Video lesson “Division by two-digit and three-digit numbers. The secret of an experienced teacher: how to explain long division to a child

>> Lesson 13. Division by two-digit and three-digit numbers

Divide 876 by 24. Calculating 800: 20 = 40 shows that the answer should be a number close to 40.

As with division by a single-digit number, we will sequentially move from dividing larger counting units to dividing smaller units.

The number of hundreds 8 is single-digit, so we divide 87 tens by 24. You get 3 tens and another 15 tens remain (87 - 3 24 = 15). 15 tens and 6 units is 156. And if 156 is divided by 24, you get 6 and 12 as a remainder (156 - 24 6 = 12). In total you get 3 tens and 6 units, that is, 36, and the remainder is 12. This is written like this:

10*. Find the sum of all possible two-digit numbers all of whose digits are odd.

Peterson Lyudmila Georgievna. Mathematics. 4th grade. Part 1. - M.: Yuventa Publishing House, 2005, - 64 p.: ill.

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Column? How can you independently practice the skill of long division at home if your child did not learn something at school? Dividing by columns is taught in grades 2-3; for parents, of course, this is a passed stage, but if you wish, you can remember the correct notation and explain in an understandable way to your student what he will need in life.

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What should a 2nd-3rd grade child know to learn to do long division?

How to correctly explain division to a 2-3 grade child so that he doesn’t have problems in the future? First, let's check if there are any gaps in knowledge. Make sure that:

• the child can freely perform addition and subtraction operations;
• knows the digits of numbers;
• knows by heart.

How to explain to a child the meaning of the action “division”?

• Everything needs to be explained to the child using a clear example.

Ask to share something among family members or friends. For example, candy, pieces of cake, etc. It is important that the child understands the essence - you need to divide equally, i.e. without a trace. Practice with different examples.

Let's say 2 groups of athletes must take seats on the bus. We know how many athletes are in each group and how many seats there are on the bus. You need to find out how many tickets one and the other group need to buy. Or 24 notebooks should be distributed to 12 students, as many as each gets.

• When the child understands the essence of the principle of division, show the mathematical notation of this operation and name the components.
• Explain that Division is the opposite operation of multiplication, multiplication inside out.

It is convenient to show the relationship between division and multiplication using a table as an example.

For example, 3 times 4 equals 12.
3 is the first multiplier;
4 - second factor;
12 is the product (the result of multiplication).

If 12 (the product) is divided by 3 (the first factor), we get 4 (the second factor).

Components when divided are called differently:

12 - dividend;
3 - divider;
4 - quotient (result of division).

How to explain to a child the division of a two-digit number by a single-digit number not in a column?

For us adults, it’s easier to write “in the corner” the old fashioned way – and that’s the end of it. BUT! Children have not yet completed long division, what should they do? How to teach a child to divide a two-digit number by a single-digit number without using column notation?

Let's take 72:3 as an example.

It's simple! We break down 72 into numbers that can easily be divided verbally by 3:
72=30+30+12.

Everything immediately became clear: we can divide 30 by 3, and a child can easily divide 12 by 3.
All that remains is to add up the results, i.e. 72:3=10 (obtained when 30 was 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 completed the calculations without difficulty.

After simple examples, you can move on to studying long division and teach your child to correctly write examples in a “corner”. To begin with, use only examples of division without a remainder.

How to explain long division to a child: solution algorithm

Large numbers are difficult to divide in your head; it is easier to use column division notation. To teach your child to perform calculations correctly, follow the algorithm:

• Determine where the dividend and divisor are in the example. Ask your child to name the numbers (what we will divide by what).

213:3
213 - dividend
3 - divider

• Write down the dividend - "corner" - divisor.

• Determine which part of the dividend we can use to divide by a given number.

We reason 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”.

7 multiplied by 3 - we get 21. Write it down.

• Find the difference (remainder).

At this stage of reasoning, teach your child to check himself. It is important that he understands that the result of a subtraction must ALWAYS be less than the divisor. If it doesn't work out, 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 2-3 grade child to divide by column

How to explain division to a child 204:12=?
1. Write it down 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, take 1. Write 1 under the “corner”.
4. 1 multiplied by 12 gets 12. We write it under 20.
5. 20 minus 12 gets 8.
Let's 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. How much should we multiply 12 to get 84?
It’s hard to say right away, we’ll try to use the selection method.
Let's take 8, for example, but don't write them down yet. We count verbally: 8 multiplied by 12 equals 96. And we have 84! Doesn't fit.
Let's try smaller ones... For example, let's take 6. We check ourselves verbally: 6 multiplied by 12 equals 72. 84-72=12. We got the same number as our divisor, but it should be either zero or less than 12. So the optimal number is 7!

7. We write 7 under the “corner” and perform the calculations. 7 multiplied by 12 gives 84.
8. We write the result in a column: 84 minus 84 equals zero. Hooray! We decided correctly!

So, you have taught your child to divide by column, now all that remains is to practice this skill and bring it to automatism.

Why is it difficult for children to learn long division?

Remember that problems with mathematics arise from the inability to quickly do simple arithmetic operations. In elementary school, you need to practice addition and subtraction and make it automatic, and learn the multiplication table from cover to cover. All! The rest is a matter of technique, and it is developed with practice.

Be patient, do not be lazy, once again explain to the child what he did not learn in the lesson, tediously but meticulously understand the reasoning algorithm and talk through each intermediate operation before voicing a ready answer. Give additional examples to practice skills, play math games - this will bear fruit and you will see the results and rejoice at your child’s success very soon. Be sure to show where and how you can apply the acquired knowledge in everyday life.

Dear readers! Tell us how you teach your children to do long division, what difficulties you have encountered and how you have overcome them.

Unfortunately, children nowadays practically do not know how to do mental calculations. This happened due to the fact that modern technologies offer each child to solve the problem with a couple of clicks. For many children, the Internet has replaced not only textbooks, but also certain skills. You can increasingly hear from the younger generation that it is not at all necessary to know mathematics, since you always have a calculator or phone at hand. But the true significance of this science lies in the development of thinking, and not in overcoming the fear of being deceived by a trader in the market.

Long division helps elementary school students become familiar with number operations. Thanks to it, the multiplication table is fixed in memory, and the skill of performing addition and subtraction operations is honed.

To implement this arithmetic operation, you need to become familiar with its components:

1. Dividend - a number that is divided.

2. Divisor - the number that is divided by.

3. Quotient - the result obtained by division.

4. Remainder is the part of the dividend that cannot be divided.

American and European division models

The rules for long division are the same in all countries. There is only a difference in the graphic part, that is, in its recording. In the European system, the dividing line, or the so-called corner, is placed on the right side of the number being divided. The divisor is written above the corner line, and the quotient is written below the horizontal line of the corner.

Dividing into a column according to the American model involves placing a corner on the left side. The quotient is written above the horizontal line of the angle, directly above the number being divided. The divisor is written under the horizontal line, to the left of the vertical line. The process of performing the action itself does not differ from the European model.

Divide by a two-digit number

To use a two-digit value, you need to write it down according to the diagram, and then carry out the action. Column division begins with the highest digits of the number being divided. The first two digits are taken if the number formed by them is greater in value than the divisor. Otherwise, the first three digits are separated. The number they form is divided by the divisor, the remainder goes down, and the result is written in the dividing corner. After this, the digit from the next digit of the number being divided is transferred, and the procedure is repeated. This continues until the number is completely divided.

If it is necessary to divide a number with a remainder, it is written separately. If you need to completely divide a number, then after the end of the digits of the number a comma is placed in the answer, indicating the beginning of the fractional part, and instead of the digits, a zero is moved down each time.

You will need:

Basics of mathematics

First, make sure that your child has mastered simpler operations: addition, subtraction, multiplication. Without these basics, it will be difficult for him to understand division.

If you see any gaps in knowledge, then repeat the previous material.

Division principle

Before you begin to explain the division algorithm, your child should develop an understanding of the process itself.

Explain to your little student that “division” is the division of a whole into equal parts.

Take a box of pencils that will act as one whole (you can take any objects - cubes, matches, apples, etc.), and invite your child to divide them equally between you and him. Then, ask him to count how many pencils were originally in the box and how many he gave to each person.

