,,,TACTILE ALGEBRA TILES " ,,APH^R ,AM]ICAN ,PR9T+ ,H\SE = ! ,BL1 ,9C4 #AHCI ,FRANK=T ,AV5UE ,L\ISVILLE1 ,,KY #DJBJF-#JJHE ,PUBLI%$ BY ,AM]ICAN ,PR9T+ ,H\SE = ! ,BL1 ,9C4 ,CATALOG ,NO4 #A-#JHDAJ-#JJ ,COPY"R ^C #BJAG BY ,AM]ICAN ,PR9T+ ,H\SE = ! ,BL1 ,9C4 ,,,W>N+3 *OK+ HAZ>D,' - ,SMALL "PS4 ,N 9T5D$ = ,*N AGES #E & "U )\T ADULT SUP]VI.N4 ,9 #A ,VOLUME ,BRL PAGES T#A-T#C & #A-#BD ,PR9T PAGES #A-A#AF T#A ,,,SPECIAL SYMBOLS US$ 9 ? VOLUME,' .= ,DOT ,LOCATOR = 8M5;N0 .=,,, ,CAPITALIZ$ PASSAGE 9DICATOR .=,' CAPITALS TERMINATOR .=.7 ,ITALICS PASSAGE 9DICATOR .=.' ,ITALICS T]M9ATOR .=_4 ,PRIM>Y BULLET .="6 ,PLUS .="- ,M9US .=9 ,SUP]SCRIPT .="7 ,EQUALS .="8 ,MULTIPLICA;N .="/ ,DIVI.N .=" ,3T9UA;N 9DICATOR ,,TRANSCRIB]',S ,,NOTES ,? VOLUME IS TRANSCRIB$ AC TO .7,! ,RULES ( ,UNIFI$ ,5GLI% ,BRL1.' ,SECOND ,$I;N #BJAC4 ,? VOLUME HAS BE5 TRANSCRIB$ AC TO .7,BRL ,=MATS3 ,PR9CIPLES ( ,PR9T-TO-,BRL ,TRANSCRIP;N1 #BJAA4.' T#C ,,,TACTILE ALGEBRA #A TILES,' ,ALGEBRA TILES IS A MA!MATICAL MANIPULATIVE T PROVIDES /UD5TS ) 3CRETE MODELS = APPROA*+ AB/RACT ALGEBRAIC 3CEPTS & PROC$URES4 ,! TOOL MAKES X POSSIBLE = /UD5TS TO VISUALIZE EXPRES.NS & EQUA;NS & ?US HELPS MEET _! DIV]SE NE$S 9 ALGEBRA /UDY4 ,= /UD5TS :O /RU7LE ) ! AB/RACT NATURE ( ALGEBRA1 ALGEBRA TILES C HELP !M A3ESS ! ALGEBRA RELAT$ 3T5T LAID \ 9 ! ,COMMON ,CORE ,/ATE ,/&>DS = ,MA!MATICS "<,,CCSSM">4 ,! TILES C 2 US$ BY /UD5TS 9 ELE;T>Y "? HI< S*OOL = ADD+1 SUBTRACT+1 & MULTIPLY+ 9TEG]S1 SIMPLIFY+ EXPRES.NS1 SOLV+ L9E> & QUADRATIC EQUA;NS1 & = ADD+1 SUBTRACT+1 MULTIPLY+1 & FACTOR+ POLYNOMIALS4 ,^! #A SUBJECTS RELATE TO ! A#A /&>DS ( _M ,,CCSSM DOMA9S S* Z ,OP]A;NS & ,ALGEBRAIC ,?9K+1 ,NUMB] & ,OP]A;NS1 ,! ,NUMB] ,SY/EM1 & ,EXPRES.NS & ,EQUA;NS4 ,PL1SE REF] TO ! ,,CCSSM = M DETAILS4 ,TACTILE ,ALGEBRA ,TILES IS AN A3ESSIBLE V].N ( ^! ALGEBRA TILES1 SPECIFICALLY DESIGN$ = /UD5TS ) BL;S & L[ VI.N4 ,X 9CLUDES MAGNETIC TILES T ! /UD5TS C MANIPULATE ON A /EEL ----------------------------#B BO>D1 & USES TACTILE SYMBOLS TO HELP /UD5TS DI6]5TIATE 2T TILES4 ,COLORS >E *OS5 ) 3SID]A;N = ! NE$S ( /UD5TS :O H L[ VI.N4 ,TACTILE ,ALGEBRA ,TILES IS 3SI/5T ) REGUL> ALGEBRA TILES US$ BY SIE VISUALLY IMPAIR$ TO "W TGR ) _! CLASSMATES 9 9CLUSIVE CLASSROOMS4 ,WE HOPE ? PRODUCT C HELP MAKE YR A#B T1*+ & LE>N+ EXP]I;E AN 5JOYABLE "O4 ,PRODUCT ,COMPON5TS ,! ,TACTILE ,ALGEBRA ,TILES KIT 3SI/S ( ! FOLL[+ COMPON5TS3 _4 ,/EEL BO>D - TWO FOLDABLE BO>DS ON : /UD5TS C PLACE MAGNETIC TILES4 ,/UD5TS C USE "O BO>D Z A "W+ BO>D & ! O!R = /ORAGE4 _4 ,SMALL LIME SQU>E TILE "<#CB PIECES"> - SMALL SQU>E TILE ) A TACTILE SQU>E SYMBOL AT ! C5T]4 ,^! REPRES5T ! NUM]ICAL VALUE 8"6#A40 _4 ,SMALL R$ SQU>E TILE "<#CB PIECES"> - SMALL SQU>E TILE ) A RAIS$ DOT AT ! C5T]4 ,^! REPRES5T ! NUM]ICAL VALUE 8"-#A40 #C _4 ,VIOLET RECTANGLE B#B TILE "<#AB PIECES"> - RECTANGLE TILE ) A TACTILE RECTANGUL> SYMBOL AT ! C5T]4 ----------------------------#C ,^! REPRES5T ! NUM]ICAL VALUE 8"6X40 _4 ,R$ RECTANGLE TILE "<#AB PIECES"> - RECTANGLE TILE ) A R[ ( RAIS$ DOTS AT ! C5T]4 ,^! REPRES5T ! NUM]ICAL VALUE 8"-X40 _4 ,L>GE BLUE SQU>E TILE "<#F PIECES"> - L>GE SQU>E TILE ) A TACTILE SQU>E SYMBOL AT ! C5T]4 ,^! REPRES5T ! NUM]ICAL VALUE 8"6X;9#B40 _4 ,L>GE R$ SQU>E TILE "<#F PIECES"> - L>GE SQU>E TILE ) A DOTT$ SQU>E SYMBOL AT ! C5T]4 ,^! REPRES5T ! NUM]ICAL VALUE 8"-X;9#B40 _4 ,BLACK SEP>A;N B> "<#B PIECES"> - TWO L;G1 N>R[ PIECES T C 2 US$ TO SEP>ATE TWO SIDES ( AN EQUA;N4 A#C ,EXAMPLES ( ,USE ,! FOLL[+ SEC;N PROVIDES S"EAL EXAMPLES ( H[ TO USE ,TACTILE ,ALGEBRA ,TILES TO SOLVE ALGEBRAIC PROBLEMS2 H["E1 X IS N A COMPREH5SIVE GUIDE ON ! USE ( ALGEBRA TILES4 ,= FUR!R DETAILS AB H[ TO USE ALGEBRA TILES1 PL1SE REF] TO O!R RES\RCES4 ,! 3CEPT ( 8Z]O PAIR0 IS (T5 US$ 9 ALGEBRA TILE ILLU/RA;NS4 ,2C ! SUM ( A NUMB] & XS OPPOSITE IS Z]O1 A POSITIVE TILE & A NEGATIVE TILE PAIR$ TGR REPRES5T Z]O & >E CALL$ A Z]O PAIR4 ,= EXAMPLE1 "O SMALL LIME SQU>E & "O SMALL R$ SQU>E =M A Z]O ----------------------------#D PAIR1 :ILE ?REE VIOLET RECTANGLES & ?REE R$ RECTANGLES MAKE UP ?REE #E Z]O PAIRS4 ,:5 US+ A#D ALGEBRA TILES TO SOLVE ALGEBRAIC PROBLEMS1 Z]O PAIRS C 2 ADD$ TO OR REMOV$ F A MA!MATICAL EXPRES.N )\T *ANG+ ! VALUE ( ! EXPRES.N4 ,9 ALL EXAMPLES1 LIME1 VIOLET1 & BLUE TILES >E US$ TO REPRES5T POSITIVE VALUES & R$ TILES >E US$ TO REPRES5T NEGATIVE VALUES4 #A4 ,ADD+ & SUBTRACT+ 9TEG]S ,TO ILLU/RATE ! 