The combined property of addition. Numbers. Addition of natural numbers. Properties of the addition of natural numbers. I. Organizational moment

Municipal general education state-financed organization

Bolshekchakovskaya middle comprehensive school

municipal Area Kaltasinsky district

Republic of Bashkortostan

Abstract

mathematics lesson on the topic:

« The combination property of addition. Computing skills and skills »

Grade 2.

UMC "Harmony"

Compiler: Teacher primary classes

first qualifying category

Meniyeva Razifa Pavlovna

2016 – 2017 academic year

The date of the: 11/15/2016

Thing: mathematics

Class: 2

Lesson number 39.

Theme lesson: The combination property of addition. Computational skills and skills.

Purpose: To acquaint students with a combination property of addition. Improve computational skills and skills.

Tasks:

Educational:

– study by students of the combination property of addition and use it for a rapid account;

– development of computing skills, ability to analyze, summarize and make reasonable conclusions, to think logically;

– to form skills is logical and argued to express your thoughts.

Educational:

– education in students' culture of communication when working in groups, interest in the study of mathematics;

– education of the perfection, mutual respect, mutual arms;

– the formation of the ability to work in a pair, listen and understand the point of view of the other.

Developing:

– development of ability to analyze, summarize, prove;

– memory development logical thinking, creative abilities;

– speech development (make up your thoughts orally, argue and prove your choice of solving the problem), thinking (to establish analogies, generalize and classify).

Type of lesson: opening of new knowledge.

Forms of students: Frontal, group, individual.

Equipment: PC, projector, textbook "Mathematics" N.B. Study 2 class Part 1, TPO, presentation, Pictures with tasks, drawings, rebuses, cards for reflection.

1. Organizational moment.

Teacher: Hello guys! Today we have guests at the lesson. Let's greet guests.(Hello)

Teacher: All are ready for a lesson?

Pupils:

All managed we gather

For work together to take

We will think, reason,

We can start a lesson.

Teacher:

Today we have an unusual lesson.
We will fly to space with you, my friend!
Many tasks await us ahead.
Well, now the training is needed.

2. Oral account.

Teacher: Who will say to me, what can I go into space?(On rocket) -Right. Here on this rocket we are with you and fly. (Display rocket on the board) And during our flight, each of you can get yourself a star for the correct answer. These stars are on the table.
- please, please tell me which geometric figures Is our rocket?

Pupils: The rocket consists of such figures as a rectangle, a triangle, a circle.

Who will show?(Show at the board)

Teacher: Well done!

So, let's start the countdown of the launch of our missile. Let's take it together with 10,9,8,7,6,5,4,3,2,1. Go!

In order not to waste time in flight, we will observe the stars and count.

How much will it be if 5 fuel on 2 units? (7)

What is the amount of numbers 90 and 8? (98)

In the girl 5 apples. She ate everything except three. How many apples does she have left? (3)

- 60 Pears grew on the oak. Boys came and shot down 20 pears. How much do pears left?(On oak pears do not grow)

If the sister is older brother, then brother ...(younger sister)

And now we will solve the rebuses:

7th, p1na, But 40 ".

Teacher: Well done!

Look, guys, to our rocket. What is her number?(15) So we fly on a rocket at number 15.

And what can you say about the number 15?(Double-digit). What number follows in number 15?(16) . And in front of the number 15?(14) . Of how many dozens and units consists of this number?(1 tens and 5 units). And what is the number today? (fifteen)

- During flight, cosmonauts lead onboard magazines. Since we are cosmonauts today, then our notebooks are called logbooks.
Open your on-board logs and write the flight date.

Gymnastics for hand

And to write beautifully and correctly, you need to smash our hands.

Put your hand on the elbow. Imagine that there is a painting brush in your hand, and in front of you - the fence. Let's paint his brush movement up, down, up, down, right, left, right, left. Draw a mug. We shake our brush and proceed to work.

We write a number, cool work and perform a certificate.

(Sit down correctly, observe the slopes of onboard logs)

3. Actualization of knowledge.

Flies rocket

Around the earthly light.

And soon the way we met aliens. To allow us to land on their planet, they offer to solve us the task. (Listen)

We ducklings considered

And, of course, tired.

Eight floated in a pond,

Two hid in the garden,

Five in the grass GalDyat.

Who will help from the guys

What action did we use?(Addition)

We coped with the task. Fly further?

Flies rocket

Around the earthly light.

And we ended up on the planet Smesharikov.

Look at them two constellations. In one 2 (two) stars of blue and 4 (four) stars of yellow color, and in another 4 blue and 2 (two) yellow.

