m a T e m a t i k a

VIII klasi

maswavleblis wigni

grifminiWebulia saqarTvelos ganaTlebis, mecnierebis,

kulturisa da sportis saministros mier 2020 wels

guram gogiSvili, Teimuraz vefxvaZe

ia mebonia, lamara qurCiSvili

gamomcemloba inteleqtiTbilisi 2020

guram gogiSvili, Teimuraz vefxvaZe,

ia mebonia, lamara qurCiSvili

m a T e m a t i k a

VIII klasi

maswavleblis wigni

gamomcemloba inteleqti

Tbilisi 2020

ISBN 978-9941-31-191-8

© guram gogiSvili, Teimuraz vefxvaZe, ia mebonia, lamara qurCiSvili, 2020.© gamomcemloba `inteleqti~, 2020.

gamomcemloba inteleqtiTbilisi, ilia WavWavaZDis gamz. 5. tel.: 2-25 05 22

www.intelekti.ge info@intelekti.ge

INTELEKTI PUBLISHING5 Ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 32) 2-25 05 22

redaqtori Teimuraz vefxvaZe

damkabadonebeli lika sxirtlaZe

teqstis amkrefi marTa wiklauri

3

me-8 klasis maTematikis saxelmZRvaneloskoncefcia

saxelmZRvanelo agebulia sabazo safexuris 2018-2024 wlebis standartisa da wliuri programebis moTxovnebis gaTvaliswinebiT; vicavT balanss Cvens Semoqme-

debiT Tavisuflebasa da dasaxelebuli standartis moTxovnebs Soris, romlebic

maTematikis swavla-swavlebis miznebTan, meTodikur orientirebsa da Sefasebas-

Tan aris dakavSirebuli. aq mTavaria Sesabamisoba im principebTan, romlebsac ma-

Tematikis standarti efuZneba _ maTematikis swavlebam (raSic saxelmZRvanelos

mniSvnelovani roli ekisreba) Tavisi wvlili unda Seitanos SemoqmedebiTi unare-

bisa da Rirebulebebis ganviTarebasa da CamoyalibebaSi.

saxelmZRvanelos Sinaarsi ganisazRvra rogorc praqtikuli moTxovnebiT, ise

maTematikis ganviTarebis Sinagani kanonzomierebebiT. saswavlo masalis SerCevasa

da gadmocemis formas Sesabamisi asakis moswavleTaTvis misawvdomobis principic

udevs safuZvlad. yuradReba gavamaxvileT mimarTulebebis integraciaze: maTema-

tika erTiani mecnierebaa da ara misi sxvadasxva nawilis gaerTianeba. saxelmZRva-

nelos struqturuli da Sinaarsobrivi gamarTuloba miRweulia gadmocemis iseTi

meTodikiT, roca sakiTxebi mkafio da daxvewilia da gadmocemulia urTierTkav-

SirSi, araizolirebulad. sistematuria wina masalis gameorebisa da ganmtkicebis

momentebis gaTvaliswineba.

saxelmZRvaneloSi warmodgenili mravalferovani saswavlo masala maswavle-

bels miscems saSualebas, maRal doneze warmarTos diferencirebuli swavleba

sxvadasxva akademiuri donis moswavleebTan, SearCios saswavlo masala specialu-

ri saganmanaTleblo saWiroebis moswavleebTan samuSaod; aqvea masala siRrmiseu-

li swavlebisaTvis, masala jgufuri da saproeqto aqtivobebisTvis, romlebSic

saTanado adgili eTmoba ist-is gamoyenebas.

aqve aRvniSnavT saxelmZRvanelos maRalmecnierul dones, gadmocemis misaw-

vdomobas, enis sizustesa da simartives. wigni mkafiodaa struqturirebuli, cal-

kea gamoyofili saklaso muSaobisTvis gankuTvnili amocanebi, sxvadasxva tipis

aqtivobebi. cxadia, es dayofa pirobiTia da maswavlebels SeuZlia cvlilebebi

Seitanos am dayofaSi.

masalis mravalferovneba, moculoba da misawvdomoba daexmareba pedagogs,

raTa man warmatebiT gadaWras standartiTa da wliuri programiT dasaxuli amo-

canebi.

sarCevi

Sesavali .....................................................................................................................................7maswavleblis wignis mokle mimoxilva ...........................................................................7maTematikis swavlebis miznebi; miznebis, standartis Sedegebis miRwevisa da Sinaarsis urTierTkavSiris matrica. .......................................................................9informacia moswavlis saxelmZRvanelos Sesaxeb; saxelmZRvaneloSi gamoyenebuli sxvadasxva struqturuli erTeulis mimaniSnebeli simboloebi da maTi ganmartebebi. .................................................................................14

I Tavi. gameoreba. procenti. procentis gamoyeneba............................................151.1. ricxvebi da maTze moqmedebebi................................................................................191.2. algebruli gamosaxuleba. gantoleba. amovxsnaT amocanebi gantolebis gamoyenebiT ...........................................................251.3. vagrZelebT procentis ganxilvas...........................................................................331.4. gavixsenoT zogierTi geometriuli figura. wre da misi nawilebi ...........42Semajamebeli wera #1 ..........................................................................................................54

II Tavi. monacemTa analizi. SemTxveviTi eqsperimenti. xdomilobis albaToba ......................................................................................................562.1. monacemTa mopoveba da mowesrigeba .......................................................................582.2. monacemTa warmodgenis zogierTi xerxi. histograma .....................................632.3. SemTxveviTi eqsperimenti. SemTxveviTobis warmomqmneli mowyobilobebi .682.4. SeuZlebeli xdomiloba. aucilebeli xdomiloba .............................................722.5. elementarul xdomilobaTa sivrce. albaTobis gamoTvlis magaliTebi .75Semajamebeli wera #2 .......................................................................................................88

III Tavi. geometriuli figurebi, geometriuli gardaqmnebi ...........................913.1. geometriuli gardaqmnebi. mobruneba ...................................................................953.2. paraleluli wrfeebi. paralelur wrfeTa aqsioma .......................................1053.3. wrfeTa paralelurobis niSnebi ............................................................................1143.4. samkuTxedis kuTxeebis jami ...................................................................................1213.5. marTkuTxa samkuTxedi ..............................................................................................130Semajamebeli wera #3 ........................................................................................................147

5

IV Tavi. Sesabamisoba simravleebs Soris. wrfivi damokidebuleba ...........1494.1. vagrZelebT dasabuTebis xerxebis Seswavlas ...................................................1514.2. naSTi. poziciuri sistemis gamoyeneba ................................................................1604.3. Sesabamisoba or simravles Soris. ........................................................................1654.4 erTi simravlis asaxva meore simravleSi ...........................................................1714.5. wrfivi damokidebuleba ............................................................................................176Semajamebeli wera #4 ........................................................................................................192

V Tavi. gantolebaTa sistemebi. gamosaxulebis gamartiveba ........................1945.1. wrfivi orucnobiani gantoleba ............................................................................1965.2. wrfiv orucnobian gantolebaTa sistema ...........................................................2035.3. wrfiv orucnobian gantolebaTa sitemis amoxsnis grafikuli xerxi .....2075.4. Casmis xerxiT wrfiv orucnobian gantolebaTa sistemis amoxsna ............2155.5. Sekrebis xerxiT wrfiv orucnobian gantolebaTa sistemis amoxsna .......2205.6. amocanebis amoxsna wrfiv orucnobian gantolebaTa sistemis gamoyenebiT ........................................................................................................2265.7. algebruli gamosaxulebis gamartiveba ..............................................................236Semajamebeli wera #5 ........................................................................................................255

VI Tavi. kvadratuli fesvi. kuburi fesvi. piTagoras Teorema ...................2586.1. kvadratuli fesvi. iracionaluri ricxvi .........................................................2606.2. piTagoras Teorema .....................................................................................................2656.3. kvadratuli fesvis Semcveli gamosaxulebis gardaqmna, gamartiveba ....2706.4. piTagoras Teoremis zogierTi gamoyeneba ........................................................2796.5. marTkuTxa samkuTxedis maxvili kuTxis sinusi, kosinusi, tangensi .......2856.6. kuburi fesvi ................................................................................................................292Semajamebeli wera #6 ........................................................................................................302

VII Tavi. oTxkuTxedebi. utolobisa da koordinatebis gamoyeneba ...........3047.1. oTxkuTxedi. amozneqili oTxkuTxedi .................................................................3077.2. utolobis Tvisebebi da maTi gamoyeneba ...........................................................3117.3. paralelogrami ............................................................................................................3207.4. paralelogramis kerZo saxeebi ..............................................................................329Semajamebeli wera #7 ........................................................................................................3457.5. gamoviyenoT koordinatebi da geometriuli gardaqmnebi ..........................3487.6. trapecia .........................................................................................................................3557.7. Talesis Teorema .........................................................................................................3627.8. samkuTxedis Suaxazi ..................................................................................................3677.9. trapeciis Suaxazi ......................................................................................................371Semajamebeli wera #8 ........................................................................................................382

6

VIII Tavi. farTobi. sivrculi figuris zedapiris farTobi da moculoba ....................................3848.1. farTobi. farTobis Tvisebebi ...............................................................................3868.2. samkuTxedis farTobi. 8.3. paralelogramis farTobi. 8.4. trapeciis farTobi ....................................................................................................3898.5. nebismieri mravalkuTxedis farTobi. wris farTobi ..................................4058.6. sivrculi figurebi. zedapiris farTobi da moculoba ..............................411Semajamebeli wera #9 ........................................................................................................421

IX Tavi. erTucnobiani utoloba. xarisxi mTeli maCvenebliT .....................4249.1. erTucnobiani utoloba ...........................................................................................4259.2. utolobebis gamoyenebis magaliTebi ...................................................................4299.3. xarisxi mTeli maCvenebliT .....................................................................................4349.4. ricxvis standartuli forma .................................................................................437Semajamebeli wera #10 ......................................................................................................450

7

Sesavali

maswavleblis wignis mokle mimoxilva

maswavleblis wigni Seicavs meTodikur rekomendaciebs maTematikis 2018-2024 wlebisTvis gankuTvnili standartis mixedviT Sedgenili me-8 klasis saxelmZR-vanelosTvis. wignis daniSnulebaa, daexmaros maswavlebels saswavlo procesis

dagegmvasa da warmarTvaSi. yovel Temas Tan axlavs misi mecnieruli da meTod-

ologiuri safuZvlebis mokle mimoxilva. miwodebulia Sesabamisi eleqtronuli,

beWdviTi resursebis misamarTebi; literaturis nusxa.

maswavleblis wignSi Tavebisa da paragrafebis numeracia da dasaxelebebi

emTxveva moswavlis saxelmZRvaneloSi arsebul numeraciasa da dasaxelebebs. war-

modgenilia TiTqmis yvela gakveTilis scenari.

wignSi gaecnobiT Cvens mosazrebebs axali masalis axsnis Sesaxeb, mocemuli

iqneba Sesabamisi axsna-ganmartebebi masalis wardgenis fazebis Sesaxeb _ moti-

vacia, sakiTxis dasma, amocanis gansazRvra, ZiriTadi masalis gamoyofa; sawyis

etapze Sesasrulebeli amocanebi da maTi analizi, Semajamebeli davalebebi da

maTi Sefasebis rubrika, TviTSefasebisTvis gankuTvnili Semajamebeli davale-

bebis garCevis xerxebi, sarezervo aqtivobebi.

amoxsnilia saxelmZRvaneloSi mocemuli TiTqmis yvela amocana. amasTanave,

miTiTebulia amoxsnis sxvadasxva xerxis gamoyenebis magaliTebi. xSirad xdeba

xolme, rom moswavle amoxsnis gansxvavebul xerxs irCevs. maswavlebeli mzad unda

iyos moswavlis mier SeTavazebuli amoxsnis gansxvavebuli xerxebis gansaxilve-

lad da Sesafasebladac.

amasTanave, Cven mier miwodebul rekomendaciebs gakveTilis dagegmvisa da

warmarTvis Sesaxeb maswavlebeli SemoqmedebiTad unda miudges _ swavleba, cno-

bili maTematikosis d. poias mixedviT, xelovnebaa.

SesaZlebelia, maswavlebelma gakveTilis warmarTvis, sxvadasxva aqtivobis

dagegmvis Tavisi originaluri xerxebic moifiqros.

saxelmZRvanelo ar aris saswavlo gegma da arc standartia. maswavlebels Seu-

Zlia mis mier SerCeuli strategiis realizaciisas, diferencirebuli swavlebis

mizniT, SearCios warmodgenili didi moculobis saswavlo masalidan saTanado

sakiTxebi ise, rom miaRwios standartiT dasaxul miznebs.

