van dung cap pham tru cai chung cai rieng trong day hoc toan

UAN DUNG CAP PHAM TRU "CAI CHUNG - CAI RIENG
TRONG DAY HOC TOAN 6 TIEU HOC
O

fong chuong trinh todn ndi ehung, mdn Todn
d tieu hpe ndi rieng thudng ddn ddt hpe sinh
(HS) di tir nhtfng trudng hpp rieng rdi khdi
qudt len, duo ve bdi todn (BT) tdng qudt hon. Khi
gidi bdi tdp, HS Iqi vdn dyng nhifng khdi niem
ehung, quy tde tdng qudt vdo tCrng trudng hop
rieng cy the. Ve mat phuong phdp ludn, giira edi
ehung vd edi rieng ed mdi quan he ehdt ehe vdi
nhau: - Mdt cdi rieng ed the Id trudng hop ddc
biet cuo nhieu edi ehung khde nhau; - Mdt edi
ehung dem dqc biet hdo tung bd phdn se eho
nhieu edi rieng khde nhau.

T

ThS. P H A M THI THANH TU*

Khai thdc BT: Ddc biet hdo dp ddi cqnh eCra tu
gide ABCD trong vi dy tren, chdng hqn eho cqnh AD
= 0 hoqe cqnh AB = 0 to ed ede BT rieng khde nhau:
BT rieng 1: Cho tom gide ABC, gpi P, Q Id
hai diem thude canh BC sao eho: BP = PQ = QC.
Hdy ehung td dien tieh tom gide APQ bdng -dien

tieh torn gide ABC (hinh 2j.
Dinh hudng Idi gidi: Van dyng tinh ehdt neu
hoi tom gide ed cirng ehieu coo thi dien tieh ti le

thudn vdi dp ddi ddy
hoy tl so dien tieh bdng
Dudi ddy Id mdt so vi dy minh hpo eho viee vdn ti so dp ddi cuo hai ddy.
dyng edp phqm tri) «edi ehung - edi rieng" trong
BT rieng 2: Cho tom
dqy hpe todn d tieu hpe.
gide ADC. Gpi M , N Id
1. Dqc biet hda h/ng thdnh phdn cua mdt hai diem thude cqnh AD
BT (edi ehung) se cho nhieu BT mdi (edi rieng) sao cho A M = M N =
khde nhau. Thdng thudng, trude mdt BT, HS gdp ND vd P, Q Id hai diem thude cqnh AC sao eho:
khd khdn trong viec tim Idi gidi. Viee dqc biet hdo AP = PQ = QC. Hdy ehung td dien tich tu gide
cde yeu to cuo BT se tqo ro dupe BT mdi md HS d l
ddng tim dupe Idi gidi hon; Gidi quyet BT mdi se MNQP bdng - dien tieh tam gide ADC (hinh 3j.
Id mdt su gpi y, djnh hudng eho viee tim tdi, phdt
Dinh huong Idi gidi:
hien cdeh gidi BT ban ddu.
Vf dy 1 (edi ehung): Cho tu gide ABCD. Tren
•^i.i,Kir-.
A v * D — ^kjicr^'
^ n r ^ / - ~ 3T "ADC
*^
nen
•^MNQ — ^Kif~^r./
NQD' ^AMP
MPQ- L.'O •^[X3C
cqnh AD ldy hai diem E, F sao eho: AE = EF =
FD. Tren cqnh BC ldy hoi diem H, G sao cho: BH SADQ= ySADC'^uy ro
= HG = GC. Nd'i EH, FG. Hdy chirng td dien tieh
S
- is

tu gide EFGH bdng - dien tieh tu gide ABCD ^MNQP 3 ^ADC"
Nhdn xet: BT rieng
P
Q
(hinh Ij.
2 Id m d t t r u d n g
Hinh 3
Dinh hudng Idi gidi:
hop ddc biet eua BT
ehung dd eho khi tu gide ABCD cd cqnh AB = 0
hoy triet tieu cqnh AB.
2. M d t BT (dugc coi Id cdi rieng) cd the Id
Mdtkhdc,doS^gg + S ^ =
trudng hqp dqc biet cua nhieu BT (cdi ehung)
TCf mdt BT (trudng hpp rieng), to md rdng vd'n
-S
- -S
3^ADB"'"
nen
3^BDC
3^ABCD
de thdnh cde BT mdi (mdt trudng hpp ehung, tdng
qudt). Hay ndi edeh khde, BT ban ddu Id trudng
^ E D G B - 3T ^ AABCD>
BCD(2).
hpp ddc biet eua ede BT mdi md rdng. Thye te
TCr I ), ( 2 ) s u y r a S , p 3 , = - S ^ , ^ .

