Dividing negative numbers, rules, examples. Dividing numbers with different signs, rules, examples

Lesson summary

Pedagogy and didactics

Organizational moment Teacher: Hello, sit down. Checking homework, the teacher turns on the projector with a homework slide, which also reflects the criteria for assessing the work. Teacher: Swap notebooks. students check answers Teacher: Evaluation criterion: everything is decided correctly put FIVE one minus FOUR twothree minus THREE in all other cases TWO. Oral work Table with the rule of signs on a magnetic board Teacher: let’s repeat the rule of signs for multiplication attention on the magnetic board.

Math lesson notes

Topic: “Dividing numbers with different signs».

Class: 6

Textbook: Muravin and Muravina

Date: 02/15/2010

Lesson number: 3

Kurgan 2010

Lesson objectives:

1. Educational: teach how to divide numbers with different signs.

2. Developmental: develop thinking and individual work skills.

3. Educational: to form a culture of mathematical writing.

Equipment:

1. Presentation

2. Wall table “Sign Rules”

3. Cards for oral work

4. Cards for independent work

Lesson plan:

I . Organizational moment (1 min)

II . Checking homework (2 min)

III . Oral work (3 min)

IV . Independent work (5 min)

V . Learning new material (15 min)

VI . Consolidation of what has been learned (12 min)

VII . Giving homework (1 min)

VIII . Lesson summary (1 min)

Lesson progress:

I. Organizational moment

Teacher: Hello, sit down. Open your notebooks, write down the date: February 15, lesson topic: “Dividing numbers with different signs,” Class work.

Today in the lesson we continue to get acquainted with operations on numbers with different signs. You will learn that you can divide not only positive but also negative numbers.

II. Checking homework

(the teacher turns on the projector with a homework slide, which also reflects the criteria for evaluating the work)

Teacher: Exchange notebooks. Attention to the slide. Home numbers were assigned: 515 (a, b, c, d), 517 (c, d). Check that the assignments are completed correctly and check your answers. With a red pencil we put “+” next to the answer if the task was solved correctly and “-” if an error was made.

(students check answers)

Teacher: Evaluation criterion: everything was decided correctly put FIVE, one minus FOUR, two or three minuses THREE, in all other cases TWO. Next to the assessment is the name of the inspector. Return the notebooks to your neighbor.

III. Oral work

(Table with the rule of signs on a magnetic board)

Teacher: let’s repeat the rule of signs for multiplication, pay attention to the magnetic board.

Identical signs	On +
		On -
	Different signs	On -
		On +

Teacher: We count orally.

(teacher holds up task cards)

Masha: 75 × (-1) = -75

Teacher: Explain the choice of sign.

Masha: The rule for signs for multiplication is: “Plus by minus - the result is minus.”

Valera: -36 × 2 = -72

Teacher: How much did Sasha get?

Sasha: -72

Teacher: Why is there a minus sign?

Sasha: The rule for signs for multiplication is: “Minus by plus - it turns out minus.”

Nina: 0.9 × (-3) = -2.7

Anton: -2.1 × (-5) = 10.5

×5

Gene: × 5 = 1

× (-3)

Lida: × (-3) = 1

Ira: The denominator is zero. You cannot divide by zero.

Teacher: Well done! We worked well orally, now we work independently using cards.

IV. Independent work

(before the start of the lesson, the teacher distributes cards with tasks for independent work and answer sheets)

Teacher: you have some leaves on your table. In the left corner at the top write down the last name, in the middle the option number, decide to rewrite the assignments in any order, everyone will receive a grade. Don't forget about the rule of signs.

