Combinative and distributive properties of multiplication. Distributive property of multiplication relative to addition and subtraction. What are the properties of multiplication of natural numbers?
Class: 3
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Target: learn to simplify an expression containing only multiplication operations.
Tasks(Slide 2):
- Introduce the associative property of multiplication.
- To form an idea of the possibility of using the studied property to rationalize calculations.
- To develop ideas about the possibility of solving “life” problems using the subject “mathematics”.
- Develop intellectual and communicative general educational skills.
- Develop organizational general educational skills, including the ability to independently evaluate the results of one’s actions, control oneself, find and correct one’s own mistakes.
Lesson type: learning new material.
Lesson plan:
1. Organizational moment.
2. Oral counting. Mathematical warm-up.
Penmanship line.
3. Report the topic and objectives of the lesson.
4. Preparation for studying new material.
5. Studying new material.
6. Physical education minute
7. Work on consolidating n. m. Solving the problem.
8. Repetition of the material covered.
9. Lesson summary.
10. Reflection
11. Homework.
Equipment: task cards, visual material (tables), presentation.
DURING THE CLASSES
I. Organizational moment
The bell rang and stopped.
The lesson begins.
You sat down quietly at your desk
Everyone looked at me.
II. Verbal counting
– Let’s count orally:
1) “Funny daisies” (Slides 3-7 multiplication table)
2) Mathematical warm-up. Game “Find the odd one out” (Slide 8)
- 485 45 864 947 670 134 (classification into groups EXTRA 45 - two-digit, 670 - there is no number 4 in the number record).
- 9 45 72 90 54 81 27 22 18 (9 is single digit, 22 is not divisible by 9)
Penmanship line. Write the numbers in your notebook, alternating: 45 22 670 9
– Underline the neatest notation of the number
III. Report the topic and objectives of the lesson.(Slide 9)
–
Write down the date and topic of the lesson.
– Read the objectives of our lesson
IV. Preparing to study new material
a) Is the expression correct?
Write on the board:
(23 + 490 + 17) + (13 + 44 + 7) = 23 + 490 + 17 + 13 + 44 + 7
– Name the property of addition used. (Collaborative)
– What opportunity does the combining property provide?
The combinational 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 apply in this case?
The combinational property makes it possible to write expressions containing only addition, without parentheses. In this case, calculations can be performed in any order.
– In that case, what is another property of addition called? (Commutative)
– Does this expression cause difficulty? Why? (We don’t know how to multiply a two-digit number by a one-digit number)
V. Study of new material
1) If we perform multiplication in the order in which the expressions are written, difficulties will arise. What will help us overcome these difficulties?
(2 * 6) * 3 = 2 * 3 * 6
2) Work according to the textbook p. 70, No. 305 (Make your guess about the results that the Wolf and the Hare will get. Test yourself by performing the calculations).
3) No. 305. Check whether the values of the expressions are equal. Orally.
Write on the board:
(5 2) 3 and 5 (2 3)
(4 7) 5 and 4 (7 5)
4) Draw a conclusion. Rule.
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.
– Explain the associative property of multiplication.
– Explain the associative property of multiplication with examples
5) Teamwork
On the board: (8 3) 2, (6 3) 3, 2 (4 7)
VI. Fizminutka
1) Game "Mirror". (Slide 10)
My mirror, tell me,
Tell me the whole truth.
Are we smarter than everyone else in the world?
Funniest and funniest of all?
Repeat after me
Funny movements of naughty physical exercises.
2) Physical exercise for the eyes “Keen Eyes”.
– Close your eyes for 7 seconds, look to the right, then left, up, down, then make 6 circles clockwise, 6 circles counterclockwise with your eyes.
