T9P Chi KHOA HQC DHSP TPHCM Le Thi Hoai Chau
MO HINH HOA TRONG DAY HOC KHAI NIEM DAO HAM
Lt TH! HOAl CHAU' TOM T A T
Cac khdi ni$m lien quan den vdn de mo hinh hoa trong dgy hoc Toan duac giai thieu torn hrac a phdn ddu bdi bao. De to chitc dgy hgc mgt tri thuc theo each tiep can md hinh hoa, yeu to ddu tien can tinh den Id nghia cua tri thirc nay, tuc la nhitng vdn de md viec gidi quyet doi hoi phdi co su can thiep cua tri thuc do. Phdn thit hai cua bdi bdo ddnh cho ec lam ro cdc nghia khde nhau lien kit trong khdi niem dgo hdm. Trong thuc te, nhiing I ghia do CO dicgc hgc sinh huy dong khi hg dihtg truac mgt tinh huong ngodi todn hgc hay khong? Cdu trd loi se tim thdy trong nghien cuu thuc nghiem md chung toi trinh bdy &
phdn cudi cua bdi bdo. Ket qud thu duac tic bdi bdo se cho thdy nhimg yiu to cdn tinh den khi dgy hoc khdi nl^m dgo hdm theo cdch tiep can mo hinh hoa.
Tic khoa: mo hinh hoa, dao ham, van toe tuc thai, tiep tuyen, xap xi afine.
ABSTRACT
Modeling in teaching the concept of derivative
The concepts related to modeling in mathematics teaching are introduced in the first part of this paper. To organize the teaching activity for a knowledge module following the modeling approach, the first factor to take into account is the significance of this knowledge, that is, the problem to which the solution requyres the intervention of such knowledge. The second part of the article is devoted to clarify the different meanings associated to derivative^
concept. In fact, are these meanings utilized by students in a situation beyond mathematics or not? The answer will be found in empirical studies presented at the end of the article. The results obtained from the paper will identify the factors to consider when teaching the concept of derivative following the modeling approach.
Keywords: modeling, derivative, instantaneous velocity, tangent, approximately affme.
Toan hoc bat nguon tu thuc tien, va moi li thuyet toan hoc dii trim tuoug den dau cung deu tim thay ung diing ciia chiing trong thuc tien (tu "thuc tien" 6 day dugc dung theo nghia rgng, bao gom ca thi^c te cugc song lan cac khoa hgc khac). The nhimg, xu huang d^t muc tieu day hgc (DH) Toan vao viec cung c4p nhung kien thuc pho thong, ren luy?n ki nang giai mgt s6 dang toan tieu bieu g4n lien vai chiing da khien cho kien thiic toan day a nha truang tra nen hinh thtic, kho khan, khong m4y hSp dSn va bo (ch d6i vai dai da s6 hgc sinh (HS).
Nhan thdy bit cap nay, chucmg trinh giang day Toan 6 nhieu nuac tren the gioi tu mky thap nien qua da chti trgng den muc tieu phat trien nSng luc sir dung mgt each sang tao nhCmg kiln thtic va ki nang toan hpc khac nhau vao viec giai quyet cac van de
' PGS TS, Khoa To^n - T i n hgc, Trudng DHSP TPHCM
Tap chi KHOA HOC DHSP TPHCM •»« «} nam ^vji
da dang do thuc tiln dat ra. Nhu3ig ciiucmg trinh cung nhu nhung kiiu DH thien ve IciSn thiic han lam, xa roi thirc tien dang dan dan bi loai bo.
1. 1VI6 hinh hoa toan hoc
Dd sir dung kifa thuc va ki nang toan vao viec giai quySt mot van de cua thuc tiln, nguoi ta phai trai qua cac buoc ciia qua trinh mo hinh hoa toan hpc - qua trinh chuyin vin dl thuoc linh vuc ngoai toan hoc thanh vin de cua toan hoc, roi sii dung cac cong cu toan dl tim cau tra loi cho v8n dl dugc dat ra ban diu. Duoi day, chiing toi se trinh bay ngln gon mpt vai khai niem co ban lien quan den qua trinh mo hinh hoa toan hpc.
1.1. He thong vd mo hinh
Noi vl qua trinh rao hinh hoa toan hpc, Chevallard (1989) xuat phat tit hai khai niera he thdng va mo hinh - cua he thong nay. Ong lay lai dinh nghia neu trong Encyclopaedia universalis, theo do thi moi he thong la mpt tap hgfp cac phan tu voi nhung tac dpng qua lai giQa chiing, ma nhiing tac dpng nay phai tuan theo rapt so nguyen li hay quy tac nao do dae trung cho he thong nay.
Mo hinh la mgt mau, rapt dai dien, mpt rainh hga dugc thiet ke de rao ta cau tnic Clia he thong, each van hanh cua mgt hoac cac su vat, hien tugng thuoc he thong nay.
Nguofi ta thuong su dung khai niem mo hinh vol hai nghia khac nhau.
Theo nghia thii nhat, mo hinh la mpt ban sao, mot vi du, co nhiing tinh chit dae trung cho su vat goc ma mo hinh do bieu dien. Vofi nghTa nay thi cac khdi ciu, chop, non (cu thi, vat chat) dugc sii dung trong DH hinh hgc la nhitng mo hinh ciia cac khai niera hinh cau, hinh chop, hinh non. Hay tap hgp R^ vol hai phep toan dupe dinh nghia nhu sau:
V (xi, X2), (yi, yj) 6 R2, Va € R:
d i . x a ) + (yi.yz) = (xi + yi,x2 + yi) a ixi,X2) = iaxi,ax2)
la mpt mo hinh cua khong gian vecto trim tugng dinh nghia trong Dai s6 tuyen tinh.
