilebis damzadebis unari. eileris maxasiaTeblis gamoTvla da gamoyeneba. maT.VI.7, maT.VI.9.

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sidideebs Soris damokidebulebebi

sidideebs Soris damokidebulebebis gamosaxvis, ganvrcobisa da aRweris unaris ganviTareba. maT.VI.5.

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monacemTa Segroveba, moxerxebuli warmodgenis magaliTebis ganxilva. wriuli da svetovani diagramebi

dasmuli amocanis Sesabamisad saWiro monacemebis Segrovebisa da warmodgenis Cvevebis ganviTareba. maT.VI.11, maT.VI.12.

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monacemTa ricxviTi maxasiaTeblebi

monacemTa saSualo, udidesi da umciresi mniSvnelobebi, maTi gamoyenebiT monacemTa erTobliobebis daxasiaTeba. monacemTa erTobliobaSi kanonzomierebebis aRmoCena da maTze msjelobis unaris ganviTareba. maT.VI.13.

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maswavleblis wignis mokle mimoxilva maswavleblis sarekomendacio wignis daniSnulebaa, maswavlebels miawodos saTanado meTodikuri rekomendaciebi erovnuli saswavlo gegmis VI klasis maTematikis programis mixedviT Sedgenili saxelmZRvanelosTvis, daexmaros mas saxelmZRvaneloSi warmodgenili masalis gaanalizebaSi, saswavlo procesis dagegmvasa da warmarTvaSi. maswavleblis sarekomendacio wignSi Tavebisa da paragrafebis dasaxelebebi emTxveva moswavlis saxelmZRvaneloSi SemoRebul numeraciasa da dasaxelebebs. warmodgenilia yvela gakveTilis scenari, aqtivobebisa da savarjiSoTa ganxilvis meSveobiT swavlebis rekomendaciebi, amoxsnilia TiTqmis yvela amocana, romlebic klasSi da Sin davalebis saxiT sruldeba, amasTanave, miTiTebulia amocanebis amoxsnebis sxvadasxva xerxebi, mocemulia kiTxvebis sistema, romlebic SeiZleba maswavlebelma gamoiyenos gakveTilis interaqtiuli saxiT warmarTvis dros. gansakuTrebuli yuradReba eTmoba masalis interaqtiuli formiT gadacemis xelSewyobas. Cveni rekomandaciebi gamiznulia moswavleebSi motivaciis Seqmnisa da maTematikis Seswavlisadmi interesis gazrdisken, rac uzrunvelyofilia praqtikuli amocanis saSualebiT – gamoyenebiTi aspeqtis xazgasmiT, an maTematikis Sinagani kanonzomierebebis gaazrebiT. aRwerilia saswavlo procesis warmarTvis sxvadasxva forma moswavleze orientirebuli saswavlo garemos Sesaqmnelad: muSaoba mTeli klasis monawileobiT, jgufebad (SesaZlebelia wyvilebad) dayofil klasTan, individualurad da a. S. am sarekomendacio wigniT vexmarebiT maswavleblebs warmatebiT daZlion saswavlo procesTan dakavSirebuli siZneleebi. gansakuTrebiT, im davalebebis ganxilvisas, romlebic azrovnebis zeda sami donis (analizi, sinTezi, Sefaseba) amoqmedebasTan aris dakavSirebuli. nimuSis mixedviT davalebebis Sesrulebis (gamoyenebis done) garda, gvaqvs davalebebi, romlebic moswavlis SemoqmedebiTi potencialis zrdasa da amoqmedebas iwvevs. azrovnebis zeda doneebis asamoqmedeblad miwodebulia specialuri meTodebi. magaliTad, dasazustebeli SekiTxvebis CamonaTvali, romelTa saSualebiT

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xdeba `mogzauroba~ qveda doneebidan zeda doneebisken da Semdgom ukusvliT – zeda doneebidan qveda doneebisken. maswavleblis wignSi yoveli Temis swavlebasTan dakavSirebuli sakiTxebis gadmocemisas ganvixilavT am Temis Sesabamis mecnierul safuZvlebs – vzrunavT maswavleblis kompetenciis ganviTarebaze. meeqvse klasi dawyebiTi safexuris damamTavrebeli welia, mimdinareobs gameoreba, codnis ganmtkiceba da gaRrmaveba esg-Si warmodgenili oTxive mimarTulebiT. maswavleblis wigni Seicavs miTiTebebs, Tu ra damatebiTi resursebis gamoyeneba SeuZlia maswavlebels am etapze saswavlo procesSi. mniSvnelovania, rom maswavleblis wigns axlavs sayuradRebo literaturis nusxa. TiToeul dasaxelebul wignze aris miTiTeba. maswavleblis wigns axlavs Semajamebeli sakontrolo weris nimuSebi, TandarTuli Sefasebis sqemiT. gasaTvaliswinebelia samizne asakis Taviseburebebi. ukve SesaZlebeli xdeba iseTi amocanebis ganxilva, romlebSic deduqciuri msjelobebia gamoyenebuli. amocanaTa mravalferovneba (`testuri~, variantebis daTvlaze, amowurvis da gamoricxvis xerxebis da naSTTa ariTmetikis gamoyenebaze, amocanebi mosazrebulobis gamovlenaze – `moisazre~) exmareba maswavlebels moswavleebTan individualuri swavlebis ganxorcielebaSi. mudmivad mimdinareobs mimarTeba ganvlili masalisken. maswavleblis wignis saSualebiT mudmivad vawvdiT pedagogebs rekomendaciebs sxvadasxva akademiuri donis moswavleebTan muSaobis warmarTvisTvis. saxelmZRvaneloSi masalis gadmocemis meTodika (magaliTad, xSiri mimarTeba ukve Seswavlilisadmi) exmareba maswavlebels am mimarTulebiTac. moswavleze orientirebuli swavlebis xelSemwyobi garemos uzrunvelyofa unda warimarTos saswavlo procesis moswavleTa saWiroebebze morgebis saSualebiT, swavlebis am formis Sesaxeb ukve visaubreT V klasis maswavleblis wignSi, magram mainc CavTvaleT saWirod, kidev erTxel SevexoT im ZiriTad rekomendaciebs, romlebic diferencirebuli swavlebis ganxorcielebisas aris gasaTvaliswinebeli: miznobrivi midgomebi gansxvavebuli mzaobis moswavleebTan, SedarebiT dabali mzaobis moswavleebTan swavlebis teqnikis gamoyeneba, romelic meTodikur literaturaSi skafoldingis saxeliTaa cnobili. skafondingi inglisuri terminia da xaraCos agebas niSnavs (am SemTxvevaSi, saubaria saswavlo xaraCoze). maswavlebeli qmnis swavlebis strategiis an davalebis models, xolo Semdgom pasuxismgeblobas problemis gadaWraze TandaTan gadaitans moswavleze. maswavlebeli xels uwyobs moswavleebs im dros, rodesac cnebebi da gadasaWreli amocanebi pirvelad gamoCndeba moswavleTa winaSe. amgvari mxardaWera SeiZleba gulisxmobdes wabiZgebas, CarCos miwodebas, an sxva resursebiT uzrunvelyofas, romelic Sinaarsis gagebas uwyobs xels. skafondingi moswavleebSi ganamtkicebs im unar-Cvevebs, romlebsac moswavleebi axali masalis gacnobamde unda flobdnen; rodesac moswavle Secdomas uSvebs, an davalebis Sesruleba uWirs, mas maswavlebelma saswavlo xaraCoebi unda SeaSvelos. aqtivobebi, romliTac vsargeblobT SedarebiT dabali mzaobis moswavleebTan, unda gamoirCeodes Semdegi maxasiaTeblebiT: 1) ganmeorebiT swavleba gansxvavebuli meTodiT; 2) TanaklaselTan muSaobis saSualebis micema; 3) davalebebis Sesasruleblad saklaso Canawerebis, teqstis `gadaxedvis~ saSualebis micema;

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4) 5) 6) 7) 8) 9) 10)

davalebis dayofa nawilebad, nabij-nabij instruqciebis micema; miniSnebebis micema; ZiriTadi sakiTxis gamoyofa; nawilobrivi amoxsnis warmodgenis saSualebis micema; ucnobi sityvebis axsnis saSualebis micema; TvalsaCino masalis gamoyenebis saSualeba; damatebiTi dros gamoyofa.

aqtivobebi maRali mzaobis moswavleebisTvis SeiZleba ase davaxasiaToT: 1) SedarebiT rTuli, abstraqciis maRali donis mqone davalebebi, romlebsac moswavleebi damoukideblad asruleben; 2) iseTi davalebebis ganxilva, romlebic saWiroebs kargad gaazrebas, dafiqrebas, zogjer axali informaciis moZiebas; 3) sxvadasxva gziT amocanebis amoxsna; 4) saswavlo sakiTxis dakavSireba realur situaciasTan; 5) sxvadasxva saswavlo sakiTxis erTmaneTTan dakavSireba; 6) e. w. `mravalnabijiani~ amocanebis ganxilva; 7) maRali mzaobis moswavle mzad unda iyos daexmaros Tanaklasels misces iseTi miniSneba, romelic daexmareba mas amocanis amoxsnaSi; 8) mivceT arasworad amoxsnili amocana da vTxovoT, ipovos Secdoma; 9) TviTon Seqmnas amocanis amosaxsnelad saWiro sqema, diagrama; 10) mocemuli algebruli gamosaxulebebis mixedviT moifiqros amocana, romlis amoxsnisas es gamosaxuleba gamoiyeneba. diferencirebuli swavlebis ZiriTadi principia moswavleTa moqnili, moklevadiani dajgufebebis gamoyofa, romelic mzaobas da interess emyareba. Cven mier mowodebuli mravalferovani masala da gamoyenebis sxvadasxva xerxis analizi saSualebas aZlevs maswavlebels (misi profesiuli, eTikuri, aRmzrdelobiTi unarebis aqtiuri amoqmedebiT) warmatebiT gaarTvas Tavi moswavleze orientirebuli swavlebis xelSemwyobi garemos uzrunvelyofis amocanas. maswavlebels unda axsovdes, rom Tavisi damokidebulebiT moswavleebis, gansaxilveli sakiTxis, Tu, sazogadod, mTeli saswavlo procesis mimarT, moswavleebSi amkvidrebs movlenaTa da faqtebis aRqmis garkveul stereotipebs. amitom, aucilebelia, rom es damokidebuleba iyos yovelmxriv gaazrebuli – Semoqmedi, Tanamedrove, ganaTlebuli pirovnebis aRzrdaze orientirebuli. axalgazrdebs dasabuTebis moTxovnis grZnoba uCndebaT 11-12 wlis asakidan. moswavleTa saazrovno unarebis ganviTareba erovnuli saswavlo gegmis erT-erTi umTavresi moTxovnaa – moswavle unda gacdes masalis ubralod damaxsovreba-gazepirebas. kerZod, codnis dagrovebasTan erTad, man unda SeZlos am codnis gageba, gamoyeneba, analizi, sinTezi da bolos – Sefaseba. am gziT is unda axdendes codnis konstruirebas. maswavlebeli am wignis saSualebiT mudmivad miiRebs rekomendaciebs moswavleTa datvirTvisa da davalebebis raodenobebis ise gansazRvraSi, rom moswavle Sin davalebebis Sesrulebisas ar aRmoCndes dauZleveli siZneleebis winaSe. Aamavdroulad, arc davalebebi iyos gadatvirTuli erTferovani amocanebiT. rTul amocanebze, kerZod e. w. `moisazres~ tipis amocanebze muSaoba klasSi maswavleblis daxmarebiT unda warimar-

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Tos, davalebebis ZiriTadi nawili klasSi Sesrulebuli amocanebis analogiuri SeiZleba iyos, magram unda iyos calkeuli SemTxvevebi, roca moswavlem damoukideblad unda SeZlos amoxsnis povna. gansakuTrebiT, es SeiZleba maRali mzaobis moswavleebs exebodes, romlebsac zogjer gansxvavebuli, damatebiTi amocanebic unda SevTavazoT Sesasruleblad. `moisazre~, `vip~-is rubrikiT warmodgenili amocanebi, rogorc wesi, klasSi jgufuri muSaobiT SeiZleba Sesruldes. swavlebis erT-erTi saintereso da mniSvnelovani formaa jgufuri muSaoba. es muSaoba SeiZleba saswavlo procesis erT-erTi Semadgeneli nawili iyos – daukavSirdes axali masalis gaazrebas, praqtikul saqmianobas, an Semajamebeli daskvnis gamotanas; mas SeiZleba mTeli gakveTilic daeTmos. jgufuri muSaobis warmarTvaSi saxelmZRvaneloSi warmodgenili masala, an Sesabamisi ramdenimeetapiani amocanebis sistema dagexmarebaT. jgufur muSaobas SeiZleba Sejibrebis saxec mivceT. saxelmZRvaneloSi warmodgenili TamaSebic SeiZleba jgufuri muSaobis formiT warvmarToT. Teoriuli masalis gadmocemis Cveneuli meTodika saSualebas gaZlevT airCioT jgufuri muSaobis moxerxebuli forma; magaliTad, paeqroba jgufebs Soris, romlebsac kapitnebi xelmZRvaneloben. jgufebs SeiZleba erTnairi davalebebi mivceT. maSin Tavad jgufebi aqtiurad erTvebian miRebuli Sedegebis saerTo ganxilvaSi – ukeT gauweven oponirebas sxvaTa mosazrebebs. SeiZleba jgufebSi samuSao ganawildes da moswavleTa erToblivi muSaoba raime erTi amocanis gadawyvetas gulisxmobdes; jgufis wevrebma unda gaiazron, rom maTi warmateba erTmaneTzea damokidebuli. es zrdis wevrebis pasuxismgeblobas. TanamSromlobis xarisxis asamaRleblad SeiZleba scadoT saerTo Sefaseba – jgufebi asruleben saerTo davalebas da maTi wevrebi iReben erTnair Sefasebas. Tumca, es wesi zogierTi moswavlisaTvis ar unda iqces daumsaxurebeli maRali Sefasebebis miRebis saSualebad. sasurvelia, rom jgufuri muSaobisTvis gankuTvnili amocana sxvadasxva xerxiT ixsenebodes. igi saintereso msjelobis saSualebas iZleva. jgufuri muSaobis daskvniT etapze – Sedegebis sajaro ganxilvisas Zalze mniSvnelovania pedagogis roli moswavleTa mier polemikis koreqtuli formebiT warmarTvaSi, sxvaTa mosazrebis mosmenis unaris damkvidrebaSi, warmatebisa Tu warumateblobis gamoxatvis misaRebi formebis CamoyalibebaSi. maswavleblebs vTavazobT rekomendaciebs saSinao davalebis Semowmebis CatarebaSi. igi ar unda Semoifarglebodes rveulebis gadaxedviT da merxebs Soris `CamovliT~. yoveli davalebis ramdenime gamorCeuli amocanisTvis maswavleblebs vTavazobT komentarebs da rekomendaciebs maTi sxvadasxva xerxiT amoxsnisa da ganxilvisTvis. es masala maT daexmareba moswavleTa kritikuli azrovnebis ganviTarebis kuTxiT. moswavleebi ar gadavtvirToT Sin Sesasrulebeli davalebebiT, rTuli amocanebiT. maTi umravlesoba klasSi, maswavleblis daxmarebiT unda amoixsnas. daxmarebaSi Cven vgulisxmobT, rom saorientacio kiTxvebis saSualebiT moswavlem TviTon gaakeTos aRmoCena. zogierTi cnobili specialisti (ix., mag., [20], [21], [24]) maswavlebels mebaRes adarebs, romelic uvlis baRs da xels uwyobs baRis mcenareTa zrdasa da srulyofas. moswavles ar unda dasWirdes damxmare `maswavlebeli~ davalebebis Sesasruleblad. amocanaTa mravalferovneba, romliTac gamorCeulia Cveni saxelmZRvanelo (amocanebi mosazrebulobaze, swrafad amoxsnaze, amocanebi jgufuri muSaobisTvis, amocanebi gameorebisTvis) xels uwyobs maswavlebels saintereso gaxados saswavlo procesi.

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maswavlebelma SeiZleba gamoiyenos Cven mier SemoTavazebuli didaqtikuri TamaSebi, romelTa gamoyeneba zrdis moswavleTa interess maTematikis Seswavlisadmi. maTematika sxva saskolo disciplinebs OSoris yvelaze metad xasiaTdeba iseTi specifikuri kompetenciebis saWiroebiT, rogoricaa, magaliTad, analizi, mixvedra, amocnoba, arCeva, msjeloba-dasabuTeba. maTematikis swavlebis procesSi yuradReba unda gamaxvildes iseTi pirovnuli Tvisebebis ganviTarebaze, rogoricaa: Sromismoyvareoba, akuratuloba, nebisyofis unaris gamomuSaveba, yuradRebianoba, iniciativianoba, dawyebuli saqmis bolomde miyvanis Cveva. aucilebelia, moswavles swavlis survilis unari Camouyalibdes, daeuflos damoukideblad muSaobis Cvevebs. VI klasi dawyebiTi safexuris bolo welia, roca maTematikis swavlebam unda Seqmnas myari safuZveli rogorc arsebuli codnisa da unarebis mimarT, aseve maTematikis Semdgomi SeswavlisaTvis maTi ganviTarebis mimarT. swavlebisa da aRzrdis organuli Sexameba, codnis aTviseba da SemecnebiTi SesaZleblobebis ganviTareba, Teoriuli codnis amaRleba da praqtikaSi gamoyenebis unaris ganviTareba – yovelive amisTvis aucilebeli Cvevebis gamomuSaveba aris swavlebis dawyebiTi safexuris mizani. dakvirveba da cda, Sedareba, mizez-Sedegobrivi kavSirebi TiTqos ufro bunebismetyvelebis SeswavlisaTvis aris damaxasiaTebeli, Tumca isini maTematikaSic gamoiyeneba. am unarebis gamoyeneba xSiria monacemTa damuSavebis, cxrilebis aRwerisa da Sedarebisas. geometriuli figurebis ganxilvis dros abstraqciis, idealizaciis unarebis ganviTarebasTan gvaqvs saqme – yoveldRiur cxovrebaSi SemCneuli formebis idealizacia gvaZlevs im figurebs, romlebsac swavlebis procesSi ganvixilavT. kanonzomierebebis aRmoCenis unarebis ganviTarebas ukavSirdeba sidideTa garkveuli damokidebulebisas (maT Soris, realur viTarebaSi) erTi sididis cvlilebis gavlenis aRwera meore sididis cvlilebaze, damokidebulebis cxriliT, formuliT warmodgena. swavlebis ZiriTadi meTodi, romliTac unda mimdinareobdes Cveni saxelmZRvanelos gamoyenebiT swavleba, aris gadacemis interaqtiuloba. Cven vamzadebT maswavlebels aseTi swav­ lebisTvis. saxelmZRvaneloSi warmodgenili teqstebic am meTodis gaTvaliswinebiTaa agebuli. kvlevebi adasturebs, rom moswavle yvelaze ukeT maSin swavlobs, rodesac igi aqtiuri da qmediTia da ara maSin, roca aqtiur da qmediT maswavlebels uyurebs. Cven mier SemoTavazebuli sqemebidan kargad Cans, rom gakveTilebi ar aris leqciuri; amasTanave, moswavleebi swavloben ara marto maswavleblTan, aramed TanaklaselebTan urTierTobebiTac. kerZod, amas vaRwevT gundebs Soris dagegmili SejibrebebiT. am SejibrebebSi gundis kapitani SeiZleba maRali mzaobis moswavle iyos, magram prezentaciebSi monawileoba, Sedegebis warmodgena gundis sxva wevrebsac moeTxoveba. amitom kapitnis xelmZRvanelobiT gundis yvela wevri emzadeba Sedegebis warmosadgenad, romelic saerTo Sefasebazec moqmedebs. saswavlo-saganmanaTleblo resursebSi Sedis, magaliTad, geometriuli figurebis modelebi, romlebic geometriuli cnebebis TvalsaCinod warmodgenisaTvis gamoiyeneba. gakveTilebis scenarebSi aRwerilia rekomendaciebi maT gamoyenebasTan dakavSirebiT. maswavlebelma SeiZleba gamoiyenos plakatebi, romlebzec warmodgenilia kavSirebi zomis sxvadasxva erTeulebs Soris (farTobis, sigrZis); SeiZleba gamoiyenos cxrilebi, romlebzedac gamosaxulia martivi ricxvebi (magaliTad, 1-dan 100-mde), 2-is, 3-is xarisxebi, romlebzec am xarisxebis bolo cifrebTan dakavSirebuli kanonzomie-

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rebebis ilustracia xdeba. mniSvnelovania sxvadasxva cnebebis moculobebis venis diagramebiT warmodgena. maswavlebels momzadebuli unda hqondes suraTebi sxvadasxva mravalwaxnagebiT, romelTaTvisac gamoTvlilia Sesabamisi eileris maxasiaTebeli. eileris Teoremis TvalsaCino warmodgena mniSvnelovania sivrculi figuris elementebis raodenobebis dadgenis dros (ix. maT. VI.9). maswavlebels vurCevT moamzados (SesaZloa, moswavleebs moamzadebinos) TvalsaCino masala wrewirebis urTierTganlagebis Sesaxeb, maT centrebs Soris manZilisa da radiusebis gamoyenebiT. maswavlebelma agreTve intensiurad unda mimarTos G-PriEd-is (saqarTvelos dawyebiTi ganaTlebis proeqtis) mier momzadebul saganmanaTleblo resursebs.

moswavlis Sefasebis meqanizmebi moswavlis Sefaseba eyrdnoba Sefasebis miznebsa da ZiriTad kriteriumebs, romlebic kargad aris cnobili meTodikuri literaturidan – moswavlis Zlieri da susti mxareebis gamovlena da swavlis xelSewyoba. Sefasebis miznebis Sesabamisad, rogorc cnobilia, arsebobs ori forma – ganmaviTarebeli da ganmsazRvreli, I-IV klasebSi, ZiriTadad, pirveli formiT SemovifarglebiT. V-VI klasebSi meore formis gamoyenebasTan erTad viyenebT pirvel formasac. ganmsazRvreli Sefaseba mxolod SeZenil codnasa da unarebs afasebs. misi daniSnulebaa, rom gavarkvioT, mivaRwieT Tu ara dasaxul mizans, ra niSnulzea moswavlis done saswavlo gegmasTan da miznebTan mimarTebaSi. ganmaviTarebeli Sefaseba, faqtobrivad, unda ganvixiloT rogorc procesi, romelic gulisxmobs sxvadasxva aqtivobiT, swavlebis sxvadasxva saSualebiT (e. w. ganmaviTarebeli Sefasebis instrumentiT) informaciis Segrovebas moswavlis mier sakiTxebis gagebis, gaazrebis Sesaxeb da am informaciis gamoyenebas maTi progresis xelSesawyobad. es Sefaseba moswavleTa swavlebasa da ganviTarebazea orientirebuli. gansazRvruli Sefasebis instrumentic (mag. testi) zogjer SeiZleba vaqcioT ganmaviTarebeli Sefasebis instrumentad, Tu, magaliTad, moswavleebs davyofT mcire jgufebad da vTxovT, Seadaron miRebuli pasuxebi, amoxsnis gzebi da axsnan kidec maTi mosazrebebi. ganmaviTarebeli Sefasebis mixedviT maswavlebelma SeiZleba dagegmos swavlebis procesi, individualuri midgomebi TiToeul moswavlesTan – muSaobis moculoba da strategia. amrigad, am SefasebiT aRmovaCenT moswavleTa naklsa da warmatebebs, sanam maT quliT SevafasebdeT. maswavleblis komentari (zepiri an werilobiTi) ganmaviTarebeli Sefasebis instrumentad miiCneva, roca es komentari mxolod SeqebiT ki ar Semoifargleba, aramed mas aZlevs saTanado ganmartebas am Seqebis arsis Sesaxeb. ganmaviTarebeli Sefasebis instrumentebia: romelime moswavlis komentari, an Sefaseba sxva moswavlis mier gamoTqmuli mosazrebis Sesaxeb, Semajamebeli baraTebi, TviTSefaseba. maT Soris yvelaze mZlavr instrumentad TanatolTa mosazrebebis Sefasebas davasaxelebT.

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kargad mofiqrebuli kiTxvebis dasmac ganmaviTarebeli Sefasebis instrumentia, roca is faqtebis SedarebiT analizs moiTxovs da ara faqtobrivi codnis Semowmebas. am Sefasebis instrumentad SeiZleba gamoviyenoT Seswavlili cnebebis mixedviT moswavleTa mier testebis Sedgena. kidev ufro maRalL dones ki am cnebaTa vizualizacia warmoadgens. yoveli aqtivobisas maswavleblis mier warmoebuli dakvirvebac moswavleTa CarTulobaze ganmaviTarebeli Sefasebis instrumentia, rom araferi vTqvaT sxvadasxva rCevasa da rekomendaciaze. maSasadame, ganmaviTarebeli Sefaseba uwyvetad mimdinareobs – axal masalaze gadasvlamde da axali masalis aTvisebis procesSic. Cven ukve aRvniSneT V klasis maswavleblis wignSi, rom erovnuli saswavlo gegmis axal da Zvel redaqciebs Soris ZiriTadi gansxvaveba SefasebasTan aris dakavSirebuli. axal dokumentSi aqcenti keTdeba moswavlis Sefasebis swavlis konstruqtivistul principebTan dakavSirebaze; Sefasebis amocanebis Sesaxeb gaCnda aseTi Canaweri: `... moswavlis Sefasebis ZiriTad amocanas warmoadgens: a) aCvenos, rogor mimdinareobs codnis konstruirebis procesi da mexsierebaSi codnaTa urTierTdakavSireba; b) axali sakiTxis dawyebamde daadginos moswavlis winare codna da warmodgenebi; g) gamoavlinos, ramdenad axerxebs moswavle sakuTari Zlieri da susti mxareebis damoukidebel Sefasebas, ramdenad gaazrebul da efeqtian nabijebs dgams sakuTari winsvlis xelSesawyobad~. aqve, axal redaqciaSi, miTiTebulia, rom ganmsazRvrel SefasebaSi igulisxmeba codnaTa urTierTdakavSirebis unaris Sefaseba, samive codnis erTobliobaTa funqciurad gamoyenebis unaris Sefaseba da moswavlis akademiuri miRwevis donis dadgena. siaxlea isic, rom ganmsazRvrel SefasebaSi miTiTebul qulas unda axldes komentarebi Zlieri da susti mxareebis aRweriTa da xarvezebis gamosasworebeli miTiTebebiT. axal redaqciaSi xazgasmiTaa dazustebuli, rom Semajamebeli davalebis komponentSi aucilebelia kompleqsuri davalebebis gamoyeneba (amocanebis amoxsna, konkretuli problemis gadaWra). aqve miTiTebulia, rom amgvar davalebebSi Sesrulebuli samuSaos mravalmxrivi SefasebisTvis pedagogma unda SeimuSaos Sefasebis kriteriumi. Cven vaZlevT maswavlebels Semajamebeli samuSaos nimuSebs da rekomendaciebs Sefasebis kriteriumebis Sesadgenad. VI klasSi aqtiurad gamoiyeneba ganmsazRvreli Sefaseba. ganmsazRvreli Sefaseba, faqtobrivad, aris qula, romelsac maswavlebeli moswavles uwers garkveuli Temis, an sakiTxis dasrulebis Semdeg. aq gamoiyeneba sxvadasxva Semajamebeli aqtivobebi, magaliTad, proeqtis prezentacia, jgufuri muSaobis Sefaseba, Semajamebeli sakontrolo wera. ganmaviTarebeli Sefasebisas xSirad iyeneben e. w. qvizebs – Sefasebis formas, roca moswavles ukukavSirisTvis didi lodini ar uwevs. Cvens saxelmZRvaneloSi warmodgenili daxurulboloiani amocanebi qvizebis Catarebis saSualebas iZleva. aq SeiZleba aseTi cnobili xerxic gamoviyenoT: moswavleebs vurigebT furclebs, romlebzec warmodgenilia amocanebi saTanado savaraudo pasuxebiT, isini moniSnulia asoebiT: a), b), g), d). moswavleebi Suaze gayofili furclis orive nawilSi weren pasuxebs. samuSaos dasrulebis Semdeg moswavleebi pasuxebis furclis erT nawils abareben maswavlebels. amis Semdeg maswavlebeli acxadebs swor pasuxebs,

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moswavleebi ki TviTon ajameben sakuTar Sedegebs, gaigeben ra miiRes qvizSi da CaerTvebian diskusiaSi – ratom dauSves Secdoma ama TuU im davalebis Sesrulebis dros. winare codnis Sefasebis erT-erTi forma aris diagnostikuri testireba. diagnostikuri Sefaseba. gamoyofen or saxes – skriningul tests da fokusirebul tests. am testebis daniSnuleba V klasis maswavleblis wignSi iyo aRwerili, isic iTqva, rom skrininguli testi SeiZleba gamoviyenoT orjer – saswavlo wlis dasawyisSi da saswavlo wlis dasrulebamde garkveuli droiT adre, rom darCes dro mdgomareobis gasaumjobeseblad. pirveli testirebis mizani Zlieri an susti mxareebis adreuli dadgenaa. fokusirebuli testireba tardeba maswavleblis Sexedulebisamebr: a) roca aris riski, rom moswavlem raime sakiTxis nawili ver aiTvisa da Semajamebel weramde aris dro mdgomareobis gamosasworeblad; b) roca saWiroa xarvezebis masStabebis dadgena; g) axal Tavze gadasvlis win – aris Tu ara mzaoba axal masalaze gadasasvlelad. am SemTxvevaSi mniSvnelovania isic, rom Semdgom gakveTilebze diferencirebuli swavlebis gakveTilebis Casatareblad gveqneba saWiro informacia. maswavleblis wignSi aRwerilia TiTqmis yvela aqtivobis, RonisZiebis, proeqtis CatarebasTan dakavSirebuli rekomendaciebi. es rekomendaciebi Sejamebuli saxiT SeiZleba aq ase warmovadginoT: yoveli aqtivobisadmi wayenebuli mTavari moTxovna unda iyos ZiriTadi didaqtikuri miznis arseboba – im Temis Seswavlis miznis arseboba, romlis gadawyvetis procesi mocemuli aqtivobisas mimdinareobs. am miznis gaazreba moswavleebisTvisac aris mniSvnelovani. saubari unda daviwyoT ara imiT, Tu ras SeviswavliT, aramed iseTi situaciis SeqmniT, roca moswavle motivirebulia da poulobs pasuxs kiTxvaze – `risTvisaa saWiro~. motivaciis Seqmna ki SesaZlebelia aqtualuri praqtikuli amocanis dayenebiT da misi amoxsnis xerxebis ZiebiT daviwyoT. mniSvnelovania amocanis dasma da moswavleebTan erTad misi amoxsnis Ziebis procesis ganxorcieleba. saganmanaTleblo procesis centrSi unda idges moswavle da misi interesebi. swavla niSnavs informaciis dagrovebas da unar-Cvevebis ganviTarebas. amasTanave, es procesi SeiZleba sxvadasxva formiT mimdinareobdes. es formebia: muSaoba jgufebSi (SeiZleba or-orad), mTeli klasis monawileobiT, individualurad. maswavlebelma swavleba ise unda warmarTos, rom sakiTxis Seswavlis procesis ZiriTadi Semoqmedi Tavad moswavle iyos. maswavlebeli warmmarTvelis, `diriJoris~ funqcias unda asrulebdes. aqtivobis procesSi Seswavlili faqtebi TavisTavad aris mniSvnelovani, Tumca, kidev ufro mniSvnelovania is, rom maTi Seswavlis procesi Tavis kvals tovebs – mimdinareobs sxvadasxva inteleqtualuri unaris ganviTareba. maswavlebeli unda cdilobdes, rom yoveli axali SemecnebiTi amocana TviT moswavlem Camoayalibos. maswavlebelma, rogorc wesi, Tavidan bolomde unda gaiazros yoveli aqtivoba, winaswar, drois mixedviT, unda iyos ganawilebuli mTeli samuSao. Tumca, rogorc yovel SemoqmedebiT procesSi, gakveTilis Catarebis procesSic, SeiZleba dadges Tqvens gegmebSi cvlilebis Setanis aucilebloba. kiTxvebze pasuxebis gacemis sistema ar unda iyos erTferovani – mxolod warmatebul moswavleebTan mimarTebiT ar unda SemovifargloT. moswavlis raime mosazrebas myisve nu upasuxebT, swor pasuxsac ki maSinve nu daeTanxmebiT – gaakeTeT pauza, iqneb garkveuli eWvic ki gamoTqvaT wamoyenebuli mosazrebis sisworis mimarT. amiT miaRwevT imas, rom moswavleebi daubrundebian dasmuli kiTxvis analizs da male WeSma-

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riti daskvna – swori pasuxi – klasis dominantur mosazrebad gadaiqceva. klasi, erToblivi ZalisxmeviT `gaiZulebT~ daeTanxmoT moswavlis pozicias. es axarebs, amxnevebs da aerTianebs moswavleebs. es maTi erToblivi azris gamarjvebaa, Tqveni mizanic xom es aris – moswavle Semoqmed, codniT aRWurvil, iniciativian, xalisian axalgazrdad CamoayaliboT. maTematika mZlavri emociuri muxtis matarebelia da misi amoqmedeba Tqveni ZalisxmeviT miiRweva. axali masalis ganmtkicebis procesi `testebis~ `amoxsniT~ SeiZleba daviwyoT. maTi Sesruleba, ZiriTadad swori pasuxis SerCeviT unda Semoifarglos – zogjer SeiZleba komentarebis gakeTebac gaxdes saWiro, moswavle TandaTan eCveva am tipis davalebebis swrafad Sesrulebas. jgufuri muSaobis warmarTvis procesze Cveni rekomendaciebi zemoT iyo warmodgenili.

informacia moswavlis wignis Sesaxeb moswavlis wigni agebulia yvela im moTxovnis gaTvaliswinebiT, romelsac moswavlis saxelmZRvanelo unda akmayofilebdes. daculia saxelmZRvaneloTa Sedgenis sayovelTaod aRiarebuli meTodikuri principebi (ix., mag., [20]). wignSi warmodgenil masalas axasiaTebs maRali mecnieruli done da, imavdroulad, saswavlo kursi agebulia moswavleTa asakobrivi Taviseburebebis gaTvaliswinebiT, igi xels uwyobs moswavleTa motivaciis gaZlierebas. mravladaa sxvadasxva sirTulis testebi da amocanebi, rac xels Seuwyobs maswavlebels sxvadasxva akademiuri donis moswavleebTan diferencirebuli midgomisas warmatebiT gaarTvas Tavi saswavlo process. teqsti xels uwobs maswavlebels gaaZlieros moswavleTa CarTuloba, moyvanilia saintereso istoriuli faqtebi, warmodgenilia masala sxvadasxva didaqtikuri TamaSis, Sejibrebis, viqtorinis Casatareblad. meeqvse klasi dawyebiTi safexuris damamTavrebeli welia, mimdinareobs oTxive mimarTulebiT (ricxvebi da ricxvebze moqmedebebi, algebra, geometria, monacemTa analizi, albaToba, statistika) gameoreba codnis ganmtkiceba da gaRrmaveba. maswavlebels kvlav SevaxsenebT, rom saxelmZRvanelo aris erT-erTi saSualeba da ara erTaderTi, romelic saswavlo gegmaSi mocemuli Sedegebis miRwevas emsaxureba. saxelmZRvanelo ar aris TavisTavad programa. maswavlebeli orientirebuli unda iyos ara mocemuli `masalis gavlaze~, aramed sagnobrivi standartis Sedegebis misaRwevad mocemuli testebis, savarjiSoebis gamoyenebaze. sworia, roca maswavlebeli argebs saxelmZRvanelos moswavleebis saWiroebebs da ara, piriqiT, rodesac maswavlebeli cdilobs moergos saxelmZRvanelos. maswavlebeli swavlebis process unda miudges SemoqmedebiTad, man SeiZleba arc gamoiyenos romelime teqsti an aqtivoba, Secvalos Cven mier miTiTebuli drois reglamenti, romelic romelime konkretuli TemisTvisaa gankuTvnili, gadaanacvlos paragrafebis Tanamimdevroba. magaliTad, saswavlo wlis dasawyisSi SeiZleba meti dro dauTmos V klasis saswavlo programiT gaTvaliswinebuli sakiTxebis gameorebas, vidre Cven mier SemoTavazebul saxelmZRvaneloSia warmodgenili. Tumca, isic unda aRiniSnos, rom warmodgenili masala sruliad sakmarisia im unar-Cvevebis gansaviTareblad, romlebic dawyebiTi safexuris ukanasknel wels moswavles moeTxoveba. ar aris saWiro sxva saxelmZRvaneloSi mocemuli amocanebis

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gamoyeneba (magaliTad, `mravalsafexuriani~ ariTmetikuli moqmedebebis Semcveli ricxviTi gamosaxulebebis mniSvnelobebis gamoTvla). unda gvaxsovdes, rom maTematikis swavlebis mTavari funqcia pirovnebis saerTo-kulturuli ganviTarebaa, romelic azrovnebisa da im Cvevebis ganviTarebas emsaxureba, romlebic aucilebelia sazogadoebaSi srulyofili saqmianobisTvis. `maTematikuri ganaTleba~ Tavisi arsiT unda iyos `maTematikis saSualebiT ganaTleba~. dawyebiT safexurze swavlebisas igulisxmeba SedarebiT dasrulebuli maTematikuri ganaTlebis etapi, romelmac unda Seqmnas sabazo skolaSi swavlebis baza. maswavlebelma SeiZleba gamoiCinos SemoqmedebiTi midgoma Cven mier mocemuli rekomendaciebis mimarT. yovel karg maswavlebels swavlebis sakuTari xerxebi SeiZleba hqondes. amiT yvela kargi maswavlebeli gansxvavdeba sxva kargi maswavleblisgan. Tumca, unda iTqvas, rom arsebobs zogadi meTodikuri principebi, romlebic damuSavebulia literaturaSi. maTi gacnoba daexmareba maswavlebels muSaobaSi. wignis bolos miTiTebuli literaturac amaSi dagexmarebaT. Tavisi agebulebiT da SinaarsiT Cveni VI klasis saxelmZRvanelo wina V klasis saxelmZRvanelos bunebrivi gagrZelebaa. bevri sakiTxi xelmeored ufro gaRrmavebulad aris warmodgenili. sakiTxebis ganmeorebiT ganxilvas VI klasis standartic iTvaliswinebs (zomis zogierTi erTeuli; naturaluri ricxvebis, wiladebis sxvadasxva aspeqtis warmoCena). misaRwevi Sedegebis Sesabamisi Sinaarsis SerCevas moswavleTaTvis misawvdomobis principi udevs safuZvlad; es SerCeva Camoyalibebuli stabiluri birTvis bazaze xdeba. igi gafiltrulia maTematikis swavlebis gamocdilebiT; tradiciuli Temebis Tanamimdevrobis (magaliTad, wiladebis da aTwiladebis) axleburi gaazrebiT, maTematikis erTianobidan gamomdinare integrirebis xarisxis gazrdisa da masalis axali sakiTxebiT Sevsebis aucileblobis gaTvaliswinebiT. maswavlebels mudmivad unda axsovdes, rom maTematika erTiani mecnierebaa da ara misi sxvadasxva nawilis xelovnuri gaerTianeba, amitom calkeuli fragmentebis konkretul warmodgenas Tan sdevs sakiTxebis urTierTkavSirSi gadmocemis gaTvaliswineba. sistematurad mimdinareobs wina masalis gameoreba da ganmtkiceba, yoveli siaxlis Seswavla moswavlis aqtiuri monawileobiT ganixileba; davalebebi, maTi mravalferovneba, xels uwyobs moswavleTa motivaciis gazrdas, kritikuli da SemoqmedebiTi azrovnebis ganviTarebas. davalebebi iseTi TanamimdevrobiTaa warmodgenili, rom martividan rTulisken iyos mimarTuli, calkea gamoyofili davalebebi saklaso da saSinao muSaobisTvis, savarjiSoTa mravalferovani sistema xels uwyobs sxvadasxva SesaZleblobebis mqone moswavleebTan muSaobis, maTematikis Segnebulad swavlisa da misi Seswavlisadmi interesis gaRrmavebis ganxorcielebas. saxelmZRvaneloSi mocemuli zogierTi rubrika (`s~ – sxvadasxva, `moisazre~) xels uwyobs moswavlis cnobismoyvareobis gazrdas, ubiZgebs mas SemoqmedebiTi muSaobisken, movlenebis Sesaxeb codnis SeZenisken da TviTganviTarebisken; axali faqtebis moZiebisaTvis teqnologiebis gamoyenebisken. `testebi~ kargi saSualebaa swrafi Sefasebebisa da waxalisebisTvis, Tumca SesaZlebelia, zogierTi testis ganxilvas Tan sdevdes msjeloba da pasuxis dasabuTeba. amasTanave, sTxoveT moswavleebs, wignSi nu miuTiTeben swor pasuxebs, CainiSnon rveulSi swori pasuxebis nomrebi, romlebsac SeiZleba Tan erTvodes mokle Canawerebi pasuxebis miRebis SesaZlo gzis aRwerisTvis. warmodgenili davalebebi ar saWiroebs

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wignSi Canawerebis gakeTebas – yvela davaleba mxolod rveulSi unda Sesruldes. masalis siuxviTa da mravalferovnebiT gamovricxeT damxmare, an damatebiTi samuSao rveulebisa da saxelmZRvaneloebis saWiroeba. moswavle unda mieCvios saxelmZRvanelos sufTad xmarebas, xSirad amas Cven SevaxsenebT xolme. SesaZlebeli unda iyos wignis gamoyeneba ramdenime wlis ganmavlobaSi. amitom wignze zrunva sxvebze zrunvis gamovlenacaa. moswavlis wignis agebuleba saxelmZRvanelosadmi wayenebul yvela moTxovnas iTvaliswinebs (ix., mag., [20]). wignis dasawyisSi gamovxatavT madlierebas yvelas mimarT, visac Tavisi rekomendaciebiT wvlili Seaqvs wignis daxvewaSi, gansakuTrebiT mniSvnelovania saswavlo gegmebisa da saxelmZRvaneloebis SemmuSavebeli jgufis eqspertTa Rvawli. maT mier klasebis mixedviT ganawilebuli saswavlo gegma UsaSualebas gvaZlevs SevarCioT Sesabamisi masala moswavleTa codnis gaRrmavebis, unarebis ganviTarebisa da formirebisTvis. moswavlis saxelmZRvanelo 5 Tavisgan Sedgeba, TiToeuli Tavi – paragrafebisgan, zogierTi paragrafi – ramdenime nawilisgan; sarCevi wignis dasawyisSia mocemuli; maTematikuri terminebi, cxrilebi, maTematikuri niSnebi, sagnobrivi saZiebeli – wignis boloSi. Cveni saxelmZRvaneloebis axali struqtura moswavleTaTvis advilad aRqmadi aRmoCnda – iolia wignSi orientireba, masSi masalis moZieba da misi gacnoba-damuSaveba. iseve, rogorc V klasis saxelmZRvanelos Sedgenisas, gaviTvaliswineT 2017 wlis gazafxulze Catarebul trening-seminarebze warmodgenili rekomendaciebi, magaliTad, mosazrebebi swavlebis procesSi mdgradi ganviTarebis principebis ganxorcielebaze. genderuli Tanasworobis principebis dacviT (Cvens `personaJebSi~ Tanasworad arian warmodgenili vaJebi da gogonebi) Cven xazs vusvamT samarTliani socialuri garemos erT-erT mniSvnelovan komponents, sagzao niSnebis geometriuli figurebiT warmodgeniT ki maswavlebels aqvs SesaZlebloba isaubros usafrTxo garemos Sesaxeb. sxvadasxva proeqtis Sesrulebis dros SeiZleba win wamovwioT movlenebis sxvadasxva kuTxiT danaxvis aucilebloba (magaliTad, Cven mier miTiTebuli avtomobilis siCqaris sidide). didi ricxvebis ganxilvisas saubari gvaqvs mosaxleobis raodenobiT e. w. did qveynebze. aq SeiZleba SevTavazoT moswavleebs proeqti am qveynebis socialur-kulturuli aRweris Sesaxeb; mniSvnelovania orientacia msoflioSi da msoflio rukaze, gansxvavebuli cxovrebis wesis dafaseba, qalaqisa da soflis cxovrebis aRwera. mniSvnelovania muSaoba moswavleTa pasuxismgeblobis gazrdaze Cvens saerTo momavalze, saubari energiis gamomuSavebis alternatiuli gzebis Sesaxeb.

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moswavlis wignis Sesabamisoba erovnuli saswavlo gegmis standartTan moswavlis saxelmZRvaneloSi warmodgenili Temebi 5 TavSia ganawilebuli da iTvaliswinebs standartSi miTiTebul misaRwev Sedegebs; arauaryofiTi ricxvebis gamosaxva, Sedareba da dalageba poziciuri sistemis gamoyenebiT, maT.VI.I, – Tavi I. moqmedebebi arauaryofiT racionalur ricxvebze. maT.VI.2. Tavi II, Tavi III. zomis erTeulebis erTmaneTTan dakavSireba, sigrZis, farTobis, moculobis erTeulebis dakavSireba erTmaneTTan, maT.VI.3. Tavi IV. problemis gadaWra gamoTvlebis, variantebis daTvlisa da mimarTebebis gamoyenebiT, maT.VI.4, III Tavi, $7, $8; IV Tavi, $1, $6, $8, $9, V Tavi, $5. sidideebs Soris damokidebulebis gamosaxva, ganvrcoba, aRwera, maT.VI.5, Tavi IV, $9. algebruli gamosaxulebis Sedgena, gamartiveba, gantolebis, utolobis Sedgena, maT. VI.6, Tavi II, Tavi III, $8. sivrculi figurebis amocnoba, aRwera, gamosaxva, maT.VI.7, Tavi IV, $7, $8. figurebs Soris da TviT figurebis elementebs Soris mimarTebebis dadgena, maT.VI.9, Tavi IV, $1, $7. brtyeli figuris farTobis gamoTvla, maT.VI.10. Tavi IV. $6. Tvisebrivi da raodenobrivi monacemebis mopoveba, maT.VI.11, Tavi V, $1. monacemebis mowesrigeba da xelsayreli formiT warmodgena, maT.VI.12, Tavi V, $2. monacemTa analizi, maT.IV.13, Tavi 5, $3, $4. Temebis Tanamimdevroba da paragrafebis mixedviT maTi organizeba xels uwyobs dasaxuli miznebisken gegmazomier svlas. xSiria sakiTxebisadmi `mibrunebis~ magaliTebi (magaliTad, V Tavis me-5 paragrafSi, vubrundebiT amocanis amoxsnis sxvadasxva xerxis gamoyenebas: sinjvis xerxi, amowurvis xerxi, venis diagramebis gamoyeneba). gansakuTrebuli yuradReba eTmoba moswavlis motivaciis gaZlierebas. wignSi xSiria realur cxovrebaSi maTematikis gamoyenebis magaliTebi da sxva saintereso da saxaliso masala (magaliTad, wrewirebis urTierTganlagebis Seswavlisas cnobili qarTuli TamaSis `laxtis~ warmodgena), rac udavod gaaRvivebs moswavleTa interess maTematikis Seswavlisadmi.

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moswavlis wignSi gamoyenebuli pirobiTi niSnebis ganmartebebi

es simbolo klasSi amosaxsneli amocanebis ganxilvis dawyeba­ ze migviTiTebs. am amocanebis nomrebi wreebSia

klasSi

Casmuli. magaliTad,

5 .

es simbolo Sin muSaobisTvis gankuTvnil amocanebs warmogvidgens. am amocanebis nomrebi samkuTxedebSia saSinao

sxvadasxva

moisazre

jgufuri

TamaSi

Cveni leqsikoni

!

l Si

rv

3

.

`sxvadasxva~ _ saintereso faqtebsa da istoriebs war­mo­gvidgens. am simbolos mosdevs amocanebi, romelTa amosaxsnelad xSi­rad sakmao gamWriaxobis gamovlenac mogiwevT. am amo­ 1  .

canebis nomrebi kvadratebSia Casmuli, magaliTad,

am simboloTi jgufuri muSaobis proeqtebia gamoyofili _ aseTi davalebebi TanaklaselebTan erTad muSaobis gamocdilebas SegmatebT. am simboloTi saswavlo TamaSebia aRniSnuli – maTi saSualebiT maTematikuri codnis gamoyenebis gamocdilebas SeiZenT. vin iqneba pirveli. am simboloTi warmodgenili amocanebis amoxsnisas ecadeT, gamWriaxobasTan erTad, sxa­ rti azrovnebac gamoavlinoT.

i

eu

Casmuli _

teqstis ukeT gagebis mizniT saWirod CavTvaleT zo­ gi­ erTi sityvis warmomavlobis da maTi sxvadasxva mniS­ vne­ lobis miTiTeba. aseTi Canawerebi teqstSi mo­ niS­ nulia aq gamosaxuli simboloTi. am simboloTi drodadro SegaxsenebT, rom yvela amocana unda gadaita­ noT rveulSi da DSemdeg SeasruloT. wignSi Canawerebi ar unda gakeTdes!

gaufrTxildiT wigns! masSi Tqvenma Canawerebma wigni SeiZleba sxvebisTvis gamousadegari gaxados! 21


gakveTilebis scenarebis Sesaxeb ganaTlebisa da mecnierebis saministrodan miRebul sarekomendacio werilSi (`maswavleblis wignis sarekomendacio struqtura~) moiTxoveboda gakveTilis scenarebi – pirveli Tavis yvela gakveTili da yoveli momdevno Tavis aranakleb 2 gakveTilis scenaris CarTva maswavleblis wignSi. Cven gaviTvaliswineT maTematikis swavlebis sirTuleebi, maswavleblebis survilebi da maswavleblis wignSi warmovadgineT yvela gakveTilis scenari, rekomendaciebi moswavleTaTvis masalis Sinaarsis miwodebis Sesaxeb, warmodgenili iqneba Sefasebis meqanizmebic da sxva meTodikuri resursebi. yoveli sakiTxis Seswavlas, rogorc wesi, ramdenime gakveTili eTmoba. SeiZleboda warmogvedgina sakiTxis Seswavlis dagegmvis sqema, magram Cven varCieT cal-calke warmogvedgina TiToeuli gakveTili, rogorc es aRniSnul werilSi moiTxoveboda. V klasis maswavleblis wignSi Tqven ukve gaecaniT dagegmvis zogad sqemas, romelSic dawvrilebiT iyo aRwerili konstruqtivistuli swavlebis samfaziani modeli da misi dagegmva. erovnuli saswavlo gegmis axal redaqciaSi xazgasmulia konstruqtivistuli swavlebis mniSvneloba, roca moswavleebis codnis konstruireba xdeba sakuTar gamocdilebaze dayrdnobiT da ara mxolod maswavleblis mier miwodebuli mza informaciis safuZvelze. aRniSnuli iyo am meTodsa da evristikul meTods Soris mimarTeba; dawvrilebiT aRvwereT 3 fazidan TiToeulis ganxorcielebis miznebi (`winaswar~, `ganmavlobaSi~, `Semdgom~). isic iyo naTqvami, rom samive fazis ganxorcielebas, rogorc wesi, ramdenime gakveTili sWirdeba. Cven mier warmodgenili gakveTilebis sqemebSi kargad Cans da Cvenc vuTiTebT xolme – xdeba Tu ara samfaziani modelis ganxorcieleba, romeli gakveTilebi romeli fazis ganxorcielebas gulisxmobs. V klasis maswavleblis wignSi mogawodeT masala sakiTxis samfaziani swavlebis Sesabamisi sqemis Sesadgenad. Zvirfaso maswavleblebo! warmatebas gisurvebT Tqvens rTul, sapatio da mniSvnelovan saqmianobaSi. imedi gvaqvs, rom Cveni TanamSromloba keTil nayofs gamoiRebs.

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I Tavi gameoreba. wiladebi da aTwiladebi saxelmZRvanelos pirvel Tavs gamorCeuli datvirTva aqvs – xdeba moswavleTa CarTva saswavlo ritmSi, mTeli saswavlo wlisTvis tonis micema. xangrZlivi sazafxulo ardadegebis Semdeg uaRresad aqtualuria adre Seswavlili sakiTxebis (naturaluri ricxvebis Cawera, moqmedebebi naturalur ricxvebze, Sedareba, sawyisi cnebebi wiladebis Sesaxeb, warmodgenebi wiladis yvela aspeqtis Sesaxeb) xelaxali gaazreba da SeZenili unarebis gamyareba-ganviTareba. aseTi dasawyisis Semdeg meTodikurad gamarTlebulia, CavataroT diagnostikuri wera, romlis Sedegebis gaTvaliswinebiT, davgegmavT Cvens muSaobas gasameorebeli mUasalis codnaSi SeniSnuli xarvezebis aRmosafxvrelad, rom TandaTanobiT gadavideT axali Temebis Seswavlaze (aTwiladebi). Tumca am masalasac ar amovwuravT. gadmocemis Cveneuli meTodika gulisxmobs arauaryofiTi racionaluri ricxvebis warmodgenebisa da am warmodgenebis gamoyenebiT maTze moqmedebebis integrirebul gadmocemas. pirvel nabijebs gadavdgamT aTwiladebis praqtikuli gamoyenebebiskenac. rac Seexeba arauaryofiTY racionalur ricxvTa dafuZnebis sakiTxebs, am mimarTulebiT mizanSewonilia, maswavlebeli icnobdes am Teoriis dafuZnebis elementebs (ix., mag. [11]). maswavleblebs SevaxsenebT, rom naturalur ricxvTa dafuZnebis Teoriis gacnobisas, ZiriTadad, or midgomaze miuTiTeben – aqsiomuri midgoma, romelic peanos saxelTan aris dakavSirebuli da, martivad rom vTqvaT, `momdevnos~ mimarTebis gamoyenebazea agebuli. meore midgoma kantoris saxelTan aris dakavSirebuli. aq simravlis elementebis ricxvi gaigeba, rogorc simravlis simZlavre, safuZvlad erTi simravlis meore simravleze urTierTcalsaxa Sesabamisobis cneba aiReba. dadebiT racionalur ricxvTa sistemis ageba ki naturalur ricxvTa simravleze dafuZnebiT, wiladebisa da aTwiladebis gamoyenebiT SeiZleba movaxdinoT (ix. mag. [11], [13]). skolaSi ricxvis cnebis gafarToebis sqema maTematikis istoriul ganviTarebas esadageba: arauaryofiTi mTeli ricxvebi ⇒ arauaryofiTi racionaluri ricxvebi ⇒ racionaluri ricxvebi ⇒ namdvili ricxvebi. Tumca, algebruli TvalsazrisiT ganzogadebis bunebrivi gzaa – arauaryofiTi mTeli ricxvebi ⇒ mTeli ricxvebi ⇒ racionaluri ricxvebi ⇒ namdvili ricxvebi. erovnuli saswavlo gegmis mixedviT, romelic pirveli sqemis ganxorcielebas gulisxmobs, racionalur ricxvTa simravleze gadasvla VII klasidan iwyeba. am mimarTulebiT VI klasSi `novatorobis~ gamoCena maswavleblebis mxridan ar aris gamarTlebuli, radgan gamoiwvevs sxva masalis, aucilebeli saswavlo

23


masalis, Sekvecas, an saerTod, ignorirebas. ufro metic, maswavlebeli valdebulia saqarTvelos mecnierebisa da ganaTlebis saministros mier damtkicebuli erovnuli saswavlo gegmiT ixelmZRvanelos. Cven mier miTiTebul literaturaSi kargad aris axsnili swavlebis aRniSnuli Tanamimdevrobis ganxorcielebis mizanSewonileba da meTodikuri dasabuTeba. am da sxva mravali sakiTxis Sesaxeb sayuradRebo meTodikur da praqtikul rekomendaciebs gagiwevT saqarTvelos dawyebiTi ganaTlebis proeqtis (G-PriEdEG) masalebic. aqtiurad unda gamoviyenoT maTi eleqtronuli resursebi.

§1. gavixsenoT – rogor CavwerT da valagebT naturalur ricxvebs paragrafs sami gakveTili eTmoba. 1-li gakveTili

Tema: naturaluri ricxvebis Cawera, Sedareba. mizani: moswavles SeeZlos aTobiTi poziciuri sistemis gamoyeneba ricxvebis Cawerisa da SedarebisTvis, xuTniSna, eqvsniSna, SvidniSna ricxvebis Cawera, Sedareba (maT. V.1., maT. VI. 1). winapirobebi: milionze didi ricxvebis wakiTxva, Cawera, rigis dasaxeleba, 10-is xarisxebis gamoyeneba ricxvis Cawerisas, cifris mniSvnelobis dasaxeleba. resursebi: saxelmZRvanelo, rveuli, saxazavi, kalami, fanqari, dafa. rogorc SesavalSi aRvniSneT, pirvel gakveTilebze mimdinareobs V klasSi Seswavlili sakiTxebis gameoreba, codnis ganmtkiceba da gaRrmaveba. Zalian mniSvnelovania aqtivobebi, romlebic dakavSirebulia aTobiTi poziciuri sistemis kargad gaazrebasTan. maswavlebeli yuradRebas amaxvilebs imaze, rom erTi da imave cifris mniSvneloba ganisazRvreba misi adgilis (poziciis) mixedviT. amitom am wess poziciuri hqvia. interaqtiuli reJimis gamoyenebiT moswavleebi ixseneben didi ricxvebis Caweras, wakiTxvas, damrgvalebas, Sedarebasa da dalagebas; iwafebian am sakiTxebSi. moswavleebi gaiazreben da CauRrmavdebian ricxvTa Caweris aTobiT – poziciur sistemas. am mniSvnelovani sakiTxis ganxilvas emsaxureba teqstSi SemoTavazebuli amocanebi. magaliTad, CineTis mosaxleobis raodenobis CanawerSi cifri 3 gamoyenebulia orjer – milionisa da aseuli milionis TanrigebSi. pirvel SemTxvevaSi is 3 milions warmoadgens, meore SemTxvevaSi – 300 milions, anu 3-iT warmodgenili ricxvi 3-is adgilze, poziciazea, damokidebuli. swored amitom ewoda ricxvebis Caweris am wess poziciuri wesi. radgan ricxvTa Caweras mxolod aTi cifris meSveobiT vaxdenT, aman ganapiroba, rom Tvlis am sistemas aTobiTi sistema ewoda, xolo ricxvTa Canawerebs – aTobiTi Canawerebi. Tvlis (ricxvTa Caweris) araaTobiT poziciur sistemebs moswavleebi momavalSi gaecnobian. moswavleebs ukve aqvT warmodgena ricxvTa Caweris arapoziciur sistemazec, ker­ Zod, romaul sistemaze. Tu klasma am etapzeve gamoxata interesi am sistemis gaxsenebis mimarT, mokle ganxilviT dakmayofildiT. magaliTad, XX warmogvidgens ocs. X simbolo, miuxedavad poziciisa, aT erTeuls warmoadgens. Tumca, romaul CanawerebSic aris poziciuri Caweris elementebi. X-is marjvnidan miwerili naklebi ricxvi, maga-

24


liTad V (xuTi), emateba X-s (aTs) da Canaweri XV warmoadgens 15-s; X-is marcxnidan miwerili naklebi ricxvi, magaliTad, I (erTi) akldeba X-s da IX warmoadgens 9-s. moswavleebTan erTad vixsenebT ricxvebis Sedarebisas utolobis an tolobis niSnebis gamoyenebas. ricxvebis dasaxelebisas veyrdnobiT Tanrigebis klasebad dayofas. moswavles unda SeeZlos aRniSnuli klasebis TanrigTa dasaxeleba. ricxvTa Sedarebas viwyebT umaRlesi Tanrigis cifrebis SedarebiT. sainteresoa da wasaxalisebeli, Tu moswavle ricxvebis Sedarebis process aRwers sityvierad. moswavleebs vavalebT gadaitanon rveulSi is cxrili, romelic saxelmZRvaneloSia warmodgenili. maswavlebeli am cxrilSi ricxvis Caweris mixedviT svams kiTxvebs, romlebic poziciuri sistemis Taviseburebebs ukavSirdeba. kiTxvebis nawili saxelmZRvaneloSia warmodgenili. sasurvelia gavagrZeloT cxrilis Sevseba da kidev ramdenime ricxvis Cawera vTxovoT moswavleebs. maT Sesaxebac vsvamT analogiur kiTxvebs. Semdeg SeiZleba gadavideT am ricxvebis Sedarebaze, dalagebaze, 10-is xarisxebis mixedviT warmodgenaze. amis Semdeg vxsniT 1 - 16 amocanebs. isini emsaxureba moswavleTa codnis gameorebasa da maT gawafvas ricxvis Cawera-wakiTxvaSi, ricxvis CanawerSi ama Tu im cifris mniSvnelobis dadgenaSi, ricxvebis gaSlili formiT CaweraSi, ricxvebis SedarebaSi da dalagebaSi. 13 - 16 -Si yuradReba gaamaxvileT imaze, rom 2765-is aseulebis Tanrigis cifri 7 warmoadgens 7 aseuls, xolo mTlianad ricxvi Seicavs 27 aseuls, (kidev 65 erTeuls, anu, 6 aTeuls da kidev 5 erTeuls). sasurvelia, Tu 1 - 6 amocanebs moswavleebi damoukideblad Seasruleben da am davalebis SesrulebiT SeiZleba Sefasdes dasaxuli miznis Sesruleba – mocemuli rigis mixedviT udidesis da umciresis dasaxeleba (maT. VI, 1). magaliTad, udidesi eqvsniSna ricxvi aris 999 aTas 999, umciresi eqvsniSna ricxvi aris 100 aTasi; udidesi aTniSna ricxvi, romelic yvela cifrs Seicavs, aris 9 876  543  210. 7 amocanaSi swori pasuxis povnis Semdeg (700  000 + 40 000 + 900 +80 +1), sasurvelia, rom es ricxvi 10-is xarisxebis gamoyenebiTac CavweroT: 7⋅105+4⋅104+9⋅102+8⋅10+1. es ricxvi aris 740  981. 11 amocanaSi viyenebT utolobis niSans ricxvebis Sedarebisas:

26095<26105 xazgasmis mizniT, kidev erTxel SevexoT 13 - 16 amocanebs: 14 2765 Seicavs 27 aseuls. 13 aRniSnuli ricxvis aseulebis Tanrigi ki Seicavs 7 aseuls. 15 2765 Seicavs 276 srul aTeuls. 16 ricxvi, romelic mxolod 28 srul aseuls da 7 erTeuls Seicavs, aris

2800+7, anu 2807. saSinao davalebad moswavleebs vaZlevT

1

- 18

amocanebs.

25


me-2 gakveTili

Tema: naturaluri ricxvebis gamosaxva, Sedareba, dalageba. mizani: poziciuri sistemis gamoyenebiT naturaluri ricxvebis gamosaxvaSi, Sedarebasa da dalagebaSi moswavleTa gawafva (mas. VI. 1). winapirobebi: naturaluri ricxvebis Cawera, cifris mniSvnelobis miTiTeba, ricx­ vebis Sedareba poziciuri sistemis gamoyenebiT. resursebi: saxelmZRvanelo, rveuli, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT. nu SemovifarglebiT mxolod merxebs Soris `CamovliT~ da rveulebSi Canawerebis `gadaxedviT~. moswavleebi rigrigobiT kiTxuloben `testebs~ da uTiTeben pasuxebs. sxvebi akontroleben maT Sedegebs da erTvebian komentarebiT. zogierT moswavles am asakSic ki SeiZleba hqondes teqstis wakiTxvaSi problemebi, amitom davalebis Semowmeba am kuTxiT moswavleebTan muSaobisTvisac aris mniSvnelovani. magaliTad, moswavle kiTxulobs pirvel `tests~. Tu udides samniSna ricxvs mivumatebT 1-s, miviRebT 1000-s. SeiZleba vTxovoT, rom pasuxi daasabuTos. dasabuTebisas erT-erTi varianti SeiZleba iyos aseTi: -udidesi samniSna ricxvi aris 999, misi momdevno ricxvi ki aris 1000. SeiZleba maswavlebelma miiCnios saWirod damatebiTi kiTxvebis dasmac. 1

- 14 amocanebi klasSi amoxsnili amocanebis analogiuria. cxadia, mniSvnelovania, rom moswavleebma dainaxon analogiebi amocanebs Soris da gamoiyenon es analogiebi. Tumca moswavleTa SemoqmedebiTi azrovnebis ganviTarebisTvis mniSvnelovania odnav gansxvavebul amocanebSic pasuxebis SerCeva. veZebT samniSna ricxvebs, romelTa cifrTa jami aris 3. am amocanis amoxsnisas moswavle iyenebs `sinjvisa da amowurvis~ meTods: vmsjelobT, jer is ricxvebi vipovoT, romelSic pirveli cifri aris 1 (0 ar SeiZleba), maSin danarCeni ori cifris jami unda iyos 2. aseTi samniSna ricxvebia 102, 120, 111. Semdeg veZebT ricxvebs, romelSic pirveli cifria 2. aseTia ori ricxvi: 201, 210. Semdeg – roca pirveli cifria 3. aseTi erTaderTia – 300). am wesis gamoyeneba SeiZleba Cven SevTavazoT da amiT klasificirebis unaris ganviTarebaze vizrunoT. Tan sayuradRebo zogad ganmaviTarebel aqtivobaSi CavabaT moswavleebi. 15

aq veZebT udides samniSna ricxvs, romlis cifrTa jamia 26. udides samniSna ricxvSi cifrebis jami aris 26, maSasadame, saZiebeli ricxvi Sedgeba 2 cxrianisa da 1 rvianisagan, es ricxvia 998. 16

xuTniSna ricxvSi aseulebis cifris 2-iT gazrda gamoiwvevs ricxvis 200-iT gazrdas, aTeulebis cifris 3-iT gazrda – 30-iT gazrdas, e. i. ricxvi gaizrdeba 230-iT. 17

am amocanis mixedviT, ricxvi gaizrdeba 2000-iT da Semcirdeba 500-iT. e. i. gaizrdeba 1500-iT. am process SeiZleba movandomoT 20-25 wuTi. danarCeni dro (daaxloebiT imdenive) davuTmoT 17 - 30 amocanebidan SerCeulTa amoxsnas an komentirebas. SeiZleba erToblivad amovxsnaT 17 da 18 -is ramdenime davaleba, 22 - 30 amocanebi. amis Semdeg SeiZleba moswavleebs davavaloT 17 da 18 -is danarCeni davalebebis da 19 - 23 -is amoxsna. analogiuri amocanebi maTTan ukve ganxilulia. magaliTad, 21 da 22 aris 18

26


17 -is da

18 -is analogiuri, romelic gakveTilis dasawyisSi ganvixileT, 23 amo-

canac maTi analogiuria. pirobidan Cans, rom aseulebisa da erTeulebis Tanrigebis cifrebi tolia da 0-isgan gansxvavebulia, radgan aTeulebis cifri 4-iT metia erTeulebis cifrze (minimum 1-ze), is 4-ze metia e. i. aTeulebis cifri SeiZleba iyos 5, 6, 7, 8, an 9. Sesabamisad, saZiebeli ricxvia 151, 262, 373, 484, an 595. 25 da 26 savarjiSoebi ricxvebis SedarebaSi gawafvas emsaxureba. 25 savarjiSoSi moswavle iwyebs umaRlesi Tanrigebis Sedarebas. magaliTad, a)-Si I ricxvis aTaseulis Tanrigis cifri ar unda aRematebodes II ricxvis aTaseulis Tanrigis cifrs 0-s, anu • niSnaki unda Seicvalos 0-iT; e)-Si II ricxvis aTaseulis Tanrigis cifri ar aRemateba I ricxvis aTaseulis Tanrigis cifrs 0-s, anu 0-is tolia. amrigad, am ricxvebis sami umaRlesi Tanrigis cifri daemTxveva. amocana dadis aTeulis Tanrigebis Sedarebaze: 12>•3, anu • niSnaki unda Seicvalos 0-iT. miviRebT: 60612>60603. 26 savarjiSos a), b) da g) amocanebi xuTniSna da oTxniSna ricxvebis Sedarebas

exeba _ cxadia, Sesabamisi xuTniSna ricxvi metia nebismier oTxniSnaze. d)-Si II ricxvi 9-iTac rom daiwyos, mainc I ricxvia meti; e)-Si II ricxvi 1-iTac rom daiwyos (0-iT dawyeba gamoiricxeba), mainc I ricxvia naklebi. 27 da 28 amocanebi ricxvebis CaweraSi gawafvas emsaxureba. 27 -is meore kiTxvis pasuxis sapovnelad ar aris saWiro ricxvebis gamokleba – miliardebis cifri 2-iT gaizarda, aseuli milionebis Tanrigis cifri 1-iT Semcirda, maSasadame, ricxvi gaizrdeba 2 miliardiT da Semcirdeba 100 milioniT, radgan 1 miliardi Seicavs 1000 milions, amitom ricxvi gaizarda 1 miliard 900 milioniT. 28 umciresi samniSna ricxvi, romlis cifrebis jamia 4, aris 103. masSi aseulebis

cifri nulisagan gansxvavebuli da rac SeiZleba mcire unda iyos, aTeulebis cif­ ri unda iyos rac SeiZleba mcire, xolo erTeulebis cifrisa da 1-is jami 4-s unda adgendes, aseTi cifria 3. 29 saZiebel ricxvs advilad vipoviT – 7777. toli ricxvebis jami gamravlebas niS-

navs – 4⋅7=28. 30 udidesi xuTniSna ricxvi, romelic gansxvavebuli kenti cifrebiT aris Cawe-

rili, aris 97531, umciresi oTxniSna ricxvi aris 1357. aRniSnuli amocanebi faravs maT. VI.1-is moTxovnebs. saSinao davalebad moswavleebs vaZlevT 19 - 28 amocanebs.

me-3 gakveTili

Tema: naturaluri ricxvebis gamosaxva, Sedareba da dalageba poziciuri sistemis gamoyenebiT. mizani: moswavles SeeZlos ricxvebis Caweris poziciuri sistemis gamoyenebiT iseTi amocanebis amoxsna, romlebic moiTxovs logikuri msjelobis, kombinatoruli azrovnebis gamoyenebas da aviTarebs saTanado unarebs. winapirobebi: naturaluri ricxvebis Cawera, wakiTxva, klasebis, Tanrigebis miTiTeba, cifrebis mniSvnelobis dasaxeleba. resursebi: saxelmZRvanelo, rveuli, dafa.

27


gakveTils viwyebT saSinao davalebis SemowmebiT. sasurvelia, rom es Semowmebac ar iyos zerele, SevecadoT, rom am gakveTiliT Sefasdes pirvel paragrafSi warmodgenili masalis Seswavla-damuSavebis Sedegad miRweuli Sedegebi. 19 - 26 amocanebis analogiuri amocanebis amoxsnebi ganixileboda wina gakveTilebze. am amocanebis amoxsna ar unda gasWirvebodaT moswavleebs. 4486 da 4491-s Soris moTavsebuli ricxvis aTaseulisa da aseulis TanrigebSi 4-ebia. miviRebT: 449*<4491. amrigad, erTeulebis TanrigSi unda eweros 0. es ricxvia 4490. 27

28

amocanis piroba SeiZleba ase gamovsaxoT 6132<*1*3<6143, saidanac vRebulobT, rom saZiebeli ricxvis aTaseulebis cifria 6, aTeulebis cifri _ 3. anu, saZiebeli ricxvia 6143. marTlac, 6132<6133<6143. SeiZleba aRmoCndes, rom moswavleebis nawils gauWirda zogierTi amocanis amoxsna. am SemTxvevaSi SeiZleba maTTan damatebiTi muSaobis Catareba. am dros maRali mzaobis moswavleebs SeiZleba vTxovoT, ifiqron `moisazres~ rubrikiT warmodgenili amocanebis amoxsnaze. am amocanebis garCeva erToblivi ZalisxmeviT unda moxdes. moswavleebi warmoadgenen mosazrebebs TiToeuli amocanis Sesaxeb. am dros saswavlo procesSi CarTuli unda iyos yvela moswavle da yvela maTganisaTvis gasagebi unda iyos yoveli amocanis amoxsnis procesi. SeiZleba araerTxel miubrundeT xolme amoxsnis ama Tu im etaps. SemoTavazebuli amocanebi kvlav ubrundeba ricxvebis Caweris aTobiTi sistemis arss, Tanrigebis klasebad dayofis wess. maTi amoxsnis procesSi moswavles uviTardeba logikuri azrovnebis, msjelobis unari. mas uwevs SesaZlo variantebis aRmoCena, uviTardeba kombinatoruli azrovneba. ajobebs, rom muSaobis jgufur formas mimarToT, amoxsnis variantebi ki klasSi sajarod ganixileT. gamorCeulad aRniSneT muSaobaSi warmatebulebi, saintereso ideis, varaudis avtorebi. 1 Tu rvaniSna ricxvis Tanrigebs klasebad davyofT, moswavle advilad SeniSnavs, rom pirveli 4-iani warmoadgens 40 milions. ninikos unda warmoeTqva 4-iT dawyebuli orniSna ricxvis dasaxeleba, saxeldobr, 45 milion Svidas... sazogadod yoveli 8-niSna ricxvis wakiTxva iwyeba orniSna ricxvis warmoTqmiT.

pirobiT, aTeulebis cifri 6-jer metia erTeulebis cifrze, amitom is SeiZleba iyos mxolod 6, erTeulebis cifri ki aris 1, aseulebis cifri 2-jer naklebia aTeulebis cifrze _ 3-is tolia. pasuxi: 361. 2

3

ori momdevno naturaluri ricxvi SeiZleba bolovdebodes: 0-iTa da 1-iT _ namravli dabolovdeba 0-iT; 1-iTa da 2-iT _ namravli dabolovdeba 2-iT; 2-iTa da 3-iT _ namravli dabolovdeba 6-iT da a. S. SeiZleba CamovweroT ricxvebis yvela SesaZlo daboloeba da mezobeli ricxvebis namravlis bolo cifri iqve mivuTiToT: 2

0

28

1 0

2 2

3 6

0 4

5 0

6 6

7 2

0 8

9 2

0


saidanac Cans, rom namravli dabolovdeba 0, 2 an 6-iT. es sqema moswavleTaTvis TvalsaCino da misawvdomia. SeiZleba asec imsjelon: ori momdevno naturaluri ricxvidan erTi luwia, erTi _ kenti, e. i. namravlic luwia. amasTanave, Tu kenti Tanamamravli 1-iT bolovdeba, namravli bolovdeba 0-iT an 2-iT (1-iT daboloebuli ricxvis mezoblebi bolovdeba 0-iT an 2-iT); Tu kenti Tanamamravli 3-iT bolovdeba, namravli bolovdeba 6-iT an 2-iT (2â&#x2039;&#x2026;3=6, 3â&#x2039;&#x2026;4=12); Tu kenti Tanamamravli 5-iT bolovdeba, namravli bolovdeba 0-iT; 7-iT daboloebuli ricxvis namravli mezoblebze bolovdeba 2-iT an 6-iT, 9-iT daboloebulis _ 2-iT an 0-iT. klasSi ganxilvisas SeiZleba SeadaroT es ori midgoma da moswavleebma Tavad SearCion ufro xelsayreli. 4 pirveli ori cifri SeiZleba iyos 1 da 0, 1 da 1, 2 da 0, an 3 da 0. yvela sxva SemTxvevaSi aseulisa da aTeulis TanrigebSi miRebuli ricxvi 9-ze metia, es ki SeuZlebelia. e. i. saZiebeli ricxvi SeiZleba iyos 101123, 112358, 202246 an 303369. 6 pirobis Sesabamisad, aseulisa da aTeulis Tanrigebis cifrebi tolia, amasTanave, maTi jami ar aRemateba 6-s _ erTeulebis cifri am jamze xom 3-iT metia. am pirobebs akmayofilebs ricxvebi: 115, 227 da 339. mesame pirobiT, samive cifris jami 9-ze naklebia; aseTia mxolod 115.

aTeulebis cifri 2-iT naklebia aseulebis cifrze. amitom is ar unda aRematebodes 7-s. amasTanave, es cifri 5-iT metia erTeulebis cifrze (romelic 0 ar aris _ is xom umaRlesi Tanrigis cifris tolia), amitom is 5-ze metia. miviReT, rom aTeulebis cifri SeiZleba iyos 6 an 7. Sesabamisad, xuTniSna ricxvs eqneba 1*861 an 2*972 saxe, sadac *-is nacvlad SeiZleba Caiweros nebismieri cifri: 10861, 11861, 12861, ... _ 10 SesaZlebloba, 20972, 21972, 22972, ... _ 10 SesaZlebloba. sul aseTi xuTniSna ricxvebis odenobaa 20. moswavleebs vavalebT Semdegi gakveTilisTvis warmoadginon saxelmZRvaneloSi mocemuli proeqtis Sesrulebis angariSi. proeqti planetebisa da msoflios mosaxleobis Sesaxeb miznad isaxavs, moswavles ganuviTardes cnobarebis (maT Soris internet-cnobarebis) gamoyenebisa da moZiebuli informaciis daxarisxeba-warmodgenis unari. man unda daadginos, misTvis saintereso informaciis mosapoveblad romeli cnobari an internet-gverdi gamoiyenos, da Semdeg SeZlos aRniSnuli wignis Tu veb-gverdis gamoyeneba. magaliTad, udidesi qveynebis monacemebis moZieba, garda geografiis saxelmZRvaneloebis, enciklopediuri leqsikonebisa da sxva msgavsi cnobarebisa, SeiZleba internet-gverdebze: http://en.wikipedia.org/wiki/list_of_countries_by_population (inglisur enaze); http://ka.wikipedia. org (qarTul enaze). planetebis ricxviTi maxasiaTeblebi SeiZleba moiZion misamarTze: http://www.kidcyber.com.au/topics/planets.htm (inglisur enaze); http://ka.wikipedia.org (qarTul enaze). mravalferovan informacias gvTavazobs Google-sistema. 7

29


§2. ra viciT wiladebis Sesaxeb? Pparagrafs oTxi gakveTili eTmoba. me-4 gakveTili

Tema: wiladis sxvadasxva aspeqti mizani: wiladis sxvadasxva aspeqtis gameoreba. moswavles SeeZlos wiladis gamosaxulebaSi mricxvelisa da mniSvnelis, mTeli da wiladi nawilebis miTiTeba, wiladis gamosaxva ukveci formiT (maT.VI.1). winapirobebi: warmodgena wiladebze, rogorc mTelis nawilze, erTobliobis na­ wilze, ricxvebis gayofis Sedegze. wiladis ZiriTadi Tviseba. resursebi: geometriuli figurebis modelebi, rveuli, saxelmZRvanelo, dafa. proeqtis SesrulebiT sabolood vafasebT wina gakveTilebze dasaxuli miznebis, standartis programiT gansazRvruli Sedegebis miRwevas. yuradRebiT gadaxedeT moswavleTa naSromebs. moiwoneT maTgan saukeTesoebi da aRniSneT, ra iyo dadebiTi maTSi. es kargi miTiTeba iqneba maTTvis, vinc ver gaarTva Tavi davalebas. Semdeg gadavdivarT paragrafSi warmodgenil sakiTxebze. maTi axsnis procesSi (wiladebis sxvadasxva aspeqtis ganxilvisas) SeiZleba gamoviyenoT damxmare resursebi. magaliTad, marTkuTxedis models vyofT oTx tol nawilad, gamovyofT maTgan 3-s, wris 6 models Tanabrad vanawilebT or merxze, TiToeulze 3 aRmoCndeba – 6:2=3. Tu or `wres~ vanawilebT Tanabrad 3 merxze, maSin TiToeulis sam nawilad danawilebis Semdeg yovel calkeul merxze aRmoCndeba wris 23 . wiladebis sxvadasxva aspeqtis warmoCena magaliTebisa da amocanebis saSualebiT xdeba. `a~ punqtSi warmodgenili suraTebiT moswavle wilads gaiazrebs, rogorc mTelis nawils, ixsenebs wiladis Caweris formas _ mniSvneli miuTiTebs, ramden tol nawiladaa dayofili mTeli, mricxveli _ am nawilebidan ramdenia aRebuli. 5 nawilis damxmare kiTxva: ramden tol nawilad jobs davyoT wre, Tu misi 12 gaferadeba gvsurs? cxadia, moswavleebi am SemTxvevaSi wres 12 tol nawilad dayofen da maTgan 5-s gaaferadeben. SeiZleba gavaferadebinoT amave wris (12 nawilad dayofilis) 14 , 34 , 13 , 23 , 12 , 16 , 56 nawilebi. wiladis, rogorc mTelis nawilis gaazrebisTvisaa gankuTvnili 1 - 5 savarjiSoebi. `b~ punqtSi moyvanili magaliTiT moswavleebi wilads, rogorc erTobliobis nawils ganixilaven. 17 aris moswavleTa odenobis _ 25-is 17 25 . am klasSi biWebi klasis 17 nawilia. biWebisa da gogonebis odenobaTa Sedarebidan SeiZleba wiladebis Sedar25 8 ebazec gadaxvideT. moswavleebi miiReben: 17 25 > 25 . `g~ punqtSi wiladi warmodgenilia, rogorc gayofis Sedegi. Tu or magidaze 6 xaWapurs Tanabrad vanawilebT, TiToeulze 6:2=3 xaWapuri aRmoCndeba. ori xaWapuris sam tol nawilad ganawilebis SemTxvevaSi TiToze 2:3= 23 xaWapuri aRmoCndeba. suraTebi am faqts ufro TvalsaCinosa da misawvdoms xdis. aqve, gavarjiSebis mizniT, ramdenime ganayofs CavwerT wiladis saxiT da piriqiT, wiladebs warmovadgenT ganayofis saxiT. magaliTad, 2:7= 27 , 4:6= 46 , 35 =3:5 da a. S. naturaluri ricxvic SeiZleba Caiweros wiladis saxiT. marTlac, 7=7:1= 71 , 7=21:3= 21 3 da a. S. am sakiTxs d) punqtSic SevexebiT.

30


`d~ punqtSi wiladebis ZiriTad Tvisebas vaxsenebT. gavixsenoT gayofis Tviseba: Tu gasayofs da gamyofs erTsa da imave ricxvze gavamravlebT, ganayofi ar Seicvleba. Tu wilads SevxedavT, rogorc gayofis Sedegs (mricxveli iyofa mniSvnelze), maSin wiladi ar Seicvleba, Tu mricxvelsac da mniSvnelsac erTsa da imave ricxvze gavamravlebT. wiladis es Tviseba suraTebiTaa demonstrirebuli, marTkuTxedis 34 da 68 nawilebis gaferadebis magaliTze. 34 miiReba 68 -is mricxvelis da mniSvnelis erTsa da imave ricxvze, kerZod, 2-ze gayofiT, anu, 68 -is 2-ze SekveciT. pedagogebs SevaxsenebT, rom wiladis am Tvisebidan gamomdinareobs proporciis Tviseba: Tu ab = dc , maSin ad=bc. cb marTlac, Tu ab  =  dc , maSin ad bd  =  bd , saidanac ad=cb (Tu toli wiladebis mniSvnelebi tolia, maSin mricxvelebic tolia). es Tviseba teqstSi dasabuTebis gareSe _ magaliTebiTaa demonstrirebuli. `e~ punqtSi moswavleebi kvlav suraTebis meSveobiT ixseneben Sereuli ricxvis Caweris wess, wakiTxvas, mis Caweras arawesieri wiladis saxiT da, piriqiT, arawesieri wiladis Sereuli ricxvis saxiT Caweras. suraTze 11 meoTxedia ( 11 4 ) gaferadebuli, anu 3 mTeli da 3 4. rogor gadavweroT Sereuli ricxvi 5 23 wiladis saxiT? 1 mTeli ramdeni mesamedia? 2 15 2 17 ( 33 ). 5 mTeli ramdeni mesamedia? ( 15 3 ). e. i. 5 3 = 3 + 3 = 3 . moswavleebs vTxovoT, Camoayalibon (aRweron) Sereuli ricxvis wiladis saxiT Caweris wesi: mTelis da mniSvnelis namravls emateba mricxveli da Sedegi iwereba mricxvelSi, mniSvneli ucvlelad gadadis. rogor CavweroT arawesieri wiladi Sereuli ricxvis saxiT? klasSi jer raime nimuSi ganixileT. 13 6 magaliTad, mimarTeT moswavleebs, rom 13 6 Caweron Sereuli ricxvis saxiT: 6 = 6 + + 66 + 16 =2+ 16 =2 16 . wiladis Sereuli ricxvis saxiT Cawerisas mricxvels vyofT mniSvnelze, miRebuli ganayofi aris mTeli nawili, naSTi _ mricxveli, mniSvneli ki ucvlelad gadadis. saklaso da saSinao davalebebis moculoba saSualebas iZleva, rom paragrafs daeTmos 2 gakveTili. es masala iZleva adre naswavlis gameorebis, ganmtkicebis karg saSualebas. Tu SeniSnavT, rom klasi swrafad ibrunebs maRal akademiur dones, SeiZleba aRar ganixiloT amocanebis nawili. daitoveT mxolod klasisTvis aqtualuri sakiTxebi. klasSi vxsniT 1 - 25 amocanebs. saxelmZRvaneloSi warmodgenili teqstis mixedviT wiladis yvela aspeqtis dawvrilebiTi garCevis Semdeg SeiZleba es amocanebi (`testebSi~ swori pasuxebis SerCeva) moswavleebma damoukideblad amoxsnan. gansakuTrebuli yuradReba dauTmeT me-6 klasis standartiT moTxovnili miznebis dafarvas – am moTxovnebis Sesabamisia 18 , 19 (wiladis ukveci formiT warmodgena), 22 (wiladis gamosaxulebaSi mTelis gamoyofa), 16 (wiladebis Sedareba). 2 amocanis amoxsnas wris, an marTkuTxedis modelis eqvs tol nawilad dayofa TvalsaCinos gaxdida. 20

savarjiSoSi moswavleebi kidev erTxel gaixseneben, rom arawesieri wiladis mricxveli ar aris naklebi (≥) mniSvnelze, xolo 21 savarjiSoSi gamoiyeneben am faqts. pirobiT, 5≥n da n≥4, amitom n=4 an n=5.

31


23 savarjiSoSi SeiZleba gamoiyenon e. w. jvaredini gamravlebis wesi: 40⋅x=60⋅20, saidanac x=1200:40, x=30. amave Sedegs ufro martivadac miiReben, Tu imsjeleben ase: mricxvelebi gviCvenebs, rom 40 60 Seikveca 2-ze, e. i. x=30. 1

saSinao davalebas SevarCevT

- 17

amocanebidan.

me-5 gakveTili

Tema: wiladebi. mizani: wiladis sxvadasxva aspeqtis Sesaxeb codnis ganmtkiceba. winapirobebi: wiladis mTeli da wiladi nawilis codna, wiladis ZiriTadi Tviseba, resursebi: brtyeli geometriuli figurebis modelebi, rveuli, saxelmZRvanelo, dafa. gakveTilis sawyis 10-15 wuTs vuTmobT saSinao davalebis Semowmebas. amjerad igi SeiZleba swrafad CavataroT, radgan, faqtobrivad es masala wina wlis masalis gameorebaa, xolo davalebebSi saWiro iyo swori pasuxebis SerCeva da amocanebi exeboda wiladis sxvadasxva aspeqtis gaazrebas, romelsac wina gakveTilze didi dro davuTmeT. saSinao davalebis

1

da

2

amocanebSi oTxi savaraudo pasuxidan sami tol

wiladebs warmoadgens. vamowmebT maTgan gansxvavebuls da vRebulobT pasuxs: _ amocanaSi a) da

2

saSinao davalebis

1

amocanaSi d) 3

- 7

amocanebis amoxsnis gacnobiT Tqven daadgenT, wila-

dis sxvadasxva aspeqti rogor gaiazres moswavleebma. 9

- 10 amocanebSi moswavleebma unda gamoavlinon naturaluri ricxvis wiladis saxiT Caweris codna da pasuxs advilad ipovian. 8 amocanaSi wiladis mTeli da wiladi nawilis povnaa saWiro da es amocana maT VI. 1 Sedegs ukavSirdeba. maT. VI 1.-Tan aris dakavSirebuli

12 - 17

amocanebic – wiladis gamartiveba (ukveci wiladis saxiT Cawera) da wiladebis toloba. 14 amocanaSi, imis gamo, rom n5 da 6n arawesieri wiladebia, n meti unda iyos 5-ze, an _ 5-is toli da naklebi 6-ze, an _ 6-is toli. aseTi ricxvebi ki oria, 5 da 6 (pasuxia d)). wiladis ZiriTadi Tvisebebis gamoye3 30 3 nebasTan aris dakavSirebuli x-is povna tolobidan 30 90  =  x . gaviTvaliswinebT, rom 90  =  9 . Semdeg ki toli wiladebis SedarebiT vRebulobT x=9. Semdeg gadavdivarT 26 - 36 amocanebisa da `moisazres~ rubrikiT warmodgenili 1 da 2 amocanebis amoxsnaze. es amocanebic wiladebis codnas ukavSirdeba. am amocanebis amoxsnisas SeiZleba moswavleebs davexmaroT damxmare kiTxvebis dasmiT. am kiTxvebis Sedgenis magaliTi 26 amocanaSia warmodgenili. cxadia, suraTidanac kargad Cans, rom AE monakveTi aris AB monakveTis 35 , xolo AB monakveTis 25 aris AD, CE, DF da EB monakveTebidan TiToeuli monakveTebi. 27 – ramden dReSi daamTavrebs kedlis amoyvanas kalatozi? (12 dReSi).

– ramdeni dRe moundeba naxevari kedlis amoyvanas? (6 dRe).

32


6. aq wiladis ZiriTadi Tvisebis gamoyenebiTac SeiZleba visargebloT: 12 = 12 29 am amocanaSi SekiTxvebi aRmavali sirTuliT aris dalagebuli. pirvel SekiTxvaze pasuxs zogierTi moswavle sinjvis gziTac daadgens (12 vaJi). moswavlem igive odenoba (12) unda aRiqvas, rogorc mTeli erTobliobis (aq _ 28-is) nawili _ 12 28 . mesame SekiTxvaze pasuxis povna SeiZleba ori gziT _ gogona 16-ia, amitom maTi wili 16 -ia, an ase: radgan mTeli aris 28 da vaJebi 12 , amitom danarCeni, anu 16 gogonebia. 28 28 28 28 amoxsnis meore gzis warmomdgeni gansakuTrebul waxalisebas imsaxurebs. 30 SeiZleba klasSi vaJebis odenoba x-iT aRvniSnoT, gogonebisa _ 3x-iT. vaJebi x = 1 nawilia (wiladi nulisgan gansxvavebul x-ze SevkveceT), goklasis, anu 4x-is 4x 4 3 3x gonebi _ 4x = 4 nawili. zogierTi, SesaZloa, TvalsaCino warmodgenebsac gamoiyenebs. magaliTad, aqedan iolad aRmoaCens, rom vaJebi klasis meoTxeds Seadgenen. 2 31 pirobis mixedviT, 24 km aris mTeli gzis 3 nawili, 12 km _ mTeli gzis mesamedia, mTeli gzis sigrZe 36 km-ia. ufro TvalsaCinoa sqemis gamoyeneba suraTidan Cans, rom mTeli gza 3⋅12=36 km-ia. 32 zogierTi SeiZleba amoixsnas jvaredini gamravlebis wesiT, zogierTi _ wiladis ZiriTadi Tvisebis gamoyenebiT. magaliTad, a) 8n=6⋅4, 8n=24, n=3; 14n=10⋅70, saidanac n=50, b) 39 = n3 . mniSvnelebis ganxilva gvaZlevs _ wiladi 3-zea Sekvecili, e. i. n=1. 34 b) jer SevkvecoT da Semdeg CavweroT 100-is mniSvneliani wiladis saxiT: 12 = 4 = 80 , 24 = 4 = 16 , 33 = 11 = 44 , 14 = 7 = 35 , 6 = 1 = 10 . 15 5 100 150 25 100 75 25 100 40 20 100 60 10 100 x zogierTebi am amocanebs jvaredini gamravlebis wesiTac iolad amoxsnian: 12 25 =100; 25x=1200. 36 SeiZleba es ricxvebi ganvalagoT (miaxloebiT) ricxviT RerZze da kiTxvebze pasuxi suraTis mixedviT gavceT: es amocana aviTarebs sidideebis Sefasebis unars. 12 bevrad naklebia 1500-ze, ami12 axlosaa 0-Tan. 1 =151, 153 axlosaa 1 -Tan. 285-s erTamde, 32 aklia, rac cxatom 1500 2 302 302 2 317 317 dia, 14 -ze naklebia, e. i. es ricxvi 1-Tan ufro axlosaa, vidre 12 -Tan. am msjelobebisken moswavleebs Cven unda `vubiZgoT~. magaliTad, vTxovoT Ta12 naklebia 1 -ze, xolo danarCeni ori wivdapirvelad ricxvebis 12 -Tan Sedareba ( 1500 2 ladi metia 12 -ze). Semdegi ori wiladidan ki im wiladis SerCeva, romelic 12 -Tan ufro `axlosaa~ (an 1-Tan ufro `axlosaa~). amis Semdeg SeiZleba moswavleebis nawils vTxovoT 28 da 33 amocanebis amoxsna damoukideblad, danarCenebs (maRali mzaobis) moswavleebs ki 1 da 2 amocanebis miTiTebebis gacnoba da am miTiTebebis saSualebiT amoxsnis gzebis Zieba.

33


rubrika `moisazreSi~ SemoTavazebuli amocanebi warmodgenili miTiTebebis gamoyenebiT `ukusvlis~ xerxis gamoyenebiT ixsneba. 1

amocanaSi miTiTebis mesame suraTidan Cans, rom II dRis Semdeg darCenili iyo 2(11+1)=24 tomara. meore suraTi miiRebs saxes:

I dRe

2

I dRis Semdeg darCenili

saidanac Cans, rom I dRis Semdeg darCenili iyo 2(24+1)=50 tomara. I suraTis Sevsebis Semdeg cxadi xdeba, rom Tavdapirvelad Setanili tomrebis odenobaa 2â&#x2039;&#x2026;(50+1)=102.

SevadginoT sqema:

III suraTidan Cans, rom 18 aris II dRis Semdeg darCenili kartofilis 2 , 9 aris 1 nawili, e. i. II dRis Semdeg darCa 27 tomara. SevavsoT II suraTi, 3 3 saidanac davadgenT, rom 28 aris I dRis Semdeg darCenili tomrebis 23 , 14 aris 13 nawili, e. i. I dRis Semdeg darCa 42 tomara, I suraTi miiRebs saxes, saidanac Cans, rom 44 aris mitanili tomrebis 2 , 22 aris 1 nawili, e. i. mitanili iyo 66 tomara. 3 3 saSinao davalebas virCevT

18 - 27

amocanebidan.

me-6 gakveTili

Tema: wiladebi. mizani: wiladebis Sesaxeb Seswavlili sakiTxebis gameoreba, codnis ganmtkiceba, moswavles SeeZlos ricxvis nawilis povna da Sebrunebuli amocanis amoxsna (maT. VI. 2). winapirobebi: wiladis ZiriTadi Tviseba, ricxvis mTeli da wiladis nawilebi. resursebi: brtyeli geometriuli figurebis modelebi, rveuli, saxelmZRvanelo, dafa. vamowmebT saSinao davalebas (15-20 wuTi). 18 - 20 amocanebis SesrulebiT SeiZleba Sefasdes moswavleTa codna wiladis ZiriTadi Tvisebis gamoyenebis Sesaxeb. amjerad, saSinao davalebis Semowmebisas, SeiZleba SemovifargloT imis SemowmebiT, xom ar gauWirdaT moswavleebs romelime amocanis amoxsna da gamovarkvioT, wiladebis ra Tvisebebi gamoiyenes moswavleebma amocanebis amoxsnisas. amasTanave, SeiZleba davukonkretoT sakiTxi, magaliTad, â&#x20AC;&#x201C; wiladis ra Tviseba gamoiyeneT 18 - 20 da 26 amocanebis amoxsnisas? SeamowmeT, xom ar aris saxelmZRvaneloSi gakeTebuli Canawerebi, magaliTad, 25 amocanis Sesrulebisas. moswavleebs SeiZleba SevTavazoT, 23 amocanaSi dasmuli davaleba qaRaldisgan gamoWrili marTkuTxedebiTac Seasrulon.

34


mosalodnelia, rom wina amocanis magaliTze moswavleebi 2 14 -s warmoadgenen wreebis saxiT:  , saidanac TvalnaTliv Cans, rom 2 14 = 94 . 24

25

a)-Si warmodgenilia 12 -is toli wiladebi, romelTa mricxveli TiTo-TiToTi

izrdeba; b) 38 -is toli wiladebi, romelTa mricxvelebi erTmaneTis momdevno 3-is jeradi ricxvebia: 3, 6, 9, 12 da a.S. 5 -is toli wiladebi. g) 45 -is toli wiladebi, d) 12 27

ricxviT sxivze wiladebis gamosaxvis Semdeg viTvaliswinebT, rom metia is ricxvi, romelic sxivze ufro marjvnivaa, g), d) da e) SemTxvevaSi SesaZloa wiladebi jer gavaerTmniSvnelianoT, Tumca es ar aris aucilebeli. savaraudod, moswavle am savarjiSos Sesrulebisas sxivis erTeuls 8 tol nawilad dayofs _ gamosaxavs mervedebs. magaliTad, d) SemTxvevaSi: 1< 3. , 4 8

klasSi vxsniT 37 - 42 amocanebs. Zalian mniSvnelovania im amocanebis dawvrilebiT amoxsna, romlebic ricxviT sxivze wiladebis gamosaxvas, wiladebis Sedarebas exeba. (adre amoxsnili 36 , 37 ). 41 amocana SeiZleba davukavSiroT maT. VI. 2, Sedegs _ ricxvis nawilis povna. 38 da 39 amocanebis davalebebidan TiTo davaleba Cveni monawileobiT unda Namoixsnas, danarCenebi moswavleebs mivceT damoukidebeli muSaobisaTvis. 1 5 1 4 20 9 42 amocanaSi gasaTvaliswinebelia, rom 4 = 20 , 5 = 20 , 1= 20 , maSasadame, aivso 20 nawi11 nawili. li, darCa asavsebi 20 37 suraTze erTeuli ukve dayofilia 8 tol nawilad, SeiZleboda es Tavad moswavles gaekeTebina (mimxvdariyo, rom saWiroa erTeulis 8 tol nawilad dayofa, 8 aris CamoTvlili wiladebis mniSvnelebis umciresi saerTo jeradi). ricxvebis sxivze ganlagebisas unda gaviTvaliswinoT, rom 14 = 28 , 12 = 48 , 34 = 68 . 39 a) Tanamamravli `Sekvecilia~ cifrTan (!). am wiladis gamartivebiT unda migve­ 49 = 7⋅7 = 7 ; Ro: 9⋅7 9⋅7 9 b) `Sekvecilia~ cifrebi; g) `Sekvecilia~ Sesakrebi; wiladi ar icvleba mricxvelis da mniSvnelis erTsa da imave nulisgan gansxvavebul ricxvze gayofiT an gamravlebiT. d) da e) magaliTebSi ki mricxvels da mniSvnels akldeba erTi da igive ricxvi. am SemTxvevaSi wiladi icvleba, sazogadod. 40 moxerxebulia pirobis mixedviT sqemis Sedgena: Tu danarCen meoTxedebsac davyofT sam-sam tol nawilad, TvalsaCino gaxdeba, rom 1 nawili. TiToeuli nawili aris mTelis 12 41 I dRes gadmotvirTes 600 t, II dRes _ darCenili 1200 t-is nax-

evari, anu kvlav 600 t. darCa gadmosatvirTi 600 t, rac mTeli xorblis 13 -ia.

35


SeiZleba asec imsjelon: II dRes gadmotvirTes darCenilis, anu 23 -is naxevari _ 13 . darCa gadmosatvirTi 13 nawili, anu 600 t. saSinao davalebisaTvis SeiZleba gamoviyenoT 28 - 35 amocanebi.

me-7 gakveTili

Tema: wiladebi. mizani: moswavles SeeZlos ricxvis nawilis povna da Sebrunebuli amocanis amoxsna (maT. VI., 2), wiladebis ZiriTadi Tvisebis gamoyeneba wiladebis gardaqmnis, Sedarebisa da Sefasebisas (maT. VI. 1). winapirobebi: wiladis ZiriTadi Tviseba; wiladis mTeli da wiladi nawili. resursebi: brtyeli geometriuli figurebis modelebi. ricxvis nawilis povnasTan dakavSirebuli amocanebi adrec araerTxel ganvixileT ( 14 , 31 , 41 , 5 , 11 , 28 ). magaliTad, 14 amocanaSi 36-is 23 unda vipovoT. rad2 gan 36-is 1 3 aris 12, 36-is 3 iqneba 24. misi Sebrunebuli amocanis amoxsna gvxvdeba 31 amocanaSi: turistebma gzis 13 nawili gaiares, darCa 24 km. e. i. 24 km aris gzis 23 , gzis 1 iqneba 12 km, mTeli gza – 12⋅3=36 km. 3 Cven kidev erTxel SemogTavazeT standartis am moTxovnasTan dakavSirebuli amocanebis amoxsnis meTodika swavlebis am etapze. sasurvelia, Tu yvela am amocanas xelmeored ganvixilavT da moswavleebi gaiwafebian analogiuri amocanebis amoxsnaSi, miT umetes, Tu maT saxlSi gauWirdaT 28 amocanis amoxsna. saSinao davalebis sxva amocanebi ki wiladebis sxvadasxva aspeqtisa da wiladis ZiriTadi Tvisebis codnas ukavSirdeba. davalebis Semowmebas SeiZleba movandomoT 15-20 wT. 48 km aris gzis 34 nawili, 16 km _ 14 nawili. mTeli gzis sigrZe 16⋅4=64 km-ia. SeiZleba sqemac gamoviyenoT: 1 nawili 16 km-ia, mTeli gza _ 4 nawili _ 64 km. 2 4 8 32 da 33 amocanebze muSaobisas suraTis mixedviT weren: 1= = = . 2 4 8 erTi marTkuTxedi Sedgeba ori naxevrisgan, oTxive marTkuTxedi Sedgeba 8 naxevrisgan, anu 4= 82 erTi marTkuTxedi 4 meoTxedisgan Sedgeba, oTxive marTkuTxedi _ 32 16 meoTxedisgan, 4= 16 4 . analogiurad, 4= 8 da a. S. momdevno amocanebSi mTelis wiladad Caweras gamoiyeneben. 28

7 nawili. am gamoTzurabma sandros da giorgis misca mTeli Tanxis 29 + 16 = 18 vlebisTvis moswavleebs mouwevT 29 da 16 wiladebis iseT wiladebad Cawera, romelTa 4 , 1 = 3 . Tu gaixseneben, mniSvnelebi tolia. aseT ricxvebs Soris umciresia 18: 29 = 18 6 18 18 7 18 rom 1= 18 da 18 < 18 , daaskvnian, rom zurabs fuli darCa. SeiZleba daazuston kidec 11 nawili. _ zurabs darCa mTeli Tanxis 18 34

36


35

sqemis gamoyenebiT iolad SeamCneven, rom II dRes gavlili manZili mTeli gzis

1 -ia. ajobebs, Tu yovel wilads Caweren iseTi wiladis saxiT, romlis mniSvneli 12-ia 6 6 , da ase gamoTvlian turistis mier sam dReSi gavlili gzis nawili. I dRe _ 12 = 12 2 , III dRe _ 1 = 4 . sul gaiara: 12 =1. sam dReSi turistma gaiara mTeli gza. II dRe _ 16 = 12 3 12 12 gakveTilis dinamikurad da nayofierad warmarTvis mizniT, maSin, rodesac Cven ufro dawvrilebiT vaanalizebT davalebis amocanebs ricxvis nawilis povnaze, wiladebis gamartivebaze da Sedarebaze im moswavleebTan, visac aseTi daxmareba sWirdeba, amavdroulad SedarebiT maRali `mzaobis~ moswavleebs vavalebT `moisazres~ rubrikiT mocemuli amocanebis amoxsnas. roca isini warmoadgenen TavianT mosazrebebs, Cven vexmarebiT amoxsnebis bolomde miyvanaSi, Tu amis saWiroeba warmoiSva. 1 - 3 amocanebi wiladis, rogorc mTelis nawilis gaazrebasa da sqemebis gamoyenebaSi gawafvas emsaxureba. 1

suraTze wakiTxuli nawilia daStrixuli. cxadia, rom mTeli wignis meoTxedi 16 gverdia, mTeli wigni 16⋅4=64 gverdisgan Sedgeba. 2

sqemaze warmodgenilia amocanis piroba, wignis wakiTxuli nawili daStrixulia. suraTidan Cans, rom 14 gverdi aris 25 nawili, 7 gverdi _ 15 nawili. mTeli wigni 7⋅5=35 gverdisgan Sedgeba. erTi traqtoris 8 dRis muSaobis Semdeg darCenil 8 dRis samuSaos ori traqtori 4 dReSi daasrulebs (yoveli maTgani darCenili 8 dRis samuSaos naxevars _ 4 dRis samuSaos asrulebs). SemoerTebulma traqtorma 16 dRis samuSaodan 4 dRis sa4 = 1 nawili Seasrula. muSao, anu, mTeli samuSaos 16 4 me- 4 da me- 5 amocanebi logikuri azrovnebis unaris ganviTarebisTvisaa gankuTvnili. 3

damxmare kiTxva: 5 rveulis yidvis Semdeg darCenil 40 TeTrs ramdeni unda daamatos Teonam, rom kidev erTi rveuli iyidos? 20 TeTri, e. i. rveuli Rirs 40+20=60 TeTri. 4

5 CavasxaT 7-jer 5-litriani WurWliT 35 litri wyali da amoviRoT 12-litrianiT

12 litri. saSinao davalebad mivceT I Tavis damatebiTi amocanebi 1 - 10 .

me-8 gakveTili

jgufur muSaobas win uZRvis saSinao davalebis Semowmeba da ganxilva. jgufuri muSaoba am aqtivobis gansaxorcieleblad klasi davyoT jgufebad. TiToeul jgufSi eqvsi­ moswavle CavrToT. am jgufuri muSaobis mizani moswavleebis mniSvnelovani unar-Cvevebis ganviTarebaa (samuSaos gonivruli dagegmva da ganawileba, TanaklaselebTan TanamSromloba, varaudis gamoTqma, sakuTari da sxvaTa namuSevris kritikuli Sefaseba, dakvirvebuloba, ganzogadeba, naazrevis gadmocema _ Camoyalibeba, gaRrmaveba da ganmtkiceba). am miznis gansaxorcieleblad viyenebT mimdinare saswavlo masalas _ wiladebs.

37


1 davalebis Sesasruleblad Tqven SeiZleba urCioT, rom jgufi sam nawilad ganawildes, TiToeulma warmodgenili mimdevrobebidan erT-erTi ganixilos da upasuxos dasmul SekiTxvebs, izrunos sxvaTa davalebebis gadawyvetazec. samuSaos dasrulebisas miRebuli Sedegebi unda Sejerdes _ moswavleebs unda axsovdeT, rom warmatebasac da warumateblobasac jgufis wevrebi Tanabrad inawileben. 4 , ... I. 12 , 24 , 38 , 16 a) yoveli momdevno wiladis mricxveli aRemateba winas 1-iT; yoveli wiladis mricxveli am wiladis rigiT nomers gamosaxavs mocemul erTobliobaSi. b) yoveli momdevno mniSvneli winaze orjer metia. misi mniSvnelobaa 2n, sadac n wiladis rigiTi nomeria. 5 , 6 , 7 . g) 32 64 128 n-uri wiladi SeiZleba ase Caiweros 2nn . kerZod, am erTobliobaSi me-8 wevri iqneba 8 8 28 , anu 256 . 4 , ... II. 11 , 24 , 39 , 16 a) mocemul mimdevrobaSi yoveli wiladis mricxveli am wiladis rigiT nomers warmoadgens. b) yoveli wiladis mniSvneli amave wiladis mricxvelis kvadratia; 5 , 6 , 7 . g) 25 36 49 n-uri wiladia _ nn2 . is 1n -is tolia. 9 , 16 , ... III. 12 , 45 , 10 17 a) mricxveli _ wiladis rigiTi nomris kvadratia. b) mniSvneli mricxvels 1-iT aRemateba. 36 49 g) 25 26 , 37 , 50 . 2 n-uri wiladia: _ n2n+1 .

cxadia, moswavleebs ar moeTxovebaT n-uri wiladis formulis dawera, sakmarisia sityvierad aRweron mimdevrobebi. 2

jgufis erTma wyvilma moifiqros kanonzomiereba mricxvelebisTvis, meore wyvilma _ mniSvnelebisTvis, mesamem Seajeros miRebuli Sedegebi da Seadginos wiladebi. SesaZlebelia TiToeulma wyvilma moifiqros erTiani kanonzomiereba wiladebisT17 vis. magaliTad, 21 ; 58 ; 10 27 ; 64 . 2 cxadia, aq n-uri wevri moicema formuliT: n n+1 3 . maRal Sefasebas daimsaxureben gonebamaxvilurad Sedgenili erTobliobebis avtorebi da am kanonzomierebaTa aRmomCeni moswavleebi. jgufuri muSaobis damTavrebis Semdeg moswavleebs vavalebT am paragrafis darCenili amocanebis amoxsnas. moswavleTa SesaZleblobebis gaTvaliswinebiT SeiZleba SevarCioT ramdenime amocana Tavis damatebiTi amocanebidan. mocemuli gakveTilisTvis gankuTvnili saSinao davalebis amocanebi, ZiriTadad, wiladis ZiriTadi Tvisebis gamoyenebas moiTxovda. magaliTad, 34 cxra meTormetedsac 9 4 4 4⋅5 20 warmoadgens, vinaidan 34 = 3⋅3 4⋅3 = 12 ; 3 warmoadgens 20 meTxuTmeteds, vinaidan 3 = 3⋅5 = 15 .

38


3 45 wuTi aris saaTis 45 60 nawili, anu 4 nawili, vinaidan 1 saaTi=60 wT. 250 g aris kilo250 = 1 . gramis meoTxedi, vinaidan 1 kg=1000 g da 1000 4 saSinao davalebad SeiZleba mivceT I Tavis damatebiTi amocanebidan 11 - 15 amocanebi.

§3. viwyebT aTwiladebis Seswavlas me-9 gakveTili

Tema: aTwiladebi. mizani: moswavlem SeZlos aTwiladis gamosaxva wiladis saxiT, aTwiladis wakiTxva, Tanrigebis miTiTeba, cifris mniSvnelobis miTiTeba aTwiladSi (maT. VI. 1) da mzadeba maTi gamoyenebisaTvis. winapirobebi: wiladebis Cawera, wakiTxva, mricxvelisa da mniSvnelis miTiTeba, Sereuli ricxvis gadaqceva arawesier wiladad, arawesieris – Sereul ricxvad. resursebi: kvadratis formis qaRaldis furclebi, saxelmZRvanelo, rveuli, feradi fanqrebi, dafa. saSinao davalebis Semowmebas da mis ganxilvas SeiZleba didi dro ar moandomoT (10-15 wT). amocanebSi ZiriTadi aqcenti keTdeba wiladis ZiriTad Tvisebaze, wiladebis Sekvecaze, Sereuli ricxvis arawesier wiladad Caweris da arawesieri wiladis Sereuli ricxvis saxiT warmodgenis codnaze. yvela es sakiTxi gakveTilis winapirobebSia miTiTebuli da maTi codna aucilebelia dasaxuli miznis Sesrulebis saqmeSi. axali Temis ganxilvisas SeiZleba gamoviyenoT damxmare masala – kvadratis formis furclebi, romlebic dayofilia 100 tol `patara~ kvadratad. am patara kvadratebiT Sedgenili yoveli nawili (isini SeiZleba gaferadebiT gamovyoT) SeiZleba warmovadginoT aTwiladiT. aTwiladebis Seswavla mniSvnelovania im mxrivac, rom ricxvis cnebis gafarToeba namdvil ricxvamde arsebiTad ukavSirdeba ricxvis aTwiladis saxiT warmodgenas. mravalmxrivi informacia, romelsac moswavle yoveldRiurad Rebulobs, uxvad Seicavs aTwiladis saxiT Caweril monacemebs. es saukeTeso samotivacio faqtoria masalis efeqturad asaTviseblad. maswavlebelma unda gamoiyenos es viTareba. paragrafi samotivacio informaciiT iwyeba. am paragrafSi moswavleebi gaecnobian ricxvis Caweris axal formas _ aTwilads; daeuflebian zogierTi ricxvis Caweras aTwiladis saxiT. kerZod, jerjerobiT aTwiladis saxiT Caweren im wiladebs, romelTa mniSvneli aris 10-is xarisxi. axal cnebamde misvla etapobrivad xdeba. `a~ punqtSi jer meaTedebis Caweras da wakiTxvas daeuflebian. Semdeg measedebs Caweren da kiTxuloben, miuTiTeben meaTedisa da measedis Tanrigebis cifrebis mniSvnelobebs. `b~ punqtSi euflebian erTze meti ricxvebis aTwiladis saxiT Caweras, ganasxvaveben mTel da wilad nawilebs. `g~ punqtSi ki aTwiladad Caweren nebismier wilads, romlis mniSvneli 10-is xarisxia; moswavleebi iwafebian aTwiladad Cawerili ricxvebis wakiTx-

39


vaSi, gavarjiSdebian Tanrigis mixedviT cifris mniSvnelobis dasaxelebaSi, aTwiladi Tanrigebis mixedviT am aTwiladis jamis saxiT CaweraSi. punqtebis ganxilva moswavleTa aqtiuri CarTulobiT unda mimdinareobdes. maswavlebels am SemTxvevaSiU daxmarebas gauwevs saxelmZRvaneloSi warmodgenili kiTxvebis sistema. sasurvelia, rom yvela cxrili moswavleebma saxelmZRvanelodan rveulebSi gadaitanon, da, moTxovnis Sesabamisad, iq Seavson (magaliTad, Tanrigebis mixedviT ricxvebis Cawera). cxrilebis Sevsebis Semdeg moswavleebi kiTxuloben Canawerebs, magaliTad, a) punqtSi, wiladi – xuTi measedi, aTwiladi – nuli mTeli, 5 measedi. SesaZlebelia dafasTan gamoZaxeba da Canaweris iq warmodgena. SeiZleba maswavlebelma Tvali Seavlos moswavleTa rveulebs da Tu darwmunda, rom arcerTi cxrili sworad ar aris Sevsebuli, dafaze TviTon gaakeTos swori Canaweri. 9 da 10 testebis amoxsnamde, mivmarToT moswavleebs, gaixsenon, ramdeni TeTri Seadgens 1 lars. ra nawilia 1 TeTri 1 laris? _ measedi: 1 T=0,01 lari; analogiurad 1 m =0,001 km. 11 testSi ganayofs CavwerT wiladis saxiT, gamovyofT mTels da gadavwerT 5 aTwiladis saxiT: 15 10 =1 10 =1,5. 15 , 16 testebisTvis Sesamzadeblad moswavleebma unda gaixsenon, ramdeni meaTedisgan Sedgeba 1 mTeli (aTi meaTedisgan, 1= 10 10 ). 3 mTeli Sedgeba 30 meaTedisgan. 1 mTeli asi measedia, 3 mTeli _ 300 measedi. 1 - 16 testebSi swori pasuxebis miTiTebiT SeiZleba gakveTilze dasaxuli miznebis miRwevis donis Sefaseba. am process SeiZleba 10-15 wuTi davuTmoT da igi ase CavataroT: gamoZaxebuli moswavle kiTxulobs pirvel davalebas da pasuxobs – g). sxvebi Tvals adevneben gamoZaxebuli moswavlis pasuxs da akontroleben mas. mivmarTavT kidec sxvebs, rom gamoTqvan sakuTari mosazreba am pasuxis Sesaxeb. meore moswavle kiTxulobs meore `tests~ da pasuxobs – b) (da ara d), romelic nul mTelsa da 6 meaTaseds warmogvidgens). ase gagrZeldeba 16 testis CaTvliT. saSinao davalebad vaZlevT

1

- 14

amocanebs.

me-10 gakveTili

Tema: aTwiladebi. mizani: moswavlem SeZlos dasaxelebuli wiladis Cawera aTwiladis saxiT, aTwiladSi cifris mniSvnelobis miTiTeba (maT. VI. 1), aRmoaCinos aTwiladis gamoyenebis zogierTi upiratesoba. winapirobebi: wiladebi, Cawera, wakiTxva, mTeli nawilis gamoyofa. resursebi: kvadratis formis qaRaldis furclebi, fanqrebi, saxelmZRvanelo, kalami, rveuli. vamowmebT saSinao davalebas (15 wT). moswavleebi asaxeleben pasuxebs da akontroleben sxvaTa Sedegebs. siZneleebi SeiZleba warmoiSvas 7 amocanidan, vwerT gaSlili saxiT poziciur sistemaSi 0,723=0,7+0,02+0,003 (saTanrigo Sesakrebebis jamis saxiT).

40


8

5 erTeulSi 50 meaTedia, radgan 10 meaTedi aris 1 ( 10 10 =1).

9

725 aris 1000-is 0,725.

78 . aq SeiZleba vTxovoT moswavleebs, es wiladi aTwiladis 78 aris 1000-is 1000 saxiTac Caweron. 10

12

352 =0,352. 352 kg=0,352 t, radgan 1 kg aris tonis meaTasedi 1000

13 da 14 amocanebSiO moiTxoveba wiladebis da Sereuli ricxvebis aTwiladebis saxiT Cawera. am amocanebis amoxsnebis gaTvaliswinebiT, meti yuradReba davuTmoT `dabali~ mzaobis moswavleebTan 17 - 18 amocanebis amoxsnebs. am dros `maRali~ mzaobis moswavleebs vaZlevT damoukideblad muSaobisTvis 19 - 25 amocanebs. Semdeg isini Sedegebs mTel klass warmoudgenen. danarCen amocanebs klasSi erToblivad vxsniT. 19 amocanaSi cxrilis Sevseba SeiZleba moswavleTa mier SemoTavazebuli ricxvebiT ganvagrZoT, an piriqiT, moswavleebi gavavarjiSoT cxrilSi Cawerili ricxvebis wakiTxvaSi. 26 mivmarToT moswavleebs, gaixsenon masis erTeulebs Soris kavSirebi: 1 t=1000 1 tona=0,001 t, 1 c=0,1 t. kg, 1 c=100 kg, 1 t=10 c, saidanac 1 kg= 1000 600 =3,600 tona; Tu gaviTvaliswinebT, rom 6 6 centneri 600 kg-ia, e. i. 3 t 6 c=3 1000 600 wiladis 100c=0,6 t. miviRebT 3 t 6 c=3,6 t. anu 0,600=0,6 amave Sedegs miviRebT 1000 ze Sekvecis Sedegadac. 3 c 52 kg=352 kg, e. i. 12 t 3c 52 kg=12,352 t. analogiurad, 16 c12 kg=1t 612 kg=1,612 t. 29 es amocana SeiZleba gantolebis Sedgenis gareSec amoixsnas. 1000-s gamovakloT gogonebis Warbi odenoba. darCenili moswavleebi _ gogonebi da vaJebi Tanabari odenobisaa. (1000-96):2=452. skolaSi yofila 452+96=548 gogona, rac moswavleTa 0,548-ia. 30 moswavleebma ician, rogor Caweron aTwiladis saxiT is wiladebi, romelTa mniSvneli 10-is xarisxs warmoadgens, amitom Seecdebian mocemuli wiladebis Cawe­ ras sasurveli saxiT, risTvisac gamoiyeneben wiladis ZiriTad Tvisebas, magaliTad, 7 7⋅5 35 20 =20⋅5= 100 =0,35.

saSinao davalebad moswavleebs

15 - 26

amocanebs vaZlevT.

me-11 gakveTili

Tema: aTwiladebi. mizani: moswavles SeeZlos aTwiladis daSla saTanrigo Sesakrebebis jamis saxiT (maT. VI. 1.) winapirobebi: naturaluri ricxvis daSla saTanrigo Sesakrebebis jamis saxiT. resursebi: saxelmZRvanelo, rveuli, dafa. 16 amocanis amoxsniT (es amocana klasSi amoxsnili – 19 amocanis analogiuria) da misi ganxilviT SeiZleba kargi warmodgena SeviqmnaT miTiTebuli miznis Sesrulebis doneze. sxva amocanebidan gamosarCevia 17 da 26 amocanebi.

41


saSinao davalebis 17 savarjiSos amoxsnis Semdeg moswavle ufro sarwmunod miiCnevs, rom aTwilads aqvs Tviseba: mocemul aTwiladSi marjvnidan nulebis miweriT mocemulis toli aTwiladi miiReba. sasurvelia, iqve paraleli gavavloT naturaluri ricxvebisTvis marjvnidan nulis miwerasTan. am dros ricxvi 10-jer izrdeba. magaliTebi kidev ufro naTels mofens am faqtebs. 0,9=0,90=0,900=... 9≠90, 9=9,0=9,00=... 3 , anu 375 lari, Salvam miiRo 26 giorgim miiRo 500 lari, zurabma _ 500-is 4 1000–(500+375)=125 lari. SeiZleba nawilebSi kargad garkveulma moswavlem daaskvnas, rom zurabma aiRo naxevris 34 , anu Salvas darCa naxevris 14 nawili, 500-is 14 nawili 125 laria. vagrZelebT muSaobas wiladebis sxvadasxva aspeqtis gasacnobad, wiladis aTwiladad Cawerisa da aTwiladis wiladad Caweris amocanebze, aTwiladebSi cifris mniSvnelobis miTiTebaze. gTavazobT testebs daxuruli da Riaboloiani amocanebiT. maTi saSualebiT SeiZleba Semajamebeli werisaTvis moswavleTa Semzadeba. testi 1. SearCieT swori pasuxi: 1. 1 metris ra nawilia 24 sm? 24 a) 100 24 b) 10

6 g) 25

d) 10 24 .

64 SeiZleba SevkvecoT 2. 160 a) 5-ze

b) 10-ze g) 16-ze d) 20-ze.

3. 70-is ra nawilia 21? 3 b) 10 a) 10 3

x = 15 , maSin x= 4. Tu 12 36 a) 15 b) 5

42

7 g) 10

d) 37 .

g) 3

d) 6.

5. 1 saaTis 34 nawili aris a) 15 wuTi b) 30 wuTi

g) 45 wuTi

d) 75 wuTi

6. 25 wuTi aris saaTis 5 nawili a) 14 nawili b) 12

1 nawili g) 25

5 nawili d) 10

7. 25 = a) 2:5 b) 5:2

g) 2⋅5

d) 4:50.

8. 9= a) 39 b) 27 3

g) 18 3

d) 18 1.


9. giorgim 72 laridan daxarja 48 lari. Tanxis ra nawili daxarja giorgim? 1 5 a) 72 g) 12 d) 23 . 48 b) 48 10. brigada 96 km-ian gzis monakveTs aremontebs. gzis ra nawili darCeba gasaremontebeli 60km-is garemontebis Semdeg? 96 a) 60 g) 36 d) 36 96 b) 60 60 96 . swori pasuxebi: 1

2

3

4

5

6

7

8

9

10

g

g

a

b

g

b

a

b

d

d

testi 2 3 ; 12  6 ; 2  5 ; 3  23 ; 3  4 ; 127 . 1. CawereT aTwiladis saxiT: 8 10 10 100 100 1000 1000 45 18 21 28 75 2. SekveceT da CawereT aTwiladis saxiT: 63 90 ; 90 ; 60 ; 300 ; 40 ; 50 . 3. gamoiyeneT wiladis ZiriTadi Tviseba da CawereT aTwiladis saxiT: 9 ; 3 ; 15 ; 7 . 12 ; 35 ; 25 50 50 20 4. CawereT Sereuli ricxvis saxiT: 12,5; 12,05; 1,125; 2,005; 10,12. 5. gamosaxeT kilogramebSi: 250 g; 52 g; 3 g;

3 kg 25 g;

1 kg 5 g.

testebi ZiriTadad, wiladis ZiriTadi Tvisebisa da wiladis, rogorc mTelis nawilis gameorebas exeba. magaliTad, 9 amocanaSi unda vipovoT 48 ra nawilia 72-is, vwerT 48 2 wilads 48 72 da vamartivebT: 72 = 3 . 10 amocanaSi Tavdapirvelad gasaremontebeli gzis sigrZe vipovoT: 96–60=36, Sem-

3 deg – 36 96 = 8 (12-ze SekveciT). moswavleebs gamovucxadebT, rom Semdeg gakveTilze Catardeba Semajamebeli wera gavlil masalaze (ricxvebis Cawera, wiladebi, aTwiladebi).

me-12 gakveTili

Semajamebeli wera Tema: naturaluri ricxvebis Cawera; naturaluri ricxvebis Sedareba; wiladi, rogorc mTelis nawili; aTwiladebis Cawera. (maT. VI. 1.) SearCieT swori pasuxi: 1. dawereT xazgasmuli cifris mniSvneloba ricxvSi 532 694 007 501. a) 9 b) 9000000 g) 90000000 d) 900000.

43


2. udidesi oTxniSna ricxvi, romlis CanawerSi monawileobs cifrebi 0, 5, 6, 9 (yvela), aris a) 5069 b) 9065 g) 9506 d) 9650 3. Tu ricxvi Seicavs mxolod 32 srul aseulsa da 7 aTeuls, is SeiZleba ase CavweroT: a) 32+70 b) 3200+70 g) 320+700 d) 3200+700 4. TaigulSi 62 yvavilia; maTgan mxolod 23-ia vardi. Taigulis ra nawils Seadgens danarCeni yvavilebi? 39 62 16 a) 23 62 b) 62 g) 62 d) 62 . 5. gamosaxeT 3 sT da 15 wuTi saaTebiT: 15 sT b) 3 15 sT g) 3 1 sT d) 3 2 sT. a) 3 100 45 4 3 6. gamosaxeT 2 t 7  c da 85 kg tonebiT: a) 2,785 t b) 27,85 t g) 278,5 t

d) 0,2785 t.

amoxseniT amocanebi: 7. CawereT mocemuli aTwiladebi sityvierad da miuTiTeT TiToeulSi xazgasmuli cifris mniSvneloba: a) 81,675; b) 943,0074. 8. CawereT mocemuli wiladebi aTwiladis saxiT: 7 d) 19 . a) 14 b) 45 g) 25 50 pasuxebi da miTiTebebi amocanebis amosaxsnelad 1

2

3

4

5

6

g

d

b

b

g

a

7. a) oTxmocdaerTi mTeli da eqvsas samocdaTxuTmeti meaTasedi; xazgasmuli cifris mniSvnelobaa 0,07. b) cxraas ormocdasami mTeli da samocdaToTxmeti meaTiaTasedi; xazgasmuli cifris mniSvnelobaa 0,007. 7 =0,28; d) 19 =0,38. 8. a) 14 =0,25; b) 45 =0,8; g) 25 50 Sefasebis sqema testuri amocanebidan TiToeulis swori amoxsna Sefasdes 1 quliT. me-7 da me-8 amocanebidan TiToeuli Sefasdes maqsimum 2 quliT. gTavazobT Sefasebis rubrikas maswavleblisTvis. SeiZleba moswavleebsac SevTavazoT Sefasebis rubrikebi Sedegebis ganxilvisas. es aamaRlebs maTi TviTSefasebis unaris dones. Ees aqtivoba ganmaviTarebeli Sefasebis mniSvnelovan instruments warmoadgens.

44


Sefasebis kriteriumi #7 Cawerilia aTwiladi sityvierad; miTiTebulia xazgasmuli cifris mniSvneloba

Sesrulebis doneebi

ver gaerkva amocanaSi (0 qula)

(2 qula)

Cawerilia sityvierad mxolod mTeli nawili da miTiTebulia xazgasmuli cifris mniSvneloba uxeSi SecdomebiT (magaliTad –7)

Cawerilia ricxvi sityvierad mcire SecdomiT da miTiTebulia xazgasmuli cifris mniSvneloba mxolod erTi Tanrigis cdomilebiT

Cawerilia ricxvebi sityvierad sworad da miTiTebulia xazgasmuli cifrebis mniSvnelobebi sworad

(0,5+0,5=1 qula)

(1+1=2 qula)

Sesrulebulia mricxvelis mniSvnelze gayofa umniSvnelo xarvezebiT, an gamoyenebulia wiladis ZiriTadi Tviseba mcire SecdomiT; Cawerilia wiladebi aTwiladebis saxiT mciredi xarvezebiT

sworadaa gayofili mricxveli mniSvnelze, an warmatebiTaa gamoyenebuli wiladis ZiriTadi Tviseba. yvela wiladi gadawerilia aTwiladis saxiT sworad

(0,5+0,5=1 qula)

(1+1=2 qula)

(0,5 qula) #8 Sesrulebulia mricxvelis mniSvnelze gayofa, an gamoyenebulia wiladis ZiriTadi Tviseba; wiladebi Cawerilia aTwiladebis saxiT (2 qula)

ver gaerkva amocanaSi (0 qula)

Cawerilia mricxvelis mniSvnelze gayofa SecdomiT, an arasworadaa gamoyenebuli wiladis ZiriTadi Tviseba. arcerTi wiladi araa sworad Cawerili aTwiladis saxiT (0,5 qula)

Tu maswavleblisaTvis ufro komfortulia, SeiZleba rubrikebi ar gaformdes cxrilis saxiT da daiweros punqtebad, magaliTad, me-7 amocanisaTvis a) ricxvi Cawerilia sityvierad b) miTiTebulia xazgasmuli cifris mniSvneloba. 1 qula – Tu a) da b) punqtebidan Sesrulebulia mxolod erTi. 2 qula – TuU a) da b) punqtebidan Sesrulebulia orive. Tu romelime punqtSi xarvezebi araa uxeSi, daiweros am punqtSi 0,5 qula, SeiZleba gamoiyenoT Sefasebis aseTi sqema: testuri tipis amocanebi – TiToeuli swori amoxsna Sefasdes 1 quliT. me-7 da me-8 amocanebidan TiToeuli Sefasdes maqsimum 2 quliT. me-7 amocanaSi TiToeuli ricxvis swori sityvieri Canaweri – 0,5 qula, TiToeuli xazgasmuli cifris mniSvnelobis swori miTiTeba – 0,5 qula. me-8 amocanaSi TiToeuli wiladis aTwiladad swori warmodgena – 0,5 qula.

45


§4. aTwiladebis Sedareba da dalageba me-13 gakveTili

Tema: aTwiladebis Sedareba da dalageba. mizani: moswavles SeeZlos aTwiladis CanawerSi Tanrigebis da cifrebis mniSvnelobebis miTiTeba, am codnis gamoyeneba aTwiladebis Sedarebisa da dalagebisas (maT. VI. 1). winapirobebi: aTwiladSi Tanrigebisa da cifris mniSvnelobis miTiTeba, aTwiladis gamosaxva ricxviT sxivze (maT. VI. 1). resursebi: saxazavi, rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT Semajamebeli weris Sedegebis ganxilviT, tipuri Secdomebis warmodgeniTa da gasworebiT. am process SeiZleba 15-20 wuTi davuTmoT. Semdeg gadavdivarT axal masalaze, romelic ukavSirdeba wina gakveTilebze Seswavlili masalis gamoyenebas, amitom sasurvelia Semzadebis mizniT ramdenime amocanis amoxsna aTwiladSi Tanrigebisa da cifris mniSvnelobis povnaze. magaliTad, SeiZleba SevTavazoT moswavleebs aTwiladis warmodgena saTanrigo Sesakrebebis jamis saxiT: 12,256=1⋅10+2+0,2+0,05+0,006. – daasaxeleT am aTwiladis CanawerSi aTeulebis Tanrigis cifri, miuTiTeT misi mniSvneloba; – daasaxeleT meaTedebis Tanrigis cifri misi mniSvnelobis miTiTebiT; – daasaxeleT measedebis Tanrigis cifris mniSvneloba; – daasaxeleT meaTasedebis Tanrigis cifris mniSvneloba. ramdenime magaliTis ganxilvis Semdeg SeiZleba gavixsenoT aTwiladis gamosaxva ricxviT sxivze. 0 1 2 1,2 rom gamovsaxoT ricxviT sxivze, unda warmovidginoT, rom 1-dan 2-mde Sualedi dayofilia 10 tol nawilad, 1,2-s Seesabameba 1-is marjvniv dayofis meore wertili, SeiZleba ramdenime analogiuri magaliTis ganxilva. am Semzadebas SeiZleba mohyves axali masalis warmodgena, romelic saxelmZRvaneloSi warmodgenili teqstis mixedviT SeiZleba warvmarToT. `a~ punqtSi moyvanilia samotivacio amocana. moswavle dasmul kiTxvaze pasuxs aTwiladebis SedarebiT SeZlebs. Sesadareblad amjerad ricxviTi sxivi gamoviyeneT. teqstSivea SemoTavazebuli suraTi _ danayofebiani ricxviTi sxivi. am sxivs moswavleebi gadaitanen rveulSi da mis danayofebze miaweren Sesabamis aTwiladebs (iqve nimuSicaa mocemuli), riTac gaiwafebian sxivze aTwiladis Sesabamisi wertilis moniSvnaSi da wertilis Sesabamisi aTwiladis dadgenaSi _ gaerkvevian sxivze ricxvebis ganlagebis principSi:

`b~ punqtSi aTwiladebs vadarebT naturaluri ricxvebis Sedarebis msgavsad, Tanrigebis mixedviT.

46


I magaliTSi measedis TanrigSi vwyvetT Sedarebas, radgan 4<8. II magaliTSi viyenebT aTwiladis marjvnidan nulis miweris Tvisebas, romelic wina paragrafis saSinao davalebis savarjiSoebSi iyo demonstrirebuli. amitom am punqtis ganxilvamde moswavleebs SevaxsenebT, rom marjvnidan nulis miweriT aTwiladi ar icvleba (igulisxmeba mZimis marjvniv arsebuli nawilisTvis 0-is miwera). 7 70 700 10 = 100 = 1000 =... , anu 0,7=0,70=0,700=... am faqts viyenebT ganxilul magaliTSic. am magaliTSi ilustrirebulia zrdis mixedviT ricxvebis dalagebis principi: jer virCevT umciress, Semdeg darCenilebidan umciress da a. S. aseTive wesiT SeiZleba davalagoT moswavleebi mwkrivSi simaRlis zrdis mixedviT. moswavleebi swrafad aRmoaCenen, rom aTwiladebis Sedarebisas SesaZlebelia marjvnidan nulis miweris gareSec miaRwion sasurvel Sedegs. 3

testic toli ricxvebis sxvadasxva saxiT gamosaxvaSi gawafvaa. savarjiSoSiO 3 =0,3=0,30 da 0,5= 1 . warmodgenelia ori gansxvavebuli ricxvi: 10 2 testebi 5 - 8 aTwiladis marjvnidan 0-is miweris Tvisebis gamoyenebiT ixsneba: Tu gaviTvaliswinebT, rom 10,101=10,1010 da 10,102=10,1020, moswavle SeZlebs am or aTwilads Soris ricxvebis dasaxelebas. magaliTad, 10, 1011; 10,1012 da a. S. 5 7

2,5=2,500.

8

3=3,0 da 4=4,0. maT Soris aris, magaliTad, 3,2.

10

zrdis mixedviT dalagebis erT-erTi xerxi paragrafSi garCeul magaliTSi iyo warmodgenili. moswavleebi, analogiurad, am savarjiSoSic jer yvelaze nakleb ricxvs SearCeven marcxnidan marjvniv gadaadgilebisas saTanado cifrebis Sedarebis mixedviT _ pirvelive gansxvavebuli cifris mixedviT. es ricxvia 0,04. Semdeg measedebis Tanrigebsac Seadareben da SearCeven umciress 0,4 da a. S. analogiurad, 11 savarjiSoSi varCevT udidess, Semdeg danarCenidan udidess da a. S. klasSi SeiZleba am 10 amocanis amoxsniT SemovifargloT. amasTanave, teqstSi warmodgenili amocanebis kargad garCevisa da zogierT maTganze erToblivi msjelobis Semdeg moswavleebi danarCenebis amoxsnas damoukideblad SeZleben (magaliTad, 6 da 9 -is). am muSaobis SedegebiT SeiZleba Sefasdes gakveTilze dasaxuli miznis miRwevis

done. saSinao davalebad moswavleebs vaZlevT

1

- 9

amocanebs.

me-14 gakveTili

Tema: aTwiladebis Sedareba da dalageba. mizani: moswavleTa codnis ganmtkiceba da gaRrmaveba aTwiladebis Sedarebasa da maT dalagebaSi. winapirobebi: aTwiladis Cawera wiladis saxiT, wiladis ZiriTadi Tviseba. resursebi: kalami, rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT. moswavleebi rigrigobiT kiTxuloben `testebs~, asaxeleben pasuxebs da akontroleben erTmaneTis Sedegebs. SeiZleba

47


davusvaT damatebiTi kiTxvebi. magaliTad, daasaxelos kidev erTi ricxvi 4-sa da 5-s Soris (amocana 6 ). Zalian mniSvnelovania 2 amocanaze yuradRebis gamaxvileba – erTi da igive ricxvi SeiZleba Caiweros sxvadasxva formiT, sxvadasxva wiladiT, aTwi7 ; 0,70; 0,7 – erTi da imave ricxvis sxvadasxva Canaweria, ricxviT ladiT, magaliTad, 10 sxivze es ricxvi erTi wertiliT gamoisaxeba. 14 ; 0,25; 0,250 – erTi da imave ricxvis 2 sxvadasxva Canaweria. maSasadame, amocaniT warmodgenili ricxvebis raodenoba aris 2. moswavleebs SevaxsenebT – Canaweri sxvaa da ricxvi sxva, erTi da imave ricxvis Cawera sxvadasxva saxiT SeiZleba. am ori winadadebidan meti yuradReba meoreze gavamaxviloT, pirveli winadadebis gaazreba TandaTan xdeba moswavleTa gonebaSi. klasSi vagrZelebT amocanebis ganxilvas da vxsniT 11 - 26 amocanebidan SerCeul amocanebs (moswavleTa SesaZleblobebisa da drois limitis gaTvaliswinebiT). kidev erTxel vimeorebT – amocanebis Cveneuli dayofa (saSinao, klasSi) rekomendaciis xasiaTs atarebs da maswavlebels, cxadia, SeuZlia gadauxvios am wesisgan, zogierTi amocana, romlis Sesrulebas Cven klasSi vTavazobT, SeiZleba maswavlebelma moswavles saSinao davalebad misces. 13 moswavleTa nawili igonebs kiTxvebs, nawili pasuxobs maT. savaraudoa, rom es kiTxva-pasuxi kvlav aTwiladebis SedarebaSi gavarjiSebas emsaxureba. SeiZleba vTxovoT moswavleebs, rom ricxvebi daalagon zrdis mixedviT. 10,93<10,97<11,07<11,08. 200 14 savarjiSoSi moswavleebs kvlav vaxsenebT, rom 0,2=0,20=0,200= 1000 . kidev erTxel xazs vusvamT, rom aTwiladis wilad nawilSi marjvnidan 0-ebis miweriT aTwiladis 200 = 20 = 2 . ucvleloba dakavSirebulia wiladis SekvecasTan, kerZod, 1000 100 10 21 amocanaSi moswavleebi iyeneben aTwiladis marjvnidan 0-is miweris Tvisebas da samive punqtis Sesrulebis Semdeg ivaraudeben, rom es procesi usasrulod SeiZleba gagrZeldes da nebismier or ricxvs Soris SeiZleba uamravi ricxvis dasaxeleba. es ganaviTarebs moswavlis abstraqciis unars, gaxdeba pirveli miaxloeba racionalur ricxvTa simravlis simkvrivis Tvisebisken. 22 da 23 amocanebSi CamoTvlili wiladebidan mxolod isini Caiwereba iseT wiladebad, romelTa mniSvneli 100 an 1000-ia, romelTa mniSvnelia 5, 125, 20, 50. momavalSi moswavleebi Seityoben, rom wiladi sasrul aTwiladad Caiwereba im SemTxvevaSi, Tu misi ukveci saxiT Caweris SemTxvevaSi mniSvnelSi 2-isa da 5-is xarisxebis garda sxva 7 , 7 wiladebis mniSvnelebSi 2-is da 5-is martivi Tanamamravli ar aris. marTlac, 23 , 56 , 12 15 garda gvxvdeba 3-ianic, isini sasruli aTwiladiT ver Caiwereba. danarCeni wiladebis mniSvnelebi 2-ebisa da 5-ebis (an mxolod erT-erTis) namravlebisagan Sedgeba. aseT zogad daskvnebamde moswavleebi mivlen TandaTanobiT. amjerad, sakmarisia sinjvis xerxiT dakmayofildnen. 99 24 amocanaSi SemoTavazebulia ori wiladis Sedarebis principi: 100 -s 1-mde 1 . 999 -s ki _ 1 999 99 `Seavsebs~ 100 1000 1000 , amitom 1000 ufro axlosaa 1-Tan, e. i. is 100 -ze marjvnivaa ricxviT RerZze _ is metia. marTalia, am ricxvebis Sedareba maTi aTwiladebis saxiT CaweriT gacilebiT martivia, magram am principiT SeiZleba iseTi ricxvebis Sedarebac, romlebic sasrul aTwiladad ar Caiwereba.

48


999

1

1000

1

1000

25 amocanaSi 1000 -s 1-mde `Seavsebs~ 1000 , 1001 -s _ 1001 . e. i. 1001 ufro axlosaa

1-Tan anu ufro marjvnivaa ricxviT RerZze _ is metia. ricxvebis Sedarebis es principi xSirad sxva meTodebze xelsayreli aRmoCndeba xolme. 26 sayuradRebo savarjiSoa! moswavleTa nawili, albaT, ver SeZlebs saTanado wiladebis dasaxelebas, radgan ar izrunebs am wiladebis ukveci saxiT warmodgenaze. 2 3 Tu mocemul wiladebs CavwerT ukveci saxiT, miviRebT, 54 ; 23 ; 16 15 ; 3 ; 2 . maTgan mxolod 5 da 3 aris miTiTebuli saxis: 5 = 125 ; 3 = 150 . 4 2 4 100 2 100 amjerad klasSi am periodSi gamosayeneblad gTavazobT agreTve damoukidebeli (fokusirebuli) weris nimuSs. es amocanebi SeiZleba saSinao davalebisTvisac gamoiyenoT, an personaluri davalebebisTvis romelime uaxloes gakveTilze.

SearCieT swori pasuxi: 1. 0,2; 0,202; 0,22; 0,022 ricxvebis zrdis mixedviT dalagebis SemTxvevaSi mesame adgilze aRmoCndeba a) 0,2 b) 0,202 g) 0,22 d) 0,022. 2. 2,12-sa da 2,13-s Soris aris ricxvi a) 2,105 b) 2,115 g) 2,125 d) 2,11. 3. 0,012= 12 a) 12 10 b) 100

12 g) 1000

12 . d) 10000

7= 4. 25 a) 0,7 b) 0,07

g) 0,35

d) 0,28.

5. 45 -s erTamde Seavsebs ricxvi: a) 15 b) 25 g) 35

d) 45 .

6. 0,998-s erTamde Seavsebs ricxvi: a) 0,2 b) 0,02 g) 0,002 d) 0,0002. amoxseniT amocanebi: 7. erT kolofSi aris 0,309 kg qiSmiSi, meoreSi – 0,31 kg. romelSia meti? 8. Teonam karaqis sami naWeri iyida. TiTo 0,21 kg-iani. gadaiyvaneT es monacemi gramebSi da gamoTvaleT, sul ramdeni grami karaqi iyida Teonam. Riaboloian amocanebSi (7 da 8) aTwiladisTvis marjvnidan 0-is miweris Tvisebazea gamaxvilebuli yuradReba: 0,31=0,310 309<M310 e. i. 0,309 kg<0,31 kg. me-8 amocanaSic 0-is miwera gauadvilebs moswavles monacemis gramebSi gadayvanas. 0,21 kg=0,210 kg=210 grami. samive naWeri iqneba 3⋅210=630-gramiani.

49


testebis pasuxebia: 1 – b, 2 – g, 3 – d, 4 – d, 5 – a, 6 – g. am weris Sedegebi maswavlebels gauadvilebs Semdgomi muSaobis dagegmvis process. saSinao davalebad SeiZleba SearCioT amocanebi

11 - 16

amocanebidan.

me-15 gakveTili

Tema: aTwiladebis Sedareba da dalageba. mizani: moswavleebi gaiwafon aTwiladebis CaweraSi, ricxviT sxivze gamosaxvasa da SedarebaSi (maT. VI. 1). winapirobebi: ricxvebis sxvadasxva saxiT Cawera (wiladiT, aTwiladiT), aTwiladSi Tanrigebis, cifrebis mniSvnelobebis dasaxeleba. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebisas yuradReba gaamaxvileT 11 da 16 amocanebze, danarCenebi klasSi amoxsnilebis analogiuria. 11 savarjiSoSi 58 -is aTwiladis saxiT Casawerad mricxvelic da mniSvnelic unda gavamravloT 125-ze. pedagogebs SevaxsenebT: 58 -is mniSvneli 8 aris 23. mniSvnelSi 10-is xarisxis misaRebad mricxvelsac da mniSvnelsac vamravlebT 53-ze. amiT yovel Tanamamravl 2-s saTanado 5-ze vamravlebT, rac gvaZlevs 10-is xarisxebs. analogiurad, magaliTad, 13 20 -is mricxvelsac da mniSvnelsac vamravlebT 5-ze 2 (20=2 ⋅5). a) arasworia. magaliTad, SevadaroT 2,112 da 2,31. pirvel ricxvSi aTwiladi niSnebis raodenoba metia, magram is naklebia meore ricxvze: 2,112<2,31. b) sworia. is mravaljer gamoviyeneT am paragrafis savarjiSoebis amoxsnisas. g) arasworia. magaliTad, 2,23 da 2,32 aratoli aTwiladebis wiladi nawilebis Tanrigebis odenobebi tolia. aRsaniSnavia, rom b) punqtSi sakmarisi aRmoCnda erTi magaliTis (kontrmagaliTis) miTiTeba da amiT davasabuTeT am winadadebis mcdaroba. 16

proeqtze muSaobisas olimpiuri TamaSebis Sesaxeb farTo informacias mogawvdiT Cven mier araerTxel SemoTavazebuli internetis saZiebo sistemebi: Google an Wikipedia (ixileT: Olympic games). klasSi SeiZleba amoixsnas amocanebi, romlebic aTwiladebis Sedarebisa da dalagebis codnis ganmtkicebas emsaxureba. isini SeiZleba gamoyenebuli iqnas, rogorc saswavlo procesSi mimdinare Sefaseba, damoukidebeli werisas aRmoCenili xarvezebis aRmofxvris saSualeba. 1. SeadareT ricxvebi: 2,08 5,199 3,25

50

3,012 5,8 3,2500

2,419 3,23 4,19

2,5001 3,199 4,1901


2. daalageT aTwiladebi zrdis mixedviT: 5,009; 5,1; 5,01; 5,201; 5,102. 3. CawereT yvela naturaluri ricxvi, romelic aris a) 3,02-sa da 7,9-s Soris; b) 0,102-sa da 8,3-s Soris.

4. CawereT raime sami ricxvi, romelic aris a) 12,1-sa da 12,5-s Soris; b) 12,18-sa da 12,25-s Soris; g) 12,1-sa da 12,2-s Soris; d) 12,11-sa da 12,12-s Soris. 5. gamoiyeneT 2, 3, 0, 5 cifrebidan yoveli cifri TiTojer da CawereT yvela iseTi ricxvi, romelic metia 2-ze da naklebia 2,5-ze. 4 amocanis a)-s pasuxad, savaraudod, moswavleebi Caweren ricxvebs 12,2; 12,3 da 12,4. Tu Cven vTxovT kidev aseTi ricxvebis dasaxelebas, maT mouwevT aTwiladis marjvnidan 0-ebis miwera, rac daexmareba kidev sxva ricxvebis dasaxelebaSi. magaliTad, daasaxeleben 12,11; 12,12; ... 12,49-s. es procesiM SeiZleba kidev gagrZeldes, dasaxeldes ricxvebi 12,111; 12,112; ... 12,489 da a.S. Sedegad SeiZleba moswavleebma daaskvnan, rom nebismier or ricxvs Soris usasrulo odenobis ricxvebia. saSinao davalebad SeiZleba SevurCioT amocanebi darCenili 9 - 26 , 11 - 16 amocanebidan da I Tavis damatebiTi 16 - 20 amocanebidanac.

me-16 gakveTili

gakveTilis Catarebis forma: jgufuri muSaoba. Tema: ricxvebis Cawera aTwiladis saxiT; aTwiladis TvisebaTa gamoyeneba. mizani: moswavle gaiwafos wiladis aTwiladis saxiT CaweraSi; man gaiazros aTwiladis wilad nawilSi marjvnidan nulis miweris arsi da SeZlos misi gamoyeneba ricxvebis Sedarebisas; gaiwafos im wiladebis amocnobaSi, romlebic Caiwereba sasruli aTwiladis saxiT; gaiwafos ricxvis sxvadasxva saxiT CaweraSi. winapirobebia: wiladebi da aTwiladebi, maTi Caweris formebi; wiladebisa da aTwiladebis Sedareba; wiladis warmodgena aTwiladis saxiT. resursebi: saweri saSualebebi; furclebi davalebaTa Sesasruleblad; dafa; pedagogis mier momzadebuli baraTebi. aqtivobis aRwera: gakveTili moswavleTa jgufebad dayofiT iwyeba. Tqven Tu migaCniaT, rom es procesi did dros waiRebs, Tavad _ operatiulad warmarTeT es procesi. amiT `ZalTa balanssac~ ukeT daicavT. muSaoba aucilebeli wina codnis SemowmebiT daiwyeT: jgufebs vTavazobT Caweron 11 , wiladebi aTwiladebis saxiT da Semdeg Seadaron isini. magaliTad, 58 da 35 , 35 da 16 5 da 3 . jgufebi pasuxebs sajarod warmoadgenen _ aRweren wiladis aTwiladad 7 5 gadayvanisa da aTwiladebis Sedarebis process.

51


Semdeg jgufebs vTavazobT amoxsnan a) savarjiSo da Seadginon msgavsi savarjiSoebi mowinaaRmdege gundisTvis. am savarjiSos amoxsnisas gaixseneben da gamoiyeneben codnas wiladis aTwiladad Cawerisa da aTwiladis wilad nawilSi marjvnidan nulebis miweris Sesaxeb. savaraudod, savarjiSos Sedgenisas jgufebi daweren or an sam gansxvavebul ricxvs da Caweren mas sxvadasxva saxiT. magaliTad, davweroT ricxvebi 34 da 15 . 2 =0,2=0,20=0,200= 3 da a. S. CavweroT isini sxvadasxva saxiT: 34 =0,75=0,750, 15 = 10 15 CamovayaliboT amocana mowinaaRmdege jgufisTvis: 3 ; 0,20 ramdenia gansxvavebuli? CamoTvlili ricxvebidan: 34 ; 0,75; 15 ; 0,2; 0,750; 0,200; 15 moswavleebma unda SeniSnon, rom aq aris warmodgenili mxolod ori ricxvi, gansxvavebulia maTi Caweris formebi. Sedgenili savarjiSoebis gacvlis Semdeg jgufebi warmoadgenen sxva jgufis mier SeTavazebuli savarjiSos amoxsnas. Semdeg etapze moswavleebs vTavazobT Caweron CamoTvlili ricxvebi (magaliTad, 1 , 3 , 7 , 7 , 5 ) iseTi wiladebis saxiT, romelTa mniSvneli aris 10, 100 an 1000, anu 5 8 25 15 12 10-is xarisxi. (davaleba SeiZleba mivawodoT baraTebiT da dafaze warmovadginoT). 7 da 5 aseTi wiladis saxiT ver warmodgindeba. savaraudoa, rom jgufebi daaskvnian: 15 12 Sesabamisad, es ricxvebi sasruli aTwiladiT ver Caiwereba. jgufebs SevTavazoT, Tavad moifiqron wiladebi, romlebic Caiwereba 10-is xarisxiani mniSvnelis mqone wiladis saxiT da romlebic ver Caiwereba aseTi saxiT. aq moswavleebi CavrToT miRebuli Sedegebis analizSi. moswavleebi msjeloben: 15-isa da 12-is TanamamravlebSi urevia 7 ver Caiwereba wiladiT, 3, romlis jeradebs Soris ar aris 10-is xarisxi. e. i. 15 romlis mniSvneli 10-is xarisxia. 10-is xarisxebi mxolod 5-ebisa da 2-ebis gamravlebiT miiReba. (10=2â&#x2039;&#x2026;5). moswavleebi am etapze daaskvnian: warmodgenili wiladebidan sasrul aTwiladad Caiwereba mxolod is wiladebi, romelTa mniSvnelis martiv mamravlebad daSlaSi mxolod 2-ebi da 5-ebi monawileobs. am ganxilvas yuradRebiT adevneT Tvali da saWiroebis SemTxvevaSi kiTxvebiT warmarTeT procesi. imuSaveT mTel jgufTan da ara mxolod zogierT moswavlesTan. bolo etapze jgufebs vTavazobT b) savarjiSos. 105-is martiv mamravlebad daSlaSi 63 mainc Caiwera sasrul aTwiladad, 140 _ ver 5-is garda 3 da 7-ic Sedis, magram 105 105 63 = 3 = 6 , Caiwera. imsjelon mizezebze. wiladebis Sekvecis Semdeg daadgenen, rom 105 5 10 140= 4 da is ver Caiwereba wiladis saxiT, romlis mniSvneli 10-is xarisxia. 105 3 am bolo aqtivobis Sedegad moswavleTa jgufebi wiladis aTwiladad Caweris pirobebSi mniSvnelovan dazustebas Seitanen _ Caamateben sityvas `ukveci~ da daskvna saboloo saxes miiRebs: sasrul aTwiladad Caiwereba mxolod is wiladi, romlis Sekvecis Semdeg mniSvnelis martiv mamravlebad daSlaSi 2-isa da 5-isgan gansxvavebuli martivi Tanamamravli ar monawileobs. aqtivobis Sedegad moswavle gaiwafeba wiladis Tvisebis gamoyenebaSi, aTwiladebis SedarebaSi. wiladis aTwiladad gadaweraSi. mas uviTardeba analizuri azrovnebis unari, romlis Sedegadac is qmnis produqts (sinTezi) _ ayalibebs wiladis sasrul aTwiladad Caweris SesaZleblobis pirobebs. amrigad, sakmaod maRalia am aqtivobis ganmaviTarebeli mniSvneloba. aqtivobis Sefaseba. aqtivobis dasasruls mTel klasTan erTad unda SevajamoT miRebuli Sedegebi, davasaxeloT nakli, davsaxoT misi gamosworebis gzebi.

52


pedagogma unda CainiSnos moswavleTa individualuri Sedegebi. gamoarCios dro­ ebiT CamorCenilebi, raTa Semdgom diferencirebuli midgomiT gaaZlieros maTTan muSaoba. pedagogma sajarod unda daasaxelos da waaxalisos warmatebuli moswavleebi, gaamxnevos yvela moswavle. am aqtivobam pedagogs unda ukarnaxos Semdgomi muSaobis sworad dagegmvis gzebi, SesaZloa swavlebis intensivobac Sesabamisad Secvalos.

§5. aTwiladebis damrgvaleba me-17 gakveTili

Tema: aTwiladebis damrgvaleba. mizani: moswavles SeeZlos aTwiladis damrgvaleba mocemuli sizustiT (magaliTad, meaTedis, measedis) (maT. VI. 2.). winapirobebi: naturaluri ricxvebis damrgvaleba. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebas SeiZleba 5-10 wuTi davuTmoT. axal masalaze gadasvlisTvis saWiro Semzadeba naturaluri ricxvebis Sedarebis wesis gaxseneba da ramdenime Sesabamisi magaliTis ganxilvaa. amisTvis SeiZleba gamoviyenoT amocana 20 I Tavis damatebiTi amocanebidan. paragrafSi warmodgenilia damrgvalebis kerZo magaliTebi da Semdeg aRwerilia damrgvalebis zogadi wesi: nebismier Tanrigamde damrgvalebisas vakvirdebiT am Tanrigis uaxloes marjvena cifrs, vadarebT mas 5-s da a. S. SigadaSig paragrafSi moswavleebs vTavazobT savarjiSoebs damoukideblad Sesasruleblad _ axali masalis ukeT asaTviseblad. • 84,75 _ measedis Tanrigis cifria 5, amitom meaTedis Tanrigis cifrs vzrdiT 1-iT: 84,75≈84,8. miRebuli ricxvi metia mocemulze _ damrgvaleba moxda metobiT. • 29,399 ricxvis measedebamde damrgvalebisas miviRebT 29,40-s. 29,399≈29,40   9+1=10 9>5 miRebuli ricxvis CanawerSi 0 miuTiTebs, rom damrgvaleba measedebamde SevasruleT. cxadia, aq damrgvaleba xdeba metobiT. saklaso savarjiSoebSi gaviTvaliswinoT, rom uaxloes mTel ricxvamde damrgvaleba erTeulebamde damrgvalebas gulisxmobs. es paragrafis teqstSicaa miTiTebuli. klasSi vxsniT 1 - 9 amocanebs. maTgan 9 amocanis davalebebi SeiZleba amosaxsnelad moswavleebs SevTavazoT da am davalebis SesrulebiT SevafasoT dasaxuli miznis miRwevis done. am davalebebis amoxsna moiTxovs damrgvalebis wesis kargad gaazrebas. saSinao davalebad vaZlevT 1 - 8 amocanebs.

53


me-18 gakveTili

Tema: aTwiladebis damrgvaleba. mizani: aTwiladebis damrgvalebaze codnis ganmtkiceba amocanebis gamoyenebiT. winapirobebi: aTwiladebis damrgvalebis wesis codna. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis amocanebis amoxsna damrgvalebis wesis sworad gamoyenebas moiTxovda. klasSi vagrZelebT am Temis Sesaxeb codnis ganmtkicebas, vxsniT 10 - 16 amocanebs. am amocanebidan, sasurvelia, rom 11 - 13 amocanebi erToblivi ZalisxmeviT amoixsnas. danarCenebis amoxsnas moswavleebi damoukidebladac SeZleben. 11 ricxviT sxivze damrgvalebis Sedegebis gamosaxvisas moswavlisTvis TvalsaCino gaxdeba, rom damrgvalebis Sedegi zogjer mocemul ricxvze metia, zogjer _ naklebi.

sxivze ganlagebuli ricxvebi zrdis mixedviTaa dalagebuli. 12 amocanaSi orjer damrgvalebiT miRebuli Sedegi gansxvavdeba pirdapir me­ aTedebamde damrgvalebuli Sedegisgan: 2,4>2,3. 13 amocanaSi orjer damrgvalebiT miRebuli Sedegi 2,34≈2,3 tolia pirdapir meaTedebamde damrgvalebulis. am ori amocanis Sedegebze dakvirvebiT moswavle aR­ moaCens, rom ori Tanamimdevrulad Catarebuli damrgvalebiT SeiZleba miviRoT erTi damrgvalebisgan gansxvavebuli Sedegi, anu orjer damrgvalebiT miRebuli `Secdoma~ (cdomileba) SeiZleba gansxvavdebodes erTjer damrgvalebisas miRebul cdomilebisgan. aseTi amocanebi moswavles uviTarebs praqtikuli samuSaos Sesrulebisas miaxloebiTi gamoTvlebis sworad Catarebis, misaRebi Sedegis gaanalizebis, daskvnis gamotanis unars. saSinao davalebad vaZlevT 9 - 14 amocanebs.

me-19 gakveTili

Tema: aTwiladebis damrgvaleba. mizani: aTwiladebis Sedarebis, dalagebisa da damrgvalebis Sesaxeb codnis ganmtkiceba. winapirobebi: aTwiladebSi Tanrigebis, cifrebis mniSvnelobebis dasaxeleba, gamoyeneba Sedarebisa da damrgvalebis dros. resursebi: rveuli, saxelmZRvanelo, saxazavi, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT. zogierTma moswavlem SeiZleba kompiuteris gamoyenebiT diagramis agebac ki SeZlos (amocana 11 ). xeliT diagramis agebis SemTxvevaSi moswavlem unda izrunos masebis gamomsaxveli RerZis masStabis

54


SerCevaze. man unda gaiTvaliswinos, rom udidesi masaa 49 kg. SeiZleba zogierTma 49 an meti erTeuli gamosaxos RerZze. ufro xelsayrelia TiTo erTeuliT 5 kg igulisxmos da maSin RerZze 10 erTeulis gamosaxvac sakmarisi iqneboda.

13

amocana klasSi Sesrulebuli me- 11 amocanis msgavsia. maTi amoxsnisas moswavleebi sidideebis zrdiT an klebiT dalagebaSic iwafebian. ricxvebis klebiT dalagebis principi aseTia: jer virCevT udidess, Semdeg darCenilebidan udidess da a. S. klasSi samuSaod da gakveTilze dasaxuli miznis SesrulebaSi dagvexmareba amocanebi 1. daamrgvaleT: a) erTeulebamde: 23,51; 59,6; 0,41; 0,815. b) meaTedebamde: 23,51; 23,56; 42,95; 102,109. g) measedebamde: 2,105; 13,562; 0,996; 0,138. 2. measedebamde damrgvlebisas miRebul tolobaSi erTi cifri dafarulia kvad­ ratiT. ra SeiZleba eweros kvadratis qveS? a) 41,35   2≈41,35. b) 41,37   8≈41,38 3. CawereT aTwiladis saxiT: 2  +  3  +  4 2 +  10 100 1000 2  + 4 4100 +3+  10 1000

5  +  2 3⋅10+2 +  100 1000 4 7 100  +  1000 .

4. daSaleT aTwiladi Tanrigebis mixedviT: 3,15; 2,027; 45,21; 0,102. 5. daamrgvaleT 7,4581 erTeulebamde, meaTedebamde, measedebamde, meaTasedebamde da TiToeuli damrgvalebisas miuTiTeT, metobiT moxda damrgvaleba, Tu naklebobiT. sainteresoa me-5 amocana. igi moiTxovs erTdroulad aTwiladebis damrgvalebisa da Sedarebis codnas: meaTedebamde sizustiT: 7,4581≈7,5 7,5>7,4581, damrgvaleba moxda metobiT. measedebamde sizustiT, 7,4581≈7,46 7,46>7,458, aqac damrgvaleba moxda metobiT. meaTasedebamde sizustiT, 7,4581≈7,458, 7,458<7,4581 damrgvaleba moxda naklebobiT. saSinao davalebad SeurCieT darCenili amocanebi klasSi amosaxsneli 1

- 16 -dan,

an saSinao davalebis amocanebidan, I Tavis damatebiTi 21 - 25 amocanebidan.

55


§6. aTwiladebis zogierTi gamoyeneba me-20 gakveTili

Tema: aTwiladebis zogierTi gamoyeneba. mizani: moswavlem SeZlos aTwiladebis gamoyeneba zomis sxvadasxva erTeulis erTmaneTTan dakavSirebisas. winapirobebi: aTwiladebis Cawera, Sedareba, cifrebis mniSvnelobebis dasaxeleba. resursebi: plakati, romelzec gamosaxulia zomis sxvadasxva erTeulebs Soris kavSirebis gamosaxvisas aTwiladebis gamoyeneba (sigrZis, masis, fulis), rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebas SeiZleba davuTmoT 10-15 wuTi. yuradReba gavamaxviloT metobiT da naklebobiT damrgvalebaze. magaliTad, Tu 2,0913-s davamrgvalebT meaTedebamde, miviRebT masze met ricxvs (2,1-s), Tu davamrgvalebT measedebamde, maSin mocemulze nakleb ricxvs miviRebT. axal masalaze gadasvlamde SeiZleba SevTavazoT teqsturi amocana ricxvis aTwiladis saxiT Caweris Sesaxeb. magaliTad, erTma fermerma moiyvana 7  15 tona xorbali, meorem – 8  35 tona. ramdeni tona xorbali moiyvana orivem erTad? monacemebi da pasuxi CawereT aTwiladebiT. 7  15 , 8  35 da maTi jami – 15  45 aTwiladebis saxiT ase Caiwereba: 7,2; 8,6; 15,8, radgan 1 2 3 6 8 , 7  5 =7  10 , 8  5 =8  10 , 15  45 =15  10 TiToeulis erTeulebamde damrgvalebiT miviRebT ricxvebs: 7-s, 9-s da 16-s. moswavleebTan erTad vixsenebT sigrZisa da masis sazom erTeulebs (es aris am gakveTilis nayofierad warmarTvis aucilebeli winare codna); am erTeulebs Soris arsebuli Tanafardobebis warmosadgenad viyenebT ricxvis Caweris aTwilad formas: 1 m = 0,001 km, 1 dm = 0,1 m, 1 sm =0,01 m, 1 mm =0,001 m, 1 dm = 0,0001 km da a. S. aRsaniSnavia, rom xSirad moswavleebs litri masis sazomi erTeuli hgoniaT. auxseniT, rom wyliT savse erTlitriani qila ufro msubuqia, vidre, magaliTad, TafliT savse. litri _ moculobis erTeulia. aTwiladebs Tanxebis gamosaxvisTvisac iyeneben. magaliTad, 12 lari da 25 TeTri 12,25 laria, 12 lari da 5 TeTri aris 12,05 lari (da ara 12,5), 100 dolari da 50 centi 100,5 dolaria. klasSi amosaxsnelad SerCeul amocanebs metwilad yofacxovrebaSi gamoyenebadi datvirTvac aqvs. moswavleebi Tanxebs aTwiladebis saSualebiT gamosaxaven larebSi, avtomobilis spidometrze gamosaxul informacias _ sxvadasxva erTeulebSi, sazom erTeulebs irCeven situaciis mixedviT (savarjiSo 11 ). saSinao davalebis SesrulebisTvis Sesamzadeblad SeiZleba moswavleebTan erTad gaixsenoT ricxvis 14 nawilis povna. 11 amocanaSi 500 kg-is 14 nawili 500:4=125 kgia. gasayidi darCa 500-125=375kg=0,375 t. SesaZloa daadginoT, rom 14 nawilis gayidvis Semdeg sacavSi arsebuli bostneulis 34 nawili darCeba da gamoTvaloT 500-is 34 nawili. saSinao davalebad vaZlevT

56

1

- 15

amocanebs.


21-e gakveTili

Tema: aTwiladebis gamoyeneba. mizani: aTwiladebis gamoyenebaze codnis ganmtkiceba. winapirobebi: zomis sxvadasxva erTeulebi (masis, sigrZis, fulis). resursebi: rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT (15-20 wT). gamovarCevT 15 amocanebs.

14

da

14

amocana praqtikuli xasiaTisaa, moswavleebi iwafebian gazomvebis CatarebaSi, gazomvebis Sedegebs iyeneben sasurveli zomis sufris SesarCevad. magaliTad, vTqvaT, magidas aqvs marTkuTxedis forma, romlis sigane 65 sm-ia, sigrZe _ 155 sm, maSin sufris sigane da sigrZe unda iyos 95 sm da 185 sm; metrebSi _ 0,95 m da 1,85 m. 15 amocanaSi moswavlem unda SeZlos cxriliT mocemuli informaciis gaazreba, monacemTa warmodgena sigrZis sxva erTeuliT, miRebuli ricxvebis Sedareba da saTanado Sefasebebis moZieba (cxrilis mixedviT).

sxvadasxva metruli sistemis Sesaxeb informaciis moZieba-gacnoba Zalze sasargebloa moswav­ leTa ganaTlebisTvis; erTi da imave sididis sazom erTeulTa Soris kavSirebis codna aucilebelia mravali praqtikuli amocanis amosaxsnelad. maswavleblebs mivawodebT informacias zogierTi zomis erTeulis Sesaxeb: 1 funti ≈ 453,6 g 1 angstremi ≈ 0,0000001 mm 1 uncia ≈ 28 g 1 mili ≈ 1600 m 1 mikrometri (mikroni) ≈ 0,001 mm 1 futi ≈ 30 sm 1 nanometri ≈ 0,000001 mm 1 duimi ≈ 2,5 sm Semajamebeli amocanebi Seswavlili masalis gameorebasa da ganmtkicebas emsaxureba. 3

x savarjiSos magaliTebs moswavleebi sxvadasxvanairad xsnian. magaliTad, 29 4 =8

tolobidan daadgenen, rom II wiladis mricxvelic da mniSvnelic 2-ze gamrav­lebiTaa miRebuli. e. i. x=29⋅2. zogjer SeiZleba daweron, x=8:4⋅29 _ isini gayofiT gamoTvlian ricxvs, romelzec I wiladis mricxveli da mniSvneli gamravlda, Semdeg miRebul ricxvze amravleben 29-s. SeiZleba proporciis Tviseba gamoiyenon: x⋅4=29⋅8, saidanac gamoTvlian x-s. 4 savarjiSo ar avaldebulebs moswavles, axsovdes, ramdeni litria mililitri, SeiZleba gasaxseneblad §6-is teqsti gamoiyenos. gaxsenebisaTvis SeiZleba nacnobi sazomi erTeulebi gamoviyenoT. magaliTad, metri da milimetri. metrSi 1000 milimetria, anu milimetri metris meaTasedia. Sedgenil sityvebSi, romlebic zomis erTeulebs warmogvidgens `mili~ meaTasedis aRsaniSnavad gamoiyeneba xolme. mililitric litris meaTasedia: 1 ml = 0,001 l.

