10 Describe Polya

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10 Describe Polyas Problem Solving ModelMeaning of Problem and problem- Solvings .Solving mathematics problems are activitiesinvolving problemsinthe form of mathematicslanguage, including mechanical problems, puzzles,quiz and the use of mathematicsskillsin actualsituations.

According to Lester, a prominent mathematicianin the 1970s,defined problem-solving as : Problem involving a situation whereby anindividualor a group is required tocarryout the workingsolution. Indoingso,they have todetermine the strategy and methodof problem-solving first, beforeimplementing the workingsolution. The strategyof problem solvingneeds aset of activities which willlead to the problem-solving process.Problem posing and problem solvinginvolve examiningsituations thatarise in mathematics andotherdisciplines andincommon experiences,describing these situations mathematically, formulating appropriatemathematical questions, andusing a varietyofstrategies to findsolutions. Bydeveloping their problem-solvingskills, students willcome to realize the

potentialusefulnessof mathematicsin theirlives.Problem solvingis a term that often meansdifferent things todifferentpeople.Sometimesit even meansdifferent things at different times for thesame people! It may meansolvingsimple word problems that appearinstandard textbooks, applying mathematics to real-worldsituations, solvingnonroutine problemsor puzzles, orcreating and testing mathematicalconjecturesthat maylead to the studyofnew concepts. In everycase, however, problemsolvinginvolves anindividualconfronting a situation which she hasnoguaranteed way to resolve.Some tasks are problems for everyone (like findingthe volume of a puddle), some are problems forvirtuallynoone (like countinghow many eggs are in a dozen), andsome are problems forsome people but1

not forothers (like findingout how manyballoons 4 children have if each has 3balloons, or finding the area of a circle).Problem solvinginvolves far more thansolving the word problemsincludedin the students' textbooks;it is an approach tolearning anddoingmathematics that emphasizes questioning and figuring thingsout. TheCurriculum and EvaluationStandardsof the National Councilof TeachersofMathematicsconsiders problem solving as the central focusof the mathematicscurriculum."As such, it is a primary goal of all mathematics instruction and an integral partof all mathematics activity. Problem solving is not a distinct topic but a processthat should permeate the entire program and provide the context in whichconcepts and skills can be learned." (p. 23)Thus, problem solvinginvolves allstudents a large part of the time;it isnot anincidental topicstuckon at the endof the lessonorchapter, norisit justfor those who are interestedinor have already mastered the day'slesson.Studentsshould have opportunities to pose as well as tosolve problems;not allproblemsconsideredshouldbe taken from the text orcreatedby the teacher.However, the situations explored must be interesting,engaging, andintellectuallystimulating. Worthwhile mathematical tasks are not onlyinteresting


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to the students, they alsodevelop the students' mathematicalunderstandingsandskills, stimulate them to make connections anddevelop a coherentframework for mathematicalideas, promote communication about mathematics,represent mathematics as anongoing human activity, draw on theirdiversebackground experiences andinclinations, and promote the development of all

students'dispositions todo mathematics (ProfessionalStandardsof theNational Councilof Teachersof Mathematics). As a result ofsuch activities,studentscome tounderstand mathematics anduse it effectivelyin a varietyofsituations.2

Characteristics of mathematics problemContains elements which canbe foundin the environmentItssolutionneeds properstrategyin planning, includingselectionofsuitable methods for problem-solving.Properstrategyin planning andselectionofsuitable methoddependonthe pupils acquiredknowledge and experience as well asunderstandingof the relevant problem.The abilityof problem-solvingisclosely related to the pupils levelofcognitive development, at least at its applicationlevel.The wayused for problem-solvingcannot be memorized as In the case ofreciting mathematics formula orsolving mechanical questionby meansofmemorization. Every mathematics problem ought to have itsownspecificsolution.The methodof problem-solving mayconsist of more thanone approach.The processof problem-solvingneeds toimplement by meansof a set ofsystematic activities.The processof problem-solvingneeds to apply mathematicsskills,conceptsor principles which have beenlearned and mastered.George Polya- 1887 19853

George Polya was a Hungarian whoimmigrated to the UnitedStatesin1940. His majorcontributionis for his workin problem solving. Growingup he wasvery frustrated with the practice of having to regularly memorize information. Hewas an excellent problem solver. Earlyon hisuncle tried toconvince him togointo the mathematics fieldbut he wanted tostudylaw like hislate father had. Aftera time at law school he became bored with all the legal technicalities he had tomemorize. He tiredof that andswitched to Biology and the againswitched to Latinand Literature, finallygraduating with a degree. Yet, he tiredof that quickly andwent back toschool and took math and physics. He found he loved math.

His first job was to tutor Gregor the youngsonof a baron. Gregorstruggleddue to hislackof problem solvingskills. Polya (Reimer, 1995) spent hours anddeveloped a methodof problem solving that would work for Gregor as well asothersin the same situation. Polya (Long, 1996) maintained that the skillofproblem wasnot aninborn qualitybut, something that couldbe taught. He wasinvited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. Oneday he met the doctor?sdaughterStella he began tocourt her and eventuallymarried her. Theyspent 67 years together. While inSwitzerland he loved to take


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afternoon walksin the localgarden. One day he met a youngcouple also walkingandchose another path. He continued todo thisyet he met the same couple sixmore times as he strolledin the garden. He mentioned to his wife ?how couldit bepossible to meet them so many times when he randomlychose different pathsthrough the garden?

