მათემატიკური წიგნიერება-ავტორი...
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წინამდებარე სახელმძღვანელო - ”მათემატიკური წიგნიერება” განკუთვნილია იმ მოსწავლეებისათვის, რომლებმაც სხვადასხვა მიზეზის გამო სათანადოდ ვერ აუწყვეს ფეხი სასწავლო პროცესსს და ამიტომაც გაუჭირდათ მათემატიკის სასწავლო კურსის დაძლევა.
Transcript of მათემატიკური წიგნიერება-ავტორი...
ekaterine korZaZe - maTematikuri wignierebis eqsperti.
daamTavra Tbilisis ivane javaxiS-vilis saxelobis saxelmwifo univer-sitetis meqanika-maTematikis fakul- teti. wlebis ganmavlobaSi iyo skoleb-is kompiuterizaciis programa „irmis naxtomi“-s maTematikisa da bunebisme-
tyvelebis maswavlebelTa sagnobrivi qselis koordina-tori. monawileobda zogadsaganmanaTleblo skolebis maTematikis erovnuli saswavlo gegmis SemuSavebaSi. aris maTematikis swavlebis araerTi publikaciis Tanaavtori. muSaobs ilias saxelmwifo universitetSi, ganaTlebisa da mecnierebis saministroSi maTematikis eqspertad.
ekaterine korZaZe
maTematikuri
wigniereba
ekaterine korZaZe
maTematikuri wigniereba
Tbilisi 2012
ISBN 978-9941-0-4926-2
© samoqalaqo ganviTarebis instituti,2012
proeqti ,,axali SesaZleblobebi saganmanaTleblo saWiroebis mqone bavSvebs“ dafinansebulia evrokavSirisa da Ria sazogadoebis insti-tutis mier da xorcieldeba samoqalaqo ganviTarebis institutis, aso-ciacia zekarisa da marneulis axalgazrduli centris mier.
publikaciaSi gamoTqmuli mosazrebebi SeiZleba ar gamoxatavdes evro-kavSirisa da Ria sazogadoebis institutis mosazrebebs.
Project “New Opportunities for Educationally Deprived Children” was made possible by the support of European Union and Open Society Institute. Project is implementing by the Civic development Institute, Association “Zekari” and Marneuli Youth Center.
The contents of this publication do not necessarily reflect the views of European
Union and Open Society Institute
avtori: ekaterine korZaZerecenzenti: simon janaSiadizainer-damkabadonebeli: ekaterine wiqariSvili
sakonsultacio sabWoanTaZe iaboWoriSvili mananagorgoZe sofiomosiaSvili Tamar
wignis SeqmnaSi gansakuTebuli wvlili Seitanes samoqalaqo ganviTrebis institutis TanamSromlebma, ganaTlebis eqspertebma, romlebmac SeimuSaves ganaTlebis meore Sansis qarTuli modeli. ideis warmatebiT ganxorcielebis procesSi aqtiuradmonawileobdnen:baqraZe maiaTofaZe qeTevankobalaZe qeTevankorZaZe ekaterinepapava paatasamsonia irinaWanturia Tamar daibeWda gamomcemloba mwignobariTbilisi, 2012
Sesavali
winamdebare saxelmZRvanelo gankuTvnilia im moswavleebi-
sTvis, romlebmac sxvadasxva mizezis gamo saTanadod ver au-
wyves fexi saswavlo process da gauWirdaT maTematikis sas-
wavlo kursis daZleva.
imisaTvis, rom davexmaroT am moswavleebs rac SeiZleba
swrafad da advilad gadalaxon es winaRoba, Cven maT vTavazobT
gakveTilebs, sadac gansakuTrebiT TvalsaCinodaa warmodgeni-
li yvelaze fundamenturi maTematikuri cnebebi da operacie-
bi. TiToeuli paragrafi exeba maTematikis yvelaze aqtiurad
gamoyenebad Temebs, romlebsac yoveldRiur cxovrebaSi xvde-
ba yoveli moswavle. gakveTilebi ise Sedgenili, rom moswavlem
Tavad gaiazros TiToeuli cnebis saWiroeba, gansazRvros maTi
gamoyenebis areali da adgili.
gakveTilebis teqstebi ar aris moculobiTi. dawerilia
moswavleTaTvis gasagebi eniT. yovel paragrafs erTvis mra-
valferovani savarjiSoebi. isini gaTvlilia moswavleebSi
sxvadasxva donis saazrovno unarebis ganviTarebaze. vim-
edovnebT, rom saxelmZRvanelo garkveul daxmarebas gauwevs
moswavleebs, daZlion maTematikis SeswavlasTan dakavSirebu-
li problemebi da SemdgomSi warmatebiT gaumklavdnen sxva,
ufro rTul da kompleqsur maTematikur Temebs.
saxelmZRvanelos momzadebaSi gaweuli wvlilisTvis mad-
lobas vuxdiT maTematikis maswavleblebs: nona SiukaSvils,
nana boWoriSvils, neli nasyidaSvils, asmaT CxituniZes, maia
undilaSvils, Tamar WarbaZes, ekaterine CixlaZes, marina
Tofurias, manana Tofurias, irma amiranaSvils, Tamar muru-
siZes, lali gveleTians, ekaterine WuaZes, mzia buaZes, aseve, fizika-maTematikis mecnierebaTa doqtors-nino WeliZes
ekaterine korZaZe
sarCevi
1. ricxvebi da maTi Cawera .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .5
2. ricxvebis Sedareba . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .8
3. naturaluri ricxvebis Sekreba . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .11
4. naturaluri ricxvebis gamokleba . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .15
5. gamravleba .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .18
6. gayofa .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .21
7. moqmedebebis Tanmimdevrobis Sesaxeb .. . .. . .. . .. . .. . .. . .. . .. . .. .23
8. wiladebi .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .26
9. wiladis ZiriTadi Tvisebis Sesaxeb wiladebis Sedareba . .. .29
10. aTwiladebi .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .32
11. aTwiladebis Sekreba da gamokleba . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .34
12. ricxvebis damrgvaleba .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .37
13. zomis erTeulebi . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .39
14. proporciuli damokidebuleba . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .41
15. ukuproporciuloba .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .44
16. masStabi . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .46
17. procenti . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .48
18. amocanebi procentebze .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .51
19. gantoleba . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .54
20. geometriuli figurebi .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .57
21. wertili, wrfe, sibrtye . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .60
22. simetria .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .63
23. marTkuTxedi. marTkuTxedis perimetri. .. . .. . .. . .. . .. . .. . .. . .. .66
24. marTkuTxedis farTobi .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .68
25. marTkuTxa paralelepipedis zedapiris farTobi.. . .. . .. . .. .71
26. moculoba. marTkuTxa paralelepipedis moculoba .. . .. . .. .74
27. koordinatebi .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .77
28. cxrili . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .81
29. wriuli diagrama . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .84
30. svetovani diagrama . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .87
31. monacemTa saSualo . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .90
32. kompiuteruli programa „geogebra” . .. . .. . .. . .. . .. . .. . .. . .. . .. .93
5
iciT Tu ara, rom uZveles adamianebs mx-olod oramde scodniaT Tvla. yvela sxva ufro didi raodenobis gamo-saxatavad sargeblobdnen sityviT `bevri~. albaT dame TanxmebiT, rom es sakmaod mouxerxebeli iqne boda imaTTvis, vinc ganizraxavda movlenebis odnav meti sizusTiT aR-weras da gadmocemas. dRes ar arsebobs ra-odenoba, romlis zustad gamosaxvac SeuZlebeli iyos. amisaTvis gamoiyeneba sul aTi simbolo - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, romleb-sac cifrebs uwodeben. zogi ricxvis Casawerad erTi cifria sakmarisi, zogis Casawerad ki - ramodenime. magaliTad, adamians 2 xeli aqvs, Tbilisidan baTu-mamde manZili 350 kilometria, tokioSi daaxloebiT 21 000 000 (21 milioni) adamiani cxovrobs.
ricxvis CanawerSi cifris mniSvneloba damokide-bulia mis mier dakavebul poziciaze. magaliTad, 15-Si, cifri1 aRniSnavs aTeulebis raodenobas, 5 ki - erTeulebis raodenobas:
15aTeuli erTeuli
1 5
Canaweri 357 niSnavs, rom dasaTvleli obieqtebi ise daajgufes, rom gamovida 3 aseuli (100 _ 100 ad
1 ricxvebi da maTi Cawera
6
dawyobili 3 jgufi nivTebisa), 5 aTeuli da kidev 7 cali nivTi. anu, ricxvi 357 Sedgeba 3 sruli aseu-lisagan, 5 aTeulisa da 7 erTeulisagan:
357aseuli aTeuli erTeuli
3 5 7
ricxvi yovelTvis raodenobas ar aRniSnavs. mag-aliTad, fexburTelis nomeri konkretul adamian-ze an gundSi mis poziciaze mimaniSnebeli erTgvari simboloa. mekareebs maisurze xSirad 1 aweriaT, Tavdamsxmeli ki me-10 an me-11 nomriT aris xolme aRniSnuli.
savarjioSoebi
1. CawereT cifrebiT ricxvebi: ormocdasami, Tor-
meti, asocdaxuTi, samasormocdaTxuTmeti.
2. daasaxeleT da CawereT mocemuli ricxvebis wina
da momdevno ricxvebi: 19, 51, 99, 171
3. ricxvis CanawerSi 647 romeli cifri gamoxatavs,
ramdeni sruli aseulia ricxvSi?
4. ramdeni aseuli, aTeuli da erTeulia mocemul
ricxvebSi: 135, 425, 980?
5. ramdeni orniSna ricxvi arsebobs?
6. CawereT yvela SesaZlo samniSna ricxvi cifrebiT
2, 3 da 7 ise, rom CanawerSi cifrebi ar gaimeoroT.
7. gamoiyeneT cifrebi 8 da 0 da CawereT yvela Ses-
aZlo samniSna ricxvi (cifrebis gameoreba Sesa-
Zlebelia). daasaxeleT miRebuli ricxvebi.
7
8. Tu aRniSnavs aseuls, aRniSnavs aTeuls,
aRniSnavs erTeuls, maSin
CanaweriT ra ricxvia aRniSnuli?
9. Tu aRniSnavs aseuls, aRniSnavs aTeuls,
aRniSnavs erTeuls, maSin rogor CawerdiT am
aRniSvnebis meSveobiT ricxvebs: 145, 236, 74?
10. ramdeniT gaizrdeba 77 , Tu mas marcxnidan mivu-
werT 7 - s ?
8
2 ricxvebis Sedareba
yvela adamiani xvdeba situaciaSi, roca unda gaarkvios erTi nivTi ufro mZimea, Tu meore, erT yuTSi meti sagania Tu meoreSi, erTi qalaqi ufro Sors aris, Tu meore. Tu gaviTvaliswinebT, rom ricxvi faqtiurad sididis an movlenis maxasiaTe-belia (anu, konkretuli wona, raodenoba, manZili ricxviT gamoisaxeba), zemoT dasaxelebuli proble-mebis gadaWra SesaZlebelia saTanado ricxvebis Se-darebiT.
SevadaroT ori ricxvi erTmaneTs - niSnavs, ga-varkvioT romeli maTgania meti an naklebi. naklebi (Sesadarebel ricxvebs Soris mcire) ricxvi Tvlis dros ufro adre saxeldeba, vidre meti ricxvi. mag-aliTad, Tu 1-dan movyvebiT Tvlas 15 ufro adre dasaxeldeba, vidre 21, radgan 15 naklebia 21-ze. Tu sxva mxriv mivudgebiT am ricxvebis Sedarebis sakiTxs da wina gakveTilSi ganxilur ricxvis Cawer-is wess gavixsenebT, miviRebT:
15 niSnavs raRac sagnebisagan Semdgar 1 aTeuls da kidev 5 aseTive sagans:
15aTeuli erTeuli
1 5
21 ki aris igive sagnebisagan Semdgari 2 aTeuli da kidev 1:
21aTeuli erTeuli
2 1
am cxrilebidan aSkarad Cans, rom 21 metia, vidre 15.ase TvalnaTliv Tu warmoidgenT yovel gansax-
ilvel magaliTs, ricxvebis Sedareba arasodes gagi-WirdebaT.
9
calke problebaa Sedarebis Sedegis ise Cawera, rom es yvelasaTvis naTeli iyos. amisaTvis gamoiy-eneba Semdegi sami simbolo: >, < da =.
Canawers 15<21 - nebismieri adamiani ase waikiTx-avs: TxuTmeti naklebia 21-ze.
Canaweri 100 >95 - ikiTxeba, rogorc 100 metia 95-ze.
Tu Sesadarebeli sidideebi tolia, maSin gamoiy-eneba simbolo =.
savarjioSoebi
1. SeadareT ricxvebi: 54 da 58, 97 da 90, 126 da 100, 456 da 458.
2. daalageT ricxvebi zrdis mixedviT: 36, 120 da 68.3. Tu zrdis mixedviT davalagebT ricxvebs: 26, 37,
65 da 19, romeli aRmoCndeba bolo adgilze? pirvel adgilze?
4. erTidaigive saxis kanfeti pirvel maRaziaSi Rirs 50 TeTri, meoreSi ki 65. romel maRaziaSia kan-feti ufro iafi?
