s t DUNG HINH HOC CAO CtP SOI SANG MOT SO KlfN THifC COii HiNH HOC SO CAP CHO SINH VIEN SO PHAM TOAN
TS. I R A N VIET C a O N G '
T
rong day hpc toan, co nhieu van de cua toan hgc SP cap noi chung va hinh hpc so cap (HHSC) noi rieng chi eo the dupe hieu diing ban chait neu chiing dupc nhin tir khia canh cua toan hgc cao cap. Trong qua trinh viet saeh, vl nhiiiig li do su pham nen cac tac gia thudng dua ra van de theo hudng ngudi dpe thda nhan ketqua hoae trinh bay ldi giai cac bai toan (BT) ma chira li giai dupc tai sao, xuat phat td dau, li do gi ma chiing ta lai c6 each lam do.Vi vay, viec soi sang eae kien thdc cua loan hgc sdcap bang cac kien thuc cua toan hgc eao cap ndi ehung va trong linh vuc hinh hoc n6i rieng la can thiet. Qua do, giup ngupi hpc n^m viing kien thdc cung nhu hieu dupc coi nguon cua cae kien thdc trong HHSC.
1. Sddung cac kien thdc cua hinh hpc cao cap (HHCC) de giai thich mgt so kien thuc khd trong HHSC, dong thdi, chinh xac hoa mot so kien thdc cua toan hpc pho thong cho sinh vien (SV)
Trong chuong trinhtoan HHSC, mgt so kien thuc thudng dupc cac tac gia trinh bay mot each vSn tat, Vi vay, ngudi hpc kh6 CO the hieu dugc mpt each can ke nhifng kien thuc do, Dekh5c phye dieu nay, trong qua trinh day hpe, giang vien (GV) co the su dung kien thirc cua HHCC nhiim soi sang cac kien thii'c cua HHSC, tdd6, chinh xac hoa kien thde cho SV.
Wdu.'Trong HHSC CO khai niem ve hai tam giae bkng nhau nhusau: Haitam giic bing nhau thitdn tai duy nhi't mgt phep ddi hinh bien tam giic nay thanh tam giic kia.
Thuc te eho thay, nhieu SV mdi chl nSm dupe neu ton tai mpt phep ddi hinh bien tam giae nay thanh tam giae kia thi hai tam giae do bang nhau de van dung vao giai toan ehd chua nam ro nguon goc eua khai niem nay.Tacothesddungcackien thdc cua HHCC giup SV sang to nhu sau: Trong khong gian E^, hai tam giae ABC'va A 'B'C'se tao ra hai he diem dge lap tuyen tinh [A. B, C} va [A', B', C). Do do, ton tai duy nhat mot phep A f i n / ' E ^ ^ £ ^ s a o cho: y(4} = / ! l ' / ( ^
= B'ya'f{C)=C'.
Xetphep bien d6ltuyen tinh lien ket / c u a fvdi hai
CO sa\'AD.~Ac\ va \A-B' A-ncua E^. Ta c6: fi'm^YB- vai^Ac.)^'A'c'-.\^o^ABC^^A'B'C'nen:AB.AC.cosA
= A'B'.A'C'.cosA' c:> JBAC = W~B-AX'\ nghTa la
/ b a o ton tich vo hudng trong E-; suy ra 7 la phep bien doi tare giao nen f\a phep^d&ng cutrong E^, tdc f la phep doi hinh trong mat phang.
Vay, neu hai tam giae b^ng nhau thi ton tai duy nhat mpt phep ddi hinh bien tam giae nay thanh tam giae kia.
2. S d dung kie'n thdc cua HHCC de giup SV hieu ro cgi nguon cua cac kien thdc trong qua trinh g i a l c a c B T HHSC
Khi giai mot so BT HHSC, chiing ta khong tn/c tiep giai BT ma tien hanh chdng minh mpt kien thdc nao do (chang han nhu mgt dinh li hoae sudung ket qua cua mgt BT phy.,.), Cau hoi duge dat ra la can cuva dua vao c o s d nao de chiing ta sudung cae kien thdc do trong qua trinh giai BT.
Trong HHCC, mpt tinh chat a cua hinh H ggi la batbien doivdinhom Fneu mgi hinh H'tupng dupng vdi Hdoi vdi nhom Fdeu edtinh chat a. Do d6, thay vi nghien ciru hinh H. ta chgn trong cac hinh tuong duPng afin voi hinh Hmgt hinh H ' m a tren d6tinh chat a de chdng minh, Khi do, co the xem W'laanh cua H qua mgt phep afin /nao do, Sau khi chdng minh dupc tinh chat atren hinh H',\a thuc hien phep bien d6lafin /'•'bien hinh H'thanh hinh H. Khid6,tinh chat afin atren hinh Hduoc chirng minh,
Ching han, ta xet BT sau:
BT1: Chungminhrangtrong mot tam giae bat ki, neu m6i canh cua tam giae dupc chia lam ba phan b^ng nhau va noi cac diem ehia vdi cac dinh ddi dien cua canh do ta dupc sau dudng thang tao nen mpt hinh lye giae thi cac dudng cheo cua hinh lyc giae nay se dong quy tai mpt diem.
