NATIONAL ACADEMY OF EDUCATION MANAGEMENT JournalofEducationManagement,2019, Vol. 11,No. U,pp. 98-104 This paper is available online at http://jem.naem.edu.vn
PHAT TRIEN TU DUY T H U A T TOAN CHO SINH VIEN QUA KHAI THAC LCJl GIAI M O T S O BAI TOAN SIT DUNG PHEP CHIEU PHU (HINH HOC HOA HINH)
Tran H6ng Hai^
Tom tat. Viec day hpc cac kien thiic cd sd, cd ban khong chi giiip sinh vien co nhflng hieu bi6t sau sac ve nghe nghiep, cong viec cua mmh, ma con giiip cho sinh vien biet each suy nghT de nhan thiic va giai quyet nhflng van de nay sinh trong thuc tien. Theo muc tigu do, hpc phan Hlnh hoc hpa hlnh vfla trang bi cho sinh vien nhflng kien thiic cO ban de doc hieu va thiet ke cac ban ve ky thuat, dong thbi tflng bfldc giiip ngUbi hpc biet each giai quyet vin de theo mot trinh tu nhat dinh, tflc la phat trien tfl duy thuat toan. Bai vi6t nay trinh bay mpt trong nhflng bien phap phat trien tu duy thuat toan cho sinh vien cac trUcfng Dai hoc khoi ky thuat thong qua hpc phan Hinh hpc hpa hinh.
Td khda: Tu duy thugt todn, phep chiiu, Hinh hgc hga hinh.
1. D a t v a n d e
Phat trien nang lUc ngUbi hpc, trong do co nang lUc giai quyet van de va sang tao dang la m6t trong nhflng diem mdi trong giao due Quoc te va Viet Nam. Dieu nay khong chi diing vdi giao due pho thong ma con diing vdi giao due Dai hpc.
Theo "Tieu chu^n ve giao due tii nam 2000 cua cac nha giao due Au My" [Bao Giao due thc)i dai, 3/1994]: Cii nhan tot nghiep can phai dUdc ren luyen tot kha nang giao tiep, trao doi thong tin va lam viec co hieu qua cao trong mot nhdm cong dong; phai co kien thiic va hieu biet tot vS may tinh va cac nganh ky thuat khae.
Van dung toan hpc vao giai quyet v^n de, dac biet la nhflng van de nay sinh tU thUc ti6n la mdt trong nhfing nang lUc cua ngUdi hpc dfldc nhieu nUdc tren the gidi dat ra trong the ky XXI. Theo William Lauden (1992) [5]: Bay nang lUc cd ban cua ngUcii lao dpng the ki XXI can cd nhu sau:
(1) Nang luc thu thap, phan tich va to chUc thong tin; (2) Nang lUc truyen thong y tUdng va thong tin; (3) Nang lUc lap ke hoach va to chiic hoat dpng; (4) Nang lUc lam viec vdi doi tac va theo nhdm; (5) Nang luc s^ dung tU duy toan hpc va ky thuat; (6) Nang lUc giai quylt vkn de; (7) Nang luc s^ dung cong nghe.
Ve tkm quan trpng cua tu duy thuat toan, Gerald Futschek (2006) [4] cho rang: Tu duy thuat toan dupc the hien bdi kha nang thiet ke va hieu cac thuat toan: Kha nang phan ti'ch vSn d6 dudc dUa ra; Kha nang xae djnh mpt van de chinh xae; Kha nang tim ki6m cac hanh dpng cd ban du de xae djnh cac vkn de; Kha nang xay dung mot thuat toan chinh xae de giai quyet mot van de nhat dinh bang each sir dung nhflng hanh dpng cd ban; Kha nang de suy nghi vl tat ca cac tnfbng hdp Ngay nhan bai: 15/10/2019, Ngay nhan dang: 14/11/2019.
^Trudng Dai hoc Mo - Dia chat; e-mail: HonghaitranOSS'yahoD.com.
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cd the dac biet va binh thudng ciia mpt van de; Kha nang de nang cao hieu qua cua mpt thuat toan.
Neu ai do mu6n thiet ke cac thuat toan de giai quy6t dUdc van dS, thi ngUdi do can cd kha nang tU duy thuat toan; TU duy thuat toan ham chUa mpt sU sang tao manh me.
