Meeting Criteria for Early Years Foundation Standards

Meeting Criteria for Early Years Foundation Standards Unit 3 Early years foundation standards are guidelines with in which a childcare facility work, members of staff must be checked by â€Å"Disclosure and barring service†, complete a Health declaration, least one member of staff needs a paediatric first aid certificate and should obtain a level 3 childcare qualification before they start caring for children, one member of staff must have suitable experience with under 2s, at least half of all other staff must hold a level 2 qualification in child care. The children’s health and safety is paramount. Child care facilities must work within the correct adult to child ratio, only permitting the correct number of children for the space they have available. Children under 2 1 adult -3 children Children aged 2 1 adult – 4 children Children aged 3+ 1 adult – 8 children Good. Within the facility provisions must be made for the development of every child, no child is excluded, every child must be supported in fulfilling their maximum potential, the early years stages have a long and lasting effect on the child’s future and ability to start school, no child should be left behind. Each child will be assessed so all of their needs can be met on an individual level and they have a personal plan in place so they succeed in all areas. The plan set in place should be available to parents, cares and any outside practitioner such as health visitors if needed. Every child should have equal opportunity, face no discriminatory actions and every child must be included and fully supported in their learning environment. Practitioners must remember every child is unique, they need to forge positive relationships, offer enabling environments, and plan varied activities as each child learns in a different way and at different rate , every child will have a key person, this person is the point of contact with the child’s parent or carer, they are responsible for helping the child be happy and safe, they are responsible for that child’s care, development and learning, they keep clear notes of the child’s progress and help parents with ideas for their child’s progress at home.( www.gov.uk//policies//early-years-foundation-stage) There are seven main criteria for the early year’s foundation the three prime areas are: Communication and language Physical development Personal, social and emotional development. Then there are four specific areas: Literacy Mathematics Understanding the world Expressive art and design. These core guides must be implemented into children’s everyday environment, communication and language: Children need to talk freely to one another and adults alike, roll play encourages children to interact, asking the children to describe things being big or small, talk about family members being male and female, young or old, encourage them to use descriptive language, and help them understand how it relates to them. Physical development: These activities develop the child’s skills set in handling equipment or showing control and co-ordination this is done with games or dance that involve pushing or pulling or outside games that use ball skills such as patting or kicking, throwing and catching, all of these skills help develop the child’s body and hand and eye co-ordination. Personal, social and emotional development: These skills are the ones a child needs to be confident, to talk and play with other children, to talk about themselves and their own environmen t, to talk about who they are. Doing an activity as simple as â€Å"show and tell† would give the child confidence to talk about something they love or something they have done, if you have children from many different cultures talking about their festivals and traditions encourages understanding of others. Literacy: Children love to be read to so try to use a story that is repetitive, giving the children confidence to join in, or ask the children to make up an alternative ending to a favourite story. Giving each child a word card and going round the room and using those words to build a story would help a child understand the context of the word and phonic sounds to build a word. Mathematics: Children enjoy filling and emptying bottles and containers using sand or water, doing so helps to understand, volume and size, building blocks can help them count, how many blocks can you build up before they collapse. Understanding the world: Children need to know where and how they fi t in, looking at traditions and festivals across all cultures encourages an understanding of self, making festival food , or making a garland worn at a wedding ceremony, helps the children understand the importance of different cultures, they also need to understand the natural world so going on a bug hunt, looking at the life cycle of a butterfly or creating a garden full of food for insects, this would help them understand the things they have seen and why there important to us as humans. Expressive art and design: Children like exploring and using many different materials, clay, play dough, card and paper, they can experiment with how to change their shape and form, use glue or tape and colour. Children like to be imaginative model making is a great way to encourage imagination and develop skills, mixing paint in to a variety of colours, so they can see that you only need primary colours to make any colour you wish. Respecting and valuing individuality Children and families must feel respect for who they are regardless of colour, race or sexual orientation, it’s the practitioners’ job to lead by example the children must understand that everybody is different, everybody has strengths and weaknesses, and everybody is valued. Encouraging manners, caring attitude and understanding will help children become well-rounded young adults. Children should be encouraged to celebrate their cultural differences, and to respect others family backgrounds, this needs to be done in a sharing way, having a world day were the children can look at different countries, the animals living there, the food, maybe ask a parent to ready a native story, it would bring to life the differences giving them greater understanding of where they fit in and how we all fit together. It is important for children not only to share their cultural differences but also their personal differences, some children are good at art some are good at swimming, helpi ng them celebrate their achievements helps children see that everybody is good at something but not all good at the same thing. Doing group activities helps the child learn to work as a team, encourage problem solving, help the children listen and negotiate with one another, or take time at lunch to sit together, talk politely, understanding that eating with the right cutlery and washing your hands before you sit down are all normal behaviour. Activity: Outcome: Awards, celebrate the children’s achievements in and out of care setting Reinforce positive self-image Make a class photo album, ask children to bring photos of family celebrations, and explore the different festivals across the cultures. Bring to life the different festivals and celebrations that happen around the world. Encourage understanding of other peoples culture Give children positive role models such as gold medal paralympians, put posters up around the room for the children to admire Let children understand that being disabled is not a negative and great achievements can still be made Include toys and play equipment that reflect other cultures, such as dressing up clothes, kitchen utensils in the play kitchen, puzzles and dolls By playing with these objects the children will become familiar with different cultures Cook food from around the world. Let the children make and taste different foods that they might not have everyday This activity would stimulate them and get them talking about feeling, texture, smell and taste. Cooking activities also help with measuring and science, and help children to understand health and safety and good hygiene. Sing songs and read stories from around the world, include rhymes and action songs Children love to participate; this activity is great for children whose first language is not English. Positive and negative behaviour Positive reinforcement is a technique used by care givers to modify behaviour, this involves giving positive reinforcement as often as possible, and reminding the children that negative behaviour will have consequences as a carer it is best to focus on the positive but if a negative occurs it needs to be dealt with swiftly and an explanation as to why it is negative behaviour, help the child to develop empathy, the child needs to know it is their behaviour you disapprove of not them as a person, if a child is consistently reprimanded for negative behaviour they will quickly become labelled as â€Å"the naughty child†, an action plan must be made for this child so that positive reinforcement can turn the situation around and positive behaviour becomes the norm. At no time should there be: pain, punishment, intimidation, yelling, degradation, humiliation, shame or guilt this would only hurt and confuse the child. Children need a positive environment to develop their self-esteem, emotional growth and well-being. (www.positivereinforcementforkids.com) Behaviour is not learned once but learned every day. Consistency is the key to dealing with any behavioural situation, children respond to set boundaries and feel safe knowing what is expected of them, let the children know what kind of behaviour you are looking for, with older children they could help set the consequences of their negative behaviour; it would encourage them to be more positive as they have set the rules. As the care giver you need to remain positive at all times it is your professional duty to do so, be a positive care giver, a negative attitude leads to a negative outcome! Consistency is key. Boundaries are needed to establish right from wrong. Children understand the their behaviour has consequences- both positive and negative Reward positive behaviour: give out stickers, or mark with a kind word â€Å"Thank you for being kind and helpful†, â€Å"Your team work was great†. Make the children feel valued and important. Star chart: so the children know that with consistent positive behaviour also comes reward. Praise the child for the effort not just their achievements, praise them for their strengths and remind them everyone is different. If you praise one child use the opportunity to encourage the rest of the group. Managing conflict We all encounter conflict every day, in the childcare environment it’s child to child conflict or child to adult conflict, study’s done at Texas tech uni by Dennis, Colwell and Lindsey show that girls often have child to child conflict that is more often than not resolved within their peer group, whereas boys often have child to adult conflict and that boys look to the adult to resolve the conflict.