IkfUvaHpC JNGAYNAY so 3 2012 ''''iri''''^ii i ii'''' in ii 17 Sir dung phifdng phap dien tfch de ren luyen, phat trien nang lut suy luan va chuhg minh cho hpc sinh tieu hpc L3( u> luan va chung minh la nang lu[.]
IkfUvaHpCs o 3-2012 JNGAYNAY 17 'iri'^ii.i.ii'.in.ii Sir dung phifdng phap dien tfch de ren luyen, phat trien nang lut suy luan va chuhg minh cho hpc sinh tieu hpc TS NGUYEN MANH CHUNG LE THI HOA L A M THI NGA TrUdng Dgi hpc Hong DQc u> luan va chung minh la nang luc ca ban L3(*cua nguoi hoc va lam toan, Trong d^y hpc can, CO the hieu nang lire suy luan va chung minh la kha nang van dung cac phep suy luan de tien hanh chung minh mot menh de toan hoc hoac giai quyet mot van de nao cua toan hoc, Ren luyen, phat trien nang lye suy luan va chung minh la mot nhiem vu quan trong cong tae day va hoe Toan a truong thong Viec ren luyen, phat trien nang lire suy luan va chung minh se giup hgc sinh (HS) nhan thuc the gioi ,\ung quanh mot each khach quan, bien ehung Co rat nhieu bien phap ren luyen, phat trien nang luc suy luan va ehung minh eho HS, nhien, su dung phuong phap dien tich (PPDT) de giai cac bai toan Hinh hoc eo rat nhieu thuan Igi viec ren luyen, phat trien nang luc suy luan va chung minh eho HS Tieu hgc, Khai niem vc phuong phap dien tich PPDT la mgt phuang phap giai toan, diing de giai cac bai toan ve dien tich ma khong su dung tr\rc tiep eong thuc ve tinh dien tich cae hinh PPDT la ea sa cua viec giai bai tap toan ve eat ghep hinh Ren luyen, phat then nang lure suy luan va chirng minh cho HS qua vice sir dung PPDT de giai cac bai toan Hinh hoc & Tieu hoc De ren luyen, phat trien nang luc suy luan va ehung minh eho HS Tieu hge thong qua vice giai cae bai toan Hinh hge bang PPDT, giao vien (GV) can chu trgng viec xay dung he thong bai tap nham ren luyen, phat trien nang luc suy luan va ehung minh cho HS tren ca sa eae tinh chat hinh hge sau: 1) Neu mgt hinh duge chia cac hinh nho thi tong dien tich cua eae hinh nho bang dien tich ciia hinh Ion ban dau; 2) Neu ghep cae hinh nho lai de duge mgt hinh lan thi dien tich hinh Ion thu duge bdng tdng dien tich eua eae hinh nho dem ghep lai; 3) Neu hai hinh co dien tich bang cung bot di mgt phan chung thi hai phan lai cung eo dipn tich bang nhau; 4) Neu ta them vao hai hinh co dien ti'eh bang mgt phan chung thi ta nhan duge hai hinh co dien tich bang nhau; 5) Khi so canh day khong doi thi dien tich va chieu cao ciia mgt tam giae la hai dai lugng ti le thuan; 6) Khi chieu cao khong doi thi so dien tich va so canh day la hai dai lugng ti le thuan; 7) Khi so dien tich khong doi thi so canh day va chieu cao eiia mgt tam giae la hai dai lugng ti le nghjch; 8) Neu hai tam giae co chieu dai canh day va duong eao bang thi dien tich cua chung bang nhau; 9) Neu hai tam giae eo dien bang va eo so canh day bang thi eo chieu eao bang nhau; 10) Neu hai tam giae co dien tich bang va eo chieu eao cung bang thi eo so canh day bang , Tren ea sdf nhung tinh chat tren, GV can ehu trgng viec xay dung he thong bai tap nham ren luyen, phat trien nang lire suy luan dien dich, suy luan tucmg tu, cdc phep chung minh trye tiep, gidn tiep cho HS Mot so vi du ve vice ren luyen, phat trien nang luc suy luan va chung minh cho HS qua vice sir dung PPDT de giai cac bai toan Hinh hoc if Tieu hoc Bai viet chi tap trung vao van de khai thdc mot bdi todn vd^tim nhieu lai gidi cho mot bdi todn, de thay duge y nghTa quan trgng eua van de ren luyen, phat trien nang lire suy luan va chung minh cho HS 3.1 Khai thdc mgt bdi todn Vi du 1: Xet bdi todn sau: Bdi todn 1: Cho tam giae A13C, tren