Flores And De Leon

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)

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

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

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

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

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

Laguna State Polytechnic University Siniloan Campus Siniloan, Laguna MISSION The University shall primarily provide advanced education, professional, technological and vocational instruction in agriculture, fisheries, forestry, science, engineering, industrial technologies, teacher education, medicine, law, arts and sciences, information technologies and other related fields. It shall also undertake research and extension services and provide progressive leadership in its areas of specialization. next back content

Laguna State Polytechnic University Siniloan Campus Siniloan, Laguna GOALS In pursuit of the college vision/mission the College of Education is committed to develop the full potentials of the individuals and equip them with knowledge, skills and attitudes in Teacher Education allied fields to effectively respond to the increasing demands, challenges and opportunities of changing time for global competitiveness. next back content

Laguna State Polytechnic University Siniloan Campus Siniloan, Laguna OBJECTIVES OF BSEd Program Produce graduate who can demonstrate and practice the professional and ethical requirement for the Bachelor of Secondary Education such as: 1. To serve as positive and powerful role models in the pursuit of the learning thereby maintaining high regards to professional growth. 2. Focus on the significance of providing wholesome and desirable learning environment. 3. Facilitate learning process in diverse type of learners. 4. Used varied learning approaches and activities, instructional materials and learning resources. 5. Used assessment data, plan and revise teaching – learning plans. 6. Direct and strengthen the links between school and community activities. 7. Conduct research and develo pment in Teacher Education and other related activities. next back content

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

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

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

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

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

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

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

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

The general objectives of this module are to understand the different uses and to impart knowledge about the different tools use in teaching mathematics. With this, the students familiarize their selves on what tools are appropriate to use in different activities in mathematics. It helps both teachers and students to know how a certain tools work and with that we can apply it in our study and also in our everyday living. It also impart knowledge when a certain tools use in mathematics came from and with that our minds gain more information. This module helps us a lot in our all aspects of our lives. GENERAL OBJECTIVES next back content

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.

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

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. .

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

An example illustrating the use of the Addition Table, adding 2 and 7: 1. move down the Given the first number column to a row containing 2 and ; 2. move down the To add column to a row containing 7: the 7th row, then; 3. move across to the Move bead columns and perform the operations specified: in the lower deck , count 2 beads; in the upper deck , count 1 bead. 4. The answer then presents itself on the abacus. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ACTIVITY 3 Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ next back content Test Yourself Instruction: Identify the following. 1. Napier’s bones was created by__________. 2. Napier’s bones also called___________. 3. The __________consist of strips woods, metal or heavy cardboard. 4. A set of ____________might be enclosed in a convenient carrying case. 5-6. Napier’s bones was use for calculation of________and _________of numbers. 7-8. Napier’s bones was base on ___________and__________sue by Fibonacci writing in the Liber Abaci. 9. Napier published his version of rods in a work printed in Edinburgh, Scotland, at the end of 1617 entitled___________. 10. More advanced use of ____________can even extracts square roots.

B. Instruction: Find the square root of the following using Napier’s bones. 9, 801 2. 26, 244 3. 982, 081 4. 107, 651 5. 374, 544 6. 431, 134 7. 9, 125, 761 8. 26, 568, 327 9. 101, 002, 334 10. 21, 708, 023 next back content

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

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

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

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

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

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

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

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

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. ______________________________________________________________________________________________________________________

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

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

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

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

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

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

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

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

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ next back content CHAPTER TEST Test Yourself A. Instruction: Identify the following. 1. The user of an abacus who slides the beads of the abacus by hand is called an _______. 2. The oldest known counting board was discovered on the Greek Island of ______ in 1899. 3. Napier published his version of rods in a work printed in Edinburgh, Scotland at the end of 1617 entitled __________. 4. Pascal’s Triangle is a geometric arrangement of _________ in a triangle. 5. The Arithmetic Rope is also called ___________. 6. Counting rods were used by ancient ___________ for more than 2000 years. 7. _________ were token or coin like medals produced across Europe from the 13th through the 17 centuries 8. _________ is also known as dibs. 9. _________ is also called counting frame. 10. The ________ is consisting of strips of woods, metal or heavy cardboard.

B. Instruction: Answer the following questions. In your own opinion, what is the best traditional tool? Why? What traditional tool has much purpose? Why? Differentiate Pascal’s Triangle to Counting Rods. How these different traditional tools improve mathematics? Explain the importance of these different traditional tools. C. Instruction: Find the square root of the following using knucklebones. 169 = 6. 99, 998 = 7, 205 = 7. 101, 625 = 8, 112 = 8. 925, 619 = 50, 613 = 9. 54, 234 = 5. 76, 483 = 10. 76, 908 = next back content

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

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

“ Nothing is permanent except change,” it is a quotation from _______ that tells us that everything in this world are develop as time goes by that’s why we are now move on in modern times because we improve ourselves and also the different things around us. We transform to a better one which all thing can work in just only one click. In this chapter we will enter to a world that is full of modern tools used in mathematics. General Objectives: At the end of this chapter, the student must be able to: 1. recognize the different tools used in different mathematical equation 2. appreciate the different tools for their personal use 3. use the different tools in their mathematical activities. next back content Chapter II: MODERN TOOLS IN MATHEMATICS

