. 5 1 1

. 5

1

5

2551

Backward Design

. 5

2

5

2551

.. 2551

ISBN _ _ _-_ _ _-_ _ -_ _ _ _-_

. .

2551

(CD) . 1 1 ..................................

(CD) . 1 2 ..................................

(CD) . 2 1 ..................................

(CD) . 2 2 ..................................

(CD) . 3 1 ..................................

(CD) . 3 2 ..................................

(CD) . 4 1 ........................................

(CD) . 4 2 ........................................

(CD) . 5 1 ........................................

(CD) . 5 2 ........................................

(CD) . 6 1 ........................................

(CD) . 6 2 ........................................

. 5 (Backward Design) (child-centered) /

2551 1 2 3 4 5 6 3 /

3

1

1)

2) Backward Design (BwD)

3)

2

3 /

1 1

2

Backward Design (BwD)6

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1)

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3)

4)

5)

2

1. / /

2. BwD (Backward Design Template) 3

1

2

3

3. BwD

3.1 1 1 5 3

3.2

3.3

3.4 (K) (A) / (P)

3.5

3.6

3.7

3.8 5

1

2

3

4

5

3.9

2

3.10 /

3.11

3 /

(CD)

1)

2) 3

(1)

(2)

(3) /

3)

(1) (Portfolio)

(2)

1.2

. 5 1

. 5 1

1.3

/

/

Logo 9

Logo 10

Logo 12

Logo 13

2. Backward Design (BwD)

Backward Design (BwD) (Enduring Understanding)

Backward Design

Backward Design

Backward Design 3

1

2

3

1

1.

2. (Enduring Understanding)

2551 1 Backward Design

Concept

1. (Broad Concept) 2. (Operative Concept Functional Concept)

1. ()

2.

3.

4.

5. 6.

2

Backward Design Backward Design

(Rubrics)

// (Meaningful Task) //

/

/

- 1.

23 2. 5.2 1.

2.

23 5.2 3.

23

5.2 1.

/

/

2. 23 5.2 3. 23 5.2

3

3

3

1.

2.

/

1. 23 2. 5.2 3. . 5 1 ( )

6

1.

2.

3.

4.

5.

6.

2551 5

1.

2.

3.

4.

5.

2551

1.

5.

2.

6.

3.

7.

4.

8.

6 Backward Design

Backward design 2

3

Backward Design WHERETO ()

W (Where) (What)

H (Hook) (Hold)

E (Equip) (Experience) / (Explore)

R (Rethink) (Revise)

E (Evaluate)

T (Tailored)

O (Organized)

Backward Design Template

...........................................

1

...........................................................................................................................................................

...........................................................................................................................................................

1. .................................................................

2. .................................................................

1. .................................................................

2. .................................................................

1. .................................................................

2. .................................................................

3. ................................................................. / ...

1. .................................................................

2. .................................................................

3. .................................................................

2

1.

1.1. ......................................................................................................................................

1.2. .......................................................................................................................................

2.

2.1

1) .........................................................

2) .........................................................2.2

1) .........................................................

2) .........................................................

3.

3.1..............................................................................................................................................

3.2.............................................................................................................................................

3.3.............................................................................................................................................

3

1. ...............................................................................................................................................

2. ................................................................................................................................................

Backward Design

...()

...()

...()

...( 1 )

...()

...()

...()

...()

...( 2551

(Knowledge: K)

(Affective: A)

/ (Performance: P))...(

3 )

...( )

...()...(

)

...()

/...( )

...()

/

Backward Design

3.

.. 2542 ( 2) .. 2545 24 (2) (3) (Child-Centered)

(Brain-Based Learning: BBL)

(Problem-Based Learning: PBL)

(Multiple Intelligences)

(Cooperative Learning)

6 (Six Thinking Hats)

(Inquiry Process)

(Problem Solving)

(Project Work)

(Active Learning)

(Concept Mapping)

(Experience Learning)

(Role Playing)

(Simulation Gaming)

3 / (Rubrics)

4. / 1 3 4 5 6

. 1.1. 1.2. 1.3. 1.4. 3.1. 3.2. 4.1. 5.2 6.1

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1 1

1. ( 4.1 . 46/4)2. ( 4.1 . 46/5)3. n n ( 4.2 . 46/6)4. ( 6.1 . 46/1)

5.

