These materials make up the fourth group and are intended for the gradual memorization of tables of addition, subtraction, multiplication and division of numbers. As a result of working with these materials, the child should learn to freely perform in his mind the addition and multiplication of single-digit numbers and their inverse operations: subtraction, if the subtrahend and the difference are single-digit numbers, and division without a remainder by a single-digit divisor if the dividend does not exceed 81. The materials are divided into four series corresponding to four arithmetic operations. Each series ends with test material - a blank card, which the child fills out, performing actions in his mind. To orient the child and the teacher, arithmetic operations, and therefore the materials of each series, are assigned a specific color: addition - red, subtraction - green, multiplication - yellow, division - blue.

Series 1 "Addition" includes the following materials: "snake game" for addition, a board with stripes and test cards 1 and 2 for addition, work cards 3, 4, 5 and 6 for addition exercises.

"Snake game" for addition- the child turns a snake from colored rods into a gold one, while practicing counting and replacing combinations of colored rods representing 2 or more single-valued terms with gold and black and white rods corresponding to their sum.

Board with stripes for folding; control cards 1 and 2 for addition- the material gives the child the opportunity to solve all addition examples single digit numbers from 1 + 1 to 9 + 9 using strips corresponding to numbers from 1 to 9. Control cards are used to check the correctness of the solution.

Work cards3-6 for addition exercises- the name of the material speaks for itself. Using cards 3-5, numerous exercises on adding single-digit numbers are performed; card 6 is a test one.

Subtraction Series 2 includes the following materials: subtraction snake game, stripe board and subtraction practice card 1, subtraction practice cards 2 and 3.

"Snake game" for subtraction- the material is similar to the “snake game” for addition, but gray rods have been added to it, indicating subtraction. The colored snake is turned back into gold, with the gray beads "eating" the colored ones and the child essentially practicing subtraction.

Board with stripes for subtraction; test card 1 for subtraction- the material allows, using strips corresponding to numbers from 1 to 9, to solve all subtraction examples when the subtrahend and the difference are single-digit numbers. Subtraction card 1 is used to control errors.

Worksheets 2 and 3 for subtraction exercises- designed for subtraction exercises. Card 3 is a test card.

Multiplication Series 3 includes the following materials: Multiplication with Bead Rods, Multiplication Board with Multiplication Practice Cards 1 and 2, Multiplication Practice Cards 3-5.

Multiplication with bead rods - Using this material, the child consistently solves all examples of multiplying single-digit numbers. He can also lay out a multiplication table using colored rods. The product of 2 numbers looks like a rectangle or square.

Multiplication board - Using concrete bead material, the child learns to multiply numbers up to 10 inclusive and completes all examples from 1 x 1 to 10 x 10. Control cards 1 and 2 are used to check the correctness of the results. The child again sees the result of multiplying two numbers in the form of a rectangle or square with the corresponding side lengths. He also becomes acquainted with the commutative law of multiplication.

Work cards3- 5 for multiplication exercises - give the child the opportunity to practice multiplying numbers, each of which does not exceed 10. Card 5 is a test card.

Let us finally list the materials of the 4th series: division board, work cards 1 and 2 for division exercises.

Division board- with the help of specific material made from beads, the child can solve division examples, including those with a remainder, where the divisor and quotient are single-digit numbers.

Worksheets 1 and 2 for division exercises- used for exercises on division without remainder. Card 2 is a test card.

MINISTRY OF EDUCATION OF THE REPUBLIC OF KAZAKHSTAN

KOSTANAY STATE PEDAGOGICAL INSTITUTE

Essay

On the topic of: “Adding and subtracting, dividing and multiplying ordinary fractions.”

Kostanay

1. From the history of ordinary fractions………………………………………………………..3

2. Actions with ordinary fractions..................................................5

2.1. Addition and subtraction of ordinary fractions……………………........5

2.2. Multiplication and division of ordinary fractions ………………………….7

3. Examples for adding, subtracting, multiplying and dividing fractions……. 10

4. List of references………………………………………………………...11

1. From the history of the emergence of ordinary fractions.

Fractions appeared in ancient times. When dividing up spoils, when measuring quantities, and in other similar cases, people encountered the need to introduce fractions.

The ancient Egyptians already knew how to divide 2 objects into three people; for this number -2/3- they had a special symbol. By the way, this was the only fraction used by Egyptian scribes that did not have a unit in the numerator - all other fractions certainly had a unit in the numerator (the so-called basic fractions): 1/2; 1/3; 1/28; … . If the Egyptian needed to use other fractions, he represented them as a sum of base fractions. For example, instead of 8/15 they wrote 1/3+1/5. Sometimes it was convenient. In the Ahmes papyrus there is a task:

“Divide 7 loaves among 8 people.” If you cut each loaf into 8 pieces, you will have to make 49 cuts.

And in Egyptian this problem was solved like this: The fraction 7/8 was written in the form of fractions: 1/2+1/4+1/8. This means that each person should be given half a loaf, a quarter of a loaf, and an eighth of loaf; Therefore, four loaves were cut in half, two loaves into 4 parts and one loaf into 8 shares, after which everyone was given a part of it.

But adding such fractions was inconvenient. After all, both terms can contain equal parts, and then upon addition a fraction of the form 2/n will appear. But the Egyptians did not allow such fractions. Therefore, the Ahmes papyrus begins with a table in which all fractions of this type from 2/5 to 2/99 are written as a sum of shares. This table was also used to divide numbers. Here, for example, is how 5 was divided by 21: 5/21

The Egyptians also knew how to multiply and divide fractions. But to multiply, you had to multiply fractions by fractions, and then, perhaps, use the table again. The situation with division was even more difficult.

In ancient Babylon they preferred the opposite - a constant denominator of 60. Sexagesimal fractions, inherited from Babylon, were used by Greek and Arab mathematicians and astronomers. But it was inconvenient to work on natural numbers written in the decimal system and fractions written in the sexagesimal system. But working with ordinary fractions was already quite difficult. Therefore, the Dutch mathematician Simon Stevin proposed switching to decimal fractions.

An interesting system of fractions was in ancient Rome. It was based on dividing a unit of weight into 12 parts, which was called ass. The twelfth part of an ace was called an ounce. And the path, time and other quantities were compared with a visual thing - weight. For example, a Roman might say that he walked seven ounces of a path or read five ounces of a book. In this case, of course, it was not about weighing the path or the book. This meant that 7/12 of the journey had been completed or 5/12 of the book had been read. And for fractions obtained by reducing fractions with a denominator of 12 or splitting twelfths into smaller ones, there were special names.

Even now they sometimes say: “He studied this issue meticulously.” This means that the issue has been studied to the end, that not even the smallest ambiguity remains. And the strange word “scrupulous” comes from the Roman name for 1/288 assa - “scrupulus”. The following names were also in use: “semis” - half an ass, “sextans” - a sixth of it, “semiounce” - half an ounce, i.e. 1/24 asses, etc. In total, 18 different names for fractions were used. To work with fractions, you had to remember the addition table and the multiplication table for these fractions. Therefore, the Roman merchants firmly knew that when adding triens (1/3 assa) and sextans, the result is semis, and when multiplying imp (2/3 assa) by sescunce (2/3 ounce, i.e. 1/8 assa), the result is an ounce . To facilitate the work, special tables were compiled, some of which have come down to us.

The modern system of writing fractions with a numerator and denominator was created in India. Only there they wrote the denominator at the top, and the numerator at the bottom, and did not write a fractional line. And the Arabs began to write down fractions exactly as they do now.

Common fraction is a number of the form

, Where m And n – natural numbers, for example. Number m called numerator of the fraction, n – denominator. Among ordinary fractions, a distinction is made between proper and improper fractions. The fraction is called correct, if its numerator is less than its denominator, and wrong, if its numerator is greater than or equal to the denominator.

2. Actions with ordinary fractions.

2.1. Adding and subtracting ordinary fractions.

Addition ordinary fractions is done like this:

a) if the denominators of the fractions are the same, then add the numerator of the second fraction to the numerator of the first fraction and leave the same denominator, i.e.

;

b) if the denominators of the fractions are different, then the fractions are first reduced to a common denominator, preferably to the smallest, and then the numerator of the second fraction is added to the numerator of the first fraction, i.e.

.

Subtraction ordinary fractions are performed as follows:

a) if the denominators of the fractions are the same, then subtract the numerator of the second fraction from the numerator of the first fraction and leave the same denominator, i.e.

.

b) if the denominators are different, then first the fractions are reduced to a common denominator, and then the numerator of the second fraction is subtracted from the numerator of the first fraction, i.e.

.

Adding and subtracting fractions. If the denominators of the fractions are the same, then in order to add the fractions, you need to add their numerators, and in order to subtract the fractions, you need to subtract their numerators (in the same order). The resulting sum or difference will be the numerator of the result; the denominator will remain the same.

For example:

If the denominators of the fractions are different, you must first reduce the fractions to a common denominator. When adding mixed numbers, their whole and fractional parts are added separately. When subtracting mixed numbers, we recommend first converting them to improper fractions, then subtracting one from the other, and then converting the result again, if necessary, to mixed number form.

For example:

2.2. Multiplying and dividing ordinary fractions.

Multiplication ordinary fractions is performed as follows:

,

those. The numerators are multiplied separately, the denominators separately, the first product is made the numerator, the second - the denominator.

When multiplying a fraction by a natural number, the numerator of the fraction is multiplied by this number, and the denominator is left unchanged.

If the factors are mixed numbers, then you must first write them as improper fractions, then use the rule for multiplying fractions.

Division ordinary fractions are performed as follows:

,

those. dividend

multiplied by the reciprocal of the divisor.

Multiplying a common fraction by a whole number.

To multiply a fraction by a whole number, simply multiply the numerator of the fraction by that number, leaving the same denominator.

For example:

Multiplying a mixed number by an integer.

When multiplying a mixed number by a whole number, in most cases it is easier to multiply the whole number and the fraction by the whole number separately.

For example:

Multiplying fraction by fraction

To multiply a fraction by a fraction, you multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

For example:

.

Dividing ordinary fractions by whole numbers.

