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Столярова Виктория Дмитриевна. Обучение младших школьников решению задач с геометрическими величинами

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opfaHr.Baqnfi rr YlvTer'i ocHoBHofi rrrKoJrBr TaJ{xe AoJrxHBr 6BITB 3HaKoM6r c KflaccuQl,I(arlHefi aJrr npaBHJrbHoro nocTpoeHur npone.4eBTl4K,r, arrytlJrr,Barl]M
KpbITAD) 3HaHr,ri, paqfioHaIBHOrO nCnOrB3OBaHHr )^re6HOrO BpeMeHU
KOMJrer{Ulr, rByqeHvr4
C repuunotr
v pemelJ]du
(<3aAaqa>) JIIoAI,I
xrz3Hrl, KaK Ha 6brroBoM,
3ro uoryr
BOCrrrrTaHue rroApacTi
rar
r,r
nplr
((or-
O3Ha-
lpofpaMMofi.
nocrorHHo crarKr.rBarorcs
s
nonceAnenHofi
na npotpeccr{oHaJrbHoM ypoane. KaNgouy r.r: nac
vrfii
rrHbre npo6:reurr, Korop;re 3aqacryro MBr Ha3hrBaeM
6rrt
o6qerocy4apcrBeHHr,re sagaun (ocnoenne KocMoca,
fiprrxoAr,rrc.r pernar; Te
3aAarraMra.
3a.!iaq, onpeaeneHHsx
r.r
orqero rroKoreHr,rJr, ooopoHa cTpaHBr
,qelreHHBrx KoJrreKTr,rBoB
u rpynn (coopyu<enue o6rerror,
ycraHoBJreHve cBg3ei u 3aBr.rcr,rMocrei
pe,q or,qeJrlHbrMH Jrr,r.+rocrrvr
r.r
r.r
T.n.r, 3aAaqn onpe-
BsrnycK Jrr,rreparypBr,
Ap.), a raxxe 3a.qaqr,r, KoropBre cro{T rre-
[4].
K per:renuro pasno[JlaHoBbx xIBHeHHhrx 3aaaq TIKoJIbHItKoB HaqI,IHaK)T foT08r.116
yxe
B
Pe[as
MnaArrreM rKoJrBHoM Bo3pacre B npoqecce
gaAaqrl,
]qaquecr npuo6peraror
K)T rr crrcreMarr,r3rrpyror
HoBbre
r,
o6yrexrr varerr,rarlrre.
.r
3aKpeflnrror, yruy6:rx-
pr(e rrMerorquecr MareMarr{rrecKrre snaxru. O6yuarorqa.n
EyHKrInr TeKCTOBbX 34,{aq MOXeT
ObrTb npOAeMOHCTpr,rpOBarra 3aAarrzrMl,l, B KOTO-
prx [18]:
- pacKpbrBaercq KoHxperr;rft cMBrcr aput[ueruuecrrax neficrnufi;
- BBoArrcr
parlHoHaJrEHhre npraeMbr BbrrrncJreHHfi
u coorBercrBlrorque
HM
rlpaBvnai
- BEITIoJIH.f,K)Tc.t ra6nrnHrre
- HCnOnb3)mTCr
I Itl nHeta6ru.tHlIe
COOTHOrrreHr.rr
sbr.IncJleHut;
Mexny pa3JrnqHBMr.r
eAr.rHr.rrlaMr.r
rnMepeHr,rt
BEJIIIqI-IH I{ T.A.
_boree Tofo, cyqecTB)'ro II,Ie MexlpeaMeTHhre cB.r3rr HaqajrBHofo K]?ca Ma-
TeMarriKu c apyflrMra yre6urrur.r ar,rcqunnuH{rMr,r no3BoJrrror orpa6oraru yuenue
qr,rrarb, noBTopHTb rpaMMaruqecru,re nopuu (npanour,rcaHr.re cnoBapHbx cnoB,
npr,rMer{eHrre
r-r3fraeuux nparun opQorpaQuu, npaBr{n coKparrreHnr croa n r.g.).
10
3a.Eaqll
sltrrornffor
pa3Br.rBarcq,,ro
Qyuxquro rro o?Hor]reHa]o r< yqaqrrMc,
MJIa.roD( x;l.accoy
[4g,c.774]. B npoqecce perueHrfi Ter(croBBrx
3aAaq
o,rpa6arhr_
BaIOTC' J,,Ir{eHItJI:
- BBrrroJrrurrr onepaur,ru
arranrusa
3;iuvl{:
v cuarea, a6crparrap oBalar, uroHKperri_
- npoBolr,rTB pacc)D(AeHr.r, no
aHar.rofl.rH;
- o6o6rqart cnoco6u pelrenar
rr,rnoBrrx
_ H:xoAr4Tb
nprr3HaKr,r
a6crparrntx
3a,qaq;
MareMarrrqecKr,rx
noHrrrjli B peaJr;Hbrx
ra, cJIeAoBareJ]BHo, ycrauaBJlxBarr
cB.f,36 TeoperiqecKl.rx 3HaHLr,
B 06JIACTII MATEMATIiKI, C XIZ3HbTO.
o6remax
Eonruoe
3HaqeHI.Ie IlMeer perreHlze
3aAaq
L
B
BocrlltralMlt nEqHocrI,I
IUI,IXCT:
- nprrB,Baerc, Kyn'Typa MBrrrneH,u,
o6rqena.r ri
Mbrcnefi;
-
osrpaoaTBrraercr
1^Ia_
BBrpa(eHrr.f, co6crBeHH'rx
)a,reHHe eJ,ryrraT6 MHeHrze yt{uren
Ir
o.{HOKJIaCCHI.IKOB,
aHaJII.t3I,tpoBaTs V OrIeHI.rBaTE
ycJIbt4aHHOei
- nupa6aruaaercr a(KyparHocrb
B BeA
ennu aanuceft,
- pacrrJr.rprerc, Kpyr_o3op;
-
BOCIII4
TBIBaeTc{ TIyBCTBO KOJIIeKTI_IBI,3Ma
CpeAI.t ml(on6Hl{Koa I.I T.A.
floerolry naxHo, .rro6tr
ftr{TeJrl lrMen r:ry6orr.re [peAcraBJreHx, o
reK_
croBofr sa,qaqe, o e€ crpyxrype, yMeJr
perrarB TaKr,re 3a.qaqH pa3nuqHBrMr.r
cl]oco_
6aua u nepe4aea,r 3Tr.r 3HaHrlrr
cBorzM )^reHa(:M.
flpo6reua perxeHrz, u qncro MareMaruqecKrlx
qr{x nepeA
qeJroBe*oM B
rpoqecce
3a.qaq, r.r 3aAarr, Bo3Hr,rKaro_
efo
npor.r3BOAC"BeHHOfi
uru o'rroBofi ae.f,TeJrBgocrr.r, usl'ltaLerca rlg.uasHa.
O.quaro .4o Hacrorqero BpeMeHa
Her o6Irlerrpr,rH{Tofi TpaKToBKu caMoro norurrr,r, (€agaqo).
B ulrporou cMBIcJIe gJroBa rroA 3a_
Aaqed noHl.rMaerc.{ HeKoropa, cLrryar:|Ilr,
Tpe6yoqa{ EccneAoBafil.I,r x pa3peue_
nzl 're:roaerov (r,ua penarorqefi cucrevofi)
[19,c.171].
Orgelrno
cro_f,T MareMarl,rqecKlle
3a,qaqu, peltteHl{e
cneqaaJIBHBIMI.I MareMarr4 qecKl.rMrz
cpeAcrBaMI,I
I{
Koroprlx Aocrl4faerc,
MeroAaMx. Cpe,qz nrrx atr.qe_
11
Jrrror 3aAaqr{ Ha) rHbre (HanpnMep, TeopeMa @epua, npo6nelra fonr6axa z gp.),
perxeHne roropux cnoco6crByer pa3Brrrr,rn MareMarr.r([ u ee npruoxenr,rfi,
u
sa_
gavu ;ive6une, Korop;re ctryxar am
Qopvuporannr neo6xogr,rnrurx MareMarr,rqe_
crrx
ananufi, yl"l.esufi, a HaBbrKoB
y
pa3ubrx fpyfirr o6yvaeuux (uonbHr,rKoB.
clry rareret KypcoB, cry.qeHTor u gp,) u HanpaBneHBr Ha r€MeHeHrre KaqecrB
nravnocrr.r o6yuaeuoro (ne:lraa -:naro, ne yl,re:r - yuero u r.n,).
Texcrosa.f, 3a4aqa
r3hrKe c rpe6oBaHrleM
noHeHTa
-
ofitcaHr.re uexoropofi cLrryar\rrrl, Ha esrecrBeHHoM
aarl (onutrecrBeHnfo xapaKTeprrcrriKy raxoro_ru6o rou_
grofi curya\vv, ycraHoBr.trL
HoIxeHI{t Me)K.e/ eg KoMnoHeHTaMI.I
flpzlepxnaarcr
]FraJrrr...ue
vltu orc)"TcrBr4e HcKoroporo or_
uru onpenenlrrB
BIrA
ororo orHorxeHI,tJL
coBpeMeHHofi repMliHolrorr,rn, Mo)nHo cKa3arb,
croB€ur 3agaqa npeAcraBJrrer
co6oi clorecnyro
Mo,qeJr6
qro
reK_
crrryaqulr, ,BJreHr,rr, co-
6urur, npoqecca u r.n. Kax r nro6ofi uoge.nn, g reKcrosofi
3a,qaqe oru{cbrBaerc,
ne nce co6urrae rJru r.Brenve, a JII,IITII ero KoJII{qecrBeHHbte n
$ynxqr.ronalrune
xapaKTepr,rcTr,rKH.
Marevarnqecxax 3aAa.ra
-
oro cs.f,saHHui naxonrqecrulfi paecKa3, B Koro_
poM BBeAeHbr 3HaqeHr.tr HeKoTopLx BeflnrluH LI npe,qnaraeTc, oTbrcKaTb
apyrne
HeI,BBeCTH6Ie 3HarIeHI4t BeJII-itIHH, 3aBIrCI,IMhIe OT
,qaHHbIx I,I CBr3aHH6Ie C
H'MI]
onpeAeneHHErMr.l cooTHorxeHrxrMx, yKa3aHHBtMr,r B ycJroBn}t,
Tarr,ru o6pa:otr, qerKoro,. eAzHoro onpeAeneHr.r, TeKcroBot 3aAaqr,r
B
Hacro{rqee BpeMJr Her, BBoAr.rrcr Jrntrb ee noHrrr,re.
B ra-{ecrse npr,rMepa, paccMorpr.rM HeKoropbre, cucreuarlrslrpoaanuxe Or_
sr,rHHr.rKoBofi M.B. [34]. Texcroaa.x :a.qaqa sro:
-
.
TpeooBaHr,re nJrr.r Bonpoc,
caxnrx yclorui (aaroprr:
.
Koropbri ny)(Iaerc, B orBere, c yqeroM
@pr.r.4uan
JI.M., Tlpeqxufi E.H.);
Borlpoc, oTBer Ha Koropbrfi Mo)Ker
6trrt
ap[(pMersrrecrr.rx.qeficrtr.ri (aeropn: Mopo M.2.,
. or
onr4_
nafigen 3a cqer BbrflorHeHr,ur
Ibrlrrano A.M.);
caHlre HeKoropoi clrryaqull Ha ecrecrBeHHoM g3Hxe c me6eear {eM
AaTb Kor[qecrBeHrq,'rc xaparcep]rcrr,rKy KoMnoHeHTa onlcaHHofi c],1rryarlLlu, Ltnrr
ycTaHoBETb HaJrZqUe OTHOIIIeHI.I' MeXAy ee KoMnoHeHTitMr,I, I,tJ
l
OnpeAenIrTS BI.I,4
3roro orHorxeHr.t{ (anropu: Croinona JI.[I., llurrrxa;ro A.M.).
B uaqaltnoi
ruu M,B.)
rrrKoJre reKcroBbre 3a,qaqu Ha3brBaror cK))KerHbrMH (Eoraauo_
B cBt3lz c reM,
qro
onlrcllBanr pezrnllble xLl3HeHHr,rs er.rryarJrll,
[poqeccbr, flBrreHrx, HanpnMep, TaKxe Kax: KynJrro _ npo,qaxy, flpor,t3BoauTej]b_
oHr,r
Hocr6 TpyAa, ABr.rxeHr4e u r.n, C npegcraanenuoi ToqKu 3peHr,lf, TeKcroBarr 3aAaqa
-
gro croBecHaq
MoAeJrB cr.rryaqllu
(releuur, npoqecca). flpu arou
B TercroBoft
34.{aqe (KaK I,I B MoAeJrI{) on[cblBaerct He Bct cl4Tyauuq c ue:IB.rafiIIluIr.rr,r nogpo6_
Hocr-lMl.t, a
nulx;
HeKoropble e€ croponu, B ocHoBHoM, KoJrI{qecrBeHHBIe xapax_
TEDI,ICTI,IKI-I.
Crpyrrypa rerccroeoft sa4avu
,rlro6a.r
rexcrosar
3a.qaqa
cocrolrr u: gayx vacrei: yctroBr4, ulpe6oaanlrx
(aonpoca). Vuclogue 3HaqeHHr BeJrr{qr{H r.r cyqecrByrorq}re MexAy HlrMH
3aBr.r_
cxMocrlr, T.e. KoJrr4qecrBeHHble
eaga_
.iu u orHoueHut Mexry
r.r
KarIecrBeHHEre xapaxrepl,IcrnKlr
Hr.rMr.r, Ha3brBaror
yc;roaura coo6qarorcr cBeAeHr.ir o6
pr,B)ilolllHx galruxe
ycJroBuev
o6rerrax
r,r
o6rerru, o6 u:secrHrrx L
:ruur,rn, o6 orHorrreHr,rrx Mex(.4y Hr,rMri.
(ullr
o6:rerror
ycronualru) :agavr.r. B
HeKoropbx Benrrqr4Hax, xapaKTe_
HeI,BBecrHhIx 3HaqeHri{x 3TIlx Be-
B sagave o6uqno ue
oAHo, a EecKoJrbKo
ycnoBr.ri, KoropBre Ha3bnarcT sreMeHrapnrnau[9,c. I 82].
Tpe6oaalrzr 3ag'a'tu
-
3To yKa3aHxe
rofo, qro nyNuo nafiru. Onu uoryr
obrr6 ceopMyJrr{poBaHbr KaK B BonpocuTeJr;Hoit, rax
Qopr',re,
rx raKxe Mo)r(er 6urr
us
nogecrnosarelluoi
secxonrrco. Beluvrauy, 3HaqeHr,re roropoft rpe6y_
ercfl Hai4Tti, Ha3bBaror ucrouofi genuqugofi, a qI4croBBIe 3HaqeHrzr ucKoMhIX
JII,IqI,IH _ I4CKOM6IMII, I,IJIH HEI.I3BECTHbIMI].
Be_
Cr.rcreuy rsanuocBr3aHHhrx ycJroBxfi u rpe6oaanufi ua3lrBalor BhrcKa3BrBa_
TerBHoi MoAenEn 3aAarru. .{:rx roro uro6u yrcanrr crpyKTypy
BITTIBHTB
3aAaqr4, Ha,qo
ee yclloBug pl rpe6oranzr, T.e. nocrpoI4T6 BbrcKa3;IBarenlHyro MoAeJIB
3aaarlfi.
Paccuarpura.r 3aAaqy
B y3KoM cMErcJre
JIIIT6 CJIe,qlTOIqI{e COCTaBHI'Ie SJIeMeHTEI:
3Toro
oHflTU', B Hefi MoxHo BbrAe_
-
cJIOBeCHOe IT3JIO)KeHIIe CIOXeTa, B KOTOpOM
'BHO
I.uII.I B 3aBya[]IpoBzrHHofi
Qopue lxasaxa t[ynxqnonaltnar 3aBuc]tMocrl Mex.q/ Bejrr{rrnHaMr.r, q}rcJroB;re
3HaqeHlrt KoTopEx BxoArT B 3aAatry;
-
rtHcJIoBEIe 3HarIeHIrt BeJII{qItH I,IJTH qI,ICJIOBBIe
AaHH6Ie, O KOTOpbIX rOBO_
pr.rTcr B TeKcTe 3aAaqr4;
-
3AAAHLIe, OobrrrHo cQopMynr{poBarrHoe B Br.rAe
nafaerct y3Hars HeI,BBecrHhIe
Borrpoca, B KoTopoM npeA_
3HaqeHI,Ls oAHofi l,tlta HecKoJIbKIt( BeJII,ITIHH.
Kaxgax :agaua - ero eAr.trcrBo yclroBllu u qeln. Ecm Her oaHoro n3 srr,rx
KoMrroHeHToB, To Her r.t 3aAaqr{.
AI,IT6 aHaJII.I3 TeKcra 3aAaqr c
3ro oqeul BalKHo
BaTB,
Tfi
vro6u nporo_
3lo
osHa.raer, .{To
co6lrogennelr raKofo eAI,tHcrBa.
aHaJrH3 ycJroBnr 3aAa.rr,r Heo6xo.qlrMo coorHocrrrb
por, Bonpoc
rlMETE a rr,r4y,
c BonpocoM :agavlr
3aAaqr,r aHalr[3r.rpoBarE HanpaBneHHo c ycnoBr,reM,
r-r,
xao6o-
I,Ix Her63, pa3pBr_
KaK OHI,i COCTaBI_f,IOT OIHO UeIOe.
Yuarquvc.a npe4:rararor 3aAatrlr c ao:pacraroulei crerreHlro rpyAHocrlr, KoTopbre perrrz
ra
orcr
nocJreAoBareJrbHo
-
or nepBoro K rroc:re,{HeMy. IIo xo:l.ruecrry
(aqecrBy peueHHEx 3aAaq MoxHo cy,Eur6 o HaBbrKe pe6enra, ca.neanuoro c rofi
r4.ira
vHoir reMoi4. Eur-r pe6euor< ne cMor cnpaBr,ITbcr c raruu-nr.r6o saganraeu,
oH AoJrxeH
o6rrcnurr, qro
BBBBaJTo
AaBareJrro cKoppeKTnpoBar;
ro
y Hero 3arpyAHeHue. Oro no:aonr,rr npeno-
cnolo o6yvarougno AerreJrlHocrb
orHocrrreJrBHo
xaxgoro pe6enrca,
3aaa'ru u lD( perxeHr.re 3aHr.rMaror a o6yreur.ru uftoJrbHrlxoB BecbMa cyrqecTBeHHOe MeCTO
I{ IIO
BpeMeHI-r,
}I nO IrX BJI}I'HIIIO Ha )A{CTBeHHOe
pa3BXTI.Ie pe-
6enra.
Baxno, vro6rr )^rriren6 rrMen rrry6oxr,re npeAcraBneuur o rercroeoi sagase, o ee crpyKrype, ),tren pe rarb raKue 3aAarru pa3rr,rvnrrur.r cnoco6auu r,r nepeAaBan 3TI4 3HaIIUt CBOITM rIeHI,IKaM.
Bu4u renclotrrx :agav
C ognoi croponu, [orrrrr,re rercrosofi 3aAa.rlr oAHo3HaqHo olpeAenrercs
ycnoBr.reM H Bo[pocoM, c
gpyrofi croponu, oHo MHororrJraHoBo, no3ToMy nocrpo-
r,rT6
eaHlrro KJraccllQr.rKaqrxo croxHo, refqe Kjraccu(blrrlHpoBaT6
TeKcToBr,re 3a-
AAqH IIO PA3HbIM OCHOBAHIi'M.
Ec:ru g ocnoranr-re ruaccnrpuxaquu noJroxnrb KoJrr,rqecrBo geficrrnfi. ueo6xogr.rrr,roe 4nx perrreHnq 3aAaqr4, To reKcroBBre 3a.qaqu
uoryr 6rnr [23]:
. npocrurur,r (pelraevue a o4no 4eficrnr,re),
.
cocraanrrr.,rr.r
(pe[raevue n gra u 6onee aeficrBr.rfi).
flpocrrre :agauu uoxHo K;raccuQlrqupoBarb
noMonl;ro Koropbrx
.
oHr,r
3aBnctrMocrlr or AeficrBr4fi, c
peuarorcr. flpocmre :agavu, pe[raeM;re:
cJroxeHr.reM (:agavu na HaxoxAeHr.re
c)
fMBr, 3aAaqr{ Ha yBeJrr,rqeHr,re q}rc-
JIa Ha HeCKOJI;KO eAItHI4q, 3aAarIH Ha HtrxOtKAeHUe
.
B
yMeHrlraeuoro);
BbIqI,ITar IeM (:agauu na yluenrrreHae qI,IcJra Ha HecKoJrbKo eAr4nriq,3aAa-
qI.I Ha HaXOXIeHI,Ie HeI.BBeCTHO|O CJIaIaeMOfO, 3aAarl[ Ha HaxoxAeHue Heu3BecT_
HOIO BblJl.ITaeMOIO. jaaaql.I Ha pa3HocrHoe cpaaueune);
.
yMHoxeHr,reM
(:agavr aa
yBenIrqeHI,Ie qlIcJIa B HecKoJIBKo pa3, 3aAaqr,r Ha
Haxo)I(aeHlte npor.l3BeaeHl-I_tr, 3a,4aqli Ha HaxoxIeHI.Ie HeH3BeCTHOTO ),AneHb laeMo-
ro);
.
AeJreHr,reM (sa4avr,r
ua
yuentlreH[e
qucJra B HecKonrKo pa3, 3a/{arrn Ha
Ae-
neHlle Ha paBHbIe qacTI-I, 3a.{aqll Ha KpaTIIOe CpaBHeHLte, 3AAal{Il Ha HiXOXAeHIIe
xer,rsrecruoro gen-rrenr).
flpocrrre reKcroBbre 3aAaqr.r'MoxHo xlacclrfrlrqlrponam
nouxrui, QopuupyeMrx npli
B 3aBncuMocru
or
rD( peueHr,rn, Ha 3aAaqr{:
. pacKphrBaroqfie cuucn apr,rQrraeruuecxnx geicrrr,rfi;
.
Mexly
pe3ynETaToM
/
a, veurure na
/ g (pa:nocruoe /
flpoaecru macclrQlrr<aqlno cocraBHlrx 3aAaq HaMHoro
cJro)r(Hee, T.K. Her
pac(pbrBzuorlr4e B3allMocBr3b
H
KOMTTOHeHTZTMU
apuSlrerr.r.recrrx gefi crnui;
.
HaxoxleHr,rr orHorreHrz{ 6olsue na
KpaTHoe cpaBHeHr.re).
eAnHoro ocHoBaHHr, npu KoropoM 3ala.ry uorxno 6rruo 6tr orrecrr,r
Eofi rr3 rpyn[ B npege:rax ognofi r:raccutpuxaqur.r[5, 18].
rolrro r
o4-
Ec:ru npoana;rr.r:apoBar6 pa3,qen <<Perlear,re reKcroB;rx
no MareMarr,r(e Anr 1-4 ruraccoB [33,38,42], To reKcroBsre
3a,qau>)
3aAaqr,r
[porpaMM
MoxHo rurac-
cr,rSr,rqr.rporarr na aaga.ru:
. c nponopqlroHarBHbrMr,r
BeJrrrrr{HaM[ (ABr,rxeHre (cKopocrB, BpeMr, pac-
crornue); pa6ora (npon:rogrrreJrhHocrL, apeur, o6reu pa6onr); crouuoct (qeHa, KoJruqecrBo, cror,ruocrr); pacxoA MareprrirJra (pacxog
na
I
npeglrer, xo:ruve-
crBo npeAMeroB, o6rqui pacxo4); c6op ypoxar (ypoxafinocrt, Macca ypoxar,
IUroIuaAb yqacTKa)
-
-
I.r
T,n,);
3aaaqu Ha HaxoxleHr.re qeTBepToro nponopqrioHafibHoro;
HanponoprlHor{aJrEHoe.4eneHxe;
Ha Haxolr(aeHrae Her43BecTHbrx rro AByM p:BHocTqM;
.
3aAarrr,r JrorpFrecKoro
.
Ha HaxoxneHl4e aoJII{ IIeJIOTO U UeJIO|O IIO eIO ,UOJII{.
Ec.nu paccuorperu
To K npeAcraBneHHofi
u xov6r.rnaropHoro xapaKTepa;
pat4el <feouerpuqecKne
nporpavll [38],
aulre ruraccr.rrpr,rKaqu MoxHo .qo6arurr :a4aura:
. c reoMerpr4qecKr,rMr
BervtrrtnaMtr (,anranu cropon reouerpr.rvecxofi Qury-
pBI, nepuMeTp MHOfOyronbHHKa, nJrorlaAb KBaIpaTa,
.
