Phát triển tư duy thuật giải cho học sinh trung học phổ thông trong dạy học nội dung phương pháp tọa độ trong không gian






 !"!
#$%"!

&'(!)*+,





 !"!
'/01))&2)'3#45/6)728'9:)&8';8<=0'>?@ABC)D;)
EFGH3I,J J,-J
#$%"!
K;D7K1)'9L)&<M)3J'=BN/O)'/)&
&'(!)*+,
#E 

 !"#$%&'()*+,-'./#01(
23-.45647%
%-#-8!/91(%9-(:*;*
4/494 ('<!%9=-3> ?@58
 (647%-A9-5@
%&#8!/5=' 'B=C@4*%*C
DE<:%-A9-5*(
FGHIJ0=(:%-9)K
.H1H4;=LK%-M!K;.=

&N91(%9' O*@/
K

23.
&/0P)';K>?
PQ$RST2
FU2VSW$?QXY
SZ>[\-]
]^F_$V`2>abQ
cSW$>d$V2eVQf2$g\
h2VQiS?W2eVQf2$g\
jeQRV\kXlV_UV`$
mVnT2eVo2eVQf2$g\
p>q2eeq$rU\s2?t2
u$v\wx$\s2?t2
ylXV_z$VV{$VQi2
'9:)&-3 Q#R#S!T
U!E
Q
]]|O! <-4/494 (!%9
]cS<JAO!*:A9
]h$99@!
]h]l9@!8
]jl9@!((8
]j]l9@!(
]jc('@}-9
]jhl9@!(8
]m9-O(8- (9@!
]m]F (9@!4/-5!J,03
~-%•

]p9-O(8- (€*;
]p]F (€*,€-,/JK3~-%

]u9-O(8- (8'49
]y2K• O49-O(8- (
9H (8'49H-:
]‚l.*/]
'9:)&+3VEWS!XY
Q
c]Sƒ,89
cc2<89
ch>JI-
cj$%ƒI-
cj].89
cjc.899:
cm l.A889
cpl.*/c
'9:)&Z3E[ TE \  ]  
 
 !"!
ju
h]S<J(:0 („49-O(8

hcS<J'@4944 !H44149-O(8
- (<4/494<-%
m…
hc]†@494]•-A9-5 (9%}1A#!.
.95J <5!+=49@9%
}H
hcc†@494c•-‡!HO5!

I*389!<'9999*38H
hch†@494h•l.41(ˆK@4*(@ 
98'.&#(A(-5H,8-
 (<4/494<-%
hcj†@494j•-A9-5 (8'46;&#(
A(-5 (-}4/4948'4
hhl.*/h
'9:)&.3WTEE
j]Sƒ,@!
jcD}<@!
jh2<@!
jj^€4 !IO!-
jm>949IO!-
jp>99.A8@!
jul.*/jI&MI (49-O(
8V- (<4/494
<-%
"^#
bQQi\VUSlVR_
!E_^`
Kabbcb Kabde0df
e? e9:
V V
S /-5!‰4~
? ?„/494(.
?$ ?„/Š4/
V -4D%
2 @!
>$ >J}
2&' 2&M'8

VVV5
]
EQ]
-J#RY
]]$%<D!M‰-9
ƒ @!ƒ*.}‰I=H* C#
*M*‹!94}9(:1I@4%@4HŒ
@ HM'0€449-O9:.-:.
9ƒ•LD'D!c…]…ŽA(€•/4949
ƒ4D%4849(,,=9=<=9 
•4G4‰O!‡*4=!%•'C74/
494=8*!@„H!•-B*(@‘ƒ.
}ˆ•9<.58!=„!* I!=}64
’
S%9&„!*!%A-%Š5*!<!%
•%ƒ’M4.}=‘=4/494=64H/
•1.OJ9!%9=H441&#(I8H
4D%3*<!*!4O=!+'•*“HH
H4I49-O-,@!”9#?5(@ (
9H@A8“A(.€.M*9ƒ
]c9-O(8H-+A---34D
%‰'@- (9S%9=HI 98
A(.38-.8 (K'9=K 
9H8=HA08=4#9'O85
*•<-}!<9ˆ/%A9' <=
(:1'98!14G4-5<=H*€
O8A(.'9H
|@5!+8=O8‡'9=‡ 9=“H4
416–(49-O99-,@9•#
c

