,,,/UD5T "WBOOK = GEOMETRO,' ,,,NETS ( #C;D SOLIDS3 T1*]'S MANUAL,' ,PUBLI%$ BY ,AM]ICAN ,PR9T+ ,H\SE = ! ,BL1 ,9C4 ,L\ISVILLE1 ,,KY WWW4APH4ORG ,COPY"R ^C #BJAD1 ,AM]ICAN ,PR9T+ ,H\SE = ! ,BL4 ,FUR!R REPRODUC;N OR 4TRIBU;N 9 O!R ?AN A SPECIALIZ$ =MAT IS PROHIBIT$4 ,TRANSCRIP;N ( ,"P ,NO4 #FA-#DBA-#BCD ,US$ ) ,CATALOG ,NUMB] #A-#JCJBA-#JJ ,TRANSCRIB$ #BJBB 9TO ,UNIFI$ ,5GLI% ,BRL BY ,AM]ICAN ,PR9T+ ,H\SE = ! ,BL ,L\ISVILLE1 ,,KY ,9 #A ,VOLUME ,BRL PAGES T#A-T#B1 P#A-P#D1 & #A-#FE ,PR9T PAGES ,I-,,XI & #A-#EF T#A ,AN A3ESSIBLE HTML V].N ( ? GUIDEBOOK IS AVAILABLE AT WWW4APH4ORG_/MANUALS_/ ,,W>N+ ,,*OK+ ,,HAZ>D ,- ,? PRODUCT 3TA9S SMALL "PS4 ,N = USE ) *N "U ! AGE ( #C YE>S4 ,? 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TODDL] YE>S4 ,Z WE G "? \R "D1 WE PO9T \ PLAY ?+S & 5VIRON;TAL OBJECTS T H ! %APE ( A CIRCLE1 SQU>E1 OR RECTANGLE4 ,L;G 2F A * C PUT A LABEL ON A %APE1 HE C F9D OBJECTS & TOYS T >E CIRCLES1 OR PO9T TO ! PICTURE ( ! CIRCLE1 SQU>E1 & S ON1 9 A BOOK4 ,? TYPE ( LE>N+ IS NATURAL & FUN = A "Y * & 2G9S TO BUILD ! F.DA;N = MA!MATICAL SKILLS & GEOMETRY4 ------------------------------------,,VI ,9 .7,F.DA;NS ( ,$UCA;N1 ,SECOND ,$I;N1 ,VOLUME ,,II1 ,9/RUC;NAL ,/RATEGIES = ,T1*+ ,*N & ,Y\?S ) ,VISUAL ,IMPAIR;TS1.' ,KAPP]MAN1 ,HE9ZE1 & ,/ICK5 WRITE T 8AN "U/&+ ( MA!MATICS IS ESS5TIAL = FULL "PICIPA;N 9 #A SOCIETY0 "<#BJJJ1 ;P4 #CGJ">4 A,,VI ,! AU?ORS AL PO9T \ T 8/UD5TS ) VISUAL IMPAIR;TS C GA9 A COMPREH5.N ( BASIC MA!MATICAL 3CEPTS "? ! USE ( R1L OBJECTS & MANIPULATIVES10 & /RESS T 8T1*]S ( /UD5TS ) VISUAL IMPAIR;TS T1* ! UNIQUE SKILLS RELAT$ TO MA!MATICS1 S* Z1 US+ AN ABACUS & TACTILE GRAPHICS & COLLABORATE ) G5]AL $UCA;N T1*]S TO 5SURE T EA* /UD5T GA9S S.D MA!MATICAL SKILLS0 "<;P4 #CGJ">4 ,= ! "Y * ) A VISUAL IMPAIR;T1 A3ESS TO APPROPRIATE $UCA;NAL TOOLS & ! SKILLS NEC = US+ !M >E CRUCIAL 9 ! 9/RUC;N ( MA!MATICS4 ,2C MA!MATICS IS "O ( ! M AB/RACT & HI1S1 T1*]S ( ! VISUALLY IMPAIR$ M/ PROVIDE /UD5TS ) A V>IETY ( TANGIBLE OBJECTS & MANIPULATIVES T W MAKE 9/RUC;N M1N+;L4 ,= ! /UDY ( GEOMETRY1 ? 9VOLVES TACTILE IMAGES & #C-;,D MANIPULATIVES T REPRES5T ! BASIC 3CEPTS ( GEOMETRY4 -----------------------------------,,VII ,! ,/UD5T ,"WBOOK1 MANIPULATIVE MAT]IALS1 & ,T1*]'S ,MANUAL PROVIDE ! TOOLS NEC = M1N+;L 9/RUC;N ( A,,VII GEOMETRY = /UD5TS :O >E VISUALLY IMPAIR$4 ,! ,/UD5T ,"WBOOK REVOLVES >.D ?REE K9DS ( TASKS4 ,! F/ TASK IS TO FAMILI>IZE /UD5TS ) V>I\S #C-;,D SOLIDS & _! G5]AL PROP]TIES4 ,! SECOND TASK IS TO HELP /UD5TS GRASP ! DI6ICULT 3CEPTS ( H[ #C-;,D SOLIDS RELATE TO _! #B-;,D GRAPHIC REPRES5TA;NS4 ,LA/LY1 ! ?IRD TASK IS TO HELP /UD5TS "U/& H[ #C-;,D OBJECTS >E MADE ) #B-;,D OBJECTS4 ----------------------------------,,VIII ,/UD5TS W LE>N H[ TO3 _4 ,ID5TIFY & TACTUALLY ANALYZE #B-;,D %APES _4 ,ID5TIFY1 BUILD1 EXAM9E & TACTUALLY ANALYZE #C-;,D SOLIDS _4 ,ID5TIFY & COMP>E ! PROP]TIES ( #C-;,D SOLIDS1 S* Z ! NUMB] ( FACES1 $GES1 & V]TICES _4 ,3/RUCT NETS ( #C-;,D SOLIDS _4 ,DECOMPOSE #C-;,D SOLIDS 9TO NETS _4 ,ID5TIFY & 3/RUCT #B-;,D #C REPRES5TA;NS ( A SOLID F A,,VIII V>Y+ P]SPECTIVES _4 ,ID5TIFY #B-;,D FIGURES T >E NETS ( #C-;,D SOLIDS _4 ,MAT* TACTILE DRAW+S ( #C-;,D SOLIDS ) TACTILE GRAPHIC NETS ,ACTIVITIES 9 ! ,/UD5T ,"WBOOK = ,GEOMETRO ADDRESS ! ,PR9CIPLES & ,/&>DS = ,S*OOL ,MA!MATICS "<,NA;NAL ,C\NCIL ( ,T1*]S ( ,MA!MATICS"> & ! ,COMMON ,CORE ,/ATE ,/&>DS = ,MA!MATICS4 ,/UD5T ,"WBOOK = ,GEOMETRO ACTIVITIES ADDRESS ! FOLL[+ ,NA;NAL ,C\NCIL ( ,T1*]S ( ,MA!MATICS ,/&>DS = ,GEOMETRY4 ------------------------------------,,IX ,GRADES ,PRE-;,K TO ,GRADE #H3 _4 ,ANALYZE "*I/ICS ,GRADES ,PRE-;,K TO ,GRADE #E3 _4 ,VISUALIZA;N ,ACTIVITIES 9 ! ,/UD5T ,"WBOOK = ,GEOMETRO ADDRESS ! FOLL[+ ,COMMON ,CORE ,/ATE ,/&>DS = ,GEOMETRY4 A,,IX ,K9D]G>T53 _4 ,ID5TIFY & DESCRIBE %APES _4 ,ANALYZE1 COMP>E1 CR1TE1 & COMPOSE %APES ,GRADES #A-#C3 _4 ,R1SON ) %APES & _! ATTRIBUTES ,WE HOPE T Y F9D ^! MAT]IALS TO 2 HELP;L1 FUN1 & 9VIT+ = YR /UD5TS4 ,OP5 ! PAGES ( ! ,/UD5T ,"WBOOK & WAT* YR /UD5TS LE>N6 #E ,,OV]VIEW ,X-,,XI ,! ,/UD5T ,"WBOOK = ,GEOMETRO IS AN EXT5.N & ADAPTA;N ( ! ,NET'S ( #C-;,D ,SOLIDS "WBOOK COMM]CIALLY AVAILABLE F ,GEOMETRO4 ,) ! ADDI;N ( SPECIAL EX]CISES & TACTILE GRAPHICS1 ! ,/UD5T ,"WBOOK = ,GEOMETRO IS DESIGN$ = /UD5TS :O >E BL & VISUALLY IMPAIR$4 ,A SET ( MANIPULATIVE MAT]IALS 3TA9+ ROD MODELS1 MAGNETIC TILES1 A MAGNETIC BO>D1 A ,BRL ,POCKET ,FOLD]1 & A ROLL ( ,GRAPHIC ,>T ,TAPE F ,,APH1 AL;G ) ! ,T1*]'S ,MANUAL & ! ,/UD5T ,"WBOOK1 MAKE UP ,/UD5T ,"WBOOK = ,GEOMETRO4 ,ADDI;NALLY1 ! ACTIVITIES 9 ! "WBOOK >E TO 2 US$ ) ,GEOMETRO TILES1 : M/ 2 PUR*AS$ SEP>ATELY F ,,APH4 ,,,DESCRIP;N ( #A MANIPULATIVE MAT]IALS,' ,,ROD ,,MODELS ,! ROD MODELS 7 DEVELOP$ BY ,GEOMETRO 9 ORD] TO BETT] ILLU/RATE RELA;N%IPS 2T TWO- & ?REE-DIM5.NAL ASPECTS ( SOLIDS2 3CEPTS T >E NOTORI\SLY DI6ICULT = /UD5TS :O >E VISUALLY IMPAIR$4 ,! ROD MODELS >E MADE ( PLA/IC RODS & >E 8SKELETONS0 ( #C-;,D SOLIDS ) AN $GE L5G? ( AB #D4E IN41 MAT*+ ! SIZE ( ,GEOMETRO TILES4 ,! MODELS H TWO IMPORTANT PROP]TIES4 ,F/1 ALL JO9TS >E FLEXIBLE1 : MAKES ! SOLIDS APPE> FLOPPY & UNABLE TO /& ON _! [N4 ,SECOND1 N ALL $GES H A FIX$ L5G? --------------------------------------#B "4 ,F\R ROD MODELS >E 9CLUD$3 TWO PRISMS & TWO PYRAMIDS4 ,9 ! MODELS1 V]TICES >E FLEXIBLE & "! >E ?REE K9DS ( $GES3 FIX$ L5G?1 $GES T %ORT51 & $GES T L5G!N4 ,9 ORD] TO IMPROVE COMMUNICA;N ( #G ID1S1 "S $GES ( ! ROD MODELS >E A#B LABEL$ 9 ! PHOTOS 9 ? MANUAL4 ,! $GES T C 2 %ORT5$ F #D4E IN4 >E LABEL$ ) "O /RIP ( #A/D IN4 ,GRAPHIC ,>T ,TAPE4 ,^? T C 2 L5G!N$ F #D4E IN4 >E LABEL$ ) TWO /RIPS ( #A/D IN4 ,GRAPHIC ,>T ,TAPE4 ,$GES T >E FIX$ L5G? >E N LABEL$4 ,SEE ^1,FIGURE ^1#A4 ,FIGURE #A4 ,F\R ,ROD ,MODELS @.