JOURNAL OF SCIENCE OF HNUE Educational Sci 2011, ''''ol 56, Xo 4, pp 13 23 KHAI THAC N H C N G CACH TAO TINH HUONG G d i VAN DE TRONG DAY HOC MON TOAN d T R U C J N G TRUNG HOC P H 6 THONG Le Tti^n A n[.]
JOURNAL OF SCIENCE OF HNUE Educational Sci 2011, \'ol 56, Xo 4, pp 13-23 KHAI THAC N H C N G CACH TAO TINH HUONG G d i VAN DE TRONG DAY HOC MON TOAN d T R U C J N G TRUNG HOC P H THONG Le Tti^n A n h Trudng Dai hoc Su pham Hd Noi E-mail: letuananhll (iliotniail.com Tom t&t Thuc te cho thay giao vien toan thudng gap kho khan van dung day hoc pliat hien va giai quylt van dl vao thuc tif'ii day hoc Ti'en cO sd nhung each tao tinh huong goi vin dl bai bao se phan tich va dua nhiing gpi y cu thi hdn dl co thi thilt kl dupc nhung tinh huong gpi van de, phan tich goc su pham ciia viec sir dung nhung tinh huong da\- hoc mdn Toan cf trung hoc phS tliong Md dau Thuc te cho tha}- giao vien toan thudng gap nhilu khd khan van dung day hpc phat hien va giai quylt van dl (DHPHYGQ\'D) vao thuc tiln day hpc: Trinh dp hpc sinh (HS) mdt ldp khdng ddng diu; Thdi gian day hpc van dung phupng phap DHPHVGQVD can nhilu hpn so vdi sii dung nhiing phuong phap tru}-en thong; Mdt sd giao vien chua thao thilt kl tinh huong gpi van dl (THGA'D), chua nam virng vl DHPHA'GQA'D Tren cP sP nhiing each tao THGVD [5; 197-199], chiing tdi se phan tich va dua nhiing gdi y cu thi hpn dl cd thi thilt ke dupc nhiing THG\'D, phan tich gdc dp su pham va xay dung nhiing vi du vl nhiing each tao THG\'D day hpc mdn Toan trung hpc phd thdng (THPT), 2.1 Noi d u n g nghien cu'u NhiJng each thong dung de tao tinh huong gdi van de THGVD thoa man dong thdi dilu kien: ton tai mdt van dl, gpi nhu cau nhan thiic va khpi day nilm tin d kha nang ban than [5; 186-187], Nhiing each thudng dimg di tap THGVD: du dpan nhd nhan xet true quan va thuc nghiem (tinh toan, dac ), lat ngupc van de, xem xet tUPng tu, khai quat hda, giai bai tap ma ngudi hpc chua bilt thuat giai, tim sai lam Idi giai, phat hien nguyen nhan sai lam va siia chiia sai lam [5; 197-199], 13 Le Tuan Anh 2.2 Mot s6 lu'u y ve nhiing each tao tinh huong gdi van d i \'iec phan loai nhfmg each tao T I I ( i \ T ) ndi tren chi mang tinh chat tiTdng (1(M, chang han mot TIKJX'D cd the viia x(Mn nhu dupc thiet ke nhd du doan tfr nhan xet true (luan, thirc nghiem, (long thiti dua vivo khai (luat hda Xhrmg cach tao THGX'D ndi tren la nhung gdi y dinh hudng d('' giao vien cd thi thiel ke nhfrng 141G\'I) i)liii hdp vdi dieu kif'ii cho i^hep, tu dd cd I h i van dung ) H P H \ C ; Q \ ' D (hue tien day hoc Giao vien nen cAii cfr vao trinh dp HS, thdi gian day hoc, phudng tien day hoc d('' thiet ke lira chpn va sii dung nhiing r i l G \ D phil hdj) (clni v tdi dilu kien gdi nhu can nhan thiic va khdi day nir'm tin (i kha nang) Khai thac hop ly nhung each tao T H C ; \ D ndi tren tiong fla}- hpc, giao vien cd till dat dupc nhieu miic dich: vira tao TIKiVD dv diw hoc mdt ndi dung nao dd (heo phuong phap D H P I i \ ' G Q \ ) , vira giuj) hpc sinh ren lii\en nhrmg hoat ddng tri (lie CO ban (trim tUpng hda, khai (juat hda, dac biet hoa tUcJng i U hda ) nhiing hoat ddng tri tne phiic hpp (lat ngUpc van de ); phat triln tU (tU ham, tu bien chiing, (U sang tao ); phat triln kha nang suy doan: phat hien va