Berr,rquHrn>
Ha coorHortreHHe
npruoyro:nxr.rra);
eAr.rHr.rq .qnHHLr (Mr,mnr,rMerp, caHTr{Merp, Aeqr,rMerp,
Merp, Kr.rnoMerp), MaccBt (rpaMM, KnJrorpaMM, rIeHTHep, ronna), epevenu (cexynAa, Mr.rHlTa, qac, clrTKr,r, Heaenr, Mec.rq, rog).
flpocrue
B Hr,rx
I,r
cocraBH;Ie reKcroBr,Ie 3a,{aqll B 3aBHcLIMocrL or olr4sHBaeMoro
cnxera MoxHo ruracclrrpuqupoaars ua [23]:
. HaxoxaeHlre MaccBu
. Kyruno-fipoAaJ(y;
. LBMepeHrze.qnxHr,r, paccTornlu;
. HaxoxleHrre neprrMeTpa, nnoqaAfi
. c6op ypoNax;
. pacxol MaTepuaJra;
..[BI.IXeHte
nO Cyute I-tJlI{ nO BOne,
. paoory xnh coBMecTHylo paooryi
;
. c ea[Hr,rrlaMr,r
BpeMeHr,r H T,n,
Onucugaeuue B 3aAaqa>( croxer;r Aocraroqno pa:noo6pa:nrr, nogroMy
npl,tBe,{eHHat KIaccuQHKaII}It MoxeT H He
)
O r.naccutpraxaqr.ru KaK npocr;rx, TaK
IIITLIBaTE BCeX BapI4aHTOB.
I.r
cocraBHhrx 3aaarr B 3aBr{cr{Mocru
or
cooTBeTcTBI,It rIncJIa aaHHhx I,LtcKoMbIx HHrIero He cKa3aIIo B nporpaMMax Iro
MareMarr.rKe
Anr l-4 Knaccos [52]. Ha nau B3DrrA, gro cBr3auo c reM, trro
B oc-
HoBHoM B sagauax yue6nurcoB qrrcJro AaHH;rx coorBercrByer ucnouor"ry, Oanaro
Bcrpeqi
orcr zaraqr, B Koropbrx HapyueHo 3To coorBercrsr.{e. Zcnorrl}'}orcr
oH}] rtuTeJl.f,Mtr npu pa3StcHeHHI{ noHtTlt-s (3aaaqa)), npoBepKlr ero noHIlMaHIrt,
EopMIIpoBaHIrt )'I\reHI,I.f, aHaJIH3IlpoBaTL TeKcT 34.{aq[.
K:raccur[raxaqru 3aAaq B 3aBr4cr,rMocru
or coorBercrBnt
'ur4cra AarHLrx
r4
ncxorr,Gx [9]:
.
onpeAeneHHble
-
,{aHHblx neo6xoAulvroe
u AocraroqHoe KoJIIiqecrBo
AJIt
rIoJIrIeHIrt tlcKoMSIx;
. 3aAaql c irJIBTepHaTIrBHhIM
yCJIOBITeM
-
AaHHBIX CTOJIEKO, qTO OHI{ AOnyC-
KaIOT HeC(OJrbKO BapI.IaHTOB I-ICKOMBIX;
.
HeonpeAeJreHH;re (c negocrarorqnau 4annrrtr,ru)
KOJII{qeCTBO
-
4axn6rx HeAocraroqHoe
AJI' nOJryrIeHI,It UC(OMbIX;
. flepeolpeAeneHH;re 3aAaqu (:agavallr c n:6rrro.rHrrlru ganuuur.r) - 4auarrx 6olrnre neo6xo4r,ruoro, rrosroMy oHrr He Bce ucrroJrb3yrorcr anr noJrfreHat
I-rcKoMhrx.
Comacrancx c MHeHraeM Buuorpagoaoft
E.n. tl0]
cqr.rrax)rqefi He coBceM
ToqHbrM roBopr,rrb o ruraccraSuxaqrau cocraBHBx saaaq. O,4Haxo noao6nrre x.nac-
cnQrrKarluu yaooHBr, T.K. [o3BorsroT BLI,4enr{T6 3anaqu ocHoBHBx BI,IAoB I,I ycBor,rT6
ytrarlnMcr anfopr{TMEr Ilx perreHrrr.
O6yvenre MJraarrux rnKoJlbHrrxoB peureH rl rekcroBr'rx taaaq
O4nofi ra: gaxHeirux npo6reu o6luenr.rx MareMaruKe .rnnrerc.r tpopulrpoBaHr{e y }4rarqnxcf, },MeHwI perraTB TeKcToBBIe 3a,4aql{.
Orser Ha rpe6onauue 3anaqu norf{aerc.{ B pe3yJllTare ee peurenr,rx, PeIXr4T6 3aAatry
B IXITpOKOM
CMbICJIe 3TOIO CJTOBa
-
3TO, 3HaqLIT, paCKp6IT6 CBt3Ir
L7
Mexly
AaHHLIMIi, y(a3aHHBlMU yCnOBI-reM 3AAA{I;4., U UCKOMLIML
BeJTT,ITIHHaMI,I,
onpeAenlrrb nocJreAoBarerruocrb npr,rMeHeHrxr o6nlax nonoxeHr,rfi uareuaruru
(npaara:r, saronoa, rbopuyn
u r.n.), arrnolnrarr Aeficrsrr HaA AaHHhrMr4
3a,uaqu.
ucnonb3yt grlz o6rque no;roxeur.n, u notDrllrrl orrer ua rpe6oaaa,49 3aAa.ru arfil
aoKa3aTB HeBO3MOXHOCTE
Tepuux
<<peruenr-re
efo BbIIOIHeHI4.f,.
3aIarrn> rnxpoKo rrpr.rMeHrercr B MareMarr.rxe. Orlzu
TepMvHoM o6osHaqarcr cBr3aHHBIe r.,rex4r
co6oi, Ho Bce )Ke He oAuHaKoBhre no-
Ysrut l7l:
- perxeHlreM 3a,qaql Ha3BrBaKlr peynbrar, To ecrb otaer na tpe6onat*te :a_
laqr4i
-
perxeHr,reM 3aaaqr4 Ha3brBaK)T npoqecc Haxox,{eHr,rr gTofo p$yJrbTaTa, To
ecT6 BCIO AetTeJILHOCTb rIeJIOBeKa,
pellliuoqefo 3alaqy, c MoMeHTa HarIaIa
r{TeHI,LS
34,{aqri Ao oKoHrraHvt peureHuft
-
HaA ycnoBr,r{Mu
Arr
re AetcrBHr, Koropbre rrpor.r3Bolrr
Ha ocHoBe o6urux noloNesui uareuaruru
peureHrreM 3aAaqr,r Ha36rBaror nI,I rb
}t r{x
cJreAcrBr,rrMr{
nonyrreHr.rr oTBeTa 3aaaqr,r.
Peruexue sagav
*
oro pa6ora necxonrxo ueo6rrvuar, a nMeHHo )McrBeHHa,
pa6ora. A vro6rr nayvlrrrcr raxofi-nu6o pa6ore, uyxuo npeABapnreJrbHo xopoIrro H3)AnrE Tor MarepuaJr, HaA Koroprru nplrg€rcr pa6oraru, Te r.rHcrp),\,reHTBI, c
noMorrlbro Koroplrx
3navur,
g.n.n
rou, uro co6ofi
crei
nrrno:lrrercr era pa6ora.
roro qro6u Hayqr,rr;cr perxarr
oHr,r
3aAaqr,r, uago
paro6pamcr a
lpeAcraBr.sror, Kax our.r ycrpoeHhr, H3 KaKux cocraBHbx qa-
oHIt cocrotr, KaKoBbI HI{crpyMeHTLI, c noMoIIIBIo Koropblx trpor€BoAr,rrc,
pelneHr]e 3a.4aq.
OcHonHa.s oco6euHocrE TeKcroBEx 3a,qaq
cocrour
B Torra
[11], uro a nrx ne
yKa3brBaercr np.sMo, Kaxoe nuenxo .4eficrnue (unn geficrnur) go.nxtro
rrolrHexo
An.rr
no:rfreHrrr orBera
Ha
rpe6orauue :agavr,r.
llrar, pa:nuuaror o6rqni u vacrnrr! noAxo,qbr K perneHr,lo
He cr)rqaiHbr.
rrllrt noAxol
9acrstrfi no.e(oA cBBaH c perrreHr,reM
ocHoBaH Ha roM,
6urr au-
:a4av. Ha:aanr.rr
3a.{aq qacrHbrx augor.
qro ecrl o6rqee npu pelreuuu .nro6brx
3a,qarr
-
06sra-
18
nBr pemerr.rr, Koroplre BbrqJreHr.rn
coAepxaHr4e npr,tMepno
HOM XapaKTepe
cy
A. IIofia [48,c.201]. Kor[qecrgo 3ranoB r.l rr(
oArrHa.KoBo y pa3Hhrx aBTopoB, vro roropr,rr o6 o6reKrus_
lecTBoBaHI,rJr COOTBeTCTByTOTTIHX sTarrOB B Ae{TenLHOCTr.r pe_
uaK)Iqero, Ea:oebrM cqr4Tarorcr qeubrpe gTana perxeHr.tr 3a,qarra, npeAro)KeHHLre
.{ervrlqorofi T. E. lI2, c.361
Baxuefinruu sTanoM perneHr,rr 3a1aviv flBlrtercfr neparrfi eran
Trre 3aAaqrr (ana.nns rexcra).
I{elr
srana
-
-
Boclpfis-
rror{Jrrb 3aAar{y, T.e. BbrAeJrr,rrB Bce
MHOXECTBA I,I OTHOI]ICHI-IJ{, BCJIHqI,IHbI I,I 3ABHCI,IMOCTU MC)KTO/ HXMI,I, qICJIOBBIE
AaIIHEIe! JIeKCI{qeCKOe 3HaqeHlre cJIoB.
Pesyltrarou
BEITIoJIHeHI,II
3Toto gTana _{BJrterct noHuMar{ne 3a,aagu. TaK
KaK C TOTIKII 3peHI.It nCrXOnOn I,I BOCIIplItTr,re TeKCTa
noiMelrrr 3aAa.ry
-
He perrrurxB ee.
,ftr
_ sto ero
noHI4r.IaHIre. He
roro vro6u go6urlcs nostuau u, 3a1arru,
noJle3Ho BocnoJrb3oBaTBcr pa3HBIMr.r nplreMaMr,r, KoTopbre HaKorrunlrcb B coBpe-
uennoi
rraero,araxe,
flpueuu nmolneHvt
analfi4ga 3a1aqu:
-.qpaMarn3arlbr, o6urpunanr,re :agauu;
- pa:6ueun
e
rercra
3aAaqrr Ha cMhtcJroBbre qacrr.r:
- IrocTaHoBKa cneqlraJrBHhrx BonpocoB;
- nepeQoprvrynupoBKa reKcra;
-
lepeepa3rapoBaHr,re 3aAaqr,r (3aMeHxTE TepMxH coAepxaH[eM; 3aMeHr,rTB
onr{caHr,re repMvHoM, cnoBoM; 3aMeHr.rrE cJroBo cr,rHoHnMou;
crBeHHhre cJroBa; Kou(perr3r.rporaru, go6aar,rn He
y6paru uecyqe_
Merrruue currcr
nogpo6no_
crn);
- nocrpoeHr,re MoAeru (cxeMa, pucynox, ra6naqa, trepre)K, npeAMerHar Mo.qer6, BhrpaxeHne);
-
onpeaeneHr,re Bn.{a 3a,{aql,I r.r BBrIroJrHeHr,re coorBercrB),Ionlei cxeMBr
-
Kpalxor.r 3arlrcu.
Bropofi eran
gror srarr
3TAIIA
_
Ha3brBzl
-
floncx rrJraHa peuren[q. Aonflre roAhr MeroAr,rcrlr
r
ocHoBHrrM, Ho Ao Hero HaAo enle gofirra, go6parrcx.
COOTHECTI.I BOIIDOC C YCJIOBI,IEM.
r.rMeHHo
I{elt
19
,{anurd smn rpe6yer paccyxAeHr,ri,
Ho ecJrra ux ocyulecrBlr.f,T; yc.r.Ho, KaK
uacro 6;raaer, To Muorle geru, oco6enuo (Gr.I3yaJrBl), He ocBorr
).MeHI,r, lroI€TE
nJraH perueurrr sagawr. Hyxuu npr.reuu rpaQuueclrofi rlr_rr<caqr,ru no4o6nux
pac_
cyrgen*tfi. Taxne
cyqecrByror
B
npr,rerr,rrr, rcarc rpaQuuecxar
cxeva
r_r
ra6;rr.rqa paccpr(AeHr,rr,
poccr,ricKofi Mero,qaKe 6onee 100 .ner.
flpueurr arnolnenul. ora
a:
- paccyxgexr.rr (or ycnoBrrr K Bonpocy; or Borrpoca K ycnoBltK);
rro cnoBecHoMy 3aAasuro
[o
MoAeJrr.t;
olrouresui)i
- qacrHbri [oAxoA pemeHr4r 3aAaq, Ha:]BaHr{e Br.rla, THrr a 3anaiiv.
Tperrafi sran peureHrlr
3a.qaqr,r
-
Br,rnoJrHeHHe rrJraHa
-
sau6olee oyure_
crBeHHrri srar, oco6enso upr.r apuQverr,rvecKoM perxeHrrr,r 3aAaqr,r. Uer6 grana _
BLrrIoJrHr.tTb onepaUHrr B cooTBeTcTByrcqefi
uarerr.rarrqecroi o6lacrr,r (apvQue_
rr,rxa, alre6pa, reoMerpur, non Ka r,r.qp.) ycrno unr.r nncrvenno.
flpr,reuu auno,rneHnr 3Tana:
-
apr.rQrraerr,rvecrue 4eilcrnu.r,
ooopu;rennue BbrpaKeHrreM, no AeicrBr,rrM
(6es norcuenr.rr, c rorcHeHr.reM, c Bonpocarur,l);
- rr3MepeHr,Ie, cqeT Ha Mo.4enu;
- penleHrie ypaBHeHr.rfr ;
- norlrqecKue ofi epaqlrfi
;
Alra:rug urxo.nrnoft [paKTr,tKr,r cBnlerenLcrByer, qro Ha 1?oK:x MareMarr,rKr,r
nplr perxeHr.Ir,r TeKcToBErx 3aAar{ nperaMyIrIecTBeHHOe BHrrMaI ,re ylen eTc.{ BTopo_
rrly u oco6euno Tper6eMy sranau. flepnufi 3Tan cqnraeTc, npofiIeHHbrM. ecnu
)
reHr.rKr,t
cMornu cKa3arB, qro B 3aAaqe AaHo,
9erneprrrfi orarr
-
r.r
qro rryxHo HafiTfi.
[poBepKa Br,rrroJrHeuHoro perrren[c. I{ent erana _
y6enr,rrrcr B r.rcrr.rHHocrr,r nrr6panuoro rr;raHa r,t BBrnoJrHennrx geficrrr,rfi, nocle
vero cQopuylupoBarb orBer 3aAaqn.
3ro
carr,rrrfi
ueneralrnui srat. EolrunrscrBo y.rurenefi y6exgeno n rou,
qro ecnx Aerr.r Bo BpeMr
HeHr]eM
r.rJrr,r
perrreHr,rJr 3aAaqu npoBep.f,nu
ce6x (no geficrnlrxM c rrorc_
c BorrpocaMn), To B Apyroft nporepxe onu ne Hyx,aarcTc.s.
flpnevu nrno.nneHur
gTana:
20
.{o penrennr:
-
flpr,rKnlKa oTBeTa r.u a ycTzrHoBJreHrre tpaHHq c ToqKr,r 3peHr4, 3npaBor.o
cIr,Grcna, 6eo rr.rareuarr.rxu.
Bo rpeur peuenr.r.r:
- no cM6IcJD/ nonj,4reHHhrx BBrpa(eHr,rfi;
- ocMbrcjreHr,re xoAa perneHr{r no BonpocaM.
floc:re peurenlrr sagavu:
- peureHrre ,qpyr.rrM cnoco6ou;
- perxeHr.re Apyrr.rM MeTo,qoM;
- nolcTar{oBKa pe3ynETaTa
B ycJroBr{e;
- cpaanenue c o6pa-:qou;
- cocraBneHr,re u pelrenr.re o6parnoii :agavu,
Bce .rerbrpe grana pelxeHl,rr 3aAatru oAr.rHa(oBo naxsu. Tolrr<o
BBrrIoJrHe_
HIre BCeX STanOB IIO3BOI'eT CqI.ITaTL peIIIeHIre 3aBepIrIeHHBIM IIOJIHOCTbIO.
CraHoglrrct coBepIIreHHo tcHo, qro
oBJIa.{eHI,Ie
}fieHxrMI,I BbInoJrHrrB lre_
peqlcneHHbre sranhr perreHrrr 3aAaq nporeKaer He roJrbKo n uaua;rtxofi Inxole,
Ho I,r Ha AarbHefiufix
1.2
cryrreHrx o6yuennx.
leouerpuvecr[e
Be.lrfiqrlHhr B HaqaJrbHoM Kypce MareMarrrKrr
feouerpravecrcue BeJrnqrzubr - sro caoficraa reouerpr,rvecxr,rx
Quryp, xapax_
Teprr3yroque nx $opMy r.r pa3Mephr, K r .rrr,r ornoc.src.r: Anr,IHa, nepuMerp, nJro_
qagr,
o6rer'.r
lr
Berfira 4:aa
yr:ra. B na.ralr,lrofi lrrco:re B coAepxaHr,rn yue6xoro
rlpeAMera <<Mareuaulra>> BhrAeJreHBr cneAyrorulre reoMerpnqecru.re BeJrnquHbr:
AJIuHa, rrepnMerp, o6rerrl r.r n:rorqagr, a
3peHltt
raKxe
err.rHr,rrrhr
rx
MaTeMaTI,IK 9TO CKaJItpHLIe BeJII.fiI,IHbI, TaI{ KaK OHI.I
r,r:rraepenru. C
Onpe.{ejlrroTc,
rovrn
oBoI,IMI_I
rrricJreHnBrMr4
3xatlexuruu [45].
B reoverpnu, npexAe Bcero, tr3yararor ro qncro, Koropoe rroJr) raerc,
B
pe3yJrrTare r,BMepeHar Berr{qr,rHrr, T.e. Mepy Berrzqr.rHx npra arr6paxnofi
egr,rr*rqe
BeJrr,rqr,rHBr.
floerouy qacro oro
qucJro Ea3brBaror A.nr,ruofi, rr.lrollaAtro, o6l€uorr,r.
Oruocurenrno 3Toro qlIcJIa peIualo" pa3nI,HHbIe reoperlltlecKl,Ie 34aaqr,r, B qacr_
HOCTr{, KaKUM TpeOOBaHnrM OHO IOJDKHO y,4OBJreTBOprTB Kar( Mepa BeJrr{qr,rH6r,
cyrlecrByer
Jrr.l
oHo,
KaKr.rM
o6pa3oM ero Mo)KHo onpeAeJrr,rrb. Boo6rqe npaanna
u3MepeH[c reoMerprrqecKr{x BeJrrrrrrin r.l
fix
o6ocnosaHr,re
-
saltmefi
ru
:araqa
feoMeTpr,ru.
Bonpocrr, cB{3aHHbre c r43MepeHreM BenriqrrH, AocraroqHo rpyAHbr, a 3aAa_
qu c feoMeTpnqecKl,tMrz
BeJII.TII.IHaMI,I BbI36IBaIOT
y rralIHXCC
HarIaJI6HEIX KJIaCCOB
cepb€3Hlre rpyAHocrr4 B ocBoeHr,rlr. flogroMy yqrrrenrc neo6xogr,ruo npoBo.4rrrs
KaqecrBeHH)'ro pa6ory
no noaroroBKe K rrx perxeHrrro.
.II7rx
sroro €My
caMoMy
ueo6xogr,ruo suarB TeopernqecKr.re ocHoBbr 3Toro MareprrirJra.
PaccMorpr.rM no4po6uee reoperr,rqecKr,re ocHoBBt HeKorop;rx reoMerpr{qe-
crrrx seruqru. HaqHeM c nrI.IHhI.
B reouerprau
,{nnHa
-
3To BeJrrrql,rHa, xapa(Tepra3}K)Iqar
lporrxeHuocrr
orpeoKa, a raKxe Ap),nfi nnmlfi (nouauofi, KpnBofi). AJrlHoft orpeo(a Ha3hrBaer_
ct IroJloxureJrbHat ae:ruuuua, o6.nagaroqat
I
) paBHbre orpe3Kr.r nMeK)r paBHEre AJrr,rHLr;
2) ecln orpe:or cocror.rr
Anr,rH
cneA,,IoIqIrMI,I croftcrnarrau [45]:
r,r3 AB)D(
orpeKoB, To ero AJrnHa paBHa cyMMe
ero qacrefi.
3rr,r csoficraa
r,r3Mepr,rr; AJrr{ny
,qnr.rHBr
orpe3Ka r4cnonByrorcr npra e€ r,r:vepeur,rn.
orpeKa, rrpruo xMerb eAr,rHurly
e,qnnuqefi gBIrxercs, AJrr4Ha npor,t3BonbHoro
AnnHBI orpe3Ka
B reouerptu raxofi
orp*Ka. Peynrraron [3MepeHru
AJrrrHbr.
tBJlterct rIoJIoxureJIEHoe AeficrBltrenbHoe qucJlo _ efo HtBbIBaIor
rrHcJreHHBrM 3HaqeHr,reM
AJrr,tHBt
poi
gro6u
orp$Ka npu nrr6pannofi
eAr,rHr.rrIe AnrrHEr uJrr,r Me_
Eclu o6osuavr.rrt gluny orpe:ra 6yrzoit X, enuuutry g;rvubr - t , a noryqaeMoe npr,r H3MepeHr.I[ .qeicrBl,ITenbHoe quclo 6yrnoi a,
g;rr,rnrr 4anuoro orpesKa,
To MoxHo 3ar[.rcarB:
a = nE\X)vJrra X
Iloryraevoe nplr
:
r.BMepeHr.rlr AJrr,rHLr orpesrta rroJroxnTelrbHoe aeficrgu-
TeJrbrroe qrrc:ro AorrlKHo yAoBnerBop{Tb
1. Ecnr-r 4aa
AJTT,IHLI
prgy rpe6oBarrufi
orpe:xa paBHbr, To qlcneHHrre
2. Ecna orpeson
ero
a. E,
r
cocrouT ll3 orpe3KoB
:
3HaqeHr,rJr
x 1 ut x
2
rix AnrrH Toxe paBH6r,
, To
qr,rcJreHHoe gHaqeHr,re
paBHo c)a{Me qI,IcJIeHHbD( 3HaqeHxi .qruH orpe3KoB x rt x
I
2.
22
3. flpn sauelre eAr-rHltrlbr AJrltHhr qucJreHHoe
pe3Ka ).Benlqr,rrct (yruelrsrunrcr) Bo
uenrlre (6olrnre) crapofi
3Harreur,re ,4m{Hbr AaHHoro
or-
crolrbro pa3, Bo croJrbKo HoBa, eAur{uqa
.
4. {ucnennoe gsaqeHr.re eAuHr.rrrHoro orp$Ka paBHo elnHr,rqe.
.{ora:ano, qro IroJIoxItreJIbHoe 1eficrnurelrgoe qIicJIo, tBJrrrcrqeec,
poi
Me-
Anr.rHEr 3aAaflHoro orpe3Ka, BcerAa
cyrqecrByer r,r enI,rHcrBeHHo, ,{ora:ano
TaK)r(e, qro AJrx Karrnoro rroJroxl,ITeJr;Horo geficrar.rtelluoro qucna cplecrByer
oTpBoK, anr.rHa KoToporo BbrpaKaeTc{ 3TrrM ttr,rcJroM.
3auerurvr, uro vacro paAu KparKocru pequ, q[cneHHoe 3HaqeHle arr,rHhr
or-
pe3(a r{a:}brBaror upocro 4nnunofi. Haapuuep, B 3aAaHnr{ <<HafiArre AJrrrHy .qaHr{oro oTpe3K0) noa cnoBoM (,urnHa> noapa3yr{eBaeTcr t[rcJreHHoe 3HaqeHr.re IJrr{_
HBr
orpBKa. He ueuee qacro ,{onycKaror r{ IpFyro BoJrbHocrL
-
roroptr:
<<I{s-
MepB) orl,e:or<>) BMecro (dtbMep; Afl,rHy orpesKa>.
3agava. floc, porar; orpeoK, Ajlr.rHa Koroporo 3,2 E. Kaxlu 6y4er vr.rc:reuHoe 3HaqeHr.re ,{Jrr.rH6r groro orpe3Ka, ecnn eAunuuy Anr{nrr E 1rae.nr,rvrlru r 3 pa:a?
Pewenue. flocrpourr.r npou3Bon;HBrfi orp$oK
nuqnuu. 3arev noqpor,rM nprMyro, orMerr,rM
Ha
n 6yAeu cuurarr ero
Hei ToqKy
I
eAn-
u orroxltM or
Hee
3 orpe:xa, AnuHhr Koropbrx paanu E. florqrvzu otpezox AB, gnuxa roroporo 3.E.
{ro6u nonyq rr
orpezox 3,2 E, HaAo BBecrI,I HoB},ro e.quHr,rqy ,qJtr,lru.
egunuvnrrfi orpeoK
0,2
=
l.5
rra,Eo
nclm or roqKr.r B
Et
eroro
pa:6nm ua l0 paanrrx vacrei, m.r6o na 5, nocxolsry
ouoxurr
orpesra AC 6yter panHa3,2 E.
gFt E
,ftr
orpeaox,''paannfi
'
ro aJruHa
---------]5 agrnruo.o,
23
9ro6rr gnnonnuru rropoe rpe6onanrze
3, cor.nacno KoropoMy rrptr yBennqeHxn
A
.IHhr AaHHoro
orp€Ka
3aAaqu, Bocnon*yeMc, cBoficrBoM
eAr.rHr.rqbr
),a,{eHbmr,rrc.s B 3 pasa.
r
3 pa-:a vucleruroe 3HarreHrre
pa.:.qer[M
3
,2
ua2, rlolytnv.:
3,2:3=3_:3=lt=1_1.