(www.kon.org/urc/dennis) As the practitioner it’s you job to manage conflict, like many life lessons children need to be equipped to deal with it, Vygotsky: a Russian social development theorist said â€Å" conflict provides a learning experience for children and in doing so they would learn to function better in the social context†, (www.simplypsychology.org) Vygotsky’s theory is not that uncommon amongst child behavioural theorist, his arguments are supported by the likes of Erikson who thought â€Å" life is full of conflict and in order to become a b etter person one must resolve conflict in each stage of life†. Often conflict is cause because of the developmental stage the child is at and they as children have not reached the stage where they have empathy or understanding, or sometimes children act out because it has become their learned behaviour, it is the way they have been treated or something they have seen in their everyday lives due to lack of good role models. As the care giver you need to step back and ask: Why is the child acting this way? What has made the child feel like this? Is it a cry for help? Does he/she need my understanding and empathy towards their feelings? Are the Childs need being met, are they unhappy, scared, confused or frustrated? Are they tired or hungry? Conflict is often born out of a misunderstanding, if identified quickly it can be resolved quickly, children need to feel that their side of the story is heard, as the care giver it is up to you to guide the situation from one of upset and stress on to a calmer and more positive footing, with a peaceful conflict resolution. Bibliography www.gov.uk//policies//early-years-foundation-stage http://www.positivereinforcementforkids.com/ www.simplypsychology.org www.kon.org/urc/dennis

Morality and Laws in The Trial and Death of Socrates Essay -- Morals S

Morality and Laws in The Trial and Death of Socrates Upon reading Plato, The Trial and Death of Socrates, Socrates strongly held views on the relationship between morality and laws become apparent to the reader. Equally, Socrates makes clear why laws should be followed and why disobedience to the law is rarely justified. Finally, he makes clear his views regarding civil disobedience. Socrates’ view on morality is that anyone can do wrong. It is said that injuring someone in return for injury to oneself is wrong. He follows this with the connection between morality and the city. You do badly without the cities authorization; you are doing wrong towards the city and the laws. He felt if you are behaving against peoples mind and in this way, behaving against the city. It is a way of destroying the cities laws and so you are hurting citizens by doing so. An example of this is the general understanding that you shouldn’t hurt your father. If you do so than you are disrespecting laws within your city. Of course you will get convicted for this, and it doesn’t change the idea that you acted against the city. Socrates follows by explaining what is taught to each citizen. You are told that you were born with certain laws. Your father and mother brought you to the world in which they live and thus you should respect and obey by their rules. The laws were already there. That means, that your mother and father are as important as the city and you should respect the city as so. Socrates describes the city and its laws more preciously. You don’t have the same rights as your parents. They educated you and thought you the rules in the city that you should follow. They taught you which behavior is right and which is wrong. It is immoral to treat your parents the way they treat you. You don’t have the rights to treat your parents the same way they treat you. That means that parents have a higher position in life than their children. It doesn’t really matter what they did to them. It is not moral to return the same behavior. He follows with the relationship of a person to the city. He says, â€Å"One must obey the commands of one’s city and country, or persuade it as to the nature of justice. It is impious to bring violence to bear against your mother or father, it is much more so to use it against your country.† (Crito 51c) One should not only obey their father and mother... ...iritual things?†(Apology 24c). That shows that one should believe in gods, just because the city does so. Socrates always searches for more knowledge about the city and its laws. We see that Euthyphro and him do not accept everything that is said by the city. On Socrates’ point of view either the disobedience to the law or to the civil disobedience can be justified. To justify it correctly you have to be able to argue and find reasons for every given rule. Not just believe in what others say. My personal opinion is that it is wrong to do be an object of the rules someone told you when you were born. Everyone has the right to argue about the city and laws or with his father or mother. I totally agree that Socrates found it important to research about life’s morality and not just think the same way others do. That is a way of proving the knowledge of men. Ones sitting quiet in the corner usually have more knowledge than others that talk so much about what they know. Many men with a high position in life do not always have the most knowledge. Do not believe in anyone theories without questioning it. It is important to prove that someone has the knowledge of what they speak.

Narcolepsy Essay -- Sleep Disorders Sleeping Papers

Narcolepsy Narcolepsy is a permanent sleep disorder that gets worse with age. It is estimated that about 40,000 to 135,000 people in the United States have narcolepsy. (Narcolepsy, 2002). Most people who have narcolepsy are often times misdiagnosed because the symptoms, especially if noticed at the beginning stages, are very similar to other sleep disorders. Some of the symptoms that go along with narcolepsy are excessive sleepiness, especially during the daytime or when sleep is inappropriate, cataplexy which is when a person suddenly looses control of their muscles, sleep paralysis which is when the person is unable to move for a short period of time after they wake up from sleep, and hypnagogic hallucinations which is very vivid dreams, often experienced minutes after falling asleep. (Wunder 2002). For a person with narcolepsy, the message about when to sleep and be awake often arrive at the brain at the wrong time or wrong place, causing them to fall asleep while their emotions are high or even during physical activities. According to Margan (2003), narcoleptic patients have a shortage of neurons which produce excitatory neurotransmitters such as hypocretin and orexin. With less of these neurotransmitters, the signals for when to sleep and when to be awake is sent at the wrong times. Scientists have found that narcoleptic patients lack a chemical in the brain called hypocretin. Hypocretin helps to regulate sleep and stimulates arousal. It was also discovered that the number of neurons which secrete hypocretin is also reduced. (Wunder 2002). It is uncertain why these are missing. Treatment for narcolepsy is limited, but there are possible medications to help limit the effects of this disorde... ...ness Associated With Narcolepsy. American Family Physician. pp. 973. Ault, Alicia. (1999, January). FDA Approves First New Narcolepsy Drug in USA in Decades. Lancet. pp. 131. Morgan, K. (2003, March). No Rest for the Waking. Science News, 163, 181. O'Brien, James C. (2004). Facts about narcolepsy – frequently asked questions. Retrieved April 2, 2004, from http://www.talkaboutsleep.com/sleepdisorders/ Narcolepsy_FAQS.htm Seydel, C. (2000, August). Wake Up Call for Narcoleptics. Science Now, 2. Takahashi, Joseph. (1999, September). Narcolepsy Genes Wakes Up the Sleep Field. Science. pp. 2076. Travis, J. (2000, September). Brain-Cell Loss Found In Narcolepsy. Science News, 158, 148. Wunder, Heidi T. (2002). Living with narcolepsy. Retrieved April 2, 2004, from http://www.sleepfoundation.org/publications/livingnarcolepsy.cfm

The dictionary definition

The dictionary definition of a linguistic communication is stated as a system of communicating consisting of sounds, words and grammar, or the system of communicating used by the people of a peculiar state or profession in Cambridge Advanced Learner ‘s Dictionary ( 2nd edition, 2005 ) . And that means, larning a linguistic communication to pass on decently in that linguistic communication requires get the hanging all parts of a linguistic communication system such the vocabulary, the grammar and the sound system. Harmonizing to Dawani ( 2006 ) in order to pass on in a linguistic communication one should analyze a balance of four basic linguistic communication accomplishments which are reading, listening, composing, and speech production. That ‘s why instruction and rating of these accomplishments have ever been peculiarly of import. In this paper, the rating of one of the four basic linguistic communication accomplishments, viz. â€Å" Reading Skill † will be discussed in item. However it is indispensable to put some basic facts about â€Å" reading † before get downing to discourse its rating. Reading in a different linguistic communication has traditionally stemmed from the demand of accessing the written literature of a higher civilization. Subsequently with the visual aspect of communicative attack, alternatively of high civilization literature, reading texts started to be chosen harmonizing to the reading purposes like utilizing mundane stuffs such as newspaper articles, coach agendas, etc. , for intents to develop communicative competency. As a consequence, instruction of reading and reading drills at any degrees became a cardinal portion of linguistic communication instruction. As a definition, reading accomplishment is called an expertness which turns composing into significance and enables the eloquence and comprehension of the proprietor. Agring that, Rasinski ( 2004 ) provinces, accurate and automatic decryption of the words by giving expressive readings of it to accomplish maximal comprehension means reading eloquence. Consequently, reading does non simply average accurate decryption of the words like instructors thought for old ages. And that ‘s why the appraisal of reading should affect the ability of eloquence and comprehension, for certain. From another point of position, harmonizing to Stages of Reading Development of Chall ‘s ( 1983 ) Reading is a accomplishment which develops through some phases and a uninterrupted procedure. Agring that, Beers ( 2006 ) adds that in instance of a skip in a reading development phase, pupils will fight in their reading ability and finally this will besides impact their authorship accomplishments. That ‘s why instructors should be entirely attentive to pupils ‘ comprehension of each phase of reading procedure before go oning with the following degree. Harmonizing to Carnine & A ; Silbert ( 2004 ) , â€Å" Appraisals supply the information that will enable pedagogues to supply kids with direction that starts where they are and so construct on that base to assist kids progress to the higher levels. † ( p. 13-15 ) As it is evident from all these research workers ‘ statements, the rating of reading is a important portion of learning a linguistic communication. Although most of the researches of that field unite in the importance of rating, they differ in the sentiments for the rating mediums. No affair how much a peculiar rating type is praised by pedagogues, in world all of them has some advantages and disadvantages and usage of these rating types is merely a affair of penchant. Hughes ( 2007 ) states that, all appraisals are fundamentally designed to develop the acquisition of pupils. As it is stated in Hughes ( 2007 ) , appraisal is the procedure of:Gathering information from a assortment of beginningsSupplying pupils with descriptive feedback for future developmentMiscue AnalysisIn this type of appraisal, the mistakes a kid makes during the reading procedure are diagnosed. Using this method, a instructor can be more easy cognizant of the pupils ‘ comprehension by looking at the errors he/she makes.The 6 Types of MiscuesCorrection: A child self-corrects his/her mistake and re-reads the section/word without motivating. Interpolation: A kid adds a few words which are non on the page. Omission: A kid omits a word while reading. Repeat: A kid repeats a word or part of the text. Reversal: A kid will change by reversal the order of the text or the word. Substitution: A kid inserts a different word, alternatively of reading a specific word. Some regulations about utilizing miscue analysis:Use unfamiliar text, non something the kid knows from memory.Do non utilize miscue analysis on novice readersGive the pupil some pick in the reading choice.You will necessitate a quiet topographic point without breaks, it can be really ready to hand to enter the kid which provides you with an chance to listen to the transition more than one time.Photocopy the choice the pupil will read, utilize this to enter the miscues.Record each miscue. ( Use hypens for skipped words, record each permutation ( Internet Explorer, went for when ) , use ^ for interpolation and record the word ( s ) , circle omitted words, underline repeated words, you may besides desire to utilize // for repeated words.What do the miscues tell you?Correction:This is good! We want readers to self-correct. However is the reader reading excessively fast? Is the reader mis-correcting accurate reading? If so, the reader frequently does n't see himself as a ‘good †˜ reader.InterpolationDoes the inserted word detract from intending? If non, it may merely intend the reader is doing sense but besides inserts. The reader may besides be reading excessively fast. If the interpolation is something like utilizing finished for coating, this should be addressed.Omission:When words are omitted, it may intend weaker ocular trailing. Determine if the significance of the transition is affected or non. If non, skips can besides be the consequence of non concentrating or reading excessively fast. It may besides intend the sight vocabulary is weaker.RepeatTonss of repeat may intend that the text degree is excessively hard. Sometimes readers repeat when they ‘re unsure and will reiterate the word ( s ) to do sense of the transition.Reversal:Watch for altered significance. Many reversals happen with immature readers with high frequence words – of for for etc.Substitutions:Sometimes a kid will utilize a permutation because they do n't understand th e word being read. Does the permutation make sense in the transition, is it a logical permutation?In SummaryUsing miscue analysis is an of import diagnostic tool that should be done every 6-8 hebdomads to see how the reader is bettering in the schemes used. Making sense of the miscues will assist you with following stairss to better the kid ‘s reading. It is worthwhile to hold a few inquiries prepared that Lashkar-e-Taiba you know about the kid ‘s comprehension of the transition read as miscue analysis tends to trust on reding you of the schemes used. Miscue analysis may look clip devouring ab initio, nevertheless, the more you do, the easier the procedure gets.

Fostering Youth to Become Champions of Tomorrow

Name of Essay: Fostering Nepali Youth to become Champions of Reform:  Designing a Better Future for the Country Name of Author: Sushil Kumar Mahato. Department of Pharmacy, Bahauddin Zakaria university, Multan, Pakistan. Age: 21 years Contact : [email  protected] com Mobile number: 00923326019248 I have a special dream for the country. I have a dream of New Nepal in which every citizen owes his own home as I do . Every citizen has access to basic needs of food and clothing.Every citizen is provided with opportunity of job which fits him well as per his qualification. Every citizen has access to facilities of healthcare and education for his family as I do . Every citizen has equal access for various opportunities without any discrimination based on sex, race ,religion ,regional variations ,political influences and economic status of an individual. These needs and aspirations of Nepalese citizens must be addressed by the constitution of New Nepal. I have dreamt of it today and to fulfill the dream of New Nepal is the aim of my life. Dare to dream Youth.How can Nepal line up with the developing countries when half of its population comprising of youth is facing hard times struggling to meet the needs of his family in the gulf countries? How can Nepalese society lead to peace and prosperous if majority of young children are deprived of proper education due to poverty, illiteracy and lack of awareness? How can Nepalese society march ahead in path of progress when large population of female youth is living their life in a strong feeling of hatred, insecurity and fear in their own home? The 21st century is the generation which says no discrimination based on gender .The world has progressed a lot in terms of gender development and emphasize on gender equity. Female are as competent as males in every sector of life but still a great number of Nepalese young women is facing gender discrimination in their own homes. As majority of rural society is facing gender cri ses, the agony of most rural young girls can be heard in such voices, Am I born as a Jewel and pearl, Then, why do my parents hide me from the world, My brother goes to school and is well fed, I word hard even have to struggle for food,But the fate of jewels is to give shine to all, Then why do my parents make me work in the fields all the day long, With mud and in the hot sun, Why cannot they distinguish my sweats and tears, Do I have to live all my life with fear? It’s my great aspiration to fight against all these circumstances and lead a country toward prosperity but as 21 years old I know my limits. However, knowing my limits doesn’t mean