Topic: The sum of three or more terms. Goal: - Students mastering the method of adding multi-digit numbers, based on previous knowledge of the laws of mathematics

When studying this topic, the main tasks of the teacher are to summarize and systematize students’ knowledge about the operations of addition and subtraction, consolidate oral addition and subtraction skills, and develop conscious and strong skills in written calculations. Addition and subtraction multi-digit numbers are studied simultaneously. This creates Better conditions to master knowledge, skills and abilities, since the questions of the theory of these actions are interrelated, and the calculation techniques are similar.

Preparatory work for studying the topic begins when studying the numbering of multi-digit numbers. For this purpose, first of all, they repeat the oral methods of addition and subtraction and the properties of the actions on which they rely, for example: 8400 + 600, 9800-700, 2000-1700, 740 000 + 160 000, etc. They also repeat written techniques for adding and subtracting three-digit numbers. It is useful to include in oral exercises tasks on addition and subtraction of place numbers with explanations of the form: 6 hundred. + 8 cells = 1 thousand 4 hundred; 1 cell thousand 5 des. thousand - 7 des. thousand = 15 des. thousand - 7 des. thousand = 8 des. thousand such preparatory work creates the opportunity for students to independently explain written techniques for adding and subtracting multi-digit numbers.

Next, the case of addition and subtraction is introduced with increasing difficulty: the number of transitions through a bit unit gradually increases; cases of subtraction are included when the minuend contains zeros; addition of several terms is studied, as well as addition and subtraction of named numbers. When getting acquainted with new cases, children first give detailed explanations of the calculations (name the digit units and the transformations performed).

To 9 units add 7 units, you get 16 units, or 1 ten and 6 units; We write 6 units under units, and add ten to tens. To 9 tens we add 0 tens, we get 9 tens, and another 1 ten - we get 10 tens, or 1 hundred, in place of the tens we write 0 in the total, and add 1 hundred to the hundreds.

0 cells + 0 tbsp. = 0 cell, 0 cell + 1 cell. = 1 cell To 7 thousand we add 6 thousand, we get 13 thousand, or 1 ten thousand and 3 units of a thousand. We write down 3 units of thousands, and add 1 tens of thousands to 4 tens of thousands to get 5 tens of thousands. Amount 53 1906.

After the children have mastered the calculation technique, they move on to abbreviated explanations of the solution: out loud and silently. Let's show with the same example: 9 and 7 - sixteen, 6 we write, 1 we remember; 9 yes 0 - nine, yes 1 - ten, 0 we write, 1 we remember; 0 plus 0 is zero, and 1 is one (we write it down), etc. Brief explanations help develop quick calculation skills.

Some difficulties arise in cases of subtraction, when the minuend is expressed by a digit number. The sequential fragmentation of units of the highest category into units of the lowest is conveniently illustrated on accounts (1000 can be represented as 9 hundred, 9 des., 10 units; 10,000 - as 9 thousand, 9 hundred, 9 des., 10 units and and etc.). It is also useful to include in oral exercises a solution with an explanation of such examples: 1 des. - 2 units, 1 hundred. - 5 des., 1 thousand - 7 hundred. and so on. Particular attention should be paid to cases of subtraction in which sequential fragmentation of units of the highest rank is performed repeatedly, for example: 100 100 - 205 708. It is advisable to compare such cases with the previous ones (100 00 - 4097 and 701 000 - 4097, etc.), and so or require a tentative explanation of the solution to the examples.

We cannot subtract 8 units from zero units. We take 1 hundred (put a dot over the hundreds) and split the hundred into tens. There are 10 tens in 1 hundred, take 1 ten from 10 tens (remember that there are 9 tens left). We split the ten into units, we get 10 units. From 10 units we subtract 0 tens, we get 9 tens. We cannot subtract 7 hundreds from zero hundreds. We take 1 hundred thousand, split it into tens of thousands, get 10 tens of thousands, of which we take 1 ten thousand and split it into units of thousands (remember that there are 9 tens of thousands left), etc. Later, children repeatedly explain the solution to subtraction examples. Let's give a short explanation of the example considered: take 1 hundred, subtract 8 from 10, get 2; subtract zero from 9 to get 9; take 1 hundred thousand, subtract 7 from 10, get 3; subtract 5 from 9 to get 4; subtract 0 from 9 to get 9; subtract 2 from 3 to get 1; difference 194392.

As with everything else, a variety of exercises must be included to develop computational skills. You should offer tasks as often as possible: solve and check the solutions to examples in one of the ways, or less often in two ways. This helps not only to consolidate knowledge of the relationships between results and components of actions, but also contributes to the development of computational skills and fosters the habit of self-control.

