At school these actions are studied from simple to complex. Therefore, it is imperative to thoroughly understand the algorithm for performing these operations on simple examples. So that later there will be no difficulties with dividing decimal fractions into a column. After all, this is the most difficult version of such tasks.
This subject requires consistent study. Gaps in knowledge are unacceptable here. Every student should learn this principle already in the first grade. Therefore, if you miss several lessons in a row, you will have to master the material on your own. Otherwise, later problems will arise not only with mathematics, but also with other subjects related to it.
Second required condition Successful learning of mathematics - move on to examples of long division only after you have mastered addition, subtraction and multiplication.
It will be difficult for a child to divide if he has not learned the multiplication table. By the way, it is better to teach it using the Pythagorean table. There is nothing superfluous, and multiplication is easier to learn in this case.
If difficulty arises in solving examples in a column for division and multiplication, then you should begin to solve the problem with multiplication. Since division is the inverse operation of multiplication:
Continue this multiplication in a column until the numbers in the second factor run out. Now they need to be folded. This will be the answer you are looking for.
First, you need to imagine that the given fractions are not decimals, but natural ones. That is, remove the commas from them and then proceed as described in the previous case.
The difference begins when the answer is written down. At this moment, it is necessary to count all the numbers that appear after the decimal points in both fractions. This is exactly how many of them need to be counted from the end of the answer and put a comma there.
It is convenient to illustrate this algorithm using an example: 0.25 x 0.33:
Before solving long division examples, you need to remember the names of the numbers that appear in the long division example. The first of them (the one that is divided) is divisible. The second (divided by) is the divisor. The answer is private.
After this, using a simple everyday example, we will explain the essence of this mathematical operation. For example, if you take 10 sweets, then it’s easy to divide them equally between mom and dad. But what if you need to give them to your parents and brother?
After this, you can get acquainted with the rules of division and master them in specific examples. First simple ones, and then move on to more and more complex ones.
First, let us present the procedure for natural numbers divisible by a single-digit number. They will also be the basis for multi-digit divisors or decimal fractions. Only then should you make small changes, but more on that later:
The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Now there should be at least two of them, but if they turn out to be less than the divisor, then you have to work with the first three digits.
There is one more nuance in this division. The fact is that the remainder and the number added to it are sometimes not divisible by the divisor. Then you have to add another number in order. But the answer must be zero. If you are dividing three-digit numbers into a column, you may need to remove more than two digits. Then a rule is introduced: there should be one less zero in the answer than the number of digits removed.
You can consider this division using the example - 12082: 863.
The answer in the example would be the number 14.
Or a few zeros? In this case, the remainder is zero, but the dividend still contains zeros. There is no need to despair, everything is simpler than it might seem. It is enough to simply add to the answer all the zeros that remain undivided.
For example, you need to divide 400 by 5. The incomplete dividend is 40. Five fits into it 8 times. This means that the answer should be written as 8. When subtracting, there is no remainder left. That is, the division is completed, but a zero remains in the dividend. It will have to be added to the answer. Thus, dividing 400 by 5 equals 80.
Again, this number looks like a natural number, if not for the comma separating the whole part from the fractional part. This suggests that the division of decimal fractions into a column is similar to that described above.
The only difference will be the semicolon. It is supposed to be put in the answer as soon as the first digit from the fractional part is removed. Another way to say this is this: if you have finished dividing the whole part, put a comma and continue the solution further.
When solving examples of long division with decimal fractions, you need to remember that any number of zeros can be added to the part after the decimal point. Sometimes this is necessary in order to complete the numbers.
It may seem complicated. But only at the beginning. After all, how to divide a column of fractions by a natural number is already clear. This means that we need to reduce this example to an already familiar form.
It's easy to do. You need to multiply both fractions by 10, 100, 1,000 or 10,000, and maybe by a million if the problem requires it. The multiplier is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, the result will be that you will have to divide the fraction by a natural number.
And this will be the worst case scenario. After all, it may happen that the dividend from this operation becomes an integer. Then the solution to the example with division into a column of fractions will be reduced to the very simple option: operations with natural numbers.
As an example: divide 28.4 by 3.2:
The division is complete. The result of example 28.4:3.2 is 8.875.
Just like with multiplication, long division is not needed here. It is enough to simply move the comma in the desired direction for a certain number of digits. Moreover, using this principle, you can solve examples with both integers and decimal fractions.
So, if you need to divide by 10, 100 or 1,000, then the decimal point is moved to the left by the same number of digits as there are zeros in the divisor. That is, when a number is divisible by 100, the decimal point must move to the left by two digits. If the dividend is a natural number, then it is assumed that the comma is at the end.