Theo nghia thit hai thi mo hinh la rapt bilu diin cho cac phin quan trpng cua mgt he thong (co sin hoac sip dupe xay dung) voi muc dich nghien cim he thong do. Noi each khac, mo hinh la eai thu dugc tit viec diln dat theo rapt ngon ngit nao do cac dae trung cM yeu ciia mot tinh huSng, rapt he thong ma nguM ta cin nghien cihi. Cach bieu dien nay tuan theo mgt tap hgp cac quy tic nao do. Khi cac quy tic iy la quy tic toan hgc thi mgt mo hinh toan hpc da dupe tao ra.
Bai bao nay sit dung thuat ngu mo hinh theo nghia thir hai.
1.2. Mo hinh hoa todn hpc
Mo hinh todn hoe la su giai thich bing toan hpc cho mpt he th6ng ngoai toan hoc vol nhirng eau hoi xac dinh ma nguoi ta dat ra tren he th6ng nay. Qua trinh mo hinh
Tap chi KHOA HOC DHSP TPHCM Le Thi Hoai Chau
hoa toan hoc la qua trinh thiit lap mgt mo hinh toan hgc cho van de ngoai toan hgc, giai quyet van de trong mo hinh do, roi the hien va danh gia lai giai trong ngii canh thuc te, cai tien mo hinh neu each giai quyet khong the chap nhan. Trong phan con lai Clia bai viet, de ngan ggn, chiing toi se dung thuat ngii mo hinh hoa thay cho mo hinh hoa todn hoc.
Noi ve qua trinh mo hinh hoa, L. Coulange (1997) phan biet ba khai niem khac nhau: bai toan thuc tien (probleme concret), bai toan phong thirc tien (probleme pseudo -concret) va bai toan toan hgc (probleme mathematique) (tham khao [3], Le Van Tien, 2005, tr. 93). Voi ba khai niem nay, L. Coulange (1997) dua vao cac thuat ngu mo hinh thirc tien, mo hinh phong thirc tien, mo hinh todn hoc, va chia qua trinh mo hinh hoa thanh 4 buoc.
Phong theo Coulange (1997), chiing toi dua ra duoi day sa do torn luge cac buac cua qua trinh mo hinh hoa:
M 6 HiNH THVC ITfiN PHAM VI NGOAl TOAN HQC H^ihSnghc^^i^hiAr^r^p^to&nh^c
C&tt hdi liSr^ quan d^ h4 A$ng C&n tra UA '"" ' cho bai toan tiiac ttm
(1) Rat gfii hf thoDg (gjj^ tfi nhftng th Ang tin th6a ding) Md HINH PHONG THV'C TIEN
Bdi todn phdng ihfc ti^
PHAM VI P H 6 N 0 THyC HfiN
Can tra lorl cho bai toan ph.6ffig thqrc tiln (2) Hiit b i ^ van dc bing ngdn ngO toan b^c dd lya chpn
M 6 H I N H T O A N HQC PHAM VI TO A N Hpc Bdi to^ todn hpc (3) Giai bai toan CantralM
cho bai toan tout hoc Sff 06, Qud trinh md hinh hda
Chung toi cu ihk hoa 4 bu6c ciia qua trinh mo hinh hoa nhu sau:
Chiing toi cu the hoa 4 buac ciia qua trinh mo hinh hoa nhu sau:
Bifoc L Xay dung mo hinh phong thuc tien ciia van de, tiic la xac dinh cac yeu to CO y nghTa quan trgng nhat trong he thong va xac lap nhiing quy luat ma chung ta phai tuan theo.
Bir&c 2. Xay dung mo hinh toan hgc cho van de dang xet, tuc la dien.ta lai duai dang ngon ngQ toan hgc cho mo hinh phong thuc tien. Luu y la ling vai vkn 6k dang
Tap chi KHOA HOC DHSP TPHCM Sd 65 nam 2014
xera xet co thi co nhilu mo hinh toan hpc khac nhau, tiiy theo cho cac yeu to nao ciia he thdng va m6i lien he nao giiia chiing dupe xem la quan trpng.
Bu&c 3. Sit dung cac cong cu toan hpc dl khao sat va giai quylt bai toan hinh thanh 6 bugc hai.
Bie&c 4. Phan tich va kilm dinh lai cac kit qua thu dugc trong buac ba, 6 day, nguai ta phai xac dinh rauc dp phii hgp eiia mo hinh va ket qua tinh toan vol van dl thuc tl. Nlu kit qua khong thi chip nhan dugc thi phai lap lai qua trinh dl tim cau tra lai phii hgp cho bai toan ban dau.
7.5. Mo hinh hoa trong dgy hoc todn
v i n dl la phai b6i duang cho HS nang luc giai quylt cac van de ciia thuc tiln bing nhung kiln thuc toan raa hg thu nhan dugc. Viec gin DH Toan voi giai quylt cac van de cua thuc tien mang lai nhieu lgi ich. Truac het no giiip HS hieu nghTa cua tri thuc hgc dugc, li do ton tai va Ipi ich cua no cho cupc song ca nhan ciing nhu xa hgi.
Sau do, quan diem DH gan vai mo hinh no co the tao hirng thii hgc tap, ren luyen nang lire tu duy cho HS.
Noi ve mo hinh hoa trong DH Toan, tac gia Le Van Tien (2005) phan biet hai khai niera DH mo hinh hoa va DH bdng mo hinh hoa.
DH mo hinh hoa la day hpc each thu-c xay dung mo hinh toan hoc ciia thuc tien, nhim tcri tra lai cho nhihig cau hoi, vin dl nay sinh tit thuc tien.