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 task. (There is a condition, a question)
– Select a condition, a 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 this task in terms of structure? (Compound problem, because it is impossible to immediately answer the question of the problem or the solution requires composing an expression)
– Type of task? (Compound task for sequential actions))
– Solve the problem without a short note by composing an expression. To do this, use the following card:
Help card
– In a notebook, the solution to the problem can be written 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 (using cards): (Slide 12)
– Place 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 (using a textbook)
(p. 71, No. 307 – according to options)
1st century (8 2) 2 = (6 2) 3 = (19 1) 0 =
2nd century (7 3) 3 = (9 2) 4 = (12 9) 0 =
Examination:
1st century (8 2) 2 = 32 (6 2) 3 = 36 (19 1) 0 = 0.
2nd century (7 3) 3 = 63 (9 2) 4 = 72 (12 9) 0 = 0
Properties of multiplication:(Slide 14).
- Commutative property
- Matching property
– Why do you need to know the properties of multiplication? (Slide 15).
- To count quickly
- Choose a rational method of counting
- To solve problems
VIII. Repetition of covered material. "Windmills".(Slide 16, 17)
- Increase the numbers 485, 583 and 681 by 38 and write down three numerical expressions (option 1)
- Reduce the numbers 583, 545 and 507 by 38 and write three numerical expressions (option 2)
485
+ 38
523583
+ 38
621681
+ 38
719583
– 38
545545
– 38
507507
– 38
469
Students complete assignments based on options (two students solve assignments on additional boards).
Peer review.
IX. Lesson summary
– What did you learn in class today?
– What is the meaning of the associative property of multiplication?
X. Reflection
– Who thinks that they understand the meaning of the associative property of multiplication? Who is satisfied with their work in class? Why?
– Who knows what he still 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 palm.
XI. Homework information
– What homework would you like to receive?
Optionally:
1. Learn the rule p. 70
2. Come up with and write down an expression on a new topic with a solution
(4 lessons, No. 113–135)
Lesson 1 (113–118)
Target– introduce students to the combination of their_
the ability of multiplication.
In the first lesson, it is useful to remember what properties
arithmetic operations are already known to children. For this
exercises during which schoolchildren will
use this or that property. For example, you can
Is it possible to assert that the values of the expressions in a given column_
are the same:
875 + (78 + 284)
(875 + 78) + 284
875 + (284 + 78)
(875 + 284) + 78
It makes sense to offer expressions whose meanings are
children cannot calculate, in this case they will be_
need to draw a conclusion based on reasoning.
Comparing, for example, the first and second expressions, they
note their similarities and differences; remember the matcher_
new property of addition (two adjacent terms can be
replace them with the sum), which means that the values are expressed
the marriages will be the same. The third expression is appropriate
compare differently with the first and using the commutative
property of addition, draw a conclusion. Fourth expression
can be compared with the second.
– What properties of addition are applicable for calculations?
change the meanings of these expressions? (Commutative
and associative.)
– What properties does multiplication have?
The guys remember that they know the commutative
property of multiplication. (It is reflected on p. 34 of the textbook
nickname “Try to remember!”)
- Today in class we will meet another one of ours_
multiplication!
On the board is the drawing given intask 113 . Teacher
rats in various ways. Children's proposals discussed_
are given. If difficulties arise, you can contact
to the analysis of the methods proposed by Misha and Masha.
(6 · 4) · 2: there are 6 squares in one rectangle, smart_
By pressing 6 by 4, Masha finds out how many squares contain
rectangles in one row. Multiplying the resulting re_
The result is 2, she finds out how many squares contain
rectangles in two rows, i.e. how many small ones are there?
number of squares in the picture.
Then we discuss Misha’s method: 6 · (4 · 2). You first_
we complete the action in brackets – 4 2, i.e. we find out how many
total of rectangles in two rows. In one rectangle_
nick 6 squares. Multiplying 6 by the result obtained,
We answer the question posed. Thus, both
another expression indicates how many small
squares in the picture.
This means (6 · 4) · 2 = 6 · (4 · 2).