5. mocemulia cxrili, sadac aRniSnulia 4 monawilis monacemebi: sportuli Sejibrebisas sarben bi-likze 5 wuTis ganmavlobaSi gavlili manZilebi metrebSi.
monawile manZili
giorgi 465
manana 487
elene 399
Salva 400
10
romel monawiles aqvs saukeTeso monacemi? romeli monawile aRmoCnda yvelaze susti?6. dawereT yvela ricxvi, romeli metia 24-ze, ma-
gram naklebia 30-ze.7. ra udidesi ricxvi SeiZleba Caiweros cifrebiT
2, 3 da 5?8. 0, 3, 6 cifrebis gamoyenebiT CawereT umciresi
SesaZlo samniSna ricxvi.9. Tbilisidan goramde manZili 90 kilometria,
Tbilisidan xaSuramde - 120, xolo Tbilisidan ba-Tumamde manZili 350 kilometria. romeli qalaqia TbilisTan yvelaze axlos? romelia Tbilisidan yvelaze daSorebuli qalaqi?
10. varskvlavis nacvlad CawereT romelime cifri ise, rom Canaweri swori gamovides.
23 > 23896 < 9
11
3 naturaluri ricxvebis Sekreba
yovel adamians dReSi ramodenimejer uxdeba ori an meti ricxvis Sekreba. Tu jer 50 TeTri daxarjeT, Semdeg ki 80 TeTri, mTliani danaxarjis gasagebad, cxadia, rom 50-s unda miumatoT 80. Tu maRaziam dRis pirvel naxevarSi 45 kilogrami kartofili gayida, dRis meore naxevarSi ki _ 60 kilogrami, 45-s unda miumatoT 60, Tu gsurT gaigoT mTeli dRis ganmavlobaSi ramdeni kilogrami kartofili gayida maRaziam.
Sekrebis moqmedebas `+~ niSniT aRniSnaven. vTqvaT, Sesakrebia 16 da 23. ukve iciT, rom
Canaweri 16 aRniSnavs 1 srul aTeuls da kidev 6 erTeuls. 23 ki - 2 sruli aTeuli da kidev 3 er-Teulia.
gamodis, rom gvaqvs 3 sruli aTeuli da 9 er-Teuli, es ki 39-ia.
16 23 39+ =Tu gvinda SevkriboT ricxvebi 123, 24 da 52, ase
unda moviqceT:123 aris 1 aseuli, 2 aTeuli da 3 erTeuli.24 aris 2 aTeuli da 4 erTeuli.52 aris 5 aTeuli da 2 erTeuli.sul - 1 aseuli, 9 aTeuli da 9 erTeuli, anu 199.
12
odnav ufro rTulia, magaliTad, 78-is da 67-is jamis gamoTvla.
78 aris 7 aTeuli da 8 erTeuli. 67 ki - 6 aTeuli da 7 erTeuli. daviwyoT erTeulebis SekrebiT. 8 da 7 erTeuli
15 erTeulia, rac 1 sruli aTeulia da kidev 5 er-Teuli.
axla aTeulebi SevkriboT: 7 aTeuli da 6 aTeu-li aris 13 aTeuli, anu 1 aseuli da kidev 3 aTeuli (radgan, 10 aTeuli adgens 1 aseuls).
davTvaloT aseulebis, aTeulebisa da er-Teulebis saerTo raodenoba: 1 aseuli, 4 aTeuli da 5 erTeuli. anu, miviReT 145.
13
Tu yuradRebiT waikiTxeT gakveTili, albaT SeamCnevdiT, rom Sekrebis procedura erTgvarad 2 safexurad SeiZleba daiyos.
pirveli: TiToeuli Sesakrebi unda davSaloT aseulebis, aTeulebisa da erTeulebis jamad (aseT daSlas saTanrigo erTeulebad daSlas uwodeben).
meore: SevkriboT calke erTeulebi, calke aT-eulebi. amasTanave, kargad unda gvaxsovdes, rom 10 erTeuli 1 sruli aTeulia, 10 aTeuli _ 1 sruli aseuli. miRebuli Sedegis ricxvis saxiT Cawera arc Tu ise Znelia. yoveldRiur cxovrebaSi admianebi zemoT aRweril operaciebs sakmaod swrafad da ze-pirad asruleben.
savarjioSoebi
1. nimuSis mixedviT wrmoadgineT ricxvebi: 26, 48, 35, 138, 347, 407.
nimuSi: 459 _ 4 aseuli, 5 aTeuli, 9 erTeuli.2. gamotovebul adgilSi CawereT Sesabamisi cifri: 13 erTeuli = ___ aTeuli da ___ erTeuli 17 erTeuli = ____ aTeuli da ____ erTeuli. 23 aTeuli = _____ aseuli da _____ aTeuli. 45 aTeuli = _____ aseuli da _____ aTeuli.3. SeasruleT moqmedebebi: 35 + 20; 56+30; 237+50; 235+1004. rogor Seicvleba ricxvis Canaweri, Tu am ricxvs
20-s mivumatebT? ganixileT sxvadasxva SemTxveva.5. erT skolaSi 20 axali kompiuteria, meoreSi ki _
18. ramdeni kompiuteria orive skolaSi erTad?6. puri Rirs 65 TeTri, orcxobila ki 90 TeTri. ra
Tanxa dagWirdebaT orives SesaZenad?
14
7. sacxobSi orSabaTs gamoacxves 145 funTuSa, samSabaTs ki 120. ramdeni funTuSa gamoacxves sacxobSi ori dRis ganmavlobaSi?
8. gaagrZeleT ricxvebisagan Semdgari mimdevroba (CawereT saTanado ricxvebi) 120, 140, 160, 180, 200, ____, ____, _____
9. ori qalaqidan erTmaneTis Sesaxvedrad gamovida ori avtomobili. erTma Sexvedramde gaiara 15 ki-lometri, meorem _ 25 kilometri. ra manZilia qa-laqebs Soris?
10. ori ricxvis jami metia erTerT Sesakrebze 7-iT.
risi tolia meore Sesakrebi?
15
4 naturaluri ricxvebis gamokleba
vTqvaT, maRaziaSi hqondaT 120 kilogrami Saqari, gayides 45 kilogrami. Tu gsurT gaigoT, ramdeni kilogrami Saqaria darCenili maRaziaSi, 120-s unda gamoakloT 45. an, Tu gqondaT 80 TeTri da SeiZineT 50 TeTriani funTuSa, 80-s unda gamoakloT 50, rom gaigoT ra Tanxa dagrCaT. vinc wina paragrafi gul-dasmiT waikiTxa, albaT, ar gauWirdeba gamoklebis operaciis Sesruleba. Tavidan, orive ricxvi (sak-lebic _ romelsac vaklebT da maklebic _ romelic akldeba) unda daSaloT aTeulebisa da erTeulebis jamad.
mag., 65-s gamovakloT 34. 65 aris 6 aTeuli da 5 erTeuli.34 aris 3 aTeuli da 4 erTeuli.aSkaraa, rom gamoklebis Semdeg gvrCeba 3 aTeuli
da 1 erTeuli, anu 31.
analogiurad, vTqvaT, 248-s unda gamovakloT 125.248 aris 2 aseuli, 4 Teuli da 8 erTeuli.125 aris 1 aseuli, 2 aTeuli da 5 erTeuli. gamoklebis Semdeg gvrCeba 1 aseuli, 2 aTeuli da
3 erTeuli, anu, 123.
16
axla cota rTuli magaliTi gavarCioT. 120-s gamovakloT 45.
120 aris 1 aseuli da 2 aTeuli. es igivea, rac 10 aTeuli, 1 aTeuli da 10 erTeuli, anu 11 aTeuli da 10 erTeuli.
45 aris 4 aTeuli da 5 erTeuli.gamoklebis Semdeg miviRebT: 7 aTeuls da 5 er-
Teuls, 75-s.
258-s gamovakloT 189.258 aris: 1 aseuli, 14 aTeuli da 18 erTeuli.189 aris: 1 aseuli, 8 aTeuli da 9 erTeuli.gamoklebis Semdeg vRebulobT: 6 aTeulsa da 9
erTeuls, 69-s. Tu rmodenime SemTxvevas detalurad gaarCevT
maswavlebelTan erTad, SemdegSi damoukideblad Se-ZlebT ricxvebis gamoklebas, swrafad da zepirad.
17
savarjioSoebi
1. 98 aris 8 aTeuli da ____ erTeuli, 67 aris ____ aTeuli da 17 erTeuli, 28 aris ___ aTeuli da 18 erTeuli, 127 aris _____ aTeuli da 17 er-Teuli.
2. ipoveT sxvaoba: 45 – 23, 67 – 32, 167–56, 45 – 29, 126 – 68.
3. ipoveT ucnobi Sesakrebi: ___ + 15 = 60, 45 + ___ = 120
4. ninos aqvs 3 lari da 20 TeTri. ra Tanxa darCeba mas, Tu is daxarjavs 1 larsa da 10 TeTrs?
5. milis sigrZe iyo 120 santimetri. CamoaWres 40 santimetriani nawili. ra sigrZis aris milis dar-Cenili nawili?
6. gamoklebis Sesrulebis gareSe SeadareT ricxve-bi: 70 – 15 da 70 – 25, 125 – 80 da 125 – 85, 69 – 40 da 89 – 40, 57 – 45 da 59 – 45.
7. maRaziaSi hqondaT 80 cali puri. dRis ganma-vlobaSi gaiyida 55 puri. danarCeni qarxanas daubrunda. ramdeni puri daubrunes qarxanas?
8. erT skolaSi 90 pirvelklaselia, meoreSi – 20-iT naklebi. ramdeni pirvelklaseli swavlobs meore skolaSi?
9. erTi saxis limonaTi Rirs 80 TeTri. meore saxisa 20 TeTriT iafia. ra Tanxis gadaxda mogiwevT, Tu orive saxis limonaTs SeiZenT?
10. erT ricxvs (saklebs) gamoakles meore ricxvi (maklebi) da Sedegad miiRes 20. rogor unda SecvaloT maklebi, rom sxvaoba miiRoT 25?
18
vTqvaT, maRaziaSi yoveldRe miaqvT 20 yuTi koka-kola. 4 dRis Semdeg Tu daiTvlian miRebuli koka-kolis saerTo raodenobas, cxadia mouwevT am ja-mis gamoTvla: 20 20 20 20+ + + . Tu agrovebT fuls da kviraSi erTxel yulabaSi agdebT 50 TeTrs. 7 kviris ganmavlobaSi mogrovebuli Tanxa gamoiTvleba ase: 50 50 50 50 50 50 50+ + + + + + . advilad SeamCnevT, rom orive SemTxvevaSi jamis TiToeuli Sesakrebi erTi-daigive ricxvia. aseT viTarebebSi Canaweris optimi-zaciis mizniT (an, gamartivebis mizniT) SemoiRebu-lia amgvari aRniSvna: 20 . 4; 50 . 7; pirveli Canaweri ikiTxeba - `20 gamravlebuli 4-ze~, meore Canaweri ki - `50 gamravlebuli 7-ze~. gamravlebis komponentebs Tanamamravlebs uwodeben, gamravlebis Sedegs ki _ namravls.
Zalian martivia gamravlebis ori Tvisebis SemC-neva. pirveli: Tu ricxvs 1-ze gaamravlebT, igive ricxvs miiRebT. meore: Tanamamravlebis gadanacv-
5 gamravleba
19
lebiT namravli ar icvleba (anu: 5 . 7 = 7 . 5).ricxvis 0-ze namravli 0-is tolia.
savarjioSoebi
1. CawereT gamravlebis saxiT naxatze gamosaxuli faq-ti.
2. CawereT gamravlebis saxiT: 23 23 23 23+ + + , 45 45 45 45 45 45+ + + + + , 125 125 125 125 125 125+ + + + + .
3. CawereT jamis saxiT: 34 . 4; 25 . 6; 120 . 3; 90 . 5.4. SeavseT gamotovebul adgilebi: 25 + 25 + _ + _ + _ = 25 . 5; 12 + _ + _ = _ . 3;
20
5. warmoadgineT jamis saxiT da ipoveT mniSvneloba: 20 . 4; 25 . 3; 100 . 4; 80 . 2;6. puri Rirs 60 TeTri. ra eRireba 5 cali aseTive
puri?
7. erT yuTSi 10 fanqaria. ramdeni fanqari iqneba 7
aseTive yuTSi?
8. erTi milis sigrZe 2 metria. ramdeni mili unda
SeaduRon, rom miiRon 12 metri sigrZis mili?
9. rogor Seicvleba ori ricxvis namravli, Tu
erTerT Tanamamravls 2-jer gavzrdiT?
10. rogor Seicvleba ori ricxvis namravli, Tu
orive Tanamamravls 2-jer SevamcirebT?
21
vTqvaT, 20 kanfeti Tanabrad unda gaunawiloT 4 bavSvs. jer yvelas TiTo-TiTos CamourigebT, Sem-deg kidev TiTo-TiTos. 5 jer Camorigebis Semdeg gasanawilebeli kanfetebic amoiwureba da yovel bavSvs aRmoaCndeba 5 kanfeti. axla aseTi SemTxveva ganvixiloT: gaqvT 80 kilogrami fqvili, romle-bic 10 kilogramian tomrebSi unda gadaanawiloT. gadanawilebis Sedegad miviRebT 8 savse tomaras, TiToeuls 10 kilogrami fqviliT. aRwerili pro-cesebi maTematikuri simboloebiT ase Caiwereba: 20 : 4; 80 : 10. pirveli Canaweri ikiTxeba, rogorc `20 gayofili 4-ze~, meore ki _ `80 gayofili 10-ze~. ricxvs, romelsac vyofT `gasayofs~ uwodeben, romelzedac vyofT _ `gamyofs~, gayofis Sedegs ki `ganayofi~ ewodeba. nebismieri ricxvis 1-ze gayofis Sedegi TviTon es ricxvia. sayovelTao SeTanxmebiT 0-ze gayofa ar SeiZleba.
6 gayofa
22
savarjioSoebi
1. ramdeni kilogrami xorbali Sexvdeba 5 adamiani-dan TiToeuls, Tu isini 100 kilogram xorbals Tanabrad gainawileben.
