Mathematics Educational Theories Essay

My investigation has led me to think about lots of ideas that can be used with children in Early Years classroom as well as with KS1. Numbers displayed on telephone sets and remote controls are probably the first form of numbers in an order that young children come across with. Primarily, number lines 0 – 10 & 0 – 20, as a part of the 100 square can be used where children familiarise themselves with numbers and recognise the correct order of numbers. Number lines are very versatile. They can be of any size, for individual or whole- class use. They can start on any number. Blank number lines are infinitely adaptable; they can be used for counting calculations using all four number operations. They are a good way to practice and overlearn the number bonds to 20 that children need to be able to remember fluently. Visual counting pattern on number lines can help children to understand relative numbers and number sequences. Moving a step forward from the correct order of numbers, the children can identify and colour all the odd and even numbers and establish rules for recognition. When children are confident with bigger numbers, a large 100 square is ideal to work with the whole class to learn timetables, addition & subtraction of larger numbers by counting numbers above it or below it. Children can also identify multiples of 2,3, 5, 10 and others by highlighting numbers in different colours and demonstrate sequential patterns. They can reverse the two digit numbers, read them and make new numbers. Problem solving activities such as pick a number between 0-10 or 0-20, double it and add 1, is very exciting as children manipulate with numbers at their own pace. Lots of different games can be introduced. Snakes & Ladders, number dominions, dice games, dot to dot, finding the difference between two dices; place value cards are few examples. By playing and replaying a selection of games, children can practice to grapple ideas, number facts and concepts in a way that they can enjoy and strengthen skills at the same time. Active involvement aids their learning and enhances their attitude towards the subject. But it is important to focus on the particular learning target that the game is reinforcing. Hundred squares can be cut into several parts to form a jigsaw and children can be challenged to put the pieces together again. (See some activity ideas in the appendix)

Is our Generation More Stressed than the Previous Ones?

Nowadays, stress is becoming a common problem. Our younger generation thinks that modern life is generally much more stressful than in the past. However, others say that the amount of stress people have today is overstated as older generations coped with more pressure, but they never thought about such a phenomenon as â€Å"stress†. This contradiction makes many people compare the negative aspects of stress to society today and in the past.On the one hand, the previous generations experienced great stress because of poor living conditions, food shortage and numerous wars, which were taking away millions of people’s lives. Moreover, they did not have such helpful technologies, which make the life of a modern person easier. Besides, without an advanced medical system many people died of simple illnesses and diseases because there were no necessary medicines and vaccines. That is why life expectancy in the past was much lower than it is now.On the other hand, because of th e increasing pace of life nowadays people are always in a chronic rush, which causes various mental and physical health problems, such as backache, depression, fatigue, headaches, high blood pressure, and insomnia. What is more, almost everybody in our modern society is trying to achieve high standard of living, therefore, people have to spend long hour at work. As a result, they feel overworked and burn out, and hardly have time to relax and escape from everyday routine, which leads them to losing their balance between work and personal life.Furthermore, there is no doubt that several decades ago, the environment was much better than today with fresh water, air and food, whereas today we face air, water and noise pollution, which of course contribute to the symptoms of stress. In addition, nowadays, the majority of people lives in big and overcrowded cities and suffer from such common city problems as standstill traffic congestions, unemployment, high crime rate and high cost of li ving.The unemployment rate is constantly increasing; hence, the number of job-seeker is growing, which leads to the increasing competitiveness in job opportunities. In my opinion, each generation suffers from stress; the difference is in the attitude to it. To my mind, today time is more compressed than ever and because of the speedy pace of life and everyday overload, people feel exhausted as human beings are not designed for a prolonged, high-speed activity.However, we have to admit that in a certain sense our ancestors had their own reasons of stress, which should not be underestimated. To sum up, stress has been felt by all generations in their lives. It is really hard to compare either modern generation, or the previous one is under more stress that the other one, because each generation has its own problems. People would better concentrate on reducing the number of stress and try to make future living conditions more favorable.

Religious Ethic Essay

Discuss the view that only a religious ethic can provide an acceptable basis for medical ethics. Medical ethics concerns many areas of ethical debate. Including such controversial issues such as euthanasia, abortion and human cloning, medical ethics sparks lively debates. The issue of abortion is a very relevant and controversial issue. There are opposition and supporters from both a religious ethical background and a non-religious ethical background. Those who come from a Christian ethical background tend to have a similar argument, that of the sanctity of life. Roman Catholics oppose abortion using the Christian ethical theory of Natural Law. Abortion would be going against natural law as it interferes with God’s will. Abortion is right in no circumstances, in other words it is intrinsically evil, as it involves the murder of an innocent life. Protestants do, in principle, oppose abortion on the ground that murder is wrong, as stated in the bible; â€Å"Thou shalt not kill† (Exodus) and as abortion is in effect murder, abortion must therefore also be wrong. However, some more liberal protestants allow abortion in certain circumstances, such as before the nervous system has developed (before the foetus can â€Å"feel†) if the mother’s life or health is threatened or if the pregnancy is from an exceptional situation, such as the result of rape. There seems to be no reason however, as to why these two standpoints could not be from a non-religious ethic. The appreciation of life is not something that is exclusive to religion, and therefore the view that abortion is murder could stand with anyone who believes that the foetus is a person, and this need not be a religious standpoint. Equally the Liberal protestant view could again apply to anyone who believes that murder is wrong, yet as circumstances differ, the things to be considered also differ. This is a relativist situationist standpoint but again does not have to be religious. In fact many non-religious people do tend to be situationists as they judge decisions on their consequences and not on the action itself, as no action is considered intrinsically wrong, as it is not deemed by any higher being or book, as it is in religion. These arguments also apply in relation to euthanasia. Roman Catholics view murder as wrong and as euthanasia is essentially murder, it must also be wrong. It would also be considered as going against Natural Law as it interferes with God’s will, just as abortion does. Protestants would claim that the bible condemns euthanasia as it is murder, yet come more liberal Protestants again claim that all factors should be considered in order to make a decision relative to that person’s individual situation. Many people would argue however that without a religious ethic there are no clear rules and boundaries. Deontological ethical theories, that are absolutist theories that apply in all situations, provide these boundaries as the answer will always be the same. Yet not all religious ethical theories are deontological, such as Fletcher’s Situation ethics, which is teleological. This means that religious ethics do not always provide a clear cut answer, just as non-religious ethics may not. Many religious fundamentalists would argue that atheists, for example, cannot have a similar appreciation of life as they do. Atheists do not believe in God, and some would argue that as they do not believe in anything, they cannot have the same appreciation of life. Yet it appears that without a belief in anything religious, life becomes even more precious. Without the promise of an afterlife, which is provided by Christianity, this current life becomes even more precious and amazing. As you only get one shot at life from an atheist perspective, the feeling to preserve it as much as possible, along with do the same for others would seem to be even stronger. Without the excuse that the person/foetus will go to Heaven, murder seems to be even more wrong, as it would take away the one life that that person has been offered. In conclusion I think that a religious ethic is by no means the only acceptable basis for medical ethics. Although some Christian ethics will provide a stable answer for every situation, an answer that will never change and therefore will have clear-cut boundaries, not every person in the world will ever be of the same religion, and therefore it cannot be universal. Therefore a non-religious ethic which everyone could agree on seems more acceptable, such as one that allows situations to be considered, because therefore a religious ethic could be used in certain circumstances if the people involved would like to do so, as that happens to be their own â€Å"situation†; similarly if someone does not want to apply a religious ethical theory then they are not obliged to do so, because again this option would apply to their situation. Therefore situationist ethics that are not based on religion can be made universal, allowing religious ethics to be applied or not according to the wishes of the people involved and this seems to me to be the only acceptable basis for medical ethics, an ethic that will allow for everyone’s personal beliefs.

Do cuts in UK government spending affect UK secondary education exam Essay

Do cuts in UK government spending affect UK secondary education exam results - Essay Example European models were based on the assumption that only a certain portion of the population needed and could profit from a college education. U.S. goals ambitiously sought the higher education of 25-50% of the population. The community college served all that desired an education, mostly the unserved populations of adult, (and) full-time workers with families† (Newman, Renee M, 1998). Primary education was the first to receive early government recognition. However, funds management was not encouraging. However, efforts to effectively fund primary school activities continued. Secondary education came in for recognition for support later on depending on the government of the land (Evaluation of World Bank’s Support for Primary Education, 2007). Statistically, secondary education in the United Kingdom is already facing flak due to the below average treatment secondary education is getting. On the one hand, there is below average salaries for secondary education teachers and on the other, there is drop out rate of 30% in secondary education in the UK (Curtis, Polly, 2009). Fundamentally and technically, the cut in spending does not amount to over  £100 million which is just a fraction of the  £85 billion the UK spends annually on education for the year 2010-11. However, the figure could trigger cascading effect in spending that attracts experience and talents. The UK is still counted as the Mecca for education and it holds the distinction as the best education provider in terms of facilities (UK providing assistance for tertiary education facilities, 2009). Better put, the question should be: is government spending the determinant to exam results for the UK secondary education? The motivation to study comes from students determined to do well academically for future prospects and better personality. Secondary school students are normally well versed in the need for higher studies. Educational costs cannot be the determining factor for

Principles of Aerodynamics Coursework Example | Topics and Well Written Essays - 2000 words

Principles of Aerodynamics - Coursework Example 2: Coefficient of Drag = CD = 0.054 Area = S = 15 m2 Thrust = T= 1500 N Density= A = 0.5 kg/m3 For a steady and level flight, drag force is equal to the thrust produced by engines, D = T = 1500 N D = (? A V?2) S CD = (? (0.5) (V?2)) (15) (0.054) = 1500 => V? = 86.06 m/s = 8.6 E +1 m/s Question No. 3: Question No. 4: The sketches shown below illustrate the trend of variation in CL, CD, and L/D ratios with increasing angle of attack. Question No. 5: Critical Mach number corresponds to that value of Mach number for free stream flow for which a localized mach number of ‘1’ is obtained at any point around the airfoil. When this condition arises, a shock wave is created at the point where the flow reaches the sonic speed. As the speed increases, regions of very low pressure are created. This causes the flow to separate from the airfoil thereby substantially increasing the drag forces on it. The figure illustrates this phenomenon. Some of the important design features incor porated in the aircrafts in order to contain the effects of this situation are using thin airfoil and / or super critical airfoil (Anderson Introduction 763). Making an airfoil thinner increases the value of Critical Mach Number and hence the airplane can fly at very high speeds without a significant increase in drag forces on it. ... Their unique design limits the rise of drag forces even after the critical mach number is reached. Such airfoils have successfully been utilized in TACT aircraft program run by NASA Dryden Flight Research Center (Cury). Question No. 6: In the above illustration, the triangle represents the fuselage of an aircraft as seen from front. ‘’ is the angle of bank for the turn. LV = L cos  LH = L sin  The centripetal force required by the aircraft to take the turn is provided by horizontal component of lift force given by LH, equating the two; L sin  =  †¦ (a) the component LV balances the weight of the aircraft, hence = L cos  = mg From the above equation, L = mg / cos  Putting values in (a) and simplifying; g tan  =  putting values for  = 15o and r = 1500 m gives v = 62.79 m/s Load Factor = L/W = L/L cos  = 1/ cos  = 1/cos 15o = 1.035 Question No. 7: Following are the control surfaces used to control the motion of an aircraft along different axes: (1) The longitudinal axis: Ailerons (2) The Vertical or Normal Axis: Rudder (3) The lateral Axis: Elevator The figure shows the above mentioned control surfaces and the functions they perform. All the control surfaces work on the principle of creating drag for the incoming wind thereby changing the direction of the wind. Due to this change in velocity, a momentum change occurs which causes a force to act on the control surface and the desired movement of the aircraft is achieved this way. The ailerons tilt the aircraft around the longitudinal axis. They are always installed in pairs. The opposite motion of the two ailerons creates a couple which acts about the longitudinal axis to cause the desired motion. Rudder rotates the

