Early years school life In the lower grades things are not easy for a child. Often after a math lesson they do not fully understand the topic covered. To help your child master the material covered, you will need to explain to the student yourself what he does not understand. Parents come to the rescue, and the question immediately arises: “How to explain division to a child?” This can be done in several ways, but initially you should make sure that the child has thoroughly mastered such mathematical operations as addition, subtraction and multiplication.(You can read about ways to teach children addition and multiplication And ).
Teaching your child the basics of division
It is important that the child understands the essence of such a mathematical operation as division. To do this, it is necessary to explain to him that division is the division of something into equal shares. It is recommended to turn the learning process into interesting game so that the child is concentrated.
Division in a playful way
TIP: The division table is just as important to learn as the multiplication table. It's better to do this during the holidays!
Help your child understand that division is the inverse of multiplication.
The most in a simple way explain division is a visual demonstration of the division of objects into equal shares. You can use anything as divisible items, but preferably something interesting for the child. Examples include candy and toys.
How to explain division to a child using toys?
Initially, you need to take 2 candies and ask the child to divide them between 2 plush toys. Thanks to this simple example, the child will understand the essence of mathematical division. After this, you can move on to more complex division examples.
How division occurs is shown in detail and in a playful way in the following video:
You can also take a box of colored pencils, which will act as one whole, and invite your child to divide them equally between himself and you. Afterwards, ask your child to count how many pencils were in the box at the beginning and how many he was able to give away.
As the child understands, the parent can increase the number of objects and the number of participants in the task. Then you need to tell that it is not always possible to divide something equally and some items sometimes remain “nobody’s”. For example, you can offer to divide 9 apples between grandparents, dad and mom. The child must understand that everyone will receive only 2 apples, and one will be left over.
Division in a playful way
This way, you will explain the basics of division and prepare your child for more complex school tasks.
TIP: Try to engage with your child in a playful way. Then he will be interested in studying, which means that classes will be fun and effortless.
You will also find it interesting and useful to print the division table as a picture.
The easiest way to divide single-digit numbers by single-digit numbers is to use . To do this, it is enough to explain to the child that division is the inverse action of multiplication. This can be done on any correct example division of natural numbers.
For example: 2 multiplied by 3 equals 6. Based on this example, demonstrate to your child the process of division. You should proceed as follows: divide 6 by any factor, for example, by the number 2. The answer will be 3, that is, the factor not used in the division.
In this way, you can divide multi-digit (two-digit) numbers into single-digit numbers.
Column division algorithm
Before you begin explaining long division, you need to tell your child about the meaning of dividend, divisor, and quotient. In the example 20:4=5, 20 is the dividend, 4 is the divisor, and 5 is the quotient. Each individual number in the example has one name.
Multi-digit numbers (three-digit and two-digit) are easiest to divide into columns. To do this, you need to write multi-digit numbers with a corner.
For example, you need to divide the three-digit number 369 by the single-digit number 3.
The divisor is a three-digit number number 369, and the divisor is a single-digit number 3. First of all, it is important to explain to the child that long division occurs in several stages:
- Determining the part of the dividend suitable for primary division. In this case, the number is 3. 3:3=1. The number 1 must be written in the quotient column.
- “Lower” the next divisible number. In this case it is number 6. 6:3=2 . The resulting number 2 must be written into the quotient.
- Next, you need to “lower” the next divisible number 9. 9 is divisible by 3 without a remainder, the resulting result must be written into the quotient. The result of dividing the three-digit number 369 by 3 is 123.
Dividing a decimal number by a two-digit number works in much the same way. In case of decimal It is necessary to explain to the child that the comma in the divisor is moved to as many places as it is moved in the dividend. This is followed by the usual division into a column.
It is necessary to warn the child about cases of division with a remainder. As an example, you can divide the two-digit number 26 by 5 using a column. This leaves a remainder of 1.
After the explanation, it is important to allow the child to independently solve several examples so that all the material studied remains in the child’s memory for a long time.
You can also watch a video where everything is explained in clear language.
And finally, do not teach yourself or your child to use online calculator to learn how to divide 145 by 9, 34 by 40, 100 by 4, 30 by 80, 416 by 52 and other examples. This will not benefit you or him.
