Equalities, Inequalities, Linear function, Radicals, Matrices

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2. 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, Asian countries NEXT PREVIOUS CONTENTS

3. 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 technology and other related fields. It shall also undertake research and extension services, and provide a progressive leadership in its areas of specialization. PREVIOUS NEXT CONTENTS

4. 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. PREVIOUS NEXT CONTENTS

5. Produce graduates who can demonstrate and practice the professional and ethical requirements for the Bachelor of Secondary Education such as: 1. To serve as positive and powerful role models in the pursuit of 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 types of learners. 4. Use varied learning approaches and activities, instructional materials and learning resources. 5. Use assessment data, plan and revise teaching-learning plans. 6. Direct and strengthen the links between school and community activities. 7. Conduct research and development in Teacher Education and other related activities. PREVIOUS NEXT CONTENTS

6. This Teacher’s Visual Presentation Hand-out entitled “ Developing Skills in Algebra for First Year High School Students” is part of the requirements in Educational Technology 2 under the revised Education curriculum for 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 through 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 of instructional materials. The output of the group’s effort may serve as an educational research of the institution in providing 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 FLORANTE R. DE CASTRO Module Consultant LYDIA R. CHAVEZ Dean, College of Education PREVIOUS NEXT CONTENTS

7. This module that contributes to knowledge in algebra would not be possible without friends, families, teachers and the persons who encourage us to finish this module. To Prof. Lydia R. Chavez, Dean of College of Education, for her support and motivation that lifts the spirit of the authors, To Mr. Florante R. De Castro, our module consultant, for lending the authors his time and intelligence that helped a lot in finishing the module, To Mr. For-Ian V. Sandoval, our adviser, for his guidance and help during the days that the authors find difficulties in completing the module, To Mrs. Evangeline Cruz, the university librarian, in allowing us to borrow our reference books in the university library, To Dr. Corazon San Agustin for her support and motivation that helped a lot in finishing this module, To each member of our families who loves unconditionally and supports us financially, And finally, all praises and glory be unto God whom we can’t thank enough for realizing the vision of our work. THE AUTHORS PREVIOUS NEXT CONTENTS

8. In pursuit of quality learning of high school student, we designed a module that will help the students develop their skills in Mathematics. This will also extend their learning and attain more knowledge about the topics. We have tried to bring out the basic ideas and techniques as simply and clearly as possible. Most of the topics are introduced in every chapter. The authors believe that it will help the students to encourage themselves in studying the lessons. Numerous ILLUSTRATIVE EXAMPLES and ACTIVITIES are given in every topic. The authors believe that it will give the students opportunity to practice their mathematical abilities. A CHAPTER TEST and NOTES TO REMEMBER are included at the end of every chapter. The authors’ aim is to develop the skills of the first year high school students in Algebra . PREVI O US NEXT CONTENTS

9. After reading, understanding and answering all the lessons and activities in this module, the students are expected to: 1. Understand what equality is, 2. Apply the properties of equality in solving, 3. Understand what inequality is, 4. Apply the property of inequality in solving inequalities, 5. Define what a linear function is, 6. Get the x and y intercepts of the line, 7. Identify what are the systems of linear equations, 8. Learn ways of solving radical equations, 9. Solve equations with two radical terms, 10. Understand what matrices are, 11. Identify the properties of matrices, and 12. Learn how to add and multiply matrices. PREVIOUS NEXT CONTENTS

10. PREVIOUS TABLE OF CONTENTS VMGO TITLE ACKNOWLEDGMENT INTRODUCTION GENERAL OBJECTIVES TABLE OF CONTENTS FOREWORD NEXT CONTENTS LESSON 1: EQUATIONS CHAPTER 1 - UNDERSTANDING EQUALITIES LESSON 2: PROPERTIES OF EQUALITY LESSON 3: SOLVING EQUALITIES IN ONE VARIABLE CHAPTER 2 : UNDERSTANDING INEQUALITIES LESSON 4: SOLUTION SET OF INEQUALITIES IN ONE VARIABLE LESSON 5: PROPERTIES OF INEQUALITIES

11. NEXT PREVIOUS CHAPTER 4 : UNDERSTANDING RADICAL EQUATION LESSON 11: PERFECT SQUARES AND PERFECT CUBES LESSON 12: EVALUATING EQUATIONS USING RADICALS LESSON 13: SOLVING RADICAL EQUATION LESSON 14: SOLVING RADICAL EQUATION WITH TWO RADICAL EQ... CONTENTS LESSON 9 :GRAPHING LINEAR FUNCTION CHAPTER 3 : UNDERSTANDING LINEAR FUNCTION LESSON 8: GETTING THE X AND Y INTERCEPT OF THE LINE LESSON 6: APPLYING THEPROPERTIES OF INEQUALITY LESSON 7: DEFINING LINEAR FUNCTION LESSON 10: SYSTEM OF LINEAR EQUATION