This action gives the same result as if the number were to be multiplied by 0.1, 0.01 or 0.001. In these examples, the comma is also moved to the left by the number of digits, equal to length fractional part.
When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the decimal point should move to the right by one digit (or two, three, depending on the number of zeros or the length of the fractional part).
It is worth noting that the number of digits given in the dividend may not be sufficient. Then the missing zeros can be added to the left (in the whole part) or to the right (after the decimal point).
In this case, it will not be possible to obtain an accurate answer when dividing into a column. How to solve an example if you encounter a fraction with a period? Here we need to move on to ordinary fractions. And then divide them according to the previously learned rules.
For example, you need to divide 0.(3) by 0.6. The first fraction is periodic. It converts to the fraction 3/9, which when reduced gives 1/3. The second fraction is the final decimal. It’s even easier to write it down as usual: 6/10, which is equal to 3/5. The rule for dividing ordinary fractions requires replacing division with multiplication and divisor with the reciprocal. That is, the example comes down to multiplying 1/3 by 5/3. The answer will be 5/9.
Then several solutions are possible. Firstly, common fraction You can try to convert it to decimal. Then divide two decimals using the above algorithm.
Secondly, every finite decimal can be written in ordinary form. But this is not always convenient. Most often, such fractions turn out to be huge. And the answers are cumbersome. Therefore, the first approach is considered more preferable.
If you want to learn how to multiply and divide round three-digit numbers in your head, then you are in luck, because in this lesson you will be able to do it. If you do not know, or know but poorly, how to multiply and divide round three-digit numbers, then this lesson is designed specifically for you. How great it is to be able to quickly count, do multiplication and division calculations! While everyone is thinking, you will already know the answer.
And appreciation and honor -
For everyone who loves mental arithmetic!
Sharpen your skills
In multiplication and division!
Choose the method you need -
Count quickly and have fun!
Multiplying and dividing a round three-digit number by a single-digit number can easily be replaced by hundreds and tens.
Solution: 1. Replace the number 180 with tens:
2. In the second example, we replace the number 900 with hundreds:
Let's get acquainted with another method of mental calculations and solve examples. Let's remember the rule for multiplying a sum by a number.
When multiplying a sum by a number, each term must be multiplied by that number, and the resulting products added.
Let's remember the rule for dividing a sum by a number.
When dividing a sum by a number, you must divide each term by that number and add the resulting quotients.
Solution: 1. We break down the number 240 into its components and carry out the calculations:
2. Replace the first factor in the second example with the sum of the bit terms and find the product:
3. Let's do the same technique, only to find the quotient:
4. Let's repeat the operation in the last example, only here we replace the dividend not with bit terms, but with convenient terms:
You can use another method for multiplying and dividing three-digit numbers by a single-digit number.
Solution: 1. If we multiply the divisor by three, we get the dividend ninety.
2. Let's take two hundred and four times and get eight hundred - the dividend, therefore, the selection was made correctly.
.
If you cannot find the correct answer the first time, you must continue to select numbers until the results match completely.
Solve the examples in Figure 1.
Rice. 1. Examples
Solution: 1. In the first and second examples, replace the first numbers with hundreds:
2. In the third and fourth examples, we will use the technique of decomposition into bit terms:
3. In the last pair of examples, we use the selection method to solve:
, examination
« Oral techniques for multiplying and dividing three-digit numbers."
Goals:
1. Teach how to multiply and divide multi-digit numbers;
2. Repeat the commutative property of multiplication and the property of multiplying a sum by a number;
3. Repeat units of measurement.
4. Consolidate knowledge of the multiplication tables.
5. Build computational skills and develop logical thinking.
6. Develop students’ cognitive activity when studying mathematics.
Tasks: develop the ability to search for information and work with it;
develop the ability to substantiate and defend the expressed judgment;
develop motivation for learning activities and interest in acquiring knowledge and methods of action;
cultivate interest in the subject and activity.
Org. moment
Children, today is a wonderful day. Look, I smile at you and you will smile at me. Turn to each other and smile. Well done, sit down at your desks. You can feel how warm and bright our class has become from the smiles.
Rook offers you a game called “Tangram”. Take envelopes with geometric shapes and make a silhouette drawing of a rook from them. (work in pairs).
- Look what a rook I made. Compare.
— Tell me, what figures did you use?
— How many triangles?
- What other ones? geometric figures You know?
Rook asks you to remember what you learned in previous lessons, so how will this knowledge be useful to us today?
1. Read the numbers: 540, 700, 210, 900, 650, 380,400, 820
— Indicate the number of hundreds and tens in each of them.