Option 1

1) - 5 × 6;

2) - 1 × (-7);

3) - 11 × 0;

4) 0.2 × (-8);

5) 12 × (-0.2);

6) - 2.5 × 0.4;

7) 1.2 × (-14);

8) -9.8 × (-10)

9) -1 × (-12) × (-0.5)

Option 2

1) 4 × (-7);

2) - 1 × 6;

3) 0 × (-13);

4) 0.3 × (-6);

5) 11 × (-0.1);

6) - 2.4 × 0.2;

7) 1.2 × (-14);

8) -9.8 × (-10)

9) -1 × (-14) × (-0.2)

Solution 1 option

1) - 5 × 6=-30

2) - 1 × (-7)=7

3) - 11 × 0=0

4) 0.2 × (-8)=-1.6

5) 12 × (-0.2)=-2.4

6) - 2.5 × 0.4=-1

7) 1.2 × (-14) = -16.8

8) -9.8 × (-10)=98

9) -1×(-12)×(-0.5)= 12×(-0.5)=-6

Solution 2 option

1) 4 × (-7)=-28

2) - 1 × 6=-6

3) 0 × (-13)=0

4) 0.3 × (-6)=1.8

5) 11 × (-0.1)=-1.1

6) - 2.4 × 0.2=-0.48

7) 1.2 × (-14) = -16.8

8) -9.8 × (-10)=98

9) -1 × (-14) × (-0.2) = 14 × (-0.2) = -2.8

ANSWERS Option 1

ANSWERS option 2

1) -30

2) 7

3) 0

4) -1,6

5) -2,4

6) -1

7) -16,8

8) 98

9) -6

1) -28

2) - 6

3) 0

4) -1,8

5) -1,1

6) - 0,48

7) -16,8

8) 98

9) -2,8

Teacher: We finish the work; we hand over the cards and pieces of paper. Works counting THREE will not be accepted. ONE-TWO-THREE all work has been submitted.

V. Learning new material

Teacher: Let's move on to learning new material. You already know how to multiply positive and negative numbers, in today's lesson you will learn how to divide numbers with different signs.

a: b

I write on the board, you write in your notebook.

Now this same expression is in the form of a fraction

Teacher: we replaced division with multiplication. Write it down and highlight it in color.

Teacher: write down two of your examples of replacing division with multiplication.

(pause)

Teacher: read your examples, please, Anton.

Anton: =

Teacher: right write down Anton’s example, read the second example.

Anton: - = ;

Teacher: that’s right - write it down, Sveta will read out her examples.

Sveta: -11:5 =

Teacher: right, second example.

Sveta: =

Teacher: Well done.

Teacher: write down in your notebook 5: (-7). How can I write this expression using multiplication?

Anya: 5: (-7) =

Teacher: that's right. Recording

5: (-7) = = - = -

Note that dividing plus by minus gives minus.

On -

We write -3: 8 = = - .

When you divide a minus by a plus, you get a minus.

On +

Next example:

4: (-5) = = =

When you divide minus by minus, you get plus.

On -

(the teacher posts a table of the rules of signs for division on the board)

Teacher: look carefully at the table and find differences from the table of the rules of signs for multiplication.

Katya: There are no differences, the tables are the same.

Teacher: that's right. The sign rule for division is exactly the same as for multiplication.

Identical signs	On +
		On -
	Different signs	On -
		On +

Teacher: copy the table of the rules of signs for division into your notebook, highlight the signs in color, and remember.

Teacher: numbers and inverses. Let's find their work.

- (-8) = = 1

These numbers in the product give one.

Consider the numbers a and

Highlight:

Numbers that produce one in the product are called reciprocals.

Teacher: let's give an example of mutual reciprocal numbers. and 2 reciprocal? Let's check:

Let's write one more example

Teacher: will the numbers and 3 be reciprocal?

Katya: and 3 are not reciprocals, since their product is equal to -1.

Teacher: come up with and write down 3 examples of reciprocal numbers and write them down in your notebook.

(pause)

Teacher: we read out our examples in a chain, starting from the last desk of the third row. Vasya, please.

Vasya: and 4.

Teacher: why?

Vasya: the product is equal to one.

Anya: and -7.

Pasha: and -3.

Anton: and 3.

