tn6i iMoiiliJ BOI DUeNG NANG LttC V A N DUNG T O A N HOC CAO C ^ fl^ GlAl TOAN PHO THfiNG VA N A N G LUC GIAI TOAN SO CA> CHO SINH V | £ N N G A N H SIT PHAM TOlN R NMriiiMtiir Tdm TAT ^ Ning luc da s[.]
tn6i-iMoiiliJ BOI DUeNG NANG LttC VAN DUNG TOAN HOC CAO C ^ fl^ GlAl TOAN PHO THfiNG VA N A N G LUC GIAI TOAN SO CA> CHO SINH V | £ N N G A N H SIT PHAM TOlN R-NMriiiMtiir Tdm TAT ^ Ning luc da sinh viin su pham biin abindi quan bgng bong viic phit biin ning lpc cho hpc M ttong pud binh day hgc BH viit binh biy vi mit si minh ti da ning Ivc vin dung toin hgc cao dp dim toin phi bring vi ning ivc gtdf lodn sa dp da slab van nginh su pham ding btdi chra cdc biin phla di bdi dtiOng dc ning Ivc di Tikhia: NSng Ive, toin hgc cao dp, biin phi bidng,toinsa cip, supham M i diu n i bap gim ed nhing nil dung, nhing hoat d ^ Yiu ciu chuyin dii cdn bin, todn diin gido cin thue Win vd cd nhl>ng tinh huingttpngdi cdc dgc nin gido dge tt> ehi yiu nhim bang bf kiin hoat ding dupe diin ttiic sang phii ttiin phim chit vd ndng Ivc nguii Dva vdo chuin diu ttlnh d i dai hpc khil hpe, biit vin dgng tti Uiie vdp gidi quyit cdc vin ngdnh Sv phpm dip tap gidp vlin Tmng hgc pii d i ttivc Bin dii hii nguii gido viin phii ttiay dii ttiing; chuin nghi nghiip cua gide vlin Tmng ^ ta# duy, cich tiip cin qud tinh gldp dgc Qga d i phi thing vd edc nghiin c i u Hin quan Ching t£ cin xdc djnh vdttangbj chp sinh vlin nhing ndng d i xuit mil s i ning Igc die ttii cin phdi dat dupe' Ivc nghi nghiip ddp ing yig ciu dii mil nil trin cOa sinh vlin dai hpp su phamttidnnhu sau; Ndng Bii vii sinh viin su pham h)in ngpdl cdc ndng luc ive vin dgng ttidn hpc cap cip d i giii ttidn pU Chung v i gldp dgc vd i^y hpc hp cin c i cdc ndng ttiing; Ndng Igc tiin hinh qud ttlnh day hpc loin; Igc riing gin vii Tein i ttuing phi fliing Ndng ivc thiit ki vd si> dgng phupng Binttongdsy T i dd di xdc djnh vd bii duing ning Ivc v$n dgng hpe Tpdn; Ndng Iue nghiin c i u vd thiit ki bdi iff, todn hpe cao cip di glilttidnphi ttiing vd ndng Ive Ndng Iue t i chie heat ding ngppi khia Ipdn; Utn) gidl todn sa cip che eia sinh viin ngdnh su pham Ivc giii hjdn so cip; Ning ivc vjn dgng kiin ttlOettpng qud trinh day hpc kjdn hpe vie thgc tiin; Ning luc ddnh gid kit qui Mit s i vin d i v i ndng Ivc v i ning Ivc cia hpc tip eia hpe sinh Bieo djnh huing phit trSn' ndng Ive; Ndng Ivc nghiin c i u khpa hpc gidc due; sinh vlin au pham tpin Ndng Igc Id mil khdi niim khd ttthi tugng cua Ndng ive day hpe tich hpp; Ndng ivc day hgc phSn tdm ly hpe Khdi niim ndng Ive dvpc hiiuttieonhiiu