Ci trong nudc, cac nha nghien cflu cung rk chii trong tdi viec ren luyen va phat trien tU duy thuat toan. Theo Nguyin Ba Kim (2015) [3]: Viec phat trien tu duy thuat toan trong nha trudng la can thiet vi nhflng li do sau day: i) TU duy thuat toan giiip ngudi hpc hlnh dung dUdc vi6c tU dpng hoa trong nhflng ITnh vUc boat dpng khae nhau ciia con ngudi, gdp phan khae phuc sU ngan each gifia nha trUdng va xa hoi tfl dong boa. ii) TU duy thuat toan giiip ngubi hpc lam quen vdi each lam viec trong khi giai bai toan bang may tinh dien til. iii) Tu duy thuat toan giup ngudi hpc hpc tap t6t nhflng mon hpc d nha trUdng, ro net nhat la mon Toan.iv) Tu duy thuat toan cung gdp phan phat Uien nhflng nang lUc tri tue chung nhU phan tich, tong hdp, khai quat boa,... va hlnh thanh nhflng pham chat cua ngUdi lao dpng mdi nhu tinh ngan nap, ki luat, tinh phe phan va thdi quen tU kiem tra v.v...
Trong cac trubng Dai hpc khdi ky thuat, hpc phSn Hinh hpc hpa hlnh cung clp cho sinh vien nhflng kien thiic can thiet de bieu dien cac vat the trong khong gian 0-cIit ba chieu len mat phang va giai cac bai toan thuoc khong gian ba chilu bang each ve tren cac hinh bieu dien dieu do. Nhfing kien thflc ciia mdn hpc nay la cd s6 cho viec dpc hieu va thiet ke cac ban ve ky thuat, phuc vu cho nghe nghiep sau nay cua sinh vien. Ta co the thSy hpc phan Hmh hoc hpa hinh cd lien quan trflc tiep tdi cac nang luc (5), (6). (7): Hpc phan nay gdp phan phat trien cac nang luc su: dung tu duy thuat toan, nang luc sfl dung phan mem AutoCAD, nang lUc van dung toan hpc vao thiet ke, sang tao cac chi tiet k! thuat, cac mat khdng gian trong thUc tien.
Hpc phln nay ddi hdi ngUbi hpc phai biet tU bpc (day la phong each hpc d dai hpc), biet giai quyet nhflng bai toan ve xae dinh hinh chilu va tinh toan cac kich thUdc cua vat the tren ban ve, biet hdp tac, sang tao van dung va tim ra nhflng thuat toan giai nhflng bai toan. Cac bai toan hinh hpa dat ra thubng d cac vi tri tong quat vdi nhfing ldi giai kha phfic tap va kho doi vdi ngudi hpc, trong khi cung vdi cac bai toan do khi xet d cac trudng hdp hay vi tri dac biet se cho ldi giai ngan gpn de hieu, dong thdi Ibi giai cua cac bai toan do cung giiip ngUbi hpc d l dang lien tUdng tdi cac bai toan thUc te ciia ve ky thuat. Mpt thUc te nfla d l nhan thay, do la da so sinh vien chfla cd tU duy thuat toan hoae chfla van dung tU duy thuat toan trong hpc tap hpc phan Hinh hpc hpa hinh.
Npi dung bai bao trinh bay viec khai thae Ibi giai ciia mot so bai toan Hinh hpc hpa hinh sit dung phep chieu phu vdi muc dich: Giiip ngudi hpc thay dUdc vai trd ciia phep chieu phu trong viec giai mot bai toan d trUdng hdp tong quat, qua dd phat trien tU duy thuat toan cung nhu tflng budc ren luyen kha nang tU hpc, tu giai quyet van de cho ngfldi hpc.
2. No! d u n g n g h i e n cihi 2.1. Dai citdng ve phep chieu
* Phep chieu xuyen tam [2] (Hinh 1)
Trong khdng gian lay mat phang (P) va diem S khong thuoc (P).
Chieu mpt diem A bat ky tii tam S len mp(P) la:
- Dung dubng thing SA
- Xae dinh giao diem A' cua dflbng thang SA va mp(P) Cac ten gpi:
- S: T^m chieu
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Hmh 2 - (P): Mat phang hinh chieu
- SA: Dubng thang chieu (Tia chieu)
- A': Hmh chieu cua diem A tii tam chieu S len mat phang (P).
Tinh chat: Phep chieu xuyen tam bao ton quan bS lien thuoc, bao ton thfl tu cac diem.
* Phep chieu song song (Hinh 2)
Phep chieu song song la trubng hdp dac biet cua phep chieu xuyen tam khi tam chieu d v6 cimg
Tinh chat: Ngoai cac tinh chat cua phep chieu xuyen tam, phep chieu song song con cd tinh chat dac tning: Bao ton ti so ddn cua ba diem thang hang.
* Phep chieu thang gdc: La trudng hdp dac biet cua phep chieu song song khi phUdng chieu s vudng gdc vdi mat phang hinh chieu (P).
Ngoai cac tinh chat cua phep chieu song song, phep chieu thang gdc se bien mpt goe vuong thanh mot gdc vuong neu gdc vudng chilu c6 mot canh song song vdi mat phang hinh chieu.
Nhan xet: Ngoai mot so trudng hdp (vj tri) dac biet, noi chung cac phep chilu khdng bao toan khoang each gifla cac diem (Phep chieu la mpt phep bien doi afin, khong phai la phep bien doi meuic)
* Xay dUng do thfic ciia diem trong phUdng phap hai hinh chieu thang gdc
Hinh 3
Lay hai mat phang vudng gdc vdi nhau: Mat phang (Pi) thang dflng va mat phang (P2) nam
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ngang, giao tuyen cua chung la x. Cho mpt diem A trong khong gian, gpi Ai la hinh chieu cua A tren (Pi), gpi A2 ia hinh chieu cua A tren (P2); sau dd xoay (P2) quanh true x de nfia phia trudc Clia (P2) triing vdi niia phia dudi cua (Pi). Ta dUdc do thflc ciia diem A drong phUdng phap hai hinh chieu thing gdc. Ky hieu A (Ai, A2).
* Ve phep chilu phu: Trong mpt sd trUdng hdp dac biet, ngUbi ta cdn sii dung cac phep chieu theo mpt phUdng nao dd (khong nhat thiet phai la thang gdc) len mdt mat phang hinh chilu khae vdi (Pi) hay (P2). Phep chieu nhu the dUdc gpi la phep chieu phu. Trong Hinh hpc Hpa hinh, phep chieu phu it dUdc quan t ^ va gidi thieu vdi ngUbi hpc. Cac bai toan sii dung phep chilu phu bao gib cung la cac bai toan khd, tuy nhien se cho Ibi giai hay va dpc dao.
2.2. Khai thae mot so bai toan van dung phep chieu phu
Bai toan 1. Tim giao diem cua dUdng canh AB vdi mat phang (P) = (ViP, V2P) Ciing tim hieu va khai thae Idi giai bai toan theo nhfing each giai khae nhau
Cach 1 (Hinh 4a): DUng mat phang phu trd (R) xae dinh bang hai dUbng thang song song p va q, vdi p qua A, q qua B; d day ta lay p va q la hai dUbng thang mat.
- Tim giao tuyen U = (R) D (P) trong do I = p n (P); J= q n (P). De cd I ta diing mat phang mat (M) lam mat phang phu trd. De cd J ta diing mat phang mat (N) lam mat phang phu trd
- Xae djnh giao dilm H = IJ fl AB.
Cach 2 (Hinh 4b): Dung mat phang phu trd (R) la mat phang canh: ViR = V2R = AiBi = A2B2.
- Tim giao tuyln MN = (R) n (P) vdi M = ViR n ViP; N = V2R n V2P.
- Ve hlnh chieu canh cua AB va MN. Xae dinh giao diem H = AB n MN:
H3 = A3B3 n M3N3; H3 ^ Hi e A i B i va H2 £ A2B2.
X j
'^:N=nT\l J-
c, = Hfm,
(:
s,^
y^
Hinh 4a Hinh 4b
Cach 3: Dung phep chieu phu chieu mat phang (P) va dubng canh AB len mat phang P theo hudng ViP xuong mat phang (Pg) (Hinh 5)
Nhan xet: Cung mot bai toan dat ra vdi 3 each giai Ishac nhau, co the nhan thiy viec siJ dung phep chieu phu cho ta ldi giai ngan gpn, doc dao trong ldi giai thii 3
Bai toan 2. Cho ba dudng thang cheo nhau d, e, f. Hay dung mot dUdng thing cat d tai M, cat e tai N, cat f tai K sao cho MN = NK.
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Ldi giai (Hinh 6): Chieu ba dubng thang d, e, f theo phUdng dubng thang e len mat phang hinh chilu bang (P2) ta dUdc hinh chieu phu cua d, e, flan lUdt la d', e', f. Do e' la mdt diem nen nen hinh chilu phu cua diem N thuoc e la diem N' thuoc e'. De MK = NK thi M ' N ' = N'K'. Nhu vay thuat toan de giai bai toan nhU sau:
- Budc 1: Dflng diem M' tbupc d' va K' tbupc f sao cho N' la trung diem cua M'K'.
