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NGHIEN COU Ll LUAN I REN LUYEN Ki NANG GIAI TOAN CHO HOC SINH PHO THONG QUA VIEC VAN DUNG PHEP BIEN HINH DO TH! TRINH Tnfdng flai hpe Sif pham Dai hoc Tti^l Nguyen Email dDltillrlnli@gmaJI cam Tom tdt[.]
NGHIEN COU Ll LUAN I REN LUYEN Ki NANG GIAI TOAN CHO HOC SINH PHO THONG QUA VIEC VAN DUNG PHEP BIEN HINH DO TH! TRINH Tnfdng flai hpe Sif pham - Dai hoc Tti^l Nguyen Email: dDltillrlnli@gmaJI.cam Tom tdt: Ren luyen kindng gidi todn cho hoe sinh Id mot nhdng viec Idm quan cua ngudi gido vien nham ndng cao chdt luang day - hoc d trudng thdng Trong bai tac gia dicdp tdi viec ren luyen ki ndng gidi todn cho hoc sinh bang phuang phdp bien hinh Viec thuc hien quy trinh day hoc gidi bai tdp todn hoc thudng duac tien hdnh theo bdn bude Tim hieu ndi dung bdi todn, Xdy dung chuang trinh gidi, Trinh bdy Idi giai; Kiem tra va nghiin cdu Idi gidi Qua cdc vidu minh hoa cho thay viec ren luyen cho hoc sinh kindng gidi toan noi chung vd kindng gidi cdc bdi todn hinh hoc noi neng la mot nhdng yeu cau quan trong, cdn thiit doi vdi moi gido viin de gdp phdn ndng cao hieu qua dgy - hoc mon Toan a truang thdng Ttfkhod: Ki ndng; kindng gidi todn; phep bien hinh (Nhdn bdi 28/12/2016; Nhdn kit qud phdn bien vd chinh sda 10/01/2017; Duyef ddng 25/01/2017) I.Datvande Mpt bai toln Hinh hoc ed the giai dupc bIng nhQng each khae Tuy nhien, trudc mdt bai toan mdi ve Hinh hoc, ta nen xem xet lya chon phuong phap giai thich hop de cd ket qua nhanh, gon nhat De giai mdt bai toln Hinh hoc bIng phuong phap bien hinh (PPBH), trude het ta phai nhan duoc dau hieu ciia Idp cac bai toan cd kha nang giai duac bang phuong phap khong, Cau hdi dat la Lam the nao de nhan biet duoc mot bai toan Hinh hoc co kha nang giai duoc bIng PPBH? Cosd cua viec sQdung phep bien hinh vao moi bai loan la dQ kien cua bai toan va tinh chat ciia hinh dpi hoi phai thiet lap (chQng mmh) hole dieu kien ddi hdi d hinh can dung da xuat hien nhQng yeu to cd mdi hen he den mdt phep bien hinh cu the nao dd Muon nhan biet duoe mdt bai toan Hinh hoe cd kha nang giai duoc bIng PPBH, ta phai xem xet, phan tich ndi dung bai toan de tim yeu td ed mdi lien he den mot phep bien hinh cu the Sau dd, van dung cac tinh chat cua phep bien hinh tim Idi giai hay dap sd cua bai toan duoc xet, Oac biet can luu y PPBH thuong gap va sir dung viee giai mot bai toan quy tieh hay bai toan dUng hinh, nhat la bai toan dung hinh Theo quan diem li thuyet tap hop, hinh duoc hieu la mot tap hop diem nao Viec sQ dung mdt hinh hoc nao dd quy ve dung mot sd diem hQu han de xac dmh eung co nghia la dii de tao nen hinh Trong mat phang thdng thudng, mdt diem duoc xac dinh bdi giao diem cua hai diiong, co dudng thang va duong conic ma duong tron la mdt elip dac biet Trong hai duong dung de xac dmh diem phai dung la mot cac giao diem cua chung, thuong thi mpt duong da co san dQ kien cua bai toan cdn dudng thd hai la quy tich ciia