,=D4=9=9A99=/9=—V/K=+5
K4–!M-,@•,–.=,
* =,<*4=9 =,,!!J9!499
(:•-34D%@(=MI49-O(
8A#!6!}=Šˆ-!<949=H
Š *@9:@FH=9:
 -@995J=9< (‹!49
-O(8
 ]h>H!<J9%-5:}I49-O(
8=-9%-5HHOO*9.•
?/F/S• !"# $%&'
#   ($()*+,-./0012*9.•†G?
2€•34%5%5 6 7897  :
%9# &-")*+,-# ./0012 *
 •:€••;<++=+ %# $
# +,--%'# >7'%5?4 !"
7'%@/122H%-5:}49-O
(8%A (<4/494<
-%
]jM:}4/494 <•-34D%
*:}!<9O@9@:I5•VVŽ4~
%?@Lƒ <O:}VVM*Lƒ%
ƒ JO:}VV2H99=%A!<@ <3
€KJ=K/AK9JVVK
JK!J*:@ J•K,MVVK'O
}=4/-5 J•(O'9VV'9 J@
h
8'9VV(OI*!@9'O}5}'‹9
%ƒ J
|HHOM(<4/494<-%}

II!O49-O(8
2}1!A-MI:-:=6%I
:}**• “Phát triển tư duy thuật giải cho học sinh trung
học phổ thông trong dạy học nội dung phương pháp tọa độ trong không
gian”
+JEghi
Sƒ,:}**49-O(8
-4D%- (<4/494<
-%
ZJTEhi
H@!ƒ8949#)#(•
h](8*5˜ 14849-O(8
˜
 hc>O49-O(81-:K/
•*;*˜
hh$HO49-O(8<.}4/
494<-%(%˜
hj2K'@494O-O(8-
 (4/494<-%˜
.J^"!
2.A#!6!}.4*,@49-O(
8-4D% (4/494<-
%„!<J'@494&#(5“H44149-O
j
(8C3H441#@A88 (
!%9•-34D%
jJ hi
2:}*;*
2:}9*@I*,*4/4948 (!%9=
9*@I#!*,e9ƒ=99=99:=

9*@!8H*:AO*!O!I&M9'@494
49-O(8
$9%-5:}9MIH*:A-.4.I
IJ\\S!#$
 p]V@JH/•*;*I@49-O(8

 pc"#(9'@4944 !‹!49-O(
8- (<4/494<-
%
 phl.A8:}*HO*!*@!8
9:9-4D%
kXl#$
241!•1!ƒ*@!8=*C!H'J
/
 -J Q#R#S!T
U!E
Q
]]|O! <-4/494 (!%9
]cS<JAO!9
]h$99@!I(8
]j9-O(8- (9@!
m
]m9-O(8- (€*;
]p49-O(8- (8'49
]u2K• O49-O(8- (
9H (8'4H-:
]yl.*/]
 +JVEWS!XY
Q
c]Sƒ,89

cc2<89
ch>J89
cj$%ƒI-
cml.A889
cpl.*/c
   ZJ  E  [  T      E  \  ]

 !"!
h]S<J(:0 („49-O(8

hcJS<J'@4944 !‹!49-O(8
- (<4/494<-%
chl.*$/h
 .JWTEE
j]Sƒ,@!
jcD}@!
jh2<@!
jj>99.A8@!
mml.*I@!
p
ppl.*$/j
mJ"^WT
u
'9:)&-
 Q#R#S!T
U!E
Q
1.1. Quan điểm hoạt động trong phương pháp dạy học môn toán
26'.=A9-5 (*!<A9-5IO
 <‹!@K!ƒ, ($+4

*!<A9-5&L*;%|9-5(H9}•%
=%=4#,'.D%=%-
I4JV@9}('‹K <
!5%AK <H‹!6–(49-OI-,@
!<99=,9 
"M49‡!<< (ƒO149@K 
<*:@H-C}!ƒ, (!*K 
<O4*(@!<J-K <49@
?@4#,!< <K <4164D
}.K <<4}4‡48
?@.9 <I+)K-}M
€=‰'@*-}4/4942K-}(* *.A8
!<A9-5 <9- <=.A8-B*(@•!<
!}<HHO* *IIO4*(@ .A8/FH
1484#'K <„K!}<9*!/•
€@Š A9-5 (-:/•@4#,
-:I4/494 („AO! <
:}-%D*;* (=*M(AO! <
*!I8#!*;9ƒ2<AO!(O@
!<9H!0%AK• #(•
y
™A%# $B 5+CD%'7>5%'7>
EF 5G>54 7F '# 
HIG7F 5J7> E %  %'7>
HKL4? 7M :CD? +E++
+E5&NO !"%'7>
HP: %'7>C  ? %5 7L&O9'# 
2(=-:;C•(=**M(AO! 
<*!I8=/•*;*!5
1.2. Một số quan điểm liên quan khác

