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1) The document discusses the teaching of mathematics across different grade levels, covering topics such as the nature of mathematics, scope at primary and secondary levels, strategies based on objectives like problem solving and concept attainment, theoretical basis for problem solving strategies, techniques for problem solving, and evaluating student performance.
2) Key strategies discussed include problem solving, concept attainment, and understanding goals through approaches like authority teaching, interaction, discovery and teacher-controlled presentations.
3) Evaluation of mathematics learning incorporates both testing procedures like individual/group tests and non-testing procedures such as interviews, questionnaires and anecdotal records.
Recreational Mathematics
Recreational mathematics includes puzzles, games, and problems that do not require advanced mathematical knowledge. It encompasses logic puzzles, mathematical games that can be analyzed with tools like combinatorial game theory, and mathematical puzzles that must be solved using specific rules but do not involve direct competition. Some common topics in recreational mathematics are tangrams, palindromic numbers, Rubik's cubes, and magic squares.
Islamic Mathematics
Islamic mathematics is a term used in the history of mathematics that refers to the mathematics developed in the Islamic world between 622 and 1600
Mathematical Process Standards
The Process Standards from the National Council of Teachers of Mathematics, expanded with examples, analogies and frameworks.
The teaching of mathematics
1. The document discusses various strategies for teaching mathematics, including focusing on knowledge and skill goals, understanding goals, and problem-solving goals.
2. Key strategies discussed are the problem-solving strategy, concept attainment strategy, and concept formation strategy.
3. The problem-solving strategy involves steps like restating the problem, identifying key information, estimating, and checking solutions. The concept attainment strategy helps students identify essential attributes of concepts. The concept formation strategy helps students make connections between elements of a concept.
Mcq's on Mathematics Education
1. The document contains 52 multiple choice questions related to mathematics education.
2. The questions cover topics like teaching methods, curriculum, assessment, educational theorists like Bloom and Bruner, and concepts in mathematics.
3. Answer keys are provided for each question to test understanding of core concepts and best practices in mathematics pedagogy.
TEACHING OF MATHEMATICS
Mathematics is defined as a science of patterns and relationships that reveals hidden patterns in the world and relies on logic and creativity. The two main goals of teaching mathematics are developing critical thinking and problem solving skills. Learning mathematics is most effective when done through active learning and problem solving. The document then outlines the key stage standards and grade level standards for mathematics in terms of the concepts, skills, and applications students are expected to understand at each level.
The Teaching Profession
The Nature of Teaching
Teaching is a process that facilitates learning.
Teaching is the specialized application of knowledge, skills and attributes designed to provide unique service to meet the educational needs of the individual and the society.
Teaching emphasizes the development of values and guides students in their social relationships.
What is a Profession?
A profession is an occupation that involves specialised training and formal qualification before one is allowed to practice or work.
Society and community place a great deal of trust in the professions.
A formal qualification (university or college diploma, degree) gained over time.
Specialized Knowledge (e.g. teaching secondary Mathematics)
License or permission to practice
Exhibits high agreed standards of behavior and practice
Someone with high personal standards and values
.............................................
Recommended
2b
1) The document discusses the teaching of mathematics across different grade levels, covering topics such as the nature of mathematics, scope at primary and secondary levels, strategies based on objectives like problem solving and concept attainment, theoretical basis for problem solving strategies, techniques for problem solving, and evaluating student performance.
2) Key strategies discussed include problem solving, concept attainment, and understanding goals through approaches like authority teaching, interaction, discovery and teacher-controlled presentations.
3) Evaluation of mathematics learning incorporates both testing procedures like individual/group tests and non-testing procedures such as interviews, questionnaires and anecdotal records.
Recreational Mathematics
Recreational mathematics includes puzzles, games, and problems that do not require advanced mathematical knowledge. It encompasses logic puzzles, mathematical games that can be analyzed with tools like combinatorial game theory, and mathematical puzzles that must be solved using specific rules but do not involve direct competition. Some common topics in recreational mathematics are tangrams, palindromic numbers, Rubik's cubes, and magic squares.
