Question 1: Mathematical analogy of analogy, like mathematical discovery, is usually based on exploration methods such as analogy and induction, and then tries to prove or deny the conjecture, thus achieving the purpose of solving the problem. Analogy and induction are two important ways to get a guess. The basic process of solving problems by analogy can be represented as follows: the key to using analogy is to find the appropriate analogy object. According to the different angles of looking for analogy objects, analogy methods are often divided into the following three types. If the object in three-dimensional space is reduced to the object in two-dimensional (or one-dimensional) space, this analogy method is called dimension reduction analogy. In Example 2, all sides of a regular tetrahedron with a side length of 1 are regarded as spheres, and S is the intersection of six spheres. It is proved that the distance between any pair of points in S is greater than 65433. Analyze and consider the analogy proposition on the plane: "A regular triangle with a side length of 1 is a circle with each side as its diameter, and S' is the intersection of three circles". By exploring the similar nature of s', we can seek the argumentation idea of this question. As shown in the figure, it is easy to know that s' is contained in a circle with the center of gravity of a regular triangle as the center and the radius as the center. Therefore, the distance between any two points in s' does not exceed 65438. Prove: As shown in the figure, in the regular tetrahedron ABCD, m and n are the midpoint of BC and AD respectively, g is the center of △BCD, and Mn ∩ Ag = O. Obviously, O is the center of the regular tetrahedron ABCD. Yizhi OG=? AG=, and it can be deduced that the distance between any two points in a ball with O as the center and OG as the radius is not greater than, and its ball O must contain S, which is proved as follows. According to the symmetry, we might as well look at the tetrahedron OMCG in the spatial region. Let p be any point in tetrahedral OMCG, and it is not in sphere O, which proves that p is not in S either. If the ball o passes through OC at t point. △TON，ON=，OT=，cos∠TON=cos(π-∠TOM)=-。 By cosine theorem: TN2=ON2+OT2+2ON? OT？ =,∴TN=。 In Rt△AGD, n is the midpoint of AD, ∴GN=. From GN= NT=, OG=OT, ON=ON, we get △ gon △ ton. ∴∠TON=∠GON, they are all obtuse angles. So obviously, any point P in △GOC that does not belong to the ball O has ∠ PON >; ; ∠TON, that is, there is PN & gtTN=, and point P does not belong to the area S outside the sphere with N as the center and AD as the diameter. In this way, sphere O contains the intersection S of six spheres, that is, there are no two points in S, so that its distance is greater than. There is no ready-made analogy for some problems to be solved, but we can find analogy problems by observing and relying on structural similarity, and then transform the original problems into analogy problems through appropriate replacement. Example 3 is given. The analysis shows that if any two of the seven real numbers are equal, the conclusion is obviously valid. If the seven real numbers are not equal, it is difficult to start. However, after careful observation, we can find that the tangent formula of the difference between the two angles is very similar in structure, so we choose the latter as an analogy and turn it into an analogy problem through proper substitution. For substitution, xk = tanαk(k = 1, 2, ..., 7), which proves that it must exist. Prove xk=tanαk(k =l, 2, …, 7) and αk∈(-,), then the original proposition is transformed into: prove that there are two real numbers αi, αj∈(-,), and satisfy 0≤tan(αi-αj)≤? According to pigeonhole principle, αk must have four in [0,] or in (-0), so it is better to set four in [0,]. Note that tan0=0, tan=, and in [0,], tanx is increasing function, so we only need to prove the existence of αi and αj, so that 0; αj, then 0≤αi-αj ≤, so 0 ≤tan (α i-α j) ≤In this way, with the corresponding xi=tanαi and xj=tanαj, there will be 0 ≤to simplify analogy, that is, to compare the original proposition to an analogy proposition that is simpler than the original proposition, and to seek the solution ideas and methods of the original proposition through the inspiration of the solution ideas and methods of the analogy proposition. For example, a multivariate problem can be first analogized as a problem of several elements, and a higher-order problem can be analogized as ... > >

Question 2: Senior high school mathematics, analogy, write a detailed solution process k (k+2) =1/6 [k (k+2) (k+4)-(k-2) k (k+2)].

