1

( .) (ETV) 1 1 - 3 1 - 3 2551 1 - 3 - .. 2554 2556 - 13.30 . 15.00 . (((EEETTTVVV))) . 1 4 () . () .

2554

(.) 1 1 10 1. 2. 3. 4. 5. 6. 7. GSP 8. 9. 10. 14 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. GSP 12. 13. 14.

2554

1

1 1 2 40 3 73 4 120 5 165 6 177 7 GSP 194 8 209 9 216 10 241

(. 2549) 65 (TIMSS 1999, PISA 2000 & 2003)

(.) 2551 .. 2551 .. 2553 . 3 . 1 3 1 (.. 2554) 2 (.. 2555) 3 (.. 2556)

- -

(Concept)

1 ( 1) 7 2 14 (ETV) 2 45 13.30 . 15.00 . 30 6 2554 16 2554

1 - 6

1. 1

1

2. 2 3

2

1

3. 4

5

2

- -

4.

6 7

2

5. 8

1

6. 9 10

2

7. GSP 11 GSP

1

8. 12

1

9. 13

1

10. 14

1

14 (ETV) 45 15 1

- -

1 ( 1) (ETV)

6 2554 16 2554

13.30 . 14.15 . 14.15 . 15.00 .

4 2554 - 1

-

11 2554 18 2554 25 2554 2 2554

9 2554 GSP

16 2554

15.00 . 16.00 . 15 30 (ETV) 2

- -

.

2 1) 80

( 12 ) 2)

60 ( 6 84 140 )

- -

1

1.

1.

1. (Pedagogical Content Knowledge : PCK) 2.

/ 1. (Pedagogical Content Knowledge : PCK)

(Subject Matter Knowledge : SMK) (Pedagogical Knowledge) (Knowledge about Context)

2. (Critical Thinking) (Decision Making) (Creative Thinking) (Reflective Thinking)

3.

.

1

2

1. (Pedagogical Content Knowledge : PCK)

(Ball : 1989) (Borkow & Putnam : 1995)

1) (Subject Matter Knowledge : SMK)

2) (Pedagogical Knowledge)

3) (Knowledge about Context)

(1) (2) (3) (4)

(Grossman :1990)

3

1.

1) (Subject Matter Knowledge : SMK)

2) (Pedagogical Knowledge)

Pedagogical Content Knowledge (PCK) (Discovery)

1) 4 6

Pedagogical Content Knowledge (PCK)

Community Students Schools District

Educational Aims

Learners &Learning

Instructional Principles

Classroom Management

Substantive Knowledge and beliefs

Syntactic Knowledge and beliefs

Knowledge and Beliefs about Context

influences

Pedagogical Knowledge and Beliefs

influences

Subject Matter Knowledge and Beliefs

influences

.

1

4

(1) 1 1 1

(2) 1 1 1 2 1,2,3,4,5,6,7,8,

(3) 1

(4) 100 1

(5) 1 1 2 2 3 3 4 4 7 200,000

2)

2 (Creative Discovery) (Guided Discovery)

(Intuition)

5

1.

1 2 4 8 16 16 2 3 4 5

1 1 1 1 2 2 2 2 3 4 4 4

7 8 8 15 16 31

3 1 4 1 5 1 n n ( n + 1) 1

16 1 + 2 + 4+8 +16 +32 +64 +128 +256 + 512 + 1,024 + 2,048 + 4,096 +8,192 +16,384 + 32,768 17 1

17 232,768 = 65,536 16 65,536 1 = 65,535 3)

Carl Friedrich Gauss (1777 -1855) 1 100

.

1

6

Gauss Gauss 5050

Gauss 1 + 2 + 3 + 4 + 5 + 96 + 97 + 98 + 99 + 100 1 + 100 = 101 , 2 + 99 = 101 , 3 + 98 = 101 101 50 1 100 50 101 = 5050

37 42 2 2 37 42 18 84 9 168 4 336 2 672 1 1344 37 42 = 42 + 168 + 1344 = 1554 37 2 18 92 4 42 , 168 1344 1554 37 42 = 1554

4)

4.1 2 50 100 76 23

7

1.

94 76 76 170 70 +1 = 71 94 71 23 23 4.2 311 113 198 891 1089 1089

4.3 50 100 84 54 54 138 38+1 = 39 84 39 45 45

4.4 9 27 10 270 9 36 234

+

+

+

.

1

8

23 + 4 = 27 4.5 29 2 29 2 = 58 5 58 + 5 = 63 5 63 5 = 315

(315 5 31) 2 (31 2 = 29) 4.6 51 2 51 2 = 102 10 102 +10 = 112 5 112 5 = 560 9 560 + 9 = 569 50 569 50 = 519 ( ) () 4.7 15,873

(1) 15,873 7 = (1) 111,111 (2) 15,873 14 = (2) 222,222 (3) 15,873 21 = (3) 333,333 (4) 15,873 28 = (4) 444,444

9

1.

(5) 15,873 35 = (5) 555,555 (6) 15,873 42 = (6) 666,666 (7) 15,873 49 = (7) 777,777 (8) 15,873 56 = (8) 888,888 (9) 15,873 63 = (9) 999,999

4.8

12,345,679 1 12,345,679 9 111,111,111 2 12,345,679 18 222,222,222 9 12,345,679 81 999,999,999

5)

5.1

.

1

10

1 2 3 1 a 2 b 3 c

1. 48 6 2. (4 3) 93

3. 4 + (3 4) 4. (642) (2 8) 5. [25 (72)]5 6. (4 5) 6 7. (8 14 ) + (1.5 2) 8. (7 4) 5 9. 5(1+ 82 ) 10. 5 (42 42) 11. (13 13) 169 12. 644 + 2

8 .

. . . . . . . . .

H() .

. . . . . . . . . .

12 1 8 2 12

1 16 16 25 14 5 23 25 5 1 18 1 HAPPY NEW YEAR 5.2

x = 4 y = 3

. .. .. .. .. 2 4 5 1 8 5 9

A : (2 x) (2 y) D : x + y + 1 H : (2 x) y 1 O : (2 x) y G : (4 y) x + 1 T : (3 y) (2 x) A HOT DOG

11

1.

5.3 a = 3 b = 4 c = 1

(1) 1 9 (2) (3)

15 16 * 7 a + b+ c (a + b) c 5.4 (Cross Number)

1. 2 x = 71 (22) 2. x 7 = 8 3. y + 5 = 20 3. 3 t = 45 4. y 3 = 4 4. y + 3 = 16 5. 5 r = 29 + ( 12 + 12 ) 6. 5 y = 26 ( 1 22 ) 7. 8 v = 15 (213) 7. 6 u = 68 + 4

8. y = 18 ( 923 ) 8. y 1 = 15 10. y = 14 ( 7 1+4 4 ) 9. f = 7 + (

11 12 2 )

11. k = 72 (2 2) 12. n + 5 = 94 14. 3g = 230 5 15. 2 y = 109+ 3

3

16. u = 100 (8 1) 16. 2 d = 197 (11) 18. u = 62 (8 2) 17. 3 d = 81 20. 2 y = 97 (15 1) 19. 1 y = 121 (10 4)

(3 a) b c (2 a) + b c b a + c

{(2 b) + c} a {(3 a) b}c a + b + c (3 b) a c (b a) c a + b c

4 9 2 3 5 8* 8 1 6

.

1

12

12 35 15 12 6 1 12 12 68 75 92 58 41 12 56 15 13 5 12 16 12 89 55 98 27 81 5.5 (Mystery Message) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (1) 2 x = 9 + 62 (2) 1 y3 = 1 (3) x + 5 = 16 (4) x + 5 = 6 (5) 14 x = 34 (6) + 2 = 7 (7) 3 x = 35 + (3 3) (8) + 10 = 24 (9) x + 5 = 5 + ( 12 2) (10) x - 4.6 = 3.4 (11) (10 + v) = 11 (12) 918 =

s2

(13) 5 m = 47 + 3 (14) 8 k = 14 6 (15) 14 + g = 19 (16) 2 w = 2 + (24 4)

A = 3 B = 9 C = 14 D = 4 E = 1 F = 15 G = 18 H = 6 I = 12 J = 21 K = 10 L = 23 M = 17 N = 5 O = 20 P = 16 Q = 25 R = 28 S = 22 T = 7 U = 26 V = 11 W = 8 X = 24 Y = 2 Z = 19 HAVE A NICE WEEKEN

3 5 1 5 1 2 6 5 5 3 1

1 2 1 1 2 6 8 2 7 5 9 2 5 8 8 7 4 1

1 2 3 4

5 2 6 7

8 9 10

11 12 14 15

16 17 18 19

20

1 2 3 4

5 6 7

8 9 10

11 12 14 15

16 17 18 19

20

13

1.

5.6

. . 3 4 . . .. ... .. .. 48 128 2 26 100 66 2 98 66 15 11 5 : 2x = 97 (5 5) : 1 x4 = 15 + ( 28 2+3 3 ) : 6x = 593 102 : 3 x = 197 + (9 19 ) : x 7= 63 + 2 7 : 7 m = 9 + 5 : 2 x = 102 + 1 : 3 x = 15 : 4 y = 16 : 7 x = 82 5 : 2 x = 23 + 7 : 3 x = 300 +93 : x13 = 7 5 : x8 = 25 4

.

1

14

2.

(Critical Thinking) (Decision Making) (Creative Thinking) (Reflective Thinking)

(Critical Thinking) (Decision Making) (Creative Thinking) (Reflective Thinking)

15

1.

2.1 (Classifying)

2.2 (Comparing)

2.3 (Sequencing)

2.4 (Analyzing Parts and Whole)

1 20 2 5 2 2 x 5 x 2 = 20 13 18 25

2.5 (Identifying Patterns & Relationships)

.

1

16

2, 5, 8, 11, 14,... 100 1 2 2 5 = 2 + 3 3 8 = 2 + 2 (3) 4 11 = 2 + 3 (3) 5 14 = 2 + 4 (3) 100 100 2 + 99 (3) = 2 + 297 = 299

2.6 (Induction)

1 20

2.7 (Deduction)

17

1.

2.8 (Spatial Visualization)

(Critical Thinking) (Creative Thinking) (Critical Thinking)

5 10 2 12 36 16 4 14 1 100 50

+ ( ) 1 2 3 4 5 = 100

1) 2)

2 5 6 3

18

12

14

14

16

12

112

14

18

14

56

12

512

712

16

1112

23

34

.

1

18

3 3 5

3

2 5 7 3

1 11

(Creative Thinking)

1 2 5 10 2 4 6 24 3 2 8 12 4 4 5 15

14 8

5 3

2 1 6 3

9

19

1.

3. 2544

5

3.1 4

4 1 2 3 4 4

.

1

20

1

2

3

4 (Number Sense) (Spatial Sense)

(1) (2) (3) (4) (5) (6) (7) / (8) (9)

21

1.

(1)

1) 2) 3) 4)

136

36 :

1) 100 36 ( 136) 100 + 36 = 136 100 36 = 64

2) 90 46 ( 136 ) 90 + 46 = 136 90 46 = 44

3) 80 56 ( 136 ) 80 + 56 = 136 80 56 = 24

4) 80 90 85 51 85 + 51 = 136 85 51 = 34 2

.

1

22

5) 2 86 50 86 + 50 = 136 86 50 = 36

(2)

8 5 4 4 100

5 5 8 = 40 60 604 = 15

23

1.

() () () () ()

5 6 7 8 9 10

5 8 = 40 6 8 = 48 7 8 = 56 8 8 = 64 9 8 = 72 10 8 = 80

15 13 11 9 7 5

15 4 = 60 13 4 = 52 11 4 = 44 9 4 = 36 7 4 = 28 5 4 = 20

100 100 100 100 100 100

(3)

1) 2) 3)

3 2 , 5 , 10 , 17 , 26 , 37 , 2 , 5 , 10 , 17 , 26 , 37 , 1 3 5 7 9 11 2 2 2 2 2 2 2

.