nhung diem cd mdt tinh chat dac trQng hinh hoc va dope suy tQ mdt dudng da cho dQ kien bdi mdt phep bien hinh, Bai bao nay, chung tdi de cap tdi viec ren luyen ki nang (KN) gili toan cho hoc sinh (HS) bang PPBH, Ren luyen ki nang gili toan cho hoc sinh phd thdng qua viec van dung phep bien hinh 2.1 Quy trinh dgy hge gidi bdi tap todn hoc theo tutudng cda G.Polya Trong chuong tnnh mdn Toan d trUdng phd thdng, nhieu bai tap toan chUa cd hole khdng eo thuat giai va eung khong cd mdt thuat gi^i tong quat nao de giat tat ca cac bai toan, Chiing ta chi cd the thdng qua viec day hoc giai mdt sd bai toan cu the de dan truyen thu cho HS each thQc, kinh nghiem viec suy nghT, tim toi Idi giaicho mdi bai toan Day hoc giai bai tap toan khdng cd nghTa la giao vien (GV) cung cap cho HS Idi giai bai toan 6iet Idi giai bai toan khong quan bang lam the nao de gill duoe bai toan, vl vay can trang bi nhQng hudng dan chung, goi y cac suy nghT tim tdi, phat hien each giai bai toan la can thiet DUa tren nhQng tUtUdng tong quat ciing vdi nhung gOi y chi tilt cua G Polya ve cach thdc giai toan, phuong phap tim toi Idi giai cho mdt bai toan thudng duoc tien hanh theo bdn buoc sau [1! - Budc 1- Tim hieu ndi dung bdi todn Oe tim hieu noi dung eua bai toan, can chu y cac yeu to cabin nhu Phan biet cai da cho, cai phai tim va cai phai chifng minh, Diing cdng thuc, ki hieu, hinh ve„ de dien ta de bai, Phan biel cac phan khae cua dieu kien, Cd the dien ta cac dieu kien dd cdng thQc khdng' - Budc Xdy dung chuang trinh giai Yeu td quan giai duoc bai toan la viec xay dung chuong tnnh giai cho bai toan Vi vay, thuc hien, GV can chu y Phan lich bai toan da cho nhieu bai toan don gian S013B-THANG1/ED17-63 CQ NGHIEN ClfU Li LUAN quen thudc; Lua chpn nhQng kiln thQc da hpe g i n gui hon c l vdi dQ kiln eua bai toln rdi dU doln kit q u i ; SQ — AB cac diem A, C, B' dyng nhQng phucmg phIp dac thu vdi tQng dang toln lln lupt biln cle nhuchQng minh, toan dung hinh, toan quy tich,, diemC;B,A - Bude 3: Trinh bdy Idi gidi: Trinh bay lai Idi gili sau Do dd, qua phIp da dieu ehinh nhQng cho can thilt, - BQdc 4: Kiim tra vd nghiin ctfu Idi gidi: Kiem tra lai tinh t i l n theo vec to k i t q u i v l cac lip luan q u i trinh gili; Nhin lai t o l n -AB AAC'B' biln thinh bd cle budc gili, rQt tri thQc phuong phIp de gill mdt bai toln;Tim thIm elch gill khlc (nlu cd the); Khai thIc k i t q u i cd t h i cd cua bai t o l n ; O l xuat bai toan tuong tu, b l i toln die bilt hole khai quit hoa b l i toln 2.2 Vidu rin luyin rin luyen kindng gidi todn cho hge sinh thdng qua viic van dung phep bien hinh dudng trdn ndi t i l p eua A>^C'6' bien tam 1^ dQdng Sau day la mdt sd vi dy chung tdi thiet ke dua theo trdn ndi t i l p cua AC'BA' quy trinh bude gili mdt bai toan cda Polya nhlm ren Dodd,tacd / , / , -AB luyen KN gili toln eho HS thdng qua vi^c van dung phep biln hinh HoatdpngA: Kilm tra v l nghiin cQu Idi gili Vidu: Xet bli toln: Gpi A', B', C