3CEPT ( ADD+ TWO POSITIVE NUMB]S1 S* Z #C"6#F1 H ! /UD5T %[ TWO GR\PS ( SMALL LIME SQU>ES1 "O ) ?REE TILES REPRES5T+ POSITIVE #C1 & ! O!R ) SIX TILES REPRES5T+ POSITIVE #F "<,PICTURE #A">4 ,! /UD5T W !N ADD ! TWO GR\PS BY MOV+ ! TILES TGR4 ,H ! /UD5T C.T ! TILES TO GET ! ANSW]1 #C"6#F "7 #I4 ,PICTURE #A3 #E ,MODEL+ TWO POSITIVE NUMB]S #C & #F US+ ,TACTILE ,ALGEBRA ,TILES ,TO ILLU/RATE ! 3CEPT ( ADD+ A POSITIVE NUMB] & A NEGATIVE NUMB]1 S* Z #C"6"<"-#G">1 H ! /UD5T %[ TWO GR\PS ( SMALL SQU>ES1 "O ) ?REE LIME SQU>ES REPRES5T+ POSITIVE #C1 & ! O!R ) SEV5 R$ SQU>ES REPRES5T+ NEGATIVE #G4 ,TO ADD ! TWO GR\PS1 MOVE !M TGR4 ,2C ?REE LIME SQU>ES & ?REE R$ SQU>ES MAKE UP ?REE Z]O PAIRS1 !Y C 2 REMOV$ )\T *ANG+ ! TOTAL VALUE ( ! EXPRES.N4 ,S9CE "! >E F\R SMALL R$ SQU>ES REMA9+1 ! ANSW] TO ! PROBLEM IS "-#D4 ,TO ILLU/RATE SUBTRAC;N PROBLEMS S* Z "-#E"-"<"-#B">1 F/ H ! /UD5T USE "O GR\P ( FIVE SMALL R$ SQU>ES TO REPRES5T "-#E4 ,TO SUBTRACT "-#B1 REMOVE TWO SMALL R$ #G SQU>ES4 ,?REE R$ SQU>ES A#E >E LEFT1 S ! ANSW] IS "-#C4 ----------------------------#F ,TO ILLU/RATE SUBTRAC;N PROBLEMS S* Z "-#B"-#E1 REM9D ! /UD5T T TO SUBTRACT A NUMB]1 Y C ADD XS OPPOSITE4 ,"!=E1 ! PROBLEM 2COMES "-#B"6"<"-#E">4 ,!N1 H ! /UD5T %[ TWO GR\PS ( SMALL R$ SQU>ES1 "O GR\P REPRES5T+ "-#B & ! O!R REPRES5T+ "-#E4 ,! /UD5T W MOVE ! TILES TGR & GET ! ANSW]1 "-#G4 ,ANO!R WAY TO ILLU/RATE ? PROBLEM IS TO USE Z]O PAIRS4 ,F/1 H ! /UD5T %[ TWO SMALL R$ SQU>ES : REPRES5T "-#B4 ,NEXT1 ADD FIVE Z]O PAIRS ( SMALL R$ & LIME SQU>ES S T "! >E 5 LIME SQU>ES TO SUBTRACT4 ,NOTICE T ADD+ Z]O PAIRS DOES N *ANGE ! TOTAL VALUE4 ,!N1 REMOVE FIVE LIME SQU>ES TO SUBTRACT #E4 ,! ANSW] IS / "-#G4 #B4 ,MULTIPLICA;N ( A#F 9TEG]S ,TO ILLU/RATE ! MULTIPLICA;N PROBLEM #C"8#D1 PLACE ?REE GR\PS ( F\R SMALL LIME SQU>ES TGR TO =M A RECTANGLE4 ,! V]TICAL & HORIZONTAL