sua chiia nhfrng sai lam hpc toan; hinh nhiing pham chat tri tue (tinh linh hoat, tinh ddc lap, tinh sang tao), Trong bai bao na}- chiing tdi co gang lira chpn nhiing bai toan minh hoa khdng phiic tap khdng can nhifMi thdi gian tir nhung phan mdn khac cua mdn Toan (sd hpc, dai so, hinh hpc giai tich ) nham khang dinh rang cd nhilu cd hdi d l tao THGVD day hpc mdn Toan d tnrdng T H P T 2.3 Khai thac mot s6 each tao tinh huong gdi v4n d i day hpc mon Toan of trifdng t r u n g hoc ph6 t h o n g Phan se trinh ba}- vice khai thac mdt so each tao T H G \ ' D dita vao nhan xet trirc quan vh thuc nghiem, lat ngupc ^•an de tuong tu hda, khai quat hda va phat hien, siia chua sai lAm Idi giai 2,3.1 Di/a vao tri^c q u a n , thij'c n g h i e m - Tao T H G \ ' D dua \-ao nhan xet nhd true ciuan \-a thuc nghiem cd t h i gdp phan giai qn}'et hpp ly moi quan he giua phUPng dien thirc nghiem \ a phuPng dien suy dien da}- hoc mdn Toan d trudng T H P T Theo |5: 38], ^'Phai chii y ca hai phUdng dien dd mdi cd the hudng dan hoc sinh hpc toan, mdi khai thac dupc day du tilm nang mdn Toan d l tliUc hien muc tieu giao due toan dien" Thirc te cho thay, mdt sd giao \'ien chua chii v diing miic d i n phuong dien thuc nghiem day hpc mdn Toan d trudng T H P T - Tao T H G \ ' D dua vao nhan xet nhd true quan ^•a thuc nghiem tao nhieu cd hdi cho hpc sinh ren lu}-eii cac hoat ddng tri tue cd ban: triru tUpng hda, khai quat hda, phan tfch, tong hpp, so sanh,,, 14 Khai thdc nhung cdch ta.o tinh huong gai vdn de day hoc mon Todn Giao vien can cd nhfrng vi du, bai tap giiip hpc sinh thay rang nhfrng nhan xet dua vao thuc nghiem cd thi dung hoac sai Tir dd can chiing minh hoac bac bd mOt du doan dua vao true quan va thuc nghiem: Tii nhan xet = 1'^; l-f - 2'; + I- = 3' din du doan + H- + 4(2n - 1) = n' (V/?, G A'*) la inOt kit qua diing (cd (hi cluing minh bang phUdng phap qui nap toan hpc) \'di bai toan "Gpi (C) la thi ciia ham so // - /• -|- - Tim hai dilm M, X /• tren hai nhanh khac ciia (G) cho dp dai MX nhd nhat" nhilu hpc sinh san ve (G) da khang dinh (dua vao true ciuan) la gia tri nhd nhat cua MX la khoang each giua hai dilm ci.rc (ri \B = 2\/5 vdi 4(1; 2) va B{- 1; - ) (Tuy nhien ket qua khdng diing) [7; 11] Cd thi dira \-ao qui trinh sau de tao THG\'D nhd nhan xet true quan va thuc nghiem: Budc Kiem tra bang thuc nghiem hoac true giac (hinh ve, dimg may tinh bo tiii hoac nhfrng phan mem nui}- tinh, chang han nhiing phan mim hinh hpc ddng (Dynamic Computer Software) hoac nhfrng chuong trinh dai so may tinh (Computer Algebra System) de kilm tra Budc Hinh du doan Budc Chiing minh du doan hoac bac bd du doan \'di hau hit cac bai toan phuong dien thuc nghiem chi thi hien hpc sinh suy nghl, tim tdi Idi giai ciia bai toan (thudng chi dupc trinh ba}- gia}- nhap) Tuy nhien phupng dien na}- ciing xuat hien hpc sinh trinh bay Idi giai mot bai toan (vi du A'I ldp cac bai toan tinh dao ham cap ii {n N*) cua mdt ham so) Trong mdt so trudng hpp, giao vien nen chuyin mdt so bai toan chiing iniiih sang bai toan tim tdi, tao cd hdi cho hpc sinh tim tdi kiln thiic toan hpc Gd thi ap dung each na}- doi vdi nhilu bai toan chiing minh bang phudiig phap qui nap toan hpc Vi du (Hudng dan hpc sinh ldp 11 tim Idi giai bai toan bang each tao THGVD