INTRODUCTION
WHAT IS THE PROBLEM SOLVING Problem solving means differen !ings o differen "eo"le# !e ma$ori % of "eo"le believe !a i is solving &ord "roblems' and !is in(l)des non ro) ine "roblems and real "roblems* Problem solving is !e "ro(ess of a""l%ing "revio)sl% a(+)ired ,no&ledge' s,ills' and )nders anding o ne& and )nfamiliar si )a ions* Problem solving is !e "ro(ess )sed o find an ans&er o a s a emen or a +)es ion -Hamada' R*.* / Smi !0* Problem solving s!o)ld also in(or"ora e (omm)ni(a ion and $)s ifi(a ion or defense of !e ma !ema i(al ideas !a arise from !e "roblem Problem solving means engaging in a as, for &!i(! !e sol) ion me !od is no ,no&n in advan(e* In order o find a sol) ion' s )den s m)s dra& on !eir ,no&ledge' and !ro)g! !is "ro(ess' !e% &ill of en develo" ne& ma !ema i(al )nders andings*
POLYA MODEL
Pol%a -1234 and 12560 &as !e firs o des(ribe a "roblem solving model based on (lassroom e7"erien(e • • • • )nders and !e "roblem devise a "lan (arr% o) "lan loo, ba(,
Polya First Principle: Understand the Problem T!is seems so obvio)s !a i is of en no even men ioned' %e s )den s are of en s %mied in !eir effor s o solve "roblems sim"l% be(a)se !e% don8 )nders and i f)ll%' or even in "ar * Pol%a a)g! ea(!ers o as, s )den s +)es ions s)(! as9
• • • •
:o %o) )nders and all !e &ords )sed in s a ing !e "roblem; W!a are %o) as,ed o find or s!o&; <an %o) res a e !e "roblem in %o)r o&n &ords; <an %o) !in, of a "i( )re or a diagram !a mig! !el" %o) Is !ere eno)g! informa ion o enable %o) o find a sol) ion;
)nders and !e "roblem;
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Polya Second Principle: Devise a plan Pol%a men ions -124=0 !a i is man% reasonable &a%s o solve "roblems* T!e s,ill a (!oosing an a""ro"ria e s ra eg% is bes learned b% solving man% "roblems* .o) &ill find (!oosing a s ra eg% in(reasingl% eas%* A "ar ial lis of s ra egies is in(l)ded9
• • • • • • •
G)ess and (!e(, Ma,e and orderl% lis Elimina e "ossibili ies >se s%mme r% <onsider s"e(ial (ases >se dire( reasoning Solve an e+)a ion
• • • • • • •
Loo, for a "a ern :ra& a "i( )re Solve a sim"ler "roblem >se a model Wor, ba(,&ard >se a form)la Be ingenio)s
Polya third Principle: Carry out the plan T!is s e" is )s)all% easier !an devising !e "lan* In general -124=0' all %o) need is (are and "a ien(e' given !a %o) !ave !e ne(essar% s,ills* Persis en &i ! !e "lan !a %o) !ave (!osen* If i (on in)es no o &or, dis(ard i and (!oose ano !er* :on? be misled# !is is !o& ma !ema i(s is done' even b% "rofessionals* Polya Fourth Principle: Look back • • • (onfirm res)l s and arg)men s assess effe( iveness of sol) ion assess a(()ra(% of res)l s
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assess )sef)lness of sol) ion for solving o !er "roblems
QUESTION 1
@>ESTION 1 T!e villagers &ere b)ilding a bridge* W!ile &or,ing )nder !e bridge Rodne% see onl% !e legs of !ose &al,ing b%* He (o)n ed 1A legs in one gro)"* W!a (ombina ion of s!ee" and (!ildren (o)ld !ave been in !a gro)";
S ra eg% 1
S e" 1* )nders and !e "roblem 1* W!ile &or,ing )nder !e bridge Rodne% see onl% !e legs of !ose &al,ing b%* 6* He (o)n ed 1A legs in one gro)"* B* W!a (ombina ion of s!ee" and (!ildren (o)ld !ave been in !a gro)";
S e" 69 devise a "lan I?ll (!oose g)ess and (!e(,
S e" B9 (arr% o) "lan I?ll (!oose g)ess and (!e(, Tr%9 B "eo"le and 6 s!ee" 5 leg C Dlegs -Too man% legs0 B "eo"le 5 legs • and 1 s!ee" E 1A (orre( G E 13 F false
C 3 legs
Rodne% sa& 1A legs - B "eo"le and 1 s!ee" &al,ing b% 0
S e" 39 loo, ba(, <ombina ion of s!ee" and (!ildren H 1A legs !e (o)n ed E i s B "eo"le and 1 s!ee"
S ra eg% 6
S e" 1 9 )nders and !e "roblem 1* W!ile &or,ing )nder !e bridge Rodne% see onl% !e legs of !ose &al,ing b%* 6* He (o)n ed 1A legs in one gro)"* B* W!a (ombina ion of s!ee" and (!ildren (o)ld !ave been in !a gro)";
S e" 6 9 devise a "lan I?ll (!oose o )se able*- )se a diagram0
S e" B 9 (arr% o) "lan >se a able
"eo"le 6 B 3
leg 3 5 D
s!ee" 6 1 1
