Methodical manual on mathematics for primary school teachers

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Methodical manual on mathematics for primary school teachers
Methods of teaching mathematics in primary school
Lecture number 1
 Topic №: Teaching mathematics in primary school
methodology of subjects
Plan:
  1. Methodical system of teaching.
  2. The relationship of the methodology of teaching mathematics with other disciplines.
          Methodology of teaching mathematics or didactics Mathematics is a subject that organizes the teaching of mathematics, which is part of the system of pedagogical sciences. The word "Greek" means "path". Mathematical methodology is one of the main branches of pedagogy and didactics, and at the level of development of our society is an independent discipline that teaches the laws of teaching and learning mathematics in accordance with the goals of education. Mathematics is a basic subject taught in primary grades.
         Mathematics education starts at the preschool and ends at the university. The methodology of teaching mathematics develops on the basis of the thematic psychological constitution of teaching and general pedagogical theory, as well as the technology of using psychological and pedagogical theory in the teaching of primary mathematics. In addition, in the methodology of teaching mathematics, the teaching methods of mathematics are characterized.
         In order to open the subject of methods of teaching mathematics, it is necessary to define "the content of teaching mathematics, the main components of the process of teaching mathematics." Teaching elementary school, especially math, is a complex process that controls students' thinking skills using a variety of visual aids. Taking into account the knowledge of students' thinking skills, all this information is processed and transmitted to the student. The student receives information from the teacher, textbooks, other sources, and transmits the acquired knowledge to the teacher.
         Therefore, in the process of teaching, information is carried out in two directions, that is, this direction is transmitted from the teacher to the learner (direct connection) and from the learner to the teacher (feedback).
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Thus, the methodology of teaching mathematics is a branch of pedagogical science, which is part of the system of pedagogical sciences, which studies the laws of teaching mathematics at a certain stage of development of mathematics in accordance with the goals of teaching set by society.
In order to effectively teach mathematics to primary school students, the future teacher must master the subject of mathematics teaching methodology developed for primary school and its system.
The subject of the methodology of teaching elementary mathematics can be interpreted as follows:
  1. Justifying the goals set in the fall from math teaching is why the process is taught, taught;
  2. Scientific development of the content of the teaching process:
What to learn?
How can this knowledge, science, technology and culture meet the requirements of modern development when given to children?
         How to distribute the systematized knowledge in accordance with the age characteristics of students, to ensure consistency in the study of the basics of science, to eliminate the burden on students, to ensure that the content of education corresponds to the learning abilities of students?
  1. Scientific development of teaching methods:
How to teach?
In other words, what should be the methodology of educational work in order for students to acquire the knowledge, skills, and intellectual abilities that are needed today?
  1. Development of teaching aids - textbooks, didactic materials, manuals, technical aids. What to teach?
  2. Scientific development of the organization of education.
How to conduct lessons and extracurricular forms of education, how to organize educational work, how to organize educational work, how to solve educational problems more effectively, not only the process of acquiring knowledge of the educational process, but also the process of formation and development of students' personality.
Didactic, goals, content, methods, tools and forms of teaching are the main components of the methodological system. A. M. According to Pyshkalo, the methodological system is a complex system, which can be represented by a unique graph.
The concept of mathematics teaching methods appeared in 1703. L. with the methodology of mathematics. F. Magnitskiy, P. S. Gurev, A. V. Grubya, V. A. Evtushevskiy, V. A. Latishev, A. I. Goldenberg, S. I. Shokhor, Trotsky, and later M. I. Loro, A. S. pchelka, A. M. Pyshkalo, L. I. Skatkin, M. A. Bantova, A. A. Stolyar, V. A. Drozda, A. Sh. Lebenberg, I. U. Bikbaeva and several scientists, including staff from the Research Institute.
The subject of mathematics teaching methods is divided into three depending on its structural features:
  1. General mathematics of mathematics teaching in this section reveals the purpose, content, form, methods of mathematical science, the methodological system of its means on the basis of the laws of pedagogy, psychology and didactic principles.
  2. Special Mathematics of Mathematics Teaching This section shows the ways to apply the laws and rules of general methods of teaching mathematics to specific topic materials.
  3. Specific methods of teaching mathematics.
A) Special issues of general methodology.
B) Special issues of special methodology.
For example: Planning a math lesson in 1st grade is a special issue of general methodology. If in the 1st grade students are taught to introduce the concepts of "intersection", "0 + 3"…, this is a special problem of the special methodology.
         Methods of teaching mathematics in primary school Other disciplines, first of all, the subject of "mathematics" is inextricably linked with its basic subject. The level of development of mathematics has always influenced the choice of the content of the school mathematics course.
For example: XVIII In the nineteenth century, when a natural number was called in mathematics, a set of ones was understood, and in the teaching of elementary arithmetic, great emphasis was placed on exercises to construct each of the first decimal numbers from one.
         Modern mathematics is based on the theory of sets based on the concept of natural numbers. Establishing a mutually valued compatibility between the elements of finite sets allows the separation of classes of mutually equivalent sets. However, the common denominator that characterizes each of these classes is that they allow the separation of natural numbers.
         Such an understanding of the nature of natural numbers leads to the introduction into practice of exercises of a mutually valuable compatibility between the elements of a comparable set of objects.
Example: Assignments for students on page 1 of a modern math textbook for 5st grade. The picture shows how many fruits and vegetables there are, how many of them there are, how many chickens you can get in your ashes, how many chickens you have, how many cats you can get. Which circle is bigger? 16 red, 7 blue circles are baked on the board.
         Completion of such tasks encourages children to establish a mutually valuable correspondence between the elements of the set, which is important in the formation of the concept of natural numbers.
         The methodology of teaching mathematics depends on the methodology of general mathematics. The laws defined by the general mathematics methodology are used by the primary mathematics teaching methodology, taking into account the age characteristics of young swimmers.
         The methodology of teaching primary mathematics is inextricably linked with the science of pedagogy and is based on its laws. There is a two-way connection between the methodology of teaching mathematics and pedagogy.
         On the one hand, the methodology of mathematics is based on the general theory of pedagogy and is formed on this basis, which ensures the integrity of the methodological and theoretical convergence in solving problems of teaching mathematics.
         From the second tone, pedagogy relies on the information obtained by special methodologies in the formation of general laws, which ensures its vitality and accuracy.
         It is based on the thematic material of pedagogical methods, which is used in generalization, and in turn it serves as a guide in the development of methods. Mathematical methodology is related to pedagogical psychology and youth psychology. In solving many problems of upbringing and education it is necessary to use a lot of knowledge about pedagogical psychology and youth psychology.
         Youth psychology studies the laws of formation of a person's spiritual image under the influence of education, the psychological characteristics of children of different ages, as well as the psychological laws of children's knowledge and skills, the development of their independence and creativity, the laws of personal development.
         The methodology of primary mathematics is related to the methodology of other methods of teaching the mother tongue, natural sciences, drawing, cocktails and other sciences. It is important for the teacher to take this into account in order to make interdisciplinary connections.
         It is more difficult to make interdisciplinary connections in the upper grades, as each subject is taught by a specific teacher.
         Not so in elementary grades. All subjects are taught by one teacher, and therefore he has the opportunity to make interdisciplinary connections.
         In the lessons on various subjects of primary education, students get a concrete idea of ​​the surrounding events and phenomena, their properties. The distinguishing feature of mathematics is that mathematics is abstracted from the thematic content of the events and objects studied at the same time as the study of objective existence in relation to everything that does not belong to the most general aspects of the material world and its spatial form and relations. This is the great power of mathematics, that is, the abstractness and generality of concepts, and this is the possibility of establishing all-round connections and relations with other disciplines.
         In establishing such connections can be based on general facts, such as numbers, arithmetic operations, concepts and elements of geometric figures, quantities, shapes, various skills and abilities, types of activities, forms and methods of teaching.
         Mathematics uses students' knowledge of natural sciences, geography, history, painting, drawing, labor, physical education and other subjects.
Information on these disciplines can serve as material for arithmetic problems and examples. For example, to learn about historical events, the length of the borders of our country and other countries, the faces of the occupied territories, the length of rivers, the height of mountains, the length and depth of sea ash. It can serve as a basic material in arithmetic problems and examples in mathematics lessons, in comparing and analyzing numbers.
On the other hand, mathematical knowledge should be widely used in other subjects.
For example, in an ash cocktail class, swimmers cut flowers out of paper for math lessons and make didactic materials out of plasticine. They also draw and circle geometric shapes such as squares, triangles, right triangles, circles in a pencil, learn to distinguish them and name them.
         In math classes, swimmers are introduced to the following symbols of objects, long-short, wide-narrow, thick-thin, and so on. In the ash cocktail class, swimmers reinforce various items, such as toys.
         Like math lessons, ash cocktail lessons develop students' spatial awareness. Swimmers learn to point to the middle, top, bottom, left and right sides of the paper. The knowledge of students in mathematics and drawing can be widely used in the study of certain topics in geography, for example: the calculation of scales, the plan of the school plot, a simple plan of housing: the concept of scale is formed only on a solid basis of measurement skills. In physical education classes, swimmers consolidate their knowledge of quantity. These mics find their thematic office in running, swimming in this or that distance, jumping in height or length. The connection between the teaching of mathematics and the mother tongue is unique. In a math class, the teacher develops students ’mathematical speech. Thematic, fluent mathematical speech seems to have a positive effect on the mastery of mathematical concepts. A math teacher teaches students not only to solve problems and examples correctly, but also to write correctly and to form sentences correctly. Writing numbers and other mathematical terms and expressions is reinforced in mother tongue lessons. The knowledge gained in mathematics lessons is used in training workshops, school experimental fields, as well as in industrial and agricultural enterprises, where swimmers practice internships, and is consolidated in joint stock companies.
                                   
Lecture number 2
Topic: Primary mathematics course
Plan:
  1. Tasks of teaching mathematics in primary school.
  2. The structure and content of the elementary mathematics course.
 
Basic terms: Educational, pedagogical, applied arithmetic, algebra, geometry. 
 "On reforming the system of education and training to bring up a harmoniously developed generation" and "National Program of Personnel Training" identify issues of improving the quality of teaching mathematics, as well as the formation of thinking and personal qualities, mathematical literacy and creative abilities of students.
Therefore, an elementary mathematics course is a subject of study.
The purpose of the elementary mathematics course is to help students solve the tasks set for the school, such as "to provide students with a thorough knowledge of the basics of science, to form in them a high level of consciousness, to teach them to marry, to make conscious choices." Like any subject, the elementary course of mathematics must solve educational, pedagogical, practical tasks. One of the main tasks of teaching mathematics is to create in students a certain thematic system of calculation, measurement and graphic skills.
Swimmers should learn to open up laws and relationships as independently as possible, to do as much generalization as they can, and to draw oral and written conclusions.
The primary school mathematics program has the main task of integrating theoretical knowledge with practice, teaching students the mathematical knowledge and skills necessary for their future careers and daily lives, and shaping them to be able to apply this knowledge and skills throughout their lives. Let's give an example of raising the theoretical level in the teaching of mathematics.
For example, if you compare the process of adding 2 to 1 to make 1, and adding 3 to 2 to add 1 to 6, the children's attention is drawn to the fact that each successive number is formed by adding one to the previous number. Explain how to form the numbers 7, 8,….
         This example shows the importance of comparing, contrasting, establishing connections between them, and forming appropriate generalizations: in such an approximation, it is easier to assimilate the material.
         The theoretical level of studying the topic of numbering the first decimal number increases, because along with the study of numbers, they learn the principle of formation of each consecutive number in a natural series.
         The number obtained in this way will help you to study the coefficient within 20 as well as the numbering within 100 and so on.
         Example 2 According to the previous program, in 20 and 100, addition and subtraction skills were taught based on the properties of the actions.
         As a result, it would be necessary for children to master more than 100 calculation methods to perform addition and subtraction within 20. Now, in the knowledge of adding and subtracting the sum of the four basic properties of a number and subtracting a number from the sum and the sum of numbers, different methods of solving any example of addition and subtraction of multi-digit numbers within 1000 are taught. Teaching mathematics not only considers it a task for children to acquire certain knowledge and skills, but also implies the general development of cognitive abilities in them, such as cognition, memory, thinking, imagination. Work in this direction allows them to teach methods of mental activity (analysis, synthesis, comparison, generalization, abstraction, concretization).
         In continuous connection with the problem of developing logical thinking in children, it involves the development of oral and written mathematical speech - all the qualities of speech, such as conciseness, simplicity, comprehensibility, integrity. Teaching in primary school should be carried out in close connection with upbringing. This important task of teaching is to create the most favorable conditions in the learning process for students to form a worldview, the basis of everyday behavior, the formation of valuable personality traits and qualities.
         Primary education is at the same time developmental. Educational education ensures the development of observation, thinking, speech, memory, imagination, and thus prepares a person for work. The solution of educational tasks in the teaching of elementary mathematics depends on the level of readiness of students to study this course, the level of solution of developmental and teaching problems provided for in the school curriculum.
         It is necessary to cultivate in children an interest in mathematical knowledge, the ability to use them and the ability to acquire them independently. When preparing children, it is necessary to pay attention to the formation of practical skills and abilities (drawing pictures of simple figures, forming them by folding a sheet of paper, drawing cross-section and other figures, etc.). During this period, children must learn to listen to and perform tasks that are important and necessary for the work of the teacher, adults, to follow the instructions of the teacher, to perform the task in order, to bring the results to the problem, to control their work… other skills.
         An elementary math course is an integral part of a school math course. The core of the mathematics program consists of arithmetic of natural numbers and basic quantities, around which elements of algebra and geometry are combined, and these elements are integrated into the system of arithmetic knowledge, allowing a high degree of mastery of concepts in numbers, arithmetic operations and mathematical relations.
         An elementary math course is a whole course that includes three disciplines on the structure of Google. Since the elements of arithmetic in the program of elementary classes include acquaintance with natural numbers, some important properties of the four arithmetic operations of zero numbers and the results arising from them, it is possible to consciously master the methods of calculation. This is the substitution property of addition and multiplication, the distribution law of multiplication and division is the result of the basic properties: addition to the sum, subtraction from the sum, addition to the sum, subtraction of the sum, multiplication of the sum by the sum, . Each of the basic properties is revealed on the basis of performing practical operations on sets or numbers, as a result of which swimmers must come to generalizations.
         Simultaneously with the study of the properties of arithmetic operations and appropriate calculation methods, the connections between the results of arithmetic operations and their components are revealed. The program pays great attention to the oral and written methods of characterization.
Work on written calculation methods will begin in 2nd grade. Continues in 3rd and 4th grade. In order to prepare for the study of a systematic course of mathematics, ideas about fractions are given. The concept of fraction is introduced as one of the equal parts of the whole, and is given as the formation, writing, reading of fractions, finding the fraction of a number, finding the number itself by fraction, comparing fractions.
Fractions are included as a set of fractions, fractions are replaced, comparisons are given on an instructional basis. The arithmetic material of the program includes introducing swimmers to the basic quantities of length, mass, weight, time, surface, estimate, speed, units of measurement of these quantities, methods of measurement using various measuring instruments.
When teaching the numbering of the first numbers of a natural string, cm is entered. The two decimals and the numbers within 100 are entered in cm, then d. This allows, firstly, to form in children the concept of number not only as a result of counting, but also as a result of measurement, and secondly, to acquaint children with the numbers expressed in length measurements.
Operations on named numbers are performed at the same time as operations on unnamed numbers, because the basis of both cases lies in the decimal number system itself.
The elements of algebra are taught from the 1st grade and the meaning of the concepts of variables is explained. Studying them is connected with studying arithmetic material. Simple equations are considered first, then complex equations. The equations are taught first by the selection method and then by the connections between the components and the results of the operation. In addition to solving equations, students are taught to solve problems by constructing equations.
Variable inequalities are introduced as the character that defines the letter variable. In this case, the inequalities are solved by choice.
Geometric material serves the purpose of introducing children to the simplest geometric figures, developing their spatial imagination, showing thematic connections of arithmetic laws, thematic illustrations. Geometric material introduces children to the simplest geometric figures, curves and curved sections, polygons and curved sections, polygons and their elements, angles, right angles, cross-section, polygon perimeter.
Tugri teaches them to be able to find the face of a rectangle, a square, and any figure in general. Problems are exercises that are used to solve many problems in an elementary mathematics course. Problem-solving reveals the properties of arithmetic operations, the relationship between the results of operations and their components, and the thematic content of …s.
In the process of problem solving, swimmers acquire the skills and competencies they need in life. Therefore, the content of the mathematics course is very large. It is necessary to burn such a solid foundation of mathematical knowledge in the primary grades that it is possible to confidently build further mathematical education on this foundation.
Control questions:
  1. What are the main tasks of teaching mathematics in primary school?
  2. What are the main tasks of preparing for an elementary mathematics course?
  3. List the features of an elementary math course?
  4. What is the content of the arithmetic, algebra, geometry part of the elementary school curriculum?
Lecture number 3
Topic: Teaching mathematics in primary school
methods of organization.
Plan:
  1. The concept of style (method) types it.
  2. Method of organizing educational activities.
  3. Independent work of swimmers - as teaching methods.
  4. Didactic game method in the organization of teaching.
  5. Methods used depending on the level of activity of swimmers.
  6. Methods used to determine the degree of adaptation of swimmers.
 
Key words: Style, dialogue, explanation, induction, deduction, analogy, analysis, synthesis, comparison, problem, explanatory, illustrative, reproductive. 
Examples of methods are questions about how to teach in order to achieve higher educational and pedagogical results in teaching. The concept of teaching method is one of the basic concepts of methodology. Reading methods are ways in which teachers and learners work together to gain new knowledge, skills and competencies. Teachers' ability and thinking develop. Therefore, teaching methods performed three main functions, such as coordination, nurturing, and development. In order to consciously select from certain teaching methods that are relevant to the new content of education and the new tasks, it is first necessary to study the classification of all teaching methods and existing teaching methods.
Teaching methods control the organization, motivation and control of the joint activities of the teacher and the learners. Therefore, they are divided into three groups:
  1. The method of organizing learning activities.
  2. Methods of stimulating learning activities.
  • Methods of monitoring the effectiveness of learning activities.
  1. Methods of organizing learning activities are divided into several groups:
  2. Sources of learning for swimmers at home: Oral, instructional, practical methods.
  3. In the direction of the swimmer's thought: induction, deduction, analogy.
  4. The level of pedagogical influence management, the degree of independence of students in learning: The method of educational work performed under the guidance of a teacher. The method of independent years of swimmers. By the level of independent activity of swimmers: Explanatory-illustrative, reproductive, the method of puzzling knowledge, the method of partial research and study.
         Sources of knowledge for swimmers: Oral, instructional methods.
1) Oral methods provide the most information in a short period of time, burning puzzles in front of the swimmers will show them how to solve them.
         These techniques help swimmers develop thinking skills.
A) Explanation: The method of explaining knowledge is that the teacher describes the material, and the learners receive it, that is, the knowledge is ready. The description of the study material should be clear, concise and concise. The explanatory method is used to acquaint students with theoretical materials in the field of data, to provide guidance to swimmers on the use of teaching aids. It is necessary to explain a number of issues of the elementary mathematics course with an explanation.
For example, in explaining a triangle, the teacher uses triangles of different shapes, colors, and sizes embedded in the paper. These are triangles, and if they differ from each other, they are all called triangles. A triangle has three, three, three sides, and three angles, and the angle at which the end of a triangle consists of a point and the side of an intersection is explained by cutting off one corner of the triangle.
B) Interview: this is one of the most common and leading teaching methods, which can be used at different stages of the lesson, for different purposes, that is, to describe new material, to consolidate, to repeat homework, to check independent work.
         Interview is a question-and-answer method of teaching, in which teachers solve students' educational and pedagogical problems through a system of specially selected questions and answers based on their knowledge and practical experience.
         Catechistic and heuristic dialogue is used in teaching. Catechistic dialogue is based on a system of questions that require simple recall of previously acquired knowledge and definitions. The main purpose of this conversation is to check and evaluate knowledge in the form of consolidation and repetition of new materials.
For example: How do you know how many times 7 * 5 = 35?
            How to know the divisions 7 ÷ 8 or 56 ÷ 56 without multiplication 7 * 56 = 8?
           Using the method of subtraction of 60-24, the method of subtraction of 70-18 = (70-110) -8 = 60-8 = 52 is derived.
The questions asked should force swimmers to compare, contrast, group, or search for connections between events and facts in order to activate their thinking. The following questions call for the same: "Why?", "What does it mean?", "How else can this be done?", "How to understand it?".
C) Story - The explanation of the teacher's knowledge can be done in the form of a story. It is mainly used to provide historical information about the development of the history of mathematics and the development of measurement systems.
G) Students' work with books is one of the manifestations of oral teaching methods. The printed word has great influence. The book is one of the sources of knowledge, textbooks and manuals describe a systematic course of the basics of science, provide material for students 'independent work. At all stages of the teaching process, work with textbooks and books is carried out, but this work requires students' skills and teachers. Depending on their reading skills, it is necessary to involve students in independent reading of the text given in the book.
         Reading a math text or problem text is new and difficult for learners, so it is important to check what the learner is reading from the textbook. In the textbooks, attention should be paid to reading the instructions given before each exercise.
         In mathematics teaching, the ability to read pictures, drawings and diagrams, while the ability to understand the mathematical notation that forms the main content of the textbook is of great importance. In this case, the end of the work should be to use the opportunities provided by the textbook for the independent acquisition of new knowledge through drawing, sketching, oral expressions, mathematical notation.
D) Demonstrative methods. This method of teaching allows swimmers to acquire knowledge based on their observations.
Observation is an active form of emotional thinking and is widely used in elementary school. Objects of observation are objects, objects and their various models, instruction manuals in different languages. Teaching instructional methods are inseparable from oral teaching methods. Demonstration of instruction manuals is always accompanied by explanations of the teacher and students. There are three main forms of sharing teaching aids with a teacher:
a) The teacher directs the learners ’observations using words.
b) Verbal explanations provide information about the invisible aspects of the object.
c) The instructions serve as illustrations confirming or clarifying the teacher's oral explanations.
g) The teacher summarizes the swimmer's observations and draws conclusions.
The implementation of the demonstration method in mathematics lessons is based on the perceptions of swimmers on the one hand, and their imagination on the other. The correct use of instruction in mathematics lessons allows the formation of meaningful concepts of quantitative imagination, develops logical thinking, speech, helps to come to generalizations that can be used later in practice on the basis of consideration and analysis of thematic events.
Z) Practical methods. Methods related to the process of forming and perfecting skills and competencies are practical methods. This includes written and oral exercises, practical laboratory work, some types of independent work. Exercises are mainly used as a method of consolidation and application of knowledge.
         An exercise is a planned repetition performed in order to coordinate or reinforce an action. Exercises are used to develop numeracy skills, calculation skills and abilities, arithmetic problem solving skills.
Exercises should be used in a particular system, following the principle of transition from light to complex. Exercises should develop swimmers ’independence in training, coaching, and creative exercises. The first exercises to strengthen this or that action, method, parable solving are performed under the guidance of a teacher.
The teacher will help the swimmers for a while. Therefore, the exercises are performed independently. Exercises of a creative nature include solving problems and parables in different ways, creating a parable on an expression, creating a problem based on a short writing scheme, solving problems of a perceptual nature.
Practical and laboratory work is used to get acquainted with the quantities and their measurement. Conducting practical and laboratory work allows students to actively acquire knowledge, skills and abilities, elements of independent judgment and inference develop research skills, enrich students' imagination and expand their knowledge.
Therefore, practical and laboratory work is one of the most effective methods of teaching.
2) Induction, deduction, analogy.
         The method of induction is such a way of knowing that the swimmer's thought goes from unity to generality, from particular conclusions to general conclusions. Inductive conclusion is a conclusion that goes from particular to general. Using this method, the teacher carefully selects examples, problems, instructional materials to reveal a rule or issue a rule.
         The method of deduction is also widely used in primary school in connection with the method of induction. The method of deduction is such a way of knowing that this way gives special knowledge on the basis of general knowledge. It is a transition from general rules of deduction to specific examples, thematic rules.
         First graders are taught to lead children in an inductive way to a conclusion to explain the connection between summation and addition.
         How many circles can be found before using the guide.
         0 0 0 0 0 0 0
         5+2=7                           7-5=2                   7-2=5
         Then the following exercises are performed with other numbers and other instructional materials, and the children's faces express the general conclusion: "If the first additive is lost from the sum, the second addition remains, if the second addition is lost from the sum, the first addition remains."
         Deductive conclusions are the sum of several specific conclusions. Therefore, this method forces swimmers to marry and search.
For example: Deductive reasoning is used to explain the property of dividing a sum by a number:
         For example: a) In order for a sum to be a number, it is necessary to calculate the sum and divide it by a number.
         а) (8+6):2=14:2=7                б) (8+6):2=8:2+6:2=4+3=7
         It is necessary to divide each additive into numbers and add the resulting results. The analogy concludes that in the analogy of some features of objects, it is presumed that these objects are similar in other respects.
         The analogy is a conclusion that goes from private to private.
For example, teaching the written methods of addition and subtraction of three-digit numbers to addition and subtraction of multi-digit numbers is based on the use of analogy. For this purpose, it is recommended to solve the following examples, where each successive example includes the previous one:
        
