The Analysis of subtraction of fractions the rule space method at grade 7

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T.C.

BAHÇEŞEHİR ÜNİVERSİTESİ

THE ANALYSIS OF SUBTRACTION OF FRACTIONS

THE RULE SPACE METHOD

AT GRADE 7

Master’s Thesis

H.GÜLAY TEZCAN

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T.C.

BAHÇEŞEHİR ÜNİVERSİTESİ

INSTITUTE OF SCIENCE

INFORMATION TECHNOLOGIES PROGRAM

THE ANALYSIS OF SUBTRACTION OF FRACTIONS

THE RULE SPACE METHOD

AT GRADE 7

Master’s Thesis

H.GÜLAY TEZCAN

Thesis Supervisor: ASST.PROF. DR. ORHAN GÖKÇÖL

İSTANBUL, 2008

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T.C.

BAHÇEŞEHİR ÜNİVERSİTESİ

DEPARTMENT OF COMPUTER ENGINEERING INFORMATION TECHNOLOGIES PROGRAM

Name of the Thesis: The Analysis of Subtraction of Fractions Using Rule Space Method At Grade 7

Name Last Name of the Student: H. Gülay Tezcan Date of Thesis Defense: Eylül 2008

The thesis has been approved by the Institute of Department of Computer Engineering Information Technologies Program

Prof. Dr. A. Bülent ÖZGÜLER Director

I certify that this thesis meets all the requirements as a thesis for the degree of Master of Science.

Signature

Asst. Prof. Doç. Dr. Orhan GÖKÇÖL Program Coordinator

Signature

This is to certify that we have read this thesis and that we find it fully adequate in scope, quality and content, as a thesis for the degree of Master of Science.

Examining Committee Members Signatures

Asst.Prof.Dr. Orhan GÖKÇÖL

Thesis Supervisor ---

Asst.Prof.Dr. M. Alper TUNGA

Member ---

Asst.Prof.Dr. Yalçın ÇEKİÇ

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% ( ! &! (! ,! ==> ( /M % . =4 N M P M O ==4 = O O / P M M /4 M M P O 1 =/4 == = P 1 1 M4 N = P ? > =M4 M N M / P 1 ? O4 OM M O P N =0 =O4 O / M M P N N N4 M= /M P / / =N4 = / P M ?4 ? O P 5 5 =?4 N O O = P 5 5 54 M /= P N =54 M O 5 P N N 14 M M P / / =14 = 1 O / P =0 =0 >4 M5 / P 1 =>4 O O = P M =04 O O / 5 P =/ =/ /04 = N O = P M M

(50)

M1 % 3 " 4 . =. ( " " /. - " M. - " " O. 7 8 N. & = " ?. ( " " 5. - " % * 6 7 " % /0 6 " % " " " " ! " 0 = % 0 10 "

(51)

M> 5 3, 0 1 8 + ; + :) <0 1/*1;1*8 + 8* / 1/ ++ % . . " 10 % " B . " " N0 10 % " 2 . " N0 % "

(52)

O0 5 , 0 /8 ): + 8* / + 1/ <0 ): 0 + 0 < 7) 1 + 1/ ++ # " E " ! ( " % " " 11! " 10 5 4, /+ /* /* * ;1 1)/+ > 1/ / 0 +1+ @ ): ) <) /* ; / 1 8 + :) ++ /* ) ++ 3 4 5 I J ' + * /0 /M => /O /0 // /O *+ & < 1< / 0 M 0 = / 0 ) + * / = / 0 M 0 0 ) <) 3 4 5 I J ' ++ =? MN 3 M 0=4 0 >0 3 /=4 0 >? 3 =>4 0 11 3 /N4 = 00 3 004 0 >= 3 /N4 0 >O 3 =/4 = 00 3 004 ) =? 1/ 3 M N4 0 1> 3 /=4 0 >1 3 =M4 0 15 3 /O4 0 >1 3 ==4 0 >= 3 /N4 0 >M 3 =14 0 >> 3 =/4

(53)

O= ! "# % " . $178 5 3, +< 1 1; 1/:) 1)/ ): ++ )/< 8+1)/ ): ++ , ( " % " ( " " O 5 % T " & " " .