†: @ƒAO! <=*+
AO!*;(.5J*;(.. '•59AO!
 (9*;(.(HAO!*;(.
 <„*;(.5J5*,}•'C!
CHI.ŽJ!<!%-38-K!#=
KH(!M#'‹H
S<5J3*:@KA(-5<ƒO
HS<(.J5J!(D9-€HHO#(-
(DA(-58A(.MIFH-A9-5
 (1 8-5J/}-}1 (•5
J-}H!<•6ŽH(95J(
V. <4ˆ-3/9
!%-3
„*;(.. =4* <,}3
FH (48* ( <=D}95J4+)
,}=AH. .}=C3
49-O-,@#9!5
‚
2(=.4#,-šAO! („*;(.5J
*;(.. “H44149-O4/494 (49-O
(8:!1(
-JZJ;?n';K)K(B
-JZJ-J"';K)K(Bb'/6b&KoK
V(3.4&6-MI'9‡/8.
4} 4>J!<J'9C KA(0&9€‹!!%8
A9-58‡@!%8A9-58M(=3.9@!-
9I8
•8„•-9O!<(K K
Š@!<9/-€=.6!<JK '
„!* .A8*'.D%•Q2\Ž!<*4'9

%-•_\\Ž!%8*38*4'9H›]m=-humœ
$+„98?/F/S5••8*!<A(0
,&9/-€A(€!<JK K9/M4„!<
-5&9€-:KJ!<JK K
9H.A8!!J’›c…=-]cœ
2K9@!-:IJM-‹!”8IHK
,M/'8A-•
-A9-5:}8A(.9MI™'9=3
-&N•
•$HI'9.(5!-!<98„
O9E%'.HC 9(%
•$HI'9H9O8%M4
538„9HA9*‰9I@
9HH94}
]…
•$HK'98„K984 !9
M4
‡K€-:=3M(-‹148HKD!
9@!923!•-<:–9•
,&9€,60?@!•-<,&9€J9
O@A98@A(:,6
9%+'0'<J!<J98'9=M*98
16-.=HI-343M499
83.A8J•%48*6EJŽ,
4} 4@A8$~ =.8!<'9'‹9J
+)!9(,@-+I!56HOM4
!<84941J!Š1!9(, (-(‰
3
A    Q+7J &-%%7+?7=7!
 R S !"%*7J #C  l9@!

!•-<(9!•-<6-@5!.!
4/494O8A(.9'9‰-2@!•-<,
69=8HM89,M992H
EH95}'Oˆ4469(:=J
!</MM€Š/}!<5}'OˆM€
>‰'@- (16;*4/@'Oˆ4G4
-5<.}@HO'.I8=9,
M4/@'OˆH489-5<H82%
K*4-5*'49-OH8
-J.J"';K)K(Bb9</072b9</0b'/6b&KoK
-J.J-JKhái niệm tư duy
]]
V@&AHI9!3'.2@!ƒ
<J <ˆ*%+)348O'.9
'.H(!<#0=60,&9/=48 -
K9'8MKA(*9<6|9-5
}H*(
(*!<A9-5#!*;489K<,'8M=
!J*:@A@':-H,A(*@-
@9A!-H'.
„‡O-.•ž(=84–!MMD
}!<9‰'@*'<=*A9-5489,.
9A-99@!=499=*;*(&M@-
A9-5 <8&M&<38!'8489
 !<99.4=49@K!J*:@4A(*(Š
C -!J*:@%O9-3) <*<*3
H=* <Š:'O&<*3:(
3@-!J*:@‰“*3HK.A8
(-%K:'O(*KA9
-5-‡9=4#,D4=@:*:*KMI