[...] Quy trinh DH co thi la: DH tri thirc todn hoc U thuyil —• Van d(mg cdc tri thuc ndy vdo viec gidi cdc bdi todn thirc tien vd do do vdo viec xdy dung mo hinh cua thirc tiin.
(Le Van Tiln, 2005, tr. 96)
Ve mat su phara, quy trinh nay lam mit di vai tro dpng co cua bai toan thye tien.
Ve mat khoa hpc luan, no lam rait di ngu6n g6c thuc tiln ciia tri thirc toan hgc. Quy trinh DH bSng mo hinh hoa eho phep khic phuc khilm khuylt nay.
Van de la day hpc Toan thong qua DH mo hinh hoa. Nhu vay, tri thirc toan hpc cin giang day se nay sinh qua qua trinh giai quylt cac bai toan thirc tien. Quy trinh DH tucmg img CO thi la: Boi /odn rtirc tiin -> Xdy dimg mo hinh todn hoc -^ Cdu trd Idi cho bdi loan thirc tiin -> Tri thirc cdn gidng day -» Van dung tri thirc ndy vdo gidi cdc hdi todn Ihirc tiin. (Le Van Tiln, 2005, tr. 96)
Nhimg CO so li luan trinh bay 6 tren din ta din ch6 phai thira nhan tinh cin thilt tat yeu ciia mpt nghien dm khoa hpc luan vl nghia cua tri thiic cin day (nguSn g6c nay sinh, li do ton tai ciia tri thirc, nhQng vin dl ma no cho phep giai quylt, v.v...) d6i voi viec t6 chiic DH theo each tilp can mo hinh hoa.
2. Nghia ciia khai nifm d?o ham
Thuat nga dgo hdm ma chiing toi noi din trong bai bao nay diing dl chi dgo hdm bge nhdt ciia hdm so tgi mgt diim thugc tap xdc dinh.
Tap chi KHOA HOC DHSP TPHCM LS Thi Hoai Chau
2. L Nhirng bai todn ca ban la nguon goc Mnh thanh khdi niem dqo ham Co hai bai toan ca ban la nguon goc lam nay sinh khai niem dao ham, mgt thugc ITnh vuc Hinh hgc va mgt ddn tu Vat li.
Bdi toan hinh hgc: xdc dinh tiep tuyen ciia ducmg cong
Neu nhu truoc day, nhieu bai toan cua Dai so chi co the dugc giai quySt nho cac cong cii va phuang phap ciia Hinh hgc, thi ke tii thg ki XVI, vai he thdng ki hieu do Viete de nghi vao nam 1591, Dai s6 da tach khoi Hinh hgc, phat triSn m6t each dgc lap vai nhiing phuang phap co siic manh Ion lao. Nhan thay siic manh iy, Descartes va Fermat da khai thac no vao nghien ciiu Hinh hgc bang viec xay dung nen Hinh hoc gidi tich. Sir ra dai ciia Hinh hgc giai tich khien cho van de nghien ciiu nhi^u ducmg cong phiic tap tra nen de dang han. Bai toan tim tiep tuyen ciia mgt duang cong bit ki cung dugc dat ra. Bai toan nay chi dugc cac nha toan hgc thai ki truac giai quySt d6i voi mgt so duong dSc biet (duong tron, duang conic, duang xoan 6c Archimedes) bang cong cu Clia hinh hgc co dien. Tuy nhien, voi hang loat nhiing duong cong mai xu4t Men, bai toan xac dinh tiep tuyen doi hoi mgt phuang
phap tong quat han.
Khai niem tiep tuyen liic nay dugc hiSu theo nhiing quan niem mdfi nhu la vi tri '"tai han"
cua cat tuyen hay duong thang trCing vai mgt phan vo cung nho vai duang cong tai tiep diem.
Chinh tu quan niem "vi tri tai han" nay ma he so goc Clia ti6p tuySn vai duong congy =f(x) dugc dinh nghTa (theo ngon ngii ngay nay) boi bieu thuc i i „ , / ' - ^ * ) - / M . / • ( . ) .
Bdi todn vgt li: tim van toe tire thai
Thira nhan rang co the xem van t6c tire thoi 17„ cua vat the co phuong trinh chuyen dpng 5 = S(t) la giai han ciia van toe trung binh trong khoang thoi gian (t,t + At) khi A/ -> 0, Nevrton cung da di den bieu thiic xac dinh V[{ (co ciing ban chit vai bieu thiic xac dinh he s6 goc cua tiep tuyen) ma theo ngon ngQ ngay nay ta vilt la:
S{t + At)-f(_t)
I'.fi'l)
- lim --s\t).
2.2. NhCmg nghia khac nhau ciia khdi nilm dgo ham Dgo ham-hi so gdc cua tlip luyin
Bay la nghia hinh hgc cua khai niem dao ham. Nha khai niem dao ham cua hara s6 tai mpt diem, nguai ta tra lai dugc cau hoi tdn tgi hay Ichong tiip tuyin ciia dudng cong tgi diim ndy va neu ton tai thi do la duong thing nao, dung no ra sao. Cau tra lai
T9P chi KHOA HOC DHSP TPHCM sS 65 nam 2014
cho cau hoi iy trong trucmg hpp mgt duang cong bit ki kho raa tim thiy vol cac cong cu cua hinh hpc thuan tuy.
