Excellent lecture notes on the circumference 1 First, talk about the teaching materials.
The circumference is selected from the third section of the first volume of the sixth grade mathematics of Beijing Normal University. The teaching of this course is based on the knowledge of the perimeter of rectangle and square, which deepens the previous knowledge of "circle" and is also the basis for learning the area of circle later. This course serves as a link between the past and the future, and is an important content in elementary geometry teaching.
According to the curriculum standards and the intention of compiling teaching materials, the teaching objectives of this section are established as follows:
1. Knowledge goal: Know what a circle is; Understand the meaning of pi; Understand and master the calculation formula of circumference.
2. Ability goal: I will initially use formulas to solve some simple practical problems in my life.
3. Ideological goal: To inspire students' pride as sons and daughters of China by introducing the stories of Zu Chongzhi and Pi.
Teaching emphasis: explore and discover the relationship between the circumference and diameter of a circle.
Teaching difficulties: using the knowledge of circumference to solve some simple practical problems.
Second, talk about teaching methods and learning methods.
According to the teaching content and students' cognitive rules, I first use the method of courseware demonstration to help students understand the circle, infiltrate and transform their thoughts; Then, students are guided to know and understand pi by experiments, and the calculation formula of pi is deduced, so as to cultivate students' operational skills and improve their ability of analysis, comparison, reasoning and generalization. Finally, the method of self-study guidance is adopted to guide students to think, measure and calculate themselves, and finally find out the relationship between circumference and its diameter and radius, so as to improve students' self-study level. In teaching, we should pay attention to students' independent thinking and group communication, and use various learning forms interactively to achieve the teaching goal of developing intelligence and cultivating ability.
Teaching preparation:
1. Multimedia courseware.
3. Each student should prepare three disks with different sizes and integer diameters, a line and a ruler.
Third, talk about the teaching process
(A) create a situation, passionate investment
The courseware shows the story of a race between a little black donkey with two generations of love and a little yellow donkey with a king. Guide the students to observe and think: what is it that requires the donkey to walk? Ask the students to reveal the topic: the circumference of a circle.
Using multimedia courseware to assist teaching can effectively stimulate students' interest in learning and make them have a strong desire to learn, thus forming a good learning motivation. )
(B) independent cooperation to explore new knowledge
Teaching aid demonstration, intuitive perception, combined with cognitive understanding of the circle.
Students experiment independently, measure the circumference around the coil, and the teacher guides the operation points to cultivate students' practical ability. )
2. Complete the experiment in groups.
A. Measurement record: Students measure the circumference and diameter of a circle, and then record the data to cultivate students' practical operation ability.
B. comparison: compare data and reveal relationships.
Students continue the experiment, work out the quotient of dividing the circumference of each circle by its diameter, and record the quotient. Through calculation, students find that the perimeters of these three circles are all more than three times their diameters and lengths. It is concluded that the circumference of other circles measured is also more than 3 times its diameter.
Cultivate students' practical ability and skills in the process of experimental operation, and improve students' ability of analysis, comparison, reasoning and generalization. )
3. Introduce pi.
(1) First, introduce a number that is more than 3 times. This is a definite number, which we call pi. Expressed by the formula: circumference/diameter = pi.
② Introduce the reading and writing methods of π.
(3) Finally, the story of Zu Chongzhi and ancient mathematician Pi is introduced in combination with the portrait, so as to inspire students' pride as sons and daughters of China. At the same time, it is pointed out that pi is an infinite decimal and its approximate value in primary schools is 3. 14.
④ Students summed up the formula for calculating the circumference:
The circumference of a circle = the diameter of a circle ×π, which is expressed by letters as c =π× d.
The courseware shows a circular runway with a diameter of 50 meters and a schematic diagram of its circumscribed square runway. Let the students observe and think about the diameter of the circle and the side length of the square, and then quickly calculate the perimeters of the two runways with formulas. Let's see if the game between the king and the two generations is fair.
