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INSTITUT PERGURUAN ILMU KHAS

PRA PROGRAM IJAZAH SARJANA MUDA PENDIDIKAN

(BASIC MATHEMATIC)

Name: Mohamad Amri bin Muhamad Sidid

I/C No: 910823-03-5865

Group/Unit: PRA-PISMP J2.1

Subject : MT2311D1 BASIC MATHEMATICS

Name of Lecturer: Puan Aminah bt Hj Samsudin

Date of Submission: 02/04/20

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PROBLEM SOLVING

Problem-solving in mathematics can be referred as an organized process to achieve the

goal of a problem. The aim of the problem-solving is to overcome obstacles set in the problem.

In order to overcome these obstacles, pupils need to analyze the information given, decide and

implement strategies and methods to solve the problems.

A prominent mathematician in the 1970s defined problem-solving as: Problem

involving a situation whereby an individual or a group is required to carry out the working

solution(by Lester). In doing so, they have to determine the strategy and the 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.

Solving mathematics problems are activities involving problems in the form of

mathematics language, including mechanical problems, puzzles, quiz and the use of mathematics

skills in actual situations

The word problem may sound common to you. However, it has a specific meaning in

mathematics. A problem is any task in which you are faced with a situation whereby the path to a

solution is not obvious and immediate. You may need to intergrade some of your existing

knowledge in order to overcome obstacles to get the solution. In other words, to solve a problem

is to:

1. Find a way where no way is known off-hand.

2. Find a way out of difficulty.

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3. Find a way around an obstacle.

4. Attain a desired end, which is not immediately attainable, by appropriate means.

y Routine problem

Routine problem means the problem that dont need a strategy to solve it.

It only need a simple way to solve it and just get the answer while think it.

y Non routine problem

Basically, routine problems is a :

1. Mechanical problems to training pupils especially in arithmetic skills involving :

2. Additional

3. Subtraction

4. Multiplication

5. Division

Non routine problem means the problem need a strategy or more to solve it or unusual or unique

problems. We have to do more than one way to get the answer. Non-routine problem solving can

be challenging and interesting on:

Do not know any standard procedure

Requires the application of skills, concepts or principles which have been

mastered

The method cannot be memorized

Needs a set of systematic activities. They are planning,strategy and suitable

methods.

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Polyas Models

Polyas Problem Solving Techniques

In 1945 George Polya 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 of problem solving.

1. Understand the Problem

2. Devise a plan

3.Carry outtheplan

4. Look back

1. 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 dont understand it fully, or even in part.

Polya taught teachers to ask students questions such as:

First. You have to understandthe 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?

2. Devise a plan

Polya mentions that there are 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:

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y Guess and check Use a variable

y Draw a picture Look for a pattern

y Make and orderly list Eliminate possibilities

y Consider special cases Solve an equation

y Solve a simpler problem Work backward

y Use a formula Use symmetry

y Use coordinates Use a model

y Use direct reasoning Use indirect reasoning

y Brute force Be ingenious

3. Carry out the plan

y Implement the starategy or strategies that yau have chosen until the problem is solved or

until a new course of action is suggested.

y Give yourself a reasonable amount of time in which to solve yhe problem. If you are not

successful, seek hints from other or put the problem aside for a while.

y Dont be afraid of starting over. Often a fresh start and a new strategy will lead to

success.

4. Look back

y Polya mentions that much can be gained by taking the time to reflect and look back at

what you have done, what worked and what didnt.

y Doing this will enable you to predict what strategy to use to solve future problem

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Question 1:

PROBLEM 1

Guessing Tonis Number.

Toni is thinking of a number. If you double the number and add 11, the result is 39. What

number is Toni thinking of?

STRATEGY 1:- Guess and Check

Step 1:- Understand the problem.

Assume that Tonis number as X.

2 . X+ 11 = 39

What Xshould be the number?

Step 2:- Devise a plan.

y Try to guess Xwith the number area 1 to 10.

y The answers X for 1 to 10 are too small.

y The Xnumbers are too large start from 16 to 20.

y So, the Xnumbers should be around 10 to 15.

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Step 3:- Carry out the plan.

Guess (5) 2.5 + 11 = 21 + 11 = 21 too small!

Guess (10) 2.10 + 11 = 20 + 11 = 31 this is too small!

Guess (20) 2.20 + 11 = 40 + 11 = 51 this is too large!

Guess (15) 2.15 + 11 = 30 + 11 = 41 this is a bit large.

Guess (14) 2.14 + 11 = 28 + 11 = 39 this check!

#Toni number must be 14.

Step 4:- Look back.

The Xnumber, Tonis number should be 14.

2 . X+ 11 = 39

2 . 14 + 11 = 39 #

STRATEGY 2:- Making a table and looking for a pattern.

Step 1:- Understand the problem.

Assume Tonis number as Y.

Y . 2 + 11 = 39

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Step 2:- Devise a plan.

y Make a table.

y Try and guess the number

y Look for a pattern.

y From the pattern, find the answer.

Step 3:- Carry out the plan.

Trial

Number

Result Using Tonis Number

5 2.5(2) + 11 = 21

2.6(2) + 11 = 23

2.7(2) + 11 = 25

2.8(2) + 11 = 27

2.9(2) + 11 = 29

6

7

8

9

We need to get to 39 and we jump by 2 each time we take a step of 1. Therefore, we need to

take:-

39 27 = 12 = 6

2 6

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More steps: - We should guess 8 + 6 = 14 as Tonis number as before.

Step 4:- Look back.

27 12 = 6 8 2.14 + 11 = 39#

+ 12 2 + 6

39 14

Conclusion:

There are two-type strategy solutions for question 1, first strategy is guess and check strategy and

the second strategy is making a table and looking for a pattern strategy. I think that making a

table and looking for pattern strategy is suitable for use in Question 1 and it helps me to solve

this question easily and quickly rather than guess and check strategy because the step quite

arranged then guess and check strategy.

