Independent work on nok and node. Greatest common divisor

Sections: Mathematics

Lesson type – lesson in applying knowledge and skills.

Lesson Objectives

Educational: organize student activities to update knowledge and skills on the topic: “GCD and LCM” and ensure their creative application in solving problems of finding GCD and LCM numbers.
Educational: to promote the development of mental operations in students: the ability to analyze, highlight the main thing, and present solutions to problems.
Educational: the formation of humane relationships in the classroom, independence and activity, perseverance, the ability to overcome difficulties, maximum performance.

Lesson structure

Organizational moment – 2 min.
Gymnastics of the mind. Algorithms for accelerated calculations – 6 min.
Updating previously studied material – 6 min.
Finding GCD using the Euclidean algorithm – 9 min.
Using the formula GCD (a, b) GCD (a, b) = ab and the Euclidean algorithm for finding the LCM of numbers – 7 min.
Independent work – 5 min.
Checking and discussing the results obtained – 2 min.
Homework information – 1 min.
Summing up – 2 min.

Lesson progress

1. Organizational moment.

Stage objectives: provide a normal external environment for work and psychologically prepare students for communication in the upcoming lesson.

Greetings

Teacher: Hello, please sit down. My respects and best wishes to everyone.

Checking students' readiness for the lesson: marking of absentees, state of workplaces, availability of notebooks, textbooks, pens, diaries.

Teacher: My friends! Is everyone ready for the lesson? Wonderful! Attention! Let's start work!

Disclosure of the general goals of the lesson and its plan.

Teacher: - The topic of our lesson is the greatest common divisor and the least common multiple. The lesson plan is in front of you on the board. Meet him. Does anyone have any comments?

No. Then we will try to implement it together with you.

2. Gymnastics of the mind. Algorithms for accelerated calculations.

Stage tasks: remember and consolidate accelerated calculation algorithms, definition
divisibility.

Four students perform tasks at the board, reminiscent of mental calculation techniques.

Teacher: At the beginning of the lesson we will do gymnastics. No, not a physical education session. Physical perfection is a great thing. But the beauty of a person lies primarily in the harmony of his beautiful thoughts, beautiful words and beautiful deeds. We will conduct mental gymnastics.

B 625: 25
E 1225: 35
U 7225: 85
WITH 4225: 65

(Sample answer - dividing the number 625 by the number 25 means finding a number that, multiplied by 25, gives 625. Rule: to square a two-digit number ending in 5, it is enough to multiply the number of its tens by the number increased by 1, and add 25 to the work on the right.

625: 25 = 25
1225: 35 = 35
7225: 85 = 85
4225: 65 = 65).

AND 2376: 99
ABOUT 234: 9
L 41958: 999
TO 3861: 99
A 5742: 99

(A sample answer is to divide the number 2376 by the number 99, which means finding a number that, multiplied by 99, will give 2376. Rule: to multiply by a number written in nines, you must add as many zeros to the multiplicand on the right as there are nines in the factor, and from subtract the multiplicand of the result.

2376: 99 = 24
234: 9 = 26
41958: 999 = 42
3861: 99 = 39
5742: 99 = 58).

IN 792: 11
A 693: 11
AND 748: 11
TO 649: 11

(Sample answer - dividing the number 792 by the number 11 means finding a number that, multiplied by 11, will give 792. Rule: to multiply a two-digit number by 11, the sum of its digits is less than 10, you need to write the sum of its digits between the digits of the number. To To multiply by 11 a two-digit number whose sum of digits is greater than or equal to 10, you need to write the excess of the sum of the digits of the number by 10 between the tens digit increased by 1 and the units digit.

792: 11 = 72
693: 11 = 63
748: 11 = 68
649: 11 = 59).

D 2916: 54
AND 2704: 52
Z 3249: 57
U 3136: 56

(Sample answer - dividing the number 2916 by the number 54 means finding a number that, multiplied by 54, will give 2916. Rule: to square a two-digit number that has 5 tens, it is enough to add the ones digit to 25 and add a square to the result on the right number of units so that the result is a four-digit number.

2916: 54 = 54
2704: 52 = 52
3249: 57 = 57
3136: 56 =56).

3. Updating previously studied material

Stage tasks: update the knowledge and skills that will be used in solving the proposed problems.

Frontal work on tasks written on the board. The student answers the question posed. After answering, students review their answer according to the following scheme: correctness, validity, completeness.

Determination of the greatest common divisor of natural numbers.

(A sample answer is the largest natural number by which each of the given natural numbers is divided is called the greatest common divisor of these numbers).

Determining the least common multiple of natural numbers.

(Sample answer - the smallest natural number that is divisible by each of the given natural numbers is called the least common multiple of these numbers).

Methods for finding GCD and LCM of the numbers we have studied.

(Sample answer

by definition GCD and NOC;
brute force method;
Euclidean algorithm for finding GCD numbers;
use of formula GCD (a, b) GCD (a, b) = ab)

(Sample answer - to find the GCM of natural numbers by brute force, it is advisable to sort through the divisors of the smallest number in descending order. To find the GCM of natural numbers by brute force, it is advisable to sort through the multiples of the largest number in ascending order.

Find C GCD(391,299) according to the Euclidean algorithm.

(Sample answer - to find the gcd of two numbers, sequential division is carried out. First, divide the larger number by the smaller. If a remainder is obtained, then divide the smaller number by the remainder. If a remainder is obtained again, then divide the first remainder by the second. Continue dividing in this way until until the remainder is 0. The last divisor is the gcd of these numbers. The convenience of the Euclidean algorithm becomes especially noticeable if we use a well-thought-out form of notation:

391	299	92	23
	1	3	4

In this table, the original numbers are first written down, divided in your head, the remainders are written down on the right, and the quotients are written down at the bottom, until the process is completed. The last divisor is the gcd.

4. Finding GCD using the Euclidean algorithm

Stage tasks: application of the Euclidean algorithm for solving CT problems, 2005, task B1.

Four students do tasks at the board. All tasks are taken from centralized testing materials.

Teacher: It is proposed to find the GCD using the Euclidean algorithm. Approach the task creatively.

(Sample answer - to find the gcd of three or more numbers, first find the gcd of any two of them, then the gcd of the found divisor and the third given number.

5. FindingNOC (a, c), using the Euclidean algorithm and formulaGCD (a, b) GCD (a, b) = ab.

Stage tasks: application of Euclidean algorithm and formula GCD (a, b) GCD (a, b) = ab to solve DH problems.
Contents of the stage
The student at the board and the whole class perform the following task:

6. Independent work - solving problems in groups

Stage tasks: organize the activities of students when carrying out independent work on solving problems of increased complexity in finding gcd and lcm of numbers.

There are 4 tasks written on the board. To solve these tasks, students sitting at adjacent desks unite. Each group decides to choose one of the tasks.

7. Checking the results obtained

Stage tasks: testing students’ ability to apply knowledge, skills and abilities when solving problems of increased complexity to find the LCM and GCD of numbers.

Checking the results obtained. Students mutually check their independent work, checking the board, where the solution to the independent work assignments is written, make marks and hand in the pieces of paper.

Teacher: My friends! You probably noticed the letters in front of the proposed tasks. Arrange the answers to the proposed tasks in ascending order and decipher the words of gratitude to the author of such a beautiful thought.

(Sample answer –

THANK YOU)

8. Information about homework

Stage tasks: inform students about homework, ensure understanding of content and methods of completion.

Suggested to find GCD (a, b) And NOC (a, c). Numbers A And V take it yourself arbitrarily.

9. Summing up

Stage tasks: Provide qualitative assessment of the work of the class and individual students.

Teacher: Let's summarize our lesson. I think you liked Euclid’s beautiful method of finding gcd of numbers and I have no doubt that you can handle problems of this type.

Dear friends! To summarize the lesson, I would like to hear your opinion about the lesson.

What was interesting and instructive in the lesson?
Can I be sure that you can cope with tasks of this type?
Which tasks turned out to be the most difficult?
What knowledge gaps were revealed during the lesson?
What problems did this lesson create?
How do you assess the role of a teacher? Did it help you acquire skills and knowledge?mi for solving problems of this type?

Taking into account the work throughout the lesson, students, together with the teacher, comment and evaluate the answers of their friends.

Teacher: Dear friends. Thank you very much for the pleasant communication. I thank everyone who took an active part in the work. You really helped me teach this lesson. I hope for further cooperation.

The lesson is over!

Lesson type: consolidation of the studied material.

Lesson objectives:

Develop skills in finding GCD using factorization and solving problems using GCD.

Develop the ability to independently check the correctness of a task.

Raise the level of mathematical culture.

Develop an interest in mathematics.

Develop students' logical thinking.

Teaching aids: personal computer (working in the POWER POINT environment), interactive whiteboard. (Presentation)

Lesson progress

I. Organizational moment.

Hello guys! Check if you have everything ready for the lesson: diary, textbook, notebook, pen. Drafts, for those who find it difficult to calculate in their heads.

II. Communicate the lesson topic and purpose.

What did we do in the last lesson? (We learned to find the greatest common divisor). Today we will continue working with the greatest common divisor. The topic of our lesson: “Greatest common divisor.” In this lesson we will find the greatest common divisor of several numbers and solve problems using knowledge about finding the greatest common divisor.

Open your notebooks, write down the number, class work and lesson topic: “Greatest common divisor.”

III. Oral work.

So, let's stir up your gray cells and answer the question: “Is the statement true?” You need to explain your answer. (slide 2)

A prime number has exactly two divisors. (Yes, one and this number itself)

A composite number has one divisor. (No, since a composite number must have more than 2 divisors)

The smallest two-digit prime number is 11. (Yes, 10 is a composite number)

The largest two-digit composite number is 99. (Yes, it is divisible by 1, 3, 99. And the next number is three-digit).

Some composite numbers cannot be factorized. (No, any composite number can be factorized)

The number 96 is prime. (No, it is divisible by 1, 3, 96 – 3 divisors are a composite number)

The numbers 8 and 10 are relatively prime. (No, there is a common factor of 2)

IV. Doing exercises.

Check whether the factorization into prime factors is correct. (No, 10 is a composite number, and we factor it into prime factors. 10 can be replaced by the product of prime numbers 2 and 5). (Slide 3)

Find the error. (The number 9 is composite). Tell us how to find the greatest common divisor? (Slide 4)

What's wrong? (The numbers 28 and 21 have one common divisor - 7). (Slide 5)

Find the greatest common divisor of the numbers 72, 54 and 36. While completing the task, we recite each stage. We work at the board in notebooks (Slide 6)

GCD (72, 54, 36) = 2*3*3 = 18

Are the numbers 64 and 81 coprime?

GCD (64, 81) = 1

Answer: the numbers 64 and 81 are relatively prime.

V. Problem solving.

Solve the problem. (At the board and in the notebook)

We bought 270 markers and 675 pencils for first-graders. What is the largest number of gifts that can be prepared so that they contain the same number of markers and the same number of pencils? How many markers and pencils will there be in each gift? (Slide 7)

Felt pens – 270 pcs., per? pcs. in 1 p.

Pencils – 675 pcs., per? pcs. in 1 p.

Total gifts - ? pcs.

1) 3·3·3·5=135 (p.) – will prepare

2) 270:135=2 (f.) – in 1 gift

3) 675:135=5 (k.) – in 1 gift

Answer: 135 gifts, 2 markers, 5 pencils.

VI. Physical exercise.

Sit equally. Place your hands behind your back. Without turning your head, look at the window, at the stand on the opposite side, up, at the desk, at the board. Close your eyes, imagine a blue sky. Open your eyes. Place your hands on the table. Let's continue...

Next task.

At the depot, 2 trains were formed from identical cars. The first is for 456 passengers, the second is for 494 passengers. How many cars are there in each train, if it is known that the total number of cars does not exceed 30? (Slide 8)

1 train – 456 pax., ? vag.

2nd train – 494 pax., ? vag.

Total number of cars< 30 шт.

1) 19·2=38 (m.) – in each car

2) 456:38=12 (c.) – in 1 composition

3) 494:38=13 (v.) – in 2 compositions

Check: 12+13=25 (v.)

Answer: 12 cars, 13 cars.

VII. Independent work.

When completing tasks in independent work, do not forget about the signs of divisibility and other rules. I wish you good luck! (Slide 9)

Hand in your notebooks. Now we will check whether you completed the tasks correctly. (Analysis of mistakes made.) (Slide 10)

VIII. Homework

Let's write down our homework and then summarize the lesson. So, open your diaries and write down your homework:

clause 6 p. 21, No. 161, 182, 192 (oral). (Slide 11)

IX. Summing up.

What was our goal today? (Learn to solve problems by finding gcd).

What numbers are called coprime?

How to find GCD?

Who should be recognized for good work? (Grading for work in class)

Type of work -practicing drawing techniques and displaying object images.

Target: PC 2.5 organize the productive activities of preschoolers (drawing, modeling, applique, design; PC 2.7 analyze the process and results of organizing various types of activities and communication of children; OK 2 organize their own activities, determine methods for solving professional problems, evaluate their effectiveness and quality; OK 5 use information and communication technologies for improving professional activities.

The task takes 3 hours to complete.

Assignment: Using an Internet resource (see the “Catalog of Internet Resources” for the methodological manual), get acquainted with the technique of drawing various images. Practice the technique of showing 3-4 images of birds and animals.

In the process of practicing the display technique, it is necessary to use a vertically located sheet of A3 paper, gouache paint, and a brush. Draw 3-4 images in the manual using gouache, colored pencils and felt-tip pens.

Prepare to demonstrate the technique of showing birds and animals during a practical lesson outside the GCD (you can use a faintly drawn outline with a simple pencil).

Reporting form: drawn images and readiness for practical demonstration (samples for the “Pedagogical Piggy Bank”).

Evaluation criteria:

· Quality of the resulting image (recognizability of the image, compositional correspondence to the sheet and paper);

· Verbal accompaniment;

· The process and result of the display should be clearly visible to children.

Possible tasks that allow you to study the features of pedagogical conditions for the artistic and aesthetic development of preschool children that exist in the practice of preschool educational institutions

Type of work:

Parent survey: in order to identify their ideas on the problem of artistic and aesthetic development by preschoolers.

Conclusion:
Questionnaire for parents

Dear parents _________________________________(child's name)

Please answer the questions provided in the questionnaire.

Your sincere answers will help to study the problem in more depth and outline ways to improve the pedagogical process in kindergarten.

1. At what age do you think the targeted artistic and aesthetic development of a child is necessary?________________________________________________

2. From your point of view, the artistic and aesthetic development and education of children should, to a greater extent, be aimed at (choose the statement that corresponds to your opinion):

Development of skills to feel beauty, respond to beauty

Formation of some art historical knowledge

Development of interest in art,

Developing interest in creative leisure, crafts (embroidery, weaving, designing)

Mastery of productive activities (sculpting, drawing, designing)

Self-expression, manifestation of emotions, feelings

Creative experience

Experience in working with different materials (sand, clay, sanguine, coal, etc.), experimenting with them;

Development of certain qualities (independence, organization, ability to plan activities)

Another option______________________________________________________________

3. What types of children's productive activities are most interesting to your child (mark with the + symbol)? Do you consider it mandatory to attend preschool (mark with v)?

Drawing

Application

Artistic work (embroidery, weaving, etc.)

Construction and design

Comments_______________________________________________________________

4. Which direction of design activity is more preferable for you (in the development of decorative activities in your child and are you ready to participate with him)?

Painting toys in the style of folk crafts

- “designing” puppet and carnival clothes

Making postcards, bookmarks, etc.

Decorating objects (boxes, vases, disposable glasses, etc.) and making simple objects (key chains)

Making a patchwork doll, etc.

making New Year's toys, Christmas tree models, costumes

production of city models, insolations, unusual souvenirs

Layout of visiting decorations for the holidays (garlands, etc.)

Your option_______________________________________________

5. Does your child often draw, sculpt, or design?____

6. Does your child often pay attention to “beauty” in the world around him (natural objects, beautiful little things in everyday life, etc.)______ ________________________________________

7. Does the child use interesting words (figurative comparisons, exaggerations, comparative forms) when he sees something beautiful or ugly (Name typical or favorite ones)______________________________________________________________

8. How does a child typically behave when he notices something beautiful __________________________________________________________

9. How does your child’s desire for beauty manifest itself?_________________________________________________________________

10. Does your child ask questions about art? asks for clarification of some words (for example - what is beauty? Landscape? Sculpture? Designer?)__________________________________________

11. Does your child ask to buy new pencils, paints, plasticine, books with interesting illustrations?________________________________________________________________

12. When your child brings work (drawings, applications) from kindergarten, who does he want to show it to, how does he show his “pride” or his unwillingness to show it ___________________

13. Are you involved in any artistic activity, craft, or “artistic leisure”?___________________________

14. Do you have a collection of children's works at home? Comments (who started collecting, what is presented, how the works “get” into the collection?)?__________________________________________

15. If a child gets carried away and begins to dirty a sheet of paper or “play around” with paints, your typical reaction is _____________________________________________

16. Please name the difficulties that arise in the process of drawing (sculpting, appliqué or design) for your child?_____________________________________________

17. Are you ready to take part in any events organized in kindergarten in the direction of the artistic and aesthetic development of preschoolers (making costumes together with children, drawings, creative competitions)? Which ones? _________________________ Comments_______________

18. Formulate your wishes to teachers, preschool educational institutions in terms of organization, conduct, and content of work on the artistic and aesthetic development of children _________________________

APPLICATION

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Related information.

Independent work on the topic “Greatest common divisor”

Find all the common factors of the numbers and underline their greatest common factor:

a) 50 and 70; b) 34 and 51; c) 8 and 27. Name a pair of relatively prime numbers, if such a pair exists.

2. Write down two numbers for which the greatest common divisor is the number: a) 7; b) 24.

3. Find the gcd of the numbers: a) 55 and 88; b) 72 and 96; c) 720 and 90; d) 255 and 350; e) 675 and 825.

Option 2

1. Find all common divisors of numbers and underline their greatest common divisor:

a) 30 and 40; b) 39 and 65; c)25 and 9;. Name a pair of relatively prime numbers, if such a pair exists.

2. Write down two numbers for which the greatest common divisor is the number: a) 9; b) 21.

3. Find the gcd of the numbers: a) 44 and 99; b) 630 and 70; c) 64 and 80; d) 242 and 999; e) 7920 and 594.