Rules for addition with minus. Adding and subtracting positive and negative numbers

Instructions

There are four types of mathematical operations: addition, subtraction, multiplication and division. Therefore, there will be four types of examples. 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 parentheses are opened, the “+” sign is changed to the opposite, then from the larger (modulo) number “6” the smaller one, “3,” is subtracted, after which the answer is assigned the larger sign, that is, “-”.
2) -3+6=3. This can be written according to the principle ("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 action of addition is replaced by subtraction, then the modules are summed up and the result is given a minus sign.

Subtraction.1) 8-(-5)=8+5=13. The parentheses are opened, the sign of the action is reversed, and an example of addition is obtained.
2) -9-3=-12. The elements of the example are added and get general 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 away from the answer.

Multiplication and division: When performing multiplication or division, the sign does not affect the operation itself. When multiplying or dividing numbers with the answer, a “minus” sign is assigned; if the numbers have 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 at home. How to correctly explain to a child the solution to examples of adding and subtracting multi-digit numbers? Let's try to figure this out.

You will need

1. Textbook on mathematics.
2. Paper.
3. Handle.

Instructions

Read the example. To do this, divide each multivalued into classes. Starting from the end of the number, count three digits at a time and put a dot (23.867.567). Let us remind you that the first three digits from the end of the number are to units, the next three are to class, then millions come. 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 below each other: units under units, tens under tens, hundreds under hundreds, etc.

Perform addition or subtraction. Start performing the action with units. Write down the result under the category with which you performed the action. If the result is number(), then we write the units in place of the answer, and add the number of tens to the units of the digit. 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 and perform the action.

Read the answer.

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Please note

Prohibit your child from using a calculator even to check the solution to an example. Addition is tested by subtraction, and subtraction is tested by addition.

Useful advice

If a child has a good grasp of the techniques of written calculations within 1000, then operations with multi-digit numbers, performed in an analogous manner, will not cause any difficulties.
Give your child a competition to see how many examples he can solve in 10 minutes. Such training will help automate computational techniques.

Multiplication is one of the four basic mathematical operations and underlies many more complex functions. 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 there are three main components involved in it. One of them is called the first factor and is a number that is subject to the multiplication operation. For this reason, it has a second, somewhat less common name - “multiplicable”. The second component of the multiplication operation is usually called the second factor: it represents 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 swapped: the result of the multiplication will not change. Finally, the third component of the multiplication operation, resulting from its result, is called the product.

Order of 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, 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 when calculating the desired product. It should be borne in mind that the verification necessarily assumes that the terms involved in the summation are identical and correspond to the first factor.

Solving multiplication examples

Thus, in order to solve the problem associated with the need to carry out multiplication, it may be enough to add the required number of first factors a given number of times. This method can be convenient for carrying out almost any calculations related to this operation. At the same time, in mathematics there are quite often standard ones that involve standard single-digit integers. In order to facilitate their calculation, the so-called multiplication system 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 facilitate the process of solving multiplication examples, based on the use of such numbers. However, for more complex options You will need to carry out this mathematical operation yourself.

Video on the topic

Sources:

Multiplication in 2019

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

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

Number multiplication operation

There are three main elements involved in the multiplication operation. The first of these, usually called the first factor or multiplicand, is the number that will be subject to the multiplication operation. The second, called the second factor, is the number by which the first factor will be multiplied. Finally, the result of the multiplication operation performed is most often called a product.

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

An example of solving a multiplication problem

Let's look at solutions to multiplication problems. 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 of the numbers in question, 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 the 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 to solve this way large number drawing examples of the same type 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 positive single-digit integers. Simply put, a multiplication table is a set of results of multiplying with each other from 1 to 9. Once you have learned this table, you can no longer resort to multiplication every time you need to solve an example for such simple numbers, but simply remember its result.

Video on the topic

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 look at several examples of adding numbers with different signs.

Look at this gear (see Fig. 1).

Rice. 1. Clock gear

This is not a hand that directly shows the time and not a dial (see Fig. 2). But without this part the clock does not work.

Rice. 2. Gear inside the clock

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 equivalent operations and can be performed in any order. In direct order, we can calculate: , but we can’t start with subtraction, since we haven’t yet agreed on what .