1

24

13, 15, 17 2 , 5 , 10 , 17 , 26 , 37 ,

37+13 = 50, 50+15 = 65 65+17 = 82 50, 65, 82

(4)

1) 2)

5

3 5 3.5 : 5 ( ) 3 ( 135 5 ) 3.5 ( ) 8.2

25

1.

8

(5)

1) 2) 3)

6 5 4356 9084 5471 9346 4782 7623 2420 3474 1267 12678 2094 6540 4350 4140 5330 3215 4456 9989 : 1. 2 ( 5 ) 2) 5 0 ( 5 0 ) 2420 6540 4350 4140 5330 3215

2. 6 6540 4350 4140

.

1

26

(6)

2 1 1 2 3 1 +2 = 3 1 3 4 1 +2+3 = 6 1 4 5 1 +2 +3+4 = 10

2 1 3 1 + 2 = 3 4 1 + 2 + 3 = 6

5 1 + 2 + 3 + 4 = 10 6 1 + 2 + 3 + 4 + 5 = 15 7 1 + 2 + 3 + 4 + 5 + 6 = 21 8 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 9 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

10 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

27

1.

15

(7) 8 : 8 21 1 + 2 + 3 + 4 + 5 + 6 = 21 8 (8 21 ) 6+3+5+1+4+2+3+5+1 = 30 (8 21 ) 30 = 168 30 = 138 n a x x = (n 21) a

(8)

10,000

1) 4 4

2) 100 100 3) 10

(9) 60 1,2,3,4 5

.

1

28

60 : A B A B A B 60

+ ( A B) = 60

+ 6 = 60 + 6 = 54 + 6 = 48 + 6 = 42 + 6 = 12 6 = 6

A B 60 54, 48, 42, 36, 30, 24, 18, 12 6 B A 6 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

: : 30 86 :

29

1.

1 : 1. 30 2. 2

13 17 2 : 1. 30 2.

()

()

1 4 29 58 62 5 20 25 50 70 10

11 12 13

40 44 48 52

20 19 18 17

40 38 36 34

80 82 84 86

13 17 2 1 29 3 : x 30 x

1

1

.

1

30

2x 4(30 x) 86 2x + 4(30 x) = 86 2x + 120 4x = 86 2x = 86 120 x = 234 x = 17 17 13 17 34 13 52 30 86

3.2

1)

2)

31

1.

3)

"" "" "" " ."

(Open-ended Problem) 2.85 1.30 31

.. 31 2.85 = 0.95 1.30 2.85 (0.95 + 1.30) = 0.6 0. .. 1.30 ..

.

1

32

.. 1.30 0.6 .. 1.30 0.6 0.6 3.3

1. 2.

/

33

1.

5 2

1

5 2 25 2

5 2 25

3

1 2 5 5 2 5 5 = 25

5 .

1 .

2 5 10 1

1 1 1 1

.

1

34

4 1 10 5 5 10 = 50 5 50

2 50 2 = 25

3.4

/

1) 2) 3) /

4)

5)

35

1.

2 1 2 2 1 7,000 5 3

100

.. 84,000 (7000 12 = 84000) 1 .. 2 800 80,000 .. 93 )93900

127000(

.

1

36

1 3.5

15

37

1.

1 2 3

1 2 3 4 5 6

7 6 5 4 3 2

7 6 5 4 3 2

1 3 5 7 9 11

3 + 3 < 9 2 + 2 < 11

4 1 4

/

.

1

38

. . : , 2544. Ball, D. L. and Hill,H.C.. Knowing Mathematics for Teaching : Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide?. American Educator, 2005. Bitter , Gary G. and others. Mathematics Methods for the Elementary and Middle School. Boston : Allyn and Bacon, 1989. Borkow, H.,& Pultman,R.T. Expanding Teachers knowledgebase: A cognitive psychological

Perspective on professional development. In R.T. Gusky & M.Huberman(Eds.) Professional Development in Education. New York : Teacher College Press, 1995.

Charles, Randall; Lester,Frank; Odaffer,Phases. How to Evaluate Progress in Problem Solving. Verginia: The National Council of Teacher of Mathematics Inc.,1987. Daniel T. Dolan and James Williamson. Teaching Problem-Solving Strategies. California : Addison Wesley Publishing Company,Inc.,1983. Grossman, P. The Making of a Teacher: Teacher Knowledge and Teacher and Teacher Education. New York: Teachers College Press, 1990. Gary, G. Bitter, and others. Mathematics Methods for the Elementary and Middle School A Comprehensive Approach. Boston : Allyn and Bacon, 1989. Hill, H. C. and Ball, D. L. Learning Mathematics for Teaching : Results from Californias Mathematics Professional Development Institutes. Journal for Research in Mathematics Education. Vol. 356, No. 5, pp. 330 351, 2004. Hill, H. C. and Ball, D. L. Effect of Teachers Mathematical Knowledge for Teaching on Student Achievement. American Educational Research Journal Summer, Vol. 42, No. 2.

pp. 371 406, 2004. Kennedy , G.A.,and Tipps,S. Guiding Children's Learning of Mathematics. 7 th ed. California : Wadsworth Inc.,1994. Kroll, Diana L. and Tammy Miller. "Insights from Research on Mathematical Problem Solving" in Research Ideas for the Classroom Middle Grades Mathematics . Douglas T. Owens Editor New York : Macmillan Publishing Company, 1993.

39

1.