( 6.1 . 46/2)6. ( 6.1 . 46/3)7.

( 6.1 . 46/4)8.

( 6.1 . 46/5)9. ( 6.1 . 46/6)

...

...

1. n 2. n + 1 n (common

difference) 3. n + 1 n 4. a1 , a2 , a3 , , an a1 + a2 + a3 + + an a1 + a2 + a3 + + an + a1 , a2 ,

a3 , , an , 5. 5. / ...

1. 2.

3.

4.

2

1.

1. 2.

3. 4. 5. 6. 17.

8. 9.

10.

2.

2.1

1)

2)

3) /

4)

5) / 2.2

1)

2) /3) /

4)

5) /

3.

3.1 6

3.2

3.3

3

1 25

11

24

35

45

55

6 5

1

5 1 1 1

1.

2.

3.

2

4.

(K)

3

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

5.

1.

2. 3 3. 3 4.

5.

6.

7.

1

2

1. 1 3

3

3 3

2.

3.

3 1

4. 3

5.

3

4

5

8.

9. /

1. 1 2

2. 2

33. 3

4.

10.

1. _____________________________________________

_____________________________________________________

2. / _________________________________________

____________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________

2

5 1 1 3

1.

n 2.

1. ( 4.1 . 46/4)2. ( 4.1 . 46/5)3. n n ( 4.2 . 46/6)4. ( 6.1 . 46/1)

5.

( 6.1 . 46/2)6. ( 6.1 . 46/3)7.

( 6.1 . 46/4)8.

( 6.1 . 46/5) 9. ( 6.1 . 46/6)3.

1. (K)

2. (K)

3. (A)

4.

(P)4.

(K)

1.

2. 3

3. 1 3

1

4. 1.1 1.1 3

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

3.

1

4. 1.1

5.

6.

7.

1 1. 10 (10 )

2.

3. 2

1. ((x) = 2x+1 ((x) x = 1, 2, 3, 4

2. x x(f(x))

3. ()

4.

((x) = 2x+1, x = 1, 2, 3, 4 ()

( (x) = 3, 5, 7, 9

3, 5, 7, 9

5. y = x2+1

x = 1, 2, 3

x = 1, 2, 3,

x = 1, 2, 3, , 10

x =1, 2, 3, , 10,

6.

7.

8. 1

9. 1

10. a1, a2, a3

, , an a1 , a2 , a3 , a1 1

11. an = 3x+2 1 10

12. 45 1 2 1 2 1 2

13. 3

1. 1.1 . 5 1 ( )

2. 1.1 4

5 8.

1020 ( )9. /

1.

2. 1

3. . 5 1 ( )

1.

2.

3.

4. . 5 1 ( ) 10. 1. _____________________________________________

_____________________________________________________

2. / _________________________________________

____________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________

3

5 1 1 5

1.

n + 1 n (common difference)2. 1. ( 4.1 . 46/4)2. ( 4.1 . 46/5)3. n n ( 4.2 . 46/6)4. ( 6.1 . 46/1)

5.

( 6.1 . 46/2)6. ( 6.1 . 46/3)7.

( 6.1 . 46/4)8.

( 6.1 . 46/5)9. ( 6.1 . 46/63.

1. (K)

2. (K)4. (A)5.

(P)4.

(K)

1. 3

2. 2

2

3. 1.2 3

1.2

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

3 2

2

4. 1.2 3

1.2

5.

6.

7.

1

2

1. 2 2 3, 6, 9, 12 2, 4, 16, 256

2. 2 1 3 2 4 3 2

3.