He laterdid experiments that he called the random walk problem.Severalyearslater he published a paper proving that if the walkcontinuedlong enoughthat one wassure to return to the starting point. In 1940 he and his wife moved tothe UnitedStatesbecause of theirconcern for Nazism in Germany (Long, 1996).He taught briefly at Brown University and then, for the remainderof hislife, atStanford University. He quicklybecame wellknown for his research and teachingson problem solving. He taught manyclasses to elementary andsecondary4

classroom teacherson how to motivate and teach skills to theirstudentsin thearea of problem solving.In 1945 he published the book How toSolve It which quicklybecame hismost prized publication. It soldoverone millioncopies and hasbeen translatedinto 17 languages. In this text he identifies fourbasic principles. In How ToSolveIt, G. Polya describes foursteps forsolving problems andoutlines them at theverybeginningof the book for easy reference. The stepsoutline a seriesofgeneral questions that the problem solvingstudent canuse tosuccessfully writeresolutions. Without the questions, commonsense goes through the sameprocess; the questionssimply allow students tosee the processon paper. Polyadesigned the questions tobe general enough that studentscould apply them toalmost any problem.The foursteps are: understanding the problem, devising a plan, carryingout the plan, and lookingback.This methodisverysimilar to the methodin Thinking MathematicallybyJohn Mason, except Polya separatesdevising a plan, andcarryingout the plan.This mayseem silly at first, but Polya argues that it does make a difference. Byfirst devising a plan, studentscan eliminate mistakes they might make by rushinginto the actual executionof the plan. When they planit out first and thendo themath, it is possible tocheck their work as theygo along.Polya?s First Principle: Understand the Problem

In the first principle, pupils wouldbe guided tounderstand:(a)Variablesinvolvedin the problem;(b)Relationship between the variables which have been ascertained; and(c)Variable which needs tobe thoroughlysearchedor answered.5

In theother words, Understanding the Problem canbe explainby this question :1. Canyoustate the problem inyourown words?2. What are you trying to findordo?3. What are the unknowns?4. What informationdoyouobtain from the problem?


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5. What information, if any, is missingornot needed?For the students, theyshouldbe able tostate the unknown, or the thingthey want to find to answer the question, the data the questiongives them to workwith, and the condition, orlimitingcircumstances they must work around. If theycanidentify allof these, and explain the question toother people, then they have

a goodunderstandingof what the problem is asking. Polya suggests that studentsdraw a picture if possible, orintroduce some kindofnotation tovisualize thequestion. Thisseemssoobvious that it isoftennot even mentioned, yet studentsare oftenstymiedin their efforts tosolve problemssimplybecause theydon?tunderstandit fully, or evenin part.Polya taught teachers to askstudents questionssuch as:

yDoyouunderstand all the wordsusedinstating the problem?

yWhat are you asked to findorshow?

yCanyou restate the problem inyourown words?

yCanyou thinkof a picture or a diagram that might help youunderstand theproblem?

yIs there enough information to enable you to find a solution?Some techniques that may help students with thisimportant aspect of problemsolving - understanding the problem - include restating the problem in theirownwords, drawing a picture, or actingout the problem situation.Some teachers havestudents workin pairson problems, with one student reading the problem andthen, without referring to the written text, explaining what the problem is about totheir partner.6

Polya?s Second Principle: Devise a planPolya mentions (1957) that it are many reasonable ways tosolve problems.The skill at choosing an appropriate strategyisbest learnedbysolving many

problems. We will findchoosing a strategyincreasingly easy. Todevise a plan,studentscanstart by trying to thinkof a related problem we have solvedbefore tohelp them. If the student can thinkof a problem they have solvedbefore that hada similarunknown, it could alsobe helpful.Studentscan also try to restate theproblem in an easierordifferent way, and try tosolve that. Bylooking at theserelated problems, students maybe able touse the same method, orother part ofthe planused. Afterstudents have decided which calculations, computations, orconstructions that theyneed, and have made sure that alldata andconditionswere used, theycan tryout their plan. A partiallist ofstrategiesisincluded:We alsocanuse these instructions tounderstand more details.

Find the connectionbetween the data and theunknown. You maybe obliged toconsiderauxiliary problemsif animmediate connectioncannot be found. Youshouldobtain eventually aplan of the solution.

Have youseenit before? Or have youseen thesame problem in a slightlydifferent form?Do you know a related problem? Doyouknow a


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theorem that couldbe useful?Look at the unknown!And try to thinkof a familiarproblem having the same or a similarunknown.Here is a problem related to yours and solvedbefore. Could you use it? Couldyouuse its

result? Couldyouuse its method? Shouldyouintroduce some auxiliary element inorder tomake itsuse possible?7

Guess andcheckMake anorderlylistEliminate possibilitiesUse symmetryConsiderspecialcasesUse direct reasoningSolve an equation

Alsosuggested:Look for a patternDraw a pictureSolve a simpler problemUse a modelWorkbackwardUse a formulaBe creativeUse your head/noggin

Couldyou restate the problem? Couldyou restate it stilldifferently? Goback todefinitions.

Ifyoucannot solve the proposed problem try tosolve first some related problem.Couldyouimagine a more accessible related problem? A more general problem?A more special problem? An analogous problem? Couldyousolve a part of theproblem? Keep only a part of the condition, drop the other part; how faris theunknown thendetermined, how canit vary? Couldyouderive somethingusefulfrom the data? Couldyou thinkofotherdata appropriate todetermine theunknown? Couldyouchange the unknownordata, orboth ifnecessary, so thatthe new unknown and the new data are nearer to each other?

Didyouuse all the data? Didyouuse the whole condition? Have you takenintoaccount all essentialnotionsinvolvedin the problem?Polya?s third Principle: Carry out the plan

Carryingout the planissometimes the easiest part ofsolving a problem.However, manystudentsjump to thisstep toosoon. Otherscarryoutinappropriate plans, orgive up toosoon andstop halfway through solving theproblem. To reinforce the processof making a plan andcarryingit out, teachersmight use the following technique: Divide a sheet ofnotebook paperinto twocolumns. On the left side of the page, the student solves the problem. On the rightside of the page, the student writes about what isgoingonin his/her mindconcerning the problem. Is the problem hard? How canyouget started? What


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strategy might work? How didyou feel about the problem?Thisstep isusually easier thandevising the plan. Ingeneral (1957), allyouneediscare and patience, given that you have the necessaryskills. Persistentwith the plan that you have chosen. Ifit continuesnot to workdiscardit andchoose another. Dont be misled, thisis how mathematicsisdone, evenby

professionals.8Here are some tips touse:

Implement the strategyinStep 2 and perform anynecessary actionsor

computations. Check each step of the plan asyou proceed. This maybe intuitive checkingor aformal proofof each step.. Keep an accurate recordofyour work. Canyousee clearly that the step iscorrect? Canyou prove that it iscorrect?Polyas Fourth Principle: Look backThisis the part of problem-solving that most people tend toignore. One way

forus toimprove is to review past experiences andunderstand why we succeedor fail.Soit isimportant to monitorourown performance review the wholeexercise inorder that we cando evenbetterin the future.Polya mentions (1957) that much canbe gainedby taking the time to reflectandlookback at what you have done, what worked and what didnt. Doing this willenable you to predict what strategy touse tosolve future problems. George Polyawent on to publish a two-volume set, Mathematics and Plausible Reasoning(1954) and Mathematical Discovery (1962). These texts form the basis for thecurrent thinkingin mathematics education and are as timely andimportant todayas when they were written. Polya hasbecome known as the fatherof problemsolving.