12. NEXT PREVIOUS REFERENCES CONTENTS CHAPTER 5 : UNDERSTANDING MATRICES LESSON 16: ADDITION OF MATRICES LESSON 18: BASIC PROPERTIES OF MATRICES LESSON 17: MULTIPLICATION OF MATRICES LESSON 19: PRODUCTS OF MATRICES LESSON 15 : UNDERSTANDING MATRICES

13. CHAPTER I : UNDERSTANDING EQUALITIES PREVIOUS NEXT CONTENTS

42. Properties of Equality 1. Reflexive Property of Equality A number is always equal to itself. 2. Symmetric Property of Equali ty Interchanging the left member and right member of an equation does not change the sense of equality. 3. Transitive property of Equality If one number is equal to a second number and the second number is equal to the third number, then the 1 st and 3 rd number are also equal. 4. Addition Property of Equality Adding same number to both sides of an equation does not change the sense of equality. 5. Multiplication Property of Equality Multiplying both sides of an equation by the same number does not change the sense of equality. The properties of equality are very useful in transforming or rewriting an equation into an equivalent one. NEXT PREVIOUS CONTENTS

43. NEXT AL-KHOWARIZMI Arithmetic, in its purest form, deals all the different kinds of real numbers, their properties, and the skills needed for calculating, manipulating, and utilizing them in practical situations. Algebra extends the range and power of elementary arithmetic to include not just the constant quantities called variables. PREVIOUS CONTENTS

44. PREVIOUS NEXT CHAPTER II : UNDERSTANDING INEQUALITIES CONTENTS

59. PROPERTIES OF INEQUALITY 1. Addition/ Subtraction Property of Inequality If the same quantity is added or subtracted on both sides of an inequality, the resulting inequality is equivalent to the original inequality. 2. Multiplication Property of Inequality If the same positive is multiplied to both sides of an inequality, the resulting inequality is equivalent to the original inequality. If the same negative quantity is multiplied to both sides of an inequality, the direction of the inequality should be reversed. PREVIOUS NEXT CONTENTS

60. Leonhard Euler Equations and inequalities are basic importance in science, technology, business and commerce. They are mathematical sentences of physical laws, logical relationships, or any other connection between quantities and objects. PREVIOUS NEXT CONTENTS

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86. Rene Descartes Undoubtedly one of the most ingenious and useful inventions of mathematics is the xy-coordinate system, which is formally called the Cartesian Coordinate System, named after its inventor, Rene Descartes. A thorough study of the graphical representation of linear equations in two variables using the Cartesian coordinate system. PREVIOUS NEXT CONTENTS

87. PREVIOUS NEXT CHAPTER IV : UNDERSTANDING RADICAL EQUATIONS CONTENTS

101. b. 6 - 5 = ( 6 – 5 ) 2 = ( ) 2 1 = x or x = 1 c. ( ) 2 = ( 3 ) 2 = 9 x = 63 REMEMBER ! To solve radical equations, 1. Isolate the radical, they should be on the left side of the equation, 2. Apply the power rule, 3. Solve the resulting equation, and 4. Check it. PREVIOUS NEXT

117. NEXT PREVIOUS CHAPTER V : UNDERSTANDING MATRICES CONTENTS

135. PREVIOUS NEXT We use the terms scalar and scalar multiplication because, in abstract algebra, we often have the need to consider more general scalars than real numbers. However, in this book, we restrict our attention to scalars that are real numbers. Definition 3.1 If c is a real number and A is a matrix whose (i,j) th element is a ij , then the scalar product cA is the matrix whose (i,j) th element is ca ij .

148. A matrix is a rectangular array of numbers or elements of a ring. One of the principal uses of matrices is in representing systems of equations of the first degree in several unknowns. Each matrix row represents one equation, and the entries in a row are the coefficients of the variables in the equations, in some fixed order. Addition and multiplication of matrices can be defined so that certain sets of matrices form algebraic systems. Let the elements of the matrices considered be arbitrary real numbers, although the elements could have been chosen from other fields or rings. A zero matrix is one in which all the elements are zero; an identity matrix, I m of order m, is a square matrix of order m in which all the elements are zero except those on the main diagonal, which are 1. The order of an identity matrix may be omitted if implied by the text, and I m is then shortened to I. PREVIOUS NEXT CONTENTS

150. BOOKS Alferez, M. S. Quick Math Review. Gepress Printing. Benigno, Ph. D., G. D. Basic Mathematics for College Students (Revised Ed). Rex Bookstore. Bernabe, J. G. Elementary Algebra, Textbook for First Year. JTW Corporation. Dasco, N. T. Intermediate Algebra (Mathematics II). Academic Publication. Marquez, L. Mathematics beyond 2000. Vibal Publishing House. Orines, F. B. Elementary Algebra. Phoenix Publishing House. Padua, R. N. , Adanza, E. G.Contemporary College Algebra with Applications. Rex Bookstore. Vance-Addison, E. P.. Modern Algebra (3 rd Ed.) Addison-Wesley Publishing Company, Incorporated. NEXT PREVIOUS CONTENTS

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