Cven mier warmodgenil rekomendaciebSi, romlebic yoveli gakveTilis Catare-

bis sqemas exeba, ZiriTadi aqcenti saswavlo procesSi moswavlis aqtiuri CabmisT-

vis xelSemwyobi pirobebis Seqmnaze keTdeba, raTa es procesi gaxdes mimzidveli

da nayofieri dasaxuli miznis misaRwevad. swavleba Ziebisa da dasabuTebis gziT

unda mimdinareobdes. moswavle, rogorc wesi, Tavad unda aRmoaCendes WeSmarite-

bas motivirebuli situaciis dagegmvis Semdeg; mTavaria kvlevis procesi, saswav-

8

lo procesis SemoqmedebiTi suliskveTeba, ara marto pasuxis mosmena kiTxvaze _

`ra~? aramed _ `rogor? an _ `kidev rogor~? es yvelaferi aucilebelia moswav-

leTa momavali profesiuli saqmianobisTvisa da sazogadoebriv cxovrebaSi aqti-

uri monawileobisTvis.

maswavleblis wigni Seicavs rekomendaciebs Sefasebis sqemebis gamoyenebis

Sesaxebac. gansakuTrebuli mniSvneloba eniWeba ganmaviTarebeli Sefasebis ganx-

orcielebis xelSewyobas _ rogor vawarmooT moswavleebis muSaobaze dakvirve-

ba, rogor movaxerxoT maTi aqtivobebis dafiqsireba yovel gakveTilze, rogor

gamoviyenoT es Sefaseba miRweuli Sedegebis gasaumjobeseblad.

skolaSi ori tipis Sefaseba gamoiyeneba, ganmsazRvreli da ganmaviTarebeli.

ganmsazRvreli Sefaseba akontrolebs moswavlis miRwevis dones standartiT gan-

sazRvrul miznebTan mimarTebaSi. ganmsazRvrel SefasebaSi iwereba qulebi.

ganmaviTarebeli Sefaseba akontrolebs TiToeuli moswavlis ganviTarebis

dinamikas da xels uwyobs swavlis xarisxis gaumjobesebas.

ganmaviTarebeli Sefasebisas maswavlebeli moswavleTa saqmianobas amowmebs

ara maTi miRwevis donis gansasjelad da qulebis dasawerad, aramed maT dasaxmare-

blad. maswavlebeli swavlis procesSi akvirdeba TiToeul moswavles, swavlobs

maT saWiroebebs, raTa TiToeul maTgans maqsimalurad Seuwyos xeli winsvlaSi.

aseT konstruqciul saswavlo garemoSi moswavleebs ar aferxebT warumateblobis

SiSi da maswavleblis rCevisa da mxardaWeris imedi aqvT. moswavleebi iZenen axal

codnas, iZenen gamocdilebas da iumjobeseben unarebs.

ganmaviTarebeli Sefasebis procesSi unda CaerTon maswavleblebica da mos-

wavleebic. maswavleblis daxmarebiT, moswavleebi Tavadac cdiloben, daadginon

sakuTari moTxovnilebebi, gamoavlinon Zlieri da susti mxareebi, Semaferxebeli

faqtorebi, igrZnon winsvla da irwmunon, rom SeuZliaT siZneleTa etapobrivi

gadalaxva. es faqtorebi zrdis maT qmediTobasa da pasuxismgeblobas.

amrigad, ganmaviTarebeli Sefaseba, faqtobrivad, unda ganvixiloT rogorc

procesi, romelic gulisxmobs sxvadasxva aqtivobiT (e. w. ganmaviTarebeli Sefase-

bis instrumentiT) informaciis Segrovebas moswavlis mier sakiTxebis gagebis,

gaazrebis Sesaxeb da am informaciis gamoyenebas moswavleTa swavla-swavlebaSi

progresis xelSesawyobad.

ganmsazRvreli Sefasebis instrumentic (mag. testi) zogjer SeiZleba ganmavi-

Tarebeli Sefasebis instrumentad vaqcioT, Tu, magaliTad, moswavleebs davyofT

mcire jgufebad da davavalebT miRebuli pasuxebisa da amoxsnis gzebis Sedarebas,

maTi mosazrebebis axsnas.

ganmaviTarebeli Sefasebis mixedviT, maswavlebelma SeiZleba dagegmos swav-

lebis procesi, diferencirebuli midgomebi moswavleebTan _ muSaobis moculo-

ba da strategia.

yoveli aqtivobisas moswavleTa CarTulobaze maswavleblis mier warmoebu-

li dakvirvebac ganmaviTarebeli Sefasebis instrumentia, rom araferi vTqvaT am

dros gakeTebul rCevebze.

maSasadame, ganmaviTarebeli Sefaseba uwyvetad unda mimdinareobdes _ saSinao

davalebis ganxilvisas (rac faqtobrivad, TiTqmis yoveli gakveTilis dawyebisas

unda Catardes), axal masalaze gadasvlamde da am masalis aTvisebis procesSic.

9

maTematikis swavlebis miznebi; miznebis, standartis Sedegebis miRwevisa da Sinaarsis urTierTkavSiris matrica

sabazo safexuris standarti, romlis mixedviTac aris Cveni saxelmZRvanelo

Sedgenili, Semdegi nawilebisgan Sedgeba:

a) sagnis swavla-swavlebis miznebi

b) standartis Sedegebi da Sinaarsi

g) meTodikuri orientirebi

d) Sefaseba

maswavlebeli kargad unda icnobdes 2018-2024 wlebisTvis gankuTvnil stan-dartsa da mis moTxovnebs. misi muSaoba unda warimarTos am moTxovnebis Sesaba-

misad, raTa standartSi Camoyalibebul swavla-swavlebis miznebze muSaobiT ma-

Tematikam Tavisi wvlili Seitanos erovnuli saswavlo gegmiT gaTvaliswinebuli

unarebisa da Rirebulebebis Camoyalibebasa da ganviTarebaSi.

axal standartSi Camoyalibebuli miznebi, logikuri da kritikuli azrovne-

bis xerxebisa da maTematikis enis dauflebasTan erTad, maTematikuri instrumen-

tebis gamoyenebiT realuri problemebis gadaWrasac gulisxmobs.

aqve aRvniSnavT azrovnebis maTematikuri stilis Camoyalibebis mniSvnelobas,

rac gulisxmobs:

1) obieqtTa klasifikaciis unars, 2) kanonzomierebebis aRmoCenasa da dadge-nas, 3) argumentirebis unars.

sagnis swavla-swavlebis miznebi, Sefasebis indikatorebi pasuxobs standar-

tis Sedegebs. isini pasuxobs SekiTxvas: ra unda SeZlos moswavlem maTematikaSi

yoveli Temis Seswavlis Semdeg. Cven mier Sedgenil cxrilSi warmodgenil yovel

Sedegs Tan erTvis indeqsi, romelic miuTiTebs sagans, swavlebis etapsa da stan-

dartis Sedegis nomers. es aRniSvna maTematikis standartSi miRebuli aRniSvnis

analogiuria.

standartis Sedegebis miRwevisa da Sinaarsis urTierTkavSiris matrica

Temebis CamonaTvalimiznebi: Sinaarsis gardaqmna miznad;

standartSi miTiTebuli Sesabamisi punqtebidro

gameoreba: racionaluri

ricxvebi, cvladiani gamo-

saxuleba, gantoleba, pro-

centi, wre da misi nawilebi

moqmedebebi racionalur ricxvebze, racional-

uri ricxvebis Caweris sxvadasxva xerxi da maTi

gamoyeneba amocanebis amoxsnisas (maT. sab. 1, 2, 3, 4, 7, 8, 9). verbalurad aRwerili situaciis algebruli saxiT Cawera, gantolebis SedgeniT

amocanebis amoxsna (maT. sab. 1, 2, 3, 4, 7, 8). wre da misi nawilebi; geometriul figuraTa zomebis

dadgena (maT. sab. 1, 2, 5, 6, 7)

12 sT

10

monacemTa mopoveba da

dajgufeba, monacemTa war-

modgenis xerxebi, SemTx-

veviTi eqsperimenti

dasmuli amocanis amosaxsnelad saWiro monace-

mebis mopoveba (maT. sab. 4, 7, 8, 9); monacemebis mowesrigeba da warmodgena dasmuli amocanis

amosaxsnelad xelsayreli formiT (maT. sab. 4, 5, 6); SemTxveviTobis warmomqmneli mowyobi-lobebis gamoyeneba (maT. sab. 4, 5, 6, 7, 8, 9)

7 sT

geometriuli gardaqmnebi.

mobruneba

geometriul figuraTa Tvisebebis aRwera geome-

triuli gardaqmnebis gamoyenebiT (maT. sab. 3, 4, 7, 8, 9)

5 sT

paraleluri wrfeebi. pa-

ralelur wrfeTa aqsioma.

wrfeTa paralelurobis

niSnebi

paralelurobis aqsiomis magaliTis gamoyenebiT

aqsiomasa da Teoremas Soris gansxvavebis gaaz-

reba; geometriuli debulebebis marTebulobis

dasabuTeba (maT. sab. 1, 2, 3)

5 sT

samkuTxedis kuTxeebis

jami

figurisa da misi elementebis zomebis moZebna

(maT. sab. 1, 2, 3)3 sT

marTkuTxa samkuTxedi figuraTa Tvisebebis gamoyeneba figuraTa

klasifikaciisTvisa da maTi saxeobebis Sesadar-

eblad (maT. sab. 3, 4, 7, 8, 9)4 sT

kuTxis biseqtrisis Tviseba geometriuli debulebebis marTebulobis dasa-

buTeba (maT. sab. 1, 2, 3)4 sT

dasabuTebis xerxebi;

debulebis sawinaaRmde-

go debuleba. debulebis

dasabuTeba kontrmagali-

Tis gamoyenebiT. wanamZ-

Rvrebi da daskvna

debulebebis marTebulobis dasabuTeba sxva-

dasxva xerxis gamoyenebiT (maT. sab. 1, 2, 3).10 sT.

naSTi. poziciuri sistemis

gamoyeneba

poziciuri sistemis gamoyeneba naturalur ric-

xvTa simravlis klasificirebisa da gayofadobis

niSnebis dasabuTebisTvis (maT. sab. 3, 4, 7, 8, 9).

4 sT

Sesabamisoba or simravles

Soris, asaxva

or simravles Soris Sesabamisobis ageba, gamosax-

va da gamokvleva (maT. sab. 3, 4, 7, 8, 9) 6 sT

wrfivi damokidebuleba sidideebs Soris wrfivi damokidebulebis amoc-

noba, gaanalizeba da gamosaxva (maT. sab. 3, 4, 5).4 sT

wrfivi orucnobiani gan-

toleba. wrfiv orucnobi-

an gantolebaTa sistemis

amoxsnis xerxebi (Casmis xe-

rxi, Sekrebis xerxi); wrfiv

orucnobian gantolebaTa

sistemis amoxsnis grafi-

kuli xerxi; amocanebis

amoxsna wrfiv orucnobian

gantolebaTa sistemis gam-

oyenebiT

verbalurad aRwerili situaciis algebruli ga-

mosaxulebis (formulis) saxiT Cawera (maT. sab.

4, 5, 6, 7, 8, 9). gantolebis Sedgena verbalurad mocemuli amocanis Sesabamisad, gantolebis Ses-

abamisi amocanis Sedgena (maT. sab. 3, 4, 7, 8, 9)13 sT.

11

algebruli gamosaxulebis gamartiveba

algebruli gamosaxulebebis gamartiveba da maTi ricxviTi mniSvnelobebis povna cvladTa sxvadasxva mniSvnelobisaTvis

5 sT.

kvadratuli fesvi, iracio-naluri ricxvi; kvadrat-uli fesvis Semcveli gamosaxulebebis gardaqm-na; piTagoras Teorema, piTagoris Teoremis gamoy-eneba; marTkuTxa samkuTxe-dis maxvili kuTxis sinusi, kosinusi, tangensi; kuburi fesvi

kvadratuli fesvis Semcveli gamosaxulebebis gamartiveba, gamoTvlebTan dakavSirebuli amocanebis amoxsna (maT. sab. 4, 5, 7, 8, 9); geome-triuli debulebebis marTebulobis dasabuTeba (maT. sab. 1, 2, 3); figurisa da misi elementebis zomebis moZebna (maT. sab. 1, 2, 5, 6, 7)

18 sT

oTxkuTxedi, amozneqili oTxkuTxedi; utolobis Tvisebebis gamoyeneba; paralelogrami, para-lelogramis kerZo saxee-bi (rombi, marTkuTxedi, kvadrati); koordinatTa meTodisa da geometriuli gardaqmnebis gamoyeneba; trapecia

figuraTa Tvisebebis gamoyeneba figuraTa klasificirebisaTvis da maTi saxeobebis Ses-adareblad (maT. sab. 3, 4, 7, 8, 9); figuris da misi elementebis zomebis moZebna (maT. sab. 1, 2, 5, 6, 7); sxvadasxva xerxis gamoyenebiT geometriul debulebaTa marTebulobis dasabuTeba (maT. sab. 1, 2, 3)

14 sT

farTobi, farTobis Tvise-bebi; samkuTxedis farTobi, paralelogramis farTobi, trapeciis farTobi; sivr-culi figurebi, moculoba

figurisa da misi elementebis zomebis moZebna (maT. sab. 1, 2, 5, 6, 7); geometriuli debulebebis marTebulobis dasabuTeba (maT. sab. 1, 2, 3). fig-uraTa Tvisebebis gamoyeneba figuraTa klasifi-cirebisaTvis da maTi saxeobebis Sesadareblad (maT. sab. 3, 4, 7, 8, 9)

24 sT

erTucnobiani utoloba, utolobebis gamoyenebis magaliTebi

verbalurad aRwerili situaciis algebruli ga-mosaxulebis (formulis) saxiT Cawera (maT. sab. 4, 5, 6, 7, 9); utolobis Sedgena verbalurad moce-muli amocanis Sesabamisad (maT. sab. 3, 4, 7, 8, 9)

9 sT

xarisxi mTeli maCvenebliT, ricxvis standartuli forma

mTelmaCvenebliani xarisxi, xarisxis Tvisebebi (maT. sab. 6, 7) ricxvebis Cawera standartuli formiT, moqmedebebi standartuli formiT Caweril ricxvebze (maT. sab. 3, 4, 6)

6 sT

albaToba, xdomilobebi SemTxveviTobis warmomqmneli mowyobilobebi _ moneta, urna, kamaTeli, ruleti. SemTxveviTi eqsperimenti. xdomilobaTa magaliTebi. aucile-beli da SeuZlebeli xdomilobebi. Tanabrad mosalodneli xdomilobebi. xdomilobebis ga-mosaxvis sxvadasxva forma. elementarul xdomi-lobaTa sivrce (maT. sab. 1, 3, 4, 5, 6, 7, 8, 9).

9 sT

VIII klasis masalis gameoreba da sarezervo saaTebi 10 sT

12

informacia moswavlis saxelmZRvanelos Sesaxeb

me-8 klasis saxelmZRvanelo moswavlis wignisadmi wayenebuli yvela moTx-ovnis Sesabamisadaa agebuli. sarCevi da rCevebi wigniT sargeblobisaTvis wignis dasawyisSia mocemuli, maTematikuri terminebis leqsikoni, maTematikuri niSnebi da maTi gamoyenebis magaliTebi, sagnobrivi saZiebeli _ wignis boloSi.

moswavleebSi maTematikis, rogorc zogadsakacobrio kulturis nawilis Sesax-eb warmodgenis Camoyalibebas xels uwyobs SinaarsSi CarTuli epizodebi mecnier-ebis istoriidan. kargad gvesmis, rom saxelmZRvanelos Sinaarsi, agebis stili, mis elementebs Soris logikuri kavSirebi gansazRvravs moswavleTa SemecnebiTi saqmi-anobis process. codnis aTvisebis procesSi saxelmZRvanelos mniSvnelovani fun-qcia ekisreba. am funqciis ganxorcielebaSi mniSvnelovania wignis struqturac. Tavebi dasaTaurebulia da masalis mokle anotaciiT iwyeba; Tavebi paragrafebi-sgan Sedgeba; paragrafebi SeiZleba ramdenime nawilisgan Sedgebodes. amasTanave, paragrafis bolo nawilSi, romelic specialuri simboloTi _ `s~ (sxva dasxva) _ aris gamoyofili, mocemulia damatebiTi sainformacio masala, terminebis warmo-Soba, istoriuli teqstebi.

saxelmZRvanelo 9 Tavisgan Sedgeba. yovel TavSi gadmocemuli masala, ZiriT-adad, romelime erTi mimarTulebis mixedviT aris warmodgenili; Tumca, zogjer, maTematikis erTianobisa da integrirebulad gadmocemis aucileblobis gaTval-iswinebiT, CarTulia paragrafebi sxva mimarTulebebidan, rac masalis gadmoce-mis TanamimdevrobiTac aris gamowveuli. yovel Temas mosdevs Semajamebeli amo-canebi da amocanebi TviTSefasebisTvis, romlebiTac moswavleebi Tavad afaseben sakuTar codnas. sarekomendacio xasiaTisaa Cveni miTiTeba saklaso da saSinao samuSaoebad amocanebis ganawilebis Sesaxeb.

gameorebis procesi, cxadia, yoveli axali Temis gadacemas Tan axlavs – axali codna winare codnaze dayrdnobiT Sendeba, Tumca, mainc gamovyaviT saxelmZR-vanelos dasawyisSi paragrafebi, romlebic axali saswavlo wlis procesSi CarT-vis kargi winapirobaa, gameorebulia ramdenime mniSvnelovani sakiTxi, moswavlee-bi nel-nela `Sedian formaSi~, Seswavlil ricxviT sistemebze Tvalis gadavleba, ricxvebis warmodgenis formebis gaxseneba (maT Soris _ procentiT), cvladiani gamosaxulebisa da gantolebis gamoyenebis gameoreba, proporciis, proporci-ulobis (geometriuli masalis gamoyenebiTac) da zogierTi sxva cnebis gaxseneba axal masalaze gadasvlis kargi winapirobaa; bunebrivad xdeba axali sakiTxis gac-nobac _ procenti _ 1-ze naklebi da 100-ze meti.

zogadi konceptualuri moTxovnebi, romlebsac saxelmZRvanelos Sinaarss

vuyenebT, SeiZleba ase SevajamoT: a) xeli Seuwyos codnis sxvadasxva sferos Soris urTierTkavSirebis gacno-

bierebas. paragrafebis umetesi nawili ilustrirebuli iyos faqtebiTa da savar-jiSoebiT codnis sxvadasxva sferodan.

b) emyarebodes Tanamedrove koncefciebs, romlebic Teoriul-simravluri enis zomier gamoyenebas, moswavleTa asakobrivi Taviseburebebis gaTvaliswineba-sa da sagnisadmi interesis aRZvras gulisxmobs.

g) savarjiSoTa sistema ise iyos dagegmili, rom mimdinareobdes masalis uwyve-ti gameoreba mTeli wlis ganmavlobaSi.

13

d) saSualebas aZlevdes moswavleebs, gamoumuSavdeT individualuri da jgu-furi proeqtebis ganxorcielebisTvis saWiro unar-Cvevebi.

e) mWidro kavSirs amyarebdes moswavleTaTvis nacnob sinamdvilesTan, praqti-kuli xasiaTis amocanebis ganxilvasTan; SeeZlos varaudebis gamoTqma problemis winaswari Sefasebisas.

v) yoveli siaxle emyarebodes ukve arsebul codnas, afarToebdes da avi-Tarebdes am codnas.

z) masalis gadmocemis stili (Sinaarsi da forma) maswavlebels SemoqmedebiTi, dinamiuri, xalisiani gakveTilebis Catarebis SesaZleblobas uqmnides.

saxelmZRvaneloSi gamokveTilia yoveli paragrafis mixedviT Semajamebeli daskvnebi. sxvadasxva mimarTulebebi erTmaneTTan mWidro kavSirSi gadaicema, Tumca, calkeuli fragmentebis gadmocemisas, logikuri Tanamimdevrobis dacvis principis Sesabamisad, erTi romelime Temaa gaSlili da am procesSi, SeZlebi-sdagvarad, CarTulia masala, romelic sxva Temasac ganekuTvneba.

savarjiSoTa sistema _ `testebi~, amocanebi, proeqtebi _ Teoriuli masalis Seswavlis stimulia da xSirad saswavlo gegmaSi miTiTebuli Sedegebis sistemas ukavSirdeba. Tumca, amocanebi, ZiriTadad, Tavis, paragrafis saTaurs pasuxobs.

SinaarsSi CarTulia sakiTxebi, romlebic mdgradi ganviTarebis principebis ganxorcelebas exeba.

moswavlis saxelmZRvanelo saswavlo procesis dagegmvisa da warmarTvis erT-erTi saSualebaa. moswavleebTan saubari SeiZleba wignis teqstisgan gansx-vavebuladac aagos maswavlebelma. saxelmZRvanelo exmareba maswavlebels _ masSi dafiqsirebulia SemecnebiTi saqmianoba, romelic maswavlebelma unda warmarTos. SevecadeT, rom ar Seqmniliyo savarjiSoTa sxva krebulebis gamoyenebis saWiroe-ba. maswavlebelma unda gaiTvaliswinos, rom im krebulebis umetesoba erovnuli saswavlo gegmisa da standartis moTxovnis Sesabamisad ar aris organizebuli; maTi saSualebiT SeuZlebelia sakvanZo SekiTxvebze pasuxebis moZieba da mkvid-ri warmodgenebis Camoyalibeba. Cvens saxelmZRvaneloSi warmodgenili masala (teqstebi, SekiTxvebi) xels uwyobs moswavlis etapobriv miyvanas axal codnam-de, cnebaTa gaazrebamde, mkvidri warmodgenebis Camoyalibebamde; Semajamebeli davalebebi, romlebic axlavs yovel Tematur bloks, moiTxovs SeZenil codnaTa integrirebul gamoyenebas. did yuradRebas vuTmobT saxelmZRvaneloSi warmod-genili masalis sandoobas, obieqturobas. amocanebi isea SerCeuli, rom amoxsnis sxvadasxva xerxis (algebruli, geometriuli) gamoyenebaa SesaZlebeli, rac xels uwyobs kritikuli azrovnebis ganviTarebas. struqturuli erTeulebi (Tema, para-grafi, `moisazre~ da `vip~ rubrikiT warmodgenili amocanebi, jgufuri muSao-bisTvis gankuTvnili amocanebi, amocanebi klasSi Sesasruleblad, amocanebi Sin amosaxsnelad) gamoyofilia saTanado simboloebiT, romlebsac moswavle wignis

dasawyisSive gaecnoba.

14

saxelmZRvaneloSi gamoyenebuli sxvadasxva struqturuli erTeulis mimaniSnebeli simboloebi da maTi ganmartebebi

waikiTxeT es rCevebi da gaecaniT specialuri simboloebis daniSnulebas.

saxelmZRvanelos TiToeuli Tavi masSi mocemuli ma-salis mokle aRweriT iwyeba.

es simbolo klasSi amosaxsneli amocanebis ganxilvis dawyebaze migviTiTebs. am amocanebis nomrebi wreebSia Casmuli. magaliTad, 1 .

es simbolo Sin muSaobisTvis gankuTvnil amocanebs warmogvidgens. am amocanebis nomrebi samkuTxedebSia Casmuli, magaliTad, 3 .

`sxvadasxva~ _ maTematikasTan dakavSirebul saint-ereso faqtebsa da istoriebs war mo gvidgens.

am simbolos mosdevs amocanebi, romelTa amosaxsnelad xSi rad sakmao gamWriaxobis gamovlenac mogiwevT. am amo-canebis nomrebi kvadratebSia Casmuli, magaliTad, 1 .

es simbolo axlavs davalebebs, romelTa Sesruleba informaciuli an sakomunikacio teqnologiebis gamo-yenebas moiTxovs.

SemogTavazebT jgufuri muSaobis proeqtebs, rom-lebic TanaklaselebTan erToblivad muSaobis gamoc-dilebas SegmatebT. SeiZleba dagWirdeT informaciis damoukideblad moZiebac da misi damuSavebac.

vin iqneba pirveli. am simboloTi warmodgenili amo-canebis amoxsnisas ecadeT, gamWriaxobasTan erTad, sxar ti azrovnebac gamoavlinoT.

teqstis ukeT gagebis mizniT, saWirod CavTvaleT zogierTi sityvis warmomavlobisa da misi sxvadasxva mniSvnelobis miTiTeba. aseTi Canawerebi teqstSi mo-niSnulia aq gamosaxuli simboloTi.

es simbolo miganiSnebT, rom unda gamoTqvaT Tqveni mosazreba miTiTebul sakiTxze.

es simbolo yovel paragrafSi warmogidgenT kiTxvebs ganxiluli sakiTxebis Sesaxeb.

am simboloTi drodadro SegaxsenebT, rom yvela amo-cana unda gadaitanoT rveulSi da Semdeg amoxsnaT. wignSi Canawerebi ar unda gakeTdes!

Cvenileqsikoni

saSinao

moisazre

i

klasSi

rv

eul

Si!

gaufrTxildiT wigns!

masSi Tqvenma Canawerebma wigni SeiZleba

sxvebisTvis gamousadegari gaxados.

s

j

•

15

I Tavi

gameoreba. procenti. procentis gamoyeneba

me-8 klasis saxelmZRvanelo Sedgenilia sabazo safexuris maTematikis ax-ali standartis moTxovnis Sesabamisad. saxelmZRvaneloSi asaxulia erovnul

saswavlo gegmaSi gamoyofili Temebi. TiToeul TemaSi warmodgenili masala: e.

w. testebi, proeqtebi, Semajamebeli davalebebi da sxva araerTi aqtivoba xels

uwyobs miTiTebul sakvanZo SekiTxvebze pasuxebis moZiebas. I Tavic, cxadia, yvela

am aspeqtis gaTvaliswinebiTaa Sedgenili, magram mas gansakuTrebuli datvirT-

va aqvs _ moswavleebi unda CaerTon saswavlo ritmSi, axal Temebze gadasvlas

unda uswrebdes im ZiriTadi sakiTxebis gameoreba, romlebzec axali masala unda

daSendes. mniSvnelovnad miviCnieT im sakiTxebis gameoreba, romlebic me-8 kla-sis Sesaswavl Temebs (ricxvebi; algebra, algebruli gamosaxuleba, geometriuli

obieqtebi) ukavSirdeba. adre miRebuli codnis gaxseneba da gaRrmaveba xels Seu-

wyobs axali sakiTxebis Seswavlas. mkvidri warmodgenebi, romlebic yoveli cnebis

Seswavlisas unda Camoyalibdes moswavlis mexsierebaSi, cxadia, winare warmodgen-

ebs unda eyrdnobodes. mimdinareobs yoveli cnebis Sesaxeb warmodgenebis Semd-

gomi dazustebis procesi. magaliTad, ricxvis cnebis gafarToebis process Tan

axlavs ricxvis warmodgenis axali formebis ganxilvac. Tu adre ar ganixileboda

erTze naklebi, an 100-ze meti procenti da moswavleTaTvis uCveulo iyo 200%-iT gazrdis warmodgena, amjerad am SemTxvevebsac ganvixilavT. kargad nacnobi geo-

metriuli figuris _ wris _ gaxsenebas, Tan axlavs misi gamoyeneba simravleebze

operaciebis gamosaxvisa da procentis gaazrebisas.

saxelmZRvanelo xels uwyobs maTematikis, rogorc sxvadasxva struqturis

(algebruli, geometriuli, albaTur-statistikuri) Sesaxeb mecnierebis erTiano-

bis aRqmasa da ilustrirebas (maTematika erTiani mecnierebaa da ara misi sxva-

dasxva nawilis xelovnuri gaerTianeba).

maTematikis swavlebam Tavisi wvlili unda Seitanos mdgradi ganviTarebis

miRwevis sxvadasxva aspeqtis mniSvnelobis gacnobierebaSi. I TavSi viwyebT iseTi

proeqtebis SeTavazebasac, romlebic moswavleebs Caafiqrebs adamianebis usa-

frTxo, jansaRi da mdgradi momavlis uzrunvelyofaze.

gameorebisas did adgils vuTmobT ricxvebis Sesaxeb moswavleTa codnis ga-

farToebasa da gaRrmavebas.

maswavleblebs SevaxsenebT, rom naturalur ricxvTa dafuZnebis Teoriis war-

modgenisas, ZiriTadad, or midgomaze miuTiTeben _ aqsiomuri midgoma, romelic

16

peanos saxelTan aris dakavSirebuli da simravluri koncefcia (kantoris Teori-

aze dafuZnebuli).

pirveli midgomisas naturalur ricxvTa mwkrivi aris {N; 1; F} struqtura, ro-melic Sedgeba (ix. mag., [20]):

1. N simravlisgan (am simravlis elementebs naturaluri ricxvebi hqvia).2. gamoyofilia elementi, romelsac erTi ewodeba _ 1.3. gvaqvs mimarTeba: F(x; y), x-is momdevno aris y.mocemuli struqturis ZiriTadi Tvisebebi aqsiomebiT aris warmodgenilni:

1) 1 ar aris raime naturaluri ricxvis momdevno;2) nebismieri naturaluri ricxvisTvis arsebobs erTi da mxolod erTi misi

momdevno;