Islamic Mathematics
Islamic mathematics is a term used in the history of mathematics that refers to the mathematics developed in the Islamic world between 622 and 1600
Mathematical Process Standards
The Process Standards from the National Council of Teachers of Mathematics, expanded with examples, analogies and frameworks.
The teaching of mathematics
1. The document discusses various strategies for teaching mathematics, including focusing on knowledge and skill goals, understanding goals, and problem-solving goals.
2. Key strategies discussed are the problem-solving strategy, concept attainment strategy, and concept formation strategy.
3. The problem-solving strategy involves steps like restating the problem, identifying key information, estimating, and checking solutions. The concept attainment strategy helps students identify essential attributes of concepts. The concept formation strategy helps students make connections between elements of a concept.
Mcq's on Mathematics Education
1. The document contains 52 multiple choice questions related to mathematics education.
2. The questions cover topics like teaching methods, curriculum, assessment, educational theorists like Bloom and Bruner, and concepts in mathematics.
3. Answer keys are provided for each question to test understanding of core concepts and best practices in mathematics pedagogy.
TEACHING OF MATHEMATICS
Mathematics is defined as a science of patterns and relationships that reveals hidden patterns in the world and relies on logic and creativity. The two main goals of teaching mathematics are developing critical thinking and problem solving skills. Learning mathematics is most effective when done through active learning and problem solving. The document then outlines the key stage standards and grade level standards for mathematics in terms of the concepts, skills, and applications students are expected to understand at each level.
The Teaching Profession
The Nature of Teaching
Teaching is a process that facilitates learning.
Teaching is the specialized application of knowledge, skills and attributes designed to provide unique service to meet the educational needs of the individual and the society.
Teaching emphasizes the development of values and guides students in their social relationships.
What is a Profession?
A profession is an occupation that involves specialised training and formal qualification before one is allowed to practice or work.
Society and community place a great deal of trust in the professions.
A formal qualification (university or college diploma, degree) gained over time.
Specialized Knowledge (e.g. teaching secondary Mathematics)
License or permission to practice
Exhibits high agreed standards of behavior and practice
Someone with high personal standards and values
.............................................
Principles and Strategies of Teaching
This document outlines five principles for effective teaching: 1) clear communication with students, 2) a stimulating learning environment, 3) in-depth knowledge of the subject area, 4) effective assessment and feedback, and 5) evaluation and improvement of teaching practices. For each principle, it lists best practices for teachers to follow. It also discusses goals for student learning in mathematics, including knowledge and skills, understanding, and problem solving. Finally, it outlines strategies for teaching understanding and problem solving skills in mathematics.
Foundations of curriculum
The document discusses different types of curriculum:
1. Recommended curriculum refers to curricula proposed by scholars and organizations.
2. Written curriculum includes documents and syllabi created by curriculum experts and teachers for implementation.
3. Taught curriculum is what is actually taught in classrooms which may differ from the written curriculum. Additional factors like available resources are considered.
Rme realistic-mathematics-education-literature-review
This document provides an overview of Realistic Mathematics Education (RME), including its key characteristics and principles for designing lessons based on this approach. RME stresses starting with real-world contexts that are meaningful to students, and having students explore problems and develop mathematical concepts through guided reinvention that incorporates both horizontal and vertical mathematization. Lessons based on RME should include contextual problems for student exploration, opportunities for students to develop and use their own models and strategies, and an interactive teaching process that weaves together different mathematical strands.
What is Realistic Mathematics Education? National Mathematics Conference Swak...
Realistic Mathematics Education (RME) is an approach to teaching mathematics that is based on several key principles:
1) Starting with real-life contexts that are meaningful to students and using them as a source for mathematical concepts.
2) Encouraging the use of various problem-solving strategies over rote memorization of procedures.