As the child understands, increase the number of objects and the number of participants. Further, it should be noted that it is not always possible to divide equally and some items remain “drawn”. For example, offer to divide 9 pears between grandparents, dad and mom. The child must learn that everyone will receive 2 pears, and one will be left over.

Relationship with the multiplication table

Show your child that division is the opposite of multiplication.

• Take the multiplication table and show the student the relationship between the two operations.
• For example, 4x5=20. Remind your child that the number 20 is the product of two numbers, 4 and 5.
• Then, clearly show that division is the opposite process: 20/5=4, 20/4=5.

Point out to the child that the correct answer will always be a factor not involved in division.

• Consider other examples.

If your child knows the multiplication table well and understands the relationship between two mathematical operations, he will easily master division. Whether to memorize it in reverse order is your choice.

Definition of concepts

Before starting classes, identify and learn the names of the elements that participate in the division process.

"Dividend"– the number to be divided.

"Divider" - This is the number by which the “dividend” is divided.

"Private"– this is the result that we obtain during the calculation process.

For clarity, you can give an example:

For your son/daughter's birthday, you bought 96 candies so that the child could treat his friends. Total number of invitees – 8.

Explain that a bag of 96 candies is a “divisible.” Eight children are a “divider”. And the number of sweets that each child will receive is “private”.

Column division algorithm without remainder

Now show your child the calculation algorithm using an example about candy.

• Take a blank piece of paper/notebook and write the numbers 96 and 8.
• Divide them with perpendicular lines.

• Show the elements clearly.
• Point out that the result of a calculation is written under the “divisor”, and the calculation is written under the “dividend”.
• Invite your little student to look at the number 96 and determine the number that is greater than 8.
• Of the two numbers 9 and 6, this number will be 9.
• Ask your child how many digits 8 can “fit” in 9. The child, remembering the multiplication table, can easily determine that only once. Therefore, write down the number 1 under the underscore.
• Next, multiply the divisor 8 by the result 1. Write the resulting number 8 under the first digit of the number being divided.
• Put a “subtraction” sign between them and summarize. That is, if you subtract 8 from 9, you get 1. Write down the result.

At this stage, explain to your child that the result of a subtraction must always be less than the divisor. If it turns out the other way around, it means that the baby incorrectly determined how many 8s are in 9.

• Ask your child again to identify the digit that is greater than the divisor 8. As you can see, the number 1 is less than 8. Therefore, we should combine it with the next digit of the divisible number - 6.
• Add 6 to one and get 16.
• Next, ask your child how many 8 are contained in 16. The correct answer is 2, add to the first.

• Multiply 8 by 2 again. Write the resulting result under the number 16.
• By “subtracting” (16-16) we get 0, which means that our calculation result is 12.

Schoolchildren learn column division, or, more correctly, the written method of dividing by a corner, already in the third grade of elementary school, but often so little attention is paid to this topic that by the 9th-11th grade not all students can use it fluently. Division by a column by a two-digit number is taught in the 4th grade, as is division by a three-digit number, and then this technique is used only as an auxiliary technique when solving any equations or finding the value of an expression.

Obviously, by paying more attention to long division than is included in the school curriculum, the child will make it easier for him to complete math assignments up to the 11th grade. And for this you need little - to understand the topic and study, solve, keeping the algorithm in your head, to bring the calculation skill to automatism.

Algorithm for dividing by a two-digit number

As with division by a single-digit number, we will sequentially move from dividing larger counting units to dividing smaller units.

1. Find the first incomplete dividend. This is a number that is divided by a divisor to produce a number greater than or equal to 1. This means that the first partial dividend is always greater than the divisor. When dividing by a two-digit number, the first partial dividend must have at least 2 digits.

Examples 76 8:24. First incomplete dividend 76
265 :53 26 is less than 53, which means it is not suitable. You need to add the next number (5). The first incomplete dividend is 265.

2. Determine the number of digits in the quotient. To determine the number of digits in a quotient, you should remember that the incomplete dividend corresponds to one digit of the quotient, and all other digits of the dividend correspond to one more digit of the quotient.