Similar work is being done withtask 114 . Pos_
After this, children get acquainted with the formulation of the associative
properties of multiplication and compare it with the formulation
associative properties of addition.
Targettasks 115–117 - find out if children understand
formulation of the associative property of multiplication.
By doingtasks 116 we recommend using_
get a calculator. This will allow students to repeat well_
measurement of three-digit numbers.
Problem 118It's better to decide in class.
If children find it difficult to decide independently_
research institutetasks 118 , then the teacher can use the technique of
judgments of ready-made solutions or explanations of expressions,
written down according to the conditions of this problem. For example:
10 5 8 10 8 5
(8 10) 5 8 (10 5)
(2_column),as well as tasks48, 54, 55 TPO No. 1.
Lesson 2 (119–125)
Target
multiplication in calculations; derive the multiplication rule
number by 10.
Work withtask 119 organized according to
instructions given in the textbook:
a) children use the commutative property of multiplication
tion, rearranging the factors in the product 4 10 = 10 4,
find the value of the product 10 · 4 by adding the tens.
The following entries are made in notebooks:
4 10 = 40;
6 10 = 60, etc.
b) children act in the same way as when completing the task_
nia a). In notebooks write down those equalities that do not exist
in task a): 5 10 = 50; 7 10 = 70; 9 10 = 90;
c) analyze and compare the written equalities,
draw a conclusion (when multiplying a number by 10, you must assign
to the first factor zero and write the resulting number in
result);
d) check the formulated rule using calculations_
tore.
Application of the combinatory property of multiplication and pr_
Multiplying by 10 allows students to multiply
"round" tens to a single digit number, using on_
table multiplication skills (90 · 3, 70 · 4, etc.).
For this purpose, they are carried outtasks 120, 121, 123, 124.
By doingtasks 120 children first arranging_
draw brackets in a textbook with a pencil and then comment
your actions. For example: (5 · 7) · 10 = 35 · 10 – produced here
maintaining the first and second factors replaced its values
reading. It is useful to immediately find out what the value of pro_ is
production 35 10; 5 · (7 · 10) = 5 · 70 – here is the product
the second and third factors were replaced by its value.
When calculating the value of the product 5 70 children
can reason like this: let’s use the commutative
property of multiplication - 5 · 70 = 70 · 5. Now 7 dec. Can
repeat 5 times, we get 35 des.; this number is 350.
When explaining some equalities intask 121
schoolchildren first use the commutative their_
multiplication, and then associative. For example:
4 6 10 = 40 6
(4 10) 6 = 40 6
each equality on the left and on the right.
By calculating the values of the expressions written on the left,
the guys turn to the multiplication table and then take away_
calculate the result by 10 times:
(4 6) 10 = 24 10
INtask 123 It's useful to consider different ways
would justify the answer. For example, you can in the second expression
we can replace the product with its value, and we get_
what is the first expression:
4 (7 10) = 4 70
In the third expression you need in this case first
Use the associative property of multiplication:
(4 7) 10 = 4 (7 10) and then replace the product of it
meaning.
But you can do things differently, focusing not on
the first, and the second expression. In this case, the number 70 in per_
In this expression you need to represent it as a product:
4 70 = 4 (7 10)
And in the third expression, use to transform_
calling by combining property:
(4 7) 10 = 4 (7 10)
Organizing a discussion of different courses of action
Vtask 123 , the teacher can focus on dialogue
Misha and Masha, who is brought intask 124 .
where to indicate on the diagram known and unknown values_
ranks. As a result, the diagram looks like:
For computational exercises in class, we recommend
blowingtask 125, andtasks 59, 60 from TVET No. 1 .
Lesson 3 (126–132)
Target– learn to use the associative property
multiplication for calculations, improve skills
to solve problems.
Task 126performed orally. His goal is perfection
development of computational skills and the ability to apply
the associative property of multiplication. For example, comparing
expressions a) 45 10 and 9 50, students reason: number
45 can be represented as the product of 9 5, and then
replace the product of numbers 5 10 with its value.
