Decimal fractions: definitions, recording, examples, actions with decimal fractions. Reading decimals Writing and reading decimals lesson summary

Of the many fractions found in arithmetic, those with 10, 100, 1000 in the denominator deserve special attention - in general, any power of ten. These fractions have a special name and notation.

A decimal is any number whose denominator is a power of ten.

Decimal examples:

Why was it necessary to isolate such fractions at all? Why do they need their own entry form? There are at least three reasons for this:

1. Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, bring the fractions to a common denominator. In decimal fractions, none of this is required;
2. Reduction of calculations. Decimals add and multiply according to their own rules, and with a little practice you will be able to work with them much faster than with ordinary ones;
3. Ease of recording. Unlike ordinary fractions, decimals are written in one line without loss of clarity.

Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you demand change in the amount of 2/3 rubles in a store :)

Rules for writing decimal fractions

The main advantage of decimal fractions is a convenient and visual notation. Namely:

Decimal notation is a form of decimal notation where the integer part is separated from the fractional part using a regular dot or comma. In this case, the separator itself (dot or comma) is called the decimal point.

For example, 0.3 (read: “zero integer, 3 tenths”); 7.25 (7 integers, 25 hundredths); 3.049 (3 integers, 49 thousandths). All examples are taken from the previous definition.

In writing, a comma is usually used as a decimal point. Here and below, the comma will also be used throughout the site.

To write an arbitrary decimal fraction in the specified form, you need to follow three simple steps:

1. Write out the numerator separately;
2. Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
3. If the decimal point has shifted, and after it there are zeros at the end of the record, they must be crossed out.

It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, any number of zeros can be assigned to the left of any number without harm to health. It's ugly, but sometimes useful.

At first glance, this algorithm may seem rather complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:

A task. For each fraction, indicate its decimal notation:

The numerator of the first fraction: 73. We shift the decimal point by one sign (because the denominator is 10) - we get 7.3.

The numerator of the second fraction: 9. We shift the decimal point by two digits (because the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange notation like “.09”.

The numerator of the third fraction: 10029. We shift the decimal point by three digits (because the denominator is 1000) - we get 10.029.

The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10.500. There are extra zeros at the end of the number. We cross them out - we get 10.5.

Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as is done in the last example. However, in no case should you do this with zeros that are inside the number (which are surrounded by other digits). That is why we got 10.029 and 10.5, and not 1.29 and 1.5.

So, we figured out the definition and form of recording decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.

Change from fractions to decimals

Consider a simple numerical fraction of the form a / b . You can use the basic property of a fraction and multiply the numerator and denominator by such a number that you get a power of ten below. But before doing so, please read the following:

There are denominators that are not reduced to the power of ten. Learn to recognize such fractions, because they cannot be worked with according to the algorithm described below.

That's it. Well, how to understand whether the denominator is reduced to the power of ten or not?

The answer is simple: factorize the denominator into prime factors. If only factors 2 and 5 are present in the expansion, this number can be reduced to the power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the degree of ten.

A task. Check if the specified fractions can be represented as decimals:

We write out and factorize the denominators of these fractions:

20 \u003d 4 5 \u003d 2 2 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.

12 \u003d 4 3 \u003d 2 2 3 - there is a "forbidden" factor 3. The fraction cannot be represented as a decimal.

640 \u003d 8 8 10 \u003d 2 3 2 3 2 5 \u003d 2 7 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction is represented as a decimal.

48 \u003d 6 8 \u003d 2 3 2 3 \u003d 2 4 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.

So, we figured out the denominator - now we will consider the entire algorithm for switching to decimal fractions:

1. Factorize the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
2. Count how many twos and fives are present in the decomposition (there will be no other numbers there, remember?). Choose such an additional multiplier so that the number of twos and fives is equal.
3. Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.

Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest such factor from all possible ones.

And one more thing: if there is an integer part in the original fraction, be sure to convert this fraction to an improper one - and only then apply the described algorithm.

A task. Convert these numbers to decimals:

Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, a fraction can be represented as a decimal. There are two twos and no fives in the expansion, so the additional factor is 5 2 = 25. The number of twos and fives will be equal to it. We have:

Now let's deal with the second fraction. To do this, note that 24 \u003d 3 8 \u003d 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.

The last two fractions have denominators 5 (a prime number) and 20 = 4 5 = 2 2 5 respectively - only twos and fives are present everywhere. At the same time, in the first case, “for complete happiness”, there is not enough multiplier 2, and in the second - 5. We get:

Switching from decimals to ordinary

The reverse conversion - from decimal notation to normal - is much easier. There are no restrictions and special checks, so you can always convert a decimal fraction into a classic "two-story" one.

