Rèn luyện kỹ năng giải hệ phương trình cho học sinh khá giỏi trung học phổ thông tt

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ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC GIÁO DỤC





ĐÀO THỊ PHƯƠNG THẢO




RÈN LUYỆN KỸ NĂNG GIẢI HỆ PHƯƠNG TRÌNH CHO
HỌC SINH KHÁ GIỎI TRUNG HỌC PHỔ THÔNG




LUẬN VĂN THẠC SĨ SƯ PHẠM TOÁN

HÀ NỘI-2012
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ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC GIÁO DỤC




ĐÀO THỊ PHƯƠNG THẢO


RÈN LUYỆN KỸ NĂNG GIẢI HỆ PHƯƠNG TRÌNH CHO HỌC SINH
KHÁ GIỎI TRUNG HỌC PHỔ THÔNG



LUẬN VĂN THẠC SĨ SƯ PHẠM TOÁN
Chuyên ngành: LÝ LUẬN VÀ PHƯƠNG PHÁP DẠY HỌC
(BỘ MÔN TOÁN)
Mã số: 60 14 10



Người hướng dẫn khoa học: TS. Phạm Văn Quốc

HÀ NỘI-2012

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LỜI CẢM ƠN












Bắc Ninh, tháng 11 năm 2012
Tác giả


Đào Thị Phương Thảo

4
DANH MỤC CÁC CHỮ VIẾT TẮT

Viết tắt
Viết đầy đủ

HD
HS
NXB
PT
TH
THPT
TNSP
VP
VT
XHCN







ng



















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MỤC LỤC
Trang
MỞ ĐẦU
CHƯƠNG 1: CƠ SỞ LÍ LUẬN VÀ THỰC TIỄN ……………………
  
 

.
.


.






n

.

CHƯƠNG 2: RÈN LUYỆN KĨ NĂNG GIẢI HỆ PHƯƠNG TRÌNH
CHO HỌC SINH ………………………………………………………





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

.
.
2
)






0)(
2121
2
 xxxxxx


 


.



 





             


23
23
25
28
31

31
32
33
36
36
39

42
45
45

49
57

69
69
70
76

79

85
86
7


10,10 
nm
aathìnma

2.7.3. 
2.8. .
CHƯƠNG 3: THỰC NGHIỆM SƯ PHẠM ………………………




KẾT LUẬN …………………………………………………………
TÀI LIỆU THAM KHẢO …………………………………………

87

88
89
95
99
99
99
114
116
117
118

















3

1.  

, 
“Giúp học sinh phát triển toàn diện về đạo đức, trí tuệ, thể chất, thẩm mĩ và các kĩ
năng cơ bản nhằm hình thành nhân cách con người Việt Nam Xã hội chủ nghĩa, xây
dựng tư cách và trách nhiệm công đồng, chuẩn bị cho học sinh tiếp tục học lên
hoặc đi vào cuộc sống lao động, tham gia xây dựng và bảo vệ tổ quốc”
  “phát huy tính tích cực,
tự giác, chủ động, sáng tạo của học sinh, phù hợp với đặc điểm của từng lớp học,
môn học; bồi dưỡng phương pháp tự học, rèn luyện kĩ năng vận dụng kiến thức vào
thực tiễn; tác động đến tình cảm, đem lại niềm vui, hứng thú học tập cho học sinh”.
Trong 
, cung , 

   

 


, , . 

 logarit. 
. 


. 
 c
 ,,  


4


.
2. 
 

 - 
 
     -     
-
g gian-  , K3,
-

, 
 .
3. 
- 

- 
+  
+ 
+ 
4. 
- c 

- 

5. 
 10A11 2010-2011 

6. 
+ 

5
+ 
7. 

 ,    ,       
 
, .
8. 


+ 



9. 




.
:  
1.1. K 
1.1.1. , 


 kĩ năng là năng lực sử dụng các dữ
kiện, các tri thức hay các khái niệm đã có, năng lực vận dụng chúng để phát hiện
những thuộc tính bản chất của các sự vật và giải quyết thành công nhiệm vụ lí luận
hay thực hành xác định”.
Theo2 “Kĩ năng là khả năng vận dụng tri thức

khoa học vào thực tiễn” 

6
Theo) “Kĩ năng là một nghệ thuật, là
khả năng vận dụng những hiểu biết có được ở bạn để đạt được mục đích của mình,
kĩ năng còn có thể đặc trưng như toàn bộ các thói quen nhất định, kĩ năng là khả
năng làm việc có phương pháp”.
Theo  “Trong toán học kĩ năng là khả
năng giải các bài toán, thực hiện các chứng minh cũng như phân tích có phê phán
các lời giải và chứng minh nhận được”.
  
 


1.1.2. 
 



.


      



g 
, 
 
- 





7
- 
 




- 

- 
- 



m


- 
 .
- 

- 
1.1.3. 

-  
.

-  
- 
- 
- y

1.1.4. 
8
  
- 
 

- TH

-  d  , s         

1.2. Nhy
1.2.1. 
1.2.2.   THPT
1.

1.5. 
1.5.1. 

1.5.2. 
1.5.3.   
 
sinh 





inh.

1.6. 
   

 

9
 


.
: INH
2.1. 
2.1.1. 
2.1.2. 
2.1.3. 
2.1.4. 
2.2. 
2.2.1.  

.
 








)2(.391152
)1(95
3
2
xyxyx
yxx

2. G

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
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yyx
yx

: 


 G








)2(.0222
)1(0964
22
224
yxyx
yyxx

  

 
 
4
 





2.222
)1(1
yx
yx

:t
t.
10
2.2.2. 



5. 







)2(.6)1)(1(
)1(1
2
2
yxyx
xxyx

: 
y+1 theo x.
6. 







)2(.662
)1(922
2
2234

xxyx
xyxyxx
(
:
(3)

.
7. 







)2(.2
)1(3
33
22
yxxy
xyyx







)2(.1)2(log)2(log
)1(24

32
22
yxyx
yx

 :           
)2)(2(4
22
yxyxyx 
(1).
:
.
2.2.3. 
9. 







)2(.282
)1(12
33
22
xyyx
xy

: 
1. 2y

2
-x
2
.
10. 







)2(.)1(51
)1(16411
22
33
xy
xyyx

11. 







.
1
2255

33
yxyx
yx

11

2.3. 

: c 
.
2.3.1. ( 

 







)2(.6)2(
)1(0234
22
2
yx
yxx

: 

.











)2(.
2
9
)1(
3
16
x
y
xy
y
x
xy

2.3.2. 
3. 








)2(.43
)1(54
2
2
xyy
xyx

 







.35
30
33
22
yx
xyyx

:


3
)( ba 
.

t:  nhanh  
.
. 





.23
42
22
yxyx
yxxy

2.3.3. 
. 







.32
32
2
22
xyyx
xyx

12
h: 

.










.3
1
4
1
2
2
x
xyx
x
yx


2.4. 

.
2.4.1.  



(a-v)(u-b)=0. K
. 





.1
1
2
xyx
yxxy

 



3
22
 yxyx


. 








.)1(1
2
2
22
yxyx
yx

:
.
. 







)2().9(9
)1(211
342
3
yyxyyx
yx

  

. G








.1
1
22
yx
yxyxyx

13
. 





)2(.2
)1(331
22
yx
yxxy

2.4.2. 
0)(
2121
2
 xxxxxx


0))((0)(
212121
2
 xxxxxxxxxx

21
, xx

21
, xx

. 







)2(.53
)1(0123
2
2
xyxy
yxxyyx

. 
22

2 (1)
2 1 2 2 (2)
xy x y x y
x y y x x y
   



   


(2008).
 

 

22222
)(2 yxyxyxyyxyxxy 

0)1)(())(()()(  yxyyxyxyxyxyxy

0)12)((  xyyx
.
. 








)2(342
)1(32
22
22
yyxxyx
yxx

 



'

:
0
22
 feydxcxybyax



(my+n)
2

).
    












)2(.75
)1(522
2
2
xxyy
yxyx

14
2.4.3.  

10. 







)2(.32
)1()1()12(2
2
23
yxx

yxyxx

1. 






)2(.022
)1(02
2223
yxyyxyxx
xxy
2012).
: 

(2).
:.
. 








)2(.)(2)(
)1(0)(2345

222
322
yxyxxy
yxyxyyx
2011).

2.5. 




.
2.5.1. rong 
. 








)2(22
)1(2322
22
yxyx
yxyx
(2010).
. 








)2(2
)1(
3
yxyx
yxyx
2002).
3. 







.01
015132
22
22
yxyx
yxyx

15
            


 



2.5.2. .
2.5.2.1. Đặt u = x + y, v = x – y
 







.13
12
23
22
xyx
yxy

 








.13
3
23
23
yx
xyxy

2.5.2.2. Đặt
y
yv
x
xu
1
,
1












.4
11
4

11
22
22
yx
yx
yx
yx

4. 
















.)1(2)1(
4
1
1)(
22

yxyx
xy
yx

 




















.4
1
4
1
1)(

22
22
22
xy
yx
xy
xy
yx
yx

2.5.2.3. Đặt
x
yv
y
xu
1
,
1


 




























.5
1
1)(
3
1
1)(
2
22
xy
yx
xy
yx

16
7. 




























.17
1
1)(
3
11
1)(
2
22
xy
yx
xy
xy
xy
yx

2.5.2.4. Đặt ẩn phụ là căn bậc hai của biểu thức bậc nhất đối với x, y.
. 







.423
112
yx
yxyx

. 








.1
2
22
yx
yxyx

. 







.243232
5323
yxyx
yxyx











22
527
yxyx
yxyx
2001).
2.5.3. .
     
),(),,( yxgvyxfu 
ngay trong



. 







.2
3
33
22
yxxy

xyyx

:           
xypyxS  ,


xyyx ,
22


xyvyxu  ,
22
.
: .

2
22
18
( 1)( 1) 72
x y x y
xy x y

   

  


: 
t 
xy


xy
.
17
t theo
2
xx

2
yy


: :

18
72
ab
ab





(I)
1) Thay
22
,a x x b y y   

(1)
22

18
( 1)( 1) 72
x y x y
xy x y

   

  

2.
2) Thay
22
,a x xy b y xy   

(2)
22
22
18
( ) 72
xy
xy x y








3) Thay

2
2 , 2a x x b x y   

(3)
2
4 18
( 2)(2 ) 72
x x y
x x x y

  

  


4) Thay
11
,a x b y
xy
   

(4)
22
( ) 18
( 1)( 1) 72
x y xy x y xy
x y xy
   



  


5) Thay
22
2,a x xy b y xy   

(5)
22
18
( 2 )( ) 72
x y xy
xy x y y x

  

  


6) Thay
yxb
x
y
y
x
a  ,

(6)



















72)(
18
x
y
y
x
yx
x
y
y
x
yx

7) Thay

yxb
xy
a  ,
11

(7)







xyyx
x
y
y
x
yx
4)(
18
2


8) Thay
y
x
byxa  ,

18

(8)







yyxx
y
x
yx
72)(
18

9) Thay
y
xb
y
xa
1
,
1
2
2


(9)

















32233
72
181
11
)1(
yyxxyyx
yy
xx


- ta thu
t 
  .
- 
22
7

21
ab
ab







1) Thay
22
,a x y b xy  

(1)
22
4 4 2 2
7
21
x y xy
x y x y

  


  



2) Thay

11
,a x b y
xy
   

(2)
22
22
11
7
11
21
xy
xy
xy
xy

   




   



3) Thay
1
,
x

a x b
yy
  

(3)
2 2 2
17
( 1) 21
xy x y
xy x y
  


  


4) Thay
1
,a x y b
y
  

(4)
2 2 2
( ) 1 9
( 2) 21 1
x y y y
x y y y
  



   


5) Thay
22
2 , 2a x x b y x   

19
(5)
22
4 4 2 2
47
4 ( ) 21
x y x
x y x x y
  


   




3. 






.64
9)2)(2(
2
yxx
yxxx

4. 







.58623
542
22
22
yxyx
yxyx

5. 










4
5
)1(
4
5
24
232
xxyyx
xyxyyxyx

2008).
6. 







.1
1
23
2234
xyxyx
yxyxx

7. 








.3
3
2244
22
yxyx
xyyx

8. 







.15))((
3))((
22
22
yxyx
yxyx

:
.
9. 








.41)24()24(
29
yyyxx
yxyx

10. 







yyxx
yyxyx
)2)(1(
4)(1
2
2

.
11. 








.6
191
22
333
xxyy
xyx

2. 







.01
5
)(
03)1(
2
2
x
yx
yxx

20
3. 







.67545
125)13(9
22
33
yxyx
xy


2.6.   
Tro
.
2.6.1. 
2.6.2.   f(u)=f(v)
. 








)2(.1
)1(33
33
yx
xyyx

Nh           f(x)=f(y),  
 f 
 
1;0, yx
f 

 
1;0


. 







)2(
2
1
)1(932293
22

2323
yxyx
yyyxxx
(
A 2012).

- 
f(u) f(v). 
(a;b) u = x - y, v = xy

- 
12,12  yvxu

 
2;2, vu









.
2
1
221812818128
22
2323

yxyx
yyyxxx

. 







)2(74324
)1(025)3()14(
22
2
xyx
yyxx
(
A 2010).