lln iQpt I I trung 1) HQdng dan HS gili b l i t o l n bIng elch khlc: Cach diem eua ele canh BC, CA, AB eua /^BC Gpi I^, 1^ theo {SQ dung phuong phIp tdng hpp) thQ ty II tam dudng trdn ndi tiep /sAB'C, AB^'C ChQng Do A' B' C lan lupt II trung diem eua cac canh BC, CA, AB cua A,4fiC n i n ta cd A,4C'B' = AC'BA' minh rinq / , / , =—AB a I 2 Do i,, /j lan lupt II t i m dudng trdn ndi tilp cac MB'C, ABA'C nen ta cd Al, // C'l^ va Bl^ // Cl, => AAC Hogtdpng 1:T\m hilu npidung bli toan = ACBIj Hoat ddng cua GV Hoat dong cua HS Dodd,tacd-4/,^C7j - Yeu cau HS ghi gia - Ghi g i l thiet, k i t l u i n va ve hinh => AIIC la hinh binh hinh t h i l t , k i t luan v l ve hinh ( H i n h l ) - Xac d i n h cac yeu t d da eho, khdng doi va yeu t d can chQng minh - C l e yeu t d da cho I I AABC cd A ; B ; C lan luot I I t r u n g d i e m cac canh BC, CA, AB v l I,, I^ lan luot la tam d u d n g trdn ndi tiep AAB'C, ABA'C - Yeu t d can chQng m i n h la ' ^ Hogt ddng 2: Xay dUng chuong trinh gili Hoat dong ciia GV Hoat ddng cua HS - Nhan xet dac diem cua AAB'C; - A A B ' C ' = ABA'C ABA'C? -Tj^, AAC'B'^AC-BA- Do AAB'C = ABA'C n^n t d n tai phep bien hinh nao b i l n t a m giac t h i n h t a m giac khdng? -DoT^ ^AC'B'^AC'BA' 1,1, = A C va I|, I, lan lupt la t a m d u d n g trdn ndi t i l p AAB'C, ABA'C'thi 1,, I, c d I I I n h cua qua phep b i l n hinh nao khdng? - Theo dinh nghia phep t i n h tien, ta co d i n g thQc nao? -Vay m d i quan hegiQa i, L v a A B nhuthenao? LI.=-AB ' ' Hoot ddng 3: ThQc hien chQOng trinh gili Do A', B', C lan luot la trung dilm cua cle canh BC CA, AB nen qua phep tinh tien theo vie to • KHDA HOC GIAO DUG 2) TQ bai t o l n tren, nlu gpi Ij II tam dQdng trdn ndi tiep AA'B'C (Hinh 2) Theo chung minh tren, ta cd: AC = C'B = B'A'^I,I^ ^ AB' = B'C^C'A' = I,I^ BA' = A'C = C'B' = l^ly Dodd AAC'B'= AC'BA'= AB'A'C ••AA'B'C = AI,IJy =>S,,., = S,.,^ = S,.,=S, Dodd,taed5/,7^/^ ^-^ABC- Mat khae, theo chQng minh bai toan gde, ta ed C\ ^B'l^vaCI,=A'l^ Ta cd AC'IJ, = AI^B'A' Do do, ta ed ket q u i b l i toan nhU sau: Bai toln 2: Goi A', B' Clan lupt la trung dilm cda cac canh 8C, CA, AB cua A,48C, Gpi /,, 1^ 1^ theo thQ tQ la ta dudng trdn ndi t i l p cle tam gile AAB'C, ABA'C, ACA'B ChQng minh ring: a)5i - - S ABC 'h'l': NGHIEN CDU Lf LUAN b)AC'lj, = Aip'A' c) [ACS') = [CBA') = [B'AO = Cfl'S'O = [1,1J) 3) TQ b l i t o l n gde, ta cd phep tjnh tien T-^,: D va BCAH II hinh binh hinh nen -Nhan xet vec to AH -Nhan xet v l mdi quan 'AH = 20i (Hinh 4) he giOa dilm A va dilm quy tich H, tQ xle dinh quy - Do r ^ : A^Hv'a tich diem H diem diem A la dudng trdn (0; /?) nen quy tich H la dUdng trdn (0'; Pti A di ddng tren dudng II Inh cda dudng trdn (0; R) qua trdn (6; R) td thudc iAB'C'thlnh y l u to tuong Qng thudc AC'B^: Do dd, neu cle tam gile AAB'C, ABA'C, ACA'B' phep tinh tiln theo verto 201 ta lan lUpt thay cac yeu td /,, 1^ /^ cle ylu td O, O^ lan lupt 11 t i m dudng trdn ngoai t i l p cac