Addition of multidigit numbers. Multi-digit numbers. Cards for the correction of knowledge

For fitting and assembly work 21.11.2021

For fitting and assembly work

Single-digit numbers are added using an addition table. The addition table, or rather the results of the addition of single-digit numbers, must be remembered by heart.

Example... Add the single-digit numbers 4 and 9:

Adding multidigit numbers

Multidigit numbers are added by place using the displacement and combination laws of addition.

Example... Add the two-digit numbers 26 and 48:

26 + 48 = (20 + 6) + (40 + 8) = 20 + 6 + 40 + 8 = (20 + 40) + (6 + 8) = 60 + 14 = 60 + (10 + 4) = 60 + 10 + 4 = (60 + 10) + 4 = 70 + 4 = 74

First, we decomposed the terms into digits, then we grouped tens into one group, and ones into another, and performed addition by digits, that is, we added tens with tens and ones with ones, then one ten, resulting from the addition of ones, was added to tens, which we had 6 from adding tens, and at the end added tens with ones.

The addition form that we used is too long and therefore inconvenient, therefore, when adding multi-digit numbers, another, more convenient form of notation is usually used, which is called column addition.

Column fold

It is more convenient to add multi-valued natural numbers in a column.

Column fold is a notation and addition method used when adding multi-digit numbers. Column addition is also called stacking.

Consider column addition using the example of adding the numbers 7056 and 483.

Column addition is written as follows: one term is written under the other so that the numbers of the same digits stand under each other (units under units, tens under tens, etc.). For convenience, the lower number is usually written under the higher number. On the left, a plus sign is put between the terms, and a horizontal line is drawn under the lower term:

The resulting record can be mentally divided into columns as shown in the figure:

All further actions are reduced to the addition of single-digit numbers that are in one column. The calculation is performed bitwise from right to left, starting from the ones place.

If, as a result of addition, a number less than 10 is obtained, then it is written under the line in the same place.

We start the calculation with the ones: add the numbers 6 and 3. As a result, we have the number 9. Since 9< 10, то записываем это число под чертой, в том же разряде:

If, as a result of the addition, a number equal to 10 or greater than 10 is obtained, then under the line in the same place the value of the digit of the units of the received number is written, and the value of the digit of the tens of the received number is stored (it is used in the next step).

We proceed to the addition of numbers in the next digit, that is, to the addition of the values of the tens digit. We add the numbers 5 and 8, we get the number 13. Since 13> 10, then under the line, in the same place, we write down the number 3 (this is the value of the digit of units of 13), and we remember the number 1 (this is the value of the tens place of 13), at the same time they say we write three, and one in the mind... In order not to forget about the memorized number, it is usually written above the next (left) digit:

The memorized number is added to the sum of the numbers of the next digit.

Go to the next digit and add the numbers 0 and 4. As a result, we have 4. To the resulting number we add the memorized number 1, we get 5. Since 5< 10, то под чертой, в том же разряде, записываем число 5:

After that, there is a transition one bit to the left and the actions are repeated. This process continues until the numbers run out.

If the column contains only one number, and we do not have a memorized number (from the previous addition), in this case we simply write this number under the line, in the same place.

Since the next column contains only one number - 7, and we do not have a memorized number in memory, we simply write 7 under the line, in the same place:

There are no further numbers and there are no numbers in memory either. This completes the addition process. The natural number under the line is the result of the addition of these numbers. Now you can write the sum of these numbers in the usual form:

7056 + 483 = 7539

Let's look at a couple more examples of column addition to deal with the remaining nuances.

Example... Add numbers 29 and 6 in a column.

We add 9 and 6, as a result we get the number 15. Since 15> 10, we write down the number 5, and remember the number 1:

If the column contains only one number, and we have a memorized number (from the previous addition), then the memorized number is simply added to this one number.

The next column contains only one number - 2. Since we have the number 1 in our memory, we need to add it to 2. As a result, we get the number 3:

Example... Add the numbers 43 and 94 in a column.

Add 3 and 4. As a result, we have the number 7. Since 7< 10, то записываем это число под чертой, в том же разряде:

If in the last digit, as a result of addition, a number equal to 10 or greater than 10 is obtained, then under the line in the same digit the value of the digit of units of the received number is written, and the value of the digit of tens of the received number is written under the line in the next digit.

In the next digit we add the numbers 4 and 9, we get the number 13. Since 13> 10, then under the line, in the same digit, we write the number 3, and the number 1 we write under the line in the next digit:

The convenience of column addition lies in the fact that the addition of multi-digit natural numbers is actually reduced to the addition of single-digit numbers and the recording of the addition process takes up less space.

