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A column? How to work out the skill of division in a column at home if the child did not learn something at school? Dividing by a column is taught in grades 2-3, for parents, of course, this is a passed stage, but if you wish, you can remember the correct entry and explain in an accessible way to your student what he will need in life.
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What should a child in grades 2-3 know in order to learn how to divide in a column?
How to properly explain to a child in grades 2-3 the division by a column so that he does not have problems in the future? First, let's check if there are any gaps in knowledge. Make sure that:
- the child freely performs addition and subtraction operations;
- knows the digits of numbers;
- knows by heart.
How to explain to the child the meaning of the action "division"?
- The child needs to explain everything with a good example.
Ask to share something between family members or friends. For example, sweets, cake pieces, etc. It is important that the child understands the essence - you need to share equally, i.e. without a trace. Practice with different examples.
Let's say 2 groups of athletes have to take seats on the bus. It is known how many athletes are in each group and how many seats are in the bus. You need to find out how many tickets you need to buy one and the second group. Or 24 notebooks need to be distributed to 12 students, how many will get each.
- When the child learns the essence of the principle of division, show the mathematical notation of this operation, name the components.
- Explain what division is the opposite of multiplication, multiplication inside out.
It is convenient to show the relationship between division and multiplication using the example of a table.
For example, 3 times 4 equals 12.
3 is the first multiplier;
4 - second multiplier;
12 - product (the result of multiplication).
If 12 (the product) is divided by 3 (the first factor), we get 4 (the second factor).
Components when dividing called differently:
12 - divisible;
3 - divider;
4 - quotient (the result of division).
How to explain to a child the division of a two-digit number by a single number is not in a column?
It is easier for us, adults, to write down “in the old fashioned way” with a “corner” - and that's it. BUT! Children have not yet passed the division in a column, what should I do? How to teach a child to share two-digit number on unambiguous without using record by a column?
Let's take 72:3 as an example.
Everything is simple! We decompose 72 into such numbers that are easy to verbally divide by 3:
72=30+30+12.
Everything immediately became clear: we can divide 30 by 3, and the child can easily divide 12 by 3.
All that remains is to add up the results, i.e. 72:3=10 (obtained when 30 divided by 3) + 10 (30 divided by 3) + 4 (12 divided by 3).
72:3=24
We did not use long division, but the child understood the reasoning and performed the calculations without difficulty.
After simple examples you can proceed to the study of division in a column, teach your child to correctly write down examples in a “corner”. To begin with, use only examples for division without a remainder.
How to explain to a child the division into a column: a solution algorithm
Large numbers are difficult to divide in the mind, it is easier to use the notation of division by a column. To teach a child to perform calculations correctly, follow the algorithm:
- Determine where the dividend and divisor are in the example. Ask the child to name the numbers (what will we divide by).
213:3
213 - divisible
3 - divider
- Write down the dividend - "corner" - divisor.
- Determine which part of the dividend we can use to divide by a given number.
We argue like this: 2 is not divisible by 3, which means we take 21.
- Determine how many times the divisor "fits" in the selected part.
21 divided by 3 - take 7.
- Multiply the divisor by the selected number, write the result under the "corner".
Multiply 7 by 3 - we get 21. We write it down.
- Find the difference (remainder).
At this stage of reasoning, teach the child to check himself. It is important that he understands that the result of the subtraction should ALWAYS be less divisor. If it turned out wrong, you need to increase the selected number and perform the action again.
- Repeat the steps until the remainder is 0.
How to reason correctly to teach a child in grades 2-3 to divide in a column
How to explain division to a child 204:12=?
1.
We write in a column.
204 is the dividend, 12 is the divisor.
2.
2 is not divisible by 12, so we take 20.
3.
To divide 20 by 12, we take 1. We write 1 under the “corner”.
4.
Multiply 1 by 12, we get 12. We write under 20.
5.
20 minus 12 is 8.
We check ourselves. Is 8 less than 12 (divisor)? Ok, that's right, let's move on.
6.
Next to 8 we write 4. 84 divided by 12. By how much do you need to multiply 12 to get 84?
It’s hard to say right away, let’s try to act by the selection method.
Take, for example, 8, but do not write down yet. We count verbally: 8 times 12 will be 96. And we have 84! Not suitable.