The translation algorithm is as follows:

1. Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing - do not overdo it and do not cross out the internal zeros surrounded by other numbers;
2. Calculate how many digits are in the original decimal fraction after the decimal point. Take the number 1 and add as many zeros to the right as you counted the characters. This will be the denominator;
3. Actually, write down the fraction whose numerator and denominator we just found. Reduce if possible. If there was an integer part in the original fraction, now we will get an improper fraction, which is very convenient for further calculations.

A task. Convert decimals to ordinary: 0.008; 3.107; 2.25; 7,2008.

We cross out the zeros on the left and the commas - we get the following numbers (these will be numerators): 8; 3107; 225; 72008.

In the first and second fractions after the decimal point there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.

Finally, let's combine the numerators and denominators into ordinary fractions:

As can be seen from the examples, the resulting fraction can very often be reduced. Once again, I note that any decimal fraction can be represented as an ordinary one. The reverse transformation is not always possible.

We will devote this material to such an important topic as decimal fractions. First, let's define the basic definitions, give examples and dwell on the rules of decimal notation, as well as what the digits of decimal fractions are. Next, we highlight the main types: finite and infinite, periodic and non-periodic fractions. In the final part, we will show how the points corresponding to fractional numbers are located on the coordinate axis.

What is decimal notation for fractional numbers

The so-called decimal notation for fractional numbers can be used for both natural and fractional numbers. It looks like a set of two or more numbers with a comma between them.

The decimal point is used to separate the integer part from the fractional part. As a rule, the last digit of a decimal is never a zero, unless the decimal point is immediately after the first zero.

What are some examples of fractional numbers in decimal notation? It can be 34 , 21 , 0 , 35035044 , 0 , 0001 , 11 231 552 , 9 etc.

In some textbooks, you can find the use of a dot instead of a comma (5. 67, 6789. 1011, etc.). This option is considered equivalent, but it is more typical for English-language sources.

Definition of decimals

Based on the above concept of decimal notation, we can formulate the following definition of decimal fractions:

Definition 1

Decimals are fractional numbers in decimal notation.

Why do we need to write fractions in this form? It gives us some advantages over ordinary ones, for example, a more compact notation, especially in cases where the denominator is 1000, 100, 10, etc. or a mixed number. For example, instead of 6 10 we can specify 0 , 6 , instead of 25 10000 - 0 , 0023 , instead of 512 3 100 - 512 , 03 .

How to correctly represent ordinary fractions with tens, hundreds, thousands in the denominator in decimal form will be described in a separate material.

How to read decimals correctly

There are some rules for reading records of decimals. So, those decimal fractions that correspond to their correct ordinary equivalents are read almost the same, but with the addition of the words "zero tenths" at the beginning. So, the entry 0 , 14 , which corresponds to 14 100 , is read as "zero point fourteen hundredths."

If a decimal fraction can be associated with a mixed number, then it is read in the same way as this number. So, if we have a fraction 56, 002, which corresponds to 56 2 1000, we read such an entry as "fifty-six point two thousandths."

The value of a digit in a decimal notation depends on where it is located (just like in the case of natural numbers). So, in decimal fraction 0, 7, seven is tenths, in 0, 0007 it is ten thousandths, and in fraction 70,000, 345 it means seven tens of thousands of whole units. Thus, in decimal fractions, there is also the concept of a number digit.

The names of the digits located before the comma are similar to those that exist in natural numbers. The names of those that are located after are clearly presented in the table:

Let's take an example.

Example 1

We have decimal 43, 098. She has a four in the tens place, a three in the units place, zero in the tenth place, 9 in the hundredth place, and 8 in the thousandth place.

It is customary to distinguish the digits of decimal fractions by seniority. If we move through the numbers from left to right, then we will go from high to low digits. It turns out that hundreds are older than tens, and millionths are younger than hundredths. If we take that final decimal fraction, which we cited as an example above, then in it the senior, or highest, will be the digit of hundreds, and the lowest, or lowest, will be the digit of 10 thousandths.

Any decimal fraction can be decomposed into separate digits, that is, represented as a sum. This operation is performed in the same way as for natural numbers.

Example 2

Let's try to expand the fraction 56, 0455 into digits.