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

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

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

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

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

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

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

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

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

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

B. Solve the given number using calculator. 1,723 + 2,574 = 2. 15,323 + 18,335 = 3. √ 4,563 – 10 = 4. (36 + 152) – (244 + 165) = 5. 12 x 144 = 6. 21.3 x 183. 32 = 7. 125 ÷ 5 = 8. (35 + 112) ÷ (258 – 132) 9 3 + 28= 18 3 – 180= next back content

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

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

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

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

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

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

 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

 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

 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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In such cases, the user may slide the upper scale to the left until its right index aligns with the 2, effectively multiplying by 0.2 instead of by 2, as in the illustration below: Here the user of the slide rule must remember to adjust the decimal point appropriately to correct the final answer. We wanted to find 2×7, but instead we calculated 0.2×7=1.4. So the true answer is not 1.4 but 14. Resetting the slide is not the only way to handle multiplications that would result in off-scale results, such as 2×7; some other methods are: Use the double-decade scales A and B. Use the folded scales. In this example, set the left 1 of C opposite the 2 of D. Move the cursor to 7 on CF, and read the result from DF. 3.Use the CI inverted scale. Position the 7 on the CI scale above the 2 on the D scale, and then read the result off of the D scale, below the 1 on the CI scale. Since 1 occurs in two places on the CI scale, one of them will always be on-scale. 4.Use both the CI inverted scale and the C scale. Line up the 2 of CI with the 1 of D, and read the result from D, below the 7 on the C scale. next back content

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

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

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

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

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 trig functions, for angles in degrees. For angles from around 5.7 up to 90 degrees, sines are found by comparing the S scale with C. The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with C or, for angles greater than 45 degrees, CI. Common forms such as k sin x can be read directly from x on the S scale to the result on the D scale, when the C-scale index is set at k. For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/radian. Inverse trigonometric functions are found by reversing the process. next back content

Many slide rules have S, T, and ST scales marked with degrees and minutes. So-called decitrig models use decimal fractions of degrees instead. Logarithms and exponentials Base-10 logarithms and exponentials are found using the L scale, which is linear. Some slide rules have a Ln scale, which is for base e. The Ln scale was invented by an 11th grade student, Stephen B. Cohen, in 1958. The original intent was to allow the user to select an exponent x (in the range 0 to 2.3) on the Ln scale and read e x on the C (or D) scale and e– x on the CI (or DI) scale. Pickett, Inc. was given exclusive rights to the scale. Later, the inventor created a set of "marks" on the Ln scale to extend the range beyond the 2.3 limit, but Pickett never incorporated these marks on any of its slide rules. next back content

Addition and subtraction Slide rules are not typically used for addition and subtraction, but it is nevertheless possible to do so using two different techniques. The first method to perform addition and subtraction on the C and D (or any comparable scales) requires converting the problem into one of division. For addition, the quotient of the two variables plus one times the divisor equals their sum: For subtraction, the quotient of the two variables minus one times the divisor equals their difference: next back content

This method is similar to the addition/subtraction technique used for high-speed electronic circuits with the logarithmic number system in specialized computer applications like the Gravity Pipe (GRAPE) supercomputer and hidden Markov models. The second method utilizes a sliding linear L scale available on some models. Addition and subtraction are performed by sliding the cursor left (for subtraction) or right (for addition) then returning the slide to 0 to read the result Different kinds of slide rule Circular slide rules Pickett circular slide rule with two cursors. (4.25 in./10.9 cm diameter) Reverse has additional scale and one cursor. next back content

A simple circular slide rule, made by Concise Co., Ltd., Tokyo, Japan, with only inverse, square and cubic scales. On the reverse is a handy list of 38 metric/imperial conversion factors. Breitling Navitimer wristwatch with circular slide rule. next back content

Circular slide rules are mechanically more rugged and smoother-moving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm off-centre of the pivot can result in a 0.2 mm worst case alignment error. The pivot, however, does prevent scratching of the face and cursors. The highest accuracy scales are placed on the outer rings. Rather than "split" scales, high-end circular rules use spiral scales for more complex operations like log-of-log scales. One eight-inch premium circular rule had a 50-inch spiral log-log scale. next back content

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ ACTIVITY 9 Test Yourself Instruction: Answer the following questions. 1. Discuss briefly the uses and importance of slide rule. ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. What are the advantages of using slide rule? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

3. What are the disadvantages of using slide rule? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. How these slide rules help us in our study? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content Instruction: Express as a sum or difference of their logarithm. Log xyz = Log xy = Z 3. Log x 2 = 4. Log x 2 y 2 = 5. Log x 2 = y 2 6. Log 3 √ xy = 7. 2 Log x 2 y = 8. Log x 3 = y 9. 2 Log xyz = 10. Log x 3 y 3 z 3 =

PROTRACTOR LESSON 5 Protractor In geometry, a protractor is a circular or semicircular tool for measuring an angle or a circle. The units of measurement utilized are usually degrees invented by Joseph Huddard in 1801. Some protractors are simple half-discs; these have existed since ancient times. More advanced protractors, such as the Bevel Protractor have one or two swinging arms, which can be used to help measure the angle. Objectives: 1. To give the meaning of protractor. 2. To recognize the uses of protractor. 3. To use it in measuring angles. next back content

Bevel Protractor A Bevel Protractor is a graduated circular protractor having a pivoted arm; used for measuring or marking off angles. Sometimes veneer scales are attached to give more accurate readings. It has wide application in architectural and mechanical drawing although with the availability of modern drawing software or CAD the tool is less commonly used in that sphere. Universal Bevel Protractors are also used by toolmakers, as they measure angles by mechanical contact they are classed as mechanical protractors. Gallery next back content A half circle protractor marked in degrees (180 degrees).

next back content A half circle protractor marked in degrees (180 degrees). A 360° protractor marked in degrees. A "Cras Navigation Plotter" double-protractor, in foreground.

