Add Maths Project Work - 1182 Words

PROJECT WORK FOR ADDITIONAL MATHEMATICS 2012 POLYGONS IN OUR LIFE Name: Class: Teacher: I/C number: CONTENT |No. |Title |Pages | |1 |Objectives | | |2 |Introduction | | |3 |Part 1 | | |4 |Part 2†¦show more content†¦Some other generalizations of polygons are described below. HISTORY OF POLYGONS Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C.[citation needed] Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine.[20] In 1952, Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.[21] [pic] The historical image of polygons 1969 SOME PICTURES OF POLYGONS METHODS OF FINDING AREA OF TRIANGLE Method 1: h b If you know base (b) and height (h) of the triangle, the following formula can be applied. Area =  ½ x b x h Method 2: a c b If you know three sides (a, b and c) of the triangle, Heron’s Method can be applied. s = (a+b+c) / 2 Area =( s (s-a) (s-b) (s-c) Method 3: a ( b If you know twoShow MoreRelatedUsing Math Vocabulary With Intentionality While Incorporating A Comprehensive Vocabulary Program1538 Words   |  7 PagesStudents will use math vocabulary with intentionality while incorporating a comprehensive vocabulary program that will be updated and utilized throughout the year as a main component of my instruction. Throughout the ELL certification program, vocabulary has been consistently emphasized as an invaluable strategy for the teaching and learning of ELL students. Although this has been stressed, it has not become a focus in my instruction; my plan is to develop a structure that will ensure that vocabularyRead MoreDifferent Kinds Of Energy And Which Ramp Will Be Faster1183 Words   |  5 Pagespotential energy, and Mechanical energy. I would also like to find out if the tallest or the lowest ramp will be faster. I will also tell you what my hypothesis is and the materials I will be using. Plus ill add a statistical question about what ramp is faster. I will also be talking about math and how to find the missing angle in a trangle. There are a few different kinds a energy and they are kinetic energy, potential energy, gravitational potential energy, elastic potential energy, chemical potentialRead MoreComparison Of My Math Strengths And Weaknesses866 Words   |  4 Pages My Math Strengths and Weaknesses As I am approaching the end of my educational career, it is time for me to evaluate how well I will do teaching a subject I struggled in elementary school, math. Math is a subject I had a love hate relationship with. I liked it when I was good and hated it when I struggled. I noticed this pattern not only because of my abilities, but also of my teacher’s abilities to teach the subject. My goal is not to be a genius in math but I do want to be able to helpRead MoreUsing Internet Sites With Sixth Graders1653 Words   |  7 PagesUsing Internet Sites with Seventh Graders My idea for having my seventh graders do a research project came from the session of my Walsh University class in which we were given the assignment of researching information on anything we were interested in with our electronic devices. This sparked an idea: How much more interested could my students be, than being given the opportunity to select their own topic to research? I knew I would be able to let them do their research at school on chromebooksRead MoreWhy Is A Manipulative?1223 Words   |  5 Pagesfractions bars and algebra tiles to name a few. These manipulatives are in the form of Java or Flash applets, a web base technology. Normal playing cards have so many uses in teaching mathematical lessons. When teaching your students how to add, subtract, sort, compute or compare numbers, using a deck of cards gets the students working with numbers while playing a game. There are two advantages of using a deck of cards as a hands-on manipulative: First advantage; cards are inexpensive, canRead MoreProgressivism : Teaching As A Future Educator896 Words   |  4 Pagesengage the students and ensure that what they are learning will stick with them and add to their educational goals. Throughout my education, I remember how the majority of my teachers were always the main focus of the classroom environment. I had to make sure I paid close attention in hopes of being able to understand what was being taught to me. There were times that I felt rushed to understand things such as math problems and before I knew it, I was being tested on it. With the progressivism styleRead MoreJob Report1125 Words   |  5 PagesDeveloper? †¢ A Senior Developer tends to oversee the development of current project’s they tend to be the leader of the project sometimes they have to perform coding assignments set by the company, Depending on the size of the project if they’re short on time the senior developer will have to write more or less code since they normally work in teams and do little bits for each project and piece them all together. Senior Developers also tend to do a lot of analysing because they have to look through allRead MoreDuring My Undergraduate Tenure, I Was Extremely Active1142 Words   |  5 Pagestenure, I was extremely active within the community as well as on campus. I worked with a wide spectrum of individuals, from fellow peers to Vice Chancellors. Several of the activities included tutoring, lab work and presentations. I believe, these works are the initial stones that laid the ground work for my current path to a graduate degree in the biomedical sciences. While at the University of Arkansas at Pine Bluff, one place I volunteered my time was within the STEM program. This program held SaturdayRead MoreMarketing Plan For A Business Plan1159 Words   |  5 Pagespopulation-wise. For instance if you are a College Graduate planning to offer Math tutoring to students in your community, then your business scope will be small and so will your startup costs. In fact, you could probably launch your business for next to nothing. You could market via word-of-mouth and do your tutoring at home, in the school library or even online via Skype. If on the other hand, you plan to expand your scope to tutoring Math, Science and English to any students within your city, then your costsRead MoreMusic Lessons Improve Life Essay1540 Words   |  7 Pagesall play the piano. Do they do it for fun or is it something else? Some recent studies have shown that there are improvements that people haven’t considered before. â€Å"The studies have shown th at children in music lessons scored higher in English and math than students who had no music lessons at all†¦Ã¢â‚¬  (Baedeker). So with lessons in music, students could do better in life than without music lessons. With improved academics, they are able to solve problems better and get to college and thus get better

The Fascinating Life and Times of Alexander Hamilton

Alexander Hamilton was born in the British West Indies in 1755 or 1757. There is some dispute of his birth year due to early records and Hamiltons own claims. He was born out of wedlock to James A. Hamilton and Rachel Faucett Lavien. His mother died in 1768 leaving him largely an orphan. He worked for Beekman and Cruger as a clerk and was adopted by a local merchant, Thomas Stevens, a man some believe to be his biological father. His intellect prompted leaders on the island to want him to be educated in the American colonies. A fund was collected to send him there to further his education. Education Hamilton was extremely smart. He went to a grammar school in Elizabethtown, New Jersey from 1772-1773. He then enrolled at Kings College, New York (now Columbia University) either late in 1773 or early in 1774. He later practiced law along with being a huge part in the founding of the United States. Personal Life Hamilton married Elizabeth Schuyler on December 14, 1780. Elizabeth was one of the three Schuyler sisters that were influential during the American Revolution. Hamilton and his wife  remained very close despite his having an affair with Maria Reynolds, a married woman. Together they built and lived in the Grange in New York City. Hamilton and Elizabeth had eight children: Philip (killed in a duel in 1801) Angelica, Alexander, James Alexander, John Church, William Stephen, Eliza, and Philip (born soon after the first Philip was killed.) Revolutionary War Activities In 1775, Hamilton joined the local militia to help fight in the Revolutionary War like many students from Kings College. His study of military tactics led him to the rank of lieutenant. His continued efforts and friendship to prominent patriots like John Jay led him to raise a company of men and become their captain. He was soon appointed to George Washingtons staff. He served as Washingtons untitled Chief of Staff for four years. He was a trusted officer and enjoyed a great deal of respect and confidence from Washington. Hamilton made many connections and was instrumental in the war effort. Hamilton and the Federalist Papers Hamilton was a New York delegate to the Constitutional Convention in 1787. After the Constitutional Convention, he worked  with John Jay and James Madison to try and persuade New York to join in ratifying the new constitution. They jointly wrote the Federalist Papers. These consisted of 85 essays of which Hamilton wrote 51. These had a huge impact not only on ratification but also on Constitutional law. First Secretary of the Treasury Alexander Hamilton was selected by George Washington to be the first Secretary of the Treasury on September 11, 1789. In this role, he had a huge impact in the formation of the U.S. Government including the following items: Assuming all the states debts from the war thereby increasing federal power.Creating the U.S. MintCreating the first national bankProposing an excise tax on whiskey to raise revenue for the federal governmentFighting for a stronger federal government Hamilton resigned from the Treasury in January, 1795. Life After the Treasury Although Hamilton left the Treasury in 1795, he was not removed from political life. He remained a close friend of Washington and influenced his farewell address. In the election of 1796, he schemed to have Thomas Pinckney elected president over John Adams. However, his intrigue backfired and Adams won the presidency. In 1798 with the endorsement of Washington, Hamilton became a major general in the Army, to help lead in case of hostilities with France. Hamiltons machinations in the Election of 1800 unwittingly led to Thomas Jeffersons election as president and Hamiltons hated rival, Aaron Burr, as vice president. Death After Burrs term as Vice President, he desired the office of governor of New York which Hamilton again worked to oppose. This constant rivalry eventually led to Aaron Burr challenging Hamilton to a duel in 1804. Hamilton accepted and the Burr-Hamilton duel occurred on July 11, 1804, at the Heights of Weehawken in New Jersey. It is believed that Hamilton fired first and probably honored his pre-duel pledge to throw away his shot. However, Burr fired at and shot Hamilton in the abdomen. He died from his wounds a day later. Burr would never again occupy a political office in large part due to the fallout from the duel.

