JOURNAL OF SCIENCE OF HNUE 2012, Vol 57, No 10, pp 19 25 REN LUYEN KHAI QUAT HOA CUNG VCll DAC BIET HOA VA HE THONG HOA CHO HOC SINH TRONG KHI DAY HOC M O N TOAN 6 TRUCtNG PHO THONG T h a i Th j H o n[.]
JOURNAL OF SCIENCE OF HNUE 2012, Vol 57, No 10, pp 19-25 REN LUYEN KHAI QUAT HOA CUNG VCll DAC BIET HOA VA HE THONG HOA CHO HOC SINH TRONG KHI DAY HOC M O N TOAN TRUCtNG PHO THONG Thai Thj H o n g Lam*, Nguyen Thj M y HSng Truifng Dgi Iwc Vinh "Email: hlamdhv@gmail.com Tom tat Tap luyen cho hoc sinh khai thac cac hoal ddng khai quat hoa, dac biet hoa, phat bien moi quan he chung - rieng qua Irinh day hpc toiin trudng Ihong la viec can thiet va co the thuc hien dupc Bai bao neu mot so vi du va khai quat hda diTdc sd day hoc tbeo hUdng khai thac mdi quan be gifla cac hoal dpng thong qua gdp phan phat trien cho bpc sinh kha nang he tbdng bda Tifkhda: Khai quat bda dac biet hoa, boat dpng tri tue Mcf dau Viec ren luyen cac boat ddng trf tue cd ban cbo bpc sinh la mpt nhflng muc dich can dat dfldc day hpc Toan d tmdng tbdng Khai quat hda la boat ddng tri tue CO vai tro quan trpng boat ddng nban tbflc ndi chung va hpc tap mdn Toan ndi rieng Vi vay, giao vien cin khai thac triet de cac tinb budng cd the de tap luyen boat dpng cbo bpc sinb qua irinb day hpc Tuy nhien, viec tap luyen boat dpng khong the tiSn banh mdt each ddc lap, rieng biet, ma phai ket hdp ciing vdi mpt so boat ddng khac lien quan thi mdi dat hieu qua cao Tap luyen cho hpc sinb kbai thac cac boat dpng kbai quat bda, dac biSt bda, phat hien mdi quan he chung - rieng qua trinh day bpc Toan d tnidng pbo tbdng la viec can thiet va cd the thifc hien dUdc Trong bai bao chung tdi quan tam viec ren luyen boat dpng phat bien mdi quan be cbung - rieng, khai quat hda va dac biet bda, qua gdp pbSn pbat trien cho bpc sinb kha nang he thdng bda Noi dung nghien cihi 2.1 Mot so khai niem He thdng hda la sfl sap xep tfl cac doi tfldng va hien tfldng theo nhdm va nhdm nbd, biy theo cac mat gidng va khac cua chung [4;55] Mpt nbflng phfldng dien cua be thdng bda la lam ro nbiJng moi quan be gifla nbflng kien tbflc khac lien quan den kbai quat hda Tren cd sd cac qua trinh khai quat boa va be thong hda bpc sinh linh bpi he thdng tri thflc - dd la he thong bieu tflpng va khai niem Tbieu khai quat boa va he thong bda thi hpc sinb khdng the linh hdi tot tri thflc Thai Thi Hong Lam, Nguyen Thj My Ihiiig K.D.Usinxki cung da khang djnh rdng: In thflc ma khdng cd he thong tUa nhu mdl cai kho dd mpi thfl dfldc quang nem vao lpn xpn Vii ban than dng chil kho cung khong lim thay dflUc gl [4;.55| Khdi qudi Itiiii (KQH) la chuyen ICI mpt lap hdp doi tupng sang mpt tap hpp ldn hdn chfla lap hop ban dau bang each neu bat mpt so ddc diem chung cua cac phan tfl tap xuat phdt (tfle la di tfl cdi rieng den cai chung) Ddc hiel