4. 2

5. 112

6.

7. 6, 10, 14, 50

8. 100 100 n n

9. n an = a1 + (n-1)d

10. n 5, 10, 15, 100 10, 15, 20,

11. n . 5 1 . 5 1 ( )

12. 45 n 10

13. 2 10

14. 1 3

1. 1.2 . 5 1 ( )

2. 1.2 4

5

8.

1020

9. /

1. 2. 2 3. 4. . 5 1 ( )

1. 2. 3.

4. . 5 1 ( ) 10. 1. _____________________________________________

_____________________________________________________

2. / _________________________________________

____________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________

4 5 1 1 5

1.

n + 1 n (common ratio)2.

1. ( 4.1 . 46/4)2. ( 4.1 . 46/5)3. n n ( 4.2 . 46/6)4. ( 6.1 . 46/1)

5.

( 6.1 . 46/2)6. ( 6.1 . 46/3)7.

( 6.1 . 46/4)8.

( 6.1 . 46/5)9. ( 6.1 . 46/6)3.

1. (K)

2. (K)3. (A)4.

(P)4.

(K)

1. 3

2. 3

3

3. 1.3 3

1.3

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

3.

3

4. 1.3 3

1.3

5.

6.

7. 1

2 1. 3 3 6, 8, 10, 12 2, 4, 8, 16 1, 4, 9, 162. 2 1, 3 2 4 3 3 3. 4. 3 5. 112

6. 7. 2, 4, 8, 16, 50 8. 100 100 n n

9. n an = a1rn-110. n 5, 25, 125, 625 100 5, -25, 125, -625 11. n . 5 1 ( )

12. n 10 13. 5 14. 3

1. 1.3 . 5 1 ( )

2. 1.3 4 5

8. 2030

9. /

1. 3 2. 3 3. . 5 1 ( )

1. 2. 3.

4. . 5 1 ( ) 10. 1. _____________________________________________

_____________________________________________________

2. / _________________________________________

____________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________

5 5 1 1 6

1.

a1 , a2 , a3 , , an a1 + a2 + a3 + + an a1 + a2 + a3 + + an + a1 , a2 , a3 , ,

an,

2. 1. ( 4.1 . 46/4)2. ( 4.1 . 46/5)3. n n ( 4.2 . 46/6)4. ( 6.1 . 46/1)

5.

( 6.1 . 46/2)6. ( 6.1 . 46/3)7.

( 6.1 . 46/4)8.

( 6.1 . 46/5)9. ( 6.1 . 46/6)3. 1. n (K)2. n (K)3. (K)

4. (A)

5.

(P)

4.

(K)

1. 3

2. 4 5 3

4 5

3. 1.4 1.4 3

1.5 1.5

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

3.

4 5

4. 1.4

1.5

5. 6.

7.

1

1. 2. 2 1.

2.

3. ( ()

4. 1, 4, 9, 16 1 + 4 + 9 + 16 205. 4 (()

6. 4 (()

7.

= [2(1) + 2] + [2(2) + 2] + [2(3) + 2] ]

= 188. 5 (()

9. 5 1

10. 1 2, 3 4 5 11. (()

12. 23 13.

14. n 15. n

= 16. n

=

= =

17. 18. 2, 4, 6, , 500

19. 1 100 100 n

20. n Sn = (2a1+ (n-1) d) Sn = (a1+an)21. n

22. 23. 3

1. 1.4 1.5 () . 5 1 ( )

2. 1.4 1.5 () 4

5

8.

34 2030 9. /

1. 4 (() 2. 5 (() 3. . 5 1 ( )

1. 2. 3. 4. . 5 1 ( )10.

1. _____________________________________________

_____________________________________________________

2. / _________________________________________

____________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________

6

5 1 1 5

1.

2. 1. ( 4.1 . 46/4)2. ( 4.1 . 46/5)3. n n ( 4.2 . 46/6)4. ( 6.1 . 46/1)

5.

