,,APH^R #A ,A4I;N @& ,SUBTRAC;N ,,GUIDEBOOK --------------------------------------#B ,A4I;N @& ,SUBTRAC;N ,TA# ,9TRODUC;N ,',? BOOKLET HAS BE5 TRANSCRIB$ US+ ..,! ,NEME? ,BRL ,CODE = ,MA!MATICS & ,SCI;E ,NOT,N1 #1972 .,REVI.N #2007 ,UPDATE4,' ,! ,A4I;N & ,SUBTRAC;N ,TA# C 2 US$ 6F9D ! SUM ( TWO :OLE NUMB]S F #1 "? #10 OR ! DI6];E (A RELAT$ SUBTRAC;N PRO#M4 ,X RE9=CES ! BASIC A4I;N & SUBTRAC;N FACTS4 ,= A4I;N1 "O A45D IS LOCAT$ ON ! TOP R[ (! *>T &! O!R IS LOCAT$ ON ! LEFT1 9 ! F/ COLUMN (! *>T4 ,! PO9T AT : ! R[ & COLUMN (! TWO A45DS -E TGR ON ! *>T IS ! SUM4 ,3V]SELY1 ! *>T C 2 US$ = SUBTRAC;N4 ,/UD5TS C USE ! TA# 6*ECK _! "W #A AF !Y H PRODUC$ A TA# OR A "P (A A#B TA# ON _! [N OR H "W$ A PRO#M OR A GR\P ( PRO#MS !MVS4 ,X IS RECOMM5D$ T ? *>T 2 US$ .ONLY 0/UD5TS :O H _H REP1T$ OPPORTUNITIES 6USE 3CRETE MANIPULATIVES & SEMI 3CRETE DIAGRAMS & DRAW+S ( _! [N 6SOLVE A4I;N & SUBTRAC;N PRO#MS4 --------------------------------------#C ,PURPOSE ,! ,A4I;N & ,SUBTRAC;N ,TA# C 2 US$ 6F9D ! SUM ( TWO :OLE NUMB]S F #1 "? #10_4 ,LIKEWISE1 ! TA# C 2 US$ 6F9D ! DI6];E 2T :OLE NUMB]S F #2 "? #20 :5 ! M9U5D IS GIV5 &! SUBTRAH5D IS AL A :OLE NUMB] F #2 "? #20_4 ,US+ ! TA# 6DET]M9E ! SUM OR DI6];E PROVIDES PRACTICE & RE9=CES LE>N+ ! BASIC A4I;N & SUBTRAC;N FACTS4 ,US+ ! ,A4I;N ,TA#3 ,ME?ODS #1 & #2 ,"! >E S"EAL WAYS 6USE ! ,A4I;N ,TA# 9 ORD] 6ADD TWO NUMB]S4 ,! F/ NUMB] A#C C 2 F.D ACR ! TOP R[ &! SECOND NUMB] D[N ! F/ COLUMN OR VICE-V]SA4 ,9 EI CASE1 ! ANSW] OR ! SUM (! TWO NUMB]S IS ! PO9T ": ! R[ &! COLUMN MEET :5 Y FOLL[ ! NUMB] AT ! TOP (! F/ R[ /RAIE A4$ DOES N A6ECT ! SUM1 I4E41 A+B .K B+A_4 #C ,',KEY3 A#D FC ,F/ COLUMN ;FR ,F/ R[,' 7777777777777777777777777777777777777777 FC "33 ;FR + #1 #2 #3 #4 _#5 #6 #7 #8 #1 #2 #3 #4 #5 .#6 #7 #8 #9 #2 #3 #4 #5 #6 .#7 #8 #9 #10 _#3 .#4 .#5 .#6 .#7 _#8 #9 #10 #11 #4 #5 #6 #7 #8 #9 #10 #11 #12 #5 #6 #7 #8 #9 #10 #11 #12 #13 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG --------------------------------------#E 7777777777777777777777777777777777777777 + #1 #2 _#3 #4 #5 #6 #7 #8 #1 #2 #3 .#4 #5 #6 #7 #8 #9 #2 #3 #4 .#5 #6 #7 #8 #9 #10 #3 #4 #5 .#6 #7 #8 #9 #10 #11 #4 #5 #6 .#7 #8 #9 #10 #11 #12 _#5 .#6 .