§ 1 Multiplication of positive and negative numbers
In this lesson we will learn 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 equal to -6.
The numbers 6 and -6 are opposite numbers.
Thus, we can conclude:
When multiplying -1 by natural number the opposite number will be obtained.
For negative numbers, as well as for positive ones, the commutative law of multiplication is satisfied:
If you multiply a natural number by -1, you also get the opposite number
When you multiply any non-negative number by 1, you get the same number.
For example:
For negative numbers this statement is also true: -5 ∙1 = -5; -2 ∙ 1 = -2.
When you multiply any number by 1, you get the same number.
We have already seen that when you multiply minus 1 by a natural number, you get its opposite number. 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 combinatory multiplication law can be applied:
1 ∙ 21 = -21, i.e. the product of minus 3 and 7 is equal to minus 21.
When two numbers with different signs are multiplied, a negative number is obtained whose modulus is equal to the product of the moduli of the factors.
What is the product of numbers with the same signs?
We know that when two positive numbers are multiplied, the result is a positive number. Let's find the product of two negative numbers.
Let's replace one of the factors with a product with a factor of minus 1.
Let's apply the rule we 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,
it will turn out to be -80.
Let's formulate a rule:
When two numbers with the same signs are multiplied, a positive number is obtained whose modulus 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 will find the roots of the following equations:
y ∙ (-2) = 10. 5 ∙ 2 = 10, which means x = 5; 5 ∙ (-2) = -10, which means a = 5; -5 ∙ (-2) = 10, which means y = -5.
Let's write down the solutions to the equations. The factor in each equation 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 it.
When dividing numbers with the same signs (and these are the first and second equations), a positive number is obtained whose modulus 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 whose modulus 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 moduli of the dividend and the divisor.
Thus, we have formulated the rules for multiplying and dividing positive and negative numbers.
List of used literature:
- Mathematics. Grade 6: lesson plans for I.I.’s textbook. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilina. – Mnemosyne, 2009.
- Mathematics. 6th grade: textbook for students educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
- Mathematics. 6th grade: textbook for students of general education institutions./N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. – M.: Mnemosyne, 2013.
- Handbook of mathematics - http://lyudmilanik.com.ua
- Student's Guide to high school http://shkolo.ru
Topic of the open lesson: "Multiplying Negative and Positive Numbers"
Date of: 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 the development of mathematical speech, working memory, 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 initial presentation 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
-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 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 express your thoughts 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.
During the classes
Structural elements of the 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 a frontal survey 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.
5,2-6,7=-1,52,9+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.
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.
(Annex 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 about what numbers 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"
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, 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)
Rules for working in groups.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 exercises, 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 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 of 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 this topic cause you?
- 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 that you understand everything. 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.
Openness of students to transfer and comprehend their actions, to identify positive and negative aspects of the 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 assigned 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 dividing 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 number and vice versa, you need to divide the module of the dividend by the module of the divisor, and write the result with a minus sign.
Literally it looks like this:
a ÷ - b = - a ÷ b
A ÷ b = - a ÷ b.
The result of dividing numbers with different signs is always 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 numbers a and b that have different signs, you need to multiply the number a by the inverse of the number b, that is, b - 1. This formulation is applicable to the set of rational and real numbers; it allows you to move 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 will look at several typical examples.
Example 1. How to divide numbers with different signs?
Divide - 35 by 7.
First, let's write down the modules of the dividend and divisor:
35 = 35 , 7 = 7 .
Now let's separate the modules:
35 7 = 35 7 = 5 .
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 divide fractional numbers with rational signs, the dividend and divisor must be represented as ordinary fractions.
Example 3. How to divide numbers with different signs?
Divide the mixed number - 3 3 22 by the decimal fraction 0, (23).
The modules of the dividend and divisor are respectively equal to 3 3 22 and 0, (23). Converting 3 3 22 into a common fraction, we get:
3 3 22 = 3 22 + 3 22 = 69 22.
