Goals:
educational: develop the ability to reduce fractions to the lowest common denominator and find an additional factor in more difficult cases; develop the ability to convert ordinary fractions to decimals;
developing: develop logical thinking, memory,students' computing skills
Educational: to cultivate cognitive interest in the subject
During the classes
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 after 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 reduced fractions to a new denominator. Today we will find the common denominator for several fractions and find out what the least common denominator of fractions is.
IV. Learning new material
1. Any 2 fractions can be reduced to the same denominator, or, in other words, to a common denominator.
- Find several common denominators of fractions. Name 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 select the lowest common denominator (LCD) - then calculations with fractions turn out to be simpler. 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 you can find the NC of fractions.
1) Let's bring the fractions 7/21 and 2/7 to a common denominator.
- What is special about the numbers 21 and 7? (21 is divisible by 7.)
(The teacher gives the reasoning.)
- The larger denominator - the number 21 - is divided by the smaller denominator 7, therefore, it can be taken as the common denominator of these fractions. This common denominator is the lowest possible.
This means that we only need to bring 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 divided by another, then N3 will be the larger denominator.)
2) Let's 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 an infinite number of common multiples of 4 and 5: 20, 40, 60, 80, etc. The smallest multiple of 20 is the product of 4 and 5.
This means that you need to bring each of the fractions to a denominator of 20:
- What conclusion can be drawn? (If the denominators of the fractions are mutually prime, then the lowest common denominator is their product.)
V. Physical education minute
VI. Working on a task
VII. Reinforcing the material learned
1. No. 279 p. 45 (oral). Work in pairs.
One person from the pair 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 (a-e) p. 46 (with a detailed commentary at the board 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 needs to be done to complete the task? (Bring the fractions to the new denominator.)
4. No. 281 p. 46 (c) (one student per back side boards, the rest in notebooks, self-test).
Solution:
VIII. Independent work
Option I
1. Reduce the fractions to the new denominator 24:
2. Reduce the fraction 3/5 to a new denominator: 15; 25; 40; 55; 250; 300.
Option II
1. Reduce the fractions to the new denominator 48:
2. Reduce the fraction 4/7 to a new denominator: 14; 28; 49; 70; 210; 350.
3. Express the fraction in hundredths:
Option III (for more advanced students)
1. Reduce the fractions to the new denominator 84:
2. Reduce the fraction 5/8 to a new denominator: 16; 24; 56; 80; 240; 3200.
3. Express the fraction in hundredths:
IX. Reinforcing the material learned
1. No. 290 p. 47 (oral). Work in pairs.
- What did you use to solve it? (The main property of a fraction.)
- State 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 (independent, mutual verification).
- What number is the greatest common divisor of the numerator and denominator?
X. Lesson summary
- What number can serve as the common denominator of two fractions?
- How do you reduce fractions to their lowest common denominator?
- What property is the rule for reducing fractions to a common denominator based on?
Homework:
I originally wanted to include common denominator techniques in the Adding and Subtracting Fractions section. But there turned out to be so much information, and its importance is so great (after all, not only numerical fractions have common denominators), that it is better to study this issue separately.
So, let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of a fraction comes to the rescue, which, let me remind you, sounds like this:
A fraction will not change if its numerator and denominator are multiplied by the same number other than zero.
Thus, if you choose the factors correctly, the denominators of the fractions will become equal - this process is called reduction to a common denominator. And the required numbers, “evening out” the denominators, are called additional factors.
Why do we need to reduce fractions to a common denominator? Here are just a few reasons:
- Adding and subtracting fractions with different denominators. There is no other way to perform this operation;
- Comparing fractions. Sometimes reduction to a common denominator greatly simplifies this task;
- Solving problems involving fractions and percentages. Percentages are essentially ordinary expressions that contain fractions.
There are many ways to find numbers that, when multiplied by them, will make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, effectiveness.
Criss-cross multiplication
The simplest and reliable way, which is guaranteed to equalize the denominators. We will act “in a headlong manner”: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:
As additional factors, consider the denominators of neighboring fractions. We get:
Yes, it's that simple. If you are just starting to study fractions, it is better to work using this method - this way you will insure yourself against many mistakes and are guaranteed to get the result.
The only drawback this method- you have to do a lot of counting, because the denominators are multiplied “over and over”, and the result can be very large numbers. This is the price to pay for reliability.
Common Divisor Method
This technique helps to significantly reduce calculations, but, unfortunately, it is used quite rarely. The method is as follows:
- Before you go straight ahead (i.e., using the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided into the other.
- The number resulting from this division will be an additional factor for the fraction with a smaller denominator.
- In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is where the savings lie. At the same time, the probability of error is sharply reduced.
Task. Find the meanings of the expressions:
Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divided without a remainder by the other, we use the method of common factors. We have:
Note that the second fraction was not multiplied by anything at all. In fact, we cut the amount of computation in half!
By the way, I didn’t take the fractions in this example by chance. If you're interested, try counting them using the criss-cross method. After reduction, the answers will be the same, but there will be much more work.
This is the power of the common divisors method, but, again, it can only be used when one of the denominators is divisible by the other without a remainder. Which happens quite rarely.
Least common multiple method
When we reduce fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.
There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the original fractions, as is assumed in the “criss-cross” method.
For example, for denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24: 12 = 2. This number is much less than the product 8 · 12 = 96.
The smallest number that is divisible by each of the denominators is called their least common multiple (LCM).
Notation: The least common multiple of a and b is denoted by LCM(a ; b) . For example, LCM(16, 24) = 48 ; LCM(8; 12) = 24 .
If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:
Task. Find the meanings of the expressions:
Note that 234 = 117 2; 351 = 117 3. Factors 2 and 3 are coprime (have no common factors other than 1), and factor 117 is common. Therefore LCM(234, 351) = 117 2 3 = 702.
Likewise, 15 = 5 3; 20 = 5 · 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15, 20) = 5 3 4 = 60.
Now let's bring the fractions to common denominators:
Notice how useful it was to factorize the original denominators:
- Having discovered identical factors, we immediately arrived at the least common multiple, which, generally speaking, is a non-trivial problem;
- From the resulting expansion you can find out which factors are “missing” in each fraction. For example, 234 · 3 = 702, therefore, for the first fraction the additional factor is 3.
To appreciate how much of a difference the least common multiple method makes, try calculating these same examples using the criss-cross method. Of course, without a calculator. I think after this comments will be unnecessary.
Don't think that there won't be such complex fractions in the real examples. They meet all the time, and the above tasks are not the limit!
The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally “by eye,” but in general this is a complex computational task that requires separate consideration. We won't touch on that here.
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PUBLIC LESSON
5 CLASS
Mathematic teacher
Municipal educational
institution "Basic
comprehensive school No. 6" in the village of Donskoy, Trunovsky district, Baltser (Sedina) Natalya Sergeevna
Reducing fractions to a common denominator.
Goals:
- introduce students to the algorithm for reducing fractions to a common denominator and show practical orientation;
- develop students’ cognitive interest, the ability to see connections with mathematics and the world around them;
- to form the information culture of students;
- Foster a culture of communication with computers.
Equipment:
The teacher has a computer, a multimedia projector,Power Point, handouts for working in pairs.
Students have notebooks, textbooks, pencils, colored pencils, and rulers.
During the classes
I. Organizational moment.Teacher's introduction: 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 down comfortably, relax, look into each other's eyes, smile at each other, wish your neighbor on your desk a good mood with your eyes. I also wish you a good mood and active work.
Guys, please look at the slide (Slide 2)
I came to you with this mood, raise your hands if your mood matches mine.
Who's in a different mood...
I will try to keep your spirits up during class.I wish you good luck, good luck.
II. Updating knowledge.
Guys, the Germans still have this saying “getting into fractions”, which means getting into a difficult situation. And so that you and I don’t get into fractions, i.e. in a difficult situation and must know and be able to do a lot. Let us define the area of “knowledge”. What you already know and can do using fractions.
Repetition of 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 did the tractor driver plow? (Slide 6)
3. How much of the road did the bus travel? (Slide 7)
4. What part of the plums was left on the plates? (Slide 8)
5. (Slide 9) Reduce to the denominator 36 those of these fractions that are possible:
, , , , , , , , , , .
III.Learning 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. Are there more boys or girls in the class?
What fractions can you compare, what do we need to do for this?Reduce fractions to the same denominator.
- What do you think we will do in class?
Reduce fractions to a common denominator.
Yes, the topic of our lesson is “Reducing fractions to a common denominator.”
(Slide 11).
Write down the date and topic of the lesson in your notebooks: “Reducing fractions to a common denominator.”
Why do we need this?
To compare, perform operations with fractions, solve practical problems.
The goal of our lesson is to learn how to reduce fractions to a common denominator.
Let's reduce the fractions to the same denominator.
To what denominator can they be reduced?
Which one is more convenient and why?
(Slide 12).
So, that > means there are more girls in the class
Answer : There are more girls in the class.
So we made sure that we decided this task We can only be able to reduce fractions to a common denominator.
Let's try together to formulate a rule for bringing fractions to a common denominator.
Get acquainted with the “algorithm” - the rule for bringing fractions to a common denominator.
(Slide 13).
Rule:
additional multiplier;
Here we have a rule that turns 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 reminder, let's follow it step by step.
How to reduce fractions to a common denominator?
IV. Physical education minute.(Slide 15).
Come on, do it with me
The exercise is like this:
Once - we stood up, stretched,
Two - bent over, straightened up,
Three - clap your hands three times
Three nods of the head.
Four - arms wider,
Five, six, sit down quietly.
Let's discard seven, eight laziness.
V. Work on the topic of the lesson.
No. 806 (Slide 16).
Students work independently in pairs. A frontal inspection is being organized.
Find several 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). 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 your notebook, and write the fractions with new denominators in the table.
A) ; b) ; V) ; G) ;
d) ; b) ; V) ; G) .
answers: (Slide 18, 19).
Which pair completed it without errors? Well done! Fine!
And who has one mistake? And for those who were unable to complete it without errors, don’t worry, we are just starting to study the topic and you will work on it in the next lessons.
VI. Summarizing.(Slide 20).
Teacher asks students the following questions:
What goal did we set for ourselves at the beginning of the lesson?
Do you think we have achieved this goal?
How to reduce fractions to the lowest 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 from the lesson?
All sorts of fractions are needed
All fractions are important.
Learn fractions, then
good luck will shine on you.
If you know fractions,
Exactly the meaning of understanding them,
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 done in the lesson, please select the red rectangle, put it aside andwrite D/Z to “5”
Guys who think that the lesson was interesting, to a certain extent useful for you, you were quite comfortable during the lesson, please select the yellow rectangle, put it aside andwrite D/Z to “4”
Guys who think that you understood what was discussed in the lesson, but you should get advice from the teacher, please select the green rectangle, put it aside andwrite D/Z to “3”.
VII. Homework(Slide 22):
clause 8.4, No. 809, No. 812, at “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 note and give 5 for active work in the lesson. When leaving class, guys, attach the card you chose to the board. Thanks for the lesson. Good luck! (Slide 23) Thank you for the lesson!
Application
№ 808 | |||||||
№ 808 Reduce to the lowest common denominator of the fraction. | |||||||
№ 808 Reduce to the lowest common denominator of the fraction.№ 808 Reduce to the lowest common denominator of the fraction. | |||||||
Application
Rule:
To reduce fractions to a common denominator, you need to:
1) choose the lowest common denominator;
2) divide the lowest common denominator by 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:
To reduce fractions to a common denominator, you need to:
1) choose the lowest common denominator;
2) divide the lowest common denominator by 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.
In this lesson we will look at reducing fractions to a common denominator and solve problems on this topic. Let's define the concept of a common denominator and an additional factor, and remember about relatively prime numbers. Let's define the concept of the lowest 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. The main property of a fraction.
If the numerator and denominator of a fraction are multiplied or divided by the same natural number, you get an equal fraction.
For example, the numerator and denominator of a fraction can be divided by 2. We get the 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 an additional factor.
Conclusion. A fraction can be reduced to any denominator that is a multiple of the denominator of the given fraction. To bring a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.
1. Reduce the fraction to the denominator 35.
The number 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. Reduce the fraction to denominator 18.
Let's find an additional factor. To do this, divide the new denominator by the original one. We get 3. Multiply the numerator and denominator of this fraction by 3.
3. Reduce the fraction to a denominator of 60.
Dividing 60 by 15 gives an additional factor. It is equal to 4. Multiply the numerator and denominator by 4.
