basic math again
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BASIC MATHEMATICS
COURSE WORKPOLYAS MODEL
STUDENTS NAME : ZAHIDAH BINTI MD YUSSOF
I/C NUMBER : 900407-06-!"4
INDE# NUMBER :$00%$!$"!0046
&ROUP : PPISMP MATHEMATIS COHORT '(ULY0% INTAKE)
SEMESTER : TWO
LECTURERS NAME : PN NOOR KHALIZA BINTI MOHD KHAIRUDDIN
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ACKNOWLED&EMENT
F*+, . 112 I 3.15 1* . 8 . &.5 ;, I 8+.1? +=+5*= . Polyas Model,;;,,11@ A+ 112 I
?= . **,8 ,;8 * 5**;1 ,
I 1,. 3.15 1* . 8 . ?@ ,*; ?8?*; 1;++2 P K81* 38.
=1*. . 8*, , N. .+=. ?@ 1.
??+,2 F+82 N*882H, 5 (,?* I 1,. 3 . 8 . ?@
,*.+ 8 13@, 81> ? 5 ?@ =+.> ??+, * .+5+ . **,8 8*, , 5
=* >1*. 18.=8 8@ 1,. 8.>1 38. + *
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CHAPTER !: INTRODUCTION
C.?>1 ?8?*;1 ,, ,;8 , >+.1? ,.1+.>.+**, . 5 8*=8+ .+5+ ?8?*;1 >+.;,,,
,;8 , +>+,*.2 ,+;*.2 5 =+1**. A ,;**= ,>; .
8? 8.=8 *, 8 *1*@ . =+1* +.? ,>;**; >+*;, 5 . .+?
3 5 ?.+ ,+; ;.;>, T8 P+*;*>1, 5 S5+5, .+ S;8..1
M8?*;, 'N*.1 C.;*1 . T;8+, . M8?*;,2 $000) ;11, .+
*,+;*.1 >+.=+?, 8 ?>8,* >+.1? ,.1*=
,5, 5 ,.>8*,*;*. 3*8 ?8?*;1 >+.;,,, ,;8 ,
+>+,*.2 ?8?*;1 +,.*=2 ,+;*.2 5 =+1**. I =.,
. . >+.;1*? 8 ,5, ,8.15 5 *;+,5 ,.>8*,*;*. 3*8?8?*;1 >+.;,,,2 ,>;*11@ >+.1? ,.1+,*.2 5
+,.*=2 5 8*+ *;+,5 *1*@ . +1; . 5 ?.*.+ 8*+ 3.+ ,8.15
15 . =++ ,+;*. 5 ;>*1*@ .+ =+1**. T8,2 8 *1*@ .
=+1* *, 8 +,1 . ;+* ?8?*;1 >+*;, 5 *, *?>.+
;.?>. . ?8?*;1 *1*@ T8 5? . 8*, *1*@ *, .;*1 >+.1?-,.1 * 8*+ *1**, . .+?1 >+.1?,
,>..,1@2 8*+ 1**1*@ * 5 ?=?2 5 8*+ *1*@ . ,+;
5 =+1* T8+ *, 1,. ?>*+*;1 +,;8..1 1
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CHAPTER $: LITERATURE REIEW
A 311-.3 >+.1?-,.1+.+.1?-,.1+.+?; . .8,*, . 8+*,*;
,*11, 38*1 *=.+*= J?=+*1 ,*11, ;,,+@ . +=1 ., ;*1*=2
*?>1?*=2 5 1..*= ; 3+ +1115 , .+**.2 .+=**.2
;*.2 5 ;*+8,*. ,+=*,2
1@,*, . *.+?*.2 ***1 5 ,, +>+,*.2 5 ,,,,? .
11*=2 5 1.;1 >1*= E;*. ++, . +=1*. .
81, I *;15, >+.+?; . 1.;1 ;*.,2
?.*.+*= >+.=+,, 5 ;.,*,;@ . 1.;1 >1,2 5 +5-. 5;*,*.,
',>5
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;8+;+*,*;, 8 ?@ *1; >+.+?; T, +.;,, F.+
?>12 *5*+.+?;
F*11@2 ,+=@ +8,*.2 .+=**=2 ;*= >1,2 5 ;8;*= 5
+*;1+ .; .