( 6.1 . 46/2)6. ( 6.1 . 46/3)7.

( 6.1 . 46/4)8.

( 6.1 . 46/5)9. ( 6.1 . 46/6)3.

1. n (K)2. n (K)3. (K)

4. (A)

5.

(P)

4.

(K)

1. 3

2. 1.5 1.5 3

3. 75%

4. 50%

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

3. 1.5

4. 75%

5. 3

3

5.

6.

7.

1

Sn = (2a1+ (n-1) d) Sn = (a1+an) 2 1. 2, 4, 8, 16, , 1,024

2. 2, 4, 8, , 1,024

3. 2, 4, 8, 16, 100 100

4.

5. n Sn =

Sn = a1-an r Sn =

6. n

7. n

8. . 5 1 ( ) 9. n

10.

11.

12. 1.5 . 5 1 ( )13. 1.514. 5 n 5 5

15.

16.

3

1. 1.5 . 5 1 ( )

2. 1.5 4 5

1. 2. 10 (10 ) 3. 1

1. 34 2030

2.

3. 9. / 1. 2. n 3. . 5 1 (

)

1. 2. . 5 1 ( ) 3.

4. 10.

1. _____________________________________________

_____________________________________________________

2. / _________________________________________

____________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________ 2

15

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2 1

1. ( 5.2 . 46/2)

2. ( 5.3 . 46/2)

3. ( 6.1 . 46/1)4. ( 6.1 . 46/2)

5. ( 6.1 . 46/3)

6. ( 6.1 . 46/4)

7. ( 6.1 . 46/5)8. ( 6.1 . 46/6)

...

...

1. k k 2. 3. / ...

1. 2. 3.

4.

2

1.

1. 2. 3. 4. 25.

6. 7. 8.

2.

2.1 1)

2)

3) /

4)

5) / 2.2

1)

2) /3) /

4)

5) /

3.

3.1 6

3.2

3.3

3

3 15

75

85

95

7 5 1 2 5 1.

k k 2. 1. ( 5.2 . 46/2)

2. ( 5.3 . 46/2)

3. ( 6.1 . 46/1)4. ( 6.1 . 46/2)

5. ( 6.1 . 46/3)

6. ( 6.1 . 46/4)

7. ( 6.1 . 46/5)8. ( 6.1 . 46/6)3.

1. (K)

2. (K) 3. (K)

4. (A)

5.

(P)

4.

(K)

1.

2. 3

3. 2.1 3

2.1

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

3.

2.1

5. 1.

2. 6.

7. 1

1. 30 (30 )

2.

3. 2

1. 2. (1) 3 2

3. 3 2

1 1

1 2 2

3 3

1 4

2 2 5

3 6

4. 6

5. 6 1 () 2 2 () 3 2 3 = 6 1

6. (2) 5 3 5 2 2 5

7. 5 3 5 2 2 5 1

3

5 2

2

2

3 (5 (2 ( 2(2 = 120 8. 120 1 () 3 2 () 5 3 () 2 4 ( ) 2 4 () 2 3 5 2 2 2 = 120 2

9. 27 (12)

(Multiplication principle) k 10. 27 11. (Addition principle) k

12. (Addition principle) 4 0, 1, 2, 3, 4, 5 4 0 0, 2, 4 1 0 1 5 4 3 1 (5 (4 (3 = 60 2 0 2 4 4 3 2 (4 (4 (4 = 96 4 0, 1, 2, 3, 4, 5 60 + 96 = 156 13. 11 (Addition principle) k

14. 12 15. 9, 12 16. 9, 12 3

1. 2.1 . 5 1 ( ) 2. 2.1

. 5 1 ( )

4

5

8.

45 2030 ()

9. /

1. (1)

2. (2)

3. (Addition principle)

4. . 5 1 ( )

1. 2 3. 4. . 5 1 ( )10.

1. _________________________________________________

_____________________________________________________

2. / _________________________________________

_________________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________

8

5 1 2 5 1.

