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Common factor and greatest common factor of six aspects of cognition
Someone called me a nerd when I was studying! I didn't think so then. I know what I am doing, so I won't stay. Now think about it, many books I read before are fools' books, and many previous learning methods are fools' methods. In order to prevent students from following in my footsteps, I have to reflect on myself and my current education.

Why do some schoolmasters feel that they have forgotten everything they have learned after graduation, and there is nothing to remember and know?

Knowing is understanding, preliminary understanding, shallow and low-order, and it can be said that there may be no thinking. What the child said is understood, not necessarily really understood. Words such as nerd and pedant show that simple knowledge may be a wrong understanding. Because most knowledge is rote learning, scores and grades are obtained in a short time (the evaluation criteria also point to some simple knowledge and rote learning), but they will not be applied, analyzed and evaluated. (Bloom's classification of teaching objectives: from low to high, the educational objectives in the cognitive field can be divided into six levels: cognition (knowledge)-understanding (understanding)-application-analysis-synthesis-evaluation. )

If you can teach, use, prove, associate, explain, analyze what you have learned and understand the meaning, you will understand.

Take the common factor and the greatest common factor in the fifth grade of Qingdao Edition as an example. In the past, I would look for the common factor and the greatest common divisor, and I would look for the common factor and the greatest common divisor to solve 80% of the doubts.

After learning the common multiple and the least common multiple of this unit, after comprehensive practice, 80% can find the common factor and the greatest common factor correctly, and 60% can do the bouquet problem.

At the end of the period, or at the beginning, the direct common factor and the greatest common factor are not examined. Originally, children learned 80 in the window and 70 in the unit. Now it is difficult to learn 50! Have you forgotten the original knowledge? There must be something to forget. What is the main reason? It turned out to be mechanical imitation, simple repetition, and no understanding. Finally, the problem of bouquets was comprehensively investigated, and even one more step was taken. I can only sigh at the sea.

There are six aspects to understanding: explanation, clarification, application, epiphany, ecstasy and self-knowledge.

What is understanding? Take Common Factor and Maximum Common Factor, the second volume of the fifth grade of Qingdao Edition, as an example, let us enter the voyage of understanding.

1 will explain.

"Interpretation" is a Chinese word, pronounced jiě, which means to think on the basis of observation and reasonably explain the reasons for the changes of things, the relationship between things or the laws of their development. From the biography of Chen Yuan in the later Han Dynasty.

Can you give an example to explain what is the common factor and how to find it? How to find the greatest common factor? Why did you cut this piece of paper with a length of 24 cm and a width of 18 cm into a square with a whole centimeter? After cutting, there is nothing extra. The side length of a square is several centimeters. Why do we need to solve it by common factor?

Why is the greatest common factor obtained by short division?

What's the use of learning the common factor and the greatest common factor?

Students should use evidence to prove their conclusions. Mathematics makes sense!

This is to prove your conclusion with evidence and apply it to the situation. 555SS

2 will clarify

Explain the phenomenon, clarify the meaning, and imply something.

In the model 1, there are 24 squares and 18 squares, of which 24 squares can be arranged in rows 1, 2, 3, 4, 6, 8, 12 and 24, while 18 squares can be arranged in rows/kloc. 24 and 18 can be arranged in rows 1, 2, 3 and 6. 1, 2, 3, 6 are the common factors of 24 and 18, which can be arranged in six rows at most, and 6 is the greatest common factor of these two numbers.

This model can clarify the meaning of common factor and greatest common factor.

3 can be applied

Learning situation: the previous understanding factors are not diverse enough. In order to enter this course quickly, it is suggested to solve the problems before class:

A bunch of lilies (12) should be evenly inserted in the vase, with nothing left. How many insertion methods are there? A bunch of roses, 16, should be evenly inserted in the vase, and there should be no excess. How many insertion methods are there?

1. There are 96 apples in the basket. If you don't take it out one by one, you don't take it one by one. You need to take the same number every time. When you finish it, it's nothing more or less. How many ways can you hold them?

2.( 1996 Japanese arithmetic olympiad) There are 50 cards with the numbers 1- 50 written on both sides. One side of the card is red and the other side is blue. There are 50 students in a class. The teacher put the blue side of 50 cards on the table. He said to the students, "Please follow the order of your student numbers.

1. There are four children, each of whom is older than the other 1 year. The product of their ages is 360. How old is the oldest child?

Teacher Wang of a class leads the whole class to plant trees. The students were evenly divided into three groups. If teachers and classmates each plant as many trees. * * * 572 trees have been planted, so how many students are there in this class and how many trees have each been planted?

Discover the application of mathematics from life or the corresponding abstract mathematical knowledge from application, and discover the profound meaning in the topic of common factor and greatest common factor.

4. knowledgeable.

What is the essence behind short division, shift division and polyphase subtraction?

The method of enumerating and screening to find the greatest common factor is the application of enumerating one by one, which is widely used in primary schools. The advantage is that there is no repetition or omission.

The essence of decomposing prime factor and short division is the same, which is to decompose prime factor first and then extract common factor. Because you can find short division at the same time, which is more convenient.

One of phase subtraction and phase division uses division, and the other uses subtraction.

From the algorithmic point of view, there is no essential difference between the two, but in the calculation process, if one number is large and the other number is small, it may take several subtractions to achieve the effect of division. In contrast, the time complexity of partition is stable.

Essence is the common factor of two numbers, and it is also the common factor of the sum and difference of two numbers. These two numbers are multiples of m, and the difference between these two numbers is also a multiple of m.

After removing the same multiple, the rest is also a multiple of this number.

If children can find the principle of short division, why can they find the greatest common factor like this, and why should they multiply by the factor on the left? Why don't you quote until there are only two factors left: 1 and yourself?

5 will be fascinated.

Teachers think from students' perspective, and students think from each other's perspective.

6 will know.

What are the advantages or disadvantages of your own method? You have self-knowledge.