Moringa Oleifera as an Alternative Detergent

1 Chapter 1 THE PROBLEM Introduction This study about Moringa Oleifera enables the creation of a new homemade detergent and its target to clean fabric and furniture. This study will prove that even vegetables like Malunggay can also be used as house cleaners. This study will try to improve the cleaning powers of the other commercial detergents by adding malunggay extract to it. It will prove that the power of malunggay is not limited and this will prove people wrong. They think malunggay can only be used as medicines or food, but the researchers study, Malunggay’s purpose is to be a liquid detergent.There is a great possibility that malunggay’s power can also clean items in the household. The target of this study is to improve the progress of cleanliness in our economy. Malunggay can clean and it can dissolve dirt in a great speed. Chemically speaking, there is a great possibility that malunggay’s chlorophyll might have the power not the leaves itself. The decisi on in doing this study was made during this late year of 2011 since herbal products such as malunggay shampoos and conditioners.These herbal items are greatly needed by the public market not because only it has a small price, but also it has an effect. Malunggay can clean stains and dissolve them in a great time. Testing malunggay’s power but sooner and sooner, malunggay’s power can be soon unveiled and discovered. When this happens, there will be a great upgrade in the whole economy. 2 Statement of the Problem Moringa Oleifera as an alternative detergent This study will be conducted to develop a horseradish (Moringacae/Moringa Oleifera) liquid detergent.Specifically, this study will attempt to answer the following questions. 1. How long does it take for the moringa detergent to remove stains? 1. 1 Catsup 1. 2Coffee 1. Is moringa detergent better than the stain removing capacity of the commercial detergent? Significance of the study This study will contribute to a grow ing number of demands to a commercial detergent by helping people work easier and finish earlier. People need many detergents for their works such as cleaning clothes but with malunggay detergents, you can clean in just 2-5 wipes the furniture.This can also motivate and challenge school authorities to a cleaner and have a less stress and it’s only for a low cost price. This can contribute to household maids and moms so that they would not have a hard time in removing stains that need to be washed out for a very long time. Scope and Delimitation This study is focused on making an alternative liquid detergent from malunggay. This study is limited to only using malunggay as a detergent not as using this as shampoos and conditioners.The investigation is concerned on the economy of the Philippines may sink down to poverty because of high prices of items in the public market which are needed in great amount have a high cost. This study does not only want the economy of the Philippi nes but also the world’s economy because it can contribute to world safety because it does not have poisonous and hazardous to health chemicals. In the way, we can contribute to the economy, we can save lives, and we can save Mother Earth. 8 Chapter 3 METHODOLOGY This chapter presents the method and procedure in conducting the study.This includes the Research Design, Materials and Instrumentation, Collection of materials, Preparation of Treatments, Detergent Making, Testing and Evaluation of the Detergent’s Quality and, Statistical Treatment. Research Design The researchers used CRD or Completely Randomized Design because this study has replications of treatments (R) was assigned completely at random to independent experimental subjects. This study has 2 treatments; T1 = Control and T2 = Malunggay (Moringa Oleifera). Materials and InstrumentationThis experimentation used 1 mL of Malunggay extract, 1 mL of Malunggay’s chlorophyll, 10 samples of Malunggay seeds. C ollection of Materials This experiment was done in General Santos City. 1 mL of Malunggay extract and 1 mL of Malunggay’s chlorophyll was from General Santos City. The 10 samples of Malunggay seeds are from the local residences in one of our researchers. 9 Preparation of Treatments Malunggay Extract (mL) Treatments T1 R2 R1 T2 R2 R1 LEGEND T- Treatments R- Replication 10 Experimental Layout T2R1 T1R1 T2R2 T1R2 Legend T – Treatment T2 – Malunggay E. T1 – ControlR – Replication 11 Detergent MakingIn making this detergent, you will need 2 cups of distilled water, Fragrance, 4 packs of unflavored gelatin, an empty jar and, Red food coloring. Heat one cup of distilled water until it boils. Then, stir in 4 packages of unflavored gelatin until dissolved. Remove the mixture from heat and stir in another 1 cup of distilled water. Add 10-20 drops of fragrance and coloring. Pour Malunggay extract and Chlorophyll into the mixture. Stir the mixture then place i n a cold temperature for approximately 5-10 minutes. Pour the gel into a clean jar or a small container. Testing and Evaluation of the Detergent’s QualityThe Malunggay detergent was compared to a commercial detergent by sinking them in the same size of a pail and at the same time with the same number of water and limit of detergent. The same stains were also added. Photographs were taken for public documentations. Statistical Analysis (ANOVA) It used the t-test dependent to determine the significant differences in the two treatments and replications and to understand the statistical independence of one treatment to another treatment on another material and t-test was used to determine which detergent was faster enough to clean the same amount of stains. 2 Chapter IV Results and Discussion This chapter presents and explains the data gathered in texture, graphical and tabular manner. Table 4. 1 Mean Duration in Removing Catsup Stains Treatments| R1| R2| R3| Mean| T1(Commercial) | 13. 00s| 17. 05s| 17. 00s| 15. 22s| T2(Malunggay)| 11. 25s| 12. 35s| 15. 17s| 12. 59s| Table 4. 2 Mean Duration in Removing Coffee Stains Treatments| R1| R2| R3| Mean| T1(Commercial)| 17. 25s| 18. 14s| 17. 25s| 15. 51s| T2(Malunggay)| 18. 35s| 25. 17| 15. 21s| 17. 91s|From the table, it could be seen that replicates 1, 2 and, 3 reflect the seconds in cleaning the stains by each detergent. The mean duration in removing catsup stains using T2 (Malunggay) is much faster at 12. 59s compared to T1 (commercial detergent) which is at 15. 22s. However, the mean duration in removing coffee stains using Malunggay was a bit higher at 17. 91s. The above data prove that Malunggay is indeed an effective household material in removing stains. 13 Chapter V CONCLUSIONS AND RECOMMENDATIONSThis chapter states in concise form the generalization in the form of conclusions and the solutions to the problem in the form of recommendations. Conclusions Based on the analysis of the results of this study, th e researchers arrived at the following conclusions: 1. For the Moringa Oleifera detergent to remove the stains, it takes: 1. 1) 12. 59s to remove the Catsup Stains 1. 2) 17. 91s to remove the Coffee Stains 2. Moringa detergent is better than the stain removing capacity of the commercial detergent because it can clean faster without much scrubbing.Recommendations There were factors of the study that need further analysis and consideration. To this, extent, the researchers came up with the following recommendations: 1. Antibacterial Activity Test of Moringa detergent should be conducted. 2. Further studies to the potency of Moringa detergent in removing stains such as chocolate, soy sauce and alike should be done. 14 Abstract This study reveals the use of Moringa Oleifera or locally known as Malunggay, as an alternative detergent in cleaning fabrics and furniture.The alternative detergent was tested against the commercial detergent, by measuring the estimated time both detergents coul d clean. The outcome of the test was that the newly made detergent almost has the same estimated time with the commercial detergent, but cleaning the stains needs more scrubbing if the commercial detergent is used, while on the other hand, cleaning the stains with the alternative detergent provides less scrubbing. Thus, Moringa liquid detergent is an effective household material in removing stains.