2. 1. ( 5.2 . 46/2)

2. ( 5.3 . 46/2)

3. ( 6.1 . 46/1)4. ( 6.1 . 46/2)

5. ( 6.1 . 46/3)

6. ( 6.1 . 46/4)

7. ( 6.1 . 46/5) 8. ( 6.1 . 46/6)3.

1. (K)

2. (K)

3. (K) 4. (K)

5. (A)

6.

(P)

4.

(K)

1. 3

2.

2.1

3. 2.2 3

2.2

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

4.

2.1

3.

2.2

5.

6.

7. 1

1. 1 1 2. 2 1. 45 . 5 1

( ) 2. 3. 4. (Random Experiment)

5. (14)

1

1 1

(H) (T)

S = {H, T}

n (S) = 2 2

2 2

(HH) (HT) (TH) (TT) = {HH, HT, TH, TT}

n (S) = 2 (2 = 4 3

1 1

1, 2, 3, 4, 5, 6

S = {1, 2, 3, 4, 5, 6}

n (S) = 6 4

2 1

(1, 1), (1, 2), (1, 3), ..., (1, 6), (2, 1), ..., (2, 6), ..., (6, 6)

S = {(1, 1), (1, 2), (1, 3), ..., (1, 6), (2, 1), ..., (2, 6), ..., (6, 6)}

n (S) = 6 (6 = 36 6. 4 7. (Sample Space) S n(S)

8. (Event) (14)

1

1 1

(H) (T)

(H) = {1} 2

2 2

(HH) (HT) (TH) (TT) = {HH, HT, TH, TT}

(HH) (HT) (TH) = {3} 3

1 1

1, 2, 3, 4, 5, 6 6 6 = {1} 4

2 1

(1, 1), (1, 2), (1, 3), ..., (1, 6), (2, 1), ..., (2, 6), ..., (6, 6)

6 (1, 5), (2, 4), (3, 3), (5, 1), (4, 2) = {5}9. S

E E V S10. 2 1 1. 2. 3. 4. 5. 9

6. 11.

3

1. 2.2 . 5 1

( ) 2. 2.2

. 5 1 ( ) 4

5

8.

45 1520 510 ( )

9. /

1. (14)

2.

3. . 5 1 ( )

1. 2.

3. 4. . 5 1 ( )10. 1. _________________________________________________

_____________________________________________________

2. /__________________________________________

_________________________________________________________

3. _________________________________________________

_____________________________________________________________

4. __________________________________________

__________________________________________________________________

__________________________

_______ / ________ / ________

9

5 1 2 5

1.

S E1, E2 E1 ( S E2 ( S

1 E1 E2 S P (E1 ( E2) = P (E1) + P (E2) P (E1 ( E2) 2 E1 E2 S P (E1 ( E2) = P (E1) + P (E2) 3 E S P (E( ) = 1 P (E)

4 E1, E2 S P (E1 E2) = P (E2) P (E1 ( E2)2. 1. ( 5.2 . 46/2)

2. ( 5.3 . 46/2)

3. ( 6.1 . 46/1)4. ( 6.1 . 46/2)

5. ( 6.1 . 46/3)

6. ( 6.1 . 46/4)

7. ( 6.1 . 46/5)8. ( 6.1 . 46/6)3.

1. (K)

2. (K)

3. (K)

4. (K) 5. (K)

6. (A)

7.

(P)

4.

(K)

1. 3

2. 2.2 3

2.2 2.2 2.2

3. 75%

4. 50%

(A)

1. 3

2. 3

/ (P)

1. / 3

2.

/

3.

2.2 2.2

4. 3

3

5.

1.

2. 6.

7.

1 1. 2. 2 1. 4 . 5 1 ( ) . 5 1

2. 3. 4. N S K E S E

P (E) P (E) =

5. 6. (14)

1. E E

2. 3 1 E1 3 E2 2 1 E3

3. 4 2 5 3 1 1. 2. 3. 1

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

7. 8.

