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7ALGORITHMIC FOUNDATIONS OF EXPECTATIONS OF SYSTEMATICITY AND CONSISTENCY IN TEACHING DIVISION OPERATIONS OVER NATURAL NUMBERS
Ibrahimov F.,
doctor of sciences in pedagogy, professor, Sheki branch of Azerbaijan State Pedagogical University
Azerbaijan, Sheki Imanova A.
doctor of philosophy in mathematics, senior lecturer, Sheki branch of Azerbaijan State Pedagogical University
Azerbaijan, Sheki ORCID: 0000-0003-1566-6465 DOI: 10.5281/zenodo.6575790
Abstract
The article substantiates the relevance of the algorithmic basis of the expectation of systematicity and consistency in teaching division actions over natural numbers, based on the expansion of the range of natural numbers on educational units it adequately explains the algorithmic basis for anticipating and systematizing the sequence of elements reflected in the learners' ability to perform division operation.
Keywords: decimal composition of numbers; divisible, divisor, quotient and remainder; incomplete divisible; sequential subtraction; division algorithm; open description of the algorithm; division into equal parts; division by composition; special quotients.
The actuality of the subject. Improving the pedagogical process and its subsystems is one of the eternal problems of didactics. The development of algorithmic culture is the central link in these processes, because in these processes the mental activity of students is controlled. Algorithms and algorithmic construction practices are invaluable in both analytical and heuristic types of mental activity. The topic "Algorithmic foundations of expectations of systematicity and consistency in teaching division operations over natural numbers" is an element related to the algorithmic culture of students, in other words, it is included in the everlasting problem of didactics (learning cognitive control). Therefore, we claim the relevance of the research topic.
Methodological basis of the research: L. Ber-talanfs idea of system analysis, the "system-structural" approach, formed as a branch of dialectics.
Interpretation of the research work. The framework document of the National Curriculum of General Education in the Republic of Azerbaijan stipulates that through the teaching of mathematics, primary school students are able to perform arithmetic operations, master oral and written algorithms. [10;55]. The analysis of the activities of primary school students related to the practical implementation of division operations shows that many of them have difficulty mastering the execution algorithm of this operation. Of course, one of the reasons for this is the complexity of the division algorithm, while other reasons are due to the lack of methodological work related to the mastery of the algorithm.
Students' performance in mastering the written division algorithm can be regulated if the implementation of the following is elementary for them, or rather, the preparation of students meets the following requirements:
a) must know the decimal structure of numbers and be able to separate all floor units from numbers;
b) must be able to multiply single-digit numbers correctly and quickly, and must be able to subtract multi-digit numbers quickly and accurately;
c) have good skills in verbal arithmetic - be able to quickly multiply two-digit and three-digit numbers by one-digit numbers;
d) they must know the relationship between multiplication and division, and be able to use that connection.
Indeed, students use the knowledge, skills, and habits mentioned above to perform the necessary, elementary, sequential steps in the division algorithm. It is not difficult to see this in the open form of the division algorithm. Let's look at it.
Let's first accept the following symbolic symbols.
Fn=xiX2 x3...xn- divisible, Q=yiy2y3...ym - divider, n>m.
Fl=Xl, F2=XlX2, F3=X1 X2 X3... Fn= Xl X2 X3...Xn
Let the numbers obtained by transferring the numbers after Xm (Xm + 1) belonging to Fn to the difference used in the algorithm be: Ti, T2, T3...
1. If Fm>Q is conditionally paid, it is transferred to point 3, if it is Fm<Q, it is transferred to point 2.
2. The number F m+1 is separated and the process is carried out in accordance with point 3, that is, it goes to point 3.
3. In Fm (F m+1) it is determined how many times Q is found, i.e. the first digit of the quotient is found and this number is multiplied by Q, the product obtained from Fm (F m+1) is subtracted and transmitted to the 4th point.
4. If Fn =Fm (Fn =Fm+1) or if the difference specified in the third point is equal to zero, and the numbers following Fn's numbers covered by Fm (Fm+1) are equal to zero, they are written in the quotient, go to point 9, otherwise go to point 5.
5. To the right of the obtained difference the last digit of Fm+1 (F m+2) is transferred, in the number compiled by this rule - (in T1) it is determined how many times the divisor-Q is located, ie the next digit of
the quotient is found and this number is multiplied to Q, the product obtained is subtracted from T\, go to point 6.
6. If Fn=Fm+\ (Fn=Fm+2), or if the difference specified in paragraph 5 is equal to zero, and the numbers following the numbers of Fn surrounded by Fm+\ (Fm+2) are equal to zero, they is written in the quotient, otherwise go to point 7.
7. The last digit of Fm+\ (Fm+2) is transferred to the end of the obtained number, the number T2 is formed, how many times Q is located in T2, i.e. the next digit of the quotient is determined, this number is multiplied by Q, the product is subtracted from T2, go to point 8.
8. The process continues with the analogous application of points 5,6,7 until all the digits of Fn are used.
9. The process ends, quotient and remainder are determined. [4; 93-94]
As can be seen, in fact, mastering the algorithm of written division begins on the first day of training, and this work is carried out at all stages of teaching mathematics in the primary grades. The process of mastering the skills to perform the necessary steps of the algorithm is a system with a complex structure, in general, "intellectual reflection is a complex process" [1; 64].
There is no doubt that the basis of the system of learning the written division be the numerical structure of the divider. Accordingly, the division algorithm can be taught by distinguishing the following cases:
1. Mastering the division algorithm by table;
2. Mastering algorithms for dividing a two-digit number into a single-digit and a two-digit number;
3. Mastering the algorithm of dividing a three-digit and multi-digit number by a single-digit number;
4. Mastering the algorithm of division by a number expressed by ones and zeros;
5. Mastering of the division algorithm for round numbers;
6. Mastering the algorithm of dividing by a two-digit number and, as a preparation for it, dividing a three-digit number by a two -digit number, provided that a single-digit number is obtained;
7. Mastering the algorithm of dividing a multi-digit number by a three-digit number and, as a preparation for it, dividing a multi-digit number by a three-digit number, provided that a single-digit number is obtained;
8. Mastering the algorithm of dividing numbers with zeros at the end.
Now let's take a closer look at each case and try to explain some of its features.
There is no doubt that the question of the sequence in which the two types of division should be taught is of particular importance. Most Methodists believe that it is desirable to teach division into equal parts first. Because children are still familiar with the division into equal parts in their life experiences from kindergarten; division by composition is difficult for students to perceive.
Scientific sources emphasize that the recording of division into equal parts is both simple and understandable for a child; division by composition is
difficult and complex to write; it takes a long time to teach the correct spelling of division by composition [11; 173]
Indeed, this method is quite specific, clear and can be easily demonstrated with the help of visual aids. Thus, it is more appropriate to start the division operation without dividing it into equal parts; then students should be introduced with the division by composition.
Multiplication and division can be done together or separately; when teaching separately, it may be more appropriate to include the multiplication first and then the division. We support the idea that a separate system of these actions is appreciated. In this case, students' attention is focused on only one action for a certain period of time: the field of study becomes more intensive; as a result, students can learn more about these issues.
When teaching multiplication, it is possible to further clarify its relationship between addition and subtraction. The relationship between multiplication and division must be explained at a certain stage in the study of division.
When dividing into equal parts for the first time, you can use the following algorithmic form:
1. From the group of things to be divided into equal parts, something equal to the number of parts is taken, so that when divided, each part has one thing, one unit;
2. From the rest of the group of things, again dividing the given amount into parts, one thing is taken for each part, so much that the second unit falls out.
3. From the rest of the group of things, again dividing the given amount into parts, one thing is taken for each part, so much that the third unit falls out and etc.
4. This operation is continued until the end of the given items and the result is determined.
Here, the implementation of division with subtraction of equal numbers comes to the fore. So that:
1. When dividing the divisible into any equal parts, the number of parts is first subtracted from the divisible, subtracted is divided into units.
2. The number of parts divided by the remainder is subtracted, subtracted is divided into units and is added to the singular singles separated in the first step, two equal groups are formed;
3. The number of redistributed parts is deducted from the remainder, divided by the deducted singles, and the number of singles in the groups created in step 2 is taken as the result of the division.
