Although mathematics seems difficult to most people, it is far from true. Many mathematical operations are quite easy to understand, especially if you know the rules and formulas. So, knowing the multiplication table, you can quickly multiply in your head. The main thing is to constantly train and not forget the rules of multiplication. The same can be said about division.
Let's look at the division of integers, fractions and negatives. Let's remember the basic rules, techniques and methods.
Division operation
Let's start, perhaps, with the very definition and name of the numbers that participate in this operation. This will greatly facilitate further presentation and perception of information.
Division is one of the four basic mathematical operations. Its study begins in elementary school. It is then that the children are shown the first example of dividing a number by a number and the rules are explained.
The operation involves two numbers: the dividend and the divisor. The first is the number that is being divided, the second is the number that is being divided by. The result of division is the quotient.
There are several notations for writing this operation: “:”, “/” and a horizontal bar - writing in the form of a fraction, when the dividend is at the top, and the divisor is below, below the line.
Rules
When studying a particular mathematical operation, the teacher is obliged to introduce students to the basic rules that they should know. True, they are not always remembered as well as we would like. That's why we decided to refresh your memory a little on the four fundamental rules.
Basic rules for dividing numbers that you should always remember:
1. You cannot divide by zero. This rule should be remembered first.
2. You can divide zero by any number, but the result will always be zero.
3. If a number is divided by one, we get the same number.
4. If a number is divided by itself, we get one.
As you can see, the rules are quite simple and easy to remember. Although some may forget such a simple rule as impossibility or confuse the division of zero by a number with it.
per number
One of the most useful rules is the sign by which the possibility of division is determined natural number for the other without any reserve. Thus, the signs of divisibility by 2, 3, 5, 6, 9, 10 are distinguished. Let us consider them in more detail. They make it much easier to perform operations on numbers. We also give an example for each rule of dividing a number by a number.
These rules-signs are quite widely used by mathematicians.
Test for divisibility by 2
The easiest sign to remember. A number that ends in an even digit (2, 4, 6, 8) or 0 is always divisible by two. Quite easy to remember and use. So, the number 236 ends in an even digit, which means it is divisible by two.
Let's check: 236:2 = 118. Indeed, 236 is divisible by 2 without a remainder.
This rule is best known not only to adults, but also to children.
Test for divisibility by 3
How to correctly divide numbers by 3? Remember the following rule.
A number is divisible by 3 if the sum of its digits is a multiple of three. For example, let's take the number 381. The sum of all digits will be 12. This is three, which means it is divisible by 3 without a remainder.
Let's also check this example. 381: 3 = 127, then everything is correct.
Divisibility test for numbers by 5
Everything is simple here too. You can divide by 5 without a remainder only those numbers that end in 5 or 0. For example, let’s take numbers such as 705 or 800. The first ends in 5, the second in zero, therefore they are both divisible by 5. This is one one of the simplest rules that allows you to quickly divide by a single-digit number 5.
Let's check this sign using the following examples: 405:5 = 81; 600:5 = 120. As you can see, the sign works.
Divisibility by 6
If you want to find out whether a number is divisible by 6, then you first need to find out whether it is divisible by 2, and then by 3. If so, then the number can be divided by 6 without a remainder. For example, the number 216 is divisible by 2 , since it ends with an even digit, and with 3, since the sum of the digits is 9.
Let's check: 216:6 = 36. The example shows that this sign is valid.
Divisibility by 9
Let's also talk about how to divide numbers by 9. given number The sum of digits whose sum is a multiple of 9 is divisible. Similar to the rule of dividing by 3. For example, the number 918. Let's add all the digits and get 18 - a number that is a multiple of 9. This means that it is divisible by 9 without a remainder.
Let's solve this example to check: 918:9 = 102.
Divisibility by 10
One last sign to know. Only those numbers that end in 0 are divisible by 10. This pattern is quite simple and easy to remember. So, 500:10 = 50.
That's all the main signs. By remembering them, you can make your life easier. Of course, there are other numbers for which there are signs of divisibility, but we have highlighted only the main ones.
