!4 SO 7 2011 DOUVAHQC —« « JSGKY \AY Thiet VA cac bai toan nho de day hpc sinh tieu hpc giai mot bai toan TA NGOC TRI Trfldng DHSP Ha Ndi 2 A Ldi gidi thipu Con trai tdi hien dang hpc Idp 4 nam hpc 20[.]
!4 SO 7-2011 DOUVAHQC —«.« JSGKY \AY Thiet VA cac bai toan nho de day hpc sinh tieu hpc giai mot bai toan TA NGOC TRI Trfldng DHSP Ha Ndi A Ldi gidi thipu: Con trai tdi hien dang hpc Idp nam hpc 2010-2011 Khi chdu tudi vi tdi di lam nghien cflu sinh nen chdu cung tdi sang Anh de hpe Chdu da vao hpc ludn ehUdng trinh hpe ky eua Idp (Year 1) mdt trfldng tieu hpe ben Anh (ben Anh hpe sinh vao Idp d nam hpc dau tien sau sinh nhat lan thU ndm, mdi nam hpe eua hp thUdng chia lam ba ky) Khi hodn thdnh chUdng trinh nghien cflu sinh va trd ve nude chau tdi bat dau vdo hpc chUdng trinh hpc ky Idp cua chung ta Thfnh thoang bay gid chdu hay nhd tdi giang giai eho bai todn nay, bai toan np, rdi bdi giang cija ed gido chdu day tren Idp Chi'nh vi vay tdi ed dilu kien tiep xue vdi toan d bae tieu hpe nhieu hdn Vd tdi quyet djnh Idm mdt nghien cflu ve day todn d bae tieu hpc, cu the Id cho hpe sinh Idp Cau hdi cua tdi la "Khi day hoe sinh tieu hge mdt bai todn, ed the Id mdt bdi toan khd ddi vdi hpe sinh, Idm the ndo di ngudi thay ed the giup ede chdu?' Trong cau hdi "hpc sinh tieu hod' dupe hieu Id hpc sinh Idp (9-10 tudi) "Bai toan" hoac "bai toan khd" ddi vdi ede chdu nghien cflu dfldc hieu Id mdt bai todn ma ede chdu gap khd khan hpc tap, cd the dd Id mdt bdi todn ma cde chdu ehua giai dupe sau ed gang de Idm, eung ed the la mdt bdi tddn xuat hien cde de thi hpe sinh gidi ma ehung ta, vdi tu each Id "thay" nghien cflu nay, song khdng Id gido vien dUng Idp tha'y can phai giup dd ehung vi tha'y rang day khdng phai Id bdi todn ma da sd hpe sinh ed the giai dfldc Cde em can sfl giup dd cua ehung ta Trong nghien cflu tdi da thfle hien vdi mdt hpe sinh, dd eung Id eon trai tdi nhfl da de cap d tren Chdu Id mdt hpc sinh dfldc danh gia ket thue Idp la hpe toan d tdp tren Idp Ldp eua chau, 4A1 -Tidu hpe Xuan hda APhuc yen-VTnh Phuc, theo danh gid eua ed gido day Pham Thu Hien, la mdt Idp cd du cde ddi tUdng hpe sinh: tfl gidi, khd, den trung binh ve todn Nghien cflu tien hanh tfl dau nam hpe Idp (thdng 9/2010) den thdi diem viet bao cdo (thang 1/2011) Bdi bdo ndy chfnh la ket qua cua nghien cflu tren, la nhflng dieu qua thfle te tdi da diJng de day chau eung vdi nhflng ed sd ly thuyet cija phUdng phdp md tdi dd sfl dyng Mye dich: thdng qua bai viet ndy.vd cde bdi viet.sap tdi tdi mudn gdp mdt tieng ndi, cd cd sd ca ly ludn va thfle tiin, de gdp ptian xay dflng nhflng each day va hpe toan d bdc tilu hpc dat hieu qua giao due tdt nha't B Polya (1887-1985): Ong Id ngfldi My gdc Hung-ga-rl (Hungaria), tren thfle te