hia; Ning ivc phdt Iriin chupng trinh luc nghla khdc Tiin ttii giii, cing nhu p Viit Ndng Ivc 1dm viic Irin cic ciu tnic (khing glan) Nam, nhiiu nhd nghiin ciu gidp dgc hpc hiin dai ttidn hpe tr£m lupng dang rii quan tdm din vin d i phdt ttiin ndng Igc, Thsp cdc nhd Tdm 11 hpc, "Ndng luc Id mil Thdnh t i c i a ning Ivc vin dgng tpdn hpc cao l i hpp phic tap nhing Biuic Bnh Idm II cia mil cip d i gidl tPin p h i thing v i ndng Ivc glil ttiln nguii, phi hpp vii nhing yiu ciu cua mil hoat s v c i p 3.1 Thinh t6 cia ning Ige vin dgng toin ding nhit djnh, ddm bdo chp hppl ding d i diin ci kit qui" (Pham Minh Hpc, 198g) Tuy nhiin, hgc cao cip di glil loin phi thing Xic lip mil quan h i glia chupng trinh toUi Iheo John Erpenback, "Ndng Ive dupe xdy dung IrSn cp s i t i ttiic, Biiit lip qua gidttj,ciu ttie nhu phi thing vil kiin ttiic ttidn dupe ttang bj ttoK B cic khd ndng, hlnhttidnhqua ttii nghlim/ cung truing su pham e i qua kinh nghlim, hiin ttive hia qua y chP (Jphn S i dgng mil s i kiin ttiic cia Tpdn phi ttiAig Erpenback,1998) Hpdc De Keteie™ chp ring- d i minh hpa chp cde khii niim eia Tcdn hpc cao •Ndng ivc id lip hpp cie H ndng tdc ding lin cdc cip, Tein hpc hiin dpi nil dungttongmil leal Bnh huing chp ttuie d i glil S i dgng cdc khdi nlim cia Tcdn cap cip li quyit edc vin d i dp Bnh huing ndy ddt ra' cing cg d i nhin nhin Tpdn phi ttiing ttieP quaiv Nhu vjy ning luc Id mil khdi niim tich hpp diim ttiing nhit, diy d i vd siu s i e Su dgng kiin thic Tpin cao cip d i gidi ttiU • Tiw*no * hflc Hftnfl etc mil s i kiin thic ttpng chupng ttlnh Tpin Jii 36>TiiaiiwliiLteUoDvc ^ r Xir Bling ding ttioi chinh xdc hpa mit s i kiin Blue ggi] TPdn phi Bling -mu '•^ii til,, *»» USl ^ « y*'p edc bdi ttidn ttwng tv vd bdi ttidn ting qudt, sdng tpp cdc bdi !» mii; c i kiin ttiic chuyin sdu d i bii duing hpc ttidn mii i^i sinh glil C i ttii k i m i l s i cdch ttiic, duing H i Bling hia cdc dang vd cdc cdch gidl chp sing tap nhing bilttidnmil nhu sau; Vin dgng hai if Ong dang bii ttidn, due kit vd blil cich ttang bi W i n ly ccr ban cua Triit hpc vdt biin ching I I nhing tti ttiic phuong phdp ttfpng ing vii mil nil id nguyin ly v i mii Gin h i vd nguyin ly v i sv phdt dung day hpc »! Ndng iue luing ttuie nhing khin, sai am ttiin; Vdn dgng suy ludn c i ly v i suy ludn kigle- Vin K cia hpc sinh ttong qui ttlnh linh hii nhing kiin dgng cic cip pham tti v i cil chung vd cdi riing bdn chdt vd hiin trpng, nil dung vd hlnh ttiie Biic,ttpngqudttlnhgidl lodn; Vin dgng tdm ly hpc liin ttring, phuung phdp ,Vin dgng cdc quy ludt ttlit hpc