How to calculate the integer part from an improper fraction. Isolating the whole part from an improper fraction

It is customary to write without the $ "+" $ sign in the form $ n \\ frac (a) (b) $.

Example 1

For example, the sum $ 4 + \\ frac (3) (5) $ is written $ 4 \\ frac (3) (5) $. Such notation is called a mixed fraction, and the number that corresponds to it is called a mixed number.

Definition 1

Mixed numberis a number that is equal to the sum of the natural number $ n $ and the regular fraction $ \\ frac (a) (b) $, and is written as $ n \\ frac (a) (b) $. In this case, the number $ n $ is called $ n \\ frac (a) (b) $, and the number $ \\ frac (a) (b) $ is called the fractional part of the number /

For mixed numbers, the equalities $ n \\ frac (a) (b) \u003d n + \\ frac (a) (b) $ and $ n + \\ frac (a) (b) \u003d n \\ frac (a) (b) $ hold.

Example 2

For example, the number $ 7 \\ frac (4) (9) $ is a mixed number, where the natural number $ 7 $ is its integer part, $ \\ frac (4) (9) $ is its fractional part. Examples of mixed numbers: $ 17 \\ frac (1) (2) $, $ 456 \\ frac (111) (500) $, $ 23000 \\ frac (4) (5) $.

There are numbers in mixed notation that contain an improper fraction in the fractional part. For example, $ 3 \\ frac (54) (5) $, $ 56 \\ frac (9) (2) $. The recording of these numbers can be represented as the sum of their integer and fractional parts. For example, $ 3 \\ frac (54) (5) \u003d 3 + \\ frac (54) (5) $ and $ 56 \\ frac (9) (2) \u003d 56 + \\ frac (9) (2) $. Such numbers are not suitable for the definition of a mixed number, because the fractional part of the mixed numbers must be a regular fraction.

The number $ 0 \\ frac (2) (7) $ is also not a mixed number, because $ 0 $ is not a natural number.

Converting a mixed number to an improper fraction

Algorithm for converting a mixed number to an improper fraction:

Write the mixed number $ n \\ frac (a) (b) $ as the sum of the integer and fractional parts of this number, i.e. as $ n + \\ frac (a) (b) $.

Replace the whole part of the original mixed number with a fraction with the denominator $ 1 $.

Add the fractions $ \\ frac (n) (1) $ and $ \\ frac (a) (b) $ to get the desired improper fraction equal to the original mixed number.

Example 3

Expand mixed number $ 7 \\ frac (3) (5) $ as improper fraction.

Decision.

Let's use the algorithm for converting a mixed number into an improper fraction.

Mixed number $ 7 \\ frac (3) (5) \u003d 7 + \\ frac (3) (5) $.

Let's write the number $ 7 $ as $ \\ frac (7) (1) $.

Add up the fractions $ \\ frac (7) (1) + \\ frac (3) (5) \u003d \\ frac (35) (5) + \\ frac (3) (5) \u003d \\ frac (38) (5) $.

Let's write a short record of this solution:

Answer: $ 7 \\ frac (3) (5) \u003d \\ frac (38) (5) $

The whole algorithm for converting a mixed number $ n \\ frac (a) (b) $ into an improper fraction is reduced to \\ textit (a formula for converting a mixed number into an improper fraction):

Example 4

Write mixed number $ 14 \\ frac (3) (5) $ as improper fraction.

Decision.

Let's use the formula $ n \\ frac (a) (b) \u003d \\ frac (n \\ cdot b + a) (b) $ to convert the mixed number to an improper fraction. In this example, $ n \u003d 14 $, $ a \u003d 3 $, $ b \u003d 5 $.

We get $ 14 \\ frac (3) (5) \u003d \\ frac (14 \\ cdot 5 + 3) (5) \u003d \\ frac (73) (5) $.

Answer: $ 14 \\ frac (3) (5) \u003d \\ frac (73) (5) $

Isolating the whole part from an improper fraction

When receiving a numerical solution, it is not customary to leave an answer in the form of an incorrect fraction. An improper fraction is converted to an equal natural number (if the numerator is completely divisible by the denominator), or the integer part is separated from the improper fraction (if the numerator is not completely divisible by the denominator).

