Rule for adding numbers with opposite signs. Addition and subtraction of whole numbers

Instruction

There are four types of mathematical operations: addition, subtraction, multiplication and division. Therefore, there will be four types of examples with. Negative numbers within the example are highlighted so as not to confuse the mathematical operation. For example, 6-(-7), 5+(-9), -4*(-3) or 34:(-17).

Addition. This action can look like: 1) 3+(-6)=3-6=-3. Replacing the action: first, the brackets are opened, the "+" sign is reversed, then the smaller "3" is subtracted from the larger (modulo) number "6", after which the answer is assigned the larger sign, that is, "-".
2) -3+6=3. This one can be written as - ("6-3") or according to the principle "subtract the smaller from the larger and assign the sign of the larger to the answer."
3) -3+(-6)=-3-6=-9. When opening, the replacement of the action of addition by subtraction, then the modules are summed up and the result is given a minus sign.

Subtraction.1) 8-(-5)=8+5=13. The brackets are opened, the sign of the action is reversed, and an addition example is obtained.
2) -9-3=-12. The elements of the example are added together and given a common "-" sign.
3) -10-(-5)=-10+5=-5. When opening the brackets, the sign changes again to "+", then the smaller number is subtracted from the larger number and the sign of the larger number is taken from the answer.

Multiplication and division. When performing multiplication or division, the sign does not affect the operation itself. When multiplying or dividing numbers, a minus sign is assigned to the answer, if numbers with the same signs, the result always has a plus sign. 1)-4*9=-36; -6:2=-3.
2)6*(-5)=-30; 45:(-5)=-9.
3)-7*(-8)=56; -44:(-11)=4.

Sources:

• table with cons

How to decide examples? Children often turn to their parents with this question if homework needs to be done. How to correctly explain to a child the solution of examples for addition and subtraction of multi-digit numbers? Let's try to figure this out.

You will need

• 1. Mathematics textbook.
• 2. Paper.
• 3. Handle.

Instruction

Read the example. To do this, each multivalued is divided into classes. Starting from the end of the number, count off three digits and put a dot (23.867.567). Recall that the first three digits from the end of the number to units, the next three - to the class, then there are millions. We read the number: twenty-three eight hundred sixty-seven thousand sixty-seven.

Write down an example. Please note that the units of each digit are written strictly under each other: units under units, tens under tens, hundreds under hundreds, etc.

Perform addition or subtraction. Start doing the action with units. Write the result under the category with which the action was performed. If it turned out to be a number (), then we write the units at the place of the answer, and add the number of tens to the units of the discharge. If the number of units of any digit in the minuend is less than in the subtrahend, we take 10 units of the next digit, perform the action.

Read the answer.

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note

Forbid your child to use a calculator, even to check the solution of an example. Addition is tested by subtraction, and subtraction is tested by addition.

Useful advice

If a child learns well the techniques of written calculations within 1000, then actions with multi-digit numbers performed by analogy will not cause difficulties.
Arrange a competition for your child: how many examples can he solve in 10 minutes. Such training will help automate computational techniques.

Multiplication is one of the four basic mathematical operations and is the basis of many more complex functions. In this case, in fact, multiplication is based on the operation of addition: knowledge of this allows you to correctly solve any example.

To understand the essence of the multiplication operation, it is necessary to take into account that three main components are involved in it. One of them is called the first factor and represents the number that is subjected to the multiplication operation. For this reason, it has a second, somewhat less common name - "multiplier". The second component of the multiplication operation is called the second factor: it is the number by which the multiplicand is multiplied. Thus, both of these components are called multipliers, which emphasizes their equal status, as well as the fact that they can be interchanged: the result of multiplication will not change from this. Finally, the third component of the multiplication operation, resulting from it, is called the product.

The order of the multiplication operation

The essence of the multiplication operation is based on a simpler arithmetic operation -. In fact, multiplication is the summation of the first factor, or multiplicand, such a number of times that corresponds to the second factor. For example, in order to multiply 8 by 4, you need to add the number 8 4 times, resulting in 32. This method, in addition to providing an understanding of the essence of the multiplication operation, can be used to check the result obtained by calculating the desired product. It should be borne in mind that the verification necessarily assumes that the terms involved in the summation are the same and correspond to the first factor.

Solving multiplication examples

Thus, in order to solve , associated with the need to carry out multiplication, it may be sufficient to add the required number of first factors a given number of times. Such a method can be convenient for performing almost any calculations associated with this operation. At the same time, in mathematics quite often there are typical ones, in which standard single-digit integers participate. In order to facilitate their calculation, the so-called multiplication was created, which includes a complete list of products of positive integer single-digit numbers, that is, numbers from 1 to 9. Thus, once you have learned, you can significantly simplify the process of solving multiplication examples, based on the use of such numbers. However, for more complex options, it will be necessary to carry out this mathematical operation yourself.