Id mot nha toan hpc thuan tUy Ong nghien cflu rd't nhieu eae ITnh vuc khac eua Todn hpc nhU: Ly thuyet Sd, Hinh hpe, Giai tieh To hdp, Xac sud't, Nhieu ket qua nghien cflu ve toan da dfldc gan vdi ten tudi cua dng nhfl Djnh ly dem Polya Td hpp Bat dang thfle Polya-Vinogradov Xde suat, Nhieu giai thfldng eao quy eua edng ddng todn hpe the gidi mang ten dng nhfl: giai thfldng Polya eua Hdi Todn hpc Cdng nghiep va Qng dyng Hoa ky (SIAM) vdi mot ngfldi Viet Nam Id GS Vu Hd Van da doat giai gan day (2008), giai thfldng Polya eho bdi viet xua't sac tren Tap chi Toan d Dai hpe (College Mathematical Journal) eiJa Hiep hdj Todn hpc ^^y (MAA), hay giai thUdng Polya eua Hdi Todn hpc Hodng gia Ludn ddn (LMS) (Dpc them d [M]) Tuy nhien Polya cung rat ndi tieng tren the gidi vdi b6 sdeh ban ve day va hpc todn Bd sdeh dd da dfldc djch rat nhilu thfl tieng (15) vd cho den ndm dng qua ddi (1985) da dfldc bdn dflpe hdn mdt trieu cudn (xem [M]) Bd sdeh cOa dng da dupe djch tieng Viet; dd la eae cudn "Giai bdi toan nhUthe nao?', "Sang tao toan hod'.va "Todn hoc vd cacsuy luan cd ly' Cd the ndi cac tuf tudng v l giai toan vd day hpc todn ede cudn sdeh tren cua Polya cd anh hudng rat Idn den cde nha gido due hpe mdn todn (xem [S]) Se khdng qud ngae nhien neu chUdng trinh giang day ve Giao dye hpe Todn (Mathematics Education) d mot dai hpc cija My, Anh, Uc, lai yeu eau ngUdi hpe phai dpc hoac nghien cflu eae cudn sdeh eua Polya Ngay tfl hpc phd thdng va sau dd Id Dai hpc SU pham Todn, tdi da cd djp dpc cde cudn sdeh Nhu ede tde gia dich cde cudn sdeh gidi thieu ban djch tieng Viet eae cudn sdeh tren eua Polya tu ban tieng Nga Khi cd dieu kien hpe d nude ngoai tdi da de y va may man dfldc dpc ban gdc tieng Anh eua cac cudn sdeh dd Chinh vi vay bai viet trfeh ddn sdeh cua Polya tdi se trfeh tfl nguyen gdc va sau dd djch van tdt ndi dung sang tieng Viet C Quan diem cilia Polya vdi d^y va hpc toan: Day todn, theo Polyd (xem [P]) Id " teach to think" (day each nghi) Chinh vi vdy khdng phai ddn thuSn Id ta day hpc sinh DauvaHpc SO 7-2011 JSCAY SAY ldi giai eCia bai toan Tide rang vi nhieu dp lue; chang han ve thdi gian, thi cfl, ndng lue, cho nen nhieu chflng ta thudng chf dflng lai viee day cho hpe sinh Idi giai cua bdi todn dd Todn hpe Id mdn the thao eua tu eho nen ed rat nhilu ed hdi cho ngUdi thay cd the giup eho hpe sinh cija minh ren luyen sfl sdng tao Neu gap mot bai todn phai day eho hpc sinh, nhat Id nhflng bai todn khd ehung ta chf dpc bdi todn len eho hpe sinh rdi day cho cde em Idi giai, tham chf hfldng ddn cde em giai thi v i n chfla du Phan Idn cac bai todn khd ddi hdi ngfldi thay phai ed ndng Ifle phan chia thdnh eae bdiioan nhd hdn de giup eho hpc sinh ed nhflng "nen tang" nghT ve bai todn can giai dd