When learning the addition and subtraction of multi-digit numbers, it is important to pay attention to oral techniques for performing these actions, otherwise, having mastered written calculation techniques, children begin to use them for both written and oral cases. For this purpose, when solving examples, it is necessary to invite students to choose examples that they can solve orally (with writing in a line), and only solve the most difficult examples using written techniques (with writing in a column). In oral exercises, you should systematically reinforce the techniques of oral addition and subtraction of 2-3-digit numbers, as well as multi-digit ones, using the techniques of rearrangement and grouping when adding several numbers, using, where appropriate, the technique of rounding one of the components of addition and subtraction. Following the study of addition and subtraction of multi-digit numbers, proceed to addition and subtraction of composite named numbers expressed in metric measures, since the methods of these calculations are similar. The ability to perform operations on named numbers is necessary to solve problems. Actions on compound named numbers can be performed in different ways: either immediately add (subtract) units of identical names, starting with the lowest ones, simultaneously performing the corresponding transformations, or first transform these numbers into simple named numbers with the same names, perform operations on them as abstract ones numbers and express the result in larger units of measurement. Both techniques are demonstrated by students. The first method is economical in recording, well illustrates the analogy of actions on abstract and named numbers, but is somewhat difficult for children. Its use should be limited to 2-3 exercises, the purpose of which is to compare calculation techniques with abstract and named numbers:

• 12647 12m 647 kg 12 km 647 m 13086 13 km 086 m
• 5384 5m 384 kg 5 km 384 m 8265 8 km 265 m
• (10 hundreds form 1 thousand, which we add to thousands, ... 10 hundred kilograms form 1 thousand kilograms, or 1 ton, which we add to tons, etc.; ... from 0 hundreds 2 hundreds cannot be subtracted, we take 1 thousand, 1 a thousand is 10 hundreds, from 10 hundreds we subtract 2 hundreds and similarly; ... we occupy 1 km, in 1 km - 1000 m or 10 hundred meters, from 10 hundred meters we subtract 2 hundred meters). As you can see, here children have to operate with numbers of the form 10 hundred kilograms, 10 hundred meters, 10 tens of kopecks, etc., which have double names - units of counting and units of measurement, which, of course, complicates their transformations and actions on them.

The second method of computing over named numbers is much simpler, although more cumbersome to write, and is most widely used when solving examples and problems. To shorten the notation, named number conversions can be done orally and not written down:

124 rub. - 78 rub. 50 kopecks = 45 rub. 50 kopecks 12400

Somewhat later (at the end of the second half of the third grade), addition and subtraction of named numbers expressed in measures of time are studied. These calculations are much more complex because time units are in non-decimal ratios. Children are specifically drawn to this by asking them to compare the solutions to the examples (i.e., find similar and different methods of calculation):

• 13 m 54 cm 13 h 54 min 12 m 34 cm 12 h 34 min
• 6 m 46 cm 6 h 46 min 8 m 56 cm 8 h 56 min

It is advisable to perform addition and subtraction of composite named numbers expressed in time units without replacing them with simple named numbers, for example:

• 12 years 10 months
• 5 years 11 months
• 6 years 11 months

From 10 months Do not subtract 11 months, take 1 year and express it in months - 12 months. 12 months yes 10 months - this is 22 months. From 22 months. subtract 11 months, we get 11 months, from 11 years, subtract 5 years, we get 6 years.

Exercises on addition and subtraction of named numbers, expressed in units of time, with small numbers should be performed orally without writing down the calculations in a column.

In the process of studying the addition and subtraction of multi-digit numbers, they repeat and consolidate knowledge about actions: names of components and results of actions, properties, finding unknown components, the issue of changing the sum and difference when measuring one of the components is considered.

M.A. Bantova identifies the following mistakes students make when adding and subtracting multi-digit numbers:

1. Errors caused by incorrectly recording examples in a column when writing addition and subtraction.

In order to prevent such errors, it is necessary to discuss such incorrect decisions with students, as a result of which they should notice that in this example the numbers are signed incorrectly, so they added tens with ones, hundreds with tens, but the numbers should be signed so that the units are under the ones, tens under tens, etc., and adding ones with ones, tens with tens, etc. In addition, you need to teach students to check the solutions to examples. This error can be easily detected by checking the result by estimating the result. So, in relation to the given example of addition, the student’s reasoning will be as follows: “To 5 hundreds they added a number that is less than 1 hundred, and in total they got 9 hundreds, which means an error was made in the solution.”

2. Errors when performing written addition, caused by forgetting the units of one or another category that needed to be remembered, and when subtracting - the units that were occupied.