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Topic: Addition of multidigit numbers.
Lesson type: opening new material.
Purpose: To teach a written algorithm for adding multi-digit numbers.
Objectives: 1) Repeat the numbering of multi-digit numbers; 2) Teach children to add multi-digit numbers based on the addition of three-digit numbers; 3) Development of mathematical speech, logical thinking; 4) Raise interest in the lesson, the ability to work in an organized lesson;

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Lesson summary

Mathematics

in the 4th grade.

N.F. Vinogradov "Primary school of the XXI century".

Primary school teacher

Lyceum No. 15 in Saratov

Lukyanova Elena Anatolyevna

Saratov 2011

Topic: Addition of multidigit numbers.

Lesson type: discovery of new material.

Target: Teach a written algorithm for adding multi-digit numbers.

Tasks: 1) Repeat the numbering of multi-digit numbers; 2) Teach children to add multi-digit numbers based on the addition of three-digit numbers; 3) Development of mathematical speech, logical thinking; 4) Raise interest in the lesson, the ability to work in an organized lesson;

Equipment: presentation, textbook by V.N. Rudnitskaya "Mathematics Grade 4", a workbook on mathematics, notes on the board;

During the classes:

Organizing time

Hello guys! Sit down.

We open notebooks, write down the number, great work.

2. Verbal counting. Repeating numbering of multi-digit numbers.

Look at the board, numbers are given here, but they are missing.

Your task is to restore the number series:

9 000 ___ ___ ___ ___ ___ ___ 9 007;

(9 000, 9 001, 9 002, 9 003, 9 004, 9 005, 9 006, 9 007)

99 998 ___ ___ ___ ___ ___ ___ 100 005.

(99 998, 99 999, 100 000, 100 001, 100 002, 100 003, 100 004, 100 005)

Next task, be careful:

1) To what number must 1 be added to get the number 100,000? (99 999).

Write down this number. Give it to me.

Let's check if this is so. (Response to presentations)

2) To what number do you need to add 1 to make a million? (999 999)

- What is this number?

Let's check. (Response to presentations)

3) From which four-digit number do you need to subtract 1 to get a three-digit number? (1 000)

- Call this number.

Let's see if this is so. (Response to presentations)

Let's count with you orally. Let's repeat the oral techniques of adding three-digit numbers.

320+70=390, 120+120=240,

260+40=300, 605+5=610,

300+90=390, 400+250=650,

500+200=700, 715+20=735.

Preparation for learning new material.

Guys, now we have repeated the oral addition techniques, and now let's remember the written addition techniques.

We open the workbook on page 13 №43

Add three-digit numbers.

436 308 732 296

251 167 196 487

687 475 928 783

Here, numerical expressions are already written in a column.

Where do we start adding up? (We start adding from the ones place)

We decide with commenting.

3. Explanation of the new material.

We are now adding three-digit numbers. Guys, look at the board, here is a numerical expression with multi-digit numbers:

296 375 + 38 007

How are we going to sign these numbers? (Class under class, category under category).

Okay, let's add these two numbers together with me.

We sign the numbers under each other.

We look, in the second number in the class of thousands, what category is missing? (Hundreds of thousands are missing)

This means that we begin to sign the second number under the tens of thousands, since we do not have the hundreds of thousands.

296 375

38 007

334 382

Do not forget that there should always be 3 digits in the unit class, check.

Let's calculate, where do we start adding? (From the class of units, from the category of units)

5 + 7 = 12, 2 we write under units, 1 we remember; 7 + 0 and 1 more we get 8, we write down under tens; 3 + 0 = 3, we write under hundreds; We pass to the class of thousands, add the units of thousands 6 + 8 = 14, write 4 under the units, remember 1; 9 + 3 and 1 more we get 13, 3 we write under tens, 1 goes to the next digit; 2 + 1 + 3, write 3 under hundreds.

Read the answer. (334 382)

So how do we add two multi-digit numbers? (Just like three-digit numbers, in a column, bitwise).

What class do we start adding up with? (From class units)

From what level? (From the category of units).

Guys, what do you think we are going to do in class today? What is the topic of our lesson? (We will add multi-digit numbers)

Quite right, the topic of our lesson is "Addition of Multiple Numbers".

4. Primary consolidation of new material.

Open the tutorial on page 27 # 91. We carry out this task, read and find the sum of the numbers.

Find the sum of the numbers.

