A Modular Workbook in Education Technology 2

1. next different tools use in Mathematics Franz Arjon E. Flores Jonel V. De Leon (Authors) Mrs. Arlene Advento (Consultant) Mr. For-Ian V. Sandoval (Adviser)

2. Vision Mission Goals Objectives of BSE program Foreword (authors) Foreword (recommendation) Acknowledgement Introduction General objectives CHAPTER I: TRADITIONAL TOOLS IN MATHEMATICS Lesson 1 Abacus How to use abacus Different kinds of abacus Contribution of abacus in the development of mathematics Test Yourself Lesson 2 Knucklebones How to play knucklebones Example of playing knucklebones Test Yourself next back Table of contents

3. Lesson 4 Pascal’s triangle Using Pascal's Triangle Contributions of Pascal’s triangle in the development of Mathematics Test Yourself Lesson 5 Counting rods Using counting rods Test Yourself CHAPTER TEST CHAPTER II: MODERN TOOLS USE IN MATHEMATICS Lesson 1 Calculator Different kinds of calculator Basic operations in calculator Test Yourself Lesson 2 Compass Different kinds of compass How to use a drafting compass Test Yourself Lesson 3 Graph paper Types of graph paper Graph paper for games Test Yourself next back Lesson 3 Napier’s bones Use of Napier’s bone in different operation Test Yourself

4. Lesson 4 Slide rule Use of slide rule in different operation Different kinds of slide rule Test Yourself Lesson 5 Protractor Gallery Steps in using protractor Test Yourself CHAPTER TEST CHAPTER III: HIGH TECH. TOOLS IN MATHEMATICS Lesson 1 Speed score How it works Special features Test Yourself Lesson 2 Graphing calculator Different kinds of graphing calculator Test Yourself Lesson 3 DigiMemoL2 Portable and Compact Efficient Test Yourself next back

5. Lesson 4 Interactive Whiteboard How smart board use in the classroom Using the interactive whiteboard Using power point in smart board Test Yourself CHAPTER TEST Bibliography About the Authors next back

6. Laguna State Polytechnic University Siniloan Campus Siniloan, Laguna VISION A premier university in CALABARZON, offering academic programs and related services designed to respond to the requirements of the Philippines and the global economy, particularly in Asian Countries. next back content

10. This Teacher’s “ Developing the Reading Comprehension of First Year High School students” is part of the requirements in Educational Technology 2 under the revised Education curriculum based on CHED Memorandum Order (CMO)-30, Series of 2004. Educational Technology 2 is a three (3)-unit course designed to introduce both traditional and innovative technologies to facilitate and foster meaningful and effective learning where students are expected to demonstrate a sound understanding of the nature, application and production of the various types of educational technologies. The students are provided with guidance and assistance of selected faculty members of the College on the selection, production and utilization of appropriate technology tools in developing technology-based teacher support materials. Through the role and functions of computers especially the Internet, the student researchers and the advisers are able to design and develop various types of alternative delivery systems. These kinds of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation and utilization of instructional materials. FOREWORD next back content

11. The output of the group’s effort on this enterprise may serve as a contribution to the existing body of instructional materials that the institution may utilize in order to provide effective and quality education. The lessons and evaluations presented in this module may also function as a supplementary reference for secondary teachers and students. FRANZ ARJON E. FLORES Module Developer JONEL V. DE LEON Module Developer next back content

12. This Teacher’s “ Developing the Reading Comprehension of First Year High School students” is part of the requirements in Educational Technology 2 under the revised Education curriculum based on CHED Memorandum Order (CMO)-30, Series of 2004. Educational Technology 2 is a three (3)-unit course designed to introduce both traditional and innovative technologies to facilitate and foster meaningful and effective learning where students are expected to demonstrate a sound understanding of the nature, application and production of the various types of educational technologies. The students are provided with guidance and assistance of selected faculty members of the College on the selection, production and utilization of appropriate technology tools in developing technology-based teacher support materials. Through the role and functions of computers especially the Internet, the student researchers and the advisers are able to design and develop various types of alternative delivery systems. These kinds of activities offer a remarkable learning experience for the education students as future mentors especially in the preparation and utilization of instructional materials. next back content

13. The output of the group’s effort on this enterprise may serve as a contribution to the existing body of instructional materials that the institution may utilize in order to provide effective and quality education. The lessons and evaluations presented in this module may also function as a supplementary reference for secondary teachers and students. FOR-IAN V. SANDOVAL Computer Instructor / Adviser Educational Technology 2 MRS. VICTORIA I. CABILDO Module Consultant LYDIA R. CHAVEZ Dean College of Education next back content

14. Acknowledgement We would like to thank all our friends and classmates for the help and moral support that was given for the accomplishment of this module. We also wish to thank Ms. Micah Ela A. Monsalve for the inspiration and moral support that was given for the fulfillment of this module. Mrs. Arlene Advento, our module consultant, for the editing, lay out, encoding and other technical support. Mr. For-Ian Sandoval, our computer instructor, for giving us a chance to experience this kind of activity that challenge our skills and knowledge and also for the encouragement and guidance next back content

15. Mrs. Chavez, DEAN of the College of Education, for her moral support and guidance for the fulfillment of this module our family, for the inspiration and financial support They were robbed of many precious times as we locked ourselves in our rooms when our minds went prolific and our hands itched to write And finally, we thank our Almighty God, the source of all knowledge, understanding and wisdom. From him we owe all that we have. They are all part for the fulfillment of this module next back content

16. INTRODUCTION Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions." Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.“ Through the use of abstraction and logical reasoning, tools and gadgets, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. next back content

17. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, in fitful bursts, until the Renaissance, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day. Today, mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later. This module aims to give the students the information about the tools used in the past and in the present to make mathematics easier to understand. next back content

19. next back content All things in this world begin in simple thing and they have an origin why they evolve in this world and also a history how they develop and recognize by people. We are all part of history that’s why it is very important to study where a certain thing came from. This chapter will help you to enlighten your mind how this different traditional tool evolves and what are their contributions in development of the world of numbers. Objectives : At the end of this chapter, the students must be able to: 1. familiarize the different tools used in the past in teaching mathematics in the past 2. give importance to what is used in the past mathematics 3. apply their knowledge in their daily lives.

20. ABACUS LESSON 1 Objectives: 1. To familiarize oneself on the different kinds of abacus. 2. To appreciate the usefulness of abacus in improving mathematics. 3. To apply the knowledge gained in problem solving using abacus. next back content Abacus

21. How to use an abacus The Abacus utilizes a combination of two bases (base-2 and base-5) to represent decimal numbers. It is held horizontally with the smaller deck at the top. Each bead on the top deck has the value 5 and each bead on the lower deck has the value 1. The beads are pushed towards the central crossbar to show numbers. Working from right to left, the first vertical line represents units, the next tens, the next hundreds and so on. Addition on the abacus involves registering the numbers on the beads in the straight-forward left-to-right sequence they are written down in. As long as the digits are placed correctly, and the carry’s noted properly, the answer to the operation immediately presents itself right on the abacus. There are 4 approaches to performing additions, each applied to a particular situation. Each of these techniques is explained in tabular form in the sections that follow. next back content The abacus , also called a counting frame , is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abacuses are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal. The abacus was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia, Africa, and else where. .

22. Simple Addition When performing the addition 6+2, one would move 1 bead from the upper deck down (value = 5) and one bead from the lower deck up (value = 1); this represents 6. Moving 2 beads from the lower deck (in the same column) up (value = 1 * 2 beads = 2) would complete the operation. The answer is then obtained by reading resultant bead positions. next back content +1 +1 6 0, 1,2 or 3 +1 +4 9 0 +1 +3 8 0 or 1 +1 +2 7 0, 1 or 2 +1 5 0, 1,2 3, or 4 +4 4 0, or 3 +3 3 0,1,5 or 6 +2 2 0,1,2,5,6, or 7 +1 1 0,1,2,3,4,5,6,7 or 8 in the upper deck in the lower deck Move beads) To add Given the first number

24. Combined Adding-up And Taking Off When the original number registered on a rod is smaller than 5, but will become greater than 5 after the addition, one bead from the upper-deck is moved down (added on to the beam) and one or more beads from the lower deck removed from the beam. When a sum greater than 10 occurs on a certain rod, beads are removed from either or both the upper and lower decks and 1 bead is added to the rod directly to the left. Example: When adding 9 (10-1) to 8, one bead from the lower deck is removed (-1) and one bead from the lower deck on the row directly to the left is added (+10). next back content +1 -1 4 (+5 -1) 4,3,2 or 1 +1 -2 3 (+5 -2) 4,3 or 2 +1 -3 2 (+5 -3) 4 or 3 +1 -4 1 (+5 -4) 4 in the upper deck in the lower deck Move bead(s) To add (formula) Given the first number

25. Combined Taking-off And Place Advancement When a sum greater than 10 occurs on a certain rod, beads are removed from either or both the upper and lower decks and 1 bead is added to the rod directly to the left. Example: when adding 9 (10-1) to 8, one bead from the lower deck is removed (-1) and one bead from the lower deck on the row directly to the left is added (+10). next back content in the lower deck +1 -1 9 (-1+10) 1,2,3,4,6,7,8 or 9 +1 -2 8 (-2+10) 2,3,4,7,8 or 9 +1 -3 7 (-3 +10) 3,4,8 or 9 +1 -4 6 (-4 +10) 4 or 9 +1 -1 5 (-5 +10) 5,6,7,8, or 9 +1 -1 -1 4 (-6 +10) 6,7,8 or 9 +1 +1 -2 3 (-7 +10) 7,8 or 9 +1 -1 -3 2 (-8 +10) 8 or 9 +1 -1 -4 1(-9 +10) 9 lower deck, adjacent (left) column in the upper deck Move bead(s) To add (formula) Given the first number