Find out how much stars in the first and second constellation?

How did you consider? Who will write on the board the expression of the first constellation (2+4=6), and who is the second constellation (4+2=6).

What can be said about expressions?(They are the same)

What rule we remember?(The amount does not change from the rearrangement of the terms)

What is the priority of addition? (This embodiment is called Movement)

4. Work on new material.

Flies rocket

Around the earthly light.

And on our way another planet on which the gnomes live. They prepared a task for us. Look at the screen. (Slide 1)

How many groups can be divided balls?(3) (Slide 2)

Make an expression on this picture. Who will write on the board? (3 + 4 + 5 \u003d 12)

What signs can these balls be divided into two groups?(In color and form)

Let's split them in color. That's what we did.(Slide 3)

Now according to this picture, we will make an expression. We combined the red balls in one group. How many red balls? (7) How did you know? (K 3 + 4) and then to this amount add orange balls. How many orange balls do we have? (five). Guys, we have grown red balls into one group, so we will replace them with the amount, for this we will write them in brackets, and to this amount weighing the number of orange balls. And that's what we did. (Slide 4)

Now we divide these balls in shape and drink another expression.(Slide 5) . Here we have connected in one group of 4 red and orange balls, so we will replace them here and write them in brackets. It means to number 3 add the amount of red and orange balls. And this expression it turned out.(Slide 6)

Write down these two expressions in logbooks.

Now decide next task Dwarfs. (Slide 7)

What are the signs you can decompose apples?(In color and size)

First we divide them in color. How many red apples? (7) How did you know? (2 + 6) These red apples we combined into one group, so we will replace them with the amount and write them in brackets, and then to the sum of the red apples will add green apples.(Slide 8)

Write down the expression in the logbooks.(2+6)+4=12

Let's check.(Slide 9) Read the expression.

Now we divide the apples in size. What are we connected in one group? (Little apples) How many small apples have become? (10) How did you know? (6 + 4), so we will replace them with the amount and write them in brackets. And we have such an expression: to a 2-minded apple add the sum of small red and green apples. Write down the expression.

Let's check.(Slide 10) Read the expression.

To get these expressions, we have two neighboring terms replaced the value of their sum and the third number added to this amount.

Now compare these expressions. See the result of these expressions. In the first and in the second expression, the result was the same.

What number turned out in these expressions?(12)

We can write such equality: (2 + 6) + 4 \u003d 2 + (6 + 4)( Write on blackboard)

This property is called the combination property of addition.

Fizminutka.

And now, we together
We fly on the rocket. (Hands up, Palm connect - "Dome of the Rocket".)
On the socks rose.
Quickly, quickly hands down.
One two three four -
Here flies a rocket sweating. (Pull head up, shoulders down.)

Open your tutorials on page 69 Read the rule. (read the rule) (two neighboring terms can be replaced by the value of their sum. This is a combatant property of addition (10 + 5) + 3 \u003d 10 + (5 + 3) with a combatant property of addition, you can use when calculating expressions)

So two neighboring terms we replace the value of their sum and add the third number to this amount. This is a combination property of addition. So we got acquainted with another property of addition.

Flies rocket

Around the earthly light.

And now we flew on our rocket near the stars so close that each of you can get a star. On these stars, the task you want to perform is written.

Task: "Decide these expressions. Use the combinated property of addition

1) 9+3+4 2) 8+4+5

(Two work at the board)

Teacher: Let's continue our journey.

Flies rocket

Around the earthly light.

And we have an unknown planet on which Luntik lives. He will allow us to land on your planet if we decide the next task. In the tutorial on page 69 you need to solve the task number 227. The first pair of examples we will analyze together. (The student writes an example on the board (21 + 9) +7) So we define the procedure, first we will perform an action in the bracket, the sum of two numbers 21 and 9 will be adding 7 it turns out 37. I decide the second example (the board decides the other student, writes Example 21+ (9 + 7)) We first find the value of the amount in the bracket, will be 16 then this amount will add to the number 21 it turns out 37.

Compare the result. The value in two expressions turned out to be the same. And what expression it was more convenient and easier to solve? (21 + 9) +7. And why? (Since there is a convenient number in brackets for addition). So, a combination property can be used for convenient computing.

And now we work in a pair. When solving this task, you can consult with a subsidement on the desk.

Let's check now what expression it was more convenient to solve. Consider who of you will answer.

Gymnastics for eyes

- Guys fell to me on the table. She wants our eyes a little rested.