3) yoveli naturaluri ricxvi araumetes erTi naturaluri ricxvis momdevnoa;4) (induqciis aqsioma) N-is yoveli qvesmravle, romelSic Sedis 1 da yovel x-Tan

erTad Sedis misi momdevno naturaluri ricxvi, N-s emTxveva.aqsiomuri meTodis gatarebisas asec amboben: elementebis buneba, romlisganac

N Sedgeba, ar aris arsebiTi.naturaluri ricxvis cnebis dafuZnebis meore midgoma kantoris saxelTan aris

dakavSirebuli. aq elementebis ricxvi gaigeba, rogorc simravlis simZlavris ker-

Zo SemTxveva. safuZvlad erTi simravlis meore simravleze urTierTcalsaxa Sesa-

bamisobis cneba aiReba. am mimarTebiT xdeba simravleTa faqtorizacia _ klasebad

dayofa. am koncefciis Sesabamisad, TiToeuli klasi xasiaTdeba misi simZlavriT.

aracarieli sasruli simravlis simZlavre, gansazRvrebis Tanaxmad, naturaluri

ricxvia.

cxadia, skolaSi ariTmetikis sawyisi Seswavla TvalsaCino unda iyos. igi ar

unda misdevdes arc aqsiomur meTods da arc sxva sqemas. Tumca, es imas ar niSnavs,

rom ricxvTa sistemis swavlebas ar unda hqondes Tavisi logikuri agebuleba. Cven

mier SemoTavazebuli sqemebi axlos aris naturaluri ricxvis Sesaxeb pirvel war-

modgenebTan. is ki naturaluri ricxvis rigobiT aspeqtTan ufroa dakavSirebuli.

am aspeqtis mecnieruli safuZveli ufro pirveli midgomaa. kolmogorovic [20] ki am SemTxvevaSi Teoriul-simravlur koncefcias uaryofs, rom araferi vTqvaT

simravluri koncefciis namdvil mowinaaRmdegeebze.

ricxvis cnebis gafarToebas meTodikur literaturaSi xSirad sqematurad ase

warmoadgenen:

(1) arauaryofiTi mTeli ricxvebi

(2) mTeli ricxvebi (5) arauaryofiTi racionaluri ricxvebi

(3) racionaluri ricxvebi (6) arauaryofiTi namdvili ricxvebi

(4) algebruli ricxvebi (7) namdvili ricxvebi

(8) kompleqsuri ricxvebi

17

saSualo skolaSi Tanamimdevroba (erovnuli saswavlo gegmis mixedviT) aseTia:

(1) → (5) → (3) → (7)umaRlesi skola am Tanamimdevrobas ase agrZelebs:

(7) → (8) → (4)(bolo etapi mxolod specialur fakultetebze ganixileba).

algebris TvalsazrisiT ganzogadebis bunebrivi gza aseTia:

(1) → (2) → (3) → (4).skolaSi Tanamimdevroba istoriul aspeqtebTan da genetizmis principebTan

aris dakavSirebuli. uaryofiTi ricxvebis Semotanis istoriac amas adasturebs.

me-7 klasSi mTavrdeba pirveli sameulis Seswavla. gavlilia gza:(1) → (5) → (3).

me-8 klasSi, gameorebisas, SeiZleba yuradReba gavamaxviloT algebruli gafar-Toebis momentebze da gameorebas Semdegi sqemiT warmovadgenT:

(1) → (2) → (3);amasTanave, yuradRebas vamaxvilebT ricxvebsa da maT sxvadasxva Canawerebs So-

ris gansxvavebaze. ricxvi erTia da misi Caweris saxeebi sxvadasxvaa (magaliTad,

procentuli saxe racionaluri ricxvis Caweris erT-erT formad aris warmodge-

nili). pirveli Tavis amocanebSi am sakiTxs didi yuradReba eTmoba.

pirvel TavSi warmodgenili masala ganxilvis moTxovnebis urTierTkavSiris

gaTvaliswinebiT, SeiZleba ase warmovadginoT:

Sinaarsi

ricxvebi

ricxvebze moqmedebebi

algebruli gamosaxuleba, gantoleba

gantolebis gamoyenebiT amocanebis amoxsna

Semajamebeli da damatebiTi amocanebi,

amocanebi TviTSefasebisTvis

mkvidri warmodgenebi:

ra zogadi warmodgenebi unda Ca-

mou yalibdes moswavles xangrZliv

mexsie rebaSi?

sakvanZo SekiTxvebi:

ra konceptualur kiTxvebze minda

moswavleebma gascen pasuxi, rom mox-

des maTi provocireba axali codnis

SesaZenad?

18

yoveli Temis sakiTxis Seswavlis dros miTiTebuli iqneba Sesabamisi mkvidri

warmodgenebi da sakvanZo SekiTxvebi.

maTematikuri cnebis aTvisebis procesi maTematikis erT-erTi meTodis _

abstraqciis gamoyenebas ukavSirdeba; abstraqcia xdeba realuri obieqtebis

garkveuli Tvisebebidan. gansakuTrebiT uWirT moswavleebs ricxvebidan cvladeb-

ze gadasvla, rac asoebis gamoyenebas ukavSirdeba. aq warmoSobili siZneleebis

daZleva mxolod xangrZlivi varjiSis procesSia SesaZlebeli. yuradReba unda

gamaxvildes cvladis dasaSvebi mniSvnelobebis miTiTebis aucileblobaze. mag-

aliTad, cvladiT SeiZleba warmodgenili iyos nebismieri racionaluri ricxvi,

cvladi SeiZleba iyos mxolod mTeli an naturaluri ricxvi.

mizanSewonilad vTvliT zogadi saxis zogierTi rekomendaciis Sesaxeb sau-

barsac, romlebic momavalSi SeiZleba ar gavimeoroT:

• maswavlebeli unda cdilobdes, rom yoveli axali SemecnebiTi amocana, garkveuli Semzadebis Semdeg, TviT moswavlem Camoayalibos.

• maswavleblisa da moswavlis erToblivi ZalisxmeviT (dakvirvebiT, konkret-uli SemTxvevebis analiziT) iqmneba garkveuli warmodgena arsebul maTematikur

kanonzomierebaze.

• maswavleblis xelmZRvanelobiT moswavleebi eZeben amocanis amoxsnis gegmas da xSirad TviT axdenen gegmis realizacias.

maswavleblebs kargad moexsenebaT, rom saswavlo procesisTvis mzadebis zo-

gadi sqema gulisxmobs:

• saswavlo wlis dawyebamde Catarebul muSaobas.• gakveTilebis sistemis gaazrebas saswavlo wlis win.• konkretuli gakveTilisTvis momzadebas.

axali saxelmZRvaneloebis SemoRebis procesSi saswavlo wlis dawyebamde

muSaobaSi igulisxmeba maTematikis standartis moTxovnebis kargad gaazreba.

2018 wlis standartis gacnoba SeiZleboda ganaTlebis, mecnierebis, kultur-isa da sportis saministros veb-gverdze. 2019 wlis oqtomberSi Setanilia mcireo-deni cvlilebebi. magaliTad, me-8 klasis wliur programaSi daemata albaTobis sakiTxebi, rac, cxadia, Cvenc gaviTvaliswineT. SemoTavazebulia axali formebi

maswavleblis wignis asagebad.

ZiriTadi masala, romelsac unda daeyrdnos maswavlebeli saswavlo procesSi,

erovnuli saswavlo gegmaa. saxelmZRvanelo damxmare saSualebaa erovnuli sas-

wavlo gegmis Sesasruleblad. savarjiSoebis mravalferovani sistema, saintereso

CanarTebi, kompleqsuri/proeqtuli da sxva saxis davalebebi maswavlebels saSua-

lebas aZlevs aRar eZebos damatebiTi savarjiSoebi umaRles saswavleblebSi Sems-

vlelTaTvis gankuTvnil wignebSi, romlebic, rogorc wesi, saxelmZRvanelosTvis

wardgenil moTxovnebs ar akmayofileben da ar gadian iseT mkacr Semowmebas, ro-

gorc konkursze wardgenili saskolo saxelmZRvaneloebi. moswavlis saxelmZRva-

nelo, maswavleblis sarekomendacio wigni maswavlebels exmareba moswavlesa da

Sedegze orientirebuli saswavlo procesis dagegmvasa da warmarTvaSi.

19

1.1. ricxvebi da maTze moqmedebebi

Tema: ricxvebi da maTi gamoyeneba yoveldRiur cxovrebaSi da mecnierebis

sxvadasxva dargSi.

TemasTan dakavSirebuli mkvidri warmodgenebi: forma, warmodgena.

Tematuri sakvanZo SekiTxva: rogor SeiZleba gamoviyenoT ricxvebis Tvisebe-

bi sxvadasxva saxis praqtikuli amocanidan momdinare gamoTvlebisas? amocanebis amoxsnisas?

Temis farglebSi dasamuSavebeli sakiTxebi: ricxvis cnebis gafarToebis

procesi _ naturaluri ricxvebi, raodenobis gamosaxva naturaluri ricxve-

biT, racionaluri ricxvebi, gazomvis Sedegis gamosaxva dadebiTi racionaluri

ricxviT,^ racionaluri ricxvis Cawera wiladisa da aTwiladis saxiT. mimarTebebi

naturalur ricxvTa (N) mTel ricxvTa (Z) da racionalur ricxvTa simravleebs Soris.

saaTebis savaraudo raodenoba: 3 sT

samizne cnebebi da

maTTan dakavSirebu-

li mkvidri

warmodgenebi:

I etapi

davalebebi:

sakiTxebi: sakvanZo SekiTxvebi:

naturaluri, mTeli,

racionaluri ricx-

vebi; Caweris xerxebi;

Canawerebis gamoyene-

ba: wiladi, aTwiladi

(sasruli, usasrulo

perioduli), procen-

ti. mimarTebebi am

cnebebis moculobebs

Soris.

naturaluri

ricxvis Caw-

eris sistemebi

(poziciuri da

arapoziciuri),

mTeli ricxve-

bis gamoyeneba

yoveldRiur cx-

ovrebaSi, racio-

naluri ricxve-

bis praqtikuli

gamoyeneba.

ra SemTxvevaSi Cai-

wereba sasruli

aTwiladis saxiT

wiladiT warmodge-

nili racionaluri

ricxvi? ra ewode-ba ricxvs, romlis

mimatebiT mocemuli

ricxvisadmi miiReba

nuli? aris Tu ara nuli naturaluri

ricxvi? mTeli ric-xvi? racionaluri ricxvi? SesaZlebelia Tu ara |a|=–a toloba iyos swori romelime

racionaluri ricx-

visTvis?

amocanebi

saxelmZR-

vanelodan, rom-

lebic klasSi

Sesasruleblad

aris gamoyofili

( 1 - 11 ) saSinao davalebad

miecemaT analogi-

uri amocanebi:

1 - 13

20

aqtivobebi: maswavlebelma SeiZleba gamoiyenos SesavalSi da saxelmZRvane-

loSi warmodgenili masala ricxvebis Sesaxeb sakiTxebis gadmocemis dalagebis,

Tanamimdevrobebisa da istoriuli aspeqtebis Sesaxeb. masala nacnobi unda iyos

moswavleebisTvis. mimdinareobs gameorebis, codnis ganmtkicebisa da momavali

muSaobisTvis fsiqologiuri ganwyobis Seqmnis procesi. saxelmZRvanelos teqs-

ti racionaluri ricxvebis warmodgeniT iwyeba, magram sasurvelia naturaluri

ricxvebiT daviwyoT _ naturaluri ricxvebis Cawera, raodenobiTi da rigobiTi

aspeqtebi, dalageba. saswavlo procesi unda warimarTos moswavleTa aqtiuri Car-

TulobiT.

• viTvliT sagnebs, ra ricxvebs viyenebT? ras davadgenT Tvlisas?• raodenobis garkvevisas aqvs Tu ara mniSvneloba, romeli sagnidan viw-

yebT? ra kiTxvas vsvamT am dros? ras warmogvidgens sagnebis gadaTvlis bolos dasaxelebuli ricxvi?

• gadavdivarT rigobiT aspeqtze, roca vsvamT kiTxvas _ romeli? meramdene?• geometriul warmodgenaze gadasvlas mosdevs gazomvis saWiroebis gaazre-

bac da ukve dadebiTi racionaluri ricxvebis Semotanaze vzrunavT.

moswavleebi ixseneben damrgvalebis wesebs, msjeloben maTi gamoyenebis mniS-

vnelobaze,. geometriuli warmodgena da praqtikuli gamoyenebebi (magaliTad,

haeris temperaturis warmodgenisas) gvexmareba uaryofiTi racionaluri ricxve-

bis warmodgenaSi.

I etaps vasrulebT 1 - 11 `testebze~ pasuxebis SerCeviTa da mokle, sajaro

ganxilviT. 1 - 5 `testebze~ pasuxebis SerCeva ar iqneba Zneli. yuradReba mi-aqcieT yvela moswavlis CarTulobas saswavlo procesSi; gansakuTrebiT sWirde-

ba Tqveni Tanadgoma specialuri saganmanaTleblo saWiroebis mqone moswavleebs;

isini ar unda grZnobdnen Tavs saswavlo procesidan gariyulad.

SeiZleba meti yuradReba davuTmoT 6 - 7 davalebebs _ asoebis gamoyeneba da maTi saSualebiT faqtebis Cawera, rogorc wesi, uWirT moswavleebs.

• gavixsenoT ricxvis modulis cneba. ra saxiT Caiwereba a ricxvis moduli (a nebismieri racionaluri ricxvia)? ra saxiT Caiwereba a ricxvis mopirdapire ricxvi? ra SemTxvevaSia a ricxvis moduli a ricxvis mopirdapire ricxvis toli? sworia Tu ara winadadeba: nebismieri racionaluri ricxvis moduli dadebiTi

ricxvia.

am kiTxvebze pasuxebis analizis Semdeg SeiZleba 6 - 7 `teqstebze~ pasuxebis SerCeva.