5
15
15
Taxr,rlr o6pa:ou, fiplr ear.r'rrrle .qnr{nu 3 E
crpoeHHoro orpe3ra
lC
6yaer paono
I
-I
qrcrer*oe
3HaqeHr{e .rIJrrrHBr no_
.
Kaxgrrft venorer npe,qcraBJrrer, qro raKoe nnoua,{B
KoMHarr,r, Irnoua,{b
yqacTKa 3eMJrr.t, nnorqa,{B noBepxxocTu, KoTopyo Ha"qo rroKpacr,rTb.
oH TaKxe rro_
Hr,rMaeT, qTo ecJru 3eMeJrbHBre yqacTKn o.qr.rHaxoBBr,
trno IaIB KBapTI.Ipbr cKrraaBrBaeTc{ Il3 nno
m Ilnorla|qlr
r4x paBHbr; trTo
Ia,4r.r KoMHaT r.r njlorrl4.qr,r
apyrux
ee
noMeueHufi,
3ro o6u4eunoe
upeAcraBJreHue o flnorqagfi ncnonmyerc, rrpn e6 onpeAe_
reoMerpnu, r,{e roBop{T o rrnoua"{u Qurypu. Ho reorrrerpr,rvecxue 6urv_
Pbr ycTpoeHbr rro-pa3HoMy, r.r nosToMy, r(oraa roBoprT o nno Iaar4, BhraeJrrroT
neHr,rr,r B
onpe4en€nnrrft xlacc $r.ryp. Hanprzuep, paccMarprrBzlror rrnorqaft, MHoroyroJrb_
HHKa,
nnorla'qr [pou3BorbHofi nlocxoi r[r.tryprr, flnorqaAl lroBepxuocrn MHoro_
rparHrrKa
r,r
ap.
xe xax rr npr4 paccMorpeHrru .qltHhr orpgKa, 6ygelr r.rcnolrsorarr no_
Hrfie (cocrorrE ll3), olpeAenri ero cneA)noqr,rM o6parorr,r: Qurlpa F cocror,rr
Tar<
(cocraanexa) ns Snryp
F,
ra
F,
uer o6rqr,rx rlryrpenurx rorreK.
ecJrr,r
oHa,Br.,Ierc.,
llx o6leguueuueu u v
nr,rx
24
B arofi xe curyaqur.r Mo)r(Ho roBopfirr, uro Surypa F pa:6lrra na
Qurypu
F, n Fr.
@urypu F, u Frna clegyrolleM pucyHKe aMepr BHyrpeHHr.re o6rque rovKH.
B srou c;qrvae uelr':r yrBepxAarb, vro rfurypa F cocrolt lr: Qlrryp F, a Fr.
Ec:rN Qr.rrypa F cocror.rr r.t3 Sraryp
F, u F,, ro nH
I.lorqagsm {nrypr,r Hasuaaerct
nag Ars Kaxaoft 4uryprr ran, vro [45]:
rlr F =4 O4.
noJro)KI.ITeJIbHarr BeruquHa, onDe,4eneH-
l) panuue $vrypu xMeror paBH;re rnouaAr.r;
2) ecnlr rfurlpa cocroltr ri3 AByx.racrefi, To ee flrorua.{b paBHa cyMMe
nrorqaAei
3Tr-rx
qacrei.
rlro6u r.r:uepurl nnouaAb
Qlrryprr, nyNuo
npaBr4Jro,
rrMerb e.qr.rHrlrly n:rou1a4u. Kar
TaKoi e,4uHuue[ f,BJrrerc.{ rrJroqa.qL KBaApara co cropoHofi, paBHofi
eALrHIrqHoMy
orpe:xy. Yc.noruuc.f, nJroqaAr elrlHlrqHoro KBaApara o6o3Haqarb
6yxnofi E, a qficJro, Koropoe noJ]yqaercr B pe3yJrLTare H3MepeHr4r n:rouaau 0urypst - S(F), 3ro qucno Ha3brBarcT qlcJreHHLrM 3HaqeHueM nnouaArr Qlrrypu i'
npu nrrOpaunofi eArrHr.iqe rrJrorqa.{H E. Ono go:rxuo y4oBJrerBoprr; ycroBnrM:
l. r{zcno S(F)
2.
-
Ec:li Qurypu
noJroxr.rreJrr,Hoe.
paBHbr, To
r.r
paBHbr qr.rcJreHHbre 3HaqeHrls r.rx
3. Ec:ra rfurypa F cocrozr H3 $xr)? F, u
F,
nroqarefi.
To qxcJreHHoe
3Har{eHr.te
rrJrollaAu @uryptr F paBHo cyMMe qr.rcreHH;rx 3HaqeHufi flnorlaAefi rpuryp
F,.
F,
r.l
4. llpr,r 3aMeHe eAnr{trrlbr nnorlaAH qr{cJreHHoe
r[urypu F yBerHqrBaeTcr (yrr.reurflraercr)
3Ha.reHr,re nnonla.qr,r ,{aHHoii
Bo croJrbKo
xe
pa3, Bo cKoJrEKo HoBa,
egnnuqa vensure (6oilule) crapoi.
5. Iluc.neuuoe
parnuu l, r.e. Sf) =
3HaqeHre nnoqa,qH e,qr.rHr,rqHoro KBaApara rrpr.rHr,rMaerc,
1.
6, Ecnu Qurypa
F, lmrerc.e vactro Qurypu F' ro.{ucleunoe
3HaqeH[e
Qurypu F, ne 6onrl:re qrcreHHoro 3HaqeHHr nJroqagn Qrarlprr F, , r.e.
,l cr,3 s(ri )< s(4).
B reouerpnr,r aora:aHo, qro Anr
rrx
MEoroyroJrBHr4KoB r.r npor43BoJr6H6rx nJroc_
Qraryp raxoe vficno BcerAa cyqecrByer u elrrHcrBeuHo g:rr
(Durypx, y r<oroprrx rnoqa.{H
@opuy:ru
pa: relorpaurraa
LI3 OnpeAeneHlxt
glr
paBHLr
raxgofi tpuryprr.
, :rLasbrBatorcr panHoaeruote.
Bbrqr,rcJleH[r rrJrorrlaaa rrprMoyronlHr{Ka, TpeyroJrbHr{Ka, na_
6brnr,r BbrBeAeHbr AaBno.
B reoverpur.r rx o6ocuogrraaror, ncxo,{-s
nnoqaal.I, npI,I 3TOM qIrcJIeHHOe 3HaqeHI,Ie I[JIoIIla,4u Ha3hIBaroT
finolqa.4Bro, a q]rcneHHoe 3Har{eHI,Ie AJII{HBI OTpe3Ka _
AnnHoi.
Teopeua. fl;rorqa4t nprMolroJrsHr{Ka paBHa npor,BBeAeHnro Anrdr{ cocer_
HIiX efo cTopoH,
I,k srofi reopevu
BbrreKaer cJreAcrBxe: nnot4adt npr"r4oyzonbHozo mpe_
yzo.lrbHuKa paaHa nonoeuue npouseedeHua ezo Kamemos.
Taxzu o6pa:ov,
B AaHHoM laparpaQe Mhr gaJrr.r reoperaqecroe o6ocHosa_
Hne ,qBlx Hari6oree rr3yqaeMBx B HaqanExof rrrKoJre reoMerpr,rrtecKr4
ANI4HA I.I
x
BeJrrltIHH
_
IIJIOUAIE.
1.3 Teopemvecrne tr MeroA[qecnrre rpe6onannn
r
ra4aqaM c reoMerprr_
qecKItM[ BeJr[qrrHaM[ rlpll l3yqeHllrr MareMaruKrr B Ha.ranbHofi
[trone
,{anna-r reua r.reyvaerc.f, B TeqeHrre acex :rer o6y.reur-ra B Ha.ra,,rbHofi
[Kore
[38], varepua.n pa3,qena (<Bnnerel)) B ocHoBHoe coAepxzrHrre rgrpca MareMarrrKrz,
BBEIEHIIE EAI.IHI4II I,BMCPEHI4JI BENI,iIII4H CONPffKCHO C I.I3)^IEHI,IEM I{YMEPAq}IX qE.
nEIx HeorpllrlarenBHbx qvceJl, nocKollBKy coorHolrleHne MexIy HrIMH (:a lrcr<no_
rIeHI4eM eAr{Hr,ru Bper'renu)
rupaxeuo r gecxruuuofi czcreMe cqr]cllenr,rq
fl
M
= 10
AM
=
100 cM). KpoMe roro, B Kypce HaqaJ'rbHofi MareMarnKn
nufi Qarr crr:an
JI.f. flerepcos.rau-
c pealr,rsaquefi HAer,r napa;rJreJrenu3Ma B r,r3)^reHnr,r qr.rcel
r.r
gei-
crBr,rfi Haa Hr,rMr,r [35], npr.r roropona cyrb noHrrr4r HaryparrHoe qucno pacrpbrBa-
ercr c reoperHKo-MHoxecrBeHHof ToqKr{ 3peHur, B aKcHoMar[qecKofi uHrepnpeTaIIUU I,I TeOplII' t3MepeHut BeJIIrqItH.
O6yrexue r,BMepeHnro xaxgoi nemvr.ruu B HaqalBHoM r$?ce
MareMarr,rKr,r
npor,rcxogur noeranuo [49].
I
gman. Vmoqueuue npedcmaeneuui, tysadwa utKonbHukoa
o
leruquHe.
Beedeuue mepuuxa.
Uenr ,uansoro gtana
-
crpopvlrponars y fraIquxc.f, npe.qcraBneHne o roM,
qro Bce oKpyxarorqne nac o6rerrrr o6:ragaror cnoficrsaMr.r
MoxHo cpaBHr,rBar6: pytr(a Kopoqe yrasrr,r, ap6y: rxxelee
r.urr{ nprr3HaKaMr.r,
ux
x6lora, nprMoyroJr;-
ABCD 6olrure Kpyra, [epeMeHa,qrr.rrcq MeHE[re, qell lpor u r.n. Cpaeuexue
pa3nr,rrrHbrx npelMeroB no nporrxeHHocrn, yroqHeHne cMBrcra cJroB <<6nrxe>>,
nr.rr<
.((KOpOqe)),
urx
(yxe),
(<Bbr[re)) no3BoJrreT BBec"r{ TepMuH aJruHa; cpaBHeHr4e pa3nr{q-
no rpoplre r{ Ha3HaqeHvro npe,qMeroB (uarqe
ryp pa:nofi Qoplrrr) ruxercr oclronofi gnr
Ba)KHO OCO3HaTb
pa:luvurruu
-
rnocrra< reouerpuuecrrx
BBeAeHHT
tpr,r-
repulrna nnorqa4r. 3gect
npaftTl,Iqeolg,lo 3HaqI-IMOCTb I.I3J IaeMOIO nOIUITI,IJI, CB{3aTb eIO
o6reKTaMI,I, nepeBecru Nl,Iteficrr-re
nogltuq
Ha
C
q3ltK MareMarltxll
Ha gannou grarre yMecrHo. r{crroJrl3oBarb yre6uo-npo6levu6re cr.rryaqxr,l
,qB),x Br4,4oB:
I
Bhre
. Yqarqratvrcc fipeAnararorc.{ AJrr aHalrv3a
no BceM BHerrrHr,tM
v
cpaBHeHHr
o6lexru,
oAHHaKo-
rrpr,r3Ha(aM (qBery, pa3Mepy, ua:nauenr,rro). Tpe6yercr
Hariru orrtI.r.rr,re. Hanprauep, gne oArrHaxoB;re ruelrtle xopo6ru uoryT orJruqarbct
uaccoi.
2, Vuarqlrucr npe,qJrararorc.,r
H6re rro BHerrrHr.JM npr.r:naxau
o6n1ee
!r5
a*urvga lr cpaBHerur o6lexru, pa-snr,rv-
(nanpuvep, KplD{Ga
r,r
Kopo6ra). Tpe6yercr yra:ara
cooficrro. Taroarn',r uoxer 6rrrr o6aeu (evecruvoct), tucora,
Onrrr o64errzr frarqlxcr c 3aAaHlrrMr,r
qlMocrorreJrbHo sHftru Ha rryxnoe caoicrro v
rraacca.
yKa3aHHbD( Br.rnoB no3BoJrr,tr r,rM
:arpr-rxcr.rpoeart ero aep6alrno
-
Macca, .qJrr4Ha, o66eM. OcHoBy AerrenbHocTr,r
)^IarUnxc, Ha .qar{HoM erane cocTaB_
I,
OT NPAKfl,IqECKVE ACNCTBII', CAMOCTO'TEJIbHO B6IIIOJII{.'IEMLIE
B PZI3JII.IqHEIX,
(AK
npaBr.rJro, fi rpoBLIx cr.rTyaqr,rJrx.
II
eman. HenocpedcmaeHtroe cpalHeHue aeJruvuu.
flelr
rannoro grana
-
c$opur,rponarr y )qaqrlxc,r [peAcr:rBneHue o roM,
rrro BeJrr{rraHLr MoxHo cpaBHrrBarB, ycraHaBJflrBas.
orHorrrenue noprgxa (xax npa_
Bnno, Hecrpororo, r.16o tenr.rqrnu uoryr 6rrrr u paonn) Ha
MuoxecrBe
oAHopo,{_
HBIX BEJII{qI]H.
Jlorr.rra yue6nrrx curyaqufi onpeAen eTc, cnoco6orr.r HenocpeAcTBeHHoro
cpaBHeHHr BeJrrrqr,rH,
roropui
nognexr.rr ycBoeHrro
_
cHaqaJra Bra3yanBHbrfi
(3Aec6 pa3rrzqu.f,
4o:rxnrr 6rrrr oveaugxu), 3areM npr{noxeurrev (glnnn), uano_
xenlreu (nlroqagu), c nouorquo Mycrgin;Hbx ycxnufi (rr.raccu), orr1yrrleHni
(Bpe-
ur, reunepaqrpa). B
6relruar sagaua -
:areplJrennra AaHHoro gTarra ]4raqrrMc, npeAnaraeTc, npo_
3a.qaHr,re,
unnroe.rpupyoqee r{eBo3MoxHocr; rrpr.rMeHeHr,r, ri3_
necrnux cnocobog cpaBHeHr.rr BeJrr{q}ru. Hanpuuep, cpaBHrrrb rro
KOTOpbre yaaJreHBr.{pyr
or rpyra (orpexu, paclonoxeHHBle
AJrraHe
o6rerru,
Ha pa3HED( qaCT.f,x
ruraccuofi gocru). Pa:peruenne yxa:anHhrx nporr,rBopeqfii 3aKnroqaerc, gtr6ooe
s
MepKr,r
-
nocpeAHr,rKa,
III tman.
OnocpeooaaHHoe cpasHeHue aenuquH,
Uenr .uanHoro srana
-
oQopulzpoaarr y yqaquxc, npeAcraBJreHr{e o roM,
qTo:
l. vepxa gonxna 6rnr
z, qlcJleHHoe
o.4nopognofi c usvep.reuofi nenr,r.rranofi, ygo6uofi;
3HaqeHI,Ie BeJII{qI{HEI 3ztBI,IcIrr
peuu.a (uepxu): veu 6o.nrure uepxa,
3. cpanur,Barr MoxHo
or alr6paxuofi
eAI.IHIrqu r.BMe_
rev vr.rclo (vepa) venru:e n uao6opor;
roJrBKo BeJrr4qHHBr, rBMepeHHBre oAHo, eAlrHxrlefi
n3MepeHHr.
34ecr y"laqr,recr BbrrroJrHrror ylpaxHeHx, ua ax6op noAxoArruei
MepKu
(gna u:uepeur,rx AnrrH
-
BepeBorrKr.r, rroJrocKr{
6yvarr.r, xycovxr.r npoBoJroKrr, na_
noqK[ pa3HOfO pa.:vepa). ocyrqecrrnxr r.€MepeHr{e BenrrtrnH
(aMI4, )Aarquecr ycraHaBnr.rBaror 3aBHcr.rMocrB
prBnr4qHBrMra Mep-
Mexly BenxqrrHofi ra e,ur.rnuueft
28
Belriqr.rH6r, oco3HtlroT HeooxoAr,rMocTb BBeAer ,r, eauHofi (o6qenpr4n
Tofi)
qhr.
IJenecoo6pa:uo o3HaKoMrrrB
f{alluxc,
e.qr,rHr.r_
co craprrHH;rMr4 eAriHrruaMr.r u3Me_
peHrxr Benr,rqr.rH
-
caxenr, $yr, Syrr, xpg, JroKorB, rrar. Mero.qrzsecxu gepHo no_
go6pannar cncreua ga,qanui HarJr.f,AHo rrJUrrocrpr{pyer
)AarrlriMcr, rrro Bce r,Ic_
rronuyeMlre paHee eAriur.rqlr (4o aregeur,rx cran4aprntx) 6u:rr.r
cal:aHu, Karc
npaBrlJro, c qacrrMn TeJra qeJroBeKa, a, snavrn,
nocullt
cy6beKTl.rBHbti xapaKTep.
IV eman. Beedeuue cmaudapmuux edunu4 usuepeuua eenuuun.
flponcxogur gro MeroIoM .qeMoHcrparllru: yqr.rrenb npeAnaraer
yqaruuxcr pa3nFrH6re npe,{Mer6r (o6rer<ru) _ Hocr,rreJrr{ e,urrH[qHoi
IIaMr.r BeJrr,Irrr4H.
BHr.rMaHrrro
BeII,IqI{HBI;
] {aqfiect
I0JIXH6I
OCO3HaTB, qTO He3aBITCI,IMO OT MaTepHiula, H3 KO_
TOpOK) r,BroToBneu .qanuufi o6pa:eq (61-rVara, npoBolroKa,
rnacrra.nrzx, uurxa),
rce o6rexru o6lagaror o6u[M cgoicraolr _ gnranofi, uanpuuep
[35-37].
.(.runa (cv, gM). Macca lrcr). O6reu 1n.1.
fl;rorqagr Qurypu (rcr.cu,
ry6.4rr.r, xy6.u). .(nuna
[email protected]
(u)
,{-tnna (uu, xv). Bperrar (cex, r,ll,rn, q, cyrKx, ro1, seK). Macca (r,
flnoqagr (ap, ra). fpa,4ycuax rr,repffia
[email protected]
V eman. @opnupoeaHue u3Mepumerwrux ytveHuu..
Llenr aanlroro grana
Hr,rro Anr{H
-
cQoprr,rupooaru
y yqaqrD(c, cnoco6nocru
(
ltsMeDe_
orp*KoB, n:roqagefi Qnryp, uacc rel, BMecrr.tMocrr,r coeyaoB c noMo_
trlBIo craH,qaprHBrx e,qrrHrrq BelrrirrnH. perueuue yre6nofi :a.qauu
HiurpaBJreHo Ha
oco3Hauue Wanvruuc\ He3(oHoMI,rrIHocrI,I Bo BpeMeHL r.rcnortrsosaHll
o6Da3ua _
3TirnoHa Ant I,I3MepeHI.Ig Be]IIIIfLLH u, KaK cIe.qcTBI.Ie, BBeAeHIle
IBMepI,ITeJIBHOIO
r.rucrpyr,renra (nr,urefi xu, nalerrr,r, rpaxcnoprl,Ipa).
IV emen, Bwnonuetue apueuemuuecntx deicmeui c
rxal u.
utveHoaaHHbt
ru suc_
29
Sopvr,rpoaauue npeAcraBreHr4fi o caoficrnax nerlr.tux, tpopuzponalrae y {aqux_
)
cs cnoco6Hocru r npeo6pa:oaatnro, cpaBHeHl.rrc, cJro)KeHI.IK), BEIqhraHHIo,
yMHo_
xeHI4IO I{ AeJIeHI,tK} BeJIIttIt4H, BbIpaXeHHbIX B e.4I_THVIIaX
CHaqa]Ia
onHofo, 3aTeM
pa3Hbx Har.rMeHoBaulrfi. 3gecr aaxno,
vro6u
rrq
)4{arrlr4ec, oco3H:urr4, {To
BEr_
norHeHl4r AetcrBl-ri c ljuel{onannuuv grrcJraMu,I{x HyxHo BBrpa3l]_r.b
B e.ql,tHr,Iqax
oAHoro HitrdMeHoBasus. Tax
MexAy AetcrBI-rtMH c
xe HyxHo o6parurr
OTBJIeqeHHEIMI.I
BHr.rMaHue
) raquxc,
Ha cBc3b
qlrcjlaMll lI MeHoaanuuun (:a ncrmovena_
14
eM ITMeHOBaHH;IX ql4cen, BhIpa(eHHBIX A egUnr,rqax
npeuexu).
flpou:rnocrpupyeM Mero,r,rKy
npr.rMepe Ben[qr4Htr <<Buecuauocrt.
O6beu
-
r,r3) {eHr4.,r reoMerpr{qecKr.rx BeJrr,HrrH Ha
O6reu> [10].
sro cBoficrBo Marepuan;Hbrx reJr, pacrroJro)KeHHBrx
B TpexMepHoM
npocrpaHcrBe, Koropoe 3aK;rK)qaetcq s cnoco6uocrn 3aHrltvrarb qasrb
npocrpaH_
CTBA.
B usyreur,ru .qaunoi tevrr BErAeJIrerc, .qBe ocHoBHBrx cryneuz. Cnavala
yqaulrxcr Qopurpyercr npercraBJreHr{e o6 06reue reJra Kar( 1MeCmUJvtOCmu,
eAr4Hr.rq:x fi3MepeHr.n
o6leua; cloco6nocrb
uLIo pa3rrqHBIX ealzHl-lu, pemeHHro
o6:relroa
lel.
K r,BMepeHr.rK) BMecrr,rMocrr,r
3a,4aq Ha cpaBHeHr4e, cJIoxeHI4e H BBIqI.ITaHI4e
3areu yrovurerc.n npeAcraBneHxe o6 o6reue
cBoicrBax rr BbrBo,qr{Tc{ $op}ryna
c noMo-
.{,,Ls
r<aK tscn qr.rHe, ero
B6rrrncreHr4, o6relra npruoyroJrrrHoro na_
pa[Jlenenurreaa.
C o6leuou rar< cgoficraou pa3rrzrlHrrx npe.qMeroB y.{au[ec.f, 3HaKoMrrc.s
nocJre I,I3)4teHIzt IJIT,IHLI rz Maccbl, noSToMy v3f{eHvlo BeJIHqITHrI
<o6reu>
Ho rlpeluecrBoBarb noBTopeHlle ocHoBHBIX cae4eulzi o BeJII{qLtHax
!1
aolx-
lrx
I,I3Mepe_
r.r3yqeHfic
apyrgx
HTIU.
Z:yueur,re o6reua npoucxoguT no roMy xe lraHy, r{To
BeJrr{qr.rH.
K erovy ypoxy HyxHo nolroroBr.rrE
IlaMepeHr.rr
o6reua
-
pa3Jrr4qH6re evr<ocru
6auxn, craranvuxu, KpyxKr{, valrr<u.
ycBoeHl-rt aerEMI.I naHHOfO
pa6ory xaNloro yveur.rxa.
14
(cocy4tr) g:rr
.ftr
s$Qer<rznuoro
nOnrrzr neo6xogl,IMo OpfaHIz3oBarE
npaKTIZqecKyIo
t0
wcHoBol{
IIIKOJIbHI,IKOB
(
ant
BBe.qeHI,It HOBOfr BeJII{qI,IHBLBJITeTC' CnOCO6HOCTb MJIaIIIII_iX
CPAAHEHI-IIO NPC,EMETOB I,I KJIACCU(bI,IKAIIUI,I TP)/nN NPC.qMETOB
NO
pa3Ir,rqHBIM [pI.BHar(aM
-
UBeTy, oopMe, Macce, pa3Mepy, qTo rro3BoJrreT uM Bhr_
,4enrrTB npeIMeTBr, HanpHMep, oar,rHaKoBbre
[o
oopMe, Ho pa3HBre Ilo pa3Mepy.
O6rerralru gla cpaBHeHr4r Moryr 6rrru no6rre npe4uerrr _ ropo6rz, MarpernKrl,
cocygu. Baxno, vro6u pasrulvux B pa:|Mepzx 6u.nr,r overtgnu u onr,r o6laga;ru
cBotcrBoM at tecmuuocmu, to ecru 6lt,ru ronbre BHyrpri. vqurem
o6paqaer
BHr,rMaHr,re r{a
To, qTo B cJDAae, KOr,4a
OAI,TH
IIpeIMeT
IOJTHOCTBTO
rroMe IaeTc, B
ApyroM, r.oBoprr, v.to ero a.tuecmutrocmb vrrrl o6oeu 6olttte.,IIanHoe s6rcra3H_
saur,re o6ssaremHo conpoBoxAaercr geuoucrpaqzefi .
Ha srane HenocpeAcrBeHHoro
cpalBHeHr,r,
o6levoa npeg;rar alora, 3aRalua
6olruero no o6reuy cocyga, ropo6xr.r, npuveu pasnuvzr r o6re_
uax go:rxnrr 6trrr otIeBruIHLr.
