>>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. Radii of geographical parallels 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; V)

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 guidelines discussion programs Integrated Lessons

    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 competition. 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 event were Minus Troikin with three balloons and Plus Chetverikov, who has 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 next 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 - Addition laws in physical phenomena. Think of examples for 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!

Practically the entire course of mathematics is based on operations with positive and negative numbers. Indeed, 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 occur.

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 greater and which is smaller, and subtract from greater value lesser.
  • 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 modulus "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 place - 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.

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 amount, 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 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 arise from 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)

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