DOCSLIB.ORG

Pythagorean Theorem Project

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Pythagorean Theorem Project Wheel of Theodorus Project Grading Rubric Grading rubric allows student to track their progress and calculate total possible points earned for the completion of this assigned project. The points listed represent the maximum total points available for satisfactory completion of each assigned task. Based on the student’s answers and quality of work on these assigned tasks; all, part or none of the points may be earned. Construction and Labeling of the Wheel: (25 points) Wheel contains a minimum of 17 triangles. (5 pts) _________ True center is marked clearly with minimal radius. (5 pts) _________ All triangles constructed properly with no gaps or overlays. (5 pts) _________ All right angles and legs are marked correctly. (5 pts) _________ Each hypotenuse length is marked correctly (radical form). (5 pts) _________ Calculation Chart: (40 points) Pythagorean Theorem is written at top of chart (10 pts) _________ All entries completed for the first 17 triangles. (10 pts) _________ Measure for hypotenuse, shown in radical form. (10 pts) _________ Measure for hypotenuse, shown in decimal form . (10 pts) _________ Project Questions: (20 points) Question #1 (4 pts) _________ Question #2 (4 pts) _________ Question #3 (4 pts) _________ Question #4 (4 pts) _________ Question #5 (4 pts) _________ Creativity: (15 points) Neatness and showing all work for each step. (3 pts) _________ Design is original. (3 pts) _________ Design is decorated with attention to detail (3 pts) _________ Effective incorporation of the wheel into the theme of the design. (3 pts) _________ Color is used effectively unless B&W contrast is part of intentional design. (3 pts) _________ Total Points (out of 100) _________ THEODORUS [OF CYRENE] FACT SHEET Birth Theodorus [of Cyrene], was born in the 5th century around 425 BC, in Cyrene Greece Theodorus moved to Athens, Greece where he spent most of his life and was certainly in Athens at a time when Socrates was alive. Education and Career Theodorus [of Cyrene], was a Greek Mathematician and a Pythagorean, (a member of devoted followers of Pythagoras), and a pupil of Pythagoras and Protagoras, but also a teacher and tutor of mathematics to Plato and Theatus. Our knowledge of Theodorus comes from Plato who wrote about him in his work Theaetetus and the Sophist. He was distinguished in astronomy, arithmetic, music and educational subjects, and is best remembered by mathematicians for his contribution to the development of irrational numbers and it is this aspect of his work which Plato referred to. Circumstances of Death Theodorus [of Cyrene] was a philosopher of the Cyranic School. He lived in both Greece and Alexandria, before ending his days in his native city of Cyrene where he died in 398 BC. He was condemned to die by being forced to drink poison. THEODORUS [OF CYRENE] PROJECT QUESTIONS 1. When and where was Theodorus born? ______________________________________________ ______________________________________________ 2. Where did Theodorus spend most of his working life? ______________________________________________ 3. Name one of his most influential teachers. ______________________________________________ 4. Name one of his most influential pupils. ______________________________________________ 5. According to legend; how did Theodorus die? ______________________________________________ WHEEL OF THEODORUS CALCULATION CHART 1. Determine the length of each hypotenuse using the Pythagorean Theorem. 2. Record the measures of both (leg a) and (leg b) 3. Record hypotenuse value in its radical form 4. Using a calculator; find and record the equivalent decimal number to the radical. 5. State the Pythagorean Theorem: __________________________________ Triangle Measure Measure Measure of Hypotenuse Measure of Hypotenuse of Leg a of Leg b (in radical form) (in decimal form) 1 1 1 2 1.414213562 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Directions: Start by marking a central point on your paper that all triangles will rotate around and draw a right triangle with both legs being 1 inch in length. Use a ruler, to be exact, and a note card to help make right angles. Draw the hypotenuse. Now use the hypotenuse of this triangle as the leg of the next triangle and make the second leg 1 inch in length. Again, make sure to use a ruler and note card for accuracy. Draw in the hypotenuse. Continue this process until all triangles have been created. Now outline your wheel and turn it into a picture. Be creative! Be colorful, and use your imagination. .