Abortion – “the Wrong of Abortion”

Abortion is one of the most controversial topics of all times. The definition most people associate with abortion is the termination of unwanted pregnancy. In their essay, â€Å"The Wrong of Abortion†, Patrick Lee and Robert P. George argue that intentional abortion is unjust and therefore objectively immoral no matter the circumstances. Also, they argue that â€Å"the burden of carrying the baby is significantly less than the harm the baby would suffer by being killed; the mother and father have a special responsibility to the child; it follows that intentional abortion (even in few cases where the baby’s death is an unintended but foreseen side effect) is unjust † (24). I am personally in between pro-life and pro-choice. On the one hand, I agree with their argument in that the mother and the father are responsible for their baby and that abortion should not be a choice. However, I disagree with the part where they say that abortion is unjust even if the baby (fetus) may have a defect. Yet, I believe that the choice of abortion is immoral if women use it as their last resort- contraceptive purposes, but I think that abortion should be allowed if the baby (fetus), which is still in the womb, is predicted to have a side effect such as deformation or diseases like Down’s syndrome. For example, if I were to bear a child and I find out later on that my baby has Down’s syndrome; then, in this case, I will choose to get aborted, not for selfish reasons, but because this defect may hurt my baby in the long run. Besides, my baby is the one that has to live with it for the rest of his/her life and it will definitely have a big effect on them in the future. In short, I am pro-life in most cases, especially if women do not take responsibility for their actions, but I am pro-choice if and only if there are side effects with the baby or the mother that might endanger their lives and of course, abortion is permissible in case of incest and rape. Lee and George claim that human embryos (fetuses) are complete human beings that have not fully developed to its mature stage; therefore, a human being is what is killed in abortion. I agree completely that the fetus, or the human embryo, is in fact a living being. Moreover, human embryo is the â€Å"same† as human beings except, the difference between these two is that the embryo is not a full human person because the fetus is not fully developed yet. Every new life, whether it be animal or human, begins at conception. With this being said, no matter what the circumstances of conception, no matter how far along in the pregnancy, abortion, in my opinion, always ends the life of an individual human being. Abortion destroys the lives of helpless and innocent babies that have not done anything wrong. Everyone is raised knowing the difference between right and wrong. Murder is wrong, so why is not abortion? Defenders of abortion argue that it is not murder if the child is unborn. So, why is it that if an infant is destroyed a month before the birth, there is no problem, but if killed a month after birth, it is considered as inhumane murder? Lee and George support their argument by providing three important facts that differentiate a human embryo is, in fact, a human being. First, they say that sex cells and somatic cells are part of a larger organism while the human embryo is a complete or whole organism, though immature (14). Secondly, they say that the embryo is human and has all the characteristics of a human being but the sex and somatic cells are genetically and functionally different because they cannot develop separately while the embryo can. Last but not least, they claim that embryo develops all of the organs and organ systems that are necessary to turn themselves into a mature human being. Above all, the human embryo, from conception onward, is fully programmed actively to develop himself or herself to the mature stage of a human being, unless prevented by disease or violence (14). With these reasons, it can be said that abortion results in the death of a human being. As a result, abortion is murder since the fetus being destroyed is breathing, has a human form, and has feelings. Carol Everett, who is a former abortionist, once said at the conference Meet the Abortion Providers, â€Å"the product abortion, is skillfully marketed and sold to the women at a crisis time in her life. She buys the product, finds it defective and wants to return it for a refund, but it is too late. † In most cases, abortion is intentional killing. Most women use aborting as an easy â€Å"way out† because they want to avoid in becoming a parent. Parents do have a responsibility to make sacrifices for their children, even if they have not voluntary assumed such responsibilities, or given their consent to the personal relationship with the child- this is the authors’ claim (22). I completely agree with their claim because a person should accept the consequences of risks that one knowingly and willingly takes. I believe that it is common sense that both women and men should know that contraceptives are not 100 percent effective; for this reason, if they are willingly having sexual intercourse, then they should know that they are taking the risk in possibly becoming pregnant. Therefore, a woman who becomes pregnant should accept her pregnancy as the consequence of taking the risk involved in sexual intercourse. This means that the woman has a duty or a responsibility of taking care for her child regardless if she wanted the baby or not. Since we have special responsibilities to those with whom we are closely untied, it follows that we in fact do have a special responsibility to our children anterior to our having voluntarily assumed such responsibility or consented to the relationship† (23). Abortion is clearly used to avoid responsibility and the authors call this unjust or intentional killing. Nevertheless, while the authors argue that abortion is intentional killing most of the tim e, they also claim that causing death as a side effect is morally permissible. For example, if the pregnant woman has cancer in her uterus that will surely endanger the woman’s life, then Lee and George claim that, in this case, it can be morally right to remove the cancer with the baby still in her womb, even if the baby (fetus) dies as a result. They consider the baby’s death as a side effect as well as the ending of the pregnancy itself but they claim that the mother’s life is more important. This type of abortion is known as indirect or non-intentional killing (21). However, they also assert that not every death that is caused because of side effects is right. For instance, if the mother or the father have a bad habit of smoking when they know for a fact that this will endanger the baby’s (fetus) development, and for this reason, the woman wants to get an abortion because they find out that their baby has a defect- this choice she is making is an unjust act since she could have avoided it but instead, did not do anything to change; therefore, this is the consequence they have to face. It was immoral for them to continue with their actions when they know this will or might cause harm to their child. The act that would cause the child’s death would avoid harm to the parent but cause a significantly worse harm to his child (21). All in all, the parents have a special responsibility to the child even if they did not want or were not expecting a baby in the first place, they should act responsibly in virtue of being their biological parents. I, however, only partially agree with their argument mentioned above. I agree completely in that abortion should be performed if the woman has a disease that will endanger her life as well as the baby’s. Nonetheless, in the second example, although it was their fault for causing their child to not develop properly, I think that the parents should be given the choice to perform abortion or to keep their child. Like I mentioned in the beginning, if I were to have a child that is deformed or is mentally unstable, then I would get an abortion even if it is 100 percent my fault. I want my baby to be happy, and I know for a fact that my baby might not be happy in the future because of their defect and I will never forgive myself because my child does not deserve to go through hardship because of the actions that I’ve done. For this reason, I would not call it unjust killing in this case. After critically analyzing Lee and George’s argument, I come to a conclusion that it is very difficult to draw a line between keeping one’s life or being responsible for one’s actions. On the one hand, if the woman voluntarily put herself into a situation where it might bring her the existence of a person, then in this case no matter what, she is held responsible and accountable for her actions since to make that ‘choice' after a pregnancy is underway, merely as a matter of birth control, is an immoral act. So, abortion is morally wrong since the mother had sexual intercourse of her own free will. On the other hand, the situation becomes complicated when one has to choose whether it is better to get an abortion if there is something wrong with the baby due to the parent’s actions. Would one save the life or choose to abort although this was also their responsibility? With all my aforementioned reasons, I am still in between pro-life and pro-choice because I believe that abortion can be permissible depending on the situation.

