Goals:

educational: form the ability to reduce fractions to the lowest common denominator and find an additional factor in more complex cases; to form the ability to convert ordinary fractions into decimal;

developing: develop logical thinking, memory,computational skills of students

Educational: to foster a 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 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 one 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. Which part of the pool will the pump fill in 1 minute?

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

III ... Lesson topic message

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

IV. Learning new material

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

- Find some common denominators of fractions. What is their lowest common denominator?

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

In this case, as a rule, they try to choose the lowest common denominator (LCN) - then calculations with fractions are easier. The lowest common denominator is the lowest common multiple of the denominators of the given fractions.

2. Consider with examples how you can find the NOZ of fractions.

1) Bring the fractions 7/21 and 2/7 to a common denominator.

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

(The reasoning is given by the teacher.)

- The larger denominator - the number 21 - is divisible by the smaller denominator 7, therefore, it can be taken as the 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 the other, then NOZ will be the larger denominator.)

2) Bring the fractions 3/4 and 2/5 to a common denominator.

- What can you say about the numbers 4 and 5? (The 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 numbers, then their product will be the lowest common denominator.)

V. Physical education

Vi. Working on a task

Vii. Consolidation of the studied material

1. No. 279, p. 45 (orally). Work in pairs.

The teacher is answered by one of the couple.

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

2. № 283 (a-e) p. 46 (with a detailed commentary at the blackboard and in notebooks, a) b) write down the solution 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 do you need to do to complete the task? (Reduce 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. Reduce fractions to the new denominator 24:

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

Option II

1. Bring the fractions to the new denominator 48:

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

3. Express in hundredths of a fraction:

Option III (for more advanced students)

1. Bring the fractions to the new denominator 84:

2. Bring the fraction 5/8 to the 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 basic 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 check).

- What number is called the greatest common factor of a numerator and denominator?

X. Lesson summary

- What number can serve as the common denominator of two fractions?

- How do you bring fractions to the lowest common denominator?

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

Homework:

Common denominator of fractions

Fractions AND have the same denominator. They say they have common denominator 25. Fractions and have different denominators, but they can be brought to a common denominator using the basic property of fractions. To do this, we find a number that is divisible by 8 and 3, for example, 24. Let us bring the fractions to the denominator 24, for this we multiply the numerator and denominator of the fraction by additional factor 3. The additional factor is usually written on the left above the numerator:

Multiply the numerator and denominator of the fraction by an additional factor of 8:

Let us bring the fractions to a common denominator. Most often, fractions result in the lowest common denominator, which is the lowest common multiple of the fraction's denominator. Since the LCM (8, 12) = 24, then the fractions can be reduced to the denominator 24. Find the additional factors of the fractions: 24: 8 = 3, 24:12 = 2. Then

Several fractions can be brought to a common denominator.

Example. Let us bring the fractions to a common denominator. Since 25 = 5 2, 10 = 2 5, 6 = 2 3, then LCM (25, 10, 6) = 2 3 5 2 = 150.

Let's find additional factors of fractions and bring them to the denominator 150:

Comparison of fractions

In fig. 4.7 shows a segment AB of length 1. It is divided into 7 equal parts. The segment AC has a length and the segment AD has a length.

The length of the segment AD is greater than the length of the segment AC, i.e. the fraction is greater than the fraction

Of the two fractions with a common denominator, the one with the larger numerator is larger, i.e.

For example, or

To compare any two fractions, they are brought to a common denominator, and then the rule for comparing fractions with a common denominator is applied.

Example. Compare fractions

Solution. LCM (8, 14) = 56. Then Since 21> 20, then

If the first fraction is less than the second, and the second is less than the third, then the first is less than the third.

Proof. Let three fractions be given. Let's bring them to a common denominator. Let after that they have the form Since the first fraction is less

second, then r< s. Так как вторая дробь меньше третьей, то s < t. Из полученных неравенств для натуральных чисел следует, что r < t, тогда первая дробь меньше третьей.

The fraction is called correct if its numerator is less than the denominator.

The fraction is called wrong if its numerator is greater than or equal to the denominator.

For example, fractions are correct and fractions are incorrect.

The correct fraction is less than 1 and the improper fraction is greater than or equal to 1.

To use the preview of presentations, create yourself a Google account (account) and log into it: https://accounts.google.com

Slide captions:

Preview:

PUBLIC LESSON

CLASS 5

Mathematic teacher

Municipal general education

institutions "Main

secondary school №6 "village of Donsky Trunovsky district Baltser (Syedina) Natalya Sergeevna

Bringing fractions to a common denominator.

Goals:

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

Equipment:

the teacher has a computer, a multimedia projector,Power Point, handout for working in pairs.

for students - notebooks, textbooks, pencils, colored pencils, rulers.

During the classes

I. Organizational moment.Teacher introduction: emotional attitude, motivation of students.

- Good day! Today I will give the lesson, Natalya Sergeevna. I am very glad to see you, it is interesting for me to meet you and work with you. Please sit down more comfortably, relax, look into each other's eyes, smile at each other, with your eyes wish your neighbor on the desk a good mood. I also wish you a 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 coincides with mine.