vdp vide sdng giai ttidn cua tdc gid G Pplya; Thing qua day hpc if tao cdc bdi todn chp hpe sinh ching minh edc djnh II Todn hpe, day hpe gidi cie tf"-ratal tstii 37 tn£N-TMOB({lt>di tdp todn luydn tdp cho sinh iridn cdch t»ln dii chung Gido vidn cd thi t l chi>c cho hgc sinh Ham hrcmg doong nhin nh$n ^nh 11 b^ todn theo nhiiu cdc bdi todn mfli thflng qua cdc hoat dflng sau: cdch khdc din din cdc cdch chi>ng minh, gidi - Xdt ttuflng hyp tam gidc SAB diu canh a Ta todn khdc Tir dfl luy$n @p cdc cdch huy dflng doyc t^i todn sau: Idin ttiirc khdc cho sinh vidn Khi ttiyc hi$n Bdi todn: Cho A SAB diu canh a M Id ttun bi$n phdp ndy d n quan tdm cdc dfli toyng quan h$ dilm cua AB, mflt dofl^g thing d bit kl cIt cfc fating bdi todn doyc xem xdt cdi d$t ttang cdc mfl canh SA, SM, SB lln loyt tai A', M', B' Cht>ng mnli hinh khdc nhau; Ching han xem h> di^n Id bfl phdn ^ 1 ^ cua hinh hflp, h;j' theo cdc kiai h> di#n d l cfl cdc '^"S '^'*"SB'-"'SM''' teai hlnh h ^ toang i>ng ngoai tiip nfl 0#c bi|t chd Y A , , A, i j, ^ i ttyng diin dat cdc djnh II, cdc bditodnttieocdc cdch 'JA \!^"^ '''^ "^ ^"^ d di qua di4m^ toong doang toongttilchvfli cdch gidi khdc ^ °' f f ? '^! *° „L Ching han khaittidcmil lidn hd gi[>a cdi lidng, , f *' *°^"= ^ho A SAB Gpi M Idttungdilm cdi Chung dirge nhllu tdc gia nghidn ci>u vdn dyng ^\^°; ""^^ T * " ^ *^^"9 *^ ™* '^' ^^^ A d t cdc vdo day hyc khdm phd 'Khdm phd phdt ttiln tCr "^"^ ^l, *^' '^"' °"" ^^^'^^ """^ ^"^ mflt t)di toin ttidnh nhiiu bdi h>dn mfli ttieo quan ^^^ diim mil cdi riing nimttpngnhiiu cdi chung khdc nhau* vd Tim til cic kiin ttiic mil, bdi ttidn mii - Xdt tnj^ng hyp dotmg ttiing d dl qua ttpng t i nhiiu ttving hpp riing Biep hr Bring nhiiu cil tdm tam gidc ta cfl ttil ddt bdi ttidn: riing dupe bao ttim bii mil cdi Chung cdi ting Bdi todn Cho tam gidc ABC Mflt dodng ttili^ qudr" "Mil cdi riing e i ttii Id ttuing hpp die biil d bit kl d t cdc c^nh SA, SB lln li>yt tai A' B' SM cia nhiiu cdi chung khdc nhau" vd "Mil cdi chung, dem ddc biil had ling b i phin khdc nhau, bing Co ^^^,3 SA' SB' nhing cdch khdc s i chp nhiiu cdi riing khic nhaui*i a, Ching minh ring duing ttiing d luin di qua Qua vide Bip cin phupng phdp day hpc khdm mil diim ^ u c i ,djnh.