PROBLEM 2

(a) Dad workshop has 25 damaged vehicles consisting of motorcycles and cars. The

Total number of tyres of both motorcycles and cars equal 70. Find the number of

damaged motorcycles and cars in Dads workshop.

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Make a chart

Step 1

Understand the problem

Determine the number of motorcycles and cars in the Dad workshop.

Dad workshop has 35 vehicles consisting motorcycles and cars.

How many tyres for one car? =4

How many tyres for one motorcycle ? =2

How many tyres for motorcycles and cars in the station? =70

Step 2

Devise a strategy for solving it.

Sometimes we could model this on paper, but accuracy must suffer. We could also use

equations. But to get a result,make a table and see what happen at the last.

Step 3

Carry out

t

he

st

rate

gy.

Make a table with row and column and try to make improvement to solve the problem

based on the table that we do.

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Motorcycle Car Vehicles Tyres

19=(19 x 2) 6=(6 x 4) 25 62

18=(18 x 2) 7=(7 x 4) 25 64

17=(17 x 2) 8=(8 x 4) 25 66

16=(16 x 2) 9=(9 x 4) 25 68

15=(15 x 2) 10=(10 x 4) 25 70

Finally , we found same number of tyres equal to 70. So, we can see how many tyres for

25 vehicles consisting motorcycles and cars in Dads workshop at the column that shows

we get 70 tyres,. The numbers of motorcycles is 15 and for cars is 10.

Step 4

Look back

Did we answer the question asked ? Yes

Does our answer seem reasonable ? yes

Did we confident the answer that we do?yes,of course.

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Useequation

Step 1

Understand the problem

Determine the number of damaged motorcycles and cars in Dads workshop.

Dads workshop has 25 vehicles consisting motorcycles and cars.

How many tyres for one motorcycle ? 2

How many tyres for one car? 4

How many tyres for motorcycles and cars in the station? 70

Step 2

Devise a strategy for solving it

We know how much the numbers of tyre for vehicles consist motorcycles and cars . The

numbers of vehicles is 25. So,we can write an equation that models the situation. We use

algebra equation.

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Step 3

Carry outthestrategy

Letp be the numbers of motorcycle and qbe the numbers of cars.

We know the total for motorcycles and cars are 25.

The first equation is

p + q = 25 ___________________ 1

Letp be the numbers of motorcycle and qbe the numbers of cars

We know a motorcycle has 2 tyres and a car has 4 tyres. Beside that , we know the

numbers of tyres is 70.

The second equation is

2p + 4q = 70_________________ 2

p + q = 25 ___________ 1

2p + 4q = 70 ___________ 2

Solve q in 1

p + q = 25

p = 25-q ___________3

Put 3 in 2

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2p + 4q = 70 (simplfy)

p + 2q = 35

25 - q +2q = 35

q = 35-25

q = 10

Replace q =10 in 1

p + q = 25

p + 10 = 25

p = 25 10

p = 15

At the end, we getp = 15 and q = 10 . We knowp = motorcycle and q = car, so the

numbers of motorcycle is 15 and for car is 10.

Step 4

Look back

Did we answer the question asked ? Yes.Does our answer seem reasonable ? yes, when we

multiply the motorcycle 15(2) =30 and car 10(4)=40. Sum of the motorcycle and car is 70. The

numbers of tyres is equal with questions.

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REFLECTION

First time I learn this topic I feel very happy because I studied with my tutor. She is a

kind teacher and a good lecturer. This assignment quite difficult but I have many source of

information and the best is from the internet. Puan Aminah also always give me a moral support

and advise to make a perfect work and she also give me and my friend many knowledge about

this task.

I do this work and I get much moral value in this assignment. I also have learned how to

find the question and try to solve the question with use more strategy. This make me think that I

can do this in my life too for solve my problem. I can find much strategy to solve the same

problem.

My friends also give me their co-operation when I need their help to help me solve the

question and give me idea to this task. It make me feel more enjoy and happy to do this task.

Here I would like to thank to them because give support.

I hope that, I will more interested to learn more about Mathematic and try to more

discipline and focus in my study. I also hope that, I can improve my attitude in my study by

doing some assignment like that need me to find out something I do not know about that

sometime. So, thats all for my reflection. Thank you.

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BIBLIOGRAPHY

Ee Teck Ee, (2002) CHALLENGING EXERCISES ON ELEMENTARY MATHEMATICS

. Singapore Asian Publications.

Robyn Zevenbergen, Shelley Dole & Robert J. Wright (2004), Teaching

Mathematics in Primary Schools.

http://mathforum.org/~sarah/Discussion.Sessions/Polya.html

www.google.maths

http://en.wikipedia.org/wiki

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Content

Appreciation ..3

Problem soving...4-5

Polya model.6-7

Question 1..8-10

Question 212-16

Reflection ..17

Biblioghraphy 18

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Appreciation

Assalamualaikum warahmatullahi wabarakatuh. Thanks to God, because He give me

chance to finish this assignment. I feel so grateful.Firstly I would like to thanks to Puan Aminah

Hj Samsudin, my lecture for Basic Mathematic. She help me to solve the problem in a way to

finish this task. she helping will together with our class discussed about the related topic about

this short task. Thank you for her guidance and advices

Then, I want to thanks to all my friends from class PRA_PISMP J2.1 which gave me a lot

of support and an ideas to completed this task. I also want to thanks to my family, which gave

me a strong support and spirit to complete this task.

Finally, I hope Madam Aminah Haji Samsudin will satisfy with my coursework.

Thank You.