National Council of Teacher of Mathematics. . An agenda for Action :Recommendations for School Mathematics of 1980s. Verginia :The National Council of Teacher of Mathematics Inc., 1973. National Council of Teacher of Mathematics. . Curriculum and Evaluation Standard for School Mathematics . Verginia :The National Council of Teacher of Mathematics Inc., 1989. . Principles and Standards for School Mathematics . Reston :The National Council of Teacher of Mathematics Inc., 2000. . Problem Solving in School Mathematics. Fift Printing , Verginia :The National Council of Teacher of Mathematics Inc., 1980. Nicholas A. Branca. "Problem Solving as a Goal , Process, and Basic Skill" in Problem Solving in School Mathematics 1980 Yearbook, Fift Printing , Verginia :The National Council of Teacher of Mathematics Inc., 1980. Polya ,G. How To Solve It . Second Edition , New York : Doubleday & Company,Inc.,1957. Post, Thomas. R. Teaching Mathematics in Grades K-8 : Research Based Methods. Massachusetts : Allyn and Bacon , 1988. Shulman, L. S. Those who understand : Knowledge growth in teaching. Educational Researcher, Number 15 , Vol. 2 , pp. 4 14 , 1986. Thomas P. Carpenter and others. Children's Mathematics Cognitively Guided Instruction. Reston : The National Council OF Teachers of Mathematics Inc., 1999. Wilson J. and Murdoch K. How to Succeed with Thinking. Australia : Curriculum Corporation, 2006.

.

1

40

2.

1. ...

2.

/

1.

2.

3. 1, 2, 3, 4, 5, 6, 7, 8, 9 0

4. 5.

1 0

6. 1, 2, 3, 4 0

7.

2.

41

T E 1, 2, 3, 4, 5, 6, 7, 8, 9,T, E, 0

8. 9. 10 1 2

3 10 1 10 1

10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1

10. 10

0 1 2 3 4 5 6 7 8 9

11. 1, 2, 3, ... 12. 10

...

13. 10

.

1

42

1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000

14. 5,743,521

5 5 1,000,000 = 5,000,000 5 5 100 = 500

15.

16. 17.

18.

19.

2.

43

4 (1) (undefined term)

(2) (defined term) (3) (axiom postulate) (4) (theorem)

.

1

44

()

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

17

: : XVII : 17

()

2.

45

()

()

() ()

() ()

()

I V X L C D M

.

1

46

IV VI IX XII

40 2 = 38

(260) + 57 = 177

(4160) + 55 = 2,515

(13602) + (4160) + 55 = 49,315

2.

47

(1603) + (13602) + (4160) + 55 = 265,315

2 61 3660

20 1820 .... 18202, 18203, 18204, ...

5(18202) 36,000 15(1820) 5,400 7(20) 140 6 6 41,546

.

1

48

2(18202) 14,400 6(1820) 2,160

0(20) 0 3 3 16,563

1, 2, 3, 4, 5, 6, 7, 8, 9 0

....

1 0 1, 2, 3, 4 0 T E 1, 2, 3, 4, 5, 6, 7, 8, 9,T, E, 0

10 1 10 1 10 10 1 10 1 10 1

2.

49

1 0

1

10

11

100

101

110

111

1000

1001

1010

1011

1100

1101

1110

1111

.

1

50

...

2 4

(16) 2 3 (8)

2 2 (4)

2 1 (2)

2 0 (1)

101112 10111

10111 = (124) + (023) + (122) + (121) + (120)

= (116) + (08) + (14) + (12) + (11) = 16 + 0 + 4 + 2 + 1 = 23 1, 2, 3, 4 0

1

2

3

4

10

11

12

13

2.

51

14

20

21

22

30

40

100

103

120

124

.

1

52

...

5 4

(625) 5 3

(125) 5 2

(25) 5 1 (5)

5 0 (1)

13425 1342

124 = (152) + (251) + (450)

= (125) + (25) + (41) = 25 + 10 + 4 = 39

1 2 3 4 5 6 7 8 9 T E 0

2.

53

1

2

3

4

5

6

7

8

9

T

E

10

11

19

1T

1E

20

.

1

54

90

T0

E0

EE

100

2.

55

13E

...

12 4

(20,736) 12 3

(1,728) 12 2

(144) 12 1 (12)

12 0 (1)

13E = (1122) + (3121) + (11120)

= (1144) + (312) + (111) = 144 + 36 + 11 = 191

10 10 1 2 3 10 10 1

10 1

10 1

10

1

.

1

56

10 1

1, 2, 3, 4, 5, 6, 7, 8, 9 0 ... 10 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1

10 1, 2, 3, 4, 5, 6, 7, 8, 9 0 1, 2, 3, 4, 5, 6, 7, 8, 9 0 , , , , , , , ,

10 1

2.

57

1,258 , 1 1 2, 3, 4, 5,....

2 5 10 2 , 5 10

10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ... 5, 49, 172, 3,096 0 37,042

1 2 5 8

0 1 . .

.

1

58

3 30,000 7 7,000 0 0 4 40 2 2

1 10 (1 )

16 5 2 1 16 10 (1 )

1 6 5 3 1 6 5 1, 6 5 165 165

2.

59

3 5 9

3 5 9 0 3 0 5 9 3, 0, 5 9 3,059

2 9 4 1 6 5 0 0 29,401,605 2 1 9 3 5 8 2 1 0 0 2 0 1 9 3 0 5 8 2 1 2,019,305,821

.

1

60

0 1 2 1 415,962 2,906,813 15,027,421

92,571,840,000

(1) 1,372 , (2) 731,624,835 ,,

2.

61

(3) 4,250,397,618 ,,, (1) 926,148 , (2) 37,100,502,004 ,,,

(1) 39,025 (2) 725,910 (3) 461,083,507

10 ...

10

1 10 10 1 10 10 10 10 100 10 10 100 1,000

.

1

62

10 10 1,000 10,000 10 10 10,000 100,000 10 10 100,000 1,000,000 10 10 1,000,000 10,000,000

10

9 1,000,000,000

108 100,000,000

107 10,000,000

106 1,000,000

105 100,000

104 10,000

103 1,000

102 100

101 10

100 1

125,837,504 1 1 100,000,000 100,000,000 2 2 10,000,000 20,000,000 5 5 1,000,000 5,000,000 8 8 100,000 800,000 3 3 10,000 30,000 7 7 1,000 7,000 5 5 100 500 0 0 10 0 4 4 1 4

0 0 42,915,073 4 4 10,000,000 40,000,000 2 2 1,000,000 2,000,000 9 9 100,000 900,000 1 1 10,000 10,000

2.