^ , phd, mii liin hi biin ching cua cip pham tti "Cil "• ™ ' ' ™ ^ ''' "" ' ^ " S '™"9 " ctio riing vd cil chung" vd ling hpp y kiin cia cdo tie ' " , ' " ^ ^'^^ ^*'^' "''* " • ' = ' " - " gid ching lii xin niu mil s i djnh huing nhim T" " '^'^ ''*' '"*" * * " '* '^' '""S- " ' * * khdm phd phdt hiin cde kiin ttiie ttiing qua day hpe " ' " l ' * " ' * " '''*"9 trung diim cua duingttiingAB chii di hinh hpc; Dv dpin phdt hiin djnh huing '* " " ' '*"' '^ ''* ^ * * " ' 'eng ttpng Idm lam iii giii cdc bdi tcdn hlnh hpc ttiep ttr ttring di t i * ° ' '^'\* "*" " ' * " ' < " " "^"S' '="' 9lic, t i * ttuing hppriingdin ttuing hpp Chung liyttuing "*'>''9sv khdp sdt ? ! " ' ' " ' 9'*° ABC, mit mil phing (o) cil cdc wnh mil lip hpp dii lupng sang mil Idp nhi han - hav =*• SB, SO, SG lin lupt tai A', B', C, G' Ching •hi m A t ,.Al k _ _ -1 , -1 ' chl mil diittrpng- chia ttpng mit tdp diu; Khdm phd, phit ttiin ICr mit bdi lodnttidnhnhiiu bdi toin mil theo quan diim mit cdi Hing nim ttong nhiiu C- minh ring; CD CVT ,.r- £±+.SS.^.^,,^ Bdi todn niy e i nhiiu huing giii; s i dgng Bdi ™ ? ^ " S ^ ^ i r t i * •=*' * " " * " ^ "*= blil hp4thoptt>ngbi phin khdc ta ttiu dupe r S ^.^ i ^ l r * ' "^ " ' * ' "^"5 "^ ""• "'^ " " ' chua cdc bdi loan VI dg Cho ASAB Gpi M id ttung diim cua S ' "sJf'^JS ' ^ J ' h " ''^' " ^ •*"= '=""' ^ !.M, SB ldn lupt tpi A, M', B' Ching minh ring ^ * ~ ~ "1 SB' SM' "^ P * "^^"S khil hinh hpc khing gian di gUt cing cg vie to; s i dgng Bii Bch vv Ta tiip tgc xem bdi todn ndy id cil chung, gHo Viin t i chie cho hpc sinh phdt hiin bdittidnm« Bling qua edc heal ding: • ^ * ' » " » " SABC diu ta dupe bdi ttidn; Bil todn Cho t i diin SABC diu cpnh a Gel Q l i ^"9 Iim lam gidc ABC, mil mil phing (a) t * "'' • - * » - - — — _ _ _ cde canh SA, SB, SC, SG lin lupt tai A', B', C, S Bdittidnndy c i cdch gidl; k i ttiim cde duing _ S ' Ching minh ling — + — + M n g phg qua A vd B spng spng vii duing Uiing d SA' SB' s •SSG' rii su dgng dinh 11 Taiet; s i dgng cing cg diin tich • Xit mit phing (a) di qua dinh A cua t i difs oe chimg minh; su dgng kiin ttiie vecttr ta e i bdi tcdn; Niu ta xem kit qui bdi todn ttin i i mil cdi Bil tPin Cho hinh chip SABC Gpl Q Id ttgo) U-TuciigoliiittUoiigc i-i tn^-TMOBlIl ^'!^iI^«°L^ ABC m i l m i l phing (a) qua A d t edc cann SB, SC, SG ldn lupt tai B', C, G' Chung minh D i ttivc hiin dupe cdc hoat ding t i n ttpng qud tlnh day hpe giang viin cin luu y nhung vin ã^"'^ "Đ-Đ'-Đã ã * - ^ ^ dvng mit s i Bnh huing che sinh viin lip °> 8i4 tri khang dii v.v Tip duyit cho sinh vlin khai Bide ttim ndng * ^ "*= * ^ ' ^ **'' H ^ ">*' lien h i glia kiin Sdch gjde khoatodnphi ttiing bing cdch ting qudt 1^*^ * ° ^ hpc cao c ^ vdtodnhpc phi Bling hpd m i ring tt> m i l bii ttidn Bidnh chuii cdc bdi tiinttinhhlnh Bldnh vd vin dgng kiin Bluetodnhpc ttidn cao cip VI dg 2: Tu bai toin, Ching minh ling ^ ^ u '^'* » i tii sung hi