maswmath10

7/23/2019 Maswmath10

1/64

gamomcemloba inteleqtiTbilisi 2012

guram gogiSvili, Teimuraz vefxvaZe

ia mebonia, lamara qurCiSvili

maTematika X

maswavleblis wigni

(meTodikuri rekomendaciebi, sakontrolo weris

variantebis nimuSebi da miTiTebebi amocanebis amosaxsnelad)

redaqtori Teimuraz vefxvaZe

grifi mieniWa 2012 wels ssip ganaTlebis xarisxis ganviTarebis

erovnuli centris (brZaneba # 375, 18. 05. 2012) mier


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ISBN 978-9941-439-11-7

gamomcemloba inteleqti~, 2012.

guram gogiSvili, Teimuraz vefxvaZe,

ia mebonia, lamara qurCiSvili

maTematika

X klasi

maswavleblis wignigamomcemloba inteleqti

Tbilisi 2012

Guram Gogishvili, Teimuraz Vepkhvadze,

Ia Mebonia, Lamara Kurchishvili

MATHS X

Teacher's BookINTELEKTI Publishers

Tbilisi 2012

gamomcemloba inteleqti

Tbilisi, ilia WavWavaZis gamz. 5. tel.: 2-25 05 22, 2-91 22 83.

www.intelekti.ge [email protected] [email protected] PUBLISHERS

5 Ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 32) 2-25 05 22, 2-91 22 83.


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3

s a r G e v i

Sesavali.......................................................................................................................................... 5

X klasis erovnuli saswavlo gegma ..................... ...................... ..................... .................. 9

Sinaarsisa da miznebis ruka .................. ...................... ..................... ..................... ............ 17

saswavlo masalis wardgenis fazebi da gakveTilis

dagegmvis zogadi principebi ..................... ...................... ..................... ..................... ....... 22

sanimuSo gakveTilebi .................... ...................... ..................... ..................... ..................... .. 30

I Tavi

gameoreba. simravle. ricxviTi simravleebi ............................. 39

1.1. simravle. qvesimravle. simravleTa mocemis xerxebi .................. .................. 41

1.2. moLqmedebebi simravleebze. venis diagramebi.................................................... 44

$1.3. amocanebis amoxsna (venis diagramebis gamoyeneba) ................... ..................... 47

$1.4. racionaluri ricxvebi.

racionaluri ricxvebis Caweris sxvadasxva forma .................... ................... 52

$1.5. namdvil ricxvTa simravle................... ..................... ...................... ..................... .. 54

sakontrolo wera ..................... ..................... ..................... ...................... ................. 55

$1.6. fesvi. wiladmaCvenebliani xarisxi .................... ..................... ..................... ........ 57

$1.7. ricxviTi Sualedebi ...................... ..................... ..................... ..................... ............ 59$1.8. wrewiris sigrZe da wris farTobi ................... ..................... ...................... ........ 60

II Tavi funqcia. monacemTa analizi da statistika ............................. 65

$2.1. sidideebs Soris damokidebulebebis magaliTebi. funqcia ................... ..... 65

$2.2. funqciis grafiki ...................... ..................... ..................... ..................... ................. 67

$2.3. simravleebs Soris damokidebulebebisa da

funqciis mocemis xerxebi ..................... ..................... ...................... ..................... . 69

$2.4. funqciis Tvisebebi ...................... ..................... ..................... ..................... .............. 70

$2.5. wrfivi funqcia. kvadratuli funqcia. y= xk

( k0) funqcia.......................... 73$2.6. modulis Secveli funqciebi .................... ..................... ..................... ................... 75

sakontrolo wera ................... ..................... ..................... ..................... .................... 82

$2.7. geometriuli gardaqmnebi sibrtyeze ................... ..................... ..................... .... 85

$2.8. monacemTa analizi da statistika ................... ..................... ..................... .......... 94

$2.9. statistikuri monacemebis Semajamebeli ricxviTi maxasiaTeblebi ...... 100

sakontrolo wera ..................... ...................... ..................... ..................... ............... 106

$2.10. mimdevroba .................. ..................... ...................... ..................... ..................... .......... 109

$2.11. rekurentuli wesiT mocemuli mimdevrobebi ................... ..................... ........ 111

sakontrolo wera ..................... ..................... ..................... ...................... ............... 113


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5

Sesavali

wigni Seicavs meTodikur rekomendaciebs axali erovnuli saswavlo gegmis mixed-

viT Sedgenili maTematikis X klasis saxelZRvanelosTvis (guram gogiSvili, Teimu-

raz vefxvaZe, ia mebonia, lamara qurCiSvili, maTematika X).

am wignSi maswavleblebs vTavazobT rekomendaciebs masalis SerCevaSi, saswav-

lo procesis dagegmvasa da warmarTvaSi. moswavleTa codnis SefasebaTa kriteri-

umebis Sedgenisa da damatebiTi samecniero da meTodologiuri Sinaarsis informa-

ciis mopovebaSi. mocemuli iqneba Sesabamisi axsna-ganmartebebi masalis wardgenisfazebis Sesaxeb _ motivacia, sakiTxis dasma, amocanis gansazRvra, ZiriTadi saswav-

lo masalis gamoyofa, sawyis etapze Sesasrulebeli savarjiSoebi da maTi analizi,

damagvirgvinebel etapze Sesasrulebeli amocanebi da maTi analizi, damoukidebeli

samuSaoebis gansazRvra da maTi Semowmebis formebi. mocemuli iqneba gakveTilis

dagegmvis sqemebi da ramdenime sanimuSo gakveTilis scenari.

wignSi Tavebisa da paragrafebis numeracia da dasaxeleba emTxveva moswavlis

saxelmZRvaneloSi SemoRebul numeraciasa da dasaxelebas.

mocemuli iqneba komentarebi TiToeuli Tavisa da paragrafis mixedviT.

X klasis saswavlo gegma, Sinaarsisa da miznebis ruka, Sefasebis ZiriTadi md-

genelebi da maTi saSualebiT Sefasebis kriteriumebis ganmsazRvreli rekomenda-ciebi, romlebic wignSia mocemuli, mTeli saswavlo wlis ganmavlobaSi gamoiyeneba;

aZlevs maswavlebels masalis gadacemis orientirebsa da Sefasebis formebs. unda

aRiniSinos, rom moswavleebis muSaobaze dakvirveba, maTi aqtivobebis Sefaseba yov-

el gakveTilze unda mimdinareobdes.

maswavlebels SevaxsenebT, rom Cven yovelTvis vcdilobdiT Segveqmna saxelmZ-

Rvaneloebi, romlebic yovel moswavleze iqneboda gaTvlili. Tumca, aris masala,

romelic gamorCeulad warmatebul moswavleebTan damatebiTi muSaobisTvis Sei-

Zleba gamoviyenoT. SevecdebiT am muSaobaSi maswavleblis wigniT dagexmaroT, raTa

moiZioT saWiro damatebiTi informacia.swavlebis procesSi meTodikuri orientirebis garkvevis sakiTxSi dagexmare-

baT literaturis CamonaTvali, romelic wignis bolos aris mocemuli. magaliTad,

dRevandel situacias kargad exmaureba gamonaTqvamebi h. froidentalis wignidan

([38] gv. 39): `maswavlebels unda SeeZlos swavlebis warmarTva programiT, romelic

kardinalurad gansxvavdeba imisgan, romliTac mas aswavlidnen... moTxovna, rom

swrafad cvalebad sazogadoebaSi ganaTleba gardaqmnis Canasaxsac Seicavdes ara

marto maTematikas exeba... feliqs klaini laparakobda ormag daviwyebaze: univer-

sitetSi swavlisas saskolo maTematikis daviwyebaze, Semdeg _ sauniversiteto

maTematikis daviwyebaze skolaSi maswavleblad dabrunebisas~.


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6

SevecdebiT yoveli Temis warmodgenisas mogawodoT detaluri komentarebi da

masalis mecnieruli da meTodologiuri safuZvlebi.

swavlebis procesSi mTavaria kvlevis procesi; ara marto pasuxebis mosmena kiTx-

vaze `ra~, aramed kiTxvaze `rogor~, an `kidev rogor~, amasTanave, mniSvnelovania

imis codnac - `ratom~ da pasuxebi kiTxvaze - `es risTvis mWirdeba~. yvelaferi es

aucilebelia moswavleTa momavali profesiuli saqmianobisTvis, demokratiul pro-

cesebSi aqtiuri monawileobisTvis dRevandel sazogadoebaSi, romelic sul ufro

metad xdeba maTematikaze damokidebuli.

wigni dagexmarebaT zemoxsenebuli amocanebis ganxorcielebis meTodikuri xe-

rxebis gacnobaSi kvlevis procesi SeiZleba individualuri iyos, SeiZleba jgu-

furi muSaobiT xorcieldebodes. mocemuli iqneba rekomendaciebi sxvadasxva tipis

saswavlo procesis ganxorcielebisTvis. moswavlis saxelmZRvanelo am procesis

erT-erTi damxmare saSualebaa. amitom saWirod vTvliT misi agebis principebi da

masalis struqturirebis sakiTxebi gagacnoT.

moswavlis saxelmZRvanelos Sedgenisas gaviTvaliswineT is ZiriTadi orientire-

bi, romlebic dRes moqmed standartebSia miTiTebuli orientireba Sedegze da amSedegis demonstrirebaze, kavSiri sagnebs Soris da TviT erTi sagnis SigniT mis

sxvadasxva nawilebs Soris (maTematikaSi warmodgenil mimarTulebebs Soris); sak-

laso da klasgareSe muSaobis saqmianobis erTianoba; orientacia uklebliv yvela

moswavleze (sakuTari SesaZleblobebidan gamomdinare, yoveli moswavle sxvadasxva

xarisxiT miaRwevs Sedegebs). viTvaliswinebT, rom maswavlebelTa individualuri

Taviseburebebi mniSvnelovnad aisaxeba saswavlo procesebze erovnuli saswavlo

gegmis Sesruleba sxvadasxva gziT SeiZleba.

me-10 klasis saxelmZRvanelos Sedgenisas gaviTvaliswineT, rom igi saSualo

skolis swavlebis mesame safexuris pirvel etaps warmodgens; maTematika erTianikursis moculobiT iswavleba da moswavleebi emzadebian profesiuli arCevanis ga-

sakeTeblad. maT unda miiRon ganaTleba, romelic xels Seuwyobs ukeT gaerkvnen

TavianT midrekilebebsa da interesebSi.

saxelmZRvanelos Sinaarss axali saswavlo programis moTxovnebi gansazRvravs.

igi maTematikis erTianobis suliskveTebiT da praqtikuli gamoyenebis win wamow-

eviT xasiaTdeba. saswavlo programaSi mocemuli geometriuli nawili saSualebas

gvaZlevs ganvaxorcieloT am nawilis axali ideebiT Sevsebis procesi. yoveli axali

Temis axsna, metwilad, intuiciur doneze, praqtikasTan mimarTebiT da TvalsaC-

inoebis gamoyenebiT iwyeba. maTematikis swavlebaSi praqtikuli da gamoyenebiTi

aspeqtebis gaZliereba mimdinareobs iseTi masalis SerCeviT, romelic aqtualuriadRevandelobisTvis _ axloa moswavlis yofasTan.

moswavleebSi maTematikas warmovadgenT, rogorc zogadsakacobrio kulturis

nawils. amas xels uwyobs SinaarsSi organulad CarTuli emociurad mdidari epizo-

debi mecnierebis istoriidan, romlebic moswavleebs acnobs im adamianTa saxelebs,

romlebic mecnierebas qmnidnen. es yovelive ukavSirdeba maTematikis swavlebis

erT-erT ZiriTad princips _ sagnisadmi interesis gaRvivebis princips. es intere-

si, sxva stimulebTan erTad, mkveTrad zrdis mecadineobebis nayofierebas. maTema-

tikiT dainteresebas aZlierebs moswavleebTan erTad misi ideebisa da meTodebis

silamazis aRmoCena, am meTodebis efeqtianad gamoyenebis unar-Cvevebis floba.


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7

rogorc aRvniSneT, yoveli axali Temis axsna metwilad intuiciur doneze,

praqtikasTan mimarTebiT da TvalsaCinoebis gamoyenebiT iwyeba. Tumca, SemdegSi

vcdilobT, simkacris garkveuli donis SenarCunebiT, isini formaluri dasabuTebe-

biTac gavamagroT. TvalsaCinoeba xSirad met informacias Seicavs, vidre sityvebi.