Load more

Recommended publications
  • Plato As "Architectof Science"
    Plato as "Architectof Science" LEONID ZHMUD ABSTRACT The figureof the cordialhost of the Academy,who invitedthe mostgifted math- ematiciansand cultivatedpure research, whose keen intellectwas able if not to solve the particularproblem then at least to show the methodfor its solution: this figureis quite familiarto studentsof Greekscience. But was the Academy as such a centerof scientificresearch, and did Plato really set for mathemati- cians and astronomersthe problemsthey shouldstudy and methodsthey should use? Oursources tell aboutPlato's friendship or at leastacquaintance with many brilliantmathematicians of his day (Theodorus,Archytas, Theaetetus), but they were neverhis pupils,rather vice versa- he learnedmuch from them and actively used this knowledgein developinghis philosophy.There is no reliableevidence that Eudoxus,Menaechmus, Dinostratus, Theudius, and others, whom many scholarsunite into the groupof so-called"Academic mathematicians," ever were his pupilsor close associates.Our analysis of therelevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' Catalogue of geometers, and Philodemus'History of the Academy,etc.) shows thatthe very tendencyof por- trayingPlato as the architectof sciencegoes back to the earlyAcademy and is bornout of interpretationsof his dialogues. I Plato's relationship to the exact sciences used to be one of the traditional problems in the history of ancient Greek science and philosophy.' From the nineteenth century on it was examined in various aspects, the most significant of which were the historical, philosophical and methodological. In the last century and at the beginning of this century attention was paid peredominantly, although not exclusively, to the first of these aspects, especially to the questions how great Plato's contribution to specific math- ematical research really was, and how reliable our sources are in ascrib- ing to him particular scientific discoveries.
    [Show full text]
  • Chapter 1. Classical Greek Mathematics
    Chapter 1. Classical Greek Mathematics Greek science and mathematics is distinguished from that of earlier cultures by its desire to know, in contrast to a need to make purely utilitarian advances or improvements. Greek geometry displays abstract and deductive elements which were largely lost during the Dark Ages, following the collapse of the Roman Empire, and only gradually recovered in the 16th and 17th centuries. It must be understood that many of the great discoveries in geometry were made about two and half thousand years ago. Given the difficulty of preserving fragile manuscripts, written on parchment or papyrus, over centuries when warfare could wipe out civilizations, it is not too surprising to find that we do not have many reliable records about the origin of Greek geometry or of its practitioners. We may count ourselves lucky that a few commentaries on Greek geometry, written in the fourth or fifth centuries of the present era, have survived to provide us with what details we have. Greeks from Ionia had settled in Asia Minor and there they had contact with two ancient civilizations, those of Babylon and Egypt. Although knowledge of science was elementary among the Babylonians and Egyptians, nonetheless they supplied the initial impetus that directed the Greeks towards the pursuit of systematic science. All authorities trace the beginnings of Greek geometry to Egypt. The rudimentary study of geometry in Egypt arose out of practical needs. Revenue was raised by taxation of landed prop- erty, and its assessment depended on the accurate fixing of the boundaries of fields in the possession of the landowners.