tam giac Hoofddng3:ThUe hiln chuong trinh gili nay, ta cd cac k i t q u i tUdng tu nhQ d bli t o l n gde va Gpi C'= CO n (0), gpi I la trung dilm ciia 6C{Hinh 4) Do 01II dudng trung binh cua ACC'B nin BC = 201 bli toan Do dd, ta cd k i t Do BC'AH\h hinh binh hinh n i n ta ed BC = AH qui II bli toln nhQ sau (Hinh 3): => AH = 201 hay IH ^201 Bai toln 3: Gpi A', B' Dodd,taed T~:A^H C lan iQcJt la trung dilm Do diem A thudc dUdng ciia cle canh BC CA, AB trdn (0; R) n i n quy tieh dilm A la cda AABC Gpi 0,, 0^ 0, dudng trdn (0; R] Hinh lan luot II t i m dUdng trdn Suy ra, quy tich diem H II ngoai tilp cae tam giac dudng trdn (0'; R) II Inh eua Hinh AAB'C ABA'C ACA'B' ChQng minh ring: dudng trdn (O; R) qua phep tinh a}>lfi = 20,0, t i l n theo veeto 201 Hootddng 4: Kilm tra v l nghien cQu Idi gili 1) HUdng dan HS tim elch gili khlc: Cach (sQ c) (ACfi") = [CBA-) = {B'A'O - [A'B'C) = {0,0p,) TQ bli toln va bai t o l n 3, ta thu dupc k i t q u i bai dyng phIp ddi xQng tim) Goi / la trung dilm canh BC goi A'^AOr-^ (0) (Hinh toln nhQ sau: Bai toan 4: Goi A', 8', C lan iQpt la trung diem eua cac 5) Do 6H//CA'vlCH//BA'nen taedBHCA'II hinh binh canh BC CA AB cua AABC Gpi O, O^ O, v l (,, 1^ I, theo thQ hinh Suy / la trung dilm eua tu 11 tim dudng trdn ngoai t i l p va ndi tilp cle tam giac doan HA' AAB'C, ABA'C, AO^'B' ChQng minh ring AOpp^ = Ai,ijy =>B-A'-^H Vidu 1: Xet bli toln: Cho dudng trdn (0; R) va BC la Do diem A thudc dudng day eung ed djnh Mdt dilm A di ddng tren dudng trdn trdn (0; /?) va A' = AOr^ (0) nen (0; R) Tim quy tich trUc tam H eua AABC dilm A chay khap dudng De td chQc day hoc cho HS, GV ed t h i hudng dan trdn (0; R) thi A' eung chay khIp dudng trdn (0; /?) Suy ra, quy HS nhusau: tich dilm A' la dudng trdn (0; Hogtdpng l-.Tim hieu d e t o a n Hinh R) Hoat dong cOaHS Hoat d6ng cua GV Suy quy tieh H la dudng trdn {0'; ft) II Inh eua - Yeu cau HS ghi g i l - Ghi g i l t h i l t k i t luan va ve dudng trdn (0;fl) qua phep ddi xQng tam / thiet, ket luan, ve hinh hinh Cach (SQ dung phep ddi xQng tryc) bai t o l n - Yeu t d cd d i n h : O u d n g t r d n ( ; Gpi A^ AH n (O), A, = AH n BC Do - Xac dinh cac yeu t d khong doi, yeu t d co dinh, yeu t d sinh quy tich, yeu t d quy tich (Hinh 4) R)va day c u n g BC - Yeu t d k h d n g d d i : Ban kinh R -Yeu t d sinh q u y tich: D i l m A - Y e u t d q u y tich:TrUc t a m H cua AABC Hoat dpng 2: Xay dyng chUOng trinh gili Hoat d6ng ci^a GV Hoat dong cija HS - Dudodn quy tieh: Cho diem A m o t sd vi t r i dac biet de xac d m h VI tri ciia d i e m H (Hinh 4) - D y d o a n quy tich eua d i e m H d i e m A thay ddi la thuoc loai trdn - Ve t h e m d u d n g kinh CC; Do 01 la d u d n g trung binh cita ACCS ^A^ = CAA, = CBB] nen AHBA^ cd BA, vQa II dQdng phan gile, vQa la dQdng cao Do dd, diem H ddi xQng vdi diem Aj qua dudng thang BC Do diem A thudc dudng trdn {0;R)vaA^=AH (BA'Q hay ta cd (BHQ = thi khdng chi giup ele em hieu rd hOn ndi dung