^ NGHIEN ClJll GIAO DUG NUflC NGOAI REN LUYEN VA PHAT TRIEN CAC NANG LlfC TRI TUE QUA DAY HOC TOAN LOfP 11 CUA NlTdC CONG HOA DAN CHU NHAN DAN LAO SOMGHAY SONGSAMAYVONG Bd Gido due Lao Emaii somchav23[.]
^ NGHIEN ClJll GIAO DUG NUflC NGOAI REN LUYEN VA PHAT TRIEN CAC NANG LlfC TRI TUE QUA DAY HOC TOAN LOfP 11 CUA NlTdC CONG HOA DAN CHU NHAN DAN LAO SOMGHAY SONGSAMAYVONG - Bd Gido due Lao Emaii: somchav23135g8@gniaii.Gom NGUYEN NGpG GIANG - TrifOng Dai hoc Ngdn lidng TP Ho Ctii Minii Emaii: nguyenngoGgiang.net@gmaM.coni Torn tdt: Ren luyen vd phdt triin cdc ndng lUe tri tue Id nhiem vu rdt quan trpng viec day hoc todn dnudc Cdng hda Ddn chu Nhdn ddn Lao hien Hoc smh khong nhdng cdn duac trang bi kien thtfc todn hoc md cdn cdn duoc trang bl cdch thtfc tu duy, phdt hien gidl quyet vdn de De Idm dugc dieu ndy, cdc em cdn dugc boi dudng phuong phdp hoe nhdm tdng cudng tinh ddc Idp, sdng tgo Hoe smh sau hoc se hinh thdnh duge the gidi quan bien chtfng, nhin nhdn vdn de tinh thong nhdt va moi lien he hOu ed, gdn bd khdng khit vdi Bdi viet di cdp den van de ren luyen va phdt trien cae ndng luc tri tue qua day hoc Todn Idp 11 cua nuac Cdng hoa Ddn chu Nhdn ddn Lao Ttf khoa: Ren luyen, phdt tnen, ndng lUc tri tue, day hoc; todn, Ldo (Nhan bdi 22/12/2016; Nhan kit qud phdn bien vd chinh stfa ngdy 07/01/2017; Duyet ddng 25/4/2017) Oat van de Hien nay, Lao dang bUdc vao nhdng nim ban le CLia cdng nghiep hda - hien dai hda nhim dUa dat nudc tien theo va hda nhap vdi nen kmh te the gidi, Giao due ndi chung va giao due thdng ndi rieng ddng vai trd quan trong phat tnen ngudn nhan lUc chat lUOng cao Nghi quyet dai hdi Ding Nhan dan Cach mang Lio lan thu X da khing dmh "Kien dinh dudng ldi ddi mdi toan dien"trongddcd ddi mdi ve chat lUong day hoc, De ddi mdl va nang cao chat lUOng day hoc thi ren luyen v i phat tnen nang lUc (NL) tn tue cho hoc smh (HS) la cdng viec cap thiet va quan nhat d bac Phd thdng, Cac tU quan nhu phan tich, tdng hop, so sanh, tUong tu hda, khil quat hoa, dac biet hda, trdu tuang hda gid vai trd then chdt ddi vdi viec hinh cac pham chat tn tue cua HS npoc Cdng hda Dan chu Nhan dan Lao, 2, Khai niem tri tue Khai mem tri tue la mdt khii mem cd td lau ddi va phuc tap Cd rit nhieu cach hieu khae ve tri tue In tue la "Phan suy nghi, tU ciia ngUdi bao gdm nhdng kha nang tudng tuong, ghi nhd, phe phan, thu nhan tn thdc cd the tien tdi phat mmh khoa hoc, sang tao nghe thuat"[1] Theo Piaget,"Trf tue khdng phai susaochep, tUc la su bat chuoc hien thuc, ma la ket qua cua su ddng hda tich cue ciia thuc te vai cac dang thUc tam li cua ehii the, cac dang hoat ddng cua tu duy, tdc la cd cae sU chuyen hoa thuc hay tuong trUng chinh chii the tac ddng tren khach the, gid vai trd chii dao phat tnen tri tuevatri thuc (kien thdcV'M] Theo nha tam li hoc ngUdi Nga B G Ananhev, tri tue la dac diem tam li phuc tap ciia nguoi ma ket qui cua cdng viec lao ddng va hoc tap phu thudc vao nd [2] p.aget dmh nghia tuduy nhu mot dang tri tue npi • KHOA HOC GIAO OUC tdm, khae vdi tri tue giac - ddng (cam glic - van ddng), khdng chi dUa tren hanh ddng va tn giac true tiep, ma tren su goi lai (phan tinh - tai tao) cd tinh tuang trUng (hay"ki bieu") bang ngdn ngd, hinh inh tinh than, Tu gin vdi sU