    [Show full text]
  • Irrationals Or Incommensurables III: the Greek Solution
    Irrationals or Incommensurables III: The Greek Solution PH I L L I P S. J 0 N E S CBoTH THE PYTHAGOREAN number-theoretic with 17; but the fact that he treated only these proof of the irrationality of ;f2 (or of the incom­ cases indicates that he still did not have a general mensurability of the side and diagonal of a square) process.2 Theaetetus, on the other hand, is supposed and the successive subtraction process which we to have been the original source for such general applied to the square and the pentagon in earlier theorems as Euclid, Book X, Proposition 8: If two articles' required generalization before one could magnitudes have not to one another the ratio which a claim that mathematicians had a theory of number has to a number, the magnitudes will be in­ irrationals. Some generalizations developed quite commensurable; and Book X, 9: The squares on rapidly. We shall merely sketch in the steps in this straight lines commensurable in length have to one development and explain the finished product as it another the ratio which a square number has to a appeared in Euclid. square number; . .. ; and Squares which have not to The chief contributors to this development one another the ratio which a square number has to a were Theodorus of Cyrene (ca. 400 B.C.), Theae­ square number will not have their sides commensu­ tetus (ca. 375 B.C.), and Eudoxus of Cnidos (ca. rable in length either.3 40Q-347 B.C.). Theodorus showed the irrationality The use in these propositions of the words of the non-square integers from 3 to 17.
    [Show full text]
  • Some Important Historical Names, Dates,* and Events
    Some Important Historical Names, Dates,* and Events MATHEMATICAL GENERAL Early Beginnings (Before the Sixth Century B.C.) B.C. 30,000 Notched wolf bone B.C. 3300 Menes unites Egypt 8000 Ishango bone 2600 Great Pyramid at Gizeh 2500 Table tablets from Nippur 2100 Code of Hammurabi 1900 Plimpton 322 1500 Phoenician alphabet 1850 Moscow Papyrus 1200 Trojan War 1650 Rhind Papyrus 700 Homer: The Odyssey Classical Period (Sixth Century B.C. to Fifth Century) B.C. 622-547 Thales of Miletus B.C. 558-486 Darius the Great 585-500 Pythagoras of Samos 485-430 Herodotus ca. 470 Theodorus of Cyrene 480 Battle of Thermopylae 460-380 Hippocrates of Chios 469-399 Socrates ca. 420 Hippias of Elis 431 Peloponnesian War 408-355 Euxodus of Cnidos 388 Plato founds Academy 323-285 Euclid 356-323 Alexander the Great 287-212 Archimedes ca. 370 Eudemus of Rhodes 262-190 Apollonius of Perga 331 Foundation of Alexandria ca. 240 Nicomedes 213 Books burned in China ca. 230 Eratosthenes of Cyrene 212 Fall of Syracuse to Romans A.D. ca. 75 Heron of Alexandria 195 Rosetta Stone engraved ca. 100 Nicomachus of Gerasa 106-43 Cicero 85-160 Claudius Ptolemy 44 Assassination of Caesar ca. 250 Diophantus 27 Beginning of Roman Empire ca. 260 Liu Hui A.D. 100 Paper made in China ca. 300 Pappus of Alexandria 272-337 Constantine the Great 365-395 Theon of Alexandria 286 Division of the Empire / d. 415 Hypatia 324 Constantinople founded 410-485 Proclus 455 Vandals sack Rome *Most dates before 600 B.C.
    [Show full text]
  • Introduction
    In this Mastery Mission students will have the opportunity to explore the historical development of the concept of irrational numbers. They will research the life and times of the Greek mathematician Theodorus and will construct and analyze his famous “Wheel of Theodorus." Introduction When early Greek mathematicians attempted to measure the hypotenuse of right angle triangles, they discovered something shocking. They realized that it wasn’t always possible to do so using the numbers that they had available to them at the time. Pythagoras, one of their most influential leaders, believed that all mathematical puzzles could be solved by using integers and their ratios (called rational numbers). In fact, legend has it that the Pythagoreans, as they were known, were so sure that they were right, that when one of their students told his teachers that he disagreed; he was promptly put to death! One of the great pioneers in the early work showing the existence of irrational numbers was Theodorus of Cyrene. In this Mastery Mission you will construct his famous “Wheel of Theodorus” and explore the interesting dilemma he faced when trying to measure what he had created. So, choose a partner to work with, and move onto the Task section to find out the details of your mission. Task You and your partner will complete the following tasks: 1. Research Theodorus of Crete by using the links that are provided. 2. Use the information that you have gathered to complete the “Who was Theodorus?” worksheet. 3. Construct the Wheel of Theodorus by following these instructions. 4. Use your wheel to explore some interesting mathematical ideas on the number system by completing the “Analyzing the Wheel” worksheet.