(54)

O/ 5 3 &! !#& # !& ! ,

% ( ! &! (! ,! ==> ( & /O % . =4 N M P M O ==4 = O O / P M M /4 M M P O 1 =/4 == = P 1 1 M4 N = P ? > =M4 M N M / P 1 ? O4 OM M O P N =0 =O4 O / M M P N N N4 M= /M P / / =N4 = / P M ?4 ? O P 5 5 =?4 N O O = P 5 5 54 M /= P N =54 M O 5 P N N 14 M M P / / =14 = 1 O / P =0 =0 >4 M5 / P 1 =>4 O O = P M =04 O O / 5 P =/ =/ /04 = N O = P M M

(55)

OM % 3 " 4 . =. ( " " /. - " M. - " " O. 7 8 N. & = " ?. ( " " 5. - " % * 6 7 " % /0 6 " % " " " " ! " 0 = % 0 10 "

(56)

OO 5 5, 0 1 8 + ; + :) <0 1/*1;1*8 + 8* / 1/ ++ % . . " 10 % " B . " " N0 10 % " 2 . " N0 % "

(57)

ON 5 I, 0 /8 ): + 8* / + 1/ <0 ): 0 + 0 < 7) 1 + 1/ ++ # " E " ! ( & " % " " 1=! " 10 5 J, /+ /* /* * ;1 1)/+ > 1/ / 0 +1+ @ ): ) <) /* ; / 1 8 + :) ++ /* ) ( & 3 4 5 I J ' + * => /M => // /= => /M *+ & < 1< M / O / / / 0 ) + * M 0 / = / O / ) <) 3 4 5 I J ' ++ =O 5> 3 O 1M4 1= 3 M04 >? 3 =O4 1/ 3 /14 >/ 3 /M4 15 3 M04 1M 3 M04 >M 3 /N4 ) =? 1/ 3 M N4 0 1> 3 /=4 0 >1 3 =M4 0 15 3 /O4 0 >1 3 ==4 0 >= 3 /N4 0 >M 3 =14 0 >> 3 =/4

(58)

O? "# % " . $178 5 , +< 1 1; 1/:) 1)/ ): ++ )/< 8+1)/ :) ++ , ( & " % " " " ( ( ( , ? % / % T " & " " .

(59)

O5 5 3 4 &! !#& # ! & ! ,

% ( ! &! (! ,! ==> ( ( /O % . =4 N M P M O ==4 = O O / P M M /4 M M P O 1 =/4 == = P 1 1 M4 N = P ? > =M4 M N M / P 1 ? O4 OM M O P N =0 =O4 O / M M P N N N4 M= /M P / / =N4 = / P M ?4 ? O P 5 5 =?4 N O O = P 5 5 54 M /= P N =54 M O 5 P N N 14 M M P / / =14 = 1 O / P =0 =0 >4 M5 / P 1 =>4 O O = P M =04 O O / 5 P =/ =/ /04 = N O = P M M

(60)

O1 % 3 " 4 . =. ( " " /. - " M. - " " O. 7 8 N. & = " ?. ( " " 5. - " % * 6 7 " % /0 6 " % " " " " ! " 0 = % 0 10 "

(61)

O> 5 ', 0 1 8 + ; + :) <0 1/*1;1*8 + 8* / 1/ ++ % . . " 10 % " B . " " N0 10 % " 2 . " N0 % " $ ! %

(62)