<,PHOTO %[S A 4PLAY ( ! F\R ROD MODELS 9 #B-;,D1 M>K$ ) ,GRAPHIC ,>T ,TAPE Z DESCRIB$4@.> --------------------------------------#C ,! $GES ( ! ROD MODELS 9CLUD$ ) ! ,/UD5T ,"WBOOK >E N LABEL$4 ,WE TRU/ T T1*]S W QKLY LE>N TO RECOGNIZE ! ?REE TYPES ( $GES BY _! APPE>.E & W 2 ABLE TO USE ! MODELS )\T ! LABELS4 ,IF DESIR$1 "O C LABEL ! $GES FOLL[+ ! EXAMPLE & 9/RUC;N 3TA9$ "H4 ,A ROLL ( ,GRAPHIC ,>T ,TAPE IS 9CLUD$ & C 2 US$ TO LABEL ROD MODEL $GES IF DET]M9$ TO 2 NE$$ BY A /UD5T4 ,A SU7E/$ ME?OD = LABEL+1 Z SE5 9 PHOTOS 9 ? ,T1*]'S ,MANUAL3 WRAP TWO PIECES ( #A/D IN4 ,GRAPHIC ,>T ,TAPE >.D $GES ( ROD MODEL T C 2 L5G!N$ F A#C #D4E IN41 PLAC$ APPROXIMATELY #A/D IN4 A"P2 WRAP #A PIECE ( #A/D IN4 ,GRAPHIC ,>T ,TAPE >.D $GES ( ROD MODEL T C 2 %ORT5$ F #D4E IN4 ,COLOR$ ELECTRICAL TAPE1 AVAILABLE 9 H>DW>E /ORES1 IS AL E6ICI5T = ? PURPOSE4 ,ROD MODELS H FLEXIBLE JO9TS4 ,! L5G!N+ & %ORT5+ ( $GES IS A*IEV$ BY US+ TELESCOPIC JUNC;NS4 ,9 ^1,FIGURE ^2#B ! $GE M>K$ ) "O /RIP ( TAPE IS MADE ( TWO %ORT RODS2 ! $GE M>K$ ) TWO /RIPS ( TAPE IS MADE ( TWO L;G] RODS4 ,FIGURE #B4 ,ROD ,MODEL @.<,PHOTO %[S A CLOSE-UP ( "O ROD MODEL 9 #B-;,D1 M>K$ ) ,GRAPHIC ,>T ,TAPE Z DESCRIB$4@.> --------------------------------------#D ,! $GES C AL 2 EASILY ID5TIFI$ BY ! L5G? ( ! ?ICK] RODS 9/1D ( _! TAPE LABELS2 %ORT = ! COLLAPSIBLE $GE ) "O /RIP ( TAPE1 & L;G = ! EXT5DABLE $GE ) TWO /RIPS ( TAPE4 ,! MODEL'S FLEXIBLE JO9TS & ! V>IABLE $GE L5G?S S]VE "O PURPOSE3 !Y MAKE #I ! MODELS FUNC;NAL 9 BO? TWO & ?REE A#D DIM5.NS4 ,9 O!R ^WS1 ! MODELS C 2 POSI;N$ UP"R Z #C-;,D SOLIDS & !N PLAC$ FLAT ON A PAGE TO REPRES5T #B-;,D DRAW+S ( ! SAME SOLID4 ,! PURPOSE IS TO ILLU/RATE1 TACTILELY & K9ES!TICALLY1 ! RELA;N%IP 2T A #C-;,D SOLID & XS TACTILE DRAW+4 ,ROD MODELS ( ! F\R SIMPLE/ #C-;,D SOLIDS >E 9CLUD$3 TWO PRISMS & TWO PYRAMIDS4 ,2C ! DETAILS ( $GE %ORT5+ & L5G!N+ T A3OMPANY ! *ANGE F #C-;,D TO #B-;,D >E SLIE DESCRIB$ 9DIVIDUALLY 9 ! NEXT SEC;N4 --------------------------------------#E ,,,MAGNETIC TILES & BO>D,' ,MAGNETIC TILES COME Z A SET ( SIX TYPES ( POLYGONS,-! SAME T >E US$ "?\T ! ,/UD5T ,"WBOOK,-SQU>E1 RECTANGLE1 P5TAGON1 HEXAGON & TWO TYPES ( TRIANGLES1 EQUILAT]AL & ISOSCELES4 ,! MAGNETIC TILES >E DESIGN$ ) SLID ON : ! TILES C 2 ASSEMBL$1 MANIPULAT$1 & /OR$ = A P]IOD ( "T4 ,AN ,,APH ,BRL ,POCKET ,FOLD] IS 9CLUD$ & C 2 US$ = /OR+ ! MAGNETIC BO>D ) ! /UD5T'S "W4 ,! PURPOSE ( 9CLUD+ ! MAGNETIC TILES & BO>D IS TO HELP /UD5TS DOCU;T _! "W1 ) ! FUNC;NAL;Y SIMIL> TO T ( A DRAW+ BO>D1 SKET*PAD1 OR A NOTEBOOK4 ,FIGURE #D4 ,MAGNETIC ,TILE #AA ,POLYGONS A#F ^1,FIGURE ^2#D 4PLAYS MAGNETIC TILE POLYGONS >RANG$ ON A MAGNETIC BO>D = DOCU;TA;N & /ORAGE ( /UD5T'S "W4 --------------------------------------#G ,,GEOMETRO ,,TILES ,GEOMETRO TILES >E TRANSP>5T PLA/IC POLYGONS ) HOOK-&-LOOP MAT]IAL ATTA*$ TO _! $GES4 ,! POLYGONS JO9 "O TO ANO!R BY T\*+ ! HOOK TO LOOP & LOOP TO HOOK1 RESPECTIVELY4 ,! #C-;,D SOLIDS C 2 EASILY BUILT US+ COMB9A;NS ( ,GEOMETRO TILES4 --------------------------------------#H ,FIGURE #E4 ,GEOMETRO ,TILES ^1,FIGURE ^2#E %[S A L>GE SET ( ,GEOMETRO TILES1 AVAILABLE F ,,APH4 ,= ! LESSONS DESCRIB$ 9 ? MANUAL1 ! FOLL[+ POLYGONS W 2 US$3 REGUL> " TRIANGLES1 SQU>ES1 P5TAGONS1 HEXAGONS1 ISOSCELES TRIANGLES1 & RECTANGLES4 ,REGUL> OCTAGONS & DECAGONS >E AL AVAILABLE & C 2 US$ = EXT5.NS ( ! PROPOS$ LESSONS2 B S* A#H EX]CISES >E N 9CLUD$ 9 ? ,/UD5T ,"WBOOK4 --------------------------------------#I ,,,GUIDE TO "W+ ) ROD MODELS,' ,,,:Y >E ROD MODELS NEC8,' ,AL? ALL ! $GES H EQUAL L5G? 9 ! PYRAMID %[N 9 ^1,FIGURE ^1#F1 !Y D N APPE> EQUAL 9 ! PHOTOGRAPH4 ,X IS EASY TO OBS]VE T ! PYRAMID 3SI/S ( F\R EQUILAT]AL TRIANGLES1 & ?US1 ALL ! $GES H ! SAME L5G?4 ,B LOOK AG6 ,Y MAY EV5 USE A RUL]4 ,! $GES 9 ! PICTURE >E N EQUAL 9 L5G? AT ALL4 ,"O MI ,TRIANGUL> ,PYRAMID @.<,PHOTO %[S A REGUL> TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES4@.> ,Z ^1,FIGURE ^2#F ILLU/RATES1 WE #AC >E US$ TO SEE+ OBJECTS 9 A#AJ P]SPECTIVE2 ! FACT T "S $GES 2COME %ORT] &_/OR ANGLES *ANGE DOES N 4TURB U4 ,\R BRA9S H LE>N$ TO 9T]PRET ) EASE ! %ORT5+ & L5G!N+ ( $GES & *ANGES 9 ANGLES Z A NATURAL EXP]I;E ( VIEW+ A #C-;,D OBJECT4 ,WE LE>N$ TO RECOGNIZE :AT ! OBJECT IS BY ! F1TURES X HAS1 AL? X MIE VISUALLY IMPAIR$4 -------------------------------------#AA ,TO HELP SOLVE ? PROBLEM1 WE DECID$ TO APPROA* X F A LIT]AL /&PO9T4 8,VIEW+0 R1LLY M1NS TO MAKE A PROJEC;N ( #C-;,D OBJECTS ONTO \R RET9A4 ,DRAW+ A #C-;,D OBJECT IS V M* ! SAME2 WE MAKE A #B-;,D REPRES5TA;N ON A PIECE ( PAP]4 ,! 