BC lay M cho BM = MC, N la di^m tren canh AC eho NC = 2xNA Keo dai MN cdt BA tai P Chiing to rdng AP - AB 18i SO 3-2012 iijfe«a-i DmvoHc - _ — JNGAY Nt Bai giai: Noi BN i:\\ ta c6: S|.UM =" SMPC (Vi BM = MC v;\ chung cliicii cao tu P) SBMN = SMNC (Vi BM = MC vii chun^ chieu cao tu N) Do SpuM • SHIMM '^ SMI'C • S MNC, tuc la S PUN = S PNC ( I ) Lai cd S PNC " X S AI'N (Vi NC = x NA va cluing chieu cao tir P) (2) Tu (1) va (2) co: x S AI'N = S PUN = S AUN Su\ AP = PB (Do tam giae co chung chieu cao h^i tu N c6 dien tich bang nen da> bdng nhau) b) Tinh ti so dg dai PN va PB Thay Joi \ i tri cua M N la cd bai toan sau: Bdi todn 2: Cho tam giae ABC co AB = 2cm, tren BC lay M cho BM =^ x MC N la diem tren AC cho AN = 2xNC Keo dai MN cdt BA tai P a) Tinh AP Hudng dan giai: Tuang ti,r bai I ta chung minh duge S BCM = 2XSMCA va S BPM ~ 2XSMPA- SfPN ~ b) So sanh PN vdi NM, 3XSNPA ; SDPC = 3XSAPU Tu dd, so sanh duge S MI, va Spoc va tinh ti so dp dai PN va PB, Huang dan giai: Bai loan 5: Cho tam giae ABC, tren BC la> M cho Tuang tu" bai i ta chung minh duge 3x S PNC = S PBN va dua vao cac tinh chat, dac diem cua tam giae cd MC = x BM, N la dicrn tren canh AC cho AN = 3xNC dien tich bang de giai bai toan Keo dai MN cdt BA tai P biet AB = 6cm,Tinh PB Bdi todn 6: Cho tam giae ABC tren AB lay M Thay doi vj tri cua M, N ta cd bai toan sau: Bai todn 3: Cho tam giae ABC, tren BC ISy M sao cho BM = 3xMA N la diciii tren canh AC cho eho MC= 2xMB, N la diem tren AC cho AN = NC = x AN K e o d a i N B c a t M C t a i P 4xNC.Keo dai MN cdt BA tai P a) So sanh SAPB va S MV b) Tinh ti s6 dp dai PM va PC a) So sanh S APM va SMPC b) So sanh AB vdi PB Nhu vay, tu mgt bai loan chi ean iha> doi vi tri cac diem ta duge cac bai toan khac nhung each Hudng dan giai: _ giai lai lien quan den nhau, dieu na> giup HS cd kha Tuang tu" bai I ta chung minh duge S APM ndng vgn dgng de khai thac mgi bai toan nhdm hieu sau 4xSMPcvaAB = 8xPB sdc bai toan cung nhu ren luyen tu logic cho HS Thay doi vj tri cua M, N,P ta cd bai toan sau: /'/ du 2: XL'I bdi loan dud Bai tudn -/: Cho tam giae ABC, tren AB lay M Bai todn I: Cho tam giae ABC, tren cac canh AB, cho MB = x AM, N la diem tren canh AC cho AC lan lugt ISy cac 6\km E, D cho AB = x AE; NC = x AN Keo dai BN cdt CM tai P AC = X AD Cac doan thdng BD va CE cdt tai O, a) So sanh S ABC va SpucAO cat BC tgi M Chirng minh rdng BM = MC Giai: AB = X AE; AC = X AD nen ta co: ^AHI) - ^ACE ~ T ^ /!ô(ã ( V ^ ^i:iiO ~ ^IXO ( ^ } ' Tu (1), (2), (3) va (4) suy S^^^^ =S^m-, tir dd suy duang cao xuat phat tu B den OA bang duong cao xuat phat tQ C dSn OA, do S,MOH 'MOC MB = MC DmjvaHoc SO 3-2012 ^NGAYNAY Tu ket qua bai todn ta cd: chung minh dupe M triing N dd ba doan thdng AM, BD vd CE ciing cdt t^i O AB = AEx3 Tu nhung ket qua tren, neu thay doi ti so d gid thiet eua bai toan ta se ed nhifng ket qua tuang tu kha thu vi Nhu vay, tir mgt bai toan chi can thay ddi mgt so diem gia thiet hoac ket luan cua bai toan goc ta duge cac bai toan khde nhung each giai lai lien quan den nhau, dieu giup HS ed kha nang van dung cac phep suy luan dien dich, suy luan tuang tu de khai thac mgt bai toan nhdm hieu sau sac bai toan cung nhu ren luyen tu logic, tu sang tao cho HS 3.2 Tim cdc cdch gidi khac ciia mgt bdi todn Xet bdi todn: Cho hinh thang vuong A BCD vuong gdc d A va D, day be AB = 12cm, day lan CD = 16cm, canh dai AD = 8cm, M la mot diem tren AD each D la 2cin Tu M ke dudng thdng song song vdi day hinh thang cat BC tai N Tinh dien tich tu giae MNCD, biet MN vuong gdc vdi AD * Phan tich: Do MN song song vai AB va CD nen tur giae MNCD la hinh thang do de tinh di?n tich MNCD ta cd hudng: Huang thic nhdt: tinh dp dai MN Hudng thu 2: dien tich MNCD bang tong dien tich tam giae * Hudng thir nhat ta cd cae each giai sau: Cdchl: VI MNCD la hinh thang nen dien tich tam giae NCD la: 16x2:2 = 16(cm ) Vi ABMN la hinh thang nen dien tfch tam giae ABN la: 12x (8-2):2 = 36(em^) Dien tich hinh thang ABCD la: (12+16) x8:2 = 112 (cm^) Dien tich tam giae AND la: 12 - (16+36) = 60 (cm^) 20 so 3-2012 l».', Chieu cao MN ciia tam gidc ADN la: 60x2:8 = 15(cm) DivMi tich hinh thang MNCD la: (15+16) x2:2 = 31 (cm^) Cdch 2: Tir B ke dudng thdng song song vdi AD cdt MN tsii 11 va eat DC tai E tu N ke Nl vuong goc vdi DC Noi N vai i;, Tacd:DE = Mll = A B - 12em BE = AD = 8cm E C - 16-12 = 4cm; NI = MD = 2cm SBI:C = ECx BE:2 = 4x8:2 = 16 (cm^) SNI;C- - IX^- 1N:2 = 4x2:2 = 4(cm^) H D SuNii=16-4=12(cm') Chieu cao NH cua tam giae BNI: Id: 12x2:8 = 3cm B A Kf y\ ( ; • E ][ SMNCD = (a+16) Do dai MNIa: 12 + 3= 15 cm X 2:2 = a+16 (cm') SAnNM = (12+a) X (8-2):2 = 3x (12+a) (cm^) Dien tich hinh thang MNCD la: (15+16) x 2:2 = 31 (cm^) Md SABCD ~ SMNCD + SABNM Cdch 3: nena+I6+3x(I2+a)= 112=>a= 15(cm) Dien tich hinh thang ABCD la: (12 + 16) x 8:2 = 112 (cm^) Dipn tich hinh thang MNCD Id: 15+16 = 31 (cm") Gpi dp dai MN la a, ta cd: * Hudng thu hai ta cd cae each giai sau: Cdch 4: Tinh nhu each ta cd: SADN = 60cm^ y sd eua MD va AD la: 2:8 = 1/4 nen: SMND= 1/4 SADN => SMND= 1/4,60 = 15 cm^ Dien tich hinh thang MNCD la: 15+16 - 31 (cm^) Cdch 5: Dien tich tam giae MDC la: 16x2:2 = 16 (em*) Dien tich hinh thang ABCD la: (12 +16) x 8:2 = 112 (em^) Dien tich tam gidc BMC la: 112 - (16 + 36) = 60 (cm^) AM = BH = - - c m ; MD = CE = 2em BH gap CE so lan la: 6:2= (lan) nen SBMN =3XSNMC do: SBMC=4XSNMC Dien tich tam gidc MDC la: 60:4 = 15(cm^) Dien tich hinh thang MNCD la: 15+16 = 31 (cm^) Vdri hudng suy nghT ma cd tai each giai kliac nhau, mdi each giai deu van dung cac kien thue da hgc mgt each linh hoat Viec giup HS tim cae hudng giai mgt bai toan cung nhu tim nhieu each giai cho bai toan la rat quan trgng, giup HS ren luyen tu logic, sir sang tao Trong day hgc Toan d Tieu hge, viec ren luyen, phat triln nang luc suy luan va chung minh cho HS la mgt muc tieu quan trgng, de dat duge dieu dd, GV cdn c6 phuang phap hgp ly va su dau tu ky ludng, nhdt la viec bdi duong HS gidi Tren day, bai viet dd gioi thi^u mpt so vi du ve ren liiy?n, phat trien ndng \\xc suy lugn va chirng minh cho HS Tieu hgc thong qua giai cac bai toan Hinh hge bdng PPDT Qua dd, cho thdy GV can quan tam tdi viec boi dudng ndng lyre suy luan va ehumg minh cho HS Tieu hge thi PPDT la mgt cong cy kha hOru hieu dl dat duge myc tieu Tai iifu tham khao: Tran Dien Hien, Bdi dudng hoc sinh gidi todn Tiiu hgc, NXB Dai hge Su pham, H 2009 Do Trung Hipu, D6 Dinh Hoan, Vu Duong Thyy, Vu Qudc Chung, Phuang phdp day hoc mon Todn a Tiiu hoc NXB Dai hgc Su pham, H 2009 Sdeh gido khoa Todn 1, 2, 3, 4, 5, NXB Giao dye, H 2009 ... ndng \\xc suy lugn va chirng minh cho HS Tieu hgc thong qua giai cac bai toan Hinh hge bdng PPDT Qua dd, cho thdy GV can quan tam tdi viec boi dudng ndng lyre suy luan va ehumg minh cho HS Tieu... loan dud Bai tudn -/: Cho tam giae ABC, tren AB lay M Bai todn I: Cho tam giae ABC, tren cac canh AB, cho MB = x AM, N la diem tren canh AC cho AC lan lugt ISy cac 6\km E, D cho AB = x AE; NC =... PB Bdi todn 6: Cho tam giae ABC tren AB lay M Thay doi vj tri cua M, N ta cd bai toan sau: Bai todn 3: Cho tam giae ABC, tren BC ISy M sao cho BM = 3xMA N la diciii tren canh AC cho eho MC= 2xMB,