EntranceHow to add with different signs. Adding and subtracting rational numbers
Almost the entire mathematics course is based on operations with positive and negative numbers. After all, as soon as we begin to study the coordinate line, numbers with plus and minus signs begin to appear 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. Let us recall the rules by which these actions occur.
Adding numbers with different signs
If to solve a problem we need to add a negative number “-b” to some number “a”, then we need to act as follows.
- Let's take the modules of both numbers - |a| and |b| - and compare these absolute values with each other.
- Let us note which module is larger and which is smaller, and subtract the smaller value from the larger value.
- Let us put in front of the resulting number the sign of the number whose modulus is greater.
This will be the answer. We can put it 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 turn out to be equal. If so, then we can stop at this point - we are talking about opposite numbers, and their sum will always be equal to zero.
Subtracting numbers with different signs
We've dealt with addition, now let's look at the rule for subtraction. It is also quite simple - and in addition, 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 you need to subtract “c” from “a”, then we write it like this: a – (-c) = a + c.
- If “a” is a negative number, and “c” is positive, and “c” needs to be subtracted from “a”, then we write it as follows: (- a)– c = - a+ (-c).
Thus, when subtracting numbers with different signs, we end up returning to the rules of addition, and when adding numbers with different signs, we return to the rules of subtraction. Memorizing these rules allows you to solve problems quickly and easily.
This lesson covers addition and subtraction of rational numbers. The topic is classified as complex. Here it is necessary to use the entire arsenal of previously acquired knowledge.
The rules for adding and subtracting integers also apply to rational numbers. Recall that rational numbers are numbers that can be represented as a fraction, where a – this is the numerator of the fraction, b is the denominator of the fraction. Wherein, b should not be zero.
In this lesson, we will increasingly call fractions and mixed numbers by one common phrase - rational numbers.
Lesson navigation:Example 1. Find the meaning of the expression:
Let's enclose each rational number in brackets along with its signs. We take into account that the plus given in the expression is an operation sign and does not apply to the fraction. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract the smaller module from the larger module, and before the resulting answer put the sign of the rational number whose module is larger. And in order to understand which modulus is greater and which is smaller, you need to be able to compare the moduli of these fractions before calculating them:
The modulus of a rational number is greater than the modulus of a rational number. Therefore, we subtracted from . We received an answer. Then, reducing this fraction by 2, we got the final answer.
Some primitive actions, such as putting numbers in brackets and adding modules, can be skipped. This example can be written briefly:
Example 2. Find the meaning of the expression:
Let's enclose each rational number in brackets along with its signs. We take into account that the minus standing between rational numbers is a sign of the operation and does not apply to the fraction. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
Let's replace subtraction with addition. Let us remind you that to do this you need to add to the minuend the number opposite to the subtrahend:
We obtained the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus in front of the resulting answer:
Note. It is not necessary to enclose every rational number in parentheses. This is done for convenience, in order to clearly see which signs the rational numbers have.
Example 3. Find the meaning of the expression:
In this expression, the fractions have different denominators. To make our task easier, let's reduce these fractions to a common denominator. We will not dwell in detail on how to do this. If you experience difficulties, be sure to repeat the lesson.
After reducing the fractions to a common denominator, the expression will take the following form:
This is the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the resulting answer we put the sign of the rational number whose module is greater:
Let's write down the solution to this example in short:
Example 4. Find the value of an expression
Let's calculate this expression as follows: add the rational numbers and, then subtract the rational number from the resulting result. 
First action:
Second action:
Example 5. Find the meaning of the expression:
Let's represent the integer −1 as a fraction, and convert the mixed number into an improper fraction:
Let's enclose each rational number in brackets along with its signs:
We obtained the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the resulting answer we put the sign of the rational number whose module is greater:
We received an answer.
There is a second solution. It consists of putting whole parts together separately.
