nawili

moswavlis wigni

maswavleblis wignimoswavlis wigni

dafinansebulia

`moswavleebis

saxelmZRvaneloebiT

uzrunvelyofis programis~

farglebSi

maTematika

�maswavleblis wigni

inteleqti

3

rCevebi wigniT sargeblobisTvis .................................................................................... 4

sidideebis gazomva. geometriuli figurebi3

1. dro. drois erTeulebi .............................................................................................. 52. sigrZis erTeulebi ..................................................................................................... 15 amocanebi gameorebisTvis ........................................................................................ 213. masis erTeulebi .......................................................................................................... 224. amocanebis amoxsna (ganayofisa da naSTis gamoyeneba) ............................... 285. mravalkuTxedi.............................................................................................................. 356. kuTxis gazomva. marTobuli wrfeebi ................................................................. 417. samkuTxedi ..................................................................................................................... 50 amocanebi gameorebisTvis ........................................................................................ 588. farTobi .......................................................................................................................... 60 amocanebis nimuSebi maTematikuri SejibrebisTvis ...................................... 689. zogierTi sivrculi figura .................................................................................. 71 III Tavis Semajamebeli amocanebi .......................................................................... 77amocanebi TviTSefasebisTvis ......................................................................................... 81

amovxsnaT amocanebi. movipovoT da SeviswavloT

monacemebi4

1. movipovoT da SeviswavloT monacemebi. amovxsnaT amocanebi ................. 842. warmovadgenT monacemebs cxrilebis gamoyenebiT ........................................ 883. viyenebT cxrils amocanebis amoxsnisas ............................................................ 964. piqtograma ................................................................................................................... 1005. svetovani diagrama ................................................................................................... 104 V klasis saswavlo masalis Semajamebeli amocanebi ................................. 108

pasuxebi ................................................................................................................................. 116maTematikuri simboloebi da maTi gamoyenebis magaliTebi ........................... 129sidideTa zomis erTeulebi .......................................................................................... 129maTematikuri terminebis leqsikoni ......................................................................... 130sagnobrivi saZiebeli ....................................................................................................... 132martivi ricxvebis cxrili (500-mde) ......................................................................... 132

sarCevi

4

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 simboloebis daniSnulebas.

es simbolo klasSi Sesasrulebeli samuSaoebis dawyeba-

ze migviTiTebs. am amocanebis nomrebi wreebSia Cas-

muli, feradicaa, magaliTad, 4 .

es simbolo Sin muSaobisTvis gankuTvnil amocanebs

warmogvidgens. am amocanebis nomrebi samkuTxedebSia

Casmuli, magaliTad, 5 .

`sxvadasxva~ _ saintereso faqtebsa da istoriebs

warmogvidgens.

am simbolos mosdevs amocanebi, romelTa amosaxsnelad

sakmao gamWriaxobis gamovlenac mogiwevT. am amo canebis

nomrebi kvadratebSia Casmuli, magaliTad, 1 .

am simboloTi jgufuri muSaobis proeqtebia gamo-

yofili _ aseTi davalebebi TanaklaselebTan erTo-

blivad muSaobis gamocdilebas SegmatebT.

am simboloTi TamaSebia aRniSnuli – maTi saSuale-

biT maTematikuri codnis gamoyenebis gamocdilebas

SeiZenT.

vin iqneba pirveli. am simboloTi warmodgenili amo-

canebis amoxsnisas ecadeT, gamWriaxobasTan erTad

sxa rti azrovnebac gamoavlinoT.

teqstis ukeT gagebis mizniT, saWirod CavTvaleT

zogierTi sityvis warmomavlobis, misi msgavsi mniSvn-

elobebis miTiTeba. aseTi Canawerebi teqstSi moniSnu-

lia aq gamosaxuli simboloTi.

am simboloTi gamoyofili yvela amocana unda gadaita-

noT rveulSi da DSemdeg SeasruloT.

Cveni

leqsikoni

jgufuri

saSinao

moisazre

sxvadasxva

TamaSi

i

klasSi

rv

eul

Si!

gaufrTxildiT wigns!

masSi Tqvenma Canawerebma (amoxsnebma, miTiTebebma) wigni SeiZleba sxvebisTvis

gamousadegari gaxados!