Trong [P], Polya eung eho rang "If there is a problem you can't solve, then there is an easier problem you can solve: find it."; tfle la Neu cd mdt bdi todn ma ban chfla giai dfldc, thi ed bdi todn d i hdn ma ban ed the giai" IX huyen, thj xa dfldc lay tfl trang web http://dethi.violet.vn/ Bai todn 1: Cho day sd vdi cac ptian tfl nhfl sau 1, 5, 9, 13, , Hay xem sd dflng d vj tri 2000 la sd nao? Thay cho Bai toan chung ta se thiet ke vf dy mdt Bai todn 1' eho dich cua ehung ta vdn la giai dflpe Bai todn nhien Bai todn 1' se giup cho hpc sinh cd ddng Iflc de giai Bai todn Bai toan 1': Cho day sd 1, 5, 9, 13, ,.Chung ta hay 1.1 Tim quy ludt cua day nay? 1.2 Sd 13 day tren dflpe tao tfl phan tfl dau tien eua day tren (sd 1) nhfl the ndo? 1.3 Sd d vj tri thfl 6, 8, 10 cua day sd tren Id sd nao? 1.4 Sd dflng d vj trf thfl 2000 eija day sd tren la sd ndo? Nhfl tren da de cap den Bai toan 1' khdng phai la nha't; Bdi todn 1' d tren chf la mdt vi dy ve mot cdeh thiet ke d l ed the day dflpe Bai toan Neu hpc sinh cd hpc Iflc yeu Cung theo Polya ([P]) "The student should acquire as hdn, thi thay eho 1.1 ta lai ed the thay jnhait'^^^y bang ede much experience of independent work as possible But if cdu hdi phy de hpe sinh cd the phdt hien quy luat cua day he is left alone with his problem without any help or with unsd sufficient help, he may make no progress at all"; tarn djch Trong nghien cflu ma tdi thfle hien chdu da biet dfldc Id "Hpc sinh nen de dpc lap Idm cac edng viec nhieu nha't ed edu tra idi eho 1.1 song lai khdng biet la se bat dau giai Bdi the Tuy nhien, neu chf ed hpe sinh vdi bai todn md khdng todn nhu the nao Tdi da bat dau bang cdu hdi 1.2, rdi 1.3 cd giup dd gi hoac giup chfla du thi hpc sinh dd ed the chang de giup cho chau cang hieu them dupe quy luat phdt tien bd dfldc gi" Tfl quan diem cho tha'y ddi vdi da sd hien mot sd cua day nhd sd dau tien (1) Tfl dd tao sfl hpc sinh, viee "giup dd" eua ngfldi thay mdt cdeh "vfla du" pha'n khich va tfl tin eua tre de tim dflpe sd eua day dflng d cde em hpc todn Id mdt viec can thiet va quan trpng vj trf thfl 2000 Polya eung cho rang "The teacher should help, but not too Bdi todn 2: Cho day ede sd tfl nhien 1,2,3 n Tim n much, not too little", tfle Id "Thay nen giup, song khdng qud , de sd cac chfl sd eua day tren gap ba lan n? nhieu vd eung khdng qud ft." Ddy Id diem quan trpng vi the Ddi vdi hpe sinh Idp thi day la mot bai toan qua khd; ndo Id "khdng qud nhieu" vd "khdng qua ft"? Theo quan khd tfl ngdn ngO the hien bdi todn dd trd di! Trong diem cua chung tdi dieu phy thude vao nhieu yeu td vd trfldng hdp eua tdi thi chdu tfl dau cho biet Id chau khdng cd mdt each ehung eho mpi Idp hpc sinh Cd t h i vdi khdng hieu bai toan nhfl the nao nfla