<^>

*TrUoingDaihpcVinli

Tap ehi Giao due 56 2 5 6 (ki a • a/aon)


eho thdy, ede phdt minh todn hpe phdn Idn Id sy
md rdng ttr nhtfng cdi rieng dd biet.
Vl du 1 (BT riengj: Cho tam gide deu ABC.
Diem I ndm trong tam gide Id dinh ehung eua ba
tom gide bdng nhau lAB, lAC, IBC. Cde diem M,
N, P Idn luqt Id trung diem cuo ede cqnh IA, IB,
IC. Xdc dinh ti sd dien tieh gitfa hai tam gide
MNPvdABCf/7/n/j4j.
Dinh hudng Idi gidi: " "

67 ehung 2: Cho torn gide deu ABC. Diem I
ndm trong tam gide Id dinh ehung eua bo tam
gide bdng nhau lAB, lAC, IBC. Tren IA ldy diem
M sao eho: IM = - IA. Tren IB ldy diem N sao eho
IN = ^ I B . Tren IC ldy diem P sao cho IP = y IC.
Hdy xdc dinh ti sd dien tieh gitfa hoi tam gide
MNP vd ABC.

Theo bdi ro to ed: S^, = S^^ = S^^i = j S ^ c - ^^'^
C

Dinhhuextglagidi:Tacd.S^
°do^^=

-0^^=

QC

= -^S^^
^ "MBI = — S ^ ;

4^ABI= 12 ABC- . . . .
c

Tuong ty:S^p,= - S ^ c = SMM-

S

•

ce

ce

- —S - — S

^NPI ~

t/

BPI ~

^

^CBI

- — S •S
df

- - S

-

ABI

ac , ce , ae ^^

^^y- ^MNP= T2 ABc~ 4^ABc-

^y

bf

Phdn tich «edi rieng"
A
edn md rdng d tren theo
ede yeu to nhu: tam gide
deu ABC, vj tri diem I,...
to ed ede BT ehung dupe
md rdng. Cy the, to ed
Hinh 4
ede BT ehung dupe md
"
rdng tCf BT r/eng dd biet
nhusou:

BT ehung 1: Cho tom gide deu ABC. Diem I
ndm trong tam gide Id dinh ehung eua ba tom
gide bdng nhau lAB, lAC, IBC. Tren IA, IB, IC Idn
I _. I-' J-'
kA kl D ''A/_ /A^ /P a
luot lay eoe diem M, N, P: —
=—= '^

'

'

IA

IB

IC

b

(vdi 0< a < b). Hdy xdc djnh ti sd dien tich gitfa
hai tam gide MNP vd tam gide ABC (hinh 5j.
Dinh hudng Idi gidi: Theo bdi ra, ta dupe:
s

- ^ s

•-^MNl

- — s

jj^tM\

''•' '"'""':'•:'

t>^

Tuong tu: S

Mr

-

"'NPI-

-,

,.,;.;•;

•'•':
^.

. ; .

,

0 ^
^ c
c
'^ c

' i ' ^ B P l " p"^CBI'"^MPI" " ^ ^ A P | -

A

^s

-^-

• '

Suya:S^^, + S^„+S^p,
= ^TyS...

hay S„^p =

Hinh 5
^S
Nhdn xet: BT dd eho Id mdt trudng hpp tdng
,. ,
' J 1 1 L- I' -' /JW IN IP a 1
quat eua w dy 1 khi ti so: —= — = — = _ = _ .

Tap ehi Giao due so 2 5 6 (ki a • avaoi i)

Ka=Vf^mACI bf ABI Suy ra: S^^p
MNP = (^bd

df
bf
Mt

—+—+-)S.