We close our eyes, that's what miracles(Close both eyes)
Our eyes are resting, exercise perform(Continue to stand with closed eyes)
And now we will open them, build a bridge through the river.(Open eyes, look the bridge)
Draw the letter "O", it turns out easy(Eyes draw the letter "O")
Up raise, look down(Eyes raise up, lowered down)
Right, left (Eyes moving to the right and left)
Start again.(Eyes watch up and down)

Star back Invites us to work in working notebooks. Open the workingbook on page 45 Find No. 109. Show with the help of brackets which two terms replaced the value of the amount. (Check)

5. The results of the lesson.

Our space journey ends. We finally return home to your planet. What new learned in the lesson? (Got acquainted the combatant property of addition) .

6. Homework.

Write down homework: № 228, page 69.: "You need to show with the help of brackets, what 2 terms you will replace the value of their sum to find the value of each expression." So you need to use the combination property of addition.

7. Evaluation, reflection.

Today you were real cosmonauts. Let's consider how many stars you have collected during space travel. Well done. Estimation.

You have a sprocket on your desks. If you liked the lesson, then draw, a cheerful star, if not - sad.

Thank you for the lesson.

Add one number to another is quite simple. Consider an example, 4 + 3 \u003d 7. This expression means that three units added to four units and in the end they received seven units.
Numbers 3 and 4, which we have arisen called speed. And the result is addition number 7 is called sum.

Sum - This is the addition of numbers. Plus sign "+".
In an alphabent form, this example will look like this:

a +.b \u003d.c.

Administration components:
a. - Speed, b. - terms, c. - Amount.
If we add 4 units to 3 units, then as a result of the addition, we will get the same result it will be 7.

From this example, we conclude that no matter how we changed places the terms of the answer remains unchanged:

Called this property of the components movement Act of Addition.

Move the law of addition.

From the change of places of the terms, the amount does not change.

In an alphabet record, the Movement Law looks like this:

a +.b \u003d.b +.a.

If we look at the three terms, for example, take the numbers 1, 2 and 4. And we will perform addition in this order, first add 1 + 2, and then we will perform addition to the resulting amount 4, then we obtain the expression:

(1+2)+4=7

We can do on the contrary, first fold 2 + 4, and then add the amount to the amount received 1. The example will look like this:

1+(2+4)=7

The answer remained the same. In both types of addition of the same example, the answer is the same. We conclude:

(1+2)+4=1+(2+4)

This property of addition is called combining Law of Addition.

Movement and the combination of addition law works for all non-negative numbers.

Combining law of addition.

To add the third number to the sum of two numbers, you can add the amount of the second and third number to the first number.

(a +.b) +.c \u003d.a + (b +.c)

The combatant law works for any number of components. We use this law when we need to add numbers in a convenient order. For example, lay three numbers 12, 6, 8 and 4. It will be more convenient to first add 12 and 8, and then add the amount of two numbers 6 and 4 to the resulting sum.
(12+8)+(6+4)=30

Property of addition with zero.

When adding a number with zero, as a result, the amount will be the same number.

3+0=3
0+3=3
3+0=0+3

In an alphabent expression, addition with zero will look like this:

a + 0 \u003da.
0+ a \u003d.a.

Questions on the topic natural numbers:
Addition Table, make up and see how the property of the transmission law works?
The folding table from 1 to 10 may look like this:

The second variant of the folding table.

If we look at the folding tables, it can be seen how the transition law works.

In the expression A + B \u003d C amount, what will happen?
Answer: The amount is the result of the addition of the components. a + b and p.

In the expression A + B \u003d C terms, what will happen?
Answer: a and b. The components are the numbers that we fold.

What happens to a number if adding 0 to it?
Answer: Nothing, the number will not change. When adding with zero, the number remains the same, because zero is the absence of units.

How many of the terms should be in the example, so that a combination of the combat law of addition can be applied?
Answer: From the three terms and more.

Write down the transcendent law in alphabet?
Answer: a + b \u003d b + a

Examples for tasks.
Example number 1:
Record the answer in the following expressions: a) 15 + 7 b) 7 + 15
Answer: a) 22 b) 22

Example number 2:
Apply a combination law to the term: 1 + 3 + 5 + 2 + 9
1+3+5+2+9=(1+9)+(5+2)+3=10+7+3=10+(7+3)=10+10=20
Answer: 20.

Example number 3:
Decide the expression:
a) 5921 + 0 b) 0 + 5921
Decision:
a) 5921 + 0 \u003d 5921
b) 0 + 5921 \u003d 5921

Addition has two properties: Movement and blend.