Discussing incorrectly solved examples with students also helps prevent such errors. After this, it is important to emphasize that you should always check yourself to see if you forgot to add a number that you should have remembered, or if you forgot that you took units of some rank. Students themselves can help identify such errors by performing tests of addition by subtraction and subtraction by addition.

Note that in some methodological manuals and articles to prevent the mentioned errors in written addition with the transition through ten, it is recommended to start addition with units that were memorized. For example, when solving the above example, the student must then reason: “Add 5 to ten, you get 14, write four, and remember 1: 1 yes 3 - four, yes 2, total 6”, etc. This should not be done, because some students transfer this technique to written multiplication, which will cause an error, for example, when multiplying the numbers 354 and 6, they reason like this: “4 multiplied by 6, you get 24, write four, remember 2; 2 yes 5 - 7, 7 multiplied by 6, you get 42”, etc.

3. Errors in oral techniques for adding and subtracting numbers greater than a hundred (540±300, 1600±700, etc.) are the same as when adding and subtracting numbers within a hundred. To eliminate them, use methodological techniques, which were mentioned above.

Column addition, or as they also say, column addition, is a method widely used for adding multivalued natural numbers. The essence of this method is that the addition of two or more multi-digit numbers is reduced to several simple operations of adding single-digit numbers.

The article describes in detail how to perform the addition of two or more multi-digit natural numbers. The rule for adding numbers into a column and examples of solutions with an analysis of all the most typical situations that arise when adding numbers into a column are given.

Adding two numbers in a column: what do you need to know?

Before we move directly to the operation of column addition, we will consider some important points. To quickly master the material, it is advisable to:

1. Know and have a good understanding of the addition table. So, when carrying out intermediate calculations, you don’t have to waste time and constantly refer to the addition table.
2. Remember the properties of addition of natural numbers. Especially properties related to adding zeros. Let us recall them briefly. If one of the two terms is equal to zero, then the sum is equal to the other term. The sum of two zeros is zero.
3. Know the rules for comparing natural numbers.
4. Know what the digit of a natural number is. Recall that the digit is the position and value of the digit in the notation of the number. The digit determines the meaning of a digit in a number - units, tens, hundreds, thousands, etc.

Let us describe the algorithm for adding numbers in a column using concrete example. Let us add the numbers 724980032 and 30095. First, you should write down these numbers according to the rules for writing addition in a column.

Numbers are written one below the other, the digits of each digit are located, respectively, one below the other. We put a plus sign on the left, and draw a horizontal line under the numbers.

Now we mentally divide the record into columns by digits.

All that remains to be done is to add up the single-digit numbers in each column.

We start with the rightmost column (the units digit). We add up the numbers and write the value of the units under the line. If, when adding, the value of tens turns out to be different from zero, remember this number.

Add up the numbers in the second column. To the result we add the number of tens that we remembered in the previous step.

We repeat the entire process with each column, up to the far left.

This presentation is a simplified diagram of the algorithm for adding natural numbers in a column. Now that we understand the essence of the method, let's look at each step in detail.

First we add up the units, that is, the numbers in the right column. If we get a number less than 10, write it in the same column and move on to the next one. If the result of addition is greater than or equal to 10, then under the line in the first column we write down the value of the units place, and remember the value of the tens place. For example, the number turned out to be 17. Then we write down the number 7 - the value of units, and the value of tens - 1 - we remember. They usually say: “we write seven, one in mind.”

In our example, when adding the numbers in the first column, we get the number 7.

7 < 10 , поэтому записываем это число в разряд единиц результата, а запоминать нам ничего не нужно.

Next, we add the numbers in the next column, that is, in the tens place. We carry out the same actions, only we need to add the number that we kept in mind to the amount. If the amount is less than 10, simply write the number under the second column. If the result is greater than or equal to 10, we write down the value of the units of this number in the second column, and remember the number from the tens place.

In our case, we add the numbers 3 and 9, resulting in 3 + 9 = 12. We didn’t remember anything in the previous step, so we don’t need to add anything to this result.

12 > 10, so in the second column we write down the number 2 from the units place, and keep the number 1 from the tens place in mind. For convenience, you can write this number above the next column in a different color.

In the third column, the sum of the digits is zero (0 + 0 = 0). To this sum we add the number that we previously kept in mind, and we get 0 + 1 = 1. write down:

Moving on to the next column, we also add 0 + 0 = 0 and write the result as 0, since we did not remember anything in the previous step.

The next step gives 8 + 3 = 11. In the column we write the number 1 from the units digit. We keep the number 1 from the tens place in mind and move on to the next column.

