Tgp chi Khoa hgc Trudng Dgi hgc Can Tho Phdn C: Khoa hgc Xd hgi. Nhdn van vd Gido due: 33 (2014): '.
Tap chl Khoa hoc Tru'dng Oai hoc Can Thd website: sj.ctu.edu.vn
V^N D U N G P H i r O N G P H A P T O A D O D E GIAI B A I T O A N H I N H H O C K H O N G G I A N
Nguyen Thi Tuyen'
' HQC vien cao hgc lap Ly ludn vd phuang phdp dgy hgc bg mon Todn, khda 19, Khoa Suphgm Thong tin chung:
Ngdy nhgn: 29/04/2014 Ngdy chdp nhgn: 29/08/2014
TUle:
Applying the coordinate method toward the stereometric problems
Td khda:
Phucmg phdp tga do, tga dg hoa
Keywords:
Coordinate method, coordinates chemical
ABSTRACT
Stereometry is an important part of the mathematics curriculum high school today.The stereometric problems are pretty complicated, requiring learners to have good and critical thinking. Solving some stereometric problems is relatively difficult and takes more time, but the use of coordinate method will make them much simpler. In this article, we would like to introduce how to apply coordinates method toward the stereometric problems.
T6M TAT
Hinh hgc khdng gian la mgt bd phgn quan trgng cua chucmg trinh todn trung hgc phd thong hien nay. Cdc bdi todn hinh hgc khong gian khd phuc tap, ddi hdi ngudi hgc phdi cd tu duy tdt. Viec gidi mdt sd bdi todn hinh hgc khdng gian tuang ddi kho vd tdn nhiiu thdi gian nhung neu gidi theo phucmg phdp tga dg se dan gidn hon. Trong bdi viit ndy, chiing tdi xin gidi thiiu cdch van dung phucmg phdp tga do di gidi bdi tgdn hinh hgc khdng gian.
1 DAT VAN DE
,.i,::lHmhvhpp^khdh^iaff'la mdn hinh hpc kha trOru tugng nen da sd hpc sinh e ngai khi hpc vd phdn ndy. Trong cac dd thi tuydn sinh Dai hpc - Cao ddng gdn ddy, phdn hinh hpc khdng gian dugc ra dudi dgng ma hgc sinh cd thd gidi bang hai phuang phdp: Phuong p h ^ hinh hoc thudn tiiy vd phuang pbap tpa dO. Vide gidi bai todn hinh hgc klidng gian bdng phuang phap hinh hpc thuan tfty gap nhidu khd khdn ddi vdi hpc sinh vfta hgc xong ldp 12 vi da phdn cde em it nhidu da quen gidi cdc bdi todn tpa dO trong khdng gian.
Vipc gidi bdi toan hinh hpc khdng gian bdng phuong pbap tpa do cd rdt nhidu uu vidt, tuy nhidn hpc sinh cung gap khdng it khd khdn. Bdi vi, phuong phap ndy chim dugc dd cgp nhidu ttong cac sdeh giao khoa, hpc suih phd tiidng it dugc tidp can, va phuang phdp ndy chi tdi uu vdi mOt ldp bdi
todn ndo dd ehu khdng phdi luc ndo nd ciing td ra hieu qua.
Dd eac em hpc sinh Idp 12 cd thdm phuong phap gidi toan hinh hpc khdng gian, chuan bj cho ki thi cudi cdp. Trong khudn khd bdi bdo, chung tdi chft ydu tap trung vao cac van dd sau:
Ddu hieu nhdn bidt vd cae budc gidi bai toan hinh hpc khdng gian bang phuang phap tpa dO-
- Dua ra mgt sd each dgt hd ttuc tga dg vdi mOt sd hinh dac bidt.
Trinh bay mot so bdi tap hinh hgc khdn^
gian dugc giai theo phuong phap tpa dd va mdt sd bdi tap dugc gidi theo hai phuong phap: Phucmg phap tdng hgp va phuang phdp tpa dO. Dieu ndy giup hgc sinh ren luydn ki ndng gidi todn bang tpa dO vd cd thd ttd ndn linh hoat trong viec lya chgn phuang phap giai sao cho phu hop vdi timg bdi toan.
