Distributive property of addition and multiplication. Basic properties of multiplication of integers

Lesson Objectives:

Obtain equalities expressing the distributive property of multiplication with respect to addition and subtraction.
Teach students to apply this property from left to right.
Show important practical value this property.
Develop in students logical thinking. Strengthen your computer skills.

Equipment: computers, posters with the properties of multiplication, with images of cars and apples, cards.

During the classes

1. Introductory speech of the teacher.

Today in the lesson we will consider another property of multiplication, which is of great practical importance, it helps to quickly multiply multi-digit numbers. Let us repeat the previously studied properties of multiplication. As we study a new topic, we will check our homework.

2. Solution of oral exercises.

I. Write on the board:

1 - Monday
2 - Tuesday
3 - Wednesday
4 - Thursday
5 - Friday
6 - Saturday
7 – Sunday

Exercise. Consider the day of the week. Multiply the number of the planned day by 2. Add 5 to the product. Multiply the sum by 5. Increase the product by 10 times. name the result. You have guessed... a day.

(№ * 2 + 5) * 5 * 10

II. Task from electronic textbook"Mathematics 5-11kl. New opportunities for mastering the course of mathematics. Practicum". Drofa LLC 2004, DOS LLC 2004, CD-ROM, NFPK. Section “Mathematics. Integers". Task number 8. Express control. Fill in the empty cells in the chain. Option 1.

III. On the desk:

a+b
(a+b)*c
m-n
m * c – n * c

2) Simplify:

5*x*6*y
3*2*a
a * 8 * 7
3*a*b

3) For what values of x does the equality become true:

x + 3 = 3 + x
407 * x = x * 407? Why?

What properties of multiplication were used?

3. Learning new material.

On the board is a poster with pictures of cars.

Picture 1.

Task for 1 group of students (boys).

In the garage in 2 rows there are trucks and cars. Write expressions.

How trucks in the 1st row? How many cars?
How many trucks are in the 2nd row? How many cars?
How many cars are in the garage?
How many trucks are in lane 1? How many trucks are in two rows?
How many cars are in the 1st row? How many cars are in two rows?
How many cars are in the garage?

Find the values of expressions 3 and 6. Compare these values. Write expressions in a notebook. Read equality.

Task for 2 groups of students (boys).

In the garage in 2 rows there are trucks and cars. What do the expressions mean:

4 – 3
4 * 2
3 * 2
(4 – 3) * 2
4 * 2 – 3 * 2

Find the values of the last two expressions.

So, between these expressions, you can put the sign =.

Let's read the equality: (4 - 3) * 2 = 4 * 2 - 3 * 2.

Poster with images of red and green apples.

Figure 2.

Task for the 3rd group of students (girls).

Compose expressions.

What is the mass of one red and one green apple together?
What is the mass of all the apples together?
What is the mass of all red apples together?
What is the mass of all green apples together?
What is the mass of all apples?

Find the values of expressions 2 and 5 and compare them. Write this expression in your notebook. Read.

Task for 4 groups of students (girls).

The mass of one red apple is 100 g, one green apple is 80 g.

Compose expressions.

How many g is the mass of one red apple greater than that of a green one?
What is the mass of all red apples?
What is the mass of all green apples?
By how many g is the mass of all red apples greater than that of green ones?

Find the values of expressions 2 and 5. Compare them. Read equality. Are the equalities true only for these numbers?

4. Checking homework.

Exercise. By abbreviation conditions of the problem to put the main question, compose an expression and find its value for the given values of the variables.

1 group

Find the value of the expression for a = 82, b = 21, c = 2.

2 group

Find the value of the expression at a = 82, b = 21, c = 2.

3 group

Find the value of the expression for a = 60, b = 40, c = 3.

4 group

Find the value of the expression at a = 60, b = 40, c = 3.

Class work.

Compare expression values.