Related videos

Sources:

• Multiplication in 2019

Multiplication is one of the four basic arithmetic operations, which is often used both in school and in Everyday life. How can you quickly multiply two numbers?

The basis of the most complex mathematical calculations are four basic arithmetic operations: subtraction, addition, multiplication and division. At the same time, despite their independence, these operations, upon closer examination, turn out to be interconnected. Such a relationship exists, for example, between addition and multiplication.

Number multiplication operation

There are three main elements involved in the multiplication operation. The first of these, which is commonly referred to as the first factor or multiplicand, is the number that will be subjected to the multiplication operation. The second, which is called the second factor, is the number by which the first factor will be multiplied. Finally, the result of the multiplication operation carried out is most often called the product.

It should be remembered that the essence of the multiplication operation is actually based on addition: for its implementation, it is necessary to add together a certain number of first factors, and the number of terms in this sum must be equal to the second factor. In addition to calculating the product of the two factors under consideration, this algorithm can also be used to check the resulting result.

An example of solving a multiplication task

Consider solutions to the multiplication problem. Suppose, according to the conditions of the task, it is necessary to calculate the product of two numbers, among which the first factor is 8, and the second is 4. In accordance with the definition of the multiplication operation, this actually means that you need to add the number 8 4 times. The result is 32 - this is the product considered numbers, that is, the result of their multiplication.

In addition, it must be remembered that the so-called commutative law applies to the multiplication operation, which establishes that changing the places of factors in the original example will not change its result. Thus, you can add the number 4 8 times, resulting in the same product - 32.

Multiplication table

It is clear that solving a large number of examples of the same type in this way is a rather tedious task. In order to facilitate this task, the so-called multiplication was invented. In fact, it is a list of products of integer positive single-digit numbers. Simply put, a multiplication table is a collection of results of multiplication between each other from 1 to 9. Once you have learned this table, you can no longer resort to multiplication whenever you need to solve an example for such prime numbers, but simply remember its result.

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Practically the entire course of mathematics is based on operations with positive and negative numbers. After all, as soon as we begin to study the coordinate line, numbers with plus and minus signs begin to meet us everywhere, in every new topic. There is nothing easier than adding ordinary positive numbers together, it is not difficult to subtract one from the other. Even arithmetic with two negative numbers is rarely a problem.

However, many people get confused in adding and subtracting numbers with different signs. Recall the rules by which these actions take place.

Addition of numbers with different signs

If to solve the problem we need to add a negative number "-b" to a certain number "a", then we need to act as follows.

• Let's take modules of both numbers - |a| and |b| - and compare these absolute values ​​with each other.
• Note which of the modules is larger and which is smaller, and subtract the smaller value from the larger value.
• We put before the resulting number the sign of the number whose modulus is greater.

This will be the answer. It can be put more simply: if in the expression a + (-b) the modulus of the number "b" is greater than the modulus of "a", then we subtract "a" from "b" and put a "minus" in front of the result. If the module "a" is greater, then "b" is subtracted from "a" - and the solution is obtained with a "plus" sign.

It also happens that the modules are equal. If so, then you can stop at this point - we are talking about opposite numbers, and their sum will always be zero.

Subtraction of numbers with different signs

We figured out the addition, now consider the rule for subtraction. It is also quite simple - and besides, it completely repeats a similar rule for subtracting two negative numbers.

In order to subtract from a certain number "a" - arbitrary, that is, with any sign - a negative number "c", you need to add to our arbitrary number "a" the number opposite to "c". For example:

• If “a” is a positive number, and “c” is negative, and “c” must be subtracted from “a”, then we write it like this: a - (-c) \u003d a + c.
• If “a” is a negative number, and “c” is positive, and “c” must be subtracted from “a”, then we write as follows: (- a) - c \u003d - a + (-c).

Thus, when subtracting numbers with different signs, we eventually return to the rules of addition, and when adding numbers with different signs, we return to the rules of subtraction. Remembering these rules allows you to solve problems quickly and easily.

the formation of knowledge about the rule for adding numbers with different signs, the ability to apply it in the simplest cases;

development of skills to compare, identify patterns, generalize;

education of a responsible attitude to educational work.

Equipment: multimedia projector, screen.

Lesson type: lesson learning new material.

DURING THE CLASSES

1. Organizational moment.

Stand up straight

They sat down quietly.

