The division of natural numbers, especially multi-digit ones, is conveniently carried out by a special method, which is called division by a column (in a column). You can also find the name corner division. Let us immediately note that the column can be used to both divide natural numbers without a remainder and divide natural numbers with a remainder.
In this article we will look at how long division is performed. Here we will talk about recording rules and all intermediate calculations. First, let's focus on dividing a multi-digit natural number by a single-digit number with a column. After this, we will focus on cases when both the dividend and the divisor are multi-valued natural numbers. The entire theory of this article is provided with typical 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 do column division in writing on paper with a checkered line - this way there is less chance of straying from the desired row and column.
First, the dividend and divisor are written in one line from left to right, after which a symbol of the form is drawn between the written numbers. For example, if the dividend is the number 6 105 and the divisor is 5 5, then their correct recording when dividing into a column will be as follows:
Look at the following diagram, illustrating places to write the dividend, divisor, quotient, remainder, and intermediate calculations when dividing by a column.
From the above diagram it is clear that the required 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 in advance about the availability of space on the page. In this case, you 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 will be required. For example, when dividing by a column the natural number 614,808 by 51,234 (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), intermediate calculations will require less space than when dividing the numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3). To confirm our words, we present complete records of division by a column of these natural numbers:
Now you can proceed directly to the process of dividing natural numbers by a column.
Column division of a natural number by a single-digit natural number, column division algorithm
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 helpful to practice your initial long division skills with these simple examples.
Example.
Let us need to divide with a column of 8 by 2.
Solution.
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 with a column.
First, we write down the dividend 8 and the divisor 2 as required by the method:
Now we begin to find out how many times the divisor is contained in the dividend. To do this, we sequentially 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 the place of the quotient we write the number by which we multiplied the divisor. If we get a number greater than the dividend, 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 have received a number equal to the dividend, so we write it under the dividend, and in place of the quotient we write the number 4. In this case, the record will take the following form:
The final stage of dividing single-digit natural numbers with a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract the numbers above this line in the same way as is done when subtracting natural numbers in a column. The resulting number 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 before us a completed recording of the column division of the number 8 by 2. We see that the quotient of 8:2 is 4 (and the remainder is 0).
Answer:
8:2=4 .
Now let's look at how a column divides single-digit natural numbers with a remainder.
Example.
Divide with a column 7 by 3.
Solution.
On initial stage the entry looks like this:
We begin to find out how many times the dividend contains the 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 comparing 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 (the multiplication was carried out by 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.
Thus, the partial quotient is 2 and the remainder is 1.
Answer:
7:3=2 (rest. 1) .
Now you can move on to dividing multi-digit natural numbers by columns into single-digit natural numbers.
Now we'll figure it out long division algorithm. At each stage, we will present the results obtained by dividing the multi-digit natural number 140,288 by the single-digit natural number 4. This example was not chosen by chance, since when solving it we will encounter all possible nuances and will be able to analyze them in detail.
First we look at the first digit on the left in the dividend notation. 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 to the consideration the next digit on the left in the notation of the dividend, and continue to work with the number determined by the two digits under consideration. For convenience, we highlight in our notation the number with which we will work.
The first digit from the left in the notation of the dividend 140288 is the digit 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the notation of the dividend. At the same time, we see the number 14, with which we have to work further. We highlight this number in the notation of the dividend.
The following steps 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 sequentially 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, we write it under the highlighted number according to the recording rules used when subtracting natural numbers in a column. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (in 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 highlighted 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).
Multiply the divisor 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>14 . Since at the last step we received the number 16, which is greater than 14, then under the highlighted number we write the number 12, which was obtained at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate point the multiplication was carried out precisely by it.
At this stage, from the selected number, subtract the number located under it using a column. The result of the subtraction is written under the horizontal line. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at that point is the very last action that completely completes the process of long division). Here, for your own control, it would not be amiss to compare the result of the subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake was made somewhere.
We need to subtract the number 12 from the number 14 with a column (for the correctness of the recording, we must remember to put a minus sign to the left of the numbers being subtracted). After completing this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with the divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next point.
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 down the zero), we write down the number located in the same column in the notation of the dividend. If there are no numbers in the record of the dividend in this column, then the division by column ends there. After this, we select the number formed under the horizontal line, accept it as a working number, and repeat points 2 to 4 of the algorithm with it.
Under the horizontal line to the right of the number 2 already there, we write down 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 with it the actions of the second, third and fourth points of the algorithm.
Multiply the divisor 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 the subtraction in a column. Since we are subtracting equal natural numbers, then by virtue of the property of subtracting equal natural numbers, the result is zero. We do not write down the zero (since this is not the final stage of division with a column), but we remember the place where we could write it (for convenience, we will mark this place with a black rectangle).
Under the horizontal line to the right of the remembered place we write down the number 2, since it is precisely it that is in the record of the dividend 140,288 in this column. Thus, under the horizontal line we have the number 2.
We take the number 2 as the working number, mark it, and we will once again have to perform the actions of 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 the 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 the subtraction in 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, add the number 8 (since it is in this column in the entry for the dividend 140 288). Thus, the number 28 appears under the horizontal line.
We take this number as a working number, mark it, and repeat steps 2-4.
There shouldn't be any problems here if you have been careful up to now. Having completed all the necessary steps, the following result is obtained.
All that remains is to carry out the steps from points 2, 3, 4 one last time (we leave this to you), after which you will get a complete picture of dividing the natural numbers 140,288 and 4 into a column:
Please note that the number 0 is written in the very bottom line. If this was not the last step of division by a column (that is, if in the record of the dividend there were numbers left in the columns on the right), then we would not write this zero.
Thus, looking at the completed record of dividing the multi-digit natural number 140,288 by the single-digit natural number 4, we see that the quotient is the number 35,072 (and the remainder of the division is zero, it is in 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 7 136 and the divisor is a single-digit natural number 9.
Solution.
At the first step of the algorithm for dividing natural numbers by columns, we get a record of the form
After performing the actions from the second, third and fourth points of the algorithm, the column division record will take the form
Repeating the cycle, we will have
One more pass will give us a complete picture of the column division of the natural numbers 7,136 and 9
Thus, the partial quotient is 792, and the remainder is 8.
Answer:
7 136:9=792 (rest. 8) .
And this example demonstrates what long division should look like.
Example.
Divide the natural number 7,042,035 by the single-digit natural number 7.
Solution.
The most convenient way to do division is by column. 
Answer:
7 042 035:7=1 006 005 .
Column division of multi-digit natural numbers
We hasten to please you: if you have thoroughly mastered the column division algorithm from the previous paragraph of this article, then you almost already know how to perform column division of multi-digit natural numbers. This is true, since stages 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first point.
At the first stage of dividing multi-digit natural numbers into a column, you need to look not at the first digit on the left in the notation of the dividend, but at the number of them equal to the number of digits contained in the notation of the divisor. 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 notation of the dividend. After this, the actions specified in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.
All that remains is to see the application of the column division algorithm for multi-valued natural numbers in practice when solving examples.
Example.
Let's perform column division of multi-digit natural numbers 5,562 and 206.
Solution.
Since the divisor 206 contains 3 digits, we look at the first 3 digits on the left in 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 number, select it, and move on 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 multiplication is difficult, then it is better to multiply natural numbers in a column): 206 0 = 0<556
, 206·1=206<556
, 206·2=412<556
, 206·3=618>556. Since we received a number that is greater than the number 556, then under the highlighted 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 we multiplied by it at the penultimate step). The column division entry takes the following form:
We perform column subtraction. We get the difference 144, this number is less than the divisor, so you can safely continue performing the required actions.
Under the horizontal line to the right of the number there we write the number 2, since it is in the record of the dividend 5562 in this column:
Now we work with the number 1,442, select it, and go through steps two through four again.
Multiply the divisor 206 by 0, 1, 2, 3, ... until you 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 carry out the subtraction in a column, we get zero, but we don’t write it down right away, we just remember its position, because we don’t know whether the division ends here, or whether we’ll have to repeat the steps of the algorithm again:
Now we see that we cannot write any number under the horizontal line to the right of the remembered position, since there are no digits in the record of the dividend in this column. Therefore, this completes the division by column, and we complete the entry:
- Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
- Mathematics. Any textbooks for 5th grade of general education institutions.
After a short break, I will return to methods of teaching mathematics, this time for older students.