Parts of a protractor There are two sets of numbers on a protractor, so you can measure angles from either side of the protractor. next back content

Steps in using a protractor: 1. Look at the angle you are measuring. 2. If the angles are greater than 90 degrees (an obtuse angle) use the scale that is greater than 90. 3. If the angles are less than 90 degrees (an acute angle) use the scale that is less than 90. 4. Place the protractor on the angle so that the center point is on the vertex and the zero line is along one side of the angle. 5. Follow the scale beginning at zero and find the number where the other side of the angle intersects the scale on the protractor. 6. The number where the line intersects is the degree of the angle. next back content

Example One: Right Angle A right angle measures actually 90 degrees . next back content

Example Two: Obtuse Angle An obtuse angle measures greater than 90 degrees. next back content

Example Three: Acute Angle An acute angle measures less than 90 degrees. next back content

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ ACTIVITY 10 Test Yourself A. Instruction. Using a protractor, measure the following angles. 1. 2. 3. next back content

4. 5. next back content 6. 7. 8. 9. 10.

B. Instruction: Draw the following angles given its measures. Use a protractor. 1. m<LMN=115 6. m<LUV=234 2. m<HIJ=18 7. m<MIC=200 3. m<STU=23 8. m<FAM=123 4. m<NOK=270 9. m<ELA=213 5. m<JON=143 10.mARJ<=214 next back content

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ CHAPTER TEST next back content Test Yourself A. Instruction: Define the following. 1. Calculator ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. Slide rule ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

next back content 3. Compass ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. Protractor ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 5. Graph paper ________________________________________________________________________________________________________________________________________________________________________________________________________________________________

next back content B. Instruction: Solve the following numbers using calculator. 9 3 = 6. Sec 75° = 2,456,712 = 7. Cot 83° = 2 3. Tan 270° = 8. Csc -95° = 4. Sin 35° = 9. 5. Cos 150° = √ 1,892 = 10. 8 2 + 9 3 = √ 125 C. Instruction: Graph the following ordered pairs on a graphing paper and connect all the points orderly. State the shape being formed. A (3,5) E (-3, -5) B (-3,5) F (3, -5) C (-6, 2 ½) G (6, -2.5) D (-6, -5/2) H (6, 2 ½)

next back content Instruction: Transform the following operations of logarithm according to its rules. 1. Log 8 + Log x = 2. Log (x-2) + Log (x + 3) = 3. Log x + Log 4 = 2 2 4. – 2 Log x + Log (x – 4) = 2 5. 2 Log + Log y – Log 9 = 2 Instruction: Draw the following angles given its measure. Use a protractor. 1. m<ABC = 135° 2. m<DEF = 10° 3. m<GHI = 93° 4 m<JKL = 75° 5. m<MNO = 43°

ChapterIII: HIGH TECH. TOOLS IN MATHEMATICS We are now in modern times but we want more that’s why we are still exploring and inventing something new, standard, high quality, and a better one than the things we have today that can really satisfied the needs and also the wants of the people. But the question is “are we satisfied?” This chapter will help us to answer this question because of the new, latest, hottest, and high technology tools use in mathematics. next back content General Objectives : At the end of the chapter, the students must be able to: 1. enumerate the different high tech. tools 2. appreciate the importance of different high tech tools to us 3. apply the knowledge we gain in our study and also in our everyday living.

LESSON 1 SPEED SCORE Objectives 1. To recognize the uses of speed score. 2. To appreciate the importance of speed score. 3. To examine the special features of speed score. Speed Score next back content

SpeedScore is the ultimate timesaving device for busy teachers. It includes 75 perforated Answer Templates (4 formats...T/F, Multiple Choice, combination T/F and Multiple Choice, and an alternate Multiple Choice where every other row uses the same answer letter designations... designed to help students avoid placing two answers in the same row.) SpeedScore 100 reproducible Student Answer Sheets (4 formats...T/F, Multiple Choice, combination T/F and Multiple Choice, and an alternate Multiple Choice where every other row uses the same answer letter designations...designed to help students avoid placing two answers in the same row. How it works Teacher punches out correct answer circle for each test/quiz item on perforated Answer Template...slide this Answer Template into the slot on the front of the SpeedScorer....turn on Speed Scorer and light from SpeedScorer illuminates the correct answer circles on the Answer Template. next back content

