RlN UlYiN 1NA0 lAC TO ROY POANIICH VA IRNG OOP GHO HOC SINH I t P 9 IHRNG OUA GAC RAI TOAN GIAI RANG P H I N G PHAP Vt IHtM HINH PHO ThS BACH PHUONG VINH* Mdt bai toan (BT) hinh hpc ed the cd nhieu ea[.]
Trang 1RlN UlYiN 1NA0 lAC TO ROY POANIICH VA IRNG OOP GHO HOC SINH ItP 9 IHRNG OUA GAC RAI TOAN GIAI RANG PHING PHAP Vt IHtM HINH PHO
ThS BACH PHUONG VINH*
Mdt bai toan (BT) hinh hpc ed the cd nhieu
each giai khae nhau, nliung cd nhQng BT
vdi dQ kien da cho hpc sinh (HS) cd t h i
gap bitdc, khdng tim ra each giai nlu khdng ve ttidm
hinh phij Odi vdi nhung BT dd.vlgc ve them hinh
phy giup HS tim dupe lijl giai d l ddng va thudn Ipi
hon, thdm chi cd BT phai ve them hinh phy mdi tim
dupe Idi giai Vd'n d l ddt ra Id lieu ed phuong phap
(PP) ve them hinh phy ehung cho eac BT hinh hpc
hay khdng, ve them hinh phu nhu f h l nao de thudn
Ipi cho vide giai toan vd phat trien tu duy eho HS?
Oe tao ra dupe mdt hinh phu lien kit tudng minh cdc
mdi quan he toan hpc giua dilu kign da cho (gia ttiilt)
vdi dilu kien phai tim (kit luan), ddi hdi HS phai thyc
hidn cde thao tac tu duy: phdn tich (PT), td'ng hpp
(TH), so sdnh, tuong ty hoa, ddc bigt hod, khai quat
hod ; trong dd, ttiao tdc tu duy ptidn tich va td'ng
hpp (PTVTH) giu vai trd nin tang Giao vien (GV)
ein trang bj eho HS nhung kiln thQc, kT ndng can
thilt va eac bien phap PT de tim each ve hinh phu
phu hpp cho mdi BT; tQ dd, HS cd t h i chu ddng tim
dupc tiudng giai quydt vd'n d l vd ren luyen eac thao
tac tar duy PTVTH
1 Phan tich va tdng hop
'PTIi dung tride chia cai toin there thanh tung
phan, hoac tach ra tdng thude tinh hay khia canh
rieng biet nim trong eai toan the do"; "TH la dung
tri dc htfp lai cie phan cua eii toan the, hoac ket
hgp lai nhung thude tinh hay khia canh khic nhau
nim tmng eii loan thedd"{\) Theo Tu dien Tieng
Viet: VTIaphin chia that su hay bing tudng tugng
mgt dd'i tugng nhan Me ra thanh cic ye'u to, trai
vdi TH; THIi to hgp bing tudng tugng hay thatsu,^
cic yiu td'rieng re nio dd lim thanh mdt ehinh the,
trai vdiPr(2) Theo Nguyen Ba Kim: "PTla tieh
(trong tu tudng) mdt hg thd'ng thinh nhung vat, tieh
mot vat thinh nhung bd phin rieng le; TH la lien ket
(trong tu tudng) nhung bd phin thanh mdt vat, lien
ket nhieu vit thinh mdt he thong"{3)
Cdttilhiaj, ttong ho^dgng giai toan, PTVTH dupc
ttie hien: PT BT la ndu rd gia thilt (ylu td da cho) va
42 Tap chi Gido due so 302
kit ludn (ylu td phai tim), tim mdi lien he glQa chiing
o l giai quy^ BT, ta cd f h l tach, chia BT ttianh taing tmdng hpp lidng le, vlec giai quylt taing van de nhd se
d l ddng hon TH ode ylu td, mdi quan he glQa eac yd'u td vQa PT ehung ta se nJt ra dupc kit ludn mdi;
