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Question 1
Problem Solving
Problem solving has long been recognized as one of the hallmarks of
mathematics. The greatest goal of learning mathematics is to have people
become good problem solvers. We do not mean doing exercises that are routine
practice for skill building.
Definition of problem solving
What does problem solving mean? Problem solving is a process. It is the means
by which an individual uses previously acquired knowledge, skills, and
understanding to satisfy the demands of an unfamiliar situation. The student
must synthesize what he or she has learned and apply it to the new situation.
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Compare and contrast characteristics of routine and non-routine problems
There are two kinds of problems in our life. There are routine problem and non-
routine problem. There is comparing of routine and non-routine problems:
Routine problems Non-routine problems
Real life problem Complex problem
Involve one mathematical operation Require more than one mathematical
operation
Basic skills and sequence steps Using critical and creative thinking skills
Need understanding, retrieve Need understanding, retrieve
information, choose the operation and information, choose the operation and
algorithms algorithms
Solving through story telling and relate Various strategies and methods to
it to real situation solve it
Example problem: Example problem:
Each puppet act takes 15 minutes. Adult : RM8.29 each
How long do 4 acts take? Youth : RM5.49 each
The Vaughn family takes a train trip.
They buy 3 adult tickets and 3 youth
tickets. How much does the Vaughn
family spend for tickets?
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Solution:
4 acts take 1 hour
Solution:
Adult Ticket costs RM8
Youth Ticket costs RM5
Try the simpler problem:
Adult Ticket : 3 x RM8 = RM24
Youth Ticket : 3 x RM5 = RM15
Total ticket costs:RM24+RM15 = RM39
Now, solve the problem the same way:
3 x RM8.29 = RM24.87
3 x RM5.49 = RM16.47
Total ticket cost: RM41.34
The Vaughn family spends RM41.34 on
train tickets.
Strategies / heuristics for problem solving
There are many ways to solve word problems. Several word problems will be
given and students can choose the method they feel most comfortable with.
Some methods are:
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• Choose the operation
• Guess, check and improve
• Look for a pattern
• Draw a diagram
• Draw a table
• Logical reasoning
• Make an analogy
• Act a situation
• Working backwards
• Make a graph
(i) Choose the operation
Sample problem:
The second grades pick 32 pumpkins. The first graders pick 17 pumpkins. How
many more pumpkins do the second graders pick than the first graders?
Solution:
The question asks how many more, so we can subtract.
32 – 17 = 15
The second graders pick 15 more pumpkins than the graders.
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(ii) Guess, check and improve
In the strategy guess and check, we first guess at a solution using as reasonable
a guess as possible. Then we check to see whether the guess is correct. If not,
the step is to learn as much as possible about the solution based of the guess
before making the next guess. This strategy can be regarded as a form of trial
and error, where the information about the error helps us choose what trial to
make next.
Sample problem:
A worker at Putra Minigolf has a basket of 30 balls for a minigolf party. There are
more them 5 people in the party. The golf balls are passed equally to people in
the party. After the golf balls are passed around, 6 are left.
How many people are in the party?
How many golf balls does each person get?
Solution:
One way to solve the problem is to make a guess, check it, and revise you guess
until you find the correct answer.
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Guess: 6 people in the party
Check: 30/6 = 5 There are no golf balls left.
Try are different number.
Guess: 7 people in the party
Check: 30/7 = 4 R2 There are 2 golf balls are left.
Try a different number.
Guess: 8 people in the party
Check: 30/8 = 3 R3 One possible answer is that 8
6 golf balls are left
people in the party. Each person
gets 3 golf balls and 6 balls are
left.
(iii) Look for a pattern
When using the look for a pattern strategy, on usually lists several specific
instances of a problem and then look to see if a pattern emerges that suggest a
solution to entire problem.
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Sample problem:
Sara arranged pine cones into different groups. She started with one pine cone in
the first group. In the next group she put 2 pine cones. She put 4 pine cones in
the third group, 8 pine cones in the fourth, and so on. If the number in each row
continues to increase in the same way, how many pine cones does she put in the
eighth group?
