To multiply numbers with different signs. Multiplication of numbers with different signs (6th grade)

Educational:

• Activity education;

Lesson type

Equipment:

1. Projector and computer.

Lesson plan

1. Organizational moment

2. Updating knowledge

3. Mathematical dictation

4.Performing the test

5. Solution of exercises

6. Summary of the lesson

7. Homework.

During the classes

1. Organizing moment

Today we will continue to work on multiplication and division of 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 still not quite working out. In addition, you will learn a lot of interesting things about the first month of spring - March. (Slide1)

2. Actualization of knowledge.

3x=27; -5x=-45; x:(2,5)=5.

3.Mathematical dictation(slide 6.7)

Option 1

Option 2

4. Test execution ( 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. Summary of the lesson

7. Homework:

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"Multiplication and division of 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”, development of skills in applying the operations of multiplication and division of a positive number by a negative number and vice versa, as well as a negative number on a negative number.

Lesson objectives:

Educational:

Fixing the rules on this topic;

Formation of skills and abilities to work with operations of multiplication and division of numbers with different signs.

Developing:

Development of cognitive interest;

Development logical thinking, memory, attention;

Educational:

Activity education;

Teaching students the skills of independent work;

Education of love for nature, instilling interest in folk signs.

Lesson type. Lesson-repetitions and generalizations.

Equipment:

Projector and computer.

Lesson plan

1. Organizational moment

2. Updating knowledge

3. Mathematical dictation

4.Performing the test

5. Solution of exercises

6. Summary of the lesson

7. Homework.

During the classes

1. Organizing moment

Hello guys! What did we do in previous lessons? (by multiplication and division rational numbers.)

Today we will continue to work on multiplication and division of 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 still not quite working out. In addition, you will learn a lot of interesting things about the first month of spring - March. (Slide1)

2. Actualization of knowledge.

Review the rules for multiplying and dividing positive and negative numbers.

Remember the mnemonic rule. (Slide 2)

Perform multiplication: (slide 3)

5×3; 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; -5x=-45; x:(2,5)=5.

3.Mathematical dictation(slide 6.7)

Option 1

Option 2

Students exchange notebooks, check and grade.

4. Test execution ( slide 8)

Once upon a time in Russia, years were counted from March 1, from the beginning of the agricultural spring, from the first spring drop. March was the "beginner" of the year. The name of the month "March" comes from the Romans. They named this month in honor of one of their gods, to find out what kind of god it is, the test will help you.

Answer : Martius

The Romans named one month of the year in honor of Mars, the god of war, called Martius. In Russia, this name was simplified, taking only the first four letters. (Slide 9).

People say: "Mart is unfaithful, now he cries, now he laughs." There are many folk signs associated with March. Some of its days have their own names. Let's now all together we will make a folk calendar for March.

5. Solution of exercises

Students at the blackboard solve examples whose answers are the days of the month. An example appears on the board, followed by the day of the month with the name and folk omen.

(Slides 10 to 19)

March 4 - Arkhip. On Arkhip, women were supposed to spend the whole day in the kitchen. The more she prepares any food, the richer the house will be.

2) y×(-2.5)=-15

March, 6- Timothy-spring. If on Timofeev's day there is snow with zadulina, then the harvest is for spring crops.

3) -50, 4:x=-4, 2

4) -0.25:5×(-260)

March 13- Vasily the dropper: drops from the roofs. Nest birds curl, and migratory birds fly from warm places.

5) -29,12: (-2,08)

March 14th- Evdokia (Avdotya-plushcha) - the snow flattens the infusion. The second meeting of spring (the first on Stretenie). What is Evdokia - such is the 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 - drove the rooks. Rooks sit on arable land, and if they fly directly to the nests, there will be a friendly spring.

8) 7.15×(-4): (-1.3)

March 22- Magpies - day equals night. Winter ends, spring begins, larks arrive. According to an old custom, larks and waders are baked from dough.

9) -12.5×50: (-25)

10) 100+(-2,1:0,03)

30th of March- Alexey is warm. Water from the mountains, and fish from the camp (from the winter hut). What are the streams on this day (large or small), such is the floodplain (overflow).

6. Summary of the lesson

Guys, did you like today's lesson? What new did you learn today? What did we repeat? I suggest that you prepare the calendar for April yourself. You must find signs of April and make up examples with answers corresponding to the day of the month.

7. Homework: pp. 218 No. 1174, 1179(1) (Slide 20)

AT this lesson considers multiplication and division of rational numbers.

Lesson content

Multiplication of rational numbers

The rules for multiplying integers are also valid for 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 before the answer.

To see clearly that we are dealing with numbers that have different signs, we enclose each rational number in brackets along with its signs.

The modulus of a number is , and the modulus of a number is . Multiplying the resulting modules as positive fractions, we received the 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 carried out in brackets, before which the minus is placed.

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 answer.

Solution for this example can be written shorter:

Example 4 Find the value of an expression

The solution for this example can be written shorter:

Example 5 Find the value of an expression

This is the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus before the received 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. Rewrite the rest as is

We got the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus in front of the received answer. The entry with modules can be omitted so as not to clutter up the expression

The solution for this example can be written shorter

Example 7 Find the value of an expression

This is the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus before the received answer

At first, the answer turned out to be an improper fraction, but we singled out the whole part in it. note that whole part was isolated from the fraction module. The resulting mixed number was enclosed in brackets preceded by a minus. This is done in order to fulfill the requirement of the rule. And the rule required that the received answer be preceded by a minus sign.

The solution for this example can be written shorter:

Example 8 Find the value of an expression

First, we multiply and and multiply the resulting number with the remaining number 5. We will skip the entry with modules so as not to clutter up the expression.

Answer: expression value equals −2.

Example 9 Find the value of an expression:

Let's translate mixed numbers into improper fractions:

We got the multiplication of negative rational numbers. We multiply the modules of these numbers and put a plus in front of the received answer. The entry with modules can be omitted so as not to clutter up the expression

Example 10 Find the value of an expression

The expression consists of several factors. According to the associative law of multiplication, if an expression consists of several factors, then the product will not depend on the order of operations. This allows us to evaluate the given expression in any order.

We will not reinvent the wheel, but calculate this expression from left to right in the order of the factors. We skip the entry with modules so as not to clutter up the expression

Third action:

Fourth action:

Answer: the value of the expression is

Example 11. Find the value of an expression

Remember the law of multiplication by zero. This law states that 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, therefore, without wasting time, we answer that the value of the expression is 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, therefore, 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 procedure 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 results. We will use this method.

According to the order of operations, if the expression contains addition and multiplication, then the first thing to do is to perform the multiplication. Therefore, in the resulting new expression, we take in brackets those parameters that must be multiplied. So 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 can be seen that this example could be solved even in the mind. Therefore, one should develop the skill of analyzing an expression before starting to solve it. It is likely that it can be solved in the mind and save a lot of time and nerves. And on control and exams, as you know, time is very expensive.

Example 14 Find the value of the expression −4.2 × 3.2

This is the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus before the received answer

Notice how the modules of rational numbers were multiplied. AT this case, to multiply the modules 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. We multiply the modules of these numbers and put a minus before the received answer

Notice how the modules of rational numbers were multiplied. In this case, in order to multiply the modules of rational numbers, it took to be able to.

Example 16 Find the value of the expression −4.2 × (−7.5)

This is the multiplication of negative rational numbers. We multiply the modules of these numbers and put a plus in front of the received answer

Division of rational numbers

The rules for dividing integers are also valid for 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.

And to share decimal to another decimal fraction, you need to move the comma to the right in the dividend and in the divisor by as many digits as there are after the decimal point in the divisor, then divide as by a regular number.

Example 1 Find the value of an 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 got 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 modules of these rational numbers and put a minus before the answer.

Let's complete this example. The entry with modules can be omitted so as not to clutter up the expression

Thus, 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 . We 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, again, you need to multiply the first fraction by the reciprocal of the second.

The reciprocal of the second fraction is the fraction . We multiply the first fraction by it:

We got the multiplication of negative rational numbers. We already know how such an expression is calculated. It is necessary to multiply the modules of rational numbers and put a plus in front of the answer.

Let's complete this example. You can skip the entry with modules to avoid cluttering up the expression:

Example 4 Find the value of an expression

To calculate this expression, you need to multiply the first number -3 by the reciprocal of the fraction.

The reciprocal of a fraction is a fraction. By it and multiply the first number −3

Example 6 Find the value of an expression

To calculate this expression, you need to multiply the first fraction by the reciprocal of 4.

The reciprocal of 4 is a fraction. We 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 reciprocal of −3

The reciprocal of −3 is a fraction. We 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 dividend modulus by the divisor modulus and put a minus before the received answer

Note how the modulus of the dividend has been divided into the modulus of the divisor. In this case, to do it right, it took to be able to.

If there is no desire to mess around with decimal fractions (and this happens often), then these, then convert these mixed numbers to improper fractions, and then go directly to division.

Let's calculate the previous expression -14.4: 1.8 in this way. Convert decimals to mixed numbers:

Now let's translate the resulting mixed numbers into improper fractions:

Now you can deal directly with division, namely divide a fraction by a fraction. To do this, you need to multiply the first fraction by the reciprocal of the second:

Example 7 Find the value of an expression

Let's convert the decimal -2.06 to an improper fraction, and multiply this fraction by the reciprocal of the second:

Multistoried fractions

You can often find an expression in which the division of fractions is written using a fractional line. For example, an expression could be written like this:

What is the difference between expressions and ? Actually there is no difference. These two expressions carry the same meaning and you 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 fractional line. The result is a fraction, which the people agreed to call multistory.