Now the bell has rung

Let's start our lesson.

Guys! Today we have guests at our lesson. Let's turn to them and smile at each other. So we start our lesson.

slide 2- The epigraph of the lesson: “He who does not notice anything does not study anything.

Whoever studies nothing is always whining and bored.

Roman Sef (children's writer)

Sweet 3 - I suggest you play the reverse game. Rules of the game: you need to divide the words into two groups: gain, lie, warmth, gave, truth, good, loss, took, evil, cold, positive, negative.

There are many contradictions in life. With their help, we define the surrounding reality. For our lesson, I need the latter: positive - negative.

What are we talking about in mathematics when we use these words? (About numbers.)

The great Pythagoras said: "Numbers rule the world." I propose to talk about the most mysterious numbers in science - numbers with different signs. - Negative numbers appeared in science as the opposite of positive ones. Their path to science was difficult, because even many scientists did not support the idea of ​​their existence.

What concepts and quantities do people measure with positive and negative numbers? (charges of elementary particles, temperature, losses, height and depth, etc.)

slide 4- Words opposite in meaning - antonyms (table).

2. Setting the topic of the lesson.

Slide 5 (work with the table) What numbers did you learn in previous lessons?
– What tasks related to positive and negative numbers can you perform?
- Attention to the screen. (Slide 5)
What numbers are in the table?
- Name the modules of numbers written horizontally.
– Specify the largest number, specify the number with the largest modulus.
- Answer the same questions for numbers written vertically.
– Do the largest number and the number with the largest modulus always coincide?
Find the sum of positive numbers, the sum of negative numbers.
- Formulate the rule for adding positive numbers and the rule for adding negative numbers.
What numbers are left to add?
- Can you put them together?
Do you know the rule for adding numbers with different signs?
- Formulate the topic of the lesson.
- What is your goal? .Think what we will do today? (Answers of children). Today we continue to get acquainted with positive and negative numbers. The topic of our lesson is “Addition of numbers with different signs.” And our goal: to learn without errors, to add numbers with different signs. Write down the number and topic of the lesson in your notebook.

3. Work on the topic of the lesson.

slide 6.– Using these concepts, find the results of adding numbers with different signs on the screen.
What numbers are the result of adding positive numbers, negative numbers?
What numbers are the result of adding numbers with different signs?
What determines the sign of the sum of numbers with different signs? (Slide 5)
– From the term with the largest modulus.
“It's like pulling a rope. The strongest wins.

Slide 7- Let's play. Imagine that you are pulling a rope. . Teacher. Rivals usually meet in competitions. And today we will visit several tournaments with you. The first thing that awaits us is the final of the tug-of-war contest. There are Ivan Minusov at number -7 and Petr Plusov at number +5. Who do you think will win? Why? So, Ivan Minusov won, he really turned out to be stronger than his opponent, and was able to drag him to his negative side by exactly two steps.

Slide 8.- . And now we will visit other competitions. Here is the final of the shooting competition. The best in this form were Minus Troikin with three balloons and Plus Chetverikov, who had four balloons in stock. And here guys, what do you think, who will be the winner?

Slide 9- Competitions have shown that the strongest wins. So when adding numbers with different signs: -7 + 5 = -2 and -3 + 4 = +1. Guys, how do numbers with different signs add up? Students offer their own options.

The teacher formulates the rule, gives examples.

10 + 12 = +(12 – 10) = +2

4 + 3,6 = -(4 – 3,6) = -0,4

Students during the demonstration can comment on the solution that appears on the slide.

slide 10- Teacher, let's play one more game "Sea battle". An enemy ship is approaching our coast, it must be knocked out and sunk. For this we have a gun. But to hit the target, you need to make accurate calculations. What will you see now. Ready? Then go ahead! Please do not be distracted, the examples change exactly after 3 seconds. Is everyone ready?

Students take turns going to the board and calculating the examples that appear on the slide. - List the steps to complete the task.

slide 11- Textbook work: p.180 p.33, read the rule for adding numbers with different signs. Comments on a rule.
- What is the difference between the rule proposed in the textbook and the algorithm you compiled? Consider examples in the textbook with commentary.

slide 12- Teacher-Now guys, let's have a experiment. But not chemical, but mathematical! Take the numbers 6 and 8, the plus and minus signs, and mix everything well. Let's get four examples-experience. Do them in your notebook. (two students decide on the wings of the board, then the answers are checked). What conclusions can be drawn from this experiment?(The role of signs). Let's do 2 more experiments. , but with your numbers (one person goes out to the board). Let's invent numbers for each other and check the results of the experiment (mutual verification).

slide 13 .- The rule is displayed on the screen in verse form. .