Let's try less... For example, let's take 6. We check ourselves verbally: 6 times 12 equals 72. 84-72=12. We got the same number as our divisor, but it must be either zero or less than 12. So, the optimal number is 7!
7.
We write 7 under the "corner" and perform the calculations. Multiply 7 by 12 to get 84.
8.
We write the result in a column: 84 minus 84 equals zero. Hooray! We made the right decision!
So, you have taught the child to divide in a column, now it remains to work out this skill, bring it to automatism.
Why is it difficult for children to learn to divide in a column?
Remember that problems with mathematics arise from the inability to quickly do simple arithmetic operations. AT primary school you need to work out and bring addition and subtraction to automaticity, learn the multiplication table “from cover to cover”. All! The rest is a matter of technique, and it is developed with practice.
Be patient, don't be lazy once again to explain to the child what he did not learn in the lesson, it is tedious, but meticulous to understand the reasoning algorithm and say each intermediate operation before voicing the finished answer. Give additional examples to practice skills, play games math games- this will bear fruit and you will see the results and rejoice at the success of the child very soon. Be sure to show where and how you can apply the knowledge gained in Everyday life.
Dear readers! Tell us how you teach your children to divide in a column, what difficulties you had to face and how you overcame them.
Division natural numbers, especially multi-valued ones, it is convenient to carry out a special method, which is called division by a column (in a column). You can also see the name corner division. Immediately, we note that the column can be carried out both division of natural numbers without a remainder, and division of natural numbers with a remainder.
In this article, we will understand how division by a column is performed. Here we will talk about the writing rules, and about all intermediate calculations. First, let us dwell on the division of a multi-valued natural number by a single-digit number by a column. After that, we will focus on cases where both the dividend and the divisor are multi-valued natural numbers. The whole theory of this article is provided with characteristic examples of division by a column of natural numbers with detailed explanations of the solution and illustrations.
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Rules for recording when dividing by a column
Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results when dividing natural numbers by a column. Let's say right away that it is most convenient to divide in a column in writing on paper with a checkered line - so there is less chance of going astray from the desired row and column.
First, the dividend and the divisor are written in one line from left to right, after which a symbol of the form is displayed between the written numbers. For example, if the dividend is the number 6 105, and the divisor is 5 5, then their correct notation when divided into a column will be:
Look at the following scheme, illustrating the places for writing the dividend, divisor, quotient, remainder and intermediate calculations when dividing by a column.
It can be seen from the above diagram that the desired quotient (or incomplete quotient when dividing with a remainder) will be written below the divisor under the horizontal line. And intermediate calculations will be carried out below the dividend, and you need to take care of the availability of space on the page in advance. In this case, one should be guided by the rule: the greater the difference in the number of characters in the entries of the dividend and divisor, the more space is required. For example, when dividing a natural number 614 808 by 51 234 by a column (614 808 is a six-digit number, 51 234 is a five-digit number, the difference in the number of characters in the records is 6−5=1 ) for intermediate calculations, you will need less space than when dividing the numbers 8058 and 4 (here the difference in the number of characters is 4−1=3). To confirm our words, we present the completed records of division by a column of these natural numbers:
Now you can go directly to the process of dividing natural numbers by a column.
Division by a column of a natural number by a single-digit natural number, division algorithm by a column
It is clear that dividing one single-digit natural number by another is quite simple, and there is no reason to divide these numbers into a column. However, it will be useful to practice the initial skills of division by a column on these simple examples.
Example.
Let us need to divide by a column 8 by 2.
Decision.
Of course, we can perform division using the multiplication table, and immediately write down the answer 8:2=4.
But we are interested in how to divide these numbers by a column.
First, we write the dividend 8 and the divisor 2 as required by the method:
Now we start to figure out how many times the divisor is in the dividend. To do this, we successively multiply the divisor by the numbers 0, 1, 2, 3, ... until the result is a number equal to the dividend (or a number greater than the dividend, if there is a division with a remainder). If we get a number equal to the dividend, then we immediately write it under the dividend, and in place of the private we write the number by which we multiplied the divisor. If we get a number greater than the divisible, then under the divisor we write the number calculated at the penultimate step, and in place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.
