+ All Categories
Transcript
Page 1: Esemen Matematik Penyelesaian Masalah

INSTITUT PENGURUAN KAMPUS ILMU KHAS

SHORT COURSEWORK

PRA PERSEDIAAN IJAZAH SARJANA MUDA PERGURUAN

NAME : ANIS SYAFIQAH BINTI MOHAMED NOOR

GROUP/ UNIT : J2.1 ( PEND. KHAS ( BM, KS, PJ )

SUBJECT : MT 2310 D1

NAME OF LECTURER : PUAN AMINAH HJ.SAMSUDIN

DATE OF SUBMISSION : 2 APRIL 2010

Page 2: Esemen Matematik Penyelesaian Masalah

Contents

1.0 What is mathematics problem? .......................................................

2.0 Problem solving in mathematic.........................................................

3.0 Types of problems..............................................................................

3.1 Routine problem....................................................................

3.2 Non-rotine problem...............................................................

4.0 Polya’s modle...................................................................................

4.1 Understand the problem.......................................................

4.2 Devise a plan........................................................................

4.3 Carry out the plan..................................................................

4.4 Look back..............................................................................

5.0 Question 1..........................................................................................

5.1 Understand the problem........................................................

5.2 Devise a plan.........................................................................

5.3 Carry out the plan..................................................................

5.4 Look back..............................................................................

6.0 Question 2.........................................................................................

6.1 Understand the problem.........................................................

6.2 Devise a plan.........................................................................

6.3 Carry out the plan..................................................................

6.4 Look back...................................................................................

7.0 Reflection ............................................................................................

8.0 References ............................................................................................

9.0 Lampiran ....................................................................................................

Page 3: Esemen Matematik Penyelesaian Masalah

1.0What is mathematics problem?

(Kantowski 1977)

Problem develops when students are given mathematics question which they cannot answer

directly or cannot apply their acquired knowledge or given information within a very short

time interval.

(Hayes 1978)

He stated that students are said to face mathematics problem when they try to solve it, but

cannot find ways to achieve to goal directly.

10.0 Problem solving in mathematic.

Problem solving in mathematics is an organised process that needs to achieve the goal of

problem. The aim of problem solving is to overcome obstacles set in the problem. In addition

to overcome the obstacles, students need to analyze the information given, decide and use

much kind of strategies and methods to solve the problems.

Brownell (1942) says

...‘problem solving refers only to perceptual and conceptual tasks, the nature of which the

subject by reason of original nature of previous learning, or of organization of the task, is

able to understand, but for which at the time he knows no direct means of satisfaction. The

subject experiences perplexity in the problem situation, but he does not experiences utter

confusion...problem solving becomes the process by which the subject extricates himself from

his problem’

Page 4: Esemen Matematik Penyelesaian Masalah

3.0 Types of problems.

There are two types of problems. Problems can be divided to a routine and non-routine

problem.

3.1 Routine problem.

Routine problem is a problem that can be easily solve f problem. Routine problems consist

of a direct question where is problem solver can answer the question directly without use all

sort of strategy. Besides that, in a routine problem, problem solver can initially know the

method to solve the problem. Routine problems sometime can be a non-routine problem for

certain person. For example, 678 × 25 = ___ is a non- routine problem for primary student

in standard one. It is because, they does not know the procedure for multicolumn

multiplication. Routine problem are those that merely involved an arithmetic operation. These

arithmetic operation consist of a several characteristics. The arithmetic operation should

present a question to be answered, gives facts or number to use, and can be solved by direct

application of previously learned algorithms and basic task is to identify the operation

appropriate for solving problem.

3.2Non-rotine problem.

Non- routine problems are totally different from routine problems. In fact, non-routine

problem is a kind of unique problem solving which requires the application of skills, concept

or principle which have been learned and mastered. The question for non-routine problem is

not directly as routine problems. It is also meant that whenever we are facing an unusual

problem or situation which we don’t know the procedures to solve it. Besides that, non-

routine problem stresses the uses of heuristics which a problem solver need to learn by

discovering many things for themselves. Heuristics in non-routine problem do not guarantee a

solution but provide a more highly probable method for solving problems. In fact non-routine

Page 5: Esemen Matematik Penyelesaian Masalah

problems can be divided into two, static and active. Static mean fixed which problem solver

known goal and known element. Active can be simply to three element which have fixed

goal with changing elements, fixed element with alternative goal and changing elements with

alternative goal. Non-routine problem also serves a different purpose than routine problem

solving. Furthermore, non-routine problem are useful for daily life (in the present or in the

future). Non - routine problem mostly concerned with developing students’ mathematical

reasoning power. It also fostering the understanding that mathematics is a creative endeavor.

Non - routine can be seen as evoking an ‘I tried this and I tried that, and eureka, I finally

figured it out.’ reaction. In non-routine problem no convenient model or solution path that is

readily available to apply to solving a problem. That is in sharp contrast to routine problem

solving where there are readily identifiable models (the meanings of the arithmetic operations

and the associated templates) to apply to problem situations.

4.0 Polya’s modle

In mathematics have a model called Polya’s model that use for solve the problems solving.

George Polya is the one who successfully established this model in 1957. According to him,

problem solving in mathematics could be implemented in four stages as shown below:

1) Understand the problem

The initial step in problem solving is the student need to understand the problem in hand. The

student needs to identify:

What is given, what are the entities, numbers, connections and values involve?

What are they looking for?

To understand complex problem students need to post questions, relate it to other

similar problem, focus on the important parts of the problem, develop a model and

draw a diagram.

Page 6: Esemen Matematik Penyelesaian Masalah

2) Devise a plan

Students need to identify the following aspects when devising a strategy:

What are the operations involved?

What are the heuristic/ strategy/ algorithm required?

There are several heuristics/ strategies that students can apply in problem solving and they

are:

Guess and trial & error

Develop a model

Sketch a diagram

Simplify the problem

Construct a table

Experiment and simulation

Working backward

Investigate all possibilities

Identify sub goal

Making analogy

3) Carry out the plan.

What is to be conducted depends on what has been planned earlier and that includes:

Interpreting the information given into mathematical form

Carry the strategies that have chosen with calculation and processes.

Checking every step use.

4) Look back

Page 7: Esemen Matematik Penyelesaian Masalah

Check all the calculations involved

Check again all the important information that has been given

Consider whether the solution is logical

Look other alternative solution

Read the question again and make sure the question has really been answered.

5.0 Question 1

At the bus station have 7 girl that wait for the bus. Each girl carries 7 bags. In each

bag have 7 cats. Each cat has 7 kittens. Hence, calculate the total of legs there.

Page 8: Esemen Matematik Penyelesaian Masalah

1) Understand the problems

Information given:

There are 7 girls wait for bus at the bus station.

Each of them carries 7 bags that contain 7 cats.

Each cat has 7 kittens.

Find out the total of legs.

What question asks for? :

How much the total legs at all?

How much cats legs?

How much kitten legs?

How much girl legs?

2) Devise a plan

Draw a diagram.

3) Carry out the plan

1 person carries 7 bags.

1 begs have 7 cats.

1 cat has 7 kittens.

7

7

7

beg

cat

kitten

Page 9: Esemen Matematik Penyelesaian Masalah

We could actually continue drawing the tree diagram until we had show finish shows the

branches. However, this would be complicated to draw and might not very accurate instead

let’s draw the simple one diagram to describe the relationship between the layers and

branches as shown in the next diagram.

Girl

Cat

Kitten

Page 10: Esemen Matematik Penyelesaian Masalah

1 2 3 4 5 6 7

cats

Kitten

7 × 8 = 56 ( cats and kitten in one bag )

Total beg seven ( 7) :

Hence, 56 × 7 beg = 392 ( Kitten and cats in 7 beg for one person )

392 392 392 392 392 392392

Page 11: Esemen Matematik Penyelesaian Masalah

1 person carry 392 cat and kitten.

Hence, 7 person carry how much cat and kitten at that time???

Alternative one use addition:

392 + 392 + 392 +392 + 392 + 392 + 392 = 2744 cats and kitten

Alternative two use multiplication:

1 person = 392 total of cats and kitten

7 person =???????

= (392 × 7) ÷ 1

= 2744 total of cats and kitten.

The questions ask how much the total leg at all?

