- Co moi lien he nao giua cai phai tim vdi cai da cho?
- Dieu kien cd dii de' xac dinh dugc an khong? Thieu ? Hay thCra?
- Ddi vdi bai toan hinh hoc ndi chung, yeu cau HS phai ve hinh (nhdt Id
ddi vdi bdi todn hinh hgc khong gian). Sau khi ve hinh xong va the hien dugc
cac yeu td da cho, ye'u td c^n tlm tren hinh ve, liic dd hgc sinh se hieu rd bai toan hon. Mdt dieu c^n luu y la: hinh ve phai mang tinh tdng quat, khdng nen ve hinh trong trudng hgp dac biet.
De dat dugc nhung yeu cau ndi tren trong budc tim hilu dau bai toan, ngodi yeu td tdc ddng bdi ngudi Thdy budc hpc sinh phdi cd nhimg khd nang
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ve tu duy nhdt dinh.
Budc 2: Lap ke hoach gidi toan
Lap ke' hoach gidi toan la xdc dinh trinh tu cho viec gidi quyet nhung ddi hdi cua bai toan, Ndi mot each khac la day cho hpc sinh tim ra nhirng budc di tie'p can each gidi cua bdi todn. Cd rat nhieu each de tim ra Idi gidi bai toan. Ngudi day ed the sir dung cac cau hdi phan tich di len, tong hap hoac cac phep suy luan, quy nap de giiip hpc sinh tu tim ra Idi gidi cua bai toan. d budc nay, rdt cdn den thao tac phan tich bai todn da cho thanh nhieu bai toan dan gian han. Hpc sinh phai huy dong kien thiic cd lien quan den nhirng khai niem, nhirng quan he trong de toan, rdi lira chpn trong sd dd nhung kien thiic gdn giii han vdi nhirng dir kien cua bai toan. Md mdm, du doan, thir xet mot vai khd nang, ke ca mot vai trudng hap dac biet; ddi khi cdn xet mot bai toan tuang tu hoac bai todn khai quat cua bai todn da cho..
Mudn hpc sinh dat duac ke't qua trong day bpc gidi todn, cdn cdn rdt nhieu dg'n nghe thuat hudng ddn eua ngudi thdy. Ngoai ra, hpc sinh cung phdi biet huy ddng nhiing kien thu:c sdn cd, biet tap trung suy nghi vdi Idng say me hpc hdi kham pha trong gidi toan .
Budc 3: Thuc hien ke' hoach gidi toan
Hoat dong thuc hien ke' hoaeh gidi todn bao gdm: viec chpn mot cdch gidi va trinh bay loi gidi bai todn de hie'u nhdt, phii hop nhdt vdi bac hpc. Ldi gidi bai toan duac hieu la tap hap cac thao tae sdp theo thii tu de di den muc dich
yeu cau ddi hdi ciia bai todn. Thao tac dd cd the la phep tinh ca ban, phep
dung hinh ca ban, hoac m6t day cdc suy luan...
Cdn phai luu y rdng: Ciing mot vdn de nhung each trinh bay ldi gidi d mdi cdp la khac nhau. Tuy nhien, du trinh bay theo each nao thi Idi gidi mot bai toan khdng cho phep cd sai Idm. Yeu cdu nay cd nghla la ldi gidi bdi toan phai ddm bdo dd chinh xdc ve kie'n thiic, hap I6gic ve quy tdc suy luan, ng6n ngir dien dat trong sang. Vdi nhirng ddi hdi tren, khdng the khdng can den cdc thao tdc tu duy cua ngudi gidi todn.