ttiing bdi tip niu Id Idm ving ttin nil Bip tam gide ABC ttil; oic bii giang ttidn hpc cao dip nhim tdng cuing i,M*blB+cJc.On C i ttii biin dii (1) ttrpng J _ _ _ ^ dupng vil -r^JA*-rJ>jB+^rtJC=H2l, (Trong d i , , K i I,, K ^ r Id bdn kinh ving ttin nil t i p lam gidc ABC) Khi di (2) lupng dupng v i i S|i4+S,ffl+s,/C-6 (3), vii S,; S,; S, id diin tieh cde tam gidc IBC- ICA lAB •fidiinliieicbiltodnlSn^mdlphinp^ ttpng khing gian Ching hpn: "Niu id d i i S S tu ttupng i ttuing dpi hpe v i edc Bii Nhing bdi ttidn, djnh II cua hlnh hpc cae cio hi|n cg Ihi cia n i i t u i n g phi Biing ttiep quan ttiuing dupe phdt biiu che Wiing glan n - chiiu diim biin ching s i gip phin giip hp y ttiic dinh ting qudt Dp d i Mil la xit n = n = ttil nhino buing gidi tpdn phi Biing bing ttr ttring tpdn hpc bdi todn, djnh U d i tti Bldnh bii todn dinh 11 pi1= cao cip, todn hpc hiin dpi vd sau d i chuyin tdi hlnh hpc sp eip sang cdch giii phi ttiing Trin cp s i 11 luin d i , niu HS dd biit mit dinh HI L ' ^ t M " * ° "H * ' ' / ' « " ] « * b huing 11, mil biltodnttengmil phing (ttte Id ing vil n = glil c*: bdi l'*" Ket £^t"S'i;r'titnp"kS^"''*'?.'^*^«'^'*" qud ttrpng t r teng khing gian (ing vil n = 3) sia-nuiniaiii.jii tnfii-iiuslii •^ CQng chua dung hodn hidn cic bit biin cua nhim Seminar l i m i l hlnh ttiuc hpc lip, nghiin p), biin dii trpng i>ng ritttilchhpp vii phuong phdp hpc lip, nghiin cOa Nhlhig ktldi niim cua hlnh hpc cap cip c i Bii o bdc cao ding, dai hpc vi n i phdt huy dupcUi ting quit ttr khdng gian - chiiu lin khing glan ttch cue heat ding cua sinh viin vd phupng g,] 3-chiiu Id: ddp tpp cua nhd hiring, c i tdc dgng phdtttiinM • Bpn hlnh; Tam gide »• T i diin hii vd hlnh Bldnh phuong phdp 1dm viic khoa In cho sinh vlin • Hip; Hlnh binh hdnh » Hlnh hip Hinh vuing >• Hlnh lip phupng Hpc tip, nghiin eiru Bloc hlnh ttiic semia Hlnh c h i nhdt > Hinh hip chi nhit ttiec ding nghla s i tao sp ttii, say ni • SJiu ciu; Duing t i n »- Mit c^u nghiin ciu Bm hiiu chdn ly vd c i tdc dgi^ lifa • Phuong Bch cua mit dilm dii vii mil siiu ciu: din sv sing tao nhl>ng phuong phdp, 1dm cho i M Phvongttchcua »-Phuong Bch ciia v»n t u i n g thdnh cd v i lip t u i n g khpa hgc at mil diim dii vii mil diim dii vii bil duing ning Iue tu phi phdn, y chl kitnti mil duing ttin mil mil ciu bin bl, die tlnh thin ttpng vd khiim lin, dfc biii & mic d i cao hon, t i djnh 11, bdi hidn eua hlnh Id phim chit trung ttiue wM kit qui vd mpi nguji Ching han, ehi d i sau ddy cua hlnli hot hpc so cip, ci ttii ting qudtttidnhdinh II, bdi ttidn ing cOa hlnh hpe cap