8 `testiT~ vixsenebT ricxvis gaSlas saTanrigo Sesakrebebis jamis saxiT, geometriuli warmodgenebi gvexmareba 10 da 11 amocanebis amoxsnaSi.

gadavdivarT naturalur ricxvTa, mTel ricxvTa da racionalur ricxvTa

simravleebis aRniSvnebze da Sesabamisi cnebebis moculobebs Soris mimarTebebis

ganxilvaze. I etaps zogierTma maswavlebelma SeiZleba 1 gakveTili dauTmos, zo-gierTma _ ori (gasaTvaliswinebelia sazafxulo xangrZlivi ardadegebis Semd-

gomi ganwyobis specifikac); unda gaviTvaliswinoT klasis momzadebis donec.

21

II etapi kompleqsuri davaleba,

davalebebi:

sakiTxebi: ricxvebis gamosaxva ricxviT wrfeze. ra-

cionaluri ricxvis warmodgena sxvadasxva saxiT: eq-

vivalenturi (toli) wiladebiT, aTwiladebiT, pro-

centiT. racionaluri ricxvebis dalageba zrdis, an

klebis mixedviT; mimarTebebi ricxvTa simravleebs So-

ris, racionalur ricxvebze moqmedebTa Tvisebebi. ra-

cionaluri ricxvis Cawera.

klasSi _ 12 - 15 ;

saSinao: 14 - 15 , 16

aqtivobebi: II etapi iwyeba saSinao davalebis SemowmebiT: amocanebi:

1 - 10 . yuradReba gavamaxviloT 10 - 13 amocanebze, ar davkmayofildeT swori pasuxebis miTiTebiT. moswavleebs vTxovoT axsnan, rom –a SeiZleba aRniSna-vdes dadebiT an uaryofiT ricxvs, – a-Ti aRiniSneba a ricxvis mopirdapire ricx-vi. maSasadame, Tu, magaliTad, a= –15 (a ricxvi uaryofiTia), maSin –a=15, –a dadebi-Ti ricxvia. ar aris swori mimarTebebi: Q⊂N, Z⊂N, N⊂Q⊂Z. moswavleebi asaxeleben:

racionalur ricxvs, magaliTad, 35-s, romelic ar aris naturaluri ricxvi; mTel

ricxvs, magaliTad, (–2)-s, romelic ar aris naturaluri ricxvi; (– 57 )-s, romelic ar aris mTeli ricxvi.

sasurvelia, moswavleebma Tavad, ricxviTi wrfis gamoyenebiT, warmoadginon

12 -is amoxsna. 13 amocanis amoxsna moiTxovs damrgvalebis wesebis codnas, wi-naswari Sefasebis gamoyenebas. am aqtivobis Sesrulebas praqtikuli mniSvneloba

aqvs.

aRniSnuli amocanebis amoxsnebis garCeva dagvexmareba ganmaviTarebeli Se-

fasebis ganxorcielebaSi; sasurvelia CaviniSnoT ganmaviTarebeli Sefasebis Se-

degebi specialur wignakSi.

saSinao davalebis amocanebis garCevis Semdeg gadavdivarT klasSi Sesas-

rulebel amocanebze: 12 - 15 . amoxsnebze msjeloba mimdinareobs yvela moswavlis monawileobiT.

amocanebis amoxsnis procesSi SeiZleba davexmaroT moswavleebs Semdegi kiTx-

vebiT:

• mocemul or ricxvs Soris romeli gamoisaxeba ricxviT wrfeze ufro mar-jvniv mdebare wertiliT?

• ra geometriuli azri aqvs a ricxvis moduls?• SesaZlebelia Tu ara, romelime racionaluri n ricxvisTvis swori iyos

toloba |n|= –n?• uaryofiTia Tu dadebiTi ricxvi a, Tu misi Sesabamisi wertili saTavis

marcxniv aris?

22

• rogor SeiZleba SevadaroT tolmniSvneliani wiladebi? sxvadasxvamniSvne-liani wiladebi?

• ra aTwiladiT Caiwereba 0,47 da 0,48 aTwiladebi, Tu mZimis Semdgom sam aTwilad niSans gamoviyenebT?

15 amocanis amoxsna gulisxmobs konkretuli magaliTebis moZiebas. moswav-leebi msjeloben, asaxeleben magaliTebs, SearCeven maTgan swor nimuSebs.

III etapi davalebebi:

sakiTxebi: ricxviTi gamosaxulebis mniSvnelobis povna

(ricxvebze moqmedebebi, moqmedebebis Tvisebebi).

cvladiani gamosaxulebis mniSvnelobis povna cvladis

mocemuli mniSvnelobisTvis, racionaluri ricxvebis Se-

darebisa da dalagebis wesebi

klasSi: 16 - 20

saSinao: 17 - 19 .

aqtivobebi: saswavlo process viwyebT saSinao davalebis 16 amocanis amox-snis ganxilviT.

17 - 19 amocanebis amoxsnebi, ZiriTadad, ricxvebis Sedarebis wesebis gam-oyenebas moiTxovs. analogiuri amocanebi amoxsnili iyo wina gakveTilzec.

klasSi, erToblivi ZalisxmeviT, vxsniT 16 - 20 amocanebs. 16 davalebis amo-canebis amoxsnisas SeiZleba gamoviyenoT kontrmagaliTiT dasabuTebis xerxi.

magaliTad, nebismieri ori mTeli ricxvis jami rom ar aris naturaluri ricxvi,

SeiZleba davasabuToT (–7) da 5 mTeli ricxvebis miTiTebiT. maTi jami aris –2. –2 ar aris naturaluri ricxvi.

SeiZleba CavataroT damoukidebeli wera. moswavleebis erTi nawili poulobs

17 davalebis a) gamosaxulebis mniSvnelobas,^ meore nawili _ b) gamosaxulebis mniSvnelobas. winaswar ki moswavleebs daavaleT daasaxelon ricxviT gamosax-

ulebaSi warmodgenili moqmedebebis Tanamimdevroba. aqtivobebi, romlebic daka-

vSirebulia 18 - 20 amocanebis amoxsnasTan, moiTxovs wiladis ZiriTadi Tvisebis codnas. klasSi kidev erTxel ganixileT, ra saxis wiladebi SeiZleba Caiweros sas-

ruli aTwiladis saxiT. ratom aris racionaluri ricxvis warmomdgeni usasrulo

aTwiladi perioduli?resursebi: carci an markeri, dafa, moswavlis saxelmZRvanelo, samuSao

rveulebi. SeiZleba davamzadoT da gamoviyenoT plakatebi, romlebzec warmod-

genilia: naturaluri ricxvis aTobiTi Canaweri saTanrigo Sesakrebebis jamis sax-

iT, procentis Cawera ricxvis saxiT, ricxvis Cawera procentis saxiT, mimarTebebi

N, Z da Q simravleebs Soris.Sefasebis indikatorebi: moswavlem unda SeZlos poziciuri sistemis gamoy-

eneba ricxvebis Cawerisas (maT. sab. 4, 5).• racionalur ricxvebze moqmedebebis Sesruleba da maTi Sedegis Sefaseba

(maT. sab. 1, 2, 3, 4, 5);• msjeloba-dasabuTebis zogierTi xerxis gamoyeneba (maT. sab. 1, 2,^ 3);• gamoTvlebTan dakavSirebuli amocanebis amoxsna (maT. sab. 7, 8,^ 9).

23

miTiTebebi zogierTi amocanis amosaxsnelad:

7 a da –a mopirdapire ricxvebis gamomsaxvel wertilebs Soris manZili aris 2|a|. gaviTvaliswinoT, rom TiToeuli es wertili saTavis sxvadasxva mxaresaa da saTavidan |a|-s toli manZiliTaa daSorebuli (modulis geometriuli azri).

11 AB=2,7+1,7=4,4; B (1,7)0A (–2,7)

12 b0a

suraTis mixedviT, a0, |a|>|b|. cxadia, a–b, a+2

24

3 1%=0,01. 4 10%=0,1.

7 3125,351 _ am ricxvSi xazgasmuli cifris mniSvnelobaa 5100 .

13 1) 15,82+1,4+8,1>15+1+8=24 2) 7,66+9,11+6,939=22+0,77+0,939 es ricxvi aris 23-sa da 24-s Soris. 3) 6,99+3,89+15,39>6+3+15=24 4) 4,673+12,15+5,008=21+0,673+0,15+0,008

25

1.2. algebruli gamosaxuleba. gantoleba.

amovxsnaT amocanebi gantolebis gamoyenebiT

amovxsnaT amocanebi gantolebis gamoyenebiT.

Tema: algebra, algebruli gamosaxulebebi

TemasTan dakavSirebuli mkvidri warmodgenebi: gamartiveba, kavSirebi

TemasTan dakavSirebuli sakvanZo SekiTxvebi: riT gansxvavdeba cvladebis

Semcveli toloba gantolebisgan?rogor unda SevadginoT da amovxsnaT erTucnobiani gantoleba amocanebis

amoxsnisas?Temis farglebSi dasamuSavebeli sakiTxebi: algebruli gamosaxuleba, gan-

toleba, amocanebis amoxsna gantolebis gamoyenebiT

saaTebis savaraudo raodenoba: 3 sT

samizne cnebebi da maTTan

dakavSirebuli mkvidri

warmodgenebi:

I etapi davalebebi:

sakiTxebi: sakvanZo SekiTxvebi:

cvladi, cvladis Semcveli

gamosaxuleba (algebruli

gamosaxuleba), igiveoba,

gantoleba, algebruli ga-

mosaxulebis mniSvnelobis

povna cvladis miTiTebuli

mniSvnelobisTvis. gansx-

vaveba igiveobasa da gan-

tolebas Soris.

algebruli

gamosaxulebis

Sedgena da mniSvn-

elobis gamoTvla;

amocanis pirobis

mixedviT gan-

tolebis Sedgena

da misi amoxsna

ra gansxvavebaa

igiveobasa da gan-

tolebas Soris? ra gansxvavebaa ricxviT

gamosaxulebasa da

cvladian gamosa-

xulebas Soris?

amocanebi

klasSi:

1 - 12

saSinao:

1 - 12 .

aqtivobebi: gakveTili iwyeba saSinao davalebis SemowmebiT. miTiTebebi ukve

mogawodeT.

saSinao davalebis Semowmebis Semdeg saxelmZRvanelos teqstis mixedviT vmsje-

lobT warmodgenili cnebebis (gantoleba, cvladi, cvladis Semcveli gamosaxule-

ba) Sesaxeb. teqsti saSualebas gvaZlevs moviZioT pasuxebi sakvanZo SekiTxvebze.

magaliTad, gantolebis mocema niSnavs amocanis dasmasac: perimetris mixedviT,

ucnobi gverdis x-iT aRniSvnis Semdeg, SevadgenT tolobas: 12+5+x=26. dasmulia amocana: vipovoT mesame gverdi. iqve miniSnebulia, rom mesame gverds vipoviT, Tu

amovxsniT gantolebas. SegviZlia visaubroT moswavleebTan, rom cvladis (cvla-

debis) Semcveli tolobis mocema ar niSnavs gantolebis mocemas. cvladebis Semc-

veli toloba SeiZleba igiveoba iyos.

vaxsenebT moswavleebs ramdenime cnobil igiveobas (Semoklebuli gamravlebis

formulebs). aRniSnuli tolobebi WeSmaritia maTSi Semavali cvladebis nebismeri

mniSvnelobebisTvis. cvladi mocemuli simravlis nebismieri elemenetia, am SemT-

26

xvevaSi mocemul simravled SeiZleba miviCnioT CvenTvis cnobili ricxvebis sim-

ravle (racionalur ricxvTa simravle). cvladis, igiveobisa da gantolebis cne-

bebi kargad aris aRwerili cnobili qarTveli mecnieris Salva fxakaZis naSromSi

[11]. maswavleblebs vurCevT, Tu SeZleben, gaecnon am saxelmZRvanelos.1 gantolebis Sesaxeb saubrisas SeiZleba gamoviyenoT cnobili maTematiko-

sis fuSes [38] gamonaTqvami: `gantoleba aris toloba, romelic jer kidev ar aris WeSmariti, magram iswrafvis, rom iyos WeSmariti, Tumca, winaswar Znelia Tqma,

SeiZleba Tu ara amis miRweva~... `amovxsnaT gantoleba _ niSnavs, ganvsazRvroT,

Tu es SesaZlebelia, masSi Semavali elementebi ise, rom miviRoT swori toloba~.

saxelmZRvaneloSi warmodgenil sakontrolo kiTxvebze pasuxebis moZieba da-

exmareba moswavleebs winare codnisa da gamocdilebis gaaqtiurebaSi: algebru-

li gamosaxuleba SeiZleba Seicavdes ramdenime cvlads. gantoleba aris toloba,

romelic Seicavs cvlads da dasmulia amocana am gantolebis amoxsnis Sesaxeb.

gantolebis amoxsna niSnavs vipovoT cvladis is mniSvneloba (mniSvnelobebi),

romlis (romelTa) CasmiT miiReba WeSmariti toloba. SeiZleba damatebiTi kiTxve-

biT davazustoT moswavleTa warmodgenebi algebruli gamosaxulebis, igiveobis,

gantolebis Sesaxeb. SeiZleba movamzadoT plakati Sesabamisi gamosaxulebebiT

da vTxovoT moswavleebs gaarCion, romel cnebas aRwers plakatze warmodgeni-

li gamosaxulebebi. SeiZleba davexmaroT maT cnebebs Soris kavSirebis da mimar-

Tebebis gaazrebaSi. es sakontrolo kiTxvebi da saxelmZRvaneloSi warmodgenili

sxva davalebebi xels uwyobs ganmaviTarebeli Sefasebis gamokveTas, romelsac

maswavlebeli sistematurad unda awarmoebdes. dakvirveba, Sedegebis fiqsireba

gansakuTrebiT advilia `testebze~ pasuxebis SerCevis procesSi. ZiriTadi aqcen-

ti keTdeba gantolebis fesvis, tolfasobis, igiveobis, cvladiani gamosaxulebis

mniSvnelobis cnebebis kargad dauflebaze; `testebze~ pasuxebis moZiebac SeiZle-

ba msjelobisa da dasabuTebis procesiT mimdinareobdes. magaliTad, moviTxovoT

pasuxis dasabuTeba 3 amocanaSi (tolfasoba), 4 -Si (ricxviTi wrfis gamoyene-ba), 5 amocanaSi (amocanis amosaxsnelad gantolebis Sedgena), 6 amocanaSi (pro-porciuli sidideebis gamosaxva, samkuTxedis utolobis gaxseneba), 8 amocanaSi (igiveobis cnebis gaazreba), 1 , 2 , 3 , 9 da 10 amocanebSi gantolebis fesvis cnebis gaazreba. `testebis~ mravalferovani sistema yvela samizne cnebas moicavs.

amasTanave, nu mosTxovT moswavleebs sxvadasxva cnebis gansazRvrebebis Camoya-

libebas, SesTavazeT maT fsevdo gansazRvrebebi. sakmarisia moswavlem SeZlos Se-

sabamisi obieqtis dasaxeleba. gana SeiZleba moswavles vkiTxoT, magaliTad, ras

ewodeba naturaluri ricxvi? mas unda SeeZlos naturaluri ricxvis dasaxeleba, unda SeeZlos igiveobis, gantolebis magaliTebis miTiTeba. moswavleebs Tanda-

TanobiT, Cveni daxmarebiTa da ZalisxmeviT, xangrZlivi da SemoqmedebiTi swavlis

procesSi uyalibdebaT warmodgenebi maTematikur cnebebze, maT kavSirebsa da ga-

moyenebebze.

zogierTi `testis~ pasuxis dasabuTebisas SeiZleba kidev erTxel SevamowmoT

Sesabamisi cnebis aTvisebis done.