(

)

9.

10. 14

11.

12. ( )

S E1, E2 E1 ( S E2 ( S

1 E1 E2 S P (E1 ( E2) = P (E1) + P (E2) P (E1 ( E2) 2 E1 E2 S P (E1 ( E2) = P (E1) + P (E2) 3 E S P (E( ) = 1 P (E)

4 E1, E2 S P (E1 E2) = P (E2) P (E1 ( E2)13. 4 1 14.

15.

16.

3

1. 2.2 2.2 . 5 1 ( )

2. 2.2 2.2 4

5

25 1 (30 ) 8.

() 2030

9. /

1. (14)2.

3. . 5 1 ( )

1. 2. 3. 4. . 5 1 (

)10.

1. _______________________________________

________________________________________________

2. / ____________________________________

___________________________________________________

3. ___________________________________________

_________________________________________________________

4. _______________________________________

______________________________________________________________

_______________________________

_______ / ________ / ________ 3

/

1

2

1

2

.

...... ..............

................................ ........ ..............................

-

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

= ()

X =

Y =

N =

1

1.

1) 1, 2, 4, 7, 11

2) a, 2a + 3, 3a + 4, 4a + 5

3) 2, 4, 8, 16, 32

4) 2b 1, 3b + 1, 4b + 3, 5b + 5

2.

1) 1, 6, 11, 16, ...

2) 0, 7 , 5, 2 , ...

3) -5, -3, -1, 1, 3, ...

4) 12, 8, 5, 3, 0, 3, ...

3. -1, 1, 3, 6, ... 1) a 5 = 8

3) a 6 = 11

2) a 7 = 13

4) a 8 = 15

4. 12, 6, 0, ...

1) a 25 = -142

3) a 75 = -484

2) a 100 = -582

4) a 200 = -2,460

5. 3 9 6 18 (a 1, 2d)

1) (2, 2)

3) (2, 4)

2) (4, 4)

4) (4, 8)

6. -4, a, 4, b (2a, 3b)

1) (0, 8)

3) (2, 8)

2) (0, 24)

4) (4, 24)

7. a 1 a n 2 35 d = 3 n2 n

1) 132

3) 96

2) 12

4) 0

8. -1, 1, 3, 5, 7, 9 6

1) -3 3

2) -3 2

3) 8 11 3

4) 8 11 2

9. 0,,, 4

1) 1 +

2) 1 + ( ) n

3) 1

4) 1 ( ) n10. a n = 1 + (-1)n

1) a1

3) a32) a4

4) a5

11. 7 36, -12, 4, ...

1)

3)

2)

4)

12. 3 7 192

1) 2

3) 3

2) 4

4) 6

13. x 1, x + 1, 2x + 5 x

1) 1, -2

3) 2, -3

2) 1, -3

4) 3, -2

14. 3 2

1) 210 31

3) 210 24

2) 210

4) 24

15. 5 5 1) 14,400

3) 28,800

2) 36,600

4) 42,20016. 7 3 2 2

1) 48

3) 72

2) 288

4) 45617. 9 2 10

1) 10 8!

3) 10 9!

2) 80 8!

4) 80 9!18. 2 3 7

1) 120

3) 360

2) 420

4) 72019. number

1) 120

3) 240

2) 280

4) 36020. 7

1) 120

3) 240

2) 720

4) 96021. 8

1) 5,040

3) 10,080

2) 40,320

4) 80,64022. 7 6

1) 2! 5! 6!

3) 2! 6! 7!

2) 5! 6!

4) 6! 7!23. m n

1) m! n!

3) 2! m! n!

2) (m + n)!

4) m !+ n!24. 5 5

1) 2! 5! 5!

3) 2! 10!

2) 2! 5!

4) 1025. 10

1) 8! 2!

3) 9! 2!

2) 18! 2!