Naturally, the subtraction operation is used in the first acquaintance with the division process, i.e. in the division operation by visual means. Then the link between the division and the multiplication should be used in its entirety. It should not be forgotten that when students first use the first decimal numbers, they easily master the results of the division, and then in the second decimal or hundred circle, the devision is developed on those obtained at the initial stage. In this case, it is not difficult to use the existing knowledge to quickly find quotient. Therefore, the following conditional stages in
the study of the division can be distinguished: 1) The operation of the division in equal parts is clarified by visual means in accordance with the algorithm presented above; 2) Division of second decimal numbers are studied according to an algorithm based on the distribution property of multiplication.
As you know, multiplication and division are performed by table and not by table. All cases of multiplication of single-digit numbers and cases of division corresponding to them belong to multiplication and division according to the table. Multiplication of two-digit numbers by single-digit numbers whose product does not exceed the number 100, and the corresponding cases of division are included in non-tabular multiplication and division. Multiplication of two-digit numbers by one-digit numbers without deriving more than 100 numbers and corresponding cases of the division. [2;98]. Off-table multiplication and division learning are essentially different from table multiplication and division learning; multiplication by table and division are mastered with memorization of all calculation results; with extra-tabular multiplication and division , none of them is memorized by heart; the main task here is to master the methods of calculation.
Multiplication and division methods based on the distribution and displacement properties of operations are the main purpose of off-table multiplication and division studies. It is true that students have encountered these properties of operations before, but now the application of these properties is clearer and more consistent.
When dividing a two-digit number into a single-digit number, it is necessary to divide the divisible number into two numbers so that when dividing one of them, tens should be obtained, and when dividing the other, units. To do this, sometimes it is necessary to divide the number into tens and ones (64:2), and sometimes one or several tens should be divided into units (36:2). We need to start with the simplest case first; then it is necessary to divide the round tens, which give in the quotient a number consisting of tens and ones (30:2) and after that, it is also necessary to move on to the general form of division, which requires the division of tens into units. A general method that can be applied equally to all these situations is to find tens of quotients first, and then ones. Its algorithmic description is as follows:
1. Determine how many tens and units a two-digit divisible number consists of; go to point 2.
2. If the tens of the given divisible are divisible into the required parts, execute it and determine the tens of the quotient; go through point 3; otherwise go to point 4.
3. Divide the units of the divisible number into the necessary parts, determine the units of the quotient; go to point 8.
4. Perform the division into the required parts of the tens of a divisible number, separate the maximum possible number of tens and divide it into the required parts to determine the tens of the quotient, go to point 5.
5. Convert the subtracted tens of the divisor into singulars and combine them with the singulars of the given number, go to point 6.
6. Divide the sum of ones into the required parts, separate the tens and ones in the result, go to point 7.
7. Sum up the result obtained in point 4 with the result obtained in point 6, determine the tens and units of the quotient, go to point 8.
8. Write quotient according to tens and ones.
In this process, a system of tasks and exercises should be applied in such a way that students do not have difficulty in performing their activities according to the presented algorithm, as if a sequence of the necessary, elementary steps of a given algorithm should be engraved in their memory. Of course, only in this case can each student divide a two-digit number by a one-digit number. It is necessary to increase the algorithmic activity of students in this direction to the level of "automation", to act as an elementary step in the algorithm of the written division.
Now let's try to explain our idea of teaching the algorithm for dividing a two-digit number by a one-digit number.
In this case, the quotient of the division is found by trial (trial) and is checked by multiplication. Thus, this case of division is completely based on the multiplication operation and does not depend on whether it is a division into components or equal parts.
In this process, a system of tasks and exercises should be used so that students can master the algorithm for finding quotient faster. The following steps are required in this algorithm:
1. Determine the number of tens of the divisible and divisor, go point 2.
2. Determine how many times the tens of the divisor are in the tens of the divisible, that is, find the "test number" and go to step 3.
3. Multiply the test number by the divisor, compare the product with the divisible, and if equality is determined, go to point 6, otherwise to point 4.
4. If the product of the test number is greater than the divisible, reduce it (the test number) by one, multiply it by the divisor, go to point 6, otherwise go to point 5.
5. Increase the test number by one, multiply it by the divisor, and go to point 6.
6. To accept that the quotient is a found test number.
It is a good idea to introduce students to the methods of "sequential subtraction", "sequential addition" and "sequential testing" before introducing them to the method of finding the quotient faster. For this purpose, it would be useful to refer to the solution of relevant issues. [2; 207]
Students are well aware of the usefulness of finding a test number based on the ratio of the tens of the divisible and divisor after sequential addition, sequential addition and sequential test methods, as well as performing the process based on this number. [5; 91]
It should be noted that the various methods used are fully compatible with the preparation of students. They mastered the concepts of fractions, obtaining and comparing parts, finding numbers by parts, "many
times", "dividing a two-digit number by a single digit number", "division with remainder", "round tens", "division by ten", "division by table" and so on. Therefore, students determine the relationship between multiplication and division, and do not have difficulty in performing the operation quickly. It is not difficult to guide the student to rely on the judgment of "how many times are 2 tens in 8 tens" to determine how many times 21 is in 84? It is not difficult to see this from experience. To get results quickly, you should choose exercises for the student that will teach him to perform the following sequence of algorithmic steps:
1. Separate tens in divisible two-digit numbers;
2. Distinguish the tens in the divisor;
3. Divide the tens of the divisible by the tens of the divisor;
4. Multiply the result by the divisor and compare the product with the divisible.
Dividing three-digit numbers into single digits is one of the most important stages in mastering the division of multi-digit numbers.. At this stage, it is necessary to separate the oral and written division . Division practical skills require students to be able to analyze the composition of a number, as well as the ability to divide a number into components depending on the divisor. [7; 114]. Oral division, which is also explained in detail and mastered in a thoughtful way, is a good preparation for the written division. In fact, there are many commonalities between these two images of the division. It is necessary to move from the oral division to the written division so that students feel the commonalities that unite them, "see" the meaning of division in the symbols of the written mechanism. [6; 117]
It is necessary to try to enable students to master the following two important groups of oral division algorithms of three-digit numbers: 1) division by a single digit number based on the ability to divide round hundreds, tens and single digits, which are obtained by dividing a number into decimal components, and the property of dividing the sum by numbers applies here;
2) Division by single digits of numbers, the hundreds, tens and units of which are not divisible by a single digit, but can be divided on favorable terms.
If these two important groups are having difficulty, it may be helpful to focus on the specific cases included in the following sequence:
1) Oral division of round hundreds or decimals resulting in division by a single-digit number (800: 8; 8 hundred: 8 = 1 hundred; 800: 8 = 100); 2) Divide the number formed from the hundred and tens (240: 2; 300:
3); 3) Divide numbers, hundreds and tens of which are not individually divisible by a divisor (120: 3; 360: 9);
4) Divide round hundreds, in the case when the number of hundreds is not divisible by a divisor and in the quotient hundreds and tens are obtained (600: 4; 900: 6); 5) Divide a three-digit number formed by hundreds and tens.
In this case, both the hundreds and the tens of the three-digit number are not divisible by divisor, but these floor of it together form tens, which are completely divisible by divisors (420: 3; 560: 4), here two methods of division can be distinguished: We
divide the number 420 separately into 2 pieces, each of which is divided into 3. These numbers are 300 and 120. Dividing each of them by 3, we get 100 + 40 = 140; 2) We consider 420 as 42 tens. Divide 42 tens by 3 and get 14 tens or 140. This method of division can be demonstrated by visual means - with the help of wand sets. [2; 229-230]
Both of these methods have the same degree of difficulty. After studying each of these situations separately, it is helpful to give students a variety of mixed-type exercises. Undoubtedly, at this stage of teaching, the main goal is to teach students to use oral division techniques; therefore, the teacher should not be content with receiving answers from students, but should ask them how they found the answer and, in this case, what methods and techniques of division were used.
Now let's move on to interpret the algorithmic basis of the three-digit numbers written division, or rather try. We believe that, as in the case of other acts, it is necessary to close the written division with the oral division, to arrange the various cases of the division on the condition that the difficulty increases.
Experience shows that it is necessary to start with the image of the division, which is known to students, and move to the image that is unusual for them. We support the inclusion of exercises on dividing three-digit numbers by single-digit numbers in the educational process in the following sequential order.
1. Each floor of the divisor is completely divisible by the divisor (846: 2; 936: 3).
2. Hundreds are not completely divided, and the remainder must be broken down into units (575: 5).
2. Hundreds are not completely divided, and the remainder must be broken down into tens (728: 4; 429: 3).
3. The remnants of both hundreds and tens must be broken down to the lower floors in succession (685: 5).
4.When dividing three-digit numbers into single-digit numbers, in a special case two-digit quotient is obtained (168: 2; 546: 6).
5. In the complicated form of the above special case, the tens are not completely divided, and the remainder of the tens must be broken down into units (258:9; 450:6).
Undoubtedly, the most important thing is that students to consciously master the technique of performing a single-digit written division. It is necessary to try so that each student can correctly and quickly perform the operation in all possible cases when dividing three (or more) digit numbers by a single digit number. Of course, the initial case is that all floor units are divided into divisors, and experience shows that in this case, the division is performed by students without difficulty. Other cases are a bit difficult to master. Therefore, in these cases, it is necessary to use visual aids effectively. First, you can complete the division by referring to visual aids and past experience, and then move on to a form of judgment.
In this process, students must learn that the division operation begins on the top floor, that each
floor must be divided, and that when dividing each floor, a quotient equal to the units of this floor is taken.
When students complete the "thousandth" concentration, they should know that division of three-digit number into one-digit number consists of a system of necessary steps performed in the following sequence, and be able to perform it:
1. Separate the hundredth of a higher floor unit from a three-digit number and compare it with the divisor; if that floor unit is not smaller than the divisor, it must pass to point 2, otherwise to point 3.
2. To determine how many times the divisor is located in the hundredth, that is, to determine the first digit of the quotient, to multiply it by the divisor, to subtract product from the hundredth floor unit, to go to the 5th point.
3. To determine how many tens a three-digit number consists of, i.e. to separate the first two units of the digit together, go to the 4th point.
4. Determine how many times the divisor is in the number (F) formed on the basis of two separated numbers, i.e. in the number of tens, i.e. determine the 1st digit - the ten of the quotient (the quotient will be a two-digit number) and its product with the divisor, subtract from F (from the number of tens), go to the 6th point.
5. If the units of the tens and units are zero in the residual division, then the digits of zero corresponding to the tens and units should be moved to the quotient and go to point 10, otherwise to point 7.
6. If the resulting remainder and the last digit is zero, the last zero digit must be written to the quotient, go to item 9, otherwise go to item 10.
7. To the right of the difference, move the unit of the digit showing tens, determine how many times the divisor is in the resulting number (in TO, i.e. determine the tens of the quotient and, multiplying it by the divisor, subtract the resulting product from T1 and go to point 8.
8. If the last digit of the residual divisible is zero, one zero is written to the quotient and go to point 10, otherwise go to point 9.
9. To the right of the remainder, move the last digit of the divisible, determine how many times the divisor is located in the resulting number - (T2) , i.e. determine the last digit of the quotient, multiply it by the divisor and subtract from T2.
10. Check the correctness of the solution by comparing the product of the quotient by the divisor with the divisible.
Mastering the mechanism of dividing a three-digit number by a one-digit number is a particularly important step in the development of the practice of division. [7; 115]. Of course, at the end of a single-digit number, it is necessary to explain the state of the residual section. Experience shows that this process is not so difficult.
As you know, dividing a multi-digit number into two-digit numbers turns several times into dividing a three-digit number into two-digit numbers, obtaining a single-digit number in the quotient. Therefore, if we want to instill in students a good habit of dividing a multi-digit number by a two-digit number, we must
first give them the opportunity to divide a three-digit number by a two-digit number, with a single-digit number in the quotient. [9; 47]
By mastering the division algorithm, we also mean that the necessary steps are elementary for students. Otherwise, the student will not be able to perform his algorithmic activity. The main difficulty in division is to select and find the quotient numbers. If students do it well, then later they will not face great difficulties. You need to perform a series of exercises to master the process of dividing a three-digit number by a two-digit number, with obtaining a single-digit number in the quotient. [8; 121]
As a result of the above work, students are ready to divide any multi-digit number into two-digit numbers: students study the principle of gradual division of numbers, starting from the upper floors, and learn to find the digit of the quotient experimentally; this is enough to acquire a new habit. [4; 109]
The division of a three-digit number by a two-digit number should be included in the educational process on the basis of an appropriate explanation. It is proposed to start by explaining the division of 488 by 61. The explanation is as follows: "To choose the digits of the quotient, first rounding the divisor we get 60, then divide 488 by 60; For this we divide 48 by 6 and get 8, the number 8 is not the last, it is a test number, we need to divide 488 by 61, not by 60. we need to check this figure. Multiplying 61 by 8, we get 488. So, the number 8 is true."
After strengthening the mastery of algorithmic steps, which are reflected in the explanation of the implementation of examples of this type, it is supposed to include exercises related to obtaining two-digit numbers in the quotient. Therefore, in order to teach students the algorithm of dividing any multi-digit number into two-digit numbers and to form their habit of using this algorithm quickly and correctly, it is necessary to teach division into numbers expressed by ones and zeros or into round tens. There is no doubt that division by 10 can be considered as division into equal parts, and division by composition. Taking this division as a division into 10 equal parts, we say that when dividing each ten by 10, one unit is obtained. Explaining this division operation as division by composition, we say how many times there are 10 (divisors) in a given number (divisible). It is useful to teach the methodology of division by 10 to students both full and remainder division. Division into round tens is, after all, nothing more than the division of two-digit and three-digit numbers into round tens. Therefore, it is natural to start this case of division by dividing three-digit numbers into round tens to obtain a single-digit number in the quotient. It is necessary to perform many exercises to help students master the method of finding the digit of quotient by dividing the number of tens of a divisible number by the number of tens of a divisor. [8; 47]
It is possible to distinguish the following stages of work aimed at mastering the algorithm of multivalued division:
1. Division by a single digit number;
2. Division by 20, 100, 1000, etc.;
3. Division by round tens;
4. Division into two-digit and three-digit numbers.
At the stage of the division operation, when
finding the digits of the quotient is relatively easy, students need to be introduced to a more complex case of the division operation, that is, with the division of numbers that give zeros in the middle or at the end of the quotient. These special cases help to better understand the general rule of division. It is well known that in such examples, some students do not pay attention to getting the residuals and, depending on the process of splitting them each time, getting the quotient, and skip the zeros. To prevent such errors, the following methods should be used:
a) show that it is possible to determine the total number of numbers in the quotient based on the value of the first digit of the quotient;
b) mark with dots the places of the digits that will be received in the quotient;
c) specify which digit is divided and which digit is obtained in the quotient when performing the division;
d) explain the function and the value of zero in the quotient and show how the quotient changes when by mistake not to write zero next to the quotient. [12; 226]
There is no doubt that one more should be added to these methods. After the division operation is completed, its accuracy must be checked by multiplication;
Experience has shown that at the end of dividing a multi-digit number by a single digit, as in the case of dividing a three-digit number by one digit, it is advisable to explain and put into practice the following case of division with a remainder: the last digit of the divisible is not divisible by the divisor, and this digit is the remainder, and zero is written in the quotient. When explaining this case of division, one more should be added to the above methods: in the division operation, the number obtained in the quotient must first be approximately calculated in the mind.
As you know, until they teach the practice of multiplication and division in the millionth circle, children consider it possible to consider division as division into equal parts and division by composition. This work begins with the repetition of knowledge.
After solving several examples , the execution algorithm is defined as follows: "To divide a number with zeros at the end by 10, you need to throw out the last zero of this number." Then examples of division with remainder are solved: 142:10, 6114:10, etc. Focusing students' attention on the private, and here on the remainder, is considered an important methodological approach. This methodology is applicable to 10, 100, 1000, and several explanatory examples are solved. Finally, the algorithm for dividing by a number expressed in ones and zeros is generally expressed: "To divide a number into numbers consisting of ones and zeros, it is enough to discard as many digits to the right of this number as there are zeros in the divisor." [3; 226]
Students have some knowledge and skills about dividing into round tens in the "thousandth" circle of
multiplication and division, and as the process continues, the range of numbers expands. It is natural to apply this process to round hundreds, etc. [11; 262263]
It is necessary to distinguish two cases in the work of dividing into round hundreds:
1) divide by round hundreds, to obtain a single digit number in the quotient;
2) divide any number into round hundreds.
The first case forms the basis of the second.
Students should be able to use terminology related
to the division algorithm (names of floor units, names of components, etc.). The importance of this issue should be properly assessed.
When analyzing the process of dividing a multi-digit number by a two-digit number (for example, 18792: 54 = 348), it becomes clear that dividing these numbers several times turns into dividing a three-digit number by a two-digit number, with obtaining a single-digit number in the quotient (187: 54; 259: 54; 432: 54). Therefore, if we want to instill in students a good habit of dividing a multi-digit number by a two-digit number, we must first give them the opportunity to divide a three-digit number by a two-digit number, with a single-digit number in the quotient.
Of course, in order to divide a multi-digit number into a three-digit number, it is always necessary to be able to find the digit of the quotient obtained by dividing three-digit and four-digit numbers into three-digit numbers. This is due to the fact that it is necessary to isolate this stage of the division operation and study it independently and comprehensively; this part of the division forms the basis of the operation of dividing any number by a three-digit number.
Here, too, it is necessary to follow the path of division into two-digit numbers, that is, round the divisor to hundreds, divide hundreds by hundreds to find the digit of the quotient; it is necessary to check this number and, having come to the conclusion that this number is correctly found, enter it into the quotient.
Here it is also necessary to draw students' attention to the following question: sometimes it is more useful to supplement the divisor to a large round number in order to find the quotient digit faster. This is necessary when the second digit of the divisor is 8 or 9.
In the methodological literature, it is recommended to organize the exercises at this stage in the following system:
a) Solve examples whose coefficients are easy to find at the first trialis
b) Solve the examples whose quotient is after the second sample, and where this trial digit of the quotient decreases;
c) Solve the examples by checking the found trial digit of the quotient several times (1326: 166);
d) Solve the examples related to division with remainder [6; 264].
Particular attention should be paid to division into three-digit numbers with a multi-digit number in the quotient.
The ability to divide a four-digit number by a three-digit number, the quotient of which is a single-digit number, provides a solid foundation for instilling
the habit of dividing any number by a three-digit number. When performing exercises, it is useful to follow the following sequence:
a) first, examples of division with remainder;
b) then examples of the same type, provided that zeros are obtained in the quotient ;
c) and finally, examples of division with remainder with or without zeros in the quotient.
During the exercises, a brief explanation is given, and the student is asked to explain in detail the solution of the example only in case of difficulties.
In order to master the algorithm of dividing numbers with zeros at the end, students must be motivated to solve the corresponding tasks. By drawing the attention of students to the solution of expedient tasks, in this case it is possible to come to the appropriate result.
Tasks should require an explanation of the solution. By drawing students' attention to the decision-making process, we can say that here it is necessary to perform the operation of dividing numbers with zeros at the end, just as if we were dividing any multi-digit numbers into 3- and 4-digit numbers. Here division can be briefly depicted by dropping the same number of zeros in the divisible and divisor. This method is based on the following property: if we reduce the divisible and the divisor by the same number of times, the quotient will not change.
One important issue should not be overlooked here. In the case of residual division, the removed zeros should be added to the remainder (to the right of the significant numbers). In this case, it would be useful to solve a few examples.
Scientific innovations. A technology has been developed to implement the expectations of systematicity and consistency when teaching division operations on natural numbers.
Practical significance. The development of technology to realize the expectations of systematicity and consistency in teaching division operations over
natural numbers will have a positive impact on the formation of an environment that excludes possible errors in the teaching activities of practical teachers.
Result. The use of technology to realize the expectations of systematicity and consistency in the teaching of the division operation over natural numbers is a more effective methodological approach from the point of view of forming the algorithmic culture of students and improving the quality of learning.
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