M€5!98A(.=@I&MK8.=K;
@!l.A8A9-5('3E*!<;•Hž
‡HHO-6K‰O!/'8(
™(*84–!'<3*!<A9-548
9,.9A
™l.A8A9-5('3E*!<;•O
@A%K= <
™†8M(*•4#'@=C <*4J
48958}A8 <
]c
3‹!489J
™(*A9-549-O<9 
™l9O-(489I!}<9‡
<,(.<,9=H4ƒ<O*3
1.4.2. Tư duy biện chứng trong toán học
--.=4/494('@}*4/
494 O8 .4N4'@}(l%K
(=-.*M9:O8-9
%O%H*J( (
2‹!49-O-B*(@*J('@}-9
E48-B*(@859J-9
-:!J*:@6=8•-'<*V
1&„!&N9J•- 9<='.D49-O
„!<(9J„KA(*#
<6$H(!HO* -(=9
 !I!Ÿ-@94ƒ9-}9('@
}+6439H95II=95
DO=9A9!<*4J%AO@H•
I4/@=I!‰V/K+643-•:H9
996'8M9A!@

-<J=JG3-•:8/
1.4.3. Khái niệm tư duy thuật giải
(9*5}'O*<('@}-A9
-53}9(%A5}94ƒ
9992(=(9*('@
}
KC>C%'9? %# <C+E
]h
? :8&N  %'7>$"T
K

TKB D"% U%>9B6 78+VJ+5G
>
K

TPF >O9D"% 7J B U%
9B6 78
K
W
TXO<">O9Y"R>$(7(JRCZ
>O9Y"R>CG+7(J
K

T[- F6 O9N>%'7>
K

T(7ON:%
-H=•
]
ŽO@*@8=•

c
™
m
ŽO@
*&#(8
l9@!(8&9€-:*4G
4K.A8:}I5H898
S%4:*:K41H8'C!•
™VO'8M8K,MH•O'8M
%K*4/@'Oˆ8
™20!K94/49494/@'Oˆ8
™VO,M894/49499}
ƒ6•0!K989-594D%
™VOK/•/M4I*4-5!9(,@L
2(=49-O(8*!<I@1.H4
41549-OH8
‡9@!I(8M(-‹O49-O(
8- (9=9:48D}=IO9
 <(8%A <H640!K=
J9A(0C349-O(8
]j
#(*!<J,ƒI49-O(8-!%9 (
/4/494<-%•-3-4D%
-#(5!OI5%5-!<J-3
4!9(.JIO!=3~=!‰4~!H(.JN
5@89.%48E*!S
-'€4/494„/5!<%<!<4N4
9/}%<!59!JA@KO!=
3~=!‰4~-%
#(*!<J,ƒO@9 <(8•

p<q-JI%'7>
]
K

5!O!3~3
x 1 y 1 z
1 1 2
+ +
= =
−

!4
•Ž
3
c j …6  \+ − − =

9L)&<M)3
>J'9(.9„!O(85-M4} 4=:
9:48(:1(O4/-5,0I4/
-5!J=H9„!!M('9A„<I4-
99
?.4/-5!J3~
>3~AO! 
( )
]• ]•… [ − −
 >3~H„/Š
4/
( )
d
a 1;1; 2= −

uur

/-5!J*3
x 1 t
y 1 t
z 2t
= − +


= − +


= −

r
 ]
∈
lH94ƒ8•
]m
5!O!3~•
…
…
…
= +


= +


= +


6 6 "
  :
\ \ 

!4•Ž•
 …6 ^ A\ _+ + + =

9L?-35!<O!•Ž
e8@4•
…
…
…
U&•†(•$ •F¡…
6 6 "
  :
\ \ 
= +


= +


= +



(&=(= '4/-514/-5J4/
-5
( ) ( ) ( )

… … …
U •† •$ •F¡…
+ + +
6 "  : \ 
stue84•¢Ž
5!
⇒
&=(= 
9L?+3l.*<15!
( )
• •I 6  \

