How to subtract numbers with negative signs. Addition of numbers with different signs

Are you here: Family and relationships

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 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, 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 for 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 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 - Fizkulminutka. -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!

Addition of negative numbers.

The sum of negative numbers is a negative number. The module of the sum is equal to the sum of the modules of the terms.

Let's see why the sum of negative numbers will also be a negative number. The coordinate line will help us with this, on which we will perform the addition of the numbers -3 and -5. Let's mark a point on the coordinate line corresponding to the number -3.

To the number -3 we need to add the number -5. Where do we go from the point corresponding to the number -3? That's right, to the left! For 5 single segments. We mark the point and write the number corresponding to it. This number is -8.

So, when adding negative numbers using a coordinate line, we are always to the left of the reference point, therefore, it is clear that the result of adding negative numbers is also a negative number.

Note. We added the numbers -3 and -5, i.e. found the value of the expression -3+(-5). Usually, when adding rational numbers, they simply write down these numbers with their signs, as if listing all the numbers that need to be added. Such a notation is called an algebraic sum. Apply (in our example) record: -3-5=-8.

Example. Find the sum of negative numbers: -23-42-54. (Agree that this entry is shorter and more convenient like this: -23+(-42)+(-54))?

We decide according to the rule of adding negative numbers: we add the modules of the terms: 23+42+54=119. The result will be with a minus sign.

They usually write it down like this: -23-42-54 \u003d -119.

Addition of numbers with different signs.

The sum of two numbers with different signs has the sign of the addend with a large modulus. To find the modulus of the sum, you need to subtract the smaller modulus from the larger modulus.

Let's perform the addition of numbers with different signs using the coordinate line.

1) -4+6. It is required to add the number -4 to the number 6. We mark the number -4 with a point on the coordinate line. The number 6 is positive, which means that from the point with coordinate -4 we need to go to the right by 6 unit segments. We ended up to the right of the origin (from zero) by 2 unit segments.

The result of the sum of the numbers -4 and 6 is the positive number 2:

— 4+6=2. How could you get the number 2? Subtract 4 from 6, i.e. subtract the smaller one from the larger one. The result has the same sign as the term with a large modulus.

2) Let's calculate: -7+3 using the coordinate line. We mark the point corresponding to the number -7. We go to the right by 3 unit segments and get a point with coordinate -4. We were and remained to the left of the origin: the answer is a negative number.

— 7+3=-4. We could get this result as follows: we subtracted the smaller one from the larger module, i.e. 7-3=4. As a result, the sign of the term with a larger module was set: |-7|>|3|.

Examples. Calculate: a) -4+5-9+2-6-3; b) -10-20+15-25.

In this lesson we will learn addition and subtraction of whole numbers, 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, this cannot be said about negative numbers, which confuse many beginners with their minuses before each digit. As practice shows, mistakes made due to negative numbers upset students the most.

Lesson content

Integer addition and subtraction examples

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

Consider the simplest expression: 1 + 3. The value of this expression is 4:

This example can be understood using the 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 the coordinate line. To do this, from the point where the number 1 is located, you need to move three steps to the left. As a result, we will find ourselves at the point where the negative number −2 is located. The figure shows 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, we must remember that if addition is carried out, then we 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 the 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 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 three steps to the left. 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 the left by 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 two steps to the right. 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 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, let alone 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 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, the addition of numbers with different signs is carried out. −2 is negative and 5 is positive. 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 put the sign of the number whose module is greater in front of the answer.

So, let's see which module is larger:

The modulus of 5 is greater than the modulus of −2. The rule requires subtracting the smaller from the larger module. Therefore, we must subtract 2 from 5, and before the received 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, the addition of numbers with different signs is carried out. 3 is positive and -2 is negative. Note that the number -2 is enclosed in parentheses to make the expression clearer. This expression is much easier to understand than the expression 3+−2.

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

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 put the sign of the greater modulus number before the answer. The number 3 has a larger module, so the sign of this number is put in the answer. That is, the answer is yes.

Usually written shorter 3 + (−2) = 1

Example 3 Find the value of the expression 3 − 7

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

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

3 − 7 = 7 − 3 = −4

There is a slight snag in this expression. Recall that the equal sign (=) is placed between values 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 the expression 7 − 3 is located at the second stage, which is not equal to −4.

To correct this situation, the expression 7 − 3 must be put in brackets and put a minus before this bracket:

3 − 7 = − (7 − 3) = − (4) = −4

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

After the expression is evaluated, the brackets can be removed, which 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 brackets and operation signs can complicate the solution of a seemingly very simple task, so it is more expedient to learn how to write such examples briefly, for example 3 − 7 = − 4.