68 305 and 9 286 673 and 12 869

18,000 and 6,375 1,480 and 260,387

306 250 and 18 998 458 207 and 207 954

How are we going to decide? (In a column, writing down category under category, class under class)

With what place will we start adding? (From the least significant bit, the ones category, from the ones class).

68 305 18 000 306 250 673 1 480 458 207

9 286 6 375 18 998 12 869 260 387 207 954

77 591 24 375 325 248 13 552 261 867 666 161

Guys, let's draw a conclusion, so how do we add two multi-digit numbers? (Just like three-digit numbers, in a column, bitwise).

How do we write numbers? (Class under class, categories under category).

5. Physical education

Exercise for the eyes: Guys, close your eyes, I am counting to ten, now open; just look with your eyes to the right, left, down, now draw a figure eight with your eyes. - We continue to work.

6. Independent work.

Now you are working on your own. We open the workbook on page 13 №45. - You need to find the values of numerical expressions.

The 1st variant performs the first two columns, the 2nd variant performs the second two columns.

Be careful, we'll check later.

Perform the addition.

48 356 209 366 2 874 687

12 974 1 793 19 057 29 630

61 330 211 159 21 931 30 317

Checking the answers. You name only the result.

7. Problem.

Guys, but after all, polydigit numbers are found here not only in numerical expressions, they can also be found in problems.

Let's solve the problem with you now, listen to it carefully:

The Vesuvius volcano in the Apennines is located at an altitude of 1,277 m above sea level. The Etna volcano in Sicily is 2,063 m higher than Vesuvius, and the Tolima volcano is 1,875 m higher than the Etna volcano. What is the height of the Tolima volcano above sea level?

What is the problem about? (About volcanoes)

What volcanoes? (About Vesuvius, Etna and Tolima)

What do you know about them, where are they?

Look at the blackboard.

- Vesuvius is the only activevolcano on South Of Italy, about 15 km fromNaples... Included in Apennine mountain system.

- Etna - actingvolcanolocated on the east coastSicily... In Arabic, Etna is called the "Mountain of Fire". Etna is the highest active volcano inEurope... It should be noted that the height of Etna varies from eruption to eruption.

Tolima is located in South America, it is active, reaches a tremendous height, since the mountains on which it is located themselves rise high above sea level.

(drawings depicting volcanoes)

Let's solve the problem. Let's write down the condition:

What words do we write out for a short condition?

V. - 1 277 m.,

This - ? m, 2,063 m higher

T.- ? m, 1875 m. higher

What is the main question in the problem? (What is the height of Tolima volcano?)

Let's circle it in an oval.

Can we answer the main question of the problem? (No, because we do not know the answer to the first question)

What arithmetic operation will we answer the first question? (Folded)

We record the first action.

We write down in a column, class under class, category under category.

1) 1 277

2 063

3 340

From what level? (From the category of units)

What do we write in the name? (Height of Mount Etna)

Now can we answer the main question of the problem? (Yes)

What is the arithmetic operation? (addition)

What are we going to add with what? (3340 and 1875)

We record the second action.

2) 3 340

1 875

5 215

Where do we start adding? (from the class of units, from the category of units)

How do we write down the answer? Let's read the problem question again. (The height of the Tolima volcano is 5,215 meters). - We write down the answer.

8. Lesson summary.

Guys, what new things did you learn in the lesson today? (Adding multidigit numbers).

How do I add two multidigit numbers? (Just like three-digit numbers, bitwise)

How do we write numbers? (class under class, rank under rank).

With what category do we start adding? (from the category of units).

What class? (from unit class).

9. Homework.

Uch. p.28 No. 97, r.t. p.13 # 45

Problem learning

Theme: "Adding multidigit numbers"

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

Tasks:

- practicing the skills of adding multi-digit numbers;

Strengthen the ability to solve problems of different types;

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

Write expressions in two steps.

Planned results:

Item skills:

Be able to order polydigit natural numbers;

Be able to name the components of four arithmetic operations;

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

Be able to name the categories.

Personal UUD:

Accepting the image of a "good student";

Respect for other opinions;

The ability to overcome difficulties, to bring the work started to its completion.

Regulatory UUD:

Determine and formulate the goal of the lesson;

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

Carry out step-by-step control when solving an educational problem;

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

Cognitive UUD:

To navigate in a textbook, notebook;

To navigate in your knowledge system (to determine the boundaries of knowledge / ignorance);

Find answers to questions using your life experience.

Communicative UUD:

Hear and understand the speech of others;

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

During the classes:

Org. moment. (Greetings).

