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Presentation on the topic: magical decimals

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On the most ordinary day after school, two best friends, fifth grade students Anna and Tanya did homework mathematics. They opened the textbook and saw decimal fractions... On an ordinary day after school, two best friends, fifth-grade students Anna and Tanya were doing their homework in mathematics. They opened the textbook and saw decimal fractions... I don't understand anything! What's happened? These ... like them ... but ... decimal fractions. We didn't pass them! Tanya was outraged. Solve the problem with decimal fractions - Anna reads. - In the spring, they sowed 0.9 fields, and harvested only 0.6 fields. How much crop was not harvested from the field?

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All the same, they sowed 0 or 9? Tanya asked. All the same, they sowed 0 or 9? Tanya asked. Maybe add 9 to 0? Anna suggested. No, we should probably choose 0 or 9 ourselves! Anna agreed. And just as the girls wanted to write it down, the textbooks began to dance and sing: We really need decimal fractions. What is a crooked letter? Or is it a comma? But what does the comma have to do with it, Maya the fairy will tell us!

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Decimal fractions appeared in the works of Arab mathematicians in the Middle Ages and independently in ancient China. But even earlier, in ancient Babylon, fractions of the same type were used, but of course sexagesimal. Decimal fractions appeared in the works of Arab mathematicians in the Middle Ages and independently in ancient China. But even earlier, in ancient Babylon, fractions of the same type were used, but of course sexagesimal. Later, the scientist Hartmann Beyer (1563-1625) published the essay “Decimal Logistics” where he wrote: “... I noticed that technicians and artisans, when measuring any length, very rarely and only in exceptional cases express it in whole numbers of the same name; usually they have to either take small measures, or turn to fractions, in the same way astronomers measure quantities not only in degrees, but also in fractions of a degree, i.e. minutes, seconds, etc., but it seems to me that dividing them into 60 parts is not as convenient as dividing by 10, into 100 parts, etc., because in the latter case it is much easier to add, subtract and generally perform arithmetic operations ; It seems to me that decimal parts, if introduced instead of sexagesimal, would be useful not only for astronomy, but also for all kinds of calculations. Simon Stevin introduced decimal fractions into European practice. Until then, anyone who dealt with non-integer numbers had to fiddle with numerators and denominators.

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Today we use decimals naturally and freely. However, what seems natural to us served as a real stumbling block for the scientists of the Middle Ages. IN Western Europe 16th century along with the widespread decimal system for representing integers, sexagesimal fractions were used everywhere in calculations, dating back to the ancient tradition of the Babylonians. It took the bright mind of the Dutch mathematician Simon Stevin to bring the record of both integer and fractional numbers into a single system. Apparently, the impetus for the creation of decimal fractions was the tables of compound interest compiled by him. In 1585 he published the book Tithing, in which he explained decimal fractions. Stevin's notation was not perfect, just like the notation of his colleagues and followers. This is how they would write the number 3.1415: Today we use decimals naturally and freely. However, what seems natural to us served as a real stumbling block for the scientists of the Middle Ages. Western Europe in the 16th century along with the widespread decimal system for representing integers, sexagesimal fractions were used everywhere in calculations, dating back to the ancient tradition of the Babylonians. It took the bright mind of the Dutch mathematician Simon Stevin to bring the record of both integer and fractional numbers into a single system. Apparently, the impetus for the creation of decimal fractions was the tables of compound interest compiled by him. In 1585 he published the book Tithing, in which he explained decimal fractions. Stevin's notation was not perfect, just like the notation of his colleagues and followers. This is how they would write the number 3.1415:

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We have heard a lot about air. Air is 99.96% composed of three gases: nitrogen, oxygen and argon. Carbon dioxide contains 0.03%, the rest accounts for 0.01%. We have heard a lot about air. Air is 99.96% composed of three gases: nitrogen, oxygen and argon. Carbon dioxide contains 0.03%, the rest accounts for 0.01%.

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Of great importance for the knowledge of the world is the problem of the numerical ratio between the atoms of various elements. Of great importance for the knowledge of the world is the problem of the numerical ratio between the atoms of various elements. If we compare the iron, cobalt and nickel available on the whole Earth, it turns out that Earth consists of: Iron 92% Cobalt 0.5% Nickel 7.5% Finest chemical analyzes a huge number of meteorites that fell to Earth gave remarkable results. It turned out that in iron meteorites percentage iron, cobalt and nickel amazingly coincides with their content on our planet.

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You can tell me a lot, You can tell me a lot, About what decimal fractions are, About what you can at the end of the fractional part, To the right, discard or insert zeros. Well, how to compare them, you tell me. Well, it's certainly easier than ever. Compare the whole parts of the decimal fraction, And the one that has more of it, Of course, there will be more. Well, if those parts are just equal, Then what should I do, you tell me. If two decimal fractions have equal integer parts, You look at the first of the mismatched digits, And the one with the larger one, of course, will also have the larger one. Do you remember everything, you tell me?

