A decimal fraction must contain a comma. The numerical part of the fraction that is located to the left of the decimal point is called the whole part; to the right - fractional:
5.28 5 - integer part 28 - fractional part
The fractional part of a decimal consists 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 thousandths);
- millionths - 0.000001 (one millionth);
- ten millionths - 0.0000001 (one ten millionth);
- hundred millionths - 0.00000001 (one hundred millionths);
- billionths - 0.000000001 (one billionth), etc.
- read the number that makes up the whole 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 - one hundred point two thousand eight hundred thirty-four ten thousandths.
Writing Decimals
To write a decimal fraction:
- write down the whole part of the fraction and put a comma (the number meaning the whole 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 with zeros).
For example:
- twenty point nine - 20.9 - in this example everything is simple;
- five point one one hundredth - 5.01 - the word “hundredth” means that there should be two digits after the decimal point, but since the number 1 does not have a tenth place, it is replaced by zero;
- zero point eight hundred eight thousandths - 0.808;
- three point fifteen tenths - such a decimal fraction cannot be written down, because there was an error in the pronunciation of the fractional part - the number 15 contains two digits, and the word “tenths” implies only one. Correct would be three point fifteen hundredths (or thousandths, ten thousandths, etc.).
Comparison of decimals
Comparison of decimal fractions is carried out similarly to comparison of natural numbers.
- first, the whole parts of fractions are compared - the decimal fraction whose whole part is larger will be greater;
- if the whole parts of fractions are equal, compare the fractional parts bit by bit, from left to right, starting from the decimal point: tenths, hundredths, thousandths, etc. The comparison is carried out until the first discrepancy - the greater will be the decimal fraction which has a larger unequal digit in the corresponding digit of the fractional part. For example: 1,2 8 3 > 1,27 9, because in the hundredths place the first fraction has 8, and the second has 7.
Subject:
Target: introduce students to new numbers - decimal fractions, build knowledge and
Lesson type:
Equipment:
tasks.
View document contents
"Lesson summary on the topic "The concept of a decimal fraction. Reading and writing decimal fractions.""
Subject: The concept of a decimal fraction. Reading and writing decimals.
Target: introduce students to new numbers - decimal fractions, build knowledge and
mastery of mathematics methods; cultivate a culture of mathematical thinking.
Lesson type: lesson of learning new material.
Equipment: teacher's computer, screen, multimedia projector; on the tables: sheets with
tasks.
Lesson structure:
Organizing time.
Guys, today in class you must discover new knowledge, but, as you know, every new knowledge is related to what we have already learned. So let's start with a review.
Preparing to study new material.
Solve the anagram: fraction, angle, numerator, denominator.
Read the numbers in the table of digits.
From the numbers given, choose: natural numbers, proper fractions, improper fractions, mixed numbers.
Familiarization with new material.
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| Our lesson will be dedicated to The topic of the lesson is “The concept of a decimal fraction. Reading and writing decimals." Lesson motto: Have excellent knowledge on the topic “Decimal Fractions.” |
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| Let's remember how the decimal number system works. Let's look at the table of categories and answer the questions: Questions: Read the numbers written in the table. How does the position of the unit change in each subsequent line compared to the previous one? How does the value of the corresponding number change? What arithmetic operation corresponds to this change? Conclusion : moving the unit one digit to the right, each time we decreased the corresponding number by 10 times and did this until we reached the last digit - the units digit. Is it possible to reduce one by 10 times? Problem: But there is no place for this number in our table of digits yet. Think about how you need to change the table of digits so that you can write the number in it. |
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| We reason that we need to move the number 1 to the right by one place. But there are no digits to the right of the units digit, which means we need to add another column. Come up with a name for this column: tenths. Reasoning similarly: (hundredths) and: 10t. = (thousandths), etc. |
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| Since we reasoned correctly, we get the following table: |
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| 2 units 3 tenths. And in order to write numbers outside the table, we need to separate the whole part from the fractional part with some sign. We agreed to do this using a comma or period. In our country, as a rule, a comma is used, and in the USA and some other countries, a period is used. We read the numbers as follows: a) 2,3 or 2.3 (two point three or two, comma, three or two, point, three) |