Have students darken an answer circle for each test/quiz item on their Student Answer Sheet...place each Student Answer Sheet on top of illuminated Answer Template...each student's incorrect answer circle will become illuminated on their Student Answer Sheet for your quick grading of each answer (a correct student answer circle will block out the light from the illuminated answer circle on the Answer Template). Special features The SpeedScorer also has an adjustable (from 10 to 60 minutes) auto shut-off; so, if you should forget to turn off your SpeedScorer, it will automatically shut off to save your battery power. next back content

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ ACTIVITY 11 Test Yourself Instruction: Answer the following questions. 1. What is speed score? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. What are the special features of speed scorer? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

3. Discuss briefly the advantages of speed score to other tools. __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. How this speed score helps us in our studying math? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 5. Explain how this tool works? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

LESSON 2 GRAPHING CALCULATOR Objectives: 1. To recognize graphing calculator. 2. To appreciate the help of graphing calculator to us. 3. To differentiate the kinds of graphing calculator. Graphing Calculator next back content

The TI-84 Plus is an enhanced version of the TI-83 Plus with 3 times more memory and included USB computer cable. Fully compatible with the TI-83 Plus. Requires 4 AAA batteries. Back-up lithium battery. CBL2 and CBR compatible. The TI-84 Plus is fully loaded with a built-in USB port, an improved display, many preloaded Apps, and much more, the maximum performance TI-84 Plus Silver Edition is 100% compatible, keystroke-for-keystroke with the TI-83 Plus family. Integrating it into your classroom couldn’t be easier! With the TI-84 Plus Silver Edition, all students can now share their work by connecting their handheld to any TI presentation tools for the whole class to see, fostering a collaborative learning environment. Plus, a built-in USB port makes data transfer to your computer and between handhelds easier than ever. next back content

Different kinds of graphing calculator TI-84 Plus Graphing Calculator - Teacher's Kit Yellow Teacher Pack includes (10) TI-84 PLUS calculators, each with a bright, easy-to-spot "school bus yellow" back cover and slide case. In addition, each units faceplate is inscribed with the words "School Property". Also includes (10) USB cables, (10) unit-to-unit cables and manuals; (40) AAA batteries; (1) product CD, poster and transparency.With the TI-84 Plus, all students can now share their work by connecting their TI-84 Plus to any TI presentation tools for the whole class to see, fostering a collaborative learning environment. Plus, a built-in USB port makes data transfer to your computer and between handhelds easier than ever. The TI-84 Plus handheld is fully loaded with convenient features and accessories, including a kickstand slide case, compatibility with TI presentation tools on every unit. TI-89 Graphing Calculator The most advanced TI Graphing Handheld. Ideal for calculus! Now includes USB cable for computer connectivity. 4 AAA batteries. Back-up lithium battery. CBL2 and CBR compatible. Clam Shell Design. next back content

TI-89 Titanium Graphics Calculator Introducing TI's most powerful graphing calculator: The TI-89 Titanium, offering new features, pre-loaded Apps, and even more versatility. A built-in USB port makes data transfer ultra-convenient. Plus, with three times the memory of the TI-89, you can store more Apps, data, and programs. The TI-89 Titanium's advanced functionality and 3-D graphing make problem-solving for AP advanced mathematics and engineering courses infinitely easier. It is the most powerful TI graphing calculator allowed for use on the AP Calculus, AP Statistics, AP Physics, AP Chemistry, PSAT/NMSQT, SAT I , SAT II, and Math IC & IIC exams. Includes calculator with preloaded applications, USB cable and a Product CD. next back content

Combines the functionality of the TI92PLUS with the added benefit of 3x more Flash ROM memory and improved interface. Voyage200 Personal Learning Tool Complete with a built-in QWERTY keyboard, a Computer Algebra System (CAS), and "Pretty Print", which shows mathematical expressions the way they are traditionally written. Voyage200 has 8 pre-loaded software applications including the Geometer's Sketchpad, Cabri Geometry. TI-Voyage200 Calculator The most powerful handheld functionality from Texas Instruments Voyage 200 lets you easily customize your handheld by adding new functionality through software applications. Choose the Voyage 200 Personal Learning Tool (the great update to the TI-92 Plus) with its large display and QWERTY keyboard.The Voyage 200 PLT features a built-in QWERTY keyboard, new icon desktop for easy navigation and organization of your Apps, a viewscreen port and a clock to keep track of time or to use for timing experiments. You will also enjoy computer connectivity with the TI-GRAPH LINK USB cable and TI Connect software already in the box. next back content

Discovering Math on the Voyage200 New Exploration Workbook Teacher resources that help integrate TI technology from kindergarten through college. Get the most out of your Voyage200 calculator. These explorations expand the usefulness of this valuable mathematics learning tool. Guided investigations are designed to help teachers direct lessons into a variety of mathematical areas. Geometric Investigation with Voyage200 & Cabri The explorations in this book are intended to promote an environment of inquiry through thought provoking, open-ended problems. There is no intended order of topics. Any exploration can be used where it fits into your program. There is no suggested time for an exploration. Some students may think about a given problem for weeks or months, coming up with new insights and interconnections as they consider other problems. next back content