5

1 dro, drois erTeulebi

sidideebis gazomva.geometriuli figurebi3

kiTxvaze pasuxis gasacemad drois dasaxelebul erTeulebs Soris kavSiris gaxseneba dagWirdebaT:

• ramdeni wuTia 1 saaTSi?

• ramdeni wuTia 3 saaTSi?

am kiTxvebze pasuxebis gacemis Sem-deg maisurebis saZiebel odenobasac ad vi lad ipoviT. axla cxrilis saxiT warmovadginoT kavSirebi drois erTeu-lebs Soris.

• ra dros uCvenebs suraTze gamosaxuli isrebiani sa-aTi?

• didi da patara isrebidan romelia wuTebis isari da romelia saaTebis isari?

• ciferblatis 60 danayofidan drois ra erTeulSi gaivlis TiToeul danayofs didi isari?

• ramden wuTSi gaivlis TiToeul danayofs patara isari?

• am saaTis mixedviT ra dro iqneboda 12 saaTis win, ra dro iqneba 12 saaTis Semdeg?

• vTqvaT, amJamad, roca am saaTs vakvirdebiT, axalgaTenebulia. ra sityvebis damatebiT davazustebT saaTis mier naCvenebi drois dasa-xe lebas?

• ra sityvebiT davazustebT 12 saaTis Semdeg (saRamos) naCvenebi dro is dasaxelebas?

• drois dasaxelebisas ratom viyenebT xolme sityvebs: `dRis~, `Ramis~?

• ra droidan iwyeba saaTis aTvla dRis dasawyisSi? ramdeni saa-Tis Semdeg iwyeba Semdegi _ meore dRis aTvla, mesame dRis aTvla?

fabrika yovel wuTSi 1 cal maisurs amza debs. ramden maisurs daamzadebs fabrika 3 saaTSi?

60 wami (wm) = 1 wuTs (wT)

60 wuTi (wT) = 1 saaTs (sT)

24 saaTi (sT) = 1 dRe-Rames7 dRe-Rame = 1 kviras12 Tve = 1 wels100 weli = 1 saukunes

1)

am Tavis Seswavlisas Tqven kidev erTxel safuZvlianad ganixilavT drois, sigrZisa da masis erTeulebs, maT gamoyenebas. geometriul figurebsac daubrundebiT. SeiswavliT kuTxis gazomvis amocanebs. gaarkvevT, ras niSnavs ipovoT figuris farTobi; rogor unda ipovoT, magaliTad, Tqveni ezos farTobi. am sakiTxebis codna gaaRrmavebs Tqvens warmodgenebs gare-moze da sxvadasxva movlenaze.

6

2) SeiZleba mogviwios drois erTeulebis Sekreba, an gamokleba. zog jer SeiZleba saWiro gaxdes axal erTeulze gadasvla.

magaliTi 1

magaliTi 2

vTqvaT, gakveTilebi 45-wuTiania. pirveli gakveTili iwyeba, roca 9-s aklia 15 wuTi, Sesvenebebi 10-wuTiania. romel saaTze dasrulde-ba meo re gakveTili?

+ -1 sT 30 wT4 sT 50 wT

5 sT 80 wT = 6 sT 20 wT(radgan 80 wT = 1 sT 20 wT)

19 wT 28 wm 9 wT 45 wm

9 wT 43 wm(radgan 18 wT = 17 wT 60 wm)

18 88

gakveTilis dawyebis droa 8 sT 45 wT.

(radgan wuTebis Sekrebisas miRebuli 145 wuTi = 2 sT 25 wT)

+

8 sT 45 wT 45 wT 10 wT 45 wT

8 sT 145 wT = 10 sT 25 wT

SearCieT swori pasuxi (imuSaveT rveulSi):

1 `15 saaTi da 15 wuTi~ niSnavs imaves, rasac a) `dRis 3-is 15 wuTi~ b) `dRis 4-is 15 wuTi~ g) `Ramis 3-is 15 wuTi~ d) `Ramis 4-is 15 wuTi~.

2 `15 saaTi da 20 wuTi~ niSnavs imaves, rasac a) `dRis ori saaTi da 20 wuTi~ b) `dRis sami saaTi da 20 wuTi~ g) `Ramis ori saaTi da 20 wuTi~ d) `Ramis sami saaTi da 20 wuTi~.