57


or aTwilads Soris moTavsebuli aTwiladis SesarCevad iwyeben Sedarebas 6 umaRlesi Tanrigidan, a) mTeli nawilebi tolia, wilad nawilSi unda eweros raime naturaluri ricxvi 314-sa da 431-s Soris. d) nulebis miweris Semdeg moswavle advilad SearCevs wilad nawilSi raime ricxvs 100-sa da 105-s Soris. 8 â&#x20AC;&#x2030; a) 58 dm + 32 sm + 4 mm = 5800 mm + 320 mm + 4 mm = 6124 mm =6,124 m b) 3 km + 410 sm = 3000 m + 4 m + 10 sm = 3004 m + 0,1 m = 3004,1 m. danarCen amocanebs moswavleebs saSinao davalebad vaZlevT.

22-e gakveTili

Tema: aTwiladebis Sedareba da dalageba, aTwiladebis damrgvaleba. aTwiladebis gamoyeneba. mizani: miTiTebuli Temebis Sesaxeb codnis ganmtkiceba, masalis gameoreba. resursebi: saxelmZRvanelo, rveuli, dafa. miTiTebuli Temebis irgvliv masalis gameoreba, codnis ganmtkiceba mimdinareobs Semajamebeli amocanebis garCeviT (maTgan nawili saSinao davalebad iyo micemuli). gTavazobT testur davalebas, romlebzec swori pasuxebis SerCevas da Sedegebis ganxilvas 10-15 wuTi dasWirdeba. 1. sazRvao mili aris 1,853 km. sigrZis am erTeulis meaTedebamde damrgvalebiT miiReba a) 2 km

b) 1,8 km

g) 1,9 km

d) 1,7 km.

2. ricxvi 9,2542 damrgvaldeba metobiT, Tu mas davamrgvalebT a) erTeulebamde b) meaTedebamde g) measedebamde g) meaTasedebamde. 3. 2,12-isa da 2,13-s Soris aris a) 2,112 b) 2,113 g) 2,126 d) 2,132. 4. romeli winadadebaa swori? a) 1,5 sT metia 1 sT 40 wuTze b) 1,05 lari aris 1 lari da 50 TeTri g) 2,1 km aris 210 metri d) 2,01 km aris 2010 metri. 3= 5. 2â&#x20AC;&#x2030;25 a) 2,75

b) 2,3

g) 2,03

pasuxebi

58

1

2

3

4

5

g

b

g

d

d

d) 2,12.


Ees iqneba Semzadebac Semajamebeli werisTvis. Aam SemzadebisTvis SeiZleba I Tavis damatebiTi 26 - 31 amocanebis gamoyenebac. moswavleebs gamovucxadebT, rom Semdeg mecadineobaze Catardeba Semajamebeli wera ganxiluli Temebis irgvliv.

23-e gakveTili

Semajamebeli wera Tema: aTwiladebis Sedareba da dalageba; aTwiladebis damrgvaleba; aTwiladebis gamoyeneba. 1. Tu aTwiladebs: 0,064; 1,231; 0,97; 1,083 davalagebT klebiT, maSin mesame aRmoCndeba a) 1,231 b) 0,064 g) 1,083 d) 0,97 2. utoloba 5,9*89<5,9188 sworia, roca * niSnakis nacvlad Cawerilia a) 1 b) 9 g) 0 d) nebismieri cifri 3. vTqvaT, a ricxvi 9,201-sa da 9,202-s Sorisaa. maSin a SeiZleba iyos a) 9,2005 b) 9,2015 g) 9,2025 d) 9,251 4. damrgvalebis Sedegad gvaqvs: 15,675≈15,68. damrgvaleba moxda a) erTeulebamde b) meaTedebamde g) measedebamde d) meaTasedebamde 5. 7,5 sT aris b) 7 sT 50 wT

g) 7 sT 5 wT

d) 7 sT 12 wT

6. 1324 grami aris a) 13,24 kg b) 1,324 kg

g) 13,24 t

d) 132,4 c

a) 7 sT da 30 wT

7. • daamrgvaleT 3,0763: a) erTeulebamde; b) meaTedebamde; g) measedebamde; d) mea­ Tasedebamde. • daalageT mocemuli da damrgvalebiT miRebuli aTwiladebi zrdis mixedviT. 8. ramdeni lari unda gadaixados dedam 6 qila rZeSi, Tu TiTo qila 135 TeTri Rirs? pasuxebi da miTiTebebi amocanebis amosaxsnelad 1

2

3

4

5

6

d

g

b

g

a

b

7. a) 3,0763≈3 b) 3,0763≈3,1 g) 3,0763≈3,08 • zrdiT dalagebisas: 3; 3,076; 3,0763; 3,08; 3,1

d) 3,0763≈3,076

8. 1) 6⋅135=810 (mTeli rZis Rirebuleba TeTrebSi) 2) 810 TeTri = 8,1 lari (Rirebuleba larebSi) anu 6⋅135 TeTri = 810 TeTri = 8,1 lari

59


Sefasebis sqema testuri tipis amocanebi _ TiToeuli swori amoxsna Sefasdes 1 quliT. me-7 amocana _ maqsimum 2 quliT, me-8 amocana _ maqsimum 2 quliT.

rubrikebi maswavleblisTvis #7 Ria davalebisTvis 0 qula

amocanis arsi ver gaigo, an mxolod gadmowera amocanis piroba

0,5 qula

mxolod erTi damrgvalebaa sworad Sesrulebuli. zrdis mixedviT dalagebaSic Secdomebia

1 qula

mxolod ori damrgvalebaa sworad Sesrulebuli; sworad daalaga zrdis mixedviT, an erTi damrgvaleba Seasrula sworad da Sedegebis zrdiT dalagebisas dauSva erTi Secdoma.

1,5 qula

sami damrgvaleba Seasrula sworad da Sedegebi sworad daalaga zrdis mixedviT

2 qula

oTxive damrgvaleba Seasrula sworad da Sedegebi daalaga zrdis mixedviT

#8 Ria davalebisaTvis

60

0 qula

amocanis arsi ver gaigo, an mxolod gadmowera amocanis piroba

0,5 qula

gamravlebis moqmedeba SearCia sworad, magram ver gaamravla, an SeZlo mxolod 1 qilis Rirebulebis Cawera larebSi

1 qula

sworad SearCia moqmedeba (gamravleba) da Seasrula mciredi xarveziT. Sedegi ver warmoadgina larebSi

1,5 qula

sworad SearCia moqmedeba (gamravleba), Seasrula da Sedegi warmoadgina larebSi SecdomiT an mciredi Secdoma dauSva gamravlebaSi da Sedegi sworad Cawera larebSi

2 qula

sworad SearCia moqmedeba, Seasrula sworad da Sedegi Cawera larebSi sworad


II Tavi. moqmedebebi aTwiladebze

§1. aTwiladebis Sekreba 24-e gakveTili

Tema: aTwiladebis Sekreba. mizani: moswavles SeeZlos aTwiladebis Sekreba; aTwiladebis Sekrebis procesis gaazreba-dakavSireba ricxvis Caweris poziciur sistemasTan; poziciuri sistemis Tvisebebis gamoyeneba aTwiladi ricxvebis Sekrebisas; aTwiladebis SekrebaSi gavar­ jiSeba; mocemuli sizustiT aTwiladis damrgvalebis saSualebiT Sekrebis Sedegis Sefasebis unaris ganviTareba; aTwiladebis Sekrebisas kalkulatoris gamoyenebaSi gawafva; dauflebuli codnis gamoyeneba sxvadasxva saxis praqtikuli amocanis amoxsnisas (maT. VI. 2). winapirobebi: aTwiladebis damrgvaleba, saTanrigo Sesakrebebis jamad Cawera. resursebi: kalkulatori, rveuli, saxelmZRvanelo, dafa. I gakveTiliT iwyeba dasaxelebuli Temis Seswavlis pirveli faza `winaswar~. amave gakveTilze iwyeba faza `ganmavlobaSi~, romelic Semdeg gakveTilzec gagrZeldeba. gakveTils viwyebT Semajamebeli weris Sedegebis warmodgeniT. SeiZleba saWirod CavTvaloT klasSi zogierTi amocanis ganxilva. sayuradReboa Tavad moswavleTa komentarebi miRebuli Sefasebebis Sesaxeb. am aqtivobas mniSvnelovani ganmaviTarebeli efeqti aqvs. am process dasWirdeba TxuTmetiode wuTi. gakveTilis Semdeg monakveTs im wina masalis gaxsenebiT gavagrZelebT, romelic aucilebelia dasaxuli miznis misaRwevad. magaliTad, miznis miRwevisTvis aucilebeli wina codnaa naturaluri ricxvebis qveSmiweriT Sekrebis, naturaluri ricxvebis damrgvalebis wesebis, sazomi erTeulebis erTmaneTTan kavSirisa da Tanrigis erTeulebis mniSvnelobis codna. paragrafs praqtikuli (samotivacio) amocanis SeTavazebiT viwyebT. amocanis amosaxsnelad moswavleebs aTwiladebis Sekreba sWirdebaT. amjerad isini masebs jer gramebiT gamosaxaven da Sekreben naturalur ricxvebs: 82463 grami + 84805 grami, Semdeg miRebul jams _ 167268 grams gamosaxaven kilogramebiT: 167,268 kg. imave aTwiladebis qveSmiweriT SekrebiT (ise, rom Sesabamisi Tanrigebi erTmaneTis qveS eweros) igive Sedegi miiReba. qveSmiweriT Sekrebis algoriTmis (moqmedebaTa mimdevrobis) aRweris Semdeg mos­ wavleebs magaliTebiT gavwafavT/gavavarjiSebT aTwiladebis SekrebaSi. Sekrebis procesi teqstSi dawvrilebiT aris warmodgenili sailustracio magaliTiT. savarjiSod saklaso amocanebs viyenebT.

61


yuradReba gamaxvilebulia im SemTxvevaze, roca Sesakrebi aTwiladebis wilad nawilebSi sxvadasxva odenobis Tanrigebia da amitom marjvnidan nulebis miweras vaxdenT, an nulebis Caweris gareSe _ mxolod maTi warmosaxviT davkmayofildebiT. SeiZleba ramdenime Sekreba kalkulatoriT SevasrulebinoT. saklaso savarjiSoebSi ramdenime Sesakrebis Sekrebisas mouwevT Sekrebis Tvisebebis gamoyeneba. saxelmZRvaneloSi warmodgenili magaliTebis kargad garCevis Semdeg (cxadia, moswavleebis CarTulobiT), dasaxuli miznis miRwevis done SeiZleba nawilobriv mainc Sefasdes 1 - 10 `testebis~ amoxsniT, romlebsac moswavleebs vavalebT. isini rveulebSi muSaoben am amocanebze da dafasTan warmoadgenen pasuxebs sajaro aRiarebis 1 - 10 9 misaRebad. saSinao davalebad vaZlevT amocanebs. am amocanebidan amocanaSi 2004 wlis olimpiur TamaSebze qarTveli sportsmenis, giorgi asaniZis warmatebebzea saubari. misi Sedegis gasagebad aTwiladebis Sekreba gviwevs. SeiZleba moswavleebs davavaloT 2016 wlis olimpiur TamaSebze giorgi asaniZis moswavlis, laSa talaxaZis saocari Sedegebi moiZios internetiT. qarTvel sportsmenTa miRwevebis asaxva amocanebSi zrdis moswavleTa motivacias swavlis procesSi.

25-e gakveTili

Tema: aTwiladebis Sekreba. mizani: moswavles SeeZlos aTwiladebis damrgvalebis gamoyenebiT aTwiladebis jamis miaxloebiTi mniSvnelobis povna (maT. VI. 2.). davalebis Semowmebisas meti yuradReba davuTmoT

10

amocanis amoxsnas, romelic gakveTilze dasaxuli miznis SesrulebasTan aris dakavSirebuli. 10

jamis Sefasebisas moswavlem SeiZleba jer uaxloes mTelebamde daamrgvalos, yoveli damrgvaleba metobiT ganxorcieldeba, amitom SeiZleba daaskvnas, rom jami 27-ze naklebia. sxvanairad, mxolod mTelebis SekrebiTac ki davadgenT, rom jami 24-ze metia. meaTedebis SekrebiT TiTqmis 2 erTeuli grovdeba, e. i. jami miaxloebiT 26-ia, zusti jamia 25,867. im amocanebidan, romlebic klasSi ixsneba, zogierT amocanaSi jamis Sefaseba mo­ iTxoveba – 11 - 20 . magaliTad, 12 amocanaSi moswavleebs mouwevT Sekrebis Sedegebis Sefasebisas damrgvalebis gamoyeneba. a) mTeli nawilebis jamia 9, wiladi nawilebis jami 1-ze metia, e. i. jami metia 10-ze. b) mTeli nawilebis jamia 9, wiladi nawilebis jami 1-ze naklebia, e. i. jami naklebia 10-ze. g) mTelebis jami 11-ia, cxadia, wiladi nawilebis jami 2-ze naklebia, e. i. jami naklebia 13-ze, miT umetes, is naklebi iqneba 15-ze. ricxviTi gamosaxulebis mniSvnelobis Sefasebis unaris ganviTareba Zalze mniSvnelovania praqtikuli saqmianobisas nebismieri profesiis adamianisTvis. zogierT SemTxvevaSi martivad Casatarebeli SefasebiT SeiZleba Tavidan aviciloT Sromatevadi gamoTvlebi. 14 savarjiSoTi moswavles uviTardeba monacemebis raime kanonzomierebiT dala­ gebis unari.

62


15 viyenebT Sekrebis gadanacvlebadobis, anu komutaciurobis (a+b=b+a) da jufTebadobis, anu asociaciurobis (a+b)+c=a+(b+c) Tvisebebs da vajgufebT Sesakrebebs CvenTvis xelsayreli mimdevrobiT: a) (3,47+35,53)+(8,12+11,88)+24,5=39+20+24,5=83,5; b) (4,007+3,023)+(51,12+18,88)+(7+9,17)=7,03+70+16,17=(7,03+16,17)+70=23,2+70=93,2. 16 a) da b) amocanebSi mTeli nawilebis Sekrebac ki sakmarisia, rom moswavleebi darwmundnen _ Sekreba arasworadaa Sesrulebuli. a)-Si 8+4 bevrad naklebia 87-ze, b)Si 37+64>92. g) da d) amocanebSi moswavleebi, albaT zepirad Sekrebas amjobineben. es saukeTeso arCevani iqneba. 17 - 20 amocanebis amoxsnisas moswavleebi gaiazreben aTwiladebis Sekrebis codnis mniSvnelobas sayofacxovrebo situaciebSi gamosayeneblad. es axal, sayuradRebo samotivacio muxts matebs maT damokidebulebas ganxiluli Temisadmi. saSinao davalebad vaZlevs 11 - 20 amocanebs.

26-e gakveTili

Tema: aTwiladebis Sekreba. mizani: aTwiladebis Sekrebasa da gamoyenebaze codnis gantkiceba. winapirobebi: aTwiladebis Sekrebis sxvadasxva xerxis gamoyeneba (zepiri, SefasebiT, qveSmiweriT). resursebi: kalami, rveuli, saxelmZRvanelo, dafa. es gakveTili warmoadgens aRniSnuli Temis Seswavlis bolo fazas â&#x20AC;&#x201C; `Semdgom~, romelic amave gakveTilze dasruldeba. saSinao davalebis Semowmebas SeiZleba 20 â&#x20AC;&#x201C; 25 wT davuTmoT (SeiZleba metic). zogierTi amocanis amoxsna SeiZleba moswavleebs gauWirdeT. am mxriv, yuradsaRebia 10 , 14 , 17 , 19 da 20 amocanebi. SeiZleba mogviwios am amocanebis amoxsnebis garCeva klasSi. 14 SevasruloT Sekreba qveSmiweriT: 3*2,*50 + 27,1** *3*,323 II Sesakrebis meaTasedebis cifri 3-ia, measedis _ 7. Sesakrebebis measedebis SekrebiT 12 miiReba, 2-s vwerT, 1 emateba meaTedebis Tanrigis erTeuls. e. i. Sesakrebebis meaTedebis Tanrigebis jami SeiZleba iyos mxolod 2. I Sesakrebis meaTedi 1 yofila. I Sesakrebis aTeulebSic 1 unda eweros. miviRebT: 312,150 + 27,173 b) *57,*6 339,323 + *5**,1* 2800,15 II Sesakrebis measedebis TanrigSi mxolod 9 SeiZleba eweros. jamSi 5-s vwerT, 1 emateba meaTedebis Tanrigs, e. i. Sesakrebebis meaTedebis Tanrigebis jami unda iyos 10, amitom I Sesakrebis meaTedi yofila 9. erTi emateba erTeulebis Tanrigs, e. i. erTeulebis Tanrigebis jami unda iyos 9 _ II Sesakrebis erTeulebi yofila 2. aTeulis

63


Tanrigsac 1 emateba, II Sesakrebis aTeulis TanrigSi 4 unda eweros, I-is aseulis TanrigSi _ 2, II-is aTaseulis TanrigSic _ 2. miviRebT: 257,96 + 2542,19 2800,15 17 amocana momavalSi ariTmetikuli progresiis Tvisebebis propedevtikaa. maswavlebelma icis, rom amocanis pirobiT (erTi da imave ricxvis mimatebiT) Camowerili ricxvebi aris ariTmetikuli progresiis wevrebi da ar aris SemTxveviTi, rom I da V wevrebis jami II da IV wevrebis jams daemTxveva. cnobilia, rom sazogadod ariTmetikul progresiaSi 2an=an-k+an+k nebismieri k da n naturaluri ricxvebisTvis, roca k<n. SeiZleba moswavleebma Seadginon aseTive tipis sxva mimdevrobac da Sekribon erTi da imave wevrisagan tolad daSorebuli wevrebi, magaliTad, 1; 3; 5; 7; 9; 11; 13; 15; 17 ricxvebidan yoveli momdevno miiReba winasTvis 2-is mimatebiT. 1+17=3+15=5+13=7+11, 1+15=3+13=5+11=7+9, da a. S. SeiZleba gamoTqvan varaudi _ SeamCnion kanonzomiereba, romelsac momavalSi daasabuTeben. 19

a) (6,19+20,81)+(14,4+0,6)=27+15=42, b) (42,79+0,21)+(25,7+35,17+9,13)+6=43+70+6=119. 20 es amocana moswavles uviTarebs praqtikul Cvevebs: qviTris monacemebiT gamoTvalos komunaluri gadasaxadebis odenoba. magaliTad, ori sulisgan Semdgari ojaxisTvis: dasufTavebis mosakrebeli erT sulze 1,2 laria, ojaxi 2,4 lars ixdis. wylis gadasaxadi sulze 2,4 laria, or sulze _ 4,8 lari. eleqtroenergiis gadasaxadi ar aris damokidebuli ojaxis suladobaze, is kvt. sT-ebis odenobazea damokidebuli. vTqvaT, am ojaxs daxarjuli 307 kvt. sT-isTvis qviTris mixedviT 54,33 lari aqvs gadasaxdeli. bunebrivi airis gadasaxadi 28,8 laria. mTliani Tanxa SekrebiT unda iangariSon: 2,4+4,8+54,33+22,8=84,3 lari. aseTi amocanebiT Zlierdeba moswavleTa CarTuloba saojaxo biujetis marTvaSi, maT uviTardebaT yairaTiani cxovrebis upiratesobis gancda, kerZod, rogor SeiZleba daizogos komunaluri xarjebi da am ekonomiiT miRebuli Tanxa moaxmaron ojaxis sxva aucilebel moTxovnebs. darCenili dro SeiZleba `vip~-is rubrikiT warmodgenili amocanebis amoxsnas davuTmoT. am rubrikiT warmodgenili amocanebi sxart pasuxebs moiTxovs, amitom saxelmZRva­ neloSi maTi pasuxebi araa warmodgenili. am amocanebiT kidev erTxel vubrundebiT Seswavlil masalas, vamowmebT misi aTvisebis xarisxs, operaciaTa swrafad Sesrulebis unars (rac zogjer gadamwyvet mniSvnelobasac ki iZens). davalebas Sejibris saxec aqvs, rac zrdis moswavleTa motivacias, axalisebs maT, mopovebuli warmateba ki sixaruls aniWebs, ganawyobs Semdgomi warmatebebisTvis. 23 amocanis pasuxebi: a) 9,279 b) 1,16382 + + 0,684 9,51458 9,963 10,67840

saSinao davalebisTvis viyenebT II Tavis damatebiT 1

64

-

5

amocanebs.


§2. aTwiladebis gamokleba 27-e gakveTili

Tema: aTwiladebis gamokleba. mizani: moswavles SeeZlos aTwiladebis gamokleba, aTwiladebis damrgvalebis gamoyenebiT aTwiladebis gamoklebis Sedegis miaxloebiTi mniSvnelobis povna (maT. VI, 2.) winapirobebi: sxvadasxva xerxiT aTwiladebis Sekreba. naturaluri ricxvebis gamokleba. resursebi: rveuli, saxelmZRvanelo, kalkulatori, dafa. saSinao davalebis Semowmebas SeiZleba 10-15 wT davuTmoT. aqve SeiZleba Sesakrebebis damrgvalebis gamoyenebiT jamis miaxloebiTi mniSvnelobis povnazec gavarjiSeba, Sekrebis Sesrulebisas Sekrebis Tvisebebis gamoyeneba. es yvelaferi gamoklebis aTvisebis winapirobebad SeiZleba ganvixiloT. aTwiladebis gamoklebis ganxilvisas SeiZleba gamoviyenoT saxelmZRvaneloSi mocemuli teqsti; iq dawvrilebiT aris aRwerili gamoklebis procesi, damrgvalebis gamoyenebiT gamoklebis Sedegis Sefaseba, gamoklebisas poziciuri sistemis gamoyeneba, kavSiri naturaluri ricxvebis gamoklebis wesebTan. klasSi SeiZleba amovxsnaT 1

-

8

amocanebi. saxelmZRvaneloSi warmodgenili teqstis mixedviT gamoklebis ramdenime magaliTis ganxilvis Semdeg es `testebi~ SeiZleba moswavleebma damoukideblad daZlion. zogjer mimarTeT moswavleebs, rom miRebuli Sedegebi kalkulatoriT gadaamowmon. gamoklebis dros, rodesac saklebis romelime Tanrigis cifriT gamosaxuli ricxvi naklebia maklebis Sesabamisi Tanrigis cifriT gamosaxul ricxvze, gviwevs mezobeli _ ufro maRali Tanrigidan Tanrigis 1 erTeulis gadatana dabal TanrigSi da misi daSla am Tanrigis erTeulebad. am dros unda gaviTvaliswinoT poziciuri sistemis Tvisebebi. kerZod, 9 1 9 10 9 9 1 9 9 10 1= 10 10 = 10 + 10 = 10 + 100 = 10 + 100 + 100 = 10 + 100 +1000, anu 1=9⋅0,1+9⋅0,01+10⋅0,001. am faqtzea yuradReba gamaxvilebuli saklaso savarjiSoebSi, #1-4. gamoklebis dros im Tvisebebs viyenebT, romlebsac naturaluri ricxvebis gamoklebis drosac viyenebdiT: a–a=0; a–0=a; a–(b–c)=(a+c)–b; a–(b+c)=(a–b)–c _ am bolo Tvisebis ilustracia aseTi martivi magaliTiTac SeiZleba: Tinikos hqonda 100 lari, man SeiZina 20-lariani wigni da 30-lariani CanTa. ra Tanxa darCa Tinikos? SeiZleba jer daxarjuli Tanxa SevkriboT da gamovakloT 100-s: 100–(20+30)=100–50=50; an 100-s gamovakloT jer wignis fasi, Semdeg _ CanTis fasi: (100–20)–30. Sedegi erTi da igivea. (a+b)–c=(a–c)+b. vaxtangs erT jibeSi 50 lari aqvs, meoreSi _ 20 lari, man SeiZina 10-lariani wigni. ramdeni lari darCa mas? 10 lari vaxtangis Tanxas akldeba: (50+20)–10. man SeiZleba es Tanxa 50-larianiT gaistumros, maSin darCeba (50–10)+20. klasSi 1 - 4 savarjiSoebi ar gulisxmobs 1-is miRebas mxolod SekrebiT. moswavlem unda gaiazros, rom yoveli Tanrigis 1 erTeuli mezobeli (ufro dabali) Tanrigis 10 10 10 erTeulis tolia: 1= 10 10 ; 0,1= 100 =10⋅0,01; 0,01=1000=10⋅0,001 da a. S. 0,10=0,09+0,01=0,09+0,010=0,09+0,009+0,001. 4 savarjiSoSi saSinao davalebad vaZlevT 1 - 10 amocanebs.

65


28-e gakveTili

Tema: aTwiladebis gamokleba. mizani: moswavles SeeZlos Sekrebisa da gamoklebis Semcveli ariTmetikuli gamosaxulebis mniSvnelobis povna, aTwiladebis damrgvalebis gamoyenebiT gamosaxulebis miaxloebiTi mniSvnelobis povna, an Sefaseba. winapirobebi: aTwiladebis damrgvaleba, aTwiladebis Sekreba, naturaluri ricxvebis Sekreba da gamokleba. Sekrebisa da gamoklebis Tvisebebi. resursebi: rveuli, saxelmZRvanelo, dafa, kalkulatori. saSinao davalebis Semowmebas didi dro ar dasWirdeba. moswavleebi rigrigobiT kiTxuloben `testebs~ da asaxeleben pasuxebs, zogjer msjeloben pasuxis miRebis wesebze da aRniSnaven sxvaTa Secdomebs (Tu is SeniSnes). merxebs Soris yuradRebiT Camovlisas SeiZleba SevamowoT 9 , 10 amocanebSi davalebebis Sesrulebis xarisxi. Semdeg gadavdivarT leebTan erTad.

9

- 17 amocanebis garCevaze. amoxsnebze vmsjelobT moswav-

11 - 13 amocanebSi moswavleebi, garda amocanis amoxsnisTvis saWiro moqmedebebis

SerCevasa da gamoklebaSi gawafvisa, sayuradRebo informaciasac iZenen. kerZod, olimpiuri TamaSebis, cxovelebis Sesaxeb. amaSi maT rubrika `Cveni leqsikonic~ daexmareba. 11 38,07–(9,5+9,6+9,5)=9,47 wm. 12 10 000 m =10 km. pasuxi: 42,195–10=32,195 (km). 14 a) sxvaobis measedis TanrigSi 9-is misaRebad 11-s unda gamovakloT 2. e. i. saklebis am TanrigSi 1 unda eweros. meaTedebidan erTi erTeulis aRebis Semdeg saklebis meaTedis TanrigSi 3 rCeba. sxvaobis meaTedis TanrigSi 3-is misaRebad maklebis meaTedis TanrigSi 0 unda eweros. miviRebT: 22,41 5,02 17,39 16 erTi SexedviT ubralo amocana SeiZleba moswavleTaTvis gansjis sagani gaxdes.

magaliTad, ratom izrdeba yovel momdevno 50 welSi mosaxleobis odenoba metiT, vidre wina 50-Si. (moswavleebi mixvdebian, rom ukve gazrdili mosaxleoba momdevno TaobebSi met STamomavlobas iZleva); ram ganapiroba mosaxleobis mkveTri mateba bolo 50 welSi (globaluri omebis ararseboba, miRwevebi medicinaSi, soflis meurneobaSi)? amocanis b) kiTxvis pasuxi SeiZleba ornairad miviRoT: 6,45–1,13, an (a)-Si miRebuli Sedegebis gamoyenebiT): 0,47+0,91+3,94. 17 a) maSikos Tavdapirveli Tanxis naxevaria 12,8–1,65=11,15 lari. b) Tavdapirveli Tanxaa 11,15+11,15=22,3 lari. (SevkribeT, radgan aTwiladis gamravleba jer ar uswavliaT). saSinao davalebad vaZlevT

66

11 - 21

amocanebs.


29-e gakveTili

Tema: aTwiladebis Sekreba da gamokleba. mizani: moswavleebi gaiwafon Sekrebisa da gamoklebis Semcveli gamosaxulebis mniSvnelobis povnaSi, gamosaxulebis miaxloebiTi mniSvnelobis povnaSi (maT. VI. 2.), gaifarTovon gamoyenebaTa areali. winapirobebi: aTwiladebis Sekreba da gamokleba, Sekrebisa da gamoklebis Tvisebebi. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis amocanebidan gamovyoT 16 , 18 da snebi gakveTilze dasaxuli miznis miRwevasac emsaxureba.

21

amocanebi, maTi amox-

16

gamoviyenoT Sekrebisa da gamoklebis Tvisebebi: a) (2,68–1,68)+3,15; b) (4,15+5,85)+(8,42–3,42)=10+5. 18

moswavle iRebs nebismier xuT aTwilads maklebebad, saTanado saklebebis sapovnelad yovel maTgans umatebs 0,3-s. magaliTad, maklebia 2,182, saklebi iqneba 2,182+0,3=2,482. marTlac, 2,482–2,182=0,3. SeiZleba aviRoT xuTi saklebi. TiToeuli _ 0,3-ze meti aTwiladi. maklebis sapovnelad yovel maTgans vaklebT 0,3-s. magaliTad, saklebia 1,14, maklebi iqneba 1,14–0,3=0,84. marTlac, 1,14–0,84=0,3. mivceT moswavleebs saSualeba aRweron am sxvaobebis mofiqrebis maT mier SerCeuli strategia da wavaxalisoT isini. 21 b) sxvaoba 6-is tolia, roca x=7,1–6=1,1. TuU x metia am mniSvnelobaze, maSin

sxvaoba naklebi iqneba 6-ze, radgan maklebis gazrda iwvevs sxvaobis Semcirebas. amis gaTvaliswinebiT moswavlem unda daasaxelos 1,1-ze meti raime ricxvi. ar aris gamoricxuli, rom moswavlem SeZlos sityvierad aRweros, an utolobis saxiT Caweros pasuxi: 1,1-ze meti raime ricxvi; an 7,1-ze naklebi da 1,1-ze meti ricxvi; an 1,1<x<7,1 (isini uaryofiT ricxvebs jer ar icnoben). g) ori erTnairi ricxvis SekrebiT unda miiRos 1-ze meti ricxvi. cxadia, Sesakrebebi 0,5-ze meti toli ricxvebia. moswavlem SeiZleba daasaxelos 0,5-ze meti raime ricxvi. ar aris gamoricxuli, rom SeZlos pasuxis sityvierad Camoyalibeba: 0,5-ze meti nebismieri ricxvi; an pasuxis Cawera utolobis saxiT: x>0,5. saSinao davalebis Semowmebas da maTgan zogierTi amocanis garCevas 15-20 wT SeiZleba movandomoT. gakveTilis darCenili monakveTi SeiZleba davuTmoT jgufur mecadineobas. davalebebi saxelmZRvaneloSia warmodgenili. SemoTavazebuli amocanebi sirTulis sxvadasxva donis davalebebs Seicavs da, amrigad, saSualebas gvaZlevs moswavleTa jgufuri muSaobisas xazi gavusvaT samuSaos gonivruli ganawilebis mniSvnelobas. pirveli davaleba saSualo sirTulisaa _ marTalia, moiTxoveba zogadi wesis aRmoCena-Camoyalibeba, magram SeTavazebuli nimuSi mniSvnelovnad amartivebs dasmul amocanas _ is daiyvaneba wesis ganzogadebaze: 9-iT (8-iT) daboloebuli wiladi nawilis mqone aTwiladis mimatebisas SeiZleba mivumatoT uaxloesi `mrgvali~ ricxvi da Semdeg gamovakloT gansxvaveba:

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2+0,9=2+1–0,1 3,5+0,98=3,5+(1–0,02)=3,5+1–0,02. meore davaleba ukve moZiebuli da Camoyalibebuli wesis ilustracias eTmoba, rac, bunebrivia, mowafisgan gacilebiT nakleb kompetencias moiTxovs. a) 6,4+1,9=6,4+2–0,1=8,4–0,1=8,3 b) 7,5+1,9=7,5+2–0,1=9,4–0,1=9,3; ................ z) 6,45+1,99=6,45+2–0,01=8,45–0,01=8,44. mesame davaleba moswavlisgan azrovnebis maRal dones moiTxovs; is SeiZleba davSaloT ramdenime etapad _ ramdenime kerZo magaliTis ganxilva _ es dakvirvebaa; kanonzomierebis SemCneva da misi Camoyalibeba _ es ganzogadebaa; gamoTqmuli varaudis kritikuli gansja da, calkeul SemTxvevebSi, praqtikuli magaliTebiT Semowmeba _ es Sefasebaa. Tu gaviTvaliswinebT, rom a–(b–c)=a–b+c, miviRebT 6–1,99=6–(2–0,01)=6–2+0,01=4+0,01=4,01. am, an Semdeg gakveTilze SeiZleba CavataroT fokusirebuli wera, romlis saSualebiT davadgenT Temis Seswavlis xarisxs. gTavazobT amocanebs damoukidebeli samuSaosTvis. 3 + 4 , b=12+ 2 + 5 . ipoveT a+b. 1. a=10+ 10 100 10 100 2. SeasruleT moqmedeba: 15,83–10,61 15,83–10,9

15,83–10,19 15,8–10,61

3. ipoveT gamosaxulebis mniSvneloba: 28,15–20,2+31,35–,7,9. 4. SekribeT da jami daamrgvaleT measedebamde: 18,426+22,89. 5. SarSan skolis laboratoriebis ganaxlebisTvis daixarja 925,5 lari, rac 120,8 lariT naklebia wels amave saqmisTvis gamoyofil Tanxaze. ra Tanxa gamoiyo wels laboratoriebis ganaxlebisTvis? saSinao davalebisTvis SeiZleba gamoviyenoT II Tavis damatebiTi 6 - 10 amocanebi. SeiZleba damatebiT maRali mzaobis moswavleebs vTxovoT ifiqron `moisazres~ rubrikiT warmodgenil amocanebze.

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30-e gakveTili

Tema: aTwiladebis Sekreba da gamokleba. mizani: moswavleebi gaiwafon aTwiladebis Sekrebasa da gamoklebis Semcveli gamosaxulebebis mniSvnelobebis povnaSi, damrgvalebis gamoyenebiT miaxloebiTi mniSvnelobis povnaSi (maT. VI. 2.). winapirobebi: aTwiladebis Sekreba da gamokleba. resursebi: rveuli, saxelmZRvanelo, dafa. gakveTilis dro pirobiTad (miaxloebiT) or tol nawilad davyoT. pirveli nawili davuTmoT saSinao davalebis Semowmebas, `moisazres~ rubrikiT warmodgenili amocanebis amoxsnas, meore nawili ki didaqtikur TamaSs `jamis Sefaseba ar meSleba~. rubrika `moisazres~ amocanebi amjerad aTwiladebze moqmedebebs eZRvneba. 1 b) miTiTebis mixedviT, saZiebeli ricxvis mTeli nawili unda iyos 24, g)

miTiTebis mixedviT, saZiebel ricxvs aqvs saxe: 24,*6, d) miTiTebis mixedviT, 2+4+*+6=13, saidanac miviRebT *=1. pasuxi: 24,16. 2 II ricxvi miiReba pirveli ricxvidan 0,07-is mimatebiT; III miiReba II-dan 0,08is mimatebis mixedviT, IV ki III-dan 0,09-is mimatebiT. savaraudod, momdevno wevrebis misaRebad wina wevrs unda mivumatoT mimdevrobiT, 0,10; 0,11; 0,12. miviRebT momdevno sam wevrs: 0,43; 0,54; 0,62.

I gamosaxulebis mniSvnelobaa 21,84. II gamosaxulebis yoveli Sesakrebi I gamosaxulebis Sesabamis Sesakrebs `avsebs~ 10-mde (7,82=10–2,18, 5,64=10–4,36 da a. S.), amitom II gamosaxulebis mniSvneloba iqneba 40–21,84=18,16. 3

es amocana sayuradReboa mravali TvalsazrisiT: moswavlem unda miagnos 6 amocanis amoxsnis `gasaRebs~. misi meSveobiT ki Tanmimdevrulad moZebnos sxva ricxvebic (gasaRebs warmogvidgens erT-erT diagonalze ganlagebuli oTxi ricxvis jami). es kvadrati sainteresoa erT-erTi diagonalis mimarT erTgvari `simetriiT~, meore diagonalze ganlagebuli ricxvebis CanawerebiT. aseTi cxrilebis analizi moswavles garkveulad aaxloebs matricis cnebasTanac. moswavleebma SeiZleba gamoiyenon koordinatebic da miuTiTon, rom, magaliTad, (a; 2) da (b; 1) kvadratebSi ricxvebi erTi da imave cifrebiTaa Cawerili, maTi Tanamimdevroba ki gansxvavebulia.

TamaSi _ `jamis Sefaseba ar meSleba~ am TamaSis mizani miRebuli codnis praqtikuli gamoyenebis unaris ganmtkicebaa; amasTanave, principuli mniSvneloba eniWeba moqmedebis siswrafesac _ kargi iqneba, Tu daawesebT yoveli svlisTvis drois limits. TamaSis msvleloba detalurad aris aRwerili saxelmZRvaneloSi. SevniSnoT, rom aseTi TamaSebis Catareba Zalze sasargebloa maswavleblis muSaobis Sedegebis Sesa­ fasebladac _ TamaSis dros moswavle maqsimalurad moti­ virebulia warmatebis misaRwevad da mis mier daSvebuli SecdomebiT SesaZlebelia swavlebis xarvezebis aRmoCena. unda gaviTvaliswinoT

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agreTve moswavleTa indivi­dualuri Tvisebebic. zogierTisTvis forsirebuli reJimiT muSaoba damabnevelia. aseTebs ufro damzogavi reJimi SeurCieT. Semdegi gakveTilisTvis SeiZleba davavaloT saxelmZRvaneloSi mocemuli proeqtis Sesruleba. proeqti SemoTavazebuli proeqti Seicavs sxvadasxva donis davalebebs. gazeTis moZieba, wiladuri monacemebis SerCeva, amowera, damrgvaleba. moswavlisTvis iolad daZ­ le­ vadia yvela aqtivoba. Tumca, msgavsi davalebebis Sesruleba Zalze sasargebloa mos­ wavleTaTvis, zrdis maT damoukideblobas, ganaTlebuloba-gacnobierebas, ayalibebs informaciis moZieba-klasificireba-damuSavebis unar-Cvevebs. proeqtis ufro Znel nawils warmoadgens Catarebuli muSaobis Sefasebis etapi _ moswavlem unda gaaanalizos Sedegebi da gamoarCios is SemTxvevebi, roca monacemTa damrgvaleba ama Tu im Tanrigamde arsebiTad amaxinjebs informacias da gamousadegars xdis mas. magaliTad, azri ara aqvs kalaTburTelTa gundis wevrebis simaRleTa metrebiT warmodgenisas mTelamde damrgvalebas, iseve, rogorc 100-metrian distanciaze rbenis ukanaskneli ramdenime rekorduli monacemis wamis meaTedebamde damrgvalebas.

§3. aTwiladebis gamravleba 31-e gakveTili

Tema: aTwiladebis gamravleba. mizani: moswavles SeeZlos aTwiladis naturalur ricxvze, aTwiladis aTwiladze gamravlebis procesis aRwera, ricxvebze moqmedebebis gamoyeneba aTwiladebis gamravlebisas, 10-is xarisxze gamravlebis wesis gamoyeneba, ricxvebis damrgvalebis gamoyenebiT namravlis povna miaxloebiT (maT. VI. 2). aTwiladze gamravlebis gamoyeneba zomis erTi erTeulis meoresTan dakavSirebisas (maT. VI. 3). winapirobebi: naturaluri ricxvebis gamravlebis algoriTmis codna, 10-is xarisxebze gamravlebis codna. resursebi: plakati, romelzec gamosaxulia gamravlebis Tvisebebi da am Tvisebebis gamoyenebis magaliTebi; rveuli, saxelmZRvanelo, dafa, kalkulatori. gakveTili SeiZleba daviwyoT moswavleebis mier Sesrulebuli im proeqtis prezentaciiT, romelic davalebad wina gakveTilze miiRes. Tumca, zogierTma maswavlebelma SeiZleba prezentaciebis mowyoba Semdegi gakveTilisTvis gadados, radgan am gakveTiliT iwyeba aTwiladebis gamravlebis Seswavlis I faza `winaswar~ da aqve gadadis is fazaSi `ganmavlobaSi~. axal masalaze gadasvlisTvis aucilebeli winapirobaa naturaluri ricxvebis gamravlebis algoriTmis gaxseneba. ganvixilavT naturaluri ricxvebis gamravlebis magaliTebs, vxsniT gamravlebis wess.

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magaliTad, 23⋅42 gamravlebisas 23 jer mravldeba 2-ze, Semdeg 40-ze Cven ki al­ goriTmSi warmovadgenT 4-ze gamravlebas, magram poziciaTa wanacvlebiT vaRwevT 40-ze gamravlebas (0-s ar vwerT, mas vgulisxmobT). analogiuri principiT xdeba aTwiladebis gamravlebac. Sedegad, namravlSi mZimis marjvniv imdeni cifri unda iyos, ramdenicaa mZimis marjvniv orive TanamamravlSi erTad. amitom aTwiladebis gamravlebis procesSi mZimeebs mxolod namravlSi iTva­ liswineben. cxadia, moswavleebi am moqmedebebs SemdgomSi meqanikurad Seasruleben. sasurvelia gamoviyenoT teqstSi SemoTavazebuli amocana an msgavsi amocana, romlis amoxsnis procesSi Tanamamravlebis damrgvalebiT namravlis Sefasebas praqtikuli gamoyeneba aqvs, raTa moswavlem gaiazros damrgvalebis praqtikuli mniSvneloba. zogierT SemTxvevaSi ki mxolod SefasebiT Semoifarglos. saxelmZRvaneloSi mocemuli teqstis mixedviT masalis axsna moswavleTa aqtiuri monawileobiT unda mimdinareobdes. moswavleebma TviTon unda SeZlon ganxiluli magaliTebis saSualebiT aTwiladebis gamravlebis wesis Camoyalibeba. moswavleebma unda SeZlon im plakatze warmodgenili Canawerebis axsnac, romelzec gamravlebis Tvisebebi da am Tvisebebis gamoyenebebia gamosaxuli. gakveTilis am nawilis xangrZlivoba SeiZleba 15 wuTiT ganisazRvros. Semdegi 20 wuTi davuTmoT jgufur muSaobas im ori davalebis mixedviT, romelic saxelmZRvaneloSia warmodgenili. teqstSi SemoTavazebulia jgufebad gaerTianebul moswavleebTan nimuSebis safuZ­ velze (magaliTebze dakvirvebiT) aTwiladis 10-is xarisxebze gamravlebis wesis migneba, aRwera, Camoyalibeba. klasSi SemoTavazebuli savarjiSoebiT moswavleebi gaiwafebian miRebuli codnis gamoyenebaSic. 5 - 7 savarjiSoebiT moswavleebi gaiwafebian namravlSi mZimis adgilis SerCevaSi. Tanamamravlebis mTeli nawilebis gadamravlebiT daaskvnian, rom namravli 9 metia 120-ze. Tanamamravlebis umaRles Tanrigamde damrgvalebisas miiReben: 4⋅30=120. yoveli damrgvaleba naklebobiT moxda, e. i. zusti namravli metia 120-ze. 10 - 13 savarjiSoebSi iyeneben gamravlebis Tvisebebs, romlebic paragrafis teq­ stSia warmodgenili: 0,2⋅0,3⋅5=0,3⋅0,2⋅5=0,3⋅ (0,2⋅5)=0,3⋅1=0,3 10   gadanacvlebadoba jufdebadoba erTis Tviseba 13 4⋅3⋅1,5=3⋅4⋅1,5=3⋅(4⋅1,5)=3⋅6=18

savarjiSoebSi moswavleebi gamoiyeneben aTwiladis 10-is xarisxze gam­ rav­ lebis wess (jgufuri muSaobisas ukve aqvT mignebuli). cxadia, moswavleebTan ar vaxsenebT 10-is mTel uaryofiT xarisxebs, maT Sesaxeb vambobT: `aTwiladebi, romlis aTwiladi Tanrigebis bolo Tanrigis cifria 1, danarCeni Tanrigebis cifrebi ki mTelebis CaTvliT nulebia~. am amocanebidan 10 da 17 amocanebi exmianeba standartis maT. VI. 3-is misaRwev Sedegs – iyenebs aTwiladebze gamravlebas zomis mcire erTeulis did erTeulTan Tanafardobis gamosaxvisTvis: 5,49 metri=5,49⋅100sm=549 sm, radgan 1 m=100 sm. 0,04 c=4000 g, radgan 1 c=100000 g (1 c=100 kg=100000 g). klasSi 20 amocanasac vxsniT, xolo saSinao davalebas moswavleebs 1 - 20 amocanebidan SevurCevT. 14 - 19

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32-e gakveTili

Tema: aTwiladebis gamravleba. mizani: moswavles SeeZlos aTwiladebze gamravlebis wesis gamoyeneba, aTwiladebis gamravlebisas gamoTvlebis gasamartiveblad gamravlebis wesebis gamoyeneba, aTwiladebze gamravlebis gamoyeneba zomis erTi erTeulis meore erTeulTan Tanafardobis gamosaxvisTvis, gamravlebisa da gayofis Semcvel tolobebSi ucnobi komponentis povna (maT. VI. 2, maT. VI. 3). winapirobebi: naturaluri ricxvebis gamravleba, naturaluri ricxvebis namravlis miaxloebiTi mniSvnelobis povna Tanamamravlebis damrgvalebis gamoyenebiT. resursebi: rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT, romelsac SeiZleba miaxloebiT 15 wT davuTmoT. yuradReba gavamaxviloT 16 - 19 amocanebze, romlebic maT. VI. 3-Si miTiTebul misaRwev Sedegs ukavSirdeba; 0,4 kg=0,4⋅1000 g=400 g; 4,8 dm=4,8⋅10 sm=48 sm. klasSi vxsniT 21 - 31 amocanebs. 31 amocanaSi gayofis Semcvel tolobebSi ucnobi gasayofis povna moiTxoveba –

ucnobi gasayofi ganayofisa da gamyofis namravlia; x:3=1,7, x=1,7⋅3=5,1. vxsniT amocanebs, romelSic proporciuli damokidebulebebis codna gamoiyeneba. 25 Tu 1 saaTis ganmavlobaSi 1 naTura xarjavs 1,26 TeTris eleqtroenergias, maSin 1 naTura 10 saaTSi daxarjavs 1,26⋅10=12,6 (TeTrs) 12 saaTSi – 1,26⋅12=12,6+2,52=15,12 (TeTri) 27 Tu 1 dolari 2,38 larSi icvleba, maSin 100 dolari gadaicvleba 238 larSi. 150 dolari – 238+119=357 (larSi).

ramdenime amocana gamravlebis wesis da Tvisebebis gamoyenebazea, magaliTad, 22 a) 1,25⋅8,78⋅80 = 8,78⋅1,25⋅80 = 8,78⋅ (1,25⋅80)= 8,78⋅100=878;   viyenebT: gadanacvlebadobas jufTebadobas

b) 0,4⋅19,807⋅2500=19,807⋅ (0,4⋅2500)=19,807⋅1000=19807 sasurvelia, moswavleebma yoveli Tvisebis gamoyeneba daafiqsiron. amiT kidev erTxel warmoaCenen sakuTar codnas, an aRmoaCenen xarvezebs da izruneben maT aRmofxvraze. saSinao davalebad vaZlevT

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21 - 25

amocanebs.


33-e gakveTili

Tema: aTwiladebis gamravleba. mizani: moswavles SeeZlos aTwiladebis gamravlebisa da gayofis Semcvel tolobebSi ucnobi komponentis povna, aTwiladebis namravlis miaxloebiTi mniSvnelobis povna, proporciulobis gamoyeneba amocanebis amoxsnisas (maT. VI. 2. maT. VI. 3). resursebi: rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT. es amocanebi gakveTilze dasaxul yvela mizans exmianeba. 21

Tu y:0,2=14,48, maSin y=14,48⋅0,2=2,896

22 Tu 1 pirvelklaselisTvis 1 dReSi gamoyofilia 1,4 lari, maSin 46 pirvelklaselisTvis 22 dReSi saWiro iqneba. 22⋅46⋅1,4 lari=1416,8 lari, anu 1416 lari da 80 TeTri. 23

radgan 1 dm=10 sm, amitom

25,6 dm=25,6⋅10=256 sm radgan 1 km=1000 m=10000 dm=100000 sm, amitom 0,45km=0,45⋅100000=45000sm. I ricxvia 9,67, II ricxvi _ 9,76⋅9=87,03. maTi jamia 96,7. waxalisebas daimsaxurebs moswavle, romelic mixvdeba, rom saZiebeli jami I ricxvze 10-jer metia da amiT amocanas amoxsnis II ricxvis 9-ze gamravlebis gareSe: 9,67⋅10=96,7. 24

25

gamoviyenoT saxelmZRvanelo: III striqonSi erTeulebis Tanrigi 0-ia, IV striqonSi erTeulebis Tanrigi _5. amitom I Tanamamravlis measedi aris 5. cxadia, I Tanamamravlis erTeulis TanrigSi 3 weria, radgan misi 2-ze gamravlebiT miRebuli ricxvi (III striqoni) 7-iT iwyeba. miviReT: I Tanamamravlia 3,85. maSin vRebulobT, rom II Tanamamravlis erTeulis TanrigSi 1-ia. pasuxi:

3,85 41,2 770 385    1540   158,620 ×

saSinao davalebis Semowmebis Semdeg vatarebT jgufur mecadineobas saxelmZRvaneloSi warmodgenili masalis gamoyenebiT. amocanebi jgufuri muSaobisTvis es amocanebi namravlis an Tanamamravlebis SefasebaSi gawafvas emsaxureba. maTi amoxsna moswavles uviTarebs logikuri azrovnebis, analizis unars. jgufSi muSaoba mas ufro komunikabelurs xdis, eCveva sxvisi azris mosmenas, Sefasebas, oponirebas, sakuTari azris gamoTqmas, dacvas, gansjis procesSi warmoSobili mosazrebebidan racionaluri ideis aRmoCenas da mis ganviTarebas, sxvaTa darwmunebas sakuTari mosazrebis sisworeSi. 1

mxolod mTeli nawilebis gamravlebiTac daadgenen, rom udidesia b) namravli, umciresi _ g) namravli. erTze naklebi ori ricxvis namravli 1-ze naklebia, amitom g) namravli 1-ze naklebia; mxolod mTeli nawilebis gadamravlebiTac davadgenT, rom b) namravli 1-ze

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metia; a) namravlis naklebobiT damrgvalebuli Tanamamravlebis namravli 0,3⋅4=1,2 aris 1-ze meti, amitom a) namravli miT umetes metia 1-ze. jgufebi Sefasdeba ara mxolod pasuxis warmodgeniT, aramed am pasuxis dasabuTebiT. pasuxebis Semowmeba weriTi gamravlebiT moswavleTa romelime jgufs daevaleba. a) ori ricxvis namravli, romelTagan TiToeuli metia 1-ze, metia erTze. g) ori ricxvis namravli, romelTagan TiToeuli naklebia 1-ze, naklebia 1-ze. konkretuli magaliTebiT SeiZleba Seamowmon TavianTi pasuxebi. magaliTad, a) SemTxvevaSi 1,2⋅1,05=1,26>1, g) SemTxvevaSi 0,9⋅0,95=0,855<1. sazogadod, raime a>0 ricxvis 1-ze met ricxvze gamravlebisas miiReba a-ze meti ricxvi, 1-ze nakleb ricxvze gamravlebisas miiReba a-ze naklebi ricxvi. b) SemTxvevaSi namravls ver SevafasebT, is SeiZleba 1-ze metic aRmoCndes, 1-ze naklebic da erTis tolic. amis dasabuTeba martivad SeiZleba: magaliTad, 0,9⋅3,2=2,88>1; 0,6⋅1,2=0,72<1; 0,4⋅2,5=1. 2

es amocana winas ganzogadebaa, a) SemTxvevaSi m-s ver SevadarebT 1-s. amis dasabuTeba magaliTebiT SeiZleba: I, m=1,2, n=2, k=2,4; II. m=0,9, n=2, k=1,8; III. m=1, n=2,5; k=2,5. am SemTxvevebSi n>1 da k>1. am SemTxvevebSi m aRmoCnda 1-ze metic, 1-ze naklebic da 1-is tolic. b) erTze meti ricxvi (n>1) unda gavamravloT 1-ze naklebze, rom miviRoT 1-ze naklebi ricxvi (k<1). e. i. m<1. g) 1-is (k=1) misaRebad 1-ze meti ricxvi (n>1) unda gavamravloT 1-ze naklebze. e. i. m<1. d) m-s ver SevadarebT 1-s. marTlac ganvixiloT sami magaliTi: m⋅0,9=0,1, saidanac m<1; m⋅0,1=0,9, saidanac m>1; m⋅0,7=0,7, saidanac m=1. e) 1-ze meti ricxvis (k>1) misaRebad 1-ze naklebi ricxvi (n<1) unda gamravldes 1-ze met ricxvze, e. i. m>1. v) SemTxveva g)-s msgavsia. jgufebs SevTavazoT TiToeuli SemTxvevisTvis moifiqron magaliTebi. a) SemTxve­ visTvis ukve SemogTavazeT 2 magaliTi. b) SemTxveva: vTqvaT, n=3,5, k=0,5. maSin m SeiZleba iyos, magaliTad, 0,1. marTlac, 0,1⋅3,5=0,35<1. g) SemTxvevaSi m da n Sebrunebuli ricxvebia. magaliTad, Tu n=1,25= 54 , maSin m= 45 =0,8. 3

saSinao davalebisTvis gamoviyenebT 11 - 15 davalebebs II Tavis damatebiTi amocanebidan. maRali mzaobis moswavleebs daavaleT Semdegi – me-4 paragrafis pirvel gverdze warmodgenili masalis gacnoba.

§4. aTwiladis gayofa naturalur ricxvze 34-e gakveTili

Tema: aTwiladis gayofa naturalur ricxvze. mizani: moswavleebis mier aTwiladis naturalur ricxvze gayofis gaazreba da Sesabamisi wesis aRwera, erTianiTa da mis marjvniv miwerili nulebiT Caweril ricxvze gayofis wesis aRmoCena da gamoyeneba, aTwiladis naturalur ricxvze gayofis gamoyeneba zomis erTeulebs Soris Tanafardobis gamosaxvisTvis (maT. VI. 2.).

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winapirobebi: naturaluri ricxvebis gamravleba da gayofa. resursebi: plakati, romelzec gamosaxulia aTwiladis 10-is xarisxebze gamrav­ lebisa da gayofis magaliTebi, rveuli, saxelmZRvanelo, dafa. saSinao davalebis SemowmebiT vajamebT aTwiladebis gamravlebis wesis gamoyenebas, 10-is xarisxebze gamravlebis wesis gamoyenebas. es procesi axali masalis asaxsnelad saWiro Semamzadebeli samuSaoc aris. SeiZleba CavTvaloT, rom aq sruldeba wina Temis Seswavlis bolo faza da iwyeba axali Temis Seswavlis sawyisi faza `winaswar~. Semdeg, ZiriTadad, saxelmZRvaneloSi warmodgenili teqstis mixedviT ganvixilavT aTwiladis naturalur ricxvze gayofis magaliTebs, 10-is xarisxebze gayofis wesi ki jgufuri muSaobis Catarebis Sedegad Camoyalibdeba. am process mosdevs klasSi 1 12 amocanebis amoxsna, romliTac SevafasebT dasaxuli miznis miRwevis dones. paragrafs viwyebT samotivacio praqtikuli amocaniT, romlis amoxsna dakavSi­ rebulia aTwiladis naturalur ricxvze gayofasTan. am da sxva magaliTebSi gayofis process saxelmZRvanelo dawvrilebiT aRwers. maTze dakvirvebis Sedegad savaraudoa, rom moswavle SeZlebs gayofis gaazrebas da saTanado wesis aRweras. es wesic Tvlis aTobiT poziciur sistemasTanaa dakavSirebuli. teqstSi gayofis Sedegs gamravlebiT vamowmebT xolme, anu moswavle gayofas aRiqvams, rogorc gamravlebis Sebrunebul operacias. amitom mosalodnelia, rom moswavleTa jgufebi 10-is xarisxebze gayofisas, gamravlebis msgavsad, mxolod mZimis gadanacvlebiT dakmayofildebian. maswavleblis mier SeTavazebul magaliTebze dakvirvebiT klasi aRmoaCens 1-ianiTa da mis marjvniv miwerili nulebiT gamosaxul naturalur ricxvze gayofis wess. Tu gamravlebis dros mZime gadagvqonda marjvniv, gayofisas gadagvaqvs marcxniv imdeni TanrigiT, ramdeni nulicaa gamyofSi. am wesiT Sesrulebuli gayofis Sedegi moswavleTa jgufebma unda Seamowmon Cveulebrivi gayofisa da gamravlebis algoriTmebis gamoyenebiT. klasSi Sesasrulebeli savarjiSoebiT moswavle gaiwafeba aTwiladebis naturalur ricxvze gayofaSi, amocanebis amoxsnisTvis saWiro moqmedebebis SerCevaSi. saSinao davalebad vaZlevT 1 - 13 amocanebs.

35-e gakveTili

Tema: aTwiladis naturalur ricxvze gayofa. mizani: aTwiladis naturalur ricxvze gayofis gamoyeneba gamravlebisa da gayofis Semcvel tolobebSi ucnobi komponentis povnisas (maT. VI. 2.). moswavlem ganimtkicos codna aTwiladis naturalur ricxvze gayofis SesrulebaSi da gamoiyenos es codna praqtikul amocanebSi (maT. VI. 2.). winapirobebi: naturaluri ricxvebis gamravleba da gayofa. resursebi: rveuli, saxelmZRvanelo, dafa, kalkulatori. saSinao davalebis Semowmebas SeiZleba 15 wuTi dasWirdes. saSinao davalebis amo8 9 da amocanebze. maTSi 10-is xarisxebze canebidan yuradReba gavamaxviloT gayofa gamoiyeneba zomis erTeulebs Soris Tanafardobebis gasarkvevad. gakveTilze dasaxul erT-erT mizans Seesabameba 15 amocanis amoxsna, romelic moswavleebTan erTad mimdinareobs. klasSi vxsniT 13 - 19 amocanebs.

75


13 amocanaSi SeiZleba jer 1 detalis masa gaigon (4,4 kg), Semdeg _ 15 detalis. SeiZleba daweron proporcia: 26:15=114,4:x, an 114,4:26=x:15, sadac x aris 15 cali detalis masa. 16 amocanebSi zogjer (praqtikuli mosazrebidan gamomdinare) viTxovT moqmedebis

Sedegis winaswar Sefasebas. am amocanaSi winaswari SefasebiT wigni 4-ze meti da 5-ze naklebi lari unda Rirdes, miaxloebiT _ 4,5 lari. wignis zusti Rirebuleba 4,65 laria. SeiZleboda 23,25-is gayofis nacvlad gamoTvlebis gamartivebis mizniT es ricxvi 10-ze gagveyo da Sedegi 2-ze gagvemravlebina: 23,25:10⋅2=2,325⋅2=4,65. 17 moswavlem unda gaiazros, rom Tanabrad ganawilebuli 16 sakidiT fardis dasamagreblad farda 15 nawilad daiyofa (da ara 16-ad). e. i. sakidebs Soris aris 2,4 m : 15 = 0,16 m. 18 da 19 amocanebi moiTxovs naturalur ricxvze gayofis Sesrulebas. 18 50-jer Semcirebul gegmaze bilikis sigrZe 50-jer mcirdeba da unda Sevasru-

loT gayofa: 4,8:50. 19 ramdenjer mZimea 10,3 kg zRvis wyali 8 kg navTobze?

15 Tu 30,6:x=15, maSin

x=30,6:15

30,6:15=2,04 30    60 60 0

10,3:8=1,2875 8 23 16 70 64 60 56 40 40 0

saSinao davalebad moswavleebs vaZlevT 14 - 20 amocanebs; amasTanave vTxovT gaecnon `moisazres~ rubrikiT warmodgenil amocanebs da moifiqron maTi amoxsnis gzebi (maRali mzaobis moswavleebis davaleba).

36-e gakveTili

Tema: aTwiladis naturalur ricxvze gayofa. mizani: moswavleebi gaiwafon aTwiladis naturalur ricxvze gayofasa da aTwiladebis gamravlebaSi; amocanebis amoxsnisas sxvadasxva sqemebis gamoyenebaSi. winapirobebi: naturaluri ricxvebis gamravleba da gayofa, aTwiladebis gamravleba; wiladis Cawera aTwiladad; wiladis aRqma, rogorc mTelis nawilis. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebisas (10-15 wT) yuradRebas vamaxvilebT Semdeg amocanebze: saSinao davalebis 14 savarjiSoTi moswavle gaiwafeba garkveuli kanonzomiere­ bebiT aRwerili ricxvebis povnasa da cxrilSi ganTavsebaSi. 18 pirobiT, (3,56+x)4=17,8, saidanac x=0,89.

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19

es amocana klasSi amoxsnili 17 amocanis msgavsia. yvavilebs Soris Sualedebis odenobaa 2,5:0,5=5. e. i. dargulia 6 yvavili:



yovel mwkrivSi yvavilebs Soris aris 30:0,6=50 Sualedi, e. i. yovel mwkrivSi 51 yvavilia darguli, sam mwkrivSi _ 153 yvavili. 20

saSinao davalebis Semowmebis Semdeg moswavleebTan erTad vmsjelobT `moisazres~ 1 amocanaze, SevTavazebT maT amoxsnis sqemas. amis Semdeg SedarebiT maRali mzaobis moswavleebs vTxovT gamoiyenon warmodgenili sqema da Seecadon analogiurad amoxsnan `moisazres~ sxva amocanebi. Cven ki danarCenebTan erTad vxsniT amocanebs. gTavazobT ramdenime amocanas. maTgan, moswavleTa SesaZleblobebis mixedviT, arCevans Tqven gaakeTebT. 1. SeasruleT gayofa: 7,14:6 21,15:5 6,5:4 8,4:5

12,824:4 25,2:12

2. SeasruleT gayofa da SeamowmeT gamravlebiT: 11,6:5 8,4:7 6,15:15 15:4 10:8 12:5 3. amoxseniT gantoleba: 2x=1,12 4x=8,4

3x=10,2

4. Tu zogierTi darRvevisas jarimas droulad ar gadaixdiT, gadasaxadi 10-jer izrdeba. a) ra Tanxa geqnebaT gadasaxdeli aseTi darRvevisas 23,4 lariani jarimis nacvlad, Tu mas droze ar gadaixdiT? b) Teonam jarimis gadaxda daagviana da 42,5 laris gadaxda mouwia. ramdeni lari iyo Teonas jarima Tavdapirvelad? 5. avtomobilis dasaqiraveblad 12 adamians 126 lari sWirdeba. ramdeni unda gadaixados TiToeulma, Tu am Tanxas Tanabrad gainawileben? bolo ori amocanis amoxsnis procesSi SeiZleba mTeli klasi CavrToT. moswavleTa nawilTan aseTi individualuri muSaobis Semdeg erToblivad ganvixilavT `moisazres~ rubrikiT warmodgenil yvela amocanas. es amocanebi mTelis nawilis gaazrebasa da gamoTvlas, amoca­nebis amoxsnisas sqemis gamoyenebis unaris ganviTarebas da aTwiladebze moqmedebebis gamoyenebaSi gavarjiSebas eZRvneba. 1

27,6 kg SeiZleba sqemis gamoyeneba:

I yuTi II yuTi

I yuTSi meoreze 27,6 kg-iT metia, amitom 1 marTkuTxedi warmoadgens 27,6 kg-s. pasuxi: I yuTSi aris 2⋅27,6=55,2 kg, II yuTSi _ 27,6 kg. (aseTi sqema, faqtobrivad amocanis gantolebiT modelirebis propedevtikaa).

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2

kvlav sqemas viyenebT:

I yuTi II yuTi

pirobis Tanaxmad, I yuTSi 27,6 kg-iT metia, e. i. 3 marTkuTxedi 27,6 kg-s warmoadgens 1 marTkuTxedi _ 9,2 kg-s. pasuxi: I yuTSi aris 9,2⋅4=36,8 kg, II yuTSi _ 9,2 kg. amocanis Sedgenas zogierTi moswavle sqemis daxazviT, sqemaze sidideebis 3 miTiTebiT daiwyebs. Semdgom ki saTanado teqstsac Seadgens. Tu didi yuTebidan amoviRebT 136,2 kg vaSls, maSin sul yuTebSi darCeba 4 430,2–136,2=294 kg vaSli da patara da did yuTebSi vaSlebis masebi gaTanabrdeba. patara yuTebSi yofila 294:2=147 kg vaSli; TiToO yuTSi _ 12,25 kg. e. i. patara yuTebis odenobaa 147:12,25=12. did yuTebSi yofila 430,2-147=283,2 kg vaSli, TiTo yuTSi _ 17,7 kg. didi yuTebis odenobaa 283,2:17,7=16. moswavleebs vavalebT Seasrulon davalebebi, romlebic individualuri muSaobis proeqtSi aris warmodgenili.

37-e gakveTili

Tema: aTwiladebis gamravleba, aTwiladis naturalur ricxvze gayofa. mizani: moswavle gaiwafos aTwiladebze Seswavlili moqmedebebis gamoyenebaSi praqtikuli amocanebis amoxsnisas. winapirobebi: aTwiladebze moqmedebebi _ Sekreba, gamokleba, gamravleba, naturalur ricxvze gayofa. resursebi: kalami, rveuli, saxelmZRvanelo. moswavleebi warmoadgenen TavianT mosazrebebs proeqtiT gansazRvruli davalebis Sesrulebaze, ris Semdegac, amocanebis amoxsniT, vagrZelebT muSaobas aTwiladebze ariTmetikuli moqmedebebis gamoyenebaSi. proeqtis mizania, moswavle gaiwafos saqonlis yidva-gayidvisas mogebis gamoTvlaSi, sxvadasxva variantidan ufro momgebianis SerCevaSi; praqtikuli saqmianobisas ricxvebze moqmedebebis gamoyenebaSi. moswavlem jer unda gamoTvalos, 9 tona tvirTis (erTi avtomobiliT) TbilisSi gadazidva ramdenad sarfiania, anu Seadaros TbilisSi da foTSi am tvirTis realizebis Sedegebi: TbilisSi yoveli tona 160 lariT ufro Zvirad iyideba, vidre foTSi. 9 tonaSi movigebT 160⋅9=1440 lars. satransporto xarjebis gaTvaliswinebiT erTi manqaniT miRebuli mogebaa 1440–350=1090 lari. 170 tona tvirTis gadazidvisas 18 avtomobili mTlianad Seivseba, me-19-Si 8 tona tvirTs CavtvirTavT. 8 tona tvirTi, gadazidvis 350 laris gaTvaliswinebiT, iZleva mogebas _ 8⋅160–350=930 lari. e. i. Tu gadasazidia 170 t tvirTi, ufro momgebiania mTliani tvirTis TbilisSi Catana, risTvisac 19 avtomobilia (reisia) saWiro. Tu 200 tona tvirTs gadavzidavT, 22 avtomobili mTlianad Seivseba da darCeba kidev 2 tona. am SemTxvevaSic (wina SemTxvevis msgavsad) mTeli tvirTis TbilisSi realizeba ufro momgebiania, vidre foTSi realizeba. Tumca moswavleebi albaT SeniSnaven, rom

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darCenili 2 tonisTvis 23-e avtomobilis daqiraveba ar Rirs, radgan 2 tona TbilisSi mxolod 320 lariT metad iyideba, vidre foTSi. mis gadmosazidad ki 350 laria saWiro, e. i. 200 tona tvirTis SemTxvevaSi jobs 198 t TbilisSi gadaizidos 22 avtomobiliT, 2 t ki foTSive gaiyidos. gTavazobT amocanebs klasSi amosaxsnelad: 1. TiTo parkSi eteva 2,4 kg Saqris fxvnili. gveyofa Tu ara 150 parki 350 kg Saqris fxvnilis parkebSi gasanawileblad? 2. turistma 113,6 lari kvebisTvis 4 dRes unda imyofinos; amasTanave, yoveldRiurad erTi da igive Tanxa unda daxarjos. ramdeni lari eqneba turists daxarjuli kvebaze mesame dRis bolos? 3. marTa 25 gverdis dabeWdvas 3,75 sT andomebs. ra dros andomebs marTa erTi gverdis dabeWdvas? 4. liam Tavisi xelfasis 0,06 nawili koncertis 3 bileTSi gadaixada. mTeli xelfasis ra nawili gadaixada liam erT bileTSi? 5. qeTim iyida 32 erTnairi rveuli 35,2 larad, Tornikem – 25 erTnairi kalami 42,5 larad. romeli ufro Zviria, qeTis nayidi 1 rveuli, Tu Tornikes nayidi 1 kalami? ramdeniT? #1 amocanis amosaxsnelad SeiZleba moswavleebma 150⋅2,4 namravliT gaigon ramdeni kg Saqari eteva 150 parkSi da Seadaron 350 kg-s. (360>350, e. i. parkebis raodenoba sakmarisia). SeiZleba zogierTma 350-is 2,4-ze gayofiT daadginos saWiro parkebis odenoba, aTwiladebze gayofa jer ar uswavliaT da gamoiyenos kalkulatori. 2. turists kvebaze SeuZlia daxarjos dReSi 113,6:4=28,4 lari. mesame dRis bolos mas daxarjuli eqneba 28,4⋅3 anu 113,6–28,4=85,2 lari. 3. TiTo gverdis beWdvas sWirdeba 3,75:25=0,15 sT. 0,15 sT=60 wT⋅0,15=9 wT. 4. erT bileTSi liam gadaixada xelfasis 0,06:3=0,02 nawili. 5. rveulis fasia 35,2:32=1,1lari; kalmis fasia 42,5:25=1,7 lari. kalami rveulze 0,6 lariT Zviria, anu 60 TeTriT. saSinao davaleba SeiZleba SevarCioT II Tavis damatebiTi 16 da 20 amocanebidan.

79


§5. aTwiladze gayofa 38-e gakveTili

Tema: aTwiladze gayofa. mizani: moswavlem SeZlos aTwiladze gayofis moqmedebis Sesruleba, sizustis miTiTebis gareSe aTwiladze gayofis Semcveli gamosaxulebis miaxloebiTi mniSvnelobis povna (maT. VI. 2.) winapirobebi: aTwiladis naturalur ricxvze gayofis codna. resursebi: baraTebi, romlebzec gamosaxulia ricxvebis `1-ianiTa da marcxniv miwerili nulebiT~ warmodgenil ricxvze (10-is uaryofiT xarisxebze) gayofis magaliTebis nimuSebi; rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT. saSinao davalebis amocanebis amoxsnebi aTwiladis naturalur ricxvze gayofis SesrulebasTan iyo dakavSirebuli. vinaidan aTwiladze gayofa daiyvaneba naturalur ricxvze gayofaze, amitom am amocanebis garCeva axal masalaze gadasvlisTvis saWiro Semamzadebel samuSaosac Seicavs, am procesiT ukve iwyeba faza `winaswar~. magaliTad, 16 amocanis amoxsnaSi dasmul kiTxvas rom vupasuxoT, unda SevasruloT gayofa: 0,72:6=0,12. 17 amocanaSi ki ori moqmedebis Catareba mogviwevs: 1,8 × 1 rveuli – 1,8 lari, 15 rveuli – 1,8 15 es Tanxa 20 rveuls eyofa, Tu 1 rveuli eRireba 27:20 lari, 90 27:20=1,35 (lari) 18 20 27,0 70 60 100 100 0 anu 1 lari da 35 TeTri. davalebis Semowmebasa da aTwiladis ricxvze gayofis ramdenime magaliTis ganxilvas (Semamzadebeli samuSao) SeiZleba 15 wuTi dasWirdes. amis Semdeg SeiZleba saubari gavagrZeloT saxelmZRvaneloSi warmodgenili teqstis mixedviT, aq me-3 magaliTi erTerT dasaxul mizans exmianeba – gayofis Sedegis Sefaseba damrgvalebis gamoyenebiT. teqstSi SemoTavazebul amocanaSi aTwiladebiT gamosaxuli Tanxa (larebSi) gadagvyavs TeTrebSi da aTwiladis nacvlad naturalur ricxvze gayofa gviwevs. aseT SemTxvevamde dagvyavs nebismier aTwiladze gayofis SemTxvevac. kerZod, gasayofsac da gamyofsac vamravlebT iseT ricxvze (ufro xSirad 10-is xarisxze), rom gamyofSi mTeli ricxvi miviRoT. teqstSi es procesi demonstrirebulia magaliTebiT. aTwiladze gayofis wesis aRweris Semdeg, jgufebad gaerTianebul moswavleebs vurigebT ricxvebis 10-is uaryofiT xarisxze (anu 1-ianiTa da marcxniv miwerili nulebiT gamosaxul aTwiladze) gayofis magaliTebis nimuSebs. maTze dakvirvebiT mosalodnelia, rom moswavleebi `aRmoaCenen~ 10-is uaryofiT xarisxze gayofis wess: ramdeni cifricaa gamyofSi mZimis marjvniv, gasayofSi mZime imdenive TanrigiT marjvniv gadaiwevs. mZimis marjvniv gadaweva ki 10-is naturalurmaCveneblian xarisxze gamravlebas niSnavs. savaraudod, moswavleTa jgufebi daakavSireben 10-is uaryofiT xarisxze gayofas 10-is naturalur xarisxze gamravlebasTan. magaliTad,

80


0,1-ze gayofa 10-ze gamravlebas niSnavs; 0,01-ze gayofa _ 100-ze gamravlebas; 0,001-ze gayofa _ 1000-ze gamravlebas da a. S. gakveTilze dasaxuli miznis Sesrulebis Sefasebis mizniT SeiZleba moswavleebs davavaloT 1 - 11 amocanebSi swori pasuxebis miReba. yvela es `testi~ aTwiladze gayofas Seicavs. 10 amocaniT ki vamowmebT moswavlis codnas ricxvis nawilis povnisa da am amocanis Sebrunebuli amocanis amoxsnis Sesaxeb (maT. VI. 2.). analogiuri amocanebi adrec ganvixileT. rom vipovoT 0,9-is ra nawilia 0,18, unda SevasruloT gayofa 0,18:0,9=1,8:9=0,2. saSinao davalebad vaZlevT

1

- 10

amocanebs.

39-e gakveTili

Tema: aTwiladebze ariTmetikuli moqmedebebi. mizani: moswavles SeeZlos amocanis amoxsnisas sworad SearCios ricxvebze moqmedeba da sworad Seasrulos aTwiladebze ariTmetikuli moqmedebebi. winapirobebi: aTwiladebis Sekreba, gamokleba, aTwiladis naturalur ricxvze gamravleba da gayofa, aTwiladze gayofa. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebas SeiZleba mxolod 10 wuTi movandomoT. SeiZleba gamovyoT 6 amocana – rom vipovoT 0,7-is ra nawilia 0,14, unda SevasruloT gayofa: 0,14:0,7. aq SeiZleba SemovifargloT kiTxvis dasmiT da pasuxis miRebiT: – ra moqmedebas vasrulebT 6 `testis~ pasuxis misaRebad? amis Semdeg gadavdivarT amocanebis amoxsnaze, 12 - 18 . yoveli amocana unda amoixsnas moswavleTa aqtiuri CarTulobiT, am CarTulobis uzrunvelyofa `damxmare~ kiTxvebis gamoyenebiT SeiZleba. 12 - 15 amocanebSi moswavleebi unda mixvdnen, rom gayofis Sesrulebaa saWiro: 12 Tu bilikis sigrZe 14,6 km-ia da gasarbenia 58,4 km, maSin am manZilis gavlisTvis saWiroa, rom biliks SemovuaroT 58,4:14,6=4-jer. 13 Tu 19,6 lariT vyidulobT 4,9-larian modelebs, maSin SesaZlebelia 19,6:4,9=4 modelis yidva. 14 mariamma Zmebs Tanabrad 54,4 lari gaunawila. TiToeuls Sexvda 13,6 lari, mariams hyolia (54,4:13,6) Zma – 4 Zma. klasSi 16 amocanis amoxsnisTvis SeiZleba damxmare kiTxvebi SevaSveloT: ramden wT-Si daicala I avzi? (114:9,5=12 wT-Si); ramden wT-Si daicala II avzi? (amave droSi _ 12 wT-Si). e. i. II avzidan wT-Si gaedineboda 222:12=18,5 l wyali. 17 amocana winas msgavsia, amitom damxmare kiTxvebi aRar dagvWirdeba. muSa dReSi iRebda 8⋅2,7=21,6 lars. Tu is dReSi 6 sT-s imuSavebs da imave Tanxas miiRebs, saaTSi gadauxdian 21,6:6=3,6 lars. 17 xorblisgan miRebuli fqvilis masa xorblis masis 0,8 nawilia. ra ricxvis 0,8 nawilia 20 tona, anu 20000 kg? x⋅0,8=20000, saidanac x=20000:0,8=25000 kg = 25 t. an x:10⋅8=20000, saidanac x:10=20000:8.

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am asakis moswavleTaTvis sayuradReboa saxelmZRvanelos miTiTebaSi SemoTa­ vazebuli amoxsnac: amocanis pirobis mixedviT vagebT diagramas. xorblis raodenobas 10 marTkuTxedi Seesabameba. 20000 kg fqvils am diagramiT Seesabameba 8 marTkuTxedi. e. i. yoveli marTkuTxedi warmoadgens 20000:8=2500 kg-s. sul yofila 2500⋅10=25000 kg = 25 t xorbali. amocanis pirobam SeiZleba zogierTi moswavlis gakvirveba gamoiwvios _ sad `qreba~ dafqvisas xorblis 0,2 nawili? Tqven auxsniT, rom am nawils qmnis xorblis minarevebi, qato, dafqvis Tanmdevi danakargebi da a. S. aseTi komentarebiT Tqven sayuradRebo ganmaviTarebel aqtivobas SesTavazebT moswavleebs. saSinao davalebad moswavleebs vaZlevT

11 - 16

amocanebs.

me-40 gakveTili

Tema: aTwiladebze ariTmetikuli moqmedebebi. mizani: moswavles SeeZlos aTwiladebze moqmedebebis gamoyenebiT gamravlebisa da gayofis Semcvel tolobebSi ucnobi komponentis povna da pasuxis Semowmeba (maT. VI. 2.); mocemuli ricxvis nawilis povna, Sebrunebuli amocanis amoxsna (maT. VI. 2). winapirobebi: aTwiladebze ariTmetikuli moqmedebebis Sesruleba. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebisas yuradRebas vamaxvilebT gamravlebisa da gayofis Semcvel tolobebSi ucnobi komponentis povnisas moqmedebis sworad SerCevaze, amocana 11 . unda aRiniSnos, rom moswavleebs ufro uadvildebaT ricxvis nawilis povnis amocanis amoxsna, amitom Sebrunebuli amocanis amoxsnisas SeiZleba saZiebeli mTeli odenobis aRniSvna x-iT da x-is Semcveli gantolebis dawera, saidanac gamoCndeba, rom x-is sapovnelad gayofaa saWiro. magaliTad, 14 amocanaSi SeiZleba detalebis mTeli odenoba aRvniSnoT x-iT. miviRebT, rom 80 aris x-is 0,16 nawili, anu x⋅0,16=80, saidanac x=500. xaratma unda daamzados kidev 500–80=420 detali. klasSi amoxsnili 17 -is msgavsia: muSa dReSi iRebda 3,6⋅7=25,2 lars. axla am Tanxas mas 6 sT-Si uxdian, sT-Si 25,2:6=4,2 lars. 15

gakveTilze dasaxuli miznis SesrulebaSi dagvexmareba analogiuri amocanebiT Sedgenili damoukidebeli samuSao, romelic Temis Seswavlis donis dadgenazea fokusirebuli. 1. amoxseniT gantoleba: 2,4x=300 1,8x=4,5

8,1:x=1,35

2. ipoveT ricxvi, Tu misi 0,3 nawili aris 12. 3. ianvris TveSi laSam bavSvTa saxlis sasargeblod 300 lari gadaricxa, rac misi xelfasis 0,2 nawilia. gamoTvaleT laSas xelfasi.

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4. nucas xelafasis 0,16 nawili da lelas xelfasis 0,4 nawili toli aRmoCnda. romlis xelfasia meti? 5. gzis 0,6 nawilis SekeTebis Semdeg SesakeTebeli darCa 84 km. ipoveT mTeli gzis sigrZe. Sefasebis sqema 1. davalebaSi TiToeuli gantolebis amoxsna oretapiania: a) x-is sapovnelad moqmedebis SerCeva; b) moqmedebis Sesruleba mxolod a) punqtis Sesruleba 1 quliT Sefasdeba, a) da b) punqti erTad _ 2 quliT. b) punqtis Sefasebisas SeiZleba 0,5 qulis gamoyenebac moqmedebis mcire xarveziT Sesrulebis SemTxvevaSi. am amocanaSi maswavlebeli, savaraudod, sami gantolebidan SearCevs erT-erTs (gantolebebis fesvebia 125; 2,5 da 6). 2. a) b) a) a)

aqac amoxsna or etapad SeiZleba davyoT: sworad SearCia moqmedeba (12:0,3), an Seadgina gantoleba x⋅0,3=12; gamoTvala 12:0,3. – 1 qula; da b) – 2 qula.

3. ori etapi: a) b) a) a)

Seadgina gamosaxuleba 300 0,2 , an Seadgina gantoleba x⋅0,2=300; gamoTvala 300:0,2 – 1 qula da b) – 2 qula.

4. amoxsnis etapebi: a) dawera, rom nucas xelfasis 0,16 nawilia: x⋅0,16, an dawera, rom lelas xelfasis 0,4 nawilia _ y⋅0,4; b) gautola nucas xelfasis 0,16 lelas xelfasis 0,4-s (0,16x=0,4y) da gamoitana daskvna: y mravldeba ufro did ricxvze, x _ naklebze da Sedegebi tolia, e. i. y<x, _ metia nucas xelfasi; g) cvladebis Semotanis da gantolebis Sedgenis gareSe daadgina: nucas xelfasis mcire nawili lelas xelfasis meti nawilis tolia. amrigad, nucas meti xelfasi aqvs. SeiZleba gaamartivoT kidec es amocana meti TvalsaCinoebisTvis. a-s 13 nawili tolia b-s 12 nawilis. SeadareT a da b. sqematuradac warmoadgineT: a a) – 1 qula a) da b), an g) – 2 qula

b

5. amoxsnis etapebi: a) daadgina, rom SesakeTebeli darCa 0,4 nawili, anu 84 km aris mTeli gzis 0,4 nawili; 84 an Seadgina gantoleba x⋅0,4=84 da amoxsna (am saxis ganb) dawera gamosaxuleba 0,4 tolebis amoxsnis codna wina amocanebSic moeTxoveboda).

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g) ipova 8,4:0,4 a) – 1 qula a) da b) – 2 qula. b) da g) – 2 qula. mTeli testis maqsimaluri qula: 2+2+2+2+2=10 Tu klasis samuSao Sedegi 60%-ze naklebi araa, SeiZleba CavTvaloT, rom miznis miRwevis done damakmayofilebelia. Tumca maswavlebeli yuradRebas aqcevs calkeuli moswavleebis Sedegebs da maTTan gegmavs individualur muSaobas. saSinao davalebad SeiZleba SevarCioT II Tavis damatebiTi amocanebidan 21 - 25 amocanebi. damoukidebeli weris Sedegebis gacnoba da ganxilva Semdegi gakveTilisTvis gadavdoT.