2K&KoK3
9L?-3e
( )
• •I 6  \
*O!3~!‰4~•Ž
e8@4/-5
c j
]
]
…
c
6 
 
6 
\
\



= − +


+ − − =

= − +



= −





(&=(= ='4/-514/-5J
4/-5•
( )
] ] c c j …  − + − + − − − =


⇔ −2 + 2
⇔


⇒



t+4t-4=0

t=1
x=-1+1=0
y=-1+1=0
z=-2.1=-2
9L?+3l.*
−H(0;0; 2)
]p
e))'L32.-I'H4/-5!J5.
4/-5!J-=H!94ƒ8-:
p<q+JV*!'/3
$O!
( ) ( )
…•c•] =  ]• ]•h ^ −
!4•Ž•
c h …6  \+ + =
5!O!
3~U†!4•uJ
2K&KoK
?.4/-5!J3~U†
9L?-35!!<O!!3~A
>3~U†AO!
( )
…•c•]
‰AO!
( )
]• ]•h ^ −

9L?+35!„/Š4/3~
>3~U†H„/Š4/*•
=

r uuur
AB
u AB
( )
]• h•c = −

9L?Z3
l.*4/-5!JU†*•


= +

= +


= + ⇒ = −
 
 
= +
= +


0
0
0
x x at
x 0 t
y y bt y 2 3t
z 1 2t
z z ct


 ]∈

• .4/-5!JU†5@5!O!
3~U†!4•ŽH8
™ e
( )
• •I 6  \
*O!3~U†!‰4~•Ž
™ "N4•
( )
c c h h ] c …  + − + + =


( )
( )
t+2-3t+3+6t=0
5t+5=0
5t=-5 t=-1
x=-1
y=2-3 -1 5 H( 1;5; 1)
z 1 2 1 1
⇔ 2
⇔
⇔ ⇔


⇒ = ⇒ − −



= + − = −

]u
Sƒ,'4(*(:1@ <•
]
Ž
FH19„!ƒ@6„-59'
:-A(0$HOG!<41'8-5'(A(08'
9-‡-34=41'8+* -5'(*384G4
‡A(0.MA9(-!<3M€“
5•A(08'9 -:=C349-O•
9„!*@8
p<qZJ I%'7>
c
K

|9 9I!‰4~=3~=!‰1-/
-599=419„!8-MJ=IH9„!„!<
Š@!<9H@A8-892(A9 
9H9:(:1:9'8'9=‡H9
:€5(8O*!A„C3
*!'
?.4/-5!4•ŽAcO!
( )
h•]• ] = c• ]• •j ^− −
Ž%H
!4
( )
• c h ] …` 6  \− + − =


9L)&<M)3
 VOA(-5*44/-5!‰4~=5!
„/494(.‡8.=9:4#,* O0!K
/
S‰4~AO!U=†5!‰4~“
^
uuur
*!!<
„/Š4/
S‰4~•Ž%H!‰4~•|Ž5!‰4~•Ž„/
494(.!‰4~•|Ž*!!<„/Š4/
2(!‰4~•Ž“H„/494(.
 
=
 
uur uuur uur
P Q
: n AB,n
2K&KoK3
]y
™ S‰4~•ŽAO!
( )
h•]• ]  −

™ ?„/494(.!4•|Ž
( )
= −
uur
Q
n 2; 1;3

™ V„/%G4/H9‰‹!-:•Ž*•

( )
( )
Q
AB 1; 2;5
n 2; 1;3
= − −
= −
uuur
uur
™ S‰4~•ŽH„/494(.*
( )
P Q
: n AB,n 1;13;5
 
= = −
 
uur uuur uur
™
( ) ( ) ( )
− + − + − =
0 0 0
mp(P) : A x x B y y C z z 0

( ) ( ) ( )
⇒ −1 − + − + + =
⇔
x 3 13 y 1 5 z 1 0
x-13y-5z+15=0

„*;(.5J=-A9-5 (=-+48 <
,=<9 =!58!@!A9-58A(.M
I. -K.}HOG-K-34
93 (48643&9.}H
2(=@4#,-:*!99
6'8M'9=5#01(I'9=E‡
H.*49 <•9Ž„!<-54G4
A9-5ˆ-'9
‡5-:=H9'8'9 (•
Thuật giải
9L?-•5!!<O!!!4•ŽA
9L?+35!„/494(.!‰4~
V„/H9‰‹!-:!4•Ž*•
Q
AB n = =
uuur uur
S4•ŽH„/494(.•
Q
n AB,n
 
=
 
r uuur uur
9L?Z•l.*4/-5!4•Ž3

( ) ( ) ( )
0 0 0
A x x B y y C z z 0− + − + − =