This column contains only one number 9. If we didn't have the number 1 in memory, we would simply rewrite the number 9 under the horizontal line. However, given that we remembered the number 1 in the previous step, we need to add 9 + 1 and write down the result.

Therefore, under the horizontal line we write 0, and again keep one in mind.

Moving to the next column, add 4 and 1, write the result under the line.

The next column contains only the number 2. So in the previous step we didn’t remember anything, we just rewrote this number under the line.

We do the same with the last column containing the number 7.

There are no more columns, and there is also nothing in memory, so we can say that the column addition operation is over. The number written below the line is the result of adding the two upper numbers.

To understand all the possible nuances, let's look at a few more examples.

Example 1. Addition of natural numbers in a column

Let's add two natural numbers: 21 and 36.

First, let's write these numbers according to the rule for writing addition in a column:

Starting from the right column, we proceed to adding numbers.

Since 7< 10 , записываем 7 под чертой.

Add up the numbers in the second column.

Since 5< 10 , а в памяти с предыдущего шага ничего нет, записываем результат

There are no more numbers in the memory and in the next column, the addition is completed. 21 + 36 = 57

Example 2. Addition of natural numbers in a column

What is 47 + 38?

7 + 8 = 15, so let's write 5 in the first column under the line, and keep 1 in mind.

Now we add the values ​​from the tens place: 4 + 3 = 7. Don't forget about one and add it to the result:

7 + 1 = 8. We write the resulting number below the line.

This is the result of addition.

Example 3. Adding natural numbers in a column

Now let's take two three-digit numbers and add them.

3 + 9 = 12 ; 12 > 10

Write 2 below the line, keep 1 in mind.

8 + 5 = 13 ; 13 > 10

We add 13 and the memorized unit, we get:

13 + 1 = 14 ; 14 > 10

We write 4 below the line, keep 1 in mind.

Don't forget that in the previous step we remembered 1.

We write 0 below the line, keep 1 in mind.

In the last column we move the unit that we remembered earlier under the line and get the final result of the addition.

783 + 259 = 1042

Example 4. Addition of natural numbers in a column

Let's find the sum of the numbers 56927 and 90.

As always, first we write down the condition:

7 + 0 = 7 ; 7 < 10

2 + 9 = 11 ; 11 > 10

We write down 1 below the line, keep 1 in mind and move on to the next column.

We write 0 below the line, keep 1 in mind and move on to the next column.

The column contains one number 6. We add it with the remembered unit.

6 + 1 = 7 ; 7 < 10

We write 7 under the line and move on to the next column.

The column contains one number 5​​​​​. We move it under the line and finish the addition operation.

Problem-based learning

Subject: "Addition of multi-digit numbers"

Target: developing the skill of adding multi-digit numbers.

Tasks:

- practicing the skills of adding multi-digit numbers;

Strengthen problem solving skills different types;

To consolidate knowledge of the rules about the order of actions and the ability

Write expressions in two steps.

Planned results:

Subject Skills:

Be able to order natural multi-digit numbers;

Be able to name the components of four arithmetic operations;

Be able to add multi-digit numbers and use appropriate terms;

Be able to name categories.

Personal UUD:

Accepting the image " good student»;

Respect for other opinions;

The ability to overcome difficulties and bring work started to completion.

Regulatory UUD:

Determine and formulate the purpose of the activity in the lesson;

Explain the sequence of actions in the lesson; work according to the algorithm, instructions;

Carry out step-by-step control when solving a learning task;

Establish a connection between the purpose of an activity and its result.

Cognitive UUD:

Find your way through a textbook or notebook;

Navigate your knowledge system (define the boundaries of knowledge/ignorance);

Find answers to questions using your life experience.

Communication UUD:

Listen and understand the speech of others;

- be able to express your thoughts with sufficient completeness and accuracy.

During the classes:

Org. moment. (Greetings).

Math, friends,

It's impossible not to love,

Very strict science

A very exact science

Interesting science –

This is MATH!

Updating knowledge. ( Combined stage.)

CHALLENGE PHASE.

I'm in a hurry to get up quickly,

Then I search all day long,

Everyone has a sheet of work on their desk. Complete it.

(There is a card with examples on the table:( 48+37; 56+85; 528+165; 253+614; 208+549)

(One student goes to the blackboard and works at the blackboard. Examples are written on the blackboard, he needs to solve them.)

Let's check the student at the board and ourselves. (85, 141, 688, 867, 757)

How did you add the numbers? (in writing, by rank)

Explain your actions using the algorithm for adding two-digit and three-digit numbers (they wrote units under ones, tens under tens, hundreds under hundreds; first added ones and wrote under ones, then added tens and wrote under tens; then added hundreds and wrote under hundreds).

