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 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 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 identical 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 lesson covers multiplication and division. rational numbers.
Lesson contentMultiplying rational numbers
The rules for multiplying integers also apply to rational numbers. In other words, to multiply rational numbers, you need to be able to
Also, you need to know the basic laws of multiplication, such as: the commutative law of multiplication, the associative law of multiplication, the distributive law of multiplication and multiplication by zero.
Example 1. Find the value of an expression
This is the multiplication of rational numbers with different signs. To multiply rational numbers with different signs, you need to multiply their modules and put a minus in front of the resulting answer.
To clearly see that we are dealing with numbers that have different signs, we enclose each rational number in brackets along with its signs
The modulus of the number is equal to , and the modulus of the number is equal to . Multiplying the resulting modules as positive fractions, we received an answer, but before the answer we put a minus, as the rule required of us. To ensure this minus before the answer, the multiplication of modules was performed in parentheses, preceded by a minus.
The short solution looks like this:
Example 2. Find the value of an expression 
Example 3. Find the value of an expression 
This is the multiplication of negative rational numbers. To multiply negative rational numbers, you need to multiply their modules and put a plus in front of the resulting answer
Solution for this example can be written briefly:
Example 4. Find the value of an expression 
The solution for this example can be written briefly:
Example 5. Find the value of an expression
This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer
The short solution will look much simpler:
Example 6. Find the value of an expression
Let's convert the mixed number to improper fraction. Let's rewrite the rest as it is
We obtained the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer. The entry with modules can be skipped so as not to clutter the expression
The solution for this example can be written briefly
Example 7. Find the value of an expression
This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer
At first the answer turned out to be an improper fraction, but we highlighted the whole part in it. note that whole part was separated from the fraction module. The resulting mixed number was enclosed in parentheses preceded by a minus sign. This is done to ensure that the requirement of the rule is fulfilled. And the rule required that the answer received be preceded by a minus.
The solution for this example can be written briefly:
Example 8. Find the value of an expression 
First, let's multiply and and multiply the resulting number with the remaining number 5. We'll skip the entry with modules so as not to clutter the expression.
Answer: expression value equals −2.
Example 9. Find the meaning of the expression: 
Let's translate mixed numbers to improper fractions:
We got the multiplication of negative rational numbers. Let's multiply the modules of these numbers and put a plus in front of the resulting answer. The entry with modules can be skipped so as not to clutter the expression
Example 10. Find the value of an expression
The expression consists of several factors. According to combinational law multiplication, if the expression consists of several factors, then the product will not depend on the order of operations. This allows us to calculate this expression in any order.
Let's not reinvent the wheel, but calculate this expression from left to right in the order of the factors. Let's skip the entry with modules so as not to clutter the expression
Third action:
Fourth action:
Answer: the value of the expression is
Example 11. Find the value of an expression
Let's remember the law of multiplication by zero. This law states that a product is equal to zero if at least one of the factors equal to zero.
In our example, one of the factors is equal to zero, so without wasting time we answer that the value of the expression is equal to zero:
Example 12. Find the value of an expression 
The product is equal to zero if at least one of the factors is equal to zero.
In our example, one of the factors is equal to zero, so without wasting time we answer that the value of the expression equals zero:
Example 13. Find the value of an expression 
You can use the order of actions and first calculate the expression in brackets and multiply the resulting answer with a fraction.
You can also use the distributive law of multiplication - multiply each term of the sum by a fraction and add the resulting results. We will use this method.
According to the order of operations, if an expression contains addition and multiplication, then the multiplication must be performed first. Therefore, in the resulting new expression, let’s put in brackets those parameters that must be multiplied. This way we can clearly see which actions to perform earlier and which later:
Third action:
Answer: expression value equals
The solution for this example can be written much shorter. It will look like this:
It is clear that this example could be solved even in one’s mind. Therefore, you should develop the skill of analyzing an expression before solving it. It is likely that it can be solved mentally and save a lot of time and nerves. And in tests and exams, as you know, time is very valuable.
Example 14. Find the value of the expression −4.2 × 3.2
This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer
Notice how the modules of rational numbers were multiplied. In this case, to multiply the moduli of rational numbers, it took .
Example 15. Find the value of the expression −0.15 × 4
This is the multiplication of rational numbers with different signs. Let's multiply the modules of these numbers and put a minus in front of the resulting answer
Notice how the modules of rational numbers were multiplied. In this case, in order to multiply the moduli of rational numbers, it was necessary to be able to.
Example 16. Find the value of the expression −4.2 × (−7.5)
This is the multiplication of negative rational numbers. Let's multiply the modules of these numbers and put a plus in front of the resulting answer
Division of rational numbers
The rules for dividing integers also apply to rational numbers. In other words, to be able to divide rational numbers, you need to be able to
Otherwise, the same methods for dividing ordinary and decimal fractions are used. To divide a common fraction by another fraction, you need to multiply the first fraction by the reciprocal of the second fraction.
And to divide decimal to another decimal fraction, you need to move the decimal point in the dividend and in the divisor to the right by as many digits as there are after the decimal point in the divisor, then perform the division as with a regular number.
Example 1. Find the meaning of the expression:
This is the division of rational numbers with different signs. To calculate such an expression, you need to multiply the first fraction by the reciprocal of the second.
So, let's multiply the first fraction by the reciprocal of the second.
We obtained the multiplication of rational numbers with different signs. And we already know how to calculate such expressions. To do this, you need to multiply the moduli of these rational numbers and put a minus in front of the resulting answer.
Let's complete this example to the end. The entry with modules can be skipped so as not to clutter the expression
So the value of the expression is
The detailed solution is as follows:
A short solution would look like this:
Example 2. Find the value of an expression
This is the division of rational numbers with different signs. To calculate this expression, you need to multiply the first fraction by the reciprocal of the second.
The reciprocal of the second fraction is the fraction . Let's multiply the first fraction by it:
A short solution would look like this:
Example 3. Find the value of an expression
This is the division of negative rational numbers. To calculate this expression, you again need to multiply the first fraction by the reciprocal of the second.
The reciprocal of the second fraction is the fraction . Let's multiply the first fraction by it:
We got the multiplication of negative rational numbers. How is it calculated similar expression we already know. You need to multiply the moduli of rational numbers and put a plus in front of the resulting answer.
Let's finish this example to the end. You can skip the entry with modules so as not to clutter the expression:
Example 4. Find the value of an expression
To calculate this expression, you need to multiply the first number −3 by the fraction, reciprocal fraction.
The inverse of a fraction is the fraction . Multiply the first number −3 by it
Example 6. Find the value of an expression
To calculate this expression, you need to multiply the first fraction by the number reciprocal of number 4.
The reciprocal of the number 4 is a fraction. Multiply the first fraction by it
Example 5. Find the value of an expression
To calculate this expression, you need to multiply the first fraction by the inverse of −3
The inverse of −3 is a fraction. Let's multiply the first fraction by it:
Example 6. Find the value of the expression −14.4: 1.8
This is the division of rational numbers with different signs. To calculate this expression, you need to divide the module of the dividend by the module of the divisor and put a minus before the resulting answer.
Notice how the module of the dividend was divided by the module of the divisor. In this case, to do it correctly, it was necessary to be able to.
If you don't want to mess around with decimals (and this happens often), then these, then convert these mixed numbers into improper fractions, and then do the division itself.
Let's calculate the previous expression −14.4: 1.8 this way. Let's convert decimals to mixed numbers:
Now let’s convert the resulting mixed numbers into improper fractions:
Now you can do division directly, namely, divide a fraction by a fraction. To do this, you need to multiply the first fraction by the inverse fraction of the second:
Example 7. Find the value of an expression 
Let's convert the decimal fraction −2.06 to an improper fraction, and multiply this fraction by the reciprocal of the second fraction:
Multistory fractions
You can often come across an expression in which the division of fractions is written using a fraction line. For example, the expression could be written as follows:
What is the difference between the expressions and ? There's really no difference. These two expressions carry the same meaning and we can put an equal sign between them:
In the first case, the division sign is a colon and the expression is written on one line. In the second case, the division of fractions is written using a fraction line. The result is a fraction that people agree to call multi-storey.