9T]PRETA;N ( ^! PROJEC;NS Z A#AA #C-;,D OBJECTS HAPP5S 9 ! BRA94 ,S1 TO MIMIC ? PROCESS1 ? SET ( MAT]IALS 9CLUDES #C-;,D MODELS T >E FLEXIBLE 5 TO 2 PLAC$ FLAT ON A SURFACE4 ,:5 PLAC$ FLAT1 Z ) ! PROCESS ( DRAW+1 "S $GES GET %ORT]1 "S L;G]1 & ANGLES W *ANGE4 ,? IS :AT HAPP5S ) ! ROD MODELS4 ,MO/ ( ! $GES >E 3/RUCT$ ) TELESCOPIC JUNC;NS1 S !Y C EI %ORT5 OR EXT5D4 ,! V]TICES >E FULLY FLEXIBLE TO ALL[ *ANGES 9 ANGLES4 ,? SKELETON C 2 POSI;N$ UP"R Z A #C-;,D OBJECT OR PLAC$ FLAT ON A SURFACE TO LOOK L A #B-;,D DRAW+4 ,! PHYSICAL PROCESS ( FLATT5+ ! #C-;,D OBJECT GIVES K9ES!TIC "U/&+ ( :Y "S $GES H TO *ANGE L5G?S & :Y ANGLES H TO *ANGE,-2C WE WANT TO MAKE X FLAT4 ,? IS H[ A /UD5T ) A VISUAL IMPAIR;T COMPREH5DS ! IDEA ( A DRAW+2 F ! K9ES!TIC & TACTILE 9PUT ( REP1T$LY FLATT5+ & PROPP+ UP ! MODEL4 ,T IS1 SWIT*+ 2T #B-;,D & #C-;,D4 -------------------------------------#AB ,( C\RSE1 WE C VIEW #C-;,D #AE OBJECTS F DI6]5T P]SPECTIVES1 & A#AB ACLY1 WE DRAW !M DI6]5TLY4 ,! SAME APPLIES TO FOLD+2 ! ROD MODELS C 2 FOLD$ 9 DI6]5T WAYS TO REFLECT DI6]5T P]SPECTIVES4 ,H["E1 X IS "O ?+ TO LOOK AT AN OBJECT & =M A M5TAL IMAGE1 & A TOTALLY DI6]5T MATT] TO PHYSICALLY FOLD AN OBJECT4 ,! FACT T WE >E D1L+ ) PHYSICAL OBJECTS " ADDS "S 3/RA9TS4 ,EV5 M S IF WE >E TRY+ TO HELP "Y *N :O >E LE>N+ TO 8SEE40 ,WE H TO KEEP X SIMPLE4 ,? IS :Y ! MODELS >E N /RICTLY DO+ ALL T A TRUE P]SPECTIVE DEM&S4 ,AL1 !Y D N FOLD 9 E POSSIBLE WAY "4 ,EV5 ) ^! LIMITA;NS1 "! >E _M WAYS TO FOLD & UNFOLD ! MODELS2 & ! PROCESS W / HELP /UD5TS DEVELOP ID1S & 3CEPTS4 ,,,H[ TO USE ! ROD MODELS,' ,? SEC;N FOCUSES 9 DETAIL ON H[ TO USE EA* MODEL S ! T1*] HAS ! BROADE/ POSSIBLE "U/&+ ( A MODEL'S USE & LIMITA;NS4 ,?US1 ! T1*] C M B#AB EASILY TAILOR ! LESSON TO ! SPECIFIC NE$S ( A /UD5T4 -------------------------------------#AC ,"! IS A SIMIL> SEQU;E ( /EPS T WE SU7E/ = ALL MODELS4 ^7,ASK /UD5T TO3^' #A4 ,BUILD A GIV5 #C-;,D SOLID US+ ,GEOMETRO TILES4 #B4 ,FIT ! ROD MODELS OV] ! ,GEOMETRO TILES 9 A SPECIFIC WAY SU7E/$ = EA* SOLID & %[N 9 ! PHOTOGRAPHS4 #C4 ,REMOVE ! ROD MODEL F ! ,GEOMETRO TILE MODEL & POSI;N X 2S ! TILE MODEL2 SUPPORT = "S ( ! ROD MODELS W 2 NEC TO KEEP !M 9 A #C-;,D 3FIGURA;N4 #D4 ,PLACE ! ROD MODEL 9TO A #B-;,D 3FIGURA;N ) SIMULTANE\S EXT5D+ OR %ORT5+ ( APPROPRIATE $GES :5 NEC4 #E4 ,EXAM9E ! #B-;,D 3FIGURA;N & COMP>E X TO TACTILE GRAPHICS = A GIV5 SOLID4 #F4 ,EXPLORE & 4CUSS ! ORIG9 ( SOLID & DOTT$ L9ES 9 ! TACTILE DRAW+S4 ,! MODELS >E DESCRIB$ "H 9 A #AG SEQU;E4 ,= EA* MODEL1 "! >E TWO A#AC OR ?REE POSSIBLE WAYS TO =M A #B-;,D 3FIGURA;N2 !Y RELATE TO ! WAYS A SOLID C 2 VIEW$ F DI6]5T DIREC;NS4 -------------------------------------#AD ,! EXAMPLES ( DI6]5T POSI;NS = A GIV5 SOLID >E %[N 9 A S]IES ( PHOTOGRAPHS & DESCRIP;NS4 ,! EXAMPLES REFLECT T "! >E _M POSSIBLE #B-;,D REPRES5TA;NS ( A GIV5 #C-;,D %APE4 ,SIE US$ TO RECOGNIZ+ ! #C-;,D %APES F _! V>I\S #B-;,D REPRES5TA;NS4 ,S9CE ? EXP]I;E IS N TYPICALLY A3ESSIBLE TO /UD5TS :O >E VISUALLY IMPAIR$1 ! TACTILE EX]CISE ( MANIPULAT+ A #B-;,D ROD MODEL MIIABIL;Y ( S* #B-;,D REPRES5TA;NS4 ,! DETAILS ( EA* 3FIGURA;N >E N Z IMPORTANT Z ! G5]AL IDEA T F/1 "! >E _M POSSIBILITIES & !Y CORRESPOND TO _M POSSIBLE VIEWS ( ! SAME OBJECT2 & SECOND1 ! #B-;,D 3FIGURA;NS 8LOOK_/FEEL0 Q DI6]5T ?AN ! R1L #C-;,D OBJECTS4 ,? IS 2C ! $GE L5G?S & ANGLES >E *ANG$4 ,( C\RSE1 :ILE A#AD RECOGNI;N ( "S REPRES5TA;NS MIE4 ,9/1D1 WE LOOK SLIE "!1 & WE REFLECT ? FACT BY DRAW+ ! 89VISIBLE0 $GES ) BROK5 OR DOTT$ L9ES4 ,^! ID1S & OBS]VA;NS >E 9A3ESSIBLE TO A VISUALLY IMPAIR$ P]SON1 Z T1*]S "K V WELL4 ,H["E1 X IS NEC TO HELP A * COMPREH5D & 9T]PRET A TACTILE DRAW+ T 3TA9S S* BROK5 L9ES4 ,! IDEA T "S $GES >E 8OBSCUR$0 BY ! FRONT FACES MIE PRISM " ,! ROD MODEL = ! CUBE IS ! SIMPLE/ ( ! MODELS4 ,ALL #AB $GES H ! SAME FIX$ L5G?1 & ALL JO9TS >E FLEXIBLE4 ,! CUBE MODEL IS AL 9T]E/+ 9 T X %[S1 3TR>Y TO \R 9TUITIVE EXPECTA;N1 ! CUBE IS N /IFF IF ! JO9TS >E FLEXIBLE4 ,? ORIG9ATES F ! FACT T A SQU>E IS N /IFF IF X HAS FLEXIBLE JO9TS4 ,9 FACT1 A TRIANGLE IS ! ONLY POLYGON T IS A#AF /IFF ) FLEXIBLE JO9TS4 ,? OBS]VA;N IS "S"TS MISS$ S9CE WE T5D TO MAKE CUBES US+ MAT]IALS T /I65 JO9TS S* Z M>%MALL[S1 GUMMIES1 OR P1S4 ,S1 ! CU/OM>Y /IFF JO9TS HIDE ! FACT T A CUBE IS 9TR9SICALLY FLEXIBLE6 ,9 A TRUE P]SPECTIVE VIEW ( A CUBE1 "S ( XS $GES R1LLY D *ANGE L5G?4 ,Y C SEE ? DEPICT$ 9 ANY PHOTOGRAPH ( A CUBE2 = EXAMPLE1 9 />T+ POSI;N1 ^1,FIGURE ^1#H4 ,! ROD MODEL IS SIMPLIFI$ & DOES N 9CLUDE ? SUBTLE *ANGE4 ,=TUNATELY1 ! #B-;,D REPRES5TA;N ( ! CUBE C 2 ILLU/RAT$ EV5 ) ? SIMPLIFICA;N4 -------------------------------------#AG ,SQU>E ,PRISM "<,CUBE">1 ,/>T+ ,POSI;N ,/>T+ POSI;N = ! CUBE IS ) A SQU>E FACE 9 FRONT & P>ALLEL TO ! VIEW]4 ,H["E1 ? POSI;N IS V SYMMETRICAL & A LOT ( ! CUBE IS N SE5 CLE>LY4 ,SEE ^1,FIGURE ^1#G4 ,X IS BETT] TO ROTATE ! CUBE A #BA LL BIT AWAY F ? SYMMETRICAL A#AG POSI;N & USE ! ROTAT$ POSI;N Z A />T+ VIEW1 Z ILLU/RAT$ 9 .-,FIGURE .-#H = ! VIEW F ! LEFT & ^1,FIGURE ^2#I = ! VIEW F ! "R4 ,FIGURE #G4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 /RAI ,FIGURE #H4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 VIEW$ F ! LEFT4@.> ,FIGURE #I4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 VIEW$ F ! "R4@.> -------------------------------------#AH ,SQU>E ,PRISM "<,CUBE">1 ,POSI;N #A ,! CUBE IS SLIT+1 FACE-ON A#AH POSI;N4 ,SEE ^1,FIGURE ^1#AJ4 ,FIGURE #AJ4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 9 ,POSI;N #A4 ,T IS1 SLIT+1 FACE-ON POSI;N "4@.> ,FIGURE #AA4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES1 9 #C-;,D1 /&+ 2S ! CORRESPOND+ ROD MODEL 2+ HELD 9 #C-;,D4@.