2. ramden xuTeuls gamoyofT 20 erTeulisagan Sem-dgari grovidan? (ramden 5 _ erTeulian grovebad daSliT 20 erTeulisagan Semdgar grovas?)
3. 45 metri sigrZis kabeli 9 Tanabari zomis naWrad daWres. ra sigrZis aris TiToeuli naWeri?
4. ramdeni 5 kilogramis tevadobis yuTi dag-WirdebaT 30 kilogrami burRulis Tanabrad ga-dasanawileblad?
5. Sokoladis fila 24 Tanabari zomis kvadratisa-gan Sedgeba. ramdeni kvadrati Sexvdeba 6 gogo-nadan TiToeuls, Tu isini filas Tanabrad gai-nawileben?
6. puri 800 grams iwonis. ra wona eqneba puris TiToeul naWers, Tu purs 4 Tanabari zomis naW-rad daWrian?
7. milis sigrZe 20 metria. ramden nawilad unda daWraT mili, rom miiRoT Tanabari sigrZis 5 naWeri?
8. ramden Tanabari zomis naWers miiRebT, Tu mils jer Suaze gaWriT, Semdeg kidev TiToeul nawils 3 Tanabar nawilad gaWriT?
9. SeasruleT gayofa: 80 : 4; 60 : 3; 120 : 4; 48 : 4;10. SeavseT gamotovebuli adgilebi: 80 : _ = 20;
50 : _ = 5.
23
7 moqmedebebis Tanmimdevrobis Sesaxeb
ar aris gamoricxuli, rom aRmoCndeT amgvar situaciaSi: Tqven da Tqveni 3 megobari SebrZandiT kafeSi, aiReT 4 naWeri namcxvari, 2 limonaTi da 2 finjani Cai. danaxarji oTxivem Tanabrad unda ga-dainawiloT. namcxvris 1 naWeri Rirs 80 TeTri, 1 boTli limonaTi 70 TeTri, 1 finjani Cais Rire-buleba 40 TeTria. Tanxa, romlis gadaxdac mouw-evs TiToeul Tqvengans ase gamoiTvleba: jer unda gaigoT 4 naWeri namcxvris Rirebuleba, Semdeg 2 li-monaTisaTvis saWiro Tanxa, gamoiTvaloT ra Rirs 2 finjani Cai, SekroboT miRebuli ricxvebi da jami gayoT 4-ze. aRwerili qmedebebis maTematikuri Canaweri aseTia:
(80 . 4 + 70 . 2 + 40 . 2) : 4. SeiZleba iTqvas, rom aseTi Canawerebi ikiTxeba
iseve, rogorc Cveulebriv enaze dawerili raime te-qsti. mTavaria icodeT moqmedebebis Tanmimdevroba
24
da maTematikuri simboloebis mniSvneloba. magal-iTad, gamravleba da gayofa ufro prioritetuli moqmedebebia, vidre Sekreba da gamokleba. anu, jer gamravleba sruldeba da Semdeg Sekreba. frCxilebi, am SemTxevvaSi, jams aRniSnavs da miuTiTebs, rom 4-ze unda gaiyos samive namravlis jamis mniSvneloba da ara, magaliTad, marto 40-isa da 2-is namravli.
maTematikuri Canaweri ise unda gakeTdes, rom is erTnairad iqnas gagebuli yvela `mkiTxvelis~ mier.
savarjioSoebi
1. aRwereT moqmedebebis Tanmimdevroba CanawerebSi: (30 . 5 + 2) : 2; 16 : 4 + 30 : 5; (20 + 15) . 60 : (5 + 2)
2. gamoTvaleT gamosaxulebis mniSvneloba:
(125 + 45) : 10 + 3;
3. gamoTvaleT gamosaxulebis mniSvneloba:
(120 : 4 + 90 : 3) . 2;
4. SeadareT gamosaxulebebis mniSvnelobebi:
(30 + 15) : 5; da 30 + 15 : 3;
5. CasviT ricxvebs Soris moqmedebebis niSnebi, frCx-ilebi da tolobis niSani ise, rom miiRoT WeSmariti toloba: 50 40 9 10
6. oTxi aTianis meSveobiT CawereT 4 sxvadasxva mniSvn-elobis mqone gamosaxuleba.
7. vTqvaT, sacxobs 1 puris gamocxoba ujdeba 50 Te-Tri. ra Tanxa darCeba sacxobs mogebis saxiT 1000 puris gayidvis Semdeg, Tu puri gaiyideba 70 TeT-rad? qvemoT moyvanil gamosaxulebebidan airCieT
25
is, romliTac amoixsneba amocana: (70 + 50) . 1000; 70 . 1000 + 50 . 1000; (70 _ 50) . 1000; 1000 : (70 _ 50).
8. SeadgineT gamosaxuleba da amoxseniT amocana: erTerTi satelefono kompaniis pirobebi aseTia: zaris wamowyeba Rirs 5 TeTri, yoveli nala-parakebi wuTisaTvis ixdiT 2 TeTrs. ra Tanxis gadaxda mogiwevT 15 wuTiani saubrisaTvis?
9. SeadgineT gamosaxuleba da amoxseniT amocana: erTi tomara fqvili iwonis 50 kilograms. 2 tomara fqvili Tanabrad unda gadaunawilon 4 ojaxs. ramdeni kilogrami fqvili Sexvdeba TiToeul ojaxs?
10. qvemoT moyvanilia amonaridi maTematikis Cinuri saxelmZRvanelodan. Tqveni azriT, ra SeiZleba eweros aq?
无论是交换虚电路或是永久虚电路 (20 + 30) : 5
分组中的虚电路 20 + 30 : 5
26
gTavazobT ramodenime amonarids qarTul veb-gverdebze ganTavsebuli informaciebidan:
• `12 wlamde asakis bavSvebma, saWiroebis Sem-TxvevaSi, unda miiRon abis meoTxedi dReSi 1-jer~.
• `Semowiruli Tanxis naxevari moxmardeba bavS-vTa saxlisaTvis kompiuterebis SeZenas~.
• `gamokiTxul moswavleTa mesamedi nawili isurvebda, rom gakveTilebze ufro meti six-SiriT yofiliyo gamoyenebuli kompiuteri~.
samive winadadebaSi gvxvdeba sityvebi, romle-bic raRac raodenobas Seesatyviseba. es raodenoba aris mTlianis nawili. ricxvebs, romlebic mTlianis nawils gamoxataven wiladebi ewodebaT. meoTxedi ase Caiwereba 1
4. es niSnavs, rom mTeli dayofilia 4
tol nawilad da amJamad ganvixilavT am nawilebidan mxolod 1-s. wiladi 5
6 (ikiTxeba _ xuTi meeqvsedi)
Seesabameba Semdeg raodenobas _ 6 Tanabar nawilad dayofili mTelis 5 nawili. wiladis xazs zemoT gamosaxul ricxvs ewodeba mricxveli, xazs qvemoT
8 wiladebi
27
gamosaxuls ki _ mniSvneli. mniSvneli gviCvenebs, ramden tol nawiladaa mTeli gayofili. mricxveli ki miuTiTebs, nawilebis mTliani raodenobidan ram-denia aRebuli (an, ramdenze vmsjelobT amJamad).
savarjioSoebi
1. figurebis meramdenedi nawilebia gaferadebuli?
2. ramdeni meoTxedia mTelSi? meoredi? mesamedi?
3. romel figurazea iasamnisfrad gaferadebuli am
figuris 34 nawili?
4. romeli figura Seesabameba wilads 810
?
28
5. CawereT naxazis Sesabamisi wiladi:
6. gaaferadeT figuris 13 nawili
7. gaaferadeT konfiguraciis (figuraTa erTo-
bliobis) 113 nawili
8. klasis moswavleTa 35 nawils vaJebi Seadgenen.
klasis ra nawilia gogonaTa raodenoba?
9. gamoSrobisas vaSli kargavs wonis 89 nawils. vaS-
lis wonis ra nawili rCeba gamoSrobis Semdeg?
10. daalageT zrdis mixedviT: 89; 19; 79; 39; 49.
29
9 wiladis ZiriTadi Tvisebis Sesaxeb. wiladebis Sedareba
tolmniSvneliani wiladebis Sedareba martivia da
ar Txoulobs gansakuTrebuli wesebis codnas. 78
metia 68 -ze, radgan, wiladis ganmartebis Tanxmad,
78 aris 8 tol naWrad „daWrili“ raRac mTelis 7
nawili; 68 ki _ 8 tol naWrad „daWrili“ mTelis 6
nawili. cxadia, 7 metia 6-ze, Sesabamisad _ 78 > 6
8 .
Tu zemoT moyvanil msjelobas mainc mivcemT wesis
saxes, maSin es wesi ase Camoyalibdeba:
tolmniSvneliani wiladebidan metia is, romlis mricxvelic metia.
modiT axla SevadaroT erTmaneTs wiladebi: 12 , 2
4 da
36. ilustraciaze TvalnaTliv Cans, rom es wiladebi
tolia. aseve tolia wiladebi: 23 da 4
6.
12
24
36
23 4
6
30
tol wiladebze dakvirvebiT SeiZleba wiladis ZiriTadi Tvisebis aRmoCena: Tu wiladis mricxvelsa
da mniSvnels gavamravlebT (an gavyofT) erTsa
da imave 0-isagan gansxvavebul ricxvze, miviRebT
Tavdapirveli wiladis tol wilads. wiladis am
Tvisebas xSirad iyeneben sxvadasxva mniSvneliani
wiladebis Sedarebisas. magaliTad, Tu unda
davadginoT, romelia meti 34 Tu7
8, 34-is mricxvelica
da mniSvnelic unda gavamravloT 2-ze. miviRebT 68-s.
axla, vfiqrobT, aRaravis gauWirdeba imis mixvedra,
rom 78 metia 3
4 -ze.
savarjioSoebi
1. CawereT 35 -is toli ramodenime wiladi.
2. gamoiyeneT TvalsaCinoeba da amoiwereT toli wiladebi
31
3. ujraSi CawereT ricxvebi ise, rom swori tolo-bebi miiRoT:
810
= 4
23
= 6
68
= 3
4. daalageT zrdis mixedviT ricxvebi: 35, 210
; 815
da 1.
5. figuraTa mTliani raodenobis meramdened nawils Seadgenen mwvane figurebi?
6. SeadareT wiladebi: 57 da 3
7 ; 8
9 da 7
9; 1
5 da 1
7.
7. ra ricxvi SeiZleba eweros *-is nacvlad, rom utoloba swori iyos?
*4 < 3
8; 1
* > 1
6;
8. savaWro jixurSi miitanes 45 kilogrami karto-fili. dRis pirvel naxevarSi gaiyida am raode-nobis mesamedi. mTliani raodenobis ra nawili darCa gasayidi? ramdeni kilogrami kartofili darCa gasayidi?
9. pirvel dRes moaasfaltes 200 metri sigrZis gza, rac mTeli gzis meoTxedi nawilia. ra sigrZisaa mTliani gza?
10. giorgim Seagrova 150 lari. am Tanxis 23 iT man
SeiZina wignebi. darCenili Tanxis naxevari dax-
arja zooparkSi. ra Tanxa darCa giorgis?
32
albaT SeamCnevdiT, rom zomis erTeulebis gamoy-
enebisas yvelaze xSirad gvxvdeba wiladebi: 110
, 1100
, 1
1000. 1 metri kilometris 1
1000 nawilia, 20 sant-
ometri metris 20100
nawilia, 600 grami kilogramis 60
1000 nawilia, 40 TeTri laris 40
100 nawilia _ rogorc
wesi, am saxis mimarTebebi ufro sWirdebaT adamian-
ebs yoveldRiur cxovrebaSi. maTematikosebi SeTanx-
mdnen, rom aseTi saxis ricxvebi (anu, wiladebi, rom-
lebsac mniSvnelad aqvT 10, 100, 1000 da a.S.) Caweron
mokled, mniSvnelis gareSe. mTel nawils wiladis
mricxvelSi mocemuli ricxvisagan gamoyofen mZimiT.
amasTan, wiladis mricxvels weren ise, rom igi Sei-
cavdes imden cifrs, ramdeni nulic aris mniSvnelSi.
magaliTad, 4 25100
-is nacvlad weren 4,25 (ikiTxeba: 4
mTeli da 25 measedi). Tu mricxvelSi cifrebis ra-
odenoba naklebia mniSvnelSi nulebis raodenobaze,
maSin mricxvelis win weren nulebs. magaliTad, 2 1100
Caiwereba ase: 2,01 (ikiTxeba: 2 mTeli da 1 measedi).
wiladi 67100
Caiwereba ase: 0,67 (radgan, mTeli nawili
0–is tolia) da ikiTxeba: 0 mTeli da 67 measedi.
savarjioSoebi
1. figuris ra nawilia gaferadebuli? CawereT aTwiladis saxiT.
10 aTwiladebi
33
2. CawereT aTwiladis saxiT: 2 210
; 4 910
; 24 25100
; 2100
.
3. CawereT aTwiladebi: 5 mTeli da 45 measedi, 3
mTeli da 5 measedi, 0 mTeli da 35 measedi.
4. gamosaxeT larebSi: 3 lari da 5 TeTri, 20 TeTri,
80 TeTri, 5 lari da 75 TeTri.
5. metris meramdenedi nawili 5 sm, 20 sm, 85 sm?
CawereT aTwiladebis saxiT.
6. raime modelis meSveobiT gamosaxeT ricxvebi:
0,2; 0,4; 1; 1,4.
7. CawereT wiladis saxiT mocemuli aTwiladebi:
2,7; 31,4; 4,076.
8. gamosaxeT metrebSi da CawereT aTwiladis saxiT:
15 santimetri, 60 santimetri, 7 decimetri.
