,,,TACTILE ALGEBRA TILES,' "<,NEME? ,V].N"> ,,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 T]M9ATOR .=.7 ,ITALICS PASSAGE 9DICATOR .=.' ,ITALICS T]M9ATOR .=_4 ,PRIM>Y BULLET .=_% ,OP5+ ,NEME? 9DICATOR .=_: ,CLOS+ ,NEME? 9DICATOR ,,TRANSCRIB]',S ,,NOTES ,? VOLUME IS TRANSCRIB$ AC TO .7,! ,RULES ( ,UNIFI$ ,5GLI% ,BRL1.' ,SECOND ,$I;N #BJAC1 ) #BJAD ,UPDATES4 ,MA!MATICAL 3T5T IS TRANSCRIB$ AC TO .7,! ,NEME? ,BRL ,CODE = ,MA!MATICS & ,SCI;E ,NOTA;N1.' #AIGB ,REVI.N1 #BJJG-#BJAE ,UPDATES 9CLUD+ ! ,GUIDEL9ES = US+ ,NEME? )9 ,,UEB 3TEXT4 ,? 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+#1_40 _: _4 ,SMALL R$ SQU>E TILE "<#CB PIECES"> - SMALL SQU>E TILE ) A RAIS$ DOT AT ! C5T]4 ,^! REPRES5T ! NUM]ICAL VALUE _% 8-#1_40 _: #C _4 ,VIOLET RECTANGLE B#B TILE "<#AB PIECES"> - RECTANGLE TILE ) A TACTILE RECTANGUL> SYMBOL AT ! C5T]4 ----------------------------#C ,^! REPRES5T ! NUM]ICAL VALUE _% 8+X_40 _: _4 ,R$ RECTANGLE TILE "<#AB PIECES"> - RECTANGLE TILE ) A R[ ( RAIS$ DOTS AT ! C5T]4 ,^! REPRES5T ! NUM]ICAL VALUE _% 8-X_40 _: _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+X^2_40 _: _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^2_40 _: _4 ,BLACK SEP>A;N B> "<#B PIECES"> - TWO L;G1 A#C N>R[ PIECES T C 2 US$ TO SEP>ATE TWO SIDES ( AN EQUA;N4 ,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 #E PAIR1 :ILE ?REE VIOLET #D RECTANGLES & ?REE R$ RECTANGLES MAKE UP ?REE Z]O PAIRS4 ,:5 US+ 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 _% #3+6 _:1 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+ A#D ! TILES TGR4 ,H ! /UD5T C.T ! TILES TO GET ! ANSW]1 _% #3+6 .K #9 _:4 ----------------------------#E ,PICTURE #A3 ,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 _% #3+(-7) _: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 _% -#4 _:4 #G ,TO ILLU/RATE A#E SUBTRAC;N PROBLEMS S* Z _% -#5-(-2) _:1 F/ H ! /UD5T USE "O GR\P ( FIVE SMALL R$ SQU>ES TO REPRES5T _% -#5 _:4 ,TO SUBTRACT _% -#2 _:1 REMOVE TWO SMALL R$ SQU>ES4 ,?REE R$ SQU>ES >E LEFT1 S ! ANSW] IS _% -#3 _:4 ----------------------------#F ,TO ILLU/RATE SUBTRAC;N PROBLEMS S* Z _% -#2-5 _:1 REM9D ! /UD5T T TO SUBTRACT A NUMB]1 Y C ADD XS OPPOSITE4 ,"!