If you subtract zero from a natural number. Subtraction of natural numbers - copy from Antoshka

Lesson on the topic: Lesson on the topic: "Rules for subtracting natural numbers. Examples"

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What numbers are called natural numbers?

Remember!
Number 0 and negative numbers-1, -2, -3, ... are not natural numbers.
The smallest natural number is the number 1. Each subsequent number in the series natural numbers more than the previous one by one. There is no greatest natural number, so the series of natural numbers is said to be infinite.

Subtraction- this is an action inverse of addition. Using the subtraction operation, one of two terms is determined if their sum is known.
Using this arithmetic operation, you can determine how much one number is greater or less than another.

Let's look at an example: 5 - 4 = 1.
In this example:
5 is the number being reduced;
4 is the number to be subtracted;
1 is the difference of two numbers.

What subtraction is can be explained using a coordinate ray.

Relationship between arithmetic operations “addition” and “subtraction”

For example, you need to find the difference between two numbers: 78 - 18 = ?
78 - 18 = 60.
We check the result of solving the example using the addition operation: 60 + 18 = 78.

Rules for subtracting natural numbers

Let's explain the third rule with an example: 48 - (14 + 12) = 48 - 14 - 12 = 22.

4. If you need to subtract a number from the sum of numbers, then you can first subtract the number from the first term, and then add the second term to the resulting difference.

Let's explain this rule with an example: (37 + 43) - 17 = 37 - 17 + 43 = 63.

If addition is associated with combining two sets into one, then subtraction is associated with separating a given set into two or more sets. Suppose we have a certain number of plastic sausages on a plate. Let's take one or more plastics from this set and put it aside, or better yet, eat it. We removed, that is, we took away several plastics from the initial set of sausage plastics, and the result on the plate changed downward. This is the meaning of subtraction.

Schematically, subtracting two natural numbers looks like this:

minuend − subtrahend = difference.

To indicate subtraction in writing, use the minus sign “−”.

First write down the minuend, then the minus sign, then the subtrahend. For example, writing 9 − 5 means that 5 is subtracted from 9.

Minuend is the number from which it is subtracted. In our example this is the number "9"

Subtrahend is the number that is subtracted from the minuend. In our example this is the number "5"

Difference is the number that is the result of subtraction.

Phrases "find the difference", "calculate the difference", “subtract the number 9 from the natural number 86” is understood as follows: you need to determine the number that is the result of subtracting these natural numbers.

PROPERTIES OF SUBTRACTING NATURAL NUMBERS

Property 1.

a − a = 0, where a is any natural number.

Property 2.

If a and b are unequal natural numbers, then a − b ≠ b − a

45 − 20 ≠ 20 − 45.

Property 3. Subtract from a given natural number this amount two natural numbers is the same as subtracting the first term of a given sum from a given natural number, and then subtracting the second term from the resulting difference.

a − (b + c) = (a − b) − c, where a, b and c are some natural numbers, and the conditions a > b + c or a = b+c are satisfied.

10 - (2+1) = (10 - 2) - 1 = 7

Property 4. Subtracting a given natural number from a given sum of two numbers is the same as subtracting given number from one of the terms, then add the resulting difference and the other term. It should be noted that the number being subtracted must NOT be greater than the term from which this number is being subtracted.

Topic: “Subtraction of natural numbers.”

Lesson type : lesson to improve knowledge, skills and abilities.

Lesson Objectives :

1. strengthening the property of subtraction;

2. solving problems that use the action of subtraction.

3. Test students' knowledge on the following topics:

A. solving problems that use the action of subtraction.

B. subtracting a sum from a number, and subtracting a number from a sum.

4. develop cognitive interests students, independent thinking, ability to navigate the text of a problem, speech;

Lesson objectives:

1. Educational:

Summarize knowledge on the topic "Subtraction of natural numbers";

Strengthen the ability to apply the properties of subtraction in the process of completing tasks;

Monitoring the level of knowledge, skills and abilities of students on the topic “Subtraction of natural numbers”.

