1. To divide the first fraction by the second, you need to multiply the dividend by the number that is the inverse of the divisor.
For proper and improper fractions, the division rule is as follows:
To divide a common fraction, you must multiply the numerator of the dividend by the denominator of the divisor, and multiply the denominator of the dividend by the numerator of the divisor. We take the first product as the numerator, and the second as the denominator.
Dividing a fraction by a fraction.
To divide the 1st ordinary fraction by the second one, which is not equal to zero, you need to:
- multiply the numerator of the 1st fraction by the denominator of the 2nd fraction and write the product in the numerator of the resulting fraction;
- multiply the denominator of the 1st fraction by the numerator of the 2nd fraction and write the product in the denominator of the resulting fraction.
In other words, dividing fractions leads to multiplication.
To divide the 1st fraction by the second, you need to multiply the dividend (1st fraction) by the reciprocal fraction of the divisor.
Dividing a fraction by a number.
Schematically, dividing a fraction by a natural number looks like this:
To divide a fraction by a natural number, use the following method:
We express a natural number as an improper fraction with a numerator that is equal to the number itself and a denominator that is equal to 1.
Class: 6
Presentation for the lesson
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The purpose of the lesson: Summarize and systematize students’ knowledge on the topic “Division of ordinary fractions” using multimedia technologies.
Lesson objectives:
Educational:
- consolidate theoretical knowledge: determination of reciprocal numbers; rules for adding, subtracting, multiplying, dividing ordinary fractions; rule for finding a fraction from a number.
- develop the ability to apply acquired theoretical knowledge to solve problems;
- carry out knowledge control using a computer test.
Educational:
- develop cognitive interest, intellectual and creative abilities of students;
- to form an information culture, mastering the skills of searching and analyzing information;
Educational:
- teach independent activities to acquire knowledge;
- to form conscious motives for learning, self-improvement, self-education;
- cultivate dedication and perseverance in achieving goals;
- foster mutual assistance.
Lesson plan:
- Organizational and motivation, setting lesson goals. generalization and consolidation of concepts, definitions, rules. (I – oral counting)
- Testing. (II)
- Deepening, applying knowledge, developing thinking. (III-VIII)
- Results. (IX)
- Homework. (X)
During the classes
Today our math lesson will be related to literature. An unusual journey awaits us. Since we have a math lesson, the trip will be mathematical. The topic of our lesson is “Dividing fractions.” Before you set off, you need to check that everyone is ready.
I. Oral counting
(slide 2)
| - | * | : 4 |
| 3 - 1 | * | : |
| + | 1 * | : |
| * 5 | : | 6: |
We repeat:
- What numbers are called reciprocals?
- rules for adding, subtracting, multiplying and dividing fractions.
And so, we hit the road. And as you may have guessed, we will travel according to the fairy tales of A.S. Pushkin. In which fairy tale we will make our first stop, you will learn from the words that you will receive when solving division examples. Students are given task cards and key cards. If it is possible to work on computers, then students take a multiple-choice test created in Microsoft Excel. As a result, they will receive the necessary words.
II. Programmed (differentiated) control. (test)
| Option I | Option II | Option III | IV option |
Key cards
| I century | R | O | e | m |
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | 1 | 9 | 10 | 8 |
| II century | s | b | A | To | R |
| 1 | |||||
| 2 | |||||
| 3 | 40 | 42 | 41 | 43 | 44 |
| 4 | |||||
| 5 | 7 |
| III century | R | A | T | To | And | With |
| 1 | ||||||
| 2 | 60 | 61 | 62 | 63 | 64 | 65 |
| 3 | ||||||
| 4 | ||||||
| 5 | ||||||
| 6 | 1 |
| IV century | T | R | s | O | To |
| 1 | |||||
| 2 | |||||
| 3 | 60 | 65 | 61 | 63 | 64 |
| 4 | |||||
| 5 | |||||
| 6 |
We received the words: trough, fish, old man, sea. What fairy tale have we found ourselves in? In a fairy tale about a fisherman and a fish. Who remembers the beginning of this fairy tale? ( slide 3)
An old man lived with his old woman
By the bluest sea;
They lived in a dilapidated dugout
Exactly thirty years and three years.