The Umbrella by Yasunari Kawabata Free Essay Example, 1500 words

Sometimes love can be expressed by merely being silent. It is as though lovers speak rather telepathically. Love is so strong, it leaves one imagining things as though they were real even when they are not. The story revolves around the object umbrella. In the first instance, the boy is holding the umbrella, apparently shielding himself from the strokes of the rain. The author, from this instance, paints a picture of two young people, who are strongly attracted towards each other but have no ingenious ways of expressing these mutual feeling of attraction. Thus the umbrella is symbolic of the protection the boy is supposed to over. However, it is ironical since instead of using the umbrella as a symbol of outward protection for the girl in this case it is symbolic of inward protection as the boy uses it to hide his shyness. This is understandable since the teenagers are still afraid of what the society comments in relation to their union. For example, in the opening graph, when the girl comes out of the house and sees the boy with the umbrella, she shouts, "is it raining? † as though she cannot see that it is raining. We will write a custom essay sample on The Umbrella by Yasunari Kawabata or any topic specifically for you Only $17.96 $11.86/pageorder now The point being given is that she was keen on igniting a conversation, by first getting under the umbrella besides the boy. The boy on the other hand, though evidently attracted to the girl, remains silent and covers himself more with the umbrella, not to cover himself from the downpour but from betraying his shame. This is how kids express their love, through such simple and plain acts of shame and hesitation. When the girl gets under the umbrella, the boy was too ashamed to stick the whole of his body in the umbrella. The boy was afraid of getting too close, and for that, he let himself to be beaten up by the rain. The same feelings run through the girl’s body and mind. She really wants to get close to the boy by getting hold of the handle of the umbrella, but she could not bring herself to that point. Instead she looked as though she was about to run away. It is as though the boy does not really ache for the girl. Nevertheless, far from it, the boy is burning with a strong sensational feeling. However, the feeling is always there and so deep. As the story goes by, so does the author reveal this suppressed magnetic feeling between the boy and the girl. At the photographer, the boy chose not to sit with the girl and take the photograph together. Nevertheless, he stood behind her touching her cloak, and imagining themselves together. He imagined their bodies touching.

Analysis Of The Book The Slave Community - 1127 Words

Darien Wellman Age of Jackson to 1900 Dr. Gershenhorn September 1, 2015 Blassingame, John W. The Slave Community: Plantation Life in the Antebellum South. New York: Oxford University Press, 1972. In the book titled The Slave Community: Plantation Life in the Antebellum South, author John Blassingame’s theme, focused on the history of African slave experience throughout the American South. After much research, the author said in the preface that most historians focused more on the planter instead of the slave. He also pointed out that most of the research on slaves by previous historians was based on stereotypes, and do not tell the real history of slave life and a slave’s inner self. Most of these historians, who focused on antebellum southern history, left out the African-American slave experience on purpose. Through much gathering of research, Blassingame hoped to correct this injustice to the history of African-American slaves, and show how slavery affected slaves, but also American life, culture, and thought. In the first half of the bo ok, Blassingame did a fantastic job of explaining the conditions of African slaves coming to the American south, and being exposed to Western culture for the first time. The exposure to Western culture by Africans was something that was not focused on as much by other historians. One crucial example the author used was that of one African slave known as Olaudah Equiano, who was sent to a plantation in Virginia. Based on his own accounts,Show MoreRelatedReligious Practices Of African Culture Essay1405 Words   |  6 PagesThe driving theme of this book is healing in the context of religious practices, but not only in the way one would assume---the restoration of physical or spiritual health--- it includes more than just that. Rather, it encompasses the building of a community and the preservation of the African culture in the Americas. By using the evidence that Domingos Alvares employed his own knowledge and experiences from Africa, instead of the practices of colonial Portuguese institutions, Sweet reconstructsRead MoreArnt I a Woman? Essay examples1659 Words   |  7 PagesDeborah Gray White’s Ar’n’t I a Woman? details the