Written numbering.
In the decimal number system, ten digits are used to write numbers: 1,2,3,4,5,6,7,8,9,0. The signs for writing numbers are called figures.
Discharge- a place for writing digits in a number. Each category has its own name. The name of the digits coincides with the name of the counting units - the digit of units, tens, hundreds, etc. In addition, the digits are given names that match the number of the place occupied by the digit in the notation of the number. The ranks are numbered from right to left. Accordingly: 1st digit - units digit; 2nd digit - tens digit; 3rd digit is the hundreds digit, 4th digit is the thousands digit, etc.
Numbers are recorded on based on the principle of the local value of numbers: the value of a digit depends on the place occupied by this digit in the notation of the number
In oral numbering, special words are not required to designate categories or classes that do not contain a single unit, because the names of these bit units are simply omitted. In written numbering, the number 0 is put in place of the missing units in any category or class. Let's depict the facts discussed above in the form of a diagram (see Diagram 1).
When studying numbering, students get acquainted with the characteristics of the number:
2. Indicate how many counting units of each kind are in it (units, tens, hundreds, etc.).
3. How many units are in each category.
4. Name directly the next and previous numbers for a given number (neighbors of the number).
5. Present the number as a sum of bit terms.
In mathematics, there are 3 approaches to the formation of the concept of number: axiomatic, set-theoretic and through the measurement of quantities.
In traditional and some other educational systems (“Harmony”, the system of L.V. Zankov, etc.), the concept of a number is formed on the basis of a set-theoretic approach with elements of an axiomatic one, which allows one to assimilate the properties of a number of natural numbers.
Consider now the order studying numbering in the L.V. Zankov.
In this system, the following sections are distinguished: “Single-digit numbers”, “Two-digit numbers”, “Three-digit numbers”, “Multi-digit numbers”, “Numbers within a million”. The study of numbering takes place in two stages: the preparatory (pre-numerical) stage and the study of numbers.
At the preparatory stage students reinforce the concepts of "more", "less", "equal", spatial representations of students are specified.
The study of the natural series of numbers begins with introducing students to the history of the emergence of numbers (when people did not know numbers, how they thought, and other questions). The initial basis of acquaintance with natural numbers is the set-theoretic approach. The number arises as an invariant characteristic of the class of equivalent sets, and the main tool for understanding the relations between them is the establishment of a one-to-one correspondence between the elements of the compared sets. On this basis, concepts are formed about relations more, less, equal, unequal both between sets and between the numbers corresponding to them. At this stage, students relate the number to specific finite sets.
Children get acquainted with numbers and numbers outside of their orderly arrangement. Writing numbers is studied in order of increasing difficulty of their image: 1, 4, 6, 9, 5, 3, 2, 7, 8.
At the next stage, single-digit natural numbers, which the children met in the process of comparing sets, are ordered to the beginning of the natural series of numbers and they get acquainted with its basic properties.
Work plan at this stage:
1. Activation of children's ideas about putting things in order in the most general sense of the word and about the variety of ways to put things in order (Assignment: In the picture you see a lot of different geometric shapes. Do you think there is order in this picture? Tell me, how would you put things in order among these figures. Make a drawing.)
2. Formation of ideas about some ways of ordering in mathematics, focusing on ordering in ascending and descending order.
3. Ordering the location of several diverse sets in order of increasing (decreasing) the number of elements.
Task: What can you say about the rows of circles? Can we say that they are arranged in order of increasing? Write down the number of circles in each row. Insert comparison marks.
4. Ordering of the numbers corresponding to the sets, both differing by the same number, and by different numbers.
5. Ordering of all single-valued natural numbers and the introduction of the concept of a natural series of numbers.
6. Acquaintance with the properties of the natural series of numbers (starts from 1, each next is 1 more than the previous one, infinite).
7. The concept of a segment of the natural series of numbers, the similarity and difference between the natural series of numbers and its segment.
Then students get acquainted with the number 0 (the number 0 characterizes the absence of recalculation objects).
Studying the concentr "Double figures" starts with the number 10.
Algorithm for learning two-digit numbers:
Formation of a new counting unit - ten by combining ten previous units.
Formation of ten as the next number of the natural series.
· 10 record and record analysis.
Counting in tens up to 90.
Recording the resulting numbers.
· Acquaintance with the names of round tens and analysis of their formation.
· Filling in the gaps between round tens in the natural series of numbers.
