B U6NG DAI HQC DONG THAP ^J Tap chi Khoa hpc s6 16 (11 2015) KHAI T H A C C A C M 6 I LIEN HE GIlfA CAC NOI DUNG MON TOAN VA LIEN HE TOAN HQC Vdl TH^/C TII N NHAM H 6 TR0 HQC SINH PHAT HIEN VA GIAI QU[.]
B.U6NG DAI HQC DONG THAP Tap chi Khoa hpc s6 16 (11-2015) ^JKHAI T H A C C A C M I LIEN HE GIlfA CAC NOI DUNG MON TOAN VA LIEN HE TOAN HQC Vdl TH^/C TII:N NHAM H TR0 HQC SINH PHAT HIEN VA GIAI QUYET VAN âE ã GS.TS Dao Tam*"' Tdm t^t Bdi viet trinh bdy cdc tinh hudng di hgc sinh hogt dgng khai thdc cdc m&i liin h? nhdm phdt ien cdc quy lugt todn hgc, khdc sdu-^nghia cua tri thUc vd phdt hi$n cdch gidi quyet vdn di ly hgc mdn Todn d trudng phS thdng TU khda: Hogt dgng phdt hien, hogt ddng gidi quyit vdn di Md d^u Bdi viet htfdng vdo muc tidu phdt tridn la ndng giai quyet vdn d l nhd luydn tdp cho >e smh ede hoat ddng khai thac cdc mdi lidn ^ bdn giffa cdc npi dung Iddn thffc mdn Ddn, lien he todn hpe vdi thtfc tiln, thdng qua dc sff dung mdt sd thih hudng day hpc gdi lu c l u khdm phd kien thffc mdi Ndi dung I Vide d l cdc htfdng boat dpng Ichai thac c md'i Udn hd giffa cdc npi dung kidn thffc vd :n hd tri thffc todn hpc vdi thtfc tiln nhdm ho phdt hien tri thtfc mdi, ldm sang td J nghia a tri thtfc da dtfdc hpc, cdch gidi quyd't van dtfdc dinh htfdng bdi cac tri thffc then chd't %ddy: - - Tri thffc v l phtfdng phdp luan nhdn thtfc ^n hpc: [2], [3] [7], [8] ^ - Quan diem tich hdp day hpc todn "^n cung nhtf cdc ludn diem v l kd't ndi tri 're U-ong linh vtfc tim tdi hi tud: [1], [4], [6]; t ndi tri thffc todn hpc vdi cude sdng [5] Dtfdi ddy chffng tdi trinh bdy cde htfdng at dpng theo cdc muc tidu ndu trdn: Htfdng 1: Tao cac tinh hudng dd hpc sinh at ddng phat hidn quy lufit toan hpc Y nghia ci5a cde boat dpng theo htfdng 1: tfc hidn cdc boat ddng theo htfdng trdn nh^m St tridn eho hpc smh khd ndng phdt hidn vd'n thdng qua khai thdc cdc mdi Udn hd giffa rUOng Dai hoc Vinh cde kidn thffc ciia cdc phan mdn khae mdn Toan Tff dd gdp phin gidp hpe sinh khde sdu ^ nghia cua tri thffc ding thdi gdp phSn phdt tridn kha ndng dinh htfdng gidi quydt van de day hpc todn Sau day Id cac tinh hudng gdi nhu c^u cho hpc sinh thtfc hidn cac hoat ddng theo htfdng trdn; Hnh hudng 1: Xet hinh vuong cd di$n tich S; phdn hogch hinh vudng ndy thdnh bdn hinh vuong bang cd cdng di$n tich Sj nhd cdc dudng thdng di qua hai trung diem cua cdc cgp egnh ddi Sau dd phdn chia hinh vudng cd dien tich Si thdnh hinh vudng cd diin tich S2 tuang tu nhu cdch phdn chia ddu tiin Tii'p tuc qud trinh ndy ta co cdc hinh vuong ldn lugt cd di$n tich S3; S4; ;St Ta td den cde hinh vudng cd didn tich 5;; S2; ;Si, (Hinh 1) Hinhl Khi dd nhd tinh chd't: Ti sd didn tich eua hai liinh ddng dang theo tis6 k bdng k^, ta cd 29 TRUONG DAI HOC DONG THAP diy o: —^ ' S' S — , chinh la day so J_ J_ _L _ vdi k la so nguyen dtfdng 2''2'''2'''"2C6 the nhan xet tnfc ic tong cic so (Tong dien tich cac hang c^a day trdn b^ng — hinh vuong dUdc to den bing —Sc6 nglua la — diOn tich hinh vuong ban dau) Cd the ki^m chtfng tinh dung din ciia tong tren nhtf sau: Ta goi T li tong can Iim Khi dd 7' = Um- -M' Nhff vay day sd dffPe rut tff viee khai thdc J nghia hinh hpc cua