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JOURNAL OF SCIENCE OF HNUE Interdisciplinary Sci,, 2014, Vol 59, No 6, pp 10 19 BIEN PHAP KHAC PHUC SAI LAM T H U 6 N G GAP CUA SINH VIEN KHI Sir DUNG CAC PHEP CHIJNG MINH TOAN HOC Dao Thj Hoa Khoa To[.]
JOURNAL OF SCIENCE OF HNUE Interdisciplinary Sci,, 2014, Vol 59, No 6, pp 10-19 BIEN PHAP KHAC PHUC SAI LAM T H U N G GAP CUA SINH VIEN KHI Sir DUNG CAC PHEP CHIJNG MINH TOAN HOC Dao Thj Hoa Khoa Todn, Trudng Dai hgc Su pham Hd Noi Tom tat Cac phep chiing minh piian tich, chiing minh tdng hop, chiing minh phan chiing va chiing minh loai dan la nhiing phep chiing minh thudng ducJc sii dung giai cic bai loan trirdng piid thong Tuy nhien, sii' dung nhiing phep chiJng minh niiy qua trinh giai toan, sinh vien thudng mac phai mpi so sai lam Bai bao phan lich mot sd sai lam thifdng gap ciia sinh vien sii dung cac phep chiing minh loan hoc, nguyen nhan dan den sai lim va bien phap khac phuc nhiing sai lam dd Tit khda: Phep chiing minii, sai lam, bai loan, giai toan Md dau Trong day hpc loan d nha trudng phd thdng, viec hudng din hpc sinh lim Idi giai ciia bai toan va trinh bay Idi giai bai toan la mdt cdng viec rit thudng xuyen va hit siic cin Ihiel cua mdi giao vien loan Vdi dang toan chiing minh, thay va trd ihudng xuyen sii dung cac phep chiing minh loan hpc nhU phep chiing minh phan tfch, phep chiing minh long hdp, phep chiing minh phan chiing, phep chiing minh loai dan, ,., NhU vay, nhifng tri thiic va ki nang k! xao ve cac phep chiing minh loan hpe la mdt nhiing hanh Irang quan ma mdi giao vien loan tUdng lai cin phai dUdc trang bi va ren luyen (Cd the tim hieu vl cac phep chiing minh loan hoe irong nhilu lai lieu nhU [2, 4, 5J) Trong qua trinh day hoc cac phep chiing minh loan hpc, kiem ira hil'u biit cua sinh vien ve cac phep chiing minh thdng qua cac bai tap, chiing tdi nhan Ihly sinh vien thudng mac phai mdt so sai lim khdng dang cd Vin di dat la lam thi nao de han chi loi da nhifng sai lim dd? NhU vay, mdt nghien ciiu cu the nham khac phuc sai lam ciia sinh vien sif dung cac phep chting minh loan hpc la cin thill va cd y nghia, gdp phin nang cao nang liic chuyen mdn nghiep vu cho sinh vien khoa Toan - Dai hpc Su pham Ngay nhan biii: 12/2/2014 Ngay nhan diing 15/5/2014 Tac gia lien lac Dao Thi Hoa e-mail: daothihoa sp2@iTioel,cdu.v Bien phdp khdc phuc sai ldm thudng gap ciia ainh vien klii su dung cdc phe'p chitng minh todn ho Noi dung nghien cu'u 2.1 Sai l a m sur d u n g p h e p chiing minh p h a n tich va tong h d p Nhu ta da biet, phan tich va long hpp la hai cac phep chiing minh true tilp thudng dUdc sil dung irong day hpc loan d phd thdng Ve miit li ihuyll, hai phuong phap chiing minh rll rd rang va de hid'u, Mac dii vay, sinh vien van mac phai mdt sd sai lam Nhiing sai lam se dUpc phan tfch thdng qua vf du cu the sau: Vf du: Cho bai loan sau' "Chiing minh rang nlu ba gdc ciia tam giac ADC thda man he thiic sin A =2 siu B vas C Ihi tain giac ADC can tai 4'" a) Trinh bay mdt Idi giai ciia biii loan tren b) Trinh bay nhiing hieu bill ciia minh vl phep chiJng minh da sii dung de giai bai loan tren Trong bai toan tren, d phin