Dgo hdm - tde dp bien thien cda hdm so
Phuong phap giai bai toan tim van t6c tire thai cung dugc Newton vjin dung dl giai nhilu bai toan khac cua vat li. Nara 1687, trong cu6n sach nil tiing nhit ciia minh NhCmg nguyen li todn hgc cua triit hgc tu nhien Newton da phat bieu ba djnh lu^t quan trpng vl chuyin dpng, trong do dinh luat thir hai cho thiy mpi thay doi ve trang thai chuyin dpng dlu dugc gay ra bai luc tac dpng len vat, va moi quan he giua lire vol chuyin dpng tuan theo quy tic: lire tac dgng ti le vai dao ham cip hai cua ham tga do chit dilra. Luc nay, nlu biit truac luc tac dgng thi nguoi ta co the tinh toan dugc quy dao cung nhu du doan truac duge tucmg lai cua mot su kien. NhiJng nghien cuu ciia Newton da khiln dao ham mang lai ragt quyen luc 1cm lao cho cac nha vat li.
Tir cai nhin cua vat li, Newton di den y tucmg xay dung Giai tich hgc tren ca sir cua '"chuyen dpng", trong do dao ham dugc djnh nghia nhu la toe dp bien thien titc thcri cua rapt dai lugng nao do theo "thoi gian". "Thcri gian" a day khong chi hilu theo nghia den, raa theo nghia tong quat, la rapt bien bat ki x nao do bien thien deu theo thoi gian, nghia la sao cho (;c)' = 1. Y tucmg ciia Newton da raang lai cho d?o ham rapt dae tnmg rat true quan va huu ich: dao ham la thuac do toe dp bien thien cua ham so so vai toe do bien thien eua doi so. Quan niem nay da raa ducmg cho nhihig ling dung 6 at, manh rae va vo ciing hieu qua ciia dao ham noi rieng, Giai tich noi chung, trong viec giai quyet nhieu van de khac nhau ciia vat 11 cung nhu toan hgc, de roi tu do ma rgng ra cac linh vuc khac ciia thuc tien. Chang han, vol dao ham, nguai ta da giai quyet dugc van de nghien cim su bien thien, cue tri ciia cac ham so, va img dung do d3 dugc sir dung cho cac hien tu(;mg cua kinh te hpc, xa hoi hpc...
Chiing toi gpi day la nghia vgt li cua khai niem dao ham. Can phai noi ro rang tinh tit vgt li trong truang hgp nay muon noi den linh vuc von la mgt nguon goc cua khai niem dao ham. No khong co nghia la chi giai han cho van toe ciia cac chuyen dpng CO hpc. Theo nghia vat li nay, dao ham phan anh tde do biin thien cua mpt ham s6 so voi t6c dp biln thien cila bien so j : khi x -» x^.
Dgo hdm — cong cu xap xi hdm so
The ki XVIII, sau khi khai niem dao ham va ham so dao ham da dugc dinh nghia tucmg minh, cac nha bac hpc da diing no de giai quylt nhilu van dl cua vjt li va toan hpc. Nam 1715, Taylor dua ra mpt ket qua raa ngay nay ta ggi la cong thiic khai triln Taylor: neu x = XQ + h, nghia la ft = x - XQ thi
/ W = /(Xo) + ^ f t + ^ h ^ + - + ^ ^ ^ h " + 0(ft'') trong do 0(h'') la rapt vp ciing be bac cao hon ft".
Cong thuc Taylor cho phep xip xi f(x) vai mpt ham da thiic. Trong nhilu trucmg hgp, viec nghien cim / ( x ) tra nen dl dang han nhieu nlu ta chuyen vl mpt ham da 10
Tap chi KHOA HOC DHSP TPHCM Le Thi Hoai Chau
thirc xap xi \ai fix). Trucmg hpp don gian nhat, nguoi ta co the xap xi f{x) veil mpt ham tuyen tinh (gpi la xip xi affine). Nlu chuyen sang ngon ngQ hinh hgc thi dieu nay CO nghia la phin ducmg cong / ( x ) trong mpt lan can (kha be) cua x m the dupe xip xi viii mpt doan thing (chinh la tiep tuyIn tai x). Chung toi gpi day la nghia cong cu xdp xi ciia khai niem dao ham.
3. Nghien eiru thirc trang day hoc khai niem dao ham
Trong mpt chudi nghien cim dai hai duge thuc hien tu nhieu nam qua, chiing toi muon tim hieu thuc trang DH khai niem dao hara a bac THPT va sau do tim each tac dgng vao thuc trang nay bing nhimg tinh hu6ng DH co tinh din qua trinh mo hinh hoa.
Ba cau hoi nghien cim dugc dat ra la:
Nghia eiia khai nipm dao ham dupe hinh thanh ra sao trong cac sach giao khoa (SGK) Toan Icip 11, 12 cung nhu trong thuc hanh DH cua giao vien?
Sir lira chpn ciia SGK va giao vien anh huong nhu the nao len kien thirc cua HS?
Tir quan diem tich hgp va mo hinh hoa trong DH toan, nen t6 chirc DH nhu the nao de HS co the co nhQng kien thirc day du hon ve khai niem dao ham?
Vai khuon kho co han cua bai bao, trong phan nay chiing t6i chi trinh bay mgt vai ket qua tra lai cho cau hoi thir hai.