hda (DBH) la chuyen tfl vice nghien cflu mpt tap hpp doi tupng da cho sang viec nghien cflu mpt tap hpp nhi) hOn chfla tap hdp da cho (iflc la di ttt cdi chung den cai rieng) 11 S11 Thdng thflflng, chiing ta ticn hiinh dac biet hda chuyen lii ca mdl ldp doi tfldng den mpl ddi tfldng cua Iflp dd Theo chung tdi, hoat dpng phdt hien nidi quan he chung • rieng (I'HCR) la nghien cflu hai tap hop doi tupng, quan he cho trUdc cd mdi quan he vflj nhu the nao ('lap nao nam tap nao va dua theo ddu hieu gi?) Moj quan he gifla ba hoal dpng tren dupc the hien fl bang sau f2; 12]: Bang I Quan he giUa ba hoal dpng: KQH, Hoat dong Cho trirdc Phiit hien moi quan he chung -A - neng -B -A Khai quat hda - Moi quan he cbung - rieng: B long quat hrtn A -B Dac biet boa - Moi quan be chung - rieng: B long quat bdn A DBH, PHCR Tim - Mdl quan be chung - rieng gifla A va B -B -A Trong dd A, B khdng phai la cac tap hpp doi tupng, quan he bat ky, ma mdt so phan tii ciia A vd B Cling thoa man mpt difiu kien nao dd Khai thdc moi quan he gifla ba hoat dpng tren viec tap luyen cho hpc sinh khdi quat hda, khdng chi yeu cau hp di ttt cdi rieng den cai chung (khai quat hda) ma cdn ddi hdi hp di tfl cdi chung den cdi rieng (dac biet hda) va lirni rd mdi quan he chung - rieng gifla cai dat dflpc va cdi xuat phdt Tap luyen cac hoat ddng theo hudng la phu hpp vfli quy luat chung nhat cua hoat ddng nhan thflc ma V I Lenin da tong ket; "TU true quan sinh dpng den tu trilu tupng va lil tu trilu tutmg den thuc tien, dd la dudng hien chiing cua sil nhdn Ihitc chdn ly, cUa sil nhan thiic thuc tgi, khdch quan " Chiing tdi de xudt mdt phuong thflc de boi duflng cho hpc sinh cac hoat ddng tren bao gdm cdc budc sau day: Budc 1: To chflc quan sdt cac sU vat, hien tupng rieng le (xet cac truflng hpp rieng dua tren cac vi du cu thi) Buflc 2: Xdc dinh cac dau hieu, cdc dac diem, cdc thupc tinh, cac mfli lien he cua cdc doi tupng rieng Ie 20 Ren luyen khdi qudt hda ciing vdi ddc biit hda vd he thdng hda cbo hgc sinh Bfldc 3: So sanb cac d^u bieu, cac dac diem, cac thupc tinh, cac mdi lien he dd de tim cac dau bieu chung va ban chat Bfldc 4: Tuy theo muc dicb, tach cac dau bieu cac dac diem, cac thudc linb cbung va ban chat khdi cac ddi tfldng rieng le Bfldc 5: Chuyen tfl viec nghien cflu cac doi tfldng rieng le sang nghien cflu mdt tap ldn hdn chfla cac doi tfldng rieng le dd Ket qua cua qua trinb tren ta tbu dflpc mdt cai long quat (mpt khai niem, mpt dinh li, mpt qui tac bay mpt bai loan long quat, ) Bfldc 6: Dac biet hda de cd dflpc cai xuat phat, lam rd m6i quan he cbung - rieng gifla cai dat dfldc va cai xuat phat Tiep tuc dac biet bda de cd cae trfldng bdp rieng kbac nfla 2.2 M o t s6 VI d u Vl du 1: Day bpc quy tac tim tap gia tri cua bam so Mac du khai niem tap gia tri kbdng dilpc day Ifldng