Subtraction of mixed fractions. Addition and subtraction of mixed numbers (Wolfson G.I.)

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Slides captions:

Mathematics teacher Kuznetsova Marina Nikolaevna Addition and subtraction mixed numbers

Homework

Astrid Lindgren

Mental arithmetic 1 0

What groups can we divide these fractions into?

What groups can we divide these fractions into? Proper fractions Improper fractions

Find an extra example:

Addition and subtraction of mixed numbers. Lesson Objective: To learn how to add and subtract mixed numbers.

Help 1. Add an integer part to the whole part. Add the fractional part to the resulting integer part. Formulate the rule for adding a mixed number with a natural number. 2. Add the whole part to the whole part. Add the fractional part to the fractional part Add the resulting fractional part to the resulting integer part. Formulate a rule for adding mixed numbers. 3. Subtract the whole part from the whole part. Subtract the fractional part from the fractional part Add the remaining fractional part to the remaining integer part. Formulate a rule for subtracting mixed numbers. 4. If the fractional part of the minuend is less than the fractional part of the subtrahend. We borrow one from the integer part of the reduced and represent it as an improper fraction. The resulting fraction is added to the fractional part of the reduced. Subtract whole parts and fractional parts separately. To the remaining integer part, add the remaining fractional part. Formulate a rule for subtracting a fraction from a mixed number, and the fraction of the reduced is greater than the fraction of the subtrahend.

To add two mixed numbers, you need to add their integer and fractional parts separately, add the results. To subtract a mixed number from a mixed number, you need to separately subtract their integer and fractional parts, add the results.

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

Fizkultminutka Work hard - let's have a rest, Let's get up, take a deep breath. Hands to the sides, forward, left, right turn. Three bends, stand up straight. Raise your hands up and down. Hands smoothly lowered, All smiles were presented.

4 - V 7 - O 3 - U 4 - E 5 - X 4 - P 5 - S U P E V X O

Problem Solving Page 175, No. 1115 Page 175, no. 1116

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

Homework: P. 29 (learn the rules) Pg. 178, No. 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.

Goals:

Training:

1. To introduce students to the algorithms for adding and subtracting mixed numbers by including students in practical activities.
2. Continue to work on the development of computing skills.

Developing:

1. Development of the ability to solve problems of the studied types.
2. Creation of conditions for the formation of mental operations.

Educational:

1. Cultivate a sense of camaraderie and mutual assistance.

During the classes

I. Organizational moment.

See if everything is ok:

Book, pens and notebooks.

The bell has rung now.

The lesson starts.

II. Checking homework.

Date, great job.

You have completed the task at home. You solved the puzzle. (Slide 1) And what is the answer? (Astrid Lindgren) (Slide 2)

D / s. 1. Select the whole part and arrange in ascending order. 18 -I 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 A S T R I D 2. Write it down as an improper fraction and decipher it. 41/2-D 2 3/7-N 4 9/10-R 32/5-I 14/6-G 2 2/8-E 3 ¾ -L 5 1/6-N 15 4 17 5 17 7 9 2 10 6 49 10 20 8 31 6 L AND H D G R E H

Who is Astrid Lindgren? What fairy tale did this Swedish writer write? ("Baby and Carlson") (Slide 3)

But unfortunately Carlson flew away, but left a letter.

Letter: Guys, I flew to look for diligent, attentive, hardworking, friendly guys who can come to the rescue. If I find it, I'll be back.)

Guys, let's meet a friend faster, for this we will complete math tasks. If we do them correctly, then we will have a big common cake by the return of Carlson - the sweet tooth. And everyone has their own little one.

First task.

III. Verbal counting

1. Solving chains (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. What groups can we divide these fractions into: (proper and improper fractions) (Slide 6)

9 5 8 10 24 15 7 12

8 12 11 6 13 16 7 25

What fractions are right?

What fractions are called improper?

What is another way to represent improper fractions?

What is a mixed number?