What disparity is there between defendants outcomes when you retain a Essay

What disparity is there between defendants outcomes when you retain a private attorney versus a public defender in a criminal trial - Essay Example While the presiding judge’s job is to ensure that the prosecution presents the case in a manner which is consistent with the law he is under no duty to ensure that council for either side is competent. As a result representation of a defendant at trial is very important if the defendant is going to benefit from a trial of the issues. It is important to remember that not all criminal trials involve jurors and not all parts of a jury trial require the jurors to remain a part of the proceedings. In Powell v Alabama Justice Sutherland explained the absolute requirement for legal representation at a criminal trial. He said that ‘the right to be heard would be, in many cases, of little avail if it did not comprehend the right to be heard by counsel. Even the intelligent and educated layman has small and sometimes no skill in the science of law. If charged with crimes, he is incapable, generally, of determining for himself whether the indictment is good or bad. He is unfamiliar with the rules of evidence. Left without the aid of counsel he may be put on trial without a proper charge, and convicted upon incompetent evidence, or evidence irrelevant to the issue or otherwise inadmissible. He lacks both the skill and knowledge adequately to prepare his defense, even though he have a perfect one. He requires the guiding hand of counsel at every step in the proceedings against him. Without it, though he be not guilty, he faces the danger of conviction because he does not know h ow to establish his innocence.’ (Powell v Alabama 287 U.S. 45 (1932)) In delivering the judgment for the court, Justice Sutherland went on to explain that in the event a defendant is charged with a capital offense and could not afford to retain council the court had a residual duty to appoint council for the defendant. Failing to appoint council was tantamount to a denial of the right to due process of law under the fourteenth Amendment to the

Daves Success With the Wendys Restaurant Case Study - 2

Daves Success With the Wendys Restaurant - Case Study Example Dave afterward opened his own restaurant and called it Wendy’s restaurant and started selling the food that he loves-fresh hamburgers, inspite of being advised that the market was saturated. This is a true spirit of an entrepreneur; an individual who sees an opportunity where there is none and creates something completely new and different from what everyone is offering. In addition, Dave Thomas comes up with fresh ideas and starts a strategy that is different from other businesses by providing fresh and quality food. Earlier on, he had worked at KFC selling fried chicken, but then he saw an opportunity in the market for selling fresh hamburgers. This is also something that he loved and entrepreneurs usually should start something that they love. All Dave’s ideas are successful which proves that he has the makings of a great entrepreneur.  Dave’s success with the first Wendy’s restaurant came from the fact that he offered something completely new, appeal ing and different from what others were doing –Dave offered quality and freshness and allowed customers to make orders for what they wanted. Dave was able to differentiate his product offering by selling made-to-order hamburgers, chili, French fries, (real) milkshakes and soft drinks. For an entrepreneur to succeed in any business s/he must cut a market niche for himself/herself and offer customers unique and quality products/services not available anywhere else. This unique competitive advantage became a game changer in the industry and by the sixth week, Dave had broken even. Dave simply implies that for a business to be successful it has to satisfy the needs of its customers by offering what the customer wants; in other words, providing value as well as satisfying the customer. He also implies that the business owner should aim at achieving his key objectives/reasons for starting the business.  

3D Graphing Engine Term Paper Example | Topics and Well Written Essays - 10000 words

3D Graphing Engine - Term Paper Example Nowadays geometry engines are called Vertex Shaders since they are programmable and can run so-called (vertex) shader programs to compute and animate the geometry of a scene. Every vertex that needs to be calculated can contain a lot of information, such as an x, y, z coordinate (a 3-dimensional position), texture coordinates, Normal Information (what direction the vertex faces), an identifier (which triangle it belongs to), Skinning parameters, lighting values or just about anything else. However, vertex processing alone does not result in a visible picture.   So as to see all the triangles made up of all the vertices that the vertex shaders have calculated, they need to be colored. Certainly, the invisible object that is a result of the geometry processing needs to be â€Å"wallpapered† so it becomes visible.   To be able to do this, the polygons need to be converted into pixels; this is done during the triangle setup. The pixels are then dealt with in the pixel shaders and pixel pipelines. The color value of a pixel is looked upon a texture; this texture exists in graphics memory as a bitmap that was designed by the 3D artist.   Textures can be available in different resolutions. Higher resolution textures look better; however, use more memory space and more memory bandwidth than lower resolution textures.   For far away objects, this would not only result in wasted processing cycles, but it could also lead to display anomalies. As a result of this, t extures are usually available in different resolutions.   If textures of different resolutions are combined on one object then this is referred to as mip-mapping. This mip-mapping can produce visible borders between the two textures of different resolutions, called mipmap banding.   Moreover, this mipmap banding can be minimized using different filtering techniques. Filtering means that for every pixel to be colored, more than one texel on the texture is looked up and the average is calculated and applied to the pixel.  

What is the Status of Moral Claims Essay Example | Topics and Well Written Essays - 250 words

What is the Status of Moral Claims - Essay Example f is fundamentally and diametrically opposed to such a practice and does not consider any argument for it as being ethically or morally convincing (Nussbaum 1). As such, Nussbaum presents a very simple yet well reasoned approach that defines the process of FGM as reprehensible regardless of the cultural determinants that claim to the contrary. It is this author’s belief that the strong position that Nussbaum takes is absolutely defensible. Firstly, it should be a commonly held view across humanity that violence is not the solution to any perceived problem and that humans should most certainly not be subjected to a violent form of clitoral mutilation; especially prior to the age where they can make a determination for themselves how they would like their bodies to develop and represent their own unique understanding of culture. Whereas this view may be seen as xenophobic, the truth of the matter is that it should be understood across culture as patently wrong to inflict willful damage on any member of society; especially those that are unaware with regards to the pain and trauma that such a situation will effect later in their

Case Analysis on Human Resources at Hewlett Packard Essay

Case Analysis on Human Resources at Hewlett Packard - Essay Example HP Way in conducting its business is based on a set of ethical principles. Honesty and integrity form the very core of this approach. Its customers must be sure of the deals that they make with the company. For this it’s necessary to inculcate ethical principles in HR personnel. Teamwork at HP is emphasized since stakeholders, particularly the shareholders, depend on company’s policy in generating value for all concerned. Thus a sharing responsibility for organizational outcomes among HR personnel would be desirable. The company encourages flexibility and innovation among its employees. Their primary goal should be the development of innovative and flexible ways to carry out daily tasks so that functional processes become smooth. Teamwork enables horizontal communication function to be carried out with a singular focus on organizational goals. While communication structures have been overhauled to support a more democratic and less centralized vertical functional network within the organization there is also a well planned strategy to achieve results. There are both advantages and disadvantages associated with working for a company like HP. In the first place advantages include such benefits like good future prospects, career development, performance related pay, a positive and independent work environment and opportunities for innovation and meaningful contribution. Future prospects lie with the diversity and expansion of the organization (Becker, Andrew & Dave, 2001). A fast growing organization with ever rising net revenues must be any employee’s dream because the future of the company and its employees is basically determined by the current pace of positive progress that the company is making. Career development at HP is possible depending on the scope of the job. High tech companies like HP with a fair degree of diversification into other fields through merger and acquisition (M&A) are more likely to produce positive synergies that would

Chicago Fire in December,22,2010 Research Paper Example | Topics and Well Written Essays - 750 words

Chicago Fire in December,22,2010 - Research Paper Example The fire is said to have been started by squatters burning debris to keep warm in the cold winter in a building whose utilities had been cut off. The fire is believed to have spread and caught the roof that was weighed down by snow and ice leading to its collapse on the firefighters on duty. The ensuing mishaps by the fire department, according to a federal report, are attributed to the tragic deaths and casualties that occurred during the firefighting exercise. According to the report, the firefighters sent into the vacant building had radios, yet their supervisors outside did not get to know the happenings inside: the building’s wooden pillars were letting the fire crawl up to the ceiling so fast. As a group of the fire fighters wrestled the fire from inside the building, another group was on the roof, cutting it up and spraying water on the fire. The cutting was supposed to allow smoke to escape, but it led to the roof collapsing, killing the two firefighters and injuring n ineteen more.  According to Tim Merinar, the lead federal investigator in the Chicago fire incident, the Fire Department’s supervisors ought to have taken a defensive approach in tackling the fire by ordering more firefighters to get out of the building after it was ascertained no persons were trapped inside, as had been previously feared on arrival at the scene. The report faulted the Fire Department for the deaths and injuries on the firefighters. Apparently, the department overlooked several things that ought to have been part of its strategy.

