NGHIEN CUU & UNG DUNG
Id do thay ddi thi the tich se ihay ddi nhu the ndo?
Trong trudng hgp ndy co hai yeu to quyet dinh den the lich do Id cgnh day vd cgnh ben Nhu vdy HS se cd hai hudng de mo mdm vd du dodn. ta
Hudng 1: Cho a ed djnh, b bign thign khi do
Din Luuin nHnc uic TIT DUU SHHG THO CHO HOC SIOH OOH VIEC SHRG TOO BfJI TOHO Cl|C TR! HJIIH HOC KHOHC GIHO
ThS. Biii Thdnh Trung Trudng Cao dang Kinh te - Ky thuSt Diin Bien SUMMARY
Train capacity for creative thinking high school students through innovative problem extremum stereometry
This article presents a summary of some operators towards common geometric problems to create the problem of extremum stereometry. This problem helps teachers guide students creative problem in extremum stereometry lo formation and development capacity for creative thinking in students.
Keywords: Creative thinking capacity: Creative problem; extremum stereometry Nlt^n bdi ngay: 15/5/2014. Ngay duy^t dSng: 30/5/2014
\. Dat vin dl
Sdng tao bdi todn mdi Id mpt bude quan trpng eua qud trinh gidi todn, mgt phuong thde ren luyen nang lye sdng tao todn hpc, mdt trong nhUng myc tidu chinh eua hgc tap sdng tgo. Viec phat hien, khai thdc vd de xudt bdi todn mdi giup HS rgn luyen khd ndng khdi qudt hda tri thdc cung nhu phuang rgn luy^n cho HS nhin nh§n mpt edeh tdng qudt edc tri thde dd hgc, phdt triln nang lyc vdn dyng linh hogt cde thao tdc tri tu§ td dd phdt triln nang lyc tu duy sdng tgo cho HS. Bdi bdo sS trinh bdy mpt sd hudng sdng tgo bdi todn eye tri hinh hpc khdng gian nhdm hinh thdnh vd phdt trien ndng lyc tu duy sang tgo cho HS nhu sau:
II. Nghien cuu sdu bdi todn da biet bdng cdch biln si hda
Tir nhirng bdi loan dinh lugng don gidn. da biet cdch gidi ta cho mgt so dgi lugng bien thien de dugc bdi loan eye iri hinh hgc khdng gian
Vidu 1: Ta xit bdi todn dinh lugng sau Cho hinh ehdp dlu SABC cd cgnh ddy AB - BC
= CA = a. cgnh ben bang b. tinh the tteh hinh ehdp.
Ddy Id mdt bdi todn rdt quen thupe, vigc giai bdi todn ndy rIt don gidn phii hgp vdi hdu het HS THPT.
HS tfnh duge thl tfch hinh ehdp
phdp; cd hdm s6 V, (b)
12 -a
HS cd the su dyng dao hdm dg kilm tra xera hdm sd cd eye trj theo biln b hay khdng. Do V hdm ddng bien theo b tren
^
;-l-co ngn V khdng ed GTLN vd GTNN. Do dd ta khdng cd bai todn cue tri ndo.Hudne 2: Cho b cd dinh cdn a biln thign khi dd ta cd ham sd V..
12 -a HS cdthe sir dung dgo hdm d6 xgt xem V cd eye tri theo a hay khdng hoac cd the su dyng bat ddng thue cdsi nhu 12V2 ( 6 b ' - 2 a ' ) ap dung
ib^ - 2a^ ta duoc:
rSO.S., 12
2b Tir bai todn dan gian nay nguai thdy co the dinli hlt^ng cho HS khai thac de sdng tgo ra cdc hdi todn c^tc Iri hinh hQC khong gian mdi Trtrdc het BS cdn nhin thdy bdn chdt ciia vdn de dd la. The lich cua hinh ehdp ph\t thudc vdo yiu Id ndo? Khi cdc yiu
^ a l a ' . ( 6 b ' - 2 a ' )
= > a ' . a - . ( 6 b ' - 2 a ' ) < ( 2 b - ) '
Ddu bang xay ra khi a = 6v2 tCr do ta c6 Vay f^,| CO GTLN theo a.
, . ± . 2 . b ' . ^
' 12 6
TAP CHl THifr BI GIAO DgC - sd 106 - 6/2014 • 21
NGHIEN cuu & UNG DUNG
Td ddy ta ed bdi todn thd nhit
Bdi todn 1: Cho hinh ehdp dlu SABC cd cgnh ddy AB = BC = CA = x. Cgnh ben bing a. Tinh thl tieh V hinh ehdp theo a vd x td dd tim GTLN eiia V khi X thay ddi.
hudng thd 3.
Hudng 3: Ta dat them mdi quan hg gidng bude giifa canh ben vd eanh day khi ehdng thay ddi, ehing hgn a + b = 1. Khi do ta cd bai todn md nhu sau:
Bdi todn 5: Cho hinh ehdp tam gide dlu SABC cd tdng eac cgnh bing 3. Tim GTLN cua the tich hinh ehdp.
Khi gidi bdi todn ndy gpi canh ddy Id x khi d6 cgnh ben Id 1 -x, vdi 0< x < 1. Tinh toan ta dupe thl tich V = — x^ yjlx^ — 6x + 3 . Dimg dgo hdm tim eye tri ta duge kgt qua V/^
Nhu vdy ta thdy dupe trong hinh chop tam gide deu ed cgnh bgn cd dinh thi thg tich hinh chop sg cd GTLN vi the ta ed the phdt bigu bdi todn md nhu sau:
Bdi todn 2: Cho hinh ehdp tam gide deu SABC cd cgnh ben bing b. Tim GTLN thg tich hinh ehdp.
Vdi bai toan 2 ndy HS ed the thdy dupe dl tinh dupe thl tieh hinh ehdp ta edn phdi dSt them bien cho bdi todn. The tich hinh ehdp phu thudc vdo ede yeu td nhu cgnh ddy, dudng cao, gdc giu^ eanh ben vdi mdt ddy...Tu dd HS lan lupt dat eac bidn cho bdi todn theo cdc yiu td co ban trong hinh ehdp sg phdt bilu dugc cdc bdi toan nhu sau:
Ngu dat dudng cao SO = h ta dupe bai todn 3 Bdi todn 3: Cho hinh ehdp tam gide deu SABC ed canh ben bdng a. dudng cao SH = x. tinh thl tieh hinh ehdp theo x va a. Tim x de the tieh hinh ehdp Idn nhdt.
NIU dat gde gitlra cgnh bgn va mat day (ABC) la X ta ed bdi toan 4
Bdi todn 4: Cho hinh chop tam giac dgu SABC ed canh ben bing a. Gdc gida canh bgn vd mdt day Id X. Tinh thl tich hinh ehdp theo x va a. Tim x dl thl tieh hinh ehdp Idn nhit.
Nhu vgy vdi bai todn gdc ngudi thdy khdng nhimg day dugc cho HS ky nang gidi toan md edn giup HS ed khd nang md mim du dodn, ed kha ndng tdng hgp kiln thdc, nhin nhdn duge bdn chat van dg dd Id nlu hinh ehdp tam gide dlu cd cgnh bgn cd dinh thi sg cd thl tieh Idn nhdt, td do HS bude ddu dupe sang tao bdi todn mdi, gdp phin hinh thdnh vd phdt triln ndng lyc tu duy sang tgo cho HS.
Ddi vdi HS cd kha ndng tu duy tdt hon ngudi thiy cd thl hudng din cho HS khai thdc bdi todn theo
Tuong ty ta cd thl cho a, b thay ddi thda mSn a^ +b^ = k ( khdng ddi) khi dd ta cd bdi todn sau:
Bdi todn 6: Cho hinh ehdp tam gide dlu SABC ed cgnh ddy a, cgnh ben b thay ddi ludn thda mSn (a^ -Hb^ = k khdng ddi). Tim GTLN cua thl tich hinh ehdp theo k.
Gidi: HS tinh dugc
V = — V 3 b ' - a ' = — - J 2 a ' . 2 a ' . ( 3 b ' - a ^ ) 12 1 2 2 ^ ^ Ap dyng bat dang thue c6 si cho 3 s6 2 a ^ 2 a ^ 3 b ^ - a ' taduoc:
a ' + b ^ > ^ 2 a ^ 2 a ^ ( 3 b ' - a ^ ) c=. (a^ + b ' y > 2a^2a^(3b^ - a' ) tir M ta c6 V < — k ^ Ichi di5 GTLN
24 k^/k . . . . >/2k ,,x = khia = b =
24 2
! • TAP CHi THIET BI GIAO DUC-S6106-6/2014
NGHIEN CUU & UNG DUNG
Nhir v$y vdi vi§c bi€n s6 h6a cdc dai lirgmg quen thuOc (canh bin, canh (My, gcSc...) du4i Sir din dSt cua ngudi thiy HS kh6ng nhChig dtrgc rfen luy§n ky nftng tinh thd tich hinh ehdp mS c6n du^c trvc tidp s ^ g tao ra bai toSn mdi. TCr d6 HS dugc hinh th^nh va rin luy^n nSng lyc tu duy sang t ^ .