Let's go: 2 0=0 ; 2 1=2; 2 2=4 ; 2 3=6 ; 2 4=8 . We got a number equal to the dividend, so we write it under the dividend, and in place of the private we write the number 4. The record will then look like this:
The final stage of dividing single-digit natural numbers by a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract numbers above this line in the same way as it is done when subtracting natural numbers with a column. The number obtained after subtraction will be the remainder of the division. If it is equal to zero, then the original numbers are divided without a remainder.
In our example, we get
Now we have a finished record of division by a column of the number 8 by 2. We see that the quotient 8:2 is 4 (and the remainder is 0 ).
Answer:
8:2=4 .
Now consider how the division by a column of single-digit natural numbers with a remainder is carried out.
Example.
Divide by a column 7 by 3.
Decision.
On the initial stage the entry looks like this:
We begin to find out how many times the dividend contains a divisor. We will multiply 3 by 0, 1, 2, 3, etc. until we get a number equal to or greater than the dividend 7. We get 3 0=0<7
; 3·1=3<7
; 3·2=6<7
; 3·3=9>7 (if necessary, refer to the article comparison of natural numbers). Under the dividend we write the number 6 (it was obtained at the penultimate step), and in place of the incomplete quotient we write the number 2 (multiplication was carried out on it at the penultimate step).
It remains to carry out the subtraction, and the division by a column of single-digit natural numbers 7 and 3 will be completed.
So the partial quotient is 2 , and the remainder is 1 .
Answer:
7:3=2 (rest. 1) .
Now we can move on to dividing multi-valued natural numbers by single-digit natural numbers by a column.
Now we will analyze column division algorithm. At each stage, we will present the results obtained by dividing the many-valued natural number 140 288 by the single-valued natural number 4 . This example was not chosen by chance, since when solving it, we will encounter all possible nuances, we will be able to analyze them in detail.
First, we look at the first digit from the left in the dividend entry. If the number defined by this figure is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add the next digit to the left in the dividend record, and work further with the number determined by the two digits in question. For convenience, we select in our record the number with which we will work.
The first digit from the left in the dividend 140,288 is the number 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the dividend record. At the same time, we see the number 14, with which we have to work further. We select this number in the notation of the dividend.
The following points from the second to the fourth are repeated cyclically until the division of natural numbers by a column is completed.
Now we need to determine how many times the divisor is contained in the number we are working with (for convenience, let's denote this number as x ). To do this, we successively multiply the divisor by 0, 1, 2, 3, ... until we get the number x or a number greater than x. When the number x is obtained, then we write it under the selected number according to the notation rules used when subtracting by a column of natural numbers. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (during subsequent passes of 2-4 points of the algorithm, this number is written to the right of the numbers already there). When a number is obtained that is greater than the number x, then under the selected number we write the number obtained at the penultimate step, and in place of the quotient (or to the right of the numbers already there) we write the number by which the multiplication was carried out at the penultimate step. (We carried out similar actions in the two examples discussed above).
We multiply the divisor of 4 by the numbers 0 , 1 , 2 , ... until we get a number that is equal to 14 or greater than 14 . We have 4 0=0<14
, 4·1=4<14
, 4·2=8<14
, 4·3=12<14
, 4·4=16>fourteen . Since at the last step we got the number 16, which is greater than 14, then under the selected number we write the number 12, which turned out at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate paragraph the multiplication was carried out precisely on it.
At this stage, from the selected number, subtract the number below it in a column. Below the horizontal line is the result of the subtraction. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at this point is the very last action that completely completes the division by a column). Here, for your control, it will not be superfluous to compare the result of subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake has been made somewhere.
We need to subtract the number 12 from the number 14 in a column (for the correct notation, you must not forget to put a minus sign to the left of the subtracted numbers). After the completion of this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with a divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next item.
Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we did not write zero), we write down the number located in the same column in the record of the dividend. If there are no numbers in the record of the dividend in this column, then the division by a column ends here. After that, we select the number formed under the horizontal line, take it as a working number, and repeat with it from 2 to 4 points of the algorithm.
Under the horizontal line to the right of the number 2 already there, we write the number 0, since it is the number 0 that is in the record of the dividend 140 288 in this column. Thus, the number 20 is formed under the horizontal line.
We select this number 20, take it as a working number, and repeat the actions of the second, third and fourth points of the algorithm with it.