SIDES ( ! RECTANGLE REPRES5T ! TWO FACTORS #C & #D "<,PICTURE #B">4 ,C.T ! NUMB] ( TILES 9 ! RECTANGLE TO GET XS >EA1 #AB1 : IS ! ANSW] TO ! PROBLEM4 ----------------------------#G ,PICTURE #B3 ,ILLU/RA;N ( #C"8#D US+ ,TACTILE ,ALGEBRA ,TILES ,TO ILLU/RATE ! MULTIPLICA;N PROBLEM #C"8"<"-#D">1 USE ?REE GR\PS ( F\R SMALL R$ SQU>ES TO =M A ?REE-BY-F\R RECTANGLE4 ,C.T ! TOTAL NUMB] ( TILES 9 ! RECTANGLE4 ,? "T1 ! ANSW] IS "-#AB4 ,MULTIPLICA;N PROBLEMS S* Z "<"-#C">"8#D C 2 TR1T$ Z #D"8"<"-#C"> BY REM9D+ ! #I /UD5T T ! ORD] 9 : WE A#G MULTIPLY DOES N MATT]4 ,TO ILLU/RATE ! MULTIPLICA;N PROBLEM "<"-#C">"8"<"-#D">1 REM9D ! /UD5T T ! 8"-0 SIGN C M1N 8OPPOSITE (40 ,!N1 ! PROBLEM C 2 TR1T$ Z ! OPPOSITE ( #C"8"<"-#D">4 ,?US1 ! ANSW] IS #AB4 #C4 ,SIMPLIFY+ ALGEBRAIC EXPRES.NS ,MODEL ! EXPRES.N "-#CX"-#B"-#BX"6#D"6X US+ ALGEBRA TILES4 ,2C SUBTRACT+ IS ! ----------------------------#H SAME Z ADD+ ! OPPOSITE1 ! EXPRES.N C 2 TR1T$ Z "-#CX"6"<"-#B">"6"<"-#BX">" "6#D"6X4 ,?US1 H ! /UD5T PLACE ?REE R$ RECTANGLES1 TWO SMALL R$ SQU>ES1 TWO R$ RECTANGLES1 F\R SMALL LIME SQU>ES1 & "O VIOLET RECTANGLE TGR "<,PICTURE #C">4 ,!N1 H A#H ! /UD5T MOVE L T]MS TGR4 ,T IS1 MOVE ?REE R$ RECTANGLES1 TWO R$ RECTANGLES1 & "O VIOLET RECTANGLE TGR & TWO SMALL R$ SQU>ES & F\R SMALL LIME SQU>ES TGR4 ,2C ! VIOLET RECTANGLE & "O ( ! R$ RECTANGLES =M A Z]O PAIR1 !Y C 2 REMOV$ )\T *ANG+ ! TOTAL VALUE ( ! EXPRES.N4 ,SIMIL>LY1 TWO SMALL R$ SQU>ES :5 GR\P$ ) TWO LIME SQU>ES =M TWO Z]O PAIRS & C 2 REMOV$ TOO4 ,2C "! >E ONLY F\R R$ RECTANGLES & TWO SMALL LIME SQU>ES LEFT1 ! SIMPLIFI$ EXPRES.N IS "-#DX"6#B4 ,PICTURE #C3 ,MODEL+ EXPRES.N "-#CX"6"<"-#B">" "6"<"-#BX">"6#D"6X US+ ,TACTILE ,ALGEBRA ,TILES #AA #D4 ,SOLV+ L9E> #I EQUA;NS 9 "O V>IABLE ,TAKE ! EQUA;N #BX"-#A "7 #E = AN EXAMPLE4 ,:5 MODEL+ AN EQUA;N US+ TILES1 /UD5TS C USE A BLACK SEP>A;N B> TO SEP>ATE ! TWO SIDES ( ! EQUA;N4 ,H ! /UD5T PLACE TWO VIOLET RECTANGLES & "O SMALL R$ SQU>E TO ! LEFT ( ! B>1 & FIVE SMALL LIME SQU>ES TO ! "R ( ! B> "<,PICTURE #D">4 ,ADD "O SMALL LIME SQU>E TO EA* SIDE ( ! B>4 ,2C ! NEWLY