dua vao nhan xet true quan, thuc nghiem hpc day so) Trong mdt tai lieu, cd bai toan va Idi giai sau da^': "Cho d?n- sd ( r„) dupe xac + ^-n dinh nhu sau Xi = 2003, Xn+i = -^ Vn G N* Hav tinh i2oo51 - r„ Ldi gidi TT ^ ^ TT TT Dat T,„, = ta.ny.n{ < yn < ij) =^ tany„+i = tan —h tan (/,, A _ ^ ^—^ •4 = tan {yn + -) TT Tii dd suy yn+i = yn + - • nen y„+4 = y^ -+- ^''T^{^' € ^)Day (yn) cd yn+4 = yn + kT:{k e Z),\/n G A^* Suy ra: :Cn+4 = tanyn4.4 = tan (yn + ^'TT) = tan yn = Xryn G N* Suy X200.5 = a^i = 2003." Ldi giai tren la ldi giai hay, doe dao, nhung khdng tu nhien Gd thi hudng din 15 Lc Tuan Anh hoc sinh giai bai (oan tren bang each tao THGVD dua vao nhan xet thiic nghiem nhu sau: Ld'i giai 2: B u d c 1: lla}' t.fnh Cv ;.),./:,], r.r,, /•(;, ry, + /'i I ;.; -1 • ;-| /, ./i II ^lieo :/:i .r:i ''i ':) •'•10 —•••= '.H,.-' I _ '1 -f 1 + -'4 1- 'I B u d c 2: Dn (loan ke( (iiui ''1 •'',^1 r.-j = (-7 = ' ! ) = • • • = Cii = -''lA'I ! • - ' = •'(; '-.lA-1 : ( ; • ' • • ! '>< = •'• 12 = • • • = •l'\h\2 •'•|/.-4.i(V/.' € A') Budc 3: Gluing minh d\t doan Du doan cd (he chiing minh bang j^hudng phap qui nap t(;an hoc Ldi giiii la ldi giai (U nhien cd dupe nhd tao i I G \ D dua vao (juan, thi.rc nghiem Gd (he aj) dung each giai cho nhieu bai t(jan tinh chat cua da}- so Vi du (Chuyin tir bai toan chiing minh sang bai toan tim hpc sinh Idp 11 hpc phUPng phap qui nap (^oan hpc) \'di bai toan "Chiing minh rang vdi mpi so nguyen dudng n > 2"- > 2n + 1" [11; 101], cd t h i tao THGVD bang each de toan nhu gia tri ciia so nguyen duong n, hay so sanh 2" va 2ii + 1, nhan xet true lien ciuan tdi tdi danh cho ta ludn cd sau: Tiiy theo Dl tim ldi giai cua bai toan, hpc sinh cd t h i tien hanh theo cac budc: Budc Clio n nhfrng gia tri cu t h i 1, 3, 4, va ,so sanh 2" vdi 2/i + 1, Budc 2, Du doan vdi mpi so nguyen dupng n > '•] ta ludn cd 2" > 2n + Budc 3, Chiing minh bai toan bang pliUdng phap qui nap toan hpc 2.3.2 D\ia vao lat ngifdc van de Muon khang dinh kit qua cd dupc tir lat ngupc \-an d l la dung thi can chiing minh kit cjua la sai thi phai dua dupe mdt phan vi du MQt dang thudng gap cua lat ngupc \'an d l day hpc mdn Toan la xet nienh de dao cua inQt dinh li Tu\- nhien, lat ngupc \'an de khdng ddnti, nhat vin xet menh d l dao cua mot dinh li (xem vi du [8; 134]) Khai niem menh de dao va cac dang menh de dao ciia nipt dinh li dupc trinh bay da}- dii trpiig [3:46-53] Giao vien nen tao cho hpc sinh thdi quen lat ngupc van de \a dUa nhfrng trudng hpp da dang d l hpc sinh thay rang kit qua cd du'dc tu lat ngupc \-An d l cd the diing hoac sai dd can chiing minh hoac dua phan vi du Tao THGVD dua vao lat ngUdc van d l giiip hpc sinh nam \-frng kiln thiic, liifMi ki liPn ve menh de dao dieu kien can dieu kien du Trong day hpc mdn Toan d trUdng T H P T , giao vien cd nhieu cd hdi tao THGVD nhd lat ngupc \'an de Sau da}- la mdt sd vi du vl tap T H G V D bang lat ngupc ^-an dl: Vi du (Danh cho hpc sinh ldp 10 hpc ve bat d i n g thitc) a) Xlu a, va c la dp dai canh ciia tam giac thi a-\-b > c 16 Khai thdc nhiing cdch tao tmh huong gai vdn de day hoc mon Todn Xet xem mdi menh c\i sau day diing ha}- sai: + Nlu a, h va c la cac so thirc thoa man dien kien a -\- b > c thi a, b va c la dp dai canh ciia mOt tam giac (Menh dl dao