TI-73 Explorer Graphing Calculator For grades 4-7, the TI-73 Graphing Calculator is specially designed for math and science. Includes fractions, constants and pie charts. Electronically upgradable. Link cable for data sharing with another TI-73. 4 AAA batteries. Back-up lithium battery. TI-30XIIB Student Calculator Soar in math and science classes with this powerful tool. The TI-30X combines statistics and advanced scientific functions in a durable and affordable calculator. The 2-Line display helps students explore math and science concepts in the classroom. This calculator is perfect for middle school students (grades 6-8). next back content

next back content x < 5 Or x ≥ 16 Inequality Graphing in 1D: Solve Inequalities in One Variable Smiley Figure 8 Circles, Ellipses, Hyperbolas Lines, Parabolas See Examples Advanced Intermediate Basic Equation Plotting Calculators: with Table of Values for Algebra & Analytic Geometry

next back content Find Points of Intersection Graph & Solve Systems of Nonlinear Equations Plot Families of Curves Graph & Solve Systems of Linear Equations See Examples Advanced Intermediate Basic Multiple Equation Plotting Calculators: Solve Systems of Equations

To get the GRAPH screen: green diamond and F2 after entering expressions into Y=. F2 - ZOOM #6 Zoom Standard creates a 10 x 10 pixel window. F3 Trace, F4 Re-draw next back content

F5 - Math F6 - Draw Commands next back content

Example: Graph next back content It is possible to enter more than two sections by "nesting" the when command. Nesting means placing one function inside of another function. Here is the graph in dot mode. DOT style is chosen to avoid having the sections of the graph connected. The function is entered using the WHEN command. Method 2: Using WHEN (for two pieces). To get when, type it in or use CATALOG. when( condition, trueExpression, falseExpression ) When entering more than 2 entries, it will be necessary to utilize more than one Y= position. The | command, which can be read “with”, is needed. TI-89: | is its own key Voyage: | is located above the K Method 1: Entering separate sections Piecewise Defined Functions

Example: Graph the inverse of f ( x ) = 2 x + 3. Enter f ( x ) into Y1. From the Graph Screen, choose F6, #3 DrawInv (or use Catalog ) From the Home Screen, DrawInv Y1(x). ENTER. Remember that Y1 is obtained from 2nd VARS above the subtraction key. Or check to see if your algebraic inverse answer is correct. 1. Enter f ( x ) into Y1. 2. Enter your algebraic inverse answer into Y2. 3. Turn off Y2 4. DrawInv Y1(x) 5. Turn on Y2 with bubble animation 6. Graph If your answer is correct the bubble will follow the path of the drawn inverse. next back content Note the syntax for entering the functions. Algebra and Composition of Functions Remember to get the Y1 variables from the VAR menu. Inverse Functions

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ ACTIVITY 12 Test Yourself Instruction: Answer the following questions. 1. What are the differences of graphing calculator to other calculator? ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. Explain briefly the importance of graphing calculator. ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

3. What are the special features of graphing calculator? ________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. What are the advantages and disadvantages of graphing calculator? ________________________________________________________________________________________________________________________________________________________________________________________________________________ 5. Discuss how this graphing calculator helps us? ________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

DIGIMEMOL2 LESSON 3 Objectives 1. To identify what is DigimemoL2. 2. To know the importance of DigimemoL2. 3. To formulate a generalization about the DigiMemoL2. DigiMemo L2 next back content

The DigiMemo L2 is a digital note pad that can digitally record your handwritten notes using ordinary paper. With a new larger writing area (8 " x 11"), the DigiMemo L2 will also function as a graphics tablet when plugged into the computer. Write with an actual inking pen on ordinary paper and will record both a hard and soft copy. You can scan documents or forms with a scanner and insert as a background in the DigiMemo software. The DigiMemo L2 is great for form filling applications. The DigiMemo L2 connects with USB cable to PC (for Windows 2000, XP & up only) DigiMemo - Digitally Capture and Store Everything you Write with Ink on Paper. The DigiMemo 692 is a stand-alone device with storage capability that digitally captures and stores everything you write or draw with ink on ordinary paper, without the use of computer and special paper. Then you can easily view, edit, organize and share your handwritten notes in Windows. next back content

Take Notes Place any ordinary paper or notepad on the digital pad. Write on the paper with the digital inking pen. The digital pad digitally records anything you write in its built-in storage device or an optional CF card in real time. Each page you write is stored as a digital page. next back content

Connect to the PC Connect the digital pad to your PC by any USB port. Organize Your Notes With its DigiMemo Manager software, you can easily view, edit and organize your digital pages in Windows. Save any digital pages you arbitrarily select as a book file (e-Book). Send via e-Mail Share your notes with others via e-Mail. Portable and Compact The digital pad can be effortlessly held in one hand or on the lap and easily operated while you are standing, sitting or reclining. An ultra-thin & light digital pen with a regular and replaceable ink cartridge. Only 1.24 lbs (560 g) for the digital pad and 0.03 lbs(13.8 g) for the digital pen. next back content

Familiar and Natural Feel as comfortable as you normally write with a regular pen on paper. Immediately get both a digital record and a hardcopy duplicate of your handwritten notes even when you are on the move. Instantly & digitally record your notes, ideas, sketches, drawings and flowcharts, without scanning. Any ordinary paper or notepad (up to 6.0" x 9.1" / 153 mm x 232 mm) can be applied. You cannot remember what you typed 10 years ago; however, you will always remember what you wrote. Powerful and Unforgettable Digitally capture & store everything you write with ink on ordinary paper in real time. No computer and special paper is needed. A stand-alone device with storage capability, that can be both the built-in 32 Mb storage device in the digital pad and the expandable storage capacity by an optional Compact Flash (CF) memory card. The more you write on paper, the bigger file size of the digital page will be. next back content