3 fexburTis matCi 22 saaTze daiwyo. ra ricxvs miuTiTebda am dros saaTis patara isari?

a) 22-s b) 12-s g) 10-s d) 2-s.

4 koncerti 19 saaTze daiwyo. ra ricxvs miuTiTebda am dros saaTis didi (wuTebis) isari?

a) 19-s b) 10-s g) 7-s d) 12-s.

klasSi

7

5 suraTze cifruli eleqtrosaaTia gamosaxuli. ra dros miu Ti Tebs es saaTi?

a) 11-is 10 wuTs b) 11-s aklia 10 wuTi g) 10 saaTsa da 10 wuTs d) 12-is 10 wuTs.

6 Tu amJamad 18 saaTia, maSin yofila a) dilis 6 saaTi b) dRis 4 saaTi g) dRis 8 saaTi d) saRamos 6 saaTi.

7 vTqvaT, amJamad a saaTia (a aRniSnavs orniSna ricxvs, romelic me-tia 12-ze da naklebia 24-ze). ra ricxvis tolia is ricxvi, romelsac am dros saaTis patara isari miuTiTebs?

a) a–24 b) a–10 g) a–9 d) a–12.

8 ramdeni wuTia saaTis 14

?

a) 45 wT b) 75 wT g) 4 wT d) 15 wT.

9 ramdeni wuTia saaTis 34

?

a) 45 wT b) 75 wT g) 105 wT d) 15 wT.

10 3 sT 25 wT=

a) 28 wT b) 180 wT g) 205 wT d) 503 wT.

11 5 sT 15 wT + 4 sT 50 wT=

a) 1 sT 5 wT b) 8 sT 25 wT g) 9 sT 25 wT d) 10 sT 5 wT.

12 200 weli aris a) 1 saukune b) 2 saukune g) 3 saukune d) 1

2 saukune.

13 ramdeni saaTia guSindeli dilis 9 saaTidan dRevandeli dilis 9 saaTamde?

a) 12 b) 24 g) 48 d) 36.

14 ramdeni saaTia guSindeli dilis 9 saaTidan dRevandeli saRamos 9 saaTamde?

a) 12 b) 24 g) 48 d) 36.

15 ramdeni wuTia dilis 7 saaTidan imave dilis 11 saaTamde?

a) 240 b) 4 g) 60 d) 120.

16 ramdeni wamia 1 wuTSi?

a) 80 b) 60 g) 30 d) 45.

17 ramdeni wamia 1 saaTSi?

a) 60 b) 3600 g) 360 d) 100.

11  :10

8

18 fexburTis matCis TiToeuli naxevari 45 wuTia. matCis (orive na-xevris) xangrZlivobaa

a) 1 saaTi b) 1 12

saaTi g) 1 34

saaTi d) 1 14

saaTi.

19 `9-is naxevari~ igivea, rac a) `7 saaTi da 30 wuTi~ b) `8 sT da 30 wT~ g) `8-s aklia 25 wuTi~ d) `9 sT da 30 wT.~

20 romel eleqtrosaaTze SeiZleba iyos gamosaxuli am isrebian saaTze miTiTebuli dro?

a) 20:55 b) 8:55

g) 19:55 d) 8:05 .

21 9 weli 8 Tve – 7 weli 11 Tve =

a) 1 w 9 Tve b) 2 w 3 Tve g) 2 w 9 Tve d) 17 w 7 Tve.

22 21 saukune aris a) 210 weli b) 21000 weli g) 2001 weli d) 2100 weli.

23 Tu TviTmfrinavi 19 saaTze gafrinda, SeiZleba asec vTqvaT: TviT-mfrinavi gafrinda

a) dilis 7 saaTze b) saRamos 7 saaTze g) saRamos 9 saaTze d) saRamos 11 saaTze.

amoxseniT amocanebi:

24 eqimma mariams 8 abis miReba urCia _ yovel 6 saaTSi TiTo abis. pirveli abi mariamma dilis 11 saaTze miiRo, ra dros mouwevs mariams danarCeni abebis miReba?

25 eqimi pacients 8 saaTSi erTxel nemss ukeTebs. sul gasakeTebelia 12 nemsi. pirveli nemsi 15 dekembris 16 saaTze gakeTda. rodis mouwevs eqims meoTxe nemsis gakeTeba, bolo nemsis gakeTeba?

