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1.0: INTRODUCTION OF PROBLEM

SOLVING

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1.0: INTRODUCTION OF PROBLEM SOLVING

Actually, problem involving a situation whereby an individual or a group is

required to carry out the working solution. In doing so, they have to determined the

strategy and method of problem-solving first, before implementing the working

solution. The strategy of problem-solving needs a set of activities which will lead to

the problem-solving process.

1.1.1: Types of Problem Solving

Basically, there are two types of problem-solving:

1) Routine problem

2) Non-routine problem

1.1.1.1: Routine Problem

Routine problem is actually a type of mechanical mathematics problem. It is

aimed at training the pupils so that they are able to master basic skills, especially the

arithmetic skills involving the four mathematics operations or direct application of

using mathematics formulae, laws, theorems or equations.

1.1.1.2: Non-Routine Problem

Non-routine problem is a kind of unique problem-solving which requires the

application of skills, concepts or principle which have been learned and mastered.

Thus, the method for solving non-routine problem in mathematics cannot be

memorized, and is not the same as answering certain mechanical question. The

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process of problem-solving needs a set of systematic activities with logical planning,

including proper strategy and selection of suitable method for implementation.

1.1.2: Five Stages of Problem Solving

i. To identify the problem

ii. To look for clues / information

iii. To set up hypothesis

iv. To test the hypothesis

v. To evaluate and record final conclusion

1.1.3: Strategies of Problem-Solving

To ascertain the pattern of the solution

To construct a problem-solving table

To study all possible solutions

To implement the strategies

To plan a solution model

To guess for the answer and checking

To work backwards from the ending stage to the beginning stage

To draw pictures, diagrams or graph

To select relevant mathematics notations

To explain the problem by using individual’s own words

To identify important information given

To check assumptions which are not clearly stated

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1.1.4: Methods of Problem-Solving

Figure 1: Methods of problem-solving

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2.0: GEORGE POLYA BIODATA

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2.0: GEORGE POLYA BIODATA

George Polya was a Hungarian who immigrated to the United States in 1940.

His major contribution is for his work in problem solving.

Growing up he was very frustrated with the practice of having to regularly

memorize information. He was an excellent problem solver. Early on his uncle tried

to convince him to go into the mathematics field but he wanted to study law like his

late father had. After a time at law school he became bored with all the legal

technicalities he had to memorize. He tired of that and switched to Biology and the

again 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 he

loved math.

His first job was to tutor Gregor the young son of a baron. Gregor struggled

due to his lack of problem solving skills. Polya (Reimer, 1995) spent hours and

developed a method of problem solving that would work for Gregor as well as others

in the same situation. Polya (Long, 1996) maintained that the skill of problem was

not an inborn quality but, something that could be taught.

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

to take afternoon walks in the local garden. One day he met a young couple also

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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 it could be

possible to meet them so many times when he randomly chose different paths

through the garden?.

He later did experiment 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 moved to the United States because of their concern

for Nazism in Germany (Long, 1996). 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.

George Polya went on to publish a two-volume set, Mathematics and

Plausible Reasoning (1954) and Mathematical Discovery (1962). These texts form

the basis for the current thinking in mathematics education and are as timely and

important today as when they were written. Polya has become known as the father

of problem solving.

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2.1.1: Polya’s Model

2.1.1.1: Polya’s 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. Polya taught teachers to ask students questions such as:

Do you understand all the words used in stating the problem?

What are you asked to find or show?

Can you restate the problem in your own words?

Can you think of a picture or a diagram that might help you understand the

problem?

Is there enough information to enable you to find a solution?

2.1.1.2: Polya’s Second Principle: Devise a plan

Polya mentions (1957) that it is 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 and orderly list

Eliminate possibilities

Use symmetry

Consider special cases

Use direct reasoning

Solve an equation

Look for a pattern

Draw a picture

Solve a simpler problem

Use a model

Work backward

Use a formula

Be ingenious

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2.1.1.3: Polya’s 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. You must

persistent 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.

2.1.4: Polya’s Fourth Principle: Look Back

Polya 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.

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3.0: NON-ROUTINE PROBLEM’S QUESTIONS

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3.0: NON-ROUTINE PROBLEM’S QUESTIONS

3.1.1: Question 1 (The Bottom Line)

Look carefully at each line of numbers in the number pyramid. What number should

replace the question mark in the middle of the bottom line? Explain why.

82 5 2

1 2 4 2 11 2 1 3 1 2 1

1 2 1 1 ? 1 1 2 1

How to solve it?

According to Polya’s Model, there are four steps in the problem solving process.

First step: Understanding the problem

1) Can you state the problem in your own words?

There is given a line number of pyramid. The question asked what the number is

should be replaced the question mark.

2) What are you trying to find or do?

We are finding the number that should be replaced by the question mark. To find that

number, we must observe the pattern for each line number in the pyramid.

3) What information do you obtain from the problem?

We noticed that there are five line number in the pyramid. Each line number we can

mark as 1st horizontal line until 5th horizontal line. We also know that each line consist

of one and more numbers.

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4) What are the unknown?

The unknown that we are going to find in this question is represented by question

mark.

5) State all information.

Each horizontal line consist of

1st line: 8

2nd line: 2, 5, 2

3rd line: 1, 2, 4, 2, 1

4th line: 1, 2, 1, 3, 1, 2, 1

5th line: 1, 2, 1, 1, ?, 1, 1, 2, 1

Second step: Devising a plan

1) Find the connection between data and the unknown

It seems that when the line number is goes down, the total numbers for each line is

increasing. The question mark is placed in the 5th line of the pyramid.

2) Consider auxiliary problem if an immediate connection can be found

What operation should we use to show the connection between the ? and the line

number in each row in the pyramid.

3) What strategies are you going to use?

First strategy: Look for a pattern

Second strategy: Analysis Strategy

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4) Try a strategy that seem its work.

Use a first strategy and second strategy to solve the problem.

Third step: Carrying out the plan

1) Use strategy you selected and work the problem

First Strategy: Look for a pattern

Each horizontal line,

Total for the 1st line: 8

Total for the 2nd line: 2+5+2 = 9

Total for the 3rd line: 1+2+4+2+1 = 10

Total for the 4th line: 1+2+1+3+1+2+1 = 11

Total for the 5th line: 1+2+1+1+? +1+1+2+1 = K

Let K represented the total number for the 5th line.

K is unknown. From the number pattern, we can guess that K=12 because when the

line number is goes down the pyramid, the total for each line number is increasing by

1.

Hence,

1+2+1+1+?+1+1+2+1 = 12

? = 2.

2) Check each step of the plan

We can check by subtraction method.

12-1-2-1-1-1-1-2-1 = 2

Answer: ? = 2

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3) Ensure that the steps are corrected.

The solution answer and the checking answer is the same that is 2. So, we are

confidently that ? = 2. So, 2 is the answer

Answer: The number 2 should replace the question mark. Each horizontal row adds

up to one more than the one above it, so the last row must add up to 12.

Second strategy: Analysis Strategy

i. Problem-solving strategy

a) What are given?

Each line of numbers that filled in the number of pyramid.

b) What to find?

The number that should replace the question mark in the middle

of the bottom line of the number of pyramid.

c) How to find?

Use Analysis Strategy.

ii. Solution:

We know that each line in the number of pyramid is increasing

by one when goes down the line number of pyramid number.

So, the total number for 5th line number of pyramid is 12.

Therefore, we need to add all the number in the 5th line number

of pyramid together and then the answer that we got is then we

minus by 12.

iii. Answer:

Hence, the answer that we can obtain is 2.

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Four step: Working back

1) Reread the question

“What number should replace the question mark in the middle of the bottom line?

Explain why.”

2) Did you answer the question?

Yes.

3.1.2: Question 2 (Six Daughters)

Mr. Seibold has 6 daughters. Each daughter is 4 years older than her next younger

sister. The oldest daughter is 3 times as old than her youngest sister .How old is

each of the daughters?

How to solve it?




The question asked us to find the age of all Mr Seibold’s daughter.


We must find the age of each Mr. Seibold’s daughters by using the information

given. The clue here is six daughters. So, we can create six unknown to represent

each daughters.

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There are stated we must find the age of six daughters.

We can create 6 unknown to represent the ages of each daughters. We use a, b, c,

d, e, f as our unknowns.

4) What are the unknown?

a, b, c, d, e, f

5) State all information.

The question stated that ‘each daughter is 4 years older than her next younger sister’

Then, we can build equations that related to these unknowns.

+4 +4 +4 +4 +4

a, b, c, d, e, f

a= represent the oldest daughter

f = represent the youngest daughter

The question stated that ‘the oldest daughter is 3 times as old than her youngest

sister’.