We will be able to:

56 , 0455 = 50 + 6 + 0 , 4 + 0 , 005 + 0 , 0005

If we remember the properties of addition, we can represent this fraction in other forms, for example, as the sum 56 + 0, 0455, or 56, 0055 + 0, 4, etc.

What are trailing decimals

All the fractions we talked about above are trailing decimals. This means that the number of digits after the decimal point is finite. Let's get the definition:

Definition 1

Trailing decimals are a type of decimal that has a finite number of digits after the comma.

Examples of such fractions can be 0, 367, 3, 7, 55, 102567958, 231032, 49, etc.

Any of these fractions can be converted either into a mixed number (if the value of their fractional part is different from zero), or into an ordinary fraction (if the integer part is zero). We have devoted a separate material to how this is done. Let's just point out a couple of examples here: for example, we can bring the final decimal fraction 5 , 63 to the form 5 63 100 , and 0 , 2 corresponds to 2 10 (or any other fraction equal to it, for example, 4 20 or 1 5 .)

But the reverse process, i.e. writing an ordinary fraction in decimal form may not always be performed. So, 5 13 cannot be replaced by an equal fraction with a denominator of 100, 10, etc., which means that the final decimal fraction will not work out of it.

The main types of infinite decimal fractions: periodic and non-periodic fractions

We pointed out above that finite fractions are called so because they have a finite number of digits after the decimal point. However, it may well be infinite, in which case the fractions themselves will also be called infinite.

Definition 2

Infinite decimals are those that have an infinite number of digits after the decimal point.

Obviously, such numbers simply cannot be written completely, so we indicate only a part of them and then put ellipsis. This sign indicates an infinite continuation of the sequence of decimal places. Examples of infinite decimals would be 0 , 143346732 ... , 3 , 1415989032 ... , 153 , 0245005 ... , 2 , 66666666666 ... , 69 , 748768152 ... . etc.

In the "tail" of such a fraction, there can be not only seemingly random sequences of numbers, but a constant repetition of the same character or group of characters. Fractions with alternation after the decimal point are called periodic.

Definition 3

Periodic decimal fractions are such infinite decimal fractions in which one digit or a group of several digits is repeated after the decimal point. The repeating part is called the period of the fraction.

For example, for the fraction 3, 444444 ... . the period will be the number 4, and for 76, 134134134134 ... - the group 134.

What is the minimum number of characters allowed in a periodic fraction? For periodic fractions, it will be sufficient to write the entire period once in parentheses. So, the fraction is 3, 444444 ... . it will be correct to write as 3, (4) , and 76, 134134134134 ... - as 76, (134) .

In general, entries with multiple periods in brackets will have exactly the same meaning: for example, the periodic fraction 0.677777 is the same as 0.6 (7) and 0.6 (77), etc. Entries like 0 , 67777 (7) , 0 , 67 (7777) and others are also allowed.

In order to avoid errors, we introduce the uniformity of notation. Let's agree to write only one period (the shortest possible sequence of digits), which is closest to the decimal point, and enclose it in parentheses.

That is, for the above fraction, we will consider the entry 0, 6 (7) as the main one, and, for example, in the case of the fraction 8, 9134343434, we will write 8, 91 (34) .

If the denominator of an ordinary fraction contains prime factors that are not equal to 5 and 2, then when converted to decimal notation, infinite fractions will be obtained from them.

In principle, we can write any finite fraction as a periodic one. To do this, we just need to add an infinite number of zeros to the right. How does it look on the record? Let's say we have a final fraction 45, 32. In periodic form, it will look like 45 , 32 (0) . This action is possible because adding zeros to the right of any decimal fraction gives us a fraction equal to it as a result.

Separately, one should dwell on periodic fractions with a period of 9, for example, 4, 89 (9), 31, 6 (9) . They are an alternative notation for similar fractions with a period of 0, so they are often replaced when writing with fractions with a zero period. At the same time, one is added to the value of the next digit, and (0) is indicated in parentheses. The equality of the resulting numbers is easy to check by presenting them as ordinary fractions.

For example, the fraction 8, 31 (9) can be replaced by the corresponding fraction 8, 32 (0) . Or 4 , (9) = 5 , (0) = 5 .

Infinite decimal periodic fractions are rational numbers. In other words, any periodic fraction can be represented as an ordinary fraction, and vice versa.

There are also fractions in which there is no infinitely repeating sequence after the decimal point. In this case, they are called non-periodic fractions.

Definition 4

Non-periodic decimal fractions include those infinite decimal fractions that do not contain a period after the decimal point, i.e. repeating group of numbers.