When encountering such multi-story expressions, you need to apply the same rules of division ordinary fractions. The first fraction must be multiplied by the reciprocal of the second.
Use in solution similar fractions extremely inconvenient, so you can write them in an understandable form, using a colon rather than a slash as a division sign.
For example, let's write a multi-story fraction in an understandable form. To do this, you first need to figure out where the first fraction is and where the second is, because it is not always possible to do this correctly. Multistory fractions have several fraction lines that can be confusing. The main fraction line, which separates the first fraction from the second, is usually longer than the rest.
After determining the main fractional line, you can easily understand where the first fraction is and where the second is:
Example 2.
We find the main fraction line (it is the longest) and see that the integer −3 is divided by a common fraction
And if we mistakenly took the second fractional line as the main one (the one that is shorter), then it would turn out that we are dividing the fraction by the integer 5. In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the dividend in this In this case, the number is −3, and the divisor is the fraction .
Example 3. Let's write the multi-level fraction in an understandable form
We find the main fraction line (it is the longest) and see that the fraction is divided by the integer 2
And if we mistakenly took the first fractional line as the leading one (the one that is shorter), then it would turn out that we are dividing the integer −5 by the fraction. In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the dividend in this case the fraction is , and the divisor is the integer 2.
Despite the fact that multi-level fractions are inconvenient to work with, we will encounter them very often, especially when studying higher mathematics.
Naturally, it takes Extra time and place. Therefore, you can use more quick method. This method is convenient and the output allows you to get a ready-made expression in which the first fraction has already been multiplied by the reciprocal fraction of the second.
This method is implemented as follows:
If the fraction is four-story, for example, then the number located on the first floor is raised to the top floor. And the figure located on the second floor is raised to the third floor. The resulting numbers must be connected with multiplication signs (×)
As a result, bypassing the intermediate notation, we obtain a new expression in which the first fraction has already been multiplied by the reciprocal fraction of the second. Convenience and that's it!
To avoid errors when using this method, you can be guided by the following rule:
From first to fourth. From second to third.
In the rule we're talking about about the floors. The figure from the first floor must be raised to the fourth floor. And the figure from the second floor needs to be raised to the third floor.
Let's try to calculate a multi-story fraction using the above rule.
So, we raise the number located on the first floor to the fourth floor, and raise the number located on the second floor to the third floor
As a result, bypassing the intermediate notation, we obtain a new expression in which the first fraction has already been multiplied by the reciprocal fraction of the second. Next, you can use your existing knowledge:
Let's try to calculate a multi-level fraction using a new scheme.
There are only the first, second and fourth floors. There is no third floor. But we do not deviate from the basic scheme: we raise the figure from the first floor to the fourth floor. And since there is no third floor, we leave the number located on the second floor as is
As a result, bypassing the intermediate notation, we received a new expression in which the first number −3 has already been multiplied by the reciprocal fraction of the second. Next, you can use your existing knowledge:
Let's try to calculate the multi-story fraction using the new scheme.
There are only the second, third and fourth floors. There is no first floor. Since there is no first floor, there is nothing to go up to the fourth floor, but we can raise the figure from the second floor to the third:
As a result, bypassing the intermediate notation, we received a new expression in which the first fraction has already been multiplied by the inverse of the divisor. Next, you can use your existing knowledge:
Using Variables
If the expression is complex and it seems to you that it will confuse you in the process of solving the problem, then part of the expression can be put into a variable and then work with this variable.
Mathematicians often do this. A difficult task break them down into easier subtasks and solve them. Then the solved subtasks are collected into one single whole. This creative process and this is something one learns over the years through hard training.
The use of variables is justified when working with multi-level fractions. For example:
Find the value of an expression
So, there is a fractional expression in the numerator and in the denominator of which fractional expressions. In other words, we are again faced with a multi-story fraction, which we do not like so much.
The expression in the numerator can be entered into a variable with any name, for example:
But in mathematics, in such a case, it is customary to name variables using capital Latin letters. Let's not break this tradition, and denote the first expression with a big Latin letter A
And the expression in the denominator can be denoted by the capital letter B
Now our original expression takes the form . That is, we made a replacement numerical expression to a letter, having previously entered the numerator and denominator into the variables A and B.
Now we can separately calculate the value of variable A and the value of variable B. Ready values we will insert .
Let's find the value of the variable A
Let's find the value of the variable B
Now let’s substitute their values into the main expression instead of variables A and B:
We have obtained a multi-story fraction in which we can use the scheme “from the first to the fourth, from the second to the third,” that is, raise the number located on the first floor to the fourth floor, and raise the number located on the second floor to the third floor. Further calculations will not be difficult:
Thus, the value of the expression is −1.
Of course we have considered simplest example, but our goal was to learn how we can use variables to make things easier for ourselves, to minimize errors.
Note also that the solution for this example can be written without using variables. It will look like
This solution is faster and shorter, and in this case it makes more sense to write it this way, but if the expression turns out to be complex, consisting of several parameters, brackets, roots and powers, then it is advisable to calculate it in several stages, entering part of its expressions into variables.
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Educational:
- Fostering activity;
Lesson type
Equipment:
- Projector and computer.
Lesson Plan
1.Organizational moment
2. Updating knowledge
3. Mathematical dictation
4.Test execution
5. Solution of exercises
6. Lesson summary
7. Homework.
During the classes
1. Organizational moment
Today we will continue to work on multiplying and dividing positive and negative numbers. The task of each of you is to figure out how he mastered this topic, and if necessary, to refine what is not yet completely working out. In addition, you will learn a lot of interesting things about the first month of spring - March. (Slide1)
2. Updating knowledge.
3x=27; -5 x=-45; x:(2.5)=5.
3. Mathematical dictation(slide 6.7)
Option 1
Option 2
4. Running the test ( slide 8)
Answer : Martius
5.Solution of exercises
(Slides 10 to 19)
March 4 -
2) y×(-2.5)=-15
March, 6
3) -50, 4:x=-4, 2
4) -0.25:5×(-260)
March 13
5) -29,12: (-2,08)
March 14th
6) (-6-3.6×2.5) ×(-1)
7) -81.6:48×(-10)
March 17
8) 7.15×(-4): (-1.3)
March 22
9) -12.5×50: (-25)
10) 100+(-2,1:0,03)
30th of March
6. Lesson summary
7. Homework:
View document contents
“Multiplying and dividing numbers with different signs”
Lesson topic: “Multiplication and division of numbers with different signs.”
Lesson objectives: repetition of the studied material on the topic “Multiplication and division of numbers with different signs”, practicing skills in using multiplication and division operations positive number to a negative number and vice versa, as well as a negative number to a negative number.
Lesson objectives:
Educational:
Consolidation of rules on this topic;
Formation of skills and abilities to work with operations of multiplication and division of numbers with different signs.
Educational:
Development of cognitive interest;
Development logical thinking, memory, attention;
Educational:
Fostering activity;
Instilling skills in students independent work;
Fostering a love of nature, instilling an interest in folk signs.
Lesson type. Lesson-repetition and generalization.
Equipment:
Projector and computer.
Lesson Plan
1.Organizational moment
2. Updating knowledge
3. Mathematical dictation
4.Test execution
5. Solution of exercises
6. Lesson summary
7. Homework.
During the classes
1. Organizational moment
Hello guys! What did we do in previous lessons? (Multiplying and dividing rational numbers.)
Today we will continue to work on multiplying and dividing positive and negative numbers. The task of each of you is to figure out how he mastered this topic, and if necessary, to refine what is not yet completely working out. In addition, you will learn a lot of interesting things about the first month of spring – March. (Slide1)
2. Updating knowledge.
Review the rules for multiplying and dividing positive and negative numbers.
Recall mnemonic rule. (Slide 2)
Perform multiplication: (slide 3)
5x3; 9×(-4); -10×(-8); 36×(-0.1); -20×0.5; -13×(-0.2).
2. Perform division: (slide 4)
48:(-8); -24: (-2); -200:4; -4,9:7; -8,4: (-7); 15:(- 0,3).
3. Solve the equation: (slide 5)
3x=27; -5 x=-45; x:(2.5)=5.
3. Mathematical dictation(slide 6.7)
Option 1
Option 2
Students exchange notebooks, complete the test and give a grade.
4. Running the test ( slide 8)
Once upon a time in Rus', years were counted from March 1, from the beginning of agricultural spring, from the first spring drop. March was the “starter” of the year. The name of the month “March” comes from the Romans. They named this month after one of their gods, a test will help you find out what kind of god it is.
Answer : Martius
The Romans named one month of the year Martius in honor of the god of war Mars. In Rus', this name was simplified by taking only the first four letters (Slide 9).
People say: “March is unfaithful, sometimes it cries, sometimes it laughs.” There are many folk signs associated with March. Some of its days have their own names. Let us all together now compile a folk month book for March.
5.Solution of exercises
Students at the board solve examples whose answers are the days of the month. An example appears on the board, and then the day of the month with the name and folk sign.
(Slides 10 to 19)
March 4 - Arkhip. On Arkhip, women were supposed to spend the whole day in the kitchen. The more food she prepares, the richer the house will be.
2) y×(-2.5)=-15
March, 6- Timofey-spring. If there is snow on Timofey's day, then the harvest is for spring.
3) -50, 4:x=-4, 2
4) -0.25:5×(-260)
March 13- Vasily the drip maker: drips from the roofs. Birds curl their nests, and migratory birds fly from warm places.
5) -29,12: (-2,08)
March 14th- Evdokia (Avdotya the Ivy) - the snow flattens with infusion. The second meeting of spring (the first on Meeting). As Evdokia is, so is summer. Evdokia is red - and spring is red; snow on Evdokia - for the harvest.
6) (-6-3.6×2.5) ×(-1)
7) -81.6:48×(-10)
March 17- Gerasim the rooker brought the rooks. Rooks land on arable land, and if they fly straight to their nests, there will be a friendly spring.
8) 7.15×(-4): (-1.3)
March 22- Magpies - day is equal to night. Winter ends, spring begins, the larks arrive. By old custom Larks and waders are baked from the dough.
9) -12.5×50: (-25)
10) 100+(-2,1:0,03)
30th of March- Alexey is warm. The water comes from the mountains, and the fish come from the camp (from the winter hut). Whatever the streams are like on this day (large or small), so is the floodplain (flood).
6. Lesson summary
Guys, did you like today's lesson? What new did you learn today? What did we repeat? I suggest you prepare your own month book for April. You must find the signs of April and create examples with answers corresponding to the day of the month.
7. Homework: p. 218 No. 1174, 1179(1) (Slide20)
Lesson objectives:
Educational:
- formulating rules for multiplying numbers with the same and different signs;
- mastering and improving the skills of multiplying numbers with different signs.
Educational:
- development mental operations: comparison, generalization, analysis, analogy;
- development of independent work skills;
- broadening the horizons of students.
Educational:
- fostering a record-keeping culture;
- education of responsibility, attention;
- nurturing interest in the subject.
Lesson type: learning new material.
Equipment: computer, multimedia projector, cards for the game “Mathematical Combat”, tests, knowledge cards.
Posters on the walls:
- Knowledge is the most excellent of possessions. Everyone strives for it, but it does not come on its own.
Al-Biruni - In everything I want to get to the very essence...
B. Pasternak
Lesson Plan
- Organizational moment (1 min).
- introduction teachers (3 min).
- Oral work(10 min).
- Presentation of the material (15 min).
- Mathematical chain (5 min).
- Homework (2 min).
- Test (6 min).
- Lesson summary (3 min).
During the classes
I. Organizational moment
students' readiness for the lesson.
II. Teacher's opening speech
Guys, we met with you today not in vain, but for fruitful work: gaining knowledge.
Since the universe has existed,
There is no one who does not need knowledge.
Whatever language and age we choose,
Man has always strived for knowledge...
Rudaki
In class we will study new material, consolidate it, work independently, evaluate yourself and your comrades. Everyone has a knowledge record card on their desk, in which our lesson is divided into stages. The points you earned on different stages you yourself will enter the lesson into this map. And at the end of the lesson we will summarize. Place these cards in a visible place.
III. Oral work (in the form of the game “Mathematical Combat”)
Guys, before we start new topic, let's repeat what we learned earlier. Everyone has a sheet of paper with the game “Mathematical Combat” on their desk. The vertical and horizontal columns contain the numbers that need to be added. These numbers are marked with dots. We will write the answers in those cells on the field where the dots are.
Three minutes to complete. We started work.
Now we exchanged works with our desk neighbor and check them with each other. If you think that the answer is incorrect, then carefully cross it out and write the correct one next to it. Let's check.
Now let’s check the answers with the screen ( The correct answers are projected on the screen).
For correctly solved
5 tasks are given 5 points;
4 tasks – 4 points;
3 tasks – 3 points;
2 tasks – 2 points;
1 task – 1 point.
Well done. They put everything aside. Guys, let’s enter the number of points scored for the “Mathematical Battle” into our knowledge cards ( Annex 1).
IV. Presentation of the material
Open the workbooks. Write down the number, great job.
- What operations on positive and negative numbers do you know?
- How to add two negative numbers?
- How to add two numbers with different signs?
- How to subtract numbers with different signs?
- You always use the word "module". What is the modulus of a number? A?
Today's lesson topic is also related to the operation of numbers of different signs. But it was hidden in an anagram, in which you need to swap letters and get a familiar word. Let's try to figure it out.
ENOZHEUMNI
We write down the topic of the lesson: “Multiplication.”
The purpose of our lesson: to get acquainted with the multiplication of positive and negative numbers and to formulate rules for multiplying numbers with both the same and different signs.
All attention to the board. Before you is a table with problems, solving which we will formulate the rules for multiplying positive and negative numbers.
- 2*3 = 6°C;
- –2*3 = –6°С;
- –2*(–3) = 6°С;
- 2*(–3) = –6°С;
1. The air temperature rises by 2°C every hour. Now the thermometer shows 0°C ( Appendix 2– Thermometer) (slide 1 on the computer).
- How much did you receive?(6 ° WITH).
- Someone will write the solution on the board, and we are all in notebooks.
- Let's look at the thermometer, did we get the correct answer? (slide 2 on the computer).
2. The air temperature drops by 2°C every hour. The thermometer now shows 0°C (slide 3 on the computer). What air temperature will the thermometer show after 3 hours?
- How much did you receive?(–6 ° WITH).
- We write down the corresponding solution on the board and in notebooks. Analogy with task 1.
- .(slide 4 on the computer).
3. The air temperature drops by 2°C every hour. The thermometer now shows 0°C (slide 5 on the computer).
- How much did you receive?(6 ° WITH).
- We write down the corresponding solution on the board and in notebooks. Analogy with tasks 1 and 2.
- Let's compare the result with the thermometer reading.(slide 6 on the computer).
4. The air temperature rises by 2°C every hour. The thermometer now shows 0°C (slide 7 on the computer). What air temperature did the thermometer show 3 hours ago?
- How much did you receive?(–6 ° WITH).
- We write down the corresponding solution on the board and in notebooks. Analogy with tasks 1-3.
- Let's compare the result with the thermometer reading.(slide 8 on the computer).
Look at your results. When multiplying numbers with the same signs (examples 1 and 3), what sign did you get the answer? (positive).
Fine. But in example 3, both factors are negative, and the answer is positive. Which mathematical concept allows you to move from negative numbers to positive ones? (module).