26. Combined Adding-up, Taking-off And Place Advancement There are 4 cases when beads are added to the lower-deck, removed from the upper-deck and one bead added to the adjacent rod. Example: When adding 6 (+1-5+10) to 7, one bead is added to the lower-deck, one bead removed from the upper-deck and one bead is added to the left rod (lower-deck). next back content +1 -1 +2 7 (+2 -5 +10) 5,6 or 7 +1 -1 +4 9 (+4 -5 +10) 5 +1 -1 +3 8 (+3 -5 +10) 5 or 6 +1 -1 +1 6 (+1 -5 +10) 5,6,7 or 8 lower deck, adjacent (left) column in the upper deck in the lower deck Move bead(s) To add (formula) Given the first number

27. Subtraction Subtraction is performed by first registering the minuend and then subtracting, starting from the left, by removing beads form either or both the lower or upper decks. The final bead-positions represent the answer. Simple Taking-off This is achieved by simply taking off one or more beads from the lower deck, or sometimes both. Example: When subtracting 7 (represented by -5-2= -7) from 9, remove 1 bead from the upper-deck (-5) and 2 beads from the lower deck (-2). The remaining 2 beads represent the result. Note : The “-“ symbol in the Move bead(s) columns represents moving the bead(s) away from the middle beam. next back content -1 1 1,2,3,4,5,6,7 or 8 -2 2 2,3,4,7 or 8 -3 3 3,4,8 or 9 -1 -4 9 9 -1 -3 8 8 or 9 -1 -2 7 7,8 or 9 -1 -1 6 6,7,8 or 9 -1 5 5,6,7,8, or 9 -4 4 4 or 9 in the upper deck in the lower deck Move bead(s) To subtract Given the first number

28. Combined Adding-up And Taking Off When the number of beads in the lower deck is less than the subtracter (the number being subtracted), one or more beads are added in the lower deck and 1 bead is removed from the upper-deck. Example: When subtracting 4 (+1-5 = -4) from 7 (represented by 1 bead in the upper-deck and 2 beads in the lower deck (less than 4, the subtracter), one bead is added to the lower deck (+1) and 1 bead is removed from the upper-deck (-5) leaving 3 beads, representing the result. Note : The “+” symbol in the Move bead(s) columns represents moving the bead(s) towards the middle beam; the “-“ symbol indicates that the bead(s) should be moved away from the middle beam. next back content -1 +1 4 (-5 +1) 5,6,7 or 8 -1 +2 3 (-5 +2) 5,6 or 7 -1 +3 2 (-5 +3) 5 or 6 -1 +4 1 (-5 +4) 5 in the upper deck in the lower deck Move bead(s) To subtract (formula) Given the first number

29. Combined Adding-up And Taking Off When the number of beads in the lower deck is less than the subtracter (the number being subtracted), one or more beads are added in the lower deck and 1 bead is removed from the upper-deck. Example: When subtracting 4 (+1-5 = -4) from 7 (represented by 1 bead in the upper-deck and 2 beads in the lower deck (less than 4, the subtracter), one bead is added to the lower deck (+1) and 1 bead is removed from the upper-deck (-5) leaving 3 beads, representing the result. Note : The “+” symbol in the Move bead(s) columns represents moving the bead(s) towards the middle beam; the “-“ symbol indicates that the bead(s) should be moved away from the middle beam. next back content -1 +1 4 (-5 +1) 5,6,7 or 8 -1 +2 3 (-5 +2) 5,6 or 7 -1 +3 2 (-5 +3) 5 or 6 -1 +4 1 (-5 +4) 5 in the upper deck in the lower deck Move bead(s) To subtract (formula) Given the first number

30. Taking-off From A Rod Of Higher Order And Adding-up When a number on a specific rod is smaller than the subtrahend (4 is the subtrahend when performing 13 – 4; note that in the ones column, the 3 is less than the 4) one bead for the order of tens and one bead from the lower-deck has to be taken off, and one bead from the upper-deck is counted. next back content +1 +4 -1 1(+9 -10) 0 +1 -1 5 (+5 -10) 0, 1,2 3, or 4 +4 -1 6 (+4 -10) 0 or 5 +1 -1 9 (+1 -10) 0,1,2,3,5,6,7 or 8 +2 -1 8 (+2 -10) 0,1,2,5,6, or 7 +3 -1 7 (+3 -10) 0,5 or 6 +1 +1 -1 4 (+6 -10) 0, 1,2 or 9 +1 +2 -1 3 (+7 -10) 0, 1 or 2 +1 +3 -1 2 (+8 -10) 0 or 1 in the upper deck in the lower deck lower deck, adjacent (right) column Move bead(s) To subtract Given the first number

31. Combined Taking-off From A Rod Of Higher Order, Adding-up in the Upper-deck and Taking-off in the Lower-Deck This technique is called for when a number on a specific rod is smaller than the supposed subtrahend (I have no idea what this means), but only in such cases as exemplified by 12 – 6. next back content 9 (-4 +5 - +1 -4 -1 10) 4 +1 -3 -1 8 (-3 +5 -10) 4 or 4 +1 -2 -1 7 (-2 +5 -10) 2,3 or 4 +1 -1 -1 6 (-1 +5 -10) 1,2,3 or 4 in the upper deck in the lower deck lower deck, adjacent (right) column Move bead(s) To subtract Given the first number

32. The period 2700–2300 BC saw the first appearance of the Sumerian abacus, a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system Some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus. It is the belief of Carruccio (and other Old Babalonian scholars) that Old Babylonians "may have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations". Different kind of abacus Mesopotamian abacus The use of the abacus in Ancient Egypt is mentioned by the Greek historian Herodotus, who writes that the manner of this disk's usage by the Egyptians was opposite in direction when compared with the Greek method. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, wall depictions of this instrument have not been discovered, casting some doubt over the extent to which this instrument was used. Egyptian abacus next back content

33. The Chinese abacus migrated from China to Korea around the year 1400 AD. Koreans call it jupan ( 주판 ), supan ( 수판 ) or jusan ( 주산 ). Korean abacus The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC. The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations. This Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world. A tablet found on the Greek island Salamis in 1846 AD dates back to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble 149 cm (59 in) long, 75 cm (30 in) wide, and 4.5 cm (2 in) thick, on which are 5 groups of markings. In the center of the tablet is a set of 5 parallel lines equally divided by a vertical line, capped with a semicircle at the intersection of the bottom-most horizontal line and the single vertical line. Below these lines is a wide space with a horizontal crack dividing it. Below this crack is another group of eleven parallel lines, again divided into two sections by a line perpendicular to them, but with the semicircle at the top of the intersection; the third, sixth and ninth of these lines are marked with a cross where they intersect with the vertical line. Greek abacus next back content

34. The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. Originally pebbles, calculi, were used. Later, and in medieval Europe, jetons were manufactured. Marked lines indicated units, fives, tens etc. as in the Roman numeral system. Roman abacus Japanese abacus In Japanese, the abacus is called soroban ( 算盤 , そろばん , lit. "Counting tray"), imported from China around 1600. The 1/4 abacus appeared circa 1930, and it is preferred and still manufactured in Japan today even with the proliferation, practicality, and affordability of pocket electronic calculators. The use of the soroban is still taught in Japanese primary schools as a part of mathematics. next back content

35. The Chinese abacus, known as the suànpán ( 算盤 , lit. "Counting tray"), is typically 20 cm (8 in) tall and comes in various widths depending on the operator. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads each in the bottom for both decimal and hexadecimal computation. Modern abacuses have one bead on the top deck and four beads on the bottom deck. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam. If you move them toward the beam, you count their value. If you move away, you don't count their value. The suanpan can be reset to the starting position instantly by a quick jerk along the horizontal axis to spin all the beads away from the horizontal beam at the center. Suanpans can be used for functions other than counting. Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed. There currently are schools teaching students how to use it.In the famous long scroll Along the River During the Qingming Festival painted by Zhang Zeduan (1085–1145 AD) during the Song Dynasty (960–1297 AD), a suanpan is clearly seen lying beside an account book and doctor's prescriptions on the counter of an apothecary's (Feibao).. Chinese abacus next back content

36. Russian abacus The Russian abacus, the schety (счёты), usually has a single slanted deck, with ten beads on each wire (except one wire which has four beads, for quarter-ruble fractions. This wire is usually near the user). (Older models have another 4-bead wire for quarter-kopeks, which were minted until 1916.) The Russian abacus is often used vertically, with wires from left to right in the manner of a book. The wires are usually bowed to bulge upward in the center, in order to keep the beads pinned to either of the two sides. It is cleared when all the beads are moved to the right. During manipulation, beads are moved to the left. For easy viewing, the middle 2 beads on each wire (the 5th and 6th bead) usually are of a different colour than the other eight beads. Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color. The Russian abacus was in use in all shops and markets throughout the former Soviet Union, and the usage of it was taught in most schools until the 1990s. Indian abacus First century sources, such as the Abhidharmakosa describe the knowledge and use of abacus in India. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the Abacus. Hindu texts used the term shunya (zero) to indicate the empty column on the abacus. next back content

