Additional Mathematics Project Work 2012

LIVE WITH POLYGONS ADDITIONAL MATHEMATICS PROJECT WORK 1/2012

Name

: Chen Yee Sian

I/C No

: 951228-12-6244

Class

: 5N1

Teacher‟s Name

: Cik Norhayati Bt Mohd Nor

School

: SMK Convent Klang

CONTENT Acknowledgement Objectives Introduction Part 1 Part 2 Part 3 Further Exploration Conclusion Reflection

ACKNOWLEDGEMENT First of all, I would like to say thank you, for giving me the strength to do this project work. Not forgotten my parents for providing everything, such as money, to buy anything that are related to this project work and their advise, support which are the most needed for this project. Internet, books, computers and all that. They also supported me and encouraged me to complete this task so that I will not procrastinate in doing it. Then I would like to thank my teacher, Cik Norhayati for guiding me and my friends throughout this project. We had some difficulties in doing this task, but she taught us patiently until we knew what to do. She tried and tried to teach us until we understand what we supposed to do with the project work. Last but not least, my friends who were doing this project with me and sharing our ideas. They were helpful that when we combined and discussed together, we had this task done.

OBJECTIVES The aims of carrying out this project work are:  to apply and adapt a variety of problem-solving strategies to solve problems.  to improve thinking skills.  to promote effective mathematical communication.  to develop mathematical knowledge through problem solving in a way that increases students‟ interest and confidence.  to use the language of mathematics to express mathematical ideas precisely.  to provide learning environment that stimulates and enhances effective learning.  to develop positive attitude towards mathematics.

INTRODUCTION In geometry a polygon (/pɒlɪɡɒn/) is a flat shape consisting of straight lines that are joined to form a closed chain or circuit. A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The word "polygon" derives from the Greek πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". (The word γόνυ gónu, with a short o, is unrelated and means "knee".) Today a polygon is more usually understood in terms of sides. The basic geometrical notion has been adapted in various ways to suit Historical image of polygons (1699) particular purposes. Mathematicians are often concerned only with the closed polygonal chain and with simple polygons which do not selfintersect, and may define a polygon accordingly. Geometrically two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed. In fields relating to computation, the term polygon has taken on a slightly altered meaning derived from the way the shape is stored and manipulated in computer graphics (image generation). Some other generalizations of polygons are described below.

History Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to

the 7th century B.C. Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine. In 1952, Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.

Polygons in nature Numerous regular polygons may be seen in nature. In the world of geology, crystals have flat faces, or facets, which are polygons. Quasicrystals can even have regular pentagons as faces. Another fascinating example of regular polygons occurs when the cooling of lava forms areas of tightly packed hexagonal columns of basalt, which may be seen at the Giant's Causeway in Ireland, or at the Devil's Postpile in California.

The Giant's Causeway, in Northern Ireland

The most famous hexagons in nature are found in the animal kingdom. The wax honeycomb made by bees is an array of hexagons used to store honey and pollen, and as a secure place for the larvae to grow. There also exist animals that themselves take the approximate form of regular polygons, or at least have the same symmetry. For example, sea stars display the symmetry of a pentagon or, less Starfruit, a popular fruit in Southeast Asia frequently, the heptagon or other polygons. Other echinoderms, such as sea urchins, sometimes display similar symmetries. Though echinoderms do not exhibit exact radial symmetry, jellyfish and comb jellies do, usually fourfold or eightfold. Radial symmetry (and other symmetry) is also widely observed in the plant kingdom, particularly amongst flowers, and (to a lesser extent) seeds and fruit, the most common form of such symmetry being pentagonal. A particularly striking example is the starfruit, a slightly tangy fruit popular in Southeast Asia, whose cross-section is shaped like a pentagonal star.

Moving off the earth into space, early mathematicians doing calculations using Newton's law of gravitation discovered that if two bodies (such as the sun and the earth) are orbiting one another, there exist certain points in space, called Lagrangian points, where a smaller body (such as an asteroid or a space station) will remain in a stable orbit. The sun-earth system has five Lagrangian points. The two most stable are exactly 60 degrees ahead and behind the earth in its orbit; that is, joining the center of the sun and the earth and one of these stable Lagrangian points forms an equilateral triangle. Astronomers have already found asteroids at these points. It is still debated whether it is practical to keep a space station at the Lagrangian point – although it would never need course corrections, it would have to frequently dodge the asteroids that are already present there. There are already satellites and space observatories at the less stable Lagrangian points.

Polygons are fun in Math Many shapes they have They are geometrical.

PART 1 Question (a)

Question (b) A closed shape consisting of line segments that has at least three sides. Triangles, quadrilaterals, rectangles, and squares are all types of polygons. The word "polygon" comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral or quadrangle, and nonagon are exceptions. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula. Some special polygons also have their own names; for example the regular star pentagon is also known as the pentagram. Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a nonconvex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C. Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine. In 1952, Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.