Move property of addition

If the components are changed in places, the amount will not change. Indeed, when the terms of the components are permitted, the number of units consisting in each of them will not change, and therefore the number of units consisting in the amount will also not change. This can be easily convinced, considering the following example.

We calculate the sum of two numbers 3 and 4 in two ways. We can first take the number 3 and add the number 4 to it, the result is the number 7:

Or take first number 4 and add number 3 to it, in the amount it turns out again number 7:

Thus, there can be a sign of equality between expressions 3 + 4 and 4 + 3, since they are equal to the same meaning:

move property of addition:

From the permutation of the terms, the amount does not change.

movement Act of Addition.

IN generalWith the help of letters, the prolonged property of the addition can be written as follows:

a. + b. = b. + a.

where a. and b.

The combination property of addition

The result of three or more numbers does not depend on the sequence of action. This means that the components can be as accommodated for the convenience of computing. This can be easily convinced, considering the following example.

We calculate the sum of three terms 1, 3 and 4 in two ways:

To calculate the value of the expression, we can first add numbers 1 and 3 and to add a number 4. For clarity, the sum of numbers 1 and 3 can enter into brackets to indicate that this amount will be calculated primarily:

1 + 3 + 4 = (1 + 3) + 4 = 4 + 4 = 8

Either first add numbers 3 and 4 and the result obtained add to the number 1:

1 + 3 + 4 = 1 + (3 + 4) = 1 + 7 = 8

Thus, between expressions (1 + 3) + 4 and 1 + (3 + 4), you can put a sign of equality, since they are equal to the same meaning:

(1 + 3) + 4 = 1 + (3 + 4)

The same will be if other natural numbers would be taken as the components.

The considered example allows you to formulate the combination property of addition:

The amount of three or more components does not depend on the sequence of action.

This property is differently called combining Law of Addition.

In general, with the help of letters, the combination property of addition can be written as follows:

a. + (b. + c.) = (a. + b.) + c.

where a., b. and c. - Arbitrary natural numbers.

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The properties of addition are the first step to accelerate the account. The student who owns all the rapid additions has more time for complex tasks and checks its decision. Therefore, it makes sense to consider the properties of the addition again to correctly apply them in practice.

What is addition?

To begin with, let's remember what is generally addressed? Addition is one of the first operations that are studying at school, and sometimes even in children's garden. As a rule, addition is explained by the example of fruit.

If you take 3 pears and 2 apples, fold them in the basket, then pears are the first term, the apples are the second, but the total number of fruit in the basket is the amount. This definition cannot be called wrong, but students grow, how the numbers used are growing. It is difficult to imagine the addition of hundreds of thousands of fruit.

Therefore, in mathematics use another definition, which states that addition is the movement of a point on a numerical direct right.

Many knowledge is complicated with time. So if in primary school Pupils say that a negative result of addition is a mistake, then in grade 5, everyone already knows that such an answer is possible. So with the definition of the properties of addition. Ordinary fruits just not enough to imagine big numbers. Therefore, in high schools, they go to theoretical definitions.

Properties of addition

Allocate a transitional and combination property. The transfer property tells us that the amount of the terms of the terms will not be changed.

The combination property claims that in the examples, where two or more factor, addition can be made in any order. The main thing in this case is correctly grouped the components to speed up the calculations, and not make it even stronger. The easiest option is to look at the number of units. First of all, it is necessary to add those numbers, the sum of the units in which is equal to 10, for example, 29 and 31 in the amount will be given 60.

After that, there are whole dozens and only then everything else. This is the easiest and fastest way to solve examples.

In fact, not even every professor will be able to distinguish the use of a combatious property from the transition. They are extremely similar, some mathematics consider even that the combatant property is a continuation of the transitional. For the same reason, the teachers are rarely asked to distinguish the application in the task of one property from the other. You just need to use both.

Example

Examples of the combination property of addition is not difficult to find. In almost every example, this property is used.

15 * 3 + 5-13-17-2-16-2 - To begin with multiplication.

45 + 5-13-17-2-16-2 - now grouped members so as to calculate the result as quickly as possible. For this you need to remember that the difference can be represented as the sum of negative numbers. In our case, I just bring minus for the sign of the brackets.

45 + 5-13-17-2-16-2 \u003d (45 + 5) - (13 + 17) - (2 + 2 + 16) - now perform calculations in brackets and find the final result

45+5-13-17-2-16-2=(45+5)-(13+17)-(2+2+16)=50-30-0=0

This is the answer turned out to be quite a large example. You should not be afraid of simple answers like 0 or 1. Sometimes the compilers of examples are thus confused by students.