3) Promoting the use of models and diagrams to help students understand mathematical relationships.
4) Emphasizing interaction between teacher and students, and among students, rather than just instruction from teacher to students.
5) Utilizing structured teaching materials to help build students' conceptual understanding of quantities.
INDICES & LOGARITHMS
This document provides information on indices, logarithms, and their applications. It defines indices and logarithms, outlines their basic properties and laws, and provides examples of using logarithms to perform calculations like multiplication, division, evaluating powers and roots. Logarithm tables are introduced as a tool to lookup logarithms and anti-logarithms before calculators. Worked examples demonstrate how to use logarithm tables to solve problems and determine unknown values.
Discrete Mathematics Course Outline
The goal of this course is to introduce students to ideas and techniques from discrete mathematics that are widely used in computer science. Ultimately, students are expected to understand and use (abstract) discrete structures that are the backbones of computer science. In particular, this class is meant to introduce logic, proofs, sets, functions, relations, counting, graphs and trees and with an emphasis on applications in computer science.
Real numbers - Euclid’s Division Algorithm for class 10th/grade X maths 2014
Real numbers - Euclid’s Division Algorithm for class 10th/grade X maths 2014.
Lets tute is an online learning centre. We provide quality education for all learners and 24/7 academic guidance through E-tutoring.
Our Mission-
Our aspiration is to be a renowned unpaid school on Web-World
Presentation binomial theorem
The document discusses the binomial theorem, which provides a method for expanding binomial expressions to a power. Specifically:
1) The binomial theorem states that the terms in the expansion of (a + b)n follow a predictable pattern, with the power of the first term being n and decreasing by 1 for each subsequent term as the power of the second term increases from 0 to n.
2) Pascal's triangle can be used to find the coefficients of the terms in the expansion. For example, the coefficients of the terms in (a + b)3 are the numbers in the 4th row of Pascal's triangle.
3) A general formula is provided to calculate the coefficient of any term in the
Characteristics of educational research
Educational research refers to systematic investigation of significant problems in teaching and learning inside and outside of schools. It aims to develop a science of behavior in educational situations to provide knowledge to help educators achieve their goals through effective methods. Some key characteristics of educational research include that it develops general principles and theories, takes an interdisciplinary approach, employs deductive reasoning, and strives to improve education. However, educational research is not as exact as research in physical sciences due to studying dynamic human behavior and social phenomena that are difficult to measure.
Curriculum
The document defines curriculum based on its Latin origins and various interpretations from scholars. It provides definitions from sources such as dictionaries of education from 1985, Tanner from 1980, and scholars such as Schubert, Pratt, Goodlad and Su, and Cronbleth. Curriculum is defined as the planned interaction between students and instructional content/resources/processes to achieve objectives. It also outlines five elements of curriculum: content, skills, instruction, assessment, and organization. Key features of Pakistan's 2006 National Curriculum are described such as being standards-driven and focusing on life skills and analytical thinking.
Foundation of education
Here, just a little explanation of the Foundation of Education, I made this for a presentation of MA class.
Hope that can be useful for all learners.
All the best.
Thanks
Systematic approach in Teaching( report in edtech1)
The document discusses the systematic approach to teaching, which views the entire educational program as an interconnected system. It involves defining student-centered objectives, selecting appropriate teaching methods and learning experiences, and refining the process based on evaluations to achieve the objectives. The key aspects are: (1) focusing on students and their needs, (2) planning instruction using objectives and aligned methods/materials, and (3) evaluating and refining to improve outcomes. All elements are interrelated - if one fails, learning is affected. The goal is to harmoniously integrate all parts into an effective whole.
Objectives of Mathematics Curriculum & Curricular choices at different stages...