that I can’t try to dream about changing the situation . We are the present Youth of our nation and we have the power to change the country more than any government.If this happen the change along with the impact would be massive. Earlier the question used to be, ’what can the country do for youth? But at present ag e the question should be turned into ‘what youth can do for the country? ’ Presently Nepal is undergoing through internal conflicts, which has been emerged from the unrealized rights, demands and aspirations of citizens. Constitutions drafting has been halted and the constituent assembly had gone unnatural death as it was dissolved without drafting constitution on 28th May 2012.About 25 percent of the members of constituent assembly were among youth categories but they could not play better roles in constitutional drafting process They were just used as the vote bank by the political parties. Imbalances of power, sense of ideological supremacies, identity politics and absence of strong law enforcing mechanism has led the nation to internal conflict . In these situation the influences of foreign interference has been increased in last few months especially from neighbours of Nepal i. e India and China.Nepalese youth are responsible for all these critical situations direc tly or indirectly as they are largely involved in politics. Youth should be able to convince their leaders and entire Nepalese community for the right options and suitable role models for the issues such as state reconstructing which is hanging out . One of the most easiest and most powerful ways for youth population to participate in important decisions and issues is by exercising their voting rights . Youth must promise today that they will not select any corrupt, killer, dishonest leader to represent the people for drafting the new constitution.An important role of every young generation is to vote for those who think is right to lead the country toward a better future. The youth have the benefit of being able to learn from past mistakes, therefore, the political choices – and voices – of every generation should become smarter, sharper and more critical. In theory this will lead us towards a more equal and democratic world. The problem facing by youth is the lack of technical education. If youth is trained and educated in the right manner, they will play a very constructive role in nation building.They will help the government and state and private institutions in implementation of national policies. Whereas , if Youth is not in the right direction and is unconcerned about the future of the Nation, it will just be a burden for the nation as a whole and will play no productive role. Properly educated and given right opportunities they can play a significant part in lifting themselves, their families and communities out of poverty. Currently Nepalese Youth are facing several problems. The most critical problem being faced by our youth is unemployment.Unemployment is a multi-dimensional and complex issue which starts a vicious circle of associated problems like involvement of youth in politics, bank and household burglaries, social insecurity, lawlessness, use of drugs etc. Youth are the building blocks of nation. Employment sector should be crea ted as per need of youth. The situation of Nepal is quite different now. In history of the country the good news is that a few year ago new ministry called Youth and sports ministry has been formed to address the problems facing by the youths.A few years ago the then government has launched self employment fund for the needy people and various skill training were given to youth to generate their own jobs, but it could not be implemented effectively and little or no progress was achieved. The youth community should unite together for the improvement of employment sector and also they should press the government to make strategies to develop industrial sector. Also Proper agreement should be made between the government and foreign employment providers to reduce and end the violence Nepalese youth are facing aboard in employment.A number of civil service reform commissions have been constituted and recommendations made for strengthening Nepal Civil Service. Despite five decades of expe riments with democratic practices, the behavior and mind-set of politicians and bureaucrats have not changed significantly to introduce civil service reforms on a result-oriented basis. Youth have incredible energies, talents and unquestionable love for their country. The passion of our youth in the time of disaster like flood, earthquake is the best examples of unity and patriotism for the country .A separate civil service for youth should be established and youth should be emphasized to involve in it. Youth can bring major changes in the efficiency of the services provided to the public in quick and reliable way. Another serious drawback which is hindering development process is the Corruption. Youth should be actively participation in the developmental activities to correct the irresponsible behavior shown by the authority. Questions such as,’ How much money is being spent and how is it being spent? Is that road being built correctly and transparently?Is that community of Muslims being treated fairly? How can the Government of Nepal in Kathmandu help the rural community? ’ can be aroused which would largely control the corruption at local level. Youth today have the perfect opportunity to emulate his approach, especially as the country now moves towards state restructuring. There can be more opportunities to engage in local development as NGO workers and in local Political body in DDCs and VDCs, which although have atrophied over the past two decades are the country’s great hope.That’s where Nepali youth can play a great role in decision making and seek accountability. Poverty is another major problem of our nation. Today youth must promise today that they will not let any poor people to die from hunger and will try to find the needy people and help them as per one’s capability. The youth hopes for a world free of poverty, unemployment, inequality and exploitation of man by man, a world free of discrimination on the ground s of race, language and gender, a world full of creative challenges and opportunities to conquer them.Youth must promise today that they will try to help each other and will clean their mind from any kind of prejudice, bias, hatred and will treat all others as equal to them. Further they must make a commitment today that they will help the society for the promotion of education, honest and conscious. Let’s convert these hopes into reality. Our youth in the grim situation finds no other consolation resort to the use of drugs. Growing use of narcotics is becoming a major problem of our youth. Such practices should be strongly discouraged through youth campaigns.Youth are the hopes of the country for tomorrow. To contribute in nation building they should study hard now so that they could become a skilled person in the future and can participate in the developmental works of the country. The youth can learn a lot from past mistakes and with time will become more intelligent and m ore prepared. Last but not least, the role of media in upbringing our youth has been minimal throughout the history of Nepal. Most of the youth find our Tvprogrammes non-entertaining and unattractive.The influences of western fashion and lifestyles have influenced the Nepalese society to a large extent. As a consequence of cultural invasion, our youth’s role in promoting our country has been dismal. Youth of Nepal should strongly discourage such practices through youth compaings and media can play an important role in this regard. The youth of today can do a marvellous job for the nation. With all the modern means of this computer age, where everything seems to be possible, why can’t the youth override the things done by the people in the past?We always consider the heroes of the past to be the evergreen characters. But today the youth with the proper use of the modern facilities should be evergreen characters for the times to come. Nepal can become a developed nation only if youth contribute to the best of his or her capacity and ability. Youth is wholly experimental and with the full utilization of their talents they will certainly be the champions of reforms designing a better future of tomorrow and Nepal will become a complete nation. Today there is a hope, we have opportunity. et’s give clear message to youth community, ‘Let’s involve together. We are strong’. Let me end my writing here with an inspiring quote said by the great leader Mahatma Gandhi,’ You may never know what results come of your action, but if u do nothing there will be no results’ . I am confident, with the active participation of youth the country will emerge stronger, more stable and ultimately more prosperous in the history of Nepal and the Youth of Nepal will be the example for the developing world. References: 1. www. usaid. ov. com. article titled,’ Nepali Youth and the Political Process – From the â€Å"Insideâ €  as well as â€Å"Outside† 2. www. usaid. com. article titled,’ Optimistic about the Role of Youth in Nepal’ 3. www. weitz-centre. org. Article/handbook. Youth participation in development:strategies and best practices. 4. A debate compiled by me on the topic,’Supporting women to join law enforcement and judicial system’organised by Women social organization Multan Pakistan dated 20th,April 2012 under gender equity program by support of USAIDS.A copy of proof of my participation is attached with this. 5. www. moys. gov. np. articletitled,’National Youth policy 2010’. Ministry of Youth and sports. 6. Eassy compiled by me as called by National Human right commission. titled,’Justice and dignity for all of us’. A written appreciation letter for my work is attached with this. Attached Alongwith: 1. A copy of my citizenship. 2. Appreciation letter (as mentioned in the reference. ) 3. certificate of participation in debat e(as mentioned in the reference)

