Bentuk dan ruang / Geometri dikaji kerana ia dianggap ilmu yang sangat cantik, sempurna, dan masih banyak rahsia ciptaan Ilahi yang masih belum dirungkai. Sifat-sifat inilah yang merangsang akal fikiran kita untuk terus mengkaji ilmu Bentuk dan ruang / Geometri dan mensyukuri kebesaran Ilahi. Mempelajari ilmu geometri mendedahkan kita tentang kewujudan alam ini dengan mendalam. Mengajar ilmu Bentuk dan ruang / Geometri pula melatih akal fikiran kita untuk menjana pemikiran yang kritis dan terperinci. Terdapat alasan lain kenapa kita harus belajar manipulasi Bentuk dan ruang / Geometri iaitu minat terhadap geometri sentiasa ada apabila kita memerlukan jawapan tentang peristiwa dan fungsi tentang kejadian alam sejagat. Ironinya, minat terhadap kepelbagaian bentuk dan objek seperti garisan, bulatan, segi tiga, dan segi empat yang begitu dekat dengan kehidupan manusia secara semulajadi selari dengan fenomena memandu di jalan raya, melihat kestabilan bangunan dan lain-lain lagi sering menjadi asas kepada pengembangan terhadap pengetahuan Bentuk dan ruang / Geometri. Sebenarnya pengetahuan objek geometri telah ada dalam masyarakat primitif dan pada awal ketamadunan manusia. Banyak ahli falsafah matematik memberikan pandangan mereka yang tersendiri tentang geometri. Mari kita tinjau pandangan beberapa sarjana matematik di bawah :

"Geometri adalah alat atau kaedah yang terperinci untuk menjelaskan keadaan dua bahagian alam" -Plato- |

"Geometri bukan sahaja penaakulan logik atau deduktif tetapi ia juga berhubungan, disusun dalam tatatingkat dan ditakrifkan dengan sempurna dari titik permulaan" -Aristotle- |

"Setiap kali mengkaji geometri merupakan detik berhubung dengan pemikiran Tuhan seperti mengetahui model geometri tentang pergerakan planet-planet" -Kepler- |

"Geometri menerangkan dengan sempurna atau mengkategorikan alam sejagat; bagaimana alam bertindak atau menjelma" -Galileo- |

"Geometri sebagai sesuatu yang agung, sempurna dan pengalaman yang empiris" -Descartes- |

Matematik ialah satu mata pelajaran teras di peringkat sekolah menengah dan mencakupi banyak aspek. Mata pelajaran ini bertujuan untuk melahirkan individu yang berketerampilan serta mengaplikasikan pengetahuan matematik dalam kehidupan harian secara berkesan dan bertanggungjawab semasa menyelesaikan masalah dan membuat keputusan. Kandungan sukatan pelajaran Matematik Kurikulum Bersepadu Sekolah Menengah ini merangkumi pengetahuan dan kemahiran daripada tiga bidang yang saling berkait iaitu Nombor, Bentuk dan Ruang, dan Perkaitan. Matematik merupakan jentera atau penggerak kepada pembangunan dan perkembangan dalam bidang sains dan teknologi. Dengan itu, penguasaan ilmu matematik perlu dipertingkatkan dari semasa ke semasa bagi menyediakan tenaga kerja yang sesuai dengan perkembangan dan keperluan membentuk sebuah negara maju. Kefahaman dalam geometri dapat membekalkan pengalaman yang dapat membantu pelajar membina kefahaman terhadap bentuk, ruang, garisan serta fungsi setiap bentuk, ruang dan garisan tersebut. Ia membolehkan pelajar menyelesaikan masalah dan mengaplikasikannya dalam kehidupan seharian mereka. Adalah menjadi satu tugas yang besar bagi guru untuk merealisasikan kepentingan geometri dalam kehidupan.

Sebagai contoh dalam topik transformasi yang dipelajari oleh pelajar tingkatan dua, pelajar mestilah faham dengan konsep geometri yang asas sehingga mereka faham mengapa setiap bangunan yang dibina dengan bentuk-bentuk yang berlainan tetapi masih mempunyai fungsi yang sama. Begitu juga dengan topik-topik geometri yang lain seperti sudut, transformasi, poligon, pembentangan, putaran dan lokus dua dimensi.

