Division Strategies: Partial Quotient Strategy Fold-Up & Practice | Math

# Division Strategies: Partial Quotient Strategy Fold-Up & Practice

Teaching a New Division Strategy

MATHEMATIC HISTORY

Mathematics is one of the oldest sciences in human history. In ancient times, Mathematics was defined whilst the science of numbers and shapes. Mathematics, like other branches of science, has evolved over time; it’s no longer possible to explain it in a few sentences. What I have to express now is going to be words that emphasize its various aspects, as opposed to describe mathematics. In one aspect, mathematics is a skill like painting and music. A large proportion of mathematicians perform it being an art. Out of this viewpoint, the fact a work done, a developed theory works in one of the ways or another besides mathematics does not concern them much. What matters to them is the depth of the job done, the novelty of the strategy used, the aesthetic value and the usefulness of mathematics in itself. Mathematics, in another aspect, is really a language. If the objective of science is the universe; If it is to comprehend, rule and direct everything in the universe, we ought to have the ability to browse the book of nature. The book of nature is written in the language of mathematics, with the highly cited words of Galile; its letters are shapes of geometry. To be able to understand and interpret them, we have to know the language of mathematics. In another aspect, mathematics is an intellectual game like chess.

Some mathematicians also see it as a game. Mathematics is merely a tool for the user. After entering it, we understand and perceive what mathematics is in your knowledge and in the direction of our interest. Mathematics is currently far beyond the dimensions any human can rule. Therefore, I do not believe people who cope with mathematics tend to be more than we understand and perceive it from mathematics compared to blind touched net understands and perceives the elephant. The word mathematics, for the first time, BC. It absolutely was employed by the members of the Pythagorean school in the 550s. His entry into the written literature, with Plato BC. It absolutely was in the 380s. The word meaning is “what needs to be learned”, that is, information. In the years before these dates, as opposed to the word mathematics, words which means that geometry, equivalent to it in geometry or old languages ​​were used.

It’s difficult to say anything definite about where and how mathematics started. If we take documents that are not centered on archaeological findings that need interpretation, but open enough to require interpretation, We could say so it started between 3000 and 2000 in Egypt and Mesopotamia. Based on Heredotus (485-415 BC), mathematics were only available in Egypt. As you know, 97% of the Egyptian lands aren’t suited to agriculture; It is the 3% portion that offers life to Egypt and forms the Nile delta. Therefore, these lands are really valuable. However, at the conclusion of the floods caused by the Nile river every year, the boundaries of the landowners’lands become obscure. Because the landowners also pay taxes in proportion to the land they own, after each and every flood, the “geometricists” of their state, who’re in charge of these works, should come to take the required measurements and give the landowners just as much land as they’d in the earlier year. Herodotus says that geometry has begun to emerge consequently of those measurements and calculations. A second opinion in regards to the birth of mathematics is the one put forward by Aristotle (384-322 BC). According to Aristotle, mathematics was created in Egypt. However it was born from the boredom of clergymen and priests, not the requirement for measurement-calculation caused by Nile floods. In those days, the only real intellectual class of countries such as for instance Egypt was the priest class. Because the livelihood of this class is supplied by the general public or the state, they have much time for you to give to intellectual pursuits. To help keep them busy, they invented geometry and arithmetic, the mathematics of that time, just as others invented games like chess, bridge, and go&hellip ;.Both these views may be true; priests wished to simplify the job of the geometric, or they found out how to calculate the areas of some geometric shapes such as for instance triangular and trapezoidal to test that the distribution was fair, and in this manner generated the birth of geometry.

We shall divide the written history of mathematics into five periods. The very first period is likely to be Egypt and Mesopotamia; this period BC In 2000s BC. It will cover an amount of 1500-2000 years between 500s. The 2nd period, BC. 500-M.S. It’ll cover a period of 1000 years, referred to as the Greek Mathematics period, between 500 years. The next term, M.S. It will cover a 1200-year period from the 500’s until the start of calculus and will mainly cover European mathematics in the Hind, Islam and Renaissance era. The fourth semester will cover the classical mathematics era, referred to as the golden age of mathematics, dating from 1700-1900. The time we’re surviving in, dating back once again to early 1900s, called age modern mathematics, will be the fifth period. I will endeavour to provide details about the development of mathematics in that period, contributing mathematicians, the place of mathematics in social life and the essential top features of mathematics because period.

