"Bringing fractions to a common denominator" (Grade 5). Reducing fractions to the lowest common denominator, rule, examples, solutions Reducing fractions to a common denominator 1 5

This article explains how to reduce fractions to a common denominator and how to find the smallest common denominator. Definitions are given, a rule for reducing fractions to a common denominator is given, and practical examples are considered.

What is reducing a fraction to a common denominator?

Ordinary fractions consist of a numerator - the upper part, and a denominator - the lower part. If fractions have the same denominator, they are said to have a common denominator. For example, fractions 11 14 , 17 14 , 9 14 have the same denominator 14 . In other words, they are reduced to a common denominator.

If fractions have different denominators, then they can always be reduced to a common denominator with the help of simple actions. To do this, you need to multiply the numerator and denominator by certain additional factors.

Obviously, the fractions 4 5 and 3 4 are not reduced to a common denominator. To do this, you need to use additional factors 5 and 4 to bring them to a denominator of 20. How exactly to do this? Multiply the numerator and denominator of 45 by 4, and multiply the numerator and denominator of 34 by 5. Instead of fractions 4 5 and 3 4 we get 16 20 and 15 20 respectively.

Bringing fractions to a common denominator

Reducing fractions to a common denominator is the multiplication of the numerators and denominators of fractions by factors such that the result is identical fractions with the same denominator.

Common denominator: definition, examples

What is a common denominator?

Common denominator

The common denominator of a fraction is any positive number that is a common multiple of all the given fractions.

In other words, the common denominator of some set of fractions will be such a natural number that is divisible without a remainder by all the denominators of these fractions.

The set of natural numbers is infinite, and therefore, by definition, every set of common fractions has an infinite number of common denominators. In other words, there are infinitely many common multiples for all denominators of the original set of fractions.

The common denominator for several fractions is easy to find using the definition. Let there be fractions 1 6 and 3 5 . The common denominator of the fractions will be any positive common multiple of the numbers 6 and 5. Such positive common multiples are 30, 60, 90, 120, 150, 180, 210, and so on.

Consider an example.

Example 1. Common denominator

Can di fractions 1 3, 21 6, 5 12 be reduced to a common denominator, which is equal to 150?

To find out if this is the case, you need to check if 150 is a common multiple of the denominators of the fractions, that is, for the numbers 3, 6, 12. In other words, the number 150 must be divisible by 3, 6, 12 without a remainder. Let's check:

150 ÷ 3 = 50 , 150 ÷ 6 = 25 , 150 ÷ 12 = 12 , 5

This means that 150 is not a common denominator of the indicated fractions.

Lowest common denominator

The smallest natural number from the set of common denominators of some set of fractions is called the least common denominator.

Lowest common denominator

The smallest common denominator of fractions is smallest number among all the common denominators of these fractions.

The least common divisor of a given set of numbers is the least common multiple (LCM). The LCM of all denominators of fractions is the least common denominator of those fractions.

How to find the lowest common denominator? Finding it comes down to finding the least common multiple of fractions. Let's look at an example:

Example 2: Find the lowest common denominator

We need to find the smallest common denominator for the fractions 1 10 and 127 28 .

We are looking for the LCM of numbers 10 and 28. We decompose them into simple factors and get:

10 \u003d 2 5 28 \u003d 2 2 7 N O K (15, 28) \u003d 2 2 5 7 \u003d 140

How to bring fractions to the lowest common denominator

There is a rule that explains how to reduce fractions to a common denominator. The rule consists of three points.

The rule for reducing fractions to a common denominator

Find the smallest common denominator of fractions.
For each fraction, find an additional factor. To find the multiplier, you need to divide the least common denominator by the denominator of each fraction.
Multiply the numerator and denominator by the found additional factor.

Consider the application of this rule on a specific example.

Example 3. Reducing fractions to a common denominator

There are fractions 3 14 and 5 18. Let's bring them to the lowest common denominator.

As a rule, we first find the LCM of the denominators of the fractions.