SeavseT gamotovebuli adgilebi:

26 300 wm= wT 27 600 wT= sT

28 48 sT= dRe-Rame 29 180 wT= sT.

30 144 sT= dRe-Rame 31 480 wm= wT

rv

eul

Si!

9

32 36 sT= dRe-Rame 33 16 Tve= weli

34 150 wm= wT 35 1 weli 1 Tve = Tve

36 3 wT 32 wm=2 wT wm.

SeasruleT moqmedebebi:

37 38 39

40 21 sT 30 wT 41 23 sT 23 wT 42 15 sT 20 wT 15 sT 6 wT 15 sT 20 wT 4 sT 32 wT

43 8 sT 34 wT 52 wm 2 sT 26 wT 15 wm

44 firmis muSakTa samuSao dRis xangrZlivoba 7 12

saaTia. daukvirdiT cxrils da SeecadeT SeavsoT igi:

firmis muSakebi mosvlis dro wasvlis dro

niko

kote

gurami

malxazi

levani

8:00

8:30

8L:15

15:30

15:50

16:20

45 gakveTili 45 wuTs grZeldeba, yoveli dasveneba 10-wuTiania. pirveli gakveTili 8 saaTsa da 30 wuTze iwyeba.

• ramden wuTs Seadgens 5 gakveTili?

• ramden saaTsa da wuTs Seadgens 5 gakveTili?

• ramden wuTs Seadgens yvela dasveneba 5 gakveTilis SemTxveva-Si?

• romel saaTze damTavrdeba 5 gakveTili?

46 erT-erT skolaSi gakveTilebi 9 saaTze iwyeba, 5 gakveTili 2 sa -aTze mTavrdeba, yoveli Sesveneba 15 wuTiania. gakveTilebi erT nairi xangrZlivobisaa. risi tolia es xangrZlivoba?

47 cxriliT warmodgenilia erT-erTi muSakis monacemebi samuSao drois aRricxvis Jurnalidan. samuSao kviris xangrZlivoba 35 saaTia. ramdeni wuTiT gadaaWarba dasaxelebul normas am muSakma?

+

- -

+

+ +

-

7 wT 38 wm

4 wT 16 wm

3 sT 2 wT

11 sT 36 wT

2 sT 34 wT

7 sT 28 wT

rv

eul

Si!

rv

eul

Si!

10

TariRiorSabaTi

7.XII

samSabaTi

8.XII

oTxSabaTi

9.XII

xuTSabaTi

10.XII

paraskevi

11.XII

muSaobis

xan grZ li­

voba

6 sT 35 wT 7 sT 35 wT 7 sT 25 wT 6 sT 55 wT 6 sT 35 wT

48 givim Tavisi 15000 lari garkveuli vadiT bankSi Seinaxa. am drois

manZilze banki givis yovelTviurad uxdida dasaxelebuli Tanxis 1

100-s, vadis dasasruls givis sawyisi Tanxac daubrunda. bankis mier

gacemulma mTelma Tanxam Seadgina 17250 lari. ramdeni weli iyo Tanxa bankSi Senaxuli?

SearCieT swori pasuxi (imuSaveT rveulSi):

1 1 saaTSi aris a) 360 wm b) 3600 wm g) 60 wm d) 3600 wT.

2 `16 saaTi da 20 wuTi~ niSnavs imaves, rasac a) `dRis 4-is 20 wuTi~ b) `dRis 5-is 20 wuTi~ g) `dRis 6-is 20 wuTi~ d) `dilis 5 saaTi da 20 wuTi~.

3 saskolo karnavali 18 saaTze daiwyo. am droisTvis isrebian sa aT ze saaTis isari miuTiTebda

a) 6-s b) 8-s g) 7-s d) 10-s.

4 es saaTi miuTiTebs a) 12-is 20 wT-s b) 12 saaTsa da 20 wuTs g) 2-is 20 wuTs d) 4-is 20 wuTs.

5 vTqvaT, amJamad 19 saaTia, maSin a) saRamos 7 saaTi yofila b) saRamos 6 saaTi yofila g) saRamos 9 saaTi yofila d) dilis 7 saaTi yofila.