t i i tao tmh than hay chdc nang tUOng trUng khdi dau d mdc tien hoat ddng vdi su dat tdi nhdng tUdng trUng dau tien va nhdng dang thdc bSng ngdn ngd[l] Hien nay, cd nhieu kieu phan loai khae ve tn tue Dd la md hinh ba chieu ve tri tue ciia J.R Guilford, mo hinh cic dang tri tue ciia H Gardner, md hinh phan loai cac kieu hoc tap ciia R, M Gagne, mohinh nhan thdc ciia Brien, thang nhan thdc ciia Benjamin Bloom Su phan loai cdn cd nhdng ten goi khae, ching han md hinh ba cbieu ve tri tue cua Guilford cdn cd ten goi la mohinh cau true In tue hay khdi vudng tri tue, md hinh cac dang tri tue ciia H Gardner duoc nhieu ngudi goi la thuyet da tri tue (Ml) Guilford tu nim 1950 den 1973 da dUa hai quan mem mdi Dd la, nang khieu sang tao cd san d moi ca nhan binh thuong [tdc la co he than kmh, khdng c6 benh li hay cd khuyet tat) va qui trinh sing tao cd the day va hoc dUOc Quan diem ddi lap vdi cac quan niem trudc Guilford, la sang tao chi xuat hien d cac thien tai (tdc la cd tinh bam smh) Md hinh ve tri tue Guilford gdm ba yeu td Mdt la, ndi dung (cac yeu td lam nen cho hoat ddng) gdm ndi dung tUong hmh, ndi dung tuong trung (con sd, chd viet, bieu tuong,,.,), ndi dung ngd nghTa (van ban viet hoac van ban ndi), ndi dung dng xd (cac quan hexa hdO- Hai la, hoat ddng (qui trinh hoat dong tri tue) gdm kha nang nhan thdc (nhan dang cac sU kien), tuduy hdi tu (thanh phin logic ciia tri tue), tUduy phan kl (la loai tU sang tao co dac trung la tinh Imh hoat, tinh mem deo, tmh ddc dao, tinh nhay cam van NBUfiN COU GIAO DUG NUOC N6OAI ® de), phan xet (danh gia), tri nhd Odu gitJT thong tin) Ba ia cdng trinh cua mmh sau nam nghien cdu v l khd nang sdn ph^m (ket qud cua hoat dpng) gom dan vi (cac y^u nhdn thdc Ody cd t h i coi Id mot sU kien chan dpng v l to dem gian}, idp (to^n bp cac yeu ^ cd dac tfnh gidng gido due hoc Bloom dUa mdc dp khdc Mpt Id, nhau), quan he (nguyen nhan, he qua, ddi nghich), he biet (hay tri giac) gom nhd va lap lai nguyen dang thdng th&ig (toan bo cac yeu td ducfc to chdc lai vdi nhau), tin Hai Id, hilu gom hdi tfc ve thdng tin (chda de cdp chuyen hoa (chuyen t d trang thai sang trang thai den dng dung Cd hai mdc: The hien, chuyin djch va gidi Ithac), bao ham (quan he nhan quS, suy iu|n) [1] thi'ch, cdt nghTa thdng tin) Ba la, vdn dung (sddung quy VIK> cudi the Id XIX, nhan loai da cd mot i^Udc tien t£c nguyin if, thuat toan 3i giai quyet van d l (hay bdi ]dn ve nhan thtfc cac ehdc nang cua tri tue Ngdcri ta todn) md quy t i e khdng cd sin de bdi Bon la, phdn kham pha rSng bp may xtjf if thdng tin nSm d dc vung tfch (tim cac thdnh phan cau td tdng t h i de phdn khae cda bd nao Moi ca nhan deu cd nhdng thien biet cac •jr) Ndm id, tdng hpp (kit hpp hoac to hpp cdc hudng khae Cd ngudi gidi ve linh vUe nSy nhutig thdnh phln thdnh mdt tdng the) Sdu la, ddnh gid (cdng khong gidi ve ITnh vdc khdc Chinh vi the, H Gandner da thtfc hda cae phan xet djnh ti'nh va djnh iUpng) [1] dda md hinh cac dang tri tuf (nhilu ngddi gpi la thuyet Phep phan ioai ve tinh cam gdm bdc Mdt la, tiep da tri tue) nhlm giup phat huy het tiem nang cua tdng nhan (ilng nghe mdt each b| ddng, tiep nhan vdt kich ca nhan Cau cham ngdn ndi tieng lien quan m i t thiet thich) Hai la, ddp lai (ddng