14 \u003d 2 7 18 \u003d 2 3 3 N O K (14, 18) \u003d 2 3 3 7 \u003d 126

We calculate additional factors for each fraction. For 3 14 the additional factor is 126 ÷ 14 = 9 , and for the fraction 5 18 the additional factor is 126 ÷ 18 = 7 .

We multiply the numerator and denominator of fractions by additional factors and get:

3 9 14 9 \u003d 27 126, 5 7 18 7 \u003d 35 126.

Bringing Multiple Fractions to the Least Common Denominator

According to the considered rule, not only pairs of fractions, but also more of them can be reduced to a common denominator.

Let's take another example.

Example 4. Reducing fractions to a common denominator

Bring the fractions 3 2 , 5 6 , 3 8 and 17 18 to the lowest common denominator.

Calculate the LCM of the denominators. Find the LCM of three or more numbers:

N O C (2, 6) = 6 N O C (6, 8) = 24 N O C (24, 18) = 72 N O C (2, 6, 8, 18) = 72

For 3 2 the additional factor is 72 ÷ 2 = 36 , for 5 6 the additional factor is 72 ÷ 6 = 12 , for 3 8 the additional factor is 72 ÷ 8 = 9 , finally, for 17 18 the additional factor is 72 ÷ 18 = 4 .

We multiply the fractions by additional factors and go to the lowest common denominator:

3 2 36 = 108 72 5 6 12 = 60 72 3 8 9 = 27 72 17 18 4 = 68 72

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PUBLIC LESSON

5th class

Mathematic teacher

Municipal General Education

institution "Main

secondary school No. 6, village of Donskoy, Trunovsky district, Baltser (Sedina) Natalya Sergeevna

Bringing fractions to a common denominator.

Goals:

to introduce students to the algorithm for reducing fractions to a common denominator and show a practical focus;
to develop the cognitive interest of students, the ability to see the connection with mathematics and the outside world;
to form the information culture of students;
Cultivate a culture of communication with the computer.

Equipment:

the teacher has a computer, a multimedia projector,Power point handout for pair work.

students have notebooks, textbooks, pencils, colored pencils, rulers.

During the classes

I. Organizational moment.Introduction of the teacher: emotional mood, motivation of students.

- Good afternoon! Today I will teach the lesson, Natalya Sergeevna. I am very glad to see you, I am interested in getting to know you and working with you. Please sit comfortably, relax, look into each other's eyes, smile at each other, wish your neighbor on the desk a good mood with your eyes. I also wish you good mood and active work.

Guys, please look at the slide (Slide 2)

I came to you with such a mood, raise your hands whose mood matches mine.

And who has a different mood ...

I will try my best to keep you in the mood.I wish you good luck, good luck.

II. Knowledge update.

Guys, the Germans have preserved such a saying “to get into fractions”, which means to get into a difficult situation. And so that you and I do not get into fractions, i.e. in a difficult position and must know and be able to do a lot. Let's define the area of "knowledge" with you. What you already know and can do with common fractions.

Repetition of the material from the previous lesson.

1. What part of the hour has passed since the beginning of the day? (Slide 3, 4, 5)

2. What part of the field did the tractor driver plow? (Slide 6)

3. What part of the road did the bus cover? (Slide 7)

4. What part of the plums remained on the plates? (Slide 8)

5. (Slide 9) Bring to the denominator 36 those of these fractions that are possible:

, , , , , , , , , , .

III. Studying new material. (Slide 10)

In grade 5 "A", girls make up all the students in the class, and boys make up all the students in the class. Who has more boys or girls in the class?

And what fractions can you compare, what do we need to do for this?Bring fractions to the same denominator.

- What do you think we will do in class?

Bring fractions to a common denominator.

Yes, the topic of our lesson is “Bringing fractions to a common denominator”.

(Slide 11).

Write down in your notebooks the number and the topic of the lesson: "Bringing fractions to a common denominator."

Why do we need it?

To compare, perform actions with fractions, solve practical problems.