TvalsaCino warmodgenebiT miRebuli informacia saSualebas gvaZlevs vifiqroT,

gamovTqvaT varaudi, daskvnebi gavakeToT; aseT daxmarebaze uaris Tqma ar arisgamarTlebuli.

geometriul figuraTa tolobisa da msgavsebis sakiTxs geometriul gardaqmnebs

vukavSirebT.

geometriuli faqtebis da saerTod, me-10 klasis sxva sakiTxebis gadacemi-

sas mniSvnelovani adgili aqvs daTmobili deduqciuri msjelobis nimuSebs; vizi-

arebT nikola burbakis cnobil gamonaTqvams: `berZnebidan moyolebuli _ vityviT

`maTematika~ _ vgulisxmobT damtkicebas~.

maTematikuri winadadebebis Sebrunebuli, mopirdapire, Sebrunebulis mopirda-

pire winadadebebis erTdrouli ganxilva da analizi xels uwyobs Tavad debulebisa

da misi dasabuTebis arsis gagebas.maTematikuri Semoqmedebis safuZveli yovel axal debulebasTan erTad adre

SeswavlilTan kavSirebis danaxva da ganxilvaa. did yuradRebas vuTmobT maTemati-

kuri winadadebebis sxvadasxva magaliTebiT ilustrirebas, daSvebuli Secdomis an

raime hipoTezis mcdarobis saCveneblad kontrmagaliTis miTiTebas.

masalis SerCeva da ganawileba isea mofiqrebuli, rom TiTqmis yoveli sagakveTi-

lo cikli moicavs konkretuli gamocdilebis miRebas, mis mimoxilvas, abstraqtuli

Sedegis miRebas da eqsperimentirebasac. am mxriv mniSvnelovania jgufuri muSao-

bis proeqtebi _ Sesasrulebeli saqmianobis xasiaTis mixedviT liderobas Tavis

Tavze iRebs Sesabamisi tipis erTi-ori moswavle, danarCenebi SemoTavazebuli sqe-mis mixedviT moqmedeben da ase axdenen jgufSi adaptacias, rac jgufis TiToeul

wevrs umsubuqebs saswavlo saqmianobiT ganpirobebul datvirTvas; amasTanave, yov-

el maTgans Tavis gamoCenis saSualebasac aZlevs, uviTarebs jgufuri pasuxismge-

blobis grZnobas.

moswavleTa maTematikuri ganswavlulobis Sefasebisas, Camoyalibebuli moTxov-

nebis (ix. mag. [43]) Sesabamisad, vcdilobT saxelmZRvanelos saSualebiT im unarebis

ganviTarebas, romlebic maTematikuri codnis mravalferovan cxovrebiseul situa-

ciebSi gamoyenebebTan aris dakavSirebuli. am moTxovnebis Sesabamisad aris savarji-

Soebi, romlis pasuxebi `orobiTi logikis~ CarCoebSi ar Tavsdeba; didi yuradRe-

ba eTmoba e. w. struqturirebul savarjiSoebs, roca amocanas mosdevs kiTxvaTasistema, romelSic yoveli Semdgomi kiTxva wina kiTxvebis analizs efuZneba. war-

modgenili amocanebis nawili ar aris Cveni skolisTvis tipuri. magaliTad, logi-

kuri amocanebi, romlebic sxvadasxva diagramis (cxrilebis, grafikebis, venis dia-

gramebis) gamoyenebiT ixsneba.

Teoriul-simravluri ena da simbolika, rogorc wesi, masalis gadmocemis damx-

mare saSualebaa. venis diagramebi simravleebze moqmedebebis TvalsaCino warmod-

genisa da zogierTi amocanis amosaxsnelad gamoiyeneba. vcdilobT calkeuli frag-

mentebi, saswavlo programis moTxovnaTa Sesabamisad, logikuri TanamimdevrobiT,

deduqciuri msjelobebis gamoyenebiT, analizisa da sinTezis, ganzogadebisa daspecializaciis, abstraqciisa da konkretizaciis meTodebiT gadmovceT (am meTo-


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8

debis mokle daxasiaTeba ixileT, magaliTad, [6]-Si). magaliTad, kvadratuli funq-

ciis konkretizaciis nimuSebia sxvadasxva fizikuri movlenis aRmweri funqciebi,

xolo kvadratuli funqciis ricxviTkoeficientebiani SemTxvevebis ganxilva spe-

cializaciis magaliTebis ganxilvaa (cnebis moculoba am dros mcirdeba, konkret-

izaciisas cnebis moculoba ar mcirdeba). gantolebebisa da utolobebis ganxilva

funqciuri damokidebulebebisa da grafikuli gamosaxvis saSualebiT xels uwyobsSemdgomi ganzogadebisTvis Semzadebas.

geometriuli TvalsaCinoeba moswavles saSualebas aZlevs kargad gaiazros

Sedegi, romelic gantolebis an utolobis amoxsnisas miiReba.

cnebebi xSirad formaluri gansazRvrebebis gareSe, aRweriT Semogvaqvs. vcdi-

lobT Teoriuli nawili ar iyos gadatvirTuli. cnebis aTvisebis procesis ganx-

orcieleba konkretuli amocanebis amoxsniT mimdinareobs. cnebis Semotanas ki win

uZRvis propedevtikuli muSaoba, romlis mizania konkretuli warmodgenis Seqmna

da formalizaciisTvis momzadeba. ase _ mTeli kursis ganmavlobaSi mimdinareobs

funqciis cnebis Camoyalibeba _ TandaTan xdeba misi ganviTareba. yoveli cnebis

aTviseba mis gamoyenebasac niSnavs. cnebis aTviseba ar Tavdeba gansazRvrebis cod-niT an aRweriT (roca zusti gansazRvreba ar aris mocemuli). aq mniSvnelovania ara

marto kerZodan zogadisken svla, aramed zogadidan kerZosken dabrunebac. masala

isea gadmocemuli, rom SevZloT davinaxoT Seswavlili obieqtebi im konkretul

SemTxvevaSi, romelTanac saqme gvaqvs mocemul momentSi.

maTematikis erTianobis Tvalsazrisis ganviTarebiT funqciuri ena gamoyenebu-

lia geometriuli gardaqmnebis aRwerisas. xSiria mimarTeba ara marto sabunebism-

etyvelo, aramed humanitaruli jgufis sagnebis saxelmZRvaneloebisadmi (magali-

Tad, teqstSi qarTuli anbanis asoTa gamoyenebis sixSireebis ganxilva, simetriis

nimuSebi qarTul mwerlobaSi, terminologiuri sakiTxebi da sxva).zogadi konceptualuri moTxovnebi, romlebsac saxelmZRvanelos Sinaarss vuy-

enebT, SeiZleba ase SevajamoT:

a) xeli Seuwyos codnis sxvadasxva sferos Soris urTierTkavSiris gacnobiere-

bas. paragrafebis umetesi nawili ilustrirebuli iyos faqtebiT da savarjiSoebiT

codnis sxvadasxva sferodan.

b) emyarebodes Tanamedrove koncefciebs, romelic Teoriul-simravluri enis

zomier gamoyenebas, moswavleTa asakobrivi Taviseburebebis gaTvaliswinebas, sa-

gnisadmi interesis aRZvras, sagnis dauflebas da am codnis praqtikuli da yofiTi

amocanebis amoxsnis unar-Cvevebis Camoyalibebas gulisxmobs.

g) savarjiSoebi ise iyos dagegmili, rom mimdinareobdes masalis uwyveti gameo-reba.

d) saSualebas iZleodes moswavleebs individualuri da jgufuri proeqtebis

ganxorcielebisa da maTi prezentaciis Cvevebi gamoumuSavdes. dasaxelebuli pro-

eqtebis dacvisas SeeZlos debatebSi aqtiuri monawileoba.


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9

maTematikaSi X klasis saswavlo gegma

:

,

. X1.

.

.X2.

/

.

.X3.

.

. X4.

-

.

. X5.

.

. X6.

.

. X7.

.

. X8.

.

. X9.

.

. X10.

.

. X11.

.

. X12.

.

.X13.

.

. X14.

.

. X15.

.

. X16.

.


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10

:

.X.1.

.

, :

( ,

); /

/ -

;

;

; -

;

;

/ (, 1-

[0, 1] ).

. X.2. /

.

, :

; -

;

(, , , , ;

);

(,

);

(, ; ;

, ;

).

. X.3.

.

, :

( ) -

,

;

; ;

, ;

;

( ) ;

;


11/64

11

(, , ).

.X.4. - .

, :

;

;

, ;

(,

23455 ?);

(, );

.

.X.5. . , :

/

, , ;

;

-

;

/

(, ).

:

.X.6.

.

, :

(

) (, , ,

f (x) =kx

)

; ,

, , /, /

;

;

, (,

- S (t) =vt+S0,

?);

, (

/ , )

.


12/64

12

.X.7.

.

, :

;

;

/ (,, );

;

(,

2000 . 10

. 20

, 10 .).

.X.8.

. , :

, , /

, (

) (, );

(

. ,

);

;

(,

) (,, , ),

.

:

.X.9.

.

, :

;

; : , , , ,

, ;

- /

, /.

.X.10. .

, :

/;


13/64

13

,

;

;

.

.X.11. .

, :

(

) (, /)

(,

, ,

);

( );

.

.X.12.

.

, :

;

( , /

) ;

( , ,

, )

;

/

;

() .

: ,

.X.13.

.

, :

(, ,

/);

( , ) ;

(, , , ).

.X.14.

. , :


14/64

14

()

, /;

,

;

/, / -.

.X.15. .

, :

,

(,

);

( ), ()

() ;

, ,

.

.X.16.

.

, : , (

, , ), -

/ (

);

, -

, ;

- -

,

.

1. .

.

2. :

( );

(

).


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15

3. ;

.

4. /.

5. .

6. .

7.- .

8. , , f (x) =kx

.

9. ; : , ,

, ; .

10. .

11. / .

12. / .

13. ,

.

14. .

15. : sin(x)=a, cos(x)=a, tg(x)=a .

16. .

17. .

18.

(/ ).

19. . .

20. : ,

.

21. .

22. : , , , , ;

.

23. , .

24. .

25. .

26. . .

27. -.

28. ()

.

29. (,

, , ), , , ,

, , , : ,

, .

30. : ;

- .

31. :

, ;

.


16/64

16

32. (

): , , ;

(, , , , ).

33.

: (, , );

( , ).34. : , (

); (, ,

, ); ,

.

35. .


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17

Sinaarsisa da miznebis ruka

Temebis CamonaTvaliTemebis kavSiri miznebTan, ra punqtebs

faravs Tema

savaraudo

saswavlo

dro

simravle, simravliselementebi; qvesimravle;simravlis mocemis xerxebi.simravleTa gaerTianeba,

TanakveTa da sxvaoba.simravleebze moqmedebebisTvisebebi.venis diagramebisgamoyenebiT amocanebisamoxsna

simravleTa Teoriis elementebiswarmodgena da am Teoriis terminebis,cnebebis gamoyenebis unaris gamomuSavebaSesabamisi savarjiSoTa sistemis, realuri

viTarebis modelirebisa da venisdiagramebis gamoyenebis saSualebiT;geometriuli interpretaciebisgamoyenebiT simravleebze moqmedebebisilustrireba.X.8, X.4,

5 sT

racionaluri ricxvebi.racionaluri ricxvisCaweris sxvadasxva forma;namdvil ricxvTa simravle.

fesvi, wiladmaCveneblianixarisxi.ricxviTi Sualedebi.

ricxviTi simravleebis warmodgena, maTSoris mimarTebebis aRwera; ricxvebisCaweris sxvadasxva formebi (aTwiladi,wiladi, procenti, xarisxi, fesvi);

geometriuli interpretaciebis gamoyeneba,simravleTa Teoriis gamoyeneba; namdvilricxvebze moqmedebebis Sesruleba daSedegis Sefaseba. X.1, X.2, X.3.

10 sT

wrewiris sigrZe da wrisfarTobi

geometriuli figuris TvisebebisaRwera da grafikuli gamosaxuleba;koordinatebis gamoyeneba geometriuliobieqtis Tvisebebis gamosaxvisas da misielementebis zomebis dasadgenad; brtyeli

figuris farTobis povna.X.9, X.10, X.11.

3 sT

sidideebs SorisdamokidebulebebismagaliTebi. Ffunqcia.Ffunqciis grafiki.sidideebs Sorisdamokidebulebebis dafunqciis mocemis xerxebi. Ffunqciis Tvisebebi.

jgufuri muSaoba(grafebis gamoyeneba)

funqciis cnebis aRwera, am aRwerisassimravleTa Teoriis elementebis, venisdiagramebis, grafebis gamoyenebis unarisgamomuSaveba Sesabamisi savarjiSoebiTa daproeqtiT.geometriuli interpretaciebiTa da sxvameTodebiT funqciis mocemis xerxebisganxilva. funqciis cvlilebaze dakvirveba

funqciis parametrebis cvlilebisas;X.6, X.8.

10 sT


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wrfivi funqcia.

kvadratuli funqcia. y= kx

(k0) funqcia.jgufuri mecadineoba:sidideebs Soris

damokidebulebiskvadratuli funqciiTmodelireba daoptimizaciis amocana.modulis Semcvelifunqciebi

sidideebs Soris damokidebulebisaRmweri funqcii sTvis funqciis tipismiTiTeba am funqciis gamosaxvis xerxisgandamoukideblad.praqtikuli amocanis modelireba

kvadratuli funqciiT, kvadratulifunqciis Tvisebebis gamoyenebaoptimizaciis amocanis gadawyvetisas.X.6.

5 sT

figuraTa asaxva;centruli da RerZulisimetriebi; mobruneba;mobrunebaTa kompozicia;

radianuli zomajgufuri muSaoba:RerZuli da centrulisimetriebis, mobrunebisda maTi kompoziciebisTvisebebis gamokvleva.Pproeqti: simetria

geometriuli gardaqmnebis gamoyenebisunaris gamomuSaveba, geometriuli garda-qmnebis kompoziciebis daxasiaTeba da ga-moyeneba obieqtTa zomebis, figuraTa

elementebs Soris damokidebulebebis dasa-dgenad. geometriuli gardaqmnebis Sesru-lebisa da gamosaxvisTvis koordinatebisgamoyenebis unaris gamomuSaveba. mob-runebis kuTxis zomis erTeulis war-modgena da gamoyeneba mobrunebis SedegadgadaadgilebasTan dakavSirebuli amo-canebis amoxsnisas; brtyeli figurisfarTobis povna da misi gamoyeneba.X.12, X.11, X.5.