So, let's return to the original expression:
Let's enclose each number in parentheses. To do this, the mixed number is temporary:
Let's calculate the integer parts:
(−1) + (+2) = 1
In the main expression, instead of (−1) + (+2), we write the resulting unit:
The resulting expression is . To do this, write the unit and the fraction together:
Let's write the solution this way in a shorter way:
Example 6. Find the value of an expression
Let's convert the mixed number to an improper fraction. Let's rewrite the rest without changing:
Let's enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
Let's write down the solution to this example in short:
Example 7. Find the value of an expression
Let's represent the integer −5 as a fraction, and convert the mixed number into an improper fraction:
Let's bring these fractions to a common denominator. After they are reduced to a common denominator, they will take the following form:
Let's enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
We obtained the addition of negative rational numbers. Let’s add the modules of these numbers and put a minus in front of the resulting answer:
Thus, the value of the expression is .
Let's solve this example in the second way. Let's return to the original expression:
Let's write the mixed number in expanded form. Let's rewrite the rest without changes:
We enclose each rational number in brackets together with its signs:
Let's calculate the integer parts:
In the main expression, instead of writing the resulting number −7
The expression is an expanded form of writing a mixed number. We write the number −7 and the fraction together to form the final answer:
Let's write this solution briefly:
Example 8. Find the value of an expression
We enclose each rational number in brackets together with its signs:
Let's replace subtraction with addition:
We obtained the addition of negative rational numbers. Let’s add the modules of these numbers and put a minus in front of the resulting answer:
So the value of the expression is
This example can be solved in the second way. It consists of adding whole and fractional parts separately. Let's return to the original expression:
Let's enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
We obtained the addition of negative rational numbers. Let's add the modules of these numbers and put a minus in front of the resulting answer. But this time we will add the whole parts (−1 and −2), both fractional and
Let's write this solution briefly:
Example 9. Find expression expressions 
Let's convert mixed numbers to improper fractions:
Let's enclose a rational number in brackets together with its sign. There is no need to put a rational number in parentheses, since it is already in parentheses:
We obtained the addition of negative rational numbers. Let’s add the modules of these numbers and put a minus in front of the resulting answer:
So the value of the expression is
Now let's try to solve the same example in the second way, namely by adding integer and fractional parts separately.
This time, in order to get a short solution, let's try to skip some steps, such as writing a mixed number in expanded form and replacing subtraction with addition:
Please note that fractional parts have been reduced to a common denominator.
Example 10. Find the value of an expression 
Let's replace subtraction with addition:
The resulting expression does not contain negative numbers, which are the main reason for errors. And since there are no negative numbers, we can remove the plus in front of the subtrahend and also remove the parentheses:
The result is a simple expression that is easy to calculate. Let's calculate it in any way convenient for us:
Example 11. Find the value of an expression
This is the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and before the resulting answer we put the sign of the rational number whose module is greater:
Example 12. Find the value of an expression 
The expression consists of several rational numbers. According to, first of all you need to perform the steps in brackets.
First, we calculate the expression, then we add the obtained results.
First action:
Second action:
Third action:
Answer: expression value equals
Example 13. Find the value of an expression 
Let's convert mixed numbers to improper fractions:
Let's put the rational number in brackets along with its sign. There is no need to put the rational number in parentheses, since it is already in parentheses:
Let's bring these fractions to a common denominator. After they are reduced to a common denominator, they will take the following form:
Let's replace subtraction with addition:
We obtained the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and before the resulting answer we put the sign of the rational number whose module is greater:
Thus, the meaning of the expression equals
Let's look at adding and subtracting decimals, which are also rational numbers and can be either positive or negative.