The purpose of our lesson is to learn how to reduce fractions to a common denominator.

Let's bring the fractions to the same denominator.

To what denominator can they be reduced?

Which one is more convenient and why?

(Slide 12).

So then > means there are more girls in the class

Answer : there are more girls in the class.

Thus, we made sure that we can solve this problem only by being able to bring fractions to a common denominator.

Let's try together with you to formulate the rule for reducing fractions to a common denominator.

To get acquainted with the "algorithm" of the rule of reducing fractions to a common denominator.

(Slide 13).

Rule:

additional multiplier;

Here we have a rule with you turned out to be a rule, using this rule you can always bring fractions to a common denominator.

What fractions can be reduced to any new denominator?

Give examples.

(Slide 14). Let's do it together. Paying attention to the memo, let's do it step by step.

How to bring fractions to a common denominator?

IV. Fizkultminutka.(Slide 15).

So do it with me

The exercise is:

Once - got up, stretched,

Two - bent, unbent,

Three - three hand claps

Three head nods.

Four - arms wider,

Five, six, sit quietly.

Seven, eight laziness discard.

v. Work on the topic of the lesson.

No. 806 (Slide 16).

Students work independently in pairs. A frontal check is organized.

Find multiple numbers that are multiples of two given numbers. Give the least common multiple of these numbers:is a number that is divisible by both 3 and 7

a) 3 and 7; b) 4 and 5; c) 6 and 12; d) 4 and 6.

No. 808. (Slide 17). And now you will work in pairs, while completing the task, be careful.

Bring the fractions to a common denominator, you have a table for answers on your desks, complete the solution in your notebook, and write down fractions with new denominators in the table.

A) ; b) ; V) ; G) ;

e) ; b) ; V) ; G) .

answers: (Slide 18, 19).

Which pair performed without errors? Well done! Fine!

And who with one mistake? And those who failed to complete without errors, do not worry, we are just starting to study the topic and you will work it out in the next lessons.

VI. Summarizing.(Slide 20).

Teacher ask students the following questions:

What was our goal at the beginning of the lesson?

Do you think we have achieved this goal?

How to bring fractions to the smallest denominator?

So, to bring fractions to a common denominator, what needs to be done

Where do we need fractions?(Slide 21)

What do you remember in the lesson?

All sorts of shots are needed
Fractions are important.
Learn the fraction, then

your luck will shine.
If you know fractions
To understand their exact meaning
It will even become easy

difficult task!

Guys, who think that the lesson was useful for you, and you understood everything that was said and what was done in the lesson, please select the red rectangle, put it aside andwrite D / Z on "5"

Guys, who think that the lesson was interesting, to a certain extent useful for you, you were quite comfortable in the lesson, please select the yellow rectangle, put it aside andwrite D / Z on "4"

Guys who think that they understood what was discussed at the lesson, but you should get advice from the teacher, please select the green rectangle, put it aside andwrite D / Z on "3".

VII. Homework(Slide 22):

p.8.4, No. 809, No. 812, on "5" - No. 813.

I was very pleased to work with you, my mood is good. Did your mood change during the lesson? I would like to note and put 5 for active work in the lesson. When the children leave the class, attach the card that you have chosen to the board. Thanks for the tutorial, good luck! (Slide 23) Thank you for the lesson!

Application

№ 808

№ 808 Reduce to the least common denominator of the fraction.

№ 808 Reduce to the least common denominator of the fraction.№ 808 Reduce to the least common denominator of the fraction.

Application

Rule:

To bring fractions to a common denominator:
1) choose the lowest common denominator;
2) divide the least common denominator into the denominators of these fractions, i.e. find for each fractionadditional multiplier;
3) multiply the numerator and denominator of each fraction by its additional factor.