The circumference of the circle is great. 2. Speaking of textbooks
The circumference of a circle is based on the general concept of the circumference in Book One of Grade Three and the calculation of the circumference of a rectangle and a square. At the same time, it is also the beginning of students' initial study of curves and graphs, which lays a good foundation for future study of cylinders and cones. Therefore, it plays an important role in connecting the past with the future and is an important content in elementary geometry teaching.
Second, the analysis of learning situation
Because the sixth-grade students are experiencing the transition from concrete thinking in images to abstract thinking in logic, in teaching, I pay attention to deriving the formula for calculating the circumference from the students' existing knowledge and life experience through independent inquiry, guess and verification, so that students can understand how to get the fixed value "π" in the formula.
Third, the teaching objectives
1, so that students can know the circumference, master the meaning and approximate value of pi, initially understand and master the calculation formula of circumference, and calculate the circumference correctly.
2. Through hands-on operation and practical exploration activities, cultivate and develop students' spatial concept, improve students' abstract generalization ability, and infiltrate the mathematical thinking method of "turning joy into straightness"; Cultivate students' sense of cooperation through group cooperative learning.
3. Cultivate students' patriotic feelings and stimulate students' national pride by infiltrating mathematical culture.
Fourthly, the importance and difficulty of teaching.
1, key point: calculate the circumference of the circle correctly.
2. Difficulties: Understand the meaning of pi, and deduce the calculation formula of pi.
Five, teaching preparation
A set of multimedia courseware, several CDs of different sizes, a ruler, a rope and a calculator.
Sixth, the theory of teaching process
(1) Create a situation and ask questions.
I regard the Shanghai World Expo as a main thread that runs through the classroom. When creating the situation, I introduced the earth model "Blue Planet" in the Urban Earth Pavilion to my classmates, and naturally put forward the math problem of this lesson: Students, we can approximately regard the largest cross section of the "Blue Planet" as a circle, so how do you find the circumference of this huge circle?
Design intention: The creation of Shanghai World Expo broke through the teaching materials, based on students' interests, made students full of enthusiasm and desire for learning new knowledge, stimulated students' desire for exploration, and paved the way for later study.
Some students will learn something about the circumference before they think about it, so as to see the big picture from the small. Since there is no good way to find the perimeter of a big circle, we can find some smaller circles to find their perimeter. At this time, I will give affirmation to the students' ideas in time. "Your idea opens the door to wisdom for students. The teacher is really happy for you!" If no students think of this floor, I will help them recall that they have learned the calculation of the perimeter of rectangles and squares before. It is not only to connect the rectangular playground with rectangular pieces of paper, so as to inspire students to solve problems with small circles instead of big ones.
(2) Self-study and explore new knowledge.
1, independent query
(1) Make students familiar with the concept of circumference.
Because of the previous knowledge, let the students point to the circumference of a circle first, and then say what is the circumference of a circle in their own words.
(2) Measure the circumference of the circle.
There are several ways for students to think independently and then try to solve problems in their own way. At this time, I patrolled in time to investigate the learning situation. If some students don't come up with a way, I will infiltrate a learning method for them in this link, that is, it is difficult to ask for books and advice.
Design intention: to cultivate students' thinking habits of independent thinking, improve students' hands-on operation ability, and invisibly infiltrate the self-study method-seeking books.
2. Cooperation and communication
Students communicate in groups of four or discuss problems. At this time, I will participate in their communication as a participant.
Design intention: Group cooperation aims to enhance students' sense of cooperation. In this process, through constant communication and questioning, the collision of ideas and the complementarity of thinking modes are realized, and students gradually develop a good habit of learning to listen, and learn to "take" and "give up" in the process of listening, that is, learn to analyze.
3. Introduction meeting; exhibition
(1) Some students wound a rope around the disc, then held both ends, straightened the rope, and measured the length with a ruler, which is the circumference of the circle.
(2) Some students mark any point on the circle and aim at the zero scale of the ruler. Then, they wound the original film around the ruler until this point coincides with the ruler again. The distance between these two points is the circumference of the circle.
Teacher's comment: Your methods are all ingenious. When it is not convenient to measure the circumference directly with a ruler, you turn the curve into a straight line, and then measure the length of this line segment to get the circumference of the circle.