Root Causes of Terrorism Essay Example for Free

Root Causes of Terrorism Essay Fieldman and Shapir (2004) noted that â€Å"some of the latest figures involving international suicide terrorists, their actions and the outcome of these actions, mentioning that there were eleven suicide attacks in Chechnya since 2003 involving fifteen terrorists, eleven of which are women. Their actions killing 200 and sixty one suicide attacks until June 2004 in Iraq involving seventy nine suicide terrorists that killed 840 people, and that suicide terrorism is significant in Israeli-Palestinian conflict as anti-Israel suicide attacks were numbered to 26 2003 to June 2004 (Fieldman, Shapir, 2004, p. 46). † This is proof of one of the many similar characteristics and nature of these different suicide terrorist groups: the measure and extent of their resolve. â€Å"In other areas around the world, fifty-four terrorists identified with Al Qaeda or its affiliated took part in nineteen suicide attacks in which over 280 people were killed. Overall, more than 170 suicide terrorists carried out 117 suicide attacks, a figure that does not include the numerous frustrated suicide attacks, mostly but not exclusively in Israel (Feldman, Shapir, 2004, p. 46). † Suicide terrorist groups around the world and the shared parallelism in management, nature and characteristics Despite the differences present between different suicide terrorist groups operating in different parts of the world, there is a noticeable set of parallelisms that render each group closely similar with one another, creating in a certain degree a sense of uniformity in the manner by which suicide terrorist groups are being managed, run and operated by its leaders. Suicide terrorist groups often share the same set of motivation that allows for the effective management of the personality and overall state of being of its suicide terrorists; they also share nearly the same M. O. , as well as target preferences and the penchant for an attack the lethality of which is guaranteed to extend to a large number of people victimized by the attack; politically, the goals of suicide terrorist groups (the creation of pressure to remove usurpers and modern day colonial rulers and exercise genuine governance and sovereignty over their own lands) are radically aligned with each other. Suicide terrorists and the personality management skill of its leaders An important aspect of suicide terrorism and of the suicide terrorists groups that needs to be discussed is the ‘management’ of these particular individuals. Because the tasks of suicide terrorists are not similar to any other ordinary tasks since their job has a significant level of certainty of death and the finality of life in it, leaders of either small, independent terrorist cells or national level commanders of an umbrella organization for terrorist group (or any leadership or management-level individual belonging to a terrorist group that has in its method of operation suicidal attacks) should have an intensive knowledge on how suicide terrorists should be handled and how each aspect of the suicide terrorist’s personality – emotional, psychological, spiritual, mental and physical – is ‘managed’. Recruiting suicide terrorist may appear either easy or difficult depending on the person who holds the particular opinion. People who see how emotional, psychological and other types motivation easily convinces a prospective suicide bomber to take on a mission and end his/her life for something which the suicide terrorist believes is worth it will think that all leaders need to do is manipulate people into becoming suicide terrorists by using pressure points to which the prospective recruit will easily yield or bend. Religion and militant fanaticism is one of the popular notions that people used to believe as the main reason for the recruitment of suicide bombers. There are those who believe that patriotism or the love for the country is the key motivation for suicide terrorists, while others, who recruit from areas which are economically and financially depressed, pay money or promise a better life for the family that they will leave behind. There are some reported cases of blackmail or the use of threat, as well as the presence of pressure coming from political groups to which a person belongs to, or the pressure of socially-shared political ideology that compels members to undertake such tasks. â€Å"Men become suicide terrorists in extraordinary cases, and each case requires its own explanation. There are some indirect data to suggest that a psychological trauma is an obligatory component. The trauma should be related not only to the death of relatives and/or personal threat to life but also to humiliation, personal but most particularly to a perception of national dignity (Wessely, Krasnov, 2006, p. 112). † Regardless of the particular motivation at play in a particular organization in a particular situation (since the motivation of suicide terrorists around the world varies in detail from one another in different occasions), the important thing that this illustrates is the role of effective management of the terrorist group leaders of its own suicide terrorist squads. They seldom retreat or abort missions because of last minute realizations, but this does not mean that they are crazy – professional researchers and analysts simply believes that the motivation and pressure existing is too much that it is impossible to dissuade suicide terrorists from undertaking their tasks of taking not just their lives, but that of other people as well. â€Å"Although research indicates that individual suicide attackers make choices and are not technically crazy, according to experts they are often manipulated by the pressures and belief structures of the group. Because of this, it is important to study the role of the organization in the phenomenon (Linden, 2002, p. 82). † The political significance, role and impact of suicide terrorist groups – The nature of the different suicide terrorist groups around the world is reflected in how they assert and create political significance for their group. Through their efforts towards political significance, observers can identify the essence of these groups, enumerate them and in the process allow the creation of paradigms from which terrorist groups are the nature of future terrorists groups are characterized. This aspect shows the core nature of suicide terrorist groups – a group of individuals that resorted to the use of lethal force and deadly weapon in the effort to effect change in the political landscape, a band of radical fighters who use the lives of innocent individuals as leverage to gain such power and influence, people who are motivated by different things (but nonetheless equally highly motivated, enough to die and to kill), these are the people who rely on scare and fear tactics to create pressure. Their actions create pressure – towards their enemies and sometimes towards themselves when the end product of an attack backfires and becomes a cause of problem for terrorist groups themselves – while at the same time allowing for the style and personality of this particular group to change over time, altering as well the manner by which the world responds and reacts to terrorism, particularly to suicide terrorists and their actions. â€Å"Al Qaeda’s coordinated multiple suicide airborne attacks aimed at destroying three outstanding US landmark targets the World Trade Center in New York City, the Pentagon, and Congress on September 11, 2001, demonstrated the escalating terrorist threat posed to civilian and military infrastructure With 9/11, the worst terrorist attack ever, the threshold for terrorism world-wide increased. Terrorism has moved from tactical to a strategic threat, prompting governments to invest unprecedented resources both in pre-empting terrorist attacks and in managing mass casualty attacks (Steven, Gunaratna, 2004, p. 2). † Because of the group’s political significance, analysts now grow more observant towards the efficiency by which suicide terrorist group leaders handle the organizations to make protracted and long battles like the ones being fought by terrorist group possible and sustainable over the years and over the number of suicide bombers lost and needs to be replaced time and again. Nature of suicide terrorism: political strategy versus religious undertaking – One of the most common and distorted ideas about suicide terrorism is that it is an act resulting from radical Islamist beliefs or the manifestation of the will of certain religious beliefs that glorify and moralize the use of lethal force and the taking of innocent lives. Historians clear out the confusion in this particular aspect of the nature of suicide terrorist groups around the world: historians and analysts believe that religious will is not the motivation of suicide terrorist groups; rather, it is the means by which strategic political actions are justified to the people and most especially to the suicide terrorists. For actions that carry with it severe moral stigma, there is a need for the presence of moral righteousness for the undertaking to be enacted without doubt, and here, religious and other non-political aspect of the set of beliefs comes into play; sometimes, suicide bombers are not even fully aware of the political repercussions of his/her actions nor the real political reasons why such act is necessary; for the suicide terrorist, it is enough that there is a sufficient reason wherein he can directly relate to and understand. â€Å"Suicide terrorist attacks are not so much driven by religion as by a clear strategic objective: to compel modern democracies to withdraw military forces from the territory that the terrorists view as their homeland. From Lebanon to Sri Lanka to Chechnya to Kashmir to the West Bank, every major suicide terrorist campaign over 95 percent of all incidents has had as its central objective to compel a democratic state to withdraw (LeMay, 2006, p. 252). † Religion is the balm that soothes the wounds and calms the fears of suicide bombers. At this level, the understanding and appreciation of goals is still shared by most since everyone understands and experiences the state of spirituality and religious belief necessary to make religion a powerful influence and motivation for the suicide bomber. â€Å"Other Muslims would argue the Koran supports actions of suicide bombers. These latter groups include radical and/or militant terrorist organizations with which we are familiar (Williams, Waltrip, 2004, p. 139). † The leader might not bother the suicide bomber/terrorist with the political complexities of such actions since this usually requires a level of mental capability similar to that of the learned and the educated, and not every suicide terrorist that pulls plunges himself/herself and the bomb inside a crowded place are educated, or did not have the chance to be one. â€Å"The suicide terrorist no longer thinks; he finds his inspiration and refuge in prayer, as outlined in Mohammad Atta’s letter to his comrades before the attack of September 11, 2001. There should be a ritual of washing and shaving, and the night is to be spent in prayer trying to forget the world. Then there should be further prayers, and after this Atta advises the others, ‘Smile and feel secure. For God is with the believers and the angels are guarding him without him feeling it. ’ We do not know, however, whether all or any of the suicide bombers did indeed spend the last night praying; some reportedly spent it in a bar or a night club (Laquer, 2003, p. 95). † References Bjorgo, Tore (December 2005). Root Causes of Terrorism: Myths, Reality and Ways Forward. Taylor Francis, Inc.

Examine functionalism explanations of socialisation

Examine functionalism explanations of socialisation Examine functionalism explanations of socialisation within the family Functionalists believe in the concept that of consensus and that that the family is at the heart of society, and is the key attribute in keeping society in an orderly and efficient manner. They also believe that hold the view that the nuclear family best suits the for anan advanced industrial society because it is more mobile to move from different areas thsan the traditional extended family because there a ferwer people making it economically viable,(Parsons nuclear isolated family theory.There view of the nuclear family comprises of the economi provider breadwinner husband (instrumental superior) and dependant wife (expressive superior) and children. Sociologists such as Durkheim have developed the functionalistic approach towards the family. Most of the sociloisdts coming from the United States. Functionalism is a theory that explains the role of the family within society- functionalists acknowledge the positive aspects of society and look at society on a macro scale. The family is a unit which workers for the efficiency and stabilisation of society. Functionalism is based on the belief of consensus, for example we are socialised on the norms of society- from what is right and from what is wrong. Early sociologists such as Durkheim (1858-1917) recognized how organic solidarity could be used to explain some social institutions and behaviours Durkheim stated that there were two types of solidarity, (mechanical and organic) mechanical solidarity being characterised by the division of labour yet this was deemed problematic by Durkheim. Organic solidarity on the other hand is when individuals see society as interdependent which according to Durkheim is essential for modern society to avoid anomie. Functionalist therefore are of the belief that every institution contributes to the smooth running of society- in particular the family being the most prominent function that keeps society from anarchy by producing social actors that are socialised stable individuals. The American social scientist Murdock thoroughly analysed two hundred and fifty societies and claimed that some sort of family existed in every form of society and therefore concluded that from his evidence that the family is universal. He refers to the family as a social group characterised by common residence, economic co-operation and reproduction. It includes adults of both sexes, at least two of whom maintain a socially approved relationship, and one or more children, own or adopted, of sexually cohabiting adults. He argues that the sexual and reproductive functions are essential. If reproduction did not take place there would be no members of society; it prevents disruptive effects on social order that would normally result if the sexual urges were allowed free-play. The family provides control and expression. Murdock also states that without the economic function no preparation and provision of food would take place his believes that the economic aspect of the family is enhan ced when the persons live together. Despite Murdocks thorough research of societies and observations of the family, his theory of the universal functions of the family have undergone numerous criticisms; many having the similar tone that there are many different types of families for example single parented families which are just as functional as the ideal nuclear family and in some circumstances have a better relationship with the child as there is more trust and responsibility delegated to the child. One critic stated the contents of values is culturally and historically specific and variable, and is no way universal. However Murdocks era also has to be taken into account when considering his theory, as the nuclear family was deemed the norm and he would have been socialised to be compatible with that concept. Parson is of the belief that the basic function of the family is the primary socialization of children so that they can become civilised members of the society he also believes that the family is essential for the stabilization of adult personalities- as it gives the adult opportunity to express signs of stress for example the family could relieve the adult of a stressful day at work. Parsons view was that these two functions were positively correlated because a child can be socialized into a society only if the society was institutionalized and organized into expected role structures which gives the adult stability. Parson despite criticism, believed that the patriarchal family was a strong and stable and consisted of a hierarchy which consisted of the instrumental superior father (husband), the instrumental inferior son (brother), the expressive superior mother (wife) and the inferior expressive superior daughter (sister).This role structure is the normal nuclear family. Parsons makes some assumptions regarding the family, especially regarding specific genders for example the woman is better fitted for the home and the man for more manual labour. Radical feminists argue that Parsons view of the female role in the family is an inaccurate perspective regarding the lives of females and believe that they are oppressed within the home and have to be the stereotypical emotive nurturer due to such theorists as Parson which make them confined to those roles- without there being an actual choice. Parson views the nuclear family as being best fitted for industrialisation for example jobs of today demand a desire for increased education, which means too many children on the become economic liabilities because the time taken to nurture children and the costs of schooling is very expensive. Too many children would not be economically viable which is why the industrialised family has been reduced. The reduction in size means the nuclear family is more geographically and socially mobile. Extended kinship is not needed resulting that extended kin are visited out of choice and not obligation. However this has been debated, it has been argued that specialised jobs such as politicians have to move relocate on demand, the rest of the population which is the majority, can easily remain situated in the same location for a number of years, and that the extended kin remain a fundamental part of the family as they often are in the third generation and pass down their knowledge and culture which in turn is internalised by the children and adults. Parsons work has been criticised by some theorists such as Robert Merton ( who worked in close proximity with Parsons) he argued that the social sciences were not yet ready for such sweeping theory but still had a tremendous amount of work to do gathering data and summarizing research findings with more modest theory. Interactionist theorists criticized functionalism for failing to conceptualize adequately the nature of actors and the process of interaction. Parsons work has been criticized, but has influenced generations of family scholars. Due to Durkheim, the grounds of functionalism were founded, and such social scientists as Murdock and Parsons expanded further of his ideas of the family- that the family socialises an individual through both primary and secondary socialisation, aiding the development of a stable adult character- however the individual is never fully socialised as society is continually evolving society.