We shall start the first semester with Egyptian mathematics. Written documents about ancient Egyptian mathematics and generally Egyptian history – I don’t mean the remains of archaeological works – are almost nonexistent. You can find two major causes for this. The foremost is that the ancient Egyptians wrote the writing on papyrus; The 2nd reason is the 3 big fires of the Alexandria libraries, the final of those fires happened throughout the conquest of Egypt by 641 Muslims, the written documents disappeared. Papyrus is the leaves of a reddish, reed type plant growing in the Nile delta, an average of 15-25 meters long and 30-50 inches wide. These leaves were used to create text as opposed to paper after cutting, joining, pressing and undergoing some simple operations. Words in western languages ​​such as “Paper”, “papier” are derived from the word papyrus. The common lifespan of a papyrus is 300 years; 300 years later, it is flaky as a result of moisture, heat and similar reasons. Up to now, two papyrus related to mathematics appear to have been hidden under exceptional circumstances. The key sourced elements of our knowledge of Egyptian mathematics are these two papyri. The very first of the papyrus is really a 6-meter long and 35-cm wide papyrus referred to as the Ahmes (or Rhind) papyrus. This papyrus, BC. You’re a puree written in 2000s, BC. It is a copy compiled by a “mathematician” named Ahmes in the 1650s. This papyrus was bought by the Irish antiquarian H. Rhind in the 1850s, now in the British museum. This papyrus is a book written to instruct math. In the introduction part, after a few exercises given to instruct operations with fractional numbers, 87 questions are made making use of their solutions. They are the sort of questions people can encounter in daily life, such as for instance sharing account, interest calculation, or finding the area of ​​some geometric shapes. This is just about our 8th grade mathematics. The next papyrus, referred to as the Moscow papyrus and now in the Moscow museum, can also be BC. It is really a booklet written in the 1600s. This papyrus contains 25 questions. These questions are of the sort of questions in the Ahmes papyrus, with the exception of the two. As for the other two questions, one of them could be the calculation of the volume and area of ​​the surface of the sphere part cut with a plane. Another could be the question of finding the amount of a pyramid cut by a plane. Both questions were solved correctly. Both of these questions are accepted as the pinnacle of Egyptian mathematics. The Egyptians realized that the region of ​​the circle was proportional to its diameter and found how many pi to be 4x (8/9) squared, ie 256/81 = 3.16. It’s understood that Egyptian mathematics has remained as of this level for 2000 years and has not made any significant progress.

B.C. 600s are the years once the Persians started initially to dominate the center east. B.C. By the 550s, Persians are the sole rulers of the whole middle east, including Anatolia and Egypt. The Persians organize three trips to Greece between 500-480 BC; They captured Athens in 480, but burned it, annually later, in 479, Greeks expelled the Persians from Greece. This date BC. 479 may be the date that was accepted as the beginning of Greek civilization. This date is the beginning of an extremely bright period in science, art and literature. Greek mathematics actually started prior to when this period. Two different people, Tales (624-547 BC) and Pythagoras (569-475 BC), are regarded as the daddy of Greek mathematics. Tales Milet (Aydın) was also born. It is famous he visited Egypt, stayed there for some time and learned geometry in Egypt. While in Egypt, it is described in books where he calculates the height of the fantastic pyramid by measuring the size of the shadow of the truly amazing pyramid, multiplying this number by the ratio of its length to the size of the current shadow. After time for Tales Milet, he taught them geometry by forming friends around him to instruct what he learned. It’s assumed that abstract proof centered on reasoning, which is not centered on mathematics – experimental verification, entered into Tales. Additionally, Tales is the person who is recognized as the first philosopher in human history. He was created on the island of Pythagoras Samos (Samos), another father of Greek mathematics. Pythagoras stayed with Tales for a time, visited Egypt following his advice, learned geometry there, visited Egyptian temples, learned religious information, and was taken to Babylon by capturing the Persians during the occupation of Egypt by the Persians. it’s known. During his 5 years in Babylon, he learned mathematics, music and religious information, and after time for Samos, he created a college and tried to show individuals he gathered around. For political reasons, BC. He left 518 Samos, settled in southern Italy, in the town of Crotone, where he created a semi-mystical-semi-scientific, cult-like school. The senior folks of this school called “mathematics” live together and they’re connected to one another with oath. The 2nd group contains students attending school. Pythagoras school is founded on number cult. According to them, everything could be reduced to numbers; It has an unusually perfect harmony among numbers, and harmony is a reflection of the divine harmony. Known numbers for that day are integers indicating the plurality such as for example 1,2,3,…; and kes, ¾,… will be the fractional numbers that indicate the ratio of the part to the whole. The emergence of irrational numbers with the theorem referred to as the Pythagorean theorem (the square of the best sides of the right triangle equals the square of the hypotenuse) put the Pythagorean school in a deep crisis. The discovery of irrational numbers is the initial major crisis of mathematics. Lots of the members of the Pythagorean school were massacred by a raid led by way of a big cyber named Cylon. Pythagoras saved his life, but after many years he died. Pythagoras’thoughts, the Pythagorean school lived for several years under this or that name. As can be understood from these records, Egyptian and Mesopotamian mathematics are the basis of Greek mathematics.