The abstract of the lesson "Addition and subtraction of mixed numbers". Subtraction of mixed fractions

The solution of complex examples is correct - an unbearable task for those who do not understand in mathematics of elementary rules and laws. Addition and subtraction mixed numbers rightly can be attributed to complex examples. However, when primary analysis The numbers themselves can easily carry out any actions.

What it is?

Mixed number is a combination of the whole part and fractional. For example, there are 2 and 3, of which 2 is a simple number, but 3 is already mixed, where 3 - whole part, and - fractional. Presented varieties add up and deducted in different ways, but do not entail difficulties in independent decision examples.

Full analysis of the example

For a full representation of the essence of the mixed value, it should be brought as an example a task that will help to display the meaning of the narrative of the intended. So, Vasya drove the circle around the school by bike per 1 minute and 30 seconds, and then another circle was walking in 3 minutes and 30 seconds. How much time spent Vasya for the whole walk around the school?

This example is directed to the addition of mixed numbers, which in this case do not even have to be translated per second. It turns out that the addition is carried out by separately adding minutes and seconds. As a result, we obtain the following result:

1. Addition of minutes - 1 + 3 \u003d 4.
2. Addition Seconds \u003d 30 + 30 \u003d 60 seconds \u003d 1 minute.
3. General value 4 minutes + 1 minute \u003d 5 minutes.

If you proceed from the mathematical display, the presented actions can be allocated in one expression:

From the above, it becomes clear that it is necessary to add mixed numbers individually in parts - first integers, and then fractional. If a fractional number gives an integer value, it is also folded with the whole previously obtained value. The fractional part is added to the resulting whole value - a mixed number is obtained.

Rules of addition

To secure the studied, the rule of addition of mixed numbers should be brought. Here you should use the following sequence:

1. To begin with, separating from the value of the part - to the whole and fractional.
2. Now folded the whole parts.
3. Further fold fractional.
4. If one can extract from a fractional number - to translate into a mixed value - it means that the breakdown is carried out.
5. The resulting whole part of the fractional value is folded with an integer previous value.
6. The whole part is added fractional.

For explanation, you should bring some examples:

The addition of mixed numbers occurs on the same algorithm as the subtraction, so the next action will be considered in detail.

Substitution rules

As in the first case, to subtract mixed values \u200b\u200bthere is a rule, but it is radically different from the previous sequence. So, here should adhere to the sequence:

1. An example for subtraction is represented in the form: Reduced - subtractable \u003d difference.
2. Due to the reduced equation, fractional parts of the numbers presented should be compared.
3. If a decreasing fractional part is greater, it means that subtraction is carried out on the same basis as when adding - the whole are first subtracted, and then fractional values. Both results are folded.
4. If a decreasing fractional value is less, it means that they are pre-transferred to the wrong fraction and carry out a standard subtraction.
5. Of the difference, the whole part and fractional are determined.

To explain to clarify the following examples:

From the submitted article, it became clear how to make addition and subtract mixed numbers. In the example described above, it can be seen that it does not always have to modify the numbers - to translate them from simple fractions to complex. It is often enough to simply fold or subtract entire and fractional values \u200b\u200bseparately that for a person with extensive experience can be easily held in mind.

The article describes in detail the examples of which are presented in full compliance with the mathematical rules and bases. Separate situations are disassembled, each given an example of modifications that can be encountered in solving tasks and complex examples.

\u003e\u003e Mathematics: addition and subtraction of mixed numbers-grade 6

12. Addition and subtraction of mixed numbers

Movement and combination properties of addition make it possible to reduce the addition of sour cream numbers to the addition of their integer parts and the addition of their fractional parts.
Example 1. Find the value of the amount
Decision. Let us give fractional parts of the numbers to the smallest total 8, then imagine mixed numbers in the form of their whole and fractional part:

Example 2. Find the value of the amount.
Decision. First, we give fractional parts of these numbers to the smallest common denominator 12, after separately we fold entire and fractional parts:

To fold mixed numbers, it is necessary:

1) bring fractional parts of these numbers to the smallest common denominator;

2) separately perform addition of integer parts and separate fractional parts.

If when the fractional parts are addition, it turned out the wrong fraction, to highlight the whole part of this fraction and add it to the resulting integer part.

When subtracting mixed numbers, use the properties of subtracting the amount from the number and subtraction of the number from amount .

Example 3. Find the difference value.
Decision. Let us give fractional parts to the smallest common denominator 18 and present the data of the number in the form of the amount of the whole and fractional part:

Written in short:

If the fractional part of the decreased will be less than the fractional part of the subtracted, then one unit of the whole part of the reduced is to turn into a fraction with the same denominator.

