Subtraction with different signs of the rule. Adding numbers with different signs – Knowledge Hypermarket
Read also
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 figure out why the sum of negative numbers will also be a negative number. The coordinate line will help us with this, on which we will add the numbers -3 and -5. Let us 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, left! For 5 unit segments. We mark a 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 origin, 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. This notation is called an algebraic sum. Apply (in our example) the entry: -3-5=-8.
Example. Find the sum of negative numbers: -23-42-54. (Do you agree that this entry is shorter and more convenient like this: -23+(-42)+(-54))?
Let's decide according to the rule for adding negative numbers: we add the modules of the terms: 23+42+54=119. The result will have a minus sign.
They usually write it like this: -23-42-54=-119.
Adding numbers with different signs.
The sum of two numbers with different signs has the sign of a term with a large absolute value. To find the modulus of a sum, you need to subtract the smaller modulus from the larger modulus..
Let's perform the addition of numbers with different signs using a coordinate line.
1) -4+6. You need to add the number 6 to the number -4. Let's mark the number -4 with a dot 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 found ourselves 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 module. The result has the same sign as the term with a large modulus.
2) Let's calculate: -7+3 using the coordinate line. Mark the point corresponding to the number -7. We go to the right for 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 this way: from the larger module we subtracted the smaller one, i.e. 7-3=4. As a result, we put the sign of the term with the larger modulus: |-7|>|3|.
Examples. Calculate: A) -4+5-9+2-6-3; b) -10-20+15-25.
Instructions
There are four types of mathematical operations: addition, subtraction, multiplication and division. Therefore, there will be four types of examples. Negative numbers within the example are highlighted so as not to confuse the mathematical operation. For example, 6-(-7), 5+(-9), -4*(-3) or 34:(-17).
Addition. This action can look like: 1) 3+(-6)=3-6=-3. Replacement action: first, the parentheses are opened, the “+” sign is changed to the opposite, then from the larger (modulo) number “6” the smaller one, “3,” is subtracted, after which the answer is assigned the larger sign, that is, “-”.
2) -3+6=3. This can be written according to the principle ("6-3") or according to the principle "subtract the smaller from the larger and assign the sign of the larger to the answer."
3) -3+(-6)=-3-6=-9. When opening, the action of addition is replaced by subtraction, then the modules are summed up and the result is given a minus sign.
Subtraction.1) 8-(-5)=8+5=13. The parentheses are opened, the sign of the action is reversed, and an example of addition is obtained.
2) -9-3=-12. The elements of the example are added together and receive a common sign "-".
3) -10-(-5)=-10+5=-5. When opening the brackets, the sign changes again to “+”, then the smaller number is subtracted from the larger number and the sign of the larger number is taken away from the answer.
Multiplication and division: When performing multiplication or division, the sign does not affect the operation itself. When multiplying or dividing numbers with the answer, a “minus” sign is assigned; if the numbers have the same signs, the result always has a “plus” sign. 1) -4*9=-36; -6:2=-3.
2)6*(-5)=-30; 45:(-5)=-9.
3)-7*(-8)=56; -44:(-11)=4.
Sources:
- table with cons
How to decide examples? Children often turn to their parents with this question if homework needs to be done at home. How to correctly explain to a child the solution to examples of adding and subtracting multi-digit numbers? Let's try to figure this out.
You will need
- 1. Textbook on mathematics.
- 2. Paper.
- 3. Handle.
Instructions
Read the example. To do this, divide each multivalued into classes. Starting from the end of the number, count three digits at a time and put a dot (23.867.567). Let us recall that the first three digits from the end of the number are to units, the next three are to class, then millions come. We read the number: twenty-three eight hundred sixty-seven thousand sixty-seven.
Write down an example. Please note that the units of each digit are written strictly below each other: units under units, tens under tens, hundreds under hundreds, etc.
Perform addition or subtraction. Start performing the action with units. Write down the result under the category with which you performed the action. If the result is number(), then we write the units in place of the answer, and add the number of tens to the units of the digit. If the number of units of any digit in the minuend is less than in the subtrahend, we take 10 units of the next digit and perform the action.
Read the answer.
