Multiplying negative numbers: rules, examples. Multiplying positive and negative numbers

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 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 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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Lesson objectives.

Subject:

formulate a rule for multiplying negative numbers and numbers with different signs,
teach students how to apply this rule.

Metasubject:

develop the ability to work in accordance with the proposed algorithm, draw up a plan for your actions,
develop self-control skills.

Personal:

develop communication skills,
to form the cognitive interest of students.

Equipment: computer, screen, multimedia projector, PowerPoint presentation, handouts: table for recording rules, tests.

(Textbook by N.Ya. Vilenkin “Mathematics. 6th grade”, M: “Mnemosyne”, 2013.)

During the classes

I. Organizational moment.

Communicating the topic of the lesson and recording the topic in notebooks by students.

II. Motivation.

Slide No. 2. (Lesson goal. Lesson plan).

Today we will continue to study an important arithmetic property - multiplication.

You already know how to multiply natural numbers - verbally and columnarly,

Learned how to multiply decimals and ordinary fractions. Today you will have to formulate the multiplication rule for negative numbers and numbers with different signs. And not only formulate it, but also learn to apply it.

III. Updating knowledge.

1) Slide number 3.

Solve the equations: a) x: 1.8 = 0.15; b) y: = . (Student at the blackboard)

Conclusion: to solve such equations you need to be able to multiply different numbers.

2) Checking homework independently. Review rules for multiplying decimals, fractions and mixed numbers. (Slides No. 4 and No. 5).

IV. Formulation of the rule.

Consider task 1 (slide number 6).

Consider task 2 (slide number 7).

In the process of solving problems, we had to multiply numbers with different signs and negative numbers. Let's take a closer look at this multiplication and its results.

By multiplying numbers with different signs, we get a negative number.

Let's look at another example. Find the product (–2) * 3, replacing the multiplication with the sum of identical terms. Similarly, find the product 3 * (–2). (Check - slide No. 8).

Questions:

1) What is the sign of the result when multiplying numbers with different signs?

2) How is the result module obtained? We formulate a rule for multiplying numbers with different signs and write the rule in the left column of the table. (Slide No. 9 and Appendix 1).

Rule for multiplying negative numbers and numbers with different signs.

Let's return to the second problem, in which we multiplied two negative numbers. It is quite difficult to explain such multiplication in another way.

Let's use the explanation that was given back in the 18th century by the great Russian scientist (born in Switzerland), mathematician and mechanic Leonhard Euler. (Leonard Euler left behind not only scientific works, but also wrote a number of textbooks on mathematics intended for students of the academic gymnasium).

So Euler explained the result roughly as follows. (Slide number 10).

It is clear that –2 · 3 = – 6. Therefore, the product (–2) · (–3) cannot be equal to –6. However, it must be somehow related to the number 6. There remains one possibility: (–2) · (–3) = 6. .

Questions:

1) What is the sign of the product?

2) How was the product modulus obtained?

We formulate the rule for multiplying negative numbers and fill in the right column of the table. (Slide No. 11).

To make it easier to remember the rule of signs when multiplying, you can use its formulation in verse. (Slide No. 12).

Plus by minus, multiplying,
We put a minus without yawning.
Multiply minus by minus
We'll give you a plus in response!

V. Formation of skills.

Let's learn how to apply this rule for calculations. Today in the lesson we will perform calculations only with whole numbers and decimal fractions.

1) Drawing up an action plan.

A scheme for applying the rule is drawn up. Notes are made on the board. An approximate diagram on slide No. 13.

2) Carrying out actions according to the scheme.

We solve from textbook No. 1121 (b, c, i, j, p, p). We carry out the solution in accordance with the drawn up diagram. Each example is explained by one of the students. At the same time, the solution is shown on slide No. 14.

3) Work in pairs.

Task on slide number 15.

Students work on options. First, the student from option 1 solves and explains the solution to option 2, the student from option 2 listens carefully, helps and corrects if necessary, and then the students change roles.

Additional task for those pairs who finish work earlier: No. 1125.

At the end of the work, verification is carried out using a ready-made solution located on slide No. 15 (animation is used).

If many people managed to solve No. 1125, then the conclusion is made that the sign of the number changes when multiplied by (?1).

4) Psychological relief.

5) Independent work.

Independent work - text on slide No. 17. After completing the work - self-test using a ready-made solution (slide No. 17 - animation, hyperlink to slide No. 18).

