I nawili

inteleqti

I nawili

gamomcemloba

inteleqti

guram gogiSvili, Teimuraz vefxvaZe,

ia mebonia, lamara qurCiSvili

maTematika

III klasis saxelmZRvanelo

I nawili

gamomcemloba inteleqti

Tbilisi 2011

redaqtori Teimuraz vefxvaZe

guram gogiSvili, Teimuraz vefxvaZe,

ia mebonia, lamara qurCiSvili

m a T e m a t i k a

I nawili

III klasis saxelmZRvanelo

gamomcemloba inteleqti

Tbilisi 2011

Guram Gogishvili, Teimuraz Vepkhvadze,

Ia Mebonia, Lamara Kurchishvili

MaTHS II I

Part I

InTeLeKTI Publishers

Tbilisi 2011

gamomcemloba inteleqtiTbilisi, ilia WavWavaZDis gamz. 17b. tel.: 25 05 22, 91 22 83.

www.intelekti.ge info@intelekti.ge inteleqti@caucasus.net

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

ISBN 978-9941-430-51-0 (`maTematika~, III klasi)ISBN 978-9941-430-42-8 (`maTematika~, III klasi, I nawili)

© gamomcemloba `inteleqti~, 2011.

3

Cveno patara megobaro!

gilocavT axali saswavlo wlis dadgomas. am dRes Cvenc Tqven-saviT mouTmenlad velodiT. Tqven, albaT, mogenatraT mSobliuri skola, saskolo cxovreba, megobrebi. Cven ki axali saxelmZRvanelo mogimzadeT. is dagexmarebaT wina wlebSi naswavlis gaxsenebaSi, codnis gaRrmavebaSi, SemogTavazebT mraval saintereso amocanas da dagexmarebiT kidec maTi amoxsnis gzebis ZiebaSi. Cven erTad gavagrZelebT mogzaurobas maTematikis saocar samyaroSi.

gisurvebT warmatebebs swavlasa da yoveldRiur saqmianobaSi.

guram gogiSvili, Teimuraz vefxvaZeia mebonia, lamara qurCiSvili

avtorTa madloba

kidev erTxel gvinda aRvniSnoT erovnuli saswavlo gegmebisa da Se-fasebis centris eqspertTa Rvawli maTematikis swavlebis reformirebis sakiTxSi. maTTan erToblivma muSaobam didi gamocdileba SegvZina. maTi SeniSvnebi da rekomendaciebi gaTvaliswinebulia Cven mier Sedgenil IV-XII klasis saxelmZRvaneloebSi. am wignebma advilad gaikvlies gza mo-swavleebisken da maswavlebelTa mowonebac daimsaxures.

maswavlebleblTan mravalwliani TanamSromlobis kvalobaze, maTi Txovnis gaTvaliswinebiT gadavwyviteT Segveqmna dawyebiTi safexuris pirveli naxevris (I-III klasi) saxelmZRvaneloebic. gvinda madloba ga-davuxadoT pedagogebs aseTi didi ndobisa da mxar daWerisTvis. imedia, es saxelmZRvaneloebi maT imedebs gaamarTlebs.

did madlobas movaxsenebT gamomcemloba `inteleqts~, uaRresad mWidro SemoqmedebiTi TanamSromlobisTvis.

4

sarCevi

rCevebi wigniT sargeblobisTvis ............................................6

I. ricxvebi 100-mde

1 vkiTxulobT da gamovsaxavT ricxvebs ..........................8

2 vadarebT ricxvebs ................................................................16

3 davasaxeloT uaxloesi aTeuli, oceuli .....................21

4 viyenebT Tvlas .......................................................................25

5 SevkriboT ricxvebi sxvadasxva xerxiT .........................31

6 vkrebT orniSna ricxvebs ....................................................35

7 vaklebT sxvadasxva xerxiT ................................................41

8 vaklebT qveSmiweriT ............................................................44

9 Sekrebisa da gamoklebis xerxebi ....................................50

10 vipovoT Sekrebisa da gamoklebis ucnobi wevri ......56

11 gameoreba ..................................................................................60

II. geometriuli figurebi

1 visaubroT brtyel geometriul figurebze ...............66

2 sigrZis erTeulebi ...............................................................70

3 davasaxeloT geometriuli figurebi asoebis gamoyenebiT ..............................................................................75

4 SevadginoT figura (jgufuri muSaoba) ........................80

5 sivrculi figurebi ..............................................................82

6 perioduli mimdevroba ........................................................87

7 SevadaroT mimdevrobebi .....................................................92

8 gameoreba ..................................................................................95

5

III. 100-ze meti ricxvebi. ricxvebis Sekreba da gamokleba

1 vkiTxulobT da gamovsaxavT ricxvebs ..........................100

2 1000-ze meti ricxvebi ..........................................................108

3 ramden aseuls Seicavs ricxvi? ramdeni aTeulia ricxvSi? .................................................111

4 vadarebT ricxvebs ................................................................114