American Eagle Company Essay Example | Topics and Well Written Essays - 1500 words

American Eagle Company - Essay Example It also ships them worldwide through the website. AE has introduced a new label called â€Å"aerie by American Eagle† targeting young female customers with a collection of dormwear and intimates that includes bras, undies, camis, hoodies, robes, boxers, sweats and leggings. They are specifically designed to be sweetly sexy for everyday stylish wear. These are currently sold in 19 exclusive stores as well as on its website called Apart from this AE has now targeted 25 to 40 year old customers with yet another brand called MARTIN+OSA using denim and sportswear as a base covering apparel, accessories and footwear. This is sold in 13 stores as well through its website www.martinandosa.com. AE had a sale of $ 2.98523 billion with profits of $ 411 million during the last financial year and expects a slight fall this year due to extended warm weather. A SWOT analysis displays the company outlook in detail. 6. Compared with competition it has fared better in the 2nd quarter of 2007 as it suffered a fall in sales of only 2% compared to industry suffering between 2 and 11 per cent. This shows that it is more popular than others. 1. Due to Global Warming warm weather has been extended in 2007 resulting in a longer summer. This resulted in a fall in sale of abut 2% in the 2nd quarter and the forecast of sales in 3rd quarter has been reduced. This will impact year on year sales and profitability. 3. It went on expansion through the takeover and merger route and in one case it backfired badly. Bluenotes of Canada with 100 stores was acquired in 2001 but since this too served nearly the same segment, 12-22 year olds, it failed and had to be sold off in 2004. 6. By sticking to one demographic segment (15-25 year) predominantly has made the company kaleidoscopic and the market also perceives it as a one segment player. Fresh competition will eat away

Business Management and Organizational Behavior Assignment

Business Management and Organizational Behavior - Assignment Example The factors that Jasper Hennings need to think about while settling on his course of action are culture of the business, values and beliefs which are being followed, management and employee aspects of the organization. Jasper Hennings also needs to consider the mental status of the employees while committing the offense. He requires holding up a session with the guilty employee, talking about the issues and the motive behind the actions. Jasper Hennings needs to judge all of his employees similarly and biasness should be strictly avoided in the prevailing scenario while judging and making a decision about the course of action with Henry Darger (Richard, L. D. & Marcic, D., â€Å"Understanding Management†). In the Rio Grande Supply Company, the expressed cultural values and beliefs include honesty, integrity and a reverence for every individual employee. In addition to the expressed culture values and beliefs, other subconscious values and beliefs include interest, motivation, trust and the adopted norms. In Rio Grande Supply Company, conflicting norms and values are present. Henry Darger, the chief of operations, has been found to be guilty as he violated company’s internet policy and used internet to surf certain unauthorized sites. The policy clearly stated that none of the employees are allowed to use the Rio Grange’s computers for everything except the business related purposes. Beside this, Henry Darger is being hypocritical towards an employee as he fired the lady employee for her offense. However, he himself is identified to be violating the company’s policy and it raised conflict between the norms and the ethical values of working culture, as watching adul t pornography related sites in working hours at office is strictly offensive. And, any failure to follow the rules and policy would hamper the image of the company. Henry Darger being one of the administrative members should