1 testis pasuxis SerCevis Semdeg SeiZleba moswavlem daazustos _ namravli nulis tolia mxolod im SemTxvevaSi, roca erT-erTi Tanamamravli mainc nulis

tolia.

27

maSasadame, gveqneba x–1=0 an x–1,5=0.2 am testSi moqmedebaTa urTierTSebrunebadobis Tviseba unda gamoviyenoT;

bolo moqmedebis win gveqneboda mocemuli gantolebis 3-ze gamravlebis Sedegi _ 3x=27. es gantoleba mosdevs gantolebas, romelic miiReba orive nawilisadmi 8-is mimatebiT: 3x+8=27+8, 3x+8=35. SeiZleba miRebuli gantolebebi ganvixiloT da maTze dakvirvebis Sedegad SevarCioT pasuxi.

es xerxi ufro xelsayrelia 3 `testSi~ pasuxis SerCevisas. aq ar aris aucilebeli gantolebebis amoxsna, wina wlis masalidan moswavles unda axsovdes

gantolebis igivuri gardaqmnis wesebi, roca gadavdivarT tolfas gantolebaze.

magaliTad, 8x= –4 da 8x+x=x– 4 tolfasi gantolebebia, pirveli miiReba meore gan-tolebis orive nawilidan x-is gamoklebiT. yvela moswavlis CarTvis mizniT Sei-Zleba naklebi mzaobis moswavles davavaloT orive gantolebis amoxsna; igi dar-

wmundeba, rom orive gantolebis fesvi erTi da igive ricxvia.

4 moswavlem swrafad unda gaiazros, rom b ricxvis moduli aris (–b) da sa-Tanado wertilebs Soris manZili am wertilebis saTavemde manZilebis (anu mod-

ulebi) jamis anu (a–b)-s tolia.

5 proporciis Sedgena xdeba 1 vagonis tevadobebis povniTa da saTanado gan-tolebis mixedviT. _ Tu 4 erTnairi vagoniT 240 t tvirTis gadazidvaa SesaZle-

beli, maSin 1 vagoniT ramdeni t tvirTi gadagvaqvs ( 2404 )?

_ Tu 300 t tvirTis gadasazidad x vagonia saWiro, maSin yoveli vagoniT ram-

deni t tvirTis gadazidva xdeba ( 300x )?SesaZlebelia, zogierTma moswavlem asec imsjelos: ramdenjerac 300 t me-

tia 240 t-ze, imdenjer meti iqneba saTanado vagonebis x ricxvi 4-ze da daweros 300240 =

x4 . maSin mas mouwevs am gantolebis tolfasis SerCeva amocanaSi miTiTebul

4 gantolebas Soris.

6 am amocanis amoxsnas sWirdeba samkuTxedis utolobis codna da propor-ciuli sidideebis gamosaxva 1 nawilis Sesabamisi x erTeuliT.

radgan x+2x=3x, amitom ar arsebobs samkuTxedi, romlis gverdebi 1-is, 2-isa da 3-is proporciulia.

ar aris gamoricxuli, rom moswavlem swor pasuxs Tavad ricxvebis analiziT

miagnos.

7 2 nawili _ oqro, 13 nawili _ vercxli, 15 nawili _ Senadnobi.

Senadnobis 1315 vercxlia. zogierTma moswavlem SeiZleba x cvladiT gamosaxos vercxlisa da Senadnobis masebi: 13x da 15x.

8 Semoklebuli gamravlebis formulebis codna aadvilebs swori pasuxis SerCevas.

28

9 gantolebis fesvis cnebis codna gvexmareba 4,5.a=18 tolobis mignebaSi, saidanac vpoulobT a-s.

10 am gantolebas fesvi eqneba, Tu a–1≠0, anu a≠1. es fesvi (2a+3) ricxvis (a–1)-ze, gayofiT miiReba. Tu vigulisxmebT, rom a=1, maSin 2a+3=4 da miviRebT gan-tolebas: 0.x=4, romelsac amonaxsni ara aqvs. am amocanis ganmaviTarebeli efeqti gaZlierdeba, Tu mis amoxsnas daukavSirebT ax=b saxis wrfivi gantolebis gamokv-levas da klass miiyvanT Semdeg daskvnamde: roca a=0 da b≠0, gantolebas ara aqvs fesvi; roca a=0 da b=0, maSin miiReba 0⋅x=0 gantoleba da mas nebismieri ricxvi akmayofilebs; roca a≠0, gantolebas erTi fesvi aqvs.

13 amocana moiTxovs verbalurad warmodgenili situaciis formulis saxiT Caweras (maT. sab. 4, 5, 6, 7, 8, 9). saxeldobr, Tu 2x–3 gamosaxulebis mniSvneloba 12-iT naklebia 3x+5 gamosaxulebis mniSvnelobaze, maSin

(2x–3)+12=3x+5.SevadgineT gantoleba, romlis amoxsna mogvcems x-is saZiebel mniSvnelobas.saSinao davalebad micemuli amocanebi zemoT ganxiluli `testebis~ analo-

giuria.

14 amocana moiTxovs gantolebis Sesabamisi amocanis Sedgenas (maT. sab. 3, 4, 7, 8, 9). aqtivoba SeiZleba jgufuri muSaobis formiT Catardes da jgufebs Soris Sejibrebis saxe mivceT _ romeli ufro met da saintereso amocanas Seadgens. es

procesi dagvexmareba ganmaviTarebeli Sefasebis ganxorcielebaSi.

II etapidavalebebi:

sakiTxebi: sakvanZo SekiTxvebi:

amocanebis amoxsna gan-tolebis gamoyenebiT

rogor unda SevadginoT da amovxsnaT gantoleba teqsturi amocanis amox-snisas da realuri viTare-bis modeli rebisas?

klasSi: 15 - 18

saSinao: 10 - 17 , 20 kompleqsuri davalebebi:

19 , 20 , 18 , 19

aqtivobebi: TemaSi mocemuli masala (amocanebi 15 - 20 ) xels uwyobs sakvanZo SekiTxvebze pasuxis Ziebas.

gantolebis gamoyenebiT amocanis amoxsnis win unda SevadginoT pirobis sqe-

maturi modeli. magaliTad, 15 amocanis piroba ase Caiwereba:zezva

Teona 17 lariT meti.orive _ 55 lari

x cvladis Semotana da sidideebis gamosaxva x-iT, gantolebis Sedgena:^zezva _ x, Teona _ x+17, orive _ x+(x+17), maSasadame _ 2x+17=55.gantolebis amoxsna: 2x=38, x=19.zezva _ 19, Teona 19+17=36.Semowmeba: orive _ 19+36=55, eTanxmeba pirobas.

29

16 umciresi gverdi _ x, masze 2-iT meti _ x+2, um-ciresze 9-iT meti _ x+9.

amocanis pirobis Tanaxmad, x+(x+2)+(x+9)=38. 3x+11=38, 3x=27.

x=9, x+2=11, x+9=18.amocanebis amoxsnis procesi interaqtiuli meTodiT unda mimdinareobdes.

magaliTad, am amocanis amoxsnis procesi SeiZleba ase warimarTos:

• romeli gverdis sigrZe aRvniSnoT x-iT?• rogor gamoisaxeba x-iT danarCeni gverdebi?• amocanis ra pirobis mixedviT Sedgeba gantoleba?• saWiroa Tu ara miRebuli pasuxebis Semowmeba?• SeiZleba Tu ara, rom erTi gverdi umcires gverdze 2-jer meti iyos, meore

gverdi _ 9-jer? SeecadeT ganixiloT es SemTxvevac.

17 es amocanac moswavleTa erToblivi ZalisxmeviT unda amoixsnas. nu viCqa-rebT x-is Semotanis dros. kargad davukvirdeT pirobis mokle Canawers:

I:II=5:6, II:III=4:9, I+II+III=2940.

moswavleebi TviTon unda mividnen im daskvnamde, rom x-is moxerxebul Semota-nas sWirdeba pirobis `modificireba~. am daskvnamde moswavleebi SeiZleba Semdeg

SekiTxvebze dafiqrebiT miviyvanoT.

• ra aris I da II-is Sefardeba (5:6)?• ra aris II da III-is Sefardeba (4:9)?• SeiZleba Tu ara am monacemebiT davweroT, ra iqneba I:II:III? rogor SevZloT

es? yuradReba gavamaxviloT, rom II dRes mosulTa raodenobas sxvadasxva `nawili~ Seesabameba.

moswavleebma erToblivi ganxilviTa da Tqveni SesaZlo CarTulobiT unda

miagnon gamosavals _ Sefardebebi ise unda Seicvalos, rom SefardebebSi II erTi

da imave ricxviT monawileobdes. unda gaviTvaliswinoT ricxvebis Sefardebis

Tviseba _ Sefardeba ar Seicvleba, Tu orive ricxvs erTsa da imave ricxvze gavam-

ravlebT. am meTodis aseTi xangrZlivi ganxilva, msjeloba-dasabuTebis Catareba

xels Seuwyobs meTodis `Camoyalibebas~ moswavlis xangrZliv mexsierebaSi; unda

vipovoT 4-isa da 6-is umciresi saerTo jeradi, Semdeg _ `damatebiTi~ mamravlebi da maTze gavamravloT. Sesabamisad 5:6 da 4:9 Sefardebebi SevcvaloT maTi toli da CvenTvis xelsayreli SefardebebiT: I Sefardebis orive mxare, gavamravloT

2-ze, meorisa _ 3-ze. miviRebT: I:II=10:12, II:III=12:27.maSasadame, I _ 10x, II – 12x, III – 27x. 10x+12x+27x=2940, 49x=2940, x=60. 10x=600, 12x=720, 27x=1620. marTlac, 600+720+1620=2940.SeiZleba romelime moswavlem aseTi amoxsna SemogvTavazos (igi unda wavaxa-

lisoT):

A

B

C

30

I _ x, maSin II – 65 x, III:65 x=9:4, saidanac III –

65 x⋅

94 =

2710 x.

x+ 65 x+2710 x=2940,

(1+1,2+2,7)x=2940, 4,9x=2940, x=2940:4,9=29400:49

x=600, 65 x=720, 2710 x=1620.

18 amocanis amoxsnisas moswavleebi ixseneben samkuTxedis perimetris cnebas.

)

EC+BC+BE=17 AE+AB+BE=14 AB+BC+AC=23pirobis ori mocemulobis Caweris Semdeg mivmar-

TavT moswavleebs:

_ samkuTxedis romeli gverdebis sigrZeebi monaw-

ileobs orive tolobaSi?_ ra nawilebisgan Sedgeba AC?es kiTxvebi daexmareba moswavleebs amoxsnis strategiis ZiebaSi.

_ SeiZleba Tu ara pirveli ori tolobisa da mesame tolobis gamoyenebiT mi-

viRoT gantoleba BE-s mimarT? es kiTxva SeiZleba TviTon moswavleebma moifiqron da pirveli ori tolobis SekrebiTa da mesame tolobis gamoyenebiT, gansazRvron

BE-s sigrZe. SeiZleba BE-s x-iT aRniSvnam gaaadvilos amocanis amoxsnis procesi. 19 da 20 amocanebi SeiZleba kompleqsur davalebebad miviCnioT: moiTxovs sxva-

dasxva codnis (farTobi, Semoklebuli gamravlebis formulebi, miaxloebiTi ga-

moTvlebi, ricxvebis damrgvaleba, gamoTvlebi kalkulatoriT) integrirebulad

gamoyenebas.

19 amocanis amoxsnis procesi SeiZleba Cveni daxmarebiT warimarTos. analo-giur amocanebs ( 18 da 19 ) moswavleebi Sin damoukideblad amoxsnian.

_ gavixsenoT marTkuTxedisa da kvadratis farTobis formulebi.

_ tolobebis marcxena mxareebi ra marTkuTxedis farTobs iZleva?_ rogor `davWraT~ suraTze gamosaxuli (a+b)-gverd-

iani kvadrati or kvadratad, romelTa gverdebia, Sesabam-

isad, a da b da kidev or marTkuTxedad, romelTa gver-debia a da b?

_ meore tolobis misaRebad

ra kvadratebis gamoyofa iqneba

saWiro? ra marTkuTxedebis gam-oyofa xdeba?