Not only the child goes to 1st grade - parents start with him and finish with him educational institution. The teacher at school does not always have time to explain this or that discipline to each individual student. Therefore, it has its advantages. You can explain to the child yourself, individually and slowly, what he did not understand. During this difficult period, the main thing is to be patient and not scold the student because of wrong decisions. Then everything will work out for you.
Column division is an integral part educational material junior school student. Further success in mathematics will depend on how correctly he learns to perform this action.
How to properly prepare a child to perceive new material?
Column division is a complex process that requires certain knowledge from the child. To perform division, you need to know and be able to quickly subtract, add, and multiply. Knowledge of number digits is also important.
Each of these actions should be brought to automaticity. The child should not have to think for a long time, and also be able to subtract and add not only numbers from the first ten, but within a hundred in a few seconds.
It is important to form the correct concept of division as a mathematical operation. Even when studying multiplication and division tables, the child must clearly understand that the dividend is a number that will be divided into equal parts, the divisor indicates how many parts the number should be divided into, and the quotient is the answer itself.
How to explain the algorithm of a mathematical operation step by step?
Each mathematical operation requires strict adherence to a specific algorithm. Examples of long division should be performed in this order:
- Write the example in a corner, and the places of the dividend and divisor must be strictly observed. To help the child not get confused in the first stages, we can say that we write on the left larger number, and on the right is the smaller one.
- Select a part for the first division. It must be divisible by the dividend with a remainder.
- Using the multiplication table, we determine how many times the divisor can fit in the selected part. It is important to indicate to the child that the answer should not exceed 9.
- Multiply the resulting number by the divisor and write it on the left side of the corner.
- Next, you need to find the difference between the part of the dividend and the resulting product.
- The resulting number is written below the line and the next digit number is taken down. Such actions are performed until the remainder is 0.
A clear example for students and parents
Column division can be clearly explained using this example.
- Write down 2 numbers in a column: the dividend is 536 and the divisor is 4.
- The first part for division must be divisible by 4 and the quotient must be less than 9. The number 5 is suitable for this.
- 4 fits into 5 only once, so we write 1 in the answer, and 4 under 5.
- Next, subtraction is performed: 4 is subtracted from 5 and 1 is written under the line.
- The next digit number is added to one - 3. In thirteen (13) - 4 fits 3 times. 4x3 = 12. Twelve is written under the 13th, and 3 is written as the quotient, as the next digit number.
- 12 is subtracted from 13, the answer is 1. The next digit number is taken away again - 6.
- 16 is again divided by 4. The answer is written as 4, and in the division column - 16, and the difference is drawn as 0.
By solving long division examples with your child several times, you can achieve success in quickly completing problems in middle school.
One of important stages in teaching a child mathematical operations - teaching division operations prime numbers. How to explain division to a child, when can you start mastering this topic?
In order to teach a child division, it is necessary that by the time he learns he has already mastered such mathematical operations, like addition, subtraction, and also had a clear understanding of the very essence of the operations of multiplication and division. That is, he must understand that division is the division of something into equal parts. It is also necessary to teach multiplication operations and learn the multiplication table.
I have already written about this. This article may be useful to you.
We master the operation of division (division) into parts in a playful way
At this stage, it is necessary to form in the child an understanding that division is the division of something into equal parts. The easiest way to teach a child this is to invite him to share a certain number of items among his friends or family members.
Let's say you take 8 identical cubes and ask your child to divide them into two equal parts - for him and for another person. Vary and complicate the task, invite the child to divide 8 cubes not between two, but into four people. Analyze the result with him. Change the components, try with a different number of objects and people into whom these objects need to be divided.
Important: Make sure that at first the child operates with an even number of objects, so that the result of division is the same number of parts. This will be useful at the next stage, when the child needs to understand that division is the inverse operation of multiplication.
Multiply and divide using the multiplication table
Explain to your child that in mathematics, the opposite of multiplication is called division. Using the multiplication table, demonstrate to the student the relationship between multiplication and division using any example.
Example: 4x2=8. Remind your child that the result of multiplication is the product of two numbers. After this, explain that division is the inverse of multiplication and illustrate this clearly.
Divide the resulting product “8” from the example by any of the factors “2” or “4”, and the result will always be a different factor that was not used in the operation.
You also need to teach the young student the names of the categories that describe the operation of division - “dividend”, “divisor” and “quotient”. Using an example, show which numbers are the dividend, divisor and quotient. Consolidate this knowledge, it is necessary for further training!