ADD$ SMALL LIME SQU>E COMB9$ ) ! SMALL R$ SQU>E =MS A Z]O PAIR1 BO? TILES C 2 REMOV$4 ,?US1 ! EQUA;N 2COMES #BX "7 #F4 ,TWO VIOLET RECTANGLES %D N[ 2 ON ! LEFT SIDE ( ! B> & TWO GR\PS ( ?REE SMALL LIME SQU>ES %D 2 ON ! "R SIDE ( ! B>4 ,DIVIDE EA* SIDE ( ! EQUA;N BY #B "1 & REMOVE "O GR\P ( TILES F EA* SIDE4 ,! ANSW] TO ? EQUA;N IS A#I ;X "7 #C4 ,PICTURE #D3 ,MODEL+ EQUA;N #BX"-#A "7 #E US+ ,TACTILE ,ALGEBRA ,TILES ---------------------------#AJ #E4 ,ADD+ & SUBTRACT+ POLYNOMIALS ,TO ADD TWO POLYNOMIALS1 S* Z X;9#B"-#BX"6#C & #BX9#B"6X"-#E1 F/ MODEL ! POLYNOMIALS US+ TILES4 ,H ! /UD5T MODEL "O L>GE BLUE SQU>E1 TWO R$ RECTANGLES1 & ?REE SMALL LIME SQU>ES TO REPRES5T ! F/ POLYNOMIAL1 & TWO L>GE BLUE SQU>ES1 "O VIOLET RECTANGLE1 & FIVE SMALL R$ SQU>ES TO REPRES5T ! SECOND POLYNOMIAL "<,PICTURE #E">4 ,TO ADD ^! TWO POLYNOMIALS1 JO9 ! TWO GR\PS ( TILES & REMOVE ANY Z]O PAIRS4 ,SPECIFICALLY1 ! /UD5T C #AC REMOVE "O PAIR ( R$ & A#AJ VIOLET RECTANGLES & ?REE PAIRS ( SMALL R$ & LIME SQU>ES4 ,C.T ! TILES REMA9+ & GET ! SUM ( ! TWO POLYNOMIALS : IS #CX9#B"-X"-#B4 ,PICTURE #E3 ,MODEL+ POLYNOMIALS X;9#B"-#BX"6#C & #BX9#B"6X"-#E US+ ,TACTILE ,ALGEBRA ,TILES ---------------------------#AA ,TAKE ! FOLL[+ Z AN EXAMPLE ( SUBTRAC;N ( POLYNOMIALS3 X;9#B"-#BX"-#C" "-"<#BX9#B"6#CX"-#D">4 ,2C SUBTRACT+ A POLYNOMIAL C 2 D"O BY ADD+ XS 9V]SE1 ! PROBLEM 2COMES X;9#B"-#BX"-#C" "6"<"-#BX9#B"-#CX"6#D">4 ,!N FOLL[ ! /EPS M5;N$ ABV TO MODEL ! POLYNOMIALS1 JO9 !M1 & REMOVE ANY Z]O PAIRS4 ,! ANSW] TO ? PROBLEM IS "-X;9#B"-#EX"6#A4 #F4 ,MULTIPLY+ A#AA POLYNOMIALS ,TAKE ""<#BX"-#C"> Z AN EXAMPLE4 ,Z ) ! MULTIPLICA;N ( TWO 9TEG]S PROBLEM1 :5 MULTIPLY+ TWO POLYNOMIALS1 2G9 BY SETT+ UP A RECTANGLE4 ,! F/ FACTOR1 X"6#B1 IS ! L5G? ( ! V]TICAL SIDE ( ! RECTANGLE1 & ! SECOND FACTOR1 #BX"-#C1 IS ! L5G? ( ! HORIZONTAL SIDE4 ,!N1 USE TILES TO FILL 9 ! RECTANGLE4 ,REM9D ! /UD5T T ! PRODUCT ( TWO VALUES ) ! SAME SIGN IS POSITIVE2 IF ! SIGNS >E OPPOSITE1 ! PRODUCT IS NEGATIVE4 ,= 9/.E1 X"8X "7 X;9#B "GE BLUE SQU>E">1 & X"8"<"-#A"> "7 "-X "4 ,SPECIFICALLY1 F LEFT TO "R & F TOP TO BOTTOM "1 ! TILES FILL+ UP ! RECTANGLE >E ! #AE FOLL[+3 B#AA _4 TOP L9E3 L>GE BLUE SQU>E1 L>GE BLUE SQU>E1 R$ RECTANGLE1 R$ RECTANGLE1 & R$ RECTANGLE "E PLAC$ V]TICALLY">2 _4 SECOND L9E3 VIOLET RECTANGLE1 VIOLET RECTANGLE1 ---------------------------#AB SMALL R$ SQU>E1 SMALL R$ SQU>E1 & SMALL R$ SQU>E "E PLAC$ HORIZONTALLY">2 _4 BOTTOM L9E3 VIOLET RECTANGLE1 VIOLET RECTANGLE1 SMALL R$ SQU>E1 SMALL R$ SQU>E1 & SMALL R$ SQU>E "E PLAC$ HORIZONTALLY">4 ,TO F9D ! ANSW] TO ? MULTIPLICA;N PROBLEM1 ! /UD5T NE$S TO REMOVE ANY Z]O PAIRS 9 ! RECTANGLE & C.T ALL ! TILES REMA9+4 ,?REE PAIRS ( A#AB VIOLET & R$ RECTANGLES %D 2 REMOV$4 ,! ANSW] IS #BX9#B"6X"-#F4 ,PICTURE #F3 ,ILLU/RA;N ( POLYNOMIAL MULTIPLICA;N ""<#BX"-#C"> US+ ,TACTILE ,ALGEBRA ,TILES #G4 ,FACTOR+ POLYNOMIALS ,FACTOR+ A POLYNOMIAL ) ALGEBRA TILES 9VOLVES SETT+ UP A RECTANGLE PROP]LY & R1D+ XS DIM5.NS4 ,TAKE ! POLYNOMIAL X;9#B"6#EX"6#F Z AN EXAMPLE4 ,F/1 MODEL ! POLYNOMIAL US+ ---------------------------#AC "O L>GE BLUE SQU>E1 FIVE VIOLET RECTANGLES1 & SIX SMALL LIME SQU>ES4 ,NEXT1 >RANGE ^! TILES 9TO A RECTANGLE4 ,ALW 2G9 ) ! F/ & LA/ T]MS ( A POLYNOMIAL4 ,PLACE ! L>GE BLUE TILE AT ! UPP] LEFT CORN] ( ! RECTANGLE4 ,>RANGE ! SIX #AG SMALL LIME SQU>ES 9TO A A#AC TWO-BY-?REE >RAY & PLACE !M AT ! L[] "R CORN]4 ,!N1 FILL 9 ! RE/ ( ! RECTANGLE US+ ! VIOLET RECTANGLES "<,PICTURE #G">4 ,:5 COMPLET$1 >RANGE;T ( ! TILES 9 ! RECTANGLE IS ! FOLL[+3 _4 TOP L9E3 L>GE BLUE SQU>E1 VIOLET RECTANGLE1 VIOLET RECTANGLE1 & VIOLET RECTANGLE "E PLAC$ V]TICALLY">2 _4 SECOND L9E3 VIOLET RECTANGLE1 SMALL LIME SQU>E1 SMALL LIME SQU>E1 & SMALL LIME SQU>E "2 _4 BOTTOM L9E3 VIOLET RECTANGLE1 SMALL LIME SQU>E1 SMALL LIME SQU>E1 & SMALL LIME SQU>E "4 ,PICTURE #G3 B#AC ,FACTOR+ POLYNOMIAL X;9#B"6#EX"6#F US+ ,TACTILE ,ALGEBRA ,TILES ---------------------------#AD ,Y MAY AL WANT ! /UD5T TO TRY >RANG+ ! SIX SMALL LIME SQU>ES 9TO A SIX-BY-"O >RAY4 ,! /UD5T W F9D T ) ? >RANGE;T X IS IMPOSSIBLE TO USE FIVE VIOLET RECTANGLES TO COMPLETE ! RECTANGLE4 ,LA/1 R1D ! DIM5.NS ( ! RECTANGLE CR1T$ & F9D T XS V]TICAL SIDE IS X"6#B & XS HORIZONTAL SIDE IS X"6#C4 ,"!