Definition 2

Selecting the whole part from an improper fraction is called replacing a fraction with a mixed number equal to it.

To isolate the whole part from an improper fraction, you need to represent the improper fraction $ \\ frac (a) (b) $ as a mixed number $ q \\ frac (r) (b) $, where $ q $ is an incomplete quotient, $ r $ is remainder of dividing $ a $ by $ b $. Thus, the integer part is equal to the incomplete quotient of $ a $ divided by $ b $, and the remainder is equal to the numerator of the fractional part.

Let us prove this statement. To do this, it suffices to show that $ q \\ frac (r) (b) \u003d \\ frac (a) (b) $.

Let's convert the mixed number $ q \\ frac (r) (b) $ into an improper fraction using the formula:

Because $ q $ is an incomplete quotient, $ r $ is the remainder of dividing $ a $ by $ b $, then the equality $ a \u003d b \\ cdot q + r $ is valid. Thus, $ \\ frac (q \\ cdot b + r) (b) \u003d \\ frac (a) (b) $, whence $ q \\ frac (r) (b) \u003d \\ frac (a) (b) $, as required to show.

Thus, we formulate \\ textit (the rule for separating the integer part from an improper fraction) $ \\ frac (a) (b) $:

Divide $ a $ by $ b $ with the remainder, while determining the incomplete quotient $ q $ and the remainder $ r $.

Write down the mixed number $ q \\ frac (r) (b) $, equal to the original fraction $ \\ frac (a) (b) $.

Example 5

Select the integer part from $ \\ frac (107) (4) $.

Decision.

Let's do long division:

Picture 1.

So, as a result of dividing the numerator $ a \u003d 107 $ by the denominator $ b \u003d 4 $, we get the incomplete quotient $ q \u003d 26 $ and the remainder $ r \u003d 3 $.

We get that the improper fraction $ \\ frac (107) (4) $ is equal to the mixed number $ q \\ frac (r) (b) \u003d 26 \\ frac (3) (4) $.

Answer: $ \\ frac ((\\ rm 107)) ((\\ rm 4)) (\\ rm \u003d 26) \\ frac ((\\ rm 3)) ((\\ rm 4)) $.

Adding a mixed number and a natural number

Rule of addition of mixed and natural numbers:

To add a mixed and natural number, you need to add this natural number to the integer part of the mixed number, the fractional part remains unchanged:

where $ a \\ frac (b) (c) $ is a mixed number,

$ n $ is a natural number.

Example 6

Add mixed $ 23 \\ frac (4) (7) $ and $ 3 $.

Decision.

Answer:$ 23 \\ frac (4) (7) + 3 \u003d 26 \\ frac (4) (7). $

Adding two mixed numbers

When adding two mixed numbers, their whole parts and fractional parts are added.

Example 7

Add mixed numbers $ 3 \\ frac (1) (5) $ and $ 7 \\ frac (4) (7) $.

Decision.

Let's use the formula:

\ \

Answer: $ 10 \\ frac (27) (35). $

In this article we will talk about mixed numbers... First, we give a definition of mixed numbers and give examples. Further we will dwell on the connection between mixed numbers and improper fractions. After that, we'll show you how to convert a mixed number to an improper fraction. Finally, let's look at the reverse process, which is called separating the whole part from an improper fraction.

Page navigation.

Mixed numbers, definition, examples

Mathematicians agreed that the sum n + a / b, where n is a natural number, a / b is a regular fraction, can be written without the addition sign in the form. For example, 28 + 5/7 can be abbreviated as. Such a record was called a mixed number, and the number that corresponds to a given mixed record was called a mixed number.

So we come to the definition of a mixed number.

Definition.

Mixed number Is a number equal to the sum of a natural number n and a regular fraction a / b, and written in the form. Moreover, the number n is called whole number, and the number a / b is called fractional part of number.

By definition, a mixed number is equal to the sum of its integer and fractional parts, that is, equality is true, which can also be written like this:.

Let us give examples of mixed numbers... Number is a mixed number, natural number 5 is the integer part of the number, and the fractional part of the number. Other examples of mixed numbers are .

Sometimes you can find numbers in a mixed notation, but having a fractional part of an irregular fraction, for example, or. These numbers are understood as the sum of their integer and fractional parts, for example, and ... But such numbers do not fit the definition of a mixed number, since the fractional part of the mixed numbers must be a regular fraction.