In order to prepare for the study of fractions, you need to start with the signs of divisibility and decomposition of numbers into prime factors. After this, you can move on to GCD, LCM and the fractions themselves. Again, these skills are extremely useful for understanding composition and how to manipulate numbers.
As usual, using the method, you can take the course in electronic form: Signs of divisibility, decomposition into prime factors.
The divisibility table includes basic, frequently used characteristics. The rest are usually hardly used.
Sequence of study
At the entrance, the child should be fairly confident in division.
Step 1. First, let's repeat the division with and without remainder
Step 2. Let's repeat which numbers are prime and which are composite
Step 3. Divisibility by 10 and 5
This is the easiest way, we determine it by the last digit - 0 or 5.
Step 4. Divisibility by 2
We determine by the last digit - it must be even.
Step 5. Divisibility by 3 and 9
First, we add up all the digits of the number, then check whether the resulting sum is divisible by 3/9.
Step 6. Divisibility by 4 and 6
Here it is most convenient to use composite division signs.
To divide by 4, we take only the number from the last 2 numbers (without hundreds, thousands, etc.). Divide by 2 and check the result to see if the last digit is even.
To divide by 6, the number must be simultaneously divisible by 2 (the last digit is even) and by 3 (the sum of the digits is divisible by 3).
Step 7 Divisibility by 7
There is a test for divisibility by 7, but it is complex. And often you have to apply it several times in a row.
As an alternative, you can use a simple divisibility test - subtracting multiples and checking the divisibility of the remainder. Let's work on this method at the same time.
Step 8 Let's practice finding the divisors of a number and decomposing
We have learned to check whether a number can be completely divided by a prime. Now that we know how to do this, we can start factoring the number.
The algorithm is simple:
- to do this, we sequentially begin checking the divisibility of a number by all primes, starting with the smallest one (from 2)
- as soon as we find the divisor, we get the result of division
- then we take the result and again try to divide into simple ones (starting from where we stopped)
- and so on until, as a result of the next division, we get a prime number
Step 9 Ways to lay out faster and more conveniently
Now that we have learned how to do it according to the algorithm, we can think about how to do it more conveniently and quickly.
To do this, you can take obvious divisors and divide by them first.
How can you apply the technique?
you can train your child yourself, give him examples
you can start passing
Sections: Mathematics
Class: 5
Subject: Division with remainder.
Lesson objectives:
Repeat division with a remainder, derive a rule on how to find the dividend when dividing with a remainder, and write it down as a literal expression;
- develop attention, logical thinking, mathematical speech;
- fostering a culture of speech and perseverance.
During the classes
The lesson is accompanied by a computer presentation. (Application)
I. Organizing time
II. Verbal counting. Lesson topic message
By solving the examples and filling out the table, you will be able to read the topic of the lesson.
On the desk:
Read the topic of the lesson.
We opened our notebooks, wrote down the date and topic of the lesson. (Slide 1)
III. Work on the topic of the lesson
We'll decide orally. (Slide 2)
1. Read the expressions:
30: 5
103: 10
34: 5
60: 7
47: 6
131: 11
42: 6
What two groups can they be divided into? Write down and solve those in which division has a remainder.
2. Let's check. (Slide 3)
Without remainder: |
With the remainder: |
|
30: 5 |
103: 10 = 10 (rest 3) |
Tell us how you did division with a remainder?
One natural number is not always divisible by another number. But you can always divide with a remainder.
What does it mean to divide with the remainder? To answer this question, let's solve the problem. ( Slide 4)
4 grandchildren came to visit their grandmother. The grandmother decided to treat her grandchildren to sweets. There were 23 candies in the bowl. How many candies will each grandchild get if grandma offers to divide the candies equally?
Let's reason.
How many sweets does grandma have? (23)
How many grandchildren came to visit their grandmother? (4)
What needs to be done according to the problem? (The candies must be divided equally, 23 must be divided by 4; 23 is divided by 4 with a remainder; the quotient will be 5, and the remainder will be 3.)
How many candies will each grandchild get? (Each grandchild will receive 5 candies, and there will be 3 candies left in the vase.)
Let's write down the solution. (Slide 5)
23: 4=5 (ost 3)
What is the name of the number that is being divided? (Divisible.)
What is a divisor? (The number being divided by.)
What is the result of division with a remainder called? (Incomplete quotient.)