Example 14. Find the value of the expression −3.2 + 4.3
Let's enclose each rational number in brackets along with its signs. We take into account that the plus given in the expression is an operation sign and does not apply to the decimal fraction 4.3. This decimal fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
(−3,2) + (+4,3)
This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract the smaller module from the larger module, and before the resulting answer put the sign of the rational number whose module is larger. And in order to understand which module is larger and which is smaller, you need to be able to compare the modules of these decimal fractions before calculating them:
(−3,2) + (+4,3) = |+4,3| − |−3,2| = 1,1
The modulus of the number 4.3 is greater than the modulus of the number −3.2, so we subtracted 3.2 from 4.3. We received the answer 1.1. The answer is positive, since the answer must be preceded by the sign of the rational number whose modulus is greater. And the modulus of the number 4.3 is greater than the modulus of the number −3.2
Thus, the value of the expression −3.2 + (+4.3) is 1.1
−3,2 + (+4,3) = 1,1
Example 15. Find the value of the expression 3.5 + (−8.3)
This is the addition of rational numbers with different signs. As in the previous example, we subtract the smaller one from the larger module and before the answer we put the sign of the rational number whose module is greater:
3,5 + (−8,3) = −(|−8,3| − |3,5|) = −(8,3 − 3,5) = −(4,8) = −4,8
Thus, the value of the expression 3.5 + (−8.3) is −4.8
This example can be written briefly:
3,5 + (−8,3) = −4,8
Example 16. Find the value of the expression −7.2 + (−3.11)
This is the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus in front of the resulting answer.
You can skip the entry with modules so as not to clutter the expression:
−7,2 + (−3,11) = −7,20 + (−3,11) = −(7,20 + 3,11) = −(10,31) = −10,31
Thus, the value of the expression −7.2 + (−3.11) is −10.31
This example can be written briefly:
−7,2 + (−3,11) = −10,31
Example 17. Find the value of the expression −0.48 + (−2.7)
This is the addition of negative rational numbers. Let's add their modules and put a minus in front of the resulting answer. You can skip the entry with modules so as not to clutter the expression:
−0,48 + (−2,7) = (−0,48) + (−2,70) = −(0,48 + 2,70) = −(3,18) = −3,18
Example 18. Find the value of the expression −4.9 − 5.9
Let's enclose each rational number in brackets along with its signs. We take into account that the minus, which is located between the rational numbers −4.9 and 5.9, is an operation sign and does not belong to the number 5.9. This rational number has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
(−4,9) − (+5,9)
Let's replace subtraction with addition:
(−4,9) + (−5,9)
We obtained the addition of negative rational numbers. Let’s add their modules and put a minus in front of the resulting answer:
(−4,9) + (−5,9) = −(4,9 + 5,9) = −(10,8) = −10,8
Thus, the value of the expression −4.9 − 5.9 is −10.8
−4,9 − 5,9 = −10,8
Example 19. Find the value of the expression 7 − 9.3
Let's put each number in brackets along with its signs.
(+7) − (+9,3)
Let's replace subtraction with addition
(+7) + (−9,3)
(+7) + (−9,3) = −(9,3 − 7) = −(2,3) = −2,3
Thus, the value of the expression 7 − 9.3 is −2.3
Let's write down the solution to this example in short:
7 − 9,3 = −2,3
Example 20. Find the value of the expression −0.25 − (−1.2)
Let's replace subtraction with addition:
−0,25 + (+1,2)
We obtained the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and before the answer we put the sign of the number whose module is greater:
−0,25 + (+1,2) = 1,2 − 0,25 = 0,95
Let's write down the solution to this example in short:
−0,25 − (−1,2) = 0,95
Example 21. Find the value of the expression −3.5 + (4.1 − 7.1)
Let's perform the actions in brackets, then add the resulting answer with the number −3.5
First action:
4,1 − 7,1 = (+4,1) − (+7,1) = (+4,1) + (−7,1) = −(7,1 − 4,1) = −(3,0) = −3,0
Second action:
−3,5 + (−3,0) = −(3,5 + 3,0) = −(6,5) = −6,5
Answer: the value of the expression −3.5 + (4.1 − 7.1) is −6.5.
Example 22. Find the value of the expression (3.5 − 2.9) − (3.7 − 9.1)
Let's do the steps in parentheses. Then, from the number that was obtained as a result of executing the first brackets, subtract the number that was obtained as a result of executing the second brackets:
First action:
3,5 − 2,9 = (+3,5) − (+2,9) = (+3,5) + (−2,9) = 3,5 − 2,9 = 0,6
Second action:
3,7 − 9,1 = (+3,7) − (+9,1) = (+3,7) + (−9,1) = −(9,1 − 3,7) = −(5,4) = −5,4
Third act
0,6 − (−5,4) = (+0,6) + (+5,4) = 0,6 + 5,4 = 6,0 = 6
Answer: the value of the expression (3.5 − 2.9) − (3.7 − 9.1) is 6.