Design intention: Through the demonstration of individual students, let students deeply understand the mathematical thinking method of "turning joy into straightness", so as to highlight key points and break through difficulties.
At this time, the teacher questioned, we can use similar methods to measure the perimeter of these small circles. Then, can we still use these methods to measure the perimeter of the largest cross section of the "blue planet", such as the length of the equator? Obviously not.
Design intention: bring students back to the situation at the beginning of class again, stimulate students' interest in learning in questioning, and urge them to have an urgent desire to explore general methods.
Step 4 guess and verify
(1) Observation Multimedia Courseware: Draw five circles with different sizes with five line segments with different lengths as diameters. Ask the students to guess what the circumference is related to.
(2) Discuss the relationship between the circumference and diameter of a circle.
① Group cooperation
Students are required to do their homework in groups of four. The team leader is responsible for assigning tasks. Two people cooperated in measuring the diameter. A person uses a calculator to calculate the ratio of the circumference to the diameter of a circle. The fourth person fills in the relevant data in the table below as required. See what you can find.
Ratio of circumference to diameter (two decimal places are reserved)
1 wafer
No.2 wafer
No.3 wafer
Design Intention: This link encourages students to make reasonable guesses in a well-founded situation, and then verify them according to the guesses.
② Learning "pi"
On this basis, the teacher further pointed out that due to various reasons, the ratio of circumference to diameter calculated by different circles may not be exactly the same, but in fact, this ratio is a fixed number, which we usually call "π", which is expressed by the Greek letter "π", and π is an infinite acyclic decimal. For the convenience of calculation, we generally only take its approximate value π≈3. 14。 (blackboard writing: π, π≈3. 14)
(3) Infiltrating mathematical culture
First, the teacher introduces the contents related to the circumference in Zhouyi Suan Jing and the story of Zu Chongzhi, a great mathematician and astronomer in ancient China, and then asks the students to talk about their own ideas.
Design intention: The infiltration of mathematical culture is to stimulate students' patriotic feelings and cultivate students' national pride from an early age.
5. Derive the formula
According to the relationship between the circumference and diameter of a circle, students derive a formula for calculating the circumference of a circle: circumference of a circle = diameter × pi ratio, which is expressed in letters as c = π d. If the radius is known, students will think of c = 2π r (blackboard formula: c = π d, c = 2π r). At this time, the teacher leads to the topic conveniently. (Title on the blackboard: circumference)
The circumference of the circle is great. 3. Speaking of textbooks
The circumference of a circle is the second section of Unit 4 in the first volume of the sixth grade of People's Education Press, which is based on the general concept of the circumference of the first volume of the third grade and the calculation of the circumference of rectangles and squares. Through a series of operation activities, the textbook tries to make students understand the meaning of pi through observation, analysis and induction, experience the formation process of pi, and deduce the calculation method of pi, which lays the foundation for learning the knowledge of circle area, cylinder and cone. At the same time, through the study of this class, students' practical ability, unity and cooperation ability and problem-solving ability are further cultivated, and students are educated in ideology and morality.
Second, the teaching objectives
According to the analysis of the above structural characteristics and students' cognitive rules, the teaching objectives of this lesson are determined as follows:
1, knowledge goal: let students know the circumference of a circle and understand the meaning of pi in a specific situation; Only by understanding and mastering the formula for calculating the circumference can the circumference be calculated correctly.
2. Ability goal: To cultivate students' observation ability, hands-on operation ability, analysis ability, generalization ability and cooperative learning ability through the measurement of pi, the exploration of pi and the derivation of pi formula.
3. Emotional goal: to educate students on dialectical materialism by exploring pi; Combine the story of Zu Chongzhi, an ancient mathematician in China, and educate students in patriotism.
Third, teaching focuses on difficulties.
The teaching emphasis of this lesson is: to understand and master the calculation method of the circumference of a circle; Circle is a curve figure and a new plane geometry, which deepens the teaching of perimeter calculation of plane figures. In particular, the concept of pi is abstract, so I take exploring the meaning of pi and deducing the calculation formula of pi as the difficulty of this lesson.
