Ppisem Sem 3 Assignment Mathematic:history Of Polya Model

Acknowledgement

Alhamdulillah… Thank to Allah S.W.T because give us an effort to do and finish our assignment on our Mathematic.

We have been given a task from our lecture that is,

write an essay about George Polya and The solving strategy.. All we have to do is searching the information and using the information that is given by my lecture, Mr. Ahmad Rizal Bin Che Rahim ,thank a lot to him because he already give us many information and advice that is very useful. He help us a lot in order to finish the assignment. We also want to thank to our entire friend who help us in the process of making and finishing our assignment. Thank to their information and advice in order to help out to make my assignment better. By doing this assignment we have learned many thing about the George Polya and Problem Solving Strategy. Including, the step and the skill that need to be apply. Any kind of question can be solved and resolve by applying the concept and method that is suitable.

Task 1 Write a simple article with your own word about 1. The concept of POLYA’S MODEL 2. Routine and Non-Routine Problem 3. Multiples strategies used for solving various types of problems and give an example for each strategies. 1

You are advice to include in your articles at least 3 varieties of references.

Task 2 Elaborate the questions given using two types of problems solving strategies. Select one strategy that is deemed to be the most efficient and justify their selection. a) Suppose a pair rabbits will produce a new pair of rabbits in their second month, and thereafter will produce a new pair every month. The new rabbits will do exactly the same. Start with one pair. How many pairs will there be in 10 months? b) Johana has RM 90.00 and Mariam has RM 36.00. They each bought a toy at the same price. Johana subsequently has 7 times as much as Mariam. How much does the toy cost?

History of Polya Model

He was born as Pólya György in Budapest, Hungary, and died in Palo Alto, California, USA. He was an excellent problem solver. Early on his uncle tried to suggest him to go into the mathematics field but he wanted to study law like his late father had. However, he became bored with all the study about law. He tired of that and switched to Biology. Then he getting bored again and switched to Latin and Literature, finally graduating with a degree. Yet, he tired of that quickly and went back to school and took math and physics. He found that he loved math. He was invited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. One day he met the doctor’s daughter Stella he began to court her and eventually married her. They spent 67 years together. While in Switzerland he loved 2

to take afternoon walks in the local garden. One day he met a young couple also walking and chose another path. He continued to do this yet he met the same couple six more times as he strolled in the garden. He mentioned to his wife “how could it be possible to meet them so many times when he randomly chose different paths through the garden”. He later did experiment according to the situation in the garden that he called the random walk problem. Several years later he published a paper proving that if the walk continued long enough that one was sure to return to the starting point. In 1940 he and his wife migrate to the United States because of their concern for Nazism in Germany. He taught briefly at Brown University and then, for the remainder of his life, at Stanford University. He quickly became well known for his research and teachings on problem solving. He taught many classes to elementary and secondary classroom teachers on how to motivate and teach skills to their students in the area of problem solving. In 1945 he published the book how to Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this text he identifies four basic principles.

Polya’s Four Principles First principle: Understand the problem This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don't understand it fully, or even in part. Pólya taught teachers to ask students questions such as: •

Can you state the problem in your own words?

•

What are you trying to find or do?

•

What information do you obtain from the problem

•

What are the unknown?

•

What information , if any is missing or not needed? 3

Do you need to ask a question to get the answer? Second principle: Devise a plan Pólya mentions (1957) that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included: •

Guess and check

•

Make an orderly list

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Eliminate possibilities

•

Use symmetry

•

Consider special cases

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Use direct reasoning

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Solve an equation

Also suggested: •

Look for a pattern

•

Draw a picture

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Solve a simpler problem

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Use a model

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Work backward

•

Use a formula

•

Be creative

•

Use your head/noggen

Third principle: Carry out the plan This step is usually easier than devising the plan. In general (1957), all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled, this is how mathematics is done, even by professionals. 4

•

Use the strategy you selected and work the problem

•

Check each step of the plan as you proceed

•

Ensure that the steps are correct

Fourth principle: Review/extend Pólya mentions (1957) that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem. •

Reread the question

•

Did you answer the question asked?

•

Is your answer correct?