37. School abacus School abacus used in Danish elementary school. Early 20th century. Around the world, abaci have been used in pre-schools and elementary schools as an aid in teaching the numeral system and arithmetic. In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame has been common (see image). It is still often seen as a plastic or wooden toy. The type of abacus shown here is often used to represent numbers without the use of place value. Each bead and each wire has the same value and used in this way it can represent numbers up to 100. Contribution of abacus in the development of mathematics In the history of mathematics, the contributions of the Roman Empire are sometimes overlooked. Roman Numerals are considered cumbersome and the Roman’s lacks of contributions to mathematics, and the lack of the Zero, are held in low esteem. And yet, the Roman Empire was likely the largest when viewed as a percent of world population. Their empire consistently built engineering marvels: roads that survive and are used to this day, homes and bath houses with indirect heating emulated today, plumbed sewer and water lines in and out of homes and public buildings, indoor toilets, aquaducts that included long tunnels and bridges, and huge, beautiful buildings. next back content

38. ACTIVITY 1 Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ Test Yourself Instruction: Encircle the letter of the correct answer. 1. Abacus also called_________. a.)Counting board b.)Counting frame c.)Counting rod d.)Counting beads 2.The period________saw the first appearance o the Sumerian abacus. a.)2700-2300BC b.)2200-1800BC c.)1800-1200BC d.)2500-1700BC 3.During the____________empire, around 600BC, Iranians first began to use abacus. a.)Ancient Egypt b.)Roman c.)Irian Persian d.)Achaemenid Persian 4.The Chinese abacus, known as the__________. a.)duansap b.)suanpan c.)pasuan d.)suanpad 5.___________describe the use and knowledge of abacus in India. a.)Amdhiramakosa b.)Adhirhamakosa c.)Abhidharmakosa d.)Rhamadimakosa next back content

39. 6.Hindu texts used the term___________to indicate the empty column of the abacus. a.)shunya b.)shimha c.)shayen d.)shanun 7.Koreans called abacus as___________. a.)hupan b.)jupan c.)julam d.)sampan 8.In Japanese, the abacus is called_____________. a.)soraban b.)taroban c.)olaman d.)premilan 9.School use in__________school. a.)Venish elementary b.)Turnish elementary c.)Branish elementary d.)Danish elementary 10.The Russian abacus was brought to france around 1820 by the mathematician__________. a.)Jose Victor Poncelet b.)Jean Victor Poncelet c.)John Victor Poncelet d.Jjack Victor Poncelet next back content

40. B. Instruction. Complete the table using the combined adding-up and taking off process in abacus. next back content To subtract (formula) Given the first number -1 ? 4 (-5 +1) 5,6,7 or 8 -1 +2 ? 5,6 or 7 ? +3 2 (-5 +3) ? -1 ? 1 (-5 +4) 5 in the upper deck in the lower deck Move bead(s)

41. C. Instruction. Complete the table using the combined taking-off from a rod of higher order, adding-up in the upper-deck and taking-off in the lower-deck process in abacus next back content in the upper deck in the lower deck lower deck, adjacent (right) column To subtract Given the first number +1 -1 -1 ? 1,2,3 or 4 +1 -2 -1 7 (-2 +5 -10) ? +1 -4 ? 9 (-4 +5 -10) 4 ? ? -1 8 (-3 +5 -10) 4 or 4 Move bead(s)

42. knucklebones Objectives: 1. To summarize the history of knucklebones. 2. To appreciate the importance of knucklebones. 3. To apply the knowledge of using knucklebones in daily living in the world of numbers. . next back content Knucklebones LESSON 2

43. The simplest consists in tossing up one stone, the jack, and picking up one or more from the table while it is in the air; and so on until all five stones have been picked up. Another consists in tossing up first one stone, then two, then three and so on, and catching them on the back of the hand. Different throws have received distinctive names, such as riding the elephant, peas in the pod, and horses in the stable. The origin of knucklebones is closely connected with that of dice, of which it is probably a primitive form, and is doubtless Asiatic. Sophocles, in a fragment, ascribed the invention of draughts and knucklebones ( astragaloi ) to Palamedes, who taught them to his Greek countrymen during the Trojan War. Both the Iliad and the Odyssey contain allusions to games similar in character to knucklebones, and the Palamedes tradition, as flattering to the national pride, was generally accepted throughout Greece, as is indicated by numerous literary and plastic evidences. Thus Pausanias mentions a temple of Fortune in which Palamedes made an offering of his newly invented game. According to a still more ancient tradition, Zeus, perceiving that Ganymede longed for his playmates upon Mount Ida, gave him Eros for a companion and golden dibs with which to play, and even condescended sometimes to join in the game (Apollonius). It is significant, however, that both Herodotus and Plato ascribe to the game a foreign origin. Plato (Phaedrus) names the Egyptian god Thoth as its inventor, while Herodotus relates that the Lydians, during a period of famine in the days of King Atys, originated this game and indeed almost all other games except chess. next back content

44. How to Play Knucklebones To start a turn, the player throws five stones into the air with one hand and tries to catch as many as possible on the back of the same hand. The stones that were caught are then thrown up again from the back of the hand where they came to rest and as many as possible are caught in the palm of the same hand. If no stones end up being caught, the player's turn is over.If, however, at least one stone was caught, the player prepares for the next throw by keeping one of the caught stones in the same hand and throwing all remaining stones on the ground. The player then tosses the single stone into the air, attempts to pick up one of the stones that was missed and then catches the stone that was tossed, all with the same hand. The player repeats this until all the stones have been picked up. That done, the player throws down four of the stones again, throws the single stone in the air, attempts to pick up two stones with the same hand before catching the tossed stone. This is repeated again and a final toss sees the player picking up the last stone. The process is then repeated for three stones followed by one stone and finally, all four stones are picked up before catching the single tossed stone. next back content

45. For skilful players, the game can continue in an agreed way with further permutations and challenges according to the player's whims. For instance, the other hand could be used to throw, the player may have to clap hands before doing the pick up or perhaps slap both knees. Example of playing knucklebones Step 1 Step 2 Step 3 Step 4 next back content

46. ACTIVITY 2 Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ Test Yourself Instruction: True or false. Write B if it is true and Q if it is false. ______1.Knucklebones also called jackstones. ______2.Knuclebones is closely connected with bones. ______3.Modern .knucklebones are consist of 8 points or knobs. ______4. Knucklebones played with 3 small objects. ______5. Knucklebones came from the bones of a sheep. ______6. The last player is the winner to successfully complete a prescribe serves of throws. ______7.Different throws have received distinctive names. ______8.Plato names the Egyptian god thoth as the inventor of knucklebones. ______9. Knucklebones have many uses. ______10. Knucklebones originated from Egypt. next back content

47. Napier's Bones LESSON 3 Objectives 1. To enumerate the purpose of Napier’s bone. 2. To recognize the uses of Napier’s bone. 3. To apply Napier’s bone in actual life situation.. next back content Napier’s bone

48. More advanced use of the rods can even extract square roots. Note that Napier's bones are not the same as logarithms, with which Napier's name is also associated. The abacus consists of a board with a rim; the user places Napier's rods in the rim to conduct multiplication or division. The board's left edge is divided into 9 squares, holding the numbers 1 to 9. The Napier's rods consist of strips of wood, metal or heavy cardboard. Napier's bones are three dimensional, square in cross section, with four different rods engraved on each one. next back content

49. A set of such bones might be enclosed in a convenient carrying case.A rod's surface comprises 9 squares, and each square, except for the top one, comprises two halves divided by a diagonal line. The first square of each rod holds a single-digit, and the other squares hold this number's double, triple, quadruple and so on until the last square contains nine times the number in the top square. The digits of each product are written one to each side of the diagonal; numbers less than 10 occupy the lower triangle, with a zero in the top half.A set consists of 10 rods corresponding to digits 0 to 9. The rod 0, although it may look unnecessary, is obviously still needed for multipliers or multiplicands having 0 in them. Use of Napier’s bone in different operation Multiplication next back content

50. From right to left, we obtain the units place (3), the tens (6+3=9), the hundreds (6+1=7), etc. Note that in the hundred thousands place, where 5+9=14, we note '4' and carry '1' to the next addition (similarly with 4+8=12 in the ten millions place).In cases where a digit of the multiplicand is 0, we leave a space between the rods corresponding to where a 0 rod would be. Let us suppose that we want to multiply the previous number by 96431 ; operating analogously to the previous case, we will calculate partial products of the number by multiplying 46785399 by 9, 6, 4, 3 and 1. Then we place these products in the appropriate positions, and add them using the simple pencil-and-paper method. next back content