It is clear that increasing the number by and then decreasing by means ultimately decreasing by three. Why not designate this object and count like that: adding means subtracting. Then .

The number can mean, for example, an apple. 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 make calculations easier.

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

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

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

Negative numbers can be defined in another way.

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

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

The opposite of number ;

Subtract the larger number from the smaller number: . Let's add to this expression: . We got zero. However, according to the property: the number that adds zero to five 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 zero: .

2. If you subtract a positive number from zero, 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 already covered adding numbers like these in the previous lesson, but let's make sure we understand what to do with them. For example: .

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

3. One number can be positive and the other negative.

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

Another example: . Again we write the amount as the difference. You can subtract a larger number from a smaller number by subtracting a smaller number from a larger one, but using a minus sign.

We can swap the terms: .

Another similar example: .

In all cases, the result is a subtraction.

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

To add two negative numbers, you need to add their modules 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, we add their modules and put a minus sign:

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

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

Positive and negative numbers have historically had different roles.

First we introduced natural numbers to count objects:

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

Negative numbers appeared as a tool to simplify calculations. It was not like there were any quantities in life that we could not count, and we invented negative numbers.

That is, negative numbers did not arise from real world. They just turned out to be so convenient that in some places they found application in life. For example, we often hear about negative temperature. However, we never encounter a negative number of apples. What's the difference?

The difference is that in life, negative quantities are used only for comparison, but not for quantities. If a hotel has a basement and an elevator is installed there, then in order to maintain the usual numbering of regular floors, a minus first floor may appear. This first minus means only one floor below ground level (see Fig. 1).

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

A negative temperature is only negative 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 apples, but six. 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 are used in life in the last sense(minus the first floor below the zero and first floors)

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

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Homework

In this lesson we will learn adding and subtracting integers, as well as rules for their addition and subtraction.

Recall that integers are all positive and negative numbers, as well as the number 0. For example, the following numbers are integers:

−3, −2, −1, 0, 1, 2, 3

Positive numbers are easy, and. Unfortunately, the same cannot be said about negative numbers, which confuse many beginners with their minuses in front of each number. As practice shows, mistakes made due to negative numbers frustrate students the most.

Lesson content

Examples of adding and subtracting integers

The first thing you should learn is to add and subtract integers using a coordinate line. It is not at all necessary to draw a coordinate line. It is enough to imagine it in your thoughts and see where the negative numbers are located and where the positive ones are.

Let's consider the simplest expression: 1 + 3. The value of this expression is 4:

This example can be understood using a coordinate line. To do this, from the point where the number 1 is located, you need to move three steps to the right. As a result, we will find ourselves at the point where the number 4 is located. In the figure you can see how this happens:

The plus sign in the expression 1 + 3 tells us that we should move to the right in the direction of increasing numbers.

Example 2. Let's find the value of the expression 1 − 3.

The value of this expression is −2

This example can again be understood using a coordinate line. To do this, from the point where the number 1 is located, you need to move to the left three steps. As a result, we will find ourselves at the point where the negative number −2 is located. In the picture you can see how this happens:

The minus sign in the expression 1 − 3 tells us that we should move to the left in the direction of decreasing numbers.

In general, you need to remember that if addition is carried out, then you need to move to the right in the direction of increase. If subtraction is carried out, then you need to move to the left in the direction of decrease.

Example 3. Find the value of the expression −2 + 4

The value of this expression is 2

This example can again be understood using a coordinate line. To do this, from the point where the negative number −2 is located, you need to move four steps to the right. As a result, we will find ourselves at the point where the positive number 2 is located.

It can be seen that we have moved from the point where the negative number −2 is located to the right side by four steps, and ended up at the point where the positive number 2 is located.

The plus sign in the expression −2 + 4 tells us that we should move to the right in the direction of increasing numbers.

Example 4. Find the value of the expression −1 − 3

The value of this expression is −4

This example can again be solved using a coordinate line. To do this, from the point where the negative number −1 is located, you need to move to the left three steps. As a result, we will find ourselves at the point where the negative number −4 is located

It can be seen that we have moved from the point where the negative number −1 is located to left side three steps, and ended up at the point where the negative number −4 is located.

The minus sign in the expression −1 − 3 tells us that we should move to the left in the direction of decreasing numbers.