Example 23. Find the value of an expression −3,8 + 17,15 − 6,2 − 6,15
Let us enclose each rational number in brackets along with its signs
(−3,8) + (+17,15) − (+6,2) − (+6,15)
Let's replace subtraction with addition where possible:
(−3,8) + (+17,15) + (−6,2) + (−6,15)
The expression consists of several terms. According to the combinatory law of addition, if an expression consists of several terms, then the sum will not depend on the order of actions. This means that the terms can be added in any order.
Let's not reinvent the wheel, but add all the terms from left to right in the order they appear:
First action:
(−3,8) + (+17,15) = 17,15 − 3,80 = 13,35
Second action:
13,35 + (−6,2) = 13,35 − −6,20 = 7,15
Third action:
7,15 + (−6,15) = 7,15 − 6,15 = 1,00 = 1
Answer: the value of the expression −3.8 + 17.15 − 6.2 − 6.15 is 1.
Example 24. Find the value of an expression
Let's convert the decimal fraction −1.8 to a mixed number. Let's rewrite the rest without changing:
In this lesson we will learn what a negative number is and what numbers are called opposites. We will also learn how to add negative and positive numbers (numbers with different signs) and look at several examples of adding numbers with different signs.
Look at this gear (see Fig. 1).
Rice. 1. Clock gear
This is not a hand that directly shows the time and not a dial (see Fig. 2). But without this part the clock does not work.
Rice. 2. Gear inside the clock
What does the letter Y stand for? Nothing but the sound Y. But without it, many words will not “work”. For example, the word "mouse". So are negative numbers: they do not show any quantity, but without them the calculation mechanism would be much more difficult.
We know that addition and subtraction are equivalent operations and can be performed in any order. In direct order, we can calculate: , but we can’t start with subtraction, since we haven’t yet agreed on what .
It is clear that increasing the number by and then decreasing by means ultimately decreasing by three. Why not designate this object and count like that: adding means subtracting. Then .
The number can mean, for example, an apple. The new number does not represent any real quantity. By itself, it does not mean anything like the letter Y. It's just a new tool to make calculations easier.
Let's name new numbers negative. Now we can subtract the larger number from the smaller number. Technically, you still need to subtract the smaller number from the larger number, but put a minus sign in your answer: .
Let's look at another example: 
. You can do all the actions in a row: .
However, it is easier to subtract the third 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 , we introduce a new number, which we denote , and determine that it has the following property: the sum of the number and is equal to : .
We will call the number negative, and the numbers and - opposite. Thus, we got an infinite number of new numbers, for example:
The opposite of number ;
The opposite of number ;
The opposite of number ; 
The opposite of number ;
Subtract the larger number from the smaller number: . Let's add to this expression: . We got zero. However, according to the property: the number that adds zero to five is denoted minus five: . Therefore, the expression can be denoted as .
Every positive number has a twin number, which differs only in that it is preceded by a minus sign. Such numbers are called opposite(see Fig. 3).
Rice. 3. Examples of opposite numbers
Properties of opposite numbers
1. The sum of opposite numbers is zero: .
2. If you subtract a positive number from zero, the result will be the opposite negative number: .
1. Both numbers can be positive, and we already know how to add them: .
2. Both numbers can be negative.
We already covered adding numbers like these in the previous lesson, but let's make sure we understand what to do with them. For example: .
To find this sum, add the opposite positive numbers and put a minus sign.
3. One number can be positive and the other negative.
If it is convenient for us, we can replace the addition of a negative number with the subtraction of a positive one: .
One more example: . Again we write the amount as the difference. You can subtract a larger number from a smaller number by subtracting a smaller number from a larger one, but using a minus sign.
We can swap the terms: .
Another similar example: .
In all cases, the result is a subtraction.
To briefly formulate these rules, let's remember one more term. Opposite numbers are, of course, not equal to each other. But it would be strange not to notice what they have in common. We called this common modulo number. The modulus of opposite numbers is the same: for a positive number it is equal to the number itself, and for a negative number it is equal to the opposite, positive. For example: , .
