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