The heroes of the fairy tale offer us to solve the problem.
III.
(slide 4)
Pike, crucian carp and perch together weigh 1 kg. How much does each fish weigh if the pike is 1 times heavier than the crucian carp, and the mass of the perch is equal to the mass of the crucian carp.
IV. To find out the name of the next fairy tale by A.S. Pushkin, you need to open 2 chests.
To do this, you need to solve 2 equations. The equations are solved according to the options, then the students change notebooks and the solutions are checked. ( slides 5-9)
Option I
Option II
The chests open and the title appears: The Tale of Tsar Saltan. (full title of the tale: The Tale of Tsar Saltan, of his son, the glorious and mighty hero Prince Guidon Saltanovich, and of the beautiful swan princess.)
V.
(slides 10-12)
An island lies on the sea,
There is a city on the island,
With golden-domed churches,
With towers and gardens;
This city is ruled by Prince Guidon. We will find out who we can meet there by completing the following task:
Before you is a chain of three numbers; in each line you need to eliminate the extra number.
Find the sum of the extra numbers. + 32 + = 33
There are several wonders in this city.
One of them -
The sea will swell violently,
It will boil, it will howl,
It rushes onto the empty shore,
It will splash in the fast bank,
And they will find themselves on the shore
In scales, like the heat of grief,
Thirty-three heroes.
VI. The next fairy tale by A.S. Pushkin will tell you the answer that we will get when solving the example for all actions.
(slide 13)
1 : ((slides 16-17)
The king to the window - en on the knitting needle,
He sees a cockerel beating,
Facing east.What fairy tale are we in? In the fairy tale about the golden cockerel. Our journey is coming to an end and we will end it with the words that end the fairy tale about the golden cockerel.
To find out the phrase, arrange the numbers in ascending order!
The result was the phrase: “The fairy tale is a lie, but there is a hint in it!” What does this phrase mean?
6th grade
SUBJECT: “Division of ordinary fractions”, 6th grade.
THE PURPOSE OF THE LESSON: Summarize and systematize theoretical and practical
knowledge, skills and abilities of students. Organize work on
closing gaps in students' knowledge. Improve, expand
and deepen students' knowledge of the topic.
TYPE OF LESSON: Lesson of generalization and systematization of knowledge, skills and abilities.
Equipment: On the board is the topic, purpose, lesson plan.
DURING THE CLASSES.
Each student has a “Check Sheet” on their desk.
1. homework –
2. review questions –
3. oral counting –
4. class work –
5. independent work –
1. Checking homework:
a) work in pairs on the following questions:
1) Addition, subtraction of ordinary fractions;
2) How to multiply a fraction by a fraction;
3) Multiplication of two fractions;
4) Multiplication of mixed fractions;
5) The rule for dividing fractions;
6) Division of mixed fractions;
7) What is called. reducing fractions.
b) checking homework using a ready-made solution on the board:
No. 620 (a), 624, 619 (d).
Purpose: to identify the degree of mastery of homework. Identify typical deficiencies.
Put your grades on the control sheet
Announce the purpose of the lesson: Summarize and systematize knowledge, skills and abilities in
topic: “Division of ordinary fractions.”
We repeated the theory, let's test our knowledge in practice.
2. Verbal counting.
a) Using cards: 1) Reduce the fraction: ; ; ; ...
2) Convert to an improper fraction: ; ; ...
3) Select the whole part: ; ; ...
b) Number ladder. Whoever gets to the 6th floor faster will find out:
construction of geometry (Euclid)
Option 2 - a person who wanted to be a lawyer, an officer, and a philosopher, but
became a mathematician (Descartes)
l 0.1: ½ 0.4: 0.1 a
and d e l k k a v r e t
Marks on the control sheet, for: 2" - "5", 3" - "4", 4" - "3".