minb cac sach giao kboa hien hanb, nbflng vi nd cd nhieu flng dung nen chung ta can thiel va cd the day cbo bpc sinb tfl ldp 10 Chung ta cd the djnh ngbia kbai niem tap gia tri cua bam so y = / ( x ) nhu sau: "Cho bam soy = f{x) xac dinb tren D, tap hdp tSt ea cae sd tbflc y cbo ton tai X £ D de / ( r ) ' = y dfldc gpi la tap gia tri cua bam soy = f{x)" Qua trinh day hpc quy tac tim tap gia tri cua ham s6 va flng dung de tim gia tri ldn nhat, gia tri nho nhat can quan tam khai thac mdi quan he cua ba boat ddng tren nham nang cao ky nang giai bai tap dang cbo bpc sinb Xet mdt sd bai toan cu the sau: Bdi todn 1.1: Tim tap gia tri cua cac ham sd sau: 2i)y= JT{\ h)y = x + -\c)y = x X X Chung ta se bfldng din bpc sinb lam cac bai tap b-en, chang ban eau a) vdi cac cau bdi gdi y nhfl sau: - Neu )/o lu mgt gid tri bdt ky cua hdm sdy = \/x{l — x) thi mdi quan he gida XQ thupc tap xdc dinh ciia hdm sd vd yo si nhu thi ndo ? Dfla vao dinh nghia tap gia tri cac em se tra ldi dfldc: se ton tai XQ € [0; 1] cho yO= \/.To(l - -X)\ XQ) - Viic tdn tgi XQ G [0; 1] cho y/xQ{T-^xo) Nghia la phfldng trinb ^ x ( l -x) = yo CM the han nhu the ndo? = yo cd nghiem XQ e [0; 1] • Vuy yo phdi thda mdn dieu kien gi de'nd Id mgt gid tri bdt ky cua hdm sd y — y/x{\-x)? yo phai lam cho phfldng trinh \/x{\ -x) = yo cd nghiem - Hay tim dieu kiin cdn vd du cua yo de phuang trinh >/x{l — x) = yo cd nghiem - Hdy suy tdp gid tri cila hdm sdy = \/x{l - x) - Tudng tu hdy gidi tiep cdu b) vd cdu c) u Thai Thj Hdng Lam, NguySn Thj My Hang - yii- I'ipc lim lap gid tri ctia ciic hdm id tren hdy ne'u dau hieu chung cua cdc ldi gidi ircn Id gi? Thuc chal ciia bfldc niiy la gido vicn dd yeu cau hpc sinh thuc hien hoat ddng "Khdi qudt hda mdt qud liinh di6ii Iicn mdl s6 ddi IUdng rieng le mpt qua trinh dicn tren mpt ldp ddi tupng" |2;.'i61 Ta mong dpi hoc sinh tra Ifli: Tim dicu kicn ciia yi, de phuong trinh i/n = fix) co nghiem Ircn mien \ac dinh ciiu ham so Khai qudt hda qua irinh Ircn, sin pham cua chiing la thu dupc chinh la mpt quy tac lim lap gid tri ctia ham so (phuong phdp sfl dung dinh nghia lap gid Iri) bao gom cdc budc sau: nudc I: Tiiii mien xac djnh D cua hiim so Budc 2: Coi T la tap gid Iri cQa ham so, lay phan tfl y,, bat ki thupc T Khi dd, ton tai r„ t D cho /'(.ro) ^ ija, hay phUOng trinh / ( i ) - :(/,, cd nghiem thudc D litci'ic J.- Tim dieu kicn ciin va dii ciia y,, dc phuong trinh / ( / ) = yo cd nghiem r t I) Budc 4: Kcl ludn ve Sau hpc sinh dd khdi quat hda dupc quy tac tim tap gid in d tren, vdi muc dich kiein tra viec khdi qudt dd, cd the yeu cau hpc sinh dac biet hda cdi tim dupc vdi cdc ham sd cu the de nhan dupc cai xuat phdt, thdng qua dd nhan manh mdi quan he chung - rieng gitta cdi tim dupc va cai xudt phdt: (IJ san pham tim dflpc, neu chpn /(.r) = ^J:{1 - x) thi dupc