2. Developmental:

Work on the development of the conceptual apparatus;

Develop cognitive activity;

Develop a culture of educational activities;

Develop a meaningful attitude towards your activities;

Develop the ability to highlight the main thing;

Promote the development of interest in the subject, organization, responsibility;

Develop independent thinking, seeing general pattern and draw general conclusions.

3. Educational:

Foster a responsible attitude towards learning;

Cultivate the will and perseverance to achieve final results;

Cultivate neatness;

Foster a culture of communication.

During the classes

I. Organizational moment.

Collect homework notebooks. Write down the number in your notebooks, Classwork, topic of the lesson.

II. Updating basic knowledge.

Students are asked to answer the following questions.

a) What action is called subtraction? (an action that uses the sum and one of the terms to find another term)

b) What are numbers called when subtracting? (minuend, subtrahend and difference)

c) What number is called the minuend? (the number from which to subtract)

d) Which number is called subtrahend? (the number that is being subtracted)

d) What number is called the difference? (result of subtraction)

f) How can you find out how much more one number is than another? (you need to find their difference)

g) How many properties of subtraction are there? Formulate them, give an example.

Consider an example: 64 – (5 + 4) =

How can you get the result?

Two students come to the board and write down 2 ways to solve this example.

Method I: 64 – (5 + 4) = 64 – 9 = 55. Method II: (64–4) – 5 = 55

The teacher gives a statementGeorgeAPolia: « If you want to learn how to swim, then boldly enter the water, and if you want to learn how to solve problems, then solve them!!»

Today in the lesson we will continue to study the topic “Subtraction of natural numbers” and analyzeproblems that use the action of subtraction.

I I I. Problem solving. Working with the textbook .

All tasks this lesson can be divided into 2 groups:

1) № 247, 263.

2) 249, 250, 286, 291.

Six students take turns solving problems at the board, the rest of the students solve these problems in notebooks.

Problem No. 247.

DotClies on the segmentAB. Find the length of the segmentA.C., IfAB=38 cm, andC.B.=29 cm.

Problem No. 263.

Section lengthABequal to 37 cm. PointsCAndDlie on the segmentAB, and the pointDlies between the pointsCAndB. Find the length of the segmentCD, If

A)AС=12 cm,BD=17 cm; b)AD=26 cm,C.B.=18 cm.

Problem No. 249.

One automatic machine produced 1235 parts, and the second - 1645 parts. How many more parts did the second machine produce than the first?

Problem No. 250.

96 bags of potatoes were collected from two plots of land. 54 bags were collected from the first site. How many fewer bags of potatoes were collected from the second plot than from the first?

Problem No. 286.

37 m were cut from a skein of fishing line. How many meters more fishing line was cut off than was left in the skein, if initially there were 54 m of fishing line in the skein?

Problem No. 291.

The passenger train consists of 12 carriages with 58 seats each. How much is left free seats, if there are 667 passengers on the train?

IV. Physical education minute for fingers, eyes and back (Slide 11 ).

V. Independent work(15 minutes). (Slide 12)

Option I

properties of subtraction :

a) (6571 +3455) – 2571; c) 3457 – (2457 + 349);

b) (2397 +6831) – 6831; d) 9522 – (3989 + 4522).

2) The TV tower model consists of three blocks. The height of the lower block is 1 m 35 cm, the middle one is 45 cm shorter than the lower one. What is the height of the top block if the height of the model is 4 m?

3) Perform the subtraction:

a) 8003565440 – 6989128416; b) 9000551000 – 8797496.

Option II

1) Follow the steps most in a simple way usingproperties of subtraction :

a) (6574 + 3359) – 2359; c) 5456 – (2456 + 728);

b) (1234 +2587) – 1234; d) 8289 – (2623 + 3289).

2) The armor of a medieval knight weighs 27 kg 500 g, and the sword is 18 kg 400 g lighter. How much does a shield weigh if the knight's full armor weighs 50 kg?

3) Perform the subtraction:

a) 8103096320 – 7387809278; b) 3400300200 – 5987574.

VI . Summing up the lesson. Giving grades for work in class.