For example:
+
635
+
4635
254
3254
899
7889
         After solving such examples, swimmers conclude that the addition of multi-digit numbers is done as a written addition and subtraction. The use of methods of induction, deduction, analogy is based on the analysis of mental operations, synthesis, comparison, generalization.
The method of thinking that focuses on dividing the whole into its constituent parts is called analysis. A method of thinking that focuses on the study of connections between objects or events is called synthesis.
For example, in answering the teacher's question about the name of a number consisting of one decimal and five units, swimmers use synthesis (the number consisting of one decimal and five units is 15).
         In teachers, no concept is interrelated without analysis and synthesis. These two interrelated methods of thinking are used in solving mathematical problems.
The analysis of the problem is to divide it into those given and those sought. The synthesis is to answer the question.
The method of comparison is well mastered by swimmers when the concepts under consideration, arithmetic examples, new concepts consisting of distinguishing similar and different signs of problems are struck by comparison and contrast burns. There are many similarities and contradictions in the mathematics course.
For example, the opposite concepts are similar to the operations of multiplicity, multiplication and division, multiplication and division, multiplication and division, multiplication and division, multiplication and division, multiplication and division, multiplication and division. reduce the number several times, divide into equal sources, and divide according to content.
         An elementary course in mathematics opens up great possibilities for the use of the method of comparison: comparison of numbers, expressions and numbers, comparison of two expressions, comparison of problems.
         Generalization is the separation of the most important aspects from the studied objects and their separation from the non-essential ones. A necessary condition for the formation of generalization is the assimilation of insignificant features without changing the essential features of the concepts and the essential features of the facts.
For example, in order to give children an idea of ​​a right rectangle, it is necessary to vary the important features of the concept under consideration, such as the color of the material from which it is made, its position in the plane, the relationship of the lengths of the sides. The essential features must be left unchanged, that is, all the angles must remain at right angles and the opposite sides must be equal.
  1. Teacher-led learning is the independent work of swimmers.
         In the first stage of teaching in the primary grades, the educational work carried out under the direct guidance of the teacher is widely used, the teacher must guide the students in the right direction.
         At present, as a method that allows to increase the effectiveness of teaching, much attention is paid to the independent work of swimmers. Independent work: "Independent work of students involved in the learning process is the work done on his assignments during a special time without the direct participation of the teacher, in which students consciously strive to achieve the goal set in the assignment, expressing the results of mental or physical activity in a form" - .
         Independent work is distinguished by the following:
         A) For didactic purposes.
         This work can be aimed at encouraging swimmers to accept new material, prepare it, pass on new knowledge, consolidate it, and repeat previously learned material.
         B) Work with a textbook on the material on which swimmers work, on didactic material, on a printed notebook.
C) According to the nature of the activity required from the swimmers: from this point of view, the work is differentiated according to the given pattern, the given procedure and….
G) Depending on the method of organization.
General class work in which all swimmers of the class do the same work, group work in which different groups of swimmers work on different tasks, individual work, in which each swimmer works on a specific task.
In almost every lesson in mathematics it is possible to carry out 2-3 short independent work. At the same time, giving swimmers independence in completing assignments without adequately preparing them for independent work often leads to wasted study time.
  1. Methods classified according to the level of independent activity of swimmers.
         1) Isolation-illustrative method.
         Through this method, the teacher provides ready-made information by various means, and the learners receive, understand and remember this information. The teacher provides information orally (narration, explanation), written (textbook, additional manuals), instructional (showing pictures, drawings, diagrams, methods of movement).
         Swimmers perform activities that are necessary for a high level of knowledge transfer, listening, feeling, reading, observing, comparing and remembering new information with previously learned material.
         2) Reproductive method.
         The main feature of this method is the restoration of the method of activity and repetition on the instructions of the teacher. Using this method, swimmers acquire skills and competencies.
             3) Enigmatic presentation of knowledge.
In such a statement, the teacher not only states this or that rule, but also makes a sound, puzzles and shows the process of solving it, the teacher's explanation is more convincing, teaches children to think, teaches to conduct cognitive research.
4) Partial research and heuristic method.
In this case, the teacher puts a puzzle in front of the swimmers, and he himself explains the training material, but during this narration, the students are asked questions. These burnt questions require them to join the search process and solve a cognitive problem.
5) Research method of teaching.
When working with this method, swimmers assume that they have understood the burnt puzzle, invent a method of verification, make observations, generalize, and draw conclusions.
  1. II. Methods of stimulating learning activities.
Methods of motivation and justification of teachings include games of a cognitive nature, the creation of successful learning situations, the method of rewarding and other methods.
It is necessary to separate the house, which is one of the most effective methods of awakening learning activities. In the preschool age, games that play an important role in the lives of children are divided into creative, dynamic, didactic games.
At the heart of teaching or didactic games in primary education are cognitive content, mental and will power, actions and rules that determine the course of the child's home, aimed at solving problems.
In didactic games, the main processes of thinking are analysis, comparison, inference and… development. Positive games that appear during didactic games in the learning process activate children's activities, develop their free attention and memory.
Positive games that appear during didactic games in the learning process activate children's activities, develop their free attention, memory.
At home, swimmers do a lot of math, exercises, counting, comparing numbers, and solving problems without noticing each other.
A large number of games have been created in elementary mathematics that develop children's quantitative and spatial imagination. These include "Magazin", "Zinacha", "Jim", "Arithmetic lotto",….
         III. Checking students' knowledge and skills in mathematics. Assessment and assessment of swimmers' knowledge, learning and skills is an integral part of the learning process in the primary grades.
         the process of teaching mathematics is constantly monitored. Monitoring determines the level of knowledge and the quality of knowledge transfer in swimmers, identifies gaps in knowledge, skills and competencies, and helps to prevent it.
There are 3 types of control in math classes: initial, daily, and final. The initial review is conducted at the beginning of the academic year or before learning a new topic to determine what knowledge needs to be recalled in order to learn new material.
Prior to the initial consolidation of daily inspection knowledge, the swimmers are conducted to determine whether they have correctly understood the new topic or not, and what challenges they are facing. The final examination is conducted with swimmers at the end of the study of topics, sections or quarters, at the end of the academic year.
Its purpose is to determine the results of training, to check the quality of knowledge, training and skills acquired by students. The method of controlling knowledge in mathematics is different. These methods are oral inquiry and written, practical work. Oral interrogation can be frontal and individual. In frontal questioning, questions are given to the class but the level of complexity of the questions is not the same. Taking into account the potential of each child and at the same time involving everyone in active work, the teacher takes a stratified approach to the class swimmers.
The teacher often puts the student in front of the board in order to draw the whole class's attention to the student's answer. When the teacher asks individually, the student can be given a card with the assignments and take the time to complete it. During the oral questioning, the teacher checks how well the children have mastered the learning material and tries to involve the learners in active work as much as possible.
Oral questioning allows to fully determine the knowledge of swimmers, but it takes a lot of time, which limits the ability to check swimmers. In addition, the teacher's questions and the student's answers are not recorded anywhere in the oral questioning. This deprives the teacher of the opportunity to compare the answers of different swimmers to the same question. Independent written work is carried out for the purpose of daily and final examination of knowledge, rights and skills. In the daily inspection, the independent work is not very large in size and consists mainly of assignments on the topic being studied.
In this case, the examination is inextricably linked with the teaching process in the classroom and is subject to it. Therefore, independent work can be divided into parts and given two or three times during the lesson.
Exercises and tasks for independent work are designed, checked and evaluated by the teacher, taking into account the specific characteristics of the swimmers.
Written examinations are conducted at the end of the academic quarter or academic year after the topic or section has been covered. Quarterly or year-end test questions are asked on a variety of math subjects. Quarterly or annual audits usually consist of issues and examples.
The inspection should be carried out independently by the student, without the assistance of the teacher. The teacher should carefully and qualitatively carry out the inspection work, pointing out the mistakes, difficulties and reasons of each class swimmer.
Each written work should be evaluated.
Control questions:
  1. What is meant by teaching methods?
  2. What is the classification of teaching methods, name them?
  3. What oral teaching methods are used in primary school?
  4. How do instructional and oral teaching methods relate to each other?
  5. What is the essence of the methods of induction, deduction and analogy?
  6. What mental operations underlie the use of induction, deduction and analogy methods?
  7. What is meant by independent teaching?
  8. What types of independent work are there?
  9. What is the value of a didactic house?
  10. Justify the need to use different teaching methods in the lesson?
                           
Lecture number 4
Topic: Coverage of the lesson process in mathematics
 Learning tools used for and their functions.
Plan:
  1. The structure and system of mathematics lessons in primary school, the requirements for it.
  2. Types of mathematics lessons and its stages.
  3. Scheme of lesson analysis.
  4. Homework of swimmers.
 
Basic phrases: tool, textbook, printed notebook, cards (tables: instructive tutor). 
Reference: Models: Coins, counting sticks, numbers, geometric figures; Tools: roulette, clock, ruler, compass; Instruments: Abacus, class chuti, scales. 
Teaching aids fully or partially describe the concept being taught, providing new knowledge about the concept being studied. Teaching aids can be divided into 2 classes:
The first is the class of ideal models and the material-object model. Stable textbooks in mathematics, didactic materials, study guides, various recommendations, problems and sets of exercises, tables, which are issued as a teacher's aid, belong to the class of ideal models. Various counting sticks, object pictures, pictures, diagrams, drawings, coin model, geometric figure model sets, number sets, instruments (measurements), abacuses, class chutes, filmstrips, slides and others can be included in the material-object class.
These teaching aids are called instructional manuals, they are a source of new knowledge, they take into account the extent to which knowledge is integrated, they organize the independent individual work of students.
Let’s take a look at the features of these teaching aids. The textbook is a book that clearly explains the main content of the elementary mathematics course. The main task of the textbook is to help swimmers to acquire independent knowledge and to consolidate and deepen the knowledge acquired in the course. Textbooks are the basic and necessary teaching aids for swimmers.
The mathematics textbook is structured according to the program and explains the requirements of the program. The textbook defines the system of studying some issues, reveals the general methodological directions about the program and its explanation.
The structure of the textbook is determined by the program, the sections correspond to the sections allocated in the program. Each section is divided into topics. The textbook helps the teacher to plan his / her work rationally, as it tells him / her how to reinforce the learning material on any topic, prepares him / her for learning new material, and reinforces and repeats previously learned material.
Textbook teaching is conducted in two directions: one is organizational work; the second is to work with the textbook on its content and essence.
Organizational work. From the very first lessons in school, students should learn the skills related to working with the textbook, including how to handle the book, how to store it carefully, how to open it, how to find the appropriate pages, how to use the page layouts. it is necessary to explain whether the omitted examples or the empty cells do not fill in the tables, which must burn a number.
One of the main tasks of a teacher in teaching to work with the textbook on its content and essence is to teach students to use the textbook as a source of knowledge. It is known that the textbook contains theoretical and practical materials that can be used at different stages of the lesson.
Initially, the work on the textbook is used as a reinforcement of previous oral explanations. The teacher explains a rule to the children in clear examples that give them strength, and then instructs them to look at how the problem itself is described in the textbook.
In teaching mathematics, children are explained the essence of the mathematical notes, pictures, diagrams, drawings that are available in the textbook. The materials given in the textbook of mathematics in many respects allow to solve educational problems of primary education.
For example, mathematics allows children to get acquainted with different aspects of the environment through the work of people through textbooks, pictures, etc.
The textual questions given in the textbook can be used not only for the purposes of mathematics education, but also in the upbringing of children. Mathematics of matter reflects the life and work of people, their struggle to increase labor productivity, and the socially useful labor of swimmers to save raw materials and time. In the textbook exercises, children have the opportunity to develop the skills of analyzing observation, drawing conclusions and generalization. The textbook fosters children's independence in teaching mathematics, opens wide opportunities for the development of independent work skills.
In order to increase the effectiveness of the process of teaching mathematics, in addition to textbooks, there are flashcards with math assignments, printed notebooks, manuals and instructions for teachers.
Among the teaching aids for mathematics are cards with math assignments, which are published in addition to textbooks. Their purpose is to help the teacher to carefully coordinate the main material of the program in the organization of children's independent work on individual assignments. The teacher can use the cards to conduct independent and control work, to fill in the gaps in the knowledge of swimmers, to organize knowledge, to systematize, record and control knowledge in the organization of frontal, group and individual work.
The printed math notebook, like the cards, is based on a system of exercises given in the textbook and is designed to organize students' frontal independent work. Print-based notebooks free up mechanical copying of assignment texts, thus allowing more efficient use of reading time. Instructions for teachers for primary school textbooks have been developed and published. The purpose is to help the teacher improve the quality of math teaching. At the same time, you can get a lot of useful knowledge and advice in the journals "Primary Education".
We have reviewed the learning tasks above, such as textbooks, math assignments, printed notebooks, textbook instructions, and recommendations. Now we come to the part where we talk about the middle ground.
The use of instruction stimulates the activity, attention, attention of swimmers, develops abstract thinking, allows you to carefully combine the material studied, saves time. Different types of manuals are used in teaching elementary mathematics.
Knowing the types of instructional materials allows you to choose them correctly and use them, to use them in the learning process to improve teaching.
Instructional applications can be divided into two types, namely, natural and visual instructional applications. Natural instruction manuals include things that happen in marriage, things around us, trees, pens, toys, chopsticks, buildings, and more. From the first days of school, the teacher draws the children's attention to the surrounding subjects.
For example: How many items, desks, windows, cabinets and doors in the methods? questions can be asked to swimmers.
But these objects cannot be reduced to ashes, they can be seen and felt in the fall. For this reason, small objects such as pens, pencils, counting sticks and other items can be used for counting. Sanok chups are one of the most widely used natural instruction manuals. These chups are made of wood, plastic. Each teacher and swimmer should have a set of numbered chups. During the first school year, counting sticks are used to count numbers, number numbers, create ideas about figures, and perform operations.
Now let’s look at the pictorial instructions. These include these.
A) Numbers, sign, attitude sign:
(+, -, *, / =,>, <) (0, 1, 2, 3, 4, 5, 6…)
B) Demonstration pictures. This includes pictures of each object, including toys, fruits, vegetables, flowers, birds, animals, animals, utensils, and more.
V) Model of geometric figures.
2+1+3        1+2=3
G) Numerical figures
D) 1, 2, 3, 5, 10, 20 penny coin models.
E) Graphic models, drawings, diagrams.
I) Instruments: class scissors, abacus, scales and scales, drawing and measuring instruments: class ruler, wooden meter, roulette, compass, clock model, pallet.
K) Tables: 1) Instructive; 2) Reference; 3) Teaching tables. Teaching aids.
Control questions:
  1. What is meant by teaching aids and what are their main functions?
  2. What is a textbook task and how does it relate to the program?
  3. In what direction can work with the textbook be carried out?
4. What types of manuals are available in teaching mathematics?
5. What are the natural guidelines?
6. What are the descriptive instructions? Give examples.
        
Lecture number 5
Topic: Teaching mathematics in primary school
form of organization.
Mathematics is taught in primary school in the form of lessons in school and extracurricular activities, in the form of independent homework at home, in the form of excursions in nature.
The main form of organization of educational work in mathematics is a lesson. The peculiarities of the mathematics lesson stem primarily from the characteristics of the subject.
It is known that the basic course of mathematics is structured in such a way that in addition to the study of arithmetic material, the elements of algebra and geometry are introduced. Therefore, in addition to arithmetic, geometry and algebra are considered in each lesson.
The combination of materials from different sections of the mathematics course affects the structure of the mathematics lesson and the methodology of its conduct. Another distinctive feature of the elementary course in mathematics is the combination of theoretical and practical issues. Therefore, the transfer of knowledge in each mathematics lesson is carried out simultaneously with the development of training and skills.
Preliminary preparation for one material is carried out in order to introduce the second material, to generalize, systematize, consolidate knowledge and skills in relation to the third material.
At the same time, the knowledge and skills of swimmers are monitored and recorded. The characteristics of mathematics lessons depend on the ability of students to master mathematical material. The abstract nature of the material requires the correct choice of active methods of teaching the teaching aids, the individual and differential approach to the diversity of learning activities during the lesson, and in addition to educational tasks in mathematics lessons are considered educational tasks.
The teacher plays a leading role in achieving the educational nature of the educational work, because the teacher himself determines the content, method and organization of the lesson. Mathematics teaches students to be observant, alert, critical of life, initiative in work, the formation of a clear conscience, the development of accuracy and consistency in measurements, writing, the ability to overcome difficulties.
The lessons focus on instilling in children an interest in mathematics and educating them to work independently. If the lesson is interesting for children, then they will become more active and independent in their studies, didactic games and interesting exercises will be included in the lessons in order to arouse interest in mathematics. In preparing for the lesson, the teacher must first identify the main objectives of the lesson. After defining the goals and objectives of the lesson, the teacher should determine the content of the work to be done in the lesson.
To determine the content of the lesson, the teacher must follow the requirements for modern lesson content:
  1. The content of the course should be appropriate to the syllabus;
  2. each lesson should be structured with the thematic content and purpose in mind;
  3. The content of the study material should be clear to the student, relevant to the topic, the purpose of the lesson, and should be related to life and work;
The course should cover the theory of arithmetic, algebra, geometry materials, practical activities, calculation exercises, problem solving.
  1. The methodology of work in mathematics should be able to respond to the age characteristics of the student, to correct and develop their cognitive activity, mental and practical analysis, synthesis, the formation of generalization activities;
  2. At each stage of the mathematics lesson, it is necessary to check how the students transfer lessons and knowledge;
  3. All teaching aids, textbooks, notebooks, visual aids required for the lesson should be provided with didactic materials, measuring and drawing tools;
  4. Each mathematics lesson should be distinguished by organizational accuracy, that is, each part of the lesson should have a specific purpose and be subordinated to the main purpose of the lesson, the lesson should be carefully planned and the time should be distributed among each part;
Frontal work is carried out individually and combined with a stratification approach.
  1. Repetition of what is lost in mathematics lessons should be carried out in each lesson, that is, the principle of continuous repetition should be followed;
  2. In each lesson it is necessary to enrich the vocabulary of the learner with new mathematical terms, phrases, to determine the child's speech, to observe the structure of the grammatical structure;
  3. The training material should be understandable to swimmers and within their reach;
  4. The alternation of one type of activity in the course with another should be carried out taking into account the performance skills and rapid fatigue of the swimmers;
  5. the lesson should be linked to the personal experiences of married swimmers. The main types of work performed in mathematics lessons are: Oral exercises, written calculations and problem-solving, construction and measurement exercises.
One of the most important requirements in a modern lesson is to require students to activate their cognitive and creative activities. Each lesson should be a lesson in thinking in its own way, a lesson in participation in creativity.
Subject to the basic requirements of the lesson, the teacher also influences the implementation of these requirements with the method of uz method, which depends on the nature of the class and its individual characteristics.
When preparing for a lesson, a teacher should complete a number of tasks according to a plan, with a plan. The plan should include the following elements:
  1. Draslik conduction time and its number according to the mathematical plan;
  2. The name of the course topic;
  3. The main didactic goals of the lesson, educational, pedagogical tasks;
  4. Equipment used in the lesson;
  5. The content of the work on the introduction of new material, consolidation and repetition, and the study of the next topic;
  6. Methods and techniques of study work performed in each part of the lesson;
  7. The names of the swimmers to be asked during the course;
  8. Homework.
The level of perfection of the plan depends on many factors, for example, the teacher's experience, the level of difficulty of the lesson, the complexity of the exercises that need to be considered in the lesson.
The teacher organizes the lesson according to this plan, let's look at the main types of math lessons in primary school. Depending on the didactic purposes, these types of mathematics lessons differ from each other.
  1. Lesson of learning new material;
  2. Advanced course;
  3. Lessons to strengthen knowledge, skills and abilities;
  4. Lessons of repetition of losses;
  5. Knowledge testing and assessment lessons (written work lesson);
Each math lesson has its own structure. The lesson can consist of the following main parts: organizational part, homework check, statement of the topic and purpose of the lesson, preparation of students for the reception of new material by repetition, special oral exercises, learning new material, initial consolidation of knowledge and skills, exercises use in performance, independent work of swimmers and its verification, repetition of previously passed material, assignment of yuga, completion of the lesson and completion of the lesson. Depending on the type of course, these components can vary and can be implemented in different ways.
The structure of a mixed, complex course is as follows:
  1. Organizational part;
  2. Homework check;
  3. Repetition of the missed topic;
  4. Preparation for learning new material;
  5. New topic statement;
  6. Strengthen a new theme;
  7. Repetition and consolidation of the past;
  8. Homework assignment;
  9. Completion of the lesson.
Lessons on learning new material:
  1. Organizational part;
  2. Homework check;
  3. Repetition of the passed material: a) verbal calculation exercise; b) independent work;
  4. Preparing to learn new material;
  5. Explain a new topic;
  6. Initial consolidation of a new topic;
  7. Assignment of homework and assessment of swimmers' knowledge;
  8. Completion of the lesson.
In addition to these lessons, the main parts of them will be played to consolidate the acquired knowledge. Such classes are called knowledge, skills and competency-building classes.
Exercises, practical and independent work are the main means of strengthening knowledge. The structure of this lesson can be as follows:
  1. Organizational part;
  2. Homework check;
  3. Burning lesson objectives;
  4. Repetition of the topic: a) independent work or mathematical dictation; b) questions on the topic; c) exercises on the topic;
  5. Assignment of homework, assessment of students' knowledge, ie completion of the lesson;
  6. Completion of the lesson.
Repetition lessons. The structure of the review lesson will be the same as the structure of the reinforcement lesson. Reinforcement with repetition is similar in many ways, but there are differences in the organization of lessons. Usually, some rules and regulations are reinforced by the direct adoption of new material. During consolidation, initial skills and competencies are formed. In the review lesson, the learning material is mainly systematized and generalized. The types of review lessons can be distinguished from:
  1. At the beginning of the school year and daily review lessons: Review lessons are held in all classes except the first grade for about two weeks. The purpose of revision lessons is to recall the knowledge and skills acquired in the previous academic year.
  2. Thematic review lessons. As you know, the mathematics program is divided into sections, topics. By repeating the material on the topic, swimmers distinguish the basic theoretical rules, solve a system of exercises.
  3. Generalized repetition lessons quarterly repetition, half-yearly repetition, one-year repetition.
Lessons in checking and accounting for knowledge, skills and competencies.
Systematic testing of students' knowledge is performed in each lesson. In addition, there are separate lessons for testing knowledge. The structure of such lessons is as follows:
  1. Organizational part;
  2. State the purpose of the lesson;
  3. Introduction to the content of the written work;
  4. Provide a brief guide to the work to be done;
  5. Ensuring that swimmers do their work independently;
  6. Getting the job done.
LESSON ANALYSIS
Attending and analyzing the lessons of experienced teachers, as well as the analysis of their own lessons, have a great impact on the acquisition of teaching methods. The analysis of the mathematics lesson can be done in the following areas:
  1. Determining the place and role of the lesson in the system of lessons on a given topic, it helps to accurately assess the content of the lesson, its structure, methods and techniques.
  2. Identify and justify the main didactic goals, educational and pedagogical goals of the lesson.
  3. Analysis of the content of each part of the lesson and its teaching methods, the relevance of the course material to the educational goals, the relevance of the program to the age of students, the level of development and mastery of mathematical knowledge, the activation of students' independence and intellectual activity.
  4. Evaluation of the organization of swimmers' activities, individual and collective work of swimmers, differential approach to swimmers.
  5. Defining the role of didactic materials in teaching a variety of teaching aids.
  6. Teacher's image.
  7. General grade of the course.
STUDENT HOMEWORK
Homework is one of the forms of organizing independent, individual work of swimmers outside of school hours. In the performance of homework, not only this or that material is repeated, but also important skills and abilities are formed, which is the most important part of the independent activity of swimmers.
In the course and as a result of well-organized and independently performed homework, a person's sense of safety, diligence, discipline, integrity, responsibility for the work assigned is formed and developed, the ability to plan activities, self-control skills are improved. The organization of his work must meet the following requirements:
  1. Homework assignments should be commensurate with the strength and knowledge of the swimmers. Therefore, homework is not assigned to first-graders during the first half of the school year, given that it takes time to develop independent work skills, and homework from the second semester onwards should be simpler and more manageable than what is done in class.
  2. Homework should be given systematically. The last days of the week and the days before the holiday are excluded.
  3. The amount of homework should not exceed the norm of time allotted for their completion in all subjects.
  4. Young school swimmers should be instructed on how to do their homework.
  5. Any homework should be checked by a teacher.
Homework check is an important part of the lesson, and if the checking system is set up well, the student should not think about not doing homework or doing it without conic. Checking students' homework is not only a teacher's job, it is a necessary thing. Without doing this, it is impossible to have a clear idea of ​​how the swimmers will hand over the lost material.
If homework is not checked systematically, they lose their meaning. By regularly checking the homework, the swimmer shows interest in the swimmer's learning activities, shows the importance of completing assignments, shows respect for the swimmers' hard work, and thus instills in the swimmers a positive attitude towards homework.
Depending on the nature of the assignments, the form of homework check can be different. If the homework is not connected with the previous lesson material and the lesson tasks, then it is possible to limit the review not only at the beginning of the lesson, but at any stage.
If the homework depends on the content of the lesson being taught or is based on new material taught in the previous lesson, it is important not only to check the accuracy of the answers, but also to listen to the students' explanations of the actions taken. If the swimmer is confident that even the bush swimmers will be able to do the homework, then it is possible not to check the homework at all.
Another of the homework check formulas is selective checking, in which the homework given in this step refers to the checking of the most basic places. There are other forms of homework checking as well. For example, cross-checking that similar tasks were completed only in a given task, and verification of homework related to verbal arithmetic were common in the primary grades.
  1. An important requirement for the organization of homework is its diversity in appearance and content.
Homework should include not only examples and problem solving, but also other types of homework. These types of expressions include comparing equations, solving equations of a geometric nature, and giving creative character to homework and arousing students' interest in it.
Swimmers' homework is a natural continuation of the work done in the classroom and serves to consolidate the knowledge acquired in it.
  1. It is advisable to individualize homework so that all swimmers can always get homework according to their ability. The size of the task, the purpose, the method of execution can be individualized.
  2. An important condition for swimmers to do their homework successfully is to advise parents that they can help swimmers to the best of their ability and purpose.
Homework is required in the form of a form of organizing students' independent work. At the same time, the requirement that homework should be appropriate to the strength of the swimmers is particularly important. Homework can be given by the student at the end of the lesson or in another part, the teacher writes the task on the board in the form of a short, the students write it in their diaries.
        