grueling experiences of the African American female slaves on Southern plantations. White resented the fact that African American women were nearly invisible throughout historical text, because many historians failed to see them as important contributors to America’s social, economic, or political development (3). Despite limited historical sources, she was determined to establish the African American woman as an intricate part of American historyRead More Deborah Gray White’s Ar’n’t I a Woman? Essay examples1629 Words   |  7 PagesDeborah Gray White’s Ar’n’t I a Woman? details the grueling experiences of the Afri can American female slaves on Southern plantations. White resented the fact that African American women were nearly invisible throughout historical text, because many historians failed to see them as important contributors to America’s social, economic, or political development (3). Despite limited historical sources, she was determined to establish the African American woman as an intricate part of American historyRead MoreAnnotated Bibliography Of The Life Of Frederick Douglass Essay858 Words   |  4 Pagesrealized in a community of individuals who felt strongly about one another. The author’s idea is to explore how Douglass faced these challenges and liberal statesmanship. Douglass, Frederick. Narrative of the Life of Frederick Douglass: an American Slave. New York: Barnes and Noble, 2005. Print. This a book about Frederick Douglass’s remarkable life. He was born a slave in 1818 on a plantation in Maryland. He taught himself how to read and write becoming a renowned writer and orator. This book accountsRead MoreThe Fall Of The House Of Dixie956 Words   |  4 Pagesthe Negro community. The South’s great monopoly, including cotton and tobacco, clutched African American slaves tightly, digging in maliciously and squeezing the profitability out of each man and woman born into such misfortune. Levine says, â€Å"slaves in her [Katherine Stone] family’s cotton fields†¦ ‘pick five or six hundred pounds each day for maybe a week at a time’’’ (Levine 9). Slaves worked long, exhausting hours in fields, but did not have their personal needs met because slave holders deemedRead MoreAfrican American Women : An Examination Of Female Slavery1204 Words   |  5 PagesRobert Fogel, Stanley Engerman, Eugene Genovese, and Herbert Gutman have had a profound influence on research that uncovers the experiences of slaves in the antebellum South. Yet, these historians have only done so through the centered analysis of enslaved black men – this review will focus on two stereotypes and solidarity of women. Ar’n’t I A Woman? Female Slaves in the Plantation South by Deborah Gray White provides an examination of female slavery, in which, she analyzes the situation of the mostRead MoreAnalysis Of Myne Owne Ground And Its Implications For Race Today1291 Words   |  6 PagesFebruary 3, 2017 An Analysis of â€Å"Myne Owne Ground† and its Implications for Race Today In their thought-provoking but generally well-received book, Myne Owne Ground: Race and Freedom on Virginia’s Eastern Shore, 1640-1676 ¸ authors T.H. Breen and Stephen Innes investigate a heretofore little-known community of free blacks. Despite the burgeoning slave trade and generally held racist beliefs in pre-Colonial Virginia, the authors argue, convincingly, that a community based more on land ownershipRead MoreBioliographic Information on Slavery and Colonial Rule in French West Africa by Martin Klein1117 Words   |  5 PagesBibliographic Information: Klein, Martin A., Slavery And Colonial Rule in French West Africa, 1998, Cambridge University Press The book bears importance in being one of the few studies about domestic slavery within the French West Africa. It contributes to the field of study, by elaborating on the importance of slavery in Senegal, Sudan and Guinea in Africa’s development. The period of the study lies from 1876 to 1922, wherein Klein ultimately shows the evolution of slavery. In the years prior toRead MoreBorder War By Stanley Harrold1500 Words   |  6 PagesIn the book Border War, Stanley Harrold specifically searches the ideas of social clashes between the North and South before the civil war actually began. Harrold s research solely states the history leading up to the war, which clearly emphasizes the role of slavery and its importance in the history. Stanley Harrold stresses the real challenge of slavery especially in the south and in areas in which it did not exist. According to the book, Harrold specifically states that the U.S had a problemRead MoreThe Triumph Of Seretse Khama And His Natio n924 Words   |  4 Pagestheir marriage. In the book Serets encourages Africans to record their history for it is part of their soul. The book was chosen because he is one of the African leaders who are calling for written accounts of Africa’s history. His story exposes the ignorance and arrogance that are still excisting in twentierth-century African societies. These ignorance and arrogance are actively suppressing the continent’s growth and dividing the nations.Using the data collected from this book will awaken the reader