Question (c) 1. 2. 3.

√ (

4.

|

)(

)( |

)

(

)

PART 2 Question (a) Cost = RM 20 × 300 = RM 6000

Question (b) p(m) 60 70 80 90 100 110 120 130 140

q(m) 140 130 120 110 100 90 80 70 60

θ(degree) 38.21 49.58 55.77 58.99 60.00 58.99 55.77 49.58 38.21

Area(m²) 2597.89 3463.97 3968.57 4242.53 4330.13 4242.53 3968.57 3463.97 2597.89

Question (c) The herb garden is an equilateral triangle of sides 100m with a maximum area of 4330.13m².

CONJECTURE Regular polygon will give maximum area. Eg: Square

Rectangle

Question (d) i. ii. iii.

60 < p < 140, 60 < q < 140 When p increases, q decreases, When q increases, p decreases The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Triangle Inequality Theorem.

PART 3 Question (a) i.

Quadrilateral 2 m

+2

= 300 m²

+

= 150 m²

Area = m

10 20 30 40 50 60 70 75

Area = 1400 2600 3600 4400 5000 5400 5600 5625

140 130 120 110 100 90 80 75

The maximum area is 5625 m².

ii.

Regular Pentagon

tan (

)

iii.

Regular Hexagon

tan (

iv.

)

Regular Octagon

tan 67.5° tan (

v.

)

Regular Decagon

tan tan (

)

Question (b)(i) LOCAL HERBS 1. Misai Kucing (Orthosiphon stamineus) Orthosiphon is a medicinal plant native to South East Asia and some parts of tropical Australia. It is an herbaceous shrub which grows to a height of 1.5 meters. Orthosiphon is a popular garden plant because of its unique flower, which is white and bluish with filaments resembling a cat's whiskers. In the wild, the plant can be seen growing in the forests and along roadsides. Common names in Southeast Asia are Misai Kucing (Malaysia), Kumis Kucing and Remujung (Indonesia), and Yaa Nuat Maeo (Thailand). The scientific names are Orthosiphon stamineus Benth, Ocimum aristatum BI and Orthosiphon aristatus (Blume).

Medicinal Uses Orthosiphon is used for treating the ailments of the kidney, since it has a mild diuretic effect. It is also claimed to have anti-allergenic, anti-hypertensive and anti-inflammatory properties, and is commonly used for kidney stones and nephritis. Orthosiphon is sometimes used to treat gout, diabetes, hypertension and rheumatism. It is reportedly effective for anti-fungal and anti-bacterial purposes.

How To Cure In Malaysia, people eat the leaves raw. They take a few leaves, heat them with water to make the water bitter, and then mix it with tea bags.

Research Orthosiphon began to interest researchers early in the 20th century, when it was introduced to Europe as a popular herbal health tea.

Commercial products Orthosiphon is available in many products treating for detoxification, water retention, hypertension, obesity or kidney stones. It comes in tablets, capsules, tea sachets, bottled drinks, raw herbs, dried leaves or extracts.

2. Kacip Fatimah (Labisia Pumila) Kacip fatimah (Labisa pumila) is the female version of Tongkat Ali. Kacip Fatimah is a small woody and leafy plant that grows and can be found widely in the shade of forest floors. The leaves are about 20 centimetres long, and they are traditionally used as a kind of tea by women who experience a loss of libido. Despite its long history of traditional use, the active components and mode of action have not been well studied, though some preliminary research has been published. Uses Extract from these herbs is usually ground into powder substances and are made into capsules and pills. A concoction made from boiling the plant in water is given to women in labour to hasten delivery of their babies. After childbirth, it may still be consumed by mothers to regain their strength. In other medicinal preparations, it can treat gonorrhoea, dysentery and eliminate excessive gas in the body. Traditionally, it is used in Borneo for enhancing vitality, overcome tiredness and help to tone vaginal muscles for women. The claimed uses of Kacip Fatimah include:           

Helps establish a regular menstrual cycle when periods fail to appear for reasons like stress, illness or when the pill is discontinued Prevents cramping, water retention and irritability for those with painful periods. Balances, builds and harmonizes the female reproductive system to encourage healthy conception Supports healthy vaginal flora to prevent irritation and infections. Alleviates fatigue, smooths menopausal symptoms and promote emotional well being. Prolong energy during Playtime. Helps to solve the problems related to constipation Tightens vaginal skin and walls. Anti-dysmenorrhoea; cleansing and avoiding painful or difficult menstruation Anti-flatulence, drive away and prevent the formation of gas. Firming and toning of abdominal muscles.