N0 5 , 0 /8 ): + 8* / + 1/ <0 ): 0 + 0 < 7) 1 + 1/ ++ # " E " ! ( ( " % " " 1=! " 10 5 K, /+ /* /* * ;1 1)/+ > 1/ / 0 +1+ @ ): ) <) /* ; / 1 8 + :) ++ /* ) ( ( 3 4 5 I J ' ) /= /O => /O // /M /O E / 0 / 0 0 = 0 E ) = 0 M 0 / 0 0 ) <) 3 4 5 I J ' ++ =5 0O 3 / O14 >= 3 /04 = 3 004 1? 3 /14 >> 3 /4 >/ 3 /N4 >5 3 014 = 0 3 004 ) =? 1/ 3 M N4 0 1> 3 /=4 0 >1 3 =M4 0 15 3 /O4 0 >1 3 ==4 0 >= 3 /N4 0 >M 3 =14 0 >> 3 =/4

(63)

N= "# % " . $178 5 4, +< 1 1; 1/:) 1)/ ): ++ )/< 8+1)/ ): ++ , ( ( " % " " " ( & ? " % " /! O 5 % T " & " " .

(64)

N/ 5 3 5 &! !#& # ! & ! ,

% ( ! &! (! ,! ==> ( /O % . =4 N M P M O ==4 = O O / P M M /4 M M P O 1 =/4 == = P 1 1 M4 N = P ? > =M4 M N M / P 1 ? O4 OM M O P N =0 =O4 O / M M P N N N4 M= /M P / / =N4 = / P M ?4 ? O P 5 5 =?4 N O O = P 5 5 54 M /= P N =54 M O 5 P N N 14 M M P / / =14 = 1 O / P =0 =0 >4 M5 / P 1 =>4 O O = P M =04 O O / 5 P =/ =/ /04 = N O = P M M

(65)

NM % 3 " 4 . =. ( " " /. - " M. - " " O. 7 8 N. & = " ?. ( " " 5. - " % * 6 7 " % /0 6 " % " " " " ! " 0 = % 0 10 "

(66)

NO 5 3 , 0 1 8 + ; + :) <0 1/*1;1*8 + 8* / 1/ )8 ++ % . . " 10 % " B . " " N0 10 % " 2 . " N0 % "

(67)

NN 5 33, 0 /8 ): + 8* / + 1/ <0 ): 0 + 0 < 7) 1 +1/ ++ # " E " ! ( " % " " 1=! " 10 5 3 , /+ /* /* * ;1 1)/+ > 1/ / 0 +1+ @ ): ) <) /* ; / 1 8 + :) ++ /* ) ( 3 4 5 I J ' + * // /O /= /O // // /O *+ & < 1< / 0 = 0 0 / 0 ) + * 0 0 / 0 / 0 0 ) <) 3 4 5 I J ' ++ =5 >/ 3 / N4 >O 3 =M4 = 00 3 004 >= 3 /?4 = 00 3 004 >M 3 /N4 >M 3 =14 = 00 3 004 ) =? 1/ 3 M N4 0 1> 3 /=4 0 >1 3 =M4 0 15 3 /O4 0 >1 3 ==4 0 >= 3 /N4 0 >M 3 =14 0 >> 3 =/4

(68)

N? "# % " . $178 5 5, +< 1 1; 1/:) 1)/ ): ++ )/< 8+1)/ ): ++ , ( " % " ( " % " /! O 5 % T "

(69)

N5 5 3 I &! !#& # ! & ! ,

% ( ! &! (! ,! ==> ( , /O % . =4 N M P M O ==4 = O O / P M M /4 M M P O 1 =/4 == = P 1 1 M4 N = P ? > =M4 M N M / P 1 ? O4 OM M O P N =0 =O4 O / M M P N N N4 M= /M P / / =N4 = / P M ?4 ? O P 5 5 =?4 N O O = P 5 5 54 M /= P N =54 M O 5 P N N 14 M M P / / =14 = 1 O / P =0 =0 >4 M5 / P 1 =>4 O O = P M =04 O O / 5 P =/ =/ /04 = N O = P M M

(70)

N1 % 3 " 4 . =. ( " " /. - " M. - " " O. 7 8 N. & = " ?. ( " " 5. - " % * 6 7 " % /0 6 " % " " " " ! " 0 = % 0 10 "

(71)