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Vasya found sunken treasures in the river and brought them home. He decided to sell them to a rich man. But the rich man deceived him for 1,234,567 rubles. How much are treasures really worth if 0.5 grams of treasure costs $120.5 and their weight is 564.67 grams? Vasya found sunken treasures in the river and brought them home. He decided to sell them to a rich man. But the rich man deceived him for 1,234,567 rubles. How much are treasures really worth if 0.5 grams of treasure costs $120.5 and their weight is 564.67 grams?

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The cabbage butterfly caterpillar eats 10g per month. cabbage. The tit eats 100 caterpillars daily. Calculate how much cabbage "saves" for 1 month (30 days) a family of tits, consisting of a female, a male and 4 chicks, if we assume that the chick eats 2 times less than an adult tit. The cabbage butterfly caterpillar eats 10g per month. cabbage. The tit eats 100 caterpillars daily. Calculate how much cabbage "saves" for 1 month (30 days) a family of tits, consisting of a female, a male and 4 chicks, if we assume that the chick eats 2 times less than an adult tit.

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Kolya dreamed of a chocolate bar that was 3.7 meters long and 2.1 meters wide. Tolya dreamed of a chocolate bar of the same length, but three times as large as Kolya's. By how many meters is the width of the chocolate that Tolya dreamed of longer than the width that Kolya dreamed of? Kolya dreamed of a chocolate bar that was 3.7 meters long and 2.1 meters wide. Tolya dreamed of a chocolate bar of the same length, but three times as large as Kolya's. By how many meters is the width of the chocolate that Tolya dreamed of longer than the width that Kolya dreamed of?

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The inscription remained on the empty container: GROSS - 21.8 kg, NET - 20.6 kg. 19.9 kg of oil was put into it. What should be written on the container now? The inscription remained on the empty container: GROSS - 21.8 kg, NET - 20.6 kg. 19.9 kg of oil was put into it. What should be written on the container now?

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Duck Donna Duck decided to make an apple pie. For this, she took: 0.57 kg of apples, 2 cups of flour, 0.25 kg each, 0.01 kg of butter, 2 cups of milk and 2 eggs. How much will the cake weigh when Donna Duck takes it out of the oven? How much will the pie weigh when Donna Duck's nephews eat 1/3 of the pie? Duck Donna Duck decided to make an apple pie. For this, she took: 0.57 kg of apples, 2 cups of flour, 0.25 kg each, 0.01 kg of butter, 2 cups of milk and 2 eggs. How much will the cake weigh when Donna Duck takes it out of the oven? How much will the pie weigh when Donna Duck's nephews eat 1/3 of the pie?

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In the city where fractions lived, such as 1 2/10, 2 98/100, 1872/10000, 5/100 and in general with denominators 10, 100, 1000, etc., everyone lived very friendly. No one beat anyone, did not offend, and no one argued. There were beautiful houses in this city, and there were beautiful flowers on the windows. Each fraction had its own house and garden. Bulk apples, cherries, pears, and various other flowers grew in the garden. In the city where fractions lived, such as 1 2/10, 2 98/100, 1872/10000, 5/100 and in general with denominators 10, 100, 1000, etc., everyone lived very friendly. No one beat anyone, did not offend, and no one argued. There were beautiful houses in this city, and there were beautiful flowers on the windows. Each fraction had its own house and garden. Bulk apples, cherries, pears, and various other flowers grew in the garden. There were also schools there. Small fractions went there with a denominator of 10. There were also adult fractions with denominators from 100 to 100,000 and very old ones with a denominator from 100,000 to infinity. Adult fractions ran to work.

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Well, the old men and women sat all day in rocking chairs and read books, and sometimes they spanked the bottoms of the baby fractions for disobedience or pranks, or read fairy tales to them Well, the old men and old women sat all day in rocking chairs and read books , and sometimes spanked on the bottoms of fractions-babies for disobedience or pranks, or read fairy tales to them. But one day, Shtrih attacked the city with his army. He mercilessly killed everyone, burned houses, robbed them. The war lasted for ten years. First one won, then the other, but no one could win the war. But one kind Wizard helped the helpless fractions. He extinguished the burning houses, returned the loot and drove the stroke away. Only one question worried the Wizard: "How to cure the wounded shots?". He thought for a long time, and finally came up with. Instead of a fractional line, he gave fractions commas, removed denominators, and such fractions as 1/100, 32/1000, etc. added after the integer part on the right 1, 2, 3, etc. zeros, depending on how many there were in the denominator.

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So the girls' journey through the kingdom of decimal fractions ended. On this journey, they learned a lot of new things, and now they can do any problem with decimal fractions! So the girls' journey through the kingdom of decimal fractions ended. On this journey, they learned a lot of new things, and now they can do any problem with decimal fractions!

Vilkova Angela, Vilkova Vera, Galikhina Lena, Ladoshin Sergey, Trukhanova Marina

Mathematics project in the 6th grade "The world of decimal fractions". The project shows the history of decimal fractions, the basic rules, test tasks, text tasks, crossword puzzle. Students systematize knowledge on a given topic, broaden their horizons and instill interest.

