Đang tải... (xem toàn văn)
Tgp chl KHQA HQC DHSP TPHCM SS 37 nim 2012 TIM HIEU KHA NANG CUA HQC SINH L O P 12 VE VI$C GIAI Q U V £ T BAI TOAN XET TINH DON DIEU CUA HAM SO W T H 6 N G Q U A M Q T THV''''C NGHIf M SI/PHAM NGUY£N HCr[.]
Tgp chl KHQA HQC DHSP TPHCM SS 37 nim 2012 TIM HIEU KHA N A N G CUA HQC SINH L O P 12 VE VI$C GIAI Q U V £ T BAI TOAN XET TINH DON DIEU CUA HAM SO W T H N G Q U A M Q T THV'C NGHIf M SI/PHAM NGUY£N HCru LQT T6M TAT Bdi todn xet tinh dan diiu ciia mgt hdm si khd phi bien Irong chuang Irinh todnphi Ihdng Di gidi quyit bdi todn ndy cd nhiing cdng cu gidi khdc nhau: dimg dinh nghia, di/a vdo cdc yiu Id die trung cua hdm sS dugc cho, dua vdo dd Ihi cua hdm sd hay tinh iai hdm cip I cua hdm si dd Trong bdi bao ndy, chiing tdi thiit ki mgt tinh huing dgy hfc nhdm tim hieu khd ndng cua hgc sinh ldp 12 Irong viic vdn dung cdc cdng cu gidi bdi toan xit tinh dan diiu ciia hdm si mil Ddng thdi Ihdng qua dd phdt hiin nhiing sai lim hpc sinh mdc phdi gidi bdi lodn ndy Tii khda: tinh don dieu, dao him, him sd mu ABSTRACT A research on twelfth graders' ability in solving the problem of examining the monotonicity of an exponential function through an educational experiment The problem of examining Ihe monotonicity of an exponential fijnction is quite common in high school math curriculum To solve this problem, there are various tools such as: definition, characteristics of the given function, fitenlion graphs, or the first derivative of Ihe function In this article we designed a teaching scenario to examine the gbility oftwelflh groders in applying mathematical tools lo solve the problem of examining the monotonicity of an exponential function At the same time Me also wish to detect mistakes students often make when solving this type of problem Keywords: monoticity, derivative, exponential hmction D|t van de Chung tdi bit diu nghign cim tu viec phan tich sich giio khoa (SGK) Toin 12 (ning cao) Mpt dilu thii vj chung tdi cd dupc lign quan din bii toin xet tinh don difu cua him sd mu Cic ham si xgt tinh ddng biln, nghjch biln diu cd dang y=a'' hoic cd thi dua dupc vg dgng y=a'' Ldi giai mong dpi cua SGK cho thay hpc sinh chi cin dya vao CO sd cua him sd di cho dl dua kit lufn Lifu SGK di gidi hgn viec khao sit ThS, Sd Gigo dgc vS BSo tgo TPHCM 122 nhihig him so mu d dgng y=a'' hoac co thi dua dupc vl dgng y=a'' cd giip hgc sinh khai thic dupc hit cic cdng cy de giii quylt bii toin xet tinh don difu ciia him sd mu hay khdng? Nhihig sai lim hpc sinh mic phii giii quylt cic bai toin dgng Chung tdi thilt kl mot tinh huong dgy hpc nhim tim hilu kha ning cua hpc sinh ttong vifc vgn d\iiig cic cdng cy giai bii toin xet tinh don difu cua him so mu Ding thdi thdng qua dd phit hifn nhimg sai lim hpc sinh mic phii giii cic bii toan