The document introduces the concepts of problem solving in mathematics learning. It states that problem solving skills are more important than just acquiring knowledge, as problem solving is required in daily life. It defines a problem as an obstacle between the current state and the desired goal, classifying problems as routine or non-routine. Problem solving is described as the process of overcoming obstacles through critical thinking and strategies to find a solution. A commonly used 4-step model for problem solving proposed by George Polya is introduced, involving understanding the problem, devising a strategy, carrying out the strategy, and checking the result. Sample problems and their solutions using this model are provided.

1. INTRODUCING
Mathematics learning involves the acquisition of knowledge and skills
especially problem solving skills. In real life, problem solving becomes the focus
while knowledge is only the accessory. This is because, not a single day passes
without we having to solve problems. So, the need for the problem solving approach
in teaching mathematics.
WHAT IS A PROBLEM?
A problem is a statement or a situation where there is an obstacle between us
and what we want. Problems are generally classified as routine and non-routine.
WHAT IS PROBLEM SOLVING?
Problem solving is the ability to overcome or remove the obstacle so that we
can get what we want. Problem solving is a process. It requires critical thinking,
ability to make decisions, use the correct strategy to find the solution and check the
result.
MODEL FOR PROBLEM SOLVING
The most commonly used model is that of George Polya (1973), who proposed 4
stages in problem solving, namely :
1. Understand the problem
2. Devise a strategy for solving it
3. Carry out the strategy
4. Check the result

2. QUESTION 1
In your words give the definition for :
a) Routine problem
b) Non-routine problem
QUESTION 2
Solve the following problems using Polya’s model in 2 different strategies.
A police station has 25 vehicles consisting of motorcycles and cars. The total
number of tyres of both motorcycles and cars equal to 70. Find the number of
motorcycles and cars the station has.
QUESTION 3
A T-shirt costs RM 6.50. A pair of socks RM 0.85. Azmer bought three pairs of socks and a
T-shirt. How much did he pay?
Answer for question 1:
a) Routine problem is definied as a problem in mathematic lesson that involves
easy and simple problem solving. It present a question to be answered with
out need certain strategies. It means, the routine problem can be solved by
direct application of previously learned algorithms. Example:
Ali eat 2 piece of cakes. 5 minutes later, he eat 1 more piece of cakes. How
many piece of cakes that Ali eat?
Solution: 2 piece of cakes + 1 pieces of cakes = 3 piece of cakes

3. b) Non-routine is definied as a problem in mathematic lesson that involves
difficult problem solving. It means, solving the non-routine problem need us to
think analytically based on the problem. It requires us to use our cognitive by
using the critical and creative thinking skills. It also need a solution in which
applying the skills, acquired knowledge and understanding to a new and
unfamiliar situations in order to solve it.
Answer for question 2
1. For this question, the first strategy that we use is the Drawing or Sketches’s
strategy.
Step 1: FIND OUT
First, we must draw the vehicles with two tyres. Then, we must add the tyres
until the number of tyres equal to 70. After that, we can see how much
motorcycles and cars.
Step 2: CHOOSE A STRATEGY
How should we approach this problem? We can make skatches.
Step 3: SOLVE IT
Before we add 2 more tyres to make the number of tyres become 70:
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO

4. After we add 2 more tyres to make the number of tyres become 70:
OOOO OOOO OOOO OOOO OOOO
OOOO OOOO OOOO OOOO OOOO
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO
Sign: OOOO – Car OO - Motorcycle
From this sketches, we can see how much number of motorcycles and cars at
the police station. There are 15 motorcycles and 10 cars in the police
station.
Step 4: LOOK BACK
Did we answer the correct question, and does our answer seem reasonable?
Yes. (If you want to know our answer is correct or not, you must count the
number of vehicle’s tyres).
2. The second strategy that we use is Make a Chart’s strategy.
Step 1: FIND OUT

5. What is the question we have to answer? How many motorcycles and cars in
the police station.
How many vehicles in the police station? 25 vehicles.
How many number of vehicle’s tyres in police station? 70 tyres.
How many tyres that motorcycles have? 2 tyres.
How many tyres that cars have? 4 tyres.
Step 2: CHOOSE A STRATEGY
What strategy will help here? We could model this on paper, but accuracy
would suffer. We could also use equations. But, let’s make a table.
Step 3: SOLVE IT
Firstly, we make a table with 5 rows and 3 columns. Then, we choose our
target. For example, in the police station have 6 cars and 19 motorcycles. So
we can see the total of vehicles in the police station is 62 vehicles. Then, we
try and error with the same ratio until we get the answer which are 15
motorcycles and 10 cars:
CARS
( 4 TYRES)
MOTORCYCLES
(2 TYRES)
TOTAL OF VEHICLES
(70 TYRES)
6 19 62
7 18 64
8 17 66
9 16 68
10 15 70
Calculation:
1 car have 4 tyres. 1 motorcycles have 2 tyres.
So, if there has 6 cars and 19 motorcycles...

6. 6 x 4 = 24 and 19 x 2 = 38
And the total of vehicles are 62 (24+38). But, the answer is wrong.
So, we try and error with the same ratio until we get the correct answer which
are 10 cars and 15 motocycles:
10 x 4 = 40 and 15 x 2 = 30. The total of vehicles are 70 (40 + 30). The
answer is correct.
Step 4: LOOK BACK
Did we answer the question asked? Yes.
Does our answer seem reasonable? Yes.
Answer for question 3:
Step 1
Understanding the problems
- Given : The price of a T-shirt = RM 6.50
- The price of pair of socks = RM 0.85
- Find the total costs of three pairs of socks and a T-shirt.
Step 2
Devising a plan
- Find the price of three pairs of socks by using addition and then add with a T-shirt.
Step 3
Carry out the plan
First strategy

7. Price of three of socks
RM 0.85 + RM 0.85 + RM 0.85 = RM 2.55
Add with the price of a T-shirt = RM 2.55 + RM 6.50
= RM 9.05
Second strategy
Find the price of three pairs of socks by using multiplication and add with the price of the T-
shirt.
Price of three of socks
- RM 0.85 x 3 = RM 2.55
- Add with the price of a T-shirt = RM 2.55 + RM 6.50
= RM 9.05
Step 4
Looking back
Check: By using an addition, check the answer.
Three socks: RM 0.85 + RM 0.85 + RM 0.85 + RM 6.50
= RM 9.05
Suggested strategy
The most efficient strategy to get the answer of this question is the second strategy. The
second strategy is using multiplication strategy then adds with the other price.

8. A T-shirt costs RM 6.50. A pair of socks is cost RM 0.85. Azmer bought three pairs of socks
and a T-shirt. How much did he pay?
- RM 0.85 x 3 = RM 2.55
- Add with the price of a T-shirt = RM 2.55 + RM 6.50
= RM 9.05
This strategy is efficient because we can find the answer quickly. The step is also easier and
no need to use many step. We just use to times the price of 3 pair of socks and then add
with the T-shirt price.