Name the dividend, divisor, partial quotient and remainder in our solution (23 - dividend, 4 - divisor, 5 - incomplete quotient, 3 - remainder.)
Guys, think and write down how to find the dividend of 23, knowing the divisor, partial quotient and remainder?
Let's check.
Guys, let's formulate a rule on how to find the dividend if the divisor, partial quotient and remainder are known.
Rule. (Slide 6)
The dividend is equal to the product of the divisor and the incomplete quotient added to the remainder.
a = sun + d , a - dividend, b - divisor, c - incomplete quotient, d - remainder.
When doing division with a remainder, what should we remember?
That's right, the remainder is always less than the divisor.
And if the remainder is zero, the dividend is divided by the divisor without a remainder, completely.
IV. Reinforcing the material learned
Slide 7
Find the dividend if:
A) the partial quotient is 7, the remainder is 3, and the divisor is 6.
B) the partial quotient is 11, the remainder is 1, and the divisor is 9.
C) the partial quotient is 20, the remainder is 13, and the divisor is 15.
V. Working with the textbook
1.
Working on a task.
2.
Formulation of the solution to the problem.
№ 516 (The student solves the problem at the board.)
20 x 10: 18 = 11 (rest 2)
Answer: 11 parts of 18 kg each can be cast from 10 blanks, 2 kg of cast iron will remain.
№ 519 (Workbook, p. 52 No. 1.)
Slide 8, 9
The first task is completed by the student at the blackboard. Students perform the second and third tasks independently with self-test.
We solve problems orally. (Slide 10)
VI. Lesson summary
There are 17 students in your class. You were lined up. It turned out to be several lines of 5 students and one incomplete line. How many full ranks are there and how many people are there in an incomplete rank?
Your class at the physical education lesson was again lined up. This time there were 4 identical full ranks and one incomplete? How many people are in each line? What about incomplete?
We answer the questions:
Can the remainder be greater than the divisor? Can the remainder be equal to the divisor?
How to find the dividend using the incomplete quotient, divisor and remainder?
What remainders can there be when divided by 5? Give examples.
How to check if division with remainder is correct?
Oksana thought of a number. If you increase this number by 7 times and add 17 to the product, you get 108. What number did Oksana have in mind?
VII. Homework
Point 13, No. 537, 538, workbook, p. 42, No. 4.
Bibliography
1. Mathematics: Textbook. for 5th grade. general education institutions / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. – 9th ed., stereotype. – M.: Mnemosyne, 2001. – 384 p.: ill.
2. Mathematics. 5th grade. Workbook No. 1. natural numbers / V.N. Rudnitskaya. – 7th ed. – M.: Mnemosyne, 2008. – 87 p.: ill.
3. Chesnokov A.S., Neshkov K.I. Didactic materials on mathematics for grade 5. – M.: Classics Style, 2007. – 144 p.: ill.
Sections: Mathematics
Class: 6
Lesson Objectives:
1. Educational: repetition, generalization and testing of knowledge on the topic: “Divisibility of natural numbers”; development of basic skills.
2. Developmental: develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
3. Educational: through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.
Lesson objectives:
To develop the ability to apply the concept of divisors and multiples; develop thinking and elements of creative activity; apply divisibility criteria in the simplest situations; finding GCD and LCM numbers, developing observation and logical thinking.
Lesson type- combined.
Lesson form– lesson with computer support.
Equipment:
1. Board and chalk.
2. Computer and projector.
3. Paper version of all tasks.
During the classes.
Numbers rule the world.
Pythagoras.