In fact, the addition and subtraction of integers is reduced to just addition. This means that if you want to subtract numbers, this operation can be replaced by addition.

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

To subtract one number from another means to add to the minuend a number that will be the opposite of the subtracted one.

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 learning, so we need to adapt to the new rules. The new rule says that to subtract one number from another means to add to the minuend a number that will be subtracted.

Using the expression 5 − 3 as an example, let's try to understand this rule. What is being reduced in this expression is 5, and what is being subtracted is 3. The rule says that in order to subtract 3 from 5, you need to add to 5 a number that will be opposite to 3. The opposite number for the number 3 is −3. We write a new expression:

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

5 + (−3) = |5| − |−3| = 5 − 3 = 2

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

At first, not everyone succeeds in quickly replacing subtraction with addition. This is due to the fact that positive numbers are written without a plus sign.

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

And so, for clarity, this expression can be written as follows:

(+3) − (+1)

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

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

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

(+3) − (+1) = (+3) + (−1)

Further calculation will not be difficult.

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

At first glance, it would seem what is the point of these extra gestures, 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, placing each number with its signs.

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

Before us is the subtraction operation again. 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 received answer.

So, let's add the modules of numbers, as the rule requires us to, and put a minus in front of the received answer:

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

The entry with modules must be enclosed in brackets and put a minus before these brackets. So we provide a minus, which should come before the answer:

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

The solution for this example can be written shorter:

−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 the number −3 are positive, so they will have plus signs:

(−3) − (+5) − (+7) − (+9)

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

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

Now 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 received 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 shorter:

−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. Addition is left unchanged, and subtraction is replaced by addition:

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

Observing, we will perform each action in turn, based on the previously studied 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 necessary to bring the expression to a clear form by enclosing numbers in brackets. When getting used to negative numbers, this action can be skipped, as it takes time and can be confusing.

So, for adding and subtracting integers, you need to remember the following rules:

Join our new Vkontakte group and start receiving notifications of new lessons

>>Math: Adding numbers with different signs

33. Addition of numbers with different signs

If the air temperature was equal to 9 °С, and then it changed by -6 °С (i.e., decreased by 6 °С), then it became equal to 9 + (- 6) degrees (Fig. 83).

To add the numbers 9 and - 6 with the help, you need to move point A (9) to the left by 6 unit segments (Fig. 84). We get point B (3).

Hence, 9+(- 6) = 3. The number 3 has the same sign as the term 9, and its module is equal to the difference between the modules of the terms 9 and -6.

Indeed, |3| =3 and |9| - |- 6| == 9 - 6 = 3.

If the same air temperature of 9 °С changed by -12 °С (i.e., decreased by 12 °С), then it became equal to 9 + (-12) degrees (Fig. 85). Adding the numbers 9 and -12 using the coordinate line (Fig. 86), we get 9 + (-12) \u003d -3. The number -3 has the same sign as the term -12, and its modulus is equal to the difference between the modules of the terms -12 and 9.

Indeed, | - 3| = 3 and | -12| - | -9| \u003d 12 - 9 \u003d 3.

To add two numbers with different signs:

1) subtract the smaller one from the larger module of terms;

2) put in front of the resulting number the sign of the term, the modulus of which is greater.

Usually, the sign of the sum is first determined and written down, and then the difference of the modules is found.

For example:

1) 6,1+(- 4,2)= +(6,1 - 4,2)= 1,9,
or shorter than 6.1+(-4.2) = 6.1 - 4.2 = 1.9;

When adding positive and negative numbers, you can use calculator. To enter a negative number into the calculator, you must enter the modulus of this number, then press the "sign change" key |/-/|. For example, to enter the number -56.81, you must press the keys in sequence: | 5 |, | 6 |, | ¦ |, | 8 |, | 1 |, |/-/|. Operations on numbers of any sign are performed on a microcalculator in the same way as on positive numbers.

For example, the sum -6.1 + 3.8 is calculated from program

? The numbers a and b have different signs. What sign will the sum of these numbers have if the larger modulus has a negative number?

if the smaller modulus has a negative number?

if the larger modulus has a positive number?

if the smaller modulus has a positive number?

Formulate a rule for adding numbers with different signs. How to enter a negative number into a microcalculator?

To 1045. The number 6 was changed to -10. On which side of the origin is the resulting number? How far from the origin is it? What is equal to sum 6 and -10?