From this statement, we can create an equation.

a = 3f

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1) Find the connection between data and the unknown

From the alphabet sequence that we are creating just now, we can build some

equations.

a = 3f

b = a+4

c = b+4

d = c+4

e = d+4

f = e+4


The problem that occurs in our equations is that there is no value to be substitute in

our equation and there has six different unknown to be solved.


First strategy: Create the unknowns relationship from the given information

Second strategy: Draw a picture



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First strategy: Create the unknowns relationship from the given information

From the information given:

We can create 6 unknown. We use a, b, c, d, e, f as our unknowns.

Then, we can build equations that related to these unknowns.

+4 +4 +4 +4 +4

a, b, c, d, e, f

f = 3a

b= a+4

c=b+4

d=c+4

e=d+4

f=e+4

3a = e+4

3a = (d+4)+4

3a = (c+4)+8

3a = (b+4)+12

3a = (a+4)+16

3a = a+20

2a = 20

a = 10

The answers are:

a= 10

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b= 10+4 = 14

c= 14+4 = 18

d= 18+4 = 22

e= 22+4 = 26

f = 26+4 = 30

Where a = the age of the youngest daughter and f = the age of the oldest daughter.

This prove that the sentence in the question that stated “The oldest daughter is 3

times as old than her youngest sister ‘’.

2) Check each step of the plan

The steps are been checking. No error mistake occurs.


Use all the answers we get from building equations.

+4 +4 +4 +4 +4

a, b, c, d, e, f - 1st equation

+4 +4 +4 +4 +4

10, 14, 18, 22, 26, 30 - 2nd equation

From comparison between both equation above, we know that the number sequence

for equation 2nd is same as the number sequence for equation 1st due to the same

value of addition from previous number to next number.

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Second strategy: Draw a picture

f e d c b a

1st Daughter 2nd Daughter 3rd Daughter 4th Daughter 5th Daughter 6th Daughter

f = 3a

b= a+4

c=b+4

d=c+4

e=d+4

f=e+4

3a = e+4

3a = (d+4)+4

3a = (c+4)+8

3a = (b+4)+12

3a = (a+4)+16

3a = a+20

2a = 20

a = 10

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The answers are:

a= 10

b= 10+4 = 14

c= 14+4 = 18

d= 18+4 = 22

e= 22+4 = 26

f = 26+4 = 30

Where a = the age of the youngest daughter and f = the age of the oldest daughter.

This prove that the sentence in the question that stated “The oldest daughter is 3

times as old than her youngest sister ‘’.



“Mr. Seibold has 6 daughters. Each daughter is 4 years older than her next younger

sister. The oldest daughter is 3 times as old than her youngest sister .How old is

each of the daughters?”


Yes.

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3.1.3: Question 3 (Odd One Out)

Which number on this square is the odd one out? Why?

How to solve it?




We are given a series of numbers in a square.


We must find the odd one.


The series of numbers are 3, 33, 15,36, 12 27, 34, 18, 72, 39, 6, 24, 21, 9, 42.



All the numbers seems that can be divided by 3 except 34


First strategy: Trial and Error Strategy

Second strategy: Construct a Table Strategy

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3 33 15 3612 27 34 1872 39 30 624 21 9 42





First strategy: Trial and Error Strategy

From the number sequence, we can know that all the numbers in the boxes can be

divided by 3 except 34.

We can check it out.

3 / 3 = 1

33 / 3 = 11

15 / 3 = 5

36 / 3 = 21

12 / 3 = 4

27 / 3 = 9

34 / 3 = 11 remainder 1

18 / 3 = 6

72 / 3 = 24

39 / 3 = 13

30 / 3 = 10

6 / 3 = 2

24 / 3 = 8

21 / 3 = 7

9 / 3 = 3

42 / 3 = 14

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We can check the answer by using multiplication,

Formula:

Quotient X divider (3) = the number in each boxes

By referring to the first step, we know that all the quotients that we divide by 3 can

get the actual number in each box except 34 / 3 = 11 remainder 1. This is because

when 11 x 3 = 33. Hence, 33 is not the actual number in boxes (34).

3X1 = 3

3X11 = 33

5X3 = 15

21X3 = 56

4X3 = 12

9X3 = 27

6X3 = 18

24X3 =72

13X3 =39

10X3 =30

2X3 =6

8X3 =24

7X3 =21

3X3 =9

14X3 =42

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Second strategy: Construct a Table Strategy

Assume f = number in the boxes and x = 3

f f/x

3 1

33 11

15 5

36 21

12 4

27 9

34 11 remainder 1

18 6

72 24

39 13

30 10

6 2

24 8

21 7

9 3

42 14

The number in boxes, f divided by 3, x to find out which is not an odd number.