41-e gakveTili

Tema: ariTmetikuli moqmedebebi aTwiladebze. mizani: aTwiladebze ariTmetikuli moqmedebebis Sesaxeb codnis ganmtkiceba, `moisazres~ rubrikiT warmodgenili amocanebis gamoyenebiT moswavleTa msjelobadasabuTebis unaris ganviTareba. winapirobebi: ariTmetikuli moqmedebebi aTwiladebze. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebas da damoukidebeli weris Sedegebis ganxilvas SeiZleba 15-20 wT movandomoT. SevamowmoT – aiTvises Tu ara moswavleebma ricxvis nawilis povna, nawiliT ricxvis povna, ucnobi komponentis povna gamravlebis da gayofis Semcvel tolobebSi. am amocanaTa analizi zemoT ukve iyo warmodgenili. darCenili dro SeiZleba `moisazres~ rubrikiT warmodgenili amocanebis amoxsnas davuTmoT. rubrika `moisazreSi~ amjeradac moswavleTa logikuri azrovnebis ganviTarebaze vzrunavT. 1

aTi gansxvavebuli aso aT gansxvavebul cifrs aRniSnavs. maSin maT Soris aucileblad 0-ic aris, amitom namravlic 0-ia. 2

da 3 amocanebi momavalSi dirixles principiT da albaTobis TeoriaSi birTvebis amocanebis amoxsnisTvis Semzadebaa. moswavle msjelobs CamoTvlili situaciebidan/SemTxvevebidan romelia SesaZlebeli, romelia aucilebeli da romeli SeuZlebeli. 2 amocanaSi a) b) da d) SemTxvevebi SeiZleba moxdes, SeiZleba ara. momavalSi maT SemTxveviT xdomilobebs uwodeben. radgan orze meti feris winda yuTSi ar aris, g) aucileblad moxdeba (aucilebeli xdomilobaa), e) ver moxdeba (SeuZlebeli xdomilobaaa).

amocanis pirobiT, yuTSi mxolod 6 gansxvavebuli feria, amitom Svidive gansxvavebuli feris ver iqneba (momavalSi am SemTxvevas SeuZlebel xdomilobas vuwodebT). anu Svid burTSi 2 mainc iqneba erTnairi feris (aucilebeli xdomiloba). 3

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g) ar aris gamoricxuli rom es 7 burTi Seadginos mxolod TeTri da mwvane feris burTebma _ maTi odenoba xom metia 7-ze. 4 SeiZleba xisebri diagramis gamoyeneba 2 5 6 8 5   6 8  2 6 8  2 5 8  2 5 6 diagramidan Cans, rom aris 12 SesaZlebloba. TiTqos 4 totidan TiToeulze 3 toti gamoeba da miviReT 4⋅3 toti. zogjer am meTods gamravlebis meTods uwodeben: ricxvis I adgilze (aTeulebis TanrigSi) SeiZleba Caiweros nebismieri erTi mocemuli oTxi cifridan da I adgilze Caweril yovel cifrs SeiZleba mieweros darCenili sami cifridan romelime. miviReT 4⋅3 SesaZlebloba. 5

miTiTebis mixedviT: 1 lari = 1:1,6 = 0,625 dolars; 1 evro = 2,34⋅0,625=1,4625 dolars. SeiZleba proporciac gamoiyenon: 1 evro imdenjeraa meti 1 dolarze, ramdenjerac 2,34 lari metia 1,6 larze, anu 2,34:1,16=1,4625-jer. saSinao davalebisaTvis viyenebT II Tavis damatebiTi amocanebis 26 - 30 davalebebs.

§6. gayofis Sedegis damrgvaleba 42-e gakveTili

Tema: aTwiladebis damrgvaleba. mizani: moswavles SeeZlos mocemuli sizustiT aTwiladis damrgvaleba; maT Soris, aTwiladebis gayofis Sedegis damrgvaleba (maT. VI. 2.). moswavle gaiwafos im amocanebis amocnobasa da amoxsnaSi, sadac gayofis Sedegis damrgvaleba an mxolod misi miaxloebiTi codnaa saWiro. Aam unarebis ganviTarebiT moswavle ukeT SeZlebs problemebSi swrafad orientirebas da swori gadawyvetilebis miRebas.E winapirobebi: aTwiladebis damrgvaleba. resursebi: rveuli, saxelmZRvanelo, dafa. mocemuli sizustiT aTwiladebis damrgvaleba I Tavis §5-Si iyo ganxiluli. amjerad vubrundebiT am sakiTxs, viyenebT aTwiladebis damrgvalebas gayofis Sedegis damrgvalebis algoriTmis Sedgenis dros, praqtikuli amocanebis amoxsnisas. teqstSi SemoTavazebulia praqtikuli mniSvnelobis amocana, romlis amoxsnisas aTwiladze gayofa da gayofis Sedegis damrgvalebaa saWiro. moswavle, romelic flobs damrgvalebis wesebs, advilad upasuxebs paragrafSi dasmul kiTxvebs: ganayofis measedebamde dasamrgvaleblad (rac gvaZlevs Sedegis miaxloebiT _ 0,01 sizustiT warmodgenas) _ gayofa unda Sesruldes meaTasedebis CaTvliT Tanrigis miTiTebiT. meaTedebamde dasamrgvaleblad gayofas SevasrulebT measedis Tanrigis miTiTebiT. paragrafSi dasmuli amocanis pasuxis (16,195 lari) damrgvalebiT measedebamdec da meaTedebamdec miiReba 16,2 dolari. klasSi amosaxsnel 5 savarjiSoSi moswavleebma miaqcion yuradReba im faqts, rom measedebamde damrgvalebisas meaTasedebis 5-ianma Secvala measedebis 9-iani.

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Sedegad erTi meaTedi daemata meaTedebis Tanrigs. amrigad, meaTedebis da measedebis TanrigebSi Caiwereba 0-ebi _ miviRebT naturalur ricxvs. am savarjiSos Sesrulebis Semdeg moswavleebi advilad daasaxeleben aTwilads, romlis measedebamde damrgvalebisas miiReba 2 (savarjiSo 7 ). am ricxvis erTeulis TanrigSi unda iyos 1, measedis da meaTedis TanrigebSi _ 9-ebi, meaTasedis TanrigSi 4-ze meti cifri. magaliTad, 1,995; 1,99701 da sxva. 1 km-is gavlaze ixarjeba miaxloebiT 5,5:62≈0,0887 litri benzini, 100 km-is gavlaze _ 8,87≈8,9 litri. 10

1

- 11

amocanebi saSinao davalebad miecemaT moswavleebs.

43-e gakveTili

Tema: aTwiladebis damrgvalebis gamoyeneba amocanebis amoxsnisas. mizani: moswavles SeeZlos amocanebis amoxsnisas damrgvalebis wesis gamoyeneba (maT. VI. 2.), gamravlebis nimuSSi gamotovebuli cifrebis Casma, gamoyeneba amocanebis amoxsnisas; sinjvis, variantebis daTvlis meTodis gamoyeneba (maT. VI. 4). winapirobebi: aTwiladis damrgvaleba. gakveTils viwyebT saSinao davalebis SemowmebiT; saWiro iyo gayofis Sedegis damrgvaleba, magaliTad, 2 amocanaSi 1:6=1,666...; measedebamde damrgvalebiT 1:6≈1,67. 9

rom gavigoT ramdenjer mZimea vercxliswyali rZeze, unda SevasruloT gay-

ofa. 13,6:1,03=1360:103=13,20. 103 330 309 13,2-jer (0,1 sizustiT). 210 206 40 Semdeg gadavdivarT `moisazres~ rubrikiT warmodgenil amocanebze, romelTa amoxsna gakveTilze dasaxuli erT-erTi miznis miRwevas ukavSirdeba. moisazre. 1 Tu erT-erTi Tanamamravli 1-is tolia (a⋅1=a). 2 es amocana moswavleebSi SemoqmedebiT unars aviTarebs. is miRebuli codnisa da unarebis safuZvelze qmnis axal produqts. cxadia, mocemuli pirobebiT amocana ar ixsneba, amitom moswavle msjelobs _ ra piroba daamatos. mosalodnelia, rom umravlesoba erTi boTli borjomis fass daamatebs. SeiZleba vinmem ramdenime boTlis, an eqvsive boTlis Rirebuleba daamatos: • 1 boTli borjomi Rirs 0,8 lari da a. S. • 5 boTli borjomi Rirs 4 lari da a. S. • 9 boTli borjomi Rirs 7,2 lari da a. S. samive SemTxvevaSi SeiZleba amoxsna sxvadasxvanairad warmarTon.

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I SemTxvevaSi pasuxia 0,8⋅6+3,18 lari. II SemTxvevaSi jer 1 boTlis fass daadgenen: (4:5)⋅6+3,18. III SemTxvevaSi SeiZleba ipovon 9 boTlis fasis 23 nawili: (7,2:3)⋅2+3,18. 3

SeiZleba moswavleebma qveSmiweriT gamravleba gamoiyenon da msjelobiT daadgenen, rom: ... • • • × • III striqonSi erTeulis TanrigSi unda 3,7 iyos 8, Sesabamisad, I striqonSi amave TanrigSi ... • • • unda iyos 4. ... • • • ... 8 8 8 • IV striqonis erTeulSia 2 (3⋅4=12). • III striqonis aTeulebSi 6. • I Tanamamravlis aTeulebSi 2. Tu aq SewyvetT I Tanamamravlis TanrigebiT Sevsebas, namravli ukve 8-ianebiT Cawerili ricxvi iqneba: 24⋅3,7=88,8 SeiZleba gavagrZeloT es procesi da analogiurad miviRebT Semdeg mamravlebs: 0,24; 2,4; 2,4024; 0,24024 da a. S. savaraudod, moswavleTa umravlesoba amocanas sinjviT amoxsnis: gayofen 3,7-ze jer 88-s, Semdeg 888-s da a. S. maTgan umravlesoba 888-is Semdeg gayofas Sewyvets. 4 saxaliso, Tan sayuradRebo amocanaa. moqcevisas iaxta wyalTan erTad aiwevs, anu iluminatori kvlav wylis donidan 3,5 m-is simaRleze iqneba _ arasdros wyliT ar daifareba. 5

sinjviT, analiziT moswavleebi aRmoaCenen warmodgenili saxis raime nimuSs.

285 × 39 2565  855 11115 saSinao davalebad SeiZleba mivceT II Tavis damatebiTi 31 - 35 amocanebi. SemdgomSi Cven sxvadasxva xerxiT amocanebis amoxsnas kvlav davubrundebiT. magaliTad,

§7. ricxviTi da asoiTi gamosaxulebis Sedgena 44-e gakveTili

Tema: ricxviTi gamosaxuleba. asoiTi gamosaxuleba. gantoleba. gantolebis amoxsna. mizani: moswavles SeeZlos problemis gadawyvetisas ariTmetikuli da asoiTi gamosaxulebis Sedgena, realuri viTarebis, an sityvieri aRweris Sesabamisi utolobis an gantolebis Sedgena; komutaciurobis, asociaciurobisa da distribuciulobis Tvisebebis gamoyeneba gamosaxulebis gasamartiveblad (maT. VI. 6). winapirobebi: aTwiladebze da naturalur ricxvebze ariTmetikuli moqmedebebi; ricxvis nawilis da nawiliT ricxvis povna; ricxvebze moqmedebebis Tvisebebi. resursebi: rveuli, saxelmZRvanelo, dafa, plakati, romelzec gamosaxulia ariTmetikuli moqmedebaTa Tvisebebi.

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saSinao davalebis amocanebi moiTxovda aTwiladebze ariTmetikuli moqmedebebis Sesrulebas. ucnobi sididis x-iT aRniSvna da x-is Semcveli tolobis miReba, romlis saSualebiTac vpoulobdiT ucnob sidides, moswavleebisTvis nacnobi situaciaa, analogiuri tipis martivi amocanebi V klasSic ganixileboda. saSinao davalebis 32 amocana SeiZleba amoixsnas sqemis gamoyenebiTac. 3-jer mZime 1,4 kg yvelis mcire naWeri 4-jer aRebuli mcire naWeri – 1,4 kg mcire naWeri – 1,4:4=0,35 (kg) didi naWeri – 0,35⋅3=1,05 (kg) gantolebis gamoyenebiT: mcire naWeri – x didi naWeri – 3x x+3⋅x=1,4 (1+3)x=1,4 4.x=1,4 x=1,4:4

x=0,35

3x=1,05.

saSinao davalebis Semowmebis Semdeg (15-20 wT) gadavdivarT axali masalis axsnaze, romelic aucileblad unda mimdinareobdes moswavleebis aqtiuri CarTulobiT, Cveni daxmarebiT, im wesebis gaxsenebiT, romlebic maT adre ganixiles (magaliTad, ricxvebze moqmedebaTa Tvisebebi, ariTmetikul gamosaxulebaSi moqmedebaTa Sesrulebis Tanamimdevrobis wesebi). teqstSi ganxilul amocanebTan erTad, romlebsac mivyavarT ricxviTi gamosaxulebis, gantolebis, utolobis Sedgenamde, vxsniT 1 - 15 amocanebs. teqstSi warmodgenili amocanebis klasSi garCevis Semdeg am amocanebis amoxsnas moswavleebi damoukidebladac SeZleben. am paragrafis masalis ganxilva nayofieri rom iyos, moswavleebi unda iyvnen gawafulni gantolebebis amoxsnaSi; ricxviTi gamosaxulebis mniSvnelobis gamoTvlisas moqme­debebis mimdevrobis sworad SerCevaSi, ricxviTi utolobis sisworis SemowmebaSi. am paragrafis mizania moswavle gaiwafos amocanis pirobis (realuri viTarebis, situaciis aRweris) gantolebiT an utolobiT modelirebaSi; anu romelime ucnobi sididis asoiTi aRniSvnis Semdeg danarCeni sidideebis asoiTi gamosaxulebiT gamosaxvaSi da am gamosaxulebebis erTmaneTTan dakavSirebaSi, miRebuli gantolebis (an utolobis) amoxsnaSi. moswavlem unda SeZlos Sekrebis da gamravlebis wesebis gaTvaliswinebiT, gaamar­ tivos algebruli gamosaxuleba. paragrafSi erTi da igive realuri viTareba (amocana) sxvadasxva variantSia ganxiluli. mkiTxvels vTavazobT sxvadasxva variantSi Sedgenil algebrul gamosa­ xulebebs, algebruli gamosaxulebebis erTmaneTTan dakavSirebas amocanis pirobis mixedviT, miRebuli gamosaxulebebis gamartivebas gamravlebis da Sekrebis Tvisebebis gamoyenebiT. `a~ punqtSi sityvierad Camoyalibebuli pirobis mixedviT Sedgenil ricxviT gamo­ saxulebaSi sxvadasxva ricxvebis CasmiT xdeba sidideebs Soris damokidebulebis ganzogadeba, rac gamoixateba asoiTi gamosaxulebis SedgenaSi. amave punqtSi warmod­ genilia gamravlebis ganrigebadobis Tviseba Sekrebisa da gamoklebis mimarT da

88


sailustracio magaliTi. asoiTi gamosaxulebiT sxva Tvisebebis Caweras da gamoyenebas Tavad moswavleebs vandobT. magaliTad, a) magaliTSi iyenebs Sekrebis gadanacvlebadobis (komutaciurobis) _ a+b=b+a da Semdeg jufTebadobis (asociaciurobis) _ (a+b)+c=a+(b+c) Tvisebebs: 3,7+5,7+2,3=5,7+(3,7+2,3)=5,7+6=12,7. b) magaliTSi iyenebs gamravlebis gadanacvlebadobisa da jufTebadobis Tvisebebs: 2,4x⋅5=(2,4⋅5)x=12x. g) magaliTSi _ gamravlebis ganrigebadobis (distribuciulobis) Tvisebas Sekrebis mimarT: 4,5⋅x+3,5x=(4,5+3,5)x=8x. `b~ punqtSi utolobis amoxsna amjerad sinjvis meTodiT xdeba. klasSi SemoTavazebuli savarjiSoebiT moswavle iwafeba gamravlebis, Sekrebis da gamoklebis Tvisebebis gamoyenebaSi jer ricxviTi, Semdeg asoiTi gamosaxulebebis mimarT. 10 savarjiSoSi SeiZleba Sesruldes jer gamravleba 0,5⋅1,8 da Semdeg Sekreba. an gamoyenebuli iqnas distribuciulobis Tviseba: 0,5⋅1,8+1,8=(0,5+1)⋅1,8=1,5⋅1,8. 11 cvladis nacvlad ricxvis CasmiT utolobis sisworis SemowmebaSi teqstze muSaobisas moswavle ukve gaiwafa. 14 paragrafis teqstSi ganxiluli amocanis gagrZelebaa. iTvleba, rom `b~ punqtis Seswavlisas moswavleebma ukve SeZles amocanis pirobis mixedviT Sedgenili asoiTi da ricxviTi gamosaxulebebis utolobiT dakavSireba. am SemTxvevaSic, Tu Taroebis raodenobas x-iT aRvniSnavT, miviRebT utolobas: 5,8–0,42x<1, romlis amonaxsnebi moswavlem SeiZleba sinjvis meTodiT daadginos, radgan x naturaluri ricxvia. 15 savarjiSoSi SemoTavazebuli x-is mniSvnelobebidan amonaxsnebia x=12 da x=13. maswavlebelma icis, rom am utolobis naturaluri amonaxsni unda eZebos 11,42<x<13,81 SualedSi. saSinao davalebas moswavleebs

1

- 15

amocanebidan SevurCevT.

45-e gakveTili

Tema: ricxviTi da asoiTi gamosaxulebis, gantolebis, utolobis Sedgena. mizani: sityvieri Camoyalibebis mixedviT gantolebis, utolobis Sedgena, maTi saSualebiT amocanebis amoxsna. winapirobebi: ricxvebze moqmedebebis Tvisebebi. resursebi: plakati, romelzec gamosaxulia ariTmetikul moqmedebaTa Tvisebebi. saSinao davalebis Semowmebas SeiZleba mxolod 15 wT dasWirdes. 2 - 4 , 6 , 9 saSinao davalebis savarjiSoebSi distribuciu­lo­bis Tviseba unda gamoiyenon. am paragrafis savarjiSoebis Sesrulebis Semdeg moswavle gawafuli unda iyos gamravlebis, Sekrebis da gamoklebis mimarT distribuciulobis/ ganrigebadobis Tvisebis gamoyenebaSi, anu mamravlis frCxilebs gareT gatanaSi.

89


3

(1,8–0,9+1–0,1)a=1,8a

6

0,247(0,56+0,44)=0,247.

4

14,8(36+64)=1480.

9 jer gavamartivoT asoiTi gamosaxuleba da Semdeg CavsvaT y-is mniSvneloba: (0,48–0,2+0,52)y=(0,48+0,52–0,2)y=(1–0,2)y=0,8y roca y=7, maSin 0,8y=5,6.

amocanaSi, Tu amxanagebis raodenobas aRvniSnavT x-iT, maSin mariams mouwevs (Tavisi Tavis CaTvliT) x+1 cali bileTis yidva. 13

klasSi vxsniT 16 - 24 amocanebs. maswavleblebs SevaxsenebT, rom saxelmZRvaneloSi Canawerebis gakeTeba dauSvebelia. magaliTad, 17 da 18 amocanebi moswavleebma rveulSi unda gadaitanon da iq gaakeTon Canawerebi. 16 da 18 savarjiSoebi Sekrebisa da gamravlebis TvisebebSi gawafvas emsaxureba. a) 5⋅3,9=5⋅(4–0,1)=5⋅4–5⋅0,1=20–0,5=19,5. SeiZleba moswavlem 3,9 jer 10-ze gaamravlos da Semdeg gayos 2-ze; b) (10+0,3)⋅4=40+1,2=41,2; d) 0,9⋅(0,25⋅4)⋅0,2=0,9⋅1⋅0,2=0,18; e) (1,5⋅0,6)⋅(0,5⋅4)=0,9⋅2=1,8; v) (2,5⋅8)⋅(4⋅0,25)=20⋅1=20; z) da T) gamosaxulebebSi erT-erTi Tanamamravli 0-ia, namravlic 0-ia. 21 SeiZleba amoixsnas gantolebis SedgeniT. Tu ucnob ricxvs x-iT aRvniSnavT, miviRebT: 15,6⋅x+3,4=11,2, saidanac x=0,5. 22 SeiZleba SeadginoT gantoleba: 21,08–x⋅2=12,6, saidanac x=4,24. 23 pirobiT, 3,4x+7,8<15. sinjvis meTodiT vrwmundebiT, rom mxolod x=1 da x=2 akmayofilebs utolobas, x=3 da x-is meti mniSvnelobebis CasmiT marcxena mxare marjvenaze meti xdeba. 24 pirobiT, 3,4y+0,5 naklebia 24,8-ze 17,5-iT. moswavleebi damoukidebeli ganxilviT

daadgenen am gamosaxulebaTa kavSirs, rac SeiZleba asec Caiweros: 3,4y+0,5+17,5=24,8, saidanac y=2. saSinao davalebad moswavleebs

90

15 - 25

amocanebs vaZlevT.


46-e da 47-e gakveTilebi

aqtivoba. jgufuri muSaoba: gayofadobis niSnebi. es jgufuri muSaoba sami nawilisgan Sedgeba, mas SeiZleba ori gakveTilic daeTmos. aqtivobis mizani. moswavle gaiwafos mocemuli ricxvis jeradi ricxvebis, ricxvis saTanrigo Sesakrebebis jamad warmodgenisas asoiTi gamosaxulebebis gamoyenebaSi. gaiwafos asoiTi gamosaxulebebis gamartivebisas Sekrebisa da gamravlebis Tvisebebis gamoyenebaSi; Sedegis gaanalizebaSi da daskvnebis gamotanaSi. Teoriuli masala gamoiyenos praqtikuli amocanebis gadawyvetisas. aucilebeli winare codna. naturaluri ricxvis jeradi da gamyofi; luwi da kenti ricxvebi; gayofisas naSTis dasaxeleba; Sekrebis da gamravlebis Tvisebebi (gadanacvlebadoba, jufTebadoba, gamravle­bis ganrigebadoba Sekrebis mimarT). pirveli gakveTilis garkveuli monakveTi wina gakveTilze micemuli davalebis Semowmebas SeiZleba davuTmoT – 20 wT. II kalaTSi sokos masa aRvniSnoT x-iT, maSin I kalaTSi iqneba x+0,2. pirobis Tanaxmad, x+x+0,2=13,16, saidanac x=6,48. pasuxi: II kalaTSi 6,48 kg sokoa, I-Si _ 6,68 kg. dabal klasebSi aseT amocanebs moswavleebi xsnidnen gantolebis gareSe. gamovakloT vaSlis mTel raodenobas I kalaTSi arsebuli `zedmeti~ 0,2 kg. darCenili 12,96 kg or kalaTSi Tanabrad gavanawiloT (TiToSi _ 6,48 kg) da `zedmeti~ 0,2 kg kvlav I kalaTs `davubrunoT~. anu, II kalaTSi vaSlis raodenoba gamoiTvleba ricxviTi gamosaxulebiT: 17

(13,16–0,2):2, I-Si: (13,16–0,2):2+0,2. 18

cxadia, erTi da igive ricxvebi erTi da imave cifrebiT bolovdeba. ar arsebobs cifri, romliTac Cawerili ricxvis Tavis Tavze namravli (kvadrati) 7-iT bolovdeba. e. i. giorgim dauSva Secdoma gamravlebisas. am amocanaze muSaobisas SeiZleba visargebloT SemTxveviT da gavaxsenoT moswavleebs, ra cifriT SeiZleba bolovdebodes naturaluri ricxvis kvadrati. amisaTvis SevamowmoT Cawerili ricxvebis kvadratebi (1⋅1=1, 2⋅2=4, 3⋅3=9, 4⋅4=16, 5⋅5=25, 6⋅6=36, 7⋅7=49, 8⋅8=64, 9⋅9=81, 10⋅10=100), saidanac vaskvniT, rom nebismieri ricxvis kvadrati SeiZleba bolovdebodes 1, 4, 6, 5, 9 an 0-iT. 2, 3, 7 da 8-iT arcerTi ricxvis kvadrati ar bolovdeba. 19

a) (15,48+16,92)⋅4,7=32,4⋅4,7=152,28. b) (24,18-5,98)⋅2,6=18,2⋅2,6=47,32.

vTqvaT, II dRes gasayidi darCa x t, I dRes gayidula 3,4⋅x t. pirobiT, x+x⋅3,4=6,6, saidanac x=1,5 t. I dRes gaiyida 1,5⋅3,4=5,1 tona vaSli. 20

22 SeiZleba jer gavamartivoT asoiTi gamosaxuleba, miviRebT: (5,8+8,3)⋅p:4,7=14,1⋅p:4,7=(14,1:4,7)⋅p=3p.

ucnobi ricxvi x-iT aRvniSnoT. pirobiT, (x+5,2)⋅0,5=17,5 saidanac x=29,8. 23

24

wina amocanis analogiurad, SevadginoT gantoleba (x-10,5)⋅0,5=2,5, saidanac x=15,5.

91


Tu ucnob naturalur ricxvs x-iT aRvniSnavT, pirobis Tanaxmad, miviRebT 5,6x+3,4<25,9 utolobas. SeiZleba am utolobis naturaluri amonaxsnebis mosaZebnad sinjvis meTodi gamoiyenon: x=1, x=2, x=3 da x=4-isTvis utoloba samarTliania. 4-ze meti x-isTvis marcxena mxare meti xdeba marjvenaze. 25

aqtivobis aRwera. gakveTils aucilebeli wina codnis SemowmebiT viwyebT. Cveni mimarTvis pasuxad moswavleebi adgilidan asaxeleben 7-is, 10-is jerad ricxvebs, luw da kent ricxvebs; 12-is, 24-is, 30-is gamyofebs. Semdeg vTxovT moswavleebs daasaxelon 5-is jeradi ricxvebi. (10, 15, 20, 25,...), daukvirdnen am ricxvebis struqturas. SeiZleba kiTxvebiTac davexmaroT: • 5 ra ricxvze gavamravleT 5-is, 10-is, 15-is, 20-is da a. S. 5-is jeradi naturaluri ricxvebis misaRebad? (1-ze, 2-ze, 3-ze, da a. S. _ gavamravleT naturalur ricxvebze). • ra asoiTi gamosaxulebiT SeiZleba 5-is jeradi ricxvebis warmodgena? (5n, sadac n naturaluri ricxvia). • 5n+2 gamosaxulebaSi CasviT n-is nacvlad ramdenime mniSvneloba da miRebuli ricxvebi gayaviT 5-ze. ra naSTi miiReba 5n+2 saxis ricxvebis 5-ze gayofisas? (2) • ra naSTi miiReba 5n+4 saxis ricxvebis 5-ze gayofisas? (4) analogiurad Seadgenen asoiT gamosaxulebas, romelic 3-is jerad ricxvebs warmogvidgens. aRweren luw ricxvebs da warmoadgenen maT asoiTi gamosaxulebiT (2n). kiTxvaze _ rogoria luwi ricxvis momdevno ricxvi, wina ricxvi _ luwi Tu kenti? _ pasuxi ubiZgebs maT warmoadginon asoiTi gamosaxulebiT kenti ricxvebic (2n+1 an 2n-1). Tu gvsurs, rom mocemuli gamosaxulebiT aRiwerebodes yvela naturaluri ricxvi, virCevT (2n-1)-s, sadac n naturaluri ricxvia; an (2n+1)-s sadac n nulia, an naturaluri ricxvia. jgufuri muSaobis I nawilSi xdeba jeradi ricxvebis asoiTi gamosaxulebiT warmodgenis gamoyeneba debulebebis dasabuTebisas. magaliTad, asoiTi gamosaxulebis da ganrigebadobis Tvisebis gamoyenebisTvis (I nawilis a) – e) amocanebi) Sesamzadeblad jer vTxovT moswavleebs warmoadginon asoiTi gamosaxulebiT 5-is jeradi ori gansxvavebuli ricxvi (5n da 5m). amis Semdeg jgufebs vunawilebT davalebebs. a) d) e), b) d) e), g) d) e). SeiZleba davalebebis meti jgufebisTvisac ganawileba. prezentaciisas jgufebis warmomadgenlebi aRweren Sedgenili gamosaxulebebis gamartivebis process, asaxeleben moqmedebebis gamoyenebul Tvisebebs. magaliTad, b) amocanaSi gamoiyeneben an gamravlebis Sekrebis mimarT ganrigebadobis (distri­ buciulobis) Tvisebas, romlis mixedviTac, 4n+4m=4(n+m). amrigad, miiReben kvlav 4-is jerad ricxvs. d) da e) debulebebis dasabuTebisas moswavleebs uviTardebaT ganzogadebis, msjelobis, daskvnebis gamotanis unari. am amocanebis amoxsna am wignis II Tavis da me-7 paragrafSia warmodgenili. SeiZleba romelime jgufma ver SeZlos debulebis dasabuTeba da mxolod kerZo magaliTebiT Semoifarglos. cxadia, aseTi jgufi mainc Sefasdeba, Tumca maRal Sefasebas ver daimsaxurebs. jgufuri muSaobis II nawili daviwyoT ricxvis gaSlili formiT warmodgenis SexsenebiT: Caweron gaSlili formiT ricxvebi: 23; 15; ricxvi, romelSic m aTeuli da k erTeulia (10m+k). ganixilon jgufuri muSaobis I etapze dasabuTebuli e) debulebis kerZo SemTxveva; vTqvaT, 15+x=605. iyofa Tu ara 5-ze x? upasuxeT gantolebis amoxsnis gareSe.

92


amis Semdeg jgufebs vunawilebT a), b) da g) davalebebs. 10m+k saxiT Caweril ricxvebze dakvirvebiT mixvdebian: radgan I Sesakrebi iyofa 2-ze, jami gaiyofa 2-ze im SemTxvevaSi, Tu k gaiyofa 2-ze (d) Tvisebis Tanaxmad. iqneb maSinac iyofa jami 2-ze, roca k ar iyofa 2-ze? es SeuZlebelia e) Tvisebis Tanaxmad. amrigad, mxolod luwi k-s SemTxvevaSi gaiyofa 10m+k ricxvi 2-ze. analogiurad imsjeleben 5-ze da 10-ze gayofis SemTxvevaSic. am analizis safuZvelze moswavleTa jgufebi ayalibeben 2-ze, 5-ze, 10-ze gayofadobis niSnebs, mohyavT magaliTebi. naSromTa prezentaciisas yvela jgufi unda iyos CarTuli ganxilvaSi. yuradReba mivaqcioT saboloo Sedegebis Camoyalibebis formas, arss, sailustracio amocanebs, jgufebis oponirebas. pasuxebis sisrules. jgufuri muSaobis III nawili SeiZleba aqtivobis gafarToebad CaiTvalos da mxolod I da II nawilis warmatebiT Sesrulebis SemTxvevaSi ganxorcieldes. Sesamzadeblad SeiZleba moswavleebs vTxovoT: • Secvalon varskvlavebi ricxvebiT swori tolobis misaRebad: 245=2⋅100+*, 367=3⋅100+*, 2145=2⋅1000+* da a. S. • Caweron ricxvi, romelSic m aseulia da bolo ori cifriT Sedgenili ricxvi aris k (100m+k). • Caweron ricxvi, romelSic a aTaseulia da bolo sami cifriT Cawerili ricxvi aris b(1000a+b). amis Semdeg jgufebi asabuTeben 4-ze da 8-ze gayofadobis niSnebs kvlav I nawilis d) da e) debulebebis gamoyenebiT: 100m+k gamosaxulebaSi I Sesakrebi iyofa 4-ze. vTqvaT, es jamic iyofa 4-ze, maSin k iyofa 4-ze (e) debulebis Tanaxmad. Tu vigulisxmebT, rom k iyofa 4-ze, maSin d) debulebis Tanaxmad 100m+k gaiyofa 4-ze. amrigad 100m+k mxolod maSin iyofa oTxze, roca k iyofa oTxze. amave gziT SeiZleba naturaluri ricxvis 1000a+b gamosaxulebis 8-ze gayofadobis niSnis dasabuTeba. muSaobis Sejamebis mizniT SeiZleba davusaxeloT ricxvebi da moswavleebma praqtikulad gamoiyenon maT mier Camoyalibebuli gayofadobis niSnebi, magaliTad, Seavson cxrili. ricxvi

iyofa 2-ze

iyofa 5-ze

iyofa 10-ze

iyofa 4-ze

iyofa 8-ze

24

+

_

_

+

+

104 114 1025 2032 ... aqtivoba unda davasruloT yvela Teoriuli Sedegis moswavleebis mier kidev erTxel CamoyalibebiT, moswavleTa wvlilis ganxilviT. warmatebulTa gamorCeviT da waxalisebiT, yvela moswavlis gamxnevebiT.

93


aqtivobis Sefaseba. jgufuri muSaobis warmatebiT ganxorcielebis SemTxvevaSi moswavle iwafeba mi­Rebu­li codnis (asoiTi gamosaxulebis Sedgena, gamartiveba) gamoyenebaSi; kerZo magaliTebze dakvirvebis Sedegebis ganzogadebaSi; varaudis gamoTqmaSi da dasabuTebaSi. moswavle qmnis produqts (azrovnebis umaRlesi done _ sinTezi) _ ayalibebs/aRwers ricxvebis 2-ze, 5-ze, 10-ze da Semdgom 4-ze da 8-ze gayofadobis niSnebs; mohyavs magaliTebi. iZens unars swrafad, Sromatevadi gamoTvlebis gareSe, daadginos ricxvTa gayofadobis sakiTxebi. gamoiyenos es codna amocanebis amoxsnisas. aqtivobis dasasruls pedagogma unda CainiSnos yvela gamarTlebuli pedagogi­kuri midgoma, aRmoCenili xarvezebi da dasaxos maTi gamosworebis gegma. Semdegi mecadineobisTvis moswavleebs evalebaT informaciis moZieba da individualuri muSaobis proeqtis Sesruleba.

48-e gakveTili

Tema: ricxvebze moqmedebebis gamoyeneba. mizani: moswavles SeeZlos ricxvebze moqmedebebis gamoyeneba; drois sartyelebis Sesaxeb codnis gamoyeneba drois monakveTebis povnisaTvis (maT. VI. 3); aTwiladebze ariTmetikuli moqmedebebis gamoyeneba amocanebis amosaxsnelad. winapirobebi: ariTmetikuli moqmedebebi aTwiladebze. resursebi: rveuli, saxelmZRvanelo, dafa. gakveTilis ZiriTadi nawili (20-25 wT) eTmoba individualuri proeqtis Sesrulebis Semowmebas; moswavleebs unda SeedginaT saxelmZRvaneloSi dasmul kiTxvebze pasuxebis saSualebiT cnobari sasaaToO zolebis Sesaxeb, gaerkviaT sxvadasxva qalaqSi droebs Soris arsebuli Tanafardobebi da maTi mizezebi. moswavleebi kiTxuloben sakuTar pasuxebs, maswavlebeli afasebs maT monaTxrobs. pedagogebma ician, rom sasaaTo zoli dedamiwis zedapiris nawilia, romelSic garkveuli SeTanxmebis Tanaxmad dadgenilia garkveuli dro (zoluri dro). magaliTad, nulovan zolSia londoni, pirvelSi _ romi da parizi, meoreSi _ kievi, mesameSia moskovi, peterburgi, meoTxeSi _ Tbilisi (da mTeli saqarTvelo). garkveuli ekonomikuri mosazrebebis mixedviT ama Tu im zolSi SeiZleba dadgindes specialuri, dekretuli dro, romelic gansxvavdeba zolis Sesabamisi droisgan. swored amiT SeiZleba aixsnas, magaliTad, fexburTis matCebis telegadacemaTa dawyebis droebs Soris gansxvaveba. kompiuteris da zogierTi tipis mobiluri telefonis saaTis dayenebisas naTdeba fanjara, sadac miTiTebulia sasaaTo zolebi (Time Zone). darCenili dro davuTmoT 1 - 16 Semajamebeli amocanebidan ramdenime amocanis amoxsnas. danarCenebidan moswavleebs saSinao davaleba SevurCioT.

94


49-e gakveTili (Semajamebeli)

Tema: ariTmetikuli moqmedebebi aTwiladebze. mizani: II Tavis mixedviT Seswavlili masalis gameoreba, codnis ganmtkiceba. winapirobebi: ariTmetikuli moqmedebebi aTwiladebze. resursebi: rveuli, saxelmZRvanelo, dafa. II Tavis Semajamebeli amocanebisa da `moisazres~ rubrikiT warmodgenili amocanebis gamoyenebiT mimdinareobs gavlili masalis gameoreba, codnis ganmtkiceba. 3

SevadginoT gantoleba:

x(1,25–0,75)=2, saidanac x=4.

4

amovxsnaT gantoleba: (3,15x+12,4)+(5,75x+18,6)=122,67, saidanac Sekrebis gadanacvlebadobis da jufTebadobis Tvisebebis gamoyenebiT miviRebT:L (3,15x+5,75x)+(12,4+18,6)=122,67, 8,9x+31=122,67, x=10,3. 5

x+9=30. saidanac x=21.

8

viyenebT ganrigebadobis (distribuciulobis) Tvisebas: a) 6,1(3,4+2,6)=6,1⋅6=36,6 b) gadavweroT gamosaxuleba ase: 9,1⋅1–9,1⋅0,377–0,623=9,1⋅(1–0,377)–0,623=9,1⋅0,623–1⋅0,623= =0,623⋅(9,1–1)=0,623⋅81=5,0463. 9

mZimis marjvniv gadataniT Tanamamravlic da masTan erTad namravlic gaizarda 10-jer, sami cifris marcxniv gadataniT _ Semcirda 1000-jer, sabolood namravli Semcirdeba 100-jer. 11 aRvniSnoT II dRes SekeTebuli monakveTis sigrZe x-iT, I dRes _ 6,5x-iT. pirobiT, x+6,5x=28,5, saidanac x=3,8 km. msxlis raodenoba aRvniSnoT x-iT, vaSlis _ 1,3x-iT. pirobiT, x+1,3x=33,12, saidanac x=14,4 kg. 12

14 4-ze 2-jer meti ricxvia 8. pirobiT, 2x+3=8, saidanac x=2,5.

`moisazre~ amocanebi masalis gameorebis da logikuri azrovnebis ganviTarebas emsaxureba. 1 a) yvelaze uares SemTxvevaSi amosul windebSi aTive Savia. 12 windis amoRebis SemTxvevaSi 2 aucileblad iqneba TeTri. b) yvelaze uares SemTxvevaSi 8 winda amova TeTri. 10 windis amoRebis SemTxvevaSi 2 aucileblad iqneba Savi. g) yvelaze uares SemTxvevaSi 10 winda amova erTi feris _ Savi. me-11 winda aucileblad iqneba TeTri _ gansxvavebuli feris. 2

zurabi ar zis arc Tornikes da arc sandros gverdiT. e. i. zurabi zis napiras nikas gverdiT. nikas meore mxridan Tornike uzis, darCa _ sandro zis Tornikes gverdiT: zurabi _ nika _ Tornike _ sandro

95


3

TvalsaCinoebisTvis piroba gamovsaxoT sqemiT: swavlobs ar swavlobs x mosw. y mosw. ar swavlobs x mosw.

swavlobs z mosw.

pirobiT, x+z=21, e. i. VI

b

VI

a

 VI

b

klasSi 21 moswavlea.

a) 9-ze iyofa 9n saxis ricxvebi, 9n=3⋅3n _ iyofa 3-zec. b) 4-ze iyofa 4n saxis ricxvebi, 4n=2⋅2n _ iyofa 2-zec. winadadebis mcdarobis dasamtkiceblad erTi magaliTic (kontrmagaliTi) kmara: b) 12 iyofa 3-ze da ar iyofa 9-ze; g) 6 iyofa 2-ze da ar iyofa 4-ze; d) 6 ar iyofa 4-ze da iyofa 2-ze. 4

b) g) da d) winadadebebi arasworia. 5

amocanaSi gaiTvaliswinon, rom, kenti da luwi ricxvis jami kentia, kentis momdevno ricxvi luwia, luwis momdevno _ kenti. naturaluri ricxvisTvis 1-is mimatebiT miiReba misi momdevno. meris kanfetebis raodenoba 4-ze gayofisas naSTSi 1-s iZleva da aris 15-is jeradi, 50-ze naklebi 15-is jeradi ricxvebia 15, 30, 45. maTgan 4-ze gayofisas naSTSi 1-s mxolod 45 iZleva. pasuxi: 45. 6

50-e gakveTili

Semajamebeli wera Tema: ricxviTi da asoiTi gamosaxulebis Sedgena; amocanebis amoxsna, aTwiladebze moqmedebebi, moqmedebaTa Tvisebebi da maTi gamoyeneba. SearCieT swori pasuxi: 1. x+12,305=0,21+22,49 gantolebis amonaxsnia b) x=35,005 g) x=34,795 a) x=10,395

d) x=34,375.

2. kaxim 6,4-lariani wignis sayidlad gamyidvels 10-lariani miawoda. ra Tanxa unda daubrunon kaxis? a) 4,6 lari b) 3,4 lari g) 4,4 lari d) 3,6 lari. 3. 0,4⋅0,12⋅5= a) 0,24

b) 0,024

g) 2,4

d) 24.

4. erTi ricxvi 3,4-is tolia, meore _ masze 4-jer metia. am ori ricxvis jamia a) 3,4⋅4+1 b) 3,4⋅5 g) 3,4⋅4 d) 3,4⋅4+4.

96


5. 4,2x+7,2x gamosaxulebis mniSvneloba, roca x=5, aris a) 5,7 b) 28,2 g) 57 d) 40,2 6. CamoTvlili ricxvebidan 4-ze iyofa a) 2042 b) 2045 g) 2046

d) 2048

amoxseniT amocanebi: 7. vTqvaT, 1 dolari 1,75 larze icvleba. ramdeni dolari dagvrCeba 100 dolaridan 42 lariani perangis yidvis Semdeg? 8. televizori 440 lari Rirs, am Tanxis meaTedi Rirs televizoris dasadgmeli. ra Rirs televizori da dasadgmeli erTad? pasuxebi da miTiTebebi 1

2

3

4

5

6

a

d

a

b

g

d

4. SeiZleba moswavlem pirobiT mocemuli jami gamosaxos ricxviTi gamosaxulebiT 3,4+3,4⋅4 da Semdeg gamoiyenos ganrigebadobis Tviseba, miiRebs 3,4(1+4)=3,4⋅5. SeiZleba gamoTvalos 3,4+3,4⋅4 gamosaxulebis mniSvneloba da Seadaros savaraudo pasuxebis mniSvnelobebs. SeiZleba sqemac gamoiyenos:   I ricxvi II ricxvi. sqemidan Cans, rom orive ricxvis jami I ricxvze 5-jer metia. 6. SeiZleba Seasrulos gayofa, an gamoiyenos 4-ze gayofadobis niSani. 7. 42 lariani perangi Rirs 42:1,75=24 dolari. perangis yidvis Semdeg dagvrCeba 100-24=76 dolari. pasuxi: 76 dolari. SeiZleba asec vimsjeloT: faqtobrivad gvaqvs 175 lari. yidvis Semdeg dagvrCeba 133 lari, rac Seadgens 133:1,75=76 dolars. 8. dasadgmeli 440:10=44 lari Rirs, orive erTad 440+44=484 lari. pasuxi 484 lari. Sefasebis sqema SeiZleba aseTi iyos: testuri tipis amocanebidan TiToeulSi swori pasuxis miTiTeba Sefasdes 1 quliT, araswori _ 0O quliT.

97


Sefasebis rubrika maswavlebelTaTvis #7 davalebisTvis 0 qula

ver gaerkva amocanis arsSi, an gadmowera amocanis pirobebi. SearCia moqmedeba (gayofa) imis dasadgenad, Tu ramdeni dolaria 1 lari, magram ver Seasrula es moqmedeba;

0,5 qula

an SearCia moqmedeba (gayofa) perangis fasis dasadgenad, magram ver Seasrula es moqmedeba; an 100 dolari gamosaxa larebSi. daadgina, rom 1 lari aris 47 dolari, an Cawera es ricxvi miaxloebiT;

1 qula

an daadgina perangis fasi dolarebSi, gayofisas araarsebiTi SecdomiT; an gamoTvala perangis yidvis Semdeg darCenili Tanxa larebSi.

1,5 qula

daadgina perangis fasi dolarebSi; an SearCia moqmedeba larebSi darCenili Tanxis dolarebSi gadasayvanad da ver Seasrula es moqmedeba.

2 qula

daadgina perangis fasi da gamoTvala darCenili Tanxis odenoba dolarebSi.

#8 davalebisTvis 0 qula

ver gaerkva amocanis arsSi, an gadmowera amocanis pirobebi.

0,5 qula

SearCia moqmedeba dasadgmelis fasis gamosaTvlelad, magram is Seasrula arasworad, an sul ar Seasrula.

1 qula

gamoTvala dasadgmelis fasi.

1,5 qula

SearCia moqmedeba dasadgmelis da televizoris mTliani fasis gamosaTvlelad, magram, ver (ar) Seasrula es moqmedeba.

2 qula

ipova dasadgmelis da televizoris fasi erTad.

Sefasebis rubrikebi maswavleblisTvis SeiZleba aseTic iyos: kriteriumebi CamovayaliboT punqtebad da Semdeg davweroT Sefasebis sqema. #7 davalebisaTvis a) gamosaxa laris fasi dolarSi b) gamosaxa perangis fasi dolarSi g) gamosaxa 100 dolari larSi d) gamoTvala perangis yidvis Semdeg darCenili Tanxa larebSi. e) gamoTvala perangis yidvis Semdeg darCenili Tanxa dolarSi.

98


0,5 qula: raime racionaluri moqmedebisTvis 1 qula: d), an a), an b) 2 qula: b) da e), an d) da e) punqtebis SesrulebisTvis. Sefasebis rubrika moswavleTaTvis, romelsac gamoiyeneben Sedegebis analizis procesSi. es daexmareba maT TviTSefasebaSi. #7 davalebisaTvis 0 qula

ar weria araferi, an gadmowerilia piroba.

0,5 qula

laris fasis dasadgenad, an perangis fasis dasadgenad SerCeulia moqmedeba: 1:1,75 an 4,2:1,75 da ar aris Sesrulebuli; an 100 dolari gadayvanilia larebSi.

1 qula

gamoTvlilia perangis fasi dolarSi, an 1 laris fasi dolarSi araarsebiTi SecdomebiT; an gamoTvlilia perangis yidvis Semdeg darCenili Tanxa larebSi.

1,5 qula

gamoTvlilia perangis fasi dolarSi; an perangis yidvis Semdeg larebSi darCenili Tanxis dolarSi gadasayvanad sworad SerCeuli moqmedeba Sesrulebulia umniSvnelo SecdomiT.

2 qula

amocana amoxsnilia sworad.

#8 davalebisaTvis 1 qula

gamoTvlilia dasadgmelis fasi.

2 qula

gamoTvlilia dasadgmelis da televizoris fasi erTad.

99


III Tavi.   moqmedebebi wiladebze

§1. wiladis Cawera aTwiladis saxiT 51-e gakveTili

Tema: wiladis Cawera aTwiladis saxiT. mizani: moswavles unda SeeZlos sasruli aTwiladis Cawera wiladis saxiT da wiladis Cawera sasruli aTwiladis, an perioduli aTwiladis saxiT (maT. VI. 1). winapirobebi: wiladis sxvadasxva interpretaciebi, naturaluri ricxvebis gayofa, gayofisas miRebuli naSTebis dasaxeleba, ricxvis daSla martiv mamravlebad. resursebi: kalkulatori (SeiZleba mobilurebis gamoyeneba, rogorc wesi, aseT telefonebs aqvs kalkulatori), rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT Semajamebeli weris Sedegebis warmodgeniT, Secdomebis sajaro analiziT (15 wT). am aqtivobiT iwyeba Temis damuSavebis faza `winaswar~. standartSi miTiTebulia, rom moswavles unda SeeZlos wiladis Cawera sasruli aTwiladis saxiT Sesabamis SemTxvevaSi. Cven gvsurs, moswavlem gaiazros – rodisaa es `Sesabamisi SemTxvevebi~ da SeZlos SearCios wiladebi, romlebic warmoidgineba sasruli aTwiladis saxiT, perioduli aTwiladis saxiT. amisaTvis saWiroa, rom moswavle gawafuli iyos naturaluri ricxvebis gayofaSi, SeeZlos gayofisas miRebuli naSTis dasaxeleba. wiladebis aTwiladis saxiT Caweras viwyebT etapobrivad. ase gadavdivarT fazaze `ganmavlobaSi~. jer aTwiladis saxiT CavwerT wiladebs, romelTa mniSvnelebi 10-is xarisxebia; Semdeg sxva wiladebs, wiladis ZiriTadi Tvisebis gamoyenebiT, gadavwerT ise, rom mniSvneli 10-is xarisxi iyos; Semdeg usasrulo perioduli aTwiladis saxiT vwerT wiladebs, romelTa Cawera ar xerxdeba iseTi wiladebis saxiT, romelTa mniSvnelebi 10-is xarisxebia. ramdenime wiladis usasrulo perioduli aTwiladis saxiT Caweris Semdeg unda davadginoT pirobebi: romeli wiladi Caiwereba sasruli da romeli usasrulo aTwiladis saxiT. teqstSi, 16 -is aTwiladis saxiT Caweris magaliTze, damxare kiTxvebia warmodgenili. kiTxva-pasuxis reJimSi es procesi daaxloebiT aseT saxes miiRebs: • 10-is nebismieri xarisxis martiv mamravlebad daSlaSi ra ricxvebi monawileobs? _ mxolod 5 da 2.

100


• rodisaa SesaZlebeli wiladis iseT ricxvze gamravleba, rom mniSvnelSi 10-is xarisxi miviRoT? roca mniSvnelis daSlaSi 5-isa da 2-isgan gansxvavebuli martivi ricxvi ar monawileobs. 16 sasruli aTwiladis saxiT ar Caiwereba. cxadia, yvelgan igulisxmeba ukveci wiladi. 1:6=0,166...=0,1(6) 1 -is aTwiladad CawerisTvis 1-s vyofT 6-ze. 0 6 10 • ramdeni gansxvavebuli naSTi SeiZleba miviRoT 6-ze 6 gayofisas? _ gayofis procesSi 6 gansxvavebuli naSTi SeiZleba 40 miviRoT, amitom gayofisas naSTebi daiwyebs gameorebas (sul 36 didi me-5 naSTis Semdeg) da miiReba perioduli aTwiladi: 40 jgufuri muSaobisas moswavleebi kvlav am sakiTxebze agrZeleben msjelobas. maT ukve ician, rom sasruli aTwiladis saxiT warmodgindeba is wiladebi, romelTa ukveci saxiT warmodgenisas mniSvnelis martiv mamravlebad daSlaSi 5-isa da 2-isgan 1 , 11 . danarCeni wiladebi Caiwereba gansxvavebuli mamravli ar Sedis. aseTebia 18 , 80 25 perioduli aTwiladis saxiT. aTwiladebad gadawerisas, savaraudod, moswavleebi mricxvelis mniSvnelze gayofis xerxiT isargebleben. Tumca SeiZleba sasruli aTwiladis SemTxvevaSi gamoiyenon wiladis ZiriTadi Tviseba: gaamravlon mricxvelic da mniSvnelic erTsa da imave ricxvze _ 2-isa da 5-is iseT xarisxebze, romlebic mniSvnels 10-is xarisxad warmoadgens. 1 = 1 . aq sakmarisia, mricxvelica da mniSvnelic gavamravloT 53-ze. magaliTad, 80 24⋅5 miviRebT: 1 53 53 80 = 24⋅54 = 104 =0,0125. saxelmZRvanelos teqstSi mocemuli magaliTebis erToblivi garCeviTa da jgufuri muSaobis Catarebis Semdeg (am process SeiZleba 20 wT dasWirdes), SeiZleba gakveTilze dasaxuli miznis miRwevis donis Sefasebis mizniT, darCenili dro davuTmoT moswavleebis mier 1 - 10 `testebSi~ swori pasuxebis SerCevis process. yuradReba gaamaxvileT im amocanebis amoxsnebze, roca SesaZlebelia moswavlem pasuxi zepiri angariSiT miiRos, magaliTad, 6 2 2⋅4 8 75 = 25 = 25⋅4 = 100 =0,08 SeiZleboda Segvesrulebina gayofa: 6 :75=0,08. 600 600 0 SeiZleba vkiTxoT moswavleebs: – ratom moxda, rom wiladi Caiwera sasruli aTwiladis saxiT, miuxedavad imisa, rom mniSvneli Seicavs 2 da 5-isgan gansxvavebul martiv gamyofs? (radgan Sekvecisas Seikveca 2-is da 5-isgan gansxvavebuli mamravli 3). saSinao davalebad vaZlevT

1

- 13

amocanebs.

101


52-e gakveTili

Tema: aTwiladis Cawera wiladis saxiT da wiladis Cawera aTwiladis saxiT. mizani: moswavles unda SeeZlos Sesabamis SemTxvevaSi wiladis aTwiladis saxiT Cawera (maT. VI. 1.). winapirobebi: naturalur ricxvze gayofa; wiladis ZiriTadi Tviseba. resursebi: kalkulatori, rveuli, saxelmZRvanelo, dafa. saSinao davalebis amocanebi Seicavda davalebebs, romlebSic moiTxoveboda `Sesabamis SemTxvevaSi~ wiladis aTwiladis saxiT Cawera. analogiuri amocanebi amoxsnili 6 8 da amocanebSi ki is SemTxvevebia warmodgenili, roca ar iyo gakveTilze. xerxdeba wiladis sasruli aTwiladis saxiT Cawera; moswavleebs SeiZleba vTxovoT axsnan es garemoebebi. davalebis Semowmeba SeiZleba 15 wuTSi davamTavroT. amis Semdeg gadavdivarT 11

- 18 amocanebis erTobliv ganxilvaze. aTwiladebis damrgvalebaSi am droisaTvis moswavleebi ukve gawafulni arian. klasSi savarjiSoebiT gaiwafebian wiladis CaweraSi aTwiladis saxiT da piriqiT, aTwiladis wiladis saxiT gadaweraSi (savarjiSo 12 ). 154000 77 13 I qalaqSi 18 wlamde axalgazrdebi mosaxleobis 460000 = 230 ≈0,33 nawils Seadgenen, 6 II qalaqSi _ 180000 570000 = 19 ≈0,32 nawils. moswavleebma unda gaixsenon, rom measedebamde damrgvalebuli pasuxis misaRebad sakmarisia gayofa Sesruldes meaTasedis Tanrigis CaTvliT. 25 1 16 wundebulia 1050 = 42 ≈0,02. SeiZleba davamatoT: Tu televizorebis 0,02 nawili aris wundebuli, vargisi ra nawili iqneboda? ≈0,98. es ricxvi axasiaTebs qarxnis nawarmis xarisxs. 18 100 sm-s moaWres 46 sm _ 0,46 nawili. saSinao davalebad moswavleebs vaZlevT 14 - 19 amocanebs.

53-e gakveTili (Semajamebeli)

Tema: aTwiladebis damrgvaleba, wiladis Cawera aTwiladis saxiT, gantolebis Sedgena da amoxsna. mizani: moswavleebi gaiwafon wiladis aTwiladis saxiT CaweraSi, aTwiladebis damrgvalebasa da SedarebaSi, amocanebis gantolebis Sedgenis saSualebiT amoxsnaSi. winapirobebi: aTwiladebze ariTmetikuli moqmedebebi, aTwiladebis damrgvaleba, aTwiladebis Sedareba. resursebi: kalkulatori, saxelmZRvanelo, rveuli, dafa. miTiTebuli Temebis mixedviT vatarebT Semajamebel mecadineobas, vimeorebT gavlil masalas. am gakveTiliT iwyeba dasaxelebuli Temis Seswavlis faza `Semdgom~. gakveTilis sawyisi monakveTi saSinao davalebis Semowmebas eTmoba. igi kvlav aTwiladis saxiT wiladis Caweris sakiTxebs ukavSirdeba. moswavleebs uwevT natural-

102


uri ricxvebis gayofa meaTasedebamde, rac sakmarisia imisTvis, rom mocemuli wiladi CavweroT 0,01 sizustiT aTwiladis saxiT. 2:11=0,181 20 11 2 90 11 ≈0,18 88 20 14

15

amocanis amoxsnisas moswavleebi iyeneben kalkulators, romelic ganayofs ase warmogvidgens: 2:9=0,2222222222... maSasadame, 29 ≈0,2 (0,1 sizustiT, meaTedebamde damrgvalebiT) perioduli aTwiladis saxiT ase Caiwereba: 29  =0,(2). 65 da 13 . SeiZleba CavweroT ori wiladi, romlebic 0,65-is tolia, magaliTad, 100 20 65 13 maswavleblebs SevaxsenebT, rom 100 , 20 , 0,65 aris erTi da imave ricxvis sxvadasxva Canaweri. saSinao davalebis Semowmebas 15-20 wuTi dasWirdeba. Semdeg gadavdivarT gav19

lili masalis gameorebaze III Tavis 1 - 4 damatebiTi amocanebis amoxsnis saSualebiT. maRali `mzaobis~ moswavleebs SeiZleba davavaloT `moisazres~ rubrikiT warmodgenil amocanebze fiqri, Sedegebs isini klass warmoudgenen da Cveni daxmarebiT, Tu amis saWiroeba warmoiSva, amocanebis amoxsnebs bolomde miviyvanT. Tavis damatebiTi amocanebidan vxsniT 6 6

0,15

-

8

amocanebs.

0,35 21 km

maSasadame, 21 km aris gzis 0,35 nawili. maSasadame, `nawiliT ricxvis povnasTan~ gvaqvs saqme. es amocana SeiZleba ase amovxsnaT: x⋅0,35=21, x=21:0,35=60 (km) 7 amosaxsnelia gantoleba: x⋅0,46+82,7=200 0,8⋅n>14 0,9⋅n<17 am utolobebis mixedviT, romlebsac amocanis piroba gvaZlevs, vpoulobT n naturalur ricxvs: 170 veZebT n naturalur ricxvs wiladebs Soris: 140 8 da 9 . 140 =17,5; 170 ≈18,9. 8 9 amrigad n naturaluri ricxvi SeiZleba iyos mxolod 18. aqve SeiZleba SesTavazoT mocemulis msgavsi, Tumca ufro sayuradRebo amocana, romelSic veZebT n naturalur ricxvs, romlis 0,9 nawili metia 74-ze, xolo 0,19 nawili naklebia 16-ze. amrigad, veZebT n naturalur ricxvs, romlisTvisac 0,9⋅n>74; 0,19⋅n<16, e. i. n aris 740 da 1600 wiladebs Soris. 9 19 8

103


740 =82  2 , 1600 =84  4 . 9 9 19 19 aq kalkulatoris gamoyenebiT miviRebdiT 82,222... da 84,222... . maSasadame, n SeiZleba iyos 83 an 84. moswavleebi imasac unda mieCvion, rom amocanis pirobas SeiZleba ramdenime ricxvi akmayofilebdes – pasuxs SeiZleba ramdenime ricxvi warmoadgendes.

moisazre am amocanebis amoxsnisas moswavle iwafeba problemaSi SesaZlo variantebis kvlevaSi, movlenis kanonzomierebis aRmoCenaSi, praqtikuli amocanis amoxsnisas am kanonzomierebis gamoyenebaSi. moswavle iyenebs drois sartyelebis Sesaxeb codnas, iyenebs moculobis (tevadobis) erTeuls praqtikuli amocanis amoxsnisas; uviTardeba logikuri azrovnebis da msjelobis unari. 1

am amocanis amoxsnis ramdenime variants SemogTavazebT. I. yovel Tavs 30-dan `CamovurigoT~ aucilebeli or-ori fexi. darCeba 24 fexi, romelTagan 12 Tavs `davumatebT~ kvlav or-or fexs. e. i. 12 cxoveli gaxdeba oTxfexa _ cxvari, danarCeni 18 iqneba orfexa _ indauri. am lamaz amoxsnas moswavleebi sinjvis meTods amjobineben xolme. sinjvis etapebs warmovadgenT cxriliT. cxvari

indauri

fexebi

15

15

4⋅15+2⋅15=90 metia

14

16

4⋅14+2⋅16=88 metia

13

17

4⋅13+2⋅17=86 metia

12

18

4⋅12+2⋅18=84

pasuxi: 12 cxvari, 18 indauri. SeiZleba moswavlem `zedmeti~ fexebis orze gayofiT gamoicnos cxvrebis raodenoba. ufro maRal klasSi am amocanas gantolebis SedgeniTac amoxsnian: cxvrebis raodenoba iyos x, indaurebis _ (30–x). fexebis raodenoba iqneba 4x+2(30–x)=84, saidanac x=12. 2

pirobidan Cans, rom Tbilisidan berlinisken mgzavrobis dros akldeba droSi Tbilissa da berlins Soris arsebuli gansxvaveba (Tbilisis dro uswrebs berlinis dros) da viRebT 2 sT-s, berlinidan Tbilisisken _ emateba igive dro da viRebT 6 sT-s. e. i. mgzavrobis xangrZlivoba 4 sT-ia da droSi gansxvavebaa 2 sT. maRal klasSi am amocanis amosaxsnelad SeiZleba sistema Seadginon T aris frenis dro, t _ gansxvaveba droSi.

T – t=2 , sadac T + t=6

Tbilisidan 10 sT-ze gasuli TviTmfrinavi berlinSi Tbilisis droiT 14 saaTze Cafrinda, am dros berlinSi 12 sT iyo. e. i. berlinis dro Tbilisisas 2 sT-iT CamorCeba, roca TbilisSi 9 sT-ia, berlinSi 7 sT-ia.

104


aseTi stilis amocanebis amoxsnas zogjer cxriliT warmoadgenen xolme, 3 romelSic aRwerilia gadasxmebis Tanmimdevroba. magram am SemTxvevaSi sityvieri aRwera ufro moklea. umartivesia a) SemTxveva. 6-jer CavasxaT kasrSi 5-litriani WurWliT wyali da 2-jer amoviRoT 12 litrianiT: 6⋅5-2⋅12=6. b) SeiZleba 3-jer CavasxaT kasrSi 5-litrianiT wyali, Semdeg amoviRoT 12 _ litri. kvlav 3-jer CavasxaT 5-litrianiT wyali da miRebul 18 litrs movakloT 12 litri. miiReba 6 litri wyali. g) 12-litrianSi CavasxaT ori xuTlitrianiT 10 l wyali, mesamedan SevavsoT 12 litriani. am 5-litrianSi dagvrCeba 3 litri. es 3 litri CavasxaT kasrSi. analogiuri proceduris gameorebiT kvlav miviRebT 3 litrs. amiT kasrSi SevagrovebT 6 litrs. cxadia, am wels TebervalSi 29 dRe iyo. 28 dRes qmnis 4 kvira. 29-Si unda yofiliyo mxolod mexuTe orSabaTi, e. i. orSabaTi iyo 22-Si, 15-Si,. 8-Si da 1 Tebervals. saSinao davalebisTvis gamoviyenoT davalebebi 5 , 9 - 12 III Tavis damatebiTi amocanebidan. 4

§2. wiladebis saerTo mniSvnelamde miyvana 54-e gakveTili

Tema: wiladebis miyvana saerTo mniSvnelamde. mizani: moswavles unda SeeZlos wiladis ZiriTadi Tvisebis saSualebiT wiladebis miyvana saerTo mniSvnelamde, rom momavalSi wiladebze Sekreba-gamoklebis moqmedebebi Caataros (maT VI. 2.) da gamoiyenos saTanado amocanebis amoxsnisas. winapirobebi: wiladis ZiriTadi Tviseba da misi gamoyeneba wiladebis gardaqmnis dros. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davaleba aTwiladebze ariTmetikuli moqmedebebis CatarebaSi moswavleTa gawafvas emsaxureboda. aseTi iyo, magaliTad, 5 da 9 amocanebi. 9 amocanaSi warmodgenil amocanebs moswavleebi moqmedebaTa komponentebs Soris kavSirebis gamoyenebiT xsnian, magaliTad, ucnobi Sesakrebis povna, ucnobi gasayofis povna, ucnobi gamyofis povna da a. S. es wesebi moqmedebaTa urTierTSebrunebadobis codnas efuZneba. 10 - 12 amocanebis mixedviT ki Sedgeba martivi erTucnobiani gantolebebi, romelTa amoxsna did siZneles ar unda warmoadgendes. rveuli – x 10 wigni – x+1,205 x+(x+1,205)=1,625 (x+x)+1,205=1,625 2⋅x =1,625-1,205 2x=0,420 x=0,210 pasuxi: rveuli – 0,21 kg, anu 210 grami. wigni – 1,415 kg.

105


11

qarTuls dauTmo 3⋅0,3=0,9 sT inglisurs da maTematikas _ 3–0,9=2,1 (sT) inglisurs – x maTematikas – x–0,3 x+x–0,3=2,1 2⋅x=2,4 x=1,2 (sT) inglisurze daxarja 1,2 sT, anu 60 wT⋅1,2=72 wT=1 sT 12 wT; maTematikaze – 72–60⋅0,3=72–18=54 (wT); qarTulze – 60⋅0,9=54 (wT). SevamowmoT, 54+54+72=108+72=180 wT=3 sT.

saSinao davalebis Semowmebis Semdeg gadavdivarT axali Temis damuSavebaze, romelic mimdinareobs saxelmZRaneloSi warmodgenili teqstis mixedviT. saerTo mniSvnelamde miyvanas win uZRvis winapirobebis (winare codnis) gameoreba da ganmtkiceba; wiladis ZiriTadi Tvisebis gaxseneba, gamoyeneba. moswavleebi Tavad ixseneben am Tvisebas da iyeneben Cven mier micemuli davalebebis Sesrulebis saSualebiT. aqve maswavleblebs SevaxsenebT, rom VI klasis standartis mixedviT, ar aris gaTvaliswinebuli naturaluri ricxvebis umciresi saerTo jeradis povnis algoriTmis Seswavla. es sakiTxi gadatanilia VII klasSi (aseve gadatanilia VII klasSi udidesi saerTo gamyofis povna, martiv mamravlebad daSlis algoriTmi). amitom ar aris saWiro am standartidan gadaxveva, 50 wlis winandeli saxelmZRvaneloebis gamoyeneba gakveTilebze. VI klasis masala sakmaod didia. misi damuSaveba siRrmiseulad Tu xdeba, Tu auCqareblad, yvela moswavlis CarTulobiT miiwevT win da regularulad ubrundebiT gavlil masalas, ar tovebT sxvadasxva rubrikiT warmodgenil aqtivobebs, erToblivad aanalizebT miRweul Sedegebs, iTvaliswinebT moswavleTa individualur SesaZleblobebs, yuradRebis gareSe ar tovebT damatebiT da Semajamebel amocanebs, saswavlo drois deficitic ki SeiZleba aRmoaCinoT. winare codnis gameorebis Semdeg gadavdivarT saerTo mniSvnelze miyvanis amocanis ganxilvaze. moswavleebi gamoiyeneben wiladis ZiriTad Tvisebas wiladebis saerTo mniSvnelze misayvanad. saerTo mniSvnelze miyvanil wiladebze ki advilad xorcieldeba Sekrebis da gamoklebis operaciebi, ricxvebis Sedareba. teqstSi aRwerilia gaerTmniSvnelianebis procesi sinjvis meTodiT. mniSvnelebidan metis jeradebidan varCevT meore _ ufro naklebi mniSvnelis jerads. garCeulia magaliTebi. moswavleebs vTavazobT amocanebs, romelTa amoxsna ukavSirdeba SeZenili codnis demonstrirebas. zogjer moswavleebi saerTo mniSvnelad SearCeven mniSvnelebis romelime saerTo jerads, gansxvavebuls umciresisgan (magaliTad, 16 da 18 -is gaerTmniSvnelianebisas 8 da 6 ). es sworia da amitom naSromis Sefasebaze uaryofiT gavlenas ar daweren 48 48 unda axdendes (cxadia, Tu moiTxoveba umciresi saerTo mniSvnelis moZebna, maSin miRebuli Sedegi xarveziania). savarjiSoebiT moswavleebi gaiwafebian wiladebis gaerTmniSvnelianebaSi.

106


1

da

2

savarjiSoebSi SeiZleba gamoiyenon proporciis Tviseba da gamoTvalon x, magram paragrafis miznebidan gamomdinare umjobesia daadginon 7x -is da 6x -is `damatebiTi~ mamravlebi da ise daadginon x-is mniSvneloba. magaliTad, 1 savarjiSoSi: ra ricxvze gavamravloT 7, rom miviRoT 35? _ 5-ze. e. i. x⋅5=15, saidanac x=3. cxadia, am moqmedebebs moswavleebi zepirad asruleben. 7 7 1 6 ramdeni meTxuTmetedia 5 -Si? pasuxi 5 -is 15 -ze gayofiTac SeiZleba davadginoT. magram am etapze igulisxmeba, rom moswavle 75 -s Cawers iseTi wiladis saxiT, romlis 7 mniSvnelic 15-ia _ 21 15 da daaskvnis: 5 aris 21 meTxuTmetedi. 25 = 25 . klasSi vxsniT 7 amocanasac – 14 = 4⋅25 100 saSinao davalebad vaZlevT

1

- 10

1 Seicavs 25 measeds. 4

amocanebs.

55-e gakveTili

Tema: wiladebis miyvana saerTo mniSvnelamde. mizani: moswavles SeeZlos wiladis ZiriTadi Tvisebis gamoyenebiT wiladebis gamartiveba da wiladebis saerTo mniSvnelamde miyvana (maT. VI. 2.) am proceduris gamoyeneba praqtikuli amocanebis amoxsnisas. winapirobebi: wiladis ZiriTadi Tviseba da misi gamoyeneba wiladis gasamartiveblad. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmeba dakavSirebuli iyo jeradis cnebisa da wiladis ZiriTadi Tvisebis gamoyenebasTan. moswavles SeuZlia gamoiyenos sinjvis xerxi umciresi 2 saerTo jeradis sapovnelad: 18 ar iyofa 4-ze, 18⋅2=36, 36 iyofa 4-ze, 36 iqneba 4 da 18-is umciresi saerTo jeradi, masze naklebi ricxvi ar gaiyofa orive mocemul ricxvze. 9

39 = 39⋅2 = 78 =0,78. 50 50⋅2 100

10

15 = 15⋅3 = 45 . 17 17⋅3 51

1 - 10 amocanebi SeiZleba CaiTvalos wiladebis saerTo saSinao davalebis mniSvnelamde miyvanisTvis Semamzadebel amocanebad. gakveTilze dasaxul mizanTan ki

SeiZleba davakavSiroT 8 - 15 amocanebis ganxilva. maTi nawili, magaliTad, 8 , 9 , 10 , 11 da 12 -is calkeuli davalebebi SeiZleba moswavleebs damoukideblad SesrulebisTvis davavaloT. maT mier aRniSnuli samuSaos SesrulebiT SeiZleba SevafasoT dasaxuli miznis miRwevis done. am amocanebis sxva davalebebi da 13 - 15 erToblivi msjelobiT amovxsnaT. 160 16 64

13 musikaze dadis moswavleTa 250 = 25 = 100 =0,64 nawili.

107


Teona 1 dReSi asrulebs mTeli samuSaos 13 nawils, mzia _ 15 nawils. am 5 nawils, ricxvebis saerTo mniSvnelia 15. pasuxi: Teona asrulebs mTeli samuSaos 15 3 nawils. mzia _ 15 1 3 15 elenem _ 6 nawili, giorgim _ 8 . am wiladebis gaerTmniSvnelianebiT miiReba 4 9 24 da 24 . saSinao davalebad moswavleebs mivcemT 11 - 18 amocanebs. 14

56-e gakveTili

Tema: wiladebis miyvana saerTo mniSvnelamde. mizani: moswavleebi gaiwafon wiladebis saerTo mniSvnelamde miyvanaSi; `moisazres~ rubrikiT warmodgenili amocanebis gamoyenebiT poziciuri sistemis gamoyenebasa da weriTi algoriTmiT Sesrulebuli gamravlebis magaliTSi gamotovebuli cifrebis Casmasa da pasuxebis dasabuTebaSi (maT. VI. 4). ganiviTaron masalis praqtikuli gamoyenebis unarebi. winapirobebi: wiladis ZiriTadi Tviseba, ricxvebis qveSmiweriT gamravleba, naSTiani gayofa. resursebi: rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT, romelsac SeiZleba 15 wuTi movandomoT. 11 amocanaSi moswavle unda moxvdes, rom sakmarisia wiladebi daviyvanoT 60-is tolmniSvnelian wiladebamde; 3 sT = 3⋅15 sT = 45 sT = 45 wT 5 sT = 5⋅10 sT = 50 sT = 50 wT. 4 4⋅15 60 6 6⋅10 60 12

wiladebis Sekvecis Semdeg isini SeiZleba miviyvanoT wiladebamde, romlis mniSvnelia 36: 6 2 2⋅4 8 27 = 9 = 9⋅4 = 36 5 -s ver miviyvanT wiladamde, romlis mniSvnelia 36, radgan 36 ar iyofa 7-ze. 13 7 15

amocana yvelas unda hqondes amoxsnili – wiladebi unda miviyvanoT saerTo mniSvnelamde. 9 da 5 98 56 98 2 49 7 7

56 2

28 2

7

2

98=2⋅7⋅7; 56=2⋅2⋅2⋅7 cxadia, umciresi saerTo mniSvneli aris 4⋅98. 9 9⋅15 36 5 5⋅7 35 98 = 98⋅4 = 392 , 56 = 56⋅7 = 392 .

108


17

256 = 128 = 32 = 2 640 320 80 5

2 = 2⋅2 = 4 =0,4. 5 5⋅2 10 nu davuSliT moswavleebs wiladi ase `TandaTanobiT~ Sekvecon.

18

3 1 24 = 8 ,

6 1 24 = 4

125 125 1 250 250 8⋅125= 1000 =0,125 4 =4⋅250= 1000 =0,25

amave gakveTilze SeiZleba amoixsnas `moisazres~ rubrikiT warmodgenili amocanebi. am amocanebidan gansakuTrebuli yuradReba miaqcieT 5 amocanis amoxsnas, romelic standartSi miTiTebul erT-erT mizans ukavSirdeba. aseTi etapis amocanebs momavalSic amovxsniT. `moisazres~ amocanebze muSaoba SeiZleba davavaloT maRali mzaobis moswavleebs, romlebic Semdeg mTel klass warmoudgenen amoxsnebis sakuTar variantebs, an Cveni daxmarebiT yvela moswavlis monawileobiT moxdeba amoxsnebis bolomde miyvana. SeiZleba am gakveTilze dasaxuli miznis miRwevis donis dasadgenad fokusirebuli damoukidebeli wera Catardes. gTavazobT amocanebs: 1. 15-is gamyofebia: b) 8 da 7 g) 1, 3, 5 da 15 d) 2, 3, 6 da 4. a) 10 d 5 2. 48-is jeradi udidesi orniSna ricxvia a) 12 b) 24 g) 99

d) 96

3. sabam rac SeiZleba bevr bavSvs Tanabrad gaunawila 48 kakalic da 32 kanfetic. ramden bavSvs gaunawilebda saba? a) 12-s b) 16-s g) 18-s d) 24-s 4. 21-is jeradia ricxvi a) 7 b) 14 g) 28

d) 42.

5. ipoveT 320-is udidesi orniSna gamyofi a) 160 b) 80 g) 40 d) 96 6. ipoveT 48-is umciresi samniSna jeradi a) 96 b) 144 g) 192 d) 240. 7. CawereT wiladi 56 iseTi wiladis saxiT, romlis mniSvnelia 48. 5 b) 40 a) 48 g) 10 d) 25 48 48 48 . 8. CawereT 21 28 iseTi wiladis saxiT, romlis mniSvnelia 20. 14 7 a) 15 g) 20 d) 13 20 b) 20 20 . 9. 56 da 34 -is umciresi saerTo mniSvnelia a) 10 b) 12

g) 24

d) 36.

109


10. velosipedis fasi 720 laria. sadResaswaulod velosipedi 120 lariT gaiafda. fass ra nawili daaklda? 5 a) 16 b) 56 g) 72 d) 13 . pasuxebi:

1 g

2 d

3 b

4 d

5 b

6 b

7 b

8 a

9 b

10 a

SesaZloa pasuxebi dafaze amoiweros da moswavleebma TviTSefaseba am gziT moaxdinon. SeiZleba pasuxebis dasafiqsireblad moswavleebs ori are gamouyoT. isini or adgilas afiqsireben pasuxebs, naSromebis Cabarebis Semdeg itoveben erT variants da Sedegebis ganxilvisas iyeneben mas. es xerxi drosac dagvazoginebs da moswavlis TviTSefasebasac ganaxorcielebs. 13 - 18 amocanebs III Tavis damatebiTi davalebebidan moswavleebs saSinao davalebad mivcemT. gTavazobT miTiTebebs `moisazres~ rubrikiT mocemuli amocanebis amosaxsnelad. saZiebel ricxvze 1-iT naklebi ricxvi 11-zec iyofa da 12-zec. aseTi umciresi ricxvia 11⋅12=132. pasuxi: 133. 1

saZiebel ricxvze 1-iT meti ricxvi 11-zec iyofa da 12-zec. aseTi ricxvebidan umciresia 11⋅12=132. pasuxi: 131. 2

usj (55; 65)=715. e. i. paatas da vatos nabijebi yoveli 715 sm-is Semdeg emTxveva erTmaneTs. 20-jer damTxveul nabijebs Soris 19 cali 715 sm-iani Sualedia, e. i. distanciis sigrZea 715⋅19=13585 sm=135,85 m. 3

I awonviT aiwoneba 1 kg brinji. II awonvisas sawonad 1 kg-ian sawonTan erTad 1 kg brinjsac gamoviyenebT da avwoniT 2 kg brinjs. III awonvisas cal mxares davdebT 1 kg-ian sawons da 3 kg brinjs da avwoniT 4 kg brinjs. sul sami awonviT avwoniT 7 kg brinjs. 4

radgan 6-iT dawyebuli orniSna ricxvis mxolod 1-ze namravlia kvlav orniSna, amitom II Tanamamravli aris 11. III striqonis erTeulebis TanrigSi 6-ia, e. i. I Tanamamrvalis erTeulebis Tanrigic 6-ia da miviRebT: 6,6 × 11 66 66 7  2,6 5

6 yuradReba gasamaxvilebelia im faqtze, rom gasayofis da gamyofis erTi da imave

ricxvze gamravlebisas ganayofi ar icvleba, naSTi ki icvleba. marTlac, maswavlebelma icis, rom Tu a ricxvis b-ze gayofisas ganayofia q da naSTi r, maSin a=bq+r da tolobis orive mxaris raime n-ze gamravlebiT miviRebT: an=bnq+nr, saidanac Cans, rom an-is bn-ze gayofisas ganayofi ar icvleba _ isev aris q, xolo naSTi mravldeba n-ze.

110


swored amitom miiRo mariamma irmas Sedegze _ 0,5-ze 10-jer meti naSTi, man xom gasayofica da gamyofic 10-ze gaamravla. am etapze moswavleebs rTuli dasabuTebiT ar gadavtvirTavT, mxolod magaliTebiT vaxdenT faqtis demonstrirebas. magaliTad, 23-is 5-ze gayofiT miiReba ganayofi 4 da naSTi 3. gasayofic da gamyofic gavamravloT 4-ze. maSin 92-is 20-ze gayofiT miiReba ganayofi kvlav 4, naSTi ki 12, rac 4-jer metia 3-ze. SeiZleba cxrili gamoviyenoT:

gavamravloT 10-ze gavamravloT 5-ze gavamravloT 4-ze da a. S.

gasayofi

gamyofi

ganayofi

naSTi

23

5

4

3

230

50

4

30

115

25

4

15

92

20

4

12

saidanac kargad Cans, rom ganayofi ar icvleba, naSTi icvleba _ mravldeba mamravlze (10-ze, 5-ze, 4-ze da a. S.)

§3. wiladebis da Sereuli ricxvebis Sedareba 57-e gakveTili

Tema: wiladebis da Sereuli ricxvebis Sedareba. mizani: moswavles SeeZlos wiladebis Sedarebis sxvadasxva xerxis gamoyeneba; wiladis gamosaxulebaSi mTeli da wiladi nawilebis, mricxvelisa da mniSvnelis miTiTeba da am codnis gamoyeneba wiladebis Sedarebisas (maT. VI. 1). winapirobebi: wiladebis gaerTmniSvnelianeba, gamosaxva ricxviT wrfeze. resursebi: makrateli, qaRaldis furclebi, rveuli, saxelmZRvanelo, dafa. saSinao davalebad micemuli iyo III Tavis damatebiTi amocanebidan 13 - 18 (wiladebis saerTo mniSvnelamde miyvana). wina sami gakveTili am Temis ganxilvas eZRvneboda, amitom am amocanebis amoxsna moswavleebs ar unda gauWirdeT. davalebis Semowmebis dros kidev erTxel vimeorebT wiladebis saerTo mniSvnelamde miyvanis wess, rac axal masalaze gadasvlis Semamzadebeli etapic aris (15-20 wT). wiladebis Sedarebis procesSi SeiZleba gamoviyenoT TvalsaCino masala, gamovWraT erTnairi marTkuTxedebi da moswavleebTan erTad mivyveT saxelmZRvanelos teqsts. moswavleebi ukve gawafulni arian wiladebis gaerTmniSvnelianebaSi. suraTze warmodgenili orive marTkuTxedis daStrixuli nawilebi erTnairmniSvneliani wila­ debiT gamoisaxeba: 58 da 78 -iT. maTi Sedareba mricxvelebis Sedarebaze daiyvaneba: 5<7, e. i. 58 < 78 erTnairmniSvneliani wiladebidan is aris meti, romlis mricxvelic metia. ramdenime erTnairmniSvneliani wiladis Sedarebis Semdeg moswavleebi kvlav unda daukvirdnen teqst­ Si warmodgenil marTkuTxedebs. maTi daStrixuli nawilis gamomsaxveli wiladebis mri­­ cx­ veli mniSvnelze naklebia. aseTi wiladebi wesieri wiladebia. wesieri wiladi 1-ze na­klebia. sxva SemTxvevaSi, e. i. roca mricxveli mniSvnelis tolia, an mniSvnelze me­tia, wiladi arawesieria. aseTi wiladi erTis tolia, an erTze

111


metia. magaliTad, 23 , 57 we­­ si­ eri wiladebia _ TiToeuli 1-ze naklebia. arawesieri 5 5 wiladebia, magaliTad, 5 =1, 4 >1. teqstSi ganxilulia sxvadasxvamniSvneliani wiladebis Sedarebis xerxi da Sesabamisi magaliTebi _ wiladebis gaerTmniSvnelianeba, Semdeg mricxvelebis Sedareba da ori ricxvis 1-Tan siaxlove, romelic dakavSirebualia ricxviT sxivze ricxvebis ganlagebasTan. 9 -s. orive wesieria _ naklebia 1-ze, orive ganxilul magaliTSi vadarebT 34 -sa da 10 9 -s aklia 1 . 1 > 1 , e. i. 3 ufro Sorsaa 1-gan, 1-is marcxnivaa. 34 -s erTamde aklia 14 , 10 10 4 10 4 3 3 9 anu 4 ufro marcxnivaa ricxviT sxivze: 4 < 10 . bolos vadarebT Sereul ricxvebs. savarjiSoebiT moswavleebi gaiwafebian gaerTmniSvnelianebuli wiladebis Seda­ rebaSi, ricxvebis Sedarebisas ricxviTi sxivis gamoyenebaSi. savarjiSoebi gaTvlilia ara marto gavarjiSebaze, aramed logikuri azrovnebis, msjelobis unaris ganviTarebaze. klasSi vxsniT 1 - 7 amocanebs. 1 - 4 amocanebi SeiZleba moswavleebma damoukideblad amoxsnan da maT mier samuSaos Sesrulebis mixedviT SeiZleba SevafasoT gakveTilze dasaxuli miznis Sesruleba. 5 5 8 5 5, 8 da 9 ricxvebiT Sedgeba 3 wesieri wiladi: 8 , 9 da 9 . maTi gaerTmniSvne­li­ 40 64 5 5 8 anebiT miiReba, Sesabamisad, 45 72 , 72 da 72 . zrdiT dalagebis Semdeg miviRebT: 9 , 8 , 9 . II adgilzea 58 . 6 amocanis amoxsnisas SeiZleba davsvaT kiTxva – mTeli gzis ra nawili darCa gasavleli? 7 amocanis amoxsnisas SeiZleba saWiro gaxdes Cveni Careva; – mas Semdeg, rac moswavleebi ipovnian swor pasuxebs – 25 da 35 , SeiZleba gavarTuloT kiTxva: – SeiZleba Tu ara davasaxeloT ricxvebi, romlebic 25 -sa da 35 -s Sorisaa? 25 -ze metia da 35 -ze aris naklebi. moswavleebma SeiZleba ifiqron, rom am ricxvebs Soris SeuZlebelia ricxvis povna. maSin maT SevTavazoT, es wiladebi sxva wiladebiT warmoadginon, romelTa mniSvneli 4 , 3= 6 . iqneba 5-ze meti, magaliTad, 25 = 10 5 10 5 am or ricxvs Soris aris 10 , romelic 12 -is tolia. 8 da 3 = 12 da a. S. SeiZleba meti ricxvis aRmoCenac, Tu, magaliTad, CavwerT 25 = 20 5 20 saSinao davalebad moswavleebs vaZlevT 1 - 8 amocanebs. am amocanebidan. magaliTad, 5 amocana zemoTMmoyvanili msjelobis Catarebas moiTxovs: 8 = 16 79 = 14 18 , 9 18 15 14 15 < 16 . 18 > 18 , 18 18

112


58-e gakveTili

Tema: wiladebis da Sereuli ricxvebis Sedareba. mizani: wiladebisa da Sereuli ricxvebis Sedarebis wesebis Sesaxeb codnis ganmtkiceba da saTanado gamoyenebiTi unarebis ganviTareba. winapirobebi: wiladis sxvadasxva interpretacia, gamosaxva ricxviT sxivze, wiladis ZiriTadi Tviseba resursebi: rveuli, saxelmZRvanelo, qaRaldis furclebi, makrateli, dafa. saSinao davalebis Semowmeba SeiZleba swrafad CavataroT (10-15 wT). analogiuri amocanebi klasSi iyo amoxsnili. vagrZelebT wiladebisa da Sereuli ricxvebis Sedar-

- 19 amocanebs. am amocanebidan 8 - 11 SeiZleba damoukidebel samuSaod SevTavazoT moswavleebs. 12 - 19 amocanebi erToblivi Zalebis amocanebis ganxilvas. vxsniT

8

isxmeviT amoixsneba, dafasTan gamoZaxebuli moswavlis aqtiuri CarTulobiT. 12 amocanis amoxsnisas SeiZleba damxmare modelis gamoyeneba. marTkuTxedis xuT tol nawilad dayofisas, TiToeuli nawili metia, vidre amave zomis marTkuTxedis 7 tol nawilad dayofisas miRebuli TiToeuli nawili, amitom, cxadia, 47 < 45 . 12 da 13 savarjiSoebiT moswavle ara marto adarebs wiladebs sxvadasxva xerxiT,

aramed damoukideblad ayalibebs maTematikur debulebebs _ Sedarebis wesebs. kerZod, # 12 savarjiSoSi: erTnairmricxveliani wiladebidan metia is, romlis mniSvnelic 1<1. naklebia; 13 savarjiSoSi: Tu m da n naturaluri ricxvebia, m>n, maSin m n 3

2

14 Teonas finjanSi 5 nawili rZea, 5 nawili yava. 15 gamoviyenoT wiladis ZiriTadi Tviseba:

15 = 30 , 16 = 32 , maT Sorisaa 31 17 34 17 34 34 15 45 16 48 46 17 = 51 , 17 = 51 , maT Sorisaa 51 da a. S. 16 SeiZleba yvela wiladi aTwiladis saxiT gadavweroT. ar aris aucilebeli gayofis dasruleba, periodis dadgena, an damrgvaleba. mZimis marjvniv ori-sami Tanrigis miRebis Semdeg vwyvetT gayofas: 4 =0,571...; 3 =0,6; 0,56; 0,69; 9 =0,64... 7 5 14 klebiT dalagebis Semdeg miviRebT: 9 , 3 , 4 , 0,56. 0,69, 14 5 7

17 ar aris saWiro gverdebis sigrZeebis gamoTvla. sakmarisia nawilebis Sedareba.