The number is also not a mixed number, since 0 is not a natural number.

The relationship between mixed numbers and improper fractions

Trace connection between mixed numbers and improper fractions best with examples.

Have a cake on the tray and another 3/4 of the same cake. That is, according to the sense of addition, there is 1 + 3/4 cake on the tray. Having written down the last amount as a mixed number, we state that there is a cake on the tray. Now cut the whole cake into 4 equal parts. As a result, 7/4 of the cake will be on the tray. It is clear that the "quantity" of the cake did not change, therefore.

From the considered example, the following connection is clearly visible: any mixed number can be represented as an improper fraction.

Now let's have 7/4 of the cake on the tray. Having folded the whole cake from four parts, there will be 1 + 3/4 on the tray, that is, cake. From this it is clear that.

It is clear from this example that an improper fraction can be represented as a mixed number... (In the special case when the numerator of an improper fraction is divided entirely by the denominator, the improper fraction can be represented as a natural number, for example, since 8: 4 \u003d 2).

Converting a mixed number to an improper fraction

The skill of representing mixed numbers as improper fractions is useful for performing various actions with mixed numbers. In the previous paragraph, we found out that any mixed number can be converted to an improper fraction. It's time to figure out how such a translation is carried out.

Let's write an algorithm showing how to convert a mixed number to an improper fraction:

Consider an example of converting a mixed number to an improper fraction.

Example.

Present the mixed number as an improper fraction.

Decision.

Let's perform all the necessary steps of the algorithm.

The mixed number is equal to the sum of its integer and fractional parts:.

Having written the number 5 as 5/1, the last sum will take the form.

To complete the conversion of the original mixed number into an improper fraction, it remains to add fractions with different denominators: .

The summary of the entire solution is as follows: .

Answer:

So, in order to translate a mixed number into an improper fraction, you need to perform the following chain of actions:. As a result, received , which we will use in the future.

Example.

Write the mixed number down as an improper fraction.

Decision.

Let's use the formula to convert a mixed number to an improper fraction. In this example, n \u003d 15, a \u003d 2, b \u003d 5. In this way, .

Answer:

Isolating the whole part from an improper fraction

It is not customary to write an incorrect fraction in the answer. The incorrect fraction is first replaced either by an equal natural number (when the numerator is divided entirely by the denominator), or the so-called separation of the whole part from the incorrect fraction is carried out (when the numerator is not completely divisible by the denominator).

Definition.

Isolating the whole part from an improper fraction Is the replacement of a fraction with a mixed number equal to it.

It remains to find out how you can select the whole part from the incorrect fraction.

It's very simple: an improper fraction a / b is equal to a mixed number of the form, where q is an incomplete quotient and r is the remainder of a divided by b. That is, the integer part is equal to the incomplete quotient of dividing a by b, and the remainder is equal to the numerator of the fractional part.

Let us prove this statement.

For this it is enough to show that. Let's translate the mixed into an improper fraction, as we did in the previous paragraph:. Since q is an incomplete quotient, and r is the remainder of dividing a by b, the equality a \u003d b q + r is true (if necessary, see

Sections: Mathematics

Class: 4

Basic goals:

1. Form the ability to select a whole part from an irregular fraction.
2. Review the concepts of numerator and denominator, correct and incorrect fractions, mixed numbers.
3. To actualize the ability to select a whole part from an incorrect fraction.

Thinking operations required at the design stage: action by analogy, analysis, generalization.

Equipment:

Demo material:

1) Formula of division with remainder.

Handout:

1) pieces of paper with the task (to stage 2)

2) Detailed sample for self-test (to step 6)

During the classes.

1 Self-determination for learning activities.

Objectives:

1. Motivate students for learning activities by reinforcing the situation of success achieved in the previous lesson.
2. Determine the content of the lesson.

Organization of the educational process at stage 1.

Over the course of several lessons, we have worked with some numbers. What numbers did we work with? (With fractional numbers).

What knowledge do we have about these numbers? (We know how to read them, write them down, compare them, solve problems).

I propose to continue our fruitful work. You are ready? (Yes).