Math friends

There is no way not to love

Very strict science,

A very exact science,

Interesting science –

This is MATH!

Knowledge update. ( Combined stage.)

CALL PHASE.

I hasten to get up quickly

Then I search all day

Each of them has a piece of paper with assignments on their desk. Execute it.

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

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

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

How were the numbers added? (in writing, by category)

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

What is the name of this method of addition? (bitwise addition)

Creation of 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 have caused difficulty.)

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

Formulation of 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? (Multi-valued.)

What will be the topic of the lesson? Who can formulate it? ("Adding multidigit numbers " )

Discovery of new knowledge by children and its formulation. (Work from a textbook in a notebook.)

THINKING PHASE.

Open the tutorial on p. 27, no. 90. Read the assignment. How does the textbook suggest us to complete this task? (Suggest to use the bitwise addition method)

And what should be done for this? (Recall the algorithm for the bitwise addition of three-digit numbers: we write the digit under the digit; it is necessary to add by digits, starting with ones: etc.)

Formulate an algorithm for adding multidigit numbers.

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

(The opinions of children are heard)

Primary application of new knowledge.

Complete task number 91 in the textbook. (One student comes to the board and comments on their actions when solving examples)

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

(On the charade board: prepositionPER and the picture"Dachas" .)

- The first is a preposition,

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 our luck with us

To work hard!

1. - Open the textbook on page 28, z.98. Will read the problem ...

What is known from the problem statement? (After 128509 rubles were issued from the cash register, 14902 rubles remained in it)

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

What short note can we make? (It was. Issued. Remaining.)

Will go to the blackboard ..., fill in a short note.

What is unknown? (It was.)

How to find? (To find how muchIt was , it must be said thatleft add whatissued. )

What kind of task?

Let's write it down in a notebook. (Will comment ...)

Make up two inverse problems orally.

2. - P.28, z.96. Read the problem.

What is known from the problem statement?

What do you need to find out?

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

EXAMINATION.

What is your answer

Physical minute.

One - sat down, two - got up,

Three - bent down and took out

Right hand sock

The left one is the ceiling.

And then - the other way around.

And they sat down quietly.

3. - P.29, z.102. Read the problem.

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

What do you need to find out? (Field perimeter)

Each of them has a card on the table - an assistant.

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

EXAMINATION at the blackboard.

INDIVIDUAL WORK.

Who can solve the problem right away?

Proceed with a decision, and for whom it is difficult to work with the teacher.

Working with expressions. (Group work.)

REFLEXION PHASE.

- who can compose expressions for our problem in any of the suggested ways?

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

2.850 2 + 625 2 = 2550 (km)

3.850 + 625 + 850 + 625 = 2550 (km)

(Those children who want to go to the blackboard.)

(CHECK in progress.)

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

Guys, today I was in a hurry to the lesson, I brought you cards with expressions, but I stumbled and dropped them. The cards were scattered. 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 compose an expression.

The responsible person from each group comes to the board with his own expression, is performedexamination.

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 have you learned 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.)

Oral calculation methods

Oral techniques for adding and subtracting multidigit numbers are taught in grade 4 of a four-year elementary school in the following order:

1. Numbering cases

a) Cases of the form:

99 999 + 1 345 000 - 1 560 999 + 1

560 000 - 1 399 999 + 1 40 000 - 1

When performing calculations of this type, they refer to the principle of constructing a natural series of numbers: adding one to a number gives the next number; subtracting one gives the number that precedes the count.

For example: 399 999 + 1 - adding 1 to the number, we get the following number. The number 400,000 following the number 399,999 means 399,999 + 1 = 400,000.

b) Cases of the form:

30 000 + 1 000 650 999 - 900 600 000 + 5

60 345 - 5 345 000 - 45 000 800 700 + 1 000

When performing calculations of this type, the child should know well the principle of the bitwise structure of numbers in the decimal number system.

650 999 - 900 - 650 099

2. Addition and Subtraction of Whole Thousands

Addition and subtraction of the form 32,000 + 2,000, 690,000 - 50,000 is the first computational technique that begins the formation of oral calculations in the volume of multi-digit numbers.

To master this technique, the child must have a good idea of the bit composition of a multi-digit number. Considering 32,000 as 32K and 2,000 as 2K, the reception of 32,000 + 2,000 is calculated as 32K + 2K. The answer of 34K is then treated as 34,000 and the result is written. Thus, actions in whole thousands are considered as actions by bit units, calculations in this case are reduced to tabular calculations within 10, 20 or 100.