........... Hence,

1 girl = 2 legs = 1 pairs of legs.

1 cat and kitten = 4 legs = 2 pairs of legs.

Total person that wait for the bus is 7

Hence, 7 × 2 = 14 = 7 pairs of legs of that girl.

Total cats and kitten = 2744 × 4 legs = 10976 legs = 5488 pairs of legs.

The total legs at the bus stop = the total legs of kitten and cats + the total legs of

person.

= 10976 + 14 = 10990 legs.

Page 12: Esemen Matematik Penyelesaian Masalah

The answer: 10990 legs.

4) Look back

To solve this question we need to calculate the total amount of legs for cat, kitten and a

girl that carries 7 begs each . When we look back we need to pay attention on this. It is

because went someone do this question they might be do a mistakes. For example they might

ignore the girls leg that carries that beg.

To check whether this answer correct or not we can use table as shown below:

Bag Cat Kitten The total of cat

and kitten

1 7 49 56

2 14 98 112

3 21 147 168

4 28 196 224

5 35 245 280

6 42 294 336

7 49 343 392

To get the number of kitten we need to use multiplication. In addition, we need to time

the number of cats with 7. We need to time it with 7 because each cat has 7 kittens at all. This

concept applies same to when we wanted to calculate the number of cat. It is because in each

bag also have 7 cat. But we use addition when we want to calculate the total number of cat

and kitten.

Page 13: Esemen Matematik Penyelesaian Masalah

No of girl No. Of beg No. Of cat No. Of kitten The total number of cat and kitten

1 7 49 343 392

2 14 98 686 784

3 21 147 1029 1176

4 28 196 1372 1568

5 35 245 1715 1960

6 42 294 2058 2352

7 49 343 2401 2744

Hence, from the total of cat and kitten we can calculate the total legs at the bus station by

times 2744 with 4. We times that total with 4 because one cat or kitten has 4 legs. However,

we also need to calculate the total number of girl legs. As at the bus station has 7 people, we

need to time 7 with 2 that represent to the total legs that 1 person have.

....... 2744×4 = 10976

.......7×2 = 14

......10976 + 14 = 10990

6.0 Question 2:

Page 14: Esemen Matematik Penyelesaian Masalah

Zaki wanted to know the age of a tiger at the zoo. The zoo keeper told Zaki that if he added

10 years to the age of the tiger and then doubled it, the tiger would be 90 years old. How

old is the tiger ?

1) Understanding the Problem.

Information given:

Did the zoo keeper tell Zaki the tiger's age? (no)

What was the last thing the zoo keeper did to the tiger's age? (He doubled it.)

What was the first thing the zoo keeper did to the tiger's age? (He added 10.)

What question ask for?

How old is the tiger?

2) Devise a plan

Work Backwards

3) Carry out the plan

By using work backwards strategy I need to takes the opposite turn to solve these question. I

need to start to solve this problem with the end result of the problem (90), and I need to carry

the action backward to find conditions at the beginning.

Start with 90, the final number given by zoo keeper.

Then, divide it by 2 to get the number that was doubled .

Page 15: Esemen Matematik Penyelesaian Masalah

........... 90 ÷ 2 = 45.

Subtract the answer with 10 to get the age of the tiger before 10 years was added

........... 45 - 10 = 35

Answer: The tiger was 35 years old.

Look back.

As the answer has been found , I can checked by starting with that answer and carrying the

action through from start to finish. This is the one strategy that ‘‘ advertise’’ itself by stating

the end conditions of the problem and asking to find starting conditions. The steps showed as

below:

If you double a number and get 10, what number did you double? (5)

What operation did you use to get 5?

The operation that use it division.

........... 10 ÷ 2 = 5

The zoo keeper doubled a number and got 90.

What operation could you use to get the number he doubled?

Page 16: Esemen Matematik Penyelesaian Masalah

Hence, we can use division by divide the number with 2

.......... 90 ÷ 2 = 45

Is the tiger 45 years old?

....... no, it is because zoo keeper had add 10 to the tiger’s age

What did the zoo keeper do before he doubled the tiger's age?

........ He added 10 to the tiger's age.

Which operation would you use to find out how old the tiger is?