We can also represent the divisor as an ordinary 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 in the form of roots, logarithms, powers, etc.
In such a situation, the quotient is written in the form of 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?
Let's 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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In this article we will give a definition of dividing a negative number by a negative one, formulate and justify the rule, give examples of dividing negative numbers and analyze the process of solving them.
Dividing negative numbers. Rule
Let us recall what the essence of the division operation is. This action represents finding unknown multiplier by a known product and a known other factor. A number c is called the quotient of numbers a and b if the product c · b = a is true. In this case, a ÷ b = c.
Rule for dividing negative numbers
The quotient of dividing one negative number by another negative number is equal to the quotient of dividing the moduli of these numbers.
Let a and b be negative numbers. Then
a ÷ b = a ÷ b.
This rule reduces the division of two negative numbers to the division of positive numbers. This is true not only for integers, but also for rational and real numbers. The result of dividing a negative number by a negative number is always a positive number.
Let us give another formulation of this rule, suitable for rational and real numbers. It is given using reciprocal numbers and says: to divide a negative number a by the number undefined, multiply by the number b - 1, the inverse of b.
a ÷ b = a · b - 1 .
The same rule, which reduces division to multiplication, can also be used to divide numbers with different signs.
The equality a ÷ b = a · b - 1 can be proven using the property of multiplication of real numbers and the definition of reciprocal numbers. Let's write down the equalities:
a · b - 1 · b = a · b - 1 · b = a · 1 = a .
Due to the definition of the division operation, this equality proves that there is a quotient of dividing a number by the number b.
Let's move on to consider examples.
Let's start with simple cases and move on to more complex ones.
Example 1: How to divide negative numbers
Divide - 18 by - 3.
The moduli of the divisor and dividend are respectively 3 and 18. Let's write down:
18 ÷ - 3 = - 18 ÷ - 3 = 18 ÷ 3 = 6.
Example 2: How to divide negative numbers
Divide - 5 by - 2.
Similarly, we write according to the rule:
5 ÷ - 2 = - 5 ÷ - 2 = 5 ÷ 2 = 5 2 = 2 1 2.
The same result will be obtained if we use the second formulation of the rule with the inverse number.
5 ÷ - 2 = - 5 · - 1 2 = 5 · 1 2 = 5 2 = 2 1 2 .
When dividing fractional rational numbers, it is most convenient to represent them in the form of ordinary fractions. However, finite decimal fractions can also be divided.
Example 3. How to divide negative numbers
Let's divide - 0.004 by - 0.25.
First, we write down the moduli of these numbers: 0.004 and 0.25.
Now you can choose one of two ways:
- Separate decimal fractions using a column.
- Go to fractions and do division.
Let's look at both methods.
1. When dividing decimal fractions with a column, move the decimal point two digits to the right.
Answer: - 0.004 ÷ 0.25 = 0.016
2. Now we will give a solution with the conversion of decimal fractions to ordinary ones.
0.004 = 4 1000; 0.25 = 25 100 0.004 ÷ 0.25 = 4 1000 ÷ 25 100 = 4 1000 100 25 = 4 250 = 0.016
The results obtained are consistent.
In conclusion, we note that if the dividend and divisor are irrational numbers and are given in terms of roots, powers, logarithms, etc., the result of division is written as a numerical expression, the approximate value of which is calculated if necessary.
Example 4: How to divide negative numbers
Let's calculate the quotient of dividing the numbers - 0, 5 and - 5.
0, 5 ÷ - 5 = - 0, 5 ÷ - 5 = 0, 5 ÷ 5 = 1 2 1 5 = 1 2 5 = 5 10.
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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.
Multiplying Negative Numbers
Definition 1Rule 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 is applicable for multiplying real numbers, rational numbers, and 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 of the given numbers is 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 factors are irrational numbers, then their product can be written as a 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.
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