3. Addition and subtraction of whole thousands based on the rules of arithmetic operations

The mathematics textbook for grade 4 practically does not offer calculations of the corresponding type, however, teachers often use them in oral counting.

These cases include calculations of the form: 70 200 + 400, 600 100 - 99, 3 008 + 351.425 100 - 24 100, etc.

In the calculations, knowledge of the decimal composition of multidigit numbers is used and the understanding that in all cases the actions affect only a part of the first number (the first number can be considered as a sum). Thus, actions can only be performed on a part of the first number.

For example:

Calculating the sum 70 200 + 400, you can separately add 400 and 200, and then add their sum to the number 70 000. In fact, the rule of adding the number to the sum is used.

When performing calculations in the case of 425 100 - 24 100, the rule for subtracting the number from the sum is used. 425,100 is considered as the sum of 400,000 and 25,100. 24,100 is subtracted from one of the terms (25,100 - 24,100 = 1,000), and the result is added to the first term: 400,000 + 1,000 = 401,000.

All these cases are based on a good knowledge of the bit composition of multi-digit numbers and the ability to perform oral calculations in whole digits.

Written calculation methods (columnar)

Written addition and subtraction are the main computational activities in multidigit volume calculations, since mental calculations with multidigit numbers are too complex a problem for all children. The use of written computational algorithms under these conditions is psychologically and methodologically justified.

The assimilation by children of the numbering of four-digit and multi-digit numbers allows them to carry out the transfer of the ability to add and subtract numbers "in a column" from the area of three-digit numbers to the area of multi-digit numbers.

When familiarizing with the written techniques of addition and subtraction in the volume of multidigit numbers, an analogy is drawn with the algorithm for written addition and subtraction within 1000:

1) Written addition and subtraction of any multi-digit numbers is performed in the same way as addition and subtraction of three-digit numbers.

2) When writing in a column, as in adding three-digit numbers, you should write down the category under the corresponding category, and add first units, then tens, and then hundreds, then thousands, etc. (from right to left).

It is believed that children are well taught to perform the actions of addition and subtraction in a column, therefore, the 4th grade textbook does not provide for the distribution of addition and subtraction cases by difficulty levels.

The first to consider are various cases with transitions through the discharge both during addition and subtraction: 3 126 + 4 232; 25 346 - 13 407.

Then the cases of subtraction with zeros in the diminished are considered:

600 - 25; 1 000 - 124; 30 007 - 648.

These cases are the most difficult, since they require "borrowing" of bit units not from neighboring, but from far-distant bits. It is useful to first accompany these cases with a detailed explanatory note on the chalkboard so that the children understand and see where the nines in the "empty" digits come from.

For example:

30 007 Subtract units. You cannot subtract 8 from 7. 648 I am trying to occupy one in the adjacent digit.

There are no bit units in the category of tens, hundreds and thousands, therefore, a "loan" can only be made from the category of tens of thousands: 30 thousand - 1 thousand = 29 thousand. We sign 29 over 30.

We represent the “employed” thousand as the sum of 1 thousand = 1000 = = 990 + 10.

We sign nines over the digits of hundreds and tens, and subtract 8 from 10 units, we get 2 units. But there were 7 units in the category of ones. Add them to the resulting 2 units and write 9 in the ones place.

Subtract: 9 dess. - 4 dec. = 5 dec. We write 5 in the tens place. 9 cells - 6 cells = 3 cells We write 3 in the hundreds place.

From tens of thousands there are 29 thousand left. We write 9 in the category of thousands, 2 - in the category of tens of thousands.

When studying the addition and subtraction of multi-digit numbers, it is recommended to repeat and fix the names of the components and the results of actions; properties of finding unknown components of actions when checking the results of calculations; consider the patterns of change in the sum and difference when one of the action components changes.

Many children use calculators both when performing calculations with multidigit numbers and when checking the results. In high school, it is not forbidden to use calculators if it is necessary to perform cumbersome calculations (in physics, chemistry, geometry lessons).

To stimulate the child to use the ability to independently calculate in a column, tasks should be offered that do not allow the mechanical use of a calculator to calculate the result. These are various tasks for finding errors in records or digits of calculations, for estimating rounded results of calculations, for restoring missing digits in components of actions, for choosing the correct answers from the proposed ones, etc. numbers quickly lead to fatigue of children, which provokes the appearance of errors. Therefore, you should not set more than three examples in a row for calculations with multi-digit numbers.

Lecture 10. Multiplication

1. The meaning of the multiplication action.

2. Tabular multiplication.

3. Techniques for memorizing the multiplication table.

The meaning of the multiplication action

The multiplication action is considered as the summation of the same terms.

By definition, multiplication of non-negative integers (natural numbers) is an action performed according to the following rules:

a b = a + a + a + a + a ... + a, for b> 1

b terms

a 1 = a, for b = 1

a 0 = 0, for b = 0

The use of the symbolism of multiplication makes it possible to shorten the notation of the addition of identical terms.

The record of the form 2-4 = 8 implies a reduction of the record of the form 2 + 2 + 2 + 2 = 8. It is read as follows: “take 2 4 times, it will turn out to be 8”; or: "2 times 4 is 8".

The action of multiplication in all mathematics textbooks for elementary grades is considered earlier than the action of division.

From a set-theoretic point of view, multiplication corresponds to such object actions with aggregates (sets, groups of objects) as a union of equal (equal) aggregates. Therefore, before getting acquainted with the symbolism of recording actions and calculating the results of actions, the child must learn to model all these situations on object aggregates, understand (i.e. correctly represent) them from the teacher's words, be able to show with his hands both the process and the result of the object. actions, and then describe them verbally.

Types of tasks that are offered to children before they become familiar with the symbolism of the action of multiplication (in grades 1 and 2):

1. Count in twos (threes, fives).

2. Draw a picture: "There are 2 oranges on three plates." Count how many oranges there are.

3. Find an extra entry:

Find the meaning of each expression in the most convenient way.

4. Make a record of the expression according to the picture:

The types of tasks used for the child's assimilation of the meaning of multiplication when getting acquainted with this action:

a) To correlate a drawing and a mathematical notation:

Review the drawing and explain the entries:

2 + 2 + 2 + 2 + 2 = 10 and 2.5 = 10 5 + 5 = 10 and 5-2 = 10

4 + 4 + 4 = 12 4-3=12

b) Finding the sum of the same terms: Look at the pictures and finish writing:

c) To replace addition by multiplication:

Replaces where addition by multiplication is possible and computed the results:

5+5+5+5 1+1+1+1+1 5+6+3

42 + 42 0 + 0+0 + 0 + 0 4 + 6 + 8

d) Understanding the meaning of defining the action of multiplication:

Look at the entries and explain which number is taken by the term and how many times this number is taken by the term: 6-4 = 24 9-3 = ...

6 + 6 + 6 + 6 = 24 9 + 9 + 9 =...

An expression like 3 5 is called a product. The numbers 3 and 5 in this record are called factors (factors).

An entry of the form 3 5 = 15 is called equality. The number 15 is called the value of the expression. Since the number 15 in this case is obtained as a result of multiplication, it is also often called the product.

For example:

Find the product of the numbers 4 and 6. (The product of 4 and 6 is 24.)

Since the names of the components of the multiplication action are introduced by agreement (these names are given to children and must be remembered), the teacher actively uses tasks that require recognition of the components of actions and the use of their names in speech.

For example:

1. Among these expressions, find those in which the first factor is 3 (the second factor is 2, etc.):

2-2 7-3 6-2 1.6 3-5 3-2 7-3 3-4 3-1

2. Make a product in which the second factor is 5. Find its value.

3. Select examples in which the product is 6. Underline them in red. Select examples where the work is 12. Underline them in blue.

7-3 6-1 2-2 2-3 6-2 3-2 2-6

4. What is the name of the number 4 in the expression 5 4? What is the number 5 called? Find a piece. Create an example in which the product is the same number and the factors are different.

5. The factors are 8 and 2. Find the product.

In the third grade, children get acquainted with the rule of interconnection of the components of multiplication, which is the basis for teaching how to find the unknown components of multiplication when solving equations:

If the product is divided by one factor, then you get a different factor.

For example:

Solve the equation 6 * x = 24. (The factor is unknown in the equation. To find the unknown factor, you need to divide the product by the known factor.x = 24: 6, x = 4.)

However, this rule in the 3rd grade mathematics textbook is not a generalization of the child's ideas about how to check the multiplication action. The rule for checking the results of multiplication is considered in the textbook much later - after acquaintance with out-of-table multiplication and division (familiarity with multiplication and division of two-digit numbers by single-digit numbers, which are not included in the multiplication and division table). This is due to the fact that the rule of interconnection of multiplication components is the basis for compiling the division table. Since it is assumed that the child knows by heart the table multiplication cases by this time, there is no need to check the results. There is only a need to quickly recover (remember) the required third number from two data.