mpt truflng hpp rieng, dd la mdt nhu^g cai xuat phdt Bay gifl hpc sinh da dupc trang bi mdt phUPng phdp tdng quat de tim tap gid tri cua ham sd Ci thdi diem nay, hpc smh chi mdi tim dupe tap gid tri cua cdc ham sd / ( x ) ma viec tim yo de phUdng trinh /(.r) — i/o cd nghiem cd the dua ve vice tim dieu kien co nghiem cua phUPng trinh bac hai Khi hpc chUOng trinh Dai sd - Giai U'ch 11, sau hpc sinh hpc bai "PhUPng trinh bac nhat doi vdi ••^itt.r va cosd", giao vien can thifit va cd the yfiu cau hpc sinh tim tap gid tri, gid tri ldn nhat, gid tri nhd nhat cua cac ham so f{x) ma viec tim difiu kien cua t/o de" phUPng trinh f{x) = yo cd nghiem cd the dua vfi viec tim dieu kien cd nghifim cua phUPng trinh a sin r + iicos r - c Bdi todn 1.2: Tim tap gia trj, gia tri Ifln nhat, gid tri nhd nhat cua cac ham sd sau: , , , , COS.T + sin + : ) a) y = J cos ,'r + sm x + 2:h)y = : cos X — sin ;r + , , „ , „ „ , „ siii^2.r - c o s ^ i + 3sin2xcos23: c)!/ = s u r ,r " cos'' r + sm 2j- + 1; d) y = = - " 2cos2 i + De' giai dupc cdc bai tap tren (cdi rieng), hpc sinh phai lien he vdi phUPng phap tdng quat (cdi chung) de tim tap gid tri cua ham so, tfle la ho phai khai quat hda Sau khdi qudt hda, hp lai phdi ddc bifit hda flng vdi la mflt nhflng ham s6 cu the' neu trfin, Neu hpc sinh bifit dfldc tat cd cac gia tri cua ham sd (tap gia tri) thi hp cd the fim dUdc gid tri Ifln nhat va gia tri nhfl nhat cua ham so dd (neu cfl) Do vay, vfli bai todn tim gia tri ldn nhat, gia trj nhfl nhat cua ham sd, chung ta cd the giai dfldc, bang each tim tap gid tri cua ham sd dua vao qui tac tim tap gia tri da chi fl tren Ren luyen khdi qudi hda cung vdi ddc biil hda vd hi thdng hda dm hgc sinh Chung ta cung cd the md rdng phfldng phap tim tap gia tri cua ham so hai bi6n thdng qua cac vi du sau: Bdi todn L3: a) Cho hai so thflc x va y thda man :i + 2y - Hay tim gia trj ldn nhat, gia tri nbd nhat ciia bieu thflc A ~ x? H- 2i)^ + xy -f 5.T b) Cbo hai so tbflc x va y thoa man (,r - 2f + (y I-1)^ =- Hay tim gia iri ldn nhSi, gia tri nhd nhat cua bie'u thflc A = x? -{• y^ c) Cho hai sd ihflc dfldng x va y thda man [x + yf = Hay tim gia tri ldn nhat, gia tri nhd nhat cua bieu tbflc A = x^ -^ y^ Be giai dfldc cac bai tap tren, cd the gdi y nhfl sau, chang ban vdi cau a): - Gid tri ldn nhdt gid tri nhd nhdt cung Id mdl gid tri ndo dd cda A Do dd vdn di dgt Id ta di tim tdt cd cdc gid tri ed tbied cua A - Niu ggi m Id mdt gid tri hdt ky eua A thi se nhu thi ndo ? Tdn tai bai sd r va y cbo ,r + 2y = va (-^ -I- 2y''^ + ry + J = m - Viec tdn tgi hai sd x vd y cho ,r + 2y = vd ,T^ \- 2y'^ + xy + 5,-c = m co the dugc phdl bieu nhu thi ndo ? ,T + 2y = ( - Tif do, tim dieu kien cdn vd dCi ctia m de he phuong trinb trin co nghiem - Cd diiu kiin cUa m ta suy lap gid tri cua A, cd tap gid tri cua A ta tim dugc gid tri ldn nhdt, gid tri nhd nhdt cda A Nbflng tbao tac tren boan loan tfldng tu nhif quy tac tim tap gia trj cua ham mpl bien Vi du 2: Tinb cac tich pban sau: ^'^' = „{55TFn''^"'^'"^^ c)^3 = / J ^ ^ ; d ) / = /-(^,^^)p.