Example 4. Find the difference value

Decision. We give fractional parts of these numbers to the smallest general denominator 18:

Since the fractional part of the reduced less fractional part of the subtracted, then the reduced is written as follows:

To subtract mixed numbers, it is necessary: \u200b\u200b1) to bring fractional parts of these numbers to the smallest common denominator; If a fractional part of a decreased less fractional part of subtractable, turn it into the wrong fraction, reducing the unit to the unit; 2) separately perform the subtraction of integers and separate fractional parts.

? Tell me how to fold mixed numbers And on what properties of the addition is the addition of mixed numbers. Tell us how to perform the subtraction of mixed numbers and on what properties is based on the subtraction rule of mixed numbers.

TO 363. Perform addition:

364. Perform subtraction:

365. Find the value of the expression:

366. Perform an action:

368. Find according to the formula :

369. The school pool is filled through the first pipe for 4 hours, and through the second for 6 hours. What part of the basin will remain filled after the joint work of both pipes within an hour?

370. New car Can dig a ditch for 8 hours, and old - for 12 hours. The new car worked for 3 hours, and the old 5 hours. What part of the ditch left to dig?

371. From the tape with a length of 8 m cut off a piece of length m. Find length The remaining part.

372. One chess party lasted hours, and the other hour. How long did the third party lasted if all three parties were spent 3 hours?

373. When the piece was cut off the rope, the remaining part had a length of 2 m. What length would be the remaining part, if it were cut from the rope to M less? on M greater?

374. Record all the numbers, the denominator of the fractional part of which is 12, large and smaller.

375. The coordinate beam marked the point (Fig. 17). Mark on the beam point, coordinates which are equal:

376. Find the perimeter of the ABC triangle if Av \u003d m, .

377. On one car T cargo, and on the other by t less. How many tons of cargo on two machines?

378. In one box kg grapes, which is less than in another drawer. How many kilograms of grapes in two boxes?

379. CG Paints spent on coloring windows. On the color of the doors went to kg less than on the color of the floor. How much did the paints spent if kg row on the color of the floor?

380. Three collective farm areas rated peas on square ha. The first and second links were growing peas in the Square, and the second and third - on the square hectare. Find the area of \u200b\u200beach site.

381. At the Sugar Plant on Monday, the beets were brought, on Tuesday - 2 tons are more than on Monday, and on Wednesday, it is less than on Tuesday and Monday together. Of the 7 tons of beets, 1 t sugar is obtained. How much sugar will come from brought beet?

382. In three bidones 10 liters of milk. In the first and second bidone there were l, and in the second and third l of milk. How many milk liters was in each bidon?

383. The motor ship for the river takes place km in 1 hour. The flow rate of the KM / h. Find the speed of the ship against the flow.

384 Boat speed for the river KM / H, and against the flow of km / h. What is the flow rate?

385. Fedya and Vasya went towards each other. Each hour the distance between them has decreased by km. Find the speed of Fedi if Vasi speed

386. The first cyclist caught up with the second, and the distance between them decreased every hour per km. What speed was the first cyclist, if the second was driving at a speed y km / h?

P 388. Calculate orally:

389. Find the passed numbers:

390. Include natural values \u200b\u200bM, at which the inequality is true:

391. How much percent will increase the volume of the cube, if the length of each of its rib increase by 20%?

392. The postal aircraft rose from the airfield at 10 h 40 min in the morning, stayed in flight 5 h 15 min, and on Earth during landing 1 h 37 min. When is the plane returned to the airfield?

M.393. Quadrangle S. equal parties They call rombe visas (Fig. 18). Think whether the rhombus is the right polygon. What is the similarity of solving this problem with finding solutions of double inequality 0< у<. 10 среди чисел 0,12; 15; 2,7; 10,5?

394. Prove the moving and combination of addition properties for fractions with the same denominators based on the same properties for natural numbers.

395. Perform an action:

396. In the kiosk for sale, the brands of 3 k., 5 to 5 k. And 10 k. The number of stamps of each species was equally. What is the cost of all brands of 5 k., If: a) the total value of all grades 21 p. 60 k., B) the cost of all brands of 10 k. More cost All brands of 3d to 6 r. 30 k.?

397. Perform calculations using the microcalculator and the result round up to thousandth:

3,281 0,57 + 4,356 0,278 -13,758:6,83.

398. Decide the task:

1) A lime-sulfur decoction is prepared to combat the pests of the gardens, consisting of 6 parts of sulfur, 3 parts for the oversized lime and 50 parts of water (by weight). How much will it turn out kilograms Beam, if water take 8.8 kg more than sulfur?

2) For the preparation of porcelain on 1 part of the gypsum take 2 parts sand and 25 parts of clay (by weight). How much will the porcelain kilograms turn out if you take clays by 6.9 kg more than the sand?

399. Perform actions:

1) 7225:85 + 64 2345-248 838:619;
2) 54 3465-9025:95 + 360 272:712.

D. 400. Perform an action:

but
401. Find the difference value:

402. Decide equation:

404. One tractor driver plowed the field, and the other field. What part of the field remained?

406. Fuel barrels enough for work one engine for 7 hours, and another 5 hours. What part of the fuel will remain from the full barrel after 2 hours of operation of the first engine and 3 hours of operation of the second engine?

406. For an expedition running in the taiga, the packaging with products that fell to the ground through 3 s was dropped from a helicopter. Which height was dropped by this packaging, if it flew in first second, and in every next second she flew on M greater than in the previous one?

407. How long did it go to the manufacture of the part if it was treated on a turning machine h, on the milling machine h and on the drilling machine h?

408. Find the value of the expression:

409. Of the two villages at the same time, two pedestrians came to each other and met after 1.5 hours. The distance between the villages is 12.3 km away. The speed of one pedestrian is 4.4 km / h. Find the speed of another pedestrian.

410. For the preparation of cherry jam on 3 parts of the sugar take 2 parts of the berries (by mass). How many kilograms of sugar and how many kilograms of berries need to take to get 10 kg of jam if during cooking it will decrease by 1.5 times?

411. Find the value of the expression:

a) (44.96 + 28.84: (13,7-10.9)): 1.8;

b) 102,816: (3.2 6.3) + 3.84.

412. Decide equation:

a) (x-4,7) 7.3 \u003d 38.69; c) 23,5- (2, for + 1,2A) \u003d 19.3;
b) (3,6-a) 5.8 \u003d 14.5; d) 12.98- (3.8x-1.3x) \u003d 11.23.

BUT The mathematics section in which the properties of numbers and actions are studied over them are called the theory of numbers.

The beginning of the creation of the theory of numbers was put ancient Greek scientists Pythagoras, Euclidean, Eratosthene and others.

Some problems of the theory of numbers are formulated very simply - they can understand any sixth grader. But the solution of these problems is sometimes so difficult that the century leaves for him, and there are still no answers to some answers. For example, ancient Greek mathematicians, only one pair of friendly numbers was known - 220 and 284. And only in the XVIII century. The famous mathematician, a member of the St. Petersburg Academy of Sciences Leonard Euler, found another 65 couples of friendly numbers (one of them 17 296 and 18,416). However, there is still no familiar way of finding steam friendly numbers.

Almost 250 years ago, the member of the St. Petersburg Academy of Sciences Christians Goldbach suggested that any odd number, more than 5, can be represented as the sum of three simple numbers. For example: 21 \u003d 3 + 7 + 11, 23 \u003d 5 + 7 + 11, etc.

To prove this assumption was only 200 years later, a wonderful Soviet mathematician, Academician Ivan Matveevich Vinogradov (1891-1983). But the statement "Any even number, greater than 2 can be represented as the sum of two simple numbers" (for example: 28 \u003d 11 + 17, 56 \u003d 19 + 37, 924 \u003d 311 + 613, etc.) has not yet been proven .

Objectives lesson:

• Repetition and fixing the main software, expressed in standard examples and non-standard tasks.
• Improving the skills of arithmetic operations folding and subtracting mixed numbers;
• Develop a mixture, thinking, speech, memory.
• Educating cognitive interest in the subject, love for search solutions.

Tasks lesson:

• Educational

- generalization and systematization of knowledge; development of the speed of thinking; develop the ability to analyze; Develop computing skills.

Developing

- develop cognitive processes in students, creative activity; Acquisition of research experience, the development of commutative qualities.

Educational

- formation of self-organization and independence skills; respectful relationship to each other.

Type of lesson: a lesson of generalization and systematization of knowledge.

Form of the lesson: partially searching with the elements of the didactic game.

Intergovernmental ties: biology.

Equipment lesson:

• poster;
• distribution material: cards with task;
• presentation on the subject of the lesson.

Application of health-saving technologies in the lesson:

• change of activities;
• the development of auditory and visual analyzers for each child.

Lesson plan

I. Organizational moment.

Hello. Sit down.

Presentation. Slide 1. Theme of the lesson: "Addition and subtraction of mixed numbers."

Objectives lesson:

• Repetition and fixing the main software, expressed in standard examples and non-standard tasks.
• Improving arithmetic action skills Folding and subtraction of mixed numbers, preparation for test work.

II. Actualization of reference knowledge.

On the board poster with the words of Laue.

Our lesson will pass under the motto of the French engineer - Physics Laue: "Education is what remains when everything learned is already forgotten."

