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- 1. KEPENTINGAN MEMPELAJARI BENTUK DAN RUANG ( SK ) / GEOMETRI ( SM)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 yangmerangsang akal fikiran kita untuk terus mengkaji ilmu Bentuk dan ruang / Geometri danmensyukuri kebesaran Ilahi. Mempelajari ilmu geometri mendedahkan kita tentangkewujudan alam ini dengan mendalam. Mengajar ilmu Bentuk dan ruang / Geometri pulamelatih akal fikiran kita untuk menjana pemikiran yang kritis dan terperinci. Terdapat alasanlain kenapa kita harus belajar manipulasi Bentuk dan ruang / Geometri iaitu minat terhadapgeometri sentiasa ada apabila kita memerlukan jawapan tentang peristiwa dan fungsi tentangkejadian alam sejagat. Ironinya, minat terhadap kepelbagaian bentuk dan objek sepertigarisan, bulatan, segi tiga, dan segi empat yang begitu dekat dengan kehidupan manusiasecara semulajadi selari dengan fenomena memandu di jalan raya, melihat kestabilanbangunan dan lain-lain lagi sering menjadi asas kepada pengembangan terhadap pengetahuanBentuk dan ruang / Geometri. Sebenarnya pengetahuan objek geometri telah ada dalammasyarakat primitif dan pada awal ketamadunan manusia. Banyak ahli falsafah matematikmemberikan pandangan mereka yang tersendiri tentang geometri. Mari kita tinjau pandanganbeberapa 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~
- 2. 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~ Rajah 1: Dari atas, karikatur Plato, Aristotle, Kepler, Galileo dan Descartes.Matematik ialah satu mata pelajaran teras di peringkat sekolah menengah dan mencakupi banyakaspek. Mata pelajaran ini bertujuan untuk melahirkan individu yang berketerampilan sertamengaplikasikan pengetahuan matematik dalam kehidupan harian secara berkesan danbertanggungjawab semasa menyelesaikan masalah dan membuat keputusan. Kandungan sukatanpelajaran Matematik Kurikulum Bersepadu Sekolah Menengah ini merangkumi pengetahuan dankemahiran daripada tiga bidang yang saling berkait iaitu Nombor, Bentuk dan Ruang, danPerkaitan. Matematik merupakan jentera atau penggerak kepada pembangunan danperkembangan dalam bidang sains dan teknologi. Dengan itu, penguasaan ilmu matematik perludipertingkatkan dari semasa ke semasa bagi menyediakan tenaga kerja yang sesuai denganperkembangan dan keperluan membentuk sebuah negara maju. Kefahaman dalam geometri dapatmembekalkan pengalaman yang dapat membantu pelajar membina kefahaman terhadap bentuk,ruang, garisan serta fungsi setiap bentuk, ruang dan garisan tersebut. Ia membolehkan pelajarmenyelesaikan masalah dan mengaplikasikannya dalam kehidupan seharian mereka. Adalahmenjadi satu tugas yang besar bagi guru untuk merealisasikan kepentingan geometri dalamkehidupan. Sebagai contoh dalam topik transformasi yang dipelajari oleh pelajar tingkatan dua,pelajar mestilah faham dengan konsep geometri yang asas sehingga mereka faham mengapasetiap bangunan yang dibina dengan bentuk-bentuk yang berlainan tetapi masih mempunyaifungsi yang sama. Begitu juga dengan topik-topik geometri yang lain seperti sudut, transformasi,poligon, pembentangan, putaran dan lokus dua dimensi. Nasional Concul of Supervisor ofMathematics, NTCM (1989) mengesahkan bahawa kemahiran dalam bidang geometri adalah
- 3. salah satu kemahiran asas daripada sepuluh kemahiran asas Matematik. Seharusnyalahkemahiran ini dapat disampaikan kepada pelajar dengan cara yang betul. Namun begitu, dalamsituasi sebenar yang berlaku di sekolah, sering kali terjadi kegagalan dalam kurikulumMatematik terutama dalam topik geometri bagi pelajar sekolah menengah. Ini kerana berlakusalah faham dalam konsep geometri semasa proses pengajaran dan pembelajaran tajuk geometriini.Why learn with us?The National Council of Teachers of Mathematics recognizes the importance of geometry andspatial sense in its publication Curriculum and Evaluation Standards for School Mathematics(1989). Spatial understandings are necessary for interpreting, understanding, and appreciatingour inherently geometric world. Insights and intuitions about two- and three-dimensional shapesand their characteristics, the interrelationships of shapes, and the effects of changes to shapes areimportant aspects of spatial sense. Children who develop a strong sense of spatial relationshipsand who master the concepts and language of geometry are better prepared to learn number andmeasurement ideas, as well as other advanced mathematical topics.Arithmetic is an important corner of mathematics, but too often we neglect the rest ofthe field. Geometry suffers because we have the mistaken impression that it doesntbecome real, serious mathematics until it gets abstract and we deal with proof. Butgeometry is important, even in its less formal form. Heres 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 studentsenough background in measurement and visualization, for example, only those studentswho get practice outside of school (through play, hobbies, daily life, or jobs) areguaranteed a fair shot at understanding formal geometry when it appears.Consider this: Children who play with Tinkertoy®, the construction system, developinformal experience and understanding of isosceles right triangles. They know that if thelegs are blue, the hypotenuse is red. When they study geometry or learn thePythagorean theorem, they already have the background textbook writers and teachers