=E1 ! PROBLEM 2COMES _% -#2+(-5) _:4 ,!N1 H ! /UD5T %[ TWO GR\PS ( SMALL R$ SQU>ES1 "O GR\P REPRES5T+ _% -#2 _: & ! O!R REPRES5T+ _% -#5 _:4 ,! /UD5T W MOVE ! TILES TGR & GET ! ANSW]1 _% -#7 _:4 ,ANO!R WAY TO ILLU/RATE ? PROBLEM IS TO USE Z]O PAIRS4 ,F/1 H ! /UD5T %[ TWO SMALL R$ SQU>ES : REPRES5T _% -#2 _:4 ,NEXT1 ADD A#F 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 / _% -#7 _:4 #B4 ,MULTIPLICA;N ( 9TEG]S ,TO ILLU/RATE ! MULTIPLICA;N PROBLEM _% #3@*4 _:1 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 ( _% #3@*4 _: US+ ,TACTILE ,ALGEBRA ,TILES ,TO ILLU/RATE ! #I MULTIPLICA;N PROBLEM A#G _% #3@*(-4) _: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 _% -#12 _:4 ,MULTIPLICA;N PROBLEMS S* Z _% (-3)@*4 _: C 2 TR1T$ Z _% #4@*(-3) _: BY REM9D+ ! /UD5T T ! ORD] 9 : WE MULTIPLY DOES N MATT]4 ,TO ILLU/RATE ! MULTIPLICA;N PROBLEM _% (-3)@*(-4) _:1 REM9D ! /UD5T T ! _% 8-_0 _: SIGN C M1N 8OPPOSITE (40 ,!N1 ! PROBLEM C 2 TR1T$ Z ! OPPOSITE ( _% #3@*(-4) _:4 ,?US1 ! ANSW] IS #AB4 #C4 ,SIMPLIFY+ ALGEBRAIC EXPRES.NS ,MODEL ! EXPRES.N _% -#3X-2-2X+4+X _: US+ ALGEBRA TILES4 ,2C SUBTRACT+ IS ! SAME Z ADD+ ! OPPOSITE1 ! #H EXPRES.N C 2 TR1T$ Z _% -#3X+(-2)+(-2X)+4+X _:4 ,?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 ! /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 _% -#4X+2 _:4 #AA ,PICTURE #C3 A#H ,MODEL+ EXPRES.N _% -#3X+(-2)+(-2X)+4+X _: US+ ,TACTILE ,ALGEBRA ,TILES ----------------------------#I #D4 ,SOLV+ L9E> EQUA;NS 9 "O V>IABLE ,TAKE ! EQUA;N _% #2X-1 .K #5 _: = 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$4 ,?US1 ! EQUA;N 2COMES _% #2X .K #6 _:4 ,TWO VIOLET RECTANGLES A#I %D N[ 2 ON ! LEFT SIDE ( ! B> & TWO GR\PS ( ?REE SMALL LIME SQU>ES %D 2 ON ! "R SIDE ( ! B>4 ,DIVIDE EA* SIDE ( ! EQUA;N BY #B "GE BLUE SQU>E1 TWO R$ RECTANGLES1 & ?REE SMALL LIME SQU>ES #AC TO REPRES5T ! F/ A#AJ 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 REMOVE "O PAIR ( R$ & VIOLET RECTANGLES & ?REE PAIRS ( SMALL R$ & LIME SQU>ES4 ,C.T ! TILES REMA9+ & GET ! SUM ( ! TWO POLYNOMIALS : IS _% #3X^2"-X-2 _:4 ,PICTURE #E3 ,MODEL+ POLYNOMIALS _% X^2"-2X+3 _: & _% #2X^2"+X-5 _: US+ ,TACTILE ,ALGEBRA ,TILES ---------------------------#AA ,TAKE ! FOLL[+ Z AN EXAMPLE ( SUBTRAC;N ( POLYNOMIALS3 _% X^2"-2X-3-(2X^2"+3X-4) _:4 ,2C SUBTRACT+ A POLYNOMIAL C 2 D"O BY ADD+ XS 9V]SE1 ! A#AA PROBLEM 2COMES _% X^2"-2X-3+(-2X^2"-3X+4) _:4 ,!N FOLL[ ! /EPS M5;N$ ABV TO MODEL ! POLYNOMIALS1 JO9 !M1 & REMOVE ANY Z]O PAIRS4 ,! ANSW] TO ? PROBLEM IS _% -X^2"-5X+1 _:4 #F4 ,MULTIPLY+ POLYNOMIALS ,TAKE _% (X+2)(2X-3) _: Z AN EXAMPLE4 ,Z ) ! MULTIPLICA;N ( TWO 9TEG]S PROBLEM1 :5 MULTIPLY+ TWO POLYNOMIALS1 2G9 BY SETT+ UP A RECTANGLE4 ,! F/ FACTOR1 _% X+2 _:1 IS ! L5G? ( ! V]TICAL SIDE ( ! RECTANGLE1 & ! SECOND FACTOR1 _% #2X-3 _:1 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 #AE NEGATIVE4 ,= 9/.E1 B#AA _% X@*X .K X^2 _: "GE BLUE SQU>E">1 & _% X@*(-1) .K -X _: "4 ,SPECIFICALLY1 F LEFT TO "R & F TOP TO BOTTOM "1 ! TILES FILL+ UP ! RECTANGLE >E ! FOLL[+3 _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$ A#AB SQU>E "E PLAC$ HORIZONTALLY">4 ,TO F9D ! ANSW] TO ? MULTIPLICA;N PROBLEM1 ! /UD5T NE$S TO REMOVE ANY Z]O PAIRS 9 ! RECTANGLE & C.T ALL ! TILES REMA9+4 ,?REE PAIRS ( VIOLET & R$ RECTANGLES %D 2 REMOV$4 ,! ANSW] IS _% #2X^2"+X-6 _:4 ,PICTURE #F3 ,ILLU/RA;N ( POLYNOMIAL MULTIPLICA;N _% (X+2)(2X-3) _: 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^2"+5X+6 _: Z AN EXAMPLE4 ,F/1 MODEL ! POLYNOMIAL US+ ---------------------------#AC "O L>GE BLUE SQU>E1 FIVE #AG VIOLET RECTANGLES1 & A#AC 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 SMALL LIME SQU>ES 9TO A 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 B#AC "2 _4 BOTTOM L9E3 VIOLET RECTANGLE1 SMALL LIME SQU>E1 SMALL LIME SQU>E1 & SMALL LIME SQU>E "4 ,PICTURE #G3 ,FACTOR+ POLYNOMIAL _% X^2"+5X+6 _: 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+2 _: & XS HORIZONTAL SIDE IS #AI _% X+3 _:4 ,"!=E1 ! A#AD ANSW] TO ? FACTOR+ PROBLEM IS _% (X+2)(X+3) _:4 ,"S"TS X IS NEC TO ADD Z]O PAIRS 9 ORD] TO >RANGE GIV5 TILES 9TO A RECTANGLE1 Z 9 FACTOR+ _% X^2"+5X-6 _:4 ,F/1 MODEL ! POLYNOMIAL US+ "O L>GE BLUE SQU>E1 FIVE VIOLET RECTANGLES1 & SIX SMALL 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$1 ! >RANGE;T ( ! TILES 9 ! RECTANGLE IS ! FOLL[+3 _4 TOP L9E3 "O L>GE BLUE SQU>E & SIX VIOLET B#AD RECTANGLES "E PLAC$ V]TICALLY">2 _4 BOTTOM L9E3 "O R$ RECTANGLE & SIX SMALL R$ SQU>ES "4 ---------------------------#AE ,R1D ! DIM5.NS ( ! RECTANGLE CR1T$ & F9D \ T XS V]TICAL SIDE IS _% X-1 _: & XS HORIZONTAL SIDE IS _% X+6 _:4 ,"!=E1 ! ANSW] TO ? FACTOR+ PROBLEM IS _% (X-1)(X+6) _:4 ,PICTURE #H3 ,FACTOR+ POLYNOMIAL _% X^2"+5X-6 _: 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*+ & #BA LE>N+ PROCESS4 ,^! A#AE 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 ,CALCULATOR "<,CATALOG _?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/ A#AF ,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>* ,AN?ONY ,J"OS1 ,>T ,DESIGN] ,,APH ?ANKS ! FOLL[+ T1*]S1 & _! /UD5TS1 = _! ?"\;L REVIEWS & SU7ES;NS = REVI.N ( ? PRODUCT3 ,%>ON ,CL>K1 ,MELISSA ,DONALDSON1 ,TIM ,FAHLB]G1 ,*RI/9A VON ,REYN1 ,AM&A ,RODDA-,TYL]1 ,LISA ,/>RFIELD1 ,GL5DA ,TORR;E1 & ,L\ISE ,:ITWOR?4 #BC ,,APH^R ,AM]ICAN ,PR9T+ ,H\SE = ! ,BL1 ,9C4 #AHCI ,FRANK=T ,AV5UE ,L\ISVILLE1 ,,KY #DJBJF-#JJHE ,PH"O3 #EJB-#HIE-#BDJE ,TOLL ,FREE3 #HJJ-#BBC-#AHCI ,FAX3 #EJB-#HII-#BBHD ;,E-MAIL3 9FO@AAPH4ORG ,WEB SITE3 WWW4APH4ORG ,CATALOG ,NO4 #A-#JHDAJ-#JJ ,,! ,,5D