The number of pine cones doubles each time.
x2 x2 x2
1 2 4 8
Continue the pattern to find how many pine cones will be in the eight group.
x2 x2 x2 x2 x2 x2
1 2 4 8 16 32
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x2
64 128
Sara puts 128 pine cones in the eight group.
(iv) Draw a diagram
In geometry, drawing a picture often provides the insight necessary to solve a
problem. The following problem is a non-geometric problem that can be solved
by drawing a diagram or could be done using a piece of paper.
Sample problem:
Ahmad and his father are talking a tour of his state. He lives in Taman Jaya. The
list show the distances to Taman Aru, Taman Damak, and Uda City. He wants to
travels to the cities in order from east to west. Going from east to west, put the
cities in order.
Taman Aru is 4.5 km east of Taman Jaya
Taman Damak is 3.5 km east of Taman Aru
Uda City is 3.1 km east of Taman Aru
We want to know the distance between the cities and the order of the cities.
Solution:
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• We use the problem solving is to draw a diagram.
• Use the centimeter ruler to draw a diagram.
• Draw a diagram of the distance from Taman Jaya to Taman Aru using 1
cm for each km.
• Then, draw the distances of Uda city and Taman Damak from Taman Aru.
You must know that 3.5 km is equivalent to 3.5 cm.
• The cities in order from west to east Taman Jaya, Taman Aru, Uda City
and Taman Damak.
• The cities from east to west are Taman Damak, Uda City, Taman Aru, and
Taman Jaya.
4.5 cm 3.1 cm
3.5 cm
Taman Jaya Taman Aru Uda City Taman Damak
(v) Draw a table
A table can be used to summarize data or to help us see a pattern. It also helps
us to consider all possible cases in a given problem.
Sample problem:
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Mr. Singh’s class voted on the type of fish they would to buy to go into the new
aquarium. They will buy the fish chosen by the greatest number of student.
Which type of fish is most popular?
Solution:
Find which type of fish received the most votes.
Organize the votes in a table.
Type of fish Tally Number
Angelfish 8
Clownfish 11
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Puffer 5
Mr. Singh’s class will choose the clownfish for the aquarium.
(vi) Simplify the problem
Sample problem:
Fatimah is making a deli platter for a party. She wants to have the same amount
of meat and cheese on the platter. Look at her list. Does she have the same
amount of meat and cheese?
Solution:
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Add to find how much meat.
1 + 2 + 1 +1 = 5 pounds
Add to find how much cheese.
1 + 1 + 1 = 3 pounds
Compare the amount.
5 > 3, so there is more meat.
Now solve the problem the same way, using the fractions.
Add to find how much meat.
1/4 + 1/2 + 1/4 + 1/4 = 1 ¼ pounds
Add to find how much cheese.
1/3 + 1/3 + 1/3 = 1 pounds
Compare the amounts.
1 ¼ > 1 so there is more meat than cheese.
(vii) Logical reasoning
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A carnival worker needs to fill an animal’s drinking tank with 6 gallons of water.
He has a 5-gallon pail and 8-gallon pail. How can he use these pails to get
exactly 6 gallons of water into the tank?
Solution:
How much water in each:
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• Fill 8-gallon pail.
• Pour water from 8-gallon pail to fill 5-gallon pail.
• Pour 3 gallons left in the 8-gallon pail into the tank.
Now we can empty the water from the 5-gallon pail and repeat steps 1 through 3
to get 6 gallons in the tank.
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(viii) Act a situation
We may find it helpful to act out a problem. This will help us create a visual
image of the problem. By taking an active role in finding a solution, we are more
likely to remember the process we used and be able to use it again for solving
similar problems.
Sample problem:
The Marble Collectors’ Club luncheon is today! The club president wants to seat
24 members so that every table is filled. Each round table seats 5 people. Each
rectangular table seats 6. Which shape tables should she use? How many tables
will she need?
Solution:
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Model using round tables.
Four tables are filled and 4 people are left over.
Model using rectangular tables.
The president of the rectangular tables. She will need 4 tables. Zero people will
be left over.
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(viiii) Working backwards
The strategy of working backwards entails starting with the end results and
reversing the steps you need to get those results, in order to figure out the
answer to the problem.
When do we use this strategy? What are real life examples?
There are at least two different types of problems which can best be solved by
this strategy:
(1) When the goal is singular and there are a variety of alternative routes to take.
In this situation, the strategy of working backwards allows us to ascertain which
of the alternative routes was optimal.
An example of this is when you are trying to figure out the best route to take to
get from your house to a store. You would first look at what neighborhood the
store is in and trace the optimal route backwards on a map to your home.
(2) When end results are given or known in the problem and you're asked for the
initial conditions.
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An example of this is when we are trying to figure out how much money we
started with at the beginning of the day, if we know how much money we have at
the end of the day and all of the transactions we made during the day.
Sample problem:
Ali and his father are going to the Athletic Club to play in a basketball
tournament. The game starts at 7:30 p.m. It takes those about 35 minutes to walk
to the Market Street bus stop. Which bus should they leave home?
Bus Route p.m. Schedule
Bus Stop Time Time Time
Market Street 3:00 6:30 7:00
Mini Mall 3:10 6:40 7:10
Athletic Club 3:32 7:02 7:32
Solution:
We can work backward to solve this problem. We can use the bus schedule
given. Start by finding the last bus that can arrive at the Athletic Club before 7:30
p.m.
Think: A bus arrives at the Athletic Club at 7:02 p.m. It leaves Market Street at
6:30 p.m.
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Then find the time Ali and his father need to leave home.
Think: End time : 6:30 p.m.
Elapsed time : 35 minutes ------ Time to walk to bus stop
Start time : 5:55 p.m.
They should take the 6:30 p.m. bus and leave home by 5:55 p.m.
(x) Make a graph
Each year, San Francisco, California, hosts the Aquatic Beach Sandcastle
Competition. How could a newspaper display this data to show the age group
that had the most people in the contest?
Solution:
Make the pictograph. Then compare the number of symbols for each age group.
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The age group 16 and up has the most symbols. So more participants were 16
years and older than any other age group.
Question 2
Three non-routine and the solution using the Polya’s strategies and they
have an alternative method to solve them.
Polya spelled out the thought processes of a problem solver step by step. The
bold face segment following each step is a summary for the reference process.
Each part of Polya's reflections have great bearing on the daily job of information
problem solving. As such, they are thought processes that need to be modeled
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for, and inculcated in, library users. Problem solving process has common
elements across domains:
UNDERSTANDING THE PROBLEM
• First. You have to understand the problem.
• What is the unknown? What are the data? What is the condition?
• Is it possible to satisfy the condition? Is the condition sufficient to
determine the unknown? Or is it insufficient? Or redundant? Or
contradictory?
• Draw a figure. Introduce suitable notation.
• Separate the various parts of the condition. Can you write them down?
DEVISING A PLAN
• Second. Find the connection between the data and the unknown. You
may be obliged to consider auxiliary problems if an immediate connection
cannot be found. You should obtain eventually a plan of the solution.
• Have you seen it before? Or have you seen the same problem in a slightly
different form?
• Do you know a related problem? Do you know a theorem that could be
useful?
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• Look at the unknown! And try to think of a familiar problem having the
same or a similar unknown.
• Here is a problem related to yours and solved before. Could you use it?
Could you use its result? Could you use its method? Should you introduce
some auxiliary element in order to make its use possible?
• Could you restate the problem? Could you restate it still differently? Go
back to definitions.
• If you cannot solve the proposed problem try to solve first some related
problem. Could you imagine a more accessible related problem? A more
general problem? A more special problem? An analogous problem? Could
you solve a part of the problem? Keep only a part of the condition, drop
the other part; how far is the unknown then determined, how can it vary?