63

5 5 1,000 5,000 0 0 100 0 7 7 10 70 3 3 1 3 42,915,073 42,915,073 (410,000,000)+ (21,000,000)+ (9100,000)+ (110,000)+ (51,000) + (0100)+ (710)+ (31) 40,000,000 + 2,000,000 + 900,000 + 10,000 + 5,000 + 0 + 70 + 3 42,915,073 40,000,000 + 2,000,000 + 900,000 + 10,000 + 5,000 + 70 + 3

160,924 97,358 160,924 97,358 100,000 90,000 160,924 97,358 97,358 160,924

1 100,000 9 90,000

.

1

64

513,862 514,021 513,862 514,021

500,000 500,000 513,862 514,021 10,000 10,000 513,862 514,021 3,000 4,000 513,862 514,021 514,021 513,862

301,492 93,648 299,053 94,091 310,284 5 2 93,648 94,091 6 3 301,492 299,053 310,284 5 6 93,648 94,049 299,053 301,492 310,284 310,284 301,492 299,053 94,049 93,648

5 500,000 5 500,000

1 10,000 1 10,000

3 3,000 4 4,000

2.

65

2 ...

0 10, 20, 30, 40, 50, ...,110, 120, 130, ... 0 100, 200, 300, 400, 500, ...,1100, 1200, 1300, ... 0 1000, 2000, 3000, 4000, 5000, ...,11000, 12000, 13000, ... 0 10000, 20000, 30000, 40000, 50000, ...,110000, 120000, 130000, ... 0 100000, 200000, 300000, 400000, 500000, ...,1100000, 1200000, 1300000, ... 1 2 3 4 0 10 1 2 3 4 0 6 7 8 9 10 0 6, 7, 8, 9 10 ( 1 ) 5 0 10 5 10

0

0 1 2 3 4 5 6 7 8 9 10

.

1

66

43 43 40 50 43 40 50 43 40

176 176 170 180 176 180 170 176 180 315 315 310 320 315 320

41 42 43 44 45 46 47 48 49 50 40

170 171 172 173 174 175 176 177 178 179 180

310 311 312 313 314 315 316 317 318 319 320

10 10 5 1, 2, 3 4 0 1, 2, 3 4 0 5, 6, 7, 8 9 1 1

2.

67

50 100 1 49 0 100 1, 2, 3, ...., 49 0 51 99 100 0 51, 52, 53, ..., 99 100 ( 1 ) 50 0 100 50 100 ( 1 ) 247 247 200 300 247 200 300 247 200

1,384

1,384 1,300 1,400 1,384 1,400 1,300 1,384 1,400

0 10 20 30 40 50 60 70 80 90 100

200 210 220 230 240 250 260 270 280 290 300 247

1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1384

.

1

68

1,150

1,150 1,100 1,200 1,150 1,200

500 1,000 1 499 0 1,000 1, 2, 3, ...., 499 0 501 999 1,000 0 501, 502, 503, ..., 999 1,000 ( 1 ) 500 0 1,000 500 1,000 ( 1 )

1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200

100 100 50 1, 2, 3,..., 49 0 1, 2, 3,..., 49 0 50, 51, 52, ..., 99 1 1

0 100 200 300 400 500 600 700 800 900 1000

2.

69

2,485 2,485 2,000 3,000 2,485 2,000 3,000 2,485 2,000 2,850

2,850 2,000 3,000 2,850 3,000 2,000 2,850 3,000 4,500 4,500 4,000 5,000 4,500 5,000

2,000 2,100 2,200 2,300 2,400 2,500 2,600 2,700 2,800 2,900 3,000 2,485

2,000 2,100 2,200 2,300 2,400 2,500 2,600 2,700 2,800 2,900 3,000 2,850

4,000 4,100 4,200 4,300 4,400 4,500 4,600 4,700 4,800 4,900 5,000

1,000 1,000 500 1, 2, 3,..., 499 0 0 500, 501, 502, ..., 999 1 1

.

1

70

0 1 2 3 4 5 6 7 8 9 10 1

0 1 2 3 4 5 6 7 8 9 10 1

0 1 2 3 4 5 6 7 8 9 10 1

0 1 2 3 4 5 6 7 8 9 10 1

0 1 2 3 4 5 6 7 8 9 10 1

0 1 2 3 4 5 6 7 8 9 10 1

1 1 1 1

1 1

2.

71

1. 1 5 1 5 1 5 1 5 1 5 1 5

2. 0, 1, 2, 3 4 1 1 1 1 1 1 ... 0 0 5 6, 7, 8, 9 1 1 1 1 1 1 ... 1. 2. 0, 1, 2, 3 4 0 1 5, 6, 7, 8 9 1 1 1 1 1 1 ... 1 1 1 1 1 1 1 ...

1,249,538 12 1,200,000 4 49,538 1 50,000 49,538 0 1,249,538 1,200,000

.

1

72

11,605,284 11 11,000,000 6 605,284 1 500,000 605,284 1 11,605,284 11 1 12 12,000,000

3.

73

3.

1.

2. /

1. 2. 3.

4.

5.

6. 7.

8.

.

1

74

(Learning Hierarchy)

3 2 5 2 4 6

2 4

2 6 4 4 6

4 + 3 4 + 3 = 6 () 4 + 3 = 8 () 9

3.

75

9 9

1. + 2. = 3.