N> 5 34, 0 1 8 + ; + :) <0 1/*1;1*8 + 8* / 1/ )8 ++ % . . " 10 % " B . " " N0 10 % " 2 . " N0 % "

(72)

?0 5 35, 0 /8 ): + 8* / + 1/ <0 ): 0 + 0 < 7) 1 + 1/ ++ # " E " ! ( , " % " " 1=! " 10 5 3I, /+ /* /* * ;1 1)/+ > 1/ / 0 +1+ @ ): ) <) /* ; / 1 8 + :) ++ /* ) ( , 3 4 5 I J ' + * /= // /0 /O /= /M /O *+ & < 1< / = M 0 = = 0 ) + * = = = 0 / 0 0 ) <) 3 4 5 I J ' ++ =5 5N 3 M M14 >/ 3 =14 >? 3 =54 >0 3 /=4 = 00 3 004 >M 3 //4 >1 3 0?4 = 00 3 004 ) =? 1/ 3 M N4 0 1> 3 /=4 0 >1 3 =M4 0 15 3 /O4 0 >1 3 ==4 0 >= 3 /N4 0 >M 3 =14 0 >> 3 =/4

(73)

?= "# % " . $178 5I , +< 1 1; 1/:) 1)/ ): ++ )/< 8+1)/ ): ++ , ( , " % " " " ( , " % " O 5 % T " & " " .

(74)

?/ 5 ! & ! $ ! ! ! ' ! % " % % " ! ! ! % " - " : " % " ==> 5 & - % 7 ==> 6 ! " % " 5 /0! /0 8 M1 % /0 5 " " ! " " /5S =/1

(75)

?M 5 3 ! & ! $ ! ! ! 8 ): 1 +, /0 +< 1 1)/ ): 1 8 +, 3, ( " " , - " 4, - " " 5, 7 8 I, & = " J, ( " " ', - " ! $178 5 J, ++ <0 ) - . 3 " " " \ 0 N0 4 " . 3 0 N0 \ " " \ 0 104 ) . 3 " " " ` 0 104

(76)

?O 5 ! & ! $ ! ! ! 8 ): 1 +, /0 +< 1 1)/ ): 1 8 +, 3, ( " " , - " 4, - " " 5, 7 8 I, & = " J, ( " " ', - " ! $178 5 ', ++ <0 ) - . 3 " " " \ 0 N0 4 " . 3 0 N0 \ " " \ 0 104 ) . 3 " " " ` 0 104

(77)

?N 5 4 ! & ! $ ! ! ! 8 ): 1 +, /0 +< 1 1)/ ): 1 8 +, 3, ( " " , - " 4, - " " 5, 7 8 I, & = " J, ( " " ', - " ! $178 5 , ++ <0 ) - . 3 " " " \ 0 N0 4 " . 3 0 N0 \ " " \ 0 104 ) . 3 " " " ` 0 104

(78)

?? 5 5 ! & ! $ ! ! ! 8 ): 1 +, /0 +< 1 1)/ ): 1 8 +, 3, ( " " , - " 4, - " " 5, 7 8 I, & = " J, ( " " ', - " ! $178 5 K, ++ <0 ) - . 3 " " " \ 0 N0 4 " . 3 0 N0 \ " " \ 0 104 ) . 3 " " " ` 0 104

(79)

?5 5 I ! & ! $ ! ! ! 8 ): 1 +, /0 +< 1 1)/ ): 1 8 +, 3, ( " " , - " 4, - " " 5, 7 8 I, & = " J, ( " " ', - " $ & ! $178 5 3 , ++ <0 ) - . 3 " " " \ 0 N0 4 " . 3 0 N0 \ " " \ 0 104 ) . 3 " " " ` 0 104

(80)