9. ramdeni kilogramia 20 grami, 300 grami, 5 kilo-
grami da 400 grami?.
10. SeadareT aTwiladebi: 0,55 da 0,12; 0,8 da 0,2;
4,25 da 7,15. 3.9 da 3.19999; 2 da 1.99999999
34
11 aTwiladebis Sekreba da gamokleba
vTqvaT, pijakis Sesakerad daixarja 3,14 metri qsovili, kabas ki sWirdeba - 2,83 metri (did samker-valo sawarmoebSi, sadac seriulad uSveben tansac-mels, sul aseTi sizustiTaa aRricxuli romeli saxis Casacmels ramdeni qsovili sWirdeba). rom gavigoT, sul ra masala daixarjeba pijakisa da kabis Sekervaze, unda SevkriboT ricxvebi - 3,14 da 2,83.
3,14 + 2,83 = 314100
+ 283100
= 597100
= 5,97
maSasadame, sul daixarjeba 5,97 metri qsovili (anu, 5 metri da 97 santimetri).
modiT, axla aseTi magaliTi ganvixiloT: erTi sazamTro iwonis 6,4 kilograms, meore ki - 5,28-s. ras iwonis orive sazamTro erTad?
6,4 +5,28 = 6410
+ 528100
. gavixsenoT, rom wiladis
ZiriTadi Tvisebis Tanaxmad 410
= 40100
. amis gaTval-
iswinebiT, 640100
+ 528100
= 1168100
= 11,68 .
amgvarad, davadgineT, rom ori sazamTro erTad iwonis 11,68 kilograms, rac aris 11 kilogrami da 680 grami.
uTuod SeamCnevdiT, rom aTwiladebis Sekreba
35
iseve SeiZleba moxdes `qveSmiweriT~, rogorc amas vakeTebT naturaluri ricxvebis SemTxvevisas, anu, Cveulebriv wiladad warmodgena yovelTvis aucile-beli ar aris.
3,142,835,97
+
6,405,2811,68
+
axla wesis saxe mivceT aTwiladebis Sekrebas:ori aTwiladis Sesakrebad saWiroa: 1) Sesakre-
bebSi gavaTanabroT mZimis Semdeg niSnebis ricxvi; 2) Sesakrebebi erTi meoris qveS ise davweroT, rom mZime mZimis qveS aRmoCndes; 3) miRebuli ricxvebi ise SevkriboT, rogorc naturaluri ricxvebi; 4) miRe-bul jamSi mZime davsvaT Sesakrebebis mZimeebis qveS.
aTwiladebis gamoklebis SemTxveva Sekrebis analogiuria.
114,8 _ 71,35 = 114,80 _ 71,35 = 114 80100
+ 7135100
= 4345100
= 43,45
`qveSmiweriT~ igive magaliTi ase Caiwereba: 114,8071,3543,45
+
savarjioSoebi
1. SekribeT aTwiladebi:
0,125+2,355; 3,25+7,128; 12,36+45,98.
2. gamoakeliT:
9,4-7,3; 11,1-2,8; 88,256-4,69.
3. 30 metri sigrZis mavTuls CamoaWres 4,75 metri.
36
ra sigrZis mavTuli darCa?
4. velosipedistis siCqarea 15 kilometri saaTSi, fexiT mosiarulisa ki 6,3 kilometri saaTSi. ram-deniT metia velosipedistis siCqare fexiT mo-siarulis siCqareze? ra siCqariT miuaxlovdebi-an isini erTmaneTs, Tu erTmaneTis Sesaxvedrad imoZraveben?
5. Turme, paraSutisti paraSutis gaxsnamde 4 wamis ganmavlobaSi vardeba simaRlidan. pirvel wamSi is gadis 4,9 metrs, yovel Semdeg wamSi ki - 9,8 metriT mets, vidre winaSi. ramden metrs gadis haerSi paraSutisti am 4 wamis ganmavlobaSi?
37
12 ricxvebis damrgvaleba
vTqvaT, erT skolaSi 87 meoTxeklaselia. calkeu-li SemTxvevebis garda, roca sizuste gansakuTreb-ulad mniSvnelovania, SegviZlia `davamrgvaloT~ ricxvi da vTqvaT, rom im skolaSi daaxloebiT 90 meoTxe klaselia. aseve iqcevian, roca amboben, rom Tbilisidan baTumamde manZili daaxloebiT 400 ki-lometria, rom lomi 190 kilogramamde iwonis, rom eifelis koSkis simaRle daaxloebiT 300 metria. ricxvebis damrgvaleba SeiZleba moxdes sxvadasxva sizustiT. magaliTad, davamrgvaloT 529 aseule-bamde niSnavs, rom vipovoT dasaxelebuli ricxvis uaxloesi aseuli. Tu aTeulebamde gvsurs damrgva-leba, maSin unda vipovoT 529-is uaxloesi aTeuli.
modiT, orive SemTxveva ganvixiloT detalu-rad. 529 aris 5 aseuli da kidev 29 erTeuli. 529 metia 5 aseulze da naklebia 6 aseulze, Tumca 5 aseuls aRemateba mxolod 29 erTeuliT, maSin roca 6 aseulamde rom SevavsoT, 529-s unda daematos 71 erTeuli. maSasadame, 529 ufro axloa 5 aseulTan, anu 500-Tan.
529 aris 52 aTeuli da kidev 9 erTeuli. 529 metia 52 aTeulze da naklebia 53 aTeulze. Tumca, 53 aT-eulamde Sesavsebad mxolod 1 erTeulia sakmarisi. amgvarad, 529-is uaxloesi aTeulia 53 aTeuli, anu 530.
calke ganvixiloT SemTxveva, roca dasamrgvale-beli ricxvi `zustad SuaSia~, magaliTad, rogorc 550, roca vamrgvalebT measedebamde sizustiT. aseT SemTxvevaSi, sayovelTao SeTanxmebis safuZvelze,
38
virCevT met ricxvs, anu 600. analogiurad, 75-is aT-eulebamde damrgvalebiT miiReba 80.
savarjioSoebi
1. daamrgvaleT aTeulebamde: 51, 78, 86, 91.
2. daamrgvaleT aseulebamde: 248, 456, 871, 918.
3. gTavazobT cxrils, romelSic mocemulia Tbili-sidan saqarTvelos zogierT qalaqamde manZile-bi. daamrgvaleT TiToeuli ricxvi aseulebamde sizustiT da upasuxeT SekiTxvebs:
Tbilisi-quTaisi 276 km
Tbilisi-Telavi 147 km
Tbilisi-axalcixe 207 km
Tbilisi-zugdidi 339 km
Tbilisi-ozurgeTi 321
romeli qalaqia Tbilisidan daaxloebiT 200 ki-lometris manZilze?
39
adamians xSirad sWirdeba raimes gazomva. gazom-vis gareSe igi ver SeZlebs saxlis aSenebas, tansac-mlis Sekervas, manqanis gakeTebas. Cven vzomavT iseT sidideebs, rogoricaa dro, temperatura, sigrZe, moculoba. sidide rom gazomon, amisaTvis mas adare-ben zomis raime erTeuls. magaliTad, Tu raime milis sigrZis gazomva gindaT, unda aiRoT cnobili sig-rZis mqone joxi, an Zafi da daakvirdeT, ramdenjer moTavsdeba joxi an Zafi milis sigrZis Sesabamis monakveTSi. Tu gindaT, rom Tqvens mier gazomili sididis Sesaxeb yvelas erTgvarovani warmodgenaa Seeqmnas, saWiroa isargebloT zomis sayovelTaod aRiarebuli erTeulebiT.
sazomTa metrul sistemaSi sigrZis ZiriTad er-Teulad miRebulia metri. danarCeni erTeulebi mi-iReba ZiriTadi erTeulisagan 10-jer, 100-jer, 1000-jer da a.S. gadidebiT an SemcirebiT. am erTeulebis saxelwodebebi iqmneba TavsarTebis gamoyenebiT. erTeulis 10-jer Semcirebisas iyeneben TavsarTs `deci~, 100-jer Semcirebisas _ `santi~, 1000-jer Semcirebisas _ `mili~. erTeulis 10-jer gadidebi-
13 zomis erTeulebi
40
sas iyeneben TavsarTs `deka~, 100-jer gadidebisas _ `heqto~, 1000-jer gadidebisas _ `kilo~. magaliTad, metris meaTed nawils ewodeba `decimetri~, meased nawils ki _ `santimetri~. metrze 1000-jer did si-dides ewodeba `kilometri~.
TiTqmis igive wesebi moqmedebs masis erTeulebis saxelwarmoqmnaSic. yvelaze gavrcelebuli erTeu-lia kilogrami. mis meaTased nawils ewodeba grami, xolo gramze 1000-jer nakleb erTeuls miligrami. didi masebis gasazomad iyeneben erTeulebs _ tona (1000 kilogrami) da centneri (100 kilogrami).
savarjioSoebi
1. ramdeni santimetria: 2 metri, 3 metri da 20 san-timetri, 5 decimetri.
2. ramdeni santimetria: 20 milimetri, 50 milime-tri, 125 milimetri.
3. ramdeni metria: 2 kilometri, 8 kilometri, 5 ki-lometri da 50 metri.
4. metris meramdenedi nawilia santimetri? milime-tri?
5. kilometris meramdenedi nawilia metri? 100 me-tri, decimetri?
6. ramdeni gramia: 2 kilogrami, 12 kilogrami? 3 ki-logrami da 400 grami?
7. kilogramis meramdenedi nawilia 1 grami? 10 grami? 200 grami?
8. iyides sami 250-gramiani da ramdenime 500-grami-ani qila mawoni. ramdeni 500-gramiani qila uy-idiaT, Tu sul SeiZines 2 kilogrami da 250 grami mawoni?
9. 1 kilogrami Saqari Rirs 2 lari. ra eRireba 400 grami Saqari?
41
erTi nayini Rirs 40 TeTri. arcerT Tqvengans ar gauWirdeba gamoTvalos, ra eRireba imave saxis 3 nayini (120 TeTri, anu 1 lari da 20 TeTri). modiT sqematurad gamovsaxoT es monacemebi.
nayinis raodenoba nayinis fasi
1 40 TeTri
3 120 TeTri
dakvirvebuli adamiani aucileblad amoicnobs Semdeg kanonzomierebas: ramdenjerac gaizarda nay-inis raodenoba, imdenjerve gaizarda gadasaxdeli Tanxac. 6 nayinis yidvas Tu moviwadinebT mogviwevs 240 TeTris, anu 2 larisa da 40 TeTris gadaxda. aseve martivad SeiZleba gamoviTvaloT nebismieri sxva raodenobis nayinis safasuric. or sidides Soris amgvar damokidebulebas proporciuli damokideb-uleba ewodeba. davalebis pirobis sqematuri gamo-saxva araerTi amocanis amoxsnaSi gvexmareba. modiT, ganvixiloT ramodenime maTgani.
davaleba 1: 10 kilogrami fqvilisagan cxveba 15 kilogrami puri. ramdeni fqvilia saWiro 60 kilo-grami puris gamosacxobad?
SevadginoT cxrili.
fqvilis wona gamomcxvari puris wona
10 kg 15 kg
? 60 kg
14 proporciuli damokidebuleba
42
radgan 60 4-jer metia 15-ze, Sesabamisad, 4-jer meti fqvilia saWiro. 10 unda gavamravloT 4-ze. mi-viRebT 40-s.
davaleba 2: 120 kilogrami qliavisagan amzadeben 9 kilogram Cirs. ramdeni Ciri miiReba 40 kilogrami qliavisagan?
qliavis wona Ciris wona
120 kg 9 kg
40 kg ?
Ciris dasamzadeblad aRebuli qliavis masa 3-jer Semcirda. Sesabamisad, 3-jerve Semcirdeba miRebuli Ciris masac. anu, miviRebT mxolod 3 kilogram qlia-vis Cirs.
gogonebi kargad erkvevian amgvar sakiTxebSi: vTqvaT, namcxvrebis receptebis wignSi weria, rom mTeli porcia namcxvris dasamzadeblad saWiroa 2 kg fqvili, 4 kvercxi da 1 Wiqa Saqari. Tu gaqvT mx-olod 1 kilogrami fqvili (anu, saWiro raodenobis mxolod naxevari), maSin Sesabamisad unda SeamciroT namcxvris sxva ingredientebic, anu, aiRoT 2 kvercxi da naxevari Wiqa Saqari da miiRebT igive namcxvris naxevar porcias.
raodenobasa da fass Soris proporciuli damokidebuleba ar aris yvela savaWro dawesebule-baSi. magaliTad bazarSi, 1 kona qinZs yidian 15 TeT-rad, magram 2 konas xSirad 20 TeTradac gviTmoben (nacvlad 30 TeTrisa).
43
savarjioSoebi
1. manqana 1 saaTSi gadis saSualod 80 kilometrs. ra manZils gaivlis manqana 4 saaTSi (aseTive siCqariT moZraobisas)?
2. 1 brigada asfalts agebs dReSi 200 metrze. ramdeni dRis muSaobaa saWiro 1 kilometri (1000 metri) sigrZis gzis dasagebad?
3. rveuli Rirs 20 TeTri. ramdeni rveuli mogivaT 1 larad da 20 TeTrad?
4. 1 tomara fqvilisagan cxveba 120 cali puri. ramdeni tomara fqvili unda SeiZinon sacxobSi, Tu maT daukveTes 600 cali puri?
5. oTaxis iatakis SesaRebad 2 qila saRebavia saWiro. ramdeni qila saRebavi dasWirdeba 5 daaxloebiT igive zomis oTaxis iataki SeRebvas?