Lacsap’s Fractions Free Essays

Lacsap’s Fractions IB Math 20 Portfolio By: Lorenzo Ravani Lacsap’s Fractions Lacsap is backward for Pascal. If we use Pascal’s triangle we can identify patterns in Lacsap’s fractions. The goal of this portfolio is to ? nd an equation that describes the pattern presented in Lacsap’s fraction. We will write a custom essay sample on Lacsap’s Fractions or any similar topic only for you Order Now This equation must determine the numerator and the denominator for every row possible. Numerator Elements of the Pascal’s triangle form multiple horizontal rows (n) and diagonal rows (r). The elements of the ? rst diagonal row (r = 1) are a linear function of the row number n. For every other row, each element is a parabolic function of n. Where r represents the element number and n represents the row number. The row numbers that represents the same sets of numbers as the numerators in Lacsap’s triangle, are the second row (r = 2) and the seventh row (r = 7). These rows are respectively the third element in the triangle, and equal to each other because the triangle is symmetrical. In this portfolio we will formulate an equation for only these two rows to ? nd Lacsap’s pattern. The equation for the numerator of the second and seventh row can be represented by the equation: (1/2)n * (n+1) = Nn (r) When n represents the row number. And Nn(r) represents the numerator Therefore the numerator of the sixth row is Nn(r) = (1/2)n * (n+1) Nn(r) = (1/2)6 * (6+1) Nn(r) = (3) * (7) Nn(r) = 21 Figure 2: Lacsap’s fractions. The numbers that are underlined are the numerators. Which are the same as the elements in the second and seventh row of Pascal’s triangle. Figure 1: Pascal’s triangle. The circled sets of numbers are the same as the numerators in Lacsap’s fractions. Graphical Representation The plot of the pattern represents the relationship between numerator and row number. The graph goes up to the ninth row. The rows are represented on the x-axis, and the numerator on the y-axis. The plot forms a parabolic curve, representing an exponential increase of the numerator compared to the row number. Let Nn be the numerator of the interior fraction of the nth row. The graph takes the shape of a parabola. The graph is parabolical and the equation is in the form: Nn = an2 + bn + c The parabola passes through the points (0,0) (1,1) and (5,15) At (0,0): 0 = 0 + 0 + c ! ! At (1,1): 1 = a + b ! ! ! At (5,15): 15 = 25a + 5b ! ! ! 15 = 25a + 5(1 – a) ! 15 = 25a + 5 – 5a ! 15 = 20a + 5 ! 10 = 20a! ! ! ! ! ! ! therefore c = 0 therefore b = 1 – a Check with other row numbers At (2,3): 3 = (1/2)n * (n+1) ! (1/2)(2) * (2+1) ! (1) * (3) ! N3 = (3) therefore a = (1/2) Hence b = (1/2) as well The equation for this graph therefore is Nn = (1/2)n2 + (1/2)n ! which simpli? es into ! Nn = (1/2)n * (n+1) Denominator The difference between the numerator and the denominator of the same fraction w ill be the difference between the denominator of the current fraction and the previous fraction. Ex. If you take (6/4) the difference is 2. Therefore the difference between the previous denominator of (3/2) and (6/4) is 2. ! Figure 3: Lacsap’s fractions showing differences between denominators Therefore the general statement for ? nding the denominator of the (r+1)th element in the nth row is: Dn (r) = (1/2)n * (n+1) – r ( n – r ) Where n represents the row number, r represents the the element number and Dn (r) represents the denominator. Let us use the formula we have obtained to ?nd the interior fractions in the 6th row. Finding the 6th row – First denominator ! ! ! ! ! ! ! ! ! ! ! ! – Second denominator ! ! ! ! ! ! ! ! ! ! ! ! ! denominator = 6 ( 6/2 + 1/2 ) – 1 ( 6 – 1 ) ! = 6 ( 3. 5 ) – 1 ( 5 ) ! 21 – 5 = 16 denominator = 6 ( 6/2 + 1/2 ) – 2 ( 6 – 2 ) ! = 6 ( 3. 5 ) – 2 ( 4 ) ! = 21 – 8 = 13 ! ! -Third denominator ! ! ! ! ! ! ! ! ! ! ! ! – Fourth denominator ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! – Fifth denominator ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! denominator = 6 ( 6/2 + 1/2 ) – 3 ( 6 – 3 ) ! = 6 ( 3. 5 ) – 3 ( 3 ) ! = 21 – 9 = 12 denominator = 6 ( 6/2 + 1/2 ) – 2 ( 6 – 2 ) ! = 6 ( 3. 5 ) – 2 ( 4 ) ! = 21 – 8 = 13 denominator = 6 ( 6/2 + 1/2 ) – 1 ( 6 – 1 ) ! = 6 ( 3. 5 ) – 1 ( 5 ) ! = 21 – 5 = 16 ! ! We already know from the previous investigation that the numerator is 21 for all interior fractions of the sixth row. Using these patterns, the elements of the 6th row are 1! (21/16)! (21/13)! (21/12)! (21/13)! (21/16)! 