And who has a different mood ...

I will try to maintain your mood during the lesson.I wish you good luck, good hour.

II. Knowledge update.

Guys, the Germans still have such a saying "get into fractions", which means getting into a difficult situation. And so that you and I do not get into fractions, i.e. in a difficult situation and should know a lot and be able to. Let's define the area of ​​"knowledge" with you. What you already know and can do using ordinary fractions.

Repetition of the material from the previous lesson.

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

2. What part of the field has the tractor driver plowed? (Slide 6)

3. What part of the road has the bus traveled? (Slide 7)

4. How much of the plums are left on the plates? (Slide 8)

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

, , , , , , , , , , .

III. Learning new material... (Slide 10)

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

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

- What do you think we will do in the lesson?

Bring fractions to a common denominator.

Yes, the topic of our lesson is "Reducing fractions to a common denominator."

(Slide 11).

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

Why do we need this?

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

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

Let us reduce the fractions to the same denominator.

What denominator can they be brought to?

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 a rule for reducing fractions to a common denominator.

Get acquainted with the "algorithm" of the rule for reducing fractions to a common denominator.

(Slide 13).

Rule:

additional factor;

Here we have a rule with you, the rule turned out, 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, we will follow the memo step by step.

How to bring fractions to a common denominator?

IV. Physical education.(Slide 15).

Well do with me

The exercise is:

Once - got up, stretched,

Two - bent down, unbent,

Three - three claps in your hands

Head three nods.

Four - arms wider

Five, six, sit down quietly.

Seven, eight, we will discard laziness.

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. What is the least common multiple of these numbers:this 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, be careful when completing the task.

Bring the fractions to a common denominator, you have a table for answers on your desks, complete the solution in a 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 error? Well done! OK!

And who has one mistake? And those who failed to complete it without mistakes, 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 asks students the following questions:

What was the goal we set for ourselves at the beginning of the lesson?

Do you think we have achieved this goal?

How to bring fractions to the lowest denominator?

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

Where do we need fractions?(Slide 21)

What do you remember during the lesson?

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

luck will flash for you.
If you know fractions
To understand exactly the meaning of them,
It will become easy even

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 a red rectangle, put it aside andwrite D / Z to "5"

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

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

Vii. Homework(Slide 22):

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

I was very pleased to work with you, I am in a good mood. Did your mood change during the lesson? I would like to mark and give 5 for active work in the lesson. Children leaving the class, attach the card that you have chosen to the board. Thanks for the lesson. Good luck! (Slide 23) Thank you for the lesson!

Application

Application

Rule:

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

Rule:

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

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

Topic: Addition and Subtraction of Fractions with Different Denominators

Lesson: Converting Fractions to a Common Denominator

Repetition. The main property of a fraction.

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

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 inverse 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 the complementary factor.

Output. A fraction can be reduced to any denominator, a multiple of the denominator of a 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.

35 is a multiple of 7, that is, 35 is divisible by 7 without a remainder. This means that this transformation is possible. Let's find an additional factor. To do this, divide 35 by 7. We get 5. 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. 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 4. Multiply the numerator and denominator by 4.

4. Bring the fraction to the denominator 24

In simple cases, the reduction to a new denominator is performed in the mind. It is only accepted to indicate an additional multiplier outside the bracket just 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 also have a common denominator of 15.

The common denominator of fractions can be any common multiple of their denominators. For simplicity, fractions result in the lowest common denominator. It is equal to the least common multiple of the denominators of these fractions.

Example. Reduce to the lowest 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 for the second fraction. 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. Let us bring the fractions to the denominator 12.

We brought fractions to a common denominator, that is, we found fractions equal to them, which have the same denominator.

Rule. To bring fractions to the lowest common denominator, you need

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

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

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

a) Reduce the fraction and to a common denominator.

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

b) Reduce the fraction and to a common denominator.

The lowest common denominator is 45. Dividing 45 by 9 by 15 gives 5 and 3, respectively. Bring the fractions to the denominator 45.

c) Reduce the fraction and to a common denominator.

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

Sometimes it is difficult to orally find the lowest common multiple for the denominators of given fractions. Then the common denominator and additional factors are found using prime factorization.

Reduce the fraction and to a common denominator.

Let's expand the numbers 60 and 168 into prime factors. Let's write the decomposition of 60 and add the missing factors 2 and 7 from the second decomposition. Multiply 60 by 14 to get a common denominator of 840. The complementary factor for the first fraction is 14. The complementary factor for the second fraction is 5. Bring the fractions to a common denominator of 840.

Bibliography

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

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

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

4. Rurukin A.N., Tchaikovsky I.V. Assignments for the course mathematics grade 5-6. - ZSH MEPhI, 2011.

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

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

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

Homework

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. et al. Mathematics 6. - M .: Mnemosina, 2012. (see link 1.2)

Homework: # 297, # 298, # 300.

Other assignments: # 270, # 290