a eiia di biii ta cd: Menh dl da eho la "ba gdc ciia tam giac /\DC thda man he Ihiic sin.4 = 2.sini?COMC" menh dl cin ehiing mmh la "'tam giac ABC can tai 4" De trinh bay phin a sinh vien cd nhilu each khac nhau, cdn phan b lai phu Ihupc vao phin a Vdi de bai sinh vien ihifdng mac phai nhiing sai lam Irong Idi giai nhU sau: Ldi giai 1: a) Vl tam giac ABC can lai >1 ^ = C => - TT - 2C =^ sin - sin(7r 2B) =^ sin = sin 2D => sin = 2sin Bcos B =^ siu.4 = 2sin BcosC(B = C iheo tren) Ma sin /I — siii B co.s C la menh de da eho nen ta cd dieu phai chiing minh b) Phep ehiing minh da sif dung de giai bai loan tren la phep chiJng minh phan tfch di xudng Phdn tick lai gidi: Idi giai nay, phin a, sinh vien sii dung phep phan lich di xudng, nhien irudng hdp niiy menh de da cho "sin = sin B cos C" la menh dl diing nen ehUa the kll luan gi vi menh dl can chiing minh Phep phan lich di xuong irong IrUdng hdp khdng phai la la phep chiing minh (sai lam ve luan chiing), dd phan trinh bay tren khdng phai lii Idi giai diing ciia bai loan Nhu vay, d Idi giai nl\y ca phin a va phin b deu sai Ldi giai 2: a) De chiing minh lam j i a c ,4/JC^can lai 4, la chiJn^ minh B = C De chiing minh B = C\ ta chiing minh = ;r - 2D De chiing minh A = n - 2B, la chiing minh sill = sin(7r—2B) De chiing minh sin = siu(7r-2C), la chiing minh sin = siii2Z? De chiing minh sin •= sin 2B, ta chiing minh sin = smi? cos/? De chiing minh sin = 2,sin Z?cos5, ta chiing minh sin/I = s i n c o s r ' {B = C theo tren) Ma sill = sill D cos C la menh di da bill nen ta cd diiu phai chiing minh b) Phep ehiing minh da sii dung degiai bai loan iren la phep chiing minh phan lich di len, Phdn tich Idi gidi: O Idi giai nay, phin a sinh vien sii dung phep phan lich dl len, luy nhien trUdng hpp menh dl siii.4 = 2Kin B c o s C keo Iheo menh dl 11 Diio Thi Hoa sin ^ sin D cos D dUpc giai Ihi'ch la " ^ C theo tren" la khdng co co sd, vi neu da cd B = C ih) hien nhien tam giac ABC la can lai A (sai lim vl luan cii) NhU vay, d sai lam ca phin a va phin b diu sai Ldi giai 3: a) De chting minh tam giac ABC can tai A, la chiing minh B ^C De chiing minh B = C,ta chiing minh sin(B - C) = 0, De chiing minh sm{B - C) = 0, ta chiing minh sin BcosC - cos Bsin C = De chiing minh sin BcosC - cos BsinC, ta ehiJng minh sin C cos B = cos C sin B De chiing minh sin C cos i? = cos C sin B, ta chiing minh sin C cos B -I- sin B cos C = cos C sin B -{- sin B cos C Dl' chiing minh sin C cos B + sin B cos C = cos C sin B + sm B cos C, ta chiing minh siii(B + C) =2 sin B cos C De chiing minh sin(B4-C) = 2.sm B cos C, la chiing minh,sin[7r-(B4-C)] = s i n B c o s C De chang mmh sin[7r- (B-l-C)] = sin Boos C7, ta chiing minh sin A = s i n B c o R C Ma sin A = bin B cos C la menh di da biet nen ta cd dieu phai chiing minh b) Phep chiing minh da sif dung de giai bai loan irln la phep chiing minh phan tich di xudng Phdn tich Idi gidi: Idi giai phin a, sinh vien sii dung phep phan lich di len, va trUdng hdp menh de sin A = sin B cos C la menh dl diing nen kit luan dupc menh de "tam giac ABC can" la menh dl diing Liic phep phan tich di len la phep chiing minh phan tfch di