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MBOU "Kriushinskaya secondary school"

Project

"The World of Decimals"

math in 6th grade

Supervisor: teacher of mathematics Pogodina G.B.Participants: Vilkova A.

Vilkov V.

Galikhina E.

Ladoshin S.

Trukhanova M.

2016

Project on the topic "The world of decimal fractions"

Objective of the project :

creation of conditions for deepening and systematization of knowledge about decimal fractions, actions on them;

formation of independent activity and skills of information analysis, the ability to make reasoned judgments;

Raising interest in the subject, expanding the horizons of students.

Project objectives:

study historical information on the topic;

systematize definitions and rules;

systematize tasks on a given topic;

make materials for the math classroom that can be used in the classroom.

Textbook: Mathematics. Arithmetic. Geometry. Grade 6: textbook for general education. organizations / E.A. Bunimovich, L.V. Kuznetsova, S.S. Minieva and others - 5th ed. - M.: Prosveshchenieil. - (Spheres), 2016 – 240 s.

Project type : informational, practice-oriented.

Average duration: 1 month

Project product:materials for the cabinet on the topic “Decimal fractions. Actions on decimal fractions.

Research methods:research, analysis, generalization.

1. The initial stage of the project.

At this stage, the goals and objectives of the project are formed, the creative name of the project is determined, creative groups are formed.

An important point on initial stage The action of the project is to familiarize students with the goals, objectives and progress of the project.

2. Main stage.

At this stage of the project, there will be independent work in groups and food preparation project activities. The class will be divided into five groups, each group has a specific task:

1 group collected historical facts on the topic "History of decimal fractions".

Group 2 systematized the main definitions and rules.

Group 3 worked on tasks. Students selected interesting problems using decimal fractions.

Group 4 worked on compiling tests on this topic.

Group 5 worked on compiling a crossword puzzle on the topic "Decimal Fractions"

In group 1, informational activity was dominant, 2, 3, 4, 5 - practice-oriented.

Forms of work of students: work with literature, processing and generalization of the received materials. Forms of work of the teacher: advising students.

3. Final stage.

On final stage project will show the presentation and defense of the project.

Forms of work of students: presentation of the results, answers to questions from those present, evaluation of the work of each member of their group and the work of other groups. Forms of work of the teacher: evaluation of the work of the group, summing up.

Conclusions:

After completing the project, students acquire the following knowledge, skills and abilities:

know how to read and write decimal fractions;
how to switch from a common fraction to a decimal and vice versa;
actions with decimal fractions;
able to solve problems with decimal fractions.

"History of the origin of decimal fractions"

In the 15th century the complete theory of decimal fractions was developed by the Samarkand astronomer Jemshid al-Kashi in the treatise "The Key to Arithmetic" (1427). He detailed the rules for dealing with decimal fractions. It is possible that al-Kashi did not know that decimal fractions were used in China. He himself considered them his invention. His treatises were not known to European scholars. They independently developed the theory of decimal fractions.The French scientist Francois Viet in 1579 published in Paris his work "Mathematical Canon", in which he cited trigonometric tables, in which he used decimal fractions.

2 , 5 = 2 | 5 0,13 = 0 | 13

When writing decimal fractions, he did not adhere to any specific method: sometimes he separated the integer part from the fractional vertical line, sometimes he depicted the numbers of the integer part in bold type, sometimes he wrote the numbers of the fractional part smaller. So, thanks to Vieta, decimal fractions began to penetrate into scientific calculations, but they did not enter into everyday practice.

The Dutch scientist Simon Stevin believed that decimal fractions should be used in all practical calculations. He devoted his work "The Tenth" (1585) to this, in which he introduced decimal fractions, developed rules for arithmetic operations with them and proposed a decimal system of monetary units, measures and weights.

The use of a comma as a separator, like a period, was proposed in 1616-1617. famous English mathematician John Napier.

Astronomer Johannes Kepler used the decimal point in his work

In Russia, the doctrine of decimal fractions was first expounded by Leonty Filippovich Magnitsky in his Arithmetic.

2 Section

Rules and definitions

Definition.

Fractions whose denominators are powers of ten, that is, 10, 100, 1000, and so on, are called decimal fractions. They are written with a comma, which separates the integer part from the fractional part. The record of the fractional part contains as many digits as there are zeros in the record of the denominator of the corresponding ordinary fraction.

Operations with decimals.

At addition (subtraction) of decimal fractions necessary:

1) if necessary, equalize the number of decimal places,
adding zeros to the corresponding fraction.
2) Write the fractions so that their commas are one under the other.
3) Add (subtract), ignoring the comma.
4) Put a comma in the sum (difference) under the commas, added (subtracted) fractions.

Multiplying two decimalsis done like this:
1) numbers are multiplied without taking into account commas.
2) the comma in the product is placed in such a way as to separate as many characters on the right as it is separated in both factors
taken together.
1,1 0,2 = 0,22 ; 1,1 1,1 = 1,21 ; 2,2 0,1 = 0,22 .

When multiplying a decimal by 10, 100,

1000, etc., in this fraction it is necessary to move the comma to the right by as many digits as there are zeros in the multiplier.
0,065 1000 = 0065, = 65 ;

To divide a decimal by a natural number, necessary:
1) divide the fraction by this number, ignoring the comma;
2) put a comma in the private when the division of the whole part ends.

When dividing a decimal by 10, 100, 1000, ...,
it is necessary to move the comma in this fraction to the left by as many characters as
how many zeros are in the divisor.
34,9: 10 = 3,49 ;

At dividing a decimal by a decimal, first we move the comma in the dividend and divisor to the right by as many characters as there are after the decimal point in the divisor. And then we perform division by a natural number.
543,96: 0,3 = 5439,6: 3 = 1813,2 ;
Atcomparing decimalsFirst of all, we compare the integer parts (located to the left of the comma). If the integer parts are equal, then compare the fractional parts. If the number of characters after the decimal point for the compared fractions does not match, then we assign zeros to the fraction with fewer characters and compare the resulting numbers of fractional parts.

At rounding a number to some decimal place, the numbers in all the following digits are replaced with zeros, and those after the decimal point are discarded. If the digit following the remaining digit is 5, 6, 7, 8, or 9, then the remaining digit is increased by 1. If it is 0, 1, 2, 3, or 4, then the remaining digit is left unchanged.

Round to tenths 12 8 ≈ 130; 5678 ≈5680

Section 3

Interesting tasks

Using decimals

The student went to visit his grandfather during the holidays. By railway he rode 8.5 hours, and from the station on horseback 1.5 hours. In total, he traveled 440 km. At what speed did the student ride on the railroad if he was riding horses at a speed of 10 km per hour?

Solution:

Scheme and analysis

Total 440 km

Solution

10 * 1.5 = 15 (km) - rode from the station on horseback
440-15=425(km) - traveled by rail
425: 8.5 = 50 (km / h) - traveled at a speed on the railroad

Answer: at a speed of 50 km / h he was traveling on a railway

2) The collective farmer had to be at a point located at a distance of 134.7 km from his house. 2.4 hours he rode the bus with average speed 55 km per hour, and the rest of the way he walked at a speed of 4.5 km per hour. How long did he walk?

Solution:

Scheme and analysis

Total 134.7km

II.

Solution

55*2.4=132(km) – went by bus
134.7 - 132 = 2.7 (km) - walking
2.7: 4.5 \u003d 0.6 (h) - walked
0.6=36min

Answer: 0.6 hours or 36 minutes he walked

Section 4

Test "Decimal fractions"

1. Write 12/1000 as a decimal

0,0012
0,012
0,12
0,120

2. In what bit of the number 1, 0359 is the number 3 written?

tenths
hundredths
thousandths
4.ten thousandths

3. How many minutes are in 1.5 hours?

1. 150 min
2. 120 min
3. 90 min

4. There are 6 kg of cereal in a jar. Pour out 0.2 of the contents of the jar. How much cereal is left in the jar

1. 5.8 kg
2. 4.8 kg
3. 1.2 kg

5. Without performing the action, find the product of the numbers 26.48 and 4.25 among the given answers:

1. 1,1254
2. 11,254
3. 112,54

6. Put a comma in the number 67809 so that the number 9 is in the hundredth place.

678,09
6780,9
67,809
6,7809

7. By how much do you need to increase the number 10.36 to get 17.467

16,431
16,11
7,031
7,107

8. The sides of the triangle are 10.6 dm, 7.23 dm, 11.5 dm. What is the perimeter of this triangle?

29.33 cm
94.4 dm 2
29.33 dm
17.83 dm 2

9. What number is represented as the sum of the bit terms 1 + 0.05 + 0.0007?

1. 1,0507
2. 1,057
3. 1,57
4. 1,0057
5. 1,507

10. What number should be put instead of * so that the inequality 32, * 87\u003e 32.887 becomes true

1. 0
2. 9
3. 8

Section 4

Crossword "These decimal fractions"

Horizontally

Replacing a decimal fraction with the one closest to it natural number

or zero is called ... this number up to integers.

With the help of what mathematical operation is an ordinary fraction

converted to decimal?

Where does the comma move when multiplying a decimal fraction by

10,100, 1000,...?

What sign "greater than" or "less than" will be put when comparing

fractions 16.17 and 16.2?

What category does the 7 in 51.3678 belong to?

The most significant digit in the number 975.63

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Slides captions:

Math Project WORLD OF DECIMAL FRACTIONS

History of decimal fractions.

The French scientist Francois Viet in 1579 published in Paris his work "Mathematical Canon", in which he cited trigonometric tables, in which he used decimal fractions.

Stevin wrote fractions differently than now. A circled 0 was used to indicate the fractional part. For the first time, a comma was used when writing fractions in 1592. In England, instead of a comma, they began to use a dot, in the USA it is still used. The use of a comma as a separator, like a period, was proposed in 1616-1617. famous English mathematician John Napier. Astronomer Johannes Kepler used the decimal point in his work. John Napier 1550-1617 Johannes Kepler 1571-1630 54, 789 -- 54 ⓪ 789

In Russia, the doctrine of decimal fractions was first expounded by Leonty Filippovich Magnitsky in his Arithmetic.

Rules and definitions

Definition. Fractions whose denominators are powers of ten, that is, 10, 100, 1000, and so on, are called decimal fractions. They are written with a comma, which separates the integer part from the fractional part. When adding (subtracting) decimal fractions, it is necessary: 1) if necessary, equalize the number of decimal places by adding zeros to the corresponding fraction. 2) Write the fractions so that their commas are one under the other. 3) Add (subtract), ignoring the comma. 4) Put a comma in the sum (difference) under the commas, added (subtracted) fractions.

The multiplication of two decimal fractions is performed as follows: 1) the numbers are multiplied without taking into account commas. 2) the comma in the product is placed in such a way as to separate as many characters on the right as it is separated in both factors combined. 1.1 0.2 = 0.22; 1.1 1.1 = 1.21; 2.2 0.1 = 0.22. Examples of multiplying decimal fractions in a column: When multiplying a decimal fraction by 10, 100, 1000, etc., it is necessary to move the decimal point to the right in this fraction by as many digits as there are zeros in the multiplier. 0.065 1000 = 0065, = 65 ; The multiplication of two decimal fractions is performed as follows: 1) the numbers are multiplied without taking into account commas. 2) the comma in the product is placed in such a way as to separate as many characters on the right as it is separated in both factors combined. For example: 1.1 0.2 = 0.22 ; 1.1 1.1 = 1.21; 2.2 0.1 = 0.22. Examples of multiplying decimal fractions in a column:

To divide a decimal fraction by a natural number, you must: 1) divide the fraction by this number, ignoring the comma; 2) put a comma in the private when the division of the whole part ends. When dividing a decimal fraction by 10, 100, 1000, ..., you need to move the comma in this fraction to the left by as many digits as there are zeros in the divisor. 34.9: 10 = 3.49; When dividing by a decimal fraction, first move the comma in the dividend and divisor to the right by as many digits as there are after the decimal point in the divisor. And then we perform division by a natural number. 543.96: 0.3 = 5439.6: 3 = 1813.2;

Interesting problems using decimal fractions

1) The length of the Suez Canal is 165.8 km, the length of the Panama Canal is 84.7 km less than the Suez Canal, and the length of the White Sea-Baltic Canal is 145.9 km longer than the length of the Panama Canal. What is the length of the White Sea-Baltic Canal? 1) 165.8 - 84.7 \u003d 81.1 (km) - The length of the Panama Canal. 2) 145.9 + 81.1 \u003d 227 (km) - The length of the White Sea - Baltic Canal. Answer: 227 km.

The student went to visit his grandfather during the holidays. By rail, he rode 8.5 hours, and from the station on horseback 1.5 hours. In total, he traveled 440 km. At what speed did the student ride on the railroad if he was riding horses at a speed of 10 km per hour? Solution 10 * 1.5 = 15 (km) - rode from the station on horseback 2) 440-15 = 425 (km) - rode by rail 3) 425: 8.5 \u003d 50 (km / h) - rode at a speed by rail Answer: at a speed of 50 km / h I was traveling by rail

2) Moscow Metro(by 1959) was built in 5 phases. The length of the first line of the metro is 11.6 km, the second - 14.9 km, the length of the third is 1.1 km less than the length of the second line, the length of the fourth line is 9.6 km more than the third line, and the length of the fifth line is 11.5 km less fourth. What is the length of the Moscow Metro by the beginning of 1959? 1) 14.9 - 1.1 = 13.8 (km) - Third stage. 2) 13.8 + 9.6 = 23.2 (km) - The fourth stage. 3) 23.2 - 11.5 = 11.7 (km) - Fifth stage. 4) 11.6 + 14.9 + 13.8 + 23.2 + 11.7 \u003d 75.2 (km) - The length of the Moscow metro. Answer: 75.2 km.

Test on the topic "Decimal fractions"

Write 12/1000 as a decimal fraction 0.0012 0.012 0.12 0.120 In what digit of the number 1, 0359 is the number 3 written? ten thousandths ten thousandths

how many minutes in 1.5 hours 1. 150 min 2. 120 min 3. 90 min There are 6 kg of cereal in a jar. Pour out 0.2 of the contents of the jar. How much cereal is left in the jar 1. 5.8 kg 2. 4.8 kg 3. 1.2 kg .112.54

Place a comma in the number 67809 so that the number 9 is in the hundredths place. , 7.23 dm, 11.5 dm. What is the perimeter of this triangle? 29.33 cm 94.4 in 2 29.33 in 17.83 in 2

What number is represented as the sum of the digit terms 1 + 0.05 + 0.0007? 1. 1.0507 2. 1.057 3. 1.57 4. 1.0057 5. 1.507

Crossword "These decimal fractions"

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 \ \ \ \ \ \ \ \ \ \ \

Horizontally 1. Replacing a decimal fraction with the nearest natural number or zero is called ... this number up to integers. 3. With the help of what mathematical operation does an ordinary fraction turn into a decimal? 5. Where does the comma move when multiplying a decimal fraction by 10,100, 1000,...? 7. What sign "more" or "less" will be put when comparing fractions 16.17 and 16.2? 8. What category does the number 7 belong to in the number 51.3678? 9. The most senior category in the number 975.63

Vertically 2. What arithmetic operation is performed in the example: 0.05__100=5 4. What sign can be put between the numbers 7 and 8 so that the resulting number is greater than 7 and less than 8? 5. Where does the comma move when dividing a decimal fraction by 10,100,1000,...? 6. How many digits should a comma be moved when dividing a decimal fraction by 1000000? 10. How many digits after the decimal point in the decimal notation of the fraction are eighteen point seven thousandths?

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 ROUND

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 P I T A Z A

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 P I T A Z A V P R V O

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 P I T A Z A V P R V O L E V O

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 5th

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 F I T A Z

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 F ITH

Vilkova Vera 1 2 3 4 5 6 7 8 9 10 1 2 3 6 7 9 10 4 5 5 8 F F I T A I A C O N R O L E O S SIX M N I S H Y S I C H N R O N R I

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Magic decimals. 5th class project INTRODUCTION On a typical day after school, my two best friends, fifth-grade students Annika and Lilya, were doing their math homework. They opened the textbook and saw decimals...

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Decimals Grade 5 Project

On a typical day after school, my two best friends, fifth-grade students Annika and Lilya, were doing their math homework. They opened the textbook and saw decimal fractions... I don't understand anything! What's happened? These ... like them ... but ... decimal fractions. We didn't pass them! Lily was outraged. Solve the problem with decimal fractions - Annika reads. - In the spring, they sowed 0.9 fields, and harvested only 0.6 fields. How much crop was not harvested from the field?

Lily asked. Maybe add 9 to 0? Annika suggested. No, we should probably choose 0 or 9 ourselves! Annika agreed. And just as the girls wanted to write it down, the textbooks began to dance and sing: We really need decimal fractions. What is a crooked letter? Or is it a comma? But what does the comma have to do with it, Maya the fairy will tell us!

Please to my kingdom! I found out that you do not know what decimal fractions are? And after visiting my castles, you will learn all about decimal fractions. We agree! - the girls said in unison and ended up in the kingdom.

1st castle, where you will be introduced to the history of decimal fractions 3rd castle, where you will be taught how to perform actions with decimal fractions 5th castle, where you will be told a fairy tale about decimal fractions Exit from the kingdom 4th castle, where you meet exciting tasks that have decimal fractions 2nd lock in which you will learn Interesting Facts with decimals

Decimal fractions appeared in the works of Arab mathematicians in the Middle Ages and independently in ancient China. But even earlier, in ancient Babylon, fractions of the same type were used, but of course sexagesimal. Later, the scientist Hartmann Beyer (1563-1625) published the essay “Decimal Logistics” where he wrote: “... I noticed that technicians and artisans, when measuring any length, very rarely and only in exceptional cases express it in whole numbers of the same name; usually they have to either take small measures, or turn to fractions, in the same way astronomers measure quantities not only in degrees, but also in fractions of a degree, i.e. minutes, seconds, etc., but it seems to me that dividing them into 60 parts is not as convenient as dividing by 10, into 100 parts, etc., because in the latter case it is much easier to add, subtract and generally perform arithmetic operations ; It seems to me that decimal parts, if introduced instead of sexagesimal, would be useful not only for astronomy, but also for all kinds of calculations. Simon Stevin introduced decimal fractions into European practice. Until then, anyone who dealt with non-integer numbers had to fiddle with numerators and denominators. (Material provided by Egor Gorokhov)

Why did people switch from ordinary fractions to decimals? Yes, because the actions with them are simpler, especially addition and subtraction. Add the fractions 3/50 and 7/40. First you need to find the least common multiple of their denominators (this is the number 200), then divide it by 50 and multiply the result (the number 4) by the numerator and the denominator of the first fraction. It turns out 12/200. Then you need to divide 200 by 40 and multiply the quotient (number 5) by the numerator and denominator of the second fraction. It turns out 35/200. We reduced fractions to a common denominator. Only now can we add up the numerators and get the answer: 47/200. And if these fractions are presented as a decimal notation: 3/50=0.06; 7/40 \u003d 0.175, the amount is instantly - this is 0.235. Of course, the number 1/7 has to be written only with a certain accuracy, 0.143 or 0.14287, but everything in life has its limits of accuracy. Only in the first quarter of the 18th century. fractional numbers began to be written using a simple decimal point. In some countries, and in particular in Russia, a comma is used instead of a dot. It was introduced by the German mathematician Georg Andreas Böckler in 1661.

5 3 4 1 S. Stevin 0 I II III IV 3. 1 4 1 5 4 3 1 0 2 J. H. Beyer 3 1415 A. Girard From the history of decimal fractions Today we use decimal fractions naturally and freely. However, what seems natural to us served as a real stumbling block for the scientists of the Middle Ages. Western Europe in the 16th century along with the widespread decimal system for representing integers, sexagesimal fractions were used everywhere in calculations, dating back to the ancient tradition of the Babylonians. It took the bright mind of the Dutch mathematician Simon Stevin to bring the record of both integer and fractional numbers into a single system. Apparently, the impetus for the creation of decimal fractions was the tables of compound interest compiled by him. In 1585 he published the book Tithing, in which he explained decimal fractions. Stevin's notation was not perfect, just like the notation of his colleagues and followers. Here is how they would write the number 3.1415: (Material provided by Dmitry Kruglikov)

We have heard a lot about air. Air is 99.96% composed of three gases: nitrogen, oxygen and argon. Carbon dioxide contains 0.03%, the rest accounts for 0.01%.

Of great importance for the knowledge of the world is the problem of the numerical ratio between the atoms of various elements. If we compare the iron, cobalt and nickel available on the whole Earth, it turns out that the globe consists of: Iron 92% Cobalt 0.5% Nickel 7.5% The most accurate chemical analyzes of a huge number of meteorites that fell to Earth gave wonderful results. It turned out that in iron meteorites the percentage of iron, cobalt and nickel amazingly coincides with their content on our planet. (Material provided by Gleb Ivshin)

How to add and subtract ask her. She will answer: “Remember the algorithm for adding or subtracting decimal fractions.” To begin with, the number of decimal places, you equalize, Write them in a column and of course, know That the comma should be under the comma, And then just decide. Do the addition or subtraction first, without paying any attention to the comma. Well, in your answer, of course, you put a comma under the comma in these fractions. You remember these rules forever, so that in your memory they remain like twice two! You can tell me a lot, About what decimal fractions are, About what you can at the end of the fractional part, To the right, discard or insert zeros. Well, how to compare them, you tell me. Well, it's certainly easier than ever. Compare the whole parts of the decimal fraction, And the one that has more of it, Of course, there will be more. Well, if those parts are just equal, Then what should I do, you tell me. If two decimal fractions have equal integer parts, You look at the first of the mismatched digits, And the one with the larger one, of course, will also have the larger one. Do you remember everything, you tell me? If not, ask Galina Vasilievna, (Verse provided by Kristina Nichiporuk)

Katya) The cabbage butterfly caterpillar eats 10g per month. cabbage. The tit eats 100 caterpillars daily. Calculate how much cabbage "saves" for 1 month (30 days) a family of tits, consisting of a female, a male and 4 chicks, if we assume that the chick eats 2 times less than an adult tit.

Biyanova Masha) Kolya dreamed of a chocolate bar, which is 3.7 m long and 2.1 m wide. Tolya dreamed of a chocolate bar of the same length, but three times as large as Kolya's. By how many meters is the width of the chocolate that Tolya dreamed of longer than the width that Kolya dreamed of? fractions? Authors: Masha Volkova, Lisa Vasilyeva In the city where fractions lived, such as 1 2/10, 2 98/100, 1872/10000, 5/100 and generally with denominators 10, 100, 1000, etc., everyone lived very friendly. No one beat anyone, did not offend, and no one argued. There were beautiful houses in this city, and there were beautiful flowers on the windows. Each fraction had its own house and garden. Bulk apples, cherries, pears, and various other flowers grew in the garden. There were also schools there. Small fractions went there with a denominator of 10. There were also adult fractions with denominators from 100 to 100,000 and very old ones with a denominator from 100,000 to infinity. Adult fractions ran to work.

For a day they sat in rocking chairs and read books, and sometimes slapped on the bottoms of fractions-babies for disobedience or pranks, or read fairy tales to them. But one day Shtrikh attacked the city with his army. He mercilessly killed everyone, burned houses, robbed them. The war lasted for ten years. First one won, then the other, but no one could win the war. But one kind Wizard helped the helpless fractions. He extinguished the burning houses, returned the loot and drove the stroke away. Only one question worried the Wizard: "How to cure the wounded shots?". He thought for a long time, and finally came up with. Instead of a fractional line, he gave fractions commas, removed denominators, and such fractions as 1/100, 32/1000, etc. added after the integer part on the right 1, 2, 3, etc. zeros, depending on how many there were in the denominator.

Girls in the kingdom of decimal fractions. On this journey, they learned a lot of new things, and now they can do any problem with decimal fractions!

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Presentation on the topic: Magic Decimals

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On a typical day after school, my two best friends, fifth-grade students Anna and Tanya, were doing their math homework. They opened the textbook and saw decimal fractions... On an ordinary day after school, two best friends, fifth-grade students Anna and Tanya were doing their homework in mathematics. They opened the textbook and saw decimal fractions... I don't understand anything! What's happened? These ... like them ... but ... decimal fractions. We didn't pass them! Tanya was outraged. Solve the problem with decimal fractions - Anna reads. - In the spring, they sowed 0.9 fields, and harvested only 0.6 fields. How much crop was not harvested from the field?

