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

information, choose the operation and

algorithms

Need understanding, retrieve

information, choose the operation and

algorithms

Solving through story telling and relate

it to real situation

Various strategies and methods to

solve it

Example problem:

Each puppet act takes 15 minutes.

How long do 4 acts take?

Example problem:

Adult : RM8.29 each

Youth : RM5.49 each

The Vaughn family takes a train trip.

They buy 3 adult tickets and 3 youth

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Solution:

4 acts take 1 hour

tickets. How much does the Vaughn

family spend for tickets?

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.

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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:

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?

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Solution:

The question asks how many more, so we can subtract.

32 – 17 = 15

The second graders pick 15 more pumpkins than the graders.

(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?

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Solution:

One way to solve the problem is to make a guess, check it, and revise you guess

until you find the correct answer.

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

people in the party. Each person

gets 3 golf balls and 6 balls are

left.

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6 golf 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.

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

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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

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

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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:

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

Taman Jaya Taman Aru Uda City Taman Damak

(v) Draw a table

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3.5 cm

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:

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:

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Find which type of fish received the most votes.

Organize the votes in a table.

Type of fish Tally Number

Angelfish 8

Clownfish 11

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?

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Solution:

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

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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

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?

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Solution:

How much water in each:

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.

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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:

Model using round tables.

Four tables are filled and 4 people are left over.

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Model using rectangular tables.

The president of the rectangular tables. She will need 4 tables. Zero people will

be left over.

(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.

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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.

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

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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.

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?

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Solution:

Make the pictograph. Then compare the number of symbols for each age group.

The age group 16 and up has the most symbols. So more participants were 16

years and older than any other age group.

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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

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?

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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?

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

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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.

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?

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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.

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.

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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

8 ½ - 2 ½ = 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

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.

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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?

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

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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.

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.

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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.

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.

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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?

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.

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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

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

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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.

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.

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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

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.

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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

McGraw-Hill (2002), Mathematics Level 5, McGraw-Hill School Division

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