To add two negative numbers, you need to add their modules and put a minus sign:
To add a negative and a positive number, you need to subtract the smaller module from the larger module and put the sign of the number with the larger module:
Both numbers are negative, therefore, we add their modules and put a minus sign:
Two numbers with different signs, therefore, from the modulus of the number (the larger modulus), we subtract the modulus of the number and put a minus sign (the sign of the number with the larger modulus):
Two numbers with different signs, therefore, from the modulus of the number (the larger modulus), we subtract the modulus of the number and put a minus sign (the sign of the number with the larger modulus): .
Two numbers with different signs, therefore, from the modulus of the number (the larger modulus), we subtract the modulus of the number and put a plus sign (the sign of the number with the larger modulus): .
Positive and negative numbers have historically had different roles.
First we introduced natural numbers to count objects:
Then we introduced other positive numbers - fractions, for counting non-integer quantities, parts: .
Negative numbers appeared as a tool to simplify calculations. It was not like there were any quantities in life that we could not count, and we invented negative numbers.
That is, negative numbers did not arise from the real world. They just turned out to be so convenient that in some places they found application in life. For example, we often hear about negative temperatures. However, we never encounter a negative number of apples. What's the difference?
The difference is that in life, negative quantities are used only for comparison, but not for quantities. If a hotel has a basement and an elevator is installed there, then in order to maintain the usual numbering of regular floors, a minus first floor may appear. This first minus means only one floor below ground level (see Fig. 1).
Rice. 4. Minus the first and minus the second floors
A negative temperature is 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 apples, but six. Thus, in life, positive numbers are used to determine quantities (apples, cake).
We also use them instead of names. Each phone could be given its own name, but the number of names is limited and there are no numbers. That's why we use phone numbers. Also for ordering (century follows century).
Negative numbers in life are used in the latter sense (minus the first floor below the zero and first floors)
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- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. "Gymnasium", 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. M.: Education, 1989.
- Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. M.: ZSh MEPhI, 2011.
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Homework
Lesson plan:
I. Organizational moment
Checking individual homework.
II. Updating students' basic knowledge
1. Mutual training. Control questions (pair organizational form of work - mutual testing).
2. Oral work with commenting (group organizational form of work).
3. Independent work (individual organizational form of work, self-test).
III. Lesson topic message
Group organizational form of work, putting forward a hypothesis, formulating a rule.
1. Completing training tasks according to the textbook (group organizational form of work).
2. Work of strong students using cards (individual organizational form of work).
VI. Physical pause
IX. Homework.
Target: developing 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.
- Develop the ability to work in pairs and mutual respect.
Material for the lesson: cards for mutual training, tables of work results, individual cards for repetition and reinforcement 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 needed to think about his words. How do you understand it? (“Consider unhappy 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 author's words? (If we don’t learn anything new, don’t gain new knowledge, then this day can be considered lost or unhappy. We must strive to gain new knowledge).
– And today will not be unhappy because we will again learn something new.
II. Updating students' basic knowledge
– In order to learn new material, you need to repeat what you have covered.
There was a task at home - to repeat the rules and now you will show your knowledge by working with test questions.
(Test questions on the topic “Positive and Negative Numbers”)
Work in pairs. Peer review. The results of the work are noted in the table)
| What are the numbers located to the right of the origin called? | Positive |
| What numbers are called opposites? | Two numbers that differ from each other only in signs are called opposites |
| 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 |
| How do you denote the modulus of a number? | Straight brackets |
| Formulate the rule for adding negative numbers? | To add two negative numbers you need to: add their modules and put a minus sign |
| What are the numbers located to the left of the origin called? | Negative |
| What number is opposite to zero? | 0 |
| Can the modulus of any number be a negative number? | No. Distance is never negative |
| State the rule for comparing negative numbers | Of two negative numbers, the one whose modulus is smaller is greater and the one whose modulus is greater is smaller. |
| What is the sum of opposite numbers? | 0 |
Answers to questions “+” are correct, “–” are incorrect Evaluation criteria: 5 – “5”; 4 – “4”;3 – “3”
| 1 | 2 | 3 | 4 | 5 | Grade | |
| Q/questions | ||||||
| Self/work | ||||||
| Ind/ work | ||||||
| Bottom line |
– Which questions were the most difficult?