The file size of the digital page for one paper (6.0" x 9.1") with completely full handwritten notes is only about 200 KB. Manage up to 999 digital pages for each storage device. Capture your handwritten notes up to 0.47" (12 mm) thickness of paper (about 120 sheets of paper). Can continue to write into the digital page with contents. An excellent solution for form-filling application which needs a form hardcopy and digital filling data. Efficient The battery of the digital inking pen can last approximately 14 months. With continuous use of the digital pad, four AAA alkaline batteries can last approximately 100 hours. 30 minutes automatic shutdown. next back content

Simple and Manageable With its DigiMemo Manager software, you can easily organize your digital pages in Windows. Share them with others via e-Mail. Save the digital pages you arbitrarily select as a book file (e-Book). Page thumbnails function. Easily import and remove your digital pages in a book file. Highlighting, annotating, editing (select, copy, move, erase …etc.), typing, viewing, changing color and drawing functions. Easily copy or move to another program files such as Outlook e-mail message, Word file or Excel file …etc Can be saved as BMP, JPG, GIF or PNG format. e-Mail Easily copy or move to Outlook e-mail message. Send the handwritten e-Mail anywhere anytime. Share your e-Book with friends who just need to download the DigiMemo Manager software from ACECAD website so that they can view or edit your e-Book. next back content

ACTIVITY 13 Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ Test Yourself Instruction: Answer the following questions. 1. What is DigiMemoL2? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. Discuss the purpose of DigiMemoL2. ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

3. What are the special features of DigiMemoL2? ________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. Explain the importance of DigiMemoL2. ______________________________________________________ ____________________________________________________________________________________________________________ ______________________________________________________ 5. How this DigiMemoL2 help us in our everyday living? ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

LESSON 4 interactive whiteboard Interactive whiteboard next back content Objectives 1. To identify the uses of Interactive whiteboard. 2. To know the importance of interactive whiteboard. 3. To formulate a generalization about interactive whiteboard.

Smart Boards are a special type of interactive touch-sensitive white boards that can be used to collaborate and communicate with students in the classroom, with faculty or staff in conference rooms, with colleagues over the Internet, or just about anywhere that has electricity and an outlet to plug in a Smart Board, a computer and, if you really want to benefit from the Smart Board's capabilities, a projector as well. A Smart Board can be used to enhance pedagogy and increase clarity and degree of detail through presentations. The Smart Board and computer can be used with or without a projector, however, in order to take full advantage of the Smart Board, a projector is recommended. If you choose to use the projector, you have the added functionality of pressing on the touch-sensitive surface of the board to access and control any point-and click software or product on your machine. . These include but are not limited to Internet browsers like IE and Netscape, E-mail programs like Outlook Express, Netscape, Eudora and Pegasus, and Office Suite Programs like Microsoft Office, WordPerfect Office and Star Office Suites, as well as multimedia applications. The Smart Board software allows the user to save, print, and e-mail their presentation to others or to keep it as a record in the Smart Notebook. next back content

The Smart Board can be used as both an electronic whiteboard (without a projector), a product that is used to save dry-erase marker notes to a computer or an interactive whiteboard (with a projector) which is a product that can be used to show the computer screen and interact with that information. How Are Smart Boards Used in the Classroom? In modern classrooms, the Smart Board is becoming as regular a feature as desks. Smart Boards meld high-tech functionality and tradition by acting as a computer monitor and a chalkboard at the same time. In our wired society, we can now show videos, write equations and check homework all on the same board in the classroom. Smart Boards represent an exciting technological step forward for presenters and teachers. Operation Smart Boards are touch-sensitive input devices. Via a series of cables connecting the classroom projector, the source computer and the Smart Board, the board functions as a sort of outboard mouse and monitor, allowing the user to "manipulate" the information that is being displayed via the board's touch surface. The information displayed by the source computer is projected onto the front of the board. next back content

Slide Shows 2. One function of the Smart Board is as a slide advance. Slide show presentations, such as the kind created using PowerPoint, can be projected onto Smart Boards. Presentations that do not require timing can be advanced by a tap on the board's surface; specific regions of the board are designated as slide advance/reverse to allow the presenter use of the included markers without changing the displayed slide. The advantage of using the Smart Board is that the presenter is not tied to the source computer, allowing movement during the presentation. Digital Blackboards 3. Smart Boards can function as dry-erase boards through the use of the included Smart Board pens. The pens come in standard colors and allow the presenter to make digital marks on either the subject being displayed or a new, white "page." The "ink" can also be erased with an included digital eraser that deletes the marks but does not affect the subject underneath. next back content

Peer Review 4. Especially in computer labs, but also in classrooms, students' work can be displayed on the Smart Board. Traditional homework can be scanned into the source computer, and digital work may be uploaded or transported via portable storage devices or network/Internet transport. Using the Smart Board to display assignments allows everyone in the room to see examples of good work or trouble areas, and corrections and suggestions can be made in class. Additional Suggested Uses 5. Classroom uses of the Smart Board are only limited by the instructor's imagination; there are multiple artistic possibilities for the smart board, including silhouettes and tracing. Students may use the multimedia capabilities of the board to present videos or musical pieces, and the instructor can use the board to provide materials for students who may have forgotten to bring assigned reading pieces. next back content