5 vadarebT da valagebT ricxvebs ......................................117

6 viTvliT aTeulebiT, aseulebiT .....................................120

7 davasaxeloT uaxloesi aTeuli, aseuli .......................126

gameoreba ..................................................................................129

8 SevkriboT aseulebi. SevkriboT aTaseulebi ..............132

9 SevkriboT qveSmiweriT. gviwevs aTeulis gamoyofa .................................................135

10 SevkriboT qveSmiweriT. gamovyofT aseuls, aTeuls ...............................................139

11 gamovakloT aseulebi, gamovakloT aTaseulebi .......144

gameoreba ..................................................................................147

12 gamovakloT qveSmiweriT .....................................................148

maTematikuri niSnebi da maTi gamoyenebis magaliTebi ...............................................................................153

geometriuli figurebi .......................................................153

maTematikuri terminebi ......................................................153

pasuxebi .....................................................................................154

6

r C e v e b i w i g n i T s a r g e b l o b i s T v i s

waikiTxeT es rCevebi, gaecaniT specialuri niSnakebis dan-

iSnulebas.

es niSnaki klasSi Sesasrulebeli samuSaoe-

bis dawyeba ze migviTiTebs. am amocanebis

nom rebi wreebSia Casmuli, feradicaa, maga-

li Tad, 4 .

es niSnaki Sin muSaobisas gankuTvnil amoca-

nebs warmogvidgens. am amocanebis nomrebi

samkuTxedebSia Casmuli, magaliTad, 1 .

am suraTs mosdevs amocanebi, romelTa amo-

saxsnelad xSirad sakmao gamWriaxobis gamov-

lenac mogviwevs. am amocanebis nomrebi oTx-

kuTxedSia Casmuli, magaliTad, 4 .

am niSnakiT jgufuri muSaobisTvis gankuT-

vnili amocanebia gamoyofili.

am niSnakiT TamaSebia aRniSnuli. maTSi ga-

mar jveba SeuZlebelia maTematikuri cod-

nis ga reSe.

vin iqneba pirveli. am niSnakiT warmodge-

nili amocanebis amoxsnisas ecadeT, gam-

WriaxobasTan erTad sxa rti azrovnebac ga-

moavlinoT.

am niSnakiT gamoyofili yvela amocana unda

gadaita noT rveulSi da DSemdeg SeasruloT.

jgufuri

saSinao

moisazre

i

klasSi

rv

eul

Si!

gaufrTxildiT wigns!

masSi Tqvenma Canawerebma wigni SeiZleba sxvebisTvis gamousadegari gaxados!

ricxvebi 100-mde

I Tavi

8

vkiTxulobT da gamovsaxavT ricxvebs1klasSi

suraTze aTeulebia gamosaxuli.

• sul ramdeni aTeulia?• ramdeni erTeulia 2 aTeuli?• radeni aTeulia 30 erTeuli?• daasaxeleT ricxvebi: 40, 50, 60, 70, 80, 90.• ramdeni erTeulia 90?• ramdeni aTeulia 90?

gamoiyeneT kvadratebi ( ) da warmoadgineT40 aTeulebad. aCveneT, rom

40 erTeuli = 4 aTeuli = 40.

1

2

3

suraTze 10 aTeulia gamosaxuli.

10 aTeuli aris asi erTeuli, anu 1 aseuli: 100

• ramdeni aTeulia aTi erTeuli?

• ramdeni aTeulia oci erTeuli?

1 2

9

ra ricxvebia warmodgenili suraTebiT?

• mocemuli raodenobidan sworadaa Tu ara gamoyofili aTe-ulebi da darCenili erTeulebi?

• gadaitaneR rveulSi cxri li da suraTis mixed-viT Se av seT:

• gamoiyeneT wreebi da warmoadgineT 36 aTeulebiTa da darCe nili erTeulebiT.

• gadaitaneT rveulSi daSeavseT cxrili:

SearCieT ricxvebi suraTis mixedviT:

5

6

7

4

aTeulebi erTeulebi

? ?

23

aTeulebi erTeulebi

? ?

36

23 erTeuli = ? aTeuli = ? erTeuli

? aTeuli ? erTeuli20 + 4 = ?

? aTeuli = ? erTeuli = ?

1 2

10

ra ricxvia warmodgenili?

ipoveT jami:

daasaxeleT da CawereT ricxvi, romelic Seicavs:

• waikiTxe ricxvebi: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.• ramdeni oceulisgan Sedgeba 40; 60; 80; 100? • SeiZleba 50-dan gamoiyos 1 oceuli, 2 oceuli, 3 oceuli?• SeiZleba 90-dan gamoiyos 3 oceuli, 4 oceuli, 5 oceuli?

• waikiTxeT ricxvebi: 31, 34, 38, 36.• daasaxeleT gamotovebuli ricxvebi:

31 = 1 oceuli erTeuli, 31 = 3 aTeuli erTeuli,

34 = 3 aTeuli erTeuli, 34 = 1 oceuli erTeuli,

38 = 1 oceuli erTeuli, 38 = 3 aTeuli erTeuli.

• waikiTxeT ricxvebi: 57, 78, 80, 90, 96.• TiToeul ricxvSi oceulebis gamoyofis Semdeg ramdeni

erTeuli darCeba?