De giai BT 1, can sd dyng ket qua cua BT 2 (BT phy) sau: Cho AABC va hai diem M, N lan lugt
* Tnrdng Dai hoc sir pham - Dai hoc Thai Nguyen
(ki 2-7/2014)
Tap chi Gido due so 338 4<Hlnhl
MB_ NC • minh ring giao diem cua BN va CM nim tren dudng trung tuyen xuat phii tddinhA.
That vay, gpi /Cia tnjng diem cua doan thang BC ( f t ; n f t r ) . K h i d 6 : f . ^ . ^ = M v i f i / f = f C v a
AM _AN MB ~ NC''
Ap dyng dmh Ceva cho AABC:
ba dudng thang AK, BN va CM dong quy tai mpt diem hay diem cua SA/ va
CM nkm tren dudng trung tuyen xuat phat td dinh A cua AABC.
TrdlaiSry,trongtam giae-4SC,lay cac diem nhu AB, AC, I
hinh ve {hmh 2). Khi ( J o : " ^ " ' c ^ " 2 ^^
AB, AC, ^
'j^ = - ^ = ^(doAB^=B^B,= B,CvaAC,= C,C^^
C^B). Ap dyng BT2, ta co: Ova Mthuge dudng trung tuyen >4/cLia A^SC. Chung minh tuong tu, ta 6uac N va flthugc dudng trung tuyen C/Ccua tABC; Sva P thupc dupng trung tuyen SJcua ISABC.
f\/lat khac, do Ai, BJ va CK\a ba dudng trung tuyen cua ^ABC nen Al, BJva C/Cdong quy. Do do, cac dudng cheo QM, SP va NR cua hinh lyc giae MNPQRS cung dong quy. 016^ giai nay. ehung ta thay mau chot de giai dupc BT 1 la phai chdng minh d u g c S r ^ .
C6then6i,vdi cac kien thdc cua HHSC, ngudi hpc kh6 e6 the giai thich dugc tai sao lai nghT den viec sd dyng ket qua BT2. Vdi kien thdc cua HHCC, co the gial thich dieu nay^ nhu sau: do tam giae la 2 - don hinh trong mat phang afin 2 chieu thong thudng nen la khai niem afin.
b i n g - va ti so don la khai niem afin. Khai niem luc giae va tinh chat dong quy deu co the mo ta bang ngon ngiJ cua hinh hpc afin. Do do, BT chua hoan toan cac bat bien afin. Vi tam glac thudng va tam qiae deu la hai khai niem tuong dUPng afin trong mat phang afin 2 chieu th6ng thudng nen eo the giai BT trong tn/dng hpp tam glac deu.
Chpn tam giae deu >^'S'C'tuong duong vdi tam glac ABC 6a eho qua mpt phep Afin /. Theo gia thiet, tren cac canh B'C, CA', A'B', lan lupt lay cac cap diem ehia(.4|'. A'^J; (B,. B^)ya(C\; C i ) sao cho B'/i,=/), A^ = AX' = r'fl, = s; s, = B,/(' = /i'c,=c, c > c ; f l ' (hinhS).
Khi do, ta CO lyc glac M'N'P'Q'R'S'. Can ehung minh cae diemQ' M 'nim tren dudng trung true eua doan thing B'C. De thay dudng tomg true nay di qua diem /4'(do A/l'6'C'latam glac deu).
That vay, ta c6: AB'CC,=ACB'B^ (vi canh B'C Chung, Z/^'S'C'=Z>4'C'e'= 60" v a e ' C ' , = C ' 5 ; } . D o d6: Z B'^B'C= ZC^'B', suy ra AO'e'C'can t?i Q' nen dinh 0 ' t h u p c
dudng trung tore eua doan thang S ' C .
Chirng minh tuong tu: M' thupc dudng trung tn/c cua doan thang S ' C . Do do, O ' v a JW'cung thugc dudng tnjng true canh S ' C cua
A ^ ' S ' C Hai dinh Hinh 3 S', P' cung thugc
dudng tnjng true cua canh A'Ccua AA'B'Cva hai dinh R', Wthugc ducmg trung true cua canh >4'S'cua AA'B'C Trong AA'B'Cdeu, cae dudng trung tn/c A'l', B'J'vk C K ' d o n g quy. Do do, cac dudng cheo Q'M', S'P'va N'R'cua hinh lyc giae M'N'P'Q'R'S' dong quy tai mgt diem.
Thuc hien phep afin /-'bien A4'S'Cthanh AABC.
Khi do, cac dudng cheo Q'M'. S'P'va N'R'cua hinh luc giae M'N'P'Q'R'S'tuang ung bien thanh cac dudng eheo QM, SPva NRcua hinh lue giae MNPQRS.Mat khac, do cac dudng cheo Q'M', S'P', A/'fl'dong quy tai mgt diem X', nen cae dudng cheo QM, SPva NR dong quy tai diem X = f-'{X).