1. Organizational moment.
2. Communicate the purpose of the lesson.
3. Updating of basic knowledge.
1. What is a number divisor? A?
2. What is a multiple of a number? A?
3. Is there a greatest multiple?
4. Formulate the signs of divisibility?
5. Which numbers are called prime and which are composite?
(Students’ report about Pythagoras, Eratosthenes, Euclid)
Historical information:
| Euclid - ancient Greek scientist (365 - 300 BC). Very little is known about the life of this great scientist. He lived and worked in Alexandria, the city founded by Alexander the Great. Many legends are associated with the name of Euclid. One of them says that King Ptolemy asked Euclid: “Is there a shorter way to knowledge of geometry?”, to which the scientist replied: “There is no royal road to geometry!” Euclid studied number theory a lot: it was he who proved that there are infinitely many prime numbers. The algorithm for finding the gcd of two numbers is called the Euclidean algorithm. The ancient Greek mathematician Euclid, in his book Elements, which was the main textbook in mathematics for two thousand years, proved that there are infinitely many prime numbers, i.e. Behind every prime number there is an even prime number. |
| Pythagoras (6th century BC) and his students studied the question of the divisibility of numbers. They called a number equal to the sum of all its divisors (without the number itself) a perfect number. For example, the number 6 (6 = 1 + 2 + 3), 28 (28 = 1 + 2 + 4 + 7 + 14) are perfect. The following perfect numbers are 496, 8128, 33550336 The Pythagoreans only knew the first three perfect numbers. The fourth 8128 became known in the 1st century BC. The fifth number, 33550336, was found in the 15th century. By 1983, 27 perfect numbers were already known. But scientists still don’t know whether there is an odd perfect number or whether there is a largest perfect number. The interest of ancient mathematicians in prime numbers stems from the fact that any natural number greater than 1 is either a prime number or can be composed as a product of prime numbers: 14 = 2∙7, 16 = 2∙2 ∙2∙2 The question arises: is there a last (largest) prime number? |
Task: A prime number has been conceived. The next natural number is also prime. What numbers are we talking about?
Answer: 2.3.
6. What numbers are called relatively prime?
7. Explain how to find GCD (LCD) of two numbers.
(Student’s message about finding the gcd of two numbers)
One day the numbers 24 and 60 argued about how to find a gcd. The number 24 stated that you first need to find the common numbers among all the divisors, and then choose the largest number from them. And number 60 objected:
- Well, what are you talking about! I don't like this method. I have too many divisors, and in listing them I might miss one. What if it turns out to be the largest? No, I don't like this method. And they decided to turn to the Master of Business Sciences for help. And the master answered them:
- Yes, 24, your method of finding gcd of numbers can be used, but it is not always convenient. But you can find GCD in a different way.
You need to factor 24 and 60 into prime factors.
| 24 | 2 |
| 12 | 2 |
| 6 | 2 |
| 3 | 3 |
| 1 |
| 60 | 2 |
| 30 | 2 |
| 15 | 3 |
| 5 | 5 |
| 1 |
24 = 2³ ∙ 3
60 = 2² ∙ 3 ∙ 5
You need to take the common divisors of numbers with a smaller exponent.
GCD (24;60) = 2² ∙ 3 = 12.
And to find the LCM of two numbers you need:
- Factor into prime factors;
- Write down all the prime factors that are included in the first number and in the second number with the largest exponent.
Means:
24 = 2³ ∙ 3 60 = 2² ∙ 3 ∙ 5 NOC (24;60) = 2³∙ 3 ∙ 5 = 120.
LESSON SUMMARY
MATHEMATICS
3rd grade
Plohotnyuk Victoria Nikolaevna,
primary school teacher
MBOU "Secondary School No. 6" Usinsk
Komi Republic
SUBJECT: Repeating division (technique for calculating the quotient)
TASKS:
continue work on division techniques based on operating with specific objects;
consolidate the names of numbers when dividing and multiplying;
develop mental counting skills;
continue to work on interaction skills
DURING THE CLASSES:
The lesson begins.
It will be useful for the guys.
I'll try to understand everything
I will decide correctly.
I. And now we have not just a lesson, but a cosmic lesson. We'll take a journey to the stars. During the flight, we will repeat division, remember what numbers are called when multiplying, adding, and subtracting.
And for the flight to be successful, you need to listen carefully, think, and count correctly.
But first you need to get permission to take off.
So: we give only the answer.
Difference of numbers 60 and 8 (52)
1 term 32, 2 terms 8 – sum (40)
96 decrease by 90 (6)
Sum of numbers 16 and 12 (28)
37 increase by 1 (38)
Does the number 27 contain 3 des and 7 units? (2 d 7 f)
Is the number 38 in the number series between the numbers 37 and 40? (37.39)
7 dec. Is it 70? Yes
5 dec. Is it 15? No.
What are the numbers at +/at – called?
We did a good job, but tell me, what actions did we repeat?
(+ and -)
Take your seats and check your readiness for flight.Let's read.
We are flying to other planets
We inform you about this.
II. While our rocket is gaining speed, open the logbooks and write down the flight date.
We are already there and have flown to 1 star “Hurry up”. Tempting tasks await us here:
5 ,10,11,15,20
40,30, 19 ,20,80 which number is the odd one out?
22,23, 42 ,25,26
40,42,44,46…
35,40,45,50... what number comes next?
10,20,30,40…
And this work must be done quickly and accurately. The task is: solve and check.
38+27 52-29 63-44 51+29 91-55
How can we check +, -?
The guys tried very hard, I liked it, we won’t linger here, we’ll continue the flight.Fizminutka
We stood up together once, 2,3
We are now heroes
We will put our palms to our eyes.
Let's spread our strong legs.
Turned to the right
Let's look around majestically,
And you need to go left too.
Look from under your palms.
And to the right and again
Over the right shoulder.
III. We didn’t even notice how we flew up to the “Divide-ka” star. Let's remember what numbers are called when dividing?
What is the name of the number we are dividing? (dividend)
What is the name of the number we are dividing by? (divider)
What is the result of division called? (private)
Let's write in the logbook: Dividend 10, divisor 2. What do you need to find? (private)
And we will look for the quotient using a drawing.
Who will draw?
O O O O O O O O O O
(rocket →)
How many circles are we grouping (2 each)
sk. 2 times contained in 10? (5 times)
So, what is the quotient equal to? (5)
Now let's look around, look left, right, up. What's wrong with our rocket?
In my opinion, she has lost control and urgently needs to make calculations. Who will help us? (place the cards on the board)
12:3 8:2 6:3
And here is the expression itself:
12:2
Let's repeat.
What are numbers called when divided?
What results from division?
(Seated exercise)
We stretched, raised our right and left shoulders.
IV. We flew to the star "Zapasayka". We need to check our stocks:
For the flight we took 5 bottles of lemonade, 1 liter each. How many liters of lemonade did you take? (10l)
They also took 3 boxes of cookies, 2 kg each. How many kilograms of cookies did you take?
We were thinking of taking 20 large boars and 10 small ones on the flight. When they found out what it was, they threw everything away. How many pigs were thrown away?
Let's split into crews. Look what color is the star on your desk?
Orange
Take your seats. Before work, guess what it is?
Grandfather sits wearing 100 fur coats
Whoever undresses him sheds tears
Yes, it's an onion. Do you know that onions in Ancient Rus' were considered the best remedy for diseases? And in Ancient Greece it was a sacred plant. And in Germany, heroes were decorated with onion flowers. Shall we take it on a flight?
And what's that?The child grew up without diapers,
Became an old man
100 diapers on it.
Of course it's cabbage. It has been used since ancient times as a remedy for insomnia and headaches. Its juice was used to lubricate the wounds.
Shall we take cabbage with us too?
Now let's get down to business.
Time has passed.
15 bulbs were planted 3 in a row. How many rows did you get?
15 heads of cabbage were planted equally on 3 rows. How many heads of cabbage are there in each row?
2) Give a solution.
What did you notice?
(there is one solution, but we are looking for different ones)
We decided, we decided
We're very tired,
We're about to drown
Let's clap our hands
Once - let's sit down
Let's get up quickly
Let's smile
Let's sit quietly.
And what's that? Unidentified objects are approaching us; to avoid a collision, we urgently need to find out their parameters.
∆ O
What objects did you see?
By what criteria will we group?
by color
to size
according to form
Name them.
What else is this? Some strange faces.
From which geom. consist of figures?
Which one is extra?
Very good.
We have avoided collision and our journey is coming to an end. And in order to return safely, you need to solve the crossword puzzle.
1) What do you get when you add? (sum)
2) What is the name of the number resulting from division? (private)
3) Is it straight and sharp? (corner)
4) The number that is being divided? (dividend)
5) The number to divide by? (divider)
Very bad rating? (unit)
7) How can we check “+” (subtraction)
Our flight has come to an end. We follow the pointer. What word did you get? Well done.
Yes, of course we are great.
What did they repeat today?
What did you like?
What did you find difficult?
What rating should we give ourselves?
And in order to successfully continue working on division in the next lesson and write a good independent work, you need to consolidate the material at home.
Let's write down the task:
p.54 table No. 3.
The lesson is over.