6 ramdeni wuTia 1 saaTis 15

?

a) 15 wT b) 12 wT g) 10 wT d) 5 wT.

7 4 sT 15 wT=

a) 240 wT b) 245 wT g) 260 wT d) 255 wT.

saSinao

12:20

11

8 1000 weli aris a) 10 saukune b) 1 saukune g) 20 saukune d) 100 saukune.

9 ramdeni saaTia guSindeli dilis 9 saaTidan dRevandeli dilis 10 saaTamde?

a) 12 sT b) 13 sT g) 25 sT Dd) 23 sT.

10 ramdeni saaTia guSindeli dilis 9 saaTidan xvalis dilis 9 sa a-Tamde?

a) 24 sT b) 36 sT g) 48 sT Dd) 108 sT.

11 ramdeni saaTia guSindeli dilis 9 saaTidan xvalindeli dilis 11 sa a Tamde?

a) 26 sT b) 38 sT g) 50 sT Dd) 110 sT.

12 ramdeni saaTia dilis 9-is naxevridan imave dilis 10 saaTamde?

a) 12

sT b) 1 sT g) 1 14

sT Dd) 1 12

sT.

13 koncerti 18 saaTze daiwyo da 20 sa-aTsa da 15 wuTze damTavrda. koncer ti grZeldeboda

a) 2 saaTs b) 2 14

saaTs

g) 2 12

saaTs d) 2 18

saaTs.

14 `8-is naxevari~ igivea, rac a) `7 saaTi da 30 wuTi~ b) `8 saaTi da 30 wuTi~

g) `8 saaTi da 20 wuTi~ d) `7 saaTi da 12

wuTi~.

15 18 Tve aris

a) 1 weli b) 1 13

weli g) 1 14

weli d) 1 12

weli.

16 27 Tve aris

a) 2 12

weli b) 2 14

weli g) 2 13

weli d) 2 15

weli.

17 vato mecadineobda samis naxevridan 5-is 15 wuTamde. ramdeni wuTi imecadina vatom?

a) 120 wT b) 100 wT g) 105 wT d) 115 wT.

18 ani da luka Wadraks TamaSobdnen 4-is naxev ridan 6-is 15 wu Tamde. ramden saaTs gagrZel da es TamaSi?

a) 2 sT b) 1 34

sT

g) 1 14

sT d) 1 18

sT.

12

19 2 sT 15 wT – 1 sT 45 wT=

a) 15 wT b) 30 wT g) 45 wT d) 75 wT.

SeavseT gamotovebuli adgilebi:

20 300 wT = sT 21 600 wm = wT

22 72 sT = dRe-Rame 23 180 wm = wT

24 240 wT = sT 25 168 sT = dRe-Rame

26 480 wT = sT 27 48 sT = dRe-Rame

28 2 weli 1 Tve = Tve

SeasruleT moqmedebebi:

29 8 wT 48 wm 30 3 sT 32 wT 4 wT 46 wm 12 sT 38 wT

31 23 sT 43 wT 32 7 sT 32 wT 15 wm 15 sT 45 wT 3 sT 28 wT 55 wm

amoxseniT amocanebi

33 gakveTilis xangrZlivoba 45 wuTia.

• ramden saaTs grZeldeba 6 gakveTili?

• vTqvaT,M gakveTili 9 saaTze daiwyo, TiToeuli Sesveneba 15 wu-Tia. romel saaTze damTavrdeba me-5 gakveTili, me-6 gakveTili?

34 • romel saaTze iwyeba TqvenTan gakveTilebi?

• romel saaTze mTavrdeba me-5 gakveTili?

• ramden wuTs giwevT skolaSi yofna gakveTilebis damTav re bamde?

• ramden saaTs giwevT skolaSi yofna gakveTilebis damTav rebamde?

• romel saaTze midixarT xolme saxlSi?

35 zamTris periodSi, roca Tbi -lisSi 9 saaTia, milanSi 6 sa aTia. de kember Si milanSi fex bur-Tis matCi adgilobrivi dro-iT saRamos 9 saa Tamde 15 wu-TiT adre daiwyo. ra dro iyo TbilisSi?

+ +

+_

rv

eul

Si!