For groups 1 and 2: (a + b) * c and a * c + b * c

For groups 3 and 4: (a - b) * c and a * c - b * c

(a + b) * c = a * c + b * c
(a - b) * c \u003d a * c - b * c

So, for any numbers a, b, c, it is true:

When multiplying a sum by a number, you can multiply each term by this number and add the resulting products.
When multiplying the difference by a number, you can multiply the minuend and subtracted by this number and subtract the second from the first product.
When multiplying the sum or difference by a number, the multiplication is distributed over each number enclosed in brackets. Therefore, this property of multiplication is called the distributive property of multiplication with respect to addition and subtraction.

Let's read the property statement from the textbook.

5. Consolidation of new material.

Complete #548. Apply the distributive property of multiplication.

(68 + a) * 2
17 * (14 - x)
(b-7) * 5
13*(2+y)

1) Choose tasks for assessment.

Assignments for the assessment of "5".

Example 1. Let's find the value of the product 42 * 50. Let's represent the number 42 as the sum of the numbers 40 and 2.

We get: 42 * 50 = (40 + 2) * 50. Now we apply the distribution property:

42 * 50 = (40 + 2) * 50 = 40 * 50 + 2 * 50 = 2 000 +100 = 2 100.

Similarly solve #546:

a) 91 * 8
c) 6 * 52
e) 202 * 3
g) 24 * 11
h) 35 * 12
i) 4 * 505

Represent the numbers 91.52, 202, 11, 12, 505 as a sum of tens and ones and apply the distributive property of multiplication with respect to addition.

Example 2. Find the value of the product 39 * 80.

Let's represent the number 39 as the difference between 40 and 1.

We get: 39 * 80 \u003d (40 - 1) \u003d 40 * 80 - 1 * 80 \u003d 3200 - 80 \u003d 3120.

Solve from #546:

b) 7 * 59
e) 397 * 5
d) 198 * 4
j) 25 * 399

Represent the numbers 59, 397, 198, 399 as the difference between tens and ones and apply the distributive property of multiplication with respect to subtraction.

Tasks for the assessment of "4".

Solve from No. 546 (a, c, e, g, h, i). Apply the distributive property of multiplication with respect to addition.

Solve from No. 546 (b, d, f, j). Apply the distributive property of multiplication with respect to subtraction.

Tasks for the assessment "3".

Solve No. 546 (a, c, e, g, h, i). Apply the distributive property of multiplication with respect to addition.

Solve No. 546 (b, d, f, j).

To solve problem No. 552, make an expression and draw a picture.

The distance between the two villages is 18 km. Of them went to different sides two cyclists. One travels m km per hour, and the other n km. How far apart will they be after 4 hours?

(Oral. Examples are recorded on reverse side boards.)

Replace with the missing numbers:

Assignment from the electronic textbook "Mathematics 5-11kl. New opportunities for mastering the course of mathematics. Practicum". Drofa LLC 2004, DOS LLC 2004, CD-ROM, NFPK. Section “Mathematics. Integers". Task number 7. Express control. Restore missing numbers.

6. Summing up the lesson.

So, we have considered the distributive property of multiplication with respect to addition and subtraction. Let us repeat the formulation of the property, read the equalities expressing the property. The application of the distributive property of multiplication from left to right can be expressed by the “open brackets” condition, since the expression was enclosed in brackets on the left side of the equality, but there are no brackets on the right. When solving oral exercises for guessing the day of the week, we also used the distributive property of multiplication with respect to addition.

(No. * 2 + 5) * 5 * 10 = 100 * No. + 250, and then solve an equation of the form:
100 * no + 250 = a

We have defined addition, multiplication, subtraction and division of integers. These actions (operations) have a number of characteristic results, which are called properties. In this article, we will consider the basic properties of addition and multiplication of integers, from which all other properties of these operations follow, as well as the properties of subtraction and division of integers.

Page navigation.

Integer addition has several other very important properties.