6. 1 porcia namcxvars sWirdeba: 600 grami fqvili, 1 Wiqa Tafli, 3 Wiqa rZe. ra raodenobiT unda aviRo saWiro ingredientebi, rom gamovacxo 3 porcia namcxvari?
7. moifiqreT sidideebs Soris araproporciuli damokidebulebis ramodenime nimuSi.
44
15 ukuproporciuloba
vTqvaT, muSebis jgufi erTi dRis ganmavlobaSi asfalts agebs 5 kilometr sigrZis gzaze. ramden kilometr gzas moaasfaltebs es jgufi 4 dRis gan-mavlobaSi? cxadia, rom 4-jer mets, vidre 1 dRis ganmavlobaSi, anu 5 . 4 = 20 kilometrs.
axla aseTi amocana ganvixiloT: muSebis erTi jgufi gzis garkveuli monakveTis moasfaltebas an-domebs 8 saaTs. ra dro dasWirdeba igive monakveTis moasfaltebas, Tu erTdroulad imuSavebs muSebis 2 aseTive jgufi? gavyveT `logikis jaWvs~: Tu ori jgufi imuSavebs, samuSao ufro adre dasruldeba. amasTan, radgan raodenoba 2-jer gaizarda, saWiro dro zustad 2-jer Semcirdeba. anu, 2 jgufi saqmes daasrulebs 5 : 4 = 20 saaTSi. Tu muSaTa jgufebis raodenoba 4-mde gaizrdeba, maSin samuSao dasrul-deba 2 saaTSi. 8 jgufi samuSaos daasrulebs 1 saa-TSi.
modiT, monacemebi warmovadginoT cxrilis saxiT da davakvirdeT.
jgufebis raodenoba davalebis Sesasruleblad saWiro dro
1 8 saaTi2 4 saaTi4 2 saaTi8 1 saaTi
ramdenjerac izrdeba muSaTa jgufebis raodenoba,
imdenjer mcirdeba davalebis Sesasruleblad saWiro dro. sidideebs Soris amgvar damokidebulebas ukuproporciuli damokidebuleba ewodeba, TviTon
45
sidideebs ki _ ukuproporciuli sidideebi.
savarjioSoebi
1. erTi Tivis zvini erT cxens yofnis sakvebad daaxloebiT erTi Tve. ramden xans eyofa igive zomis Tivis zvini 3 cxens?
2. xuTi adamiani yurZnis mokrefas andomebs 3 dRes. ramden dReSi mokrefs igive raodenobis yurZens 15 kaci?
3. mSeneblebis erTi jgufi saxls aSenebs 8 TveSi. ramdeni jgufi iqneba saWiro, rom saxlis aSeneba dasruldes 2 TveSi?
4. avtomobili Tbilisidan quTaisSi Casvlas ando-mebs 4 saaTs, Tu is moZraobs 60km/sT siCqariT (anu, yovel saaTSi gadis 60 kilometrs). ra dro-Si Cavidoda avtomobili quTaisSi, 120 km/sT siC-qariT rom emoZrava?
5. cxrilSi mocemulia ukuproporciuli damokide-bulebis mqone sidideebi. SeavseT cxrilis gamo-tovebuli ujrebi
1 2 16
32 8 4 1
6. moifiqreT sidideebs Soris ukuproporciuli damokidebulebis magaliTebi.
46
yovel Tqvengans unaxavs, albaT, ruka. is dedami-wis calkeuli adgilebis erTgvar `models~ war-moadgens. dedamiwis zedapiris garkveuli nawilebis realur zomebs qaRaldze, cxadia, Semcirebulad gamosaxaven. magaliTad, 1000 metri sigrZis monakve-Ti zogierT rukaze 1 santimetri sigrZis monakveTiT aris gamosaxuli. 1000 metri = 100 000 santimetris. maSasadame, realuri zomebi 100 000-jer aris Sem-cirebuli. ricxv 1
100000-s rukis masStabs uwodeben. yovel rukazea miTiTebuli masStabi. is damokide-bulia rukis sasurvel zomaze da gamosasaxavi teri-toriis zomebze.
vTqvaT, rukaze A da B soflebs Soris manZili 8 santimetria. vipovoT manZili am soflebs Soris, Tu rukis masStabia 1
100000 . masStabis mniSvneloba migviTiTebs, rom rukaze 1 santimetrian monakveTs realur viTarebaSi Seesabameba 100 000 santimetri,
16 masStabi
47
anu 1 kilometri. proporciiT advilad daadgenT, rom 8 santimetrs Seesabameba 8 kilometri. maSasa-dame, or sofels Soris manZili 8 kilometri yo-fila.
savarjioSoebi
1. realur viTarebaSi 100000 santimetriani mona-kveTi rukaze 2 santimetrs ikavebs. ipoveT rukis masStabi.
2. Tbilissa da sagarejos Soris 36 kilometria, rukaze ki es manZili 2 santimetriani monakveTi-Taa gamosaxuli. ipoveT rukis masStabi.
3. or qalaqs Soris manZili 650 kilometria. ra sig-rZis iqneba rukaze am qalaqebs Soris sarkinigzo xazis aRmniSvneli monakveTi, Tu rukis masStabad
avirCevT 1
10000000 -s?
4. rukaze or qalaqs Soris manZili 5 santimetria. ra manZilia am qalaqebs Soris sinamdvileSi, Tu
rukis masStabia 110000000
?
5. gazomeT saklaso oTaxis sigrZe da sigane. daxa-zeT rveulSi am oTaxis gegma (oTaxis gamosaxeT marTkuTxedis saxiT, romlis zomebi 100-jeraa Semcirebuli).
48
yoveldRiur cxovrebaSi xSirad gvWirdeba sx-vadasxva sididis meased nawilze saubari an maTze moqmedebebi. TeTri laris measedi nawilia, santime-tri aseve measedi nawilia metris, farTobis erTeu-li kvadratuli metri sxva, aseve Zalian gavrcele-buli farTobis erTeulis - aris measedi nawilia. albaT aman ganapiroba, rom measedi nawilis aRsan-iSnad specialuri sityva SemoiRes - `procenti~. es sitya laTinuri warmomavlobisaa da niSnavs swored measeds. (SeiZleba istoriidan TqvenTvis cnobilia kidec, rom Zvel romSi 100-100 kaciani samxedro legionebi hyavdaT, romlebsac `centuriebs~ uwo-debdnen, maT meTaurs ki `centurioni~ erqva. dola-ris meased nawil `centi~ qvia. am sityvebis fuZec igive laTinuri sityvisganaa miRebuli). mokled:
17 procenti
49
raimes erT meased nawils procenti ewodeba. pro-
centis aRsaniSnad iyeneben amgvar simbolos: % . bevr Tqvengans eqneba uTuod nanaxi maRaziebis vi-trinebze warwera - `fasdakleba 20%-iT~. es niSnavs, rom maRaziaSi sadResaswaulod, an sxva raime mizez-iT saqonlis didi nawili iyideba pirvendel fasze iafad, amasTan fasdakleba Seadgens sawyisi Rireb-
ulebis 2100
nawils.
savarjioSoebi
1. ramdeni procentia 25100
; 60100
; 5100
?2. gamosaxeT wiladis da aTwiladis saxiT: 30%,
45%, 100%.3. ipoveT 50-is 10%, 20%, 25%.4. daxazeT kvadrati, romlis gverdi 10 ujris
tolia. gaaferadeT kvadratis 40%.5. romel figurazea iasamnisfrad gaferadebuli
am figuris 75%?
6. romeli figuraa gaferadebuli 80%_iT?
50
7. CawereT wiladebi procentis saxiT: 12
; 14
;
110
; 15
.
8. ricxvis ramden procents Seadgens misi
naxevari? meaTedi? meoTxedi?
9. skolis moswavleTa 55% vaJia. ramden procents
Seadgenen gogonebi?
10. biblioTekaSi wignebis 70% qarTulia. ramden
procents Seadgens ucxoenovani wignebi?
51
18 amocanebi procentebze
albaT iciT, rom saSobaod TiTqmis yvela maRa-
zia aiafebs saqonels. dResaswaulebis dros araer-
Ti maRaziis vitrinaze amoikiTxavT gaiafebis miman-
iSnebel warweras. modiT, warmovidginoT amgvari
situacia da ganvixiloT ramodenime amocana.
amocana #1. vTqvaT fex-
sacmeli Rirda 80 lari.
ra eRireba is 25%-iani ga-
iafebis Semdeg?
25% aris ricxvis 25100
nawili. wiladis ZiriTadi
Tvisebis Tanaxmad 25100 =
14 anu, fexsacmeli gaiafe-
bula pirvandeli Rirebulebis meoTxedi nawiliT.
80-is meoTxedi ki 20 laria. 80 _ 20 = 60. gamodis,
rom amJamad fexsacmlis Rirebuleba 60 laria.
amocana # 2. albaT iciT, rom Ciris mosaRebad
xils aSroben specialuri teqnologiiT. amasTanave,
cnobilia, rom gamoSrobisas xili kargavs Tavisi wo-
nis 80%-s. ramdeni kilogrami gargari unda gamovaS-
roT, rom miviRoT 2
kg. Ciri?
amocana amovxsnaT
etapobrivad.
52
pirveli daskvna: Tu xili gamoSrobisas kargavs
Tavisi wonis 80%-s, maSasadame gamoSrobis Semdeg mi-
iReba xilis Tavdapirveli wonis 20% (radgan 100% _
80% = 20%).
meore daskvna: 20% aris 15
nawili. gamodis, rom
sabolood miRebuli 2 kg Ciri xilis Tavdapirveli
wonis mxolod mexuTedi nawilia.
mesame daskvna: ricxvi, romlis mexuTedi nawilia
2, aris 10. ese igi, unda aviRoT 10 kg gargari, rom
gamoSrobis procesis Semdeg dagvrCes 2 kg Ciri.
savarjioSoebi
1. ricxvebi: x = 20 -is 10%, y =50 -is 10%, z =15 - is 10%, t = 5 -is 10%, daalageT zrdis mixedviT
2. samusiko skolis moswavleebis 70%-s gogonebi Seadgenen. moswavleTa ramden procents Sead-genen vaJebi?
3. 20 kilogram zRvis wyalSi 1 kilogrami marilia. ramden procents Seadgens zRvis wyalSi marilis Semcveloba?
4. erT maRaziaSi saSobaod mobiluri telefonebi gaaiafes 25%-iT. ra eRireba fasdaklebis Semdeg telefoni, romlis Tavdapirveli fasi iyo 120 lari?
5. puris fasma moimata 10%-iT. ra eRireba puri gaZ-virebis Semdeg, Tu adre is 50 TeTri Rirda?
6. adre puri Rirda 40 TeTri. amJamad igive puri Rirs 70 TeTri. ramdeni procentiT gaZvirda puri?
7. qalaqis mosaxleoba 10%-iT gaizarda. romeli
53
gamosaxuleba Seesabameba qalaqis mosaxleobis amJamindel raodenobas, Tu qalaqis mosaxleo-bis raodenobas gazrdamde iyo 1 milioni?
8. SeecadeT axsnaT Semdegi winadadebebis azri: ̀ nav-Tobis gadamuSavebiT miiReba 30% navTi~, `Warxa-li 20% Saqars Seicavs~.
9. xili gamoSrobisas (Cirad damzadebisas) kargavs Tavisi wonis 80%-s. ramdeni kilogrami Ciri mi-iReba 20 kilogrami xilisagan?
10. WurWelSi asxia zRvis wyali, romelic, rogorc cnobilia Seicavs marils. rogor Seicvleba wy-alSi marilis Semcvelobis procentuli maCvene-beli, Tu zRvis wyals davumatebT mtknar (umari-lo) wyals?
54
19 gantoleba
vTqvaT, sasworis erT TefSze devs erTkilo-gramiani Svidi sawoni, meoreze ki _ or kilogrami-ani erTi sawoni. ramden kilogramiani sagani unda davdoT meore TefSze, rom saswori gawonaswordes?
amgvari amocana martivad amoixsneba, Tu ucnob sidides aRvniSnavT x-iT. maSin, am TefSze sagnisa da sawonis wona iqneba (x + 2) kg. radgan amis Semdeg sas-wori gawonaswordeba, SegviZlia davweroT toloba:
x + 2 = 7saidanac gamomdinareobs, rom x = 5pasuxi: meore TefSze unda daidos xuTkilo-
gramiani sagani.ganxiluli amocanis amosaxsnelad Cven Sevadgi-
neT toloba, romelic Seicavs ucnob sidides. amgvar tolobas gantoleba ewodeba. maSasadame, x + 2 = 7 toloba gantolebaa. ricxvi 5 am gantolebis amonax-snia (zogjer amonaxsns gantolebis fesvs uwodeben). es niSnavs, rom Tu gantolebaSi x -is nacvlad Cavs-vamT 5-s, gantoleba WeSmarit ricxviT tolobad ga-daiqceva. zogierT gantolebas erTze meti amonaxsni aqvs, zogs ki SeiZleba ar hqondes amonaxsni.
sityva `gantoleba~ indoeli maTematikosebisagan momdinareobs. isini gantolebas uwodebdnen sity-vas _ `wonasworobaSi~. ucnobis x-iT aRniSvna ki SemoiRo saxelganTqmulma frangma maTematikosma rene dekartma 1637 wels.
55
savarjioSoebi
1. ricxvebidan 2, 5 da 4 romelia 2x _ 3 = 5 ganto-
lebis amonaxsni?
2. amoxseniT gantolebebi:
a) x + 10 = 45 b) 2x + 5 = 25 g) x . 2 = 50
3. SeadgineT gantoleba amocanisaTvis: gogonam
Caifiqra ricxvi, gasamkecebul Cafiqrebul ricx-
vs gamoaklo 20 da miiRo 70. ra ricxvi Caifiqra
gogonam?