Tgp ehi Khoa hgc Trudng Dgi hgc Cdn Tho Phdn C: Khoa hgc Xd hgi. Nhdn van vd Gido dvc: 33 (2014): 98-105
2 NQI DUNG NGHIEN CUtJ
2.1 Mpt sd dau bi|u nhan biet bai toan hinh hpc khoDg gian cd thd giai bdng phuong phap tpa dp
Hinh da cho cd mOt dinh la tam didn vuong.
Hinh chdp cd mOt canh bdn vudng gdc vdi day va day la cdc tam gidc vudng, tam gidc deu, hinh vudng, hinh chft nhdt,...
Hinh lap phuang, hinh hOp chft nhat.
- Hinh da cho cd mdt dudng thdng vudng gdc vdi mat phdng, trong mat phdng dd cd nhtrng da gidc ddc biet: Tam giac vudng, tam giac ddu, hinh thoi,...
MOt vdi hinh chua cd sin tam dien vudng nhung cd the tao dugc tam didn vudng chang han:
Hai dudng thdng eheo nhau ma vudng gdc, hoac hai mdt phdng vudng gdc.
Ngodi ra, vdi mOt so bdi todn md gia thiet khdng cho nhftng hinh quen thudc nhu dd ndu d ttdn thi ta cd thd dya vdo tinh chdt song song, vudng gdc efta cdc doan thdng hay dudng thdng ttong hinh ve dd thiet lap he true tga dO-
2.2 Cac dgng todn thudng gdp
Tinh dO ddi dogn thdng, kliodng cdch tu diem den mgt phSng, khodng cdch tft diem ddn dudng thing, khodng cdch gifta hai dudng thdng.
- Tinh gdc gifta hai dudng thdng, gdc gifta hai dudng thdng, gdc gifta hai m|lt phdng.
Tinh thd tich khdi da didn, dipn tich thidt didn.
Chiing minh quan he song song, vudng gdc.
2.3 Cdc budc giai bai todn hinh hpc khdng gian bang phirong phdp tpa dp
Budc 1: Chpn hd true tpa dO Oxyz thich hgp vd tim tpa dO cdc didm cd lien quan ddn ydu cau bdi toan.
- Budc 2: Chuydn bdi toan dd cho vd bdi toan hinh hgc gidi tich vd gidi.
Birdc 3: Gidi bdi toan hinh hpc gidi tich ttdn.
Budc 4: Chuyen kdt lugn efta bai toan hinh hpc gidi tieh sang tinh chdt hinh hpc tuang ftng.
2.4 Thidt lap h$ tryc tpa dd
Vdn de quan ttpi^ nhdt ttong vide gidi bdi toan hinh khdng gian bdng phuong phjqi tpa do Id thidt lap he tga dO cho phu hgp. Sau ddy chftr^ tdi xm gidi thieu mdt sd phuong phdp d% thidt Igp hd tpa dO-
(1) Thiit Igp hi tga dg doi vdi tam dien Vdi gdc tam didn vide tpa dd hda thudng dugc thyc hidn khd dan gidn, dac bidt vdi;
- Tam dien vudng thi bd tryc tpa dO vuong gdc dugc thidt ldp ngay ttdn tam didn dd.
- Tam didn cd mOt gdc phdng vudng, khi dd ta thidt lap mOt mat cua he true tga dO ehfta gdc phdng dd.
(2) Thiet lap h^ tga dg cho hinh chdp Vdi hinh chdp, vide tpa dO hda thudng dugc thyc bidn dya ttdn d^c tinb hinh hpc ciia chftng. Ta cd cde trudng hgp thirdng gdp sau:
Hinh chdp ddu thi he tpa dO dugc thidt ldp dya trdn gdc 0 ttftng vdi tdm efta ddy va true Oz trimg vdi dudng cao ciia hinh chdp.