One of them is related to the existence of zero. This property of integer addition states that adding zero to any whole number does not change that number. Let's write down given property addition using letters: a+0=a and 0+a=a (this equality is valid due to the commutative property of addition), a is any integer. You may hear that the integer zero is called the neutral element in addition. Let's give a couple of examples. The sum of an integer −78 and zero is −78 ; if you add an integer to zero positive number 999 , then as a result we get the number 999 .

We will now formulate another property of integer addition, which is related to the existence of an opposite number for any integer. The sum of any whole number with its opposite number is zero. Here is the literal form of this property: a+(−a)=0 , where a and −a are opposite integers. For example, the sum 901+(−901) is zero; similarly, the sum of the opposite integers −97 and 97 is zero.

Basic properties of multiplication of integers

The multiplication of integers has all the properties of multiplication of natural numbers. We list the main of these properties.

Just as zero is a neutral integer with respect to addition, one is a neutral integer with respect to multiplication of integers. That is, multiplying any whole number by one does not change the number being multiplied. So 1·a=a , where a is any integer. The last equality can be rewritten as a 1=a , this allows us to make the commutative property of multiplication. Let's give two examples. The product of the integer 556 by 1 is 556; product of a unit and a whole negative number−78 is equal to −78 .

The next property of integer multiplication is related to multiplication by zero. The result of multiplying any integer a by zero zero , that is, a 0=0 . The equality 0·a=0 is also true due to the commutative property of multiplication of integers. In a particular case, when a=0, the product of zero and zero is equal to zero.

For the multiplication of integers, the property opposite to the previous one is also true. It claims that the product of two integers is equal to zero if at least one of the factors is equal to zero. In literal form, this property can be written as follows: a·b=0 , if either a=0 , or b=0 , or both a and b are equal to zero at the same time.

Distributive property of multiplication of integers with respect to addition

Together, the addition and multiplication of integers allows us to consider the distributive property of multiplication with respect to addition, which connects the two indicated actions. Using addition and multiplication together opens additional features, which we would be deprived of considering addition separately from multiplication.

So, the distributive property of multiplication with respect to addition says that the product of an integer a and the sum of two integers a and b is equal to the sum of the products of a b and a c, that is, a (b+c)=a b+a c. The same property can be written in another form: (a+b) c=a c+b c .

distribution property multiplication of integers with respect to addition, together with the associative property of addition, allow us to determine the multiplication of an integer by the sum of three and more integers, and then - and the multiplication of the sum of integers by the sum.

Also note that all other properties of addition and multiplication of integers can be obtained from the properties we have indicated, that is, they are consequences of the above properties.

Integer subtraction properties

From the obtained equality, as well as from the properties of addition and multiplication of integers, the following properties of subtraction of integers follow (a, b and c are arbitrary integers):

Subtraction of integers in general case does NOT have the commutative property: a−b≠b−a .
The difference of equal integers is equal to zero: a−a=0 .
The property of subtracting the sum of two integers from a given integer: a−(b+c)=(a−b)−c .
The property of subtracting an integer from the sum of two integers: (a+b)−c=(a−c)+b=a+(b−c) .
The distributive property of multiplication with respect to subtraction: a (b−c)=a b−a c and (a−b) c=a c−b c.
And all other properties of integer subtraction.

Integer division properties

Arguing about the meaning of division of integers, we found out that the division of integers is an action, reciprocal of multiplication. We have given the following definition: division of integers is finding unknown multiplier on famous work and a known multiplier. That is, we call the integer c the quotient of the integer a divided by the integer b when the product c·b is equal to a .