AE

B

C

← →← →←a →

← →

← →

← →a

(a+ b2 )2

(a+ b2 )⋅b2

(a+b2)b2

( b2 )2

b2

b2

b2

b2

a2

←a → ←b →

ab b2

ab

← →

←

→

b

a

31

20 amocanis amoxsnisas viyenebT kalkulators da kalkulatoriT miRebul Sedegs vadarebT orwevris kvadratis formulis gamoyenebiT miRebul Sedegs.

moswavleebs moeTxovebaT, rom imsjelon miaxloebiTi tolobebis gamoyenebis mi-

zanSewonilobaze. ra SemTxvevaSia es gamoyeneba gamarTlebuli?amave gakveTilze, saSinao davalebis Semowmebisas, rac saswavlo procesis

dasawyisSi mimdinareobs, vmsjelobT saSinao davalebis `testebis~ Sesaxeb. es

`testebi~ wina gakveTilze klasSi amoxsnili testebis analogiuria (magaliTad,

1 , 2 , 3 , 7 , 8 ).

9 `testis~ amoxsnisas, moswavlem unda SeZlos verbalurad aRwerili situ-aciis algebruli gamosaxulebis (formulis) saxiT Cawera.

dagegmili aqtivobebi aaqtiurebs moswavleTa winare codnas (algebruli ga-

mosaxulebis Sedgena, gantolebis Sedgena, geometriuli da algebruli faqtebis

dakavSireba), ganamtkicebs da aviTarebs SeZenil codnas, xels uwyobs codnaTa

urTierTdakavSirebas (gantolebis amoxsna, proporcia, proporciuli sididee-

bi, Semoklebuli gamravlebis formulebi, kvadratisa da marTkuTxedis farTobi.

SeTavazebulia zogierTi davalebis amoxsnis sxvadasxva xerxi, romelic exmareba

moswavles gaiazros ama Tu im sakiTxis Seswavlisas strategiis SerCevis mniSvne-

loba, aviTarebs moswavlis kritikul azrovnebas.

resursebi: carci (an markeri), dafa, saxelmZRvanelo, kalkulatori.

mesame gakveTili saSinao davalebis amocanebis ganxilvas eZRvneba. mniSvnelo-

vania codnis ganmtkiceba algebruli gamosaxulebebis Sedgenasa da gamartiveba-

ze, gantolebis gamoyenebiT amocanebis amoxsnaze.

saSinao davalebis amocanebi, rogorc wesi, klasSi Sesrulebuli davalebebis

analogiuria, mimdinareobs codnaTa urTierTdakavSirebisa da sistemaSi moyvanis

procesi, cnebebs Soris kavSirisa da damokidebulebebis Sesaxeb warmodgenebis

gaRrmavebis procesi. es procesi axal codnaze organulad gadasvlasac uwyobs

xels. procentebze amocanebis amoxsnis erT-erTi meTodi cvladis Semotanasa da

gantolebis Sedgenas ukavSirdeba. proporciulobisa da proporciis Tvisebebi

procentze amocanebis amoxsnis drosac xSirad gamoiyeneba.

magaliTad, 14 amocanaSi gogonebisa da biWebis raodenobebi 3x-iTa da 5x-iT SeiZleba gamovsaxoT. amocanis pirobis Tanaxmad, 5x–3x=6. yuradRebiT ganixileT

10 amocanaSi mocemuli davaleba. misi Sesruleba gantolebis Sesabamisi amoca-nis Sedgenas ukavSirdeba (maT. sab. 3, 4, 7, 8, 9). moswavleebi sajarod kiTxuloben Sedgenil amocanebs, zogierTma SeiZleba ramdenime variantic ki warmoadginos,

romlebic Sinaarsis mravalferovnebiT gamoirCeva. es momentebi ganmaviTarebeli

Sefasebis ganxorcielebaSi dagvexmareba. CainiSneT Tqveni dakvirvebis Sedege-

bi. 16 amocanis analogiuri amocanebi adrec hqondaT moswavleebs amoxsnili. SeiZleba venis diagramis gamoyenebac. 7x–2x=25 gantolebis mixedviT vpoulobT inglisuris SemswavlelTa raodenobas da frangulis SemswavlelTa raodenobas

(Sesabamisad, 35 da 10). orive enas, cxadia, swavlobs (35+10)–45 studenti.

32

maSasadame, arc erTi studenti ar swavlobs orive enas. am SemTxvevaSi simravlee-

bi aragadamkveTi wreebiT gamoisaxeba _ simravleebis TanakveTa carieli simrav-

lea. Tumca, moswavlem SeiZleba ase imsjelos, radgan 35+10=45, yoveli studenti swavlobs mxolod erT enas, an franguls, an inglisurs.

TvalsaCino warmodgenis mizniT venis diagramis gamoyenebisas, Tu diagramas

Tavdapirvelad mieca saxe , amoxsnis dasrulebisas orive enis Semswav-

lelTa raodenobis adgilze 0 Caiwereba: 35 0 10 .

17 amocana ukavSirdeba manZilis Tvisebas. Tumca, Tavdapirvelad, sasurvelia, manZilebi erTi cvladiT gamovsavxoT.

a:b=2:3, anu a:b=8:12. b:c=4:9, anu b:c=12:27. maSasadame, SeiZleba davweroT: a=8x, b=12x, c=27x. radgan 8x+12x

33

Tu Tqven saWiro CaTvliT, SeiZleba kidev erTi analogiuri

amocana dausvaT moswavleebs aq gamosaxuli figuris SemTxveva-

Si.

rac migiyvanT aseT igiveobamde: 3a(b–c)+2ac=2ab+a(b–c).

20 amocanis davalebebi moiTxovs Semoklebuli gamrav-lebis formulebis da algebruli gamosaxulebis gamartivebis codnis. sasarge-

bloa am mimarTulebiT codnis ganmtkicebisTvis.

resursebi: carci, dafa, saxelmZRvanelo, kalkulatori.

Sefasebis indikatorebi: moswavlem unda SeZlos racionalur ricxvebze mo-

qmedebebis Sesruleba da maTi Sedegis Sefaseba (maT. sab. 1, 2, 3, 4, 5); msjelo-ba-dasabuTebis zogierTi xerxis gamoyeneba (maT. sab. 1, 2, 3̂);

gantolebis Sedgena verbalurad mocemuli amocanis Sesabamisad; gantolebis

Sesabamisi amocanis Sedgena da amoxsna (maT. sab. 3, 4, 7, 8, 9).

1.3. vagrZelebT procentis ganxilvas

Tema: ricxvebi da maTi gamoyeneba yoveldRiur cxovrebaSi da mecnierebis dar-

gebSi

TemasTan dakavSirebuli mkvidri warmodgenebi: forma, warmodgena.

Tematuri sakvanZo SekiTxvebi: rogor SeiZleba gamoviyenoT ricxvebis Tvisebe-

bi da Caweris saxeebi sxvadasxva saxis praqtikuli amocanidan momdinare gamoTvle-

bisas? amocanebis amoxsnisas?

Temis farglebSi dasamuSavebeli sakiTxebi: ricxvis Cawera procentis saxiT,

ricxvis procentis povna, proporciis gamoyeneba procentze amocanebis amoxsnisas.

procentis daricxva.

saaTebis savaraudo raodenoba: 4 sT

a

ac

b

a

34

samizne cnebebi da mkvidri

warmodgenebi:

I etapi davalebebi:

sakiTxebi: sakvanZo SekiTxvebi:

procenti, procentis warmodgena wiladiT da aTwiladiT; wila-dis da aTwiladis warmodgena procen-tis saxiT;

procenti, maT So-ris, 1%-ze naklebi, 100%-ze meti; aris Tu ara orjer 10%-iT zrda 20%-iT zrdis toli?

xelfasis gaormage-biT miRebuli Tanxa ufro metia, Tu xel-fasis 100%-iT gazr-diT miRebuli Tanxa? fasis zrda 3-jer (gasammageba) ramdeni procentiT zrdas niSnavs? ricxvis ra nawilia misi 0,1%?

proeqti: `damatebuli Rire-bulebis gadasaxa-di~ (dRg).

amocanebi: klasSi:

1 - 11

saSinao: 1 - 10

aqtivobebi: pirveli aqtivoba procentis Sesaxeb codnis gameorebiT iwye-

ba. gavaxsenebT moswavleebs, rom procentic SeiZleba miviCnioT ricxvis Cawer-

is erT-erT formad. moswavles unda SeeZlos 25%-is warmodgena 0,25-isa an 14 -is saxiT, rom 25% aris ricxvis meoTxedi. 100-ze naklebi da 1-ze meti procentis gameorebis Semdeg gadavdivarT axal Temaze _ procenti _ 1-ze naklebi da 100-ze meti. yuradReba gavamaxviloT ricxvis procentiT gazrdis magaliTebze. mag-

aliTad, 200%-iT gazrda niSnavs ricxvis 3-jer gazrdas (gasammagebas); msjeloba SeiZleba ase warimarTos.

_ vTqvaT, ricxvi izrdeba 100%-iT, ris tolia ricxvis 100%?_ amave ricxvis.

_ maSasadame, ramdeniT unda gaizardos mocemuli ricxvi?_ amave ricxviT!

_ ramdenjer gaizrdeba ricxvi?_ 2-jer._ ris tolia ricxvis 200%?_ ricxvis 200% 2-jer metia am ricxvze._ maSasadame, ramdeniT izrdeba mocemuli ricxvi?_ am ricxvze 2-jer meti ricxviT, miviRebT am ricxvze 3-jer met ricxvs._ ramdenjer gaizarda ricxvi 200%-iT gazrdisas?_ ricxvi gaizarda 3-jer._ ramdenjer gaizrdeba ricxvi, Tu mas gavadidebT 400%-iT?_ gaizrdeba 5-jer._ ramdeni procentiT gaZvirdeba kalami, Tu misi fasi gaizrdeba 4-jer?_ gaizrdeba 300%-iT.moswavleebi swrafad gaiazreben gazrdis process da ar gauWirdebaT pasuxis

gacema).

35

mimdinareobs moswavleTa etapobrivi miyvana axal codnamde, axali cnebis

gaazrebamde, sakvanZo SekiTxvebze pasuxebis moZieba.

`testebi~ (daxurulboloiani amocanebi) pirveli warmodgenebis Seqmnis kar-

gi saSualebaa. SeiZleba ar SemovifargloT mxolod pasuxebis miTiTebiT,^ zog-

jer moviTxovoT arCevanis dasabuTeba da es dasabuTebac sajarod ganvixiloT.

davalebebis es tipi gansakuTrebiT mosaxerxebelia specialuri saganmanaTleblo

saWiroebis mqone moswavleebTan muSaobis dros. Tumca, ar unda vagrZnobinoT

maT, rom maT mimarT gansakuTrebul yuradRebas viCenT _ maTi integracia bune-

brivi unda iyos. Tumca, SeuZliaT `testze~ pasuxi nomris miniSnebiT gvacnobon.

`testebze~ gadasvla saxelmZRvaneloSi warmodgenili magaliTebis ganxilvis Sem-

deg xdeba. amis Semdeg moswavleebs ar gauWirdebaT aqtiuri CarTulobisas swra-

fad miuTiTon 1 - 8 amocanebis pasuxebi. 9 amocanas SeiZleba meti dro davuTmoT da amoxsnis sxvadasxva xerxi gan-

vixiloT:

I xerxi: erTi Tanamamravlis 200%-iT gazrda niSnavs namravlis 3-jer gazrdas, meore Tanamamravlis 2-jer SemcirebiT namravli 2-jer Semcirdeba. maSasadame, namravli orive moqmedebis Sesrulebis Semdeg 1,5-jer gaizrdeba, anu gaizrdeba 50%-iT.

II xerxi: I Tanamamravli – x, meore Tanamamravli – y. orive moqmedebis Catare-

bis Semdeg namravli miiRebs saxes: (x+2x) y2 =32 xy.

xy – 100%

32 xy – n% n%=

32

xy ⋅100%

xy =150%.

gaizarda 150%-100%=50%-iT.

SeiZleba asec vimsjeloT: 32 xy–xy = 12 xy

12 xy aris (xy)-is 50%. gaizarda 50%-iT.

10 480+480⋅0,12=480⋅1,12=537,6.

11 narCenebi: 100%–(30%+54%)=16%. 20⋅0,3=6 (t), 20⋅0,54=10,8 (t).

II etapi davalebebi:

sakiTxebi: sakvanZo SekiTxva:

procenti, sididis cvlileba misi raime procentiT cvli-lebisas

rogor icvleba ricxvebis namravli, roca erT-erTi 10%-iT izrdeba, meore _ 10%-iT mcirdeba?

kompleqsuri davalebebi:

19 , 21 ;

davalebebi: klasSi: 12 - 18

saSinao: 10 - 20

proeqti.

36

aqtivobebi: gakveTilis erTi nawili SeiZleba davuTmoT proeqtis Sesrulebis

ganxilvas.

proeqti moswavlis damoukidebeli SemoqmedebiTi saqmianobaa (proeqti Sei-

Zleba iyos raime problemis gamokvlevac ki). dasrulebuli proeqtebis Sefase-

ba warmodgenili referatebis mixedviT, moxsenebebis ganxilviT mimdinareobs. es

ganxilva ganmaviTarebeli Sefasebis ganxorcielebis kargi saSualebaa. amjerad,

moswavleebs moeTxovebodaT informaciis moZieba damatebuli Rirebulebis ga-

dasaxadis (dRg-s) Sesaxeb, am gadasaxadis gadaxdisaTvis saWiro procedurebis, ga-

dasaxadis procentuli odenobis Sesaxeb; mocemul konkretul SemTxvevaSi _ misi

odenobis Sesaxeb. moswavleebs evalebaT am rTuli wesebiT dadgenili gadasaxa-

dis mniSvnelovnad gamartivebuli modelis warmodgena. mivaqcioT yuradReba

imas, rom damatebuli Rirebulebis gadasaxadi, arapirdapiri gadasaxadia _ yve-

la dasabegri saqonlis Rirebulebis nawilis saxelmwifo biujetSi savaldebulo

Senatani. mas, faqtobrivad, saqonlis SemZeni piri ixdis.

proeqtis ganxilvis Semdeg gadavdivarT saSinao davalebis amocanebze _

`testebze~ pasuxebis SerCevis sakiTxze. maTi amoxsnis procesSi moswavles Seu-

Zlia moiZios sakvanZo SekiTxvaze pasuxi _ igi dakavSirebulia 8 `testTan~. I xerxi: erTi ricxvis 20%-iT gazrdisas, vRebulobT pirvel ricxvze 1,2-jer met ricxvs, meore ricxvis 20%-iT Semcirebisas vRebulobT am ricxvis 0,8-ze gamrav-lebiT miRebul ricxvs. sabolood, mocemuli ricxvi mravldeba 1,2⋅0,8-ze, anu 0,96-ze da miiReba mocemuli ricxvis 0,96. amrigad,

gvaqvs 4%-iT Semcireba.