Example 5. Find the value of the expression −2 + 2

The value of this expression is 0

This example can be solved using a coordinate line. To do this, from the point where the negative number −2 is located, you need to move to the right two steps. As a result, we will find ourselves at the point where the number 0 is located

It can be seen that we have moved from the point where the negative number −2 is located to the right side by two steps and ended up at the point where the number 0 is located.

The plus sign in the expression −2 + 2 tells us that we should move to the right in the direction of increasing numbers.

Rules for adding and subtracting integers

To add or subtract integers, it is not at all necessary to imagine a coordinate line every time, much less draw it. It is more convenient to use ready-made rules.

When applying the rules, you need to pay attention to the sign of the operation and the signs of the numbers that need to be added or subtracted. This will determine which rule to apply.

Example 1. Find the value of the expression −2 + 5

Here a positive number is added to a negative number. In other words, numbers with different signs are added. −2 is a negative number, and 5 is a positive number. For such cases, the following rule applies:

To add numbers with different signs, you need to subtract the smaller module from the larger module, and before the resulting answer put the sign of the number whose module is larger.

So, let's see which module is bigger:

The modulus of the number 5 is greater than the modulus of the number −2. The rule requires subtracting the smaller one from the larger module. Therefore, we must subtract 2 from 5, and before the resulting answer put the sign of the number whose modulus is greater.

The number 5 has a larger modulus, so the sign of this number will be in the answer. That is, the answer will be positive:

−2 + 5 = 5 − 2 = 3

Usually written shorter: −2 + 5 = 3

Example 2. Find the value of the expression 3 + (−2)

Here, as in the previous example, numbers with different signs are added. 3 is a positive number, and −2 is a negative number. Note that −2 is enclosed in parentheses to make the expression clearer. This expression is much easier to understand than the expression 3+−2.

So, let's apply the rule for adding numbers with different signs. As in the previous example, subtract the smaller module from the larger module and before the answer we put the sign of the number whose module is greater:

3 + (−2) = |3| − |−2| = 3 − 2 = 1

The modulus of the number 3 is greater than the modulus of the number −2, so we subtracted 2 from 3, and before the resulting answer we put the sign of the number whose modulus is greater. The number 3 has a larger modulus, which is why the sign of this number is included in the answer. That is, the answer is positive.

Usually written shorter 3 + (−2) = 1

Example 3. Find the value of the expression 3 − 7

In this expression, a larger number is subtracted from a smaller number. In such a case the following rule applies:

To subtract a larger number from a smaller number, you need to subtract the smaller number from the larger number, and put a minus in front of the resulting answer.

3 − 7 = 7 − 3 = −4

There is a slight catch to this expression. Let us remember that the equal sign (=) is placed between quantities and expressions when they are equal to each other.

The value of the expression 3 − 7, as we learned, is −4. This means that any transformations that we will perform in this expression must be equal to −4

But we see that at the second stage there is an expression 7 − 3, which is not equal to −4.

To correct this situation, you need to put the expression 7 − 3 in brackets and put a minus in front of this bracket:

3 − 7 = − (7 − 3) = − (4) = −4

In this case, equality will be observed at each stage:

After the expression has been calculated, the parentheses can be removed, which is what we did.

So to be more precise the solution should look like this:

3 − 7 = − (7 − 3) = − (4) = − 4

This rule can be written using variables. It will look like this:

a − b = − (b − a)

A large number of parentheses and operation signs can complicate the solution of a seemingly simple problem, so it is more advisable to learn how to write such examples briefly, for example 3 − 7 = − 4.

In fact, adding and subtracting integers comes down to nothing more than addition. This means that if you need to subtract numbers, this operation can be replaced by addition.

So, let's get acquainted with the new rule:

Subtracting one number from another means adding to the minuend a number that is opposite to the one being subtracted.

For example, consider the simplest expression 5 − 3. At the initial stages of studying mathematics, we put an equal sign and wrote down the answer:

But now we are progressing in our study, so we need to adapt to the new rules. The new rule says that subtracting one number from another means adding to the minuend the same number as the subtrahend.