Buffering Region of Histidine Monohydrochloride

Buffering Region of Histidine Monohydrochloride The objective of this experiment is to determine the buffering region of histidine monohydrochloride by titrating histidine with a base, NaOH. By plotting a suitable graph, the pKa values of histidine can be observed. Normally, a titration curve is constructed to illustrate the relationship between the pH of the mixture and the number of moles of base added to it. However in this experiment, the graph of pH against the number of moles of NaOH per mole of histidine is plotted. This is to ensure that the graph is independent of the volume and concentrations of the solutions used. After determining the pKa values of histidine, the maximal buffering capacity of the histidine-NaOH mixture, as well as the effective buffering range can be determined. Materials and Methods To prepare 20mM solution of histidine monohydrochloride, 0.196g of histidine monohydrochloride was dissolved in 46.8mL of water, according to the calculations below: No. of moles of histidine = = 9.35 10-4 mol = 46.8 mL Upon complete mixing of the 20mM histidine monohydrochloride solution using a magnetic stirrer, 20mL of the solution was transferred into a beaker. The burette was washed with distilled water followed by NaOH and subsequently filled with 0.05M NaOH. The original pH of histidine solution was measured using the pH meter before proceeding with titration. Titration was carried out by adding NaOH to the histidine solution at 0.5mL increments. After each increment, the pH value of the resulting acid-base mixture was recorded. Titration was stopped when the acid-base mixture reached pH 11.5. Results Calculations Calculation of no. of moles of histidine present in solution = = Plotting graph of pH against no. of moles of NaOH per mol of histidine Table: pH of histidine-NaOH solution with every 0.5mL of NaOH added Determining pKa values of histidine (i) Based on Graph 1, the two rectangles indicate the two regions where the curve approaches the point of inflection. The maximum and minimum points of the regions are marked with the yellow circle. By finding the average values of each set of maximum and minimum points, the respective pKa values can be determined. pKa1 = = 6.12 pKa2 = = 9.45 (ii) pKa1 is the point where = 0.5 pKa2 is the point where = 1.5 Based on Graph 1, pKa1 and pKa2 are points marked with the red cross. pKa1 = 6.16 pKa2 = 9.30 Maximal buffering capacity Effective buffering range Based on Graph 1, the acid-base mixture shows maximal buffering capacity at pH 6.12 and pH 9.45. The effective buffering range of a buffer is between  ±1 of the maximal buffering capacity. Thus, the effective buffering range of histidine is pH 5.12 to pH 7.12 and pH 8.45 to pH 10.45. If NaOH has not been accurately prepared, method used in (c)(i) will give a more reliable estimate of the pKa values. If NaOH has not been accurately prepared, the number of moles of NaOH will be different, changing the ratio of number of moles of NaOH per mole of histidine. Method (c)(ii) depends on this ratio to determine the two pKa values. Hence, inaccurate ratios will cause the resulting pKa values to vary, leading to less reliable estimate of pKa values. On the other hand, method (c)(i) does not depend on the ratio between number of moles of NaOH and histidine. Thus, an inaccurate ratio will not affect the pKa values being determined. Instead, method (c)(i) relies on the point of inflection of the graph, which plots pH against the number of moles of NaOH per mole of histidine. Plotting the graph in this manner ensures that it is independent of the volume and concentrations of the solutions used. In other words, even if NaOH has been inaccurately prepared, changing the concentration of the NaOH solution, the shape of the curve remains similar. Since the shape of the curve does not change, the point of inflection will be almost at the same point. pKa values obtained by method (c)(i) will be similar to the original values when NaOH was prepared accurately. Calculation of pH of the solution after addition of: 5mL of NaOH No. of moles of NaOH added = ÃÆ'- 0.05 = 2.5 x 10-4 mol NaOH †°Ã‚ ¡ Histidine No. of moles of histidine reacted = 2.5 x 10-4 mol Initial no. of moles of histidine = 4 x 10-4 mol No. of moles of histidine left = 4 x 10-4 2.5 x 10-4 mol = 1.5 x 10-4 mol pH = pKa + log pH = 6.12+ log = 6.34 (ii) 12mL of NaOH No. of moles of NaOH added = ÃÆ'- 0.05 = 6.0 x 10-4 mol No. of moles of NaOH left = 6.0 x 10-4 4 x 10-4 = 2.0 x 10-4 mol NaOH †°Ã‚ ¡ Histidine No. of moles of histidine reacted = 2.0 x 10-4 mol Initial no. of moles of histidine = 4 x 10-4 mol No. of moles of histidine left = 4 x 10-4 2.0 x 10-4 mol = 2.0 x 10-4 mol pH = pKa + log pH = 9.45 + log = 9.45 (i) Three ionisable groups are present in histidine at the initial pH of the experiment. The three groups are: carboxyl group, amino group and the R group (imidazole group). (ii) The amino group is responsible for the observed pKa value of 6.12 and the imidazole group is responsible for the pKa value of 9.45. Structures of ionic species of histidine that participate in cellular buffering Discussion Histidine is an amino acid that acts as a buffer and it has three ionisable groups: carboxyl group, amino group and imidazole group. In this experiment, the focus is on the dissociation constant of the amino and imidazole group. The titration curve (as shown in Graph 1) has two steps, or two points of inflection because the amino group dissociates first followed by the dissociation of imidazole group. Hence, the amino group is responsible for the observed pKa value of 6.12 and the imidazole group is responsible for the pKa value of 9.45. Two methods were used to determine the pKa values of histidine. However these calculated values are only estimates and may deviate from the actual values due to the following experimental errors: Parallax error occurs during the reading of the burette, resulting in inconsistent increment of NaOH added to the histidine solution. In other words, each increment of NaOH was not maintained at 0.5mL. This directly affects the precision of the experiment. Possible solution to minimise error: To avoid parallax error, ensure that the burette reading is taken from eye level at the bottom of the meniscus. The burette should also be placed in an upright position, perpendicular to the table. For a more precise burette reading, a black burette reading card can be placed behind the burette so as to get a clearer view, especially when colourless solutions are used. The beaker containing the histidine-NaOH mixture is placed on the magnetic stirrer throughout the titration to ensure a homogenous mixture for more accurate pH readings. After every 0.5mL of NaOH added to the mixture, the pH of the resulting mixture is recorded by using the pH meter. However, it takes time for the pH meter to generate a final pH reading that does not fluctuate. If the pH value is recorded too quickly after the addition of NaOH, the pH reading may be inaccurate. Possible solution to minimise error: To obtain greater accuracy in pH reading, ensure that an appropriate waiting time (about 2min) is maintained between the addition of NaOH and the recording of pH value. Conclusion From this experiment, it can be concluded from the titration curve that the amino group of histidine is responsible for the observed pKa value of 6.12 and the imidazole group is responsible for the pKa value of 9.45. These two pKa values correspond to the pH at which the acid-base mixture shows maximal buffering capacity. The effective buffering range of histidine is pH 5.12 to pH 7.12 and pH 8.45 to pH 10.45. EXPERIMENT 2: Effect of Buffer pKa on Buffering Capacity Introduction Buffers are solutions that are able to maintain a fairly constant pH when a small amount of acid or base is added. This experiment examines the effect of buffers pKa on buffering capacity by studying how well the two buffers of different pKa resist pH changes when acid or base is added. In scientific experiments, it is advisable to choose a buffer system in which the pKa of the weak acid is nearer to the pH of the interest. It will be ineffective for a buffer to resist pH changes if its pKa value is more than 1 pH unit from the pH of interest. Thus the study of the effect of pKa on buffering capacity is important in making a suitable choice of pH buffers for a specific experiment. Materials and Methods We study the effect of buffers pKa on buffering capacity by using 2 different buffers, potassium phosphate buffer and Tris-HCl, with pKa value 6.8 and 8.1 respectively. 3mL of 0.01M potassium phosphate buffer was pipetted into two test tubes, labelled A and B. 3mL of 0.01M Tris-HCl was also pipetted into two test tubes, labelled C and D. Three drops of universal pH indicator were added into each test tube, causing the solutions to turn green in colour (pH 7.0). HCl was added to test tubes A and C until the solutions turned pink (pH 4.0). KOH was added to test tubes B and D until the solutions turned purple (pH 10.0). The number of drops required for the solutions on each test tube to turn pink or purple in colour is recorded. The pH colour chart is used as it shows the colours of the solution at each pH level. Results Questions Table : Number of drops of acid or base needed for buffer solution to deviate from its initial neutrality (pH 7.0) pH Buffer pKa of buffer Initial pH No. of drops of HCl required to become acidic (pH 4.0) No. of drops of KOH required to become alkaline (pH 10.0) 0.01M potassium phosphate buffer 6.8 7.0 5 11 M Tris-HCl 8.1 7.0 2 20 Conclusions drawn from experiments According to Table 2, potassium phosphate buffer requires five drops of HCl to reach pH 4.0, compared to Tris-HCl which requires only two drops of HCl to reach pH 4.0. This shows that potassium phosphate buffer is a more effective buffer against acids. Potassium phosphate buffer requires eleven drops of KOH to reach pH 10.0 while Tris-HCl requires twenty drops of KOH to reach pH 10.0. Based on the results, Tris-HCl behaves as a more efficient buffer under basic conditions as it requires more amount of KOH than that of potassium phosphate to reach pH 10.0. This means that Tris-HCl has greater ability to resist increases in pH but not decreases in pH. On the other hand, potassium phosphate buffer is a more efficient buffer under acidic conditions as it requires lesser amount of HCl to reach pH 4.0. Similarly, this means that potassium phosphate buffer has greater ability to resist decreases in pH but not increases in pH. It can be deduced that a buffer with greater pKa value is a more efficient buffer in basic conditions while a buffer with smaller pKa value is a more efficient buffer in acidic conditions. Choosing a suitable buffer to study the properties of a phosphatase which functions optimally at pH 7.2 I would use the 0.01M Tris-HCl to study the properties of a phosphatase. It is more appropriate to use a buffer with effective buffering range nearer to the pH of phosphatase. Tris-HCl has an effective buffering range of pH 7.1 to 9.1 while potassium phosphatase buffer has an effective buffering range of pH 5.8 to 7.8. Simply by considering the effective buffering range of the two buffers, it can be concluded that both buffers can be used to study the properties of phosphatase which functions optimally at pH 7.2. However, considering the effective buffering range of the buffers is not sufficient to come to a sound conclusion. In this case, phosphatase is an enzyme that functions to hydrolyse phosphate groups. By adding potassium phosphate buffer to phosphatase, phosphatase will break down the phosphate group in the potassium phosphate buffer. This changes the chemical properties and hence the buffering capability of the potassium phosphate buffer. Therefore, Tris-HCl is a more suitable buffer for the studying of phosphatase. Discussion In Experiment 1, the endpoint of the reactions is determined using a pH meter and construction a titration curve. However in this experiment, the endpoint is visually observed by the help of a pH colour chart. Possible sources of experimental errors arising from this method and ways to improve the experiment are discussed below: In this experiment, only two types of buffers, Tris-HCl and potassium phosphate buffer, were used. The experiment can be improved by using more types of pH buffers to obtain more data. This will allow more accurate evaluation of the relationship between the pKa value and the buffering capacity, and thus the effect of pKa value on the buffering capacity. Although the pH colour chart is used to compare the colours of the solutions, personal judgment comes into play when determining the colour change in the chemical reactions. Possible solution to minimise error: Be consistent in deciding the point of colour change and the endpoint of the experiment. Conclusion From this experiment, it can be concluded that a buffer with greater pKa value is a more efficient buffer in basic conditions and a buffer with smaller pKa value is a more efficient buffer in acidic conditions. Though a buffers pKa can affect its buffering capacity, however when choosing a suitable buffer for an experiment, we cannot simply rely on the pKa of a buffer. It is also crucial to consider the chemical properties and structure of the buffer and other reagents to be used in the experiment. EXPERIMENT 3: Effect of Temperature on the pH of a buffer Introduction The aim of this experiment is to examine the effect of temperature on the pH of a buffer. This can be done by observing the changes in pH of two different buffers when temperature of the buffer solution decreases from room temperature to 4 °C. pH of the buffers that are used to maintain the pH of the lab samples can change during changes in temperature due to cooling process. Changes in pH of buffers upon temperature changes can be explained by the Le Chateliers Principle. The study of the effect of temperature on pH of a buffer is crucial in choosing the right pH buffer that is able to show minimum changes in buffer pH, to maintain the properties of the biological samples that requires specific pH environment. Materials and Methods We study the effect of temperature on the pH of a buffer by using two different buffers, 0.01M potassium phosphate buffer and 0.01M Tris-HCl. 3mL of each buffer solution were pipetted into two separate test tubes. The initial pH values of the two buffers at room temperature are measured using the pH meter and recorded. Subsequently, both test tubes were placed into the ice box to cool to 4 °C. After 20 minutes, the test tubes were taken out of the ice box and placed in an ice bath to maintain the temperature of the buffer solutions at 4 °C. The pH of the cooled buffer solutions were measured again and recorded to obtain the results as seen in Table 3. By evaluating the pH changes (either increase or decrease) and the extent of these changes from the original pH value, we can observe the effect of temperature on the pH of a buffer. Results Questions Table : The changes in the pH of the buffer solution as temperature is decreased to 4 °C Buffer pH at room temperature pH at 4 °C Difference in pH change (unit) 0.01M potassium phosphate buffer 7.03 7.49 0.46 0.01M Tris-HCl 7.01 8.16 1.15 Effect of temperature on the pH of Tris-HCl and potassium phosphate buffer According to Table 3, at low temperature of 4 °C, both buffer solutions become more alkaline. As temperature decreased from the room temperature to 4 °C, the pH potassium phosphate buffer increased from 7.03 to 7.49, with a difference in pH change of 0.46. With the same change in temperature, the pH of Tris-HCl increased from 7.01 to 8.16, with a difference in pH change of 1.15. This shows that Tris-HCl exhibits greater changes in pH than potassium phosphate buffer, upon a given change in temperature. In conclusion, temperature has a greater effect on the pH of Tris-HCl compared to potassium phosphate buffer. HA A Ã‚ » + H Ã‚ º ΆH = -ve As illustrated by the chemical equation above, the dissociation of buffers are endothermic processes. Being an endothermic process, heat is being absorbed and temperature decreases. Based on Le Chateliers Principle, when temperature decreases, the system will react to result in an increase in temperature. Hence, decreasing temperature to 4 °C favours the backward reaction, which is an exothermic reaction that produces heat. The position of equilibrium shifts to the left, more H Ã‚ º reacts with A Ã‚ » to form HA. Thus, the concentration of H Ã‚ º decreases and causes the pH of the buffer to increase. Discussion Based on the experimental results, it is clear that temperature changes the pH of the buffer. Though this is not a complicated experiment, it is still subjected to experimental errors and can be improved by the following ways: Only two types of buffers, Tris-HCl and potassium phosphate buffer, were used in this experiment. The experiment was also conducted at only one temperature. Using several buffers over a range of temperatures will allow us to observe the pH of a variety of buffers at different temperatures. In addition, both buffers used in this experiment showed an increase in alkalinity. Hence, including more variety of buffers will allow us to evaluate which type of buffer has tendency to become more alkaline or acidic with the changes in temperature. This experiment was conducted without the use of a thermometer, hence there was uncertainty in determining the temperature of the buffer solutions. It was assumed that by placing the test tubes in the ice box for 20 minutes and then transferring into an ice bath, the buffer solutions would be maintained at 4ËÅ ¡C. However, it is difficult to maintain ice baths at 4ËÅ ¡C for a long period of time due to heat gain from the surroundings. Possible solution to minimise error: Keep a thermometer in the ice bath and consistently check the temperature of the ice bath. Add in more ice when the ice melts. It was difficult to identify the endpoint of the experiment. Even after a long period of time (about 30 minutes), the pH reading shown on the pH meter still continued to increase slowly. Hence, stopping the experiment too early may result in an inaccurate pH reading. Possible solution to minimise error: Since it is difficult to identify the endpoint of the experiment, it is perhaps more logical to standardise the duration of the experiment for both buffer solutions. For example, 30 minutes for each buffer solution. Conclusion From this experiment, it can be concluded that a decrease in temperature will cause a change in pH of a buffer. However, the pH of the buffer does not always increase when temperature decreases. This depends on whether the dissociation process is endothermic or exothermic. In the case of an endothermic dissociation process, pH of the buffer will increase when temperature decreases. This can be explained by Le Chateliers Principle which states that the backward exothermic reaction will occur so as to counteract the change. Hence, the Tris-HCl and potassium phosphate buffers become more alkaline as temperature decreases.

catcher in the rye glass menagerie Essay -- essays research papers

The Catcher in the Rye and The Glass Menagerie The person someone becomes is influenced by the losses they have experienced in their life. In Catcher in the Rye the main character Holden Caulfield is devastated by the loss of his younger brother Allie to leukemia. The loss of Allie never leaves Holden’s mind. It changes his perception of the world. In The Glass Menagerie Amanda Wingfield’s husband abandons her and their two children Tom and Laura. For Amanda the only way to deal with the loss is to escape into a dream world. She forces this delusion onto Tom and Laura Loss of anything a loved one, a friend or a cherished possession can affect the way a person feels. Loss can change a person’s reactions or attitude toward something. This may cause the person to change the way they live their life or the decisions they make. Loss leaves an impression on a person.   Ã‚  Ã‚  Ã‚  Ã‚  Holden Caulfield is tormented by Allie’s death. The night Allie died Holden slept in the garage and broke all the windows with his fists. This was the only way Holden could deal with the loss of Allie. Allie becomes the model of the ideal person for Holden. Throughout Catcher in the Rye whenever Holden talks about Allie it’s what a great person he was. He describes Allie as extremely intelligent, someone you would have liked and as someone that never got angry. Everyone Holden meets or sees he compares them to this image of Allie but almost no one meets the standards. As Holden is eating breakfast in a diner he sees two nuns come in and helps them with their suitcases. He thinks nothing of them until he notices that their suitcases seem inexpensive. Holden talks about how he hates it when someone has cheap suitcases and how he can just get to hate someone for having them. Holden never gets over Allie’s death. It causes him to become restless. He is searching for something to replace the gap in his life that Allie left. He looked up to Allie. Whenever Holden finds something that can replace Allie he discovers flaws about it. Holden feels that if he finds something he is betraying Allie. He never truly finds something to fill the void. When Phoebe asks Holden to name something he likes a lot he can’t think of anything besides Allie. Phoebe says Allie doesn't really count because he’s dead. Holden argues â€Å"I can still like him, though, can’t I? Just because somebody’s dea... ...inessman. She blames her husband for her children’s bad qualities. When Tom stays out late or smokes Amanda says â€Å"you got it from you father†. Holden and Amanda were both traumatized by the loss they experienced. Holden’s view of the world changes because of Allie’s death. In Holden’s mind no one will ever live up to Allie. Holden can’t see good in the world anymore. There is something wrong with everything when matched up against Allie. Even though this is true Holden is on an endless quest to find something that can replace Allie. The loss of Allie causes Holden to feel only something negative can result from change. The loss of her husband throws Amanda into a fantasy world. She will not accept that she should be anything other than the southern lady she was brought up to be, that Laura is peculiar or that Tom is not a budding businessman, and that she herself might be in some ways responsible for the sorrows and flaws of her children. Loss affects people in different ways. It can tear them apart or make them stronger. For both Holden and Amanda loss forces them to struggle with their emotions in an attempt to find ways of dealing with the loss.   Ã‚  Ã‚  Ã‚  Ã‚  

Against an Alcohol Age Limit Essay -- Argumentative Persuasive Drinkin

Drinking Age: Lowered or Not   Ã‚  Ã‚  Ã‚  Ã‚  The drinking age in the United States is 21. How ridiculous is that? You are able to do so many things at the age of 18. When a person reaches the age of 18, he or she can leave their homes or be kicked out, and become their own legal guardian. They no longer are required to have their parents sign their name to any documents pertaining to them, and are now considered an adult except when it comes to alcohol. When a person 18 or older commits a crime, they will be tried as an adult. Now the 18 year old no longer goes to Juvenile Hall when convicted of a crime, but instead they go to jail, state prison, federal prison, or even death row. At 18, they are held fully responsible for their own actions, and must accept the consequences. Don’t you think we s...