– What do you need to successfully pass the test questions? (Know the rules)
2. Oral work with commenting
– 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 answers while checking)
– Why did the last example cause you difficulty?
– The sum of what numbers needs to be found, and the sum of what numbers do we know how to find?
III. Lesson topic message
– Today in class we will learn the rule for adding numbers with different signs. We will learn to add numbers with different signs. Independent work at the end of the lesson will show your progress.
IV. Learning new material
– Let’s open the notebooks, write down the date, class work, lesson topic “Adding numbers with different signs.”
– What is shown on the board? (Coordinate line)
– Prove that this is a coordinate line? (There is a reference point, a reference direction, a unit segment)
– Now we will learn together to add numbers with different signs using a coordinate line.
(Explanation by students under the guidance of the teacher.)
– Let’s find the number 0 on the coordinate line. We need to add the number 6 to 0. We take 6 steps to the right side of the origin, because the number 6 is positive (we put a colored magnet on the resulting number 6). To 6 we add the number (– 10), take 10 steps to the left of the origin, since (– 10) is a negative number (we put a colored magnet on the resulting number (– 4).)
– What answer did you receive? (- 4)
– How did you get the number 4? (10 – 6)
Draw a conclusion: From a number with a larger modulus, subtract a number with a smaller modulus.
– How did you get the minus sign in the answer?
Draw a conclusion: We took the sign of a number with a large modulus.
– Let’s write an example in a notebook:
6 + (–10) = – (10 – 6) = – 4
10 + (–3) = + (10 – 3) = 7 (Solve similarly)
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'll tell you your guesses hypothesis. You have done very important intellectual work. Like scientists, they put forward a hypothesis and discovered a new rule. Let's compare your hypothesis with the rule (a piece of paper with a printed rule is on the desk). Let's read in chorus rule adding numbers with different signs
– The rule is very important! It allows you to add numbers of different signs without using a coordinate line.
- What's not clear?
– Where can you make a mistake?
– In order to calculate tasks with positive and negative numbers correctly and without errors, 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 using a coordinate line, so we will use the rule you discovered to solve it.
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 consolidate this topic with an additional card.)
VI. Physical pause(Perform while standing)
– A person has positive and negative qualities. Distribute these qualities on the coordinate line.
(Positive qualities are to the right of the starting point, negative qualities are to the left of the starting point.)
– If the quality is negative, clap once, if it is positive, clap twice. Be careful!
– Kindness, anger, greed , mutual assistance,
understanding, rudeness, and, of course, strength of will And desire to win, which you will need now, since you have independent work ahead)
VII. Individual work followed by 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 = |
Individual work (for strong students) followed by 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: receiving ready-made information or thinking for yourself?
– What new did we learn in the lesson? (We learned to add numbers with different signs.)
– Name the rule for adding numbers with different signs.
– Tell me, was our lesson today not in vain?
- Why? (We gained new knowledge.)
- Let's return to the motto. This means that Jan Amos Kamensky was right when he said: “Consider unhappy that day or that 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 assignment - as 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 appreciate the level of his ignorance?
“Adding 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, you will learn 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 straight line and count using it? Therefore, you need to learn how to fold without it.
Let's try with you to add a negative number to a positive number, for example eight add minus six: 8+(-6). You already know that adding a negative number reduces the original number by a negative value. This means that eight must be reduced by six, that is, six must be subtracted from eight: 8-6 = 2, which gives 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 a positive number by the value of a negative one, we get six subtract eight is minus two: -8+6=-2.
As you noticed, in both the first and second examples with numbers, the action of subtraction is performed. Why? Because they have different signs (plus and minus). To avoid making mistakes when adding numbers with different signs, you should perform the following algorithm:
1. find the modules of numbers;
2. subtract the smaller module from the larger module;
3. Before the result obtained, put a number sign with a large absolute value (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.