Effective communication unit 1 D2 Essay

(D2) – Whist taking part in both group interactions and one to one interactions there are many factors that influence the effectiveness of each. This essay will therefore evaluate the factors in which I came across which were an influence to my effectiveness in the interactions. Firstly in the one to one interaction I feel the factors that came across which made my interaction effective was my speech as I feel that I used a clear pronunciations to the service user which allowed her to understand me better so that they weren’t confused with the words I spoke. I also took my time when speaking to her so that she didn’t feel I was rushing her, I did this in order to try and make her open up slightly as she would know I am there to help and spend time focusing on her and her needs so that she didn’t feel that I had better things to do with my time and that I didn’t want to be with her. Also this may make her gain a level of trust with me during the duration which will make her feel like she has someone she can rely on is not alone. I also feel that in my interaction making myself aware of her needs/situation before hand made it easier for me to understand why she was in that state as a service user in the first place and so therefore I didn’t have to recap over the bad parts of her life as that may of got her upset and made her not want to talk even more. This also helped me as I was able to think of advice beforehand so it could be better thought out advice which would hopefully be helpful to her and if she felt the advice was good then this may make her feel that I actually can help her and that there is hope in overcoming her situation. Whilst talking to her I showed my full interests into the communication and interpersonal skills that were taking place and I did this by looking interested and being focused on just her and not having any distractions around me. This would hopefully have made her feel like I am not interested in something else as that may have made her feel that I am not there to help. I felt by doing this it was a strength as it made her open up slightly due to acting as though I want to hear what she has to say and giving her the time to say it and that I was defiantly listening to her. Throughout the period of talking to her I showed assertiveness which made me portray that I was positive and confident. This hopefully then made her feel like I am not going to give up on her and that  would make her have hope in herself that there is a positive outcome. This then would hopefully have made her open up and talk due to wanting to give me information which I can then help her with in order to get her back on track and to have the positive outcome she wants. She spoke about her past in the interaction and about how her mother and father were in prison, at this point she showed a sense of embarrassment and disappointment in them and so therefore I portrayed an appropriate attitude by not judging what is being said to me and saying that this is all in confidence. So that she felt that there is nothing to be ashamed of. When speaking to her I used reflective listening as this would make her understand that I have taken into account what she has been telling me due to me repeating it in a different way, this also lets her know that I understand what she is trying to tell me and will get her to speak more in depth so that I understand exactly what she wants to happen and how she feels. When we were talking I was sitting upright in order to show that I was alert and listening as I would need to look alert in order for the service user to feel that I am aware of what they are saying and that I am bothered about the conversation. I feel that weaknesses to my one to one interaction was that I repeated myself slightly in some conversations due to her body language looking as though she wasn’t listening and this made her feel frustrated as her tone changed acting as though I was patronizing her and that she heard it the first time and wasn’t stupid, to improve this I would therefore give her time to answer what was said and not repeat myself even if she looks like she isn’t listening, if she still doesn’t answer after some time I will repeat the question again although word it in a different way so that it is not the same. I also feel I could of improved on a part of the interaction as I entered a sensitive subject quite early on in the interaction about the death of her auntie which seemed to have put her in an upset frame of mind as I feel that she hadn’t got comfortable enough with me at that time of the conversation to talking about a close subject so I feel that this was a weakness to the interaction, although I brought it back to being effective by showing her the support networks available to her which put her mind at rest and made her feel like she wasn’t alone. In the group interaction I feel the factors that came across which made my interaction effective was the way in which I spoke out in the group to make my point clear to the  other professionals, this also broke the barrier of silence between one another and this brought the others to speak out and share what they feel would be best so I felt that I was an influence on the group here. As a group I feel are strength was that we had good eye contact with one another when speaking which made us realise that we were all listening and interacting so therefore it kept the conversation going. We expressed our thought on things that we felt were negative and explained why so that we understood why we shouldn’t do certain things in certain ways. I feel that I had good use of body language in the group as I sat up straight when speaking and used hand gestures to bring the others to understand more about what I was explaining. Also I feel that my use of nodding my head to indicate that I agree with what people were saying was a good effective interaction as then people in the group knew that I was keeping on track with what they were saying and understood. Also the way in which we worked effective as a group was the way in which we were all aware of the needs and preferences of Mrs.Signh so that we understood about her past and therefore could go straight into what we are going to do with her. I also feel that I made the communication and interpersonal skills effective in the group due to asking questions in order to verify things so that I understood them correctly so that no faults in the system occurred when the meeting was over, as if we go the wrong idea about something then Mrs.Signh’s health may be at risk. In this meeting jargon was used due to it being with 3 professionals in the same profession of health care so they therefore understood the jargon used in the conversation as it was medical terminology. This helped us understand certain aspects of her health in more depth. I also feel that an effective way of communication in the meeting was that we respected each other’s opinions on Mrs.Signh and then came to a conclusion as a group with the final decision, also if we did not feel that someone else’s thoughts were very good we did not put them down we just suggested how we could improve them. I also feel in the meeting that I had a good clear voice with a calm tone so that the meeting stayed at a calm level and stopped it from becoming an intense discussion when disagreements on the care plan occurred, as an intense discussion could of lead to an argument and that would not have been professional. This also helped the other people in the group understand me clearly so that there was no miscommunication or  misinterpretations as these could of lead to the wrong inappropriate care plan which may not have benefitted Mrs. Singh. Ways to improve the meeting as a group in order to gain more effective communication and interpersonal skills may be to not rush the meeting as much as we did and to make it go on for longer stating more information in it. Also we could have introduced ourselves and our care profession at the start of the meeting and shook one another’s hand showing courtesy and politeness. Other ways which we could of improved it is by not having our coats on and so that we are more comfortable and can move about easily for use of good body language. Another way which we could have improved it in our group would be by sitting in more of a closed circle so that we can hear one another efficiently and can see each other clearly in order to see body language and facial expressions. This will help make communication effective and avoid misinterpretations so that no mistakes in the meeting are made. I feel that I could of improved the way that I acted with the group as I feel that I took a direct approach which made me out to be the main person/leader of the group who was in charge due to be speaking up and setting up the topic for discussion. By speaking up in the group with my opinions this may have made the others in the group shy and they may not feel as confident in expressing their opinions. Therefore I would improve this by making it all equal between one another so that others feel that they are on the same level and can speak up and say what they would like to discuss. References: – Robinson, L Segal, J . (2012). Effective Communication. Available: http://www.helpguide.org/mental/effective_communication_skills.htm. Last accessed 17th October 2012. – Skills you need . (2011-2012). Interpersonal Communication Skills . Available: http://www.skillsyouneed.co.uk/IPS/Interpersonal_Communication.html. Last accessed 17th October . – Stretch, B Whitehouse, M (2010). Health and Social Care Level 3 Book 1. England and Wales: Pearson Limited Education – Collins Education . (2011). Developing effective communication in health and social care. Available: http://www.collinseducation.com/resources/hscbtecnationalch01.pdf. Last accessed 17th October 2012.