TH each giai ciia cde BT bd phdn, lien kit thanh ldi giai eua BT ban dlu PTVTH la hal thao tae tu duy trai ngupc nhau, chung la hai mat dd'i lap cua mdt qua trinh ttidng nhat Tuy nhien, trong qua trinh giai toan,
HS cd f h l ttiyc hign lien tilp eac thao tac PTVTH d l tim ldi giai, khai ttiac, phdt triln BT Qua dd, HS ed ttie ndng eao ndng lyc giai toan cho HS
2 PP ve them hinh phu va mot sd'loai hinh phu thudng dupc sir dung b'ong giai toan hinh hpc
1} Yiu ci'u cua vlec vi them hinh phy Vige
ve them hinh phu nhdm mue dich tim each giai BT, lam cho BT bd nen don gian han, giup HS dua ra dupc nhQng ldi giai hay, ddc dao Mudn vdy, nd phai
Id kit qua cua eae hoat ddng tri tug: PT, TH, tuong ty d l gan kit mdi quan he giQa kiln thQc da cd vdi gia thilt da cho vd kit ludn phai tim d l giai quyet BT Vethem hinh phy ludn phai tudn theo cac phep dyng hinh CO ban va can chu y Id khdng dyng hinh phu mdt cdeh tuy tidn ma phai lya ehpn each dyng thich hpp, thda man cac tinh chat nao dd cua BT
2} Mgt so' loai hinh phu thudng duac sir dung trong giai toin hinh hgc: • d m f m n g dilm
cua dogn ttidng; dilm eh'ia trong hay chia ngoai doan ttidng dd cho theo mdt ti sd thieh hpp;jiao diem eua
cac hinh, ; - Doan thang, dudng thing, fra.'ndi hai
did'm cho bude; keo dai mpt doan thing cho tmde; tai mdt diem cho tmde, dyng dudng thdng song song (hode vudng gdc) vdi mpt duiimg ttiang da cho; dung dudng phdn giac cua mdt gdc cho budc; dyng dudng ttidng di qua mdt diem cho tmde hep ttianh vdi ducrig thing khae mdt gdc bdng gdc eho tmde; tQ mpt did'm
cho ttudc dyng tiep bjyin vdi dudng bdn da cho; dyng
ddy cung chung eua hai dudng trdn giao nhau; vd tia ddi Clia mpt tia; dung eac dudng dac bidt trong
* Tniaag Bai kpc U pka* - Bfi kf c ^ai N W i i
•(ki2-1/2013)
Trang 2tam giac; ; - Oa gOctam giac, tam giac dlu, higiac,
hinh vudng, hinh binh hdnh ; - Oudng trdn:\ie eac
dudng trdn hode cung chQa gdc di qua cac diem da
cho; ve dudng trdn tilp xuc vdi mdt dudng trdn hode
dudng ttidng da cho; ve dudng trdn ndi hode ngogi
tiep da giac
3) Mgt so bign phip PT tim ra cich ve hinh
phu thich hgp: • Qua vecac BT da biet: PT gia ttiilt,
kit ludn cua BT, tim ra cae ylu td tuong ddng vdi
kiln thde dd cd cua HS de xac djnh hinh phu thich
hpp ein ve them, dua BT cin giai quylt v l BT quen
ttiupc; • Tao ra yiu to trung gian: PT kit ludn cila
BT, tim mdi lien he giQa gia thilt vdi ylu td tmng gian
de tdylu tdtmng gian suy ra kit ludn; - Bien doi ket
kj$n eua BTdua veminh de tuang duang: PT biln
dd'i kit ludn cua BT thanh cac mdnh d l tuong duang
cd kha ndng gpi ra hudng ve hinh phy de giai BT;
- Dua vio biin doi dai so: PT kit ludn ciia BT, tim
mdi lidn he vdi cae djnh li, tinh chat, cdng ttiQc cd
lien quan, sau dd lien kit cdc mdi quan he de xac
djnh hinh phu cin ve