· Acquaintance with the name of two-digit numbers standing between tens. Establishment of the general principle of formation of these names.
Comparison of all studied natural numbers.
Before studying a new counting unit, preparatory work takes place: At home, children are given the task of finding out when and what objects are considered different groups and why they do it (a pair of shoes, gloves, a box of pencils 6 (12, 18), etc.).
Familiarization with the numbers of the second, third, etc. ten goes gradually. Each new ten is considered separately (first, the formation of the numbers of the second ten, after several lessons, the formation of the numbers of the third ten, etc.). The study of two-digit numbers is significantly extended in time. This is done so that children have the opportunity to deeply understand the principle of constructing the number system that we use.
Study of three-digit numbers starts at the end of class 2 and goes in accordance with the algorithm that we wrote for two-digit numbers.
In grades 3 and 4, students continue to get acquainted with the natural series of numbers. Consideration of the topic "multi-digit numbers»is divided into 2 stages: first, children learn numbers within the first two classes (the class of units and the class of thousands), and then they get acquainted with the numbers of the millions class.
The central moment of each new expansion of the set of natural numbers is the formation of a new counting unit (thousands, tens of thousands, hundreds of thousands, etc.). Each such unit arises primarily as a result of combining ten previous units into a single whole: ten hundred - one thousand, ten thousand - one tens of thousands, etc.
Although initially a natural number appears before students in the set-theoretic approach, already in the first grade, children are also introduced to the interpretation of the number as a result of the ratio of magnitude to the chosen measure. This happens when studying such quantities as length, mass, capacity, etc. These two approaches continue to coexist in the future, culminating in a generalization, as a result of which the concepts of exact and approximate numbers appear. The expansion of the concept of number occurs due to acquaintance with fractional, as well as positive and negative numbers.
wedge numbering. Even the Chaldeans and Babylonians had written signs for depicting numbers. Their numbering is called wedge-shaped and is found on the tombs of ancient Persian kings.
Hieroglyphic numbering. The Egyptians attribute the invention of arithmetic to the mythical person Thoth (Phot). They had decimal reckoning even under Fra Sesostris. Egyptian numbering is called hieroglyphic. The Egyptians denoted the unit, ten, hundred and thousand with special signs, hieroglyphs. Several units, tens, hundreds and thousands were depicted by the simple construction of these signs.
Chinese numbering. Numbering should also be included among the most ancient Chinese. According to the Chinese, they have been using it since the time of Fugue, the Chinese emperor, who lived 300 years BC. In this numbering, the first nine numbers are represented by special characters. There were also signs for 10, 100, 1000. Large numbers were written in columns from top to bottom.
Phoenician numbering. Finally, numbering must also be attributed to the most ancient Phoenician. The Phoenicians, in comparison with the Egyptians, made a reform in numbering in the sense that they replaced the hieroglyphs with the letters of their alphabet. The Jews also used this numbering.
The Phoenicians and Jews represented the first nine numbers and the first nine tens with the 18 initial letters of their alphabet and wrote large numbers from the right hand to the left.
In Egypt itself, hieroglyphic numbering was abandoned and first hieratic, and then demotic letters were introduced for general use (600 years before Christ). AT hieratic numbering, the first three numbers are similar to real numbers.
Greek, Roman and Church Slavonic numbering. The Greeks adopted from the Phoenicians the system of representing numbers with letters. Some say that until then they represented numbers by the very signs that are known by the name Roman numbering, and that Roman numbering is thus ancient Greek. Church Slavonic is nothing other than Greek, expressed only in Slavic letters.
The Romans used the following signs when depicting numbers:
1 - I, 5 - V, 10 - X, 50 - L, 100 - C, 500 - D, 1000 - M.
When depicting the remaining numbers, they were guided by the following rule:
If a smaller number follows a larger one, it increases the number by its magnitude; if the smaller number precedes the larger one, it reduces the number by its own amount.
In accordance with this rule, they depicted numbers as follows:
1 - I, 2 - II, 3 - III, 4 - IV, 5 - V, 6 - VI, 7 - VII, 8 - VIII, 9 - IX, 10 - X, 11 - XI, 12 - XII, 13 - XIII, 14 - XIV, 15 - XV, 16 - XVI, 17 - XVII, 18 - XVIII, 19 - XIX, 20 - XX, ... 27 - XXVII, ... 40 - XL, 60 - LX, 90 - XC, 100 - C, 110 - CX, 150 - CL, 400 - CD, 600 - DC, 900 - CM, 1100 - MC.
Numbers consisting of several thousand were written as numbers up to a thousand are written, with the only difference being that after the number of thousands on the lower right side, the letter m (mille - thousand) was assigned. Thus, 505197 = DV m CXCVII.
In Slavic and Greek numerals, the first nine numbers, nine tens and nine hundreds were designated by special letters.
In Slavic reckoning, they put on the letter titlo (¯), to indicate that the letter represents a number.
The following table shows Greek and Slavic numbering in parallel:
To designate thousands, a sign was placed in front of the number of thousands in Slavic reckoning, and in Greek reckoning, a dash was added to the number denoting thousands.
Thus,
Origin and distribution of decimal numbering
Although it is not yet possible to draw a final conclusion regarding the representation, introduction and distribution in Europe of the decimal numbering system, however, the literature provides many very important indications on this issue. Some call this system Arabic. Indeed, history shows that the decimal system was borrowed from the Arabs. Thus, it is known that at the beginning of the 13th century, the Tuscan merchant Leonard introduced his compatriots to the techniques of the decimal system after his travels in Syria and Egypt. Sarco-Bosco, a famous teacher of mathematics in Paris (died 1256), and Roger Bacon, by their writings, were most instrumental in spreading this system throughout Europe. They already point out that the decimal numbering was borrowed by the Arabs from the Indians. From the monuments of Arabic literature, it is authentically known that Abu-Abdallah-Mohammed-Ibn-Muza, originally from Koraism, traveled a long time in India in the 9th century and introduced Arabic scientists to Indian numbering after his return. The Arabic writers Avicena Aben-Ragel and Alsefadi also attribute the invention of numbering to the Indians.
Written records of Sanskrit, the language of ancient India, confirm the indications of Arab writers.
From the work of Baskara, an Indian writer of the 12th century, it is clear that the Indians knew several centuries before Baskara the representation of numbers by ten signs, because this work outlines a coherent theory of four arithmetic operations and even the extraction of square roots. Both Baskara and the more ancient writer Bramegupta consider the fact of the invention of numbering to be very ancient. In the writer of an even more ancient Ariabgat, we find the solution of many remarkable mathematical questions.
These indications seem to make it unlikely that the French geometer Chall asserted that the decimal system was a development of the Roman way of using the calculation table (Abacus) in calculations and that one introduction of zero was enough to get a real decimal system.
Arithmetic and logistics among the Greeks. The Greeks called arithmetic the doctrine of the general properties of numbers. The art of counting, or a set of practical methods for calculating, the Greeks called logistics.
The method of naming (naming) with the help of a few words of any natural number is called oral numbering.
When a person knew only the first few natural numbers, it is natural that he called each number by his own special name: "one", "two", "three", etc.
The method of oral numbering that we currently use was developed by people gradually in the process of centuries of counting practice. Modern oral numbering is based on the following principles:
The principle of bitwise counting.
To name some natural number is the same as to name the result of counting the units contained in this number. Obviously, if a given number contains a lot of units, then it is difficult to count them and it is difficult to name the result of the count.
Imagine that you need to count a huge pile of some items (buttons, matches, etc.). If you count them in one subject, it will take a very long time. Then they do so. Let's put all the items into boxes so that each box contains the same number of items. Then, if there are many of these boxes, then we will arrange them in boxes, and so that in each box there are as many boxes as there are items in one box. If there are too many boxes, then we divide them in the same way into even larger packages, and so on.
With this method of counting, not one counting unit is used, but many different ones: first, the object itself is used as a counting unit - this is the first counting unit, then the box is the second unit, the box is the third unit, etc.
These counting units are called digits, and the number of units of one digit that make up the unit of the next digit is called the base of the numbering system.
In the numbering that we use, the base is the number 10 - the number of fingers on both hands of a person. Therefore, our numbering is called decimal.
To name any number using the principle of bitwise counting, you need to name how many units of each digit are contained in this number. For example, 4 units of the 3rd category, 5 units of the 2nd category and 7 units of the 1st category - four hundred and fifty seven.
However, when you have to deal with large numbers, get by with one principle
bitwise calculation is difficult, because the number of digits may be too large. To further reduce the number of different words, it is necessary to name numbers by introducing another principle.
The principle of class association of ranks.
According to this principle, every three digits, starting from the 1st, are combined into one class: the first three digits (ones, tens and hundreds) are combined into the first class of units, the next Written numbering.