Therefore:

1x 3 = 1/6[ 1x3x 5-(- 1)x 1x 3]

2x4= 1/6(2x4x6-0x2x4)

3x 5 = 1/6(3x5x 7- 1x3x 5)

4x6= 1/6(4x6x8-2x4x6)

.......

n(n+2)= 1/6[n(n+2)(n+4)-(n-2)n(n+2)]

Total:

1x3+2x4+.。 . +n(n+2)

= 1/6[-(- 1)x 1x 3-0x2x 4+(n- 1)(n+ 1)(n+3)+n(n+2)(n+4)]

= 1/6[n(n+ 1)(2n+7)]

Question 3: How to carry out effective mathematics teaching activities through analogy in senior high school mathematics teaching is an important goal of mathematics classroom teaching reform, and it is also a key link in building a quality-oriented education mathematics classroom teaching model. Improving the effectiveness of mathematics teaching activities is an important content of mathematics classroom teaching reform. To this end, we must actively change the concept of education through teaching reflection and truly establish a new curriculum.

Question 4: What is the use of induction, explanation and analogy in primary school life? There are many rules and formulas in primary school mathematics textbooks. According to the cognitive law from special to general, through the observation, analysis and experiment of special circumstances, the general conclusion is summarized, that is, induction.

In the process of mathematical knowledge extension, analogy is often inspired and induced by comparison and association in order to seek the variation and divergence of thinking. When summarizing the knowledge system, it can also be used to connect similar content at different levels to help understand and remember. When solving problems, both the proposition itself and the method of solving problems are the driving force to generate speculation and obtain the promotion or extension of the proposition. Therefore, induction and analogy are not only important methods of mathematics learning, but also effective methods of mathematics discovery.

Induction and analogy belong to reasonable reasoning, and their conclusions need to be proved by deduction. Conjecture is the result of induction and analogy, both of which contain elements of conjecture, so conjecture itself is a reasonable reasoning. To put it bluntly, reasonable reasoning is conjecture. Newton said, "Without bold guesses, great discoveries are impossible." Therefore, designing a reasonable teaching process full of guesses can not only organize teaching well, but also improve students' interest in learning and cultivate their innovative ability.

First, induction.

Induction is a method to draw general conclusions by studying the special objects of the same kind of things, that is, the reasoning method from special to general.

1. Induction has the function of discovering and exploring truth.

Many famous theorems in mathematics are first discovered by incomplete induction, and then proved.

For example, the famous German mathematician Goldbach observed from the formulas 3+7= 10, 3+ 17=20, 13+ 17=30 that the sum of two odd prime numbers is equal to an even number. He made further experiments and found that

6=3+3,

8=3+5,

10=3+7=5+5,

12=5+7,

14=3+ 1 1=7+7,

16=3+ 13=5+ 1 1,

Therefore, he came to the conclusion that any even number that is neither prime nor prime square (that is, even number greater than 4) is the sum of two odd prime numbers. This is the famous Goldbach conjecture. Although it is still a conjecture, mathematicians found and invented many mathematical theorems in the process of proving this conjecture, which made great contributions to the development of mathematics and even the development of society.

2. Induction is of great significance in primary school mathematics education.

Almost all formulas, rules and properties in primary school mathematics are understood by incomplete induction. Therefore, teachers should seriously study the mathematics curriculum standards, thoroughly understand the teaching materials, give students the opportunity to divergent thinking, give them more guidance, more inspiration, more encouragement, give them enough time and space, and let them gradually master induction in class. For example, when teaching "average score", the teacher can give the students the problem of how many apples to give to several students, let them solve it themselves, and provide students with time and space to use their imagination, and then sum up the fairest score-how much each person has, so as to get the concept of average score. This not only cultivates students' divergent thinking, but also enables students to understand and master the concept of "average score" more deeply in this activity. When explaining concepts, laws, properties, formulas and examples, teachers should let students associate and popularize them from different sides and angles. For another example, when teaching rectangles, students can give full play to their imagination and draw rectangles with different shapes and different placement positions. Then, guide them to conclude that these figures have the same characteristics: (1) are quadrangles; (2) All four corners are right angles; (3) The opposing sides are equal. This not only cultivates students' divergent thinking ability, but also enables students to have a deeper understanding of rectangles. When teaching squares, students will not make the mistake that squares are not rectangles.

As "reasonable reasoning", incomplete induction is easy for pupils to accept and master. Therefore, incomplete induction is everywhere in primary school mathematics teaching. Students' study of definition, operation nature (law), divisibility of numbers, etc. , are understood and mastered through incomplete induction. This unique atmosphere has brought great convenience to cultivate the inductive ability of primary school students. Therefore, incomplete induction is considered to be an effective and important method to cultivate primary school students' creative thinking ability in mathematics teaching. Teachers should seize this advantage and help primary school students master incomplete induction. Let students give full play to their imagination, let them ask questions, make bold guesses, break through the general mindset and dare to guess. At the same time, it should ... >>

Question 5: Analogical thinking of mathematical thinking compares two (or two) different mathematical objects. If they are found to be similar or similar in some aspects, it is inferred that they may be similar or similar in other aspects.