Division table
In mathematics, there is not only a multiplication table, but also a division table. Once you learn it, you can easily perform operations. Essentially, a division table is a multiplication table in reverse. Compiling it yourself is not difficult. To do this, you should rewrite each line from the multiplication table in this way:
1. Put the product of the number in first place.
2. Put a division sign and write down the second factor from the table.
3. After the equal sign, write down the first factor.
For example, take the following line from the multiplication table: 2*3= 6. Now we rewrite it according to the algorithm and get: 6 ÷ 3 = 2.
Quite often, children are asked to create a table on their own, thus developing their memory and attention.
If you don’t have time to write it, you can use the one presented in the article.
Types of division
Let's talk a little about the types of division.
Let's start with the fact that we can distinguish between division of integers and fractions. Moreover, in the first case we can talk about operations with integers and decimals, and in the second - only about fractional numbers. In this case, a fraction can be either the dividend or the divisor, or both at the same time. This is due to the fact that operations on fractions are different from operations on integers.
Based on the numbers that participate in the operation, two types of division can be distinguished: into single-digit numbers and into multi-digit ones. The simplest is division by a single digit number. Here you will not need to carry out cumbersome calculations. In addition, a division table can be a good help. Dividing by other - two-, three-digit numbers - is harder.
Let's look at examples for these types of division:
14:7 = 2 (division by a single digit number).
240:12 = 20 (division by a two-digit number).
45387: 123 = 369 (division by a three-digit number).
The last one can be distinguished by division, which involves positive and negative numbers. When working with the latter, you should know the rules by which a result is assigned a positive or negative value.
When dividing numbers with different signs(the dividend is a positive number, the divisor is negative, or vice versa) we get a negative number. When dividing numbers with the same sign (both the dividend and the divisor are positive or vice versa), we get a positive number.
For clarity, consider the following examples:
Division of fractions
So, we have looked at the basic rules, given an example of dividing a number by a number, now let’s talk about how to correctly perform the same operations with fractions.
Although dividing fractions may seem like a lot of work at first, working with them is actually not that difficult. Dividing a fraction is done in much the same way as multiplying, but with one difference.
In order to divide a fraction, you must first multiply the numerator of the dividend by the denominator of the divisor and record the resulting result as the numerator of the quotient. Then multiply the denominator of the dividend by the numerator of the divisor and write the result as the denominator of the quotient.
It can be done simpler. Rewrite the divisor fraction by swapping the numerator with the denominator, and then multiply the resulting numbers.
For example, let's divide two fractions: 4/5:3/9. First, let's turn the divisor over and get 9/3. Now let's multiply the fractions: 4/5 * 9/3 = 36/15.
As you can see, everything is quite easy and no more difficult than dividing by a single-digit number. The examples are not easy to solve if you do not forget this rule.
Conclusions
Division is one of the mathematical operations that every child learns in elementary school. There are certain rules that you should know, techniques that make this operation easier. Division can be with or without a remainder; there can be division of negative and fractional numbers.
It is quite easy to remember the features of this mathematical operation. We have sorted out the most important points, we looked at more than one example of dividing a number by a number, we even talked about how to work with fractional numbers.
If you want to improve your knowledge of mathematics, we advise you to remember these simple rules. In addition, we can advise you to develop memory and mental arithmetic skills by performing mathematical dictations or simply trying to verbally calculate the quotient of two random numbers. Believe me, these skills will never be superfluous.
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“Relationship Problems” - “Mathematics has its own beauty, just like painting and poetry.” N. Zhukovsky. Method 2: algebraic Let x be the coefficient of proportionality of numbers. Every person is born internally not free. Creative task: where the proportion applies (per week). Society uses attitude, society uses mathematics.
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There are a total of 26 presentations in the topic
Division is one of the four most common arithmetic operations. There are rarely complex calculations that can do without it. Excel has a wide range of functionality for using this arithmetic operation. Let's find out how you can perform division in Excel.
IN Microsoft Excel division can be done both using formulas and using functions. The dividend and divisor are numbers and cell addresses.