    [Show full text]
  • A Concise History of the Philosophy of Mathematics
    A Thumbnail History of the Philosophy of Mathematics "It is beyond a doubt that all our knowledge begins with experience." - Imannuel Kant ( 1724 – 1804 ) ( However naïve realism is no substitute for truth [1] ) [1] " ... concepts have reference to sensible experience, but they are never, in a logical sense, deducible from them. For this reason I have never been able to comprehend the problem of the á priori as posed by Kant", from "The Problem of Space, Ether, and the Field in Physics" ( "Das Raum-, Äether- und Feld-Problem der Physik." ), by Albert Einstein, 1934 - source: "Beyond Geometry: Classic Papers from Riemann to Einstein", Dover Publications, by Peter Pesic, St. John's College, Sante Fe, New Mexico Mathematics does not have a universally accepted definition during any period of its development throughout the history of human thought. However for the last 2,500 years, beginning first with the pre - Hellenic Egyptians and Babylonians, mathematics encompasses possible deductive relationships concerned solely with logical truths derived by accepted philosophic methods of logic which the classical Greek thinkers of antiquity pioneered. Although it is normally associated with formulaic algorithms ( i.e., mechanical methods ), mathematics somehow arises in the human mind by the correspondence of observation and inductive experiential thinking together with its practical predictive powers in interpreting future as well as "seemingly" ephemeral phenomena. Why all of this is true in human progress, no one can answer faithfully. In other words, human experiences and intuitive thinking first suggest to the human mind the abstract symbols for which the economy of human thinking mathematics is well known; but it is those parts of mathematics most disconnected from observation and experience and therefore relying almost wholly upon its internal, self - consistent, deductive logics giving mathematics an independent reified, almost ontological, reality, that mathematics most powerfully interprets the ultimate hidden mysteries of nature.
    [Show full text]
  • Plato Was NOT a Mathematical Platonist
    Plato Was NOT A Mathematical Platonist INTRODUCTION In this paper I will argue that Plato was not a mathematical Platonist.1 My arguments will be based primarily on the evidence found in the Republic’s Divided Line analogy and Book 7. While perhaps not as reader-friendly as one might like2, I will present Plato’s view as it develops. First I will state what I take to be an accurate translation of the text3, next I will move to critically consider which claims remain the same and which change, and, finally, I will bring these claims together into a consistent picture of Plato’s view of mathematics, demonstrating that he was not a mathematical Platonist. Typically, the mathematical Platonist story is told on the basis of two realist components: a) that mathematical objects, like Platonic forms, exist independently of us in some metaphysical realm and the way things are in this realm fixes the truth of mathematical statements; and, b) we come to know such truths by, somehow or other, “recollecting” the way things are in the metaphysical realm. Against b), I have demonstrated, in XXXX [2012], that recollection, in the Meno, is not offered as a method for mathematical knowledge. What is offered as the mathematician’s method for attaining knowledge is the hypothetical method. There I also argued, though mostly in footnotes, against Benson’s [2003; 2006; 2008; 2010] claim, that the mathematician’s hypothetical method cannot be part of the philosopher’s dialectical method. I now turn to reconsider, on the basis of what Plato says in the Republic and Book 7, why these methods must be taken as distinct and further consider what the ontological consequences of this distinction must be4.