What did we know?

We talked about the addition, allocated the combinative and moving properties of addition. Talked about the differences in these properties, as well as the proper use of the combination property of addition. Solved a small example to show the use of a combination property in practice.

Test on the topic

Evaluation of the article

Average rating: 4.6. Total ratings obtained: 111.

The topic that this lesson is dedicated is "the properties of addition". You will get acquainted with the moving and combination properties of the addition, considering them on specific examples. Learn, in what cases you can use them to make the process of calculating more simple. Checking examples will help determine how well you learned the material learned.

Lesson: Properties of Addition

Look carefully on the expression:

9 + 6 + 8 + 7 + 2 + 4 + 1 + 3

We need to find its meaning. Let's do it.

9 + 6 = 15
15 + 8 = 23
23 + 7 = 30
30 + 2 = 32
32 + 4 = 36
36 + 1 = 37
37 + 3 = 40

The result of expression 9 + 6 + 8 + 7 + 2 + 4 + 1 + 3 \u003d 40.
Tell me, was it convenient to calculate? It was not quite convenient to calculate. Look again once in the number of this expression. Would they not be changed in place so that the calculations were more comfortable?

If we regroup the numbers in a different way:

9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 = …
9 + 1 = 10
10 + 8 = 18
18 + 2 = 20
20 + 7 = 27
27 + 3 = 30
30 + 6 = 36
36 + 4 = 40

The final result of expression 9 + 1 + 8 + 2 + 7 + 3 + 6 + 4 \u003d 40.
We see that the results of expressions turned out the same.

The components can be changed in places, if it is convenient for computing, and the value of this will not change.

In mathematics there is a law: Movement of the Additional Law. He says that the amount of terms does not change from the permutation.

Uncle Fedor and the ball argued. The ball found the value of the expression as it is recorded, and Uncle Fedor said that he knows another, a more convenient way to calculate. Do you see a more convenient way to calculate?

The ball has solved the expression as it is recorded. And Uncle Fedor, said he knows the law that allows the change in places to change in places, and changed the numbers of 25 and 3.

37 + 25 + 3 = 65 37 + 25 = 62

37 + 3 + 25 = 65 37 + 3 = 40

We see that the result remained the same, but it became much easier to consider.

Look at the following expressions and read them.

6 + (24 + 51) \u003d 81 (to 6 add the amount 24 and 51)
Is there a convenient way to calculate?
We see that if you add 6 and 24, then we will get a round number. To the round number is always easier to add something. Take in brackets the sum of numbers 6 and 24.
(6 + 24) + 51 = …
(to the sum of numbers 6 and 24 add 51)

Calculate the value of the expression and see if the expression value changed?

6 + 24 = 30
30 + 51 = 81

We see that the value of the expression remains the same.

Practice on another example.

(27 + 19) + 1 \u003d 47 (to the sum of numbers 27 and 19 add 1)
What numbers are convenient to group so that it turns out a convenient way?
You guessed that these are numbers 19 and 1. The sum of the numbers 19 and 1 take in brackets.
27 + (19 + 1) = …
(to 27 add the number of numbers 19 and 1)
Find the value of this expression. We remember that first is performed in brackets.
19 + 1 = 20
27 + 20 = 47

The value of our expression remains the same.

Full Completion Law: Two neighboring terms can be replaced by their sum.

Now it is practicing to use both laws. We need to calculate the value of the expression:

38 + 14 + 2 + 6 = …

First, we use the prolonged property of addition, which allows you to change the components of the places. We change places the terms 14 and 2.

38 + 14 + 2 + 6 = 38 + 2 + 14 + 6 = …

Now we use a compatibility, which allows us two neighboring terms to replace them with the amount.

38 + 14 + 2 + 6 = 38 + 2 + 14 + 6 = (38 + 2) + (14 + 6) =…

First learn the value of the amount 38 and 2.

Now the amount is 14 and 6.

3. Festival of pedagogical ideas " Public lesson» ().

Make home

1. Calculate the sum of the terms in different ways:

a) 5 + 3 + 5 b) 7 + 8 + 13 V) 24 + 9 + 16

2. Calculate the results of expressions:

a) 19 + 4 + 16 + 1 b) 8 + 15 + 12 + 5 V) 20 + 9 + 30 + 1

3. Calculate the amount in a convenient way:

a) 10 + 12 + 8 + 20 b) 17 + 4 + 3 + 16 V) 9 + 7 + 21 + 13