Method 1: Divide a number by a number
An Excel sheet can be used as a kind of calculator, simply dividing one number by another. The division sign is a slash (backslash) - «/» .
After this, Excel will calculate the formula and display the calculation result in the specified cell.
If a calculation is performed with several signs, then the order of their execution is carried out by the program according to the laws of mathematics. That is, first of all, division and multiplication are performed, and only then addition and subtraction.
As you know, division by 0 is an incorrect action. Therefore, if you try to make a similar calculation in Excel, the result will appear in the cell “#DIV/0!”.
Method 2: Dividing Cell Contents
You can also divide data in cells in Excel.
You can also combine, using both cell addresses and static numbers as a dividend or divisor.
Method 3: Dividing Column by Column
Calculation in tables often requires the values of one column to be divided by the data of the second column. Of course, you can divide the value of each cell in the same way as indicated above, but this procedure can be done much faster.
As you can see, after this action the procedure of dividing one column by the second will be completely completed, and the result will be displayed in a separate column. The fact is that the fill marker is used to copy the formula into the lower cells. But, taking into account the fact that by default all links are relative and not absolute, then in the formula, as you move down, the cell addresses change relative to the original coordinates. And this is exactly what we need for a specific case.
Method 4: Divide a column by a constant
There are times when you need to divide a column by the same constant number, and display the sum of the division in a separate column.
As you can see, this time the division was also performed correctly. In this case, when copying data with a fill marker, the links again remained relative. The dividend address for each line was automatically changed. But the divisor is in in this case constant number, which means that the property of relativity does not apply to it. Thus, we divided the contents of the column cells by a constant.
Method 5: Dividing a Column by Cell
But what if you need to split a column into the contents of one cell. After all, according to the principle of relativity of references, the coordinates of the dividend and the divisor will shift. We need to make the address of the cell with the divisor fixed.
After this, the result for the entire column is ready. As you can see, in this case the column was divided into a cell with a fixed address.
Method 6: PRIVATE function
Division in Excel can also be performed using a special function called PRIVATE. The peculiarity of this function is that it divides, but without a remainder. That is, when using this method When dividing, the result will always be an integer. In this case, rounding is performed not according to generally accepted mathematical rules to the nearest integer, but to a smaller one in absolute value. That is, the function will round the number 5.8 not to 6, but to 5.
Let's see the use of this function using an example.
After these steps the function PRIVATE processes the data and produces a response in the cell that was specified in the first step of this division method.
This function can also be entered manually without using the Wizard. Its syntax looks like this:
QUANTIATE(numerator,denominator)
As you can see, the main method of division in the Microsoft Office program is the use of formulas. The division symbol in them is a slash - «/» . At the same time, for certain purposes, you can use the function in the division process PRIVATE. But, you need to take into account that when calculating in this way, the difference is obtained without a remainder, as an integer. In this case, rounding is performed not according to generally accepted norms, but to a smaller integer in absolute value.
Lesson No. 9 (09/15/16)
Item: mathematics, 6-B class.
Lesson topic: Dividing numbers into in this regard. Solution of exercises (2 th lesson on topic)
Lesson type:Lesson in applying knowledge
Lesson objectives for the teacher:
1. Create conditions for practicing the skill of dividing a number in this regard (subject)
2. Develop skills in analyzing and comparing methods for solving similar types of problems (intellectual skills)
3. To develop the skills of determining activity goals and drawing up an action plan (organizational skills)
4. Learn to convey your position to others and accept other people’s positions (communication skills)
5.
Check the level of mastery of the topic
Subject Skills:
Perform all operations with natural and fractional numbers. Create mathematical models of solved problems: diagram, expression. Solve word problems with the condition of the ratio of quantities.
Organizational skills:
Determine and formulate activity goals
Make a plan to solve the problem
Act according to plan
Correlate the results of your activities with your goal
Organize independent activities to select and solve problems
Intellectual skills:
To navigate your knowledge system and recognize the need for new knowledge
Put forward hypotheses for solving the problem
Communication skills:
Practice techniques of monologue and dialogic speech
Assessment skills:
Compare your own results with the presented sample
Mandatory minimum content:
Concepts, rules, patterns:
algorithm for dividing a quantity in a given ratio
Subject Skills:
Divide a quantity in a given ratioseveral numbers, solve word problems with a given ratio of quantities,
Lesson progress:
Time:2 minutes
Organizational moment. Greetings, identification of absentees.