8 sT

monacemTa Segroveba;monacemTa klasifikacia;sixSireTa cxrili;monacemTa warmodgenisxerxebi: wertilovani dasvetovani diagramebi, wri-uli diagrama, piqtograma.proeqti: ekologiuriproblemebis Seswavla monacemebis Segroveba,

analizi.statistikuri monacemebisSemajamebeli ricxviTimaxasiaTeblebi: saSualo,Mmoda, mediana, gabnevisdiapazoni, saSualokvadratuli gadaxra.jgufuri muSaoba:saSualo kvadratuligadaxris warmodgena

zogadi formulis saxiT.

monacenTa Segrovebis Sesaferisi xerxisSerCevis, cnobarebisa da teqnologiebisgamoyenebis da arCevanis dasabuTebisunaris gamomuSaveba; monacemTawarmodgenis xerxebis floba, Sesaferisixerxis SerCevis unaris gamomuSavebaSesabamisi savarjiSoebis sistemissaSualebiT.monacemTa dasaxasiaTeblad SemajamebeliricxviTi maxasiaTeblebis gamoTvlis da

miRebuli Sedegebis gamoyenebis unarisgamomuSaveba praqtikuli amocanisgadawyvetisas.X.13, X.14, X.16.

14 sT


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mimdevroba. rekurentuliwesiT mocemulimimdevroba.jgufuri muSaoba: bankSianabarze procentis

daricxvis wesi;perioduli procesebi.

rekurentuli wesiT mocemuli mimdevrobisganxilva, rekursiis gamoyenebis aTvisebaSesabamisi savarjiSoebiTa da realuriprocesebis ganxilviT.bankSi anabarze daricxvis sxvadasxva

wesiT (martivi da rTuli saprocentodaricxvisas) daricxuli saprocentoganakveTis gamoTvlis daufleba.X.8, X.5.

8 sT

WeSmariti da mcdariwinadadebebi; Sebrunebulida mopirdapire debuleba.jgufuri muSaoba:amocanebis amoxsnasxvadasxva xerxiT

debulebaTa dasabuTebisxerxebi: kontrmagaliTi,sawinaaRmdegos daSvebis,dirixles principi.proeqti: aqsiomurimeTodis gaazreba.jgufuri muSaoba:muSaobis dagegmva;optimaluri variantisSerCeva.

jgufuri muSaoba:pirobiTi winadadebismoZieba.

msjeloba-dasabuTebis sxvadasxvaxerxis gacnoba, aTviseba, ganmtkiceba,gamoyenebis unaris ganviTareba. msjeloba-dasabuTebis procesSi simravleebs Sorisdamokidebulebebis gamosaxvis xerxebisgamoyenebaSi gawafva.

aqsiomuri meTodis gacnoba da gaazreba. msjeloba-dasabuTebisas mocemulipirobioTi winadadebisgan warmoebuliwinadadebebis/debulebebis gamoyenebisunaris daufleba.Ppraqtikuli amocanis gadawyvetisassamuSaos dagegmvisa da optimalurivariantebis SerCevis unaris ganviTareba.qarTuli mwerlobis nimuSebidan pirobiTiwinadadebebis moZieba da analizi;

prezentacia.X.4, X.9. X.10

14 sT

RerZul simetriaTakompozicia. paralelurgadatanaTa kompoziciajgufuri muSaoba:paraleluri gadatanisTvisebebiparaleluri

gadatanis warmodgenakoordinatebiT.jgufuri muSaoba:praqtikuli amocanisganxilva.

geometriuli gardaqmnebis kompoziciebsSoris mimarTebebis aRwera da maTigamoyenebis unaris gamomuSaveba;koordinatebis gamoyeneba; sibrtyisdafarvis demonstrireba.X.12.

12 sT


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geometriul figuraTawarmodgena algebrulidamokidebulebebissaSualebiT.wrfivi orucnobiani

utoloba;wrfiv orucnobianutolobaTasistema;jgufuri muSaoba:amocanis maTematikurimodelis Seqmna da amoxsna.

gantolebebiT, utolobebiT geometriulfiguraTa warmodgena.wrfivi orucnobiani utolobisa dautolobaTa sistemis geometriuliinterpretaciebi; praqtikuli amocanebis

amoxsna utolobebisa da utolobaTa sis-temebis daxmarebiT.X.7

14 sT

orucnobian gantolebaTasistemajgufuri muSaoba:

xarjTaRricxvis amocana.

teqsturi amocanebis amosaxsneladorucnobian gantolebaTa si sTemisSedgena da amoxsna, amonaxsnis

interpretacia, amoxsnis xerxebis SerCeva.Ppraqtikuli (kerZod, xarjTaRricxvis)amocanis amoxsnisas gantolebebiTmodelireba da amocanis pirobaSimocemuli SezRudvebis utolobiT anutolobaTa sistemiT gamosaxva.X.7.

8 sT

msgavsi figurebi.homoTetiamsgavsebis asaxva.

samkuTxedebis msgavsebisniSnebiproeqti: maTematikurimogzauroba harmoniissamyaroSi.

obieqtTa zomebis dasadgenad figuraTamsgavsebis Tvisebebis gamoyenebis unarisgamomuSaveba.

geometriuli gardaqmnis-homoTetiisTvisebebis gamoyeneba proeqtzemuSaobisas, harmoniis samyarosmaTematikur kanonzomierebebTandakavSireba; teqnologiebis, cnobarebis,saxelmZRvaneloebis gamoyenebiTTxzulebis Sedgena, prezentacia. X.11, X.12

12 sT

sivrceSi wertilebis,wrfeebis, sibrtyeebis

urTierTganlagebamravalwaxnagebiOor wertils SorismanZili

sivrculi figurebis amocnoba.wrfeebis, wertilebis, sibrtyeebis

urTierTganlagebis aRweraSi gawafva. Oorwertils Soris manZilis gamosaTvleliformulis daufleba, figuraTa zomebisdasadgenad koordinatebis gamoyeneba.X.11

6 sT


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sinusebis Teoremakosinusebis TeoremasamkuTxedis kuTxeebsada gverdebs Sorisdamokidebulebebi

jgufuri muSaoba:samkuTxedis amoxsna.samkuTxedis farTobisformulebi

samkuTxedis kuTxeebsa da gverdebs Soristrigonometriuli Tanafardobebi dapraqtikuli xasiaTis amocanebis amoxsnisasmaTi gamoyenebis unaris gamomuSaveba.amocanebis (maT Soris optimizaciis)

amoxsna samkuTxedis farTobisgamosaTvleli formulis gamoyenebiT.X.10, X.11.

16 sT

albaTobis formulaxdomilobaTa sivrcefardobiTi sixSire daalbaTobaalbaTobis gamoTvlismagaliTebi

SemTxveviTobis aRwera albaTurimodelebiT; TvalsaCinoebebis (diagramebis)gamoyeneba; SemTxveviTobis aRmwerixelsawyoebis gamoyeneba.ganasxvaos fardobiTi sixSire daalbaToba;

albaTobis gamosaTvleli formulisgamoyenebaSi gawafva.X.15, X.16.

6 sT

jgufuri muSaoba: naStTaariTmetikis Tvisebebisgaazreba

naStTa ariTmetikis praqtikuliinterpretaciebi;deduqcias, ganzogadebas da analogiasiyenebs mTel ricxvebs Sorisdamokidebulebebis, kanonzomierebebisdasadgenad daaxdens maT Semdgomganzogadebas.

X.4, X.10,

5 sT

mTeli ricxvebisgayofadoba. Tvlissistemebijgufuri muSaoba:daSifvris magaliTi.

mTel ricxvTa gayofadobis Tvisebebisgamoyenebis unari, Tvlis sistemebi damaTi gamoyeneba teqnologiebSi.cifrulikodirebis magaliTebis ganxilva.kavSiri poziciur sistemebs Soris.teqstis daSifvra da gaSifrva.X.2, X.4, X.5

8 sT

gameoreba 8 sT


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saswavlo masalis wardgenis fazebi da gakveTilis

dagegmvis zogadi principebi

saswavlo procesis organizaciis ZiriTadi forma gakve Tilia; saganmanaTleblo,

aRmzrdelobiTi da praqtikuli miznebis ganxorcieleba gakveTilze xdeba. amitom

maTematikis swavlebis ZiriTadi sakiTxi gakveTilis kargi momzadeba da Cata rebaa.

meTodikur literaturaSi didaqtikuri cneba _ `gakveTili~ ZiriTadad ase

aRiwereba: gakveTili logikurad dasrulebuli, mTliani saswavlo-aRmzrdelo-

biTi procesis garkveuli SemosazRvruli monakveTia. masSi rTul urTierTdamo-

kidebulebaSia procesis yvela ZiriTadi elementi: Sinaarsi, mizani, saSualebebi,meTodebi, organizacia; yovel gakveTilze gansazRvruli saganmanaTleblo da

aRmzrdelobiTi amocanebi wydeba; am amocanebis gadawyveta konkretuli saswavlo

masalis ganxilvis procesSi mimdinareobs.

maTematikis gakveTilisadmi wayenebuli mTavari moTxovnaa ZiriTadi didaqti-

kuri amocanis arseboba _ im Temis Seswavlis miznis arseboba, romlis gadawyvetis

procesi mocemul gakveTilze mimdinareobs. yoveli gakveTilis win kargad unda

gaiazroT misi Sinaarsi da mizani, amaSi Cven mier mowodebuli ruka dagexmarebaT.

miznis Sesabamisi unda iyos kar gad gaazrebuli da dagegmili saswavlo masalis

wardgenis fazebi: motivacia; ar aris sakmarisi, rom maswavlebels gaaz rebuli hqondes mizani;

saWiroa, rom igi moswavleebisTvisac gaxdes ZiriTadi mizani. moswavleebTan sau-

bari unda daviwyoT ara imiT, Tu ras vaswavliT, aramed mniSvnelovania Tavidanve

interesis aRZvra da iseTi situaciis Seqmna, roca moswavle motivirebulia da pou-

lobs pasuxebs kiTxvaze _ `risTvisaa saWiro~. motivacia SesaZlebelia praqtikuli

amocanis dayenebiTa da misi amoxsnis xerxebis ZiebiT daviwyoT, an maTematikis Siga

kanonzomierebis gaazrebiT, problemuri situaciis Seqmnis xelSewyobiT; mxolod

amis Semdeg xdeba Sesabamisi amocanis dasma da misi amoxsnis Zieba.

amocanis gansazRvris Semdeg mimdinareobs misi amoxsnis gzis Ziebis procesi.

amoxsnis Ziebis procesis warmarTvis sxvadasxva meTodikuri saSualeba arsebobs.

yovel amocanas, rogorc wesi, amoxsnis Ziebis garkveuli forma miesadageba. es

formebia _ muSaoba jgufebad (SesaZlebelia or-oradac), mTeli klasis erToblivi

monawileobiT, individualuri muSaoba da a. S. im SemTxvevaSic ki, roca amocanis

amoxsnis ZiebaSi mTeli klasi erTdrouladaa Cabmuli, maswavlebelma swavleba ise

unda warmarTos (moxerxubuli kiTxvis dasmis saSualebiT), rom Temis Seswavlis

procesis ZiriTadi Semoqmedebi Tavad moswavleebi aRmoCndnen; maswavlebeli am

SemTxvevaSi warmarTvelis, diriJoris funqcias unda asrulebdes.

moswavlis saxelmZRvanelos teqsti mogcemT aseTi gakveTilebis Catarebis sa-

Sualebas. igi ZiriTadi saswavlo saSua lebaa da amitom gakveTilisTvis mzadebisprocesSi teqstebis, ilustraciebis, savarjiSoebis gacnobas, daxarisxebas didi


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dro unda dauTmoT. Tumca, maswavleblis wignic dagex ma rebaT TiToeul etapze

(sawyisi, damagvirgvinebeli) Sesasrulebeli savarjiSoebis SerCevis sakiTxSi.

