Addition and subtraction of mixed numbers: features and examples. Lesson summary "Adding and subtracting mixed numbers"

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Slide captions:

Mathematics teacher Marina Nikolaevna Kuznetsova Addition and subtraction mixed numbers

Homework

Astrid Lindgren

Verbal count 1 0

What groups can we divide these fractions into?

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

Find another example:

Addition and subtraction of mixed numbers. Objective of the lesson: Learn to add and subtract mixed numbers.

Help 1. Add the whole part to the whole part. Add the fractional part to the resulting whole 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 whole part. Formulate the 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 whole part. Formulate the rule for subtracting mixed numbers. 4. If the fractional part of what is being reduced is less than the fractional part of what is being subtracted. We borrow one from the whole part of the minuend and represent it as an improper fraction. We add the resulting fraction with the fractional part of the minuend. We subtract the whole parts and fractional parts separately. To the remaining integer part we add the remaining fractional part. Formulate a rule for subtracting a fraction from a mixed number, where the fraction of the minuend is greater than the fraction of the subtrahend.

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

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

Physical education minute We’ve worked hard - let’s rest, Let’s get up and take a deep breath. Hands to the sides, forward, left, right turn. Three bends, stand up straight. Raise your arms up and down. They lowered their hands smoothly and gave everyone smiles.

4 – B 7 – O 3 – U 4 – E 5 – X 4 – P 5 – S U S P E V X O

Problem solving Page 175, No. 1115 Page. 175, No. 1116

What is a mixed number? What did you learn today? How to add mixed numbers? How to subtract mixed numbers?

Homework: P. 29 (learn the rules) Page. 178, No. 1136, 1137

Thank you for the lesson!

Preview:

Mathematics teacher Kuznetsova M.N.

Lesson in 5th grade on the topic:

Addition and subtraction of mixed numbers.

Goals:

Educational:

Introduce students to algorithms for adding and subtracting mixed numbers by involving students in hands-on activities.
Continue work on developing computing skills.

Educational:

Development of the ability to solve problems of the studied types.
Creating conditions for the formation of mental operations.

Educational:

Foster a sense of camaraderie and mutual assistance.

During the classes

I. Organizational moment.

See if everything is okay:

Book, pens and notebooks.

The bell has now rung.

The lesson begins.

II. Checking homework.

Date, great job.

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

D/z.

1. Select the whole part and arrange it in ascending order.

18 -I 7 -A 14 -R 11 -T 9 -S 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 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

AND

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 know how to come to the rescue. I'll find it and come back.)

Guys, let's meet a friend quickly, for this we will complete mathematical tasks. If we carry them out correctly, then by the time Carlson, the sweet tooth, returns, we will have a big communal cake. 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 called proper?

What fractions are called improper?

How to represent improper fractions differently?

What does a mixed number consist of?

(Piece of cake.)

IV. Updating knowledge.

Find another 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 add and subtract mixed numbers; to achieve this goal we will formulate rules.

V. Research

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

Exercise 1 . Performing addition 2 ½ + 3

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

Task 3 . Performing subtraction 3 5/6 – 3/6

Task 4. Performing subtraction 5 1/4 - 3 2/4

Reference

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

Add a whole part to a whole part.
Add the fractional part to the fractional part
Add the resulting fractional part to the resulting whole part.
Formulate the rule for adding mixed numbers.

Subtract the whole part from the whole part.
Subtract the fractional part from the fractional part
Add the remaining fractional part to the remaining whole part.
Formulate the rule for subtracting mixed numbers.

If the fractional part of the minuend is less than the fractional part of the subtrahend.
We borrow one from the whole part of the minuend and represent it as an improper fraction.
We add the resulting fraction with the fractional part of the minuend.
We subtract the whole parts and fractional parts separately.
To the remaining integer part we add the remaining fractional part.
Formulate a rule for subtracting a fraction from a mixed number, where the fraction of the minuend is greater than the fraction of the subtrahend.

VI. Information exchange.

You have reviewed 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 the 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 can you make sure the addition is done correctly? (by subtraction). Do 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 worked hard - let's rest,

Let's stand up and take a deep breath.

Hands to the sides, forward,

Left, right turn.

Three bends, stand up straight.

Raise your arms up and down.

Hands slowly lowered,

They brought smiles to everyone.

IX. Reinforcing the material learned

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

3 7/13 – 4/13= 4 – V

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.)

"Hunt for Fives"

2. Working on tasks.

a) Page 175, No. 1115.

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

b) Page 175, No. 1116.

What is the length of the red ribbon?
What is said about the length of the white one?
What does 2 1/5 m shorter mean?
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, and help each other.

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

X. Lesson summary (Carloson's questions).

What is a mixed number?
What did you learn today? (Add and subtract mixed numbers.)
How to add mixed numbers?
How to subtract mixed numbers?

This will help you cope with your homework.

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

XII. Reflection.

Collect the earned pieces into a cake. (3-5 parts – “5”)

The teacher evaluates the students' work. (Face). (Slide 13)

Lesson objectives:

Repetition and consolidation of basic 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.
Bring up cognitive interest to the subject, love for search solutions.