Today we will continue to work with fractional numbers. I am sure that we will succeed perfectly well. But first, let's review the material from the previous lessons.

2 Updating knowledge and fixing difficulties in individual activities.

Objectives:

1. To update the ability to find right and wrong fractions, mixed numbers, determination of right and wrong fractions, mixed numbers.
2. Actualize mental operations necessary and sufficient for the perception of new material.
3. Record a situation where students are unable to select the whole part from an incorrect fraction.

Organization of the educational process at stage 2.

What numbers did we meet in the previous lesson? (With mixed numbers).
- What does the mixed number consist of? (From integer and fractional parts).

Fractions and mixed numbers are written on the board.

What groups can the presented numbers be divided into?

Regular fractions ().

What fractions are called correct? (The fraction with the numerator less than the denominator. The regular fraction is less than one).

Incorrect fractions. (… ..)

What fractions are called incorrect? (The fraction with the numerator greater than the denominator or the numerator equal to the denominator).

Which of the irregular fractions can be represented as a natural number?

()

What fraction can be represented as a mixed number? (Incorrect fraction, where the numerator is greater than the denominator).

Determine with the help of the number ray what mixed number is the fraction

Students have a sheet with a task (P-1), one student works at the blackboard, comments.

What is the smallest mixed number? ()

The greatest? ()

What arithmetic operation helped you? (Division. Division with remainder).

Prove. (On the board: D-1).

12: 7 \u003d 1 (rest 5); 15: 7 \u003d 2 (rest 1); 25: 7 \u003d 3 (rest 4); 31: 7 \u003d 4 (rest 3)

Select the whole part of the fraction, write down the mixed number. Children work on the back of the paper. Different answer options are put on the board.

How did you proceed?

3 Identifying the causes of difficulty and setting the goal of the activity.

Objectives:

1. Organize communicative interaction to identify the distinctive property of the task to isolate a whole part from an incorrect fraction.
2. Agree on the topic and purpose of the lesson.

Organization of the educational process at stage 3.

What task did you complete? (It is necessary to select the whole part from the fraction).

How is this task different from the previous one? (The method that helped us to isolate the whole part from an improper fraction is not suitable for a fraction. It is inconvenient to show this fraction on a number ray).

What do we see? (We got different answers).

Why? (We used different methods. We do not have an algorithm for separating the whole part from an improper fraction).

What is the purpose of our lesson? (Build an algorithm and learn how to separate the whole part from an improper fraction).

Think and formulate the topic of our lesson. ("Isolation of the whole part from an improper fraction").

Well done!

The title of the lesson topic opens on the board.

4 Building a project for getting out of a difficulty.

Goal:

1. Organize communicative interaction to build a new way of action to isolate the whole part from the wrong fraction.
2. To fix a new way in a sign and verbal form and with the help of a standard.

Organization of the educational process at stage 4

In what way do you propose to find how many whole units are in a fractional number? (Numerator divided by denominator).

What sign in the notation of the fraction told you how to act? (A slash is a division sign).

On the desk:

Let's write the fraction as a quotient: 65: 7.

What kind of division is this? (Division with remainder. On the board: D-1).

Find the result. (65: 7 \u003d 9) (rest 2)

What does the quotient 9 and remainder 2 mean in the resulting equality? (The quotient 9 means that 65 contains 9 times 7 and 2 remains).

What will the quotient 9 stand for in a mixed number? (9 is the integer part of the mixed number).

On the desk:

What is the remainder of 2 in the mixed number? (2 is the numerator of the mixed fraction).

On the desk:

What about the denominator? (It remains, does not change).

On the desk:

What mixed number did we get?

Have we completed the task? (Yes).

What math action helped us? (Division with remainder. On the board: D-1).

The teacher returns to the answers on the pieces of paper, summarizes, encourages with words those who did it correctly. In a group form, students display a new method in a symbolic form on pieces of paper. The correct option is selected.

Write down, using the formula for division with remainder (D-1), what mixed number is the fraction?

On the board: D-3

How to select a whole part from an incorrect fraction?

To select the whole part from an improper fraction, you need to divide its numerator by the denominator. The quotient will be the whole part, the remainder will be the numerator, and the denominator will not change.

Well done! Thanks!

Let's check our opinion with the opinion of the textbook. Turn to page 26, Math 4 (Part 2), and read the rule silently first and then aloud.

Were we right? (Yes).

Well done!

Physical minutes (at the teacher's choice).

5 Primary reinforcement in external speech.

Goal:

Fix the way of separating the whole part from an irregular fraction in external speech.

Organization of the educational process at stage 5.

Let's repeat the algorithm for separating the whole part from an improper fraction again. D 2

We have compiled an algorithm for separating the whole part from an improper fraction. What is the purpose of our future activities? (Practice).

No. 4 (a, b, c) p. 26 - with commentary on the model.

No. 4 (d, e) page 26 - in pairs.

6 Self-test with self-test.

Goal:

1. Organize the students' independent fulfillment of tasks to isolate an entire part from an incorrect fraction.
2. Train the ability for self-control and self-esteem.
3. Test your ability to isolate the whole part from an incorrect fraction.
4. Contribute to creating a situation of success.

Organization of the educational process at stage 6.

You have managed to deduce an algorithm for separating the whole part from an improper fraction and have practiced solving examples. I think now you can complete the task yourself.

Do it yourself:

No. 3, page 26 - option 1 - columns 1 and 2;

Option 2 - 3 and 4 columns;

Anyone who wishes can complete the task and another option.

Students perform work, at the end of which they test themselves on a sample for self-examination. Card P-2 is used.

Test yourself using the self-test pattern and record the test result using the "+" or "?" green handle.

Who made mistakes while completing the assignment? (...)

What is the reason? (...)

Who's got it right?

Well done!

You can organize work on error correction in groups or frontally. Students who have not made mistakes are appointed as counselors.

7 Incorporation and repetition.

Goal:

To train the ability to separate the whole part from the wrong fraction.

Organization of the educational process at stage 7.

Let's try to apply our knowledge when comparing fractions and mixed numbers.

Find the inequality in which you want to compare the right fraction with the wrong one.

What do we do?

Select the whole part from the improper fraction.

Means ?!

An incorrect fraction is more correct. We proved this by highlighting the whole part.

Well done!

Finish the task, compare.

Let's check.

8 Reflection of educational activities in the lesson.

Objectives:

1. Fix in speech the algorithm for separating the integer part from the incorrect fraction.
2. Record the remaining difficulties and ways to overcome them.
3. Assess your own activities in the lesson.
4. Agree on homework.

Organization of the educational process at stage 8.

What have you learned in the lesson? (Select whole part from improper fraction).

What algorithm did we build? (Algorithm D-2 can be spoken).

Who had difficulties? How will you act?

Who is pleased with themselves today? Why?

It was difficult for me in class.
- I understood the lesson, but I need training.
- I understood the lesson well, but I need help.
- I'm great, I understood the lesson perfectly well.

Homework: come up with five irregular fractions and select the whole part; No. 10, No. 11, page 28 - by choice; No. 15, page 28 (a or b) - optional.

Well done! Thanks for the work in the lesson!

Lesson summary in grade 5

“Mixed numbers. Isolation of the whole part from an improper fraction "

During the classes

Organizing time. Greeting.

We will hold a verbal score and break all records

Verbal counting.

Find mistakes

Correct fractions.

b)

Let's write on the blackboard what we can't compare yet.

2. Perform division:

45: 9=5 ; 0: 67=0; 234: 1=234;

567: 567 \u003d 1; 34: 17 \u003d 2; a: a \u003d 1;

3. Perform division with remainder:

6 \u003d 2 (rest 2)

3 \u003d 8 (rest 1)

48: 9 \u003d 5 (rest 3)

Follow the steps:

We cannot solve the last example, let us write it out.

Explanation of the new material

What is shown in the picture? How many pieces are the cake divided into? How many parts did you take? Present as a fraction.

What's in this picture? It can be seen that the cake is on different trays. How many pieces are on the first tray? Second?

It can be denoted as such a number:

1 - whole part, - fractional part.

The sum of the integer and fractional parts is calledmixed number .

Determine from the figure what mixed number is equal to a fraction?

That is, we saw a connection between an improper fraction and a mixed number.

Let's draw conclusions: we can turn an improper fraction into a mixed number, i.e. as they say in mathematics, select the whole part from an improper fraction.