Whoever completed the “ladder” does No. 606 in the notebooks. The first of the students on the wing of the board does No. 606. Then he checks the class.
3.
A) No. 581 (b,d), 587 (with commentary), 591 (l,m,k), 600, 602, 593 (g,k,d,i)
The task is completed in notebooks and on the board.
b) solve the problem: Thousands of rubles were paid for a kg of sweets. How much are
Kg of these sweets?
4.
№ 1 . Follow these steps:
: answers: 1) 2) 3) 4) .
№ 2 . Represent the fraction as a fraction and do the following:
0.375: answers: 1) 2) 3) 4)
№ 3 . Solve the equation: answers: 1) 2) 3) 4) 2
№ 4 . On the first day, the tourist walked the entire route, and on the second, the rest. In
how many times more is the part of the road covered by a tourist on the first day than on
second? Answers: 1) 2) 5 3) 4)
№ 5. Present as a fraction:
: answer: 1) 2) 3) 4)
Check the solution using the template: No. 1 -4; No. 2 – 1; No. 3 – 4; No. 4 – 4; No. 5 – 3.
Put your grades on the control sheet.
Collect control sheets. Summarize. Announce grades for the lesson.
5. Lesson summary:
What basic rules did we repeat today?
6. Homework:
No. 619 (c), 620 (b), 627, individual task No. 617 (a, d, g).
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Municipal educational institution "Gymnasium No. 7"
Torzhok, Tver region.
OPEN LESSON ON THE TOPIC:
"DIVISION OF ORDINARY FRACTIONS"
6th grade
Open lesson at the city municipal district of Torzhok
(certification, 2001)
Mathematics teacher: Ufimtseva N.A.
2001
SUBJECT : " Division of ordinary fractions", 6th grade.
THE PURPOSE OF THE LESSON : Summarize and systematize theoretical and practical
Knowledge, abilities, and skills of students. Organize work on
Closing gaps in students' knowledge. Improve, expand
And deepen students' knowledge on the topic.
TYPE OF LESSON : Lesson of generalization and systematization of knowledge, skills and abilities.
Equipment : On the board is the topic, purpose, lesson plan.
DURING THE CLASSES.
Each student has a “Check Sheet” on their desk.
- Homework -
- review questions -
- verbal counting -
- classroom work -
- independent work -
- Checking homework:
A) work in pairs on the following questions:
1) Addition, subtraction of ordinary fractions;
2) How to multiply a fraction by a fraction;
3) Multiplication of two fractions;
4) Multiplication of mixed fractions;
5) The rule for dividing fractions;
6) Division of mixed fractions;
7) What is called. reducing fractions.
B) checking homework using a ready-made solution on the board:
No. 620 (a), 624, 619 (d).
Target : identify the degree of mastery of homework. Identify typical deficiencies.
Put your grades on the control sheet
Announce the purpose of the lesson: Summarize and systematize knowledge, skills and abilities in
Topic: “Dividing ordinary fractions.”
We repeated the theory, let's test our knowledge in practice.
- Verbal counting.
A) Using cards: 1) Reduce the fraction: ; ; ; ...
2) Convert to an improper fraction: ; ; ...
3) Select the whole part: ; ; ...
B) Number ladder. Whoever gets to the 6th floor faster will find out:
Geometry constructions (Euclid)
Option 2 - a person who wanted to be a lawyer, an officer, and a philosopher, but
Became a mathematician (Descartes)
D t
And r
L 0.1: ½ 0.4: 0.1 a
K k
V e
E d
3 2 4 5
I d e l k a v e r t
Marks on the control sheet, for: 2" - "5", 3" - "4", 4" - "3".
Whoever completed the “ladder” does No. 606 in the notebooks. The first of the students on the wing of the board does No. 606. Then he checks the class.