(Xem tiep trang 56)
50 Tap chi Gido dye so 338
(ki 2 - 7/2014)
gian. Qua 66, giup HS kh6ng nhung xay dyng duge moiquan he giua kien thiic cu va kien thdc mcfl ma con ren luyen cho cac em khanang sang tao. Vivay, DHKP cong thiic tinh khoang each tii mot die'm den mpt mat ph§ng thong qua phep SLTT la hoan toan kha thi.
PPDHKP mang lai nhieu loi ich doi vdi HS boi n6 giup cac em phat tnen tu duy, tri nho va tao duoc moi lien he giua kien thdc mdi vdi kien thiic cu. Bac biet, DHKPb§ngSLTTdaehdng minh dupc hieu qua cua no trong qua trinh hpc tap, kham pha tri thiic mdi ciia HS. Vivay, viec nghien cifu, phattrien vavan dyng phuong phap nayvao qua trinh DH laeanthiet.Q
{]) Nguyen Phu Lflc. Gido tnnh Xii Im&ng day hoe khdng truyin ihd'ng. Truiyng Dai hoc Can Tha, 2010.(2) Hoang Chung. L o g k hpc pho thiing. NXB Gido ducH. 1994.
(3) Nirah Halivah. Teaching for effective iearning in higher education. Ktuwer Acadeniie Publishers The Nelherland.s. 2000.
{4} Mana Salih. A Proposed Model of Self-Generated Analogical Reasoning for the Concept of Translation, Joumal of Scienee and IWalhemmalic Educalion in Southeast Asia. Vol. 31 2 No. 164-177. Faculty of Science & Technology, Sultan Idris University of Educalion. Malaysia, 2008.
Tai lieu t h a m Ithuo
Tran Van Hao {tOng chu bien). Hinh hoc 12. NXB Giaoduc VietNam. H.2010.
SUMMARY
Currently, tenching by discovery and analogy are applied a lot in leaching maths Moreover. If teach- ers combine these elements into teachmg. they not only review the old knowledge but also help students to build the new knowledge easily. The General Model of Analogy Teaching (GMAT) is used to discover the new knowledge effectively. In this article, we applied the GMAT model m teaching the formula of calculat- ing the distance from a point to a plane through an pedagogical experiment.
Thiet 1(6 tiettra bai...
(Tiep theo trang 46)
Tai liSu tham Ichao I. Dir an ViCl - Bi. Day va hpc tich circ, mpt so p h u o n g p h a p va lii thuat day hpc NXB Dai hoe supham, H, 2010.2 Nguyfin Thiinh Thi. Chuyen de btii d u ^ n g giao vien t r u n g hpc pho thong. Soc Trang, 2011.
3. Nguyen H6ng Nam. 'Thia't k^
cau hoi day hoc Van - M6t thir thach vai giao viCn". Tap chi Gido due, %6 147,2006.
SUMMARY In the current schools, pay- ing for student tests is not enough attention. The imple- mentation of this class a per- functory way. the process of in- consistency has led many stu- dents miss the opportunity to fix weaknesses, strengths, and promote the process of writing.
Posts oriented design into a post office hours pay discourse to promote the highest efficiency of this class.
Sir dung hinh hoc cao cap...
(Tiep Iheo trang 50)
_ 6 ldi giai tren ta thay, cac cap diem ( 0 ' ; M), {N'; R) va {S'\ P) lan lugt nam tren cac ducftig trung tuyen eua A/1'S'C'nen anh cua chiing qua anh xa afin / ' la cac cap diem { 0 ; l\4), (A/; fl), (S; P) ciJng n^m tren cac dudng trung tuyen eua AABC vi phep bien doi afin bao toan cac dudng tmng tuyen. Day ehinh la co sd cho viec si) dyng S r 2 d e tim Idi giai cho BT1.
Trong qua trinh day hpc, neu GV thudng xuyen khai thae cae kien thdc cua HHCC nhim soisang cac kien thde cua HHSC sectiiipSV hieu dupc mpt each ehinh xac, hieu diing ban chat eiing nhu nam dupc cpi nguon cac kien thde eua HHSC; tudo, SV thay dUPc mdi quan he giua HHCCvdiHHSC.Q
Tai lieu tham khao
1. v a n Nhu Cuong - Ta Man. Hinh hpc Afin va hinh hpc Euclid. NXB D(ii hoc quoc gia Hd Noi, 1998.
2. Trdn Viet Circ>ng - Nguyin Danh Nam. G i a o trinh Hinh hpc so-cSip.
NXB Gidodite VietNam, 2013.
3. Dito Tam. Giao trinh Hinh hoc so" cap. NXB Dgi hpc suphgm, H 2004.
4. Ng6 Viet Trung. Giao trinh Dai so tuyen ti'nh. NXB Dai hpc qud'c gia Hd Npi. 2002.
SUMMARY
This paper presents some ideas of using advanced geometry in sup- porting students learning mathematics. From advanced point of view, students would gel insight into some difficult problems in elementary geometry and make it clear the relationship between advanced geom- etry and elementary geometry.