Nasional Concul of Supervisor of Mathematics, NTCM (1989) mengesahkan bahawa kemahiran dalam bidang geometri adalah salah satu kemahiran asas daripada sepuluh kemahiran asas Matematik. Seharusnyalah kemahiran ini dapat disampaikan kepada pelajar dengan cara yang betul. Namun begitu, dalam situasi sebenar yang berlaku di sekolah, sering kali terjadi kegagalan dalam kurikulum Matematik terutama dalam topik geometri bagi pelajar sekolah menengah. Ini kerana berlaku salah faham dalam konsep geometri semasa proses pengajaran dan pembelajaran tajuk geometri ini.

**Why learn with us?**

*Curriculum and Evaluation Standards for School Mathematics*(1989). Spatial understandings are necessary for interpreting, understanding, and appreciating our inherently geometric world. Insights and intuitions about two- and three-dimensional shapes and their characteristics, the interrelationships of shapes, and the effects of changes to shapes are important aspects of spatial sense. Children who develop a strong sense of spatial relationships and who master the concepts and language of geometry are better prepared to learn number and measurement ideas, as well as other advanced mathematical topics.

**Why Geometry Is Important ?**

Arithmetic is an important corner of mathematics, but too often we neglect the rest of the field. Geometry suffers because we have the mistaken impression that it doesn't become real, serious mathematics until it gets abstract and we deal with proof. But geometry is important, even in its less formal form. Here's why.

*First, the world is built of shape and space, and geometry is its mathematics.**Second, informal geometry is good preparation. Students have trouble with abstraction if they lack sufficient experience with more concrete materials and activities.**Third, geometry has more applications than just within the field itself. Often students can solve problems from other fields more easily when they represent the problems geometrically.**And finally—a related point—many people think well visually. Geometry can be a doorway to their success in mathematics.*

Informal geometry has an equity component as well. When schools fail to give students enough background in measurement and visualization, for example, only those students who get practice outside of school (through play, hobbies, daily life, or jobs) are guaranteed a fair shot at understanding formal geometry when it appears.

Consider this: Children who play with Tinkertoy

^{®}, the construction system, develop informal experience and understanding of isosceles right triangles. They know that if the legs are blue, the hypotenuse is red. When they study geometry or learn the Pythagorean theorem, they already have the background textbook writers and teachers may unconsciously take for granted. Children who miss out on playing with triangles—for whatever reason—must get this experience and understanding somewhere else.So teachers, be watchful. When you see a student who "just doesn't get it," you might ask yourself, is it a lack of talent or a lack of experience? Think about the out-of-school experiences that might have given the student the needed background—and try to provide something that serves the same purpose in the classroom.

**Is it important that we learn Geometry? Why or why not?**

**Best Answer - Chosen by Voters**

No, it is not particularly important that you learn geometry as geometry - most people will have little use for the "facts" of geometry. However, there are a number of aspects to mathematics and mathematical thinking that it is important to learn, and learning geometry is seen as a road to that end. There are a number of reasons for this:

1.

2.

3.

Note how when you move on to algebra, the emphasis is on the manipulation rather than the mechanism of the proofs.

4.

It should be possible to devise a math curriculum that doesn't begin with geometry but still covers the important concepts.

On the other hand, it is likely that people will complain just as much about that curriculum as about geometry. After all, understanding proofs, reasoning from axioms, etc. is what people really don't like about geometry - it isn't the lines and circles. Abstract thinking doesn't come easily to most people - but that is what Mathematics is all about.

1.

*People have some intuition about plane geometry, so studying geometry can tap into that.*2.

*Geometry was developed earlier than most other forms of mathematics, so there is the idea that it might be easier to learn.*3.

*The concept of "proof" in mathematics is very important and the essence of geometry is learning about proofs and how to prove theorems in terms of axioms, etc.*Note how when you move on to algebra, the emphasis is on the manipulation rather than the mechanism of the proofs.

4.