What is this method of addition called? (bitwise addition)

Creating a problematic situation.

And now we are working in pairs: you need to solve these examples in your notebooks (four examples are written on the board): 1253+2614; 36208+54926; 4758+324; 2267+9841.

What answers did you get? (Children name their answers and find out that many have different answers, since the examples caused difficulty.)

How can you check the correctness of your answers? (Children express various assumptions, try to identify the correct one among them and come to the conclusion that they cannot do this because they do not know which of the proposed action algorithms is correct.)

Formulating the problem (topic).

What question do you have? (How to add four-digit and five-digit numbers.)

How can we call three-digit, four-digit, five-digit numbers in one word? (Multiple meanings.)

What will be the topic of the lesson? Who can formulate it? ("Addition of multi-digit numbers" )

Children's discovery of new knowledge and its formulation. (Work from the textbook in a notebook.)

CONSIDERATION PHASE.

Open the textbook on p. 27, No. 90. Read the assignment. How do they suggest we complete this task in the textbook? (It is suggested to use the bitwise addition method)

What needs to be done for this? (Remember the algorithm for bitwise addition of three-digit numbers: write the digit under the digit; add by digits, starting with ones: etc.)

Formulate an algorithm for adding multi-digit numbers.

How is it similar and how is it different from the algorithm for adding three-digit numbers?

(Children's opinions are listened to)

Primary application of new knowledge.

Complete task No. 91 in the textbook. (One student comes to the board and comments on his actions when solving examples)

To find out what we will do next, we need to guess the charade.

(On the board there is a charade: prepositionBEHIND and a picture"dachas" .)

- The first is a pretext,

The second is a summer house.

And sometimes the whole

It is difficult to solve.

( TASK ) (This inscription appears on the board.)

And now we have tasks:

Complex, simple.

We take luck with us,

To work hard!

1. - Open the textbook on page 28, section 98. Read the problem...

What is known from the conditions of the problem? (After 128,509 rubles were given out from the cash register, 14,902 rubles remained in it)

What do you need to find? (How much money was in the cash register.)

What short summary can we make? (It was. Issued. Remains.)

... will go to the board and fill out a short entry.

What is unknown? (Was.)

How to find? (To find how manywas , it is necessary to the fact thatleft add whatissued. )

What type of task?

Let's write it down in our notebooks. (There will be comments...)

Write two inverse problems orally.

2. – P.28, section 96. Read the problem.

What is known from the conditions of the problem?

What do you need to know?

Write down the solution to the problem yourself in your notebook.

EXAMINATION.

What answer did you get?

Physical exercise.

Once - they sat down, twice - they stood up,

Three - bent down and took it out

Right hand sock,

Left - ceiling.

And then - vice versa.

And they sat down quietly.

3. – P.29, section 102. Read the problem.

What is known from the conditions of the problem? (The rectangular field is 850 m long and 625 m wide)

What do you need to know? (field perimeter)

Everyone has a helper card on their table.

You must fill out the cards yourself. (I will write on the board.)

EXAMINATION at the blackboard.

INDIVIDUAL WORK.

Who can solve the problem right away?

Start deciding who finds it difficult to work with a teacher.

Working with expressions. (Group work.)

REFLECTION PHASE.

- who can create expressions for our problem using any of the proposed methods?

1. (850+625) 2 = 2550(km)

2. 850 2 + 625 2 = 2550(km)

3. 850 + 625 + 850 + 625 = 2550 (km)

(Those children who wish come to the board.)

(CHECKING in progress.)

Choose any of the methods convenient for you and write it down in your notebook.

Guys, I was in a hurry to get to class today, bringing you cards with expressions, but I tripped and dropped them. The cards fell apart. Now I need your help. We will work in groups.

I distribute cards with numbers and signs to groups of 5–6 people.

- (, +, :,), 27, 15, 7, = (27+15):7 = 6

19, (, 9,), +, =, 4, : (19+9):4 = 7

37, -, :, 24, 3, = 37-24:3 = 29

- +, :, 22, =, 36, 4 22+36:4 = 31

TASK stage: Each group must create an expression.

The person in charge from each group comes to the board with his expression,examination.

What was the difficulty?

Lesson summary.

1. What was the most important thing for you in the lesson?

2. What goals were set at the beginning of the lesson?

3. Have they been achieved?

4. What did you learn in this lesson?

5. What knowledge did you gain in the lesson? ?

6. What would you like to devote to the next lesson?

Homework. (Optionally.)

Rice. 1. Classes and ranks of numbers

Let's name the number of ones in each digit using some numbers as an example.

72439 - this number includes nine units, three tens, four hundreds, two units of thousands, seven tens of thousands.