Examples 768:24. The first incomplete dividend is 76. It corresponds to 1 digit of the quotient. After the first partial divisor there is one more digit. This means that the quotient will only have 2 digits.
265:53. The first incomplete dividend is 265. It will give 1 digit of the quotient. There are no more digits in the dividend. This means that the quotient will only have 1 digit.
15344:56. The first partial dividend is 153, and after it there are 2 more digits. This means that the quotient will only have 3 digits.

3. Find the numbers in each digit of the quotient. First, let's find the first digit of the quotient. We select an integer such that when multiplied by our divisor we get a number that is as close as possible to the first incomplete dividend. We write the quotient number under the corner, and subtract the value of the product in a column from the partial divisor. We write down the remainder. We check that it is less than the divisor.

Then we find the second digit of the quotient. We rewrite the number following the first partial divisor in the dividend into the line with the remainder. The resulting incomplete dividend is again divided by the divisor and so we find each subsequent number of the quotient until the digits of the divisor run out.

4. Find the remainder(if there is).

If the digits of the quotient run out and the remainder is 0, then the division is performed without a remainder. Otherwise, the quotient value is written with a remainder.

Division by any multi-digit number (three-digit, four-digit, etc.) is also performed.

Analysis of examples of dividing by a column by a two-digit number

First, let's look at simple cases of division, when the quotient results in a single-digit number.

Let's find the value of the quotient numbers 265 and 53.

The first incomplete dividend is 265. There are no more digits in the dividend. This means that the quotient will be a single digit number.

To make it easier to choose the quotient number, let's divide 265 not by 53, but by a close round number 50. To do this, divide 265 by 10, the result will be 26 (the remainder is 5). And divide 26 by 5, there will be 5 (remainder 1). The number 5 cannot be immediately written down in the quotient, since it is a trial number. First you need to check if it fits. Let's multiply 53*5=265. We see that the number 5 has come up. And now we can write it down in a private corner. 265-265=0. The division is completed without remainder.

The quotient of 265 and 53 is 5.

Sometimes when dividing, the test digit of the quotient does not fit, and then it needs to be changed.

Let's find the value of the quotient numbers 184 and 23.

The quotient will be a single digit number.

To make it easier to choose the quotient number, let's divide 184 not by 23, but by 20. To do this, divide 184 by 10, the result will be 18 (remainder 4). And we divide 18 by 2, the result is 9. 9 is a test number, we won’t immediately write it in the quotient, but we’ll check if it’s suitable. Let's multiply 23*9=207. 207 is greater than 184. We see that the number 9 is not suitable. The quotient will be less than 9. Let's try to see if the number 8 is suitable. Let's multiply 23*8=184. We see that the number 8 is suitable. We can write it down privately. 184-184=0. The division is completed without remainder.

The quotient of 184 and 23 is 8.

Let's consider more complex cases of division.

Let's find the value of the quotient of 768 and 24.

The first incomplete dividend is 76 tens. This means that the quotient will have 2 digits.

Let's determine the first digit of the quotient. Let's divide 76 by 24. To make it easier to choose the quotient number, let's divide 76 not by 24, but by 20. That is, you need to divide 76 by 10, there will be 7 (the remainder is 6). And divide 7 by 2, you get 3 (remainder 1). 3 is the test digit of the quotient. First let's check if it fits. Let's multiply 24*3=72. 76-72=4. The remainder is less than the divisor. This means that the number 3 is suitable and now we can write it in place of the tens of the quotient. We write 72 under the first incomplete dividend, put a minus sign between them, and write the remainder under the line.

Let's continue the division. Let's rewrite the number 8 following the first incomplete dividend into the line with the remainder. We get the following incomplete dividend – 48 units. Let's divide 48 by 24. To make it easier to find the quotient, let's divide 48 not by 24, but by 20. That is, if we divide 48 by 10, there will be 4 (the remainder is 8). And we divide 4 by 2, it becomes 2. This is the test digit of the quotient. We must first check if it will fit. Let's multiply 24*2=48. We see that the number 2 fits and, therefore, we can write it in place of the units of the quotient. 48-48=0, division is performed without remainder.

