fflNSHItNCDUULUM t r i SUf DUNG TiniNG TIT GiffA CAC BAI TOAN DE BOI DirONG NANG LUC GIAI TOAN CHO HOC SINH TRONG DAY HOC TOAN 6 TRI/0NG TRUNG HOC PH6 THONG PHAN ANH TAI TnrimgallifcSiiGdn Email pAHmM[.]
.-_- t r - i fflNSHItNCDUULUM SUf DUNG TiniNG TIT GiffA CAC BAI TOAN DE BOI DirONG NANG LUC GIAI TOAN CHO HOC SINH TRONG DAY HOC TOAN TRI/0NG TRUNG HOC PH6 THONG PHAN ANH TAI TnrimgallifcSiiGdn Email: pAHmMil@sgu.sdu.n Tdm tit Trong dgy hgc Todn d trudng phd thdng, mdt tiong ede dgng hogt ddng gidi todn id ttf bdi todn cdn gldtl tudng vdi mdt bdi todn tuang ttfdd ed cdch gidi diphdt hifn cdch gidi bdi todn da cha Bil viit di edp din edeh nhdn b Stf tutmg ttf gida ede bdi todn eua mdt sd dgng todn ehuang trinh Trung hge phd thdng di td chdC hogt ddng g fodn O l fim edeh gidi bdi todn hge sinh phdi nhdn biit bdi todn tuang ttf gdm dgng: Bdi todn ed tlnh ehdt ttfong ttfi B todn cd edu true tucmg ttf, Bdi todn cd ddu hifu tutmg ttf khdng tUdng minh Qua dd, ndng Itfc gidi todn cOa hgc sinh (A bdi dudng dgy hge todn d trudng phd thdng Tdkhda: Tuang ttf ndng Itfc gidi todn; hge sinh; trung hge phd thdng (Nh^n bdi ngdy 08/11/2016; Nhdn kit qud phdn bifn vd chlnh stfa ngdy 18/01/2017; Duyit ddng ngdy 25/02/201 I.OJM v i n d l 2J Phin log! tdong td Hogt ddng giii toin (hlOGT) eung vdl vifc sddyng Theo Nguyen Phfl Ldc, tflong ty dflpc chia lim 'vdn' kiln thfle ki ning; ngudi gili toin thudng 'huy logi: ddng' hf thdng tri thfle phuong phip v l hf thdng cie a Tuang ttf theo thudc tlnh: Khi d l u hifu dflpc nit b i i toin d l dupc 'phuong phip hda' d l gili quyet v i n k i t lufn bilu thj thudc tfnh TUcffig ti/theo thufc d l Dgy hpc (DH) Toin d trUdng phd thdng, m f t tfnh ed d u tnie nhusau: cic dgng HEXJT, l i tfl b l i toin d n gili lifn tudng vdi m f t AviBcd cung tinb chil i^.ij,.-,/' bii toin tucmg ty da ed d e h giii d l phit hifn d c h gilL Trong bii vilt niy, chflng tdi d l eip d i n d e h nhln b i l t Ac(itialiehil^_i sutuong ty gifla d c b i i toin cfla mdt sd dang toin chucmg trinh Tmng hpe phd thdng d l td chflc HE>GT Sadd 1: Tuang tutheo thudc tfnh TUtfng tU v i phin loai tUffng t y b Tuang ttf theo quan hf: Khi d l u hifu dUpc ritt 2.1 Khii nifm tuang ttf k i t luin bilu thj quan hf Tuong ty theo quan hf Theo G Polya "Tuong t y l i mdt kilu gidng nio cd d u trflc nhu sau: dd nhung d mflc dd xic djnh hem, v l d mfle dp dupe vl B ti dug lo«i (biy cnagdn t phin Inh bing khli n i f m „ Nhflng ddi tupng gidng c6 quin bi Vlh C phu h ^ vdi tiong mpt quan h i nio dd" Diln giii rd hcjn, tie g i i v i f t "Hai hf l i tUOng ty nlu ehung Sadd 2: Tuang ttf theo quan hf phil hop vdi d e mdi quan hf xic djnh ro N h i n b i l t b i i t o i n tUOng t y d l td ehtfc boat ring gifla nhflng bf phin tucmg flng" [1, tr.20] ddng g l i i t o i n d phuong