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.
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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 this rule down in letter form. For any positive real number a and any negative real number −b, the equality 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 for 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.
Multiply the negative number −4 by the positive number 5.
According to the rule for multiplying numbers with different signs, we first need to multiply the absolute values of the original factors. The modulus of −4 is 4, and the modulus of 5 is 5, and multiplying the 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.
(−4)·5=−20.
When multiplying fractional numbers with different signs, you need to be able to perform multiplication ordinary fractions, multiplication of decimal fractions and their combinations with natural and mixed numbers.
Multiply numbers with different signs 0, (2) and.
Having translated the periodical decimal into a common fraction, and also by moving from a mixed number to improper fraction, from the original product we will come to the product of ordinary fractions with different signs of the form. This product is equal to the rule for multiplying numbers with different signs. All that remains is to multiply the ordinary fractions in brackets, we have 
.
.
Separately, it is worth mentioning the multiplication of numbers with different signs, when one or both factors are
Now let's deal with multiplication and division.
Let's say we need to multiply +3 by -4. How to do it?
Let's consider such a case. Three people got into debt and each had $4 in debt. What is the total debt? In order to find it, you need to add up all three debts: 4 dollars + 4 dollars + 4 dollars = 12 dollars. We decided that the addition of three numbers 4 is denoted as 3x4. Since in in this case we are talking about debt, there is a “-” sign before the 4. We know that the total debt is $12, so our problem now becomes 3x(-4)=-12.
We will get the same result if, according to the problem, 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 number and one negative number, the result will always be a negative number. 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 only affects the sign, but does not affect the numerical value.
How to multiply two negative numbers?
Unfortunately, it is very difficult to come up with a suitable real-life example on this topic. It is easy to imagine a debt of 3 or 4 dollars, but it is absolutely impossible to imagine -4 or -3 people who got into debt.
Perhaps we will go a different 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.
Sign position when multiplied it 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 signs, 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 running inverse multiplication operations. In each of the examples above, if you multiply the quotient by the divisor, you will get the dividend and make sure it has the same sign, for example (-3)x(-4)=(+12).
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This article gives detailed review 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, a rule for dividing integers with different signs was obtained. It can be extended to both rational numbers and real numbers by repeating all the reasoning from the above 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, you need to divide the dividend by the modulus of the divisor, and put a minus sign in front of the resulting number.
Let's 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 stated 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 positive numbers, their quotient is a positive number, and the minus sign makes this number negative.
Note that the rule considered 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 inverse 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 applies to the set of rational numbers as well as the set of real numbers.
It is clear that this rule for dividing numbers with different signs allows you to move 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 when solving examples.
Examples of dividing numbers with different signs
Let us consider solutions to several characteristic examples of dividing numbers with different signs to understand the principle of applying the rules from the previous paragraph.
Divide the negative number −35 by the positive number 7.
The rule for dividing numbers with different signs prescribes first finding 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 module of the dividend by the module of the divisor, that is, we need to divide 35 by 7. Remembering how division of natural numbers is performed, we get 35:7=5. The last step left in the rule for dividing numbers with different signs is to put a minus in front of the resulting number, we have −5.
Here's the whole solution: .
It was possible to proceed from a different formulation of the rule for dividing numbers with different signs. In this case, we first find the inverse of the divisor 7. This number is the common fraction 1/7. Thus, . It remains to multiply numbers with different signs: . Obviously, we came to the same result.
(−35):7=−5 .
Calculate the quotient 8:(−60) .
According to the rule for 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 
.
Let's write down the whole solution briefly: .
.
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 other notation (for example, in decimal).
The modulus of the dividend is equal, 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.
Task 1. A point moves in a straight line from left to right at a speed of 4 dm. per second and per currently passes through point A. Where will the moving point be after 5 seconds?
It is not difficult to figure out that the point will be at 20 dm. to the right of A. Let's write the solution to this problem in relative numbers. To do this, we agree on the following symbols:
1) the speed to the right will be denoted by the sign +, and to the left by the sign –, 2) the distance of the moving point from A to the right will be denoted by the sign + and to the left by the sign –, 3) the period of time after the present moment by the sign + and before the present moment by the sign –. In our problem, the following numbers are given: speed = + 4 dm. per second, time = + 5 seconds and it turned out, as we figured out arithmetically, the number + 20 dm., expressing the distance of the moving point from A after 5 seconds. Based on the meaning of the problem, we see that it relates to multiplication. Therefore, it is convenient to write the solution to the problem:
(+ 4) ∙ (+ 5) = + 20.
Task 2. A 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, keeping in mind that the meaning of the problem has not changed, write it like this:
(+ 4) ∙ (– 5) = – 20.
Task 3. A 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 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 dm. to the right of A. Therefore, the solution to this problem should be written as follows:
(– 4) ∙ (– 5) = + 20.
The problems considered indicate how the action of multiplication should be extended to relative numbers. In the problems we have 4 cases of multiplying 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 must have a + sign when the factors identical signs(1st and 4th cases) and sign –, when the factors have different signs(cases 2 and 3).
From here we see that the product does not change from rearranging the multiplicand and the multiplier.
Exercises.
Let's do one example of a calculation that involves addition, subtraction, and multiplication.
In order not to confuse the order of actions, let us pay attention to the formula
Here is written the sum of the products of two pairs of numbers: 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 Eq.
You must first multiply the number b by c and then subtract the resulting product from a.
If it were necessary to add the product of numbers a and b with c and multiply the resulting sum by d, then one should write: (ab + c)d (compare with the formula ab + cd).
If we had to multiply the difference between the numbers a and b by c, we would write (a – b)c (compare with the formula a – bc).
Therefore, let us establish in general that if the order of actions is not indicated by parentheses, then we must first perform multiplication, and then add or subtract.
Let's start calculating our expression: let's first perform the additions written inside all the small brackets, we get:
Now we need to do the multiplication inside square brackets and then subtract the resulting product from:
Now let's perform the operations 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 you need to multiply the first number by the second, the resulting product by the 3rd, etc. It is not difficult to establish on the basis of the previous one that the absolute values of all numbers must be multiplied among themselves.
If all the factors were positive, then based on the previous one we will find that the product must also have a + sign. If any one factor were negative
e.g., (+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) the new product would have a - sign; multiplying it by the next positive factor (in our example –24 by +5), we again obtain a negative number; since all other factors are assumed to be positive, the sign of the product cannot change any more.
If there were two negative factors, then, reasoning as above, we would find that at first, until we reached the first negative factor, the product would be positive; by multiplying it by the first negative factor, the new product would turn out to be negative, and so would it be. remained until we reach the second negative factor; Then, by multiplying a negative number by a negative, the new product would be positive, which will remain so in the future if the remaining factors are positive.
If there were a third negative factor, then the resulting positive product 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 a fourth negative factor, then multiplying by it will make the product positive. Reasoning in the same way, we find that in general:
To find out the sign of the product of several factors, you need to look at how many of these factors are negative: if there are none at all, or if there are even number, then the product is positive: if negative multipliers 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 not difficult 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 the work immediately:
This is convenient because you don’t have to do useless multiplications, since the previously obtained fractional expression is reduced as much as possible.
Table 5
Table 6
With some stretch, the same explanation is valid for the product 1-5, if we assume that the “sum” is from one single
term is equal to this term. But the product 0 5 or (-3) 5 cannot be explained this way: what does the sum of zero or minus three terms mean?
However, you can rearrange the factors
If we want the product not to change when the factors are rearranged - as was the case for positive numbers - then we must assume that
Now let's move on to the product (-3) (-5). What is it equal to: -15 or +15? Both options have a reason. On the one hand, a minus in one factor already makes the product negative - all the more so it should be negative if both factors are negative. On the other hand, in table. 7 already has two minuses, but only one plus, and “in fairness” (-3)-(-5) should be equal to +15. So which should you prefer?
Table 7
Of course, you won’t be confused by such talk: from school course Mathematicians You have firmly learned that minus times minus gives a plus. But imagine that your younger brother or sister asks you: why? What is this - a teacher’s whim, an order from higher authorities, or a theorem that can be proven?
Usually the multiplication rule negative numbers explain with examples like those presented in table. 8.
Table 8
It can be explained differently. Let's write the numbers in a row
Now let's write the same numbers multiplied by 3:
It is easy to notice that each number is 3 more than the previous one. Now let's write the same numbers in reverse order(starting, for example, with 5 and 15):
Moreover, under the number -5 there was a number -15, so 3 (-5) = -15: plus by minus gives a minus.
Now let's repeat the same procedure, multiplying the numbers 1,2,3,4,5 ... by -3 (we already know that plus by minus gives minus):
Each next number the bottom row is 3 less than the previous one. Write the numbers in reverse order
and continue:
Under the number -5 there are 15, so (-3) (-5) = 15.
Perhaps these explanations would satisfy your younger brother or sister. But you have the right to ask how things really are and is it possible to prove that (-3) (-5) = 15?
The answer here is that we can prove that (-3) (-5) must equal 15 if we want the ordinary properties of addition, subtraction and multiplication to remain true for all numbers, including negative ones. The outline of this proof is as follows.
Let us first prove that 3 (-5) = -15. What is -15? This is the opposite number of 15, that is, the number that when added to 15 gives 0. So we need to prove that
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 development mathematical speech, random access 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 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
-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 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 of 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 lesson
Didactic tasks
Designed teacher activity
Designed student activities
Result
1.Organizational moment
Motivation to 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 from great pleasure, because they 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 correct assimilation of new material on 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 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.
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.
Realization of the need to achieve conditions for successful implementation homework by all students, in accordance with the task and the level of development of the students
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
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 opposite numbers, thanks 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 modulus of the two numbers being multiplied are equal 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, applying the discussed rule, you can mobilize to multiply fractions, multiply mixed numbers, multiplying 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 obtain the result, you must multiply the decimal fraction by natural number 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.
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