51. This method can also be used for multiplying decimals. For a decimal value multiplied by an integer (whole number) value ensure that the decimal number is written along the top of the grid. From this position the decimal point simply drops down the vertical line and 'falls' into the answer.When multiplying two decimal numbers together, the decimal points travel horizontally and vertically until they 'meet' at a diagonal line, the point then travels out of the grid in the same method and again 'falls' into the answer. The form of multiplication was also used in the 1202 Liber Abaci and 800 AD Islamic mathematics and known under the name of lattice multiplication. "Crest of the Peacock", by G.G, Joseph, suggests that Napier learned the details of this method from "Treviso Arithmetic", written in 1499. Division Division can be performed in a similar fashion. Let's divide 46785399 by 96431, the two numbers we used in the earlier example. Put the bars for the divisor (96431) on the board, as shown in the graphic below. Using the abacus, find all the products of the divisor from 1 to 9 by reading the displayed numbers. Note that the dividend has eight digits, whereas the partial products (save for the first one) all have six. next back content

52. So you must temporarily ignore the final two digits of 46785399, namely the '99', leaving the number 467853. Next, look for the greatest partial product that is less than the truncated dividend. In this case, it's 385724. You must mark down two things, as seen in the diagram: since 385724 is in the '4' row of the abacus, mark down a '4' as the left-most digit of the quotient; also write the partial product, left-aligned, under the original dividend, and subtract the two terms. You get the difference as 8212999. Repeat the same steps as above: truncate the number to six digits, chose the partial product immediately less than the truncated number, write the row number as the next digit of the quotient, and subtract the partial product from the difference found in the first repetition. Following the diagram should clarify this. Repeat this cycle until the result of subtraction is less than the divisor. The number left is the remainder. next back content

53. So in this example, we get a quotient of 485 with a remainder of 16364. We can just stop here and use the fractional form of the answer If you prefer, we can also find as many decimal points as we need by continuing the cycle as in standard long division. Mark a decimal point after the last digit of the quotient and append a zero to the remainder so we now have 163640. Continue the cycle, but each time appending a zero to the result after the subtraction. Let's work through a couple of digits. The first digit after the decimal point is 1, because the biggest partial product less than 163640 is 96431, from row 1. Subtracting 96431 from 163640, we're left with 67209. Appending a zero, we have 672090 to consider for the next cycle (with the partial result 485.1) The second digit after the decimal point is 6, as the biggest partial product less than 672090 is 578586 from row 6. The partial result is now 485.16, and so on. next back content

54. Extracting square roots Extracting the square root uses an additional bone which looks a bit different from the others as it has three columns on it. The first column has the first nine squares 1, 4, 9, ... 64, 81, the second column has the even numbers 2 through 18, and the last column just has the numbers 1 through 9. next back content

55. Let's find the square root of 46785399 with the bones. First, group its digits in twos starting from the right so it looks like this: 46 78 53 99 Note: A number like 85399 would be grouped as 8 53 99 Start with the leftmost group 46. Pick the largest square on the square root bone less than 46, which is 36 from the sixth row. Because we picked the sixth row, the first digit of the solution is 6. next back content 8 / 1 18 9 8 / 1 7 / 2 6 / 3 5 / 4 4 / 5 ³/ 6 ²/ 7 1 / 8 0 / 9 9 6 / 4 16 8 7 / 2 6 / 4 5 / 6 4 / 8 4 / 0 ³/ 2 ²/ 4 1 / 6 0 / 8 8 4 / 9 14 7 6 / 3 5 / 6 4 / 9 4 / 2 ³/ 5 ²/ 8 ²/ 1 ¼ 0 / 7 7 ³/ 6 12 6 5 / 4 4 / 8 4 / 2 ³/ 6 ³/ 0 ²/ 4 1 / 8 ½ 0 / 6 6 ²/ 5 10 5 4 / 5 4 / 0 ³/ 5 ³/ 0 ²/ 5 ²/ 0 1 / 5 1 / 0 0 / 5 5 1 / 6 8 4 ³/ 6 ³/ 2 ²/ 8 ²/ 4 ²/ 0 1 / 6 1 / 2 0 / 8 0 / 4 4 0 / 9 6 3 ²/ 7 ²/ 4 ²/ 1 1 / 8 1 / 5 1 / 2 0 / 9 0 / 6 0 / 3 3 0 / 4 4 2 1 / 8 1 / 6 1 / 4 1 / 2 1 / 0 0 / 8 0 / 6 0 / 4 0 / 2 2 0 / 1 2 1 0 / 9 0 / 8 0 / 7 0 / 6 0 / 5 0 / 4 0 / 3 0 / 2 0 / 1 1 √ 9 8 7 6 5 4 3 2 1 Napier's rods with the square root bone

56. Now read the second column from the sixth row on the square root bone, 12, and set 12 on the board. Then subtract the value in the first column of the sixth row, 36, from 46.Append to this the next group of digits in the number 78, to get the remainder 1078.At the end of this step, the board and intermediate calculations should look like this: next back content _____________ √ 46 78 53 99 = 6 36 -- 10 78 8 / 1 18 9 1 / 8 0 / 9 9 6 / 4 16 8 1 / 6 0 / 8 8 4 / 9 14 7 1 / 4 0 / 7 7 ³/ 6 12 6 1 / 2 0 / 6 6 ²/ 5 10 5 1 / 0 0 / 5 5 1 / 6 8 4 0 / 8 0 / 4 4 0 / 9 6 3 0 / 6 0 / 3 3 0 / 4 4 2 0 / 4 0 / 2 2 0 / 1 2 1 0 / 2 0 / 1 1 √ 2 1

57. Now, "read" the number in each row (ignore the second and third columns from the square root bone.) For example, read the sixth row as 0/6 1/2 3/6 -> 756 Now find the largest number less than the current remainder, 1078. You should find that 1024 from the eighth row is the largest value less than 1078. next back content _____________ √ 46 78 53 99 = 6 8 36 -- 10 78 10 24 ----- 54 1161 8 / 1 18 9 1 / 8 0 / 9 9 1024 6 / 4 16 8 1 / 6 0 / 8 8 889 4 / 9 14 7 1 / 4 0 / 7 7 756 ³/ 6 12 6 1 / 2 0 / 6 6 625 ²/ 5 10 5 1 / 0 0 / 5 5 496 1 / 6 8 4 0 / 8 0 / 4 4 369 0 / 9 6 3 0 / 6 0 / 3 3 244 0 / 4 4 2 0 / 4 0 / 2 2 121 0 / 1 2 1 0 / 2 0 / 1 1 (value) √ 2 1

58. As before, append 8 to get the next digit of the square root and subtract the value of the eighth row 1024 from the current remainder 1078 to get 54. Read the second column of the eighth row on the square root bone, 16, and set the number on the board as follows. The current number on the board is 12. Add to it the first digit of 16, and append the second digit of 16 to the result. So you should set the board to 12 + 1 = 13 -> append 6 -> 136 Note: If the second column of the square root bone has only one digit, just append it to the current number on board. The board and intermediate calculations now look like this. 136√1 0/10/30/60/1 2 1 2 0/20/61/20/4 4 2 3 0/30/91/80/9 6 3 4 0/41/2²/41/6 8 4 5 0/51/5³/0²/5 10 5 6 0/61/8³/6³/6 12 6 7 0/7²/14/24/9 14 7 8 0/8²/44/86/4 16 8 9 0/9²/75/48/1 18 9 _____________ √ 46 78 53 99 = 68 36 -- 10 78 10 24 ----- 54 53 next back content

59. Once again, find the row with the largest value less than the current partial remainder 5453. This time, it is the third row with 4089. _____________ √ 46 78 53 99 = 68 3 36 -- 10 78 10 24 ----- 54 53 40 89 ----- 13 64 The next digit of the square root is 3. Repeat the same steps as before and subtract 4089 from the current remainder 5453 to get 1364 as the next remainder. When you rearrange the board, notice that the second column of the square root bone is 6, a single digit. So just append 6 to the current number on the board 136 next back content 12321 8 / 1 18 9 5 / 4 ²/ 7 0 / 9 9 10944 6 / 4 16 8 4 / 8 ²/ 4 0 / 8 8 9569 4 / 9 14 7 4 / 2 ²/ 1 0 / 7 7 8196 ³/ 6 12 6 ³/ 6 1 / 8 0 / 6 6 6825 ²/ 5 10 5 ³/ 0 1 / 5 0 / 5 5 5456 1 / 6 8 4 ²/ 4 1 / 2 0 / 4 4 4089 0 / 9 6 3 1 / 8 0 / 9 0 / 3 3 2724 0 / 4 4 2 1 / 2 0 / 6 0 / 2 2 1361 0 / 1 2 1 0 / 6 0 / 3 0 / 1 1 √ 6 3 1

60. To set 1366 on the board. Remainder 136499 is 123021 from the ninth row. next back content _____________ √ 46 78 53 99 = 683 36 -- 10 78 10 24 ----- 54 53 40 89 ----- 13 64 99 8 / 1 18 9 5 / 4 5 / 4 ²/ 7 0 / 9 9 6 / 4 16 8 4 / 8 4 / 8 ²/ 4 0 / 8 8 4 / 9 14 7 4 / 2 4 / 2 ²/ 1 0 / 7 7 ³/ 6 12 6 ³/ 6 ³/ 6 1 / 8 0 / 6 6 ²/ 5 10 5 ³/ 0 ³/ 0 1 / 5 0 / 5 5 1 / 6 8 4 ²/ 4 ²/ 4 1 / 2 0 / 4 4 0 / 9 6 3 1 / 8 1 / 8 0 / 9 0 / 3 3 0 / 4 4 2 1 / 2 1 / 2 0 / 6 0 / 2 2 0 / 1 2 1 0 / 6 0 / 6 0 / 3 0 / 1 1 √ 6 6 3 1