Task 128also applies to computing
exercises that require active use
analysis and synthesis, comparison, generalization. Formulating the right
When constructing each row, most children used_
They use the concept of “increase by...”. For example: for row – 6,
12, 18, ... – “each next number increases by 6”;
for the series – 4, 8, 12, ... – “each next number is increased_
ends at 4”, etc.
But the following option is also possible: “To get a loan_
the first number in each row is increased
2 times, to get the third number in the series, the first
the number of rows was increased by 3 times, the fourth by 4 times,
fifth - 5 times, etc.
By lining up in rows according to this rule, students actually_
They literally repeat all cases of table multiplication.
reading, students can either draw
scheme, or “revive” the scheme that the teacher prepared in advance
will depict it on the board.
Children will write down the solution to the problem in a notebook on their own.
In case of difficulties in solvingtasks 129 reko_
We recommend using the technique of discussing ready-made solutions_
explanations or explanations of expressions written according to the condition
of this task:
10 · 3 3 · 4 10 · 4 (10 · 3) · 4 10 · (3 · 4)
Problem 133It is also advisable to discuss it in class.
(1) 14 + 7 = 21 (days) 2) 21 2 = 42 (days))
tasks 61, 62 TPO No. 1.
Lesson 4 (134–135)
Target– check the mastery of table skills
knowledge and problem solving skills.
134, 135 .
Targettasks 134 – summarize children’s knowledge about the table
multiplication, which can be represented as a table
Pythagoras. Therefore, after the task is completed_
No, it’s useful to find out:
a) In which cells of the table can the same be inserted?
What numbers and why? (These cells are in the bottom row_
ke and in the right column, which is due to the commutative
property of multiplication.)
b) Is it possible, without performing calculations, to say
how much is the next number greater than the previous one in each
row (column) of the table? (In the top (first) line –
by 1, in the second - by 2, in the third - by 3, etc.) This is conditional_
defined by the definition: “multiplication is the addition of one_
kov terms".
Students should also be reminded that
the entire table contains 81 cells. This corresponds to the number
which should be written in its lower right cell.
To test the knowledge, skills and abilities of students
Shmyreva G.G. Test papers. 3rd grade. – Smolensk,
Association XXI Century, 2004.
Definition. Multiplication is the action of finding the sum of identical terms. Multiply number A per number b means find the sum b terms, each of which is equal to a.
The numbers that are multiplied are called factors (or factors), and the result of the multiplication is called a product.
At multiplication The product of natural numbers is always a positive number. If one of the factors is equal to 0 (zero), then the product is equal to 0. If the product is equal to zero, then at least one of the factors is equal to 0.
If one of the two factors is equal to 1 (one), then work equal to the second factor.
- For example:
- 5 * 6 * 8 * 0 = 0
- 132 * 1 = 132
Multiplication laws
Combination law
Rule. To multiply the product of two factors by a third factor, you can multiply the first factor by the product of the second and third factors.
- For example:
- (7 * 6) * 5 = 7 * (6 * 5) = 210
- (a * b) * c = a * (b * c)
Travel law
Rule. Rearranging the factors does not change the product.
- For example:
- 7 * 6 * 5 = 5 * 6 * 7 = 210
- a * b * c = c * b * a
Distributive law
Rule. To multiply a number by a sum, you can multiply this number by each of the terms and add the resulting products.
- For example:
- 7 * (6 + 5) = 7 * 6 + 7 * 5 = 77
- a * (b + c) = ab + ac
The distributive law also applies to the action of subtraction.
- For example:
- 7 * (6 — 5) = 7 * 6 — 7 * 5 = 7
The laws of multiplication apply to any number of factors in numerical or alphabetic expression. The distributive law of multiplication is used to take the common factor out of brackets.
Rule. To convert a sum (difference) into a product, it is enough to take the same factor of the terms out of brackets, and write the remaining factors in brackets as the sum (difference).