Essay on Edwin Chadwick

Edwin Chadwick was a man that made a huge different in our world a long time ago. Mr. Edwin Chadwick was born January 24th, 1800 and passed away July 16th, 1890. He lived a very long life and it was all due to the changes in the world that he made that prolonged his life as well as our lives today. I know everyone is asking the same question, â€Å"What did a man that died about 120 years ago do that helps prolong my life today? † I will tell you and explain how we still live by Mr. Edwin Chadwick’s changes and use his invents to this day. Mr.  Chadwick was born in Manchester, England; but was raised by both his parents in London. Mr. Chadwick was an attorney. Edwin Chadwick received his earliest education in Longsight and Stockport. In 1810, his parent decided to more to London where then begun to receive education by private tutors. Mr. Edwin Chadwick was an attorney that was all about making changes in the reforming of the poor laws and making sanitary conditions better so it could improve the public health. Mr. Edwin Chadwick believed that his inventions would not only improve the health of people, but it would also save money. In 1834, Edwin Chadwick was given a job as the secretary of the Poor Law Commissioners. Mr. Chadwick’s sanitary ideas made the government fierce and sustain criticism. Mr. Chadwick paid out of his own expense for researches and publishing papers about the unsafe sanitary conditions. Mr. Edwin Chadwick had a lot of problems with his superiors and there was a lot of disagreement that lead to the dissolution in 1847 of The Poor Law Commission. At the same time that Edwin Chadwick was working for The Poor Law Commission, he was answering questions about sanitary and trying to find away to improve it. Mr. Chadwick was all about making changes in reforming of the poor laws and sanitary conditions. Mr. Chadwick strongly believed that the unsafe sanitary conditions were causing illnesses and lots of unnecessary deaths. Mr. Edwin Chadwick was putting together ideas that later was known as the â€Å"sanitary idea†. In 1842, Edwin Chadwick invented what we know today as the sewer tank. He found a healthier and proper way for removal of trash and sewage. Mr. Edwin Chadwick knew that the odor from the sewage and trash was harming the public. Mr.  Edwin Chadwick also knew that by putting the sewage underground and running it through a special pipeline into a special designed tank would be healthier for the public and by doing so the unpleasant odor (miasmas) from the above ground sewage and trash would disappear. After Mr. Chadwick introduced the sewage tank and proper trash removal the life expectancy increased. Edwin Chadwick invented the self flushing toilet; it was lined in glazed bricks. We use these several times today and never really thought about where they came from or what life would be like without them. Mr. Chadwick invented and changed our sanitary conditions; by doing so there are few illnesses and deaths. We can flush our toilet and do not have to smell that nasty odor from the gas that would one day kill us. Edwin Chadwick had the idea of separating the sewage lines and water line a distances away from each other. By doing so it proved clean fresh drinking water that we enjoy today. In 1847, Edwin Chadwick changed the amount of smoke and soot that comes from chimneys. He created a law that protects the public from the harm of hazard materials. By doing so Edwin Chadwick helped control the pollution in our environment. This made it a lot easier and safer for people to breath fresh air and enjoy being outside. Edwin Chadwick forced the government to get health inspectors to see that our community stays safe, so that we could have healthier lives. He also pushed the government for better-ventilated and less crowded housing for families, wider streets for traveling to and from, workplace health and safety legislation for employees, increased use of indoor plumbing, and stops the children employment in factories. In 1854, Edwin Chadwick pushed the government to registration of all births, all deaths, and all marriage. It is amazing that if it was not for Edwin Chadwick we would not have birth certificates of our children, death certificates of our loved ones, or marriage licenses to our spouses. A driver’s license is needed to get a job and/or any government help that anyone might need. In 1854, Edwin Chadwick pushed the government to registration of all births, all deaths, and all marriage. It is amazing that if it was not for Edwin Chadwick we would not have birth certificates of our children, death certificates of our loved ones, or marriage licenses to our spouses. By inventing these certificates we can keep count of the population of people today. Because of Edwin Chadwick, We have birth certificate for our husbands, sons, daughters, and ourselves. We have marriage licenses to show our husbands when they acts crazy; okay and the men can show them to the wives too. We need death certificates to claim life insurance policies of loved ones.

Adolescence Is the Unhappiest Time in Most People?S Lives

Adolescence is the unhappiest condemnation in just al nigh hatful? s lives Most race agrees that adolesece may be the gravelyest sentence in most pecks life. Just bid e real period of life, being youthful can be a reasonably rough judgment of conviction. both(prenominal) mountain says it is the unhappies season, mend other says it is the most happiest clock in our life. but not everyone agrees that is the unhappiest condemnation. So is the adolecense the unhappiest time in most peoples lives? , Im going to verbalize about the arguments for and against this statement and give my profess opinion.It is true that adolesence is a very hard age, may be the hardest. Firstly because, is the time in wich the person is growing up, and has to lounge about used to stomach more than responsabilities. Also, in that respect atomic number 18 mostwhat people that trust this is because the adolecent atomic number 18 more sensitive than sr. or younger people and this is a reazon why to demoralise upset for very silly razons. And, finally, some people thinks this is because, the adolecents atomic number 18nt as free as they wished to be, and this make them want to pass this part of their lives quicker.However, there argon some people that desagree with this statement, and one of the principally arguments is that is a very happy time for people because they wear upont have to worry about the resoult of their actions because they dont have as umteen responsabilities as they entrust have later when they get older. Another argument against this is that they, normaly, have more free time, and less stress. a time when they begin to discover who they are, they are seemly more independent, they are establishing friendships, and their bodies start developing. For some, ledger entry into puberty can be a very emotional, stressful, confusing, and frightening time.Some make the switch over from childhood into adolescence with solo a few minor problems , others yet, may have a more difficult time intervention the pressures and some may develop consume disorders as a way to cope. Some may enter into puberty proterozoic and be subjected to teasing by their peers. some(prenominal) fear the exercising weight that is gained during this time is permanent, leave behind panic, and desperately try to take the weight off. They are not aware that at a time the physical changes during this time cease, their weight will usually stable off and go to their bodies natural set point, without the need for dieting.Adolescence Is the Unhappiest season in Most Peoples LivesAdolescence is the unhappiest time in most peoples lives. It seems to me adolescence is a very in-chief(postnominal) period for everyone as it forms their views on life. When you are young, parents bring up and teach you, you attend something by your own and become some life fuck. Then you go to schooltime where you get basic knowledge, find refreshful friends, communic ate and share your opinions with them. This time could be divided into three periods persons childhood, school time and the last is an university or college time. Each of them is extremely all-important(a) as it influences on the further life of an individual.Some people state that adolescence is the unhappiest time for most people. It brings many accidents, frightful experiences and, as a result, person can go, for example, in a aversion world or drug addiction. These facts are followed after the bad upbringing and some problem families. Thats why these people state thar the adolescence is a very important time for a person and during this time person is touchy so they(person is thouchy so they(he or she, leave it open)) could have a bad experience and it would influence a lot to his future. However, others are assured that adolescence is the happiest time in everybodys life.When you are young, you are not responsible for anything and you dont have serious problems that should b e solved. You dont care about anything bad and enjoy spending your time with friends. mortal goes to school where he gains some experience and finds new friends. They say, in most cases children have a good upbringing and only enjoy their life. It goes without saying, that the world of today is dangerous, however it mostly doesnt influence on the person. If you ask me, I think adolescence is the most enjoyable time for everyone as it gives many opportunities for relaxation, education and entertainment.