> ,! ROD MODEL FITS OV] ! CUBE MADE ) ,GEOMETRO TILES4 ,! ROD MODEL IS !N REMOV$ F ! TILES4 ,! #C-;,D CUBE MADE ( ! ROD MODELS IS V FLOPPY2 Z M5;N$ PREVI\SLY1 X W N /& ON XS [N & NE$S TO 2 HELD 9 ! V]TICAL POSI;N4 ,SEE ^7,FIGURES #AJ & #AA4^' ,FIGURE #AB4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO #BC TILES1 9 #C-;,D1 /&+ 2S ! B#AH CORRESPOND+ ROD MODEL HAV+ ! BASE & TOP SQU>ES AL;G "O ( ! DIAGONALS EL;GAT$1 / 9 #C-;,D1 ,POSI;N #A4@.> ,FIGURE #AC4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES1 9 #C-;,D1 ,POSI;N #A1 /&+ 2S ! CORRESPOND+ ROD MODEL N[ 9 #B-;,D4@.> ,FIGURE #AD4 @.<,PHOTO %[S ! ROD MODEL ( ! SQU>E PRISM " 9 #B-;,D4@.> ,TO OBTA9 ! MO/ TYPICAL #B-;,D REPRES5TA;N ( ! CUBE1 X IS BE/ TO *ANGE ! %APE ( ! BASE & TOP SQU>ES 9TO RHOMBI 2F PLAC+ ! MODEL 9TO ! #B-;,D POSI;N4 ,EL;GATE ! SQU>ES AL;G "O ( ! -------------------------------------#AI DIAGONALS Z %[N 9 ^1,FIGURE ^1#AB2 !N1 PLACE ! CUBE ON ! SURFACE BY ROTAT+ ! V]TICAL $GES AWAY F ! VIEW]4 ,SEE ^1,FIGURE ^1#AC4 ,! RESULT+ 3FIGURA;N W LOOK L T %[N 9 ^1,FIGURE ^1#AD4 ,SQU>E ,PRISM "<,CUBE">1 A#AI ,POSI;N #B ,! CUBE MADE ( ,GEOMETRO TILES IS SLIT+1 FACE-ON POSI;N4 ,SEE ^1,FIGURE ^1#AE4 ,FIGURE #AE4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 9 ,POSI;N #B4 ,T IS1 SLIT+1 FACE-ON POSI;N "4@.> ,FIGURE #AF4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES1 9 #C-;,D1 /&+ 2S ! CORRESPOND+ ROD MODEL 2+ HELD 9 #C-;,D4@.> ,! ROD MODEL FITS OV] ! CUBE MADE ) ,GEOMETRO TILES4 ,! ROD MODEL IS REMOV$ F ! TILES4 ,! #C-;,D CUBE MADE ( ! ROD MODELS IS V FLOPPY2 Z M5;N$ PREVI\SLY1 X W N /& ON XS [N & NE$S TO 2 HELD 9 ! V]TICAL POSI;N4 #BE ,FIGURE #AG4 #BJ @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES1 9 #C-;,D1 /&+ 2S ! CORRESPOND+ ROD MODEL HAV+ ! BASE & TOP SQU>ES AL;G "O ( ! DIAGONALS EL;GAT$1 / 9 #C-;,D1 ,POSI;N #B4@.> ,FIGURE #AH4 @.<,PHOTO %[S A SQU>E PRISM " 3/RUCT$ F ,GEOMETRO TILES1 9 #C-;,D1 ,POSI;N #B1 /&+ 2S ! CORRESPOND+ ROD MODEL N[ 9 #B-;,D4@.> ,FIGURE #AI4 @.<,PHOTO %[S ! ROD MODEL ( ! SQU>E PRISM " 9 #B-;,D4@.> ,TO OBTA9 ! MO/ TYPICAL #B-;,D REPRES5TA;N ( ! CUBE1 X IS BE/ TO *ANGE ! %APE ( ! BASE & TOP SQU>ES 9TO RHOMBI 2F PLAC+ ! MODEL 9TO ! #B-;,D POSI;N4 ,EL;GATE ! SQU>ES AL;G "O ( ! DIAGONALS Z %[N 9 ^1,FIGURE ^1#AG4 ,!N1 PLACE ! CUBE ON ! SURFACE BY ROTAT+ ! V]TICAL $GES AWAY F ! VIEW]4 ,SEE ^1,FIGURE ^1#AH4 ,! RESULT+ 3FIGURA;N W LOOK L T %[N 9 ^1,FIGURE ^1#AI4 A#BJ ,,SQU>E ,,PYRAMID ,! SQU>E PYRAMID HAS EIE H FIX$ L5G? & >E UNM>K$4 ,! ?REE $GES T C EXT5D >E M>K$ ) TWO /RIPS ( ,GRAPHIC ,>T ,TAPE 9 ! PHOTOS1 & ! "O T C 3TRACT IS M>K$ ) "O /RIP ( ,GRAPHIC ,>T ,TAPE4 -------------------------------------#BA ,SQU>E ,PYRAMID1 ,/>T+ ,POSI;N ,! />T+ POSI;N = ! SQU>E PYRAMID IS ) A V]TICAL $GE 9 FRONT ( ! VIEW] & C5TRAL4 ,! ROD MODEL IS FITT$ OV] ! ,GEOMETRO MODEL S ! FIX$ L5G? $GES =M ! BASE SQU>E & ! ONLY COLLAPSIBLE $GE1 M>K$ ) "O /RIP ( TAPE1 IS AT ! BACK ( ! PYRAMID1 N VISIBLE 9 ^1,FIGURE ^1#BJ4 ,H["E1 ? POSI;N IS V SYMMETRICAL & N ALL ( ! PYRAMID C 2 SE5 CLE>LY4 ,X IS BETT] TO ROTATE ! PYRAMID A LL BIT AWAY F ? SYMMETRICAL POSI;N & USE ! ROTAT$ POSI;N Z A />T+ VIEW1 Z ILLU/RAT$ 9 ^1,FIGURE ^1#BA "1 & ^1,FIGURE ^1#BB A#BA "4 ,FIGURE #BJ4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 /RAI ,FIGURE #BA4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 9 C.T]-CLOCKWISE ROTA;N4@.> ,FIGURE #BB4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 9 CLOCKWISE ROTA;N4@.> -------------------------------------#BB ,SQU>E ,PYRAMID1 ,POSI;N #A ,! SQU>E PYRAMID MADE ( ,GEOMETRO TILES IS SLIT+1 $GE-ON POSI;N4 ,SEE ^1,FIGURE ^1#BC4 ,FIGURE #BC4 A#BB @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 9 ,POSI;N #A1 C.T]-CLOCKWISE ROTA;N4@.> ,TO DEMON/RATE ! #B-;,D REPRES5TA;N1 ! ROD PYRAMID C 2 TAK5 (F ! ,GEOMETRO MODEL & LAID FLAT BY ROTAT+ ! TOP V]TEX & PLAC+ X ON ! SURFACE ABV ! SQU>E4 ,! $GES T MEET AT ! TOP V]TEX W *ANGE L5G? 9 ! PROCESS2 ! ?REE AT ! FRONT EXT5D1 ! "O AT ! BACK COLLAPSES1 & ! BASE SQU>E *ANGES %APE 9TO A RHOMBUS4 ,SEE ^1,FIGURES ^1#BD-#BF4 ,FIGURE #BD4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 9 ,POSI;N #A1 C.T]-CLOCKWISE ROTA;N4@.> ,FIGURE #BE4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 : IS N[ 9 #B-;,D4@.> #BI ,FIGURE #BF4 B#BB @.<,PHOTO %[S ! ROD MODEL ( ! SQU>E PYRAMID 9 #B-;,D1 9 ,POSI;N #A1 C.T]-CLOCKWISE ROTA;N4@.> -------------------------------------#BC ,SQU>E ,PYRAMID1 ,POSI;N #B ,! SQU>E PYRAMID MADE ( ,GEOMETRO TILES IS SLIT+1 $GE-ON POSI;N4 ,SEE ^1,FIGURE ^1#BG4 ,TO DEMON/RATE ! #B-;,D REPRES5TA;N1 ! ROD PYRAMID IS TAK5 (F ! ,GEOMETRO MODEL & LAID FLAT BY ROTAT+ ! TOP V]TEX & PLAC+ X ON ! SURFACE ABV ! SQU>E4 ,! $GES T MEET AT ! TOP V]TEX W *ANGE L5G? 9 ! PROCESS2 ! ?REE AT ! FRONT EXT5D1 ! "O AT ! BACK COLLAPSES1 & ! BASE SQU>E *ANGES %APE 9TO A RHOMBUS4 ,SEE ^1,FIGURES ^1#BH-#CJ4 ,FIGURE #BG4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 #C-;,D1 9 ,POSI;N #B1 CLOCKWISE ROTA;N4@.> A#BC ,FIGURE #BH4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 9 ,POSI;N #B1 CLOCKWISE ROTA;N4@.> ,FIGURE #BI4 @.<,PHOTO %[S A SQU>E PYRAMID 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 : IS N[ 9 #B-;,D4@.> ,FIGURE #CJ4 @.