When dividing a fraction by a whole number, it is enough to divide the numerator by the whole number, leaving the same denominator.

For example:

What to do if the numerator of a given fraction is not divisible by a whole number. Then there is the following rule

To divide a fraction by a whole number, simply multiply the denominator of the fraction by that number, leaving the numerator the same.

For example:

Dividing a fraction by a fraction.

To divide a fraction into a fraction, you need to rewrite the first fraction, turn the second fraction over (this is important!) and multiply them, i.e. denominator to denominator, numerator to numerator:

For example:

.

3. Examples on addition, subtraction, multiplication and division of ordinary fractions.

Decide for yourself:

4.

5.

6.

Urbanskaya Margarita Viktorovna.

Studying the meaning of arithmetic operations in lessons on discovering new knowledge in primary school.

Content.

Introduction.

1. The meaning of addition and subtraction.

   The meaning of multiplication and division.

Chapter 2. Learning activities in the lesson of discovering new knowledge.

Conclusion

Bibliography

Introduction. Learning the meaning of arithmetic operations is a basic, basic skill that is acquired in the process of learning mathematics. The meaning of arithmetic operations is prepared from the beginning of the mathematics course practical exercises in combining two sets, in establishing connections between elements of two sets, in determining part of the set of presented objects.All four basic arithmetic operations in the minds of students have a direct connection with the practical problems in which they are used. The meaning of the actions of addition and subtraction, multiplication and division is revealed on the basis of practical actions with sets of objects and in a system of word problems. Determining from two numbers the third one corresponding to the given conditions, the student performs a mathematical operation. Modern systems of teaching mathematics are based on a set-theoretic approach in revealing and forming the meaning of arithmetic operations (12), (13).Amongtasks of general education, school mathematics education, it should be noted the task of student development. The process of children's thinking, the transition from practical operations to abstract, logical actions with numbers and concepts, develops most effectively in the course of studying mathematics. This is confirmed by both historical experience and the modern demand of society for the formation in students of not only practice-oriented, but also initial scientific and theoretical thinking.The relevance of this problem in the practice of primary schools made it possible to determine the topic of the thesis research:“Studying the meaning of arithmetic operations in lessons for discovering new knowledge in elementary school.”

Purpose of the study : identify pedagogical opportunitiesrevealing the meaning of arithmetic operations in the lesson of discovering new knowledge. Object of study : methodology for studying arithmetic operations.Subject of study : pedagogical resource for a lesson in discovering new knowledge when mastering arithmetic operations.Research hypothesis: Students will successfully master arithmetic operations if

Propaedeutic work on ideas about mathematical operations will be sufficient, practice-oriented, taking into account the age-related psychological characteristics and difficulties of students in the class.

The lesson material for discovering new knowledge with a problematic presentation will be accessible and structured.

In the lesson of discovering new knowledge, arithmetic operations will be directly applied in practice, in solving problems.

In accordance with the purpose and hypothesis, the study settasks : 1) analyze methodological literature on the problem of teaching arithmetic operations. 2) analyze data on the psychological prerequisites for mastering arithmetic operations. 3) reveal the features of the lesson of discovering new knowledge when students master arithmetic operations. 4) identify the features of propaedeutics of arithmetic operations.

Methodological basis The research included a system-activity approach to the education and development of a child, a set-theoretic approach to the disclosure and formation of the meaning of arithmetic operations, a cultural-historical concept of the development of a child’s personality, a psychological and pedagogical justification age characteristics students (Averin V.A., Bozhovich L.I., Vygotsky L.S., Davydov V.V., Istomina N.B., Bantova M.A., Moro M.I., Vinogradova N.F.) The study was carried out usingmethods observation, comparison and analysis of students’ activities in mathematics lessons, theoretical analysis of scientific literature on the problem, practical work of a teacher developing skills in arithmetic operations.Theoretical significance : theoretical and methodological material on this issue, the content is determined educational material in programs primary classes. Practical significance research: the types of problems used to reveal the specific meaning of arithmetic operations are identified, methods for applying the properties of arithmetic operations used to rationally solve examples are identified; The teacher’s activities in propaedeutics of students’ mastery of arithmetic operations are analyzed. Testing of the research was carried out during the practical work of the teacher. The reliability of the study is determined by the analysis of theoretical material.Research structure: the work consists of an introduction, two chapters, a conclusion, and a list of references.

Chapter 1. Arithmetic Operations

1. Psychological basis of arithmetic operations

“It is a big mistake to think that a child acquires the concept of number and other mathematical concepts directly through instruction. On the contrary, to a large extent he develops them on his own, independently and spontaneously. When adults try to impose mathematical concepts on a child prematurely, he learns them only verbally; real understanding comes only with his mental growth.” - Jean Piaget(1)

K. D. Ushinsky defined the leading role of “... the conscious activity of students guided by the teacher...” in their assimilation of the concept of number and operations with numbers. Arithmetic should be taught in schools so that each student operates with signs of actions and numerical quantities “... with the same clear consciousness with which he writes the most ordinary words that are understandable to him.” Also, in the conscious assimilation of arithmetic knowledge by students, the importance of clarity and mastery of arithmetic language was attached, “unaccustomment to which...mainly leads to failure in the study of arithmetic,” and the role of the formation of numerical associations functioning in linguistic form was emphasized. (2)

The principle of consciousness in revealing the meaning of arithmetic operations was also brought to the fore by advanced Russian methodologists of arithmetic: P. S. Guryev, F. I. Egorov, K. A. Arzhenikov, A. I. Goldenberg and others. They set the teacher the requirement to achieve students' understanding of arithmetic operations, using for this purpose visualization in arithmetic lessons, a skillful transition from operations on sets of objects to abstract operations on numbers. Today, these principles are not outdated, but have found their development in a problem-based approach to presenting new material, in the discovery of new knowledge for students in the classroom.

The concept of number, like all other concepts, is a “mental reflection” of things that arises on the basis of experience. Both the concept of number and the concept of figure are borrowed only from the external world; they do not arise in consciousness from pure thinking. Empirical experience and mastery of sensory standards underlie human counting skills. Numerical concepts arose and developed historically, in connection with the development in man of the ability to abstract from all other properties of things, except for their quantitative relations. This ability is the result of a long, experiential, historical

development and is characterized as abstract thinking. The leading role in the development process was played by language, which is a means of exchanging thoughts between people and a means of abstracting the work of human thinking. The classical provisions of Russian science about language as a social phenomenon, about the inextricable connection between speech and thinking, and the role of language in the development of thinking are fundamental for a correct understanding of the historical emergence of numerical concepts and their assimilation by children.

The formation of the concept of number in children begins even before school. A significant role in this process is played by children’s empirical, sensory knowledge of many specific objects, their perception, practical operation with them and the assimilation of numerals. With the help of numerals, the abstraction of quantitative relationships between things is accomplished. Cognition of these relationships is a complex analytical-synthetic process of sensation, perception and thinking of a child. It begins in the first signaling system, continues and develops in the second signaling system in their interaction. At different stages of this process there is a different ratio of perception of many objects, practical handling of them and counting. The role of counting gradually increases as children master it, then giving way to the leading role of arithmetic operations in the process of further development of the concept of number.

A prerequisite for understanding the meaning of mathematical operations is the child’s maturity, the timely start of schooling, which underlies school readiness.

The experience of J. Piaget is indicative, which is confirmed by observations of today's students. A 5 or 6 year old child can easily be taught to name numbers from 1 to 10. If you put 10 pebbles in a row, the child can count them correctly. But if the pebbles are laid out in a more complex pattern or in disarray, he can no longer count them accurately. Although the child knows the names of numbers, he has not yet grasped the essential idea of ​​number, namely, that the number of objects in a group remains the same, regardless of arrangement.

It can be observed that a child of 6 or 7 years old spontaneously formed the concept of number, although he had not previously been taught to count. If he is given 8 red and 8 blue pieces of cardboard, he will establish, by arranging them in pairs “1” to “1”, that the number of red ones is the same as the number of blue ones, and that both groups remain equal in number regardless of the shape they are given. attached.

Experiments with correlating “1” to “1” are also useful for studying how children develop the concept of number. Let's lay out a row of 8 red pieces at a distance of about a centimeter from each other and ask our little test subjects to take the same number of blue pieces from the box. Children's reactions will depend on age, and we can outline three stages of development. A child aged 5 years and younger will arrange the blue pieces to make a row exactly the same length as the red row, placing the red pieces close together rather than spaced apart. He thinks that the number remains the same if the length of the series is the same. At about 6 years of age, children enter the second stage; they put one blue piece against each red piece and get the correct number. But this does not always mean that children have acquired the concept of the number itself. If we move the red pieces apart, making the distance between them larger, a six-year-old child will think that there are now more pieces in a longer row, even though we have not changed their number. Between the ages of 6 and 7 they reach the third stage: they now know that whether we move or move a row apart, the number of pieces in it remains the same as in the other row.

Thus, J. Piaget concludes that children must grasp the principle of conservation of quantity before they can form the concept of number. But of course, conservation of quantity is not itself a numerical concept, it is rather a logical concept. Thus, these experiments from the field of child psychology reveal how the concept of number, which has been the subject of study by many mathematicians and logicians, is formed. (1)

Systematic schooling is a qualitatively new stage in development cognitive activity child, it determines the next stage in the formation of the concept of number and operations with numbers. The pedagogical task is to strengthen direct counting, which children are not yet proficient in, teach them to count from any point in a natural number series, and help them master reverse counting. This will strengthen the foundation of counting and move from counting and counting by one to counting in groups. It will allow you to move from the decomposition of sets of objects to the decomposition of numbers, knowledge of their composition, mutual relations to each other, to operations on numbers, oral and written calculations, their variation and the development of the most rational methods of calculation.

When moving from counting to an arithmetic operation, the teacher’s ability to correctly update the purpose of this action is important. Not just “count”, but “add”, “subtract”, “find the sum of two or more numbers”, teach children to correctly use the action sign and develop, based on achievements in group counting, appropriate methods of action.

Mastering arithmetic operations by first-graders requires their further development of analysis and synthesis, the development of systems of temporary neural connections, associations in the first signal system and their generalization in the second signal system. Performing an arithmetic operation requires the presence of certain means with the help of which it is carried out. These are previously developed systems of associations that underlie the verbal knowledge of the composition of numbers, the results of their addition and decomposition, and the arithmetic skills necessary for mastering new, more complex operations.

Arithmetic operations are formed in children on the basis of counting and their practical actions with sets of objects through the gradual abstraction of quantitative relationships between them. The successful completion of this process depends on how the gradual transition of children from concrete actions on objects to abstract calculations in the mind is ensured, the main means of performing which is the word.

1.2 The meaning of addition and subtraction.

The basis for studying the action of addition is the practical action of combining two given sets of objects. The basis of the action of subtraction is exercises on selecting a certain part of a set according to a certain criterion and then removing this part.

By discovering the specific meaning of actions, students must establish a connection between a given operation and the corresponding arithmetic operation, and become familiar with terminology and symbolism.

Addition - the operation of combining finite disjoint sets.

Addition - an arithmetic operation indicated by a plus sign ().

In the field of integers positive numbers As a result of the action of addition over these numbers - terms, a new number is obtained - a sum that contains as many units as there are in all terms together. So the amountac there is some number c - a finite number of the union of setsA AndV .

Addendum, sum - the name of the components and the result of the addition action. The double value of the sum is given: both the operational value and the result of the action.

Subtraction This is the arithmetic operation inverse to addition.Subtraction is indicated by a minus sign ().

From the numberA subtract, it decreases, called minuend. The number b is subtracted and is called the subtrahend. Expressiona-c orWith indicates the difference by how much the numberA different from numberV . Therefore, this difference is called the difference d. The difference also has a double meaning: an operation with numbers and the result of an action.

When studying the specific meaning of arithmetic operations, the main ones are traditionally identifiedstages .

On preparatory stage the specific meaning of addition and subtraction operations is revealed, examples are solved, and the method of writing them is revealed. Cases of adding and subtracting 1 are considered, when the results are obtained based on knowledge of the natural sequence of numbers.

The second stage includes the development and application of counting and counting techniques one by one and in groups for cases of adding and subtracting 2, 3, 4. It is based on the specific meaning of the actions of addition and subtraction.

The logic of the next stage offers training in the technique of rearranging terms for the cases of adding 5, 6, 7, 8, 9. The theoretical basis of this stage is the commutative property of addition. Students draw up an addition table and learn the composition of numbers from addends.

Finally, at the fourth stage, children learn subtraction techniques based on knowledge of the relationship between the sum and the terms for the cases of subtracting the numbers 5, 6, 7, 8, 9.

To consolidate, you need to create a general table of addition and subtraction that allows you to apply all the techniques you have learned.

Further, when developing computational skills, various approaches are used: learn addition and subtraction tables; consolidate them in the process of solving examples. Make tables yourself and involuntarily remember them during the exercises. After using subject actions and various computational techniques, the student is given the task of memorizing tables and the composition of each number.

So,The specific meaning of the action of addition is determined by the following:

Actions with collections of objects - combining and increasing by several elements

Increasing a given subject set by several items

Increase by several objects of a set equal to a given one

Compiling one subject set from two data

The meaning of the subtraction action is determined by the following:

Action associated with reducing the number of items

Reducing a given set by several items

Reducing a set equal to a given one by several objects

Comparison of two subject sets. (3)

1.3 The meaning of multiplication and division .

The action of multiplication is revealed in elementary school as the summation of identical terms.

Multiplication - is a mathematical operation by which a new number or quantity is obtained from two numbers or quantities. For integers, it includes the first number as a term as many times as there are units in the second. (4)

Multiplication of non-negative integer natural numbers is the action of a×b= a+ a+ a+ a+ a+…+ a, atb>1×1=a, atb=1×0=0, withb=0

ApplicationMultiplication symbolism allows you to shorten the notation for adding identical terms. The notation of the form 2 ×4=8 implies the abbreviation of the notation of the form 2+2+2+2=8. It is read like this: “take 2 4 times, you get 8”; or: “2 times 4 equals 8.” The action of multiplication in all mathematics textbooks for primary grades is considered before the action of division.

From a set-theoretic point of view, multiplication corresponds to such objective actions with aggregates, sets, groups of objects as the union of equal, equal-numbered aggregates. Therefore, before becoming acquainted with the symbolism of recording actions and calculating the results of actions, the child must learn to model all these situations on objective aggregates. To reveal the meaning of an action, you need to understand and correctly represent them from the words of the teacher, be able to show with your hands both the process and the result of an objective action, and then characterize them verbally.

Learning the multiplication tables is the main objective of teaching mathematics in 2nd and 3rd grade. Knowledge of tabular cases of the action of multiplication is learned by heart. Understanding-based learning allows students to master multiplication and division later double digit number to single-digit, two-digit numbers to two-digit, written cases of multiplication and division with the least difficulty.

When mastering multiplication and division, it is necessary to develop awareness of the activity. The child must understand how a mathematical result is obtained while understanding the practical meaning of actions, the commutative property of multiplication and the connection between the components of the multiplication action. (6) Table multiplication includes cases of multiplying single-digit natural numbers by single-digit natural numbers, the results of which are found based on the specific meaning of the multiplication action - the sums of identical terms are found.results table multiplication in accordance with the program requirements for the content of primary general education, children must know by heart. Techniques for compiling multiplication tables are based on the meaning of the action of multiplication. The results of these tables are obtained by sequential addition of identical terms. For example, multiplying the number 2: 2+2=2*2, 2+2+2=2*3, 2+2+2+2=2*4, 2+2+2+2+2=2*5

The illustration helps the child get the result by counting the figures. For small values ​​of the factors, the method of recalculation to obtain the tabular value of the product is acceptable, and the teacher often uses it when multiplying numbers 2, 3, 4.

When the value of the second multiplier is greater than 5, it is more convenient to use the method of adding to the previous result to obtain the results of tabular values. For example:2×6=2×5+2=..., 2×7=2×6+2=…,2×8= 2 ×7+2=…,2×9=2×8+2=...

A table of multiplication values ​​for the number 3 is compiled in a similar way.

The next technique on the basis of which tables of values ​​for multiplying numbers are compiled is the technique of rearranging factors. This technique is actually the firstmathematical law regarding the operation of multiplication in elementary school:Rearranging the factors does not change the product.

The way children are introduced to this rule and law is determined by the previously introduced meaning of the action of multiplication. Using object models of sets, children recalculate the results of grouping their elements different ways, making sure that the results do not change by changing the grouping methods. For example: 2 ×3 = 6, 3 × 2 = 6

Counting the elements of a picture - a set in pairs horizontally - coincides with counting elements in triplets vertically. Consideration of several variants of similar cases gives the teacher the basis to make an inductive generalization - a generalization of several special cases into a generalized rule that rearranging factors does not change the value of the product.

To memorize the multiplication tables, there are such techniques as:

Counting in twos, threes, fives;

The technique of sequential addition is the main technique for obtaining results of tabular multiplication. This technique is associated with the meaning of the action of multiplication as the addition of identical terms;

The technique of adding a term to the previous result (subtracting from the previous result).

Reception of an interconnected pair: 2 × 6 6 ×2 (rearrangement of factors);

A technique for memorizing a sequence of cases with a focus on increasing the second factor;

Taking a “portion”;

Using a memorable incident as a reference. For example, 5 × 6 =30, which means 5 ×7 =30+5 =35;

Reception of external support; A drawing or rectangular table of numbers is used as a support. By tracing a rectangle with a given number of cells on the sides on a checkered field, the child uses this model to control the result obtained or simply count the cells as best he can. For example: 4x5 = 20

Method of memorizing a table “from the end”;

Finger counting when memorizing multiplication tables. For example, you need to multiply 6 by 7. We clench the fingers on both hands into a fist, and then on each hand we bend as many fingers as each factor is greater than five. On two hands, three fingers are bent - this is the number of tens in the desired number. On one hand three fingers remained pressed to the palm, on the other - four fingers. We multiply these numbers 3 × 4 = 12 and add 30 + 12 = 42 to the number of available tens. Answer: 6 × 7 = 42.

The meaning of division

The action of division is studied in elementary school as the inverse action of multiplication.Division is the inverse mathematical operation of multiplication: finding one of the factors from the product and another factor.(7) From a set-theoretic point of view, the meaning of division corresponds to the operation of partitioning a set into equal subsets. Thus, the process of finding the results of the action of division is associated with objective actions of two types:

Splitting a set into equal parts. For example, 8 circles are divided equally into 4 boxes: put 8 circles one at a time into 4 boxes, and then count how many circles are in each box.

Dividing a set into parts with some in each part. For example, 8 circles are laid out in boxes of 4 pieces - put 8 circles of 4 pieces in boxes, and then count how many boxes there are. Division according to this principle is called “division by content.”

Using object actions and drawings, children find division results.

An expression like 12:6 is called a quotient. The number 12 in this notation is called the dividend, and the number 6 is the divisor. A notation of the form 12:6=2 is called equality. The number 2 is called the value of the expression. Since the number 2 is in this case obtained by division, it is also called quotient.

In elementary school, the action of division is the inverse action of multiplication. Therefore, students are first introduced to cases of division without a remainder within 100: “tabular” division. Children are introduced to the operation of division after they have already memorized the multiplication tables for numbers 2 and 3. Based on knowledge of these tables, already in the fourth lesson after learning about division, the first table of division by 2 is compiled. To obtain its values, an object drawing is used.

4:2=… 8:2=… 14:2=…

6:2=… 10:2=… 16:2=…

20:2=… 12:2=… 18:2=…

The values ​​of the quotients are obtained by counting the elements of the picture in the picture.

Techniques for memorizing tabular division cases are associated with methods for obtaining a division table from the corresponding tabular multiplication cases:

A technique related to the meaning of the action of division. With small values ​​of the dividend and divisor, the child can either perform objective actions to directly obtain the result of division, or perform these actions mentally, or use a finger model.

A technique associated with the rule of relationship between the components of multiplication and division. In this case, the child focuses on memorizing an interconnected triple of cases, for example: 3×7=21 21:7=3 21:3=7

If the child manages to remember well one of these cases, where usually the reference case is the case of multiplication, or he can get the result using any of the methods of memorizing the multiplication table, then, using the rule: if the product is divided by one of the factors, then the second factor will be obtained, easy to get the second and third tabular cases. (8)

Thus, when studying the action of multiplication and division, students need to know the meaning of the action of multiplication and division, table cases of multiplication and division.

Chapter 2. Teacher’s activities during the lesson of discovering new knowledge.

2.1 Lesson for discovering new knowledge in the lesson system

One of the main goals of modern Russian education is the full formation and development of the student’s abilities to independently outline an educational problem, formulate an algorithm for solving it, control the process and evaluate the result obtained - to teach how to learn, and not the reproductive transfer of knowledge, skills and abilities from teacher to student.

And this leads to changes in the tasks and conditions of the educational process, which is based on the ideas of developing the student’s personality. The active abilities of students are formed only when children do not passively learn new tasks, but are included in independent educational and cognitive activities.

Structuring lessons in primary school today reduced to a number of stages: classroom organization; updating or repetition of previously acquired knowledge and skills; formation of new knowledge and skills; discovery of new knowledge; primary fixation; independent work with self-test according to the standard, Self-analysis and self-control; inclusion of new knowledge into the knowledge system and repetition and reflection of activities.

If earlier child acted as a passive listener, then, in accordance with new trends, he should become a researcher who knows how to obtain knowledge himself, working in a group with other children or independently. The main lesson types remain the same, but changes have been made.

Gmain methodological goal of the lessonin a new typology of lessonsin systemic-activity learning - creating conditions for the manifestation of cognitive activity of students. This goalis achieved as follows:

The progress of knowledge is “from the students.” The teacher draws up and discusses the lesson plan together with the students, uses didactic material during the lesson, allowing the student to choose the most significant type and form of educational content for him. The transformative nature of students’ activities: observe, compare, group, classify, draw conclusions, find out patterns. Intensive independent activity students, associated with emotional experiences, which is accompanied by the effect of surprise. Collective search directed by the teacher (questions that awaken students’ independent thoughts, preliminary homework). The teacher creates an atmosphere of interest for each student in the work of the class. Creating pedagogical situations of communication in the classroom that allow each student to show initiative, independence, and selectivity in ways of working. Flexible structure. The teacher uses various forms and methods of organization educational activities, allowing to reveal the subjective experience of students.

Among the activity-oriented lessons, along with activitiesAnd, consider the “lesson of learning new things.” Its place in a series of lessons revealing new material, the first, introductory, in which the “discovery” of knowledge by children should occur. This requires special preparation for such a lesson. In form it can be traditional or combined lesson, lecture, excursion, research, educational and labor workshop. “The lesson of discovering new knowledge” is aimed at studying and initially consolidating new knowledge.

2.2 Modeling a lesson on discovering new knowledge.

Let's consider the structure of a lesson on discovering new knowledge using the example of studying the specific meaning of the action of subtraction.Activitytarget lesson:developing in students the ability to implement new methods of action: reducing a number by several units, subtracting a part from a whole.
Contenttarget lesson:expansion of the conceptual base by including new elements into it: the name of the components of the subtraction action and the designation of the operation with the mathematical symbol “minus”

Lesson structure for discovering new knowledge :

Subject “The meaning of the action of subtraction” Lesson type - OZ

Tasks : Explain the meaning of the action of subtraction, introduce the terminology: “difference”, “minuend”, “subtrahend”, “difference value”.

Lesson Resources : Moro M.I. etc. Mathematics: 1st grade, 1st part; workbook grade 1, part 1, electronic application.

1 Stage of motivation for educational activities (development at a personally significant level of internal readiness to fulfill the requirements of educational activities)

Teacher's actions: organizes the children's work, offers to open the textbook, prepare a notebook, offers to guess the riddle.

2 Stage of updating the necessary knowledge and trial educational action: preparation of students’ thinking, organization of the need to construct educational actions and recording by each of them an individual difficulty in a trial action.

Teacher's actions: suggests subtracting some objects from those offered, crossing out some objects from the whole on the board, making sure that there are fewer objects than there were.

Offers to add up the subject material and write down the action of addition as an example. Updates children's knowledge about the operation of addition.

Offers to compare what happened to the number of objects during addition and subtraction in different operations.

3 Stage of identifying the cause of the difficulty - help students find what exactly is the lack of their knowledge.

Teacher actions: Offers subject material (put 5 circles, remove 2 circles; put 7 squares, move 3 squares)

The name of the operation is determined - subtraction. A number of synonyms are proposed: subtract, remove, cross out, erase, etc. It is clarified which mathematical signs are known to children (+ - =), which sign is suitable for writing this operation. A recording of the subtraction action is proposed.

4 Open knowledge implementation stage: students’ construction of a new way of action and the formation of skills to apply it.

Teacher actions: It is suggested to make a drawing geometric shapes in your notebook on a line and cross out the given amount. They say how much was there, crossed out, what is left. An example with a minus sign is written under the figure. Variants of reading the expression are spoken out (subtract 3 from 7, you get 4, 5 minus 2 equals 3)

Children can be asked to find the difference in writing examples for addition and subtraction, in writing mathematical symbols, and clarify what the symbols mean.

After dynamic pause we implement:

5 Primary fixation stage with pronunciation in external speech

The goal of the stage is for students to master a new method of action when solving standard problems.

Teacher's actions: A task is proposed to reduce the number. For example, 5 bullfinches were sitting, two bullfinches flew away. How much is left? Several tasks are discussed and performed orally by the children; Next, pairs are asked to complete the diagram and write down the subtraction action. The work can be offered to one of the children at the board.

6 Independent work stage with self-test against standard .

At the stage of students’ independent work, internalization of a new method of action and executive reflection, collective and individual, occurs, achieving the goal of a trial educational action, applying new knowledge in standard tasks.

Teacher’s actions: organize students’ independent completion of standard tasks for new way actions. Tasks are offered on cards that are completed together with neighbors. In the cards, the action is subtraction and its result, which needs to be connected; connect the subject diagram and the solution using the action of subtraction; other tasks that reinforce the specific meaning of the subtraction action. Mutual checking is carried out frontally, the team reacts to the correct and incorrect answers of groups of children, and is offered the opportunity to correct the decision.

At this stage, it is important to create, if possible, a situation of success for each child.

The purpose of the stage is to repeat and consolidate what has been previously learned and prepare for the formation of a stable skill in subtraction. They identify in which case new knowledge is applicable, in particular, subtraction; its use in a system of previously studied knowledge, that is, along with the action of addition. Preparations are being made to reveal the connection between the components of action and the connection between addition and subtraction.

To do this, it is necessary to further bring the action of subtraction to the level of an automated skill; repeat learning content necessary to ensure meaningful continuity

8

When conducting reflection on educational activities, students self-assess the results of their work, understand the specific meaning of the subtraction action, determine what is known and new about the operation being studied; application of a new method of action.
To achieve this goal, students organize reflection and self-assessment of their own learning activities in the classroom. Students correlate the goals and results of their learning activities and record the degree of their compliance. The goals of further activity are outlined, guesses are made what kind of information there may be if we continue to study the action of subtraction. You can update the students’ reasoning about the purpose and when this action is needed.

Thus, by following the stages common to the structure of a lesson for discovering new knowledge, the teacher can model lessons in which the specific meaning of arithmetic operations is revealed, and other classes.

Modern methods also propose an algorithm for constructing new types of lessons.

Algorithm for designing a lesson for discovering new knowledge :

Identify and formulate new knowledge.

Model a way to discover new knowledge.

Identify the mental operations used in the discovery of new knowledge.

Determine the necessary training and methods of its repetition.

Select exercises for the actualization stage, based on the list of necessary mental operations and learning skills.

Model the difficulty and how to fix it

Model a problem situation and dialogue

Compose independent work and an objectively justified standard.

9. Determine methods for organizing and conducting primary consolidation.
10. Select tasks for the repetition stage by level.
11. Analyze the lesson based on the notes.12. Make adjustments to the outline plan if necessary

By following it when preparing a lesson, a teacher can significantly optimize his work without missing the main point of its content.

Conclusion

Bibliography

1 Jean Piaget “How children form mathematical concepts”Questions of psychology, 1966, No. 4, pp. 121-126

2 Using the pedagogical heritage of K.D. Ushinsky in teacher traininghttp://www.yspu.yar.ru

3 Formation of ideas about the specific meaning of arithmetic operationsGlagoleva Yulia Igorevna, Ph.D., Associate Professor of the Department of Primary Education of the APPO St. Petersburg (lecture materials)

4 Istomina N.B. and others. Workshop on methods of teaching mathematics in primary school: Tutorial for students of pedagogical institutes in the specialty “Pedagogy and methods of primary education” / N.B. Istomina, L.G. Latokhina, G.G. Shmyreva. - M.: Education, 1986. - 176 pp. ill.

5 Mathematics. Textbook for 2nd grade. beginning school At 2 p.m. Part 2. (Second half of the year)/ M.I. Moreau, M.A. Bantova, G.V. Beltyukova and others - 3rd ed. - M.: Education, 2004. - 96 p.: ill.

6 Primary school No. 9 - 2001, from 74.

7. Ozhegov S.I. and Shvedova N.Yu. Explanatory dictionary of the Russian language: 80,000 words and phraseological expressions / Russian Academy of Sciences. Institute of Russian Language named after V.V. Vinogradov - 4th edition updated. M.: LLC "ITI Technologies" 2006, 944 pp.

8. Beloshistaya A.V. Methods of teaching mathematics in elementary school: a course of lectures: A textbook for students of higher pedagogical educational institutions. - M.: Humanitarian Publishing Center VLADOS, 2005. - 138 pp.: ill. - (University education).

Peterson L.G. Mathematics. 2nd grade: Guidelines for teachers. - Ed. 2nd, revised And additional - M: Yuventa Publishing House, 2005. - 336 pp.: ill. About the magazine › › )