Hinh chdp cd mOt canh bdn vudng gdc ven day thi ta thudng chgn ttyc Oz la canh ben vudng gdc vdi day, gdc tga dO trftng vdi chan ducmg vudng gdc.
- Trong cdc trudng hgp khdc ta dya v^o dudng eao cua hinh chdp va tinh chdt da gidc ddy de chgn hd tpa dp phft hgp.
(3) Thiit lap hi tryc tga do cho hinh hop chU nhdt
Vdi hinh hOp chft nhdt thi vide thidt ldp he tpa dO kha dan gidn, thudng cd hai each:
Chpn mOt dinh ldm gdc tpa dO vd ba true ttung vdi ba cartii efta hinh hop chft nhat.
Chpn tdm cua day lam gdc tpa dd vd ba true song song vdi ba cgnh efta hinh hOp chft nhdt.
(4) Thiit lap hi tga dg cho hinh lang Py Vdi lang try dftng thi ta chpn tryc Oz thing dftng, goe tpa dO la mOt dinh ndo dd efta day hoac tam cua day ho§c didm ndm ttong m$t ddy Id giao efta hai dudng thdng vudng gdc. Cdc true Oy, Ox thi dya vdo tinh chdt efta da gidc ddy ma chpn cho phu hgp.
Tgp chi Khoa hgc Trudng Dgi hgc Can Tha Phdn C: Khoa hgc Xd hgi, Nhdn van vd Gido due: 33 (2014): 98-105 Vdi lang try xien, ta dya ttdn dudng cao va
tinh chdt cua ddy dd chgn he tpa dO thich hgp.
Ngoai cdc trudng hgp ttdn, ttong cdc trudng hgp khac ta dya vao quan he song song, vudng gdc va cdc tinh chdt efta dudng cao, ddy,... dk tiiidt lap he tpa do cho thich hgp.
2.5 Hf true tpa dp Oxyz
Hp true tga dO vudng gdc Oxyz ttong khdng gian Id hd gdm ba true x'0x,y'0y,z'0z ddi mOt vudng gdc.
Diem O la gdc tga dO Ox gpi Id tryc hoartii Oy gpi Id tryc tung Oz gpi Id tryc eao
Tren cdc tryc Ox, Oy, Oz Idn lugt chua ba vecta don vi I, J, k.
Cdc mat phdng (0xy),(0yz'),(0xz) ddi mOt vudng gdc nhau.
Tpa dO eua vecta: u = (x; y; z) <=^ u = (x\y;z) ^=^ u = Xi + yJ + zk
Tpa dO cua didm: OM = ;t:? + y; + zfe <=*
M(x;y;z')
Cdch xac dinh tpa dO didm l^(XM',yM''^M) ttong he tpa dp Oxyz
Tim hinh chidu M' efta M ttdn mdt phdng tga dO Oxy
Tft M' ke M'l vudng gdc vdi true x'Ox tgi / Tft M' kd M'} vudng gdc vdi tryc y'Ox tgi / Tu M ke MK vudng gdc vdi true z'Oz tgi if
y' I ^
K
O
f
\ M
>
j^x'
\^1 ,-''
M'J y.
N6u /,/, K Ian luat thuoc cac tia Ox, Oy, Oz C'u = 01
thi y„ = o ; (z„ = OK = MM'
Ngu l,J,K lin luat thuOc cac tia I x„ = ~0I
Ox', Oy', Oz' M\y„ = -0]
(z„ = -OK = -MM' 2.6 Mdt so bai toan
Bai toin 1: (SGK Hinh hoc NC lap 12). Cho liinh chop 5.^BC CO duong cao 5i4 = h.i&y litam giac ABC vuong ^C,AC~b,BC = a. Gpi M la tnmg diim cua AC va N \h diim sao cho 5W =
\rB.
a) Tinh dp ddi doan thang MN.
b) Fun sy lien he gifta a, b, h dh MN vudng gdc
Ke Ax II BC. Chpn he tryc tpa dd Oxyz nhu hinh ve sao cho X = 0, C G Oy, S 6 Oz.