1046. The number 10 was changed to -6. On which side of the origin is the resulting number? How far from the origin is it? What is the sum of 10 and -6?

1047. The number -10 was changed to 3. On which side from the origin is the resulting number? How far from the origin is it? What is the sum of -10 and 3?

1048. The number -10 was changed to 15. On which side of the origin is the resulting number? How far from the origin is it? What is the sum of -10 and 15?

1049. In the first half of the day the temperature changed by - 4 °C, and in the second - by + 12 °C. By how many degrees did the temperature change during the day?

1050. Perform addition:

1051. Add:

a) to the sum of -6 and -12 the number 20;
b) to the number 2.6 the sum is -1.8 and 5.2;
c) to the sum of -10 and -1.3 the sum of 5 and 8.7;
d) to the sum of 11 and -6.5 the sum of -3.2 and -6.

1052. Which of the numbers 8; 7.1; -7.1; -7; -0.5 is the root equations- 6 + x \u003d -13.1?

1053. Guess the root of the equation and check:

a) x + (-3) = -11; c) m + (-12) = 2;
b) - 5 + y=15; d) 3 + n = -10.

1054. Find the value of the expression:

1055. Perform actions with the help of a microcalculator:

a) - 3.2579 + (-12.308); d) -3.8564+ (-0.8397) +7.84;
b) 7.8547+ (- 9.239); e) -0.083 + (-6.378) + 3.9834;
c) -0.00154 + 0.0837; f) -0.0085+ 0.00354+ (-0.00921).

P 1056. Find the value of the sum:

1057. Find the value of the expression:

1058. How many integers are located between numbers:

a) 0 and 24; b) -12 and -3; c) -20 and 7?

1059. Express the number -10 as the sum of two negative terms so that:

a) both terms were integers;
b) both terms were decimal fractions;
c) one of the terms was a regular ordinary shot.

1060. What is the distance (in unit segments) between the points of the coordinate line with coordinates:

a) 0 and a; b) -a and a; c) -a and 0; d) a and -za?

M 1061. The radii of the geographic parallels of the earth's surface, on which the cities of Athens and Moscow are located, are respectively 5040 km and 3580 km (Fig. 87). How much shorter is the Moscow parallel than the Athens parallel?

1062. Make an equation for solving the problem: “A field with an area of 2.4 hectares was divided into two sections. Find square each section, if it is known that one of the sections:

a) 0.8 ha more than the other;
b) 0.2 ha less than the other;
c) 3 times more than the other;
d) 1.5 times less than the other;
e) constitutes another;
f) is 0.2 of another;
g) is 60% of the other;
h) is 140% of the other.”

1063. Solve the problem:

1) On the first day, the travelers traveled 240 km, on the second day 140 km, on the third day they traveled 3 times more than on the second, and on the fourth day they rested. How many kilometers did they drive on the fifth day if they averaged 230 kilometers a day in 5 days?

2) Father's monthly income is 280 rubles. The daughter's scholarship is 4 times less. How much does a mother earn per month if there are 4 people in the family, the youngest son is a schoolboy and each has an average of 135 rubles?

1064. Do the following:

1) (2,35 + 4,65) 5,3:(40-2,9);

2) (7,63-5,13) 0,4:(3,17 + 6,83).

1066. Express as the sum of two equal terms each of the numbers:

1067. Find the value a + b if:

a) a = -1.6, b = 3.2; b) a = - 2.6, b = 1.9; in)

1068. There were 8 apartments on one floor of a residential building. 2 apartments had a living area of 22.8 m 2, 3 apartments - 16.2 m 2 each, 2 apartments - 34 m 2 each. What living area did the eighth apartment have if on this floor, on average, each apartment had 24.7 m 2 of living space?

1069. There were 42 wagons in the freight train. There were 1.2 times more covered wagons than platforms, and the number of tanks was equal to the number of platforms. How many wagons of each type were in the train?

1070. Find the value of the expression

N.Ya.Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school

Mathematics planning, textbooks and books online, courses and tasks in mathematics for grade 6 download

Lesson content lesson summary support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-examination workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures graphics, tables, schemes humor, anecdotes, jokes, comics parables, sayings, crossword puzzles, quotes Add-ons abstracts articles chips for inquisitive cheat sheets textbooks basic and additional glossary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in the textbook elements of innovation in the lesson replacing obsolete knowledge with new ones Only for teachers perfect lessons calendar plan for the year methodological recommendations of the discussion program Integrated Lessons

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 the brackets are opened, 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.

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.

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.