Yang Hui Triangle (1)
Destination requirements
1. Learn about the brief history of Yang Hui triangle, master the basic nature of Yang Hui triangle.
2. Cultivating students by specializing in the general point of guessing.
3. Cultivate students discover problems through the group discussion. Inquiry knowledge, constructing knowledge research habits and team spirit of cooperative learning.
content analysis
The main content of this lesson is to summarize the three basic nature of Yang Hui triangle and study to discover some of the laws of Yang Hui triangulation.
The three basic nature of Yang Hui triangle is mainly the nature of the two expansion of the two-way coefficients, which is the basis for studying the other laws of Yang Hui triangles. Yang Hui triangle ranking digital law mainly includes a size relationship between ranging numbers. Combined relationships and links between different horizontal numbers.
Research topics are mainly for in-depth discussions of certain mathematical problems, or from mathematical perspectives to research in some daily lives and other disciplines. The purpose is to cultivate students' innovative spirit and creativity. It requires teachers to provide students with research and background, so that students have developed their knowledge. From the proposal of the problem, the process of exploration and the establishment of the conjecture is mainly completed by students, and teachers can not replace, but as organizers, they can provide necessary guidance.
Teacher first introduced the history of Yang Hui triangle, stimulating students' national pride and creating desires, and then guiding students to summarize the basic knowledge of Yang Hui triangles (the basis of research) and introduce the main methods of discovering digital laws (research strategy), and ratio The number of general items and questions, let students conduct preliminary research attempts to study the N-order Yanghui triangle, so that students fully launched their thinking into the research status.
The following main sub-group cooperation studies the horizontal digital law of Yang Hui triangle, focusing on the law, does not have to prove in the classroom.
Teaching process
(1) Review the old knowledge
1. Show the Jiaxian Triangular map, the ancient seven passenger map of Zhu Lijie, Pasca Triangle (attached), while playing music with ancient national musical instruments.
Teacher introduces the brief history of Yang Hui triangle: The Northern Song Dynasty, Jia Xindian, first using the "Jia Xian Triangle" to conduct a high-end operation, Southern Mathematician Yang Hui recorded and saved in "Detailed Nine Algorithm" (1961) The constitutional triangle ", tapped Yang Hui triangle. Yuan Dynasty mathematician Zhu Shijie expanded the "Jiaxian Triangle" into a "Ancient French Seven". Until the European until 1623, French mathematician Pasca found the "Pasca Triangle" at the age of 13.
2. Show the 15th-order Yanghui triangle or printed 15th-order Yanghui triangles to students. Control Yang Hui triangle, review the construction and basic nature of Yanghui triangle studied in the high school semester, and is described by students.
The relationship between 1 ° and binary theorem: the Nth line of Yang Hui triangle is two items
Expanded coefficient column
.
2 ° symmetry: the number of the number in the Yanghui triangle, right symmetry, the symmetry shaft is "high" on the side of the Yanghui triangle, ie
.
3 ° Structural Features: Yang Hui triangulation except for the number outside the oblique edge, it is equal to the sum of the two numbers of "shoulders", ie
.
(2) Grouping research Yang Hui triangulation ranking rules (divided into a group of research groups according to the four or five people in front and rear)
1. Introduction Mathematics Discovery: The Yang Hui triangle has a lot of beautiful laws. Ancient and modern China and foreign countries, many mathematicians such as Jia Xian, Yang Hui, Zhu Shijie, Pasca, Hua Lugeng, etc. have been studied in depth, and the results of the research have been applied to other work. The methods they study can be summarized as:
15th-order Yanghui triangle
2. Students try to explore activities.
(1) How many numbers in the n-order Yanghui triangle?
(2) What is the general formula of the n-order Yanghui triangle? That is, what is the number of rows in the n-order Yanghui triangle?