 
Lecture number 5
Topic: From math to elementary school
foreign affairs.
One of the most important tasks facing secondary schools is to achieve the maximum intellectual development of the younger generation by equipping them with the basics of modern science.
In order for students to master mathematics, physics, chemistry and other subjects in the upper grades, they need to master mathematics and develop practical skills in the primary grades. In addition to classroom activities, extracurricular activities should be conducted in primary school to help primary school learners improve their knowledge and anticipate the level of instruction.
While extracurricular and extracurricular activities are an integral part of educational work with children, they increase students ’interest in knowledge and hard work, as well as improve the quality of learning and improve their behavior. Extracurricular activities in mathematics are activities designed to expand and deepen students' mathematical knowledge.
The main purpose of extracurricular activities is to develop students' interest in science, to equip them with mathematical knowledge, skills and abilities that complement and deepen the knowledge acquired in the classroom.
In general, in elementary school, extracurricular activities are closely linked to classroom work, which is a continuation of classroom work, and sometimes deepens it.
Two types of extracurricular activities should be distinguished from each other. The first is to work with swimmers who are left behind in handing out program material, which includes additional lessons and consultations. The second is classes for swimmers interested in learning math.
It is known that the first round of classes is currently available in all schools. In this case, it is advisable to conduct classes once a week in small groups of 3-4 swimmers. Usually, extracurricular activities refer to the second type of work, and they mainly serve the following purposes:
  1. To arouse students' interest in mathematics and its applications;
  2. Expanding students' knowledge of mathematics in the program;
  3. Fostering a culture of mathematical thinking;
  4. To teach students to work with popular science literature in mathematics;
  5. Expanding students' understanding of the historical and scientific value of mathematics, the role of the school of mathematics in world science.
Some of these goals are achieved during the lesson, but due to time constraints, much of it has to be done in extracurricular activities. In school practice, the following types of extracurricular activities are performed in mathematics with younger swimmers:
  1. Hours and minutes of fun math;
  2. Organization of mathematical circles;
  3. Mathematical newspaper issue;
  4. Excursion;
  5. Creating a mathematical corner;
  6. Spending math nights;
  7. Conducting Mathematical Olympiads in primary schools.
The following rules are the basis for the organization and conduct of extracurricular activities:
  1. Extracurricular activities are conducted taking into account the knowledge, skills and abilities of students in the classroom;
  2. Extracurricular activities are based on the principles of voluntariness, initiative and the actions of swimmers, as well as to meet the individual needs of swimmers;
  3. Extracurricular activities differ from the Kura classes in the form of training and are often of an interesting nature.
Oral rehearsals are often conducted with the desire of all swimmers in the class to "again, again." At the request of the students, the continuation of the work started in class can be postponed to extracurricular time. Extracurricular activities with swimmers can be conducted twice a month, depending on the needs of the swimmers, and then, depending on the example, the problem, the game and the increase in interest.
Because in the implementation of the program there is no opportunity to solve such interesting problems, to organize games, to find riddles, to make quick calculations in the classroom.
Experiments convince us that swimmers are less tired than usual classes during fun math classes and work with a waiter-deputy.
The organization and equipment of such training should be interesting and clear. Instructional math tutorials, figure counting, figure counting, poster games, table games, mazes, making geometric shapes out of cardboard, crossword puzzles, and more can be a great help to swimmers.
The time spent on the training is determined by the purpose for which it is conducted. If the meeting with the swimmers is held after school and the goal is to get acquainted with a game, then in the beginning 10-15 minutes will be enough for such training. After the swimmers get acquainted with the house, they usually do the same house with their parents, siblings and others, that is, they are attracted to it.
If the session becomes more complicated, it may take about an hour to complete. The materials are always chosen according to the students' calculation skills, and when it comes to problems, they may be different in appearance and type of problems indicated in the program for this class. Intelligence issues, on the other hand, can go beyond this limit and at the same time help to successfully learn to solve the problem.
  1. Analytical synopsis of 1-hour extracurricular activities in mathematics in the classroom.
Today we are going to have an interesting math class. You will find out later what we are dealing with. You have to be very smart. Please answer my questions carefully. The first person to answer my questions quickly and correctly will win.
  1. I. Find a home.
Three geese flew over us. Three more flew over the cloud. Two fell into the water. How many of these geese are left in the air?
  1. II. Find the following puzzles and solve interesting problems.
  2. Two lanterns illuminate my path, a spiked pen on the lantern. What are these? (autumn, kosh).
  3. It has two ends of a pipe, two rings, and one nail in the middle. What is this? (scissors).
  4. He has three teeth in his mouth and eats hay. What is this? (panshaxa).
  5. One burns, the second burns, and the third burns. What are children? (rain, land, vegetation).
  6. The bulls are equal. A hat on his forehead. He runs in front of two brothers. The other two chase.
    What is this? (table).
  1. The boy threw flour from it and flour was formed again. How did he do it? (removing the caps).
  2. There are four candies in one plate, give these candies to each of the 4 swimmers and let one candy stand in the plate. (given to a swimmer with a plate of candy).
  3. I have eight friends, all less than me. If you count to it, you won’t burn without telling me. (tukkiz).
III. Listen to the problem told in the poem and count how many fish the fishermen caught.
         The Sultan caught - 13 churtan,
         Azam caught - 4 carp,
         The ax caught - 2 lacquers.
How many fish came out of the kirgok. (Answer - 19 points).
  1. IV. Hold the game "Right side, left side".
The purpose of the house is to strengthen the concept of right and left. The number of players is not limited.
CONTENT OF THE GAME
         Players are divided into two groups. Both rows move in opposite directions according to the manager's command. With the manager’s command to “left” or “to him,” all players turn to the appropriate side and stop. Whoever makes a mistake will leave the house. And the house will continue. Whichever group has the least players removed from that group wins.
  1. V. What right triangle can be made from two equal squares?
  2. VI. How to make an envelope from a square sheet of paper?
A summary should be made at the end of the session.
The main form of extracurricular work in mathematics at school is a math circle. If the school has a math circle, no other form of extracurricular activities (math olympiad, math night, and math newspaper publication) will be possible, as the assets that make up the math work at the school will consist of the circle members.
Experience shows that it is possible to organize and conduct roundtables with young swimmers from the 1st grade (II quarter). But usually this work is carried out with swimmers of II-IV classes.
The work of the Mathematical Circle, when properly organized and used in the methodology of its teaching, allows students to develop an interest in mathematics and develop this interest, their cognitive assets and mathematical abilities. It absorbs the skills of independent work, cultivates confidence in the strength of the face, the ability to overcome difficulties independently. It is important for children to realize that they have grown mathematically, gained new knowledge and skills in the process of working in a circle. It is necessary to make a detailed analysis of the results of independent work, emphasizing the general and individual achievements of swimmers.
Parents of swimmers can also be invited to some of the circle's activities. Despite the variety of mathematical questions and issues, the content of the circle lessons with young swimmers must meet the following basic requirements.
  1. The planning material has to do with the application material. In this case, the calculation operations do not exceed the requirements of the class program, the connection between practice and theory should be provided for calculations, problem solving, construction of geometric figures.
  2. The problems studied can have future goals, ie they can prepare students to solve arithmetic problems and so on with the help of mathematical problems to be studied in the future, such as sets, functional connections, algebraic symbolism, equations, graphs.
  3. The content of the issues to be studied should be within the reach of the children of the age in question, to enable them to solve basic educational and pedagogical problems that arouse their love for mathematics and interest in learning it.
The content of the circle includes the development of the ability to solve complex examples and problems, the ability of swimmers to think, to move from concrete to abstract, to make the necessary generalizations, and so on. Exercises, arithmetic tricks, "wonderful" squares, riddles, fun games, poems and so on play an important role in the character of fun. At the same time, the fact that the material is interesting allows you to explain in depth the mathematical rules, laws, etc., which are not the only goal.
A lot of attention is paid to the conversations of teachers, the speeches of members of the circle, some theoretical material is presented in the conversations of teachers, interesting mathematical problems are given.
The participation of a group of children in a math circle and their work is very important not only for the participants of the circle, but also for all the classmates.
The members of the circle help the teacher in preparing a joint circle, conducting excursions, publishing a math newspaper, organizing a math corner, and so on. In the circle, teachers develop skills in quick calculations and ground measurements using an arithmetic or chute along with problem solving.
The teacher plans in advance the weekly sessions with the members of the circle.
It is advisable to hold classes in the 2nd grade for 30-35 minutes, in the 3rd and 4th grades for 35-40 minutes.
When planning the work of a mathematical circle, it should be taken into account that a separate lesson does not completely solve the problem. A pre-arranged system is required, along with a complete elaboration of the questions to be covered in all planned sessions.
In this regard, it is necessary to make a plan for half a year or a year at a time. In this case, the whole material should be distributed in such a way that it is relevant to the topics covered in the lesson at this time. At the beginning of the training, changes and additions are made to the plan.
It is useful to solve problems that make the whole study of the subject more difficult, as well as to solve problems that require ingenuity, intelligence, attention, and to exchange small interesting questions with karsh.
The following classes can be held in primary school:
1 - Exercise
  1. Interview with Uzbek mathematicians about Al-Khwarizmi's childhood and how he found numbers.
  2. Find the house you want.
2 - Exercise
  1. About the structure and drawing of geometric figures (paper and cardboard).
  2. Counting house in order.
3 - Exercise
  1. Ulugbek's works on childhood and mathematics.
  2. Interesting issues.
4 - Exercise
  1. Problems solved by the method of conjecture.
  2. Working with scales.
5 - Exercise
  1. Solve the problem of "mathematics in the family."
  2. It's a joke.
6 - Exercise
  1. Conversation about the life of Umar Khayyam.
  2. Can you create a calendar.
7 - Exercise
  1. A conversation about dwarfs and giant numbers.
  2. Solve logical problems.
8 - Exercise
  1. The work of Abu Ali ibn Sina.
  2. Complete 9 assignments related to 9.
9 - Exercise
  1. Addressing issues related to school life.
  2. Learning equality, inequality with the help of visual aids.
10 - Exercise
  1. Working with matchsticks.
  2. Issues related to paid settlements.
11 - Exercise
  1. Solve cognitive problems.
  2. Learning to write numbers using Roman numerals.
12 - Exercise
  1. Conversation about the history of mathematical symbols.
  2. Information about the months of the year.
13 - Exercise
  1. Solve humorous problems.
  2. Mathematical puzzles.
14 - Exercise
  1. How did people learn to count.
  2. Logical issues.
15 - Exercise
  1. Solve geometric problems.
  2. Mathematical rebuses.
16 - Exercise
  1. Mathematics course and the use of mathematical symbols in mathematical discourse.
  2. Mathematical tricks.
17 - Exercise
  1. Teach them to perform face finding tasks.
  2. It's a joke.
18 - Exercise
  1. Interesting questions about addition and subtraction.
  2. Creating assumptions.
19 - Exercise
  1. House of arithmetic mazes.
  2. Interesting questions.
20 - Exercise
  1. games, puzzles, puzzles and fun puzzles with matchsticks.
21 - Exercise
  1. Find the deleted number.
  2. Remember the numbered essays.
Here are some questions and games that can be used in the above exercises, as well as some interesting examples and puzzles.
  1. I. Interesting issues and questions.
  2. How many kg do you need to make a billion? How many tons do you need? (1000000 kg, 1000 t).
  3. If a person drinks 8 glasses of water every day, how many liters, how many buckets, how many barrels of water will he drink in 50 years?
Note: 1 year - 365 days, 1 bucket - 12 liters, 1 barrel - 40 buckets.
  1. If a person walks 100 meters every day, how many meters does he walk in 50 days? What about 5 years?
  2. The seller sells a 36-meter-long fence to any buyer for 3 meters. How many times did the seller cut the fence? (11 times).
  3. Ahmad drew 7 flowers on the paper. Her sisters, who built it, ask her to give her 1 flower. He has 7 sisters. To fulfill the sisters' request, she took a pair of scissors and cut a sheet of paper into 3 straight lines, leaving 11 pictures of flowers in each section. How did he do it?
  4. A boy went out and found a sum of money on the road. If 2 children came to power, how much money would they have earned? There are 6 apples in the basket. Give these apples to 6 children so that there is 1 apple left in the basket. If 1 km is 1 times larger than 1000 meter, how many times is 50 km larger than 50 meters? (1000 times)
  5. A rabbit weighs 4 kg if it stands on 5 legs, how many kg if it stands on 2 legs? (5 kg).
  6. If 1 stick has 2 ends, how many ends does 1 half stick have?
  7. The birch tree has 8 branches. There are 8 branches on each branch and 1 apples on each branch. How much are all the apples? (There are no apples on the birch tree).
  8. One child divided 20 by 20 to make 88. How did he do it?
-
XX
22
88
  1. As the swimmer divided the number 18 by 2, he slipped out of the flour. How was he?
  2. Multiply the number 666 by 1 and a half times without doing any arithmetic (999. 180 0).
  3. Make 3 out of 4 matchsticks without breaking anything? (IV).
  4. The wolf climbs 1 meters high in 5 day and descends 1 meter. How many days does it take to climb a 10-foot tree? (6 days).
         One form of extracurricular activities was this math morning. These math circles are made from swimmers and taken to the stage.
Do you know?
  1. The ostrich is the largest bird on earth, weighing up to 90 kg.
  2. There are more than 800 different types of insects on the surface.
  3. The shortest man was 2 m 83 cm tall and the shortest man was 42 cm tall.
  4. So far, the heaviest man weighs 404 kg and the lightest man weighs 905 kg.
  5. One bee has to fly 1 meters to collect 300000 kg of honey and land on 9 million flowers.
  6. Philologists point out that the peoples of the earth speak about 2796 languages ​​(excluding the various dialects within several languages).
  7. A billion minutes is more than nine centuries. If we count from the beginning of our era, we will see that in 1902, the billionth minute passed.
  8. It takes more than 95 years to breathe a billion times.
  9. The lifespan of a person under the age of 70 was determined to be about 23 years of sleep, 18 years of talking, 6 years of eating, and 1,5 years of bathing.
Use of humor questions presented in oral quizzes:
  1. What sign between 2 and 3 is a number greater than 2 and less than 3? (comma 2,3).
  2. II. Arithmetic puzzles.
  3. Write the number 5 using 3 37 digits. 37 = 33 + 3 + 3: 3.
  4. Write the number 5 with 9 10 digits and using the arithmetic operation sign. 10 = 99: 9-9: 9.
  5. Write the number 100 using 5 5 5 3 and 5 1 and the action sign.
         100=5*5*5-5-5; 100=111-11; 100=33*3+3:3.  
  1. Write a number whose sum does not exceed 3 and consists of 3 different numbers. 0 + 1 + 2 = 3.
  2. What is the sum of four consecutive numbers equal to 78? 18 + 19 + 20 + 21 = 78.
  3. What is the sum and product of four numbers equal to 8? 1 + 1 + 2 + 4 = 1 * 1 * 2 * 4.
III. Writing numbers using Roman numerals.
  1. X. Olimjon MCМIX born in MCMXLIV died in (1909-1944).
  2. A. Navoi MCDXLI He was born in (1441), MDI Died in (1501).
  3. Oybek MCMV He was born in (1905), MCMLXVIII Died in (1968).
  4. X. X, Niyazi MDCCCLXXXIXX born in MCMXXIX died in (1889-1929).
Bunda M-1000, C-100, D-500, L-50 is equal.
MCMIX - 1909, MCMXLI - 1941, XNUMX - 1998, MDCCCLXXXIX - 1889.
  1. IV. Enter the required numbers instead of asterisks:
1.
+
3 ** 4 *
-
37*02
*
* 2 *
* 43 * 2
** 3 **
  57
112097
  8194
22*8
  ***
8 ***
 
  1. Create a true equation:
***** - **** = 1; 10000-9999 = 1
*** + *** = 1980; 990 + 990 = 1980
Replace the 3.5 * 6 * 7 * 8 asterisks with action signs so that the result is an expression with a value of 39 (5 ​​+ 6 * 7 * 8 = 39).  
  1. V. Working with matchsticks.
  2. Place 3 and 4 matchsticks so that the numbers 4 and 7 are formed. (IV va VII)
  3. Make 5 triangles out of 2 matchsticks.
  4. Make a 9-room house shape out of 2 matchsticks.
  5. How to make the number 4 without breaking from 15 sticks? (XIV).
  6. Replace 1 matchstick from the following incorrect equation so that the result is a true equation?
VI-IV = IX 
  1. a) VI + IV = X; b) V-IX = IX.
A math newspaper in elementary school
         The poster reflects on school life as well as the struggle for knowledge and discipline. Simultaneously with the poster in schools, it is possible to organize children's free time in a fun, boring way and to publish a mathematical newspaper in instilling in them a love for the science of mathematics.
         The names of the newspaper:
         It could be “Young Mathematician,” “Intelligence,” “Read, Calculate, Solve,” “In Bush Times,” and bucks. Particular attention should be paid to making the 1st issue of the newspaper interesting and meaningful. This will help to prepare the next issues of the newspaper in a quality way.
         The mathematical newspaper may contain information about the life and work of great mathematicians, innovations in mathematics, but some theoretical material close to it, some complex, interesting skills, interesting mathematical elements, mathematical tricks, rebuses and games, arithmetic puzzles.
         In addition, materials about students who are actively involved in the mathematical life of the school, the math circle and study with excellent grades, their photos, as well as students who learn mathematics, the typical mistakes in their answers, all the ways to correct these mistakes should be provided.
         For elementary school swimmers, the newspaper should be color-coded, issues and examples should be illustrated, and should be of an interesting nature. The poetic form of the statement is especially appealing to children. It is advisable to involve students in the creation of newspaper tasks and puzzles.
         Various news and information collected for the newspaper are interesting and humorous examples, the results of the contest are given under such headings as "Did you know?", "Find the mistake", "Guess", "Quickly solve".
         Where a mathematical newspaper is not published or there are no complete conditions for its publication, a mathematics section may be established in a class or school newspaper. This section covers math puzzles, rebuses, great examples, and problems. It is advisable that the issues presented in the newspaper are conditionally short and memorable. It is important to ensure that the newspaper is published regularly.
Mathematical excursion
One of the fun extracurricular activities in math is an excursion. Excursions are held in order to connect the school with life, theory and practice, and to acquaint students with the latest science. Math excursions are dedicated to moving games in grades 1 and 2, held outdoors or in the gym. Depending on the conditions around the school, there may be other excursions. Excursions to the house can be organized to determine the size of building materials, to determine the size of the wagon, to determine the size of the rails and other things.
The swimmer is required to carefully prepare the teacher for the tours. It is necessary for the teacher to go to the place of the tour in advance, to instruct the guide on how to explain, to set the time of the tour.
It is important that the content of the tour is clear to the swimmers, they need to know in advance what to do and how to behave. The content of the excursion The teacher should explain the new words that occur to the swimmers before going on the excursion.
During the tour, swimmers record numerical information on these questions, and using this information, swimmers create problems in the classroom and at home. in order to expand and deepen children's geometric knowledge related to land surveying, they can be introduced to the simplest way to determine the heights of buildings, towers, trees.
In addition, autumn will be introduced with the task of estimating, it is recommended to hold mobile and sitting games, entertaining relay games and a great numbering during the excursions in order to occupy the leisure time of swimmers.
Excursion time is 1,5-2 hours during the study period. During the tour there are 15-20 breaks of 2-3 minutes each, and the tour is conducted according to a specific plan, like a lesson. Using information from the field trip, the directory is used for other similar purposes to prepare visual aids for creating tables. At the end of the tour, the necessary conclusions and results are made, and the swimmers are given specific tasks, the tour is concluded.
Math corner
Having a math corner helps with extracurricular activities in math. In the math corner, the results of in-class and out-of-class work are collected. The organization of the math corner is carried out by the swimmer with the active participation of swimmers and parents.
It includes an exhibition of children's notebooks in mathematics, a digital album of information from the newspaper for problem solving, grades, velocities, a set of independently structured problems, models of geometric figures, didactic games, math competitions and math plans, tutorials, reference books, math tables, reference books, list and others.
In addition, in the mathematical corner there will be a beautifully decorated table with examples, examples and assignments for solving various exercises. This allows swimmers to take on and complete new assignments between extracurricular activities. This table is given the mathematical names that attract attention.
The table can be divided into a separate envelope or box for a list of readings, a weekly assignment, and students ’answers. After the deadline, the teacher checks the students' work and evaluates it with the participation of children, writes the results in a table. Errors are analyzed in an extracurricular activity or lesson.
 
 
 
Lecture number 6
Topic: Methods of teaching numbering in flour.
Plan:
  1. The preparatory stage of teaching numbering.
  2. Numerical introduction methodology.
 
Basic terms: number, count, ordinal number, number of items, quantity, multiple, less, more, less, cube, equal, as much, as much, high, low, thickness, length, number, number formation, order, composition, number spelling
         Children's mastery of counting skills in 10, the structure of units of order and counting numbers, the structure of two small numbers, the understanding of the relationship between numbers, the concepts of positive and inverse counting are taught in the curriculum of kindergartens and schools. Therefore, the primary task of the teacher is to determine the levels of mathematical preparation of the children who come to the first grade. Such an examination can be performed before the start of classes, during the admission of children to school or during the first week of school. The following questions can be used to identify and test children's knowledge, skills and abilities:
  1. Can you date? Count?
It is known that according to the kindergarten program, children should get up to 10 dates. Most first graders can count to 10, some to over 10. This is not yet a reason to say that children consciously count. The following questions are used to check the level of consciousness of the counter.
  1. Count these apples, pears, carrots. How many circles are there? (6-8). The swimmer's true answer is something like this. One, two, three, four, five, six. All 6 apples. This swimmer matches the last unspoken number with the total and the swimmer understands. If the child cannot match the last said number with the total, then the child cannot count. In this case, "How many apples are there?" In answering the question, the child may make other mistakes in counting all the objects. For example, they miss one of the items without counting or count it twice.
  2. Take as many pencils as there are on the table (4-5).
  3. Who knows which toys are the most: balls or dolls?
These two questions are aimed at testing the practical skills of comparing two sets of subjects in terms of the number of elements that make them up. Comparing the two sets can be done by the children matching each set element to the second set element (burning on top, burning next to it). For example: by burning one small cube on top of each large cube.
  1. See picture: for example, “Look at the picture made in the fairy tale of the turnip and tell what is between the granddaughter and the cat after the kitten, in front of the puppy. The main task of this exercise is to determine the children's perception of the relationship between the first and second order, "after", "standing in front", between "behind". However, each of the objects, all, one, a few, the same and different quantities, the rest of the quantities left, right, middle, top to bottom, bottom to top, top to bottom, high, low, to compare the size of things, comparison of width, thickness, less, before, after, longer, closer, faster, slower, morning, day, night, evening and other expressions related to the correct understanding of expressions. During the test, it is determined that children can recognize geometric shapes and solve problems. The identified knowledge, skills and abilities of children entering the first grade should be taken into account from the first days of their schooling, with special attention to the shortcomings that some children have for some reason. When studying the first decimal numbers, the preparatory period and the period of acquaintance with the corresponding numbers and numbers are counted.
         The main task of the preparatory period is to identify, supplement and systematize the knowledge, skills and abilities necessary for the transition to the study of numbering. During the preparation, the following exercises are performed:
  1. 1. Count objects, sounds and movements.
 The first exercises should be about the objects in the classroom, such as doors, windows, desks, girls in a row, boys, and counting exercises. But these objects cannot be thrown into the ashes. In performing such exercises, the building organ works. Therefore, small objects (pencils, counting sticks, toys) can be used for counting. In the process of counting, the children were asked "how much" of different information, if possible? with suzi kuprok questions are practiced to burn. During the counting exercises, it is important to explain how many items are in the group where the last number in the count is counted. Counting objects from right to left or from left to right, from bottom to top or from top to bottom does not change the counting result. In subject counting lessons, children can be taught to count objects one, two, or five.
  1. 2. Comparison and equalization of two sets by the number of elements that make them up.
In the process of performing the exercises, it is necessary to explain the meaning of the relationship of big (more, more), less (less), equal (as much). This can be done by doing a couple of practical exercises to compare groups of subjects. For example, to compare groups of large and small cubes, we place one small cube on top of each large cube. If a large cube is unpaired, then the large cubes are overpaid. The following exercises can be used for comparison:
a) Pour a few squares on the counter. Burn as many circles without counting the squares. How can this be done?
b) The package contains nuts and candies. How do you know if there are nuts or candies in the package?
A good way to compare the two sets in this exercise is to take one piece of candy from the package and burn it in a row, and put one nut in each candy and pour it into the second row. This work is continued until the nuts or candies are left unpaired. In performing such exercises, it is important to consider the relationship of excess and deficiency in relation to each other. When developing the ability to compare two groups of objects, children should be taught to determine how many objects are more (or less) in one of the groups being compared and to solve the problem in two ways by equalizing the number of objects in both groups (adding or subtracting). allows to compile the concepts of comparison of numbers, develops children's mathematical speech.
  1. Ordinal relations and ordinal values ​​of numbers.
Children were more likely to encounter disciplinary relationships (… standing in front, standing in between, coming from behind) in their pre-school experiences. At school, a variety of didactic materials can be used to supplement and systematize children's knowledge of disciplinary relations. Children can be asked such questions from the 7 pictures on page 2 of the textbook. What's going on ahead? You can ask questions about the order values ​​in Figure 3. How old is Kuzichok? What's going on in the first place? How many camels are there? What is going on in the third place? In certain lessons of the preparatory period (pages 6, 8, 9) exercises are performed to determine the spatial relationships (left, right, high, below, above, below, high, low, wide, narrow).
  1. Preparing to learn addition and subtraction.
In order to prepare children for the addition and subtraction operations, practical exercises are performed to combine the two sets and to separate the part of the set. For example: Nodira's sister drew a picture of 3 green leaves and 4 yellow leaves. How many leaf pictures are there in Nodira?
  1. Preparing to write a number.
Exercises related to drawing pictures of borders allow you to prepare for writing numbers. Such exercises are given on each page of the 1st grade textbook. By doing these exercises, swimmers learn to hold the pen correctly, draw a line, and place a note on a page. During the preparation children are introduced to notebooks, textbooks, didactic materials, rulers. The first topic in 1st grade math in the program is numbering the first decimal place. This topic is to develop children's numeracy skills, to form in them an idea of ​​the first ten numbers, to establish the ability to match the number with its name, nomenclature, print and written designation with the help of numbers.
It consists of acquainting swimmers with some properties of the natural series of numbers, the composition of numbers. The following questions can be used to get acquainted with each number of the first ten according to these tasks.
  1. How can this or that number be formed? Each number in the first decimal must be formed by adding one to the number before it and subtracting one from the number after it. This allows swimmers to combine the sequence of numbers in ascending and descending order, while the first decimal places can be two-digit or separate units.
  2. What is a number called and how is it written in print and written numbers? The children are first introduced to the print number. They are installed and burned next to the appropriate sets of objects. Writing a number that corresponds to the number being taught is taught in the lesson in question. Examples of writing numbers are given on the relevant pages of the textbook.
  3. What place does a given number occupy in the natural series of numbers? Children are taught to find answers to the following questions: What number occurs after a given number, which number comes before it, what is the place of the given number in the number line, what numbers precede it in the count, and what numbers occur after it? For example: Say the number that comes after the number 4. What is the number between 4 and 6 in a row?
  4. What is the relationship between a given number and the numbers added to it by a series of numbers? These relationships are defined in records that are executed using relationship symbols (<,>, =). A given number is more than the number before it and less than the number after it. Children are taught that the number in question is greater than all the numbers that come before it and less than the number that comes after it.
For example:
  • Compare the given numbers and burn the <,>, = characters as needed.
6*9   5*4   8*8  
  • Read the notes and write the numbers instead of the cells so that the correct entry is formed:
1<4     1>5     6=1      4+1>1     3+1>3-1
  • Correct incorrect entries.
8<9     7<5     6=4
The main issues above will be addressed in the introduction to each issue. In getting acquainted with the first numbers of the natural string, swimmers first work with the surrounding objects and their images (For example: cards with circles, sticks, apples, cars and other things). In the acquaintance of large numbers with numbers 6, 7, 8, 9, 10 there is a gradual transition from natural and image representation to abstract forms, to the use of numerical ladders. When learning the first decimal numbers, the contents of those numbers are taught. Didactic materials, pictures, various tables can be used to show different aspects of the composition of numbers.
 
