First, the teaching objectives:
1, knowledge and skills
(1) Understand the concepts of random events, inevitable events, impossible events and certain events;
(2) Correctly understand the frequency and significance of event A;
2. Process and method
In the teaching of discovery method, we can learn and improve by collecting data from coin and dice tossing experiments, summarizing experimental results, discovering laws and exploring.
3. Emotions and values
Through students' hands-on, brains and personal experiments, we can understand knowledge and the relationship between mathematical knowledge and the real world. Cultivate students' dialectical materialism and enhance their scientific consciousness.
Second, the teaching emphasis and difficulty:
Focus: (1) event classification; ⑵ Correctly understand the significance of the frequency of A events.
Difficulties: (1) Understand the difference and connection between frequency and probability; ⑵ Explain specific problems in real life with probability knowledge.
Third, the teaching process:
(A) create scenarios and introduce topics
In daily life, some questions can be answered accurately. For example, when the room temperature is below, can the water in the basin turn into ice? Will the sun rise in the east tomorrow? Wait, these things are inevitable. At the same time, some questions are difficult to give accurate answers. For example, when will you get up tomorrow? How many people eat in the school cafeteria? Can the current welfare lottery win? Wait, the results of these questions are accidental and uncertain, so it is difficult to give accurate answers.
Some things happen by accident and some things are inevitable, but there is often some internal relationship between accident and necessity.
For example, the changes in our county have definite and inevitable laws throughout the year, but which day of the year is the hottest, coldest, with the largest rainfall and snowfall in our county is uncertain and accidental.
(blackboard writing topic)
(B) teacher-student interaction, explaining the new lesson
1. Related concepts
(1) inevitable event: the event that will happen under condition S is called the inevitable event relative to condition S;
(2) Impossible events: events that will not happen under condition S are called impossible events relative to condition S;
(3) Deterministic events: inevitable events and impossible events are collectively referred to as deterministic events relative to condition S;
(4) Random events: events that may or may not occur under condition S are called random events relative to condition S;
Deterministic events and random events are collectively called events, which are generally represented by capital letters A, B, C ...
2. In the dice experiment, we can define many events, such as:
C 1 ={ 1 minute occurrence}; C2 ={ 2 minutes};
C3 ={ 3 minutes}; C4 ={ 4 minutes};
C5 = {appears at 5 o'clock}; C6 = {occurred at 6 minutes};
D 1 ={ The number of points appearing does not exceed1}; ? D2 ={ the number of occurrences is greater than 3};
D3 ={ occurrences less than 5};
E ={ the number of points appearing is less than 7}; F ={ the number of points that appear is greater than 6};
G ={ the number of points appearing is even}; H ={ the number of points appearing is odd};
……
Are they likely to happen?
3. Check the following events:
(1) The average temperature in summer in Shanghai is higher than that in winter.
(2) Stones thrown upward on the ground will fall;
The sun will rise in the east tomorrow.
Will these events happen? What is their project?
An inevitable event determines an event.
4. Check the following events:
(1) water at 50 degrees will boil at standard atmospheric pressure;
(2) melting steel at normal temperature and pressure;
(3) taking drugs makes people young forever.
Will these events happen? What event?
Impossible, impossible event? Decisive event
5. Check the following events:
(1) Someone hit the target once;
(2) Select any TV channel that is broadcasting news;
(3) The number of dice is odd.
Are these events bound to happen? What is their project?
It may or may not happen, and it is a random event.
Example 1 Determine which of the following events is inevitable, which is impossible and which is random?
(1) "Throw a stone and fall".
(2) "Ice melts at standard atmospheric pressure and the temperature is lower than 0℃";
(3) "Someone hit the target with one shot";
(4) "If A > B, A-B > 0";
(5) "Flip a coin and heads will appear";
(6) "The conductor generates heat when electrified";
(7) "Take any one of the five labels marked with the numbers 1, 2, 3, 4 and 5 respectively to get the No.4 label";
(8) "One phone receives two calls within 1 minute";
(9) "Seeds can germinate without water";
(10) "Solder melts at room temperature".
A: According to the definition, events (1), (4) and (6) are inevitable events; Events (2), (9) and (10) are impossible events; Events (3), (5), (7) and (8) are random events.
(C) hands-on experiments, found the law
1. Frequency and times: repeat the test for n times under the same condition S, and observe whether there is an event A, and call the number of times that the event A appears in the n tests as the number of times that the event A appears; Let the ratio (a) of event A = the frequency of event A. 。
What is the frequency range?
The frequency of inevitable events is 1, and the frequency of impossible events is 0. So the frequency range is 0, 1.
Some test results of coin toss in history (see textbook P 1 12).
What is the stable value of the head-up frequency of the coin toss test?
We see that the positive frequency swings around 0.5 when there are many tests.
The above experiments show that it is impossible to predict whether the random event A will occur in every experiment, but after a large number of repeated experiments, with the increase of the number of experiments, the frequency of the event A shows a certain regularity. How is this regularity reflected?
The frequency of event A is relatively stable and is constant in the interval 0, 1.
The closer this constant is to 1, it means that the greater the frequency of event A, the more frequent it is, so it is more likely to happen.
On the contrary, the smaller the possibility of an event, the lower the frequency, the smaller the frequency and the smaller the constant.
The frequency of event A is relatively stable and is constant in the interval 0, 1.
Therefore, we can use this constant to measure the possibility of event A.
For a given random event A, if the frequency (a) of event A is stable at a certain constant with the increase of test times, this constant is called P(A) and the probability of event A. 。
So what is the probability of raising your head in the above coin toss experiment?
P (face up) =0.5
2. The video "Stability of Frequency" was inserted into the micro course of Onion College.
The difference and connection between frequency and probability is the difficulty of this class. The video of micro-class in Onion College explains the relationship between frequency and probability in its unique way.
3. Difference and connection between frequency and probability: The frequency of a random event refers to the ratio of the number of times the event occurs to the total number of tests N, which has certain stability and always swings around a certain constant, and the amplitude of this swing becomes smaller and smaller with the increase of the number of tests. We call this constant the probability of random events, which quantitatively reflects the probability of random events. Frequency can be approximated as the probability of the event under the premise of repeated tests.
In practical problems, the probability of random event A is often unknown (such as the probability of shooting a target under certain conditions). How do you get the probability of event A?
Through a large number of repeated experiments, the stable value of the frequency of event A, that is, probability, is obtained.
We study random events that can be repeatedly tested under the same conditions, and they all have frequency stability.
(4) Summary
1, inevitable event, impossible event, definite event, random event, frequency, frequency and probability.
2. Probability is the stable value of frequency, and only the estimated value of probability can be obtained according to the frequency of random events.
3. It is unpredictable whether the random event A will occur in each experiment, but after a large number of repeated experiments, with the increase of the number of experiments, the frequency of the event A will gradually stabilize at a constant in the interval [0, 1] (that is, the probability of the event A). The closer this constant is to 1, the greater the probability of event A, that is, the greater the possibility of event A; On the contrary, the closer the probability is to 0, the less likely the event A will happen. Therefore, probability is a measure of the possibility of an event.
(5) Transfer
P 1 13 Exercise: 1, 2, 3.