Tacd:A(0;0;0),5(0;0;h),
SN = (x^: y„; z«), SB = (a; b; -h)
SN=ISB-
''» = 3 = ' * " 3 ' 3 ' T ^ b a b 2h 2/1Tgp cin Khoa hgc Trucmg Dgi hgc (2in Tha Phdn C: Khoa hgc Xa hoi, Nhm van vd Gida due: 33 (2014): 98-105
a) Ta co: JlfW = |A*N| = J - + - + — = i V 4 a M ^ P T l 6 F
b)Tac6: MN = ( | ; ^ ; ^ ) , S B = ( a ; 6 : - / i ) . _ a' i' 2h' MN 1SB<=>MN.SB = 0*^- ; r - = 0
3 6 3 Vay MN ISBOii 2 a ' - 4 ' - 4*^ = 0.
Bai toan 2; Cho hinh lang uu ABC.A'B'C co day ia tam giac deu canh bing a,AA' = /i va vuong goe voi (ABC). Bilt r ^ g khoang each giiJa A'B' va BC bing d. Chung minh ting a =
Giai
M
•
a C
c
Gpi M Id tning didm AB. Chon he ttyc tpa do Oaryz nhu hinh vd sao cho M = O.B G Ox.C G Oy. Ta cd:
A'{-^:0:k).B-(^:0:h).
B[-;0;0).C(0;-^:K) WP = (o; 0; OXA^ = (a; 0-, -h)
d = d(.A'B'.
>/4hZ+3a^
lf^-«^l jr^ft2+^
dy/WT3d^ = ahy/3 <=> d=^(4ft2 + 30^) = 3a^h^ ' V3(h^-d^) (dpcm).
Bdi toan 1 ndu gjai theo phuong phdp hinh hpc thudn tuy thi g ^ tid ngai d cau b. Vide tim khoai^
each gifta hai dudng thdng eheo nhau cua bdi todn 2 gdp nhieu khd khan doi vdi mdt sd hpc smh chua ndm vung phuang phdp tim khodi^ each ^ua hai dudng thang eheo nhau. Ldi gjdi bdng phuong phdp tpa dp cung ngan gpn vd kha Aaa ^dn.
Bdi toan 3: (DH khdi B 2007). Cho hinh chdp tft giac ddu S.ABCD cd ddy la hinh vudng cgnh a.
Gpi E la diem ddi xftng ciia D qua trung di&n 5^4, M Id trung didm AE, N \k tnn^ didm BC. Chiing minh MN vudng gdc vdi BD vd tinh khodi^ each giiia hm dudi^ thang MN va AC.
Giai
Tgp chi Khoa hgc Trudng Dgi hgc Cdn Tho Phdn C. Khoa hgc Xahgi, Nhdn van vd Gido d\ic: 33 (2014): 98-105 Phmmg phdp tdng hgp Phirong phdp tpa dy
Suy ra tft giac MICN Id hinh binh hdnh
=> MN II IC (1) m^:[ll\%'^BDUSAC)
=i BD lie (2)
TCt (1) v4 (2) suy ra: MN ± BD (dpcm)
(
IC cziSAC-) = • " " " (^''C).'1C c (SAC) MN II IC=> d(MN,AC) = d{MN,(SAC))
Ggi 0 Id tam cua hinh vuong ABCD
Chon he true tpa dp Oxyz nhu hinh vg sao cha C G Ox,D e O y . S e O z . T a c o :
0(0; 0 ; 0 ) , / l ( = f ^ ; 0 ; 0 ) , 8 ( 0 : ^ : 0 ) C (2|5; 0: o),£) (0;2j?; o) ,5(0i 0; ft),ft > 0 Goi / la trung diim SA. Ta co:
-ai/2 -aV2 h\
N(-^;^;0) - d(N, (SAC))
Gpi P 14 trung diim OC. Ta c6:
=> d(W, (SAC)) = WP = ^—
Vkyd(MN.AC)=~.