,, ;*., A,+;*. *, >+.;,, 8 .;;+, 38 8 ,;
.;,, *. . ,>;**; >+.>+*, . =*+.>+*, * *,.1*. +.? 8 .+*=*1 T8*, ?*=8 5. . 5+,5
8 ,,; . ;+* >8.? 5 . 1+ >>1@ 8 ,? 8.+@ . .8+
>>1*;1 ;,,
E+1@ +,+;8 . =+1**. .;,5 . 1?+@ ,;8..1 ;8*15+,
*1**, . =+1* ?+ ;.;>, 'D+.;,, . =+1**.
?.= ?8?*;, 5;*. +,+;8+, * 8 .+?+ S.
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Shapiro'!96) 3+. 8 ?.= ?8?*;11@ =*5 ,5,2 8
5? . =+1**., .;;+, +.? 8 *+, ?>1,2 8 ***1
,=, . 1+*= T+,+ * 8 =+1 .+? *, 1?., ?+=5 * *? 3*8
=+1**. 5 *, ;;.?>1*,85 *??5*1@ .+ 38.1 ;1,, . >+.1?, .
,*=1 @> ?.= 1,, ;>1 ,5, =+1**., +*> =+511@ 5
+ ?*,5 1+ ,=, .+ 8@ 5. . 5 11
Krutetskii'!976) 1@5 8 =+1**. *1*@ . .8 J.+?1J 5
J;>1J '=*5) ,5, * ,+*, . >+*?, H 8@>.8,*5 8
J,5, 3*8 5*+ *1**, + ;8+;+*5 @ 5*+;, * 5=+ .
5? . .8 8 *1*@ . =+1* ?8?*;1 ?+*1 5 8
*1*@ . +??+ =+1**.,J K+,** ,5*5 !9 ,5, 3*8 +*?, 3*8 8 !9 ,5,2 K+,**
;.;155 8 ?.+ J;>1J '=*5) ,5, 3+ 1 . .+? ?8?*;1
=+1**., +>*51@ 5 +.51@ H .5 8 8, J;>1J ,5, 3+
1 . 5*,;+ 8 =+1 ,+;+ . 8 >+.1?, .+ 8@ ,.11J ,5, *+5 >..+1@ * 8*, , I .+5+ .+,5, . .+?1 =+1**., ;.++;1@2 8@ 85 . ,+; +.? 8
,>;**; ;. 5 ,*=1 . ,*?*1+**,2 ,+;+,2 5 +1*. ,8*>,
M., . 8 *, 1*++ . 8 >+.;,, . =+1**. *, * 8
;. . ?+ ;.;>,2 +*8?*;2 5 1=+ T8+ *, 1; . +,+;8 .
=+1**. * 8 ;. . 8*=8+ .+5+ ?8?*;1 >+.;,,, ,;8 ,
>+.1? ,.1+.1?-,.1
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CHAPTER ": METHODOLO&Y
A, >+ . 8*, 3.+ . >+.1? ,.1 >+.1?-
,.1+.;,,, 38*;8 + 5+,5 8 >+.1?2 512 ;++@ . 8 >1
5 1.. ;
1. Understand the prole!" #See$
M@ ,5, + .+.1? A 8*, >.* =+.>
5*,;,,*. *, *;*1 T8 ,*., 8 ?@ 81> . 5+,5 8 >+.1?
+:
W8 + @. ,5 . *5 . .+ ,8.3
C @. 5+3 >*;+ .+ 5*=+? . 81> @. 5+,5 8 >+.1?
C @. +, 8 >+.1? * @.+ .3 3.+5,
C @. 3.+ . ,.? ?+*;1 ?>1, 8 3.15 81> ? 8
>+.1? ;1++
%. Devise a plan" #Plan$
T8 >1 *, ,11@ ;115 ,+=@ T8 .11.3*= 1*, . ,+=*,2 18.=8 .
8,*+.1?, 5 5+?* * 8 ,? ;8* ;
>>1*5 E?* ,*?>1+ .+ ,>;*1 ;, . 8 >+.1? . =* *,*=8 *. 8
,.1*. . 8 .+*=*1 >+.1?