Administrative Ethics Essay Example | Topics and Well Written Essays - 500 words

Administrative Ethics - Essay Example However, once that is established, the first dictum of the ASPA code of ethics tells us that service to the public is above service to oneself and that certainly applies in this case. If one is to remain quiet about this it is certainly possible that the boss would be very pleased and Tristate would also be very happy. Of course, there are plenty of vested interests in the construction of the mall simply because a lot of people have a lot to gain from the construction. It is basically a project worth billions of dollars to all the stakeholders and they are essentially serving themselves. Therefore, if we are to follow the code of ethics given by the ASPA, we have to consider what the best course of action would be to preserve the interest of the public. In this case, we can come with two different options of which the first option would be to report the matter to the higher authorities and see what they decide to do with it. Reports to the committee or reports to the council about the incident could be very useful in making sure that the right people have the right information about what is going on concerning the megamall project. In fact, the information could even be taken to the media since that would be more than likely to put an immediate halt to the process. The second option would be to see if the construction of the mall is really towards the benefit of the public. For example, there are many jobs to be created in the mall for the local community with white collar positions such store managers, financial accountants, legal advisors, logistics managers and several other managerial positions that would be needed by companies who setup operations in the mall. For the blue collared workers, there would be jobs in food courts, restaurants, mall security and other store positions that would do a lot for the economy of the region. Thus the public certainly has something to gain from the mall being there. The decision therefore becomes to tell or not

Where does Europe end and why Essay Example | Topics and Well Written Essays - 2500 words

Where does Europe end and why - Essay Example The boundaries which constitute Europe are thus vague, rather dynamic as Delanty and Rumsfeld (2005) indicate, i.e. Europe is still going through an economic as well as political transformation whereby it is struggling to incorporate the spaces surrounding the traditional Europe as buffer regions, stability of which is crucial for the sustainability of European states. The European Union is consistently going through a transition which is further expected to enlarge after the incorporation of Turkey into the European Union. In this case, according to the social theory model of EU proposed by Delanty and Rumsfeld (2005), the boundary in further going to be enhanced creating a further confusion about the end of Europe. The paper aims at answering the question in geographical as well as in economical, political and sociological perspectives to give the most valid explanations about where does Europe actually end. The paper intend on arguing that the boundaries of Europe in its true sens e extend beyond the Russian as well as Turkish borders. To further draw patterns and conclusions across variables, it is important to assess the traditional borders of Europe which the geographers have relied upon for years. According to the geographers, Europe is divided from Asia at the East from Ural Mountains and seas, whereas it is surrounded by watersheds on other sides. By this explanation, Europe ends somewhere around Russia at its East, and in Turkey around it`s South East where Asia or rather Middle East begins. This is where the problems still lie. Russia and Turkey both are located about halfway in Europe, thus it becomes difficult to assess where the other continents begin. Geography can`t be studied keeping political and sociological angles aside, thus when these factors are also considered it becomes difficult to decide where Europe ends. It can however be concluded, for the purpose of eliminating confusion that Europe ends