Sinarsisa da miznebis rukaSi ver mivuTiTeT Temis Seswavlis yvela mizani,

romelic saganmanaTleblo da aRmzrdelobiTi amocanebis gadawyvetis komponen-

tebisgan Sedgeba. ar unda gavaigivoT mizani SinaarsTan. magaliTad, samkuTxedis

kuTxeebis jamis formulis gamoyvana saxelmZRvaneloSi isea warmodgenili, rom

igi am Temis Sinaarsis miznad gardaqmnis process kargad warmogvidgens; dasmuli

amocanis amoxsna kritikuli azrovnebis, msjelobisa da dasabuTebis unaris gamo-

muSaveba, geometriul figuraTa gamosaxvis da am figuraTa Tvisebebis SeswavlaSi

eqsperimentis gamoyenebis unaris gamomuSavebaa. am miznebis gansaxorcieleblad qa-

Raldisgan gamoWril samkuTxedze (modelze) Catarebuli eqsperimentis (gamoiyeneba

ramdenime RerZuli simetria) an dasabuTebis xer xia mowodebuli. meore maTgani

dasmul kiTxvebze pasuxis SerCevis, msjelobisa da dasabuTebebis fonze mimdin-

areobs. aq maswavlebelma SeiZleba sxva eqsperimentic daamatos (transportiris

gamoyenebiT). ufro dawvrilebiT am gakveTilis Sesaxeb qvemoT mogaxsenebT.

am konkretuli magaliTidan Cans, rom aqtivobis mizani saganmanaTleblo daaRmzrdelobiTi miznebis erTobliobaa.

gakveTilze Seswavlili faqtebi TavisTavad aris mniSvnelovani; Tumca, kidev

ufro mniSvnelovania is, rom maTi Seswavlis dros ganxorcielebuli procesi Ta-

vis kvals tovebs _ kritikuli azrovnebis ganviTareba, msjeloba da dasabuTebis

unaris gamomuSaveba. amasTanave, moswavleTa gonebaSi gaiazreba mniSvnelovani

sakiTxebi: adamianis mier faqtebis aRmoCena, aRmocenebuli amocanebis amoxsna da

Sedegebis dafiqsireba azrovnebaSi. konkretuli deduqciuri msjeloba, dasabuTeba

im SemTxvevaSi atarebs aRmzrdelobiT funqcias, roca moswavles gavagebinebT misi

Catarebis mniSvnelobas.axla davasaxelebT ramodenime sasargeblo zogad reko mendacias, romelic

meTodikis sakiTxebs ganekuTvneba:

maswavlebeli cdilobs, rom yoveli axali SemecnebiTi amo cana TviT moswav-

lem Camoayalibos;

maswavleblis xelmZRvanelobiT da moswavleTa Za lis xmeviT _ dakvirvebis,

cdis, konkretuli SemTxvevebis analizis Sedegad iqmneba warmodgena, hipoTeza

arsebul kanonzomierebaze.

maswavleblis xelmZRvanelobiT mimdinareobs dasa buTebiT gzis Zieba, amo-

canis amoxsnis gegmis Sedgena; rasac xSirad mosdevs TviT moswavleebis mier am

gegmis realizacia.axla SevexoT saswavlo procesisTvis mzadebis zogad sqemas:

maswavleblis muSaoba saswavlo wlis win.

gakveTilTa sistemis gaazreba saswavlo wlis win;

konkretuli gakveTilisTvis momzadeba.

momzadebisas maswavlebelma unda gamoiyenos moswavlis wigni, maswavlebelis

sarekomendacio wigni da, saWiroebis SemTxvevaSi, iq miTiTebuli literatura.

sarekomendacio wignis mixedviT, sasurvelia, maswavlebeli Tavdapirvelad saswav-

lo gegmas, Sefasebis sistemas da gakveTilebis Semo Tavazebul nimuSebs gaecnos.

sarekomendacio wignis Sesavali mas gaacnobs avtorTa mier saxelmZRvanelos agebisZiriTad principebs.


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maswavlebeli SemoqmedebiTad unda miudges Cvens rekomendaciebs; igi

safuZvlad iRebs Cven mier SemoTavazebul gegmas da azustebs mas sakuTari

gamocdilebiTa da klasis Taviseburebebis gaTvaliswinebiT. es dazustebebi

gansakuTrebiT mniSvnelovania sawyis etapze Sesasrulebeli savarjiSoebis da

damamTavrebel etapze Sesasrulebeli savarjiSoebisa da sakontrolo weris

variantebis SerCevisas.gakveTilis dagegmva iTvaliswinebs Casatarebeli procesis tips:

axali masalis gacnoba

masalis ganmtkiceba

codnis Semowmeba

sxva tipis gakveTilebi (gakveTili bunebaSi, gakveTili-proeqti ...)

Tumca, zogierTi tipis procesi (axali masalis gacnoba, ganmtkiceba, Semow-

meba), rogorc wesi, yovel gakveTilze mimdinareobs, SesaZlebelia _ sxvadasxva

moculobiT.

moswavleTa codnis Semowmeba, moswavleTa muSaobaze dakvirveba yovel gakveTil-

ze mimdinareobs. gakveTilze mTavaria vaswavloT da aRvzardoT. swavleba arniSnavs mxolod codnis gadacemas _ swavleba codnis SemoqmedebiTad dauf lebas

unda niSnavdes, misi gamoyenebis unaris ganviTarebaze unda iyos mogebuli. moswav-

leTa Sefaseba swavlis procesze maswavleblis dakvirvebebiT, moswavleTa mier

sakontrolo da damoukidebeli samuSaoebis Sesrulebis xarisxiT ganisazRvreba.

yoveli gakveTilis Semdeg sakuTar wignakSi CainiSneT moswavleebze dakvirvebebis

Sedegebi, gaiTvaliswineT moswavleTa SemoqmedebiTi aqtiuroba (masalis aTvisebis

done kargad Cans savarjiSoebis amoxsnis drosac _ ganmtkicebis pro cesSi).

saswavlo masalis dasabuTebuli SerCeva iTvaliswinebs Semdeg moTxovnebs:

saswavlo masalis Sesabamisoba Temis mizanTan gakveTilze Sesasrulebeli samuSaos moculobis swori gansazRvra

optimaluri Tanafardoba konkretulsa da zogads Soris

Teoriasa da praqtikas Soris aucilebeli urTierT kavSiris ganxorcieleba

maswavlebelma, rogorc wesi, Tavidan bolomde detalurad unda gaiazros

gakveTili, winaswar, drois mixedviT unda iyos ganawilebuli mTeli samuSao.

magaliTad, Tu gakveTilze axal Temaze gadasvlac aris gaTvaliswinebuli,

maSin SeiZleba im sakiTxebis Sesaxeb msjeloba, romelTa bunebriv da kanonzomier

gagrZelebis axali sakiTxebi Seicavs, SeiZleba gakveTili pirdapir im praqtikuli

amocanis ganxilviT daviwyoT, romlis maTematikuri modelis Seswavla axali maTema-

tikuri faqtebis aRmoCenas, hipoTezis Camoyalibebasa da dasabuTebebs moiTxovs.am procesis bunebrivi gagrZeleba Sesabamisi savarjiSoTa sistemis ganxilvaa.

kiTxvebze pasuxebis gacemis sistema ar unda iyos erTferovani _ mxolod

warmatebuli moswavleebTan mimarTebiT ar unda SemoifargloT; moswavlis raime

mosazrebas myisve nu upasuxebT. swor pasuxsac ki maSinve nu daeTanxmebiT xolme

_ gaakeTeT pauza, iqneb garkveuli eWvic ki gamoTqvaT misi mosazrebis sisworis

mimarT. amiT miaRwevT imas, rom bavSvebi daubrundebian dasmuli kiTxvis analizs

da male WeSmariti daskvna _ swori pasuxi klasis dominantur mosazrebad gadaiq-

ceva. klasi, erToblivi ZalisxmeviT, `gaiZulebT~ daeTanxmoT mis pozicias. es ax-

arebs, amxnevebs da aerTianebs axalgazrdebs. es maTi erToblivi azris gamarjvebaa.Tqvens mizansac xom es warmoadgens _ moswavle CamoayaliboT Semoqmed, codniT


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aRWurvil, iniciativian, xalisian axalgazrdad. maTematika mZlavri emociuri mux-

tis matarebelia da misi amoqmedeba Tqveni ZalisxmeviT miiRweva~.

yuradRebiT unda movisminoT yvela pasuxi, uxeSi Secdomis SemTxvevaSic ki

dauSvebelia mkacri uaryofiTi Sefa sebebis gamoTqma.

yuradReba miaqcieT, rom terminebi, cnebebi da movlenebi sworad iyos ganmarte-

buli.gakveTilebis dagegmvasa da warmarTvaSi xels SegiwyobT sanimuSo gakveTilebis

scenarebi.

axali masalis ganmtkicebis procesi SeiZleba e. w. `testuri~ amocanebis `amox-

sniT~ daviwyoT, maTi Sesruleba swori pasuxis SerCeviT unda Semoifarglos _

zogjer SeiZleba komentarebis gakeTebac gaxdes saWiro. yovel paragrafSi mo ce-

muli masala, rogorc wesi, 2 gakveTilzea gaTvaliswinebuli; meore gakveTilze

codnis ganmtkicebaze zrunviT SemovifarglebiT.

swavlebis erT-erTi saintereso da mniSvnelovani forma jgufuri muSaobaa.

es muSaoba SeiZleba gakveTilis procesis erT-erTi Semadgeneli nawili iyos _

daukavSirdes axali masalis gaazrebas, praqtikuli saqmianobis (eqsperimentis) anSemajamebeli daskvnebis gamotanas, an, SesaZlebelia, mas mTeli gakveTilic da-

vuTmoT. mis warmarTvaSi moswavlis saxelmZRvaneloSi warmodgenili amocanebis

sistema dagexmarebaT (magaliTad, amocanebi jgufuri muSaobisTvis). am Sem TxvevaSi

jgufur muSaobas SeiZleba Sejibris saxec ki mivceT.

organizaciulad jgufuri muSaobis es varianti _ paeqroba _ jgufuri muS-

aoba SeiZleba ase movawyoT:

winaswar vacxadebT Catarebis dRes; moswavleebs vavalebT samuSao rveulis

ormagi furceli iqonion. paeqroba moswavleTagan kapitnebisa da maTi TanaSemweebis

dasaxelebiT iwyeba. optimaluria 4-5 moswavlisgan Semdgari jgufebi _ gundebi.gundebis dakompleqteba SeiZleba kapitnebsac miandoT. mTa varia, `arCevnebma~ didi

dro ar wagarTvaT.

mas Semdeg, rac gundebi dakompleqtdeba, winaswar gamravlebuli amocanebi

daurigeT gundebs (an dafaze amowereT pirobebi). am SemTxvevaSi yvela gunds

erTnairi davaleba miecema.

gakveTilis dasrulebamde 10-12 wuTiT adre kapitnebs evalebaT warmoadginon

maTi gundebis Sedegebi _ maTi amocanebis amoxsnebi. CaibareT es amoxsnebi winas-

wari komentarebis gareSe da saukeTeso amoxsnebis avtorebi rigrigobiT miipatiJeT

dafasTan naSromTa mokle prezentaciisTvis. cxadia, kritika da polemika, Tu amis

safuZveli arsebobs, unda iyos uSeRa vaTo, magram koreqtuli. am bWobaSi TqvencmogiwevT xandaxan Cabma; zogjer mediatoris rolis Sesrulebac. es procedura

aRniSnul droze mets ar moiTxovs, radgan amocanebi yvelas kar gad aqvs gaazre-

buli da mxolod sakvanZo punqtebia xaz gasasmeli.

am gansjis dasrulebisTanave unda aRdges merxebis Tavda pirveli ganlageba,

Semdeg acxadebT gundebis mier mopovebul qulebs (TiToeuli amocana SeiZleba

2-quliani skaliT Sefasdes) da dakavebul adgilebs am paeqrobaSi. SeiZleba daawe-

soT damatebiTi qulebi prezentaciis Sesafaseblad.

jgufuri muSaobisTvis amocanebi SeiZleba SeirCes saxelmZRvaneloSi Sesabamisi

niSnakiT gamoyofili adgilidan. Tumca, winaswar yvela amocana rom ar iyos `gaSi-


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fruli~, iqneb maTi pirobebi odnav SecvaloT, an, zogierTi amocana paragrafis

damatebiTi savarjiSoebis krebulidan airCioT.

moswavlis saxelmZRvaneloSi SemoTavazebuli jgufuri muSaobis zogierTi

proeqti raime erTi axali Temis Sesabamisi struqturirebuli kiTxvebisgan Sedge-

nili amocanaa _ yoveli kiTxva winas ukavSirdeba _ kiTxvebze pasuxebis sistema

raime erTi axali Temis Sesabamisi problemis dasmas da amoxsnas gulisxmobs. amSemTxvevaSi pasuxebis sistema, romelic mos wavleTa jgufis erToblivi Zalisxmevis

Sedegia, maTi koleqtiuri SemoqmedebiTi naSromia da misi Sefaseba prezen taciisas

ganxorcieldeba.

zogjer jgufebs sxvadasxva saxis davalebebi SeiZleba mivceT _ gansxvavebuli

eqsperimentis Catareba (magaliTad, geometriuli obieqtis Tvisebebis dasadgenad

an dasabuTebis gziT geometriuli faqtis ganxilva-dasabuTeba, Sedegis warmodgena.

am SemTxvevaSi moswavleebi msjeloben warmodgenili varaudebis koreqtulobaze da

adareben maT. aseTi tipis jgufuri muSaobebi, rogorc zemoT aRvniSneT, gakveTi-

lis fragmenti SeiZleba iyos da xSirad unda gamoviyenoT.

Teoriuli masalis gadmocemis Cveneuli meTodika saSualebas gaZlevT airCioT,yvelaze ufro mosaxerxebeli forma Sinaarsisa da miznebis rukaSi miTiTebuli

moTxovnebis Sesasruleblad.

amrigad, gakveTilis dagegmva SesaZlebelia Semdegi sqemiT mimdinareobdes.

saswavlo miznebi _ gakveTilis ideuri Sinaarsi; is, rasac unda miaRwioT, ra

codna da unar-Cvevebi unda SeiZinon moswavleebma am gakveTilze;

Sefaseba _ ra kriteriumebiT SevafasebT dagegmilis Sesrulebis xarisxs da

ra SemTxvevaSi vityviT, rom gakveTilma warmatebulad Caiara;

miRwevis saSualebebi _ ra meTodebiT, moswavleTa ra organizebiTa da

ra damatebiTi resursebis saSualebiT apirebT dasaxuli amocanebis gadaWras;sasurvelia winaswar gaTvaloT drois savaraudo xarji; aucileblad winaswar

gansazRvreT saSinao davalebis xasiaTi da moculoba.