^j) Chung ta se hfldng dSn bpc sinb giai cac bai tap tren, chang ban cau a) vdi cac gdi y nbfl sau: - Hdy nhdn xet can cua tich phdn vd hdm so dudi ddu tich phdn? Can doi xflng Ham sd lay tfch pban la ticb cua x^ (bam chin) vdi - —- - Tit tinh chdt can ddi xiing vd tinh chan cda hdm sd cd the"chgn phep doi bien nhU thi ndo? C d t h e d a t z ' = -t - Vdi phep doi biin nhU vdy thi tich phdn duoc viet dudi dgng nhu the ndo ? 23 Thiii Thi Hong Lam, Nguyen Thj My Hang -1 - (.-I)' ;, 2()12'/.'-' „ :', 2()12'-l '' ' i 2I>I2-' I - Tich phdn khong phti thuoc mo hien sii lay lich phdn dd /, cd the dupc viet itltil the ndo? ,, i ô!i2::^_,,,.(2) ' ã', 2(112' I -Tit (I) vd (2) cd ihestiy diiu gi? Hdy cnng (I) vd (2) ve theo ve! 2/, f.i'ilt ^h = 1-f.,'',l.r 2Tich phan cuoi cung dUdc tinh mdl edch dc dang - Tuong IU, lidv tinh cdc lich phdn d cdu h), c), d) - Cdc Itch phdn iren co diein chung gi? Can lay lich phan ddi xflng, mau sd cua ham lay tich phan deu chfla bieu thflc m'+l, lfl sd Clia ham lay u'ch phan la hiim chdn /(.x) - Hdy khdi qudt hda lich phdn long qudt vd phuong phdp gidi! Sdn pham cKiing ta cd dupc la bai loan td'ng qudt sau va phflpng phdp giai: Bdi todn long qudt: Tinh tich phdn I = j —; TIIX (I), dd f(x.) la ham chdn, lien tuc tren [~a,a],iv la sd thuc dUPng khdc Phuong phdp gidi: Dat x = -t, tich phdn da cho dflpc viet dfldi dang: / = ?!!!^Md,= j!!!!ZMrf,.(2) _„ m' + i, m' + Cdng ve vdi ve cua (1) va (2) ta suy ra: / = - / /(a )dx Sau dd cd bai todn tdng qudt trfin, tiep tuc yeu cau hpc sinh dac biet hda de' cd cdi xuat phdt (chang han vfli a = 1, f{x) = \x\ in = S thi ta cd cau c), lam rd moi quan he chung - rifing gitta cdi dat dupc va cdi xuat phat Tiep tuc dac biet hoa de' cd dupc cdc truflng hpp rieng khdc nfla Chang han, ta cd dflpc cac bai toan tinh tich phan sau: sin" x + cos" X' , ?,!••+ 3.r'" + , ' \/l - r^ , d.f, ax; dx \ 2' + l ,^2 2012'+ \ 10'+ Hoat ddng khdi qudt hda, dac biet hoa, phdt hien moi quan he chung - rieng difin sudt qua trinh day hpc Giao vifin can thik va cd the' tao co hpi cho hpc sinh dupc tap luyen thuflng xuyfin Chang ban, sau hoc sinh hoc ham so dong bien, ham so nghich bien d ldp 10, cd thfi' cho hpc sinh giai cac phuong trinh nhU: \/x + l = - i ' + 29; v'7 - Zx = X? + 3.T - 13, khdi quat lfin phuong phap giai phuang trinh dang J{x) = g(x),x s (a, 6), dd f[x) va g(x) dpn dieu ngflpc chieu trfin (a, b) Tai thfli Ren luyen khiii qudt hda cung vdi due biil htki vd he ihdng htki cho Iwc \-inh diem cd it cd hpi de hpc sinh dfldc tap luyen giai nhflng phfldng irinh dang tren Tuy nhicn, len