len, NhUng d phin b) lai tra Idi la phep chiing minh phan lich di xuong Sai lim d day la nham Iln giua phep phan tich di len va phan tfch di xudng Nhu vay, d sai lam 3, phan a diing va phan b sai Lbi giai 4: a) am A = sin Bcos C ^ sin[7r - (B + C)] = 2.smBcosC Tam giac ABC can b) Phep chiing minh da sii dung de giai bai loan tren la phep chiing minh phan tich di xudng Phdn tich Idl gidi: O Idi giai nay, phan a, sinh vien sii dung phep chiing minh tdng hdp, nhUng phin b) lai tra Idi dd la phep chiing minh phan tfch di xudng chi quan tam din kf hieu " ^ " ma khong quan lam den menh d i xull phat la menh dl da cho hay menh dl cin chiing minh NhU vay, sai lim 5, phin a diing va phan b sai Bieu phdp khdc phuc sai ldm thudng gap cim sinh vien st? dung cdc phep chung mmh todn hoc Cd the thay ring, hai phep chiing minh phan tfch va tdng hdp la hai phep ehiing minh rit cd ban ma sinh vien nganh SU pham Toan can nam viing Tuy nhien IhUc hanh sinh vien vSn mac sai lam nhu chua hieu rd ve phep ehiing minh phan tfch va phep chiing minh idng hdp; chifa phan biet dupe phep chiing minh phan tfch vdi phep chiing minh tdng hdp; chua nam dUpc nao thi phep phan tfch trd phep chiing minh phan tfch Dae biei la siJ dung phep phan tfch di xudng: nlu menh de da cho, da biet la diing thi menh de cin phai chiing mmh chua chac da diing Mac dii vay, sinh vien van thifa nhan rang neu menh de da cho, da biet la dung thi menh de can phai chiing minh la diing, cho nen dan din nhu'ng sai lam 2.2 Sai lam sur dung phep chirng minh phan chiing va phep chiing minh loai dan Khi chiing minh mpt menh de loan hpc, ngoai cac phep chiing minh Iriic tiep, ta cdn cd the sii dung phep chiing minh gian tiep Nhiing phep chiJng minh gian liep ihudng dupc sii dung la phep chiing minh phan chiing va phep chiing minh loai dan Khi sii dung hai phep chiing minh gian liep nay, sinh vien ihudng mac phai mdt so sai lam, Cac sai lam se dUdc phan tfch thdng qua cac vi du cu the sau day: Vl du: Cho bai toan: "ChiJng minh rang neu mot dUdng thang cat mdt irong hai mat phang song song ihi nd cat mat phang cdn lai" a) Trinh bay mdt Idi giai ciia bai toan tren b) Trinh bay nhiing hieu biet ciia minh ve phUdng phap chiing minh da sii dung de giai bai loan tren Ci bai loan nay, menh dl da cho: " ( P ) / / ( Q ) va o cat (P)", menh di phai chiing minh: "(/ cai (Q)" Khi giai bai sinh vien cd the mac phai mdt sd sai lam Idi giai nhu sau: Lcli giai 1: a) Gia sii (P)//{Q), a cit (P) nhung a khdng cat (Q) Vi a khdng ell (Q) nen fi//(Q) ma iheo gia thilt {P)//{Q) nen a / / ( P ) (mau thuin gia thill a cat (P)) Vay dilu gia su" la sai, suy a va (Q) ell b) Phep chiing minh da sii dung de giai bai loan tren la phep ehiing minh phan chiing (Sau dd trinh bay ve phep chiing minh phan chiing) Phdn tich Idi gidi: Vdi each irinh bay nay, sinh vien mac sai lam d phan a, dd la lap luan: "Vi a khdng cat (Q) nen a//{Q)" va lap luan: "a//{Q) ma theo gia ihilt {P)//{Q) nen (///{P)" lap luan ihii nhll thilu trUdng hpp a C (Q); lap luan ihii hai thieu trUdng h d p n C (P) Loii giai 2: a) Gia sfi { P ) / / ( g ) , a cat (P) nhUng a//{Q) Vi a//{Q) ma iheo gia thilt {P)//{Q) nen