- Repetition and systematization of the main theoretical principles:
A) No. 581 (b,d), 587 (with commentary), 591 (l,m,k), 600, 602, 593 (g,k,d,i)
The task is completed in notebooks and on the board.
B) solve the problem: Thousands of rubles were paid for a kg of sweets. How much are
Kg of these sweets?
- Independent work. Purpose: to check your understanding of this topic.
№ 1 . Follow these steps:
: answers: 1) 2) 3) 4) .
№ 2 . Represent the fraction as a fraction and do the following:
0.375: answers: 1) 2) 3) 4)
№ 3 . Solve the equation: answers: 1) 2) 3) 4) 2
№ 4 . On the first day, the tourist walked the entire route, and on the second, the rest. In
How many times more is the part of the road covered by a tourist on the first day than on
Second? Answers: 1) 2) 5 3) 4)
№ 5. Present as a fraction:
: answer: 1) 2) 3) 4)
Check the solution using the template: No. 1 -4; No. 2 – 1; No. 3 – 4; No. 4 – 4; No. 5 – 3.
Put your grades on the control sheet.
Collect control sheets. Summarize. Announce grades for the lesson.
- Lesson summary:
What basic rules did we repeat today?
- Homework:
No. 619 (c), 620 (b), 627, individual assignment No. 617 (a, e, g)
COURSE WORK
ON ALGEBRA AND PRINCIPLES OF ANALYSIS
ON THIS TOPIC
"TRIGONOMETRIC FUNCTIONS"
Creative group of the Department of Mathematics
"Gymnasium No. 3" Udomlya.
Lesson No. 3-4 developed by a math teacher
Ufimtseva N.A.
2000
Municipal educational institution "Gymnasium No. 7"
Torzhok, Tver region.
PUBLIC LESSON
Multiplying Decimals
The decimal notation allows you to multiply fractions using almost the same rules that you use to multiply natural numbers. The difference is that it is necessary to determine the place of the comma in the resulting product.
Let us explain this with an example; Let's calculate the product 2.5 1.02.
Let's move the comma in the first factor one digit to the right, and in the second factor two digits to the right. Thus, the first factor will increase by 10 times, the second by 10 2 = 100 times, and the product by 10 100 = 1000 times.
Let's define the product of natural numbers 25 and 102:
25 102 = 2550.
This number is 1000 times greater than the required product. Therefore, it is necessary to reduce the number 2550 by 1000 = 10 3 times, that is, move the comma in this number to the left by 3 digits. Thus,
2,5 1,02 = 2,550 = 2,55.
You can think differently:
Thus, in order to multiply two decimal fractions9, it is enough, without paying attention to commas, to multiply them as natural numbers9 and then in the resulting product on the right, separate as many digits with a comma as there were after the commas in both factors together.
For example,
Decimal division
Let's look at the example of dividing a decimal fraction by a natural number.
Example. Calculate 46.8:2.
Solution. Divide 4 tens by 2 - we get the quotient number 2 (2 tens).
We divide 6 units by 2 - we get the quotient number 3 (3 units).
The division of the integer part is complete; we separate the whole part in the quotient with a comma.
We divide 8 tenths by 2 - we get the quotient number 4 (4 tenths). The remainder is 0—the division is complete.
Dividing a decimal by a decimal is reduced to dividing by a natural number by moving the commas in the dividend and divisor so many digits to the right that the divisor becomes a natural number.
Example. Calculate 4.42:0.2.
Solution. Since the divisor has one digit after the decimal point, it is enough to move the commas in the dividend and divisor one digit to the right. Thus, the dividend and divisor increase by 10 times, so the quotient will not change. In this case, the divisor will be a natural number.
You can reason this way:
But the exact result is not always obtained when dividing decimal fractions. More often you have to be content with an approximate private.
Example. Find the quotient 1.723:0.03.
Solution. Let's get rid of the comma in the divisor: 1.723:0.03= 172.3:3. Let's do the division.