Rule:

Lesson topic: Reducing fractions to a common denominator

Goals:

educational: to form the ability to bring fractions to the lowest common denominator and find an additional factor in more complex cases; to form the ability to translate ordinary fractions into decimals;

developing: develop logical thinking, memory,students' computing skills

Educational: to cultivate cognitive interest in the subject

During the classes

I. Organizational moment

II. Verbal counting

1. Find the greatest common divisor and least common multiple of the numbers: 10 and 12; 12 and 8; 15 and 9; 6 and 4; 6 and 8; 12 and 15; 12 and 10; 16 and 20; 11 and 7.

2. Two tourists left the same point at the same time in different directions. The speed of the first tourist is 6 km/h, the speed of the second is 7 km/h. How far apart will they be in 3 hours?

3. The pump fills the pool in 48 minutes. What part of the pool will the pump fill in 1 minute?

4. There are five sons in the family, each of them has one sister. How many children are in the family? (6 children.)

III . Lesson topic message

- In the last lesson, we brought fractions to a new denominator. Today we will find a common denominator for several fractions and find out what is the smallest common denominator of fractions.

IV. Learning new material

1. Any 2 fractions can be reduced to the same denominator, or, in other words, to a common denominator.

- Find some common denominators of fractions. Name their lowest common denominator.

The common denominator of fractions can be any common multiple of their denominators. .

At the same time, as a rule, they try to choose the smallest common denominator (LCD) - then calculations with fractions turn out to be easier. The least common denominator is equal to the least common multiple of the denominators of the given fractions.

2. Let's look at examples of how to find NOZ of fractions.

1) Let's reduce the fractions 7/21 and 2/7 to a common denominator.

- What is special about the numbers 21 and 7? (21 is evenly divisible by 7.)

(Reasoning leads the teacher.)

- The larger denominator - the number 21 - is divisible by the smaller denominator 7, therefore, it can be taken as a common denominator of these fractions. This common denominator is the smallest possible.

This means that you only need to reduce the fraction 2/7 to the denominator 21. To do this, we will find an additional factor: 21: 7 = 3.

- What conclusion can be drawn? (If one denominator of a fraction is divisible by another, then NOZ will have a larger denominator.)

2) Let's reduce the fractions 3/4 and 2/5 to a common denominator.

- What can you say about the numbers 4 and 5? (Numbers are relatively prime.) The common denominator of these fractions must be divisible by both 4 and 5, i.e. be their common multiple. There are infinitely many common multiples of 4 and 5: 20, 40, 60, 80, etc. The smallest multiple of 20 is the product of 4 and 5.

So, you need to bring each of the fractions to the denominator 20:

- What conclusion can be drawn? (If the denominators of the fractions are coprime, then the least common denominator is their product.)

V. Physical education

VI. Working on a task

VII. Consolidation of the studied material

1. No. 279 p. 45 (oral). Work in pairs.

Someone from a couple answers the teacher.

- Why can't the fraction 3/5 be reduced to a denominator of 36? (36 is not a multiple of 5.)

2. No. 283 (а-е) p. 46 (with a detailed commentary at the blackboard and in notebooks, a) b) write down the decision in detail, then pronounce it all orally, write down only fractions with a new denominator).

Solution:

Additional multipliers: 24:6 = 4, 24:8 = 3.

Additional multipliers: 45:9 = 5, 45:15 = 3.

3. Name the numbers that:

a) more than 4/7, but less than 5/7; b) more than 1/6, but less than 2/6; c) more than 5/8, but less than 3/4.

- What needs to be done to complete the task? (Bring fractions to a new denominator.)

4. No. 281 p. 46 (c) (one student on the back of the board, the rest in notebooks, self-examination).

Solution:

VIII. Independent work

Option I

1. Bring fractions to a new denominator 24:

2. Bring the fraction 3/5 to a new denominator: 15; 25; 40; 55; 250; 300.

Option II

1. Bring fractions to a new denominator 48:

2. Bring the fraction 4/7 to a new denominator: 14; 28; 49; 70; 210; 350.

3. Express in hundredths of a fraction:

Option III (for more advanced students)

1. Bring fractions to a new denominator 84:

2. Bring the fraction 5/8 to a new denominator: 16; 24; 56; 80; 240; 3200.

3. Express in hundredths of a fraction:

IX. Consolidation of the studied material

1. No. 290 p. 47 (oral). Work in pairs.

- What was used in the solution? (The main property of a fraction.)