2. Name the number in which: 87dec., 5hundred, 64dec., 3hundred, 25dec., 49dec.,
7 hundred, 11 des.
3. Increase the numbers by 10 times: 42, 27, 91, 65, 73, 58.
2. Blitz survey
1.Volodya stayed with his grandmother for two weeks and another 4 days. How many days did Volodya stay with his grandmother? (18 days)
2. Vitya swam 26 meters. He swam 4 meters less than Seryozha. How many meters did Seryozha swim? (30 meters)
3. There are 38 old apple trees and 19 young ones in the garden. How many fewer young apple trees are there than old ones? (for 19 apple trees)
- Well done! Well done. Let `s have some rest.
3. Physical exercise
4. Introduction to the topic.
What groups can the following expressions be divided into:
15 ∙ 4 200 ∙ 4
320 ∙ 2 25 ∙ 3
Write them down in 2 columns and find the value.
— What groups did you divide these expressions into?
— Which tasks are more difficult for you to cope with? (Why do you think?)
- What was the difficulty?
(In that one column contains three-digit numbers)
— Try to set a learning task for today’s lesson yourself.
(Learn to multiply and divide three-digit numbers orally)
5. Report the topic of the lesson. Setting educational objectives.
The topic of today's lesson: “Techniques for mental calculations within 1000”
— What do we need to do to make it easier to solve such examples? ( Listen to the teacher’s explanation, read the information in the textbook, listen to classmates, remember the multiplication and division tables, practice solving such examples, etc.)
6. Getting to know new material.
Let's try to solve the expression: 120*4. To orally multiply a number by a single-digit factor, perform the action, starting the multiplication not with units, as in written multiplication, but differently: first multiply hundreds, 100 * 4 = 400, then tens 20 * 4 = 80, after one, but we will study this later As a result, we add the resulting numbers 400+80=480
Let's try to solve the division expression: 820:2. To verbally divide a number into a single-digit factor, perform the same action as in the multiplication method. First we divide the hundreds 800:2=400, then the tens 20:2=10, then we add the results 400+10=410 Let's try to do it together:
230 * 4 = 200 * 4 + 30 * 4=920; 360: 4 =300:4(75)+60:4(15)=90
150 * 4 =100*4+50*4=600; 680: 4 =600:4(150)+80:4(20)=170
TASK. One rook, following a tractor plow, is capable of destroying 420 plant pests in a day. How many worms will a rook eat in 2 days?
— What does the problem statement say?
- What question needs to be answered?
— How many actions do you need to perform to do this?
— How can you find out how many worms a rook will eat in two days?
— Write down the solution to the problem in your notebook.
- What answer did you get?
- Who agrees with... show me.
- How did you think?
— Guys, you coped very well with the tasks that the birds offered you.
Lesson summary. Reflection.
— Guys, have we completed our tasks?
Division is one of the four basic mathematical operations (addition, subtraction, multiplication). Division, like other operations, is important not only in mathematics, but also in Everyday life. For example, you as a whole class (25 people) donate money and buy a gift for the teacher, but you don’t spend it all, there will be change left over. So you will need to divide the change among everyone. The division operation comes into play to help you solve this problem.
Division is an interesting operation, as we will see in this article!
So, a little theory, and then practice! What is division? Division is breaking something into equal parts. That is, it could be a bag of sweets that needs to be divided into equal parts. For example, there are 9 candies in a bag, and the person who wants to receive them is three. Then you need to divide these 9 candies among three people.
It is written like this: 9:3, the answer will be the number 3. That is, dividing the number 9 by the number 3 shows the number of three numbers contained in the number 9. The reverse action, a check, will be multiplication. 3*3=9. Right? Absolutely.
So let's look at example 12:6. First, let's name each component of the example. 12 – dividend, that is. a number that can be divided into parts. 6 is a divisor, this is the number of parts into which the dividend is divided. And the result will be a number called “quotient”.
Let's divide 12 by 6, the answer will be the number 2. You can check the solution by multiplying: 2*6=12. It turns out that the number 6 is contained 2 times in the number 12.
What is division with a remainder? This is the same division, only the result is not an even number, as shown above.
For example, let's divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, then the answer will be 3 and the remainder is 2, and is written like this: 17:5 = 3(2).
For example, 22:7. In the same way, we determine the maximum number divisible by 7 to 22. This number is 21. The answer then will be: 3 and the remainder 1. And it is written: 22:7 = 3 (1).
A special case of division would be division by the number 3 and the number 9. If you want to find out whether a number is divisible by 3 or 9 without a remainder, then you will need:
Find the sum of the digits of the dividend.