3 Giai mpt so BT hinh hpc bdng each ve
them hinh phu nhdm ren luyen cho HS thao tac
tuduy PTVTH
Cd nhilu loai hinh phu trong giai toan hinh hpc
ndn vigc ve thdm hinh phu rd't da dang, khdng theo
khudn mdu nhd't djnh ndo Do vdy, khdng ed PP
chung eho vide ve them hinh phy Mudn vd dupc
hinh phy thieti hpp, thuan Ipl cho qud trinh giai toan
ddi hdi HS phai liilt dy doan tdt tren co sd cde suy
Iugn, ed cdi nhin td'ng quat, bilt nhin nhdn BT mpt
each tdng the dudi nhilu khia egnh khae nhau, khai
ttiac cac du kign tilm an trong cde yd'u td da eho cua
BT Sau dd, lidn kit mdi quan hg glQa cde ye'u td dd
va dua ra Idi giai, phat triln BT
Vi dij 1: Cho dudrng
bdn (0) va did'm A ndm
ben trong dudng tnjn Ve
day BC vudng gdc vdi OA
tai A, ve day EF bd't ki dl
qua A va khdng vudng gdc
vdi OA Hay so sanh dp dai
haidayBCvaEFC/j//i/i/;
PTBTlim cich gia: De
so sanh dp ddi hai dogn
ttidng khdng bdng nhau cd rat nhilu each, chdng han:
dya vao quan he giua canh va gdc ddi dien trong ^ m
gide; quan he giua dudng xien va hinh chilu; bat dang
ttiuc tam giac; quan hg dudng kinh va day cung;
khoang each tai tdm din day trong dudng bdn,
' ^ L
Hinh I
(ki2-1/2013)
Theo gia thiet, BC ± 0A tgi A nen OA Id khoang each tQtdm 0 de'n ddy BC Oilu nay hudng cho HS suy nghT: mudn so sanh hai ddy BC va EF edttie su dyng kiln thQc v l khoang each tQtdm de'n ddy eung ttong dudng trdn DQ kien cua BT da cd khoang each tQtdm 0 din ddy BC, chua cd khoang each hi tdm
0 din ddy EF nen HS phai tao ra yd'u td dd bdng each ke O K I EF(K6EF)
Dya vao quan he canh huyen vdi egnh gde vudng trong tam gide OAK va khoang each tQtdm din ddy,
HS de dang suy ra dupc BC < EF Do dd, dudng vudng gde OK ha tu O den day EF la dudng phu can ve
Ldigikye OK 1 EF (K e EF) Tam giac AOK
vudng tgi K, ed OA la canh huyin va OK la mdt canh gde vudng ndn OA > OK Theo djnh li ve khoang each tQ tdm din ddy cung cua dudng trdn suy ra BC<EF
TH lai cde kit qua, ta cd: EF la ddy bat ki di qua
A vd khdng vudng gdc vdi OA, ddy BC dl qua A vd vudng gdc vdi OA thi khi dd BC < EF HS di din kit ludn: trong eac ddy eung di qua A, ddy cung vudng gde vdi OA tgi A cd dp damhd nhat
Vidu2: Cho doan thing AB, hai dudng thing d
vd d' lan lupt vudng gdc vdi AB tgi A vd B M Id tmng diem^ cua AB Lay C, D lln lupt tren d, d' sao cho CMD = 90° Chiitig minh rdng CD Id tilp tuyln eua dudng trdn dudng kinh AB
PT cae yeu to cua BT: O l CD la tie'p tuyln eua
dudng trdn dudng kinh AB thi CD phai vudng gdc vdi ban kinh cila dudng trdn Vi trong d l bdi chua xud't hien ban kinh dd cua dudng trdn ndn ta cin tao
ra bdng each ke MH I C D (He CD) va chiing minh
MH = MA
M udn si} dyng tinh chd't v l tilp tuye'n cila dudng trdn, ta phai chung m i n h : ^ M C = AHMC, nghTa la phai ed them ylu td 6, = 63