Whenstudentslookbackon the problem and the plan theycarriedout,theycanincrease theirunderstandingof the solution. It is alwaysgood to recheckthe result and argument used, and to make sure that it is possible tocheck them.Thenstudentsshould ask, "Can I get the result in a different way?"and "Can I usethis for another problem?" The last chapterof the bookis a very helpfulencyclopedia of the termsusedin the explanationof the first chapter.A partiallist ofstrategiesisincluded:1. Check the resultsin the original problem. Insome cases, this will require a proof.9

2. Interpret the solutionin termsof the original problem. Doesyour answer makesense? Isit reasonable?3. Determine whether there is another methodof finding the solution.4. If possible, determine other relatedor more general problems for which thetechniques will work.While it might seem most logical tobegin problem solving with Polya's firstactivity and proceed through each activityuntil the end, not allsuccessful problemsolversdoso. Manysuccessful problem solversbeginbyunderstanding theproblem and making a plan. But then as theystart carryingout their plan, theymay find that they have not completelyunderstood the problem, in which case


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theygoback tostep one. Or they may find that theiroriginal planis extremelydifficult to pursue, so theygoback tostep two andselect another approach. Byusing these four activities as a generalguide, however, studentscanbecomemore adept at monitoring theirown thinking. This "thinking about their thinking"can help them toimprove their problem solvingskills.

Students move through a continuum ofstagesin theirdevelopment asproblem solvers (Kantowski, 1980). Initially, they have little ornounderstandingofwhat problem solvingis, of what a strategyis, orof the mathematicalstructure ofa problem.Such studentsusuallydonot know where tobegin tosolve a problem;the teacher must model the problem solving process for these students. At thesecondlevel, students are able to follow someone else'ssolution and maysuggest strategies forsimilar problems.They may participate activelyingroup problem solvingsituationsbut feelinsecure about independent activities, requiring the teacher'scontinuedsupport.At the thirdlevel, studentsbegin tobe comfortable with solving problems,suggestingstrategiesdifferent from those they have seenusedbefore. They

understand and appreciate that problems may have multiple solutionsor perhapsevennosolution at all. Finally, at the last level, students are not only adept atsolving problems, they are alsointerestedin finding elegant and efficient solutionsandin exploring alternate solutions to the same problem. In teaching problem10

solving, it isimportant to address the needsofstudents at each of these levelswithin the classroom.Insummary, the real test of whether a student knows mathematicsiswhethershe canuse it in a problem situation.Studentsshould experienceproblems asintroductions tolearning about new topics, as applicationsofcontentalreadystudied, as puzzlesornon-routine problems that have manysolutions,and assituations that have noone best answer. Theyshouldnot onlysolveproblemsbut also pose them. Theyshould focusonunderstanding a problem,making a plan forsolvingit, carryingout their plan, and thenlookingback at whatthey have done.20 Explain routine and Non routine ProblemsFuturistscontinue tostress that our future isgoing toundergochange at arate evengreater than present generations have experienced. Thisimplies thattodays and future problems will have a dynamiccomponent.Such problemschange or evolve as they are beingstudied. It is evident then that a fundamentalskill fordealing with the future is active problem solving, i.e., the ability tosolveproblems which are undergoingchange during the processof resolution.11

Problem solvingcanbe dividedinto twocategories, routine andnon routinewhich isRoutine problem solving and Non routine problem solving.ROUTINE PROBLEMRoutine problem solvingisstresses the use ofsetsofknownor prescribedprocedures (algorithms) tosolve problems. In a routine problem, the problemsolverknows a solution method andonlyneeds tocarryit out. The strength ofthis approach is that it is easily accessedby paper-pencil tests.Since todays


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computers andcalculatorscan quickly and accurately perform the most complexarrangementsof algorithms for multi-step routine problems, the typical workplacedoesnot require a high levelof proficiencyin routine problem solving. However,todays workplace does require many employees tobe proficient in Non routineproblem solving. Routine problems are sometimescalled exercises, and

technicallydonot fit the definitionof problem stated above. When the goalof aneducational activityis to promote all the aspectsof problem solving (includingdevising a solution plan), thennon routine problems (or exercises) are

appropriate.Routine problem is actuallyis a type of mechanical mathematic problem. Itaimed at training the students for able to masterbasicskills, especially thearithmeticskills which involving the fouroperations, addition, subtraction,multiplication anddivision (+, -, , ), ordirects applicationsofusing mathematicsformulae, laws, theoremsor equations. Generallyspeaking, routine problems arethe most basicsimple type of problem-solvingin mathematics, asitsgoalexpressioncanbe achievedby meansofcertain algorithm.