13

36 zafxulis periodSi, roca TbilisSi 9 saaTia, milanSi 7 saaTia. ivnisSi milanSi saopero speqtakli saRamos 9-is naxevarze daiwyo. ra dro iyo maSin TbilisSi?

37 irakli yoveldRiurad muSaobas dilis 8 saaTze iwyebs da 15 saaTsa da 30 wuTze amTavrebs. ramden saaTs Seadgens misi samuSao kvira (5 samuSao dRe)?

38 Tbilisis drosa da pari-zis dros Soris gasxvaveba 3 sa a Tia _ roca TbilisSi 4 sa aTia, parizSi pirveli sa-aTia. 25 dekembers ni kom Tbi-lisidan parizSi daureka ko-tes Tbilisis droiT 14 sa -aTsa da 47 wuTze; maT 18 wuTi isa ub res; saubris dasrulebidan 3 saaTisa da 58 wuTis Semdeg kote sax lidan gavida. ra dro iyo am dros parizSi?

SeadgineT Tqveni samuSao dRis ganri gi _ romel saaTze dge biT, romel saaT ze ga mo-dixarT saxlidan, romel saaT ze iwye ba Tqve ni gakveTilebi, romel sa aT ze brundebiT saxlSi, ramden xans isve nebT, amzadebT gakveTilebs, kiT xu lobT, uyu rebT televizors, iye nebT kom piu ters, exmarebiT mSoblebs, rodis iZi-nebT, ramden xans gZinavT, ramden saaTs Sead-gens gakveTilebis momzadebisTvis gan kuTv-nili dro, samuSao dRis ra nawils Sea dgens skolaSi me cadineobebze daswreba, ga kveTi lebis momzadeba. gaqvT Tu ara rai me gata ceba _ magaliTad, mis devT sports, cekvavT, mReriT, swav lobT musikas, an xatvas? ra dros andomebT am saqmianobas?

aRwereT agreTve, ra saxis cvlilebebi xdeba xolme Tqvens gan rigSi da rogor axerxebT am cvlilebebisas drois xelaxal ga na wi lebas? ra saqmianobas dauTmob diT met dros (ramden saaTs), amis Se saZlebloba rom mogeceT?

proeqti

14

moisazre

1 yoveli dRe-Ramis ganmavlobaSi anis isrebiani saaTi ramdenime wuTiT win midis (ramdenime wuTiT met dros uCvenebs).

• ramdeni saaTiT unda wavides win saaTi, rom kvlav swori dro gvi Cvenos?

• ramdeni wuTiT unda wavides win saaTi, rom kvlav swori dro gvi Cvenos?

• vTqvaT, saaTi dRe-RameSi 3 wuTiT midis win. ramdeni dRe-Ra mis Semdeg uCvenebs es saaTi swor dros?

• aseTive amocana ganixileT cifruli eleqtrosaaTis SemTx ve vaSi.

2 amindis biurom 15 saaTze gamoacxada, rom 7 dRis ganmavlobaSi mziani dRee-bia mosalodneli. SeiZleba, Tu ara viva-raudoT, rom 60 saaTis Semdeg mziani amindi iqneba?

3 Tbilisidan londonSi Cafrenas TviTmfrinavi 3 saaTsa da 40 wuTs andomebs. nikolozi gaemgzavra Tbilisidan londonSi TviTmfrinaviT Tbilisis droiT 13 sT da 55 wT-ze da Cafrinda londonSi londonis droiT 13 sT da 35 wT-ze. romeli saaTia londonSi, roca TbilisSi 21 saaTia?

4 ninos saaTi zustia, giorgis saaTi 10 wuTiT winaa, nikolozis _ 7 wuTiT ukan. samiveni 16 saaTze unda Sexvedrodnen erTmaneTs rusTavelis metrosTan. isini Tavisi saaTis mixedviT daTqmul dros mividnen metrosTan.

• miuTiTeT TiToeulis mosvlis dro.

• ramden wuTs eloda pirveli mosuli mesames?

5 ereklem filmis dasawyisSi daxeda saaTs da Tqva: filmi 17 sa aTsa da 12 wuTze daiwyo. filmis dasasruls naTias saaTi 18 sT da 45 wT-s aCvenebda. erekles saaTi 10 wuTiT win iyo, naTias saaTi 2 wT-iT CamorCeboda. ramden xans gagrZelda filmi?