Marketing Communication Based on Video Research Paper

Marketing Communication Based on Video - Research Paper Example Absolut is a business entity which was originated as a brain child of Lars Olsson Smith, the king of Vodka. The Absolut organization exists to provide the best quality vodka in the world. Today it is one of the leading brands of Vodka. The originators suggest that the main ingredient required to produce Absolut is a grain grown in Ahus, in Southern Sweden. The producers have coined a â€Å"one source concept†. It is this one source concept that the manufacturers believe causes Absolut vodka to be absolute. The Absolut website states that it is made of only natural ingredients, namely, winter wheat and water. Absolut is viewed as a perfect unit in that its bottle, its taste and its price combine to form one package. Over the years Absolut has added many flavors to its original and premium product. Flavors include Peppar, Citron, Mandarin, Rasberri, Vanilla, Kurant and Pears. Absolut on their website proposes that their Mandarin and Orange are two of the most popular flavors in the world. The brand’s origins and its background The brand Absolut was established since 1889 when Lars Olsson Smith registered the brand name and began to revolutionize the manner in which vodka was made. By 1979, it was imperative for Absolut to begin to export to countries around the world in order to survive. The genesis for Absolut was the idea of the legendary Lars Olsson Smith who revolutionized the manner in which distillation was done. He created the rectification method of distillation and to this day producers of alcoholic beverages still use this method. For Smith, rectification allowed all the impurities involved in the production of vodka to be removed. Thus, Smith called the product of his rectification Absolut rent branvin which means â€Å"Absolute pure vodka† in Swedish. For this reason, Lars Olsson Smith is known as the King of Vodka. In 1992, Absolut advertising campaign was inducted into the American Marketing Association’s Marketing Hall of Fame. Amazingly, this feat was won without the use of te levision as an advertising medium. Richard Lewis (1996) the mastermind behind Absolut’s advertising campaign claims that the major purpose of the campaign was â€Å"to build a healthy and enduring brand for Absolut.† Throughout the first hundred years of its existence Absolut was perfected in its taste, its texture and its packaging. 1.3 The brand’s popularity Although Absolut in its website views itself as the number one selling vodka in the United States, other sources such as Impact International believe that in 2010 Absolut was the fourth largest â€Å"international premium spirit in the world. Impact International noted that Absolut is available in 126 markets. Lewis (1996) notes that in 1981 at the beginning of the Absolut advertising campaign, 20,000 cases of vodka were sold to the United States each year by 1995; sales had reached to 3 million cases per year. Sarah Edmunds of Reuters in her interview with Bengt Baron posits, â€Å"In the United States †¦ Absolut has about 10-1/2 to 11 percent of the total vodka market, sales rose to 4.9 million nine-litre cases in 2006, up from 4.7 million in 2005. Also at this interview Baron observed that Absolut has â€Å"98 percent brand awareness in the (U.S.) consumer (market) and we're still growing faster than the market in general†. Nonetheless, Absolut still advertises itself on its website as the leading brand of vodka in the world. A positioning- perceptual map is an

Law and Business Ethics and Social Responsibility Essay

Law and Business Ethics and Social Responsibility - Essay Example states. Thus, the major U.S. law primarily consists of state law. The U.S. law is framed from various sources including statutory law, constitutional law, administrative regulations, common law and treaties. At the state and federal levels, the U.S. law was mostly derived from U.K. Common law, enforced at the instant of the Revolutionary War. Although the American law has deviated significantly from its ancestor in terms of procedure as well as substance, and has integrated a great number of innovations in the civil law. The main aim of the law is to rehabilitate people and organizations violating the law. In the U.S. law stare decisis pertains to the  sharing  of a case than to  obiter dicta i.e., things supposed by the way, as it has been decided by the American Supreme Court that dicta are not binding but might be pursued if it is adequately persuasive. The U.S Supreme Court has made the stare decisis principle most flexible in cases of constitutional nature. It has been stated by the Supreme Court that is the court provides various reasons for a verdict then each reason unambiguously marked as an independent ground, by the court for the verdict should not be simply treated as a dictum (Burnham, 2006). Microsoft is one of the leading software companies. In terms of CSR, Microsoft has framed mission to serve global communities and play its role in addressing public causes. One such step taken by Microsoft in the fiscal year 2011 involves reaching almost 250 million teachers and students around the globe in collaboration with their partners in learning program by the year 2013 to provide technology to their classrooms (Microsoft 2011 Citizenship report,