sai, cd (hi dua mot phan vi du, chang han a = 1, = va c = - ) , 4- Nlu a b va c la cac sd thuc dUPng \-a a + b > c thi a b va c la dp dai canh ciia mot tam giac (Menh dao sai, cd the du'a mdt phan vi du, ch/nig han a ^2 b = A c - 1), b) Xlu a b va c la dai canh ciia mdi tain giac (hi a + b > c.b + c > a c+a > b Xet xem menh d l sau da}- dimg hay sai: Xlu a + b > r, b+r > «, c + a > b thi a b va c la dp dai canh ciia mdt tam giac (kit qua n;i}- diiiig vi cdng timg vl cua hai ba bat dang thiic ndi tien thi la se suy dupc a, b \h c la cac sd dUdng), Vi du 4- (Danh cho hoc sinh ldp 12 hoc khao sat va \'e thi cua ham sd) a) Tfr dinh li: Xlu mOt ham so /(,r) cd dao ham trfMi khoang / va f'{x) > vdi mpi X G / thi ham sd f{.v) dong biln tren khoang / Tao T H G \ ' D : Phai chang mdt ham so f{.v) cd dao ham va dong bien tren khoang / thi /'(.r) > vdi mpi r G / K i t qua na}- khdng diing, chang han ham so f{.v) = r'^ dong biln \-a cd dao ham tren mpi khoang / cd dang (—a; a) (vdi a > 0) chiia 0, nhung /''(O) = b) Ham so chi cd the dat cue tri tai mot dilm ma tai dd dao ham cua ham so bang hoac tai dd ham so khdng cd dao ham Tao T H G \ ' D : Phai chang mdt ham sd cd dao ham bang hoac khdng cd dao ham tai mot dilm thi dat cue tri tai dilm dd K i t qua khdng dung, cd t h i dua mot phan vi du: chang han ham sd y = x^ cd dao ham bang tai dilm x = nhung ham sd khdng dat cue tri tai dilm dd 2.3.3 Di^a vao tvfdng ti^ hoa Phep su}' luan tUdng tU dupc sii dung rat da dang mdn Toan va thudng dupc d l cap dudi nhung gdc dp sau: hai phep chiing minh tUPng tu, hai hinh tUPng tu va hai tinh chat tUPng tu [2; 12-13], Theo Polya ([10]), tuong tu thudc ve nhfrng suy luan cd li, nhfrng kit luan rut tUOng tu hda thudng cd tinh chat gia thuylt, du doan, Trong lich sir toan hpc, su}' luan tUPng tu la ngudn gdc ciia nhilu phat minh, Cd t h i tham khao them vl suy luan tUdng tu cac tai lieu [2, 10], - Trong day hpc mdn Toan d trudng T H P T , thdng qua nhfrng vi du, nhiing bai tap cu t h i , giao vien can giiip hpc sinh thay rang mac dii kit qua riit tii tUPng tu thudng cd tinh gia thuylt, du dpan, nhien nd cd vai trd quan trpng viec tim ldi giai bai toan va kham pha kiln thiic Nlu muon khang dinh kit qua 17 Le Tuan Anh nit tfr suy luan tUdng tu diing thi cfui cluing minh, neu muon khang dinh khdng dung thi cAn dUa phan vi du Tudng tu hda cd the khdng cho kit (lua diing, chang han mot tam giac thi dudng cao ddng qui, nhimg mdt tii dien thi dirdng cao khdng ddng qui (dien dd chi diing vdi mot, ldp lit dien dac biet: tii dir>n cd hai cap canh doi dien vudng gdc); mac dii (rong mo(, tam giac, cac (.rung tuyen ddng qui tai trpng tam ciia (am giac va nid(, tii dicMi thi dudng tuyIn (noi dinh va trpng tam eiia mat doi dien) cung ddng (pii t,ai trpng t,am ciia tii dien Mdt sd vi du vl tao THGVD dira vao tUcJng tu hda: Sir tuong tu mdn 4i)an d tritdng phd thdng rat phong phii: sir tUdng tir gifra hinh hpc phang (HHP) va hinh hpc khdng gian (HHKG), giiTa tam giac va tii giac (giiia hai trudng hdp rieng ciia mdt trudng hpp tong quat), giua cap sd cdng va cap sd nhan, sin.r \'a cos.;-, tan.r va cot v; Vi du (Danh cho hpc sinh ldp 11 hpc HHKG) Sir tUPng tir gifra HHP va HHKG rat phong phu Giao vien cd thi khai thac sir tUdng tu na}- dl tao