Teacher: Well done. Enough. Reciprocal numbers are numbers that produce one in the product.

VI . Consolidation of what has been learned

Teacher: we solve orally along the chain and comment on No. 520 we need to replace division with multiplication and explain the sign, we start from the first desk of the first row, please, Vova, under the letter “a”.

Vova: 6: 3 = 6 = 2 plus on plus gives plus

Katya: 63: (-3) = 63 -63 = - 21 plus and minus gives a minus.

Teacher: the following examples under the letters “g” and “d” with reverse side The boards are solved by Petya and Masha, the rest are solved in notebooks.

(pause)

Teacher: pay attention to the board. Let's check.

Petya: -23: (-) = -23 = 232 = 46

Teacher: explain the choice of sign.

Petya: according to the rule: minus for minus gives plus.

Masha: - : = - = - = -1.5

Teacher: why the minus sign?

Masha: minus plus plus gives minus.

Teacher: Let's solve No. 521. Anton will do the solution with an explanation at the board. Please, Anton under the letter "a". All the rest are in the notebook.

Anton: - : = - = - = - = -2

Teacher: I got a different sign, what about you?

Katya: the sign is correct, because according to the rule: minus plus plus gives minus.

Teacher: Well done, sit down. The next example is solved by Lena from the other side of the board. We work independently.

(pause)

Teacher: Lena, explain how you solved it.

Lena: - : = - = = =

Teacher: thank you, Lena, sit down. Under the letters “c” and “d” you decide on your own, someone will comment on the solution at the end.

(pause)

Teacher: Kostya, please give me the floor.

Kostya: - : = - : 0. You cannot divide by zero.

1: (-) = -1)= 1 = 3

Teacher: Kostya, why plus?

Kostya: minus for minus gives a plus.

VII . Giving homework

Teacher: homework on the side board No. 521 (d, f), 522 (d, f). Don't forget about the rule of signs. Learn definitions.

VIII. Lesson summary

Teacher: today we learned to divide numbers with different signs, repeated the rule of signs for multiplication, checked its validity for division and became familiar with reciprocal numbers. Katya, what numbers are called reciprocals?

Katya: A pair of numbers that gives one in the product is called reciprocal.

Teacher: thank you, Katya. The following grades are awarded for work in class:

Anton five, Katya five, Sveta five.

In addition to these grades, everyone will receive grades for independent work, you will learn the results in the next lesson.

Appendix 1.

Slide with homework and evaluation criterion

№515

a) 2 ⋅ (0.2+1) = 2 ⋅ 1.2 = 2.4

b) 0.8 ⋅ (27 29) = 0.8 ⋅ (-2) = -1.6

c) (99.9 100.9) ⋅ (-1.7 0.3) = -1 ⋅ (-2) = 2

d) (2009-2000) ⋅ (-0.8) ⋅ (2.4 5.8)= 9 ⋅ (-0.8) ⋅ (-3.4)=24.48

№517

Evaluation criterion:

everything was decided correctly put FIVE,

one minus FOUR,

two or three minuses THREE,

in all other cases TWO.

Appendix 2.

Homework.

№521

e) - : = - = - = - = -15

e) - : (- = - = = = 84

№522

e) : (= : (- = - = - = - = -20

e) - : (- = - : (- = - : 0 cannot be divided by zero!

Appendix 3.

Board design.

Identical signs	On +
		On -
	Different signs	On -
		On +

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Now let's deal with multiplication and division.

Let's say we need to multiply +3 by -4. How to do this?

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 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).

Since winter is coming, it’s time to think about what to change your iron horse’s shoes into so as not to slip on the ice and feel confident on winter roads. You can, for example, buy Yokohama tires on the website: mvo.ru or some others, the main thing is that they are of high quality, more information and prices you can find out on the website Mvo.ru.

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 to positive.

Page navigation.

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 dividing numbers with different signs is used 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.

Example.

Divide the negative number −35 by the positive number 7.

Solution.

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.

Answer:

(−35):7=−5 .

Example.

Calculate the quotient 8:(−60) .

Solution.

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: .

Answer:

When dividing fractions 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).

Example.

Solution.

The module of the dividend is equal to , and the module of the divisor is equal to 0,(23) . To divide the modulus of the dividend by the modulus of the divisor, let's move on to ordinary fractions.

Let's convert a mixed number to an ordinary fraction: , and also

The focus of this article is division of negative numbers. First, the rule for dividing a negative number by a negative is given, its justification is given, and after that examples of dividing negative numbers with detailed description decisions.

Page navigation.

Rule for dividing negative numbers

Before giving the rule for dividing negative numbers, let us recall the meaning of the operation of division. Division inherently represents finding an unknown factor by famous work and a known other factor. That is, the number c is the quotient of a divided by b when c·b=a, and vice versa, if c·b=a, then a:b=c.

Rule for dividing negative numbers the following: the quotient of dividing one negative number by another is equal to the quotient of dividing the numerator by the modulus of the denominator.

Let's write down the sounded rule using letters. If a and b are negative numbers, then the equality is true a:b=|a|:|b| .

The equality a:b=a b −1 is easy to prove, starting from properties of multiplication of real numbers and definitions of reciprocal numbers. Indeed, on this basis we can write a chain of equalities of the form (a b −1) b=a (b −1 b)=a 1=a, which, due to the meaning of division mentioned at the beginning of the article, proves that a·b −1 is the quotient of a divided by b.

And this rule allows you to move from dividing negative numbers to multiplying.

It remains to consider the application of the considered rules for dividing negative numbers when solving examples.

Examples of dividing negative numbers

Let's sort it out examples of dividing negative numbers. Let's start with simple cases on which we will work out the application of the division rule.

Example.

Divide negative −18 by negative −3, then calculate the quotient (−5):(−2) .

Solution.

According to the rule for dividing negative numbers, the quotient of dividing −18 by −3 is equal to the quotient of dividing the absolute values of these numbers. Since |−18|=18 and |−3|=3, then (−18):(−3)=|−18|:|−3|=18:3 , all that remains is to divide the natural numbers, we have 18:3=6.

We solve the second part of the task in the same way. Since |−5|=5 and |−2|=2 , then (−5):(−2)=|−5|:|−2|=5:2 . This quotient corresponds to the common fraction 5/2, which can be written as a mixed number.

The same results are obtained if we use a different rule for dividing negative numbers. Indeed, the number −3 is the inverse number , then , now we multiply negative numbers: . Likewise, .

Answer:

(−18):(−3)=6 and .

When dividing fractional rational numbers, it is most convenient to work with ordinary fractions. But, if convenient, you can also divide finite decimal fractions.

Example.

Divide the number −0.004 by −0.25.

Solution.

The modules of the dividend and divisor are equal to 0.004 and 0.25, respectively, then according to the rule for dividing negative numbers we have (−0,004):(−0,25)=0,004:0,25 .

or perform column division of decimal fractions,
or go from decimals to ordinary fractions, then divide the corresponding ordinary fractions.

Let's look at both approaches.

To divide 0.004 by 0.25 with a column, first move the decimal point 2 digits to the right, and we will arrive at dividing 0.4 by 25. Now we do the division by column:

Thus, 0.004:0.25=0.016.

Now let’s show what the solution would look like if we decided to convert decimal fractions to ordinary fractions. Because and , then , and execute

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.

Lesson progress

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 quite working. 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. 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 14

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)

March 30

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 in students the skills of 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.

Lesson progress

1. Organizational moment

Hello guys! What did we do in previous lessons? (Multiplying and dividing rational numbers.)

2. Updating knowledge.

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

Remember the 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. Test execution ( 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 in honor of 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 14- 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 equal to night. Winter ends, spring begins, the larks arrive. According to an ancient custom, larks and waders are baked from the dough.

9) -12.5×50: (-25)

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

March 30- 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)