Division of numbers with different signs, as a rule, examples. Division of negative numbers: rule and examples Multiplication and division of opposite numbers

§ 1 Multiplication of positive and negative numbers

In this lesson, we will get acquainted with the rules for multiplying and dividing positive and negative numbers.

It is known that any product can be represented as a sum of identical terms.

The term -1 must be added 6 times:

(-1)+(-1)+(-1) +(-1) +(-1) + (-1) =-6

So the product of -1 and 6 is -6.

The numbers 6 and -6 are opposite numbers.

Thus, we can conclude:

When you multiply -1 by a natural number, you get its opposite number.

For negative numbers, as well as for positive ones, the commutative law of multiplication is fulfilled:

If a natural number is multiplied by -1, then the opposite number will also be obtained.

Multiplying any non-negative number by 1 results in the same number.

For example:

For negative numbers, this statement is also true: -5 ∙1 = -5; -2 ∙ 1 = -2.

Multiplying any number by 1 results in the same number.

We have already seen that when minus 1 is multiplied by a natural number, the opposite number will be obtained. When multiplying a negative number, this statement is also true.

For example: (-1) ∙ (-4) = 4.

Also -1 ∙ 0 = 0, the number 0 is the opposite of itself.

When you multiply any number by minus 1, you get its opposite number.

Let's move on to other cases of multiplication. Let's find the product of the numbers -3 and 7.

The negative factor -3 can be replaced by the product of -1 and 3. Then the associative multiplication law can be applied:

1 ∙ 21 = -21, i.e. the product of minus 3 and 7 is minus 21.

When multiplying two numbers with different signs, a negative number is obtained, the modulus of which is equal to the product of the moduli of the factors.

What is the product of numbers with the same sign?

We know that when you multiply two positive numbers, you get a positive number. Find the product of two negative numbers.

Let's replace one of the factors with a product with a factor minus 1.

We apply the rule we have derived, when multiplying two numbers with different signs, a negative number is obtained, the modulus of which is equal to the product of the moduli of the factors,

get -80.

Let's formulate the rule:

When multiplying two numbers with the same signs, a positive number is obtained, the modulus of which is equal to the product of the moduli of the factors.

§ 2 Division of positive and negative numbers

Let's move on to division.

By selection we find the roots of the following equations:

y ∙ (-2) = 10. 5 ∙ 2 = 10, so x = 5; 5 ∙ (-2) = -10, so a = 5; -5 ∙ (-2) = 10, so y = -5.

Let us write down the solutions of the equations. In each equation, the factor is unknown. We find the unknown factor by dividing the product by the known factor, we have already selected the values ​​of the unknown factors.

Let's analyze.

When dividing numbers with the same signs (and these are the first and second equations), a positive number is obtained, the modulus of which is equal to the quotient of the moduli of the dividend and divisor.

When dividing numbers with different signs (this is the third equation), a negative number is obtained, the modulus of which is equal to the quotient of the moduli of the dividend and divisor. Those. when dividing positive and negative numbers, the sign of the quotient is determined by the same rules as the sign of the product. And the modulus of the quotient is equal to the quotient of the modulus of the dividend and divisor.

Thus, we have formulated the rules for multiplication and division of positive and negative numbers.

List of used literature:

1. Maths. Grade 6: lesson plans for the textbook by I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilin. – Mnemosyne, 2009.
2. Maths. Grade 6: a textbook for students of educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
3. Maths. Grade 6: textbook for students of educational institutions./N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M.: Mnemosyne, 2013.
4. Mathematics Handbook - http://lyudmilanik.com.ua
5. Handbook for students in high school http://shkolo.ru

This article provides a detailed overview dividing numbers with different signs. First, the rule for dividing numbers with different signs is given. Below are examples of dividing positive numbers by negative and negative numbers by positive.

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Rule for dividing numbers with different signs

In the article division of integers, the rule for dividing integers with different signs was obtained. It can be extended to both rational numbers and real numbers by repeating all the arguments from the specified article.

So, rule for dividing numbers with different signs has the following formulation: to divide a positive number by a negative or a negative number by a positive one, it is necessary to divide the dividend by the modulus of the divisor, and put a minus sign in front of the resulting number.

We write this division rule using letters. If the numbers a and b have different signs, then the formula is valid a:b=−|a|:|b| .