*But it is clear that one reason for learning geometry is that at one time it really was more important than it is today, and so we have the tradition of learning it before other areas of math.*It should be possible to devise a math curriculum that doesn't begin with geometry but still covers the important concepts.

On the other hand, it is likely that people will complain just as much about that curriculum as about geometry. After all, understanding proofs, reasoning from axioms, etc. is what people really don't like about geometry - it isn't the lines and circles. Abstract thinking doesn't come easily to most people - but that is what Mathematics is all about.

**Using This Lab**

The activities in this lab will help you bring this practice to your teaching. Before you try them, read the introduction to each category of activities—shape and space. It outlines the rationale for teaching the topic, briefly describes the activities, explains how the activities relate to different grade levels or to daily life, and connects the topic to national standards. Then follow the links to the activities themselves. There you can access a background page that elaborates on the rationale and the grade-level information. You may also find additional connections to standards for that specific activity as well as related resources for investigating the topic further.

Collectively, the activities explore sophisticated mathematics without using formal geometry. All you have to do is think about shape and space—and maybe do a

*little*calculation.Are you ready? Then start your exploration with either activities

*or***about shape***.***about space**We first meet geometry through shapes and their properties. The activities in this category touch upon many aspects of shape.

Geometry and spatial sense are vast; developing deep understanding takes years and encompasses many subfields. The mathematics here spans a range as well, but by no means "covers" geometry in grades K–8. Visualization is an important part of geometrical thinking. It's the skill you use when you pretend to be somewhere else and imagine how that place looks, or when you fancy how a situation would look if things were just a little bit different. But visualization is especially problematic in three dimensions—perhaps because math curricula do not emphasize three-dimensional geometry. Some people have a hard time, for example, rotating an object in their minds to see how it would look from a different angle. When looking at a map, others find it hard both to imagine where they are on the map and to grasp the relationships of the map objects around them.

**Geometry**

Geometry, the study of space and spatial relationships, is an important and essential branch of the mathematics curriculum at all grade levels. The ability to apply geometric concepts is a life skill used in many occupations. The study of geometry provides the student with a vehicle for enhancing logical reasoning and deductive thinking for modeling abstract problems.

The study of Geometry develops logical reasoning and deductive thinking, which helps us expand both mentally and mathematically. Euclidean Geometry is a branch of mathematics where one must understand the material, and apply the understood material to discover patterns and relationships. **Why Is Euclidean Geometry Important to Understand ?**

The importance of Euclidean geometry is one of historical and practical use for the study of mathematics in today's society. Euclidean geometry is one of the oldest branches of mathematics, developed by Euclid in 300BC, and serves as the basis of modern mathematics that governs our world.

"Euclid's Elements written in 300BC, ranks second only to the bible as the most published book in history. It has been studied virtually unchanged to this day, as a geometry textbook and as a model of deductive logic. Euclid listed five axioms that he viewed as general truths and five postulates, which are truths about a particular field. These ten statements and basic rules of logic, serve as a model of deductive reasoning." (Charles D Miller, Vern E Heeren, E John Hornsby Jr. Mathematics Ideas for Memorial University, customedition Harper Collins College Publishers, 1994).

Properties of planes, for example, appear in our daily life. If we are given any three points that are not in a straight line, then a plane can be passed through these points. That is why camera, telescope, and surveying equipment tripods have three legs; no matter how irregular the surface, the tips of these legs determine a plane. On the other hand, if a camera support had four legs, the legs would wobble unless each leg was carefully extended just the right amount. Since the surface of the Earth is not flat, angles play a key role the study of geodesy, the measurement of distances on the Earth's surface. Without such geometric knowledge of angles we would lose an extremely important field of mathematics know as trigonometry. Geometry is a field, which play an important role in the careers of engineers, physicists and mathematicians alike.

Since the beginning of time, man has pondered about the universe and the stars contained in it. It was the use of geometry that helped man develop a working model of our solar system, which helped to predict accurately the motion of our planets in our solar system. Through the study of geometry man has learned about ellipses, which is the path of motion of the planets around the sun. Man has found uses of parabolas through the study of geometry. Parabolic mirrors are used in telescopes, which we use to study the motion of planets, moons, the sun and other stars in our universe.