Preach the law
Students are open and creative individuals, and they will participate in the communication between teachers and students in the classroom with their existing knowledge, experience, inspiration and interest. They will enrich the classroom with their own guesses and verifications. Let the math classroom be full of vitality. Therefore, let students experience the formation process of mathematical knowledge, experience the creativity of mathematical learning, feel the rigor of mathematics and the accuracy of conclusions, and then cultivate students' consciousness of thinking about problems with mathematical thinking methods. This is the basic quality that students must have to adapt to future life. With these understandings of the new curriculum, in the teaching of the circle, I used methods such as operation, guessing and verification, as follows:
1, using the mathematical thinking method of guessing and verifying: in teaching, I first let students guess boldly, and then guide students to find ways to verify guessing with language, so that students can feel the mathematical thinking method and feel the rigor of mathematics.
2. Hands-on operation, active interaction: Teachers let students find problems and explore independently through activities such as measurement, experiment and calculation. The specific method is: let students use learning tools to calculate and find the law, so as to deduce the calculation method of the circumference of the circle. In the process of exploration, the teacher gives guidance and acts as a guide for students' learning.
3. Observation, discussion, communication and cooperation: In teaching, teachers organize students to communicate in groups on the basis of independent thinking, and put forward communication methods and steps according to the age characteristics of students, so that students can communicate in an orderly, purposeful and orderly manner. Improve the timeliness of communication.
teaching program
The whole teaching process is divided into four links.
The first link: create the situation and establish the concept of circumference.
1. Show me a clock. What is the track of the small second hand on the clock in one minute? So how far does the top of the small second hand go in an hour? Lead the topic (circle)
2. Let the students take out the circular learning tool to have a look, touch and say which part the circumference refers to. Experience and understand the meaning of the circle by yourself. Effective touch experience and sufficient rational generalization make the construction process of the concept of circumference substantial and effective. )
The second link: hands-on practice, feeling the measurement method.
1. Question: The circumference is a curve. How to measure the circumference of a circle?
2. Guide students to operate, cooperate and communicate, and find out the method of measuring circumference.
3. Students report and demonstrate the measurement method. (rope winding method, rolling method)
4. These circles are relatively small. If there is a big circle, can you measure its circumference by winding and rolling the rope? What if it can't be measured directly?
Through deliberate reflection and free evaluation of the two measurement methods, students dialectically felt the limitations of the "winding" and "rolling" methods, which stimulated their enthusiasm for exploring the "calculation formula" and made a "psychological" foreshadowing for further study of the calculation of pi. )
The third link: put forward a reasonable guess and verify it.
1. The circumference of a square is four times the length of its side, so is the circumference of a circle also related to line segments? (Encourage students to make bold guesses)
2. As a group, take out the circular learning tools prepared before class, measure their perimeters and diameters respectively, calculate the ratio of perimeters and diameters, and fill in the following table: (The measured values are accurate to the millimeter)
Trade name circumference diameter 1 circle
The second lap
Third circle
3. Through observation and comparison, students can find that no matter how the size of the circle changes,
The ratio of the circumference to the diameter of a circle is probably more than three times.
4. The teacher introduced pi.
5. Students read relevant information about pi, and teachers carry out patriotic education in time.
6. Can the formula for calculating the circumference be obtained by analyzing the table? because
In front of layers of bedding, students can easily get the calculation method of the circumference of the circle:
C=πdC=2πr
(This part of the content is mainly to let students operate independently, explore independently, and through observation, find problems, participate in cooperation and exchange, summarize and get solutions to problems, so that students can get certain emotional experience and enjoy the pleasure of success. It improves students' ability of analysis, reasoning and generalization, and develops students' concept of space. )
The fourth link: using knowledge to solve problems.
1. Example 1 (Design purpose: Through example calculation, students can better understand the role of mathematics in life, solve real life problems, and lay a good foundation for final practice)
Now, can you tell us the distance that the tireless little second hand traveled in an hour? What kind of data do you want to get to solve this problem (design purpose: let students find the conditions to solve the problem themselves and cultivate their independent thinking ability. This problem echoes the previous introductory question, thus solving the problem from beginning to end. )