ldp 12, hpc sinh dfldc hpc moi quan he gifla dau ciia dao ham va tfnh dong bien, nghjch bien cfla bam sd, cac bam so mu va logaril Ibi se cd nhicu cd hdi dc luyen lap hdn, Khai quat bda qua trinh iren ta thu dfldc sd dd day hpc gdp phan ren luyen kha nang kbai quat hda cung vdi he Ihong hda va dac bicl hda: Ciil rieng KQii ,-• DBll Ciii chung omi ~7 ^^""l Cai rieng khiic Ket luan Nbfl vay, vice tap luyen va khai thac cac boat ddng khai quat hda, dac bicl bda, phal bien mdi quan be chung - rieng la can Ihiet va cd the Ihflc hicn dfldc qua trinh day hpc Toan d trfldng pbo thdng Tbflc hien cac boat dpng tbeo bfldng la pbCi bdp vdi mot Irong nhflng nguyen lac cua phfldng pbap nban tbflc bicn chflng: Pbfldng pbap di lfl cai chung den cai rieng chi cd the pbat huy tac dung neu nd lien he hflu cd va k6t hdp vdi phfldng phap tien tfl cai rieng den cai chung Hai phfldng phap kbdng pbai la hai phep bien chflng - bien chflng ti^n va bien chflng thoai, ma chfnh la bai mat cua cung mdt phep bien chflng, bai nban td cua phfldng pbap nban tbflc bien chflng thong nhat [3;73] Hdn nfla, tbdng qua vice lap luyen va khai thac cac boat ddng tren ciing gdp pbSn phal trien cho HS kha nang he tbdng hda va dam bao ti'nh tbdng nbSt cua suy luan quy nap va suy luan dien dich TAI LIEU THAM KHAO [ ] Nguyin Ba Kim, 2002 PhUOng phdp dgy hgc mdn Todn Nxb Dai hpc Sfl pham Ha Npi [2] Nguyin Ba Kim, Tdn Than, Vfldng Dfldng Minh, 1990 Khuyin khich mgt sd hogt dpng tri lue cua hgc sinh qua mdn todn Nxb Giao due Ha Ndi [3] A P Sep - Tu - Lin Phuong phdp nhdn thdc bien chdng Nxb Sach Giao kboa Mac Lenin [4J M Alecxeep, V Onbisuc, M Crugliac, V Zabdlin, X Vecxcle, 1976 Phdt trien tu hgc sinh Nxb Giao due Ha Ndi ABSTRACT Teaching generalization, specialization and systemization when teaching mathematics Teaching mathematics students that exploring generalized and specialized activities in order to discover relationships between that which is general and that which is specific is both necessary and possible This paper provides examples and a generalized diagram which illustrates the relationship between tbe general and specific in order to improve student's ability to make use of systemization 25 ... can thik va cd the'' tao co hpi cho hpc sinh dupc tap luyen thuflng xuyfin Chang ban, sau hoc sinh hoc ham so dong bien, ham so nghich bien d ldp 10, cd thfi'' cho hpc sinh giai cac phuong trinh nhU:... luyen khiii qudt hda cung vdi due biil htki vd he ihdng htki cho Iwc \-inh diem cd it cd hpi de hpc sinh dfldc tap luyen giai nhflng phfldng irinh dang tren Tuy nhicn, len ldp 12, hpc sinh dfldc... phUPng trinh bac hai Khi hpc chUOng trinh Dai sd - Giai U''ch 11, sau hpc sinh hpc bai "PhUPng trinh bac nhat doi vdi ••^itt.r va cosd", giao vien can thifit va cd the yfiu cau hpc sinh tim tap gid