Ha BbrrBJIer re
a<6
a=6
a>6
3gect l:xolrnurr,r cpaBr{rrBarcT npe,qMerbr BH3yaJI6Ho, ycraHaBJrr,rB:1, orHoutenue <<6ollIlre>>, (MeHblIe)), (paBHo>r Mexly ux o6reuarr.ru, 3anucbrBaK)T pe_
3ynbrarbr cpaBHeHr.rr c noMorq;ro coorBercrByrorqux gnaxon: o6rerr,r JroxKr{.
oqe_
o6reua qalrrr (n < v), o6tev KpylrcrI.r pareH o6teuy uncrn (x =
u), o6leu 6ovru 6olllre o6teua oe.qpa (6 > l).
BrlAHo, MeHLrre
flpr nocralrolre
npo6neurr }^{auuMcr
'qlur cpaBHeHr,r, [pelnararorc, 06!_
eKThr, o06eMbr KoTopbrx He3Haqr{Tejrhrro oTJrrrqaroTcf,
lpyr
oT ap}Ta, no3ToMy Brr_
3yan6Hoe nx cpaBHenr,re HeBo3MoxHo. Karc upaaulo, 3necb Hc[oJrb3yrorcf,
IBa co_
cy,qa
upoKlrf, Ho Hr.r3xlifi (cocya A) ra y3xufi, Ho rucoxr.rfi(cocy4
-
,4yrqufi yre6nuft onnr, cn.lsauuufi
E). IIpegH_
c
)4{aur,rMcr cnoco6 pelrerr,rx npo6leuu
r.r3) {eHr.reM AJtr.rHbr r.r Maccr,r, rro,qcnan<er
-
o6reurr HyxHo [3Mepr,rr;, ,{,lrr onpege_
reHur 6orburero no o6beMy (nlrecrnuocru) cocy,qa
.qocraroqHo HanorHr,rr' .rlo
xpaen cocyg A aogofi, a:areu lepelnrr ao4y a cocyg
E. Bo3r,roxHtr sapuaHTr,r:
a) acr ao4a a cocyg B ue yl.recrunact: A > E;
6) :anolnr.uacr rolrbKo rracrh cocyAa E: A < B;
a) cocyA E
:anolnzlcr
rpaer: A = E.
llnt 3aKpenJreHl,I.f, gannoro cnoco6a onocpe.qoBaHuot-o cpaBHeHIu nore3Ho
B6rrroJrHr.rrl [oAo6Hoe 3a,{arre, Ho He c cocy,qaMx,
a xopo6rcaulr, Oqesulua npo_
6reua
.4o
-
noaa B KaqecrBe nocpeaHlrKa Ircno,rb3oBarBct He Moxer. yqa
ruec,
npeanafarcT ricnoJrb3oBaTb ,qpyr{e BeuIecTBa, KaK npaBr,rno,
cbrnyque _ necoK,
coru, xpyna, 6ycllrnu.
r(
3ToMy ypoKy HyxHo nonpocr.trb Aerefi npr,ruecrr.r
cnora cocy.uu
qarxKI.I, KpyxKI,I, craKaHBI, nlraltlr,
MI,rcKI_I. C
gruva
caMocrorrelrhHyro pa6ory: cpanur,rru o napaM,
qefr
MeHb[e, cAeJrarb
-
TaKx(e BHoBb
HOCTB
Ii
o6parlnr
B TerpaAr{
qei
cocy.q no o6lerr.ry
r,r
6olture,
3anIlcI,I.
y.rauuxc{ (r nexnuou ruge) ua
TpaH3LI TI.IBHOCTb OTHOIIIeHI,i,
6anxu,
elvrrcoctsMlr MO)(HO [poBecTH
coorBercrB)floq[e p]rcyHKr.r
BHr.rMaHr.re
-
a
floleguo
cr{MMerpr,rq_
(paBHo)) II aHTI,rCI.IMMelpFIHoqTb
L TpaI3I,I_
TIiBHocrb orHorrremafi <<6olrrlre-velrtlre>
4lx o6levoa.
Ec,rna>6,6>c,roa>c
floc;re roro, KaK )nrarlaecr :aal.l
4nr4c;t orrpe,Ien
rb
6onb1111.1fi
no o6reuy
rlpeAMer Blr3yrurLHo r,r c rroMolrlbrc nocpe.qHriKa (ao4u,
xpynu), 3anrrcErBarb pe_
3yJlbrarbr cpaBHeHHt o6leuog ren c noMorublo 3HaKoB (( >
gTo
), ( < )) (
u
= ).
no3BoJlter cAeJIarB BbIBo,a o r.oM, qro HoBoe cBoficrBo _ o6leM _
ncaxe:tcs aeru_
qxHoi.
llJlt
nocraHoBKr.r npo6.nelru cpannenfi.{ o6reMog c nor\,ror{hlo
Mepxu MoxHo
npeA:roxr.rrb yqaur.rMc.[ flpu4yMarl cr4.ryarlr4rt, KorAa
a3BecrHhre cnoco6u cpa.a_
HeHr4r ucnoJrb3oBarb 3arpyAHr.rren6Ho. Hanpr.rvep, 6o.nrurue no
neHHr,re cocyaEr nepeJrrrTb
3agaur,re
l.
apyr
B
Apyfa HeBo3Mo)rGO.
Hesuairca, umodnuquficn ua Jlyue, ue uoucem cpalHltmb c6oto
qauKy A c uaarco L{, ocmaeuteficz e qeemouuou
zopode.
Kar 6rrru r nogo6nrrx
y
o6reuy uJrrr y1a_
[36]
c.rq,.rarx? Vvarqr-rec.a, ocHoBbrBalrcb Ha r.rMero[IeMc,
HI-IX OnbITe CpaBHeHI-I.f, BeII_rrrHH, AoraaaroTcr, qTo Ir)rKHo H3MepHTb
O6beM co_
cy,4oB t4 cpaBHHrr,
ror)
reHHbre
qucra _ p*yJrBTaTBr
florpe6nocrr B r.BMepeHr,ru y )qau[xcr
r.r3MepeHr,rr.
Bo3Hr]Kaer r.r B
curyaqurx, KorAa
HyxHo He npocro onpeAerr{Tb 6onbIrrxfi (uentrur,rfi) npelMer, a ycraHoBr,rrr,, Ha
cKoJrBKo oArzH cocy4 6o:rrrue (rr.reurflre) .qpyroro.
B raqecrge o6lexros .q:t{ r43MepeHrz, B;r6uparorc, cocyABr, Brr3yaJrbHo He
orJrr,rrlz olqnecr ro o6terray. Yvlrrell 3apaHee roroBr.rr pasurre
ueprlr_euxocrn. B
3aBucr,rMocru
or
xoAcul)4o MepKy
roxru
o6aeMa lr3Mep.seMoro cocyAa,
-
orreBr,r,4Ho,
yqaqlrec, go.nxuu alr6oarr
no_
rrro z3Mep{Tb o6teu aegpa c norr.rorqtro .rafiuoft
ueygo6uo, sro safiMer cjlurrrKoM MHoro BpeMeHI-I. yqutelr cneuuanlno
3arryTbrBaer
gerei, no6yN4ar ux
r
ntr6opy a,qeKBarHoro cnryar\uA rrocpeAHuKa.
He c,nelyer noAcxa3brBarb fraqr4Mcr, qro AJn cpaBHeHr,rr o6relros cocyAoB r,rx
rryxHo rr3Meprrrl oAHofr Mepxofi
-
oHz
laorp arr6part
r.r
pasnue evrcocru. B erou
CITIAC B XHO BbIIIOJIHHTF I.BMEPCHI.IC HCOAHOKPATHO I, NON)^II,ITb IIPOTI,BOPEqI4BbIe B6IBo,{bI o KoJILrqecrBeHHoi xapaKTepucr Ke o6beMoB cocy.qoB
nr,rfi
x
I,Ix orHotue_
. Vuarquecx rroAMeur npr.rqnxy nporr{Bopeqr.rBbrx BbrBoAoB: cocyabr
H6r pa3HbrMr.r MepKaMH, npxrreM r,BMeHeHrre MepKrj r3MeH{eT
6o*'re
egr'rnnqa r.BMepeHlit o6leua,
uutalr o6:reurt MoxHo JIIiIIIr TorAa,
ur.r. flo.nyveuntre pesynLTarbr
reu
rz3Mepe_
u Mepy cocyAa: qeM
MeHBITIe ee 3HaqeHlie.
floslouv
cnan-
KorAa oHI,r I.BMepeHhI oAIrHaxoBbrMI,I MepKa_
voxuo :arpr.rrcupoBarb c noMorrlBro ollopHoro cur-
33
qeM 6o,tbure effiEt,|Ila
r3Mepe lq MaccHt TeM
MeBbI'Ie ee 3xaqeHfie
Cpaouloarr, cuaanrratr
u S6tqllTaTb Maccbt
MO)|({O TOTLKO TOrAlt.
KOrIa Ofir t{3MeDel|Dl
oarruKoBt tMu Mep|(.llMll
B uerrx
s(oHoMr.ru )Ae6Horo BpeMeHr4 nporleAypy u3MepeHr,r, o6beMoB
PA3HBIMH MCPKAMI.I MOXHO 3AMEHI.ITB PCINCHNEM CJICAYIOIII]1r:1* 3AAA'9..
3aaauue 2. B eedpo exodum 8 6auorc, a e ,aaiHux
o6t'eu 6on'we
-
eedpa uru vafinurca?
-
9 marcux
xe
6auox. Vei
f36)
3aganue 3. B oduu 6o4ouorc souno 5 eedep eodu, a e dpyzofi _ 9 notoeuu_
rcoe.
Ksrcoi 6ououox ta.teem 6ottwufr ofuett? [36]
3al.anze 4. Mawa uzuepma o6teu 6auxu 6 KpytcKax u 3anucuta paseH_
=
cmao 6
5rc.
3ametv o6aeu emofi cre 6aurcu Maaa u3Mepura a uautkcrx: 6
=
ga.
V
Mawuuozo 6o,tourca ugueuuttcn o6teu? Hem nu oau6rcu e ee paccycrcdeuusat?
Iloveuy
e pezytomame u3wepeHut o6t
eua 6ououxa nonyqurucb pa3Hle uucna?
t36l
O6o6rqar BbrnoJrHeHHbre geitctzur' no H3MepeHHro o6:rervror cocy,aon, yua-
ullec{
OopMynnp
ytor tpazuno:
umo6at u3^iepumb
o6t'eu npeduema, uado ett-
6pama ueprcy (eduuu4y uzttepeuun) u y3Hamb, cKoJtbKo MepoK codepcrcumcn e
emou npedueme,
,{aunrri arrrog naoxno :aQrrcuporarb c loMorrlEro onopHoro c}rrHaJra.
34
r{To6br [3Mepkrb o66eM cocy.qa, HyxHo
BLr6parb MepKy
cKor'tbKo aaK[x MepoK
Ar{
colepx{Tcq
l.l
y3HaTL,
B 3TOV COCVne.
3aKpeIIJIeHuq yMeHr.tt I43MeptTb o6beM pa3r[.{HbrMr_.I MepKaMI4 MOXHO
IlpeanoxrITb cIe,4yloulue 3aaaHr.tt.
3aaanue 5. I4zuepo o6ueu cmaKaHa cHaLtarru npu nououlu uctiuoii aotrcrcu,
sameu - cmonoeo . B xarcou cnyuae uucrxo noryuurocb 6otawe? Iloueny?
3aaanlre 6.
IIpu
l37f
tLsuepeuuu o6t eua 6ouourca pasttuuuatuu ueprcauu Mawa
noryut&q qucra 5 u 3. Llseecmuo, cmo a Kauecmee uepox Maaa ucnotrb3oecua
eedpo u 6aury. Kqrcoe qucno coomeemcmqtem rcatcdoi uaprce?
[37)
3ag.al:ue
7
,
Ateua ctet
3a 3a6mpaKou 3 ,.taiiuarc toucrcu ucuuHoaozo Bape_
uon u 2 uqiude notrcxu rcny6uuuuozo eapeuaa. Crconurco ou c.uert eapeuan? Karcan
sdecu eduuu1a usuepeuun? l37l
Cne4Jroqui 3Tan B H3yqeHriri
o6.heMa cB.{3aH c BBeaeHHeM
ealtHllrlhr r.I3MepeHtu o6r,euoe Nltgrux !I cbrrlyqxx
o6lrlenpltrlroft
rer. yqHTerL coo6rqaer
y^{a-
rIIlMc-,{, qro e dpeeuue epeueua dnn tzt4epeHuz o6ueuct ucnonosoeeiu pa3ruuHbte
uepxu
-
6ouxy, eedpo, rcado, noroaHuK. B pasuutx MecmHocmrx omu JvepKu
uume.rbHo omruqarucb dpyz om d14tza. Muuu_1ecedy yuecmuo cua6dumu
cmpaquzuu
c uso6palceuueu
gmux ttepoK,
edwtu4ot o6veua dotcrcuu 6otma pe3ruqHd
3Ha_
tutn_
npulle p(Buepbt oduoi u moi lce
-
u3aecmHo, ,!mo
uocroecxan xadu
6atna npuuepuo e notmopa pasa 6onowe xueecrcoi.
,{a,ree
yrure,rr o6parqaer
BHrrMaHre y.raquxc.s Ha pa3Jrr.t.rHEre
no Qopue
eurocru (cpe4n Koropbrx ecrb n[TpoBas 6anra) u coo6rqaer, qro o6.leu scex
3TLrx cocyAoB oAr.rHaroarrfi
-
I
Jr.
B navecrae olopHoro clifHaJra MoxHo llcrroJrb3oBarb ljJrJrrocrpau[rc,
flocre
aneAeHus craHAaprHofi eAr,rHrirlbr
yqau[ec, y[pa(Hrrorc,
B
r,r3Me_
peuulr o6teua c noMorrlbro 1 l, peuraror rexcroBBre 3a,qaqu pa3Hbrx BrrAoB _
Ha
HaxoxAeHI,Ie C)'I{MbI, OCTaTKa, yBeJrLlqeHI,L H
JA{eHbIIIeHr.L .{hcrIa Ha HecKoJIBKO
eAxHt4rI, TpeHr,rpyrcTcr B BbrnoJrHeHr.tu apr,rQueru.recrtx
4eficTBr,rfi
c
r{MeHoBaH_
IIbIMI,I qIICJIAMI,I.
LneayeT HanoMHr.rTb yrra4.tMc{, qTo nepea perxeHr.reM npr,rMepos Heo6xo_
AuMo npoBeprrb, oArrHaKoBbrMr,r nr,r MepKaMu BbrpaKeHbr KoMnoHeHTLt lleitc:IaanB :aganuxx Ha cpalHeHxe rrana 5 cu * 3 xr yrarquecx yxe AoJrxHbr Bulerb (cro_
aylrrcp>. B :arepueHltI.r .qaHHoro gTana aep6alrno
Qr.rrcupyrorc, Br,tBo,qbr:
-
o6reM .{grqerc.f, Ber}t.ruHofi _ oH xapaKTepr,r3yer BMecrr.lMocrb cocyAoB
(6o;trure r.r:ru uenure xu.qKocrr.r rlJl{ chrrryqux BerrlecrB a ulrx aofi.qer);
- vro6u usuepurr o6:beM cocyaa, qDrno arr6paru naepxy H y3HarB, cKoJrBKo
TaKrrx MepoK co.qepxrrTcr B sToM COCy.qe;
-c
yBeJILrqeHI,IeM MepKI,I qficJreHHoe 3HaqeHI_Ie
o6teua yuenruraercr
r.r
nao6opor;
- cpaBHlrBarb, cKJIaA;Barb
oHr.r H3MepeHLr
-
Arr
LI
Bblrluratt o6tellu uoxHo roJIbKo .r.oraa, Korna
oAHofi n rofi xe MepKoi;
o6lelra ucno;r:yror elr.rH),ro MepKy (sralon) _ rlt,trp.
Paclpocrpauetuoi olru6roft gerefi .ruxercr cMerrreHrle nolstr4ir o6.beu u
tump. llozTouy eue pa3 cne.{yer o6parr]rr }rx BHr.rMaHr.re Ha ro, rrro o6reu ero
r,r3MepeHrr.s
cBofrcrBo, xapaKTepr,t3yrorqee BMecrr.rMocrb cocyAa, a nr,rrp _ sro MepKa, gTaJroH,
c
noMo[Ibro Koroporo usuep.nror o6:r,eu.
Y,tzrrrea-x to, qro
o6r,eu gro He nepB:ur BenLl.tr4Ha, c xotopofi :rtaxoultc_r
nepBoKJraccHI-lKI-I, nore3Ho nocJre ee Ll3yt{eHr4s cHcTeMaTr.r3I-IpoBaTr, 3HaHI,tt yqa-
rquxct o clrr{
BeJII{qLtH r.r
o6rqeu npuuqune Hx rz3MepeHllt. {auntre nonxrlrx ue
ycBar4Barorcr 6rrcrpo, rpe6yror Anr4TeJr;Hofi orpa6orKn.
Iloerouy B ycrHlre
yrrpaxHeHrrJ{ Ha nocJre.4yrotqxx ypoKax cne.{yeT cucTeMHo BK.rrrcqaTE cjreA}.roulHe
Bonpoc6r
-
14
3aAaHHt.
Kax[e
aerr4quHr,r Bbr 3Haere?
({:runa, uacca, o6rerrl.)
- floveuy AnuHa llBruerar' gerr4quHofi? ({.nllrrr uoNuo cpaBHr.rBalb, u3Mc-
prrr lr r.n.)
- 9ro :naqur r43MepHTb Maccy npe.qMera?
cKoJrLKo raKr.rx MepoK ypaBHoBecrrr naunsrfi
- -fsrqercq
rrlbro u3Meprror
lll
(Bu6parr MepKy
r,r y3Harb,
npeguer.)
gennqznofi :rmp? Ilouer',ry? (Jlzrp
- :ro uepra, c efo rroMoBMecrr4Mocrh (o6relr) cocyaoa, a BeJrr.tqr.rHa - 3To cBoficrBo
npe4rleron.)
- -fsrslorcs
Ha3HarreHr.re,
-
Jrrr BeJIHTII-rHaMr4
Qopua? (Her, ux
taKue caoictsa npeAMeroB, KaK uBer.. 3a[ax.
nell:r usvepurt,)
JHzur, rrro Macca cJroHeHKa paBHa Macce .{Byx
uacce 4ayx 6ereltorxKoB, MoxeM
rx
MbI
rurprr, a lracca Nupa$a _
yrBepxrarl, qro
Macchr cJIoHeHKa
v
xfl.I-
pa$a parnu? (Her, rax KaK Macca cJroHeHKa ra xupa$a H3MepeHbr pa3rvr{HbrMu
vepraun.)
Ilonerur,r rar Ne ynparrueHltr:
. Ha cootrecenue npeAMera
o6lagaer:
yKa3Ka
JIEHTA'
-{[r.rHa
KpyxKa
I-i
cBoficrBa, roroprru 4annlrfi npe4ver
37
. Ha ycraHoBreHrie coorBercrBl-it Mex.qy BenIrqLIHoft
r{ I-.tHcTpyMeHTOM AJUr ee Il3MepeHrrr, Berr.[tllHoii
I_I
MepKor, BerrrtuHo,
ri cBoftcTBoM,
KOTOpOe OHa 060_
3HaqaeT.
3agalrlre 8. Bo6epu us cJroa, 3anucquHrJtx e cxo1rcax, nodxodnwue no cMbrc_
tty 2 ctoea.'
fiLtuua (.nuneirca, caHmlarcmp, KuJroepq j4, rcacmpn,ta).
Macca (mourca, KLLqozpa
eecat, cyuua).
^L
O6t'eu (eedpo, rxump, 6epe6Ka, ozopod). l37l
v qrlrblBal, qro rr 3)/qeHrre Berv qriH cBr3aHo c
Qopurapoaauueu
y
Mraarxr_tx
rnKoJrLHr..r(oB npeAcraB.[eHu[ o
cyrr{ qtrcJra, ero aeoficraeunoi npupole (pey:Ib-
rar cqera
t-l
npe,4MeroB B
rpynle
pe3ynbrar H3MepeHru ne;rv vuu), c,reayer o6pa_
TrrTb BHrrMaHIle r{atul4xc,I Ha aHaJlotxK) cBoficTB oTHOIIIeHnfi <<60;ruure>>, <<uenr,_
ue)), ((paBHo)) Irq BerrirrriH It
ycraHoBneHHbrx paHee
a:
6, ro
qllceJr, coorHoueHr.Ifi Me)KAy qacTblo
x rlerbrM,
ant fpytttt [peAMeroB, Onopnue crifHalrbr Moryr BLIrJrr_
4err cre4yorqulr o6pa:ou.
Ecnn
Ajr_g
6:
a
Ecnua)6,ro61a
38
Ectw
a:6.
a 6:c.To
a:
c
Ec'lru a> 6, a 6> c,To a> c
O6o6r{ennufi ofiopHBrfi curHaJr no3BoJrrer )nrarqr4Mce 6ee rpyga onpe,ae-
rltrb,
KaKoe cBoicrBo tBJl-f,erct BeJII{qIIHot, a KaKoe
3aAaHI-re
9. Vmo us nepeuucneHtTo?o
- Her.
tsrtemct aeruqutTctvu: druua,
yzonbHuK, mel4nepamypa, Jt4acca, nump. annemum, KoHq)ema, gpel4t, 3anax,
mpe?16em,
o6teu, utuputa.
Brrnonnrs sro gaaalrr{e, yqau[ecr o6ocHosbrsaror crofi nrr6op, nor*yxcr
onpe,EeneHrreM BeJrnqr.rHbr
-
caoicrao, Koropoe MoxHo
III{H r{atq}Iec{ cTalJr(uBaroTc{
c
r{oBbrMrr BeJrHrrI4HaMH
Buecre c reu rcoxr[era He rBJrrercr serlrquHofi
-
r,r3Meprrrb.
-
B grov
BpeMf,, TeMnepaTypa.
sro soo6rqe He cBo[crBo,
npeAMeT, a BeJrrnq[HaMr.r .rBrreTc.r Macca KoHQeTbr n ee ,qJrnHa.
lJ Betuaua
3ro KonaqecrBeHHar xapaKreprcrrKa HeKoroporo cBoicBa np€a'
-
Benx,rrHy MoxHo r3Meprrbx pBynhrar mMep€Hxq BHpa3rrh lMeHoBaHHLrM ificnov, 3Ha,{en[t oaroi BenrrqtHbr Moxto cpaBr{lBarr c nouorrlbKr3FaKos >, <, =.
2) qTo6H r.{3Meprrh BenvqrHy, HyxHo Bbrbpa, L MepKy
colepxrfrcr a ft 3MepreMoi Bei{qhH€.
r,r
FuaTb, c(orlKo pa3 oHa
.4,rzopunn4 t 3iieperlt4e Belly.quR:
o
o
BH6parb MepKy
Y3Harb, crcn6Ko pa3
B
a=3e
r3MepseMoi Beru,{rHe.
3) C yBerx,re reM MepKx 3HalreHxe Bern'rxHH yMeHburaercr, x Hao6opor. no3mMy cpaBHHBarb,
bl MoxHo roirhKo mr[a, l(orla
oHn r€MepeHbr olBoi Lr rotr xe MepKofi.
cxran6rB
4) Anr rcMepeHnr Eenx,ilrH Bbr6rpat0r
o6ure
!r,
saaa-
Bc€x MepKx-3raroHbr:
a
39
C:re.4yrorqui
Itrle
IITKoJTIHITKI-I
rrar
o6leua a cnoco6os ero
B ocBoeHr4r4
r.r3MepeHr,tJl, MnaA-
Aenalor B 3 ruracce. 3gecr u:yleuue 4auuoft
xeno c $opuaponaHr4eM
H pa3Brrrr,reM npe,qcraBneur.rfi
BeJIT,IqITHBI
COnp{-
yrarqzxcr o6 o6reusrx
reoMerpr{qecxrrx Qr,rrypax, rzx csoficrsax n ocHoBHBrx xaparcrepucrurax. floero_
My cHaqiura cne.{yer yroqHr.tr; 3HaHr,rr ,aerei o reoMerpuqecKr4x
Qurypax, rx ar,r_
.qix, pa3Becrr.r noHrrr.rJr nJrocKr.rx u o6leunrx Qnryp, ncnounurr na:saHx.s o6:b_
eMHEIX
Quryp,
BhrAeJrHTb Lrx npv3HaKrl r.r cocraBH;re 3JreMeHTEr. Oco6oe gszua_
Hue HyxHo y.qeJrr,rT6 np.f,MoyfonbHoMy
nap:r renenr4neay _
KoJrr{qecTBo
yflIoB, cropoH (pe6ep), rpauen, lepuzu. 3gect o6q:arelrso'ToqHr.rTb
lrcrroJrE3oBar;
JrH reoMerpuqecxr,rx re.n
(lre orpaHr.rquBaqcb pa6oroi c
BTopr.rrB e.qr.rHr,rrlbr H3Mepexur AnrrHbr
(cu, gu,
rra)
rx
Mo.I1e_
r,r:o6paxeuueu). no_
u coorHorleHri, Me)Kry HHMli.
B nepaou Kracce y fraqrlxc.r cQopvr.rpora.:rocb npe.qcraBneHve o6 o6teue
KaK BMecruMocru cocygor. Ha ,qaHHoM 3Tane ueo6xoguuo pacmnpurb
) reHr4qecKoe npeAcraBneHue o6 o6r,eue nyreM BbrAeJreHr.iJr uogoro cgoficrga npeAMeroB _
3aHriMarL Mecro B npocrpaHcrse.