4) 19! 2!26. 12 3 3, 4 5 12

1) 1,001

3) 1,550

2) 2,310

4) 3,03027. 5 5 3 3

1) 120

3) 1,200

2) 12,000

4) 120,00028. 5

1) 24

3) 120

2) 720

4) 76829. 5 10

1) 50

3) 1,050

2) 25,200

4) 126,00030. 10

1) 215,040

3) 362,880

2) 2,150,400

4) 3,628,80031. 5 5

1) 362,880

3) 14,400

2) 2,880

4) 1,08032. 10

1) 181,440

3) 281,350

2) 361,420

4) 362,88033. 8

1) 5,040

3) 10,080

2) 40,320

4) 80,64034. 5 5

1) 12

3) 24

2) 60

4) 12035. 7

1) 120

3) 240

2) 720

4) 1,44036. m n

(1)(m 1)! (n 1)!

(2)m! (n 1)!

(3)(m 1)! n!

1(1 ) (2)

2(3) (1)

3(2) (3)

4(1), (2) (3)

37. PEACE

1) 15

3) 20

2) 60

4) 12038. tobacco

1) 1,260

3) 1,080

2) 840

4) 63039. assassination

1) 10,810,800

3) 1,081,080

2) 108,108

4) 10,81040. statistics

1) 8,400

3) 12,600

2) 25,200

4) 50,40041. a2b3c4

1) 720

3) 1,260

2) 1,890

4) 2,42042. Mississippi 1) 11,550

3) 17,325

2) 34,650

4) 37,82543. = n 1) 16

3) 18

2) 20

4) 2144. = 930 n 1) 30

3) 31

2) 33

4) 3445. n

1) (2n)! = n !2n

2) 1-3-5(2n 1) =

3) 2-4-6(2n) =

4) 1-3-5(2n 1) =

46. r I + 1 d r d n

1) n (n 1) (n 2)(n r + 1) (n r)

2) (n r + 1) (n r) 3(2(13) n (n 1) (n 2)(n r + 1)

4) n (n 1) (n 2) 3(2(147.

1) (n r)! =

2) = (r 2)!

3) = r

4) = n r + 1

48.

1) (3(4)! = 12! 3) 3! 4 != 12!

2) (3!)2 = 36

4) (32)! = 9!49. K

1) 630

3) 1,260

2) 1,540

4) 3,08050. C n, 14 = C n, 10 n

1) 4

3) 8

2) 18

4) 2451. = 6 n

1) 6

3) 7

2) 8

4) 952. = C n, 2

1) 153

3) 128

2) 98

4) 7253. 5!

1) C 8, 6

3) C 10, 32) C 6, 1 . C 6, 4

4) C 5, 1 . C 6, 254.

1) C 7, 3 = 35

3) C 12 ,8 = 495

2) C 40, 2 = 7804) C 22 ,3 = 1,45055.

1) C n+1 ,k = C n, k + C n, k 1

2) C n - 1, k = C n ,k + C n, k - 1

3) C n+1 ,k = Cn , k + C n , k 1

4) C n - 1,k = C n,k + Cn,k 1

56. C6,0 + C6,1 + C6,2 + C6,3 + C6,4 + C6,5 + C6,6

1) 26

3) 26 1

2) 26 + 1

4) 62 157. 2 1 S = {(a, b) |a b } 2

1) {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)}

2) {(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)}

3) {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

4) {(1, 6), )2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

58.

1) {(x, y) |x y }

2) {(x, y)| x, y }

3) {(x, y) |1 d x d 6 1 d yd 6}

4) {(x, y)| xy = 36}

59. 10 5 6

1) 21

3) 441

2) 1,003

4) 1,33060. 1 52

1) 13

3) 26

2) 39

4) 52

61. 1 5

1) 25

3) 32

2) 64

4) 12562. 4 1

1) 4

3) 16

2) 24

4) 3263. 3 1

1) 36

3) 108

2) 180

4) 21664. 1 1

1) 3

3) 8

2) 9

4) 1265. 1 2

1) {H, T}

3) {HH, TT}

2) {HH, HT, TT}4) {HH, HT, TH, TT}66. a b b a b a

1)

3) 2)

4)

67. 15 3 5 2

1)

3)

2)

4)

68. 8 100 1 20 2 10 5 1 10

1) 0.125

3) 0.375

2) 0.625

4) 0.87569. 1 9 9 1

1) ,, 3) ,,

2) ,,

4) ,,

70.

