(A) Teaching objectives
1. Knowledge and skill objectives:
(1) Grasp the meaning of the logical conjunction and.
(2) Correct application of logical conjunctions to solve problems
(3) Master the truth table and apply it to solve problems.
2. Process and method objectives:
In observation and thinking, in solving and proving problems, this course should pay special attention to the cultivation of students' rigorous thinking quality.
3. Emotions, attitudes and values:
Stimulate students' enthusiasm, curiosity, rigorous learning attitude and enterprising spirit.
(B) Teaching focus and difficulties
Key points: Understand the meaning of logical conjunctions through mathematical examples, so that students can correctly express relevant mathematical contents.
Difficulties:
1, correctly understand the stipulation and decisiveness of proposition Pq.
2. Express the proposition Pq concisely and accurately.
Teaching aid preparation: materials related to the content of teaching materials.
Teaching assumption: In observation and thinking, in solving and proving problems, this class should pay special attention to the cultivation of students' rigorous thinking quality.
(3) Teaching process
Students' questioning process:
1, Introduction
In today's society, people can't do any work or study without logic. Having certain logical knowledge is an important aspect of citizens' cultural quality. Mathematics is characterized by strong logic, especially after entering high school. The mathematics they study emphasizes logic more than junior high school. If they don't learn some logical knowledge, they will often make logical mistakes unconsciously in the process of our study. In fact, students have been exposed to some simple logic knowledge since junior high school.
In mathematics, conjunctions are sometimes used, such as and or not. We also use these conjunctions in our daily life, but their meanings and usages are different from those in mathematics. The meaning and usage of conjunctions and non-conjunctions in mathematics are introduced below.
For the sake of simplicity, lowercase letters p, q, r and s are often used to represent propositions in the future. (Pay attention to the difference between the condition P and the conclusion Q of the learning proposition in the previous section)
2, thinking, analysis
Question 1: What is the relationship between the following three propositions?
① 12 can be divisible by 3;
② 12 can be divisible by 4;
③ 12 is divisible by 3 and 4.
Students can easily see that in the (1) group of propositions, proposition ③ is a new proposition derived from propositions ① and ② by conjunctions.
Question 2: Have we studied the proposition of using conjunctions and conjunctions like this before? Can you give some examples?
For example: Proposition P: The diagonals of diamonds are equal, and the diagonals of diamonds are equally divided.
3. Inductive definition
Generally speaking, a new proposition, named pq and pronounced P and Q, is obtained by using conjunctions and connecting Proposition P and Proposition Q. ..
Is "He" in the proposition pq the same word as "He" in the following proposition?
If xA and xB, xB.
The meaning of the word and in the definition is similar to that in the proposition. But the logical conjunctions here are equivalent to "and", "and" and "in everyday language, and they are equal, which means that both are satisfied at the same time. Note: Symbols and openings are downward.
Note: P and Q in P and Q propositions are two propositions, while P and Q in original proposition, inverse proposition, negative proposition and inverse proposition are two parts of the same proposition.
4. Provisions on the Truth and False of Proposition pq
Can you determine the true value of the proposition pq? What is the connection between proposition pq and proposition P and Q?
Guide students to analyze the truth and falsehood of propositions P, Q and pq in the previous examples, and summarize the general law of the relationship between the truth and falsehood of these three propositions.
For example, in the above example, among the propositions in the (1) group, ① ② are all true propositions, so the proposition ③ is true.
Generally speaking, we stipulate that:
When P and Q are true propositions, pq is true propositions; When one of the two propositions P and Q is false, pq is false.
Step 5: Example
Example 1: Combine the following propositions into a new proposition pq, and judge whether it is true or false.
(1)p: the diagonal of the parallelogram is equally divided, and q: the diagonal of the parallelogram is equal.
(2)p: the diagonal lines of the rhombus are perpendicular to each other, and q: the diagonal lines of the rhombus are equally divided;
(3) P: 35 is a multiple of 15, and Q: 35 is a multiple of 7.
Solution: (1)pq: The diagonals of the parallelogram are equally divided.
Since P is a true proposition and Q is a true proposition, pq is a true proposition.
(2)pq: The diagonals of the rhombus are perpendicular to each other and equally divided.
Since P is a true proposition and Q is a true proposition, pq is a true proposition.
(3) PQ: 35 is a multiple of 15, and 35 is a multiple of 7. It can also be abbreviated as 35 is a multiple of 15 and a multiple of 7.
Since P is a false proposition and Q is a true proposition, pq is a false proposition.
Explain that when using and contacting new propositions, if you simplify them, you should pay attention to keeping the meaning of the propositions unchanged.
Example 2: Rewrite the following propositions with logical conjunctions to judge whether they are true or false.
(1) 1 is both odd and prime;
(2)2 is a prime number and 3 is a prime number;
6. Consolidation exercise: P20 exercise 1, 2
7. Teaching reflection:
(1) Grasp the meaning of the logical conjunction and.
(2) Correct application of logical conjunctions to solve problems