vide xdt day cdc d sd didn tieh Dilu dd ehiShg td ed thd phdt hidn kidn thffc dai sd' nhd sff dung cdc tri thffc tiinh hpe, sff dung cdc hinh bieu diln eua ede hinh liinh hpc Dieu btfdc dau sdng td y nghia cAa viec Ueh hdp kidn thtfc day hpc todn Tinh hudng 2: Xdt xrb chdi sau: Hai ngudi ldn lu0 dgt cdc ddng xu dong chdt nhu lin mdt bdn chd nhdt, mdi ldn dgt mgt ddng xu, ldn lugt ngudi dgt trude, ngudi dgt sau; khdng dugc dgt chdng lin Ngudi ndo dgt ddng xu cudi ciing se thdng Hdy tim chiin luge ddt ddng xu de thdng (Ddt dong xu cudi cilng ed nglua Id ngtfdi tidp theo Ichong cdn eh5 de dat nffa) Hinh2 Dtfdi day Id htfdng ttf phdt hidn chidn Itfdc chdi - quy iudt chdi dd th^ng 30 Tap chf Khoa hoc so 16 (11-2015) - Xdt trffdng hdp ddc bidt: Gia sff mdt bin nhd ddn mtfc chi ddt dtfdc mot ddng xu thdi, dd ngtfdi di dau se thiJng Bdi vi ngtfdi di dau ciing Id ngtfdi di cudi cilng - Ndu gia thuyet: Dd th^ng can phai di dau - Kilm tra tinh diing ddn eua gia thuydt tren- Ngtfdi di dau ddt ddng xu vdo tdm cda hinh chff nhat - tdm ddi xtfng iChi dd ngtfdi thtf hai ddt ba't ky vi tri ndo thi ngtfdi di dau tidn ddt dtfdc dong xu d vi tri ddi xffng tren mat bdn Nhtf vay chtfng ndo ngtfdi thff hai cdn ddt dtfdc ding xu len ban thi ngtfdi thtf nhd't v5n c6 vi tri ddi xtfng dd dat (xem Hinh 2) va nhtf vdy chtfng mtfc ndo ngtfdi thtf hai khdng dat dtfdc nffa thi kdt thuc Do vay ngtfdi di d i u Id ngtfdi ddt dong xu cudi cilng - ngtfdi thang Hnh hudng 3: Yeu cdu hgc sinh gidi thick Tgi ldm cdn cdu, ddm cdu, ddm di trdn nhd lgi ket ndi cdc thip theo hinh tam gidc (Hinh 3) Hinh De giai thieh hidn ttfdng ndu trdn hpe suih phdi huy dpng kidn thffc v l dilu kidn xdc dinh mdt tam giac: Mdt tam gidc hodn todn xdc dinh nhat cho biet dp ddi ba canh Vi vay kdt ndi cde thep theo edc hinh tam gidc thi kdt can se khdng hi bidn dang dtfdi tae dpng ei5a gid ldn, bao to, tdc ddng cd trpng Itfc ldn Htfdng 2: Tao cd hdi dd hpc smh ho?t ddng kh^c phuc cac m§u t h u i n nhdn thffc Y nghia cua htfdng hoat dpng 2: De kh^c phuc man thuan giffa mdt ben Id tinh hudng tri thffc mdi, bdn khdc Id kien thtfc va kinh nghiem da cd cua hpc sinh budc hp phai hoat dpng tim toi cdc kien thffc trung gian dffdc tao TRUCflSTG DAI HQC DONG T H A P nen tff cdc tri thffc da cd nham tao sd dd nhdn thffc mdi ttfpng thieh vdi tinh hudng mdi Tff mau thuin dtfdc kh^c phuc, gdp phan phdt tridn khd nang gidi quydt vdn d l ciia hpe smb C6 thd md ta htfdng hoat ddng trdn qua tinh hudng sau ddy: Tinh hudng 4: Cho tii diin ABCD cd AB = CD = p; AC = BD = q; AD = BC = r Hdy tinh khodng cdch tU dinh A din mat phdng (BCD) Vdi cdc kid'n thtfc da cd sdch gido khoa (SGK) lien quan dd'n tpa dd, tich vd htfdng vd cac cdch tinh khoang each da biet Irong hinh hpc phIng khdng the vdn dung de giai quydt bdi todn trdn Dd gidi bdi todn trdn hpe sinh phdi bidt cdch thidt lap mdi Udn hd giffa tff didn gin deu vd hinh hop chff nhat nhd sff dung cdch kidn tao sau day: Qua cdc cdp canh ddi ei5a tff didn ABCD ta dffng cdc cap mdt phIng song song lan Itfdt chffa cdc canh ddi ndu trdn Ba cdp mat phlng song song ndy ddi mdt cdt tao mpt hmh hop chff nhdt Kilm tra dilu nhd sff dung cdc mdnh d l