THGVD day HHKG Gach lam na\- giiip hpc sinh thay dupc lien he giiia HHP (dupc hpc chii ylu P trung hpc cP sd (THCS)) va HHKG (dupc gidi thieu d cudi cap THCS va hpc chii veu d THPT), tao cP hdi cho hpc sinh dn lai kiln thiic hinh hpc P THCS va ap dung nhiing kiln thiic hpc HHKG dong thdi giup hpc sinh phat hien nhiing kiln thiic ve HHKG tfr nhfrng kiln thiic ve HHP Sau day la su tUPng tu giiia mot so }'lu td HHP va ylu td tUdng ling HHKG: Bdng Cdc yeu td tiXdng tU giiia HHP vd HHKG Hinh hoc khong gian Hinh hoc phang Phifdng phap Dirdng thing Dudng thang Dudng thang Mat phang Tii dien Tam giac Tii dien vudng Tam giac vudng PhUdng phap "tdng hpp' Tam gicic diu Tii dien diu Hinh hop Hinh binh hanh Hinh chu nhat - Hinh hop chfr nhat Hinh vudng - Hinh lap phuong Dudng trdn (hinh trdn) Mat cau (hinh cau) Phudng trinh dudng thing Phudng trinh dudng thing mat phang trpng khdng gian Phuong trinh dudng thing PhUdng trinh dudng thing PhuPng phap tpa dp mat phing khdng gian PhUdng trinh dudng trdn PhUdng trinh mat can mat phing khdng gian 18 Khai thdc nhiing cdch tao tmh huong gai vdn de day hoc mon Todn Do khudn kho cua bai bao, chiing tdi chi xin minh hpa mdt phin nhd nhfrng ndi dung ndi tren: sU tUPng tU gifra tam giac vudng va tii dien vudng Trong bang dudi day, giao vien cd thi tao THGVD bang suy luan tUdng tu dl hudng din hpc sinh ldp 11 THPT tim nhfrng kit qua tii dien vudng (c) cdt ben phai) tii nhiing tinh chit tUdng tu doi vdi tam giac vudng (HS dupc hpc tfr THCS) Bdng Cdc yeu to tiicfng tii giiia tam gidc vudng vd tii dien vudng Tinh chat cua tuf dien vuong Tinh chat cua tam giac vuong Cho tii dien ABCD cd gdc tam dien dinh Cho AABC cd = 90" cd ba gdc d dinh deu vudng H la chan dudng cao tff xuong H la chan dudng cao tfr A xuong mat canh BC phing {BCD) Dien tich tam giac bang 1 AH-^~ BC -AB.AC AB-^ ' AC-2 = AB' + AC2(Dinh li Pitago) AC = BC.CH] AB' = BC.BH cos'B + cos'C = sm'B + sin'C = 2.3.4 Thi tich tii dien bing -AB.AC.AD 1 1 n Am~ AB^ ' AC^ ' AD^ Binh phuong dien tich ABCD (mat huyin ciia tii dien) bang tdng binh phUdng dien tich cua cac tam giac ABC, ACD va ADB (cac mat vudng ciia tii dien) Binh phudng dien tich AACD bing dien tich ABCD nhan vdi dien tich ciia ACHD cos'{ABC, BCD) + cos'{ABD, BCD)+ cos'{ACD,BCD) =1 sin '{ABC, BCD) + sin '{ABD, BCD)+ sm'{ACD,BCD) = Difa vao khai quat hoa Khai quat hda la mOt hoat ddng tri tue cd ban Theo Kd-ru-tee-xki, khai quat hda nhanh chdng va rdng rai cac doi tUpng, quan he va cac phep toan la mOt phin cot ldi cua cau true nang luc toan hpc ([6]) Nang luc khai quat hda dupc Hiep hdi qudc te vi danh gid kit qua hoc tap lEA ciia UNESCO chpn la mdt 10 chi tieu cd ban ciia nang luc toan hpc Giong nhu suy luan tuong tu, kit qua cho dupc tii khai quat hda thudng cd tinh gia thuylt du dpan Mudn khing dinh dupc kit qua CP dupc tfr khai quat hda la diing thi phai chiing minh, mudn khing dinh la sai thi cin dua phan vi du Gd the tim hieu them vl khai quat hda cac tai lieu [4, 5, 10], Vi du (Danh cho hpc sinh kha gidi ldp 10 hpc phUdng trinh bac cao quy ve bac 2) 19 Le 4\ian Anh Tao T H ( ; \ I) dv hpc sinh kha gidi kham pha dang phUdng trinh tdng quat hdn ca phUdng trinh a.r' I br^ + cr' I /;.;• a - (1) va a./'' + br^ + cr' Irx + a = (2) (.