Objectives of Mathematics Curriculum & Curricular choices at different stages...Dr.Jaganmohana Rao Gurugubelli
As per the Individual Differences and levels of understanding, curriculum should be changed from one stage to another stage.MATHEMATICS
The document discusses the origins and nature of mathematics. It defines mathematics as the science of quantity, measurement and special relations. The history of mathematics is described as investigating the origin of discoveries and methods from the past. Key contributions include the Chinese place value system and early Greek concepts of number and magnitude. The nature of mathematics is explained as a science of discovery, intellectual puzzle, tool, intuitive art with its own language/symbols, abstract concepts, and basis in logic and drawing conclusions. Needs, significance, and values of teaching mathematics are provided along with areas of study and contributions of great mathematicians like Euclid, Pythagoras, Aryabhatta, and Ramanujan. Notable mathematics-related days are
Mathematics Laboratory Approach
The document describes the math laboratory approach, which is a form of inductive and guided discovery learning by doing. It leads students to discover mathematical facts through hands-on activities. The procedure involves providing materials, clear instructions, experiments, and conclusions. An example demonstrates using this approach to show the relationship between a cylinder and cone by filling different shapes with rice. Advantages are that it presents math as practical, knowledge is more meaningful, it builds confidence, and students enjoy remaining active in the laboratory using different equipment.
Curriculum evaluation models
Curriculum evaluation: The assessment of the merit and worth of any program curriculum.
Curriculum evaluation is an attempt to toss light on two questions: Do planned programs, courses, activities, and learning opportunities as developed and organized actually produce desired results/learning outcomes? How can the curriculum offerings best be improved?
Curriculum Evaluation Models: How can the merits and worth of such aspects of curriculum is determined? Evaluation specialists have proposed an array of models, an examination of which can provide useful background for the process curriculum evaluation.
Fundamental principle of curriculum development and instruction
This document discusses the principles of curriculum development and instruction. It begins by stating the specific objectives of appreciating the importance of principles in curriculum development, learning curriculum development through examples, and knowing principles that can improve school systems. It introduces principles as the base of school programs and discusses how integrating principles can improve education. It then lists and describes 7 fundamental principles of curriculum development and instruction, including making teaching the purpose of curriculum, reflecting human aspirations, perpetuating universal education, using truthful concepts, embedding values, recognizing teaching/learning as limitless, and relating principles to the school environment. It analyzes how the external environment of trends and the internal environment of a school system's values and culture influence curriculum and instruction.
CTET Mathematics Pedagogy part -1
The document discusses key topics in mathematics pedagogy for CTET exams, including:
- Defining pedagogy and mathematics.
- The nature of mathematics as both a science of discovery and logical processes.
- Guiding principles and vision for mathematics in the NCF-2005 curriculum.
- Strategies for teaching mathematics like written work, oral work, group work and homework.
- Reasons for keeping mathematics in school curriculums like its basis in other sciences and role in developing logical thinking.
- The language of mathematics including concepts, terminology, symbols and algorithms.
- Approaches like community mathematics and mathematical communication to engage students.
Curriculum development in the philippines
The document discusses the development of curriculum in the Philippines under different periods of history. During colonial rule, the curriculum served colonial goals and objectives. After independence, reforms were made including introducing the vernacular as the medium of instruction in primary schools and emphasizing a community school concept. Curriculum continued to be revised to meet the needs of the times and include more Philippine-oriented materials, vocational education, and use of new instructional technologies.
Problem Solving in Mathematics Education
A major focus on current mathematics education is "problem solving." But "problem solving" means something very different from "Doing the exercises at the end of the chapter." An explanation of what problem solving is, and how it can be implemented.
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Principles and Strategies of Teaching
This document outlines five principles for effective teaching: 1) clear communication with students, 2) a stimulating learning environment, 3) in-depth knowledge of the subject area, 4) effective assessment and feedback, and 5) evaluation and improvement of teaching practices. For each principle, it lists best practices for teachers to follow. It also discusses goals for student learning in mathematics, including knowledge and skills, understanding, and problem solving. Finally, it outlines strategies for teaching understanding and problem solving skills in mathematics.