Geometry is also used in the early stages of our life, to help develop the mind in determining differences. We all had the game as a child where we must place the different shapes (square, triangles, circles, and so on) in the right slots. These games help us as toddlers to make deductions that in return expand our mind, and are the first exposure to mathematics we as individuals will encounter.

Geometry holds a great deal of importance tin fields such as engineering and architecture. For example, many bridges that play an important role in our lives in terms of travel show congruent and similar triangles. These triangles help make the bridge more stable and enables the bridges to withstand great amounts of stress and strain placed on them. In the construction of buildings, geometry can play two roles; one in making the structure more stable and one in enhancing beauty. Geometric shapes can turn buildings and other structures such as the Taj Mahal into great landmarks admired by all. We can use geometry to study such historical landmarks as the leaning tower of Pisa (the leaning bell tower at the cathedral at Pisa), to calculate the angle of its lean and why the structure still stands.

Geometry holds great importance in the forever-expanding world of mathematics. It enables us to picture what is happening in problems we may encounter in the study of mathematics. The study of geometry helps us develop the ability to visualize shapes, volume, area, and so on. Geometric proofs play an important role in the expansion and understanding of many branches of mathematics, from Venn diagrams in set theory to area under the graph in calculus.

One must realize that probably the most important reason a mathematician and/or non-mathematician should understand geometry is the use of deductive thinking and logic. For the mathematician, the use of logic and deductive thinking is important especially in such courses as finite mathematics. For the non-mathematician, logic and deductive reasoning could play a role in doing such courses as Philosophy.

These are reasons why geometry is important in our lives as citizens of the modern world. It is important to understand Euclidean geometry when studying a course because Euclidean geometry does not follow any set pattern. In a course such as calculus, if one knows the pattern or steps to doing a specific type question, then one can easily do these types of questions. But for Euclidean geometry, one can only learn the axioms and results proven from these axioms. The student must apply these axioms with no set pattern or list of steps for solving such problems. Therefore, each problem can have one, two, three, four or infinitely many solutions.

We seem to find aspects of Euclidean geometry everywhere in life. These aspects appear in one's career, in things we take for granted such as bridges, and even the homes we live in. Man used circles and angles to create the sundial, an elementary form of timepiece. The introduction of time made mankind more aware of seasons, age, the concept of motion, and so on. It is important to understand Euclidean geometry even in our entertainment. In such games as pool one uses angles and triangles indirectly to place the balls into the pockets of the pool table. Euclidean geometry is important to understand even for the artist, since most drawings include shapes such as triangles, squares, rhombus, and so on. In modern art these geometric shapes appear as the main concept of the art. So even for artists, an understanding of the very basic concepts of Euclidean geometry is important.

In physics, there is a great deal of importance for the understanding of the aspects of Euclidean geometry. Similar triangles are often used in order to calculate height (similar ratios) and to determine the values of angles (really important in such branches as optics) when solving problems. The use of diagrams, created through the use of knowledge of Euclidean geometry, help the physicist see the details of problems that in return guide him or her to the solutions.

It is extremely important to understand Euclidean geometry because it plays such an important role in our lives. For the engineer and the architect, geometry plays a role in making structures safe and sturdy enough to be able to withstand strains and stress placed on them by nature and man. Besides the practical use of the understanding of Euclidean geometry, it helps us develop deductive reasoning with the use of logic, which helps us expand both mentally and mathematically. Euclidean geometry is a course where one must understand the material, and apply the understood material to questions that may appear in one's mathematical career. Since there is no set pattern when taking on such a problem, it is really important to be able to understand the material. The knowledge in Euclidean geometry also plays an important role in our leisure pleasure, since it is found in children games, pool, and the art world. One can easily see that the concepts of Euclidean geometry directly or indirectly govern our world and our way of thinking about our world. Since our calendar year, for example, is based on one complete revolution of the Earth around the sun (this path is an ellipse), the importance of understanding Euclidean geometry is endless because it plays such an important role governing the aspects of our lives.

Sangat membantu saya dalam melakukan beberapa kajian. Terima kasih :)

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