We multiply the divisor of 4 by 0 , 1 , 2 , ... until we get the number 20 or a number that is greater than 20 . We have 4 0=0<20
, 4·1=4<20
, 4·2=8<20
, 4·3=12<20
, 4·4=16<20
, 4·5=20
. Так как мы получили число, равное числу 20
, то записываем его под отмеченным числом, а на месте частного, справа от уже имеющегося там числа 3
записываем число 5
(на него производилось умножение).
We carry out subtraction by a column. Since we subtract equal natural numbers, then, due to the property of subtracting equal natural numbers, we get zero as a result. We do not write zero (since this is not the final stage of dividing by a column), but we remember the place where we could write it down (for convenience, we will mark this place with a black rectangle).
Under the horizontal line to the right of the memorized place, we write the number 2, since it is she who is in the entry of the dividend 140 288 in this column. Thus, under the horizontal line we have the number 2 .
We take the number 2 as a working number, mark it, and once again we will have to perform the steps from 2-4 points of the algorithm.
We multiply the divisor by 0 , 1 , 2 and so on, and compare the resulting numbers with the marked number 2 . We have 4 0=0<2
, 4·1=4>2. Therefore, under the marked number, we write the number 0 (it was obtained at the penultimate step), and in place of the quotient to the right of the number already there, we write the number 0 (we multiplied by 0 at the penultimate step).
We perform subtraction by a column, we get the number 2 under the horizontal line. We check ourselves by comparing the resulting number with the divisor 4 . Since 2<4
, то можно спокойно двигаться дальше.
Under the horizontal line to the right of the number 2, we add the number 8 (since it is in this column in the record of the dividend 140 288). Thus, under the horizontal line is the number 28.
We accept this number as a worker, mark it, and repeat steps 2-4 of paragraphs.
There shouldn't be any problems here if you've been careful up to now. Having done all the necessary actions, the following result is obtained.
It remains for the last time to carry out the actions from points 2, 3, 4 (we provide it to you), after which you will get a complete picture of dividing natural numbers 140 288 and 4 in a column:
Please note that the number 0 is written at the very bottom of the line. If this were not the last step of dividing by a column (that is, if there were numbers in the columns on the right in the record of the dividend), then we would not write this zero.
Thus, looking at the completed record of dividing the multi-valued natural number 140 288 by the single-valued natural number 4, we see that the number 35 072 is private (and the remainder of the division is zero, it is on the very bottom line).
Of course, when dividing natural numbers by a column, you will not describe all your actions in such detail. Your solutions will look something like the following examples.
Example.
Perform long division if the dividend is 7136 and the divisor is a single natural number 9.
Decision.
At the first step of the algorithm for dividing natural numbers by a column, we get a record of the form
After performing the actions from the second, third and fourth points of the algorithm, the record of division by a column will take the form
Repeating the cycle, we will have
One more pass will give us a complete picture of division by a column of natural numbers 7 136 and 9
Thus, the partial quotient is 792 , and the remainder of the division is 8 .
Answer:
7 136:9=792 (rest 8) .
And this example demonstrates how long division should look like.
Example.
Divide the natural number 7 042 035 by the single digit natural number 7 .
Decision.
It is most convenient to perform division by a column. 
Answer:
7 042 035:7=1 006 005 .
Division by a column of multivalued natural numbers
We hasten to please you: if you have well mastered the algorithm for dividing by a column from the previous paragraph of this article, then you already almost know how to perform division by a column of multivalued natural numbers. This is true, since steps 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first step.
At the first stage of dividing into a column of multi-valued natural numbers, you need to look not at the first digit on the left in the dividend entry, but at as many of them as there are digits in the divisor entry. If the number defined by these numbers is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the record of the dividend. After that, the actions indicated in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.
It remains only to see the application of the algorithm for dividing by a column of multi-valued natural numbers in practice when solving examples.
Example.
Let's perform division by a column of multivalued natural numbers 5562 and 206.
Decision.
Since 3 characters are involved in the record of the divisor 206, we look at the first 3 digits on the left in the record of the dividend 5 562. These numbers correspond to the number 556. Since 556 is greater than the divisor 206, we take the number 556 as a working one, select it, and proceed to the next stage of the algorithm.