Phd research propsal - Impact of Diabetes among the Lebanese Community Essay

Phd research propsal - Impact of Diabetes among the Lebanese Community in Sydney - Essay Example The potential research questions for this study include; According to a study conducted by World Health Organization (WHO)1, the number of people with diabetes is sharply on the rise in recent years. This study found out that in the year 2000 the number of people affected with the disease was about 171 million worldwide. But more alarming is the projection for the year 2030, if we continue to adopt the existing lifestyle. It has been projected that by 2030 this figure might reach a whopping 366 million. Though, India and China top the list of countries with maximum number of diabetes cases, the share of developing nations is certainly on the rise in the coming years. The increasing cases of diabetes amongst Lebanese people too are a cause of worry. Kristensen et al. (2007) find out that Lebanese population too has a high prevalence of diabetes, and their cultural and belief systems about healthcare worsen the situation of glycaemic regulation. Sydney is host to a large number of Lebanese people, who have been there in search for better job opportunities and better living standards. Now the pertinent question is why to have a focus on the Lebanese population in Sydney. Well, the foremost reason is - to have a focused approach while conducting the research study. In addition, a number of other reasons makes it an interesting topic. Bautista & Engler (2005) state that the Lebanese population in Sydney tends not to have any acculturation to the native Australian culture. There are many studies indicating increased prevalence of diabetes mellitus in these families. These have been ascribed to hereditary factors, food habits, prevalence of metabolic syndrome, and increased consanguinity over many generations (Abou-Daoud, 1969). Acculturation is a social phenomenon where family values tend to play important roles, and social and cultural factors related to the immigrant race determine the patterns of acceptance or resistance of newer cultural