Number 25346 contains six ones, four tens, three hundreds, five thousands and two tens of thousands.

State the number of units of each digit using the example of a number 3126 . Let's check: six ones, two tens, one hundred, three thousand units.

Let's fill in the blanks together (see Figure 2).

Rice. 2. Illustration for the problem

1 ten = 10 units

1 hundred = 10 tens

1 thousand = 10 hundreds

1 ten thousand = 10 thousand units

1 hundred thousand = 10 tens of thousands

1 million = 10 hundred thousand

The purpose of our lesson is to learn how to perform written addition and subtraction of multi-digit numbers. You already know how to add and subtract three-digit numbers in a column. Adding and subtracting multi-digit numbers is done in exactly the same way.

Let's compare two columns of calculations (see Fig. 3).

Rice. 3. Addition of multi-digit numbers in a column

Did you notice that on the right appeared new rank, thousand digit. Let's explain how the calculations are made: 6 units + 2 units = 8 units.

Then add the tens: 2 tens + 9 tens = 11 tens. 11 tens is 1 ten and 1 hundred. Let's add a hundred to hundreds. 1 hundred + 2 hundreds = 3 hundreds, but we also added one, so under hundreds we write 4. We calculate the units of thousands: 3 thousand + 4 thousand = 7 thousand. So the answer is 7418.

Let's consider subtraction (see Fig. 4).

Rice. 4. Subtracting multi-digit numbers in a column

Compare the two columns of calculations. The units of thousands and tens of thousands appeared on the right. Let us explain how the subtraction is performed. It is impossible to subtract 7 from 6 ones, so let’s take one ten from the previous digit: 16 - 7 = 9, write 9 under the ones. We calculate tens: 4 - 0 = 4, but we took one ten, so we write 3. Subtract hundreds. It is impossible to subtract 4 hundreds from 3 hundreds, so we take one unit of thousands, this is 10 hundreds, 13 hundreds - 4 hundreds = 9 hundreds. Subtract units of thousands. We took one unit of thousands, so we subtract 4 - 3 = 1. We rewrite two, since the tens of thousands digit is missing. Answer: 21939.

Task 1. Perform the calculation, writing the solution in a column: 528047+106875. And check addition using subtraction.

Let us explain how we performed the addition of multi-digit numbers: 7 units + 5 units = 12. 12 is 2 units and 1 ten. We write 2 under the units, and add the ten to the tens. We calculate tens: 4 tens + 7 tens = 11 tens, and 1 ten was added, it turned out 12 tens. Under the tens we write 2, and we add one hundred to the hundreds. We calculate hundreds: 0 + 8 = 8, but one hundred was added, so we wrote 9 under hundreds. Let’s find the number of thousand units: 8 + 6 = 14. 14 thousand units are 4 thousand units and 1 ten thousand, write to tens. We count tens of thousands: 2 tens of thousands + 0 and 1 tens of thousands added, we get 3 tens of thousands. Add up hundreds of thousands: 5 + 1 = 6.

We read the answer: 634922 (six hundred thirty-four thousand nine hundred twenty-two) (see Fig. 5).

Rice. 5. Illustration for task 1

To perform the check, subtract one of the terms from the sum value. Let's explain how the subtraction is done: you can't subtract 7 from 2, so we'll take 1 ten. 12 - 7 = 5. We calculate tens: we took 1 ten, so there is 1 left. We cannot subtract 4 from 1, so we take 1 hundred, 1 hundred is 10 tens. 11 - 4 = 7. Calculate hundreds: since we took 1 hundred, there are 8 left. 8 - 0 = 8 hundreds. We calculate the units of thousands: you cannot subtract eight from four, so we take 1 ten thousand. 14 - 8 = 6. We write it under units of thousands. We calculate tens of thousands. We borrowed one ten, there are 2 left. 2 - 2 = 0. We calculate hundreds of thousands: 6 - 5 = 1. We read the answer: 106875 (one hundred six thousand eight hundred seventy-five) (see Fig. 6).

Rice. 7. Illustration for task 2

Let's explain how subtraction is performed: you cannot subtract 6 from 0, so we take one ten, 10 - 6 = 4. There are 5 tens left. It is impossible to subtract 7 from 5, so we take one hundred, one hundred is 10 tens. 15 - 7 = 8 tens. 4 hundred left. 4 hundreds - 4 hundreds = 0. We calculate units of thousands: 2 - 1 = 1. We calculate tens of thousands: 2 - 2 = 0. We rewrite 3, since the hundreds of thousands place is missing in the subtrahend. We read the answer: 301084 (three hundred one thousand eighty four).

To check subtraction by addition, you need to add the subtrahend to the difference value (see Fig. 8).

Rice. 8. Illustration for task 2

Let's explain how addition is done: 4 + 6 = 10, under the units we write 0, and the ten is added to the tens. We calculate tens: 8 + 7 = 15 and add 1 ten, we get 16 tens. We write 6 in place of tens, and add 1 hundred to hundreds. 0 + 4 = 4 yes 1 hundred = 5 hundreds. We calculate units of thousands: 1 + 1 = 2. We add tens of thousands: 0 + 2 = 2. We rewrite hundreds of thousands. We read the result: 322560 (three hundred twenty-two thousand five hundred sixty).

We compare with the minuend and see that the numbers coincide, which means the subtraction was performed correctly. Let's write down the result: 301084 (three hundred one thousand eighty four).

Let's solve a mathematical puzzle (see Fig. 9).

Rice. 9. Rebus

Let's determine which digits are missing in the numbers. It is impossible to subtract a number from 4 and get 9, so we will take one ten. From 14 you need to subtract 5 to get 9. Subtract 8 and get 0. This means that in the place of tens there is the number 8, but one ten was taken, so we write 9. We determine the number of hundreds: from three you need to subtract two to get one. We write 2 hundreds in place (see Fig. 10).

Rice. 10. Solution math puzzle

Today we learned to perform written addition and subtraction of multi-digit numbers.

1. Bashmakov M.I. Nefedova M.G. Mathematics. 4th grade. M.: Astrel, 2009.
2. M. I. Moro, M. A. Bantova, G. V. Beltyukova and others. Mathematics. 4th grade. Part 1 of 2, 2011.
3. Demidova T. E. Kozlova S. A. Tonkikh A. P. Mathematics. 4th grade 2nd ed., rev. - M.: Balass, 2013.

Dhomework assignment

1) Task: write it down in a column and solve.

2) The maximum depth of the ocean is 11,022 m. Calculate the difference between the depth of the ocean and the highest point on Earth, if the height of the highest point high mountain in the world (Everest) is 8,848 m above sea level.

3) The weed plant cornflower produces 6,680 seeds per year, and a plant such as rye brome produces 5,260 fewer, field sow thistle produces 12,920 more than cornflower. How many seeds do these plants together produce per year?

Sorokin A. S.

C65 Counting technique (Methods of rational calculations*
numbers). M., “Knowledge”, 1976.

120 s. (National University, Faculty of Natural Sciences)

The book presents in popular science form one of
interesting branches of computational mathematics.

The book is intended for students of technical universities, engineering
ners and economists. It may be useful for secondary school teachers
her school when organizing lectures on mental arithmetic, as well as
students of people's universities in natural sciences
niy and everyone who has to deal with computing
operations.

20200-126,„
073(02Р76 B3 ~ 16 -3-76 b1

(C) Publishing house "Knowledge", 1976

INTRODUCTION

The current level of development of socialist
National economy characterized by widespread introduction
the use of electronic computer technology and economics
co-mathematical methods in all branches of Soviet
economy. More and more mathematical calculations
are included as a necessary component in the work
Worker, engineer, economist, specialists,
Having never previously encountered the need to
complete computational work. But despite the fact that
mathematical culture of modern production
Nika became disproportionately higher compared to the level
worker of the first five-year plans, for arithmetic calculations
you, when you have to carry them out, the waste is unreasonable
given a lot of time. “Inability to count quickly and proficiently”
one hundred is such a common and modern flaw-
com that we don’t notice him, despite all
the harm they cause,” wrote I. F. Sludsky in 1925
year. Unfortunately, this quote is not outdated today,
however, taking into account the fact that now under the ability to quickly and
just to consider is understood somewhat differently than it was
in mind at the time. Lack of quick skills
close calculations often forces one to refuse

from evaluation calculations, from considering a number of options,
so necessary for making an informed decision.

Admiration for mathematics as the most accurate
knowledge often turns into the belief of infallibility and opti-
|the smallness of those counting methods that we learn in
high school. Any interference with routine, but
|counting methods that we have mastered are most often called
there is a protest (sometimes unconscious) that was previously

manifests itself in relation to new methods,
Mastery of rational, fast and elegant technology

Which account requires certain efforts from a person, and |
the main thing is a creative attitude towards computing
process, because most effective methods, giving the most
greater gain in computational work, based
on the conscious use of the main features
numbers used in calculations. Knowledge of these is important
properties of specific numbers sometimes gives exceptional
new results. For example, even in the presence of arithm
meters perform multiplication of numbers 0.9999997-0.9999998-
this is not an easy task (similar and even more complex calculations
changes have to be made when calculating reliability
elements and systems). But the calculation is done verbally
easier and faster than any mathematical machine
Once you become familiar with the addition method, you will be able to
to be convinced of the correctness of this statement.