2. Tir dSng thue hinh hpc chuyin sang bit ding thDt;
Xudt phdi lit cdc bdi todn ddng thitc hinh hoc, la dp difng cdc bdt ddng thirc quen ihudc vd tit do phdi bieu thdnh cdc bdi todn ctfc tr] hinh hoc.
Villi/ 2: Ta xit bai toan sau: Cho til dien OABC c6 OA, OB, OC ddi mOt vudng gdc. Gui a , |3, X lin lugt la gdc giaa OA, OB, OC vdi mjt phSng (ABC).
CMR: Sin^a + Sin^p + Sin^X = 1.(1) Giai: GQI H la hinh chi^u vudng gdc cua O tr€n mp(ABC) khi dd ta cd:
O H '
° 0 A ' Smp = Sin6BH=>Sin'P= ° ^
> Sina5inp5mX • 3x/3
Sina = SinOAH => Sin V = -
SinX = SinOCH=>Sin'X =
Sin'a+Sin'p-i-Sin'X = Otf(- O B ' O H ' OC 1
"^OB' T)
ddu bdng xdy ra khi Sina = Sinp = SinX hay OA = OB = OC. Tit ddy ta cd bdi todn sau:
Bdi todn I: Cho td di?n OABC cd OA, OB, OC ddi mgt vudng gdc. Ggi a , P, X lin lupt Id gdc giiia OA, OB, OC vdi mat phing (ABC). Tim GTNN ciia bilu thde
T = Sina.Smp.Siia
Ap dyng bat ding thde bunhiacopxki cho vl trdi eiia(l)taed:
(Sin^a-i-Sin^p-i-Sin^>.)^<3(Sin*a+Sin*p+Sin*X) oSin''a-i-Sin''p-i-Sin*A.>
M$t khde ta Igi cd:
O H ' O A ' "^ O B ' ^ O C do dd Sin^a + Sin'p -i- Sin'X = 1 Trude hit ngudi thiy cin cho HS thiy dupe kit qud cua bdi todn Id mdt ding thirc hinh hpc, tde Id mdt dgi lupng khdng ddi. Khi cd mpt dgi lupng khdng dli th) ta sS cd dgi lir^g Idn nhdt, nhd nhdt, dd chinh Id y nghTa cda hai bit ding thiix: quen thudc:
Cdsi vd Bunhiacopxki. Nhu v§y ta sg dp dyng hai bit ding thdc ndy vdo v i ^ sdng tgo bdi todn mdi.
Ap dyng BDT cd si cho vl trdi cua (1) ta dupe:
Sin'a-(-Sin'p-(-Sin'^>3Vsin'a5in^P5in'>.
Ddu bing xdy ra khi Sina = SinP = SinX Hay OA = OB = OC. Tir ddy ta cd bdi todn sau:
Bdi todn 2: Cho td di§n OABC cd OA, OB, OC ddi m$t vudng gdc. Ggi a,p,X lin lupt Id gde gida OA, OB, OC vdi m$t phing (ABC). Tim GTNN eua bilu thde T = Sin''a + S!n''p + Sm*l
Lgi dp dung bit ding thirc bunhiacopxki ta ed:
(Sina + SiAp-t-SinX)'<3(Sia'a-i-Sin'p-FSin'>.)
<=> Sina -HSinp + Sin^ < Vs
Diu bing xdy ra khi Sina = Sinp = SinX. Hay OA = OB = OC.
3. Tuong ty tfr hinh hpc phing sang hinh hpe khdng gian
Xudt phdt tir nhihig tinh chat hay nhdng bdi todn don gidn trong hinh hgc phing, bang cdch x^t tuong ty ta eung sg tgo ra dugc cdc bdi todn eye trj trong hinh hpc khdng gian.