4. Fixing the topic of the lesson.

Slide 14 - Teacher - “All kinds of signs are needed, all kinds of signs are important!” Now, guys, we will divide with you into two teams. The boys will be in the team of Santa Claus, and the girls will be in the team of the Sun. Your task, without calculating the examples, is to determine in which of them negative answers will be obtained, and in which positive ones, and write out the letters of these examples in a notebook. Boys, respectively, are negative, and girls are positive (cards are issued from the application). A self-check is in progress.

Well done! You have an excellent sense for signs. This will help you complete the following task

Slide 15 - Physical education. -10, 0,15,18, -5,14,0, -8, -5, etc. (negative numbers - squat, positive numbers - pull up, jump up)

slide 16-Solve 9 examples on your own (task on cards in the application). 1 person at the board. Do a self test. Answers are displayed on the screen, students correct errors in their notebooks. Raise your hands who's right. (Marks are given only for good and excellent results)

Slide 17- Rules help us to solve examples correctly. Let's repeat them On the screen, the algorithm for adding numbers with different signs.

5. Organization of independent work.

Slide 18-FRontal work through the game "Guess the word"(task on cards in the application).

Slide 19 - You should get a score for the game - "five"

Slide 20-A now, attention. Homework. Homework shouldn't be difficult for you.

Slide 21 - The laws of addition in physical phenomena. Come up with examples of adding numbers with different signs and ask them to each other. What new did you learn? Have we achieved our goal?

Slide 22 - So the lesson is over, let's summarize now. Reflection. The teacher comments and grades the lesson.

Slide 23 - Thank you for your attention!

I wish you to have more positive and less negative in your life, I want to tell you guys, thank you for your active work. I think that you can easily apply what you have learned in subsequent lessons. The lesson is over. Thank you all very much. Goodbye!

In this lesson, we will learn what a negative number is and what numbers are called opposites. We will also learn how to add negative and positive numbers (numbers with different signs) and analyze several examples of adding numbers with different signs.

Look at this gear (see Fig. 1).

Rice. 1. Clock gear

This is not an arrow that directly shows the time and not a dial (see Fig. 2). But without this detail, the clock does not work.

Rice. 2. Gear inside the watch

What does the letter Y stand for? Nothing but the sound Y. But without it, many words will not “work”. For example, the word "mouse". So are negative numbers: they do not show any quantity, but without them the calculation mechanism would be much more difficult.

We know that addition and subtraction are equal operations, and they can be performed in any order. In the record in direct order, we can calculate: , but there is no way to start with subtraction, since we have not agreed yet, but what is .

It is clear that increasing the number by and then decreasing by means, as a result, a decrease by three. Why not designate this object and count it this way: to add is to subtract. Then .

The number can mean, for example, apples. The new number does not represent any real quantity. By itself, it does not mean anything, like the letter Y. It's just a new tool to simplify calculations.

Let's name new numbers negative. Now we can subtract a larger number from a smaller number. Technically, you still need to subtract the smaller number from the larger number, but put a minus sign in the answer: .

Let's look at another example: . You can do all the actions in a row:.

However, it is easier to subtract the third number from the first number, and then add the second number:

Negative numbers can be defined in another way.

For each natural number, for example , let's introduce a new number, which we denote , and determine that it has the following property: the sum of the number and is equal to : .

The number will be called negative, and the numbers and - opposite. Thus, we got an infinite number of new numbers, for example:

The opposite of number ;

The opposite of ;

The opposite of ;

The opposite of ;

Subtract the larger number from the smaller number: Let's add to this expression: . We got zero. However, according to the property: a number that adds up to five gives zero is denoted minus five:. Therefore, the expression can be denoted as .

Every positive number has a twin number, which differs only in that it is preceded by a minus sign. Such numbers are called opposite(See Fig. 3).

Rice. 3. Examples of opposite numbers

Properties of opposite numbers

1. The sum of opposite numbers is equal to zero:.

2. If you subtract a positive number from zero, then the result will be the opposite negative number: .

1. Both numbers can be positive, and we already know how to add them: .

2. Both numbers can be negative.

We have already covered the addition of such numbers in the previous lesson, but we will make sure that we understand what to do with them. For example: .

To find this sum, add opposite positive numbers and put a minus sign.

3. One number can be positive and another negative.

We can replace the addition of a negative number, if it is convenient for us, with the subtraction of a positive one:.

One more example: . Again, write the sum as a difference. You can subtract a larger number from a smaller number by subtracting a smaller number from a larger one, but putting a minus sign.