Sections: Mathematics
Lesson objectives:
- Obtain equalities expressing the distributive property of multiplication relative to addition and subtraction.
- Teach students to apply this property from left to right.
- Show the important practical significance of this property.
- To develop logical thinking in students. Strengthen computer skills.
Equipment: computers, posters with multiplication properties, with images of cars and apples, cards.
During the classes
1. Introductory speech by the teacher.
Today in the lesson we will look at 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. Solving oral exercises.
I. Write on the board:
1 – Monday
2 – Tuesday
3 – Wednesday
4 – Thursday
5 – Friday
6 – Saturday
7 – Sunday
Exercise. Think about the day of the week. Multiply the number of the planned day by 2. Add 5 to the product. Multiply the amount by 5. Increase the product by 10 times. Name the result. You wished for... a day.
(№ * 2 + 5) * 5 * 10
II. Assignment from the electronic textbook “Mathematics 5-11 grades. New opportunities for mastering a mathematics course. Workshop". "Drofa" LLC 2004, "DOS" LLC 2004, CD - ROM, NFPC." Section “Mathematics. Integers". Task No. 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) At 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. Studying new material.
There is a poster with pictures of cars on the board.
Picture 1.
Assignment for 1 group of students (boys).
In the garage there are 2 rows of trucks and cars. Write down expressions.
- How many trucks are there in the 1st row? How many cars?
- How many trucks are there in the 2nd row? How many cars?
- How many cars are there in total in the garage?
- How many trucks are there in the 1st row? How many trucks are there in two rows?
- How many cars are there in the 1st row? How many cars are there in two rows?
- How many cars are there in the garage?
Find the values of expressions 3 and 6. Compare these values. Write down the expressions in your notebook. Read equality.
Assignment for group 2 of students (boys).
In the garage there are 2 rows of 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.
This means that you can put an = sign between these expressions.
Let's read the equality: (4 – 3) * 2 = 4 * 2 – 3 * 2.
Poster with images of red and green apples.
Figure 2.
Assignment for group 3 students (girls).
Make up 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.
Assignment for group 4 students (girls).
The mass of one red apple is 100 g, one green apple is 80 g.
Make up 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?
- How many grams are the mass of all red apples greater than the mass of green apples?
Find the meanings of expressions 2 and 5. Compare them. Read equality. Are the equalities true only for these numbers?
4. Checking homework.
Exercise. Based on a brief description of the problem conditions, pose the main question, compose an expression and find its value for given values of the variables.
1 group
Find the value of the expression when a = 82, b = 21, c = 2.
2nd group
Find the value of the expression for 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 for a = 60, b =40, c = 3.
Work in the classroom.
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 = a * c – b * c
So, for any numbers a, b, c, the following is true:
- When multiplying a sum by a number, you can multiply each term by that number and add the resulting products.
- When multiplying the difference by a number, you can multiply the minuend and subtrahend by this number and subtract the second from the first product.
- When multiplying a sum or difference by a number, the multiplication is distributed over each number enclosed in parentheses. Therefore, this property of multiplication is called the distributive property of multiplication with respect to addition and subtraction.
Let's read the formulation of the property 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) Select assignments for assessment.
Tasks graded “5”.
Example 1. Let's find the value of the product 42 * 50. Let's imagine 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.
Solve No. 546 in a similar way:
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 relative to addition.
Example 2. Let's find the value of the product 39 * 80.
Let's imagine the number 39 as the difference between 40 and 1.
We get: 39 * 80 = (40 – 1) = 40 * 80 – 1 * 80 = 3,200 – 80 = 3,120.
Solve from No. 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 relative to subtraction.
Tasks graded “4”.
Solve from No. 546 (a, c, d, g, h, i). Apply the distributive property of multiplication relative to addition.
Solve from No. 546 (b, d, f, j). Apply the distributive property of multiplication relative to subtraction.
Tasks graded “3”.
Solve No. 546 (a, c, d, g, h, i). Apply the distributive property of multiplication relative to addition.
Solve No. 546 (b, d, f, j).
To solve problem No. 552, compose an expression and make a drawing.
The distance between the two villages is 18 km. Two cyclists rode out from them in different directions. One travels m km per hour, and the other n km. What will be the distance between them after 4 hours?
Fill in the squares.
For what values of x is the equality true:
a) 3 * (x + 5) = 3 * x + 15
b) (3 + 5) * x = 3 * x + 5 * x
c) (7 + x) * 5 = 7 * 5 + 8 * 5
d) (x + 2) * 4 = 2 * 4 + 2 * 4
e) (5 – 3) * x = 5 * x – 3 * x
f) (5 – 3) * x = 5 * x – 3 * 2
The distributive property of multiplication allows us to quickly multiply multi-digit numbers.
2) Let's continue checking your homework.
1) Perform multiplication:
2) Find the error:
Why should the multiplication of these numbers be written as in the penultimate example?
It turns out that column multiplication of multi-digit numbers is also based on the distributive property of multiplication.
Let's look at an example:
Therefore, we begin to write the product 423 by 50 under tens.
(Orally. Examples are written on the back of the board.)
Replace with the missing numbers:
Assignment from the electronic textbook “Mathematics 5-11 grades. New opportunities for mastering a mathematics course. Workshop". "Drofa" LLC 2004, "DOS" LLC 2004, CD - ROM, NFPC." Section “Mathematics. Integers". Task No. 7. Express control. Recover missing numbers.
6. Summing up the lesson.
So, we have looked at the distributive property of multiplication relative to addition and subtraction. Let's repeat the formulation of the property, read the equalities expressing the property. The application of the distributive property of left-to-right multiplication can be expressed by the “open parentheses” condition, since on the left side of the equality the expression was enclosed in parentheses, but on the right side there were no parentheses. When solving oral exercises for guessing the day of the week, we also used the distributive property of multiplication relative to addition.
(No. * 2 + 5) * 5 * 10 = 100 * No. + 250, and then solved an equation of the form:
100 * No + 250 = a
Let's consider an example that confirms the validity of the commutative property of multiplying two natural numbers. Starting from the meaning of multiplying two natural numbers, let's calculate the product of numbers 2 and 6, as well as the product of 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 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 equal to 12 (if necessary, see the article on the addition of three or more numbers). Therefore, 6·2=2·6.
Here is a picture illustrating the commutative property of multiplying two natural numbers.
Combinative property of multiplication of natural numbers.
Let's voice 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 resulting result by the second factor. That is, a·(b·c)=(a·b)·c, where a , b and c can be any natural numbers (the expressions whose values are calculated first are enclosed in parentheses).
Let us give an example to confirm the associative property of multiplying natural numbers. Let's calculate the product 4·(3·2) . According to 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 multiply (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, confirming the validity of the property in question.
Let us show a drawing 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 relative to addition.
The following property connects 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 a given number with the product of the second term and a given number. This is the so-called distributive property of multiplication relative to addition.
Using letters, the distributive property of multiplication relative to addition is written as (a+b)c=ac+bc(in the expression a·c+b·c, multiplication is performed first, after which addition is performed; more details about this are written in the article), where a, b and c are arbitrary natural numbers. Note that the force of the commutative property of multiplication, the distributive property of multiplication can be written in the 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 validity of 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 a figure corresponding to the distributive property of multiplication relative to addition.
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. Thus, 
. The properties of addition allow us to say that the final 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 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 paragraph. The product of the numbers 45 and 0 is equal to zero. If we multiply 0 by 45,970, we also get zero.
Now you can safely begin studying the rules by which multiplication of natural numbers is carried out.
Bibliography.
- Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
- Mathematics. Any textbooks for 5th grade of general education institutions.