Sometimes non-periodic fractions look very similar to periodic ones. For example, 9 , 03003000300003 ... at first glance seems to have a period, but a detailed analysis of the decimal places confirms that this is still a non-periodic fraction. You have to be very careful with numbers like this.

Non-periodic fractions are irrational numbers. They are not converted to ordinary fractions.

Basic operations with decimals

The following operations can be performed with decimal fractions: comparison, subtraction, addition, division and multiplication. Let's analyze each of them separately.

Comparing decimals can be reduced to comparing ordinary fractions that correspond to the original decimals. But infinite non-periodic fractions cannot be reduced to this form, and converting decimal fractions to ordinary ones is often a laborious task. How to quickly perform a comparison action if we need to do it in the course of solving the problem? It is convenient to compare decimal fractions by digits in the same way as we compare natural numbers. We will devote a separate article to this method.

To add one decimal fraction to another, it is convenient to use the column addition method, as for natural numbers. To add periodic decimal fractions, you must first replace them with ordinary ones and count according to the standard scheme. If, according to the conditions of the problem, we need to add infinite non-periodic fractions, then we must first round them up to a certain digit, and then add them. The smaller the digit to which we round, the higher the accuracy of the calculation will be. For subtraction, multiplication and division of infinite fractions, preliminary rounding is also necessary.

Finding the difference of decimal fractions is the opposite of addition. In fact, with the help of subtraction, we can find a number whose sum with the subtracted fraction will give us the reduced one. We will talk about this in more detail in a separate article.

Multiplication of decimal fractions is done in the same way as for natural numbers. The method of calculation by a column is also suitable for this. We again reduce this action with periodic fractions to the multiplication of ordinary fractions according to the rules already studied. Infinite fractions, as we remember, must be rounded before counting.

The process of dividing decimals is the reverse of the multiplication process. When solving problems, we also use column counts.

You can set an exact correspondence between the end decimal and a point on the coordinate axis. Let's figure out how to mark a point on the axis that will exactly correspond to the required decimal fraction.

We have already studied how to construct points corresponding to ordinary fractions, and decimal fractions can be reduced to this form. For example, an ordinary fraction 14 10 is the same as 1 , 4 , so the point corresponding to it will be removed from the origin in the positive direction by exactly the same distance:

You can do without replacing the decimal fraction with an ordinary one, and take the digit expansion method as a basis. So, if we need to mark a point whose coordinate will be equal to 15 , 4008 , then we will first represent this number as a sum 15 + 0 , 4 + , 0008 . To begin with, we set aside 15 whole unit segments in the positive direction from the origin, then 4 tenths of one segment, and then 8 ten-thousandths of one segment. As a result, we will get a coordinate point, which corresponds to the fraction 15, 4008.

For an infinite decimal fraction, it is better to use this particular method, since it allows you to approach the desired point as close as you like. In some cases, it is possible to build an exact correspondence of an infinite fraction on the coordinate axis: for example, 2 = 1, 41421. . . , and this fraction can be associated with a point on the coordinate ray, remote from 0 by the length of the diagonal of the square, the side of which will be equal to one unit segment.

If we find not a point on the axis, but a decimal fraction corresponding to it, then this action is called the decimal measurement of the segment. Let's see how to do it right.

Suppose we need to get from zero to a given point on the coordinate axis (or get as close as possible in the case of an infinite fraction). To do this, we gradually set aside unit segments from the origin of coordinates until we get to the desired point. After whole segments, if necessary, we measure tenths, hundredths and smaller parts so that the correspondence is as accurate as possible. As a result, we got a decimal fraction that corresponds to a given point on the coordinate axis.

Above we gave a picture with a point M. Look at it again: to get to this point, you need to measure one unit segment from zero and four tenths of it, since this point corresponds to the decimal fraction 1, 4.

If we cannot hit a point in the process of decimal measurement, then it means that an infinite decimal fraction corresponds to it.

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Lessonmathematics in the 5th grade on the topic "Decimal notation of fractional numbers"

Topic: The concept of decimal fraction. Reading and writing decimals.

The purpose of the lesson: introduce the concept of decimal fractions, their correct reading and writing.

Tasks:

To organize the work of students in the study and primary consolidation of the concept of "decimal fraction", the algorithm for recording decimal fractions.

Create conditions for the formation of UUD:

Communicative UUD: listening skills, discipline, independent thinking.