#7 _#8 #9 #10 #11 #12 #13 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ,= /UD5TS ) GD TRACK+ SKILLS1 ,ME?OD #1 OR #2 %D 2 F9E4 ,= /UD5TS :O MAY H DI6ICULTY TRACK+1 H["E1 ANO!R BASIC ME?OD MAY 2 US$4 A#E ,US+ ! ,A4I;N ,TA#3 ,ME?ODS #3 & #4 ,US+ ^! ME?ODS1 H ! /UD5T LOCATE ! F/ NUMB] EI ON ! TOP R[ OR 9 ! F/ COLUMN4 ,!N EI MOVE D[N ! L9E OR MOVE ACR ! L9E ! SAME NUMB] ( "TS Z YR SECOND NUMB] 6BE A4$4 ,= EXAMPLE1 6F9D #5+3_3 #1_4 ,F9D ! #5 9 ! TOP R[4 #2_4 ,/AY 9 ! FIVES COLUMN1 & MOVE D[N #3 "TS4 #3_4 ,! ANSW] IS #8_2 #5+3 .K #8_4 --------------------------------------#F 7777777777777777777777777777777777777777 + #1 #2 #3 #4 _#5 #6 #7 #8 #1 #2 #3 #4 #5 _#6 #7 #8 #9 #2 #3 #4 #5 #6 _#7 #8 #9 #10 #3 #4 #5 #6 #7 _#8 #9 #10 #11 #4 #5 #6 #7 #8 #9 #10 #11 #12 #5 #6 #7 #8 #9 #10 #11 #12 #13 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG #E ,ALT]NATIVELY1 USE ! REV]SE A#F PROCESS 6F9D #5+3_3 #1_4 ,F9D ! #5 9 ! F/ COLUMN4 #2_4 ,/AY 9 ! FIVES R[1 & MOVE OV] #3 "TS4 #3_4 ,! ANSW] IS #8_2 #5+3 .K #8_4 7777777777777777777777777777777777777777 + #1 #2 #3 #4 #5 #6 #7 #8 #1 #2 #3 #4 #5 #6 #7 #8 #9 #2 #3 #4 #5 #6 #7 #8 #9 #10 #3 #4 #5 #6 #7 #8 #9 #10 #11 #4 #5 #6 #7 #8 #9 #10 #11 #12 _#5 _#6 _#7 _#8 #9 #10 #11 #12 #13 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG --------------------------------------#G ,H ! /UD5T USE EI ,ME?OD #3 OR #4 63FIRM T #3+5 AL EQUALS #8_2 #5+3 .K #3+5_4 ,J Z ) ,ME?ODS #1 & #2, ? C L1D 6A 4CUS.N (! ,-MUTATIVE ,PROP]TY ( ,A4I;N4 ,GUID$ ,PRACTICE ,H ! /UD5T PRACTICE F9D+ ! SUM ( #2+3 US+ "O OR M (! ME?ODS 4CUSS$4 A#G ,ME?OD #1 #1_4 ,H ! /UD5T F9D ! TOP R[ (! A4I;N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] #2 9 T R[4 #3_4 ,H ! /UD5T F9D ! F/ COLUMN FUR!/ 6! LEFT4 #4_4 ,ASK ! /UD5T 6F9D ! NUMB] #3 9 T COLUMN4 #5_4 ,H ! /UD5T MOVE D[N ! R[ T IS H1D$ BY #2 & ACR ! COLUMN T />TS ) #3 UNTIL ^! TWO L9ES -E TGR4 ,! NUMB] ": ^! L9ES MEET OR 9T]SECT IS ! ANSW] OR SUM4 #6_4 ,H ! /UD5T R1D ! ANSW]4 (5) --------------------------------------#H ,ME?OD #2 #1_4 ,H ! /UD5T F9D ! F/ COLUMN FUR!