a C (P) hoac ajj{P) (mau thuan gia ihiet n eat (P)) Vay dieu gia sii la sai, suy n va [Q] cat b) Phep chiing minh da sii dung de giai bai loan iren la phep chiing minh phan 13 chiing (Sau dd trinh bay vi phep chiing minh phan chiing) Phdn tich Idi gidi: Vdi each trinh bay nay, sinh vien mic sai lim d phin a, dd la xac dinh menh dl phii dinh chUa diing, bdi vl menh dl ein chiJng minh la "o cli (Q)" nen menh dl phii dinh khdng phai la "a//{Qy\ Menh dl phii dinh diing phai la: "n C (Q) hoac (i//{Qy\ Nhu vay la thilu Irifdng hpp o C (Q) Ldi giai 3: a) Gia sii {P)//{Q), a ck (P) nhung a c (Q) Vi a c (Q) ma theo gia ihill a eat (P) nen (P) n (Q) ^ (man thuan gia thill ( P ) / / ( Q ) ) Vay dieu gia sif la sai, suy a va (C^) cai b) Phep chiing minh da sii dung de giai bai loan tren la phep chiing mmh phan chiing (Sau dd irinh bay ve phep chiing minh phan chiing) Phdn tich [di gidi: Vdi each trinh bay nay, sinh vien mac sai llm lUdng tif nhu d sai lam 2, la xac dinh menh dl phii djnh chUa diing, bdi vi menh de cin chiing minh la "a ck (Q)" nen menh dl phii dinh khdng phai la "(/ C ((?]" Menh dl phii dinh diing phai la: -a c iQ) hoac {a)//{Q)"- Nhu vay la thilu trudng hpp a//{Q) Ldi giai 4: a) Gia sii a cai (P) nhUng (i//{Q) VI a//{Q) ma theo gia thilt {P)//{Q) nen a//{P) (mau ihuln gia thiet a ck (P)) Vay diln gia sii la sai, suy cCing xay =* a{2 - b)b{2 - c)c{2 - a) > (*) Mat khac iip dung bit ding thiic Cauchy cho cac sd khong am a, ~ b,b,2 - c,c.,2 - a, ta cd a(2 - h)b{2 - (•)c{2 — a) < 1, man ihuin vdi {*) Vay gia sii la sai, nen la cd dieu phai chiing minh 2.3.3 Yeu cau sinh vien lay cac vi du ve mdi loai chifng minh Viec sinh vien lly dUdc eac vi du vi mdi loai ehiing minh, se giiip giao vien nim dupc miic dp nhan bill kiln ihiic cua sinh viln vl cac phep chiing minh de kjp Ihdi dieu chinh cho phii hdp Ddng ihdi qua sinh vien dUpc ciing co vl cac loai chiing minh Cd Ihe yeu cau sinh vien lu tim hieu ve cac bai loan chiing minh d phd thdng vii su" dung cac phep chiing minh phii hdp Cung cd Ihe dUa mdt bai toan cu the d thdng va yeu ciu sinh vien sii dung nhilu phep chiing minh de giai bai toan Vi du: Sii dung cac phep chiing minh loan hoc giai bai loiin sau bang nhilu each va chi rd phep chiJng minh da sif dung moi each: "Cho hai sd thuc x, y thda man •I + y = Chiing minh ring i y < 1" Bien phdp khac phuc sai lam thudng gap cua smh vien sd dung cue phep chdng minh todn hoc 2.3.4 Tao cac tinh hudng co suf dung cac phep chiing minh toan hpc de sinh vien trao doi, thao luan De cd the khac phuc nhiing sai him cho sinh vien siJ dung cac phep chiing minh loan hpe, giao vien cd the tao cac tinh hudng cd sai lam, hoac khdng cd sai liim de sinh vien lU phan tich, tii xoay xd, lU lim each giai quylt Tren cd sd dd giao vien nhan xel, danh gia, gdp y Tif dd sinh vien thay dUdc linh diing, sai each nghT, each lam, tranh dUdc nhu'ng sai llm, sinh vien se co dUdc nhiing kT nang sii dung cac phep chiing minh nay, cung nhU hudng din hpc smh giai cac bai loan chiing minh sau trudng Vf du 1: Cho bai toan: "Trong mat phang (P) cho hai dUdng thang cat a vit h Hai dudng thing A, b ciing song song vdi