Now you will show your knowledge of the addition and subtraction of ordinary fractions with different denominators, as well as the addition and subtraction of mixed numbers.

1) Remember the famous fables of I. Krylova "Dragonfly and Ant".

Dragonfly jump, summer red lost
Lookage did not have time, like winter rolls into the eyes.

A task. The dragonfly half of the red summer sleeve slept, the third part of the time was danced, the sixth part - sang. The rest of the time she decided to devote preparation for the winter. What part of the summer dragonfly was preparing for winter?

Answer: In the summer, the dragonfly did not prepare at all.

And now, remember the reduction of fractions:

Write out of these fractions those that can be reduced, and reduce:

Remember which fractions are called correct and what incorrect?

- Right fractions, those whose numerator is less than the denominator.
- Incorrect fractions, those whose numerator is more or equal to the denominator.

(Cards: read the fraction and call - the correct or incorrect fraction.)

How to highlight a whole part of the wrong fraction?

- The numerator must be divided into the denominator.

(Oral cards: allocate the whole part of the wrong fraction.)

III. Systematization of knowledge. Cards. Perform the addition and subtraction of ordinary fractions. Left examples, the answers on the right are recorded. Solving the example of the arrow of the relationship with the answer.

Slides 2-7. This amazing tree refers to the number of trees - giants. It grows in India and Malaysia.

The most unusual thing is how its branches grow. Numerous and heavy, they scatter in all directions from the trunk, although the mighty, but, nevertheless, not able to withstand them all on their own.

The whole focus is that the branches themselves take part of the load from it: each of them has thick processes, hanging out to the earth itself and representing nothing but the aerial roots of the tree.

Conscreasing in the ground, they not only provide branches with additional support, but also supply nutrients and water in them. Gradually, they turn into new trunks and ring-shaped "galleries" are formed around the main trunk, the diameter of which sometimes reaches 450 m.

Deciding the tasks, as well as the calculation of the values \u200b\u200bof expressions, replace the number with the corresponding letters and you will learn the name of this tree.

Solve the task:

Calculate the values \u200b\u200bof the expression:

Answer: Banyan.

Total lesson: We prepared for control work. For this, we and we repeated the addition and subtraction of fractions, as well as mixed numbers. Do not forget to cut the fractions that turned out as a result of addition and subtraction, and do not forget to allocate the whole part.

House. Task: § 2, paragraph 12 № 392.

If you have time, perform additional tasks.

Additional task:

• Solve the equation:

Cards:

Perform the addition and subtraction of ordinary fractions.

_________________________________________

Solve the task:

Calculate the values \u200b\u200bof the expression:

Samoenalysis of the lesson of mathematics in 6 "a" class.

The subject of the lesson: addition and subtraction of mixed numbers.

Type of lesson: Generalization lesson and systematization of knowledge.

Form of the lesson: partially searching with the elements of the didactic game.

1) This is a lesson for repetition and fixing the main software, but only expressed in solving standard examples and non-standard tasks. In this lesson, we repeated arithmetic actions (addition, subtraction) over ordinary fractions and above mixed numbers. These topics are studied in the course of grade 6 mathematics. When studying mathematics, there is a lot of time to spend on the workout of various skills. During this period, students lose interest in the subject. To support this interest, I use various techniques to activate students in the lesson. One of these techniques is the didactic game. It allows you to make the learning process exciting, create high activity in the lesson. The next lesson will be a test. I believe that this lesson "gave" positive emotions from the guys, the arithmetic actions over mixed numbers were worked out and tuned to the test work.

2) in the class on the list - 19 students, attended by the lesson - 16 students. Sleepy - 4, strong - 1.

3) educational - generalization and systematization of knowledge; development of the speed of thinking; introducing a gaming situation to remove nervously - mental stress; develop the ability to analyze; Develop computing skills.
Developing - develop cognitive processes in students, creative activity; Acquisition of research experience, the development of commutative qualities.
Educational- formation of self-organization and independence skills; respectful relationship to each other.
The attention of the guys is being activated unobtrusively, interest in the subject, creative fantasy develops.

4) One of the successful stages of the lesson, I consider the solution of tasks and examples where it was necessary to make a word Banyan. Students, as it were, are engaged in mathematics and at the same time expand their horizons.

5) The lesson was saturated. The lesson is very logical built.

6) The lesson was made by me, as a teacher a lot of handouts, which I printed on the computer.

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Signatures for slides:

Mathematics teacher Kuznetsova Marina Nikolaevna Adjustment and subtraction of mixed numbers

Homework

Astrid Lindgren

Oral account 1 0

What groups can we divide these fractions?