The terms can be interchanged: .

Another similar example: .

In all cases, the result is a subtraction.

To briefly formulate these rules, let's recall another term. Opposite numbers, of course, are not equal to each other. But it would be strange not to notice they have something in common. This common we called modulus of number. The modulus of opposite numbers is the same: for a positive number it is equal to the number itself, and for a negative one it is the opposite, positive. For example: , .

To add two negative numbers, add their modulus and put a minus sign:

To add a negative and a positive number, you need to subtract the smaller module from the larger module and put the sign of the number with the larger module:

Both numbers are negative, therefore, add their modules and put a minus sign:

Two numbers with different signs, therefore, from the modulus of the number (larger modulus) we subtract the modulus of the number and put a minus sign (the sign of the number with a larger modulus):

Two numbers with different signs, therefore, from the modulus of the number (larger modulus) we subtract the modulus of the number and put a minus sign (the sign of the number with a large modulus): .

Two numbers with different signs, therefore, subtract the module of the number from the module of the number (larger module) and put a plus sign (sign of the number with a large module): .

Positive and negative numbers have historically different roles.

First, we introduced natural numbers for counting objects:

Then we introduced other positive numbers - fractions, for counting non-integer quantities, parts: .

Negative numbers appeared as a tool to simplify calculations. There was no such thing that in life there were some quantities that we could not count, and we invented negative numbers.

That is, negative numbers did not originate from the real world. They just turned out to be so convenient that in some places they were used in life. For example, we often hear about negative temperatures. In this case, we never encounter a negative number of apples. What is the difference?

The difference is that in real life negative values ​​are only used for comparison, not for quantities. If a basement was equipped in the hotel and an elevator was launched there, then in order to leave the usual numbering of ordinary floors, a minus the first floor may appear. This minus one means only one floor below ground level (see Fig. 1).

Rice. 4. Minus the first and minus the second floors

A negative temperature is negative only compared to zero, which was chosen by the author of the scale, Anders Celsius. There are other scales, and the same temperature may no longer be negative there.

At the same time, we understand that it is impossible to change the starting point so that there are not five, but six apples. Thus, in life, positive numbers are used to determine quantities (apples, cake).

We also use them instead of names. Each phone could be given its own name, but the number of names is limited, and there are no numbers. That's why we use phone numbers. Also for ordering (century follows century).

Negative numbers in life are used in the last sense (minus the first floor below the zero and first floors)

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. "Gymnasium", 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. M.: Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. M.: ZSh MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A guide for students in grade 6 of the MEPhI correspondence school. M.: ZSh MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. M .: Education, Mathematics Teacher Library, 1989.

1. Math-prosto.ru ().
2. youtube().
3. School-assistant.ru ().
4. Allforchildren.ru ().

Homework

In this article, we will deal with adding numbers with different signs. Here we give a rule for adding a positive and a negative number, and consider examples of the application of this rule when adding numbers with different signs.

Page navigation.

Rule for adding numbers with different signs

Examples of adding numbers with different signs

Consider examples of adding numbers with different signs according to the rule discussed in the previous paragraph. Let's start with a simple example.

Example.

Add the numbers −5 and 2 .

Solution.

We need to add numbers with different signs. Let's follow all the steps prescribed by the rule of adding positive and negative numbers.

First, we find the modules of the terms, they are equal to 5 and 2, respectively.

The modulus of the number −5 is greater than the modulus of the number 2, so remember the minus sign.

It remains to put the memorized minus sign in front of the resulting number, we get −3. This completes the addition of numbers with different signs.

Answer:

(−5)+2=−3 .

To add rational numbers with different signs that are not integers, they should be represented as ordinary fractions (you can work with decimal fractions, if it is convenient). Let's take a look at this point in the next example.

Example.

Add a positive number and a negative number −1.25.

Solution.

Let's represent the numbers in the form of ordinary fractions, for this we will perform the transition from a mixed number to an improper fraction: , and translate the decimal fraction into an ordinary one: .

Now you can use the rule for adding numbers with different signs.

The modules of the added numbers are 17/8 and 5/4. For the convenience of performing further actions, we reduce the fractions to a common denominator, as a result we have 17/8 and 10/8.

Now we need to compare the common fractions 17/8 and 10/8. Since 17>10 , then . Thus, the term with a plus sign has a larger modulus, therefore, remember the plus sign.

Now we subtract the smaller one from the larger module, that is, we subtract fractions with the same denominators: .

It remains to put a memorized plus sign in front of the resulting number, we get, but - this is the number 7/8.