sau: - Ndu hai mdt ph^ng song song hi c^t bdi mdt mdt phang thtf ba thi giao tuyd'n nhan dtfpe Id hai dtfdng thdng song song - Ndu mdt hinh binh hanh ed hai dtfdng chdo b^ng thi hinh binh hdnh dd Id hinh chff nhdt Tgp chi Khoa hpc so 16 (11-2015) tich cua ttf didn Id V dtfdc tinh theo thd tich cua hmh hop theo cdng thtfc sau: V = xyz-4—xyz = —xyz Cdc sd x\ y; z dtfdc tinh theo p; q; r theo he phtfPng trinh sau: W+/=p' Khi khodng each h tff dinh A dd'n mat phang (BCD) dtfdc tinh theo edng thffc h= — ; vdi S Id dien tich eua tam gidc BCD tinh theo ba canh p; q; r Htfdng 3: T^o cd hdi dd hpc sinh kham pha cac ffhg dung mdi cua cac kidn thffb SGK Muc tidu effa nhffng boat ddng khai thdc cdc ffng dung c i a nhffng khai nidm, dinh IJ, quy tae Id dd kidn thffc SGK xam nhap tdt hdn vao cdc linh vtfc kidn thffc khac cua mdn Toan Tff dd tao mdi Udn he nhilu mdt, da dang cua cdc kidn thffc dtfdc day, tao dieu kidn de hpc sinh Udn ttfdng, huy ddng Icidn thffc gidi quydt van d l todn hpc phong phd hdn Thtfc hien cdc boat ddng ndy se gdp phin dtf tinh quan didm tich hdp day hpc todn Dtfdi day ehdng ta xdt each xdy dtfng cdc quy trinh sff dung dinh 1^ Thales tam gidc d l phdt hidn cdch chtfng minh ba diem thing hdng theo mdt phtfdng phdp mdi Vide xdy dtfng quy trinh dtfdc tien hanh theo cac btfdc sau: Budc 1: Khdo sdt ede trtfdng hdp ridng a Cho tam gidc /iBC Cac didm M, N ttfdng tfng la trung diem cdc canh AB; AC Gpi /, / lan Itfdt la trung diem cua doan MN vd canh BC Hay sff dung dinh 1^ Thales chffng minh ba diim A; /; J thing hdng b Cho tam gidc ABC Dtfdng thang d song song vdi BC elt edc canh AB; AC tai cdc diim Hinh Tff dd ta cd hinh hOp chff nhdt ttfdng ffng M; N Gpi 7; J i l n Itfdt Id toing didm M/iNB.CEDF ngoai tidp ttf didn da cho (Hinh cdc doan MN vd canh BC Chffng minh ba diem 4) Ta ddt MA = x\BM = y; AfC = Khi dd the A; / ; y thing hang 31 Tap chr Khoa hpc so 16 (11-2015) TRUONG DAI HOC DONG THAP ciing vd ding thffc (*) vd hai tia DB va EC cung htfdng ndn C =Cj c^;, Biiih6 Hai trtfdng hpp tren cd cdch lap ludn chung nhtf sau: Gid sff A/cdt BC tai AT (Hinh 5) Bdng cdch ttfdng ttf ta cd quy trinh chfftig Khi dd theo dinh IJ Thales edc tam gidc minh ba diem A; B; C thing hdng gdm ba btflk IM IN ABK v i ACK ta cd: (1) vi cdng sau day: KB~ KC Ve dtfdng thing d qua B cho cdc AI bSng ^ ^ Ttt (1) v i IM=IN suy BK=KC didm A; C ndm khdc phia vdi dtfdng thing d Ve edc dtfdng thing AE song song vdi Tttdd JsK CF; cic diem £; F thudc dtfdng thing d Budc 2: Cho hoc sinh khai qu^t stf ki6n sau: Chtfng mmh = (Hinh 7) Ban Do MN song song vdi BC nfin theo dinh IJ BC CF Thales ta cd: dpc ed thd Idem tra quy tilnh nhd sff dung dinh IJ Thales ttfdng ttf nhtf d quy tnnh AM MN -NM MI AB ' ' BC ' ' BJ Budc 3- Phdt bieu quy trinh 1: Chffng minh ba diim A; B; C thing hdng De chffng minh ba diem A; B; C thing hdng ta thtfc hien theo quy trinh ba btfdc sau: Ve dtfdng thing d qua A cho cdc didm B; C ndm ve mdt nffa mdt phlng cd bd Id dtfdng thing d Ve cdc dtfdng thing BD; CE song song ffinh? vdi nhau; D vd £ thupc dtfdng thing d J1-1,»_ 1, ^ ^ Chung minh - ^ = (•) CElinh AE Cd the kiem tra dung ddn cua quy trinh nhtf sau: Kdo