cip bing cdch ndng s i so cip bie dai hpe e i ttii tiin hdnh Olio iula chiiu tCr n =2 n = lin Bldnh si chiiu n ting qudt nhim vd seminar chp sinh viin; Vin di xiy dm vdttiaycic khdi niim cua hlnh hpc so cip bil khdi hinh hgc i tniing phi thing niim cia hlnh hpc cao cip; Cdc chi d i c i thi t i chie Blip ludn nhim vl • Tam gidc, h> diin ^ Don hlnh seminar eia nil dung niy Id; • Hlnh binh hdnh, hlnh hip > Hip - Ching minh m i l s i djnh 11 bing cdch suy bta • Duing ttin, mil ciu k- Siiu ciu Ipgie dva vdp h i tiin d i Hilbert; • Phuong Bch cua r Phuong Bch cia - Tim vd phinttchBii hiin cia hi ttin di Img mil diim dii vii mit mil ciu dii vil Tein i ttuing phi Biing; mil duing ttin diim mil siiu ciu, - LI Bluyil v i phip dp dopn thing vd II fliuyil Sau Ihu doge bditodntrong khing glan n - vi.diin tieh Hlnh hpe Eudlde; chiiu, su dgng cdc bit ding ttiic ling qudt vd edc - Cie hlnh hpc phi Eudlde Win ttiic cia khing gian n - chiiu di biin dii bdi Cdc ehi d i cia nil dung ndy s i giilp sinh vHn lodn vi cdc bdittidnting qudt mii, V i i cie bil todn nim dupe co s i eua Hlnh hpe day i tui»( mil niy, cho n = vd n = ta si duoc cie bii todn phi thing, rin luyin cic than tdc tu dinjIMt Iim quen vil phuong phdp Bin d i , chlnh xdc H i hlnh hpc so cip miittrpngOng [6], Vldg4: Xit bdi Ipdn "Chp tam gide ABC nil dupe khdi niim hlnh hpc tiring phi thing, tiip duing ttin Idm Duing tung ttiyin CC, pit Cdu lac bi khpa hpe Id m i l hinh Oiile hpc lip, duing ttin tpl diim ttii hal D Ching minh ring nghiin ciu c i tdc dgng m i ring vd di sdu Bm A CA" t CB" = CC, CD", ta ci bdi todn m i ring mil vin d i khoa hpc Irong chupng ttlnh ddo t^a chott>diin; "Cho t i diin ABCD nil ttip hlnh ciu Hpat ding edu lpc b i khpa hpc tao cho sinh i«ii s;0;R) Duing ttung tuyin DD, (D, Id tpng tim ndng ivc hpp tdc, tinh ttiin ning ding, tdc phonj Un cua tam gIdc ABC) cit mit ciu tal diim ttii hai E viic khoa hpc N i glip sinh viin nam ving nr"~ nil dung cin Biiil thing qua nhing din ching Chung minh ring DA' + 03^ + DC*= DD DE" hpa hip din, sinh ding Ding ttiii dam din ' T i bdittidnphing ban diu vd kit qui'cia bdi viin nhing hiiu biit mii mi vdttivcl i r todn tu ttong khing gian ta c i gii ttiuyit Nil dung hoat ding edu lac b i e i Ihi id: Tin »ng quit sau; Trong khing glan Eucllde n-chiiu hiiu siu v i lodn phi ttiing nhing vin di chuyii e cho n-don hlnh vii cde dinh A , A A , A sdu cia todn Win dai;ttiigiii vd hi Uiing hia tt* phi ttiing; hii thio, ttanh luin cic vin d i phgc w cil siiu ciu lai diim ttii hai B Ching minh i?ng cho idm bdi tip lin, I t,'