Maintenance 6. Keep dry-erase markers far away from the Smart Board; many presenters have made the mistake of grabbing an ink marker and writing on the board's surface. If stray ink marks are made on the board, they are difficult to clean and require the attention of a technician. Make sure that your Smart Board is calibrated. If a Smart Board is not calibrated, it is not capable of accurately representing input. If you put your finger or marker on one spot, the board might display input at another location. Calibration software is included with the board. If you are not comfortable calibrating the board yourself, contact technical support. Special Considerations 7. Since the image is projected onto the Smart Board from the computer, presenters should be sure that there is nothing visible on the monitor of the source computer that the audience should not see. Web sites should be opened and checked for pop-ups and inappropriate content prior to commencing the presentation. If necessary, they can be saved for offline access, saving the presenter from possible embarrassment if the site cannot be found, or the Internet connection goes down. next back content

Getting Started 1. After the computer (push the middle-top button on the front of the desktop computer) and projector (push the red button in the back) are powered up, you more than likely will need to use the focus and zoom features on the projector. The picture will probably be very fuzzy. These buttons are located at the top left of the projector if you are facing it like this. The Smart Board software will automatically load at the startup and your screen should look like this. next back content

However, if it doesn't, no worries. Just click on the Smart Notebook 2.2 icon on the desktop. next back content

2. Now, the most important step of this whole process is to calibrate the software . I can't emphasize this enough, and it only takes a few seconds to do. If the board is not calibrated, you will become extremely frustrated because you won't be able to write or use the mouse like you normally do. I know this from experience! Once the projector is showing clearly on the board, go to the computer screen and single-click on the icon with the blue screen at the bottom right. This icon is located in the task bar at the bottom right of the screen. 3. Once you have clicked on the icon you will want to choose "Orient Board", and click on that. Don't worry about the "Pen Tray Settings" or "Select Boards" tabs. They should already be working correctly. next back content

4. Another popup box will emerge that is entitled "Pick the Orientation Precision". You can choose "Quick", "Standard", or "Fine". Standard works well and is recommended but if you are in a hurry, "Quick" will work well too. next back content

5. Position yourself next to the board like you will be presenting. Press your finger to the board in the middle of the red + on the board. After completing this step, the software will move to the next red +, press on that, etc. Keep going until there are no more red + symbols. 6. Once completed, close the popup box and the board will be calibrated correctly. You are now ready to begin your presentation! Please remember that after calibrating the software with the projector and the board, if any of the equipment gets moved accidentally or turned off, etc. you will have to recalibrate the board in order for it work correctly again. next back content

Using the Interactive Whiteboard You can communicate and work together in many ways by using the Smart Board. You can write with the colored markers located in the Pen Tray, use your finger as a mouse, or use the the interactive keyboard. You can go between the mouse and the interactive keyboard by pressing either button. These buttons are located on the right-hand side of the pen tray. If you need to double-click, tap your finger twice quickly on the Board. You can also right click by pressing the "right mouse" button in the Pen Tray. After clicking on the board, be patient, it sometimes takes a few extra seconds for the right-click menu to appear. If you want to write on the board, pickup a stylus from the Pen Tray and write over any part of a displayed image. When you put the stylus back in the pen tray, the mouse cursor will reappear on screen. next back content

However, if you want to save your document with the new material written on it you must first save it BEFORE putting the stylus back in the pen tray or all of your work will disappear. If you want to erase your work, just pick up the eraser on the left-hand side of the Pen Tray. While you are writing on screen a box like this will appear. This are the buttons you will use to either save or print. a. Click the camera button if you would like to save the written notes AND the application background. b. Click the area capture button/middle icon if you would like to save a selected area of your notes. Then drag your finger over the selected area. c. Click the print button if you would like to send the notes and application background to the printer next back content

If you would like to edit your notes, then go back to your Notebook window by clicking on it. It should be open in the taskbar at the bottom of the screen. Using PowerPoint with Smart Board 2. Next, open your PowerPoint presentation. 3. Go to View, Slide Show. 1. First, you need to have the Notebook program open. next back content

4. Use this toolbar to move forward and backwards through your PowerPoint presentation. 5. If you would like to make additions or explanations in PowerPoint, pick-up a stylus and write over the slide show. Remember, DO NOT return the stylus to the pen tray until you have saved your work. 6. You can save directly into PowerPoint or into Notebook. Press the middle button on the toolbar and a popup menu will appear. Note: On this particular thumbnail some of the options are grayed out because there was nothing edited on the PowerPoint slide. 7. To save into Notebook, choose Smart Notebook from the options or if you would like to save into PowerPoint choose File, Save from the File menu. next back content

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ ACTIVITY 14 Test Yourself Instruction: Answer the following questions. next back content 1. Differentiate interactive whiteboard to other high technology tools? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. How interactive whiteboard is used in a classroom discussion? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3. What are the advantages of this interactive whiteboard? ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. Explain briefly the importance of interactive whiteboard? ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 5. Discuss briefly how interactive whiteboard is used in a power point presentation? ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

Name: ___________________________ Year &Section: ___________ Teacher: _________________________ Score: ______Date:_______ CHAPTER TEST Test Yourself A. Define the following. 1. Graphing calculator ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 2. DigiMemoL2 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

3. Speed score ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 4. Interactive whiteboard ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ B. Instruction: Answer the following questions. 1. What is the best high tech. tool? Why? ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________ next back content

2. What are the advantages of the different high tech. tools to the tools that we are using today? ___________________________________________________________________________________________________________________________________________________________________________ 3. For you what is the effect of these different high tech. tools? ___________________________________________________________________________________________________________________________________________________________________________ 4. What generalization can you formulate about these different high tech. tools? ___________________________________________________________________________________________________________________________________________________________________________ 5. What is the difference between graphing calculator to an ordinary calculator? ___________________________________________________________________________________________________________________________________________________________________________ next back content

URL http:// www.venturaes.com /curriculum\. September 20, 2009 http:// en.wikipedia.org/wiki/Nomogram . December 18, 2009 http://www.mastersgames.com/cat/table/knucklebones-osselets.htm http://webgraphing.com. December 29, 2009 http:// search.yahoo.com/search?p = abacus&toggle =1&cop= mss&ei =UTF-8&fr=yfp-t-701. Jan 18, 2010 http://en.wikipedia.org/wiki/Knucklebones. Jan 19, 2010 http:// search.yahoo.com/search;_ylt =A0oGk9fIrm5LeHoA.utXNyoA?p=calculator&fr2= sb-top&fr =yfp-t-701&sao=1 . Jan 21, 2010 http:// search.yahoo.com/search;_ylt =A0oGkwUPr25LASYAGkRXNyoA?fr2= sg-gac&sado =1&p=pascal%27s%20triangle&fr=yfp-t-701&pqstr=pascal%27&gprid=Fvla.MwmQV24i.hgvveG9A&sac=1&sao=1 . Jan 25, 2010 next back content

http://en.wikipedia.org/wiki/Napier%27s_bones Jan 26, 2010 http:// search.yahoo.com/search;_ylt =A0oGklRlr25LAcsAXARXNyoA?fr2= sg-gac&sado =1&p=napier%27s%20bones&fr=yfp-t-701&pqstr=napier%27s&gprid=j465pDQYQ9upx.Pltkhk1A&sac=1&sao=1 . Jan 29, 2010 http;//www.Ehow.com/curriculum/. Feb 2, 2010 http;//www.answers.com/logic-1/. Feb 3, 2010 http;//www.answers.com/Albert-einstein/. Feb 3, 2010 http;//www.answers.com/Benjamin-pierce/. Feb 3, 2010 http;//www.answers.com/mathematician/. Feb 3, 2010 http:// mathbits.com/MathBits/TISection/Voyage/MoreGraph.htm . Feb 4, 2010 next back content

http://www.studyzone.org/mtestprep/math8/e/protractor6l.cfm http://www.ehow.co.uk/about_5101358_smart-boards-used-classroom.html next back content

http:// images.google.com.ph/images?hl = tl&gbv =2&tbs=isch%3A1&sa=1&q= isometric+graph+paper&btnG = Maghanap&aq = f&oq =&start=0 . Feb 3, 2010 http://www.thg.ru/technews/images/smart_board-260407.jpg IMAGES http:// images.google.com.ph/images?hl = tl&source = hp&q = tools+in+teaching+mathematics&btnG = Maghanap+ng+Mga+larawan&gbv =2&aq= f&oq = Jan 3, 2010 http:// images.google.com.ph/images?gbv =2&hl= tl&q = tools+in+teaching+mathematics&sa = N&start =20&ndsp=20. Jan 5, 2010 http:// images.google.com.ph/images?gbv =2&hl= tl&q = tools+in+teaching+mathematics&sa = N&start =60&ndsp=20. Jan 10, 2010 http:// images.google.com.ph/images?gbv =2&hl= tl&q = tools+in+teaching+mathematics&sa = N&start =180&ndsp=20/ Jan. 11, 2010 http:// images.google.com.ph/images?gbv =2&hl= tl&q = tools+in+teaching+mathematics&sa = N&start =500&ndsp=20 . Jan 14, 2010 next back content

about the author Franz Arjon E. Flores is the eldest son of Mr. Aries V. Flores and Mrs. Joan E. Flores(+). He was Born on December 3, 1991 at Siniloan, Laguna. He had finished his basic education at Siniloan Elementary School in 2004. He Continue his secondary education at Siniloan National High School in 2008. He finished his tertiary level on 2012 at Laguna State Polythecnic University with the degrre of Bachelor of Secondary Education, major in Mathematics. next back content

about the author Jonel V. De Leon is the youngest son of Mr. Juanito S. De Leon and Mrs. Eliza V. De Leon. He was Born on 28th of October 1991 at Siniloan, Laguna. He had finished his elementary level at Halayhayin Elementary School in 2004. He Continue his secondary education at Siniloan National High School (STVSHS its present name) in 2008. He graduated at Laguna State Polythecnic University last 2012 with the degree of Bachelor of Secondary Education major in mathematics. next back content

Flores And De Leon

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