12NON ROUTINE PROBLEMNon routine problem solvingisstresses the use of heuristics andoftenrequireslittle tonouse of algorithms. The problem solverdoesnot initiallyknow amethod forsolving the problem. Unlike algorithms, heuristics are proceduresorstrategies that donot guarantee a solution to a problem but provide a more highlyprobable method fordiscovering the solution. Building a model anddrawing apicture of a problem are twobasic problem-solving heuristics. Other heuristicsinclude describing the problem situation, making the problem simpler, findingirrelevant information, workingbackwards, andclassifyinginformation.Actually, Non-routine problem is a unique problem-solving which requires

the applicationofskills, conceptsor principles which have beenlearned andmastered. Method forsolvingnon-routine problem in mathematicsisdifferent fromanswering mechanical question. It needssystematic activities with logicalplanning, including properstrategy andselectionofsuitable method forimplementation. Most of the non-routine problems required a heuristic approachsuch as the applicationof experiences and practical effort, or plannedstrategy, toattainitsgoal expression..There are two typesofnon routine problem solvingsituations which isstatic and Active.Staticnon routine problems have a fixedknowngoal and fixed,known elements that are used to resolve the problem.Solving a jigsaw puzzle isan example of a staticnon routine problem. Given all pieces to a puzzle and apicture of the goal, learners are challenged to arrange the pieces tocomplete thepicture. Various heuristicssuch asclassifying the piecesbycolor, connecting thepieces which form the border, orconnecting the pieces which form a salientfeature to the puzzle, such as a flag pole, are typical waysin which peopleattempt to resolve such problems.13

Active non routine problem solving may have a fixedgoal with changingelements, a changinggoalor alternative goals with fixed elements, orchangingor


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alternative goals with changing elements. The heuristicsusedin this form ofproblem solving are known asstrategies. People whostudysuch problems mustlearn tochange or adapt theirstrategies as the problem unfolds.According to John Dewey, learning through problem-solvingincludes fivestages. They are brieflyshownin the following figure.

PROCE

SS OF PROBLE

M-SOLVING

The processofidentifying the problem involves activities tounderstand andascertainimportant aspectscontainedin the problem. The stage oflooking forinformationinvolves activities tocollect materialsor facts related to the problem.The stage that followsis the settingup of a hypothesis tosuggest strategies andmethods tosolve the problem identified. The next stage is testing the hypothesiswhereby the suggestedstrategies and methods are implementedin the process.Finally an evaluationis made on the techniquesusedduring the processofsolving the problem with decisionon finalconclusion and records.14

30 Gather information and select three non routine problems and solve each ofthese problems using two or more types of problem solving strategies.Elaborate on the different strategies. Select one strategy that is deemed tobe most efficient and justify selection.Question: Amy and Judysold 12 show tickets altogether. Amysold 2 more ticketsthan Judy. How many ticketsdid each girlsell?Strategy:1) UNDERSTAND:What doyouneed to find?Youneed toknow that 12 tickets were soldin all. You alsoneed toknow that Amysold 2 more tickets than Judy.2) PLAN:How canyousolve the problem?15

a) Guess and CheckYoucanguess andcheck to find twonumbers with a sum of 12 and a differenceof 2. Ifyour first guessdoesnot work, try twodifferent numbers.3) SOLVE:First Guess:Amy = 8 ticketsJudy = 4 ticketsCheck8 + 4 = 128 - 4 = 4 ( Amysold 4 more tickets)

These numbersdonot work!Second Guess:Amy = 7 ticketsJudy = 5 ticketsCheck7 + 5 = 127- 5 = 2 ( Amysold 2 more tickets)These numbersdo work!


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Amysold 7 tickets and Judysold 5 tickets.b) Draw a Picture12 tickets were soldLets we divide halfEach others

Amy(6) Judy(6)16* Amysold 2 more tickets than Judy. Decrease 1 tickets from JudyNow, we got the answer.Amysold 7 tickets andJudy(5) Judysold 5 tickets.Amy(7)Justify Selection:In this question, I think that draw a picture strategyis more suitable.Studentscandirectlysee that there are how to arrange the strategy. Then, they willsee how to findthe answer. On the sport theycanget the true answer, which is 7 show ticketsissoldbyAmy andonly 5 show ticketsissoldby Judy.Besides, thisstrategyis easy tocarryout. It onlyneeds todraw it on a paper.Sometimes, we are lackofideas how can the question act. Even though we are able toget the idea, we stillneed to aware on the relationship on the diagram.So, I prefer the strategyofdraw a picture more because it issave and easier tobecarryout.17

Question: Laura has 3 greenchips, 4 blue chips and 1 redchip in herbag. Whatfractional part of the bagofchipsisgreen?Strategy:1) UNDERSTAND:What doyouneed to find?Youneed to find how manychips are in all. Thenyouneed to find how manyofthe chips are green.2) PLAN:How canyousolve the problem?a) Draw a PictureYoucandraw a picture toshow the information. Thenyoucanuse the picture tofind the answer.3) SOLVE:Draw 8 chips.3/8 of the chips are green.18

b) Make a diagram3/8 of the chips are green.Justify Selection:In this question, I thinkstrategydraw a diagram is more suitable. Thisisbecause itis easier tounderstand andcarryout. Besides, studentscango through to the answer..Therefore, in myopinionstrategy 2 is more suitable for me because it iseasier tounderstand andcarryout.Question: Judyis taking picturesof Jim, Karen and Mike.She asks them, " How


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manydifferent wayscouldyou three childrenstandin a line?"Strategy:1) UNDERSTAND:What doyouneed toknow?19

Youneed toknow that anyof the studentscanbe first, secondor third.2) PLAN:How canyousolve the problem?

a) Make a ListYoucan make a list to help you find all the different ways. Choose one student tobe first, and another tobe second. The last one willbe third.3) SOLVE:Whenyou make yourlist, you willnotice that there are 2 ways for Jim tobe first, 2ways for Karen tobe first and 2 ways for Mike tobe first.First Second ThirdJim Karen Mike

Jim Mike KarenKaren Jim MikeKaren Mike JimMike Karen JimMike Jim KarenSo, there are 6 ways that the childrencouldstandinline.b) Guess and CheckFirstly, let saystart with:Jim, follow by Karen andlastlyis MikeKaren, follow by Jim andlastlyis MikeMike, follow by Karen andlastlyis Jim

But, there have another ways that the childrencouldstandinline.Jim, follow by Mike andlastlyis KarenKaren, follow by Mike andlastlyis JimMike, follow by Jim andlastlyis Karen.* Thats mean there are 6 ways that the childrencouldstandinline.20

Justify Selection:In this question, I thinkstrategy 1 make a list is more suitable. Thisisbecause itis easier tounderstand andcarryout.Sometimes, we might face on problems whenuseguess andcheck, like unable todetermine the answer accurately. Then, students willwaste their time toguessuntilget the right answerTherefore, in myopinionstrategy 1 which is make a list is more suitable for mebecause it is easier tounderstand , save time and easier tocarryout.40 Create three new but similar problems and solve it using the suggestedstrategies.

a)Make a list1.Doug has 2 pairsof pants: a black pair and a green pair. He has 4 shirts: awhite shirt, a redshirt, a greyshirt, and a stripedshirt. How manydifferentoutfitscan he put together? (Hint: Cornplete the organisedlist.)


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Make a listPantsShirts21

BlackWhiteBlackRed

BlackBlackPantsShirtsGreenGreenGreenGreenUnderstanding the Problem How many pairsof pantsdoes Doug have? (2) How manyshirtsdoes Doug have? (4)Planning a Solution Suppose Doug wears hisblack pants. What colorshirt can he wear? (white, red,grey, orstriped) If Doug wears hisstripedshirt, how manydifferent outfitscan he wear? (2:stripedblack andstripedgreen) If Doug wears the green pants, can he wear all 4 shirts? (yes)Finding the AnswerMake an Organized ListBlackWhiteBlackRedBlackGrayBlackStripedGreenWhiteGreenRedGreenGrayGreenStripedDougcan make 8 different outfits.22

2.Theres only one bicycle at the IPGMCampus Tuanku Bainun that Sara, TirahandEzzaty can borrow. How many times could each of them borrow thebicycle?Strategy:1) UNDERSTAND:What do you need to know?

You need to know that any of the students can be first, second or third.2) PLAN:How can you solve the problem?You can make a list to help you find all the times they can borrow the bicycle.Choose one student to be first, and another to be second. The last one will bethird.3) SOLVE:When you make your list, you will notice that there are 2 times for Sara to be


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first, 2 times forTirah to be first and 2 times forEzzaty to be first.First SecondThirdSara Tirah EzzatySara EzzatyTirahTirah Sara EzzatyT

irahEzz

aty SaraEzzatyTirah SaraEzzaty Sara TirahSo, there are 6 times that the students couldborrow the bicycle.23

3.The letters ABCD, canbe put into a different order: DCBA or BADC. Howmanydifferent combinationsof the letters ABCD canyou make?i.Strategy:a) UNDERSTAND:What do you need to know?You need to know that any of the combination letters ABCD can can be first,second or third.b) PLAN:How can you solve the problem?You can make a list to help you find all the different combination letters ABCD.Choose one combination letterst to be first, and another to be second. The lastone will be third.c) SOLVE:To answer this question, obviously, you have to make a list.Teach yourstudents to make a SYSTEMATIC list. For example:ABCDABDCACBDACDBADBCADCBBACDBADCBCADBCDABDACBDCACABDCADBCBADCBDACDABCDBADABCDACBDBACDBCA


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DCABDCBABy making a SYSTEMATIC list, students willsee every possible combination.24

(Later, perhaps, they willlearn that the numberof permutationsofsize 4 taken

from a set of 4 canbe representedby the formula 4 * 3 * 2 * 1 =24).b)Draw a picture

1. In a sunnyday, Johnny walksinto the school hall, he saw 4 girls and 3boysstandingin front the gate. Each girlcarrying 3 schoolbags and each boycarrying 2. How manyshoppingbagsdid Johnnysaw?First, draw a picture so that we cancount.Count the schoolbags. The fourgirls have 12 bags. The twoboys have 4 bags.12 + 4 = 16Johnnysaw 16 schoolbagsin front the gate.2. Jennysaw 2 big houses and 3 small houses are buildbesides the lake.Each of the big houses had 4 windows and each small houses had 2 windows. How

many windowsdid Jennysaw?25

First, draw a picture so that we cancount.Count the windows. Big houses have eight windows.Small houses have six windows.8 + 6 = 14Jennysaw 14 windowson the housesby the lake.3.In a restaurant, Kellysaw 6 rectangle tables and 3 triangle tables. Each ofthe rectangle tables had 4 legs and the triangle tables had 3 legs each. How manytable legsdid Kellysee in the restaurant?First, draw a picture so that we cancount.Count the legsof the tables. Rectangle tables have 24 legs. Triangle tables have 9 legs.

24 + 9 = 33Kellysaw 33 table legsin the restaurant.

c)Make a Diagram26

A coin has two face, head and tail. Ifyou toss 3 coins together, determine that howmanydifferent combinationsof heads and tailsyou wouldget.

yThere are 6 combinationsof heads and tails.27


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Ygke dua

AcknowledgementAlhamdulillah

Thank to Allah S.W.T because give us an effort todo and finish ourassignment onour Mathematic. We have beengiven a task from ourlecture that is,write an essay about George Polya and The solvingstrategy..All we have todoissearching the information andusing the information thatisgivenby mylecture, Mr. Ahmad Rizal Bin Che Rahim ,thank a lot to him becausehe alreadygive us manyinformation and advice that isveryuseful. He help us a lotinorder to finish the assignment.We also want to thank toour entire friend who help usin the processofmaking and finishingour assignment. Thank to theirinformation and advice inorderto help out to make my assignment better.Bydoing this assignment we have learned many thing about the George

Polya and Problem SolvingStrategy. Including, the step and the skill that need tobeapply. Anykindof questioncanbe solved and resolve by applying the concept andmethod that issuitable.Task 1Write a simple article with yourown word about1. The concept of POLYAS MODEL2. Routine and Non-Routine Problem3. Multiplesstrategiesused forsolvingvarious typesof problems andgive anexample for each strategies.1

You are advice toinclude inyour articles at least 3 varietiesof references.

Task2Elaborate the questionsgivenusing two typesof problemssolvingstrategies.Select

one strategy that isdeemed tobe the most efficient andjustify theirselection.a) Suppose a pair rabbits will produce a new pairof rabbitsin theirsecondmonth, and thereafter will produce a new pair every month. The new rabbitswilldo exactly the same.Start with one pair. How many pairs will there be in10 months?b) Johana has RM 90.00 and Mariam has RM 36.00. They each bought a toy atthe same price. Johana subsequently has 7 times as much as Mariam. Howmuch does the toycost?History of Polya Model

He wasborn asPlyaG

yrgyin Budapest, Hungary, anddiedin Palo Alto,California, USA. He was an excellent problem solver. Earlyon hisuncle tried tosuggest him togointo the mathematics fieldbut he wanted tostudylaw like hislatefather had. However, he became bored with all the study about law. He tiredof thatandswitched to Biology. Then he gettingbored again andswitched to Latin andLiterature, finallygraduating with a degree. Yet, he tiredof that quickly and went backtoschool and took math and physics. He found that he loved math.He wasinvited to teach in Zurich, Switzerland. There he worked with a Dr.


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Weber. One day he met the doctorsdaughterStella he began tocourt her andeventually married her. Theyspent 67 years together. While inSwitzerland he loved2

to take afternoon walksin the localgarden. One day he met a youngcouple alsowalking andchose another path. He continued todo thisyet he met the same couple

six more times as he strolledin the garden. He mentioned to his wife how couldit bepossible to meet them so many times when he randomlychose different pathsthrough the garden.He laterdid experiment according to the situationin the garden that he calledthe random walk problem.Severalyearslater he published a paper proving that ifthe walkcontinuedlong enough that one wassure to return to the starting point.In 1940 he and his wife migrate to the UnitedStatesbecause of theirconcernfor Nazism in Germany. He taught briefly at Brown University and then, for theremainderof hislife, at Stanford University. He quicklybecame wellknown for hisresearch and teachingson problem solving. He taught manyclasses to elementaryandsecondaryclassroom teacherson how to motivate and teach skills to theirstudentsin the area of problem solving.In 1945 he published the book how toSolve It which quicklybecame his mostprized publication. It soldoverone millioncopies and hasbeen translatedinto 17languages. In this text he identifies fourbasic principles.Polyas Four PrinciplesFirst principle: Understand the problemThisseemssoobvious that it isoftennot even mentioned, yet students are oftenstymiedin their efforts tosolve problemssimplybecause theydon't understanditfully, or evenin part. Plya taught teachers to askstudents questionssuch as:

yCanyoustate the problem inyourown words?

yWhat are you trying to findordo?

y

What informationdoyouobtain from the problemyWhat are the unknown?yWhat information , if anyis missingornot needed?3

Doyouneed to ask a question toget the answer?Second principle: Devise a plan

Plya mentions (1957) that there are many reasonable ways tosolve problems. Theskill at choosing an appropriate strategyisbest learnedbysolving many problems.You will findchoosing a strategyincreasingly easy. A partiallist ofstrategiesisincluded:

yGuess andcheck

y

Make anorderlylistyEliminate possibilities

yUse symmetry

yConsiderspecialcases

yUse direct reasoning

ySolve an equationAlsosuggested:

yLook for a pattern


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yDraw a picture

ySolve a simpler problem

yUse a model

yWorkbackward

yUse a formula

yBe creativeyUse your head/noggenThird principle: Carry out the planThisstep isusually easier thandevising the plan. Ingeneral (1957), allyouneediscare and patience, given that you have the necessaryskills. Persist with the plan thatyou have chosen. Ifit continuesnot to workdiscardit andchoose another. Don't bemisled, thisis how mathematicsisdone, evenby professionals.4

yUse the strategyyouselected and work the problem

yCheck each step of the plan asyou proceed

yEnsure that the steps are correct

Fourth principle: Review/extendPlya mentions (1957) that much canbe gainedby taking the time to reflect andlookback at what you have done, what worked and what didn't. Doing this will enable youto predict what strategy touse tosolve future problems, if these relate to the originalproblem.

yReread the question

yDidyou answer the question asked?

yIsyour answercorrect?

yDoesyour answerseems reasonable5

Routine and Non-Routine Problem

Routine andnon-routine are one type of problems that we learnin thissemesterinBasic Mathematics. As we allknow, a problem is a task for which the personconfrontingit want orneed to find a solution and must make an attempt to find asolution.From ourdiscussion and previouslesson that we alreadylearninclassroom, weconclude that routine problem problems are those that merelyinvolved an arithmeticoperation with the characteristicscanbe solvedbydirect applicationof previouslylearned algorithms and the basic taskis toidentify the operation appropriate forsolving problem, gives the factsornumbers touse and presents a question tobeanswered.Inother word, routine problem solvinginvolvesusing at least one of four arithmetic

operations and/or ratio tosolve problems that are practicalinnature. Routineproblem solvingconcerns to a large degree the kindof problem solving that serves asociallyuseful function that hasimmediate and future payoff. The critical matterknows what arithmetic todoin the first place. Actuallydoing the arithmeticissecondary to the matter.Fornon-routine problem, it occurs when anindividualisconfronted with anunusualproblem situation, andisnot aware of a standard procedure forsolvingit. Theindividual has tocreate a procedure. Todoso, we must become familiar with the


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problem situation, collect appropriate information, identify an efficient strategy, anduse the strategy tosolve the problem.Non-routine problem are also those that call for the use of processes far more thanthose of routine problems with the characteristicsuse ofstrategiesinvolvingsomenon-algorithmic approaches andcanbe solvedin manydistinct in many ways

requiringdifferent thinking process.This problem solving alsoserves a different purpose than routine problem solving.While routine problem solvingconcernssolving problems that are useful fordailyliving (in the present orin the future), non-routine problem solvingconcerns that onlyindirectly. Non-routine problem solvingis mostlyconcerned with developingstudentsmathematical reasoning power and fostering the understanding that mathematicsis6

a creative Endeavour. From the point ofview ofstudents, non-routine problemsolvingcanbe challenging andinteresting.It isimportant that we share how tosolve problemsso that our friends are exposedto a varietyofstrategies as well as the idea that there maybe more thanone way toreach a solution. It isunwise to force other people touse one particularstrategy fortwoimportant reasons. First, often more thanone strategycanbe applied tosolvinga problem.Second, the goalis forstudents tosearch for and applyusefulstrategies,not to trainstudents to make use of a particularstrategy.Finally, non-routine problem solvingshouldnot be reserved forspecialstudentssuchas those who finish the regular work early. Allofusshould participate in andbeencouraged tosucceed at non-routine problem solving. Allstudentscanbenefit fromthe kindsof thinking that isinvolvedinnon-routine problem solving.3. Multiply strategy used for solving various types of problem and give anexample for each strategy.Making a listFirst, inorder tosolve the problem byusing a method that is making a list.Making a list is a systematic methodoforganizinginformationin rowsorcolumns. Byputtinggiveninformationin anorderedlist, youcanclearly analyze thisinformationand thensolve the problem bycompleting the list. Example, whenlooking for apatternor rule in a problem, when we listing the problem, the data canbe easilygenerated andorganized the information. We can alsodo a listing result from aguess and test method.Example of question.7

Ali and his entire friend are willbe going to the schoolcampingin HutanSimpan.Histeacher ask Ali tolist out the thing that are need tobring when theygo to thecamping. List out possible things that Ali and his friendneed tobringduring the

camping.Step 1 Understanding the problem.

1. Ali and his want togo to the camping.2. Their teacher asks Ali tolist out things tobring.Step 2 Plan the answer1. Findout the things that isneed forcamping2. List the basic and personal things.3. List the things according to the type


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Step 3 Acting outList the things that isneed forcampingNo Personal things Basic things1 Shirt /Trousers/Tracksuit

Food2 Bags Water3 Water Bottle Fuel4 Medicine Matches5 Shoes Cooking Utensil6 Gloves Wood/Gasstove7 Knife Plate8 Watch Glass9 Compass10 Tent11 Cap

12 Matches13 Map14 Torchlight15 Candle16 Rope17 Mat188

1920Step 4 Look Back1. Determine whether the list is relevant.2.The thingsissuitable for the purposeUsing DiagramThe other method that maybe using tosolve a problem, making a drawingis anexcellent strategyby which youcanvisualize the problem you are asked tosolve bymaking a drawingof the giveninformation. Thisstrategyis especially exceptionalifyou are unable tovisualize the problem inyour mind. Example, we draw the situationof an event, we cansee the situationclearly, such a mapped problem we need toshow the route togo to a placed, so tosolve it we need todraw the route tosee itcrearly.Example of question:I have 4 shirtsone is red, one yellow, one white, andone blue. I have 2 pairsofpants that are black andkhaki andone skirt that isdarkblue. I can wear all thesewith all 4 shirts. How manydifferent outfitsdo I have?Step 1 Understanding the problem1.To find manydifferent outfit from 4 different shirt and 2 trousers 1 skirtStep 2 Devising the Plan

1.Using a diagram inorder tosolve the problem.Step 3 Acting out9


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Results 3 outfits with the redshirt 3 outfits with the yellow shirt 3 outfits with the blue shirt 3 outfits with the white shirt

I have 12 outfits with the clothes that I have in mycloset.10B l a ck p a n t s B lu e ski r t K a h ki p a n t sB lu e S h i r tB l a ck p a n t s B lu e ski r t K a h ki p a n t sR e dS h ir t

B l a ck p a n t s B lu e ski r t K a h ki p a n t sW h i t e S h i r tB l a ck p a n t s B lu e ski r t K a h ki p a n t sY e llo w S h i r tStep 4 Look back

1.The 12 outfit canbe calculatedFinding A Pattern11

Finding a patternis a strategy wherebyyoucanobserve giveninformationsuch aspictures, numbers, letters, words, colours, orsounds. Byobserving each givenelement, one at a time inconsecutive sequence, youcansolve the problem bydeciding what the next element and elements willbe in the pattern. Byusing thismethod also, we can estimate the answer andusingit asinformationsosolve theproblem.Example: Find the next three termsof each sequence byusingconstant differences.A. 1, 3, 5, 7, 9, Step 1 Understanding the problem1.To findnext three termsusingconstant differentStep 2 Devising the Plan

1.Determine the constant different2.Determine the patternof the commondifferentStep 3 Acting outA. 1, 3, 5, 7, 9, 1 3 5 7 9 11 13 15+2 +2 +2 +2 +2 +2 +2Answer:11,13,15The commondifferent is +2Step 4 Look Back1. 15 -13 = 22. 13 - 11= 23. 13 9 = 2Allof the remainingis 2,therefore the pattern and the commondifferent is 2.12

Using TableIn the other hand, making a chart or table is a verygoodstrategy whereby


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informationisorganizedin a clear, readable format; we cansee the result clearly andsee it more reliable. By analyzinginformationin a clear, concise chart, youcaninterpret information andsee what the problem is and how it canbe solved.Oftentimes, after placinggiveninformationin a chart or table, a guide canbedetected this makes the problem easy tosolve.For example rather thanyoulisting a

verylonginformation that issame andkeep repeatingisbetter tousing a table orchart to make it easier tointerpret.Example:In the farm of Pak Hassan, there are about 32 legsof animal, it consist ofbuffaloandduck.How many animal are Pak Hassan have if at least the numberofbothanimalis 2.Step 1 Understanding the problem1.Tocalculate the numberofcow andduck.13

2.At least 2 number each of the animal.Step 2 Devising the plan1. Using the table tosolve the problem2. Applying multiply and addition.Step 3 acting out.Buffalo(4 legs) BuffaloLegsDuck(2 legs) Duck Legs Buffalo+DuckLegs5 20 6 12 322 8 12 24 323 12 10 20 326 24 4 8 327 28 2 4 328 32 0 0 320 0 16 32 32The possiblynumber for Pak Hassan animalin his farm is,Buffalo(4 legs)Duck(2 legs)5 62 123 106 414

7 2Step 4 Look Back

Buffalo(4 legs) Buffalo


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LegsDuck(2 legs) Duck Legs Buffalo+DuckLegs5

20 6 12 3228 1224 323 12 1020 326 24 4 8 327 28 24 321.32 - (62) = 2020 4 = 52.32 (12 2) = 88 4 = 23.32 (102) = 1212 4 = 3