75 nawili, b aris 72 nawili, c _ 70 , gavaerTmniSvnelianoT isini. a aris 120 120 120 a, umciresia c. 10

udidesia 12

18 gaerTmniSvnelianebiT vRebulobT: I dRes waikiTxa 45 nawili, II dRes _ 45 .

cxadia, II dRes waikiTxa meti. sul wakiTxulia 22 45 nawili, rac wasakiTxze naklebia.

113


19 18 tol nawilad dayofili erTeulovani monakveTis yoveli danayofis sigrZe 1 -is tolia. 18

1 = 6 . A aris 0-dan me-6 danayofis marjvena wertili. A-dan marjvniv 5 -iT 3 18 18 11 -s. C-dan marcxniv 7 danayofiT ( 7 -iT) daSorebulia C wertili, is gamosaxavs 18 18 4 daSorebulia D wer­tili. is gamosaxavs 18 -s. SeiZleba suraTis mixedviT upasuxon kiTxvebs: ramdeniTaa daSorebuli D wertili A-sgan? C wertili U-Usgan, D _ U-sgan? saSinao davalebad moswavleebs vaZlevT

9

- 18

amocanebs.

59-e gakveTili (Semajamebeli)

Tema: wiladebisa da Sereuli ricxvebis Sedareba. mizani: moswavleebi gaiwafon wiladebisa da Sereuli ricxvebis Sedarebasa da dalagebaSi da amiT gaiRrmaon codna am ricxvebisa da maTiO gamoyenebis Sesaxeb. winapirobebi: wiladis ZiriTadi Tviseba, saerTo mniSvnelamde miyvana. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis 11 da 12 savarjiSoebi erTnairmricxveliani wiladebis Sedarebis wesis gamoyenebaSi gawafvaa. 3 , 3 , 3 , 5 , 5 , 7 . sasurvelia moswavleebma wiladebis aseTi erTobliobis 5 7 11 7 11 11 Sedgenis kanonzomierebazec gamoTqvan sakuTari mosazrebebi. gavaerTmniSvnelianoT: 3 = 231 , 3 = 165 , 3 = 105 , 5 = 275 , 5 = 175 , 7 = 245 da davalagoT zrdiT: 5 385 7 385 11 385 7 385 11 385 11 385 3 , 3, 5 , 3, 7 , 5. 11 7 11 5 11 7 SeiZleba umjobesi aRmoCndes am ricxvebis aTwiladebad Cawera. gayofa gavagrZeloT 3 =0,27...; 5 =0,71...; 5 =0,45...; 7 =0,63... . measedebamde (CaTvliT). marTlac, 35 =0,6; 37 =0,42...; 11 7 11 11 13

moswavleebs urCieT, rom am donis gamoTvlebisas kalkulators nu moixmaren. ufro rTuli gamoTvlebis SemTxvevaSi ki unda daixmaron kalkulatori. isini unda flobdnen misi gamoyenebis teqnikas. 14 klasSi amoxsnili 18 amocanis msgavsia. 15

umjobesia yvela ricxvi CavweroT aTwiladis saxiT: 7 =1,28; 1  1 =1,25 da 1,32. 1,3; 1  25 =1,4; 1 25 4 davalagoT zrdiT, miviRebT 1 , 1 7 , 1,3, 1,32, 1  2 . 1  4 25 5

114


1 nawili, II miliT _ 1 nawili. 7 sT-Si I miliT 1 sT-Si I miliT ivseba auzis 10 4 7 28 3 30 7. aivseba auzis 10 = 40 nawili, 3 saaTSi II miliT aivseba 4 = 40 nawili. 34 > 10 16

4 m. sammetrianis TiTo nawili 37 m-ia, oTxmetrianis _ 10 3 = 30 , 4 = 28 , 3 > 4 _ I ficris nawilebia ufro grZeli. 7 70 10 70 7 10 17

9 , 2 = 10 , 4 = 12 . gavaerTmniSvnelianoT wiladebi: 15 = 45 9 45 15 45 1 2 4. davalagoT es wiladebi zrdis mixedviT: 5 < 9 < 15 18

pirobiT, davalebis Sesrulebas yvelaze cota dro _ gakveTilis 15 nawili moandoma 4 nawili _ mariamma, ninikom moandoma 2 nawili. vatom. yvelaze meti dro _ 15 9 radgan gakveTili aRniSnuli Temis SeswavlisTvis Semajamebeli saswavlo procesis rols asrulebs, amitom am davalebis Semowmebisas ar unda viCqaroT – kargad gavarkvioT, aiTvises Tu ara moswavleebma sakiTxi, Sesrulebulia Tu ara dasaxuli mizani. saWiroebis SemTxvevaSi zogierTi amocana SeiZleba klasSi amoixsnas. damatebiT amave Temaze – damoukidebeli muSaobisTvis gTavazobT III Tavis damatebiTi amocanebidan davalebebs 19, 20, 21, 22, 24; saSinao davalebad ki amocanebs 23, 25-30. am amocanebidan 28 - 30 , sasurvelia, klasSi amoixsnas. 28 m 28 cnobilia, rom 12 naklebia 48 -ze. risi toli SeiZleba iyos m? – gvaqvs sxvadasxvamniSvneliani wiladebi. – xom ar SeiZleba meore wiladis Sekveca? gaaiolebda es wiladebis Sedarebas? m< 7 ; – ra utolobas miviRebT m asos Semcvels? 12 12 – ra naturaluri ricxvebi akmayofilebs am utolobas? (m=1, 2, 3, 4, 5, 6). 29 SeiZleba asoiTi gamosaxulebis Sedgena. Tu `lobiani~ a lari Rirs, xaWapuri

3a eRireba 3a lari. naxevari lobiani a2 = 2a 4 (lari), xaWapuris meoTxedi – 4 (lari). cxadia, xaWapuris meoTxedSi mogviwevs metis gadaxda. am amocanebze pasuxs zogierTi moswavle zepirad miagnebs. 30 swrafad muSaobs is, romelic saaTSi samuSaos met nawils asrulebs, Sesadare3 da 4 . belia 20 25 3 15 4 16 20 = 100 , 25 = 100 .

§4. wiladebis Sekreba da gamokleba me-60 gakveTili

Tema: wiladebis Sekreba da gamokleba. mizani: moswavles SeeZlos wiladis ZiriTadi Tvisebis gamoyeneba wiladebze Sekreba-gamoklebis moqmedebebis Sesrulebisas (maT. VI. 2.), gaiazros am moqmedebis arsi da miRebuli codniT gaamdidros amocanaTa gadawyvetis maTematikuri aparati.

115


winapirobebi: wiladis ZiriTadi Tviseba. resursebi: qaRaldis furclebi, makrateli, rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebisas kidev erTxel davrwmundeT – aiTvises Tu ara moswavleebma wiladis ZiriTadi Tvisebis gamoyeneba saerTo mniSvnelamde miyvanis ganxorcielebis procesSi. teqstSi warmodgenili suraTebis ukeT gaazrebis mizniT, sasurvelia gamoviyenoT toli marTkuTxedebis modelebi. teqstis mixedviT Sedgenili am modelebis mixedviT moswavle martivad SeamCnevs, rom 3–2=1. 38 + 18 = 48 , 8 8 8 tolmniSvneliani wiladebis Sekrebisa da gamoklebis ramdenime magaliTis ganxilvis Semdeg savaraudoa, rom moswavle SeZlebs aseTi wiladebis Sekreba-gamoklebis wesis Camoyalibebas: tolmniSvneliani wiladebis Sekrebisas (gamoklebisas), mniSvneli ga­da­ dis ucvlelad, mricxvelebi ikribeba (akldeba). teqstSi detaluradaa ganxiluli Sekrebis da gamoklebis magaliTebi, roca: mniSv­ nelebi sxvadasxvaa, SesakrebTa raodenoba 2-ze metia, wiladebis Sekrebisas Sekrebis Tvisebebis gamoyenebaa saWiro. teqstSi ganxilul me-5 da me-6 magaliTebSi moqmedebebis Sesrulebisas Sesakrebebis arawesier wiladebad warmodgenis gamoyenebasac vTavazobT. zogjer moswavleebi am xerxsac mimarTaven, gansakuTrebiT, gamoklebisas, roca saklebis wiladi nawili maklebis wilad nawilze naklebia. magram SevniSnavT, rom arawesier wiladebad warmodgena, roca Sesakrebebis mTeli nawilebi `didi~ ricxvebia, mouxerxebelia. aseT SemTxvevaSi teqstSi warmodgenili wesia (me-5 da me-6 magaliTebSi) umjobesi. klasSi vxsniT 1 - 10 amocanebs. klasSi amosaxsneli savarjiSoebidan yuradReba gavamaxviloT nebze.

9

da 10 amoca­

a+b udidesia, Tu Sesakrebebia udidesi. radgan isini gansxvavebulia, erTi 13 27 maTgani 14 15 -ia, meore _ 15 , jami _ 15 . 10 1 10 a–b udidesia, roca a udidesia _ udris 11 -s, b umciresia _ udris 11 -s. 1 9 a–b= 10 11 – 11 = 11 . 9

danarCeni amocanebis amoxsnas moswavleebi damoukidebladac SeZleben. maT mier Sesrulebuli samuSaos mixedviT SeiZleba SevafasoT gakveTilze dasaxuli miznis Sesruleba. davalebad vaZlevT 1 - 9 amocanebs.

61-e gakveTili

Tema: wiladebis Sekreba da gamokleba. mizani: moswavle gaiwafos wiladebis Sekrebasa da gamoklebSi saerTo mniSvnelamde miyvanis gamoyenebiT (maT. VI. 2.). winapirobebi: wiladis ZiriTadi Tvisebis gamoyeneba wiladebis gardaqmnis dros. resursebi: rveuli, saxelmZRvanelo, dafa.

116


7 8 saSinao davalebis Semowmebisas yuradReba gaamaxvileT da amocanebis amoxsnaze. SeiZleba am amocanebma garkveuli siZneleebi Seuqmnas moswavleebs, miuxedavad imisa, rom isini 9 da 10 -is analogiuria. amitom sasurvelia maTi klasSi dawvrilebiTi garCeva:

a+b udidesia, roca a da b udidesia. isini gansxvavebulia, amitom erTi 15 31 maTgani 16 17 -ia, meore _ 17 , a+b= 17 . 7

1 a-b udidesia, roca a udidesia, tolia 12 # 8 13 -is da b umciresia, tolia 13 -is. 11 . a–b= 13 davalebis Semowmebis process SeiZleba 15-20 wT dasWirdes. Semdeg gadavdivarT 11

- 20 amocanebze. SeiZleba maswavlebelma miiCnios, rom am amocanebidan nawili saSinao davalebad gamoiyenos 10 - 19 amocanebTan erTad. Tumca, sasurvelia, 15 -is ramdenime savarjiSo da 16 - 20 klasSi amoixsnas, amasTanave - moswavleTa aqtiuri monawileobiT. 16 da 17 amocanebs SeiZleba vuwodoT 25 39 1 `amocanebi nawilebze~; roca SeRebilia kedlis 14 40 + 40 = 40 , darCenili iqneba kedlis 40 40 39 1 nawili (1– 39 40 = 40 – 40 = 40 ). 1 1 1 41 17 10, 8 da 7 qula miiRo meTormeteklaselTa 7 + 3 + 2 = 42 nawilma. pirobiT, dana­ 1 nawilma miiRo Sefaseba 6. Tu meTormeteklaselTa 1 aris erTi r­Cenebma, anu 42 42 moswavle, skolaSi yofila 42 meTormeteklaseli. 1

1

20 1 sT-Si mxolod I miliT ivseba auzis 4 nawili, mxolod II miliT _ 6 nawili,

5 nawili. Sesavsebi darCeba 1– 5 = 7 nawili. oriveTi 1 sT-Si ivseba 14 + 16 = 12 12 12 saSinao davalebad vaZlevT

10 - 19

amocanebs.

62-e da 63-e gakveTilebi

Tema: wiladebis da Sereuli ricxvebis Sekreba da gamokleba. mizani: moswavleebi gaiwafon wiladebisa da Sereuli ricxvebis Sekrebasa da gamoklebaSi, wiladebis aTwiladis saxiT CaweraSi, ganiviTaron saTanado unarebi. winapirobebi: wiladis ZiriTadi Tviseba da misi gamoyeneba, wiladis warmodgena aTwiladis saxiT. resursebi: kalkulatori, rveuli, saxelmZRvanelo, dafa. am gakveTilebze vajamebT muSaobas wiladebis aTwiladebad Caweraze, wiladebis Sekrebasa da gamoklebaze. yuradReba gaamaxvileT – aiTvises Tu ara wiladebisa da Sereuli ricxvebis Sekrebisa da gamoklebis yvela SemTxveva moswavleebma. magaliTad, roca mTels akldeba wiladi, roca saklebis wiladi nawili naklebia maklebis wilad nawilze. saSinao davalebis Semowmebas SeiZleba pirveli gakveTilis 15-20 wT movandomoT. gakveTilis Semdegi monakveTi SeiZleba codnaSi xarvezebis aRmofxvras mivuZRvnaT. amisaTvis SeiZleba mokle fokusirebuli weris Catareba, romelic dasrulebisTanave unda gaanalizdes.

117


gTavazobT amocanebs: 1. 34  +  45  = 11 a) 79 b) 1 20

g) 1  29

1 d) 1 20

11  –  1  = 2. 24 4 a)   12 b) 43 48

5 g) 24

d) 38

3. 12  +  56  –  13 = a) 1 b) 15

g) 25

d) 23

4. 0,65 –  12 = a) 0,25

g) 0,35

d) 0,45

g)  13

d) 19 75 .

b) 0,15

2= 5. 0,12 +  15 6 a)   18 b) 25 pasuxebi: 1

2

3

4

5

b

g

a

b

d

jgufur muSaobas meore gakveTili eTmoba. saSinao davalebad SeiZleba mivceT 31 38 davalebebi III Tavis damatebiTi amocanebidan, romelic or davalebad ganawildeba. axla gTavazobT miTiTebebs saSinao davalebis zogierTi amocanisTvis: 7 2 5 7 3 3 3 14 a) (1 +5 )+(2 + )+28+4 =7+3+28+4 =42 . 9 9 12 12 4 4 4 7 5 1 1 1 11 b) (8 12 –7 12 )+3 5 =1 6 +3 5 =4 30 . g) (8 59 –4 29 )–2 16 =4 13 –2 16 =2 16 . d)-Si moqmedebebi SeiZleba TanamimdevrobiT SevasruloT, umjobesia, ricxvebi aTwila­debis saxiT CavweroT: (5 35 –3,15)+7 12 25 =(5,60–3,15)+7,48=9,93. 10–6 34 sxvaobiT gavigebT ramdeni l Rvinoa III WurWelSi, III WurWelSi 3 14 (l) 1 (l), pirvelSi _ 6 3 –2 1 =4 2 (l). Rvinoa. meoreSi _ 5 13 –3 14 =2 12 4 12 3 15

16

c=18–12 12 =5 12 (sm);

b=12 23 –5 12 =7 16 (sm);

a=18–12 23 =5 13 (sm) (an 12 12 –7 16 ).

m = 4m . 4m aris 9-ze naklebi, roca m=1 an m=2. a) 13 52 m 5m b) 6 = 30 , m-is arc erTi naturaluri mniSvnelobisTvis 5m ar aris 5-ze naklebi. pasuxi: aseTi m ar arsebobs. m < 10 , amitom m aris 10-ze naklebi nebismieri naturaluri ricxvi. g) 85 85 7 <2. m-is zrdisas 7 wiladi mcirdeba, e. i. 7 >2, roca m<4, anu m=3, 2 d) Tu m=4, m m m an 1. 17

118


rogorc zemoT aRvniSneT, Semdeg gakveTilze vatarebT jgufur muSaobas, romlis saSualebiT mimdinareobs gavlili masalis gameoreba, codnis ganmtkiceba. saxelmZRvaneloSi damatebiT warmodgenilia amocanebi rubrikiT `vemzadebiT viqtorinisTvis~. gTavazobT miTiTebebs jgufuri muSaobis Catarebisa da klasgareSe muSaobisTvis gankuTvnili amocanebis amosaxsnelad. jgufuri muSaoba am jgufuri muSaobisas moswavleebs kvlav racionaluri ricxvis aTwiladad warmodgenis formebze vumaxvilebT yuradRebas. am sakiTxs wina paragrafebis savar­ ji­ SoebSic SevexeT. amjerad mizani masalis gameorebaze, ganmtkicebaze, gaRmavebaze, siste­maSi moyvanaze zrunvaa. jgufuri muSaoba ramdenime etapadaa dayofili: I etapze moswavleTa jgufebi we­ riTi gayofiT an kalkulatoriT Caweren aTwiladis saxiT maswavleblis mier war­ mo­dgenil wiladebs, romlebic or jgufad aris dayofili (magaliTad, teqstSi a) da b) jgufebSi gaerTianebul wiladebs). moswavleebi miuTiTeben yoveli wiladis mniSvnelis martiv gamyofebs. es daexmareba moswavleebs, jgufuri muSaobis II etapze Seadginon wiladebi, romelTa mniSvnelis martiv gamyofebs Soris 2-isa da 5-is garda sxva martivi gamyofi ar aris da Caweron isini aTwiladis saxiT. III etapze Seadginon ukveci wiladebi, romelTa mniSvnelebis martiv gamyofebSi Sedis 2-isa da 5-isgan gansxvavebuli erTi martivi gamyofi mainc da Caweron es wiladebic aTwiladis saxiT. bolo etapze moswavleTa jgufebi cdiloben gamoTqvan varaudi, aRmoaCinon kanonzomiereba (gaaanalizon miRebuli Sedegebi) da Camoayalibon wesi _ ra SemTxvevaSi Caiwereba ricxvi perioduli aTwiladis saxiT (Seqmnan produqti analizis safuZvelze _ sinTezi): Tu wiladis mniSvnelis martiv gamyofebSi 2-isa da 5-is garda martivi ricxvi ar aris, wiladi Caiwereba sasruli aTwiladis saxiT. Tu ukveci wiladis mniSvnelis martiv gamyofebSi 2-isa da 5-isgan gansxvavebuli 1 martivi gamyofi mainc aris, maSin wiladi Caiwereba usasrulo perioduli aTwiladis saxiT. vemzadebiT viqtorinisTvis amocanebi SerCeulia viqtorinebisTvis saxasiaTo specifikis gaTvaliswinebiT. magaliTad, #1 amocanaSi I ricxvi luwia da metia 2-ze, amitom is ar aris martivi. mxolod is ar aris martivi? vamowmebT II-s. is iyofa 5-ze, III _ iyofa 3-ze. e. i. martivi SeiZleba iyos mxolod meoTxe. Cven swrafad (Sejibris pirobebSi) ver vamowmebT am pasuxis siswores, magram am informaciiT da pasuxebis Camoyalibebis formis gaTvaliswinebiT virCevT d)-s. #2 amocanaSi namravli 6-iT bolovdeba. b) da g)-dan virCevT b)-s. radgan Tanamamravlebis mTeli nawilebis namravli 156 ver iqneba. arCevanis gakeTebis aseTi `aramaTematikuri~ forma mxolod viqtorinebis an msgavsi konkursebis drosaa dasaSvebi.

119


3 moswavle daadgens, rom ciferblati, masze moniSnuli 12 danayofis Sesabamisad iyofa 12 tol nawilad (seqtorad). maTgan 4 nawili adgens mcire `ares~, 8 nawili _ dids. amrigad mcire nawili aris didis 12 . 7 4 cifrebidan mxolod 9-is 9 SeiZleba iyos erTeulebis cifri. yvela sxva cifris 7 (0-is garda) wiladia. e. i. aTeulebis cifria 9, erTeulebis _ 7. 9

21 iyofa 7-ze; 7-ic iyofa 7-ze, amitom bolo ori cifriT Sedgenili ricxvic unda gaiyos 7-ze. es ricxvia 42. pasuxi: g) 4. 5

6

dRe-Ramis I meoTxedSi 24:4=6 saaTia, misi naxevari _ 3 sT.

gamravlebisas boloSi 0 miiReba mxolod 5-is luw ricxvebze gamravlebiT. 1-dan 25-is CaTvliT ricxvebSi ricxvi 5 Tanamamravlad gvxvdeba 6-jer (5, 10-Si, 15-Si, 20-Si, 25-Si ki _ 2-jer). luwi ricxvi ki Tanamamravlad gacilebiT metjer gvxvdeba. e. i. 1-dan 25-is CaTvliT ricxvebis namravli bolovdeba 6 nuliT (ramdeni 5-ianic Segvxvda Tanamamravlad). 2000 8 moswavle SeiZleba zepirad mixvdes, rom 10 =200 da mricxvelis cifrTa jamia 2. SeiZleba imsjelos ase: saZiebeli wiladis Sekvecis Sedegad miviReT 200 1 , e. i. 7

mricxveli ise Seefardeba mniSvnels, rogorc 200:1. mricxveli aRvniSnoT 200x-iT, mniSvneli _ x-iT. pirobiT, 200x+x=2010, saidanac x=10. mricxveli tolia 200⋅10=2000is, cifrTa jamia 2. SeiZleba zepirad: mricxveli da mniSvneli pirobiTad, 201 nawilia, 1 nawili 10-ia, mricxvelia 200⋅10. 9

wina amocanis analogiuria. Sekvecis Semdeg miviReT 91 . mricxveli aRvniSnoT

9x-iT, mniSvneli _ x-iT. pirobiT, 9x+x=2010, saidanac x=201. mricxvelia 201⋅9=1809, misi cifrTa jamia 18. zepirad: mricxveli da mniSvneli pirobiTad 9+1=10 nawilia, 1 nawili 201-ia, mricxveli _ 201⋅9=1809 10 Tu sami momdevno naturaluri ricxvidan pirvels aRvniSnavT x-iT, miviRebT

x+x+1+x+2=108, saidanac x=35. cifrebis namravlia 15. SeiZleba moswavlem ricxvebi Casvas. magaliTad, b=10, a=7 da Seamowmos tolobebi, 10a–7b=0 aRmoCndeba swori. moswavleebisTvis savsebiT daZlevadia aseTi zogadi midgomac: a= 7b 10 ; 10a=7b, 10a–7b=0. 11

12 ivnisSi 30 dRea. 28 dReSi kviris yoveli dRe, maT Soris paraskevic aris 4-jer (2-jer-kent ricxvebSi). pirobiT,M paraskevi unda iyos darCenili ori dRidan kentSi _ 29-Si. paraskevia 22-Sic, 19-Si _ samSabaTi.

a) yvelaze araxelsayrel SemTxvevaSi amoRebuli 20 burTulidan 12 iqneba wiTeli, 8 _ mwvane. 21-e aucileblad aRmoCndeba lurji. pasuxi: 21. b) yvelaze araxelsayrel SemTxvevaSi amoRebuli sami burTula sxvadasxva ferisaa, meoTxes feri aucileblad am samidan romelimes fers daemTxveva. 13

120


14 sainteresoa, rom monakveTis or nawilad nebismierad gayofis dros am nawilebis Sua wertilebs Soris manZili monakveTis naxevaria. marTlac, suraTze O wertiliT monakveTi gaiyo or nawilad, AO-s naxevari iyos x, OB-s naxevari _ y.

x A

x

y O

y B

miviRebT 2(x+y)=AB. Cvens amocanaSi 12,6 dm aris monakveTis naxevari (x+y), monakveTis sigrZea 25,2 dm. 15 sinjvis meTodiT: aTeulebi

erTeulebi

1 4 2 8 pirobiT, cifrTa jami 10-ia, pasuxi _ 28. x es amocana maRal klasSi SeiZleba amoixsnas gantolebiT: x+30 y+40 = y , saidanac 3 y= 4x 3 , x nebismieri 3-is jeradi naturaluri ricxvia. aseTi wiladebi 4 -is tolia. am etapze SeiZleba moswavlem imsjelos ase: mocemuli wiladis toli wiladi miiReba misi mricxvelis da mniSvnelis erTsa da imave ricxvze gamravlebiT, magaliTad, 2-ze, 2x anu x+30 y+40 = 2y . 30-is mimatebiT 2-ze gamravldeba 30, 40-is mimatebiT _ 40. e. i. wiladi 3 iyo 30 40 = 4 . 16

17 Tu Tavs avaridebT gantolebis Sedgenas, SevamCnevT, rom mniSvneli mricxvelze

4⋅5 = 20 . `7 nawiliT~ metia, 7 nawili 35-ia, 1 nawili _ 5. mocemuli wiladia 11⋅5 55 18 mricxveli da mniSvneli erTad `13 nawilia~. (am wiladis toli wiladebi aris

5x saxis, 5x+8x=13x). 13 nawili 130-ia, 1 nawili _ 10. es wiladia 5⋅10 = 50 . 8⋅10 80 8x 19

wigni Rirs 56,3–40,9=15,4 lari; albomi Rirs 56,3–38,1=18,2 lari; CanTa _ 29,4–18,2=11,2 lari; Cogani _ 40,9–29,4=11,5 lari.

20 SevniSnoT, rom yoveli Sesakrebi 1-is tolia. sul 99 Sesakrebia, jami iqneba 99. 21 29⋅3=87. sul miRebulia 87 werili _ kenti raodenobis, gagzavnili werilebis raodenoba ki luwia. es SeuZlebelia. 3+x 9 9 18 22 pirobiT, 2x = x . cxadia, x = 2x . amrigad, 3+x=18, x=15. 2 20 5 50 20 50 23 9 = 90 , 9 = 90 . 90 da 90 -s Soris veZebT wiladebs, romelTa mniSvneli aris 10, anu 3 36 4 45 5 wiladebs, romlebic ikveceba 9-ze. esenia 27 90 = 10 , 90 = 10 , 90 = 10 .

121


§5. wiladebis gamravleba 64-e gakveTili

Tema: wiladebis gamravleba. mizani: moswavlem SeZlos arauaryofiT racionalur ricxvebze ariTmetikuli moqmedebebis Catareba (maT. VI. 2.) da maTi gamoyeneba. winapirobebi: ricxvis nawilis povna, nawiliT ricxvis povna. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebas SeiZleba 15-20 wT davuTmoT. moswavleebs SedarebiT uWirT e. w. `teqsturi~ amocanebis amoxsna. amitom, sasurvelia 36 - 38 (III Tavis damatebiTi) amocanebis amoxsnebis kargi analizi. saWiroebis SemTxvevaSi maTi klasSi amoxsnis Catareba. 1 37 YTu kombaini 6 dReSi iRebs mosavals, maSin is yoveldRiurad mosavlis 6 nawilis 2 + 3 = 5 (nawilis). aRebas SeZlebs, meore ki _ 14 nawilis aRebas, orive _ 16 + 14 = 12 12 12 aseTi amocanebi dResac popularulia skolebSi, maTematikur skolebSi misaReb gamocdebze (me-7 klasSi Sesasvlelad). am dros, cxadia, igulisxmeba, rom kombainis muSaoba `Tanabaria~ (Tanabari siCqaris analogiurad), drois tol SualedebSi samuSaos toli nawilebi sruldeba. cxadia, amaze moswavleebTan yuradRebas ar vamaxvilebT. 38 amocanis amoxsna moiTxovs mTelidan Sereuli ricxvis gamoklebas, kidev erTx-

el vamowmebT – asruleben Tu ara moswavleebi aRniSnul moqmedebas sworad. axal masalaze gadasvla mimdinareobs saxelmZRvaneloSi warmodgenili amocanis amoxsniT, roca mTelis wiladze gamravlebas vukavSirebT mTelis nawilis povnas. magaliTad, 120-is 23 -is povnisas 120-is mesameds vpoulobT. dasmul kiTxvebze moswavleebi pasuxoben: – 120-is mesamedi aris 40, radgan 120:3=40. – 120-is 23 ki orjer metia 40-ze da 80-is tolia. 120 wT aris 2 sT. – e. i. 2 saaTis 23 aris 43 sT. miviReT, rom 2-is 23 aris 43 . Cven vajamebT: – 2-is 23 -is povnisas vasrulebT gamravlebas 2⋅ 23 , maSasadame, 2⋅ 23 = 43 . gamravlebis wesis Camoyalibebas TviTon moswavleebi SeZleben. klasSi dawvrilebiT ganixileT teqstSi warmodgenili magaliTebi. teqstis mixedviT 2-is 23 -ze gamravlebis ilustraciis Semdeg ricxviTi sxivi gamoiyeneba. moswavle ricxviTi sxivis gamoyenebiTac miiRebs amave Sedegs. 2-is 23 -is povnisas vasrulebT gamravlebas 2⋅ 23 = 43 . maSasadame, 23 ⋅2 namravlis povnasac vukavSirebT ricxvis nawilis povnas, rac gvaZlevs 23 ⋅2= 23 + 23 = 43 . 7 ⋅10= 7 + 7 +...+ 7 = 70 . analogiurad, 50 50 50 50 50 am magaliTebis ganxilvis Semdeg, savaraudod, moswavleebi SeZleben ricxvis wiladze gamravlebis wesis Camoyalibebas:

122


naturaluri ricxvis wiladze gamravlebisas wiladis mricxvels vamravlebT mocemul ricxvze, mniSvnels gadavitanT ucvlelad. amrigad, moswavleebi ricxvis nawilis povnisas SesaZloa ricxvs gaamravleben nawilis gamomsaxvel wiladze; SesaZloa ricxvs gayofen am wiladis mniSvnelze da ganayofs gaamravleben mricxvelze _ Sedegi erTi da igivea. magaliTad, 160-is 34 = 160⋅ 34 an 160:4⋅3. TvalsaCinoebisTvis wiladebis gamravlebisas ricxviT sxivs viyenebT, sadac kargad Cans, rom, magaliTad, 34 ⋅ 12 = 38 anu, 34 -is 12 nawili ( 34 -is naxevari) aris 38 . kidev ramdenime magaliTis ganxilvis Semdeg moswavleebs vTxovT aRweron wiladebis gamravlebis wesi: wiladebis gamravlebisas miiReba wiladi, romlis mricxveli Tanamamravlebis mricxvelebis namravlia, mniSvneli _ Tanamamravlebis mniSvnelebis namravli. wiladebis gamravlebis aRwerili wesis kerZo SemTxvevaa wiladis naturalur ricxv­ ze gamravlebac. marTlac, gakveTilis dasawyisSi Sesrulebuli gamravleba SeiZleba ase gadaiweros: 4 2⋅ 23 = 21 ⋅ 23 = 2⋅2 1⋅3 = 3 . teqstSi gamravlebis erT-erTi Tviseba _ 0-ze gamravlebis Tvisebaa ilustrire­ buli. SeiZleba moswavleebma analogiurad Camoayalibon 1-ze gamravlebis Tvisebac: a⋅1=1⋅a=a da moiyvanon magaliTebi; gamravlebis gadanacvlebadobis Tviseba, romelic teqstSicaa ilustrirebuli. a⋅b=b⋅a; jufTebadobis Tviseba; a(bc)=(ab)c. 4 - 10 testebi erTi SexedviT martivia, magram mniSvnelovania yoveli maTganis

Sesrulebisas davafiqsiroT gamravlebis Sesabamisi Tvisebebi, gavukeToT komentari. magaliTad, 4 nulis Tvisebaa, 6 1-ze gamravlebis, 7 amocanis Sesrulebis Sedegi SeiZleba maTematikuri debulebis saxiT Camoayalibon: Sebrunebuli ricxvebis namravli n 1-is toli. SevaxsenebT moswavleebs, rom m n da m Sebrunebuli ricxvebia (cxadia, mn≠0). 8 -Si viyenebT jufTebadobis Tvisebas da Semdeg 6 da 7 savarjiSoebSi gamoyenebul 1-ze gamravlebis da Sebrunebuli ricxvebis gamravlebis Tvisebebs: 1 12 ⋅( 23 ⋅ 45 )=( 32 ⋅ 23 )⋅ 45 =1⋅ 45 = 45 . 10 zogierTi moswavlisTvis uCveuloa, roca nawili aTwiladis saxiTaa Cawerili.

4 , e. i. 16-is 0,4 nawilis sapovnelad gamravleba unda Sesruldes: SevaxsenebT,M rom 0,4= 10 16⋅0,4. saSinao davalebad vaZlevT 1 - 10 savarjiSoebs.

123


65-e gakveTili

Tema: wiladebis gamravleba. mizani: moswavles SeeZlos ricxvis nawilis povnisa da Sebrunebuli amocanis amoxsnisas wiladebze moqmedebebis swori SerCeva da Sesruleba. winapirobebi: Sereuli ricxvis gardaqmna arawesier wiladad. wiladebis sxvadasxva interpretacia. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebas amjerad SeiZleba didi dro ar dasWirdes – saWiro iyo gamravlebis martiv SemTxvevebSi swori pasuxis miTiTeba. klasSi vagrZelebT muSaobas gakveTilze dasaxuli miznis misaRwevad, raSic gvexmareba 11 - 22 amocanebi, romlebic, ZiriTadad, ricxvis nawilis povnisa da Sebrunebuli amocanis amoxsnas ukavSirdeba. magaliTad, 18 amocanaSi miTiTebulia, rom a+2,05 4 aris 6  58 -is 45 , amitom, a+2,05=6  58  ⋅  45 , saidanac a+2,05=  53 8  ⋅  5 ; a+2,05=5,3 da a=3,25. 1 20 I dRes gaiyida 4,32⋅ 4 =1,08 tona, darCa 4,32–1,08=3,24 t. II dRes gaiyida 3,24⋅0,75=2,43 t, darCa 3,24–2,43=0,81 t, rac gaiyida III dRes. pasuxi: 0,81 t. zogierTma moswavlem SeiZleba ase imsjelos: I dRes gaiyida 14 nawili, darCa 34 nawili, anu (4,32⋅ 34 ) tona. II dRes gaiyida darCenilis 0,75, anu darCa darCenilis 0,25, romelic gaiyida III dRes. pasuxi: 4,32⋅ 34 ⋅0,25=0,81 (t). 21 SeiZleba nedli yavis raodenoba aRniSnon x-iT da Seadginon gantoleba:

x:1 16 =750, saidanac x=875. pasuxi: 875 grami. SeiZleba imsjelon ase: moxaluli yavis odenoba aris nedli yavis odenobaze 1 16 -jer naklebi, Sesabamisad, nedli yavis odenoba moxalulze 1 16 -jer metia: 750⋅1 16 . saSinao davalebisTvis gaTvaliswinebulia

11 - 22

amocanebi.

66-e gakveTili

Tema: wiladebis gamravleba. mizani: moswavleebi gaiwafon wiladebis gamravlebaSi, ricxvis nawilis povnaze amocanebis amoxsnaSi. winapirobebi: ricxvis nawilis povna da Sebrunebuli amocanis amoxsna. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis Semowmebisas SeiZleba davadginoT _ aiTvises Tu ara moswavleebma wiladebis gamravleba da misi gamoyeneba ricxvis nawilis povnisa da Sebrunebuli amocanis amoxsnisas. saSinao davalebis amocanebi am procedurebis ganxorcielebas moiTxovda. magaliTad,

124


12

a) 5 saaTis 0,3 nawili aris 5⋅0,3=1,5 saaTi.

15 a) marcxena nawilSi miiReba 1-ze meti ricxvi, marjvenaSi _ 1-ze naklebi, anu Tanafardobis dasadgenad gamravleba ar aris saWiro.

b) 15⋅ 35 =9, marjvena mxare ki 1-ze naklebia. 9> 34 ⋅ 79 . 18

SeiZleba moswavlem Sereuli ricxvi (Tanamamravli) Caweros arawesieri wi­

ladis saxiT da Seasrulos gamravleba. meore xerxi 17 amocanaSia ilustrirebuli. davalebaTa analizis dros maswavlebelma unda gamoikvlios klasis moswavleTa umravlesoba romel xerxs aniWebs upiratesobas da ratom? 15 1 15 jer gadmoasxes 52 ⋅ 14 = 58 t, darCa 2 12 - 58 = 15 8 t. Semdeg gadmoasxes 8 ⋅ 2 = 16 t, 15 15 darCa 15 8 - 16 = 16 t. SeiZleba ase imsjelon: 3 I gadmosxmis Semdeg avzSi darCa 2 1 2 ⋅ 4 t benzini. II gadmosxmis Semdeg _ darCenilis 1 nawili, anu 2 1 ⋅ 3 ⋅ 1 = 15 t. 2 2 4 2 16 2 22 SeiZleba lelas xelfasi x-iT aRvniSnoT da SevadginoT gantoleba x⋅ =80, 9 2 1 9 saidanac x=360 (vmsjelobT ase: Tu 80 aris 9 nawili, 9 nawili iqneba 40, 9 ki _ 9⋅40=360). 20

Tu aRmovaCenT xarvezebs moswavleTa mier masalis aTvisebis saqmeSi, maSin SeiZleba gavagrZeloT muSaoba wiladebis gamravlebaze da Sesabamisi amocanebis amoxsnaze III Tavis damatebiTi amocanebidan: 39 - 56 . Aam amocanebis nawili skolaSi SeiZleba amovxsnaT, danarCeni amocanebi saSinao davalebad mivceT. amasTanave, dabali mzaobis moswavleebs SeiZleba SevTavazoT amocanebi 39 - 46 , SedarebiT maRali mzaobis moswavleebs ki _ 47 - 49 amocanebi. Cven vakvirdebiT maT muSaobas, saWiroebis SemTxvevaSi, vexmarebiT maT. 47 - 49 amocanebis amoxsnebi, sasurvelia, rom moswavleebma mTeli klasis winaSe warmoadginon. 47 900-is 0,6 aris 900⋅0,6=540 12,6⋅ 2 3

12,6 48

A

B

C

D

CD-s sigrZe SeiZleba ornairad vipovoT: 12,6:3=4,2, 4,2⋅2=8,4; an 6 ⋅ 2 =12  2 ⋅ 2 = 63 ⋅ 2 =  21⋅2 = 42 =8,4. 12,6⋅ 23 =12 10 3 5 3 5 3 5 5 1

49 I dRes gaiyida – 300⋅ 4 =75 (kg)

darCa – 300–75=225

II dRes gaiyida 225⋅ 25 =45⋅2=90 (kg)

125


67-e gakveTili

Tema: wiladebis gamravleba. mizani: moswavleTa logikuri azrovnebis, sivrculi warmodgenebis unarebis ganviTareba. winapirobebi: mTelis nawilis povna da nawiliT mTelis povna. resursebi: qaRaldis furclebi, makrateli, fanqari, kalami, saxazavi, rveuli, saxelmZRvanelo, dafa. dasaxlebuli Tema dakavSirebulia saSinao davalebasTan, romlis Semowmebis Semdeg SeiZleba CavataroT jgufuri mecadineobebi `vip~-is da `moisazres~ rubrikiT mocemuli amocanebis gamoyenebiT. aseTi jgufuri muSaobis Catarebis sqemebi Cven araerTxel detalurad aRvwereT. axla ki gTavazobT miTiTebebs am amocanebis amosaxsnelad. moisazre 20–n unda iyofodes 6-ze: 20–n=6, saidanac n=14; 20–n=12, saidanac n=8; 20–n=18, saidanac n=2. 1

n n+1 10 da 25 rom naturaluri iyos erTdroulad, n da n+1 erTdroulad unda iyofodes 5-ze. es SeuZlebelia. 2

naTiam mosva da rZiT Seavso finjnis 16  +  13  +  12 nawili. 16  +  13  +  12 =1, e. i. naTias mier nawil-nawil Casxmuli rZiT zustad ivseba erTi finjani. bolos naTiam gamocala mTeli finjani. sabolood man dalia 2 finjani _ 1 finjani yava da 1 finjani rZe. 3

    

    

4

mocemuli figura warmovadginoT aseTi saxiT. igi 12 kvadratisagan Sedgeba. 4 tol nawilad dayofis SemTxvevaSi TiTo nawils unda qmnides 3 kvadrati. miviRebT. 4 dReSi _ 20 avtomobili 2 dReSi _ 10 avtomobili 10 dReSi _ 50 avtomobili 5

i

am rubrikiT warmodgenili amocanebi ZiriTadad moswavleTa sivrculi

warmodge­nebis gaRrmavebis, mosazrebulobis, kombinatoruli azrovnebis ganviTarebis xelSe­mwyo­bia. 1) amocanaSi kubebis daTvla xelsayrelia Sreebis mixedviT: 15+13+5+4+1=38. M 2) gamoWeriT PQRS marTkuTxedis formis furceli da Q R aq warmodgenili suraTis mixedviT masze moniSneT ABCD a marTkuTxedi da MN monakveTi. Semdeg furceli gaWeriT AB, CD B C da MN monakveTebis gaswvriv. ABCD marTkuTxedi `wamoayeneT~ _ N misi fuZe gaxdes BC monakveTi, xolo mis marjvniv darCenili a figura SeabruneT NC monakveTze 1800-iT. miiRebT sasurvel P S figuras. A D

126


2 6 5

4 1 3

3) moswavleebi Seecdebian modelis damzadebas, albaT yuradRebas miaqceven cifrebis orientaciasac. pasuxi SeiZleba ase warmoadginon.

4) variantebis warmodgenisas daTvaleT mxolod arsebiTad gansxva­ve­bulebi (amaze saubaria amocanis miTiTebaSic). unda miiRoT aseTi tipis SemTxvevebi:

naSromebis Sefasebisas gaiTvaliswineT rogorc Sesrulebis dro, aseve Sesrulebis xarisxi (SemTxvevaTa raodenoba), prezentaciis damajerebloba. saSinao davalebad moswavleebs SeiZleba mivceT amocanebi rubrikidan `Seamowme Seni codna~.

§6. wiladebis gayofa 68-e gakveTili

Tema: wiladebis gayofa. mizani: moswavleebs SeeZloT wiladebis gayofis Sesruleba (maT. VI. 2.) da am codnis praqtikuli gamoyeneba. winapirobebi: wiladebis gamravleba, gayofis dakavSireba jerad SekrebasTan. resursebi: qaRaldis furclebi, makrateli, fanqari, rveuli, saxelmZRvanelo. saSinao davalebis amocanebidan yuradRebas vamaxvilebT 55 , 56 amocanebze. im amocanebis amoxsna, romlebic ucnobis aRmniSvnel asoebs Seicavs, gansakuTrebiT uWirT moswavleebs. 2 x x⋅2 x 55 Tu  60 -is 5 aris 2, maSin 60⋅5 =2 wiladi SeiZleba SevkvecoT 2-ze, 150 =2, gasayofi x=2⋅150=300. m

5

m⋅5

m

56  20 -is 9 ase Caiwereba: 20⋅9 , Sekvecis Semdeg gvaqvs:  36 >7, naturaluri m ricxvi

unda akmayofilebdes pirobas: m:36>7, m>36⋅7, m>252. am ricxvebidan umciresi aris 253.

am process SeiZleba 15-20 wT movandomoT. wiladebis gayofis Sesrulebamde SeiZleba naturaluri ricxvebis gayofis magaliTze gavixsenoT gayofis kavSiri toli jgufebis gamoyofasTan, jerad gamoklebasTan.

127


SeiZleba aseTi martivi amocanac ki ganvixiloT: gvaqvs Sokoladis fila, romelic eqvs tol nawiladaa dayofili da igi gvinda Tanabrad gavunawiloT 3 bavSvs, TiToeuls ramdeni Sexvdeba? eqvsi nawili 3 cali ornawiliani danayofia, TiToeuls Sexvdeba 2 nawili. rogorc am amocanis, ise teqstSi ganxiluli amocanis ganxilvis dros SeiZleba visargebloT qaRaldis furcliT, romelic dayofilia 6 tol marTkuTxedad da mas makratliT vanawilebT 3 tol nawilad. axla 94  :  34 gayofis geometriuli (TvalsaCino) warmodgena SevTavazoT moswavleebs. viRebT marTkuTxedis formis or naWers da TiToeuls vanawilebT 4 tol marTkuTxedad. aviRoT am meoTxedis toli kidev erTi marTkuTxedi, sul gveqneba 9 mcire marTkuTxedi. es erToblioba SeiZleba warmovadginoT, rogorc 3 cali   34 marTkuTxedisgan Semdgari figura. 94  :  34 =3, anu 2  14  :  34 =3. am moqmedebis TvalsaCinod Cvenebidan gadavdivarT gayofis wesis Camoyalibebisas Sebrunebuli wiladis cnebis gamoyenebaze. ufro advilad gadaiWreba mTelis wiladze gayofis ilustraciis warmodgena: 4 :  13 . ramdeni mesamedia am 4 erTeulSi? TiToeulSi 3 mesamedia, sul – 12 aseTi mesamedia. 4=12⋅  13 . 4:  13 =12. kvlav SevaxsenebT maswavleblebs, rom dadebiTi racionaluri ricxvebs gamravlebisa da gayofis sistematuri Seswavla daiwyeba me-7 klasidan. amjerad SeiZleba SemovifargloT mxolod im magaliTebiT, romlebic saxelmZRvaneloSia warmodgenili. 1 - 10 `testebSi~ pasuxebis SerCeva SeiZleba moswavleebma damoukidebladac SeZlon. saSinao davalebad moswavleebs vaZlevT 1 - 10 amocanebs.

69-e gakveTili

Tema: wiladebis gayofa. mizani: moswavle gaiwafos ricxvis nawilis povnis gamoyenebaSi nawilebze sxvadasxva tipis amocanebis amoxsnisas (maT. VI. 2.). winapirobebi: ricxvis nawilis povna. resursebi: rveuli, saxelmZRvanelo, dafa. 7

- 16 amocanebs da vatarebT jgufur muSaobas. 128

da

8

amocanebze. 1 7 8 da savarjiSoebSi SeiZleba Seadginon gantoleba: x-is 4 aris 12, anu 1 x⋅ 4 =12, x=48 da x-is 23 aris 12, anu x⋅ 23 =12, x=18. moswavleebma swori pasuxi savaraudo pasuxebis sinjviTac SeiZleba aRmoaCinon. magaliTad, 7 savarjiSoSi 3-is 14 ≠12, 48-is 14 = 12. pasuxi: 48. 2 ≠12, 72-is 2 ≠12, 60-is 2 ≠12, 18-is 2 =12. swori pasuxia d) 18. Tumca, 8 -Si 8-is 3 3 3 3 aseT SemTxvevebSi upiratesoba pasuxebis Semowmebas ar mieniWeba. saSinao davalebis Semowmebas SeiZleba 10-15 wT dasWirdes. amis Semdeg vxsniT 11 saSinao davalebis amocanebidan yuradRebas vamaxvilebT


5

12 SeiZleba moswavleebma amocanis pirobiT SearCion wiladebi, magaliTad, 12 da

5 da gayofiT daadginon, ramdenjeraa meti II wiladi I-ze. 5 : 5 =4. pasuxi: II wiladi 3 3 12 I-ze metia 4-jer. SeiZleba gayofis gareSec daaskvnan: mniSvnelis 4-jer gazrda wilads amcirebs 4-jer. 2 16 mTeli gzis TiToeul meoTxeds TviTmfrinavi 3 sT-Si faravs. mTel gzas dafaravs 4⋅ 23 = 83 =2 23 sT=2 sT 40 wT-Si. saklaso samuSaoebSi CarTulia jgufuri muSaoba, romlis mizania moswavle gaiwafos ricxvis nawilis povnis wesis gamoyenebaSi nawilebze sxva tipis amocanebis amoxsnisas. I davalebaSi moswavleTa jgufebi pouloben ricxvis nawilis mixedviT ricxvs nebismieri xerxiT. SeiZleba, magaliTad, ucnobi ricxvi aRniSnon x-iT da Seadginon gantoleba a) SemTxvevisTvis: x ⋅  17 =40, saidanac x=280; b) SemTxvevisTvis: x ⋅  13 =20, saidanac 9 aris x=60; g) SemTxvevisTvis: x ⋅  57 =40, x=56 da a. S. SeiZleba imsjelon ase: d) Tu ricxvis 16 1 iqneba 108:9=12, saZiebeli ricxvi _ 16⋅12=192; e) Tu ricxvis 7 aris 9 , 1 iqneba 108, 16 15 25 15 9 :7= 9 , saZiebeli ricxvi iqneba 9 ⋅15= 27 . yvela SemTxvevaSi ucnobi ricxvis 25 175 175 35 sapovnelad ricxvis mocemul nawils yofen nawilebis ricxvze. moswavleTa daskvna, savaraudod, aseTi saxiT Caiwereba: Tu a aris x ricxvis p nawili, maSin x=a:p. II davalebaSi moswavleebs vTxovT ipovon erTi ricxvi meoris ra nawilia. am SemTxvevaSi ucnobi aris nawili. Tu mas x-iT aRvniSnavT: a) amocanas (12-is ra nawilia 4?) ase warmovadgenT: 12-is x nawili aris 4, anu 12⋅x=4, 4. saidanac x= 12 3. b) 16-is ra nawilia 3? 16-is x nawili aris 3, anu 16⋅x=3, saidanac x= 16 1 , anu 4 ⋅x= 1 , x= 1 : 4 = 1 . 3) 45 -is x nawili aris 15 5 15 15 5 12 savaraudoa, rom moswavleebi daaskvnian: nebismieri a da b≠0 ricxvebisTvis a ricxvi aris b ricxvis a:b nawili. SeiZleba yuradReba gavamaxviloT im faqtzec, rom a ricxvis 1-ze naklebi nawili a-ze naklebia, a ricxvis 1-ze meti nawili a-ze metia, a ricxvi aris Tavisi Tavis 1 nawili. (radgan swavlebis am etapze uaryofiT ricxvebs ar ganvixilavT, amitom Camoyalibebuli debulebebi sworia). III etapze moswavleebi maT mier nawilebze Camoyalibebul debulebebs gamoiyeneben amocanebis amoxsnisas: 5 nawili, darCa 7 . pirobiT, x-is 7 aris 7c, anu 1) vTqvaT, iyo x c. gaiyida 12 12 12 7 x⋅ 12 =7, saidanac x=12 c. 2) gaiares 17 nawili, darCaT 67 nawili _ 57 -iT meti, vidre gaiares. pirobiT mTeli 5 gzis _ x-is 5 7 nawili aris 1,5, anu x⋅ 7 =1,5; x=2,1 km. 7 , II klasma 3) nergebis mTeli raodenoba aRvniSnoT x-iT. I klasma dargo x⋅ 20 5 13 3 darCenilis, anu x⋅ 13 20 -is 8 nawili. darCa x⋅ 20 -is 8 nawili, rac dargo III klasma da, 3 39 pirobiT, x⋅ 13 20 ⋅ 8 =195, saidanac x⋅ 160 =195, x=800. saSinao davalebad moswavleebs vaZlevT

11 - 23 amocanebs.

129


70-e gakveTili (Semajamebeli)

Tema: wiladebis gayofa. mizani: moswavles SeeZlos wiladebis gayofisa da gamravlebis gamoyeneba amocanebis amoxsnisas, moqmedebebis swori SerCeva da Sesruleba. winapirobebi: wiladebis gamravleba da gayofa. resursebi: rveuli, saxelmZRvanelo, dafa. moswavleebs uWirT e. w. `teqstur~ amocanebSi moqmedebebis SerCeva. amitom saSinao davalebis amocanebis garCevaze dro ar unda daviSuroT, gakveTilis ZiriTadi nawili SeiZleba saSinao davalebis Semowmebas davuTmoT, danarCeni dro – `moisazres~ rubrikiT mocemuli amocanebis amoxsnas. gTavazobT rekomendaciebs am amocanebis amoxsnebsa da maT axsnaze. 11

amocana klasSi amoxsnili 12 amocanis msgavsia: II wiladis mniSvneli I-ze 3-jer naklebia, e. i. II wiladi I-ze 3-jer metia. cxrilis mixedviT Cans, rom Sebrunebuli ricxvebis jami 2-ia, roca es ricxvebi aris 1 da 1. yvela sxva warmodgenil SemTxvevaSi _ 2-ze metia. marTlac, momavalSi daamtkiceben, rom nebismieri dadebiTi ricxvisa da misi Sebrunebulis jami 2 (a–b)2 ≥0, saidanac vRebulobT, rom ar aris naklebi 2-ze: ab  +  ba  – 2 =  (a–b) . radgan ab>0, ab ab a  +  b  – 2 ≥ 0, a  +  b  ≥ 2, tolobas adgili aqvs, roca a=b, anu roca a  =  b  = 1. am Sedegis b a b a b a misaRebad SeiZleboda ariTmetikul da geometriul saSualoebs Soris arsebuli Tanafardobac gamogveyenebina. 13

14

cxrilis mixedviT SeiZleba vivaraudoT, rom wesieri wiladi ufro axlosaa erTTan, vidre misi Sebrunebuli. momavalSi moswavleebi SeZleben am faqtis dasabuTebas: vTqvaT, a<1, maSin 1a >1, ( 1a  – 1) – (1–a)=  1a  + a–2, saidanac 1a  + a ≥ 2 (amocana 13 ) utolobis gaTvaliswinebiT miviRebT 1a  – 1 ≥ 1 – a. am asakisTvis Sesaferisi msjeloba SeiZleba iyos: vTqvaT m n aris wesieri wiladi, n–m n n–m m<n. maSin m n -s aklia 1-mde n , xolo misi Sebrunebuli _ m daSorebulia 1-sgan m n–m -iT. m<n utolobis gaTvaliswinebiT n–m n < m , e. i. wesieri wiladi ufro axlosaa erTTan, vidre misi Sebrunebuli. 1 1 16 gzis sigrZe aRvniSnoT x-iT. I dRis Semdeg darCa x ⋅  . II dRes gaiara x ⋅  -is 2 2 3 , darCa _ x ⋅  1 -is 2 . pirobiT, x ⋅  1  ⋅  2 =24, saidanac x=120. 5 2 5 2 5 SeakeTa gzis 16 , darCa 56 nawili, rac 46 -iT metia dagebulze. gzis sigrZe aRvniSnoT x-iT. pirobiT, x-is 46 =4,8 km, saidanac x=7,2 km. 17

mTeli Tanxa iyos x. pirobiT, x-is 23 =500, saidanac x=750. SeiZleba moswavlem asec imsjelos: Sesasrulebeli darCa Sesrulebulis naxevari, e. i. gadasaxdelia gadaxdilis naxevari _ 250 lari, sul 500+250=750 lari. 18

19

130

furclebis raodenobis 3 8 aris 36.


1 =65 moswavle, maTgan 65⋅ 2 =26 gogonaa. me-6 klaseli me-6 klasSi swavlobs 910⋅ 14 5 26 1 gogonebis odenoba aris skolis moswavleTa 910 = 35 nawili. 1 -is 2 aris me-6 klaseli gogonebis odenoba, anu me-6 SeiZleba imsjelon ase: 910⋅ 14 5 1 2 1 =910-is 1 nawili, 26. klaseli gogonebis odenobaa 910⋅ 14 ⋅ 5 =910⋅ 35 35 3 2 ; x⋅ 3 ⋅ 2 =x⋅ 1 _ wris 1 23 wris farTobi aRvniSnoT x-iT. lurjia wris x⋅ -is 8 3 8 3 4 4 3 1 1 nawili, wiTelia x⋅ 8 -is 3 , anu wris 8 nawili. 22

am amocanebis amoxsnis Semdeg moswavleebma SeiZleba daaskvnan, rom ricxvis nawilis nawili mocemuli nawilebis gamomsaxveli ricxvebis namravlis tolia. moisazre 1

2 CaZinebis momentSi mgzavrs darCenili hqonda gzis 2 . radgan 2-jer meti gaiara

CaZinebulma, vidre darCa gasavleli, CaZinebuls gauvlia 2 nawili, darCa 1 nawili, e. i. CaZinebulma gaiara II naxevris 23 nawili: 12 ⋅ 23 = 13 (mTeli gzis mesamedi). SeiZleba sqema gamoiyenon: CaZinebuli

darCa

gzis II naxevari

gzis I naxevari CaZinebuls gauvlia 6 nawilidan 2 _ mTeli gzis 26  =  13 nawili. 1

2

3 80:7 2 =10 3 .

gvWirdeba 11 yuTi. 10 aivseba, me-11-Si unda iyos 7 12 -is 23 nawili, anu 5 kg kartofili. SeiZleboda kartofilis es raodenoba 80–7 12 ⋅10 ricxviTi gamosaxulebiTac gamogveTvala. 7 8

7 8

7 8

4 a) mricxvelebi udidesia, mniSvnelebi umciresi: 1 + 3 an 3 + 1 . udidesia 3 + 1 .

b) unda davweroT ori wesieri wiladi, romlebic 1-Tan yvelaze axlosaa. CamoTvlili cifrebiT Sedgenili wesieri wiladebidan 1-Tan yvelaze axlos aris 1-dan 18 -iT daSorebuli wiladi _ 78 . darCenili cifrebiT Sedgenili udidesi wesieri wiladia 34 . g) SeiZleba CamovweroT gaerTmniSvnelianebuli yvela wesieri wiladi: 1 = 28 , 1 = 21 , 1 = 14 , 1 = 12 , 3 = 63 , 3 = 42 , 3 = 36 , 4 = 56 , 4 = 48 , 6 = 72 . 3 84 4 84 6 84 7 84 4 84 6 84 7 84 6 84 7 84 7 84 amaTgan ori wiladis mricxvelebis jami unda iyos rac SeiZleba axlos 84-Tan. umjobesia mricxvelebi CamovweroT zrdiT da yovel maTgans movuZebnoT udidesi SesaZlo Sesakrebi. 76 78

12

14

21

28

36

42

48

56

63

72

77 77 75

131


42 3 1 udidesia 36 84 -isa da 84 -is jami, pasuxi: 7 da 2 . d) mricxvelebi umciresia, mniSvnelebi _ udidesi. esenia 18 da 37 an 38 da 17 . 3 + 1 jami umciresia. 8 7 moswavleebs gamovucxadebT, rom Semdeg mecadineobaze Sedgeba Semajamebeli wera: wiladis Cawera aTwiladis saxiT, wiladebis miyvana saerTo mniSvnelamde, wiladebis Sedareba, wiladebis gamravleba da gayofa. urCieT moswavleebs, gaimeoron erTxel ukve ganxiluli am Tavis damatebiTi amocanebi 1 - 43 .

71-e gakveTili

Semajamebeli wera #4 Tema: wiladis Cawera aTwiladis saxiT, wiladebis saerTo mniSvnelamde miyvana, wiladebis gamravleba da gayofa. mizani: moswavleebis mier masalis aTvisebis Semowmeba. SearCieT swori pasuxi 1. CamoTvlili wiladebidan mxolod erTi ar Caiwereba sasruli aTwiladis saxiT. 7 , 5. CawereT es wiladi perioduli aTwiladis saxiT: 12 , 38 , 80 6 a) 0,(5) b) 0,3(75) g) 0,0(8) d) 0,8(3). 8 -is saerTo mniSvneli SeiZleba iyos 2. 38 -is da 18 a) 180 b) 72 g) 90 d) 128. 3. TuU wiladebs: 37 , 38 , 25 , 12 zrdiT davalagebT, III adgilze iqneba a) 37 b) 38 g) 25 d) 12 . 4. 3 25 â&#x20AC;&#x201C;1 45 = a) 1 35 b) 2 25

g) 2 15

d) 1 25 .

5. Tu marTkuTxedis sigane 2 17 dm-ia, sigrZe siganeze 2,8-jer metia, maSin am marTkuTxedis sigrZe aris 8 dm a) 4 70 b) 6 dm g) 5 17 dm d) 4,56 dm. 6. ramdeni meTxuTmetedia 15 -Si? a) 5 b) 4 g) 3

d) 75.

amoxseniT amocanebi. 7. natom erT dRes waikiTxa wignis 25 nawili, II dRes _ wignis naxevari. amis Semdeg mas wasakiTxi darCa 16 gverdi. ramdengverdian wigns kiTxulobs nato?

132


8. miwis nakveTis farTobi 720 m2-ia. mis 23 nawilze baRia, am nawilis 38 nawilis tolia bostnis farTobi. ramdeni m2 uWiravs bostans? pasuxebi da miTiTebebi 1

2

3

4

5

6

d

b

a

a

b

g

1. gaixsenon, rom sasrul aTwiladad Caiwereba is ukveci wiladi, romlis mniSvnelis martiv mamravlebad daSlaSi ar Sedis 2-isa da 5-isgan gansxvavebuli ricxvi. amrigad, CamoTvlili wiladebidan sasrul aTwiladad mxolod 56 ar Caiwereba, 56 =0,8(3). 3. CamoTvlili wiladebidan 12 -s arc erTi ar aRemateba. 38 < 37 . SevadaroT 25 da 37 . 2 = 14 , 3 = 15 , 2 < 3 . 2 da 3 -is Sedareba aRaraa saWiro, III adgilzea 3 . 5 35 7 35 5 7 5 8 7 SeiZleba es ricxvebi oTxive wiladis gaerTmniSvnelianebis Semdeg davalagoT zrdiT. 3 . 6. 15 = 15 9 nawili. darCenili 1 nawili 16 7. natom or dReSi waikiTxa wignis 25 + 12 = 10 10 gverdia, e. i. wignSi 160 gverdia. 8. bostans ukavia nakveTis 23 ⋅ 38 = 14 nawili, anu 180 m2. SeiZleba moswavlem gamoTvalos jer nargavebis farTobi (720⋅ 12 =480 m2), Semdeg _ misi 38 nawili (480⋅ 38 =180 m2). Sefasebis sqema. testuri tipis amocanebidan TiToeulis swori amoxsna Sefasdes 1 quliT. Sefasebis rubrikebis Caweris sxvadasxva formidan Tqven SegiZliaT gamoiyenoT TqvenTvis ufro misaRebi forma. Sefasebis rubrika #7 davalebisTvis 0 qula

ver gaigo amocanis arsi, araferi dawera, an Cawera mxolod amocanis piroba.

0,5 qula

wignis wakiTxuli nawilis dasadgenad SearCia moqmedeba.

1 qula

1,5 qula

2 qula

9 ). ipova wignis wakiTxuli nawili ( 10 wignis wakiTxuli nawilTan erTad ipova wignis wasakiTxi (darCenili) nawilic, amasTanave, yvela moqmedeba warmoadgina naSromSi; 1 an raime formiT daafiqsira, rom wignis 10 nawili 16 gverdia. srulyofilad amoxsna amocana.

133


#8 davalebisTvis 0 qula 0,5 qula 1 qula 1,5 qula 2 qula

ver gaigo amocanis arsi, araferi dawera, an Cawera mxolod amocanis pirobebi. SearCia moqmedeba baRis farTobis, an bostnis nawilis gamosaTvlelad (720⋅ 23 an 23 ⋅ 38 ) ipova baRis farTobi ipova baRis farTobi da SearCia moqmedeba bostnis farTobis sapovnelad, an gamoTvala, rom bostans ukavia mTeli nakveTis 38 nawili. srulyofilad amoxsna amocana.

SeiZleba msgavsi rubrikebi davurigoT moswavleebs naSromis analizis procesSi.

§7. amocanebis amoxsna 72-e da 73-e gakveTilebi

Tema: amocanebis amoxsnis sxvadasxva xerxi. mizani: moswavle iyenebdes amocanebis amoxsnis sxvadasxva xerxs – gantolebis, an utolobis Sedgena da amoxsna, amowurvisa da gamoricxvis xerxebi, kanonzomierebaTa aRmoCenisa da ganvrcobis xerxi, e. w. `ukusvlis~ meTodi. (maT. VI. 4., maT. VI. 6.) moswavles gauRrmavdes codnis praqtikaSi gamoyenebis unari. winapirobebi: arauaryofiT ricxvebze ariTmetikuli moqmedebebi, ariTmetikuli moqmedebebis Semcvel tolobebSi ucnobi komponentis povna. resursebi: rveuli, saxelmZRvanelo, saaTis modeli, dafa. amocanebis amoxsnis sxvadasxva xerxis gamoyeneba aviTarebs moswavleTa kritikuli azrovnebisa da msjeloba-dasabuTebis unars, codnis praqtikaSi gamoyenebis uaRresad mniSvnelovan unars. paragrafSi warmodgenili masala SeiZleba gavanawiloT or gakveTilze. moswavleebi gaecnobian amocanebis amoxsnis xerxebs: ukusvliT amoxsna, gantolebis Sedgena, kanonzomierebebis aRmoCena da maTi gamoyeneba. gTavazobT rekomendaciebs teqstSi warmodgenili saklasod da saSinaod gankuTvnili magaliTebis ganxilvisa da amocanebis amosaxsnelad. teqstis I amocanis amoxsnisas saaTis modelic gamoviyeneT. amocanis amoxsnas bolo pirobidan viwyebT: II suraTze filarmoniaSi misvlis droa dafiqsirebuli _ 640; III suraTze _ mariamis saxlidan gamosvlis momenti _ 6 sT 40 wT _ 30 wT=6 sT 10 wT; IV suraTze _ sadilis dawyebis dro _ 6 sT 10 wT _ 45 wT=5 sT 25 wT. 5 sT, amave amocanis amosaxsnelad gantolebacaa Sedgenili, romlis amonaxsnia x=5 12 anu 5 sT 25 wT. teqstis II amocana ukusvliT, kiTxva-pasuxis reJimiTac SeiZleba amovxsnaT: • ra gaamravla nucam 8-ze, rom mieRo 128? 128:8=16. • ras gamoaklo 4, rom mieRo 16? 16+4=20. pasuxi 20.

134


SeiZleba sqemac SevadginoT. – 4

⋅ 8

128

cxadia, Cafiqrebuli ricxvis mosaZebnad sqemas ukusvliT unda gavyveT. + 4

: 8

128

ukusvlisas gamravleba Seicvala gayofiT, gamokleba _ mimatebiT. moswavleebs vTavazobT, es amocana gantolebis SedgeniTac amoxsnan. Tu Cafiqrebul ricxvs x-iT aRniSnaven, pirobiT (x–4)⋅8=128, saidanac x=20. teqstis 3 magaliTSi 4-is xarisxebis cxrilze dakvirvebiT moswavleebi daaskvnian, rom 4-is kenti xarisxi 4-iT bolovdeba. luwi xarisxi _ 6-iT. 415 dabolovdeba 4-iT. klasSi savarjiSo. 6

5-is nebismieri xarisxi bolovdeba 5-iT.

7

517 bolovdeba 5-iT, 25=32. maTi jamis erTeulebis cifria 7.

9 dafiqsirebuli ricxvi aRvniSnoT x-iT da SevadginoT gantoleba: (x+6)⋅9=162, saidanac x=12. SeiZleba ukusvliT amovxsnaT.

pirobiT ukusvliT

+ 6

⋅ 9

: 9

– 6

162

162

10

ricxvi

2

22

23

24

25

26

27

28

29

210

...

bolo cifri

2

4

8

6

2

4

8

6

2

4

...

savaraudoa, rom moswavleebi SeamCneven: 2-is yoveli xarisxi, romlis maCvenebeli 4-is jeradia, 6-iT bolovdeba. Tu maCveneblis 4-ze gayofisas naSTi 3-ia, ricxvi bolovdeba 8-iT, naSTi 2-ia _ bolovdeba 4-iT, naSTi 1-ia _ 2-iT. pasuxi: 2n-is bolo cifria 2, Tu n-is 4-ze gayofisas miiReba naSTi 1; 296 bolovdeba 6-iT. 11 vxsniT ukusvliT: Tu erekles marjvena jibeSi 1 lariT meti aRmoaCndeba, vidre marcxenaSi, maSin mis marcxena jibeSi iqneba 12, marjvenaSi _13. marcxenaSi 12 imis Semdeg gaxdeba, rac marjvnidan amoiReben imdens, ramdenic aris marcxenaSi, e. i. Tavdapirvelad marcxenaSi ido 6 da marjvnidan daamates kidev 6. Tavdapirvelad marcxenaSi yofila 6 lari, marjvenaSi _ 19 lari. 12 maRaziaSi Sesvlisas ilias Tanxis naxevari aris 1 lari da ukanaskneli 4 lari,

anu ilias maRaziaSi Sesvlisas hqonda (4+1)⋅2=10 lari, es fuli bazridan gamosuli ilias darCenili Tanxaa. bazarSi Sesvlisas ilias Tanxis naxevari aris 1 lari da kidev 10 lari. e. i. Tavdapirvelad ilias hqonda (10+1)⋅2=22 lari.

135


sqema:

maRazia

bazari

am amocanis amosaxsnelad moswavleebi Semzadebulebi arian amocanebiT: • iliam daxarja naxevari Tanxa da darCa 7 lari. ramdeni hqonda Tavdapirvelad? 14. • iliam daxarja mTeli Tavisi Tanxis naxevari da kidev 1 lari. amis Semdeg mas darCa 7 lari. ramdeni hqonda Tavdapirvelad? (7+1)2=16 lari. 2

gamoiyeneT sqema:

hqonda 40⋅5=200 lari. an sqemis gareSe: kaxis darCa Tavisi fulis 25 ; 25 nawili aris 80 lari. 1 nawili _ 40 lari. 5 mTeli Tanxa _ 40⋅5=200 lari. 4

TiToeuls miemata 10 weli, anu maTi asakebis jami gaizarda 20-iT. pasuxi 28.

6

gamoiyeneT klasSi # 10 amocanisTvis agebuli cxrili.

9

512 bolovdeba 5-iT, 113 _ 1-iT. sxvaoba dabolovdeba 4-iT.

klasebidan wavida 8 moswavle. darCenili 38 moswavle klasebSi Tanabradaa ganawilebuli, TiToO klasSi _ 19 moswavle. moswavleTa wasvlamde erT klasSi iyo 19+5=24 moswavle, meoreSi 19+3=22 moswavle. 10

11

fqvilis 78 aris 28 (kg), 18 _ 4 (kg). Tavdapirvelad iyo 8⋅4=32 (kg).

gadawyobis Semdeg I Taroze 4x wignia, meoreze _ x. pirobiT, 5x=55, x=11. SeiZleba sqema gamoiyenon. gadawyobis Semdeg: 12

I Taro II Taro

55 wigni

pasuxi: a) gadawyobis Semdeg I Taroze iyo 44 wigni, II Taroze _ 11. b) II Tarodan wignebis naxevari gadauwyviaT, e. i. gadawyobamde II-ze wigni, I-ze _ 33. 13 sqemiT:

suraTidan Cans, rom ingas Tanxis naxevari 11+1=12 laria, hqonda 24 lari.

3-jer dasaxelebuli 10 l Seadgens mTeli Tanxis mesameds. amrigad, 30 lari mesamedia, mTeli Tanxaa 90 lari. 14

136

iyo 11⋅2=22


15

SevadginoT bolo cifrebis (erTeulebis cifrebis) cxrili 3-is xaris­

xebisTvis: ricxvi

31

32

33

34

35

36

37

38

39

...

bolo cifri

3

9

7

1

3

9

7

1

3

...

roca 3-is xarisxis maCvenebeli 4-ze gayofisas naSTSi 1-s iZleva, 3-is xarisxi bolovdeba 3-iT _ 321 bolovdeba 3-iT. 83-is 4-ze gayofisas naSTi 3-ia, 383 bolovdeba 7-iT. 16 a) cxrilis yovel striqonSi toli wiladebia. pirvel da meore striqonebSi mesame wiladisgan II wiladi miiReba 3-ze SekveciT, I wiladi miiReba 6-ze SekveciT. 18 -gan 3-ze SekveciT miiReba 6 , 6-ze SekveciT _ 3 . 24 8 4 3 +x=1, b) yovel striqonSi III adgilze I da II ricxvebis jamia. III striqonisTvis 14 11 . saidanac x= 14 35 laris Semadgenel fulis erTeulebs Soris unda iyos 10-larianebic. sxvagvarad 35 lari fulis aseTi erTeulebiT ver Segrovdeba. 17

sinjvis meTodi SeiZleba aRvweroT cxriliT: 10

5

3

2

1

2

1

SeiZleba

2

3

ar SeiZleba

1

5

SeiZleba

1

4

2

1

ar SeiZleba

pasuxi: molarem SeiZleba gamoiyenos 3 cali 10-lariani, 2 cali 2-lariani da erTi 1-lariani; an 1 cali 10-lariani da 5 cali 5-lariani.

§8. ricxviTi utolobis Tvisebebi 74-e gakveTili

Tema: ricxviTi utoloba. mizani: moswavles SeeZlos asoiTi gamosaxulebis Sedgena da gamartiveba, ariTmetikuli moqmedebebis Sedegis Sefaseba, utolobebis Tvisebebis gamoyeneba (maT. VI. 2, maT. VI. 4. maT. VI. 6.). winapirobebi: ariTmetikuli moqmedebebi arauaryofiT racionalur ricxvebze. resursebi: geometriuli figurebis – kubebis modelebi, rveuli, saxelmZRvanelo. ricxviTi utolobebis Tvisebebis gacnoba da Camoyalibeba, sasurvelia, TvalsaCino masalis gamoyenebiT mimdinareobdes. am gamoyenebis magaliTebi saxelmZRvaneloSia mocemuli.

137


iq mxolod erTi Tvisebis Cvenebisas gamoiyeneba kubis modelebi. Tumca, es SesaZlebelia sxva Tvisebebis ganxilvis drosac. magaliTad, `maRal~ kubs da dabal `kubs~ Tavze davadgamT tol kubebs (meore Tviseba). klasSi SeiZleba amoixsnas 1 - 9 amocanebi. saSinao davalebad vaZlevT 1 - 9 amocanebs. teqsti praqtikuli amocanebiT iwyeba. marTkuTxedis formis nakveTis Robis sigrZis gamoTvlis Semdeg Robis sigrZes vafasebT nakveTis gverdebis sigrZeebis naklebobiT damrgvalebiT. nakveTis Robis sigrZis SefasebisTvis vadgenT ricxviT utolobas ric­ x­ viT utolobis Tvisebebis gamoyenebiT. es Tvisebebi teqstSive suraTebiTaa demonstrirebuli. am etapze Tavs avaridebT aramkacri utolobebis gamoyenebas. Tumca, amocanebis amoxsnisas, saWiroebis SemTxvevaSi, a≥b utolobis nacvlad viyenebT `a>b an a=b~ Canawers.

75-e gakveTili

Tema: asoiTi gamosaxulebis Sedgena da gamartiveba. mizani: moswavles SeeZlos asoiTi gamosaxulebis Sedgena da gamartiveba, utolobebisa da moqmedebebis Tvisebebis gamoyeneba ricxviTi gamosaxulebis Sefasebisas (maT. VI. 2., maT. VI. 4., maT. VI. 6.). winapirobebi: ariTmetikuli moqmedebebi arauaryofiT racionalur ricxvebze. resursebi: rveuli, saxelmZRvanelo, dafa. 2 saSinao davalebis amocanis amoxsnisas, sasurvelia, rom moswavlem gamravlebis ganrigebadobis Tviseba gamoiyenos (Sekrebis mimarT). amave Tvisebis gamoyenebas eyrdnoba 7 - 9 amocanebi. sxva amocanebSi utolobebis Tvisebebis gamoyeneba iyo saWiro. 6

moswavleebi swrafad miagneben swor Tanafardobas _ a)-s: utolobaSi meti ricxvi mravldeba mamravlze, romelic metia naklebi ricxvis mamravlze. marTlac, utolobis Tvisebebis mixedviT, 7,5>7,23 utolobidan gvaqvs: 7,5⋅6 14 >7,23⋅6 14 (1) 6 14 >5 utolobidan gvaqvs: 7,23⋅6 14 >7,23⋅5 (2) (1) da (2) utolobebidan miviRebT a)-s. faqtobrivad, am savarjiSoSi moswavle iyenebs maswavleblisTvis utolobis kargad cnobil Tvisebas: Tu a>b>0 da c>d>0. maSin ac>bd. klasSi vxsniT 10 - 14 amocanebs. es amocanebic utolobebis Tvisebebis gamoyenebiT ixsneba. 10 ricxviT sxivze ganlagebuli ricxvebi zrdis mixedviTaa dalagebuli. mosaxer­ 6 xebelia yvela ricxvi aTwiladis saxiT gadavweroT: 13 2 =6,5; 5 =1,2. 12 cxadia, a–b<a+b (Cven xom uaryofiT ricxvebs ar ganvixilavT). a+b naklebia (2a+b)-zec da (a+2b)-zec. SevadaroT 2a+b da a+2b utolobis Tvisebebis gamoyenebiT: mocemuli a>b utolobis orive mxares mivumatoT a+b, miviRebT: a+b+a>a+b+b

138


2a+b>a+2b. pasuxi: a-b<a+b<a+2b<2a+b. 13 piroba sqematurad ase CavweroT:

luka+ Tebrole=Tornike+irma, luka>Tornike. marcxena mxares Tu mets gamovaklebT, marjvena mxares _ naklebs, cxadia, marcxniv naklebi darCeba: Tebrole <irma. pasuxi: irma ufrosia Tebroleze. 14 moswavleebma unda Seadaron:

1,2+8,4+5,4, 1,2+8,4+6 da 1,2+8,4+5,2 gamosaxulebebi 15-s. am samidan I da III gamosaxulebebi ar aRemeteba 15-s. e. i. elenes SeuZlia iyidos 5,2 an 5,4-lariani saqaRalde. SeiZleba saqaRaldis fasi moswavleebma x-iT aRniSnon da amocanis piroba Caweron 1,2+8,4+x<15 an 1,2+8,4+x=15 (anu 1,2+8,4+x≤15) damokidebulebebis saxiT. utolobis TvisebebiT, utolobis orive mxares gamoakleben 9,6-s. miiReben x<5,4 an x=5,4. daaskvnian, rom saqaRaldis fasi 5,4 lars ar unda aRematebodes. saSinao davalebad vaZlevT

10 - 15

amocanebs.

76-e gakveTili

Tema: ramdenime terminis gamoyeneba. mizani: moswavle sworad iyenebdes terminebs – `yvela~, `yoveli~, `TiToeuli~, `zogierTi~, `erT-erTi~, `arc erTi”, `erTi mainc~, `nebismieri~ (maT. VI. 4.). winapirobebi: wiladebis sxvadasxva interpretacia, naturaluri ricxvi, martivi ricxvi, aTwiladi. resursebi: plakati, romelzec gamosaxulia sxvadasxva termini (yvela, yoveli, TiToeuli, zogierTi, erT-erTi, erTi mainc, nebismieri) da maTiO Sinaarsis aRwera, rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT. gTavazobT rekomendaciebs zogierTi amocanisTvis. suraTidan davadgenT: b>c; suraTze ar aris gamosaxuli 855, magram moswavle am ricxvs 1000-is marcxniv warmoidgens da daadgens: b>855; a<b utolobis orive mxaris 2-ze gamravlebiT, miiReben 2a<2b; a<c utolobis orive mxarisadmi b-s damatebiT, miiReben a+b<b+c. suraTidan Cans, rom a<b da c<b. I utolobaSi naklebs naklebi mivumatoT, mets meti, miviRebT: a+c<b+b, anu a+c<2b. paragrafSi warmodgenili utolobis Tvisebebis gamoyenebiT, gvaqvs: a<b ⇒ a+c<b+c ⇒ a+c<2b. c<b ⇒ c+b<b+b am savarjiSoSi moswavle iyenebs maswavleblisTvis utolobis kargad cnobil Tvisebas: TuU a<b da c<d, maSin a+c<b+d. 11

139


12

I awonviT SevadaroT ori sagani, II awonviT I awonvis Sedegad gamovlenili msubuqi sagani SevadaroT mesame sagans. maTgan msubuqi davdoT pirvel adgilze _ is umCatesia. darCenili oridan III awonviT SevarCevT ufro msubuqs da davdebT meore adgilze. maTgan mZime ki am sameulSi yvelaze mZime iqneba. am gziT sagnebs davalagebT masebis zrdis mixedviT. wina amocanis amoxsnis msgavsad 3 awonviT davalagebT raime sam sagans masebis zrdiT. IV awonviT meoTxe sagans SevadarebT Suas, V awonviT sabolood davadgenT meoTxe sagnis adgils. pasuxi: 5 awonva. mama deda 14 luka 13

AN

ani

70

mama

120

deda

pirobis mixedviT Sedgenili sqemidan TvalnaTliv Cans, rom lukas dedas anis dedaze 190 lariT meti Semosavali aqvs. 15

2,4+4,3+4,2<11. ver SeZleben.

amis Semdeg vatarebT jgufur muSaobas gakveTilze dasaxuli miznis misaRwevad. viyenebT `Cvens leqsikons~, romelic, sasurvelia, rom maswavlebelma klasSi plakatis saxiT warmoadginos da moswavleebs sTxovos, saTiTaod yvela terminis axsnis wakiTxva da komentarebi. jgufuri muSaoba Cvens leqsikonSi axsnili sityvebis (terminebis) gamoyenebis nimuSebi warmodgenilia amocanebis saxiT, romlebic jgufebad gaerTianebulma moswavleebma unda amoxsnan leqsikonis daxmarebiT. pedagogma didi sifrTxiliT unda ganixilos es sakiTxebi klasSi. moswavleebi am aqtivobiT arsebiTad gaiRrmaveben da gaifarToeben leqsikas. am aqtivobas aucileblad unda axldes erToblivi, sakmaod aqtiuri gansja. 1

yoveli wiladi SeiZleba CavweroT sasruli aTwiladiT _ am winadadebis mcdarobis damtkiceba erTi magaliTiTac (gamonaklisiT/kontrmagaliTiT) SeiZleba. magaliTad, 23 ar Caiwereba sasruli aTwiladis saxiT. • yoveli naturaluri ricxvi SeiZleba CavweroT wiladis saxiT _ WeSmaritia. yoveli n ricxvi iwereba n1 , 2n 2 da a. S. saxiT. • yoveli martivi ricxvi kentia _ mcdaria. magaliTad, 2 _ aris luwi martivi ricxvi. • yoveli naturaluri ricxvi iyofa 3-ze _ mcdaria. magaliTad, 5. • zogierTi naturaluri ricxvi iyofa 3-ze. igivea rac: arsebobs erTi mainc naturaluri ricxvi, romelic iyofa 3-ze. WeSmaritia, aseTi ricxvi arsebobs, magaliTad, 3. • zogierTi naturaluri ricxvi ar iyofa 3-ze. igivea, rac: arsebobs erTi mainc naturaluri ricxvi, romelic ar iyofa 3-ze. WeSmaritia, magaliTad, 5. • luwi martivi ricxvi erTaderTia _ WemSaritia, es ricxvia 2, 2-isgan gansxvavebuli luwi ricxvi 1-is da Tavis Tavis garda 2-zec iyofa, e. i. is aucileblad Sedgenilia.