Divide by 3 or 9 (depending on what you need).
If the answer is obtained without a remainder, then the number will be divided without a remainder.
For example, the number 18. The sum of the digits is 1+8 = 9. The sum of the digits is divisible by both 3 and 9. The number 18:9=2, 18:3=6. Divided without remainder.
For example, the number 63. The sum of the digits is 6+3 = 9. Divisible by both 9 and 3. 63:9 = 7, and 63:3 = 21. Such operations are carried out with any number to find out whether it is divisible with the remainder by 3 or 9, or not.
Multiplication and division are opposite operations. Multiplication can be used as a test for division, and division can be used as a test for multiplication. You can learn more about multiplication and master the operation in our article about multiplication. Which describes multiplication in detail and how to do it correctly. There you will also find the multiplication table and examples for training.
Here is an example of checking division and multiplication. Let's say the example is 6*4. Answer: 24. Then let's check the answer by division: 24:4=6, 24:6=4. It was decided correctly. In this case, the check is performed by dividing the answer by one of the factors.
Or an example is given for the division 56:8. Answer: 7. Then the test will be 8*7=56. Right? Yes. IN in this case verification is done by multiplying the answer by the divisor.
In third grade they are just starting to go through division. Therefore, third graders solve the simplest problems:
Problem 1. A factory worker was given the task of putting 56 cakes into 8 packages. How many cakes should be put in each package to make the same amount in each?
Problem 2. On New Year's Eve at school, children in a class of 15 students were given 75 candies. How many candies should each child receive?
Problem 3. Roma, Sasha and Misha picked 27 apples from the apple tree. How many apples will each person get if they need to be divided equally?
Problem 4. Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many additional cookies do the kids need to buy so that each gets 15?
The division in the fourth grade is more serious than in the third. All calculations are carried out using the column division method, and the numbers involved in the division are not small. What is long division? You can find the answer below:
What is long division? This is a method that allows you to find the answer to division. large numbers. If prime numbers like 16 and 4, can be divided, and the answer is clear - 4. That 512:8 in the mind is not easy for a child. And it’s our task to talk about the technique for solving such examples.
Let's look at an example, 512:8.
1 step. Let's write the dividend and divisor as follows:
The quotient will ultimately be written under the divisor, and the calculations under the dividend.
Step 2. We start dividing from left to right. First we take the number 5: 
Step 3. The number 5 is less than the number 8, which means it will not be possible to divide. Therefore, we take another digit of the dividend:
Now 51 is greater than 8. This is an incomplete quotient.
Step 4. We put a dot under the divisor.
Step 5. After 51 there is another number 2, which means there will be one more number in the answer, that is. quotient is a two-digit number. Let's put the second point:
Step 6. We begin the division operation. The largest number divisible by 8 without a remainder to 51 is 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:
Step 7. Then write the number exactly below the number 51 and put a “-” sign:
Step 8. Then we subtract 48 from 51 and get the answer 3.
* 9 step*. We take down the number 2 and write it next to the number 3:
Step 10 We divide the resulting number 32 by 8 and get the second digit of the answer – 4.
So the answer is 64, without remainder. If we divided the number 513, then the remainder would be one.
Dividing three-digit numbers is done using the long division method, which was explained in the example above. An example of just a three-digit number.
Dividing fractions is not as difficult as it seems at first glance. For example, (2/3):(1/4). The method of this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but to do this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3)*4, this is equal to 8/3 or 2 integers and 2/3. Let's give another example, with an illustration for better understanding. Consider the fractions (4/7):(2/5):
As in the previous example, we reverse the 2/5 divisor and get 5/2, replacing division with multiplication. We then get (4/7)*(5/2). We make a reduction and answer: 10/7, then take out the whole part: 1 whole and 3/7.
Let's imagine the number 148951784296, and divide it into three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is units, 9 is tens, 2 is hundreds.
Division of natural numbers is the simplest division described in this article. It can be either with or without a remainder. The divisor and dividend can be any non-fractional, integer numbers. 
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Presentation is another way to visualize the topic of division. Below we will find a link to an excellent presentation that does a good job of explaining how to divide, what division is, what dividend, divisor and quotient are. Don’t waste your time, but consolidate your knowledge!
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve mental arithmetic skills in an interesting game form.
The game “Guess the Operation” develops thinking and memory. The main point game, you need to choose a mathematical sign for the equality to be true. Examples are given on the screen, look carefully and put the required “+” or “-” sign so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.
The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.
The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.
The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.
The Piggy Bank game develops thinking and memory. The main point of the game is to choose which piggy bank to use more money.In this game there are four piggy banks, you need to count which piggy bank has the most money and show this piggy bank with the mouse. If you answered correctly, then you score points and continue playing.
