INTRODUCTION
Coming to
history of mathematics, several civilizations — in China, India, Egypt in
Central America and Mesopotamia — contributed to mathematics as we know it
today.As civilizations developed, mathematicians began to work with geometry,
which computes areas and volumes to make angular measurements and has many
practical applications. Geometry is used in everything from home construction
to fashion and interior design.Mathematician who works in the field of geometry
is called a geometer. It is one of the oldest branches of mathematics, having
arisen in response to such practical problems as those found in surveying, and
its name is derived from Greek words meaning “Earth measurement.”
The great
advancement in geometry came from Euclid in 300 BC when he wrote a text titled
'Elements.' The next tremendous advancement in the field of geometry occurred
in the 17th century when René Descartes discovered coordinate geometry. Some
specific branches of geometry are Euclidean geometry, analytic geometry,
projective geometry, differential geometry, non-Euclidean geometries, and
topology. Congruence and similarity are more relevant topics among geometry.
Traditionally,
similarity is defined as the same shape which is not necessarily in the same
size. The conceptualization of another
similarity which is more based on mathematics involves the observation of a
proportional relation when the relevant lengths in similar shapes or within a
shape are compared.
In
geometry, two figures or objects are congruent if they have the same shape and
size, or if one has the same shape and size as the mirror image of the other.
When two plane figures are congruent implies that their corresponding
characteristics are "congruent" or "equal" including not
just their corresponding sides and angles, but also their corresponding
diagonals, perimeters, and areas.
NEED AND SIGNIFICANCE OF THE STUDY
According to Baykul, two planar shapes are
congruent if their sizes and forms are
the same, and congruence is a special case of similarity.
All congruent
shapes are similar but similar shapes may not be congruent. Once we know that
two are congruent, then we can just "copy over'' the information from the
figure we know about to the other figure. Similarity is math speak for "
these things are the same except that one might be bigger or smaller than the
other ''.
Everyone
is living through situations involving congruence and similarity. Some are
aware about it and some are not. The people who are aware can appreciate its
beauty and handle the situations by applying their comprehensions, at its
optimum and others can't. The individuals working as surveyors and jewellery
makers are exactly using these concepts in their daily life, and their
creations become unique and gorgeous when these concepts reflect in their work.
If everyone came across on comprehending these concepts in their life
situations, they will begin to appreciate their aesthetic face of life and
scope for new innovations widens.
HYPOTHESES
● There may be applications of
geometric congruence and similarity in life situations and man made things.
● There may be possibilities that the
geometric congruence and similarity nurture or enhance the qualities of real
life situations and man made things.
● Common people may nor be aware of
life situations including man made things in which geometric congruence and
similarity are applied.
OBJECTIVES
● To identify the concept, principle
and procedures related to geometric congruence and similarity.
● To identify the life situations in
which the concept, principle and procedures in geometric congruence and
similarity are applied.
● To analyse the benefits of concept,
principle and procedures in geometric congruence and similarity in nurturing
the real life situations as well as man made things.
● To appreciate the presence of
concept, principle and procedures in geometric congruence and similarity in
life situations or man made things by comprehending it's benefits.
● To analyse the comprehension of
common people on mathematical congruence and similarity.
METHODOLOGY
Data was
primarily collected by referring to books and internet sites. Mathematical
journals helped me to gather information on my topic.
For
analysing the comprehension of common people on this topic, a survey was
conducted. A questionnaire under the guidance of Dr. Asha O.S was prepared
and arranged as a Google form. The data
was collected from parents of students
of SNM Training College Moothakunnam through google forms.
Also data
from common people were collected directly from my neighbours through
interviews.
A total of
20 responses were taken as samples for analysis. The data obtained through Ms
Excel and others has analysed and interpreted through statistical tools.
DATA COLLECTION AND ANALYSIS
All
congruent figures are similar, but not all similar figures are congruent.
Congruence
means two objects (whether two dimensional or three dimensional) are identical
in size and shape.In geometry, congruent figures have three properties ; Same
size, Same shape and Corresponding parts are congruent
Similar figures have the same shape and
proportions but are not necessarily the same size. Angles of similar figures
will be equal, but lengths of sides usually are not equal.
The
presence of congruence and Similarity could be seen in many situations
including construction, designing, packings, human body, nature, science field etc…
|
No. |
Situation |
Presence of concept and procedure felt |
Implications |
|
1 |
Scale drawing |
●
The
concept of similarity is used to represent countries through maps by
decreasing ratio. ●
The
concept of similarity is used for representing extremely small cells by
enlargement. ●
The
concept of similarity is applied in architecture in drawing plans by decreasing
ratio of original measurements of building ●
Similarity
concept is applied in representing distant galaxies as photographs |
●
Extremely
large as well as microscopic organisms can be represented through this
concept. ●
Application
of this concept is necessary for further studies and findings on these
enlarged things
|
|
2 |
Construction |
●
The
concept of similarity is applied in constructing A-frame houses in which the
second floor is the dilation of the house. ●
Decorations
can follow the principles of similarity to have more beauty. |
●
Create
variety in building structure ●
Can
be built easily and beautifully. |
|
3 |
Fluid mechanics |
●
The
principle of Similarity is used in creating models from prototype ●
Principle
of geometric similarity is used in finding the ratio of linear dimensions of
model and prototype ●
The
principles of kinematic similarity are applied on motion of prototypes and
models by finding the ratio of their velocity and acceleration at a point. ●
Similarity
on forces over prototype and model are tested to find if there exist dynamic
similarity. |
●
Useful
in experimenting on models with low cost. ●
Compared
with the size of prototypes, models are easy to handle.