4.32 (4 2) =2424 4 = 6

5.32 ( 2 2) = 2828 4 = 715

Task 2Elaborate the questionsgivenusing two typesof problemssolvingstrategies.Selectone strategy that isdeemed tobe the most efficient andjustify theirselection.c) Suppose a pair rabbits will produce a new pairof rabbitsin theirsecondmonth, and thereafter will produce a new pair every month. The new rabbitswilldo exactly the same.Start with one pair. How many pairs will there be in10 months?Method 1Step 1 Understanding the problem.1. To find the numberof rabbit between 10 month2. To find the totalnumberof rabbitStep 2 Plan the answer4. Each pairof rabbit has to wait forsecond month togive born.5. Calculate the rabbit according to the condition that isgiven.Step 3 Acting out1 (xy)16

2 (xy+xy)

3 (xy+xy)(xy)4 (xy+xy)(xy+xy)(xy)5 (xy+xy)(xy+xy)(xy+xy)(xy)(xy)6 (xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy)(xy)(xy)7 (xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy)(xy)(xy)(xy)(xy)8 (xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy)(xy)(xy)(xy)(xy)


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(xy)(xy)(xy)9 (xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)

10 (xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy+xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)(xy)xy= pair of rabbitTotalnumberof pair rabbit is89 pair.Step 4 Look Back

1. Determine whether the totalnumberof pair rabbit is recalculated.2. Every purpose is applyinorder to findnumberof rabbit.17

Method 2Step 1 Understanding the problem1. To find the numberof rabbit between 10 month2. To find the totalnumberof rabbitStep 2 Devising the Plan1. Each pairof rabbit has to wait forsecond month togive born.2. Calculate the rabbit according to the condition that isgiven18

Step 3 Acting out1st month2ndmonth19Results 1st month: 1 pairof rabbit 2ndmonth: 1 pairof rabbit 3rdmonth: 1 pairof rabbit 4th month: 2 pairof rabbit 5th month: 3 pairof rabbit 6th month: 5 pairof rabbit 7th month: 8 pairof rabbit 8th month: 13 pairof rabbit 9th month:21 pairof rabbit

10th month:34 pairof rabbitTotal pairof rabbit in all 10th month is89 .Step 4 Look back1. Determine whether the totalnumberof pair rabbit is recalculated.2. Every purpose is appliedinorder to findnumberof rabbit.20

a) Johana has RM 90.00 and Mariam has RM 36.00. They each bought a toy atthe same price. Johana subsequently has 7 times as much as Mariam. How


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much does the toycost?Method 1Step 1 Understanding the problem1.To find the cost of the toy.2.The data given Johana has RM90.00,Mariam has RM36.00.

3.The balance of Johana moneyis 7 times as much as Mariam.Step 2 Devising the Plan1.Use strategyofguess andcheckby applyingin form of table.2.List the balance both of them, begin from the lowest numberofsubsequencewhich is the ratio from Joanna toStep 3 Acting outJOHANA (RM 90.00 = x) MARIAM (RM 36.00 = y)PRICEx- 7

*she spent 7 timesthan mariam

SUBSEQUENT PRICEy-1*SUBSEQUENT83 7 35 121

76 14 34 269 21 33 362 28 32 455 35 31 548 42 30 641 49 29 734 56 28 827 53 27 920 70 26 1013 77 11 256 84 12 24Step 4 Look back1. Cost of the toy are obtain2. The balance for Johana is 7 times more than Mariam3. The answer are acceptable and rasionalMethod 2Step 1 Understanding the problem1. To find the cost of the toy2. Both Johana and Mariam have RM90 and RM36 each.Step 2 Devising Plan1. Johana and Mariam use their money tobuy the toy at the same price2. Use Simultaneous equationstrategyStep 3 Acting OutMaking equation90 7y = x -------------1


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36 y = x --------------222

From equation 236 y = x36 x = y --------------3

Substitute equation 3 to equation 190 7( 36 x ) = x90 252 + 7x = x-162 = -6x6x = 162 sub x = 27 into equation 336 27 = yx = 27 soy = 9Balance for Juana Balance for Mariam= 9 x 7 = 9= 63x = 27 price of the toysStep 4 Looking back4. Cost of the toy are obtain5. The balance for Johana is 7 times more than Mariam6. The answer are acceptable and rasional23

ReflectionFirst and foremost, praise to The Almighty God forgivingusgood health andsafety while finishing this math assignment for thissemester.We have face many problems whendo this assignment. First, I donot know whattodo and write. We always make group discussioninorder tocomplete our task.Find the informationusinginternet alsogive usobstacle. The obstacles that wemust face is we found that whenusing this way, we got many pages that related tothis topicbut, for find the accurate andsuitable page, we must read all pages. Notonly that, when we found the information, it give problem indownloading them. But,allof that not breakup ourspirit to finish the assignment.We also read more books to find research about the topic. Although, we hadgotarticles from internet but we alsouse books togain more knowledge. Not only that,this assignment givesus a lot ofknowledge andgrows the positive attitude inourheart such as working as a group. Besides that, we wish we can read the notesonceandimmediatelyunderstand andgrabbed the point easily. We also hope that wecould expressbetterinunderstanding problem solving.Although we face manyobstaclesincompleting this task, we felt verysatisfiedand really thankful. We also feelvery relief and happy when finish this assignment

and hope this assignment willsatisfiedourlecturer andget better result incomingexam.24

BibilografiWebhttp://pred.boun.edu.tr/ps/ps3.htmlhttp://www.geocities.com/polyapower/http://en.wikipedia.org/wiki/George_P%C3%B3lya


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www.lexington1.net/technology/instruct/ppts/mathppts/Numeracy%20&%20Concepts/Problem%20Solving%20II.pptwww.instruction.greenriver.edu/reising/Problem%20Solving%20Strategies.pptwww.oglethorpe.edu/faculty/~k_sorenson/documents/EDPThinkingandproblemsolving.p

ptwww.lessonplanet.com/search?keywords=problem+solving+-math&rating=3 - 31k www.math.twsu.edu/history/Men/polya.htmlBookGeoge Polya,How ToSolve It.25

10 Describe Polya

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