For example:

9-2 = ... 5-4 = ... 1*7 = ...

18:2 = ... 20:4 = ... 7:7 = ...

When performing oral out-of-table multiplication, which requires a rather complex algorithm, verification is necessary, since many children often make mistakes in these cases.

The rule for checking the action of multiplication:

1) The product is divided by a factor.

2) Compare the result with another factor. If these numbers are equal, the multiplication is correct.

For example: 18 4 = 72. Checking: 1) 72: 4 = 18; 2) 18 = 18.

Table multiplication

Learning the multiplication table is the central task of teaching mathematics in grades 2 and 3.

Table multiplication includes the cases of multiplying single-digit natural numbers by single-digit natural numbers, the results of which are found on the basis of the specific meaning of the multiplication action (the sums of the same terms are found).

The results of multiplication in tables, in accordance with the program requirements for knowledge, skills and abilities, children must know by heart. Multiplication with the number zero, multiplication with the numbers 1 and 10 belong to special cases.

The first techniques for compiling multiplication tables are related to the meaning of the multiplication action (see the previous paragraph). The results of these tables are obtained by sequential addition of the same terms.

For example:

The drawing next to it helps the child get the result by recounting the figures. For small values of the multipliers, the counting technique for obtaining the tabular value of the product is quite acceptable, and the teacher often uses it when obtaining the results of tables of multiplication values for the numbers 2, 3, 4. This example shows that this technique is convenient only for small values of the second factor.

When the value of the second factor is greater than 5, it is more convenient to use another technique to obtain the results of tabular values: the technique of adding to the previous result. For example:

Calculate and remember: 2-6 = 2.5 + 2 = ... 2-7 = 2.6 + 2 = ... 2-8 = 2.7 + 2 2.9 = 2-8 + 2 = ...

In a mathematics textbook for grade 2, this technique is given more extensively, and therefore it is not always correctly understood from the point of view of the execution technique:

2 + 2 + 2 + 2 + 2 + 2 + 2 2-7it.p.

In a similar way, a table of values for multiplying the number 3 is compiled.

The next technique, on the basis of which tables of values for multiplying numbers are compiled, is the technique of rearranging the factors.

This technique is actually the first mathematical law regarding the operation of multiplication in elementary school:

The product does not change from the rearrangement of the factors.

The way children get to know this rule (law) is due to the previously introduced sense of the multiplication action. Using object models of sets, children count the results of grouping their elements in different ways, making sure that the results do not change as the methods of grouping change.

For example:

The counting of elements of the picture (set) in pairs horizontally coincides with the counting of elements in triplets vertically. Consideration of several options for such cases gives the teacher a reason to make an inductive generalization (that is, a generalization of several special cases in a generalized rule) that permuting the factors does not change the value of the product.

Based on this rule, used as a counting method, a multiplication table by 2 is compiled.

For example:

Using the multiplication table of the number 2, calculate and remember the multiplication table by 2:

2 = 2 = 2 = 2 = 2 = 2 = 2 =

Based on the same technique, a multiplication table by 3 is compiled:

3-4 = 12 3-7 = 21 4-3 = ... 7-3=...

3-5= 15 3-8 = 24 5-3 = ... 8-3 = ...

3-6 = 18 3-9 = 27 6-3=... 9-3 = ...

The compilation of the first two tables is divided into two lessons, which accordingly increases the time allotted for memorizing them. Each of the last two tables is compiled in one lesson, since it is assumed that children, knowing the original table, do not have to separately memorize the results of the tables obtained by rearranging the factors. In fact, many children learn each table separately because the lack of mental flexibility makes it difficult for them to easily reverse engineer the memorized table case schema model. When calculating cases of the form 9 2 or 8 3, the children again return to the method of successive addition, which naturally takes time to get the result. This situation is most likely due to the fact that for a significant number of children, such a time spacing of interrelated multiplication cases (those connected by the rule of permutation of factors) does not allow the formation of an associative chain focused specifically on the relationship. The same situation was observed in a number of children when using the property of permutation of terms to compile addition tables: having memorized case 3 + 5, such a child learns separately case 5 + 3, since the requirement to learn this case comes from the teacher 16 lessons after the requirement to memorize the first one, and when In this case, a table of the form □ + 4, □ - 4. was memorized in the interval. In other words, the delay in the formation of an associative connection, focused on the relationship of these cases, turned out to be too great for the child, which prevented the formation of such a connection. Therefore, each case from an actually interconnected pair is learned by the child separately by heart.

