additional mathematics project work Kelantan 2/2012

PROJECT WORK FOR ADDITIONAL MATHEMATICS 2012 NAME : AFIF MURSYIDI BIN MOHD HURI CLASS : 5 BUHTURI SCHOOL : SMK(A) WATANIAH TEACHER : Pn. ZURAIFAH BT NO. I/C : 951104-29-5055

Prefaces………………………3 Introduction………………….5 Objective…………………….11 Method Investigation..……….13 Task 1……………………….15 Task 2……………………….19 Task 3……………………….21 Task 4………………………..24 Conclusion…………………..26 Reflection……………………28

2

3

In the Name Of Allah , the Most Gracious and the Most Merciful … 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 mean encouraged me to complete this task so that I will not procrastinate in doing it. Then I would like to thank my teacher, Pn.Zuraifah 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.

...

4

5

INTRODUCTION The purpose of add math project work is to provide an opportunity for students to apply mathematical concepts and skills in problem-solving that they have learnt in classroom. This project work can help students to understand add math more easily and aid students in visualising certain mathematical concepts which are difficult to show clearly through pen and paper In addition, this project work is essential to help students to learn how to cope with future challenges using their mathematical abilities. It also aims to foster moral values in line with a student's academic development. besides this, the students are encouraged to do their own research. As a result students become more independent. It also makes the learning process more fun and effective. This project work also makes add math an enjoyable and exciting subject encouraging the students to learn math more skills in more heuristic manner. Therefore it is beneficial to all students who take add math

6

History of differentiation The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid (c. 300 BC), Archimedes (c. 287–212 BC) and Apollonius of Perga (c. 262–190 BC).[1] Archimedes also introduced the use of infinitesimals, although these were primarily used to study areas and volumes rather than derivatives and tangents; see Archimedes' use of infinitesimals. The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as early as 500 AD, when the astronomer and mathematician Aryabhata (476–550) used infinitesimals to study the motion of the moon.[2] The use of infinitesimals to compute rates of change was developed significantly by Bhāskara II (1114–1185); indeed, it has been argued[3] that many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem".[4] The Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), was the first to discover the derivative of cubic polynomials, an important result in differential calculus;[5] his Treatise on Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions.[6] The modern development of calculus is usually credited to Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716), who provided independent[7] and unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes,[8] which had not been significantly extended since the time of Ibn al-Haytham (Alhazen).[9] For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such as Isaac Barrow (1630–1677), René Descartes (1596–1650), Christiaan Huygens (1629–1695), Blaise Pascal (1623–1662) and John Wallis (1616–1703). Isaac Barrow is generally given credit for the early development of the derivative.[10] Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation to theoretical physics, while Leibniz systematically developed much of the notation still used today. Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789–1857), Bernhard Riemann (1826–1866), and Karl Weierstrass (1815–1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.

7

History of arithmetic progression Euler stated that every arithmetic progression beginning with 1 contains an infinite number of primes. The theorem in the above form was first conjectured by Legendre in his attempted unsuccessful proofs of quadratic reciprocity and proved by Dirichlet in (Dirichlet 1837) with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution of primes. The theorem represents the beginning of rigorous analytic number theory.

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference of 2. If the initial term of an arithmetic progression is and the common difference of successive members is d, then the nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series. The behavior of the arithmetic progression depends on the common difference d. If the common difference is:  

Positive, the members (terms) will grow towards positive infinity. Negative, the members (terms) will grow towards negative infinity.

8

Sum This section is about Finite arithmetic series. For Infinite arithmetic series, see Infinite arithmetic series.

The sum of the members of a finite arithmetic progression is called an arithmetic series. Expressing the arithmetic series in two different ways:

Adding both sides of the two equations, all terms involving d cancel:

Dividing both sides by 2 produces a common form of the equation:

An alternate form results from re-inserting the substitution:

:

In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya (section 2.18) .[1] So, for example, the sum of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up to the 50th term is

9

Product The product of the members of a finite arithmetic progression with an initial element a1, common differences d, and n elements in total is determined in a closed expression

where

denotes the rising factorial and

formula is not valid when

denotes the Gamma function. (Note however that the

is a negative integer or zero.)