^°Y^t^,= l^^^^^P^MCNhdn xet: Vf dy 1 Id mdt trudng hpp ddc biet
cuo BT ehung 2 khi: 7 ^ 7 ^ T ^ y Vi dy 2 (BT riengj: Cho 2 sd ty nhien a = 2, b
= 3. Tim them so ty nhien e sao eho tdng eua 3 so
bdng tieh eua ehung. Cd the tim dupe bao nhieu
sd'ty nhien e nhu the?
Dinh hudng Idi gidi: - D i thdy c = 1 thda mdn
BT; - Vdi c > 1 thi khdng cd so e ndo de 2 + 3 + e
= 2 X 3 x e hoy 5 = 5 xc.
Khai thdc BT: M d rdng theo gid tri so eua a, b
ta ed:
BT ehung 1: Cho hoi sd ty nhien a, b khde 0,
ed: ab - (a +b) = 1. Tim them so ty nhien e de bo
so a, b, e ed tdng bdng tieh.
Dinh hudng Idi gidi: Bd bo so edn tim Id: (m,
n, c) trong dd: mne = m + n +e hay e = (m + n) :
(mn - 1) = 1 vi theo bdi ra: n + m = m n - 1.
Nhdn xet: BT d vf dy 2 Id mdt trudng hpp rieng
eua BT ehung / khi m = 2, n = 3.
BT ehung 2: Cho hai sd'ty nhien a, b khde 0: a
= m, b = n. Tim them ede so ty nhien de dupe mdt
bd sd ed tdng bdng tieh.
Dinh hudng Idi gidi: Bd sd edn tim Id: m, n,
1 , . . . , 1 (ed mn - (m + n ) sd 1).
Nhdn xet: Vf dy 2 Id mdt trudng hpp rieng
cuo BT ehung 2 khi m = 2, n = 3.
BT ehung 3: Cho ba sd'ty nhien a, b, c khde 0:
a = m, b = n, e = p. Tim them edc sd ty nhien de
dupe mdt bd sd cd tdng bdng tieh.

v,;i-, Ij
(Xem fiep trang 44j


cdng thue tinh dien tieh hinh binh hdnh. HD ndy
ehua 2 HD nhd Id «gidi thieu ehieu eao hinh
binh hdnh" vd «Xdy dung edng thue tfnh dien
tfch hinh binh hdnh".
3j Do dung DH: Dya tren eo sd ede HDDH,
GV tdng ket Ipi cde dd dung DH edn dung, dy
kien edeh thue td chuc su dyng.
3. Vi dy chudn bj bdi dqy «Dien tich hinh

binh hdnh"
A. Myc tieu: Giup HS: - Biet dudng eao vd
ve dupe dudng coo cuo hinh binh hdnh; - Hinh
thdnh edng thdc tinh dien tieh hinh binh hdnh;
- Bude ddu biet van dyng edng thire tinh dien tieh
hinh binh hdnh de gidi ede bdi tdp lien quan.
6. Do dung DH: GV chudn bj ede mdnh bio
ed hinh dqng nhu hinh ve trong SGK. HS chudn bi
gidy ke d vudng (d vudng cqnh 1 em), eke vd keo.
C. Cac HDDH ehu yeu
aj Cung ed cdng thuc tfnh dien tfch hinh ehu
nhdt: GV yeu edu 2 HS len bdng ldm 2 bdi tdp
(Bdi 1: Tinh dien tich hinh ehtf nhdt. Bdi 2: Tinh
dien tieh hinh ehtf nhdt ed ehieu ddi 8 em, ehieu
rdng 5 em (xem hinh ben), HS dudi Idp eung ldm
rdi nhdn xet. GV yeu edu HS neu edeh ldm tu dd
neu edng thirc tinh dien tich hinh ehtf nhdt. GV

nhdc Iqi edng thue tinh dien tieh hinh ehtf nhdt.
bj Gidi thieu chieu cao eua hinh binh hdnh.
Cde HD cua CV: - Yeu edu HS ve hinh binh hdnh
ABCD vdo vd; GV ve tren bdng; - Hudng ddn HS
ve dudng coo AH cuo hinh binh hdnh (ehi nen
neu mdt dpng dudng cao), sou dd gidi thieu
dudng coo cua hinh binh hdnh; - Gidi thieu ehieu
coo cuo hinh binh hdnh; iuu y HS phdn biet dudng
coo vd ehieu coo.
cj Hinh thdnh cdng thue tfnh dien tfch hinh binh
hdnh. Cde HD cuo GV: - Hudng ddn HS cdt phdn
tom gide ADH (HS dd ed hinh binh hdnh ABCD
vd ve dupe dudng coo AH); - Gpi y de HS ghep
de dupe hinh ehtf nhdt; - Yeu edu HS nhdn xet
dien tich hinh binh hdnh ban ddu vd hinh ehtf
nhdt vua nhdn dupe; GV ke't ludn; - Yeu edu HS
nhqn xet mdi quan he gitfa cde yeu to cuo 2 hinh
rdi rut ro cdng thue tinh dien tieh hinh binh hdnh;
- GV ke't ludn, ghi edng thtrc; - GV hudng ddn HS
thude edng thifc.
dj Thue hdnh: Bdi 1. Vdn dyng tryc tiep edng
thue dien tieh hinh binh hdnh khi biet dp ddi ddy
vd chieu coo (Yeu edu: - HS ldm bdi; - Gpi HS
duo ro ke't qud, HS khde nhdn xet vd neu rd edeh
ldm; - GV nhdn xet vd ket ludn vd nhdc Iqi edeh

^

tinh dien tich hinh binh hdnh). Bdi 2. Vein dyng
tinh dien tieh hinh ehtf nhdt vd hinh binh hdnh

ddng thdi nhdn mqnh edeh hinh thdnh edng thue
tinh dien tieh hinh binh hdnh (Yeu edu: - HS ldm
bdi; - HS duo ro ke't qud; - GV yeu edu HS nhdn
xet dien tieh hoi hinh; - GV nhdn xet vd ke't ludn).
6d/ 3. Bdi todn ed Idi vdn vdn dyng edng thue
tinh dien tieh hinh binh hdnh. (Yeu edu: - HS neu
yeu edu bdi todn; - HS ldm vd chtfo bdi; - GV
nhdn xet vd ke't ludn).
De gid dqy dqt ke't qud tdt, edng viee chudn
bj eua GV phdi ehu ddo, ti mi vd khoa hpc. Sou
mdi tiet dqy, GV cdn ghi chep ede vd'n de ndy
sinh de ed tu lieu hodn thien ke hoqch bdi dqy
hoqe dieu ehinh, bd sung trong ede tiet hqe ke
tiep. Ben cqnh dd, GV eung rdt edn sy chi dqo
sdt sao ve chuyen mdn d trudng, phdng gido dye,
sd GD-DT (sy ehi dqo ndy khdng nen qud mdy
mdc, thien ve hinh thifc md edn khuyen khich GV
ed nhtfng dot phd trong gidng dqy ndi ehung vd
xdy dyng ke hoqch bdi hqe ndi rieng). •
Tai lieu tham khao
1. Toan 4. NXB Gido due, H 2005.
2. Toan 4 (SGV). NXB Gido due, H 2005.
3. Kifiu Due Thanh - Hoang Ngoc Hung - Le Tie'n
Thanh - NguySn Van Tuan. IVIot so van de ve noi
dung va phuung phap day hoc mon Toan tieu hoc.
NXB Gido due, UlOO].
4. D6 Trung Hieu - D6 Dinh Hoan - Ha Sy H6. Phuwng
phap day hpc Toan tieu hoc. NXB Gido due, H 1993.
5. Bo GD-DT. Tdi lieu boi dudng gidng viin cdt cdn
cap tinh, thdnh phd mdn Todn lap 3, 4, 5.

Van dung cap pham tru...
(Tiep theo trang 41 j
Dinh hudng Idi gidi:
M d t b d so edn tim Id: m, n , q , 1 , . . . , 1 (ed mnp
- (m + n + p) so ] ) . •
Tai lieu tham khao
I . G . I . Ruzavin - A. Nusanbaev - G. Shlialchin. IVIot
so quan diem triet hpc trong toan hpc. NXB Gido
di^cH. 1983.
2. Thai Duy Tuyen. Triet hpc giao due Viet Nam.
NXB Dqi hpc supham, H. 2007.
3. NguySn Canh Toan. Phmmg phap luan duy vat
bien chirng bien chiing vdi viec hpc, day, nghien cuu
toan hpc, tap 1. NXB Dqi hpc qudc gia, H. 1997. i

Tap chi Giao due s6 2 5 6 (ki a - a/aon)