What groups can we divide these fractions? Correct fractions incorrect fractions

Find an extra example:

Addition and subtraction of mixed numbers. The purpose of the lesson: learn to perform addition and subtract mixed numbers.

Help 1. To the whole part of adding a whole part. To the resulting whole part of adding a fractional part. Formulate the rule of addition of a mixed number with natural. 2. To the whole part of adding a whole part. To the fractional part, add a fractional part to the resulting integer part to add the resulting fractional part. Formulate the rule of addition of mixed numbers. 3. From the whole part of the deduction integer. From the fractional part of the fractional part to the remaining whole part of adding the remaining fractional part. Formulate a rule of subtraction of mixed numbers. 4. If the fractional part of the reduced less fractional part is subtracted. We occupy in the whole part of a reduced unit and present it as an incorrect fraction. The resulting fraction is folded with a fractional part of the reduced. We deduct separately parts and fractional parts. By the remaining whole part, we add the remaining fractional part. Formulate a rule of subtraction from a mixed number of fraction, and the fraction of the reduced more fraction is subtracted.

To fold two mixed numbers, you need to fold separately their entire and fractional parts, fold the results obtained. To subtract a mixed number from a mixed number, you need to separately deduct their entire and fractional parts, fold the results obtained.

= (3 + 2) + () = 5 + = 5 – = (5 – 3) + ()= 2 + = 2

Fizkultminutka was worked out - rest, stand up, deeply sigh. Hands to the side, forward, left, right turn. Three tilt, straight up. Hands down and up lift. Hands smoothly lowered, all the smiles presented.

4 - in 7 - about 3 - in 4 - e 5 - x 4 - p 5 - with y with p e in x about

Solving tasks page. 175, No. 1115 p. 175, № 1116

What is a mixed number? What have you learned today? How to fold mixed numbers? How to subtract mixed numbers?

Homework: P. 29 (Learning Rules) p. 178, № 1136, 1137

Thank you for the lesson!

Preview:

Mathematics teacher Kuznetsova M.N.

Lesson in grade 5 on the topic:

Addition and subtraction of mixed numbers.

Objectives:

Training:

1. Generate students with algorithms of addition and subtraction of mixed numbers by incorporating students into practical activities.
2. Continue work on the development of computing skills.

Developing:

1. Development of the ability to solve the tasks of the studied species.
2. Creating conditions for the formation of thought operations.

Educational:

1. Relieve a sense of partnership and mutual execution.

During the classes

I. Organizational moment.

Look, all right:

Book, handles and notebooks.

The call now called.

The lesson begins.

II. Check your homework.

Date, cool job.

At home you performed a task. You solved the rebus. (Slide 1) and what is the answer? (Astrid Lindgren) (Slide 2)

D / s. 1. Allocate the whole part and arrange in ascending order. 18th 7 -A 14 -R 11 -T 9 -C 21 -D 5 5 5 5 5 5 1 2/5 1 4/5 2 1/5 2 4/5 3 3/5 4 1/5 And with t p and d 2. Write in the form of incorrect fraction and decipher. 41/2-d 2 3/7-H 4 9/10-p 32/5-and 14/6-g 2 2/8-E 3 ¾ -L 5 1/6-H 15 4 17 5 17 7 9 2 10 6 49 10 20 8 31 6 L. AND N. D. G. R E. N.

And who is Astrid Lindgren? What fairy tale wrote this Swedish writer? ("Kid and Carlson") (Slide 3)

But unfortunately, Carlson flew away, but left a letter.

Letter: Guys, I flew to look for diligent, attentive, hardworking, friendly, who know how to help the guys. I will find - come back.)

Guys, let's meet faster with the other, for this we will perform mathematical tasks. If we fulfill them correctly, we have to return Carlson - sweet tooths will get a large total cake. And everyone has their own.

First task.

III. Verbal counting

1. Chain solution (p. 175, No. 1111).

2/5 + 1/5 + 2/5 – 3/7 – 1/7 = 3/7

5/17 + 7/17 – 12/17 + 7/9 – 4/9 = 3/9

2. On which groups we can split these fractions: (correct and incorrect fractions) (slide 6)

9 5 8 10 24 15 7 12

8 12 11 6 13 16 7 25

What fractions are called correct?

What fractions are called wrong?

How to imagine the wrong fraction?

What makes a mixed number?

(Piece of cake.)

IV. Actualization of knowledge.

Find an extra example:

2/8 + 3/8 14/12 – 7/12 7/9 + 1/9 3 1/7 + 2 3/7 18/27 -5/27

Try to formulate theme lesson (addition of mixed numbers) (slide8)

Today, at the lesson, we will learn to perform the addition and subtraction of mixed numbers, to achieve this goal we formulate the rules.