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Today we use decimals naturally and freely. However, what seems natural to us served as a real stumbling block for the scientists of the Middle Ages. Western Europe in the 16th century along with the widespread decimal system for representing integers, sexagesimal fractions were used everywhere in calculations, dating back to the ancient tradition of the Babylonians. It took the bright mind of the Dutch mathematician Simon Stevin to bring the record of both integer and fractional numbers into a single system. Apparently, the impetus for the creation of decimal fractions was the tables of compound interest compiled by him. In 1585 he published the book Tithing, in which he explained decimal fractions. Stevin's notation was not perfect, just like the notation of his colleagues and followers. This is how they would write the number 3.1415: Today we use decimals naturally and freely. However, what seems natural to us served as a real stumbling block for the scientists of the Middle Ages. Western Europe in the 16th century along with the widespread decimal system for representing integers, sexagesimal fractions were used everywhere in calculations, dating back to the ancient tradition of the Babylonians. It took the bright mind of the Dutch mathematician Simon Stevin to bring the record of both integer and fractional numbers into a single system. Apparently, the impetus for the creation of decimal fractions was the tables of compound interest compiled by him. In 1585 he published the book Tithing, in which he explained decimal fractions. Stevin's notation was not perfect, just like the notation of his colleagues and followers. This is how they would write the number 3.1415:

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Of great importance for the knowledge of the world is the problem of the numerical ratio between the atoms of various elements. Of great importance for the knowledge of the world is the problem of the numerical ratio between the atoms of various elements. If we compare the iron, cobalt and nickel available on the whole Earth, it turns out that the globe consists of: Iron 92% Cobalt 0.5% Nickel 7.5% The most accurate chemical analyzes of a huge number of meteorites that fell to Earth gave wonderful results. It turned out that in iron meteorites the percentage of iron, cobalt and nickel amazingly coincides with their content on our planet.

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Magic Decimals

The project was completed by a student

Inozemtseva Elizabeth

Mathematics teacher Voronenko I. E.

Introduction

On the most ordinary day after school, two best friends, fifth-grade students Katya and Ira, were doing their homework in mathematics. They opened the textbook and saw decimals...

I don't understand anything! What's happened? These… like their… a… decimals. We did not pass them! - Ira was indignant.

Solve the problem with decimal fractions - Katya reads. - “In the spring they sowed 0.9 fields, and harvested only 0.6 fields. How much crop was not harvested from the field?

All the same, did they sow 0 or 9? - Ira asked.

Maybe you need to add 9 to 0? - Katya suggested.

No, we should probably choose 0 or 9 ourselves!

Katya agreed. And just as the girls wanted to write it down, the textbooks began to dance and sang:

Decimals

We really need.

What is a crooked letter?

Or is it a comma?

But what does the comma have to do with it, the fairy Maya will tell us!

Here comes the fairy!

Please to my kingdom! I found out that you don't know what decimals are?

And after visiting my castles, you will learn all about decimal fractions.

We agree! - the girls said in chorus and ended up in the kingdom.

Kingdom of decimals

The first castle where you will be introduced to the history of decimal fractions.

2nd castle, in which you will learn interesting facts about decimal fractions.

The 3rd castle, in which you will be taught how to perform actions with decimal fractions.

4th castle, where you will meet with exciting tasks that have decimal fractions.

5th castle, where you will be told a fairy tale about decimal fractions.

Lock 1 From the history of decimals

Decimal fractions appeared in the works of Arab mathematicians in the Middle Ages and independently in Ancient China. But before, in Ancient Babylon, used fractions of the same type, but of course sexagesimal.

Later, the scientist Hartmann Beyer published the essay “Decimal Logistics” where he wrote: “... I noticed that technicians and artisans, when they measure any length, very rarely, in exceptional cases, express it in integers of one name; usually they have to either take small measures, or turn to fractions, in the same way astronomers measure quantities not only in degrees, but also in fractions of a degree, i.e. minutes, seconds, etc., but it seems to me that their division into 60 parts is not as convenient as the division into 10,100 parts, etc., because in the latter case it is much easier to add, subtract and generally perform arithmetic operations; It seems to me that decimal parts, if introduced instead of sexagesimal, would be useful not only for astronomy but also for all kinds of calculations.

Simon Stevin introduced decimal fractions into European practice. Until then, anyone who dealt with non-integer numbers had to fiddle with numerators and denominators.

Lock 2 Decimals in a person's life

We have heard a lot about air. Air is 99.96% composed of 3 gases: nitrogen, oxygen and argon.

Castle 3 Interesting

Of great importance for the knowledge of the world is the problem of the numerical ratio between the atoms of various elements.

If we compare the iron, cobalt and nickel available on the whole Earth, it turns out that the globe consists of:

Iron by 92%

Cobalt at 0.5%

Nickel by 7.5%

The most accurate chemical analyzes of a huge number of meteorites that fell to Earth gave remarkable results. It turned out that in iron meteorites the percentage of iron, cobalt and nickel coincides with their content on our planet.