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INTRODUCTION

Problem solving is a process by which an individual required prior end recentknowledge, thinking skills, relevant strategies and understanding to reach the

demand or the goal of the unfamiliar situation. Problem solving involving a situation

whereby an individual or a group is required to carry out the working solution.

Mathematical problems should comes from various contexts : real life contexts,

mathematical contexts, imaginary contexts or physical contexts. Pupils should

understand mathematical concepts first, before acquiring the problem-solving skills.

According to Smith ( 2001,5 ), problem-solving skills can only be developedafter pupils have gone through different level of problems. The levels of problem are

shown in table below. In schools, pupils are exposed to different types and levels of

problems. This is the process of how pupils developed different level of problem-

solvong skills before they are able to solve non-routine problems. Problems can be

posed according to type and level as follows : (a) puzzles, (b) quizzes, (c) drill

exercises, (d) simple translation, (e) multiple-step translation, (f) applied problems,

(g) routine problems and (h) non-routine problems.

Table: Levels of Problem

Level Criteria

Own words problems The problems require you to discuss or rephrase main ideas or

procedures using your own words.

Level 1 Problems These are mechanical and drill problems, and are directly

related to examples in the book.

Level 2 Problems These problems required understanding of the concept and

closely related to the example in the book.

Level 3 Problems These problems are the extension of the examples, but

generally do not have corresponding examples in the book.

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Problem Solving These problems require problem solving skills or original

thinking and generally do not have direct examples in the book.

Research Problems These problems required internet research or library work.

Most are intended for individual research but a few are group

research projects.

LITERATURE REVIEW

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George Polba was born in Hungary in 1887. After receiving his PhD at

University of Budapest, where his dissertation involved questions in probability, he

taught at the Swiss Federal Institute of Technology in Zurich. In 1940, he came to

Brown University in the United States and then joined the faculty at Stanford

University in 1942.

In his studies, he became interested in the process of discovery, or how

mathematical results were derived. He felt that to understand a theory, one must

know how it was discovered. Thus his teaching emphasized the process of discovery

rather than simply the development of appropriate skills. To promote the problem

solving approach, he developed the following 4 steps.1) Understand the problem.

2) Devise a plan.

3) Carry out the plan.

4) Look back.

Polyas accomplishments include over 250 mathematical papers and three

books that promote his popular approach to problem solving. His famous book How

to Solve It, which has been translated into 15 languages, introduced his four-step

approach together with heuristics, or strategies, which are helpful in solving

problems. Other important works of Polya are Mathematical Discovery, Volumes I

and II, and Mathematics and Plausible Reasoning, Volumes I and II.

Polya, who died in 1985 at the age of 97, left mathematics with an important

legacy of teaching for problem solving. In addition, he left the following Ten

Commandments for Teachers.

1) Be interested in your subject.

2) Know your subject.

3) Try to read the faces of your students, try to see their expectations and

difficulties; put yourself in their place.

4) Realize that the best way to learn anything is to discover it by yourself.

5) Give your students not only information but know-how, mental attitudes, thehabit of methodical work.

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6) Let them learn guessing.

7) Let them learn proving.

8) Look out for such features of the problem at hand as may be useful in solving

the problems to come try to disclose the general pattern that lies behind the

present concrete situation.

9) Do not give away your whole secret at once let the students guess before

you tell it let them find out by themselves as much as is feasible.

10)Suggest it; do not force it down their throats.

METHODOLOGY

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A famous mathematician, George Polya, devoted much of his teaching to

helping students become better problem solvers. His major contribution is what has

been known as the four- steps process for solving problems.

Step 1: Understand the Problem

a) Do you understand all the words?

b) Can you restate the problem in your own words?

c) Do you know what is given?

d) Do you know what the goal is?

e) Is there enough information?

f) Is there extraneous information?

g) Is this problem similar to another problem you have solved?

S tep 2: Devise a plan

(Can one of the following strategies (heuristics) be used? (A strategy is

defined as an artful means to an end.)

1) Guess and test.

2) Use a variable.

3) Look for a pattern.

4) Make a list.

5) Solve a simpler problem.

6) Draw a picture.

7) Draw a diagram.

8) Use direct reasoning.

9) Use indirect reasoning.10)Use properties of numbers.

11)Solve an equivalent problem.

12)Work backward.

13)Use cases.

14)Solve an equation.

15)Look for a formula.

16)Do a simulation.

17)Use a model.

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18)Use dimensional analysis.

19)Use sub goals.

20)Use coordinates.

21)Use symmetry.

Step 3: Carry out the plan.

a) Implement the strategy or strategies that you have chosen until the

problem is solved or until the new course of action is suggested.

b) Give yourself a reasonable amount of time in which to solve the

problem. If you are not successful, seek hints from others or put the

problem aside for a while. (You may have a flash of insight when you

least expect it!)

c) Do not be afraid starting over. Often, a fresh start and a new strategy

will lead to success.

Step 4: Look back.

a) Is your solution is correct? Does your answer is satisfy the statement

of the problem?

b) Can you see the easier solution?c) Can you see how you can extend your solution to more general

cases?

Routine and Non-Routine Problem

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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 a creative Endeavour. From the point of view of students, non-routine

problem solving can be challenging and interesting.

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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.

ANALYZING PROBLEM

PROBLEM 1

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On Ariannas way to visit her friend, she leave her house at 2:45 P.M. and travel 1

miles to the train, 12 miles on the train, and mile to her friend's house from the

train station. If she get there at 4:45 P.M., how many miles per hour did she travel?

By using the Polyas Model, we choose the SIMPLIFY THE

PROBLEM as our strategy to solve the Problem 1.

Step 1: Understand The Problem.

Arianna left her house at 2:45 P.M.

She travelled for 1 miles to train, 12 miles on train, and miles to her

friends house.

She arrived at her friends house at 4:45 P.M.

Step 2: Devise a Plan

For this problem, it might be helpful for students to use simpler numbers to learn the

steps they need to follow to solve it. Have students change the problem to:

Arianna left the house at 2:45 P.M.

She travelled for 15 miles

She arrived at 4:45 P.M.

How many miles per hour did she travelled?

Step 3: Carry Out The Plan.

Find the distance travelled.

1 + 12 + = 15 miles

Find the time spent.

The time from 2:45 to 4:45 is 2 hours.

Divide to find the miles per hour.

15 divided by 2 hours = 7 miles per hour or 7.5 miles per hour.

Step 4: Check out the answer.

Then, they should check the math to be sure it is correct.

1 + 12 + = 15 miles 2:45 P.M. to 4:45 P.M. is 2 hours.

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15 miles divided by 2 hours = 7.5 or 7 miles per hour

PROBLEM 2

Alissa is wrapping presents for her friends. She has made 10 rings for 10 friends

using brightly colored polymer clay. She has bought 10 little jewelry boxes and now

she is shopping for wrapping paper and ribbon. She estimates that she needs a

rectangle of paper 20cm by 15cm to wrap each box. She finds lovely silver

wrapping paper that is sold in 60cm x 60cm sheets. Since the paper is expensive,she does not want to buy too much. How many sheets should she buy?

By using the Polyas Model, we choose the USING A DIAGRAM as our

strategy to solve the Problem 2.


Alissa made 10 rings for her 10 friends.

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She bought 10 jewellery box and she want to wrap the box with wrapping

papers and ribbon.

She estimates that she needs a rectangle of paper 20cm by 15cm to wrap

each box.

The wrapping paper sold is in 60cm x 60cm sheets.

How many sheet of wrapping papers should she buy?


For this problem, it might be helpful for students to use diagram to simplify the

information and the requirement of the students.

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60 cm

20 cm 20 cm 20 cm

60 cm

15 cm

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Step 3: Carry Out The Plan.

Find the area of the 60 cm 60 cm sheet wrapping paper

60 cm 60 cm = 3600 cm2

Find the area for each piece of small wrapping paper that is 20 cm by 15 cm,

needed to wrap each box.

20 cm 15 cm = 300 cm2

Calculate the number of small wrapping paper that can be obtained from a

piece of silver wrapping paper.

3600 cm2 300 cm2 = 12 pieces of small wrapping paper.

Since we need only 10 pieces of small wrapping paper, therefore she only

have to buy one piece of silver wrapping paper.

12

15 cm

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13

60 cm

20 cm 20 cm 20 cm

60 cm

15 cm

15 cm

15 cm


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Step 4: Check out the answer.

Then, they should check the math to be sure it is correct.

Multiply the area of one piece of small wrapping paper with 10.

300 cm2 10 = 3000 cm2

Compare with the area of one piece of silver wrapping paper.

Therefore, it is shown that the answer is correct.

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PROBLEM 3

In the farm of Pak Hassan, there are about 32 legs of animal, it consist of buffaloand duck. How many animals is Pak Hassan having if at least the number of both

animals is 2?

By using the Polyas Model, we choose the USING A TABLE as our

strategy to solve the Problem 3.


1) To calculate the number of cow and duck.

2) At least 2 number each of the animals.


1) Using the table to solve the problem.

2) Applying multiply and addition.

Step 3: Acting out.

Buffalo

(4 legs)

Buffalo

Legs

Duck

(2 legs)

Duck Legs Buffalo

+Duck

Legs

5 20 6 12 32

2 8 12 24 32

3 12 10 20 32

6 24 4 8 32

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7 28 2 4 32

8 32 0 0 32

0 0 16 32 32

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

Buffalo

(4 legs)

Duck

(2 legs)

5 6

2 12

3 10

6 4

7 2

Step 4: Look Back

Buffalo

(4 legs)

Buffalo

Legs

Duck

(2 legs)

Duck Legs Buffalo

+Duck Legs

5 20 6 12 32

2 8 12 24 32

3 12 10 20 32

6 24 4 8 32

7 28 2 4 32

1. 32 - (62) = 20

20 4 = 5

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2. 32 (12 2) = 8

8 4 = 2

3. 32 (102) = 12

12 4 = 3

1. 32 (4 2) = 24

24 4 = 6

2. 32 ( 2 2) = 28

28 4 = 7

CONCLUSION

We can conclude that there are many type of problem solving that can be

used in solving daily problem. In PROBLEM 1, we decide to solve our created

problem by using SIMPLIFY THE PROBLEM method. This method helps us discover

relationships and patterns among data. It encourages us to organize information in a

logical way and to look critically at the data to find patterns and develop a solution.

DRAWING A DIAGRAM is the most common problem solving strategy. We

use the strategy of drawing a diagram again and again as we show in PROBLEM 2.

First we need to learn how to interpret a problem and draw a useful diagram. Very

often, we need to draw a diagram just to understand the meaning of the problem.

The diagram represents the problem in a way we can see it, understand it, and

think about it while we look for the next step.

Problem solving using tables might seem complicated, but it is easily mastered

with some instruction. Not all of the types of question need us to CONSTRUCT A

TABLE. It depends on the question. For PROBLEM 3, we use a table so that we can

solve the problem easily.

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In conclusion, we realize that there are many strategies that we can use in

solving a problem. All of these strategies can be stretched when combined with other

strategies such as looking for patterns or drawing a picture. By combining this

strategy with others, we can analyze the data that is given to find more complex

relationships.

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