difn d u tnic D Centner cho ring: 3.1 Cdc bil toin ci tfnh chit tddng td "Tuong tu I I mdt tfl mft d u tnie co sd hoic ngudn d f n Mdt sd b i i t o i n ndi dung khie nhung mft d u tnic khic hay mye tiiu hf thdng" (2, tr.20] VI phupng difn logic Nguyfn Phu Ldc eho ring: "tuong ty chflng cd tfnh chit tuong ty nhau, O l tim cich gili mOt b i i toin; tim hifu, phin tfch g i i t h i l t v i k i t luin cfla bU l i suy luin dd kft luin vi stf gidng cua d e toin niy, ngudi gtii lifn tudng d i n b i i toin cd tlnh chft d l u hifu khic cua eic ddi tupng" [3, tr.82 - 83], Tfllflufn trfn, ehung tdi quan nif m: Tuang tU l i suy tuong ty vdi b i i t o i n d i cho Tfl d c h glii efla nd djnh luin dya vio d l u hifu gidng efla dot tupng mye hudng phit hifn cich giii b i i toin Vf du: Cho tfl difn ABCD cd cic canh AB AC ti*u (dupe suy hofc phit hifn) v i ddi tupng co sd (da dupe bilt d i n hole da hilu) Trong DH Toin, hai vin d l AD ddi mdt vudng gdc vdi Chflng minh dupe xem l i tifOng tu nfu chflng ed cflng tinh chit hay cd vai trd nhu hay gifla cic bf phin tuong flng cfla SflCD = S ^ f l c + ^ 0 + ^ (1) Chung ed mdi quan hf gidng d mdt sd d l u hifu Tfl difn ABCD AB±AC, AC AD ADIAB 6 KMA Hpc GUo DVC -^m N H I ^ COU Ll LUAN C " ^ H l i hlnh chilu efla A tifn mp(SCOJ(Hlnhl) Hf thfle (1) goi sy ' l i f n tudng'tdi Ojnh If Pitago tiong tam giie vudng (ed nhifu phuong phap chflng minh) » ^ Can luu y HS, "lifn tudng" tucmg ty gida tfnh chit cua bii toin htnh hpe khdng gian v l bii t o i n hinh hpc phing cd t h i l i s y i i f n tudmg^ve d e h gili quylt v i n d l Mdt d c phuc/ng phip chflng minh Ojnh li Pitago l i sfl dyng tam giie B'~ ddng dgng Xft tam giic Hlnh vudng ABC dUdng cao AH (Hlnh 2) Tfl hai tam giie vudng ddng dgng AHB v i CAS, tacd AB BC HB~ AB • AB^ = BH3C (2) Tuongtytaed AC^ = HC.BC Cdng (2) v i (3), tacd (3) AB^ + AC^ = BHMC + HC£C = BC^ (4) "Lifn tudng" t ^ phucmg phip tren diy d l chflng minh hf thfle (1), tfl g i i thilt ta ed tam giic EAO vudng tgi A ed dudng cao AH (Hinh 1), ta ct EA'= nfn hay EH£D -BC^£A^=-BC^£H£D > - ! = ( -BC£H\i-BC£D\ dodd S^ (5) Rd ring cd su tucmg ty gifla hf thfle (5) v i hf thfle (2) Hem nfla, d l cd hf thfle (S), ta da sfl dyng hf thfle (2) tiong chflng minh Chflng minh tuong ty, ta ed ^ACD ~ ^HDC-^BCD Vidtf 1: Cho gdc tam difn (Oxyz) v i d i l m G thudc mien gdc tam difn Dyng m i t phing a qua G va l l n luat d t Ox, Oy, O? tgi A S, C cho G l i trpng t i m cua tam giic A5C Cau true cua bai toan khdng gian, g ^ lifn tudng tdi d u trflc b i i toan phing sau: Bdi todn I.