Video on the topic
note
Prohibit your child from using a calculator even to check the solution to an example. Addition is tested by subtraction, and subtraction is tested by addition.
If the child masters well the techniques of written calculations within 1000, then actions with multi-digit numbers, performed in a similar way, will not cause difficulties.
Give your child a competition to see how many examples he can solve in 10 minutes. Such training will help automate computational techniques.
Multiplication is one of the four basic mathematical operations that underlies many more complex functions. In fact, multiplication is based on the operation of addition: knowledge of this allows you to correctly solve any example.
To understand the essence of the multiplication operation, it is necessary to take into account that there are three main components involved in it. One of them is called the first factor and is a number that is subject to the multiplication operation. For this reason, it has a second, somewhat less common name - “multiplicable”. The second component of the multiplication operation is usually called the second factor: it represents the number by which the multiplicand is multiplied. Thus, both of these components are called multipliers, which emphasizes their equal status, as well as the fact that they can be swapped: the result of the multiplication will not change. Finally, the third component of the multiplication operation, resulting from its result, is called the product.
Order of multiplication operation
The essence of the multiplication operation is based on a simpler arithmetic operation -. In fact, multiplication is the sum of the first factor, or multiplicand, a number of times that corresponds to the second factor. For example, in order to multiply 8 by 4, you need to add the number 8 4 times, resulting in 32. This method, in addition to providing an understanding of the essence of the multiplication operation, can be used to check the result obtained when calculating the desired product. It should be borne in mind that the verification necessarily assumes that the terms involved in the summation are identical and correspond to the first factor.Solving multiplication examples
Thus, in order to solve the problem associated with the need to carry out multiplication, it may be enough to add the required number of first factors a given number of times. This method can be convenient for carrying out almost any calculations related to this operation. At the same time, in mathematics there are quite often standard numbers that involve standard single-digit integers. In order to facilitate their calculation, the so-called multiplication was created, which includes full list products of positive integers single digit numbers, that is, numbers from 1 to 9. Thus, once you have learned , you can significantly facilitate the process of solving multiplication examples based on the use of such numbers. However, for more complex options it will be necessary to carry out this mathematical operation yourself.Video on the topic
Sources:
- Multiplication in 2019
Multiplication is one of the four basic arithmetic operations, which is often used both in school and in Everyday life. How can you quickly multiply two numbers?
The basis of the most complex mathematical calculations There are four basic arithmetic operations: subtraction, addition, multiplication and division. Moreover, despite their independence, these operations, upon closer examination, turn out to be interconnected. Such a connection exists, for example, between addition and multiplication.
Number multiplication operation
There are three main elements involved in the multiplication operation. The first of these, usually called the first factor or multiplicand, is the number that will be subject to the multiplication operation. The second, called the second factor, is the number by which the first factor will be multiplied. Finally, the result of the multiplication operation performed is most often called a product.It should be remembered that the essence of the multiplication operation is actually based on addition: to carry it out, it is necessary to add together a certain number of the first factors, and the number of terms of this sum must be equal to the second factor. In addition to calculating the product of the two factors in question, this algorithm can also be used to check the resulting result.
An example of solving a multiplication problem
Let's look at solutions to multiplication problems. Suppose, according to the conditions of the task, it is necessary to calculate the product of two numbers, among which the first factor is 8, and the second is 4. In accordance with the definition of the multiplication operation, this actually means that you need to add the number 8 4 times. The result is 32 - this is the product of the numbers in question, that is, the result of their multiplication.In addition, it must be remembered that the so-called commutative law applies to the multiplication operation, which states that changing the places of the factors in the original example will not change its result. Thus, you can add the number 4 8 times, resulting in the same product - 32.
Multiplication table
It is clear that to solve this way a large number of drawing examples of the same type is a rather tedious task. In order to facilitate this task, the so-called multiplication was invented. In fact, it is a list of products of positive single-digit integers. Simply put, a multiplication table is a set of results of multiplying with each other from 1 to 9. Once you have learned this table, you can no longer resort to multiplication every time you need to solve an example for such simple numbers, but simply remember its result.Video on the topic
Lesson plan:
Individual verification homework.
II. Update background knowledge students
1. Mutual training. Control questions(steam room organizational form work - mutual verification).
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 study new material, you need to repeat what you have learned.
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 for successful completion security 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?
In this article we will look in detail at how it is done addition of integers. First we will form general idea about the addition of integers, and let's see what the addition of integers on a coordinate line is. This knowledge will help us formulate rules for adding positive, negative, and integers with different signs. Here we will examine in detail the application of addition rules when solving examples and learn how to check the results obtained. At the end of the article we will talk about the addition of three and more integers.
Page navigation.
Understanding addition of integers
Here are examples of adding integer opposite numbers. The sum of the numbers −5 and 5 is zero, the sum of 901+(−901) is zero, and the result of adding the opposite integers 1,567,893 and −1,567,893 is also zero.
Addition of an arbitrary integer and zero
Let's use the coordinate line to understand what the result of adding two integers, one of which is zero, is.
Adding an arbitrary integer a to zero means moving unit segments from the origin to a distance a. Thus, we find ourselves at the point with coordinate a. Therefore, the result of adding zero and an arbitrary integer is the added integer.
On the other hand, adding zero to an arbitrary integer means moving from the point whose coordinate is specified by a given integer to a distance of zero. In other words, we will remain at the starting point. Therefore, the result of adding an arbitrary integer and zero is the given integer.
So, the sum of two integers, one of which is zero, is equal to the other integer. In particular, zero plus zero is zero.
Let's give a few examples. The sum of the integers 78 and 0 is 78; the result of adding zero and −903 is −903 ; also 0+0=0 .
Checking the result of addition
After adding two integers, it is useful to check the result. We already know that to check the result of adding two natural numbers, we need to subtract any of the terms from the resulting sum, and this should result in another term. Checking the result of adding integers performed similarly. But subtracting integers comes down to adding to the minuend the number opposite to the one being subtracted. Thus, to check the result of adding two integers, you need to add to the resulting sum the number opposite to any of the terms, which should result in another term.
Let's look at examples of checking the result of adding two integers.
Example.
When adding two integers 13 and −9, the number 4 was obtained, check the result.
Solution.
Let's add to the resulting sum 4 the number −13, opposite to the term 13, and see if we get another term −9.
So, let's calculate the sum 4+(−13) . This is the sum of integers with opposite signs. The modules of the terms are 4 and 13, respectively. The term whose modulus is greater has a minus sign, which we remember. Now subtract from the larger module and subtract the smaller one: 13−4=9. All that remains is to put the remembered minus sign in front of the resulting number, we have −9.
When checking, we received a number equal to another term, therefore, the original sum was calculated correctly.−19. Since we received a number equal to another term, the addition of the numbers −35 and −19 was performed correctly.
Adding three or more integers
Up to this point we have talked about adding two integers. In other words, we considered sums consisting of two terms. However, the combinative property of adding integers allows us to uniquely determine the sum of three, four, or more integers.
Based on the properties of addition of integers, we can assert that the sum of three, four, and so on numbers does not depend on the way the parentheses are placed indicating the order in which actions are performed, as well as on the order of the terms in the sum. We substantiated these statements when we talked about the addition of three or more natural numbers. For integers, all reasoning is completely the same, and we will not repeat ourselves.0+(−101) +(−17)+5 . After this, placing the parentheses in any acceptable way, we will still get the number −113.
Answer:
5+(−17)+0+(−101)=−113 .
Bibliography.
- Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
In this article we will deal with adding numbers with different signs. Here we will give a rule for adding positive and negative numbers, 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
Let's 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 for 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 remembered 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 fold rational numbers with different signs that are not integers, they should be represented as ordinary fractions (you can also work with decimals, if this is convenient). Let's look at this point when solving 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 convert the decimal fraction to an ordinary fraction: 
.
Now you can use the rule for adding numbers with different signs.
The modules of the numbers being added are 17/8 and 5/4. For ease of execution further actions, let's bring 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 module, therefore, remember the plus sign.
Now we subtract the smaller one from the larger module, that is, we subtract fractions with the same denominators: 
.
All that remains is to put the remembered plus sign in front of the resulting number, we get , but - this is the number 7/8.