Essentially, you need to teach your child the multiplication table in reverse, and it is necessary to memorize it just as well as the multiplication table itself, because this will be necessary when you start learning long division.
Divide by column - let's give an example
Before starting the lesson, remember with your child what the numbers are called during the division operation. What is a “divisor”, “divisible”, “quotient”? Teach how to accurately and quickly identify these categories. This will be very useful when teaching your child how to divide prime numbers.
We explain clearly
Let's divide 938 by 7. In this example, 938 is the dividend, 7 is the divisor. The result will be a quotient, and that is what needs to be calculated.
Step 1. We write down the numbers, separating them with a “corner”.
Step 2. Show the student the numbers of the dividend and ask him to choose from them the smallest number that is greater than the divisor. Of the three numbers 9, 3 and 8, this number will be 9. Invite your child to analyze how many times the number 7 can be contained in the number 9? That's right, just once. Therefore, the first result we recorded will be 1.
Step 3. Let's move on to the design of division by column:
We multiply the divisor 7x1 and get 7. We write the resulting result under the first number of our dividend 938 and subtract it, as usual, in a column. That is, from 9 we subtract 7 and get 2.
We write down the result.
Step 4. The number we see is less than the divisor, so we need to increase it. To do this, we combine it with the next unused number of our dividend - it will be 3. We assign 3 to the resulting number 2.
Step 5. Next, we proceed according to the already known algorithm. Let's analyze how many times our divisor 7 is contained in the resulting number 23? That's right, three times. We fix the number 3 in the quotient. And the result of the product - 21 (7 * 3) is written below under the number 23 in a column.
Step.6 Now all that remains is to find the last number of our quotient. Using the already familiar algorithm, we continue to do calculations in the column. By subtracting in column (23-21) we get the difference. It equals 2.
From the dividend we have one number left unused - 8. We combine it with the number 2 obtained as a result of subtraction, we get - 28.
Step.7 Let's analyze how many times our divisor 7 is contained in the resulting number? That's right, 4 times. We write the resulting number into the result. So, we get the quotient obtained by dividing by a column = 134.
How to teach a child division - reinforcing the skill
The main reason why many schoolchildren have problems with mathematics is the inability to quickly do simple arithmetic calculations. And on this basis all mathematics is built. primary school. Especially often the problem is in multiplication and division.
In order for a child to learn how to quickly and efficiently carry out division calculations in his head, the correct teaching methods and consolidation of the skill are necessary. To do this, we advise you to use today’s popular textbooks on learning division skills. Some are designed for children to study with their parents, others for independent work.
- "Division. Level 3. Workbook" from the largest international center for additional education Kumon
- "Division. Level 4. Workbook" from Kumon
- “Not Mental Arithmetic. A system for teaching a child fast multiplication and division. In 21 days. Notepad-simulator." from Sh. Akhmadulin - author of best-selling educational books
The most important thing when you teach a child long division is to master the algorithm, which, in general, is quite simple.
If a child is good at using the multiplication table and “reverse” division, he will not have any difficulties. However, it is very important to constantly practice the acquired skill. Don't stop there once you realize that your child has grasped the essence of the method.
In order to easily teach your child division operations you need:
- So that at the age of two or three years he masters the whole-part relationship. He must develop an understanding of the whole as an inseparable category and the perception of a separate part of the whole as an independent object. For example, a toy truck is a whole, and its body, wheels, doors are parts of this whole.
- So that in the younger school age the child could freely operate with the addition and subtraction of numbers and understood the essence of the processes of multiplication and division.
In order for a child to enjoy mathematics, it is necessary to arouse his interest in mathematics and mathematical operations, not only during learning, but also in everyday situations.
Therefore, encourage and develop your child’s observation skills, draw analogies with mathematical operations (counting and division operations, analysis of “part-whole” relationships, etc.) during construction, games and observations of nature.
Teacher, child development center specialist
Druzhinina Elena
website specifically for the project
Video story for parents on how to correctly explain long division to a child:
How to teach a child division? The simplest method is learn long division. This is much easier than carrying out calculations in your head; it helps you avoid getting confused, not “losing” the numbers, and developing 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 give a simple example how to divide with remainder:
There is a jar for 5 liters of water and 2 jars of 2 liters each. When water is poured from a five-liter jar into two-liter jars, 1 liter of unused water will remain in the five-liter jar. This is the remainder. In digital form it looks like this:
5:2=2 rest (1). Where is 1 from? 2x2=4, 5-4=1.