0 9 (+) (=) 4 3 4 + 3 = 7 9 0+0 = 0 0+1 = 1 0+2 = 2 0+3 = 3 0+4 = 4 0+5 = 5 0+6 = 6 0+7 = 7 0+8 = 8 0+9 = 9 1+0 = 1 1+1 = 2 1+2 = 3 1+3 = 4 1+4 = 5 1+5 = 6 1+6 = 7 1+7 = 8 1+8 = 9 2+0 = 2 2+1 = 3 2+2 = 4 2+3 = 5 2+4 = 6 2+5 = 7 2+6 = 8 2+7 = 9 3+0 = 3 3+1 = 4 3+2 = 5 3+3 = 6 3+4 = 7 3+5 = 8 3+6 = 9 4+0 = 4 4+1 = 5 4+2 = 6 4+3 = 7 4+4 = 8 4+5 = 9 5+0 = 5 5+1 = 6 5+2 = 7 5+3 = 8 5+4 = 9 6+0 = 6 6+1 = 7 6+2 = 8 6+3 = 9 7+0 = 7 7+1 = 8 7+2 = 9 8+0 = 8 8+1 = 9 9+0 = 9 9

1) 9 0+9 9+0 1+8 8+1 2+7 7+2 3+6 6+3 4+5 5+4

2) 0 + 4 = 4 + 0 = 4 2 + 6 = 6 + 2 = 8 3 + 2 = 2 + 3 = 5

4 + 3 = 7

.

1

76

3) 0 + 0 = 0 6 + 0 = 6 0 + 8 = 8 20

1 + 9 = 10 , 2 + 8 = 10 , 3 + 7 = 10 , 4 + 6 = 10 , 5 + 5 = 10 , 6 + 4 = 10 , 7 + 3 = 10 , 8 + 2 = 10 9 + 1 = 10 20 8 + 3

8 + 3 8 + 3

8 8 2 1

10 10 1

11

9+3 , 8+5 , 7+4 , 6+5

1 12 + 5 12 13, 14, 15, 16, 17 12 + 5 = 17 2 10 12 + 5 = 1 10 12 + 5 = 10 + 2 + 5 2 10 = 10 + 7 3 = 17

3.

77

1 2 20

1)

2) cursor[ ]

3)

4)

100 4 5 3 4 6 3 8 1 8 1 7 1 3 4 2 7

+ + 5 + 8 = 13 4 + 3 = 7

6 + 1 = 7 4 + 8 = 12 ( 2) 3 + 1 = 4

.

1

78

100 () 13 + 24 = (1) 1 3 2 4

(2) 1 3 2 4 3 7 13 + 24 = 37 100 () 38 + 27 = (1) 3 8 2 7

+

+

+

3.

79

(2) 15 10 1 5 1 3 8 2 7 5 (3) 3 8 2 7 6 5

(4)

3 8 2 7 6 5 38 + 27 = 65

+

+

1

1 +

1

.

1

80

1,000 1 149 + 135 = (1) 1 4 9 1 3 5

(2) 14 10 1 4 1 1 4 9 1 3 5 2 8 4 (3) 1 4 9 1 3 5 2 8 4

+

+

1

1 +

3.

81

2 189 + 98 = (1) 1 8 9 9 8

(2) 17 10 1 7 1 17 1 18 10

1 8 1

1 8 9 9 8 2 8 7 (3) 1 8 9 9 8 2 8 7

+

+

+ 1 1

1 1

.

1

82

0 0 2 0 4 1 5 2 3 0 6 3 202 + 158 = (1) 2 0 2 1 5 8 (2) 10 10 1 0 1 5 1 6 3 2 0 2 1 5 8 3 6 0

+ 4 + 2 = 6 0 + 5 = 0 2 + 1 = 3

+

+ 1

3.

83

(3)

2 0 2 1 5 8 3 6 0

100,000

100,000 100, 1,000 1 , 10 , 100 1,000

1 1 10 1 100 1 1000 1

1 3752 + 2424 = (1)

1000 1000 100 100 10 10 1 1 1000 100 100 10 10 100 100 10 100 3 7 5 2

1000 1000 100 100 10 10 1 1 2 4 2 4 100 100 1 1

+ 1

+

.

1

84

(2) 1 6

10 7 100 11 10 1 1 1 1000 5 1 1000 6

1000 1000 100 100 10 1 1 1000 100 100 100 100 100 3 7 5 2

1000 1000 100 100 10 10 1 1 2 4 2 4 1000 100 100 1 1 6 1 7 6

(3) 3 7 5 2

2 4 2 4 6 1 7 6 2 4486 + 2377 = 1000 1000 100 100 10 10 1 1 1000 1000 100 100 10 10 1 1 10 10 1 1 10 10 4 4 8 6

1000 1000 100 100 10 10 1 1 2 3 7 7 100 100 10 10 1 1 10 10 1 1 6 8 6 3 10 10 1

+

+ 1

1

+

1. 10 1 2. 10 1

1 1

10 10 10

10

3.

85

4 4 8 6 2 3 7 7 6 8 6 3

3 2766 + 5243 =

1000 1000 100 100 10 10 1 1 100 100 10 10 1 1 100 100 10 10 1 1 100 2 7 6 6

1000 1000 100 100 10 10 1 1 5 2 4 3 1000 1000 100 10 10 1 1000 1000 8 0 0 9

2 7 6 6 5 2 4 3 8 0 0 9

5 2 3

+ 1 1

+

1. 10 1 2. 10 1

+ 1 1

1 1

.

1

86

2 5

2 3 9 4 9 4 = 6 () 9 4 = 4 () 5 2 3 (1) (2) (3) = (4)

=

3.

87

9 () (=) 5 5 5 5 4 5 3 5 2 5 1 5 0

4 4 4 3 4 2 4 1 4 0

3 3 3 2 3 1 3 0

2 2 2 1 2 0

1 1 1 0

0 0 3 2 3 2 = 3 3 3 =

.