|
|
4 |
Measuring heights |
●
The
principle of congruence in angles and proportion of corresponding sides are
used in finding heights in different situations. ●
The
above principle can be applied for finding heights of tall trees from
shadows. ●
The
principle of similarity is used for finding heights of huge buildings. ●
The
concept of similarity can be used for finding the width of a river without
crossing it. |
●
This
makes calculation easier ●
Desired
length can be found out from minimum available information ●
This
method can be used to avoid dangers in several situations.
|
|
5 |
Design Method of the Inflatable Buffer Landing System
|
●
Similarity
relation is used to ensure safety of the landing attenuation system of an
interstellar probe. ●
Similarity
relation is used for obtaining theoretical derivation between the full-scale
airbag landing attenuation system in the Martian atmosphere and its Earth
prototype. |
●
Reliability
of the air bag can be checked. ●
Precautions
can be taken from impacts
|
|
6 |
Construction |
●
The
concept of congruent figures are used in all constructions ●
The
principles of Congruence applied in constructing pillars gave support to the
construction. ●
The
congruent figures can be used for increasing aesthetic beauty of
constructions. |
●
New
patterns in structures can be introduced through congruent figures. ●
The
aesthetic nature can be nurtured.
|
|
7 |
Arrangement |
●
The
congruent products can be arranged in well order. ●
The
pages of a book follow the concept of congruence. Hence their arrangement of
the book gave a perfect order. |
●
Arrangement
gets a structure through this concept. ●
It
beautifies the packing. ●
It
standardised distribution of products.
|
|
8 |
Designing |
●
Congruent
and similar geometrical figures are
exclusively used in jewellery. ●
The
principles of congruence are applied in making of earrings. ●
The
concept of Similarity is applied in introducing the different sizes in
clothing like S, L , XL , XXL etc… ●
Congruent
figures as well as similar figures are used in designing patterns in clothes. |
●
This
improves the aesthetic beauty ●
Different
sizes of clothes increases choices of customers.
|
|
9 |
Nature |
●
The
concept on Similarity can be seen in newborn animals with their parent ●
The
leaves in a tree are similar to each other ●
The
concept of congruence can be seen in leaves opposite to midrib. ●
The
concept of congruence can be seen in seeds and beans. |
●
This
gave beauty to the things ●
Things
can be separated on the basis of older and younger.
|
|
10 |
Human body |
●
The
concept of congruence and Similarity can be seen in different body parts,
internal as well as external. ●
The
congruence is visible on DNA
replication structure. ●
The
concept of congruent figures is seen in chromosome replication. ●
In
the Skeletal system similarity in bones are visible. |
●
This
gave the perfect structure for the human body. ●
Congruent
body parts beautifies humans ●
Properties
can be analysed easily.
|
Perception of the people in
congruence and Similarity
In the
second phase of data collection, a survey was conducted to evaluate the
comprehension of common people about congruence and Similarity in their
respective domains. Questions were asked to analyse their opinions on values that can
be imparted through teaching mathematics and teaching these topics related to
daily life.
- Values imparted through teaching mathematics
|
What are
the values that can be imparted through teaching mathematics? |
||
|
Practical value |
11 |
35.5% |
|
Vocational value |
4 |
12.9% |
|
Cultural value |
1 |
3.2% |
|
Disciplinary value |
2 |
6.5% |
|
Aesthetic value |
2 |
6.5% |
|
Social values |
1 |
3.2% |
|
All the above |
10 |
32.3% |
Most
of the people, about 35.5% responded that practical values are mainly
imparted through teaching mathematics. 12.9% respond that vocational values are
imparted through mathematics. A 6.5% of people equally agree that mathematics
impart disciplinary values and aesthetic values. Also 3.2% equally says that cultural values and social values
can be imparted through teaching mathematics. Nearly 32.3% people agree that
all the above values can be imparted through mathematics teaching.
|
Have you heard about mathematical congruence and
Similarity? |
||
|
Yes |
12 |
60% |
|
No |
8 |
40% |
Majority of the people about 60% are
heard about the mathematical congruence and similarity. There are 40% who are
not aware about these concepts.
|
The biscuits in a packet have the same shape and size.