<,PHOTO %[S ! ROD MODEL ( ! SQU>E PYRAMID 9 #B-;,D1 9 ,POSI;N #B1 CLOCKWISE ROTA;N4@.> -------------------------------------#BD ,,TRIANGUL> ,,PRISM ,! TRIANGUL> PRISM HAS N9E $GES4 ,9 XS ROD MODEL1 F\R ( ! N9E $GES C 2 %ORT5$ & >E LABEL$ ) "O /RIP ( #A_/#D IN4 ,GRAPHIC ,>T ,TAPE 9 ! PHOTOS4 ,! REMA9+ FIVE $GES H FIX$ L5G?S & >E UNLABEL$4 ,WE 4CUSS ?REE WAYS TO LOOK AT ! TRIANGUL> PRISM1 Z %[N 2L = ?REE #CA POSI;NS4 A#BD ,TRIANGUL> ,PRISM1 ,POSI;N #A ,! PRISM MADE ( ,GEOMETRO TILES IS POSI;N$ ) A SQU>E FACE 9 FRONT1 V]TICAL1 & P>ALLEL TO ! VIEW]4 ,TWO TRIANGUL> FACES >E HORIZONTAL1 & "O ( ?REE V]TICAL $GES IS AT ! BACK & C5TRAL4 ,FIGURE #CA4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #A1 ) A SQU>E FACE 9 FRONT1 V]TICAL1 & P>ALLEL TO ! VIEW]4@.> ,! ROD MODEL IS FITT$ OV] ! ,GEOMETRO TILES ) ?REE UNLABEL$ $GES POSI;N$ V]TICALLY & TWO UNLABEL$ $GES HORIZONTAL TO COMPLETE ! FRONT SQU>E4 ,9 ! RESULT+ 3FIGURA;N1 EA* ( ! -------------------------------------#BE BASE TRIANGLES HAS "O FIX$ L5G? $GE1 P>ALLEL TO ! VIEW]1 & TWO V>IABLE L5G? $GES1 EXT5D+ T[>DS ! BACK4 ,SEE ^1,FIGURE ^1#CA4 ,TO COLLAPSE ! #C-;,D ROD MODEL 9TO #B-;,D1 ! MODEL HAS TO F/ 2 TAK5 A#BE (F ! ,GEOMETRO TILES4 ,S9CE ! JO9TS >E FLEXIBLE1 ! ROD MODEL IS N ABLE TO /& ON XS [N Z A #C-;,D /RUCTURE ONCE REMOV$ F ! ,GEOMETRO TILES1 & "!=E NE$S SUPPORT TO REMA9 V]TICAL4 ,SEE ^1,FIGURE ^1#CB4 ,FIGURE #CB4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL 2+ HELD 9 #C-;,D1 9 ,POSI;N #A4@.> -------------------------------------#BF ,! ROD MODEL IS N[ R1DY TO 2 PUT 9TO XS #B-;,D 3FIGURA;N BY ROTAT+ ! V]TICAL $GES 9TO HORIZONTAL POSI;NS & 9 A DIREC;N AWAY F ! VIEW]4 ,BY DO+ ?1 ! TOP TRIANGUL> FACE IS PLAC$ ABV ! BOTTOM TRIANGUL> FACE4 ,9 ! #B-;,D 3FIGURA;N1 ! ?REE P>ALLEL $GES T 7 V]TICAL 9 #C-;,D POSI;N M/ REMA9 P>ALLEL4 ,! RESPECTIVE $GES ( ! BOTTOM & TOP BASE TRIANGLES AL H TO REMA9 EQUAL & P>ALLEL4 ,SEE ^1,FIGURE ^1#CC4 ,FIGURE #CC4 @.<,PHOTO %[S A TRIANGUL> #CC PRISM 3/RUCT$ F ,GEOMETRO A#BF TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 : IS N[ 9 #B-;,D1 9 ,POSI;N #A4@.> ,PL1SE NOTE T ! TRANSI;N F V]TICAL TO HORIZONTAL POSI;N IS ALW P]=M$ AWAY F ! VIEW]1 N T[>D ! VIEW]4 ,? 9TUITIVELY CORRESPONDS ) ! FACT T :5 WE LOOK AT AN OBJECT1 ! SIDE T FACES U ALW REMA9S 9 VIEW4 -------------------------------------#BG ,S9CE TWO $GES ( EA* ( ! BASE TRIANGLES H V>IABLE L5G?1 X IS AL POSSIBLE TO *ANGE ! 8LOOK0 ( ! PRISM BY *ANG+ ! L5G?S ( ^! $GES4 ,? HAS TO 2 D"O 9 PAIRS S T ! FRONT SQU>E KEEPS XS %APE4 ,! RESULT+ 3FIGURA;NS CORRESPOND TO DI6]5T VIEWS ( ! PRISM1 Z IF VIEW$ F SLI PRISM 9 #B-;,D1 /RAIIABLE L5G? $GES EL;GAT$4@.> ,VIEW ,/RAI PRISM 9 #B-;,D1 /RAIIABLE L5G? $GES %ORT5$4@.> ,VIEW ,F ,"R,-,FIGURE #CF4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 VIEW F ! "R4@.> ,VIEW ,F ,LEFT,-,FIGURE #CG4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 VIEW F ! LEFT4@.> -------------------------------------#BH ,TRIANGUL> ,PRISM1 ,POSI;N #B ,! PRISM IS POSI;N$ ) A SQU>E V]TICAL FACE P>ALLEL TO ! VIEW] & AT ! BACK4 ,"O ( ?REE V]TICAL $GES IS 9 FRONT & C5TRAL4 ,SEE ^1,FIGURE ^1#CH4 ,FIGURE #CH4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #B1 ) A V]TICAL $GE #CE FRONT & C5T]4@.> A#BH ,TO OBTA9 ! #B-;,D 3FIGURA;N1 ! SAME /EPS = ! PREVI\S POSI;N %D 2 P]=M$4 ,>RANGE ! ROD MODEL OV] ! ,GEOMETRO TILE MODEL1 TAKE ! ROD MODEL (F1 !N ROTATE ! V]TICAL $GES AWAY F ! VIEW] & PLACE !M 9 HORIZONTAL POSI;N4 ,SEE ^7,FIGURES #CI & #DJ4^' ,FIGURE #CI4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL 2+ HELD 9 #C-;,D1 9 ,POSI;N #B4@.> ,FIGURE #DJ4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 N[ 9 #B-;,D1 9 ,POSI;N #B4@.> -------------------------------------#BI ,! RESULT+ #B-;,D 3FIGURA;N CORRESPONDS TO ! VIEW ( ! PRISM :5 VIEW$ /RAIE OBTA9$ BY PAIR-WISE %ORT5+ ( ! $GES M>K$ ) "O /RIP ( TAPE4 ,SEE ^1,FIGURES ^1#DA-#DD4 ,VIEW ,/RAI PRISM 9 #B-;,D1 /RAIIABLE L5G? $GES EL;GAT$4@.> ,VIEW ,/RAI PRISM 9 #B-;,D1 /RAIIABLE L5G? $GES %ORT5$4@.> ,VIEW ,F ,"R,-,FIGURE #DC4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 VIEW F ! "R4@.> ,VIEW ,F ,LEFT,-,FIGURE #DD4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 VIEW F ! LEFT4@.> ,TRIANGUL> ,PRISM1 ,POSI;N #C ,? IS A R DI6]5T & UNUSUAL />T+ POSI;N ( ! PRISM4 ,! PRISM LIES ON A SQU>E SIDE1 ) ! TWO TRIANGUL> FACES V]TICAL & FAC+ ! VIEW]4 ,? POSI;N IS N US$ 9 #CG ! ,/UD5T ,"WBOOK4 ,H["E1 X IS B#BI WELL WOR? 9TRODUC+ TO ! /UD5T ) VISUAL -------------------------------------#CJ IMPAIR;T S9CE ! TRIANGUL> PRISM CORRESPONDS TO IMPORTANT "EY"D OBJECTS S* Z RO(S ( BUILD+S1 & X HELPS TO GIVE A M COMPREH5SIVE A3.T4 ,SEE ^1,FIGURE ^1#DE4 ,VIEW ,/RAI PRISM 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #C1 ) ! TWO TRIANGUL> FACES V]TICAL & FAC+ ! VIEW]4@.> ,VIEW ,F ,"R,-,FIGURE #DF4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #C1 VIEW F ! "R4@.> ,VIEW ,F ,"R,-,FIGURE #DG4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL 2+ HELD 9 #C-;,D1 9 ,POSI;N #C1 VIEW$ F ! "R4@.> ,VIEW ,F ,"R,-,FIGURE #DH4 A#CJ @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 N[ 9 #B-;,D1 9 ,POSI;N #C1 Z VIEW$ F ! "R4@.> ,VIEW ,F ,"R,-,FIGURE #DI4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 9 ,POSI;N #C1 VIEW F ! "R4@.