Babylonian Mathematics Essay

1 Introduction Our first knowledge of mankind’s use of mathematics comes from the Egyptians and Babylonians. Both civilizations developed mathematics that was similar in scope but different in particulars. There can be no denying the fact that the totality of their mathematics was profoundly elementary2 , but their astronomy of later times did achieve a level comparable to the Greeks. Assyria 2 Basic Facts The Babylonian civilization has its roots dating to 4000BCE with the Sumerians in Mesopotamia. Yet little is known about the Sumerians. Sumer was first settled between 4500 and 4000 BC by a non-Semitic 1  °2002, c 2 Neugebauer, G. Donald Allen 1951 Babylonian Mathematics 2 people who did not speak the Sumerian language. These people now are called Ubaidians, for the village Al-Ubaid, where their remains were first uncovered. Even less is known about their mathematics. Of the little that is known, the Sumerians of the Mesopotamian valley built homes and temples and decorated them with artistic pottery and mosaics in geometric patterns. The Ubaidians were the first civilizing force in the region. They drained marshes for agriculture, developed trade and established industries including weaving, leatherwork, metalwork, masonry, and pottery. The people called Sumerians, whose language prevailed in the territory, probably came from around Anatolia, probably arriving in Sumer about 3300 BC. For a brief chronological outline of Mesopotamia see http://www.gatewaystobabylon.com/introduction/briefchonology.htm. See also  http://www.wsu.edu:8080/ËÅ"dee/MESO/TIMELINE.HTM for more detailed information. The early Sumerians did have writing for numbers as shown below. Owing to the scarcity of resources, the Sumerians adapted the ubiquitous clay in the region developing a writing that required the use of a stylus to carve into a soft clay tablet. It predated the 1 10 60 600 3,600 36,000 cuneiform (wedge) pattern of writing that the Sumerians had developed during the fourth millennium. It probably antedates the Egyptian hieroglyphic may have been the earliest form of written communication. The Babylonians, and other cultures including the Assyrians, and Hittites, inherited Sumerian law and literature and importantly their style of writing. Here we focus on the later period of the Mesopotamian civilization which engulfed the Sumerian civilization. The Mesopotamian civilizations are often called Babylonian, though this is not correct. Actually, Babylon3 was not the first great city, though the whole civilization is called Babylonian. Babylon, even during its existence, was not always 3 The first reference to the Babylon site of a temple occurs in about 2200 BCE. The name means â€Å"gate of God.† It became an independent city-state in 1894 BCE and Babylonia was the surrounding area. Its location is about 56 miles south of modern Baghdad. Babylonian Mathematics 3 the center of Mesopotamian culture. The region, at least that between the two rivers, the Tigris and the Euphrates, is also called Chaldea. The dates of the Mesopotamian civilizations date from 2000-600 BCE. Somewhat earlier we see the unification of local principates by powerful leaders — not unlike that in China. One of the most powerful was Sargon the Great (ca. 2276-2221 BC). Under his rule the region was forged into an empire called the dynasty of Akkad and the Akkadian language began to replace Sumerian. Vast public works, such as irrigation canals and embankment fortifications, were completed about this time. These were needed because of the nature of the geography combined with the need to feed a large population. Because the Trigris and Euphrates would flood in heavy rains and the clay soil was not very absorptive, such constructions were necessary if a large civilization was to flourish. Later in about 2218 BCE tribesmen from the eastern hills, the Gutians, overthrew Akkadian rule giving rise to the 3rd Dynasty of Ur. They ruled much of Mesopotamia. However, this dynasty was soon to perish by the influx of Elamites from the north, which eventually destroyed the city of Ur in about 2000 BC. These tribes took command of all the ancient cities and mixed with the local people. No city gained overall control until Hammurabi of Babylon (reigned about 1792-1750 BCE) united the country for a few years toward the end of his reign. The Babylonian â€Å"texts† come to us in the form of clay tablets, usually about the size of a hand. They were inscribed in cuneiform, a wedge-shaped writing owing its appearance to the stylus that was used to make it. Two types of mathematical tablets are generally found, table-texts and problem texts. Table-texts are just that, tables of values for some purpose, such as multiplication tables, weights and measures tables, reciprocal tables, and the like. Many of the table texts are clearly â€Å"school texts†, written by apprentice scribes. The second class of tab lets are concerned with the solutions or methods of solution to algebraic or geometrical problems. Some tables contain up to two hundred problems, of gradual increasing difficulty. No doubt, the role of the teacher was significant. Babylon fell to Cyrus of Persia in 538 BC, but the city was spared. Babylonian Mathematics 4 The Darius inscription on cliff near Bisotun The great empire was finished. However, another period of Babylonian mathematical history occurred in about 300BCE, when the Seleucids, successors of Alexander the Great came into command. The 300 year period has furnished a great number of astronomical records which are remarkably mathematical — comparable to Ptolemy’s Almagest. Mathematical texts though are rare from this period. This points to the acuity and survival of the mathematical texts from the old-Babylonian period (about 1800 to 1600 BCE), and it is the old period we will focus on. The use of cuneiform script formed a strong bond. Laws, tax accounts, stories, school lessons, personal letters were impressed on soft clay tablets and then were baked in the hot sun or in ovens. From one region, the site of ancient Nippur, there have been recovered some 50,000 tablets. Many university libraries have large collections of cuneiform tablets. The largest collections from t he Nippur excavations, for example, are to be found at Philadelphia, Jena, and Istanbul. All total, at least 500,000 tablets have been recovered to date. Even still, it is estimated that the vast bulk of existing tablets is still buried in the ruins of ancient cities. Babylonian Mathematics 5 Deciphering cuneiform succeeded the Egyptian hieroglyphic. Indeed, just as for hieroglyphics, the key to deciphering was a trilingual inscription found by a British office, Henry Rawlinson (1810-1895), stationed as an advisor to the Shah. In 516 BCE Darius the Great, who reigned in 522-486 BCE, caused a lasting monument4 to his rule to be engraved in bas relief on a 100 Ãâ€" 150 foot surface on a rock cliff, the â€Å"Mountain of the Gods† at Behistun5 at the foot of the Zagros Mountains in the Kermanshah region of modern Iran along the road between modern Hamadan (Iran) and Baghdad, near the town of Bisotun. In antiquity, the name of the village was Bagastà ¢na, which means ‘place where the gods dwell’. Like the Rosetta stone, it was inscribed in  three languages — Old Persian, Elamite, and Akkadian (Babylonian). However, all three were then unknown. Only because Old Persian has only 43 signs and had been the subject of serious investigation since the beginning of the century was the deciphering possible. Progress was very slow. Rawlinson was able to correctly assign correct values to 246 characters, and moreover, he discovered that the same sign could stand for different consonantal sounds, depending on the vowel that followed. (polyphony) It has only been in the 20th century that substantial publications have appeared. Rawlinson published the completed translation and grammar in 1846-1851. He was eventually knighted and served in parliament (1858, 1865-68). For more details on this inscription, see the article by Jona Lendering at http://www.livius.org/be-bm/behistun/behistun01.html. A translation is included. Babylonian Numbers 3 In mathematics, the Babylonians (Sumerians) were somewhat more advanced than the Egyptians. †¢ Their mathematical notation was positional but sexagesimal. to some sources, the actual events described in the monument took place between 522 and 520 BCE. 5 also spelled Bistoun  ¯  ¯ 4 According Babylonian Mathematics †¢ They used no zero. 6 †¢ More general fractions, though not all fractions, were admitted. †¢ They could extract square roots. †¢ They could solve linear systems. †¢ They worked with Pythagorean triples. †¢ They studied circular measurement. †¢ They solved cubic equations with the help of tables. †¢ Their geometry was sometimes incorrect. For enumeration the Babylonians used symbols for 1, 10, 60, 600, 3,600, 36,000, and 216,000, similar to the earlier period. Below are four of the symbols. They did arithmetic in base 60, sexagesimal. 