Tgp chi KHOA HQC DHSP TPHCM Thyc nghifm dii vdi hpc sinh Thyc nghifm dupc tiln hinh trgn hpc sinh ldp 12 ban khoa hpc ty nhign vdi chuong trinh toan nang cao Thdi dilm thyc hifn la sau hpc sinh da hpc xong bii ham si mu Thdi gian thyc nghifm dinh cho bai toin li 15 phut Hpc sinh sg lim viec ci nhin Hpc sinh se dupc phit giiy lim bii fren dd cd in dg bii toin Giiy nhip cung Ham so i)y = '1^ 2) Dirpc Khdng Nguyen Huu Lai dupc phit cho hpc sinh vi thu lgi sau gid lim Dilu niy cho phgp chung tdi thu thip thgm diu vit thi hifn mil quan hf ci nhin cua hpc sinh Bii toin thyc nghifm: Cd thg bigt dupe tinh ddng bign vi nghieh bign cua cic him sd cho bing sau diy hay khdng? (Dinh diu X vio d mi em lya chpn vi giii thich hoic cho ldi giai tuang irng) - Neu khdng, giii thich vi sao? - Neu c6, trinh biy ldi giii cua em ' V)y = K" c)y = 3'' d ) r = 2>-' 21 Phin tich mpt so yiu to tru&c th^ nghidm 2.1.1 Cdc bien Vifc chpn cic bai toin thyc nghifm dupc dgt frgn CO sd lya chpn gid tti cdc biln didactic sau diy VI: "Him si li him si mu hofc cd thi biin dii dupc vi ham si mii bien x hay khdng?" Hai gia tti cua biln: - Hdm si Id hdm sd mU hogc cd thi biin dii dugc vi him si mO biin x - Hdm si khdng Id hdm sd mil hogc khdng thi biin ddi dugc ve hdm sd mii biin X • V2: "Biiu thi:c mU li tuyin tinh hay khdng tuyin tinh theo x?" Biiu thiic mii Id tuyin tinh theo x Biiu thirc mU khong Id tuyin tinh theox Ta bilt ring, him sd mu y = a" cd miln xic djnh li R Vi vay, him sd dang y = a"''' chi la him sd mu cua biln t = u(x) nlu nhu miln gii ttj cua u(x) la R Trudng hpp die biet: u(x) bilu diln tuyln tinh theo x thi a"'"' li him si mu Ngupc lai, him si di cbo chi li "mdt phin" ciia him sd mu (dd thi cua nd chi li mpt tfp thyc sy cua bim sd mu), hoic khdng li ham sd mu • V3: "Bi thi him si qua (0 ,1) hogc (1, a) hay khong?" Dd thi hdm si qua (0 ,1) hogc (I, ">• Dd thi ham sd khdng qua (0 ,1) vd (1, a) Trong a li co si cda ham sd da cho 2.1.2 Dgc trung cua bdi tadn dugc lua chon 123 si 37 nim 2012 Tgp chl KHOA HQC DHSP TPHCM Bii niy dupc cho vdi nhilu him s6 khic nhau, dd cd nhttng him si quen thupc (dupc cho SGK vi SBT) vi khdng quen thufc Dieu niy cho phgp chung tdi tim hieu ung xtt cua hpc sinh trude nhttng him so khdng quen thupc doi vdi kilu nhifm vy xgt tinh ddng bien, nghjch bien cua him so Dfc bift, gii trj cua biln VI dupc chpn cau c) li hdm sd khdng Id hdm sd mu, sg cho phgp lim rd moi quan bf ci nhan cua hpc sinh dii vdi vifc xgt tinh don difu him sd Chdng tdi dy dodn ring cic ldi giai cua hpc sinh sg su dyng kT thuft xgt co si dg suy tinh don difu cua cic him so di cho 1.3 Cdc chiin luge cd thi Vdi cic him sd dupe lya chpn, bii toin bao gdm nhimg dang him sd khic lien quan din ham si mu Tinh chit cic him niy it nhigu dgu cd lign quan din cic tinh chit