Foundations of curriculum
The document discusses different types of curriculum:
1. Recommended curriculum refers to curricula proposed by scholars and organizations.
2. Written curriculum includes documents and syllabi created by curriculum experts and teachers for implementation.
3. Taught curriculum is what is actually taught in classrooms which may differ from the written curriculum. Additional factors like available resources are considered.
Rme realistic-mathematics-education-literature-review
This document provides an overview of Realistic Mathematics Education (RME), including its key characteristics and principles for designing lessons based on this approach. RME stresses starting with real-world contexts that are meaningful to students, and having students explore problems and develop mathematical concepts through guided reinvention that incorporates both horizontal and vertical mathematization. Lessons based on RME should include contextual problems for student exploration, opportunities for students to develop and use their own models and strategies, and an interactive teaching process that weaves together different mathematical strands.
What is Realistic Mathematics Education? National Mathematics Conference Swak...
Realistic Mathematics Education (RME) is an approach to teaching mathematics that is based on several key principles:
1) Starting with real-life contexts that are meaningful to students and using them as a source for mathematical concepts.
2) Encouraging the use of various problem-solving strategies over rote memorization of procedures.
3) Promoting the use of models and diagrams to help students understand mathematical relationships.
4) Emphasizing interaction between teacher and students, and among students, rather than just instruction from teacher to students.
5) Utilizing structured teaching materials to help build students' conceptual understanding of quantities.
INDICES & LOGARITHMS
This document provides information on indices, logarithms, and their applications. It defines indices and logarithms, outlines their basic properties and laws, and provides examples of using logarithms to perform calculations like multiplication, division, evaluating powers and roots. Logarithm tables are introduced as a tool to lookup logarithms and anti-logarithms before calculators. Worked examples demonstrate how to use logarithm tables to solve problems and determine unknown values.
Discrete Mathematics Course Outline
The goal of this course is to introduce students to ideas and techniques from discrete mathematics that are widely used in computer science. Ultimately, students are expected to understand and use (abstract) discrete structures that are the backbones of computer science. In particular, this class is meant to introduce logic, proofs, sets, functions, relations, counting, graphs and trees and with an emphasis on applications in computer science.
Real numbers - Euclid’s Division Algorithm for class 10th/grade X maths 2014
Real numbers - Euclid’s Division Algorithm for class 10th/grade X maths 2014.
Lets tute is an online learning centre. We provide quality education for all learners and 24/7 academic guidance through E-tutoring.
Our Mission-
Our aspiration is to be a renowned unpaid school on Web-World
Presentation binomial theorem
The document discusses the binomial theorem, which provides a method for expanding binomial expressions to a power. Specifically:
1) The binomial theorem states that the terms in the expansion of (a + b)n follow a predictable pattern, with the power of the first term being n and decreasing by 1 for each subsequent term as the power of the second term increases from 0 to n.
2) Pascal's triangle can be used to find the coefficients of the terms in the expansion. For example, the coefficients of the terms in (a + b)3 are the numbers in the 4th row of Pascal's triangle.
3) A general formula is provided to calculate the coefficient of any term in the
Characteristics of educational research
Educational research refers to systematic investigation of significant problems in teaching and learning inside and outside of schools. It aims to develop a science of behavior in educational situations to provide knowledge to help educators achieve their goals through effective methods. Some key characteristics of educational research include that it develops general principles and theories, takes an interdisciplinary approach, employs deductive reasoning, and strives to improve education. However, educational research is not as exact as research in physical sciences due to studying dynamic human behavior and social phenomena that are difficult to measure.
Curriculum
The document defines curriculum based on its Latin origins and various interpretations from scholars. It provides definitions from sources such as dictionaries of education from 1985, Tanner from 1980, and scholars such as Schubert, Pratt, Goodlad and Su, and Cronbleth. Curriculum is defined as the planned interaction between students and instructional content/resources/processes to achieve objectives. It also outlines five elements of curriculum: content, skills, instruction, assessment, and organization. Key features of Pakistan's 2006 National Curriculum are described such as being standards-driven and focusing on life skills and analytical thinking.