The game “Fast addition reboot” develops thinking, memory and attention. The main point of the game is to choose the correct terms, the sum of which will be equal to the given number. In this game, three numbers are given on the screen and a task is given, add the number, the screen indicates which number needs to be added. You select the desired numbers from three numbers and press them. If you answered correctly, then you score points and continue playing.
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Mathematics lesson on the topic "Multiplying and dividing three-digit numbers by a single-digit number without going through the place value."
Target: consolidate the knowledge, skills and abilities of multiplying and dividing a three-digit number by a single-digit number without going through a digit; develop the ability to apply theoretical knowledge and problem-solving skills in practice; develop verbal and logical thinking through posing problematic questions, attentiveness, intelligence, independence; bring up moral qualities by organizing mutual assistance, discussing the qualities needed in the lesson. positive lesson motivation.
Equipment: computer, overhead projector, presentation, cards.
DURING THE CLASSES
Breathing exercise “New lesson”.
For an entertaining lesson
A loud bell started.
Are you ready to count?
Divide and multiply quickly.
- What qualities and learning skills will we need in the classroom? Select.
(slide No. 2)
Quick wits
Savvy
Laziness
Attention
Noise
Perseverance
- Do we take them with us to class?
II. Checking homework
Attention! Attention!
We start the lesson by checking homework.
Homework: No. 745, p. 160.
(slide No. 3)
"Find the extra number"
321, 222, 243, 212, 444, 221, 214, 211, 311, 142, 123
(slide 2)
- Who agrees with the number?
Children raise their hands.
Create an example whose answer can be 444.
What else was assigned at home?
2. Mathematical dictation.
Product of numbers 8 and 9;
quotient of 36 and 4;
increase 8 by 6 times;
reduce 27 by 3 times;
How many times is 15 greater than 3?
1 factor is 9, the second is the same, what is the product equal to;
dividend 42, quotient 7, what is the divisor;
What number cannot be divided by?
Now check yourself!(Slide No. 4)
b) You answer either “yes” or “no” to the following questions
All three-digit numbers are odd;
All three-digit numbers are greater than 9;
If a number is multiplied by 1, it becomes 1;
If a number is divided by itself, the result is 0;
All even numbers are divisible by 2
Some three-digit numbers are less than 9;
You cannot divide by 0;
When you multiply a number by 1, you get the same number;
Test yourself!(Slide No. 4)
III. Verbal counting
(slide 5)
1. One T-shirt in the store costs 80 rubles. How much money do you need to pay to buy T-shirts for all the boys in our class?(80 rub. x 8 = 640 rub.)
2. We bought skirts for the girls in our class. We paid 250 rubles for the entire purchase. How much does one skirt cost?(250r.:1=250r.)
3. The school purchased 200 packs of laundry soap. Each pack costs 5 rubles. Calculate the total purchase price.(5 rubles x 200 = 1000 rubles)
- What did we repeat when solving this problem?(We repeated the multiplication and division tables.)
IV. State the topic and purpose of the lesson.
V. Fixing the material.
a) Solving the problem using short notation
(slide No. 6)
- Think and compose a problem, starting with the words:
In a week our school spends...
- What is this task about?(This problem is about vegetables: potatoes and carrots.)
- What is known in the problem?(It is known that potatoes488 kg consumed.)
- What is said about carrots?(Carrots are consumed 4 times less than potatoes.)
- How do we find out how many carrots have been used?(Division action 488: 4 = 122 kg)
- Is it possible to answer the problem question now?(Let's add potatoes and carrots together and answer the question in the problem.)
Solving the problem on the board and in notebooks with comments
Physical exercise.
a) Game “Sharing - not sharing”
(Slide No. 7)
- I name a couple of numbers. Your task: if the numbers are divided among themselves, then you quietly get up; If they don’t share, then clap your hands.
248: 2 = ;
367: 3 = ;
848: 4 = ;
481: 2 = ;
936: 3 = ;
695: 3 = .
b) Exercise for the eyes. (Slide No. 8,9)
Watch carefully the movement of the multi-colored circles!
VI. Consolidation
a) Write down only the answers. (Slide No. 10)
Check (Slide No. 11).
b) Working with the textbook.
Page 160 No. 741 - at the blackboard.
Analysis and analysis of the problem.
c) Independent work
223
450
101
777
684
969
Peer review.
VII. Homework. (slide No. 12)
- At home you should solve No. 747p. 160.
(Analysis of d/z).
VII. Lesson summary. Grading.
Reflection (Today in class I….).