61. In practice, you often don't need to find the value of every row to get the answer. You may be able to guess which row has the answer by looking at the number on the first few bones on the board and comparing it with the first few digits of the remainder. But in these diagrams, we show the values of all rows to make it easier to understand. _____________ √ 46 78 53 99 = 683 9 36 -- 10 78 10 24 ----- 54 53 40 89 ----- 13 64 99 12 30 21 -------- 1 34 78 next back content 123021 8 / 1 18 9 5 / 4 5 / 4 ²/ 7 0 / 9 9 109344 6 / 4 16 8 4 / 8 4 / 8 ²/ 4 0 / 8 8 95669 4 / 9 14 7 4 / 2 4 / 2 ²/ 1 0 / 7 7 81996 ³/ 6 12 6 ³/ 6 ³/ 6 1 / 8 0 / 6 6 68325 ²/ 5 10 5 ³/ 0 ³/ 0 1 / 5 0 / 5 5 54656 1 / 6 8 4 ²/ 4 ²/ 4 1 / 2 0 / 4 4 40989 0 / 9 6 3 1 / 8 1 / 8 0 / 9 0 / 3 3 27324 0 / 4 4 2 1 / 2 1 / 2 0 / 6 0 / 2 2 13661 0 / 1 2 1 0 / 6 0 / 6 0 / 3 0 / 1 1 √ 6 6 3 1

62. You've now "used up" all the digits of our number, and you still have a remainder. This means you've got the integer portion of the square root but there's some fractional bit still left. Notice that if we've really got the integer part of the square root, the current result squared (6839² = 46771921) must be the largest perfect square smaller than 46785899. Why? The square root of 46785399 is going to be something like 6839.xxxx... This means 6839² is smaller than 46785399, but 6840² is bigger than 46785399 -- the same thing as saying that 6839² is the largest perfect square smaller than 46785399. This idea is used later on to understand how the technique works, but for now let's continue to generate more digits of the square root. This idea is used later on to understand how the technique works, but for now let's continue to generate more digits of the square root. Similar to finding the fractional portion of the answer in long division, append two zeros to the remainder to get the new remainder 1347800. The second column of the ninth row of the square root bone is 18 and the current number on the board is 1366. So compute 1366 + 1 -> 1367 -> append 8 -> 13678 next back content

63. To set 13678 on the board. The board and intermediate computations now look like this. _____________ √ 46 78 53 99 = 6839. 36 -- 10 78 10 24 ----- 54 53 40 89 ----- 13 64 99 12 30 21 -------- 1 34 78 00 next back content 8 / 1 18 9 7 / 2 6 / 3 5 / 4 ²/ 7 0 / 9 9 6 / 4 16 8 6 / 4 5 / 6 4 / 8 ²/ 4 0 / 8 8 4 / 9 14 7 5 / 6 4 / 9 4 / 2 ²/ 1 0 / 7 7 ³/ 6 12 6 4 / 8 4 / 2 ³/ 6 1 / 8 0 / 6 6 ²/ 5 10 5 4 / 0 ³/ 5 ³/ 0 1 / 5 0 / 5 5 1 / 6 8 4 ³/ 2 ²/ 8 ²/ 4 1 / 2 0 / 4 4 0 / 9 6 3 ²/ 4 ²/ 1 1 / 8 0 / 9 0 / 3 3 0 / 4 4 2 1 / 6 1 / 4 1 / 2 0 / 6 0 / 2 2 0 / 1 2 1 0 / 8 0 / 7 0 / 6 0 / 3 0 / 1 1 √ 8 7 6 3 1

64. The ninth row with 1231101 is the largest value smaller than the remainder, so the first digit of the fractional part of the square root is 9. _____________ √ 46 78 53 99 = 6839. 9 36 -- 10 78 10 24 ----- 54 53 40 89 ----- 13 64 99 12 30 21 -------- 1 34 78 00 1 23 11 01 ---------- 11 66 99 next back content 1231101 8 / 1 18 9 7 / 2 6 / 3 5 / 4 ²/ 7 0 / 9 9 1094304 6 / 4 16 8 6 / 4 5 / 6 4 / 8 ²/ 4 0 / 8 8 957509 4 / 9 14 7 5 / 6 4 / 9 4 / 2 ²/ 1 0 / 7 7 820716 ³/ 6 12 6 4 / 8 4 / 2 ³/ 6 1 / 8 0 / 6 6 683925 ²/ 5 10 5 4 / 0 ³/ 5 ³/ 0 1 / 5 0 / 5 5 547136 1 / 6 8 4 ³/ 2 ²/ 8 ²/ 4 1 / 2 0 / 4 4 410349 0 / 9 6 3 ²/ 4 ²/ 1 1 / 8 0 / 9 0 / 3 3 273564 0 / 4 4 2 1 / 6 1 / 4 1 / 2 0 / 6 0 / 2 2 136781 0 / 1 2 1 0 / 8 0 / 7 0 / 6 0 / 3 0 / 1 1 √ 8 7 6 3 1

65. Subtract the value of the ninth row from the remainder and append a couple more zeros to get the new remainder 11669900. The second column on the ninth row is 18 with 13678 on the board, so compute 13678 + 1 -> 13679 -> append 8 -> 136798 And set 136798 on the board. _____________ √ 46 78 53 99 = 6839.9 36 -- 10 78 10 24 ----- 54 53 40 89 ----- 13 64 99 12 30 21 -------- 1 34 78 00 1 23 11 01 ---------- 11 66 99 00 next back content 8 / 1 18 9 7 / 2 8 / 1 6 / 3 5 / 4 ²/ 7 0 / 9 9 6 / 4 16 8 6 / 4 7 / 2 5 / 6 4 / 8 ²/ 4 0 / 8 8 4 / 9 14 7 5 / 6 6 / 3 4 / 9 4 / 2 ²/ 1 0 / 7 7 ³/ 6 12 6 4 / 8 5 / 4 4 / 2 ³/ 6 1 / 8 0 / 6 6 ²/ 5 10 5 4 / 0 4 / 5 ³/ 5 ³/ 0 1 / 5 0 / 5 5 1 / 6 8 4 ³/ 2 ³/ 6 ²/ 8 ²/ 4 1 / 2 0 / 4 4 0 / 9 6 3 ²/ 4 ²/ 7 ²/ 1 1 / 8 0 / 9 0 / 3 3 0 / 4 4 2 1 / 6 1 / 8 1 / 4 1 / 2 0 / 6 0 / 2 2 0 / 1 2 1 0 / 8 0 / 9 0 / 7 0 / 6 0 / 3 0 / 1 1 √ 8 9 7 6 3 1

66. You can continue these steps to find as many digits as you need and you stop when you have the precision you want, or if you find that the reminder becomes zero which means you have the exact square root. Having found the desired number of digits, you can easily determine whether or not you need to round up; i.e., increment the last digit. You don't need to find another digit to see if it is equal to or greater than five. Simply append 25 to the root and compare that to the remainder; if it is less than or equal to the remainder, then the next digit will be at least five and round up is needed. In the example above, we see that 6839925 is less than 11669900, so we need to round up the root to 6840.0. There's only one more trick left to describe. If you want to find the square root of a number that isn't an integer, say 54782.917. Everything is the same, except you start out by grouping the digits to the left and right of the decimal point in groups of two. That is, group 54782.917 as 5 47 82 . 91 7 And proceed to extract the square root from these groups of digits. next back content

67. Contributions of Napier's bones in the development of Mathematics John Napier's rods (1617) (Napier Rechenstäbchen), also known as Napier's bones, are one of his most important contributions to the world of mathematics and alongside William Oughtred's invention of the Slide Rule (1615), represents one of the most revolutionary developments in calculation devices since the abacus. The rods were basically multiplication tables inscribed on sticks of wood or bone. In addition to multiplication, the rods were also used in taking square roots and cube roots. next back content

70. PASCAL’S TRIANGLE LESSON 4 Objectives 1. To the use Pascal's triangle. 2. To appreciate the importance of Pascal's triangle. 3. To apply knowledge gained in using Pascal’s triangle in studying mathematics. Pascal's Triangle next back content

71. In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row 0 , and the numbers in each row are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. next back content

72. Combinations The triangle also shows you how many Combinations of objects are possible. Example, if you have 16 pool balls, how many different ways could you choose just 3 of them (ignoring the order that you select them)? Answer: go down to row 16 (the top row is 0), and then along 3 places and the value there is your answer, 560 . Here is an extract at row 16: 1 14 91 364 ... 1 15 105 455 1365 ... 1 16 120 560 1820 4368 ... In fact there is a formula from Combinations for working out the value at any place in Pascal's triangle: It is commonly called "n choose k" and written C (n, k). The "!" means "factorial", for example 4! = 1×2×3×4 = 24) next back content

73. Example: Row 4, term 2 in Pascal's Triangle is "6". Let's see if the formula works: Polynomials Pascal's Triangle can also show you the coefficients in binomial expansion: next back content 1, 3, 3, 1 (x + 1) 3 = 1 x 3 + 3 x 2 + 3 x + 1 3 1, 4, 6, 4, 1 (x + 1) 4 = 1 x 4 + 4 x 3 + 6 x 2 + 4 x + 1 4 1, 2, 1 (x + 1) 2 = 1 x 2 + 2 x + 1 2 Pascal's Triangle Binomial Expansion Power