Updating knowledge.
9 minutes
Students (expected actions)
UUD
Hello guys! Please open your notebooks, write down the date - today is September 15, 2016. Sit back and let's remember what we talked about in the last lesson and what tasks we learned to do?
Do you have any questions while solving your homework? (If “yes”, then I call someone who wants to show the solution to the board, if “no” - we move on)
Let's see how you learned to do the tasks you just talked about.
And we will try to answer the following questions:
What is an attitude?
Read the ratios: 15:6; 3:5; 5/7; ½: ¾ ; 0.5:0.3
Which of the recorded relationships do you think can be simplified? Simplify
Now let's look at the solutions on the board
If during the solution there were errors when using the algorithm, we recite it again, pay attention to the presence of a support with the algorithm on the board
Possible answers:
We learned to solve problems and examples of dividing numbers in this regard.
1 person writes down the solution to a homework problem on the board
1 student works independently at the board
All students answer questions, complete assignments orally, and, if necessary, do calculations in notebooks.
Students read the problem and tell its solution, the class makes comments, evaluates the work
Possible answers:
Regulatory: understand the level and quality of learning the material.
Communicative: expressing your thoughts.
Cognitive: conscious construction of a speech utterance, summing up a concept.
Learning new material
10 minutes
Teacher's actions (content of dialogue)
Students (expected actions)
Learning Tools
Creating a problem situation
Now please divide the number 120 into in the following respects: a) 1:5; b) 1/3:2/3; c) 3:2:5
Complete task a), give explanations for completion. (100.20) (40.80) (36.24.60).
Complete task b) with the help of the teacher, placing emphasis on the need to first simplify the relationship.
Have difficulty completing c) all or many students
Regulatory: goal setting
Communicative: asking questions
Cognitive: independent identification and formulation of a cognitive goal
Formulation
problems
(topics and objectives of the lesson)
What question did you have while completing this assignment? Try to define your difficulties in one sentence
Formulate difficulties in the form of questions
Determine the topic, edit it with the help of the teacher, write it down in a notebook
Define goals:
Create an algorithm for dividing a number in a relation containing more than two terms
Learn to use a rule to solve problems
Regulatory: formulate and maintain a learning task;
Communicative: the ability to express one’s thoughts;
Cognitive:
bringing under the rule;
Formulation
new knowledge
We have divided the number in a given ratio.
They conclude:
to divide a number in a given relation, you need to divide this number by the sum of the terms of the relation and multiply the result by each member of the relation.
Regulatory:
highlight what has been learned and what needs to be learned.
Communicative:
ability to express one's thoughts, argumentation.
Consolidating new material
20 minutes
Teacher's actions (content of dialogue)
Students (expected actions)
Application of new knowledge
Let's solve several problems involving dividing a number in a given ratio.
Divide:
Number 42 in ratio 5:2
Number 28 in ratio 2:5:1
Number 27 in the ratio 0.2:0.3:0.4
(we are working on checking the second answer by adding the obtained values)
Solving problems with control at the board:
№ 40, 43*.
Work in pairs, self-test according to the model.
They find an error in the answers given and prove they are right in two ways.
If desired, at the board, the class works independently, controls the solution
Regulatory:
draw up a plan and sequence of actions;
Communicative:
perceive the text taking into account the assigned educational task, find in the text the information necessary for the solution.
Cognitive: put forward hypotheses for solving a problem
Lesson summary
4 minutes
Teacher's actions (content of dialogue)
Students (expected actions)
Reflection
Answer questions, justifying your answer
Cognitive: reflection on methods and conditions of action, adequate understanding reasons for success and failure, control and evaluation of the process and results of activities
P 1.3, No. 44 (a, b, d).
write in a diary, look at it in a textbook