From the voiced rule, it is clear that the result of dividing numbers with different signs is a negative number. Indeed, since the modulus of the dividend and the modulus of the divisor are more positive than the number, then their quotient is a positive number, and the minus sign makes this number negative.

Note that the considered rule reduces the division of numbers with different signs to the division of positive numbers.

You can give another formulation of the rule for dividing numbers with different signs: to divide the number a by the number b, you need to multiply the number a by the number b −1, the reciprocal of the number b. That is, a:b=a b −1 .

This rule can be used when it is possible to go beyond the set of integers (since not every integer has an inverse). In other words, it is applicable on the set of rational numbers as well as on the set of real numbers.

It is clear that this rule for dividing numbers with different signs allows you to go from division to multiplication.

The same rule is used when dividing negative numbers.

It remains to consider how this rule for dividing numbers with different signs is applied in solving examples.

Examples of dividing numbers with different signs

Let us consider solutions of several characteristic examples of dividing numbers with different signs to grasp the principle of applying the rules from the previous paragraph.

Example.

Divide the negative number −35 by the positive number 7 .

Solution.

The rule for dividing numbers with different signs prescribes first to find the modules of the dividend and divisor. The modulus of −35 is 35 and the modulus of 7 is 7. Now we need to divide the modulus of the dividend by the modulus of the divisor, that is, we need to divide 35 by 7. Remembering how the division of natural numbers is performed, we get 35:7=5. The last step of the rule for dividing numbers with different signs remains - put a minus in front of the resulting number, we have -5.

Here is the whole solution: .

One could proceed from a different formulation of the rule for dividing numbers with different signs. In this case, we first find the number that is the reciprocal of the divisor 7. This number is the common fraction 1/7. In this way, . It remains to perform the multiplication of numbers with different signs: . Obviously, we came to the same result.

Answer:

(−35):7=−5 .

Example.

Calculate the quotient 8:(−60) .

Solution.

By the rule of dividing numbers with different signs, we have 8:(−60)=−(|8|:|−60|)=−(8:60) . The resulting expression corresponds to a negative ordinary fraction (see the division sign as a fraction bar), you can reduce the fraction by 4, we get .

We write down the whole solution briefly: .

Answer:

.

When dividing fractional rational numbers with different signs, their dividend and divisor are usually represented as ordinary fractions. This is due to the fact that it is not always convenient to perform division with numbers in a different notation (for example, in decimal).

Example.

Solution.

The modulus of the dividend is , and the modulus of the divisor is 0,(23) . To divide the modulus of the dividend by the modulus of the divisor, let's move on to ordinary fractions.

Let's translate a mixed number into an ordinary fraction: , as well as

Topic of the open lesson: "Multiplication of negative and positive numbers"

The date: 03/17/2017

Teacher: Kuts V.V.

Class: 6 g

The purpose and objectives of the lesson:

introduce rules for multiplying two negative numbers and numbers with different signs;

promote the development of mathematical speech, working memory, voluntary attention, visual-effective thinking;

formation of internal processes of intellectual, personal, emotional development.

to cultivate a culture of behavior in frontal work, individual and group work.

Lesson type: lesson of primary presentation of new knowledge

Forms of study: frontal, work in pairs, work in groups, individual work.

Teaching methods: verbal (conversation, dialogue); visual (work with didactic material); deductive (analysis, application of knowledge, generalization, project activities).

Concepts and terms : modulus of number, positive and negative numbers, multiplication.

Planned results learning

- be able to multiply numbers with different signs, multiply negative numbers;

Apply the rule for multiplying positive and negative numbers when solving exercises, fix the rules for multiplying decimal and ordinary fractions.

Regulatory - be able to determine and formulate the goal in the lesson with the help of a teacher; pronounce the sequence of actions in the lesson; work according to a collective plan; evaluate the correctness of the action. Plan your action in accordance with the task; make the necessary adjustments to the action after its completion based on its assessment and taking into account the mistakes made; express your guess.Communicative - be able to formulate their thoughts orally; listen and understand the speech of others; jointly agree on the rules of behavior and communication at school and follow them.

Cognitive - to be able to navigate in their system of knowledge, to distinguish new knowledge from already known with the help of a teacher; acquire new knowledge; find answers to questions using the textbook, your life experience and information learned in class.