Lesson objectives:

Educational

– generalization and systematization of knowledge; development of quick thinking; develop the ability to analyze; develop computing skills.
Developmental
– develop in students cognitive processes, creative activity; acquiring experience in research activities, developing commutative qualities.
Educational
– formation of self-organization and independence skills; respectful attitude towards each other.

Lesson type: lesson of generalization and systematization of knowledge.

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

Interdisciplinary connections: biology.

Lesson equipment:

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

Application 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: “Adding and subtracting mixed numbers.”

Lesson objectives:

Repetition and consolidation of basic program material, expressed in standard examples and non-standard tasks.
Improving arithmetic skills, adding and subtracting mixed numbers, preparing for tests.

II. Updating basic knowledge.

There is a poster with Laue's words on the board.

Our lesson will be held under the motto of the French engineer and 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 addition and subtraction of mixed numbers.

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

The jumping dragonfly sang the red summer
Before I even had time to look back, winter was rolling in 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 of the time. She decided to devote the rest of her time to preparing for winter. How much of the summer did Dragonfly spend preparing for winter?

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

Now let's remember the reduction of fractions:

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

Remember which fractions are called proper and which are improper?

– Proper fractions are those whose numerator is less than the denominator.
– Improper fractions are those whose numerator is greater than or equal to the denominator.

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

How to separate the whole part from an improper fraction?

– The numerator must be divided by the denominator.

(Oral cards: isolate the whole part from an improper fraction.)

III. Systematization of knowledge. Cards. Perform addition and subtraction ordinary fractions. Examples on the left, answers written on the right. Having solved the example, match it with the answer with an arrow.

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 the way its branches grow. Numerous and heavy, they scatter in all directions from the trunk, which, although powerful, is nevertheless not capable of supporting them all on its own.

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

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

Having solved the problems, as well as calculated the meanings of the expressions, 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. For this purpose, we repeated the addition and subtraction of fractions, as well as mixed numbers. Remember to cancel fractions resulting from addition and subtraction, and remember to highlight the whole part.

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

If you have time, complete additional tasks.

Additional task:

Solve the equation:

Cards:

Perform addition and subtraction of ordinary fractions.

_________________________________________

Solve the problem:

Calculate the values of the expression:

Self-analysis of a mathematics lesson in 6th grade.

Lesson topic: Addition and subtraction of mixed numbers.

Lesson type: lesson of generalization and systematization of knowledge.

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

1) This is a lesson on repetition and consolidation of basic program material, but only expressed in solving standard examples and non-standard problems. On this lesson we repeated arithmetic operations(addition, subtraction) over ordinary fractions and mixed numbers. These topics are studied in the 6th grade mathematics course. When studying mathematics, you have to spend a lot of time practicing various skills. During this period, students lose interest in the subject. To maintain this interest, I use various techniques to activate students in the lesson. One of these methods is didactic game. It allows you to make the learning process fun and create high activity in the lesson. The next lesson will be test. I think that this lesson “gave” positive emotions to the children, they practiced arithmetic operations on mixed numbers and got ready for the test.

2) There were 19 students in the class according to the list, 16 students were present at the lesson. Low achievers – 4, strong – 1.

3) Educational – generalization and systematization of knowledge; development of quick thinking; by introducing a game situation, relieve nervous and mental stress; develop the ability to analyze; develop computing skills.
Developmental– develop students’ cognitive processes and creative activity; acquiring experience in research activities, developing commutative qualities.
Educational– formation of self-organization and independence skills; respectful attitude towards each other.
Games unobtrusively activate children's attention, instill interest in the subject, and develop creative imagination.

4) I consider one of the successful stages of the lesson to be solving problems and examples where it was necessary to form the word BANYAN. Students seem to be doing mathematics and at the same time expanding their horizons.

5) The lesson was intense. The lesson is structured very logically.

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

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

Subtracting mixed fractions with like denominators.

Let's consider an example with the condition that the integer and fractional parts being reduced are greater than the integer and fractional parts being subtracted, respectively. Under such conditions, subtraction occurs separately. We subtract the integer part from the whole part, and the fractional part from the fractional part.

Let's look at 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:

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

Let's consider an example with the condition when the fractional part of the minuend is less than the corresponding fractional part of the subtrahend. In this case, we borrow one from the whole in the minuend.

Let's look at an example:

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

The minuend \(6\frac(1)(4)\) has a smaller fractional part than the fractional part of the subtrahend \(3\frac(3)(4)\). 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 minuend 3 does not have a fractional part, so we cannot immediately subtract. Let's borrow one from the whole part of 3, and then do the subtraction. We will write the unit as \(3 = 2 + 1 = 2 + \frac(5)(5) = 2\frac(5)(5)\)

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

Subtracting mixed fractions with different denominators.

Let's consider an example with the condition that the fractional parts of the minuend and subtrahend have different denominators. You need to bring it to a common denominator, and then perform subtraction.