When encountering such multi-story expressions, the same division rules must be applied. ordinary fractions. The first fraction must be multiplied by the reciprocal of the second.

It is extremely inconvenient to use such fractions in a solution, so you can write them in an understandable form, using not a fractional bar, but a colon as a division sign.

For example, let's write a multi-storey 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. Multistoried fractions have several fractional features that can be confusing. The main fractional bar, which separates the first fraction from the second, is usually longer than the others.

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 fractional line (it is the longest) and we see that the integer number −3 is divided by an ordinary fraction

And if we mistakenly took the second fractional line for the main one (the one that is shorter), then it would turn out that we divide the fraction by an integer 5 In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the divisible in this case is the number −3, and the divisor is a fraction.

Example 3 We write in an understandable form a multi-storey fraction

We find the main fractional line (it is the longest) and we see that the fraction is divided by an integer 2

And if we mistakenly took the first fractional line for the main one (the one that is shorter), then it would turn out that we divide the integer −5 by a fraction. In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the divisible in this case is a fraction, and the divisor is an integer 2.

Despite the fact that multi-storey fractions are inconvenient in work, we will encounter them very often, especially when studying higher mathematics.

Naturally, to convert a multi-storey fraction into clear view takes extra time and space. Therefore, you can use a faster method. This method is convenient and at the output allows you to get a ready-made expression in which the first fraction has already been multiplied by the reciprocal of the second.

This method is implemented as follows:

If the fraction is four-storied, for example, as, then the figure located on the first floor is raised to the highest floor. And the number located on the second floor is raised to the third floor. The resulting numbers must be connected with multiplication icons (×)

As a result, bypassing the intermediate notation, we get a new expression in which the first fraction has already been multiplied by the reciprocal of the second. Convenience and more!

To avoid mistakes when using this method, you can use the following rule:

From first to fourth. From the second to the third.

In rule we are talking about floors. The figure from the first floor must be raised to the fourth floor. And the figure from the second floor must be raised to the third floor.

Let's try to calculate a multi-storey fraction using the above rule.

So, the number located on the first floor is raised to the fourth floor, and the number located on the second floor is raised to the third floor.

As a result, bypassing the intermediate notation, we get a new expression in which the first fraction has already been multiplied by the reciprocal of the second. You can use what you already know:

Let's try to calculate a multi-storey fraction using a new scheme.

There is only the first, second and fourth floors. The third floor is missing. But we do not deviate from the main 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 it is

As a result, bypassing the intermediate notation, we got a new expression , in which the first number −3 has already been multiplied by the fraction that is the reciprocal of the second. You can use what you already know:

Let's try to calculate a multi-storey fraction using a new scheme.

There are only the second, third and fourth floors. The first floor is missing. Since the first floor is missing, 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 got a new expression in which the first fraction has already been multiplied by the reciprocal of the divisor. You can use what you already know:

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 entered into a variable and then work with this variable.

Mathematicians often do this. difficult task break it down into smaller subtasks and solve them. Then they collect the solved subtasks into one single whole. This is creative process and this is learned over the years, hard training.

The use of variables is justified when working with multi-storey 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 again have 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 give the name of the variables from capital Latin letters. Let's not break this tradition, and denote the first expression through a big latin letter A

And the expression in the denominator can be denoted by a capital Latin letter B

Now our original expression becomes . That is, we have made the substitution numeric expression to alphabetic, having previously entered the numerator and denominator in variables A and B.

Now we can separately calculate the values ​​of variable A and the value of variable B. We will insert the finished values ​​into the expression.

Find the value of a variable A

Find the value of a variable B

Now let's substitute in the main expression instead of variables A and B their values:

We got a multi-story fraction in which you can use the “from the first to the fourth, from the second to the third” scheme, 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 calculation will not be difficult:

Thus, the value of the expression is −1.

Of course we have considered the simplest example, but our goal was to find out how you can use variables to make things easier for yourself, 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 is more expedient 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, putting some of its expressions into variables.

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Now let's deal with multiplication and division.

Suppose we need to multiply +3 by -4. How to do it?

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

We will get the same result if, according to the condition of the problem, each of the four people has a debt of 3 dollars. 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 multiplying one positive 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 do you multiply two negative numbers?

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

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

Therefore, it is quite logical, although a bit 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 sign, we get a positive number. Multiplying two numbers with different signs, we get a negative number.

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

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

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