Could you derive something useful from the data? Could you think of other
data appropriate to determine the unknown? Could you change the
unknown or data, or both if necessary, so that the new unknown and the
new data are nearer to each other?
• Did you use all the data? Did you use the whole condition? Have you
taken into account all essential notions involved in the problem?
CARRYING OUT THE PLAN
• Third. Carry out your plan.
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• Carrying out your plan of the solution, check each step. Can you see
clearly that the step is correct? Can you prove that it is correct?
LOOKING BACK
• Fourth. Examine the solution obtained.
• Can you check the result? Can you check the argument?
• Can you derive the solution differently? Can you see it at a glance?
• Can you use the result, or the method, for some other problem?
Sample question 1
Strategy: Draw a diagram
It has often been said that a picture is worth a thousand words. This is
particularly true in problem solving. In geometry, drawing often provides the
insight necessary to solve a problem. The following problem is a non geometric
problem that can be solved by drawing a diagram or could be done using a piece
of paper; means by using act it out strategy.
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Problem:
Amin has a single sheet of 8 ½-by-11in. paper. He needs to measure exactly 6
in. Can he do it using the sheet paper?
Understanding the problem
The problem is to use only an 8 ½-by-11 in. paper to measure something of
exactly 6 in. There are the two edge lengths with which to work.
Devising a plan
A natural thing to ask is what combinations of lengths can be made from the two
given lengths? We could fold to halve the lengths, but this would lead to such
fractional lengths as 4 ¼ in., 5 ½ in., 2 1/8 in., and so on, that appear unhelpful.
Another idea is to consider other folds that could be used to combine 8 ½ in. and
11 in. to get 6 in. For example, consider how to obtain 6 from 8 ½ and 11.
11 – 8 ½ = 2 ½
8½-2½=6
If we could fold the paper to obtain 2 ½ in. from the two given lengths and the
‘subtract’ the 2 ½ from 8 ½, then we could obtain the desired result of 6. One
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strategy to investigate how to obtain 2 ½ in. from the 11 in. and 8 ½ in. lengths is
to draw a diagram.
Carrying out the plan
By folding the paper as shown by the arrows in figure below, we can obtain a
length of 6 in.
With the fold in figure, we obtain a length of 11 – 8 1/2, or 2 ½ in. With the fold
figure (c), we obtain 8 ½ - 2 ½, or 6 in.
Looking back
Though there are other ways to solve this problem, drawing a diagram showing
the folds of the paper combines notions of geometry and gives a way to fold a
square from a rectangle. An entirely different way to solve the problem is
possible if have more than one sheet of paper. Other than that, we also can act it
out and fold the paper.
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Sample question 2
Strategy: Use a variable
Problem:
In a small town, three children deliver all the newspapers. Abu delivers three
times as many papers as Busu, and Chuan delivers 13 more than Abu. If the
three children delivered a total of 496 papers, how many papers does each
deliver?
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Understanding the problem
The problem asks for the number of papers that each child delivers. It gives
information that compares the number of papers that each child delivers as well
as the total numbers of papers delivered in the town.
Devising a plan
Let a, b, and c is the number of papers delivered by Abu, Busu and Chuan,
respectively. We translate the given information into equations as follows:
Abu delivers three times as many papers as Busu : a = 3b
Chuan delivers 13 more papers than Abu : c = a + 13
Total delivery is 496 : a + b + c =496
To reduce the number of variable, substitute 3b for a in the second and third
equations:
c = a + 13 becomes c = 3b +13
a + b + c = 496 becomes 3b + b + c = 496
Next, make an equation in one variable, b, by substituting 3b + 13 for c in the
equation 3b + b + c = 496, solve for b, and then find a and c.
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Carrying out the plan
3b + b + 3b + 13 = 496
7b + 13 = 496
7b = 483
b = 69
Thus a = 3b = (3 x 69) = 207.
Also, c = a + 13 = 207 + 13 = 220.
So, Abu delivers 207 papers, Busu delivers 69 papers, and Chuan delivers 220
papers.
Looking back
To check the answers, follow the original information, by using make a sentence,
start by using
a = 207, b = 69, and c = 220.