/ 6! LEFT ON ! A4I;N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] #2 9 T COLUMN4 #3_4 ,H ! /UD5T F9D ! TOP R[ (! TA#4 #4_4 ,ASK ! /UD5T 6F9D ! NUMB] #3 9 #G T R[4 A#H #5_4 ,H ! /UD5T MOVE ACR ! COLUMN T />TS ) #2 & D[N ! R[ T IS H1D$ BY #3 UNTIL ^! TWO L9ES -E TGR4 ,! NUMB] ": ! TWO L9ES MEET OR 9T]SECT IS ! ANSW] OR SUM4 #6_4 ,H ! /UD5T R1D ! ANSW]4 (5) ,ME?OD #3 #1_4 ,H ! /UD5T F9D ! TOP R[ (! A4I;N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] #2 9 T R[4 #3_4 ,/>T+ ON T NUMB]1 ASK ! /UD5T 6MOVE D[N T L9E #3 "TS4 #4_4 ,ASK ! /UD5T 6R1D ! ANSW]4 (5) --------------------------------------#I ,ME?OD #4 #1_4 ,H ! /UD5T F9D ! F/ COLUMN ON ! A4I;N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] #2 9 T COLUMN4 #3_4 ,/>T+ ON T NUMB] & /AY+ ON ! SAME L9E1 ASK ! /UD5T 6MOVE #3 COLUMNS 6! "R4 #4_4 ,ASK ! /UD5T 6R1D ! ANSW]4 A#I (5) ,A4I;NAL ,PRACTICE ,H ! /UD5TS "W ! FOLL[+ PRO#MS = PRACTICE4 ,!Y %D USE :I*"E ME?OD IS MO/ APPROPRIATE4 #1) #1+3 .K (4) #2) #2+4 .K (6) #3) #5+5 .K (10) #4) #7+6 .K (13) #5) #9+8 .K (17) #6) #6+4 .K (10) #7) #5+9 .K (14) #8) #9+9 .K (18) #9) #8+6 .K (14) #10) #9+3 .K (12) -------------------------------------#AJ ,Y MAY WANT 6H ! /UD5TS F9D ! SUMS :5 ! ORD] (! A45DS IS REV]S$4 #1) #3+1 .K (4) #2) #4+2 .K (6) #I #3) #5+5 .K (10) A#AJ #4) #6+7 .K (13) #5) #8+9 .K (17) #6) #4+6 .K (10) #7) #9+5 .K (14) #8) #9+9 .K (18) #9) #6+8 .K (14) #10) #9+3 .K (12) ,? CD L1D 96A 4CUS.N (! ,-MUTATIVE ,PROP]TY ( ,A4I;N4 ,IF /UD5TS >E V FAMILI> )! V>I,NS 9TRODUC$ S F>1 TRY GIV+ !M "S A4I;N PRO#MS ) MISS+ A45DS1 & H !M USE ! TA# 6F9D ! MISS+ NUMB]4 ,? IS A GD PRECURSOR 6SUBTRAC;N4 #1) ---- +3 .K #7 (4) #2) #5+ ---- .K #11 (6) #3) ---- +7 .K #17 (10) #4) ---- +2 .K #9 (7) #5) #6+ ---- .K #9 (3) #6) #8+ ---- .K #12 (4) #7) ---- +9 .K #10 (1) #8) #4+ ---- .K #6 (2) #9) #7+ ---- .K #16 (9) B#AJ #10) ---- +8 .K #13 (5) #11) #9+ ---- .K #15 (6) #12) ---- +5 .K #14 (9) -------------------------------------#AA ,US+ ! ,SUBTRAC;N ,TA#3 ,ME?ODS #1 & #2 ,SIMIL> 6A4I;N1 "! >E S"EAL WAYS 6USE ! ,SUBTRAC;N ,TA# 9 ORD] 6SUBTRACT TWO :OLE NUMB]S :5 ! M9U5D (/>T+ NUMB]) IS NO L>G] ?AN #20 &! SUBTRAH5D (NUMB] 6BE SUBTRACT$) IS NO L>G] ?AN #10_4 ,! SUBTRAH5D (NUMB] 6BE SUBTRACT$) IS LOCAT$ EI 9 ! TOP R[ OR 9 ! F/ COLUMN4 ,! M9U5D (/>T+ NUMB]) IS F.D EI 9 ! SAME R[ OR COLUMN Z ! GIV5 SUBTRAH5D (NUMB] 6BE SUBTRACT$4) ,IF ! SUBTRAH5D IS F.D 9 ! F/ COLUMN &! M9U5D IS LOCAT$ /RAI 6US+ ! ,A4I;N ,TA#1 ! #7 C 2 LOCAT$ 9 ! F/ COLUMN &! #16 9 ! SAME L9E 6! "R4 ,/AY+ 9 ! SAME COLUMN Z ! #16 & MOV+ UPW>D 6! TOP (! COLUMN IS ! NUMB] #9_4 ,AG1 US+ ? ME?OD #16-7 .K #9_4 ,',9 ! TA# 2L1 NUMB] SIGNS A#AB H BE5 OMITT$4,' 7777777777777777777777777777777777777777 + 1 2 3 4 5 6 _7 8 9 1 2 3 4 5 6 7 .8 9 10 2 3 4 5 6 7 8 .9 10 11 3 4 5 6 7 8 9 .10 11 12 4 5 6 7 8 9 10 .11 12 13 5 6 7 8 9 10 11 .12 13 14 6 7 8 9 10 11 12 .13 14 15 7 8 9 10 11 12 13 .14 15 16 8 9 10 11 12 13 14 .15 16 17 _9 .10 .AA .12 .AC .14 .15 _16 17 18 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG #AC ,',9 ! TA# 2L1 NUMB] SIGNS H #AC BE5 OMITT$4,' 7777777777777777777777777777777777777777 + 1 2 3 4 5 6 7 8 _9 1 2 3 4 5 6 7 8 9 .10 2 3 4 5 6 7 8 9 10 .11 3 4 5 6 7 8 9 10 11 .12 4 5 6 7 8 9 10 11 12 .13 5 6 7 8 9 10 11 12 13 .14 6 7 8 9 10 11 12 13 14 .15 _7 .8 .9 .10 .11 .12 .13 .14 .15 _16 8 9 10 11 12 13 14 15 16 17 9 10 11 12 13 14 15 16 17 18 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ,US+ ! ,SUBTRAC;N ,TA#3 ,ME?ODS #3 & #4 ,US+ ^! ME?ODS1 H ! /UD5T LOCATE ! SUBTRAH5D EI ON ! TOP R[ OR 9 ! F/ COLUMN4 ,/>T+ AT T NUMB]1 C.T ACR OR D[N 6! M9U5D4 ,! NUMB] ( MOVES ACR OR D[N EQUALS ! DI6];E4 -------------------------------------#AD ,= EXAMPLE 6F9D ! DI6];E ( #16-7_3 A#AD #1_4 ,F9D ! #7 ON ! TOP R[4 #2_4 ,/AY 9 ! SEV5S COLUMN1 & C.T D[N 6! #16 : IS ! M9U5D4 #3_4 ,! ANSW] IS #9_2 #16-7 .K #9_4 7777777777777777777777777777777777777777 + #1 #2 #3 #4 #5 #6 _#7 #8 #9 #1 #2 #3 #4 #5 #6 #7 _#8 #9 #10 #2 #3 #4 #5 #6 #7 #8 _#9 #10 #11 #3 #4 #5 #6 #7 #8 #9 _#10 #11 #12 #4 #5 #6 #7 #8 #9 #10 _#11 #12 #13 #5 #6 #7 #8 #9 #10 #11 _#12 #13 #14 #6 #7 #8 #9 #10 #11 #12 _#13 #14 #15 #7 #8 #9 #10 #11 #12 #13 _#14 #15 #16 #8 #9 #10 #11 #12 #13 #14 _#15 #16 #17 #9 #10 #11 #12 #13 #14 #15 _#16 #17 #18 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG #AE ,',9 ! TA# 2L1 NUMB] SIGNS H #AE BE5 OMITT$4,' 7777777777777777777777777777777777777777 + 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12 13 5 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14 15 _7 _8 _9 _10 _11 _12 _13 _14 _15 _16 8 9 10 11 12 13 14 15 16 17 9 10 11 12 13 14 15 16 17 18 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG ,H ! /UD5T USE EI ,ME?OD #3 OR #4 63FIRM T #16-7 .K #9_4 -------------------------------------#AF ,GUID$ ,PRACTICE ,H ! /UD5T PRACTICE F9D+ ! DI6];E ( #13-6 US+ "O OR M (! ME?ODS 