Attention rule: To multiply two numbers with the same signs, you need to multiply their absolute values and put a plus sign in front of the result. (2 people repeat).
Let's return to example 3. What are the modules (–2) and (–3) equal to? Let's multiply these modules. How much did you receive? With what sign?
When multiplying numbers with different signs (examples 2 and 4), what sign did you get the answer? (negative).
Formulate your own rules for multiplying numbers with different signs.
Rule: When multiplying numbers with different signs, you need to multiply their modules and put a minus sign in front of the result. (2 people repeat).
Let's return to examples No. 2 and No. 4. What are the magnitudes of their factors? Let's multiply these modules. How much did you receive? What sign should be given as a result?
Using these two rules, you can also multiply fractions: decimal, mixed, ordinary.
There are several examples on the board in front of you. We will decide three together with me, and the rest on our own. Pay attention to the recording and design.
Well done. Let's open the textbooks and mark the rules that need to be learned for the next lesson (page 190, §7 (point 35)). Knowing these rules will help you quickly master the division of positive and negative numbers in the future.
V. Mathematical chain
And now Dunno wants to check how you have learned the new material and will ask you a few questions. We must write down the solution and answers in notebooks ( Appendix 3– Mathematical chain).
Computer presentation
Hello guys. I see you are very smart and inquisitive, so I want to ask you a few questions. Be careful, especially with signs.
My first question is: multiply (–3) by (–13).
Second question: multiply what you got in the first task by (–0,1).
Third question: multiply the result of the second task by (–2).
Fourth question: multiply (-1/3) by the result of the third task.
And the last, fifth question: calculate the freezing point of mercury by multiplying the result of the fourth task by 15.
Thanks for the work. I wish you success.
Guys, let's check how we completed the tasks. Everyone got up.
How much did you get in the first task?
Those who have a different answer, sit down, and those who sit down, we give ourselves 0 points for the mathematical chain on the knowledge record card. The rest don't put anything.
How much did you get in the second task?
If you have a different answer, sit down and add 1 point to your knowledge card for the mathematical chain.
How much did you get in the third task?
For those who have a different answer, sit down and add 2 points to your knowledge record card for the mathematical chain.
How much did you get in the fourth task?
For those who have a different answer, sit down and add 3 points to your knowledge record card for the mathematical chain.
How much did you get in the fifth task?
For those who have a different answer, sit down and add 4 points to your knowledge record card for the mathematical chain. The remaining guys solved all 5 tasks correctly. Sit down, you give yourself 5 points for the mathematical chain on your knowledge record card.
What is the freezing point of mercury?(–39 °C).
VI. Homework
§7 (clause 35, page 190), No. 1121 – textbook: Mathematics. 6th grade: [N.Ya.Vilenkin and others]
Creative task: Write a problem on multiplying positive and negative numbers.
VII. Test
Let's move on to next stage lesson: taking a test ( Appendix 4).
You need to solve the tasks and circle the number of the correct answer. For the first two correctly completed tasks you will receive 1 point, for the 3rd task - 2 points, for the 4th task - 3 points. We started work.
Δ –1 point;
o –2 points;
–3 points.
Now let’s write down the numbers of the correct answers in the table below the test. Let's check the results. You should get the number 1418 in the empty cells (I write on the board). Whoever received it puts 7 points on the knowledge card. Those who made mistakes put the number of points scored only for correctly completed tasks on the knowledge record card.
The Great Great War lasted exactly 1418 days. Patriotic War, a victory in which the Russian people came at a heavy price. And on May 9, 2010 we will celebrate the 65th anniversary of the Victory over Nazi Germany.
VIII. Lesson summary
Now let’s calculate the total number of points you scored for the lesson and enter the results into the students’ knowledge record card. Then we deal these cards.
15 – 17 points – score “5”;
10 – 14 points – score “4”;
less than 10 points – score “3”.
Raise your hands who received “5”, “4”, “3”.
- What topic did we cover today?
- How to multiply numbers with the same signs; with different signs?
So, our lesson has come to an end. I want to say THANK YOU for your work in this lesson.