> ,! POSI;N SE5 9 ^1,FIGURE ^1#DE IS SYMMETRICAL & A LOT ( ! SOLID _C 2 SE5 CLE>LY2 X IS BETT] TO ROTATE ! MODEL A LL BIT AWAY F ? SYMMETRICAL POSI;N & USE ! ROTAT$ POSI;N Z A />T+ VIEW4 ,? IS ILLU/RAT$ 9 ^1,FIGURE ^1#DF1 VIEW F ! "R & ^1,FIGURE ^1#EJ1 VIEW F ! LEFT4 -------------------------------------#CA ,VIEW ,F ,LEFT,-,FIGURE #EJ4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #C1 VIEW F ! LEFT4@.> ,VIEW ,F ,LEFT,-,FIGURE #EA4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES /&+ #CI 2S ! CORRESPOND+ ROD MODEL 2+ A#CA HELD 9 #C-;,D1 9 ,POSI;N #C1 VIEW F LEFT4@.> ,VIEW ,F ,LEFT,-,FIGURE #EB4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 N[ 9 #B-;,D1 9 ,POSI;N #C1 VIEW F LEFT4@.> ,VIEW ,F ,LEFT,-,FIGURE #EC4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 9 ,POSI;N #C1 VIEW F LEFT4@.> ,O!R /EPS TO OBTA9 ! #B-;,D 3FIGURA;NS >E Z 2F3 ,>RANGE ! ROD MODEL OV] ! ,GEOMETRO TILE MODEL1 TAKE ! ROD MODEL (F1 !N ROTATE ! V]TICAL $GES AWAY F ! VIEW] TO PLACE !M 9 HORIZONTAL POSI;N4 ,SEE ^7,FIGURES #DF-#DH & #EJ-#EB4^' ,"! IS "O ADDI;NAL /EP AF ^7,FIGURES #DG & #EA^' T IS N EXPLICITLY %[N 9 ! PHOTOGRAPHS2 "NLY1 ! BASE SQU>E IS *ANG$ 9TO A RHOMBUS BY EL;GAT+ "O ( ! DIAGONALS & %ORT5+ ! O!R4 ,AF T1 ! PRISM IS PLAC$ 9 ! HORIZONTAL POSI;N1 %[N 9 ^7,FIGURES #DH & #EB1^' RESPECTIVELY4 ,,TRIANGUL> ,,PYRAMID #CB ,! TRIANGUL> PYRAMID HAS SIX $GES4 ,9 ! ROD MODEL1 ALL $GES H V>IABLE L5G?2 ?REE $GES1 LABEL$ ) "O /RIP ( ,GRAPHIC ,>T ,TAPE 9 ! PHOTOS1 C 2 %ORT5$1 & ?REE1 LABEL$ ) TWO /RIPS ( ,GRAPHIC ,>T ,TAPE1 C EXT5D4 ,WE 4CUSS TWO WAYS TO LOOK AT ! TRIANGUL> PYRAMID4 ,TRIANGUL> ,PYRAMID1 ,POSI;N #A ,! TRIANGUL> PYRAMID IS POSI;N$ ) A FACE P>ALLEL TO ! VIEW] & 9 FRONT4 ,AN $GE IS AT ! BACK & C5T]4 ,! ROD MODEL %D 2 FITT$ OV] ! ,GEOMETRO MODEL S ! ?REE EXT5DABLE $GES FIT OV] ! FRONT TRIANGLE :ILE ! COLLAPSIBLE $GES G T[>D ! BACK4 ,SEE ^1,FIGURE ^1#ED4 ,FIGURE #ED4 @.<,PHOTO %[S A TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #A1 ) A V]TICAL $GE BACK & C5T]4@.> #DA ,TO TRANS=M ! #C-;,D ROD MODEL #CC 9TO #B-;,D1 ! ROD MODEL M/ 2 TAK5 (F ! ,GEOMETRO TILES &1 SUBSEQU5TLY1 ! TOP V]TEX PLAC$ ON ! SURFACE ABV ! BASE TRIANGLE4 ,9 ! PROCESS1 ! $GES M>K$ ) "O /RIP ( TAPE W %ORT5 :ILE ! $GES M>K$ ) TWO /RIPS ( TAPE EXT5D A LL4 ,SEE ^1,FIGURES ^1#EE-#EG4 ,FIGURE #EE4 @.<,PHOTO %[S A TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 9 ,POSI;N #A4@.> ,FIGURE #EF4 @.<,PHOTO %[S A TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES 2S ! CORRESPOND+ ROD MODEL1 N[ 9 #B-;,D1 9 ,POSI;N #A4@.> ,VIEW ,/RAI PYRAMID 9 #B-;,D1 ,POSI;N #A1 VIEW$ /RAI ,! LESS SYMMETRICAL VIEWS ( ! PYRAMID C 2 OBTA9$ BY FUR!R %ORT5+ "O ( ! COLLAPSIBLE $GES1 Z %[N 9 ^7,FIGURES #EH & #EI4^' A#CC ,VIEW ,F ,"R,-,FIGURE #EH4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PYRAMID 9 #B-;,D1 ,POSI;N #A1 VIEW$ F ! "R4@.> ,VIEW ,F ,LEFT,-,FIGURE #EI4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PYRAMID 9 #B-;,D1 ,POSI;N #A1 VIEW$ F ! LEFT4@.> -------------------------------------#CD ,TRIANGUL> ,PYRAMID1 ,POSI;N #B ,! TRIANGUL> PYRAMID IS POSI;N$ ) A FACE P>ALLEL TO ! VIEW] & AT ! BACK4 ,AN $GE IS 9 FRONT & C5T]4 ,! ROD MODEL IS PLAC$ OV] ! ,GEOMETRO MODEL ) AN EXT5DABLE $GE1 M>K$ ) TWO /RIPS ( TAPE 9 ! PHOTOS1 FRONT & C5T]4 ,! REMA9+ $GES MAY 2 >RANG$ Z %[N 9 ^7,FIGURE #FJ OR #FA4^' ,FIGURE #FJ4 @.<,PHOTO %[S A TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #B4@.> #DC ,FIGURE #FA4 A#CD @.<,PHOTO %[S A TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X1 9 ,POSI;N #B1 ) EXT5DABLE & 3TRACTIBLE $GES REV]S$ F ,FIGURE #FJ4@.> ,FIGURE #FB4 @.<,PHOTO %[S A TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES /&+ 2S ! CORRESPOND+ ROD MODEL1 9 #C-;,D1 9 ,POSI;N #B4@.> ,FIGURE #FC4 @.<,PHOTO %[S A TRIANGUL> PYRAMID 3/RUCT$ F ,GEOMETRO TILES1 9 ,POSI;N #B1 /&+ 2S ! CORRESPOND+ ROD MODEL1 N[ 9 #B-;,D4@.> ,VIEW ,/RAI PYRAMID 9 #B-;,D1 ,POSI;N #B1 VIEW$ /RAI -------------------------------------#CE ,TO =M ! #B-;,D PROJEC;N1 ! TOP V]TEX IS PLAC$ ON ! PLANE ABV ! BASE TRIANGLE1 = : ! FRONT $GE "K$ ) TWO /RIPS ( TAPE"> NE$S TO 2 MAXIMALLY EXT5D$4 ,! BACK HORIZONTAL $GE "K$ ) "O A#CE /RIP ( TAPE"> MAY 2 SLIK$ ) TWO /RIPS ( TAPE1 Z %[N 9 ^1,FIGURE ^1#FE4 ,VIEW ,F ,LEFT,-,FIGURE #FE4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PYRAMID 9 #B-;,D1 ,POSI;N #B1 Z DESCRIB$4@.> #DE ,,,! P$AGOGICAL #CF-#CG PROCESS,- F\R LESSONS ) ! TRIANGUL> PRISM,' ,! OV]ALL GOAL ( ! ,/UD5T ,"WBOOK IS TO HELP /UD5TS :O >E VISUALLY IMPAIR$ =M ! BE/ POSSIBLE 9TUITIVE "U/&+ ( BASIC SPATIAL 3CEPTS1 ESPECIALLY ! RELA;N%IPS 2T ASPECTS ( ?REE- & TWO-DIM5.NAL OBJECTS4 ,! ,/UD5T ,"WBOOK IS DIVID$ 9TO F\R TYPES ( LESSONS1 R\ SET ( ACTIVITIES = DI6]5T #C-;,D SOLIDS4 ,! F\R LESSONS >E SPECIFI$ 9 /EP-BY-/EP DETAIL 9 ! ,T1*]'S ,GUIDE SEC;N ( ! ,/UD5T ,"WBOOK4 ,"H ! LESSONS >E 4CUSS$ 9 DETAIL 9 ORD] TO GIVE ! T1*] NEC -------------------------------------#CH BACKGR.D & SUPPORT+ 9=MA;N = PROVID+ 9/RUC;N4 ,"! >E OPPORTUNITIES = REPETI;N & AL FLEXIBIL;Y = ! T1*] TO *OOSE A LESSON &_/OR A SOLID AT A LEVEL T IS APPROPRIATE = A /UD5T4 ,? ALL[S = A#CH T1*]S TO (F] 5 PRACTICE & TO *ECK /UD5T PROGRESS4 ,LET'S LOOK AT ! F\R LESSONS US+ "O ( ! SOLIDS 9 ! "WBOOK,-! TRIANGUL> PRISM,-Z AN EXAMPLE4 -------------------------------------#CI ,,LESSON #A ,"U/&+ #B-;,D ,TACTILE ,DRAW+S ( #C-;,D ,SOLIDS #A4 ,ASK ! /UD5T TO MAKE A TRIANGUL> PRISM US+ ,GEOMETRO TILES2 OR ALT]NATIVELY1 (F] A R1DY-MADE SOLID = /UD5T'S EXAM9A;N4 #B4 ,POSI;N ! ,GEOMETRO MODEL ( ! TRIANGUL> PRISM 9 "O ( ! POSI;NS DESCRIB$ 9 ! SEC;N D1L+ ) ? SOLID2 T IS1 POSI;N #A OR #B4 ,"H X IS %[N 9 POSI;N #A1 9 ^1,FIGURE ^1#FF4 ,FIGURE #FF4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES1 ,POSI;N #A4@.