Objective 1. The point moves in a straight line from left to right at a speed of 4 dm. per second and is currently passing through point A. Where will the moving point be after 5 seconds?

It is easy to figure out that the point will be at 20 inches. to the right of A. Let us write the solution of this problem in relative numbers. To do this, we will agree in the following indications:

1) the speed to the right will be denoted by a + sign, and to the left by a - sign, 2) the distance of a moving point from A to the right will be denoted by a + sign and to the left by a - sign, 3) the time interval after the present moment by a + sign and up to the present moment by a - sign. In our problem, the following numbers are given: speed = + 4 dm. per second, time = + 5 seconds and it turned out, as they figured out arithmetically, the number + 20 dm., expressing the distance of the moving point from A in 5 seconds. According to the meaning of the problem, we see that it refers to multiplication. Therefore, it is convenient to write the solution to the problem:

(+ 4) ∙ (+ 5) = + 20.

Objective 2. The point moves in a straight line from left to right at a speed of 4 dm. per second and is currently passing through point A. Where was this point 5 seconds ago?

The answer is clear: the point was to the left of A at a distance of 20 dm.

The solution is convenient, according to the conditions regarding the signs, and, bearing in mind that the meaning of the problem has not changed, it can be written as follows:

(+ 4) ∙ (– 5) = – 20.

Objective 3. The point moves in a straight line from right to left at a speed of 4 dm. per second and is currently passing through point A. Where will the moving point be after 5 seconds?

The answer is clear: 20 dm. to the left of A. Therefore, according to the same conditions regarding the signs, we can write the solution to this problem as follows:

(– 4) ∙ (+ 5) = – 20.

Task 4. The point moves in a straight line from right to left at a speed of 4 dm. per second and is currently passing through point A. Where was the moving point 5 seconds ago?

The answer is clear: at a distance of 20 inches. to the right of A. Therefore, the solution to this problem should be written as follows:

(– 4) ∙ (– 5) = + 20.

The problems considered indicate how to extend the action of multiplication to relative numbers. We have in problems 4 cases of multiplication of numbers with all possible combinations of signs:

1) (+ 4) ∙ (+ 5) = + 20;
2) (+ 4) ∙ (– 5) = – 20;
3) (– 4) ∙ (+ 5) = – 20;
4) (– 4) ∙ (– 5) = + 20.

In all four cases, the absolute values ​​of these numbers should be multiplied, the product has to be given a + sign when the factors have the same signs (1st and 4th cases) and the sign - when the multipliers have different signs(cases 2 and 3).

From here we see that the product does not change from the permutation of the multiplier and the multiplier.

Exercises.

Let's perform one example for a calculation, which includes addition and subtraction and multiplication.

In order not to confuse the order of actions, let us pay attention to the formula

The sum of the products of two pairs of numbers is written here: therefore, you must first multiply the number a by the number b, then multiply the number c by the number d and then add the resulting products. Also in the formula

you must first multiply the number b by c and then subtract the resulting product from a.

If it were required to add the product of numbers a and b to c and multiply the resulting sum by d, then one would write: (ab + c) d (compare with the formula ab + cd).

If it was necessary to multiply the difference between the numbers a and b by c, then they would write (a - b) c (compare with the formula a - bc).

Therefore, we will establish in general that if the order of actions is not indicated by brackets, then we must first perform multiplication, and then addition or subtraction.

Let's start calculating our expression: we first execute the additions written inside all the small brackets, we get:

Now we need to perform the multiplication inside the square brackets and then subtract the resulting product from:

Now let's perform the actions inside the twisted brackets: first multiplication and then subtraction:

Now all that remains is to perform multiplication and subtraction:

16. Product of several factors. Let it be required to find

(–5) ∙ (+4) ∙ (–2) ∙ (–3) ∙ (+7) ∙ (–1) ∙ (+5).

Here the first number must be multiplied by the second, the resulting product by the third, etc. It is not difficult to establish on the basis of the previous one that the absolute values ​​of all numbers must be multiplied with each other.

If all the factors were positive, then on the basis of the previous one we find that the product must also have a + sign. If any one factor were negative

eg (+2) ∙ (+3) ∙ (+4) ∙ (–1) ∙ (+5) ∙ (+6),

then the product of all the factors preceding it would give a + sign (in our example, (+2) ∙ (+3) ∙ (+4) = +24, from multiplying the resulting product by a negative number (in our example +24 multiplied by –1) would get the sign of the new product -; multiplying it by the next positive factor (in our example –24 by +5), we get again a negative number; since all other factors are assumed to be positive, the sign of the product can no longer change.

If there were two negative factors, then, arguing as above, they would find that at first, until he reached the first negative factor, the product would be positive, from multiplying it by the first negative factor, the new product would turn out to be negative and so it would be and remained until we reach the second negative factor; then from multiplying a negative number by a negative, the new product would turn out to be positive, which will remain so in the future if the other factors are positive.

If there were still a third negative factor, then the product obtained positively from multiplying it by this third negative factor would become negative; it would remain so if the other factors were all positive. But if there is still a fourth negative factor, then multiplying by it will make the product positive. Arguing in the same way, we find that in general:

To find out the sign of the product of several factors, you need to see how many of these factors are negative: if there are none at all, or if their number is even, then the product is positive: if there are negative factors odd number, then the product is negative.

So now we can easily find out that

(–5) ∙ (+4) ∙ (–2) ∙ (–3) ∙ (+7) ∙ (–1) ∙ (+5) = +4200.

(+3) ∙ (–2) ∙ (+7) ∙ (+3) ∙ (–5) ∙ (–1) = –630.

Now it is easy to see that the sign of the work, as well as its absolute value, do not depend on the order of the factors.

Convenient when dealing with fractional numbers, find a work immediately:

This is convenient because you do not have to do useless multiplications, since the previously obtained fractional expression is reduced as much as possible.

This article focuses on division of negative numbers... First, a rule for dividing a negative number by a negative is given, its justification is given, and after that examples of dividing negative numbers with detailed description solutions.

Page navigation.

The rule for dividing negative numbers

Before giving the rule for dividing negative numbers, let us recall the meaning of the action of division. Division in essence represents finding an unknown factor by famous work and a known other factor. That is, the number c is the quotient of dividing a by b when c b = a, and vice versa, if c b = a, then a: b = c.

The rule for dividing negative numbers the following: the quotient of dividing one negative number by another is equal to the quotient of dividing the numerator by the modulus of the denominator.

Let's write down the voiced rule using letters. If a and b are negative numbers, then the equality a: b = | a |: | b | .

The equality a: b = a b −1 is easy to prove, starting from multiplication properties real numbers and definitions of reciprocal numbers. Indeed, on this basis, we can write down a chain of equalities of the form (a b −1) b = a (b −1 b) = a 1 = a, which, by virtue of the meaning of division mentioned at the beginning of the article, proves that a · b −1 is the quotient of the division of a by b.

And this rule allows you to go from dividing negative numbers to multiplication.

It remains to consider the application of the considered rules for dividing negative numbers when solving examples.

Examples of dividing negative numbers

Let's analyze examples of dividing negative numbers... Let's start with simple cases where we work out the division rule.

Example.

Divide the negative number −18 by the negative number −3, then calculate the quotient (−5): (- 2).

Solution.

According to the rule of dividing negative numbers, the quotient of dividing −18 by −3 is equal to the quotient of dividing the absolute values ​​of these numbers. Since | −18 | = 18 and | −3 | = 3, then (−18):(−3)=|−18|:|−3|=18:3 , it remains only to perform the division of natural numbers, we have 18: 3 = 6.

Similarly, we solve the second part of the task. Since | −5 | = 5 and | −2 | = 2, then (−5):(−2)=|−5|:|−2|=5:2 ... This quotient corresponds to the ordinary fraction 5/2, which can be written as a mixed number.

The same results are obtained if you use a different rule for dividing negative numbers. Indeed, the number −3 is inversely the number, then , now we perform multiplication of negative numbers: ... Similarly,.

Answer:

(−18): (- 3) = 6 and .

When dividing fractional rational numbers it is most convenient to work with ordinary fractions... But, if convenient, then you can divide the final decimal fractions.

Example.

Divide −0.004 by −0.25.

Solution.

The moduli of the dividend and the divisor are 0.004 and 0.25, respectively, then, according to the rule for dividing negative numbers, we have (−0,004):(−0,25)=0,004:0,25 .

• or perform division of decimal fractions with a column,
• either go from decimal fractions to ordinary, and then divide the corresponding ordinary fractions.

Let's take a look at both approaches.

To divide 0.004 by 0.25 in a column, first move the comma 2 digits to the right, thus we come to dividing 0.4 by 25. Now we do long division:

So 0.004: 0.25 = 0.016.

Now let's show what the solution would look like if we decided to convert decimal fractions to ordinary ones. Because and then , and execute

Open lesson topic: "Multiplication of negative and positive numbers"

Date: 17.03.2017

Teacher: V.V. Kuts

Class: 6 g

The purpose and objectives of the lesson:

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

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

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

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

Lesson type: a lesson in the primary presentation of new knowledge

Forms of training: 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 numbers, 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, consolidate the rules for multiplying decimal and ordinary fractions.

Regulatory - be able to define and formulate a goal in the lesson with the help of a teacher; to pronounce the sequence of actions in the lesson; work according to a collectively drawn up plan; evaluate the correctness of the action. Plan your action in accordance with the task at hand; make the necessary adjustments to the action after its completion, based on its assessment and taking into account the mistakes made; make your guess.Communicative - be able to form your thoughts in verbally; listen to and understand the speech of others; jointly agree on and follow the rules of conduct and communication at school.

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

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

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

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

During the classes

Structural elements lesson

Didactic tasks

Projected teacher activity

Projected student activities

Result

1.Organizational moment

Motivation for successful activity

Checking readiness for the lesson.

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

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

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

I wish you a good job today too.

Guys, the motto for 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 appetite. "

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

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

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

Emotional attitude

-With interest, with pleasure.

Willingness to start a lesson

Positive motivation to learn new topic

2. Activation cognitive activities

Prepare them for the assimilation of new knowledge and methods of action.

Organize a frontal survey based on the material covered.

Guys, who will tell me which one is main skill in math? ( Check). Right.

So I'll check you now how well you can count.

We will now perform a mathematical warm-up with you.

We work as usual, count verbally, 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 stamp your feet.

Well done boys.

Tell me, what actions did we perform with the numbers?

What rule did we use when invoicing?

Formulate these rules.

Answer questions by solving small examples.

Addition and subtraction.

Add numbers with different signs, add numbers with negative signs, and subtract positive and negative numbers.

The readiness of students to pose a problematic question, to find ways to solve the problem.

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

Stimulate students to formulate the topic and purpose of the lesson.

Organize work in pairs.

Well, it's time to move on to learning new material, but first, let's review the material from the previous lessons. A math crossword puzzle will help us with this.

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

Guys, the crossword puzzle is 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 are starting to solve the crossword puzzle, I give you 1.5 minutes. Who will do everything, put down the pens for me to see.

(Annex 1)

1. What numbers are used for counting?

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

3.The numbers that are represented by a fraction are called?

4. Two numbers that differ from each other only in signs are called?

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

6. Are natural numbers, opposite numbers and zero 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 it out.

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 let's get back to our crossword puzzle. At the very beginning, I said that it contains an encrypted word that will tell us the topic of the lesson.

So what will be the topic of our lesson?

And what are we going to multiply with you today?

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

Let's think, what numbers can we already multiply?

What numbers will we learn to multiply today?

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

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

Please tell me 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.

They work in pairs, set the topic and the purpose of the lesson.

1.Natural

2.Module

3.Rational

4.Opposite

5.Positive

6.Integer

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.

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

4. Learning new knowledge and ways of acting

Master new knowledge on the topic.

-Organize group work (learning new material)

- Now, in order to achieve our goal, we will proceed to the first task, derive the 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 of "Multiplication of positive and negative numbers."

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

They repeated in my head how we should work in a group. If someone has forgotten, 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 number 2, in task number 1 you have 4 tasks in total. And in order to solve these problems, our thermometer will help you for this, each group has one.

You make all your notes on a piece of paper.

As soon as the group has a solution to the first problem, you show it on the board.

You are given 5-7 minutes to work.

(Appendix 2 )

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

Rules for working in groups.

Working in groups is very easy

Be able to observe five rules:

first: do not interrupt,

when tells

friend, there must be silence around;

second: do not shout loudly,

and give the arguments;

and the third rule is simple:

decide what is important to you;

fourthly: it is not enough to know verbally,

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.Fizzy

Establish the correctness of assimilation of new material on 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 the Presentation)

What conclusions can we draw when examining 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 do I 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)

By multiplying positive numbers, the answer is always a positive number.

Multiplying a negative number by a positive one always gives a negative number in the answer.

By multiplying negative numbers, the answer is always a positive number.

Multiplying a positive number by a negative number produces a negative number.

To multiply two numbers with different signs, you needmultiply 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 do physical exercises, reinforcing the rules.

Prevent fatigue

7.Initial securing of new material

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

Organize frontal and independent work on the passed material.

Let's fix the rules, and tell each other these same rules as a pair. I'll give you a minute for that.

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

Opening page 192 # 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 are given 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 equals the number on the roof of the house.

Solve examples by applying the knowledge gained

Raise your hands who have not had any mistakes, well done….

Active actions of students to apply knowledge in life.

9. Reflection (lesson summary, assessment of students' performance results)

Provide reflection of students, i.e. their assessment of their performance

Organize a wrap-up of the lesson

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

Let's remember the topic of our lesson again? What goal did we set? - Did we achieve this goal?

What difficulties caused you this topic?

- 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. (I give half a minute)

Well guys, are you ready to show how you did your lesson today? So, we raise and, I also raise a smiley for you.

I am very pleased with you in class today! I see that everyone has understood the material. Guys, you are great!

The lesson is over, thank you for your attention!

Answer questions, evaluate their work

Yes, we did.

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

10 .Homework information

Provide an understanding of purpose, content and ways of accomplishment homework

Provides an understanding of the purpose of the homework.

Homework:

1. Learn the rules of multiplication
2.No. 1121 (3 columns).
3. Creative task: make a test of 5 questions with multiple answers.

They write down their homework, trying to comprehend and understand.

Realization 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 lesson, we will review the rules for adding positive and negative numbers. We will also learn how to multiply numbers with different signs and learn the rules of signs for multiplication. Let's look at examples of multiplying positive and negative numbers.

The property of multiplying by zero remains true in the case of negative numbers. Zero multiplied by any number - there will be zero.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M .: Mnemosina, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics grade 6. - Gymnasium. 2006.
3. Depman I. Ya., Vilenkin N. Ya. Behind the pages of a mathematics textbook. - M .: Education, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Assignments for the course mathematics grade 5-6. - Moscow: ZSH MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. Grade 6 Student Handbook correspondence school MEPhI. - Moscow: ZSH MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-companion for grades 5-6 of high school. - M .: Education, Library of the teacher of mathematics, 1989.

Homework

1. Internet portal Mnemonica.ru ().
2. Internet portal Youtube.com ().
3. School-assistant.ru Internet portal ().
4. Internet portal Bymath.net ().

Now let's deal with multiplication and division.

Let's say we want to multiply +3 by -4. How to do it?

Let's consider this case. Three people are in debt, and each has $ 4 in debt. What is the total debt? In order to find it, you need to add up all three debts: $ 4 + $ 4 + $ 4 = $ 12. We decided that the addition of three numbers 4 is denoted as 3 × 4. Since we are talking about debt in this case, there is a "-" in front of 4. We know that the total debt is $ 12, so our problem now looks like 3x (-4) = - 12.

We will get the same result if, according to the problem statement, each of the four people has a debt of $ 3. In other words, (+4) x (-3) = - 12. And since the order of the factors does not matter, we get (-4) x (+3) = - 12 and (+4) x (-3) = - 12.

Let's summarize the results. When you multiply one positive and one negative number, the result will always be negative. The numerical value of the answer will be the same as in the case of positive numbers. Product (+4) x (+3) = + 12. The presence of the "-" sign affects only the sign, but does not affect the numerical value.

How do you multiply two negative numbers?

Unfortunately, it is very difficult to come up with a suitable example from life on this topic. It is easy to imagine a debt of $ 3 or $ 4, but it is completely impossible to imagine a -4 or -3 person going into debt.

Perhaps we will go the other way. In multiplication, when the sign of one of the factors changes, the sign of the product changes. If we change the signs of both factors, we must change twice work mark, first from positive to negative, and then vice versa, from negative to positive, that is, the product will have an initial sign.

Therefore, it is quite logical, although a little strange, that (-3) x (-4) = + 12.

Position of the sign when multiplied, changes like this:

• positive number x positive number = positive number;
• negative number x positive number = negative number;
• positive number x negative number = negative number;
• negative number x negative number = positive number.

In other words, multiplying two numbers with the same sign, we get a positive number. Multiplying two numbers with different signs, we get a negative number.

The same rule is true for the action opposite to multiplication - for.

You can easily verify this by holding inverse multiplication operations... If in each of the examples above, you multiply the quotient by the divisor, you get the dividend, and make sure it has the same sign, for example (-3) x (-4) = (+ 12).

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