mat phang (Q) Chiing minh rang (P) song song vdi {Qy\ a) Trinh bay ldi giai cua biii loiin Iren b) Trinh bay nhiing hieu bill ciia minh vl phUdng phap chiing minh da sii dung de giai bai loan tren Trong nhiJng ldi giai sau, Idi giai nao diing, Idi giai nao sai, vi sao? Ldi giai 1: a) Gia sii mat phang (P) chiJa hai dudng thang a, b cat va ciing song song vdi mat phIng {Q) nhung (P) cat (Q) Gpi (• la giao tuyin ciia (P) va (Q), a//{Q) va a C (P) nen c//a TUdng IU c//b Suy a//b (mau Ihuan vdi gia Ihili) Vay gia sii la sai, tif dd suy.ra dieu phai chiing minh b) PhUdng phap chting minh da sii dung ia chiing minh phan chifng Ldi giai 2: a) Mai phang (P) chi cd the song song vdi map phang {Q) hoac mat phang (P) cit mat phang {Q) Neu mat phang (P) cit mat phlng (Q) theo giao tuyen c thi a//{Q) va a C (P) nen c//a TUdng tu c//b Suy a//b (man thuin vdi gia Ihilt) Vay mat phang (P) song song vdi map phang (Q) b) PhUdng phap chiing minh da sif dung la chiing minh loai din Ldi giai 3: a) Mai phang (P) chi cd the song song vdi map phlng [Q) hoac mai phang (P) ell mat phang (Q) Nlu mat phlng (P) cit mat phang [Q) iheo giao tuyen c ihi a//{Q) va a C (P) nen c//a TUdng tu c//b Suy a//b (mau thuan vdi gia thilt) Vay mat phang (P) song song vdi map phang {Q) b) PhUdng phap chiing minh da su" dung la chiJng minh phan chiing , Nhan xet: Ca ba Idi giai Iren deu sai, cu the: ldi giai thiJ nhll, sai llm d phin a xac dinh menh d& phii dinh irong chiing minh phan chiing chUa diing va lap luan c//a, cjjb suy 0//6 la chua dii, vi (/ ed the triing b ldi giai thiJ hai, sai lam cung d phin a xac djnh cac kha nang xay cua ( P ) va {Q) ehUa du va phan cdn lai sai llm nhU d ldi giai thii nhat ldi giai thii ba, sai lam cung tUdng tii nhU Idi giai 2, ngoai cdn xac dinh phep chiJng minh chUa diing Vidu 2: Giai bai loan sau bing phep chiing minh phan chiing: 'Cho2"'-l(Tn G A') la mpi so nguyen to Chiing minh ring m la mdt so nguyen to" (1) Ldi giai sau diing hay sai vi sao? Ldi giai: De chiing minh bai loan tren ta ta chi viee chiing minh bai loan: "Cho m la hdp sd Chiing minh rang 2'" - l(m £ A'^) la hdp so" (2) That vay: m lii hpp so O- m = pq vdi Vpg € /V, p, q > Ta cd: 2"' - = 2"" - = (2? - 1)(2'''''-'' + 2P(''-2) + + l ) Cac thifa so cua 2"' — diu nguyen dUdng va ldn hdn dd 2'" — la hpp so Vay nlu 2"' — l(m £ A'^) la mdt sd nguyen id thi m phai la so nguyin to Nhdn xet: Sai iam d Idi giai bai loan tren la khdng suT dung phep chiing minh phan chiing theo diing yeu cau debai Hon nOa, chiing minhdupc biii toan (2) chUa dii dS'khang dinh la ehiing minh dUdc bai toan (1), 2.3.5 Yeu cau sinh vien du kien nhiing sai lam thifdng gap giai cac bai toan cu the cd SUT dung cac phep chiing minh Ciing vdi viec tao eac tinh hudng cd sai llm hoac khdng cd sai lam de sinh vien IU hpc lap, ta cd the xay diing cac de loan cu Ihl va yeu cau sinh vien dii kiln nhitng sai lam cd the xay giai cac bai loan dd Vi'du 1: Dif kien nhiing sai lamed the xay rakhi giai bai toan sau, phan tfch nguyen nhan sai llm va cho ldi giai chinh xac: "Trinh bay ldi giai bai toan: Cho4sd a,b,:i:.y thda man d' -i-H^ = vh:r^ +y'^ = 2.' Chiing minh ring: —\/2