Let's try to understand this rule using the example of expression 5 − 3. The minuend in this expression is 5, and the subtrahend is 3. The rule says that in order to subtract 3 from 5, you need to add to 5 a number that is the opposite of 3. The opposite of the number 3 is −3. Let's write a new expression:

And we already know how to find meanings for such expressions. This is the addition of numbers with different signs, which we looked at earlier. To add numbers with different signs, we subtract the smaller module from the larger module, and before the resulting answer we put the sign of the number whose module is greater:

5 + (−3) = |5| − |−3| = 5 − 3 = 2

The modulus of the number 5 is greater than the modulus of the number −3. Therefore, we subtracted 3 from 5 and got 2. The number 5 has a larger modulus, so we put the sign of this number in the answer. That is, the answer is positive.

At first, not everyone is able to quickly replace subtraction with addition. This is because positive numbers are written without the plus sign.

For example, in the expression 3 − 1, the minus sign indicating subtraction is an operation sign and does not refer to one. One in this case is a positive number, and it has its own plus sign, but we don’t see it, since a plus is not written before positive numbers.

Therefore, for clarity, this expression can be written as follows:

(+3) − (+1)

For convenience, numbers with their own signs are placed in brackets. In this case, replacing subtraction with addition is much easier.

In the expression (+3) − (+1), the number being subtracted is (+1), and the opposite number is (−1).

Let's replace subtraction with addition and instead of the subtrahend (+1) we write the opposite number (−1)

(+3) − (+1) = (+3) + (−1)

Further calculations will not be difficult.

(+3) − (+1) = (+3) + (−1) = |3| − |−1| = 3 − 1 = 2

At first glance, it might seem like there’s no point in these extra movements if you can use the good old method to put an equal sign and immediately write down the answer 2. In fact, this rule will help us out more than once.

Let's solve the previous example 3 − 7 using the subtraction rule. First, let's bring the expression to a clear form, assigning each number its own signs.

Three has a plus sign because it is a positive number. The minus sign indicating subtraction does not apply to seven. Seven has a plus sign because it is a positive number:

Let's replace subtraction with addition:

(+3) − (+7) = (+3) + (−7)

Further calculation is not difficult:

(+3) − (−7) = (+3) + (-7) = −(|−7| − |+3|) = −(7 − 3) = −(4) = −4

Example 7. Find the value of the expression −4 − 5

Again we have a subtraction operation. This operation must be replaced by addition. To the minuend (−4) we add the number opposite to the subtrahend (+5). The opposite number for the subtrahend (+5) is the number (−5).

(−4) − (+5) = (−4) + (−5)

We have come to a situation where we need to add negative numbers. For such cases, the following rule applies:

To add negative numbers, you need to add their modules and put a minus in front of the resulting answer.

So, let’s add up the modules of numbers, as the rule requires us to do, and put a minus in front of the resulting answer:

(−4) − (+5) = (−4) + (−5) = |−4| + |−5| = 4 + 5 = −9

The entry with modules must be enclosed in brackets and a minus sign must be placed before these brackets. This way we will provide a minus that should appear before the answer:

(−4) − (+5) = (−4) + (−5) = −(|−4| + |−5|) = −(4 + 5) = −(9) = −9

The solution for this example can be written briefly:

−4 − 5 = −(4 + 5) = −9

or even shorter:

−4 − 5 = −9

Example 8. Find the value of the expression −3 − 5 − 7 − 9

Let's bring the expression to a clear form. Here, all numbers except −3 are positive, so they will have plus signs:

(−3) − (+5) − (+7) − (+9)

Let's replace subtractions with additions. All minuses, except the minus in front of the three, will change to pluses, and all positive numbers will change to the opposite:

(−3) − (+5) − (+7) − (+9) = (−3) + (−5) + (−7) + (−9)

Now let's apply the rule for adding negative numbers. To add negative numbers, you need to add their modules and put a minus in front of the resulting answer:

(−3) − (+5) − (+7) − (+9) = (−3) + (−5) + (−7) + (−9) =

= −(|−3| + |−5| + |−7| + |−9|) = −(3 + 5 + 7 + 9) = −(24) = −24

The solution to this example can be written briefly:

−3 − 5 − 7 − 9 = −(3 + 5 + 7 + 9) = −24

or even shorter:

−3 − 5 − 7 − 9 = −24

Example 9. Find the value of the expression −10 + 6 − 15 + 11 − 7

Let's bring the expression to a clear form:

(−10) + (+6) − (+15) + (+11) − (+7)

There are two operations here: addition and subtraction. We leave addition unchanged, and replace subtraction with addition:

(−10) + (+6) − (+15) + (+11) − (+7) = (−10) + (+6) + (−15) + (+11) + (−7)

Observing, we will perform each action in turn, based on the previously learned rules. Entries with modules can be skipped:

First action:

(−10) + (+6) = − (10 − 6) = − (4) = − 4

Second action:

(−4) + (−15) = − (4 + 15) = − (19) = − 19

Third action:

(−19) + (+11) = − (19 − 11) = − (8) = −8

Fourth action:

(−8) + (−7) = − (8 + 7) = − (15) = − 15

Thus, the value of the expression −10 + 6 − 15 + 11 − 7 is −15

Note. It is not at all necessary to bring the expression into a understandable form by enclosing numbers in parentheses. When habituation to negative numbers occurs, this step can be skipped because it is time-consuming and can be confusing.

So, to add and subtract integers, you need to remember the following rules:

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In this article we will look in detail at how it is done addition of integers. First we will form general idea about the addition of integers, and let's see what the addition of integers on a coordinate line is. This knowledge will help us formulate rules for adding positive, negative, and integers with different signs. Here we will examine in detail the application of addition rules when solving examples and learn how to check the results obtained. At the end of the article, we will talk about adding three or more integers.

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Understanding addition of integers

Here are examples of adding integer opposite numbers. The sum of the numbers −5 and 5 is zero, the sum of 901+(−901) is zero, and the result of adding the opposite integers 1,567,893 and −1,567,893 is also zero.

Addition of an arbitrary integer and zero

Let's use the coordinate line to understand what is the result of adding two integers, one of which is zero.

Adding an arbitrary integer a to zero means moving unit segments from the origin to a distance a. Thus, we find ourselves at the point with coordinate a. Therefore, the result of adding zero and an arbitrary integer is the added integer.

On the other hand, adding zero to an arbitrary integer means moving from the point whose coordinate is specified by a given integer to a distance of zero. In other words, we will remain at the starting point. Therefore, the result of adding an arbitrary integer and zero is the given integer.

So, the sum of two integers, one of which is zero, is equal to the other integer. In particular, zero plus zero is zero.

Let's give a few examples. The sum of the integers 78 and 0 is 78; the result of adding zero and −903 is −903 ; also 0+0=0 .

Checking the result of addition

After adding two integers, it is useful to check the result. We already know that to check the result of adding two natural numbers, we need to subtract any of the terms from the resulting sum, and this should result in another term. Checking the result of adding integers performed similarly. But subtracting integers comes down to adding to the minuend the number opposite to the one being subtracted. Thus, to check the result of adding two integers, you need to add to the resulting sum the number opposite to any of the terms, which should result in another term.

Let's look at examples of checking the result of adding two integers.

Example.

When adding two integers 13 and −9, the number 4 was obtained, check the result.

Solution.

Let's add to the resulting sum 4 the number −13, opposite to the term 13, and see if we get another term −9.

So, let's calculate the sum 4+(−13) . This is the sum of integers with opposite signs. The modules of the terms are 4 and 13, respectively. The term whose modulus is greater has a minus sign, which we remember. Now subtract from the larger module and subtract the smaller one: 13−4=9. All that remains is to put the remembered minus sign in front of the resulting number, we have −9.

When checking, we received a number equal to another term, therefore, the original sum was calculated correctly.−19. Since we received a number equal to another term, the addition of the numbers −35 and −19 was performed correctly.

Adding three or more integers

Up to this point we have talked about adding two integers. In other words, we considered sums consisting of two terms. However, the combinatory property of adding integers allows us to uniquely determine the sum of three, four, or more integers.

Based on the properties of addition of integers, we can state that the sum of three, four, and so on numbers does not depend on the way the parentheses are placed indicating the order in which actions are performed, as well as on the order of the terms in the sum. We substantiated these statements when we talked about the addition of three or more natural numbers. For integers, all reasoning is completely the same, and we will not repeat ourselves.0+(−101) +(−17)+5 . After this, placing the parentheses in any acceptable way, we will still get the number −113.

Answer:

5+(−17)+0+(−101)=−113 .