cua ham sd mu Vi nhu viy chung sg li ham sd mu hoic li mpt phan cua him sd mu Cic chien lupc sau diy dya ttgn co sd cac tinh chit cua him simu • ST„: "Chiin lupc ca si": doi vdi him so y = a"'"', ip dyng ki thugt so sinh ca sd vdi Nlu a > thi him sd ddng biln Nlu a < thi him sd nghjch biln • STai: "Chiin lupc djnh li": - Nlu (V X, > X2 => f(x,) > fl;x2)) thi f dong biln Nlu (V x, > X2 => f(x,) < fc)) thi f nghjch biln SThh: "Chien lupc him hpp": 124 Xgt tinh ddng biln, nghjch biln ciia cic him thinh phin Xdt tinh ding biln, nghjch biln ciia him hpp di cho • STai,: "Chiin lupc dpo him": Xgt tinh dong biln, nghjch biln ciia ham si dya vio dau cua dgo him 2.1.4 Nhirng quan sdt cd thi ã"=âã n Su lua chgn hdm sd • Him sd niy tucmg ung vdi gia trj thu nhit cua tit ci cic bien VI, V2, V3 Diy la dgng him sd hoin toin quen thupc doi vdi hpc sinh ma SGK di dl cap Chung tdi chpn bii niy vdi myc dich lim CO sd dl so sinh iing xu cua hpc sinh ddi vdi nhihig dgng him sd khic Ttt cung thiy dupc ring bufc cua thi chl len hpc sinh ttong vifc xgt tinh dan difu cua him so mu • Cic chiin lupc cd thg: STtj: "Chiin luge ca si": nhu tren di ndi, diy li dgng him sd him co ban nhit cua him mu, rit sit vdi djnh nghia dupc ttinh biy ttong SGK, vi vfy mii chiin lupc CO sd chic chin se dupc hgc sinh lya chpn STji; "Chiin luge dinh li"; STa,: "Chiin luge dgo hdm " Hai chiin lupc STj, vi STa, cd thi giii quylt bii toin, nhign chung s5 khdng cd ca hfi dl xay vdi him si vi chiin lupc ca sd di thdng Imh D Cdi cd the quan sdt dugc ti hgc sinh: Tgp chi KHOA HQC DHSP TPHCM Ldi gidi tuang Ong vdi chiin luge casdSTcs: =ei Vi — < ngn y bien b)y = 7r' nghjch D Su lua chgn hdm sd • Gii tri cua cie biln dupc chpn: Bii dupc xiy dyng dya trgn biln V2 vdi gii ttj: Bieu thiic mu Id tuyin tinh theo x • Cic chiin lupc cd thi: - ST„: "Chiin luge ca sd": diy li chiin lupc dupc uu tign doi vdi him sd dang mii Vi vgy, cd nhigu co hfi dg xiy chien lupc niy cho du ham sd di cho CO thda man digu kien cua ham sd mu hay khdng - STji: "Chiin luge dinh li": cung cd I the xiy chiin lupc niy vi dang him sd chua that sy dung vdi dang di djnh nghia, bdi vi chung tdi di chpn gii tri thu I nhat cua bign V2 SThi,: "Chiin luge hdm hgp": ca I I hfi xiy chiin lupc niy cung bing nhu i "chien luge dinh li" I Him y = Jt^" li hpp cua hai him phin u(x) = 3x vi y = it" Him u(x) I la dg dang bilt dupc tinh dan difu cua nd Do dd tinh dom difu cua him Jt" cung i dupe dl ding xic djnh i! ST*: "Chiin luge daa hdm " ' D Cdi cd thi quan sdt dugc tir hgc sinh: t - Ldi gidi tucmg Ong vai chiin luge easiSTr,: Nguyen Huu Lai Cd hai trudng hpp tuang img vdi chien lupc niy: THI: Tacd y = 7C^' = {7!