MW = ( ^ ; 0 ; ^ ) , I C = (0;aV2;0) MW.BD = 5 ^ . 0 + 0. aV2 - ^. 0 = 0
AC = (ay/2; 0; O), MA = (0; - — ; —-) , . _ , -ah^f2
\MN,AC] = (0;—-—;0) _ , -^ -a^k IMN,AC].MA = (0; - ; 0 )
d(MN.AC) _ HMNMJMA^ _ I 4 I _ aVz
\\i^M\\
Ldi gidi cua bdi todn bdng phucmg phdp tdng hgp ta thdy nd cOng ngdn ggn vd di hiiu, nhung khi dgc de di tim ddp dn thi rdt khd phdt hien dugc tu gidc MICN Id hinh binh hdnh, ddy Id mdu chdt chinh di tim ra ldi gidi. Viic chung minh vd tinh khodng theo phucmg phdp tga dg rdt di ddng niu viec tim ra tga dg cdc diim chinh xdc, nhin cd ve ddi dong nhung phuong phdp ndy khdng ddi hdi hgc sinh phdi tu duy cao. Do dd, phuong phdp tga dg phii hop vdi ddi tupng hgc sinh khdng co ky ndng gidi todn hinh hgc khdng gian theo phuang
phdp hinh hgc thudn tuy.
Bai toan 4: (DH khdi A ndm 2012). Cho hinh chdp S. ABC cd day la tam gidc ddu cgnh a. Hinh chidu vudng gdc ciia S trdn (ABC) la diem H thupc cgnh AB sao cho HA ~ 2HB. Gdc giua dudng tiidng SC vd mat phdng (ABC) bdng 60°. Tinh thd tich cua khdi chdp S.ABC va tinh khoang each gifta hai dudng thang SA vd BC theo a.
Giai
Tap chi Khoa hoc TrudngDai hoc Can Tha Ph&n C: Khoa hoc Xa hpi, Nhdn van va Gido due: 33 (2014): 98-105 Phinrng phdp tong h ^ Phtnmg phitp toa d$
HC la hinh chieu cua SC len (ABC)
=> G6c giiia SC va (ABC) la JCH = 60°
Gpi D la trung diem canh ^ S . Ta co: CD = 2 3 ~ 6
Xet AH DC vuong tai H, ta co:
-•,HD--
HC =
-JHD^+ CD' = J(f)' + [if]' = i f
Xet ASHC vuong tai H, ta co:
SH = HC. tan 60° = 2
;,V3=221
Gpi D la trung diem AB;kiHE H DC, E e BC Chpn h^ tryc tpa dO Oxyz nhu hinh ve sao cho H = 0,B eOx,EeOy,Se Oz. TIL CO:
H ( 0 ; 0 ; 0 ) , y 4 ( ^ ; 0 ; o ) , B ( | ; 0 ; 0 ) ,
SC = (-f ; ^ ; - 2 ) . * = CO;0:1). GOC giiiaiC va (ABC) bang 60° nen ta co:
|sc.S| Ml sin 60° =
ThS tich ciia khoi chop S.ABC la Vs,tBc = -SH.S^^
1 a ^ aV3 _ a^/7 _ 3 ' 3 • 4 12 ^ ^ •*
K6 Ax II BC, Ice HW II Ax, N £Ax
Ta co: BC II (iXN) nen ii(5>4, BC) = d(BC, (SAN)) = d(B, (SAN))
mm'
^ ) , M = ( f ; 0 ; ^ )
Vi HB n (i/lN) = ,4 nen ta co: d^aXSAN)) _ AB d(H.{SAN)) ~ AH~
= -m AW 1 (SHN)
a = ( ^ ; 0 ; [S2,S5] = ( 0 ; ^ ; 0 )
, ^ 6 2 ' . 3 -'
The tieh ciia khoi chop S.ABC la
''^•^??#-ilMFli^l#-^ <^™'
BC = ( Y ; 2 f ; 0 ) , I s = (a; 0;0)
=» (SAN) 1 (SHN) MSt khac: (SAN) n (SHN) = SN
Tir H UHK 1 SN (K E SN). Khi do: HK 1 (SAN) h a y r a g f l = f-a'^.-a'^m.-a'j3\
d(H,(SAN))=HK <- • ' ( 2 ' e • 3 J
AH = ^AHN = AH.sin60<>=i^ d(SA,BC) = ^^^Si^ J ^ 'J^
Xet ASHN vudng tgi H, ta cd: HK.SN = NH.SH =^
(*)=> d(B, (S>17V)) = |d(W, (^SAN)) = ^ V d y d ( S > l , S C ) = ^
Z.OT gidi bdi todn trin cOng chung minh dugc uu diim cOa phuang phdp tga dg.
