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Target: learn to simplify an expression containing only multiplication actions.
Tasks(Slide 2):
- Introduce the combinative property of multiplication.
- To form an idea of the possibility of using the studied property to rationalize computations.
- To develop ideas in the possibility of solving "life" problems by means of the subject "mathematics".
- Develop intellectual and communicative general educational skills.
- To develop organizational general educational skills, including the ability to independently evaluate the result of one's actions, to control oneself, to find and correct one's own mistakes.
Lesson type: learning new material.
Lesson plan:
1. Organizational moment.
2. Verbal counting. Mathematical warm-up.
Calligraphy string.
3. Communication of the topic and objectives of the lesson.
4. Preparation for the study of new material.
5. Learning new material.
6. Physical education
7. Work to consolidate n. m. Solution of the problem.
8. Repetition of the passed material.
9. Lesson summary.
10. Reflection
11. Homework.
Equipment: task cards, visual material (tables), presentation.
DURING THE CLASSES
I. Organizational moment
The bell rang and fell silent.
The lesson begins.
You sat down quietly at your desks
Everyone looked at me.
II. Verbal counting
- Let's count orally:
1) "Cheerful daisies" (Slides 3-7 multiplication table)
2) Mathematical warm-up. Find unnecessary game (Slide 8)
- 485 45 864 947 670 134 (classification into groups OVER 45 - two-digit, 670 - there is no number 4 in the number record).
- 9 45 72 90 54 81 27 22 18 (9 is one-digit, 22 is not divisible by 9)
Calligraphy string. Write down numbers in a notebook, alternating: 45 22 670 9
- Underline the tidiest number notation
III. Communication of the topic and objectives of the lesson.(Slide 9)
–
Write down the number, the topic of the lesson.
- Read the objectives of our lesson
IV. Preparing to learn new material
a) Is the expression true
Writing on the board:
(23 + 490 + 17) + (13 + 44 + 7) = 23 + 490 + 17 + 13 + 44 + 7
- Name the addition property used. (Conjoint)
- What opportunity does the combination property give?
The combination property makes it possible to write expressions containing only addition, without parentheses.
43 + 17 + (45 + 65 + 91) = 91 + 65 + 45 + 43 + 17
- What properties of addition do we use in this case?
The combination property makes it possible to write expressions containing only addition, without parentheses. In this case, the calculations can be performed in any order.
- In that case, what is the name of one more property of addition? (Translocative)
- Does this expression cause difficulty? Why? (We do not know how to multiply a two-digit number by a single-digit number)
V. Study of new material
1) If we carry out the multiplication in the order in which the expressions are written, then difficulties will arise. What will help us to remove these difficulties?
(2 * 6) * 3 = 2 * 3 * 6
2) Work on the textbook p. 70, № 305 (Give your guess about the results that the Wolf and the Hare will get. Check yourself by performing the calculations).
3) No. 305. Check if the values of the expressions are equal. Orally.
Writing on the board:
(5 2) 3 and 5 (2 3)
(4 7) 5 and 4 (7 5)
4) Make a conclusion. Rule.
To multiply the product of two numbers by the third number, you can multiply the first number by the product of the second and third.
- Tell us the combination property of multiplication.
- Explain the combinatory property of multiplication by examples
5) Teamwork
On the board: (8 3) 2, (6 3) 3, 2 (4 7)
Vi. Fizminutka
1) Game "Mirror". (Slide 10)
Light my mirror, tell me
Yes, report the whole truth.
We are smarter than everyone else
All the funniest and funniest?
Repeat everything after me
Cheerful movements of a naughty physical minute.
2) Fizminutka for eyes "Keen eyes".
- Close your eyes for 7 seconds, look to the right, then to the left, up, down, then make your eyes 6 circles clockwise, 6 circles counterclockwise.
Vii. Consolidation of what has been learned
1) Work according to the textbook. the solution of the problem. (Slide 11)
(p. 71, no. 308) Read the text. Prove that this is a challenge. (There is a condition, a question)
- Highlight the condition, question.
- Name the numerical data. (Three, 6, three-liter)
- What do they mean? (Three boxes. 6 cans, each can contains 3 liters of juice)
- What is the task in terms of structure? (Composite problem, because you cannot immediately answer the question of the problem or the solution requires the compilation of an expression)
- Task type? (Composite task for sequential actions))
- Solve the problem without a short note drawing up an expression. To do this, use the following card:
Assistant card
- In a notebook, the solution to the problem can be formulated as follows: (3 6) 3
- Can we solve the problem in this order?
(3 6) 3 = (3 3) 6 = 9 6 = 54 (l).
3 (3 6) = (3 3) 6 = 9 6 = 54 (l)
Answer: 54 liters of juice in all boxes.
2) Work in pairs (on cards): (Slide 12)
- Put the signs without calculating:
(15 * 2) * 4 15 * (2 * 4) (–What property?)
(8 * 9) * 6 7 * (9 * 6)
(428 * 2) * 0 1 * (2 * 3)
(3 * 4) * 2 3 + 4 + 2
(2 * 3) * 4 (4 * 2) * 3
Check: (Slide 13)
(15 * 2) * 4 = 15 * (2 * 4)
(8 * 9) * 6 > 7 * (9 * 6)
(428 * 2) * 0 < 1 * (2 * 3)
(3 * 4) * 2 > 3 + 4 + 2
(2 * 3) * 4 = (4 * 2) * 3
3) Independent work (according to the textbook)
(p. 71, No. 307 - by options)
1 c. (8 2) 2 = (6 2) 3 = (19 1) 0 =
2 c. (7 3) 3 = (9 2) 4 = (12 9) 0 =
Examination:
1 c. (8 2) 2 = 32 (6 2) 3 = 36 (19 1) 0 = 0.
2 c. (7 3) 3 = 63 (9 2) 4 = 72 (12 9) 0 = 0
Multiplication properties:(Slide 14).
- Displacement property
- Combination property
- Why do you need to know the properties of multiplication? (Slide 15).
- To count quickly
- Choose a rational way of counting
- To solve problems
VIII. Repetition of the passed material. "Windmills".(Slide 16, 17)
- Increase the numbers 485, 583 and 681 by 38 and write down three numerical expressions (1 option)
- Reduce the numbers 583, 545 and 507 by 38 and write down three numerical expressions (option 2)
485
+ 38
523583
+ 38
621681
+ 38
719583
– 38
545545
– 38
507507
– 38
469
Students complete assignments according to options (two students solve assignments on additional boards).
Mutual verification.
IX. Lesson summary
- What did you learn in the lesson today?
- What is the meaning of the combination property of multiplication?
X. Reflection
- Who believes that they have understood the meaning of the combination property of multiplication? Who is satisfied with their work in the lesson? Why?
- Who knows what else he needs to work on?
- Guys, if you liked the lesson, if you are satisfied with your work, then put your hands on your elbows and show me your palms. And if you were upset about something, then show me the back of your hand.
XI. Homework information
- What homework would you like to receive?
Optionally:
1. Learn the rule with. 70
2. Come up with and write down an expression on a new topic with a solution
Combination property of multiplication
Goals: to acquaint students with the combinatory property of multiplication; to teach how to use the combined property of multiplication in the analysis of numerical expressions; repeat the properties of addition and the displacement property of multiplication; improve computing skills; develop the ability to analyze, reason.
Subject results:
to get acquainted with the combined property of multiplication, to form ideas about the possibility of using the studied property to rationalize computations.
Metasubject results:
Regulatory: plan your action in accordance with the assigned task, accept and save the educational task.
Cognitive: use symbolic means, models and schemes for solving problems, focus on a variety of ways to solve problems; establish analogies.
Communicative: build speech utterances in oral and writing, form their own opinion, ask and answer questions, proving the correctness of their opinion.
Personal: develop the ability to self-esteem, promote success in mastering the material.
Lesson type: learning new material.
Equipment: cards with the task, visual material (tables), presentation.
DURING THE CLASSES
I ... Organizing time(emotional attitude)
The long-awaited call is given
The lesson begins.
Have you all had time to rest?
And now - go ahead, get down to business!
Guys, let's wish each other in the lesson to be attentive, collected, diligent. Let's greet each other with smiles and start the lesson.
II. Basic knowledge actualization + Goal setting
On the board incomplete record of the topic ______________________ property of multiplication
Looking at the incomplete transcript, think about what we will be doing in the lesson and what the topic of today's lesson is. (Children's reasoning)
Today we will get acquainted with the new property of multiplication, the name of which we will learn by completing the tasks of oral counting and the tasks included in your sheets - lesson cards, we will learn how to use the new property of multiplication when analyzing numerical expressions; we repeat the properties of addition and the displacement property of multiplication ;; we will develop computational skills, the ability to analyze, reason.
