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PS 4305 CT Mathematics I 2008
Semester 3 2008/2009
BA PRIMARY EDUCATION
PS4305 CURRICULUM & TEACHING:
Lower Primary Mathematics 3
REPORT
Topic :
The use of mathematics in other subjects or fields in primary schools will be
examined.
Izyan Izzati bte Ismail (08B0527)Diana bte Simpai (08B0529)Dyg. Suhana bte Hj. Abg. Ali (08B0532)Nurul Khairunnisa bte Morni (08B0534)
Lecturer: Dr. See Kin Hai
Sultan Hassanal Bolkiah Institute of Education
University Brunei Darussalam
Saturday, 15th November 2008
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PS 4305 CT Mathematics I 2008
INTRODUCTION
According to Leonard M. Keenedy and Steve Tipps (2000): “Mathematics is a
language for describing common events in everyday life and complex events in business,
science and technology”
Below are some examples of application of Mathematics in traveling, astronomy,
engineering and manufacturing.
When a family plans a vacation, they use mathematics to estimate distances, times
for departure and return, fuel needed, food and other supplies required and costs of
maintaining the family vehicle. As astronomers and engineers plan space travel,
mathematics is used to calculate distances, times for departure and return, fuel needed,
food and other supplies and costs of maintaining the space vehicle. When manufacturers
plan for distribution across the Brunei Darussalam, they employ mathematics to calculate
distances, time for departure and return, fuel needed, food and other supplies required,
and costs of maintaining the vehicle.
Three different problems, three levels of mathematics and three needs for
precision, and all these three require the same thinking process. As a result of this,
mathematics is a tool and a language for solving problems great and small.
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PS 4305 CT Mathematics I 2008
This report will focus on the use of mathematics in other subject including
different aspect of mathematics in different area or field. New mathematics courses have
been developed that focus on “nontraditional” topics and applications in real world. As
today’s student learn mathematical concepts and thinking, they must apply, adapt and
extend old concepts to new tasks and existing ideas into real world. In the 21st century,
children will need mathematics for complex and common applications.
The report of the Cockcroft Committee Mathematics Count (HMSO, 1982)
comments that although there are some books linking mathematics to other subjects such
as art and science, more are needed. Paragraph 292 of the Report states:
Almost all children find pleasure in working with shapes, and work of this can
encourage the development of positive attitudes towards mathematics in those who are
finding difficulty with number work.
In later chapter, we will discuss about mathematics in different fields, namely
mathematics through art and design, mathematics in everyday life, mathematics in
medicine, mathematics in sports, mathematics in cooking, mathematics in psychology,
mathematics in architecture and mathematics in nature.
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PS 4305 CT Mathematics I 2008
MATHEMATICS IN ART AND DESIGN
W W Sawyer cited by Anne Woodman and Eric Albany defined mathematics as
the classification and study of all possible patterns and relationships. Pattern is anything
the mind can recognize as regularity. Mathematical patterns or relationships in a
colourful way add to a child’s understanding and appreciation of them. From there,
children may come to realize that inherent in many artistic forms exists a mathematical
precision too often taken for granted. Symmetry is a dominant characteristic of both the
natural and manmade world.
By linking Mathematics to other subjects such as Arts and Science, children can
find pleasure in working with shapes, and work of this kind can encourage the
development of positive attitudes towards Mathematics in those who are finding
difficulty with number work.
Children will be able to draw their attention between functional and aesthetic
aspects if they see mathematics as a more attractive subject. Mathematics has its own
intrinsic beauty and aesthetic appeal, but its cultural role is determined mainly by its
perceived educational qualities.
I’ve decided to choose Islamic pattern and Chinese Tangram as examples of
mathematics in art and design. This is because in both of this examples, lies between it, is
the beauty and the mathematical concept
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PS 4305 CT Mathematics I 2008
Pattern
Patterns abound in everyday life. Pattern is a fundamental concept in both
Mathematics and Art. The inclusion of patterns in schools encourages students to develop
habits of looking for and using Mathematics. Patterns take many forms in both natural
and designed situations. Leaves and trees are examples of recognized patterns.
The octagonal geometrical structure of the building is as shown in the diagram
below.
Although the greatest bulk of Islamic patterns are built on hexagonal structures,
the octagonal structure is the basis of the most sophisticated, complex and beautiful
Islamic patterns of the kind that occur in the Alhambra. The two highlighted shapes in the
center are the most prominent and instantly recognizable shapes in Islamic Geometrical
Art and can be seen in abundance throughout the Islamic World.
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PS 4305 CT Mathematics I 2008
Islamic Patterns
The use of geometric design in mosque, Arabic architecture and artifacts has great
religious significance. It is a form of sacred geometry.
1) Draw in the axes of symmetry.
2) By using the intersections of grid lines and axes of symmetry, it result in a symmetrical pattern which is the bold lines.
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PS 4305 CT Mathematics I 2008
3) The dotted line will be the master pattern for the next step.
Islamic pattern involves with axes of symmetry, and from there, we can build up a
symmetrical pattern, using the intersections of grid lines and axes of symmetry as stated
above by using step by step examples. Shapes such as Polygon that occur within the
design are part of Mathematics content. From here, children can construct their own
design by using reflection, rotation and translation.
At the most practical level, mathematics tools have been used to create beautiful
designs realized in the architecture and decoration of palaces and mosques. Isometries,
similarities, and transformations can transform images exactly or with purposeful
distortion, projections can represent three-dimensional forms on two-dimensional picture
surfaces, even curved ones. Special transformations can distort or unscramble a distorted
image, producing anamorphic art. Compasses, rulers, grids and mechanical devices are
physical tools for the creation of art, but without the power of mathematical relationships
and processes these tools would have little creative power.
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PS 4305 CT Mathematics I 2008
Through the study of patterns, children can learn to see relationships and make
connections, generalizations, and predictions about the world around them. Working with
patterns empowers children to solve problems confidently and relate new situations to
previous experiences. Children should deal with both repeating patterns and patterns that
can reflect, rotate and translate. Below are examples of Islamic Pattern.
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PS 4305 CT Mathematics I 2008
Chinese Tangram
The Tangram puzzle was invented in China over 4000 years ago. The puzzle
requires the seven pieces to be used to create an arrangement as all seven pieces must be
used and no pieces should be overlap. The mathematically inclined are interested in the
puzzle because of the geometry and ratios of the pieces. Tangrams can be found in
classrooms around the world to teach basic math ideas in an interesting way.
The Rules of Tangrams are it needs to be creating different patterns using all
seven tangrams. The tangrams must lay flat, they must touch and none may overlap.
There are many patterns available, which you may recreate or you can come up with your
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PS 4305 CT Mathematics I 2008
own creations. Below is the picture of Chinese tangram.
Student can learn the names of all the polygonal pieces of Tangram namely
square, parallelogram, quadrilateral, polygon and triangle. Do you know that chinese
tangrams provide an excellent source of investigational material which can be linked with
several mathematical topics such as symmetry, angles, fractions and areas. Basically it is
up to the teacher itself to extract as much mathematics as possible while at the same time
encouraging the creative aspects of the activities.
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PS 4305 CT Mathematics I 2008
Below are example of mathematical topics that arise in Chinese tangram are
angles in the shape of triangle.
90°
arc
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PS 4305 CT Mathematics I 2008
MATHEMATICS IN EVERYDAY LIFE
Cooking & Eating food
There’s a range of Mathematics you can do during these times, involving
activities such as measurement, finding fractions, dividing and comparing. Food comes in
pieces and division involving counting and distributing things into fair shapes. Both kinds
of division are mathematically important, and both can lead into thinking about fraction.
After eating, the leftover food will need to be fit in different sized containers, and this
intuition is built on an understanding of conservation of volume. Figuring out which
containers to use for leftovers is a natural way to help children to develop these skills.
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PS 4305 CT Mathematics I 2008
Travelling
“Where are we now? When will we get there?” are questions that are commonly ask
by children. Whether you’re traveling cross country or are stuck in a traffic jam on the
way to school, traveling presents straightforward and ready-made opportunities for doing
Mathematics.
When traveling, questions about when you’ll get there are actually questions
about time, distance, and space. All of these are critical mathematics concepts. When
traveling, calculating, comparing distances and how long it takes to cover distances at
given speeds are part of mathematics. Traveling also gives opportunities to make sense
of directions, maps and spatial relationships. Traveling also involves with units of
measure like block, kilometer, and mile. These types of units are used to figure out
distances which involve with mathematical concepts.
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PS 4305 CT Mathematics I 2008
Decision making
In decision making whether to continue to invest money in a companies or to stay
with the swimming club , graphs and other representation are more likely to use to
describe changes over time and relate the story they want to tell. It is the shape of the
graph as it bumps, slopes, drops and level spots that convey the story. The Mathematical
focus is on thinking about and representing change over time. Much of higher
mathematics is about describing the current state of something, as well as how gradually
or suddenly something is changing. This can be done by constructing graphs.
By making a line that tells the story of how he/she felt about the issue from the
starting point until now. The better he/she felt, the higher the line should go. The more
negative he/she felt, the lower the lines go. This type of line graph tells the story by itself.
Questions like this help in decision making:
“Did you notice a gradual increase in your interest, or was it pretty sudden? How
could you show that on the graph?”
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PS 4305 CT Mathematics I 2008
MATHEMATICS IN SPORTS
Often, mathematics and sports do not linked very well together. Many people
think that both subjects are far away from each other.
As stated by Sadovskii and Sadovskii (1993), “it may seem, at first sight, that mathematics and
sports are very far apart. Indeed, many young people have mistakenly believe that learning math is
one thing and going in for sports is quite another.”
Therefore, in schools sometimes we see that those who are very good in
mathematics do not tend to sports and vice versa. Admittedly, too many bright pupils
look down on games and physical training. Similarly, those who are actively in sports and
physical activities tend to look down on the importance of mathematics. Most of them
didn’t see the relevance of mathematics in sports. This is because of their little
knowledge of the diverse linkage between mathematics and sports.
The use of mathematics in sports can be seen in different types of sports. Almost
every sport have mathematical application in it. Some sports that uses math are tennis,
ballet, gymnastics, basketball as well as running. For this particular report, we’ll see how
math is used in two of this sports which is running and basketball.
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PS 4305 CT Mathematics I 2008
Mathematics in running
In all running activities or also known as sprints, the objective is to cover a given
distance in as short a time as possible relative to the opposition. According to Townend
(1984), the time here is determined by the athlete’s speed which in turn depends on two
quantities which is the length and the stride of the frequency. Therefore, in order to
enhance a sprinter performance, one needs to pay attention of these two mathematical
qualities. Townend (1984) states that the values of these two quatities depend upon the
nature of the event itself. This explains why a sprinter has a much longer stride compared
to a marathoner.
In a sprint event, once the starting pistol has been fired the sprinter’s main
concern is to accelerate to his maximum speed as quickly as possible and then maintain it
as long as possible. In order to do this, there are quite variety of technique that needed to
take into consideration. One of the most important techniques begins at the starting point.
It is the toe-to-toe distance of the athlete at the starting point as shown in Figure A.
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toe-to-toe distance
PS 4305 CT Mathematics I 2008
Figure A
According to Townend (1984), it consists of three different techniques are as
follows (applied for college athletes):
i. The bullet start, toe-to-toe distance 0.28 m approximately,
ii. The medium start, toe-to-toe distance in the range 0.4 m – 0.53 m
approximately,
iii. The elongated start, toe-to-toe distance in the range 0.61 m – 0.71 m
approximately
According to Townend (1984), in order to know which technique suits a sprinter
best. It can be done by using some mathematical approaches. This can be done by using
the slope of tangent to determine the athlete’s velocity (rate of change). Then the average
of these results is taken accordingly to the three techniques. The result of the average will
determine which technique suits the athlete the best as shown in Figure B.
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PS 4305 CT Mathematics I 2008
Fig. B
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PS 4305 CT Mathematics I 2008
Mathematics in basketball
In basketball, in order to make a shoot, a player needs to throw the ball in the
basket. Here, the release position of the ball is some height h above ground level.
However, this time the purpose of the throw is to make the ball pass through the hoop
instead of attempting to maximize the range of the throw. Most importantly is the angle
of entry of the ball into the hoop is correct since otherwise the ball will not pass through
the hoop.
Angle of entry
According to Townend, if the basketball, of diameter 24.6 cm (the midpoint of the range
of permited diameters), approaches the hop from directly above then it has a circular
open-ing of diameter 45 cm through which it can pass. If the ball approaches from any
other angle, say ß to the horizontal as shown in the Figure C below, then the opening
presented at right angles to the ball’s path is an ellipse having one axis of length 45 cm
and the other of length d cm where d=45 sin ß. Clearly, the use of mathematical aspects
such as trigonometry is used in order to find out the angle of entry in a basketball game.
Figure C
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PS 4305 CT Mathematics I 2008
Free Shots
While it is possible to shoot the ball from any one of countless positions on a
basketball court, the throw which is rather popular is free shot. This is the shot awarded
to a player who has been fouled and is taken from a fixed position. To take the shot, the
player stands behind the free throw line and attempts to throw the ball through the hoop.
For instance, according to Townend (1984), in order a free shot released from free
throw line at a height of 2.15 m (refer Figure D) with a velocity of 8 m/s will score, it
must be released at an angle of 64º 28’ to the horizontal. This explains the very acute
angles of release seen in basketball games when free throw is awarded.
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Figure D
PS 4305 CT Mathematics I 2008
MATHEMATICS IN MEDICINE
Mathematics in Body Mass Index (BMI)
According to www.cdc.gov, a site for centres for diseases control and prevention,
Body Mass Index (BMI) is a number calculated from a person’s weight and height. BMI
is a reliable indicator of body fatness for people and also as a screening for any
possibilities leading to health problems. Here, the use of mathematics is that it
implements the use of algorithms, graphs and also the use of measurements units as
shown in Figure E.
A person weight and height is measured, which is again another aspects of
mathematical procedure. Then this height and weight will be used in the calculation of
the BMI according to the BMI formula (Figure E). The outcomes will then determine the
category of which a person belongs to (Figure F). These data can also be obtained by
using another mathematical aspects which is graphs as shown in Figure G.
Figure E Figure F
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PS 4305 CT Mathematics I 2008
Figure G
Mathematics in Medical Images
Medical images can be obtained through computerized axial tomography (CAT),
magnetic resonance imaging (MRI), or positron emission tomography (PET). These
equipments (Figure H) enable us to see any abnormalities in the lungs, brain such as
tumors. According to http://www.mathaware.org, this involves the use of math such as;
the use of reconstructive mathematical techniques where thousands of separate
measurements are mathematically combined to create a single image and also the use of
measurements to read the output (refer Figure I) such as the size of the tumor.
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PS 4305 CT Mathematics I 2008
Figure H Figure I
Mathematics in DNA Technology
In DNA technology, strands of DNA are examined through techniques of
topology and differential geometry. Databases of human genome are very extensive and
complex. According to http://www.mathaware.org, mathematical simulation and
modeling are also key to visualizing and understanding recombinant DNA technology.
Therefore, mathematical approaches such as pattern recognition and sequence
comparison are required. Hence, we need to know how the sequence and pattern to read
and know the genetic code.
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PS 4305 CT Mathematics I 2008
Mahematics in Pharmaceuticals
In pharmaceutical industry, math is used to determine how much medication one
should take and how to manufacture the correct dosage as shown in Figure J. New drugs
are also being designed as new mathematical algorithms are created.
Figure J
Mathematics in Medical Statistics
Health statistics have long been collected and analyzed for a variety of purposes,
including cost control, public policy research, demography, and disease trends correlated
with other variables such as environmental factors. A very good example will be the
observation chart that is commonly and widely used in the hospital where different
variables such as the temperature, blood sugar, blood pressure and urination is plotted
down in a chart in relation to time or else as shown in Figure K.
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PS 4305 CT Mathematics I 2008
Figure K
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PS 4305 CT Mathematics I 2008
MATHEMATICS IN COOKING
When we want to buy a particular food, most of the time we have a particular
place to buy the food though it is also available at other places having the looks of almost
the same. It is not more about the matter of the price of the food. It’s the flavour!! How
the food differs in taste is because of how the cook measure the amount of ingredients
used. This is where cooking has a relationship with mathematics.
The cook has mathematical skills in cooking but it would not be that complex.
Cooking involves mathematical calculations like addition, subtraction, multiplication,
division, ratios and so forth. The mathematical skills used in cooking bring about the
quality of the cook. Whether it tastes delicious or sour, it all depends on how the
ingredients are being measured.
Most of the time, recipe guideline provides us with a single batch of food. If we
have the single batch recipe guidelines, it is totally not a problem if we want to prepare
more than the single batch. We do not have to waste our time by producing a single batch
of food at a time. Easily, we could just make it at one time by adjusting the amount of the
ingredients.
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PS 4305 CT Mathematics I 2008
An example of a recipe is Mouth-watering cookie:
1 cup flour
½ tsp. baking soda,
½ tsp. salt
½ cup butter, 1
1/3 cup brown sugar
1/3 cup sugar
1 egg
½ tsp. vanilla
1 cup chocolate chips
If I want to produce double of this batch, I could just multiply the amount of each
ingredient by two to produce at a single time. For example, 1 cup of flour will become 2
cups of flour as I am producing the doubled batch of this recipe. Apparently,
mathematical operation is involved.
Moreover, catering service is a very popular service among Bruneians for some
occasions like wedding, birthday party and so forth. When we deal with the caterer, they
would ask the expected number of people will attend for the occasion. This is for reason
that the caterer has the basis in estimating how much food they need to prepare. They
need to estimate how much each person will eat so that they can estimate the amount of
food to be cooked to match the number of estimated people.
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PS 4305 CT Mathematics I 2008
However, there are limitations to this matter. The estimated food might not be
sufficient to meet the number of people eating because people might eat more than once
especially if they stay at the particular place for a longer time. Aside from that, the
number of people come might go beyond the expectation or lesser due to certain
circumstances. This will affect the amount of food prepared.
Some of the time, we might not get the desired ingredients for cooking. For
instance, I need to use big marshmallow but unfortunately, I could not get the big one. So
I just buy the small one and this will alter the amount used. So, I need to use my
mathematical skill in estimating how much small marshmallow is proportion to one big
marshmallow. If I need 5 big marshmallows and one big marshmallow is proportion to 2
small marshmallows, therefore I need to use 10 small marshmallows to meet the amount
of ingredients used.
Considering how long our food cook is also counted in cooking for it affects the
quality of the food. For instance, a packet of noodles take 2 minutes to cook. If I want to
cook three packets of noodles, I do not have to cook them one by one in order to get the
right time of cooking. I could just mix the three and multiply the time that is, 2 minutes
multiply by three packets.
However, the time estimated might alter due to some consequences. The amount
of fire used might not be exactly the same which therefore will affect the time for the
foods to cook. In addition to that, the size of the food might not be the same; the bigger
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PS 4305 CT Mathematics I 2008
the food, the bigger the surface area, the long the time to cook. The food might be a
mixed of small and big sizes and therefore is related to the timing for the food to cook.
In relation to cooking and the curriculum, it is clearly shown that they are related.
In order for the pupils to see the use of mathematics in cooking, it is the teacher’s role
that is important here. The teacher needs to emphasize to the pupils that we also use
mathematics calculation in cooking. By opening up their minds, the pupils then can apply
their knowledge at home. To some pupils who did not involve themselves with work in
the kitchen, the teacher could ‘bring’ them in cooking situation through solving word
problems. For instance:
a) I have 900g of sugar. To bake a cake, I only need 150g. How much grams of
sugar do I need to take?
b) A 2 litres of water boils for 10 minutes. How much time will a 5 litres of water
will boil?
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PS 4305 CT Mathematics I 2008
MATHEMATICS IN PSYCHOLOGY
Researchers in psychology quantify and analyze their research findings. In order
for them to come up quantifying and analyzing their research findings, they use
mathematical calculation as their statistical tool. For instance, they use addition,
multiplication and division in calculating mean, median, standard deviation and so forth.
Normal distribution or Gaussian curve is required for researches in psychologist
as it helps to estimate whether or not a sample mean represents a population mean.
According to Shavelson cited by Best & Kahn (1998; p253) normal distribution is a
“Mathematical model, an idealization that can be used to represent data collected in
behavioral research.” From this definition, it is clearly tell us that the work of
psychologists have relationship with mathematics.
In order to construct a normal distribution, it has characteristics which are:
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PS 4305 CT Mathematics I 2008
1) The curve is symmetrical
2) The mean, median and mode stands at a point in the middle of the curve.
3) There are no boundaries: the tail of the curve on the left and the right side will not
touch the line of the x-axis.
4) The scores are near the mean, median and mode.
5) The curve is a version of a bar graph.
Researchers will only come up with normal distributions by undergoing some
mathematical calculations for instance in calculating z-score where a score minus its
mean, the whole things is divided by standard deviation of a distribution. The purpose of
z-score in research is to locate a sample mean from the population mean that is, whether
it is above or below a population mean. If the mean of the sample is the same as the mean
of the population, it will form a normal distribution.
Apparently, the mathematics curriculum also blends in psychology area.
Therefore, primary school teacher must not forget to highlight to the pupils that not only
the pupils studying mathematics and the teachers of mathematics do mathematics, but
psychologist also involves with mathematics though from the term psychologist itself, it
tells us that they are people who study about people’s minds and behaviour.
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MATHEMATICS IN ARCHITECTURE
Architecture has in the past done great things for geometry. Together with the need to
measure the land they lived on, it was people's need to build their buildings that caused
them to first investigate the theory of form and shape. But today, 4500 years after the
great pyramids were built in Egypt, what can mathematics do for architecture? For
thousand of years mathematics has been an invaluable tool for design and construction.
Below is a partial list of some mathematical concepts which have been used in
architecture over the centuries:
1. Pyramids
2. Prisms
3. Golden rectangle
4. Optical illusions
5. Cubes
6. Polyhedra
7. Geodestic dome
8. Triangles
9. Phytogorean theorem
10. squares, rectangles
11. Parallelograms
12. Circles, semi-circles
13. Spheres, hemisphere
14. Polygons
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PS 4305 CT Mathematics I 2008
15. Angles
16. Symmetry
17. Parbolic curves
18. Catenary curves
The design of a structure is influenced by its surroundings, by the availability and
type of materials and by the imagination and resource upon which architect can draw.
Historical Architecture
Pyramids of Egypt, Mexico and the Yucatan used the mathematical knowledge of
right angles, squares, Phytagorean theorem, volume and estimation in its construction.
Whereas the Machu Picchu in Peru are well known for its unique geometric plans. The
famous Pathenon in Greece uses the golden rectangle, optical illusions, precision
measurements and knowledge of proportion. Moreover, the Ancient Theater at Epidaurus
emphasize on its geometric plans. The Roman architects rely mostly on the use of circles,
semicircles, hemispheres and arches. The Architects of the Byzantine mostly uses
squares, circles, cubes, hemisphere and arches for its construction.
Berger, makes a study of the way that the Pythagorean ideas of ratios of small
numbers were used in the construction of the Temple of Athena Parthenos. The ratio 2 : 3
and its square 4 : 9 were fundamental to the construction. A basic rectangle of sides 4 : 9
was constructed from three rectangles of sides 3 and 4 with diagonal 5. This form of
construction also meant that the 3 : 4 : 5 Pythagorean triangle could be used to good
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PS 4305 CT Mathematics I 2008
effect to ensure that right angles in the building were accurately determined. The length
of the Temple is 69.5 m, its width is 30.88 m and the height at the cornice is 13.72 m. To
a fairly high degree of accuracy this means that the ratio width : length = 4 : 9 while also
the ratio height : width = 4 : 9. Berger took the greatest common denominator of these
measurements to arrive at the ratios
height : width : length = 16 : 36 : 81
which gives a basic module of length 0.858 m
Then the length of the Temple is 92 modules, its width is 62 modules and its height
is 42 modules. The module length is used throughout, for example the overall height of
the Temple is 21 modules, and the columns are 12 modules high. The naos, which in
Greek temples is the inner area containing the statue of the god, is 21.44 m wide and 48.3
m long which again is in the ratio 4 : 9. Berger notes the amazing fact that the columns
are 1.905 m in diameter and the distance between their axes is 4.293 m, again the ratio of
4 : 9 is being used.
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PS 4305 CT Mathematics I 2008
The famous Phatenon
Modern Architecture
With the discovery of new building materials, new mathematical ideas were adopted
and used to maximize the potential of these materials. Using wide range of available
building materials architects has been able to design virtually any shape.
Formation of the hyperbolic paraboloid, the geodestic structures of Buckminster
Fuller, the module designs of Paolo Soleri, the parabolic airplane hanger, solid synthetic
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PS 4305 CT Mathematics I 2008
structures mimicking the tents of the nomads, catenary curve cables supporting the
Olympic Sports Hall in Tokyo and Octagonal home with an elliptical dome ceiling. One
good example of a modern architecture is The Gherkin. There are three main features that
make The Gherkin stands out from most other sky-scrapers: it's round rather than square,
it bulges in the middle and tapers to a thin end towards the top, and it's based on a
spiralling design. All these could easily be taken as purely aesthetic features, yet they all
cater to specific constraints.
The Gherkin
Another example is the London City Hall. As with the Gherkin, the shape of the
London City Hall was not only chosen for its looks, but also to maximise energy
efficiency. One way of doing this is to minimise the surface area of the building, so that
unwanted heat loss or gain can be prevented. As the mathematicians amongst you will
know, of all solid shapes, the sphere has the least surface area compared to volume. This
is why the London City Hall has a near-spherical shape.
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PS 4305 CT Mathematics I 2008
The London City Hall
Architecture is an evolving field. Architects study, refine, enhance, reuse ideas
form the past as well as create new ones. In the final analysis, an architect is free to
imagine any design as long as the mathematics and materials exist to support the
structure.
ARCHITECTURE IN NATURE
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Lobachevsky quoted, “There is no branch of mathematics, however abstract,
which may not someday be applied to phenomena of the real world”. As nature puts forth
its wonders, most of us are oblivious to the massive calculations and mathematical work
needed to explain something very routine to nature. In the orb spider’s web the
Mathematical ideas that appear in the web are radii, chords, parallel segments, triangles,
congruent corresponding angles, the logarithmic spiral, the catenary curve and the
transcendental number .
According to Pappas T. (1994), ‘Bees have not studied tessellation theory nor orb
spiders logarithmic spirals. Nature uses the most efficient and effective forms that require
the least expenditure of energy and materials. Nature has mastered the art of solving
maximum / minimum problems, linear algebra problems and finding optimum solutions
involving constraints. By focusing on the magical facts about honeybee we can see how
nature binds in mathematics in the most natural way.’
The honeybee
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Among the square, the triangle and hexagon (the only three self-tessellating
regular polygon),the hexagon has the smallest perimeter of a given area. This is why the
honeybee are hexagonal in shape. This makes bees use less wax and do less work to
enclose the same space as in square or triangle. The comb’s walls are made up of cells
which are about 1/80 of an inch thick, yet can support 30 times their own weight. A
honeycomb of about 14.5’’ x 8.8’’ can hold more than 5 pounds of honey, while it only
requires about 1.5 ounces of wax to construct. The bees form the hexagonal prisms in
three rhombic sections and the walls of the cell meet at exactly 1200 angles. The comb is
built vertically downward and the bees use parts of their bodies as measuring instrument.
Their heads act as plummets.
Bees have its own compass when searching for food. Bees’ orientation is influenced
by the Earth’s magnetic field. Bees can detect minute fluctuations in the Earth’s magnetic
field. This is why the bees occupying a new location simultaneously begin to build the
hive in different parts of the new area without any bee directing them. All the bees orient
their new comb in the same direction as their old hive.
Bees also have their own way of communication. Bees communicate the direction of
the food and the distance by transmitting codes in a form of a “dance”. The orientation
of the dance in relation to the sun gives the direction of the food, while the duration of
the dance indicates the distance. Honeybees “know” that the shortest distance between
two point is a straight line and mathematician call this line as the Beeline. The honeybee
gets its mathematical training via its genetic codes.
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Another example is the flock of birds. Have you ever wondered why a flock of birds
in flight as they swooped through the air don’t collide? Heppner F.H. established 4
simple rules based on avian behaviour and used triangles for birds on why flock of birds
doesn’t collide:
1. Birds are attached to a focal point or roost.
2. Birds are attached to each other.
3. Birds want to maintain a fixed velocity.
4. Flight paths are altered by random occurrences such as a gust of wind.
The flock of birds
Fractals have come to be referred to as the geometry of nature. Fractals can appear as
symmetrically changing or growing objects or as randomly asymmetrically changing
objects. In either case, fractals are changing according to Mathematical rules or patterns
used to describe and dictate the growth of an initial object. Think of a geometric fractal as
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an endless generating pattern. Pattern continually replicates itself but in a smaller version.
Thus when a portion of a geometric fractal is magnified it looks exactly like the original
version. In contrast, when a portion of a Euclidean object as circle is magnified it begins
to appear less curve. A fern is an ideal example of fractal replication. If you zero in on
any portion of the fractal fern, it appears as the original fern leaf. A fractal fern can be
created on a computer.
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CONCLUSION
By acquiring these knowledge on architecture and nature will contribute many
advantages to the primary pupils and the school curriculum. Some of it are it will widen
the pupil’s knowledge because their learning is not textbook based anymore. Pupils will
also have more interest in the learning of Mathematics as it will become more meaningful
to them. Mathematics is seen as a subject closely related to real life. Moreover, pupils
who have the ambition of a profession related to architecture and nature will extent and
deepen their interests towards these fields.
As Galileo mentioned. “… the universe stands continually open to our gaze, but it
cannot be understood unless one first learns to comprehend the language and interpret the
characters in which it is written. It is written in the language of mathematics, and its
characters are triangles, circles and other geometric figures, without which it is humanly
impossible to understand a single word of it…”
Mathematics is not just about learning the basic four operations or counting
numbers but it is more than that. It evolves in the world around us in almost every fields /
subjects / professions in this world. Mathematics is so powerful that from the day it was
first used until today it is still considered as one of the fundamental subjects and fields.
Mathematics is everywhere and in almost every field such as in everyday life, Art &
design, Architecture, Nature, Cooking, Psychology, Sports and Medicine as mentioned in
this report.
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REFERENCES
Best, J.W., & Kahn, J.V. (8th edition). (1998). Research in Education. Needham
Heights: Allyn & Bacon
Cockcroft Committee Mathematics Count (1982) Mathematics in other subjects. London:
Zed Books
Leonard M. Kennedy, Steve, T (2000) Guiding Children’s learning of Mathematics.
California: Wadsworth
Pappas, T. (1994). The Magic of Mathematics: discovering the spell of mathematics. San
Carlos, CA: Wide World Publishing.
Sadovskii L.E. & Sadovskii A.L. (1993). Mathematics and Sports. USA: American
Mathematical Society
Townend, M.S. (1984). Mathematics in Sports. Chichester: Ellis Horwood Limited
Woodman, A & Albany E (1988) Mathematics through Arts and Design. London: Unwin
Hyman
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http://www.cdc.gov/nccdphp/dnpa/healthyweight/assessing/bmi/index.htm
Retrieved on 20 October 2008.
http://www.mathaware.org/mam/94/essay.html
Retrieved on 20 October 2008.
http://www.bookrags.com/research/mathematics-and-psychology-wom/
Retrieved on 21 October 2008
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