Bdi todn 5: (Gidi todn Hinh hgc 11, NXB Gido Due). Cho hmh ldp phuang ABCD.A'B'C'D' canh
a. Gpi M,N Id hai didm ndm ttdn hai canh B'C vd CD sao B'M =^B'C'.CN = 'CD. Chftng minh AM_ J, BN vd tinh khodng cdch giu:a AM vd BN.
Giai
Tgp chi Khoa hgc Trudng Dgt hgc Can Tha Phdn C: Khoa hgc Xd hdi. Nhdn vdn vd Gido due: 33 (2014): 98-105
Pbmmg phap tong hyp Phirong phap tpa dp
- — i J ^ 4i \
c
E
Ke ME II CC,(EeBC);I = BNnAE.
MBE = ABCN (c. c. c) => BSC = 3 I B
=»Tll giic INCE nOi tilp => NIE = 90°
Hay BN 1 AE (1). Mat khac:
{M;;rc"'^^--«c.Bco)
=>ME IBN (2)
TCr(l)va(2)suyra:BW 1 (AME) => BN LAM . MJM c (AME)
°-\(AME) IBN,(ISBN) Trong (AME), tir lUIH 1 AM, H E AM S\iyr^:d(AM,BN) = IH
Xet AABE vuong tai E, ta co:
AE = 4AmTBE^ = ^!^;AB'=AI.AE
3
" ' Vl3o m Xft AAME vuong tai £, ta co:
AM = 4APTJm= | H £ ! + a 2 = J!H
\ 9 3
A>i;// - tiAME(g. o) =s. — = —
^ " ^-^ ME AM _ MEAM _ 9 a
'z
" \ X i i^
\
D
o- V~-^^
r
y,
Chpn he true tpa dO Oxyz nhu hinh ve sao cho A = 0,B' E0X,D' EOy,A 6 Oz. Taco:
yl(0; 0; 0), Jl« fa; Y : o) ,B(a; 0; o),
AM
-{a:f,-a)
.BN = (—-,a;a) —. -2a— , — -2a' -2a' AM.BN =—:r- + —— = 0
=>AM IBN AB = (a;0;0), ,—. — ^ ,2a' 13a^
\AM,m\ = (a';-^:-^)
. _ \[AM,B!'i].'AB\ _ \a^ _ 9 a
d(AM,BN) •-
\[AM.BN\\ |2B6^^ V286'
ilH =
Ddi vdi hinh lap phuang thi viic gidi bdi todn bdng phuang phdp tga do cd nhieu thudn lai nhdt.
Viic chgn hi true tga do vd tim tga do cdc diem -- cung dan gidn. Do dd, ldi gidi hang phucmg phdp ndy ngdn ggn han. Di gidi duoc bdi todn ndy bang phuang phip tdng hgp thi doi hdi hgc sinh ndm viing kien thiic cua hinh hoc phdng vd hinh hpc Idiong gian.