Games such as "Find", "Relay", "Arithmetic maze" can be used to consolidate and repeat the composition of numbers. For example, when conducting a “Find a House” game, children are asked to find out how the number 7 can be formed from two conjunctions. The swimmer with the most points is declared the winner.
After getting acquainted with the numbers 1-10, the children get acquainted with the number 0 and the number 0 used to write it. This can be taught as follows. Pour 3 horse counts into the pan. Take a chup. How many chupi are left? (2) We write this as 3-1 = 2. Get another one. How many chups are left? (1). We write this as 2-1 = 1. Get another one. How many chups are left? Not a single one was left. Writing 1-1 = 0 indicates that the result of the last example is not a single chup, that is, if there is nothing left in our ashes, on the table, in the bowl, it is said to write a number called 0 and the number 0 to denote it. Then the number 0 is compared with the number 1 and it is said that 0 is less than 1, that is, any number is smaller than the number that comes after it, and it is taught to write 0 <1. Then swimmers are taught to conclude that the number "0" must be ahead of 1 in the sequence of numbers. This means that as a result of learning to number numbers within 10, swimmers should acquire the following knowledge, skills, and abilities.
1) Carefully combine the names, the sequence (in reverse and reverse order) of the numbers 1-10. Teach them to read and write correctly.
2) Know the position of any number in the sequence of numbers.
3) Compare numbers and make appropriate entries using the <,>, = symbols.
4) Thorough knowledge of the composition of numbers.
Control questions:
  1. What questions are initially used to study the numbers in flour?
  2. At what stage is the numbering in flour taught?
  3. What concepts are used in the preparatory phase of learning to number numbers?
  4. How is the number presented?
  5. How many numbers are involved in numbering?
  6. How is each of the unta numbers formed?
  7. What didactic games are used to study the composition of numbers with two additions?
  8. What is the order of numbers?
  9. How to enter the number zero?
  10. How to compare numbers?       
Lecture number 7
Topic: Methods of learning to number numbers on the face.
Plan:
  1. Verbal numbering of numbers.
  2. Written numbering of numbers.
 
Basic terms: number, number, numbering, oral, written, number of rooms, two-digit number, first room units, second room units, vowel composition, first and second vowels, place value of numbers.
When learning to number numbers on the face, swimmers become acquainted with the new unit of decimal and the important concept of the decimal number system - the concept of room. The name and writing of the principles of formation of two-digit numbers, oral and written numbering of numbers are the basis of coordination. The teacher's task in learning to number numbers on the face is to teach children to count objects individually and in groups, to teach children to read and write numbers on the face, to determine which units are written from right to left (room I units), decimals (room II units). to show how to determine the load, to achieve concepts and terms such as first and second room units, room number, sum of room additions, one- and two-digit numbers that swimmers can master. There are two stages in the numbering process: numbering numbers 11-20 and numbers 21-100. The numbering of two-digit numbers up to 20 (11-20) and two-digit numbers larger than 20 (21-100) is similar, the oral and written numbering of these numbers is based on the principle of grouping units in numbers and the place values ​​of numbers in writing numbers. Therefore, the process of assigning the decimal composition of the second decimal numbers and writing these numbers serves as a preparatory stage for assigning numbers within a hundred. Separating the second decimal in the study of numbering allows you to better understand the decimal composition of numbers and the principle of the place value of numbers. Introduction to numbers within 20 and then within 100 is done according to this plan. Before a) preparation; b) verbal; c) written numbering is taught. The work on the study of the second decimal number, that is, the preparatory work is carried out in the repetition of the topic "Decimal". It is shown to children that it is not enough to know the first decimal, ie the numbers from 1 to 10, and that it is necessary to count numbers greater than 10. This includes exercises for counting objects by decimals. For example: How many swimmers are in the first row of a class? What about the second row? How many swimmers are there in the class? Exercises for counting a group of objects (how many pairs of children are next to the board?) Are also included. In the same way, you can count the pieces of chup in pairs, three, five, and the buttons on the cardboard can be counted in pairs, five, five. As an example, you can use exercises to say the name of the second decimal number: Which number is said after the number 4 in the counter? After the number 40? Which number is said before the number 7? What about the number 17? What number is formed by adding 20 to 1? Such exercises convince swimmers that in addition to the first decimal numbers, there are numbers, that there are many, that there is a certain similarity between the numbers that are familiar to children in the order of their occurrence in the order of designation. For example: I class 94, pictures on page 95.
Learning to verbally number the second decimal number begins with developing an understanding of flour in children. The children try to make a flour by tying the sticks into 10 pieces. (Page 94, Fig. 1). Then, by doing the counting exercises of the flour using the chups, adding and subtracting the flours, the children become convinced that they can add and subtract the flour as well as the ones (p. 94, Fig. 3). Then the formation of numbers from 11 to 20 from flours and ones, their name is taught.
Teacher: How do you get the number that comes after 9 in the count?
Swimmer: 9 to 1 should be added.        
Teacher: Add 9 chup to 1 chups, how many chups are there?
Swimmer: 10 chup or one untalik.        
Teacher: How do you get the number that comes after 10 in the count?
Swimmer: You need to add one to 10.        
Teacher: Tie one flour and burn another chup. What is the total number of chups?
Swimmer: 11 chup.
Teacher: How many flours and how many separate chups did you have in total?
Swimmer: 1 flour and again buta chup.
Teacher: So how many tens and how many in the number 11?
Swimmer: The number 11 has 1 vowel and one vowel.
9+1=10
10+1=11
11 = 1 un 1 a
10 = 1 flour
1 un + 1 = 11
1 un 1 bir = 11
The work on the next numbers is carried out in a similar way, that is, the formation of other numbers in the second decimal, as well as the order of their arrival in the count. As a guide, in addition to the sticks, strips with 10 circles each are used. This instruction includes exercises to strengthen the knowledge of the phonetic structure of numbers, based on and without reference to the manuals:
  1. Count to 15 chups. Determine how many flours and how many separate chups there are?
  2. Separate 1 flour chup and 4 horse chup. How many chups were taken in total?
  3. How many vowels and ones are there in number 18?
  4. What number consists of 1 decimal and 9 units?
  5. Burn 12 chups, burn one (20-25) chups next to it and say how many chups there are?
  6. Sing 17 chups, go aerata one by one from them. (7-8) and how many chups are left?
  7. Subtract one by one from 20 to 10.
Written numbering
Written numbering of numbers greater than 10 is based on the grouping of units into vowels and the application of the principle of place values ​​of numbers: when counting from right to left, units are written in the first place, decimals in the second place. An abacus is used to explain the proper principle of writing two-digit numbers, which is a table with two rows of pockets, one row for chups and the other row for chirma numbers.
The teacher shows the students how to put 5, 6, 8, 11, 10, 15 in the above pockets, and then tells the students to burn 17 sticks in the pockets.
Teacher: How many chups are there in all?
Swimmer: And seven.        
Teacher: How many tens?
Swimmer: Bitta.        
Teacher: Let's denote this by a number. (Burns number 1 to the lower left pocket). How many units are there in the number 17? Let's denote this by a number. (The bottom flour burns the number 7 in the pocket). The number 17 is written. What does the first 7 digits on the right mean?
Swimmer: Seven units.
Teacher: What does the number 1 in the second place mean?
Swimmer: One flour.
Several similar numbers are constructed. Then the children write the numbers in their notebooks on tables with “decimals” and “units” and explain the value of each number. The spelling of the numbers 20, 10 is taught separately. The number (1, 2) indicates that the number has 1, 2 vowels, and the number 0 indicates that the number has no unit. To strengthen the skills of writing numbers, an individual guide is used, ie a table, in which verbal numbering is repeated. For example: Specify the number 17. How many decimals and how many units are there in this number? Specify the number that comes before the number 18 that comes after the number 13? It is taught to write more than 15 numbers 1, to solve examples 12 + 1, 18-1 and to write the answer, to explain how to find the result. The explanation of 12 + 1 is as follows. If we add 12 to 1, we get 13, because if we add 1 to the number, we get the next number in the count. As swimmers compare numbers, they see that one number (one character) is needed to write numbers consisting of units, and two numbers (two characters) are needed to write numbers consisting of decimals and ones.
One-digit and two-digit number terms are introduced. Exercises on distinguishing one-digit and two-digit numbers are performed.
  1. Write one-digit, then two-digit numbers before this sequence of numbers.
2, 13, 8, 17, 15, 6, 11, 10           
  1. Write 4 arbitrary one-digit numbers and multiply each number by 10, what numbers are formed? What can you call them.
  2. Using numbers 1 and 2, write first one-digit numbers and then two-digit numbers.
  3. Just use the number 2 itself and write the two-digit mon. 2, 22.
Learning to number numbers in a hundred is done according to a plan, as in 20 ogzaki, sungra written numbering is taught and the numbering of numbers within 20 is done in the learned order:
  1. Number of decimals 10, 20, 30, 40, 50,… Formation and naming of numbers.
  2. The formation of numbers from decimals and units. The vowel composition of two-digit numbers, the natural sequence of numbers within 100.
  3. Written numbering, writing and reading of two-digit numbers, first and second room units.
  4. Addition and subtraction methods based on knowing the numbering of numbers.
  5. replacing a two-digit number with the sum of the room numbers.
So, the method of numbering the numbers inside the face is similar to the method of teaching the numbering of numbers inside 20. In this case, the composition of the room and room numbers is a novelty. The first room units, the second room units are introduced in practice to analyze the flour content of numbers. For example: 56 has 5 vowels and 6 ones. It can be said differently: the number 56 consists of 1 units of 6 room and 2 units of 4 rooms. To understand the concept of room number, cards with numbers such as 1, 2, 3,… 9, 10, 20, 30, soni 90 are used. Using these cards, they can mark any two-digit number. For example: 6 cards are formed from cards with the numbers 20 and 26. The reverse assignment can also be given. Which room numbers are 18 and 81, 43 and 34? 10, 8,… 18. this practical work done with cards helps to represent any number in the form of the sum of the joins of the room moments. 97 = 90 + 7, 80 + 5 = 85. the swimmers ’knowledge of numbering is then consolidated as they learn to add and subtract within 100. As a result of learning to number numbers on the face, swimmers should acquire the following knowledge, skills, and abilities.
  1. Matching the names of numbers in a face to understand how they are formed from decimals and units.
  2. Know the order of arrival of numbers in the counter. Be able to compare numbers based on knowledge of the positions of numbers in a natural sequence, as well as knowledge of the vowel structure of numbers.
  3. Write and read numbers in the face, learn how to write units (room I units) and decimals (room units II) when counting from right to left.
  4. Know how to add and subtract numbers based on knowledge of natural sequences. To be able to add and subtract numbers on the basis of the vowel components of numbers, to acquire the ability to replace numbers with the sum of room additions, depending on the pattern, without using the term room sum.
Control questions:       
  1. How many steps does it take to learn to number numbers on a face?
  2. How to verbally number the numbers on the face?
  3. Do you have a written numbering?
  4. Writing the numbers on the face is subject to the Canadian procedure?
  5. How is the comparison of the numbers inside the face done?
  6. How many hundreds, how many units are there in 25?
  7. Which number consists of 3 decimals and 7 ones?
Lecture number 8
Topic: Numbering numbers in thousands.
Plan:
  1. Preparatory work for learning numbering.
  2. The new unit of account is the introduction of the thousand.
  3. Verbal numbering.
  4. Written numbering.
 
Basic terms: numbering, series of numbers, thousandths, oral, written numbering, three-digit number, number, number, third room, sequence, vowel structure of a number.
The teacher's task in learning to number numbers in thousands is to teach children the following.
  1. Count the objects one by one, in pairs, and in groups of hundreds.
  2. Know how to read and write numbers in thousands and how they come in natural order.
  3. Be able to form numbers from hundreds, tens, and ones.
  4. Determine to which units, decimals, and hundreds are written from right to left.
  5. Express the number as the sum of the room additions and find the total number of any room unit in the given number.
The work on the verbal numbering of numbers in a thousand can be divided into several stages:
  1. I. Preparatory work.
The main task of this step is to repeat the part of the numbering material within 100 that helps to number numbers within 1000. For this purpose, swimmers can be offered approximately such exercises.
  1. Say the numbers in order from 18 to 23, 36 to 45, 77 to 89, respectively.
  2. Say 4-5 more numbers in each row: 76, 77, 78,… 45, 46, 47,… 20, 30, 40,….
  3. Say a number consisting of 3 units of 3 decimal places. Say the previous number. How to form the next number? How many numbers do you need to write this number? What number 83 can be represented by the sum of the room additions?
  4. What numbers 79, 85, 92 stand between the numbers?
  5. Write a number consisting of 5 units of 4 units and 8 units of 0 units.
  6. How many different numbers are 62, 44, 70?
  7. II. Introduce swimmers to the new unit of counting - the thousand.
This introductory tutorial can be done using 10 bundles of chups and a bunch of chups (10 separate chups, a bunch of 9 chups with 100 chups in each bundle) with 9 chups in each. This is how you can start the introduction of a new counting unit hundred. Numbers 1 to 10 are counted separately, and 10 chups are tied together with a rubber band. Next to the 9 bunches of flour chups, 1 bunch of flour is burned and 10 bunches of flour are formed into 1 bunch of flour, 2 bunches of flour… 10 bunches. How can you count how many units there are in all of these stacks (flour, twenty, thirty, yuz. Hundred). Then 10 bonds of flour are connected with rubber as a bond - a hundred, and hundreds of counts are made by tying: 1 hundred - a hundred, 2 hundred - two hundred,… 10 hundred - a thousand is explained and thousands can be counted. (III class - 27 pages).
III. Verbal numbering.
The next step in learning verbal numbering is to introduce swimmers to the numbers in the natural range from 100 to 1000. In the previous step, the children were introduced to three-digit numbers ending in zeros and 1000 in the following order: 100… 200… 300… 400… 500… 600… 700… 800… 900…. Now it is necessary to fill in the gap between every two three-digit numbers ending in zeros, that is, to fill the natural series of numbers from 100 to 1000. For this purpose, first of all, how to form each number in the next row in the row, and how many more than the previous one, it is repeated by doing a few exercises with the children. The following exercises can be used to create and reinforce ideas about the natural sequence of numbers from 1 to 1000:
  1. Count from 335 to 405, from 768 to 786, from 992 to 1000, one by one.
  2. Count from 800 to 789, from 400 to 375, from 421 to 40, from 1000 to 985 one by one.
  3. What numbers are between 293 and 315, between 576 and 566?
  4. How many numbers are there between 300, 400, 700-800, 100-1000?
  5. IV. At this stage, the vowel components of three-digit numbers, that is, their formation from hundreds, tens, units, are taught. For this purpose, instructions - guides are used chuplar, chuplar handle (class III, page 29). They describe numbers consisting of room numbers using instruction manuals. For example: 3 faces with 5 units of 2 units, 7 units with 9 units of XNUMX units were fired.
Reverse Exercises - Indicate how many hundreds, tens, and ones are in the said numbers. Numbers of units, tens or numbers in the room of both units at the same time are much more difficult for swimmers. An index is used to look at these numbers. 601, 705, 560….
  1. V. Exercises involving the replacement of numbers expressed in large units with numbers expressed in smaller units also helped to harmonize the vowel compositions of three-digit numbers. The following exercises are performed:
    • 2 м necha cmga teng? 3 мwhat about
    • 800 cm how many meters
At this stage, children should be taught to determine the total number of units in a given three-digit number, the total number of decimals. Written numbering: In order to prepare for the study of written numbering of three-digit numbers, the problems of written numbering of two-digit numbers are repeated: "number", the meanings of number terms, the differences between them, the role of numbers in writing numbers. Emphasis is placed on the use of zeros in writing numbers. Here the children are introduced to the first room units, which are familiar to them, based on the concepts of the first room units, the second room units, and the new concept is the third room units. The units of room II are called) the hundreds are written in the third place (these are called the units of room III) and then it is understood how to write the number 1000. The following exercises will strengthen your knowledge of written numbering
  1. Explain how the number three hundred and one hundred and ten are written and why they are written that way.
  2. Write all of the numbers lying between 696 and 703
  3. Write all three-digit numbers that can be written using the numbers 5,7,9, use each number only once to write each number.
  4. What does the number 635,67,306,666 mean when you write these numbers 6.
  5. 7 How many numbers and numbers do you need to write the numbers 1 and 701, 333 and 33, 500 and 501, 600, 601, 610, 160?
         As a result of learning to number numbers within 1000, swimmers should acquire the following knowledge and skills
  1. Know the names of numbers in 1000, how to form each successive number in a series of numbers, how much is greater than the number that precedes it, and how much is less than the number that comes after it.
  2. know the position of each number in the sequence of numbers.
  3. Be able to read and write knowing the place value of numbers.
  4. Using numbers to know the contents of a room, be able to compare two numbers according to their positions in the number series.
  5. get the number to replace its khan with the sum of its additions.
  6. Addition and subtraction of numbers based on knowledge of the natural sequence of numbers and the composition of flour.
  7. Three-digit number know the terms of the third room units.
Control questions:
  1. How many steps are used to number numbers in a thousand?
  2. What is the position of units, tens, and hundreds in three-digit numbers from right to left?
  3. How to read a number with three digits, knowing the numerical values ​​of the number?
  4. How is voice numbering done?
  5. How is written numbering done?
  6. What is the purpose of teaching you to count to hundreds?
  7. What is the purpose of a set of cards with numbers?
  8. What is being done in preparation for the numbering of thousands?
Lecture №9
Topic: Methods of studying the numbering of multi-digit numbers.
Plan:
  1. The preparatory stage of teaching numbering.
  2. Introduce the concept of class.
  3. Introduce 6-digit numbers to form, read, and write.
  4. Strengthen the knowledge and skills of swimmers.
 
Basic terms: number, room, number of rooms, concept of class, class of ones, thousands, millions, multi-digit number, sum of room additions.
   
The main task of the teacher in solving the numbering of multi-digit numbers is to reveal the essence of the concept of a new unit of number - the concept of thousandths and on this basis to teach children to read and write multi-digit numbers, to determine their knowledge of the natural sequence, generalization. Oral and written learning of numbering of multi-digit numbers can be divided into several stages.
  1. I. Preparatory work.
The task of this step is to repeat the basic problems of numbering one-, two-, and three-digit numbers. For this purpose, a system of exercises developed in class III is used.
  1. Say the number that comes after each of the numbers 28, 90, 999.
  2. Count from 25 to 32, from 243 to 251, from 987 to 1000. Count from 30 to 90, from 250 to 340.
  3. Read the numbers: 426, 803, 600, 111, 999, 1000, 528, 808. How many units, decimals, hundreds are there in each of these numbers?
  4. Write the following numbers. 9 faces 5 flour 6 units, 8 faces 4 units, 5 faces 9 flour 7 units.
  5. How many hundreds, tens, units are there in a thousand?
  6. Write all three-digit numbers that can be used using the numbers 1, 3, 4. Express these numbers as the sum of the room additions.
The following questions are also available.
a) How many units are there in a flour?
c) How many tens are there in one hundred?
g) How many times larger than the decimal unit?
d) How many times less than a hundredth of a tenth?
It is also possible to repeat the natural sequence of numbers 1-1000. From 200 numbers, add and subtract, 50, 100, add and subtract. In the count, say the number that comes after the number 399, the number that comes before the number 600. When repeating the numbering in a thousand, children are introduced to the representation of numbers in chut.
  1. II. Learning to number.
This stage consists of acquainting children with the first class - the class of units and the second class with the class of migils, their structures, the names of the rooms of each class. It is also important to make children aware of how upper class room units are formed from lower class room units. In this case, the table of rooms and classrooms is the main instructional tool. The explanation begins with a repetition of how the teaching work is formed. Therefore, children can be asked to count, for example, from 995. The teacher replaces 10 chut pieces on the III wire with hundreds, and one piece on the IV wire - a thousand. Calculations are made in thousands and tens of thousands are generated. Calculations are made in tens of thousands. Calculations are made by replacing 10 tens of thousands with hundreds of thousands, and finally 10 hundred thousand is replaced by millions, then the formation of the class of units, tens, and hundreds of units is taught using the table of thousands, tens of thousands, hundreds of thousands.
III. Introduction to the formation, reading and writing of second grade numbers.
In this case, the table of rooms and classrooms with chutes will be a visual guide. Teaching can begin with brushing numbers. First brush first class numbers (for example: 5, 25, 375…). Then the numbers of class II (for example: 3 thousand, 43 thousand, 543 thousand… 900 thousand) are added. Swimmers' attention is drawn to the notation of the numbers in the table (at the end, three zeros indicate the absence of first-class units), then the number of digits in the number is determined by the position of the upper cell of these numbers. For example: in the number 47000, the upper room is in the 5th place. This means that this number consists of 5 digits and it is taught that it has five digits. Hence: Class II numbers are formed from thousands, just as Class I numbers are formed from units. When reading the numbers of the second grade, the word "thousand" is added, and in writing it is written in the class of thousands, that is, from the right to the left, in the fourth, fifth and sixth places with numbers.
  1. IV. Introduction to the formation, reading and writing of six-digit numbers.
At this stage, the numbering table with chups was the main guide. Using a set of numbers, we determine a number that is familiar from the numbering table. For example: we set the number 257000, then we put the given number from the right to the first zero, for example, a 4-digit card. The number 257004 is formed. By doing this, we get two more numbers, for example, 257084, 257684. Several more numbers are assigned to the numbering table. Children learn to read them correctly and write numbers without a table, first with the help of a teacher and then independently. In this case, one class is separated from the second class by a small interval, and then it is proposed to perform reverse exercises, ie exercises to replace a multi-digit number with the sum of the numbers of classes I and II. 24605 = 24000 + 600 + 5.
  1. V. Strengthening the knowledge and skills of swimmers.
These include reading and writing multi-digit numbers, comparing numbers, replacing multi-digit numbers with the sum of room additions, multiplying numbers by 10, 100, 1000 times, and subtracting numbers ending in zeros by 10, 100, 1000 times. units, exercises for finding the total number of units, decimals, hundreds of given multi-digit numbers for the conversion of large units into small units.
For example:
  1. Write the numbers below with numbers. Four hundred and sixty, four thousand and one units, III 420 units of class, II 5 units of class, I 56 units of class.
  2. Compare the numbers: 20007 and 200007; 6004 and 5030.
  3. Write a single number that comes directly after the number 699997, 50089, before the number 600801, 300100.
  4. Name the neighbors of the following numbers: 20000, 50000, 800000.
  5. Describe the following numbers as the sum of the room numbers: 8506, 2500, 4897, 98001.
  6. Decrease the number 268000 by 100 times, increase 800 by 10 times.
In performing these exercises, swimmers rely on knowing the place values ​​of numbers in writing numbers.
  1. Write the numbers: 2815, 5182, 8125, how many tens are there in each of them? How many thousands are there in each of these?
  2. Express in larger units: 7031 cm, 842 dm, 340 м.
  3. Express in smaller units: 25 м 60 cm, 5 ton, 8 kg.
  4. VI. Introduction to the formation of the class of millions.
At this stage, swimmers practice reading and writing 7-9-digit numbers. A new class of numbers is introduced in the same way as an acquaintance with a class of millions is introduced in a class of thousands. It focuses on the numbering of 4-6-digit numbers: the formation of the next upper room unit from 10 units of the lower room, the ability to multiply and read numbers, the table of rooms and classes to write numbers, to write numbers without this table, the value of numbers in writing numbers. , know the room content of numbers and…
As a result of learning to number multi-digit numbers, swimmers:
  1. Within a class of millions, they must be able to match the names of natural numbers, understand how they are formed, and know their phonetic composition.
  2. You need to know the names of the classes and the rooms within each class.
  3. Within a class of millions, every Canadian must be able to read and write numbers.
  4. They should be able to compare numbers.
  5. To be able to describe any number as the sum of room additions, to find the given number of units, decimals and… total number, to replace small units with large units and vice versa, large units with small units, to increase numbers by 10, 100, 1000 times and to end with zeros they must be able to reduce the numbers by 10, 100, 1000 times.
Control questions:
  1. The preparation phase for digitizing multi-digit numbers puts Canadian goals in front of you?
  2. The concept of class is introduced in Canada?
  3. How many room units are there in a class?
  4. Say the room names of the one class.
  5. How many rooms are there in a class of thousands?
  6. How is the comparison of multi-digit numbers done?
  7. What is meant by room addicts?
  8. When studying multi-digit numbers, do you pay attention to the value of the numbers?
 