When compiling the multiplication table for the number 5 in grade 3, only the first product is obtained by adding the same terms: 5-5 = 5 + 5 + 5 + 5 + 5 = 25. The remaining cases are obtained by adding five to the previous result:

5-6 = 5- 5 + 5 = 30 5-7 = 5-6 + 5 = 35 5-8 = 5-7 + 5 = 40 5-9 = 5- 8 + 5 = 45

Simultaneously with this table, an interconnected multiplication table by 5: 6 5 is compiled; 7 5; 8 5; 9 5.

The multiplication table of the number 6 contains four cases: 6 6; 6 7; 6-8; 6-9.

The multiplication table by 6 contains three cases: 7 6; 8 6; 9 6.

The multiplication table of the number 7 contains three cases: 7 7; 7 8; 7 9.

The multiplication table by 7 contains two cases: 8 7; 9 7.

The multiplication table of the number 8 contains two cases: 8 8; 8 9.

The multiplication table by 8 contains one case: 9 8.

The multiplication table of the number 9 contains, only one case: 9 9.

The theoretical approach to such a construction of the system for studying table multiplication assumes that it is in this correspondence that the child will remember the cases of table multiplication.

The greatest number of cases contains the multiplication table of the number 2, which is the easiest to remember, and the multiplication table of 9, which is the most difficult to remember, contains only one case. In reality, considering each new "portion" of the multiplication table, the teacher usually restores the entire volume of each table (all cases). Even if the teacher draws the attention of the children to the fact that, for example, only the case 9 9 is a new case in this lesson, and 9 8, 9 7, etc. were studied in previous lessons, most of the children perceive the entire proposed volume as material for a new memorization. Thus, in fact, for many children, the multiplication table of the number 9 is the largest and most complex (and this is really so, if we bear in mind the list of all cases that refers to it).

A large amount of material that requires memorization by heart, the difficulty in the formation of associative links when memorizing interrelated cases, the need for all children to achieve a strong memorization of all table cases by heart within the time frame established by the program - all this makes the topic of studying table multiplication in primary grades one of the most methodologically difficult. In this regard, important are the issues related to the methods of memorizing the multiplication table by the child.

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

The article describes in detail how to add two or more multi-digit natural numbers. A 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.

Column addition of two numbers: what you need to know?

Before we move on directly to the column addition operation, let's look at some important points. For a quick mastery of the material, it is desirable:

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

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

The numbers are written one below the other, the numbers of each category are located, respectively, one below the other. On the left we put a plus sign, and under the numbers we draw a horizontal line.

Now we mentally divide the record into columns by category.

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

We start from the rightmost column (units place). We add the numbers, and under the line we write down the value of the units. If, when adding up, the value of tens as a result turned out to be non-zero, remember this number.

We add the numbers of the second column. We add to the result the number of tens, which we memorized in the previous step.

We repeat the whole process with each column, up to the extreme left.

This presentation is a simplified diagram of the algorithm for adding natural numbers in a column. Now that we have figured out the essence of the method, we will consider each step in detail.

First, add the units, that is, the numbers in the right column. If we get a number less than 10, write it down in the same column and move on to the next. 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 ones place, and the value of the tens place is memorized. For example, the number is 17. Then we write down the number 7 - the value of units, and the value of tens - 1 - we remember. Usually they say: "We write seven, one in the mind."

In our example, when adding the numbers of 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, add the numbers in the next column, that is, in the tens place. We carry out the same actions, only the number that we had in mind must be added to the amount. If the amount is less than 10, just write the number under the second column. If the result is greater than or equal to 10, write in the second column the value of the units of this number, and remember the number from the tens place.

In our case, we add the numbers 3 and 9, as a result we have 3 + 9 = 12. At the previous step, we did not remember anything, so nothing needs to be added to this result.

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

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

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

The next step gives 8 + 3 = 11. In the column, write down the number 1 from the category of units. Keep the number 1 in the tens in mind and move on to the next column.

This column contains only one number 9. If we did not 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 bar, we write 0, and again we keep one in mind.

Moving on 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 did not remember anything, we just rewrite 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 nothing in memory either, so we can say that the column addition operation is over. The number written below the line is the result of the addition of the two upper numbers.

To deal with all the possible nuances, consider a few more examples.

Example 1. Column addition of natural numbers

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

First, we write down these numbers according to the rule of writing when adding in a column:

Starting from the right column, we proceed to the addition of numbers.

Since 7< 10 , записываем 7 под чертой.