1.0: INTRODUCTION OF PROBLEM SOLVING 1
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?
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?
The question asked us to find the age of all Mr Seibold’s daughter.
2) What are you trying to find or do?
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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3) What information do you obtain from the problem?
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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Second step: Devising a plan
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
2) Consider auxiliary problem if an immediate connection can be found
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.
3) What strategies are you going to use?
First strategy: Create the unknowns relationship from the given information
Second strategy: Draw a picture
4) Try a strategy that seem its work.
Use a first strategy and second strategy to solve the problem.
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Third step: Carrying out the plan
1) Use strategy you selected and work the problem
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.
3) Ensure that the steps are corrected.
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 ‘’.
Four step: Working back
1) Reread the question
“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?”
2) Did you answer the question?
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?
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?
We are given a series of numbers in a square.
2) What are you trying to find or do?
We must find the odd one.
3) What information do you obtain from the problem?
The series of numbers are 3, 33, 15,36, 12 27, 34, 18, 72, 39, 6, 24, 21, 9, 42.
Second step: Devising a plan
1) Consider auxiliary problem if an immediate connection can be found
All the numbers seems that can be divided by 3 except 34
2) What strategies are you going to use?
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
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: 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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2) Ensure that the steps are corrected.
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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Four step: Working back
1) Reread the question
“Which number on this square is the odd one out? Why?”
2) Did you answer the question?
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.
First step: Understanding the problem
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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Second step: Devising a plan
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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Four step: Working back
1) Reread the question
“Change the 2 triangles below (as shown in the diagram above) to 4 triangles by
removing only one stick.”
2) Did you answer the question?
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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