Formation of a responsible attitude to learning based on motivation for learning new things;

Formation of communicative competence in the process of communication and cooperation with peers in learning activities;

To be able to carry out self-assessment based on the criterion of success of educational activities; focus on learning success.

During the classes

Structural elements of the lesson

Didactic tasks

Projected teacher activity

Projected student activity

Result

1. Organizational moment

Motivation for successful activity

Check readiness for the lesson.

- Good afternoon guys! Have a seat! Check if you have everything ready for the lesson: notebook and textbook, diary and writing materials.

I am glad to see you at the lesson today in a good mood.

Look into each other's eyes, smile, wish your comrade a good working mood with your eyes.

I also wish you good work today.

Guys, the motto of today's lesson will be a quote from the French writer Anatole France:

“Learning can only be fun. To digest knowledge, one must absorb it with gusto.”

Guys, who will tell me what it means to absorb knowledge with an appetite?

So today we will absorb knowledge with great pleasure at the lesson, because they will be useful to us in the future.

Therefore, we rather open notebooks and write down the number, cool work.

Emotional mood

- With interest, with pleasure.

Ready to start the lesson

Positive motivation to study new topic

2. Activation cognitive activity

Prepare them to learn new knowledge and ways of doing things.

Organize a face-to-face survey on the material covered.

Guys, who will tell me what is the most important skill in mathematics? ( Check). Correctly.

So I'll test you now, how well you can count.

We will now do a math exercise.

We work as usual, we count orally, and write down the answer in writing. I give you 1 min.

5,2-6,7=-1,5

2,9+0,3=-2,6

9+0,3=9,3

6+7,21=13,21

15,22-3,34=-18,56

Let's check the answers.

We will check the answers, if you agree with the answer, then clap your hands, if you do not agree, then stomp your feet.

Well done boys.

Tell me, what actions did we perform with numbers?

What rule did we use when counting?

Formulate these rules.

Answer questions by solving small examples.

Addition and subtraction.

Adding numbers with different signs, adding numbers with negative signs, and subtracting positive and negative numbers.

The readiness of students to formulate a problematic issue, to find ways to solve the problem.

3. Motivation for setting the topic and purpose of the lesson

Encourage students to set the topic and purpose of the lesson.

Organize work in pairs.

Well, it's time to move on to the study of new material, but first, let's repeat the material of the previous lessons. A mathematical crossword puzzle will help us with this.

But this crossword puzzle is not ordinary, it contains a keyword that will tell us the topic of today's lesson.

The crossword puzzle lies on your tables, we will work with it in pairs. And once in pairs, then remind me how it is in pairs?

We remembered the rule of working in pairs, but now we start solving the crossword puzzle, I give you 1.5 minutes. Whoever does everything, put your pens so I can see.

(Attachment 1)

1. What numbers are used in counting?

2. The distance from the origin to any point is called?

3. Are the numbers that are represented by a fraction called?

4. Are two numbers that differ from each other only in signs called?

5. What numbers lie to the right of zero on the coordinate line?

6. Natural numbers, their opposite numbers and zero are called?

7. What number is called neutral?

8. A number showing the position of a point on a straight line?

9. What numbers lie to the left of zero on the coordinate line?

So, the time is up. Let's check.

We have solved the whole crossword puzzle and thus repeated the material of the previous lessons. Raise your hand, who made only one mistake, and who made two? (So ​​you guys are great).

Well, now back to our crossword puzzle. At the very beginning, I said that it contained a word that would tell us the topic of the lesson.

So what is the topic of our lesson?

And what are we going to multiply today?

Let's think, for this we recall the types of numbers that we already know.

Let's think about what numbers we already know how to multiply?

What numbers will we learn to multiply today?

Write in your notebook the topic of the lesson: "Multiplying positive and negative numbers."

So, guys, figured out what we will talk about today in the lesson.

Tell me, please, the purpose of our lesson, what should each of you learn and what should you try to learn by the end of the lesson?

Guys, well, in order to achieve this goal, what tasks will we have to solve with you?

Quite right. These are the two tasks that we will have to solve with you today.

Work in pairs, set the topic and purpose of the lesson.