> #DG ,FIT ! ROD MODEL ( ! TRIANGUL> A#CI PRISM SNU7LY ON ! PRISM MADE ( ,GEOMETRO TILES1 Z 9 ^1,FIGURE ^1#FG4 ,LET ! /UD5T EXAM9E ? 3FIGURA;N4 ,FIGURE #FG4 @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES1 ,POSI;N #A1 ) ! CORRESPOND+ ROD MODEL PLAC$ OV] X4@.> -------------------------------------#DJ #C4 ,REMOVE ! ROD MODEL F ! ,GEOMETRO MODEL & POSI;N ! TWO SIDE BY SIDE4 ,! ROD MODEL MAY H "S DI6ICULTY /&+ ON XS [N1 S Y MI PRISM 3/RUCT$ F ,GEOMETRO TILES1 ,POSI;N #A1 /&+ 2S ! CORRESPOND+ ROD MODEL1 2+ HELD 9 #C-;,D4@.> #D4 ,,N[1 ! MO/ IMPORTANT "P6 ,SL[LY PLACE ! ROD MODEL ( ! PRISM FLAT ON ! TABLE4 ,! FLEXIBLE JO9TS W ALL[ ! RANGE ( MO;N4 ,SEE ^1,FIGURE ^1#FI4 ,FIGURE #FI4 A#DJ @.<,PHOTO %[S A TRIANGUL> PRISM 3/RUCT$ F ,GEOMETRO TILES1 ,POSI;N #A1 /&+ 2S ! CORRESPOND+ ROD MODEL1 N[ 9 #B-;,D4@.> #E4 ,RAISE ! PRISM 9TO #C-;,D BY REV]S+ ! ABV /EPS4 ,PLACE X FLAT AG1 9TO #B-;,D4 ,REP1T Z _M "TS Z REQUIR$ = ! /UD5T TO "U/& ,,:Y ! #B-;,D DRAW+ FUNC;NS ! WAY X DOES4 ,! TRIANGUL> PRISM C 2 PUT 9TO #B-;,D )\T *ANGES 9 ! L5G? ( ! RODS "4 ,H["E1 -------------------------------------#DA X DOES N LOOK ! WAY WE NORMALLY SEE X2 ! $GES M>K$ ) "O /RIP ( TAPE %D 2 %ORT5$ "4 ,AF ! /UD5T 2COMES FAMILI> ) ! IDEA ( ! #B-;,D APPE>.E ( ! PRISM1 ! T1*] C 9TRODUCE ! /EP TO %ORT5 ! $GES M>K$ ) "O /RIP ( TAPE4 ,H["E1 = O!R SOLIDS1 "! W 2 $GES T H TO 2 %ORT5$ OR EL;GAT$ TO 2 ABLE TO PUT ! SOLID 9TO #B-;,D4 ,9 EA* CASE1 X IS V IMPORTANT TO #DI LET ! /UD5T FEEL H[ "S $GES H TO A#DA %ORT5 & "S EL;GATE 9 ! PROCESS ( GO+ F #C-;,D TO #B-;,D4 ,FIGURE #GJ4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 )\T *ANGES 9 ! L5G? ( ! RODS1 Z DESCRIB$4@.> ,FIGURE #GA4 @.<,PHOTO %[S ! ROD MODEL ( ! TRIANGUL> PRISM 9 #B-;,D1 ) $GES "K$ ) "O /RIP ( TAPE"> %ORT5$1 Z DESCRIB$4@.> -------------------------------------#DB #F4 ,ASK ! /UD5T TO EXAM9E ! L>GE TACTILE DRAW+ ( ! SOLID ON PAGE #AC ( ! ,/UD5T ,"WBOOK4 ,NOTICE T "! >E SOLID & DOTT$ L9ES ON ! DRAW+4 #G4 ,EXAM9E ! #B-;,D ROD MODEL ( ! SOLID & DET]M9E : $GES MAY 2 8HIDD50 9 ! FLAT 3FIGURA;N &1 "!=E1 CORRESPOND ) DOTT$ L9ES ON ! TACTILE DRAW+4 #H4 ,! L>GE TACTILE DRAW+ ON PAGE #AC ( ! ,/UD5T ,"WBOOK MAT*ES ! DIM5.NS ( ! ROD MODEL :5 9 #B-;,D 3FIGURA;N4 ,! TASK = ! /UD5T IS TO SUP]IMPOSE A#DB ! #B-;,D ROD MODEL ONTO ! L>GE TACTILE DRAW+4 ,ADJU/+ ! L5G?S ( ! $GES MI PRISM1 DEMON/RAT+ ! ORIG9 ( A #B-;,D TACTILE DRAW+4 -------------------------------------#DC ,FIGURE #GB4 #I4 ,AF ! /UD5T 2COMES FAMILI> ) ! PROCESS ( FOLD+ & UNFOLD+ ! #C-;,D TO #B-;,D1 & ?US1 ! ORIG9 ( ! L>GE TACTILE REPRES5TA;N1 ASK ! /UD5T TO EXAM9E ! SAME1 B SMALL]1 TACTILE DRAW+S ON PAGE #AE ( ! ,/UD5T ,"WBOOK4 ,NOTICE ! ORI5TA;N ( ! TWO SOLIDS ON ! PAGE2 ONLY "O CORRESPONDS TO ! DRAW+ ON PAGE #AC4 -------------------------------------#DD #AJ4 ,PAGE #AD ( ! ,/UD5T ,"WBOOK IS ESS5TIALLY ! SAME EX]CISE2 H["E1 #EA N[ ! #C-;,D PRISM IS FOLD$ TO A#DD #B-;,D A BIT DI6]5TLY BY FOLL[+ ! 9/RUC;NS = POSI;N #B ( ! TRIANGUL> PRISM4 ,VISUALLY1 WE RECOGNIZE ) NO DI6ICULTY AT ALL T ! DRAW+ REPRES5TS ! SAME PRISM4 ,A /UD5T ) VISUAL IMPAIR;T1 ESPECIALLY "O :O IS 3G5ITALLY VISUALLY IMPAIR$1 MAY N COMPREH5D H[ ! TWO REPRES5TA;NS ON PAGES #AC & #AD ( ! ,/UD5T ,"WBOOK %[ ! SAME SOLID4 ,! TWO EX]CISES 3VEY ! IDEA T A #C-;,D OBJECT W LOOK DI6]5T :5 VIEW$ F DI6]5T DIREC;NS2 ! SAME WAY ! ROD MODEL MAY LOOK & FEEL DI6]5T :5 FOLD$ 9 V>I\S WAYS4 ,ASK ! /UD5T TO EXAM9E ! TWO SMALL TACTILE DRAW+S ON PAGE #AE ( ! ,/UD5T ,"WBOOK4 ,COMP>E ! ORI5TA;N ( BO? SOLIDS ON PAGE #AE TO CORRESPOND+ DRAW+S ON PAGES #AC & #AD4 ,,LESSON #B #DE ,DECOMPOS+ #C-;,D ,SOLIDS 9TO ,NETS #A4 ,ASK ! /UD5T TO ID5TIFY ! #C-;,D SOLID REPRES5T$ BY ! TACTILE DRAW+ ON PAGE #AC ( ! ,/UD5T ,"WBOOK Z A TRIANGUL> PRISM4 ,OR ALT]NATIVELY1 PRES5T ! /UD5T ) A R1DY-MADE MODEL ( A TRIANGUL> PRISM4 #B4 ,ASK ! /UD5T TO DECOMPOSE ! ,GEOMETRO TILE MODEL S X LIES FLAT1 =M+ A NET4 #C4 ,ASK ! /UD5T TO RECORD ! NET ON ! MAGNETIC BO>D US+ ! MAGNETIC " POLYGONS4 #D4 ,ASK ! /UD5T TO PUT ! ,GEOMETRO MODEL BACK 9TO #C-;,D & !N DECOMPOSE AG1 =M+ A DI6]5T NET4 #E4 ,2F RECORD+ EA* NEW NET ON ! BO>D1 ! /UD5T M/ MAKE SURE T ! SAME NET ISN'T RECORD$ "! ALR4 ,= ! SIMPLE/ SOLID1 S* Z ! TRIANGUL> PYRAMID1 "! IS LL *ALL5GE 9 ? EX]CISE S9CE "! >E #EC FEW DI6]5T NETS POSSIBLE4 ,H["E1 #DF = MO/ SOLIDS1 _M NETS >E POSSIBLE4 ,= EXAMPLE1 A CUBE HAS #AA POSSIBLE NETS4 ,! *ALL5GE IS 9 4COV]+ !M ALL & MAK+ SURE T !Y >E ALL UNIQUE4 ,X MAY 2 A *ALL5GE = ! /UD5T TO DECIDE ON ! 