Now let's look at the order of division into a column with a remainder. This visually simplifies 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. Dividend (17) – located according to left side. To the right of the dividend, write the divisor (5). A vertical line is drawn between them (indicating the division sign), and then, from this line, a horizontal line is drawn, emphasizing the divisor. The main features are indicated in orange.
- Search for the whole. Next, the first and simplest calculation is carried out - how many divisors fit into 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 the fourth five does not fit. Let's go back to three. At 17 liter jar will fit 3 five-liter ones. We write the result in the form: 3 is written under the line, under the divisor. 3 is an incomplete quotient.
- Definition of remainder. 3*5=15. We write 15 under the dividend. We draw a line (indicated by the “=” sign). Subtract the resulting number from the dividend: 17-15=2. We write the result below 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
Difficulty arises when the divisor is larger than the dividend. Decimals they are not yet studied in the 3rd grade curriculum, but, following the logic, the answer should 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 limited by 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 partial quotient is equal to 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 cans will fit into a 5-liter jar? , because 6 is greater than 5.
The assignment requires filling 5 liters - not a single one has been filled. This means that all 5 remain. Answer: partial quotient = 0, remainder = 5.
Division begins to be studied in the third grade of school. By this time, students should already be able to do the division of two-digit numbers by single-digit numbers.
Solve the problem: 18 sweets need to be distributed to five children. How many candies will be left?
Examples:
We find the incomplete quotient: 3*1=3, 3*2=6, 3*3=9, 3*4=12, 3*5=15. 5 – overkill. Let's go back 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 of one.
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 process of division with increasing calculated numbers. If in the third grade calculations were carried out on the basis of a basic multiplication table ranging from 1 to 10, then fourth graders carry out calculations with multi-digit numbers over 100.
It is most convenient to perform this action in a column, since the incomplete quotient will also be a two-digit number (in most cases), and the column algorithm simplifies the calculations and makes them more visual.
Let's divide multi-digit numbers to double digits: 386:25
This example differs from the previous ones in the number of calculation levels, although 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 select the remainder.
First level
The divisor is a two-digit number. The dividend is three-digit. We select the first two left digits of the dividend - this is 38. We compare them with the divisor. Is 38 more than 25? Yes, that means 38 can be divided by 25. How many whole 25 are in 38?
25*1=25, 25*2=50. 50 is more than 38, let's go back one step.
Answer - 1. Write the unit to the zone not completely private.
38-25=13. Write the number 13 below the line.
Second level
Is 13 more than 25? No - that means you can “lower” the number 6 down by adding it next to 13, on the right. It turned out to be 136. Is 136 more than 25? Yes - that means you can subtract it. How many times can 25 fit into 136?
25*1=25, 25*2=50, 25*3=75, 25*4=100, 25*5=125, 256*=150. 150 is more than 136 – we go back one step. We write the number 5 in the incomplete quotient zone, to the right of one.
Calculate the remainder:
136-125=11. Write it below the line. Is 11 more than 25? No - division cannot be carried out. Does the dividend have digits left? No - there is nothing more to share. The calculations are completed.
Answer: the partial quotient is 15, the remainder is 11.
What if such a division is proposed, when the two-digit divisor is greater than the first two digits of the multi-digit dividend? In this case, the third (fourth, fifth and subsequent) digit of the dividend takes part in the calculations immediately.
Let's give examples for 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 is more than 75? No - division cannot be carried out. We take all 3 numbers. Is 386 more than 75? Yes, division can be done. We carry out calculations.
75*1=75, 75*2=150, 75*3=225, 75*4=300, 75*5= 375, 75*6=450. 450 is more than 386 – we go back a step. We write 5 in the incomplete quotient zone.
Single-digit natural numbers are easy to divide in your head. But how to divide multi-digit numbers? If a number already has more than two digits, mental counting can take a lot of time, and the likelihood of errors when operating with multi-digit numbers increases.
Column division is a convenient method often used for dividing multi-digit natural numbers. It is this method that this article is devoted to. Below we will look at how to perform long division. First, let's look at the algorithm for dividing a multi-digit number by a single-digit number into a column, and then multi-digit by multi-digit number. In addition to theory, the article provides practical examples of long division.