C grofi
ueJrBK)
MoxHo npe.qloxlrrr yre6no-
npo6.;reuntre cvryat\un, cBq3aHHBre c flortblTKofi pa3Mecrl_lrr, Ha Mcore nperMera
HeooJrbrrroro pa3Mepa (Hanpr.rMep, KHr.lxKu Ha rrorKe, urpyurrr,r n
nraQy)
npe4rraer
6o:rrruero pa:naepa. B p*ynbrare aua,rz:a nogo6xrx crarya1iafi
f{a ryrec., cKa_
xyr, qro 3aAaHrre BbrnorHt4T; He y,qalrocb noroMy, rrro npegver o6relrgee, :ara_
uaem 6onawe aecma.
Vvnrenr coo6rqaer, .rro cBoiicrBo
npeAMeroB 3aHu amb Mecmo
a
npo_
cmpaHcmae Ha36lBaerc{ o6.aeaou. VroqHqrorcq u3BecrHble :Hanuq o6 o6.reue
r.r
o6rqr,x cnoco6ax ero H3MepeHlis: Henocpe.qcrBeHHoe (nusya:rsno, aloxenueu) n
oflocpe.qoBaHHoe
(c nouorqrro xnlKocrn, cbrn)/qux BenlecrB, craHAaprHErx
Me-
oor<).
Ilo;reno o6cylzrb c yqanlrrMr.tc.{,
KaKxe npeAMerbr
reouerpu,recrne $u_
ryprr uuelor o6reM, a KaKr,re Her. Olrn cgelaror nrnoA o roM, rrro y Bcex rurocKHX
npeAMeroB
r.r
reoMerprzr{ecKr,rx Quryp
o6reu orc},TcrByer. 3ro yrovueuue ueo6_
xogxMo An.f, noHuMaHLl\ cyrr.r BeJrrrr{HHBr. O6terrauu
yxe rBJrrrcTc{ He cocy.{6r, a ropo6xu
r.r
g:rl cpatuer*rl o6reuoa
pa3nr,rqHbrx npeaMeroB
_
MoaeJru nprMo_
40
yronbHoro napa:Urenerrr4rre,{a. Csaqa;ra )nra IHMc{ npeAJraxaercn spaBH[T6 no
o6rcr"ry xopo6rn, pa3Jrrlrrar B pa3Mepfi Koropbrx orreBr.rAnrr (nanpruep, o6yanax
xopo6xa u ynaxoaxa g.nx gp<or), sareu
yxe He raK
cpaBHr,rrL
oqeBr,rAHrr (nnsxa"a,
o6reutr xopo6or
He rrpr4Bo.qrr K
p*ynbrary.
-
ropo6r<n, pa-rnr.rur n o6reuax xoroprrx
uo rIrlrpoKru{ r.r BlrcoKaf,, no ysrar). flonrrrxr,r
Bx3ya:rbHo r.r BJroxeur,reM oguofi
<Doprvry.nupyercx
qelt
xopo6ru B Apyryo
ga.rrueftlreft Ae{reJr;Hocrr,r _
Hayqurbcr l43Meprrb o6reu ,4pyruu cnoco6ou. 3aueruu, uro xopo6ru golNnu
6nrr :Haqurelturruu no pa3Mepy, vro6rr y yrarqr,rxc, He Bo3Hr,rKJro x(eJraHrre
3a-
nonnurr ropo6xu cbm1r.rnMu BeruecrBaMn (colrro, necnorlr n r.n.). Kpoue :roro,
cMbIcJI TepMUHa t3n4epumb
wanJuecl
CB'3BIBaIOT I,IMeHHO C lr3MepgHrreM, To ecT6
KoJII{qecrBeHHhIM onpeAeJIeHI{eM BeJr}IqI,tHhI, ee qrrcJleHHoro 3HaqeHI,Ic.
flogroMy
frarqnMcf, npeAJlat?ror arr6pa:m uepry, 3anolrur,rrb xopo6xlr npe4rrlerauu_
MepKzrMu r,r cpaBHr.tr; r,rx Kolu,rqecrBo B
,{arB
yqarq!
c.s
r
KaxAofi rcopo6xe. O.rerragno, vro no6yx_
ax6opy pa3nriqHhrx MepoK He cneAyer, nocxonbKy BecB npel_
necrB),Toqufr oTIBIT no lr3MepeHl4ro AnItHbI, Maccbl, BMecrI,IMocrt{, rrJIoIrIaAr4 co_
3.qaJr
ycnoBv{ Ma rroHr'IMaHIdA B3ar.rMocB{3l4
HbIM B
psynBrare r.reirlepeuu.r
I I 9]
vex4l ueproi u q}IcJIoM, non) reH_
.
O4raxo c 1ntarquMxcr uoxno o6cy4rarr, rarcoft npeAuer uoxHo llcnoJrb3oBarb B KaqecrBe MepKrz Anr r,BMepeur.r.f, o6reua rcopo6xu (np.ruoyroruroro na_
par:relenznega) - cnr,rvevnrrfi ropo6ox, ry6nr lacrur rpg.nofi mocxoi
Qopllu,
KycoqeK npoBonoKu, Kpyrrr]'ro 6ycnny? Ouerr,rguo, qro
HBre r,BMep{eMoMy
o6lexry uepxn
-
aerr nu6eplr
agexaar_
ry6r.rx, cnuvevnnfi r<opo6ox, o6ocnognnas
cnofi ax6op cxoxecrBro o6rerros c r,BMepreMhrM no Qoprr.re. ,{alee yuurelr co_
o6rqaer, uro anr r.BMepeHr,Ir o6reuoa npuuxro [crroJrB3oBarB B KarrecrBe MepKr,r
ry6 c pe6por*1, panubrM eAnHlrqe Inlrurr
Hrre 3Tlrx
o6reua
-
-
1
ry6oa a narypa"rrlryro Benr,Frr.rty
I
crra3,l
gv3, I lr3.
cu, I gu, 1 u, noxa:r,raaer u:o6paxe_
r,r
3HaKoMr,rr
c o6ognaqenueu erunr.rq
4I
Ky6ruecrcnfi
c""t".etp,
sy6nuecrxfi aerlxMerp:
."rp: I t nd
Ky6xvecrcrfi
@l
I
|l?-I
1 ivr
ItrM
Iiy6ruecrcnfi AerlrrMerp paBes no o6remy 1 :rrrpy.
Yuecrno noKa3ar; uo,qe.nt
I
cu3
uI
Alt3
-
oHa ecr6
r
<<aputlueruuecrou
rrqHKe)) r.rJrlr lrx MoxHo r{3foToBrrTb rra ypoKax Tpyaa.
Cle4yroqar yue6nax aagaua
MepKaMn
-
KaK cAeJrarb
ceq3aHa c 3anoJrHeHueu r,r:uepreuoro o6rerua
ero 6rrcrpo u nparr.rruuo? yqr,rrer; caeufia,'r;Ho 3arro;r-
nrer xopo6ry (:ryvrue -
npo3paqH),ro, cleJraHnyn r.rs
uepxauu ne6pexuo, (€achrrrrlr) xy6uru n xopo6xy, a He
nlacrura) ry6uraur-
BbrKJraAbrB{ur HX B
AeTIi 3aMeTtT, qTO BHyTpU KOpOOKI' KyOLIKI-I JIeXaT He IIJIOTHO,
[oJrHeHHbre npocrpaHcrBa, a, 3Haqr,rr,
prrhr, noxa
He 6yaer
OCTaJrr,rCb He3a-
o6reu onpege:ren He roqHo.
ToqHoro oflpeAereHl-I-s q[creHHofo 3HatleHl,tfl o6teva
norpur nepBlri crot
prA;r.
!ls,
6orce
xy6urt Hyxtro yJroxrrrb
(ocnoeaar.re),
:areu aropofi
r.r
B
raK Aa-
nee, uoxa xopo6ra He 3anoJrHr{Tc{ AoBepxy.
AJruna 5 cM
flocle roro, rar ropo6xa 3anoJIHeHa,
nrrxux ry6oe, Ecru
H)axHo cocqllTarb Korl,IqecrBo e,qn-
xopo6xa ne6oJrbrrrar, To caeJrarb sro MoxHo npocrrrM nepe-
c.reroM, oaHaxo llcHo, qro aJIt lptMoyfoJlBltofo rapaJlJle:IenuneAa 6olruroro
42
pa3Mepa gToT cnoco6 onpe.[eneHur o6beMa HenpueMreM, [ocKon6(y,
Bo-nepBbrx,
HeT TaXOTO KOJMqeCTBa (y6r,rKoB-MepoK, BO-BTOpLX, CnOCO6 nepecrreTa
HeSKOHO_
Mrrqerr Bo BpeMeHr,t.
Y.rnrorr o6pallaer
cren 6ucrprri cnoco6 sbr.rucreHx{
Qoplryru!),
TeM caMhrM
Brrr.rMaurae
rrJrorqa.qr,r
no6yxaar ) rarrrqxcr
)ryauaxcr, qro
nprMo),Tonr,Hrr(a
r4M
pl<e u3Be_
(c nouoqrro
K BBrABr.rx(eHrrK) flrnore36r o c\,, re_
crBoBaHr.rx Qopuy;ru AJrr BhrrrlrcJreHr,rr
o6teua npauoyroJrbr{oro flapaJrJreJrenuueEa. aua.'.ana Ha KoHKperHoM npr.rMepe, gareu g o6rqenr auge yuarquecx y6eNga_
rcTC.{, .rTO OOseM nprMoyroJrLHoro napaJrneJrenuneaa paBeH rrpor43tsoaenurc
Tpex
ero rr3MepeHuft
-
A nr4r:r,r,
rnrpunlor rr BrrcorLr, nocKoJlbKy cnaua,ta xy6nrcra rurcna_
.qhrBarorcr Ha ocHoBarr{[ Qurypu
- zx KoJrr.rqecrBo palHo rrpou3BeAoHr4ro ,EJruubr
Ha rxxpnHy (no eopMyJre nJrorrlaAr.r flprMoyronbrrrr(a), gare^,r onpeAenqerc,
Kojru_
rrecrBo raKux cJroeB
-
oHo paBHo q[cneHHoMy 3Harreuuro
Onoputrfi curna,r npeAcraBJreH
aucoru Qurypu [lg].
Ha pr4cyHKe.
Ero6rr nafiru o6teu fiapannenelnlreAa, Ea,uo
lnolqaab oclloBaErt.fi I/lUFOtrCrrTb Ea BLTCOTy.
v
:
'b)' c
(a,
I
oo 6eM
nJrorqaAb
BEICOT
OCEOBAHITff
flocle groro Monuo BepHyrbcr x
npo6nerr.rnouy 3a,qar{r{rc
}r
B;rqr,rcrr.rrb
o6reu ropo6ox 6o.nruroro pa3Mepa yxe c noMorquo
Sopuymr, r,r3MepxB ToJrBKo
gnxHy, rnupr.rHy u rtrcory. 3auerrarvr, r{To AaHHat rpoplryla a
nporparvn,re JI.f. fle_
TepcoH .f,Bnrercr ocHoBot
Anr
r43) reHur coqerarer6Horo csoficrsa ynHoxeHr4r,
ero reoMerprrlecKofi r.rnrepnperaquefi.
{m
(a. a).
c=
a. (a. c)
:axpenleunr IBy.reHHoi $oprraynu yqaur,rMc, npelJrarax)Tc, 3araHr,r,
pa3Hofi creneHr.r cnoxGocru u npr,rKna.qHoft HanpaBneHHocru.
43
3aaaune 10.,[nuua KoMHamu 5
u, utupuna 4 u, a ewcoma 3 n. Hafidu
ee
o6teu, rutouqada nona, nomorKa, cmeu, l37l
3anaHr.re i
sbrcoma pa6Ha
L
Ocuoeauue xopo6rcu aannemcn xeadpam co cmopouofi 8 du, a
I n. Haimu o6'ueu
rcopo6xu.
l37l
3aaauue 12. 3anonuume ma6nu4y l37l:
S
S
OCHOBAHI{JI
rroBepxrocTIl
B
4
3agaxue 13.
IIs
depeennnozo 6pycrca, uuent4ezo Qopuy npanoyzontuozo
napantenenuneda, dnuua xomopozo 24
ebrcoma
,|
r20
20
?
II c , adpe3anu
rcy6 c
cu, wupuua e 3 pasa ueruawe dnw*t, a
pe6pou 6 cu. Hafidume o6z'eu ocmaeueitcn ua-
cmu. [371
Sa4atue 14. Karc usueuumc.s o6ueu nptJrtoyeonbHozo napatmenenuneda, ecnu eeo sucomy yseruuumo: a) e
3a,qaHr,re 15.
2
pasa?; 6) ua
Macca rcupnuta 3
rc2.
I cu? l37l
Raxoea uacca uzpweqHozo KupnuquKa,
cdenautozo us moeo tce Mamepuara, ecru ace ezo u3 lepeHuz (dnuua, uupuua u
earcoma) neuuwe e 10 paz? (3ma sadaqa uJrrncmprq)yem s3auJrtocat3b
uaccat
- o6tett u2pweqHozo
tweHbute ao cmorbKo
KulmuuuKa tleHbute e 1000 pas, 3Haqum, ezo Macca
Jte po3, marc xarc ouu cdenanat ug oduozo uamepuaaa). l37l
Orgelrnrri lpor creAyer flocBrrurB
ycraHoBJreHr,[o coorHorxeHr.rfi Mex.S/
lIMeHoBaHHhrMr,r qlrcJraMr,r, BBrpaxeHHbrMr{ B e,{rrHr,tqax o6rcrua
ro
ocHoBaHr,re
o6teua u
(puc. 40). ,{,rx sro-
npo:pavuoi r<opo6xu rpopvu (y6a c pe6poM 1 4u eanonnlrru ry-
6nxaur.r c pe6polr 1 cM
-
nocraBr,rrB Apyr Ha Apyra
pa:ou:1au3:1000cu3.
r.rx KorrjqecrBo paBHo 100
eue l0 xy6unor -
(10
x
10),
:areu rro
BErcore
xo.lr,rqecrBo rzlKrr( cJroeg. Taxr.fi.,r o6-
44
1 Au3 = 10(X) crug
I ua = 10OO muo
Coornolreuue ueNgy xy6u.recxr.rM MerpoM, aerlr.rMerpoM n caHTliMerpoM
MoxHo ycrurHoBur; JrorlrqecKr,rM nyreM, ucfionb3y,
o6lelra npxuo_
$opuyly
).ronbHoro napaJrne:renane4a (ry6a), Bce r,r3MepeHr,rr Koroporo paBHhr
I
M
V=(a.r).c
V = (10 . l0) . l0 =
1
000 au3
V = (100 . 100) . 100 = I 000 000 cu3
B sroru naparpaQe M6r onpeaenrrirrr
BaHI,tt K 3a"qaqaM
c
reoperr.rtrecKr,re
n
Meroara.recxr.re me6o-
reoMelpzqecKr,IMlt BeJtrtqllHzlMr,I rIpI,I r.r3yr{eHr.In MaTeMaTuKr,t
B
Bslso4sl no neproft r.laae
Taruu o6pa:olr, rrerxoro, eAr,rHoro onpeAeneHu, TeKcroBofr 3rutaqlr
B
HaqarbHofr lIIKore.
EacTo.f,ulee BpeM{ HeT, BBo,U.rTcr rr4urL ee nOHrTxe. Mr,r no4
TeKcToBOfi 3ala.refi
OyAelr nounrr,rarr rpe6orarrue xnu Bonpoc,
ToM onr.rcaHHbrx
roroprrfi nyxlaerc.f,
yclozuir. J[o6as reKcronar 3a.{aqa cocror,tr
B orBere, c yqe_
t43 AByx .racrer:
ycnoBxfl rr rpe6onanur (ronpoca). Iloumrae rexcrogofi :a4avu
MHoronJraHoBo,
no3roMy nocrpohr; e.qr.rHyro xnaccr.r$urcaur,no croxHo, reFte r:raccutpuquponarr
TeKCTOBbIe 3aaaql{
no
34.{aq cocror.rr B ToM,
pa3HEIM OCHOBaHI.TSM. OCHOSHa' Oco6engocrr, TeKcroBhlx
qro
B Ht4x He ),1(a3ErBaerc, np_sMo, KaKoe xMeHHo
AeficrBl_Ie
45
(utu
aeficrnur') gorxuo
6urr
BbrrrorHeHo Am nony.reHfi.f, oraera na rpe6oaanue
:agavra. Peureurae 3aAaqu fipeAnoflafaer BrIIroJrHeHI,Ie .rerblpex gTanoB: aHanu3
TeKCTa 3aAaqll, nOllCK IIJIaHa peIXeHI,tt, BI'InOJIHeHT{e IIJIaHa LI npoBepKa BBIITOJIHeHHOTO
B
pe reHLl{.
HaqanrHofi rxKone B colepxaxzn yue6aoro npeAMera <Mareuarr,rxo>
KpoMe reKcroBbrx 3a,qaq npeAcraBneH pa3.qen <<feolrerpravecrne gelzqusu>>. K
HI.IM
orHoctrct:
AnI-rHa, neplzMerp,
o6teu u n.norqa4r, a raKxe eAlrH[qbr nx
r,BMe_
penur. Coveranue pelxeHl{t reKcroBltx 3aAaq I{ reoMerpvqecKlD( BcJrntr4H npeAcTaBJIteT OCOO)'IO
CJTOXHOCTS
AJI'
MJIa,qIIInX IIKOJIbHIiKOB, IIpeX,IIe BCerO.
cJroxHocrhro r.rx reopernqecKoro o6ocnonaslrx. B g:rou
vu
CO
y6e1vrr4cl, paccMar_
pI,IBiU TeOpeTI{qeCKxe acneKTbr ,{aHHoro MaTepI.IaJra HarIilJIBHO KJpCa MaTeMaTI_IKr.1.
Paccuorpea Mero,{}rqecKr.re rpe6oaanux K 3a,qaqaM c reoMerpuqecKuMu BeIruIjLlIJAl'lLt, MbI OTMeqaeM, qTo pa6oTa HaA 3aaaqaMr.I
vnnauu rpe6yer oT ) rr.rTenr
reoMeTpI{qecKr_tMI,I BeJII{_
HaqaJrbHErx KnaccoB cepbe3Hofi
MaHHocrI4 B npoBeAeHIrI,t e€ c
Tr.rK, TaK
c
ulal
nolrorosKr,r u o6lv-
:Iaun IrIKoJrbHHKaMI.t, KaK Ha ypoKax MareMa-
u Bo BHe)4)oqHofr Ae.sren6Hocrr.r.
46
UIABA
METOAI{KA OEYqEHI4S MJIAAIIIUX IIIKOJIbHI4KOB PEIXDHI{IO 3AAAq C TEOMETPI{TIECKI4MI{ BEJII{TIIIHAMI{
2.
2.1
Aualur y.re6llo-MeroArrqecKrix
Ira.fart[oft utrcorsr
KoMrrJreKToB no
MareMarrixe rrl
uo npo6,reue ucarreAoBaHrrfl
flpoana,ru:upyeu HecKoJrbKo yte6no-uerogu.recKrrx (oMnJreKToB lo Mare_
uarure, pa-rpa6orauHhrx B coorBercrrur.r c rpe6oranusr{n OfOC HOO. HaqHeM c
freoHo-MeroluqecKoro
KoMrrJreKTa
<Ilko:ra
poccr.ru>r.
Arropou
Kypqa MareMa_
YMK rnrsrorcx Mopo M.I4., Bolnona C.ll, Crenanosa C.B.,
Eaarona M.A., Eelrtorosa f.B. r.r Ap [33]. paccuorpuv I KaKoi neplrol r-r KaKxM
rI.IKr4 B .qaHHoM
o6pa3oM MnaA rLIe uIK0JIbHI-iKLI 3HaKoMJITct
c
reoMeTprqecKl.IMli nerutrrrHaMll,
e,4ltHl-IuaMI-I IIX It3MepeHI,I{ I.I pemaJoT c H}iMI{ 3alaqU.
3naKorr.rcrso
rcracce
c Beru.ruHofi
s 1 ron)ToAr.ru
I
I
((IJII4HD) B groM KoMrIJreKTe fipoucxo.ql.IT
B
Ha crpanr,rqe 66 [25].
Bygeu y<aruca s3ueptrb OlPeSKs A CaHtrta€tpAx
(npr, qscrax: 'l cn, 5 crr) 9 rreprhtb orpeoK|
:agaxroi
,4rrnxur.
lcM lcM lcM
lcu
lcr.l
il
'|
2
3
4
Il
lcu lcu
5
o,r
cu
6
Icu
7
9
n
crr
Ao 9Toro ypoKa Ha grarre no.qroroBr,rrelulofi pa6orur frxrenb yroqHrer
I,IMeroq ec{ y Aerefi [pe,4craBreHr.rr ua ypore <,{luuuee, Kopoqe).
I
47
Bnava,re ylalrluecx peuarcT 3aAaql,t Ha HaxoxAeHrre IJIIIHII o.rpe3Ka c no_
Morrlbrc r.r3MepeHr.rii, u ronbKo ao aropofi vacru y.re6nlrxa uareManrru I rlacca
(c,18) [26] ]iM rrpe.qJraraercr 3a,qaqa Ha BLrr{ricreHrze IJrr{Hr,r orpe3ra
5. Havepu 2 orpe:xa: AnhHa o.qHoro 5 crq, a
AnHHa ,qpyroro Ha
2 cM MeHbue.
C,legyrorqax enr4Hriqa r,r3MepeHat Errr4lbr,
Kypce MareMarr.iKrr - 3ro AeuraMerp (M.1. V. 2 c.
c xoropofi
3HaKoMsrc.{
Ieru
B
5I) [26]
[eqnnaerp
vro ecru 6onee KpynHar, qeM caHTrMeTp,<
eAHHxqa A,yHbr
- AeqHMerP.
Ysxaeu,
lgM=10cM
012345678910
11
12
Taxxe n rypce MareMarr.rKrr 2 rlacca no yMK <Ilkola poccr4u)
Kr.r
3HaKoMrrcr
c
Mr.r.JrJrr4MerpoM
(M.2
q.l. c.l}) [27] 14 ero coorHoueHr.teM
CAHTUMETDOM.
Mnnnuuerp
y3HaeM HoByto eAr,tHhqy
AnHHbt
yqHTbcg EbtnorHtTb H3MepeHrr.
rbr )ry(e 3Haeulb TaKre
urKoJrbHri_
H
npoAonxlM
eaHHHttbt MXHbt: AeuxMeTp,
CAHTXMETO.
6onee lteaxag egrnriqa AntHbr
B I caxrrxerpe l0
- trlnnxMerp,
uxlaxrerpoe.
c
48
9epe: necxolrxo ypoKoB
uu
-
rrrerp (M.2,
BBo,qLITcq cJle.4yroua, eAltHxua I_I3McpeHx{ AJut_
9.1, c.13) l27l
Merp
Bcnounr,r eAHHHrlbr An,l.Hbt, Koropbre rbr y)t<e
Merp
euJ,i-; o,qxa eAHHlrqa A.nHHbt.
3HaeLUb.
-
96 96 1616-sG
83 84 85
ffid4
cg__iq- iz--99--99--i
41 42 43 44
45
46 47 48
23 24 25 26
flocregnei eAnslruei
cr
B Ha.rarbHot rIIKoJre,
H3MepeHr.rr aJrnHr,r, c
49
27
roropofi yqarqrrec, tsqrpeqaror_
tBJlrerct KZJroMerp. 3garoucrgo c nei lpoucxoAnr
notyroauu 4 KJracca pu rz3freHuu pa-:.qela <BenrvlrHu>
I
f3 11.
H3MepeHHr 6oasurxx paccroexri xcnonb3ytor 6onee
KpynHyo, qeM Merp, eAhHHtly AnhHbl _ KxIo eTp.
Ann
1
49
Eue oasoi reoMerpu.tecKoE BerriqHHorl, ecrpevaroqefi
ruru naqa,truoft [IKo.[bI, rBrrercq
c.r B Kypce MareMa-
rrepriMerp. flo.q neplruerpou noHHMaroT cyM-
My Inr.iH Bcex cropou
Qurypu. yuarqaecr BrrepBbre B 3Tofi Beru.tr.tHol1 3HaKoM{T_
ct Bo BTopoM KJracce Bo 2 nosryrort4u [2g]. 3ra BeJrr.iqr4 Ha
BBo.qr4Tc{ Ha npriMeDe
peueHur roroBofi 3a.qaqu.
6ygeu yrxrucs BbrrlHcrerb neprMerp
HHKA.
l.
hsuepu cropoxbr npeMoyrorbxHKa
nppMoyronb_
11 o6bscHH, KaK
no-pa3HoMy MO)I<HO abtqHclHTb ero nepHMerp:
11 2+5+2+ 5 = 14 (cm)
Oraer: 14 cM.
2l (2.2t + (s . 2) = 14 (o.r)
Oraer: 14 cr'r.