1

88

3 0 = 9

9 9 9 8 9 7 9 6 9 5 9 4 9 3 9 2 9 1 9 0

8 8 8 7 8 6 8 5 8 4 8 3 8 2 8 1 8 0

7 7 7 6 7 5 7 4 7 3 7 2 7 1 7 0

6 6 6 5 6 4 6 3 6 2 6 1 6 0 10 10 1 = 9 , 10 2 = 8 , 10 3 = 7 , 10 4 = 6 , 10 6 = 4 , 10 7 = 3 , 10 8 = 2 , 10 9 = 1 10 0 = 10 10 1 17 6 17 16 ,15 ,14 ,13,12 ,11 17 6 = 11 2 10 13 8

13 8

3 10 8 = 16 3 , 11 9 , 14 8 , 15 9 , 16 -7 , 13 4 , 17 8 , 11 2

3.

89

20 1)

2) cursor [ ] 3) 4)

100 5 6 4 9 1 3 () 25 13 = (1) 2 5 1 3

9 6 = 3 5 4 = 1

.

1

90

(2) 3 1 2 5 1 3 1 2 (3) 2 5 1 3 1 2 100 45 27 = (1) 4 5 2 7 (2) 5 7 1 10 5 15 7 2 4 5 2 7 1 8 (3) 4 5 2 7 1 8

3 15

3 15

3.

91

1,000 100,000 4 3 5 4 3 5 5 3 2 3 1 8 3 1 8 2 9 8 1 7 1 2 7 3 4 4 0 0 4 6 0 5 1 5 9 2 3 1 0 3 0 1, 10, 100, 1,000 1 314 109 = ( ) (1) 3 1 4 1 0 9

3 2 1 1 1 1 15 8 = 7

2 1 = 1 3 3 = 0

15 8 = 7 3 1 = 2 4 3 = 1

12 8 = 4 13 9 = 4 5 2 = 3

5 2 = 3 0 9 = 0 6 5 = 1 4 1 = 3

.

1

92

(2) 1 9 1 3 1 4 1 0 9

2 0 5 (3) 3 1 4 1 0 9 2 0 5

2 4755 2689 = () (1) 4 7 5 5 2 6 8 9

0 14

0 14

1000 1000 1000 1000

100 10 1 100 100

100 100 100 100

10 10

1 1 1 10

10 1

3.

93

(2) 1 1 9 1 10 8 100 6 1000 2

4 7 5 5 2 6 8 9

2 0 6 6 (3) 4 7 5 5 2 6 8 9 2 0 6 6

6 14 15

6 14 15

1000 100 10 1 1000 1000 1000

100 100 100 100 100 100

1010 1010

1010 10 10 10 10 10 10

1 1 1 1 1 1 1 1 1 1

1

1 1

1

1010

.

1

94

9 9 1

3 + 6 = 6 + 3 = 9 3 = 9 6 = 2

2 + 5 = 5 + 2 = 7 2 = 7 5 =

9

7

3.

95

1 2

. 3 1. 2. 3.

. 3 1. 2. 3.

.

1

96

1 :

2 :

3 : 1 2 3 1

3.

97

4

1 3 1 6 2 6 2 6 + 2 =

1) 9 6 2) 30 25 3) 470

31

2 6 8 6 8 6 + = 8 8 6 =

6 8

?

?

( 2 )

.

1

98

6 + = 8

6 6 + 2 = 8 2 2 6 8

8 6 8 6 8 6 = 8 6 = 2 2 6 8

1) 5 9 2) 3,600

5,600 3) 157,000

2 499,000

3 2 8 2 8 + 2 = 8 8 2 =

( 2 )

?

3.

99

2

2 8

+ 2 = 8 2 2 + 6 = 8 6 + 2 = 8 6 6 2 8

8 8 2 = 6 8 2 8 2 8 2 = 8 2 = 6 6 2 8

1) 15

78 2) 2,284

3,929 3) 1,500,000

3,420,800

( 6 )

( 6 )

.

1

100

2 3 1 8 2 8 2 8 2 =

1) 84 54 2) 8,500 2,700 3) 60,000 34,072

2 8 6 8 6 8 = 6 8 6 = 2 8 8 2 = 6 2 2

8 6

?

?

3.

101

8 6 8 6 8 6 = 8 6 = 2 2 6

1) 34 22 2) 8,000 4,800 3) 21,750 18,250

3 2 6 2 6 2 = 6 6 + 2 = 2 + 6 =

2 8 2 6

6 + 2 = 8 2 + 6 = 8

?

.

1

102

1) 29 7 2) 1,050 305 3) 195,000

1,500,000 3 2 2 1 2 3 4 () () () 3 + 4 =

1) 12 27 2) 575 135 3) 11,106,000 13,095,000

2 1 1

?

3.

103

1 7 3 () () ()

7 = 3 + 7 3 = 7 = 3 + 7 2 3 4 4 7 3

7 3 7 3 = 7 3 = 4 4 7

1) 680 325 2) 625 417 3) 8,000 2,550

4 2

?

.

1

104

2 1 2 5 3 5 3 =

2 2

1) 423 250

2) 7,200 1,440 3) 10,072 18,906

2 1

3.

105

2.1 3 5 3

5 + 3 = 5 3

5 + 3 = 8 5

3 3

5 + 3 = 8

2.2 3 8 3 8 3 =

5 3

5 3

.

1

106

8 3

8 3 = 5 8 3

8 3 = 5 5

1) 350 50 2) 746 144

3) 10,000 3,518

8

3

8

3

3.

107

.

1

108

3.

109

.

1

110

3.

111

.

1

112

3.

113

.

1

114

3.

115

.

1

116

3.

117

.

1

118

3.

119

.

1

120

4.

1.

2.

/ 1. 2. 3. 1 0 0 4.

5.

6.

7.

8.

9.

4.

121

10.

11. 2

12.

13. 14.

15.

16.

= 17.

= ( ) +

.

1

122

(1) o 3 + 3 + 3 + 3 = 4 3 4 3 o 3 5 35 = 15

(2) o 4 60

460 = 240 o 8 3 38 = 24

(3) o 4 3 34 = 12

(4) o 1 1

3 7 37 = 21

3

7

(5) o 2 3

23 = 6

(1) (2) (3) ( )

4.