Fame in Cinema and Television Essay Example for Free

Fame in Cinema and Television Essay The â€Å"star phenomenon† began in theatrical advertising of certain actors’ names in the 1820s. It was not immediately transferred to Hollywood, nor to the many other film industries developing in parallel across the glove. Hollywood studios at first, from about 1909 to 1914, ignored â€Å"stars† – actors in whose offscreen lifestyle and personalities audiences demonstrated a particular interest. This was partly because of the costs involved in â€Å"manufacturing stardom† on a scale which the studies could translate into measureable box-office revenue, and for fear of the power which stars might then wield. Stars need all kinds of resources lavished on their construction such as privileged access to screen and narrative space, to lighting, to the care of costumers, make-up workers, voice coaches, personal trainers, etc. , as well as to audience interest through previews, supply of publicity materials, etc. Skillful casting is also important, though rarely discussed in work on stars, perhaps because it is seen to detract from the star’s own intentions in a performance. Key career decisions involve a star’s choice of casting agency or the choices made by a particular film’s casting director. Once established, the star system worked lucratively for the studios. Stars were used as part of the studio’s â€Å"branding† or promise of certain kinds of narrative and production values. They were useful in â€Å"differentiating† studios’ films. Stars were literally part of the studio’s capital, like plant and equipment, and could be traded as such. James Stewart, making an interesting comparison with sports celebrities, said once â€Å"Your studio could trade you around like ball player like when I was traded once to Universal for the use of their back lot for three weeks. † Stars’ large salaries, said to be due to nebulous qualities such as â€Å"talent† or â€Å"charisma†, worked to negate the powers of acting unions, who might otherwise have been able to calculate acting labor and ask for more equal distribution of profits (Branston and Stafford 2003). And stars have always functioned as a key part of Hollywood’s relationship to broader capitalist structures. In the 1930s, for example, over-production of manufactured goods had reached crisis point in North America, and the large banks funding Hollywood sought its help in shifting goods from warehouses to consumers. In addition to this, the celebrity is part of the public sphere, essentially an actor or, to use Robert Altman’s 1992 film characterization of Hollywood denizens, a â€Å"player. † In the contemporary public sphere, divisions exist between different types of players: politicians are made to seem distinctly different from entertainment figures; businesspeople are distinguished from sports stars. And yet in the mediated representation of this panoply of players, they begin to blend together. Film stars like Arnold Schwarzenegger share the stage with politicians like George Bush; Gorbachev appears in a film by Wenders; Michael Jackson hangs out on the White House lawn with Ronald Reagan; Nelson Mandela fills an entire issue of Vogue. The celebrity is a category that identifies these slippages in identification and differentiation. Leadership, a concept that is often used to provide a definitional distance from vulgarity of celebrity status, provides the last discursive location for understanding the public individual. The argument I want to advance here is that in contemporary culture, there is a convergence in the source of power between the political leader and other forms of celebrity. Both are forms of subjectivity that are sanctioned by the culture and enter the symbolic realm of providing meaning and significance for the culture. The categorical distinction of forms of power is dissolving in favor of a unified system of celebrity status, in which the sanctioning of power is based on similar emotive and irrational, yet culturally deeply embedded, sentiments (Marshall 1997). Of course, depending on the type of media where actors and actresses appear, their power and charisma varies. In addition to this, depending on the type of media used, individual’s star quality or qualities of being a celebrity varies. On television, an individual can become a star without ceasing to be his or her anonymous self, because the medium celebrates innocuous, domestic normality. Once on the â€Å"The Tonight Show† Jack Paar maddened the studio audience by attentively quizzing one of its number and ignoring Cary Grant, who’d been planted in the adjoining seats. As