cac kien thQc ve phep biln hinh ma cdn phIt triln dupe nlng lUc (0) gili toln Hinh hoc eho HS, tQ dd gdp phan nlng cao hifu Do dd, ta ed bai toln nhu sau: ' Bli toln 6: Cho tam giac ABC ndi tiep dudng trdn q u i day - hpe mdn Toln d trUdng phd thdng (O) va H Jl true t i m cua tam gile ABC ChQng minh rang (0) = (BHQ = (BHA) ={AHQ TAILIIUTHAMKHAO -TQ Idi gili elch 3, ta cd 0^^ (BHQ -> (BA'Q Suy [1], G,Polya, (1997), Gidi mpt bdi todn nhu thi ndo Dj^ 0, -> 0, vdi 0, 0, lan lupt la tam dudng ngoai HdThuan - BuiTUdng djch, NXB Gilo dye t i l p ABA'C, A6WC Do dd, ta cd / II trung dilm cCia 00,, vdi [21 Tran Vilt CUdng - Nguyen Danh Nam, (2013), I I I trung dilm eua BC Gido trinh Hinh hge sa cdp NXB Gilo due Vilt Nam Vdi vai trd tuong ty, ta ed: Vdi J, K lan luat I I trung [3], Bill Van Nghj, (2009), I'dn dung lilugn vdo thUc diem cda AC, AB v l 0^, 0^ lan lupt II t i m dudng trdn tien dgy hoc mdn Todn d trudng thdng, NXB Oai hoc ngoai tiep AAHC, AAHS thi J, K lan lupt II trung diem eua Supham, Ha Ndi 00,, 00, [4] Bui Van Nghj - Thi Trinh - NguyIn Tiln Trung Do dd, ta cd AB = 0,0^ op^ = BC va Op, = AC - HoIng Ngpc Anh, (2016), Phdt triin ndng Itfc sUphgm Do do, ta cd bai toan nhu sau: cho sinh viin dgi hge ngdnh sUphgm Todn, NXB Gilo due Bai toan 7: Cho tam giac ABC ndi t i l p dudng trdn Vidt Nam (0) v l H II true tam cua tam gile ABC Gpi 0^ 0^ 0^ lan [5], G,Polya, (1997), Sdng tao todn hoc, NguyIn Sy luot II tam dudng trdn ngoai tiep ABHC, AAHC, AAHB Tuyln - Phan Tit Olc - Hd Thuan (dich), NXB Gilo due PRACTISING STUDENTS'SKILLAT DOING MATHS BY APPLYING TRANSFIGURATION METHOD Do Thi Trinh Thai Nguyen University of Education Emaih dothitrinh@gmail.com Abstract: Practising students' skill at doing Maths Is an important task of teachers in order to improve teachin learning quality at general schools The author refers to practising students' skill at doing Maths by transfiguration w steps: Find out content of exercise; Develop solving process; Present solution; Check and research solutions Th illustration, we can see that this practice m general Maths solving and skills at doing Geometry In particular is one of requirements, it is necessary for each teacher to contribute improving the efficiency of Maths teaching at general sch Keywords: Skill; skill at doing Maths; transfiguration 6 • KHOA HOC GlAO DUC ... die bilt hole khai quit hoa b l i toln 2.2 Vidu rin luyin rin luyen kindng gidi todn cho hge sinh thdng qua viic van dung phep bien hinh dudng trdn ndi t i l p eua A>^C''6'' bien tam 1^ dQdng... (2016), Phdt triin ndng Itfc sUphgm Do do, ta cd bai toan nhu sau: cho sinh viin dgi hge ngdnh sUphgm Todn, NXB Gilo due Bai toan 7: Cho tam giac ABC ndi t i l p dudng trdn Vidt Nam (0) v l H II true... p AAB''C, ABA''C''thi 1,, I, c d I I I n h cua qua phep b i l n hinh nao khdng? - Theo dinh nghia phep t i n h tien, ta co d i n g thQc nao? -Vay m d i quan hegiQa i, L v a A B nhuthenao? LI.=-AB