Foundation of education
Here, just a little explanation of the Foundation of Education, I made this for a presentation of MA class.
Hope that can be useful for all learners.
All the best.
Thanks
Systematic approach in Teaching( report in edtech1)
The document discusses the systematic approach to teaching, which views the entire educational program as an interconnected system. It involves defining student-centered objectives, selecting appropriate teaching methods and learning experiences, and refining the process based on evaluations to achieve the objectives. The key aspects are: (1) focusing on students and their needs, (2) planning instruction using objectives and aligned methods/materials, and (3) evaluating and refining to improve outcomes. All elements are interrelated - if one fails, learning is affected. The goal is to harmoniously integrate all parts into an effective whole.
Objectives of Mathematics Curriculum & Curricular choices at different stages...
Objectives of Mathematics Curriculum & Curricular choices at different stages...Dr.Jaganmohana Rao Gurugubelli
As per the Individual Differences and levels of understanding, curriculum should be changed from one stage to another stage.MATHEMATICS
The document discusses the origins and nature of mathematics. It defines mathematics as the science of quantity, measurement and special relations. The history of mathematics is described as investigating the origin of discoveries and methods from the past. Key contributions include the Chinese place value system and early Greek concepts of number and magnitude. The nature of mathematics is explained as a science of discovery, intellectual puzzle, tool, intuitive art with its own language/symbols, abstract concepts, and basis in logic and drawing conclusions. Needs, significance, and values of teaching mathematics are provided along with areas of study and contributions of great mathematicians like Euclid, Pythagoras, Aryabhatta, and Ramanujan. Notable mathematics-related days are
Mathematics Laboratory Approach
The document describes the math laboratory approach, which is a form of inductive and guided discovery learning by doing. It leads students to discover mathematical facts through hands-on activities. The procedure involves providing materials, clear instructions, experiments, and conclusions. An example demonstrates using this approach to show the relationship between a cylinder and cone by filling different shapes with rice. Advantages are that it presents math as practical, knowledge is more meaningful, it builds confidence, and students enjoy remaining active in the laboratory using different equipment.
Curriculum evaluation models
Curriculum evaluation: The assessment of the merit and worth of any program curriculum.
Curriculum evaluation is an attempt to toss light on two questions: Do planned programs, courses, activities, and learning opportunities as developed and organized actually produce desired results/learning outcomes? How can the curriculum offerings best be improved?
Curriculum Evaluation Models: How can the merits and worth of such aspects of curriculum is determined? Evaluation specialists have proposed an array of models, an examination of which can provide useful background for the process curriculum evaluation.
Fundamental principle of curriculum development and instruction
This document discusses the principles of curriculum development and instruction. It begins by stating the specific objectives of appreciating the importance of principles in curriculum development, learning curriculum development through examples, and knowing principles that can improve school systems. It introduces principles as the base of school programs and discusses how integrating principles can improve education. It then lists and describes 7 fundamental principles of curriculum development and instruction, including making teaching the purpose of curriculum, reflecting human aspirations, perpetuating universal education, using truthful concepts, embedding values, recognizing teaching/learning as limitless, and relating principles to the school environment. It analyzes how the external environment of trends and the internal environment of a school system's values and culture influence curriculum and instruction.
CTET Mathematics Pedagogy part -1
The document discusses key topics in mathematics pedagogy for CTET exams, including:
- Defining pedagogy and mathematics.
- The nature of mathematics as both a science of discovery and logical processes.
- Guiding principles and vision for mathematics in the NCF-2005 curriculum.
- Strategies for teaching mathematics like written work, oral work, group work and homework.
- Reasons for keeping mathematics in school curriculums like its basis in other sciences and role in developing logical thinking.