1) 2 1 = 0.75

2) 1 PORCHASE = 0.375

3) 2 = 0.75

4) 3 2 0, 1, 2, 3 =

71. 6 4 4

1)

3)

2)

4)

72. 6 2

1)

3)

2)

4)

73. 6 12 12 6

1)

3)

2)

4)

74. 52 4 4 1)

3)

2)

4)

75. 10 6 4 3 2 1

1) 0.4

3) 0.5

2) 0.6

4) 0.776. 2 1

1) 0.25

3) 0.40

2) 0.75

4) 0.8077. 70% A 40% B 30% C A B 20% A C 15% B C 10% 5% 1 1

1) 0.35

3) 0.45

2) 0.55

4) 0.6578. 52 2 2

1)

3) 2)

4)

79. 2 52 2

1)

3)

2)

4)

80. 4 3 3 3

1)

3)

2)

4)

2

1. 31, 28, 25, ..., -32 n

d=28 31 = -3

an=a1 + (n 1)d

-32=31 + (n 1) (-3)

-32=31 3n + 3

3n=66

N=22

22

2. 4 0, 1, 2, 3, 4, 5

4 0 0, 2, 4

1 0

1

5

4

3

1 5 4 3 = 60

2 0

2

4

4

3

2 4 4 3 = 96

4 0, 1, 2, 3, 4, 5 60 + 96 = 156

//

4321

1.

2.

3.

4.

5.

6.

4 = 3 = 2 = 1 =

.

/

4321

1.

2.

3.

4.

5.

4 = 3 = 2 = 1 =

.

/

.

.

() . . .

4

4

3

2

1

.

.

() . . .

4

4

3

2

1

..........................................................................................................

...........................................................................................

-

1

2

3

4

5

6

7

8

9

10

1

-

4

-

3

-

2

-

1

2

- 4

-

3

-

2

-

1

3

-

4

-

3

-

2

- 1

4

-

4

-

3

-

2

-

1

3 3 4

1

1.1

. 46 1.

2.

3.

1.2

. 461.

1.3

. 461.

1.4

. 461.

2

2.1

. 461.

2.2

. 461.

3

3.1

. 46

3.2 (visualization) (spatial reasoning)

(geometric model)

. 46

4

4.1 (pattern)

. 461.

2.

3.

4.

5.

4.2 (mathematical model)

. 461. - -

2. -

3.

4.

5.

6. n n

5

5.1

. 461.

2.

3.

5.2

. 461.

2.

5.3

. 461.

2.

6

6.1

. 461.

2.

3.

4.

5.

6. -

(Portfolio)

1.

2.

3.

4.

5.

6.

10

1.

2.

3.

4.

5.

6.

7. 2

(Scoring rubrics)

8.

9.

10.

10

1.

2. Rubrics

3.

(Portfolio)

......................................................................... ...... ............. ............

.............................................................................

1. .............................................................

.............................................................

2. .............................................................

.............................................................

3. .............................................................

.............................................................

.............................................................

4.

( 10 ).............................................................

.............................................................

.............................................................

..............................................................

..........................................................................................................

..........................................................................

1234

1.

2.

3.

4.

4

4

3

2

1

4

3

2

1

4

3

2

1

4

3

2

1

4

3

2

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2.

3.

4.

5.

6. 1

7.

8.

9.

10.

1.

2.

3.

1.

2.

3.

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1.

2.

3.

4.

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1.

2.

3.

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