Budc 4: Nhdn dinh, ddnh gid bdi todn
d budc ndy, ngudi gido vien cdn luyen cho hpc sinh cd thdi quen kiem tra
lai tiing budc gidi, xem xet ldi gidi cd phu hop vdi cap hpc khdng, dap sd eiia
bdi todn cd hcfp ly hay khdng?. Ben canh dd, can phdi ludn Iu6n giiip hpc sinh khai thdc phat trien bai todn theo nhieu hudng khac nhau hoac nit ra nhirng kinh nghiem cdn thie't mdi khi gidi xong mot bai todn, De 1dm dupe dieu dd
khdng phdi Id de dang, bdi ngudi hpc ngodi kien thiic todn hpc eiia minh cdn phai huy ddng cae thao tdc tu duy: phan tich, tdng hpp, so sdnh, trim tuong hda, khdt qudt hda, dac biet hda, tuomg tu hda mdi cd the cho nhan bdi todn
moi cd the sang tae duac nhirng bai todn tuomg tu hoac sdng tao ban bdi todn da timg lam.
2.2.2. Hinh thdnh, ren luyen, boi duorng phdm chdt tri tue todn hgc cho hgc sinh thong qua dgy hgc gidi todn
Bdi duang, phat trien nang lire tu duy cho hpc sinh thdng qua cac boat ddng gidi toan la nhiem vu quan trpng va cap thiet trong doi mdi each day hpc toan d pho thong hien nay. Mudn bdi dudng va phdt trien nang luc tu duy cho hpc sinh tnrdc bet phdi hinh thanh va phat trien pham chdt todn hpc cho hpc sinh. Pham chdt toan hpc ndm trong tri thong minh, ma tri th6ng minh la ket qua phiie hap cua kie'n thiic, kl nang todn hpc ciing nang lire tri tue va dp nhay cam todn hpc cua ngudi hpc. Tuy nhien, tri thPng minh todn hpc kh6ng han do bam sinh ma cd. Tri thdng minh todn hpc dupe hinh thanh va phat trien phdn Idn trong qud trinh hoat dong toan hpc va dac biet la hoat dong gidi toan. Ngudi ta cd the do dupe tri thdng minh con ngudi nhd cdc may mdc hien dai
nhung hp cung de dang ddnh gid tri thdng minh cung cdc ndng luc tri tue
qua hogt dgng gidi todn. Mot hpc sinh dupe coi la thPng minh hay la ngudi ed
tu duy toan hpc tdt thudng dupe bieu bien qua boat ddng giai todn nhu sau:
- Cd su quan sat tinh te', mau phdt hien ra eae diem niit trong mdi bai toan
de gidi quye't theo chieu hudng hop ly, dpc ddo,
- Suy luan cd can cii, ro rang, nhdt qudn logic, cd dc td md va ludn mudn
tim tdi cdi mdi. (adsbygoogle = window.adsbygoogle || []).push({});
- Khd nang suy luan theo hai hudng xudi ngupc nhanh nhay,
- Tri tudng tuang phdt trien nen khd nang chuyen tir cu the sang khai quat
hod rdt tdt.
- Cd khd nang tu hpc va hiing thu vdi viec tu tim tdi cae bdi todn de
gidi.Khd nang du kiln hudng gidi quyet mdi bdi todn cung nhu viec van cdc
Nhu vay, tri thdng minh toan hpc chinh Id sdn pham cua kien thiic, kl nang toan hpc ma chii the ed dupe thdng qua boat dong trong mot he thdng tu duy ddc lap, sang tao. Tinh d6c lap tu duy ciia chii the dupe the hien d khd nang chii ddng, tich cue, bie't quan sat tinh te cac bai toan, vdn de todn bpc gap phdi d^ mau phat hien, tim ra cdc mdu chdt; tir dd de xuat cdc each gidi quyet. Tri thdng minh cdn the hien d khd nang sang tao cua mdi chii the thdng qua dc phe phan, nhan xet, danh gid nhirng each gidi da ed de lira chpn each gidi mdi d6c dao, biet each tu minh kiem tra va hoan thien vdn de dat ra, dac biet la cdn biet sdng tao vdn de mdi.