The information in the first sentence, “Abu delivers three times as many papers
as Busu” checks,
Since 207 = 3 x 69.
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The second sentence, “Chuan delivers 13 more papers than Abu” is true
because
220 = 207 + 13
The information on the total delivery checks, since
207 + 69 + 220 = 496.
Sample question 3
Strategy: Guess, check and improve
Problem:
A farmer had some cows and ducks. One day he counted 20 heads and 56 legs.
How many cows and ducks did he have?
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Step 1: Understanding the problem
How many the cows and the ducks that the farmer’s have?
• Cows have four legs
• Ducks have two legs
Step 2: Devising a plan
We make a plan to shows various combination of 20 ducks and cows and how
many legs they have altogether. We use guess and checks strategy.
Step 3: Carrying out the plan
Numbers of cows Numbers of ducks Numbers of legs
12 12 80 + 24 = 104
11 9 44 + 18 = 62
10 10 40 + 20 = 60
9 11 36 + 22 = 58
8 12 32 + 24 = 56
Step 4: Looking back
Checks for the answers by using a variable
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Lets a = duck, b = cow
2a + 4b = 56 --------------------------------(i)
a + b = 20 -----------------------------------(ii)
From (ii),
a = 20 – b ----------------------------------(iii)
Replace (iii) into (i),
2(20-b) + 4b = 56
40 - 2b + 4b = 56
2b = 16
b=8
Replace b = 8 into (iii),
a = 20 - 8
=12
Therefore, the answer 8 cows and 12 ducks are correct.
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REFLECTION
We received this task about Strategic Problem Solving in Basic Mathematics by
our lecturer, Puan Azizan Yeop Zaharie on 5 March 2007. After the briefing about
the task, we find our group which is Syaza Yasmin, Nurzehan and Hasnurfarisha.
We divided our jobs to three persons. All of us must search any strategies in
Polya’s Problem Solving.
In this tack, we need to write up definition of problem solving, compare
and contrast characteristics of routine and non-routine problems and must give
an example for that problem. Other than that, we also must write up ten
strategies or heuristics that used for problem solving.
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After did this task, our perception of problem solving now as compared to
previous experiences with solving problem. When we using this problem solving,
our work look more regular from using previous experiences. We also easy to
understand because when we used strategies to solve the problem, we can see
the steps one by one and the step are regular and easy to understand for
students.
By using Problem Solving Strategies, we find that there are so easily to
solve a mathematical problem. Student did not facing any difficulties when solve
any problem in Mathematics. So, Mathematics learning can becomes more fun.
By first time learning Problem Solving in our class, we can define that
Polya spelled out the thought processes of a problem solver step by step. The
bold face segment following each step is a summary for the reference process.
Each part of Polya's reflections has great bearing on the daily job of information
problem solving. As such, they are thought processes that need to be modeled
for, and inculcated in, library users. There is four easy steps in Polya’s Problem
solving which is step one; understanding the problem, step two; devising a plan,
step three; carrying out the plan, step four; looking back. These four step usually
used in mathematic problem solving. Actually, these steps were commonly used
for us and other students, but we did not know what the process is called by.
Polya’s problem solving makes the mathematic problem easier to solve.
Finally, thanks very much to everyone was helping us in process doing
this assignment especially to Puan Azizan Yeop Zaharie, other lecturers, friends
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and whose were helps us. Thanks very much from us. Without all of you, we
faced many difficulties and problems during doing this task. Thanks very much.
REFERENCES
http://rhlschool.com
Max A. Sobel, Evan M. Malestsky (1970), Teaching Mathematics A Sourcebook
of Aids, Activities and Strategies, Second Education, Prentice – Hall
Rick Billstein, Shlomo Libeskind (2004), A Problem Solving Approach to
Mathematics, Pearson Education
McGraw-Hill (2002), Mathematics Level 9, McGraw-Hill School Division
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McGraw-Hill (2002), Mathematics Level 5, McGraw-Hill School Division
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