Second-term theorem public course teaching case
(First teach)
Tucao: Lu Ming
2002
year
4
month
30
day
First, teaching objectives
1. Understand the Yanghui triangle. Its behavior is: (1) Writing Yang Hui triangles with incomplete summation; (2) according to Yang Hui triangle
The binary use is expanded.
2, master the two-term theorem. Its behavior is: (1) can guess the two exhibition coefficients based on combined ideas and incomplete summary law
And two expansion
(2) Correctly distinguish between the two-term coefficient and a factor; (3) The can make an extension of any given two items and find its specific item or coefficient.
Second, the key points and difficulties of teaching
1. Key: Discovery, understanding and preliminary application of binary theorem.
2, difficult point: the discovery of the binary theorem.
(Teaching aid: multimedia courseware)
Third, the teaching process
1, scene settings
Question 1: If today is Monday, a few days later? How is calculated?
Expected answer: Wednesday, the transformation of the problem is how much "30 30 is calculated after the remainder".
Question 2: If today is Monday,
What is the week? How is calculated?
Expected answer: Transformation of problems into seeking "
What is the remainder of 7 after 7, that is, research
What is the expansion? This is the content of this class, after learning this lesson, this question is not difficult to solve.
(Design intentions, make students express their purpose, use suspense to inspire their learning motivation. Ousbell believes that motivation is a prerequisite for learning, and cognitive power, that is, students are eager to cognition, understand and master knowledge, and correctly The statement, the tendency to successfully solve the problem is an important driving force for students to learn.
2, new teaching
Step 1: Let students start
;
;
Teachers will organize the above extensions into the following model
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Question 1: Please find the law between the above data.
Expected answer: The numbers in the middle of the next line are equal to the sum of the neighboring numbers of the corresponding position of the previous line.
Question 2:
The expansion type is an example, and the law of each alphabetical arrangement is the relationship between the number of items and the multiplier index; the relationship between the factor of the second term and the multiplier index.
Expected answer: 1 The number of times the expansion is arranged in a letter, and the other alphabet is arranged, and the two letters and is equal to the multiplier index; 2 The number of outlets is more than the multiplier index. 3 The coefficient of the second term in the expansion is equal to the multiplier index.
Preliminary summary:
(※)
(Design intent: The above is presented to the "triangle" of the coefficient of coefficient, and the "triangle" is played, although the teacher will present this "triangle" model to the student in the form of the conclusion, but it is not The final result, but a valid tool for finding coefficient rules, which is convenient for students to link new learning materials to their original cognitive structures and have the meaning of the original cognitive structure. Such learning is It is a psychological process that makes sense rather than mechanical, is a psychological process of active construction rather than passive death.
Exercise: Expand
Teachers are staged, telling students, the above coefficient table is the masterpiece of the Mathemist of my country, called Yang Hui triangles, this invention is more than 400 years earlier than European Pasca triangle. You have done the same exploration with Yang Hui today to encourage students to explore the enthusiasm and inspire national pride and patriotic enthusiasm as an ancient future generation. Step 2: Continue termination
How to expand
as well as
?
(Design intentions: Make students feel that Yang Hui triangle is not enough, stimulate students to continue to learn new and simpler methods.)
Continue new
Teacher: In order to find the law, we will
The letters in the first parentheses are branched
The letters in the second parentheses are branched
;And so on. Please calculate with a polynomial multiplication algorithm again:
.........
.........
.........
.........
.........
(Design intention: The above presentation is to build a "cognitive bridge" to activate the knowledge and experience of the students in the student cognitive structure, which makes it easy to learn, using existing knowledge and experience. New knowledge. And migrate into a strange situation.)
Question 1:
Take the example, there are several cases multiplied
item? Letters here
Which parentheses come from?
Question 2: Since the above letters
From 4 different brackets,
Can you use the number of combinations?
Question 3: Can you change the meaning 2 described in the question 2 into a combination of propositions?
(Expected answer: There are 4 brackets, two letters in each parentheses, one is
,one is
. Each parentheses can only take one letter and take two
Two
, Then multiply, how many of the different testers? )
Question 4: Please use the method of the class ratio to find other factors in the two expansion, and will:
The coefficients in parentheses are filled in with the form of a combination.
Rendering the II Treatment - Book Topics:
.
3, deepen understanding
Ask students to summarize:
1 The difference between the two-term theorem expansion, the index, the number of items?
2 What is the structural characteristics of the two-term theorem expansion? Which one is most representative?
As a result, students have a two-term theorem, two expansion, binary factors, and two coefficients, and two expansion-based generalizations. This is the focus of this lesson.
(Design intentions: teachers use the form of asking, through let students summarize, discover the law, excavate the potential significance of learning materials, thus making learning into meaningful learning.)
4, consolidate the application
[Example 1] Expand 1
2
[Example 2] 1
Expansion 4 of the coefficients and the binary factor of the fourth item.
2
Expansion
The coefficient of item.
Variation: in the second-term theorem, order
What kind of formula gets?
Thinking:
why?
[Example 3] Solve the starting problem:
,
The front is a multiple of 7, so the remainder is
Therefore, it should be Tuesday.
Description: Solve some of the recollection issues is the application of two-term theorem.
Fourth, classroom summary
1 This section we mainly learned the development of the binary, two methods, one is Yang Hui triangle, the second is the two-way theorem, two methods have a thousand autumn.
2 binary theorem expressions and expansion types,
3 To correctly distinguish the "factor" and "binary factor",
4 A suitable value to impart the letter in the binary theorem, you can ask for some special combined polynomial values.
Five, layout homework (omitted)
