Reading decimals. Writing and reading decimals

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

Subject: The concept of a decimal fraction. Reading and writing decimals.

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

Tasks:

Organize the work of students to study and initially consolidate the concept of “decimal fraction” and the algorithm for writing decimal fractions.

Create conditions for the formation of UUD:

Communicative UUD: listening skills, discipline, independent thinking.

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

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

Lesson type: lesson on 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:

They independently identify the problem and solve it;

Independently determine the topic and goals of the lesson;

Derive a rule;

Work with the textbook text;

Answer questions;

Solve problems independently;

Evaluate themselves and each other;

They reflect.

Teaching methods: verbal, visual - illustrative, practical

Resources: multimedia projector, presentation.

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

A decimal fraction differs from an ordinary fraction in that its denominator is a place value.

For example:

Decimal fractions are separated from ordinary fractions into a separate form, which led to their own rules for comparing, adding, subtracting, multiplying and dividing these fractions. In principle, you can work with decimal fractions using the rules of ordinary fractions. Own rules for converting decimal fractions simplify calculations, and rules for converting ordinary fractions to decimals, and vice versa, serve as a link between these types of fractions.

Writing and reading decimal fractions allows you to write them down, compare them, and perform operations on them according to rules very similar to the rules for operations with natural numbers.

The system of decimal fractions and operations on them was first outlined in the 15th century. Samarkand mathematician and astronomer Dzhemshid ibn-Masudal-Kashi in the book “The Key to the Art of Counting”.

The whole part of the decimal fraction is separated from the fractional part by a comma; in some countries (the USA) they put a period. If a decimal fraction does not have an integer part, then the number 0 is placed before the decimal point.

You can add any number of zeros to the fractional part of a decimal on the right; this does not change the value of the fraction. The fractional part of a decimal is read at the last significant digit.

For example:
0.3 - three tenths
0.75 - seventy-five hundredths
0.000005 - five millionths.

Reading the whole part of a decimal is the same as reading natural numbers.

For example:
27.5 - twenty seven...;
1.57 - one...

After the whole part of the decimal fraction the word “whole” is pronounced.

For example:
10.7 - ten point seven

0.67 - zero point sixty-seven hundredths.

Decimal places are the digits of the fractional part. The fractional part is not read by digits (unlike natural numbers), but as a whole, therefore the fractional part of a decimal fraction is determined by the last significant digit on the right. The place system of the fractional part of the decimal is somewhat different than that of natural numbers.

• 1st digit after busy - tenths digit
• 2nd decimal place - hundredths place
• 3rd decimal place - thousandths place
• 4th decimal place - ten-thousandth place
• 5th decimal place - hundred thousandths place
• 6th decimal place - millionth place
• The 7th decimal place is the ten-millionth place
• The 8th decimal place is the hundred millionth place

The first three digits are most often used in calculations. The large digit capacity of the fractional part of decimals is used only in specific branches of knowledge where infinitesimal quantities are calculated.

Converting a decimal to a mixed fraction consists of the following: the number before the decimal point is written as an integer part of the mixed fraction; the number after the decimal point is the numerator of its fractional part, and in the denominator of the fractional part write a unit with as many zeros as there are digits after the decimal point.

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 of fractional numbers

The so-called decimal notation of 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 needed to separate the whole part from the fractional part. As a rule, the last digit of a decimal fraction is not a zero, unless the decimal point appears immediately after the first zero.

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

In some textbooks you can find the use of a period 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 represent 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 contains 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 discussed in a separate material.

How to read decimals correctly

There are some rules for reading decimal notations. Thus, those decimal fractions that correspond to their regular ordinary equivalents are read almost the same way, but with the addition of the words “zero tenths” at the beginning. Thus, 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 the fraction 56, 002, which corresponds to 56 2 1000, we read this entry as “fifty-six point two thousandths.”

The meaning of a digit in a decimal fraction depends on where it is located (the same as in the case of natural numbers). So, in the decimal fraction 0.7, seven is tenths, in 0.0007 it is ten thousandths, and in the fraction 70,000.345 it means seven tens of thousands of whole units. Thus, in decimal fractions there is also the concept of place value.

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

Let's look at an example.

Example 1

We have the decimal fraction 43,098. She has a four in the tens place, a three in the units place, a zero in the tenths place, 9 in the hundredths place, and 8 in the thousandths place.

It is customary to distinguish the ranks of decimal fractions by precedence. If we move through the numbers from left to right, then we will go from the most significant to the least significant. It turns out that hundreds are older than tens, and parts per million are younger than hundredths. If we take that final decimal fraction that we cited as an example above, then the highest, or highest, place in it will be the hundreds place, and the lowest, or lowest, place will be the 10-thousandth place.