Regulatory UUD: understand the learning task of the lesson, carry out the solution of the learning task under the guidance of the teacher, determine the purpose of the learning task, control your actions in the process of its implementation, detect and correct errors, answer the final questions and evaluate your achievements

Personal UUD: formation of educational motivation, the need to acquire new knowledge.

Lesson type: lesson learning new material

Lesson construction technology: problem method, work in pairs

Forms of work: individual, frontal, conversation, work in pairs.

Organization of student activities in the classroom:

Independently come to the problem and solve it;

Independently determine the topic, objectives of the lesson;

Derive the rule;

Work with the text of the textbook;

Answer questions;

Solve problems independently;

Assess themselves and each other;

Reflect.

Teaching methods: verbal, visual - illustrative, practical

Resources: multimedia projector, presentation.

Educational and methodological support: textbook"Maths. Grade 5 "author N.Ya. Vilenkin; CD "Mathematics. Teaching according to new standards. Theory. Methodology. Practice. Publishing house "Uchitel".

The decimal fraction must contain a comma. That numerical part of the fraction, which is located to the left of the decimal point, is called the whole; to the right - fractional:

5.28 5 - integer part 28 - fractional part

The fractional part of a decimal is made up of decimal places(decimal places):

• tenths - 0.1 (one tenth);
• hundredths - 0.01 (one hundredth);
• thousandths - 0.001 (one thousandth);
• ten-thousandths - 0.0001 (one ten-thousandth);
• hundred thousandths - 0.00001 (one hundred thousandth);
• millionths - 0.000001 (one millionth);
• ten millionths - 0.0000001 (one ten millionth);
• one hundred millionth - 0.00000001 (one hundred millionth);
• billionths - 0.000000001 (one billionth), etc.

• read the number that is the integer part of the fraction and add the word " whole";
• read the number that makes up the fractional part of the fraction and add the name of the least significant digit.

For example:

• 0.25 - zero point twenty-five hundredths;
• 9.1 - nine point one tenth;
• 18.013 - eighteen point thirteen thousandths;
• 100.2834 is one hundred and two thousand eight hundred and thirty-four ten thousandths.

Writing decimals

To write a decimal fraction, you must:

• write down the integer part of the fraction and put a comma (the number meaning the integer part of the fraction always ends with the word " whole");
• write the fractional part of the fraction in such a way that the last digit falls into the desired digit (if there are no significant digits in certain decimal places, they are replaced by zeros).

For example:

• twenty point nine - 20.9 - in this example, everything is simple;
• five point one hundredth - 5.01 - the word "hundredth" means that there should be two digits after the decimal point, but since there is no tenth place in the number 1, it is replaced by zero;
• zero point eight hundred and eight thousandths - 0.808;
• three point fifteen - it is impossible to write such a decimal fraction, because a mistake was made in the pronunciation of the fractional part - the number 15 contains two digits, and the word "tenths" means only one. Correct will be three point fifteen hundredths (or thousandths, ten thousandths, etc.).

Decimal Comparison

Comparison of decimal fractions is carried out similarly to comparison of natural numbers.

1. first, the integer parts of the fractions are compared - the decimal fraction with the larger integer part will be larger;
2. if the integer parts of the fractions are equal, the fractional parts are compared bit by bit, from left to right, starting from the comma: tenths, hundredths, thousandths, etc. The comparison is carried out until the first discrepancy - that decimal fraction will be larger, which will have a larger unequal digit in the corresponding digit of the fractional part. For example: 1.2 8 3 > 1,27 9, because in hundredths the first fraction has 8, and the second has 7.

Sections: Maths

Topic: Concept of decimal fraction. Reading and writing decimals.

Goals:

1. Formation of knowledge and skills to write and read decimal fractions. To introduce students to new numbers - decimal fractions (a new way of writing a number)
2. Develop intuition, conjecture, erudition and mastery of the methods of mathematics.
3. Awaken mathematical curiosity and initiative, develop a sustainable interest in mathematics.
4. Cultivate a culture of mathematical thinking.

Development goal: Formation of skills of self-assessment and self-analysis of educational activities.

Problem - developing lesson (combined)

Stages:

1) problem situation;
2) problem;
3) search for methods of its solution;
4) problem solving

Lesson motto:

Lesson objective

epigraphs:

"You can't learn math by watching your neighbor do it"
(poet Nivei)

"Learning should be fun ... To digest knowledge, you need to absorb it with appetite"
(Anatole France)

Equipment:

1. individual cards - tasks;
2. task cards for work in pairs;
3. visibility for oral work, for historical reference;
4. magnetic board

Repetition:

1. Common fractions
2. Geometric figures

During the classes

The ancient Greek poet Nivei argued that mathematics cannot be learned by watching a neighbor do it. Therefore, today we will work all actively, well and for the benefit of the mind.