M 1
M 5*=+?
W+* *.
U, =,, 5 ;8;
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W.+ ;3+5
I5*@ , =.1
&. arry out the plan" #Do$
C++@*= . 8 >1 *, ,11@ ,*+ 8 51 H.3* 5 5+,5 8 ?., >+.1?, + . ,.1 I >1 5., . 3.+ *??5*1@2 >+,*, 5 5. .
1 @.+,1 = 5*,;.+=5 I . ,+=@ *, 3.+*=2 +@ 5*+ . *1
8 ,.1*. *, .*5
(. Look ack" #heck$T8*, *, ,*?>1@ ;8;*= 11 ,>, 5 ;1;1*., I ,.? ;,,2 8*, 3*11 +*+
>+.. I+>+ 8 ,.1*. * +?, . 8 .+*=*1 >+.1? D., @.+ ,3+
? ,, I, * +,.1 T82 5+?* 388+ 8+ *, .8+ ?8.5
. *5*= 8 ,.1*. I >.,,*12 5+?* .8+ +15 .+ ?.+ =+1
>+.1?, .+ 38*;8 8 ;8*, 3*11 3.+
T8, 5 .8+ =+1 ?8?*;, >+.1?-,.1+.1? ,.1
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L.+ L .. 8, ,+5 . 31 .+ +;*, S8 315 ! ?*1 8 *+, 32
?*1, 8 ,;.5 32 9 ?*1, 8 8*+5 32 !" ?*1, 8 .+8 32 5
!7 ?*1, 8 *8 3 I ,8 ;.*, . *;+, 8+ 31*= 5*,; 8*,
3@2 * 38*;8 3 3*11 L.+ *+, 31 "7 ?*1,
By using the Polyas Model, we choose the IDENTIFYING PTTE!N as
our strategy to sol"e the Pro#le$ %.
Step 1" Understand the prole!
C1, / I.+?*. ',*= ,*.)
L.+ 8, ,+5 . 31
! ?*1 8 *+, 3
?*1, 8 ,;.5 3
9 ?*1, 8 8*+5 3
!" ?*1, 8 .+8 3
!7 ?*1, 8 *8 3
Step %" Devise a plan
L.. .+ >+
M 1 . 8 3, 5 8 ?+ . ?*1, 315 ;8 . 8
*+, .+ 3,
Step &" Do the plan
WEEK MILES WALKED
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!
$
"
4
!
9
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H.3 ?@ ?.+ ?*1, 5*5 L.+ 31 * W $ 8 * W !
' ! 4 ?.+ ?*1,)
H.3 ?@ ?*1, 5*5 L.+ 31 * W " 8 * W $
'9 4 ?.+ ?*1,)
H.3 ?@ ?*1, 5*5 L.+ 31 * W 4 8 * W "
'!" 9 4 ?.+ ?*1,)
WEEK MILES WALKED
6
7
%
9
!0
!" 4 !7
!7 4 $!
$! 4 $
$ 4 $9
$9 4 ""
"" 4 "7
L.+ 3*11 *+, 31 "7 ?*1, * 8 8 3
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Y. ?@ +1* 8 @. 5*5 8 . 8 8
3 . ,.1+.1? N.*; 82 8 ?+ . *?, 8 @. 55 4 ?*1,
*, . 1,, 8 8 3 ?+
W $: ! 4 4 55**.1 ?*1,
W ": $ 4 % 55**.1 ?*1,
W 4: " 4 !$ 55**.1 ?*1,
I 8 3 8 L.+ *+, 31, "7 ?*1,2 ,8 8, 555 "6 ?*1, . 8 .+*=*1
5*,; . ! ?*1 B;, 8+ + "6 Q 4 9 =+.>, . 4 * "62 * ?, 8
8 32 ,*; 8 3 ?+ *, . ?.+ 8 8 ?+ . =+.>, . 4
?*1, 555
Step (" heck your ans)er
S.1+.1? @ .,++, * 3. 5*+ 3@,2 5 =*= 8
,? ,3+2 >+.