74. Using Pascal's Triangle Heads and Tails Pascal's Triangle can show you how many ways heads and tails can combine. This can then show you "the odds" (or probability) of any combination. For example, if you toss a coin three times, there is only one combination that will give you three heads (HHH), but there are three that will give two heads and one tail (HHT, HTH, THH), also three that give one head and two tails (HTT, THT, TTH) and one for all Tails (TTT). This is the pattern "1,3,3,1" in Pascal's Triangle. next back content ... etc ... 1, 4, 6, 4, 1 HHHH HHHT, HHTH, HTHH, THHH HHTT, HTHT, HTTH, THHT, THTH, TTHH HTTT, THTT, TTHT, TTTH TTTT 4 1, 3, 3, 1 HHH HHT, HTH, THH HTT, THT, TTH TTT 3 1, 2, 1 HH HT TH TT 2 1, 1 H T 1 Pascal's Triangle Possible Results (Grouped) Tosses

75. Contributions of Pascal’s triangle in the development of mathematics Pascal's triangle. Each number is the sum of the two directly above it. The triangle demonstrates many mathematical properties in addition to showing binomial coefficients. Pascal continued to influence mathematics throughout his life. His Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") of 1653 described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle. The triangle can also be represented: He defines the numbers in the triangle by recursion: Call the number in the (m+1)st row and (n+1)st column tmn. Then tmn = tm-1,n + tm,n-1, for m = 0, 1, 2... and n = 0, 1, 2... next back content

76. The boundary conditions are tm, -1 = 0, t-1, n for m = 1, 2, 3... and n = 1, 2, 3... The generator t00 = 1. Pascal concludes with the proof, In 1654, prompted by a friend interested in gambling problems, he corresponded with Fermat on the subject, and from that collaboration was born the mathematical theory of probabilities. The friend was the Chevalier de Méré, and the specific problem was that of two players who want to finish a game early and, given the current circumstances of the game, want to divide the stakes fairly, based on the chance each has of winning the game from that point. From this discussion, the notion of expected value was introduced. Pascal later (in the Pensées ) used a probabilistic argument, Pascal's Wager, to justify belief in God and a virtuous life. The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz' formulation of the infinitesimal calculus.After a religious experience in 1654, Pascal mostly gave up work in mathematics. winner. next back content

77. However, after a sleepless night in 1658, he anonymously offered a prize for the quadrature of a cycloid. Solutions were offered by John Wallis, Christiaan Huygens, Christopher Wren, and others; Pascal, under the pseudonym Amos Dettonville, published his own solution. Controversy and heated argument followed after Pascal announced him the winner. next back content

78. ACTIVITY 4 Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ next back content Test Yourself A. Instruction: Answer the following questions. 1. What is the contribution of Pascal triangle in the development of mathematics? ______________________________________________________________________________________________________________________ 2. Site one advantages of Pascal triangle and explain. ______________________________________________________________________________________________________________________ 3. Discuss briefly the importance of Pascal triangle. ______________________________________________________________________________________________________________________

79. B. Expand the given binomials and write the pattern in Pascal’s triangle. 1. (x+1) 6 = 4. (2x+28) 4 = 2. (2x+1) 4 = 5. (5x-25) 2 = 3. (7x+7) 3 = C. Transform the following Pascal’s triangle into binomial expansion. 1. 1, 3, 3, 1 = 4. 2, 6, 6, 2 = 2. 1, 7, 21, 35, 35, 21, 7, 1 = 5. 8, 16, 16, 8 = 3. 4, 8, 4 = 4. Create your own Pascal triangle and explain how it works. ______________________________________________________________________________________________________________________ 5. What is the help of Pascal triangle to us? ______________________________________________________________________________________________________________________ next back content

80. COUNTING RODS LESSON 5 Objectives 1. To familiarize on the history of the counting rods in mathematics 2. To recognize the persons involved in the development of counting rods 3. To skillfully answer problems using of the tool Counting Rod next back content

81. Yang Hui (Pascal's) triangle, as depicted by Zhu Shijie in 1303, using rod numerals. Counting Rods (simplified Chinese; traditional Chinese: 籌 ; pinyin: chóu ; Japanese: 算木 , sangi) are small bars, typically 3-14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction. The written forms based on them are called Rod Numerals. They are a true positional numeral system with digits for 1-9 and later for 0. Using counting rods Counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternatingly used. Generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc., while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. Sun Tzu wrote that "one is vertical, ten is horizontal." Red rods represent positive numbers and black rods represent negative numbers. Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter. next back content

82. The Nine Chapters on the Mathematical Art, which was mainly composed in the first century CE, stated "(when subtraction) subtract same signed numbers, add different signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number." Later, a go stone was sometimes used to represent 0.In Japan, mathematicians put counting rods on a counting board, a sheet of cloth with grids, and used only vertical forms relying on the grids. next back content Horizontal Vertical 9 8 7 6 5 4 3 2 1 0 Positive numbers

83. next back content Horizontal Vertical -9 -8 -7 -6 -5 -4 -3 -2 -1 0 Negative numbers

84. Example 1: Example 2: next back content -6, 720 -407 5, 089 231 -1, 032 -197 609 2, 821

85. ACTIVITY 5 Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ Test Yourself A. instruction. Match column A to column B. Write only the correct answer before the number. A B ______1. It represents positive numbers. A) Black rods ______2. He said” a good calculator doesn’t B) Yang Hui Use counting rods.” C) Counting rods ______3. It represents negative numbers. D)Zhu Shijie ______4. Invented the counting rods. E) Red rods ______5. It is used for the positioning of units, F) Horizontal red rods hundreds, ten thousands, etc. G) Laozi ______6. It is the written forms of counting forms. H) Counting board ______7.It is used by the mathematician for the I)Vertical rod number calculation in China, Japan, Korea & Vietnam. J) Rod numerals ______8. It is used for the positioning of tens, thousands, hundred thousands, etc. ______9. He depicted the Yang Hui triangle in 1303 ______10 Are small bars, typically 3.14 cm. long. next back content

86. B. Convert the given numeral numbers into Rod Numerals. next back content -9143 -291 7123 482

89. D. Instruction: Expand the given binomials and write the pattern in Pascal’s triangle. 1. (3x+3)3 = 2. (4x+4)2 = 3. (7x+7)5 = 4. (x+1)8 = 5. (5x+1)3 = 6. (6x-3)3 = 7. (-8x+2)2 = 8. (4x-8)4 = 9. (5x-9)3 = 10. (2x+1)2 = next back content

90. E. Instruction: Convert the following numeral numbers into rod numerals. next back content 10. -40, 574 9. 201,004 8. 45, 101 7. -12, 021 6. -112, 374 5. 452, 753 4. 18, 113 3. 20, 765 2. -7, 164 1. 9, 135

92. CALCULATOR LESSON 1 Objectives: 1. To identify different kinds of calculator. 2. To recognize the people who contributed in the development of calculator. 3. To help us make computations easier. Calculator next back content

93. A calculator is a device that is used for performing mathematical calculations. It differs from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming. Modern electronic calculators are generally small, digital, (often pocket-sized) and usually inexpensive. In addition to general purpose calculators, there are those designed for specific markets. for example, there are scientific calculators which focus on operations slightly more complex than those specific to arithmetic - for instance, trigonometric and statistical calculations. Some calculators even have the ability to do computer algebra. Graphing calculators can be used to graph functions defined on the real line, or higher dimensional Euclidean space. They often serve other purposes, however. next back content

94. Different kinds of calculator Pocket calculators In early 1971 Pico Electronics and General Instrument also introduced their first collaboration in ICs, a complete single chip calculator IC for the Monroe Royal Digital III calculator. Pico was a spinout by five GI design engineers whose vision was to create single chip calculator ICs. Pico and GI went on to have significant success in the burgeoning handheld calculator market. By 1970, a calculator could be made using just a few chips of low power consumption, allowing portable models powered from rechargeable batteries. The first portable calculators appeared in Japan in 1970, and were soon marketed around the world. These included the Sanyo ICC-0081 "Mini Calculator", the Canon Pocketronic, and the Sharp QT-8B "micro Compet". The Canon Pocketronic was a development of the "Cal-Tech" project which had been started at Texas Instruments in 1965 as a research project to produce a portable calculator. The Pocketronic has no traditional display; numerical output is on thermal paper tape. next back content

95. The first American-made pocket-sized calculator, the Bowmar 901B (popularly referred to as The Bowmar Brain ), measuring 5.2×3.0×1.5 in (131×77×37 mm), came out in the fall of 1971, with four functions and an eight-digit red LED display, for $240, while in August 1972 the four-function Sinclair Executive became the first slimline pocket calculator measuring 5.4×2.2×0.35 in (138×56×9 mm) and weighing 2.5 oz (70g). It retailed for around $150. By the end of the decade, similar calculators were priced less than $10 (GB£5). Mechanical calculators Mechanical calculators continued to be sold, though in rapidly decreasing numbers, into the early 1970s, with many of the manufacturers closing down or being taken over. Comptometer type calculators were often retained for much longer to be used for adding and listing duties, especially in accounting, since a trained and skilled operator could enter all the digits of a number in one movement of the hands on a Comptometer quicker than was possible serially with a 10-key electronic calculator. The spread of the computer rather than the simple electronic calculator put an end to the Comptometer. Also, by the end of the 1970s, the slide rule had become obsolete. next back content