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It is most convenient to keep notes on squared paper, since when making calculations, the lines will prevent you from getting confused in the digits. First, the dividend and divisor are written from left to right in one line, and then separated by a special division sign in a column, which looks like:
Let's say we need to divide 6105 by 55, let's write:
We will write intermediate calculations under the dividend, and the result will be written under the divisor. IN general case The column division scheme looks like this:
It should be remembered that for calculations you will need free place On the page. Moreover, the greater the difference in the digits of the dividend and divisor, the more calculations there will be.
For example, dividing the numbers 614,808 and 51,234 will require less space than dividing the number 8,058 by 4. Even though in the second case the numbers are smaller, the difference in the number of digits is larger, and the calculations will be more cumbersome. Let's illustrate this:
It is most convenient to practice practical skills on simple examples. Therefore, let's divide the numbers 8 and 2 into a column. Of course, this operation is easy to perform in your head or using the multiplication table, but detailed analysis It will be useful for clarity, although we already know that 8 ÷ 2 = 4.
So, first we write down the dividend and divisor according to the column division method.
The next step is to find out how many divisors the dividend contains. How to do it? We successively multiply the divisor by 0, 1, 2, 3. . We do this until the result is a number equal to or greater than the dividend. If the result immediately results in a number equal to the dividend, then under the divisor we write the number by which the divisor was multiplied.
Otherwise, when we get a number greater than the dividend, under the divisor we write the number calculated at the penultimate step. In place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.
Let's go back to the example.
2 · 0 = 0 ; 2 · 1 = 2 ; 2 · 2 = 4 ; 2 · 3 = 6 ; 2 4 = 8
So, we immediately got a number equal to the dividend. We write it under the dividend, and write the number 4, by which we multiplied the divisor, in the place of the quotient.
Now all that remains is to subtract the numbers under the divisor (also using the column method). In our case, 8 - 8 = 0.
This example is dividing numbers without a remainder. The number obtained after subtraction is the remainder of the division. If it is equal to zero, then the numbers are divided without a remainder.
Now let's look at an example where numbers are divided with a remainder. Let's divide natural number 7 to the natural number 3.
In this case, sequentially multiplying three by 0, 1, 2, 3. . we get as a result:
3 0 = 0< 7 ; 3 · 1 = 3 < 7 ; 3 · 2 = 6 < 7 ; 3 · 3 = 9 > 7
Under the dividend we write the number obtained in the penultimate step. Using the divisor we write down the number 2 - the incomplete quotient obtained in the penultimate step. It was by two that we multiplied the divisor when we got 6.
To complete the operation, subtract 6 from 7 and get:
This example is dividing numbers with a remainder. The partial quotient is 2 and the remainder is 1.
Now, after considering elementary examples, let's move on to dividing multi-digit natural numbers into single-digit ones.
We will consider the column division algorithm using the example of dividing the multi-digit number 140288 by the number 4. Let’s say right away that it is much easier to understand the essence of the method using practical examples, and this example was not chosen by chance, as it illustrates all the possible nuances of dividing natural numbers in a column.
1. Write the numbers together with the division symbol in a column. Now look at the first digit on the left in the dividend notation. Two cases are possible: the number defined by this digit is greater than the divisor, and vice versa. In the first case, we work with this number, in the second, we additionally take the next digit in the dividend notation and work with the corresponding two-digit number. In accordance with this point, let’s highlight in the example record the number with which we will work initially. This number is 14 because the first digit of the dividend 1 is less than the divisor 4.
2. Determine how many times the numerator is contained in the resulting number. Let's denote this number as x = 14. We successively multiply the divisor 4 by each member of the series of natural numbers ℕ, including zero: 0, 1, 2, 3 and so on. We do this until we get x or a number greater than x as a result. When the result of multiplication is the number 14, we write it under the highlighted number according to the rules for writing subtraction in a column. The factor by which the divisor was multiplied is written under the divisor. If the result of multiplication is a number greater than x, then under the highlighted number we write the number obtained at the penultimate step, and in place of the incomplete quotient (under the divisor) we write the factor by which the multiplication was carried out at the penultimate step.
In accordance with the algorithm we have:
4 0 = 0< 14 ; 4 · 1 = 4 < 14 ; 4 · 2 = 8 < 14 ; 4 · 3 = 12 < 14 ; 4 · 4 = 16 > 14 .
Under the highlighted number we write the number 12 obtained in the penultimate step. In place of the quotient we write the factor 3.