Currently there is no literature in Russian
literature, at least relatively fully illuminating the
Themes and methods that simplify calculations. One of the most
The most famous book in this area is the mathematician G. N.]
Berman's "Counting Techniques" contains very little
number of known techniques and cannot satisfy
meet the demands of today. But she also became a bib-
lyographic rarity. Interesting job E. Kot-
Lera and R. McShane “Quick counting system for Fuck
Tenberg", published in translation from in English V
1967, includes mainly specific developments
ki of the German professor.

This work is intended to replenish, as far as possible,
thread this gap, help everyone who has to
dealing with calculations, place them at their disposal
the most rational methods of calculations, essentially
but shortening the computational process, simplifying
it and helping to increase the reliability of poly
expected results.

The work presents materials on rationalization
tions for performing basic arithmetic operations
checking the correctness of the results obtained. Most-|
more promising and general methods the author tried to illuminate
more fully, show various aspects of their application,
so that the reader can actively master them, and sometimes develop
keep going. The desire to show all the possibilities
Then they forced the author to sometimes violate the order of the premises.
understanding the material by chapter. In particular, to
show the logic of development and use of the method, ma-

material on squaring numbers of a certain vi-
Yes, it ended up in the chapter on multiplication.

When viewing the material, the question may arise:
Is it really possible to remember everything written here? Really-
Do you need to remember all this? Principles of application
New methods certainly need to be mastered. Much has been
will follow directly from these basic principles
ny (such as, for example, the method of additions). Some
methods, despite the relatively narrow range of applications
words are so simple that they are remembered involuntarily
But. As a child, I was told how to build a
the square of numbers ending in 5 is the number of tens
multiply by the following number and add 25:

65-65=? 6-(6+1) =42 65-65 = 4225.
This turned out to be sufficient for such a simple me-
Tod remained forever in memory, and entered into active art.
senal of my computational methods. But of course
a book can teach something only to those who are interested
a person reading it with a pencil and paper in hand
kah.

The vast majority of proposed methods
extremely simple, but detailed formal description
takes up a lot of space. Therefore, when faced with long
multi-step calculation methods, do not be alarmed,
take it. In the end, most likely everything will turn out to be very pro-
one hundred. Most of the techniques are designed for oral speech.
number with record final result, some
These methods make written calculations easier.

Sometimes performing arithmetic operations with
the same numbers are described using
different methods. The reader is given the opportunity
choose the one that is specifically for him
most simple.

At the beginning of the second chapter, the author gives recommendations on
recording and arrangement of numbers in calculated examples,
but in the future I myself will not benefit from these recommendations -
Yes. This is no coincidence. Unusual location of the chi-
sat down, unusual recording may interfere with perception
new material being presented and this must be taken into account
hide.

The author will be grateful to all readers for their comments.
any comments about the work that can be sent or to
Editorial address or directly to the author: Moscow,
129243, Rocket Boulevard, 15, apt. 46,

Chapter 1

METHODS THAT SIMPLIFY
ADDING AND SUBTRACTING

WITH addition and subtraction are simple
great arithmetic operations. Presumably
It is assumed that the reader performs these actions without difficulty.
opinions. Therefore, the material in this chapter should be considered
as an attempt to systematize our knowledge of
technique for performing addition and subtraction, emphasizing
pay attention to those details of the computational process
sa, which allow you to perform it somewhat faster
and with less effort, because it is difficult to name common me-
methods that provide a significant gain in the volume of computation
lazy when doing addition and subtraction.

ORAL ADDITION OF MULTIPLE-DIGITAL NUMBERS

If you need to find the sum of a series
multi-digit numbers orally, without making any notes
this, then we can recommend the following order:
numbers, illustrated with the example of addition
numbers:

5754
2315
+ 6438

We sum up the most significant digit of the terms

Adding up all the leading digits, we assign
to the amount O

and continue to add the numbers of the next digit
220+7+3+4+3=237,

again we assign 0 and add third-digit numbers -

yes 237-2370; 2370+5+1+3+1=2380,
attribute last time 0 and complete the calculation
amounts

2380-23 800; 23 800+4+5+8+3 = 23 820.

At the end of the calculations you have to remember the relative
But big number, but we add to it every
times only a single digit number. This makes it much easier
no mental calculation.
Find the amounts yourself:

1) 2374 2) 2437 3) 1234 4) 659
3943 7538 124 3541

+ + + 35+

6513 1467 2343 2413

7231 9325 594 79

Answers: 1) 20061, 2) 20,767, 3) 4330, 4) 6692.