,MULTIPLIC,N @& ,DIVI.N ,GUIDEBOOK ,,APH^R ,COPY"R ^C #BJAA ,AM]ICAN ,PR9T+ ,H\SE =! ,BL ,9 #A ,VOLUME PAGES ;I-;III & #A-BJ ,,SPECIAL ,,SYMBOLS ,,US$ ,,9 ,,? ,,VOLUME ,-PUT] ,BRL ,CODE _+ 2G9 ,-PUT] ,BRL ,CODE _: 5D ,-PUT] ,BRL ,CODE @ 7#D7 AT SIGN . 7#DF7 P]IOD ,O!R ,SYMBOLS @& AMP]S& SYMBOL ^R REGI/]$ TRADEM>K SYMBOL ^C COPY"R SYMBOL _/ SLA% SYMBOL _ BOLDFACE 9DICATOR ,,TRANSCRIB]'S ,,NOTES ;III ,? VOLUME IS TRANSCRIB$ AC 6! RULES ( ..,! ,NEME? ,BRL ,CODE = ,MA!MATICS & ,SCI;E .,NOT,N #1972 ,REVI.N1 #2007- #2011 UPDATES4 ,MULTIPLIC,N @& ,DIVI.N #A ,GUIDEBOOK ,,APH^R ,MULTIPLIC,N @& ,DIVI.N ,TA# ,9TRODUC;N ,! ,MULTIPLICATION_/,DIVISION ,TA# C 2 US$ 6F9D ! PRODUCT ( TWO :OLE NUMB]S F #1 "? #10 OR ! QUOTI5T (A RELAT$ DIVI.N PRO#M4 ,X RE9=CES ! BASIC MULTIPLIC,N & DIVI.N FACTS4 ,= MULTIPLIC,N1 "O FACTOR IS LOCAT$ ON ! TOP (! *>T &! O!R IS LOCAT$ ON ! LEFT4 ,! PO9T AT : ! TWO FACTORS -E TGR ON ! *>T IS ! PRODUCT4 ,3V]SELY1 ! *>T C 2 US$ = DIVI.N4 ,! TA# C 2 US$ 0/UD5TS 6*ECK _! "W AF !Y H PRODUC$ A TA# OR A "P (A TA# ON _! [N OR H "W$ A PRO#M OR A GR\P ( PRO#MS !MVS4 ,! TA# C AL 2 US$ 0/UD5TS 64COV] PATT]NS PRODUC$ 0! V>I\S FACTS1 I4E41 :5 "O (! FACTORS IS #2, ALL (! PRODUCTS >E EV5 NUMB]S2 :5 "O (! FACTORS IS #5, ALL (! PRODUCTS 5D 9 EI #0 OR #5_2 :5 ( ACTORS IS #9, ! TWO DIGITS 9 ! PRODUCT1 MAKE A SUM ( #9_4 ,X IS RECOMM5D$ T ? *>T 2 US$ ONLY 0/UD5TS :O H _H REP1T$ OPPORTUNITIES 6USE 3CRETE MANIPULATIVES & SEMICONCRETE DIAGRAMS & DRAW+S ( _! [N 6SOLVE MULTIPLIC,N & DIVI.N PRO#MS4 ,= MO/ /UD5TS1 ? WD O3UR >.D ! ?IRD GRADE4 ,PURPOSE ,! 5CLOS$ ,MULTIPLICATION_/,DIVISION ,TA# C 2 US$ 6F9D ! PRODUCT ( TWO :OLE NUMB]S F #1 "? #10_4 ,LIKEWISE1 ! TA# C 2 US$ 6F9D ! QUOTI5T : IS A :OLE NUMB] F #1 "? #10 :5 ! DIVID5D IS GIV5 &! DIVISOR IS AL A :OLE NUMB] F #1 "? #10_4 ,US+ ! TA# 6DET]M9E ! PRODUCT OR QUOTI5T PROVIDES PRACTICE & RE9=CES LE>N+ ! BASIC MULTIPLIC,N & DIVI.N FACTS4 ,US+ ! ,MULTIPLIC,N ,TA#3 ,ME?ODS #1 & #2 ,"! >E S"EAL WAYS 6USE ! ,MULTIPLIC,N ,TA# 9 ORD] 6MULTIPLY TWO NUMB]S4 ,! F/ NUMB] C 2 F.D ACR ! TOP R[ &! SECOND B] D[N ! F/ COLUMN OR VICE-V]SA4 ,9 #C EI CASE1 ! ANSW] OR ! PRODUCT (! TWO NUMB]S IS ! PO9T ": ! R[ &! COLUMN MEET :5 Y FOLL[ ! NUMB] AT ! TOP (! F/ R[ /RAIE MULTIPLI$ DOES N A6ECT ! PRODUCT1 I4E41 A@*B .K B@*A_4 ,',THE FOLL[+ MULTIPLICA;N TA#S H BE5 REPRODUC$ WITH\T COLUMN OR R[ H1D+S4,' @* _ #1 #2 #3 #4 .#5 33333333333333333333333333333 #1 _ #1 #2 #3 #4 #5 #2 _ #2 #4 #6 #8 #10 .#3 _ #3 #6 #9 #12 .#15 #4 _ #4 #8 #12 #16 #20 #5 _ #5 #10 #15 #20 #25 @* _ #1 #2 .#3 #4 #5 33333333333333333333333333333 #1 _ #1 #2 #3 #4 #5 #2 _ #2 #4 #6 #8 #10 #3 _ #3 #6 #9 #12 #15 #4 _ #4 #8 #12 #16 #20 .#5 _ #5 #10 .#15 #20 #25 ,= /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 IS DESCRIB$ 2L4 ,US+ ! ,MULTIPLIC,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 ! #E L9E ! SAME NUMB] ( "TS Z YR SECOND NUMB] 6BE MULTIPLI$4 ,= EXAMPLE1 6F9D #5@*3_3 #1_4 ,F9D ! #5 9 ! TOP R[4 #2_4 ,/AY 9 ! FIVES COLUMN1 & MOVE D[N #3 "TS4 #3_4 ,! ANSW] IS #15_2 #5@*3 .K #15_4 @* _ #1 #2 #3 #4 .#5 333333333333333333333333 #1 _ #1 #2 #3 #4 .#5 #2 _ #2 #4 #6 #8 .#10 #3 _ #3 #6 #9 #12 .#15 #4 _ #4 #8 #12 #16 #20 #5 _ #5 #10 #15 #20 #25 @* _ #1 #2 #3 #4 #5 333333333333333333333333333 #1 _ #1 #2 #3 #4 #5 #2 _ #2 #4 #6 #8 #10 #3 _ #3 #6 #9 #12 #15 #4 _ #4 #8 #12 #16 #20 .#5 _ .#5 .#10 .#15 #20 #25 ,H ! /UD5T USE EI ,ME?OD #3 OR #4 63FIRM T #3@*5 AL EQUALS #15_2 #5@*3 .K #3@*5_4 ,J Z ) ME?ODS #1 & #2, ? C L1D 6A 4CUS.N (! ,-MUTATIVE ,PROP]TY ( ,MULTIPLIC,N4 ,GUID$ ,PRACTICE ,H ! /UD5T PRACTICE F9D+ ! PRODUCT ( #2@*3 US+ "O OR M (! ME?ODS 4CUSS$4 ,ME?OD #1 #1_4 ,H ! /UD5T F9D ! TOP R[ (! MULTIPLIC,N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] #2 9 T R[4 #3_4 ,H ! /UD5T F9D ! F/ COLUMN FUR!/ 6! LEFT4 #4_4 ,ASK ! /UD5T 6F9D ! NUMB] #3 9 T COLUMN4 #5_4 ,H ! /UD5T MOVE D[N ! R[ T IS H1D$ BY 8#2_0 & ACR ! COLUMN T />TS ) 8#3_0 UNTIL ^! TWO L9ES -E TGR4 ,! NUMB] ": ^! L9ES MEET OR 9T]SECT IS ! ANSW] OR PRODUCT4 #G #6_4 ,H ! /UD5T R1D ! ANSW]4 (6) ,ME?OD #2 #1_4 ,H ! /UD5T F9D ! F/ COLUMN FUR!/ 6! LEFT ON ! MULTIPLIC,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 T R[4 #5_4 ,H ! /UD5T MOVE ACR ! COLUMN T />TS ) 8#2_0 & D[N ! R[ T IS H1D$ BY 8#3_0 UNTIL ^! TWO L9ES -E TGR4 ,! NUMB] ": ! TWO L9ES MEET OR 9T]SECT IS ! ANSW] OR PRODUCT4 #6_4 ,H ! /UD5T R1D ! ANSW]4 (6) ,ME?OD #3 #1_4 ,H ! /UD5T F9D ! TOP R[ (! MULTIPLIC,N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] 8#2_0 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 (6) ,ME?OD #4 #1_4 ,H ! /UD5T F9D ! F/ COLUMN ON ! MULTIPLIC,N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] 8#2_0 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 (6) ,A4I;NAL ,PRACTICE ,H ! /UD5TS "W ! FOLL[+ PRO#MS = PRACTICE4 ,!Y %D USE :I*"E ME?OD IS EASIE/ = !M OR :I*"E Y RECOMM5D4 #1) #1@*3 .K (3) #2) #2@*4 .K (8) #3) #5@*5 .K (25) #4) #7@*6 .K (42) #5) #9@*8 .K (72) #6) #6@*4 .K (24) #7) #5@*9 .K (45) #8) #9@*10 .K (90) #I #9) #8@*6 .K (48) #10) #9@*3 .K (27) ,EXT5D$ ,ACTIVITIES #1_4 ,IF Y WI% & IF "T ALL[S1 Y MAY WANT 6H ! /UD5TS F9D ! PRODUCTS :5 ! ORD] (! MULTIPLI]S IS REV]S$4 #1) #3@*1 .K (3) #2) #4@*2 .K (8) #3) #5@*5 .K (25) #4) #6@*7 .K (42) #5) #8@*9 .K (72) #6) #4@*6 .K (24) #7) #9@*5 .K (45) #8) #10@*9 .K (90) #9) #6@*8 .K (48) #10) #3@*9 .K (27) ,? CD L1D 96A 4CUS.N (! ,-MUTATIVE ,PROP]TY ( ,MULTIPLIC,N4 #2_4 ,IF /UD5TS >E V FAMILI> )! V>I,NS 9TRODUC$ S F>1 TRY GIV+ !M "S MULTIPLIC,N PRO#MS ) MISS+ MULTIPLIC&S OR MISS+ MULTIPLI]S1 & H !M USE ! TA# 6F9D ! MISS+ NUMB]4 ,? IS A GD PRECURSOR 6DIVI.N4 #1) ---- @*3 .K #12 (4) #2) #5@* ---- .K #30 (6) #3) ---- @*7 .K #49 (7) #4) ---- @*2 .K #14 (7) #5) #6@* ---- .K #18 (3) #6) #8@* ---- .K #48 (6) #7) ---- @*9 .K #27 (3) #8) #4@* ---- .K #8 (2) #9) #7@* ---- .K #63 (9) #10) ---- @*8 .K #40 (5) #11) #9@* ---- .K #54 (6) #12) ---- @*5 .K #50 (10) #3_4 ,? TA# C AL 2 US$ 64CUSS E/IM,N4 ,IF ! MULTIPLIC& OR ! MULTIPLI] IS A :OLE NUMB] F #1 "? #10 &! O!R IS EI A FRAC;N OR A DECIMAL 2T #1 & #10 (GRT] ?AN #1 & LESS ?AN #10), !N ! TA# C 2 US$ 6PRACTICE & RE9=CE ! IDEA ( E/IMAT+ AN ANSW]4 ,= EXAMPLE1 6E/IMATE #2.5@*8, LOCATE ! PRODUCT ( _#2@*8 (16) &! #AA PRODUCT ( _#3@*8 (24) S9CE #2.5 IS 2T ! :OLE NUMB]S #2 & #3_4 ,! PRODUCT ( #2.5@*8, !N1 IS "S": 2T #16 & #24_4 ,? CD L1D 6A 4CUS.N (! IMPORT.E ( E/IMAT+ OR HAV+ "S IDEA ( :AT ! ANSW] 6A PRO#M %D BE4 ,PRACTICE E/IMAT+ ! FOLL[+3 #1) #6@*7.3 (42-48) #2) #4@*8_?1_/8_# (32-36) #3) #8.9@*7 (56-63) #4) #3_?3_/4_#@*3 (9-12) #5) #7@*2_?1_/2_# (14-21) #6) #5@*6.6 (30-35) #7) #2.2@*3 (6-9) #8) #4_?1_/3_#@*8 (32-40) #9) #6_?5_/8_#@*5 (30-35) #10) #8@*8.4 (64-72) ,US+ ! ,DIVI.N ,TA#3 ,ME?ODS #1 & #2 ,SIMIL> 6MULTIPLIC,N1 "! >E S"EAL WAYS 6USE ! ,DIVI.N ,TA# 9 ORD] 6DIVIDE TWO :OLE NUMB]S :5 ! DIVID5D IS NO L>G] ?AN #100 &! DIVISOR IS NO L>G] ?AN #10_4 ,! DIVISOR IS LOCAT$ EI 9 ! TOP R[ OR 9 ! F/ COLUMN4 ,! DIVID5D IS F.D EI 9 ! SAME R[ Z ! GIV5 DIVISOR OR 9 ! SAME COLUMN Z ! GIV5 DIVISOR4 ,IF ! DIVISOR IS F.D 9 ! F/ COLUMN &! DIVID5D IS LOCAT$ /RAI 6US+ ! ,MULTIPLIC,N TA#1 ! #8 C 2 LOCAT$ 9 ! F/ LUMN &! #16 9 ! SAME SAME L9E 6! #AC "R4 ,/AY+ 9 ! SAME COLUMN Z ! #16 & MOV+ UPW>D 6! TOP (! COLUMN IS ! NUMB] #2_4 ,AG1 US+ ? ME?OD1 #16./8 .K #2_4 @* _ #1 #2 #3 #4 #5 #6 #7 .#8 3333333333333333333333333333333333333 #1 _ #1 #2 #3 #4 #5 #6 #7 #8 .#2 _ #2 #4 06 #8 #10 #12 #14 .#16 #3 _ #3 #6 #9 #12 #15 #18 #21 #24 #4 _ #4 #8 #12 #16 #20 #24 #28 #32 #5 _ #5 #10 #15 #20 #25 #30 #35 #40 #6 _ #6 #12 #18 #24 #30 #36 #42 #48 #7 _ #7 #14 #21 #28 #35 #42 #49 #56 #8 _ #8 #16 #24 #32 #40 #48 #56 #64 _ #1 .#2 #3 #4 #5 #6 #7 #8 3333333333333333333333333333333333333 #1 _ #1 #2 #3 #4 #5 #6 #7 #8 #2 _ #2 #4 #6 #8 #10 #12 #14 #16 #3 _ #3 #6 #9 #12 #15 #18 #21 #24 #4 _ #4 #8 #12 #16 #20 #24 #28 #32 #5 _ #5 #10 #15 #20 #25 #30 #35 #40 #6 _ #6 #12 #18 #24 #30 #36 #42 #48 #7 _ #7 #14 #21 #28 #35 #42 #49 #56 .