U'u diem cua phtnmg phap tpa dd Phuong phdp tpa dO giup gidi mpt sd bai todn hinli hpc khdng gian dan gian hon khi giai bdng phuang phap hinh hpc tiiudn tfty-'
Lupng kien thftc vd kT nang dd giup hpc sinh cd the gidi cac bai todn hinh hpc thdng qua phuang p h ^ ndy khdng nhidu ehu ydu la cac kidn thuc vd tpa dO vecto ttong khdng gian, phuang ttinh dudng tiidng, m$t phdng, mdi quan he gifta chftng.
Phuang phap ndy khdng qud khd ndn ddi vdi cac em hpc sinh trung binh yeu vide sft dung phuong phdp ndy dan gian han nhidu so vdi
phuang phdp tdng hpp, chft ydu Id dgy cdc em cdch dat he true tpa dO sao cho phu hop.
Nhiryc didm ciia phuang phdp tpa dp Khdng phai tdt cd cdc bdi todn ve hinh hpc khdng gian ddu co thd sft dung phuong phap tpa dO dk gidi, chi vdi nhung hinh dac biet cd nhung cgnh cd quan he vudng gdc vdi nhau thi ta mdi nen su dung phuang phdp ndy vi neu khdng vide tinh tpa dp cdc didm rdt khd khSn.
Sft dyng phuang phdp ndy ddi hdi phdi cd ki nang tinh toan khd tdt vd phdi nhd duprc cac cdng thftc vd phuang trinh ciia dudng thang, mgt phang, cdc cdng thftc vd tinh gdc vd khoang cdch. MOt sd cdng thftc khd gidng nhau ndn ddi khi dd gdy ndn sy nhdm lan.
3 T H V C N G H I ^ M
3.1 Muc dich thyc nghifm
Thye nghidm nhdm kidm tta tinh khd tiu vd hidu qud cua phuong phdp tpa dp hda qua dgy hpc
7t9» chi Khoa hgc Trudng Dgi hge Cdn Tha Phdn C: Khoa hgc Xa hgi, Nhdn vdn vd Gido thic: 33 (2014): 98-105 <
^ a i bdi tap hinh hpc khdng gian bdng phuang p h ^ tpa dO, kdt hpp dieu tra va phdng van.
3 J Npi dang va phinmg phap thyc nghidm Thye n ^ e m dupe tien hanh tai Idp 12A, trudng Trung hpc Phd thdng Hda Binh. Lap 12A gdm 40 hpc sinh cd ket qua hpc tap tucmg ddi ddng ddu do thdy Tran Ngi^dn Khai Hung ^ang day mdn toan. Thdy Himg da cd ttdn 7 nam kinh nghidm gidng day mdn todn ldp 12. Trudc kia, khi day hpc gidi cdc bdi tgp vd tpa dp troi^ khdng gian thdy Hung it gidi thidu cdc bdi tap hinh hpc khdng gian cd thd gidi bdng phuang phdp tpa do. Do dd, hpc smh khdi^ dupe ren luyen {diuong phdp tpa dO hda vd edm thdy e ng^ trong ky thi tdt nghidp vd dgi hpc vi cd bai ky thi ddu cd cdu bdi tap hinh hpc khdng ^an ma giai theo phuong phdp tdng hpp thi cdc em khdng mdy ty tin bdi cd mOt so bdi tap ttong thdi gian ngdn khdng thd tim dupe len gidi.
Qud trinh thyc nghidm duoc tidn hdnh ttong cde tidt gjdi bdi tap vd daiy hpc mdt sd bdi tap hinh hpc khdng gian theo phuong phdp tpa dp vd cdc tidt day ndy duoc phdn bd sau Idii hpc cac bdi phuang phdp tpa dO ttong khdng gian. Trudc khi thyc nghiem, chiing tdi cung vdi thdy Hung da cftng nhau ttao ddi phuang phdp day hpc smh cdch chpn he true tpa dp khdng ^an sao cho phft hpp va dd dang tim dupe tpa dO cdc didm ttong bdi todn.
Ben cgnh dd, chung tdi trao ddi xin y kidn efta rapt sd ^ao vidn, phdng 20 hpc sinh dang hpc ldp
12 vd 9 hpc sinh viia hpc xong ldp 12.