^\ n^ > ngn him so dong biln TH2: dong nhit tinh dan difu cda him y = ir^" vdi him _y = ;r' Do dd, tinh don difu ciia him dupc cho xic djnh nhu sau: Vi 71 > ngn nen him sd da cho li ddng bign Ldi gidi tuang iing vdi chiin luge dinh li STji: V X|, X2: X] > X2 ta cd: 3xi > 3x2 => tdi gidi tuang ting vdi chien luge hdm hgp ST/,/,: u(x) = 3x li ddng biln tgn R It" li ddng biln tren R ngn it""" li ddng biln fren R Ldi gidi tuang ting vdi chien luge dgo hdm ST^/,: y' = 3T^'\tm > vdi mpi x thufc R ngn him si y = it^" ddng biln ttgn R e ) y = 3'' D • Su lua chgn hdm sd Gii trj cua cie biln dupc chpn: Him s6 y = thda cic gii trj cua cic biln sau: Gii tri tiiii h^i cua biln VI: Hdm sd khdng thi biin ddi dugc vi hdm sd mS biin x(y = a') Gid ttj thd hai eua biln V2: Biiu thirc muld khdng tuyin tinh theo x 125 Tgp chi KHOA HQC DHSP TPHCM Gia trj thd nhat cua bien V3: Dd thi hdm sd qua (0,1) hodc (I,a) Him so niy thda hai digm dfc bift cua him so mu li (0,1) vi (l,fl) (trudng hpp niy a bing 3) Tuy nhien, him niy chi li "mgt phdn" cua him so mu bdi ring tfp gii trj cua nd li [l.+co) Him y = 3' khdng cd tinh chit ludn ting hoic ludn gidm nhu tinh dan difu cua him so mu Do dd, khdng thi khio sit tinh chit niy bing ki thuit so sinh CO so cua nd vdi Cic chiin lupc cd the: Ngu hpc sinh cho ring cd the biet dupc tinh ding biln, nghjch biln cua him sd y = 3^ thi cic chign lupc sau diy cd thg xiy ra: STcs: "Chiin luge ca sd ": mac du y = 3' khdng phii li him sd mQ nhung him niy cd hai digm die bift (0,1) va (l,a) vi cd dang a""" ngn cd nhilu co hpi xuit hien chign lupc niy STjih: "Chiin luge hdm hgp ": chign lupc cd thg xiy trudng hpp hpc sinh nhin dang dupc him Tuy nhign theo chung tdi, thg chl da khdng tgo co hdi cho chiin lupc xiy STah: "Chiin luge dgo hdm": mfc du cd SGK cd gidi thifu phuong phip xet tinh bign thign cua mpt him sd bing dgo him, nhign ddi vdi bai niy phuang phip dgo him khdng li ttpng tim, do chung tdi nghi ring cd rit it co hfi xiy chiin lupc niy D Cdi cd thi quan sdt dugc tii hgc sinh: 126 si 37 nam 2012 Ldi gidi tuang img vai chiin luge ca si ST„: VI > ngn y = y , ddng bign fren R Lai gidi tuang img vol chiin luge hdm hgp ST/,/,: Him so x^ dong biln trgn [0, +») vi nghjch bien trgn (-oo, 0] Do him sd y = y ddng bign trgn [0, +M) vi nghjch biln tren (-«, 0] Lai gidi tuang ling vol chiin luge dga hdm STj/,: y' = 2x3'' Khi X > thi y' > ngn ham s6 ddng biln Khi X < thi y' < ngn him so nghjch bign d)y = ' ' D Su lua chgn hdm sd • Gii trj cua cic biln dupc chpn: - Gii trj thu nhit cua biln VI: Ham sd Id hdm sd mi hogc cd thi biin ddi ve hdm sd mi biin x (y = cf) Gii trj thu nhit cua biln V2: Bien thirc mi la tuyin tinh theo x Him dupc cho vdi myc dich tim higu moi quan hf ca nhin cua hoc sinh vl ham sd mu Mpt cich rd rang hon chung tdi mudn kiim chung ring cd phai hpc sinh di thgt sy gin lign hay dong nhit him si mu vdi mft bilu dien bao gdm ca s i a vi mft bilu