-Cho gdcxOy v i d i l m Mthupc m i l n gdc dd Dyng dudng t h i n g dqua M v i l i n lupt ^Ox,Oy tgi A eho M la trung d i l m cua dogn AB Giii b i i toan phing theo eac cich sau: Cacii giii (Kmh 3) Phin tich: Gii sfl da dyng dudng thing dthda man b i i t o i n ' Khi dd ta xem A, S la hai dinh ddi dien eua hinh binh hanh OAO,B v i M l i t i m cfla hinh blnh hanh OAOfi Suy W ' la trung d i l m cua AB v l QO^ Tiidd suy eich dyng: - Dyng OOj nhin M lam Hlnh trung diem - Qua O, dyng eic dudng thing l l n lupt song song vdi Qx d t py tai v i song song vdi Oy d t Ox tai A -Dudng thing phii dyng ddi qua A, B Cich giii (Hlnh 4) Phan tich: Xft phfp ddi xflng t i m M, ky hifu D^ Qua ^ O cd Inh l i O, Theo g i i thilt d lln lupt d t Ox tgi A d t py tgi suy ^ thudc Ox B thude Oy VI A4 l i trung d i l m eua AB suy A l i I n h cfla qua D^ cJo dd A thudc oy l i i n h cfla pyquaO^ Tfl dd ta cd d c h dyng: - Dung 0,11 I n h efla Hlnh OquaO^ - Dyng 0, / I I Inh cfla Oy qua D^, - A l i giao d i l m cfla Ox v i o y , - Dudng tiling d qua A, M Bii t o i n phing tdng quit cua bai t o i n 1: Cho gdc xOy v i d i l m M b i t kl Dyng dudng thing d qua W v i l i n i u o t d t Ox, Oy tai A cho = k>Q MB Tfl d c h giii cfla b i i t o i n gcri cho ta d e h giii b i i toin tdng q u i t Phin tich: Gii sfl M thudc miln gdc xOy x i t c^ +c2 xc= -S S +S S +*s S Viy h i thfle (1) dupe chflng minh 3.2 Cie bil toin cd du trie tuWig td Sy tucmg ty cOa eie b i i toin ed t h i nhin bift duoc d d i u hifu clu true cfla ehung Mdt b i i toin ed d u trflc tuong tu vdl mdt b i i t o i n khic goi cho ngudi gtii lifn tudng tdi phucmg phip giii b i i toin niy d l tim hudng giii b i i toin d i cho phep vj tu t i m M ti sd -it, ky hifu V^' Qua V;,', cd i n h liO, Theo g i i thiet d l l n lucn d t Ox tai A d t Oy tgi suy ' MB cuaSqua V j d o d d A t h u d c O y i i l n h e u a O y q u a V;,* Tfl dd ta cd d c h dyng: - Dyng O, la Inh cua Oqua V„', sti 127-TIUNG 2/2917 67 C Q NGHI£N COU Li LUAN - Dung O y i i Inh cfla Oy qua \,;,', - Dilm A l i giao dilm cfla Ox v i oy - Dudng thing d qua A M Tuong tUM thude miln ngoii gdc xOy, dd ta x i t phf p \n ty t i m Mtisd Jt Xft quan hf tucmg ty gifla d u trflc cfla b l i t o i n hlnh hpc khdng gian vt dy v i cie bai toin v i bli toin tdng quit eua hlnh hpc phing: Bii toin hlnh h^c khdng gian (VfduD Gdc tam di^n (Oxyz) Bil toin hlnh hoc phing GdcxOy G thudc mifn gdc tam M thude miln gdc difn Dyng mit phing (a) qua G Dung dudng thing d qua M v i lln luoi elt OK Oy Oz tai v i lln luot d t Ox Oy tai A AB,C eich giii gidng Hilu cy rd d i l u niy d l t h i y d l u / \ hilu tUcmg ty cua b i i t o i n / ngudi gili phii nhln dupc b i n chit cfla sy tucmg t y mdi djnh hUdng dflng eich giii 3.3 Bil todn ci diu Hinhfi biiu tuang tulthing tddng mlnh Mdt sd b i i t o i n thogt nhln khdng d l thiy sy tuong ty efla nd vdi mdt b i i t o i n da cd d c h giii Trong ttijtjnq hop niy, d l u hifu tucmg t y cua b i i toin khdng du^c tudng minh Phli t h i t "tlnh" ngudi gill toin miA phit hifn d l u hifu tucmg