•

Does your answer seems reasonable

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Routine and Non-Routine Problem Routine and non-routine are one type of problems that we learn in this semester in Basic Mathematics. As we all know, a problem is a task for which the person confronting it want or need to find a solution and must make an attempt to find a solution. From our discussion and previous lesson that we already learn in classroom, we conclude that routine problem problems are those that merely involved an arithmetic operation with the characteristics can be solved by direct application of previously learned algorithms and the basic task is to identify the operation appropriate for solving problem, gives the facts or numbers to use and presents a question to be answered. In other word, routine problem solving involves using at least one of four arithmetic operations and/or ratio to solve problems that are practical in nature. Routine problem solving concerns to a large degree the kind of problem solving that serves a socially useful function that has immediate and future payoff. The critical matter knows what arithmetic to do in the first place. Actually doing the arithmetic is secondary to the matter. For non-routine problem, it occurs when an individual is confronted with an unusual problem situation, and is not aware of a standard procedure for solving it. The individual has to create a procedure. To do so, we must become familiar with the problem situation, collect appropriate information, identify an efficient strategy, and use the strategy to solve the problem. Non-routine problem are also those that call for the use of processes far more than those of routine problems with the characteristics use of strategies involving some non-algorithmic approaches and can be solved in many distinct in many ways requiring different thinking process. This problem solving also serves a different purpose than routine problem solving. While routine problem solving concerns solving problems that are useful for daily living (in the present or in the future), non-routine problem solving concerns that only indirectly. Non-routine problem solving is mostly concerned with developing students’ mathematical reasoning power and fostering the understanding that mathematics is 6

a creative Endeavour. From the point of view of students, non-routine problem solving can be challenging and interesting. It is important that we share how to solve problems so that our friends are exposed to a variety of strategies as well as the idea that there may be more than one way to reach a solution. It is unwise to force other people to use one particular strategy for two important reasons. First, often more than one strategy can be applied to solving a problem. Second, the goal is for students to search for and apply useful strategies, not to train students to make use of a particular strategy. Finally, non-routine problem solving should not be reserved for special students such as those who finish the regular work early. All of us should participate in and be encouraged to succeed at non-routine problem solving. All students can benefit from the kinds of thinking that is involved in non-routine problem solving.

3. Multiply strategy used for solving various types of problem and give an example for each strategy. Making a list First, in order to solve the problem by using a method that is making a list. Making a list is a systematic method of organizing information in rows or columns. By putting given information in an ordered list, you can clearly analyze this information and then solve the problem by completing the list. Example, when looking for a pattern or rule in a problem, when we listing the problem, the data can be easily generated and organized the information. We can also do a listing result from a guess and test method. Example of question. 7

Ali and his entire friend are will be going to the school camping in Hutan Simpan.His teacher ask Ali to list out the thing that are need to bring when they go to the camping. List out possible things that Ali and his friend need to bring during the camping. Step 1 Understanding the problem. 1. Ali and his want to go to the camping. 2. Their teacher asks Ali to list out things to bring. Step 2 Plan the answer 1. Find out the things that is need for camping 2. List the basic and personal things. 3. List the things according to the type

Step 3 Acting out List the things that is need for camping No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Personal things Shirt /Trousers /Track suit Bags Water Bottle Medicine Shoes Gloves Knife Watch Compass Tent Cap Matches Map Torchlight Candle Rope Mat

Basic things Food Water Fuel Matches Cooking Utensil Wood/Gas stove Plate Glass

8

19 20

Step 4 Look Back 1. Determine whether the list is relevant. 2.The things is suitable for the purpose

Using Diagram The other method that may be using to solve a problem, making a drawing is an excellent strategy by which you can visualize the problem you are asked to solve by making a drawing of the given information. This strategy is especially exceptional if you are unable to visualize the problem in your mind. Example, we draw the situation of an event, we can see the situation clearly, such a mapped problem we need to show the route to go to a placed, so to solve it we need to draw the route to see it crearly. Example of question: I have 4 shirts one is red, one yellow, one white, and one blue. I have 2 pairs of pants that are black and khaki and one skirt that is dark blue. I can wear all these with all 4 shirts. How many different outfits do I have? Step 1 Understanding the problem 1.To find many different outfit from 4 different shirt and 2 trousers 1 skirt Step 2 Devising the Plan 1.Using a diagram in order to solve the problem. Step 3 Acting out

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R e d S h ir t B la c k p a n t s

B lu e s k ir t

B lu e S h ir t K a h k i p a n ts

B la c k p a n t s

Y e llo w S h ir t B la c k p a n t s

B lu e s k ir t

B lu e s k ir t

K a h k i p a n ts

W h it e S h ir t K a h k i p a n ts

B la c k p a n t s

Results •

3 outfits with the red shirt

•

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 my closet. 10

B lu e s k ir t

K a h k i p a n ts

Step 4 Look back 1.The 12 outfit can be calculated

Finding A Pattern 11

Finding a pattern is a strategy whereby you can observe given information such as pictures, numbers, letters, words, colours, or sounds. By observing each given element, one at a time in consecutive sequence, you can solve the problem by deciding what the next element and elements will be in the pattern. By using this method also, we can estimate the answer and using it as information so solve the problem. Example: •

Find the next three terms of each sequence by using constant differences.

A. 1, 3, 5, 7, 9, …

Step 1 Understanding the problem 1.To find next three terms using constant different Step 2 Devising the Plan 1.Determine the constant different 2.Determine the pattern of the common different Step 3 Acting out A. 1, 3, 5, 7, 9, …

1 3 5 7 9 11 13 15

+2

+2

+2

+2

+2

+2

+2

Answer:11,13,15 The common different is +2

Step 4 Look Back 1. 15 -13 = 2 2. 13 - 11= 2 3. 13 – 9 = 2 All of the remaining is 2,therefore the pattern and the common different is 2.

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Using Table In the other hand, making a chart or table is a very good strategy whereby information is organized in a clear, readable format; we can see the result clearly and see it more reliable. By analyzing information in a clear, concise chart, you can interpret information and see what the problem is and how it can be solved. Oftentimes, after placing given information in a chart or table, a guide can be detected this makes the problem easy to solve.For example rather than you listing a very long information that is same and keep repeating is better to using a table or chart to make it easier to interpret. Example: In the farm of Pak Hassan, there are about 32 legs of animal, it consist of buffalo and duck.How many animal are Pak Hassan have if at least the number of both animal is 2. Step 1 Understanding the problem 1.To calculate the number of cow and duck. 13

2.At least 2 number each of the animal. Step 2 Devising the plan 1. Using the table to solve the problem 2. Applying multiply and addition.

Step 3 acting out. Buffalo (4 legs)

Duck Buffalo

(2 legs)

Duck Legs

Legs 5 2 3 6 7 8 0

Buffalo +Duck

20 8 12 24 28 32 0

6 12 10 4 2 0 16

12 24 20 8 4 0 32

The possibly number for Pak Hassan animal in his farm is,

Buffalo

Duck

(4 legs)

(2 legs) 5 2 3 6

6 12 10 4 14

Legs 32 32 32 32 32 32 32

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2

Step 4 Look Back Buffalo (4 legs)

Duck Buffalo

(2 legs)

Duck Legs

Legs 5 2 3 6 7

20 8 12 24 28

Buffalo +Duck

6 12 10 4 2

1. 32 - (6×2) = 20

20 ÷4 = 5 2. 32 – (12 × 2) = 8

8÷4=2 3. 32 –(10×2) = 12 12 ÷ 4 = 3 4. 32 –(4 × 2) = 24

24 ÷ 4 = 6 5. 32 – ( 2 × 2) = 28

28 ÷ 4 = 7

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12 24 20 8 4

Legs 32 32 32 32 32

Task 2 Elaborate the questions given using two types of problems solving strategies. Select one strategy that is deemed to be the most efficient and justify their selection. c) Suppose a pair rabbits will produce a new pair of rabbits in their second month, and thereafter will produce a new pair every month. The new rabbits will do exactly the same. Start with one pair. How many pairs will there be in 10 months? Method 1 Step 1 Understanding the problem. 1. To find the number of rabbit between 10 month 2. To find the total number of rabbit Step 2 Plan the answer 4. Each pair of rabbit has to wait for second month to give born. 5. Calculate the rabbit according to the condition that is given.

Step 3 Acting out 1

(xy)

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2

(xy+xy)

3

(xy+xy)(xy)

4

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

5

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

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

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(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+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)

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(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)(xy)(xy)(xy)(xy)(xy)(xy)

xy= pair of rabbit Total number of pair rabbit is 89 pair. Step 4 Look Back 1. Determine whether the total number of pair rabbit is recalculated. 2. Every purpose is apply in order to find number of rabbit. 17

Method 2

Step 1 Understanding the problem 1. To find the number of rabbit between 10 month 2. To find the total number of rabbit

Step 2 Devising the Plan 1. Each pair of rabbit has to wait for second month to give born. 2. Calculate the rabbit according to the condition that is given

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Step 3 Acting out

1st month

2nd month

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Results •

1st month: 1 pair of rabbit

•

2nd month: 1 pair of rabbit

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3rd month: 1 pair of rabbit

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4th month: 2 pair of rabbit

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5th month: 3 pair of rabbit

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6th month: 5 pair of rabbit

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7th month: 8 pair of rabbit

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8th month: 13 pair of rabbit

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9th month:21 pair of rabbit

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10th month:34 pair of rabbit

Total pair of rabbit in all 10th month is 89.

Step 4 Look back 1. Determine whether the total number of pair rabbit is recalculated. 2. Every purpose is applied in order to find number of rabbit.

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a) Johana has RM 90.00 and Mariam has RM 36.00. They each bought a toy at the same price. Johana subsequently has 7 times as much as Mariam. How much does the toy cost? Method 1 Step 1 Understanding the problem 1.To find the cost of the toy. 2.The data given Johana has RM90.00,Mariam has RM36.00. 3.The balance of Johana money is 7 times as much as Mariam. Step 2 Devising the Plan 1.Use strategy of guess and check by applying in form of table. 2.List the balance both of them, begin from the lowest number ofsubsequence which is the ratio from Joanna to

Step 3 Acting out JOHANA (RM 90.00 = x) PRICE SUBSEQUENT

MARIAM (RM 36.00 = y) PRICE SUBSEQUENT

x- 7

y-1

*she spent 7 times

*

than mariam 83

7

35 21

1

76 69 62 55 48 41 34 27 20 13 6

14 21 28 35 42 49 56 53 70 77 84

34 33 32 31 30 29 28 27 26 11 12

2 3 4 5 6 7 8 9 10 25 24

Step 4 Look back 1. Cost of the toy are obtain 2. The balance for Johana is 7 times more than Mariam 3. The answer are acceptable and rasional

Method 2

Step 1 Understanding the problem 1. To find the cost of the toy 2. Both Johana and Mariam have RM90 and RM36 each. Step 2 Devising Plan 1. Johana and Mariam use their money to buy the toy at the same price 2. Use Simultaneous equation strategy Step 3 Acting Out Making equation 90 – 7y = x -------------1 36 – y = x --------------2 22

From equation 2 36 – y = x 36 – x = y --------------3 Substitute equation 3 to equation 1 90 – 7( 36 – x ) = x 90 – 252 + 7x = x -162 = -6x 6x = 162

sub x = 27 into equation 3 36 – 27 = y

x = 27

so y = 9

Balance for Juana

Balance for Mariam

=9x7

=9

= 63 x = 27 price of the toys

Step 4 Looking back 4. Cost of the toy are obtain 5. The balance for Johana is 7 times more than Mariam 6. The answer are acceptable and rasional

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Reflection

First and foremost, praise to The Almighty God for giving us good health and safety while finishing this math assignment for this semester. We have face many problems when do this assignment. First, I do not know what to do and write. We always make group discussion in order to complete our task. Find the information using internet also give us obstacle. The obstacles that we must face is we found that when using this way, we got many pages that related to this topic but, for find the accurate and suitable page, we must read all pages. Not only that, when we found the information, it give problem in downloading them. But, all of that not break up our spirit to finish the assignment. We also read more books to find research about the topic. Although, we had got articles from internet but we also use books to gain more knowledge. Not only that, this assignment gives us a lot of knowledge and grows the positive attitude in our heart such as working as a group. Besides that, we wish we can read the notes once and immediately understand and grabbed the point easily. We also hope that we could express better in understanding problem solving. Although we face many obstacles in completing this task, we felt very satisfied and really thankful. We also feel very relief and happy when finish this assignment and hope this assignment will satisfied our lecturer and get better result in coming exam.

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Bibilografi Web http://pred.boun.edu.tr/ps/ps3.html http://www.geocities.com/polyapower/ http://en.wikipedia.org/wiki/George_P%C3%B3lya www.lexington1.net/technology/instruct/ppts/mathppts/Numeracy%20&%20Concepts/Proble m%20Solving%20II.ppt www.instruction.greenriver.edu/reising/Problem%20Solving%20Strategies.ppt www.oglethorpe.edu/faculty/~k_sorenson/documents/EDPThinkingandproblemsolving.ppt www.lessonplanet.com/search?keywords=problem+solving+-math&rating=3 - 31k – www.math.twsu.edu/history/Men/polya.html Book Geoge Polya,How To Solve It.

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