This definition, as well as all the properties of operations on integers considered above, allow us to establish the validity of the following properties of division of integers:

No integer can be divided by zero.
The property of dividing zero by an arbitrary non-zero integer a : 0:a=0 .
Property of dividing equal integers: a:a=1 , where a is any non-zero integer.
The property of dividing an arbitrary integer a by one: a:1=a .
In general, division of integers does NOT have the commutative property: a:b≠b:a .
The properties of dividing the sum and difference of two integers by an integer are: (a+b):c=a:c+b:c and (a−b):c=a:c−b:c , where a , b , and c are integers such that both a and b are divisible by c , and c is nonzero.
The property of dividing the product of two integers a and b by a nonzero integer c : (a b):c=(a:c) b if a is divisible by c ; (a b):c=a (b:c) if b is divisible by c ; (a b):c=(a:c) b=a (b:c) if both a and b are divisible by c .
The property of dividing an integer a by the product of two integers b and c (numbers a , b and c such that dividing a by b c is possible): a:(b c)=(a:b) c=(a :c) b .
Any other property of integer division.

Consider an example confirming the validity of the commutative property of multiplication of two natural numbers. Based on the meaning of multiplication of two natural numbers, we calculate the product of the numbers 2 and 6, as well as the product of the numbers 6 and 2, and check the equality of the multiplication results. The product of numbers 6 and 2 is equal to the sum 6+6, from the addition table we find 6+6=12. And the product of the numbers 2 and 6 is equal to the sum of 2+2+2+2+2+2, which is equal to 12 (if necessary, see the material of the article adding three or more numbers). Therefore, 6 2=2 6 .

Here is a picture illustrating the commutative property of multiplying two natural numbers.

Associative property of multiplication of natural numbers.

Let's voice the associative property of multiplication of natural numbers: multiply a given number by this work two numbers is the same as multiplying the given number by the first factor, and multiplying the result by the second factor. That is, a (b c)=(a b) c, where a , b and c can be any natural numbers (in round brackets contain expressions whose values are evaluated first).

Let us give an example to confirm the associative property of multiplication of natural numbers. Compute the product 4·(3·2) . By the meaning of multiplication, we have 3 2=3+3=6 , then 4 (3 2)=4 6=4+4+4+4+4+4=24 . Now let's do the multiplication (4 3) 2 . Since 4 3=4+4+4=12 , then (4 3) 2=12 2=12+12=24 . Thus, the equality 4·(3·2)=(4·3)·2 is true, which confirms the validity of the considered property.

Let's show a picture illustrating the associative property of multiplication of natural numbers.

In conclusion of this paragraph, we note that the associative property of multiplication allows us to uniquely determine the multiplication of three or more natural numbers.

Distributive property of multiplication with respect to addition.

The next property relates addition and multiplication. It is formulated as follows: multiplying a given sum of two numbers by a given number is the same as adding the product of the first term and given number with the product of the second term and the given number . This is the so-called distributive property of multiplication with respect to addition.

Using letters, the distributive property of multiplication with respect to addition is written as (a+b) c=a c+b c(in the expression a c + b c, multiplication is performed first, after which addition is performed, more about this is written in the article), where a, b and c are arbitrary natural numbers. Note that the strength of the commutative property of multiplication, the distributive property of multiplication can be written in following form: a (b+c)=a b+a c.

Let us give an example confirming the distributive property of multiplication of natural numbers. Let's check the equality (3+4) 2=3 2+4 2 . We have (3+4) 2=7 2=7+7=14 , and 3 2+4 2=(3+3)+(4+4)=6+8=14 , hence the equality ( 3+4) 2=3 2+4 2 is correct.

Let's show a picture corresponding to the distributive property of multiplication with respect to addition.

The distributive property of multiplication with respect to subtraction.

If we adhere to the meaning of multiplication, then the product 0 n, where n is an arbitrary natural number greater than one, is the sum of n terms, each of which is equal to zero. In this way, . The properties of addition allow us to assert that the last sum is zero.

Thus, for any natural number n, the equality 0 n=0 holds.

In order for the commutative property of multiplication to remain valid, we also accept the validity of the equality n·0=0 for any natural number n.

So, the product of zero and a natural number is zero, that is 0 n=0 and n 0=0, where n is an arbitrary natural number. The last statement is a formulation of the multiplication property of a natural number and zero.

In conclusion, we give a couple of examples related to the property of multiplication discussed in this subsection. The product of the numbers 45 and 0 is zero. If we multiply 0 by 45970, then we also get zero.

Now you can safely begin to study the rules by which the multiplication of natural numbers is carried out.

Bibliography.

Maths. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
Maths. Any textbooks for 5 classes of educational institutions.

Let's draw a rectangle on a piece of paper in a cage with sides of 5 cm and 3 cm. Let's break it into squares with a side of 1 cm ( fig. 143). Let's count the number of cells located in the rectangle. This can be done, for example, like this.

The number of squares with a side of 1 cm is 5 * 3. Each such square consists of four cells. That's why total number cells is (5 * 3 ) * 4 .

The same problem can be solved differently. Each of the five columns of the rectangle consists of three squares with a side of 1 cm. Therefore, one column contains 3 * 4 cells. Therefore, there will be 5 * (3 * 4 ) cells in total.

The cell count in Figure 143 illustrates in two ways associative property of multiplication for numbers 5, 3 and 4 . We have: (5 * 3 ) * 4 = 5 * (3 * 4 ).

To multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third numbers.

(ab)c = a(bc)

It follows from the commutative and associative properties of multiplication that when multiplying several numbers, factors can be interchanged and enclosed in brackets, thereby determining the order of calculations.

For example, the equalities are true:

abc=cba

17 * 2 * 3 * 5 = (17 * 3 ) * (2 * 5 ).

In figure 144, segment AB divides the rectangle considered above into a rectangle and a square.

We count the number of squares with a side of 1 cm in two ways.

On the one hand, there are 3 * 3 of them in the resulting square, and 3 * 2 in the rectangle. In total we get 3 * 3 + 3 * 2 squares. On the other hand, in each of the three lines given rectangle there are 3 + 2 squares. Then them total equals 3 * (3 + 2 ).

Equalsto 3 * (3 + 2 ) = 3 * 3 + 3 * 2 illustrates distributive property of multiplication with respect to addition.

To multiply a number by the sum of two numbers, you can multiply this number by each term and add the resulting products.

In literal form, this property is written as follows:

a(b + c) = ab + ac

It follows from the distributive property of multiplication with respect to addition that

ab + ac = a(b + c).

This equality allows the formula P = 2 a + 2 b to find the perimeter of a rectangle to be written as follows:

P = 2 (a + b).

Note that the distribution property is valid for three or more terms. For example:

a(m + n + p + q) = am + an + ap + aq.

The distributive property of multiplication with respect to subtraction also holds: if b > c or b = c, then

a(b − c) = ab − ac

Example 1 . Calculate convenient way:

1 ) 25 * 867 * 4 ;

2 ) 329 * 75 + 329 * 246 .

1) We use the commutative, and the eclipse associative property multiplications:

25 * 867 * 4 = 867 * (25 * 4 ) = 867 * 100 = 86 700 .

2) We have:

329 * 754 + 329 * 246 = 329 * (754 + 246 ) = 329 * 1 000 = 329 000 .

Example 2 . Simplify the expression:

1) 4 a * 3 b;

2 ) 18m − 13m.

1) Using the commutative and associative properties of multiplication, we get:

4 a * 3 b \u003d (4 * 3) * ab \u003d 12 ab.

2) Using the distributive property of multiplication with respect to subtraction, we obtain:

18m - 13m = m(18 - 13 ) = m * 5 = 5m.

Example 3 . Write the expression 5 (2 m + 7) so that it does not contain brackets.

According to the distributive property of multiplication with respect to addition, we have:

5 (2 m + 7 ) = 5 * 2 m + 5 * 7 = 10 m + 35 .

Such a transformation is called opening brackets.

Example 4 . Calculate the value of the expression 125 * 24 * 283 in a convenient way.

Solution. We have:

125 * 24 * 283 = 125 * 8 * 3 * 283 = (125 * 8 ) * (3 * 283 ) = 1 000 * 849 = 849 000 .

Example 5 . Perform the multiplication: 3 days 18 hours * 6.