- 4. 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 doesnt get it," you mightask yourself, is it a lack of talent or a lack of experience? Think about the out-of-schoolexperiences that might have given the student the needed background—and try toprovide something that serves the same purpose in the classroom.Many people have less-than-fond memories of learning geometry. What they remember mostvividly is the proofs that they had to learn in high school. For most people this was an unpleasantexperience in memorization of trivial statements in a particular sequence. Not only was this anunpleasant experience for them, but they also saw little purpose in it. If they ever thought to asktheir teacher why they had to learn it they were told something like, "Its good for you to learn tothink logically." Naturally, this answer did little to make the experience more pleasant. Peoplewith longer memories may remember their elementary school geometry experience. Theirmemories of this are usually somewhat more pleasant, as they remember learning shape namesand names of geometric objects such as points, lines, line segments, arcs, and rays. But they stilldidnt gain much of a perspective on why it was important to learn geometry.Is it important that we learn Geometry? Why or why not?Best Answer - Chosen by VotersNo, it is not particularly important that you learn geometry as geometry - most people will havelittle use for the "facts" of geometry. However, there are a number of aspects to mathematics andmathematical thinking that it is important to learn, and learning geometry is seen as a road to thatend. There are a number of reasons for this: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 ideathat it might be easier to learn.3. The concept of "proof" in mathematics is very important and the essence of geometry islearning about proofs and how to prove theorems in terms of axioms, etc.
- 5. Note how when you move on to algebra, the emphasis is on the manipulation rather than themechanism of the proofs.4. But it is clear that one reason for learning geometry is that at one time it really was moreimportant 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 doesnt begin with geometry but stillcovers the important concepts.On the other hand, it is likely that people will complain just as much about that curriculum asabout geometry. After all, understanding proofs, reasoning from axioms, etc. is what peoplereally dont like about geometry - it isnt the lines and circles. Abstract thinking doesnt comeeasily to most people - but that is what Mathematics is all about.The activities in this lab will help you bring this practice to your teaching. Before you trythem, read the introduction to each category of activities—shape and space. It outlinesthe rationale for teaching the topic, briefly describes the activities, explains how theactivities relate to different grade levels or to daily life, and connects the topic to nationalstandards. Then follow the links to the activities themselves. There you can access abackground page that elaborates on the rationale and the grade-level information. Youmay also find additional connections to standards for that specific activity as well asrelated resources for investigating the topic further.Collectively, the activities explore sophisticated mathematics without using formalgeometry. All you have to do is think about shape and space—and maybe do a littlecalculation.Are you ready? Then start your exploration with either activities about shape or aboutspace.We first meet geometry through shapes and their properties. The activities in thiscategory touch upon many aspects of shape.Geometry and spatial sense are vast; developing deep understanding takes years andencompasses many subfields. The mathematics here spans a range as well, but by no
- 6. means "covers" geometry in grades K–8. Visualization is an important part ofgeometrical thinking. Its the skill you use when you pretend to be somewhere else andimagine how that place looks, or when you fancy how a situation would look if thingswere just a little bit different. But visualization is especially problematic in threedimensions—perhaps because math curricula do not emphasize three-dimensionalgeometry. Some people have a hard time, for example, rotating an object in their mindsto see how it would look from a different angle. When looking at a map, others find ithard both to imagine where they are on the map and to grasp the relationships of themap objects around them.GeometryGeometry, the study of space and spatial relationships, is an important and essential branch ofthe mathematics curriculum at all grade levels. The ability to apply geometric concepts is a lifeskill used in many occupations. The study of geometry provides the student with a vehicle forenhancing logical reasoning and deductive thinking for modeling abstract problems.The study of Geometry develops logical reasoning and deductive thinking, which helps usexpand both mentally and mathematically. Euclidean Geometry is a branch of mathematicswhere one must understand the material, and apply the understood material to discover patternsand relationships.Why is Euclidean Geometry Important to Understand?The importance of Euclidean geometry is one of historical and practical use for the study ofmathematics in todays 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 ourworld."Euclids Elements written in 300BC, ranks second only to the bible as the most published bookin history. It has been studied virtually unchanged to this day, as a geometry textbook and as amodel of deductive logic. Euclid listed five axioms that he viewed as general truths and fivepostulates, 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 thatare 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, thelegs would wobble unless each leg was carefully extended just the right amount. Since thesurface of the Earth is not flat, angles play a key role the study of geodesy, the measurement ofdistances on the Earths surface. Without such geometric knowledge of angles we would lose anextremely important field of mathematics know as trigonometry. Geometry is a field, which playan important role in the careers of engineers, physicists and mathematicians alike.