1.Natural

2.Module

3. Rational

4.Opposite

5.Positive

6. Whole

7.Zero

8.Coordinate

9.Negative

-"Multiplication"

Positive and negative numbers

"Multiplication of Positive and Negative Numbers"

The purpose of the lesson:

Learn to multiply positive and negative numbers

First, to learn how to multiply positive and negative numbers, you need to get a rule.

Secondly, when we get the rule, then what should we do? (learn to apply it when solving examples).

4. Learning new knowledge and ways of acting

Acquire new knowledge on the topic.

-Organize work in groups (learning new material)

- Now, in order to achieve our goal, we will begin the first task, we will derive a rule for multiplying positive and negative numbers.

And research work will help us in this. And who will tell me why it is called research? - In this work, we will explore to discover the rules "Multiplication of positive and negative numbers."

Your research work will take place in groups, in total we will have 5 research groups.

We repeated in our heads how we should work in a group. If someone forgot, then the rules are in front of you on the screen.

the purpose of your research work: Exploring the tasks, gradually deduce the rule "Multiplication of negative and positive numbers" in task No. 2, in task No. 1 you have 4 tasks in total. And in order to solve these problems, our thermometer will help you, each group has one.

All entries are made on a piece of paper.

Once the group has a solution for the first problem, you show it on the board.

You are given 5-7 minutes to work.

(Annex 2 )

Work in groups (fill in the table, conduct research)

Rules for working in groups.

Working in groups is very easy

Know five rules to follow:

first: do not interrupt,

when he tells

friend, there should be silence around;

second: do not shout loudly,

and give arguments;

and the third rule is simply:

decide what is important to you;

fourthly: it is not enough to know orally

must be recorded;

and fifthly: sum up, think,

what could you do.

Mastery

the knowledge and methods of action that are determined by the objectives of the lesson

5.Fizminutka

To establish the correctness of assimilation of new material at this stage, to identify misconceptions and their correction

Okay, I put all your answers in the table, now let's look at each line in our table (see presentation)

What conclusions can we draw from the study of the table.

1 line. What numbers are we multiplying? What number is the answer?

2 line. What numbers are we multiplying? What number is the answer?

3 line. What numbers are we multiplying? What number is the answer?

4 line. What numbers are we multiplying? What number is the answer?

And so you analyzed the examples, and are ready to formulate the rules, for this you had to fill in the gaps in the second task.

How to multiply a negative number by a positive one?

- How to multiply two negative numbers?

Let's get some rest.

Positive answer - sit down, negative - get up.

5*6

2*2

7*(-4)

2*(-3)

8*(-8)

7*(-2)

5*3

4*(-9)

5*(-5)

9*(-8)

15*(-3)

7*(-6)

Multiplying positive numbers always results in a positive number.

Multiplying a negative number by a positive number always results in a negative number.

Multiplying negative numbers always results in a positive number.

Multiplying a positive number by a negative number results in a negative number.

To multiply two numbers with different signs,multiply modules of these numbers and put a "-" sign in front of the resulting number.

- To multiply two negative numbers, you needmultiply their modules and put a sign in front of the resulting number «+».

Students perform physical exercises, reinforcing the rules.

Prevent fatigue

7.Primary fixing of new material

To master the ability to apply the acquired knowledge in practice.

Organize frontal and independent work on the material covered.

We will fix the rules, and we will tell each other in pairs these same rules. I give you a minute for this.

Tell me, can we now move on to solving examples? Yes we can.

We open page 192 No. 1121

All together we will make the 1st and 2nd lines a) 5 * (-6) = 30

b) 9*(-3)=-27

g) 0.7*(-8)=-5.6

h) -0.5*6=-3

n) 1.2*(-14)=-16.8

o) -20.5*(-46)=943

three people at the blackboard

You have 5 minutes to solve the examples.

And we check everything together.

Creative task in pairs. (Appendix 3)

Insert the numbers so that on each floor their product is equal to the number on the roof of the house.

Solve examples using the knowledge gained

Raise your hands who did not have mistakes, well done ....

Active actions of students to apply knowledge in life.

9. Reflection (outcome of the lesson, assessment of the results of students' activities)

Provide students with reflection, i.e. their evaluation of their activities

Organize a lesson summary

Our lesson has come to an end, let's summarize.

Let's revisit the topic of our lesson, shall we? What was our goal? - Have we achieved this goal?