....... To find out the tiger age is we need to use subtraction by subtrac 45 with 10. Then the

answer will come out.

The answer is the tiger is 35 year’s old

Others ways to check the answer:

We can check this answer by calculate back the real answer 35 year’s old until we get 90.

The question stated that zoo keeper had add 10 year’s to the tiger’s age and double it that

makes the tiger’s age 90 year’s old.

Hence, to check the answer we need to add 35 with 10 and multiply it by 2. The answer for

this question correct when we can get 90 as the answer for the addition and multiplication

processes.

Calculation:

...... 35 + 10 = 45

Page 17: Esemen Matematik Penyelesaian Masalah

...... 45 × 2 = 90

Conclusion: The answer for this question is correct. The tiger’s age is 35 year’s old.

Reflection

Page 18: Esemen Matematik Penyelesaian Masalah

Based on this short coursework, I need to create a question regarding to non-routine

problems. Then I need to apply the polya’s model by use strategies that I have learn before.

To solve that question that I has created I used draw a diagram and working backwards

strategies. I choose these two strategies because I think these two strategies are suitable and

easy to apply on the question that I have created.

For the first question many strategies can be apply on such as logical reasoning, find a

pattern and make a table. However, I think the most suitable ways to solve this question is

draw a diagram. It is because I can clearly imagine the situation on that time. Hence, it will

make easy for me to make a calculation during solve this problem. Furthermore, it also can

avoid me for being careless when solve this question. However, this strategy also has a

weakness behind the advantages. For example, this strategy cannot be use by the person who

has low imagination as her or his cannot imagine the situation clearly as guided by the

question.

For the next question, I choose to use working backwards. It is because by refer to

this question I only have information about the end result of the problem and I need to find

the condition at the beginning refers to the clue that had given. Working backward is the one

strategies that can be use to solve the situation that only have the end result.

Then, I used make a table and reverse operation as strategies to check the answer for

both questions that I have created. I have check the first question by used the table. It is

because I can detect directly if I make a mistake on that question. For the second question I

just use the reverse operation to check the answer.

I feel very excited during do this assignment. It is because I have a chance to create a

question by my own. By doing this short coursework I arise much knowledge on how to solve

Page 19: Esemen Matematik Penyelesaian Masalah

problems. I also know how to apply polya’s model in addition to solve non – routine

problems. As a result of that I can easily solve the mathematics problems in a short period.

References

Page 20: Esemen Matematik Penyelesaian Masalah

Alfred S. Posamentier & Stephen Krulik.2009.Problem Solving in Mathematics Grades 3-6,

Powerful strategies to Deepen Understanding.United States of America:Corwin A Sage

Company.

Mok Soon Sang.2004. A Primary Education Course in Mathematics for Post Graduated

diploma (K.P.L.I).Kuala Lumpur: Kumpulan Budiman Sdn.Bhd.

Mayer, Richard E. 1992. Thinking, Problem Solving, Cognition, 2nd edition. New York: Freeman.

Noor Shah Saad & Sazelli Abdul Ghani.2008.Teaching Mathematics in secondary school:

Theories and Practices. Kuala Lumpur: Universiti Pendidikan Sultan Idris.

Mok Soon Sang. 2003. A Mathematics Course for Diploma of Education

( Semester 2 & 3).Kuala Lumpur:Kumpulan Budiman Sdn.Bhd.

Paul Lau Ngee Kiong.2004. Mathematics Education: Exploring Issues to Improve

Performance.Sarawak:Persatuan Perkembangan Profesionalisme Pendidikan Sarawak

( PROFES).

Tiada Nama Pengarang.The use of diagram in solving non routine.Dilayarai pada 24 March

2010 di URL http://www.emis.de/proceedings/PME28/RR/RR103_Pantziara.pdf.

Tiada Nama Pengarang.Learning to solve non-routine problems. Diayari pada 24 March 2010

di URL http://ilkogretim-online.org.tr/vol6say1/v6s1m5.pdf.

Tiada Nama Pengarang. Wikipedia. Dilayari pada 24 March 2010 di URL http://wiki.answers.com/Q/Definition_of_routine_and_non_routine

Page 21: Esemen Matematik Penyelesaian Masalah

Top Related