96. Programmable calculators The first desktop programmable calculators were produced in the mid-1960s by Mathatronics and Casio (AL-1000). These machines were, however, very heavy and expensive. The first programmable pocket calculator was the HP-65, in 1974; it had a capacity of 100 instructions, and could store and retrieve programs with a built-in magnetic card reader. A year later the HP-25C introduced continuous memory , i.e. programs and data were retained in CMOS memory during power-off. In 1979, HPreleased the first alphanumeric , programmable, expandable calculator, the HP-41C. It could be expanded with RAM (memory) and ROM (software) modules, as well as peripherals like bar code readers, microcassette and floppy disk drives, paper-roll thermal printers, and miscellaneous communication interfaces. The first Soviet programmable desktop calculator ISKRA 123, powered by the power grid, was released at he beginning of the 1970s. The first Soviet pocket battery-powered programmable calculator,Elektronika "B3-21", was developed by the end of 1977 and released at the beginning of 1978. The successor of B3-21, the Elektronika B3-34 wasn't backward compatible with B3-21, even if it kept the reverse Polish notation (RPN). next back content

97. Basic Operations. Most calculators today have the following operations, which you need to know how to use : next back content Inverse Sine Function, arcsine, or "the angle whose sine is" SIN -1 Natural Logarithm, take the log of LN square root y raised to the power x y x "negation" Sine Function SIN "Exponentiate this," raise e to the power x e x raised to the power ^ over, divided by, division by / times, or multiply by * minus or subtraction, Note: there is DIFFERENT key to make a positive number into a negative number, perhaps marked (-) or NEG known as - plus, or addition + English Equivalent Operation

98. next back content Parentheses, "Do this first" ( ) Cosine Function COS Get the number from memory for immediate use Recall Put a number in memory for later use Store (STO) Inverse Tangent Function, arctangent, or "the angle whose tangent is" TAN -1 Tangent Function TAN Inverse Cosine Function, arccosine, or "the angle whose cosine is" COS -1

99. Taking the Square Root of a Number Let's start with an easy one that you already should know the answer to: What is the square root of 4? Hopefully you KNOW the answer is 2 even without using the calculator. The question is: do I enter the 4 first and then hit the sqrt key (think to yourself: I have the number 4, and now I want to take its square root), or do you hit the sqrt key first and then the number 4 (think: I want to take the square root of something, and in this case, 4)? Depending on what YOUR calculator expects, you will either get 2 or not! Make sure you know which order of entry your calculator expects you to use. Taking the Power of Some Number Suppose you have a calculator with an operation key that raises some number to some power: does it raise the first number to the power of the second number, or the second number to the power of the first? This is especially important to have clear in your mind if the key is marked "yx" or "xy" for you will need to know which is x and which is y. Try using 2 and 3 as a test: if ^ represents the key that your calculator uses for raising a number to some power, enter 2^3. What did you get? If you got 8, then you just took the 3rd power of 2 (2 "cubed"). next back content

100. Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ ACTIVITY 6 Test Yourself Instruction: Answer the following questions. 1. Give the advantages of calculator to other tools. ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. What are the importance of calculator to us? ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

101. 3. Discuss briefly the development of Electronic Calculator Mid-90’s up to present. ________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. Is calculator has a bad effect to us? Why? ________________________________________________________________________________________________________________________________________________________________________________________________________________________ 5. Explain how calculator helps us in our everyday living. ________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

103. COMPASS LESSON 2 Objectives: . 1. To know the purpose of compass. 2. To use compass in drawing, constructing and measuring angle. 3. To enumerate the different kinds of compass. Compass next back content

104. A compass or, more properly, pair of compasses is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as a tool to measure distances, in particular on maps. Compasses can be used for mathematics, drafting, navigation, and other purposes. What is compass in mathematics? Compasses are usually made of metal, and consist of two parts connected by a hinge which can be adjusted. Typically one part has a spike at its end, and the other part a pencil, or sometimes a pen. Circles can be made by fastening one leg of the compasses into the paper with the spike, putting the pencil on the paper, and moving the pencil around while keeping the hinge on the same angle. The radius of the circle can be adjusted by changing the angle of the hinge. Distances can be measured on a map using compasses with two spikes, also called a dividing compass. The hinge is set in such a way that the distance between the spikes on the map represents a certain distance in reality, and by measuring how many times the compasses fit between two points on the map the distance between those points can be calculated. next back content

105. Compass and straightedge construction is used to illustrate principles of plane geometry. Although a real pair of compasses is used to draft visible illustrations, the ideal compass used in proofs is an abstract creator of perfect circles. The most rigorous definition of this abstract tool is the "collapsing compass"; having drawn a circle from a given point with a given radius, it disappears; it cannot simply be moved to another point and used to draw another circle of equal radius (unlike a real pair of compasses). Euclid showed in his second proposition that such a collapsing compass could be used to transfer a distance, proving that a collapsing compass could do anything a real compass can do. Different kinds of compass Beam compass is an instrument with a wooden or brass beam and sliding sockets, or cursors, for drawing and dividing circles larger than those made by a regular pair of compasses. Loose leg wing dividers are made of all forged steel. The pencil holder, thumb screws, brass pivot and branches are all well built. They are used for scribing circles and stepping off repetitive measurements with some accuracy. next back content

106. Scribe-compass is an instrument used by carpenters and other tradesmen. Some compasses can be used to scribe circles, bisect angles and in this case to trace a line. It is the compass in the most simple form. Both branches are crimped metal. One branch has a pencil sleeve while the other branch is crimped with a fine point protruding from the end. The wing nut serves two purposes, first it tightens the pencil and secondly it locks in the desired distance when the wing nut is turned clockwise. How to use a drafting compass: To draw arcs or circles on architectural plans or engineering plans, a compass is used. One arm of a compass has a needle, or shoulder needle. The other arm contains drafting lead. To draw a circle or arc, first adjust the width, or spread, of the compass. Then place the compass needle at the center of the circle where the two center lines intersect. Rotate the compass to draw the line. This is accomplished by twisting the thumb and forefinger. During rotation, lean the outer edge of the compass in the direction of the movement. The drafting lead in a compass should be sharpened to a chisel point. This will result in a line of even thickness during rotation. The compass lead must be soft (F or HB) so the circle can be drawn with dark object lines. next back content

107. Get a sheet of paper, preferably graphing paper, and prepare the compass. Make sure the compass pencil and pencil are both sharp. It is best that the paper be on a hard, flat surface for more accurate lines.  Step 2 Draw a circle. Use a compass to draw a perfect circle in the middle of the paper. The circle should be a minimum of 2 inches across to be easier to work with. Draw a dot at the center point of the circle and label it "O." Leave a lot of free space around the circle to draw on in the following steps. A one circle width of space all around the original circle is good. (Black circle.)  Step 3 Obtain the first vertex. Use the edge of a ruler to draw a straight, vertical line through either side of the circle so that it passes through point "O." next back content

108. This line passes through the original circle at two points on the top and bottom of the circle; label the top point "A" with a large dot. This is the top corner of the pentagon.  Step 4 Draw the point "B." Use the edge of a ruler to draw a straight line perpendicular (horizontal) to the line "OA" that passes through center point "O." This line passes through the original circle at two points on the left and right of the circle; label the right point "B" with a large dot.  Step 5 Find midpoint of line "OB." Use a ruler to measure halfway from point "O" to point "B" and label the point "C" with a large dot. next back content

109.  Step 6 Find the midpoint of the line opposite line segment "OB." Use a compass to draw a circle centered at "C" that passes through point "A." This circle passes through the line segment opposite to line segment "OA"; label this point "D" with a large dot. (Green circle.)  Step 7 Obtain the second and third vertices. Use the compass to draw a circle centered at point "A" that passes through point "D." This circle passes through the original circle at two points on the upper left and upper right; label these points as "E" and "F" with a large dot. These are the upper left and right corners of the pentagon. (Purple circle.) next back content

110.  Step 8 Obtain the fourth vertex. Use the compass to draw a circle centered at point "E" that passes through point "A." This circle passes through the original circle at one point on the bottom left; label this point "G" with a large dot. This is the lower left corner of the pentagon. (Blue circle.)  Step 9 Obtain the fifth vertex. Use the compass to draw a circle centered at point "F" that passes through point "A." This circle passes through the original circle at one point on the bottom left; label this point "H" with a large dot. This is the lower right corner of the pentagon. (Aqua circle.) next back content

111.  Step 10 Draw the pentagon. Erase all lines, circles and points except for the five points "A," "E," "G," "H" and "F." Use the edge of a ruler to connect each of these five points of the pentagon next back content

112. ACTIVITY 7 Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ Test Yourself Following directions. 1. Draw a circle using a compass and name it circle A. Inside the circle, draw one horizontal and vertical line that will pass through point A and determine how many angles are formed then measure it. next back content 2. Construct an angle using a compass, name it as angle ABC.