8SIMPLE0 FACT ( :E!R TWO NETS >E DI6]5T OR ! SAME4 ,IF "O ( TWO ID5TICAL NETS IS ROTAT$ OR FLIPP$ "1 X MAY N 2 IMMLY OBVI\S T !Y >E ID5TICAL4 ,? CD 2 A DEM&+1 B WOR?Y1 EX]CISE = A /UD5T ) VISUAL IMPAIR;T4 #F4 ,AF E/ABLI%+ T A NEW NET IS DI6]5T F ALL PREVI\SLY RECORD$ "OS1 ASK ! /UD5T TO RECORD EA* NEW NET4 #G4 ,/EPS #D-#F C 2 REP1T$1 Z _M "TS Z APPROPRIATE4 -------------------------------------#DG ,,LESSON #C ,*OOS+ ,NETS = #C-;,D ,SOLIDS ,? LESSON IS A REV]SAL ( ,LESSON #B4 ,! /UD5T IS PRES5T$ ) A S]IES ( A#DG #B-;,D TACTILE FIGURES & M/ DECIDE IF EA* C 2 FOLD$ 9TO A #C-;,D SOLID "RANGE;T IS A NET = ! SOLID">4 #A4 ,ASK ! /UD5T TO ID5TIFY ! #C-;,D SOLID REPRES5T$ BY ! TACTILE DRAW+ ON PAGE #AC ( ! ,/UD5T ,"WBOOK4 ,OR ALT]NATIVELY1 PRES5T ! /UD5T ) A R1DY-MADE MODEL ( ! SOLID4 #B4 ,ASK ! /UD5T TO EXAM9E ! TACTILE DRAW+S ( ! #B-;,D FIGURES ON PAGE #AF4 #C4 ,! TASK = ? LESSON IS TO ID5TIFY1 & 9DICATE1 : ( ! F\R #B-;,D FIGURES C 2 FOLD$ 9TO A TRIANGUL> PRISM "E ! NETS = ! TRIANGUL> PRISM">4 #D4 ,DEP5D+ ON AGE &_/OR ABILITIES1 A /UD5T MAY "W )\T MANIPULATIVES1 OR USE ,GEOMETRO TILES TO *ECK ! PR$IC;NS4 ,IF ! -------------------------------------#DH /UD5T HAS DI6ICULTY TRANSF]R+ ! PATT]N F ! TACTILE DRAW+ TO ,GEOMETRO TILES1 X IS SUITABLE TO USE ! MAGNETIC TILES Z AN 9T]M$I>Y4 ,! MAGNETIC TILES #EE FIT 9SIDE ! RESPECTIVE TACTILE A#DH DRAW+S & !Y MAY HELP A /UD5T BETT] "U/& ! PATT]N ( EA* #B-;,D FIGURE2 TRANSF] TO ,GEOMETRO TILES IF NEC4 ,FIGURE #GC4 ^1,FIGURE ^1#GC DEPICTS A PAGE ( TACTILE DRAW+S F ! ,/UD5T ,"WBOOK1 ) MAGNETIC TILES FITT$ 9SIDE "S ( ! TACTILE #B-;,D FIGURES = BETT] RECOGNI;N & TRANSF] ( ! PATT]N1 & = BETT] "U/&+ ! PATT]N ( ! #B-;,D FIGURES4 -------------------------------------#DI #E4 ,AF ! PATT]N ( ! #B-;,D FIGURE IS TRANSF]R$ F ! TACTILE DRAW+ TO ,GEOMETRO TILES1 ! /UD5T W 2 ABLE TO *ECK1 BY FOLD+ ! TILES1 IF ! PRISM C 2 OBTA9$ F ! #B-;,D FIGURE4 #F4 ,ASK ! /UD5T TO 9DICATE ! #B-;,D FIGURES T C 2 FOLD$ 9TO ! TRIANGUL> PRISM4 ,,LESSON #D #EJ ,*ANG+ #B-;,D ,FIGURES ,S ,!Y ,C ,2 ,FOLD$ 9TO A ,SOLID ,ONLY TWO ( ! F\R #B-;,D FIGURES F ,LESSON #C "<%[N ON PAGE #AF ( ! ,/UD5T ,"WBOOK"> C 2 FOLD$ 9TO A TRIANGUL> PRISM4 ,! O!R TWO #B-;,D FIGURES REQUIRE "S *ANGES 9 ORD] TO D S4 ,NEI ( ! TWO #B-;,D FIGURES ON PAGE #AG C 2 FOLD$ 9TO A TRIANGUL> PRISM2 BO? REQUIRE "S *ANGES4 ,! OBJECT ( ? LESSON IS TO >RIVE AT1 & DESCRIBE1 ! NEC *ANGES4 #A4 ,ASK ! /UD5T TO ID5TIFY ! #C-;,D SOLID REPRES5T$ BY ! TACTILE DRAW+S ON PAGE #AC ( ! ,/UD5T ,"WBOOK4 ,OR ALT]NATIVELY1 PRES5T ! /UD5T ) A R1DY-MADE MODEL ( ! SOLID4 #B4 ,ASK ! /UD5T TO EXAM9E TACTILE DRAW+S ( #B-;,D FIGURES ON PAGE #AG ( ! ,/UD5T ,"WBOOK4 #C4 ,ASK ! /UD5T TO 3/RUCT ! SAME #B-;,D FIGURES US+ ,GEOMETRO TILES & *ECK IF ! FIGURE C 2 FOLD$ 9TO ! #EG ID5TIFI$ SOLID4 A#EJ -------------------------------------#EA #D4 ,IF NE$$1 X MAY 2 USE;L = ! /UD5T TO 9ITIALLY PLACE ! MAGNETIC TILES 9SIDE ! POLYGONS ON ! TACTILE DRAW+S ( ! #B-;,D FIGURES4 #E4 ,ASK ! /UD5T TO DET]M9E :Y ! #B-;,D FIGURES W N FOLD 9TO ! #C-;,D TRIANGUL> PRISM1 & 9DICATE H[ TO *ANGE !M S !Y W FOLD CORRECTLY4 ,G5]ALLY1 "?\T ! BOOK1 ! NATURE ( NEC *ANGES C V>Y4 ,! #B-;,D FIGURES MAY NE$ M OR FEW] POLYGONS1 !Y MAY NE$ A RE->RANGE;T ( ! POLYGONS1 OR !Y MAY H WR;G POLYGONS T NE$ TO 2 EX*ANG$4 ,COMB9A;NS ( ! ABV CD AL APPE> 9 "S CASES4 ,= EXAMPLE1 IF WE LOOK AT ! #B-;,D FIGURES = ! TRIANGUL> PRISM ON PAGE #AG ( ! ,/UD5T ,"WBOOK1 ! F/ FIGURE HAS A CORRECT NUMB] ( POLYGONS "ES & TWO TRIANGLES">1 B !Y >E >RANG$ 9CORRECTLY4 ,! /UD5T NE$S TO *ANGE ! POSI;N ( "O ( ! TRIANGLES & *OOSE ": TO PUT X TO =M A#EA -------------------------------------#EB A NET4 ,! SECOND #B-;,D FIGURE1 ON ! O!R H&1 HAS TOO FEW POLYGONS2 "O TRIANGLE IS MISS+1 & ! /UD5T HAS TO DECIDE ": TO PLACE AN EXTRA TRIANGLE TO =M A NET4 #F4 ,! MAGNETIC TILES C 2 US$ = ! 3/RUC;N ( ! NET ON ! MAGNETIC BO>D4 ,! MAGNETIC BO>D W HOLD ! TILES SU6ICI5TLY & /R;GLY 9 PLACE S T X C 2 9S]T$ 9TO ! ,BRL ,POCKET ,FOLD]1 : C !N 2 PLAC$ 9 ! ,/UD5T ,"WBOOK & REF]R$ TO LAT]4 #G4 ,! F9AL /EP ( ! LESSON IS = ! /UD5T TO WRITE A DESCRIP;N ( ! *ANGES NE$$ TO MAKE ! #B-;,D FIGURES 2COME NETS4 ,? PROVIDES AN EXCELL5T OPPORTUN;Y TO DEVELOP & USE SPACE-RELAT$ VOCABUL>Y & EXPRES.NS4 ,A /UD5T C USE A J\RNAL OR O!R NOTEBOOK TO RECORD ! DESCRIP;NS1 OR USE ANY O!R ME?OD AGRE$ ^U BY /UD5T & T1*]4 #EI ,,ADDI;NAL ,,ACTIVITIES #EC #A4 ^7,EXPLOR+ G5]AL PROP]TIES ( SOLIDS3^' ,? EX]CISE CD 2 EI AN 9ITIAL OR F9AL ACTIV;Y RELAT$ TO EA* SOLID4 ,! /UD5T MAY C.T & RECORD ! NUMB] ( FACES1 $GES1 & V]TICES & SPECIFY :AT POLYGONS >E US$ TO MAKE A SOLID4 ,TO HELP ) ! C.T+ & SEQU;E ( /EPS1 ! /UD5T MAY USE M>K]S,-REMOVABLE /ICK]S = FACES & "S HOOK-&-LOOP MAT]IAL M>K]S = $GES & V]TICES4 ^1,FIGURE ^1#GD DEPICTS HOOK-&-LOOP MAT]IAL M>K]S1 AVAILABLE F ,GEOMETRO1 US$ TO HELP C.T $GES & V]TICES ( A SOLID MADE F ,GEOMETRO TILES4 ^1,FIGURE ^1#GE DEPICTS /ICK]S US$ TO HELP C.T FACES ( A SOLID4 -------------------------------------#ED ,FIGURE #GD4 @.<,PHOTO %[S A P5TAGON 3/RUCT$ F ,GEOMETRO TILES ) HOOK-&-LOOP MAT]IAL M>K]S1 Z DESCRIB$4@.> ,FIGURE #GE4 A#ED @.<,PHOTO %[S A P5TAGON 3/RUCT$ F ,GEOMETRO TILES ) /ICK]S1 Z DESCRIB$4@.> @.<,SYMBOLS US$ 2L3 "- ,M9US "6 ,PLUS "7 ,EQUAL@.> ,"! IS A SIMPLE RELA;N%IP 2T ! NUMB] ( FACES "<;,F">1 V]TICES "<;,V">1 & $GES "<;,E"> T APPLIES TO E SOLID1 Z 4COV]$ BY ! ,SWISS MA!MATICIAN1 ,LEONH>D ,EUL]4 ,! RELA;N%IP1 "KN Z ! ,EUL] EQUA;N IS3 ;,V"-,E"6,F"7#B ,Z AN ADDI;NAL TASK1 A /UD5T MAY EI *ECK ! EQUA;N = A "PICUL> SOLID OR >RIVE AT ! RELA;N%IP AF COLLECT+ ! 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