Nghien eiru thirc trang DH khai niera dao ham da dugc chiing toi tien hanh qua nhilu thuc nghiem. Chiing toi se trich ra de trinh bay 6 day mgt s6 kit qua thu dugc qua ba trong s6 nhiing thuc nghiem do. Trong ba thuc nghiem thi co mpt thuoc pham vi dl tai nghien eiru Md hinh hda trong dgy hgc Todn (Le Thi Hoai Chau, 2014) va hai thupe khuon khfi cac luan van Thac si chuyen nganh Li luan va Phuong phap DH Toan bao ve a Dai hgc su pham TP H6 Chi Minh (Biii Anh Tuin (2007), Ng6 Minh Due (2013)). Thuc nghiem dugc tiln hanh ducii hinh thirc cho HS lam viec (ca nhan hoac nhom) tren cac vin dl (toan hgc hay ngoai toan hpc) dugc xay dung theo nhung muc dich nghien ciiu khac nhau. Duoi day chiing toi chi trich ra nam trong s6 cac vin dl da sir dung cho ba thue nghiem do. Thuat ngu bdi todn dugc diing de noi ve cac vin dl nay. Nhimg phan tich trinh bay duoi day dlu thu dupe qua cac pha HS lam viec ca nhan.
3.1. Thirc nghiem 1
Bdi loan 1. Cho hdm s6 fix) = Vx vdxo = 1.
a) Tinh /(XQ) vd /'(xo).
b) Khong diing may tinh bd tiii, img dung cong thirc / ( x ) » / ( x o ) + / ' ( X o ) ( x - Xo)
di tinh gin dimg (Idy din ba cha sd thap phdn) cdc gid tri .^0,85 ; V0i9 ; ^0,95 ;
JTfl5:^fTJ•,Ji^S.
Tap chi KHOA HOC DHSP TPHCM sS 65 nam 2014
c) Goi a b. c, d. e. f ldn luai Id cdc gid tri gdn dung cua VoisS ; V0i9 ; V0.95 .- yflfiS: V U ; VLTS". Trong mgi phdng tga dg Oxy, cdc diim A (0.85: a). B (0.9; b).
C(0.95:c), D(l,05:d). E (1.1: e) vd F (1.15; J) ca Ihdng hdng hay khdng? Gidi ihich ro li do.
(Trich tir Bill Anh Tuin (2007) va Le Thi Hoai Chau (2014)) 6 day HS dl dang tinh dugc ngay /(x„) = 1 ; f (xp) = 0,5 ; V ^ S * 0,925;
Vo;9*0,95 ; Va95 =0,975; ,fijs ^1,025; V l T * l , 0 5 ; ^ U S " * 1,075. Sau do, nhimg chiln luge xet su thing hang co the dugc sir dung la:
Chiln luge "vecto": Dimg khai niem hai vecto ciing phucmg trong hinh hgc giai tich dl chimg minh it nhit 4 bp ba dilm thing hang;
Chiln luge "dilm"; Ve 6 dilm tren len mjt phing tga dg, roi ket luan;
Chiln luge "ducmg thing": Vilt phucmg trinh ducmg thing qua 2 trong 6 diem do, sau do kilra chimg cac dilm con lai co nim tren duong thang hay khong;
Chiln luge "tang dlu": 6 dilm da cho co hoanh d6 tang deu 0,05, con tung dp tang dlu 0,025, nen chiing thang hang;
Chiln luge "phuang trinh": Tit cong thiic tinh gin dung, ta suy ra cac diem A, B, C, D, E va F CO tpa dp thoa man phuong trinh y = 0,5(x - 1) + 1. Day la phuong trinh cua tilp tuyIn tai XQ. Vay chiing thing hang. Day la chiln luge toi uu cua bai toan. Lim ^ ring khi diing chiln luge nky eiing co the HS kit luan sau diem dang xit thugc duong thing y = 0,5(x + 1) ma khong nhic din "tilp tuyIn". Vi IS do tac gia Bill Anh Tuan gpi day la chien luge "phucmg trinh".
Xet s\r thang h^ng cua cac diem biet tpa dp ciia chiing la mgt kieu nhiem vu quen thugc ma HS thucmg gap trong Hinh hpc giai tich. Nhung tpa dp eac dilm a day da dugc tac gia chpn sao cho viec giai quyet bai toan bang cac chien luge hinh hgc gap nhilu kho khan. Li do eua sir lua chpn nay la de thiic day HS den vol viec sflt dung dao ham, hay noi chinh xac hon la phuong trinh tiep tuyen, de chiing to sir thing hang ciia sau diem da cho. Cong thirc xap xi dupe nhac lai (da co trong SGK Giai tich 11), trong do ve phai co lien quan den phucmg trinh cua tiep tuyen veil ducmg cong tai XQ (mpt ngi dimg dugc chu trpng trong SGK cung nhu trong cae de thi tot nghiep trung hgc pho thong (THPT) hay tuyen sinh dai hgc) cung la lira chpn nhim tag thuan lgi cho su xuit hien ciia chiln luge "phuang trinh".
Bai toan 1 se cho phep kiem tra sir ton tai hay khong trong kiln thirc cua HS nghia cdng cu xdp xi ciia dao hara. Su xip xi a day dupe dat trong phara vi hinh hpc va gan lien vol nghia hinh hpc: trong mpt lan can kha be cua XQ nguai ta co thi xip xi ducmg cong vai tilp tuyen ctia no tai XQ.
Bai toan 1 la mpt phan cua thuc nghiem dupe thuc hien vol 107 HS vira t6t nghiep THPT a Cin Tho, sau do da dupe chung toi neu ra cho 49 HS Icrp 12 thupe mpt truong tren dja ban TP H6 Chi Minh. Bang 1 ducii day th6ng ke kit qua thu dugc.