well as a practical joke, this was a boast of television’s license to bestow celebrity on those it promiscuously or fortuitously favors. But the medium can just as easily rescind that celebrity. Obsolescence is built into the television star, as it is into the sets themselves: hence those mournful commercials for American Express in which the celebrities of yesteryear- the man who lent his croaky voice to Bugs Bunny or a candidate for the Vice-Presidency in 1964- laud the company’s card, which restores to them an identity and a televisibility they’d forfeited. The game show contestants experience this brief tenure of television celebrity- Warhol’s fifteen minutes- at its most accelerated. But in order to quality for it, they have to surrender themselves to the medium. Their only way of winning games is to abase themselves, feigning hysteria on â€Å"The Price is Right,† exchanging sordid confidences on â€Å"The Newlywed Game,† incompetently acting out inane charades on Bruce Forsyth’s â€Å"Generation Game. † The cruelest of the games is â€Å"The Gong Show,† where one’s span of celebrity may not even extend to fifteen seconds. More or less, untalented contestants sing, dance, juggle or fiddle until the inevitable gong sends them back to nonentity. For some, the gong supervenes immediately. They’ve been warned this will happen, and coached to disappear with dignity, but are expected to go through with their act all the same and suffer their condemnation. Even a few seconds of television fame is worth the price of one’s self-esteem. The show pretends to be a talent quest, but is a smirking parody of that. The hosts on the game shows are, for similar reasons, parodies of geniality. A host soothes his guests and smoothes obstacles out of their way. But in homage to Groucho, the comperes subject their victims to a ritual humiliation, and their patter keeps the game-players throughout flinching and ill-at-ease (Conrad 1982). Television is good but may not be ideal for preserving important works. On the other hand, a good film can be shown anywhere in the world where there is an audience. Furthermore, the cinema will turn actors and actresses into stars. There are many well-known television actors and actresses, but they have no international fame like their big-screen counterparts. Films together with film magazines contribute directly to the formation of a star system and its attendant mythology. The stars perceived themselves to be, and were in turn also used as, icons for a modern lifestyle, especially fashion (Zhang 2005). They are given greater chances to achieve or receive international awards and become known not only in a particular state but to the whole world, unlike in the case of television stars. Those famous actors who appeared on television ten years ago have now vanished due either to lot or disintegrated videotape or a lack of interest by the contemporary audience. In Africa, there was a necessity to build more cinema theaters, instead of enforcing further use of television, because it was helping them to maintain a viable film industry. In Iran, they have more than 150 cinema houses. Their industry if progressing because they have a loyal audience who make it possible to recuperate money invested in production, which in turn is invested in the making of new films (Ukadike 2002). As a whole, it can be said that fame in cinema is more lasting than fame in television. In addition to this, the stars or celebrities appearing on cinemas rather than on televisions are the ones who are more favored by producers and stockholders. Moreover, they are preferred than the television stars to be used in magazines, especially if it is an international magazine. As such, the lifestyle of actors and actresses in cinemas are greater than those who only appear in television shows. The cinema industry as well as its actors and actresses are greatly favored and nowadays, more specifically preferred by a good number of the countries. Bibliography BRANSTON, GILL and STAFFORD, ROY, The Media Students Book (USA: Routledge, 2003). CONRAD, PETER, Television (USA: Routledge, 1983). MARSHALL, P. DAVID, Celebrity and Power: Fame in Contemporary Culture (Minneapolis: Regents of the University of Minnesota, 1997). UKADIKE, NWACHUKWU FRANK, Questioning African Cinema (Minneapolis: University of Minnesota Press, 2002). ZHANG, ZHEN, An Amorous History of the Silver Screen (London: University of Chicago Press, 2005).

Vedic Mathematics Multiplication

Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure Vedic Mathematics Multiplication Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure