Lesson plan:
Checking individual homework.
II. Updating the basic knowledge of students
1. Mutual exercise. Control questions (pair organizational form of work - mutual verification).
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 of 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 for successful delivery control questions? (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? (- four)
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?
"Addition of numbers with different signs" - Mathematics textbook Grade 6 (Vilenkin)
Short description:
In this section, you will learn the rules for adding numbers with different signs: that is, learn how to add negative and positive numbers.
You already know how to add them on a coordinate line, but in each example you won’t draw a line and count along it? Therefore, you need to learn how to add without it.
Let's try with you to add a negative number to a positive number, for example add eight minus six: 8+(-6). You already know that adding a negative number causes the original number to decrease by the value of the negative number. This means that eight must be reduced by six, that is, six should be subtracted from eight: 8-6=2, it turns out two. In this example, everything seems to be clear, we subtract six from eight.
And if we take this example: add a positive number to a negative number. For example, minus eight add six: -8+6. The essence remains the same: we reduce the positive number by the value of the negative, we get six subtracting eight will be minus two: -8+6=-2.
As you noticed, both in the first and in the second example, subtraction is performed with numbers. Why? Because they have different signs (plus and minus). In order not to make mistakes when adding numbers with different signs, you should perform the following algorithm of actions:
1. find modules of numbers;
2. subtract the smaller module from the larger module;
3. before the result, put a number sign with a large modulus (usually only a minus sign is put, and a plus sign is not put).
If you add numbers with different signs, following this algorithm, then you will have much less chance of making a mistake.
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).
Rice. 83
To add the numbers 9 and -6 using the coordinate line, you need to move the point A (9) to the left by 6 unit segments (Fig. 84). We get point B(3).
Rice. 84
Hence, 9 + (-6) = 3. The number 3 has the same sign as the term 9, and its modulus 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).
Rice. 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.
Rice. 86
Indeed, |-3| = 3 and |-12| - |-9| = 12 - 9 = 3.
Usually, the sign of the sum is first determined and written down, and then the difference of the modules is found.
For example:
When adding positive and negative numbers, you can use a calculator. To enter a negative number into the microcalculator, 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: . 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 by the program
In short, this program is written like this: 
.
Questions for self-examination
- 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?
Do the exercises
1061. The number 6 was changed to -10. On which side from the origin is the resulting number? How far from the origin is it? What is the sum of 6 and -10?
1062. The number 10 was changed to -6. On which side from the origin is the resulting number? How far from the origin is it? What is the sum of 10 and -6?
1063. The number -10 was changed to 3. On which side of the origin is the resulting number? How far from the origin is it? What is the sum of -10 and 3?
1064. 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?
1065. In the first half of the day, the temperature changed by -4°С, and in the second - by +12°С. By how many degrees did the temperature change during the day?
1066. Perform addition:
- a) 26 + (-6);
- b) -70 + 50;
- c) -17 + 30;
- d) 80 + (-120);
- e) -6.3 + 7.8;
- f) -9 + 10.2;
- g) 1 + (-0.39);
- h) 0.3 + (-1.2);
1067. 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.
1068. Which of the numbers 8; 7.1; -7.1; -7; -0.5 is the root of the equation -6 + x = -13.1?
1069. Guess the root of the equation and check:
- a) x + (-3) = -11;
- b) -5 + y = 15;
- c) t + (-12) = 2;
- d) 3 + n = -10.
1070. Find the value of the expression:
1071. Follow the steps using the calculator:
- a) -3.2579 + (-12.308);
- b) 7.8547 + (-9.239);
- c) -0.00154 + 0.0837;
- d) -3.8564 + (-0.8397) + 7.84;
- e) -0.083 + (-6.378) + 3.9834;
- f) -0.0085 + 0.00354 + (-0.00921).
1072. Find the value of the sum:
1073. Find the value of the expression:
1074. How many integers are located between the numbers:
- a) 0 and 24;
- b) -12 and -3;
- c) -20 and 7?
1075. 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 proper ordinary fraction.
1076. 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?
1077. 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?
Rice. 87
1078. Make an equation for solving the problem: “A field with an area of 2.4 hectares was divided into two sections. Find the area of each parcel if one of the parcels is known to be:
1079. Solve the problem:
- 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?
- A farmer with two sons placed the collected apples in 4 containers, on average 135 kg each. The farmer collected 280 kg of apples, and the youngest son - 4 times less. How many kilograms of apples did the eldest son collect?
1080. Follow these steps:
- (2,35 + 4,65) 5,3: (40 - 2,9);
- (7,63 - 5,13) 0,4: (3,17 + 6,83).
1081. Perform addition:
1082.
Present as a sum of two equal terms each of the numbers: 10; -eight; -6.8; 
.
1083. Find the value a + b if:
1084. There were 8 apartments on one floor of the residential building. Living area of 22.8 m 2 had 2 apartments, 16.2 m 2 - 3 apartments, 34 m 2 - 2 apartments. What living area did the eighth apartment have if on this floor, on average, each apartment had 24.7 m 2 of living space?
1085. The freight train consisted of 42 wagons. 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?
1086.
Find the value of an expression 
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 simple 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. subtract from smaller more You can subtract the smaller from the larger, but put 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 entered integers for item counting:
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 temperature. 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)
- 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. Moscow: Education, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A guide for students in grade 6 of the MEPhI correspondence school. M.: ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Interlocutor textbook for grades 5-6 high school. M .: Education, Mathematics Teacher Library, 1989.
- Math-prosto.ru ().
- youtube().
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- Allforchildren.ru ().
Homework
Addition of negative numbers.
The sum of negative numbers is a negative number. Sum modulus is equal to the sum modules of 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 added 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.
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