Sefasebis zogadi principebi

saskolo Sefasebis axali sistema, romelic erovnuli saswavlo gegmiT aris

gansazRvruli, iTvaliswinebs Semdeg aucilebel midgomebs: akademiuri moswrebis

Sefaseba unda iyos xSiri da mravalmxrivi. unda Sefasdes ara marto informaciis

floba, aramed SeZenili unar-Cvevebi. ar aris sakmarisi moswavle mxolod sakon-

trolo werebis safuZvelze Sefasdes. maswavlebeli unda afasebdes prezentaciebis,moswavlisave TviTSefasebis, jgufuri muSaobis, Tu sxva tipis aqtivobebis mixed-

viT. maswavlebelma Sefasebisas unda gaiT valiswinos saganmanaTleblo procesSi

moswavlis CarTulobis xarisxi (saxlSi micemuli davalebis Sesrulebis xarisxi,

gakveTilze aqtiuroba, SemoqmedebiToba da sxva), amasTanave, mizanSewonilia mo-

swavlesac gavacnoT winaswar Sefasebis kriteriumebi. am kriteriumebis SedgenaSi

SeiZleba moswavleTa CarTvac.

sakontrolo werebis Sefasebis sqemebs am wignSi gaec nobiT. dauSvebelia mo-

swavleTa qcevis gaTvaliswineba akademiuri moswrebis Sefasebisas _ gakveTilze

arasaTanadod moqceva, rogorc wesi, TavisTavad aisaxeba akademiur moswrebaze.

moswavlis niSani unda gamomdinareobdes mis mier sagnis Seswavlis sxvadasxvakomponentisgan _ sakontrolo weris Sesruleba, gakveTilze msjeloba, jgufuri


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27

muSaoba, prezentacia da sxva. unda gvaxsovdes, rom semestris niSa ni (umaRlesi qu-

laa 10) aucileblad sxvadasxva komponentis Sefasebebisgan unda gamomdinareobdes.

axla gTavazobT erovnuli saswavlo gegmiT warmodgenil moTxovnebs maTemti-

kaSi moswavlis Sefasebis Sesaxeb.

1)

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28/64

28

.

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29/64

29

5. ,

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, :

;

;

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,

.

:

( )

;

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,

0 - 1

,

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30/64

30

sanimuSo gakveTilebi

I gakveTili

aqtivoba: monacemTa TvalsaCinod warmodgenis saSualebiT simravleebze moqme-

debebis Tvisebebis dasabuTeba, amocanebis amoxsna ($1.2).

reziume: moswavleebi iyeneben venis diagramebs, monacemTa cxrilebiT warmodge-

nebs simravleebze moqmedebebis Tvisebebis ganxilvisas, amocanebis amoxsnisas. aya-

libeben sakuTar pozicias da adareben warmodgenil formebs. ecnobian simravleTaTeoriis gamaerTianebel rols da misi gamoyenebis SesaZleblobebs.

aqtivobis mizani

mos wavleebi mieCvion cxrilebis, venis diagramebis gamoyenebas, kidev erTxel

Seafason Teoriuli codnis gamoyenebis sferoebis gafarToebis mniSvneloba.

ga iazron gadawyvetilebebis miRebisas TvalsaCinoebebis mniSvneloba, dasabu-

Tebis sxva xerxebis gamoyenebis aucilebloba; yvela SemTxvevis ganxilvis aucileb-

loba.

aucilebeli wina codna

sim ravle, qvesimravle. moqmedebebi simravleebze sim ravleebis gamosaxva, simravleebze moqmedebebis gamosaxva venis diagramebiT

masala

saskolo sa xelmZRvanelo, $1.2 savarjiSoebi

aqtivobis aRwera

viwyebT simravlis, qvesimravlis, simravleTa TanakveTis, gaerTianebis, sxvaobis

cnebebis gaxsenebiT. sasurvelia mecadineoba kiTxvebis dasmiT warvmarToT. Tumca

es ar unda gadaiqces pedagogisa da mopasuxe moswavlis kiTxva-pasuxis `reJimSi~

mimdinare procesad. moswavlis yovel pasuxs unda amyarebdes sxva moswavleTamosazrebebic. moswavleTa met dainteresebas gamoiwvevda maTi skolis an klasis

moswavleTa simravlis sxvadasxva qvesimravleebis ganxilva, maTze moqmedebebi. Ta-

vad moswavleebs daavaleT saTanado magaliTebis moyvana. vTqvaT, moswavlem daas-

axela skola `albionis~ yvela klasis simravle (M). aq SeiZleba Tqven CaerToT da

ikiTxoT: aris Tu ara skolis X' klasi am simravlis elementi? (aris); aris Tu ara

X1 klasis moswavle salome M simravlis elementi? (ar aris); aris Tu ara skolis

V-VII klasebis simravle M simravlis elementi? (ar aris); aris Tu ara skolis V-

VII klaselTa simravle skolis moswavleTa simravlis qvesimravle? (aris) da sxva.

daavaleT moswavleebs saTanado aRniSvnebi da mimarTebebi dafaze warmoadginon.

gansakuTrebuli sifrTxiliT moekideT universaluri simravlis cnebas. ajobebs,Tu moeridebiT `wiaRsvlebs~, nu CauRrmavdebiT _ sakmarisia, ramdenime magaliTi


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daasaxeloT, saWiroebis SemTxvevaSi saxelmZRvaneloSi warmodgenil magaliTs

kidev erTi daaxmareT _ moswavleTaTvis cnobili ricxviTi simravleebisTvis

universalur simravled SeiZleba miviCnioT R (namdvil ricxva simravle).

Semdeg gavecnobiT simravleebze moqmedebebis TvalsaCino ilustraciis kidev

erT xerxs _ cxrilebis gamoyenebis xerxs. magaliTad, A da B simravleebis gaer-

Tianeba da TanakveTa cxrilebiT ase warmogvidgeba.

A B AB

1 1 1

1 0 1

0 1 1

0 0 0

aqac, cxrilis Sedgena moswavleebis monawileobiT unda moxdes. sakmarisia imis

aRniSvna, rom cxrilSi Cawerili `1~-ebi niSnavs, rom raime SerCeuli elementi ekuT-

vnis saTanado svetSi miTiTebul simravles, `0~ _ ar ekuTvnis. amis Semdeg moswav-leebi TviTon Seavseben cxrilebis mesame svetebs I da II svetebis mixedviT.

aqtiurad gamoiyeneT venis diagramebi qvesimravlis, simravleebze moqmedebebis

TvalsaCino warmodgenis mizniT. SeiZleba ar SemoifargloT `standartuli~ saxis

diagramebiT. magaliTad, TanakveTisa da gaerTianebis warmodgenisas gamoiyeneT

an

saxis simravleebic.

klasSi ganixileT agreTve saxelmZRvanelos 1.2-is 3, 6, 10, 11, 18 testebi. me-10

testze muSaobisas kidev erTxel unda gamoikveTos utolobaTa sistemis amonaxsnTa

simravlis arsi _ esaa sistemis Semadgeneli utolobebis amonaxsnTa simravleebis

TanakveTa. iqve 1.2-is 30-e amocanasac SevexebiT da simravleTa gaerTianebis

cnebas davukavSirebT. 22-e da 45-e amocanebiT geometriul figurebsac CavrTavT

gamokvlevaSi. venis diagramebi SeiZleba davixmaroT amave paragrafis 32-e amocanaSi

_ logikuri amocanebi, rogorc wesi, moswavleTa saerTo dainteresebas iwvevs.

monacemTa analizis unaris ganviTarebas emsaxureba 1.2-is 35-e da 36-e amocanebi.

maTSi Tavmoyrilia ganxiluli masalis TiTqmis yvela aspeqti.

yvela moswavle, romelic arsebiT wvlils Seitans 36-e amocanis aqtiur

ganxilvasa da gadawyvetaSi, uTuod maRal Sefasebas imsaxurebs, Tumca is moswavle,

vinc ver SeZlebs swrafad gaiazros mocemuloba da warmoadginos am amocanis

amoxsna, ar unda `daisajos~ dabali SefasebiT. es amocana, me-40 amocanasTan erTad,

iqneb xanmokle jgufuri muSaobisTvisac SearCioT (es savsebiT dasaSvebia maRali

akademiuri donis klasebSi), rasac unda moyves aseve operatiuli prezentacia da

naSromTa ganxilva.moswavleebs saSinao davalebad mivceT 1.2-is 2; 5; 8; 12; 18; 21; 26; 29; 35;

43 amocanebi. ramdenime moswavles SeiZleba personalur davalebad mivceT

A B AB

1 1 1

1 0 0

0 1 0

0 0 0


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saxelmZRvanelos mixedviT simravleebze moqmedebebis TvalsaCino ilustraciis

erT-erTi xerxis _ cxrilebis gamoyenebis SesaZleblobaTa ukeT gacnoba da Semdeg

gakveTilze klasisTvis warmodgena.

aqtivobis Semdgomi gaRrmaveba-gafarToeba SeiZleba logikuri amocanebis amoxsna

venis diagramebis gamoyenebas davukavSiroT. unda gvaxsovdes, rom diagramebiT

yvela SesaZlo SemTxveva unda amoiwuros. yuradRebas daimsaxurebs 36-e amocanis anmisi msgavsi amocanis ufro detaluri ganxilvac. aqtivobas gaaRrmavebda zogierTi

simravluri tolobis dasabuTebis warmodgena. magaliTad, ganrigebadobis Tviseba

erT-erTi moqmedebisa meore moqmedebis mimarT.

aqtivobis Sedegi da Sefaseba. moswavleebi gaiwafebian simravleebze moqmedebebis

venis diagramebiTa da cxrilebiT warmodgenaSi; daakavSireben da Seadareben am

or warmodgenas; imsjeleben ama Tu im situaciaSi romelimes upiratesobaze,

gavarjiSdebian diskusiis warmarTvaSi. gaecnobian simravleTa Teoriis mravalmxriv

gamoyenebebs, am Teoriis farTo SesaZleblobebs. aqtivoba friad nayofieri SeiZleba

gaxdes moswavleTa logikuri msjelobis, kritikuli azrovnebis, dasabuTebis

unarebis ganviTarebis TvalsazrisiT.

II gakveTili

aqtivoba: Sesyidvis saukeTeso variantis SerCeva

reziume: moswavleebi adgenen dasmuli praqtikuli amocanis maTematikur mo-

dels, romelic wrfivi funqciis saSualebiT warmoidgineba. am funqciis Tvisebe-

bis gaTvaliswinebiT _ magaliTad, grafikis saSualebiT _ akeTeben xelsayrel,

saukeTeso (optimalur) arCevans ($2.5; $2.6 amocanebi gameorebisTvis).

aqtivobis mizani:

praq tikuli amocanis amoxsnisas amocanis maTematikuri modelis Sedgenis Cve-

vebis daufleba;

fun qciuri damokidebulebis geometriuli gamosaxvis saSualebiT daskvnebis

gakeTebis unaris gamomuSaveba;

koordintTa meTodis SesaZleblobaTa ukeT gacnoba da praqtikuli gamoyeneba;

alternatiul SemTxvevaTa analizi da optimaluri arCevanis Ziebis unaris

gaRrmaveba;

masalasaskolo saxelmZRvanelo

aucilebeli wina codna

wrfi vi funqcia

wrfi vi funqciis Tvisebebi. wrfivi funqciis grafiki

aqtivobis aRwera

mecadineobas viwyebT wrfivi funqciis da misi Tvisebebis gameorebiT. SeiZleba

amocanebis amosaxsnelad propedevtikuli muSaobac CavataroT:

_ vTqvaT, taqsiT mgzavrobis Rirebuleba ase iangariSeba: winaswari gadasaxadia20 TeTri; Semdeg yovel kilometrze _ 40 TeTri; ra Tanxa iqneba gadasaxdeli x

km-ze (x km-is gavlis Semdgom)?


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_ Tu x km-ze gadasaxdel Tanxas y-iT aRvniSnavT; maSin ra formuliT Caiwereba

y-is x-ze damokidebuleba?

_ ra funqcia moicema am formuliT?

_ ra aris am funqciis gra fiki?

winaswari `moTelvis~ Semdeg klass jgufebad vyofT. I jgufs SevTavazebT $2.5,

#27 amo canas, me-2 jgufs

$2.6, amocanebi gameorebisTvis,

#1 amo canas, me-3 da me-4jgufebs _ analogiur amocanebs. magaliTad, Semdeg amocanas: wignis maRazia myid-

vels momsaxurebis 3 formas sTavazobs.