1 Finding the 7th row – First denominator ! ! ! ! ! ! ! ! ! ! ! ! – Second denominator ! ! ! ! ! ! ! ! ! ! ! ! – Third denominator ! ! ! ! ! ! ! ! ! ! ! ! – Fourth denominator ! ! ! ! ! ! ! ! ! ! ! ! ! ! denominator = 7 ( 7/2 + 1/2 ) – 1 ( 7 – 1 ) ! =7(4)–1(6) ! = 28 – 6 = 22 denominator = 7 ( 7/2 + 1/2 ) – 2 ( 7 – 2 ) ! =7(4)–2(5) ! = 28 – 10 = 18 denominator = 7 ( 7/2 + 1/2 ) – 3 ( 7 – 3 ) ! =7(4)–3(4) ! = 28 – 12 = 16 denominator = 7 ( 7/2 + 1/2 ) – 4 ( 7 – 3 ) ! =7(4)–3(4) ! = 28 – 12 = 16 ! ! ! ! ! ! Fifth denominator ! ! ! ! ! ! ! ! ! ! ! ! – Sixth denominator ! ! ! ! ! ! ! ! ! ! ! ! denominator = 7 ( 7/2 + 1/2 ) – 2 ( 7 – 2 ) ! ! =7(4)–2(5) ! ! = 28 – 10 = 18 ! ! denominator = 7 ( 7/2 + 1/2 ) – 1 ( 7 – 1 ) ! =7(4)–1(6) ! = 28 – 6 = 22 We already know from the previous investigation that the numerator is 28 for all interior fractions of the seventh row. Using these patterns, the elements of the 7th row are 1 (28/22) (28/18) (28/16) (28/16) (28/18) (28/22) 1 General Statement To ? nd a general statement we combined the two equations needed to ? nd the numerator and to ? nd the denominator. Which are (1/2)n * (n+1) to ? d the numerator and (1/2)n * (n+1) – n( r – n) to ? nd the denominator. By letting En(r) be the ( r + 1 )th element in the nth row, the general statement is: En(r) = {[ (1/2)n * (n+1) ] / [ (1/2)n * (n+1) – r( n – r) ]} Where n represents the row number and r represents the the element number. Limitations The ‘1’ at the beginning and end of each row is taken out before making calculations. Therefore the second element in each equation is now regarded as the ? rst element. Secondly, the r in the general statement should be greater than 0. Thirdly the very ? rst row of the given pattern is counted as the 1st row. Lacsap’s triangle is symmetrical like Pascal’s, therefore the elements on the left side of the line of symmetry are the same as the elements on the right side of the line of symmetry, as shown in Figure 4. Fourthly, we only formulated equations based on the second and the seventh rows in Pascal’s triangle. These rows are the only ones that have the same pattern as Lacsap’s fractions. Every other row creates either a linear equation or a different parabolic equation which doesn’t match Lacsap’s pattern. Lastly, all fractions should be kept when reduced; provided that no fractions common to the numerator and the denominator are to be cancelled. ex. 6/4 cannot be reduced to 3/2 ) Figure 4: The triangle has the same fractions on both sides. The only fractions that occur only once are the ones crossed by this line of symmetry. 1 Validity With this statement you can ? nd any fraction is Lacsap’s pattern and to prove this I will use this equa tion to ? nd the elements of the 9th row. The subscript represents the 9th row, and the number in parentheses represents the element number. – E9(1)!! ! – First element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(2)!! ! – Second element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(3)!! ! – Third element! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 1( 9 – 1) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 1( 8 ) ]} {[ 45 ] / [ 45 – 8 ]} {[ 45 ] / [ 37 ]} 45/37 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 2( 9 – 2) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 2 ( 7 ) ]} {[ 45 ] / [ 45 – 14 ]} {[ 45 ] / [ 31 ]} 45/31 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 3 ( 9 – 3) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 3( 6 ) ]} {[ 45 ] / [ 45 – 18 ]} {[ 45 ] / [ 27 ]} 45/27 E9(4)!! ! – Fourth element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(4)!! ! – Fifth element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(3)!! ! – Sixth element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(2)!! ! – Seventh element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(1)!! ! – Eighth element! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! [ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 4( 9 – 4) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 4( 5 ) ]} {[ 45 ] / [ 45 – 20 ]} {[ 45 ] / [ 25 ]} 45/25 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 4( 9 – 4) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 4( 5 ) ]} {[ 45 ] / [ 45 – 20 ]} {[ 45 ] / [ 25 ]} 45/25 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 3 ( 9 – 3) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 3( 6 ) ]} {[ 45 ] / [ 45 – 18 ]} {[ 45 ] / [ 27 ]} 45/27 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 2( 9 – 2) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 2 ( 7 ) ]} {[ 45 ] / [ 45 – 14 ]} {[ 45 ] / [ 31 ]} 45/31 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 1( 9 – 1) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 1( 8 ) ]} {[ 45 ] / [ 45 – 8 ]} {[ 45 ] / [ 37 ]} 45/37 From these calculations, derived from the general statement the 9th row is 1 (45/37)! ! (45/31)! ! (45/27)! (45/25)! (45/25)! (45/27) (45/31)! (45/37)! ! 1 Using the general statement we have obtained from Pascal’s triangle, and following the limitations stated, we will be able to produce the elements of any given row in Lacsap’s pattern. This equatio n determines the numerator and the denominator for every row possible. How to cite Lacsap’s Fractions, Essay examples