- Formulate the main property of a fraction.

(Answer: a) x = 3, b) x = 5, c) x = 5, d) x = 7.)

2. No. 289 (c, d) p. 47 (independently, mutual verification).

- What is the greatest common divisor of the numerator and denominator?

X. Summing up the lesson

- What number can be the common denominator of two fractions?

- How to bring fractions to the lowest common denominator?

- What is the rule for reducing fractions to a common denominator?

Homework:

Fractions have different or the same denominators. Same denominator or otherwise called common denominator at the fraction An example of a common denominator:

\(\frac(17)(5), \frac(1)(5)\)

An example of different denominators for fractions:

\(\frac(8)(3), \frac(2)(13)\)

How to find a common denominator of a fraction?

The first fraction has a denominator of 3, the second is 13. You need to find a number that is divisible by both 3 and 13. This number is 39.

The first fraction must be multiplied by additional multiplier 13. So that the fraction does not change, we must multiply both the numerator by 13 and the denominator.

\(\frac(8)(3) = \frac(8 \times \color(red) (13))(3 \times \color(red) (13)) = \frac(104)(39)\)

We multiply the second fraction by an additional factor of 3.

\(\frac(2)(13) = \frac(2 \times \color(red) (3))(13 \times \color(red) (3)) = \frac(6)(39)\)

We have reduced the common denominator of the fraction:

\(\frac(8)(3) = \frac(104)(39), \frac(2)(13) = \frac(6)(39)\)

Lowest common denominator.

Consider another example:

Let's bring the fractions \(\frac(5)(8)\) and \(\frac(7)(12)\) to a common denominator.

The common denominator for the numbers 8 and 12 can be the numbers 24, 48, 96, 120, ..., it is customary to choose lowest common denominator in our case, this number is 24.

Lowest common denominator is the smallest number that divides the denominator of the first and second fractions.

How to find the lowest common denominator?
By enumeration of numbers, by which the denominator of the first and second fractions is divided and choose the smallest of them.

We need to multiply a fraction with a denominator of 8 by 3, and a fraction with a denominator of 12 multiply by 2.

\(\begin(align)&\frac(5)(8) = \frac(5 \times \color(red) (3))(8 \times \color(red) (3)) = \frac(15 )(24)\\\\&\frac(7)(12) = \frac(7 \times \color(red) (2))(12 \times \color(red) (2)) = \frac( 14)(24)\\\\ \end(align)\)

If you can’t immediately bring the fractions to the lowest common denominator, there’s nothing to worry about, in the future, when solving the example, you may have to get the answer

A common denominator can be found for any two fractions; it can be the product of the denominators of these fractions.

For example:
Reduce the fractions \(\frac(1)(4)\) and \(\frac(9)(16)\) to the lowest common denominator.

The easiest way to find the common denominator is to multiply the denominators 4⋅16=64. The number 64 is not the lowest common denominator. The task is to find the smallest common denominator. So we are looking further. We need a number that is divisible by both 4 and 16, this is the number 16. Let's reduce the fraction to a common denominator, multiply the fraction with a denominator of 4 by 4, and the fraction with a denominator of 16 by one. We get:

\(\begin(align)&\frac(1)(4) = \frac(1 \times \color(red) (4))(4 \times \color(red) (4)) = \frac(4 )(16)\\\\&\frac(9)(16) = \frac(9 \times \color(red) (1))(16 \times \color(red) (1)) = \frac( 9)(16)\\\\ \end(align)\)

In this lesson, we will look at reducing fractions to a common denominator and solve problems on this topic. Let's give a definition of the concept of a common denominator and an additional factor, remember about coprime numbers. Let's define the concept of the least common denominator (LCD) and solve a number of problems to find it.