I forma _ winaswar 60 laris gadamxdeli yovel wigns 3 la rad yidulobs; II for-

ma _ winaswar 30 laris gadamxdeli yovel wigns 10 larad yidulobs; III forma _

winaswar gadaxdis gareSe yoveli wigni 20 lari Rirs. wignebis odenobis mixedviT

romeli formaa ufro xelsayreli? aRniSneT x-iT Sesyiduli wignebis odenoba da

TiToeul SemTxvevaSi warmoadgineT formuliT funqcia, romelic gamosaxavs gadax-

dili Tanxis damokidebulebas x-ze. erTsa da imave sakoordinato sibrtyeze aageT

miRebuli funqciebis grafikebi. SeadgineT is funqcia, romelic yvelaze xelsay-

rel (optimalur) arCevans Seesabameba.bolo kiTxvaze pasuxi gulisxmobs axali funqciis algebrulad Caweras da geo-

metriul gamosaxvasac.

aqtivobis Semdgomi gaRrmaveba-gafarToeba im funqciebis ganxilvas SeiZleba mi-

vuZRvnaT, romlebic `ramdenime formuliT~ warmoidgineba.

magaliTad, ganvixiloT im funqciis grafiki, romelic bolos mowodebuli amo-

canis optimalur arCevans Seesabameba

Tu

Tu

, < 3-2x utolobidan vRebulobT x> 123

7- .

34 M da K simravleebi SeiZleba asec warmovadginoT:

M={-6; -5; -4; -3; -2; -1; 0; 1; 2; 3; 4; 5; 6}.

K={x | x4 an x2, xZ}.

35

a ) b )

g)

39 a ) TuU aA da AB, maSin aB;Tu aB da BC, maSin aC.miviReT, rom a simravlis nebismieri elementi ekuTvnis C simravlesac e. i. AC.b) AB niSnavs, rom A-s yoveli elementi B-s elementicaa. BA niSnavs, rom B-s

yoveli elementi A-s elementicaa, e. i. A da B erTi da imave elementebisgan Sedgeba

_ A=B.

44 amocanis amoxsna saxelmZRvaneloSia warmodgenili. amave gziT SeiZleba

vipovoT n-elementiani simravlis qvesimravleTa odenoba _ im n-eulebis odenoba,

romlebSic yovel adgilze 0 an 1-ia. aseTi n-eulebis odenobaa 2n.

45 vTqvaT, n-elementiani simravlis An={a1; a2; ...;an} qvesimravleTa odenobaa

n. es qvesimravleebi (n+1)-elementiani A

n+1={a

1; a

2; ...; a

n; a

n+1} simravlis

qvesimravleebicaa. An+1

-is danarCeni qvesimravleebic amdenivea da isini miiReba An-

is yoveli qvesimravlisTvis an+1

elementis mierTebiT. anu, Tu n+1

-iT aRvniSnavT

An+1-is qvesimravleebis odenobas, miviRebT

n+1

=2n

, 1

=2.

sxivi, romelic I sakoordinato

meoTxedis _ kuTxis biseqtrisaa.


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44-45 amocanebze muSaobisas moswavles uviTardeba ganzogadebisa da savaraudo

debulebaTa dasabuTebis, analizisa da sinTezis unarebi. moswavlis SefasebaSi

unda aisaxos misi naSromis prezentaciis xarisxica da sxvaTa prezentaciebis dros

warmoCenili aqtivoba.

$1.2. moqmedebebi simravleebze. venis diagramebi

mizani. simravleebze moqmedebebis (gaerTianeba, TanakveTa, sxvaoba), venis dia-

gramebis gamoyenebebis unaris ganviTareba.

paragrafSi mocemuli masalis didi nawili SesaZlebelia moswavleTaTvis nacnobi

iyos. es kidev ufro met saSualebas aZlevs maswavlebels swavleba interaqtiuri

meTodiT, moswavleTa uSualo monawileobiTa da aqtiurobiT warmarTos. es

xerxi ar Semoifargleba kiTxva-pasuxis reJimiT. kiTxvebi moswavleebisganac unda

modiodes (amis miRweva klasSi SemoqmedebiTi, mastimulirebeli saswavlo garemos

SeqmniTaa SesaZlebeli) da maTi ganxilva, ZiriTadad, TviT moswvaleTa erToblivi

ZalisxmeviT unda mimdinareobdes, rac faqtobrivad gamoricxavs e. w. `gamoZaxebeli~

moswavlis xangrZliv monologs, an masTan maswavleblis xangrZliv dialogs. aseTi

aqtivoba simravleTa Teoriis saswavlo kursis Seswavlis Semajamebel etaps

unda emsaxurebodes. mniSvnelovania moswavleTa Cabma axali amocanebis Sedgenis

procesSic.

pirvel gakveTils viwyebT wina paragrafis sakiTxebis gameorebiT, axalzegadasvlis Semzadebis procesiT. gakveTilis sawyisi nawili SeiZleba paragrafis

teqstisgan gansxvavebuli iyos.

ra SemTxvevaSi vityviT, rom A da B simravleebi tolia?

Tu A da B simravleebi toli ar aris, arsebobs Tu ara A-Si elementi, romelic

B-s ar ekuTvnis?

ra SemTxvevaSi vityviT, rom A simravle B-s qvesimravlea? SeiZleba Tu ara A

da B toli iyos, roca A aris B-s qvesimravle?

rogor SeiZleba gamovsaxoT venis diagramebiT A da B-s Soris es mimarTeba

AB?yuradReba miaqcieT im faqts, rom xSirad cariel simravleze moswavleebi

saubroben, rogorc ar arsebul obieqtze carieli simravle iseve arsebobs,

rogorc nebismieri sxva simravle, `ar arsebobs~ obieqtebi, romlebic misi

elementebia.

savarjiSoebi ise unda SevarCioT (Tu yvela davalebas ar SevasrulebT), rom

yvela aspeqti gaTvaliswinebuli iyos.

paragrafSi warmodgenili masala or gakveTilze SeiZleba ganawildes.


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jgufuri muSaoba simravleebze moqmedebebis Tvisebebimizani. sxvadasxva xerxis gamoyenebiT simravleebze moqmedebebis warmodgena da

maTi dasabuTeba; kritikuli azrovnebis ganviTrebaa.

maswavlebels SevaxsnebT, rom kritikuli azrovneba, aris saazrovni unar-

Cveva, romelic moicavs erTdroulad ori an meti gansxvavebuli mosazrebis (am

SemTxvevaSi mosazrebaTa ricxvi samia). ganxilvas da maT gagebas, raime TvisebisdasabuTebas sxvadasxva midgomis safuZvelze da imis gacnobierebas, rom sxvasac

SeiZleba sakuTari, gansxvavebuli mosazreba hqondes.

jgufuri muSaoba iTvaliswinebs kompleqsuri davalebebis Sesrulebas, romelic

miznad isaxavs moswavleTa CarTvas TanamSromlobaSi, davalebebis Sesasruleblad

calkeul moswavleTa Zalisxmevis misadagebas sxvaTa ZalisxmevasTan. iqmneba saerTo

produqti, romelic im xerxebiT davalebebis Sesrulebebis warmodgenaa, romlebic

simravleebze moqmedebebis Tvisebebis Seswavlisas SeiZleba gamoviyenoT. TiToeul

jgufSi moswvaleebi inawileben davalebas: zogierTi cdilobs SeTavazebuli xerxi

gaarCios da gamoiyenos Tvisebebis dasabuTebisas (ar unda gamovricxoT mTeli

gundis erToblivi Zalisxmeva yvela mimarTulebiTac). magaliTad, am xerxiT (AB)C=(AC)(BC)Tvisebis Cveneba Semdegi cxrilis agebas gulisxmobs:

A B C AC BC AB (AB)C (AC)(BC)

1 1 1 1 1 1 1 1

1 1 0 1 1 1 1 1

1 0 1 1 1 0 1 1

1 0 0 1 0 0 0 0

0 1 1 1 1 0 1 1

0 1 0 0 1 0 0 0

0 0 1 1 1 0 1 1

0 0 0 0 0 0 0 0

bolo ori sveti emTxveva erTmaneTs es asabuTebs Tvisebis WeSmaritobas.

(AB)C=(AC)(BC)tolobis venis diagramiT Cvenebis erT-erTi magaliTi qvemoTaa mocemuli:

B B

A

A

A A

A

B

B

B B

B

C

C

C C

C (AC)(BC)

(AB)C

BC

AC


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maswavlebeli jgufs ar avaldebulebs yvela Tvisebis dasabuTebas. jgufuri

muSaoba gacilebiT ufro maRali efeqtianobiT gamoirCeva, roca mas Sejibris xasia-

Ti aqvs. moswavleebs eZlevaT garkveuli dro da is jgufi, romelic met Tvisebas

da meti xerxiT dasabuTebas warmoadgens, gamarjvebulad gamocxaddeba. muSaobis

dasasruls naSromTa prezentaciis xarisxi jgufebis Sefasebis sayuradRebo

komponents unda warmoadgendes. mniSvnelovania agreTve warCinebulTa gamoyofada sajaro xazgasma maTi wvlilisa.

miTiTebebi:

24 |x-2|3, -3x-23, -1x5. [-1; 5]Z={-1; 0; 1; 2; 3; 4; 5}.

26 a) |x+2|3, x+23 an x+2-3, x1 an x-5; A=(-; -5][1; +)

b ) |x-

4|


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42 SeiZleba diagramebis gamoyeneba: Tu feradi nawili

10-elementiania, mTliani 17 elementiani, maSin Seuferadebeli

nawili 7-elementiania. miviRebT:

44 DC={2}.

48 ganvixilavT or SesaZlo SemTxvevas: wrfeebi ikveTeba, wrfeebi paraleluria.

(Tu ori wrfe emTxveva erTmaneTs, gvaqvs erTi wrfe da ara ori, rasac gvTavazobs

amocanis piroba).

$1.3

. amocanebis amoxsna(venis diagramebis gamoyeneba

)

mizani. simravleebis, simravleebze moqmedebebis venis diagramebiT warmodgenebis

sxvadasxva gamoyenebis unaris ganviTareba.

paragrafSi warmodgenili masala sxvadasxva saxis (logikuri, praqtikuli da sxva)

amocanebis amoxsnisas simravleTa Teoriis elementebis gamoyenebas ukavSirdeba.

amasTanave, ZiriTadad, simravleebze moqmedebebis venis diagramebiT warmodgenebi

gamoiyeneba. amocanebis amoxsnisas logikis is nawili, romelic simravleebs

exeba, an rac igivea, gansaxilveli obieqtebis Tvisebebs, daiyvaneba simravleebze

moqmedebebis Tvisebebze.

paragrafSi warmodgenil masalas vuTmobT or gakveTils.pirveli gakveTili Semamzadebeli procesiT iwyeba moswavleebi ixseneben

simravleebze moqmdebebs, maT warmodgenas venis diagramebiT. Semdeg ganvixilavT

Teoriul nawilSi mocemul amocanebs. aq, rogorc wesi, sami simravlis SemTxveva

gvaqvs miiReba 8 nawilisgan Sedgenili figura da amocanis amoxsna daiyvaneba

am figurebis Sesabamisi ricxvebis povnaze. rogorc wesi, viwyebT im figuridan,

romelic samive figuris TanakveTas Seesabameba; zogjer Sesabamisi ricxvi ucnobia

mas x-iT aRvniSnavT da amocanis amoxsna x-is mimarT gantolebis Sedgenaze

daiyvaneba.

pasuxebi da miTiTebebi:

1 gamoviyenoT diagramebi:

8 studentma ar icis arc franguli, arc inglisuri. me- 3 winadadeba arasworia.

ing. frang.

7023 23 4523

100(7023+23+4523)=8


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2

sainf. sport.

250 180 40

uyurebs sainformacios da ar uyurebs sportuls 430180=250 gamokiTxuli;

220180=40 gamokiTuli uyurebs sportuls da ar uyurebs sainformacios; arcerTs ar uyurebs 800(250+180+40)=330 gamokiTxuli.

3 3510=25 moswavle dadis maTematikis da biologiis wreebidan erTze mainc,

mxolod maTematikaze dadis 2511=14 moswavle, oriveze 6 moswavle.

7 M maTematikaSi, F fizikaSi, U ucxo enaSi `da-

makmayofilebeli~ Sefasebis mqone moswavleTa simrvaleebia.

gamovsaxoT isini diagramebiT. daviwyoT im moswavleTa sim-

ravliT, romelTac samive saganSi aqvT `damakmayofilebeli~,

es simravle 5 elementiania. Semdeg CavwerT erT saganSi`damakmayofilebelis~ mqone moswavleTa odenobebs: 4; 4 da

11. Semdeg darCenili orsagnebianebis odenobebs.

a) 40(4+6+5+2+4+11+4)=4, b) 2+6+4=12.

8 SemoviRoT aRniSvna F fexburTiT dakavebul moswav-

leTa simravle, K kalaTburTiT, M mZleosnobiT. suraTis

mixedviT vwerT gantolebas

9-x+8-x+10-x+3+4+5+x=38-3, saidanac x=2.

sportis mxolod erTi saxeobiT dakavebulia9-x+8-x+10-

x=21 moswavle.

9 SemoviRoT aRniSvna: K kargi maCveneblebis mqone moswavleTa simravle, S

sportsmen moswavleTa simravle.

pirobiT, n(K)=30, n(S)=28, n(KS)=17. am monacemebidan gamomdinareobs, romn(KS)=30+28-17=41, rac aRemateba gamokiTxul moswavleTa odenobas, amrigadmonacemebi arasworia.