1. What topics did we continue to study with you today?

2. What properties of subtraction did we repeat today?

3. Can the subtrahend be greater than the minuend?

V II . Homework: clause 7, no. 293, 294, 296. (Slide 13 )

In this lesson you will learn what direct and inverse operations are in mathematics. The teacher will talk about all the components of subtraction and also show two ways to subtract a sum from a number.

In life, we are constantly faced with direct and opposite actions. You can pour water into a mug, you can pour water out. You can go into the house, then leave the house. There are many such examples.

In mathematics, we can also easily find a pair of such opposite actions. This is addition and subtraction.

Rice. 1. Illustration of addition

Subtraction: there were 5 apples, 2 were taken away, 3 remained. The result was subtraction (Fig. 2).

Rice. 2. Subtraction

It is clear that adding and subtracting is opposite actions Thus, addition and subtraction are mutually opposite operations.

To perform addition or subtraction, we do not take objects to help us and do not put them in one pile. We solve such a problem abstractly, using numbers and opposite operations.

For example, to subtract 2 from 5, we must understand what is left.

And to do this we need to imagine 5 as the sum of two parts.

And we understand that if we subtract 2, then 3 remains.

The same quantity can be represented and written different ways. All these methods are equivalent: . We can always use the one that is convenient for us in this case. Now it is convenient for us to imagine that 5 is the sum of 3 and 2. Therefore, if we remove, subtract one part (2), then the second (3) will remain.

How to subtract 7 from 15?

We immediately imagine that . This means that after subtracting 7, 8 remains.

It becomes clear that subtraction is finding unknown date decomposition.

Let's look at the example again. To subtract 2 from the number 5, you need to represent 5 as two terms and find the unknown term. This will be the result of the subtraction.

If you need to subtract a number from a number:

This means that the number must be represented as two terms and .

One term is unknown to us. We need to find him. This is the result of subtraction.

It is clear that it is impossible to take more apples from the vase than were there. Therefore, when we talk about subtracting natural numbers, we cannot smaller number subtract the larger one. Then there will be other numbers, not just natural ones, and subtracting a larger number from a smaller number will become possible.

Or here’s another reasoning: to subtract means to present it in the form of two terms, but the terms, the parts, cannot be greater than the whole.

But for now the agreement is as follows: from the number we subtract the number , only if not less than . The result will be a new number.

Rice. 3. Names of components when subtracting

The word "difference" is very similar to the word "difference". In fact, what is the difference, how different is the number 15 from the number 7, 15 apples from 7 apples? For 8 apples. That is, the difference between the numbers 15 and 7 is the difference between them.

Thus, on the one hand, the difference is the result of subtraction from more less. On the other hand, this is how much one number differs from another, the difference between them.

Dad is 36 years old, and mom is 2 years younger. How old is mom?

Subtract 2 from 36.

This is the first type of problem that we solve using subtraction: we know one number, we need to find a second one that is smaller by a known amount. That is, we immediately know the minuend and subtrahend, numbers and .

There are 25 people in the class, 14 of them are girls. How many boys are there in the class?

It is clear that there are only 25 girls and boys. There are 14 girls, an unknown number of boys.

We need to find the unknown term. And the search unknown term- this is already a subtraction task. From 25 you need to subtract 14.

There are 11 boys in the class.

This is the second type of problem, when two numbers are added, one of them is known and the other is not. But the result, the amount, is known.

Known and are highlighted in blue. It is necessary to find the unknown term. But searching for an unknown term is subtraction.

My sister is 12 years old and my brother is 9. How old is my sister? older than brother?

My sister is 3 years older than my brother.

This is the third type of task - comparison task.

There were 17 apples in the vase. Petya took 4 apples, Masha took 3. How many apples are left in the vase?

Solution

Petya took 4, Masha - 3, they took a total of apples. To find how much is left, subtract:

If you write it in one line:

Let's count how many apples were left each time Petya and Masha took apples. Petya took 4, left. Masha took 3 more, left.

Or, in one line, .

There are 10 apples left in the vase.