Mudn phat trien tri th6ng minh cho hpc sinh, ngudi Thdy cdn thudng xuyen ren luyen cdc thao tdc tu duy todn hpc eho cdc em nhu phan tich, tong hap, so sanh, tuang tu, khai quat hda, triiu tuang hda, dac biet hda, he thdng hda V v....Viec ren luyen tri thdng minh ndi tren cd the bang nhieu con dudng khac nhau nhimg con dudng tdt nhat van la con dudng thong qua cac boat ddng gidi toan.
a. Phdn tich, tdng hap trong hogt ddng gidi todn
•\ f f
Xet ve ban chat d gdc dp ly luan chung ta thay:
Phdn tich la mot thao tdc tu duy nhdm tach cac van de Idn thanh nhiing van
de nhd de timg budc gidi quye't cac van di nhd; sau dd gop lai giai quye't tong
the vdn de ldn.
Tong hap la khd nang boat dong cua tu duy biet gop nhiing bp phan cua
khdch thi lien quan ciing chu de, cung he thdng lam thanh ban chat ciia su vat mdi.
Trong boat ddng giai toan, phan tieh - tong hop cd mdi quan he mat thiet, bo xung, dan xen vao nhau trong qua trinh tu duy thdng nhat de gidi bai toan.
Phdn tich la ca sd de tim ra ldi gidi bai toan va thudng dupe tien hdnh
* Phan tieh di lcn{suy nguac liii). Phan tich ndy dupe lodi ngudi bilt din tir
300 nam trude cdng nguyen, bat dau tir Hylap, vdi phdt bilu ciia Pappus trong
cuon "nghe thuat gia toan". Pappus ndi: " Ta mudn dat dugc Vk qua mong
muon thi phai di tir ket qua dd, roi muon dat dugc ket qua nay thi phai di tir
' r f
ket qua nao trude nira.. .cho den eudi ciing ta tim dupe mot dilu gi da bilt hay da cdng nhan la diing". D I van dung phep phan tieh di len , Platon dl ra mdt bdi todn: Ldm the nao de mang 6 lit nude tir sdng vl bl nlu trong tay chi cd 2 loai thiing, mdt thimg 4 lit vd mdt thiing 9 lit
Qud trinh phan tich: 6 ^ 9-3= 9-(4-l) nen suy ra cdch ldm
Cdch ldm:
- Dong nude day ean 9 lit rdi dd 2 ldn vdo can 4 lit. Trong can 9 lit cdn I lit nude.
- Dd nude d ean 4 lit xudng sdng rdi dd Hit nude d ean 9 lit vdo cao 4 lit. Trong ean 4 lit cdn thieu 3 lit nude mdi day.
- Dong mdt lan nira nude vdo day can 9 lit rdi dd vdo can 4 lit da cd Hit nude. Sd nude cdn lai trong can 9 lit Id 6 lit.
* Phan tieh di xudng ( suy ngugc tien)
Gid sir cd A , tir A ta suy ra Ai tiic Id A=> Ai, rdi lai tir Ai => A2... ; (adsbygoogle = window.adsbygoogle || []).push({});
Ap-i => An- Khi gap An la phan dodn sai thi dimg lai vi khi dd chac ehan A sai. Cdn nlu An diing thi chua chac dd cd ket luan gi vi A ed the sai hoac dimg. Thao tdc tu duy phan tich thudng dupe dien ra nhieu nhat trong hai budc ciia qua trinh gidi toan dd la: tim hieu ddu bai toan va lap ke hoach gidi toan.
Khi tim hieu ddu bai toan cung nhu lap ke hoach giai toan, ngudi hpc sinh
da phai thuc hien mot chudi cac thao tac tu duy phan tich d trong ddu nhdm
tach bach mot each ro rang dau la dir Y\tx\{cdi dd cho), dau la dn sd(cai bdt
tim), dau la dieu kien ciia bai toan {ti(c la mdi lien he giua cdc du kien vai
Qua trinh phan tich cd the dien ra dan gian hay phu-c tap cdn phu thupc
vao ttmg bai loan cu the va nang lire phan tich cua limg em. Ke't qua dat dupe
sau phan tich la: giiip ngudi hpc phdt hien ra hudng gidi quyet bai todn hoac
nhan ra bai toan thude dang toan dien hinh da cd thuat toan. Ciing tir day khd ndng phdn tich cua ngu&i hgc duac ren luyen vd ngay cdng phdt triin
n r r f
Tong hap Id phuang phdp suy luan di tir cdi da biet den cdi chua biet.