4CUSS$4 ,ME?OD #1 #1_4 ,H ! /UD5T F9D ! TOP R[ (! TA#4 #2_4 ,ASK ! /UD5T 6F9D ! SUBTRAH5D1 #6, 9 T R[4 A#AF #3_4 ,H ! /UD5T MOVE D[N ! COLUMN H1D$ BY #6 UNTIL ! M9U5D1 #13, IS LOCAT$4 #4_4 ,H ! /UD5T MOVE 6! LEFT ACR ! L9E 6! F/ COLUMN4 #5_4 ,ASK ! /UD5T 6R1D ! NUMB] 9 ! F/ COLUMN (7)_4 ,ME?OD #2 #1_4 ,H ! /UD5T F9D ! F/ COLUMN 6! LEFT ON ! TA#4 #2_4 ,ASK ! /UD5T 6F9D ! SUBTRAH5D1 #6, 9 T COLUMN4 -------------------------------------#AG #3_4 ,H ! /UD5T MOVE ACR ! L9E UNTIL ! M9U5D1 #13, IS LOCAT$4 #4_4 ,H ! /UD5T MOVE UP ! SAME COLUMN T ! #13 IS 9 UNTIL ! NUMB] AT ! TOP (! COLUMN IS LOCAT$4 #5_4 ,ASK ! /UD5T 6R1D ! NUMB] 9 ! TOP R[ (7)_4 ,ME?OD #3 #1_4 ,H ! /UD5T F9D ! TOP R[ (! TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] #6 #AG 9 T R[4 A#AG #3_4 ,/>T+ ON T NUMB]1 ASK ! /UD5T 6MOVE D[N NUMB] 0NUMB] UNTIL #13 IS LOCAT$4 #4_4 ,ASK ! /UD5T H[ _M "TS HE OR %E _H 6MOVE D[N 6A NEW NUMB] 2F LOCAT+ ! #13_4 (7) #5_4 ,! /UD5T %D ANSW] #7_2 #13-6 .K #7_4 ,ME?OD #4 #1_4 ,H ! /UD5T F9D ! F/ COLUMN FUR!/ 6! LEFT ON ! TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] #6 9 T COLUMN4 #3_4 ,/>T+ ON T NUMB]1 ASK ! /UD5T 6MOVE ACR ! L9E NUMB] 0NUMB] UNTIL #13 IS LOCAT$4 -------------------------------------#AH #4_4 ,ASK ! /UD5T H[ _M "TS HE OR %E _H 6MOVE ACR 6A NEW NUMB] 2F LOCAT+ ! #13_4 (7) #5_4 ,! /UD5T %D ANSW] #7_2 #13-6 .K #7_4 ,A4I;NAL ,PRACTICE A#AH #1) #6-1 (5) #2) #4-2 (2) #3) #8-2 (6) #4) #10-5 (5) #5) #12-4 (8) #6) #12-3 (9) #7) #10-6 (4) #8) #17-8 (9) #9) #18-10 (8) ,IF /UD5TS >E V FAMILI> )! V>I,NS 9TRODUC$ S F>1 TRY GIV+ !M "S SUBTRAC;N PRO#MS ) MISS+ M9U5DS OR SUBTRAH5DS1 & H !M USE ! TA# 6F9D ! MISS+ NUMB]4 #1) ---- -#3 .K #4 (7) #2) ---- -#6 .K #4 (10) #3) #19- ---- .K #9 (10) #4) #15- ---- .K #8 (7) #5) ---- -#4 .K #5 (9) #6) #20- ---- .K #10 (10) #7) #9- ---- .K #7 (2) #8) ---- -#4 .K #7 (11) #AI #9) #15-8 .K ---- (7) B#AH #10) ---- -#8 .K #9 (17) #11) #16- ---- .K #9 (7) #12) ---- -#7 .K #7 (14) ,NOTES3 #AI #BA ,,APH^R ,AM]ICAN ,PR9T+ ,H\SE =! ,BL1 ,9C4 ,COPY"R ^C #2010 #1839 ,FRANK=T ,AV5UE ,P4,O4 ,BOX #6085 ,L\ISVILLE1 ,K5TUCKY #40206-0085 ,PH"O3 #502-895-2405 ,TOLL ,FREE3 #800-223-1839 ,FAX3 #502-899-2274 ,EMAIL3 _+INFO@APH.ORG_: _+WWW.APH.ORG_: ,"P ,NUMB] #7-82699-00 = USE 9 ,CATALOG ,NUMB] #5-82699-00