#8 _ #8 .#16 #24 #32 #40 #48 #56 #64 ,US+ ! ,DIVI.N ,TA#3 ,ME?ODS #3 & #4 ,US+ ^! ME?ODS1 H ! /UD5T LOCATE ! DIVISOR EI ON ! TOP R[ OR 9 ! F/ COLUMN4 ,/>T+ AT T NUMB]1 C.T ACR OR D[N 6! DIVID5D4 ,! NUMB] ( MOVES ACR OR D[N EQUALS ! QUOTI5T4 ,= EXAMPLE 6F9D ! QUOTI5T ( #16./8_3 #1_4 ,F9D ! #8 ON ! TOP R[4 #2_4 ,/AY 9 ! EIT+ ON T NUMB]1 ASK ! /UD5T 6MOVE D[N NUMB] 0NUMB] UNTIL #24 IS LOCAT$4 #4_4 ,ASK ! /UD5T H[ _M "TS HE OR %E _H 6MOVE D[N 6A NEW NUMB] 2F LOCAT+ ! #24_4 (4) #5_4 ,! /UD5T %D ANSW] 8#4_20 #24./6 .K #4_4 ,ME?OD #4 #1_4 ,H ! /UD5T F9D ! F/ COLUMN FUR!/ 6! LEFT ON ! DIVI.N TA#4 #2_4 ,ASK ! /UD5T 6F9D ! NUMB] 8#6_0 9 T COLUMN4 #3_4 ,/>T+ ON T NUMB]1 ASK ! /UD5T 6MOVE ACR ! L9E NUMB] 0NUMB] UNTIL #24 IS LOCAT$4 #4_4 ,ASK ! /UD5T H[ _M "TS HE OR %E _H 6MOVE ACR 6A NEW NUMB] 2F LOCAT+ ! #24_4 (4) #5_4 ,! /UD5T %D ANSW] 8#4_20 #24./6 .K #4_4 ,A4I;NAL ,PRACTICE #1) #6./1 (6) #2) #4./2 (2) #3) #8./2 (4) #4) #10./5 (2) #5) #16./4 (4) #6) #21./3 (7) #7) #42./6 (7) #8) #64./8 (8) #9) #81./9 (9) #10) #90./9 (10) ,EXT5D$ ,ACTIVITIES #AI #1_4 ,IF Y WI% & IF "T ALL[S1 Y MAY WANT 6H ! /UD5TS F9D ! NEW QUOTI5TS :5 ! ORIG9AL QUOTI5TS & DIVISORS >E EX*ANG$ 9 ! PRO#MS ABV4 #1) #6./6 (1) #2) #4./2 (2) #3) #8./4 (2) #4) #10./2 (5) #5) #16./4 (4) #6) #21./7 (3) #7) #42./7 (6) #8) #64./8 (8) #9) #81./9 (9) #10) #90./10 (9) #2_4 ,IF /UD5TS >E V FAMILI> )! V>I,NS 9TRODUC$ S F>1 TRY GIV+ !M "S DIVI.N PRO#MS ) MISS+ DIVID5DS OR DIVISORS1 & H !M USE ! TA# 6F9D ! MISS+ NUMB]4 #1) ---- ./3 .K #4 (12) #2) ---- ./6 .K #4 (24) #3) #36./ ---- .K #9 (4) #4) #18./ ---- .K #6 (3) #5) ---- ./4 .K #5 (20) #6) #30./ ---- .K #6 (5) #7) #35./ ---- .K #7 (5) #8) ---- ./4 .K #7 (28) #9) #56./8 .K ---- (7) #10) ---- ./8 .K #9 (72) #11) #63./ ---- .K #9 (7) #12) ---- ./7 .K #7 (49) ,,NOTES ,,APH^R #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] #17-125-010 ,= USE 9 ,CATALOG ,NUMB] #5-82700-01