140


2 yoveli naturaluri ricxvi iyofa 5-ze. es winadadeba mcdaria, radgan, magaliTad, 11 naturaluri ricxvia da ar iyofa 5-ze. 3 winadadebebi SeiZleba davamTavroT, magaliTad ase: • TiToeuli naturaluri ricxvia; • erTi mainc luwi ricxvia; erTi mainc kenti ricxvia; erTi mainc erTniSna ricxvia, da a. S. • arc erTi ar iyofa 7-ze. • erT-erTi am ricxvebidan iyofa 9-ze. • zogierTi am ricxvebidan orniSna ricxvia. (es winadadeba II winadadebis tolfasia: `zogierTi~ gulisxmobs `erTs mainc~). • zogierTi am ricxvebidan ar iyofa 5-ze. (es winadadeba IV winadadebebisgan gansxvavdeba, aq `zogierTSi~ igulisxmeba `erTi mainc~, anu erTi, an erTze meti. IV winadadebaSi igulisxmeba mxolod erTi). • nebismieri naklebia 20-ze; • arsebobs martivi ricxvi maT Soris; arsebobs kenti ricxvi maT Soris; arsebobs orniSna ricxvi maT Soris, da a. S. (`arsebobs~ igulisxmeba arsebobs erTi, an erTze meti). • ar arsebobs samniSna ricxvi maT Soris; ar arsebobs kenti martivi ricxvi maT Soris da a. S.

`M yoveli naturaluri ricxvi 1-ze iyofa~ da `nebismieri naturaluri 1-ze iyofa~. • `zogierTi naturaluri ricxvi 5-ze iyofa~ da `arsebobs naturaluri romelic 5-ze iyofa~. • `erTi mainc martivi ricxvi ar aris kenti ricxvi~ da `arsebobs martivi romelic ar aris kenti ricxvi~. • `yoveli wiladi ar Caiwereba sasruli aTwiladis saxiT~ da `nebismieri ar Caiwereba sasruli aTwiladis saxiT~A. 4

ricxvi ricxvi, ricxvi, wiladi

am jgufuri muSaobis Semdeg moswavle ukeT Cawvdeba zogierTi mniSvnelovani maTematikuri terminis Sinaarss. mixvdeba, rom yvela da nebismieri, yoveli da TiToeuli, zogierTi da erTi mainc erTi da imave azris gamosaxatavad gamoiyeneba. Tu winadadeba raime Tvisebas zogierTi (erTi mainc) obieqtisTvis WeSmaritad acxadebs, maSin am winadadebis WeSmaritobis dasadgenad erTi damadasturebeli magaliTic kmara. xolo Tu winadadeba raime Tvisebas yvela (nebismier) obieqts `mia­ wers~, maSin mxolod erTi uarmyofeli magaliTiTac ki (gamonaklisiT/kontrmaga­ liTiT) SeiZleba am winadadebis uaryofis (mcdarobis) dasabuTeba. SeiZleba maswavlebelma arCios, rom am gakveTilze masalis gameorebis mizniT amoxsnas ramdenime damatebiTi amocana (me-3 Tavis damatebiTi amocanebidan 57 - 75 ) danarCenidan saSinao davaleba SearCios, xolo jgufur muSaobas SeiZleba Semdegi gakveTili dauTmos.

141


77-e da 78-e gakveTilebi

Tema: ariTmetikuli moqmedebebi arauaryofiT racionalur ricxvebze. mizani: arauaryofiT racionalur ricxvebze moqmedebebisa da maTi gamoyenebis Sesaxeb codnis ganmtkiceba. winapirobebi: ariTmetikuli moqmedebebi arauaryofiT racionalur ricxvebze. am or gakveTilze (Semajamebeli gakveTilebi) mimdinareobs ganvlili masalis gameoreba, codnis ganmtkiceba. saxelmZRvaneloSi warmodgenil amocanebs maswavlebeli or gakveTilze gamoiyenebs. maTi nawili klasSi ixsneba, danarCenebi – Sin. gTavazobT, rekomendaciebs amocanebis amosaxsnelad. me-3 Tavis SEemajamebeli amocanebi (amocanebi TviTSefasebisTvisac) 1

b) Sekrebis mimarT gamravlebis ganrigebadobis Tvisebis gamoyenebiT SeiZleba gamoTvlebis gamartiveba: ( 74 : 23  – 74 : 54 )⋅6=( 74 ⋅ 32  – 74 ⋅ 45 )⋅6= 74 ⋅( 32  – 45 )⋅6=7 207 . II ricxvi aRvniSnoT x-iT, I iqneba 3 12 x, III – 2 12 x. pirobiT, 3 12 x+x+2 12 x=24 12 , saidanac x=3 12 . pasuxi: I ricxvia 3 12 ⋅3 12 =12 14 , II ricxvia 3 12 , III ricxvia 2 12 ⋅3 12 =8 34 . 3

vTqvaT, gasavleli manZili x km-ia. pirobiT, x⋅ 59 =90, saidanac x=162 km. II dRes gaiares 90:1,5=60 km, II dRes gasavleli darCaT 162–(90+60)=12 km. 5

pirobiT, I ricxvi miiReba meorisTvis marjvnidan 0-is miweriT. e. i. I ricxvi II-ze 10-jer metia. II ricxvi aRvniSnoT x-iT, I _ 10x-iT. 10x–x=243, saidanac x=27. 7

wina amocanis analogiurad daaskvnian, rom II ricxvi I-ze 10-jer metia. I 8 aRvniSnoT x-iT, II ki 10x-iT. pirobiT, x+10x=198, saidanac x=18. 7 8 7 1 8 1 9 8 ufro axlosa 1-Tan, vidre 7 ( 8 daSorebulia 1-dan 8 -iT, 7 daSorebulia 7 -iT). sxva ingredientebis damatebiT 87 ufro scildeba 1-s, vidre 78 . pasuxi: I namcxvris wona ufro axlos iqneba 1 kg-Tan. 12 6 x+2 6 10 20 = 10 . pirobiT, 10 < 10 , anu x+2<6 saidanac x=1; 2; 3. 1

11 moswavleTa 3 -s, anu 21 moswavles eqskursiaSi monawileoba ar miuRia.

pirvel klasSi moswavleTa odenoba 5-is jeradia (5-ze iyofa), radgan 35 nawili eqskursantia, meore klasSi odenoba 4-is jeradia. am odenobaTa jami 63-is tolia, ganvixiloT SesaZlo SemTxvevebi _ viyenebT sinjvis meTods: I klasSi moswavleTa odenoba

II klasSi moswavleTa odenoba

gavagrZelebT (+), Tu vwyvetT (_) ganxilvas

60

3

_

55

8

+

50

13

_

142

ramdenma ar miiRo monawileoba

55⋅ 25 +8⋅ 14 =22+2≠21


45

18

_

40

23

_

35

28

+

30

33

_

25

38

_

35⋅ 25 +28⋅ 14 =14+7=21

yovel saaTSi I miliT ivseba auzis 15 , meore miliT icleba auzis 16 . yovel 1 nawili. auzi aivseba 30 saaTSi. saaTSi ivseba auzis 15 – 16 = 30 1 1 13 I avtomanqana 1 sT-Si gadis mTeli gzis 6 nawils, II _ 5 -s. isini yovel saaTSi 11 nawiliT. Sexvdebian 30 ≈2,727≈2,7 saaTSi. erTmaneTs uaxlovdebian mTeli gzis 15 + 16 = 30 11 12

14 vasikos naxatebis yoveli mervedi 2-iT metia ninikos naxatebis yovel mervedze.

e. i. vasikos aqvs 8⋅2=16 naxatiT meti. SeiZleboda sqemis gamoyenebac. 15 erTi awonviT ori monetis masebi SevadaroT. Tu maTgan romelime msubuqia, is

aris yalbi. Tu saswori gaTanabrda, e. i. mesame (romelic sasworze ar davdeT) monetaa yalbi. 16

I awonviT sam-sami monetis masebi SevadaroT. Tu sasworze dadebuli sameulebidan romelime msubuqia, am samSia yalbi moneta. Tu saswori gawonasworda, yalbi moneta mesame (romelic sasworze ar davdeT) sameulSia. II awonviT wina amocanis msgavsad amovicnobT yalb monetas. 17 SeiZleba jer qalaqebs Soris manZili ipovon: 80⋅2,4=192 km-s, Semdeg avtobusis

siCqare: 80⋅ 34 =60km/sT. da bolos, avtobusis mier 192 km-is gavlaze daxarjuli dro: 192:60=3,2 sT. am amocanebis amoxsnis Sedegad moswavle iwafeba arastandartuli amocanebis amoxsnaSi. amocanebis mravalferovani Tematika, aqtualuroba zrdis moswavleTa motivacias. am amocanebiT SesaZlebelia klasSi gaimarTos saintereso ganxilvebi, polemika, SesaZlebelia Catardes paeqroba _ jgufuri muSaoba, vip aqtivoba. ar aris aucilebeli yvela amocanis Tanmimdevrulad ganxilva _ SeiZleba drodadro daubrundeT xolme am amocanebs. unda gvaxsovdes, maTi ganxilvisas mTavari `moqmedi pirebi~ Tavad moswavleebi unda iyvnen. moswavleebs gamovucxadebT, rom Semdeg gakveTilze Semajamebeli wera tardeba ricxvis nawilisa da nawiliT ricxvis povnaze, utolobebis Tvisebebis gamoyenebaze, amocanebis amoxsnis sxvadasxva xerxis gamoyenebaze, ricxviTi gamosaxulebis mniSvnelobis povnaze.

143


79-e gakveTili

Semajamebeli wera SearCieT swori pasuxi: 1. yuTSi 4 wiTeli da 6 Savi fanqaria. wiTeli fanqrebis odenoba aris Savi fanqrebis odenobis 4 nawili a) 10 b) 46 nawili g) 14 nawili d) 64 nawili. 2. 412+222 bolovdeba a) 0-iT b) 6-iT

g) 4-iT

d) 8-iT.

3. sakontrolo weraSi klasis moswavleTa 18 -ma 10-10 qula miiRo. danarCenma 21 moswavlem ufro dabali Sefasebebi. ramdenma miiRo 10 qula? a) 4 moswavle b) 3 moswavle g) 2 moswavle d) 8 moswavle. 4. 1,5-saaTiani filmi saRamos 8 saaTze daiwyo. filmis msvlelobisas 5-jer CarTes 3-wuTiani reklama. filmi dasrulda a) 930-ze b) 915-ze g) 1015-ze Dd) 945-ze 5. 7 13 :1 29 = a) 7 19

b) 7 23 g) 6 d) 7 12 .

6. Tu m<c da m>b, maSin ricxviT sxivze a) b aris m da c-s Soris b) m aris b da c-s Soris g) c aris b da m-s Soris d) b emTxveva c-s. amoxseniT amocanebi: 7. gias 8,4 lari aqvs, iraklis _ 12,1 lari, beJans _ 20,2 lari. ramdeni lari darCebaT biWebs 35,6-lariani burTis yidvis Semdeg? 8. amovxsnaT gantolebebi: a) x : 1,4+ 37 =2 b) 3 12  : x– 25 =1. pasuxebi da miTiTebebi: 1

2

3

4

5

6

b

d

b

d

g

b

2. 412 bolovdeba 4-iT, 222-ic bolovdeba 4-iT, jami bolovdeba 8-iT. 3. klasis 78 aris 21, 18 aris 3 moswavle. 4. 5 reklama 15 wuTs daikavebs. amitom filmi dasruldeba dawyebidan 1 sT 45 wTis Semdeg _ 945-ze. 7. biWebis saerTo Tanxaa 8,4+12,1+20,2=40,7 lari. burTis yidvis Semdeg darCebaT 40,7_35,6=5,1 lari. 8. a) x=2,2, b) x=2,5.

144


Sefasebis sqema. testuri tipis amocanebidan TiToeulis swori amoxsna Sefasdes 1 quliT. Sefasebis rubrikebi #7 amocanisTvis: 0 qula

ver gaigo amocanis arsi, an ar uweria araferi, rac ukavSirdeba amocanis amoxsnis process.

0,5 qula

SearCia moqmedeba biWebis saerTo Tanxis gamosaTvlelad.

1 qula

gamoTvala biWebis saerToO Tanxa.

1,5 qula

gamoTvala biWebis saerTo Tanxa da SearCia moqmedeba darCenili Tanxis gamosaTvlelad.

2 qula

amoxsna amocana srulyofilad.

#8-is a) amocanisTvis: 0 qula

ver gaigo amocanis arsi, an ar uweria araferi, rac ukavSirdeba amocanis amoxsnis process.

0,5 qula

gamoTvala I Sesakrebi.

1 qula

amoxsna amocana srulyofilad.

#8-is b) amocanisaTvis 0,5 qula

gamoTvala saklebi.

1 qula

amoxsna amocana srulyofilad.

145


IV Tavi vswavlobT geometriuli figurebis Tvisebebs

§1. geometriuli figurebi (gameoreba) me-80 gakveTili

Tema: samkuTxedebi, oTxkuTxedebi mizani: moswavlem gaimeoros samkuTxedebisa da oTxkuTxedebis klasifikacia maTi elementebis mixedviT, sworad gamoiyenos terminebi â&#x20AC;&#x201C; `yvela~, `TiToeuli~, `zogierTi~, `erT-erTi~, `arcerTi~, `erTaderTi~ (maT. VI. 4.). winapirobebi: kuTxe, kuTxis zoma, marTi kuTxe, paraleluri wrfeebi, gadamkveTi wrfeebi, kuTxeebis klasifikacia. resursebi: samkuTxedebis da oTxkuTxedebis sxvadasxva saxeebis modelebi, saxazavi, fanqari, rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT Semajamebeli weris Sedegebis gacnobiT. amis Semdeg gadavdivarT im geometriuli figurebis gameorebaze, romlebsac moswavleebi ukve icnoben. zemoT naxsenebi terminebis gamoyeneba ricxvebis Tvisebebisa an ricxviT erTobliobebs Soris mimarTebebis dadgenisas (maT. VI. 4.) ukve ganxiluli iyo. axla saSualeba gvaqvs gavimeoroT es terminebi geometriuli figurebis Tvisebebis an figurebis erTobliobebs Soris mimarTebebis aRwerisas. am sakiTxebis ganxilvisas SeiZleba visargebloT saxelmZRvanelos teqstiT da geometriul figuraTa modelebiT. moswavlem unda SeZlos teqstSi warmodgenili samkuTxedebidan aRweros TiToeuli. samkuTxedebidan erTi tolferdaa, mas ori gverdi aqvs toli. am gverdebs ferdebs uwodeben, mesame gverds _ fuZes. meore samkuTxedi tolgverdaa _ samive gverdi toli aqvs. aRvniSnoT, rom tolgverda samkuTxedi tolferdacaa, masac ori gverdi toli aqvs. am gakveTilisTvis moswavles unda SeeZlos kuTxis amocnoba, aRwera, unda ganasxvavebdes marT, maxvil da blagv kuTxeebs. oTxkuTxedebidan gavixsenoT marTkuTxedi. moswavlem unda SeZlos daasaxelos misi wveroebi, gverdebi. romelime wveros Semcveli gverdebi, marTkuTxedis gverdebis Semcveli paraleluri da gadamkveTi wrfeebi. marTkuTxedi oTxkuTxedia marTiU kuTxeebiT, e. i. kvadratic marTkuTxedia. `yvela kvadrati marTkuTxedia~, `yvela marTkuTxedi ar aris kvadrati~, `arsebobs marTkuTxedi, roemlic kvadratia~, anu, `zogierTi marTkuTxedi kvadratia~; `arsebobs marTkuTxedi, romelic kvadrati ar aris~, `ar arsebobs kvadrati, romelic marTkuTxedi ar aris~, anu `yoveli kvadrati marTkuTxedia~. am winadadebebis ganxilva, `yovelis~ `zogierTis~ arsis gaazreba uviTarebs moswavles logikur azrovnebas, umdidrebs leqsikas. es mimarTebebi dasaxelebul figurebs Soris SeiZleba venis diagramiT warmovadginoT:

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oTxkuTxedebi marTkuTxedebi kvadratebi

es mimarTebebi SeiZleba aramaTematikur obieqtebzec gaavrceloT. magaliTad, `yoveli arwivi frinvelia~, `zogierTi frinveli arwivia~, `yvela frinveli cocxali arsebaa~, ... saklaso da saSinao savarjiSoebiT moswavle gaiwafeba geometriuli figurebis amocnobaSi, maT Soris damokidebulebebis dadgenaSi, perimetris gamoTvlaSi, sayuradRebo amocanebis amoxsnaSi. klasSi vxsniT 1 - 10 amocanebs, maTi saSualebiT moswavleebi ixseneben kuTxeebis klasifikacias zomebis mixedviT, paralelur da gadamkveT wrfeebs, samkuTxedebis klasifikacias. saSinao davalebad vaZlevT 1 - 9 amocanebs.

81-e gakveTili

Tema: geometriuli figurebi mizani: moswavleebi ixseneben samkuTxedebis klasifikacias gverdebisa da kuTxeebis mixedviT; gadamkveT da paralelur wrfeebs; mosazRvre kuTxeebs, gaSlil, maxvil da marT kuTxes. winapirobebi: mravalkuTxedebis klasifikacia gverdebis raodenobis mixedviT. resursebi: rveuli, saxelmZRvanelo, saxazavi, dafa. saSinao davalebis Semowmeba da klasSi 11 - 19 amocanebis Sesruleba gakveTilze dasaxuli miznebis miRwevas emsaxureba. gTavazobT saSinao davalebis da klasSi gansaxilveli zogierTi amocanis amoxsnebs. 9 savarjiSo mravalkuTxedis saxeebs Soris mimarTebebis dadgenas eZRvneba. 9

savarjiSos a), b), g) figurebi arsebobs da moswavleebs SeiZleba vTxovoT aseTi figurebis warmodgena:

ar arsebobs kvadrati, romelic ar aris marTkuTxedi. kvadratis oTxive kuTxe marTia _ is marTkuTxedis kerZo saxea. klasSi 11 - 19 amocanebis amoxsna sasurvelia, moswavleTa aqtiuri CarTulobiT mimdinareobdes â&#x20AC;&#x201C; dafasTan viZaxebT erT-erT moswavles da, magaliTad, 13 amocanis ganxilvisas igi cdilobs mocemuli 4 wertilidan (isini dafazea gamosaxuli) yovel 2-ze gaavlos wrfe da daiTvalos maTi raodenoba. komentarebisTvis mivmarTavT kla-

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sis sxva moswavleebsac. Tumca, maTi daTvla SeiZleba dafaze gamosaxvis gareSec, msje­ lobiT: 13 yovel wertilze 3 wrfe gaivleba, 4⋅3=12, magram yoveli wrfe orjer davTvaleT (or wertilze erTaderTi wrfe gaivleba), amitom sul gaivleba 12:L2=6 wrfe: AB, AC, AD, BC, BD da CD. monakveTebic 6-ia. 14 wrfeze ganlagebuli 6 wertili 5 Sualeds qmnis. kidura wertilebs Soris iqneba 5⋅2,4=12 sm. 17 am marTkuTxedis ori mezobeli gverdis sigrZeTa jami 12,4 sm-ia, erTi gverdi 8,2 sm-a, meore _ 12,4–8,2=4,2 sm. es marTkuTxedi ar aris kvadrati. 18 mcireodeni sajaro ganxilviT moswavleebi mixvdebian, rom aq a SesaZle­belia sami SemTxveva: roca cnobilia I) 2a da a+b; II) 2b da a+b, III) 2a da 2b. b I. mopirdapire gverdebis jamia 16,2 sm, maSin maTgan TiTo gverdi 8,1 sm-ia; sxva gverdia 20,4–8,1=12,3 sm. II. mopirdapire gverdebis jamia 20,4 sm, maSin maTgan TiTo gverdi 10,2 sm-ia, sxva gverdia 16,2–10,2=6 sm. III. mopirdapire gverdebis jamia 16,2 sm, mopirdapire gverdebis sxva wyvilis jamia 20, 4 sm. e. i. gverdebia 8,1 sm da 10,2 sm. P P 19 cxadia, 2 =a+b, aqedan b= 2 –a.

saSinao davalebas virCevT

10 - 19

amocanebidan. am amocanebis nawili SeiZleba Semdegi gakveTilis davalebadac gamoiyenoT.

82-e gakveTili

Tema: geometriuli figurebi mizani: Seswavlili geometriuli faqtebis gameoreba, codnisa da unarebis ganmtkiceba. winapirobebi: gadamkveTi da paraleluri wrfeebi, samkuTxedebis sxvadasxva klasifikacia. resursebi: qaRaldis furceli, makrateli, saxazavi, rveuli, saxelmZRvanelo, dafa. gakveTils viwyebT saSinao davalebis SemowmebiT. Ddavalebis zogierTi amocana Cveni daxmarebiT SeiZleba klasSic amoixsnas. magaliTad, moswavle gamosaxavs dafaze wrfis monakveTs, masze aiRebs 8 wertils 12 amocana) da iwyebs Sualedebs Soris manZilebis ajamvas – 7⋅2,5=17,5 sm. aqve Sei( Zleba davsvaT kiTxva: – wrfeze 8 wertilia, ramdeni Sualedia am wertilebs Soris? (7) – 10 wertili rom aviRoT, ramdeni iqneba Sualedebis raodenoba? (9) Semdeg viyenebT qaRaldis furcels, makratels da vTxovT moswavleebs warmoadginon 15 amocanis amoxsna;

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magaliTad, saerTo erTi wertili aqvs Semdeg figurebs:

saerTo monakveTi aqvs figurebs:

saerTo samkuTxedi aqvs figurebs:

gTavazobT rekomendaciebs sxva amocanebis amosaxsneladac: 11

moswavle damoukideblad xazavs sxvadasxva saxis samkuTxedebs, rac daexmareba mas samkuTxedebis simravlis sxvadasxva klasifikaciaSi. 12

sainteresoa, rom winadadeba, romelic WeSmaritia `yoveli~, `nebismieri~ obieqtisTvis, WeSmaritia `zogierTi~, `erTi mainc~ obieqtisTvisac. marTlac, radgan `yoveli tolgverda samkuTxedi aris maxvilkuTxa~, aqedan vRebulobT: `arsebobs erTi mainc maxvilkuTxa tolgverda samkuTxedi~. piriqiT, sazogadod ase ar aris, radgan `zogierTi~ SeiZleba `erTsac~ niSnavdes, orsac, magram ara aucileblad nebismiers. 16 a) mosazRvre kuTxeebidan umciresis zoma aRvniSnoT x-iT, meoris zoma iqneba 3x. maTi jamia  x+3x=1800, saidanac x=450. b) mosazRvre kuTxeebidan umciresis zoma aRvniSnoT x-iT, meoris zoma iqneba x+200. maTi jami 1800-ia: x+x+200=1800, saidanac x=800. g) mosazRvre kuTxeebidan erT-erTis zoma iyos x, meoris _ 23 x, x+ 23 x=1800, saidanac 1 23 x=1800, x=1080. pasuxi: 1080 da 720. 17 es amocana klasSi amoxsnili 18 amocanis msgavsad amoixsneba. miRebuli Sedegebidan gamovyofT im SemTxvevas, roca gverdebi naturaluri ricxvebia _ 8 m da 3 m.

Sekrebili sami gverdidan SeiZleba mxolod erTi iyos 3,2 m sigrZis, maSin marTkuTxedis gverdebi iqneba 3,2 m da (22,2–3,2):2=9,5 m. Tu am sami gverdidan oria 3,2 m sigrZis, maSin mesame gverdi iqneba 22,2–2⋅3,2=15,8 m. 18

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moswavle martivad daadgens, rom saubari ar aris mosazRvre kuTxeebze, aqedan ki miiRebs, rom mocemulia ori toli kuTxis jami. maSin erT-erTia 310-iani. misi mosazRvrea 1490-iani. 21

amocanis ganxilvisas Tavi unda avaridoT sruli kuTxis cnebis gamoyenebas _ davkmayofildeT ori gaSlili kuTxiT. gakveTilis meti nawili SeiZleba saSinao davalebis amocanebis garCevas dasWirdes. amasTanave, SeiZleba maswavlebelma 11 - 21 amocanebidan yvela ar misces saSinao davalebad da nawili klasSi amosaxsnelad Semoinaxos. zogierTi maswavlebeli gadawyvets darCenili dro `vip~-is rubrikiT warmodgenili amocanebis gamoyenebiT jgufuri muSaobis Caatarebas moaxmaros. SeiZleba saswavlo procesi jgufebs Soris

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Sejibrebis formiT Catardes, gaTvaliswinebuli unda iyos amocanis swrafad da sworad amoxsna. erTgvari simartivisa da TvalsaCinoebis miuxedavad, 1 moswavleTagan saz­ rianobas, sakiTxis operatiulad ganxilvis unaris gamovlenas moiTxovs da am unaris ganviTarebasac uwyobs xels. am aqtivobiT Tqven SeZlebT gavlili masalis aTvisebis xarisxis Semowmebasac. es ki samomavlod Tqveni samoqmedo taqtikis dazustebas Seuwyobs xels. gTavazobT dayofis SesaZlo variantebs. 1)

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2 A, B da C wertilebidan SeiZleba SeirCes wertilebis 3 wyvili (A, B; A, C; B, C), Tu mesame wertilad SevarCevT jer M-s, Semdeg N-s da bolos K-s, miviRebT 3+3+3=9 samkuTxeds. Tu wyvilebs SevarCevT M, N da K wertilebs Soris, analogiuri gziT miviRebT kidev 9 samkuTxeds. amrigad, sul miiReba 18 samkuTxedi. 1 da 2 amocanebi, mosalodnelia, moswavleebma srulyofilad ver amoxsnan. pedagogma amoxsnis xarisxi mignebuli variantebis odenobiT SeiZleba daadginos. gamorCenili variantebis Zieba ki saSinao davalebad misces moswavleebs. 3 davalebiT kidev erTxel moaxdenT masalis aTvisebis diagnostikas. am davalebas ki moswavleebi, albaT, warmatebiT daZleven. masSi swori winadadebebia: a), b), d), e), v), z). ar aris swori g) da T) winadadebebi. gaiTvaliswineT, rom am amocanebis ganxilvisas ar igulisxmeba, rom moswavleebma samkuTxedebis kuTxeebis jamis Tviseba ician. am aqtivobis dasasruls uaRresad efeqtian Sedegs mogvcems warmodgenili naSromebis sajaro ganxilva da saqmiani polemika.

jgufuri muSaobis damTavrebis Semdeg SeiZleba vTxovoT moswavleebs gaecnon `moisazres~ rubrikiT warmodgenil amocanebs da moemzadon klasSi maTi erToblivi ganxilvisaTvis. davalebad SeiZleba `saSinaos~ rubrikidan darCenili amocanebi da Tavis damatebiTi 1 - 4 amocanebi gamoviyenoT. 1 amocanaSi moswavlem sasurvelia gamoiyenos saxazavi mocemuli zomebis marTkuTxedis asagebad.

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83-e gakveTili

Tema: geometriuli figurebi mizani: geometriuli figurebis Tvisebebis codnis gamoyenebiT amocanebis amoxsna, moswavleTa msjeloba-dasabuTebis, kombinatoruli msjelobis unarebis ganviTareba. winapirobebi: samkuTxedi, oTxkuTxedebi da maTi saxeebi, asoiTi gamosaxulebis Sedgena da gamartiveba. resursebi: qaRaldis furclebi, makrateli, rveuli, saxelmZRvanelo, saxazavi, dafa. sasurvelia, rom `moisazres~ rubrikiT warmodgenili amocanebi Cveni xelmZRvanelobiTa da daxmarebiT klasSi erToblivi ZalisxmeviT amoixsnas. gTavazobT rekomendaciebs am procesis gansaxorcieleblad. ReroebiT Sedgenil aT tolgverda samkuTxedSi 8 pataris garda igulisxmeba ori didi _ 4 patara samkuTxediT Sedgenili samkuTxedi. suraTze warmodgenilia arsebiTad gansxvavebuli SemTxvevebi, roca 4 asanTis aRebis Semdeg darCeba 4 tolgverda samkuTxedi (maT Soris 2 pataraa, 2 didi). 1

maswavlebelma icis, rom moniSnuli 6 wertilidan 3-is B SerCevaTa raodenoba dalagebis gaTvaliswinebis gareSe aris 6! =20, magram am sameulebidan 3 sameuli unda gamovricxoT, C63= 3!3! M N radgan maTSi wertilebi erT wrfezea ganlagebuli. pasuxi: 20–3=17. am etapze moswavle kombinatorikis formulebs ar icnobs, A C K swored maT Sesaswavlad Semzadebaa aseTi amocanebis amoxsnis erT-erTi mizani. man samkuTxedebis dasaTvlelad raime strategia unda SearCios. B magaliTad, samkuTxedebis daTvla SeiZleba ase awarmoon: A, M, N, K wveroebiT Seqmnili samkuTxedebis raodenoba aris 4. B, M, N, K wveroebiTac da C, M, N, K wveroebiTac 4-4 samkuTxedi M N Seiqmneba. am gadaTvlisas DMNK dasaxeldeba 3-jer. amitom am 12 samkuTxedSi gansxvavebulia 10. garda amisa, gvaqvs samkuTxedebi, romlebSic ABC samkuTxedis or-ori wveroa. ABN, ACN, BAK, BKC, A C K CAM, CBM da TviT ABC. pasuxi: 17. SeiZleba strategia aseTi iyos: jer davasaxelebT yvela samkuTxeds, romlis erTerTi wveroa A; Semdeg yvela samkuTxeds, romlis erT-erTi wveroa B da wveroebs Soris ar aris A; Semdeg – C-wveroians, romlis wveroebs Soris arc A da arc B ar aris da a. S. 2

3 9 patara samkuTxedi, kidev 3 cali samkuTxedi, Sedgenili 4 patara samkuTxediT

da 1 mTliani samkuTxedi _ sul 9+3+1=13.

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magaliTad, SeiZleba ase gaiWras:

Tu marTkuTxedis gverdebs a da b-Ti aRvniSnavT, amocanis piroba aseT saxes miiRebs: 2a+b=21 2b+a=14,1, saidanac 3a+3b=35,1, a+b=11,7, p=2(a+b)=23,4 (sm). moswavleebi, albaT, pirobas sistemis saxiT ar Caweren, magram sityvierad aRweren, rom daviTma da giorgim, marTkuTxedis ori mezobeli gverdidan TiToeuli sam-samjer Sekribes da miiRes 21+14,1=35,1. am ori gverdis jami yofila 35,1:3=11,7, perimetri 11,7⋅2=23,4 sm. 5

§2. wrewiri. wre 84-e gakveTili

Tema: wrewiri, wre mizani: moswavlem SeZlos wrewirisa da misi elementebis aRwera; wrewiris, wrisa da maT elementebs Soris mimarTebebis dadgena – wertilebis ganlageba wrewirze, wrewiris SigniT da wrewiris gareT (maT. VI. 9). winapirobebi: sakuTxedis utoloba, mravalkuTxedebis klasifikacia. resursebi: wris modelebi, rveuli, saxelmZRvanelo, dafa. standartSi (maT. VI. 9) wrewirTan dakavSirebul misaRwev SedegebSi miTiTebulia wrewirebis urTierTganlagebis SemTxvevebis aRwera. am sakiTxis ganxilvamde vatarebT Sesabamis muSaobas moswavleebTan wrewirisa da misi elementebis aRwerisa da maT Soris mimarTebebis dasadgenad. amasTanave, ukve viwyebT muSaobas, rom moswavleebi mieCvion deduqciur msjelobas. mniSvnelovania, rom moswavle asabuTebs geometriul debulebas _ Teoremas: centrze gamavali qorda udidesia. dasabuTebisas is iyenebs samkuTxedis utolobas, romelic aris aucilebeli winare codna am gakveTilis warmatebiT ganxorcielebisTvis. diametrisgan gansxvavebuli nebismieri ABB qordisTvis AB<AO+OB, anu AB<2r. e. i. diametri udidesi sigrZis qordaa. gaamaxvileT yuradReba, rom ori radiusiT da wrewiris rkaliT SemosazRvruli wris nebismieri nawili seqtoria. seqtorebia suraTze Seferadebuli nawilebic da darCenili nawilebic. moswavleebi xedaven dafaze an kedelze gamosaxul geometriul figurebs da amoicnoben maT Soris wrewirebs.

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vTqvaT, moswavleebi da maswavlebeli skolis ezoSi arian. sad unda dadgnen moswavleebi, rom yoveli maTgani maswavleblisgan Tanabrad iyos daSorebuli? maswavlebeli aRmoCndeba am wrewiris centrSi. davxazoT dafaze ramdenime wrewiri da moswavleebma moniSnon yoveli maTganis centri. Semdeg daStrixon/SeRebon wrewiriT SemosazRvruli sibrtyis nawili _ es wrea (wrewirTan erTad). visargebloT am suraTebiT da gavavarjiSoT moswavleebi wrewiris wertilebis, mis SigniT, mis gareT mdebare wertilebis miTiTebaSi (klasSi amosaxsneli amocana 10 ) da mxolod amis Semdeg romelime moswavles vTxovoT dafaze gamosaxuli wrewiris romelime wertili SeaerTos centrTan monakveTiT. es wrewiris radiusia. ra geometriuli figuraa radiusi? (monakveTi). aqve mivuTiToT,M rom aRniSvnaTa gamartivebis mizniT radiusis sigrZesac radiuss vuwodebT xolme (rogorc es gvqonda monakveTebis SemTxvevaSi). radiusis gamosaxvaSi da miTiTebaSi gawafvis Semdeg sxva moswavlem moniSnos wrewirze sami wertili. nebismieri oris SeerTebiT miRebuli monakveTi qordaa. (orze meti wertilis moniSvna imitom vTxoveT, rom ori wertilis moniSvnis SemTxvevaSi SeiZleba qorda diametrs daemTxves da am etapze gamoiwvios gaugebroba). moswavleebi ageben qordebs da adareben maT sigrZeebs. romelia yvelaze grZeli? romelic centrze gadis. mas diametri ewodeba. diametri qordaa. amis Semdeg gavavarjiSoT radiusiT diametris da diametriT radiusis gamoTvlaSi (amocanebi 7 , 8 , 12 ). dafaze gamosaxul wrewirze moniSnon wertilebi A, B da C. gaamuqon A da B wertilebs Soris wrewiris erT-erTi nawili, es wrewiris rkalia. wrewiris darCenili nawilic rkalia. SevTavazoT, dafaze gamosaxul wrewirze gaavlon ori radiusi OA da OB, SeRebon/daStrixon am radiusebiT da AB rkaliT SemosazRvruli sibrtyis nawili _ es seqtoria. ixseneben seqtoris msgavss stadionze, cirkSi, arenaze. wrewirSi Caxazuli mravalkuTxedebis gasacnobad SeiZleba parag­ rafis teqstis bolos warmodgenili suraTebi da saSinao davalebis 6 savarjiSos msgavsi suraTebi gamoviyenoT. amis Semdeg moswavleebTan erToblivi ZalisxmeviT vcdilobT 1 - 6 kiTxvebze pasuxebis etapobrivad Camoyalibebas. SeiZleba davaxmaroT kiTxvebi: • wrewiris wertilebi romeli wertilidan arian tolad daSorebuli? • romeli ori wertilis SeerTebiT miiReba radiusi? • ra geometriuli figuraa radiusi? • wrewirSi ramdeni radiusis gavleba SeiZleba? • romeli ori wertilis SeerTebiT miiReba qorda? • ra figurebiT aris SemosazRvruli seqtori? • ras gamoiyenebdiT wrewiris modelis Sesaqmnelad, mavTuls, Tu qaRaldis furcels? ras gamoiyenebdiT wris modelis Sesaqmnelad? da a. S. aqtivobis gafarToebis mizniT moswavleebi iyeneben miRebul codnas praqtikuli amocanebis amoxsnisas (amocana 15 ).

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aqtivobis Tema yvela moswavlisTvis iolad aRsaqmelia. pedagogma unda gamoiyenos es SesaZlebloba da yvela moswavle aqtiurad Caabas ganxilvaSi. pedagogis yoveli mimarTva moswavlisadmi ar unda iqces dialogad. aqtivobis Sefaseba. am aqtivobis Sedegad moswavle gaecnoba kidev erT geometriul figuras _ wrewirs da mis elementebs. gaifarToebs geometriuli aRqmis unars, rac safuZvels uqmnis momavalSi geometriuli debulebebis damtkicebis, geometriuli amocanebis amoxsnis SeswavlisTvis. aqtivobis dasasruls CainiSneT SeniSnuli xarvezebi, iqve dasaxeT misi gamosworebis gzebi. CainiSneT is moswavleebi da maTTan dakavSirebuli sakiTxebi, romelTa gaTvaliswineba mogiwevT momavalSi, klasSi moswavleebTan erTad SeajameT aqtivobis Sedegebi. moismineT Tavad mos­ wavleTa Sefasebebi da SeniSvnebi. 1

wrewiri aris erTi wertilidan _ centridan _ erTi da imave manZiliT daSorebul wertilTa (sibrtyis wertilTa) simravle. 2

wrewiris radiusi aris monakveTi, romelic wrewiris romelime wertils centrTan aerTebs. radiusi vuwodeT am monakveTis sigrZesac _ es amartivebs masalis gadmocemas, ar iwvevs gaugebrobas. amitomac aris damkvidrebuli maTematikur literaturaSi. 3

qorda wrewiris nebismieri ori wertilis SemaerTebeli monakveTia; diametri centrze gamavali qordaa. wre aris wrewiriT SemosazRvruli sibrtyis nawili _ r-radiusiani wre aris sibrtyis im wertilTa simravle, romelTagan mocemul wertilamde _ centramde _ manZili r-s ar aRemateba, wrewiris/sazRvris wertilebic wres ekuTvnis. 4

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wris seqtori SemosazRvrulia ori radiusiT da wrewiris rkaliT.

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mravalkuTxedi Caxazulia wrewirSi, Tu misi yoveli wvero wrewirze Zevs.

cxadia, am kiTxvebze pasuxisas ar moveliT cnebebis mkacrad Camoyalibebas. amas unda hqondes cnebis aRweris saxe, moswavleebi iyeneben teqsts da suraTebs. saSinao davalebis 8

1

- 6

savarjiSoebiT isini ufro azusteben am cnebebis Sinaarss.

savarjiSoSi kidev erTxel vamaxvilebT yuradRebas: diametri aris qorda.

9

wrewirSi Caxazuli samkuTxedis gverdebidan mxolod erTi SeiZleba iyos diametri, danarCeni ori diametrisgan gansxvavebuli qordebia _ naklebia diametrze. am daskvnamde moswavle TvalsaCinoebiT _ suraTebiT _ midis. is ar iyenebs geometriis aqsiomebs (magaliTad, or wertilze erTaderTi wrfe gaivleba, amitom wrewiris nebismier wertilze erTaderTi diametri gaivleba). klasSi moswavleTa aqtiuri monawileobiT ganixileba 1 - 11 amocanebi, 1 - 6 amocanebi ki saSinao davalebisTvis aris gankuTvnili. gaviTvaliswinoT, rom zogjer moswavleebs sWirdebaT `amosunTqvis~ saSualebis micema â&#x20AC;&#x201C; mcire moculobis davalebis micema, yuradRebis gamaxvileba saxelmZRvanelos teqstis mixedviT skolaSi Catarebuli msjelobis garCevaze. Cven mier mowodebuli masala Tqven unda moargoT Tqveni moswavleebis individualur SesaZleblobebs.

154


85-e gakveTili

Tema: wrewiri, wre. mizani: moswavle gaiwafos wrewirisa da wris elementebis dasaxelebasa da maT Soris mimarTebebis aRweraSi. winapirobebi: mravalkuTxedis saxeebze warmodgenebi resursebi: rveuli, saxelmZRvanelo, dafa 1

- 6 amocanebSi moswavleebis mier pasuxebis SerCevis Semowmebas didi dro ar dasWirdeba (10-15 wT). gakveTilze vagrZelebT wrewirTan dakavSirebuli sakiTxebis ganxilvas 12 - 16 amocanebis gamoyenebiT. 12 amocaniT vamowmebT SeuZliaT Tu ara moswavleebs yvela qordidan diametris gamoyofa, ra niSniT gamoiyofa diametri. 13 da 14 amocanebis analogiuri amocanebi moswavleebs adrec aqvT amoxsnili, isini aviTareben kanonzomierebebis aRmoCenis da ganvrcobis unarebs (wrewirebis gamoyenebiT). 15 amocana SeiZleba CaiTvalos praqtikul amocanad, romelic iyenebs wrewiris maxasiaTebel Tvisebas. figura suraTzea warmodgenili, romelic Sedgeba marTkuTxedisa da ori toli seqtorisgan (naxevarwrisgan). 16 am savarjiSoze muSaobisas moswavle iwafeba or simravles Soris Sesabamisobis damyarebaSi da ufro momzadebuli xvdeba maRal klasebSi simravleebis Seswavlas. saSinao davalebis amocanebiT xdeba saklaso muSaobisas SeZenili codnisa da unarebis gamtkiceba da ganviTareba. saSinao davalebad moswavleebs vaZlevT

7

- 12 amocanebs.

86-e gakveTili

Tema: wrewiri. wre mizani: miTiTebul TemasTan dakavSirebuli sakiTxebis gameoreba, codnis ganmtkiceba, wrewirTan dakavSirebul sayofacxovrebo sakiTxebze warmodgenebis ganviTareba. winapirobebi: wrewiri, wre da maTi elementebi. resursebi: rveuli, saxelmZRvanelo, dafa. saSinao davalebis amocanebi moswavleTa geometriul warmodgenebs, Siga, sazRvris da gare wertilebis codnas amowmebs. 11

amocanaSi dasmuli kiTxva maTematikuri sizustis TvalsazrisiT SeiZleba arakoreqtuli iyos. Tumca, gasagebia, rom is wrewiria ufro `didi~, romlis radiusic ufro didia. saSinao davalebis amocanebis garCevis Semdeg SeiZleba amovxsnaT ramdenime amocana am Tavis damatebiTi amocanebidan. 5 - 7 . 7 amocaniT iwyeba wrewirebis urTierTganlagebis SemTxvevebis ganxilva, Tumca, am amocanis amoxsniT SeiZleba CavTvaloT,

155


rom Semamzadebel samuSaos vatarebT aRniSnuli Temis dawvrilebiTi ganxilvisaTvis. anu, esaa aRniSnuli Temis Seswavlis sawyisi faza – `winaswar~. am process waadgeba damatebiTi amocanebis 8 - 10 davalebebic, romlebiTac moswavle ufro gaiwafeba naxazebis agebaSi. es amocanebi SeiZleba saSinao davalebadac gamoviyenoT. Semdegi mecadineobisaTvis SeiZleba SevTavazoT moswavleebs proeqti – `wrewiri da wre Cven garSemo~. amisTvis maT SeiZleba gamoiyenon masala, romelic warmodgenilia rubrikiT `sxvadasxva~. moswavleebma SeiZleba Seavson kidec es masala da warmoadginon Semdeg gakveTilze. Tumca, aq mocemuli masalis wakiTxva, garCeva da gagebac saswavlo procesis mniSvnelovan aspeqtad gveCveneba.

§3. ori wrewiris urTierTganlagebis SemTxvevebi 87-e gakveTili

Tema: ori wrewiris urTierTganlagebis SemTxvevebi. mizani: moswavlem SeZlos sibrtyeze wrewirebis urTierTganlagebis Sesaxeb msje­ loba maT centrebs Soris manZilisa da radiusebis gamoyenebiT da amiT gaiRrmaos geometriuli warmodgenebi. winapirobebi: wrewirisa da wris Sesaxeb mkafio warmodgenebi. resursebi: rveuli, saxelmZRvanelo, dafa, suraTebi wrewirebis urTierTganlagebis SemTxvevebis warmodgeniT. gakveTils viwyebT moswavleebis mier micemuli proeqtis prezentaciebiT, vmsje­ lobT maT mier mopovebuli masalis Sesaxeb, yoveldRiur cxovrebaSi wrewiris Tvisebebis gamoyenebaze. am process SeiZleba 15-20 wT davuTmoT. Semdeg gadavdivarT axali Temis ganxilvaze, vcdilobT gakveTilze dasaxuli miznis miRwevas. gakveTilze SeiZleba gamoviyenoT plakatebi, romlebzec gamosaxulia wrewirebis urTierTganlagebis SemTxvevebi. es plakatebi maswavlebelma saxelmZRvaneloSi mocemuli suraTebis mixedviT SeiZleba gaakeTos. isini amocanebis amoxsnisasac gamogvadgeba. paragrafis teqstSi warmodgenili TiToeuli SemTxveva sajarod unda ganvixiloT gakveTilze: • roca erTi wrewiri meoris gareTaa da wrewirebs saerTo wertili ara aqvs, maSin OO1>r1+r2 (am Sedegs moswavleebi martivad miiReben). • roca wrewirebi erTmaneTs garedan exeba, maSin OO1=r1+r2. • roca wrewirebi erTmaneTs kveTs, maSin OO1<r1+r2. (Cven vadasturebT moswavleTa intuiciur mosazrebas). • roca erTi wrewiri meores exeba Signidan, maSin OO1=r1–r2, r1>r2. • roca wrewirebs saerTo wertili ara aqvs da erTi wrewiri meoris SigniTaa, maSin OO1<r1–r2, sadac r1>r2. • koncentruli wrewirebiT Seqmnili rgolis siganea r1-r2, sadac r1>r2. SeamCnevT, rom yovel amocanas saSinao davalebidan axlavs naxazi. klasSi zogierTi amocanis amoxsnisas amocanis pirobis suraTiT warmodgenaSi maswavleblisa da

156


moswavleTa erToblivi Zalisxmeva igulisxmeba, amitom es amocanebi suraTebis gareSea mocemuli. maTi amoxsnisas moswavles geometriuli amocanis pirobis mixedviT naxazis agebis unari uviTardeba. klasSi SeiZleba SemovifargloT mxolod 1 - 5 amocanebis ganxilviT. yvela SemTxvevis dawvrilebiT aRwera Tavdapirvelad suraTebis mixedviT, Semdeg – dafaze, did dros moiTxovs. saSinao davalebad vaZlevT 1 - 7 amocanebs.

88-e gakveTili

Tema: wrewirebis urTierTganlageba. mizani: moswavle gaiwafos ori wrewiris urTierTganlagebis yvela SemTxvevis ganxilvaSi, urTierTganlagebis Sesaxeb daskvnebis gakeTebaSi (centrebs Soris manZilisa da radiusebis gamoyenebiT). winapirobebi: wrewirisa da wris Sesaxeb mkafio warmodgenebi. resursebi: rveuli, saxelmZRvanelo, dafa. davalebis SemowmebasTan erTad vatarebT moswavleTa gamokiTxvas wrewirebis urTierTganlagebis Sesaxeb. – vTqvaT wrewirebs saerTo wertilebi ara aqvs da erTi wrewiris wertilebi meore wrewiris gareTaa. gamosaxeT es SemTxveva dafaze da dawereT Tanafardoba centrebs Soris manZilsa da radiusebis jams Soris, romelia meti? – ras niSnavs, rom ori wrewiri garedan exeba erTmaneTs? SeadareT am SemTxvevaSi centrebs Soris manZili radiusebis jams. – vTqvaT, wrewirebi erTmaneTs kveTs; SeadareT centrebs Soris manZili da radiusebis jami, gamosaxeT dafaze. – vTqvaT, wrewirebs saerTo centri aqvs. SeadareT centrebs Soris manZili da radiusebis sxvaoba. – vTqvaT, erTi wrewiri meoris SigniT aris, magram maT sxvadasxva centri aqvs. SeadareT centrebs Soris manZili da radiusebis sxvaoba. – ras ewodeba koncentruli wrewirebi? risi tolia maT mier Sedgenili rgolis sigane? am kiTxvebis gamoyenebiT SeiZleba Sefasdes gakveTilebze dasaxuli miznebis miRwevis xarisxi. gakveTilze vxsniT 6 - 12 amocanebs. pirobebis TvalsaCino warmodgenebi gvexmareba yvela am amocanis kiTxvebze pasuxebis gacemaSi. a) DB=AB–AD=12 sm–8 sm=4 sm. 7 b) CD=CB–DB=5 sm–4 sm=1 sm. am amocanis amoxsnisas mniSvnelovania romelime 8 moswavlem sajarod warmoadginos amocanis piroba suraTiT (Seasrulos naxazi). amoxsnis dawyebamde vkiTxoT moswavleebs suraTze warmodgenili monakveTebidan romlebis sigrZea cnobili? OA=AP=PB=OB=4,3 dm, OP=6 dm. amis Semdeg isini

157


pasuxoben amocanis kiTxvebs: a) OAPB-s perimetria 4,3⋅4=17,2 dm; b) OAP da OBP samkuTxedebs toli gverdebi aqvs, TiToeuli maTganis perimetria 4,3+4,3+6=14,6 dm. amocanis amoxsnas kvlav pirobis TvalsaCino 10 warmodgeniT viwyebT. moswavleebi (erTmaneTis daxma­ rebiT) mixvdebian, rom es wrewirebi ar ikveTeba, erTerT wrewirze A wertilis SerCevis Semdeg vasaxelebT miRebuli OAO1 samkuTxedis yoveli gverdis sigrZes da amocanis pasuxs; OO1=4⋅2,3=9,2; perimetri tolia 2,3+9,2+9,5=21 dm-is. 12 paragrafis teqstSi warmodgenilia koncentruli wre­ wirebi. moswavle unda mixvdes: radgan AB-s sigrZe rgolis siganeze _ 0,9 dm-ze metia, amitom O, A da B wertilebi erT wrfeze ar iqneba ganlagebuli. mcire wrewiris radiusis gamoTvlis Semdeg (4,5–0,9=3,6 dm) SesaZlebelia OAB samkuTxedis perimetris gamoTvlac: 3,6+4,5+3=11,1 dm. yvela es amocana moswavleTa aqtiuri CarTulobiT unda amoixsnas. saSinao davalebad vaZlevT 8 - 12 amocanebs da vagrZelebT moswavleebis mier yoveldRiur cxovrebaSi wrewirebis gamoyenebis magaliTebis moZiebas. Aamjerad, maT SeiZleba SevTavazoT moZraobis niSnebis moZieba da am niSnebSi geometriuli figurebis gamoyenebis nimuSebis warmodgena. moswavleebi advilad miagneben internetSi am niSnebs da maT klasifikacias, romelic garkveuli kanonzomierebiT xasiaTdeba. aris SemTxvevebi, roca marTkuTxedebi gamoiyeneba da aris SemTxvevebi, roca gamoiyeneba wrewirebi.F zogierT niSanSi erTi da imave figuris gamoyenebebic (magaliTad, amkrZalavi, mimTiTebeli) gansxvavebulia (magaliTad, feriT). G

89-e gakveTili (Semajamebeli)

Tema: geometriuli figurebi. mizani: geometriuli figurebis Sesaxeb codnis gameoreba da ganmtkiceba. winapirobebi: samkuTxedis, oTxkuTxedebis klasifikaciis, wrewirisa da wris Sesaxeb mkafio warmodgenebi. resursebi: saxelmZRvanelo, rveuli, dafa. saSinao davalebis Semowmebisas SeiZleba gavamaxviloT yuradReba 11 amocanaze. pirobiT, d2=d1+3,4. amasTanave, d1+d2=10 _ didi wrewiris diametri SigniT mdebare wrewirebis diametrebis jamis tolia. miviRebT, rom d1+d1+3,4=10, saidanac d1=3,3 dm, d2=6,7 dm. pasuxi: didi wrewiris SigniT mdebare wrewirebis radiusebia 1,65 dm da 3,35 dm. aqve SeiZleba moswavleebs davusvaT aseTi amocana: vTqvaT, me11 amocanaSi ar aris moTxovnili, rom samive wrewiris centri erT wrfezea. rogori ganlageba SeiZleba hqondes am wrewirebs? saklaso ganxilvis Sedegi uTuod iqneba damakmayofilebeli. wrewirebs SeiZleba hqondes aseTi ganlageba (cxadia, Tu damatebiTi piroba ar SemovitaneT, wrewirebis radiusebs am SemTxvevaSi ver vipoviT).

158


saSinao davalebis Semowmebis Semdeg (15 wT) gadavdivarT geometriuli figurebis Tvisebebis Sesaxeb Seswavlili masalis gameorebaze. SeiZleba Cven TviTon SevadginoT amocanebi, roca mocemulia, magaliTad, marTkuTxedis perimetri, or mezobel gverds Soris raime Tanafardoba (erTi metia, magaliTad, meoreze 3-jer, an 30 sm-iT) da unda vipovoT gverdebi; roca mocemulia Tanafardoba mosazRvre kuTxeebs Soris da veZebT am kuTxeebs. wrewirebis Sesaxeb masala ki SeiZleba gavimeoroT Tavis damatebiTi amocanebiT 11 - 19 . gTavazobT miTiTebebs am amocanebis amosaxsnelad. 11 2r1=11,6 2r2=8,8 O1O2=r1+r2 O1O2=? O1O2=5,8+4,4=10,2 O1

O2

1

12 r1=12 2

r2=10 18 O1O2 gamoisaxeba mTeli ricxviT O1O2<23,2

O1

O2

O1O2=? O1O2>r1+r2 O1O2>12 48 +10 18 =22 56 . radgan O1O2<23,2, maSin pirobis gaTvaliswinebiT, O1O2=23. 13 2r=6,8 A

2r =5,2 r r1 1 O D E O1 OO =4 34 1 DE=? B

r=3,4, r1=2,6, r+r1=6 r+r1=OD+DE+DE+EO1=OO1+DE 6=4 34 +DE DE=1 14 (sm) r r1 >9 12 , gamoisaxeba mTeli ricxviT 14 OO 1

2r =12,5 1 O O1 2r2=10,8 OO1=?

OO1<r1+r2 OO1<6,25+5,4, OOI<11,65 OO1 SeiZleba iyos 10 sm an 11 sm

159


1 A 2r2=10,8 15 2r1=9 6

PAOBO =? O O1 1

P =2r1+2r2=9 16 +10 45 B AOBO 29 PAOBO =19 5+24 30 =19 30 (sm) 1

1

16

a) OO1=10 sm, r1=8sm, r2=2sm O1 O2 wrewirebi garedan exeba erTmaneTs, radgan r1+r2=OO1

b) OO1=4 dm, r1=11dm, r2=7 dm r1–r2=OO1 wrewirebi erTmaneTs Signidan exeba

O1 O2

g) O O O1=12 sm, r1=5sm, r2=3 sm r +r2<OO1 O1 O2 1 wrewirebs saerTo wertili ara aqvs. es amocana zustad pasuxobs standartSi miTiTebul misaRwev Sedegs: `akeTebs daskvnas sibrtyeze wrewirebis urTierTganlagebis Sesaxeb, maT centrebs Soris manZilisa da radiusebis gamoiyenebiT~. amitom mas gansakuTrebuli yuradReba miaqcieT. 4

17 r1= 7 r2

AB=12 sm r1=? r2=? O

A

B

r2–r1=12, 37 r2=12,

r2– 47 r2=12 r2=28 (sm),

r1=16 (sm). 18 r1=5,4 sm

r2=3,6 sm r3=? vipovoT rgolis sigane; r1–r2=1,8 r3=1,8:2 r3=0,9 (sm)

160

O O1


19

OO1=71,2 sm r1=3r2

2r1=? 2r2=?

r1+r2=71,2 3r2+r2=71,2 4r2=71,2, 2r1=106,8 sm

O

O1

r2=17,8, r1=53,4 2r2=35,6 sm

moswavleebs gamovucxadebT, rom Semdeg mecadineobaze Catardeba Semajamebeli wera Semdegi Temebis mixedviT: samkuTxedi, oTxkuTxedis saxeebi, mravalkuTxedis perimetri, wrewiri, wre, wrewirebis urTierTganlageba. O 90-e gakveTili

Tema: geometriuli figurebi Semajamebeli wera SearCieT swori pasuxi: 1. tolferda samkuTxedis perimetri 8,4 dm-ia, fuZis sigrZe ferdis sigrZeze 3-jer naklebia. am samkuTxedis fuZe aris a) 2,8 dm b) 1,4 dm g) 1,2 dm d) 3,6 dm. 2. mosazRvre metia umciresze a) 600-iT

kuTxeebidan erTi meoreze 3-jer metia. am kuTxeebidan udidesi b) 450-iT g) 300-iT d) 900-iT.

3. Tu mosazRvre kuTxeebi tolia, maSin TiToeuli maTgani aris a) 450 b) 600 g) 900 d) 1800. 4. marTkuTxedis nebismieri sami kuTxis jami aris a)1800 b) 900 g) 2700 d) 3600. 5. suraTze O _ wrewiris centria, AOB samkuTxedis perimetri 28 sm-ia, AB qordis sigrZe _ 9,4 sm. wris radiusi aris a) 18,6 sm b) 9,3 sm g) 4,7 sm d) 14 sm. 6. ori toli wrewiris A da B centrebs Soris manZili 24 sm-ia, CD=6,2 sm. am wrewirebidan TiToe­ ulis radiusi aris a) 8,9 sm b) 17,8 sm g) 9,8 sm d) 3,1 sm. amoxseniT amocanebi 7. samkuTxedis gverdebis sigrZeebi sami momdevno naturaluri ricxviT gamoisaxeba. samkuTxedis perimetria 42 sm. ipoveT gverdebi.

161


8. suraTze ori wrewiri exeba erTmaneTs da did wrewirs. patara wrewirebidan TiToeulis radiusi 3 sm-ia, didi wrewiris radiusia 15 sm. ipoveT im samkuTxedis gverdebi, romlis wveroebi am wrewirebis centrebia. pasuxebi da miTiTebebi 1

2

3

4

5

6

g

d

g

g

b

a

1. SeiZleba fuZe x-iT aRniSnon, ferdebidan TiToeuli _ 3x-iT. maSin 7x=8,4 dm, x=1,2 dm. fuZe 1,2 dm-ia. 2. SeiZleba mosazRvre kuTxeebi x da 3x-iT aRniSnon, 4x=1800, x=450. es kuTxeebia 1350 da 450. udidesi metia umciresze 1350–450=900-iT metia. an asec: 3x–x=2x; 2x=2⋅450=900. 5. OA+OB=28–9,4=18,6 sm, OA=OB=18,6:2=9,3 sm. 6. pirobiT, AB=24 sm, CD=6,2 sm. maSin radiusi iqneba (24–6,2):2=17,8:2=8,9 sm. 7. SeiZleba samkuTxedis gverdebi x, x+1 da x+2-iT aRniSnon da Seadginon gantoleba x+x+1+x+2=42, saidanac x=13. samkuTxedis gverdebia 13 sm, 14 sm, 15 sm. SeiZleba umciresi gverdis gamosaTvlelad gamoTvalon ricxviTi gamosaxulebis mniSvneloba: (42–3):3=13 (sm) an 42:3–1=13 sm. an pasuxs miagnon sinjvis meTodiT. 8. pirobiT, OA=15 sm, O1A=3 sm, O1O2O samkuTxedis gverdebia OO1=OO2=15–3=12 sm, O1O2=2⋅3=6 sm. Sefasebis sqema: testuri amocanebidan TiToeulis swori amoxsna Sefasdes 1 quliT. Sejamebis rubrika #7 amocanisTvis

162

0 qula

naSromSi araa gadadgmuli racionaluri nabiji

0,5 qula

samkuTxedis gverdebisTvis SemoiRoO adekvaturi aRniSvnebi, magaliTad x, x+1, x+2.

1 qula

Seadgina gantoleba amocanis amosaxsnelad

1,5 qula

amoxsna gantoleba mcire xarveziT

2 qula

amocana amoxsna srulyofilad


Sejamebis rubrika #8 amocanisaTvis 0 qula

naSromSi araa gadadgmuli racionaluri nabiji

0,5 qula

SeamCnia, rom OA=15 sm, an OO1=6 sm

1 qula

gamoTvala OO1 da OO2 monakveTebis sigrZeebi

1,5 qula

dawera OO1O2 samkuTxedis perimetris gamosaTvleli gamosaxuleba mcire xarveziT, an dawera sworad da ver gamoTvala mniSvneloba

2 qula

amocana amoxsnilia srulyofilad

cxadia, maswavlebelma moswavlis mier amocanis arastandartulad amoxsnis etapebs unda miusadagos sqemis safexurebi (Sefasebis rubrikebi).

91-e gakveTili

Tema: geometriuli figurebi. mizani: Semajamebeli weris Sedegebis gacnoba, wrewirebis urTierTganlagebis SemTxvevebis aRweris Sesaxeb codnis ganmtkiceba. winapirobebi: geometriuli figurebi – wrewiri, wre, paralelogramis saxeebi, samkuTxedi. resursebi: rveuli, saxelmZRvanelo, dafa, saxazavi. gakveTils mTlianad vuTmobT Semajamebeli weris Sedegebis gacnobas da wrewirebis urTierTganlagebis SemTxvevebTan dakavSirebuli amocanebis amoxsnas, codnis ganmtkicebas. SeiZleba gamoviyenoT Tavis damatebiTi amocanebi 20 - 28 . am amocanebidan nawils klasSi vxsniT, moswavleTa aqtiuri CarTulobiT, danarCenebs saSinao davalebad vaZlevT moswavleebs. gTavazobT miTiTebebs amocanaTa amosaxsnelad. 20 r1+4=2r2 OO1=6,2 r=? r1=? O O 1

r1=2r2–4 radgan wrewirebi exeba garedan, maTi radiusebis jamia OO1. 2r2–4+r2=6,2 3r2=10,2 r2=3,4 r1=6,8–4=2,8. 4

21 2r1= 15 ⋅2r2

OO1=33 sm O O1 r =? r =? 1

2

4  ⋅2r , anu r = 4 r 2r1= 15 2 1 15 2 radgan wrewirebi Signidan exeba erTmaneTs, radiusebis sxvaobaa OO1.

163


4 ⋅r =33 r2– 15 2 4 (1– 15 )r2=33 11 15 r2=33 r2=45 r1=12. 22 r1=r2=r3–2

PO O O =46 1

2

O1

3

r1=? r2=? r3=?

r1

r1+r2+r2+r3+r1+r3=46 2r1+2r2+2r3=46 r1+r2+r3=23 r3=r1+2 da r2=r1 2r1+r1+2=23 3r1=21 r1=7, r2=7. r3=9. diametrebia 14 sm, 14 sm da 18 sm.

O3 r2

r3

O2

23 2r3=r1+32,4

r1=?

2r3=4r1, amasTanave O1 O2 2r3=r1+32,4, saidanac 4r1=r1+32,4 3r1=32,4 r1=10,8 (sm) zogierTi moswavle suraTidanac SeamCnevs, rom didi wrewiris diametris sigrZe pataris radiusze 3 cali patara radiusiT metia da asoiTi gamosaxulebis Sedgenis gareSe miiRebs pasuxs: 32,4:3=10,8. B

24 r1=r2

A O1

r3=8,2

O2

PO O O =? 1

2

3

PO O O =r3–r1+r3–r1+2r1=2r3=16,4. zogierTi moswavle suraTidan SeamCnevs, rom O1O2O3 samkuTxedis perimetris naxevaria O1A+O1O3, romelic didi wrewiris radiuss udris: O1A+O1O3=O1B+O1O3=O3B. 1

2

O3

3

B1

25 A wertilze gavavloT radiusi, maSin A A wertilTan uax-

loesi iqneba B1 werili

164

B

radiusis bolo B wertili, marTlac, aviRoT sxva A

OA+AB1>RO, OA+AB=R e. i. AB1>AB AB=OB–OA AB=12,8:2–1,4=6,4–1,4=5 (sm).

O


26 r1=12,4 r2=4,2 O1O2<18 O1O2 santimetrebis mTeli ricxviT gamoisaxeba. jer vaCvenoT, rom AB<CD, anu A da B wrewire­ bis uaxloesi wertilebia: R1+AB+R2<R1+CD+R2 e. i. AB<CD O1O2>r1+r2 O1O2>16,8 O1O2<18 e. i. O1O2=17.

C D O1

A

B

O2

27 MN=8

r1=1,2r2

O

M

N

r1=? r2=? gaviTvaliswinoT, rom MN=r1–r2. 28 AC=18 sm CD=8,6 sm DB=4,2 sm

r1=?

r2=?

A

C O1

D O

B

r2=8,6:2=4,3 2r1=AC+CD+DB=30,8 r1=15,4 (sm)

sasurvelia, rom klasSi amoixsnas 22 , 25 da 27 amocanebi.

§4. RerZuli simetria 92-e gakveTili

Tema: RerZuli simetria mizani: moswavlem SeZlos mocemuli brtyeli figuris simetriuli figuris ageba, figuris simetriis RerZis povna da pasuxis dasabuTeba gadakecviT, sarkis gamoyenebiT (maT. VI. 8.). winapirobebi: geometriuli figurebi – wrewiri, wre, paralelogrami da misi sa­ xeebi. resursebi: qaRaldis ujriani furclebi, rveuli, saxelmZRvanelo. saSinao davalebis Semowmebis Semdeg (10-15 wT) gadavdivarT axali Temis ganxilvaze. gakveTilze dasaxuli miznis miRwevis saqmeSi sasurvelia, rom gamoviyenoT TvalsaCino masala – ujriani rveulebi. maTi saSualebiT warmovadginoT wrfis mimarT simetriuli figurebi, SevamowmoT Cveni varaudi gadakecvis gamoyenebiT, SeiZleba sarkec gamoviy-

165


enoT ise, rogorc es teqstSia miTiTebuli. SeiZleba gamovsaxoT ujrian rveulze figurebi (kvadrati, rombi, marTkuTxedi) da gadakecvis gamoyenebiT SevamowmoT figuraTa simetriuloba. RerZul simetrias moswavleebs suraTebiT, aRweriT vacnobT. dasabuTebis gareSe _ suraTebze dakvirvebis safuZvelze veubnebiT, rom simetriuli figurebi tolia, am faqtis dasabuTebas isini maRal klasebSi SeZleben, rodesac ecodinebaT samkuTxedebis tolobis niSnebi. maRal klasebSi daasabuTeben, rom RerZuli simetria sibrtyis gardaqmnaa, romlis drosac nebismier or wertils Soris manZili ar icvleba. sibrtyis aseT gardaqmnas moZraoba ewodeba. paragrafis teqstis 1 magaliTSi SemoTavazebuli suraTis mixedviT moswavle aRwers simetriuli figuris agebis algoriTms: raime A Awertilis simetriuli A′ wertili mocemuli l RerZis mimarT am A wertilidan l RerZisadmi gavlebul marTobul wrfezea. amasTan, mocemuli A wertili da misi simetriuli A′ Awertili l-dan toli manZilebiTaa daSorebuli. magaliT 2-Si ki piriqiT, simetriuli wertilebis mixedviT simetriis RerZis povnis algoriTmis aRweras vTavazobT: simetriis RerZi aris im monakveTis SuamarTobi, romlis boloebi simetriuli wertilebia. paragrafSi simetriuli figurebicaa warmodgenili erTi an ramdenime simetriis RerZiT. klasSi 2 savarjiSoSi kvadratis simetriis RerZebis daTvlisas sasurvelia moswavlem dafaze gamosaxos isini. maswavlebelma icis, rom kvadrati marTkuTxedicaa da rombic, amitom kvadratis simetriis RerZebi gverdebis SuamarTobebicaa (rogorc marTkuTxedis) da diagonalebis Semcveli wrfeebic (rogorc rombis). 6 moswavleTa yuradRebas mivaqcevT im faqtze, rom simetriis RerZze `sibrtyis~ gadakecvisas yoveli wertili Tavis simetriuls emTxveva, simetriis RerZis yoveli wertili Tavisi Tavis simetriulia. 7 MN monakveTis Sua wertilze gavlebuli marTobi simetriis RerZia.mas­wav­ lebelma icis, rom es SuamarTobi wrewiris centrze gaivlis. moswavleebi am faqts kargi naxazis SemTxvevaSi SeamCneven. vTqvaT, A wertilze gavlebuli MN-is paraleluri qorda wrewirs A′ wertilSi kveTs. es qorda simetriis RerZis marTobulia da A′ Aaris A-s simetriuli am RerZis mimarT. 8 A da C, M da N wertilebi simetriulia BD simetriis RerZis mimarT. B wertili Tavisi Tavis simetriulia. moswavleebma ukve ician, rom raime RerZis mimarT simetriuli figurebi tolia. AMB da CNB simetriuli samkuTxedebia BD RerZis mimarT, e. i. isini tolia. 1 - 13 amocanebi or gakveTilze SegiklasSi amosaxsneli 1 - 12 da saSinao ZliaT gaanawiloT. 7 saSinao davalebis savarjiSoSi yuradRebas vamaxvilebT: b) SemTxvevaSi wrewiri Tavisi Tavis simetriuli, e. i. l wrfe wrewiris simetriis RerZia. g) SemTxvevaSi wrewiri da misi simetriuli figura ikveTeba. 10

da 11 savarjiSo yuradRebas amaxvilebs: Tu marTkuTxedi ar aris kvadrati, diagonalebi ar aris misi simetriis RerZebi. kvadratis simetriis RerZebi gverdebis SuamarTobebis garda misi diagonalebis Semcveli wrfeebicaa.

166


moswavle ixsenebs, rom simetriuli figurebi tolia da ABC samkuTxedis agebis gareSe daaskvnis, rom A′B′C′ samkuTxedis perimetric 9,2 sm-ia. 8 saklaso 10 , 11 da saSinao savarjiSoebSi Tu ar moviTxovdiT umciresi raodenobis ujrebis damatebas, mocemuli figuris simetriulamde Sevsebis uamravi SesaZlebloba iqneboda. magaliTad, moswavlem mocemuli figura jer marTkuTxedamde Seavsos da Semdeg am marTkuTxedis simetriuli aagos. 12

13 amocanaSi moswavleebs vTavazobT Tanamimdevrulad aagon figuraTa kom­ poziciebi. es maTTvis sruliad daZlevadi da saxaliso davalebaa. amavdroulad moswavleebs vuyalibebT pirvel warmodgenebs asaxvaTa kompoziciaze, rac maRal klasebSi sagangebod Seiswavleba.

93-e gakveTili

Tema: RerZuli simetria. mizani: RerZuli simetriis Sesaxeb amocanebis amoxsniT moswavle gaiwafos mocemuli wrfis mimarT figuris simetriuli figuris agebaSi. winapirobebi: paraleluri wrfeebi, sxivi, wertili, wrfe, monakveTi. resursebi: qaRaldis ujriani furclebi, feradi fanqrebi, saxazavi, fargali, rveuli, saxelmZRvanelo. saSinao davalebis amocanebis Sesaxeb Cven ukve visaubreT. davalebis Semowmebis (15-20 wT) Semdeg vagrZelebT muSaobas imave Temis irgvliv. viyenebT Tavis damatebiT amocanebs 29 - 37 . nawili am amocanebisa moswavleebs saSinao davalebad miecemaT wina gakveTilidan darCenil amocanebTan erTad. amocanebis amoxsnis dros, sasurvelia, gamoviyenoT TvalsaCino masala, ujriani qaRaldis furclebi. magaliTad, 29 - 34 amocanebis amoxsnebis Cveneba SeiZleba ujriani furclis gadakecvis gamoyenebiT; mocemuli figurisa da misi simetriulis gamosaxva sxvadasxva feris fanqrebis gamoyenebiT ufro TvalsaCino iqneba. SeiZleba simetriuli figuris agebis procesi ase aRvweroT: 29 ujriani furclis gamoyenebiT, A wertilze advilad gaivleba a wrfis marTo-

buli wrfe, Semdeg gadavzomavT gadakveTis O wertilidan wrfis meore mxares OA1 monakveTs. maSin A1 iqneba A-s simetriuli. 30 amocanaSi sakmarisia C da D wertilebis simetriuli wertilebis ageba da maTi SeerTeba monakveTiT. 31 a wrfeze aviRoT raime A wertili da avagoT misi simetriuli b wrfis mimarT, miviRebT A1 wertils. a wrfis C wertilis simetriuli Tavad C wertili iqneba. amrigad, A1C wrfe iqneba AC-s simetriuli b wrfis mimarT. sasurvelia, rom gadakecvis gamoyenebiTac davrwmundeT amaSi. 32 es amocana wina amocanis analogiurad amovxsnaT.

a

C A b

A1

33 avagebT C daD D wertilebis simetriul C1 da D1 wertilebs. gadakecvis saSuale-

biT davrwmundebiT agebis sisworeSi.

167


34 avagebT B wertilis simetriul B1 wertils, A-s simetriuli Tavad A A wertilia. miviRebT AB1 sxivs. gadakecviT davrwmundeT pasuxis sisworeSi. 35 sakmarisia vipovoTY a wrfis mimarT O centris simetriuli O1 wertili da O1

centris mqone imave radiusis wrewiri davxazoT. gadakecviT davrwmundebiT, rom ageba sworad CavatareT. 36 sakmarisia avagoT D, A da C wertilebis simetriuli D1, A1, C1 wertilebi.

miiReba kvadratis simetriuli A1BC1D1 kvadrati. gadakecvis gamoyenebiT davrwmundebiT agebis sisworeSi. 37 ujrediani furclis gamoyenebiT vpoulobT figurebis `kidura~ wertilebis si-

metriul wertilebs da vavlebT monakveTebs. gadakecviTac davrwmundebiT, rom ageba sworad aris Sesrulebuli. miiReba A, X, H asoebi. sakmarisia, ramdenime amocana kargad gavarCioT klasSi. amis Semdeg moswavleebs danarCenebis amoxsna saxlSi ar gauWirdebaT.

§5. paraleluri gadatana 94-e gakveTili

Tema: paraleluri gadatana. mizani: moswavlem SeZlos paraleluri gadatanis demonstrireba, mocemuli brtye­ li figuris (wertili, monakveTi, texili, mravalkuTxedi) paraleluri gadatana ise, rom miTiTebuli wertili gadavides sibrtyis miTiTebul wertilSi. winapirobebi: wrfis mimarT simetriis demonstrireba. resursebi: kuTxedi (marTkuTxa sakuTxedis formis saxazavi), saxazavi, feradi fanqrebi, kalami, rveuli, saxelmZRvanelo, dafa. Cven ukve gavarCieT amocanebi, romlebic saSinao davalebad hqondaT micemuli moswavleebs. davalebis Semowmebisas gavarkvioT, aiTvises Tu ara moswavleebma brtyeli figuris simetriuli figuris arsi, agebis xerxebi, simetriis RerZebis povna da dasabuTeba raime xerxiT. davalebis amocanebi aRniSnuli unarebis gamoyenebasTan iyo dakavSirebuli. davalebis Semowmebasa da analizs SeiZleba 15-20 wT movandomoT. amis Semdeg gadavdivarT axali masalis axsnaze, paraleluri gadatanis aRweras kuTxedisa da saxazavis saSualebiT warmovadgenT TvalsaCinod, viyenebT saxelmZRvaneloSi moyvanil teqsts – TvalsaCinod warmovadgenT samkuTxedis paralelur gadatanas mocemuli mimarTulebiT, mocemul manZilze. yuradReba gavamaxviloT imaze, rom paraleluri gadatanis mocema sakmarisia ori A da B wertiliT, romelTagan B miiReba A-sgan paraleluri gadataniT. maSin am paraleluri gadataniT nebismieri sxva D wertili miiReba raime C wertilisgan AB wrfis paraleluri `moZraobiT~ (C-dan imave mimarTulebiT, romliTac A-dan B miviReT) AB monakveTis tol manZilze. SeiZleba asec vTqvaT, moxdeba paraleluri gadatana `mimarTuli~ AB monakveTiT. dawvrilebiT ganvixiloT teqstSi warmodgenili orive amocana – monakveTis paraleluri gadatana, roca miiReba misi toli da `parale-

168


luri~ monakveTi da marTkuTxedis paraleluri gadatana. orive es amocana standartSi warmodgenili miznebis SesrulebasTan aris dakavSirebuli. moswavle gaecnoba paraleluri gadatanisas figuris anasaxis agebis algoriTms. paraleluri gadatana moZraobaa _ sibrtyis gardaqmna, roca or wertils Soris manZili am wertilebis saxeebs Soris manZilis tolia. paraleluri gadatanisas figura Tavis tol figuraSi gadadis. martivi magaliTebiT, TvalsaCinoebis farTo gamoyenebiT moswavleebi pirvel, Tumca sakmaod aratravialur warmodgenas iqmnian am asaxvaze, asaxvaTa kompoziciaze. klasSi 4 savarjiSos Sesrulebis dros moswavlem unda gaiazros, rom RerZis nebismieri wertili gadadis marjvniv 3,5 erTeuliT, anu nebismieri wertilis koordinats emateba 3,5. wertili, romlis koordinatia 2,5, gadadis wertilSi, romelic gamosaxavs 2,5+3,5=6-s. 8 am savarjiSoSi moswavle ori paraleluri gadatanis kompozicias asrulebs. maswavlebelma icis, rom es kompozicia (Tv  %Tu , sadac u=(0; – 4), v =(3; 0)) SeiZleba erTi paraleluri gadataniT (Ta , sadac a=(3; –4)) Seicvalos. am Canawerebs da terminebs moswavle­ebi momavalSi gaecnobian. 10 amocanac ori paraleluri gadatanis kompoziciis erTi paraleluri gadataniTN Secvlas exeba. AB monakveTiT paraleluri gadatana (yvela paraleluri gadatanis mimarTuleba saxelmZRvaneloSi miTiTebulia) DC monakveTiT paraleluri gadatanaa. e. i. AD da Semdeg AB monakveTiT paraleluri gadatana SeiZleba Seicvalos AC monakveTiT paraleluri gadataniT. SegaxsenebT, terminiT `kompozicia~ am etapze ar vsargeblobT. 5

paraleluri gadatanisas figuris zomebi ar icvleba.

paraleluri gadatanisas A wertili gadavida 4 erTeuliT marjvniv, B-c gadava 4 erTeuliT marjvniv _ wertilSi, romlis koordinatia 5+4=9. 10 ricxviTi sxivis gaswvriv marjvniv paraleluri gadatanisas M da N wertilebis koordinatebs emateba 5, anu A-s koordinati iqneba 6,2+5=11,2, B-s koordinati _ 9+5=14. 4 erTeuliT marcxniv moZraobisas _ akldeba 4. amitom C-s koordinati iqneba 6,2– 4=2,2, D-s koordinati _ 9–4=5. AB gadadis CD-Si 9 erTeuliT marcxniv paraleluri gadataniT. 11 am savarjiSoSi moswavle ori paraleluri gadatanis kompozicias erTi para­ leluri gadataniT cvlis mimarTuli monakveTis miTiTebiT, rac momavalSi veqtoris, veqtorebis Sekrebis Seswavlis propedevtikaa. amjerad ki moswavleebi Tvals miadevneben erTi raime wertilis gada­ adgilebas am ori Tanamimdevruli gardaqmnis Sedegad. es savsebiT gansa­zRvravs sawyisi figuridan saboloo figuraze gadasvlas. 8

saSinao davalebas virCevT am rubrikiT SemoTavazebuli

1

- 11

amocanebidan.

169


95-e gakveTili

Tema: paraleluri gadatana. mizani: moswavle gaiwafos paraleluri gadatanis demonstrirebaSi, paraleluri gadatanisas mocemuli figuris anasaxis povnaSi. winapirobebi: simetria wrfis mimarT, paraleluri wrfeebi. resursebi: feradi fanqrebi, kalami, saxazavi, kuTxedi, fargali, rveuli, saxelmZRvanelo, ujriani furclebi. Cven ukve visaubreT saSinao davalebis amocanebis Sesaxeb. davalebis Semowmebas SeiZleba movandomoT 15-20 wT. amis Semdeg vxsniT amocanebs imave Temaze Tavis damatebiTi 38 - 42 amocanebidan. yvela am amocanis amoxsna mimdinareobs ujriani furclebis gamoyenebiT. SeiZleba am amocanebis amoxsna moswavleebsac davavaloT, rogorc wesi, isini ujrian rveulebs iyeneben, SeiZleba Cveni daxmarebac gaxdes saWiro. paraleluri gadatanis praqtikuli ganxorcielebisas viyenebT am gardaqmnis Tvisebebs: yoveli monakveTi mis paralelur da tol monakveTSi gadadis, R-radiusiani wrewiri aisaxeba R-radiusian wrewirSi, romlis centri miiReba paraleluri gadataniT mocemuli wrewiris centrisgan. 40 amocanis Sesrulebisas ki saWiro iqneba im kvadratis ageba (ujrian furclebze

es ar aris Zneli), romlis A1 wvero O wertilSia, xolo diagonalebis gadatanis O1 wertili daemTxveva C wertils, moxdeba paraleluri gadatana AO `mimarTuli monakveTiT~ (sawyisi A da bolo O wertilis mqone monakveTiT). analogiuria 41 amocanac. 42 amocanaSi saWiro iqneba imave radiusis wrewiris Semoxazva, romlis centri O1 wertilia. es wrewirebi simetriulia im wrfis mimarT, romelic OO1 monakveTis Sua wertilze da am monakveTis marTobulad gadis. am marTobis gavleba SeiZleba kuTxedis gamoyenebiT. saSinao davalebad moswavleebs SevurCevT amocanebs `saSinaos~ darCenili amocanebidan da IV Tavis damatebiTi 43 - 50 amocanebidan. am amocanebis amoxsniT moswavleebi gaiRrmaveben da ganimtkiceben codnas orive geometriuli gardaqmnis Sesaxeb â&#x20AC;&#x201C; RerZuli simetria, paraleluri gadatana.

96-e gakveTili

Tema: RerZuli simetria, paraleluri gadatana. mizani: RerZul simetriasa da paralelur gadatanasTan dakavSirebuli sakiTxebis gameoreba, codnis ganmtkiceba. resursebi: kalami, rveuli, saxelmZRvanelo. vamowmebT saSinao davalebas â&#x20AC;&#x201C; rogoraa naxazebi Sesrulebuli, simetriuli figurebi miTiTebuli, simetriis RerZebi aRniSnuli, paraleluri gadatana Sesrulebuli? qvemoT warmogidgenT miTiTebebs da pasuxebs. 43 am amocanis pasuxebi SegviZlia furclis simetriis RerZze gadakecviTac davadginoT. aRmoCndeba, rom mxolod 2) da 6) figurebis simetriis RerZebia gavlebuli.

170


44 amocanis pasuxebi saxelmZRvaneloSia warmodgenili da TiToeul SemTxvevaSi miTiTebulia dasamatebeli ujrebis minimaluri odenoba. dasamatebeli ujrebi ufro muqi ferisaa. 45 da 46 amocanebis pasuxebi saxelmZRvaneloSia warmodgenili. 46 amocanaSi SesaZlebelia moswavle mixvdes, rom saerTo iqneba is ujrebi, romlebis simetriulic aris figuraSi (anu ar aris saWiro misi simetriulis damateba). 47 - 50 amocanebis pasuxebi saxelmZRvaneloSia warmodgenili. davalebis Semowmebis Semdeg ganvixilavT `moisazres~ rubrikiT mocemul amocanebs – periodulobis aRmoCena da gamoyeneba, logikuri amocanebi, amocanebi, romlebic SeiZleba amoixsnas gantolebis SedgeniT, an sinjvis meTodiT. 1 nakiani weli 366-dRiania, rac aris 52 kvira da 2 dRe (366:7=52 (naSTi 2)). yoveli

7 dRis Semdeg kvlav dgeba xuTSabaTi. bolo ori dRe iqneba xuTSabaTi da paraskevi, 31 dekemberi paraskevia. 2 aranakiani weli 365-dRiania, rac aris 52 kvira da 1 dRe. xuTSabaTi 31 dekemberia. 3 radgan `wiTeli an Savi~ warwera ar aris swori, amitom am yuTSi TeTri birTvia.

`SavSi~ Savi ar SeiZleba idos, TeTri ukve `TeTrSi~ _ Savi.

ganawilebulia, e. i. `SavSi~ wiTelia,

4 I Sesakrebi metia meoreze 5-jer. es Sesakrebebi SeiZleba aRvniSnoT 5x da x-iT.

miviRebT: 5x+x=480, saidanac x=80. I Sesakrebia 400, II _ 80. SeiZleba moswavlem es amocana sinjvis meTodiT amoxsnas. mixvdes, rom unda mosinjos 0-iT daboloebuli ricxvebi, an gantolebis Sedgenis gareSe mixvdes, rom II Sesakrebi jamis 16 -ia. SeiZleba SemoviRoT aRniSvna:L mcire ricxvi aris x, didi iqneba 4x+5. pirobiT, 4x+5+x=125. saidanac x=24. erTi Sesakrebia 24, meore _ 101. SeiZleba moswavlem 125-s gamoaklos naSTi 5 da Semdeg wina amocanis msgavsad gaagrZelos amoxsna _ mixvdes, rom mcire Sesakrebi 120-is 15 -ia. 5

saSinao davalebad moswavleebs mivceT `saSinaos~ rubrikidan

12 - 17

daxurul-

boloiani amocanebi.