What property makes their packing
beautiful? |
||
|
Congruence |
11 |
55% |
|
Similarity |
4 |
20% |
|
None of the above |
0 |
0% |
|
No idea |
5 |
25% |
Above average responses , 55%
answered correctly to the question. A 20% got confused and answered the
similarity wrongly. There are 25% who don't have an idea about answering this
question.
|
Whether you can find out the presence of mathematical
congruence and similarity in daily life and in your working field? |
||
|
Definitely |
4 |
20% |
|
Sometimes |
10 |
50% |
|
Never |
1 |
5% |
|
No idea |
5 |
25% |
Around 65% respondents answered this
question correctly. There are 5% of respondents who got confused and answered
incorrectly. There are 5% respondents who think neither congruence or
Similarity is the answer. There are nearly 25% who have no idea about answering
this question.
|
Whether you can find out the presence of mathematical
congruence and similarity in daily life and in your working field? |
||
|
Definitely |
4 |
20% |
|
Sometimes |
10 |
50% |
|
Never |
1 |
10% |
|
No idea |
5 |
25% |
Nearly 20% of people could
definitely find the presence of mathematical congruence and similarity in their
daily life and in their working field. A 50% of people responded that sometimes
they could find the presence of these concepts in daily life and working
filed.There is aa 5% people among respondents who never find the presence of
these mathematical concepts around them. There are nearly 25% who don't have
any idea about these.
|
Opinion regarding the applications of the concepts of '
similarity and congruence' in daily life? |
||
|
There are applications |
8 |
40% |
|
There may be applications, but don't know about it clearly |
9 |
45% |
|
No application |
0 |
0% |
|
No idea |
3 |
15% |
40% of the respondents agreed that there are applications of the mathematical concepts congruence and similarity in daily life. Majority of respondents ,about 45%, say that there may be applications and they are not clear about it. There are nearly 15% of respondents who don't have any idea about it.
|
What is your opinion about teaching these topics
associated with daily life? |
||
|
Is required |
18 |
90% |
|
Not required |
2 |
10% |
Majority of respondents about 90% agree that teaching these topics associated with daily life is required. But there is a minority of 10% who thinks it is not required.
Geometry is a popular branch in
mathematics. But geometric congruence and Similarity are quite unknown among
many ones.Mathematics teaching can impart several values including cultural
values, social values, practical values, vocational values etc…Our daily life
is surrounded by situations involving geometric congruence and similarity. But
only a few are aware about it.Geometric congruence and Similarity are closely
related to life and can be seen in various fields, from small packing to higher
scientific experiments.Congruence and similarity are interesting topics to
learn. Even Though there is a group who doesn't have ideas about its
application.Topics like congruence and similarity, which are generally taught
only through triangles, made a veil over their visions on seeing it in other
objects. That may be the reason for opting wrong options when the questions are
asked in different contexts.
This project on Congruence and
Similarity helped me to understand the importance of this concept in daily
life. I am astonished by seeing numerous situations in life including
constructions, designing things like fabrics and jewellery, in scientific
experiments, nature, human body etc.. and understanding this concept will make
us appreciate it at an optimum level.
The survey on common people helped
me to study their comprehension of mathematical congruence and similarity.
Majority of the people are not aware about the values that can be imparted
through teaching mathematics. Every one sees only the practical values.So I
think lack of awareness about the aesthetic values obtained through congruence
and similarity prevent their vision on seeing these concepts in their
surroundings.
This survey reveals that there are
people who don't have any idea about these concepts. The percentage shows that
even the
people who heard about these concepts couldn't answer the two questions
on Similarity and congruence. That means when questions are twisted from
triangles to life some of them become confused to answer correctly.
The 60% are heard about these
concepts but only 40% say that these have applications and a 45% are not sure
about these applications clearly. This implies that teaching these concepts through daily life is
necessary to have actual comprehension. That's why only 20% can find these
concepts in daily life and there exists a total 30% who don't have any idea at
all. Majority of people support teaching mathematics related to daily life.
It's a way to increase the interest in learning.
The learner has to know the values
that he is going to gain through
learning mathematics. This will increase the curiosity of the learner and hence
interest can be created.
- https://www.livescience.com/38936-mathematics.html
- https://en.m.wikipedia.org/wiki/Congruence_(geometry)
- https://www.britannica.com/science/geometry/Ancient-geometry-abstract-and-applied
- https://ghr.nlm.nih.gov/primer/basics/dna
- https://prezi.com/j5q7qqo0ug9q/jewelers-in-geometry/
- https://www.reference.com/world-view/geometry-used-fashion-design-da1c95464a3cd91a
- http://mechanicalinventions.blogspot.com/2014/06/similitude-in-fluid-mechanics-similarity-l
- https://www.onlinemath4all.com/using-similar-triangles-to-measure-height.html
- https://www.geogebra.org/m/XupeKSf
- https://www.hindawi.com/journals/ijae/2019/5326979
- https://www.wyzant.com/resources/lessons/math/geometry/introduction/history_of_geometry