I. "Star hour of ordinary fraction" - oral work

First tour

Second round "Logic chains"

Arrange in ascending order.

Third round.

The student made a mistake while applying the basic
fraction properties. Find the mistake!

Fourth round

Exploring a new topic

Consider the table of digits and answer the questions:

Questions:

1. How does the position of the unit in each next line change compared to the previous one?
2. How does this change its significance?
3. How does the value of the corresponding number change?
4. What arithmetic operation corresponds to this change?

Conclusion: by moving the unit one digit to the right, each time we reduced the corresponding number by 10 times and did this until we reached the last digit - the units digit.

Is it possible to reduce the unit by 10 times?
Of course,

Problem: But there is no place for this number in our tables of digits yet.

Think about how you need to change the table of digits so that you can write a number in it.

We argue, we need to shift the number 1 to the right by one digit.

Similarly:

Name the categories : tenths, hundredths, thousandths, ten thousandths, etc. integer part fractional part

2 units 3 tenths
2 units 3 hundredths

And in order to write numbers outside the table, we need to separate the integer part from the fractional part with some sign. We agreed to do this with a comma or a period. In our country, as a rule, a comma is used, and in the USA and some other countries - a period. The numbers are written and read as follows:

a) 2.3 or 2.3 (two point three or two, comma, three or two, dot, three)
b) 2.03 or 2.03 (two point three hundredths or two, comma, zero, three or two, dot, zero, three)

Rule: If a comma (or a period) is used in the decimal notation of a number, then they say that the number is written as a decimal fraction.

For brevity, the numbers are simply called decimal fractions.
Note that the decimal is not a new type of number, but a new way
number entries.

So, the motto of our lesson: “Have excellent knowledge on the topic“ Decimal Fractions ”

Lesson objective: prove that fractions cannot put us in a difficult position.

And now we will visit the "Historical Village"

Fractions appeared in ancient times. When dividing the booty, when measuring quantities, and in other similar cases, people met with the need to introduce fractions. Actions on fractions in the Middle Ages were considered the most difficult area of ​​\u200b\u200bmathematics. Until now, the Germans say about a person who is in a difficult situation, that he "fell into fractions." To make it easier to work with fractions, decimal fractions were invented. In Europe, they were introduced in 1585 by a Dutch mathematician and engineer. Simon Stevin. Here is how he depicted the fraction:

14,382, 14 0 3 1 8 2 2 3
In France, decimal fractions were introduced François Viet in 1579; his fraction record: 14.382, 14/382, 14
And we have outlined the doctrine of decimal fractions Leonty Filippovich Magnitsky in 1703 in the textbook of mathematics "Arithmetic, that is, the science of numerals"
Here are some other ways to represent decimals:
14. 3. 8. 2. ;

Charger(musical accompaniment)

II. Exercises

1. Recording the topic of the lesson.
2. The first table is to write down the numbers yourself.
3. The second table is to write down the numbers by digits.

III. change- is carried out in order to maintain a good mood, good spirits, mathematical attitude.

Anatole France once said: "Learning should be fun ... To digest knowledge, you need to absorb it with appetite"

Orally:

1. Vitya Verkhoglyadkin found the correct fraction, which is greater than 1, but keeps his "discovery a secret". Why?
2. Vitya Verkhoglyadkin made 11 circle diameters. Then he counted the number of radii drawn and got the number 21. Is his answer correct?
3. There was a detachment of soldiers: ten rows of seven soldiers in a row. How?

a) they were mustachioed.
How many mustachioed soldiers were there?
How many beardless soldiers were there?
b) they were nosed.
How many nosy soldiers were there?
How many snub-nosed soldiers were there?
Record: = 0.8; = 0.4

IV. Repetition - developmental exercises (work in pairs)

Lake Rebusnoye(Application)

V. Summary of the lesson.

Reflection.

What did you learn new for yourself?
- What did you find difficult?
- What have you learned?
- What was the problem in the lesson?
- Were we able to solve it?

Evaluation of their work (on leaflets, where the tables of ranks). Write how you learned the material of the lesson.

1. Got good knowledge.
2. Got all the material.
3. Learned the material partially.

VI. Homework. No. 38.1, 38.2 , Workbook (p. 28)