§6. figuris farTobi. marTkuTxedis farTobis gamoTvla 97-e gakveTili

Tema: figuris farTobi. mizani: moswavlem SeZlos brtyeli figuris farTobis Sefaseba am figuris kvadratuli erTgvarovani badiT dafarvis gamoyenebiT, marTkuTxedis formis figuris farTobis povna – Sesabamis erTeulebSi gamosaxva, farTobis adiciurobis Tvisebis gamoyeneba (maT. VI. 10.). winapirobebi: marTkuTxedi, kvadrati, sigrZis erTeulebi da maT Soris kavSirebi.

171


resursebi: kalami, rveuli, saxazavi, saxelmZRvanelo, plakati, romelzec gamosaxulia farTobis erTeulebi, maT Soris kavSirebi, dafa. saSinao daavalebis Semowmebas SeiZleba didi dro ar dasWirdes – moswavleebi kiTxuloben amocanebs da miuTiTeben swor pasuxebs. qvemoT es pasuxebia warmodgenili. 12

13

14

15

16

17

d

b

d

a

d

d

saxelmZRvaneloSi warmodgenili magaliTebi isea Sedgenili, rom mTlianad pasuxobs standartis misaRwev Sedegebs: figurebis Sedareba axali sididiT – farTobiT; farTobis Sefaseba kvadratuli erTgvarovani badiT dafarvis gamoyenebiT; farTobis erTeulebi da maT Soris kavSirebi; farTobis Tvisebebi (magaliTad, adiciurobis Tviseba); marTkuTxedis formis mqone figuris farTobis gamosaTvleli formula. aq unda SevaxamoT axsnis leqciuri da interaqtiuli meTodi Sesabamisi kiTxvebis SerCevis gamoyenebiT; amaSi saxelmZRvaneloSi mocemuli teqstic dagexmarebaT. magaliTad, aTwiladebis gamoyenebiT moswavleebi TviTon miagneben, rom 1 dm2=0,01m2, 1 sm2=0,0001 m2, 1sm2=0,01 dm2, 1 mm2=0,01 sm2. es aqtivoba pasuxobs standartis misaRwev Sedegs: iyenebs aTwiladebze gamravlebas zomis (magaliTad, farTobis) mcire erTeulis did erTeulTan Tanafardobis gamosaxvisTvis. am Tanafardobebis migneba Cven mier warmodgenili plakatis gamoyenebiT mimdinareobs. sasurvelia, farTobis adiciurobis Tvisebis TvalsaCinod warmodgena. 1

- 10 amocanebs klasSi vxsniT.

gakveTilze ganxorcielebuli aqtivobebiT moswavlem unda gaiazros farTobis cnebis arsi; gaecnos farTobis sazom erTeulebs; SeZlos maTi dakavSireba erTmaneTTan; SeZlos brtyeli figuris farTobis Sefaseba, miaxloebiT gamoTvla misi kvadratuli erTgvarovani badiT dafarvis saSualebiT; gamoiyenos farTobis Tvisebebi figurebis farTobebis gamoTvlisas, praqtikuli amocanebis amoxsnisas; gaiwafos marTkuTxedis formis obieqtebis farTobis gamoTvlaSi. moswavle ecnoba farTobis sazom erTeuls _ kvadrats, romlis gverdi sigrZis sazomi erTeulia. Tumca, klasSi pirvelive savarjiSos Sesrulebisas xvdeba, rom farTobis sazom erTeulad SeiZleba raime sxva geometriuli figurac gamodges _ magaliTad, samkuTxedi, (iseve, rogorc sigrZis sazom erTeulad SeiZleba mtkavelic ki miviRoT). teqstSi demonstrirebulia brtyeli figuris farTobis gazomva misi erTgvarovani badiT dafarvis saSualebiT. miaqcieT yuradReba _ farTobs vzomavT naklebobiT da metobiT, viTvliT figuris mTlianad dasafarad saWiro kvadratebis umcires raodenobas, figuris SigniT moTavsebuli kvadratebis raodenobas da vadgenT figuris farTobis sazRvrebs. Semdeg farTobis sazomi ufro mcire erTeuliT SeiZleba davazustoT farTobis Sefaseba da a. S. es maswavleblisTvis kargad nacnobi ideaa, rac brtyeli figuris farTobis gansazRvras ukavSirdeba. paragrafis masalis warmatebiT asaTviseblad moswavles unda SeeZlos ricxvis nawilis gamosaxva wiladiT an aTwiladiT. magaliTad, 1m2=10 dm ⋅ 10 dm=100 dm2, saidanac

172


1 m2 =0,01 m2; 1m2=100 sm ⋅ 100 sm=10 000 sm2, saidanac 10 sm2=0,0001 m2; 1 sm2=0,01 dm2; 1dm2= 100 1 mm2=0,01 sm2. pirveli ramdenime amocanis amoxsnisas SeiZleba davasaxeloT xolme paragrafSi CamoTvlili farTobis sami Tvisebidan romels viyenebT: klasSi 1 , 3 savarjiSoebSi _ II Tviseba; 4 savarjiSoSiO _ I da II Tviseba (simetriuli samkuTxedebi tolia); 5 - 8 savarjiSoebSi _ III Tviseba da a. S. igulisxmeba, rom moswavle kvadratis farTobs _ 256 mm2-s gayofs 4-ze _ kvadrati 4 toli samkuTxedisgan Sedgeba. 7 8

marTkuTxedi ori toli samkuTxedisgan Sedgeba. TiToeuli maTganis farTobi marTkuTxedis farTobis naxevaria ((4⋅6):2=12). miRebul Sedegs moswavle momavalSi asec Camoayalibebs: marTkuTxa samkuTxedis farTobi misi kaTetebis namravlis naxevaria. teqstze muSaobisas moswavlem ukve daadgina, rom, 1 sm2=0,01 dm2. maSin 1,2 sm2=1,2⋅0,01=0,012 dm2. 9

saSinao davalebad moswavleebs SeiZleba mivceT

1

- 10

amocanebi.

98-e gakveTili

Tema: figuris farTobi. mizani: moswavleebi ukeT gaecnon brtyeli figuris farTobis Tvisebebs da gaiwafon marTkuTxedis formis figuris farTobis gamoTvlaSi. winapirobebi: sigrZis erTeulebi, marTkuTxedi, kvadrati, farTobis erTeulebi. resursebi: kalami, fanqari, saxazavi, rveuli, saxelmZRvanelo, marTkuTxedis formis qaRaldis furclebi. saSinao davalebis amocanebi moiTxovs farTobis zomis erTeulebs Soris kavSirebisa da marTkuTxedis farTobis povnis codnas. sainteresoa 7 amocana – misi amoxsna dakavSirebulia farTobis adiciurobis TvisebasTan – marTkuTxedis farTobi 2-jer metia TiToeuli samkuTxedis farTobze, romelic diagonalis gavlebiT miiReba, analogiuria 8 amocana. intuiciuri warmodgenebi am Tvisebaze, figurebis tolobaze daexmareba moswavleebs aRniSnuli amocanebis amoxsnaSi. SeiZleba klasSi marTkuTxedi gavWraT diagonalze da urTierTSeTavsebiT vaCvenoT moswavleebs adiciurobis Tvisebis gamoyenebis marTebuloba. davalebis Semowmeba SeiZleba miaxloebiT 15 wuTSi davamTavroT, ris Semdeg vagrZelebT amocanebis amoxsnas – 11 - 16 amocanebi. warmogidgenT rekomendaciebs maTi amoxsnebisTvis, farTobis Tvisebebis ukeT gasaazreblad aqtivobebis CatarebisTvis. 11 SeiZleba ABC samkuTxedis farTobi aRvniSnoT x-iT, ACD samkuTxedis _ (x+8)iT. pirobiT, x+x+8=64,8, saidanac x=28,4. pasuxi: ABC samkuTxedis farTobia 28,4sm2, ACD samkuTxedis _ 28,4+8=36,4 sm2. SeiZleba gantolebis Sedgenis gareSe ABC samkuTxedis farTobis gamosaTvleli ricxviTi gamosaxulebis Sedgena: (64,8–8):2. 12 g) yuradRebas vamaxvilebT figuris erTi fragmentis farTobis gamoTvlaze: mTliani fragmentis farTobia 4 sm2, Seferadebulia naxevari, anu 2 sm2.

173


13 pasuxi SeiZleba cxriliT warmovadginoT:

x

gverdebi

farTobi

1 da 5

5

2 da 4

8

3 da 3

9

– udidesi

miRebuli Sedegis mixedviT da intuiciuri warmodgenebiT moswavleebi ivaraudeben, rom farTobi udidesia, roca gverdebi tolia (ara mxolod am P  – x 2 cvladebis naturaluri mniSvnelobebisTvis). momavalSi daasabuTeben, rom mocemuli perimetris marTkuTxedebs Soris udidesi farTobi kvadrats aqvs (marTlac, Tu marTkuTxedis gverdebia x da P 2 –x, misi P 2 P farTobi S=x( 2 –x) udidesia, roca x aris y= –x + 2 x parabolis P wveros abscisa: x= P 4 . maSin meore gverdic 4 -ia, figura ki kvadratia). Tu am erTi amocaniT moswavleebi swor daskvnas ver gamoitanen, SevTavazoT kidev erTi msgavsi amocana. magaliTad, 12 sm SevcvaloT 16 sm-iT. miviRebT: gverdebi

farTobi

1 da 7

7

2 da 6

12

3 da 5

15

4 da 4

16

_ udidesi farTobi _ gverdebi tolia.

14 1 km2=1000 m ⋅1000 m =1000000 m2; 1,2 km2=1200000 m2=120 ha.

dairgo 1200000:10=120000 nergi. 15 kvadratis formis nakveTis gverdi 32,5 m-ia, farTobi _ 1056,25 m2, marTkuTxa nakveTis farTobi _ 2112,5m2. misi erTi gverdi (pirobiT) 32,5 m-ia. meore gverdi _ 2112,5:32,5=65 m. danarCeni sami gverdis SemosaRobad saWiro mesris sigrZea 32,5+65+65=162,5 m. amoxsnis es gza sworia, magram damoukideblad, an maswavleblis daxmarebiT moswavleebma es amocana SeiZleba ufro sxartad, farTobis gamoTvlis gareSe amoxsnan: marTkuTxa nakveTis farTobi orjer metia da misi erTi gverdi kvadratis gverdis tolia, amitom misi meore gverdi kvadratis gverdze orjer meti unda iyos. mTliani marTkuTxedis farTobs (48 sm2) 16 va­ k­ lebT Seuferadebeli samkuTxedebis farTobebs. ADB samkuTxedis farTobi ADBO marTkuTxedis farTobis naxevaria da tolia 24:2=12 sm2-is. BEC samkuTxedis farTobi BECN marTkuT­xedis farTobis naxevaria da tolia 12:2=6 sm2-is, AFC samkuTxedis farTobi _ AMCF marTkuTxedis farTobis naxevari da tolia 16:2=8 sm2-is. ABC samkuTxedis farTobia 48–(12+6+8)=22 sm2. saSinao davalebad moswavleebs vaZlevT

174

11 - 20

amocanebs.


99-e gakveTili

Tema: figuris farTobi. mizani: masalis gameoreba, codnis ganmtkiceba. winapirobebi: farTobi, farTobis Tvisebebi, farTobis erTeulebi, kanonzomierebis aRmoCena da ganvrcoba. resursebi: kalami, rveuli, saxelmZRvanelo. saSinao davalebis Semowmebas SeiZleba 15-20 wT movandomoT. SeiZleba yuradReba gavamaxviloT Semdegi amocanebis amoxsnaze: 15 1 ficris farTobia 40⋅10-=400 sm2=0,04 m2. iatakze dagebulia 36,6:0,04=915 cali ficari. 18 a) iatakis farTobia 5,2⋅3,5=18,2m2, mis SesaRebad saWiroa 18,2⋅0,2=3,64 kg saRebavi. b) am praqtikuli savarjiSos SesrulebiT moswavle gaiwafeba gazomvebis CatarebaSi, Sesabamisi sazomi erTeulis SerCevaSi da praqtikuli gadawyvetilebis miRebaSi. am kvadratis gverdi 4,5 sm-ia, farTobi _ 20,25 sm2. 18 sm-is perimetris mqone marTkuTxedis gverdebi SeiZleba iyos, magaliTad, 1 sm da 8 sm, 2 sm da 7 sm, 4 sm da 5 sm 19

da a. S. Sesabamisad, farTobebia 8 sm2, 14 sm2, 20 sm2, ... maT Soris kvadratis farTobi udidesia. es amocana klasSi amoxsnili me-13 amocanaSi gamoTqmuli varaudis dadasturebas da kidev erTxel gaazrebas emsaxureba. damoklebis Semdeg 100 m sigrZis meseri gaxda 60 m da axlad SemoRobili kvadratis gverdia 15 m, farTobi 225 m2. gakveTilis Semdegi monakveTi daeTmoba jgufur muSaobas, romelic SeiZleba jgufebs Soris Sejibrebis saxiT CavataroT. mxedvelobaSi miiReba amoxsnebis swrafad da sworad warmodgena. gamoviyenoT `vip~ rubrikis amocanebi. am aqtivobis Catareba moiTxovs kanonzomierebis aRmoCenisa da ganvrcobis, gayofadobis niSnebis gamoyenebisa da farTobis Tvisebebis codnas. pirveli amocanis amoxsnaSi moswavles TvalsaCineoba da intuiciuri warmodgenebi daexmareba. Sedegs aseTi suraTiT warmovadgenT. me-2 amocanaSi Tu gverdia x, maSin perimetria 4x. 4x–x=12, 3x=12, x=4. x2=16. cxadia, cvladis Semotanis nacvlad nawilebis ganxilviTac mividodiT saTanado daskvnamde. me-3 amocanaSi a ricxvis mcire nawilic ki (17%) metia b ricxvis met nawilze (39%-ze). aqedan vRebulobT, rom a>b. me-4 amocanaSi SeiniSneba aseTi kanonzomierebebi: a) mZimis Semdeg Cawerilia TanamimdevrobiT aRebuli naturaluri ricxvebis kvadratebi. am wesis Tanaxmad Semdegi ori cifri unda iyos 64. b) mZimis Semdeg Tanamimdevrulad Cawerilia naturaluri ricxvebi (1-dan dawyebuli). am wesis Tanaxmad Semdegi ori cifri unda iyos 13. me-5 amocaSi viyenebT 3-ze gayofadobis niSans (SesaZlebelia uSualod SemowmebiTac SemovifargloT) da aTi SesaZlo ricxvidan virCevT sams: 417, 447, 477. 20

saSinao davalebad moswavleebs vaZlevT davalebebs am Tavis damatebiTi 51 - 60 amocanebidan.

175


me-100 gakveTili

Tema: RerZuli simetria, paraleluri gadatana, figuris farTobi. mizani: miTiTebuli sakiTxebis irgvliv Semajamebeli mecadineobis Catareba, codnis ganmtkiceba, gameoreba. winapirobebi: RerZuli simetria, paraleluri gadatana, farTobi. resursebi: kalami, fanqari, saxazavi, rveuli, saxelmZRvanelo, dafa. Semajamebeli mecadineobis Catarebisas viyenebT Tavis damatebiT amocanebs. saSi­ nao davalebac am amocanebidan iyo micemuli 51 - 60 amocanebi. 63 praqtikuli amocanaa da misi amoxsna sasurvelia klasSi erToblivi ZalisxmeviT. amave amocanebidan 61 d 62 Cven mier SemoTavazebul oTx amocanasTan erTad (RerZul simetriasa da paralelur gadatanaze) SegiZliaT gamoiyenoT fikusirebuli damoukidebeli samuSaosTvis: 1) wrewirSi gavlebulia ori urTierTmarTobuli diametri. ramdeni simetriis RerZi aqvs am figuras? 2) ricxviT sxivze wertili, romlis koordinatia 5,2, paraleluri gadataniT gadadis RerZis wertilSi, romlis koordinatia 7,9. am gadataniT romel wertilSi gadadis C wertili, romlis koordinatia 12,9? 3) ujrian furcelze moniSnuli A wertili paraleluri gadataniT gadadis 5 ujriT qvemoT. Semdegi paraleluri gadataniT 3 ujriT marjvniv. miiReba B wertili. kidev ra paraleluri gadatanebiT SeiZleba gadavides A wertili B wertilSi? 4) ABCD kvadrati paraleluri gadatanisas gadadis KLBA kvadratSi, romeli monakveTiT SeiZleba ganxorcieldes es gardaqmna? axla gTavazobT miTiTebebs Tavis damatebiTi amocanebidan zogierTi amocanis amosaxsnelad. 57 erTi gverdis 2-jer SemcirebiT, marTkuTxedis farTobi 2-jer Semcirdeba; 54:2=27. meore gverdis 3-jer gazrdiT, farTobi 3-jer gaizrdeba – 27⋅3=81. kvadratis farTobi aris 81, gverdi 9, radgan 9⋅9=81. SevamowmoT miTiTeba: 58 30 7 z x 21 35 y y y 10 8 z x t t t z x xy⋅zt=yz⋅tx e. i. Tu a, b, c, d farTobebia, maSin yovel aseT oTx mezobel marTkuTxedSi, romelTa farTobebia a, b, c da d, gvaqvs ad=bc. Cven SemTxvevaSi miviRebT 30⋅35=21⋅m, saidanac m=50; 30 m k 35⋅8=10⋅n, saidanac 21 35 n n=28 10 8 da mn=35k, saidanac k= 50⋅28 35 =40.

176

a c

b d


B

C

59

TiToeuli samkuTxedis farTobi aris 64:4=16 (sm2) A

D

60 S1+S2=96,9 (viyenebT adiciurobis Tvisebas) S1=2S2 am ori pirobiT advilad vipoviT TiToeul farTobs: 3S2=96,9, S2=32,3 (sm2) S1=64,6 (sm2).

S1 S2

analogiuria 61 amocana. axla SemoTavazebuli oTxi amocanis komentarebsac mogawvdiT.

C K

1) am figuras (wrewirisa da ori AB da CD diametris erToblioba) aqvs 4 simetriis RerZi, AB, CD, MN da KL (MN da KL, Sesabamisad COB da AOC kuTxeebs Suaze yofen).

A

N B

O M

D

L

2) wertili, romlis koordinatia 12,9 gadava wertilSi, romlis koordinatia 12,9+2,7, radgan 7,9â&#x20AC;&#x201C;5,2=2,7. 3) SeiZleba, magaliTad, vTqvaT, rom A wertili gadadis B-Si paraleluri gadataniT â&#x20AC;&#x201C; 3 ujriT marjvniv da 5 ujriT qvemoT. L B C 4) es paraleluri gadatana SeiZleba ganxorcieldes, magaliTad, CB monakveTiT (C-dan B-sken). moswavleebs gamovucxadebT, rom Semdeg gakveTilze amave Temebze Catardeba Semajamebeli wera.

K

A

D

101-e gakveTili

Semajamebeli wera Tema: RerZuli simetria da paraleluri gadatana, figuris farTobi. SearCieT swori pasuxi: 1. suraTze gamosaxuli kvadratiTa da wriT Sedgenili figuris simetriis RerZebis raodenoba aris a) 8 b) 6 g) 4 d) 2. 2. Tu paraleluri gadatanisas A wertili, romlis koordinatia 4,2, gadadis B wertilSi, romlis koordinatia 6,1, maSin C wertilSi koordinatiT 10 gadadis wertili, romlis koordinatia a) 8,1 b) 11,9 g) 5,8 d) 16,1.

177


3. ujrian furcelze moniSnuli A wertili 5 ujriT marjvniv da 4 ujriT qvemoT paraleluri gadatanisas gadadis B wertilSi. A wertili B wertilSi gadadis agreTve paraleluri gadatanisas a) 9 ujriT marjvniv b) 9 ujriT qvemoT g) 5 ujriT marcxniv da 4 ujriT zemoT d) 4 ujriT qvemoT da 5 ujriT marjvniv. 4. ABCD kvadrati paraleluri gadatanisas gadavida DCFE kvadratSi. paraleluri gadatana Sesrulebulia a) AB monakveTiT b) DC monakveTiT g) AC monakveTiT d) BC monakveTiT. 5. Tu kvadratis perimetri aris 12 sm. misi farTobia a) 144 sm2 b) 36 sm2 g) 9 sm2 d) 24 sm2. 6. suraTze gamosaxuli kvadratis Seferadebuli nawilis farTobi 2,4 dm2-ia. kvadratis farTobi aris a) 6,4 dm2 b) 7,2 dm2 g) 19,2 dm2 d) 4,8 dm2. amovxsnaT amocanebi 7. suraTze gamosaxuli wrewiris radiusi 5 sm-ia. B′ wertili simetriulia B wertilis AC wrfis mimarT. ABC samkuTxedis perimetri 24 sm-ia. daasruleT suraTi da ipoveT ABCB′ oTxkuTxedis perimetri. 8. ABCDE xuTkuTxedis farTobi 100 sm2-ia. ABDE oTxkuTxedis farTobi BCD samkuTxedis farTobze 12,8 sm2-iT metia. ipoveT BCD samkuTxedis farTobi. pasuxebi da miTiTebebi 1

2

3

4

5

6

g

a

d

d

g

a

7. pirobiT, AC=2⋅5=10 sm, ABC da AB′C samkuTxedebi simetriulia AC wrfis mimarT. e. i. es samkuTxedebi tolia, gverdebic toli aqvs. AB+BC=24–AC=24–10=14 sm, AB′+B′C=AB+BC=14 sm, ABCB′ oTxkuTxedis perimetria 28 sm. 8. SeiZleba BCD samkuTxedis farTobi aRvniSnoT x-iT, maSin ABDE-s farTobi iqneba x+12,8. pirobiT, x+x+12,8=100, saidanac x=43,6 sm2. SeiZleba BCD samkuTxedis farTobis sapovnelad gamoTvalon (100–12,8):2 ricxviTi gamosaxulebis mniSvneloba.

178


Sefasebis sqema testuri amocanebidan TiToeulis swori amoxsna Sefasdes 1 quliT. me-7 da me-8 amocanebis srulyofili amoxsna SeiZleba Sefasdes or-ori quliT TiToeuli. Sejamebis rubrikebi #7 amocanisaTvis 0 qula

ver gaigo amocanis arsi, an araa gadadgmuli raime racionaluri nabiji.

0,5 qula

aago B wertilis simetriuli AC wrfis mimarT.

1 qula

aago B wertilis simetriuli AC wrfis mimarT da SeamCnia ABC da AB'C samkuTxedebis toloba.

1,5 qula

Seasrula wina punqtebi da gamoTvala AB+BC.

2 qula

amocana amoxsna srulyofilad.

Sejamebis rubrikebi #8 amocanisaTvis 0 qula

ver gaigo amocanis arsi, an araa gadadgmuli raime racionaluri nabiji.

0,5 qula

BCD samkuTxedis farTobi aRniSna x-iT, ABDE oTxkuTxedis farTobi – (x+12,8)-iT; an xuTkuTxedis farTobi gamosaxa ori cvladiT, magaliTad, x+y.

1 qula

xuTkuTxedis farTobi gamoisaxa erTi cvla­ diT (x+x+12,8), an Seadgina gantoleba BCD samkuTxedis farTobis sapovnelad.

1,5 qula

Seadgina BCD samkuTxedis farTobis sapovnelad erTcvladiani gantoleba da amoxsna is xarvezebiT.

2 qula

raime gziT amocana amoxsna srulyofilad.

§7. sivrculi figurebi 102-e gakveTili

Tema: sivrculi figurebi. mizani: moswavlem gaiRrmaos sivrculi warmodgenebi, amoicnos sivrculi figurebi; grafikuli gamosaxulebis mixedviT SeZlos maTi aRwera, elementebis (wiboebis, wax­ nagebis, wveroebis) dasaxeleba; elementebis urTierTmdebareobis aRwera. SeZlos zogierTi sivrculi figuris Slilis warmodgena, figuris Sesabamisi Slilis amocnoba da Slilis mixedviT figuris aRdgena. sivrculi figurebis gamosaxulebebze dakvir­ vebis safuZvelze gamoTqvas varaudi misi elementebis raodenobebs Soris kavSiris Sesaxeb, gaecnos eileris formulas da SeZlos misi gamoyeneba. SeZlos Teoriuli masalis gamoyeneba sayofacxovrebo Tematikis amocanebSi.

179


winapirobebi: brtyeli geometriuli figurebi – marTkuTxedi, kvadrati, mravalkuTxedebi, brtyeli figuris farTobi. marTkuTxa paralelepipedis da kubis Slilis damzadeba, Slilis mixedviT modelis damzadeba da dasaxeleba (maT. V.7.). resursebi: marTkuTxa paralelepipedis, kubis, piramidis modelebi, makrateli, qaRaldis furclebi, kalami, rveuli, saxelmZRvanelo. gakveTils viwyebT Semajamebeli weris Sedegebis garCeviT, tipuri Secdomebis dasaxelebiTa da moswavleebis mier dafasTan maTi gasworebiT (15-20 wT). Semdeg gadavdivarT axal masalaze. gadasvlisaTvis saWiroa winapirobebis gaxseneba – moswavleebs vuCvenebT sxvadasxva geometriuli figuris models da vTxovT daasaxelon am figurebis saxelwodebebi. aqve gavixsenebT wiboebis, waxnagebis, wveroebis terminebs da vTxovT maT, daasaxelon TiToeuli modelis mixedviT maTi wiboebis, waxnagebis, wveroebis raodenobebi. moswavleebi iyeneben makratelsa da qaRaldis furclebs da cdiloben jer gamosaxon, Semdeg ki gamoWran marTkuTxa paralelepipedisa da kubis Slilebi, aagon Sesabamisi modelebi. amis Semdeg moswavleebs gavacnobT marTkuTxa paralelepipedisa da kubisgan gansxvavebul sivrcul figurebs – piramidebi, prizmebi, birTvi – Sesabamisi modelebis gamoyenebiT. vuTiTebT waxnagebs – maT Soris, paralelurs, gadamkveTs. `sxvadasxva~ rubrikiT warmodgenili masalis mixedviT SeiZleba visaubroT moswavleebTan yoveldRiur cxovrebaSi sivrculi figuris formebis Sesaxeb, `platonis sxeulebis~ Sesaxeb. yvelaze nacnobi moswavleebisTvis, cxadia, marTkuTxa paralelepipedia. moswavle yoveldRiurad xedavs msgavsi formis mraval sxeuls. man unda SeZlos maTi aRmoCena da dasaxeleba, magaliTad, oTaxi, aguri, asanTis kolofi. moswavleebi asaxeleben am figuris wveroebs, wiboebs, waxnagebs, paralelur da mosazRvre waxnagebs, mocemuli wibos Semcvel waxnagebs da sxva. Semdeg sxva mravalwaxnagebis elementebis dasaxelebasa da daxasiaTebaze gadavlen. mniSvnelovania moswavleTa mier Catarebuli gamoTvlebi, kavSiris Zieba wiboebis, waxnagebis da wveroebis raodenobebs Soris. SesaZlebelia, maT ivaraudon im formulis arseboba, rac eileris formulis saxeliTaa cnobili: nebismieri mravalwaxnagas wveroebis da waxnagebis raodenobebis jami wiboebis raodenobaze 2-iT metia. teqstSi es Teorema asea warmodgenili: nebismieri mravalwaxnagasTvis v+f–e=2, sadac v aris wveroebis, f _ waxnagebis, e _ wiboebis raodenoba. b) punqtSi marTkuTxa paralelepipedis da paramidis Slilebia warmodgenili. Sli­ lis farTobi mravalwaxnagas waxnagebis farTobebis jamia, SemoklebiT mas mravalwaxnagas zedapiris farTobi ewodeba. teqstSi warmodgenili marTkuTxa paralelepipedis zedapiris farTobis gamoTvlamde ajobebs, Tu moswavle miuTiTebs wiboebis sigrZeebs da miuTiTebs tol waxnagebs. amiT gamoTvlebs gaimartivebs _ toli waxnagebis farTobebs TiTojer OiangariSebs da Semdeg gaamravlebs orze: (2⋅6+2⋅4+4⋅6)⋅2=88 sm2. klasSi vxsniT 1

180

- 12 amocanebs.


klasSi 1 - 6 savarjiSoebi mravalwaxnagas elementebis amocnobasa da aRweraSi gawafvas emsaxureba. 7 viyenebT eileris Teoremas: 6+5–x=2, sadac x wiboebis raodenobaa,

x=9. SeiZleba moswavle Seecados kidec aseTi mravalwaxnagas agebas, Tqvenc waaxaliseT moswavle aseTi interesisTvis. 8 - 10 savarjiSoebiT moswavle iRrmavebs sivrcul warmodgenebs, kerZod, ganixilavs SliliT figuris aRdgenas. moswavle warmosaxvaSi kecavs Slils, ganixilavs SesaZlo variantebs da aRadgens figuras. SeiZleba modelebis gamoyenebac. 10 amocana aratrivialur davalebas aZlevs moswavles. aq gasaTvaliswinebelia ara marto waxnagebis ganlageba, aramed maTze nomrebis orientaciac. moswavleebi am davalebas albaT modelebis daxmarebiT mainc gaarTmeven Tavs.

saSinao davalebad moswavleebs vaZlevT

1

- 12

amocanebs.

103-e gakveTili

Tema: sivrculi figurebi. mizani: moswavlem SeZlos sivrculi figuris elementebis urTierTmimarTebebis dasaxeleba, eileris formulis gamoyeneba sivrculi figuris elementebis raodenobis dasadgenad (maT. VI. 9). winapirobebi: marTkuTxa paralelepipedi, kubi, maTi Slilebi. resursebi: sivrculi figurebis modelebi, kalami, rveuli, fanqari, saxazavi, saxelmZRvanelo. 1 - 12 vamowmebT saSinao davalebis amocanebs. maTi amoxsna moiTxovs figurebis amocnobas, dasaxlebas, figuraTa elementebis dasaxelebas, piramidis fuZeSi wveroebis raodenobis mixedviT wiboebis raodenobis povnas, Slilebis mixedviT figuris amocnobas. saSinao davalebis 3 da 4 savarjiSoebis amosaxsnelad paragrafis teqstis suraTebze naxaven, romelia piramidis fuZe. 6

Tu x wiboebis raodenobaa, maSin eileris TeoremiT 16+10_x=2, saidanac x=24.

kubis waxnagebze dawerili ricxvebis jamia 1+2+3+4+5+6=21. mopirdapire waxnagebze dawerili ricxvebis jami unda iyos 21:3=7. warmodgenili kubebidan aseTi SeiZleba iyos mxolod b). 11

klasSi vxsniT 13 - 19 amocanebs. gTavazobT rekomendaciebs zogierTi maTganis amosaxsnelad. 13 I wibos sigrZe aRvniSnoT x-iT; II wibo masze 2 sm-iT naklebia, II wibo iqneba x–2; III wibo I-ze 3-jer metia _ 3x. gaviTvaliswinoT, rom marTkuTxa paralelepipeds toli wiboebis sami oTxeuli aqvs. miviRebT, 4(x+x–2+3x)=100, saidanac x=5,4 pasuxi: 5,4 sm; 3,4 sm da 16,2 sm.

181


mivaqcioT yuradReba! yvela monacemi sigrZis erTi da imave sazomi erTeuliT iyos gamosaxuli:L 1 m=100 sm. 14 10⋅3 gamravlebiT TiToeuli wibo daviTvaleT 2-jer _ TiTo wibo or wveros CaeTvala, amitom wiboebis raodenoba aris: (10⋅3):2=15. waxnagebis raodenobas (x-s) eileris TeoremiT gamovTvliT: 10+x–15=2, saidanac x=7. Tu vinme am mravalwaxnagas daxazvas moisurvebs, SeiZleba suraTi xuTkuTxa prizmas daemsgavsos. 15 sazogadod, mravalwaxnagas wiboebis raodenobis dasadgenad yoveli wverodan gamosuli wiboebis raodenobebi (wveros xarisxebi) unda SevkriboT da gavyoT 2-ze, radgan yoveli wibo orjer daviTvaleT _ or wveros CavuTvaleT. e. i. yoveli wverodan gamosuli wiboebis raodenobebis (wveroebis xarisxebis) jami unda iyos luwi. a) SemTxvevaSi 5⋅3 kentia _ SeuZlebelia, b) SemTxvevaSi wiboebis jamia (6⋅3):2=9, g) SemTxvevaSi 7⋅3 kentia _ SeuZlebelia. 16 -Si me- 14 amocanis msgavsad, wiboebis raodenobaa (14⋅3):2=21. Tu waxnagebis raodenobas aRvniSnavT x-iT, eileris TeoremiT: 14+x–21=2, saidanac x=9. maswavlebelma icis, rom es figura SeiZleba iyos SvidkuTxa prizma. 17 oTaxis kedlebis farTobebis jami aris (paralelepipedis gverdiTi zedapiris farTobi) (4⋅3+5⋅3)⋅2=54 m2. SpaleriT unda daifaros: 54–6=48 m2. 18 kedlebis farTobia (3⋅3)⋅4=36 m2. SpaleriT unda daifaros 36–7=29 m2. amisTvis saWiroa 29:2=14,5 naWeri Spaleri, unda viyidoT 15 naWeri. 19 kedlebis farTobia (5⋅5)⋅4=100 m2, unda daifaros 100–9=91 m2. unda SeviZinoT 91:4=22,75 naWeri. es SeuZlebelia, amitom vyidulobT 23 naWers. misi Rirebulebaa 23⋅8=184 lari. saSinao davalebad moswavleebs vaZlevT 13 - 17 amocanebs.

104-e gakveTili

Tema: sivrculi figurebi. mizani: moswavle gaiwafos sivrculi figurebis amocnobasa, aRwerasa da gamosaxvaSi. winapirobebi: sivrculi figurebisa da maTi elementebis (wibo, waxnagi, wvero) amocnoba da dasaxeleba. resursebi: rveuli, feradi fanqrebi, kalami, saxazavi, dafa, saxelmZRvanelo; plakati, romelzec moculobis erTeulebi da maT Soris kavSirebia gamosaxuli. saSinao davalebis Semowmebisas yuradReba gavamaxviloT canebis amoxsnebze.

13 ,

16

da

17

amo-

mocemuli marTkuTxa paralelepipedis fuZis gverdebi aRvniSnoT a-Ti da b-Ti. gverdiTi zedapiris farTobi iqneba S=(10,8⋅a+10,8⋅b)⋅2. ganrigebadobis (distribuciulobis) Tvisebis gamoyenebiT, S=10,8⋅(a+b)⋅2. 13

(a+b)⋅2 fuZis perimetria. pasuxi: S=10,8⋅16,8=181,44.

182


sazogadod, marTkuTxa paralelepipedis gverdiTi zedapiris farTobia 2(a+b)⋅h, anu p⋅h, sadac p fuZis perimetria. 16 fuZis gverdia 12 sm. marTkuTxa paralelepipedis zedapiris farTobis gamoTvla SeiZleba ricxviTi gamosaxulebiT: 12⋅15,5⋅4+12⋅12⋅2. Tumca, gverdiTi zedapiris gamosaTvlelad ar iyo saWiro fuZis gverdis povna. gverdiTi zedapiris farTobia 48⋅15,5 (ix. me-13 amocana), fuZis farTobia 144 sm2. pasuxi: 1032 sm2. 17 kedlebis farTobebis jamia S=(4,8⋅2,8+4,2⋅2,8)⋅2. am ricxviTi gamosaxulebis mniSvnelobis gamosaTvlelad umjobesia distribuciulobis Tvisebis gamoyeneba. miviRebT: S=(4,8+4,2)⋅2,8⋅2=9⋅5,6=50,4 m2. am kedlebis SesaRebad saWiroa 50,4⋅0,25=12,6 kg saRebavi, 12,6:1,8=7 qila.

saSinao davalebis Semowmebas SeiZleba 15-20 wT davuTmoT. danarCeni dro SeiZleba davuTmoT fokusirebul damoukidebel weras da misi Sedegebis analizs, romlis mixedviT SeiZleba SevafasoT wina or gakveTilze dasaxuli miznis miRwevis done – sivrculi figurebis amocnoba; wiboebis, waxnagebis wveroebis miTiTeba; eileris Teoremis codna da misi Semowmeba sxvadasxva mravalwaxnagaze. gTavazobT amocanebis nimuSebs damoukidebeli werisTvis: F G 1. suraTze marTkuTxa paralelepipedia gamosaxuli. daasaxeleT misi a) wiboebi, b) waxnagebi, g) wveroebi.

E

H B

C

A

D

2. suraTze gamosaxul marTkuTxa paralelepipedSi daasaxeleT: a) paraleluri waxnagebi, b) paraleluri wiboebi. 3. suraTze gamosaxuli figurebidan romelia prizma, romelia piramida?

4. 1) 2) 3) 4) 5) 6)

1) 2)

3)

4)

5)

sworia Tu ara, rom: kubi aris marTkuTxa paralelepipedi piramida aris prizma kubi aris prizma yvela prizmis fuZe marTkuTxedia. samkuTxa prizmas aqvs sami gverdiTi waxnagi. samkuTxa piramidas aqvs sami waxnagi.

5. CawereT eileris formula da SeamowmeT is me-3 amocanaSi mocemuli 1), 3), 4) da 5) figurebze.

183


me-4 amocanaSi sworia 1, 3, 5) winadadebebi. sasurvelia, Sedegebis analizis dros araswori winadadebebis mcdaroba moswavleebma magaliTebiT (suraTebiT) daadasturon. me-5 amocanaSi yoveli figurisTvis unda Caweron wverobis, waxnagebis, da wiboebis raodenoba da Seamowmon formula: wverobis da waxnagebis raodenobebis jami 2-iT metia wiboebis raodenobaze. magaliTad, 3) figuris wveroebis raodenoba 6-ia, waxnagebis – 5, wiboebis – 9. marTlac, 6+5–9=2. Tavis damatebiTi amocanebi 69 da 70 saSinao davalebad mivceT moswavleebs. amasTanave, moswavleebs davavalebT gaecnon proeqts da Semdegi mecadineobisTvis warmoadginon misi Sesrulebis angariSi.

105-e gakveTili

Tema: sivrculi figurebi. mizani: sivrculi figurebis Sesaxeb sakiTxebis gameoreba, codnis ganmtkiceba. winapirobebi: mravalwaxnagebis saxeebi, birTvi, maTi elementebi. resursebi: sivrculi figurebis modelebi, kalami, feradi fanqrebi, saxazavi, fargali, rveuli, saxelmZRvanelo. sawyis 15-20 wuTs vuTmobT saSinao davalebis Semowmebas (Tavis damatebiTi amocanebi 69 , 70 ), proeqtis prezentacias. proeqti proeqti iTvaliswinebs praqtikuli saqmianobisas monacemebis mopovebaSi da mopovebuli monacemebiT gadawyvetilebis miRebaSi gavarjiSebas. moswavle atarebs gazomvebs, irCevs Sesaferis sazom erTeuls da sazom xelsawyos, zomavs oTaxis sigrZes, siganes, simaRles, fanjrebis da karebis sigrZes, siganes. miRebuli SedegebiT asrulebs gamoTvlebs _ angariSobs kedlebis farTobebs, kar-fanjris farTobs da gamoTvlis SpaleriT dasafari zedapiris farTobs. moipovebs informacias Spaleris rulonebis zomebis Sesaxeb. SeiZleba garemontebul saxlSi gazomos Spaleris sigane, gamoTvalos Spaleris rulonis farTobi da remontisTvis saWiro naWrebis raodenoba. isargeblos internet-maRaziebiT da sxva. am aqtivobiT moswavle iZens sayofacxovrebo problemebis damoukideblad gada­ Wris Cvevebs. imyarebs SeZenil codnas da ojaxis wevrebsac ki gauwevs kompetentur konsultacias am sakiTxebis irgvliv. farTobis gameorebis mizniT, moswavleebs vavalebT Seadginon me-4 Tavis damatebiTi 64 - 68 amocanebis msgavsi amocanebi da amoxsnan. agreTve vurCevT, damoukideblad gaecnon me-8 paragrafis Sesaval gverds da Tavisi mosazrebebi gamoTqvan Semdeg gakveTilze.

184


§8. moculoba 106-e gakveTili

Tema: moculoba. mizani: moswavles Camouyalibdes warmodgena sivrculi figuris maxasiaTebel sidideze – moculobaze, SeZlos marTkuTxa paralelepipedis moculobis gamoTvla; SeZlos aTwiladebze gamravlebis gamoyeneba zomis (magaliTad, moculobis) mcire erTeulis did erTeulTan Tanafardobis warmodgenisas, sigrZis, farTobis da moculobis Sesabamisi erTeulebis erTmaneTTan dakavSireba (maT. VI. 3.). winapirobebi: farTobi, farTobis adiciuroba, farTobis erTeulebi. resursebi: sivrculi figurebis modelebi, kalami, rveuli, feradi fanqrebi, saxazavi, saxelmZRvanelo, dafa. farTobi brtyeli figuris maxasiaTebeli sididea. am sididis mixedviT vadarebTY erTmaneTs brtel figurebs. moculobebis mixedviT ki sivrcul figurebs vadarebT erTmaneTs. moculobaze gadasvlamde mizanSewonilia farTobis gameoreba da misi Tvisebebis gamoyenebaze amocanebis amoxsna. ZiriTadad, farTobis adiciurobis Tvisebas viyenebdiT. misi analogiuria moculobis adiciurobis Tviseba, romelic safuZvlad udevs sazomi erTeulebis gamoyenebiT moculobis dadebiT ricxvTa simravleze asaxvas. moswavleebs unda SeedginaT Semdegi amocanebis msgavsi amocanebi. 64

SeiZleba SevavsoT marTkuTxedamde, romlis farTobi mTeli marTkuTxedis farTobis meoTxedia. xolo samkuTxedis farTobi iqneba mTeli marTkuTxedis farTobis mervedi. 65 am amocanis amoxsnac farTobis adiciurobas efuZneba.

cxadia, BNK samkuTxedis farTobi toli iqneba BNKO marTkuTxedis farTobis naxevris – 2⋅3:2=3 (sm2) farTobi DANM=4⋅3:2=6 (sm2) ABCD kvadratis farTobia 6⋅6=36 (sm2) MNKCD xuTkuTxedis – 36–(3+6)=27(sm2) K 2 4 B 66

C

4 6 L 2 A

6

D

1

K

B 2 N

C

O D

6

4 A

3

3 M

D

advilad vipoviT im samkuTxedebis farTobebs, romlebic `avseben~ LKD KLD samkuTxeds kvadratamde. 4⋅4:2=8, 2⋅6:2=6 2⋅6:2=6 amitom saZiebeli sidide aris 6⋅6 – (8+6+6)=16 (sm2) B

E

C

68 BE= 10 BC

farTobi DABE=4sm2 farT. ABCD=? A

F

D

185


cxadia, saZiebeli sidide 20-jer aRemateba ABE samkuTxedis farTobs, radgan es sidide 10-jer aRemateba ABEF marTkuTxedis farTobs. amitom farT. ABCD=20⋅4=80 (sm2) danarCeni amocanebic farTobis adiciurobis gamoyenebiT ixsneba. saSinao davalebis Semowmebis (miaxloebiT – 25 wT) Semdeg gadavdivarT moculobis cnebis axsnaze. yuradRebiT vismenT moswavleTa mosazrebebs moculobis arsisa da marTkuTxa paralelepipedis moculobis formulis Sesaxeb. paragrafis teqstSi aRwerilia marTkuTxa paralelepipedis moculobis gazomva erTeulovani kubebis `CalagebiT~. moswavle etapobrivad midis marTkuTxa parale­ lepipedis moculobis gamosaTvlel formulamde, ecnoba moculobis sazom erTeulebs da akavSirebs maT erTmaneTTan: 1 m3=10⋅10⋅10 dm3=1000 dm3. (SeiZleba asec CavweroT: 1 m3=10 dm⋅10 dm⋅10 dm=1000 dm3). 1 dm3=10⋅10⋅10 sm3=1000 sm3. 1 ⋅ 1 ⋅ 1 m3 = 0,001 m3. 1 dm3 = 10 10 10 1 1 ⋅ 1 dm3 = 0,001 dm3. 1 sm3 = 10 ⋅ 10 10 1 m3 = 100⋅100⋅100 sm3 = 1000000 sm3. 1 ⋅ 1 ⋅ 1 m3 = 0,000001 m3. 1 sm3 = 100 100 100 TuU marTkuTxa paralelepipedis ganzomilebebia a, b da c (erTi wverodan gamosuli wiboebi), maSin moculoba V=abc. klasSi vxsniT 1 - 5 amocanebs. es amocanebi SeiZleba moswavleebma damoukideblad amoxsnan da am davalebis SemowmebiT SeiZleba SevafasoT gakveTilze dasaxuli miznis miRwevis done. 1 kubis wibos sigrZe 20 sm-ia, moculobaa _ 8000 sm3. 2 marTkuTxa paralelepipedis zomebia 0,2 m, 0,4 m da 0,8 m. misi moculobaa 0,064 m3.

teqstSi miTiTebulia, rom marTkuTxa paralelepipedis moculoba fuZis 5 farTobis da simaRlis namravlis tolia: V=32⋅5=160 dm3. saSinao davalebad vaZlevT

1

- 7

amocanebs.

107-e gakveTili

Tema: moculoba. mizani: moswavlem SeZlos moculobis erTeulebs Soris kavSirebis gamoyeneba, misi dakavSireba farTobis, sigrZis Sesabamis erTeulebTan (maT. VI. 3.). winapirobebi: sivrculi figurebi – kubi, marTkuTxa paralelepipedi. L resursebi: kalami, fanqrebi, saxazavi, rveuli, saxelmZRvanelo, dafa. saSinao davalebis SemowmebiT SeiZleba vimsjeloT imis Sesaxeb, aiTvises Tu ara moswavleebma moculobis erTeulebis gamoyeneba, maT Soris kavSirebi; 1 - 7 amo-

186


canebi, ZiriTadad, am codnas amowmebs. davalebis Semowmebas SeiZleba 1015 wT davuTmoT. amis Semdeg vxsniT 6 - 11 amocanebs. moswavleebs viZaxebT dafasTan, vamowmebT, SeuZliaT, Tu ara marTkuTxa paralelepipedis gamosaxva, saWiroebis SemTxvevaSi vexmarebiT maT naxazis SesrulebaSi. SeiZleba Tavdapirvelad ori erTmaneTze `acdenilad~ dadebuli marTkuTxedis gamosaxviT daviwyoT, rasac mosdevs wveroebis SeerTeba. magram zogierTi wibo, romelic ar Cans, wyvetiliT unda iyos gamosaxuli. dafaze uwyveti monakveTis wyvetiliT Secvla ar aris Zneli. Tumca, sasurvelia am wyvetili monakveTebis gamosaxva Tavidanve SeZlos moswavlem. aq viwyebT wyvetili monakveTebis gamosaxvas. am droisaTvis moswavles ukve unda SeeZlos marTkuTxa paralelepipedis gamosaxva, maT es figura V klasSic ganixiles, magram sasurvelia mivaqcioT yuradReba sivrculi figuris gamosaxvis sakiTxs. 6 amocanaSi formulas gamoviyenebT: 72=S⋅3, saidanac fuZis farTobi S=24 m2. 1-isgan gansxvavebuli sami kenti ricxvis namravli 30-ia. es ricxvebia: 2, 3 da 5. am marTkuTxa paralelepipedis zedapiris farTobia 2(2⋅3+2⋅5+3⋅5)=62 sm2. 8 vTqvaT, I kubis wibo aris a, II kubis wibo _ 5a. I-is moculoba da zedapiris farTobia, Sesabamisad, a3 da 6a2, II-is _ 125a3 da 150a2. moculoba 125-jer metia, zedapiris farTobi _ 25-jer meti. sazogadod, Tu kubis wibo gaizrdeba n-jer, moculoba gaizrdeba n3-jer (radgan moculobis gamosaTvlel formulaSi wibo monawileobs Tanamamravlad 3-jer), zedapiris farTobi izrdeba n2-jer. 9 aguris zomebia 0,25 m, 0,12 m da 0,065 m, aguris moculobaa 0,25⋅0,12⋅0,065=0,00195 m3. motanili agurebis raodenobaa 3,9:0,00195=2000. 10 Tu iatakis farTobi S-ia, maSin 45=S⋅2,5, saidanac S=18 m2. 18 m2-is parketiT da­ farva Rirs 18⋅24=432 lari. 7

11 es amocana erTgvari gafrTxilebaa moswavleTaTvis. V=abc formulas viyenebT

maSin, roca ganzomilebebi sigrZis erTi da imave erTeulebiTaa warmodgenili. amrigad, vRebulobT: V=16⋅0,4 dm3=6,4 dm3. saSinao davalebad moswavleebs vaZlevT

8

- 14

amocanebs.

108-e gakveTili

Tema: sivrculi figuris moculoba. mizani: moculobis Sesaxeb sakiTxebis gameoreba, codnis ganmtkiceba, amocanebis amoxsna. winapirobebi: marTkuTxa paralelepipedisa da kubis moculobis formulebi. resursebi: kalami, fanqrebi, saxazavi, rveuli, saxelmZRvanelo.

187


saSinao davalebis amocanebi moculobis sayofacxovrebo sakiTxebSi gamoyenebebTan aris dakavSirebuli. maTematikis gamoyenebebze amocanebis amoxsna zrdis motivacias, moswavleTa interess maTematikis Seswavlisadmi. saSinao davalebis savarjiSoebi: detalis moculobaa 480 sm3, detalis masa _ 480⋅2,7=1296 grami=1 kg 296 g. momavalSi moswavle Seityobs, rom 2,7 g/sm3 aris aluminis simkvrive. 8

konteineris moculobaa 8 m3, yoveli yuTis _ 0,04 m3. SeiZleba gavigoT ramdeni yuTi eteva konteinerSi da ise vupasuxoT kiTxvebs: 8:0,04=200 yuTi. pasuxi: 190 da 200 yuTi eteva, 210 yuTi ar eteva. SeiZleba moswavlem yuTebis raodenoba gaamravlos yuTis moculobaze da Seadaros konteineris moculobas. amocanebi am etapze isea SerCeuli, rom yuTebiT konteineri mTlianad ivseba, mag­ ram momavalSi moswavlem imazec unda izrunos, SeiZleba Tu ara konteinerSi yuTebis maqsimalurad kompaqturad Calageba. 9

gadakecviT miRebuli yuTis zomebia: sigrZe _ 1,8–0,6=1,2 m, sigane _ 1,5–0,6=0,9 m, simaRlea _ 0,3 m. moculoba iqneba 1,2⋅0,9⋅0,3=0,324 m3=324 dm3. 13

wyliT Seivso 10 dm3, rac WurWlis 14 -ia, e. i. WurWlis moculobaa 40 dm3. Tu WurWlis simaRle x dm-ia, maSin 2,5⋅4⋅x=40, saidanac x=4. pasuxi: 4 dm. 14

moculobaze Semdgomi muSaobisTvis SeiZleba gamoviyenoT Tavis damatebiTi 71 80 amocanebi. gakveTilze vxsniT 71 -is a), b), g), 72 , 77 , 80 amocanebs. danarCeni amocanebidan moswavleebs, maTi SesaZleblobebis gaTvaliswinebiT SevurCevT saSinao davalebas. gTavazobT rekomendaciebs zogierTi amocanis amosaxsnelad. 71-e amocanaSi samive SemTxvevaSi ganxilvis Semdeg, gvaqvs pasuxebi 72-e amocanis oTxive kiTxvaze. 71-Si isea SerCeuli monacemebi, rom erTniri moculobis marTkuTxa paralelepipedebs sxvadasxva zedapiris farTobi hqondeT (a da b) da piriqiT, erTnairi zedapiris farTobis mqone marTkuTxa paralelepipedebs sxvadasxva moculoba hqondeT (a da g). 75 unda vipovoT ricxvi, romlis kubi aris 125. sinjvis meTodis gamoyenebiT es

ricxvi aris 5. wibos sigrZea 5 sm, zedapiris farTobi – 6⋅25=150 sm2.

76 fuZeze ganlagdeba 6 yuTi, sul aseTi 3 `Sre~

gveqneba; 18 yuTi.

9 3

77 patara yuTi SeiZleba ar Caetios did yuTSi. mag-

2

1

2 6

aliTad, vTqvaT, `pataris~ zomebia 15 sm, 1 sm, 1 sm, `didis~ – 5 sm, 3 sm, 2 sm. `patara~ vuwodeT yuTs, romlis moculobac naklebia: V1=15 sm3, `didis~ – V2=30 sm3. moswavleebi SeiZleba Tavad mixvdnen, rom `patara~ yuTs SeiZleba `simaRle~ meti hqondes da ar Caetios did yuTSi.

188


79 aq viciT – ab=23,5 dm2, c=1,2 m, anu c=12 dm. V=abc=23,5⋅12 (dm3) 80

a= 7 dm, b=12 sm, anu b=1,2 dm. 7⋅1,2+7⋅c+1,2⋅c=197,2:2 aqedan vipoviT c-s, c=11 dm.

§9. sidideebs Soris damokidebulebebi 109-e gakveTili

Tema: damokidebulebebi sidideebs Soris. mizani: sidideebs Soris damokidebulebebis aRmoCenis, maTi aRwerisa da gamosaxvis unaris ganviTareba. moswavlem unda SeZlos martiv SemTxvevebSi erTi sididis cvlilebisas meore _ masze damokidebuli sididis _ cvlilebis aRwera, Seavsos sidideebs Soris damokidebulebis gamomsaxveli cxrilebi; ganiviTaros sidideebs Soris damokidebulebis aRmweri asoiTi gamosaxulebis Sedgenis unari. SeZlos am gamosaxulebis mniSvnelobebis gamoTvla masSi Semavali cvladebis sxvadasxva mniSvnelobisTvis. miRebuli codna gamoiyenos praqtikuli amocanebis gadawyvetisas. (maT. V. 5.). winapirobebi: algebruli gamosaxulebis Sedgena da gamartiveba. erTi cvladis Semcvel asoiT gamosaxulebaSi sxvadasxva ricxvis CasmiT cvladis mniSvnelobebsa da gamosaxulebis mniSvnelobebs Soris damokidebulebis gamomsaxveli cxrilis Sevseba (maT. V. 5.). resursebi:L ruka masze gamosaxuli masStabiT, rveuli, saxazavi, saxelmZRvanelo, dafa. saSinao davalebis amocanebze Cven ukve visaubreT. amocanaTa raodenoba ar iyo didi, Semowmeba SeiZleba 10-15 wT-Si davamTavroT da axal masalaze gadavideT. axali Temis axsna unda mimdinareobdes moswavleTa aqtiuri CarTulobiT, interaqtiuli meTodiT. am meTodis ganxorcielebaSi maswavlebels teqstSi ganxiluli magaliTebi da iq warmodgenili kiTxvaTa sistema exmareba. am kiTxvebze pasuxebis gacemiT moswavle aqtiurad aris Cabmuli axali masalis ganxilvis procesSi. cxrilebi ki moswavleebma rveulebSi unda gadaitanon da iq Seavson (magaliTi 1). am magaliTSi kargad Cans erTi sididis cvlilebis damokidebuleba meore sididesa da sxva atributebze (maT. VI 5). ruka SeiZleba meore magaliTis ganxilvis dros gamoviyenoT. teqstSi, sami praqtikuli amocanis magaliTze, moswavle ecnoba sidideebs Soris damokidebulebas, maT gamosaxvas cxriliT, formuliT, aRwers am damokidebulebebs. pirvel magaliTSi moswavleebi SeamCneven, rom saRebavis masa imdenjer izrdeba, ramdenjerac izrdeba iatakis farTobi. anu, iatakis farTobis saRebavis masasTan Sefardeba mudmivi sididea da I cxrilSi udris 0,2-s, II-Si _ 0,3-s. momavalSi aseTi saxis damokidebulebas or sidides Soris (roca sidideebis Sefardeba mudmivia) pirdapirproporciul damokidebulebas vuwodebT.

189


meore magaliTSic pirdapirproporciuli damokidebulebaa aRwerili. moswavle damoukidebladac adgens erTi SemTxvevisTvis asoiT gamosaxulebas (formulas). avsebs cxrilebs da aanalizebs kanonzomierebebs: orive cxrilSi sidideebis Sefardeba mudmivi ricxvia, I SemTxvevaSi 500 000, II SemTxvevaSi _ 1000000. es ganapirobebs meore SemTxvevaSi L-is mniSvnelobaTa ufro `swraf~ zrdas, vidre _ pirvel SemTxvevaSi. miuxedavad imisa, rom mesame magaliTSic erTi sididis (Tveebis raodenoba) gazrda meoris (Tanxis) gazrdas iwvevs, es sidideebi ar aris pirdapirproporciuli, maTi Sefardeba ar aris mudmivi sidide. formulas amjerad moswavle teqstSi dasmul kiTxvebze pasuxiT miiRebs: • 2 Tvis Semdeg bankSi iqneba 306 lari. • 3 Tvis Semdeg _ 309 lari • t TveSi emateba 3t lari • t Tvis Semdeg dagrovili Tanxa iqneba 300+3t lari. miiReba formula a=300+3t. paragrafis bolos adgens analogiur formulas sxva monacemebisTvis. aRvniSnoT am SemTxvevaSi bankSi dagrovili Tanxa t Tvis Semdeg b-Ti: b=400+2t. vupasuxoT paragrafis bolo kiTxvas: ramden TveSi iqneba am or SemTxvevaSi bankebSi dagrovili Tanxebi toli? anu, t-s ra mniSvnelobisTvis iqneba a da b toli? SeiZleba moswavlem am kiTxvis pasuxi sinjvis meTodiT aRmoaCinos: Setanili Tanxa

300

400

Tanxa 1 Tvis Semdeg

303

402

Tanxa 2 Tvis Semdeg

306

404

Tanxa 3 Tvis Semdeg

309

406

Tanxa 4 Tvis Semdeg

312

408

gansxvaveba yoveli Tvis Semdeg 1-iT mcirdeba. Tavdapirveli gansxvaveba _ 100 lari 100 TveSi gabaTildeba, 100 TveSi orive Tanxa gaizrdeba 600 laramde. SeiZleba moswavlem aseT xerxs mimarTos: Caweros a=b piroba: 300+3t=400+2t. saidanac miiRebs, t=100. darCenili dro SeiZleba davuTmoT testuri davalebebis amoxsnebs ( 1 - 9 ). saSinao davalebad vaZlevT 1 - 8 amocanebs. klasSi Sesasrulebel amocanebSi, ZiriTadad, yuradReba gamaxvilebulia erTi sididis meore sidideze proporciuli damokidebulebebis SemTxvevebze, roca erTi rveulis fasis mixedviT gamravlebis operaciis gamoyenebiT vpoulobT ramdenime rveulis Rirebulebas, roca drois yovel erTeulSi gavlili manZilis mixedviT vpoulobT drois raRac monakveTSi gavlil manZils (gamravlebiT), roca yovel saaTSi Sesrulebuli samuSaos mixedviT vpoulobT drois raime monakveTSi Sesrulebul samuSaos. yvela SemTxvevaSi vgulisxmobT, rom sidideebis Sefardeba mudmivia. gansakuTrebiT sainteresoa 9 amocana. Tu yovel 1 km-ze 20 TeTria gadasaxdeli, S km-ze gadasaxadi iqneba 20S; sul, Cajdomisas gadaxdili 50 TeTris CaTvliT, 50+20S. es ukanaskneli ar aris pirdapirproporciuli damokidebuleba, x da y dadebiT sidideebs Soris damokidebuleba pirdapirproporciulia, roca y=kx, sadac k mudmivia.

190


110-e gakveTili

Tema: sidideebs Soris damokidebulebebi. mizani: moswavles SeuZlia sidideebs Soris damokidebulebebis gamosaxva, ganvrcoba da aRwera; sidideebs Soris damokidebulebebis gamomsaxveli cxrilis Sevseba; cvladis miTiTebuli mniSvnelobebisTvis damokidebuli sididis gamotovebuli mniSvnelobebis povna (maT. VI. 5.). winapirobebi: algebruli gamosaxulebebis Sedgena da gamartiveba, moqmedebebi dadebiT racionalur ricxvebze. resursebi:L kalami, rveuli, saxelmZRvanelo. gakveTils viwyebT saSinao davalebis SemowmebiT. moswavleebs moeTxovebaT algebruli gamosaxulebebis Sedgena da gamartiveba. es Cvevebi uSualod ukavSirdeba winapirobebs. 1 Tu 1 sm3 rkina 7,8 grams iwonis, maSin V sm3-is m masa ase Caiwereba – m=7,8V; kubis moculoba aris a3, a aris kubis wibo; Tu kubis wibos 3-jer gavzrdiT, moculoba iqneba (3a)3=3a⋅3a⋅3a=27a3 (gadanacvlebadobis TvisebiT) – moculoba gaizarda 27-jer. 2

Tu or sidides Soris damokidebuleba y=5+3x formuliT aris mocemuli da x=10, maSin y=5+3⋅10=35. 3

4 SeiZleba x-is savaraudo mniSvnelobebis CasmiT miagnon im mniSvnelobas, romlisTvisac y=21. SeiZleba y-is nacvlad Casvan 21 da amoxsnan 21=7+2x gantoleba. 5

Tu tolgverda samkuTxedis perimetri aris p, gverdi – a, maSin p=3a.

6

ramdenjeraa qalaqebs Soris manZili Semcirebuli rukaze? gaviTvaliswinoT: 120 km =12000000=2000000. miviRebT 120 km:6 sm=12000000:6=2000000-jer, e. i. rukis masStabia 1:2000000. 7

yovel km-ze ixarjeba 0,1 litri benzini.

Tu yovel heqtarze 80 centner kartofils iReben, maSin S heqtarze aRebuli kartofilis masas ase CavwerT: m=80S. 8

amis Semdeg (15-20 wT) gadavdivarT axali amocanebis ganxilvaze – 10 - 17 . 14 aris Zalian saintereso praqtikuli amocana. misi analogiuri amocanebi SemdgomSi jgufuri muSaobis dros gamoiyeneba. 10 amocanis ganxilvis dros SeiZleba moswavleebs Cveni daxmareba dasWirdeT.

– Tu gvecodineba 1 m qsovilSi gadaxdili Tanxa, SevZlebT x m qsovilSiU gadaxdili Tanxis povnas? ra moqmedebis Catareba mogviwevs? (SevZlebT; mogviwevs gamravlebis Sesruleba). – ra moqmedebiT vipoviT 1 m qsovilSi gadaxdil Tanxas? rogor vipovoT? (gayofiT – 108:27=4 (lari)). – axla rogor CavweroT x m qsovilSi gadasaxdeli y Tanxa? (y=4x).

191


11 da 15 amocanebis amoxsnisas moswavleebs cxrilebi gadaaqvT rveulebSi da damoukideblad avseben maT. 11 x=1-s Seesabameba y=4,5. e. i k=4,5. cxrili saxelmZRvanelos pasuxebSia.

am qalaqebs Soris realuri manZilia 18⋅500000 sm=9000000 sm. meore rukaze manZili iqneba 9000000:1200000=7,5 sm. 12

SeiZleba moswavlem pasuxi erTi ricxviTi gamosaxulebiT miiRos: 18⋅500000:1200000. 14 momsaxurebis I formiT x cal wignSi vixdiT y=60+5x lars, II formiT _ y=20x

lars. SeiZleba gadaxdis xelsayreli formis SesarCevad moswavlem sinjvis meTodi gamoiyenos. wignebis raodenoba

I forma

II forma

1

65

20

2

70

40

3

75

60

4

80

80

5

85

100

daskvna: 4-ze naklebi wignis yidvisas umjobesia gadaxdis II formis gamoyeneba; 4 wignis dros Tanxebi tolia. 4-ze meti wignis yidvisas I formaa ufro xelsayreli. zogierTi pedagogi, albaT, analizur xerxs gamoiyenebda: SevadaroT x wignSi gadaxdili Tanxebi _ rodis aris I formiT gadaxdili Tanxa naklebi? 60+5x<20x, saidanac x>4. 16 kvadratis farTobi – S=a2, perimetri – P=4a, 17

x

1  P2. a= 14 , S= 14 P⋅ 14 P=16

P=2x+2a, a= 12 (p–2a), S= 12 (p–2x)⋅x.

a saSinao davalebad vaZlevT

9

- 12

amocanebs.

111-e gakveTili

Tema: damokidebulebebi sidideebs Soris. mizani: sidideebs Soris damokidebulebis Sesaxeb codnis gameoreba da ganmtkiceba. winapirobebi: algebruli gamosaxulebebis Sedgena, gamartiveba, moqmedebaTa Tvi­ sebebis gamoyeneba. resursebi: kalami, rveuli, saxelmZRvanelo. 9 - 12 amocanebi. ZiriTadad, moiTxoveboda vamowmebT saSinao davalebas – cxrilebis Sevseba da maTi saSualebiT sidideebs Soris damokidebulebebis aRwera, ra gavlenas axdens sxvadasxva perimetris (magaliTad, `koeficientis~) cvlileba damokidebulebaze.

192


cxrilebidan TvalsaCinoa, rom b) SemTxvevaSi y ufro swrafad izrdeba x-is zrdasTan erTad. maswavlebelma icis, rom y=ax+b gantolebiT y warmogvidgeba, rogorc wrfivi funqcia, gantoleba ki wrfis gantolebas warmoadgens. Tu a sakuTxo koeficienti dadebiTia, y=ax+b zrdadi funqciaa _ x-is zrda iwvevs y-is zrdas. rac metia a, miT metia wrfis mier OX RerZis dadebiT mimarTulebasTan Seqmnili maxvili kuTxe _ x-is zrdasTan erTa y ufro swrafad izrdeba. Tu a sakuTxo koeficienti uaryofiTia (rasac me-6 klaselebi ar ganixilaven), wrfe OX RerZis dadebiT mimarTulebasTan blagv kuTxes adgens _ funqcia klebadia _ x-is zrda iwvevs y-is klebas. 9

11

Sevsebuli svetidan Cans, rom k=5,5.

12 amocana sayuradRebo nimuSia ramdenime formuliT mocemuli funqciis mniSvnelobaTa povnis. am aqtivobiT moswavle ukeT wvdeba funqciis mniSvnelobis arss, rac SemdgomSi gauadvilebs funqciis grafikis agebis process.

saSinao davalebis Semowmebis Semdeg CavataroT jgufuri muSaoba saxelmZRvaneloSi warmodgenili masalis mixedviT. jgufuri muSaoba mizani: moswavlem ganiviTaros sidideebs Soris damokidebulebebis aRmoCenis, aRwerisa da formulis saxiT gamosaxvis unari. gamoiyenos miRebuli codna praqtikul saqmianobaSi, kerZod, momsaxurebis formebidan SearCios xelsayreli. jgufebs Soris nawildeba praqtikuli xasiaTis amocanebi, romelTa gadawyvetisas moswavleebs mouwevT sidideebs Soris damokidebulebis aRwera formulis saSualebiT. vTqvaT, nayidi kalmebis raodenobaa x, gadaxdili Tanxa _ y. maSin amocana 1)-Si gadaxdis I formiT y=60+0,5x lars gadavixdiT, II formiT _ y=30+x lars, III formiT _ 2x lars. amocana 2)-Si gadaxdis I formiT gadavixdiT y=60+0,4x lars, II formiT y=24+0,8x lars. III formiT _ y=1,4x lars. momsaxurebis am formebis Sesadareblad saxelmZRvaneloSi sinjvis meTodia SemoTavazebuli. x-is mniSvnelobebad 10-is jeradi ricxvebia dasaxelebuli. SevadgenT saTanado cxrilebs:

amocana 1. kalmebis raodenoba 10 20 30 40 50 60 70 80 90 100

gadaxdis I forma 65 70 75 80 85 90 95 100 105 110

gadaxdis II forma 40 50 60 70 80 90 100 110 120 130

gadaxdis III forma 20 40 60 80 100 120 140 160 180 200

193


amocana 2. kalmebis raodenoba 10 20 30 40 50 60 70 80 90 100

gadaxdis I forma 64 68 72 76 80 84 88 92 96 100

gadaxdis II forma 32 40 48 56 64 72 80 88 96 104

gadaxdis III forma 14 28 42 56 70 84 98 112 126 140

cxrilebidan Cans, rom amocana 1-Si 30-ze naklebi kalmis yidvisas gadaxdis III formaa ufro xelsayreli, 40-dan 60-mde _ II formaa xelsayreli, 60-ze meti kalmis yidvisas gadaxdis I forma jobs. amocana 2-Si 40-ze naklebi kalmis yidvisas gadaxdis III formaa ufro xelsayreli, 50-dan 90-mde gadaxdis II formaa xelsayreli, 90-ze meti kalmis yidvisas _ I forma. calkea gamosayofi e. w. sasazRvro mniSvnelobebi. magaliTad amocana 2-Si 40 kalmis yidvisas II da III formebi erTsa da imave Sedegs gvaZlevs. amocanis analizuri gziT amoxsnisas vadarebT gadaxdil Tanxebs. magaliTad, amocana 1-Si 30+x<2x, saidanac x>30 _ niSnavs, rom, roca kalmebis raodenoba 30-ze metia, gadaxdis II forma UIII-ze xelsayrelia; 60+0,5x<30+x, saidanac x>60 _ niSnavs, rom roca kalmebis raodenoba 60-ze metia, gadaxdis I forma II-ze xelsayrelia da a. S. momavalSi, wrfivi funqciebis Seswavlis Semdeg, moswavleebi am ekonomikuri amocanis amosaxsnelad grafikul meTodsac gamoiyeneben. jgufuri muSaobis damTavrebisa da Sedegebis sajaro ganxilvis Semdeg moswavleebs davalebad mivcemT saxelmZRvaneloSi mocemuli proeqtis Sesrulebas da Semdeg gakveTilze warmodgenas.

112-e gakveTili

Tema: sidideebs Soris damokidebulebebi. mizani: moswavlem SeZlos codnis (sidideebs Soris damokidebulebebi) gamoyeneba sayofacxovrebo amocanebis amoxsnisas. gakveTili eTmoba moswavleebis mier proeqtis prezentacias. proeqti paragrafis Temas da jgufur muSaobas exmaureba. amjerad moswavles Tavad mouwevs amocanis pirobis Camoyalibeba da kiTxvis dasma. amisTvis man unda moiZios informacia eleqtroenergiis gadasaxadis Sesaxeb. SeiZleba ufrosebs mimarTos daxmarebisTvis, gamoiyenos qviTrebi. informacias SeiZleba aseTi saxe hqondes: TveSi 100 kvt-mde (CaTvliT) daxarjuli eleqtroenergiisTvis vixdiT 1kvt-Si 13 TeTrs; 101-dan 300 kvt-mde (CaTvliT) _ vixdiT

194


16 TeTrs kvt-Si; 300-ze metisTvis _ 18 TeTrs kvt-Si. formuliT es informacia ase Caiwereba: y= 13x, Tu 0≤x≤100; y=16x, Tu 100<x≤300; y=18x, TuU x>300. TuU moswavlem ver SeZlo informaciis formuliT Cawera, formulamde etapobrivad, kiTxvebiT miviyvanT: • ramdens gadavixdiT, Tu daxarjulia x kvt, sadac x≤100 (x<100 an x=100)? x⋅13. • ramdens gadavixdiT, Tu daxarjulia 100-ze meti, magram araumetes 300 kvt (100<x≤300)? 16x TeTrs. • ramdens gadavixdiT 230 kvt-Si? 230⋅16 TeTrs. • TuU daxarjulia x kvt, sadac x>300, ra Tanxa geqneba gadasaxdeli? _ 18x TeTri. • ramdens gadavixdiT 315 kvt-Si? 315⋅18 TeTrs da a. S. Tqven SegiZliaT moswavleebs SesTavazoT gadaxdis ufro `moqnili~ wesi, romelic formulebiT ase aRiwereba: y=13x, Tu x≤100 y=1300+(x–100)16, Tu 100<x≤300 y=1300+3200+(x–300)⋅18, Tu x>300. am wesis aRwera, gaanalizeba bevr sasargeblo informacias SesZens moswavles. sayuradReboa moswavleTa mosazrebebis Sejereba, polemika wamoWril sakiTxebTan dakavSirebiT. moswavleebs gamovucxadebT, rom Semdeg gakveTilze Catardeba Semajamebeli wera Semdegi Temebis mixedviT: sivrculi figurebi da maTi elementebi, moculoba, moculobis gamoTvla, sidideebs Soris damokidebulebebi.

113-e gakveTili

Semajamebeli wera Tema: sivrculi figurebi da maTi elementebi, moculobis da zedapiris farTobis gamoTvla. sidideebs Soris damokidebulebebi. SearCieT swori pasuxi: 1. marTkuTxa paralelepipedis fuZis farTobi 18 m2-ia, simaRle _ 10 m. misi moculobaa a) 56 m3 b) 180 m3 g) 112 m3 d) 360 m3. 2. marTkuTxa paralelepipedis fuZe kvadratia, romlis perimetri 20 sm-ia, erTi gverdiTi waxnagis farTobia 15 sm2. am paralelepipedis simaRle aris a) 3 sm b) 5 sm g) 3,75 sm d) 10 sm. 3. Tu piramidas aqvs 12 wibo, is aris a) samkuTxa piramida b) oTxkuTxa piramida g) xuTkuTxa piramida d) eqvskuTxa piramida.

195


4. kubis zedapiris farTobi 96 dm2-ia. am kubis moculoba aris a) 64 dm3 b) 32 dm3 g) 256 dm3 d) 16 dm3. 5. Tu 3 kg vaSli 5,4 lari Rirs, maSin x kg vaSli Rirs a) 5,4x lari b) 2,7x lari g) 1,8x lari d) 0,9x lari. 6. y=2x+3 formulaSi qeTi svams mimdevrobiT naturalur ricxvebs. y-is yoveli momdevno mniSvneloba winaze a) 3-iT metia b) 2-iT metia g) 5-iT metia d) 1-iT metia. amoxseniT amocanebi: 7. kubis formis akvariumis wibo 50 sm-ia. ra simaRleze iqneba am akvariumSi Casxmuli 60 litri wyali? 8. manZili A soflidan B qalaqamde 1:1 000 000 masStabian rukaze 8 sm-ia. ra droSi gaivlis daviTi am manZils, Tu is 60 km/sT siCqariT moZraobs? pasuxebi da miTiTebebi 1

2

3

4

5

6

b

a

d

a

g

b

2. fuZis yoveli gverdi 20:4=5 sm-ia, gverdiTi waxnagis farTobi 15=5h, saidanac h=3 sm. 4. yoveli waxnagis farTobia 96:6=16 dm2, wibo 4 dm, moculoba 43=64 dm3. 5. 1 kg vaSli Rirs 5,4:3=1,8 lari, x kg Rirs 1,8x lari. 6. savaraudod, moswavle Casvams x-is ramdenime momdevno mniSvnelobas da daadgens swor pasuxs. SeiZleba ufro gawafulma moswavlem x-is koeficientze dakvirvebiT amoxsnas amocana. 7. akvariumSi Casxmuli wyali miiRebs marTkuTxa paralelepipedis formas, romlis ganzomilebebia 5 dm, 5 dm da x dm (wylis simaRle). gaviTvaliswinoT, rom 60 l=60 dm3. pirobiT, 5⋅5⋅x=60, saidanac x=2,4 dm. 8. gasavlelia 8 000 000 sm=80 km. daviTi am manZils 80:60=1 13 sT-Si (1 sT 20 wT-Si) gaivlis. Sefasebis sqema Sefasebis rubrika #7 amocanisTvis 0 qula 0,5 qula

naSromSi araa warmodgenili raime gonivruli nabiji amoxsnisken mimavali. daakavSira moculobis sazomi erTeulebi: 60 l=60 dm3, an gamosaxa simaRlis saSualebiT wylis moculoba akvariumSi: 50⋅50⋅x, an 5⋅5⋅x.

196


1 qula

gamosaxa wylis moculoba akvariumSi da gautola 60-s: 5⋅5⋅x=60 (Tundac araswori ganzomilebebiT: 50⋅50⋅x=60).

1,5 qula

sworad Seadgina gantoleba da amoxsna igi xarvezebiT.

2 qula

srulyofilad amoxsna amocana.

Sefasebis rubrika #8 amocanisTvis 0 qula

naSromSi araa warmodgenili raime gonivruli nabiji amoxsnisken mimavali.

0,5 qula

sworad SearCia moqmedeba realuri manZilis gamosaTvlelad, magram ver gamoTvala.

1 qula

sworad gamoTvala realuri manZili, an arasworad gamoTvlili realuri manZili gayo 60-ze drois sapovnelad.

1,5 qula

drois gamosaTvlelad realuri manZili gayo 60-ze, magram ver Seasrula gayofa sworad.

2 qula

srulyofilad amoxsna amocana.

114-e gakveTili (Semajamebeli)

Tema: sivrculi figurebi, moculoba, sidideebs Soris damokidebulebebi. mizani: miTiTebuli Temebis mixedviT gameoreba, codnis ganmtkiceba. Semajamebeli gakveTilis Semadgeneli nawilia Semajamebeli weris Sedegebis gacnoba, tipuri Secdomebis garCeva, zogierTi amocanis amoxsna, (15-20 wT). danarCeni dro daeTmoba masalis gameorebas IV Tavis Semajamebeli 1 - 21 amocanebis amoxsnis saSualebiT. maTgan nawili SeiZleba maswavlebelma moswavleebs saSinao davalebad misces. gTavazobT rekomendaciebs am amocanebis amosaxsnelad. me-4 TavisM Semajamebeli amocanebi (amocanebi TviTSefasebisTvis) 3 darbazis kedlebis farTobebis jamia 864 m2=86400 dm2. TiToeuli ujris farTobia

16 dm2. darbazis kedlebze gamosaxulia 86400:16=5400 ujra. 7 oTaxis moculobaa 9,2⋅6,2⋅5,2=296,608 m3. 42 moswavle SeiZleba iyos oTaxSi, romlis

moculoba 42⋅6=252 m3-ze naklebi ar aris, es oTaxi sanitarul normebs akmayofilebs. aucilebeli ar aris gamoTvlebis zustad Sesruleba: 42 moswavles sWirdeba oTaxi, romlis moculoba 252 m3-ze naklebi ar unda iyos. oTaxis zomebis naklebobiT damrgvalebiTac ki miiReba meti moculoba: 9⋅6⋅5=270 m3. 8

gavamartivoT asoiTi gamosaxuleba: 12x+6y+12+3x-6y+3=15x+15=15(x+1).

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amrigad, gamosaxuleba y-ze ar aris damokidebuli _ misi mniSvneloba mxolod xzea damokidebuli. CavsvaT x=2,5, miviRebT: 15(2,5+1)=52,5. 10 marika wuTSi krefs 168:4=42 sityvas, t wuTSi _ 42⋅t sityvas. Tu sityvebis raodenobas n-iT aRvniSnavT,M n=42t. Tu marika ufro Cqara daiwyebs beWdvas, t-s koeficienti gaizrdeba. 12 a laris measeds fosta itovebs, e. i. agzavnis a laris 0,99-s _ 0,99a lars, c=0,99a, saidanac a=c:0,99= 100c 99 . 15 sainteresoa, rom AD da BP wrfeebs ara aqvs saerTo wertili, magram isini paraleluri ar aris. maswavlebelma icis, rom paraleluria wrfeebi, romlebic erT sibrtyeSia da saerTo wertili ara aqvs. AD da BBP wrfeebs saerTo wertili ara aqvs, magram ar aris erT sibrtyeSi. aseT wrfeebs acdenili wrfeebi ewodeba. 18 darCenili nawilis moculoba kubis moculobis da amoWrili marTkuTxa paralelepipedis moculobis sxvaobis tolia: v=2,43–0,8⋅1⋅2,4=11,904 m3. 19 a) erTi SeRebili waxnagi mxolod didi kubis gverdiT waxnagebze `gamoCenili~ patara kubebidan imaT aqvs, romlebic didi kubis wibos wertilebs ar Seicavs. aseTia 10⋅10=100 cali TiTo waxnagTan, sul 6⋅100=600 cali. b) ori SeRebili waxnagi aqvs didi kubis wiboebze ganlagebuli patara kubebidan imaT, romlebic didi kubis wveroebs ar Seicavs. aseTi patara kubi TiTo wibosTan aris 10, sul _ 12⋅10=120. g) sami waxnagi mxolod wveroebTan mdebare patara kubebs aqvs SeRebili _ aseTia 8. d) danarCens ara aqvs SeRebili arc erTi waxnagi, aseTia 123–(600+120+8)=1000, anu yvela patara kubi, romelic didi kubis zedapirze ganlagebuli patara kubebis mocilebiT miiReba. es qmnis kubs, romlis gverdia 10 sm. anu SeiZleba zogierTma moswavlem `Semoafcqvnas~ did kubs gareTa (SeRebili) fena da darCeba SeuRebavi kubebisgan Sedgenili kubi, romlis wibo iqneba 10 sm. b) Slilis dakecviT miRebul kubSi SeRebili waxnagebi iqneba paralelur 20 sibrtyeebze, g) ar aris Slili – Sedgeba 5 kvadratisgan. a), d) da e) _ mocemuli kubis Slilebia. 21 AMOE da MBFO marTkuTxedebs erTi gverdi saerTo aqvs, MBFO-s farTobi 1,5-

jer metia, e. i. misi FO gverdi 1,5-jer metia EO-ze. amitom FCNO-s farTobic 1,5-jer metia ONDE-s farTobze. aRvniSnoT ONDE-s farTobi x-iT, miviRebT x⋅1,5=210, saidanac x=140 sm2.

198


V Tavi monacemTa analizi da statistika §1. monacemTa Segroveba 115-e da 116-e gakveTilebi

Tema: monacemTa Segrovebis xerxebi. mizani: moswavles gamoumuSavdes raime movlenis, procesis SeswavlisTvis saWiro monacemebis SerCevis, am monacemebis mopovebisTvis Sesaferisi saSualebebis SerCevis, martivi eqsperimentis dagegmvis unari. gaiwafos monacemebis mopovebisTvis kiTxvarebis SedgenaSi. (maT. VI. 11.). winapirobebi: SekiTxvebis mocemuli CamonaTvalidan saWiro monacemTa Sesagroveblad Sesaferisi kiTxvebis SerCeva da gamoyeneba (maT. V. 11.). resursebi: rveuli, saxelmZRvanelo, dafa. mexuTe klasSi ukve ganvixileT dasmuli amocanis amosaxsnelad saWiro Tvisebrivi da raodenobrivi monacemebis mopovebis zogierTi xerxi; SekiTxvebis mocemuli CamonaTvalidan Sesabamisi kiTxvebis SerCeva, mocemul TemasTan dakavSirebiT kiTxvebis dasma da am kiTxvebis saSualebiT saWiro monacemebis mopoveba. amjerad mimdinareobs am Temis ufro siRrmiseulad Seswavla – mza anketiT respondentTa gamokiTxva, martivi statistikuri eqsperimentis dagegmva da monacemTa Segrovebis Sesaferisi saSualebebis SerCeva. swored am sakiTxebTan aris dakavSirebuli aqtivobebi, romlebic saxelmZRvanelos teqstSiU warmodgenili amocanebis mixedviT SeiZleba ganxorcieldes. am sakiTxebis ganxilviT iwyeba Temis damuSavebis sawyisi faza: `winaswar~. teqstSi vTavazobT ramdenime tipis amocanas. `a~ amocanis gadasawyvetad gamo­kiTx­ vis an kiTxvaris Sevsebis meTodia xelsayreli. kiTxvebi SeiZleba iyos: romel konkursSi miviRebdiT monawileobas: cekva, simRera, mxatvruli kiTxva, Teatraluri warmodgena, ... SeiZleba kiTxvarSi ar miuTiToT konkursis saxeoba da davadginoT, romeli saxeobaa ufro popularuli. kiTxvariTve SeiZleba SevarCioT konkursebis Catarebis dro, xangrZlivoba, davazustoT sxva saorganizacio sakiT­xebic. b) remontis SemTxvevaSi, gazomvebis Catareba, gazomvis Sedegebis sasurveli masalis zomebTan Sedareba, masalis fasis gaTvaliswinebaa saWiro. es aqtivoba moswavleebma, bunebrivia, mSoblebTan TanamSromlobiT, erToblivi konsultaciebiT unda ganaxorcielon.