(Piece of cake.)

IV. Knowledge update.

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 the topic of the lesson (Addition of mixed numbers) (Slide 8)

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

V. Research

Students work in groups to complete tasks of varying difficulty. All students are divided into 4 groups. A task is distributed to the desk of each group and reference material. To solve the task, you need to select the appropriate rule.

Exercise 1 . Perform addition 2 ½ + 3

Task 2. Perform addition 2 1/4 + 1 2/4

Task 3 . Subtracting 3 5/6 – 3/6

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

Reference

1. Add the fractional part to the resulting integer part.
2. Formulate the rule for adding a mixed number with a natural number.

1. Add the whole part to the whole part.
2. Add fractional part to fractional part
3. Add the resulting fractional part to the resulting integer part.
4. Formulate a rule for adding mixed numbers.

1. Subtract the integer part from the integer part.
2. Subtract the fractional part from the fractional part
3. To the remaining integer part, add the remaining fractional part.
4. Formulate a rule for subtracting mixed numbers.

1. If the fractional part of the minuend is less than the fractional part of the subtrahend.
2. We borrow one from the integer part of the reduced and represent it as an improper fraction.
3. The resulting fraction is added to the fractional part of the reduced.
4. Subtract whole parts and fractional parts separately.
5. To the remaining integer part, add the remaining fractional part.
6. Formulate a rule for subtracting a fraction from a mixed number, and the fraction of the reduced is greater than the fraction of the subtrahend.

VI. Information exchange.

You've covered the rules for adding and subtracting mixed numbers. What do they have in common? (Actions are performed first with integers, then with fractional parts.)

Formulate a rule for adding mixed numbers. (Slide 9)

Formulate a rule for subtracting mixed numbers. (Slide 10)

Page 174 textbooks, rule

(Piece of cake.)

VII. Application

- Let's go back to the example:

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

How to make sure the addition is done correctly? (by subtraction). Make a check.

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

(Piece of cake.)

VIII. Physical education minute(Slide)

We tried - let's have a rest,

Let's get up, take a deep breath.

Hands to the side, forward

Left, right turn.

Three bends, stand up straight.

Raise your hands up and down.

Hands slowly lowered

Everyone was given smiles.

IX. Consolidation of the studied material

1. Carlson sent a telegram, but all the words were mixed up. Let's solve the examples and match them with the answers. (Slide 11)

3 7/13 - 4/13= 4 - B

5 2/5+1/5= 7 4/6 - O

10 2/3-6= 3 3/13 - U

2 2/7+2 4/7= 4 6/7 - E

8 5/9-3= 5 5/9 - X

3/6+7 1/6 = 4 2/3 - P

7 4/5-3 4/5= 5 3/5 - C

(Piece of cake.)

"The Hunt for Fives"

2. Work on tasks.

a) Page 175, No. 1115.

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

b) Page 175, no. 1116.

1. What is the length of the red ribbon?
2. What is said about the length of white?
3. What does 2 1/5 m shorter mean?
4. How will you solve this problem?

Decide. Read the answer.(The length of the white tape is 1 2/5 meters.)

(Piece of cake.)

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

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

X. Lesson Summary (questions of Carloson).

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

This will help you with your homework.

XI. Homework: Page 178, No. 1136,1137

XII. Reflection.

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

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

Solving complex examples correctly is an overwhelming task for those who do not understand elementary rules and laws in mathematics. Addition and subtraction of mixed numbers can rightfully be attributed to complex examples. However, at correct parsing numbers themselves, you can easily carry out any action.

What it is?

A mixed number is a combination of an integer part and a fractional part. For example, there are 2 and 3, of which 2 is a prime number, but 3 is already mixed, where 3 is whole part, and is fractional. The presented varieties are added and subtracted in different ways, but do not entail difficulties in independent decision examples.

Complete analysis of the example

For a full presentation of the essence of a mixed meaning, one should give an example of a task that will help to display the meaning of the narration conceived. So, Vasya cycled around the school in 1 minute and 30 seconds, and then walked another circle in 3 minutes and 30 seconds. How much time did Vasya spend on the whole walk around the school?

This example is aimed at adding mixed numbers, which in this case do not even have to be converted to seconds beforehand. It turns out that addition is carried out by separately adding minutes and seconds. As a result, we get the following result:

1. The addition of minutes is 1+3=4.
2. Adding seconds = 30+30=60 seconds = 1 minute.
3. General value 4 minutes + 1 minute = 5 minutes.

If we proceed from the mathematical display, then the presented actions can be distinguished in one expression:

From the above, it becomes clear that mixed numbers should be added separately in parts - first the whole parts, and then the fractional ones. If a fractional number still gives an integer value, it is also added to the integer value obtained earlier. The fractional part is added to the resulting integer value - a mixed number is obtained.

Addition rules

To consolidate what has been learned, the rule for adding mixed numbers should be given. Here you should use the following sequence:

1. To begin with, separate the parts from the value - into integer and fractional.
2. Now fold the whole pieces.
3. Next, add the fractions.
4. If you can still extract an integer part from a fractional number - convert it to a mixed value - then a similar breakdown is carried out.
5. The integer part obtained from the fractional value is added to the integer value previously obtained.
6. The fractional part is added to the whole part.

To clarify, a few examples should be given:

Addition of mixed numbers follows the same algorithm as subtraction, so the following action will be discussed in detail below.

subtraction rules

As in the first case, there is a rule for subtracting mixed values, but it is fundamentally different from the previous sequence. So, here you should follow the sequence:

1. An example for subtraction is presented in the form: reduced - subtracted = difference.
2. In connection with the above equation, you should first compare the fractional parts of the numbers presented.
3. If the decrement has a larger fractional part, then the subtraction is carried out according to the same criterion as during addition - first integers are subtracted, and then fractional values. Both results add up.
4. If the reduced fractional value is less, then they are first converted into an improper fraction and a standard subtraction is carried out.
5. From the difference obtained, the integer part and the fractional part are determined.

For clarification it should be following examples:

From the presented article, it became clear how to add and subtract mixed numbers. In the example described above, it can be seen that it is not always necessary to modify numbers - to transfer them from simple fractions to complex ones. Often it is enough to simply add or subtract whole and fractional values ​​​​separately, which for a person with more experience can be easily carried out in the mind.

The article discusses in detail the examples, the solution of which is presented in full accordance with the mathematical rules and foundations. Separate situations are analyzed, for each an example of modifications that can be encountered in solving problems and complex examples is given.

>>Math: Addition and subtraction of mixed numbers-Grade 6

12. Addition and subtraction of mixed numbers

The commutative and associative properties of addition make it possible to reduce the addition of sour numbers to the addition of their whole parts and to the addition of their fractional parts.
Example 1 Find the value of the sum
Solution. We reduce the fractional parts of the numbers to the least common 8, then represent the mixed numbers as the sum of their integer and fractional parts:

Example 2 Let's find the value of the sum.
Solution. First, we bring the fractional parts of these numbers to the lowest common denominator 12, then add the integer and fractional parts separately:

To add mixed numbers:

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

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

If, when adding the fractional parts, an improper fraction is obtained, select the integer part from this fraction and add it to the resulting integer part.

When subtracting mixed numbers, use the properties of subtracting a sum from a number and subtracting a number from amounts .

Example 3 Let's find the value of the difference.
Solution. We bring the fractional parts to the lowest common denominator 18 and represent these numbers as the sum of the integer and fractional parts:

Write briefly:

If the fractional part of the minuend turns out to be less than the fractional part of the subtrahend, then one unit of the integer part of the minuend must be converted into a fraction with the same denominator.

Example 4. Find the value of the difference

Solution. Let's bring the fractional parts of these numbers to the lowest common denominator 18:

Since the fractional part of the minuend is less than the fractional part of the subtrahend, we write the minuend as follows:

To perform the subtraction of mixed numbers, you must: 1) bring the fractional parts of these numbers to the lowest common denominator; if the fractional part of the minuend is less than the fractional part of the subtrahend, turn it into an improper fraction by decreasing the integer part by one; 2) separately perform the subtraction of integer parts and separate fractional parts.

? Tell me how to add mixed numbers and on what properties of addition the addition of mixed numbers is based. Explain how to subtract mixed numbers and what properties the rule for subtracting mixed numbers is based on.

TO 363. Perform addition:

364. Subtract:

365. Find the value of the expression:

366. Perform the action:

368. Find by formula :

369. The school pool is filled through the first pipe in 4 hours, and through the second in 6 hours. What part of the pool remains to be filled after the joint work of both pipes for an hour?

370. New car can dig a ditch in 8 hours, and the old one in 12 hours. The new machine worked 3 hours, and the old one 5 hours. How much of the ditch is left to dig?

371. A piece of length m was cut from a tape 8 m long. Find length the rest.

372. One chess game lasted an hour, and another hour. How long did the third game last if 3 hours were spent on all three games?

373. When a piece was cut from the rope, the remaining part had a length of 2 m. What length would the remaining part be if the rope was cut off m less? m more?

374. Write down all the numbers, the denominator of the fractional part of which is 12, greater and less.

375. A point is marked on the coordinate ray (Fig. 17). Mark points on the beam coordinates which are equal:

376. Find the perimeter of triangle ABC, if AB = m, .

377. There are tons of cargo on one machine, and tons less on the other. How many tons of cargo on two cars?

378. In one box there are kg of grapes, which is one kg less than in the other box. How many kilograms of grapes are in two boxes?

379. Kg of paint was used to paint windows. On the painting of the doors went on kg less than on the painting of the floor. How much paint was used in total if kg went to paint the floor?

380. Three collective-farm units have grown peas for area ha. The first and second links grew peas on a hectare area, and the second and third - on a hectare area. Find the area of ​​each lot.

381. Tons of beets were brought to the sugar factory on Monday, on Tuesday - 2 tons more than on Monday, and on Wednesday - ton less than on Tuesday and Monday together. From 7 tons of beets, 1 ton of sugar is obtained. How much sugar will be obtained from the imported beets?

382. There are 10 liters of milk in three cans. The first and second cans contained l, and the second and third l of milk. How many liters of milk were in each can?

383. The motor ship along the river passes km in 1 hour. The speed of the current is km / h. Find the speed of the boat against the current.

384 Boat speed downstream km/h, and against the current km/h. What is the speed of the current?

385. Fedya and Vasya walked towards each other. Every hour the distance between them decreased by km. Find Fedi's speed if Vasya's speed is

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

P 388. Calculate orally:

389. Find the missing numbers:

390. Find natural values ​​of m for which the inequality is true:

391. By what percent will the volume of a cube increase if the length of each of its edges is increased by 20%?

392. The mail plane took off from the airfield at 10:40 am, stayed in flight for 5 hours and 15 minutes, and on the ground during landings for 1 hour and 37 minutes. When did the plane return to the airfield?

M 393. Quadrilateral with equal parties called the VIZ rhombus (Fig. 18). Consider if the rhombus is a regular polygon. What is the similarity between solving this problem and finding solutions to the double inequality 0< у<. 10 среди чисел 0,12; 15; 2,7; 10,5?

394. Prove the commutative and associative properties of addition for fractions with the same denominators on the basis of the same properties for natural numbers.

395. Perform the action:

396. Stamps of 3 kopecks, 5 kopecks and 10 kopecks arrived at the kiosk for sale. The number of stamps of each type was the same. What is the value of all stamps for 5 k., if: a) the total value of all stamps is 21 rubles. 60 k., b) the cost of all stamps is 10 k. more cost all brands for 3 k. for 6 p. 30 k.?

397. Perform calculations using a microcalculator and round the result to thousandths:

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

398. Solve the problem:

1) To control garden pests, a lime-sulfur decoction is prepared, consisting of 6 parts of sulfur, 3 parts of quicklime and 50 parts of water (by weight). How much will kilograms decoction, if you take 8.8 kg more water than sulfur?

2) For the preparation of porcelain, 2 parts of sand and 25 parts of clay (by weight) are taken for 1 part of gypsum. How many kilograms of porcelain will you get if you take 6.9 kg more clay than sand?

399. Do the following:

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

D 400. Perform the action:

A
401. Find the value of the difference:

402. Solve the equation:

404. One tractor driver plowed the fields, and the other of the same field. What part of the field is left to plow?

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

406. For an expedition working in the taiga, a package with food was dropped from a helicopter, which fell to the ground after 3 s. From what height was this package dropped if in the first second it flew m, and in each subsequent second it flew m more than in the previous one?

407. How long did it take to manufacture a part if it was processed on a lathe h, on a milling machine h and on a drilling machine h?

408. Find the value of the expression:

409. Two pedestrians left two villages at the same time towards each other and met after 1.5 hours. The distance between the villages is 12.3 km. The speed of one pedestrian is 4.4 km/h. Find the speed of the other pedestrian.

410. To make cherry jam, 2 parts of berries (by weight) are taken for 3 parts of sugar. How many kilograms of sugar and how many kilograms of berries should be taken to get 10 kg of jam if it decreases by 1.5 times during cooking?

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. Solve the equation:

a) (x-4.7) 7.3 = 38.69; c) 23.5-(2,3a + 1.2a) = 19.3;
b) (3.6-a) 5.8 = 14.5; d) 12.98-(3.8x-1.3x) = 11.23.

A The branch of mathematics that studies the properties of numbers and operations on them is called number theory.

The beginning of the creation of number theory was laid by the ancient Greeks. scientists Pythagoras, Euclid, Eratosthenes and others.

Some problems of number theory are formulated very simply - they can be understood by any sixth grader. But the solution of these problems is sometimes so difficult that it takes centuries to solve, and some questions still remain unanswered. For example, ancient Greek mathematicians knew only one pair of friendly numbers - 220 and 284. And only in the 18th century. the famous mathematician, member of the St. Petersburg Academy of Sciences Leonard Euler found 65 more pairs of friendly numbers (one of them is 17296 and 18416). However, there is still no general way to find pairs of friendly numbers.

Almost 250 years ago, Christian Goldbach, a member of the St. Petersburg Academy of Sciences, suggested that any odd number greater than 5 can be represented as the sum of three prime numbers. For example: 21 = 3 + 7 + 11, 23 = 5 + 7 + 11, etc.

Only 200 years later, the remarkable Soviet mathematician, Academician Ivan Matveyevich Vinogradov (1891-1983), managed to prove this assumption. But the statement "Any even number greater than 2 can be represented as the sum of two prime numbers" (for example: 28=11 + 17, 56 = 19+37, 924 = 311 + 613, etc.) has not yet been proven .

Mixed fractions can be subtracted just like simple fractions. To subtract mixed numbers of fractions, you need to know a few subtraction rules. Let's study these rules with examples.

Subtraction of mixed fractions with the same denominators.

Consider an example with the condition that the integer and fractional part to be reduced are greater than the integer and fractional parts to be subtracted, respectively. Under such conditions, the subtraction occurs separately. The integer part is subtracted from the integer part, and the fractional part from the fractional.

Consider an example:

Subtract mixed fractions \(5\frac(3)(7)\) and \(1\frac(1)(7)\).

\(5\frac(3)(7)-1\frac(1)(7) = (5-1) + (\frac(3)(7)-\frac(1)(7)) = 4\ frac(2)(7)\)

The correctness of the subtraction is checked by addition. Let's check the subtraction:

\(4\frac(2)(7)+1\frac(1)(7) = (4 + 1) + (\frac(2)(7) + \frac(1)(7)) = 5\ frac(3)(7)\)

Consider an example with the condition that the fractional part of the minuend is less than the fractional part of the subtrahend, respectively. In this case, we borrow one from the integer in the minuend.

Consider an example:

Subtract mixed fractions \(6\frac(1)(4)\) and \(3\frac(3)(4)\).

The reduced \(6\frac(1)(4)\) has a smaller fractional part than 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 \(\frac{4}{4} = 1\)

\(\begin(align)&6\frac(1)(4)-3\frac(3)(4) = (6 + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (1) + \frac(1)(4))-3\frac(3)(4) = (5 + \color(red) (\frac(4)(4)) + \frac(1)(4))-3\frac(3)(4) = (5 + \frac(5)(4))-3\frac(3)(4) = \\\\ &= 5\frac(5)(4)-3\frac(3)(4) = 2\frac(2)(4) = 2\frac(1)(4)\\\\ \end(align)\)

Next example:

\(7\frac(8)(19)-3 = 4\frac(8)(19)\)

Subtracting a mixed fraction from a whole number.

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

The reduced 3 does not have a fractional part, so we cannot immediately subtract. Let's take the integer part of y 3 unit, and then perform the subtraction. We write the unit as \(3 = 2 + 1 = 2 + \frac(5)(5) = 2\frac(5)(5)\)

\(3-1\frac(2)(5)= (2 + \color(red) (1))-1\frac(2)(5) = (2 + \color(red) (\frac(5 )(5)))-1\frac(2)(5) = 2\frac(5)(5)-1\frac(2)(5) = 1\frac(3)(5)\)

Subtraction of mixed fractions with different denominators.

Consider an example with the condition if the fractional parts of the minuend and the subtrahend have different denominators. It is necessary to reduce to a common denominator, and then perform a subtraction.

Subtract two mixed fractions \(2\frac(2)(3)\) and \(1\frac(1)(4)\) with different denominators.

The common denominator is 12.

\(2\frac(2)(3)-1\frac(1)(4) = 2\frac(2 \times \color(red) (4))(3 \times \color(red) (4) )-1\frac(1 \times \color(red) (3))(4 \times \color(red) (3)) = 2\frac(8)(12)-1\frac(3)(12 ) = 1\frac(5)(12)\)

Related questions:
How to subtract mixed fractions? How to solve mixed fractions?
Answer: you need to decide what type the expression belongs to and apply the solution algorithm according to the type of expression. Subtract the integer from the integer part, subtract the fractional part from the fractional part.

How to subtract a fraction from a whole number? How to subtract a fraction from a whole number?
Answer: you need to take a unit from an integer and write this unit as a fraction

\(4 = 3 + 1 = 3 + \frac(7)(7) = 3\frac(7)(7)\),

and then subtract the whole from the whole, subtract the fractional part from the fractional part. Example:

\(4-2\frac(3)(7) = (3 + \color(red) (1))-2\frac(3)(7) = (3 + \color(red) (\frac(7 )(7)))-2\frac(3)(7) = 3\frac(7)(7)-2\frac(3)(7) = 1\frac(4)(7)\)

Example #1:
Subtract a proper fraction from one: a) \(1-\frac(8)(33)\) b) \(1-\frac(6)(7)\)

Solution:
a) Let's represent the unit as a fraction with a denominator of 33. We get \(1 = \frac(33)(33)\)

\(1-\frac(8)(33) = \frac(33)(33)-\frac(8)(33) = \frac(25)(33)\)

b) Let's represent the unit as a fraction with a denominator of 7. We get \(1 = \frac(7)(7)\)

\(1-\frac(6)(7) = \frac(7)(7)-\frac(6)(7) = \frac(7-6)(7) = \frac(1)(7) \)

Example #2:
Subtract a mixed fraction from an integer: a) \(21-10\frac(4)(5)\) b) \(2-1\frac(1)(3)\)

Solution:
a) Let's take 21 units from an integer and write it like this \(21 = 20 + 1 = 20 + \frac(5)(5) = 20\frac(5)(5)\)

\(21-10\frac(4)(5) = (20 + 1)-10\frac(4)(5) = (20 + \frac(5)(5))-10\frac(4)( 5) = 20\frac(5)(5)-10\frac(4)(5) = 10\frac(1)(5)\\\\\)

b) Let's take 1 from the integer 2 and write it like this \(2 = 1 + 1 = 1 + \frac(3)(3) = 1\frac(3)(3)\)

\(2-1\frac(1)(3) = (1 + 1)-1\frac(1)(3) = (1 + \frac(3)(3))-1\frac(1)( 3) = 1\frac(3)(3)-1\frac(1)(3) = \frac(2)(3)\\\\\)

Example #3:
Subtract an integer from a mixed fraction: a) \(15\frac(6)(17)-4\) b) \(23\frac(1)(2)-12\)

a) \(15\frac(6)(17)-4 = 11\frac(6)(17)\)

b) \(23\frac(1)(2)-12 = 11\frac(1)(2)\)

Example #4:
Subtract a proper fraction from a mixed fraction: a) \(1\frac(4)(5)-\frac(4)(5)\)

\(1\frac(4)(5)-\frac(4)(5) = 1\\\\\)

Example #5:
Compute \(5\frac(5)(16)-3\frac(3)(8)\)

\(\begin(align)&5\frac(5)(16)-3\frac(3)(8) = 5\frac(5)(16)-3\frac(3 \times \color(red) ( 2))(8 \times \color(red) (2)) = 5\frac(5)(16)-3\frac(6)(16) = (5 + \frac(5)(16))- 3\frac(6)(16) = (4 + \color(red) (1) + \frac(5)(16))-3\frac(6)(16) = \\\\ &= (4 + \color(red) (\frac(16)(16)) + \frac(5)(16))-3\frac(6)(16) = (4 + \color(red) (\frac(21 )(16)))-3\frac(3)(8) = 4\frac(21)(16)-3\frac(6)(16) = 1\frac(15)(16)\\\\ \end(align)\)

Lesson Objectives:

• Repetition and consolidation of the main program material, expressed in standard examples and non-standard tasks.
• Improving the skills of arithmetic operations adding and subtracting mixed numbers;
• Develop ingenuity, thinking, speech, memory.
• Cultivate cognitive interest in the subject, love for search solutions.

Lesson objectives:

• Educational

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

Educational

- to develop students' cognitive processes, creative activity; acquisition of research experience, development of commutative qualities.

Educational

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

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

Lesson form: partly search with elements of a didactic game.

Intersubject communications: biology.

Lesson equipment:

• poster;
• handout: task cards;
• presentation on the topic of the lesson.

The use of health-saving technologies in the classroom:

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

Lesson plan

I. Organizational moment.

Hello. Sit down.

Presentation. slide 1. Lesson topic: “Addition and subtraction of mixed numbers”.

Lesson Objectives:

• Repetition and consolidation of the main program material, expressed in standard examples and non-standard tasks.
• Improving the skills of arithmetic operations, adding and subtracting mixed numbers, preparing for the test.

II. Updating of basic knowledge.

On the board is a poster with the words of Laue.

Our lesson will be held under the motto of the French engineer - physicist Laue: "Education is what remains when everything learned has already been forgotten."

Now you will show your knowledge of adding and subtracting ordinary fractions with different denominators, as well as adding and subtracting mixed numbers.

1) Remember the famous fable by I. Krylov “The Dragonfly and the Ant”.

Jumping dragonfly, red summer sang
I didn’t have time to look back, as winter rolls into my eyes.

Task. The Jumping Dragonfly slept for half of the red summer, danced for a third of the time, and sang for a sixth. The rest of the time she decided to devote to preparing for the winter. What part of the summer did the Dragonfly prepare for winter?

Answer: in the summer, the Dragonfly did not prepare at all for winter.

And now, remember the reduction of fractions:

Write down from these fractions those that can be reduced, and perform the reduction:

Remember which fractions are called correct and which are incorrect?

- Proper fractions, those in which the numerator is less than the denominator.
- Improper fractions, those in which the numerator is greater than or equal to the denominator.

(Cards: read the fraction and call it a correct or improper fraction.)

How to extract the integer part from an improper fraction?

The numerator must be divided by the denominator.

(Oral flashcards: highlight the whole part from an improper fraction.)

III. Systematization of knowledge. Cards. Perform addition and subtraction of common fractions. Examples on the left, responses on the right. Having solved the example with an arrow, correlate with the answer.

Slides 2-7. This amazing tree is one of the giant trees. It grows in India and Malaysia.

The most unusual thing about it is how its branches grow. Numerous and heavy, they scatter in all directions from the trunk, though powerful, but nonetheless unable to bear them all on its own.

The whole trick is that the branches themselves remove part of the load from it: each of them has thick shoots that hang down sheer to the ground and are nothing more than aerial roots of a tree.

Once anchored in the ground, they not only provide additional support to the branches, but also supply nutrients and water to them. Gradually, they turn into new shafts and around the main shaft ring-shaped “galleries” are formed, the diameter of which sometimes reaches 450 m.

Having solved the problems, as well as calculating the values ​​of the expressions, we will replace the numbers with the corresponding letters and you will find out the name of this tree.

Solve the problem:

Calculate the values ​​of the expression:

Answer: BANYAN.

Lesson summary: We were preparing for the test. To do this, we repeated the addition and subtraction of fractions, as well as mixed numbers. Don't forget to cancel the fractions that result from addition and subtraction, and don't forget to highlight the whole part.

House. task: § 2, paragraph 12 No. 392.

If you have time, complete additional tasks.

Additional task:

• Solve the equation:

Cards:

Perform addition and subtraction of common fractions.

_________________________________________

Solve the problem:

Calculate the values ​​of the expression:

Self-analysis of a math lesson in 6 “a” grade.

Lesson topic: Addition and subtraction of mixed numbers.

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

Lesson form: partly search with elements of a didactic game.

1) This lesson is a repetition and consolidation of the main program material, but only expressed in the solution of standard examples and non-standard tasks. In this lesson, we repeated arithmetic operations (addition, subtraction) on ordinary fractions and on mixed numbers. These topics are studied in the 6th grade mathematics course. When studying mathematics, a lot of time has to be spent on practicing various skills. During this period, students lose interest in the subject. To maintain this interest, I use various methods of activating students in the lesson. One of these methods is the didactic game. It allows you to make the learning process exciting, create high activity in the lesson. There will be a test in the next lesson. I think that this lesson “gave” positive emotions to the guys, they worked out arithmetic operations on mixed numbers and tuned in to the test.

2) There are 19 students in the class according to the list, 16 students were present at the lesson. Poor performers - 4, strong - 1.

3) Educational - generalization and systematization of knowledge; development of speed of thinking; the introduction of a game situation to relieve nervous and mental stress; develop the ability to analyze; develop computing skills.
Educational- to develop students' cognitive processes, creative activity; acquisition of research experience, development of commutative qualities.
Educational– formation of skills of self-organization and independence; respectful relationship with each other.
In games, the children's attention is unobtrusively activated, interest in the subject is instilled, and creative imagination develops.

4) One of the successful stages of the lesson is solving problems and examples where it was necessary to compose the word BANYAN. Students, as it were, are engaged in mathematics and at the same time expand their horizons.

5) The lesson was busy. The lesson is very logical.

6) For the lesson, I, as a teacher, made a lot of handouts that I printed on a computer.