Now we multiply the divisor 206 by the numbers 0 , 1 , 2 , 3 , ... until we get a number that is either equal to 556 or greater than 556 . We have (if the multiplication is difficult, then it is better to perform the multiplication of natural numbers in a column): 206 0=0<556
, 206·1=206<556
, 206·2=412<556
, 206·3=618>556 . Since we got a number that is greater than 556, then under the selected number we write the number 412 (it was obtained at the penultimate step), and in place of the quotient we write the number 2 (since it was multiplied at the penultimate step). The column division entry takes the following form:
Perform column subtraction. We get the difference 144, this number is less than the divisor, so you can safely continue to perform the required actions.
Under the horizontal line to the right of the number available there, we write the number 2, since it is in the record of the dividend 5 562 in this column:
Now we work with the number 1442, select it, and go through steps two through four again.
We multiply the divisor 206 by 0 , 1 , 2 , 3 , ... until we get the number 1442 or a number that is greater than 1442 . Let's go: 206 0=0<1 442
, 206·1=206<1 442
, 206·2=412<1 332
, 206·3=618<1 442
, 206·4=824<1 442
, 206·5=1 030<1 442
, 206·6=1 236<1 442
, 206·7=1 442
. Таким образом, под отмеченным числом записываем 1 442
, а на месте частного правее уже имеющегося там числа записываем 7
:
We subtract by a column, we get zero, but we don’t write it down right away, but only remember its position, because we don’t know if the division ends here, or we will have to repeat the steps of the algorithm again:
Now we see that under the horizontal line to the right of the memorized position, we cannot write down any number, since there are no numbers in the record of the dividend in this column. Therefore, this division by a column is over, and we complete the entry:
- Mathematics. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
- Mathematics. Any textbooks for 5 classes of educational institutions.
How to teach a child to divide? The simplest method is learn division by a column. This is much easier than doing mental calculations, it helps not to get confused, not to “lose” numbers and develop a mental scheme that will work automatically in the future.
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How is it carried out
Division with a remainder is a method in which a number cannot be divided into exactly several parts. As a result of this mathematical operation, in addition to the whole part, an indivisible piece remains.
Let's take a simple example how to divide with a remainder:
There is a can of 5 liters of water and 2 cans of 2 liters. When water is poured from a five-liter jar into a two-liter jar, 1 liter of unused water will remain in the five-liter jar. This is the remainder. Digitally it looks like this:
5:2=2 rest (1). Where is 1 from? 2x2=4, 5-4=1.
Now consider the order of division into a column with a remainder. This visually facilitates the calculation process and helps not to lose numbers.
The algorithm determines the location of all elements and the sequence of actions by which the calculation is performed. As an example, let's divide 17 by 5.
Main stages:
- Correct entry. Divisible (17) - located on the left side. To the right of the dividend, write the divisor (5). A vertical line is drawn between them (indicates the sign of division), and then, from this line, a horizontal line is drawn, emphasizing the divisor. The main features are indicated in orange.
- The search for the whole. Next, the first and simplest calculation is carried out - how many divisors fit in the dividend. Let's use the multiplication table and check in order: 5*1=5 - fits, 5*2=10 - fits, 5*3=15 - fits, 5*4=20 - doesn't fit. Five times four is more than seventeen, which means that the fourth five does not fit. Back to three. A 17 liter jar will fit 3 five liter jars. We write the result in the form: 3 we write under the line, under the divisor. 3 is an incomplete quotient.
- Definition of the remainder. 3*5=15. 15 is written under the dividend. We draw a line (indicates the sign "="). Subtract the resulting number from the dividend: 17-15=2. We write the result below under the line - in a column (hence the name of the algorithm). 2 is the remainder.
Note! When dividing in this way, the remainder must always be less than the divisor.
When the divisor is greater than the dividend
There are cases when the divisor is greater than the dividend. Decimal fractions in the program for the 3rd grade have not yet been studied, but, following the logic, the answer must be written in the form of a fraction - at best a decimal, at worst a simple one. But (!) in addition to the program, the calculation method limits the task: it is necessary not to divide, but to find the remainder! some of them are not! How to solve such a problem?
Note! There is a rule for cases when the divisor is greater than the dividend: the incomplete quotient is 0, the remainder is equal to the dividend.
How to divide the number 5 by the number 6, highlighting the remainder? How many 6 liter jars will fit in a 5 liter jar? because 6 is greater than 5.
According to the task, it is necessary to fill 5 liters - not a single one is filled. So, all 5 are left. Answer: incomplete quotient = 0, remainder = 5.