(• la ;iii sd, a, b r la cac he sd va a / 0; gia sii r i n g hpc sinh da biet each giai hai phupiig trinh na}) Coao vien cd the hudng dan hoc sinh so sanh each giai cua hai phuong dinh dv lit dd (im dang (dug (iiia( cua cluing: Bang So sank hai cdch gidi hai phUdng trinh Cach giai phLfdng trinh (2) Cach giai p h u d n g trinh (I) Xf'l, / = khdng la nghiem ciia [)huOng Xci /• = kiidng la ngliiem ciia, phU(ing I null liinli Xet ;• 7^ 0, cilia ca hai ve ciia (2) Xc( r ^ chia cii hai ve ciia, (1) CIHJ cho /ã', (a, cd: r", (a, cd: ô(.r-4 A ) + K-'-+-^-) + r = : ( l ' ) •'•" , a(;r-^-f^)-f/;(./•- i ) + r - (2') •'• 13a(, ^ = r 4- - (dieu kien |/| > 2), dd r (1') CO dang: at' + bi + (c - 2a) = (1") IJat, /, = ;• - k i l l ( ) cd dang: uf- +f)f + (r + 2r;) = n (2") Giao vien cd t h i tao THGVD b i n g each dua \-eu can: can cii \-ao ldi giai cac phuong trinh (1) xk (2), ha}- xac dinh dang phuong trinh tdng (iua( eiia (1) va (2) day, hpc sinh cd t h i dupc hudng d i n de phat hien r i n g x + va ,r A- ,7- la cac trUPng lipp rieng cua x H— l u phuPiig trmh tong cpuit cua (1 ) VA (2 ) r k' k la a{.T' H—-) -I- b{x -\—) + c = \'a\' phuPiig trinh tdng cpiat ciia (1) va (2) se la x^ X a.r^ + b.r^ + ex' -f- bkx 4- ak' = 0(3)(a ^ 0), Fa da hudng dan hpc sinh tim dang tdng quat ciia phudng trinh (1) va (2) la phUdng trinh (3), Cach giai phudug trinh (3) tUdng tu nhu each giai cac phUdng trinh (1) va (2), Vi du (Danh cho hpc sinh ldp 10 hpc bat d i n g thiic) Bat ding thiic tam giac: nlu a b va c la dai canh ciia mot tam giac thi a + b> c Tao T H G \ ' D : ^ Xem tam giac la trudng hpp rieng eiia da giac (ldi) /; canh (/? G A\ n > 2) phai chang tdng dp dai ciia /) - canh ludn Idn hdn dp dai canh cdn lai (Ket ciua diing, cd the chiing minh dua vao bat d i n g thiic tam giac)-I- Xem a \ /y va c^ Ian ludt la c;ic tnrdng hpp rifMig ciia a", //' va c"{n G A^ n > 0), phai chang a" + b" > c" (Kit (jua khdng diing, ching han vdi n = va a, b \'a c la dp dai canh ciia mot tam giac vudng vdi canh huyIn c thi a' + b' ^ c') 2.3.5 Giiip hoc sinh p h a t h i e n va siJa chiJa sai l a m t r o n g ldi giai Cac }-eu can cd ban ddi vdi ldi giai mdt bai toan: kit qua diing, kl ca cac budc trung gian, lap luan chat che (luan de phai n h i t quan, luan cii phai diing, luan chiing phai hpp logic), ngdn ngii chinh xac vk trinh bay rd rang, dam bao tinh my thuat [5; 390], 20 Khai thdc nhung cdch tao tinh hudng gai vdn de day hoc m.6n Todn Nhin chung nhiing sai lam eiia hpc sinh giai toan r i t da dang Mot ldi giai cd sai lam thudng la vi pham mdt nhfrng yfMi ciu ndi tren D l tao T H G \ ' D cd t h i dua vao: 4- Nhfrng sai lam giao vien hu cau (dua vao kinh nghiem ciia ban than giai toan) H- Nhiing sai lam ciia hpc sinh lam bai t,a,p hoac bai kiem tra + Giao vien cd the can cii vao nhfrng sai lam ma hoc sinh ciia minh thudng gap tfr cac khda trudc de tao THC!