Written numbering is a method that allows using a small number of special characters to write down any natural number.
In oral numbering, we need special words for the first nine natural numbers, as well as a word for the second and third digits of each class and all classes starting with the second.
In decimal written numbering, to write any natural number, first of all, signs are needed to write the first nine natural numbers. These characters are called numbers. But there are no special signs for designating categories and classes in our system of written numbering, they are not needed, because. the recording of natural numbers is based on the following most important principle: the same sign (digit) denotes the same number of units of different digits, depending on where this sign is in the number entry.
So, for example, the number 3 denotes three units of the first digit, if this digit in the number entry is in the first place on the right, and the same number 3 denotes three units of the fifth digit, i.e. three tens of thousands, if this figure is in fifth place from the right, and three digits (from 4th to 6th) are combined into the second class of thousands, then the next three digits (from 7th to 9th) into the class of millions, the next three digits (from 10th to 12th) are in the class of billions, or billions, then there are the classes of trillions, quadrillions, and so on.
A million is 1 billion.
oral numbering.
Examples and tasks for oral calculations.
geometric material.
More complex tasks for all actions.
Examples and tasks for all actions.
Procedure. Parentheses.
Change private.
Division of multi-digit numbers.
Changing the work.
Multiplication of multi-digit numbers.
Repetition of addition and subtraction.
Difference change.
Subtraction of multi-digit numbers.
Amount change.
Written numbering.
oral numbering.
Numbering of integers of any size.
2 . Name the numbers in which:
a) 3 hundreds of millions 2 tens of millions;
b) 8 hundred million 4 tens of million 5 million;
c) 6 hundred million 9 million.
3 . How many millions, tens and hundreds of millions in numbers: 378 million; 905 million; 540 million?
5. Name the numbers in which:
a) 5 hundred billion 6 tens of billion;
b) 8 hundred billion 3 tens of billion 4 billion;
c) 6 hundred billion 5 billion;
6 . How many billions, tens of billions and hundreds of billions in numbers: 504 billion; 790 billion; 456 billion; 935 billion?
Name the digits of numbers in which:
a) 345 billion 248 million;
b) 400 billion 736 million;
c) 680 billion 24 million.
8. Name the numbers in which:
a) 385 units of the first class;
b) 508 units of the second class;
c) 743 units of the third class;
d) 214 units of the fourth class;
9. Name the numbers in which:
a) 56 units of the third class and 380 units of the second class;
b) 5 units of the fourth class and 25 units of the third class;
c) 1 unit of the fourth class, 300 units of the third class, 286 units of the second class and 85 units of the first class.
10 . Name the digits and classes of each number in the table and read the numbers.
Write each number in the table in a notebook.
14 . Read the following message:
Stargazers - winners will be awarded on the main square of the capital of the kingdom.
Stargazer A. counted 3056800000 celestial bodies,
stargazer B - 1317500000, and
stargazer C - 1845800000.
At the same time, it is asked who will receive the first, who will be the second, and who will be the third prize?
15 . Write the following numbers in numbers:
a) one billion one million;
b) three hundred twenty-five thousand six hundred eighteen;
c) eight million twenty-three thousand three hundred;
d) five hundred million five hundred units;
e) four billion ten million one thousand and one unit;
f) ten billion nine hundred six thousand;
g) eighty million seven thousand thirty units;
16 . What kind ranks represent the various digits of the following numbers:
568; 6798; 207886; 2326728; 20192837; 35796234865 ?
17 . Write as a single number:
a) 2000000 + 40000 + 400 + 30 + 5;
b) 20000000 + 3000000 + 700000 + 8000 + 200 + 5;
c) 300000000 + 4000000 + 50000 + 600 + 8;
18 . Decompose into bit terms of numbers:
32750; 148004; 250070; 2435600; 750420045;
19 . How much Total tens in the following numbers:
34560; 145634; 2000000; 34567280; 142345675; ?
20 . How much Total thousand in each of the following numbers:
32010; 60518; 212268; 504308; 760390; ?
21 . How much Total tens of thousands in each of the following numbers:
100000; 245624; 1000000; 34567310; 1000000000; 384104500000 ?
22. Write numbers in which:
a) six hundred forty-eight hundred;
b) one thousand two hundred and sixty two tens;
c) thirty-five hundred thousand;
d) seventeen tens of hundreds;
e) two thousand five hundred four hundred three units;
23 . Write:
a) a six-digit number in which there are no units of the hundreds digit;
b) an eight-digit number in which there are no units of the thousands place;
c) a ten-digit number in which there are no units of the tens of thousands place.
