Textbook description

After more than four years of mathematics study, students have mastered a lot of integer knowledge (including integer knowledge and four integer operations). This unit allows students to further explore the properties of integers on the basis of previous integer knowledge. The factors, multiples, prime numbers and composite numbers involved in this unit, as well as the greatest common factor and least common multiple of Unit 4, all belong to the basic content of elementary number theory. Number theory is a branch of mathematics with a long history. It is a study of the properties of integers and is famous for its preciseness, conciseness and abstraction. Mathematics has always been regarded as the "queen of science", and number theory is also known as the "queen of mathematics", which shows the position of number theory in mathematics. The knowledge of this unit, as the preparatory knowledge of number theory, has always been an important content in primary school mathematics textbooks. Through this part of the study, students can gain some knowledge about integers, on the other hand, it is also helpful to develop their abstract thinking.

In number theory, the divisibility of numbers is the most basic theory, and all concepts in this unit are based on the divisibility of numbers. For any integer a and b, there are integers n and r, so B = Na+R (where R < A). When R = 0, we say that B is divisible by A (or A is divisible by B). At this time, B = Na. Other concepts, such as factors and multiples, are based on this.

In previous mathematics textbooks, the concept of "divisibility of numbers" has been arranged at the beginning of this unit, followed by factors (called divisors in previous textbooks), features of multiples, multiples of 2, 5 and 3 (called features of numbers divisible by 2, 5 and 3 in previous textbooks), prime numbers, composite numbers, factorization factors and the greatest common factor (called divisors in previous textbooks) Students often have problems of confusing concepts and difficult understanding when studying. Therefore, compared with the previous textbooks, this experimental textbook has been adjusted in the following aspects.

1. In this unit, we study the phenomenon of divisibility, so it can be said that the concept of divisibility is a main thread running through this part of the textbook. But does the word "divisible" have to appear? Is it necessary for students to describe "X is divisible by X" and "X is divisible by X"? Because students have a lot of knowledge base to distinguish divisibility from division with remainder, and have a clear understanding of the meaning of divisibility, the lack of the definition of divisibility will not have any impact on students' understanding of other concepts. Therefore, the mathematical definition of "divisibility" is deleted from this set of textbooks, and the concepts of factors and multiples are directly derived by means of the divisibility model Na = B.

2. In previous textbooks, the method of finding the greatest common factor and the least common multiple is unique and fixed, that is, the method of decomposing prime factors by short division. Therefore, as the necessary basis for finding the greatest common factor and the least common multiple, "prime factor decomposition" has always been arranged as a compulsory content. In this textbook, because students are asked to find the greatest common factor and the smallest common multiple in various ways, the decomposition of prime factors has also lost its indispensable role. At the same time, in order to reduce the theoretical concepts of this unit, the textbook no longer regards it as formal teaching content, but as supplementary knowledge, arranged in "Do you know?" Introduced to.

3. The concepts of common factor, greatest common factor, common multiple and minimum common multiple are established on the basis of the concepts of factor and multiple, and also prepare for later learning divisor (common factor of numerator and denominator needs to be found as soon as possible) and total score (common multiple of denominator of two fractions needs to be found as soon as possible), which plays a connecting role in the whole knowledge chain. These two items can be arranged in this unit, or before the deduction and total score. Considering that this unit has many concepts and a high degree of abstraction, this textbook arranges these two parts in the fourth unit, which also highlights their applications.

Teaching suggestion

1. Because this part of the content is abstract, it is difficult to combine life examples or specific situations to teach, and it is difficult for students to understand. In previous teaching, some teachers often ignored the essence of concepts, but asked students to memorize related concepts or conclusions. Students can't sort out the relationship between concepts, and they can't master it. In addition, some teachers use some biased questions and difficult problems in the assessment, which leads students to feel bored when learning this part of knowledge, unable to understand the abstraction, rigor and logic of elementary number theory and unable to feel the charm of mathematics. In order to overcome the above problems in teaching, we should pay attention to the following two points.

(1) Strengthen the relationship between concepts, guide students to understand concepts in essence, and avoid rote learning. Factor and multiple are the two most basic concepts in this unit. If you understand the meaning of factors and multiples, you will naturally grasp the conclusion that the number of factors of a number is limited and the number of multiples is infinite, and the understanding of subsequent concepts such as common factors and common multiples will be logical. Students should be guided to master this knowledge from the perspective of contact, instead of mechanically memorizing a bunch of fragmented and irrelevant concepts and conclusions.

(2) Due to our unique knowledge.