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 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~

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 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

3. 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?
The National Council of Teachers of Mathematics recognizes the importance of geometry and
spatial sense in its publication 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.
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

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 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.
Many people have less-than-fond memories of learning geometry. What they remember most
vividly is the proofs that they had to learn in high school. For most people this was an unpleasant
experience in memorization of trivial statements in a particular sequence. Not only was this an
unpleasant experience for them, but they also saw little purpose in it. If they ever thought to ask
their teacher why they had to learn it they were told something like, "It's good for you to learn to
think logically." Naturally, this answer did little to make the experience more pleasant. People
with longer memories may remember their elementary school geometry experience. Their
memories of this are usually somewhat more pleasant, as they remember learning shape names
and names of geometric objects such as points, lines, line segments, arcs, and rays. But they still
didn't 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 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. 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.

5. 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.
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 about shape or 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

6. 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.

7. 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

8. 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.