(3) What is the sum of the number K row of the n-order Yanghui triangle? What is the sum of the number? After the students think out independently, they speak from the students to draw conclusions. N-order Yanghui triangle
Number
The third number of n 2 lines; the general term formula is
,
,
.
3. According to research, research work in the direction of ramping digital laws, the focus of work is to discover the law. Teachers inspecting guidance, if necessary, participate in a group discussion activity. Finally, the team represents the results of the study and the establishment of the conjecture.
(1) The kth of the 2k line of Yang Hui triangle is the largest;
The second k 1 line is the number of k numbers and the number of K L, that is,
The maximum number in the 2k-order Yanghui triangle is
The maximum number in the 2K 1 order Yanghui triangle is
.
(2) Yang Hui triangle
All the number of rows are odd (k∈n *), ie
Odd (m = 0, 1, ...,
); The first
All the number of rows (1 except for both ends) is even (k∈n *), ie
Even (r = 1, 2, ...,
); In all of the other rows, there must be both even numbers and odd numbers other than 1.
(3) The number of numbers in the number of P (P) is removed from the number 1 of the two ends can be divided by P, and its antiquity is also established. That is, there is any R ∈ {1, 2, ..., n-1}.
It is the number of prime.
(4) Multi-digits obtained together from all numbers from left to right is equal to
.
(5) The n number of the second N row is twice that of the n-1 number of the second N-1 line, ie.
.
......
(3) Small knot
(1) Please summarize the experience in the research process: How to select research clues, what methods for use have found conclusions, what difficulties encounter, how to break through innovation.
(2) Teachers' specifications on the expression of the nature of Yang Hui triangles, and inquiry thinking.
Homework
As shown in the figure, the number of circles in each small figure, the points, line segments, triangles, quadrangular, five-sided, and six-sided numbers, study Yang Hui triangle, you can find two Relationship between people?
Attachment (1): Proof: When
Time,
It is an odd number.
Certification: There is a unique natural number for any positive integer M.
Odd
,Make
. Assume
,
, ...,
.
when
Time,
∵ 上 上 式 分 分 是 是 是 数 是 数 是 是 数 是 数 是 数 数 数 整 整 整 整 整 整 整 整 整 整 整 整 整 整
Bamboo
It is an odd number.
Attachment (2):
Yang Hui Triangle (2)
Destination requirements
1. Explore the digital law of Yang Hui triangulation, and apply the law to see the previous N items of a type of number;
2. Explore the link between Yang Hui triangle and other mathematical objects, and cultivate students' ability to apply mathematical knowledge methods.
content analysis
The main content of this lesson is to continue to study the digital law of Yanghui triangle and its links between other mathematical issues.
1. From the process of studying the digital laws on the inclined edge of Yang Hui triangle "two waist", we can find Zhu Shijie et al.:
. This law is actually the basic nature of the Yanghui triangle.
Promotion form. Applying Zhu Shijie et al., You can find the value.
2. Research through two numbers
,or
The digital law on the oblique edge, you can get the famous Fiboaccai number
. Welfare formula from Fiboacci
, Have the nature of the number of combinations:
,
.
3. will
In the Yanghui triangle, remove all the even numbers, the remaining graphics is similar to the Sherbinsky triangle (as shown) in the fractal geometry, this triangle is an irregular phenomenon in the natural world (coastline traits, atmospheric sports, ocean turbulence, New teaching tools for wildlife populations, and even stock markets, etc.).
4. The Six Prismatic Wood Plate Rolling Ball Instructions in the Textbook Explanation of Yang Hui Triangle and Probability Statistics. When teaching, the teacher should make a teaching aid and use 16 small balls. Do a few experiments, then guide students to excavate the relationship between experiments and Yang Hui triangles, and explain with probability knowledge using the arrangement combination knowledge.
Teaching process
1. The 8th-order Yanghui triple map is displayed with a computer. It is mainly to study the digital law of Yanghui triangle, which first studies the number of oblique lines (as shown).