The rule for separating the whole part from an improper fraction:

Divide the numerator by the denominator with remainder

An incomplete quotient will be a whole part

Remainder gives the numerator and divisor gives the denominator of the fractional part

Work on the topic of the lesson.

Select a whole part from an improper fraction (along with the class):

Select the whole part from the incorrect fraction (at the board)

Compare

Historical information.

In the old days in Russia, coins were used in denominations of less than one kopeck:

penny - Ph. andpolushka - Ph.

Other coins also had names:

3 k. - altyn, 5 k. - penny, 15 k. - five altyn,

10 k. - dime, 20 k. Two-kopeck,

25 k. - a quarter, 50 k. - fifty dollars.

Independent work

How can you imagine

1 dime, 1 altyn, three polushki .

Reflection

What is your mood?

Write the fraction that best suits your knowledge:

2 (can not understand anything)

2 (it was interesting, but not clear)

3 (difficult, the topic is not interesting)

3 (it was difficult, but I will definitely make an effort to study the topic)

4 (some examples caused difficulties)

4 (everything is clear, but I cannot help)

5 (everything is clear, I can help others)

I hope your grade will only increase with each lesson! And to get a grade of 5, you need to work not only in the classroom, but also at home.

Homework.

§ 1 Isolation of the whole part from an improper fraction

In this lesson, you will learn how to convert an improper fraction to a mixed number by highlighting the whole part, and vice versa, get an improper fraction from a mixed number.

To begin with, let's remember what a mixed number and an improper fraction are.

A mixed number is a special form of a number that contains an integer and fractional parts.

An improper fraction is a fraction whose numerator is greater than or equal to the denominator.

Consider the problem:

Let's divide 8 candies for three guys. How much will each get?

To find out how many sweets each child will receive, you need

But it is not customary to write the wrong fraction in the answer. It is previously replaced either by a natural number equal to it (when the numerator is divided entirely by the denominator), or the so-called separation of the whole part from the improper fraction is carried out (when the numerator is not completely divisible by the denominator).

Separating the whole part from an improper fraction is replacing a fraction with its equal mixed number.

To select a whole part from an improper fraction, you need to divide the numerator by the denominator with a remainder. In this case, the incomplete quotient will be the whole part, the remainder will be the numerator, and the divisor will be the denominator.

Let's get back to the problem.

So, we divide 8 by 3 with a remainder, we get 2 in the incomplete quotient and 2 in the remainder.

§ 2 Representation of a mixed number as an improper fraction

Let's do the following task:

Divide 49 by 13, we get 3 in the incomplete quotient (this will be the integer part) and in the remainder 10 (we write this in the numerator of the fractional part).

The skill of representing mixed numbers as improper fractions is useful for performing various actions with mixed numbers. It's time to figure out how such a translation is carried out.

To represent the mixed number as an improper fraction, you need to multiply the denominator of the fraction by the whole part and add the numerator to the resulting product. As a result, we get a number that will be the numerator of the new fraction, and the denominator remains unchanged.

The first step is to multiply the integer part 5 by the denominator 7 to get 35.

The second step is to add the numerator 4 to the resulting product 35, it will be 39.

Now let's write 39 in the numerator and leave 7 in the denominator.

Thus, in this lesson you learned how to convert an improper fraction to a mixed number, for this you need to divide the numerator by the denominator with the remainder. Then the incomplete quotient will be the integer part, the remainder will be the numerator, and the divisor will be the denominator of the fractional part of the mixed number.

You also learned about the representation of a mixed number as an improper fraction. In order to represent the mixed number as an improper fraction, you need to multiply the denominator of the fractional part of the mixed number by an integer part and add the numerator to the resulting product.

List of used literature:

1. Mathematics grade 5. Vilenkin N.Ya., Zhokhov V.I. et al. 31st ed., erased. - M: 2013.
2. Didactic materials in mathematics grade 5. Author - Popov M.A. - year 2013
3. We calculate without errors. Works with self-test in mathematics 5-6 grades. Author - Minaeva S.S. - year 2014
4. Didactic materials in mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
5. Control and independent work in mathematics, grade 5. Authors - Popov M.A. - year 2012
6. Mathematics. Grade 5: textbook. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., Erased. - M .: Mnemosina, 2009