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 about 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 module "a" is greater, then "b" is subtracted from "a" - and the solution is obtained with a "plus" sign.

It also happens that the modules are equal. If so, then you can stop at this 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.

    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.Organizing time.

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!


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

Page navigation.

Rule for adding numbers with different signs

Examples of adding numbers with different signs

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

Example.

Add the numbers −5 and 2 .

Solution.

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

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

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

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

Answer:

(−5)+2=−3 .

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

Example.

Add a positive number and a negative number −1.25.

Solution.

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

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

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

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

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

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

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.

Lesson plan:

I. Organizational moment

Checking the individual homework.

II. Updating the basic knowledge of students

1. Mutual exercise. Control questions(steam room organizational form work - mutual check).
2. Oral work with commenting (group organizational form of work).
3. Independent work(individual organizational form of work, self-examination).

III. Lesson topic message

Group organizational form of work, putting forward a hypothesis, formulating a rule.

1. Fulfillment training tasks according to the textbook (group organizational form of work).
2. The work of strong students on cards (individual organizational form of work).

VI. Physical pause

IX. Homework.

Target: formation of the skill of adding numbers with different signs.

Tasks:

  • Formulate a rule for adding numbers with different signs.
  • Practice adding numbers with different signs.
  • Develop logical thinking.
  • To cultivate the ability to work in pairs, mutual respect.

Material for the lesson: cards for mutual training, tables of work results, individual cards for repetition and consolidation of material, a motto for individual work, cards with a rule.

DURING THE CLASSES

I. Organizing time

Let's start the lesson by checking individual homework. The motto of our lesson will be the words of Jan Amos Kamensky. At home, you should have thought about his words. How do you understand it? (“Consider unfortunate that day or that hour in which you did not learn anything new and did not add anything to your education”)
– How do you understand the words of the author? (If we do not learn anything new, do not receive new knowledge, then this day can be considered lost or unhappy. We must strive to acquire new knowledge).
– And today will not be unhappy because we will again learn something new.

II. Updating the basic knowledge of students

- To study new material, it is necessary to repeat the past.
At home there was a task - to repeat the rules and now you will show your knowledge by working with control questions.

(Test questions on the topic “Positive and negative numbers”)

Pair work. Mutual verification. The results of the work are noted in the table)

What are the numbers to the right of the origin called? Positive
What are the opposite numbers? Two numbers that differ from each other only in signs are called opposite numbers.
What is the modulus of a number? Distance from point A(a) before the start of the countdown, i.e. to the point O(0), called the modulus of a number
What is the modulus of a number? Brackets
What is the rule for adding negative numbers? To add two negative numbers, you need to add their modulus and put a minus sign
What are the numbers to the left of the origin called? Negative
What is the opposite of zero? 0
Can the absolute value of any number be negative? No. Distance is never negative
Name the rule for comparing negative numbers Of two negative numbers, the greater is the one whose modulus is less and less than the one whose modulus is greater
What is the sum of opposite numbers? 0

Answers to the questions "+" is correct, "-" is incorrect Evaluation criteria: 5 - "5"; 4 - "4"; 3 - "3"

1 2 3 4 5 Grade
Q/questions
Self/work
Ind/ work
Outcome

What questions were the most difficult?
What do you need to pass the test questions successfully? (Know the rules)

2. Oral work with commentary

– 45 + (– 45) = (– 90)
– 100 + (– 38) = (– 138)
– 3, 5 + (–2, 4) = (– 5,9)
– 17/70 + (– 26/70) = (– 43/70)
– 20 + (– 15) = (– 35)

– What knowledge did you need to solve 1-5 examples?

3. Independent work

– 86, 52 + (– 6, 3) = – 92,82
– 49/91 + (– 27/91) = – 76/91
– 76 + (– 99) = – 175
– 14 + (– 47) = – 61
– 123,5 + (– 25, 18) = – 148,68
6 + (– 10) =

(Self-test. Open during test answers)

Why did the last example give you a hard time?
- The sum of which numbers need to be found, and the sum of which numbers do we know how to find?

III. Lesson topic message

- Today in the lesson we will learn the rule of adding numbers with different signs. We will learn to add numbers with different signs. Self-study at the end of the lesson will show your progress.

IV. Learning new material

- Let's open notebooks, write down the date, class work, the topic of the lesson is "Addition of numbers with different signs."
- What is on the board? (Coordinate line)

- Prove that this is a coordinate line? (There is a reference point, a reference direction, a single segment)
- Now we will learn together to add numbers with different signs using a coordinate line.

(Explanation of students under the guidance of a teacher.)

- Let's find the number 0 on the coordinate line. The number 6 must be added to 0. We take 6 steps to the right of the origin, because the number 6 is positive (we put a colored magnet on the resulting number 6). We add the number (-10) to 6, take 10 steps to the left of the origin, because (- 10) is a negative number (put a colored magnet on the resulting number (- 4).)
- What was the answer? (- 4)
How did you get the number 4? (10 - 6)
Conclude: From the number with a large modulus, subtract the number with a smaller modulus.
- How did you get the minus sign in the answer?
Conclude: We took the sign of a number with a large module.
Let's write an example in a notebook:

6 + (–10) = – (10 – 6) = – 4
10 + (-3) = + (10 - 3) = 7 (Similarly solve)

Entry accepted:

6 + (– 10) = – (10 – 6) = – 4
10 + (– 3) = + (10 – 3) = 7

- Guys, you yourself have now formulated the rule for adding numbers with different signs. We will call your guesses hypothesis. You have done very important intellectual work. Like scientists put forward a hypothesis and discovered a new rule. Let's check your hypothesis with the rule (the sheet with the printed rule lies on the desk). Let's read in unison rule adding numbers with different signs

- The rule is very important! It allows you to add numbers of different signs without the help of a coordinate line.
- What's not clear?
- Where can you make a mistake?
- In order to correctly and without errors calculate tasks with positive and negative numbers, you need to know the rules.

V. Consolidation of the studied material

Can you find the sum of these numbers on the coordinate line?
- It is difficult to solve such an example with the help of a coordinate line, so we will use the rule you discovered when solving.
The task is written on the board:
Textbook - p. 45; No. 179 (c, d); No. 180 (a, b); No. 181 (b, c)
(A strong student works to reinforce this topic with an additional card.)

VI. Physical pause(Perform standing)

- A person has positive and negative qualities. Distribute these qualities on the coordinate line.
(Positive qualities are to the right of the reference point, negative qualities are to the left of the reference point.)
- If the quality is negative - clap once, positive - twice. Be careful!
– Kindness, anger, greed , mutual assistance, understanding, rudeness, and, of course, strength of will And striving for victory, which you will need now, since you have independent work ahead of you)
VII. Individual work followed by peer review

Option 1 Option 2
– 100 + (20) = – 100 + (30) =
100 + (– 20) = 100 + (– 30) =
56 + (– 28) = 73 + (– 28) =
4,61 + (– 2,2) = 5, 74 + (– 3,15) =
– 43 + 65 = – 43 + 35 =

Individual work (for strong students) with subsequent mutual verification

Option 1 Option 2
– 100 + (20) = – 100 + (30) =
100 + (– 20) = 100 + (– 30) =
56 + (– 28) = 73 + (– 28) =
4,61 + (– 2,2) = 5, 74 + (– 3,15) =
– 43 + 65 = – 43 + 35 =
100 + (– 28) = 100 + (– 39) =
56 + (– 27) = 73 + (– 24) =
– 4,61 + (– 2,22) = – 5, 74 + (– 3,15) =
– 43 + 68 = – 43 + 39 =

VIII. Summing up the lesson. Reflection

– I believe that you worked actively, diligently, participated in the discovery of new knowledge, expressed your opinion, now I can evaluate your work.
- Tell me, guys, what is more effective: to receive ready-made information or to think for yourself?
- What did we learn in the lesson? (Learned how to add numbers with different signs.)
Name the rule for adding numbers with different signs.
- Tell me, our lesson today was not in vain?
- Why? (Get new knowledge.)
Let's get back to the slogan. So Jan Amos Kamensky was right when he said: "Consider unfortunate the day or the hour in which you did not learn anything new and did not add anything to your education."

IX. Homework

Learn the rule (card), p.45, No. 184.
Individual task - how do you understand the words of Roger Bacon: “A person who does not know mathematics is not capable of any other sciences. Moreover, he is not even able to assess the level of his ignorance?