10 feradi televizorebis simravle aRvniSnoT F-iT, 54 sm

diagonaliani televizorebis simravle D-Ti, gamaZliere-blianebis U-Ti.

n(F)=24, n(D)=20, n(U)=15, n(FD)=11, n(FU)=8, n(DU)=12,n(FDU)=6.

diagramis Sevseba daviwyoT samive simravlis TanakveTidan:

sul 11+3+1+2+5+6+6=34.

11 n(AB)=n(A)+n(B)-n(AB), saidanac n(AB)=120+75-150=45.


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12 n(B)=x, maSin n(A)=x+26.

n(AB)=n(A)+n(B)-n(AB) formulidan miviRebT:144=x+26+x-22, saidanac x=70.

n(A)=96, n(B)=70.

13 24+8+4+10=46.

jgufuri muSaoba

mizani. simravleTa gaerTianebaSi elementebis odenobis gamosaTvleli formu-

lebis dadgena da maTi gamoyeneba.

1 savarjiSos mizania n(AB)=n(A)+n(B)-n(AB) formulis gaazreba: n(AB)komponenti 0-is tolia, Tu A da B araTanamkveTi simravleebia AB=.

2 sazogadod, n(ABC)=n(A)+n(B)+n(C)-n(AB)-n(AC)-n(BC)+n(ABC).

Tu A, B da C wyvil-wyvilad araTanamkveTi simravleebia, ma-

Sin am formulaSi mxolod pirveli sami Sesakrebi darCeba.

TvalsaCinoebisTvis sasurvelia venis diagramis gamoyeneba

amocana 1. SemoviRoT aRniSvnebi:

M musikiT gatacebul moswavleTa simravle,

C CogburTiT gatacebul moswavleTa simravlemaSin n(M)=16, n(C)=17, n(MC)=10.aqedan vaskvniT, rom n(MC)=16+17-10=23.23 moswavle an musikiT aris gatacebuli, an CogburTiT, an oriveTi. 7 moswavle

arc musikiTaa gatacebuli, arc CogburTiT.

amocana 2. SemoviRoT aRniSvnebi:

M im moswavleTa simravle, romlebmac CaTvla miiRes maTematikaSi, U un-

arebSi, Q qarTulSi, maSin

n(M)=800, n(U)=700, n(Q)=600,

n(MU)=600, n(MQ)=500, n(UQ)=400,n(MUQ)=300.n(MUQ)=900.

100-ma moswavlem ver miiRo verc erTi CaTvla.

amocanebi gameorebisTvis

6 roca a=0, utolobas amonaxsni ara aqvs. roca a0, a-ze gayofisas utolobis

niSani Secvlilia, amitom a


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50

7 SeiZleba moswavleebma erTi magaliTis safuZvelze (16, 36, an 256) SearCion

pasuxi 4) 6. miuxedavad amisa, moswavleebs mainc SevTavazeT danarCeni pasuxebis

ganxilva:

1) 2 ar SeiZleba iyos mTeli ricxvis kvadratis bolo cifri. mTeli ricxvis

kvadrati SeiZleba dabolovdes 1, 4, 5, 6, 9 an 0-iT.

2) Tu 4-is win kenti cifria, maSin ricxvi iyofa 2-ze da ar iyofa 4-ze ar iqnebaraime mTeli ricxvis kvadrati.

3) 5-ze SeiZleba dabolovdes mxolod 10n+5 saxis ricxvis kvadrati.

(10n+5)2=100n(n+1)+25. e. i. 5-ze daboloebuli ricxvis kvadrati bolovdeba 25-iT.

11 am amocanaSi Camoyalibebuli debulebis mcdaroba kontrmagaliTiT

dasabuTdeba. (ganixileT, magaliTad, ricxvi 72). sasurvelia aq, kidev erTxel xazi

gaesvas im faqts, rom es erTi magaliTic ki sakmarisia dasaxelebuli debulebis

mcdarobis dasadgenad.

12 pirobiT, ( )n n2

191

-= , saidanac n=14.

15 dasaxelebuli parabolis simetriis RerZi aris parabolis wveroze gamavali

wrfe, romelic ordinatTa RerZis paraleluria. wveros abscisaa x0= a2

- . pirobiT,a

2

- =4, saidanac a=-8. wveros ordinatia D4

- =13.

16 marTkuTxedis gverdebi aRvniSnoT x-iTa da (50x)-iT. pirobiT, x(50-x)

unda iyos udidesi veZebT y=-x2+50x funqciis udides mniSvnelobas. am funqciis

grafiki parabolaa (StoebiT qvemoT)da funqcia udides mniSvnelobas aRwevs, rocaargumenti parabolis wveros abscisis tolia, x=25. pasuxi: yoveli gverdi 25 sm-ia.

17 veZebT wris iseT M wertils, romlisganac A, B, C da

D wertilebamde manZilebis jami umciresia. aseTi M wertil-

ia AC da BD monakveTebis gadakveTis wertili. marTlac, M-

isgan gansxvavebuli nebismieri K wertilisTvis AK+KCAC,BK+KDBD.

aq SesaZlebelia ara umetes erTi tolobisa. magaliTad, Tu

AK+KC=AC, maSin KAC da radgan K ar ekuTvnis B BD-s, amitomBK+KD>BD. amrigad,

AK+KC+BK+KD>AM+MC+DM+MB.

18 dedamiwasa da mzes Soris manZili miaxloebiT 150 milioni km-ia, anu 1,5

108km-ia. manZili mzesa da axal planetas Soris miaxloebiT 97-jer meti, anu 1,455

1010km-ia.

19 mcire fuZe x sm, didi fuZe 4x sm,

,AB CD x

AK x

2

20 5

2

3= =

-=

BK=4.

D


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51

ABK-dan ( 20 5x2 )2

=16 +9x2

4 .

saidanac

x=2 an x=10,5 (x-is bolo mniSvnelobisTvis ,( ; ];

> ,

x x x x

x

x

x

x

3 4 1 0 3 131 0

31 1

5 0

3 03 5

3 0

5 0

b)

g)

d)

2

,

8

3 3

$ $

$

$

- + - -

-

-

-

-

-

*

*


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52

25 x2-y2=2007, (x-y)(x+y)=2007.

am gantolebis erT-erTi amonaxsniax y

x y

1

2007

- =

+ =* sistemis amonaxsni, x=1004, y=1003.

Tu gaviTvaliswinebT, rom 2007=3669, miviRebT aseT sistemasac

; , .x y

x yx y

3

669336 333

- =

+ == =

*

27 gansxvaveba ufrosis da umcrosis asakebs Soris aris 54=20 weli, amitom,3x-x=20, x=10.

pasuxi: 10, 14, 18, 22, 26, 30.

28 1 wlis Semdeg 128001,1=14080. 2 wlis Semdeg 140801,1=15488, 3 wlis Semdeg 154881,1=17036,8.

sasurvelia Tanxis gamosaTvlelad moswavlem SeZlos 128001,13 an 12800(1+0,1)3

ricxviTi gamosaxulebis Sedgena da aseTi amocanebis amoxsnisas x laris, n wlisa

da p%-is SemTxvevaSi isargeblos x P1100

n

+c m formulT.

29 1

boloSi 01 ver iqneba, e. i. aris 1 , anu a;

bolos win 10, anu k;

bolodan mesame asoa 20, anu u.

ori variantidan, aauka da luka, varCevT pasuxs: luka.

$1.4. racionaluri ricxvebi.

racionaluri ricxvebis Caweris sxvadasxva forma

mizani. racionaluri ricxvebis wiladis, sasruli an usasrulo aTwiladis, pro-

centis saxiT Cawerisa da Sesabamisi gamoyenebebis unaris ganviTareba. racionalur

ricxvebze moqmedebebisa da am moqmedebebis Tvisebebis gamoyeneba.mimdinareobs moswavleTa mier racionaluri ricxvebis Seswavlisas miRebuli

codnis gameoreba da gafarToeba. paragrafSi warmodgenil masalas SeiZleba da-

vuTmoT 2 gakveTili.

pirvel gakveTilze yuradRebas vamaxvilebT racionalur ricxvebze moqmedebaTa

Tvisebebze, am Tvisebebis geometriul interpretaciebze, moswavleTaTvis ukve nac-

nob ricxviT simravleebs Soris mimarTebebze (erovnuli saswavlo gegmis moTxovna-

Ta Sesabamisad). paragrafis ZiriTadi daniSnulebaa racionaluri ricxvebis Sesaxeb

im cnobebis sistemaSi moyvana da gameoreba, romlebic moswavleTaTvis cnobilia

wina klasebis maTematikis kursidan.

meore gakveTili racionaluri ricxvebis Caweris formebis gaxsenebasa da axaldazustebebs exeba. es dazustebebi usasrulo perioduli aTwiladebis ganxilvas


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ukavSirdeba; paragrafis meore punqtSi deduqciuri msjelobis nimuSia warmodge-

nili _ pnaturaluri ricxvis qnaturalur ricxvze gayofisas naSTi SeiZleba iyos

erT-erTi ricxvi 1, 2, ..., (q1) ricxvebidan.

Tumca, aq maswavlebelma SeiZleba gamoiyenos konkretizaciidan ganzogadebis

meTodze gadasvla _ p da q-s konkretuli mniSvnelobebis ganxilvas mosdevs p da q

cvladebze gadasvla, ganzogadebis procesi. am sakiTxebis Seswavla axali erovnulisaswavlo gegmiT me-9 klasSia gaTvaliswinebuli.

mniSvnelovania racionaluri ricxvis sxvadasxva Canaweris ganxilvisa da Sesa-

bamisi savarjiSoebis (1-17) amoxsnis gziT moswavles gamoumuSavoT ricxvisa da misi

Canaweris garCevis unari; racionaluri ricxvi SeiZleba Caiweros poziciuri siste-

mis gamoyenebiT; mniSvnelovania usasrulo aTwiladebis damrgvalebis procesis _

sasruli aTwiladebiT miaxloebiT warmodgenis Cvevebis daufleba (savarjiSoebi 31-

37), kalkulatoris gamoyeneba; mocemuli wiladis usasrulo aTwiladad warmodge-

nis procesis aRwera, perioduli aTwiladebis gaazreba;

pasuxebi da miTiTebebi

32 i ), l) da m ) SemTxvevebSi gasaTvaliswinebelia, rom periodi SeiZleba iyos

0 an 9.

35 davamrgvaloT meaTasedebamde, miviRebT .8,621. am SemTxvevaSi cdomilobaa

, ,

29250

8 62129

0 009.- = 0,0003


54/64

54

$1.5. namdvil ricxvTa simravle

mizani. namdvili ricxvebis, maTze moqmedebebis, moqmedebebis Tvisebebis gamo-

yenebis unaris ganviTareba.

paragrafSi warmodgenili masalis Seswavlisas yuradRebas vamaxvilebT namdvili

ricxvis Caweris SesaZleblobebze racionaluri ricxvebi, iracionalurisgangansxvavebiT, Caiwereba perioduli aTwiladebis saxiT, yoveli usasrulo aTwiladi

namdvili ricxvis Canaweria. moswavleebs unda SeeZloT namdvili ricxvebis Sedareba.

maTi aTwiladi miaxloebis amowera 10n sizustiT, ricxvTa Tvisebebze dayrdnobiT

ariTmetikuli operaciebis da algebruli gardaqmnebis Sesruleba.

unda aRiniSnos ricxvis cnebis kavSiri geometriul cnebebTan. swored

geometriaSi aRmoaCines iracionaluri ricxvebi. paragrafsac Sesabamisi magaliTis

ganxilviT viwyebT. es procesic SeiZleba nacnobi iyos moswavleTaTvis. mimdinareobs

Sesabamisi warmodgenebis dazustebebi, rac sakoordinato wrfeze ricxvTa gamosaxvas

ukavSirdeba. maswavleblebs SevaxsnebT, rom es wrfe namdvil ricxvTa R simravlis

gamosaxulebaa, xolo sakoordinato sibrtye namdvil ricxvTa wyvilebis R2simravlisa. aseTi midgomiT erT, an orcvladian Sesabamisobebs SeiZleba mivceT

geometriuli azri: yvela is wertili, romelTa koordinatebi akmayofilebs am

Sesabamisobebs wrfeze an sibrtyeze qmnis figuras. geometriuli warmodgena xSirad

amsubuqebs ama Tu im sakiTxis gagebas, TvalsaCinos xdis masalis gadacemas. amitom

mas xSirad mimarTaven.

am paragrafSi Semodis cnebebi: namdvili ricxvis aTwiladi miaxloeba naklebobiT

da metobiT. maswavleblebs SevaxsnebT, rom es cnebebi momavalSi ricxviTi mimdev-

robebis zRvrisa da namdvil ricxvebze moqmedebebis gansazRvrebebis formaluri

Teoriis safuZvelia.am paragrafiT vamTavrebT ricxvis cnebis gafarToebis process; iracionaluri

ricxvebis SemoRebis aRwera tradiciulia da maswavleblebisTvis nacnobi; geome-

triuli interpretaciebi gvexmareba ukeT aRiqvan masala. am da wina paragrafebSi

ver SexvdebiT winadadebebs, romlebic racionaluri an iracionaluri ricxvebis

`gansazRvrebebs~ Seicavs es cru gansazRvrebebi (Canawerebis mixedviT) samwux-

arod xSirad gvxvdeba damxmare saxelmZRvaneloebSi. amitom, cxadia, maswavlebeli

moerideba Semdegi kiTxvebis dasmas ras ewodeba racionaluri ricxvi; ras ewo-

deba iracionaluri ricxvi? moswavles unda SeeZlos maTi dasaxeleba, Sesabamis

simravleebs Soris mimarTebebis miTiTeba, dalagebis mimarTebis kargad gaazreba,

usasrulo araperioduli aTwiladebis mixedviT iracionaluri ricxvis dasaxeleba.am mxriv mniSvnelovania 10-15 amocanebis amoxsna.

pasuxebi da miTiTebebi:

18 gansxvaveba mocemul or ricxvs Soris iwyeba mZimis Semdeg me-16 Tanrigidan.

erT-erTi iracionaluri ricxvi, romelic mocemul ricxvebs Sorisaa, aris

0,0011200112001120100100010...

yoveli erTianis Semdeg (moniSnuli erTianidan dawyebuli) erTiT meti nulia,

vidre am erTianis win.