- The language of mathematics including concepts, terminology, symbols and algorithms.
- Approaches like community mathematics and mathematical communication to engage students.
Curriculum development in the philippines
The document discusses the development of curriculum in the Philippines under different periods of history. During colonial rule, the curriculum served colonial goals and objectives. After independence, reforms were made including introducing the vernacular as the medium of instruction in primary schools and emphasizing a community school concept. Curriculum continued to be revised to meet the needs of the times and include more Philippine-oriented materials, vocational education, and use of new instructional technologies.
Problem Solving in Mathematics Education
A major focus on current mathematics education is "problem solving." But "problem solving" means something very different from "Doing the exercises at the end of the chapter." An explanation of what problem solving is, and how it can be implemented.
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Rme realistic-mathematics-education-literature-review
Rme realistic-mathematics-education-literature-review
What is Realistic Mathematics Education? National Mathematics Conference Swak...
What is Realistic Mathematics Education? National Mathematics Conference Swak...
Real numbers - Euclid’s Division Algorithm for class 10th/grade X maths 2014
Real numbers - Euclid’s Division Algorithm for class 10th/grade X maths 2014
Systematic approach in Teaching( report in edtech1)
Systematic approach in Teaching( report in edtech1)
Objectives of Mathematics Curriculum & Curricular choices at different stages...
Objectives of Mathematics Curriculum & Curricular choices at different stages...
Fundamental principle of curriculum development and instruction
Fundamental principle of curriculum development and instruction
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معادلة باكمال مربع جديد
- 1. حل معادلات الدرجة الثانية في متغير واحد بطريقة إكمال المربع خطوات الحل أولاً : نجعل الحد الثابت ( المطلق ) في طرف والمتغيرات في الطرف الأخر ثانياً : نجعل معامل س 2 = 1 وذلك بالقسمة عليه . ثالثاً : نضيف مربع نضيف معامل س للطرفين رابعاً : نحلل الطرف الأيمن كمقدار ثلاثي مربع كامل على صورة ( س + ثابت ) 2 خامساً : نأخذ الجدر التربيعي للطرفين فينتج لنا معادلتان . سادساً : نكمل حل المعادلتين كلاً على حده فنحصل على حلين الصورة العامة لها هي : أ س 2 + ب س + ج = صفر
- 2. مثال (1) جد حل المعادلة التالية بطريقة إكمال المربع 2 س 2 + 4 س – 16 = صفر بإضافة + 16 للطرفين 2 س 2 + 4 س = 16 بالقسمة على معامل س 2 وهو 2 س 2 + 2 س = 8 معامل س = 2 نصفه =1 مربعه =1 بإضافة 1 للطرفين س 2 + 2 س + 1= 8 + 1 نكتب الطرف الأيمن على صورة ( س + ب ) 2 ( س + 1 ) 2 = 9 بأخذ الجذر التربيعي للطرفين ينتج لنا معادلتان هما
- 3. ( س + 1 ) 2 = 9 س + 1 = 3 بإضافة -1 للطرفين س = 2 أو س + 1 = -3 بإضافة -1 للطرفين س = -4 مجموعة الحل : { 2 ، -4}
- 4. مثال (2) جد حل المعادلة التالية بطريقة إكمال المربع س 2 - 8 س + 15 = صفر بإضافة -15 للطرفين س 2 - 8 س = -15 معامل س = -8 نصفه = -4 مربعه = 16 س 2 - 8 س + 16 = -15 + 16 نكتب الطرف الأيمن على صورة ( س + ب ) 2 ( س - 4 ) 2 = -15 + 16 ( س - 4 ) 2 = 1 بأخذ الجذر التربيعي للطرفين ينتج لنا معادلتان هما
- 5. س – 4 = 1 بإضافة +4 للطرفين س = 5 أو س – 4 = - 1 بإضافة +4 للطرفين س = 3 مجموعة الحل = { 5 ، 3 }