t y (in ting) cua nd vdi mdt bli toin da cd d e h gill, tfl dd djnh hudmg d c h giii cua bii tdn (dd 'tinh"cd dupc nhd rfn luyfn gill nhilu b i i toin) G l i trpng tim ciia tam giac M l i tmng diem eua dogn ABC AB Wdu2:Chohimsd Tfl dd cd t h i thye hifn eic cich giii sau: y =—x^- 2(sin a-cosa)x^- 3(sin 2a)j: +1 Cich gili (Hinh 5): Gpi X,, x^ l i hoinh dd d c dilm eye tri ciia him si, - Xle djnh mft phing (Oy Oz) - Xle djnh mft phing (G Ox) tim cie g i i tri efla a cho Xj+Xj = x^+xl (1) Hai m|t phing niy cd dilm ehung 0, gpi Of I I giao Diy l i mdt b i i t o i n giii tfch, v l tfnh chit hay clu tuyin Clia chung Khi dd, G thufc miln gdc xOt true mdi g i p lln d i u ta khd t h i y nd tuong ttf viA mft bii -Trong m i t phing (G, Ox), qua G dyng dudng thing toin (hay dgng toin) nio dd Nhung nhin kf ta se thiy d elt Ox tgi A v l d t Of tai M cho — = - (diy chinh GA d l u hifu tuong ty n i m d hf thfle J:, + Xj = x, + x] (tUong l i bii toin phing tdng quit) ty theo thufc tinh) Ta lifn tudng d i n b i i toin phuong - Trong miln gdc yOz chfla dilm M, diing dudng trinh bie hai cd hai nghifm thda m i n mft hf thfle dH thing d' qua M d t Oy tgi 8, d t Oz tai C eho M l i xflng v i d e h giii l i sfl dyng djnh If Vi - f t Tfl dd, ta c6 tmng dilm efla BC (diy i i b i i toin ta d l lifn tudng tdi) cich gill b i i toin nhu sau: - Mit phing (a) qua A 8, C Tfl g i i thift x^ x^ l i hoinh dd d c dilm cyc tri cua Cich gili 2:V|n dyng elch gili bii toin phing tdng h i m sd nfn chflng l i hai nghifm phin bift ciia phuong quit vio bli toin khdng gian cua vf dy (Hlnh 6) trinhy' = (2) Phin tfch: Gii sfl da dung dupe m i t phing (a) thda dilu kifn bii toin, gpi M la trung dilm cua BC suy Tacd >'' = 4jc^-4(sina-cosa)j:-3sin2ji: M thufc mit phing fOy, Oz} Theo gii thilt, G l i trpng A' = 4(sina-cosfl)^ + 12sm2o t i m cfla tam gllc ABC suy A i i i n h cfla M qua phf p v| ty t i m 6, ti sd -2, kf hifu V^' Do dd, A tiiufc m i t phing (p) l i Inh cua mit phing (Oy, Oz) qua V „ ' Tfldd tacd deh d i ^ g : - Dyng mp (p) l i inh cua mpfOy, Oz) qua - A l i giao dilm cfla Ox v i mp (p) Tuong ty ta dyng cie dilm 8, C Mit phing (a) qua ba dilm A f t C Chu y Khdng phli mpi bii toin tuong ty thi cd 61 KHM Hpc GUo Dgc Phuong trtnh (2) ed hai nghifm phin bift x^x^khl v i chi A'>0 o l + sin2a>0 vi x,+x^=sma-cosa,x,X2=—sin2a Olt w = sina-cosa = V2sinj a - — vdi -V2 i / S ^ , s u y r a s i n o = l - u ' Khidd, (1) » J:, + x j = ( x , + X j ) ' -2x,Xj N6H|£N COU U LU&N Q hieu q u i giii quyet cic tinh hudng mdi o s m f l - c o s a = (sina-cosa) +—sm2fl C5u^ + u - = TAILICUTHAMKHAO [1} G Polya (Hoing Chflng, Le Oinh Phi, Nguyfn Hflu Chucmg, H i Si Hd djeh), (2010), Todn hoc vd nhdng suy ludn edii, NXB Giio dyc Vift Nam [2] Bui Phucmg Uyen, (2016), Suy ludn fuWig ttf dgy hgc mdn Todn