Textbook: PEP Edition A, Chapter 1, Section 3, Electives 2-3.

First, the teaching objectives

1. Knowledge and skills:

(1) Understanding binomial theorem is a generalization of algebraic multiplication formula.

(2) Understand and master binomial theorem, and prove binomial theorem with counting principle.

2. Process and method:

Through students' participation and exploration of the formation process of binomial theorem, students' abilities of observation, analysis and generalization, as well as the ability of transferring consciousness and transforming methods are cultivated, and the way of thinking from special to general is realized.

3. Emotions, attitudes and values:

Cultivate students' consciousness of independent inquiry and spirit of cooperation, experience the discovery and creation of binomial theorem, and experience the simplicity and rigor of mathematical language.

Second, the focus and difficulty of teaching

Focus: analysis (a? B)3, and the binomial theorem is obtained.

Difficulties: Analyze the expansion process of binomial with counting principle, and find the law of each coefficient when binomial is expanded into the sum of monomials.

Third, the teaching process

(A) ask questions and introduce topics

Introduction: Research on Binomial Theorem (A? B) the expansion of n, such as: (a? b)2？ a2？ 2ab？ b2，

(a？ b)3 (a？ b)4 (a？ B) 100 so (a? B) What is the expansion of n?

The design is intended to take problems as the starting point of teaching, directly lead to topics, stimulate students' curiosity, and clarify the problems to be solved in this class.

(2) Guide inquiry and discover laws.

Re-understanding of 1 and polynomial multiplication.

Question 1. (a 1？ a2)(b 1？ What is the extension of b2)? How many items are there in the expansion? How is each project constituted?

Question 2. (a 1？ a2)(b 1？ b2)(c 1？ C2) How did each item in the expansion come into being? How many items are there in the expansion?

The design intention is to guide students to use the counting principle to solve the problem of the number of projects, to clarify the characteristics of each project, and to prepare for the follow-up study. B)3 Re-understanding of expansion

Explore 1: Do not operate (a? B)3。 Can you answer the following questions (please discuss in pairs):

(1) How many expansions were there before merging similar items?

(2) What are the different items in the expansion?

(3) What is the coefficient of each item?

(4) From the above three questions, can you draw (A? B) expansion of 3?

Question 2: To imitate the above process, please deduce (a? B) Extension of 4.

The design intention is to guide students to use the counting principle to (a? B) Rethink the expansion of 3 and analyze the form and number of each item, which is also a deduction (a? B) The extension of n provides a method for students to follow in the subsequent learning process.

(3) Formation theorem, reasoning and proof

Question 3: To imitate the above process, please deduce (a? B) the extension of n.

0n 1n？ 1kn？ kknn(a？ b)n？ Cna？ Cnab？ Cnab？ Cnb(n？ N *)- binomial theorem

Proof: (a? B) it's n (a? B) each multiplication (a? B) When multiplying, based on the principle of step-by-step counting, there are two choices: A or B..

n？ kkbk(k？ 0, 1,? N), for each item ab,

It is composed of k (a? B) I chose B, N-K (A? B) If a is selected, its frequency of occurrence is equivalent to that from n (a? B) There are two items (including similar items) in the expansion formula * * * of k n, which are ann? k

The binomial expansion is obtained by combining the combination number Cn of kb with similar terms, which is the binomial theorem.

By imitating the design intent of (a)? b)3 、( a？ B)4。 Exploration method, that is, through student analogy (a? B) Expansion of n. Prove binomial theorem by "reasoning", analyze the expansion process from the perspective of counting principle, and summarize the form of terms. By analyzing the number of items with the same form in the expansion with combinatorial knowledge, the expansion expressed by combinatorial number can be obtained.

(D) familiar with the theorem, simple application

The formula characteristics of binomial theorem: (inducing students to be familiar with the formula)

1. Number of items: * * Is there n? Project 1.

2. Times: the letter A is arranged in descending order, and the times decrease from n to 0; The letter b is arranged in ascending order, and the number of times increases from 0 to n.

The number of terms is equal to n.

0 12knk3。 Binomial coefficient: Cn in turn, where Cn, Cn, Cn,? ，Cn，？ ，Cn(k？ 0, 1,? , n) is called binomial coefficient.

kn？ Kk4。 General term of binomial expansion: Cnab in the formula is called the general term of binomial expansion. How about Tk? 1 means.

kn？ Kk is the first k whose general term is inflation. Item 1: Tk？ 1=Cnab

Change (1)(a? b)n (2)( 1？ x)n

For example. Find (2x? 16).x

Thinking 1: What is the coefficient of the third term of the expansion?

Thinking 2: What is the binomial coefficient of the third term of the expansion?

Thinking 3: Can you directly find the third item in the expansion?

The design intention is to be familiar with binomial expansion and cultivate students' computing ability.

(E) class summary, homework after class

Summary (students summarize the content of this lesson and the mathematical ideas embodied)

0n 1n？ 1kn？ Kknn 1。 Formula: (a? b)n？ Cna？ Cnab？ Cnab？ Cnb(n？ N*)

2. Thinking method: 1. From special to general way of thinking. 2. Analyze the development process of binomial with counting principle.

homework

Consolidation homework: exercise on page 36 of the textbook 1.3 A group 1, 2, 3.

0 12kn thinking expansion homework: what are the properties of binomial coefficient Cn? ，Cn，？ ，Cn