=E1 ! ANSW] TO ? FACTOR+ PROBLEM IS " "4 ,"S"TS X IS NEC TO ADD Z]O PAIRS 9 ORD] TO >RANGE GIV5 TILES 9TO A RECTANGLE1 Z 9 FACTOR+ X;9#B"6#EX"-#F4 ,F/1 MODEL ! POLYNOMIAL US+ "O L>GE BLUE SQU>E1 FIVE VIOLET #AI RECTANGLES1 & SIX SMALL A#AD R$ SQU>ES4 ,NEXT1 PLACE ! L>GE BLUE SQU>E AT ! UPP] LEFT CORN] ( ! RECTANGLE & ! SIX SMALL R$ SQU>ES AT ! L[] "R CORN] ( ! RECTANGLE 9 A "O-BY-SIX >RAY4 ,!N1 FILL 9 US+ ! RECTANGLES4 ,2C ! FIVE VIOLET RECTANGLES >E N 5\<1 A Z]O PAIR ( VIOLET & R$ RECTANGLES M/ 2 ADD$ "<,PICTURE #H">4 ,:5 COMPLET$1 ! >RANGE;T ( ! TILES 9 ! RECTANGLE IS ! FOLL[+3 _4 TOP L9E3 "O L>GE BLUE SQU>E & SIX VIOLET RECTANGLES "E PLAC$ V]TICALLY">2 _4 BOTTOM L9E3 "O R$ RECTANGLE & SIX SMALL R$ SQU>ES "4 ,R1D ! DIM5.NS ( ! #AE RECTANGLE CR1T$ & F9D \ T XS V]TICAL SIDE IS X"-#A & XS HORIZONTAL SIDE IS X"6#F4 ,"!=E1 ! ANSW] TO ? FACTOR+ PROBLEM IS "" "4 ,PICTURE #H3 ,FACTOR+ POLYNOMIAL X;9#B"6#EX"-#F US+ ,TACTILE ,ALGEBRA ,TILES ,EXT5.N ,TACTILE ,ALGEBRA ,TILES C 2 US$ 9 COMB9A;N ) O!R TOOLS AVAILABLE F ,,APH TO ADD M FLEXIBIL;Y TO ! T1*+ & LE>N+ PROCESS4 ,^! TOOLS 9CLUDE B >E N LIMIT$ TO ,EXP&$ ,2G9N]'S ,ABACUS "<,CATALOG _?3 #A-#JCAHA-#JJ">1 ,,EZ,'EE,,C.T ,ABACUS "<,CATALOG _?3 #A-#JCAHE-#JJ">1 ,ORION ,,TI-#CJ,,XS ,MULTI,VIEW ,TALK+ ,SCI5TIFIC #BA ,CALCULATOR "<,CATALOG A#AE _?3 #A-#JGCCJ-#JJ">1 & ,,DRAFTSMAN ,TACTILE ,DRAW+ ,BO>D "<,CATALOG _?3 #A-#JHHEG-#JJ">4 ,T1*]S & P>5TS C *OOSE A COMB9A;N ( TOOLS BAS$ ON CURRICULA & ! /UD5TS' NE$S4 ---------------------------#AF ,PRODUCT ,DEVELOP]S3 ,LI ,ZH\1 ,CORE-,CURRICULUM ,PROJECT ,L1D] ,KA!R9E ,CORCORAN1 ,MODEL_/" ,PATT]N ,MAK] ,ROD ,DIXON1 ,MANUFACTUR+ ,SPECIALI/ ,RA*EL ,:ITE1 ,RESE>* ,ASSI/ANT ,MAT!W ,POPPE1 ,>T ,DESIGN] ,FRANK ,HAYD51 ,DIRECTOR ( ,TE*NICAL @& ,MANUFACTUR+ ,RESE>* ,&REW ,M\LTON1 ,MANAG] ( ,TE*NICAL @& ,MANUFACTUR+ ,RESE>* A#AF ,AN?ONY ,J"OS1 ,>T ,DESIGN] ,,APH ?ANKS ! FOLL[+ T1*]S1 & _! /UD5TS1 = _! ?"\;L REVIEWS & SU7ES;NS = REVI.N ( ? 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