VI. Checking the level of assimilation of the studied material. Reflection.

Students take the test. On the same piece of paper, evaluate your work in class by filling out the table.

Test “Multiplication Rule”. Option 1.

1) –13 * 5

A. –75. B. – 65. V. 65. D. 650.

2) –5 * (–33)

A. 165. B. –165. V. 350 G. –265.

3) –18 * (–9)

A. –162. B. 180. C. 162. D. 172.

4) –7 * (–11) * (–1)

A. 77. B. 0. C.–77. G. 72.

Test “Multiplication Rule”. Option 2.

A. 84. B. 74. C. –84. G. 90.

2) –15 * (–6)

A. 80. B. –90. V. 60. D. 90.

A. 115. B. –165. V. 165. G. 0.

4) –6 * (–12) * (–1)

A. 60. B. –72. V. 72. G. 54.

VII. Homework.

Clause 35, rules, No. 1143 (a – h), No. 1145 (c).

Literature.

1) Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. “Mathematics 6. Textbook for general education institutions”, - M: “Mnemosyne”, 2013.

2) Chesnokov A.S., Neshkov K.I. “Didactic materials in mathematics for grade 6”, M: “Prosveshchenie”, 2013.

3) Nikolsky S.M. and others. “Arithmetic 6”: a textbook for educational institutions, M: “Prosveshchenie”, 2010.

4) Ershova A.P., Goloborodko V.V. “Independent and test work in mathematics for 6th grade.” M: “Ilexa”, 2010.

5) “365 tasks for ingenuity”, compiled by G. Golubkova, M: “AST-PRESS”, 2006.

6) “Big Encyclopedia of Cyril and Methodius 2010”, 3 CD.

In this article we will understand the process multiplying negative numbers. First, we formulate the rule for multiplying negative numbers and justify it. After this, we will move on to solving typical examples.

Page navigation.

We'll announce it right away rule for multiplying negative numbers: To multiply two negative numbers, you need to multiply their absolute values.

Let's write this rule using letters: for any negative real numbers −a and −b (in this case, the numbers a and b are positive) the following equality holds: (−a)·(−b)=a·b .

Let's prove the rule for multiplying negative numbers, that is, we'll prove the equality (−a)·(−b)=a·b.

In the article on multiplying numbers with different signs, we substantiated the validity of the equality a·(−b)=−a·b, similarly it is shown that (−a)·b=−a·b. These results and the properties of opposite numbers allow us to write the following equalities (−a)·(−b)=−(a·(−b))=−(−(a·b))=a·b. This proves the rule for multiplying negative numbers.

From the above multiplication rule it is clear that the product of two negative numbers is a positive number. Indeed, since the modulus of any number is positive, the product of moduli is also a positive number.

To conclude this point, we note that the rule discussed can be used to multiply real numbers, rational numbers and integers.

It's time to sort it out examples of multiplying two negative numbers, when solving we will use the rule obtained in the previous paragraph.

Multiply two negative numbers −3 and −5.

The moduli of the numbers being multiplied are 3 and 5, respectively. The product of these numbers is 15 (see multiplication of natural numbers if necessary), so the product of the original numbers is 15.

The entire process of multiplying initial negative numbers is briefly written as follows: (−3)·(−5)= 3·5=15.

Multiplying negative rational numbers using the disassembled rule can be reduced to multiplying ordinary fractions, multiplying mixed numbers, or multiplying decimals.

Calculate the product (−0.125)·(−6) .

According to the rule for multiplying negative numbers, we have (−0.125)·(−6)=0.125·6. All that remains is to finish the calculations; multiply the decimal fraction by a natural number in a column:

Finally, note that if one or both factors are irrational numbers, given in the form of roots, logarithms, powers, etc., then their product often has to be written as a numerical expression. The value of the resulting expression is calculated only when necessary.

Multiply a negative number by a negative number.

Let us first find the moduli of the numbers being multiplied: and (see properties of the logarithm). Then, according to the rule of multiplying negative numbers, we have. The resulting product is the answer.

You can continue studying the topic by referring to the section multiplying real numbers.

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?

Of course, you won’t be confused by such talk: from your school mathematics course you have firmly learned that minus by 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 rule for multiplying negative numbers is explained with examples like the one presented in table. 8.

It can be explained differently. Let's write the numbers in a row

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Now let's write the same numbers multiplied by 3:

It’s 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 in the bottom row is 3 less than the previous one. Write the numbers in reverse order

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's 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

(By taking 3 out of the bracket, we used the law of distributivity ab + ac = a(b + c) for - after all, we assume that it remains true for all numbers, including negative ones.) So, (The meticulous reader will ask us why. We honestly admit : we skip the proof of this fact - as well as the general discussion of what zero is.)

Let us now prove that (-3) (-5) = 15. To do this, we write

and multiply both sides of the equality by -5:

Let's open the brackets on the left side:

i.e. (-3) (-5) + (-15) = 0. Thus, the number is the opposite of the number -15, i.e. equal to 15. (There are also gaps in this reasoning: it would be necessary to prove that that there is only one number, the opposite of -15.)

Rules for multiplying negative numbers

Do we understand multiplication correctly?

“A and B were sitting on the pipe. A fell, B disappeared, what’s left on the pipe?
“Your letter I remains.”

(From the film “Youths in the Universe”)

Why does multiplying a number by zero result in zero?

Why does multiplying two negative numbers produce a positive number?

Teachers come up with everything they can to give answers to these two questions.

But no one has the courage to admit that there are three semantic errors in the formulation of multiplication!

Is it possible to make mistakes in basic arithmetic? After all, mathematics positions itself as an exact science.

School mathematics textbooks do not provide answers to these questions, replacing explanations with a set of rules that need to be memorized. Perhaps this topic is considered difficult to explain in middle school? Let's try to understand these issues.

7 is the multiplicand. 3 is the multiplier. 21-work.

According to the official wording:

to multiply a number by another number means to add as many multiplicands as the multiplier prescribes.

According to the accepted formulation, the factor 3 tells us that there should be three sevens on the right side of the equality.

7 * 3 = 7 + 7 + 7 = 21

But this formulation of multiplication cannot explain the questions posed above.

Let's correct the wording of multiplication

Usually in mathematics there is a lot that is meant, but it is not talked about or written down.

This refers to the plus sign before the first seven on the right side of the equation. Let's write down this plus.

7 * 3 = + 7 + 7 + 7 = 21

But what is the first seven added to? This means to zero, of course. Let's write down zero.

7 * 3 = 0 + 7 + 7 + 7 = 21

What if we multiply by three minus seven?

— 7 * 3 = 0 + (-7) + (-7) + (-7) = — 21

We write the addition of the multiplicand -7, but in fact we are subtracting from zero multiple times. Let's open the brackets.

— 7 * 3 = 0 — 7 — 7 — 7 = — 21

Now we can give a refined formulation of multiplication.

Multiplication is the process of repeatedly adding to (or subtracting from zero) the multiplicand (-7) as many times as the multiplier indicates. The multiplier (3) and its sign (+ or -) indicate the number of operations that are added to or subtracted from zero.

Using this clarified and slightly modified formulation of multiplication, the “sign rules” for multiplication when the multiplier is negative are easily explained.

7 * (-3) - there must be three minus signs after zero = 0 - (+7) - (+7) - (+7) = - 21

- 7 * (-3) - again there should be three minus signs after the zero =

0 — (-7) — (-7) — (-7) = 0 + 7 + 7 + 7 = + 21

Multiply by zero

7 * 0 = 0 + . there are no addition-to-zero operations.

If multiplication is an addition to zero, and the multiplier shows the number of operations of addition to zero, then the multiplier zero shows that nothing is added to zero. That's why it remains zero.

So, in the existing formulation of multiplication, we found three semantic errors that block the understanding of the two “sign rules” (when the multiplier is negative) and the multiplication of a number by zero.

You don't need to add the multiplicand, but add it to zero.
Multiplication is not only adding to zero, but also subtracting from zero.
The multiplier and its sign do not show the number of terms, but the number of plus or minus signs when decomposing the multiplication into terms (or subtracted ones).

Having somewhat clarified the formulation, we were able to explain the rules of signs for multiplication and the multiplication of a number by zero without the help of the commutative law of multiplication, without the distributive law, without involving analogies with the number line, without equations, without proof from the inverse, etc.

The sign rules for the refined formulation of multiplication are derived very simply.

7 * (+3) = 0 + (-7) + (-7) + (-7) = 0 — 7 — 7 — 7 = -21 (- + = -)

7 * (-3) = 0 — (+7) — (+7) — (+7) = 0 — 7 — 7 — 7 = -21 (+ — = -)

7 * (-3) = 0 — (-7) — (-7) — (-7) = 0 + 7 + 7 + 7 = +21 (- — = +)

The multiplier and its sign (+3 or -3) indicate the number of “+” or “-” signs on the right side of the equation.

The modified formulation of multiplication corresponds to the operation of raising a number to a power.

2^0 = 1 (one is not multiplied or divided by anything, so it remains one)

2^-2 = 1: 2: 2 = 1/4

2^-3 = 1: 2: 2: 2 = 1/8

Mathematicians agree that raising a number to a positive power is multiplying one over and over again. Raising a number to a negative power is dividing one multiple times.

The operation of multiplication should be similar to the operation of exponentiation.

2*3 = 0 + 2 + 2 + 2 = 6

2*0 = 0 (nothing is added to zero and nothing is subtracted from zero)

2*-3 = 0 — 2 — 2 — 2 = -6

The modified formulation of multiplication does not change anything in mathematics, but returns the original meaning of the multiplication operation, explains the “rules of signs”, multiplying a number by zero, and reconciles multiplication with exponentiation.

Let's check whether our formulation of multiplication is consistent with the division operation.

15: 5 = 3 (inverse of multiplication 5 * 3 = 15)

The quotient (3) corresponds to the number of operations of addition to zero (+3) during multiplication.

Dividing the number 15 by 5 means finding how many times you need to subtract 5 from 15. This is done by sequential subtraction until a zero result is obtained.

To find the result of division, you need to count the number of minus signs. There are three of them.

15: 5 = 3 operations of subtracting five from 15 to get zero.

15 - 5 - 5 - 5 = 0 (division 15:5)

0 + 5 + 5 + 5 = 15 (multiplying 5 * 3)

Division with remainder.

17 — 5 — 5 — 5 — 2 = 0

17: 5 = 3 and 2 remainder

If there is division with a remainder, why not multiplication with an appendage?

2 + 5 * 3 = 0 + 2 + 5 + 5 + 5 = 17

Let's look at the difference in wording on the calculator

Existing formulation of multiplication (three terms).

10 + 10 + 10 = 30

Corrected multiplication formulation (three additions to zero operations).

0 + 10 = = = 30

(Press “equals” three times.)

10 * 3 = 0 + 10 + 10 + 10 = 30

A multiplier of 3 indicates that the multiplicand 10 must be added to zero three times.

Try multiplying (-10) * (-3) by adding the term (-10) minus three times!

(-10) * (-3) = (-10) + (-10) + (-10) = -10 — 10 — 10 = -30 ?

What does the minus sign for three mean? Maybe so?

(-10) * (-3) = (-10) — (-10) — (-10) = — 10 + 10 + 10 = 10?

Ops. It is not possible to decompose the product into the sum (or difference) of terms (-10).

The revised wording does this correctly.

0 — (-10) = = = +30

(-10) * (-3) = 0 — (-10) — (-10) — (-10) = 0 + 10 + 10 + 10 = 30

The multiplier (-3) indicates that the multiplicand (-10) must be subtracted from zero three times.

Sign rules for addition and subtraction

Above we showed a simple way to derive the rules of signs for multiplication by changing the meaning of the wording of multiplication.

But for the conclusion we used the rules of signs for addition and subtraction. They are almost the same as for multiplication. Let's create a visualization of the rules of signs for addition and subtraction, so that even a first-grader can understand it.

What is “minus”, “negative”?

There is nothing negative in nature. No negative temperature, no negative direction, no negative mass, no negative charges. Even sine by its nature can only be positive.

But mathematicians came up with negative numbers. For what? What does "minus" mean?

A minus sign means the opposite direction. Left - right. Top bottom. Clockwise - counterclockwise. Back and forth. Cold - hot. Light heavy. Slow - fast. If you think about it, you can give many other examples where it is convenient to use negative values.

In the world we know, infinity starts from zero and goes to plus infinity.

“Minus infinity” does not exist in the real world. This is the same mathematical convention as the concept of “minus”.

So, “minus” denotes the opposite direction: movement, rotation, process, multiplication, addition. Let's analyze the different directions when adding and subtracting positive and negative (increasing in the other direction) numbers.

The difficulty in understanding the rules of signs for addition and subtraction is due to the fact that these rules are usually explained on the number line. On the number line, three different components are mixed, from which rules are derived. And due to confusion, due to lumping different concepts into one heap, difficulties of understanding are created.

To understand the rules, we need to divide:

the first term and the sum (they will be on the horizontal axis);
the second term (it will be on the vertical axis);
direction of addition and subtraction operations.

This division is clearly shown in the figure. Mentally imagine that the vertical axis can rotate, superimposing on the horizontal axis.

The addition operation is always performed by rotating the vertical axis clockwise (plus sign). The subtraction operation is always performed by rotating the vertical axis counterclockwise (minus sign).

Example. Diagram in lower right corner.

It can be seen that two adjacent minus signs (the sign of the subtraction operation and the sign of the number 3) have different meanings. The first minus shows the direction of subtraction. The second minus is the sign of the number on the vertical axis.

Find the first term (-2) on the horizontal axis. We find the second term (-3) on the vertical axis. Mentally rotate the vertical axis counterclockwise until (-3) aligns with the number (+1) on the horizontal axis. The number (+1) is the result of addition.

gives the same result as the addition operation in the diagram in the upper right corner.

Therefore, two adjacent minus signs can be replaced with one plus sign.

We are all accustomed to using ready-made rules of arithmetic without thinking about their meaning. Therefore, we often don’t even notice how the rules of signs for addition (subtraction) differ from the rules of signs for multiplication (division). Do they seem the same? Almost. A slight difference can be seen in the following illustration.

Now we have everything we need to derive the sign rules for multiplication. The output sequence is as follows.

We clearly show how the rules of signs for addition and subtraction are obtained.
We make semantic changes to the existing formulation of multiplication.
Based on the modified formulation of multiplication and the rules of signs for addition, we derive the rules of signs for multiplication.

Below are written Sign rules for addition and subtraction,obtained from the visualization. And in red, for comparison, the same rules of signs from the mathematics textbook. The gray plus in parentheses is an invisible plus, which is not written for a positive number.

There are always two signs between the terms: the operation sign and the number sign (we don’t write plus, but we mean it). The rules of signs prescribe the replacement of one pair of characters with another pair without changing the result of addition (subtraction). In fact, there are only two rules.

Rules 1 and 3 (for visualization) - duplicate rules 4 and 2.. Rules 1 and 3 in the school interpretation do not coincide with the visual scheme, therefore, they do not apply to the rules of signs for addition. These are some other rules.

School rule 1. (red) allows you to replace two pluses in a row with one plus. The rule does not apply to the replacement of signs in addition and subtraction.

School rule 3. (red) allows you not to write a plus sign for a positive number after a subtraction operation. The rule does not apply to the replacement of signs in addition and subtraction.

The meaning of the rules of signs for addition is the replacement of one PAIR of signs with another PAIR of signs without changing the result of the addition.

School methodologists mixed two rules in one rule:

— two rules of signs when adding and subtracting positive and negative numbers (replacing one pair of signs with another pair of signs);

- two rules according to which you can not write a plus sign for a positive number.

Two different rules mixed into one are similar to the rules of signs in multiplication, where two signs result in a third. They look exactly alike.

Great confusion! The same thing again, for better detangling. Let us highlight the operation signs in red to distinguish them from the number signs.

1. Addition and subtraction. Two rules of signs according to which pairs of signs between terms are interchanged. Operation sign and number sign.

2. Two rules according to which the plus sign for a positive number is allowed not to be written. These are the rules for the entry form. Does not apply to addition. For a positive number, only the sign of the operation is written.

3. Four rules of signs for multiplication. When two signs of factors result in a third sign of the product. The sign rules for multiplication contain only number signs.

Now that we have separated the form rules, it should be clear that the sign rules for addition and subtraction are not at all similar to the sign rules for multiplication.

“The rule for multiplying negative numbers and numbers with different signs.” 6th grade

Presentation for the lesson

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Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.

Lesson objectives.

Subject:

formulate a rule for multiplying negative numbers and numbers with different signs,
teach students how to apply this rule.

Metasubject:

develop the ability to work in accordance with the proposed algorithm, draw up a plan for your actions,
develop self-control skills.

Personal:

develop communication skills,
to form the cognitive interest of students.

Equipment: computer, screen, multimedia projector, PowerPoint presentation, handouts: table for recording rules, tests.

(Textbook by N.Ya. Vilenkin “Mathematics. 6th grade”, M: “Mnemosyne”, 2013.)

During the classes

I. Organizational moment.

Communicating the topic of the lesson and recording the topic in notebooks by students.

II. Motivation.

Slide No. 2. (Lesson goal. Lesson plan).

Today we will continue to study an important arithmetic property - multiplication.

You already know how to multiply natural numbers - verbally and columnarly,

III. Updating knowledge.

Solve the equations: a) x: 1.8 = 0.15; b) y: = . (Student at the blackboard)

Conclusion: to solve such equations you need to be able to multiply different numbers.

2) Checking homework independently. Review rules for multiplying decimals, fractions and mixed numbers. (Slides No. 4 and No. 5).

IV. Formulation of the rule.

Consider task 1 (slide number 6).

Consider task 2 (slide number 7).

In the process of solving problems, we had to multiply numbers with different signs and negative numbers. Let's take a closer look at this multiplication and its results.

By multiplying numbers with different signs, we get a negative number.

Let's look at another example. Find the product (–2) * 3, replacing the multiplication with the sum of identical terms. Similarly, find the product 3 * (–2). (Check - slide No. 8).

Questions:

1) What is the sign of the result when multiplying numbers with different signs?

2) How is the result module obtained? We formulate a rule for multiplying numbers with different signs and write the rule in the left column of the table. (Slide No. 9 and Appendix 1).

Rule for multiplying negative numbers and numbers with different signs.

Let's return to the second problem, in which we multiplied two negative numbers. It is quite difficult to explain such multiplication in another way.

So Euler explained the result roughly as follows. (Slide number 10).

Questions:

1) What is the sign of the product?

2) How was the product modulus obtained?

We formulate the rule for multiplying negative numbers and fill in the right column of the table. (Slide No. 11).

To make it easier to remember the rule of signs when multiplying, you can use its formulation in verse. (Slide No. 12).

Plus by minus, multiplying,
We put a minus without yawning.
Multiply minus by minus
We'll give you a plus in response!

V. Formation of skills.

Let's learn how to apply this rule for calculations. Today in the lesson we will perform calculations only with whole numbers and decimal fractions.

1) Drawing up an action plan.

A scheme for applying the rule is drawn up. Notes are made on the board. An approximate diagram on slide No. 13.

2) Carrying out actions according to the scheme.

3) Work in pairs.

Task on slide number 15.

Additional task for those pairs who finish work earlier: No. 1125.

At the end of the work, verification is carried out using a ready-made solution located on slide No. 15 (animation is used).

If many people managed to solve No. 1125, then the conclusion is made that the sign of the number changes when multiplied by (?1).

4) Psychological relief.

5) Independent work.

Independent work - text on slide No. 17. After completing the work - self-test using a ready-made solution (slide No. 17 - animation, hyperlink to slide No. 18).

VI. Checking the level of assimilation of the studied material. Reflection.

Students take the test. On the same piece of paper, evaluate your work in class by filling out the table.

Test “Multiplication Rule”. Option 1.

Multiplying negative numbers: rule, examples

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

Multiplying Negative Numbers

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

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 well as (- a) · b = - a · b. This follows from the property of opposite numbers, due to which the equalities will be written as follows:

(— a) · (— b) = — (— a · (— b)) = — (— (a · b)) = a · b .

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

This rule is applicable for multiplying real numbers, rational numbers, and integers.

Examples of multiplying negative numbers

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

Multiply the numbers - 3 and - 5.

Solution.

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

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

(– 3) · (– 5) = 3 · 5 = 15

Answer: (- 3) · (- 5) = 15.

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

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

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 the natural number of columns. It looks like this:

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

Answer: (− 0, 125) · (− 6) = 0, 75.

In the case when the factors are irrational numbers, then their product can be written as a numerical expression. The value is calculated only when necessary.

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

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.

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 both for 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 a simple case to focus on the steps of the rule rather than the computational complexity.

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 fractions with different signs, you need to be able to multiply ordinary fractions, multiply decimals and their combinations with natural and mixed numbers.

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

Having carried out the conversion of a periodic decimal fraction into an ordinary fraction, and also having carried out the transition from a mixed number to an 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 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?

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.

This article provides a detailed overview 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.

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Multiplying Negative Numbers

Definition 1

(- a) · (- b) = a · b.

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

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

This rule is applicable for multiplying real numbers, rational numbers, and integers.

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

Example 1

Multiply numbers - 3 and - 5.

Solution.

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

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

(- 3) · (- 5) = 3 · 5 = 15

Answer: (- 3) · (- 5) = 15.

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

Example 2

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

Solution.

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

Answer: (− 0, 125) · (− 6) = 0, 75.

In the case when the factors are irrational numbers, then their product can be written as a numerical expression. The value is calculated only when necessary.

Example 3

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

Solution

Finding the modules of the given numbers:

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

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

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

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

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