thdc mu Him s6 y = ' " li ham sd mu,my nhign nd li him mu vdi ca so i Vi nhir Tgp chi KHOA HOC DHSP TPHCM vfy tinh don difu cua nd dupc xet theo bieu thuc him >' = • Cic chiin lupc cd thi: STcs: "Chien luge ca si"', 57^: "Chiin luge dinh li"; SThi,: "Chiin luge hdm hgp "; STji,: "Chiin luge dgo hdm " D Cdi cd thi qugn sdt dugc tir hgc sinh: Ldi^idi tuang ting vdi chiin luge easdSTcs: Cd hai trudng hop xiy vdi chiin lupc niy: THI: y = 2'"" dupc biln ddi thinh dgng y = — , Idi giii nhu sau: Vi — l=> hdm sd ddng biin " Ngoii cd 21/74 (28,3%) hpc sinh cho ring khdng thi bilt dupc tinh don difu cua hmn niy vi sd mu li x^ Cdc giii thich tucmg dng li: "i^i chua biit gid tri 127 Tgp chi KHOA HQC DHSP TPHCM cua A nin khdng xdc dinh dugc ddng biin, nghieh biin": "khdng thi biit tinh ddng biin, nghieh biin vi khdng cd dgng y=d'": "VI khdng biit dugc gid tri cua x thugc khodng ndg nin khdng xit dugc tinh ddng biin, nghieh biin " Mpt si Idi giii thich khic cd dung dgo him dl khio sit nhung di din kit lufn li: "Viy' cdn chiro tham sd x nin dau cua y' chua xdc dinh Vl vgy khdng thi xdc dinh hdm sd ddng biin hay nghieh bien " RO ring Ii hpc sinh di khdng kllm tra him di cho cd la him so mu hay khdng, nhign hp di ip dyng tinh chat cua him so mu (dgo him ludn khdng chua tham so) dl giii Mpt trudng hpp tuang ty cd thi thiy d ciu b) nhu sau: Theo SGK thi y=7i''' li him s i mu CO sd It', nhign cd din 38/74 (51,3%) hpc sinh giai him niy vdi ca sd it Ddi vdi bam sd d ciu d) y=2'"'' cung cd den 14/74 (18,9%) hpc sinh giii him niy vdi CO sd li Dilu cho thiy hpc sinh da khdng kigm tra sy thda ding cua him so Ket luan Thyc nghifm dua din mft sd kgt qua sau: Ttt cic kgt qui cd dupc, chung tdi nhgn thiy da sd hpc sinh cua ldp dupc thyc nghifm tip trung cich giai cua minh si 37 nim 2012 vio vifc xgt cdc ca sd: a > thi him so dong biln, < o < thi ham si nghjch bien Cic em khdng cd nhifm vy di kiim tra him si dupc cho cd la him sd mu hay khdng? Do dd, cic em di khdng thi giii quylt dupc cau c vi d Ngoai ta cdn thiy, khdng nhilu hpc sinh su dyng dao him dl xgt tinh dan difu cua him si Nlu cd, cic em cung khdng thinh cdng dl dua din kit qui sau cung Dilu niy cd thi dupc giii thich li SGK chi dua nhihig dgng toin chi cin xgt dgn co sd la cd thi kit luin dupc tinh dan difu ciia him so mu Thyc nghifm cung md hudng xiy dyng bii tip cho hpc sinh ma gido vign cin phii cin nhic Khdng phii liic nio cung dg xuit cho hpc sinh cic bai tap quen thupc Do dd, giao vien cd thi tgo cic tinh hudng hpc tfp nhim giup hpc sinh cd the van dyng cic kign thuc co lign quan Ching ban, ddi vdi bii him so mu, giio vign cin xiy dyng hf thdng bai tgp cho bufc cdc em phii van dyng nhihig phuang phip giii khdc de giii quylt hit hf thdng bii tgp dd Tu gdp phin khic phyc cac ldi mic phii cua hpc sinh