leg D 3 3
o al 16 1A 16
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!e ans&er is B "eo"le and 1 s!ee"
s e" 3 9 loo, ba(, H 1A legs H I gro)" !ave (!ildren and s!ee" H i ?s means B "eo"le 1 s!ee" &al,ing b% bridge
CONCLUSION I (!oose &i ! )se a diagram be(a)se i ?s easier o )nders and and s%s ema i(* And !e ans&er more (lear !an ano !er s ra eg%* S )den (an be more in eres ing and )nders and !o& o solve !e "roblem*
QUESTION 2
@>ESTION 6 Robin Hood (o)ld aim !is arro& and s!oo do&n 6 birds in B min) es* He &as !aving Baron Io!n M(Jell% for dinner* Robin needed 1A birds as a side dis!* Ho& long &o)ld i a,e o s!oo do&n all !e birds;
S ra eg% 1
S e" 1 9 )nders and !e "roblem • • • • Robin Hood (o)ld aim !is arro& and s!oo do&n 6 birds in B min) es* He &as !aving Baron Io!n M(Jell% for dinner* Robin needed 1A birds as a side dis!* Ho& long &o)ld i a,e o s!oo do&n all !e birds;
S e" 6 9 devise a "lan I?ll (!oose )se a diagram
S e" B9 (arr% o) "lan birds 6 3 5 D 1A ime B min) es 5 min) es 2 min) es 16 min) es 14 min) es
S e" 3 9 loo, ba(, H Robin Hood s!oo do&n 6 birds in B min) es* H !e needed 1A birds H so' in 14 min) es !e (an s!oo do&n 1A birds
S ra eg% 6
S e" 1 9 )nders and !e "roblem • • • Robin Hood (o)ld aim !is arro& and s!oo do&n 6 birds in B min) es* He &as !aving Baron Io!n M(Jell% for dinner* Robin needed 1A birds as a side dis!*
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Ho& long &o)ld i a,e o s!oo do&n all !e birds;
S e" 6 9 devise a "lan I?ll (!oose )se "ro"er ies of n)mbers
S e" B9 (arr% o) "lan 6 min) esE B birds 1 bird E B K 6 E 1*4 min) es 1A birds E 1*4 min) es 71A birds E 14 min) es Robin Hood (an ge 1A birds in 14 min) es
S e" 3 9 loo, ba(, H Robin Hood s!oo do&n 6 birds in B min) es* H !e needed 1A birds H Ho& long &o)ld i a,e o s!oo do&n all !e birds;
E 1A birds needed 14 min) es
CONCLUSION I (!oose &i ! )se a "ro"er ies of n)mbers be(a)se i ?s easier o )nders and and s%s ema i(* And !e ans&er more (lear !an ano !er s ra eg%* S )den (an be more in eres ing and )nders and !o& o solve !e "roblem*
QUESTION 3
@>ESTION B T!e (as le ,i (!en servan s bro)g! in 3 "ies lef over from !e feas * 16 "ies &ere ea en a !e feas * @)een Mab oo, 6 !ome &i ! !er* Ho& man% "ies did !e servan s bring in o !e feas a !e beginning; S e" 1 9 )nders and !e "roblem • • • • T!e (as le ,i (!en servan s bro)g! in 3 "ies lef over from !e feas * 16 "ies &ere ea en a !e feas * @)een Mab oo, 6 !ome &i ! !er* Ho& man% "ies did !e servan s bring in o !e feas a !e beginning;
S e" 6 9 devise a "lan I?ll (!oose &or, ba(,&ards
S e" B 9 (arr% o) !e "lan • • • • • Lirs ' I8ll a((o)n for all !e "ies !a &ere ea en or a,en !ome* 16 C 6 E 13 T!en I8ll add !e 3 "ies !a &ere lef over* 13 C 3 C 1D T!erefore' !ere m)s !ave been 1D "ies a !e s ar of !e feas *
S e" 3 9 loo, ba(, • • • • 3 "ies lef over from !e feas * 16 "ies &ere ea en a !e feas * @)een Mab oo, 6 !ome &i ! !er* Ho& man% "ies did !e servan s bring in o !e feas a !e beginning; • So' i beginning from 1D "ies s ar !e feas
S ra eg% 6
S e" 1 9 )nders and !e "roblem
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T!e (as le ,i (!en servan s bro)g! in 3 "ies lef over from !e feas * 16 "ies &ere ea en a !e feas * @)een Mab oo, 6 !ome &i ! !er* Ho& man% "ies did !e servan s bring in o !e feas a !e beginning;
S e" 6 9 devise a "lan I?ll (!oose )se a "i( )re
S e" B 9 (arr% o) !e "lan
- 3 "ies lef over from !e feas 0 C
(16 "ies &ere ea en a !e feas 0
C
- @)een Mab oo, 6 !ome 0 E
- i ?s means s!e !ave 1D "ies a !e beginning0
S e" 3 9 loo, ba(, • • • • 3 "ies lef over from !e feas * 16 "ies &ere ea en a !e feas * @)een Mab oo, 6 !ome &i ! !er* Ho& man% "ies did !e servan s bring in o !e feas a !e beginning; • So' i beginning from 1D "ies s ar !e feas
CONCLUSION
I (!oose &i ! )se a "i( )re be(a)se i ?s easier o )nders and and s%s ema i(* And !e ans&er more (lear !an ano !er s ra eg%* S )den (an be more in eres ing and )nders and !o& o solve !e "roblem*