199


g) qveynebis mosaxleobis simWidrovis dasadgenad cnobarebSi informaciis mopovebaa saWiro: mosaxleobis raodenobas vyofT qveynis teritoriis farTobze (kv. km-Si gazomilze). d) asakobrivi jgufebis raodenobebis dasadgenad SeiZleba gamosakvlevi soflis, quCis, an sacxovrebeli korpusis yovelma ojaxma Camoweros ojaxis wevrebis sia (asakis miTiTebiT), an cxrilSi yovel asakobriv SualedSi miuTiTos am asakis ojaxis wevrebis raodenoba. magaliTad, vTqvaT, erT-erTi ojaxis mier Sevsebulia cxrili: 7-ze naklebi

7-dan 17-mde

18-dan 35-mde

36-dan 50-mde

2

1

1

1

51-dan 70-mde

70 da meti 1

ramdeni adamiania am ojaxSi? 6. savaraudod, col-qmari, sami Svili da 1 ufrosi Taobis warmomadgenelia (bebia an babua). klasSi Sesasrulebuli amocanebi SeiZleba or gakveTilze gadanawildes. klasSi 1 , 2 , 5 savarjiSoebiT iwafebian kiTxvarebisTvis kiTxvebis SedgenaSi. 1

a) SeiZleba SevadginoT kiTxvari Camowerili namcxvrebis da sasmelebis saxeobebiT, romelTa yidvac SesaZlebelia, TiToeulma moswavlem mianiSnos masze Tavisi sasurveli sasmelis an namcxvris saxeoba. b) magaliTad, SeiZleba dasvan kiTxvebi: • ra saganSi mecadineob damatebiT? • xelovnebis romel saxeobas euflebi? • sportis romel saxeobaze dadixar? 2

magaliTad, davsvaT kiTxvebi: • CamoTvaleT sigrZis TqvenTvis cnobili erTeulebi • ramdeni sm-ia 3 m, 3 km? 3 mogviwevs, cnobarebis da internet-gverdebis meSveobiT, davadginoT qveynebis farTobebi da mosaxleobis odenoba. simWidrovis dasadgenad mosaxleobis odenobas gavyofT qveynis teritoriis farTobze. 4

cxadia, moswavleebs am eqsperimentis Catareba am asakSi ar SeuZliaT, amitom eqsperimentis aRwerac sakmarisia. aseT ubnebSi mdebare quCebSi dRis sxvadasxva dros, magaliTad, 9, 12, 15, 18 da 21 saaTze unda CaviniSnoT moZravi avtomobilebis odenoba da gzis gavlaze daxarjuli dro. umjobesia, es eqsperimenti ramdenime dRes gagrZeldes. 5 yvela ojaxs urigdeba kiTxvari. sakmarisia, rom masSi iyos erTaderTi kiTxva: ramdeni Svilia Tqvens ojaxSi? 1 - 5 pirvel gakveTilze saSinao davalebad vaZlevT amocanebs, romelTa amoxsnebze meore gakveTilze vmsjelobT. amave gakveTilze, sasurvelia, `moisazres~ rubrikiT warmodgenili amocanebic amoixsnas.

saSinao savarjiSoebi:

1

kiTxvebi SeiZleba aseTi iyos: • CamoTvlilTagan romeli sagnis olimpiadaSi miiRebdiT monawileobas? qarTuli, maTematika, istoria da a. S. SeiZleba kiTxvarSi sagani ar mivuTiToT, rac saSualebas mogvcems, davadginoT ufro popularuli sagani klasSi. SeiZleba kiTxvarSi saorganizacio sakiTxebsac

200


SevexoT. magaliTad, Tqveni azriT, ramden saaTs unda grZeldebodes olimpiada? a) 1 sT-ze naklebi, b) 1-dan 2 sT-mde, g) 2-dan 3 sT-mde da sxva. 2 erTi da imave matCis, an momReralTa konkursis win davafiqsiroT megobrebidan TiToeulis prognozi da konkursis Semdeg SevamowmoT Sedegebi. es eqsperimenti ramdenjerme gavimeoroT.

italiis mosaxleobis simWidrove naxevari saukunis win iyo 47 000 000 301 000 ≈156 adamiani 1 km2-ze. 2017 wels italiis mosaxleobis simWidrove iyo (miaxloebiT) 60 000 000 301 000 ≈199 adamiani 1 km2-ze. simWidrove gaizarda miaxloebiT 199:156≈1,28-jer. 3

4 moviZioT cnobarebSi an internet-gverdebze qveynebis mosaxleobis ode­ nobebi, amovarCioT maTgan 5 udidesi (mcxovrebTa odenobiT) qveyana da moviZioT maTi teritoriebis farTobebi. mosaxleobis raodenobis farTobze gayofiT miviRebT qveynis mosaxleobis simWidroves. SeiZleba dasaxelebuli xuTi qveynis simWidrovec ki moviZioT cnobarebSi da internetSi. sasurvelia, rom Tqvenma moswavleebma irwmunon internetis farTo SesaZleblobebi uaxlesi sacnobaro masalis moZiebisas. 5

• • • •

SeiZleba davsvaT kiTxvebi: romeli sagani giyvarT yvelaze metad? romel sagnis mecadineobas andomebT met dros? romel sagans mecadineobT upirvelesad? romel sagans mecadineobT iZulebiT?

moisazre 1 a) aTi amoRebuli monetidan 9 namdvilia, 1 aris yalbi, maTi saerTo masa iqneba 101 grami. b) xuTi namdvili monetis masa 50 grami unda iyos. 2 namdvili da 3 yalbi monetis masa aris 53 grami. g) I yuTidan amoviReT 1 moneta, II yuTidan _ 2 moneta, III-dan _ 3 da a. S. X yuTidan amoviRoT 10 moneta. amoRebulia 1+2+3+...+10=55 moneta. es 55 moneta namdvili rom yofiliyo, maSin misi masa 550 grami iqneboda. realuri masa 550 grams gadaaWarbebs imdeni gramiT, ramdeni yalbi monetacaa maT Soris. yuTidan amoRebuli monetebis raodenoba ki yuTis nomers emTxveva. magaliTad, Tu awonili monetebi 557 grami aRmoCnda, maT Soris 7 moneta yofila yalbi, Svidive erTi yuTidanaa _ VII yuTidan. a) xuTi bavSvidan yvelaze ufrossa da yvelaze umcross Soris 4 Sualedia: 4⋅3=12 weli gansxvaveba. b) yvelaze umcrosis asaki aRvniSnoT x-iT, ufrosis iqneba x+12. pirobiT, x+12=7x, saidanac x=2. 2

I mimdevrobis yoveli Semdegi wevri miiReba winasTvis 5-is mimatebiT, II mim­ devrobaSi _ 3-is mimatebiT. radgan pirveli wevrebi tolia, amitom oriveSi aRmoCndeba is wevrebi, romlebic miiReba 15-is damatebiT: 2, 17, 32, 47 da a. S. bolo ricxvi 179-s ver gadaaWarbebs. 179-mde 2+15x saxis ricxvebis raodenobaa 11 (2+15x≤179, 15x≤177, x≤11 12 15 ). e. i. 2-s 15 emateba 11-jer, miiReba 12 cali ricxvi. 3

201


vTqvaT, marTkuTxa paralelepipedis wiboebia m, n da p naturaluri ricxvebi. zedapiris farTobi tolia 2(mn+mp+np). frCxilebSi gamosaxulebis mniSvneloba 3-ze naklebi ar aris. misi 2-ze gamravlebiT aucileblad miiReba Sedgenili ricxvi _ 2-ze meti luwi ricxvi. 4

33-is gamyofia 1, 3, 32 da 33 _ 4 gamyofi. sazogadod, Tu p martivi ricxvia da n naturaluri ricxvi, maSin pn-is gamyofebis raodenobaa n+1. 5

33-is gamyofebia 1, 3, 32, 34 _ 4 cali gamyofi. 5 -is gamyofebia 1, 5, 52, 53, 54 _ 5 cali gamyofi. 33⋅54-is gamyofebi miiReba 33-is yoveli gamyofis 54-is yovel gamyofze gamravlebiT: 1, 5, 52, 53, 54, 3, 3⋅5, 3⋅52, 3⋅53, 3⋅54, 32, 32⋅5, ... sul miiReba 4⋅5=20 cali gamyofi. sazogadod, Tu p da q martivi ricxvebia, n da m naturaluri ricxvebi, maSin pnqm-is gamyofTa raodenobaa (n+1)(m+1). am amocanebis ganxilvis Semdeg SeiZleba moswavleebs saSinao davalebad mivceT Ta6 4

vis damatebiTi 1 - 4 amocanebis amoxsna. `moisazres~ amocanebis kargad garCevis Semdeg am amocanebis amoxsna moswavleebs ar gauWirdebaT.

§2. monacemTa moxerxebuli warmodgena 117-e da 118-e gakveTilebi

Tema: monacemTa warmodgenis xelsayreli formebi. mizani: moswavles SeeZlos mopovebuli monacemebis mowesrigeba. kerZod, monacemebis klasifikacia raime Tvisebis an raodenobis mixedviT da Semdeg maTi warmodgena sixSireTa cxrilis an piqtogramis saxiT. ganuviTardes cxrilis da piqtogramis saSualebiT monacemebis daxasiaTebis, aRweris unari, SeeZlos arsebuli monacemebis rac SeiZleba xelsayreli formiT warmodgena (maT. VI. 12.). winapirobebi: moswavles SeuZlia monacemebis warmodgena cxrilis, piqtogramis gamoyenebiT. resursebi: rveuli, saxelmZRvanelo, dafa, saxazavi, feradi fanqrebi. miTiTebuli Tema V klasSic iyo ganxiluli. am gakveTilebze vimeorebT monacemTa warmodgenis zogierTi xerxis gamoyenebas (magaliTad piqtogramebs). siaxle Tvisebriv da raodenobriv monacemTa klasifikacia da dalagebaa, monacemTa cxrilis Sedgena dajgufebuli raodenobrivi monacemebis SemTxvevaSi. wriul da svetovan diagramebze Semdeg gakveTilebze visaubrebT. gakveTils viwyebT saSinao davalebis SemowmebiT. amocanebi klasSi Sesrulebulis analogiuria. 1) Tu marTkuTxedis gverdebi aris m da n (m da n naturaluri ricxvebia), maSin perimetri iqneba 2(m+n) – luwi ricxvi. es ricxvi ki 2-ze meti luwi ricxvia, e. i. Sedgenilia (martivi da Sedgenili ricxvebi moswavleebs naswavli aqvT).

202


2) marTkuTxedis farTobi gamoisaxeba ase: mn. es ricxvi SeiZleba iyos martivi, magaliTad, roca m=1 da n martivia, an piriqiT, roca n=1 da m martivia. 3) 53-is gamyofebia 1, 5, 52, 53 – sul oTxi gamyofi 4) 54=2⋅33 2-is gamyofebia 1 da 2 (2 ricxvi) 33-is gamyofebia: 1, 3, 32, 33 (4 ricxvi) am gamyofebis namravlebi iqneba 54-is gamyofebi, sul miviRebT 8 namravls. 54-s 8 gamyofi aqvs: 1⋅1, 1⋅3, 1⋅32, 1⋅33, 2⋅1, 2⋅3, 2⋅32, 2⋅33. am amocanebis garCevis Semdeg (15 wT) gadavdivarT teqstSi warmodgenili maga­ liTebis ganxilvaze. teqstSi monacemebis warmodgenis ramdenime magaliTia ganxiluli. `a~ punqtSi obieqtebis Tvisebrivi klasifikaciis magaliTia warmodgenili. I SemTxvevaSi aris mravalkuTxedebis sixSireTa cxrili warmodgenili, II SemTxvevaSi _ brtyeli da sivrculi figurebis. `b~ punqtSic mosaxleobis Tvisebrivi klasifikaciis magaliTia _ klasificireba xdeba asakobrivi jgufebis mixedviT. moswavleebs vTavazobT sixSireTa cxrilis analogiuri cxrilis Sedgenas, sadac adamianebis raodenoba WdeebiT aris warmodgenili. asakobrivi jgufebi 7 wlamde 7-dan 17 wlamde 18-dan 35-wlamde 36-dan 50 wlamde 51-dan 70 wlamde 70 wels zemoT

erTeulebi

raodenoba 1 11 11 19 13 5

moswavleebi darwmundebian, rom gamokiTxvisas monacemTa aseTi dafiqsireba xelsayrelia, saboloo Sedegebis daTvlac iolad xerxdeba. `g~ punqtSi piqtogramiT warmodgenilia qarxnebis raodenobrivi klasifikacia. piqtogramiT SeiZleba vupasuxoT kiTxvebs: • ramdeni puri gaiyida I qarxnis mier? 5⋅2000=10000. • ramdeni puri gaiyida II qarxnis mier? 2⋅2000+1000 (naxevari figura 1000 purs niSnavs). • romeli qarxnidan gaiyida yvelaze meti puri? da a. S. am sakiTxebis ganxilviT grZeldeba Temis damuSavebis sawyisi faza, `winaswar~. amocanebs or gakveTilze gadavanawilebT. klasSi. 1

piqtograma, metwilad, didi raodenobis monacemebis TvalsaCino warmodgenisas gamoiyeneba. savaraudod, am amocanaSi moswavleebi sixSireTa cxrils mianiWeben upiratesobas. 2

warmodgenili monacemebis mixedviT miviRebdiT, rom am moswavleebSi zogierTi swavlobs or enas mainc. 5

dasrulebuli cxrili saxelmZRvanelos pasuxebSia. mTliani xarji 6 Tvis

203


ganmavlobaSi SeiZleba bolo striqonis, an bolo svetis monacemebis SekrebiT miviRoT. es, Tavis mxriv, kontrolic aris. yvelaze meti xarji iyo TebervalSi, rac gaTbobasTanaa dakavSirebuli. samsuliani ojaxi wyalSi 9,45 lars ixdis, e. i. sulze gadasaxadi 3,15 laria. Tu sulze gadasaxadi 0,45 lariT gaizrdeba, samsulian ojaxze gadasaxadi gaizrdeba 1,35 lariT TveSi, 6â&#x2039;&#x2026;1,35=8,1 lariT 6 TveSi. saSinao 2

savarjiSos pasuxi saxelmZRvaneloSi ar mivuTiTeT, radgan mis Sesasrule­ blad moswavlem Tavi ar aaridos monacemTa Ziebas. 3

am savarjiSos Sesruleba moswavlis fantaziis unarzea damokidebuli. kompaqt-diskebis warmodgenisTvis savaraudod wres SearCeven, simRerebisTvis SeiZleba violinos gasaRebi an notis Canaweri gamoiyenon, cxovelebisTvis daTunia da a. S. magram ar aris aucilebeli, rom suraTi gamosaxavdes obieqts. SeiZleba mdinaris, an amomrCevlis gamosaxvisTvis wre an marTkuTxedi gamoiyenon.

rusuli

biologia

lomiZe

+

_

+

+

+

_

razmaZe

+

+

+

_

_

+

Samugia

_

+

+

+

+

_

gorgaZe

+

+

_

_

_

+

da a. S.

moswavle / sagani

istoria

magaliTad, SevadginoT cxrili ramdenime moswavlisTvis inglisuri

5

maTematika

davalebas klasSi Sesrulebuli me-5 amocanis msgavsad Seasruleben.

qarTuli

4

...

inglisuri

istoria

rusuli

biologia

mosw. raodenoba, romlebisTvisac es sagani sayvarelia

3

3

3

2

2

2

4 da vavaloT.

204

5

da a. S.

sagani

maTematika

sixSireTa cxrils SeiZleba aseTi saxe hqondes: qarTuli

amocanebis Sesruleba SeiZleba moswavleebs 119-e gakveTilisTvis da-


§3. svetovani da wriuli diagramebi 119-e da 120-e gakveTilebi

Tema:L svetovani da wriuli diagramebi. mizani: moswavle gaiwafos klasificirebuli monacemebis svetovani da wriuli diagramiT warmodgenaSi; diagramebiT warmodgenili monacemebis aRweraSi; gaiazros am masalis aqtualuroba; gaecnos ricxvis Caweris formas _ procents; gaiwafos ricxvis procentis saxiT CaweraSi da procentiT warmodgenili ricxvis wiladis saxiT CaweraSi (maT.VI.12). winapirobebi: monacemTa warmodgena cxrilebiT, maT Soris dajgufebuli raodenobrivi monacemebis SemTxvevaSi. resursebi: kalami, rveuli, saxelmZRvanelo. swavleba gadadis axal fazaSi: `ganmavlobaSi~. es gakveTilebi wina gakveTilebis gagrZelebaa – kvlav raodenobrivi da Tvisebrivi monacemebis mowesrigebasa da xelsayreli formiT warmodgenasTan gvaqvs saqme, magram amjerad – wriuli da svetovani diagramebis saSualebiT. pirvelad ganvixilavT ricxvis procentiT Caweris formas, es forma Zalzed mosaxerxebelia wriuli diagramebis Sedgenis dros. procenti VI klasis standartSi moxseniebuli ar aris, magram Cven saWirod CavTvaleT misi moxsenieba, igi xom ricxvis Caweris erT-erTi formaa, ricxvi SeiZleba Caiweros aTwiladis, wiladis, procentis saxiT. VI klasis standarti iTvaliswinebs ricxvis nawilisa da nawiliT ricxvis povnas. analogiuri amocanebi procentis SemTxvevaSi VII klasis standartiT aris gansazRvruli. magram ramdenime martivi magaliTis ganxilva procentze qmnis safuZvels, rom aRniSnuli amocanebis amoxsna momavalSi moswavleebma kargad aiTvison. es yvelaferi unda icodes maswavlebelma. samwuxarod, bevri maswavlebeli ar icnobs standarts da mis moTxovnebs. zogjer VI klasSi 50 wlis winandel (sabWoTa) saxelmZRvaneloebs iyeneben, romlebic ar Seesabameba axal erovnul saswavlo gegmas da mis moTxovnebs. saSinao davalebis Semowmebis Semdeg, saxelmZRvanelos teqstis gamoyenebiT, gadavdivarT axali Temis garCevaze. teqstis `a~ punqtSi erT-erT dRes gayiduli sasmelebis raodenobebis diagramiT

samosi

warmodgenis magaliTia ganxiluli. svetovani diagramis horizontalur RerZze mo­ niSnulia Tvisebrivi monacemebi, vertikalurze _ saTanado sixSireebi. yoveli svetis simaRle uCvenebs Sesabamisi monacemis raodenobas. `b~ punqtSi Tanxis ganawilebis wriuli diagramaa warmo­ dgenili. SeiZleboda es monacemebic svetovani diagramiT warmogvedgina. wriuli diagramiT warmodgenisas umjobesia monacemTa 3 , 1 . Tu am ganawileba warmovadginoT nawilebis saxiT: 34 , 20 10 wiladebs 100-is toli mniSvneliT warmovadgenT, monacemTa ganawilebas procentebis saxiTac miviRebT (SesaZlebelia aseTi gaerTmniSvnelianebis nacvlad mocemuli wiladebi

205


warmovadginoT aTwiladebis saxiT. usasrulo aTwiladebi maTi miaxloebiTi mniS­ vnelobebiT SevcvaloT xolme. am gziTac SevZlebT procentebis gamoyenebas). teqstis bolos moyvanilia wiladis procentiT da procentis wiladiT Caweris magaliTebi. klasSi. 1 - 4 savarjiSoebSi xdeba ricxvis procentis saxiT Cawera. amisaTvis es ricxvi unda gaamravlon 100-ze (teqstis `b~ punqti). 7

SevadginoT sixSireTa cxrili: 4 lari

5 lari

6 lari

7 lari

8 lari

9 lari

4

5

9

3

4

2

gamovsaxoT monacemebi svetovani diagramiT, romelic saxelmZRvanelos pasu­ xebSia warmodgenili.

8 saxelebis odenobebi warmovadginoT procentebiT:

mariami nino

es cxrili gagviadvilebs wriuli diagramis Sedgenas. diagramebi saxelmZRvanelos pasuxebSia warmodgenili.

nikolozi Tornike

6 20 =30% 4 20 =20% 4 20 =20% 2 20 =10% 2 20 =10% 2 20 =10%

sandro raodenobis 10 eka dasadgenad unda gamoTvalon 30-is 10%, anu 30-is 100 1 nawili): 30⋅ 1 =3 moswavle; kargi miiRo 50%nawili (anu 10 10 ma, anu naxevarma _ 15 moswavle; saSualo _ 30-is 30%-ma, 30 nawilma ( 3 -ma): 30⋅ 3 =9; aradamakmayofilebeli _ danarCenma 3-ma. anu 30-is 100 10 10 es gamoTvlebi SeiZleba gavamartivoT _ kargosnebis raodenoba friadosnebis raodenobaze 5-jer metia, saSualo miiRo friadosnebze 3-jer metma.

9

30-bavSvian

klasSi

friadosnebis

klasSi amosaxsneli amocanebi SeiZleba or gakveTilze ganvixiloT. saSinao dava1

lebis xolo

5

- 4 - 8

amocanebis amoxsna moswavleebs 120-e gakveTilisTvis movTxovoT, – Semdegi 121-e gakveTilisTvis.

1

- 4 amocanebi ricxvis procentis povnas, procentiT gamosaxvas da Sebrunebuli amocanebis amoxsnas exeba. aq umjobesia zogjer aTwiladebi gamoviyenoT, zogjer wiladebi, romlebsac miviyvanT 100-is tol mniSvnelian wiladze, Semdeg ki CavwerT procentis saxiT. 1

25%=0,25= 14

2

3

206

3 = 3⋅25 = 75 =75%, an 3 =0,75=75%. 4 4⋅25 100 4 gaferadebuli wreebi aris wreebis 12 , anu 50%.


4 7

1 =0,125=12,5%, an 1 = 125 = 12,5 =12,5% 8 8 1000 100 ianvarSi – vipovoT, magaliTad, xarjebi transportze da samedicino momsax-

urebaze: 800⋅0,15=120 (lari) 800⋅0,1=80 (lari) TebervalSi – 840⋅0,15=126 (lari) 840⋅0,1=84 (lari) a) 248 aris gamaTboblebis mTeli raodenobis 8%, 1% iqneba 31 gamaTbobeli, 100% iqneba 3100 gamaTbobeli. b) zamTarSi da Semodgomaze gaiyida 87%, danarCen dros 13%. zamTarSi da Semo­ dgomaze gaiyida 87 13 ≈6,7-jer meti. g) Semodgomaze gaiyida 3%-iT meti, anu 3⋅31=93 gamaTbobliT meti, vidre zamTarSi. 8

saSinao davalebad, damatebiT, SeiZleba gamoviyenoT 5 atebiTi amocanebidan.

- 14 davalebebi Tavis dam-

§4. monacemTa Semajamebeli ricxviTi maxasiaTeblebi 121-e da 122-e gakveTilebi

Tema: Semajamebeli ricxviTi maxasiaTeblebi. mizani: moswavle gaecnos monacemTa erTobliobis Semajamebel ricxviT maxasia­ Teblebs _ monacemTa udides da umcires mniSvnelobebs, ariTmetikul saSualos; ariTmetikuli saSualos gamoTvlis wess da mis Tvisebebs, gaiwafos ariTmetikuli saSualos gamoTvlaSi. gamoiyenos misi Tvisebebi amocanebis amoxsnisas. winapirobebi: moswavles SeuZlia monacemebis mowesrigeba da xelsayreli formiT warmodgena, wriuli da svetovani diagramebis ageba, cxrilebis Sedgena dajgufebuli monacemebisaTvis. resursebi: kalami, rveuli, saxelmZRvanelo. pirvel gakveTils viwyebT saSinao davalebis SemowmebiT, Tavis damatebiTi amocanebis 5 - 14 . am Semowmebas SeiZleba 15-20 wuTi movandomoT. 5 amocanis mixedviT moswavleebi yvebian im monacemebis Sesaxeb, romelTa Segroveba daainteresebdaT maT istoriaSi, an geografiaSi, rogor moiZiebdnen monacemebs, ra saxiT warmoadgendnen. xom ar gaxdeboda saWiro monacemTa klasifikacia? 7 da 8 davalebebic internetis, an cnobarebis gamoyenebiT monacemTa Segrovebasa da warmodgenas exeboda. maswavleblis drois dazogvis mizniT gawvdiT informacias davalebis Sesrulebis sisworis Sesamowmeblad: wynari okeanis zedapiris farTobi 179,7 milioni km2-ia, atlantis okeanis – 93,4 milioni km2, indoeTis – 74,9 milioni km2, CrdiloeTis yinulovani okeanis – 13,1 milioni km2; mdinare nilosis sigrZea 6695 km, amazonis 6516 km, ianZis 6380 km, misisipis – 6275 km, eniseis – 5539 km.

207


cxadia, es monacemebi moswavleebis mier mopovebul monacemebs miaxloebiT daemTxveva. 7 - 10 svetovani diagramis agebis dros moswavleebma SeiZleba raime masStabi gamoiyenon. 11 wriuli diagramis aRwera niSnavs imas, rom aRvniSnoT, magaliTad, vaxSmisas miRebuli sakvebi orjer aRemateba II sauzmis dros miRebul sakvebs, sadilis dros ki sakvebis TiTqmis naxevari miiReba. 12 am wriuli diagramis Sedgena moswavleebs ar unda gauWirdeT, amasTanave, SeiZleba procentebis gamoyenebac. magaliTad, Tu klasis 30 moswavlidan 18 gogonaa, 12 6 60 biWi, maSin 18 30 = 10 = 100 =60% gogonaa, danarCeni 40% – biWebi. 13 da 14 amocanebSic wriuli diagramebis ageba moiTxoveboda. axali masalis axsna SeiZleba saxelmZRvanelos teqstSi warmodgenili magaliTebiT CavataroT; aq winapirobis gamoyenebac dagvWirdeba – monacemebis cxriliT warmodgena.

monacemebis mopovebis da klasificirebis Semdeg moswavles unda SeeZlos mona­ cemTa erTobliobaSi arsebuli kanonzomierebis SemCneva, gaanalizeba; ricxviTi maxasiaTeblebis saSualebiT ramdenime erTobliobis Sedareba. am ganxilvebis Semdeg dgeba Temis ganxilvis mesame faza – `Semdgom~. aseTi konstruqtivistuli samfaziani swavlebis strategia maqsimaluri nayofierebiT gamoirCeva. teqstSi magaliTzea aRwerili ariTmetikuli saSualos gamoTvlis wesi. gamoTvli­ lia saSualo (miaxloebiT 22,7-ia) da Sedarebulia monacemebis udides (32) da umcires (19) mniSvnelobebTan. sazogadod, Tu monacemTa erToblioba erTi da imave ricxvebisgan Sedgeba, saSua­ lo emTxveva am ricxvs _ is umcires monacemadac SeiZleba CavTvaloT da udides monacemadac. vTqvaT, a1, a2, ..., an monacemebidan ori mainc gansxvavebulia, umciresi aRvniSnoT a-Ti, udidesi b-Ti, maSin n na<a1+a2+...+an<nb, saidanac a< a1+a2+ ... +a <b. n rac niSnavs, rom monacemTa saSualo yovelTvis metia umciresze da naklebia udides monacemze. xuTi monacemis SemTxvevaSi moswavleebi am faqts indeqsebiani Canawerebis gareSe, Tundac sityvieri aRweriT daasabuTeben. kidev erTi magaliTis ganxilvis Semdgom, saxelmZRvaneloSi gakeTebuli miniSnebis gaanalizebiT moswavleebi daaskvnian, rom saSualo, vTqvaT, a, tolia im erTobliobis saSualosi, romlis yvela monacemi a-s tolia. klasSi amosaxsnel

1

-

6

amocanebs, moswavleTa aqtiuri CarTulobiT, pirvel

gakveTilze vxsniT; danarCen 7 - 11 amocanebs – meoreze. 1 - 5 Semdegi gakveTilisTvis davalebad SeiZleba amocanebi gaviTvaliswinoT, xolo 6 - 11 amocanebis amoxsnebs moswavleebi 123-e gakveTilisTvis amzadeben. amave gakveTilisTvis aris gansazRvruli saxelmZRvaneloSi warmodgenili proeqtis Sesrulebac.

208


miTiTebebi amocanebis amosaxsnelad: 1 miaqcieT yuradReba: zogjer moswavleebi 0-s wevrad ar Tvlian da saSualos gamoTvlisas 4 monacemis jams gayofen 3-ze, rac Secdomaa. aTives jamia 260. kaxis wasvlis Semdeg darCeba 9 adamiani, romelTa asakebis jami 240-ia, saSualo aris: 240 9 >26. 4 saSualo umciressa da udidess Soris unda iyos. 3

5

da 6 testebis Sesrulebisas SeiZleba moswavleebma kerZo magaliTs mimarTon _ moifiqron 4 ricxvi, romelTa saSualo 10-ia. magaliTad, 8, 12, 9 da 11. TiToeulis 2-iT gazrdis Semdeg (amocana 5 ) miviRebT: 10, 14, 11 da 13, maTi saSualoa 12; erTerTis 4-iT gazrdis Semdeg (amocana 6 ) miviRebT: 8, 12, 9 d 15, maTi saSualoa 11. umjobesia msjeloba: 5 -Si: 4 ricxvis jami iyo 40. TiToeulis 2-iT gazrdis Semdeg gaxda 48, saSualo: 48 =12. 4 Tqven SeiZleba klass SesTavazoT (Tu amas gamarTlebulad miiCnevT) me-5 amocanis ganzogadebac: rogor Seicvleba 4 ricxvis saSualo, Tu TiToeuls gavzrdiT k ricxviT? amoxsna SeiZleba asec warmovadginoT: vTqvaT, es oTxi ricxvia a, b, c da d, maTi saSualoa m. maSin a+b+c+d =m. gazrdis Semdeg saSualo iqneba 4 a+k+b+k+c+k+d+k = a+b+c+d+4k =m+k. 4 4 amrigad, saSualoc gaizarda imave k ricxviT. 6 -Si 4 ricxvis jami iyo 40. erT-erTis 4-iT gazrdis Semdeg gaxda 44, saSualo _

44 =11. 4 11-ives asakebis jamia 18⋅11=198. mekaris garda 10 moTamaSis asakebis jamia 17,8⋅10=178. mekaris asakia 198–178=20 weli. 7 8

rvaves jamia 1152, 200-is garda darCenili 7-is jamia 1152–200=952, maTi saSualoa

952 =136. 7 9

garda gamoTvlebisa, amocanaSi igulisxmeba daskvnebis gamotanis unaris ganviTarebac. informaciis damoukideblad mopovebis (saklaso Jurnalidan) da am informaciis xelsayreli formiT (am SemTxvevaSi cxriliT) warmodgenis Cvevebis ganviTareba. moswavleebi, savaraudod, dekemberSi gacdenebis matebas grips da sicives daukavSireben. ar aris gamoricxuli, rom zogierTma mizezad winasadResaswaulo ganwyoba daasaxelos. 10 4 mcvelis asakebis jami iyo: 32⋅4=128. kalaZis 39 wlis fexburTeliT SecvliT 4

mcvelis asakebis jami gaxda 128_26+39=141, saSualo asaki: 141 4 =35,25≈35,3. saSualo xelfasia 640 lari, saSualo xelfasi aRemateba 9 TanamSromlis xelfass, anu xelmZRvanelis garda yvelas xelfass. is naklebia mxolod xelmZRvanelis xel­ fasze. saSualo xelfasi am sawarmos TanamSromelTa Semosavlebs kargad ver aRwers. es gamowveulia imiT, rom xelmZRvanelis xelfasi mniSvnelovnad aRemateba danarCenTa xelfass. amis gamo monacemTa erTobliobis SefasebisTvis sxva ricxviT 11

209


maxasiaTeblebsac iyeneben, romlebsac moswavleebi momavalSi gaecnobian (moda, mediana, diapazoni, saSualo kvadratuli gadaxra). saSinao 10-iani miiRo 8 moswavlem, e. i. am 8 moswavlis qulaTa jami aris 10⋅8=80; 8-iani _ 16-ma moswavlem, qulebis jami aris: 8⋅16=128 da a. S. 50 moswavlis mier miRebuli qulebis jami aris: 10⋅8+8⋅16+6⋅24+4⋅2=360, saSualo qula _ 360 50 =7,2. 9

rvave ricxvis jami aris: 144⋅8=1152. samis jami aris 452, danarCeni xuTis _ 700, am xuTis saSualo aris: 700 5 =140. 10

25 maswavleblis asakebis jamia 1350, direqtorianad asakebis jamia 53⋅26=1378. direqtoris asakia 1378-1350=28 weli. 11

150

100

paraskevi

xuTSabaTi

oTxSabaTi

samSabaTi

50

orSabaTi

1. svetovan diagramaze warmodgenilia orSabaTidan paraskevis CaTvliT Teonas mier mecadineobaze daxarjuli dro dReebis mixedviT. sul ramdeni dro daxarja Teonam mecadineobaze orSabaTidan paraskevis CaTvliT?

(dro, wT)

proeqtze muSaobiT moswavleebi gaecnobian sportuli samyaros saintereso mxareebs, ufro metad uqomageben qveynis wamyvan sportsmenebs, gundebs. amavdroulad, am muSaobiT moswavleebi ufro daxelovndebian teqnologiebis gamoyenebiT saWiro informaciis moZiebasa da maTi saTanado formiT warmodgenaSi. am gakveTilebis Catarebis Semdeg SeiZleba CavataroT fokusirebuli damoukidebeli wera, romlisaTvisac gTavazobT amocanebs:

(dReebi)

2. klasSi 18 gogonaa da 6 vaJi. klasSi biWebisa da gogonebis raodenobebi svetovani diagramiTaa warmodgenili. gogonebis Sesabamisi svetis simaRle 3 sm-ia. ramdeni sm-ia biWebis Sesabamisi svetis simaRle? gazafxuli

3. wriul diagramaze gamosaxulia gayiduli turistuli sagzurebis raodenobis ganawileba wlis droebis mixedviT. cnobilia, rom wlis ganmavlobaSi gaiyida 240 sagzuri. ramdeni sagzuri gaiyida zafxulSi?

210

zamTari 20%

35% zafxuli

10%

35% Semodgoma


zamTari

4. wriul diagramaze gamosaxulia gayiduli avtomobilebis raodenobis ganawileba wlis droebis mixedviT. cnobilia, rom zafxulSi gaiyida 108 avtomobili. sul ramdeni avtomobili gaiyida mTeli wlis ganmavlobaSi? biWebi 32%

15%

gazafxuli 35%

10%

Semodgoma

40% zafxuli

5. wriul diagramaze warmodgenilia klasSi biWebisa da gogonebis raodenobebis ganawileba. cnobilia, rom am klasSiU biWebis raodenoba 9-iT naklebia gogonebis raodenobaze. ramdeni moswavlea am klasSi?

gogonebi

pasuxebi da miTiTebebi: 1. Teonam daxarja (80+150+100+100+50) wT=480 wT=8 sT. 2. 18 gogonas Seesabameba 3 sm simaRlis sveti, biWebi 3-jer naklebia, Seesabameba 3-jer dabali – 1 sm simaRlis sveti. 35 nawili, 240⋅ 35 =84 sagzuri. 3. zafxulSi gaiyida 240-is 35% anu 240-is 100 100 15 nawili. x⋅ 15 =108, 4. 108 aris mTeli wlis ganmavlobaSi gayiduli avtomobilebis 100 100 saidanac x=720. 5. gogonebi klasis 68%-s adgenen, 36%-iT mets biWebze. isini, pirobiT 9-iT metni 36 =9, x=9:⋅ 36 =25. klasSi 25 moswavlea. arian, 4% aris 1, 100% aris 25; Aan ase: x⋅ 100 100 SeiZleba CavataroT diagnostikuri wera veb-portal `kargi skolis~ mier SemoTavazebuli masaliT (G-PriEd). Cvenc gTavazobT amisTvis davalebebs, V Tavis damatebiT amocanebs 15 - 50 . es davalebebi moicavs mTeli wlis masalas mimdevrobis dacviT. saswavlo procesisa da moswavleTa individualuri SesaZleblobebis mixedviT maswavlebeli Tavad gaakeTebs arCevans, gamoiyenos isini diagnostikuri werisaTvis, Tu gameorebisTvis. masalis gasameoreblad da Semajamebeli muSaobisTvis (moswavleTa TviTSefasebisTvisac) gTavazobT daskvniT davalebebsac, romlebic amave saxelwodebis rubrikiTaa warmodgenili.

§5. amocanebis amoxsnis sxvadasxva xerxi 123-e da 124-e gakveTilebi

Tema: amocanebis amoxsnis sxvadasxva xerxi. mizani: moswavle gaecnos amocanebis amoxsnis sxvadasxva xerxs, man SeZlos amocanis amoxsnisas damoukideblad SearCios xelsayreli xerxi da gamoiyenos is. moswavlem SeZlos amowurvis da gamoricxvis xerxebis gamoyeneba amocanebis amoxsnisas (magaliTad, amocanebi variantebis daTvlaze) (maT. VI. 4.); sxvadasxva terminis (`yvela~, `zogierTi~, `arcerTi~... ) sworad gamoyeneba (maT. VI. 4.). winapirobebi: amocanebis amoxsna gantolebis SedgeniT, moqmedebaTa Tvisebebis gamoyeneba algebruli gamosaxulebebis gamartivebisas (maT. V. 6.), monacemTa warmodgena cxrilis saxiT. resursebi: saxelmZRvanelo, rveuli, fanqrebi, saxazavi, dafa.

211


gakveTils viwyebT saSinao davalebis SemowmebiT, proeqtis prezentaciebiT (2025 wT). Semdeg viwyebT gakveTilze dasaxuli miznebis ganxorcielebas Sesabamisi amocanebis ganxilviT. viwyebT im amocanebiT, romlebic saxelmZRvanelos teqstSia warmodgenili. pirveli gakveTilis danarCeni monakveTi SeiZleba mxolod am amocanebis garCeviT da Sesabamisi xerxebis aRweriT Semoifarglos. `a~ punqtSi amocanis amoxsnisTvis cxrilebs viyenebT. maTi Sevsebis principi aRwerilia teqstSi. moswavleebi rveulSi ixazaven cxrils da daasruleben Sevsebas: qimiis svetSi erTis garda yvela ujra Seivso _ `–~-iT, e. i. qimikosi aris karlo da misi striqonis da karlos svetis gadakveTaze unda Caiweros `+~. karlos striqoni Seivseba `–~-ebiT. mesame wanamZRvris mixedviT, levanis striqonSi sami `-~ Caiwereba da meoTxe ujra (biologiis svetSi) Seivseba `+~-iT. axla Tornikes striqonSi erTis garda yvela ujra `–~-iT aRmoCndeba Sevsebuli da darCenili meoTxe ujra (maTematikosis svetSi) Seivseba `+~-iT. e. i. spartaki fizikosi yofila. cxrili miiRebs aseT saxes: profesia

fizikosi

qimikosi

maTematikosi

biologi

karlo

_

+

_

_

Tornike

_

_

+

_

spartaki

+

_

_

_

levani

_

_

_

+

stumari

`b~ punqtSi diagramis gamoyenebiT amocanis amoxsnis nimuSia warmodgenili. xSirad winadadeba: `18 msmeneli inglisurs swavlobs~ moswavleebis mier (xSirad mozrdilebis mierac) aRiqmeba, rogorc mxolod inglisuri enis SemswavlelTa odenoba. Tu vinme ambobs: `me kargad vmReri~, es ar gamoricxavs, rom es adamiani kargadac cekvavs, da xatavs kidec. wreebis saerTo nawili Seesabameba im msmenelebs, romlebic orive enas swavloben. inglisuris 18 msmenelidan mxolod inglisurs 8 swavlobs. amitom 10 swavlobs orive enas da saerTo nawilSic 10 Caiwereba. `g~ punqtSi warmodgenili masala diagramebis gamoyenebis unars ganaviTarebs. saxelmZRvaneloSi warmodgenili masalis ganxilva moswavleTaTvis srulad warmoaCens arsebul mimarTebebs. cxadia, pedagogis farTo ganaTleba aq wamoWris ramdenime aspeqts. winadadeba: `yvela meTormeteklaseli apirebs umaRles saswavlebelSi Cabarebas~ SeiZleba ase CamovayaliboT: `Tu adamiani meTormeteklaselia, maSin is apirebs umaRles saswavlebelSi Cabarebas~, anu meTormeteklaselebis simravle aris umaRles saswavlebelSi Cambareblebis simravlis qvesimravle. simravlis qvesimravle SeiZleba emTxveodes am simravles, SeiZleba ar emTxveodes (suraTebi saxelmZRvanelos teqstSia). sazogadod, winadadeba `Tu A, maSin B~ diagramiT ase gamoisaxeba B

212

A=B

an

A


`Tu A, maSin B~-dan ar gamomdinareobs `Tu B, maSin A~. swavlebis am etapze moswavleebTan saswavlo masalas am terminebSi ar ganvixilavT. zogierTma maswavlebelma SeiZleba dasaxuli miznebis misaRwevad nacvlad orisa sami gakveTili gamoiyenos. gTavazobT miTiTebebs da rekomendaciebs amocanebis amosaxsnelad.

klasSi 1

am amocanis amoxsnisas xelsayrelia cxrilis gamoyeneba: I adgili

II adgili

III adgili

IV adgili

CxaiZe

_

+

_

_

bibilaSvili

_

_

+

_

Citaia

_

_

lomiZe

_

_

CxaiZis striqonSi I da IV svetebSi Caiwereba `–~ebi; bibilaSvilis gaswvriv III svetSi Caiwereba `+~. am svetis da bibilaSvilis striqonis danarCeni ujrebi Seivseba `–~-ebiT. axla CxaiZis striqonSi aRmoCndeba sami `–~ da darCenili carieli ujra II svetSi Seivseba `+~-iT, II svetis danarCeni ujrebi `–~ebiT. radgan II adgilze myofma CxaiZem ver ajoba lomiZes, lomiZe I adgilzea da Citaias rCeba IV adgili. bolo winadadebidan (wanamZRvridan) qristine asakiT II adgilzea. radgan irina arc I-zea da arc IV-ze, mas III adgili darCa. irakli ar aris umcrosi _ ar aris IV adgilze. e. i. irakli I-zea, da dimitri boloze _ yvelaze umcrosia. 2

3 viyenebT diagramebs. TiToeuli saxeobis Semswavlelebs TiTo wriT gamovsaxavT. Sevsebis dawyeba wreebis saerTo nawilebidanaa umjobesi. samive cekvas arc erTi ar eufleba, amitom samive wris saerTo nawilSi 0-s vwerT. mxolod or cekvas _ qarTulsa da espanurs 6 eufleba, amitom saTanado nawilSi vwerT 6-s. qarTulisa da laTino-amerikulis Sesabamis nawilSi _ 4-s, espanuris da laTino-amerikulis Sesabamis nawilSi _ 3-s.

qarTuli

6

espanuri

6 4

0

3

laTino-amerikuli

espanuri

6

6 4

qarTuli

0

3

5 laTino-amerikuli

Semdeg etapze msjelobas ase vagrZelebT: Tu qarTuls 16 swavlobs, maTgan 6 swavlobs espanursac, 4 laTino-amerikulsac, e. i. darCenili 16–(6+4)=6 mxolod qarTuls swavlobs. analogiurad, mxolod espanurs swavlobs 15–(6+3)=6, mxolod laTino-amerikuls _ 12–(4+3)=5 amrigad, sul cekvas swavlobs 6+6+6+4+3+5=30 moswavle.

213


4

diagramebiT es ori winadadeba ase gamoisaxeba:

patarebi; tkbileulebis moyvarulebi

tkbileulebis moyvarulebi an

patarebi

• ana

suraTidan davadgenT, rom anas uyvars tkbileuli. maswavlebels am amocanis ufrosklaselebTan ganxilvisas SeeZlo aseTi terminologia gamoeyenebina: `patarebis simravle~ aris `tkbileulis moyvarulTa simravlis~ qvesimravle. ana ekuTvnis `patarebis simravles~. qvesimravlis yoveli elementi simravlesac ekuTvnis, e. i. ana ekuTvnis `tkbileulis moyvarulTa simravles~. sazogadod, Tu A⊂B da a∈A, maSin a∈B.

5 avtomobilis mflobelTa simravle aris marTvis mcodneTa simravlis qvesimravle:

avt. mflobelebi marTvis mcodneebi

marTvis mcodneebi an

marTvis mcodneebi a)

ar gamomdinareobs avt. mflobelebi • irma marTvis

b) magaliTad, mcodneebi ar gamomdinareobs avt. aucileblad mflobelebi • irma marTvis g) mcodneebi

• irma gamomdinareobs avt. mflobelebi

d)

marTvis mcodneebi

avt. ar gamomdinareobs mflobelebi

• irma

214

avt. mflobelebi


saSinao 1

pirobiT, sandros kostiumi da fexsacmelebi sxvadasxva ferisaa da Savi arcerTi ar aris. yavisferi fexsacmelebi nikas acvia, e. i. sandros acvia lurji fexsacmelebi da yavisferi kostiumi. lukas mxolod Savi feris kostiumi da Savi feris fexsacmelebi SeiZleba ecvas. nikas darCa lurji kostiumi. 2

viyenebT cxrils:

pirobis Tanaxmad, dali da lia ar arian maTematikis wris lia _ wevrebi, irma ar aris literatudali _ _ ris wris wevri da dali ar aris irma _ musikis wris wevri. saTanado ujrebSi `–~-ebis Casmis Semdeg dalis striqonSi da maTematikis wris svetSi TiTo carieli ujra darCeba, romlebic `+~-ebiT unda Seivsos. amrigad, dali literaturis wris wevria, irma maTematikis da lia musikis. maTematikis

3

literaturis

musikis

viyenebT cxrils:

pirobiT, cisferi perangi ar ac­viaT daviTs, karlos da zurabs. + daviTi _ `cisferis~ svetis Sesabamis ujrebkarlo _ _ Si `–~-ebis dasmis Semdeg davask+ beso _ _ _ vniT, rom besos acvia cisferi pezurabi _ _ rangi da misi striqonis danarCen ujrebs `–~-ebiT SevavsebT. TeTri perangi ar acviaT zurabs da karlos, TeTri perangi acvia daviTs. karlos ar acvia Wreli perangi. amrigad, karlos perangi Savia, zurabis _ Wreli. cisferi

4

Savi

TeTri

Wreli

viyenebT diagramebs: WQadraki

9

SaSi

6

4

jer vavsebT saerTo nawils ricxviT 6. marto Wadrakze dadis 15–6=9 wevri, marto SaSze 10–6=4 wevri. sul am wreebSi aris 9+6+4=19 wevri. wiTeli

5

wina amocanis analogiurad, gamoviyenebT wreebs wiTeli da Savi fanqrebis aRsaniSnavad: 22 moswavlidan mxolod 9-s aqvs wiTeli fanqari, e. i. 13-s aqvs mxolod Savi fanqari. 18 Savfanqriani moswavlidan 18–13=5-s aqvs orive feris fanqari.

4

Savi

5

13

2-is jeradebi 3-is jeradebi

6

viyenebT diagramebs _ wreebs, romlebic warmoadgens 2-is, 3-is da 5-is jeradi ricxvebis raRac erTobliobebs. Sevsebas saerTo nawilebidan viwyebT. pirobiT, samive wris saerTo nawilSi (TanakveTaSi) 0 iwereba.

4

1

8 7

0

6

5 5-is jeradebi

215


2-is da 5-is jeradebis Sesabamis nawilSi _ 7; 2-is da 3-is jeradebis Sesabamis nawilSi _ 8; 3-isa da 5-is jeradebis Sesabamis nawilSi _ 6. Semdeg etapze vavsebT `gare~ nawilebs. am erTo­ bliobis ricxvebidan 2-is jeradi ricxvebi, romlebic ar aris arc 3-is, arc 5-is jeradi, aris 19–(8+7)=4 cali; 3-is jeradi ricxvebi, romlebic ar aris arc 2-is, arc 5-is jeradi aris 15–(8+6)=1; 5-is jeradi ricxvebi, romlebic ar aris arc 2-is, arc 3-is jeradi, aris 18–(7+6)=5. sul am erTobliobaSi aris 4+8+1+7+6+5=31 ricxvi. 7 gamovsaxoT wanamZRv­ rebi diagramebiT. SeiZleba adgi­ li hqon­ des marjvena suraTze gamosaxul SemTxvevas: daskvna: `mariami dadis ingli­ suris wreSi~ am wanamZRvre­ bidan ar gamomdinare­obs. 8

SeiZleba

adgili

a)

bileTianebi parkSi namyofi isargebles saqanelaTi

friadosnebi; inglisur wreze mosiaruleebi

an

inglisur wreze mosiaruleebi

• mariami

Tbilisis moyvarulebi

hqondes

SemTx­vevas: diagramidan Cans, rom eliso Tbi­ lisis moyvarulia, magram ar aris Tbi­ li­­ seli _ daskvna ar gamomdinareobs wana­mZRvrebidan. 9

friadosnebi

Tbiliselebi; Tbilisis moyvarulebi

an Tbiliselebi • eliso

SeiZleba adgili hqondes SemTxvevas: A iyo parkSi, magram ar isargebla saqanelaTi _ daskvna ar gamomdinareobs wanamZRvrebidan.

• A

bileTianebi parkSi namyofi

b) B-m SeiZina bileTi, magram parkSi ar iyo. daskvna ar gamomdinareobs wanamZRvrebidan.

isargebles saqanelaTi

• B

bileTianebi parkSi namyofi

isargebles saqanelaTi

• C

g) C-m isargebla saqanelaTi, e. i. is aucileblad iyo parkSi da aucileblad SeiZina bileTi. daskvna gamom­ dinareobs wanamZRvrebidan.

Semdeg gakveTilze Catardeba Semajamebeli wera, Temebi: monacemTa warmodgenis xerxebi, amocanebis amoxsna diagramebis gamoyenebiT.

216


123-e gakveTili

TeTri

Semajamebeli wera: mocemulia erTi maRaziis mier erT dReSi gayiduli puris sixSireTa piqtograma:

ruxi Savi _ 20 puri

piqtogramis mixedviT 1-3 kiTxvebSi SearCieT swori pasuxi: 1. naxevari suraTi aRniSnavs a) 12 purs b) 10 purs

g) 1 purs

d) 40 purs.

2. am maRaziidan sul gaiyida a) 10 puri b) 200 puri

g) 190 puri

d) 180 puri.

3. gayiduli purebidan TeTri, ruxi da Savi puris raodenobebi aris, Sesabamisad, mTeli puris raodenobis a) 30%, 55% da 15% b) 60%, 110% da 30% g) 15%, 27,5% da 7,5% d) 3%, 5,5% da 1,5%. SearCieT swori pasuxi: 4. Tu 5 megobris saSualo asaki 11 welia, maTi asakebis jami aris g) 22 d) 45. a) 16 b) 55 5. Tu 5 megobris saSualo asaki 11 welia, 3 wlis Semdeg maTi saSualo asaki iqneba a) 11+5⋅3 b) 11+3 g) 11⋅3 d) 5⋅3 6. gundis 5 sportsmenis saSualo asaki aris 16 weli. Tu 12 wlis goCa Seicvleba 15 wlis kaxaTi da 14 wlis nodari _ 16 wlis daviTiT, maSin saSualo asaki gaxdeba a) 17 weli b) 21 weli g) 15,5 weli d) 18 weli. amoxseniT amocana: 7. jgufSi 24 mamakaci da 16 qalia, gamosaxeT jgufis Semadgenloba sixSireTa cxriliT, svetovani diagramiT, piqtogramiT da wriuli diagramiT. 8. ucxo enebis Semswavlel centrSi 40 swavlobs inglisurs, 25 _ germanuls, 18 _ rusuls. 12 swavlobs inglisurs da germanuls da ar swavlobs rusuls, 8 swavlobs inglisurs da rusuls da ar swavlobs germanuls, 5 swavlobs germanuls da rusuls da ar swavlobs inglisurs, mxolod inglisurs swavlobs 17. ramdeni swavlobs samive enas? ramdeni swavlobs mxolod germanuls da ramdeni _ mxolod rusuls? pasuxebi da miTiTebebi 1

2

3

4

5

6

b

b

a

b

b

a

217


7. sixSireTa cxrili svetebiani diagrama

kaci

qali

24

16

Tu erT naxats 4 adamians SevusabamebT, piqto­gramas eqneba saxe:

 

kaci qali

kacebis raodenoba aris jgufebis 24 40 =60%, qalebis _ 40%. qalebs wriul diagramaze 0  40 0 Seesabameba 360 ⋅  100 =144 -iani kuTxe.

8. gamoiyeneT diagrama: inglisuri

germanuli 12

8

5

rusuli

jer CavweroT Sesabamis nawilSi 12, 8 da 5; mxolod inglisurSi CavweroT 17. Semdeg gamovTvaloT, ramdeni swavlobs samives: 40–(17+8+12)=3; ramdeni swavlobs mxolod germanuls: 25–(12+5+3)=15; ramdeni swavlobs mxolod rusuls: 18–(8+5+3)=2.

Sefasebis sqema testuri amocanebidan TiToeulis swori pasuxi fasdeba 1 quliT, me-7 da me-8 amocanebis srulyofili amoxsnebidan TiToeuli fasdeba 2 quliT. Sefasebis rubrika #7 amocanisTvis 0 qula

araa warmodgenili raime racionaluri nabiji, amoxsnisaken mimarTuli. moswavlem warmoadgina monacemebi an cxrilis saxiT; an svetovani

0,5 qula

218

diagramiT; an piqtogramiT; an wriuli diagramiT procentis miTiTebis gareSe; an gamoTvala jgufis procentuli Semadgenloba.

1 qula

mowavlem warmoadgina monacemebi miTiTebuli oTxi saxidan romelime oriT; an gamoTvala jgufis procentuli Semadgenloba da aago wriuli diagrama.

1,5 qula

moswavlem gamoTvala jgufis procentuli Semadgenloba da aago wriuli diagrama da kidev danarCeni sami formidan (cxrili, svetovani diagrama, biqtograma) sworad gamoiyena erT-erTi; an gamosaxa monacemebi cxriliTac, svetovani diagramiTac da piqtogramiTac.

2 qula

amoxsna amocana srulyofilad.


Sefasebis rubrika #8 amocanisaTvis 0 qula

araa warmodgenili raime racionaluri nabiji, amoxsnisken mimarTuli.

0,5 qula

an Seavso diagramis is areebi, romlebic aris zustad ori simravlis TanakveTa (anu Cawera 12,8 d 5); an gamoTvala ramdeni swavlobs samive enas.

1 qula

Seavso diagramis is areebi, romlebic aris zustad ori simravlis TanakveTa da gamoTvala ramdeni swavlobs samive enas.

1,5 qula

diagrama Seavso srulad, umniSvnelo xarvezebiT, magram kiTxvebs ar gasca pasuxi.

2 qula

davaleba Seasrula srulyofilad.

124-140-e gakveTilebi

mizani:L Semajamebeli weris Sedegebis Sejameba, VI klasSi Seswavlili masalis gameoreba. 124-e gakveTilze jer ganvixilavT Semajamebeli weris davalebebs, tipur Secdomebze visaubrebT. moswavleebi gaecnobian Sefasebis rubrikas da Tavad Seafaseben Secdomis xarisxs. gamorCeviT SeaqeT isini, vinc am analizis Semdeg sworad gansazRvravs saTanado qulebs. Semdeg daurigeT naSromebi da amiT moswavleebisTvis am aqtivobis ganmaviTarebeli efeqti sakmaod Zlieri iqneba. es xangrZliv dros moiTxovs da amitom SeiZleba mTeli gakveTilic ki daeTmos am aqtivobas. Semdegi gakveTilebidan daviwyebT VI klasSi Seswavlilis gameorebas, amisaTvis davixmarT saxelmZRvaneloSi warmodgenil amocanebs (rubrikebi: `V Tavis damatebiTi amocanebi~ da `Semajamebeli amocanebi~). axla gTavazobT miTiTebebs saxelmZRvaneloSi Semajamebeli muSaobisTvis gankuTvnili amocanebis amosaxsnelad. am amocanebis nawili SeiZleba moswavleTa TviTSefasebisTvisac SearCioT. amocanebi gameorebisTvis (Semajamebeli) I. mgzavrebis Tanabari raodenoba rom iyos avtobusebSi, 442 unda iyofodes 7-ze. radgan ar iyofa, amitom SeuZlebelia. II. 7 cali kenti ricxvis jami kentia _ ar SeiZleba 442-is toli iyos. III. Svidive avtobusSi kenti raodenobis mgzavrebis arseboba SeuZlebelia. amitom erTSi mainc mgzavrebis raodenoba luwia. swori pasuxia d) III. 6 2 40 8 15 = 5 = 100 . pasuxi: b) 40. 10 SeiZleba wiladebis Sedareba gaerTmniSvnelianebiT: 4

105 108 I. 100 120 > 120 > 120 mcdaria, II WeSmaritia, III WeSmaritia, IV mcdaria. pasuxi: WeSmaritia II da III. 24 es ricxvebia: 12, 13, 23, 21, 31, da 32.

219


27 gaviTvaliswinoT, rom a<b, b<d, d<c. miviRebT: a<c, anu c>a. SeiZleboda ricx­viTi sxivis gamoyenebac. miviRebT: , saidanac TvalnaTliv Cans, rom a b d c savaraudo pasuxebidan sworia c>a. 31

jer daviTvaloT samkuTxedebi, romelTa erTi wveroa A: samkuTxedebi ADE, ADC, ADF, ABE, ABC, ABF, AEC, AEF _ rva. Semdeg daviTvaloT samkuTxedebi, romelTa wvero ar aris A da erTi wvero aris B: BDE, BDF, BDC, BFC, BFE — xuTi. samkuTxedebi, romelTa wvero ar aris arc A da arc B da erTi wvero aris C: CFD, CFE, CDE _ sami. samkuTxedebi, romelTa wveroebs Soris ar aris A, B da C: DDEF _ erTi. miviReT 8+5+3+1=17 samkuTxedi. es daTvlis erT-erTi strategia iyo. momavalSi SeiZleba es amocana ufro swrafad amoxsnan kombinatorikis formulebis gamoyenebiT: ramdennairad SeiZleba aRmovaCinoT 6 wertilidan 3 wertili mimdevrobis gaTvaliswinebis gareSe (ramdeni 3-elementiani qvesimravle arsebobs 6-elementiani 6! simravlis)? am jufTebaTa ricxvia C   = 3!3! =20. am ricxvs unda gamovakloT im sameulebis raodenoba, romlebic erT wrfezea da samkuTxedebs ver Seqmnis, aseTi sameulebi samia: A, D, B; A, F, C da B, E, C. pasuxi: 20–3=17. es amocana N#33-Tan erTad kombinatorikis formulebis SeswavlisTvis Semzadebaa. 41 es amocana Tvalzomis, figuris zomebis aRqmis, procentis gamoyenebis unaris

ganviTarebisTvisaa. 1) suraTze meoTxedze (25%-ze) `odnav didi~ nawilia daStrixuli. pasuxi: ≈27%. 2) suraTze daStrixuli nawili mTelTan _ 100%-Tan ufro axlosaa, vidre naxevarTan. pasuxi: 80%. 44 SeiZleba cxrili gamoviyenoT:

nika pirobebis mixedviT, CavwerT nikas gaswvriv Tornike or minuss. Tornikes da II-s gadakveTaze vano pluss. am plusis striqoni da sveti SevavsoT kote minusebiT da miviRebT, rom nika I adgilzea. radgan pirobiT, vano IV-zea, kotes darCa III adgili.

I _

II

III

IV

_

_

_

+

_

_

_

+

_

46 a) aseulebis TanrigSi 2 SesaZleblobaa, aTeulebis da erTeulebis TanrigebSi sam-sami. sul 2⋅3⋅3=18 SesaZleblobaa. SeiZleba xisebri diagramis gamoyeneba:

aseulebi aTeulebi erTeulebi TiTqos ori totidan TiToeulze sam-sami toti gamoeba, miRebuli eqvsi totidan TiToeulze kvlav sam-sami toti. sul 18 toti _ 18 SesaZlebloba.

220


47 b) 9 birTvis amoRebis SemTxvevaSi SeiZleba 10 birTvi _ aucileblad TeTri iqneba. ori TeTri 11. 3 birTvis amoRebis SemTxvevaSi SeiZleba samive _ iqneba wiTeli. ori wiTeli mainc rom amovides,

cxrave iyos wiTeli, Semdegi _ memainc rom amovides, unda amoviRoT iyos TeTri. Semdegi _ me-4 birTvi unda amoviRoT 5.

48

moswavleTa raodenobebis cxrili procentebis miTiTebiT, saxelmZRvanelos pasuxebSia. procentebSi gamosaxuli sixSireebis Sesabamisi kuTxeebi miaxloebiT aris 133,20, 25,60; 34,60 da 166,70. wriuli diagrama saxelmZRvanelos pasuxebSia. cxadia, moswavlem wre seqtorebad am zomebis miaxloebiTi mniSvnelobebis mixedviT unda dayos. 49 a) germanuls eswreba 30%. 60-is 30% aris 18.

30 =24 msmeneliT meti. b) inglisurs eswreba 30%-iT meti, vidre franguls, anu 80⋅ 100 30 =1080, franguls 360⋅ 20 =720, ing­ wriul diagramaze germanuls Seesabemeba 360⋅ 100 100 lisurs _ 1800. 50

radgan warwera `Savi da TeTri~ arasworia, am yuTSi orive birTvi erTi da imave feris unda iyos _ an orive Savi, an orive TeTri, amitom am yuTidan birTvis amoRebiT davadgenT am yuTSi birTvebis ferebs. ori SesaZlebloba gvaqvs: I. `Savi da TeTrSi~ _ ori Savi birTvia, maSin ori TeTri iqneba `ori SavSi~ da TeTri da Savi birTvebi iqneba `or TeTrSi~. II. `Savi da TeTrSi~ _ ori TeTria, maSin ori Savi birTvi iqneba `or TeTrSi~ da TeTri da Savi birTvebi iqneba `or SavSi~. sasurvelia, kidev erTi Semajamebeli wera Catardes, romelic mieZRvneba VI klasSi Seswavlil sakiTxebs – wrewirebis urTierTganlageba, ricxvis nawilis povna da Sebrunebuli amocanis amoxsna, farTobi, gantolebis Sedgena da amoxsna, monacemTa analizi. Semdeg unda moxdes am weris Sedegebis ganxilva da moswavleTa mier naSromebis TviTSefaseba.

Semajamebeli wera 1. mocemulia erTi firmis mier erT TveSi gayiduli kompiuterebis sixSireTa piqtograma: dabalfasiani saSualofasiani maRalfasiani cnobilia, rom saSualofasiani gaiyida 120-iT meti, vidre dabalfasiani. ramden kompiuters warmoadgens erTi suraTi? b) 20-s g) 24-s d) 48-s. a) 40-s 2. cnobilia, rom m da n naturaluri ricxvebia da 2m+n kentia, maSin a) n luwia b) 2n+1 luwia g) n+1 luwia d) n+2 luwia.

221


3. papunam sam sakontrolo weraSi miiRo Sefasebebi 72, 84 da 96. ramdeni qula miiRo papunam IV sakontrolo weraSi, Tu am oTx weraSi miRebuli qulebis saSualo 80 aRmoCnda. a) 80 b) 85 g) 75 d) 68. 4. 2,7x+0,15=1,5 gantolebis fesvia a) 5 b) 0,5

g) 0,05

d) 15 .

5. diagramaze warmodgenilia gayiduli xilis pro­ centuli raodenobebi. Tu cnobilia, rom komSi gaiyida 21 kg, maSin vaSli gaiyida a) 144 kg b) 48 kg g) 24kg d) 96 kg. 6. Tu ori wrewiris centrebs Soris manZili 4 dm-ia, maTi diametrebi ki aris 44 sm da 36 sm, maSin wrewirebs a) aqvs ori saerTo wertili b) aqvs erTaderTi saerTo wertili g) ara aqvs saerTo wertili d) aqvs saerTo centri. amoxseniT amocanebi: 7. kesom 800 laridan 25 nawili komunalur gadasaxadebSi gadaixada, darCenilis 58 _ bankSi Seitana. ramdeni lari darCa kesos? 8. vTqvaT, 1 m2 parketis dageba 15 lari Rirs. oTaxSi iatakze parketis dagebisTvis 360 lari gadaixades. iatakis sigane 4,8 m-ia. ipoveT kedlebis farTobebis jami, Tu oTaxis simaRle 3 m-ia. pasuxebi da miTiTebebi: 1

2

3

4

5

6

d

g

d

b

a

b

1. ori suraTiU da naxevari (anu xuTi cali naxevari) 120 kompiuters warmoadgens, naxevari suraTi warmoadgens 120:5=24 kompiuters, mTeli suraTi _ 48 kompiuters. an ase: 120:2,5=24. 2. 2m luwia, amitom n kentia, n+1 _ luwi. 3. Tu IV weris Sedegs x-iT aRvniSnavT, miviRebT: (72+84+96+x):4=80, saidanac x=68. 5. pirobiT, 7% 21 kg-ia, 1% _3 kg, vaSli aris 48%, anu 48⋅3=144 kg. 6. wrewirebis radiusebi aris 22 sm da 18 sm, maTi jami _ 40 sm _ centrebs Soris manZilis tolia, e. i. es wrewirebi exeba erTmaneTs, maT erTaderTi saerTo wertili aqvs. 7. kesom komunalur gadasaxadebSi gadaixada 800⋅ 25 =320 lari, bankSi Seitana (800–320)-is 58 , anu 480⋅ 58 =300 lari; darCa 800–(320+300)=180 lari.

222


SeiZleba imsjelon ase: kesom bankSi Seitana gadasaxadebis gadaxdis Semdeg 9 nawili, anu 800⋅ 9 =180 darCenili 35 -is 58 , anu 38 nawili. darCa 800 laris 1–( 25 + 38 )= 40 40 lari. 8. iatakis farTobi yofila 360:15=24 m2, sigrZea 24:4,8=5 m, kedlebis farTobebis jamia. (5⋅3+4,8⋅3)⋅2=58,8 (m2). ufro martivad ase CavwerT: (5+4,8)⋅3⋅2=9,8⋅6=58,8 (m2). Sefasebis sqema. testuri amocanebidan TiToeulis sworad amoxsna fasdeba 1 quliT, me-7 da me-8 amocanebis srulyofili amoxsnebi _ or-ori quliT. Sefasebis rubrika #7 amocanisTvis 0 qula

ar aris amoxsnisken mimarTuli raime racionaluri mcdeloba.

0,5 qula

gamoTvala komunalur gadasaxadebSi gadaxdili Tanxa.

1 qula

gamoTvala komunaluri gadasaxadebis gadaxdis Semdeg darCenili Tanxa, an gamosaxa bankSi Setanili Tanxa nawilebiT.

1,5 qula

ipova komunalur gadasaxadebSi gadaxdili Tanxa da bankSi Setanili Tanxa.

2 qula

amocana amoxsna srulyofilad.

Sefasebis rubrika #8 amocanisTvis 0 qula

ar aris amoxsnisken mimarTuli raime racionaluri mcde­ loba.

0,5 qula

gamoTvala iatakis farTobi.

1 qula

gamoTvala iatakis farTobi da ipova iatakis sigrZe.

1,5 qula

ipova iatakis sigrZe da dawera kedlebis farTobebis jamis gamosaTvlelad formula sworad da ver gamoTvala sworad.

2 qula

amocana amoxsna srulyofilad.

223


Semajamebeli amocanebi

I Tavi 1. ipoveT umciresi ricxvi im samniSna ricxvebs Soris, romlebis cifrebis jami aris 17. 2. ramdeni wuTia erTi saaTis 34 nawili? 3. ramdeni meocedia 25 ? 4. jgufSi moswavleTa raodenoba aris 48. gogonebis raodenoba jgufSi moswavleTa raodenobis 56 nawilia. ipoveT jgufSi biWebis raodenoba. 5. Sarvlis Sesakerad 1,8 metri sigrZis qsovilis naWeria saWiro, kabis Sesakerad ki _ imave qsovilis 2,4 metris sigrZis naWeri. ramdeni santimetriT meti sigrZis naWeria saWiro kabis Sesakerad? II Tavi 1. ramdeniT naklebia 10-ze 8,012? 2. marina zazaze 0,32 m-iT dabalia, Tikaze _ 0,06 m-iT maRali. ramdeni metriT maRalia zaza Tikaze? 3. amoxseniT gantoleba: x:2,4=1,8. 4. 27,6 kg Saqris fxvnili erTnair paketebSi Tanabrad gaanawiles, yovel paketSi _ 1,2 kg. ramdeni parki Seivso Saqris fxvniliT? 5. natos aqvs 240 lari. ramdeni lari darCeba mas, Tu iyidis 9 cal x laris Rirebulebis wigns? III Tavi 1. CawereT ricxvebi: 34 an 38 aTwiladis saxiT da SekribeT. 2. daalageT zrdis mixedviT ricxvebi: 59 ; 35 da 49 . 3. amoxseniT gantoleba: x⋅3  35 = 3. 4. mariamis simaRle 1  25 metria, mamamisi masze 1  14 -jer maRalia. ra simaRlisaa mariamis mama? 5. gasaremontebeli iyo gzis 12  34 km-is sigrZis monakveTi. gaaremontes 5  12 km sigrZis monakveTi. ramdeni km-iT meti sigrZisaa gasaremontebeli monakveTis darCenili nawili garemontebulze?

224


IV Tavi 1. kvadratis gverdis sigrZe 4â&#x20AC;&#x2030; 34 dm-ia. ramdeniT aris meti am kvadratis perimetri misi erTi gverdis sigrZeze? 2. ramdeni gradusi iqneba kuTxe saaTis isrebs Soris oTxis naxevarze? A O

O1 l

3. suraTze gamosaxuli figura simetriulia l wrfis mimarT. OO1=16 sm, OAO1 samkuTxedis perimetri 36 sm-ia. ipoveT wrewirebis radiusebis sigrZeebi. B

4. ABCD marTkuTxedis sigrZe siganeze 3-jer metia. AOD samkuTxedis farTobi 12 sm2-ia. ipoveT am marTkuTxedis gverdebis sigrZeebi.

C O

A

D

5. suraTze kubis Slilia gamosaxuli. cnobilia, rom am figuris perimetri 5,6 dm-ia. ipoveT kubis moculoba.

V Tavi 1. x da y sidideebs Soris damokidebuleba aseTi wesiTaa mocemuli: y=2x+1. ipoveT: a) y, roca x=6; b) x, roca y=21. 2. skolis moswavleTa raodenobis ganawileba gamosaxulia piqtogramaze: biWebi gogonebi erTi suraTi 50 moswavles aRniSnavs. ramdeniT metia biWebis raodenoba gogonebis raodenobaze? 3. wriul diagramaze sami saxis marcvleulis kulturis mosavlis procentuli ganawilebaa warmodgenili. cnobilia, rom simindis mosavali 32 000 tonaa. ramdeni tonaa xorblis mosavali?

simindi

Svria

xorbali

4. daviTi 8-kaciani jgufis xelmZRvanelia. jgufis wevrTa saSualo asaki xelmZRvanelis gareSe 21-ia, xelmZRvanelTan erTad _ 23. ramdeni wlisaa daviTi? 5. ucxo enebis Semswavlel jgufSi mxolod inglisuri da franguli enebi Seiswavleba. jgufis 25 wevri mxolod inglisurs swavlobs, 17 wevri _ orive enas. cnobilia, rom inglisurs 2-jer meti wevri swavlobs, vidre franguls. ramdeni wevria am jgufSi?

225


Sefasebis rubrikebi moswavleTaTvis I Tavi davaleba 1. 0 qula

ver gaerkva amocanaSi.

1 qula

Cawerilia raime samniSna ricxvi, romlis cifrTa jami aris 17, magram is ar aris umciresi.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi: 179.

davaleba 2. 0 qula

ver gaerkva amocanaSi.

1 qula

sworad aris Sedgenili 60-is 34 -is gamosaTvleli ricxviTi gamosaxuleba (60:4â&#x2039;&#x2026;3), magram gamoTvlilia arasworad; an sworad aris gamoTvlili 100-is 34 nawili, Secdomaa daSvebuli saaTis wuTebad warmodgenaSi.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi: 45 wT.

davaleba 3. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoyenebulia wiladis ZiriTadi Tviseba xarveziT.

2 qula

8. amoxsnilia srulyofilad, miRebulia pasuxi: 20

davaleba 4. 0 qula

ver gaerkva amocanaSi.

1 qula

SerCeulia jgufSi gogonebis raodenobis gamosaTvleli ricxviTi gamosaxuleba (60:6â&#x2039;&#x2026;5) da gamoTvlilia, SesaZloa umniSvnelo xarveziT, an miniSnebulia, rom biWebis raodenoba jgufis 16 -ia. amoxsnilia srulyofilad, miRebulia pasuxi: 8.

2 qula davaleba 5. 0 qula

ver gaerkva amocanaSi.

1 qula

SerCeulia amocanis amosaxsnelad moqmedeba (2,4-1,8), magram ar aris misi mniSvneloba gamoTvlili da gadayvanili santimetrebSi, an monacemebi amocanis pirobaSi gadayvanilia santimetrebSi.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi: 60 sm.

226


II Tavi davaleba 1. 0 qula

ver gaerkva amocanaSi.

1 qula

sworadaa SerCeuli ricxviTi gamosaxuleba (10–8,012), magram ar aris gamoTvlili misi mniSvneloba.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi: 1,988.

davaleba 2. 0 qula

ver gaerkva amocanaSi.

1 qula

sworadaa SerCeuli amocanis amosaxsnelad saWiro ricxviTi gamosaxuleba (0,32+0,06), magram ar aris gamoTvlili misi mniSvneloba.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi, warmodgenilia metrebSi _ 0,38 metri.

davaleba 3. 0 qula

ver gaerkva amocanaSi.

1 qula

sworadaa SerCeuli moqmedeba (2,4⋅1,8), magram ar aris gamoTvlili misi mniSvneloba.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi: 4,32.

davaleba 4. 0 qula

ver gaerkva amocanaSi.

1 qula

sworadaa SerCeuli moqmedeba (27,6:1,2), magram ar aris gamoTvlili misi mniSvneloba.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi: 23.

davaleba 5. 0 qula

ver gaerkva amocanaSi.

1 qula

warmoadgina asoiTi gamosaxuleba mcire xarveziT.

2 qula

amoxsnilia srulyofilad, miRebulia pasuxi: 240–9x.

III Tavi davaleba 1. 0 qula

ver gaerkva amocanaSi.

1 qula

Cawera ricxvebi aTwiladebis saxiT da ar Sekriba, an Sekriba SecdomiT.

2 qula

davaleba Seasrula srulyofilad, miiRo pasuxi: 1,125 (0,75+0,375).

227


davaleba 2. 0 qula

ver gaerkva amocanaSi.

1 qula

Cawera ricxvebi aTwiladebis saxiT, an gaaerTmniSvneliana wiladebi, an Seadara romelime ori ricxvi. davaleba Seasrula srulyofilad, miiRo pasuxi: 49 ; 59 ; 35 .

2 qula davaleba 3. 0 qula

ver gaerkva amocanaSi.

1 qula

sworad SearCia moqmedeba gantolebis amosaxsnelad (x=3:3  35 ) da gay-

ofa ver Seasrula. 2 qula

davaleba Seasrula srulyofilad, miiRo pasuxi: 56 .

davaleba 4. 0 qula

ver gaerkva amocanaSi.

1 qula

sworad SearCia moqmedeba amocanis amosaxsnelad (1 25 ⋅1  14 ) da ver Seasrula gamravleba.

2 qula

amocana amoxsna srulyofilad, miiRo pasuxi: 1  34 m, anu 1,75 m.

davaleba 5. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvala gzis gasaremontebeli nawilis sigrZe (12  34 – 5  12 = 7  14 ), SesaZloa, umniSvnelo xarveziT da SearCia moqmedeba amocanis saboloo pasuxis misaRebad: (7  14 –5  12 ). amocana amoxsna srulyofilad, miiRo pasuxi: 1  34 km.

2 qula

IV Tavi davaleba 1. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvala kvadratis perimetri, an migvaniSna, rom perimetri metia kvadratis gverdze 3⋅4  34 dm-iT.

2 qula

amocana amoxsna srulyofilad, miiRo pasuxi: 14  14 dm. (3⋅4 34 an 4⋅4 34 –4 34 ).

davaleba 2. 0 qula

ver gaerkva amocanaSi.

1 qula

daadgina, rom suraTze isrebs Soris kuTxe 600-ia.

2 qula

amocana amoxsna srulyofilad, miiRo pasuxi: 750; (300+300+150).

228


davaleba 3. 0 qula

ver gaerkva amocanaSi.

1 qula

naxazze moniSnulia OAO1 samkuTxedi da aRniSnulia OA da O1A monakveTebis toloba.

2 qula

amocana amoxsnilia srulyofilad, miRebulia pasuxi: 10 sm. ((36–16):2=10).

davaleba 4. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvlilia marTkuTxedis farTobi (12⋅4=48 (sm2)), aRniSnulia marT­kuTxedis gverdebi x da 3x-iT da maTi saSualebiT gamosaxulia marTkuTxedis farTobi (3x⋅x=3x2).

2 qula

amocana amoxsnilia srulyofilad, miiRo pasuxi: 4 sm da 12 sm (3x2=49, x2=16, x=4).

davaleba 5. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvlilia kubis wibo sworad (5,6:14=0,4 (dm)), an SecdomiT gamoTvlili kubis wibos gamoyenebiT da gamoTvlilia kubis moculoba.

2 qula

amocana amoxsnilia srulyofilad, miRebulia pasuxi: 0,064 dm3, anu 64 sm3 (0,43=0,064).

V Tavi davaleba 1. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvlilia y-is mniSvneloba, roca x=6 (y=2⋅6+1, y=13); gamoTvlilia x-is mniSvneloba, roca y=21 (21=2⋅x+1, saidanac 2x=20, x=10).

2 qula

orive davaleba Sesrulebulia srulyofilad.

davaleba 2. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvlilia biWebis raodenoba (5⋅50=250), an gogonebis raodenoba (3⋅50+25=175).

2 qula

davaleba Sesrulebulia srulyofilad, miRebulia pasuxi: 75. (250–175, an 1,5⋅50).

229


davaleba 3. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvlilia, rom simindi marcvleulis mTeli raodenobis 40%-ia, SeniSnulia, rom 32 000 aris mTeli raodenobis 40% da Sedgenilia gamosaxuleba mTeli raodenobis gamosaTvlelad (3200:40⋅100=8000), an mTeli raodenobis 5%-is gamosaTvlelad (3200:8=4000).

2 qula

amocana amoxsnilia srulyofilad, miRebulia pasuxi: 36 000 tona. (8000⋅0,45 an 4000⋅9).

davaleba 4. 0 qula

ver gaerkva amocanaSi.

1 qula

gamoTvlilia jgufis wevrebis asakTa jami xelmZRvanelis gareSe (21⋅9=168), an xelmZRvanelTan erTad (23⋅9=207).

2 qula

amocana amoxsnilia srulyofilad, miRebulia pasuxi: 39 (207–168=39).

davaleba 5. 0 qula 1 qula

2 qula

ver gaerkva amocanaSi. , an warmodgenilia nawilobriv Sevsebuli venis diagrama 25 17 dasaxelebulia inglisuris da frangulis SemswavlelTa raodenobebi (42 an 21). amocana amoxsnilia srulyofilad, miRebulia pasuxi: 46 (25+17+4 an 42+21–17).

230


beWduri da eleqtronuli resursebi saswavlo gegmaSi miTiTebuli moTxovnebis SesrulebaSi did daxmarebas gagi­ wevT beWduri da eleqtronuli resursebi: beWduri resursebi: `gzamkvlevi maswavleblisaTvis, I-VI klasi, maTematika, Tbilisi, 2011~, romlis Semdgenlebi arian: z. giunaSvili, e. korZaZe, j. gvazava, l. mamulaSvili, l. beriSvili. es wigni ganTavsebulia `Google~-is saZiebel sistemaSi; `saqarTvelos dawyebiTi ganaTlebis proeqtis~ farglebSi Seqmnil veb-portali `www.kargiskola~ da saswavlo meTodologiuri resursebis ori beWduri krebuli. aq SeiZleba naxoT videofilmebi: `ganmaviTarebeli Sefaseba maTematikaSi~, `maTematikis konstruqtivistuli swavleba dawyebiT klasebSi~. warmodgenilia davalebebi Semajamebeli da diagnostikuri werebis Casatareblad da sxva. SeiZleba gamoiyenoT, magaliTad, eleqtronuli resursebi: Silkschool.ge (saSinao skola) da www.Math.Exercises for Kids. pirveli maTgani qarTul enaze warmogidgenT VI klasis 30 videogakveTils maTematikaSi. I) SedixarT vebgverdze, miuTiTeT: `gakveTilebi~, Semdeg moniSneT `maTematika~, VI klasi. pirveli gakveTili eZRvneba aTwiladebis Sedarebasa da dalagebas. maswavlebeli saubars iwyebs wiladebis aTwiladebad Caweris Sesaxeb. SeiZleba komentarebis garEeSec movisminoT gakveTili. ufro dawvrilebiT, meti magaliTis ganxilviT aris warmodgenili im wiladebis aTwiladad Cawera, romelTa mniSvneli 10-is xarisxia. magaliTad, 271 200 70 1 2 7 1 1000  =  1000  +  1000  +  1000  =  10  +  100  +  1000  = 0,2 + 0,07 + 0,001 = 0,271. 441 400 40 1 4 4 1 100000  = 100000  +  100000  +  100000  =  1000  +  10000  +  100000  = 0,00441 36  = 78, . . . . 78 10000 36 104 bolo striqoni araa Sevsebuli. es faqti Tqven SegiZliaT SemoqmedebiTad gamoiyenoT. saxeldobr, gaaCeroT kadri da sTOxovoT moswavleebs, gamoTqvan mosazrebebi gamosaxulebis dasasruleblad. pasuxebis analizis Semdeg Tqven kvlav gaagrZelebT Cvenebas da moswavleebs miecemaT SesaZlebloba Seudaron TavianTi Sedegi ekranze warmodgenil Sedegs. meore gakveTili – wiladebisa da aTwiladebis TvalsaCino gamosaxva; gamosaxvis wesebi dakavSirebulia aTwiladebis Sedarebasa da dalagebasTan. winare codnis gaaqtiurebis Semdeg, ricxviTi sxivis gamoyenebiT, aTwiladebis Sedareba da dalagebaa warmodgenili. am gakveTilis mosmena da misi ganxilva gvexmareba gavideT VI klasis standartiT^ misaRwev Sedegebze, romlebic maT. VI.1-iT aris mocemuli.

231


kargad aris warmodgenili e. w. nulebiT Sevsebis sakiTxic. amitom yuradReba miaqcieT gakveTilis am nawils. 71,86=71,860=71,8600=. . . . gakveTili – wiladebis Sekreba da gamokleba, gvexmareba VI klasis standartSi dafiqsirebuli VI.2 Sedegis miRwevaSi. internetgakveTilSi ganxilva iwyeba tolmniSvneliani wiladebis SekrebiT. pedagogi iyenebs TvalsaCinoebas efeqturad. Semdeg, wiladis ZiriTadi Tvisebis gamoyenebiT, Y xorcieldeba sxvadasxvamniSvneliani wiladebis Sekreba, magaliTad, 7  +  2  =  21  +  16  =  37  = 1 13 . 8 3 24 24 24 24 wiladebis gamravlebisas maswavlebeli iyenebs jerad Sekrebas: 2⋅ 15  =  15  +  15  =  25 . gakveTilis leqciuri saxis gamoyenebac erT-erTi didaqtikuri xerxia. 3 -is 1 -ze gasamravleblad maswavlebeli iyenebs wiladis ZiriTad Tvisebas da ga4 2 3  ⋅  1  =  3 . naxevrebas:   34  =  68 ,  68 -is naxevaria   38 , 4 2 8 am gakveTilis msvlelobisas, iseve, rogorc zemoT ganxilul gakveTilze, Tqven SegiZliaT mNSivnelovani meToPOdikuri midgomebi ganaxorcieloT: yoveli dasmuli amocanis Semdeg SeaCeroT videogakveTili, sajarod ganixiloT es amocana da moswavleebs misceT SesaZlebloba TavianTi Sedegebi Seadaron videogakveTilis gagrZelebisas warmodgenil Sedegebs. aseTi midgoma gaaLZlierebs moswavleTa motivacias, saxalisos da efeqtians gaxdis gakveTils. Tu videogakveTilebis klasSi gamoviyenebT, SeiZleba SevamowmoT moswavleTa sxva­ dasxva inteleqtualuri unaric: yuradRebiT mosmenis, mosmenilis aRqmis, daskvnebis gakeTebis. amitom gakveTilis mosmenis Semdeg unda SevamowmoT, yuradRebiT ismendnen Tu ara moswavleebi? ra iyo gasagebi, ra ver gaiges? cxadia, es Cveneba Sesabamisi masalis gavlis Semdeg unda moxdes da masala TiTqmis nacnobi unda iyos moswavleebisTvis. eleqtronuli resursiT: www.Math.Exercises for Kids SeiZleba moiZioT TvalsaCinoebebi/suraTebi gakveTilebisaTvis. resursi inglisur enazea, magram advili aRsaqmelia am enis armcodne momxmareblisaTvisac. miT umetes, Tu Tqven saTanado komentarebs daurTavT xolme misi Cvenebisas. masSi maswavlebeli iolad SearCevs asakobriv jgufs da davalebis rubrikas.

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მათემატიკა. VI კლასი. მასწავლებლის წიგნი.  
მათემატიკა. VI კლასი. მასწავლებლის წიგნი.  
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