Vdi cae du kien eua BT chua du cho HS chiing minh, vi vdy, cin thdng qua mpt gdc tmng gian bdng hai gde tren Oilu nay hudng HS nghT din vide tao
ra yd'u td trung gian bang each ve them hinh phy ttiich hpp de tim each glal (ap dung bien phap tgo ylu tdtmng gian);
Tu gia thilt: CMD = 90''=> ACMD vudng Theo
tinh chat: trong tam giic vudng, trung tuyin xuat phit tddtnh gde vudng bing nua canh huyen, dilu
nay gpi y cho HS nghT dd'n vide tao ra y l u tdtmng
gian bang each ve diem phu N li tmng die'm cua
CD (CMFI la ye'u td tmng gian) Mat khae, do d //
d" ndn tdgiac ACDB la hinh thang va M, N lan lupt
Tap ehi Gido due so 302 43
Trang 3la tmng diem eua AB, CD
(hinh 2)
Khi dd, MN Id dudng
ttung binh eua hinh ttiang,
suy ra MHIIdlld' (cung ed
ttie ve ttidm dudng phu
bang each tQMkeMN//d,
sau dd suy ra N la tmng
did'm eua CD) vd CWi =
Cich 1: TH ket qua cua budc PT, fa cd Idi giai
cua BT nhu sau: Lay N la tmng did'm eua CD, suy
ra MN id dudng tmngbinh ciia hinh thang ABDC
=> MN //d => e , = Cm (so le trong) (1) Theo gia
ttiiet ACMD vudng tai M, ed MN Id dudng tmngbjyln
=>NC = NM => ANCM cdn tai Hz^t^ = Gm (2)
TQ(1) va (2) =>e, = Cj Do dd: AAMC = AHMC =>
MH = MA
Cich 2:T\&p tijc PT cac ylu cua d l bdi, ta cd:
theo gia thilt, M Id tmng die'm eua AB, d // d' nen ta
se tao ra gdc tmng gian bdng hai gdc 6, vd 6^ ttidng
qua vide keo ddi CM cdt d' tai E (pEQ la gde tmng
gian dupe tgo ratQ tam
giac edn dinh D cd
egnh ben DC, canh
day ChQa CM) Vdy, E
la die'm cin ve thdm
(hinh 3)
Keo dai CM cdt d'
tai E, ta cd MC = ME
Theo gia ttiilt CM 1
MD=>ADCEcdntgiD
(vi ro DMvia la dudng eao vQa la dudng tmng tajyln)
=> 6^ = DEC va e, = DEC (sole trong) => 6, "
W do 3.'Cho tam
giac ABC vudng tai A,
AHIddudngeao.Cho
bilt BH = a, HC = b
Chung minh ring:
Jabi^Y- (hinh4)
PT BT tim cich
fliiaA'OlgiaiBT, HS
can tim mdi lien he giQa eae bilu thQc d hai ve eua
bit ddng ttiiic vdi ede ylu td trong BT va biln ddi
cac bieu ttiQc d hai ve Xet ve phai eua bat ddng thQc
cin chiing minh, ta cd: ^ = ^ , gpi HS nghT de'n
tmng taiyin AM ciia ABC vudng tgi A, khi dd: AM =
^ , A H = V ^ v a A H < A M
44 Tap chi Gido due so 302
d
C
A
J t - ^ ^
M \
Hinh 3
•Or
TH cac kit qua tren suy ra dilu phai chung
minh Vdy, hinh phu can ve la dudng tmng tuyen
AM Clia AABC (M Id bung did'm cua BC) Ta di din
Idi giai BT
^ Ldi giii: Dyng dudng bung tuye'n AM (M la tmng
dilm eua BC) cua AABC AABC vudng tgi A, cd
AH la dutjng cao, AM la dudng tmng tuye'n nen: AH^MvaAH2 = BH.HC=^^// = ^ ; A M = ^ = ^
Vidu 4:Cho tam giac nhpn ABC ndi tilp trong
dudng trdn (0; R) Gpi H la tn/c tdm cila tam giac,
ke OK 1 BC, (K e BC) ChQng minh AH = 20K
(hinh 5)
PT cic yew td trong BTvi ma lien he vdi nhdng dieu dabi^Qk\g ttiuc AH = 20K giup HS lidn tudng