Tap chi KHOA HOC BHSP TPHCM Li Thi Hoai Chdu
Bdng 1. Thong ke ket qua thuc nghiem vdi bdi todn 2
\'ecto Duong thang Diem Phirong trinh TSng deu Khac Khong giai
Ket iuanthanghano Ket luan khong Ket luan thing hang Kci luan khong Ket luan thang hang Ket luan khong Ket luan thang hang Ket luan khong Ket l u ^ thang hang Ket luan khong
So lirong 107 HS
43 7 10 0 15 1 2 0 15 0 12 2
49 HS 22
2 S 1 3 1 4 0 5 0 3 0
Bang 1 cho thay chi co 2/107 va 4/49 HS sii dung chiln luge "phucmg trinh", dii tinh huong thuc nghiem da dugc thiet ke theo hucing gay tra ngai cho viec su dung eae chien luge hinh hgc. Can noi them rang khi quan sat HS 6 pha lam viec theo nhom (tham khao Le Thi Hoai Chau, 2014), chiing toi thay kieu lap luan sau day xuit hiSn a khong it HS: "^, B, C, D, E, F khdng thdng hdng vi chimg ndm tren dudng cong fix) = VF". Nhu vay, dii tung do cac diem da dupe tinh theo cong thirc gin dung, HS van khdng thiet lap dugc mil lien he giQa tinh hu6ng viii cong thiic da dugc dl cap trong DH vl sir xip xi affine ciia m6t ham s6. Duong nhu nghia cdng cu xdp xi affine va sir xdp xi hinh hgc cua dao ham chua thuc su hien dien trong kiln thire eiia HS.
3.2. Thirc nghiem 2
Bdi todn 2. Trong bdi "Phuang Irinh dao dgng ciia con ldc dan ", sdch gido khoa Vgt ll lap 12 CO neu nhdn xet sau. "Khi x nhd (x <1 rad) la cd Ihi coi sinx -~ x". Em hay dua ra ldi cd cdc li do cd the de gidi thich cho nhdn xel tren.
Bdi todn 3. Sd ddn cua mol thi Irdn sau t ndm ke lir ndm 1970 (chdng han, niu xel a ndm 2012 ihi t = 2012 - 1970 = 42) duac uae tinh bdi cdng thirc:
fit) = ^ ^ ^ (nghin nguai)
Tinh sd ddn cua Ihi trdn vdo ndm 1990 vd 2008.
Vdo hai thdi diem ndm 1990 vd 2008. ndm ndo ddn sd ciia thi trdn Idng nhanh han ? (Trich tir Ngo Minh Diic, 2013) Hai bai toan tren da dupe dua ra cho 35 HS lop 12 o hai trucmg THPT dja ban TP H6 Chi Minh. Yeu cau a bai toan 2 khong phai la tinh gan diing gia trj cua ham s6 tai mgt diem, raa la giai thich mgt xap xi hara da dugc thira nhan trong SGK Vat li 12.
Chiln luoc tim cau tra lai co the la:
Tap chi KHOA HOC DHSP TPHCM Sd 65 ndm 2014
Ap dung cong thirc xip xi affine (fix) » fixg) + /'(Xo)(x - XQ) khi |X - X„| . kha nho) cho ham so fix) — sin x tai Xg = 0.
Ap dung cong thuc limx_„ 5^!^ = 1: khi x kha nho (kha gin 0) thi ^ » 1 hay s i n x SB X.
Chpn cac gia tri x lin lugt ngay cang nho r6i dimg may tinh bo tui tinh gia trj sinx tuang ung. D6i ehilu va chi ra ket luan sin x «= x khi x kha nho.
Ve vong tron lupng giac roi lap luan dua vao dp dai cung AB chin goc a va dp dai AH = sina: Dp dai cung AB li AB = Ra = a (do R=l). Khi a kha nho thi AB = AH, tire la a = sina.
Giong nhu bai 1, bai toan 2 ciing dupe xay dung de kiem tra sir hien dien hay khong trong kien thuc cua HS nghia cdng cu xdp xi ciia dao ham, tire la tira hieu quan niem cua hg ve raoi quan he giua dao ham vol xap xi affine. Tuy nhien, khac vai bai toan 1, sir xap xi a day la xap xi so thuan tiiy. Chiing toi gia dinh rang so vai xap xi hinh hgc thi quan niem xap xi so thuan tiiy nay de eo idia nang dugc hinh thanh ban of HS.
Cau tra Icri mong dgi la sir dung cong thiic xap xi affine eua ham fix) - cong thitc hien dien trong SGK Giai tich 12. The nhung, trong thuc tl thi lai khong co mpt hpc sinh nao sir dung cong thitc nay.
Bai toan 3 von la bai toan co mat trong SGK Giai tich 12 (ehuang trinh nang cao). Dilm thay doi o day la tac gia da bo di thong bao "Dgo hdm ciia hdm sdfbiiu thi tde do Idng ddn sd cita thi Iran'" ma SGK dua vao de bai toan. Hem nua, thay vi dat cau hoi nhu SGK la ''linh tde do tdng ddn 50" trong timg nam, tac gia yeu ciu HS so sanh xera trong hai thoi diem tren "nam nao dan so ciia thj trin tang nhanh hem". Viec dat ra bai toan nay se giup kiem tra xem dae trung idc dp biln thien lieu co xuit hien hay khong trong quan niem ciia HS sau khi da dugc hpc khai niem dao ham.
Nhung chien luge ma HS co the su dung dl tim cau tra lai la:
Tinh dao ham: fit) = ^ °
Tai nam 1990, tj = 1990 - 1970 = 20 => f'(20) = 0,192 Vai nam 2008, \.2 = 2008 - 1970 = 38 => f'(38) = 0,065
Do / ' ( t l ) > / ' ( t p nen nam 1990 dan s6 thj trin tang nhanh hon nam 2008.