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

The common denominator will be the number 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 based on the type of expression. From the integer part we subtract the integer, from the fractional part we subtract 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 imagine unit as a fraction with a denominator 33. We get \(1 = \frac(33)(33)\)

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

b) Let's imagine one as a fraction with a denominator 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:
Perform a subtraction mixed fraction from an integer: a) \(21-10\frac(4)(5)\) b) \(2-1\frac(1)(3)\)

Solution:
a) Let’s borrow 21 units from the 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 one 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:
Calculate \(5\frac(5)(16)-3\frac(3)(8)\)

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

In this lesson you will learn the rules for adding and subtracting mixed numbers, learn to solve various problems on the topic “Adding and subtracting mixed numbers.” Addition and subtraction of mixed numbers is based on the properties of these numbers. When adding, you can use the commutative and combinative properties, and when subtracting numbers, you can use the properties of subtracting a number from a sum and subtracting a sum from a number.

First, let's remember what mixed numbers are. A mixed number is a number written in such a way that it has an integer part and a fractional part. For example, . Here 3 is the integer part and the fractional part.

Suppose we were given such a task. Vasya ran the first of two laps of the distance in 1 minute 40 seconds, and the second lap in 1 minute 20 seconds. How long did it take Vasya to run the entire distance and how much faster did he run the second lap than the first?

Solution

It is easy to see that we can add minutes with minutes, seconds with seconds. It turns out 2 minutes + 60 seconds, i.e. 3 minutes. But, on the other hand, 40 seconds are minutes, and 20 seconds are . And then, by analogy, in order to add these mixed numbers, we can not convert them into improper fractions, but immediately add whole minutes to each other, and fractional ones separately. This gives 2 minutes and , that is, another full minute. Total 3 minutes.

All this could have been done this way. Note that a mixed number is the sum of its integer and fractional parts. And then we will use the commutative property:

What about subtraction? The same. For purely practical reasons, the first lap is the same in minutes as the second, and in seconds it is 20 longer (or a third of a minute). Could be so:

I think you already understand the algorithm? From a whole we subtract (add to a whole) a whole, from a fraction - a fraction. Let's look at a few more examples.

Let's consolidate these calculations with a rule. To add two mixed numbers you need:

put their whole parts together;
add up their fractional parts;
if necessary, convert the sum of fractional parts into a mixed number;
add up the resulting numbers.

Let's move on to subtraction. Let's look at a few examples and then formulate a general algorithm.

Find errors in addition examples

Let's look carefully at the first example: the mixed number was replaced by the fraction , and the number - , but these fractions are not equal. If we decide to convert fractions to improper ones, we get the following:

Now let's move on to the second example, in which actions are performed according to the algorithm we considered. As you can see, all actions were performed correctly, but it is customary to write mixed numbers so that their fractional part is a proper fraction. Therefore, let's represent the fraction as a mixed number, and then perform the addition.

If we go according to plan, then we need to subtract . We cannot do this. Then let's do what we do when subtracting natural numbers: We’ll borrow from the senior rank. Only the role of the senior rank will be played here by the whole part. After all, a unit is , so you can write it instead. And then - according to plan:

Let's consolidate these calculations with a rule. To subtract one mixed number from another, you must:

compare the fractional parts of the minuend and subtrahend;
if the fractional part of the minuend is greater, then subtract the whole part from the whole part, the fractional part from the fractional part, and add the results;
if the fractional part of the subtrahend is larger, then we convert one unit from the whole part of the minuend into a fraction so that the fraction becomes improper, and then we subtract the whole part from the whole part, and the fractional one from the fractional part, and add the results.

Find errors in subtraction examples

Let's look at the first example. According to the algorithm, we must first represent 12 as a mixed number, and then perform the subtraction:

Let's look at the second example. There is an error when subtracting fractional parts: we need to subtract the fractional part of the subtrahend from the fractional part of the minuend, and not vice versa. To accomplish this, we will have to take 1 unit and represent it as a fraction.

In this lesson we were introduced to mixed numbers, learned how to add and subtract them, and formulated algorithms for addition and subtraction. We learned that to add and subtract mixed numbers it is not at all necessary to convert them into improper fractions, but rather simply add or subtract the whole parts and add or subtract the fractional parts, and then write down the final answer.

In each case we had one subtlety. For addition, we understood that sometimes the sum of fractional parts is obtained in the form of an improper fraction, therefore, if necessary, the resulting improper fraction must be reduced to the correct one, that is, the whole part must be isolated. And when subtracting, such subtlety appeared that it is not always possible to subtract the fractional part of the subtrahend from the fractional part of the minuend, so we had to “borrow” a unit from the whole part and convert it into a fraction in order to obtain an improper fraction, from which it was already possible to subtract the fractional part .

Bibliography

Mathematics. 5th grade. Zubareva I.I., Mordkovich A.G. 14th ed., revised. and additional - M.: 2013.
Vilenkin N.Ya. and others. Mathematics. 5 grades - M: Mnemosyne, 2013.
Erina T.M. Mathematics 5th grade. Slave. notebook for school Vilenkina 2013. - M: Mnemosyne, 2013.