Lecture number 10
Topic: Methods of studying and calculating arithmetic operations in primary school.
Plan:
  1. Preparatory stage. › ± 1 Addition and subtraction points.
  2. › ± 2, › ± 3, › ± 4 Addition and subtraction cases.
  3. › + 5, XNUMX, › + 6, XNUMX, › + 7, XNUMX, › + 8, XNUMX, › + 9 Addition points.
  4. › - 2, › - 2, › - 2, › - 2, › - Introduction to calculation methods for multiplication cases of type 2.
 
Basic terms: calculation, addition, subtraction, composition of numbers, parts of a number, sum, addition, subtraction, subtraction, subtraction, place, law of substitution, the relationship between the limits and results of the operation of addition, multiplication of numbers.
One of the areas of the mathematics program is the development of oral and written calculation skills in primary school students. Before learning arithmetic, it is necessary to convey its meaning to the minds of children. This work is carried out on the basis of practical work with different sets of subjects. Introduction of the student to the meaning of addition and subtraction is carried out on the basis of practical operations, such as the separation of parts of a given set from the combination of elements of two sets. The study of the practice of multiplication is limited to the practical combination of several sets of equal numbers. The study of the relationships between its components and the result is the basis for the study of the division. For the conscious mastery of different (oral and written) methods of calculation, the program provides an introduction to some important properties of arithmetic operations and their consequences. For example, in Grade I, when learning to add and subtract within 10, children become familiar with the substitution property of addition. In the study of addition and subtraction within 100, they learn how to add and subtract a number, how to subtract a number from a sum, and how to subtract a sum from a sum. Learned properties and rules allow simplification of calculations. For example: the method of swapping positions makes it easier for them to calculate 3 + 6, 2 + 8. In addition to learning the properties of arithmetic operations, the program aims to acquaint children with the existing connections between arithmetic operations and the relationship between operation limits and their results. All this knowledge is used in calculations and verification of the correctness of operations. For example, based on the knowledge of the connections between the components and the result of the multiplication operation, on the basis of each multiplication point they form the corresponding divisions: if 6 * 4 = 24, then 24: 6 = 4, 24: 4 = 6. The next issues in the study of arithmetic are related to the formation of calculation skills in swimmers based on the conscious use of oral and written calculation methods. The basic skills of verbal calculation are formed in I and II classes. In II, III classes work on written calculations will begin. At the same time, the skills of oral calculations in written calculations are improving, as oral calculations are an integral part of the process of written calculations. Having oral calculation skills ensures successful performance of written calculations. Oral calculation methods and written calculation methods are based on knowledge of the properties of actions and the relationship between the components of the action and the results of the resulting results.
       Oral calculations:
1. Calculations can be explained with records without records (that is, performed in the brain).
a) explanations can be given in full (ie at the initial stage of consolidation of the calculation method). 9 + 5 = 9 + (1 + 4) = (9 + 1) + 4 = 10 + 4 = 14
43+5=(40+3)+5=40+(3+5)=40+8=48.
b). It is possible to write the turns and results: 43 + 5 = 48. 9 + 5 = 14.
V). The calculation results can be numbered. 1). 14, 2) 48.
  1. Calculations are performed from the upper room units.
Масалан: 470-320=(400+70)-(300+20)=(400-300)+(70-20)=100+50=150.
  1. Intermediate results are stored in memory.
4. Calculations can be done in different ways.
Масалан: 26*12=26*(10+2)=26*10+26*2=260+56=312.
26*12=(20+6)*12=20*12+6*12=240+72=312.
26*12=26*(3*4)=(26*3)*4=78*4=312.
5. Operations are performed between 10 and 100,1000 and on some multi-horned numbers using verbal methods of calculations. 50020: 5 = 1004. 54024: 6 = 9004. 630045: 9 = 7005.
Some examples can be solved orally or in writing. In these cases, students compare solutions and understand the content of arithmetic operations and operations on numbers. In the teaching process, using a variety of methods to perform a large number of exercises of the nature of arithmetic operations to calculate the inadequacy of the table points.
 
Methods of teaching addition and subtraction of numbers on the topic of decimals.
The main goals of the teacher in working on this topic are:
  1. Introduce swimmers to the content of addition and subtraction,
  2. Ensuring teachers' conscious use of calculation methods.
a) The method of addition and subtraction of a number by parts.
b) The method of adding two numbers using the substitution property of the sum.
c) A method of subtraction based on the knowledge of the relationship between the sum and the addendum, using the skill of finding the second addition to the arc sum and one of the additions from knowing the appropriate state of addition in dividing numbers.
  1. To automate the skills of learning to add and subtract in flour. The work of learning to add and subtract in flour can be divided into several interrelated stages.
IStage: Preparation stage:
Disclosure of the thematic content of addition and subtraction: cases of addition and subtraction in the form a ± 1.
The work on revealing the thematic content of addition and subtraction begins in the first lessons devoted to the study of numbers 1-10. During this time, the children perform a series of exercises to combine the two sets and to separate the part of the set. In the numbering process, children were told that each number in the first decimal was formed by adding the number before it or by subtracting one from the number after it. This allows children to adjust the order of the numbers in ascending order. In the first lesson on addition and subtraction in 10, we need to summarize the knowledge that children have learned from learning the numbers 1-10, and conclude that when we add one to a number, we get the next number in the count, and when we subtract one number, we get the previous number in the row. Tables are created for +1, -1 cases, and these tables should be understood and memorized by children. Addition and subtraction in the form 1-1 = 0 and 0 + 1 = 1 are considered on the basis of indications.
II stage: + 2, + 3, Get acquainted with the calculation methods for +4 cases.
      Work on each of the children is carried out according to the same plan.
1) Preparation. In this case, the corresponding cases of the composition of numbers consisting of two adders and the table points of addition and subtraction are repeated.
For example: Before hitting +4  +1, + 2, +3 points are repeated.
2) Introduction to the appropriate method of calculation (ie by adding and subtracting numbers by parts).
3) Consolidation of new knowledge and application of this knowledge in different situations.
4) Work on the conscious assignment and memorization of table points corresponding to the composition of numbers and the corresponding cases of subtraction.
         Let's look at ?? + 2 and ?? - 2 of them. In preparation for this study, swimmers should be introduced to examples of addition and subtraction that require them to add 1 to 2 times. For example: 4 red circles are preceded by one blue circle and then another yellow circle. To calculate these circles, 4 is preceded by 1, then the second 1 is added, and they also give the intermediate results. If we add one to five, we get 6. If we add 6 to 1, we get 7, or in short, 5 plus 6, 6 plus 1 is equal to 7. Subtraction is also taught as follows: 4 - 1 = 3; 3 - 1 = 2.
         From the preparation it is necessary to introduce the methods of sung ?? + 2, ?? - 2. 4 + 2 = 6, 4 + 1 + 1, 4 + 1 = 5, 5 + 1 = 6. This is explained on the basis of an incomplete instruction. The swimmer had 4 postcards. (Puts 4 postcards in an envelope) He was given two more postcards, how much was his postcard? Guess how to add these 2 postcards to the previous 4 postcards? We add 4 to 1; There will be 5. Then how many more cards will we add: 1 + 5 = 1.
Conclusion To add 2, you can add 2 to 1 and then XNUMX to the resulting number. Note in the notebook:
4+2=6
4-2 2 =
4 + 1 + 1
4-1-1
4+1=5
4-1 3 =
5+1=6
3-1 2 =
Here, swimmers need to be taught to use the knowledge they have acquired to master the appropriate composition of numbers.
 
For example:    
4+2=6
6 is 4 and again 2
5+2=7
7 is 5 and again 2
7+2=9
9 is 7 and again 2
A table of sung ?? ± 2 is formed from several lessons
1 + 2 3-2
2 + 2 4-2
3 + 2 5-2
4 + 2 6-2
5 + 2 7-2
6 + 2 8-2
7 + 2 9-2
8 + 2 10-2
Once the table has been drawn up, the practice of adding swimmers is introduced with the names of the components and results, the numbers that are added are called the adders, and the result is called the sum.
For ± ± 3, ?? ± 4 cases, the calculation methods are taught according to the following plan:
4 + 3
6-3
6-3
4 + 3
4 + 2 + 1
6-1-2
6-2-1
4 + 1 + 2
4+2=6
6-1 5 =
6-2 4 =
4+1=5
6+1=7
5-2 3 =
4-1 3 =
5+2=7
The table is made up of several lessons: ± 3:
1+3=4
4-3 1 =
5 + 4
5 + 4
5 + 4
2+3=5
5-3 2 =
5 + 2 + 2
5 + 1 + 3
5 + 1 + 1
3+3=6
6-3 3 =
5+2=7
5+1=6
5+3=8
4+3=7
7-3 4 =
7+2=9
6+3=9
8+1=9
5+3=8
8-3 5 =
Then a table of ± ± 4 is created.
6+3=9
9-3 6 =
7+3=10
10-3 7 =
III stage: + 5, + 6, + 7, + 8, Get acquainted with the calculation methods for +9s.
For these cases, the substitution property of the sum is used. The substitution property of the sum helps to bring all the considered points to the previously hit points. Introducing children to the substitution property of kushing can begin with practical work
4+3=7                  3+4=7                  5+3=8                  3+5=8
each pair of these examples is compared, similarities, differences are shown, and conclusions are drawn. The sum does not change with the change of position of the joins. Instead of calculating 2 + 7, you can calculate 7 + 2. By solving such examples, it is concluded that it is easier to add a small number to a large number than to add a large number to a small number.
IV step: 6-, 7-, 8-, 9-, 10- calculation method for cases of appearance.
This type of calculation method is based on knowing the relationships between the sum and the adders. With the components of the addition operation, the following conclusion is reached: if one of these additions is subtracted from the sum, the other is derived. 9-5 = is considered as such. 9 is 5 and how many. 9 = 5 + 4. 9 is the sum. 5 is a compound I, and a sum is a compound II.
The second addition is 4, so 9-5 = 4
10-7
8-6
10 = 7 + 3
8 = 6 + 2
10-7 3 =
8-6 2 =
That is, if we subtract 10 from 7, we get 3, because 10 is 7 and 3.
Control questions:
  1. Which method is used to add and subtract, multiply, and divide non-negative integers?
  2. What is the verbal calculation method?
  3. How is the written calculation method performed?
  4. At what stages are addition and subtraction of numbers in flour taught?
  5. Explain the first step?
  6. How is the second stage carried out?
  7. What laws are used to perform the addition?
  8. How is the division of numbers in flour taught?
  9. What methods are used to teach arithmetic operations?
  10. Canadian didactic games are used to learn arithmetic operations?
 
 
Lecture number 11
Topic: Methods of teaching addition and subtraction of numbers in the face.
Plan:
  1. An oral method of adding and subtracting numbers within a face.
  2. Written method of addition and subtraction of numbers in the face (written and oral method of calculation).
 
Basic expressions: addition, subtraction, calculation of numbers, sum of room additions, addition and subtraction of integers, addition of decimals, oral, written method of calculation.
   
When learning to subtract and add numbers within 100 according to the requirements of the program, swimmers learn the methods of calculation for all cases of addition and subtraction, their theoretical knowledge. In class I the properties of arithmetic operations and methods of calculation of these properties are taught. Preparatory work is carried out before disclosing the properties and calculation methods. In the preparatory work, swimmers learn mathematical expressions such as the sum and difference of solar panels, get acquainted with bird equations. They learn to write one- and two-action expressions using parentheses and to replace two-digit numbers with the sum of room additions. Getting acquainted with the mathematical expression "sum" In the first grade, after the topic ?? + 3, the term "Separation" is taught in the topic of addition and subtraction in flour. In the process of teaching these, two different meanings of the terms sum and subtraction are revealed. For example: 4 + 5 and the sum of 4 and 5, 9 is the sum of numbers. In order to explain in writing the method of calculation when learning addition and subtraction in 10, it is taught to write with 2 equal signs: масалан: 6+4=6+2+2=10; 9-3=9-2-1=6. such writing prepares the learner to understand the writing of the substantiation of computational methods based on the understanding of the method of addition and subtraction by number sources 6+ (3 + 1) = 6 + 4 = 10.
The number in parentheses is entered during the numbering study. The “Kaws” sign suggests such an exercise in presentation. Add 5 to the sum of the numbers 3 and 2. After solving the exercise orally, the teacher explains how to write such examples: to show how to add a number to a given sum, write the sum in parentheses: (5 + 3) + 2… Before entering the properties, children are taught to read parentheses correctly and write them under dictation. For example, swimmers of 9- (2 + 3) are taught to read as follows: Subtract the sum of 9 and 2 from the number 3, then replace the 2-digit numbers with the sum of the room joins. For example: 34 = 30 + 4; 59 = 50 + 9.
These materials are the basis for disclosing the necessary calculation methods, and addition, subtraction training is carried out in the following order: addition and subtraction of numbers in the first 20, then addition and subtraction of two-digit numbers ending in zero, the sum of numbers, subtraction of numbers from the sum the methods of calculating addition and subtraction of one-digit numbers are taught, ie the first group is taught to add one-digit numbers of the form 2 + 9, 3 + 8, 7 + 5, 8 + 3, that is, we get two one-digit numbers whose sum is more than 10.
The abacus is used to perform the addition in the form 9 + 5 (1). As you know, we hit one-digit numbers in 10, but their sum was less than 10. Now, when adding numbers of this form, the principle of filling in 10 is used, ie it is necessary to replace the sum of the adders so that it fills the first additive by 10: 9 + 5 = 9 + (1 + 4) = (9 + 1) + 4 = 10 + 4 = 14 ( The sum of 10 + 4 is included in the second decimal). The second group includes examples of finding the sum of numbers in the form 20 + 5, 30 + 6, 70 + 4,… (2), ie the first additive is a two-digit integer, the second addend is a one-digit number. When calculating 20 + 5, the knowledge gained on the topic of numbering two-digit numbers is used. 20 is 2 decimals, 5 is the result of these 5 units 25, so 20 + 5 = 25. (3) 22 + 5 = (20 + 2) + 5 = 20 + (2 + 5) = 20 + 7 = 27
4) 20 + 50
40-10
2 un +5 un = 7 un
4 un-1 un = 3 un
20+50=70
40-10 30 =
4) 28+5=(28+2)+3=30+3=33
       (2 3)              
6) 30+25=30+(20+5)=(30+20)+5=50+5=55
                   (30 + 20) + 5 = 55
                   25+30                  20+30+5              (20+30)+5=55
                      (20 5)
7) 22+35=22+(30+5)=(22+30)+5=52+5=57
         8) 22+36=25+(30+6)=(25+30)+6=55+6=61
42 + 25
42 + 38
74 + 26
74 + 26
(40 + 2) + (20 + 5)
40+30=70
70+20=90
74+20=94
40+20=60
2+8=10
4+6=10
94+6=100
2+5=7
70+10=80
90+10=100
74+26=100
60+7=67
42+38=80
74+26=100
42+25=67
         Hence, the methodological order of teaching addition of numbers within 100 is 9 + 5 → 30 + 20 → 20 + 5 → 22 + 3 → 28 + 6 → 22 + 35 → 22 + 36. During the study of verbal methods of adding numbers within 100, swimmers are introduced to the associative property of addition.
         (4+2)+3=6+3=9
         (4+2)+3=(4+3)+2=7+2=9
         (4+2)+3=4+(2+3)=4+5=9
According to this rule, the study of examples of the form 34 + 2, 34 + 20 is taught, and the results of the two operations are compared with each other. The explanation is as follows: first I replace the number with the sum, the sum is added to the number, and then we solve it in the most convenient way.
34+2=(30+4)+2=30+(4+2)=36
34+20=(30+4)+20=(30+20)+4 =54
As a result of repeated processing of examples of this type, the swimmer develops skills, and then the method of calculation is shortened.
For example: 42 + 30 To add 42 to 30, we add 40 to 30. This 70 again becomes 2, 72 and is written as 42 + 30 = 72.
It is necessary to ask for full explanations from time to time.
Multiplication.
40-20
4 flour - 2 flour = 2 flour 2 flour = 20 40-20 = 20
45-5=(40+5)-5=40+(5-5)=40+0=40
         45-40=(40+5)-40=(40-40)+5=0+5=5
         45-3=(40+5)-3=40+(5-3)=40+2=42
45-3 40-5
(40+5)-3                        40=30+10
40+(5-3)=40+2=42                (30+10)-5
                                     30+(10-5)=30+5=35
45-9=45-(5-4)=(45-5)-4=40-4=36
45-30         (40+5)-30=(40-30)+5=10+5=15
  • 45-(20+3)=(45-20)-3=25-3=22
  • 45-(20+8)=(45-20)-8=25-8-17
Control questions: 
  1. What is done in the preparatory phase of learning to add and subtract numbers on the face?
  2. How many different methods of calculation are used in the study of addition and subtraction of numbers in the face?
  3. How is the verbal calculation performed (addition, subtraction)?
  4. How to use the laws of addition in performing arithmetic operations on the subject of hundreds?
  5. Why is the law of substitution used?
  6. What is considered in written addition and subtraction?
  7. How to add and subtract a number?
  8. How do you add a sum to a sum?
Lecture number 12
Topic: Teaching to multiply and divide numbers in the face
methodology.
Plan:
  1. I. Multiplication, division in the table.
         1) Explain the meaning of multiplication and division.
         2) Special cases of multiplication and division.
3) Multiply the numbers 2, 3, 4, 5, 6, 7, 8, 9 by one-digit numbers and teach them to create a table of correspondence.
  1. II. Off-table multiplication, division.
III. Residual division.
 