1 10 60 600 Cuneiform numerals For our purposes we will use just the first two symbols ∠¨ = 1 ≠º = 10 All numbers will be formed from these. Example: Note the notation was positional and sexagesimal: ≠ºÃ¢â€° º ≠ºÃ¢â€° º= 20  · 60 + 20 ≠ºÃ¢â€° º ∠¨ ∠¨ ∠¨ = 57 ≠ºÃ¢â€° ºÃ¢â€° º ∠¨ ∠¨ ∠¨Ã¢Ë† ¨ ∠¨ ∠¨ ∠¨Ã¢Ë† ¨ ≠º ∠¨ = 2  · 602 + 2  · 60 + 21 = 7, 331 The story is a little more complicated. A few shortcuts or abbreviation were allowed, many originating in the Seleucid period. Other Babylonian Mathematics 7 devices for representing numbers were used. Below see how the number 19 was expressed. Three ways to express the number 19 = 19 Old Babylonian. The symbol means subtraction. = 19 Formal = 19 Cursive form Seleucid Period(c. 320 BC to c. 620 AD) The horizontal symbol above the â€Å"1† designated subtraction. There is no clear reason why the Babylonians selected the sexagesimal system6 . It was  possibly selected in the interest of metrology, this according to Theon of Alexandria, a commentator of the fourth century A.D.: i.e. the values 2,3,5,10,12,15,20, and 30 all divide 60. Remnants still exist today with time and angular measurement. However, a number of theories have been posited for the Babylonians choosing the base of 60. For example7 1. The number of days, 360, in a year gave rise to the subdivision of the circle into 360 degrees, and that the chord of one sixth of a circle is equal to the radius gave rise to a natural division of the circle into six equal parts. This in turn made 60 a natural unit of counting. (Moritz Cantor, 1880) 2. The Babylonians used a 12 hour clock, with 60 minute hours. That is, two of our minutes is one minute for the Babylonians. (Lehmann-Haupt, 1889) Moreover, the (Mesopotamian) zodiac was divided into twelve equal sectors of 30 degrees each. 3. The base 60 provided a convenient way to express fractions from a variety of systems as may be needed in conversion of weights and measures. In the Egyptian system, we have seen the values 1/1, 1/2, 2/3, 1, 2, . . . , 10. Combining we see the factor of 6 needed in the denominator of fractions. This with the base 10 gives 60 as the base of the new system. (Neugebauer, 1927) 4. The number 60 is the product of the number of planets (5 known at the time) by the number of months in the year, 12. (D. J. Boorstin, 6 Recall, 7 See the very early use of the sexagesimal system in China. There may well be a connection. Georges Ifrah, The Universal History of Numbers, Wiley, New York, 2000. Babylonian Mathematics 1986) 8 5. The combination of the duodecimal system (base 12) and the base 10 system leads naturally to a base 60 system. Moreover, duodecimal systems have their remnants even today where we count some commodities such as eggs by the dozen. The English system of fluid measurement has numerous base twelve values. As we see in the charts below, the base twelve (base 3, 6?) and base two graduations are mixed. Similar values exist in the ancient Roman,  Sumerian, and Assyrian measurements. teaspoon tablespoon 1 teaspoon = 1 tablespoon = 1 fluid ounce = 1 gill = 1 cup = 1 pint = 1 quart = 1 gallon = 1 firkin = 1 hogshead = 1 3 6 24 48 96 192 768 6912 48384 inch 1 inch = 1 foot = 1 yard = 1 mile = 1 12 36 — fluid ounce 1/6 1/2 1 4 8 16 32 128 1152 8064 1/3 1 2 8 16 32 64 256 2304 16128 foot 1/12 1 3 5280 yard 1/36 1/3 1 1760 Note that missing in the first column of the liquid/dry measurement table is the important cooking measure 1/4 cup, which equals 12 teaspoons. 6. The explanations above have the common factor of attempting to give a plausibility argument based on some particular aspect of their society. Having witnessed various systems evolve in modern times, we are tempted to conjecture that a certain arbitrariness may be at work. To create or impose a number system and make it apply to an entire civilization must have been the work of a political system of great power and centralization. (We need only consider the failed American attempt to go metric beginning in 1971. See, http://lamar.colostate.edu/ hillger/dates.htm) The decision to adapt Babylonian Mathematics 9 the base may have been may been made by a ruler with little more than the advice merchants or generals with some vested need. Alternatively, with the consolidation of power in Sumeria, there may have been competing systems of measurement. Perhaps, the base 60 was chosen as a compromise. Because of the large base, multiplication was carried out with the aide of a table. Yet, there is no table of such a magnitude. Instead there are tables up to 20 and then selected values greater (i.e. 30, 40, and 50). The practitioner would be expected to decompose the number into a sum of smaller numbers and use multiplicative distributivity. A positional fault Which is it? ≠º ≠º = 10  · 60 + 10 = 10  · 602 + 10 = 3, 610 10 = 10 + 60 = 20() 1. There is no â€Å"gap†Ã‚  designator. 2. There is a true floating point — its location is undetermined except from context. ? The â€Å"gap† problem was overcome in the Seleucid period with the invention of a â€Å"zero† as a gap separator. We use the notation: d1 ; d2 , d3 , . . . = d1 + d2 d3 + 2 +  ·Ã‚ ·Ã‚ · 60 60 The values d1 ; d2 , d3 , d4 , . . . are all integers. Example ∠¨ ≠º ∠¨Ã¢Ë† ¨ ≠ºÃ¢â€° º ≠º ∠¨ ≠º ≠º ∠¨Ã¢Ë† ¨ ≠ºÃ¢â€° º 1; 24, 51, 10 = 1 + 24 51 10 + 2+ 3 60 60 60 = 1.41421296 Babylonian Mathematics 10 This number was found on the Old Babylonian Tablet (Yale Collection √ #7289) and is a very high precision estimate of 2. We will continue this discussion shortly, conjecturing on how such precision may have been obtained. The exact value of √ 2, to 8 decimal places is = 1.41421356. Fractions. Generally the only fractions permitted were such as 2 3 5 12 , , , , †¦ 60 60 60 60 because the sexagesimal expression was known. For example, 1 10 = = ;≠º 6 60 1 ∠¨ ∠¨ ∠¨ ≠ºÃ¢â€° º =; , 9 ∠¨ ∠¨ ∠¨ ≠ºÃ¢â€° º 1 Irregular fractions such as 1 , 11 , etc were not normally not used. 7 There are some tablets that remark, â€Å"7 does not divide†, or â€Å"11 does not divide†, etc. A table of all products equal to sixty has been found. 2 3 4 5 6 8 9 10 12 15 30 20 15 12 10 7,30 6,40 6 5 4 16 18 20 24 25 27 30 32 36 40 3, 45 3,20 3 2,30 2,25 2,13,20 2 1;52,30 1,40 1,30 Babylonian Mathematics You can see, for example that 8 Ãâ€" 7; 30 = 8 Ãâ€" (7 + 30 ) = 60 60 11 Note that we did not use the separatrix â€Å";† here. This is because the table is also used for reciprocals. Thus 7 30 1 = 0; 7, 30 = + 2 8 60 60 Contextual interpretation was critical. Remark. The corresponding table for our decimal system is shown below. Included also are the columns with 1 and the base 10. The product relation and the decimal expansion relations are valid in base 10. 1 2 5 10 10 5 2 1 Two tablets found in 1854 at Senkerah on the Euphrates date from 2000 B.C. They give squares of the numbers up to 59 and cubes up to 32. The Babylonians used the formula xy = ((x + y)2 − (x − y)2 )/4 to assist in multiplication. Division relied on multiplication, i.e. 1 x =x · y y There apparently was no long division. The Babylonians knew some approximations of irregular fractions. 1 =; 1, 1, 1 59 1 =; 0, 59, 0, 59 61 However, they do not appear to have noticed infinite periodic expansions.8 the decimal system, the analogous values are 1 = 0.1111 . . . and 9 Note the use of the units â€Å"0† here but not for the sexagesimal. Why? 8 In 1 11 = 0.090909 . . .. Babylonian Mathematics 12 They also seemed to have an elementary knowledge of logarithms. That is to say there are texts which concern the determination of the exponents of given numbers. 4 Babylonian Algebra In Greek mathematics there is a clear distinction between the geometric and algebraic. Overwhelmingly, the Greeks assumed a geometric position wherever possible. Only in the later work of Diophantus do we see algebraic methods of significance. On the other hand, the Babylonians assumed just as definitely, an algebraic viewpoint. They allowed operations that were forbidden in Greek mathematics and even later until the 16th century of our own era. For example, they would freely multiply areas and lengths, demonstrating that the units were of less importance. Their methods of designating unknowns, however, does invoke units. First, mathematical expression was strictly rhetorical, symbolism would not come for another two millenia with Diophantus, and then not significantly until Vieta in the 16th century. For example, the designation of the unknown was length. The designation of the square of the unknown was area. In solving linear systems of two dimensions, the unknowns were length and breadth, and length, breadth, and width for three dimensions. √ Square Roots. Recall the approximation of 2. How did they get it? There are two possibilities: (1) Applying the method of the mean. (2) Applying the approximation √ b a2  ± b ≈ a  ± 2a Babylonian Mathematics 13 Yale Babylonian Collection 1;24,51,10 30 42;25,35 Square with side 30 The product of 30 by 1;24,51,10 is precisely 42;25,35. Method of the mean. The method of the mean can easily be used to find the square root of any number. The idea is simple: to find the square root of 2, say, select x as a first approximation and take for another 2/x. The product of the two numbers is of course 2, and moreover, one must be less than and the other greater  than 2. Take the √ arithmetic average to get a value closer to 2. Precisely, we have 1. Take a = a1 as an initial approximation. √ √ 2. Idea: If a1 < 2 then a21 > 2. Babylonian Mathematics 3. So take a2 = (a1 + 4. Repeat the process. Example. Take a1 = 1. Then we have 2 3 a2 = (1 + )/2 = 1 2 2 17 3 )/2 = 1.41666†¦ = a3 = ( + 2 3/2 