ILLUSTRATING INTEGERSThe University of Texas at DallasINTRODUCTION TO INTEGERSThe University of Texas at DallasINTRODUCTION TO INTEGERSIntegers are positive and negative numers!"# $%# $ $'# $(# $)# $*# +# ,*# ,)# ,(# ,'# , ,%# "Ea-h negative numer is paired .ith a positive numer the same distan-e from + on a numer line!The University of Texas at Dallas$( $) $* ) + * (INTRODUCTION TO INTEGERS/e -an represent integers using red and yello. -ounters!Red tiles .ill represent negative integers# and yello. tiles .ill represent positive integers!The University of Texas at DallasNegative integer0ositive integerINTRODUCTION TO INTEGERSThe diagrams elo. sho. ) .ays to represent $(!Represent $( in ) more .ays!The University of Texas at Dallas$( $(NOTE TO TUTORThe University of Texas at DallasINTRODUCTION TO INTEGERSStudents may represent $( y using ( red tiles and any numer of groups of ,* and $* tilesINTRODUCTION TO INTEGERSTell .hi-h integer ea-h group of tiles represents!The University of Texas at Dallas$),*ANSWER,)INTRODUCTION TO INTEGERSIf there are the same numer of red tiles as yello. tiles# .hat numer is represented1It represents +!The University of Texas at Dallas2DDITION 2ND SU3TR2CTIONThe University of Texas at Dallas2DDING INTEGERS/e -an model integer addition .ith tiles!Represent $) .ith the fe.est numer of tilesRepresent ,& .ith the fe.est numer of tiles!The University of Texas at Dallas2DDING INTEGERS/hat numer is represented y -omining the ) groups of tiles1/rite the numer senten-e that is illustrated!$) , ,& 4 ,(The University of Texas at Dallas,(2DDING INTEGERSUse your red and yello. tiles to find ea-h sum!$) , $( 4 1The University of Texas at DallasANSWER,4$& $) , $( 4 $&2DDING INTEGERSThe University of Texas at Dallas $% , ,) 4 1,4$ ',4,* $( , ,' 4 1ANSWERANSWER $% , ,) 4 $' $( , ,' 4 ,*Adding Integers - Same Sign/e -an sho. this same idea using a numer line!-5 5 0 10 -10/hat is & , '15ove five 6&7 units to the right from 8ero!No. move four more units to the right!The final point is at 9 on the numer line!Therefore# & , ' 4 9!9Adding Integers - Same Sign-5 5 0 10 -10/hat is $& , 6$'715ove five 6&7 units to the left from 8ero!No. move four more units to the left!The final point is at $9 on the numer line!Therefore# $& , 6$'7 4 $9!-9Additive InverseWhat is (-7) + 7?-5 5 0 10 -10To sho. this# .e -an egin at 8ero and move seven units to the left!No.# move seven units to the right!Noti-e# .e are a-: at 8ero 6+7!;or every positive integer on the numer line# there is a -orresponding negative integer!These integer pairs are opposites or additive inverses!Additive Inverse Propert < ;or every numer a# a , 6$a7 4 +1Additive Inverse/hen using algera -hips# the additive inverses ma:e .hat is -alled a 8ero pair!;or example# the follo.ing is a 8ero pair!-1 ! + (-!) " #$Adding Integers - Different SignsAdd the %o&&o'in( inte(ers) (-*) + 7$-5 5 0 10 -10Start at 8ero and move four units to the left!No. move seven units to the right!The final position is at (!Therefore# 6$'7 , = 4 +!Adding Integers - Different SignsAdd the %o&&o'in( inte(ers) (-*) + 7$-5 5 0 10 -10Noti-e that seven minus four also e>uals three!In our example# the numer .ith the larger asolute value .as positive and our solution .as positive!?et@s try another one!Adding Integers - Different Signs-5 5 0 10 -10Add (-9) + +Start at 8ero and move nine pla-es to the left!No. move three pla-es to the right!The final position is at negative six#6$%7!Therefore# 6$97 , ( 4-,!Adding Integers - Different Signs-5 5 0 10 -10Add (-9) + +In this example# the numer .ith the larger asolute value is negative! The numer .ith the smaller asolute value is positive!/e :no. that 9$( 4 %!Ao.ever# 6$97 , ( 4 $%!% and uations using the same ( numers!;or exampleG = , F 4 *&F , = 4 *&*& < = 4 F*& < F 4 =2ll of these statements are true!SU3TR2CTING INTEGERSThe University of Texas at Dallas/e -an also use fa-t family .ith integers!Use your red and yello. tiles to verify this fa-t familyG$( , ,F4 ,&,F , $( 4 ,&,& $ , F 4 $(,& $ $ ( 4 ,FINTEGERS 2ND 5U?TI0?IC2TIONThe University of Texas at Dallas5U?TI0?IC2TIONRed and yello. tiles -an e used to model multipli-ation! Rememer that multipli-ation -an e des-ried as repeated addition!So ) x ( 4 1The University of Texas at Dallas) groups of ( tiles 4 % tiles5U?TI0?IC2TION) x $( means ) groups of $(The University of Texas at Dallas) x $( 4 $%5U?TI0?IC2TIONSin-e ) x ( 4 % and ( x ) 4 %# does it ma:e sense that $( x ) 4 $% 1 $( x ) 4 $% is true! ,) x $( 4 $% and $( x ,) 4 $% elong to a fa-t familyGThe University of Texas at Dallas,) x $( 4 $%$( x ,) 4 $%$% H ,) 4 $($% H $(4 ,)5U?TI0?IC2TIONUse your tiles to model ea-h multipli-ation prolem!,) x ,' 4 1The University of Texas at DallasANSWER) groups of ,',) x ,' 4 ,F5U?TI0?IC2TION,( x $( 4 1The University of Texas at DallasANSWER( groups of $(,( x $( 4 $95U?TI0?IC2TION$) x ,' 4 1The University of Texas at DallasANSWER' groups of $)$) x ,' 4 $FUse the fa-t family for$) x ,' 4 1 /e -an@t sho. $) groups of ,',' x $) 4 1 .e -an sho. ' groups of $)5U?TI0?IC2TION,*# $* are oppositesthe produ-ts are oppositeSin-e ,) and $) are opposites of ea-h other# ,) x $( and$) x $( have opposite produ-ts!The University of Texas at Dallas,* x ,( 4 ,($* x ,( 4 $(5U?TI0?IC2TIONTo model $) x $( use ) groups of the opposite of $(The University of Texas at Dallas$) x $( 4 ,%5U?TI0?IC2TIONUse tiles to illustrate $( x $' 41The University of Texas at DallasANSWER( groups of ,' 6the opposite of $'7$( x $' 4 ,*)INTEGERS 2ND DIIISIONThe University of Texas at DallasDIIISIONUse tiles to model ,*) H ,( 4 1The University of Texas at DallasDivide *) yello. tiles into ( e>ual groups,*) H ,( 4 ,'' yello. tiles in ea-h group!DIIISIONUse tiles to model $*& H ,& 41The University of Texas at DallasDivide $*& into & e>ual groups$*& H ,& 4 $(DIIISION/rite the fa-t family$' x ,) 4 $FCompare your ans.er!,) x $' 4 $F$F H ,) 4 $'$F H $' 4 ,)The University of Texas at DallasDIIISIONNo. try theseE Use red and yello. tiles to find ea-h ans.er!*7 $) , $F &7 ,( x $')7 ,% , $' %7 $& x $)(7 $& $ ,' =7 $*) H ,%'7 $% $ $= F7 $F H ,)The University of Texas at DallasANSWERDIIISIONThe University of Texas at Dallas*7 $) ,$F4 $*+)7 ,% ,$'4 ,)DIIISIONThe University of Texas at Dallas(7 $& $,'4 $9'7 $% $$=4 ,*DIIISIONThe University of Texas at Dallas&7 ,(x $' 4 $*)%7 $&x $)4 ,*+& groups of the opposite of $)DIIISIONThe University of Texas at Dallas=7 $*) H ,% 4 $)F7 $FH,)4 $'$F divided into ) e>ual groups$*) divided into % e>ual groups