    [Show full text]
  • HISTORY of MATH CONCEPTS Essentials, Images and Memes
    HISTORY OF MATH CONCEPTS Essentials, Images and Memes Sergey Finashin Modern Mathematics having roots in ancient Egypt and Babylonia, really flourished in ancient Greece. It is remarkable in Arithmetic (Number theory) and Deductive Geometry. Mathematics written in ancient Greek was translated into Arabic, together with some mathematics of India. Mathematicians of Islamic Middle East significantly developed Algebra. Later some of this mathematics was translated into Latin and became the mathematics of Western Europe. Over a period of several hundred years, it became the mathematics of the world. Some significant mathematics was also developed in other regions, such as China, southern India, and other places, but it had no such a great influence on the international mathematics. The most significant for development in mathematics was giving it firm logical foundations in ancient Greece which was culminated in Euclid’s Elements, a masterpiece establishing standards of rigorous presentation of proofs that influenced mathematics for many centuries till 19th. Content 1. Prehistory: from primitive counting to numeral systems 2. Archaic mathematics in Mesopotamia (Babylonia) and Egypt 3. Birth of Mathematics as a deductive science in Greece: Thales and Pythagoras 4. Important developments of ideas in the classical period, paradoxes of Zeno 5. Academy of Plato and his circle, development of Logic by Aristotle 6. Hellenistic Golden Age period, Euclid of Alexandria 7. Euclid’s Elements and its role in the history of Mathematics 8. Archimedes, Eratosthenes 9. Curves in the Greek Geometry, Apollonius the Great Geometer 10. Trigonometry and astronomy: Hipparchus and Ptolemy 11. Mathematics in the late Hellenistic period 12. Mathematics in China and India 13.
    [Show full text]
  • Origin of Irrational Numbers and Their Approximations
    computation Review Origin of Irrational Numbers and Their Approximations Ravi P. Agarwal 1,* and Hans Agarwal 2 1 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363, USA 2 749 Wyeth Street, Melbourne, FL 32904, USA; [email protected] * Correspondence: [email protected] Abstract: In this article a sincere effort has been made to address the origin of the incommensu- rability/irrationality of numbers. It is folklore that the starting point was several unsuccessful geometric attempts to compute the exact values of p2 and p. Ancient records substantiate that more than 5000 years back Vedic Ascetics were successful in approximating these numbers in terms of rational numbers and used these approximations for ritual sacrifices, they also indicated clearly that these numbers are incommensurable. Since then research continues for the known as well as unknown/expected irrational numbers, and their computation to trillions of decimal places. For the advancement of this broad mathematical field we shall chronologically show that each continent of the world has contributed. We genuinely hope students and teachers of mathematics will also be benefited with this article. Keywords: irrational numbers; transcendental numbers; rational approximations; history AMS Subject Classification: 01A05; 01A11; 01A32; 11-02; 65-03 Citation: Agarwal, R.P.; Agarwal, H. 1. Introduction Origin of Irrational Numbers and Their Approximations. Computation Almost from the last 2500 years philosophers have been unsuccessful in providing 2021, 9, 29. https://doi.org/10.3390/ satisfactory answer to the question “What is a Number”? The numbers 1, 2, 3, 4, , have ··· computation9030029 been called as natural numbers or positive integers because it is generally perceived that they have in some philosophical sense a natural existence independent of man.
