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.
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.
Back forward
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Routing lesson
| Lesson type | Combined | ||
| The purpose of the lesson | Repeat and consolidate the signs of divisibility; prime and composite numbers, develop the ability to find GCD and LCM and apply the algorithm for finding GCD and LCM to solve problems. | ||
| Lesson Objectives | educational | developing | educational |
| Update knowledge on topics: decomposition of numbers into prime factors; prime and composite numbers, GCD and LCM. Repetition and consolidation of acquired knowledge. Ability to apply mathematical knowledge to problem solving. |
Expanding students' horizons. Development of techniques mental activity, memory, attention, ability to compare, analyze, draw conclusions. Development cognitive activity, positive motivation for the subject. Development of the need for self-education. |
Upbringing personality culture, attitudes towards mathematics as part of universal human culture, playing a special role in social development. Developing responsibility, independence, and the ability to work in a team |
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| Cognitive UUD: | They develop skills of cognitive reflection as awareness of actions and thought processes, and master problem solving skills. training in the ability to independently identify and formulate a cognitive goal, search and highlight the necessary information using independent work and teacher questions. | ||
| Improve the ability to consciously and voluntarily construct a statement in oral and written form, analyze objects in order to highlight essential features for drawing up an algorithm, learning the ability to put forward a hypothesis; | Communicative UUD: | ||
| Develop the ability to participate in discussions; express your point of view clearly, accurately and logically; Regulatory UUD: |
Personal UUD: They learn to independently evaluate and make decisions that determine the strategy of behavior, taking into account civic and moral values. creating a situation for staging learning task based on knowledge about divisors and multiples of natural numbers; predicting the result of the level of mastery based on the concepts of divisors and multiples, GCD and LCM. Teaching control skills in the form of comparing the results of independent work with solving tasks on the board in order to detect deviations and differences from the sample, assessing what has already been learned and what still needs to be learned on the topic; |
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Learn the ability to conduct dialogue based on equal relationships and mutual respect
During the classes
Stage 1. Organizing time.
Stage 2. Updating knowledge and recording difficulties in activities.
Homework check (problem and equation)
Oral work (children rate their knowledge at the beginning of the lesson)
- Questions:
- What numbers are called natural numbers?
- Definition of prime and composite numbers (give examples)
- And 1 – what number is it? (neither simple nor compound) Why?
Signs of divisibility by 2, 3, 5, 9, 10 Which greatest number
Can identical gifts be made from 48 “Belochka” candies and 36 “Inspiration” chocolates, if you need to use all the candies and chocolates? GCD (36,48)=? Formulation of the problem:
Today we will summarize all the knowledge we have acquired on this topic.
Open your notebooks, write down the number, cool work, topic: “GCD and LCM of numbers.”
Stage 3.
What numbers are called coprime? (GCD = 1)
Find GCD and LCM of numbers 6 and 15
- GCD(6; 15) = 3, GCD(6; 15) = 30
- What is the product of GCD and LCM of these numbers? 3 * 30 = 90
- What is the product of numbers a and b? 6 * 15 = 90
What conclusion can we draw: gcd(a; b)·gcd(a; b) = a * b .
Problem solving.
Where do we already use our knowledge of GCD and LCM of numbers?
Students have handouts with tasks on the table.
Doing the exercise.
Exercise: Select true statements: (on screen)
GCD(13, 39) = 39
16 – multiple of 3
LCM(9.18) = 18
5 is a multiple of 6
7 – divisor of 14
GCD (2; 15) = 1
Every number has a divisor of 1
LCM(2;3) = 6
From the proposed correct answers, make the greatest natural number, a multiple of 5.
Answer: correct 3,5,6,7,8. The largest natural number divisible by 5 is 87635.
Physical education minute
If I believe, they stretch upward; if I don’t believe, they squat.
- Number 2 is a divisor of number 16.
- The number 33 is a multiple of 5.
- The number 10 is a divisor of 40.
- 60 is a multiple of 10 and 7
- 7 has two divisors.
Stage 4.
Children have cards with finding GCD and GCD (perform according to the options, then listen to them at the board)
| Task No. 1 The guys received identical gifts at the New Year tree. All the gifts together contained 123 oranges and 82 apples. How many children were present at the Christmas tree? How many oranges and how many apples did each person get? (you need to find the gcd of numbers 123 and 82 123 = 3 * 41; 82= 2 41 gcd(123, 82) = 41 |
Answer: 41 guys, 3 oranges and 2 apples.) Task No. 2 Two ships left the river port at the same time. 1) 15 = 3 *5; 24 = 2 * 2 * 2 * 3 The duration of the flight of one of them is 15 days, and the second – 24 days. In how many days will the ships depart at the same time again? How many voyages will the first ship make during this time? How much is the second one? You need to find the LCM of the numbers 15 and 24. LCM(15; 24) = 2 * 2 * 2 * 3 * 5=120 2) 120: 15 = 8 (p) first; |
3) 120: 24=5(r) second
Answer: after 120 days, the first one will make 8 flights, and the second one will make 5 flights.
Working with cards:
What is the largest number of identical gifts that can be made from 32 markers, 24 pens and 20 markers? How many markers, pens and markers will be in each set?
Buses leave from the final stop on two routes. The first returns every 30 minutes, the second every 40 minutes. In what shortest time will they reach the final stop again?
Task No. 3. (work in pairs)
| Decipher the name of one of the species of African antelope. (Springbok) 12 To do this, find the least common multiple of each pair of numbers, then write the letter corresponding to that number in the table. | 1) LCM(3,12) = 45 R |
| 5) LCM(9;15) = ___40 b | 2) LCM(4;5;8)= 60 O |
| 6) LCM(12;10)= 24 To | 3) LCM(8;12)= 18 With |
| 7) LCM(9;6) = 48 And | 4) LCM(16;12)= 20 n |
8) LCM(10;20)=
G 100 Fill in the empty column in the table, taking into account the data:
| 24 | 12 | 18 | 48 | 20 | 45 | 40 | 60 | |
| To | LOC(25,4) = | To do this, find the least common multiple of each pair of numbers, then write the letter corresponding to that number in the table. | With | And | n | R | b | O |
P
P
Now let's test your knowledge with independent work. Take a card on the table and make all the notes on it.
Find GCD and LCM of numbers in the most convenient way.
| Option 1 | Option 2 |
| a) 12 and 18; | a) 10 and 15; |
| b) 13 and 39; | b) 19 and 57; |
| c) 11 and 15; | c) 7 and 12. |
Are numbers coprime?
| 8 and 25 | 4 and 27 | |||||
| IN 1 | AT 2 | |||||
| A | R | V | A | R | V | |
| GCD | 6 | 13 | 1 | 5 | 19 | 1 |
| NOC | 36 | 39 | 165 | 30 | 57 | 84 |
| Yes | Yes | |||||
Stage 5. Summing up the lesson.
Today we have reviewed almost all the rules on the topic “Greatest common divisor and least common multiple” and are ready to write a test. I hope you handle it well.
The following grades were received for the lesson:
Stage 6. Homework information
Open your diaries and write down homework. Repeat the rules from paragraphs 2.3, perform No. 672 (1.2); 673 (1-3), 674..
Stage 7. Reflection.
Determine whether one of the following statements is true for yourself:
- “I figured out how to find the gcd of numbers”
- “I know how to find the gcd of numbers, but I still make mistakes.”
- “I still have unresolved questions”
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 logical thinking students.
Teaching aids: personal computer (working in the POWER POINT environment), interactive whiteboard. (Presentation)
During the classes
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 topic of the lesson: “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 same number 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? PC. in 1 p.
Pencils – 675 pcs., per? PC. in 1 p.
Total gifts - ? PC.
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 total number does not exceed 30 carriages? (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. 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. Summarizing.
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)