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| You and I have made a discovery. And this discovery is the rule for reading and writing decimal fractions. It coincided with the rule proposed by the author of the textbook. Rule: If a comma (or period) is used in the decimal notation of a number, then the number is said to be written as a decimal fraction. For brevity, numbers are simply called decimals. |
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| In science and industry, in agriculture, decimal fractions are used much more often than ordinary fractions. This is due to the simplicity of the rules for calculations with decimal fractions and their similarity to the rules for operations with natural numbers. 1703 - In Russia, the doctrine of decimal fractions was presented by Leonty Filippovich Magnitsky in the textbook “Arithmetic, that is, the science of numerals.” |
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| We have every reason to complete tasks on the topic of the lesson. First task. Read the number |
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| Read decimals What can you say about these three numbers? (they are equal) What can you conclude about the zeros that end a decimal? (you don’t have to write them, they don’t change the number) You can add zeros at the end of a decimal fraction or discard zeros, but this will not change the decimal fraction. The same fraction is written. |
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| A comma is placed between the whole and fractional parts. If there is no fractional digit, we replace it with 0 when writing the number. The number of digits after the decimal point must be equal to the number of zeros in the denominator of the common fraction. Write in decimal fraction: |
| Write decimal fractions from dictation. 7 point 8 2 point 25 hundredths 0 whole 92 hundredths 12 point 3 hundredths 5 point 187 thousandths 24 whole 24 thousandths 7 point 7 7 point 7 hundredths 7 point 7 thousandths 0 point 5 ten thousandths Now we are doing independent work, during which we will test our knowledge on the topic of the lesson. |
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| Independent work (5 minutes) |
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| Check yourself: Write as a decimal fraction (on a line); Check the answers in the table, putting the corresponding letter for each number (under each number without punctuation) What word did you get? WELL DONE |
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| Reflection |
| Homework: No. 647 a), 648 av), 649 a), 650 c) |
Numbers
Mixed numbers
Natural
Improper fractions
Proper fractions
NAME THE NATURAL NUMBERS
NAME mixed NUMBERS
NAME common fractions
What numbers are left?
FRACTIONAL NUMBERS
DECIMAL RECORDING.
DECIMALS.
TODAY'S LESSON TOPIC:
Decimal fractions. Reading and writing decimal fractions.
THE PURPOSE OF THE LESSON:
Introduce the concept of decimal fractions. Learn to read and write decimals Learn to translate common fractions with denominators 10, 100, 1000, etc. to decimal and vice versa Develop logical thinking in a new situation Foster independence and responsibility for one’s own activities.
Fractions
Ordinary
Decimals, fractions
Decimal fractions.
RECORDING
READING
Decimal
ACTIONS
WITH DECIMALS
COMPARE
If a comma is used in the decimal notation of a number, the number is said to be written as a decimal fraction.
Numbers with a denominator 10; 100; 1000, etc. agreed to write without a denominator
MATHEMATICAL DICCTATION
WRITE OUT THE NUMBERS
- THREE POINT SEVEN
- SIX POINT ONE HUNDREDTH
- FIVE POINT FOUR THOUSANDTHS
MATHEMATICAL DICCTATION
WRITE OUT THE NUMBERS
First write the whole part, and then the numerator of the fractional part
The integer part is separated from the fractional part by a comma
Numbers with denominators 10, 100, 1000, etc.
agreed to write without a denominator
After the decimal point, the numerator of the fractional part must have as many digits as there are zeros in the denominator
ALGORITHM
1. WRITE THE WHOLE PART OF A NUMBER
2. PUT A COMMA
3. AFTER THE decimal place put as many dots as there are zeros in the denominator
4. FROM THE LAST POINT WE WRITE THE NUMERATOR
5. REPLACE THE REMAINING POINTS WITH ZEROS
Decimal fractions consist of an integer part and a fraction
Integer digits
Fractional digits
thousandths
ten thousandths
hundred thousandths
millionths
3
4
5
2
3
4
5
2
4
5
0
2
FIVE POINT THREE
TWENTY-ONE POINT SEVEN
THREE POINT SEVEN
TWO POINT HUNDRED FIFTY-SIX THOUSANDTHS
SEVEN POINT TWENTY NINE HUNDREDTHS
SIX POINT ONE HUNDREDTH
FIVE POINT FOUR THOUSANDTHS
NINE point eight
= 9,0008
FIND AND WRITE THE MISSING NUMBERS
The origin and development of decimal fractions
Uzbekistan, XV century
Europe, 16th century
Russia, XVIII century
Ancient China, 2nd century BC.
The origin and development of decimal fractions in China was closely related to metrology (the study of measures). Already in the 2nd century BC. there was a decimal system of length measures.
IN 1427 year, mathematician
and astronomer from Uzbekistan ,
Al-Kashi wrote a book
"The Key to Arithmetic"
in which he formulated
basic
rules of action
with decimals
Uzbekistan, XV century
EUROPE,
century
IN 1579 year, decimal fractions are used in the “Canon of Mathematics” by the French mathematician François Vieta (1540-1603), published in Paris.
Wide
decimal propagation
in Europe began only after the publication of the book “The Tenth” by the Flemish mathematician Simone Stevina (1548-1620 ). He is considered the inventor of decimal fractions.
Russia, XVIII century
IN Russia first
systematic information
about decimals
found in Arithmetic
L.F. Magnitsky (1703)
2,135436
2 | 135436
Uzbekistan
France
Russia
Europe
1 cun,
3 beats,
5 serial,
4 hairs,
3 thinnest,
6 cobwebs
2,135436
China
2 135436
2 0 1 1 3 2 5 3 4 4 3 5 6 6
Are you probably tired?
Well, then everyone stood up together.
We stretch our arms, shoulders,
To make it easier for us to sit.
And don’t get tired at all.
check
Write the following fractions as decimals:
Write the following fractions as fractions or mixed numbers:
Summarize:
- What fraction can be used to replace an ordinary fraction, the denominator of the fractional part of which is expressed unit with one or several zeros?
- What separates the whole part of a decimal fraction from
fractional part?
- If the fraction is correct, then what is written before
do they write with a comma?
- How many decimal places should there be after the decimal point?
decimal notation?
Homework
clause 7.1;
answer the questions
№ 1211,№1212
(on repeat No. 1216)
Sections: Mathematics
Subject: The concept of decimal fraction. Reading and writing decimals.
Goals:
- Formation of knowledge and skills to write and read decimal fractions. Introduce students to new numbers - decimals (a new way of writing numbers)
- Develop intuition, conjecture, erudition and mastery of mathematical methods.
- Arouse mathematical curiosity and initiative, develop a sustainable interest in mathematics.
- Foster a culture of mathematical thinking.
Developmental goal: Formation of skills of self-assessment and self-analysis of educational activities.
Problem-based - developmental lesson (combined)
Stages:
1) problematic situation;
2) problem;
3) searching for ways to solve it;
4) problem solving
Lesson motto:
Lesson Objective
Epigraphs:
“You can’t learn math by watching your neighbor do it.”
(poet Nivey)
“You have to have fun learning... To digest knowledge, you have to absorb it with appetite”
(Anatole France)
Equipment:
- individual cards - tasks;
- task cards for working in pairs;
- visibility for oral work, for historical reference;
- magnetic board
Repetition:
- Common fractions
- Geometric figures
During the classes
The ancient Greek poet Niveus argued that mathematics cannot be learned by watching your neighbor do it. Therefore, today we will all work actively, well and with benefit to the mind.
I. “The Finest Hour of the Common Fraction” - oral work
First tour
| 1 | |||||||
Second round “Logical chains”
Arrange in ascending order.
Third round.
The student made a mistake when applying the basic
properties of fractions. Find the mistake!
Fourth round
Learning a new topic
Let's look at the table of categories and answer the questions:
| Class of thousands |
Unit class |
||||
Questions:
- How does the position of the unit change in each subsequent line compared to the previous one?
- How does this change its significance?
- How does the value of the corresponding number change?
- What arithmetic operation corresponds to this change?
Conclusion: by moving the unit one digit to the right, each time we decreased the corresponding number by 10 times and did this until we reached the last digit - the units digit.
Is it possible to reduce one by 10 times?
Certainly,
Problem: But there is no place for this number in our tables of ranks yet.
Think about how you need to change the table of digits so that you can write the number in it.
We reason that we need to move the number 1 to the right by one place.
Likewise:
Give names to the categories : tenths, hundredths, thousandths, ten-thousandths, etc. integer part fractional part
| hundreds | thousandths |
||||
2 units 3 tenths
2 units 3 hundredths
And in order to write numbers outside the table, we need to separate the whole part from the fractional part with some sign. We agreed to do this using a comma or period. In our country, as a rule, a comma is used, and in the USA and some other countries, a period is used. We write and read the numbers as follows:
a) 2.3 or 2.3 (two point three or two, comma, three or two, point, 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 period) is used in the decimal notation of a number, then the number is said to be written as a decimal fraction.
For brevity, the numbers are simply called in decimal fractions.
Note that the decimal fraction is not a new type of number, but a new way
recording numbers.
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.
Now let’s visit the “Historical Village”
Fractions appeared in ancient times. When dividing up spoils, when measuring quantities, and in other similar cases, people encountered the need to introduce fractions. Operations with fractions in the Middle Ages were considered the most difficult area of mathematics. To this day, the Germans say about a person who finds himself in a difficult situation that he “fell into fractions.” To make working with fractions easier, decimals were invented. They were introduced into Europe in 1585 by a Dutch mathematician and engineer. Simon Stevin. Here's how he represented the fraction:
14,382, 14 0 3 1 8 2 2 3
In France, decimal fractions were introduced Francois Viet in 1579; his fraction notation: 14.382, 14/382, 14
And we have expounded the doctrine of decimal fractions Leonty Filippovich Magnitsky in 1703 in the mathematics textbook “Arithmetic, that is, the science of numbers”
Here are some other ways to represent decimals:
14. 3. 8. 2. ;
Charger(musical accompaniment)
II. Exercises
- Record the topic of the lesson.
- The first table is to write down the numbers yourself.
- The second table is to write down the numbers by digit.
III. Recess– is carried out in order to maintain a good mood, good spirits, and a mathematical attitude.
Anatole France once said: “You have to have fun learning...To digest knowledge, you have to absorb it with appetite”
Orally:
- Vitya Verkhoglyadkin found the correct fraction, which is greater than 1, but keeps his “discovery” secret. Why?
- Vitya Verkhoglyadkin drew 11 diameters of a circle. Then he counted the number of radii drawn and got the number 21. Is his answer correct?
- A detachment of soldiers was walking: ten rows of seven soldiers in a row. How many?
a) they were mustachioed.
How many mustachioed soldiers were there?
How many mustacheless soldiers were there?
b) they were big-nosed.
How many big-nosed soldiers were there?
How many snub-nosed soldiers were there?
Write: = 0.8; = 0.4
IV. Repetition - developmental exercises (work in pairs)
Lake Rebusnoe(Application)
V. Lesson summary.
Reflection.
What new things have you learned?
- What did you find difficult?
- What have you learned?
- What problem was posed in class?
- Did we manage to solve it?
Evaluation of your work (on pieces of paper with tables of ranks). Write how you learned the lesson material.
- Got good knowledge.
- I mastered all the material.
- I partially understood the material.
VI. Homework. No. 38.1, 38.2, Workbook (page 28)
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".
| Lesson stage | Teacher activities | Student activity |
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| 1. Org. moment Determining needs and motives. 1 min | Hello guys! I would like to start the lesson with the words of the famous German poet and thinker I. Goethe: « Numbers (numbers) do not rule the world, but they show how the world is ruled." And today we will also plunge into the world of numbers and numbers. | Greeting students; checking the class's readiness for the lesson; organization of attention. | Greetings from teachers |
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| 2. Setting goals and objectives, updating knowledge | Guys, raise your hands who has ever seen recordings like: 3.5 and 1.56 Guys, where did you find these records? These entries represent fractions. The name of these fractions is encrypted. Let's formulate the topic and purpose of the lesson together. Today we are starting to study a very important, interesting and new topic for you. What interesting and new things would you like to know about decimal fractions? Today in class we will learn to write fractions in a new way. Write down the topic of the lesson “Decimal notation of fractional numbers” (slide ) . Read the fractions. What two groups can they be divided into? But the new notation can not be applied to all ordinary fractions. Who guessed which ones? | Asking questions. Offers to answer questions. | The guys solve the puzzle. Students formulate the topic of the lesson. Determine the objectives of the lesson. Write down the topic of the lesson. Read fractions. -All fractions have one and zero in the denominator. -Right and wrong |
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| 3. Learning new material | How can I write fractions differently? Look at the table ( slide ).
So, the problem was how to write ordinary fractions and mixed numbers in a new way. | Let's look at how to write a mixed number as a decimal fraction: (write in a notebook) From the examples considered, we will draw a conclusion and obtain the rule What pattern did you notice? A. 0.037 A. 3.5216 Create an algorithm for converting ordinary fractions to decimals. | the number of zeros is the same as the number of digits after the decimal point Students create an algorithm for converting fractions to decimals. |
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| 4. Physical education minute | http://videouroki.net/ | ||||||||||||||||||||||||||||||
| 5.Primary consolidation, pronunciation in external speech | In Russia, for the first time, decimal fractions were discussed in the Russian mathematics textbook - “Arithmetic”. We can find out its author if we write fractions and mixed numbers as decimals. (Mixed numbers are written on the board, and decimals are written on cards with a letter on the back. As students complete the task, they form a word.) (M) Doing exercises according to the textbook: 1117, 1120 | Primary consolidation is carried out through commenting on each sought-after situation, speaking out loud the established algorithm of action (what I’m doing, why, what’s going on, what’s happening | Students receive the word " MAGNITSKY" |
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| 6.Independent work. Standard check. | 1. Work in a notebook(on one's own). Write down the correct fractions in your notebook (in a column). Replace them with decimals. Examination (slide ) Now write out the improper fractions and replace them with decimals. Examination (slide ) | ||||||||||||||||||||||||||||||
| 7. Evaluation of the lesson results. Summing up the lesson (reflection). | What topic did we study today? What tasks did we set today? Are our tasks completed? | Answer questions. |
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| 8. Information about homework. | Homework. Find information (articles, some other data in any periodical literature) that contains decimal fractions. Execute No. 1139.1144 (a) Study paragraph 30 | Students write down homework depending on the level of mastery of the lesson topic |