trung hgc phd thdng: Nghiin ctfu Nghifm u = / thda m i n dilu kifn cfla u truimg hgp Phuang phdp tga khdng gian, Luin Dodd, V2sin[fl l = I I n tiln sT khoa hoc giio dyc, Tnidng Dgi hoc Su phgm TP.Hd Chi Minh ;r [3] Nguyin Phu Ldc (2010), Doy hpc h/fu qud mdn Gidi tieh tiong trucmg thdng, NXB Gilo dye Hi Npi ' ^ (k,l G Z) thda m i n dilu kiln (3) [4] Nguyen Bi Kim, (2006), Phuang phdp dgy hge = (2/ + l)ff mdn Todn, NXB Ogi hpc SU phgm, Kft l u i n [5] Bui Vin Nghj, (2009), Vdn dung Uludn vdo thtfc Trong DHToin, GV d n rfn luyfn cho HS v l cie quan tien dgy hgc mdn Todn d tiudng phd thdng, NXB Dgi hpc hf gifla tinh hudng dflpc xft v l "vdn" kiln thfle, kT ning, Su phgm tri thfle phucmg phip, tri thfle eic b i i t o i n da ed d HS [6] €)oin Quy^h (tdng chu bifn) - Vin NhU CUong Ning lyc giii t o i n phy thudc nhilu v i o 'chit lupng" dfl (chfl bien) - Phgm Khic Ban - Tg Min, (2007), Hlnh hge lifu trang "vdn" kiln thfle ki ning, tri thfle phuong p h i p ndng eao 11, NXB Giio dyc m i ngudi gili toin d i tfch luy dupc Hon nfla, chfnh sy [7] Oio Tam, (2004), Phuang phdp dgy hge hlnh hoc tich luy Ilu d i i v l "lypng" d c dfl lieu tgo tiln d l eho sy d truimg trung hge phd thdng, NXB Dai hpc Su pham phit triln tryc giic xic lip d e ' l l f n t u ^ g ' nhanh chdng, USING SIMILAR FEATURES AMONG MATHS EXERCISES TO FOSTER STUDENTS'DOING MATHS COMPETENCE IN TEACHING MATHS AT HIGH SCHOOLS Phan Anh Tai Saigon University Email: phananhtal®sgu.edu.vn Abstract In teaching Mathematics at high school, one of doing Maths activities is to associate that exercise with a similar one with given solution to find out the answer The article refers ways to recognize the similarity among Maths exercises in several forms at high school curriculum in order to organize solving activities Tofindways to these exercises students have to realize similar exercise, including types: exercise with similar features; exercise with similar structure; exercise witii implicit simitar signals Thereby, students' competence to Maths exercise will be improved when being fostered in teaching Maths at high schools Keyvmrds: Similarity; competence to Maths exercise; students; high schools Sil37- 62/2017* 69 ... quit cua bai t o i n 1: Cho gdc xOy v i d i l m M b i t kl Dyng dudng thing d qua W v i l i n i u o t d t Ox, Oy tai A cho = k>Q MB Tfl d c h giii cfla b i i t o i n gcri cho ta d e h giii b i... trflc b i i toan phing sau: Bdi todn I. -Cho gdcxOy v i d i l m Mthupc m i l n gdc dd Dyng dudng t h i n g dqua M v i l i n lupt ^Ox,Oy tgi A eho M la trung d i l m cua dogn AB Giii b i i toan... OAO,B v i M l i t i m cfla hinh blnh hanh OAOfi Suy W '' la trung d i l m cua AB v l QO^ Tiidd suy eich dyng: - Dyng OOj nhin M lam Hlnh trung diem - Qua O, dyng eic dudng thing l l n lupt song