Division begins to be studied in the third grade of school. By this time, students should already be, which allows them to divide two-digit numbers into single-digit ones.
Solve the problem: 18 sweets need to be distributed to five children. How many candies are left?
Examples:
Find the incomplete quotient: 3*1=3, 3*2=6, 3*3=9, 3*4=12, 3*5=15. 5 - bust. We return to 4.
Remainder: 3*4=12, 14-12=2.
Answer: incomplete quotient 4, 2 left.
You may ask why, when divided by 2, the remainder is either 1 or 0. According to the multiplication table, between digits that are multiples of two there is a difference per unit.
Another task: 3 pies must be divided into two.
Divide 4 pies between two.
Divide 5 pies between two.
Working with multi-digit numbers
The 4th grade program offers a more complex division process with an increase in calculated numbers. If in the third grade the calculations were carried out on the basis of the basic multiplication table ranging from 1 to 10, then the fourth-graders carry out calculations with multi-digit numbers over 100.
This action is most convenient to perform in a column, since the incomplete quotient will also be a two-digit number (in most cases), and the column algorithm facilitates calculations and makes them more visual.
Let's divide multi-digit numbers to two-digit: 386:25
This example differs from the previous ones in the number of calculation levels, although the calculations are carried out according to the same principle as before. Let's take a closer look:
386 is the dividend, 25 is the divisor. It is necessary to find the incomplete quotient and extract the remainder.
First level
The divisor is a two-digit number. The dividend is three-digit. We select the first two left digits from the dividend - this is 38. We compare them with the divisor. 38 over 25? Yes, so 38 can be divided by 25. How many whole 25s are in 38?
25*1=25, 25*2=50. 50 is greater than 38, go back one step.
Answer - 1. We write the unit to the zone not full private.
38-25=13. We write the number 13 under the line.
Second level
13 over 25? No - it means you can “lower” the number 6 down by adding it next to 13, on the right. It turned out 136. Is 136 more than 25? Yes, it means you can subtract it. How many times does 25 fit into 136?
25*1=25, 25*2=50, 25*3=75, 25*4=100, 25*5=125, 256*=150. 150 is greater than 136 - go back one step. We write the number 5 in the incomplete quotient zone, to the right of the unit.
We calculate the remainder:
136-125=11. We write under the line. 11 over 25? No, division is not possible. Does the dividend have digits left? No, there is nothing more to share. Calculations completed.
Answer: the incomplete quotient is 15, with a remainder of 11.
And if such a division is proposed, when the two-digit divisor is greater than the first two digits of the multi-valued dividend? In this case, the third (fourth, fifth and subsequent) digit of the dividend takes part in the calculations immediately.
Here are some examples division with three- and four-digit numbers:
75 is a two-digit number. 386 - three-digit. Compare the first two digits on the left with the divisor. 38 over 75? No, division is not possible. We take all 3 numbers. 386 over 75? Yes, division is possible. We carry out calculations.
75*1=75, 75*2=150, 75*3=225, 75*4=300, 75*5= 375, 75*6=450. 450 is greater than 386 - we go back a step. We write down 5 in the zone of incomplete quotient.
Of course, children learn the basics of mathematics in the classroom at school. But the teacher's explanations are not always clear to the kid. Or maybe the child got sick and missed the topic. In such cases, parents should remember their school years in order to help the child not miss important information, without which further education will be unrealistic.
Teaching a child with a column begins in the third grade. By this time, the student should already be able to use the multiplication table with ease. But if there are problems with this, it is worth immediately, because before you teach a child to divide by a column, there should not be any difficulties with multiplication.
How to teach column division?
Take for example the three-digit number 372 and divide it by 6. Choose any combination, but so that the division goes without a trace. At first, this can confuse a young mathematician.
We write down the numbers, separating them with a corner, and explain to the child that we will gradually divide this large number into six equal parts. Let's first try to divide the first digit of 3 by 6.
It is not divisible, which means we add the second one, that is, let's try to see if we can divide 37.
It is necessary to ask the child how many times the six will fit in the number 37. Anyone who knows mathematics without problems will immediately guess that the selection method can be used to select the desired multiplier. So, let's pick up, take, for example, 5 and multiply by 6 - it turns out 30, it seems that the result is not far from 37, but it's worth trying again. To do this, we multiply 6 by 6 - equal to 36. This suits us, and the first digit of the quotient has already been found - we write it under the divisor, behind the line.
