A common fraction (or mixed number) in which the denominator is one followed by one or more zeros (i.e. 10, 100, 1000, etc.):
can be written in a simpler form: without a denominator, separating the integer and fractional parts from each other with a comma (in this case, it is considered that the integer part of a proper fraction is equal to 0). First, the whole part is written, then a comma is placed, and after it the fractional part is written:
Common fractions (or mixed numbers) written in this form are called decimals.
Reading and writing decimals
Decimal fractions are written according to the same rules that are used to write natural numbers in the decimal number system. This means that in decimals, as in natural numbers, each digit expresses units that are ten times larger than the neighboring units to the right.
Consider the following entry:
The number 8 stands for prime units. The number 3 means units that are 10 times smaller than simple units, i.e. tenths. 4 means hundredths, 2 means thousandths, etc.
The numbers that appear to the right after the decimal point are called decimals.
Decimal fractions are read as follows: first the whole part is called, then the fractional part. When reading a whole part, it should always answer the question: how many whole units are there in the whole part? . The word whole (or integer) is added to the answer, depending on the number of whole units. For example, one integer, two integers, three integers, etc. When reading the fractional part, the number of shares is called and at the end they add the name of those shares with which the fractional part ends:
3.1 reads like this: three point one tenth.
2.017 reads like this: two point seventeen thousandths.
To better understand the rules for writing and reading decimal fractions, consider the table of digits and the examples of writing numbers given in it:
Please note that after the decimal point, there are as many digits after the decimal point as there are zeros in the denominator of the corresponding ordinary fraction:
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)
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)
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: Interdisciplinary communication 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.
| 1 team | 2nd team |
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.
| Reading | Number of digits after the decimal point | Writing as a decimal |
|
| 0 point 8 | |||
| 6 point 53 hundredths | |||
| 10 point 108 thousandths | |||
| 4 point 5 hundredths | |||
| 0 point 19 thousandths | |||
| 100 whole 1 thousandth | |||
| 14 point 305 ten thousandths | |||
| 0 point 6 ten thousandths | |||
| 0 whole 2147 hundred thousandths | |||
| 3 point 48 hundred thousandths | |||
| 1 whole 2 millionths |
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.
| A) | |||
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: