How to multiply a negative number by a positive one. Multiplying fractions with different signs

Topic of the open lesson: "Multiplying Negative and Positive Numbers"

Date: 03/17/2017

Teacher: Kuts V.V.

Class: 6 g

Purpose and objectives of the lesson:

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

promote development mathematical speech, RAM, voluntary attention, visual and effective thinking;

formation internal processes intellectual, personal, emotional development.

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

Lesson type: lesson of initial presentation of new knowledge

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

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

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

Planned results training

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

Regulatory – be able to determine and formulate a goal in a lesson with the help of a teacher; 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; make the necessary adjustments to the action after its completion based on its assessment and taking into account the errors made; express your guess.Communication - be able to formulate your thoughts into orally; listen and understand the speech of others; jointly agree on the rules of behavior and communication at school and follow them.

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

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

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

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

Lesson progress

Structural elements lesson

Didactic tasks

Designed teacher activity

Designed student activities

Result

1.Organizational moment

Motivation for successful activities

Checking 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'm glad to see you in class today in a good mood.

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

I also wish you good work today.

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

“The only way to learn is to have fun. To digest knowledge, you need to absorb it with appetite.”

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

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

So let’s quickly open our notebooks and write down the number, great job.

Emotional mood

-With interest, with pleasure.

Ready to start lesson

Positive motivation to study new topic

2. Activation cognitive activity

Prepare them to learn new knowledge and ways of acting.

Organize frontal survey based on the material covered.

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

So now I’ll test you how well you can count.

We will now do a mathematical warm-up.

We work as usual, count verbally and write down the answer in writing. I'll give you 1 minute.

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 guys.

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.

Readiness of students for production problematic issue, to find ways to solve the problem.

3. Motivation for setting the topic and goal 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 learning new material, but first, let's review the material from previous lessons. A mathematical crossword puzzle will help us with this.

But this crossword is not an ordinary one, it encrypts keyword, which 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 since it’s in pairs, then remind me how it’s like in pairs?

We remembered the rule of working in pairs, and now let’s start solving the crossword puzzle, I’ll give you 1.5 minutes. Whoever does everything, put your hands down so I can see.

(Appendix 1)

1.What numbers are used for counting?

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

3.Numbers that are represented by a fraction are called?

4. What are two numbers that differ from each other only in signs?

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

6.What are the natural numbers, their opposites and zero called?

7.What number is called neutral?

8. Number showing the position of a point on a line?

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

So, time is up. Let's check.

We have solved the entire crossword puzzle and thereby repeated the material from 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?

What are we going to multiply today?

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

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

What numbers will we learn to multiply today?

Write down the topic of the lesson in your notebook: “Multiplying positive and negative numbers.”

So, guys, we found out what we will talk about today in class.

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, in order to achieve this goal, what problems will we have to solve with you?

Absolutely 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

"Multiplying Positive and Negative Numbers"

Objective 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, once we have the rule, what should we do next? (learn to apply it when solving examples).

4. Learning new knowledge and ways of doing things

Gain new knowledge on the topic.

-Organize work in groups (learning new material)

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

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

Your research work will be carried out in groups, we will have 5 research groups in total.

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

Your goal research work: While exploring the problems, gradually derive the rule “Multiplying negative and positive numbers” in task No. 2; in task No. 1 you have a total of 4 problems. And to solve these problems, our thermometer will help you, each group has one.

Make all your notes on a piece of paper.

Once 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 out the table, conduct research)

Working in groups is very easy

Know how to follow five rules:

first of all: don’t interrupt,

when he talks

friend, there should be silence around;

second: don’t shout loudly,

and give 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: summarize, think,

what could you do.

Mastery

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

5. Physical training

Establish the correctness of assimilation of new material at this stage, identify misconceptions and correct them

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

What conclusions can we draw from examining the table?

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

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

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

4th 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 blanks in the second task.

How to multiply a negative number by a positive one?

- How to multiply two negative numbers?

Let's take a little rest.

A positive answer means we sit down, a negative answer we stand 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)

When multiplying positive numbers, the answer always results in a positive number.

When you multiply a negative number by a positive number, the answer is always a negative number.

When multiplying negative numbers, the answer always results in 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 the sign in front of the resulting number «+».

Students perform physical exercise, reinforcing the rules.

Prevents fatigue

7.Primary consolidation of new material

Master the ability to apply acquired knowledge in practice.

Organize frontal and independent work based on the material covered.

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

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

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 board

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 is equal to the number on the roof of the house.

Solve examples using acquired knowledge

Raise your hands if you haven't made any mistakes, well done...

Active actions students to apply knowledge in life.

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

Ensure student reflection, i.e. their assessment of their activities

Organize a lesson summary

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 did it cause you? this topic?

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

A smiling emoticon means you understand. Green means that you understand, but need to practice, and a sad smiley if you haven’t understood anything at all. (I'll give you half a minute)

Well, guys, are you ready to show how you worked in class today? So, let’s raise it and I’ll also raise a smiley face for you.

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

The lesson is over, thanks for your attention!

Answer questions and evaluate their work

Yes, we have achieved it.

Students’ openness to the transfer and comprehension of their actions, to identifying positive and negative points lesson

10 .Homework information

Provide an understanding of the purpose, content and methods of implementation homework

Provides understanding of the purpose of homework.

Homework:

1. Learn multiplication rules
2.No. 1121(3 column).
3.Creative task: make a test of 5 questions with answer options.

Write down your homework, trying to comprehend and understand.

Implementation of the need to achieve conditions for 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 multiplication by zero remains true in the case of negative numbers. Zero multiplied by any number equals zero.

References

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - 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 mathematics course, grades 5-6. - M.: ZSh MEPhI, 2011.
5. 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.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for grades 5-6 high school. - M.: Education, Mathematics Teacher Library, 1989.

Homework

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

In this article we will formulate the rule for multiplying negative numbers and give an explanation for it. The process of multiplying negative numbers will be discussed in detail. The examples show all possible cases.

Yandex.RTB R-A-339285-1

Multiplying Negative Numbers

Rule for multiplying negative numbers is that in order to multiply two negative numbers, it is necessary to multiply their modules. This rule is written as follows: for any negative numbers – a, - b, this equality is considered true.

(- a) · (- b) = a · b.

Above is the rule for multiplying two negative numbers. Based on it, we prove the expression: (- a) · (- b) = a · b. The article multiplying numbers with different signs says that the equalities a · (- b) = - a · b are valid, as is (- a) · b = - a · b. This follows from the property of opposite numbers, due to which the equalities will be written as follows:

(- a) · (- b) = (- a · (- b)) = - (- (a · b)) = a · b.

Here you can clearly see the proof of the rule for multiplying negative numbers. Based on the examples, it is clear that the product of two negative numbers is a positive number. When multiplying moduli of numbers, the result is always a positive number.

This rule applies to multiplication real numbers, rational numbers, integers.

Now let's look at examples of multiplying two negative numbers in detail. When calculating, you must use the rule written above.

Example 1

Multiply numbers - 3 and - 5.

Solution.

The absolute value of the two numbers being multiplied is equal to the positive numbers 3 and 5. Their product results in 15. It follows that the product given numbers equals 15

Let us briefly write down the multiplication of negative numbers itself:

(- 3) · (- 5) = 3 · 5 = 15

Answer: (- 3) · (- 5) = 15.

When multiplying negative rational numbers, using the discussed rule, you can mobilize to multiply fractions, multiply mixed numbers, multiply decimals.

Example 2

Calculate the product (- 0 , 125) · (- 6) .

Solution.

Using the rule for multiplying negative numbers, we obtain that (− 0, 125) · (− 6) = 0, 125 · 6. To get the result you need to multiply decimal by a natural number of columns. It looks like this:

We found that the expression will take the form (− 0, 125) · (− 6) = 0, 125 · 6 = 0, 75.

Answer: (− 0, 125) · (− 6) = 0, 75.

In the case when the multipliers are irrational numbers, then their product can be written in the form numerical expression. The value is calculated only when necessary.

Example 3

It is necessary to multiply negative - 2 by non-negative log 5 1 3.

Solution

Finding the modules of the given numbers:

2 = 2 and log 5 1 3 = - log 5 3 = log 5 3 .

Following from the rules for multiplying negative numbers, we get the result - 2 · log 5 1 3 = - 2 · log 5 3 = 2 · log 5 3 . This expression is the answer.

Answer: - 2 · log 5 1 3 = - 2 · log 5 3 = 2 · log 5 3 .

To continue studying the topic, you must repeat the section on multiplying real numbers.

If you notice an error in the text, please highlight it and press Ctrl+Enter

In this article we will deal with multiplying numbers with different signs. Here we will first formulate the rule for multiplying positive and negative numbers, justify it, and then consider the application of this rule when solving examples.

Page navigation.

Rule for multiplying numbers with different signs

Multiplying a positive number by a negative number, as well as a negative number by a positive number, is carried out as follows: the rule for multiplying numbers with different signs: to multiply numbers with different signs, you need to multiply and put a minus sign in front of the resulting product.

Let's write it down this rule in letter form. For any positive real number a and a real negative number −b the following equality holds: a·(−b)=−(|a|·|b|) , and also for a negative number −a and a positive number b the equality (−a)·b=−(|a|·|b|) .

The rule for multiplying numbers with different signs is fully consistent with properties of operations with real numbers. Indeed, on their basis it is easy to show that for real and positive numbers a and b a chain of equalities of the form a·(−b)+a·b=a·((−b)+b)=a·0=0, which proves that a·(−b) and a·b are opposite numbers, which implies the equality a·(−b)=−(a·b) . And from it follows the validity of the multiplication rule in question.

It should be noted that the stated rule for multiplying numbers with different signs is valid for both real numbers and rational numbers and for integers. This follows from the fact that operations with rational and integer numbers have the same properties that were used in the proof above.

It is clear that multiplying numbers with different signs according to the resulting rule comes down to multiplying positive numbers.

It remains only to consider examples of the application of the disassembled multiplication rule when multiplying numbers with different signs.

Examples of multiplying numbers with different signs

Let's look at several solutions examples of multiplying numbers with different signs. Let's start with simple case, to focus on the rule steps rather than the computational complexities.

Example.

Multiply the negative number −4 by the positive number 5.

Solution.

According to the rule for multiplying numbers with different signs, we first need to multiply the modules of the original factors. The modulus of −4 is equal to 4, and the modulus of 5 is equal to 5, and multiplication of natural numbers 4 and 5 gives 20. Finally, it remains to put a minus sign in front of the resulting number, we have −20. This completes the multiplication.

Briefly, the solution can be written as follows: (−4)·5=−(4·5)=−20.

Answer:

(−4)·5=−20.

When multiplying fractional numbers with different signs you need to be able to perform multiplying common fractions , multiplying decimals and their combinations with natural and mixed numbers.

Example.

Multiply numbers with different signs 0, (2) and .

Solution.

Having completed converting a periodic decimal fraction to a common fraction, and also by doing moving from a mixed number to an improper fraction, from the original work we will come to the product of ordinary fractions with different signs of the form. This product, according to the rule of multiplying numbers with different signs, is equal to . All that remains is to multiply common fractions in parentheses, we have .