History of Trigonometry

Running Head bill of trig level of trig Rome Fiedler autobiography of math 501 University of Akron April 29, 2012 storey of trig An Introduction trig is useful in our world. By exploring where these concepts come from provides an rationality in putting this math to use. The term trig comes from the Hellenic phrase trigon, meaning triplicity and the Hellenic word meatria meaning eyeshadement. However it is non primal to Greek in bu sinss breeze. The maths comes from multiple plurality everyplace a span of thousands of forms and has affected over every major civilization.It is a faction of geometry, and astronomy and has m either(prenominal) an(prenominal) practical coverings over story. trigonometry is a branch of math startle created by warrant century BC by the Greek mathematician Hipp cuthus. The history of trig and of trigonometric subr byines sticks to the general lines of the history of math. Early query of tri weights could be found in the 2 nd millennium BC, in Egyptian and Babylonian math. Methodical rese waiverh of trigonometric comp adeptnts started in Greek math, and it r severallyed India as lay out of Greek astronomy.In Indian astronomy, the rese flickerh of trigonometric functions flourished in the Gupta dynasty, busyly as a result of Aryabhata. Throughout the Middle Ages, the research of trig continued in Moslem math, while it was enforced as a discrete clear in the Latin West beginning in the metempsychosis with Regiomontanus. The growth of coetaneous trig shifted in the occidental Age of Enlightenment, starting with 17th-century math and reaching its contemporary type with Leonhard Euler (1748) Etymology The word trig originates from the Greek trigonometria, implying triangle measuring, from triangle + to measure.The name develop from the guinea pig of honor of chastise triangles by applying the carnal fuckledge ships amidst the measures of its sides and angles to the field of identical tri angles (Gullberg, 1996). The word was go ind by Barthoolomus ptiticus in the epithet of his hunt d sustain Trigonometria sice de soluti adept triangularumtractus brevis et perspicius in 1595. The contemporary word hell, is originated from the Latin word sinus, which implied bay, bosom or fold, translation from Arabic word jayb. The Arabic word is in origin of version of Sanskrit jiva fit in.Sanskrit jiva in learned utilize was a alike word of jya harmonize, primarily the word for bow-string. Sanskrit jiva was interpreted into Arabic as jiba (Boyer, 1991). This word was past changed into the real Arabic word jayb, implying bosom, fold, bay, either by the Arabs or err building blockaryously of the European translators such as Robert of Chester, who translated jayb into Latin as sinus. In particular Fibonaccis sinus rectus arcus senilis was significant in creating the word sinus. Early Beginnings The origin of the subject has rich diversity. trig is not the train of one part icular person or space alone rather a development over season.The primitive Egyptians and Babylonians had cognise of theorems on the ratios of the sides of analogous triangles for m whatever an(prenominal) centuries. However pre-Greek societies were deficient of the concept of an angle measure and as a result, the sides of triangles were analyzed rather, a field that would be better cognise as trilaterometry(Boyer, 1991). The Babylonian astronomers kept comprehensive records on the rising and move of stars, the movement of the categoricts, and the solar and lunar eclipses, all of which postulate association with angular distances deliberate on the aerial sphere.Founded on one explanation of the Plimpton 322 cunei shape panelt, virtually take in even claimed that the primitive Babylonians had a set back of secants. There was, on the other hand, much intelligence as to whether it is a flurry of Pythagorean triples, a solution of quadratic equations, or a trigonometri c send back. The Egyptians, in contrast, applied an ancient kind of trig for rebuildion of pyramids and surveying the land in the 2nd millennium BC. The aboriginal beginnings of trigonometry ar plan to be the firstly numerical sequences correlating shadow lengths to age of mean solar day.Shadow remits were simple sequences of numbers which applied the shadow of a vertical stick, called a gnomon, is long in the forenoon and shortens to a minimum at noon. Then becomes lengthy and longer as the afternoon progresses (Kennedy, 1969). The shadow tables would correlate a particular hour to a particular length and were used as earlier as 1500 BC by the Egyptians. Similar tables were real by other civilizations such as the Indians and Greeks. Greek mathematics Shadow tables were the primary development in cosmos of trigonometry however the Greeks really certain trig into an arrayed science.The Greeks continued as the Babylonians astronomers did and studied the relation mingle d with angles and bandings in lengths of harmonises to develop their theories on global position and motion (Mankiewicz, 2001). pic The chord of an angle subtends the arc of the angle. Ancient Greek mathematicians used the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chords perpendicular bisector traverses the center of the circle and bisects the angle. wiz half of the bisected chord is the sin of the bisected angle, that is, pic nd accordingly the sinfulness function is as well cognize as the half-chord. As a result of this relationship, several trigonometric identities and theorems that ar know at bear were likewise known to Greek mathematicians, however in their equivalent chord form. Though there is no trigonometry in the works of Euclid and Archimedes, there atomic number 18 theorems presented in a geometric system that atomic number 18 similar to particular trigonometric righteousnesss or rules. Theorems on the lengths of chords are applications of the law of sines. In addition Archimedes theorem on mazed chords is similar to rules for sines of sums and differences of angles.From the primitive landmarks of shadow tables and the Greeks gain and stretch outing upon of astronomic knowledge from the Babylonians, there was a flutter in the improvement of trigonometry until the period of Hipparchus. Hipparchus The first trigonometric table was in fact compiled by Hipparchus of, who is known as an as the commence of trigonometry(Boyer, 1991). Hipparchus was the first to put into a table the constitute values of arc and chord for a serial publication of angles. He did this by considering every triangle was engrave in a circle of fixed radius. to each one side of the triangle became a chord, a neat line drawn between two points on a circle.To point out the parts of the triangle he wanted to find the length of the chord as a function of the underlying angle. pic For Example, in the plot triangle ACB is? inscribed in circle O. So the sides of the triangle become chord? AC, chord CB and chord AB. Hipparchus would have sought to? find the length of the chord, AC, as a function of the central? angle. He deduced a trigonometric ordinance for the? length of a chord sketched from one point on the circuit of? a circle to another (Motz, 1993). This could therefore be used to help understand the positioning of the run downts on the sphere.Though it is not known when the mannerical use of the 360 circle came into math, it is known that the methodical introduction of the 360 circle introduced a teentsy after Aristarchus of Samos comprised of On the Sizes and Distances of the Sun and Moon, since he measured an angle a part of a quadrant. It seemed that the magisterial used of the 360 circle was generally as a result of Hipparchus and his table of chords. Hipparchus might have taken the idea of that division from Hypsicles who had previously carve up the day into 36 0 parts, a division of the day that might have been recommended by Babylonian astronomy.In primeval astronomy, the zodiac had been shared into cardinal signs or thirty- half-dozen decans. A recurring unit of ammunition of approximately 360 long time could have corresponded to the signs and decans of the zodiac by dividing each sign into 30 parts and each decan into 10 parts. It was as a result of the Babylonian sexagesimal numeral system that each peak was divided into 60 minutes and each minute was divided into 60 seconds. Though Hipparchus is attri barelyed as the father of trigonometry all of his work is lost except one barely we gain knowledge of his work through and through Ptolemy. pic http//www. ies. co. p/math/java/vector/menela/menela. hypertext markup language Menelaus Menelaus of Alexandria wrote in three defys his Sphaerica. In Book I, he created a basis for globose triangles analogous to the euclidian basis for plane triangles. He established a theorem that is w ithout Euclidean analogue, that two orbiculate triangles were similar if alike angles are equal, however he did not specify between congruent and symmetric spherical triangles. some other theorem that he established was that the sum of the angles of a spherical triangle is more(prenominal) than than 180. Book II of Sphaerica applied spherical geometry to astronomy.In addition Book deuce-ace contained the theorem of Menelaus(Boyer, 1991). He but gave his well-known rule of sestet quantities(Needham, 1986). This theorem came to paly a major role in spherical trigonometry and astronomy. It was in any case believed that Melaus mya have essential a second table of chords ground on Hipparchus works, however these were lost (Smith, 1958). Ptolemy Afterwards, Claudius Ptolemy developed upon Hipparchus Chords in a Circle in his Almagest, or the numeral Syntaxis. The Almagest was of importly a work on astronomy, and astronomy relied on trigonometry.The 13 books of the Almagest w ere the most prominent and important trigonometric work of ancient times. This book was a typography of both astronomy and trigonometry and was derived from the work of Hipparchus and Menelaus. Almagest contains a table of lengths of chords in a circle and a detailed set of instructions on how to construct the table. These instructions contain some of the earliest derivtions of trigonometry. Ptolemy magisterial that Menelaus started by dividing a circle into 360o, and the diameter into cxx parts. He did this because 3 x 120 = 360, development the previous application of 3 for pi.Then each part is divided into cardinal parts, each of these again into sixty parts, and so on. This system of parts was based on the Babylonian sexagesimal or base 60-numeration system, which was the moreover system available at the time for treatment fractions (Maor, 1998). This system was based on 60 so that the number of degrees corresponding to the circumference of a circle would be the homogeneo us as the number of days in a year, which the Babylonians believed to be 360 days (Ball 1960). From Menlaus Ptolemy developed the concept that the sine is half of a chord.Ptolemy took Menelaus winding _ crd 2_ and said that the complement angle could be scripted as _ crd (180 o -2_), since 180o was half the circumference of the circle. Since today, cos_ = sin(90 o -_), it can be shown that cos_ = _ crd (180 o -2_), development a similar argument as the one shown above (van Brummelen, 2009). From these two expressions, one of the enormousest identities known today was created. That is, (_ crd 2_) 2 + _ crd (180 o -2_) 2 = 1 which is exactly sin2_ + cos2_ = 1 (van Brummelen, 2009). pichttp//nrich. maths. org/6853 pic http//en. ikipedia. org/wiki/Ptolemys_table_of_chords Using his table, Ptolemy believed that one could solve any planar triangle, if given at least one side of the triangle (Maor, 1998). A theorem that was fundamental to Ptolemys tally of chords was what was s till known at present as Ptolemys theorem, that the sum of the products of the opposite sides of a recurring many-sided was equivalent to the product of the diagonals. Ptolemy used these results to develop his trigonometric tables however whether these tables were originated from Hipparchus work could not be proved.Neither the tables of Hipparchus nor those of Ptolemy had survived to the present day, though descriptions by other ancient authors exhibits they existed. In his work, Ptolemy founded formulas for the chord of? difference and an equivalent for our modern day half-angle? formulas. Because of Ptolemys discoveries, given a chord of? an arc in a circle, the chord of half an arc can be assignd as? well. Ptolemy also discovered chords of sum and difference, chords of half an arc, and chords of half degree, from which he then built up his tables to the nearest second of chords of arcs from half degree.In the Almagest, a true distinction was make up between plane and spherical trigonometry. Plane trigonometry is the branch of trigonometry which applies its principles to plane triangles Spherical trigonometry, on the other hand, is the branch of trigonometry in which its principles are applied to spherical triangles, which are triangles on the move up of the sphere. Ptolemy began with spherical trigonometry, for he worked with spherical triangles in many of his theorems and proofs. However, when calculating the chords of arcs, he unintentionally developed a theory for plane trigonometry. Trigonometry was created for use in astronomy and because spherical trigonometry was for this purpose the more useful tool, it was the first to be developed. The use of plane trigonometry is foreign to Greek mathematicians (Kline, 1972). Spherical trigonometry was developed out of necessity for the divert and application of astronomers. In fact, spherical trigonometry was the most popular branch of trigonometry until the 1450s, even though Ptolemy did introduce a basis for plane trigonometry in the Almagest in 150 A. D. IndiaThe next major parcel to trigonometry came from India. The trigonometry of Ptolemy was based on the operational relationship between chords of a circle and central angles they subtend. The Siddhantas, a book theme to be write by Hindu scholars in late fourth century, early fifth century A. D. , changed Ptolemys trigonometry to the need of the relationship between half of a chord of a circle and half of the angle subtended at he center by the livelong chord (Kennedy, 1969). This came from the basis for the modern trigonometric function known as the sine.The Siddhantas introduction to the sine function is the chief contribution from India and marks a displacement in trigonometry. Indian mathematicians also contributed by creating their own sine table. Arya-Bhata, born in 476, was a great Indian mathematician and astronomer (Ball, 1960). He composed a book called Aryabhathiya, which contained most of the essential ideas we associate with sine and cos. His most outstanding contribution to the topic, which distinguishes him from the other mathematicians of this time, was his work on sine differences (van Brummelen, 2009).His definition of sine was literally half chord and was abbreviated jya or jiva, which exactly meant, chord (Smith 615). Sines were given in minutes, at intervals of 225 minutes. This criterion was not of the sines themselves, but instead, it was the measurement of the differences between the sines. His method of calculating them was as follows. The first sine was equal to 225. The second sine was defined as any particular sine being worked with in order to calculate the sine that directly follows (Clark 29).It was found using the following pattern (225 the previous sine) + (225 + the previous sine) 225 this come in was then subtracted from 225 to obtain the sine table. Second sine 225 225 = 0 225 / 225 = 1 0 + 1= 1 225 1 = 224 Third sine? 225 224 = 1 (225 + 224) / 225 ? 