This is a generalization from the fact that the product of the progression given by the factorial and that the product

for positive integers

and

is

is given by

Taking the example from above, the product of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up to the 50th term is

10

11

OBJECTIVES We students taking Additional Mathematics are required to carry out a project work while we are in Form 5. This year the Curriculum Development Division Ministry of Education has prepared two tasks for us. We need to choose and complete only ONE task based on our area of interest. This project can be done in groups or individually, but each of us are expected to submit an individually written report. Upon completion of the Additional Mathematics Project Work, we are to gain valuable experiences and able to:  Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems;  Experience classroom environments which are challenging, interesting and meaningful and hence improve their thinking skills.  Experience classroom environments where knowledge and skills are applied in meaningful waysin solving real-life problems  Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected  Experience classroom environments that stimulates and enhances effective learning.  Acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely  Enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increase interest and confidence  Prepare ourselves for the demand of our future undertakings and in workplace  Realise that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics.  Train ourselves not only to be independent learners but also to collaborate, to cooperate, and to share knowledge in an engaging and healthy environment  Use technology especially the ICT appropriately and effectively  Train ourselves to appreciate the intrinsic values of mathematics and to become more creative and innovative  Realize the importance and the beauty of mathematics

12

13

Method Investigation In solving and finishing this project work done,some method is used :1. Communication 

Discussion with teacher and friend help in solving problem.The information from this discussion used as a reference materials to success this project.

2. Reference 

Additional of information from various of reference material help me to find the method to solve the problem.For this Additional Mathematics project,I can get the reference from library,internet,my friends,my teacher and many more.

3. Lesson session 

The lesson session in the class help me in solve problem by using heuristics what I learn in the class.

14

15

TASK 1 The problem of sufficient water supply has become a main issue among the countries around the world. Proper planning is very important to make sure the water supply is sufficient. Carry out a simple study to solve the problem. You are suggested to use various resources such as internet, printed materials, ground study and others. Discuss on how additional mathematics can be used to solve the problem.

For this coursework, I have carried out a study on water supply problem in the district inconclusive. Total of 1000 respondents villagers set. The following are water supply problems that occurred in the district Machang

16

I've been using the knowledge of Additional Mathematics to solve the problems on the water needs. Problem: The size of a small tank Methods: Differentiation If the volume of the tank to be constructed is 50 m3.

Minimum surface area required to build the tank was calculated using the 'differentiation'

17

High tank can be calculated using the volume formula

18

19

Task 2 You are assigned as an engineer of a construction firm. You are responsible to buld an enclosed water tank for a housing estate which consist of 60 houses. The rate of water entering every house is different and the rates are 1000cm3 min-1 for the first house , 990 cm3 min-1 for the third house and so on. By using at least 3 different methods, calculate the total volume of water used by all the houses in 1 minute, if all the housing estate are using the water at the same time. Method 1 : Arithmatic Progression

20

21

Task 3

Assuming that the shape of the tank to be built is a cuboid with a base dimension of site (p x p) m2 and a height h m as as shown in figure 1 and the volume of the tank is 50 m3, find the minimum surface area needed to built the given tank.

Method : Differentiation

22

23

24

Task 4 If the number of houses in the housing estate increases in the future, what is the maximum number of houses that can be built so that the supply of water from the tank can fulfil the demand of the residents.

25

26

Conclusion After doing research, answering questions, drawing graph, some problem solving, I saw that the usage of differentiation and arithmatic progression is important in daily life. It is not just widely used in markets but also in interpreting the condition of the surrounding like the air or the water. Especially in conducting an air-pollution survey. In conclusion, statistics is a daily life essecities. Without it, surveys can’t be conducted, the stock market can’t be interpret and many more. So, we should be thankful of the people who contribute in the idea of statistics.

27

REFLECTION 28

While I conducting this project, a lot of information that I found. I have learnt how tank is made 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.T his project had taught me to responsible on the works that aregiven 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 completemthis 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

29