What difficulties did this topic cause for you?

- Guys, well, in order to evaluate your work in the lesson, you must draw a smiley face in circles that are on your tables.

A smiling emoticon means that you understand everything. Green means that you understand, but you need to practice, and a sad smiley, if you don’t understand anything at all. (Give me half a minute)

Well, guys, are you ready to show how you worked in class today? So, we raise and, I also raise a smiley for you.

I am very pleased with you today at the lesson! I see that everyone understood the material. Guys, you are great!

Lesson over, thanks for reading!

Answer questions and evaluate your work

Yes, we have.

The openness of students to the transfer and understanding of their actions, to identify positive and negative aspects of the lesson

10 .Homework Information

Ensure understanding of the purpose, content and methods of implementation homework

Provides understanding of the purpose of homework.

Homework:

1. Learn the rules of multiplication
2. No. 1121 (3rd column).
3.Creative task: compose a test of 5 multiple-choice questions.

Write down homework, trying to comprehend and understand.

Implementation of the need to achieve conditions for the successful completion of homework by all students, in accordance with the task and the level of development of students

In this article, we will look at dividing positive numbers by negative numbers and vice versa. Let's give detailed analysis rules for dividing numbers with different signs, and also give examples.

Rule for dividing numbers with different signs

The rule for integers with different signs, obtained in the article on the division of integers, is also valid for rational and real numbers. Let us give a more general formulation of this rule.

Rule for dividing numbers with different signs

When dividing a positive number by a negative one and vice versa, you need to divide the dividend modulus by the divisor modulus, and write the result with a minus sign.

In literal form, it looks like this:

a ÷ - b = - a ÷ b

A ÷ b = - a ÷ b .

Dividing numbers with different signs always results in a negative number. The considered rule, in fact, reduces the division of numbers with different signs to the division of positive numbers, since the modules of the dividend and divisor are positive.

Another equivalent mathematical formulation of this rule is:

a ÷ b = a b - 1

To divide the numbers a and bhaving different signs, you need to multiply the number a by the reciprocal of the number b, that is, b - 1. This formulation is applicable on the set of rational and real numbers, it allows you to go from division to multiplication.

Let us now consider how to apply the theory described above in practice.

How to divide numbers with different signs? Examples

Below we consider a few typical examples.

Example 1. How to divide numbers with different signs?

Divide - 35 by 7.

First, let's write the modules of the dividend and divisor:

35 = 35 , 7 = 7 .

Now let's separate the modules:

35 7 = 35 7 = 5 .

We add a minus sign in front of the result and get the answer:

Now let's use a different formulation of the rule and calculate the reciprocal of 7 .

Now let's do the multiplication:

35 1 7 = - - 35 1 7 = - 35 7 = - 5 .

Example 2. How to divide numbers with different signs?

If we share fractional numbers with rational signs, the dividend and divisor must be represented as ordinary fractions.

Example 3. How to divide numbers with different signs?

Let's divide mixed number- 3 3 22 on decimal 0 , (23) .

The modules of the dividend and the divisor are respectively 3 3 22 and 0 , (23) . Converting 3 3 22 to a common fraction, we get:

3 3 22 = 3 22 + 3 22 = 69 22 .

We can also represent the divisor as a common fraction:

0 , (23) = 0 , 23 + 0 , 0023 + 0 , 000023 = 0 , 23 1 - 0 , 01 = 0 , 23 0 , 99 = 23 99 .

Now we divide ordinary fractions, perform reductions and get the result:

69 22 ÷ 23 99 = - 69 22 99 23 = - 3 2 9 1 = - 27 2 = - 13 1 2 .

In conclusion, consider the case when the dividend and divisor are irrational numbers and are written as roots, logarithms, powers, etc.

In such a situation, the quotient is written as a numerical expression, which is simplified as much as possible. If necessary, its approximate value is calculated with the required accuracy.

Example 4. How to divide numbers with different signs?

Divide the numbers 5 7 and - 2 3 .

According to the rule for dividing numbers with different signs, we write the equality:

5 7 ÷ - 2 3 = - 5 7 ÷ - 2 3 = - 5 7 ÷ 2 3 = - 5 7 2 3 .

Let's get rid of the irrationality in the denominator and get the final answer:

5 7 2 3 = - 5 4 3 14 .

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