References.

Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.

developing 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;

fostering a responsible attitude towards educational work.

Equipment: multimedia projector, screen.

Lesson type: lesson of learning new material.

PROGRESS OF THE LESSON

1. Organizational moment.

Stand up straight

They sat down quietly.

The bell has now rung,

Let's start our lesson.

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

Slide 2- Epigraph of the lesson: “He who does not notice anything does not study anything.

He who doesn’t study anything is always whining and bored.”

Roman Sef (children's writer)

Slad 3 - I suggest playing the game “On the contrary”. Rules of the game: you need to divide the words into two groups: win, 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 last one: 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 numbers. 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 with opposite meanings are antonyms (table).

2. Setting the topic of the lesson.

Slide 5 (working with a table)– What numbers were studied in previous lessons?
– What tasks related to positive and negative numbers can you perform?
– Attention to the screen. (Slide 5)
– What numbers are presented in the table?
– Name the modules of numbers written horizontally.
– Indicate the largest number, indicate the number with the largest modulus.
– Answer the same questions for numbers written vertically.
– Do the largest number and the number with the largest absolute value always coincide?
– Find the amount 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?
– Do you know how to fold them?
– Do you know the rule for adding numbers with different signs?
– Formulate the topic of the lesson.
– What goal will you set for yourself? .Think about what we will do today? (Children's answers). Today we continue to get acquainted with positive and negative numbers. The topic of our lesson is “Adding numbers with different signs.” Our goal is to learn how to add numbers with different signs without errors. Write down the date 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 and 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 a tug of war. The strongest wins.

Slide 7- Let's play. Imagine that you are in a tug of war. . 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 competition. Meet Ivan Minusov at number -7 and Petr Plyusov 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 exactly two steps.

Slide 8.- . Now let's go to other competitions. The final of the shooting competition is before you. The best in this event were Minus Troikin with three balloons and Plus Chetverikov, who has four in stock balloon. And here guys, who do you think will be the winner?

Slide 9- The competitions showed that the strongest wins. So it is 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 and gives examples.

10 + 12 = +(12 – 10) = +2

4 + 3,6 = -(4 – 3,6) = -0,4

During the demonstration, students can comment on the solution appearing on the slide.

Slide 10- Teacher, let’s play another game “Battleship”. 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. Which ones you will see now. Are you ready? Then go ahead! Please do not be distracted, the examples change exactly after 3 seconds. Is everyone ready?

Students take turns coming to the board and calculating the examples that appear on the slide. – Name the stages of completing the task.

Slide 11- Work according to the textbook: p. 180 p. 33, read the rule for adding numbers with different signs. Comments on the rule.
– What is the difference between the rule proposed in the textbook and the algorithm you compiled? Consider the examples in the textbook with commentary.

Slide 12- Teacher - Now guys, let's conduct experiment. But not chemical, but mathematical! Let's take the numbers 6 and 8, plus and minus signs and mix everything well. Let's get four experimental examples. Do them in your notebook. (two students solve 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 conduct 2 more experiments , but with your numbers (1 person at a time goes to the board). Let's come up with numbers for each other and check the results of the experiment (mutual check).

Slide 13 .- The rule is displayed on the screen in poetic form .

4. Reinforcing 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 you into two teams. Boys will be on Santa Claus's team, and girls will be on Sunny's team. Your task, without calculating the examples, is to determine which of them will have negative answers and which will have positive answers and write down the letters of these examples in a notebook. Boys are respectively negative, and girls are positive (cards from the application are issued). A self-test is being carried out.

Well done! Your sense of signs is excellent. This will help you complete the next task

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

Slide 16-Solve 9 examples yourself (task on cards in the app). 1 person at the board. Do a self-test. The answers are displayed on the screen, and students correct mistakes in their notebooks. Raise your hands if you have it right. (Marks are given only for good and excellent result)

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

5.Organization of independent work.

Slide 18 -Fonline work through the game “Guess the word”(task on cards in the appendix).

Slide 19 - The score for the game should be “A”

Slide 20 -A now, attention. Homework. Homework should not cause you any difficulties.

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

Slide 22 - That's the end of the lesson, let's sum it up 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 the acquired knowledge in subsequent lessons. The lesson is over. Thank you all very much. Goodbye!