We will work amicably and creatively, in pairs and independently, we will complete tasks and draw conclusions.
In your cards, after each assignment, you will have to evaluate your work. If you have coped with the task without mistakes, you will set yourself +, if you have not coped, then -
Why do we need it?
Where can we apply the knowledge gained?
Proverb
To teach math - sharpen the mind
How do you understand the meaning of this proverb?
"Mathematics only then must be taught, that it puts the mind in order"
M. Lomonosov
III. Verbal counting
1. Game "Truth - lie". Children show a + or - sign
6 and 5 add 12
The difference of numbers 16 and 6 is 9
9 increase by 5 equals 14
100 is the largest three-digit number
A cube is a three-dimensional figure
The rectangle is a flat shape
The letter C opens on the board
2.The Tricky Challenge
Add the number of colors of the rainbow to the student's favorite grade.
Add the number of days in a week to the number of months in a year.
The letter 0 opens on the board
3.The Logic Problem
There were 2 birches, 4 apple trees, 5 cherries in the garden. How many fruit trees were there in the garden? The letter H opens on the board
4. Into what groups the following figures can be distributed
The letter E opens on the board
The letter T opens on the board
The letter A opens on the board
7. Can we say that the area of these figures is the same?
The letter T opens on the board
8. Pair work: Break the numbers into two groups.
List each group in ascending order (Badge of Working Together) e
499 75 345 24 521 86
The letter E opens on the board
9. Independent work
Fill out the card
The letter L opens on the board
10. Select the required sign (+ or )
Increase by 6
Zoom in 3 times
The letter L opens on the board
11. ,
2 · 6 ... 6 + 6 + 6
5 6 ... 6 4
8 6 ... 6 8
The letter H opens on the board
12. Which numeric expression is redundant? Why?
(2 +7) 0 365 0
(9 2) 1 (94-26) 0
The letter O opens on the board
13.Frontal work
Insert the missing numbers:
- What properties of addition and multiplication helped you to complete the task? (Traveling and combination properties addition; the displacement property of multiplication.)The letter E opens on the board
Topic opens on the boardCombined multiplication property
Fizminutka
To begin with, we With you
For starters, you and me
We only twist our head.
(Head rotation.)
We rotate the body too.
Of course, we can do that.
(Turns left and right.)
Finally stretched
Up and to the sides.
Have caved in.
(Stretching up and to the sides.)
III. Post new material
1. Statement of the educational problem
Can we say that the values of the expressions in this column are the same?
(For expressions 1 and 2, the combination property of addition is applicable - 2 adjacent terms can be replaced with a sum and the values of the expressions will be the same;
3 and 1 expression - applied the displaceable addition property
4 and 2 expression is a displacement property.)
-What properties are applicable for calculating data
expressions?
(Displacement and combination property)
- Is it possible to assert that the values of the expressions in this column are the same?
We have to answer this question.
We will find out today is it possible to use the combination property when multiplying?)
2.Primary assimilation of new knowledge
Count up different ways the number of all small squares and write down the expression.
1 way:(6*4)*2 = 24*2=48
(There are 6 squares in one rectangle, multiplying 6 by 4, we find out how many squares are in one row. Multiplying the result by 2, we find out how many squares are in two rows).
2 way: 6*(4*2)= 6*8=48
(First, we perform the action in brackets - 4 * 2, that is, we find out how many rectangles there are in two rows. There are 6 squares in one rectangle. By multiplying 6 by the result, we answer the question.)
Conclusion: Thus, both expressions indicate how many small squares there are in the picture.
Means: (6 * 4) * 2 = 6 * (4 * 2) - the combined property of multiplication
Knowledge of the formulation of the combination property of multiplication and its comparison with the formulation of the combination property of addition.
IV... Initial test of understanding
Open the tutorial on page 50 and find # 160
Explain what the numerical equalities under each figure mean?
(4*3)*2= 4*(3*2)
(4 snowflakes were placed in 3 squares and took 2 rows or 4 snowflakes were placed in 3 squares with 2 rows.)
(6 squares took 5 rows and placed in 2 large squares or 6 squares took 5 rows in two large squares)
Let's read the rule:
Primary anchoringWorking at the blackboard
Find No. 161 (1 column)
We read the task: ( Write each expression as a product of three single-digit numbers)
Find No. 162 (1 column)
We read the task : Is it true that the values of the expressions in each column are the same?