12 17 2 577 a4 = ( + )/2 = 12 17/12 408 14 2 )/2. a1 Now carry out this process in sexagesimal, beginning with a1 = 1; 25 using Babylonian arithmetic without rounding, to get the value 1;24,51,10. √ à º Note: 2=1; 25 = 1.4166†¦ was commonly used as a brief, rough and ready, approximation. When using sexagesimal numbering, a lot of information can be compressed into one place. Solving Quadratics. The Babylonian method for solving quadratics is essentially based on completing the square. The method(s) are not as â€Å"clean† as the modern quadratic formula, because the Babylonians allowed only positive solutions. Thus equations always were set in a form for which there was a positive solution. Negative solutions (indeed negative numbers) would not be allowed until the 16th century CE. The rhetorical method of writing a problem does not require variables. As such problems have a rather intuitive feel. Anyone could understand the problem, but without the proper tools, the solution would be impossibly difficult. No doubt this rendered a sense of the mystic to the mathematician. Consider this example I added twice the side to the square; the result is 2,51,60. What is the side? In modern terms we have the simple quadratic x2 + 2x = 10300. The student would then follow his â€Å"template† for quadratics. This template was the solution of a specific problem of the correct mathematical Babylonian Mathematics 15 type, all written rhetorically. Here is a typical example given in terms of modern variables. Problem. Solve x(x + p) = q. Solution. Set y = x + p Then we have the system xy = q y−x = p This gives 4xy + (y − x)2 = p2 + 4q (y + x)2 = p2 + 4q x+y = 2x + p = q q p2 + 4q p2 + 4q √ −p + p2 + 4q x = 2 All three forms x2 + px = q x2 = px + q x2 + q = px are solved similarly. The third is solved by equating it to the nonlinear system, x + y = p, xy = q. The student’s task would be to take the problem at hand and determine which of the forms was appropriate and then to solve it by a prescribed method. What we do not know is if the student was ever instructed in principles of solution, in this case completing the square. Or was mathematical training essentially static, with solution methods available for each and every problem that the practitioner would encounter. It is striking that these methods date back 4,000 years! Solving Cubics. The Babylonians even managed to solve cubic equations, though again only those having positive solutions. However, the form of the equation was restricted tightly. For example, solving x3 = a Babylonian Mathematics was accomplished using tables and interpolation. Mixed cubics x3 + x2 = a were also solved using tables and interpolation. The general cubic ax3 + bx2 + cx = d can be reduced to the normal form y 3 + ey 2 = g 16 To do this one needs to solve a quadratic, which the Babylonians could do. But did the Babylonians know this reduction? The Babylonians must have had extraordinary manipulative skills and as well a maturity and flexibility of algebraic skills. Solving linear systems. The solution of linear systems  were solved in a particularly clever way, reducing a problem of two variables to one variable in a sort of elimination process, vaguely reminiscent of Gaussian elimination. Solve 2 1 x − y = 500 3 2 x + y = 1800 Solution. Select x = y such that ËÅ" ËÅ" x + y = 2ËÅ" = 1800 ËÅ" ËÅ" x So, x = 900. Now make the model ËÅ" x=x+d ËÅ" We get y =y−d ËÅ" 1 2 (900 + d) − (900 − d) = 500 3 2 2 1 ( + )d + 1800/3 − 900/2 = 500 3 2 7 d = 500 − 150 6 6(350) d = 7 So, d = 300 and thus x = 1200 y = 600. Babylonian Mathematics 17 Plimpton 322 tablet Yale Babylonian collection Pythagorean Triples. 5 As we have seen there is solid evidence that the ancient Chinese were aware of the Pythagorean theorem, even though they may not have had anything near to a proof. The Babylonians, too, had such an awareness. Indeed, the evidence here is very much stronger, for an entire tablet of Pythagoreantriples has been discovered. The events surrounding them reads much like a modern detective story, with the sleuth being archaeologist Otto Neugebauer. We begin in about 1945 with the Plimpton 322 tablet, which is now the Babylonian collection at Yale University, and dates from about 1700 BCE. It appears to have the left section Babylonian Mathematics 18 broken away. Indeed, the presence of glue on the broken edge indicates that it was broken after excavation. What the tablet contains is fifteen rows of numbers, numbered from 1 to 15. Below we list a few of them in decimal form. The first column is descending numerically. The deciphering of what they mean is due mainly to Otto Neugebauer in about 1945. 1.9834†¦ 1.94915 . . . 1.38716 119 169 3367 4825 56 1 2 106 15 Interpreting Plimpton 322. To see what it means, we need a model right triangle. Write the Pythagorean triples, the edge b in the column thought to be severed from the tablet. Note that they are listed c B a b decreasing cosecant. b (c/b)2 120 (169/120)2 3456 (4825/3456)2 . . . 90 (106/90)2 Right Triangle a c 119 169 3367 4825 56 106 1 2 15 c csc2 B = ( )2 b A curious fact is that the tablet contains a few errors, no doubt transcription errors made so many centuries ago. How did the Babylonian mathematicians determine these triples? Why were they listed in this order? Assuming they knew the Pythagorean relation a2 + b2 = c2 , divide by b to get c a ( )2 + 1 = ( )2 b b Babylonian Mathematics u2 + 1 = v 2 (u − v)(u + v) = 1 Choose u + v and find u − v in the table of reciprocals. 19 Example. Take u + v=2;15. Then u − v = 0; 26, 60 Solve for u and v to get u = 0; 54, 10 v = 1; 20, 50. Multiply by an appropriate integer to clear the fraction. We get a = 65, c = 97. So b = 72. This is line 5 of the table. It is tempting to think that there must have been known general principles,  nothing short of a theory, but all that has been discovered are tablets of specific numbers and worked problems. 6 Babylonian Geometry Circular Measurement. We find that the Babylonians used Ï€ = 3 for practical computation. But, in 1936 at Susa (captured by Alexander the Great in 331 BCE), a number of tablets with significant geometric results were unearthed. One tablet compares the areas and the squares Babylonian Mathematics 20 of the sides of the regular polygons of three to seven sides. For example, there is the approximation perimeter hexagon = 0; 57, 36 circumference circumscribed circle This gives an effective Ï€ ≈ 3 1 . (Not bad.) 8 Volumes. There are two forms for the volume of a frustum given Frustum b b h a a V V a+b 2 )h ÃÆ'2 ! a+b 2 1 a−b 2 = h ( ) − ( ) 2 3 2 = ( The second is correct, the first is not. There are many geometric problems in the cuneiform texts. For example, the Babylonians were aware that †¢ The altitude of an isosceles triangle bisects the base. †¢ An angle inscribed in a semicircle is a right angle. (Thales) 7 Summary of Babylonian Mathematics That Babylonian mathematics may seem to be further advanced than that of Egypt may be due to the evidence available. So, even though Babylonian Mathematics 21 we see the development as being more general and somewhat broader in scope, there remain many similarities. For example, problems contain only specific cases. There seem to be no general formulations. The lack of notation is clearly detrimental in the handling of algebraic problems. There is an absence of clear cut distinctions between exact and approximate results. Geometric considerations play a very secondary role in Babylonian algebra, even though geometric terminology may be used. Areas and lengths are freely added, something that would not be possible in Greek mathematics. Overall, the role of geometry is diminished in comparison with algebraic and numerical methods. Questions about solvability or insolvability are absent. The concept of â€Å"proof† is unclear and uncertain. Overall, there is no sense of abstraction. In sum, Babylonian mathematics, like that of the Egyptians, is mostly utilitarian — but apparently more advanced. Exercises 1. Express the numbers 7 6, 234, 1265, and 87,432 in sexagesimal. 2. Compute the products (a) 1, 23 Ãâ€" 2, 9 (b) 2, 4, 23 Ãâ€" 3, 34 8 3. A problem on one Babylonian tablets give the base and top of an isosceles trapezoid to be 50 and 40 respectively and the side length to be 30. Find the altitude and area. Can this be done without the Pythagorean theorem? 4. Solve the following system ala the Babylonian â€Å"false position†  ´ method. State clearly what steps you are taking. 2x + 3y = 1600 5x + 4y = 2600 (The solution is (200, 400).) Babylonian Mathematics 22 5. Generalize this Babylonian algorithm for solving linear systems to arbitrary linear systems in two variables? 6. Generalize this Babylonian algorithm for solving linear systems to arbitrary linear systems? √ 7. Modify the Babylonian root finding method (for 2) to find√ the square root of any number. Use your method to approximate 3. Begin with x0 = 1. √ 8. Explain how to adapt the method of the mean to determine 3 2. n n3 + n2 1 2 2 12 9. Consider the table: 3 36 Solve the following prob4 80 150 5 6 252 lems using this table and linear interpolation. Compare with the exact values. (You can obtain the exact solutions, for example, by using Maple: evalf(solve(x3 + x2 = a, x)); Here a=the right side) (a) x3 + x2 = 55 (b) x3 + x2 = 257 10. Show that the general cubic ax3 + bx2 + cx = d can be reduced to the normal form y 3 + ey 2 = g. 11. Show how the perimeter identity is used to derive the approximation for Ï€. 12. Write a lesson plan wherein you show students how to factor quadratics ala the Babylonian methods. You may use variables,  ´ but not general formulas.