nhu da chi thyc nghifm tten TAI L i f u THAM KHAO Cue Nhi gido vi Cin bp quin 11 Giio dye (2008), Hudng ddn thuc hiin chuang trinh vd sdch gido khoa ldp 12 THPT, Nxb Giio dye Ngd Vilt Diln (2001), Phuang phdp chgn lgc gidi Todn hdm si mi vd logarit, Nxb Dai bpc Qudc gia Hi Npi Tran Vin Hao, Phan Truang Din (1991), Sdch gido vien Dgi si vd gidi tich II, Nxb Giio dye Lg Thj Thign Huong, Nguyen Anh Tuin, Lg Anh Vii (2000), Todn cao cdp, Nxb Giio dye 128 rgp chi KHOA HQC DHSP TPHCM Nguyin Hiiu Lpi NguySn HOu Lpi (2008), Khdi niem hdm sd mii d trudng Trung hgc phd thdng, Lufn van Thac si chuyen nginh Phuang phip giang day Toin, Trudng Dgi hoc Su pham TPHCM Doin Quynh, Nguyen Huy Doan (2007), Dgi si vd Gidi tich 11 ndng eao, Nxb Giio due Doan Quynh, Nguyen Huy Doan (2008), Gidi tieh 12 ndng cao Nxb Giao due Doan Quynh, Nguyen Huy Doan (2008), Sich Gido vien Giai tich 12 ndng eao Nxb Giao dye Le van Tiln (2005), Phuangphdp dgy hgc mdn Todn d trudng phd thdng, Nxb Dai hpc Quic gia TPHCM (Ngiy Tda soan nh$n dugc bii 20-02-2012: ngiy chip nh$n ding: 19-6-2012) SV" CAN THIET CUA MO HINH HOA (Tiep theo trang 121) TAI LIEU THAM KHAO Nguygn Thj Tan An, Tran Ding (2009), "Su dung md hinh hoa toan hpc ttong vifc day hpc loan", Tgp chi Gido due, (219) Gabriele Kaiser, Werner Blum, Rita Borromeo Ferri, Gloria Stillman (2011), Trends in Teaching and Learning ofMathematical Modelling Springer Gabriele Kaiser, Bharath Sriraman (2006), A Global Survey of International Perspectives on Modelling in Mathematics Eduacation, ZDM Vol 38(3) Hans-Stefan Siller, Modelling in Classroom 'Classical Models' (in Mathematics Education) and recent developments www.algebra.tuwien.ac.at/kronfellner/ ESU-6/ /l-13-SilIer.pdf OECD (2003), The Pisa 2003 - Assessment Framework - Mathematics, Reading, Science and Problem Solving Knowledge and Skills, OECD, Paris, France Rita Borromeo Ferri (2006) Theoretical and Empirical Differentiations of Phases in the Modelling Process ZDM Vol.38(2) Werner Blum, Peter L Galbraith, Hans-Wolfgang Henn, Mogens Niss (2007), Modelling and Applications in Mathematics Education Springer (Hgiy Tda soan nhin dui?c bii: 01-02-2012: ngiy chip nhin ding: 19-6-2012) 129 ... TPHCM Thyc nghifm dii vdi hpc sinh Thyc nghifm dupc tiln hinh trgn hpc sinh ldp 12 ban khoa hpc ty nhign vdi chuong trinh toan nang cao Thdi dilm thyc hifn la sau hpc sinh da hpc xong bii ham si... tim, do chung tdi nghi ring cd rit it co hfi xiy chiin lupc niy D Cdi cd thi quan sdt dugc tii hgc sinh: 126 si 37 nam 2 012 Ldi gidi tuang img vai chiin luge ca si ST„: VI > ngn y = y , ddng bign... mu), hoic khdng li ham sd mu • V3: "Bi thi him si qua (0 ,1) hogc (1, a) hay khong?" Dd thi hdm si qua (0 ,1) hogc (I, ">• Dd thi ham sd khdng qua (0 ,1) vd (1, a) Trong a li co si cda ham sd