Vi d^ 3: Ta xgt bdi todn: eho tam gide ABC vudng tgi A ed dudng eao AH = I.
a. Tim tam gide ed di?n tich Idn nhdt b. Tim GTLN eiia bilu thde T = 1 1
AB AC TAP CHl THIET BI GIAO DyC - S6 106 - 6/2014 <
Ill
NGHIEN COU & CTNG D U N GTrong tam giac vudng ABC ta cd
_}___}_ _}_
A H ' " A B ' "^ A C '
Ap dung bdt dang thdc eosi eho vl phdi ta dupe:
i = - L + ^ . 2 , | : i n : « ^ _ , i
AB' AC' V A B ' . A C ' AB.AC 2
^ A B . A C > 2 ^ S „ , c a l
Vgy dien tieh tam gide ABC ludn nhit bing 1 khi AB = AC= V 2 . khi dd tam gide ABC vudng can dinh A.
Ap dung bit ding thdc bunhiacopxki ta dugc:
. 1 1 .2 . ^ . 1 1 A B ' A C ' AB AC
Vdy GTLN cda T = y/l khiAB=AC=V2 . Trudc hit ngudi thiy cdn cho HS thdy dugc sy tuong ty hinh hgc phing vdi hinh hgc khdng gian:
Tam gide trong phdng chinh Id td di?n trong khdng gian, cde tinh chit eua tam gide d-ong phdng cung tuong tu cde tinh chit cua til: dign trong khdng gian.
Tam gide vudng dudng eao c6 dinh cd didn tieh Idn nhat thi td dien vudng dudng eao cd dinh eung cd thl tich Idn nhit. Tam giac vudng ed tdng nghich ddo ede cgnh gde vudng cd GTNN thi trong td dien eung vgy.
Tu dd HS cd thl phat bilu dugc bdi todn sau:
Bdi todn 1: Cho td didn OABC cd OA, OB, OC ddi mdt vudng gdc, cd dudng eao OH = 1.
a. Tim td dien ed thl tieh Idn nhdt b. Tim GTLN ciia bilu thdc
OA OB OC
Nlu dp dung pitago cho eac mat OAB, OAC, OBC taduge
AB-HBC-HCA = T = -
Ap dyng cosi cho ve phai ta dupe:
AB -1- BC -I- CA > A / 2 0 A . 0 B + V 2 0 B . 0 C -H
7 0 A ' + OB' + VOB' ^ O C + V o C + OA'
V20C.0A > 3VV2OA.OB.V2OB.OC.V2OC.OA O AB + BC -1- CA > 3 V2 .VOA.OB.OC diu bing xdy ra khi OA = OB =OC. Ta nhdn thiy vl trdi eua bit ddng thde chinh Id chu vi tam gide ABC, vl phdi OA.OB.OC lien hp vdi cdng thdc thl tich v^y ta lai cd bai todn sau:
Bdi todn 2: Cho td dign OABC ed OA, OB, OC ddi mdt vudng gde, ehu vi tam gide ABC bdng t c6 dinh. Tim GTLN eua thl tich td dign.
NIU cho tdng cdc cgnh eua td di?n la mdt s6 khdng ddi ta sg ed bdi todn sau
Bdi todn 3: Cho td dign OABC ed OA, OB, OC ddi mgt vudng gde. Tim GTLN the tich td di^n bilt tdng dp dai cdc cgnheda td dien Id mdt sd khdng dli S.
Vdi vipe dp dyng phep tuong ty hinh hpe phSng vdi hinh hgc khdng gian ngudi thdy khdng nhttng ren luygn dugc cho HS kha nang bilt ddt cdi da bilt vdo hoan cdnh mdi( tam gide dugc dat trong khdng gian) ma cdn ren luyen cho HS kha ndng tu duy tuong tu (hinh hpc phing vdi hinh hpc khdng gian).
Tir dd hinh thdnh va phdt triln ndng lyc tu duy sdng tao cho HS.
Tdi lifu tham khdo
1. G.polya, 1976, sdng tgo todn hgc tgp 3, NXBGD.
2. Nguyin Hihi Diln (2001). Phuang phdp gidi cdc bdi todn eye tri trong hinh hgc. NXB Khoa hpc ky thuat.
3. Phgm Gia Due - Phgm Dire Quang (2007).
Gido trinh ddi mdi phuang phdp dgy hgc mdn todn a trudng THCS nhdm hinh thdnh vd phdt trien ndng lyc sdng tgo cho HS. NXB DHSP.
4. Bui Thdnh Trung (2009). Phdt triin tu duy sdng tgo cho HS qua dgy hgc cdc bdi todn eye tr\ hinh hgc khdng gian, Lu^n vdn Thgc sy GDH, DHSPHN.
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