Basic terms: multiplication, division, multiplication and division within and outside the table, residual division, multiplication and division, multiplication table, special cases of multiplication, division, multiplication and division by 1, 0, 10.
  1. Explain the meaning of multiplication and division.
Multiplication and division in the face are taught in the second grade, but preparation for teaching begins in the first grade with the teaching of numbering, addition, and subtraction in grades 10 and 100. The essence of the preparatory work provided for in the program is to perform various tasks on a demonstrative basis. These tasks require finding the sum of different additions and representing the number as the sum of the same additions. From the first day of school, children are taught not only to count the same items, but also to count in pairs, pairs and fives.
For example: burn 3 circles 2 times. How many circles did you burn? Draw 2 squares 4 times. How many squares did you draw?
Describe the numbers 12, 15, 10 in the form of the sum of the same additions.
12=3+3+3+3                 12=4+4+4            12=6+6
10=5+5                15=3+3+3+3+3             15=5+5+5
Practical exercises are performed to prepare for the study of division. For example: Take 8 circles and burn 2 of them. It is found by counting the number of times 2 circles are formed. The following issues can be used to study the meaning of the multiplication operation.
For example:                 
  1. There are 5 apples in each tray. How many apples are there in 4 trays?
  2. The housewife received 3 packets of potatoes, each weighing 3 kg. How many kg of potatoes did he buy?
The solutions of these problems are written by the swimmers in the first grade in the form 5 + 5 + 5, 4 + 4 + 4, 3 + 3 + 3, and they know that there are the same additions to the solution in the conditions of the problem. Based on the demonstration, a number of textual issues of this type are solved. The children's attention is drawn to the fact that the adders are the same, each time the adders determine what their sum is, then the children's minds are informed that the same sum can be replaced by examples of multiplication, and how to write 5 + 5 + 5 as 5 * 3 the second number indicates that the additive is added, the dot indicates that it is a sign of the multiplication operation, and it is concluded that multiplication means the addition of a derivative. In the notation 5 * 3 = 15, 5 is the I multiplier, 3 is the II multiplier, and 15 is the multiplier, and if we multiply 5 by 3, we get 15. In the study of the meaning of the act of division, it is first revealed in the solution of the problem of division into equal parts according to its content.
For example:
  1. The teacher distributed 12 notebooks to the swimmers, 2 of them. How many swimmers did you get? Answer: 6 swimmers received notebooks.
  2. 8 carrots were given equal to 4 rabbits. How many carrots were given to each rabbit?
  3. 15 carrots were given 5 to each rabbit. How many rabbits were given carrots?
  4. They put 12 balls in 4 round bags. How many balls did each type of bag put in?
  5. They put 12 balls in 3 round bags. How many types of bags will you need?
Demonstrations are used to solve these problems. The answers to these questions are first found by counting, and then the teacher reveals that the solution to these problems can be written by division. It is said that dividing 12 by 4 is written in the form 12: 4 and the solution of the last problem can be written in the form 12: 4 = 3, where 12 is called the divisor, 4 is called the divisor, and 3 is called the division. Comparing the conditions of the above problems shows the interdependence of multiplication and division.
For example:
         5*3=15                15:3=5                15:5=3
         4*3=12                12:4=3                12:3=4 and if the multiplication is divided by one of the multipliers, it is concluded that the second multiplier is derived, then the substitution property of the multiplication operation is explained on the basis of instructions.
         For example: 
1) The class has 3 windows. There are 4 flower pots in each window. How many flower pots are there in the windows?
2) The classroom has 4 windows. There are 3 flower pots in each window. How many flower pots are there in the windows? 3 * 4 = 12 4 * 3 = 12
         By comparing the resulting solutions, they are taught what they are similar to and how they differ from, and it is concluded that multiplication does not change with the replacement of multipliers, and exercises are performed to strengthen it.
         1) Burn down the omitted numbers: 3 * 4 = 3 * ??; 9 * ?? = 7 * 9; 7 * 3 = ?? * 7
         2) Compare the expressions and put the symbol <,>, = instead of the square. 6 * 3 ?? 3 * 6; 5 * 4 ?? 5 * 4, then the property is reduced to the letters a * b = b * a.
  1. Special cases of multiplication and division.
         A) Multiply and divide by 1.
For example, it is taught to find the product of the numbers 1 * 6, 1 * 8 by adding. 1 * 6 = 1 + 1 + 1 + 1 + 1 + 1 = 6.
         In this case, children see that the more a number is in the second multiplier, the more times it is added, and the product is always equal to the second multiplier. Entering the multiplication rule in 1 as a special case is explained by the substitution property of multiplying this point. Therefore 1 * 6 = 1 * 1 = 6. Based on the relationship between multiplication and division, the rule of dividing the number by 6 is introduced, ie 1: 6 = 1 because 6 * 1 = 6, 6: 8 = 1 because 8 * 1 = 8 and in general a: 8 = a because 1 * a = a .
         B) At the same time, multiplication of zero and division of zero are still shown.
         Масалан: 0*5=0+0+0+0+0=0
         It is also taught to write the rule in letters that zero is obtained by multiplying any number by zero, ie 0 * b = 0, and then to divide zero by any number that is not equal to zero on the basis of knowing the relationship between the components and the result of multiplication.
         For example:     
At 0: 5, the swimmers make such a comment. To divide 0 by 5, you need to find a number that multiplies by 5 to get 0. This number is zero because 0 * 5 = 0 means 0: 5 = 0. Hence, it is concluded that zero is obtained by dividing zero by any number that is not equal to zero, and is written as 0: a = 0. It is not possible to divide a given number by zero, because when you take any number in the division and multiply it by zero, you get zero, not a number. 3: 0,… a: 0.
C) Multiplying 10 by a one-digit number is explained as follows.
To multiply 10 by 5, you have to multiply 1 flour by 5, and it turns out 5 flours or 50. Dividing a 2-digit number ending in zero by 10 uses the relationship between the components of the multiplication operation and the result. To find 50: 100, you need to find a number that multiplies by 10 to get 50. This is 5, so 50:10 = 5.
3) Multiply the numbers 2, 3, 4, 5, 6, 7, 8, 9 by one-digit numbers and teach them to create a table of correspondence.
         In this case, the study of each point of the table begins with the creation of a table on the first constant multiplier. Different methods are used to find the result.
1) By adding the same additions. Масалан: 3*4=3+3+3+3.
2) Add the number corresponding to the result of the previous example from the table, i.e. add 3 to the previous result to find 4 * 3 using 5-3. 3 * 5 = 3 * 4 + 3 = 15.
3) The third method of constructing a multiplication table is based on the use of an additional relative distribution property of multiplication. 8 * 7 = 8 * 5 + 8 * 2. this method is handy when considering multiplication by 6, 7, 8, 9.
4) Based on the use of the substitution property of multiplication. 5 * 7 = 7 * 5.
         For example: Let's make a multiplication table for 2.
         2*2=2+2=4
         2 * 3 =2 + 2+ 2 = 6
         2 * 4 =2 + 2 + 2+ 2 = 8
         2 * 5 =2 2 + + + 2 2+ 2 = 10
         2 * 6 =2+2+2+2+2+ 2 = 12
         2*7=2*5+2*2=10+4=14
         2*8=2*5+2*3=10+6=16
         2*9=2*6+2*3=12+6=18
         2*10=2*9+2=18+2=20
         The corresponding division table is also taught at the same time.
         2*2=4                  3*2=6                  6:2=3                   6:3=2
         2*3=6                  4*2=8                  8:2=4                   8:4=2
         2*4=8                  5*2=10                10:2=3                 10:5=2
             2*5=10            6*2=12                12:2=6                 12:6=2
         2*6=12                7*2=14                14:2=7                 14:7=2
         2*7=14                8*2=16                16:2=8                 16:8=2
         2*8=16                9*2=18                18:2=9                 18:9=2
         2*9=18                10*2=20     20:2=10      20:10=2
         2 * 10 = 20
         Based on this, each multiplication table and the corresponding cases of division are considered and give an overview of the multiplication table that needs to be memorized.
         2*2
         3 * 2 3 * 3
         4*2    4*3    4*4
         5*2    5*3    5*4    5*5
         6*2    6*3    6*4    6*5    6*6
         7*2    7*3    7*4    7*5    7*6    7*7
         8*2    8*3    8*4    8*5    8*6    8*7    8*8
         9 * 2 9 * 3 9 * 4 9 * 5 9 * 6 9 * 7 9 * 8 9 * 9
  1. II. Off-table multiplication, division.
The study of cases of multiplication and division outside the table is considered in the following order.
A) The case of multiplying a number by the sum and the sum by the number, the property of dividing the sum by the number.
These properties form the basis for learning how to multiply one-digit numbers by two-digit numbers and two-digit numbers by one-digit numbers.
For example, the following problem can be used to introduce different ways of multiplying a sum by a number. There are 3 apples on the table, each with 2 apples and 4 pears. How many fruits are on the table?  To solve this problem, you are first taught to find the fruit on 1 plate and then to find the fruit in 4 plates, then how many apples are on 4 plates, then find the number of pears on 4 plates, and then find the total number of fruits. References are made to different methods of writing, i.e. (3 + 2) * 4 = 5 * 4 = 20; (3 + 2) * 4 = 3 * 4 + 2 * 4 = 12 + 8 = 20.
By comparing the results found in solving this problem in different ways, the swimmers see that these results are the same. This example explains the meaning of different ways of multiplying a sum by a number, that is, you must first calculate the sum and then multiply it by the number. (A + V) * Multiply the concentration of S by any additive and add the results obtained.A * S + V * S. Depending on the conditions of the problem, different methods can be used to multiply the sum by the number.
For example, when calculating (2 + 4) * 6, it is easy to find the sum of 2 and 4 and then multiply 6 by the number. It is convenient to use 9 * 5 + 8 * 9 to find the value of (8 + 5) * 8.
The substitution property is used to multiply a number by its sum.
For example: 6 * (2 + 4) = (2 + 4) * 6, that is, you can use (6 + 2) * 4 to find 2 * (4 + 6).
B) Multiplication and division of numbers ending in zero.
20*3                    80:2
2 un * 3 = 6 un 8 un: 2 = 4 un
6 un = 60 4 un = 40
20*3=60              80:2=40
Now it is taught to multiply two-digit numbers by one-digit numbers. This is taught as follows:
1) We replace the two-digit number with the sum of the room additions.
2) We multiply the sum using the multiplication rule.
3) The number ending in zero is multiplied by the number.
4) One-digit, ie the second multiplier is multiplied by the number.
5) The results found are added. Масалан: 26*3=(20+6)*3=20*3+6*3=60+18=78.
When multiplying a one-digit number by a two-digit number, the rule of multiplying the number by the sum is used. Масалан: 3*17=3*(10+7)=3*10+3*7=30+21=51. You can also use the substitution property. 3 * 17 = 17 * 3 = 51. This means that if the second multiplier is a two-digit number, then it can be divided into decimals and units, and then the first multiplier can be multiplied by separate decimals and units and the results can be added, or the multipliers can be swapped when multiplying a one-digit number by two-digit numbers.
         5*16=16*5=80              4*23=23*4=92
         4*23=4*(20+3)=4*20+4*3=80+12=92
         When performing extra-division, the methods of dividing two-digit numbers into one-digit numbers and methods of dividing the sum by numbers are shown. The division of the sum into numbers is explained by solving the following problem.
         For example: The first bush has 12 m of material and the second bush has 15 m of material. If 3 m of material is used for each shirt, how many shirts can be made from both tubes?
         (12+15):3=27:3=9                (12+15):3=12:3+15:3=4+5=9
that is, first determine how much material is in both tubes, then how many shirts can be sewn from it, then find how many shirts are sewn from the first ball, then find how many shirts are sewn from the second ball, and then add the results. So, method I: to divide the sum by the number, you have to calculate the sum and divide it by the number. Method II: Divide each additive by a number and add the resulting results.
         In the study of off-table division, the simplest examples are taken, that is, when the room is first divided into additions, each additive is divided into integers: it is also mentioned the division of integers.
         24:2=(20+4):2=20:2+4:2=10+2=12    
33:3=(30+3):3=30:3+3:3=10+1=11    
36:3=(30+6):3=30:3+6:3=10+2=12
then it is taught to solve examples in the form 78: 3, 32: 2, 92: 2…. In this case, the divisor is divided into such convenient conjunctions that each of these conjunctions must be divisible by a number.
For example, to find 78: 3, you can divide 78 by 21 + 57, 39 + 39, 21 + 21 + 36, 60 + 18,….
78:3=(21+57):3=21:3+57:3=7+(21+36):3=7+21:3+36:3=7+7+(30+6):3=7+7+30:3+6:3=14+10+2=26.
In such cases, let us divide the external divisor by the sum of such integers, in which one integer divisible by the divisor and the other corresponds to the multiplication and division table: 78: 3 = (60 + 18): 3 = 60: 3 + 18: 3 = 20 + 6 = 26. 96: 2 = (80 + 16): 2 = 80: 2 + 16: 2 = 40 + 8 = 48.
Dividing a two-digit number by a two-digit number is also an off-table division. In this case, the method of division based on the relationship between the components of the multiplication operation and the result is used.
For example: 81:27 such a consideration is made in the solution. Multiplying by 27, we find the number that comes out 81. Let's multiply by 2. 27 * 2-54, 2 does not fit. We multiply 27 by 3. 81 chikadi. So, 81:27 = 3.
Therefore, multiplication and division checks are also considered. Multiplication is checked by division. 27 * 3 = 81. 1) 81: 3 = 27; 2) 27 = 27.
To verify the correctness of the solution of this example, 1) we find the multiplier by the multiplier; 2) the result found is compared with the second multiplier. If these numbers are equal, then the multiplication is done correctly.
The division can be checked by multiplication 1) the division is multiplied by the divisor; 2) the result obtained is compared with the divisor. If these numbers are equal, then the division is complete.              
III. Residual division.
The residual division studied in class III is considered in the following order.
1) Swimmers are introduced to the meaning of residual division.
For example, take three swimmers to the board and offer one of them 12 squares equal to the other two swimmers. The result is written on the board 12: 2 = 6. Then, when this swimmer divides 13 squares into two swimmers, each swimmer is multiplied by 6 squares and one square is added, and the solution is written as 13: 2 = 6 (1 residual), where 13 is divisible, 2 is divisible, 6 - bulinma, 1- koldik.
2) It is taught that the residue that comes out when dividing swimmers should be smaller than the divider.
For example, under each of the numbers 10, 12, 14, 13, 15, 16 is written the remainder of the division by 2, 3, 4. On the basis of the exhibition, their results are determined:
10: 2 = 5 (0 left) 10: 3 = 3 (1 left) 10: 4 = 2 (2 left)
12: 2 = 6 (0 left) 13: 3 = 4 (1 left) 13: 4 = 4 (1 left)
14: 2 = 7 (0 residuals) 14: 3 = 4 (2 residuals) 14: 4 = 3 (2 residuals) and the following conclusion is reached. If there is a residue in the divisor, it is always smaller than the divisor.
3) Swimmers are introduced to the method of residual division.
For example, if by comparing 18: 3, 19: 3, 28: 7, 29: 7, the closest Canadian to the divisor knows that the divisor is divisible by the smallest divisor without a remainder, then the divisor can also find the remainder, that is, how many of the 26 divisors in 3: 26 3 we need to know that there is 3 * 8 = 24 less 3 * 9 = 27 cup. There are 26 times 3 times 8 times. 8- bulinma. We find the remainder: 26-24 = 2 26: 3 = 8 (2 remainders) or 37: 5 The solution is as follows. 37 cannot be 5 without a remainder. The largest number that is less than 37 and divisible by 5 without a remainder is 35, 35 can be divided by 5 to get 7. 37-35 = 2. 2 units will increase. This is written as 37: 5 = 7 (2 residues) 47: 5 = 9 (2 residues). 47: 7 explanation. The number 47 cannot be divided by 7 without a remainder. We remember what the largest number among the numbers up to 47 is divisible by 7. This is the number 42. We find the division 47: 7 = 6. We find the remainder 47-42 = 5. 47: 7 = 6 (5 left).
Control questions:
  1. How is the meaning of multiplication taught?
  2. How is the meaning of the act of division taught?
  3. What number is multiplied by 0 and 1?
  4. How many different ways is a multiplication table created?
  5. What properties are used in the study of multiplication and division outside the table?
  6. How many different ways are there to multiply and divide a sum by a number?
  7. How to divide and multiply a two-digit number by a one-digit number?
  8. How to teach multiplication and division of numbers ending in zero?
  9. How to test multiplication and division?
  10. How is the meaning of division divided?
  11. In what ways is the division of a two-digit number into a two-digit number taught?
  12. What is the relationship of the residue from the split to the divisor?
Lecture number 13
Topic: Learning arithmetic operations on the topic of millennials
methodology.
Plan:
  1. Verbal addition and subtraction of numbers in thousands.
  2. Written addition and subtraction of numbers in thousands.
  3. Multiplication and division of numbers in thousands.
 
Basic terms: written and oral calculation, number vowel structure, addition, hundreds, tens, units, tag-tag, minus, column, multiplication, division.
  1. Verbal addition and subtraction of numbers in thousands.
It is known that the addition and subtraction of one- and two-digit numbers between 10 and 100 was learned orally by swimmers. Within a thousand, the written methods of addition and subtraction are first studied orally. Oral methods of addition and subtraction are based on the sum of the numbers, the properties of adding the sum to the number, as well as the relevant rules of subtraction, as in the face. This theoretical knowledge was imparted by the children in learning the actions inside the face. Therefore, the methodology of studying the verbal methods of addition and subtraction in the millennium has many similarities with the corresponding methodology on the subject of the hundred. Similar methods of calculation are studied in comparison with each other. A variety of exercises are used to develop numeracy skills. These exercises help to strengthen theoretical knowledge. Oral methods of addition and subtraction within a thousand are considered simultaneously and in the following order. At the preparatory stage, exercises related to the application of knowledge about numbering are considered.
For example:
300+2                 305+20                320+20                302-300
300 + 20 350 + 2 320-300 325-25
300+40+5                    325-25
300 + 25                         302-2
In finding the value of these expressions, the methods of verbal addition and subtraction within the face are used, then
500 + 300 500-300
5 hundred +3 hundred = 8 hundred 5 hundred - 3 hundred = 2 hundred
500+300=800                        500-300=200
60+80=140                            170-90
6 un + 8 un = 14 un 17 un - 9 un = 8 un
14 un = 140 170-90 = 80
240 + 380 620-380
24 un + 38 un = 62 un 62 un - 38 un = 24 un
240+380=620                        620-380=240
Such calculations strengthen the knowledge of numbering and prepare children to learn more complex methods of addition and subtraction, followed by acquaintance with the methods of addition and subtraction in the form 640 ± 300 and 640 ± 30. First, the children repeat the rules of addition and subtraction of numbers by performing exercises involving two-digit numbers.
For example: Calculate in a convenient way.
(50+6)-30=(50-30)+6=20+6=26
(50+6)-4=50+(6-4)=50+2=52
Explain the calculation method.
54-20=(50+4)-20=(50-20)+4=30+4=34
54-2=(50+4)-2=50+(4-2)=50+2=52
The method of calculating the following examples is explained based on the knowledge of how to solve these examples.
640+300=(600+40)+300=(600+300)+40=900+40=940
640-300=(600+40)-300=(600-300)+40=300+40=340
640+30=(600+40)+30=600+(40+30)=600+70=670
640-30=(600+40)30=600+(40-30)=600+10=610
Then they compare these calculation methods and determine what these methods are compatible with and what they differ from.
350 + 420
360 – 250
430 + 350 = 400 + 30 +
+ 300 + 50 = (400 + 300) +
+(30+50)=700+80=780
430 + 350 =
= 430 + (300 + 50) =
= (430 + 300) + 50 =
= 730 + 50 = 780
(300 50) (400 20)
(300 60) (200 50)
300+400=700
300-200 100 =
50+20=70
60-50 10 =
700+70=770
100+10=110
350+420=770
360-250 110 =
Hundreds are added to hundreds, tens to tens.
Hundreds are separated from hundreds, tens from tens
790-350=(700-300)+(90-50)=400+40=440
790-350=(790-300)-50=490-50=440
790-350
79 un - 35 un = 44 un
44 un = 440
240+60=(200+40)+60=200+(40+60)=200+100=300
500-40=(400+100)-40=400+(100-40)=400+60=460
490 + 350
400+300=700
430-250=
= (430-200) -50 =
= 230-50 = 180
(400 90) (300 50)
90+50=140
350 – 80
700+140=840
       (200 150)
350 – 80
150-80 70 =
          (50 30)
200+70=270
350-50 300 =
300-30 270 =
800-380=(800-300)-80=500-80=420
700+230=700+(200+30)=(700+200)+30=930
90+60=90+(10+50)=(90+10)+50=150
380+70=380+(20+50)=(380+20)+50=450
500-140=500-(100+40)=(500-100)-4=360
270-130=270-(100+30)=(270-100)-30=170-30=140
140-60=140-(40+20)=(140-40)-20=100-20=80
340-160=340-(100+60)=(340-100)-60=240-60=180
270-130=(200+70)-(100+30)=(200-100)+(70-30)=100+40=140
  1. Written addition and subtraction of numbers in thousands.
Kushish
The written methods of addition and subtraction are considered separately, first the written methods of addition and then the written methods of subtraction are considered. The rule of adding the sum to the sum is the theoretical basis for the written addition. For this reason, swimmers are explained how three-digit numbers are added based on the addition rule.
256+341=(200+50+6)+(300+40+1)=(200+300)+(50+40)+(6+1)=500+90+7=597
Now it is easy to add three-digit numbers if we write this example in the form of a column, that is, if one of the adders is written under one, the other is subdivided into units, the tens are subtracted, and the hundreds are subtracted. Using the addition rule to the sum, the units are units, tens are added with tens, and hundreds are added with hundreds. In written addition, it is added starting from the units. Written addition is taught in the following order:
1) Cases where the sum of units and decimals is less than 10.
+
232
347
 We add 2 units to 7 units. 9 units are formed, i.e. 9 units are written under the units below the line. We add 3 flours to 4 flours and 7 flours are formed. In the sum we write 7 instead of tens. We add 2 hundred to 3 hundred. 5 hundred is formed. We write 5 instead of a hundred. Yigindi 579 ga teng.
2) In cases where the sum of units or the sum of tens is equal to 10.
+
354
+
563
+
346
236
246
254
5810
7109
5910
590
809
5100
600
3) In cases where the sum of units or the sum of tens is greater than 10.
+
354
+
354
528
263
8712
5117
882
617
Multiplication
Different methods of written subtraction are studied as in addition. The procedure for subtracting the sum from the sum is first disclosed after the written subtraction method. When switching from oral subtraction to written subtraction, the rule of subtraction is taught.
Масалан: 563-412= (500+60+3)-(400+10+2)=(500-400)+(60-10)+(3-2)=100+50+1=151
It is then said that it is easier to divide three-digit numbers if the divisor is written as a column below the denominator, where it is necessary to divide the units first, then the decimals and hundreds.
-
450
136
314
Then the subtraction points are considered when the unit of decrement is 0 in the room. For example: Multiplication is explained as follows. 0 is not divisible by 6, so we get 5 flour out of 1, so we put a dot on the number 5 to make sure we don’t forget it. There are 10 units in this flour. We subtract 10 units from 6 units. 4 units come out. We write 4 units under the units. Now let's separate the tens. The dot on the number 5 reminds us that when we subtract the units, we get a decimal. We separate 3 flours from four flours. 1 flour remains. We write instead of tens. We subtract 4 hundred from 1 hundred. 3 hundred left. We write instead of hundreds. The difference is 314.
Hence:
A) Subtraction cases when the units of the denominator are smaller than the units of the denominator: 873-435.
B) Subtraction cases when the decimals are less than the decimals: 726-472.
C) Subtraction cases when the units and decimals of the denominator are smaller than the units of the denominator: 963-586.
-
963
586
377
Explanation: We cannot distinguish 3 units from 6 units. We get one tenth of 6 tenths. (We get one tenth out of 6). 1 unit and 3 units are 13 units. We subtract 13 units from 6 units. 7 units remain. We write the answer 7 under the units. There are 6 vowels instead of 5 vowels. It is impossible to separate 8 flours from it. We grind 9 out of 1 hundred. There will be 10 flours, 5 flours with the previous 15 flours. We subtract 15 flours from 8 flours. We write 7 flours in the flour room. Subtracting 8 hundredths from 5 hundredths and writing 3 in the hundredths room. The result is 377 differences.
It is much more difficult to solve examples in the form of 900-547, 906-547, 1000456 in primary school. In this case, you have to switch from one room unit to another several times.
-
1000
456
544
 Explanation: in this case we take 1 thousand, divide it by hundreds. 10 hundred is formed, we get one out of 10 hundred. We burn the dot and remember that there are 9 hundred left. Divide 1 hundred by tens. 10 flours are formed. We get one out of 10 tenths that gives 10 units then 1 hundred is 9 tenths and 10 units. 1000 should indicate that it consists of 9 hundredths, 9 tens, and 10 ones. In order to develop computational skills, it is necessary to give examples of an exercise nature at each stage of learning to divide. In the process of performing these exercises, the swimmers' thinking should be short, and the calculations should be done quickly.
  1. Multiplication and division of numbers in thousands.
An oral and written method of multiplication and division within 1000 is considered.
1) Multiplying and dividing whole hundreds by one-digit numbers.
2) Appropriate cases of multiplication and division of whole tens by one-digit numbers.
In the first group of examples, the calculation methods result in multiplication and division of integers in the table.
200 * 3 800: 4
2 hundred * 3 = 6 hundred 8 hundred: 4 = 2 hundred
200*3=600          800:4=200
Solving the examples in the second group of examples results in multiplication and division of whole vowels in the table.
60*7                    240:3                   600:6
6 flour * 7 = 42 flour 24 flour: 3 = 8 flour 6 hundred: 6 = 1 hundred
60 * 7 = 420 240: 3 = 80 600: 6 = 100
260*3=(200+60)*3=200*3+60*3=600+100=780
Written method of multiplication and division
34*2=(30+4)*2=30*2+4*2=60+8=68 куринишидаги хисоблашга асосланиб ургатилади.
234*2=(200+30+4)*2=200*2+30*2+4*2=400+60+8=468
It is easy to write examples. The explanation of the written calculation is as follows: I write…
*
234
   2
468
I multiply the units… 4 units = 8 units. We write 8 units under the units. We multiply the tens. 3 decimals * 2 = 6 decimals. We write 6 tens under the tens. We multiply 2 hundredths by 2. We write 4 faces under the hundreds. Result 468. In a written calculation, the calculations are multiplied first by units, then by decimals, and finally by hundreds.
*
347
    2
694
I write…
I multiply units…
7 units * 2 = 14 units = 1 unit 4 units. I write 4 units under units. I memorize 1 flour and add it to the flours after multiplying the flours. I multiply 3 hundredths by 2 and write in the hundredths room. Result: 694.
*
182
    3
546
I write…
I multiply units…
I write 6 units in the unit room. I multiply the tens. 8 flour * 3 = 24 flour = 2 face 4 flour. I write 4 tens under the tens. I remember 2 faces and add to the hundreds after multiplying the hundreds. I multiply the hundreds. 1 face * 3 = 3 faces. I add the 2 faces that are formed when multiplying the tens. 3 faces + 2 faces = 5 faces. I write 5 under the hundreds. I'll burn the answer. Kupaytma 546 ga teng.
The method of calculating the division in writing.
69:3=60:3+9:3=20+3=23
684:2=600:2+80:2+4:2=300+40+2=342
It is easy to write an example as an example. First hundreds, then tens, and finally units. 684 should be divided by 2. Let's find the hundreds: The number 684 has 6 faces. Our find is in a 6: 2 = 3 hundredth division. Multiply: 3 * 2 = 6 hundred. We find the tens. Multiply 8 decimals by 2 = 4 decimals 4 * 2 = 8 decimals. We find the units.
684
2
764
2
6
342
6
382
  8
16
  8
16
   4
  4
   4
  4
    0
    0
Divide 764 by 2. We find hundreds. The number 764 has 7 hundredths. We find: 7: 2 = 3 faces. It will be in the division. Multiply: 8 * 2 = 16 flours - we found. Divide: 7-6 = 1 face - to be divided again. We find the tens. 1 face and 6 tens and 16 tens. We find: 16: 2 = 8 flour - is in the division. Multiply: 8 * 2 = 16 decimals. Subtract: 16-16 = 0. the rest is gone. We find the units they are 4. We find: 4: 2 = 2 units - we found. Divide: 4-4 = 0, no residue. Let's read the division: the division is 382.
978
3
276
4
9
326
24
69
  7
  36
  6
  36
   18
  0
   18
    0
276 should be divided by 4. We find hundreds. The number 276 has 2 hundred. It is not possible to make 2 faces into 4 faces. We find the tens. The number 276 has 27 vowels. We find that 27: 4 = 6 is in the decimal fraction. Multiply by 6 * 4 = 24. Divide 27-24 = 3 flours and divide again. We find the units. 3 units and 6 units make up 36 units. We find 36: 4 = 9 units - Buddha in division. The division will be 69. Then a plan is made for the written method of dividing three-digit numbers into one-digit numbers, and the swimmers are explained how to work the example according to the plan:
Finding Hundreds…
Bulaman…
Kupaytiraman…
Ayiraman…
I can find flour lik
Kupaytiraman…
Ayiraman…
I find units…
Bulaman…
Ayiraman…
I read the answer.
Control questions:
  1. How is verbal addition and subtraction taught in a thousand?
  2. How is written addition and subtraction taught in thousands?
  3. In what order is written multiplication taught on the subject of the millennium?
  4. In what order is the written addition of numbers in thousands taught?
  5. How to teach multiplication of numbers in a thousand? (oral and written)
  6. How to teach oral and written division of numbers in a thousand?
Lecture number 14
    Topic: Addition and subtraction of multi-digit numbers.
             Plan:
  1. Addition and subtraction of multi-digit numbers
  2. Addition and subtraction of named numbers
  3. Addition and subtraction of multi-digit numbers
Basic expressions: multi-digit numbers, units, tens, hundreds, thousands, columns, addition and subtraction of named numbers.
Preparations are made before adding and subtracting multi-digit numbers. Preparatory work begins when learning to number multi-digit numbers. Initially, the verbal methods of addition and subtraction, the properties of actions are repeated.
       6400 + 300                        8400 + 600                74000 + 16000
64 hundred + 3 hundred = 67 hundred 84 hundred + 6 hundred 74 thousand + 16 thousand
The written methods of addition and subtraction of three-digit numbers are also repeated. This work allows swimmers to independently understand the written methods of addition and subtraction of multi-digit numbers. When learning to add and subtract multi-digit numbers in writing, swimmers are told to take examples that include each of the previous examples, and
+
435
+
2435
+
62435
-
637
-
7637
352
6352
16352
425
3425
examples are solved. After solving these examples, swimmers conclude that the addition of multi-digit numbers is done in the same way as written addition and subtraction. In the textbook, addition and subtraction are introduced in ascending order. The number of transitions per unit of space is gradually increased, zero-entry points are added to the denominator, addition of several additions, addition and subtraction of named numbers are added, and so on.
+
756000
ni +
750 thousand
243000
243 thousand
as can be added. When swimmers are introduced to new points, they give perfect explanations of the calculations first.
+
36679
64013
We add 9 units to 3 units, 12 units or 1 unit and 2 units are formed. We write 2 units under the units. We add tens to tens. We add 7 flours to 1 flour, 8 flours are formed, we add another flour, 9 flours are formed. We write under the decimals. We add 6 faces to 0 faces, and 6 faces are formed. We write in the Hundreds room. If we add 6 to 4, we get 10, which gives a single 10. We add 3 tens of thousands to 6 tens of thousands, 9 tens of thousands are formed, and if we add it to one tenth of a thousand, 10 tens of thousands give 1 hundred thousand. The result
100692
-
100000
-
400100
-
35472
          1
205708
13290
  99999
The children then give a brief explanation in the examples of division. When learning to add and subtract multi-digit numbers, the basic properties of addition are generalized. The substitution substitution property, which is familiar to swimmers, is applied to cases where the sum of several additions is found.
Масалан:  215+78+85=215+85+78=300+78=378.
The swimmers are then introduced to the method of grouping participants when adding multiple numbers.
23-17+48+52=140
(23+17)+(48+52)=40+100=140
23+(17+48+52)=23+117=140
This is how swimmers explain this record. In the first line, the numbers are added in the order in which they were written. In the second line, these numbers themselves are divided into groups of two. By calculating the sum and adding them, we get another 140. In the third line, the last three additions are grouped together, the sum of which is calculated and added to the 23 numbers. 140 came out. In all three cases, the result is the same 140. Another conclusion can be drawn by solving two more examples of addition in different ways. When adding several numbers, two or more of them can be replaced by their sum. Then children are given exercises to use the grouping property of the sum and the substitution property of the sum at the same time. In connection with the addition and subtraction of anonymous numbers in multi-room, work is carried out on the addition and subtraction of named numbers, expressed in terms of length, mass, time and value. Operations on such numbers can be performed in two ways. Numbers must be added and subtracted as they are given. In this case, addition and subtraction begin with small units of measurement, or both numbers are previously expressed in units of the same name, and operations on them are performed as if operations on unnamed numbers, and the result is expressed in larger units.
52 м 65 cm + 32 м 24 cm = 84 м 89 cm
+
52 м 65 cm
+
5265 cm
32 м 24 cm
3224 cm
84 м 89 cm
8489 cm
         In the study of addition and subtraction of multi-digit numbers, the connections between addition and subtraction are identified, deepened, and using this knowledge to verify the calculations, the rules for performing operations and the terms of use of parentheses are repeated. Swimmers need to understand that it is possible to omit parentheses if the numerical value of the expression does not change from dropping the parentheses. Find the exercises in the textbook to help you master this.
  1. Find the value of the expressions.
50*4+60*3          (300-50)*6
300:6-280:7                  (320+120):4
Copy these expressions without parentheses and count their clothes. In which expressions is it possible not to write parentheses?
  1. Write the expressions without parentheses so that the results do not change.
65-(40+12)          (45+25)*9            (60+12):6
(84+24)-16          40*(5+4)              (75+25):10
Constant attention should be paid to the methods of performing these actions orally, along with the development of written addition and subtraction skills. In addition, some new methods of verbal calculations, in particular the method of numbering, are introduced here. Rounding a number means replacing a number with a number ending in a nearest zero.
For example: rounding 13 means replacing it with 10. Rounding 18 is to replace it with the number 20. The children are then explained how to use the rounding method to solve addition and subtraction examples.
For example:
52+19=52+20-1=72-1=71
52+19=50+19+2=69+2=71
96-38=96-40+2=56+2=58
 