A Separate Peace Three Symbols Essay Research free essay sample

A Separate Peace: Three Symbols Essay, Research Paper A Separate Peace: Three Symbols The three dichotomous symbols in A Separate Peace by John Knowles reenforce the artlessness and immorality of the chief characters, Finny and Gene. Beside the Devon School flow two rivers on opposite sides of the school, the Naguamsett and the Devon. The Devon provides amusement and felicity for Gene and Finny as they jump from the tree into the river and keep inductions into the Super Suicide Society of the Summer Session. Finny, Gene, and their friends use the Devon # 8217 ; s warm H2O to play in during the unworried summer session. The Devon brings out Finny # 8217 ; s carefree character and personality when he jumps from the limbs of the tree. Not one Upper Middler in Devon has of all time jumped from the tree ; Finny becomes the first. After come uping, Finny says that leaping from the tree causes the most merriment he has had in hebdomads. We will write a custom essay sample on A Separate Peace Three Symbols Essay Research or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page However, the Naguamsett and the Devon wholly contrast. When Gene and Finny emerge from the Devon, they experience clean and refreshed. However, Gene describes the Naguamsett as # 8220 ; ugly, saline, fringed with fen, clay and seaweed # 8221 ; ( 68 ) . When Gene starts a battle with Quackenbush and falls into the Naguamsett because Quackenbush calls Gene # 8220 ; a maimed son-of-a-bitch, # 8221 ; Gene surfaces from the Naguamsett feeling grimy, dirty and in despairing demand of a bath ( 71 ) . Much like the clean, reviewing H2O of the Devon and the ugly saline H2O of the Naguamsett, Gene # 8217 ; s unworried attitude of the summer session immensely differs from the angry, baffled attitude of the winter session. Similarly, the two Sessionss, the summer and winter, give a different sense of experiencing toward school and life at Devon School. The summer session allows Finny to utilize his creativeness. Finny invents blitzball and founds the Super Suicide Society of the Summer Session. The pupils let their carefree attitudes flow during the summer. Finny and Gene volitionally break the regulations to hold merriment during the summer by jumping category and traveling to the beach. Finny besides wears the school tie as a belt to the traditional term tea. Gene feels that Finny can non go forth the room without being disciplined, but Finny manages to speak his manner out of the muss. However, the winter session causes a sense of stringency. The discourses now exhort the idea of # 8220 ; what we owe Devon, # 8221 ; but in the summer the pupils think of # 8220 ; what Devon owes us # 8221 ; ( 65 ) . The Masterss and category leaders try to implement continuity, but Gene realizes that raising the summer session becomes impossible. Finny is non in school, no longer shall the pupils have their unworried attitudes, and the category functionaries and Masterss now implement the regulations at Devon. Gene becomes like the winter session by salvaging a cold blast for the enemy. The winter lives to destruct the heat of the summer and does so by unleashing an unpredictable cold snowstorm. Likewise, Gene destroys Finny by let go ofing an uncontrolled jouncing of the tree limb. However, the peaceable clip and the war clip clearly display the artlessness of Finny and the immorality of Gene. During the peaceable clip, non one pupil thinks about a war. Gene and Finny play blitzball and leap from the tree, doing them both happy. Finny volitionally breaks the regulations at Devon. Like the summer session, the regulations do non be, and the pupil # 8217 ; s heads run rampantly with sloppiness. Finny # 8217 ; s imaginativeness and creativeness explode during the peaceable clip with innovations like blitzball and the initiation of the Super Suicide Society of the Summer Session. However, the war, like the winter session, brings about confusion and ill will. Students like Leper and Quackenbush get down believing about inscribing in the ground forces. Even Gene considers enlisting until he realizes that Finny needs him. Finny can non manage the alterations during the winter session. When Gene explains to Finny that a war is happening, Finny admirations, # 8220 ; Is there? # 8221 ; ( 96 ) . Finny garbages to believe in the war when Gene explains that the war comes before athleticss. Finny comes to the decision that old fat work forces in Washingtion, D.C. # 8220 ; do up # 8221 ; the war to flim-flam the people, and merely the fat work forces understand the fast one. The two rivers, the two Sessionss, and the two scenes, reinforce and clearly expose the artlessness of Finny and the immorality of Gene. 344

Issues and Ideas of Polygamy by Aisha B. Lemu free essay sample