- Budc 2: Tfl hinh chieu phu M', N', K' ta tim dfldc cac diem M, N, K (each dUng xem do thflc)
Bai toan 3. Ve giao tuyen cua mat phang (P) vdl mat chdp S.ABCD Bai toan thfldng dupc ngfldi hpc giai theo cac bUdc sau:
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- Budc 1: Nhan xet ve vi tri tUdng doi cua mat phang (P) va chdp S.ABCD (Cu the: (P) khdng cat day ABCD cua chdp)
- Budc 2: Xae dinh giao diem cua mpt canh ben (chang ban canh SA) vdi mat phang (P) (Su:
dung thuat toan xae dinh giao diem cua dUdng thang vdi mat phang)
- Budc 3: ThUc hien cac thao tac trong budc 2 dli vdi 3 canh cdn lai. Khi do ta dupc giao cua mat phang (P) vdi chdp S.ABCD.
Nhan xet: Khi giai bai toan theo hUdng truyin thong, Ibi giai tuy khdng khd ve mat tU duy nhung kha rUdm ra va ddn dieu.
Sau dSy chiing ta cung khai thae Ibi giai bai toan khi van dung phep chieu phu.
Ldi giai (Hmh 7):
- Budc I: Chilu chdp va mat phang (P) theo hudng V2P len mat phang hinh chilu dflng (Pi), dua mat phang (P) tif vi tri bit ky vl vj tri mdi la mat phang chilu.
- Budc 2; Xae dinh hinh chieu mdi cua mat phang (P) la dfldng thang trung vdi vet ViP, hinh chieu mdi cua chdp la S i ' A i ' B i ' C i ' D i ' .
-Budc 3: Xae dinh hinh chilu cua giao tuyln d vi tri mdi la l i ' 2 i ' 3 i ' 4 i ' .
- Budc 4: Dua kit qua trd lai bai toan ban dau, ta cd dUdc giao tuyln cua mat phang (P) va chop SABCD la tfl giae 1234 ( each ve xem do thiic)
Nhu vay gifla hai Idi giai tren, viec sfi dung phep chieu phu giiip chung ta cd dUdc ldi giai dpc dao, giiip ngubi hoc ren luyen dUdc kha nang van dung tot tfl duy sang tao, phat trien dUdc tU duy thuat toan khi giai quyet cac bai toan phfic tap.
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3. Ket luan
Qua khai thae mdt so bai toan h:en ta thay: Viec sfl dung phep chilu phu de giai quyet mdt so bai toan dang nang cao, ta nhan dupc ldi giai hay vdi kit qua dep de, tfi do giup ngUdi hpc co cd hpi thuan ldi tim hieu va phat trien tU duy thuat toan doi vdi cac dang bai toan lien quan, ddng thdi tflng bfldc biet van dung linh hoat phep chieu phu trong qua trinh iinh hdi tri thflc cua hpc phan Ve ky thuat.
TAX LIEU THAM KHAO
[1] Nguyin Quang CU, Nguyin Manh Dung (2004), Hudng ddn gidi bdi todn hinh hgc hga hinh.
Nxb Xay dung. ' [2] Nguyin Dinh Dien, Do Manh Mon (2006), Hinh hgc hga hinh. Tap 1, Nxb Giao Due.
[3] NgnyenBa Khn (2015), Phuong phdp dgy hgc mdn Todn. Tai banian thfl 7, Nxb Dai hpc Sfl pham. Ha Npi.
[4] Gerald Futschek (2006), Algorithmic Thinking: The Key for Understanding Computer Science. Vierma University of Technology Institute of Software Technology and Interactive Systems Favoritenstrasse 9, 1040 Vienna, Austria
[5] William Lauden (1993), Competency-Based Standards in Teaching: Two Problems - One Solution. Australian Journal of Teacher Education, 17(2).
ABSTRACT
Developing algorithmic thinking for students through exploitation of some answers to problems using minor projection (graphic geometry)
Teaching fundamental and basic knowledge not only helps students gain a deep understanding of their career and job, but also helps students to think about how to recognize and solve problems in reality According to that goal, the section of Graphic geometry equips students with basic knowledge to read and design technical drawings, and at the same time helps learners solve problems in a process to develop algorithmic thinking. This article presents one of the methods to develop mathematical thinking for students of technical universities through Graphic Geometry modules.
Keywords: Algorithm thinking, projection. Graphic geometry.
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