113. B. Enumerate the different kinds of compass and describe each. ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

114. GRAPH PAPER LESSON 3 Graph paper Graph paper, graphing paper, grid paper or millimeter paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting mathematical functions or experimental data and drawing diagrams. It is commonly found in mathematics and engineering education settings and in laboratory notebooks. Objectives 1. T o gain more information about graphing paper. 2. To appreciate the uses of graphing paper. 3. To use it in allocating and creating angles. next back content

115. Format and availability of graph paper Graph paper is available either as loose leaf paper or bound in notebooks. It is becoming less common as computer software such as spreadsheets and plotting programs has supplanted many of the former uses of graph paper. Some users of graph paper now print pdf images of the grid pattern as needed rather than buying it pre-printed. Types of graph paper Quad paper is a common form of graph paper with a sparse grid printed in light blue or gray and right to the edge of the paper. This is often four squares to the inch for work not needing too much detail. It is sometimes referred to as quadrille paper . next back content

116. Engineering paper is traditionally printed on light green or tan translucent paper. The grid lines are printed on the back side of each page and show through faintly to the front side. Each page has an unprinted margin. When photocopied or scanned, the grid lines typically do not show up in the resulting copy, which often gives the work a neat, uncluttered appearance. In the US, some engineering professors require student homework to be completed on engineering paper. Hexagonal — This paper shows regular hexagons instead of squares. These can be used to map geometric tiled or tessellated designs among other uses next back content

117. Isometric graph paper or 3D graph paper — This type is a triangular graph paper which uses a series of three guidelines forming a 60° grid of small triangles. The triangles are arranged in groups of six to make hexagons. The name suggests the use for isometric views or pseudo-three dimensional views. Among other functions, they can be used in the design of trianglepoint embroidery. Logarithmic — This type of paper has rectangles drawn in varying widths corresponding to a logarithmic scales for semilog graphs or log-log graphs In general, graphs showing grids are sometimes called Cartesian graphs because the square can be used to map measurements onto a Cartesian (x vs. y) coordinatesystes. next back content

118. Normal probability paper — This type is another graph paper with rectangles of variable widths. It is designed so that "the graph of the normal distribution function is represented on it by a straight line." Polar Coordinate — This type of paper has concentric circles divided into small arcs or 'pie wedges' to allow plotting in polar coordinates next back content

119. Graph paper for games Graph paper is also used for games, such as laying out virtual worlds in role playing games and designing new levels in grid-based games such as Minesweeper. Hexagonally tiled graph paper is particularly popular for war games and role playing games because the number of hexes passed through is more constant for the same distance no matter what direction is moved, as opposed to a square tiling where distances are longer if one moves along the diagonal. 1. Mathematical graphing. This is the first and most obvious usage. Graphing is a huge part of modern mathematics, especially once you reach higher math such as statistics and calculus. There is no way to get certain assignments done without it, so make sure you print out a good supply. 2. Art projects. Graph paper is a great way to make fashionably pixilated versions of your favorite masterpieces. It can also be used for cubist modern art and a variety of other graphic projects. As a bonus, printable grid paper often comes in a variety of grid sizes, so you may be able to customize the scale of your masterpiece. next back content

120. 3. Home renovation. Are you planning to remodel but not sure how different elements will fit into your rooms? Trial and error is way too expensive for the average person, so measure the rooms you are planning to change and make a graph paper drawing. You can find the dimensions of different furniture and accessories and make graph paper versions of them to see where they fit. These can be moved around until you find the perfect configuration, saving you a lot of money and time, as well as a few backaches from moving heavy furniture. 4. Statistical charts. If you are trying to explain a set of data to someone, nothing is as helpful as a visual aid such as a graph or a chart showing the facts and figures in a graphic, easy to understand form. Sometimes a picture truly does speak louder than words. 5. Craft project planning. If you are planning a cross stitching or embroidery project, graph paper can be the best way to make the project perfectly symmetrical. The same goes for any project where the medium is permanent, or permanent enough that trial and error simply isn't an efficient idea. Graph paper can also be used to make patterns for scrap booking tags and other paper effects. next back content

121. Example: To graph, or plot, the ordered pair (4,3), start at the origin, move 4 units to the right and 3 units up, and place a dot at the point. To graph the ordered pair (-4, -3), start at the origin, move 4 units to the left and 3 units down, and place a dot on a point. next back content

122. Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ ACTIVITY 8 Test Yourself A. Instruction: Answer the following questions. 1. Site three (3) types of graph paper and differentiate each from the other. _________________________________________________________________________________________________________________________________________________________________________________ 2. Give the usefulness of graph paper. _________________________________________________________________________________________________________________________________________________________________________________ 3. The graph paper was used in what game (s)? _________________________________________________________________________________________________________________________________________________________________________________ next back content

123. B. Instruction. Answer the following questions. 1. Using graph paper, draw a Cartesian plane and plot given ordered pair and connect this points. A (0,-4) B (0, 0) & C (2, 0).What figure is formed in connecting the three points and in what quadrant does it belong? 2. Graph each ordered pair. a. (2, -3) d. (-7, -5) b. (-4, 2) e. (11, -4) c. (-5,12) f. (-5, 8) next back content

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125. A B C D E G H F B. Find the coordinates of the following points. next back content

126. SLIDE RULE LESSON 4 Slide Rule The slide rule , also known colloquially as a slips tick , is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for "scientific" functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction. Objectives: 1. To familiarize the physical design of slide rule. 2. To appreciate the importance of slide rule. 3. To apply it in trigonometry, logarithm, and exponential. next back content

127. Slide rules come in a diverse range of styles and generally appear in a linear or circular form with a standardized set of markings (scales) essential to performing mathematical computations. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations common to that field. William Oughtred and others developed the slide rule in the 1600s based on the emerging work on logarithms by John Napier. Before the advent of the pocket calculator, it was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow through the 1950s and 1960s even as digital computing devices were being gradually introduced; but around 1974 the electronic scientific calculator made it largely obsolete and most suppliers exited the business. A slide rule positioned so as to multiply by 2. Each number on the D (bottom) scale is double the number above it on the C (middle) scale. next back content

128. Use of slide rule in different operation Multiplication A logarithm transforms the operations of multiplication and division to addition and subtraction according to the rules log( xy ) = log( x ) + log( y ) and log( x / y ) = log( x ) − log( y ). Moving the top scale to the right by a distance of log( x ), by matching the beginning of the top scale with the label x on the bottom, aligns each number y , at position log( y ) on the top scale, with the number at position log( x ) + log( y ) on the bottom scale. Because log( x ) + log( y ) = log( xy ), this position on the bottom scale gives xy , the product of x and y . For example, to calculate 3*2, the 1 on the top scale is moved to the 2 on the bottom scale. The answer, 6, is read off the bottom scale where 3 is on the top scale. In general, the 1 on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top. Operations may go "off the scale;" for example, the diagram above shows that the slide rule has not positioned the 7 on the upper scale above any number on the lower scale, so it does not give any answer for 2×7. next back content

130. Division The illustration below demonstrates the computation of 5.5/2. The 2 on the top scale is placed over the 5.5 on the bottom scale. The 1 on the top scale lies above the quotient, 2.75. There is more than one method for doing division, but the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the 1 at either end. Other operations In addition to the logarithmic scales, some slide rules have other mathematical functions encoded on other auxiliary scales. The most popular were trigonometric, usually sine and tangent, common logarithm (log10) (for taking the log of a value on a multiplier scale), natural logarithm (ln) and exponential ( ex ) scales. Some rules include a Pythagorean scale, to figure sides of triangles, and a scale to figure circles. Others feature scales for calculating hyperbolic functions. On linear rules, the scales and their labeling are highly standardized, with variation usually occurring only in terms of which scales are included and in what order: next back content

131. next back content used for finding tangents and cotangents on the D and DI scales T used for finding sines and cosines on the D scale S "inverted" scales, running from right to left, used to simplify 1/ x steps CI, DI, DIF "folded" versions of the C and D scales that start from π rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 but isn't sure whether it will be slightly less or slightly more than 10, the folded scales avoid the possibility of going off the scale. Second, by making the start π rather than the square root of 10, multiplying or dividing by π (as is common in science and engineering formulas) is simplified. CF, DF three-decade logarithmic scale, used for finding cube roots and cubes of numbers K single-decade logarithmic scales C, D two-decade logarithmic scales, used for finding square roots and squares of numbers A, B

132. next back content a linear scale, used along with the C and D scales for finding natural (base e) logarithms and e x Ln a set of log-log scales, used for finding logarithms and exponentials of numbers LLn a linear scale, used along with the C and D scales for finding base-10 logarithms and powers of 10 L used for sine's and tangents of small angles and degree–radian conversion ST, SRT

133. The scales on the front and back of a K&E 4081-3 slide rule. The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions. Roots and powers There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute x 2, for example, locate x on the D scale and read its square on the A scale. Inverting this process allows square roots to be found, and similarly for the powers 3, 1/3, 2/3, and 3/2. Care must be taken when the base, x, is found in more than one place on its scale. For instance, there are two nines on the A scale; to find the square root of nine, use the first one; the second one gives the square root of 90. next back content

134. For xy problems, use the LL scales. When several LL scales are present, use the one with x on it. First, align the leftmost 1 on the C scale with x on the LL scale. Then, find y on the C scale and go down to the LL scale with x on it. That scale will indicate the answer. If y is "off the scale," locate xy / 2 and square it using the A and B scales as described above. Trigonometry The S, T, and ST scales are used for trig functions and multiples of