We work independently in rows (check at the board), using the combination property: To multiply the product of two numbers by the third, you can multiply the first number by the product of the second and third numbers.
Summing up the results of the lesson.
Evaluation
Let's go back to the numerical expressions that we met at the beginning of the lesson. Tell me, is it possible to assert that the values of the expressions in this column are the same?
What discovery did you make in class today? Where can it be applied?
(We got acquainted with the new property of multiplication) To multiply the product of two numbers by the third, you can multiply the first number by the product of the second and third numbers.
Homework: rule p.50, no. 163 * Find proverbs or sayings famous people about mathematics
Grading.
Grades "5" are given to those guys who have no minuses in the map.
Who gets 1-2 minuses "4"
3-5 minuses - "3"
More than 5 minuses - "2"
Reflection
Finish the phrase
Today in class I ... ..
The most difficult thing for me was ... ..
Today I realized ...
Today I learned ...
Decide for yourself
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 look at the basic properties of addition and multiplication of integers, from which all other properties of these actions follow, as well as the properties of subtraction and division of integers.
Page navigation.
There are several other very important properties in the addition of integers.
One of them is related to the existence of zero. This integer addition property states that adding zero to any integer does not change that number... Let us write this property of addition using letters: a + 0 = a and 0 + a = a (this equality is valid due to the displacement property of addition), a is any integer. You may hear that the integer zero is called neutral addition. Here are a couple of examples. The sum of the integer −78 and zero is −78; if you add a positive integer 999 to zero, the result is 999.
Now we will give the formulation of another property of addition of integers, which is associated with the existence of the opposite number for any integer. The sum of any integer with its opposite number is zero... Let us write this property literally: 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 equals zero.
Basic properties of integer multiplication
Multiplication of integers has all the properties of multiplication of natural numbers. Let's list the main of these properties.
Just as zero is an integer neutral with respect to addition, one is a neutral integer with respect to integer multiplication. That is, multiplying any integer by one does not change the multiplied number... 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 displacement property of multiplication. Here are two examples. The product of an integer 556 times 1 is 556; the product of one and the negative integer −78 is −78.
The next property of integer multiplication is related to multiplication by zero. The result of multiplying any integer a by zero is zero , that is, a 0 = 0. Also, the equality 0 · a = 0 is true due to the displacement property of multiplication of integers. In the particular case, for a = 0, the product of zero by zero is equal to zero.
For the multiplication of integers, the opposite property to the previous one is also true. It claims that the product of two integers is zero if at least one of the factors is 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.
Distribution property of multiplication of integers relative to addition
Joint addition and multiplication of integers allows us to consider the distribution property of multiplication with respect to addition, which connects the two indicated actions. Using addition and multiplication together opens up 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 by the sum of two integers a and b is equal to the sum of the products a b and a c, that is, a (b + c) = a b + a c... The same property can be written in a different form: (a + b) c = a c + b c .
The distributive property of multiplying integers with respect to addition together with the combination property of addition allows you to define the multiplication of an integer by the sum of three and more integers, and then 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 specified, 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 generally does NOT have the movable property: a − b ≠ b − a.
- The difference of equal integers is 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).
- Distribution property of multiplication relative to subtraction: a (b − c) = a b − a c and (a − b) c = a c − b c.
- And all other integer subtraction properties.
Integer division properties
In discussing the meaning of dividing integers, we found out that dividing integers is the opposite of multiplication. We gave this definition: dividing integers is finding an unknown factor by famous work and a known factor. That is, we call an integer c the quotient of dividing an integer a by an 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 following properties division of integers:
- No integer may be divisible by zero.
- The property of dividing zero by an arbitrary nonzero integer a: 0: a = 0.
- The division property of equal integers: a: a = 1, where a is any nonzero integer.
- The property of dividing an arbitrary integer number a by one: a: 1 = a.
- In general, division of integers does NOT have the movable property: a: b ≠ b: a.
- Properties of dividing the sum and difference of two integers by an integer: (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 integer division properties.
Consider an example confirming the validity of the displacement property of multiplication of two natural numbers... Based on the meaning of the 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 the numbers 6 and 2 is equal to the sum of 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 2 + 2 + 2 + 2 + 2 + 2, which is 12 (if necessary, see the material of the article, the addition of three or more numbers). Therefore, 6 2 = 2 6.
Here is a figure illustrating the displacement property of multiplying two natural numbers.
Combination property of multiplication of natural numbers.
Let us sound the combinatory property of multiplying natural numbers: multiplying a given number by a given product of two numbers is the same as multiplying a 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 (expressions that are evaluated first are enclosed in parentheses).
Let us give an example to confirm the combinatory property of multiplication of natural numbers. We calculate 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 property under consideration.
Let us show a figure illustrating the combinatory property of multiplying natural numbers.
In conclusion of this item, we note that the combination property of multiplication makes it possible to unambiguously determine the multiplication of three or more natural numbers.
The distributive property of multiplication relative to addition.
The next property links addition and multiplication. It is formulated as follows: multiplying the given sum of two numbers by a given number is the same as adding the product of the first term and the given number with the product of the second term and the given number. This is the so-called distributive property of multiplication relative to addition.
With the help of letters, the distribution property of multiplication relative to addition is written as (a + b) c = a c + b c(in the expression a c + b c, multiplication is performed first, then addition, for more details see the article), where a, b and c are arbitrary natural numbers. Note that by virtue of the displaceable property of multiplication, the distributive property of multiplication can be written in the following form: a (b + c) = a b + a c.
Let's give an example that confirms the distribution property of multiplication of natural numbers. Let us 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, therefore, the equality ( 3 + 4) 2 = 3 2 + 4 2 is correct.
Let us show the figure corresponding to the distribution property of multiplication with respect to addition.
The distributive property of multiplication relative 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 addition properties allow us to assert that the last sum is zero.
Thus, for any natural number n, the equality 0 · n = 0 holds.
To keep the transposable property of multiplication valid, let us also accept the equality n · 0 = 0 for any natural number n.
So, product of zero and natural number is equal to 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 property of multiplication 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 equal to zero. If we multiply 0 by 45 970, then we also get zero.
Now you can safely start studying the rules by which the multiplication of natural numbers is carried out.
Bibliography.
- Mathematics. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
- Mathematics. Any textbooks for 5 grades of general education institutions.
Draw on a piece of paper a rectangle with sides of 5 cm and 3 cm. Divide 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. So total number cells is (5 * 3) * 4.
The same problem can be solved in a different way. 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 combination 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 the third number, you can multiply the first number by the product of the second and third numbers.
(ab) c = a (bc)
From the movable and combinatory properties of multiplication, it follows that when multiplying several numbers, the factors can be interchanged and enclosed in parentheses, 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 above rectangle into a rectangle and a square.
Let's count the number of squares with a side of 1 cm in two ways.
On the one hand, the resulting square contains 3 * 3 of them, and the rectangle contains 3 * 2. In total, we get 3 * 3 + 3 * 2 squares. On the other hand, each of the three lines of this rectangle contains 3 + 2 squares. Then their total is equal to 3 * (3 + 2).
Ravensto 3 * (3 + 2) = 3 * 3 + 3 * 2 illustrates the distributive property of multiplication relative 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
From the distributive property of multiplication with respect to addition it follows that
ab + ac = a (b + c).
This equality allows the formula P = 2 a + 2 b for finding the perimeter of a rectangle to be written in this form:
P = 2 (a + b).
Note that the distribution property is valid for three or more terms. For instance:
a (m + n + p + q) = am + an + ap + aq.
The distributive property of multiplication with respect to subtraction is also true: if b> c or b = c, then
a (b - c) = ab - ac
Example 1 . Calculate in a convenient way:
1 ) 25 * 867 * 4 ;
2 ) 329 * 75 + 329 * 246 .
1) We use the displaceable, and eclipse, the combination properties of multiplication:
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) 18 m - 13 m.
1) Using the displacement and combination properties of multiplication, we get:
4 a * 3 b = (4 * 3) * ab = 12 ab.
2) Using the distribution property of multiplication with respect to subtraction, we get:
18 m - 13 m = m (18 - 13) = m * 5 = 5 m.
Example 3 . Write down the expression 5 (2 m + 7) so that it does not contain parentheses.
According to the distribution 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 in a convenient way the value of the expression 125 * 24 * 283.
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.
Solution. We have:
3 days 18 hours * 6 = 18 days 108 hours = 22 days 12 hours.
When solving the example, the distribution property of multiplication relative to addition was used:
3 days 18 hours * 6 = (3 days + 18 hours) * 6 = 3 days * 6 + 18 hours * 6 = 18 days + 108 hours = 18 days + 96 hours + 12 hours = 18 days + 4 days + 12 hours = 22 days 12 hours
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