Issues and ideas of Polygamy by Aisha B. Lemu In the essay of Defense of Polygamy by the british, feminist, activist and writer Aisha B. Lemu, explains why polygamy could be a great solution for many women without partners or for many women with partners but with difficulties in their relations. She explain why because of the large numbers of men dying in war, and others violent confrontation groups, a huge group of widows, fiances, and women in general, have left alone without any protection for themselves or their children. The quantities in general of men in these days are low and have left a large group of women alone with no or difficult chances to get married or have a partner. ?Therefore, she states that most of women rather believe in sharing a man that not having any at all. And even more if the political laws of the country establish an obligation toward the second, third or fourth wife and their respective children, making a responsible way of supporting the rights of the woman. We will write a custom essay sample on Issues and Ideas of Polygamy by Aisha B. Lemu or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page For me this first demonstration makes a logical reason, but not the answer for a truly right solution. I dont think polygamy is the true unique solution for the uniformity of our era. I think the true problem here is the violence and misunderstanding that we had made with each other and that has lead us to unnecessary wars and undesirable misbalance of the masses of the gender. Not attacking the real problem but rather settling ourselves to the problem, content to leave the wrong things we have done as they are and conforming to live with it. It’s like the problem of deforestation; we know that the problem is there but we dont resolve it, we prefer to put on oxygen masks in the future than to attack the real problem now. So then I believe that her reason of polygamy because of the war issue is not a true solution but a momentary accordance to survival, not attacking or giving a real answer (which is to me the increase of violence of everyday that has made us left with a sex misbalance). But it doesnt end there, a big question is about to expose, at it is What will happen then in the future or into a few more years where the disorder of sexes will may be so dramatic that for every man twelve women will be shared? Then what? Polygamy will maybe not be a truth solution and then maybe we will have to attack the true problems of violence and deaths of men. Another point that the author exposed is the circumstances where the wives are sick and the husbands cant manage it or where wives are unable to have children while the husband very much wants them. This argument represents a more logical thought, although not entirely correct in my opinion and own personal believes. I just cant still understand why it is necessary to go get another partner just for the benefit of a baby, when you could solve it renting a belly without resorting to commitments with other woman. But even if it happens or not, I think that the choices of a couple are personal choices and if the wife agrees then is her own personal choice. ? The other point that exposes is the cases where a marriage has not been very successful and the husband loves another woman. The author also express some things about when a husband need to divorce from the first wife to be with the second wife, but the first wife doesnt want to be divorce for the secure of the marriage, for herself and their children. Before starting to explain anything else I do think is quite understandable the worry that a wife could have for the economical and physical securities that a man provides (especially on these countries were most of the woman depends economically in the husband), but for the secure of the marriage? What secure? Now, the author needs to explain this in more details, because if the concept of secure marriage is to cover up from the society the very truth status of a relationship because of the judgments that this could bring, then I have very poor understanding the meaning of the term marriage, it seems that this definition doesn’t involucrate then a covenant of love with a partner but a commitment with cultural society. Whatever of all the case is, we must keep in mind that it is obvious that we will some how disapprove these ideals and thoughts, at least, obvious, if you are part of that kind of cultures and believes. Each culture is different and the analysis that we give others incorporating it into our own is simply wrong and insignificant. We just cant measure the same cultures with the same tool; every culture it’s unique. And the differences that demonstrate each one of them with the others doesnt gave us the right to classify them as less or more. Having said that, it was obvious that my answers were not going to be that much of acceptable. References Aisha B. Lemu, In Defense of Polygamy 2007; http://moodle. oss. cayey. upr. edu/dev/mod/resource/view. php? inpopup=trueamp;id=4434 (all ideas and references were taking of this)