As the plant contains phytoestrogen, it is not to be taken by pregnant women and periods of menstruation.

3. Tongkat Ali (Eurycoma Longifolia) Eurycoma longifolia (commonly called tongkat ali or pasak bumi) is a flowering plant in the family Simaroubaceae, native to Indonesia, Malaysia, and, to a lesser extent, Thailand, Vietnam, and Laos. It is also known under the names penawar pahit, penawar bias, bedara merah, bedara putih, lempedu pahit, payong ali, tongkat baginda, muntah bumi, petala bumi (all the above Malay); bidara laut (Indonesian); babi kurus (Javanese); cay ba binh (Vietnamese) and tho nan (Laotian). Many of the common names refer to the plant's medicinal use and extreme bitterness. "Penawar pahit" translates simply as "bitter charm" or "bitter medicine". Older literature, such as a 1953 article in the Journal of Ecology, may cite only "penawar pahit" as the plant's common Malay name.

Growth Eurycoma longifolia is a small, evergreen tree growing to 15 m (49 ft) tall with spirally arranged, pinnate leaves 20–40 cm (8–16 inches) long with 13–41 leaflets. The flowers are dioecious, with male and female flowers on different trees; they are produced in large panicles, each flower with 5–6 very small petals. The fruit is green ripening dark red, 1– 2 cm long and 0.5–1 cm broad.

Products Fake Eurycoma longifolia products have been pulled off the shelves in several countries but are still sold over the Internet, mostly shipped from the UK. In a medical journal article, published March 2010, it was noted that "estimates place the proportion of counterfeit medications sold over the Internet from 44% to 90%" with remedies for sexual dysfunction accounting for the greatest share. It is therefore recommended that buyers of Eurycoma longifolia request from Internet vendors conclusive information, and proof, on the facilities where a product has been manufactured. In Malaysia, the common use of Eurycoma longifolia as a food and drink additive, coupled with a wide distribution of products using cheaper synthetic drugs in lieu of Eurycoma longifolia quassinoids, has led to the invention of an electronic tongue to determine the presence and concentration of genuine Eurycoma longifolia in products claiming to contain it. On the other hand, consumers who lack the sophisticated electronic tongue equipment invented in Malaysia for testing the presence of Eurycoma longifolia, but want more clarity on whether the product they obtained is indeed Eurycoma longifolia or a fake, can use their own tongue to taste the content of capsules for the bitterness of the material. Quassinoids, the biologically active components of Eurycoma longifolia root, are extremely bitter. They are named after quassin, the long-isolated bitter principle of the quassia tree. Quassin is regarded the bitterest substance in nature, 50 times more bitter

than quinine. Anything that isn't bitter, and strongly so, cannot contain quassinoids from Eurycoma longifolia . In the US, the FDA has banned numerous products such as Libidus, claiming to use Eurycoma longifolia as principal ingredient, but which instead are concoctions designed around illegal prescription drugs, or even worse, analogues of prescription drugs that have not even been tested for safety in humans, such as acetildenafil. In February 2009, the FDA warned against almost 30 illegal sexual enhancement supplements, but the names of these products change quicker than the FDA can investigate them. Libidus, for example, is now sold as Maxidus, still claiming Eurycoma longifolia (tongkat ali) as principal ingredient. The government of Malaysia has banned numerous fake products which use drugs like sildenafil citrate instead of tongkat ali in their capsules. To avoid being hurt by bad publicity on one product name, those who sell fake tongkat ali from Malaysia have resorted to using many different names for their wares. Products claiming various Eurycoma longifolia extract ratios of 1:20, 1:50, 1:100, and 1:200 are sold. Traditionally Eurycoma longifolia is extracted with water and not ethanol. However, the use of selling Eurycoma longifolia extract based on extraction ratio may be confusing and is not easily verifiable. In expectation of a competitive edge, some manufacturers are claiming standardization of their extract based on specific ingredients. Alleged standards / markers are the glycosaponin content (35–45%) and eurycomanone (>2%). While eurycomanone is one of many quassinoids in Eurycoma longifolia, saponins, known in ethnobotany primarily as fish poison played no role in the academic research on the plant. A large number of Malaysian Eurycoma longifolia products (36 out of 100) have been shown to be contaminated with mercury beyond legally permitted limits.

Herbal Nutrition   

Increase strength of the body. Help to stimulate the production of testosterone that is one of the important hormones for male. Help to increase the blood rate and body metabolism.

Question (b)(ii)

FURTHER EXPLORATION POLYGONAL BUILDINGS University of Economics and Business, Vienna

The new Library & Learning Center rises as a polygonal block from the centre of the new University campus. The LLC‟s design takes the form of a cube with both inclined and straight edges. The straight lines of the building‟s exterior separate as they move inward, becoming curvilinear and fluid to generate a free-formed interior canyon that serves as the central public plaza of the centre. All the other facilities of the LLC are housed within a single volume that also divides, becoming two separate ribbons that wind around each other to enclose this glazed gathering space. The Center comprises a „Learning Center‟ with workplaces, lounges and cloakrooms, library, a language laboratory, training classrooms, administration offices, study services and central supporting services, copy shop, book shop, data center, cafeteria, event area, clubroom and auditorium.        

Vienna, Austria 2008 – 2012 28000 m2 Gross Area: 42000 m² Length: 136m Width: 76m Height: 30m (5 Floors) Zaha Hadid with Patrik Schumacher

Pittman Dowell Residence, Los Angeles

The project is a residence for two artists. Located 15 miles north of Los Angeles at the edge of Angeles forest, the site encompasses 6 acres of land originally planned as a hillside subdivision of houses designed by Richard Neutra. Three level pads were created but only one house was built, the 1952 Serulnic Residence. The current owners have over the years developed an extensive desert garden and outdoor pavilion on one of the unbuilt pads. The new residence, to be constructed on the last level area, is circumscribed by the sole winding road which ends at the Serulnic house. Five decades after the original house was constructed in this remote area, the city has grown around it and with it the visual and physical context has changed. In a similar way, the evolving contemporary needs of the artists required a new relationship between building and landscape that is more urban and contained. Inspired by geometric arrangements of interlocking polygons, the new residence takes the form of a heptagonal figure whose purity is confounded by a series of intersecting diagonal slices though space. Bounded by an introverted exterior, living spaces unfold in a moiré of shifting perspectival frames from within and throughout the house. An irregularly shaped void caught within these intersections creates an outdoor room at the center whose edges blur into the adjoining living spaces. In this way, movement and visual relationships expand and contract to respond to centrifugal nature of the site and context.    

Los Angeles, USA Residential - Houses Micheal Maltzan Architecture 3200 SF

Casa da Musica, Porto

The Casa da Musica is situated on a travertine plaza, between the city's historic quarter and a working-class neighborhood, adjacent to the Rotunda da Boavista. The square is no longer a mere hinge between the old and the new Porto, but becomes a positive encounter of two different models of the city. The chiseled sculptural form of the white concrete shell houses the main 1,300 seat concert hall, a small 350 seat hall, rehearsal rooms, and recording studios for the Oporto National Orchestra. A terrace carved out of the sloping roofline and huge cut-out in the concrete skin connects the building to city. Stairs lead from the ground level plaza to the foyer where a second staircase continues to the Main Hall and the different levels above. Heavy concrete beams criss-cross the huge light well above. The main auditorium, shaped like a simple shoebox, is enclosed at both ends by two layers of “corrugated” glass walls. The glass, corrugated for optimal acoustics and sheer beauty, brings diffused daylight into the auditorium. The structural heart of the building is formed by four massive walls that extend from the base to the roof and connect the tilted external walls with the core of the structure. The two one meter thick walls of the main auditorium act as internal diaphragms tying the shell together in the longitudinal direction. The principal materials are white concrete, corrugated glass, travertine, plywood, and aluminum.     

Porto, Portugal 22.000 square meters 2005 Concert Hall Rem Koolhaas and Ellen van Loon

CONCLUSION Polygons tend to be studied in the first few years of school, as an introduction to basic geometry and mathematics. But after doing research, answering questions, completing table and some problem solving, I found that the usage of polygons is important in our daily life. It is not just widely used in fashion design but also in other fields especially in building of modern structures. The triangle, for instance, is often used in construction because its shape makes it comparatively strong. The use of the polygon shape reduces the quantity of materials needed to make a structure, so essentially reduces costs and maximizes profits in a business environment. Another polygon is the rectangle. The rectangle is used in a number of applications, due to the fact our field of vision broadly consists of a rectangle shape. For instance, most televisions are rectangles to allow for easy and comfortable viewing. The same can be said for photo frames and mobile phones screens. In conclusion, polygon is a daily life necessity. Without it, man‟s creativity will be limited. Therefore, we should be thankful of the people who contribute in the idea of polygon.

REFLECTION While I conducting this project, a lot of information that I found. I have learnt how polygons appear in our daily life. Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication. Not only that, I had learned some moral values that I practice. This project had taught me to responsible on the works that are given to me to be completed. This project also had made me felt more confidence to do works and not to give easily when we could not find the solution for the question. I also learned to be more discipline on time, which I was given about a month to complete this project and pass up to my teacher just in time. I also enjoy doing this project I spend my time with friends to complete this project and it had tighten our friendship. Last but not least, I proposed this project should be continue because it brings a lot of moral value to the student and also test the students understanding in Additional Mathematics.