?1 5 4 # # !# ! & ! $ ! $! # % ! # # % , . 8* / # , 5>O ) <) , > /0 8* / # , N1= ) <) , =M /0 % 2-) " - S 5>O O - S N1= T = " - S N1= 3 0 10 4 % " +< 1 1)/ ): 1 8 + , 3, ( " " . , - " . 3 4, - " " . KI 5 , 7 8 . I, & = " . 3 J, ( " " . KK ', - " . 3 +< 1 1)/ ): 1 8 + , 3, ( " " . I , - " . 3 4, - " " . 4 5 , 7 8 . 3 I, & = " . J, ( " " . I4 ', - " . 3

(81)

?> & #a,b,#c2 $ G,Xc $,-c2G,2 %,-%c % E - G I 2,E(c 2 2K IC A T- A. %! ! " M. E A T -A A A. 33' T 'D % C A A. =1 =. & A A " ' * . KK /. & A 3"+ + +4 A . 3 M. aA ' . K O. " . 3 N. & ' A A " . 3 ?. aA A = / A A . 3 5. & A . 3 1 71 < 1 1 N / 1 O /P ;1 +1 "#$ $ % &' ' ' ' ' ' (' $' ' & "' % ) * -Q /<1/1/ 1 71 < 1 1 N / 1 O /P ;1 1 +P =8/ P,

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(82)

50 & #a,b,#c2 $ G,Xc $,-c2G,2 %,-%c % E - G I 2,E(c 2 2K IC A T- A. . R & # S % E A T -A A A. 3II T 'D % C A A. > =. & A A " ' * . 3 /. & A 3"+ + +4 A . 3 M. aA ' . I O. " . 3 N. & ' A A " . ?. aA A = / A A . JI 5. & A . 3 1 71 < 1 1 N / 1 O /P ;1 +1 ' ( + ," - # ' ' ' ' ' ' ' (' $' ' & "' % ) * -Q /<1/1/ 1 71 < 1 1 N / 1 O /P ;1 1 +P =8/ P,

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(83)

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(84)

5/ - ! " < % 2 : - ) * ; " % =O /0 " 7 ! - ! =O! ; " . " " " " F & =O! " = " " " / % =O % ; " " # " - =O /0P /0 /0d /0 => =1 =5 =?d ( M1 5?0 =O =Od ?d =Od ? N O M / = " 3N =4 K 2-) " 2-) " 2-) " " % " % % " ! ! ! % 2-) " " " " "

(85)

5M ! $ ! ))=+, & ! ) $ !3=>>/4 % : ( !% Y ) ! X ) !3 =>1/4 EX. , " ( ! X 7 Y % ! $ $ !3/00/4 !" % 2 ! ( & ! 2 Y # ! , !3=>5M4# E 7 !$ !3 =>>04$ %" & & # !2 L Y( ! % !3=>514$ E ! ) - ! % ! 3=>>04 $ !X % !$ $ !3=>>04 ' E 7 !2 ! G Y) ( - 3 , 4 ! ' ( NOMQN11 # EX. , " % ! $ $ !3=>>=4) * + , * - ' , ! E@. , % -% $ $ ! 3=>>M4$ # !EX

(86)

5O % $ $ !3 4 , # . / * 0 - ' , / ) ! E@. , " & 1)*1< + , & ! ) $ Y % ! $ $ !3 4 1 ' 2 X % ) # " ! 2 $ ! X ! 2 F ! 3=>>M / , ) . =/3M4! M1:O5 % ! $ $ ! 3 =>1M4 . , 3 ' X , ) ! /0! MO:M1 % !$ $ Y G !2 G !3=>1M4 $ ) !53=4 ! 1=:>? % ! $ $ !3=>1N4 # X , - ! =/3=4! NN:5M % !) ) Y % ! $ $ !3 =>1>4 . , - $ * Y E G X 3, 4 , - - 3 L 1! /=5://04 E .F % ( ! 3 /00/4 3 X 2 - - - (!N=! MM5:MN0 L G !$ !3 =>1/4) 3 4 1 X ) & ! M! M:5=

(87)

5N 0 )8 < + , ! # !3 =>114 , 2, E .TT T T( U U .TT T T U U .TT @ " T T T " T=NT/T==>

(88)

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