ddi AB cdt tia EC tai C, (Hinh 6) Khi dd theo dinh I^ Thales dp dung cho tam gidc ALCt ta ed = AE 32 Tff ding thtfc eudi EC, Van dung hai quy trinh trdn cd thd giai cdc bdi toan sau day: Bai toan 1: Cho tam gidc /WC Hinh chff nhdt MNPQ ndi tidp tam gidc dd cho M thupc canh AB; N thudc canh AC; cdc dinh P vi Q thudc canh BC Tim quy tich tdm O cua hinh chff nhdt MNPQ Af di ddng tren AB TRUONG DAI HOC e N G THAP T^p chi Khoa hoc so 16 (11-2015) Bai toan 2: Chtfng minh ring mdt K«t lu9n tam giie: Trtfc t i m H; t i m v i giao Tren day chiing tdi x«t c i c hoat ddng khai diim O c i a c i c dtfSng trung trtfc ciia ba canh thic mgi USn h« ben cfla c i c tri thtfc, thuOc mot dtfSng thing ^^^^ ^^ ^^^ ^^, ^^; ^^ ^^^ ^^ ^^^ ^^ ^^ Bai toan 3: Cho tam giie ABC nhon M l i „ „ ^iec phit hi6n c i c quy luit toin hoc v i diem dl dong tren canh BC Goi P; Q lan ItfOt t u i - ^ i.- -.,.„ L',, i» tt 1, k - ' x ;, , khic sau Ji nghia cac tn thtfc thdng qua khai li hinh chieu vuong gdc cua M ldn c i c dtfdng , , , » j Z, , thing /a, AC Ttei quj tich trung diim / cfla * " = t ^^ ' " ' "" ' '"'" ' ^ doanPfi *'''°^"Tai lieu tham khao [1] M Alecxeep v i cong stf (1976), Phdt triin tuduy cua hoc sinh, NXB Giio due [2] Pham V i n Hoin - Trin Thiic Tnnh - Nguyen Gia Cdc (1987), Gido due hoe mdn Todn, NXB Giio due [3] NguySn The NgMa (2007), NhOng chuyin di triet hgc, NXB Khoa hoc xa hdi [4] G Pdlya (1997) Gidi mQt bdi todn nhu thg ndo, NXB Giio due [5] Organization for Economic Co-operation and Development (2003), The PISA 2003 Assessment Framework - mathematie, reading, science and problem sovling knowledge arui skills, Paris [6] D i o Tam (1997), ""R^n luy?n ndng lue chuyin doi ngon ngU thdng qua vi$e khai thdc ede phuangphdp khde gidi ede bdi todn hinh hpe d trudng trung hocphS thdng', Tap chi Nghidn cflil Giio due, (sS 12) [7] D i o Tam (chfl bidn), Trin Trung (2010), TSchUe hoat dpng nhdn thUe day hpc todn cho hge sinh trung hge thdng, NXB Dai hpc Stf pham [8] Nguyen Cinh Toin (1997), Phuang phdp lugn vgt bifn chiing vdi vile hge tdp, dgy, nghien citu todn, tip 1-2, NXB Dai hoc Qud'c gia H i Noi EXPLORING THE LINKS BETWEEN MATHEMATICS CONTENTS AND PRACTICE TO HELP STUDENTS IDENTIFY AND SOLVE PROBLEMS Summary This paper presents situations in which students have to find out relevant links in order to recognize mathematics laws, internalize knowledge significance and find solutions to problems in teaching mathematics at schools Keywords: recognitions, problem solutions Ngdy nhgn bdi: 19/9/2015: Ngdy nhgn lgi: 19/10/2015; Ngdy duyit ddng: 30/10/2015 33 ... p; q; r Htfdng 3: T^o cd hdi dd hpc sinh kham pha cac ffhg dung mdi cua cac kidn thffb SGK Muc tidu effa nhffng boat ddng khai thdc cdc ffng dung c i a nhffng khai nidm, dinh IJ, quy tae Id dd... phIng khdng the vdn dung de giai quydt bdi todn trdn Dd gidi bdi todn trdn hpe sinh phdi bidt cdch thidt lap mdi Udn hd giffa tff didn gin deu vd hinh hop chff nhat nhd sff dung cdch kidn tao... d Budc 2: Cho hoc sinh khai qu^t stf ki6n sau: Chtfng mmh = (Hinh 7) Ban Do MN song song vdi BC nfin theo dinh IJ BC CF Thales ta cd: dpc ed thd Idem tra quy tilnh nhd sff dung dinh IJ Thales