123

8 3 3 8

(4) (5) ( ) 3 7 7 3

2 3

2 3 3 2

3 + 3 = 6

11 12 13 21 22 23

1 2 3 1

2

.

1

124

2 3 3 2 2 3 = 2 (2 + 1) = (2 2) + (2 1)

2 3 = 2 (2 + 1) = (2 2) + (2 1)

1, 2, 3, 4, 5, 6, 7 , 8, 9

1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9

2 2 4 6 8 10 12 14 16 18

3 3 6 9 12 15 18 21 24 27

4 4 8 12 16 20 24 28 32 36

5 5 10 15 20 25 30 35 40 45

6 6 12 18 24 30 36 42 48 54

7 7 14 21 28 35 42 49 56 63

8 8 16 24 32 40 48 56 64 72

9 9 18 27 36 45 54 63 72 81

4.

125

* (a, b, c d)

16

*

Mathematics for Elementary School 2nd grade vol. 2

1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

12 14 16 18

18 21 24 27

24 28 32 36

30 35 40 45

12 18 24 30

14 21 28 35

16 24 32 40

18 27 36 45

4 6 8 10

6 9 12 15

8 12 16 20

10 15 20 25

36 42 48 54

42 63 56 63

48 56 64 72

54 63 72 81

a b

c d

12 14

18 21

16 18

24 27

24 28

30 35

32 36

40 45

12 18

14 21

24 30

28 35

16 24

18 27

32 40

36 45

4 6

6 9

8 10

12 15

8 12

10 15

16 20

20 25

36 42

42 63

48 54

56 63

48 56

54 63

64 72

72 81

1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

.

1

126

(1) 4 4

(2) 64

(1) (2) 64 (3) 4 81(4) 1

23

(5) 6 6

(6)

22 23 24 25 26 27 28 29 32 33 34 35 36 37 38 39 42 43 44 45 46 47 48 49 52 53 54 55 56 57 58 59 62 63 64 65 66 67 68 69 72 73 74 75 76 77 78 79 82 83 84 85 86 87 88 89 92 93 94 95 96 97 98 99

4.

127

()

15 3 3 15 =

1 5 3

6 5

1

1 5 3

3 1 5

1 5 3

2 5

1

1 3 1 2 2 6 1 3 3 9

.

1

128

15 3 :

:

:

:

1 5 3

10 5

3

1 5 3

1 5

3 5

1 5 3

1 5 3 0

3 10

3 10 3 5

1 5 3

1 5 3 0 4 5

1 5 3

4 5

1

4.

129

3 45 =

4 5 3

3 45

4 5 3

1 5

10 1

3 5 3 5 = 15 1 5

3 5

3 4 3 4 = 12 1 2

10 1 4 5

3 1 5

1 2 0

3 5 3 4

4 5 3

1 5 1 2 0 1 3 5

3 5 3 4

1 2 1 5

1 3 5

.

1

130

3 45 = 3 45 3 45 3 45

1 : 3 5 = 15 1 5 15 10 1

4.

131

2 : 3 4 = 12 1 2 12 10 1

3 : 1 3 5 135

3 45 = 135

.

1

132

45 3

1 3 5 15 1 5

2 3 4 12 1 2

45 3 = 135 10

10 0 0.5 10 ( 0 10 0.50 ) 10 0 () 275 10 = 2750 (275 2 7 5 10 2 7 5 2 7 5 0 2 7 5 0 2750 100 1000 0 2 3 10, 100 1000 0 ()

4 5 3

1 3 5

4.

133

14 23 14 23

20 3

10

4

2 3 1 4 1 2 8 0 3 0

2 0 0 3 2 2

4 3 4 20

10 3 10 20

4 20 43

10 20

20 3

10

4

10 23

4 23

2 3 1 4 9 2

2 3 0 3 2 2

4 23 10 23

20 3

10

4

103

.

1

134

(grid)

14 23 = (10 + 4) (20 + 3) = [10 (20 + 3)] + [4 (20 + 3)]

= (10 20) + (10 3) + (4 20) + (4 23) = 200 + 30 + 80 + 230 14 23 = (10 + 4) 23 = (10 4) + (10 23)

2 3 1 4 1 2 8 0 3 0

2 0 0 3 2 2

4 3 4 20

10 3 10 20

43

10 20

4 20

103

10 23

4 23

2 3 1 4 9 2

2 3 0 3 2 2

4 23 10 23

4.

135

2

738 4

2.1

2.2 2.3 2.4 2.5 2.6 2.7

2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9

2 2, 3, 4, 5, 6, 7, 8 9

738 4 = 2952 7 3 8

4 2 9 5 2

1 3

.

1

136

1225 3 = 1 9

1 9 2 2 1 9 3

23 5 115

3

14 7 98

115 98 = 17 3

73 4 292

3

46 6 276

292 276 = 16

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

2 5 3

1 4 7

7 4 3

4 6 6

4.

137

(1) 4 3 () (2) 20 5 () (3) 6 2

() (4) 4 5

() (5) 3

2 ()

5 2

(1) 4 3

(1) 12 3

(2) 12 4

(2) 20 5

(1) 100 20

(2) 100 5

(3) 6 2

(1) 12 6

(2) 12 2

(4) 4 5

(1) 20 4

.

1

138

(2) 20

5

(5) 3 2

(1) 12 3

(2) 12 4

2

(1) 12 3

(2) 12 4

1 :

2 :

12 3

1 12 3 = 9

1 2 12 3 3 = 6

12 4

1 2 3 4

4.

139

1 :

2 :

1 2 3 12 3 3 3 = 3

1 2 3 4 12 3 3 3 3 = 0

1 2 3 4

1 2 3 4

1 2 3 4

12 3 4

12 3 = 4 12 3 = 4

12 4 3

12 4 = 3 12 3 = 4

1 1 1

.

1

140

(1) 15 3 15 3 =

3 3 3 = 9 4 3 4 = 12 5 3 5