Neu gpi A Id gia thiet eiia bai todn vd B Id dieu bdi toan bat tim thi so dd cua phep tdng hop trong gidi todn se nhu sau: A =:>Ai=>A2.,..=>An=>B.
Trong boat ddng gidi todn, thao tdc tu duy tdng hop cua ngudi hpc ludn dupe dien ra dan xen tren ca sd ket qud thao tdc tu duy phdn tich mdt cdch khoa hpc. Cd nhu vay ngudi giai todn mdi cd the nhanh chdng phdt hien hudng gidi quyet bdi todn. Sau dd tu duy tdng hop cdn gdp phan giup ngudi hpc hodn
f \
thien tdt viec trinh bay ldi gidi bdi todn.
Vi du v^ viec xen ke qud trinh phdn tich tong hap trong hoat dong tlm lai gidi
bdi todn sau day:
Bai toan: Cho d la \x6c cua a, b; a,b > 0 vd + GZ.
b a
Chiing minh rang ^a + b> d
Qua trinh tim lai giai:
Phan tieh:
Trudmg hop d<0 ludn dung
Trudng hgp d>0 viec chiing minh V^ + 6> d o a + b > d . Den day viec phdn
tich gap khd khan budc ngudi giai quay sang thao tdc tu duy tdng hop
Tdng hap:Tir gia thiet
b a a.b
MaaJ^vdb;^ nen a^i^'; b ' i ^ ' v d a.b;^'(2), Tir (1) va(2) ta cd a+bi j ' h a y
b.So sdnh' tuang tu trong hogt ddng gidi todn
So sdnh la mot thao tac cua tu duy de ddi chieu, can nhdc nhung diem (adsbygoogle = window.adsbygoogle || []).push({});
gidng nhau, nhutig chd khac nhau cua hai, hay nhieu hien tuang; sau dd riit ra nhiing dieu cdn thiet giiip cho nhu cdu gidi quyet vdn de dang dat ra.
Tuang tu la mot thao tdc tu duy eua ngudi hpc trong van dung nhung dieu
nhan thiie dupe tir mot ddi tuang, hien tuang nao dd vao viec gidi quyet cac ddi tuang, hien tuang cd nhiing diem gidng hoac gdn vdi ddi tuang da bie't.
Trong dgy hgc gidi todn, thao tae tu duy so sdnh va tuang tu hda cua
ngudi hpc la rdt quan trpng ddi vdi boat ddng gidi todn ddng thdi chinh nd cung dupe hinh thanh, ren luyen va phat trien trong boat ddng giai cdc bdi todn. Nhd thao tdc tu duy nay, ngudi hpc ed the so sdnh ddi tuong todn hpc A
cd cae tinh chat a,b,c.., vdi ddi tuomg todn hpc B ed cdc tinh chat khac de tim ra sir tuomg ddng hay khde biet, hoac ngupc nhau eua cdc ddi tuong todn hpc. Day chinh Id co sd, Id tien de dan ket ludn todn hpc bdi nhd nd ngudi hpc de
dang dua nhung bai todn mdi la ve cac bai todn quen thupc da ed cac gidi hoac sdng tae nhimg bai todn mdi.
Tuy nhien, can phdi Iim y hpc sinh rang: ciing nhu quy nap khdng hoan toan, phep suy luan tuomg tu cd the dan den ket ludn sai,
Vi du: Vidul:
- Ddi tuomg todn hpc A Id tam gidc trong mat phang - Ddi tugng todn hpc B Id tir dien trong khdng gian
Vdi thao tdc so sdnh vd tuomg tu ngudi hpc ed the eho ra ket qud nghien
Tam giac