II xerxi: I ricxvi _ x namravlia xy.

II ricxvi _ y

moqmedebaTa Catarebis Semdeg: I _ x+0,2x=1,2x, namravli _ 0,96xy.

II _ y–0,2y=0,8y.

Semcirda _ 0,04xy-iT, es ki xy-is 4%-ia.

SeiZleba moviTxovoT dasabuTeba 7 da 9 `testebis~ SemTxvevaSic da ganxilva moswavleTa aqtiuri CarTulobiT warvmarToT.

7 x → 2x → 2x+4x=6x, 6x–x=5x, 500%-iT gaizarda.moswavlem SeiZleba sxva xerxic SemogvTavazos:

100% → 100%+100%, 200%+200%+200%=600%.gaizarda 500%-iT.

9 mariamis xelfasi _ x. misi 12%-ia 0,12 x. naTias xelfasi _ 3x, 0,36x. 0,36x:0,12x=3 3-jer metia.

37

amave gakveTilze grZeldeba procentis Sesaxeb warmodgenis gaRrmavebisa da

procentze amocanebis amoxsnis procesi. es procesi xels uwyobs Sefasebis indi-

katorebis moTxovnaTa dakmayofilebas da mkvidri warmodgenebis Camoyalibebas.

erT-erTi indikatori verbalurad aRwerili situaciis algebruli gamosa-

xulebis (formulis) saxiT Caweris unaris ganviTarebaa (maT. sab. 4, 5, 6, 7, 8, 9). magaliTad 15 amocanis amoxsnamde sasurvelia winaswar moisminoT moswavleTa savaraudo pasuxebi. mosalodnelia, rom moswavleTa mniSvnelovanma nawilma iva-

raudos Tavdapirveli fasis aRdgena. am eqsperimentisa da Semdgomi amoxsnis Se-

degis SedarebiTi analizi sayuradRebo ganmaviTarebeli efeqtis mqonea. amocanis

amoxsnis procesSi SeiZleba SemoviRoT x cvladi da SevadginoT algebruli gamo-saxuleba:

(x–0,1x)+(x–0,1x)⋅0,1.am algebruli gamosaxulebis gamartiveba mogvcems:

0,9x⋅1,1=0,99xmaSasadame, Tavdapirveli fasi Semcirda 1%-iT.Tu amocanis pirobis mixedviT, moswavle Tavidanve mixvdeba, rom fasi Semcir-

deba (gazrda xdeba Semcirebulis 10%-iT), maSin SevaqoT es moswavle da movTxov-oT sakuTari azris dasabuTeba. Sefasebis erT-erTi indikatori msjeloba-dasa-

buTebis gamoyenebas da miRebuli daskvnis dasabuTebas ukavSirdeba (maT. sab. 2).moswavleebs aucileblad daainteresebT 14 amocanis ganxilva. 1 kg dasarCe-

vi kakali Seicavs miaxloebiT _ 450 g nigozs. am SemTxvevaSi 1 kg nigvzis fasi (x lari) SeiZleba ase vipovoT: 450 g _ 5 l

1000 g _ x l

x= 5⋅1000450 = 50045 =

1009 ≈ 11,1.

maSasadame, kaklis yidva finansurad ufro xelsayrelia.

16 I xerxi: gaxda 45 nawili.

45 _ 100%

15 _ x% x=25%-iT gazrdila.karg Sedegs mogcemT am amocanis ilustracia moaxdinoT raime konkretuli

`kargad SerCeuli~ xelfasis magaliTze. vTqvaT, xelfasi iyo 1000 lari. mexuTe-diT Semcirebisas is gaxdeboda 800 lari. 800-s unda daematos misi meoTxedi, anu 25%, rom kvlav miviRoT 1000.

17 es amocanac sakvanZo SekiTxvaze pasuxis Ziebas ukavSirdeba: iyo x/y. cvl-ilebis Semdeg gaxdeba:

x+0,2xy–0,2y = 1,2x0,8y =

32 ⋅

xy ,

32 ⋅

xy –

xy=

12 xy gvaqvs kleba 50%-iT.

amocanis amosaxsnelad SevadgineT algebruli gamosaxuleba da pirobis mixed-

viT, xdeba misi gardaqmna.

38

18 a) 80 _ 100% n%= 48⋅100%

80 = 60%.

48 _ n% am proporciis amoxsniT vupasuxeT pirvel kiTxvas _ nivTi gaZvirda 60%-iT.b) aq 100%-is rolSi gvevlineba 128;

128 _ 100% n%=

100%⋅48128

=

100%⋅38 = 37,5%. 48 _ n%

19 da 20 amocanebis amoxsnebi moiTxovs codnaTa integrirebul gamoyenebas (procentiT cvlileba, svetovani diagramebiT informaciis gamosaxva, dolaris

larze gacvliTi kursis gaazreba, simravleTa gaerTianeba).

19 gamokiTxulTa raodenoba, diagramis mixedviT, aris 26. meore kiTxvaze pa-suxis gasacemad unda amovxsnaT proporcia:

26 _ 100% n%=

1700%26

= 65,4%

17 _ n%

mesame kiTxvaze pasuxis gasacemad unda amoixsnas proporcia:

2 _ 100% n%=

100%⋅42

= 200 %-iT.

(6-2) _ n%

20 22 ivniss: 2000⋅0,999=1998 (dolari) 1998⋅2,78=5554 lari 44 TeTri. 25 ivniss: 2000⋅0,999=1998 (dolari) 1998⋅2,79=5574 lari 42 TeTri 13 ivliss: 1998⋅2,85=5694 lari 30 TeTri sul _ 5554,44+5574,42+5694,3=16 823,16 (lari)

SeiZleba asec gamoviTvaloT:

22 ivniss: 2000⋅2,78⋅0,999=5554,44 (lari) 25 ivniss: 2000⋅2,79⋅0,999=5574,42 (lari) 13 ivliss: 2000⋅2,85⋅0,999=5694,3 (lari)

mesame gakveTilze vamowmebT meore proeqtisa da saSinao davalebis

( 10 - 20 ) Sesrulebas. mimdinareobs codnis ganmtkicebis procesi. proeqti ex-eboda procentisa da procentebze moqmedebebis codnis gamoyenebas praqtikaSi,

kerZod, sabanko operaciebis warmoebisas.

proeqtze muSaobisas moswavleebs dasWirdebaT laris dolarze da dolaris

larze gacvlis operaciebze muSaoba; wliuri sargeblisa da sargeblidan saxelm-

wifo biujetis sasargeblod gadasaricxi Tanxis odenobis angariSi.

moswavleebi, albaT, gaiTvaliswineben: saboloo Sedegi damokidebulia erTi

wlis Semdeg dolarisa da laris gacvliT kursze.

axla gTavazobT miTiTebebs saSinao davalebis ( 10 - 20 ) amocanebis amosaxs-nelad. es amocanebi klasSi amoxsnili amocanebis ( 10 - 20 ) analogiuria. procenteb-ze amocanebis amoxsnisas unda gaviTvaliswinoT: mocemul konkretul SemTxvevaSi

39

ra sididis procenti ganixileba (es sidide, rogorc wesi, 100%-ia). maSasadame, ra sidide Seesabameba 100%-s, ra sidide Seesabameba procentis sxva mniSvnelobas; erTi sididis meore sididesTan fardobis namravli 100%-ze gviCvenebs _ meo-re sididis ra nawilia, ramdeni procentia pirveli sidide, Sedegs procentuli

Sefardeba ewodeba. procentebze amocanebi sxadasxva xerxiT SeiZleba amoixsnas;

zogierTi moswavlisTvis ufro gasagebia algebruli gamosaxulebis SedgeniT

amoxsna, zogierTs _ mxolod procentebiT gamosaxul ricxvebze moqmedebebiT

uadvildeba amocanebis amoxsna. sasurvelia, rom amocanebis amoxsnisas sxadasx-

va xerxi gamoviyenoT. amasTanave, mza receptebis gamoyenebisganac SeiZleba Tavi

SevikavoT _ maTematikis Seswavlis procesSi mTavaria, moswavle daeuflos ab-

straqtuli, logikuri da kritikuli azrovnebis xerxebs. Sesabamisi inteleqtu-

aluri unarebis ganviTarebas amocanis sxvadasxva xerxiT amoxsna mniSvnelovnad

uwyobs xels. amasTanave mniSvnelovania dasabuTebis xerxis SerCevisa da strate-

giis vargisianobis dadgenis unarebis ganviTareba:

10 areklili nawili _ 2% STanTqmuli nawili _ 98% dacemuli _ 100%

100%2% =50; 98%2% =44;

11 gogonebi klasis moswavleTa meoTxeds, anu 25%-s Seadgenen. amocana Sei-Zleba amoixsnas asec:

gogonebi: x – n% sul: 4x – 100% n%=25%.

12 gasavleli: 68% _ 170 13 dagegmili: 100% _ x mTeli gza: 100% _ x darguli: 12% _ 360

x= 170⋅100%68% =250 (km). x=360⋅100%

12% =3000 (nergi).

14 pirveli wlis Semdeg: 10 000+10 000⋅0,03 meore wlis Semdeg: 10 000⋅1,03+10 000⋅1,03⋅0,03=10 000⋅1,032=10 609 (lari).

15 a) Zveli fasi: 200 _ 100% axali fasi x _ 100,5% x=201 lari.zogierTma SeiZleba amoxsnas asec: 200+200⋅0,005=201 (lari).

16 a) mocemuli kvadratis farTobi: 4 ujra misi 200% aris masze 2-jer meti _ 8 ujra, misi 150% aris 6 ujra, misi 400% aris 16 ujra.

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17 x, 2x, 5x. 2x – 2⋅12,5=25(%); x – p%

5x – 5⋅12,5=62,5(%). 8x – 100%

p%= 100%8% =12,5%

SeiZleba vimsjeloT asec: 1 nawili aris 8 nawilis 18 , anu 18 ⋅100%=12,5%;

2 aris 2⋅12,5%=25%, 5 aris 5⋅12,5%=62,5%.

18 I xerxi. sigrZe _ 140% sigane _ 60% 140%⋅60%=84%nakveTis farTobi 16%-iT Semcirda.

II xerxi:^ erTi gverdi _ x axali marTkuTxedis erTi gverdi _ 1,4x meore gverdi _ y meore gverdi _ 0,6y farTobi _ xy farTobi _ 0,84 xy

xy–0,84xy=0,16xy Semcirda 16%-iT.

19 I xerxi arsebuli fasi: x 100%-iT gazrdis Semdeg: 2x damarcxebis Semdeg:^ 2x:2,5=0,8x x _ 100% Semcirda x–0,8x=0,2x-iT. 0,2x – p% 0,2x aris x-is 0,2 nawili, anu 20%; p%=20% amoxsna SeiZleba gagvegrZelebina asec: 20%-iT Semcirda.

II xerxi: meoTxedfinalis Semdeg _ 200%-ia.naxevarfinalSi damarcxebis Semdeg _ 200%:2,5=80%.e. i. Semcirda 20%-iT.

20 I xerxi: aT-aTi procentiT gaiafebis SemTxvevaSi x–0,1x=0,9x 0,9x–0,9x.0,1=0,81x 20%-iT Semcirebis SemTxvevaSi _ 0,8x 0,81x–0,8x=0,01x. (0,01x lariT iafi iqneboda).

0,01x _ p% p%= 0,01x⋅100%0,81x ≈ 1,23%. 0,81 _ 100%

II xerxi: 100%–10%=90% 90%–90%.0,1=81% 100%–20%=80% 81%–80%=1%

41

1%81% ⋅100% ≈ 1,23%, aq

1% _ p% p%= 100%⋅1%81% ≈ 1,23%.81% _ 100%

amocanebi TviTSefasebisTvis gamoiyeneba moswavleebis mier sakuTari codnis

Sesamowmeblad. maTi Sesruleba SeiZleba klasSic, magram umjobesia, rom Sin Ses-

ruldes. klasSi unda gaimarTos Sedegebis ganxilva, amocanebis amoxsnebze msje-

loba.

aRsaniSnavia aseTi aqtivobis mniSvneloba moswavleebisTvis saganmanaTleblo,

fsiqologiuri da emociuri TvalsazrisiT.

davalebis Sesruleba SeiZleba nebayoflobiTi iyos, magram maswavlebeli iT-

valiswinebs mas ganmaviTarebeli Sefasebis ganxorcielebisas. amocanebi TviT-

SefasebisTvis ajamebs pirvel sam paragrafSi warmodgenili masalis aTvisebis

dones. es Sejameba moswavleebis mier xorcieldeba da SeiZleba ganvixiloT Sema-

jamebeli werisTvis mosamzadebel samuSaod. analogiuri amocanebi moswavleebs

ukve ganxiluli aqvT.

TviTSefasebis testis Sesrulebis analizi saSualebas mogvcems davadginoT,

romeli Temis Sesabamisi davaleba gauWirdaT moswavleebs. SeiZleba gavagrZeloT

muSaoba am Temis Sesabamisi amocanebis gamoyenebiT. maTi SerCeva Tavis damatebiTi

amocanebidanac SeiZleba.

resursebi: carci (markeri), dafa, saxelmZRvanelo, sxvadasxva plakati pro-