    [Show full text]
  • The Theodorus of Cyrene Spiral Geometry and Its Application to Antenna Design Anna Papio´ Toda, Nicolaos G
    URSI RADIO SCIENCE LETTERS, VOL. 3, 2021 1 DOI: 10.46620/21-0003 The Theodorus of Cyrene Spiral Geometry and Its Application to Antenna Design Anna Papio´ Toda, Nicolaos G. Alexopoulos, and Franco De Flaviis Abstract – In this article, we introduce for the first time the radiation properties of a spiral shape that is obtained from the work on irrational numbers by Theodorus (465 BC–398 BC) of Cyrene (presently near Shahhat, Libya). In the past, this piecewise continuous spiral shape has been erroneously introduced in the literature as the Root, Pythagorean, or Einstein spiral. In addition, we use here P. J. Davis’s interpolation function to obtain the continuous version of this spiral, and we present both the discrete and the continuous Theodorus spiral antenna radiation properties. Finally, we fabricate and perform measurements on the Theodorus spiral, and we present a comparison with its theoretically obtained radiation properties as well Figure 1. The discrete Theodorus spiral. with those obtained for the standard Archimedean shape of the same diameter. already known) and continued by erecting a perpendic- 1. Introduction ular segment of unit lengthpffiffiffiffiffi on the hypotenuse and so on until he reached the 17 [3–5]. Thep obviousffiffiffiffiffi question Theodorus lived in Cyrene, an ancient city comes up as to why he stopped at the 17. The answer founded in 631 BC in Libya [1]. Very little is known is that he stopped when the triangles begin to overlap about him and his work other than a short discussion in [5]. Plato’s Dialogues Theaetetus, Sophist, and Statesman This is a spiral composed of contiguous right [2].
    [Show full text]
  • Platons Verhältnis Zur Geschichte Der Mathematik
    174 SHORTER NOTICES [Mar., matical rigor and intuitive justification and it is obvious that he knew pre­ cisely what he wished to do in this respect. The book will certainly prove to be of great value as a text for teaching the subject of differential equations to technical students. For those interested primarily in mathematics it contains much illuminating material. There is a large collection of instructive problems to which answers are given. Among the few misprints noted there is only one which might lead to mis­ understanding. On page 31, tenth line from the bottom, the third word should be "order" instead of "degree." W. R. LONGLEY Platons Verhaltnis zur Geschichte der Mathematik. By Seth Demel. Leipzig, Felix Meiner, 1929. v + 146 pp. Price 6 RM. This interesting and valuable contribution to the history of Greek mathe­ matics and philosophy appears as the first part of the fourth volume of the Forschungen zur Geschichte der Philosophie und der Pâdagogik, edited by Artur Schneider and Wilhelm Kahl. The series is setting a high standard of scholarship, and we may venture to hope that a similar set of monographs, showing evidence of the same ideals, may in due time be founded in one of the schools of education in this country. What Dr. Demel has undertaken is a new study of the relation of mathe­ matics to philosophy in the writings of Plato. The theme is by no means a new one, as he himself is at pains to show. Blass (1861), Rothlauf (1878), Cohen (1878), and various historians of mathematics have already considered it.
    [Show full text]
  • The Spiral of Theodorus, Numerical Analysis, and Special Functions
    Journal of Computational and Applied Mathematics 235 (2010) 1042–1052 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam The spiral of Theodorus, numerical analysis, and special functionsI Walter Gautschi Department of Computer Sciences, Purdue University, West Lafayette, IN 47907-2066, United States article info a b s t r a c t Article history: Theodorus of Cyrene (ca. 460–399p B.C.), teacher of Plato und Theaetetus, is known for his Received 8 September 2009 proof of the irrationality of n, n D 2; 3; 5;:::; 17. He may have known also of a discrete Received in revised form 24 November spiral, today named after him, whose construction is based on the square roots of the 2009 numbers n D 1; 2; 3;:::. The subject of this lecture is the problem of interpolating this discrete, angular spiral by a smooth, if possible analytic, spiral. An interesting solution was Dedicated to Adhemar Bultheel on his 60th proposed in 1993 by P.J. Davis, which is based on an infinite product. The computation of birthday this product gives rise to problems of numerical analysis, in particular the summation of slowly convergent series, and the identification of the product raises questions regarding MSC: 01A20 special functions. The former are solved by a method of integration, in particular Gaussian 30E05 integration, the latter by means of Dawson's integral und the Bose–Einstein distribution. 33C47 51-03 Number-theoretic questions also loom behind this work. 65B10 ' 2009 Elsevier B.V. All rights reserved. 65D30 65D32 Keywords: Spiral of Theodorus Slowly convergent series Gaussian quadrature Bose–Einstein distribution 1.
    [Show full text]