tinh, ham thich, thd) Ba den md hinh cac dang tri tue ci^a H Gardner la ' D i ^ g la, gid tn hda (hdnh vi t h i hien sd Ida chpn cua chu the |}lt ca leo cSy gidi nhd khi' Ngudi gidi ve van ddng d i ^ g vdi sd gid tri nao dd) Bdn la, td ehdc {tdc dong den hdnh ^p buoc hp phai gidi logic toan nhu nhOng ngudi khae vi cua chu the ITnh viie dang d l cap den) Nam Thay vao dd, giao vien (GV) phdi biet bdi dddng nang Id, tinh cdch hda (hinh the gidi quan, triet ii cudc khieu, sd trddng cua tdng em HS giup cac em phat huy song, tfnh each ngUc^ eiia chiJ thi) [1] het toan bo NL cua cac em Ve so cac dang tri tue Phep phan loai ITnh vUe tam ii - van ddng gdm CO tai lieu de cap 7, cd tai lieu de cap va Gardner sau dua dang tn' tue khdc nhau.Tuy nhien, bai bae Mdt la, phdn xa tU nhien (nhU may mdt, ngap ngd, viet ehung tdi dua dang tri tue nhu iuc ban ddu dudi chan, ) Hai la, NL tri gide (tinh nhay cdm, kha nang Gardner dua ra, dd la: Tri tue ngdn ngd, tri tue Idgic toan, phan biet ve tri giac (gan, xa, ben trdi, hin phdi), phdt tri tue tudng tuwig/khdng gian, tri tue giao tiep, tri tue trien cac giac quan nhU bat, nem) Ba la, NL the chat (the lUc): Sdc manh, mem deo, nhanh nhen, kheo leo Bdn la, van ddng, tri tue am nhae, tri tue ndi tam kl nang van ddng (phit then cao hPn NL tren, lam chCi Robert Mills Gagne ndm 1965 da nghien cdu va cae ed ddng eho phep van dyng cd cae tri gide phdi hc^ cho ddi ph^p phan loai hpe tap gdm ede yeu to sau (dinh miy ehd, ehai ede mdn the thao)) Nam Id, eae kT Mdt la, hoc tap ve kl xao van ddng (tat cd nhdng g'l lien nang phi ngdn ngd (nhU sd dung cdc van dong cua co quan din di#u bd, cd chi, dpng tac) Hai la, hoe tap ve tU the de dien dat trao doi thdng tin (cd chi, dieu bd, mda, the (tit cd nhdng gi lien quan tdi gia tn vi du nhu phep kjch cam)) [1] ijdi su giao tiep, thai dp tdn trpng phap ludt, khoan Sau nhiing edng bd cua Bloom, da cd nhdng dung vdi ngucri khdc) Ba td, hoc tap ve thdng tin ngdn nghien cdu vd edi t i l n v l each phan loai eda dng, die ngd (hpe von td, ngd am, ngd phdp, hoc sU kien, hoc npi biet la thang nhan thdc Hien nay, nhieu ngudi thudng dung kien thdc) Bdn la, hpe tap kT xao tri tue (hpe each sd dyng thang nhdn thdc Bloom mdi eung gdm cap phan btet, hpe eic khai niem, hpe cic dinh li, quy t^e, hpe dp nhU: Biet, hilu, van dung, phan tich, danh gia va sing cac djnh litrinh dp cao) Nam la, hpe chien lupc kha ning tao bode thang nhan thdc eiia Nikko nhU: Biet, hieu, nhan thdc (Chien lupe khd nang nhan thde la thu phip van dyng va van dung cao Thang nhan thde cua Nikko ca nhan ve xd li thdng tin) [1 ] thudng dUac sd dung dinh gii de kiem tra trie Brien phan loai tri tue gdm eic loai sau: Mdt la, tri nghiem Tuy nhien, ddi vdi cie de kiem tra tU luan thi thdc (thdng tin -* nhan thdc -»tri thde); hai la, kT nang thang nhan thdc Bloom cu v i Bloom mdi ed Uu the hon tri tue; ba la, kT nang van ddng; bon la, thai dp; nam la, bin Tuy theo vao viec dinh gia qui trinh dau eua phuong phap nhin thde (phuang phap hpe, cich giai ngudl hpe nhu the nio thi ngUdl ta se sd dung nhdng quyet van de) [1] thang phin loai phu hop vdi cich dinh g i i nhuthe ay Trong cae each thdc phin ioai ve tri tue vd nhan Khai niem nang tUc thdc tfii thang nhan thdc eua Bloom i i thang nhin thdc Cd le khdng cd khil niem nao dupe quan tam ndi tieng v i dupe i p dyng nhieu nhat d nhieu nUdctren ITnh vUe giao due nhieu nhu khai niem NL, Cic chuyen the gidl Bloom phan loai thinh ba ITnh vUc khae nhau: gia nghien cdu ve NL eung eho ring day li mdt khii - Kha nang nhin thdc (kien thdc, tri thde) niem khd nhung lai vd eung quan trpng viec day -Cam xuc (cam xue, tinh cam,y chi, thdi dd, dng xd) hoc hien nay, la viec day hpe dinh hudng phit tnen -Tam than van dpng (phin xa, tri glic, thieh dng, kT NL nang, kT xao) Theo Td dien Oxford, NL ed ba nghTa nhung chi cd Tuy vao tifng ITnh vtfe Bloom lai tiep tue phin hai nghTa lien quan den ITnh vUc giio due NghTa thd eae mdc dp khic Nim 1956, Bloom da cdng bd nhat l i kha nang lim tot mdt dieu gi (the ability to S013g-rHANGV2017*97 ® NGHI£N CUIU GlAO DUG NUttG NGOAl something w^ell) Nghia thd hai la mdt kT nang md ban can mdt cdng viec dac bi&t hay mot nhiem vu die bi#t (a skill that you need in a particular job or for a particular task) Theo Td dien tiing Vi&t, NL Id khd nang, dilu ki?n chu quan hole t d nhiin sin cd de thdc hiin mpt hoat ddng ndo (khi d l cap tdi NL cda d6i tddng ndo do) hodc id phim chat tam ii vd sinh If tao cho ngddi khd nang hodn thdnh mpt hoat ddng ndo dd vdi chit IdOng cao (khi d l cdp den NL eCia ngudi) [3] Theo F E Weinert, NL dupe hilu la nhflng ki ndng, kl xdo hoc dupe hodc sin cd cCia ca t h i nhdm gidi quylt cdc tinh hudng xae dinh, cung nhd sii sin sdng v l ddng cd, xa hdi, vd nlng vdn dung cdc each gili quylt v l n de mdt each cd trdeh nhiem va hiiu qud nhdng tmh hudng iinh ho^t (4] Theo DenyseTrembiay, nhd tam ii hoc ngddi Phdp quan niim ring, NL id ndng hdnh ddng, dat di/pc cdng va chdng minh sd tien bd nhd viic nlng huy ddng, dat dupc thdnh cdng vd sd dung hiiu q u i nhieu ngudn idc tfch hpp cda cd nhln gili quylt cdc van de cudc sdng [5] Qua nhung djnh nghia v l NL noi tren ta nhln thIy mdt dilu ring, NL ddpe djnh nghia thdng qua ki nlng ho|c dupe djnh nghia thdng qua nlng Idm viic cda chCi the va kit qud Idm viie Id tdt hodc hiiu qud Nang lac trf tue Td nhOng d l cdp v l khdi niim cung nhd tri t u i n i u tren ta hilu NL tri t u i id khd ndng thue hiln tdt nhiem vu, edng viie v l iinh vdc tri t u i Theo Nguyin Bd Kim, NLtri tuecCia HS bao gdm edc mat sau day: - Khd ndng t d logic, ngdn ngCf chinh xlc - Khd ndng suy dodn vd tddng tupng - Cdc boat ddng tri t u i eo bdn -Cdc phdm ehIt tri t u i Cac thdnh ph^n eua tri tue ndi t r i n khdng tach rdi md gin bd hdu ca vdi Vf du nhd cdc hoat ddng tn' t u i co bdn th) ludn phdi di kim vdi khd nang suy doin (nhd nha bdc hpe Albert Einstein dd ndi "Tri tudng tucmg quan trpng hdn tri thde") eung nhd phdi tdt v l tu duy, ngdn ngd va phdi cd pham chit tri tue [61 Cac dang NL can phdt trien eho HS day hpe mdn toin id: NL tu duy; NL phat hien vd gill quylt v i n di; NL giii todn; NL vdn dung toin hpe vdo thdc tien; NL tu hoc NL hpp tde, p h i phdn va dinh gid [6] Theo Mogens Niss, ede dang NLtri t u i gdm eic dang NL sau: NL tU toin hpe (Thinking mathematically); NL phit hiin vd giii quyet vln d l (Posing and solving mathematical problems); NL md hinh hda toan hoc (Modeling mathematically); NL lip luan todn hoc (Reasoning mathematically); NL bleu diln todn hoc (Representing mathematical entities); NL sd dyng eae ki hiiu va cdng thdc toan (Handling mathematical symbols and formalisms); NL giao tiep toan hpe (Communicating 98-KHOA HOC GlAO DUC in, with, and about mathematics); NL si]rdung cic cdng cu tro gidp (Making use of aids and tools) (Tl5 Ren luyen v i phit trien ndng IOC trf t u i cho hoc sinh Theo Nguyin Bl Kim, mdnToan can ddde khai thdc d l gdp phan phit triln nhijfng NLtri tue nhi/tdduytrOdi tiftfng, trf tddng tddng khong gian, t d logic, ti/ biin chdng, ren luyin cac hoat d$ng trf t u i co b i n nhi/ phan tich, t6ng hctp, so sinh, khdi qudt hda, tuong tir hda, dac biet hda, cdc pham chit t d nhd tfnh linh hoat, tfnh ddc idp, tfnh sdng tao [6\ De lam diiu nay, GV cln ed "y thdc d l y dii v l cdc m|t sau ddy: Thd nhat, r i n luyin tif Idgic v l ngon ngjJT chinh xde Thii hai, r i n luyin khd ndng suy dodn vd tddng tddng Thd ba, r i n luyin nhdng hoat ddng trf tu6 CO bdn Tlid td, hinh thdnh nhdng phim chat tri tui - Cle biin phdp ve r i n luyen t d logic va ngon ngd chinh xlc gdm: + Dpc nhilu t l i liiu liin quan ve van 3i can dua thio iuan, cdch sd dung ti^ ngi/ cCia tai lieu ehuyin ngdnh Sd dung ddng, n l m vdng nhdng lien kit idgic nhd vd, hole, neu, thi, phO dinh va nhdng Idcmg td nhd tdn tai, vdi mpi cac kf hieu chuIn cd tinh quoc t l + Phdt triln k h i nang dinh nghia, idm viec ven' cac djnh nghia + Phit trien khd ndng hieu chdng minh, trinh bdy iai chdng minh v l ddc Idp t i l n hdnh chdng minh +Tranh ludn vdi edc ehuyin gia, nhOng ngddi cOng chuyen ngdnh de bdo ve quan d i l m cCia minh Khi tranh luin, cd nghia Id chCing ta dang cd gIng ldp lu|n mot v l n di trd n i n cd logic, theo kieu nhln - q u i Chi eIn khdng di lae di chung ta se luyen dupe idi tU nhanh v l sle slo - Cac bi^n phip ve ren luyin khd nang suy dodn vi tudng tupng gdm cdc bien phdp sau: + Tim nhieu elch gili khdc T>idng qua v i ^ tim nhieu ldi gili, HS se tim dupc cdc phuong I n tiep cdn icfl gidi bdi todn, gidp hinh thdnh NL suy dodn vd tddng tupng khdng gian cOa HS + Tim cdc bdi todn liin quan den b l i todn ban dau, dua phUdng phip quy la thdnh quen Nhdng suy dodn cd the rat t l o bao nhdng phdi c6 cln cd dda tren nhdng quy tie, kinh nghiem nhat djnh chd khdng phli dodn md lam lilu + Md hinh hda thdc tien cac bai todn toan hpe va ngupc iai - Cac bien phip ren luyin nhdng hoat ddng tri t u i CO b i n gdm: Phdn tfch, tdng hop, so sdnh, tuong td hda, khdi qudt hoa, dac b i i t hoi, trdu tdpng hda, - Cae bien phap ren luyen phim chat tri tue gom: + Ren luyen tinh linh hoat nhUNL thay ddi de ddng, nhanh chdng trat t d eda he thdng tri thde, ehuyin td gde dd quan niem niy sang gde dp quan niem khlc, xdy dung phdong phip tU mdl + Ren luyen tfnh doe ldp Tfnh ddc ldp the hien eho HS tu minh biet phdt hien vd gidi quylt van de, tU NBHIEN COU GlAO DUG NUSC NGOAl ® kilm tra v l hoan thien kiln thdc bilt holi nghi vd tra idi vilt la gid stf ddng thtfc ddng m n = k budc cung cdc clu hdi vAy.ttfgid^iitminhdi A(n) dOngvdi n = k phlivietll + Ren luyen tfnh sdng tao Tfnh sing tao t h i hiin d ndng phdt hien v l n di mdi, ldi gidi mdi, k i t qud ttf gid thiit minh di A(n) ddng tdi n = k Phep quy nap mdS [1] t o l n hoc Id mdt qud trinh di tCr gia trj d i n gid tri ^ -h Vi du minh hoa rin luyin va phit tri^n cic nlng ^ tri t u i qua d^y hoc mot bhi toin sich chd khdng phdi gidn doan tCfng gia tri 1, A:, * - - ! Nhd giio khoa Toan Idp 11 c&a nildc Cdng hda Din di6 vdy, cdch gidi, GV dd giup HS ren luyin dupc Nhan din Lio tu logic v l ngdn ngd ehinh x l c Sdch gilo khoa Todn idp 11 nudc Cpng hda Dan chQ - Budc 2: Rin luyin k h i ning suy doin v i tUdng Nhan din Lao cd bdi todn sau ddy: ti/tfng Biitoini Tilp theo GV rin luyin NL mo hinh hda cdc b i i t o l n Hdy chtfng minh rdng: thyc tien eho HS bing Idi giai sau: Ldi gidi Ain) = -f- -I- -H -1- (« - ) + n = "^"'*"^\ Ta gill b l i toln bing thue t l hda bai t o l n todn hpe nhusau: Ddy id bai toan chUdng II: PhUdng phdp Bii toin chdng minh quy nap todn hoc, bat ding thdc vd dng Ban to chtfc cdn ehgn sd n + ngUdi de dung cda sleh gido khoa todn ic^ 11 GV cho HS chdng minh bing phuong phdp quy nap Trong bli vilt nly, chung tdi dua cleh day theo hudng ren luyin, phdt trien NL tri t u i vdi cdc k i t qud nghiin cuU mi^ chfnh chung tdi tim ^ l y (bai toan 2,5 va 6) CIc bude ren luyen v l phdt trien NL trf t u i cho HS gdm cdc bi/dc sau: - Budc 1: Rin Iuy|n td logic v i ngAn ngOT chinh x i c - Mdt HS Lao Idm theo cdc budc cda phep quy nap toln hpe nhusau: Lid gidi Budel.Khi n = l t a c d = ^^^'"""'^ = Bude Gid sd ding thde ddng vdi n = k, nghia id Aik) = l-i-2-\- -\-{k-i)-\-k k{k +1) Bade Td gid thiet menh d l A(n) dung vcfl' n — A, ta can chdng minh A{n) dung vdi « = ft-j-1 Thdtvdyta cd; tham gia chuang trinh "Trd chai X 'trin kenh kinh truyen h]nh y Hoi cd bao nhieu cdch ehgn ? Ta lap iuan theo hai cdch: Cdch 7: Ngudi thd nhlt cd n -t-1 edeh ehpn Sau dd cdn n ngddi nin ngddi thdhai cd n elch ehpn.Vlycd n{n + 1) cdch chpn Nhung cleh ehpn hai ngudi AB cung Id elch chpn -\-1)cdch hai ngudi BA, do sd cleh ehpn 11 n(_n chon Cdch 2: Neu ed ngddi thdi thi ed edeh ehpn Nlu ed them nguc^ thd ba thi cd them cle elch chpn ed ngudi thd ba nay.Ta phdi chpn mdt hai ngudi tuc dau thi dIu vdi ngucri thd ba ndy vd ed them each chpn Neu cd t h i m ngudi thd tu thi cd them cdc each chpn eho ngddi thdtu nay.Ta phii chpn mdttrong ba ngudi trudc thi dau v ^ ngudi thd tu nay, vd c6 t h i m each chpn, Nlu ed them ngudi thd n -I-1 thi cd them cac each chpn cd ^(it -1-1) = + -I- + A -1- ( t + I) ngdcfi thd n + Ta phdi chpn mot n ngudi da ed d l thi dau vdi ngudi thd n -I-1, vd ed them n each Theo phip quy nap todn hpe ta ed A(n) -^ -H + + + (n - ) -F « ="(" ehpn Vay sd each ehpn Id: \ + 2-\-3->r + + 1) - GV eho phep cdc HS Idp nhan x i t ve idi gili cCia bdi todn Trude h i t la each sd dung td ngd da chfnh xic chua? Ldi gidi ciia bil toan ed gi sai sdt hay khdng? - Cau trd ldi Id tat cd cac HS Idp dIu eho ring dly l i mdt ldi gidi diing - GV lue mdi ehi each dung td eua HS ndi tren Id ehua ehinh xac Cu the, tai budc 2, gid stf ddng thtfe dung vdi n = k Id mdt cleh dung td ehua dung Phdi n-\+n cich chpn Vi hai ket qud cda each lap luan thd nhlt va thd hai Id mdt, nen: l + + + + ( n - ] ) + n = n{n +1) - GV dua co sd If thuyet v l phUOng phdp tUPng tUhdaehoHS: Tuang ttf Id mpt kieu giong ndo dd Cd the ndi tuang tuld gidng nhung d mtfc dp xde dinh han, vd d mtfc dp dugc phdn dnh bdng khdi niem Tuy vgy, ehung sdi3a-TiUNGV2ei7-99 ® NGHIEN Clhl GIAQ DUC NOQlC NGOAI ta CO the dien ta chinh xac han mpt chut SU khde Budc2 Gia sd ding thdc diing tdi n = A-, nghia la cdn bdn gitfa tuang tU vd nhtfng logi giong khdc Id B(k) = I + + ,., + (2A- - 3) + (2k -1)=^ k' dy dmh eiia ngUdi dang suy nghi Nhdng ddi tugng giong phd hap vdi mpt quan he ndo Neu Budc Td gii thiet menh de B(n) dung tdi n = k, ban CO nhtfng y dinh quy mdi quan he cdc doi ta cin chdng mmh B(n) dung vdi /i = A- + 1, That vay, tdang phu hap vdi vi nhtfng khdi niem dd dinh thi bgn se xem nhtfng doi tuang giong dy nhuld nhtfng tacddoi tuang tuang ttf Vd neu ban dat tdi nhtfng khdi niem rd g(it + 1) = + + + (2A- - 1) + (2k + I) rdng, thi ttfc Id bgn Idm sdng td sd tUang tU[8] = k' +(2k + l) = ik + iy Phep tuang tutheo D.P.Gorki \a"phep suy ludn Theo phep quy nap toan hoc, ta cd ttf eho hai ddi tuang giong d mdt so ddu hieu ta riit kit ludn rdng: Cdc doi tugng ndy giong d mpt B{n) = + + + (2« - 3) + (2n - 1) - n\ so ddu hieu khdc" - GV de xuat bai toan tuong tu khic bing each phat Neu ddi tuang A v i B ed eic dau hieu (hay thudc bieu dudi dang sau; tinh) ehung a, b, c va doi tuang A cd dau hieu d thi ta rut Bai toan ket luin dU doan ring: B ed dau hieu d Hdy chtfng mmh rdng: Cd the bleu dien so dd eiia phep tuong tU nhU sau: L2 ^ 34 n.{n+\) _ n(n + \)(n + 2) A v i B cung cd cic dau hieu a, b, c "*" "^ "^'""^ ~ • Acddau hieu neng d Ldi giii GV cho HS chdng minh bing phuang phip quy nap Cd the du doin ring: B cd dau hieu d (chdng mmh tUPng td bil toan 1, bai toan 3) Ket luan ciia pheptUdngtUchila mdt dieu dU doin, Ldi giai Du doan cd the dung, cd the sai GV eho HS phat bieu bai toin thUe te eua bai toan Trong toin hoc ta hay diing den "tuang t d ' tuang tu nhu d i lam d tren: cie trudng hpp sau, cic cich dien dat sau: Bii toin - Hai ddi tuong nao dd eiia toin hoe cd tinh chat Cd {n + 2) ngUdi ddng ki tham gia choi'Trd chai E" tuong tu tren kenh truyen hinh F Ban to ehtfe muon ehgn ngUcsi - Hai quan he nio dd ciia toin hoc la tuang tU, vdo vdng chai Hai cd bao nhieu edeh chon ^ -Vdi "phuang phip chdng minh tuang td',., Ta lap luan theo hai each: "Tuang ty, ta ed ket qui ," hay "ta ed eae ket q u i Cdch 1: tuong tu.,.", Ve "tinh chat tuong td; "quan he tuang t d ' ta se de Ngudl thd nhat ed (/; + 2) cich chpn Sau dd cap dcac vidu sau Rieng vey "chdng mmh tuong td'thi {ll + 1) ngudi nen ngUdi thd hai cd (n + 1) each chon ta hieu rang vdi each chdng mmh theo cau true Idgic da trinh bay chdng mmh doi tuong A (hay trudng hop Cudi cung cdn lai n ngUdi nen cd n each chon ngUdi a) ta thay A bdi ddi tuong B (hay thay a bdi trudng thd ba Vay cd n(ii + \)(n + 2) each chon hop b) ta van cd ket luan nhu da de xuat Vi viec trinh Nhung each chon ba ngudi ABC cung la each bay lai mdt Ian nda su chdng minh tren vdi B (hay vdi chon ba ngUdi ACli, BCA BAC CAB, CBA Do so trudng hop b] tbi dii ddng them, khong cd gi mdi khae va khdng can thiet nen ta muon cham chude trinh each chon chi la -i(« +!)(/, +2)- each chpn bay VOI y "chdng mmh tuong tu ta cd,.," (chd khong trinh bay lai day du sU chdng mmh) [8], Cdch 2- Bade & 4: Ren luyen khd nang suy doan tUdng Neu cd ngudi thdi thi cd cich ebon, Neu cd tupng va hinh nhdng pha'm chat tri tue them ngudi thd tU thi co them cac each chon cd ngUdi - GV dua bai toan ciing dang bai toan nham ren t h d t u niy Ta phai chon hai ba ngUdi luc dau thi luyen cho HS tuduy tuong tu Tadey rang tong -iin) la dau vdi ngudl thdtu nay, Theo cich lap luan cua bai toan tdng cac sd tU nhien hen tiep nen tu bai toan ta de 2, ta cd -""- cich chon Neu cd them ngudi thd nam thi bai toan tuong tu ddi vdi tdng cac s6 le lien tiep sau, Bai toan Hdy chdi-ig mmh rdng Biii)^ ] ^ } ^ ^ C f i - f)-(2yi •!) ir - GV cho HS Chung mmh bai toan bing phuong phap quy nap cd them cac cich chon cho ngUdi thd nam Ta phai chon hai bdn ngUdi trudc ddthi dau vdi ngUdi thd 14 nam nay, v i cd them — cich chon,, Neu cd them ngUdi thd (/; 2) thi co them cac each chon cd ngUdi thd |jf -r 2) nay.Ta phai chon hai l;i I) ngUoi da 0 • KHOA HOC GIAO DUC NGHllN Ciiu GlAO DUG NlAllC NGOAI ® Chpn ngudi thd nhat sd n ngddi thi cd n CO de thi dIu vdi ngudi thd (n + 2) va cd t h i m •" each chpn; chpn ngddi thd hai sd n - ngddi cdn cdch chpn Vay sd cdch chpn Id: 1.2 2.3 3.4 n.(n + l) —-+ 1- — + + —^^ each chon 2 2 iai thi c6 n - each chpn; ; chpn ngUdri thd m s6 n - ( m - l ) ngddi cdn lai thi cd n - ( m - l ) cdch chpn Vdy n l u tfnh d i n thdtdcCia nhiimg ngddi dupc Ida chpn thi cd tat ed td n ( n - l X n - ) ( n - ( m - l ) ) cdch chpn m n ngddi Theo nhdn x i t trln, ta ed s6 "^ 2 '•• n(n - \tn - 2) {n-(m-1)) - GV dt/a co sd 11 thuylt cho phuang phdp khdi each ehpn thtfc sU chi la m.M quit hda: Thtf hai lOidi qudt hda Id chuyen ttf viic nghiin ctfu mpt tdp 1.2.3 (m-1) hop ddi tugng dd eho din viic nghiin ci3u mpt tdp hgp Idn Nlu chi cd m ngddi thdi thi ed hOi, bao gdm cd tap hgp ban ddu [8] Mdt djnh nghia khdc v l khdi qudt hda nhd sau: cdch chpn Qud trinh logic chuyen ttf edi dan nhdt den cdi ehung, Nlu cd t h i m ngddi thd (m +1) thi cd them cdc ttf mpt tri thtfc it chung han den tri thtfc chung hem Viic cdch ehpn ed ngudi thd (m +1) ndy Ta phdi chpn m - i thu dugc mpt tri thtfc khdi qudt cd nghia Id stf phdn dnh hiin t/iyt sdu sdc han Id stfdi sdu vdo bdn chdt eua hiin ngddi d m ngUi^ iuc dau thi dIu vdi ngddi Ta thtfc Trong Idgic hinh thtfe, ngudi ta hieu khdi qudt hda 2.3 (m-l).m - cdch chpn khdi niim Id viic chuyin ttf khdi niim logi den khdi niem M chUng Trong dd, npi hdm cua khdi niem chOng Igi hep hem, Neu ed them ngucn' thd (m + 2) thi cd them bdivinhtfngdduhieuveiogibiggtrakhoikhdiniimehpng (Ngogi diin vd npi hdm cOa khdi niim) [7] M - GV yiu clu HS phdt bilu bdi todn tdng quit eua Tilp tuc nhd t r l n cd t h i m ngddi thd n thi cd bdi todn 1, b l i toan nhlm ren luyen tuduy sdng tao: Bai t o i n , {n-{m-\)) (n-l){n-\) cleh ehpn Hly chdng minh ring: 1.2 («.-l)^23 m^3.4 (m + l)^ ^ln-{m-i)) [n-2){r,-\) Vly sd each chpn m n ngudi la: M M M ' M 1.2.„(m-l),23, m , ,(m + l) , , (n-(m-l))„,(n-2X«-l)) _n(j^-\ln-l) {n-[m-\)) M M M M mM Vl hai k i t qud id mdt, ta ed: (trongdo W = 1.2 (m-1)) l.2 (m-l)^23 m^3.4 (m^l)^ ^(n-{m-])) (n-l)(n-\) - GV eho HS phit bieu bai t o l n thue t l eOa bil toin M M M '" M nhUsau: _n{n-\ln-l) {n-(m-\)) Bai t o i n mM Ban to chtfc cdn ehgn m son ngudi detham (trongdo M =\.2 (m - 1)) gia mpt vdng chai cua chuang trinh "Trd chai X"(m