Cdc cdu hdi phong -van
- Sau khi hpc xong chuang 4 ciia chuang trinh hinh hpc ldp 12 thi em gidi cdc bdi t|p hinh hpc khdng gian bdng phuang p h ^ thudn tuy hay phuong p h ^ tpa dp?
- Phuang phap tpa dp hda de tidp thu hay khdng?
- S ^ khi dupe trang bi thdm phuong phdp tpa dp hda thi em cd an tdm hon khi ldm cdu hinh bpc khdng gian ttong ki thi dgi hpc khdng?
3 J Phan tfch ket qua thurc nghidm Sau khi tidn hdnh thyc nghiem, qua kdt qud bdi kidm tra chung minh dupe rang cac em hpc sinh trung binh khd ed the tiep thu dupe phuong phdp nay de ddng vd cung mOt b ^ tap hinh hpc khdng gian thi sd luong hpc sinh giai duoc bdng phuang phdp toa dp nhidu hon sd hpc sinh ^di bdng phuong phdp tdng hpp.
Qua trao ddi vdi mdt sd gido vien cd nhidu ndm kinh nghidm day Idp 12 thi cac gjao vidn eung nhan dinh rang van dm^ phiror^ p h ^ tpa dp de gidi cdc bai tap hinh hpc khdng gian cd nhidu thudn lgi. Y kidn ciia cde em dang hpc ldp 12 thi nhdn xet rdng phuong phdp n ^ dl hieu han phuang phdp tdng hop va cd the tiep thu dd dang, cdn cdc em viia hpc xong Idp 12 thi cho rdng phuong phdp tpa dO hda rdt hay, cdc em edm thdy ty tin hon khi budc vdo ky thi Dai hpc.
Muon qud trinh nay dat hidu qud, can phoi hpp day hpc bdng phuang phdp tpa dd dd hpc sinh cd nhidu ea hOi gidi dupe cdc bdi tap binh hpc khdng
^an. Phuong phdp nay khdng nhftng giftp hpc sinh dn lai kidn thuc tpa dp tiong khdng gian ciia chuong 4 ttong chuang trinh Hinh hpc ldp 12 ma nd cdn la cdng cu dac lyc dd hd tra cdc em ttong cdc ky thi, dac bidt Id ky thi ti^dn sinh Dai hpc.
Dieu dd da chftng td uu diem ndi bat cua phucmg phap tpa dp.
» KETLU.4N
Trong bai bao nay chftng tdi tap trung vdo vide dua hd tryc toa dp dd gidi cdc bdi todn hinh hpc khdng gian. Ddy Id phdn quan ttpng nhdt iik gidi thdnh cdng mOt bdi todn hinh hpc khdng gian bdng phuang phap tpa dp. Cdc vi dy ttdn ddy ciing vdi kdt qud thyc nghidm su pham d trudng phd thdng da khdng dinh cdc uu didm, tinh khd thi vd hidu qua cua phuong p h ^ tpa dp hda ttong dgy hpc todn.
TAI LIEU THAM KHAO 1. Dodn Quynh tdng chu bidn. Van Nhu
Cuang chu bidn, Pham Khdc Ban, Ld Huy Hung, Ta Mdn (2010), Hinh hoc 12 ndng cao, Nxb Gido dye.
2. Van Nhu Cuong (chu bien), Pham Khdc Ban, Ta Man (2007), Bdi tap hinh hgc 11 ndng cao, NXB Gido dye.
3. Tnrang Ngpc Dung (2008), Gidi todn hinh hgc lap 11, NXB Gido Due.
4. Vd Thanh Van (chu bidn), TS. Ld Hidn Duong, N^iydn Ngpc Giang (2010), Chuyen de ung dung tga do trong gidi torn hinh hoc khong gian, NXB Dai hpc su pham, Hd Chl Minh.
5. Le Hdng Ddc, Nguyen Dftc Tri (2007), Phuang ph^ gidi todn binh hgc gidi tich trong khong gian, Nxb Ha Ndi.