3) (2+ 5).1=14
(cu)
Oreer: 14 o,r.
qaqltMcr npe,qraraerc{ paccMorperL
Bce Bo3Mo)I{H6Ie cnoco6r,I naxoxleHr.rr nepr.rMerpa nprMoyroJrr,Hr.rKa, r.r Bu6parr
Alx ce6.a naz6olee y.{o6nufi.
y
flepnoe :naroucrao c nroruaAlro npou cxo.{lir no
VMK <IllKo,ra poccr,ru>> s
3 K,racce s I noryrogzu na crpaHriue 56
[29], vvarqulrcr npeararaerc,
cpaBHarB
rIJIOtUar[ Qnrlp pasauuu croco6aua: Ha fJIa3,
Ha_JroxgHHeM
po(.
flnorqagu trerurp€xyroauxlxa 6onurue, qeM nroqa'qb
Tpeyrolbllt<a. 3to el6ro ra raae.
A
Cpaellro uoqagr xpyra H KEa,qpaTa Ha r,||a3
rpy4xo, B ra(oM c,nyqae ttcnonuly|or cnoco6
Ha4oxeHrq S|ryp.
(_
__..
r.r
c uoMolrrblo
Me_
50
E
3TOM Cnyi{ae MOXHO nOACqhTaTb KBaApaTbr C
oAhHaKoBofi nnoqaAbo, Ha KoTopbre pa36rTa Ka{Aar oHrypa, h cpaBHxrb nonyqeHHbte qtcIa.
llaJree MJraarxue rnKoJIbHIiKIi 3IIaKoMtTc.{ c pZl3JII]qHblMLi eAXHrlrIaMr.r r43Me-
peHrir flJrouaAr.r: KBaIparHLre caHrr]Merpbr (cu'?- M.3,
Hbre Aerlr.rMe'rpbr
(na'-
Y.l, c.58) [29],
KgaApar-
M.3, {.1, c.66) [29], raa.4parnue uerpu (u2 - M.3, r{.1,
c.70) l29l u r.t.
Kpoue roro, yqaul.IMc.{ Ha crpaHrrrle 60 upe,ularaercr npaBr.rno Haxox.qeHrrr rrro[Iaar.r lpsMoyforr,HuKa, uclollb3yt 3HaqeHl-te aJIIH ero cTopoH.
9ro6ut surqrcnnrb nnoqaAb npeMoyronbHlKa, Hy)t(Ho
HafiTH ero lu].uHy u lllHplrHy (s ogrxaroeux
a
noToM Bbrr{HcnHTb npor3BeAeHre
nonyr.reHHbrx qhcen (nnou{aAb 6yAeT BupaxeHa
B coorsercr8ypulHx eAHHHqax nnoqaAh).
eAxHHqax),
A
AaJree .4eTrM npeaJrafaroTcr 3aaarrri Ha
HaxoxleHl{e nepl-IMeTpa It nfloua-
4u $uryp. Ho ecnu e YMK <llIxona Pocclur> sro AocraroqHo Hecroxnbre 3aaa-
q
, nocKonbKy B KaqecrBe Quryprr npeg:raranrcq, B ocHoBHoM nprMoyronbrrrrKu
pa3Hbrx pa3MepoB, To Hanpr.rMep, a
TeMarlrKr.r sBrs.ercs.
YMK <flepcneru.rBa), rAe aBTopoM K)?ca
flerepcon n.f . 135-371,3aAaqu Ha Haxox.qeHr.re nepr.rMerpa
nJlorla,4rr o6ragaror 6o.nuuefi crerrerrbro cJroxHocrr,r,
nyro Qopiuy.
Mar.r
r.r. $r.rrypu r.rMeror cJrox-
51
n€ptrMerp qerhrpexyronLslxa pasea B M. o1aa
72 c , Apyras.ua_ 16 cu 6orr,me nepBofi, a rperb, c*poia,2'paa"
MeEbEe aropoi. IlafiIr! ser!€pryro cropoEy aroro rerrrp6xyrolraira.
"- "fiiiip"r",
P:3M=300cM
D
72 cM + 16 clr
BCt2
Hufur o"nrrerp
!
rr Bnoqaiql 3aftpa.EeaEofi Onrypbri
2tlr
3lv
/t
B rlpce uareuaruxt no YMK <fapuouux> Zcrouuna H.B. [20,21]
npea_
Jraraer He npocro HaiTu rrepr.rMerp rinu nnorqalb rprarypu, a ,4oKa3arb BepHocrh
yrBepxAeHur orHocurerBHo ero 3HaqeHr.t{, r{ Bce 3To
B
2O4. gepxo nu yrBepxAeHre, '{ro
[email protected] nnruyronsH.4Ka AKMEO paBeH 1 .qM?
Ana;roroaofi dopue.
nepuM6rp
1.2
I
\
l
.
Ha.reprn orpe3oK NruHofr
cBot
oraer.
B IAM h
npoBepb
259.
aepno nu yrBepxAeHue, qro nnou{aAb npnMoyrorbHuKa B 6 pa3 6onsule nno la'rlLl
KBa+qpara,annoulaAbKBaApaTaB6pa3
MeHbUe r|noqarqt4 npflMoyronbHuKa?
Talcraru
o6pa:ou,
aHa:tr,r3 pa3rr.tqHbrx
yre6uo-uerogn.recKrrx KoMnJreKToB rro
MaTeMaTHKe IIOKa3aJI, qTO 3a,qaqll c feoMeTpI,I.IecKI.IMI,t BeJIUqIIHaMI4 B HaqaJIBr{OM
K)?ce MareMarr.tKri rrpeAcraBreHbr ,qocraroqHo
g
6ontrrrot,r o6r€ue, Creneut
52
cJroxHocTr,t 3allaq 3aBr,rcr,rT oT aBTOpOB _ pa3pa6oTqrzKoB (ypca
MaTeMaTr,IKr.r. Teo_
peTr4trecKoe v MeToanqecKoe o6ocHoBaHrre npeAcTaBJIeHbt
B KIT'sax He BcerAa,
trTo Tpe6yeT AorronHrrTen6Hoi ronroToBr.rTeJrbHoi
pa6orrr co cTopoHBr ) ruTejur
HatIaJIEHBx r<iraccos.
BZUOT
H6rx
rpyAHocTr.r B
ycII ]t
CroxHrre 3aAarm c reoMerplrtlecKllMn
BeJInqrlHaMI.I B;I3BI_
IIx perneHxu y yqaq] c{, a, cJreAoBaTeJrbHo,
r.r
aonoJrHrTeJrB_
co cropoHLr neAarora.
2.2Ulzytetne onstra yutre,refi HaqaJrbHbtx rclaccoB no opraHl3aqflH
pa6orsl c la4avauH c reoMerpllqecx[Mn BernqllHaM[
I,I:1ur.ra yre6unr<ra n yre6uo-lrerognvecrue noco6us no MeronnKe
o6vqe_
Hr{r MareMarr,rKe g naqarlruoft nKone Eenoru[cron A.B.
r.r 3y6oaot
t9]
M6r orMeqaeM,
qro
Hr,rB
C.n.Ug],
oAHo r,t3 Hrrx He npe,{craBneHa pa6ora xa4 3agaqaMr.r c
reoMeTpI{qecKI.IMI.I BeJIHq[HaMI,I.
flpoaua.nr,r:uponar MareprraJrbr rrerrarH;rx usaauufr, B qacrHocrr.t xvDHaJroB
<<Ha.ra,'rBuar rrrKona> Lr <<HaqarlHa.s rrona.
relefi
!o
r.r noc.ne>>,
rAe orpaxeH oflbrr )rwr_
-
nparrnr<oB no pa3nr,HHhrM npo6.neuarlr rrKoJrbHoro o6pa:oaaru.rx, uaruu
6un r<oucrarr.rponau ror Sarr, .rro ur.r oglroi crartu no npo6leMe Harrero lrccJreAOBaHr4t 3a IrocJIeAHHe
foabl
flo:rorvry uu neperrr:rr,
cKrx cafiTax. Tarc yvure:r
B HI{X He OmDKeHO.
(
ar{a:rrl:y o[Lrra frurenefi ua paur.ruHhrx
yqr{TeJrE_
HaqaJrbH;rx ruaccos KHefic Ereua A,'reKcaHApoBHa
(MEOY uaua;rlua.r ruxo.na <<flporzvna:u.r>>, Mocrorcrar o6nacr:t) npeAnaraer
r.rcnoJrr3oBarb naMJrrxr.r no HaxoxleHr,rK) nepr.rMerpa pa3nnqHbrx
$r,rryp. Hanpz_
Mep, naMrrKy no HaxoxneHub nepr.rMe,rpa rpeyronbxr,rxa
[15,16]:
IIaMrrKa Ani peEeHl|f, saAaq no Haxorl(IeErre nepnMerpa
OIi| yp
l.
3anaca
Haiau nepuuerp rpeyronDElKa
co croponauu 3cu, 5cu,
6cv.
O6rscEEo: 9ro6u uafiTrl nepnMerp rpeyrorltrar<4 nyxuo ace 3 croponbr cJroxurE.
ianuco
e memoadu:
P=3+5+6=141""1
I
53
Omsem: 14
cna nepunaernp mpey?onbHuka.
flauqrra rro
npeAcTaBneHBr a
HaxoxqeHr,rrc lepr.rMerpa
Apyfux
flpunoxeuau 2. Onu paccuorpeHbr
reoMerpr,rqecKrD( Or,Iryp
Ha npr,rMepe perxeHrrr roTo_
Bofi TnnoBoi 3araqfi.
Yqarelr uaqalsnrx
KnaccoB cpeAHefi o6qeo6pa3oBarenbHoi rrrxoJrbr
(O6pa3oBarenbH6rfi rIeHTp) c. Vresxa MyHr,rrlrlnzrn;Horo
pariona Herlreropcrr.rfi
Cauapcxofi o6racrr,r
-
Exruapora Haranrq Hr,rxonaeBHa npeAJlarae.r. crea),rotqrrM
o6pa3oM opmHr'r3ora* pa6ory a 4 ruracce no noBTope'xro
Marepr.rzura Ha
AeHxe rreprrMerpa
Mero4rr
u
r,r
ruroruaAu np.rMoyrolrurara [40,4
Ha(ox-
I].
npx-
Haqerunanue na I. Opraxt:aqnr yporca.
Bor u nposnenel :ronor.
)?oK, reMa.
Havnnaerc.r ypor.
Tlixo, Trxo Bce ca,qnrecb
14 pa6oran ne .nenurecr.
flora:
I4KT.
cnarigor,
II. Vqrsufi cq€r.
l. flpogonxlr vlrc.nonoi
a)
6,8, 10,
14...
p.rg:
6)18,20,24,32...
2, yqeHbte uoAc.rmanlr, qro Ha 3eMJre Jrrogu aupy6aror sa I
ur.rrpy 20 rexrapos reca. CxomKo reKTapoB Jreca Mo)Ker r.rc_
ue:nyrt :a 10 vr.rnyr? 3a I qac?
3. Hafign o4uuaxoB6re rraplr qr.rcen:
5296
66989
5296
69689
834426 834426
7354256 7354256
83238324 832238234
4. 3agawr
r cruxax.
nr.rxrulru
:rr4r,rH.
a) B Axraprnr4e
Hacuraran 20
KII
Bgpyr nrglrnn
6) Ha gnope 4 cer.rnrn
Ha xprnrqe 4 rolr_
Cuunxr,r nocqtlral
r,r
54
HOXKX.
Pacnalncl na nologlltr.
C6lrrcq nuHrguH...
flepe4 nr.rna cro,rrxo lrgun?
,{ora:aremcrao,
PaccyNgeHlle.
e)
llo rponunre a4olt
KycToB
lll.rro 11 xBocroB.
Cocqutarr, s raKNe cMor,
9ro urara:ro 30 nor.
3ro auecre unn ry4a-ro
fleryxz u nopoc.rra.
A rerepr eonpoc raxon:
Crolrro 6u,ro reryxoa?
I,I y:uarr a 6rr:1 6u paa
Crolrro 6u:ro ropoclr?
Ilonropeuze
c
IICIIOJIb3OBAHXEM
III. Axrya.nlnaquq
I{KT.
pua,'ra.
1. Pa6ora
npuMeHeHHq paHee H3yqeHHoro Mare14qecKIiMt4
L
rpeyroltnur AMK, nprMoyrorrr,Hr.rK
HMKD, pov6 AMCD, npxuoyro:rsnsrfi rpeyro:uurar MDK,
orpe:or AC. npflMar a. J'loMaHar nt,tHrt I43 Tpex ]BeH[,ea, or( r<aagpar ABCD,
Onpoc yraqlxctJ oTBeTbr Ha
De30K
xy).
BOnpocbI.
-Pe6xra, na:onute nznlrz.
(llpaua-r a, rouanar luHur tr: rpdx arenrea)
-floraxrare u ga:oBure orpe3Kl.t.
(Orpe:on AB z XY)
-floxaNIzre l-r Ha3oBIITe rpeyronbHliKu.
(Tpeyronruur AMK, npruoyrolr,ur,rfi rpeyronrnzx
-lloraNnre z na:oBlire qerbrpexytorbHLr KLl.
(r{etrp6xyro:tuuxu ABC[, AMCA, HMKA)
-Kax Ha:sreaercq $urypa ABC,{?
Mff)
(KBaApar)
Oqeuna lvlrrenx.
3gopoarec6eper.
-Kax na.:usaercl Qrarypa AMC.(?
(Pou6)
-Kar aa-ruaaercr QHrypa HMK[?
(lp{MoyronbHrr()
-Hasogr,rre croficrBa nprMoyroJrbHrrra.
(nce yrun np.rvrre, flporrrBononoxHhre ctopontr paeurr)
-Mo.no,uurI!
IV. tD[s. MHHyrKa.
Mareuatnqecmoe
-
xe ua.reuat:nqecxoe
(rragpar, cyrlQuxc, B;rpareHrre, )?aBHeHr.re,
nJrroc, nprrcraB_
Ka, fnaxon, Ilp,rMzrt, oTpe3oK, BEIC(a36IBaIILIe, 3aAaqa, oKoHrIa_
Hr,re, AeneHr,re, cla4Ma, coqr,rHeHr.re, [psMoyroJrbHr,rK, r,r3Jroxe_
Hrre, [epr,IMeTp, Hape.IHe, nIoqaAB).
V. lonropexte.
- Ceroglrr, pe6rra, M6r c BaMlr rrory/TerrrecrByeM no ropoAy
<<Ilpruoyro.ntnrar>>. A ror ero ylr,rqu.
l.Ynaua <3aaanni>.
-A aor u neproe saganne.
- Hasorure Qurypy (npxuoyrolrurar ABCA, co cropoHaMr.r
cu u 3 crr,r).
AB=DC=5crrl
AD=BC:3cu
-Hagosr,rre ocHosHue cuoficrno npluoyroJrbHr,rxa,
vrnu
nD.tl\.trre ).
-\Iro taroe
nepr,rrraer:p
npnr,rolro:rtnnxa?
utuKa
-
9mo
dnuu ezo
5
56
-O6osHaqaercr repuuerp 6yxaoi p u 3anr4cr,rBaerc, TaK:
P=a+a+B+B
('rzraer lrenun)
-Moxno:anzcarr, r"uta. p:2 .
(a+a)
-Ha qdv ocuoaHgaercl 4auuaa :anr,rcl?
oxHSle
(.ruraer yterIur<)
- Bnvuc:rzre nepuMe.rp np.sMoyrorrHlrna ABCD.
(P=l6cM)
-Cnacu6o!
-Bcnouutzv nparzJlo Ha HaxoxAeHrle nJroIrIaAI4.
lecrxu oJ'tuHy yjttHo)tcumb Ha wupuHy, mo nonyuumcn tuoryadu).
-O6o:naqaerc.s nrouaAL 6yKBofi S ra :antcmaercq
rar: S:a .
B
Brrooa
(vuraer yvennx)
-Pe6xra, auvuc:llre rrnoqalL flprMoyro,rrnura
(15 cM. KB.)
ABC{.
-Xopo[o, llolo4qsr!
3.
Bugolu
vqarunxc.s,
flpocuorp clafi- -flocuorplrre na c.naiig, Karofi uoxere c4e:ran nuno4?
(uu rponelu uccle.qoBaHue Ha HaxoxAeHr.te nepr.rMerpa
IOB.
flpo6leua.
njroualr4;
B cpaBHeHHI-t XOpO[ro BLUHO, KaK HaXO.qI,tTC'
r.t
nepu-
np.ruoyrolurura)
-Mono.4utr! BH rDai{orHo orgetzru Ha noctanreHuufi so_
nDoc. Iloctaoaiitecr, ganougtarr lt ru{ofaa He nvrarb 3TIz n
MeTp H nJroulalh
HATU.q.
4. Vnfiua <.rE,qIlnrzutr z:Ntepeuulr lepuMerpa
r.I
nrorualz)),
-Kax nu 4ytuaere, qro erle oqeHb BaxHo 3Har6 npl{ I_i3vqeHlrrr
repltverpa u n:rorqalz?
Utpa
teouuuuat usuepeHua neouuempq u ruoutadu)
-flpaan,rslo! HasoeHre ux,
Ha.|.vHag c HaHMeHbuer4.
(nepuwerp
-
ura, cM, AM, M, KM; rrJ'roqaAL
KB., M KB., KM KB.J
VI. Ilrpa: I{cruHnoe
-
MM KB., cM KB., gM
- Jloxuoe
Br,rcxa:r,rsaHriq:
l, <dl2u
Peureuue :agawI
PaccyN4euue.
113
-
ABy3Ha.{Hbre qucra))
2. <B pyccrou arQarure 33 6yr*rr>
3. <<B pyccnou x:r,rre 12 r:racnrrx 6yxn>
4. <<KrdH ,rtrcrseHHoe nepeBo>
5. <Tonna - e4uuIlrla AJIITHEI)
6. <<@eapa:r - :ulrHr4fi Mec{q)
7. <<C,roeo ((Hoc> - uaror>
8. <<20 ge,rzrcr ua 5 u ua 4>>
9. <<B .nro6ov qerupexyro,rr,Hr.rKe rpu Bepr lrHr,r))
10. <A.C.llyurun po4rucr e 20 aexe>>
I 1. (Ecr6 TaKue qfic:ra, cyMMa KoropHx 6ollue 20>>
VII.3axpennexue.
1.
Petuenze
<<Heupocrar>>:aAaqa.
flzpor npauoyrolluofi Qoputr .qayrr,I' pa3pe3aMri pa3Aejrr{Jrr{
:agav. na 4 vactn rar, rrro6El IBe In Hvx 6tun qerup€xyro:rnofi
Sopvtr. a lae - rpel r o,rlaoi. Kax rro eorr,loxHo? Cle,rai
Caltocrogtenr,uaa pa6ora
Dr.tcvHoK.
2. 3aaata sa euquclesue P u S np.ruoyromnuxa.
-Ceiiqac gr,rnolnnM 3aaarly Ha HaxoxaeHlle nepuMe.rpa
l-r
NNOTIAAI,I.
-Ha c:rafige $zrypa nprMoyfo;rLHrlK c 3aaaHHbrM[ ,qJrr.rHaMr4
cropon, Ha:oszre nprMoyronbHxK. flepeveprme a rerpagr,
Ey4tre arrypatuu n BHuMareJrr,H;I.
58
llpo6neua.
6cu 5uv
3
clr Slrv
,{r.rgQepeHquaq.
o
Pa6ora n lapax.
-9ro uyxlro c4elarl, vro6x
ar,ru r'lcllzrb nepl{Merp
MKOH?
upxuoyro.nrnura
(tepezecru cu z twu)
Il nnoua'4;
MK=HO=65uu
KO=MH=38uw
-flepg6ri aaptianr H:xo,4rir nepHMerp, BTopo,
,4r{T
nnoulaaB.
rroMeH{erecr,
::i:y
:"
cBoero coceAa.
I
rerpa.4qMx
r4
rpoBepr4re perxeHxe
BaprzaHT
flepuuerp
BapriaHT Haxo_
2 eapuaur
(ur,a)
llnoua.qt (uu
KB.)
P=65+65+38+3 8
s=65 . 38
P=206uv
P=20cv 6rr.ru
S=2470rrarvr r<e.
06o6qenue,
cpaBHeHI-Ie.
VJII. IoAseAenue firoroB ypoKa.
-9ro urr llcc:re.{oaa,u na ypoxe?
Pe$rercru
YcrHas
rrr.rrers.
orIeHKa
(ocooeHHOCTX [ptMoyronbHHXa; r43MepeHr_re
I{ Haxo)L4eHUe
nepxMerpa H fl JlOulanl] rlprMoyfo,,lbHfi Ka.).
-Karyro pa6ory lru npoBeJrri c norurrnruz
<<Ilepauerp npx_
voyro.llbHl.rKa> z <<flrouaAu nprMoyfonbHuKa),?
(M6t ooooullln', cnoco6gl etrq[cjIeHH,
nepxverpa u nrou]auu
np-gMoyfonbHvKa, cpaBHHBaJIH
AaHHbIe noHrTr,Lt, noBTopl.Inli
eAr4gr.rrlbr
rtMepeHxr nepr,rMeTpa u
rrJrorqa.ur4
)
-{m
ganounr.rrocr r,rna nonpalr,urocr roopevx
ypoGl
-Ige npNrogrrcx naM srr,I axaHI,Ir?
-Pe6sra! Bu orr[.{Ho pa6orana. Vpor o(orqen.
(,{or'ramnee saganne).
Yqr,rrert naqaJrrulfi K;raccoB Mynlrrg,rnanruoro 6rogxerxoro
o6u1eo6pa_
3oBareJrlHoro )Frpexgerure <<Jlr,rqei Ns 6 <dlapyc>, ropo4
[sepxamcxufi Mocxoncrofi o6nacrn - Anocora Enena Bnrropoura s 3 ffiacce no yMK <IIIxona 2100>
rrpu gHa(oMcrBe
c
Qoprvry;rarr,rr.r
npegnaraer nocrporrr pa6ory
I
Ha(oxAeHI,L lepr,rMerpa I,r nnoqa4r Qr.rrypru
rar [12-14]:
Opzaurn aquouuati noneum,
- Kax grr
rlonr,rMaere nocnoBltqy:
<Kpatrocrr
-
cec,rpa ra;ranro>?
( MHeHn Ie_
rei).
Kax crggana gra uocJroBllqa c ypoKou uareuarlrror?( weunx pe6.rr)
60
flonpo6yeu paso6parucr?
II
-
Mo 6 uarcupynr4ufi eman.
Cocraar yparneur,rx rt 3aAarraM
r4
peur,r ux.
a) B rapaxe crosro 460 uauun. Kor4a n3 rapaxa BBIexarIo HecKonhKo
Marrr,rH,
TO OCTaTTOCE 5 pr40a no 44
B KaiKAOM. GXO'"XO Uarnrn Bbrexuro r.,3 ra-
"'ur,"*'
paxa?(460-X=44x5)
6) B napre
Bhrcagr.rJrri HecKoJrbKo
prnoB nrr[ no 25 gepearea B KaxqoM
prry
]r
enle 43 6epeau. Bcero aucagunu 93 4epena. Cxo.nrxo pxgor nr.rn 6rno
nocaxe-
tto?(25xX+43:93)
Ha xarrae rpynnu voxno par6nrr gannse zanucu?
10x20
30 +
96 >90
axB
(a+n);2=P
Y x 9=72
axs=S
17
1g: 69
x a>a
Brrc:ryururatorcr Bce MHeHltr gerefi.
( lpanneuur, BbrpiD(eHrxr, HepaBeHcr.Ba,
Qopmyru)
Yro
tvrH Hagrrgaer\.r
lparnenr,reu? \pa6eHcmao, cooep cat4ee nepeMeHHyto, 3Hatte_
nue xomopoti uycrcuo uadmu).
9ro oguaqaer gamcr:
(a +
r) x 2:p
(Ileputtemp)
9ro o:naqaer sarlnct: axv=.5 (Ilttoqadt)
Kro nounar, ralr ulr trHoraa HanbrBalJrr,r rarr,re ranracu? ( @opuynat)
B qeu :at<rrroqaercr orJrrrqlre r[oprr,rynu or yparnenlrr? (<Dopuyna _ gmo paeeH_
cm6o, Komopoe nomozaem HaM ycmaHosumb 3aaucurrocmt nearcdy eenuauuatvu.)
IIL
llocmanoera yue6uofi sadauu yporca. @urccaqun npo1neuw
flonpo6yfire uo QopMyne flepr,rMerpa uafirr,r ogny cropony npruogontxnxa?
3ro :aganue rustrBaer 3arpyAHeHr.re y Aerefi. p= (a + B) x 2 - flpo6neua?
{euy
narr.r nJ,rytno
nayqr.rrrcr Ha cero,qnxruneu ypore?
(Hayqumbct, ucnorb3yt Qopnymt, nmoduma cmopoHbt npr.l4oyzonbHuKa
no
eecmHbt nnot4adu u nepuuemry u oduoi cmoponat)
u3_
Kax Ne 6y4er 3B)Aarb reMa Ha[rero ypoKa. <@opuytu,r nepu empa u nJrouladu
nptttoy?onbHuKa, (reMa !r ue:r, roggrrrrcrc_f, Ha crafiae Nq 6)
IV Cumya4un ycnexa.
a
axs=S
(a+a)x2:P
Ha:owtre nepalro Qopuyny? ( Qopuyra nrorqalx)
Ha:ogure aropyo Qoplryry? ( Qopvyra nepuuerpa)
,{anafire nopa6oraeu c reprofi $opuylofi. tr4ctotrssyr Sopuy:ry, orpeAeJrr4re,
KaK HaiTr.r .qjrxHy. ,{erlr o6r,rcuxror.
a=S:B
e:S:a
{ro uoxso cKa3arb
o
HoBr,rx paBeHcrBax?
( B uux dtuua u wupuua lbtpoJrceHbt
uepes ntou.lada u c,p)).yto cmopoHy)
[aaaftre zx npovlnaeu.
S =a
x s - Ifaoqadb nptLltoy?onbHuKa paeua npouzeedeuurc, dttuu ezo cmopou.
a
=S
e
:S:a
',8
-
dnuHa cmopoHbr nptLrtoyzo,,tbHuKa pasHa uacmouy naot4adu u
trseecmuofi cmopoHu,
3anuurure erlr $opMyJrEr
B
TerpaAB.
Bugelure ux pauxofi.
Tenepr nocuorplru, rar cBq3aHr,r Mex.4y cooofi nepl-Irraerp
yfoJIbHI{Ka. Hanouuure,
rr
cTpoHbr np.{Mo_
qro raKoe nepuuerp? (Cyuua ecex cmopoH
npnMo_
y2orbHuKa)
Karlro Sopny:ry uoNso :anucaG? P= a as *a
P: (a+a) x 2
*s
vtur p =a x 2 +
B
x 2, urv
,rlo6ofi u: :rr,rx $opMyJr MoxHo florb3oBarbc, Arq HaxoxreHu{ nepuMerpa.
A raxa.r $opuyra y.qo6Hee Bcero? ( tx octedunn).
lToueuy? (B uei ecezo 2 deflcmeun).
62
@usttuuymrca dan znas
flpoulrraire nocne4toro
p = (a+r) x
2
rfoprrly.ny.
- Ilepunemp rpano)EonbHuKa paeeH cyj\4ye cmopotr, yuuoaceuuofi
Ha Oaa,
V <<Orrcpsrrxe gerbMu HoBoro)
Onuparct ua sry rfopuyny, utr nonpo6yeu aunecru {oplryrry cropoHlr nprMo)tronbHr,rKa no ero [epHMerpy
r.r
]Ii]BecrHoi cropoHe.
C qero uaqneu?
1.
Brrgenrlr neu:recrrD/ro cropo[y P = (a+r) x 2
{ro
.
saM Hanor4r,ru aer sra zanuct2 (Cocmaeuoe ypaeuenue)
Perur,rre ero!
(a + e) 2=P;
a+e:P:2;
a=P:2-s
geicrrux pelreHru
Bce
ypaBHeHr,rrl noKa3lrBax)Tcr rrogTarrso na
cJreAHee paBeHcrBo BhrAenrercr paruxoft ra
HUr
CTOpOHBT
tlepe3 r.BBecTHLre [epr4MeTp
Tenepr car,rocro.nreJrbHo rax xe
$r,rrcupyercr xax rpopmy,rra naxoxge-
r.r
BETBeAETe
cTopoHy.
QopMyJry Apyrofi croponrr.
Bce armegenntre QopMynLr Aerr,r 3a[[chrBaror
V
caafire, flo-
B
TerpaAB, Ha36rBar ID( 3HaqeHr,Ie,
Ilepeuuuoe sarcperueHue
a) Pa6ora no fre6Hr.rKy.
Ilb 1 crp. 86 no pxgau (c r.rcnolr:oaaxueu QopMyJr, c nporoBapnaaxr.reu ncnyx)
2 c xouueurr.rporarrlrerr.r (racnoleyr $opr"rynu)
Ng 3 cavocrosrer;Ho c caMorlpoBepxoi no eralory.
<Dr.r:rlrUnyrra
Ne
VI
Ne7
lloemopenue uq)leHHozo.
(r, g, e, ) no prAaM
Ns8 c
88 .racrHrre cry.{ar.r AeficrBr.tfi c qucnaur-r 1 r.r 0
WI Ilodeedeuue umozoe.
Kaxar npo6neua no3Hr.rKra y Hac
B
Harrale ypoKa?
MoNuo
lu,:nar {opMylry fieprrMerpa, al,necru Sopuyly HaxoxAeHr.rrr
croDoH;r
no rl3BecrHbrM nepraMerpy u rr3BecrHofi cropoH;r.
Hasosure ed.
Cnpaaunrct vtr c npo6:reuoi?
Pe6rra, n rce Ne 3aqeM HzrM HyxHr,r $opwynr;? (Llcnonosyz
eopuyrot, uut nezxo
l.ioJrceu Haimu Heuseecmuoe)
9ro raroe Qoplryla? (paoeucmeo, Kornopoe norrczaem ycmaHoaumb 3asucu_
t4ocm b Me)rc dy 6eruuuuattu),
KTo noMHr4T rrocJroBr,rrly B Haqrure ypoKa? ( Kparxocrl _ cecrpa raJraHTa.)
naK OIIa CBt3aHa c
HaTIIHM
Cefo,qruI[Itvv yporcoru?
Kar<ofi aueog uoNuo cgetars? (3uaa
eopuynat, aocrcuo, ercouoaa epeun, 6arcm_
po uaimu suavenue.)
Kouy 6u:ro rpyano ua ypoxe? Kro npeogo:ren rce rpygnocrz?
VIII
3adauue ua
doaa.
rle elle ucnoJrr3yercr
Hafirr.r
cJroBo <Qopuylo>,
novnMo ),?oKa
MareMarr,rKr.r.
Crp. 87 J\! 6, crp.88 J\sl0 (r).
r aKr4M oopa3oM,
r{3yrrar onbrr yqurenei naqa;ltnrx xlaccos no pa6ore Han
3aAaqaMr,t c feoMerpr{qecKr.rMr.r Berr{qr.rHaMr.i, Mbr rrpr4unr{ K Br,rBoAy,
qeHv_{ MJraaIIIltx IIIKOJIbHI,TKOB peIXeHr4IO
3aAatl
c
feoMeTpI4 qecKI.IMU BeJII_iqI.IHaML
HeooxoAr4Ma AJrrrreJrlHarr nocJre,{oBareJrlHarr pa6ora
HaBLIKa
vro 4:rr o6y_
no
Qopuuponaur,rro sroro
y yqarUuxcr. A gnr ed sSQerrnnxocrz rBopqecKxi
r.r
r.rn4uaugya,tlnufi
IIO.Q(O.q.
2.3
MeroAl.tecr[e
peKoMeHAaqrrn no o6yvexxm MJaAulr{x tnxonbrrnKoB pe_
rxeHUro 3aaaq c reoMeTpuqecKHMH BeJl1|qnHaMI_I
flpoanalu:uporan yrre6Ho-MeroAr4{ecKl.Ie
HaqatrEHofi rrrKoJrBr r.r r.t3)^rrrB onBrr
HI,I_a
3a.qat{
KoMnJreKTBr
yrurelefi no Sopvr,poaauaro
c reoMerpuqecKzMu BerrvquuaM, gaur,r 6u:ra
onrnnax pa6ora Ha 6a3e trdEOy
-
no MareMarr,rKe _uls
HaBbrKa pelIre_
npoBeAeHa cneAyloqarl
Kpacuopr6uucrafl cpe1trfl o6rqeo6pa-aoaa_
renrsaq uxo:ra Xoruneqxoro paioxa Oplorcxoi o6:racru.
64
I
ro
sran
-
r<oxcrarr.rpyourfi
lpornr
onpeAeneHrrro
-
cocroqJr B npoBeAeHr,ru (oHTponbHoro cpe3a
cQopr',rupoBaHuocrr.t HzrB6rKa peureHr,rr 3aAaq c reoMer-
plFIecKI.IMI,I BeJIHTII,IHaMI'.
II
eran
-
o6y.raroqr,rfi
-
npeAnoJrafaJr BhrnoJrHeHr,re gaAanr,rfi no
pe
reHuro
3aAaq pa$Iu.rHofi cJroxHocrn Ha HaxoxleHue nepHMerpa r4 rrJrorUaAH reoMerpuqecKrax
Q[ry?.
III
sran
-
ronrpolrnlrfi
-
lpoBepr.rJr s$Qexrraanocm npogelannofi uarr,ru
pa6orrr no norrturenulo lpoanr ctlopMr,rpoBaHHocrH AaHHoro
HaBBrKa.
Onrrrna.r pa6ora npoBoArrnacr n 3 xnacce (yrurenr Curr,ronoaa E.A.). B
Knacce 23 )Harqlrxcr.
B KoHue I rroJD/roArrr 2017-2018 yre6noro roAa
)
,rreneM
6rma nporegena (oHTponEHaJt pa6ora. B ee cocraB Bxo.qllno 3aAaHr.re Ha HaxoxAeHr{e ruIorqaarr r,r nepr.rMeTpa nprMoyronBHuKa c 3aaaHHhrMr,r cTopoHaMr.r.
1 aapr.ranr. HafiTra nronlaAb u neprrMerp nprMoyroJrrHl,r(a co cropoHaMr,r
3cu
r.r
7 clr.
2 tapuatr. Haftra rnorqagr
r,r nepr.rMerp
nprMolronLHnKa co cropoxzrMr,r
4cM u 6 cM.
Pe:ylrraru nunonHeHrir
AaHHom 3a4auut 1yillll14Mvcr KJracca rar(oBbr:
7 veronex
-
cnpaaurl:lrcb c sa4aur4ev, 6es ouru6or< (30%);
8 venoaex
-
cnpaar,rJrr.rcb c 3aAaHr.reM
3
yuarqlxcr na:ruu nnouaAb,
Ho He
uacrrvno (350lo):
narulz nepuverp tlr,rryprr,
2 y,{aquxcr onpeAenr4Jrr, fieprrMerp, Ho He HarrrJrr{ nnorrlaA;,
3 yuaquxc.r .4orrycrr{Jrr,r BErqr,rcJrr4TeJrtnrre ouru6rr.r;
8 veloner
-
ne cnpaaurracr c
sagaHr,rer\,r (35olo).
florryuenxrre gannlre npe,qcraBaM
B BrrAe ,qr.rarpaMMbr:
Peryaorarur BbfnoIHeHrR SaAaHUA
I
CnpaBh^4cb
l9afir'{Ho
JlefKo
IInoIII4Ar
rz
poBaHHoCTu
3aMeTr.ITE, rITO
cnpaBrirucb
peynbTaTbl BhInOJrHeHI,L 3aAaHI4' Ha HaXOX(IeHIle
nepn'aerpa r[r,rryphr cBr,rAeTenbcTB],ror o HeBbrcoKoM ypoaue cQoprrau_
y yqaquxcr yMeHlr.s peuarb
Mr.r, nocKoJrr,Ky
3a.qaqH c feoMerprrqecKr.rMr,r BeJrr,rquHa-
ronBxo 'rperLr qacr6 yqarqIaxc{ Kracca 6egomn6oquo cnparr.rnacb
c aaHnoi 3aaa.reft. 3TI,I AaHHbIe IIO3BOJII-iJrU HaM C,{eJIarb
BbIBOA O HeO6XOArZlrOCru
npoBe,[eHl.re Aononnr.ne:rurofi Uenenanparlenuoi pa6oEr
no nalrnofi npo6.neue.
Ha o6y^raroqev grare onBrrHofi pa6oru naruz 6ul pa:pa6oran KoMrrJreKc
3alarr c reoMerpuqecKr.tMr4 BeJrriqr4Haruu pa:rmvnoft cJroxnocrr.t, rf oprr.rupyrorqua
oco3Hal{Hoe yMeHI4e pelxaTb 3a,qaqri SToro B}raa,
T,e. oco3HaHHble
xpaBrrJIbHBIe
pa3BepHyrbre aeitctuut, $opvrzpyrorquecx B nocrocHHo r43MeHrroruuxc,
ycJroBr,rrx u porBnrroquecq B perxeHHr.r HoBBrx 3aAaq.
Kouuer<c erux sagau npr,rnegEn
r
flpr,tJroxeHr,ru 3
r nolnou o6r6ue.
pa6o_
Ty c gTIlM (OMrrJreKcoM MBr npoBoar4Jlr Ha ypoK:x MaTeMaTr.rrn
u no aneypovuoi
.qetreJlrHocrr.r n 3 ueraepra sroro yre6uoro ro,Ea.3a"qa. nolpo6no pas6r,rpa:rucr
n
npoperxuBaJrr.tcE BMecTe
c rtaqr.rMr{c.g Bcero KJracca. orAentn;rivr )
raqr,rMc,
npeaJrararurcb ,qorroJrHr{TeJrBHbre 3a,qaHru aHalorurqHue pa:o6pannrrv
B Kracce
,qnr 3axpenneHr4{ c$opruuporannoro cnoco6a geficrar.rfi np}r perneHur{
3a.qarrn
KO|O Br,rAa.
ra-
66
flo oronqasr.lo npo4elannofi pa6oru
Tr,r yqr,rreJrb
lpoBer KoHTpon;Hyro pa6ory
r.I
IJrt
no.{BeAeHl.r_f,
zroroa 3 veraep_
c yt{auriMl.rct KJIaCCa no MaTeMaTI.rKe.
Ognau u: 3aAaHHfi 3roii pa6oru 6rr:ra :aaaqa Ha Haxox.qeHne
nepflMerpa
rIaAI-I np_{Moyrorbttl,rxa nogumeunoi cJIoxHocrr.r.
I
BapxaHr. Ilepuuerp [prMoyrorrbHr.rKa
36 cu. [puaa ero 5 cu.
r.r
nJro-
Hafi_nu
nnorl(aar, IIp{Moyf oJI6HLtKa.
2 rapuanr. fl:rorqa4r rsaapara 36 cu2. Hafigure
ero replrMelp.
Pe:y,rr,raru
BbIrroJIHeHrz_{
AaHHofo 3aAar{{, yqaruuMlrcs xnacca
yqeHuKa BblnoJrHr{Jrr.r 3To 3allaHl.Ie 6es
olru6or, I
y.{eHr.rK Aonycrr.rn
raroru: 22
olnxoKy
Bbr_
qzcJIr.rTer6Hof o xapaKTepa.
[o:ryveHnrre gaaHbre npeAoraBr.rM
B Bz.qe .{uafpaMMbr:
Peeyn urarur Bbtno,nHeHhr 3aAaHhR
(xoxrpoauxuri eran)
I CnpaBrrqcb
I qacrr,rHo cnpaBrrtrcb
14: anarpauurr BI-r.qHo, qro rroJlfreHHBre pByJIbrarLI
BrIrroJrHeHm aaHHoro
3aaaHr,r.{ cBuAereJrrcrByer o BbrcoKoM yporue
c$oplrr.rpoBaHHocrr.r yMeHr.l, pe_
Irarb
3a,aaqlt 3Toro BI.I.ua.
3ro o6crosrerr,crBo,
Qe*nruocrr pa:pa6orauuoro
qIzHaMI.I
B cBolo oqepeab, Aoxa3BrBaer gQ_
HaMu KoMnJre*ca 3aaaq c feoMe.rpfitrec.Ll
Mr.l
BeJrLr_
pa3finqHofi croxHocrr,r.
flpegloxenurrfi
KoMlneKc 3aAaq c reoMeIpHr{ecKr.IMU BerIU.{uHaM
,B
qacTHocTlr, 3alaq Ha HaxoxleHlle
nJro[Ia.q[ H nepxMerpa Quryprr, nosuoxno tc_
noJrb3oBarb
HaMr.r
npu o6yuenuu peueHriro
3aaa.r AaHHoro Br,r.qa
no moOrrna yve6no_
MeTOIITqeCKI,IM (OMIUIemaMII rIO MaTeMaTI,IKe, pa3pa6oTaHHbIM
AIS HatIaIbHOi
TUKOJIEI.
r aKIrM
oopa3oM, an-f, oBJIaaeHUt Mna,{uI,IMI,I IIIKOJIEHI.IKIIMT4 yI{eHI,IeM pe_
IIIaTb 3a.4aql.I c feoMeTpI,IqecKI.IMIt BeJII{qIaHaML B
COOTBeTCTBI.II_i C Tpe6oBaHfi_{MI,I
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Tarr.rr',r o6pa:orr.r, 3aaaqu
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160c.
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npenoAaBaH
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12. .(eMll.qora T.E, Ko:nona C.A.,
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Tomux A.fI. Marenaruxa 3 r.n., I v._M.:
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Kouona C.A., Tonxnx A.fI. Marelraruxa 3 ru.. 2 q._M.:
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f.B.,
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Mareuarlrra 3 xtacc.I.2.
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fapvonru)
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24. Konocxosa O.fI. Oopunpoearue peryJrrrr.rBHBrx yqe6Hbrx geicrnrafi
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30. N4arevarraxa.3
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36. flerepcon JI.f. Mareuarnxa. 3 rur. IL2.
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37. flerepcou
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(yMK flepcnerrr.ranar
74
IIPI4IOXEHVIE
flpx.roxexue
Kraccu{txaqul
3aAaq no pa3JnqHtru ocnonaxnru,{eMrlAoBoft T.E.
OcxosaHr,re
Bll.qoga.r xaparoepucfl,rra
Bugn :aga.r
Ns
rrlaccu$uraqun
cnoco6crnorarr
Hayr{Hbre
IIeru
1.
pe
I
,rKr,r
pa3BI,ITI.IK) MareMa.
H ee rrpHnoxeHr.rfr, HayKr,r B rle-
IIOM
reHr,rr 3aaaq
Qoprr,rupoaauue MareMarr.rtrecKrli
yre6nue
rnanr,rfi, yueur.rfi r,r nanrrrcor
y
o6y.
qAEMBIX
MATCMATI,IqCCKHE
Xaparrep
sce o6texrH luareMarl,rqecxlie
3aAaHI,lU
2.
o6rerron
TCKCTOB6IE
xorx 6rr ogun o6ten xalaercx peIIbI{EIM npeAMeToM
I-iJII.I
'BJIeHI.IeM
c
J.
Korn.{ecrso
ug6moqnlrlnr
AAHHbIMII
AAHHBIX
He,qocTaro[IxMn
coAepxar r.rnQopuaqr.uo, roropar
re HyxHa Art.r arrnonnerru rpe6o.
t
raIl
co.{epxar HeAocrarotrHo unQopua.
qlrr,r
AAHHbIMI,I
3al.at{tr
Anr Bhrnonuenlrx rpe6oraxnt
}a4aqtl
perxeHne 3aaarrfi cocTorrT B cTepeo-
TI.IIIOBhIE
rnrrHoM Bocnpor.t3BeaeHr.rr.r 3arreH-
nrx geficrauft
Yporenr
TBoPTIecKO-
CJIOXHOCTI'
BOCIrpOrr3BoArrque
4.
perneHxe :agavr,r rpe6yer nercoro.
pofi
r'.lo4a
Qrzraqt,tr :ay.rennux Ae[-
:TBr.ri B I,I3MeHr.IBnIr,Ixc.{ ycnoBl,Irx
TBOptrecKrae,
perxeHfie 3aAaqn rpe6yer nor,rcra
9Bpr,rcTr,rqecKr.re
HOBbX,
5on
el{e
geficrrrfi
HeII3BeCTHL[X CIOCO-
Ko
rcecrso
npocTbre
reHr.r.s 3aAaqu
rpe6yercx
BBrlorHHTb oaHo reicrBr,re
5. BBrnoJrHreMhIx nplt
perreHr,r[ AeficrBr4fi
Anr pe
Anf, perxeHnr oagaur,r rpe6yerct
COCTABHbIE
rrrnorsutr 6oree oAnoro .qeficrsut
77
llpnroxenue 2
IlaIrrsrnf, no saxorr(AeHnxl [eprrMerpa reoMerpnqeckux enryp
2.
3aAaqa
Hafign nepnuerp npsMoynMlulKa co croponaMn 2cM n 5cM.
O6rqcII-Eo:
ctroco6t qTo6bl
I
sailfi
tIepn{gTp rrp.nt oyro]rEur-rra, upnno ace 4 cropourt aJro)totr6.
3anua, e mempadu:
P
-2
+ 5 +2 + 5 = 14 (cM) rnvP =2
+2+
5+5
=
14 (cM)
Omeem: 14 cn nepuuemp npxrtoy?oJtbHuka.
9ro6u naira [epr{Msrp flprMojnonrulrra
olEoi cTopolllr, b - g.urua rropoi cropotoL
2 cnoco6:
a.lurEa
B Qopuy,ny auecro
ianua"
nprnaenro rfopuyny p = (a + b)
6).KB rrocTaBJrro ril3Becrlrrle ,UaIIIIBre p
. 2, t1e a _
= (2 + 5) . 2 = 14 (cu)
e uempadu:
P = (2 +
5)
.2=
14 (cM)
Omeem: 14 cu neputemp nptwoyeont Huka.
3.3&[aca
Hafi,qa nepnuerp reaapara co croionofi 3 cu.
O6rrcnsto:
I
cuoco6: qTo66r IIaiTIl nepnMETp xBa,4par4 uyxuo ace 4 cropouu cloloITr
3anuco e mempadu:
P=3+3+3+3=12(cM)
Omeem: 12 cu nepuuemp ksadpqma.
2 cnoco6: 9ro6u na.i'u lrepro{srp roaAltara rrprnreuro
{opuyay p = a . 4, rge a
pourr ruagpara. B Qopllyay ouecro 6y'nru a, rocraBJllo u3Becrnhre
Aaunbre
3anucv e mempadu:
P=3.4=12(cM)
- 4ruxa cro-
Omsem: 12
4.
cna
nepur.temp r1sdpsma.
3aneca
Anf,Ha olHotr u3 cropoH nprMoyronr,HnK& paBH& 6 cM, an[Hr apyrofi cropoEEr
6onBEe. HafiIu trep[Merp trpqMoyro,IbHl|Ka.
-
Hr 4 cM
y
trp-tr{oyrorbHnKa rrporD.raorroJroxEbre croporrbr paBrrEr, qTo6br HaiTn EepuMErp
flpsMolTonbEElc, ul)I(lro Bce 4 0TopoH6r cro)rolT6. Ho neulaecrna eropar sropollA na.!,q)/ cHaqana ee. B 3a,4arre cKa3aHo, rrro oEa 6oJr6me IlepBoii Ea 4cM, 6yay npn6aurerr: 1) 6 + 4 = l0
(cM) BTopas cropoEa. Bce cropoEhr r{3BegrHb[, Mory HatrTrr treprrMsrp, cJroxy Bce qgrErpg croO6:hrcs-Eo:
-
poElr: 2) 6 + 6 + l0 + l0 = 32 (cM)
3onucb a memDqdu:
l.
2.
6+4=
6+6+
l0
Broprlr cropolla.
l0 + 10 = 32 (cM) u]nt P = (6 +
(cM)
-
Omeem: 32 cn nepu,ueup
1
0). 2 = 16 . 2 = 32 (cu)
npt torzonu&ta.
5.3atraqa
AnrIEa oaHoi rc cfopol| nprMoyronbHtrKa paBHa 8 cM, rnBna.qpyrofi cropoHLr
uenrme. Haflgn nep[Merp nprMoyronbnrrKa.
-
Hr 5 cM
O6rror-no: V npn'aoyroJr6rr,r(a nporrrBorroJroxflEre cropoHbr paBEbl. qToobr Ea[Tu [9prtr'{Erp
rrp-ndoyroJcErrK4 E)r{so ace 4 cTopoHSI cno}qTb. Ho llel!3BecTEa BTopat cropoEa" EafiIy clra-
saraeg.B34qasecKa3auo,.rToosaMegrruerrepBotrEa5cM,6yryruwrarr:1)8-5=3(w)BTopa{ qrcpona. Bce cropoHbt u3BecrH!4 Mory EatrTtr rreprtr'{qp, cnoxy Bce qer64te cropoEr:
2)8+8+3+3=22(cM)
3snucb 6 mempqdu:
1. 8 - 5 = 3 (cM) -Bropar cropoEa.
2. 8+8+3+3=22 (cM)unEP=(8 + 3) .2=
Omeem:22
6.
c
ll
.2=22(cM)
nepu emp np*MotzoJtbHuka.
3anaqa
Haqeprn xBaapar, trep[Merl, Koroporo pareu 16 cu.
O6rrcu-no: 9ro6rr nacegll{rb lcaApar, Hyrcro 3Harb,ultEty cropolrbl, a oII& rrer3Becrrra. lI
3rran, qro y KBaApara Bce qgrhrpe croporrBr lxtBIIEr, rrro [eprMerp rr4.qpara panen 1 6 cu. Pa:,qeJEo neprMerp I 6 cM IIa 4 oroposEr, nony{y oArry cropoHy.
3anucc e memoalu:
16 : 4 = 4 (cu)
-
9epvy ruagpar
cropoEartBqi[)ara
Ilpx,roxenne 3
3aAaqu na Haxor(4ex[q neprrMerpa rr rrJrorqaA[
reoMerpfiqecnltx enryp pas,uvxoft cloxnocru
l. Hafigu nepuuerp (Ba,4para co croposofi g cu.
Peureuue:
8.4=32(cr,r)
Orner: nepuuerp raa4para 32 cv..
2. Hait4u nepwlenp KBa,qpara co
croposoi
16 cu.
Peuresv e:
t6.4=64(cu)
Oraer: nepr,rue,rp nra4para 64 cu.
3. flepurr.rerp rna4para 16 clr. Haft4u ero cropoHy.
Peruenue:
16:4=4(cu)
Orrer: cropona
xna.Apara 4 cM.
4. Hafi4ra nepuuerp [prMoyroJrbHrrKa co croponavu 9 z 6 cu.
Pelreuue:
(9+6) 2=30(cu)
Orner: nepr.ruerp npxlroyro:nnr,rrca 30 cv.