4. Reduce the fraction to the denominator 24
In simple cases, reduction to a new denominator is performed mentally. It is only customary to indicate the additional factor behind a bracket slightly 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 are reduced to their lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.
Example. Reduce to the lowest common denominator of the fraction and .
First, let's 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, divide 12 by 4 and 6. Three is an additional factor for the first fraction, and two is for the second. Let's bring the fractions to the denominator 12.
We brought the fractions to a common denominator, that is, we found equal fractions that have the same denominator.
Rule. To reduce fractions to their lowest common denominator, you must
First, find the least common multiple of the denominators of these fractions, it will be their least common denominator;
Secondly, divide the lowest common denominator by the denominators of these fractions, i.e. find an additional factor for each fraction.
Third, 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 reduce 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 gives 5 and 3, respectively. We reduce the fractions to the denominator 45.
c) Reduce the fractions and to a common denominator.
The common denominator is 24. Additional factors are 2 and 3, respectively.
Sometimes it can be difficult to verbally find the least common multiple of the denominators of given fractions. Then the common denominator and additional factors are found using prime factorization.
Reduce the fractions and to a common denominator.
Let's factor 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. Let's 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 bring the fractions to a common denominator of 840.
Bibliography
1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemosyne, 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., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. - ZSh MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at 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 high school. Math teacher's library. - 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. and others. Mathematics 6. - M.: Mnemosyne, 2012. (link see 1.2)
Homework: No. 297, No. 298, No. 300.
Other tasks: No. 270, No. 290
Topic: Reducing fractions to a common denominator. Class: 5 UMK: Mathematics. 5th grade / G.V. Dorofeev, I.F. Sharygin and others, publishing house "Prosveshcheniye" Place of the lesson in the lesson system: the first lesson in the block, a lesson on familiarization with the typology of tasks Purpose: to organize activities on perception, comprehension and primary memorizing new knowledge and ways of doing things. Objectives: Educational: - consolidate the ability to find the least common multiple of numbers; - introduce the concept of an additional factor; - practice the ability to find an additional factor and bring fractions to a new common denominator; - consolidate knowledge of the basic properties of fractions and the ability to reduce fractions. Developmental: - broadening the horizons of students; - development of techniques of mental activity, memory, attention, ability to compare, analyze, draw conclusions; - increasing students' information culture and interest in the subject; - development of cognitive activity, positive motivation for the subject; - develop needs for self-education. Educational: - fostering responsibility, independence, and the ability to work in a team; - show mathematics as an interesting science, turn the lesson into an unusual lesson where every student can express himself. Planned results: Personal: - show interest in studying the topic; - demonstrate a desire to put your knowledge into practice; - express your thoughts correctly; - understand the meaning of the task; - adequately perceive the assessment of the teacher and classmates. Metasubject: . Cognitive UUD: - ability to transform models in order to identify general laws that determine subject area; - continue to develop the ability to find the least common multiple;. . Regulatory UUD: - install new ones independently learning objectives by asking questions about the unknown; - carry out educational tasks in accordance with the goal; - correlate acquired knowledge with real life; - carry out educational activities in accordance with the plan, plan your own activities. Communicative UUD: - formulate a statement, opinion; - ability to justify and defend one’s opinion; - coordinate positions with a partner and find common decision ; - competently use verbal means to present the result. Subject: - bring a fraction to a new denominator; - derive the concept of an additional factor - derive a rule: how to reduce a fraction to the lowest common denominator. Structure and course of the lesson Stage of the lesson Objectives of the stage Activities of the teacher Activities of the students Time (in minutes) 1 1. Organizational stage Create a favorable psychological mood for work Include in the business rhythm of the lesson. 2. Updating knowledge Updating basic knowledge and methods of action. Greeting, checking readiness for a lesson, organizing children's attention. Organization of oral calculations Participate in repetition work: in a conversation with the teacher they answer the questions posed. 7 3. Setting the goals and objectives of the lesson. Motivation for students' learning activities. Providing motivation for children to learn and their acceptance of lesson goals. Motivates students, together with them determines the purpose of the lesson; draws students' attention to the significance of the topic. determine the topic and purpose of the lesson. 4 Formed UUD Communicative: planning educational cooperation with the teacher and peers. Regulatory: organizing your learning activities Personal: motivation to learn Cognitive: structuring your own knowledge. Communicative: organize and plan educational cooperation with the teacher and peers. Regulatory: control and evaluation of the process and results of activities. Personal: assessment of the material being learned. Cognitive: the ability to consciously and voluntarily construct a speech statement orally. Personal: self-determination. Regulatory: goal setting. Communicative: the ability to enter into dialogue, participate in a collective discussion of an issue. 4. Primary consolidation of new knowledge Show a variety of tasks 5. Physical education session Change of activity. 6. Consolidation of new knowledge and skills 6. Control of assimilation, discussion of mistakes made and their correction. 7. Reflection (summarizing the lesson) 8. Information about homework Organization and control over the process of solving tasks. They work in pairs, independently and together with the teacher on assigned tasks. 10 Change activities, provide emotional relief for students. Practice the skills of organizing and monitoring the process of solving tasks. Students have changed their activity and are ready to continue working. 2 Work in pairs, independently and together with the teacher, on assigned tasks. 10 Give a qualitative assessment of the work of the class and individual students. Identifies the quality and level of knowledge acquisition, and also establishes the causes of identified errors. 4 Quantify student work Ensure children understand the content and how to complete homework Summarize the work of the class as a whole. Students analyze their work, express their difficulties out loud, and discuss the correctness of solving problems. Students hand in assigned assignments. Gives a comment on homework. Students write down the assignment in their diaries. 4 3 Cognitive: developing interest in this topic. Personal: formation of readiness for self-education. Communicative: be able to express your thoughts orally; listen and understand the speech of others. Regulatory: planning your activities to solve a given problem and monitoring the result obtained. Cognitive: developing interest in this topic. Personal: formation of readiness for self-education. Communicative: be able to express your thoughts orally; listen and understand the speech of others. Regulatory: planning your activities to solve a given problem and monitoring the result obtained. Personal: formation of positive self-esteem Communicative: Regulatory: the ability to independently adequately analyze the correctness of actions and make the necessary adjustments. Regulatory: assessing one's own activities in the lesson Stage of the lesson Objectives of the stage Teacher's activities Students' activities Time Formed UUD 1. Organizational stage Create a favorable psychological mood for work The teacher welcomes students, checks their readiness for the lesson, organizes children's attention. Get involved in the business rhythm of the lesson. 1 Communicative: planning educational cooperation with the teacher and peers. Regulatory: organizing your learning activities Personal: motivation for learning Updating basic knowledge and methods of action. - Before starting to study a new topic, we will review the material studied in previous lessons. To do this, let's play the game "True/False". Take a sheet of paper with the task on your desk. Please answer the question: Game “True/False” 7 Cognitive: structuring your own knowledge. Communicative: organize and plan educational cooperation with the teacher and peers. Regulatory: control and evaluation of the process and results of activities. Personal: assessment of the material being learned. 2. Updating knowledge “Without knowledge of fractions, no one can be considered knowledgeable in arithmetic” T. Cicero “+” True / “-” false o Question 3 5 1. Is it true that fractions have different 4 6 denominators? 2. Is it true that the number 12 is the least common multiple of the numbers 4 and 6? 3 Complete tasks; - answer questions 5 3 orally. Is it true that fractions 4 and 6 can be reduced to a denominator of 12? 3 9 5 10 4. Is it true that the fractions 4 and 12 are equal? 5. Is it true that the fractions 6 and 12 are equal? - Guys, what basic concepts did you have to remember to answer the questions? (OK, Basic property of fractions) - mark the fractions on the coordinate line: Mark the indicated points on the coordinate line, discussing which one is necessary a) ; 1 5 3 9 2 1 b) 3 ; determine a unit segment 2 way out to the problem: what to do? (Find NOC). Now write down the fractions so that it is immediately clear which unit segment needs to be chosen 3. Setting the goals and objectives of the lesson. Motivation for students' learning activities. 4. Learning new material Ensuring that children are motivated to learn and that they accept the goals of the lesson. What rule did you use? What is it? Look at the fractions and tell me what happened? How have they changed? Reduce fractions to a common denominator. They pronounce the Main Property of Fractions - the teacher asks a series of questions necessary to: 1) formulate the topic of the lesson; 2) formulating the purpose of the lesson; 3) individual tasks. - Write down the date in a notebook, determine the topic and purpose of the lesson. Can you guess the topic of the lesson? Formulate the topic and purpose of the lesson. What task will each of you set for yourself for today’s lesson? Draw a ladder of 5 steps in the margins and mark which one you are on. at this stage lesson on this topic. Formation of ideas about solving problems into parts. They reason, answer questions, draw conclusions. What is needed for a better and easier understanding of this topic? Why is it necessary to be able to reduce fractions to a common denominator?? Can any of you now name the stages of the algorithm? Try giving 7 1 3 1 ; ; fractions to a common denominator: ; 8 4 16 2 So, what are the stages of the algorithm? Reducing fractions to the lowest common denominator (LCD) To reduce several fractions to the lowest common denominator, you need: 4 Cognitive: the ability to consciously and voluntarily construct a speech utterance in oral form. Personal: self-determination. Regulatory: goal setting. Communicative: the ability to enter into dialogue, participate in a collective discussion of an issue. The ability to express one’s point of view and argue for it 10 Cognitive: developing interest in a given topic. Personal: formation of readiness for self-education. Communicative: be able to express your thoughts orally; listen and understand the speech of others. Regulatory: planning your activities to solve a given problem and monitoring the result obtained. -Build a monologue story in accordance with the questions posed; formulate the topic and goals of the lesson. - Answer questions Create an algorithm. They answer questions and try to complete the task. Independently, mutual control Participate in drawing up the algorithm, Write the algorithm in a notebook 1) find the least common multiple of the denominators of these fractions, it will be their lowest common denominator; 2) divide the lowest common denominator by the denominators of these fractions, i.e. find an additional factor for each fraction; 3) multiply the numerator and denominator of each fraction by its additional factor. 5. Physical education 6. Application of knowledge and skills in a new situation Change activities, provide emotional relief for students. Change activities to provide emotional relief for students. Show a variety of tasks So, we have formulated an algorithm for reducing fractions to a common base, check what is written in the textbook and does the text match our algorithm? Now let's do a few tasks from the textbook. No. 806 “True/false” No. 807(a-e), according to the wording of the task, what can be said about the common denominators? 6. Control of assimilation, discussion of mistakes made and their correction. The ability to independently apply one’s knowledge in a standard but new situation, self-control, self-testing Task cards 1 125 28 a) , ; 2 150 63 c) 4 16 17 b) , ; 21 56 35 7 5 444 120, . 12 18 777 720 Students have changed their activity and are ready to continue working. 2 Work in pairs on the task, draw conclusions. -students complete the task, 10 Work in pairs Students complete in notebooks, one at the board. Carry out mutual verification. Self-assessment. 5 Cognitive: developing interest in this topic. Personal: formation of readiness for self-education. Communicative: be able to express your thoughts orally; listen and understand the speech of others; student interaction in pair work. Regulatory: planning your activities to solve a given problem and monitoring the result obtained. Personal: the formation of positive self-esteem Communicative: Regulatory: the ability to independently adequately analyze the correctness of actions and make the necessary adjustments. 7. Reflection (summarizing the lesson) Evaluation (students highlighting and realizing what has already been learned and what still needs to be learned, awareness of the quality and level of learning); What did we talk about today? What goal have we set today? Have we achieved this goal? Was everything clear, was everything done in time? Why is it necessary to be able to reduce fractions to the lowest common denominator? Now, in your notebooks, draw a ladder of five steps and note which step on this topic you are now on, have you climbed it? How to reach the top step? I want to end the lesson with this statement: “It is not enough to just understand the problem, you need the desire to solve it. It is impossible to solve a difficult problem without a strong desire, but if you have it, it is possible. Where there is a desire, there is a way” D. Polya Students answer questions 3 Cognitive: reflection on the methods and conditions of action, adequate understanding of the reasons for success and failure, control and evaluation of the process and results of activity Communicative: the ability to express their thoughts, argumentation The lesson is over! Well done to all of you! Thanks for the work! 8. Information about homework Ensuring that children understand the purpose, content and methods of completing homework Write down the homework: compose and solve the problem into parts. No. 807 (g-k) Regulatory: assessing their own activities in the lesson Students write down the task in their diaries. 2
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