din kiln ttiuc v l dutJng tmng binh eiia tam gide cd mdt canh la AH, 0 la tmng did'm cua mdt egnh, K la tmng diem eua canh kia (cin xac djnh tam giac nay) Xet yd'u td dilm 0 Id tdm ciia dudng trdn va la tmng dilm cila mdt doan thing thi dogn thing dd phai la
dudng kinh, nen ta ke dudng kinh AD di qua 0,
na HD BT phai chQng
minh K la tmng dilm cua
HD Ta cd K la bung did'm cua BC (ttieo gia ttiiet OK
I B C ) , nlu K la ttung dllm ciia HD thi tii giac BDCH
Id hinh binh hanh Na BD,
DC ta dupc taigiac BDCH
Ldi giii: Ke dudng kinh AD, ndi HD Xet AAHD
cd 0 Id tmng diem eua AD, theo gia thid't OK 1 BC
=> K la tmng die'm cila BC Nd'i BD, DC ta ed tQ giac BDCH la hinh binh hanh (vi cd DC//BB', BD//HC)
vd K la tmng diem cila BC => K la tmng did'm cua
HD (theo tinh chat ciia hinh binh hanh) ndn OK la
Hinh 4
Hinh 5
duijng tmng binh eua AAHD;
:20K
>OK=^AHhayAH
Tren ddy la mdt sd BT minh hpa eho bidn phdp ren luyen ndng lyc PTVTH cho HS thdng qua eac
BT giai bdng PP ve them hinh phy; lam sang td sy van dyng cac bidn phap ve hinh phy ttiich hpp eho mdi BT Odng thdi, nd cung chiimg td vai trd quan bong eua cac ttiao tae tu duy PTVTH ttong hoat dpng giai toan cua HS Cd PT BT mdi xac djnh dupc cac ylu td da cho, yeu td phai tim, mdi quan he gida
(ki2-1/2013)
Trang 4Chung va ttiQ ty cua ede budc giai quylt van d l
TH eac kit qua cua PT de hoan ttiien kji giai va phdt
trie'n BT ban dau
• * •
Trong qua trinh giai ede BT hinh hpc, viec ve
ttidm hinh phu khdng nhung giup HS tim dupe \d\
glal mdt each dd dang ma cdn dua ra nhQng ldi giai
hay, ngdn gpn, ddc dao mang tinh sang tgo cao
Cac BT giai bdng PP ve them hinh phu ddi hdi HS
phai ndm vQng kiln thQc vd cd kT ndng giai todn nhlt
djnh Oltao ra dupc mpt hinh phu lien ket tudng minh
mdi quan he toan hpc glQa ode dilu kidn da eho vdi
dilu klen cin tim, HS phai thye hign ede thao tac tu
duy PTVTH; qua dd, phat trie'n ndng luc tri taig va tu
duy khoa hpc eho H S a
(1) Hoang Chiing Phuong phap day hpc to^n hpc d'
truiang trung hpc CO stir NXB Gidoduc, H 1997
(2) Hodng Phe (chu bien) Tur di^n tigng Vi|t NXB
Dd Ndng vd Trung tdm tudiin hpc Ha NOi - Dd NSng
2000
(3) NguySn Ba Kim Phuong phap day hpc mon Toan
NXB Dai hpc suphpm, H 20II
Tai liSu tham khao
1 Phan.DiJc Chinh (to'ng chu biSn) - TOn Thdn
(chu bieii) Todn 9 NXB Gido difc, H 2005
2 NguySn Bd Kim - Vucmg Duong Minh - TOn Than
Khuydn khi'ch mdt s6 hoat ddng tri tai ciia hpc sinh qua mdn Toan o truong trung hoc co s* NXB Gido
rfuc, H 1999
3 BCii V3n Nghj Phuong phap day hpc nhung ndi
dung cu thd mOn Toan NXB bai hoc su pham,
H 2008
4 NguySn Due Tdn Ve them yeu tO' phu de giai mOt
stf bai todn hinh hpc 9 NXB Gido rfucH 2005