The quotient of 768 and 24 is 32.

Let's find the value of the quotient numbers 15344 and 56.

The first incomplete dividend is 153 hundreds, which means that the quotient will have three digits.

Let's determine the first digit of the quotient. Let's divide 153 by 56. To make it easier to find the quotient, let's divide 153 not by 56, but by 50. To do this, divide 153 by 10, the result will be 15 (remainder 3). And we divide 15 by 5, it becomes 3. 3 is the test digit of the quotient. Remember: you cannot immediately write it down in private, but you must first check whether it is suitable. Let's multiply 56*3=168. 168 is greater than 153. This means that the quotient will be less than 3. Let’s check if the number 2 is suitable. Multiply 56*2=112. 153-112=41. The remainder is less than the divisor, which means that the number 2 is suitable, it can be written in the place of hundreds in the quotient.

Let us form the following incomplete dividend. 153-112=41. We rewrite the number 4 following the first incomplete dividend into the same line. We get the second incomplete dividend of 414 tens. Let's divide 414 by 56. To make it more convenient to choose the quotient number, let's divide 414 not by 56, but by 50. 414:10=41(rest.4). 41:5=8(rest.1). Remember: 8 is a test number. Let's check it out. 56*8=448. 448 is greater than 414, which means that the quotient will be less than 8. Let's check if the number 7 is suitable. Multiply 56 by 7, we get 392. 414-392=22. The remainder is less than the divisor. This means that the number fits and in the quotient we can write 7 in place of tens.

We write 4 units in the line with the new remainder. This means the next incomplete dividend is 224 units. Let's continue the division. Let's divide 224 by 56. To make it easier to find the quotient number, divide 224 by 50. That is, first by 10, there will be 22 (the remainder is 4). And divide 22 by 5, there will be 4 (remainder 2). 4 is a test number, let's check it to see if it fits. 56*4=224. And we see that the number has come up. Let's write 4 in place of units in the quotient. 224-224=0, division is performed without remainder.

The quotient of 15344 and 56 is 274.

Example for division with remainder

To make an analogy, let's take an example similar to the example above, differing only in the last digit

Let's find the value of the quotient 15345:56

We first divide in the same way as in the example 15344:56, until we reach the last incomplete dividend 225. Divide 225 by 56. To make it easier to choose the quotient number, divide 225 by 50. That is, first by 10, there will be 22 (the remainder is 5 ). And divide 22 by 5, there will be 4 (remainder 2). 4 is a test number, let's check it to see if it fits. 56*4=224. And we see that the number has come up. Let's write 4 in place of units in the quotient. 225-224=1, division done with remainder.

The quotient of 15345 and 56 is 274 (remainder 1).

Division with zero in quotient

Sometimes in a quotient one of the numbers turns out to be 0, and children often miss it, hence the wrong solution. Let's look at where 0 can come from and how not to forget it.

Let's find the value of the quotient 2870:14

The first incomplete dividend is 28 hundreds. This means that the quotient will have 3 digits. Place three dots under the corner. This is an important point. If a child loses a zero, there will be an extra dot left, which will make them think that a number is missing somewhere.

Let's determine the first digit of the quotient. Let's divide 28 by 14. By selection we get 2. Let's check if the number 2 fits. Multiply 14*2=28. The number 2 is suitable; it can be written in place of hundreds in the quotient. 28-28=0.

The result was a zero remainder. We've marked it in pink for clarity, but you don't need to write it down. We rewrite the number 7 from the dividend into the line with the remainder. But 7 is not divisible by 14 to obtain an integer, so we write 0 in the place of tens in the quotient.

Now we rewrite the last digit of the dividend (number of units) into the same line.

70:14=5 We write the number 5 instead of the last point in the quotient. 70-70=0. There is no remainder.

The quotient of 2870 and 14 is 205.

Division must be checked by multiplication.

Division examples for self-test

Find the first incomplete dividend and determine the number of digits in the quotient.

3432:66 2450:98 15145:65 18354:42 17323:17

You have mastered the topic, now practice solving several examples in a column yourself.

1428: 42 30296: 56 254415: 35 16514: 718