2 225 2 = 222 (van Brummelen, 2009). Arya-Bhata concluded that dividing a quarter of the circumference of a circle (essentially one quadrant of the unit circle) into as many equal parts, with the resulting triangles and quadrilaterals would have, on the radius, the same amount of sines of equal arcs.Doing this, he was able to form a table of natural sines corresponding to the angles in the first quadrant (van Brummelen, 2009). Although much of his work had the right idea, many of Arya-Bhatas calculations were in spotless. Later, in 1150AD, an Indian mathematician known as Bhaskara gave a more accurate method of constructing a table of sines, which considered sines in every degree (van Brummelen, 2009). Although the Indian mathematicians made attempts at creating a table to help with astronomy, their table of sines was not as accurate as that of the Greeks. Muslim mathematicsThe ancient works were translated and developed in the medieval Islamic world by Muslim mathematicians of mostly Iranian and Arab descent, who explained a medium-large number of theorems which freed the subject of trigonometry from reliance upon the come quadrilateral, as was the case in Greek mathematics as a result of the application of Menelaus theorem. In accordance with E. S. Kennedy, it was following that development in Islamic math that the first real trigonometry appeared, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles (Kennedy, 1969).E. S. Kennedy pointed out that whilst it was possible in pre-Islamic math to calculate the magnitudes of a spherical figure, in theory, by use of the table of chords and Menelaus theorem, the application of the theorem to spherical problems was very complex very (Kennedy, 1969). With the aim of observing holy days on the Islamic calendar in which timings were established by phases of the moon, astronomers at first used Menalaus method to lick the place of the moon and stars, although th at method proved to be ungainly and complex.It engaged creation of two cross right triangles by applying Menelaus theorem it was possible to solve one of the 6 sides, however only if the other 5 sides were known. To tell the time from the suns elevation, for example, repeated applications of Menelaus theorem were needed. For medieval Islamic astronomers, there was a clear challenge to find a simpler trigonometric rule (Gingerich, 1986). In the early 9th century, Muhammad ibn Musa al-Khwarizmi c a Persian Mathematician, was an early pioneer in spherical trigonometry and wrote a treatise on the subject creating accurate sine and cosine tables.By the 10th century, in the work of Abu al-Wafa al-Buzjani, another Persian Mathematician established the angle addition formulas, e. g. , sin(a + b), and discovered the sine formula for spherical trigonometry. Abul-Wafa is believed to have helped introduced the concept of the topaz function. He also may have had something to do with the develo pment of secant and cosecant. His trigonometry took on a more systematic form in which he proved theorems for double and half angle formulas. The law of sines, is also attributed to Abul-Wafa, even? hough it was first introduced by Ptolemy. This is in part? due to the fact that Abul-Wafa presented a? unreserved formulation of the law of sines for? spherical triangles, which states pic where A, B, and C are surface angles of the spherical? triangle and a, b, and c are the central angles of the? spherical triangle. In 830, Habash al-Hasib al-Marwazi created the first table of cotangents. Muhammad ibn Jabir al-Harrani al-Battani found the reciprocal functions of secant and cosecant, and created the first table of cosecants for each degree from 1 to 90.By 1151 AD, the ideas of the six trigonometric functions existed, they were just not named as we know them today. Europe It is from the Arabic influence that trigonometry reached Europe. horse opera Europe favored Arabic mathematics ove r Greek geometry. Arabic arithmetic and algebra were on a more elementary level than Greek geometry had been during the time of the Roman Empire. Romans did not display much interest in Greek trigonometry or any facets of Greek math. Therefore, Arabic math appealed to them since it was easier for them to comprehend.Leonardo Fibonacci was one mathematician who became present with trigonometry during his extensive travels in Arab countries. He then presented the knowledge he gained in Practica geometriae in 1220 AD (Gullberg, 1996). The first distinction of trigonometry as a science separate from astronomy is credited to the Persian, Nasir Eddin. He helped to differentiate plane trigonometry and spherical trigonometry. other(a) than that, little development occurred from the time of the 1200s to the 1500s, aside for the developments of the Germans in the late 15th and early sixteenth century.Germany was becoming a prosperous nation at the time and was engaged in much trade. Their in terests also developed in navigation, calendar formation, and astronomy. This interest in astronomy precipitated a general interest and need for trigonometry (Kline, 1972). Included in this movement slightly the time of 1464, the German astronomer and mathematician, Regiomontanus (also known as Iohannes Molitoris) formulated a work known as De Triangulis Omnimodis, a compilation of the trigonometry of that time.When it was in the long run printed in 1533, it became an important medium of spreading the knowledge of trigonometry throughout Europe (Gullberg, 1996). The first book began with fifty propositions on the solutions of triangles using the properties of right triangles. Although the word sine was derived from the Arabs, Regiomontanus read the term in an Arabic manuscript in Vienna and was the first to use it in Europe. The second book began with a proof of the law of sines and then included problems involving how to larn sides, angles, and areas of plane triangles.The third book contained theorems found on Greek spherics before the use of trigonometry, and the fourth was based on spherical trigonometry. In the sixteenth century, Nicholas Copernicus was a revolutionary astronomer who could also be deemed as a trigonometer. He studied law, medicine and astronomy. He completed a treatise, known as De revolutionibus orbium coelestium, the year he died in 1543. This work-contained information on trigonometry and it was similar to that of Regiomontanus, although it is not clear if they were connected or not.While this was a great achievement, Copernicus student, Rheticus, an Indian mathematician, who lived during the years 1514-1576, went further and combined the work of both these men and published a two-volume work, Opus palatinum de triangulus. Trigonometry really began to expand and formalize at this point as the functions with enjoy to arcs of circles were disregarded. Francois Viete who practiced law and spent his leisure time devoted to mathematics also . contributed trigonometry approximately this time. He came to be known as the father of the generalized analytic approach to trigonometry (Boyer, 1991).He thought of trigonometry as? an independent branch of mathematics, and he worked? without direct reference to chords in a circle. He made? tables for all six trigonometric functions for angles to the? nearest minute. Viete was also one of the first to use the? formula for the law of tangents, which states the following pic Viete was one of the first mathematicians to point on analytical trigonometry, the branch of trigonometry which focuses on the relations and properties of the trigonometric functions.This form of trigonometry became more prevalent around the time of 1635 with the work of Roberval and Torricelli. They developed the first sketch of half an arch of a sine curve. This important development assisted in the progression of trigonometry from a computational fury to a functional approach. This formed the basis o f the European contribution of trigonometry. From the influence of oriental scientists, the Europeans focused on the computation of tables and the discovery of functional relations between parts of triangles.Europe developed appropriate symbols, which replaced the verbal rules and public language in which the subject was usually presented. Previously, trigonometry was expressed in lengthy passages of confusing words, but the Europeans introduced such symbols as sin, cos, tan, etc. to simplify the subject and make it more concise. Prior to the analytic approach, the main usage of trigonometry was to measure geometric figures, but the transition of its influence from geometry to calculus began with the discovery of uncounted series representations for the trigonometric functions.Trigonometric series became useful in the theory of astronomy, around the time of the eighteenth century. Since astronomical phenomena are periodic, it was useful to have trigonometric series because they ar e periodic functions as well. The use of trigonometric series was introduced to determine the positions of the planets and interpolation, which is a mathematical functioning that estimates the values of a function at positions between given values (Kline, 1972). Many continued to make contributions to Trigonometry looking for more accurate tables to determine the six functions.These works continued up until the designing of the Scientific Calculator in 1968. In indian lodge today, trigonometry is used in physics to aide in the understanding of space, engineering and chemistry. Within mathematics it is typically seen in mainly in calculus, but also in linear algebra and statistics. Despite the nominal information available on the history of Trigonometry it is still a vital part of mathematics. The storey shows progression from astronomy and geometry and the movement from spherical to plane geometry.Today, Trigonometry is used to understand space, engineering, chemistry as well as mathematics. By exploring the history of trigonometry we see the importance of it in our world. References Boyer, Carl B. (1991), A History of maths (Second ed. ). John Wiley & Sons, Inc. 3 Bressoud, D. M. (2010). historic Refelctions on Teaching Trigonometry. Mathematics Teacher, 104 (2), 106-112. Brummelen, G. V. (2009). The Mathematics of the Heavens and the Earth. Princeton, NJ Princeton University Press. Gingerich, Owen (1986), Islamic astronomy. Scientific American 254 (10) 74.Gullberg, Jan. (1996)Mathematics from the carry Of Numbers. invigorated YorkW. W. Norton and Company, Inc. Joyce, D. E. (n. d. ). History of Trigonometry Outline. Retrieved 3 21, 2012, from History of Trigonometry Outline http//aleph0. clarku. edu/djoyce/ma105/trighist. html Kennedy, E. S. (1969), The History of Trigonometry. 31st Yearbook (National Council of Teachers of Mathematics, Washington DC) (cf. Haq, Syed Nomanul. The Indian and Persian background. pp. 603, in Seyyed Hossein Nasr, Oliver L eaman (1996). History of Islamic Philosophy. Routledge. pp. 5270.Kline, Morris. (1972) Mathematical Thought from Ancient to Modern Times. new-made York Oxford University Press. Kluemper, A. (2010, 3 24). History of Trigonometry. Retrieved 3 5, 2012, from www. xtimeline. com http//www. xtimeline. com/timeline/History-of-Trigonometry Mankiewicz, Richard. (2001)The Story of Mathematics. New JersyPrincetion University Press. Maor, E. (1998). Trigonometric Delights. New Jersey Princeton University Press. Miller, S. (2001). Understanding Transformations of biweekly Functions through Art. Mathematics Teacher , 94 (8), 632-635.Moussa, Ali (2011), Mathematical Methods in Abu al-Wafas Almagest and the Qibla Determinations. Arabic attainments and Philosophy. Cambridge University Press. 21 (1) 156. Needham, Joseph (1986), Science and Civilization in China Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei Caves Books, Ltd. Rogers, L. (n. d. ). The History of Trigonom etry- Part 1. Retrieved 3 1, 2012, from Enriching Mathematics http//nrich. maths. org/6843/index Suzuki, J. (2009). Mathematics in Historical Context. Washington D. C. The Mathematical Association of America.Smith, D. E. (1958)History of Mathematics. New Yorkcapital of Delaware Publications, Inc. Toomer, G. J. (1998), Ptolemys Almagest, Princeton University Press. Weber, K. (2005). Students Understanding of Trigonometric Functions. Mathematics Education inquiry Journal , 17 (3), 91-112. www. cartage. org. (n. d. ). Trigonometry History. Retrieved 3 5, 2012, from Trigonometry History http//www. cartage. org. lb/en/themes/sciences/Mathematics/Trigonometry/history/History%20. html van Brummelen, G. (2009)The Mathematics of the Heavens and Earth. Princeton University Press. Princeton and Oxford.