2. The results of the student division may be:
(1) Number (from left to right, from top to bottom), the number of the number (from left to right, from top to bottom), the number of numbers in the n-order Yanghui triangle.
.
(2) The sum of the above numbers:
.
3. Guide students to prove the above equations and introduce the situation about Zhu Shijie to study the number of sectors described above.
(1) Procedure:
(2) Zhu Shijie problem (such as the problem of tricks): Take more than one foot in the cubic recruitment, first tricks, ..., this trick 15 days, ..., ask for sale ... Use a number of listed languages that: Karie recruiting troops
, A total of N days, how much is a common recruitment? Problem can be converted to:
Bamboo
Bamboo
.
4. Guide students to observe the 8th-order Yanghui triangle table. The relationship between the oblique lines marked in the research diagram
(1) After adding the numbers of each of the oblique balances, it is listed in the order from top to: 1, 1, 2, 3, 5, 8, 13, 21, 34.
(2) After studying the laws of the above number, it can be speculated that the number of non-ferreed Yanghui triangles is listed as:
(3) Guide students will
Indicates the sum of the number of combinations and proof
.
,
According to the basic nature of the Yanghui triangle 3 can be launched
.
(4) Point out that the above number is the Column of Fibo, which has a wide range of applications.
5. Observe what is the characteristics of the number in each small triangle? And promoted
Yang Hui triangle
(1) (from top to bottom) The number in the rigid triangle is an even number, ie
They are even (k∈n *).
(2) The number of first oblique edges outside the kth positive triangle is odd, namely
They are odd (k∈n *).
The nature of this nature and the nature launched by the previous class
There is also an odd number of odd numbers in all the numbers on the line.
(3)
In the Yanghui triangle, the even number and odd number, which one?
In the Yanghui triangle, there are
Siki, a total of even (k∈N *), try comparison
versus
Size (leaving lessons thinking).
6. Demonstration experiment
Teachers or students put 16 uniform balls one by one into the teaching machine as shown. Statistically the number of small balls in each rectangle frame. Continuous three times of experimentation, analyze statistical results; and promote the results to the N 1 layer of teaching aid,
A small ball situation and give reasonable analysis.
(1) Set the number of channels of the channel from the first layer falling below the nth layer below the n (n, k), depending on the symmetry of the teaching, can be established as follows. Recursive mode: f (1, 1) = 1, f (n, k) = f (n, n-k 1), f (n 1, k) = f (n, k-1) f (N, k), k = 1, 2, ..., n 1, specified f (n, 0) = f (n, n 1) = 0 (n∈N *).
Basic nature of Yanghui triangle:
Visit:
. (You can use the number of methods to prove that the conclusion is true, after retiring the course)
Therefore, in the ideal state,
A small ball falls from the first layer to the ninth layer, from the number of small balls in the rectangle frames of the left to right, respectively
.
(2) The ball can be seen from a layer to the lower layer to perform a random trial, of which the probability that the small ball falls into the left
. Then the ball fell from the first floor to the n 1 layer, it can be seen as an independent repetition test. The ball will finally fall into the kth rectangle in the knee. It can be seen from the left to the left. K 1 time, its probability is
.
At a large number of repetitions, the statistics are:
The number of small balls in the kth rectangore in the n 1 layer is
.
7. summary
Yang Hui triangle mystery is endless, as long as everyone uses the method of reasoning and logic reasoning from different angles, it will definitely find more laws. At the same time, you often study other mathematics or practical problems, and create capacity will greatly improve. Interested in learning more students in Yang Hui Triangle, you can check the "Mr. Hua Lugeng" From Yang Hui triangle "book or on the Internet.
Homework
1. Whether there is constant a, b, c, make the equation
Established to all positive integers and prove your life.
2. Change the number R number of the NP line in Yanghui triangle
, The triangle obtained is called the Leibniz triangle, what is the characteristic of this triangle? Write one to two laws.