V. Study

Students work in groups by performing tasks of various complexity. All students are divided into 4 groups. The desk of each group gives a task and reference material. To solve the task, you need to select a corresponding rule.

Exercise 1 . Completion of addition 2 ½ + 3

Task 2. Addition of addition 2 1/4 + 1 2/4

Task 3. . Execution 3 5/6 - 3/6

Task 4. Execution 5 1/4 - 3 2/4

reference

1. To the resulting whole part of adding a fractional part.
2. Formulate the rule of addition of a mixed number with natural.

1. To the whole part of adding a whole part.
2. To fractional part add fractional part
3. To the resulting whole part of adding the resulting fractional part.
4. Formulate the rule of addition of mixed numbers.

1. From the whole part of subtracting the whole part.
2. From the fractional part of the fractional part
3. To the remaining whole part of adding the remaining fractional part.
4. Formulate a rule of subtraction of mixed numbers.

1. If the fractional part of the reduced less fractional part is subtracted.
2. We occupy in the whole part of a reduced unit and present it as an incorrect fraction.
3. The resulting fraction is folded with a fractional part of the reduced.
4. We deduct separately parts and fractional parts.
5. By the remaining whole part, we add the remaining fractional part.
6. Formulate a rule of subtraction from a mixed number of fraction, and the fraction of the reduced more fraction is subtracted.

Vi. Information exchange.

You reviewed the rules for the addition and subtraction of mixed numbers. What is common with them? (Actions are performed first with integers, then with fractional parts.)

Formulate the rule of addition of mixed numbers. (Slide 9)

Word a rule of subtraction of mixed numbers. (Slide 10)

P. 174 textbooks, rule

(Piece of cake.)

VII. Application

- Let's go back for example:

3 1/7 + 2 3/7= (3+2)+(1/7+3/7)=5+4/7=54/7

How to make sure that the addition is done correctly? (Subtraction). Make check.

54/7-31/7=(5-3)+(4/7-1/7)= 2+3/7= 23/7

(Piece of cake.)

VIII. Fizkultminutka(Slide)

Worked out - rest

Stand up, deeply sigh.

Hands to the side, forward,

Left, right turn.

Three tilt, straight up.

Hands down and up lift.

Hands smoothly lowered

All smiles presented.

IX. Fixing the material studied

1. Carlson sent a telegram, but all the words were confused. Let's solve examples and relate them to the answers. (Slide 11)

3 7/13 - 4/13 \u003d 4 - in

5 2/5 + 1/5 \u003d 7 4/6 - about

10 2/3-6 \u003d 3 3/13 -

2 2/7 + 2 4/7 \u003d 4 6/7 - e

8 5/9-3 \u003d 5 5/9 x

3/6 + 7 1/6 \u003d 4 2/3 - n

7 4/5-3 4/5 \u003d 5 3/5 - with

(Piece of cake.)

"Hunt for fives"

2. Work on the tasks.

a) p. 175, №1115.

1. Read the task.
2. How many candies are in one box?
3. How many candies in another box?
4. How to answer the question of the task?
5. Decide the task. Read the answer. (Two boxes 4 4/8 kg of sweets.)

b) p. 175, № 1116.

1. What is the length of the red ribbon?
2. What is said about white length?
3. What does it mean by 2 1/5 m in short?
4. How will you solve this task?

Decide. Read the answer. (White tape length 1 2/5 meters.)

(Piece of cake.)

You are wonderful students: diligent, attentive, friendly, help each other.

(Carlson flew) Carlson saw that you guys were looking for, and returned. We give him a cake.

X. The result of the lesson (Karosoneon questions).

1. What is a mixed number?
2. What have you learned today? (Fold and deduct mixed numbers.)
3. How to fold mixed numbers?
4. How to subtract mixed numbers?

It will help you to cope with the homework.

Xi. Homework:P. 178, № 1136,137

XII. Reflection.

Collect earned pieces in a cake. (3-5 parts - "5")

The teacher evaluates the work of students. (Mord). (Slide 13)

Mixed fractions as well as simple fractions can be subtracted. To take away mixed numbers of fractions you need to know several deduction rules. We study these rules on the examples.

Subtraction of mixed fractions with the same denominators.

Consider an example with the condition that the reduced integer and the fractional part is more respectively submitted by the whole and fractional part. Under such conditions, subtraction occurs separately. We deduct the whole part from the integer part, and the fractional part of the fractional.

Consider an example:

Perform the subtraction of mixed fractions \\ (5 \\ FRAC (3) (7) \\) and \\ (1 \\ FRAC (1) (7) \\).

\\ (5 \\ FRAC (3) (7) -1 \\ FRAC (1) (7) \u003d (5-1) + (\\ FRAC (3) (7) - \\ FRAC (1) (7)) \u003d 4 \\ The correctness of subtraction is checked by adding. Let's check the subtraction:

\\ (4 \\ FRAC (2) (7) +1 \\ FRAC (1) (7) \u003d (4 + 1) + (\\ FRAC (2) (7) + \\ FRAC (1) (7)) \u003d 5 \\ Consider an example with the condition when the fractional part of the decreased less than the fractional part of the subtracted one respectively. In this case, we occupy a unit in the whole in a decrease.

Perform the subtraction of mixed fractions \\ (6 \\ FRAC (1) (4) \\) and \\ (3 \\ FRAC (3) (4) \\).

{!LANG-abe136259091caeaf317bfcd5118eb67!}

Consider an example:

{!LANG-aa1a06d264ee9b5ab6b9c6bdd7d67d04!}

In the reduced \\ (6 \\ FRAC (1) (4) \\), the fractional part is less than that of the fractional part of the subtracted \\ (3 \\ FRAC (3) (4) \\). That is, \\ (\\ FRAC (1) (4)< \frac{1}{3}\), поэтому сразу отнять мы не сможем. Займем у целой части у 6 единицу, а потом выполним вычитание. Единицу мы запишем как \(\frac{4}{4} = 1\)

\\ (\\ begin (align) & 6 \\ FRAC (1) (4) -3 \\ FRAC (3) (4) \u003d (6 + \\ FRAC (1) (4)) - 3 \\ FRAC (3) (4) \u003d (5 + \\ COLOR (RED) (1) + \\ FRAC (1) (4)) - 3 \\ FRAC (3) (4) \u003d (5 + \\ Color (Red) (\\ FRAC (4) (4)) + \\ FRAC (1) (4)) - 3 \\ FRAC (3) (4) \u003d (5 + \\ FRAC (5) (4)) - 3 \\ FRAC (3) (4) \u003d \\\\\\\\ & \u003d 5 \\ FRAC (5) (4) -3 \\ FRAC (3) (4) \u003d 2 \\ FRAC (2) (4) \u003d 2 \\ FRAC (1) (4) \\\\\\\\\\ End (Align) \\)

Next example:

\\ (7 \\ FRAC (8) (19) -3 \u003d 4 \\ FRAC (8) (19) \\)

Subtraction of mixed fraction from an integer.

Example: \\ (3-1 \\ FRAC (2) (5) \\)

Reduced 3 does not have a fractional part, so we cannot immediately take away. We will take the whole part of 3 units, and then perform subtraction. Unit we will write as \\ (3 \u003d 2 + 1 \u003d 2 + \\ FRAC (5) (5) \u003d 2 \\ FRAC (5) (5) \\)

\\ (3-1 \\ FRAC (2) (5) \u003d (2 + \\ Color (Red) (1)) - 1 \\ FRAC (2) (5) \u003d (2 + \\ Color (Red) (\\ FRAC (5 ) (5))) - 1 \\ FRAC (2) (5) \u003d 2 \\ FRAC (5) (5) -1 \\ FRAC (2) (5) \u003d 1 \\ FRAC (3) (5) \\)

Subtraction of mixed fractions with different denominators.

Consider an example with the condition if fractional parts of the reduced and subtracted with different denominants. You need to lead to a common denominator, and then perform subtraction.

Perform the subtraction of two mixed fractions with different denominators \\ (2 \\ FRAC (2) (3) \\) and \\ (1 \\ FRAC (1) (4) \\).

The total denominator will be the number 12.

\\ (2 \\ FRAC (2) (3) -1 \\ FRAC (1) (4) \u003d 2 \\ FRAC (2 \\ Times \\ Color (Red) (4)) (3 \\ Times \\ Color (Red) (4) ) -1 \\ FRAC (1 \\ Times \\ Color (Red) (3)) (4 \\ Times \\ Color (RED) (3)) \u003d 2 \\ FRAC (8) (12) -1 \\ FRAC (3) (12 ) \u003d 1 \\ FRAC (5) (12) \\)

Questions on the topic:
How to deduct mixed fractions? How to solve mixed fractions?
Answer: You need to decide which type of expression and according to the type of expression to apply the solution algorithm. From the whole part, we subtract the whole, in the fractional part we subtract the fractional part.

How from an integer deduction fraction? How to take a fraction from an integer?
Answer: In an integer, you need to take a unit and write this unit in the form of a fraction

\\ (4 \u003d 3 + 1 \u003d 3 + \\ FRAC (7) (7) \u003d 3 \\ FRAC (7) (7) \\),

and then a whole take away from the whole, the fractional part from the fractional part. Example:

\\ (4-2 \\ FRAC (3) (7) \u003d (3 + \\ Color (RED) (1)) - 2 \\ FRAC (3) (7) \u003d (3 + \\ Color (Red) (\\ FRAC (7 ) (7))) - 2 \\ FRAC (3) (7) \u003d 3 \\ FRAC (7) (7) -2 \\ FRAC (3) (7) \u003d 1 \\ FRAC (4) (7) \\)

Example number 1:
Perform subtraction of the correct fraction from one: a) \\ (1- \\ FRAC (8) (33) \\) b) \\ (1- \\ FRAC (6) (7) \\)

Decision:
a) Imagine a unit as a fraction with a denominator 33. We get \\ (1 \u003d \\ FRAC (33) (33) \\)

\\ (1- \\ FRAC (8) (33) \u003d \\ FRAC (33) (33) - \\ FRAC (8) (33) \u003d \\ FRAC (25) (33) \\)

b) Imagine the unit as a fraction with the denominator 7. We get \\ (1 \u003d \\ FRAC (7) (7) \\)

\\ (1- \\ FRAC (6) (7) \u003d \\ FRAC (7) (7) - \\ FRAC (6) (7) \u003d \\ FRAC (7-6) (7) \u003d \\ FRAC (1) (7) \\)

Example number 2:
Perform the subtraction of mixed fraction from an integer: a) \\ (21-10 \\ FRAC (4) (5) \\) b) \\ (2-1 \\ FRAC (1) (3) \\)

Decision:
a) We will take the integer in an integer 21 unit and the collapse so \\ (21 \u003d 20 + 1 \u003d 20 + \\ FRAC (5) (5) \u003d 20 \\ FRAC (5) (5) \\)

\\ (21-10 \\ FRAC (4) (5) \u003d (20 + 1) -10 \\ FRAC (4) (5) \u003d (20 + \\ FRAC (5) (5)) - 10 \\ FRAC (4) ( 5) \u003d 20 \\ FRAC (5) (5) -10 \\ FRAC (4) (5) \u003d 10 \\ FRAC (1) (5) \\\\\\\\\\)

b) We will take a number of 2 units and a split so \\ (2 \u003d 1 + 1 \u003d 1 + \\ FRAC (3) (3) \u003d 1 \\ FRAC (3) (3) \\)

\\ (2-1 \\ FRAC (1) (3) \u003d (1 + 1) -1 \\ FRAC (1) (3) \u003d (1 + \\ FRAC (3) (3)) - 1 \\ FRAC (1) ( 3) \u003d 1 \\ FRAC (3) (3) -1 \\ FRAC (1) (3) \u003d \\ FRAC (2) (3) \\\\\\\\\\)

Example number 3:
Perform the subtraction of an integer number of mixed fractions: a) \\ (15 \\ FRAC (6) (17) -4 \\) b) \\ (23 \\ FRAC (1) (2) -12 \\)

a) \\ (15 \\ FRAC (6) (17) -4 \u003d 11 \\ FRAC (6) (17) \\)

b) \\ (23 \\ FRAC (1) (2) -12 \u003d 11 \\ FRAC (1) (2) \\)

Example number 4:
Perform subtraction of the correct fraction of mixed fraction: a) \\ (1 \\ FRAC (4) (5) - \\ FRAC (4) (5) \\)

\\ (1 \\ FRAC (4) (5) - \\ FRAC (4) (5) \u003d 1 \\\\\\\\\\)

Example number 5:
Calculate \\ (5 \\ FRAC (5) (16) -3 \\ FRAC (3) (8) \\)

\\ (\\ begin (Align) & 5 \\ FRAC (5) (16) -3 \\ FRAC (3) (8) \u003d 5 \\ FRAC (5) (16) -3 \\ FRAC (3 \\ Times \\ Color (Red) ( 2)) (8 \\ Times \\ COLOR (RED) (2)) \u003d 5 \\ FRAC (5) (16) -3 \\ FRAC (6) (16) \u003d (5 + \\ FRAC (5) (16)) - 3 \\ FRAC (6) (16) \u003d (4 + \\ Color (Red) (1) + \\ FRAC (5) (16)) - 3 \\ FRAC (6) (16) \u003d \\\\\\\\ & \u003d (4 + \\ Color (Red) (\\ FRAC (16) (16)) + \\ FRAC (5) (16)) - 3 \\ FRAC (6) (16) \u003d (4 + \\ Color (RED) (\\ FRAC (21 ) (16))) - 3 \\ FRAC (3) (8) \u003d 4 \\ FRAC (21) (16) -3 \\ FRAC (6) (16) \u003d 1 \\ FRAC (15) (16) \\\\\\\\ Did not find an answer to your question? Look at here