Both methods are equivalent, the answer is the same. That is, subtracting an amount is the same as subtracting each term of this amount separately.

In this lesson you will learn what direct and inverse operations are in mathematics. The teacher will talk about all the components of subtraction and also show two ways to subtract a sum from a number.

In life, we are constantly faced with direct and opposite actions. You can pour water into a mug, you can pour water out. You can go into the house, then leave the house. There are many such examples.

In mathematics, we can also easily find a pair of such opposite actions. This is addition and subtraction.

Rice. 1. Illustration of addition

Subtraction: there were 5 apples, 2 were taken away, 3 remained. The result was subtraction (Fig. 2).

Rice. 2. Subtraction

It is clear that adding and subtracting are opposite actions, thus addition and subtraction are mutually opposite actions.

To perform addition or subtraction, we do not take objects to help us and do not put them in one pile. We solve such a problem abstractly, using numbers and opposite operations.

For example, to subtract 2 from 5, we must understand what is left.

And to do this we need to imagine 5 as the sum of two parts.

And we understand that if we subtract 2, then 3 remains.

The same quantity can be represented and written in different ways. All these methods are equivalent: . We can always use the one that is convenient for us in this case. Now it is convenient for us to imagine that 5 is the sum of 3 and 2. Therefore, if we remove, subtract one part (2), then the second (3) will remain.

How to subtract 7 from 15?

We immediately imagine that . This means that after subtracting 7, 8 remains.

It becomes clear that subtraction is finding an unknown expansion number.

Let's look at the example again. To subtract 2 from the number 5, you need to represent 5 as two terms and find the unknown term. This will be the result of the subtraction.

If you need to subtract a number from a number:

This means that the number must be represented as two terms and .

One term is unknown to us. We need to find him. This is the result of subtraction.

It is clear that it is impossible to take more apples from the vase than were there. Therefore, when we talk about subtracting natural numbers, we cannot subtract a larger number from a smaller number. Then there will be other numbers, not just natural ones, and subtracting a larger number from a smaller number will become possible.

Or here’s another reasoning: to subtract means to present it in the form of two terms, but the terms, the parts, cannot be greater than the whole.

But for now the agreement is as follows: from the number we subtract the number , only if not less than . The result will be a new number.

Rice. 3. Names of components when subtracting

The word "difference" is very similar to the word "difference". In fact, what is the difference, how different is the number 15 from the number 7, 15 apples from 7 apples? For 8 apples. That is, the difference between the numbers 15 and 7 is the difference between them.

Thus, on the one hand, the difference is the result of subtracting a smaller number from a larger number. On the other hand, this is how much one number differs from another, the difference between them.

Dad is 36 years old, and mom is 2 years younger. How old is mom?

Subtract 2 from 36.

This is the first type of problem that we solve using subtraction: we know one number, we need to find a second one that is smaller by a known amount. That is, we immediately know the minuend and subtrahend, numbers and .

There are 25 people in the class, 14 of them are girls. How many boys are there in the class?

It is clear that there are only 25 girls and boys. There are 14 girls, an unknown number of boys.

We need to find the unknown term. And searching for an unknown term is already a subtraction task. From 25 you need to subtract 14.

There are 11 boys in the class.

This is the second type of problem, when two numbers are added, one of them is known and the other is not. But the result, the amount, is known.

Known and are highlighted in blue. It is necessary to find the unknown term. But searching for an unknown term is subtraction.

The sister is 12 years old, and the brother is 9. How many years is the sister older than the brother?

My sister is 3 years older than my brother.

This is the third type of task - comparison task.

There were 17 apples in the vase. Petya took 4 apples, Masha took 3. How many apples are left in the vase?

Solution

Petya took 4, Masha - 3, they took a total of apples. To find how much is left, subtract:

If you write it in one line:

Let's count how many apples were left each time Petya and Masha took apples. Petya took 4, left. Masha took 3 more, left.

Or, in one line, .

There are 10 apples left in the vase.

Both methods are equivalent, the answer is the same. That is, subtracting an amount is the same as subtracting each term of this amount separately.