Control questions:   
  1. How to add multi-digit numbers?
  2. How is multiplication of multi-digit numbers taught?
  3. How to add and subtract nominal numbers?
  4. How to teach addition and subtraction of multi-digit numbers?
Lecture number 15
Topic: Methods of learning to multiply and divide multi-digit numbers.
Plan:
  1. Multiplication, division by one-digit numbers.
  2. Multiplication, division by room numbers.
  3. Multiplication and division by two-digit and three-digit numbers.
 
Basic terms: multiplication by one-digit number, division, multiplication by room numbers, division, multiplication by two-, three-digit numbers, division, incomplete multiplication, incomplete divisor.
The methods of multiplication and division of multi-digit numbers are taught in three radically different stages.
I-stage. Multiply and divide by a one-digit number.
Much attention is paid to this step, as it is the basis for the skill acquired and the three-digit number for multiplication and division. From the generalization of the knowledge that children's multiplication is the addition of the same additions in order to prepare them for learning to write multiplication in a one-digit number, that is, multiplying the number a by the number b, making the number a multiplying by b. In this connection, multiplication of 1, multiplication by 1, zero and zero multiplication are introduced and the corresponding conclusions are expressed. If one of the multipliers is equal to 1, then the multiplier is equal to the second multiplier. If one of the multipliers is zero, the multiplication is zero, that is, 1 * a = a, a * 1 = a, 0 * a = 0, b * 0 = 0. In order to prepare for the disclosure of the written multiplication method, the rule of multiplication of a number and the method of multiplication of a two-digit number by a one-digit number should be repeated, and the sum of three, four and more numbers should be shown by different methods. Swimmers can apply the distribution property of multiplication to verbal multiplication of a multi-digit number by a one-digit number.
Масалан:234*3=(200+30+4)*3=200*3+30*3+4*3=600+90+12=702
Students will then be introduced to the written multiplication of one-digit numbers. Indicates that the text is preferred and a complete explanation of the solution of this example is given.
*
324
    3
 324 should be multiplied by 3. We write the second multiplier under one of the first multipliers, draw a line. To the left we write the multiplication sign. We start with written multiplication in units. We multiply 4 units by 3 units. It consists of 12 units, 1 unit and 2 units. We write 2 units under the units. We keep 1 flour in the heart. We multiply 2 flours by 3. 6 flours are formed. We make 6 flours and 1 flour 7 flours. We write it under tens. We multiply 3 hundred by 3. We make 9 faces. We write 9 under the hundreds. Multiplication 972. After full explanations, short explanations are used. It is useful to give examples of how to compare verbal and written multiplication of a multi-digit number to a one-digit number so that swimmers do not forget the verbal methods of calculation. 387 * 6, 260 * 3. the swimmers themselves determine which of these examples is appropriate to solve orally and which in writing. Once solved, the solution methods are compared, highlighting their similarities and differences. Once swimmers have mastered the total score of a written multiplication of a multi-digit number into a single-digit number, they are introduced to the points where the first multiplier ends with one or more zeros.
For example:     
150 * 4 = 15 un * 4 = 60 un = 60
800 * 7 = 8 hundred * 7 = 56 hundred = 5600
18000 * 3 = 18 thousand * 3 = 54 thousand = 54000
27000 * 3 = 27 thousand * 3 = 81 thousand = 81000
In order to simplify the calculations, the teacher says that multiplication should be written as a priority, and children are shown that multiplication of a single-digit number 2700 into a multi-digit number can be used in solving 4 * 9687, 8 * 2084… examples.
 *
2700
  3
8100
Swimmers are then introduced to the method of multiplying nominal numbers expressed in units of measurement by one-digit numbers. To do this, the number is first expressed in smaller units of the same name, then operations are performed on unnamed numbers, and the result obtained is expressed in larger units: 8 kg 263 gr * 6 =
*
8263
      6
49578
In preparation for learning to split a multi-digit number into a single-digit number, it is necessary to first reconcile the meaning of the division operation in the swimmer's memory with its multiplication. Division is associated with multiplication. Divide 48 by 4, so when you multiply by 4, you get the number 48. This number is equal to 12. So, 48: 4 = 12. In this regard, the division rules with 1 and 0 are repeated. a: a = 1, a: 1 = a, 0: a = 0. is used to check the relationship between multiplication and division after division by multiplication.
For example:
Check that the division is done by multiplying: 95: 19 = 5. to learn written division it is necessary to strengthen the skills of numbering: to know the number of each room unit, the total number of units of each room, the upper room unit of the number, the number of digits to be assigned by the name of the upper room unit of the number.
In order to master the algorithm of written division of a single-digit number, the methods of verbal division of a multi-digit number into a single-digit number are introduced. In this case, the rule of dividing the sum by the number is the theoretical basis.
For example:
36963:3=(30000+6000+900+60+3):3=30000:3+6000:3+900:3+60:3+3:3=12321.
Then the examples are solved, which are expressed in the form of a set of divisible convenient conjunctions.
168:3=(150+18):3=150:3+18:3=50+6=56   
The algorithm for writing a one-digit number is explained as follows.
867
3
6
289
26
24
  27
  27
    0
Divisible 867 divisible 3. The first incomplete divisor is 8 hundredths. Divide 8 hundred by 3 and we get hundreds. Hundreds are written from the tenth to the third. So the upper room of the division is the room of hundreds, and there are three numbers in the division. The position of these numbers can be indicated by dots. Let's find out how many hundred there are in the division. We divide 8 hundred by 3. 2 hundred comes out. The number 8 is divisible by 3. 6 is divisible by 3 without remainder. 6: 3 = 2. we can see how many hundred there were. We multiply 2 hundred by 3. 6 hundred comes out. We find out how many hundred we are not divided. We divide 8 hundred by 6 hundred. 2 hundred comes out. Two hundred cannot be made into three hundred. We form a second incomplete divisor. We add 3 hundredths of this 2-ounce 20-ounce to 20 ounces. There will be 6 flours. Determine how many vowels there are in the division. Divide 26 flours by 26. 3 flour comes out. Let's find out how many tens we didn't find. We multiply 8 flours by 8. 3 flour comes out. Let's find out how many tens we have. We divide 24 by 24. 26 flour remains. Two flours cannot be made into 2 flours chikadiagn. We form a third incomplete divisor. 3 flours is 2 units. We add 20 units to 20 units. There will be 7 units. Determine how many units are divided in the division. We divide 27 units by 27. 3 units come out. We divide 9 units by 9. We multiply 3 units by 9. 3 units come out. We are all units. Bulinma 27.
In the explanation, special attention should be paid to the residue in the process of writing on the board, the need to grind them.
For example, when dividing 867 by 3, it is necessary to show that the divisor can be given by the sum of 6 hundred, 24 decimals and 27 units. (600 + 240 + 27 = 867). This allows the written division algorithm to be associated with dividing the sum by a number.
867:3=(600+240+27):3=200+80+9=289.
At the same time, the first incomplete divisor must have two digits, and the other divisor must have one less room than the divisor. This point of division is explained as follows. Divisible 376 divisible 4. we form the first incomplete divisor. The upper room of the divisor is the room of the hundreds. It is not possible to make 3 faces into 4 faces. We replace 3 hundredths with tens and add 7 tens. It turns out 37 flours, which means 37 flours that are divisible by the first whole. If we divide 37 flour by 4, the flours come out, so the top room of the division is the flour room. Decimals are written from the tenth to the second. So there are two numbers in the division. (They can be replaced by dots) 37 Divide 4 by 9. 4 unilik chikadi. All in all, we calculate how many flours there are. We multiply 9 by 36. 36 flour comes out. We divide 37 by 1. 4 flour comes out. One unlikdp 4 cannot be made into 1 unliks. We add 10 unit to these 6 units 10 units to 16 units. 4 units come out. Find all the units and get 94. Bulinma XNUMX.
-376
4
 36
94
 -16
  16
    0
When performing a one-digit number division, it is necessary to systematically require verification by multiplying the results. This strengthens the skill of multiplying a one-digit number. In the following lessons, the examples of division will be gradually complicated. Examples of divisions of 4-, 5-, 6-digit numbers are considered, followed by the following cases of division in which there are zeros in the middle or at the end of the division.
1) First we consider a case in which this or that incomplete divisible zero.
For example:
1509
3
15
503
  0 9
      9
      0
By dividing the first incomplete divisor (15 hundredths), it is determined that there are three numbers in the division. However, the first digit of the division is found (5 hundredths). The second incomplete divisible zero is separated by a decimal. Unit load in the flour room. They will not be found in the division. Divide 0 by 3, it turns out zero, the number of tens in this division is zero instead of the tens in the division. 9 units of the tenth incomplete divisor. We divide 9 units by 3. 3 units come out. The number 503 was formed in the division. The division of 503 * 3 = 1509 is done.
3680
4
36
920
  08
    8
     0
In this example, the first is divisible by 36, the second by 8, and the third by 0. This means that there are no units in the unit room, in which case zeros are written instead of units.
Then the following conclusion is drawn. If this or that divisor has zero, then zero must be written instead of the corresponding room in the divisor.
2) Divide the room units of the incomplete divisor by the cases when they are smaller than the divisor.
624
3
5424
6
6
208
54
904
  24
  024
  24
    24
    0
      0
A few lessons after learning to split, students are introduced to the short spelling of dividing multi-digit numbers into single-digit numbers.
9478
7
9478
7
7
1354
24
1354
24
  37
21
    28
  37
      0
  35
    28
    28
      0
Memory can be used for the written split algorithm. It specifies the order of operations:
  1. Read and write an example.
  2. Divide the first incomplete divisor, determine the number of the upper room and numbers of the division.
  3. Complete the division to find the unit of the upper chamber of the division.
  4. Perform multiplication to see how many units this room is divided into.
  5. Do the subtraction to know how many units of this room you need to know.
  6. check that the numeric value of the division is selected.
  7. If there is a residue, express it in terms of the room units that come after that room, and add to it the divisions of that room.
  8. Keep dividing until you solve the example.
  9. Check the result.
Such a scheme should be used from the first lesson, when the written division begins.
  1. II. Step. Multiplication and division by room numbers (multiplication and division by numbers ending in zero).
First, multiplication and division without residuals by 10, 100, 1000 are considered.
For example:
Let's multiply 14 by 10. 14 has 14 units. When it is multiplied by 10, each unit becomes ten. 14 units form 14 flours or 140.
After working on a few such examples, the conclusion is drawn: when any number is multiplied by 10, the multiplication produces a number with one zero written on the right side, represented by those numbers. Such an explanation is given for the division.
For example:
Divide 160 by 10. 160 This 16 is the unit of any flour divided by 10. Dividing 16 flours by 10 yields 16 units.
This means that dividing any number ending in zero by 10 yields as many units as there are tens in the division, and one zero must be left out of the divisor to form these units. Multiplication by 100, 1000 and division without remainder are explained in the same way. Then the cases of dividing any number by 10, 100, 1000 are considered.
1425: 10 = 142 (5 k)    
         1425: 100 = 14 (25 k)
1425: 1000 = 1 (425 k)
In this example, the number of zeros in the divisor is compared with the number of digits in the divisor. When dividing a remainder by 100, 1000, divide as many numbers as there are zeros in the divisor, starting from the right, and read this number as a remainder, and read the number formed by the numbers on the left as a division. The rule of multiplication by multiplication is the theoretical basis for multiplying multi-digit numbers by numbers ending in zeros, which will be explained later.
1) 6*(5*2)=6*10=60         2) 6*(5*2)=(6*5)*2=60      3) 6*(5*2)=(6*2)*10=60
it is necessary to draw the attention of swimmers to the simplest and most convenient calculations, which give numbers ending in zeros, in the performance of exercises for the expression, consolidation of this rule, and especially for solving problems in convenient ways.
For example:
25*(9*4)=(25*4)*9=100*9=900
18*(5*7)=(18*5)*7=90*7=630
25*6*7*4=(25*4)*(6*7)=100*42=4200
Then the method of multiplication of numbers ending in zeros is taught.
26*20=26*(2*10)=(26*2)*10=520
17*40=(17*4)*10=680
26*200=(26*2)*100=5200
13*300=(13*6)*100=7800
37*2000=(37*2)*1000=74000
78*70=(78*7)*10=78*10=5460
It is then used for a written calculation.
*
78
*
456
*
69
 10
    400
  8000
780
182400
552000
The case where both multipliers end in zero is of particular importance. First of all, the cases of 30 * 50, 800 * 60 and .. are considered. Such examples are easily solved orally. Such a consideration is made here. To find 800 * 60, multiply 8 faces by 6 and multiply the limit by 10. That would be 480 hundred or 48000. Writing the solution in a line will look like this.
800 * 60 = 8 hundred (6 * 10) = (8 hundred * 6) * 10 = 48 hundred * 10 = 480 hundred = 48000
Swimmers will then be introduced to the written multiplication method, where both multipliers end in zeros. Such multiplication is as follows:
*
8400
*
1370
*
4820
  70
    5000
   80
588000
6850000
385600
After solving a few of these examples, swimmers come to the rule of multiplying numbers ending in zeros. If the multipliers end in zeros, the multiplication is ignored, and the more zeros there are in both multipliers together, the more zeros are written next to the multiplication.
The rule of dividing a number by multiplication is the theoretical basis for dividing multi-digit numbers by numbers ending in zeros. Dividing a number by a multiplier can be done in three different ways.
For example:
32:(2*4)=32:8=4
32:(2*4)=32:2:4=16:4=4
32:(2*4)=32:4:2=8:2=4
In this case, this procedure is expressed. To divide a number by a product, you can find the product and divide the number by it. Divide the number by one of the multipliers and divide the result by another multiplier.
The multiplication rule is used to justify verbal division into two-digit numbers and to divide by numbers ending in zeros. In such a division, the divisor is expressed as the product of two convenient multipliers.
360:45=360:(9*5)=360:6-9:5=40:5=8
570:30=570:10:3=57:3=19
5400:900=5400:(100*9)=5400:100:9=54:9=6
31280:80=(24000+7200+80):80=300+90+1=391
31280
80
240
391
  728
  720
      80
      80
        0
Dividing into three, four, five-digit numbers ending in zeros is done in the same way as dividing into two-digit numbers ending in zeros.
III. Step. Multiplication and division by two-, three-digit numbers.
The theoretical basis for multiplication by two- and three-digit numbers is the multiplication rule, which was introduced to swimmers in Class III and was used to multiply a one-digit number by a two-digit number. Therefore, first of all, it is necessary to recall the rule of multiplication of a number by the verbal execution of multiplication by a two-digit number.
 
 
 
Масалан: 8*14=8*(10+4)=8*10+8*4=80+32=112
After that, more difficult cases will be considered. 98 * 74 = 98 * (70 + 4) = 98 * 70 + 98 * 4
*
98
*
98
*
6860
  70
  4
  392
6860
392
7252
The teacher says that the calculations can be written briefly and gives explanations about this record:
*
67
45
 Multiply 67 by 5. We form the first incomplete multiplication. 355. Then we multiply 67 by 40. To do this, multiply 67 by 4 and write zero next to the result. But we don't write it, we leave it blank, because adding zero doesn't change the number of units, we start writing the multiplication of 67 by 4 under the tens. The second incomplete product is 268 decimal or 2680. Add the incomplete product and find the final result. 3015. In this case 335 - the first complete multiplication, 268 - the second complete multiplication. 3015 The final result is the product of the numbers 67 and 45. Multiplying three, four, five-digit numbers by two-digit numbers, and then multiplying by three-digit numbers is explained in the same way. One of the main conditions for the successful formation of the skill of multiplication of multi-digit numbers by two-digit and three-digit numbers is the precise processing of each operation and their strict repetition. Special attention should be paid to the special cases of multiplication - the multiplication of numbers with zeros at the end and multiplication with zeros in the middle of the multipliers.
*
67
45
+
168
56
728
To multiply 560 by 13, you have to multiply 56 tens by 13, the tens come out, and by writing zero to the right, we convert it to units, which is equal to 7280.
*
256
208
+
2848
712
    74048
To multiply 356 by 208, multiply 356 by 8, then multiply 356 by 200 and add the results obtained, or multiply 356 by 8 to get the first incomplete multiplication. Multiply 356 by 200 to get the second incomplete product. It will be 712 hundred or 712000. Adding the results, 74048 is formed.
*
312
340
+
1248
936
106080
To multiply 312 by 340, multiply 312 by 34 and multiply by 10.
Introduction to the two-digit number division algorithm begins with a look at how to divide a three-digit number into a two-digit number in the case of a single-digit number in the division. In this case, the first two divisors are rounded to the nearest whole number. When dividing it, the counting of the division gives the necessary number, which can be incorrect, so it must be checked. When finding the number of divisions, the divisor can be rounded to the lower side or the upper side. It is advisable to replace the divisor with a small integer. Let 378 be divided by 63. First, a single number is determined in the division, because 37 flours cannot be divided into 63 flours. Then the method of division is explained as follows: we find the number of the division, we find a two-digit number ending in zero. In cases where the divisor is a two-digit number that does not end in zero, the divisor is rounded to make it easier to choose the division number, which is replaced by the nearest whole integer. We round the divisor. 60 is formed. Divide 378 by 60. How to do it? It is enough to divide 37 by 6. 6 chikadi. The number 6 is not definite, it must be counted because 378 is required to be divided by 60, not 63. This number needs to be checked. We multiply 63 by 6. 378 chikadi. So we write the number 6 in the division. It reads:
- 378
63
   378
6
    0
The method of dividing four, five, six-digit numbers into two-digit numbers is considered. Let's see how to explain writing in these cases.
-29736
56
  280
531
  -173
    168
      -56
        56
          0
The divisor is 29736, the divisor is 56. The first total divisor is 297, there are three numbers in the division (we put three dots in their place in the division). To find the first number of the division, we round the divisor and divide 297 by 50. To do this, divide 29 by 5 to get 5 in sufficient division. Number 5 is a test number, let's check it. We multiply 56 by 5. 280 chikadi. We divide 280 by 297. There are 17 hundred left in the colony. It is not possible to make 17 hundredths into 56s. So, the number 5 is selected correctly. The second incomplete division is 173 decimals. To find the second number of the division, we divide 173 by 50. It is enough to divide 17 by 5. 3 chikadi. Number 3 is the number to be tested, we will check it. Multiply 56 by 3 to get 168. We subtract 168 from 173. 5 flour remains. 5 flours cannot be divided into 56s, so the second number 3 is the third incompletely divisible 56 units. Divide 56 by 56 to find the third number of the division. 1 comes out. Division 531. Let's check 531 * 56 = 29736
*
531
56
+
3186
2655
29736
As the skill of division increases, perfect explanations are gradually replaced by shorter explanations. In all the above cases of dividing a two-digit number, the test number of the division cannot always be found with a single test. To illustrate this, let us determine that 186:26 is a single number in the division before the variant. Divide 18 by 2 to find the number of the division. 9 chikadi. Multiply 9 by 26 to make sure 9 is selected correctly.
26*9=(20+6)*9=180+54=234, демак 234>182
The number 9 does not match. We get one less of the number to be tested. We get 8. But it's big.
26*8=(20+6)*8=160+48=208. 208>182. демак, 7 ракми тугри келади, чунки 26*7=(20+6)*7=20*7+6*7=140+42=182.
In this case we found a reliable number of the division after three trials. Particular attention should be paid to the methods of dividing a two-digit number in the case of the formation of zeros in the middle of the division.
For example: Let's divide 30444 by 43.
-30444
43
  301
708
  -344
    344
      0
The first incomplete divisor is 304. There are three numbers in the division (in the division we put three dots instead). To divide 304 by 43, it is enough to divide 30 by 4. 7 comes out, this should be tested. Let's check it out. We multiply 43 by 7. 301 comes out. Divide 301 by 304. 3 hundred left. 3 hundred cannot be made into 43 hundred. So, the number 7 is chosen correctly. The second incomplete divisor 37 is not divisible by 34 into 43, so it is not possible to make one flour out of one. This means that there are no tens in the division. In the division, we write zero instead of tens. Dividing 344 by 43 is enough to divide the third incomplete divisor 34 by 4, which is a test number. Let's check it out. We multiply 8 by 43. 8 chikadi. We found all the units. The number 344 comes true. Check: Multiply the division 8 by 708. 43 * 708 = 43.
Simultaneously with the division of anonymous numbers, the division of numbers expressed in metric measurements into two-digit numbers is also considered. There are two ways to do this: one is to divide the named numbers into unnamed numbers and to divide the named numbers into named numbers. In both cases, the division of a complex named number is reduced to the division of a simple named number, and operations are performed on the corresponding anonymous numbers: 35 сум 64 teen : 18 ga = 1 сум 98 tiyin. 48 м 24 cm : 36 cm= 134
-3564
18
-4824
36
  18
198
  36
134
-176
 -122
  162
   108
  -144
    -144
    144
      144
        0
          0
The method of dividing multi-digit numbers into three-digit numbers is similar to the method of dividing two-digit numbers. The difference is that to find the number of a division, the divisor is replaced by a close integer ending in two zeros.
For example: After dividing a three-digit number, we look at the score
In this case, the number of the division is found after three tests. The first incomplete divisible 3602 flour. There are two numbers in the division. Choosing a division number is easy. We round the divisor to be divisible.
-3564
18
  18
198
-176
  162
  -144
To do this, we replace it with the nearest small three-digit integer. It will be 600. Divide 3602 ​​by 600 to divide 36 by 6. Let's check this number: 6 632 = 6. This number does not correspond to a larger number than the known one. We get 3792. Let's check.5 * 632 = 5. 3160 <3160. 3602 rakamitugri is coming. We find it divisive. Let's find out how many tens we didn't find. 5 - 3602 = 3160.
 The number of tens is less than 632, which means that we have found the first number of the division. Dividing 4424 by 600 is enough to divide 44 by 6 to get the second incomplete division. By checking, we see that the number 7 is correct. Bulinma 7.
The ability to divide a multi-digit number into two or three-digit numbers is gradually formed. Therefore, the amount of exercises that form the division skill should be large.
Control questions:
  1. At what level are multiplication and division of multi-digit numbers taught?
  2. How to teach multiplication and division of multi-digit numbers into single-digit numbers?
  3. How to multiply room numbers?
  4. How to divide by room numbers?
  5. How many ways is it taught to multiply a number by a factor?
  6. How many ways is it divided to multiply a number?
  7. How is incomplete multiplication formed?
  8. How to divide multi-digit numbers into two- and three-digit numbers?
  9. How to teach multiplication and division of nominal numbers?
 