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S@+* 3 . 8 ?+ 38+ 8 .=8 3. ,;, . =..5, 5 ;8 ,;
3, 815 .. ;?1 S@+* 5 8 >.*, 8>*1@ 11 8
3@2 38*1 8 ,;.5 ;?1 +.*= ,1.31@ 3*8 85 *. 8
3+ 3*8 8 ,;, . 8*, ; , 8 3, *+5 5 8,52 5 .11.35 @
8 ;?1 W8*1 * 8 >.52 8 ,;.5 ;?1 3, 3.5+*= 38@ 8 ,;, .
,*?*1+ ,*2 38@ 8*, ,;, 3, 85 8>>*1@ 11 8 3@ 8.?2 8 ,3 8 *+, ;?1 ,+==1 3*8 8*,
,>, 5 . 8 8
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Step %" Devise a plan
T8 ;*
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(++@ 3, ?.3*= 8*, 13 38 8 .*;5 C8+*,@ 3, ,. ?.3*= 8+ 13
5..+ T8@ ,.>>5 . 1 5 (++@ 1+5 8 C8+*,@ ?.3, 8+ 13
.;
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Step &" Do the plan
M T1:
C8+*,@ % 5@, !6 5@, 24 days "$ 5@,
(++@ 6 5@, !$ 5@, !% 5@, 24 days
Step (" heck the *ns)er
T8 5*+; @ C8+*,@ ?.3, 8+ 13, .+ *+, % 5@ *, %2 38*1 (++@
?.3, 8+ 13, .+ *+, 6 5@ *, 6
W8 3 5*5 8 ?1*>1 . 6 5 %2 3 ; = 8 ,? 5@, 8
C8+*,@ 5 (++@ 3*11 ?.3 .=8+ 5@, . $4
W8 ?*= 8 12 3 ; ;.;15 8 (++@ 5 C8+*,@ 3*11 ?.3*=
8*+ 13, .=8+ =* * $4 5@,
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+- /S+ -00-2+ S+3*+-G4"
I 85 ,5*5 11 8 ,+=*, * >+.1? ,.1+ *, 8 , 3@ . ,.1+.1? T8*, *, ;, 8*, @> . ,+=*,
; ? ? , 8 1.=*;1 8**= . ,.1+.1? 8 I 85
F+8+?.+2 ,*= 8, ,+=*, ; ;8 ? . ? ,?+ 5;*,*.
F+.? 82 I 85 5,*= 3 >+.1? . ,8.3 8 ,*= 8*, ,+=*, *, 8
,*, 3@ 5 ?.+ *;* . = ?.+ 1.=*;1 ,3+
DESI&NED PROBLEM
L, @+2 C+ C+ *+@ T8*,
@+2 ,8 *
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Step &" Do the plan
F*5 8 *;+, * 8 ?+ . ;8*15+ *
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CHAPTER : CONCLUSION
W ; ;.;15 8 8+ + ?@ @> . >+.1? ,.1+ , ,+=*, . ,.1+.1? W *5 8 8*, ,+=*, *,
8 ,*, 3@ . ,.1+.1?, T8*, *, ;,2 3 ,5 8 5 8 3
; = +.? 8 >+*; 5 3 ; ? ,?+ =,, +.? 8 >+
8 .+?5 +.? 8 >+*; B;, . 82 3 8* 8 8*, ,+=*, *,
8 , * .+5+ . ? ,?+ =,, 5 . ,.1+.1? H;2 3 85
;+5 .8+ >+.1? 5 ,.1+.1? 8 3 ; ; , 8*,
,+=*, . ,.1++ 8*, ?8.5 , +*11*
,+=@ . ,.1+.1?,
I PROBLEM "2 3 85 =*+.1? 8 ,.1+.1? 8
3 85 5 , . ;.,+; 1 I .2 3 ; . 5. 8*, @> . ,+=@
B;, . 82 3 8* 8 8*, ,+=@ *, . ,*1 . , * ?*=
,?+ =,,
I ;.;1,*.2 3 +1* 8 8+ + ?@ @> . ,+=*, ;
,5 . ,.1+.1? I 5>5, . , . ;8.., 38 @> . ?8.5 8 3
3 . , 5 ?8.5 8 3 8* *, 8 ,*, 3@
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APPENDICES
&EOR&E POLYA
THE RANDOM WALKS OF &EOR&E POLYA
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