5,
Hai4lr nepr.rverp np{Moyronbxr.rKa co cropouauz 7 u g cu.
Pelrenue:
(7+8).2=30(cru)
Orner: nepr.rverp npruoyronrur.rra 30
crra.
6. Haianu atuny nprMoyronBur,rKa, ecnu ero rrrfiprrHa 7 clr, a nepr.ruerp paseH 40 cM.
PerreHr,re:
Bapr,raur
I
V nprrraoyro:rruura
HBrX r]Ilrpr.tHBr
r,r
nporrzBorronoxHbre cropoHLr paBHBr, To earb ABe paB_
,qBe paBHhrx AJrriHBr.
81
Ecrn ogxa nzpnra (cropolra) 7
cv,,
ro u apyras (uporr.monoloxnar) roxe
7 cvr,
7+7:14(cM)
flepauerp cocror,tr rr3
cyMMbr annH qerhrpex cropoH rrprMoyronBHuKa,
c),I{My aB}4( npoTr,rBo[oJroxHhrx cTopoH Mbr yxe y3HaJrv, TofAa c]a,rMa AB),x apy_
rlrx nporr,rBorroJroxuux cropou (Alun) 6yger paara:
40 - 14 = 26
(clr)
Tenepr y:uaev 4lr,r ny oauofi cropoHbr:
26 :2 = 13 (cu)
Oraer: glnua nprvoyronruuxa 13 cu.
L4JIIL
Baplranr
P
:
II
(a + b) . 2
vru
(a+b).2=P,
rgea-
nepurr.rerp nprMoyroJrbHnKa
-
g.nr,rna
=
?,b-
utupuua= 7 cu,P- nepuruerp = 40 cu.
Cocraauu lparueuue:
(a+7)'2=40
2a+14:49
2a= 40 - 14
2a=26
a:26 :2
a=
13
Orner: 4rrana nprMolror;HnKa 13 cM.
7. Hafi.lu AJrrxry nprMoyronbHr,Ka, ecJrr{ ero .qnr,rHa 10 cv, a nepzrr,rerp paBeH 30 cM.
Peuremre:
Baplranr I
V npruoyronrnura nporr.rBonoJroxHbre cropoH;t palHbr, To ecrB ABe paBHErX rIupr.rHr,r
u
.qBe paBHbD( AJrr.rHbr.
82
Ecnu oAHa AnrlHa (cropona) l0 cu, ro
1r
npyrar (nporurouonoxnax) roxe
l0 cu.
10
+ l0 = 20 (cM)
flepnuerp cocronr
cyMMy
r,r3 cyMMlr ,qnlrH
aBlx rlpoTnBonoJroxH6rx cTopoH
Mr,r
verrrp€x cropon nprMoyfonrHr,rKa,
yxe y3Hiurr{, Tor.aa c),}.rMa AByx apy_
rurx porr.rBofloJroxnux cropon 6y4er panna:
30 - 20 = 10 (cM)
TenepE )3HaeM rxupary oauofi cropoHhr:
l0:2=5(cu)
Orner: 4:ruua np.nuoyronrnuxa 13 cu.
}ITIII
Bapuanr
P
:
II
(a + b) . 2
-
nepnuerp nprMoyronbHr.rKa
LIJIII
(a+b)'2:P,
rlea-
Anr{Ha
= 10crr,r,b- uruplrua =
?,
P-
nepuuerp = 30
crr,r.
CocraBuM ypaBHeHne:
(10+b).2:30
20+2b=30
2b=30-20
2b=
10
b=10:2
b:5
OrBer: uupr,rHa nprvoyrolruuxa 5 cu.
8. Illrapuna nprMolronbHr4Ka 14 cv'. lpuua ua 5 cu 6ontlre. Hafian ero
nepr.rMeTp r,r [noqa,4b.
PerxeHr4e:
14+5=19(cM)
(19+ 14) .2=66(cyr)
19 . t4 = 266 (cu2)
83
Orrer: neprnrerp
nprMoyronbHrrKa 66
cu; ruroqa4r npnr,royro.nrnura 266
cM2,
9. .Qrnua upxr.royrolssnxa 7 cu.Illr,puna xa 3 cM MeHbrrre. HafiAu ero ne-
plMerp u rurouaAb.
Peurenrae:
I - 3:4 (cll)
(7 +4).2=22(cu)
l'4=28(cu")
Orrer: nepr-ruerp nprMoyronbr{r.rKa 22 cu; nnorqagr upxr,royrolrnrra 2g
cM2.
10. flepnMerp KBaapma24
cv.Hafi;lu
ero rrJrorqa,q6.
Peurenrae:
24:4=6(cw)
6.6=36(cu,)
Orrer: nnorqagr rnagpara 36
1
1. flepr.rMerp KBaApara 36
crrl2.
cu. Haigu erc urorlaAl.
Pelrenue:
4:9 (cll)
36
:
9
9=81 (o"r')
Orser: nroqaAr rnagpara 81 cu2.
12. y.{eHHKy H}DIGo 6uno uavepnurs rpf,MoyronbHr,tK co croponaun 5
9 cu, a on naueprr,lr ero co croponauu 6
r,r
crr.r r,t
8 cu.
Ha cronrro cu2 on olru6cs?
Peurenue:
5.9:45(cv,)
0.8=48(cr'1,)
48-45=3(cxr2)
Oreer: oH orxlr6cs
Ha
13. Vvexrary nylxxo
3 cn2.
6ulo xaveprnrr lprMoyronlHr,r( co cmpouaura
u 8 cu, a on xauepruJr ero co cropoHauu 8 cv
r.r
6.
10
cu
;F
84
crolrro
Ha
cu2 oll oltu6cx?
Peurenrae:
l0 . 8:80 (cu,)
8
6=48(cu,)
80 - 48 = 32 (cM,)
Orner: oH olrz6cs na32 cuz.
14. flepuuerp rprMoyroJr;HxKa 36 cu.
,{lnna ero 4 cu. Haftnu nJroluaAb
nprMoyron6Hr.rKa.
Pelrenae:
4++=61"t,
36-8=28(cu)
28:2=14(cv.)
t4.4=56(cvr2)
Orrer: n:roqagr np.auoyrontnraxa 56 cu2.
15. Cropoua rcaApara 6
crra.
Haiqr,r uuplrHy nprMoyroJrbHrlxa c raKHM xe
fiepr{MerpoM a.qJrLrnoi 3 cv.
Peruenze:
4.4=24(cu)
3+::61""'
24-6:18(cM)
18:2=9(cu)
Oraer: lrupnua nprrraoyro:nunrca 9 cu.
16. Cropoxa KBaApara 18 cv.
Haigu
nepr.Me'rpoM lr rxupr.rHofi 14 cM.
PeIneHr.re:
18.
14
4:72
+
14
:
(cnr)
23
1.t;
72 - 28 = 44 (cM)
44:2=22(cv.)
Orner: 4luna lpxuoyro:rruura 22 cv.
AJrr{Hy nprMoyroJt;Hr{Ka c raKr.rM
xe
85
17. flnorqaAr npff\.{o;,rolurnr<a 40 cur. Illupr.rua ero 4 o,a.
9euy paren nepuuerp npruoyro:rrnr,rxa?
PelreHr,re:
40:4=10(cr*r)
(10+4).2=28(cu)
Orrer:
ueplrnaerp npxuoyro;txr,ma 28 cu.
18. flnorqagr npxuoyronrnuxa 40 cu2.
[4lrna ero g cu.
Yevy paaen nepuuerp npxuoyronrnr,rra?
Pelregrae:
40:8=5(cu)
(8+5)
2 =26 (cr,.r)
Orrer: nepr.ruerp npxuoyrolrur.rra 26 cu.
19. Illupuua nprrraoyronlHlrKa a = 15 crr.r, g:ruxa b = 20 cu .
HafigH gruuy Apyrorc lprMoyronsnnrca c rofi xe finouaAbrc,
CCJII,I ETO
a 3 pa:a ueHBrrre ruupr.tHBr rlepBoro nprMoyroJrbH[Ka.
IJrr.rpr.rna
Peuesrae:
B nepBoM
AefrcrBr.rz;nlra€u n:rorqagr no
15 . 20 = 300 (cMr)
-
S oguoro
r.r
Soplryle a . b :
S
gpyrofo nprMoyronbHr.rxa
Tenepb rur{pnny BToporo
15
r.t
:3 = 5 (clr)
-
urupr&a gpyroro nprMoyronlHr.rKa
orBeqaeM Ha Bonpoc 3aAaru npr.rMenur
Qoprray.rry S : a = b
300:5:60(cu)
Orner: nnrxa gpyroro nprMoyroJlbHr,rra 60
crr.r.
20. ,{nuna np.auoyrontnprra b = 32 cu. IIINpr,rna a = 4 cu.
llafiau
pSu:rry
Apyroro [p.{MoyroJrbHrKa c raxofi xe mroqaAbro,
tsupuua s 2 pasa 6oJr6rne rrrfipr.tH;I nepBoro nprMoyron;Hr,Ira.
Per:rer*re:
y3HaeM nnorlaAb nprMo)rroJrbHr,rKoB no
.
Qoprrry:re a
32 . 4= 128 (cvr)
-
S nepBoro nprMo)ronlHuKa
TenepB rrrupr.rHy BTopof o llp.f, MoyronBHr,rKa
b:
S
CCJIH ETO
85
4 . 2 = 8 (cM)
mrpr,rHa Apyroro nprMoyronBur,rKa
-
S:
nprarrlem,ra Qoprvryny
128
:8 =
a=b
ysuaeu g;rurry Apymro
16 (cr"r)
Orrer: glr.rna gpyroro fiprMoyronEHr.rr<a
16
cu,
21. Kaxofi yqacroK 3eMn[ norpe6yer 6onrnr]ro orpary: nprMo1TonBHlr,
pa.:ueparr.rz 32
rvr ra
2
lluttu rsanparutrfi, raueroqufi ry xe morqagt?
Peuesrae:
I.
flpruoyroltHui yracror
32 . 2: 64 (vl2)
(32 +
2)
-
S
npxuoyrolrnoro yracrxa = 64 (ur)
2 = 68 (cu)
-
P nprMoyronbHoro yqacrxa
:
6g (cu)
IL Kragparnufi yuacror (uuerorqni nnoqa,qb nprMoyroJrbsoro = 64 u")
Ec.nn S rnagpara = a . a, Tora4 ns Qopr,rynu,
1:u6eM croporry KBa,qparHo_
ro ytacrxa S: a=a
(y xragpara ace cropoHbr paBnbr, Tor1a
- ra6nurry lnrHoxeHrit
3HaqeHrrr a r,r 3aMeHr,rM rD( _ 8 8:S gnn 8.8:64
Mhr 3HaeM, no46ep€u
ul;u
64=8.8
r.rm
64:8=8)
64 : 8 = 8 (u)
8 ' 4 = 32 (M)
-
III.
a. a= S
rno6ax cropona rna4paruoro yuacrxa = g
fiepr.rMerp KBalparHoro yvacrra = 32 (rrl)
P npxrr,r. - P xaagp.
68 - 32 = 36 (ru)
(u)
-
:
pa.snuqa nepnMerpoB
pa3Hr.rqa nepr.rMerpoB
Orrer: norpe6yer 6o.mlryro orpa,qr npxr,,royronrntrfi ua 36 rr,r.
22. Raxas rouna:ra norpe6yer 6olslre ru[,rHTyca: npf,MoyronbHalr pa3MepaMr,r
4 M n 9 M I-rnr KBaaparHar, rrMerorqax
ry xe nnorqa4r?
Peueune:
(4 + 9) . 2 = 26 (M)
4
9 = 36 (ru,)
-
S
-
P nepnuerpr nprMoyronrHot KoMHarLr
flno raa6 nprMo)rronBHofi
KoMHarEr
a7
(ri3 ycroBr,rr 3a.qasu KBaAparHa,
KoMHara nMeer
ry xe [noua.{6 36 M2, a I.I3
onpereneHl'I{ nJIoIqaAn KBa?qpara 3HaeM, qro
Bce cropoHBr paBHbr a = a: a = a,
cuo:rpr'ru ra6llrqy yMHoXeHUr U Bri.{au .
6 6 = 36, ro ecrb nro6a, r,r3 cropoH a= 6
3anhlrleM (npnseadM) Qopuy-rry nnoualu
KBalpara S=a.a B QopMy
HaxoxleHr.rq ee cTopOl$r S: a=a
36 : 6 = 6 (r,l)
6 . 4 = 24 (u)
-
rno6a.e ua cropou xBa,qparHofi (oMHarBr
p nepriMerp KBa"qparHoi
KoMuarrr
26-24=2(u\
Orner: norpe6yer 6omure n,ruuryca npxvoyrolrnax
na 2 lr.
pcuEp
Pe6po xy6a paato 2 caxrnlaerpoa.
EepuxH
Haim n:roqa4r o""* ,p*"1 *y6r.
Pelrenr,re:
Ky6 -
vuororpaHHr.tK, rroBepxrocrr
crr.r oAr{HaKoBBrx rro IlnouaAlr KBaAparon.
(noaepxrocrefi).
IIe_
5:(a a).6 - n.norqagt acex
2.2.6=24(cu2)
Orner: nnorqa4r acex rpanei
I,r3
V ry6a 8 neplrnu, 12 p€6ep, 6 rpanei
= a .a - finoulaAb KBalpara, TorAa
rpanefi ry6a, n: yclosur :a,4aqr.r a = 2, rorAa
Ecnra S
KOTOpOfo cocTouT
r<y6a p assa 24 cu2.
8a
24.
c.
B.
trh raa4para
Bbrpe3anr4 nprMoyroJr;Hr.rK
(cv. pr.rcynor). Hafi.{ure moua,{r
nonY'rurllrefi cr QuryPlr.
fux
peurenur uorpe6),rorc{ OopMynbr:
a a; S = a': nJroula,al KBaApara (y rraapara nce cropouu paruu)
S=a b
n.trorr1aAb rrprMoyroJrbrrr4Ka (y npxuoyrolrnlrKa npornBono-
S=
roxnue cropouu paeuu)
Aaree Bce oqeHh npocro:
KBaapar A.
S:a a unna a=S Qopuyna nJronlaAr{ KBa,qpara, roraa
8 8 = 64 n:rouaAb KBa,qpara
S=a a ulua b: S $opuyna nnou{aAr.r nprMoyron;Hr.rKa,
4 l = 4 rrJror{ag; Blrpe3aHHoro nprMo}Tonr,Hr.rKa
r43
Tofga
nnoulaAl,r KBaApaTa BbITITeM nJlolqaab BEIp$aHHofo flptMoyroJrbHrrKa
64-4=60
Orser: nro[IaAr nofly.{r]Blneficr Qurypu paana 60
KBaapar B.
S:a
7
S=
a lrrr.r a'a=
7 = 49
a
-
-
Qopuyna nnoqalrr KBa.qpara, Tor,qa
ntouaab KBarpara
a uru
4 2:8
S-
a
b=S
-
OopMyra
nroqalu
nprMoyroJrbrrr.rKa, TorAa
nrorlaAr, Bbrpe3aHnoro [psMo]TonbHriKa
u3 nJroqa,4n KBaapaTa Bbrr{TeM
[IoruaAL IIptMoyfonbHIIKa
49-8:41
Orser: nroruaar nonyrlrnneftcr Qlrrypu paaua 4l
89
Ksa,qpar C.
S=a.a r.rrua.a= S _
7 . 7 = 49
S=
a
-
rorla
nnotslaAb KBaApara
a mu a.
5.1=5
r.r3
-
OopMyna nnonlaAr KBaApara,
b=S
-
Qopvy:ra nnorua.{u nprMoyronBHriKa, TorAa
nnoqaAb BBrpe3ar{Horo [prMo],r.oJr6Hr,rKa
nnoqa,4r.l KBaApaTa BLrqTeM
InorqaaE np.f,MoyfonbHriKa
49-5=44
Orser: nrorualr nolyur.mureficx
Qurypu panua 44
25.
Ilnollagr ogxofi
r:rerr<u parna I cu.
HaiaHre nrouaas
Qaryprr. N:o6paxeHHofi
Hafinure n.noruaar
r.rso6paxeuHofi
Ha pacyHKe
A.
Qurlpu,
Ha pucyHre B.
Hafi4ure nnorqaAr
Qnryprr, r,r:o6pax€uuoi ua pvcynre C.
Hariaare nloltla4r
@urypu, r:o6paxeHHo, Ha pficyHKe D.
Hafigure nnorqagr,
Quryprr, r,r:o6paxeHuofi Ha plrc),HKe E.
Onpe.qelenze:
Henpaau,rrnufi qerlrpexlronlHr.r( _
$urypa, y roropoi cTopoHbr
I{ He napaJIneJIBH6I.
He paBHbr
Henpanu.lrxrrfi qerBrpexyronBlr{K _
O[rypa, y Koropofi cTopoHbt
paBI|bI |l He I|apaJIJIeJtbHbI.
D,
E.
He
90
A, B, D sa
Peruenue: pa3o6;gM HenpaBr4Jrbubre uerrrpexyro.nrnur<u
[prMo]'ronbrrr,x rpeyrorbHl,IKa I{ nptMoyfonl'HrrK, a
yrorururu
C, E na aBa
gpzMeyIeMEIl rpeyroJlrHuKa
He[paBItJIbHEIe
.qna
qeTbrpex-
Ir KBa,qpar.
flpulrenua Qopuy:ru nJrorrlaAn rpe)Tonr'HrrKa, KBa"qpara I{ flp.f,MoyronbHxKa JrefKo perIuM nocTaBneHH),ro 3aaaqy
@u4pa A
a
3 4:
S=
S=
b
12
%'
QopuyJra
-
a
cr'42
h
nrouallr
nrouraar nprMoyroirbHlrKa a
-
Qopnyra
-
nptMoyroJlbHl4Ka, roraa
ulo
raalr rpe)T onbHlrxa, roraa
'l 5 = 2,5 ulr' nroulaar np.sMoyfon;Horo rpego;rluura
S:%'a' h- Soplry:ra nrorqaAl.I rpeyronlHHKa
%
Yz
'2 ' 4 = 4
cu2
b
nnoruaAb lpsMoyroJlbHoro rpeyronbrrr.rra c
-
Terrepb c:roxrrB nonyr{eHHbre
nrouaru,
y3HaeM nomryro
urorqagr Suryprr A
12+2,5+4=18,5cu2
Orrer: nnoqagr Slrrypbr A 18,5 cM'
cDuzypa
S=
a b-
B
Qopr'ryJra nJIorIIaaH nptMoyroJrhHliKa, Tor,qa
5'1=5 cr'r'-
n.norualr nptMolroJlbHt{Ka a
S=%'a h-
Qopnyna nroIrlaalr rpeyronr'Hl4r{a, roraa
%'6 ' 5 =
15 cM2
-
nnouaaL np.{Mo}T onbHofo rpeyrontnura b
% a h- $opruyra n:roruaau rpeyroJIBHI.rKa, roraa
% | ' | = 0,5 cru2 nloulaar nptMoyronbHoro rpeyroJrbHnxa
S=
-
Tenepb cJroxrrB IIoryqeHHbIe nnoulaarr, y3HaeM aonHyrc
c
nlorqagr Quryprr B
91
Orner: nroqa,6 Qrrypr,r B 20,5
cIr.r2
@u4'paC
a' a; S = a'
5 ' 5 :25 crr,r2
S=
QopMyra rIrolqaArl KBa,qpara, rorAa
-
nnolqag5 rra,4para a
a'
htpopuy.ra nnoqaAl4 TpeFonbHlr(a' rorAa
S=Vz
' 1 ' 6:3
%
cu2
S=%'a h-
rurorqagr npxuoyro.rrnoro peyronruura b
T
-
Qoplryra
%'I'5=2,5
cru2
-
mo IaEI TpeyroJIbHHKa, rorAa
nrolqaAr npxuoyroJllgoro rpeyroJlbul'I(a c
Tenepl cnoxr,rB rloryqeHHhle moqaAlt, y3uaeM nonrylo ntrolqa4r rfurypu C
25+3+2,5=30,5cu'?
Oraer: rnorqa'qr $Iarypu C 30,5
D
<Duzypa
S
=
a'
b
rDopuyna nnoIqaAll nptMoyronbHuKa, TorAa
-
3 ' 4 = 12 cu2
S=%' a'h
-
-
lttorlaAb rIptMoyroJIlHI.IKa a
tpopuy.na nnoqa,{}r rpeyronlHlrKa,
% 1 ' 5 = 2,5 crvr2
S=% a hVz
cru'?
'2 ' 4 : 4 cu2
-
roraa
rurorqagr nptMoyrollsHoro rpeyrolurraxa b
Qoplryla moqaau TpeyroJtBHI'IKa, rorra
-
rllrolqaab nptMoyl'onbHoro rpe)'roJlbHuKa c
Tenepb cJroxr.rB rronfreuHEre
finorlar]I, ]BHaeM noffryo nnoqags Qurypu A
12+2,5+4=18,5cM'z
Orrer: nnoqagr r[urypu A 18,5
cIlt'z
@u4'pa E
S=
a
a; S
2.2=4
:
cu2
a2
-
$oprnryra nnouqagu KBaApara, roraa
nnoqa,qB KBaapara a
'
h
tpopuyra nuorlaAlr rpelroJIlHI'IKa, roraa
Y"
a
S=
TUIo[IaaE flphMoyronrnom rpeyronrnrlra b
Yz ' 3 ' 4: 6 cxt2
S=%'a h- Sopuyna nnolqa.ql 'rpe)'rorlulrKa, ror,{a
ruIoua,ql [ptMo)roJlbHoro rpeyronlHlrKa c
Yz ' 2 2 = 2 cu2
-
TenepL cJro)KuB norytleHubre nnotrIaAlt, y3HaeM nornylo IrJloIqa.qr Ourypu E
4+6+2:12cM2
l
92
Orser: nrouaar QuryPu E 12
cu'?
26. Hafi,qure moutagu u nepuuerpu Qurypor. C4e:rafire arrno4'
c.
B,
A.
E.
D.
Onpeae.nenne:
cyMMa AnnH Bcex cropoH
Ilepxuerp -
rDl|rypu' BbIpaxeHHbIi B MIrJrJlrrMer-
pax, caHTHMerpax, aellHMerpax, MeTpax rr T.a'
IlnouaAr, {lrrypsI
-
reoMerp[qecKoe rloHrrfie' pa:uep n"rocroft Qurypu'
nutpaxexxr,rfi n uM2, cu2,
Ilycrr raN4ar
I,I3
gu raagparaS
=a
@urypa A
1.4=4
+4+
cropoH KJIerKu paBHa
a, noJrfrllM
I cu'
tor.4a. npIrMeHHB Qopvyly nloiqa-
nnouaal oAnoft r.nerxlt
I
1= I
crr'1'z
nprvoyroJllHltK, cocrotlqufi u3 '{erblpex Kreror no I
crr't2,
tor4a
1
LIMeer qerbtpe cropoHEl, TorAa
+4
@urypa B
I
Il t.A.
cu' - nrorqa4r Qurypu;
Q rypa A
1
-
AM2, M2
-
:
10
cu - nePnuerP QuryPst.
rragpar, cocrotlqufi I.t3'{erblpex K,'IeroK no I clr2, ror4a
. 4 = 4 cv'? - nnouraar
ourypu;
$urypa B LIMeer qerblpe cropoHbl' roraa
2 + 2 + 2 + 2 : 8 cu - nepr'ruerP QuryPr'r.
@urypa C - HenpaBurlHrrfi MHoroyroJIbHIrK,
crra2,
l
cocrosuli
torAa
.4=4
cu, - nJrolllaAr $nr1pu;
Qrrypa C IrMeer [ecrb cropoH, roraa
3+ l+2r | 12+l-- l0 cv - neprverp Snryput.
Il3 qerblpex KreroK no I
r
93
@urypa D - HeflpaBnr6HBti MHoroyroJrbHr,tK, cocro.slllui Ils qerrrp€x x.nerox no I
cu', ror4a
|
4= 4 cu2 - ntorqaAr Quryphr;
t[urypa D
1
+
1
urr.reer noceMB cTopoH, TofAa
+2+ I + 1 + 1 +2+ I : 10cu-nepnuerp$urypu.
@lrrypa E - HenpaBnnbHBrfi MHoroyroJrrHr.rK, cocroarqlrfi u: .rerrrpdx r<;reror no
cu2,
1
lotga
I 4=4 cM- nrorqa,qt Sury?H;
Qnrypa E uveer noceMB cropoH, rorAa
I + I +1+3+ I + I+ I+ I=
10 cM - nepurr.rerp <puryp6l.
Bunog:
(Dnrypu A, B, C, D, E raueror oAr,rHa(oByro nJrorrlaAb, Ho HarzMeH;urrfi nepuMerp
14MeeT KBaApaT.
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neprrMeTp Bcef.qa [MeeT KBaapaT.
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[email protected]
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ELIBRARY.RU. ]\TIOAYT6 NO!CTA 1,4HTEPNET, (O'IEKqhi ELLBRARY.RU,
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B.A.
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Hacaro npoepM: 11.07.2013 14:49:36
ArrrubHocrb npoeePkr: 00:00:10
3arpy:Ru: 0o:00:02
l,4Mr u, roaHoro Oanra:
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dHrrnnarrarF htt.t/!nivore antlplasiat.r!
Haqano3arpy:xor 11,07,2018 14;49:33
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