5 Bach Phuong Vinh "Khai thac cac bdi todn trong d?y hpc phdn hinh hpc lop 9 nhdm rfen luy$n thao tdc
tu duy phan tich vd tdng hpp cho hpc sinh " Tpp chi
Gidoduc, thdng 9/2011
SUMMARY
In this paper, the author presents several analy-sis measures to find out how to draw more shapes appropriate in order to solved geometry and some geometric problems class 9 solved by method draw more shapes, to train students to manipulate analyti-cal thinking and synthesis
Day chien thuat dpc hieu
(Tief> theo tmng 39)
dpc hid'u dy la gi: mpt vdn ban bi "phdn tieh", bj
chia nat, rp la viet ve eon ngudi va cudc sd'ng ma
ddi khi chdng cd mpt chut sinh khi nao - nhu vdy,
lam sao gid vdn cdn ed "chd't van"? GV mudn thj
pham mdt CT nao dd de hinh thanh kT ndng "du
doan" trong khi dpc, lam phong phu them su cam
nhdn, coi la co sd de ddnh gia each li giai v l ddi
sdng cua nha van, nhung lai chua bao gid tQng
tich cue "ddn dau" nhu vdy trong qua trinh ddc
lap dpc hie'u van ban eua minh Neu vdy, hp lam
sao ed the dy doan dupc, lai cang khdng the tien
ligu nhQng dudoan phong phu cua HS trong tinh
hud'ng ey the de du djnh phuong an dilu khie'n,
dilu chinh eho phu hpp, de ludn ludn ehu dpng
trong mpi tinh hud'ng dgy hpc GV chi ed the day
dpe hid'u vdn ban cho HS khi ban thdn cung la
mdt "ddc gia" dung nghTa; bilt "phdn thdn" de
quan sat xem minh dang dpc hie'u vdn ban nhu
t h i nao, gap khd khdn gi; biet ddt minh vao vj tri
HS - nhung dpc gia phat trie'n - de tien lieu eae
tinh hudng can thao gd Cd le it nha't dd eung la
mdt vai djnh hudng trudc khi ngudi dgy thuc su bdt tay vao thid't k l va su dyng cde CT day hpe doe hid'u vdn ban •
(ki2-1/2013)
(1) Harris, T - Hodges, R The literacy dictionary:
The vocabulary of reading and writing DE:
International Reading Association, 1995
(2) National Council of Teachers of English, International Reading Association Standards for
English Language Arts Urbana.lL:NCTE, 1996
SUMMARY
Reading comprehension strategy is a cognitive process carefully led by specific purposes or a processing way to control the reader's effort to de-coding and making meaning from text The relation-ship between strategy and skill is caused and addi-tional As a model of information theory, there are 3 kinds of strategy: cognition metacognlUon and inter-action From the study of reader's potraJt strategies are devided into some kinds as follows acting prior knowledge and experiences, predicting, making ques-tions There are two approaches in strategy train-ing, which are to provide direcUy students with ba-sic strategies, and teach students indirectly through problem solving To do that, teachers should deride skills into elements which help students observe, identify and practise those skills
Tap chi Gido due so 302 45