- 7. Since the beginning of time, man has pondered about the universe and the stars contained in it. Itwas the use of geometry that helped man develop a working model of our solar system, whichhelped to predict accurately the motion of our planets in our solar system. Through the study ofgeometry man has learned about ellipses, which is the path of motion of the planets around thesun. Man has found uses of parabolas through the study of geometry. Parabolic mirrors are usedin telescopes, which we use to study the motion of planets, moons, the sun and other stars in ouruniverse.Geometry is also used in the early stages of our life, to help develop the mind in determiningdifferences. 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 makedeductions that in return expand our mind, and are the first exposure to mathematics we asindividuals will encounter.Geometry holds a great deal of importance tin fields such as engineering and architecture. Forexample, many bridges that play an important role in our lives in terms of travel show congruentand similar triangles. These triangles help make the bridge more stable and enables the bridges towithstand 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 enhancingbeauty. Geometric shapes can turn buildings and other structures such as the Taj Mahal into greatlandmarks admired by all. We can use geometry to study such historical landmarks as the leaningtower of Pisa (the leaning bell tower at the cathedral at Pisa), to calculate the angle of its lean andwhy the structure still stands.Geometry holds great importance in the forever-expanding world of mathematics. It enables usto picture what is happening in problems we may encounter in the study of mathematics. Thestudy 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 ofmathematics, 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 themathematician, the use of logic and deductive thinking is important especially in such courses asfinite mathematics. For the non-mathematician, logic and deductive reasoning could play a rolein doing such courses as Philosophy.These are reasons why geometry is important in our lives as citizens of the modern world. It isimportant to understand Euclidean geometry when studying a course because Euclideangeometry does not follow any set pattern. In a course such as calculus, if one knows the patternor steps to doing a specific type question, then one can easily do these types of questions. But forEuclidean geometry, one can only learn the axioms and results proven from these axioms. Thestudent 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 onescareer, in things we take for granted such as bridges, and even the homes we live in. Man used
- 8. circles and angles to create the sundial, an elementary form of timepiece. The introduction oftime made mankind more aware of seasons, age, the concept of motion, and so on. It is importantto understand Euclidean geometry even in our entertainment. In such games as pool one usesangles and triangles indirectly to place the balls into the pockets of the pool table. Euclideangeometry is important to understand even for the artist, since most drawings include shapes suchas triangles, squares, rhombus, and so on. In modern art these geometric shapes appear as themain concept of the art. So even for artists, an understanding of the very basic concepts ofEuclidean geometry is important.In physics, there is a great deal of importance for the understanding of the aspects of Euclideangeometry. Similar triangles are often used in order to calculate height (similar ratios) and todetermine the values of angles (really important in such branches as optics) when solvingproblems. 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 importantrole in our lives. For the engineer and the architect, geometry plays a role in making structuressafe and sturdy enough to be able to withstand strains and stress placed on them by nature andman. Besides the practical use of the understanding of Euclidean geometry, it helps us developdeductive reasoning with the use of logic, which helps us expand both mentally andmathematically. Euclidean geometry is a course where one must understand the material, andapply the understood material to questions that may appear in ones mathematical career. Sincethere is no set pattern when taking on such a problem, it is really important to be able tounderstand the material. The knowledge in Euclidean geometry also plays an important role inour leisure pleasure, since it is found in children games, pool, and the art world. One can easilysee that the concepts of Euclidean geometry directly or indirectly govern our world and our wayof thinking about our world. Since our calendar year, for example, is based on one completerevolution of the Earth around the sun (this path is an ellipse), the importance of understandingEuclidean geometry is endless because it plays such an important role governing the aspects ofour lives.

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