- Tinh lugng dan so tang tit nam 1970 den 1990 r6i ehia dlu cho so nam la 20 nam de tinh trung binh. Tiln hanh tuong tir cho nam 2008 r6i so sanh 2 t6c dp trung binh nay de ket luan (hoac tinh s6 dan tang tir nam 1989 din 1990 va s6 dan tang tit 2007 den 2008, co nghia la tim so dan tang trong mpt nam tinh den nam dang xet rli so sanh de ket luan).
Ve do thj ham fit) roi xera xet tren d6 thj tai hai thai dilm ein xet xem tai dau ham so doc hon (nghia la se tang nhanh hon).
Tap chi KHOA HOC DHSP TPHCM Le Till Hoai Chau
Kit qua thu dugc tit thuc nghiem cho thiy chien luge diu tien (sir dung dao ham) trong thirc tl cung khong hi xuit hien. Loi giai chilm uu thi (19/35 HS) la sil dung chien luge thir hai: tinh tde do tang dan s6 trung binh roi dua vao toe dp trung binh nay dl so sanh t6c d6 tang dan s6 tire thcri tai hai thoi dilm tren. DucVi day la mgt bai giai co dugc tir chien luge do:
rj!^:,,M^^^O..'(i^--&i
Viec da so HS lua chgn chien luge nay cho thay cac em biet rang de tim thai diem dan so tang nhanh hon thi phai tinh toan toe do tang dan so. Khai niem toe do bien thien trung binh van hien dien trong quan niem ciia hgc sinh. Nhung khi c^n phai xac dinh toe do bien thien ttic thoi thi chien luge dao ham lai khong xuat hien.
3.3. Tht^c nghiem 3
6 duoi la hai bai toan trich ra tir thuc nghiem gan day nhat, dugc chiing toi thuc hien voi 47 HS lap cuoi lap 11 ciia Truang Trung hgc Thuc hanh - Dai hgc Su p h ^ TP H6 Chi Minh.
Bdi 4. Hai chdt diem chuyen dong thdng tren mot true dinh hir&ng. Vi tri tircmg img pt, p2 Clia chdng tren true phu thuoc vdo thai gian t va sir phu thuoc do dime cho bai hai dd thi (pi), (p2) trong hinh ditai.
p '
V\
• 0
-
i , t . •
i 1 .' 1
• i; - ; - | t»
1 i
a) Dua vdo dd Ihi. em hay udc linh Ihdi diem md hai chdt diim cd cimg van tde.
bj Xdc dinh thdi diem dy bdng tinh lodn, biit rdng edc hdm sd tuang ung vdi hai do Ihi ddld:piit)= t ^ P 2 W = ; t -
Tap chi KHOA HOC DHSP TPHCM Sd 65 nam 2014
Bdi 5. Phuang irinh ctia mol chuyin ddng ciia mot vdi Id p= 4t. vaip Id vi trl ciia vgl Igi Ihdi diim I Khi t = 9 = 3^ thi la linh dugc ngay vi Iri ciia vdi Id p = 3. Nhirng Igi nhimg Ihdi diim md gid tri cua I khong phdi Id sd chinh phuang Ihi, neu khdng cd mdy linh hd Hii, la khdng di lim duac sd p biiu Ihi vi Iri luang ling ciia vdt Chang hgn, tgi I = 8,96 la chi cd Ihe ndi rdng p = 3 (vi Iri ciia vdi a gdn 3).
a) Ngudi la mudn lim mol gid tri xdp xi (cua p) biiu Ihi chinh xdc han vi Iri thuc cua vdi Igi I = 8.96. Em hay di nghi mdt (hay nhiiu) cdch Ihire de dua ra mdt gid tri nhu vdy.
Nhd rdng d ddy em khdng duac sir dung mdy tinh bo liii
b) Cho mdt vdi chuyin ddng Iheo theo phuang Irinh p= t" + 3t-l (p Id vi Iri cua vgl lai Ihdi diim I). Mol ngirdi ndi rdng tai I = 1.127 Ihi p » 1. £m hay lim mol gid Iri bieu Ihi chinh xdc han vi tri Ihuc ciia vdt Igi thai diem dd.
(Trich tir Le Thj Hoai Chau, 2014) Thuc nghiem 3 ma chung toi xay dung dugc ggi len tir nghien curu cua J-Y Gantois va M. Schneider (2009). Hai bai toan deu dat trong tinh huong ca hpc, loai tinh hudng dupe SGK Giai tich 11 uu tien de dua vao khai niem dao ham theo each tilp can DH bing mo hinh hoa.
De giai bai toan thir nhat, truac het HS phai nhan ra P2 la rapt chuyen dpng thang deu va do thj cho thiy van t5c ciia no la - . Kiln thirc vl dao ham ma HS phai sii dung a day la: hai chuyen dpng se dat eiing van toe tai thoi diera ma tiep tuyen cua pj co he s6 goc la -. Nhu vay trong trucmg hpp nay, HS phai co mpt su lien kit hai nghia khac nhau ciia dao hara - nghia vat li va nghia hinh hgc.
Bai toan thu hai dugc dua ra de tim hieu xem khai niem dao ham co dupe HS huy dpng kha de tira nhQng gia trj xip xi dja phuong ciia rapt ham s6 hay khong.