8

9

10

saSinao

1

2

3

60 + 9 70 + 3

? aTeuli ? erTeuli

? + ? = ?

1 aTeuls 8 erTeuls 9 aTeuls 9 erTeuls 10 aTeuls

? aTeuli ? erTeuli

? + ? = ?

1

1

2

2

rv

eul

Si!

11

• waikiTxeT ricxvebi: 35, 56, 70, 77, 94, 100.• gamoyaviT oceulebi TiToeuli am ricxvisgan. ramdeni er-

Te uli darCa?

anisa da lukas unda gaecaT pasuxi sam kiTx vaze:• ramdeni saTvleli Cxiria TiToeul suraT ze?• sul ramdeni Cxiria orive suraTze?• sul ramdeni lurji Cxiria orive suraTze?anim daTvala `lurji~ Cxirebi jer erT,

Sem deg meore suraTze. wiTeli Cxirebic amgva-rad daTvala.

lukam daTvala Cxirebi jer erT suraTze, Semdeg _ meoreze. amis Semdeg gadaTvala oriveze lurji Cxirebi.

romeli maTgani SeZlebs samive kiTxvaze pasuxis gacemas? Tqven ra gzas airCevdiT?

10 kvadratisgan nikam miiRo kvadratebis 1 aTeuli, 10 aTeu-lisgan 1 aseuli. warmoadgineT es raodenoba ricxviT.

50 + 8

90 + 6

60 + 5

40 + 12

4

5

6

7

10

8

11

80 + 7

60 + 16

9

12

ipoveT jami:

12

klasSi

daasaxeleT `mwvane cifris~ mniSvneloba:

risi tolia 27-Si cifri 7-is mniSvneloba?

CawereT ricxvi cifrebiT:

47 orniSna ricxvia. 4 aris aTeulis Tanrigis cifri. 7 _ er-Te ulis.

Cven aT cifrs vicnobT: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.TiToeuli cifriT Caiwereba erTniSna ricxvi.ramdeni cifriT Caiwereba orniSna ricxvebi?orniSna ricxvis aTeulis Tanrigis cifri nuli ar SeiZleba

iyos.38 orniSna ricxvia. am CanawerSi 3 gviCvenebs 30-s, 3-is

mniSvnelobaa 30, 8 gviCvenebs 8 erTeuls.38-s aTeulebiTa da erTeulebiT ase CavwerT: 38 = 30 + 8.

11

12

13

1 2

3 4

4 2

5 5

4 7

3 3

Svidi ToTxmeti asi ocdaCvidmeti

samocdaori oTxmocdaxuTi ormocdaaTi

13

SearCieT `mwvane cifris~ mniSvneloba.

CawereT erTeulebiTa da aTeulebiT: 39, 97, 77.

CawereT sityvebiT ricxvebi: 52, 46, 87

mocemuli ricxvebidan daasaxeleT yvela erTniSna ricxvi. amowereT orniSna ricxvebi

1, 2, 25, 37, 56, 5, 62, 7, 69

14

15

16

83 = 80 + 3nimuSi

saSinao

13

4 5

5 50

1

5 85 50

2 9 79 90

42 32 20

3

4 88 80

5

8 58 80

8

4 74 40

7

6 99 90

10

1 31 10

6

5 22 20

9

5 3nimuSi

5 50

14

dawereT TiToeuli ricxvi cifrebiT:

CawereT aTeulebiTa da erTeulebiT: 35, 17, 45, 96.

CawereT sityvebiT ricxvebi: 21, 80, 11, 52, 39, 9, 16, 47, 70, 87.

14

15

16

oTxmocdaaTi ocdaaTi samocdaTxuTmeti

ocdaoTxi oTxmocdasami ormocdacxra

ormocdaCvidmeti samocdaCvidmeti samocdaxuTi

sul ramdeni Cxiria gamosaxuli?a) 23 b) 30 g) 32.

igivea,^ rac

a) 40 + 8 b) 4 aTeuli g) 8 aTeuli da 4 erTeuli.

7 gogona srialebda yinulze. maT 12 gogona SeuerTda. sul ramdeni gogona srialebs axla yinulze? warmoadgineT miRebuli ricxvi aTeulebiT da erTeulebiT

a) 7 aTeuli 6 erTeuli b) 3 aTeuli 1 erTeuli g) 1 aTeuli 3 erTeuli.

SearCieT swori pasuxi

1

2

3

15

24=a) 4 erTeuli 2 erTeulib) 4 aTeuli 2 erTeulig) 2 aTeuli 4 erTeuli.

ra aris 3-is mniSvneloba ricxvSi 38?a) 3 b) 30 g) 8.

8+4=a) 8+1 b) 4+8 g) 2+4+8.

45=a) 50+4 b) 40+5 g) 4+5.

43=a) 30+4 b) 40+3 g) 4+3.

98=a) 90+8 b) 80+9 g) 9+8.

71=a) 7+1 b) 10+7 g) 70+1.

4

5

6

7

9

8

10