Starting from the hundredths place, the number 3 in the quotient is repeated endlessly, because the remainder, starting from the third stage of the division process, is always equal to the same number 1.
If you leave the first two digits after the decimal point for the quotient, you get an approximate equality: 172.3:3 ≈ 57.43.
Class: 6
Presentation for the lesson
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Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.
Lesson objectives:
Educational aspect:
- repeat and deepen knowledge on the topic “Division of ordinary fractions”
Developmental aspect:
- develop skills of analysis and comparison of material;
- develop attention, memory, speech, logical thinking, independence;
- promote the development of skills to carry out self-assessment of educational activities.
Educational aspect:
- instill in students the skill of independence in work, teach hard work and accuracy;
- cultivate the need to evaluate one’s own activities and the work of classmates;
- cultivate a culture of speech, attention to precision of formulation.
Forms of organizing educational activities:
- frontal, individual, game
Technologies used:
- collaboration technology;
- information Technology;
- gaming technologies.
Equipment:
- computer;
- multimedia projector;
- Microsoft Office PowerPoint presentation;
- task cards
During the classes
I. Organizational moment
II. Verbal counting
1. Calculate the meanings of expressions, assemble the puzzle.
Teacher: Guys, do you recognize what is shown in this photo?
Usolye Sibirskoe is one of the oldest cities in the Angara region. It was founded as a settlement in 1669 thanks to the conquerors of the Siberian expanses, the Yenisei Cossacks, the Mikhalev brothers, who discovered a salt spring on the banks of the Angara River and built a salt pan.
2. Without performing any actions, compare the quotient with the dividend:
III. Repetition of previously studied material
1. Represent the decimal as a fraction. In the table, write the letters corresponding to the answers found (work in pairs).
0.4 - A 1.2 - P 0.006 - P 3.6 - I 0.9 - W 5.008 – T 0.05 - U 2.16 - O 0.37 - D 4.44 - C 5.08 - K 2.15 – M
The name of the city Irkutsk comes from the Irkut River, which flows into the Angara. The city dates back to the first Irkutsk fort, founded by the Cossacks under the leadership of Yakov Pokhabov on July 6, 1661. By September 1670, a fortress with four towers was built on the site of the fort and was named the Kremlin. Irkutsk, almost from the very beginning, was the most important stronghold for trade with China. All Russian-Chinese trade caravans passed through the city.
2. Express the fraction as a decimal. Arrange the resulting numbers in ascending order and read the word (independently, followed by checking).
Answers: 0.8; 0.5; 0.25; 0.12; 0.032; 0.07, word – Baikal (hyperlink to the unified collection of TsOR).
IV. Reinforcing the material learned
1. Fill in the blanks:
1) 
;
2) 
;
3) 
;
4) 
2. Game “Loto” (students need to solve the first example, then move on to the example that begins with the number obtained when solving the previous one, and make a sentence).
| Option I | Option II | ||||
at the source |
|||||
lichen |
coated |
||||
Answers: Shamanka Rock - marble covered with red lichen;
Shaman-stone is a rock lying at the source of the Angara.
V. Physical education minute
Hands on your sides, arms wider.
One two three four.
Now we decided to jump.
One two three four.
We stretched ourselves higher, higher...
We squat - lower, lower.
We got up and sat down...
We got up and sat down...
And now we sat down at our desks.
VI. The solution of the problem
Solve a problem: two cars left simultaneously towards each other from the cities of Usolye-Sibirskoye and Irkutsk, the distance between which is 80 km. The speed of the first car is equal to the speed of the second. Find the speeds of each car if they meet after forty minutes.
Let x (km/h)- speed of the second car
Then x (km/h)- speed of the first car
x+ x (km/h)- approach speed
Knowing that the cars met through h and drove together 80 km, Let's make an equation:
(x+X) * =80
(x+X) =80:
x = 120:1
Answer:
- 1 option FRYING
- Option 2 OMUL
VIII. Homework
Create a task