\'D, qua dd giiip hpc sinh phat hien va giai qu}-et van de -L Dua vao tai lieu tham kliiio (ching han | l , •') 5, 9]), niiic "Sai lim d dau?'' Tap chi Toan hpc va Tudi tie (danh cho hpc sinh kha gioi) Vi du (Danh cho hoc sinh kha gidi ldp 12 hpc tich plmn) Xhan xet sau da}- diing ha}- sai? Xlu sai hav- siia lai cho diing a) The tich vat the trdn xoay sinh bdi hinh p h i n g cd gidi han la cac dudng thing X ~ a.x -— b {a < 6) va cac dd thi ham so y = fi-v), y = -/(•(-) (/(•') lien tuc tren [a:b]) qua}- xung quanh true Ox la: 27r /' [f{.v)]'-dx h) The tich ^-at t h i trdn xoay sinh bdi hinh p h i n g cd gidi han la cac dudng thing X = a, X = b, {a < b) va cac thi ham so y = /'(.r), y = y(,r) {f{x, g{x) lien tuc tren [a\b]: /(,?) < g{.r),\/x G [a:b]) quay xung quanh true Ox la: '7rJ^[g{x)^f{.r)]'clx Xhfrng sai l i m la hpc sinh da dung suy luan tUPng tu tfr nhiing cdng thiic tinh dien tich hinh p h i n g b i n g tich phan - Giao vien can chii v quan dilm bien chiing giiip hpc sinh phat hien va sua chua sai lam ldi giai, Xlu cd t h i , mot so trudng hpp giao vien nen hudng d i n hpc sinh sua lai ldi giai sai ldi giai diing (ben canh viec dua cac ldi giai khac ciia bai toan) Lam nhu va}' la gdp p h i n bdi dudng mdt so ylu to eiia trilt hpc vat bien chiing cho HS: sira chfra cd kl thiia, chir khdng phai phu dinh sach iron, Sau day la mot vi du don gian minh hpa cho y tudng nay: Vi du (Danh cho hpc sinh ldp 10 hpc b i t d i n g thiic) C h i n g han cin tim dieu kien cua i n phu /^ = ;r -j—{x ^ 0), Mot hpc sinh •' 1 da lam nhu sau: Ap dung bat dang thiic Cd-si cho hai sd x va - ta cd x H— > 2, Cd t h i tao T H G \ ' D b i n g each yen ciu hpc sinh tim chd sai lap luan tren va siia lai cho dung, day, hpc sinh da ap dung b i t d i n g thiic Cd-si nhung khdng elm V din dilu kien cua ;; va — Cd nhilu each tim dilu kien cua t Tuv nhien ben X canh viec hudng d i n hpc sinh tim cac each khac di tim dieu kien ciia t, giao vien nen hudng d i n hpc sinh sua lai ldi giai tren ldi giai diing, chang han: 4- Vdi X > 0, ap dung b i t d i n g thiic Cd-si cho so dUdng la x va - , ta cd X x+ - >2 X 21 Le Tuan Anh \ d i ;• < 0, ap dung bat, d i n g (,hiic ('d-si cho so dUdng la - /: va -— , ta CO (- r) 4- > Tfr dd (,a cd ,/: I ' • ' ' , < -2 • ' ' Wi}' dieu kien ciia an plni / la |/,| > Lam nhu vay (a da ng;im hinh (,hanh cho hoc sinh tu tudng ciia cjui luat phii dinh triet, hoc vat, bii-n chiing 2.4 Thife nghiem Xhfrng (U (U()ng (rong bai bao da d\t(ic ike gia khai thac, van dimg vao da}- chii}-en de NliUng xu h.itdng kh.nn.g trv.yen thong cho ldp ca(; hpc, day mdn Pliuang pluip day hoc mdn Todn, cho 10 khda sinh vien sU pham nganh Toan, hang chiic Idj) thudc he vira lam, \ira hoc va he (u xa ciia trUdng Dai hoc Sir pham Ha Xdi va day mdn Toan cho ldp tai (rUdng T H P T \gii}en 4at Thanh, Ha Xdi Kit qua phdng \-an, (rao ddi vdi hpc vien, sinh vien, lipc sinh quan sat cac gid hpc va kit qua Cfic bai kiem tra, bai thi, (hu hoach CIKJ thay: Sinh \-ien, hpc vien hiing tliii vdi bai giang Sinh vien, hpc \'ien cd kha nang thilt kl, khai thac nhfrng THGN'D da}- hpc mdn Toan d trudng phd thdng Sinh vien, hpc vien n i m viing D H P H V G Q \ ' D \-a cd kha nang van dung hieu qua xu hudng thuc tiln day hpc Hpc sinh hiing thu vdi bai giang, tich cue hpc tap, cd kha nang phat hien va giai quylt vin dl Chat lupng del}' va hpc dupc nang cao Xgoai ra, chiing tdi cung tien hanh mdt so thuc nghiem cd doi chiing nhim budc d i u kiem nghiem tinh kha thi ^'a hieu qua cua viec khai thac nhfrng tu tiTdng, y -tudng da neu bai bao Trong cac nam hpc 2007 20()S 2008 2009 va 2009 2010, thuc nghiem dupc tiln hanh vdi ldp tliUc nghiem (gdm 121 sinh vien) va ldp ddi chiing (120 sinh vien) tfr sinh vien su pham nganh Toan nam thii cua khoa Toan Tin, trudng Dai hpc Sir pham Ha Xdi Kit qua dilm tdng hpp nhu sau: Bdng 4- Ket qua thitc Diem Thuc nghiem Ddi cluing nghiem / 10 Tong so bai 0 12 22 40 27 13 121 0 18 27 36 21 120 K i t qua thuc nghiem birdc d i u chp thi}' kit qua hpc t a p cua sinh vien cac ldp thuc nghiem cap hdn cac ldp doi chiing, chiing to kha nang thiet kl va khai thac cd hieu qua nhiing T H G V D thuc tiln day hpc cua sinh vien d cac ldp thuc nghiem cap hPn cac ldp ddi chiing 22 Khai thdc nhiing cdch tao tmh hudng gai vdn de day hgc mon Todn Ket luan Tfr nhiing each thdng dung dl tao THGVD ([5, 197-201]), bai bao da dua mdt so gpi y cu thi dl thilt kl nhfrng tinh hudng gpi vin dl va, phan tich gdc dp su pham cua viec su dung nhfrng tinh hudng day hpc mdn Toan d THPT Hy vpng bai bap se giiip ich phin nao cho viec triln khai DHPHVGQVD thuc tiln day hpc mdn Toan d tnrdng THPT TAI LIEU THAM KHAO [I] Xguyin Mnh Can, Le Thdng Nhat, Phan Thanh Quang, 2005, Sai lam phd bien gidi todn (Dimg cho hoc sinh vd giao men, day todn PTTH) Nxb Giao due, Ha Xdi [2] Hoang Chiing 1969 Urn luyen kha nd.ng sdng tao todn hoc d trUdng phd thdng Xxb Giao due Ha Xdi [3] Hoang Chiing, 1997 Nli dng vdn di ve logic mdn Todn d trudng phd thdng trung hoc ca sd Xxb Giao due [4] Xguyin Ba Kim, 1982 Tap luyen cho hoc sinh khdi quat hda tdi lieu todn hoc Tap chi Xghien ciiu Giao due, [5] Xguyen Ba Kim, 2007 Phuang phdp day hoc mdn Todn Nxb Dai hpc Su pham Ha Xdi [6] \'.A Kp-ru-tec-xki, 1973 Tdm ly ndng luc Todn hoc cua hoc sinh Nxb Giao due, Ha XOi [7] Tudng Minh Minh, 2000 True quan gidi Todn Tap chi Toan hpc va Tudi tre, sd 277 [8] Biii Van Xghi, Vupng Dupng Minh, Nguyin Anh Tuin, 2005, Tdi lieu bdi dudng thudng xuyen giao vien trung hoc phd thdng chu ki III (2004-2007) Todn hoc Xxb Dai hpc Su pham, [9] Biii \'an Xghi, 2008, Giao trinh phuang phdp day hoc nhung ndi dung cu the mdn Todn Xxb Dai hpc Su pham Ha Ndi, [10] G, Poh'a, 1995 Todn hoc vd nhiing suy luan cd li Nxb Giao due [II] Doan Quynh (Tdng Chii bien), Nguyin Huy Doan (Chu bien), Nguyin Xuan Liem, Nguyin Khic Minh, Dang Hung Thing, 2007 Dai so vd Gidi tich ndng cao 11 Xxb Giao due Ha Ndi, ABSTRACT Creating and utilizing problematic situations in teaching mathematics in upper secondary schools Generally, upper secondary mathematics teachers encounter several difficulties while Problem Solving and Posing approach is applied to teaching and learning mathematics The main aim of this article is to discuss how to design problematic situations and apply these situations to teaching mathematics in upper secondary schools, 23 ... ''{ABC, BCD) + sin ''{ABD, BCD)+ sm''{ACD,BCD) = Difa vao khai quat hoa Khai quat hda la mOt hoat ddng tri tue cd ban Theo Kd-ru-tee-xki, khai quat hda nhanh chdng va rdng rai cac doi tUpng, quan... cho hpc sinh ren lu}-eii cac hoat ddng tri tue cd ban: triru tUpng hda, khai quat hda, phan tfch, tong hpp, so sanh,,, 14 Khai thdc nhung cdch ta.o tinh huong gai vdn de day hoc mon Todn Giao... gia thuylt, du doan, Trong lich sir toan hpc, su}'' luan tUPng tu la ngudn gdc ciia nhilu phat minh, Cd t h i tham khao them vl suy luan tUdng tu cac tai lieu [2, 10], - Trong day hpc mdn Toan