3. Subtract 12 from 14 using a column, and write the result under the horizontal line. By analogy with the first point, we compare the resulting number with the divisor.
4. Number 2 less number 4, therefore we write down under the horizontal line after the two the number located in the next digit of the dividend. If there are no more digits in the dividend, then the division operation ends. In our example, after received in previous paragraph number 2, write down the next digit of the dividend - 0. As a result, we note a new working number - 20.
Important!
Points 2 - 4 are repeated cyclically until the end of the operation of dividing natural numbers by a column.
2. Let's count again how many divisors are contained in the number 20. Multiplying 4 by 0, 1, 2, 3. . we get:
Since we received a number equal to 20 as a result, we write it under the marked number, and in place of the quotient, in the next digit, we write 5 - the factor by which the multiplication was carried out.
3. We carry out the subtraction in a column. Since the numbers are equal, the result is the number zero: 20 - 20 = 0.
4. We will not write down the number zero, since this stage is not the end of division yet. Let’s just remember the place where we could write it down and write next to it the number from the next digit of the dividend. In our case, the number is 2.
We take this number as a working number and again carry out the steps of the algorithm.
2. Multiply the divisor by 0, 1, 2, 3. . and compare the result with the marked number.
4 0 = 0< 2 ; 4 · 1 = 4 > 2
Accordingly, under the marked number we write the number 0, and under the divisor in the next digit of the quotient we also write 0.
3. Perform the subtraction operation and write the result under the line.
4. To the right under the line add the number 8, since this is the next digit of the number being divided.
Thus, we get a new working number - 28. We repeat the points of the algorithm again.
Having done everything according to the rules, we get the result:
We move the last digit of the dividend below the line - 8. IN last time We repeat algorithm points 2 - 4 and get:
In the very bottom line we write the number 0. This number is written only at the last stage of division, when the operation is completed.
Thus, the result of dividing the number 140228 by 4 is the number 35072. This example has been analyzed in great detail, and when solving practical tasks there is no need to describe all the actions so thoroughly.
We will give other examples of dividing numbers into a column and examples of writing solutions.
Example 1. Column division of natural numbers
Divide the natural number 7136 by the natural number 9.
After the second, third and fourth steps of the algorithm, the record will take the form:
Let's repeat the cycle:
The last pass, and we read the result:
Answer: The partial quotient of 7136 and 9 is 792 and the remainder is 8.
When solving practical examples, it is ideal not to use explanations in the form of verbal comments at all.
Example 2. Dividing natural numbers into a column
Divide the number 7042035 by 7.
Answer: 1006005
Division algorithm multi-digit numbers in a column is very similar to the previously discussed algorithm for dividing a multi-digit number by a single-digit number. To be more precise, the changes concern only the first point, while points 2 - 4 remain unchanged.
If, when dividing by a single-digit number, we looked only at the first digit of the dividend, now we will look at as many digits as there are in the divisor. When the number determined by these digits is greater than the divisor, we take it as the working number. Otherwise, we add another digit from the next digit of the dividend. Then we follow the steps of the algorithm described above.
Let's consider the application of the algorithm for dividing multi-digit numbers using an example.
Example 3. Dividing natural numbers into a column
Let's divide 5562 by 206.
The divisor contains three signs, so let’s immediately select the number 556 in the dividend.
556 > 206, so we take this number as a working number and move on to point 2 of the agloritm.
Multiply 206 by 0, 1, 2, 3. . and we get:
206 0 = 0< 556 ; 206 · 1 = 206 < 556 ; 206 · 2 = 412 < 556 ; 206 · 3 = 618 > 556
618 > 556, so under the divisor we write the result of the penultimate action, and under the dividend we write the factor 2
Perform column subtraction
As a result of subtraction we have the number 144. To the right of the result, under the line, we write the number from the corresponding digit of the dividend and get a new working number - 1442.
We repeat points 2 - 4 with him. We get:
206 5 = 1030< 1442 ; 206 · 6 = 1236 < 1442 ; 206 · 7 = 1442
Under the marked working number we write 1442, and in the next quotient digit we write the number 7 - the multiplier.
We carry out the subtraction in a column, and we understand that this is the end of the division operation: there are no more digits in the divisor to write to the right of the subtraction result.
To conclude this topic, we will give another example of dividing multi-digit numbers into a column, without explanation.
Example 5. Column division of natural numbers
Divide the natural number 238079 by 34.
Answer: 7002
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