Zachet questions:
  1. What are the main tasks of teaching mathematics in primary school?
  2. What are the main tasks of preparing for an elementary mathematics course?
  3. List the features of an elementary math course?
  4. What is the content of the arithmetic, algebra, geometry part of the elementary school curriculum?
  5. What is meant by teaching methods?
  6. What is the classification of teaching methods, name them?
  7. What oral teaching methods are used in primary school?
  8. How do instructional and oral teaching methods relate to each other?
  9. What is the essence of the methods of induction, deduction and analogy?
  10. What mental operations underlie the use of induction, deduction and analogy methods?
  11. What is meant by independent teaching?
  12. What types of independent work are there?
  13. What is the value of a didactic house?
  14. Justify the need to use different teaching methods in the lesson?
  15. What is meant by teaching aids and what are their main functions?
  16. What is a textbook task and how does it relate to the program?
  17. In what direction can work with the textbook be carried out?
  18. What types of tutorials are available in math teaching?
  19. What are the natural guidelines?
  20. What are the visual aids? Give examples.
  21. What questions are initially used to study the numbers in flour?
  22. At what stage is the numbering in flour taught?
  23. What concepts are used in the preparatory phase of learning to number numbers?
  24. How is the number presented?
  25. How many numbers are involved in numbering?
  26. How is each of the unta numbers formed?
  27. What didactic games are used to study the composition of numbers with two additions?
  28. What is the order of numbers?
  29. How to enter the number zero?
  30. How many steps does it take to learn to number numbers on a face?
  31. How to verbally number the numbers on the face?
  32. Do you have a written numbering?
  33. Writing the numbers on the face is subject to the Canadian procedure?
  34. How is the comparison of the numbers inside the face done?
  35. How many hundreds, how many units are there in 25?
  36. Which number consists of 3 decimals and 7 ones?
  37. How many steps are used to number numbers in a thousand?
  38. What is the position of units, tens, and hundreds in three-digit numbers from right to left?
  39. How to read a number with three digits, knowing the numerical values ​​of the number?
  40. How is voice numbering done?
  41. How is written numbering done?
  42. What is the purpose of teaching you to count to hundreds?
  43. What is the purpose of a set of cards with numbers?
  44. What is being done in preparation for the numbering of thousands?
  45. The preparation phase for digitizing multi-digit numbers puts Canadian goals in front of you?
  46. The concept of class is introduced in Canada?
  47. How many room units are there in a class?
  48. Say the room names of the one class.
  49. How many rooms are there in a class of thousands?
  50. How is the comparison of multi-digit numbers done?
  51. What is meant by room addicts?
  52. When studying multi-digit numbers, do you pay attention to the value of the numbers?
  53. Which method is used to add and subtract, multiply, and divide non-negative integers?
  54. What is the verbal calculation method?
  55. How is the written calculation method performed?
  56. At what stages are addition and subtraction of numbers in flour taught?
  57. Explain the first step?
  58. How is the second stage carried out?
  59. What laws are used to perform the addition?
  60. How is the division of numbers in flour taught?
  61. What methods are used to teach arithmetic operations?
  62. Canadian didactic games are used to learn arithmetic operations?
  63. What is done in the preparatory phase of learning to add and subtract numbers on the face?
  64. How many different methods of calculation are used in the study of addition and subtraction of numbers in the face?
  65. How is the verbal calculation performed (addition, subtraction)?
  66. How to use the laws of addition in performing arithmetic operations on the subject of hundreds?
  67. Why is the law of substitution used?
  68. What is considered in written addition and subtraction?
  69. How to add and subtract a number?
  70. How do you add a sum to a sum?
  71. How is the meaning of multiplication taught?
  72. How is the meaning of the act of division taught?
  73. What number is multiplied by 0 and 1?
  74. How many different ways is a multiplication table created?
  75. What properties are used in the study of multiplication and division outside the table?
  76. How many different ways are there to multiply and divide a sum by a number?
  77. How to divide and multiply a two-digit number by a one-digit number?
  78. How to teach multiplication and division of numbers ending in zero?
  79. How to test multiplication and division?
  80. How is the meaning of division divided?
  81. In what ways is the division of a two-digit number into a two-digit number taught?
  82. What is the relationship of the residue from the split to the divisor?
  83. How is verbal addition and subtraction taught in a thousand?
  84. How is written addition and subtraction taught in thousands?
  85. In what order is written multiplication taught on the subject of the millennium?
  86. In what order is the written addition of numbers in thousands taught?
  87. How to teach multiplication of numbers in a thousand? (oral and written)
  88. How to teach oral and written division of numbers in a thousand?
  1. How to add multi-digit numbers?
  2. How is multiplication of multi-digit numbers taught?
  3. How to add and subtract nominal numbers?
  4. How to teach addition and subtraction of multi-digit numbers?
  5. At what level are multiplication and division of multi-digit numbers taught?
  6. How to teach multiplication and division of multi-digit numbers into single-digit numbers?
  7. How to multiply room numbers?
  8. How to divide by room numbers?
  9. How many ways is it taught to multiply a number by a factor?
  10. How many ways is it divided to multiply a number?
  11. How is incomplete multiplication formed?
  12. How to divide multi-digit numbers into two- and three-digit numbers?
  13. How to teach multiplication and division of nominal numbers?
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Public lesson
 
Subject: Znakomstvo uchashchixsya s prostymi zadachami
Purpose:
        Oznakomit studentov s priemami obucheniya resheniyu prostyh zadach;
       Encourage the application of teaching methods in practice;
                      The plan:
  1. Obshchie voprosy metodiki obucheniya resheniyu prostyh zadach.
  2. Podgotovitelnaya rabota k resheniyu zadach.
  3. Classification prostyh zadach.
  4. Modeling as a means of shaping the ability to solve tasks.
Basic literature.
  1. Bantova M.A., Beltyukova G.V. Methods of teaching mathematics in elementary school. - M .: «Prosveshchenie», 1984
  2. Istomina N.B. Methods of teaching mathematics in elementary school.
M. 98.
Additional literature.
  1. Volkova S.I. Kartochki s matematicheskimi zadaniyami 4 kl. M .: «Prosveshchenie», 1993
  2. Gnedenko B.V. Formation of mirovozzreniya uchashchixsya in the process of learning mathematics. - M .: «Prosveshchenie», 1982. - 144 p .- (Biblioteka uchitelya matematiki).
  3. Green R., Lakson D. Introduction to the world of numbers. - M .: 1984
  4. Dalinger V.A. Methods of realizatsii vnutripredmetnyx svyazey pri obuchenii matematike. - M .: «Prosveshchenie», 1991
  5. Jikolkina T.K. Mathematics. Книга для учителя. 2 kl. - M .: «Drofa», 2000
Obshchie voprosy metodiki obucheniya resheniyu prostyh zadach
Nauchit detey reshat zadachi - znachit nauchit ix ustanavlivat svyazi mejdu dannymi i iskomym i v sootvetstvii s etim vybirat, a zatem i vыpolnyat arifmeticheskie deystviya.
Tsentralnыm zvenom v umenii reshat zadachi, kotorыm doljnы ovladet uchashchiesya, yavlyaetsya usvoenie svyazey mejdu dannymi i iskomym. At togo, naskolko xorosho usvoenы uchashchimisya eti svyazi, zavisit ix umenie reshat zadachi. Uchityvaya eto, v nachalnyx klassax vedetsya rabota nad gruppami zadach, reshenie kotoryx osnovyvaetsya na odnix i tex je svyazyax mejdu dannymi i iskomym, a otlichayutsya oni konkretnыm soderjaniem i chislovymi dannymi. Group takix zadach nazыvayutsya zadachami odnogo screw.
According to Bantovoy M.A. rabota nad zadachami ne doljna svoditsya k nataskivaniyu uchashchixsya na reshenie zadach snachala odnogo vida, zatem drugogo i t. d. Home tsel - nauchit detey osoznanno ustanavlivat opredelennыe svyazi mejdu dannymi i iskomym v raznyx jiznennyx situatsiyax, predusmatrivaya postepennoe ix uslojnenie. Chtoby dobitsya etogo, uchitel doljen predusmotret v metodike obucheniya resheniyu zadach kajdogo vida takie stupeni:
1) podgotovitelnuyu rabotu k resheniyu zadach;
2) oznakomlenie s resheniem zadach;
3) zakreplenie umeniya reshat zadachi.
Consider the detailed method of work on each of the so-called stupeney.
 Podgotovitelnaya rabota k resheniyu zadach
Na etoy pervoy stupeni obucheniya resheniyu zadach togo ili drugogo vida doljna byt sozdana u uchashchixsya gotovnost k vыboru arifmeticheskix deystviy pri reshenii sootvetstvuyushchix zadach: oni doljny usvoit znanie tex svyazey, o ziasye kotoriya, na znutsya zadachax.
Do resheniya prostyh zadach ucheniki usvaivayut znanie sleduyushchix svyazey:
1) Svyazi operatsiy nad mnojestvami s arifmeticheskimi deystviyami, t. e. konkretnыy smysl arifmeticheskix deystviy. For example, the operation of uniting neperesekayushchixsya mnogestv connected with the action of the slozheniya: esli imeem 4 da 2 flajka, to, chtoby uznat, skolko vsego flajkov, nado k 4 pribavit 2.
2) Svyazi otnosheniy «bolshe» i «menshe» (pa neskolko edinits i v neskolko raz) s arifmeticheskimi deystviyami, t. e. konkretnыy smыsl vyrajeniy «bolshe na. . . »,« Bolshe v… raz »,« menshe na. . . »,« Menshe v. . . raz ». For example, bolshe na 2, eto stolko je. i eshche 2, znachit, chtoby poluchit na 2 bolshe, chem 5), nado k 5 pribavit 2.
3) Svyazi mezhdu komponentami i rezultatami arifmeticheskix deystviy, t. e. pravila naxojdeniya odnogo iz komponentov arifmeticheskix deystviy po izvestnym rezultatu i drugomu component. For example, if the famous sum and one of the slagaemyx, to drugoe slagaemoe naxoditsya deystviem vыchitaniya: iz summы vыchitayut izvestnoe slagaemoe.
4) Svyazi mezhdu dannymi velichinami, naxodyashchimisya v pryamo ili obratno proportsionalnoy zavisimosti, i sootvetstvuyushchimi arifmeticheskimi deystviyami. For example, if you know the value and quantity, then you can find stoimost deystviem umnojeniya.
Krome togo, pri oznakomlenii s resheniem pervyx prostykh zadach ucheniki doljny usvoit ponyatiya i terminy, otnosyashchiesya k samoy zadache i ee resheniyu (zadacha, uslovie zadachi, vopros zadachi, reshenie zadachi, otvet na vopros zapach.
 Classification prostyh zadach
Prostye zadachi mojno razdelit na gruppy v sootvetstvii s temi arifmeticheskimi deystviyami, kotorymi oni reshayutsya.
Odnako v metodicheskom otnoshenii udobnee drugaya klassifikatsiya: delenie zadach na gruppy v zavisimosti ot tex ponyatiy, kotorye formiruyutsya pri ix reshenii. Mojno vydelit tri takie gruppy. Oxarakterizuem kajduyu iz nix.
K pervoy gruppe otnosyatsya prostye zadachi, pri reshenii kotoryx deti usvaivayut konkretnыy smыsl kajdogo iz arifmeticheskix deystviy.
In this group five tasks:
1) Finding the sum of two chisels. The girl took 3 deep plates and 2 small ones. How many plates did the girl take?
2) Finding the residue. Bylo 6 yablok. Two apples s'eli. How many left?
3) Finding the sum of odinakovyx slagaemyx (proizvedeniya).
In a living corner jili kroliki in trex kletkax, po 2 krolika in kajdoy. How many rabbits in a living corner?
4) Distribution on a regular basis. U dvux malchikov bыlo 8 confetti, u kajdogo porovnu. How many candies did the little boy have?
5) Content of content.
Kajdaya brigada shkolnikov posadila po 12 derevev, a vsego oni posadili 48 derevev. How many brigades performed this work?
Ko vtoroy gruppe otnosyatsya prostye zadachi, pri reshenii kotoryx uchashchiesya usvaivayut svyaz mezhdu komponentami i rezultatami arifmeticheskix deystviy. K nim otnosyatsya zadachi na naxojdenie neizvestnyx komponentov.
1) Naxhoddenie pervogo slagaemogo po izvestnym summe i vtoromu slagaemomu.
Devochka vыmyla neskolko glubokix tarelok i 2 melkie, a vsego ona vыmyla 5 tarelok. How many deep plates did the girl take?
2) Finding the second slagaemogo on the known sum and the first slagaemomu.
The girl took out 3 deep plates and several melkix. Vsego ona vыmyla 5 tarelok. How many small plates did the girl have?
3) Nakhozdenie umenshaemogo po izvestnym vыchitaemomu i raznosti. Deti made several skvorechnikov. Kogda 2 skvorechnika oni povesili na derevo, to u nix ostalos eshche 4 skvorechnika. How many skvorechnikov did these things?
4) Nakhodnenie vыchitaemogo po izvestnym umenshaemomu i raznosti.
Deti made 6 skvorechnikov. When several skvorechnikov oni povesili na derevo, u nix eshche ostalos 4 skvorechnika. How many skvorechnikov did these children tell on the tree?
5) Finding the first mnogitelyu on izvestnym proizvedeniyu and the second mnogitelyu.
Neizvestnoe chislo umnojili na 8 i poluchili 32. Nayti neizvestnoe chislo.
6) Nakhozdenie vtorogo mnojitelya po izvestnym proizvedeniyu i pervomu mnojitelyu.
9 umnojili na neizvestnoe chislo i poluchili 27. Nayti neizvestnoe chislo.
7) Nakhoddenie delimogo po izvestnym delitelyu i chastnomu.
Neizvestnoe chislo razdelili na 9 i poluchili 4. Find neizvestnoe chislo.
8) Finding the deletion according to the known delimomu and chastnomu.
24 razdelili na neizvestnoe chislo i poluchili 6. Nayti neizvestnoe chislo.
K tretey gruppe otnosyatsya zadachi, pri reshenii kotoryx raskrыvayutsya ponyatiya raznosti i kratnogo otnosheniya. K nim otnosyatsya prostye zadachi, svyazannыe s ponyatiem raznosti (6 types), i prostye zadachi, svyazannыe s ponyatiem kratnogo otnosheniya (6 types).
1) Differential comparison of chisel or naxojdenie raznosti dvux chisel (I vid).
Odin dom postroili for 10 weeks, and drugoy for 8 weeks. How many weeks have passed since the construction of the first house?
2) Differential comparison of chisel or naxojdenie raznosti dvux chisel (type II).
Odin dom postroili za 10 nedel, a drugoy za 8. Na skolko nedel menshe zatratili na stroitelstvo vtorogo doma?
3) Uvelichenie chisla na neskolko edinits (pryamaya forma). Odin dom postroili za 8 nedel, a na stroitelstvo vtorogo doma zatratili na 2 nedeli bolshe. How many weeks have passed since the construction of the second house?
4) Uvelichenie chisla na neskolko edinits (kosvennaya forma).
On the construction of one house took 8 weeks, on this 2 weeks less, on the construction of the second house. How many weeks have passed since the construction of the second house?
5) Increase the number of several edits (pryamaya form).
On the construction of a single house zatratili 10 weeks, and drugoy postroili on 2 weeks bystree. How many weeks did you build the second house?
6) Increase in the number of units (indirect form).
Na stroitelstvo odnogo doma zatratili 10 nedel, eto na 2 nedeli bolshe, chem zatracheno na stroitelstvo vtorogo doma. How many weeks did you build the second house?
Zadachi, svyazannыe s ponyatiem kratnogo otnosheniya. (Ne privodya primery)
1) Short comparison of chisel or nakhodzhenie kratnogo otnosheniya dvux chisel (I vid). (How much more?)
2) Short comparison of chisel or nahodzhdenie kratnogo otnosheniya dvux chisel (type II). (How many times?)
3) Uvelichenie chisla v neskolko raz (pryamaya forma).
4) Increase the number in several times (indirect form).
5) Umenshenie chisla v neskolko raz (pryamaya forma).
6) Increased number of times (indirect form).
Zdes nazvanы tolko osnovnye vidy prostyh zadach. Odnako oni ne ischerpyvayut vsego mnogoobraziya zadach.
Poryadok vvedeniya prostyh zadach podchinyaetsya soderjaniyu programmnogo materiala. V I klasse izuchayutsya deystviya slozheniya i vychitaniya i v svyazi s etim rassmatrivayutsya prostye zadachi na slojenie i vychitanie. In II class v svyazi s izucheniem deystviy umnojeniya i deleniya vvodyatsya prostye zadachi, reshaemыe etimi deystviyami.
Modeling as a means of shaping the ability to solve tasks. Vidy modeling.
Graficheskoe modelirovanie as osnovnoe means
Glubina i znachimost otkrыtiy, kotorye delaet mladshiy shkolnik, reshaya zadachi, opredelyaetsya harakterom osushchestvlyaemoy im deyatelnosti i meroy ee osvoeniya, tem, kakimi sredstvami etoy deyatelnosti on vladeet. For that mine uchenik uje in nachalnyx klassax mog vydelit and osvoit sposob resheniya shirokogo klassa zadach, a ne ogranichivalsya naxojdeniem otveta v dannoy, konkretnoy zadache, on doljen ovladet nekotorыmy teoreticheskimi zneniami o zacheiy o zachey o zache
Well-known psychologist A.N. Leontev wrote: "Aktualno soznaetsya tolko to soderjanie, kotoroe yavlyaetsya predmetom tselenapravlennoy aktivnosti subъekta." Poetomu, chtoby struktura zadachi stala predmetom analiza i izucheniya, neobxodimo otdelit ee ot vsego nesushchestvennogo i predstavit v takom vide, kotoryy obespechival by neobxodimye deystviya. Sdelat it mojno putem osobyx znakovo-simvolicheskix sredstv - modeley, odnoznachno otobrajayushchix struktura zadachi i dostatochno prostyh dlya vospriyatiya mladshimi shkolnikami.
In the structure lyuboy zadachi vыdelyayut:
  1. Subject area, t. e. objects, o kotoryx idet rech v zadache.
  2. Otnosheniya, kotoryye svyazыvayut obъekty predmetnoy oblasti.
  3. Trebovanie zadachi.
Objects of the task and the relationship between the conditions of the task. For example, in the task: «Lida narisovala 5 domikov, and Vova - na 4 domika bolshe. How many houses did Vova draw? ” - Objects yavlyayutsya:
  • kolichestvo domikov, narisovannyx Lidoy (this is a well-known object in the task);
2) kolichestvo domikov, narisovannyx Vovoy (это неизвестный объек в задаче и согласно требованийу искомый).
Svyazыvaet objects otnoshenie «bolshe na».
The structure of the task can be presented with the help of different models. But first, chem sdelat eto, utochnim nekotorye voprosy, svyazannыe s klassifikatsiey modeley i terminologiey.
All models prinyato delit na schematizirovannыe i znakovыe.
In svoy ochered, schematizirovannыe models bыvayut veshchestvennymi (oni obespechivayut fizicheskoe deystvie s predmetami) and graficheskimi (oni obespechivayut graficheskoe deystvie).
K graficheskim modelyam otnosyat risunok, uslovnыy risunok, chertej, schematicheskiy chertej (ili schemu).
Znakovaya model zadachi mojet vыpolnyatsya kak na estestvennom yazyke (t. E. Imeet slovesnuyu formu), tak i na matematicheskom (t. E. Ispolzuyutsya simvolы).
For example, znakovaya model rassmatrivaemoy zadachi, vыpolnennaya na estestvennom yazyke, - eto obshcheizvestnaya kratkaya zapis:
Znakovaya model dannoy zadachi, vыpolnennaya na matematicheskom yazyke, imeet vid vyrajeniya 5 + 4.
Uroven ovladeniya modelirovaniem opredelyaet uspex reshayushchego. Poetomu obuchenie modelirovaniyu zanimaet osoboe i glavnoe mesto v formirovanii umeniya reshat zadachi.
Lavrinenko T.A. predlagaet sleduyushchie priemы predmetnogo modelirovaniya prostyh zadach na slojenie i vychitanie: s dochislovogo perioda nachinat vыpolnyat prakticheskie uprajneniya po vsem vidam zadach, obъyasnyaya poluchennыy rezultat i vыborochno zarradiovы
- Put three red mugs, and put 5 blue mugs. How many circles did you put in?
3 8 5 - Put 6 squares, and teper 2 uberite. How many squares are left? 6 2
- Put three circles, and below put 2 squares more. How many squares are there? How did you put the square? 3 2
- Put 7 yellow treugolnikov, and below the red treugolnikov put 3 menshe, chem zheltyx. How many red triangles are there? How are you? 7 3
- Put 5 squares. Nije put 3 circles. Chego bolshe? How much more? How are you? 5 3
After znakomstva so znakami «+» i «-» neobxodimo prodoljit vыpolnenie prakticheskix uprajneniy, primenyaya graficheskoe modelirovanie, vvodya teksty zadach i vybiraya nujnoe deystvie.
- On the side of the branch 8 ptichek (put 8 sticks), 3 pletichki uleteli (otodvinuli 3 sticks). How many birds are left? What action do we choose? (Otodvinuli, znachit, «vychitanie»).
8-3 = 5 (pt.)
- U Koli 5 mashinok (put 5 kvadratikov), and u Sereji na dve mashinki menshe (vыlojite mashinki Sereji krujochkami.) How many machines u Sereji? What action do we choose? Why? (My zakrыli dva kvadrata, a skolko ostalos - stolko vыlojili kruzhkov. Ubrali 2 kvadrata, znachit, vыpolnili deystvie «vychitanie»).
     5-2 = 3 (m.)
2 Uchim pravilo «Na… menshe - delaem vychitanie»
- U Kati 6 krasnyx sharov (vykladыvaem 6 krasnyx mugkov) i 4 sinix (vykladыvaem vnizu 4 sinix mugka). Na skolko u Kati krasnyx sharov bolshe, chem sinix?
- How can we find so many red sharov? (Nuzhno iz krasnyx otodvinut stolko, skolko sinix, uznaem na skolko bolshe krasnyx sharov).
- What actions do we choose? (My otodvinuli shary, znachit, deystvie «vychitanie»).
6-4 = 2 (sh). ?
I'm right, "Mine will compare, how many odo chislo bolshe drugogo, nujno iz bolshego chisla vыchest menshee".
Itak, tselenapravlennaya rabota po formirovaniyu priemov umstvennoy deyatelnosti nachinaetsya s pervyx urokov matematiki pri izuchenii temy “Otnosheniya ravenstva-neravenstva velichin”. Deystvuya s razlichnymi predmetami, pytayas zamenit odin predmet drugim, podxodyashchim po zadannomu priznaku, deti vыdelyayut parametry veshchey, yavlyayushchiesya velichinami, t.e. svoystva, for kotoryx mojno ustanovit otnosheniya ravno, neravno, bolshe, menshe. In context zadach children znakomyatsya with dlinoy, massoy, ploshchadyu, obъemom. Poluchennыe otnosheniya modeliruyutsya snachala s pomoshchyu predmetov, graficheski (otrezkami), and then - bukvennymi formulami.
Na pervyx je urokax nujno poznakomit detey s pryamoy i krivoy liniey, a zatem s ponyatiem otrezka i nauchit chertit otrezki po lineyke. For this purpose it is possible to carry out uprajnenie sleduyushchego vida:
After that, as children, they will be selected in the concept of "task", you can learn how to create tasks on the pictures, pricem vse vidy zadach. Here it is useful to use drawings and schematic drawings, block diagrams, modeling with the help of cuts, tables and matrices.
Graficheskie models i tablitsy pozvolyayut sravnivat pary ponyatiy: levaya - pravaya, verxnyaya - nijnyaya, uvyazыvat prostranstvennuyu informatsiyu (pravaya - levaya) s informatsiey mery (shirokaya - uzkaya, korotkaya - dlinnaya) tem resym formiruya umir. Primerom can serve table:
Korotkaya (levaya)
Dlinnaya (right)
Shirokaya (verxnyaya)
Uzkaya (nijnyaya)
V besede so shkolnikami po etoy matritse sleduet zadavat protivopo-lojnye po soderjaniyu voprosy.
Question: kakaya lenta narisovana v pravoy nijney kletke? Answer: long and narrow. Question: where narisovana korotkaya i shirokaya lenta? Answer: in the left upper cell.
Tablichnye primer udobny dlya bыstrogo resheniya primerov, informatsionno svyazannyx drug s drugom. Tak, naprimer, zapolnyaya kletki tablitsy, shkolniki doljny obratit vnimanie na sovpadenie parnyx summ, naprimer: 35 + 47 = 45 + 37 = 82.
 
 
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