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20 < < .16 23 25 anu <


56/64

56

4. ipoveT ricxvebis miaxloebiTi mniSvnelobebi naklebobiT da metobiT , Se saba-

misad, 0,01 da 0,0001 sizustiT:

a ) 3,141592...; b ) 0,518977...

5. daalageT zrdis mixedviT :

x=0,1828(39); y=0,(182838) da z=0,182838185868.

6. ipoveT raime iracionaluri ricxvi (aRwereT misi Canaweri), romelic metia

0,(0022)-ze da naklebia 0,0022002200(22)-ze.

II varianti

1. vTqvaT , A aris 48-is martivi gamyofebis simravle, B ki 42-is. maSin AB=1) {2; 3; 7} 2) {2; 3}

3) {1; 2; 3} 4) {2; 3; 6}.

2. suraTze A, B da C simravleebia gamosaxuli. daStrixulinawili aris

1) ABC 2) ABC3) B(AC) 4) B(AC).

3. skolis moswavleebidan TiToeuli ucxo enebidan Seiswavlis an rusuls, aninglisurs, an orive am enas. moswavleTa 80% rusuls swavlobs, 60% inglisurs.

ramdeni procenti swavlobs orive enas? gamoiyeneT venis diagramebi.

4. ipoveT ricxvebis miaxloebiTi mniSvnelobebi naklebobiT da metobiT, Sesa-bamisad, 0,01 da 0,0001 sizustiT:

a ) 3, 2526027...; b ) 0,6290884...

5. daalageT zrdis mixedviT: x=0,3848(59); y=0,(384858); z=0,384858687...

6. ipoveT iracionaluri ricxvi, romelic metia 0,(00335)-ze da naklebia

0,003350033500(33)-ze.

Sefasebis sqema

pirveli da meore amocana TiToeuli 1 qula,mesame meeqvse TiToeuli 2 qula,sul 10 qula.

mesame meeqvse amocanebidan TiToeulis nawilobrivi amoxsnisas (magaliTad,sworadaa amoxsnili meoTxis mxolod a), an nawilobrivaa amoxsnili me-3, me-5,me-6 amocanebi ) SeiZleba gamoviyenoT 0,5; 1; 1,5; qulebi.


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57

$1.6. fesvi. wiladmaCvenebliani xarisxi

mizani. wiladmaCvenebliani xarisxis cnebis gaazreba, Sesabamisi aRniSvnebis

aTviseba da gamoyeneba.

axali erovnuli saswavlo gegmis mixedviT n-uri xarisxis fesvis cneba da misi

Tvisebebi wina klasSi ganixileba; Tumca, wiladmaCvenebliani xarisxis cnebisSemotanas win unda uswrebdes am masalis gameoreba.

paragrafSi warmodgenil sakiTxebs SeiZleba sami gakveTili davuTmoT.

pirvel gakveTils viwyebT kvadratuli fesvisa da ariTmetikuli fesvis cnebebis

gaxsnebiT.

dafaze vwerT:

; ; ;9 3 9 3 9 3!= = - = x2=9, x=3; x2=9; x=3.mivmarTavT moswavleebs da vTxovT SearCion swori Canawerebi. aq yuradRebas

vamaxvilebT kvadratuli fesvis gaxsnebaze kvadratuli fesvi 9-dan aris 3 da 3;

x2=9gantolebis fesvebi aris kvadratuli fesvis mniSvnelobebi 9-dan: 3da 3; am ori

ricxvidan erT-erTi dadebiTia, am ricxvs ewodeba ariTmetikuli kvadratuli fesvi9-dan da 9 aRniSvna gamoiyeneba mxolod ariTmetikuli kvadratuli fesvisTvis.

umaRlesi algebris sxvadasxva kursSi SeiZleba sxva aRniSvnebi gamoiyenebodes.

Cven SevinarCuneT skolaSi adre arsebuli SeTanxmebebi, romlebic maswavlebel-

sac unda SevaxsenoT.

kvadratuli fesvis analogiurad ganvixilavT luwi xarisxis fesvs namdvili

ricxvidan (uaryofiTi namdvili ricxvidan luwi xarisxis fesvi ar arsebobs).

dadebiTi namdvili ricxvidan luwi xarisxis fesvis ori mniSvneloba gvaqvs, erT-

erTi dadebiTia da amisTvis viyenebT simbolos. magaliTad, sworia: 814 =3; ar

aris swori: 814

=3, 814

=3. kenti xarisxis fesvi yoveli namdvili ricxvidan erTaderTia; 83 - =2; 83 =2.

wiladmaCvenebliani xarisxis warmodgenisas yuradRebas vamaxvilebT fuZis dade-

biTobaze: ( )8 31

- =2 ar aris swori, radgan uaryofiTi ricxvis wilad maCvenebliani

xarisxi ar gvaqvs gansazRvruli.

gansazRvrebis Tanaxmad, Tu a0, maSin a2=a; Tu a


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58

35 a ) ,2 32 3 273 15 5 5= = , saidanac davaskvniT, rom

, , , ,b) g) d)2 3 7 49 40 5 625 500 4 16 3 27> > >3 5 3 6 6 8 8 3 6 6= = = = , sai-

danac ,0 3 0 027 0 00096 6=

36 b ) 1,4433, amitom ,3 1 443 . naklebobiT, 0,01 sizustiT.

37 2< 153


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59

$1.7. ricxviTi Sualedebi

mizani. ricxviTi Sualedebis Caweris wesebis gamoyeneba garkveuli Tvisebis

wertilTa simravleebis warmosadgenad.

pirvel Tavs vamTavrebT paragrafiT, romelic sakoordinato sibrtyeze

garkveuli Tvisebis mqone wertilTa simravlis Caweris xerxebis Seswavlas eTmoba.gakveTilis dasawyisi dauTmeT simravleebis gaerTianebisa da TanakveTis

gaxsnebas. ricxviTi Sualedebis warmodgenis Semdeg mniSvnelovania im amocanebze

yuradRebis gamaxvileba, romlebic ricxviTi simravleebidan axali ricxviTi

simravleebis agebas moiTxovs (erovnul saswavlo gegmaSi Sesabamisi unar-Cvevebis

ganviTarebis mniSvnloba calkea miTiTebuli).

pasuxebi da miTiTebebi

16 Sualedis centri misi Sua wertilia, maSasadame, orive ((a) da b )) SemTxvevaSi

es a+b2 -ia.

17 suraTidan TvalsaCinoa, rom es monakveTia [ c a2; c+a2

].

18 1

10n +1=10n+1, n{1; 0; 1; 2; ...}.

n-is zrdisas 10n-is mniSvnelobebi izrdeba. amrigad, umciresi mniSvneloba n=-1

SemTxvevaSia:

101+1


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60

1.8. wrewiris sigrZe da wris farTobi

mizani. wrewiris sigrZisa da wris farTobis formulebis miRebis aRwera da

gaazreba, gamoyeneba amocanebis amosaxsnelad.

Temas SeiZleba 3 gakveTili dauTmoT. I gakveTils viwyebT wesieri mraval-

kuTxedebis warmodgeniT. SeiZleba moswavleebTan erTad CavataroT wesiermravalkuTxedSi Caxazuli da mravalkuTxedze Semoxazuli wrewirebis arsebobis

damtkiceba.

aqve SeiZleba gavixsenoT wesier mravalkuTxedSi Caxazuli da mravalkuTxedze

Semoxazuli wrewirebis radiusebis am mravalkuTxedis gverdiT gamosaxvis

formulebi.

am formulebis mixedviT advilad SevadgenT wesieri mravalkuTxedis perimetris

gamosaTvlel formulas Semoxazuli wrewiris radiusis saSualebiT:

Pn=2Rnsin 180

0

n .

amas mosdevs

Pn

2R sididis Sesaxeb saubari, romlis mricxveli n-is zrdasTanerTad Sesabamisi wrewiris C sigrZes `uaxlovdeba~ (misken miiswrafis).

cxadia, aq ar viwyebT imis axsnas _ ras niSnavs `uaxlovdeba~ (SeiZleba gvexmara

sityva `miiswrafis~, Cven ufro gasagebi sityva gamoviyeneT), zRvris ganmartebis

moyvana ar aris Zneli, gagebaa Zneli. Cvens SemTxvevaSi SeiZleba gveTqva _ C

ricxvi aris wrewiris sigrZe, Tu nebismieri dadebiTi e ricxvisTvis SeiZleba

mivuTiToT naturaluri n0 ricxvi, romelsac aqvs Tviseba: wesieri Caxazuli

nebismieri n-kuTxedisTvis, romlisTvisac n>n0, n-kuTxedis perimetri gansxvavdeba

C-sgan araumetes e ricxviT. amis Semdeg, cxadia, imis damtkicebac unda, rom aseTi

C arsebobs, igi erTaderTia. es ki arc ise advilia.

Zvel saberZneTSi wrewiris sigrZes asec gansazRvravdnen _ wrewiris sigrZe

aris ricxvi, romelic ar aris naklebi nebismieri Caxazuli wesieri n-kuTxedis

perimetrze da ar aRemateba nebismieri Semoxazuli wesieri n-kuTxedis perimetrs.

Tumca, es gansazRvrebac moiTxovs ramdenime damatebiTi debulebis damtkicebas,

romelTa ganxorcielebac saswavlo kursSi ar aris mizanSewonili. Cven mier

formulebis miRebis procesi, cxadia, ar aris mkacri da, ZiriTadad, intuicias

eyrdnoba. SeiZleba aseTi Canawerebic gamoviyenoT: n-is zrdasTan erTad

Pn2R

C2R ,

Pn2R

C2R,

da radgan Pn2R =Pn2R =n sin180

0

n , amitom C2R = C2R.

aq maTematikuri simkacris dacvas intuiciaze dayrdnobili msjelobiT vcvliT,

magram es nawili me-9 klasisgan gansxvavebiT, garkveuli msjelobis Catarebis

fonze mimdinareobs. intuiciaze dayrdnobiTaa aRwerili wris farTobis miRebis

procesic. SeiZleba kalkulatoris an kompiuteris gamoyenebiT nsin1800

n sididis

mniSvnelobebis cxrilic SevadgeninoT moswavleebs _ cxriliT ricxvis miaxlo-

ebiTi mniSvnelobebi iqneba warmodgenili.


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61

miTiTebebi

13 amocanis pirobis Sesabamisad, SemoviRoT aRniSvnebi: didi wris radiusi _

5x, mcire wris radiusi _ 3x. wreebis farTobebi iqneba, Sesabamisad, (5x)2 da (3x)2;

rgolis farTobia 25x29x2=16x2, rac aris didi wris farTobis

16x2

25x2 100%=64%.

14 a ) 522202=7232=2304 (sm2)

b) am monakveTis sigrZe radiusebis sxvaobis tolia _

5220=32 (sm).

15 gavixsenoT, rom R-radiusiani wrewiris 0-iani rkalis sigrZe gamoiTvleba

formuliTR180 .

pirobiT,R125

180=R245

180, saidanac

R1R2= 9

5.

16 S=R2=d2

4

a) S=32

4 =9

4 (mm2); b ) S==0,2

2

4 =0,01 (mm2)

17 a ) 2R=5, R=25 (dm); b) 2R=8,8, R=4,4 (dm);

S=R2

=6,25

2 =6,25

(dm2

)=625

(sm2

). S=R2

=19,36

2 =19,36

(dm2

)=1936

(sm2

).

18 gaamaxvileT moswavleTa yuradReba pirobis mniSvnelovan

nawilze _ gamoWrilia maqsimaluri farTobis wre. es ki niSnavs,

rom Sesabamisi wrewiri Caxazulia kvadratSi. Tu wrewiris radiuss

aRvniSnavTR-iT, maSin kvadratis gverdia 2R. saZiebeli Sefardebaa

(2R)2R2(2R)2

=4 4 .

19 gavixsenoT seqtoris farTobis formula ( R2

360) da ganvixiloT kuTxis

SemoTavazebuli mniSvnelobebi:

) =300, S=12230

360 =12 (sm2);

b ) =600, S=12260

360 =24 (sm2);

g ) =240, S=12224

360 =9,6 (sm2).

20 am amocanis amoxsnisas unda gadaiWras ori problema

_ jer saWiroa davadginoT Tu romeli oTxkuTxedia udidesi

farTobis da Semdeg vipovoT narCenebis farTobi.bunebrivia, aRniSnuli Tvisebis oTxkuTxedi Caxazul

oTxkuTxedebs Soris veZeboT.

maswmath10

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