Chfnh vi vdy chdu Idp hpe sinh ndy ngUdi thay thdy the la vfla du, khdng td het sfle chdn nan vd khdng ed chut ddng Iflc nao de tim qud nhieu vd khdng qud it Song, eung vdi bai todn dd sang cdeh giai bdi todn dd Idp hpe sinh khde, ft nha't vi trinh dp khde nhau, tfnh vUa du Thay eho bai todn dd tdi da thiet ke Bai todn 2' de chdu dd lai khde di rdi Dilu khang djnh lai vai trd quan trpng cd the tiep can dfldc Idi giai cua Bdi todn cua ngudi thay, ndng luc sU pham cua thay qud trinh n Bdi todn 2': Cho day ede sd tfl nhien 1, 2, day vd hpe todn ndi rieng vd day hpe ndi ehung 2.1 Neu n=9 ta ed day 1,2 Ddy cd bao nhieu chfl sd? So sdnh vdi n(=9)? Chuyen sang ddi tUpng hpc sinh tieu hpe, cy the ta ban 2.2 Neu n=99 ta cd day 1, 2, 99 Day cd bao d bdi Viet Id hpe sinh Idp (9-10 tudi) Viee giup dd nhieu chfl sd? So sdnh vdi n(=99)? Neu mudn sd eae chfl sd eho khdng qud nhieu vd khdng qud it eho cde em hpe cua day ga'p ddi n (99) thi edn thieu bao nhieu chfl sd todn lai edng quan trpng Ddc diem tdm ly eua cde em d lfla tudi Id rd't d l chdn va rd't can sU dpng vien, khuyen khich nfla? Vi lai cd sfl thieu dd? tfl edng viee mang lai Da sd ede em se tap trung ban dau 2.3 Hay cho n Id mdt sdcd baehOsd (100, 101, ) neu mudn sd ede chfl sd ciJa day gap ddi n thi ehung ta can gap mdt bai todn mdt chut vd neu nhdn dflpe nhflng ket them bao nhieu sd ed ba chfl sd nhfl vay? Vdy n=? de sd qua kha quan chung se ed tdm ly phan khdi de tidp tyc tim hieu ve bdi todn Neu bai todn khdng thay ed tien trien cde chfl sd cua day 1,2,3, n gap ddi n? Cdn gid tri ndo vdi phut phan Idn cde em se khdng mudn tiep tyc nfla khac eua n hay khdng? Chfnh vl vdy day mdt bdi todn cho hpe sinh tieu hpe, 2.4 De sd edechfl sd eua day gap ba lan n thi n phai Id ngfldi thay can ed kha ndng chia bai todn thdnh ede bai todn sd cd may chfl sd? Tim n? nhd, tflng bflde de "gpi y" cho hpc sinh di den edi dich 2.5 Hay giai bai toan trfldng hdp can sd cac chfl sd cudi cung Id bdi todn ma ngfldi thay vd hpc sinh mudn day eija day 1,2,3, , n gap bdn lan n? ga'p nam lan n? vd hpc Bai toan la bai todn sd 25, [N], ddy la bai toan khd vdi da sd hpe sinh Idp 4-ldp Tdi da edng D M$t sd vidM cu the: Bdi se bdn ve viec xay dflng viec hfldng d i n tdi, mpt hpc sinh Idp tfl chd khdng mdt sd cde vf dy cy the ehfldng trinh todn tieu hpc Idp hieu vd hodn todn chdn nan khdng biet se bdt dau tfl dau Cde vf dy trfeh ddn tfl tdi lieu [N], cde bdi tap ndng eao ma dpc bdi toan nay, sau dd chdu dd cd the hieu vd tfl minh cd gido Pham Thu Hien, Trfldng TH Xuan hda A-PhUe yengiai dflpe eae bai todn dgng 2.5 VTnh phfle da sfl dyng giang day tai Idp 4A1, nam Bdi todn 3: Cd ba xe chd hang Xe thfl nhat chd dflpe hpc 2010-2011 Cung cd vi dy Id d l thi hpc sinh gidi cap 2B ^ ^ 3750 kg Xe thfl hai chd dflpe nhieu hdn xe thfl nha't 1/2 tan hang Xe thfl ba chd dfldc hdn mfle trung binh cua ea ba xe la 100 kg hang Vay xe thfl ba ehd dfldc bao nhieu kg hang? Con tdi eung da rat lung tung vdi bai todn nguyen gdc Tdi da chia nhd nd va thiet ke Bdi toan 3': Cd ba xe chd hdng Xe thU nha't ehd dfldc 3750 kg Xe thfl hai ehd dfldc nhieu hdn xe thfl nhat Vz tan hang Xe thfl ba chd dfldc hdn mfle trung binh cua ea ba xe la 100 kg hang Hdi 3.1 Xe thU hai ehd dUde bao nhieu kg hang? 3.2 Neu ea ba xe deu ehd dupe nhU xe thfl ba thi tdng sd hang ba xe chd dflpe se hdn so vdi thfle te la bao nhieu kg hang? 3.3 Sd hang ma xe thfl nha't, xe thfl hai ehd dfldc vd sd hang ehd hdn dflpe d eau hdi 3.2 gap may lan sd hang ma xe thfl ba ehd dUde? 3.4 Sd hang ma xe thfl ba chd dfldc Id bao nhieu kg? Bang cdeh ve ca sd (dung doan thang) chdu da tim dupe edu tra Idi eho 3.3 vd tfl dd d l dang tim dflpe eau tra Idi cudi cung cho bai toan ban dau Bai todn 4: Khi viet them sd 12 vdo ben trdi mdt sd tfl nhien ed hai chfl sd thi sd dd gap len 26 lan Tim sd ed hai chfl sd dd? Chdu da rat thi'eh tra Idi theo cde cau hdi md tdi thiet ke sau day, tfl dd d i ddng van dung de Idm cde bdi todn tfldng tfl Bdi todn 4': Khi viet them sd 12 vao ben trdi mdt sd tfl nhien ed hai chfl sd thi sd dd gap len 26 lan Hdi 4.1 Sd mdi nhan dfldc hdn sd eu bao nhieu? 4.2 Tfl de bai thi hieu cua sd mdi va sd eu gap bao nhieu lan sd cu? 4.3 Sd eu (ban dau) Id bao nhieu? TUong tu nhu bdi todn tren vdi Bai toan 5: Khi viet them chfl sd vdo ben phai mdt sd tfl nhien ed ba chfl sd thi sd dd tdng them 4106 Tim sd ed ba chfl sd dd Ta ed Bdi todn 5': 5.1 Neu ta viet sd vao ben phai mdt sd ndo dd thi sd mdi nhan dupe trUdng hdp so vdi sd eu thi nhu the ndo? (can thiet thi se hdi Id gap bao nhieu lan sd cu?) Trong trUdng hpp ndy hieu gifla sd mdi va sd cu se gap bao nhieu lan sd cu? 5.2 Viet them chfl sd vao ben phai mdt sd tfl nhien cd ba chfl sd thi sd dd tdng them 4106 Tim sd ed ba chfl sd dd? Vf dy cudi eung sau day eung da giup eho tdi tim dflpe ldi giai cho mot bdi todn khd ddi vdi cde chau Bdi todn 6: Mot thang nao dd cd ba ngdy thfl Tfl deu la c h i n Hdi Chu nhat cudi eung eua thang dd Id ndo? Thiet ke sau da td cd tae dyng giup hpc sinh hfldng dan em tim tdi Idi giai Bdi todn 6': 6.1 Neu thfl Tfl tuan Id mot ngdy chdn thi thfl Tfl tuan sau la ngdy le hay chdn? 6.2 Neu cd ba ngdy thfl Tfl la chin thi thdng dd ed thfl Tfl thfl nhat eua thdng ed the la ngdy le dflpe khdng? Ld ngdy chin dflpe khdng? S67-2011 LkfHBHpc 6.3 Neu mdt thdng cd ba thfl Tfl Id ngdy chdn thi thfl Tfl dau tien la ngdy mung ma'y? Thfl Tfl cudi eung la ngdy bao nhieu? Chu nhat cudi cung Id ngdy bao nhieu? E Ket luan: Khi day hpc sinh, nha't Id hpc sinh tieu hpc lam mdt bai todn ndo dd ma hpc sinh gap khd khan, chung ta can thiet ke lai bdi todn dd Cde thiet ke dd khdng phai la nha't va nd phy thude vao nhilu yeu td, dd ed trinh dp eua hpe sinh vd dac diem tdm ly eua chung Mye dfeh eua thiet ke dd la chia nhd bdi todn ban dau cac cau hdi cd tfnh chat gdi y, du de giup dd hpe sinh, "khdng qua nhieu" vd eung " khdng qud ft" nhfl quan diem cua Polya Viec giup dd dd se cd tdc dung tao dd tdm ly cho tre, dpng vien va khfch le tre qua trinh hpc tap Nhd van ngudi Phdp A France, giai thfldng Nobel vdn hpe nam 1921, da tflng ndi "Nine-tenths of education is encouragement"; ed nghTa la "Chi'n phan mUdi eua gido dye Id sU khfch le" Dieu eho thd'y sU ddng vien, khfch le quan trpng nhUthe nao gido dye NhOng cdng viee giup dd vfla du cua chung ta, du Id thay, cd gido, cha me hoac ngfldi thdn vdi eae chdu tren thfle te se cd ti'nh quyet djnh rat Idn den sfl thdnh edng eua qua trinh day vd hpe, vd giao dye eae chdu Tde gia bdi viet eung thay rang viee thiet ke cde bdi todn nhd de day va hpe mdt bdi todn cy the nhfl tren hodn toan dp dung dflpe tren Idp, gid chfnh khda Cdeh day va hpe dd eung ed the dp dung eho cde Ifla tudi khde cua hpc sinh tieu hpe, tham ehi la hpe sinh trung hpc cd sd ho$c trung hpc phd thdng Tuy nhien vdi cde em d Ifla tudi tieu hpe viee edng thay can thiet hdn vl nhflng dac trflng het sfle rieng ve tdm ly cua hpe sinh d Ifla tudi nay, Ifla tudi ma cde chdu chf can biet "an ngu, hpc hdnh Id ngoan" ALdi cam dn: Tdc gia xin trdn trpng cam dn ed gido Pham Thu Hien, TrUdng Tieu hpe Xuan hda A-Phue YenVTnh phfle ve eae bai todn tfl gido dn vd cde trao ddi cflaco; edm dn va chfle Ta Le Ngpc Sdng, ngudi da tham gia vdi tfl cdeh Id ddi tupng cija nghien cflu ndy, nhflng dieu tdt dep nha't dfldng hpe van vd sfl nghiep cua minh Tai lieu tham khdo [M] http://histsoe.stanford.edu/pdfmem/PolyaG.pdf [N] Nguyin Ang (chu bien); Toan boi dudng hpe sinh ldp 4, Nha xuat ban Giao dye Viet Nam (2009) [P] G Polya; How to solve it?, Princeton University Press (1958) [S] A.H Schoenfeld; Polya, Problem Solving, and Educatio Mathematics Magazine, Vol 60, No (1987) 283-291 ... nhieu? Chu nhat cudi cung Id ngdy bao nhieu? E Ket luan: Khi day hpc sinh, nha''t Id hpc sinh tieu hpc lam mdt bai todn ndo dd ma hpc sinh gap khd khan, chung ta can thiet ke lai bdi todn dd Cde thiet... ren luyen sfl sdng tao Neu gap mot bai todn phai day eho hpc sinh, nhat Id nhflng bai todn khd ehung ta chf dpc bdi todn len eho hpe sinh rdi day cho cde em Idi giai, tham chf hfldng ddn cde em... da biet dfldc Id "Hpc sinh nen de dpc lap Idm cac edng viec nhieu nha''t ed edu tra idi eho 1.1 song lai khdng biet la se bat dau giai Bdi the Tuy nhien, neu chf ed hpe sinh vdi bai todn md khdng