We write the number 36 under 37 and when subtracting we get one. It is again not divisible by 6, which means that we demolish the remaining deuce to it. Now the number 12 is very easy to divide by 6. As a result, we get the second private number - two. Our division result will be 62.
Let's first consider the simple cases of division, when the quotient is a single-digit number.
Let's find the value of the private numbers 265 and 53.
To make it easier to pick up the private number, we divide 265 not by 53, but by 50. To do this, we divide 265 by 10, it will be 26 (remainder 5). And we divide 26 by 5, it will be 5. The number 5 cannot be immediately written in private, since this is a trial number. First you need to check if it fits. Let's multiply . We see that the number 5 came up. And now we can record it in private.
The value of the private numbers 265 and 53 is 5. Sometimes, when dividing, the trial digit of the private does not fit, and then it needs to be changed.
Let's find the value of the private numbers 184 and 23.
The quotient will be a single digit.
To make it easier to pick up the private number, we divide 184 not by 23, but by 20. To do this, we divide 184 by 10, it will be 18 (remainder 4). And we divide 18 by 2, it will be 9. 9 is a trial number, we won’t write it in private right away, but we’ll check if it fits. Let's multiply . And 207 is greater than 184. We see that the number 9 does not fit. The quotient will be less than 9. Let's see if the number 8 is suitable. Multiply . We see that the number 8 is suitable. We can record it privately.
The value of the private numbers 184 and 23 is 8.
Let's consider more difficult cases of division. Find the value of the private numbers 768 and 24.
The first incomplete dividend is 76 tens. So, there will be 2 digits in the quotient.
Let's determine the first digit of the quotient. Let's divide 76 by 24. To make it easier to find the private number, we divide 76 not by 24, but by 20. That is, we need to divide 76 by 10, there will be 7 (remainder 6). Divide 7 by 2 to get 3 (remainder 1). 3 is the trial digit of the quotient. Let's check if it fits first. Let's multiply . . The remainder is less than the divisor. This means that the number 3 has come up and now we can write it down in place of tens of quotients.
Let's continue the division. The next incomplete dividend is 48 units. Let's divide 48 by 24. To make it easier to pick up the private number, we divide 48 not by 24, but by 20. That is, we divide 48 by 10, there will be 4 (remainder 8). And 4 divided by 2 will be 2. This is a trial digit of the private. We must first check if it will fit. Let's multiply . We see that the number 2 has come up and, therefore, we can write it down in place of the units of the quotient.
The value of the private numbers 768 and 24 is 32.
Let's find the value of the private numbers 15 344 and 56.
The first incomplete dividend is 153 hundreds, which means that there will be three digits in the private.
Let's determine the first digit of the quotient. Let's divide 153 by 56. To make it easier to find the private number, we divide 153 not by 56, but by 50. To do this, we divide 153 by 10, there will be 15 (remainder 3). And 15 divided by 5 will be 3. 3 is the trial digit of the quotient. Remember: you cannot immediately write it in private, but you must first check whether it fits. Let's multiply . And 168 is greater than 153. So, in the quotient it will be less than 3. Let's check if the number 2 is suitable. Multiply. BUT 
. The remainder is less than the divisor, which means that the number 2 is suitable, it can be written in place of hundreds in the quotient.
We form the following incomplete dividend. That's 414 tens. Let's divide 414 by 56. To make it more convenient to choose the quotient figure, we will divide 414 not by 56, but by 50. . 
. Remember: 8 is a trial number. Let's check it out. . And 448 is greater than 414, which means that in the quotient it will be less than 8. Let's check if the number 7 is suitable. Multiply 56 by 7, we get 392. 
. The remainder is less than the divisor. So, the number came up and in the quotient in place of tens we can write 7.
Let's continue the division. The next incomplete dividend is 224 units. Divide 224 by 56. To make it easier to pick up the quotient, divide 224 by 50. That is, first by 10, it will be 22 (remainder 4). And 22 divided by 5 will be 4 (remainder 2). 4 is a trial number, let's check if it works. . And we see that the figure has come up. We write 4 in place of units in the quotient.
The value of the private numbers 15 344 and 56 - 274.
Today we learned to divide in writing by a two-digit number.
Bibliography
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Homework
Perform division