4. SeadgineT gantoleba amocanisaTvis: erT qvevrSi
3-jer meti Rvinoa, vidre meoreSi. oriveSi er-
Tad ki 240 litri Rvinoa. ramdeni litri Rvinoa
TiToeul qvevrSi?
5. sacxobSi 1 cali xaWapuris fasia S lari. Ses-
aZlebelia xaWapuris SekveTa saxlSi mitaniT.
am SemTxvevaSi, damatebiT gadasaxdelia mitanis
fasi 6 lari, rac ar aris damokidebuli xaWa-
purebis raodenobaze. romeli gamosaxulebiT
SeiZleba imis dadgena Tu ra eRireba 3 xaWapuris
SekveTa saxlSi mitaniT?
3S + 6 9S 6S + 3 3S + 18
6. erTi maisuris Sekerva sawarmos ujdeba a lari,
masze suraTis daxatva ujdeba b lari. sawarmos
mogeba aris sxvaoba maisurebis gayidviT miRebul
56
Tanxasa da maisurebis damzadebaze daxarjul
Tanxas Soris. sawarmom daamzada 100 maisuri da
gayida TiToeuli c larad. romeli gamosaxuleba
SeiZleba gamoviyenoT sawarmos mogebis gamosaT-
vlelad?
100(c _ a) 100(c _ b) 100(c _ a _ b) 100(a + b _ c)
57
geometria erTerTi yvelaze Zveli mecnierebaa. es bunebrivicaa, radgan misi meSveobiT SesaZlebelia Cvens irgvliv arsebuli sxeulebis formebis Secno-ba, maTi gazomva da sxva manipulaciebi maTze. Tu Cveni garemomcveli realobis aRweras moisurvebT, aucileblad dagWirdebaT geometriuli terminebis da geometriis ZiriTadi faqtebis codna. magali-Tad, am naxatze gamosaxul nagebobebs geometriuli figuris `piramidis~ saxe aqvT:
20 geometriuli figurebi
geometriul figuras ̀ cilindrs~ moris saxe aqvs:
58
“konusi~ qviSis grovas, an nayin kornetos gavs:
yvelaze gavrcelebuli geometriuli figura `marTkuTxa paralelepipedia~. mas yuTis an aguris moyvaniloba aqvs:
ganasxvaveben prtyel da sivrcul geometriul figurebs. brtyeli figurebia marTkuTxedi, kvadra-ti, wre, samkuTxedi.
59
savarjioSoebi
1. suraTebze mocemulia mravalwaxnaga geometri-uli figurebis Slilebi. SeecadeT am Slilebis meSveobiT aagoT geometriuli figurebis mod-elebi.
2. plastelinisa da asanTis Rerebis meSveobiT daamzadeT geometriuli figurebis karkasebi.
60
21 wertili, wrfe, sibrtye
sawyisi geometriuli cnebebi, romlis meSveobiTac SeiZleba yvela geometriuli figuris gansazRvra - aris wertili, wrfe da sibrtye. geometriuli figurebi wertilebisagan Sedgeba. sibrtyeze warmodgenas gviqmnis fanjris minis an magidis zedapiri, qaRaldis furceli. Tumca, dasaxelebuli sagnebisagan gansxvavebiT, sibrtyes sazRvari ara aqvs. igi yvela mimarTulebiT `vrceldeba~ da usasruloa. wrfe ki - swori xazia, romelic aseve usasrulod grZeldeba orive mxridan.
d M
Cnaxazze gamosaxulia d wrfe. M wertili am wrfeze
mdebareobs, C wertili ki - ara. yovel or wertilze mxolod erTi wrfis gavlebaa SesaZlebeli.
AO
monakveTi wrfis nawilia. is orive mxridan SemosazRvrulia wertilebiT. AO monakveTia. A da O wertilebs monakveTis boloebi ewodebaT. wrfes, rogorc zemoT avRniSneT, monakveTisagan gansxvavebiT, ar aqvs boloebi. wrfis yoveli wertilidan am wrfeze orive mxares SeiZleba aigos nebismieri sigrZis monakveTebi. wrfis nawils, romelic mxolod erTi mxridan aris SemosazRvruli sxivi ewodeba.
61
B
naxazze sxivia gamosaxuli. B wertils sxivis saTaves uwodeben.
savarjioSoebi
1. gaavleT wrfe. aRniSneT A, B, C, D, E, F wertilebi ise, rom D, E, F ekuTvnodnen wrfes, A, B da C ki - ara. daasaxeleT miRebuli monakveTebi.
2. romel wertilebze gadis m wrfe? b wrfe? daasax-eleT wertilebi, romlebic ar ekuTvnis m wrfes. daasaxeleT wertilebi, romlebic ar ekuTvnis arc b wrfes da arc m-s.
m B C EbD
FA
3. daxazeT ori urTierTgadamkveTi wrfe. ramden nawilad yofen isini sibrtyes?
4. naxazze mocemuli paralelepipedi. daasax-eleT monakveTebi, romlebic erTmaneTis tol-ia. romel monakveTebs aqvT saerTo wertilebi? romlebs - ara?
A
B С
D
E
F
62
5. SeiZleba Tu ara or sxvadasxva wirs hqondes 2 an meti saerTo wertili? pasuxi axseniT.
6. daxazeT AX sxivi da masze saTavidan aageT er-Timeoris mimdevrobiT sami 2 sm-iani monakveTi. ramdeni aseTi monakveTis ageba SeiZleba am sx-ivze?
7. AB monakveTze aRniSnulia C da D wertilebi. daasa xeleT AB wrfeze mdebare yvela sxivi, romelTa saTave C da D wertilebSia. maTgan romeli Seicavs CD monakveTs?
8. CamoayalibeT raime debuleba wrfesTan da wertilebTan dakavSirebiT.
63
22 simetria
sityva `simetria~ berZnuli warmoSobisaa da niS-navs garkveuli wesrigis, kanonzomierebis arsebobas sagnebis an maTi nawilebis ganlagebaSi. simetrias xSirad vxvdebiT yoveldRiur cxovrebaSi. orna-mentebi dekoraciul qsovilebsa da Spalerebze, arq-iteqturuli morTulobani Senobebze da TviT Seno-bebis fasadebi, Cveulebriv, raime RerZis mimarT si-metriul figurebs warmoadgenen. bunebaSic xSirad gvxvdeba simetriuli formebi. magaliTad, xisa da yvavilebis foTlebs simetriuli forma aqvs Sua Re-ros mimarT. Tavisi tanis mimarT simetriuli forma aqvs pepelas frTebs da frTebis moxatulobas.
brtyeli simetriuli figuris modelis damzadeba arc Tu ise Znelia. unda gaavloT wrfe da qaRaldi gadakecoT am wrfeze. daxateT TqvenTvis sasurve-li figuris naxevari da makratliT gamoWeriT ise, rom gadakecvis xazi ar gaiWras. gaSaleT miRebuli figura. igi simetriulia gadakecvis xazis mimarT, gadakecvis wrfe ki am figuris simetriis RerZia.
64
zogierT figuras erTze meti simetriis RerZi aqvs. magaliTad, marTkuTxeds simetriis 2 RerZi aqvs, kvadrats _ 4, wrewirs ki _ uamravi.
savarjioSoebi
1. daxazeT figurebi: marTkuTxedi, kvadrati, wre da gaavleT am figurebis simetriis RerZebi.
2. gaavleT naxazze mocemuli figurebis simetriis RerZebi
65
3. romel naxazzea mocemuli RerZis mimarT sime-triuli figurebi?
4. SeqmeniT ramodenime RerZulad simetriuli or-namenti.
66
gadauWarbebeli naTqvamia, rom erT-erTi yvelaze gavrcelebuli geometriuli figuraa marTkuTxedi. maRaziis vitrinebs, saxlebis kedlebs, skolebisa Tu sacxovrebeli saxlebis ezoebs xSir SemTxvevaSi swored marTkuTxedebis forma aqvT. marTkuTxede-bs, romlebsac Cven amovicnobT Cvens garemomcvel ama Tu im nivTSi, erTmaneTisagan ansxvavebs albaT gverdebis sigrZe da, Sesabamisad, perimetri da far-Tobi.
modiT, jer perimetris arsi gavigoT. figuris perimetri gverdebis sigrZeTa jamia. Tu gvinda gavigoT, ra sigrZis Robea saWiro marTkuTxa formis nakveTis SemosaRobad, an ra sigrZis ficari sWird-eba suraTis CarCos damzadebas, swored perimetri unda gavigoT.
Tu marTkuTxedis ori mezobeli gverdidan erTs aRvniSnavT a asoTi, meores - b asoTi, perimetrs ki - p asoTi, maSin perimetris gamosaTvleli formula ase Caiwereba:
2 2 2( )p a b a b a b a b= + + + = + = +
kvadrati marTkuTxedis kerZo saxea. kvadrati aris iseTi marTkuTxedi, romlis yvela gverdi a-s tolia. kvadratis perimetris gamosaTvleli formulaa:
4p a= , sadac a kvadretis gverdis sigrZes aRniSnavs.
23 marTkuTxedi. marTkuTxedis perimetri.
67
savarjioSoebi
1. kvadratis gverdis sigrZe 5 santimetria. risi tolia kvadratis perimetri?
2. marTkuTxedis gverdebis sigrZeebia 12 sm da 18 sm. gamoTvaleT marTkuTxedis perimetri.
3. kvadratis perimetri 40 metria. risi tolia kvadratis gverdis sigrZe?
4. marTkuTxa nakveTis sigrZe 10 metria, sigane ki 7 metri. nakveTi mavTulbadiTaa SeRobili. ra sigrZisaa mavTulbade?
5. nakveTis sigrZe 20 metria. ra SeiZleba iyos nakveTis sigane, rom am nakveTis SemosazRvras eyos 60 metri sigrZis mavTulbade?
6. marTkuTxedis perimetri 20 santimetria. erTi gverdis sigrZe 8 santimetria. ipoveT meore gverdis sigrZe.
7. marTkuTxedis gverdebis sigrZeebia 5 sm da 7 sm. ipoveT im kvadratis gverdis sigrZe, romlis perimetric marTkuTxedis perimetris tolia.
8. romlis periametria didi, marTkuTxedis, romlis gverdebia 4 m da 6 m, Tu kvadratis, romlis gverdis sigrZec 5 metris tolia?
9. gamoTvaleT naxazze mocemuli figuris perimetri:
68
modiT, davakvirdeT naxazs. naxazze ujrebian furcelze marTkuTxedebia gamosaxuli (zogierTi maTgani kvadratia). pirvel marTkuTxeds ukavia 18 ujra, meores - 4 ujra, measmes - 4 jraze meti, ma-gram 6-ze naklebi, meoTxes - 4 ujra, mexuTes ki dakavebuli aqvs - 16 ujra.
#1
#2 #3
#4
#5
naxazidan Cans, rom yvelaze meti ujra daikava pirvelma marTkuTxedma. Tu gaviTvaliswinebT, rom farTobi, es aris figuris zoma, romelic gamosax-avs, ra sididis adgili uWiravs figuras zedapirze (zedapirze), SegviZlia davaskvnaT, rom pirveli fig-uris farTobi naxazze mocemul figuraTa farTobe-bze metia. ai meore da meoTxe figurebs ki erTnaeri farTobi aqvT.
zustad rom gavigoT yoveli marTkuTxedis far-Tobi, unda visargebloT farTobis sayovelTaod aRiarebuli erTeulebiT: kvadratuli santimetri, kvadratuli decimetri, kvadratuli metri, heq-tari. kvadratuli santimetri aris is adgili, ra-
24 marTkuTxedis farTobi
69
sac ikavebs 1 santimetri sigrZis gverdebis mqone kvadrati. kvadratuli metri ki Seesabameba im adg-ils, rasac ikavebs 1 metri sigrZis gverdebis mqone kvadrati. Semdeg am erTeulebiT zomaven nebismier marTkuTxeds.
ABCD kvadratis farTobi 1 kvadratuli santi-metria, KEFL marTkuTxedSi ki Tavsdeba 6 aseTive kvadrati. ese igi, KEFL marTkuTxedis farTobi 6 kvadratuli santimetria. am faqts maTematikuri simboloebiT ase Caweren:
26KEFLS sm= .Tu marTxkuTxedis sigrZea b , sigane ki - b , maSin
farTobi gamoiTvleba formuliT:
savarjioSoebi
1. marTkuTxedis sigrZe 10 santimetria, sigane ki - 5 santimetri. risi tolia marTkuTxedis farTobi?
2. kvadratis gverdis sigrZe 4 metria. risi to-lia kvadratis farTobi?
3. marTkuTxedi diagonalis gaswvriv gaWres. miRebuli samkuTxedis farTobi 10 kvadrat-uli santimetria. risi toli iyo gauWreli marTkuTxedis farTobi?
4. gamoTvaleT naxazze mocemuli ABC samkuTxe-dis farTobi:
A
B C
DK
E F
L
70
5. venaxi gaSenebulia nakveTze, romelsac marT-kuTxedis forma aqvs. nakveTis sigrZea 50 me-tri, sigane ki - 20 metri. ipoveT nakveTis farTobi.
6. or kvadrats toli perimetri aqvT. toli iqneba Tu ara maTi farTobebic?
7. or marTkuTxeds toli perimetrebi aqvT. toli iqneba Tu ara maTi farTobebic?
8. vTqvaT, gTavazoben or nakveTs. erTis zomebia 20 metri da 10 metri, meore nakveTs ki kva-dratis forma aqvs da misi gverdis sigrZea 15 metri. romel nakveTs airCevdiT? ganmarteT Tqveni arCevani.