Topic: Adding and subtracting fractions with different denominators

Lesson: Reducing fractions to a common denominator

Repetition. Basic property of a fraction.

If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then a fraction equal to it will be obtained.

For example, the numerator and denominator of a fraction can be divided by 2. We get a fraction. This operation is called fraction reduction. You can also perform the reverse transformation by multiplying the numerator and denominator of the fraction by 2. In this case, we say that we have reduced the fraction to a new denominator. The number 2 is called an additional factor.

Conclusion. A fraction can be reduced to any denominator that is a multiple of the denominator of the given fraction. In order to bring a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.

1. Bring the fraction to the denominator 35.

The number 35 is a multiple of 7, that is, 35 is divisible by 7 without a remainder. So this transformation is possible. Let's find an additional factor. To do this, we divide 35 by 7. We get 5. We multiply the numerator and denominator of the original fraction by 5.

2. Bring the fraction to the denominator 18.

Let's find an additional factor. To do this, we divide the new denominator by the original one. We get 3. We multiply the numerator and denominator of this fraction by 3.

3. Bring the fraction to the denominator 60.

By dividing 60 by 15, we get an additional multiplier. It is equal to 4. Let's multiply the numerator and denominator by 4.

4. Bring the fraction to the denominator 24

In simple cases, reduction to a new denominator is performed in the mind. It is customary only to indicate an additional factor behind the bracket a little to the right and above the original fraction.

A fraction can be reduced to a denominator of 15 and a fraction can be reduced to a denominator of 15. Fractions have a common denominator of 15.

The common denominator of fractions can be any common multiple of their denominators. For simplicity, fractions are reduced to the lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.

Example. Reduce to the least common denominator of the fraction and .

First, find the least common multiple of the denominators of these fractions. This number is 12. Let's find an additional factor for the first and second fractions. To do this, we divide 12 by 4 and by 6. Three is an additional factor for the first fraction, and two for the second. We bring the fractions to the denominator 12.

We reduced the fractions to a common denominator, that is, we found fractions that are equal to them and have the same denominator.

Rule. To bring fractions to the lowest common denominator,

First, find the least common multiple of the denominators of these fractions, which will be their least common denominator;

Secondly, divide the least common denominator by the denominators of these fractions, that is, find an additional factor for each fraction.

Thirdly, multiply the numerator and denominator of each fraction by its additional factor.

a) Reduce the fractions and to a common denominator.

The lowest common denominator is 12. The additional factor for the first fraction is 4, for the second - 3. We bring the fractions to the denominator 24.

b) Reduce the fractions and to a common denominator.

The lowest common denominator is 45. Dividing 45 by 9 by 15, we get 5 and 3, respectively. We bring the fractions to the denominator 45.

c) Reduce the fractions and to a common denominator.

The common denominator is 24. The additional factors are 2 and 3, respectively.

Sometimes it is difficult to verbally find the least common multiple for the denominators of given fractions. Then the common denominator and additional factors are found by factoring into prime factors.

Reduce to a common denominator of the fraction and .

Let's decompose the numbers 60 and 168 into prime factors. Let's write out the expansion of the number 60 and add the missing factors 2 and 7 from the second expansion. Multiply 60 by 14 and get a common denominator of 840. The additional factor for the first fraction is 14. The additional factor for the second fraction is 5. Let's reduce the fractions to a common denominator of 840.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemozina, 2012.

2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.

3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.

4. Rurukin A.N., Chaikovsky I.V. Tasks for the course of mathematics grade 5-6. - ZSH MEPhI, 2011.

5. Rurukin A.N., Sochilov S.V., Chaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.

6. Shevrin L.N., Gein A.G., Koryakov I.O. and others. Mathematics: A textbook-interlocutor for grades 5-6 of high school. Library of the teacher of mathematics. - Enlightenment, 1989.

You can download the books specified in clause 1.2. this lesson.

Homework

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M .: Mnemozina, 2012. (see link 1.2)

Homework: No. 297, No. 298, No. 300.

Other tasks: #270, #290