The product of f/x in the table above shows when the number 34 divided by 3, it

consist the answer with remainder 1. This shows the number 34 is not an odd

number.

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“Which number on this square is the odd one out? Why?”


Yes.

4.0: SIMILAR PROBLEMS’

QUESTION

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3 33 15 3612 27 34 1872 39 30 624 21 9 42

4.0: SIMILAR PROBLEMS’ QUESTION

Change the 2 triangles below to 4 triangles by removing only one stick.


1) We are given two triangles that build up with sticks.

2) We need to remove only one stick from any two triangles so that we can get

four triangles.

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1) We use Analysis Strategy.

Third step: Carry out the plan

Problem-solving strategy:

1) What are given?

Two triangles that build up with sticks.

2) What to find?

Four triangles by removing only one stick.

3) How to find?

Use Analysis Strategy

Solution:

The question used ‘removing’ that means we need to remove

and not to throw away the stick in order it can be 4 triangles.

We can remove only one stick from the left side to make sure

that it seems like number 4.

For the other triangle, we let it without any movements so that

the triangle in the right side still in the original place.

Answer:

So, the answer that we can obtain is shown in the diagram below.

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“Change the 2 triangles below (as shown in the diagram above) to 4 triangles by

removing only one stick.”


Yes.

4.1.1: Justification about the Most Efficient Strategy (Analysis Strategy)

Mostly, non-routine problem needs skills of thinking in order to solve the

problem and obtain the correct answer. Hence, there are several types of strategies

available to solve these kinds of problems. As we all go through six types of

strategies on solving three non-routine problems, we found that analysis strategy is

the most efficient way to solve non-routine problems.

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Analysis strategy leads us to a proper steps regarding to solve the problems

in relation to find out the correct answer. This is because, we can understand more

about the problems and at the same time, we can obtain maximum information

available on the questions given. Due to that, we can plan a better ways to find the

answer.

Compared to the other five strategies that we use to solve stated questions

such as construct a table, draw a picture and look for a pattern, analysis strategy is

the fastest and suitable one to be used. This statement can be prove related to its

rational on the efficiency of underlying steps which including analysis the problem

more deeply and give us rough imagination about the demand of the questions.

In conclusion, we agree that analysis strategy is the most appropriate way in

order to solve the non-routine problem questions. In addition, this strategy also can

be used to overcome the problems in our daily life.

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RELECTION

There are so many experience that we go through during the completion of

our coursework on basic mathematics. Problem solving is the main topic of our

coursework. First of all, we would like to share our experience on the problems that

we faced during our work. There are lacks of resourceful materials about the types of

the strategies that need to solve various problems. At the same time, source from

the internet not exactly related about our topic. To overcome this problem we always

search the sources from our institution’s library and refer to our lecturer.

Besides that, the time available for the collaboration with our lecturer is not

much since our class schedule is very peak. We always need to arrange our time

properly to this purpose. Grateful, we had done required collaboration on time

successfully. Moreover, there are some factors which hindering our development in

complete our coursework. Mainly, the arrangement of timetable of our class and

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there are also many assignment that we have to complete on time. We had

sacrificed our time a lot to complete our assignments properly.

On the other hand, we learn a lot of strategies on solving mathematics

problems. We had being exposed to several types of skills on conducting routine and

non-routine problems. We also get to know about George Polya, who is a Hungarian

mathematician. His model guides us to the steps in problem solving methods. By

study about his model, it’s getting easy to solve any types of problems.

At last, we have to thank each other in our groups because they are much of

cooperation between us during our work in gathering information, editing information

and finalize of our coursework.

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REFERENCES

Carey L. B. (1981). Mathematics in Life. United State of America: Woodburn

Publisher.

John B. (1995). Classroom Learning. Singapore: Pearson Education Publisher.

Mathematics Problem and warm-ups. (n.d). Retrieved February 23, 2009 from

http://www.geom.uiuc.edu/~lori/mathed/problems/problist.html

Non-routine Problem Solving. (n.d). Retrieved February 25, 2009 from

http://io.uwinnipeg.ca/~jameis/Math/N.nonroutine/NEY1.html

Sellars, E. (2002). Using and Applying Mathematics. London: David Fulton

Publisher.

Take a Challenge. (n.d). Retrieved February 23, 2009 from

http://www.figurethis.org/index.html

Teo, F. (2002). Skills in Problem Solving Maths. Singapore: Federal Publication.

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