A
B
C
71
rogorc ukve iciT, suraTze gamosaxulia marT-kuTxa paralelepipedi. igi SemosazRvrulia 6 waxnag-iT. zeda - EFGH da qveda - EFGH waxnagebs fuZeebs uwodeben, danarCen 4 waxnags – gverdiT waxnagebs. marTkuTxa paralelepipedis eqvsive waxnagi marT-kuTxedia. amasTanave, mopirdapire waxnagebi erTma-neTis tolia, rac imas niSnavs, rom maT farTobic erTnaeri aqvT. Tu gvinda paralelepipedis sruli zedapiris farTobis povna, unda vipovoT TiToeuli waxnagis farTobi da miRebuli Sedegebi SevkriboT. bevri Tqvengani mixvdeba, rom sul 3, erTmaneTisagan gansxvavebuli waxnagis farTobis povnac sakmarisia, radgan danarCeni sami maTi tolia. Tu SevajamebT naTqvams, miviRebT sruli xedapiris farTobis for-mulas:
a,b da c sadac a,b da c marTkuTxa paralelepipedis erTmaneTisagan gansxvavebuli gverdebis (wiboebis)sigrZeebia.
modiT warmovidginoT aseTi situacia: unda SeRe-boT oTaxis kedlebi, romlis zomebia (sigrZe, sigane
25 marTkuTxa paralelepipedis zedapiris farTobi
AB
C
G
FE
H
D
72
da simaRle) – 4m, 5m da 3 m. 1 qila saRebavi yofnis 9 kvadratuli metris SeRebvas. gamovTvaliT, ramdeni qila saRebavi dagWirdebaT.
modiT warmovidginoT aseTi situacia: unda SeReboT oTaxis kedlebi, romlis zomebia (sigrZe, sigane da simaRle) _ 4m, 5m da 3 m. 1 qila saRebavi yofnis 9 kvadratuli metris SeRebvas. gamovTvaliT, ramdeni qila saRebavi dagWirdebaT.
savarjioSoebi
1. radgan iatakisa da Weris SeRebva ar gvaqvs ganzraxuli, amitom marTkuTxa paralelepipe-dis formis mqone oTaxis mxolod gverdiTi zedapiris farTobi unda gamovTvaloT.
Sgv = 4 . 3 . 2 + 5 . 3 . 2 = 54 kvadratuli metri
2. Tu 1 qila saRebavi yofnis 9 kvadratuli me-tris SeRebvas, maSin 54 kvadratuli metri farTobis mqone zedapirs SeRebavs 54 : 9 = 6 qila.
3. ipoveT marTkuTxa paralelepipedis gverdiTi zedapiris farTobi, Tu paralelepipedis wi-boebis sigrZeebia 10sm, 2sm da 5sm.
4. ipoveT kubis sruli zedapiris farTobi, Tu kubis waxnagi 5 santimetris tolia.
5. marTkuTxa paralelepipedis simaRle 10sm-a, fuZis perimetri ki _ 16 sm. ipoveT paralel-epipedis gverdiTi zedapiris farTobi.
6. oTaxis zomebia 5m, 4m da simaRle _ 5m. oTaxis kedlebze unda gaakran specialuri filebi
73
(kafeli). 10 fila ikavebs 1 kvadratul metr farTobs. sul mcire ramdeni aseTi fila iqne-ba saWiro?
7. erTi didi dafisagan, romlis farTobic 18 kvadratuli metria, unda daamzadon yuTi (erTi mxridan Ria, anu, saxuravis gareSe). mox-erxdeba Tu ara es, Tu ki yuTis zomebi iqneba 2m, 3m da simaRle 2 m?
8. erTi didi dafisagan, romlis farTobic 18 kvadratuli metria, unda daamzadon yuTi (erTi mxridan Ria, anu, saxuravis gareSe). ise SearCieT yuTis zomebi, rom gamouyenebeli dagrCeT rac SeiZleba cota masala.
74
26 moculoba. marTkuTxa paralelepipedis moculoba
Tu erTsa da imave qilaSi CavasxamT wyals, zeTsa da Tafls da avwoniT, aRmovaCenT, rom samive niv-Tierebis masa gansxvavebuli gamova. toli mxolod maTi moculobebia. or WurWels erTnairi moculoba aqvs, Tu erTi savse WurWlidan meoreSi gadasxmuli siTxe masac gaavsebs.
Tu WurWelSi svel qviSas CavyriT da Semdeg am WurWels gadmovapirqvavebT, maSin qviSisagan miRe-bul figuras igive moculoba eqneba, rac WurWels.
figuris moculobas farTobis msgavsi Tvisebebi aqvs:• tol figurebs toli moculobebi aqvT.• Tu figura dayofilia nawilebad, maSin mTeli fig-
uris moculoba am nawilebis moculobaTa jamis tolia.moculoba izomeba kubur santimetrebSi, kubur
decimetrebSi, kubur metrebSi. 1 kubur decimetrs
75
litrs uwodeben (litri yofiT cxovrebaSi moculobis yvelaze gavrcelebuli erTeulia). ku-buri santimetri aris iseTi kubis moculoba, rom-lis wibo 1 santimetris tolia.
Tu suraTze gamosaxuli patara kubis moculoba 1 kuburi santimetria, maSin didi kubis moculoba iqneba 8 kuburi santimetri.
modiT, gamoviyvanoT marTkuTxa paralelepipe-dis moculobis gamoTvlis wesi. vTqvaT, marTkuTxa paralelepipedis sigrZea 3 metri, sigane - 2 metri, simaRle ki - 5 metri. davyoT es paralelepipedi iseT kubebad, romelTa wibo 1 metris tolia.
miviRebT 5 horizontalur fe-nas, sadac TiToeuli fena Sedgeba 2 . 3 = 6 kubikisagan. xuTive fenaze kubikebis raodenobaa 5 . 6 = 30. Se-sabamisad, mTeli paralelepipedis moculoba gamova 30 kuburi metri.
marTkuTxa paralelepipedis moculoba rom vipovoT, saWiroa misi sigrZe gavamravloT siganeze da simaRleze.
sxva sityvebiT, marTkuTxa para-lelepipedis moculoba misi 3 gan-
76
zomilebis namravlis tolia.formulis saxiT es ase Caiwereba: V = a . b . c.
savarjioSoebi
1. gamoTvaleT naxazebze mocemuli figurebis moc-ulobebi, Tu cnobilia, rom patara kubebis moc-uloba 1 kuburi santimetria.
2. marTkuTxa paralelepipedis sigrZe 13 santime-tria, sigane - 4 santimetri, simaRle ki - 25 san-timetri. ipoveT moculoba.
3. ipoveT im kubis moculoba, romlis wibo 2 me-tria.
4. ramdeni kuburi santimetria kubur decimetrSi? kubur metrSi?
5. oTaxis simaRle 3 metria, sigane - 5 metri, sigrZe - 6 metri. ramdeni kuburi metri haeria oTaxSi?
6. sabZeli gavsebulia TiviT. sabZels marTkuTxa paralelepipedis forma aqvs. misi sigrZe 10 me-tria, sigane - 6 metri, xolo simaRle - 4 metri. gansazRvreT sabZelSi moTavsebuli Tivis masa, Tu 10 kuburi metri Tiva iwonis 500 kilograms.
= 1 sm3
77
27 koordinatebi
Tqven SesaZloa iciT, rom kinoTeatris bileTi ase gamoiyureba:
kinoTeatrSi zustad rom miagnoT sakuTar adgils, ori ricxvis codnaa saWiro. es ricxvebia: rigis no-meri da rigSi adgilis nomeri. suraTze gamosaxuli bileTis mepatrone me-3 rigis me-15 adgilze mjdara. 3 da 15 kinoTeatrSi konkretuli adgilis `koor-dinatebia~. sibrtyeze Tu gavavlebT erTmaneTis gadamkveT horizontalur da vertikalur wrfeebs, SevZlebT mivuTiToT am sibrtyeze gamosaxuli nebi-smieri obieqti.
78
magaliTad, mwvane burTula mdebareobs me-4 vertikaluri da me-3 horizontaluri xazebis gada-kveTis wertilSi, wiTeli ki _ me-5 vertikaluri da pirveli horizontaluri xazis gadakveTis wertil-Si. mwvane burTulis koordinatebia 4 da 3 _ (4; 3), wiTelisa ki _ (5; 1).igive principiT aris SemoRebuli dedamiwis koor-dinatebi _ gavlebulia grZedebi da ganedebi (ekva-toris paraleluri da perpendikularuli wrewire-bi), romlis meSveobiTac zustad SeiZleba dedami-wis Tundac yvelaze mcire dasaxlebis mdebareobis miTiTeba.
savarjioSoebi
1. romeli figuris koordinatebia (5; 2)?
0 1
1
2
2
3
3
4
4
5
5
79
2. aRniSneT A wertili, koordinatiT (3; 3)
0 1
1
2
2
3
3
4
4
5
5
3. CawereT spilos Tvalebis, yurebis, kudis aRm-niSvneli wertilebis koordinatebi
y
x
80
4. romel geometriul figuras miiRebT, Tu Tanmim-devrobiT SeaerTebT wertilebs: (1; 1), (1; 4), (5; 4), (5; 1)?
5. qalaqi West Vancouver - mdebareobs B2 kvadratSi. romel kvadratSia qalaqi New Westminster?
81
informaciis warmodgenis erTerTi yvelaze gavr-celebuli formaa cxrili. cxrili iZleva monace-mebis struqturirebis saSualebas. kargad dastruq-turebul monacemebSi ki Zneli ar aris kanonzo-mierebebis SemCneva da maT safuZvelze daskvnebis gakeTeba. modiT, ganvixiloT cxrilis erTi nimuSi. warmodgenil cxrilSi aRnusxulia erTi klasis mo-swavleTa Sedegebi xuT saganSi.
moswavlis gvari da saxeli
mSobl
iuri
ena
maTem
atik
a
ucxo
ena
bune
bism
etyv
eleb
a
musi
ka
abaSiZe zurab 8 7 7 6 8
bolqvaZe nino 10 8 8 7 10
gogelia Tamaz 6 6 7 7 8
durgliSvili nino 8 7 8 8 8
ebrliZe arCil 9 9 9 10 10
maisuraZe daTo 9 10 10 10 9
qadeiSvili irakli 8 8 7 8 9
jafariZe Tamar 10 10 9 10 10
cxrilis saSualebiT martivad miviRebT CvenTvis saWiro informacias moswavleebis Sesaxeb. klasis 3 moswavles _ jafariZes, maisuraZes da ebraliZes aqvs saukeTeso Sedegebi. maTi qulebi meryeobs 9-sa
28 cxrili
82
da 10s Soris. amasTanave, yvelaze meti Sefaseba `10~ aqvs jafariZes. maTematikaSi yvelaze dabali qula aqvs gogelias, bunebismetyvelebaSi _ abaSiZes. musi-kaSi sagnebs Soris yvelaze kargi Sedegebia _ ar-cerT moswavles ara aqvs 8-ze dabali qula.
savarjioSoebi
1. cxrilSi mocemulia supermarketis bostneulis seqciis mier aprilSi realizebuli produqciis dasaxeleba, raodenoba, Sesasyidi da sareal-izacio fasebi
produqciis dasaxeleba
1-kg Sesasyidi fasi
1-kg sarealizacio
fasi
realizebuli produqciis raodenoba
kilogramebSi
kartofili 50 TeTri 65 TeTri 2000
kombosto 30 TeTri 40 TeTri 600
stafilo 60 TeTri 80 TeTri 300
Warxali 40 TeTri 60 TeTri 150
kitri 2 lari 2 l. 50 T. 90
pamidori 2l. 50 T. 3 lari 120
cxrilis mixedviT upasuxe kiTxvebs:a) ramdeni kilogrami Warxali gaiyida?b) ra mogeba aqvs supermarkets 1 kg stafiloSi?g) ramdenia kartofilis realizebiT miRebuli
mogeba?d) ramdenia pamidoris realizebiT miRebuli mogeba?
2. cxrilSi mocemulia informacia ofisis xarjebis Sesaxeb ori kvartlis (eqvsi Tvis) mixedviT.
83
Tve masala (larebSi)
sawvavi (larebSi)
ianvari 210 300
Tebervali 170 210
marti 190 330
aprili 180 400
maisi 150 450
ivnisi 120 500
cxrilis mixedviT upasuxeT kiTxvebs:a) ramdenia ofisis xarji aprilSi?b) ramdeni lariT metia ofisis xarji ianvarSi, vi-
dre TebervalSi?g) romeli TveSia ofisis xarji udidesi?d) ramdenia masalis xarji I kvartalSi?e) ramdenia masalis xarji II kvartalSiv) ramdenia masalaze danaxarji ori kvartlis gan-
mavlobaSi?