Any decimal fraction can be expanded into individual digits, that is, presented as a sum. This action 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 get:

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 finite decimals. This means that the number of digits after the decimal point is finite. Let's derive the definition:

Definition 1

Trailing decimals are a type of decimal fraction that has a finite number of decimal places after the decimal sign.

Examples of such fractions can be 0, 367, 3, 7, 55, 102567958, 231 032, 49, etc.

Any of these fractions can be converted either to a mixed number (if the value of their fractional part is different from zero) or to an ordinary fraction (if the integer part is zero). We have devoted a separate article to how this is done. Here we’ll just point out a couple of examples: for example, we can reduce 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 a common fraction in decimal form may not always be possible. So, 5 13 cannot be replaced by an equal fraction with the denominator 100, 10, etc., which means that a final decimal fraction cannot be obtained from it.

Main types of infinite decimal fractions: periodic and non-periodic fractions

We indicated above that finite fractions are so called 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 decimal fractions are those that have an infinite number of digits after the decimal point.

Obviously, such numbers simply cannot be written down in full, so we indicate only part of them and then add an ellipsis. This sign indicates an infinite continuation of the sequence of decimal places. Examples of infinite decimal fractions include 0, 143346732…, ​​3, 1415989032…, 153, 0245005…, 2, 66666666666…, 69, 748768152…. etc.

The “tail” of such a fraction may contain not only seemingly random sequences of numbers, but also a constant repetition of the same character or group of characters. Fractions with alternating numbers after the decimal point are called periodic.

Definition 3

Periodic decimal fractions are those 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 that can be left in the notation of a periodic fraction? For periodic fractions, it will be enough to write the entire period once in parentheses. So, fraction 3, 444444…. It would be correct to write it as 3, (4), and 76, 134134134134... – as 76, (134).

In general, entries with several 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. Records of the form 0, 67777 (7), 0, 67 (7777), etc. are also acceptable.

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

That is, for the above fraction, we will consider the main entry to be 0, 6 (7), 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, they will result in infinite fractions.

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. What does it look like in recording? Let's say we have the 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 results in a fraction equal to it.

Special attention should be paid to 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. In this case, one is added to the value of the next digit, and (0) is indicated in parentheses. The equality of the resulting numbers can be easily verified by representing them as ordinary fractions.

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

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

There are also fractions that do not have an endlessly 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 need to be very careful with such numbers.

Non-periodic fractions are classified as 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 look at each of them separately.

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

To add some decimal fractions with others, 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 need to first round them 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, pre-rounding is also necessary.

Finding the difference between decimal fractions is the inverse of addition. Essentially, using subtraction we can find a number whose sum with the fraction we are subtracting will give us the fraction we are minimizing. We will talk about this in more detail in a separate article.

Multiplying decimal fractions is done in the same way as for natural numbers. The column calculation method 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 calculations.

The process of dividing decimals is the inverse of multiplying. When solving problems, we also use columnar calculations.

You can establish an exact correspondence between the final decimal fraction 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, but decimal fractions can be reduced to this form. For example, the common fraction 14 10 is the same as 1, 4, so the corresponding point 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, but use the method of expansion by digits as a basis. So, if we need to mark a point whose coordinate will be equal to 15, 4008, then we will first present this number as the sum 15 + 0, 4 +, 0008. To begin with, let’s set aside 15 whole unit segments in the positive direction from the beginning of the countdown, then 4 tenths of one segment, and then 8 ten-thousandths of one segment. As a result, we get a coordinate point that corresponds to the fraction 15, 4008.

For an infinite decimal fraction, it is better to use this method, since it allows you to get as close as you like to the desired point. In some cases, it is possible to construct an exact correspondence to 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, distant 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 a segment. Let's see how to do this correctly.

Let's say 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 postpone unit segments from the origin until we get to the desired point. After whole segments, if necessary, we measure tenths, hundredths and smaller fractions so that the match is as accurate as possible. As a result, we received a decimal fraction that corresponds to a given point on the coordinate axis.

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

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

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Math lesson 5th grade

Subject: Reading and writing decimals

Lesson objectives: Secondary comprehension of already known knowledge, development of skills and abilities for their application. Through work in a group on a problem task, students will learn to convert an ordinary fraction into a decimal fraction, strengthen the skills of reading and writing decimal fractions, speaking skills through the ability to name the digits of a decimal fraction, will explain, which fractions can be converted to final decimals and which cannot.

Language goals: Understand and explain, using mathematical terminology and in your own words, which common fraction can be converted to a decimal fraction, name the decimal places.

Subject vocabulary and terminology: Decimal fraction - decimal fraction, comma - decimal point.

Decimal places, common fraction, place unit, numerator, denominator.

Fractional places: tenths, hundredths, thousandths, etc.;

Integer digits: units, tens, hundreds, etc.

A series of useful phrases for dialogue/writing:

A decimal is another notation for a fraction

To write this fraction as a decimal, you need...

The integer part is separated from the fractional part by a comma

The fraction is read: ... whole, ... (tenths, hundredths, etc.)

Educational and developmental aspect of the lesson: Develop computational skills, mathematical speech, attention, thinking; develop ethical and aesthetic standards of behavior in the classroom, a sense of responsibility through self and mutual assessment.

Lesson type: Lesson to consolidate knowledge.

Students' knowledge at the exit: Students will:

be able to name the places of a decimal fraction;

be able to convert fractions to decimals in two ways;

understand which fractions can be converted to final decimals and which cannot;

Use a microcalculator to convert fractions to decimals.

Instilling values: The inculcation of values ​​- honesty, responsibility, respect - is carried out through work in a group and through self- and mutual assessment, global citizenship through an excursion into the history of the development of the concept of a decimal fraction, familiarity with modern ways of writing decimal fractions.

Interdisciplinary connections: An interdisciplinary connection with the Russian language is possible through the development of speaking using reading decimals and expressions with decimals. Interdisciplinary integration in the lesson is realized through activities, through reading decimals and watching videos.

Prior knowledge: Common fractions, proper/improper fractions, connection between division and fractions, basic properties of fractions, mixed numbers, digits of natural numbers.

During the classes:

Organizing time. (5 minutes)

Division into 2 teams. Method "Assemble a picture". Students find their pieces and make a picture. (Can be divided into more groups, depending on the size of the class)

Picture for the first team:

Picture for the second team:

On the reverse side of the picture there is a proposed task. Teams need to solve a problem.

Task for 1 team: Before hibernation, the bear accumulated fat and began to weigh 250 kg. Over the winter he will lose his weight. How many kilograms will a bear weigh after hibernation?

Task for 1 team: The mouse family has prepared 70 kg of grain for the winter. During the winter they will eat the reserves. How many kilograms of grain will remain after wintering?

The answer is checked against the answer prepared by the teacher in the same picture.

Updating basic knowledge and correcting it. (5 minutes)

Relay game: “Who is faster?”

Students come out one at a time from each team and write a fraction or mixed number as a decimal.

Determining the boundaries (possibilities) of applying knowledge.

We consolidate the algorithms. Exercises according to the model and in similar conditions in order to develop the skills of error-free application of knowledge.

1 . Working with cards in a team. Create a single solution on the cluster:

Option 1 (for 1 team)

3, 12, 7, 14, , , 2

Write numbers as decimals

a) 5 point 7; b) 0 point 3; c) 14 point 4 hundredths; d) 0 point 72 thousandths.

Option 2 (for 2nd team)

Write numbers as decimals

5, 7, 7, 5, 2, , ,

Write numbers as decimals

a) 3 point 7; b) 0 point 11; c) 12 point 4 hundredths; d) 8 point 27 thousandths.

How many digits after the decimal point are there in the decimal notation of a fraction?

They exchange cards and convey their decisions. A mutual check is underway.

2 . Fill the table. With subsequent mutual verification.

Dictation. Self-check and team check.

a) 3 point 3; b) 15 point 55 hundredths; c) 0 point 67 hundredths;

d) 5 point 404 thousandths; e) 87 point 1 hundredth; f) 72 point 12 thousandths;

g) 6 point 62 thousandths; h) 2 whole 2 hundredths; i) 0 point 2 hundredths.

Working with models. Mutual verification in the team and teams

Given a square. Color in the indicated part of this square.

What part of the square is shaded? Express your answer first as a decimal fraction and then as a common fraction. Paint the same part of the adjacent square in some other way.

Problem task.

“How do you write a fraction as a decimal?” 1 minute to think.

After 1 minute, lead students to the first method based on the value of the fractional line - division.

1 way: Divide 1 into 2 with a corner. (You can use the video resource “Converting fractions to decimals”

Examples for consolidation. Students perform in groups and check the sample answer of one of the commands.

Write as a decimal:

Lead students to this method, relying on the basic property of a fraction and lead students to the need to reduce to a new denominator, a digit unit. First, pay attention to the component multipliers of the bit units.

Method 2: multiply the denominator by such a number that in the denominator the smallest possible product is a digit unit - 10, 100,1000 ...

or .

Convert to decimal fraction and fill out the table:

Lesson in 5th grade, teacher-Shabarshova Ekaterina Anatolyevna.

Lesson topic: Decimal fractions. Reading and writing decimals.

Lesson objectives:

Create conditions for students to study and repeat this topic;

Development of memory, logic, mathematical thinking;

Cultivating interest in the subject.

The purpose of the lesson:

Repeat writing and reading decimal fractions;

converting a decimal fraction into a common fraction and vice versa, a common fraction into a decimal.

Lesson type: combined;

Teaching method : verbal, practical, visual.

Form of organization : collective, individual;

Contents of the activity : historical information, survey using signal cards (orally), solving tasks from the textbook, oral calculation “Find a pair”, independent work.

Equipment :signal cards, stickers for reflection, cards for self-assessment, cards with tasks for independent work.

Lesson Plan :

Organizing time. Emotional mood.

Updating knowledge. Historical reference.

Oral counting “Find a pair.”

Working from the textbook

Independent work.

Student assessment.

Reflection.

Homework.

During the classes:

Organizing time.

Hello guys! Let's greet each other! Turn to face each other and smile.

Well done! And it is on this pleasant note that we begin our lesson today!

Intentional division into groups according to the individual characteristics of students.

Write the date in your notebook, great job. I would like to draw your attention to the handouts on your desks, we will put the stickers aside for now, and the assessment sheets will be useful to you from the first task, as soon as we complete the next task, you must make a self-assessment in the sheets when completing this task.

Updating knowledge.

Guys, in the last lessons we started studying the topic “Decimal fraction. Reading and writing decimals." But you and I began to study the topic without knowing its history; a student in our class, Anatoly Shabarshov, who prepared a historical background for us, will help us with this.

Historical reference.

The concept of an abstract decimal fraction first appeared in the 15th century. It was introduced by the eminent mathematician and astronomer Al-Cauchy (fullname Jemiad ibn – Masud al – Qoshi ) at work"The Key to Arithmetic" (1427) . Al-Cauchy's discovery in Europe became known only 300 years later.

Knowing nothing about Al-Cauchy's discovery, decimal fractions were discovered for the second time, approximately 150 years after him, by the Flemish scientist mathematician and engineerSimon Stevin in labor "Decimal" (1585).

In Russia, the doctrine of decimal fractions was first givenL.P. Magnitsky in his "Arithmetic" - the first Russian mathematics textbook.(1703 g)

It was proposed in different ways to separate the whole part from the fractional part. Al-Koshi wrote the whole and fractional parts in one row, although he wrote them in different inks, or put a vertical line between them. S. Stevin, to separate the whole part from the fractional part, put a zero in the circle. The comma adopted in our time was proposed by a German astronomerJ. Kepler (1571 – 1630).

Now let's remember some rules and properties of decimal fractions.

The rules are very simple, if you agree with the statement, then raise the red signal card, if not, then raise the blue one. Let's begin!

To write decimal fractions, a fraction bar is used; (no)

A comma is used to write decimal fractions; (yes)

The whole part of the fraction is before the decimal point; (yes)

If you remove the zeros at the end of a decimal fraction, the value of the fraction will change; (no)

The places after the decimal point are called decimal places. (Yes).

2.Well done! Now open your textbooks on page 197, No. 942. (work at the blackboard)

Oral counting “Find a pair”

0,1

0,5

0,2

0,75

0,04

0,05

Work according to the textbook.

№936 (1) – task of the first difficulty level

№951 (1.2) – task of the second level of difficulty

№956(1-3) – task of the third level of difficulty

The tasks are based on the individual characteristics of all group members

Independent work.

Option 1

Write as a decimal

; ; ;

Option 2

Write the quotient as a fraction and convert it to a decimal

5: 100; 5749:100; 34:1000; 324:10.

Option 3

Reduce mixed numbers to a denominator of 100 and write the corresponding decimals

Tasks in independent work are compiled taking into account the individual characteristics of students. Options correspond to difficulty levels.

Student assessment.

Students give themselves grades for the lesson on assessment sheets and submit them to the teacher.

Reflection.

Well done guys, everyone did a good job today, so let's sum it up:

What new did you learn in class today?

What knowledge and skills did you reinforce in class today?

Did you like the lesson?

Stickers are on the table, students write down their attitude to the lesson and paste them on the prepared bulletin board.

Homework

№950,№945

APPLICATIONS

Task No.

Great

Fine

Could have done better

Overall grade for the lesson:

Student evaluation sheet:__________________________________________________________

Task No.

Great

Fine

Could have done better