Chung toi chi trinh bay tom luge 6 day nhung kit qua quan sat dugc qua thuc nghiem. 12/47 HS khong dua ra lai giai eho bai toan 1. Trong 35 HS con lai chi 13 era su ditng dao ham de giai bai toan lb, trong do 9 era dua ra dap s6 dung. Trong s6 22 HS khong sit dung dao ham thi co din 10 em tira dap s6 bing each giai phuang trinh t^ = -t. Co thi giai thich la 10 HS nay da cho ring tai thcji dilra hai chuyin dpng cc ciing van toe thi quang duong di dupe cung bing nhau. 12 HS khac khi giai cau la) cij ve thera ducmg thang song song voi P2 va "tilp xue" vai Pi, sir dung cac tinh chit hinh hpc (djnh li Pythagore) va c6 ging tira quang ducmg di dugc ciia chit dilm chuyin dpng theo phuong trinh pi, nhung khong di din kit qua.
Ta thay o day HS eo nhimg kho khan trong viec sir dung dao hara dl tim van t6c titc thcri cua rapt chuyen dgng trong trucmg hgp gian tilp da biit he sd goc ciia tilp tuyIn, du y nghia hinh hgc cua dao ham la rapt npi dung DH dugc trinh bay tuang
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Tap chi KHOA HOC DHSP TPHCM Le Thi Hoai Chau
minh trong SGK Giai tich 11. Hai nghTa hinh hgc va vat li cung khong dugc HS-huy dgng 6 day trong su gan ket vol nhau.
Ddi v6i bai toan 5, chi co 11/48 HS sir dung cong thiic tinh gin diing (b^ng vi phan Clia ham s6) ma SGK Giai tich 11 da dua vao. Nhung HS con lai hoac la thii thuc hien phep khai can (nhung khong thanh cong, vi xua nay viec tim can bac hai cua mgt so luon dugc cac em thuc hien vai may tinh bo tiii), hoac mo mam lay nhimg so '"gan voi 3" rQi thii (bang each tinh binh phuang cac s6 do). Nhimg each lam nay duang nhien khong kha thi cho truang hgp cau hoi 5b).
4. Ket lu^n
Nghien ciiu thuc nghiem ciia chiing toi cho thay dii khai niem dao ham da dugc SGK Giai tich 11 trinh bay day dii ca ba nghTa (nghTa vat li - t6c do bien thien, nghTa hinh hgc - he so goc ciia tiep tuyen, nghTa giai tich - cong cu xap xi affine), HS van co kho khan trong viec huy dgng chiing vao giai quyet mgt so van de cua thuc tien. Giai thich nguyen nhan ciia nhihig kho khan nay, chiing toi da tien hanh cac phan tich SGK va thuc hanh DH ciia giao vien tren nhieu phuang dien. Mgt trong nhung nguon goc cua kho khan tim thiy a sir lira chgn ciia SGK. SGK da tinh den quan diem gSn DH toan vol v ^ de mo hinh hoa. Ca hai each tiep can DH bdng mo hinh hoa va DH mo hinh hoa deu hien dien trong SGK. Cu the la khai niem d^o ham da dugc trinh bay theo each tiep can DH bang mo hinh hoa (de hinh thanh nghia v^t li ciia khai niem). Sau do SGK neu ra cac ling dung khac ciia dao ham, qua do nghia hinh hgc va nghTa xip xi dugc de c^p den. Tuy nhien, nhung tinh huong cho phep thiet lap moi lien ket giiia ba nghia nay (vao mgt khai niem, dao ham) khong ton tai trong SGK. Nguon goc thu hai Clia nhihig liing tiing ciia HS nam a cho, vai su lira chgn he thong bai tap ciia SGK va thirc hanh DH ciia giao vien, kha nang van dung toan hgc noi chung, dao ham noi rieng vao giai quyet nhimg van de ngoai toan hgc chua dugc phat trien a HS diing muc nhu no can phai co. Quan diSm gan DH toan voi qua trinh mo hinh hoa van chua dugc tinh din mgt each day dii boi SGK ciing nhu bai thuc te DH cua GV.
Mgt do an day hgc khai niem dao ham gSn voi quan dilm tich hgp va mo hinh hoa da dugc chiing toi thiet ke va thuc nghiem, nhung khong the trinh bay trong khuon kh6 cua bai bao nay.
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Tap chi KHOA HOC DHSP TPHCM So 65 nam 2014
TAI LIEU THAM KHAO
1. Le Thi Hoai Chau (2014), Mo hinh hoa trong dgy hoc Todn, Bao cao tong kit d l tai nghien ciiu cap ca sa, Trucmg Dai hgc Su pham TP Ho Chi Minh.
2. Ngo Minh Due (2013), Khdi niem dao ham trong day hoc Todn vd Vdt li a trudng pho thong, Luan van Thac sT, Truang Dai hgc Su pham TP Ho Chi Minh.
3. Le Van Tien (2005), Phuang phdp dgy hoc mon Todn a trudng pho thong. Nxb Dai hgc Quoc gia TP Ho Chi Minh.
4. Bui Anh Tuan (2007), Bieu dien do thi hdm so vd nghien cim duomg cong qua phuang trinh cua no: trucmg hap duang thdng, Luan van Thac sT. Truong Dai hgc Su pham TP H6 Chi Minh.
5. Chevallard Y. (1989). Le passage de I'arithmetique a I'algebre dans I'enseignement des mathematiques au college. Petit x, N° 19, pp. 45-75, La Pense Sauvage.
6. Coulange L. (1997), Les problemes concrets a "mettre en equations" dans I'enseignement, Petitx, N° 47, pp. 33-58, La Pense Sauvage.
7. Gantois J-Y, Schneider M. (2009), Introduire les derivees par les vitesses. Pour qui?
Pourquoi? Comment?, Petit x, N''79, pp. 5-21, La Pense Sauvage.
(Ng^y Tda soan nh$n duQC bAi: 05-10-2014. ng^yphSn bi$n d^nh gi^. 28-10-2014;
ngay chip nh$n dSng: 22-12-2014)