84
aris SemTxvevebi, roca informaciis `gaTvalsaC-inoeba~ xdeba saWiro. gansakuTrebiT maSin, roca informacia datvirTulia ricxvebiT da msmenel-ebis did nawils uWirs mosmenilis gaazreba da damaxsovreba. cnobili faqtia, rom TvaliT nanaxi ufro martivad aRiqmeba. swored informaciis aRq-mis gaadvilebis funqcia akisria diagramasac. qvemoT gTavazobT cxrils geografiis saxelmZRvanelodan, romelSic aRnusxulia dedamiwis yvela okeanis zeda-piris farTobebi milion kvadratul kilometrobiT:
okeanis saxelwodebafarTobi, milioni
kvadratuli kilometri
wynari 179,7
atlantis 93,4
indoeTis 74,9
CrdiloeT yinulovani 13
msoflio okeane (anu, mTlianad)
361
modiT, okeaneebiT dafaruli dedamiwis farTobi,
rogorc erTi mTliani, wris saxiT warmovadginoT, calke aRebuli okeaneebi ki ise gadavanawiloT wre-Si, rom maT Soris Tanafardoba warmovaCinoT. Tu gavixsenebT, rom wris gradusuli zoma 360 gradusia, martivad daadgenT, rom wris 1 graduss Seesabam-eba daaxloebiT 1 milioni kvadratuli metri. wynar okeanes daaxloebiT 180 gradusi uWiravs, anu wris
29 wriuli diagrama
85
naxevaria. atlantis okeane wris meoTxeds daikavebs. darCenil meoTxedze ki gadanawildeba indoeTisa da CrdiloeT yinulovani okeaneebi. sabolood, daax-loebiT aseT suraTs miviRebT:
wynari
atlantis
indoeTis
Crd.yinulovani
vfiqrob, es suraTi sakmaod naTel warmodgenas gviqmnis msoflio okeanis farTobis ganawilebaze okeaneebs Soris.
savarjioSoebi
1. erTi skolis moswavleebs Soris gogonebisa da biWebis gadanawileba wriul diagramaze ase gamoisaxeba:
vaJi
gogona
86
ra SegiZliaT TqvaT am skolaSi gogonebisa da bi-Webis Tanafardobis Sesaxeb?
2. diagramaze gamosaxulia zamTarSi gripiT daa-vadebuli adamianebis monacemebi maTi asakis mixedviT:
aRwereT diagrama,upasuxeT kiTxvebs: vin sWarbobda gripiT daavadebulTa Soris, ra asakis xalxi daavadda naklebad.
3. eqimebis rCevaa, rom saWmlis dRiuri norma oTxad ase gadanawildes: sauzme 25%, sadili - 45%, samx-ari - 15% da vaxSami - 15%. aageT saWmlis dRiuri normis ganawilebis diagrama (miaxloebiT).
4. gTavazobT cxrils, romelSic mocemulia gamok-iTxul moswavleebi ra saxis nayins aniWeben upiratesi~obas. am monacemebis mixedviT aageT wriuli diagrama
Sokoladis nayini 60
vanilis nayini 30
eskimo 30
xilis nayini 15
10 wlamde asakis
20 wlamde asakis
20-dan 40 wlamdeasakis
40-dan 60wlamdeasakis
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30 svetovani diagrama
modiT, gavagrZeloT informaciis `gaTvalsaC-inoebis~ sakiTxi da amjerad svetovani diagramis ageba SeviswavloT. davuSvaT, rom romelime sawar-mos sazogadoebasTan urTierTobis menejeri xarT da gindaT sazogadoebas miawodoT masalebi Tqveni sawarmos saqvelmoqmedo aqciebSi CarTulobis Ses-axeb. vTqvaT, ianvris TveSi saqvelmoqmedo aqciebi-saTvis sawarmom gaiRo 5 aTasi lari, TebervalSi _ 6 aTasi, martSi _ 8 aTasi, aprilSi _ 8 aTasi, maisSi ki _ 6 aTasi. Tu monacemebs davakvirdebiT, aSkaraa saqvelmoqmedo miznebisaTvis gaRebuli Tanxebis odenobis zrdis tendencia ianvar-aprilis TveebSi. roca gvinda, rom aqcenti erTdroulad gakeTdes Tanxis odenobazec da zrdazec, sasurvelia monace-mebi grafikulad warmovadginoT svetovani diagra-mis saxiT. ai ase:
dameTanxmebiT albaT, rom Tveebis mixedviT ricxvebis cal-calke dasaxelebas am saxis diagra-
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mis Cveneba jobs. diagramidan aSkarad Cans, rom yvelaze mcire Tanxa gadairicxa ianvarSi, martSi da aprilSi Tanabari raodenobis Tanxa gaiRo sawarmom. garkveul situaciebSi meti detalizacia arc aris saWiro.
svetovani diagrama Zalian efeqturia, roca kat-egorizirebuli monacemebis Sedarebaa saWiro, mag-aliTad, wina magaliTis analogiurad, amjerad ukve 2 sawarmos monacemebi Tu unda SevadaroT erTma-neTs Tveebis mixedviT.
pirveli sawarmo meore sawarmo
ianvari 5000 5000
Tebervali 6000 6000
marti 8000 7000
aprili 8000 4000
maisi 6000 5000
amgvari monacemebis safuZvelze agebuli sve-tovani diagrama iqneba:
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savarjioSoebi
1. erT-erT supermarketSi erTi dRis ganmavlobaSi misul myidvelTa raodenoba dRis 9 saaTidan 21 saaTamde, or-or saaTian intervalebSi, warmod-genilia diagramis saxiT.
2. romel saaTebSi aris am monacemebis mixedviT yvelaze meti da yvelaze naklebi myidveli? daax-loebiT ramdeni momxmarebeli midis supermar-ketSi dRis ganmavlobaSi?
3. cxrilSi warmodgenilia moswavleTa gamokiTxvis Sedegebi, Tu romelia maTi sayvareli nawarmoe-bis Janri.
nawarmoebis Janri moswavleTa raodenoba
deteqtivi 19
samecniero fantastika 8
sportuli 25
saTavgadasavlo 38
am monacemebis safuZvelze aageT svetovani dia-grama.
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31 monacemTa saSualo
bevr Tqvengans uTuod eqneba mosmenili amgvari teqstebi: `sawarmos yovelTviuri mogeba saSualod 1 milioni dolaria~, `puris fasma saSualod 10 Te-TriT moimata~, `dedamiwaze temperaturam moimata saSualod 2 gradusiT~. savaraudod sawarmos mogeba yovel TveSi erTidaigive ricxvi ar iqneba, ase TiTq-mis arasodes xdeba. arc puri gaZvirdeboda yvela maRaziaSi zustad 10 TeTriT _ albaT iarsebebs maRaziebi, sadac gaZvirebam 20 TeTri Seadgina, an iseTi maRaziebi, sadac puri mxolod 5 TeTriT gaZ-virda. temperatura yvela yveyanaSi erTnairad ar moimatebda, zogan SesaZloa matebam 3 an 4 gradu-si Seadgina, zogan ki saerTod ar momatebula. maS, saidan modis es `gasaSualoebuli~ monacemebi? aris situaciebi, roca informaciis detalizacia saWiro ar aris. raRac procesis aRwerisas `tendenciis~ codnac sakmarisia _ anu, CvenTvis saintereso pro-cesis Sesaxeb iseTi miaxloebiTi informaciis flo-baa sakmarisi, romelic zogad warmodgenas Segviqm-nis procesze masSi metismetad CaRrmavebis gareSe. aseTi saxis `miaxloebiTi mniSvnelobaa~ saSualo ariTmetikuli. Tu lias gadasaxadi satelefono momsaxureobisaTvis dekemberSi Seadgenda 4 lars TeTrs, ianvarSi 8 lars, xolo TebervalSi 6 lars, maSin SegviZlia vTqvaT, rom zamTris Tveebis ganma-vlobaSi lias yovelTviuri satelefono gadasaxadi Seadgenda saSualod (4 + 8 + 6) : 3 = 18 : 3 = 6 lars. saSualo ariTmetikuli, an monacemTa saSu-alo gamoiTvleba amgvarad: unda Seikribos monace-mebi da jami gaiyos monacemebis raodenobaze.
modiT axla aseTi amocana ganvixiloT: vTqvaT gvainteresebs imis dadgena, Tu rogoria aWaruli
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xaWapuris tipiuri fasi qalaqSi. amisaTvis ase unda moviqceT: davrekoT 8 an 10 saxaWapureSi da gavigoT fasebi (mag., 7,35 lari, 7 lari, 6,75 lari, 6,50 lari, 6,10 lari, 6 lari, 6 lari, 5,50 lari). unda SevkriboT maT mier dasaxelebuli fasebi da jami gavyoT mona-cemebis raodenobaze (am amocanis SemTxvevaSi 8-ze). (7,35 + 7 + 6,75 + 6,50 + 6,10 + 6 + 6 +5,50) : 8 = 6,40 ese igi, SegviZlia vTqvaT, rom qalaqSi aWaruli xaWa-puris saSualo fasi 6 lari da 40 TeTria.
savarjioSoebi
1. gamoTvaleT ricxvebis saSualo ariTmetikuli: 5, 6, 8, 7, 8, 8.
2. ras udris sami ricxvis saSualo ariTmetikuli, Tu am ricxvebis jami 21-is tolia?
3. ras udris 5 ricxvis saSualo ariTmetikuli, Tu am ricxvebis jami 50-is tolia?
4. risi tolia 5 ricxvis jami, Tu am ricxvebis sa-Sualo ariTmetikuli 10-is tolia?
5. mSeneblis xelfasi 10 Tvis ganmavlobaSi saSua-lod 800 lars Seadgenda. sul ra Tanxa aiRo mSenebelma am 10 Tvis ganmavlobaSi?
6. sami sazamTros saSualo masa 10 kilogramia. erT-erTi sazamTro 12 kilograms iwonis. ras udris danarCeni oris saSualo masa?
7. SeiZleba Tu ara, rom ricxviTi monacemebis sa-Sualo iyos udides monacemze meti? pasuxi daas-abuTeT.
8. yovelTviurad, mobiluri telefoniT sarge-
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blobis pirveli 200 wuTi ninos eZleva ufasod. yoveli momdevno wuTisTvis igi ixdis 5 TeTrs. cxrilSi mocemulia ninos satelefono saubrebis xangrZlivobebi 3 Tvis ganmavlobaSi.
Tve wuTebis raodenoba
ianvari 230 wuTi
Tebervali 300 wuTi
marti 250 wuTiam 3 Tvis ganmavlobaSi, TveSi saSualod ra Tanxas
ixdida nino? 9. fermers surs daadginos axali sasuqis efeq-
tianoba. amisaTvis man SearCia 5 nergi da gazoma maTi simaRleebi. amis Semdeg man dauyara am mcena-reebs sasuqi da 2 kviris Semdeg gazoma maTi si-maRleebi. svetovani diagrama gviCvenebs nergebis simaRleebs sasuqis gamoyenebamde da sasuqis gam-oyenebis Semdeg.
am diagramis mixedviT, risi tolia mcenareebis
25
20
15
10
5
0I mcenare II mcenare III mcenare IV mcenare V mcenare
sasuqis dayramde
sasuqis dayris Semdeg
saSualo zrda?
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32 kompiuteruli programa „geogebra”
kompiuteri sul ufro did adgils ikavebs yove-
li adamianis cxovrebasa da saqmianobaSi. misi meSve-
obiT SesaZlebelia informaciis moZieba, damuSaveba,
komunikacia, teqsturi Tu media-failebis Seqmna da
redaqtireba. arsebobs saganmanaTleblo kompiut-
eruli programebic. maTi meSveobiT SesaZlebelia
ukeT swavla. aseTi programaa ̀ geogebra~. ̀ geogebra~
gaZlevT saSualebas ganaxorcieloT geometriuli
agebebi kompiuterSi, aRmoaCinoT kanonzomierebebi
`geometriul movlenebze~ dakvirvebisas, Camoayali-
boT hipoTezebi, moaxdinoT TqvenTvis ukve cnobili
faqtebis vizualizacia. geometriuli figurebis
daxazvis Semdeg yvela sawyisi monacemi eqvemdebareba
cvlilebas - SesaZlebelia wertilebis gadaadgileba
mausiT, monakveTebis zomebis Secvla da sxva. Tqven
SeZlebT daadginoT, magaliTad, yovelTvis izrdeba
Tu ara marTkuTxedis farTobi, roca izrdeba misi
perimetri; erTnairi perimetris marTkuTxedebs
Soris romels aqvs yvelaze didi farTobi.
`geogebras~ moxmareba Zalzed advilia. kompiut-
eruli programis gaSvebis Semdeg ekranze Cndeba
amgvari gamosaxuleba:
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naxazebis Sesasrulebel velSi daxazavT TqvenT-vis sasurvel figuras. Tu wertilis dasma gindaT, instrumentebis panelze unda CamoSaloT Sesabamisi aRniSvnis mqone meniu da mausiT daawkapoT werti-lis naxatze (rogorc qveda suraTzea naCvenebi).
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Tu ramdenime wertili moniSneT da maTi SeerTe-ba gindaT monakveTebiT, maSin is meniu unda CamoSa-loT, romelsac monakveTi axatia.
modiT, davxazoT samkuTxedi.
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kompiuteruli programis qveda marjvena kuTxe-Si aris brZanebebis nusxa, saidanac unda airCioT `perimetri~.
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naxazis gverdze, sadac wertilebis koordinate-bi da samkuTxedis gverdebis sigrZeebia mocemuli, dafiqsirdeba samkuTxedis perimetric.
instrumentebis panelze mausiT daawkapeT is-rian niSnulze. Semdeg gadaaadgileT samkuTxedis wveroebi da SecaleT samkuTxedis gverdebis sig-rZeebi. dainaxavT, rom imavdroulad Sesabamisad Seicvleba samkuTxedis perimetric.
ekaterine korZaZe - maTematikuri wignierebis eqsperti.
daamTavra Tbilisis ivane javaxiS-vilis saxelobis saxelmwifo univer-sitetis meqanika-maTematikis fakul- teti. wlebis ganmavlobaSi iyo skoleb-is kompiuterizaciis programa „irmis naxtomi“-s maTematikisa da bunebisme-
tyvelebis maswavlebelTa sagnobrivi qselis koordina-tori. monawileobda zogadsaganmanaTleblo skolebis maTematikis erovnuli saswavlo gegmis SemuSavebaSi. aris maTematikis swavlebis araerTi publikaciis Tanaavtori. muSaobs ilias saxelmwifo universitetSi, ganaTlebisa da mecnierebis saministroSi maTematikis eqspertad.
ekaterine korZaZe
maTematikuri
wigniereba