24 . Write:
a) the smallest four-digit number;
b) the largest seven-digit number;
c) the smallest five-digit number;
25 . Write a number consisting of three classes, of two classes, of four classes.
26. Write down the following data in numbers:
Radiograms from the spacecraft:
a) The flight is going well. Of the ninety-four million, one hundred and thirty-eight thousand, one hundred and fifty-nine kilometers, only ninety-one million, one hundred and thirteen thousand, one hundred and fifty-three kilometers remained to fly.
b) Caught in a meteor shower. The on-board computer counted one hundred and eighty billion three hundred million hits against the ship's hull.
27 . Write the numbers in figures: 4 million 216 thousand and 4 million 236 thousand.
28 . Round up to thousands of numbers: 145374 and 145680; 21450 and 21550; 76459 and 76511;
29. Round up to millions of numbers: 3567400; 35247000; 115620000; 115450000; 28742000; 28327000;
30 . Round up to billions of numbers: 5780000000; 6460000000; 37047560000; 84915036000;
Ticket 19
Question 1. Methodology for teaching oral and written numbering of numbers within 1000.
I. Oral numbering
Tasks:
1) Introduction of a new counting unit hundreds;
2) Introduction of new bit numbers;
3) Introduction of non-digit three-digit numbers:
By counting 1;
By forming from hundreds, tens and units;
4) Establishment of the total number of units of any category in the entire number.
Introduction of a new counting unit hundreds:
With the help of sticks or models of bit units, under the guidance of a teacher, children repeat known bit units, and then tie 10 tens into a bundle and listen to its name - a hundred. Further, hundreds are counted (1 hundred, 2 hundreds ... 10 hundreds or a thousand). A record and drawings of bit units appear on the board
1 unit 1 cm
10 units = 1 dec. 10 cm = 1 dm
10 dec. = 1 hundred. 10 dm = 1 m
Further, it is useful with children to compare counting units - bit units with measures of length and introduce a thousand tape. 1 cm acts as a simple unit on the tape, 1 dm as a ten, and 1 m as a hundred. You can repeat the count of hundreds on the tape and mark hundreds on the tape with flags or bright ribbons.
Introduction of new bit numbers (numbers of the third category - round hundreds), their formation and name, acquaintance with new numbers: one hundred, two hundred ... nine hundred, one thousand.
Visibility: models of bit units (large squares) and tape 1000.
Introduction of non-digit three-digit numbers:
a) By counting 1 to the previous one, going beyond 100: 100 and 1-101 ..
b) By forming from hundreds, tens and ones. The inverse task is immediately performed - to decompose the numbers into bit terms, to find out the decimal composition of the number.
II. Written numbering
Tasks:
1) Designation of numbers by numbers in the table of digits. Finding out the local meaning of numbers;
2) Reading and writing numbers written outside the table;
3) Consolidation of knowledge of numbering.
1.Designation of numbers by numbers in the table of digits. Learning to read numbers using a numbering table. Visibility: numbering table, vertical and horizontal abacus.
As a result of observations at this stage, children are led to the conclusion that hundreds are units of the third category, written in number in third place, counting from right to left. It also introduces the concept of a three-digit number and that zero means the absence of units of any category.
2. Reading three-digit numbers written outside the table and writing them based on knowledge of the local meaning of the numbers.
Types of exercises:
1) From these numbers, write down only those in which the number 7 stands for des, units, cells.
2) Using the numbers 3, 0, 1, write down all three-digit numbers (digits are not repeated in the number)
3) What does the number 0 mean in the records of these numbers?
3. Consolidation of knowledge of numbering:
a) In the process of studying written numbering, work continues on mastering the decimal composition of numbers. For this purpose, cards with bit numbers are now used. (Numbers are formed by superposition and vice versa)
b) Work is also underway on the assimilation of natural following, but now written exercises are also used: a record of the previous and subsequent; add 1, subtract 1; fill in the gap - write down the numbers from ... to ...
c) Identification of the largest and smallest among single-digit, two-digit and three-digit numbers.
Reverse the capture that the smallest is written as 1 and zeros, and the largest as tens.
d) When studying numbering, children learn to determine the total number of units of any category in the entire number, and not just in the corresponding category.
Visibility: models of bit units.
