Conical volume handout 1 introduction of micro-course works
This work is a micro-lesson designed for the knowledge point of "the volume of a cone" in Unit 2 of the sixth grade mathematics textbook published by Jiangsu Education Publishing House. It is suitable for students who are about to learn conical volume in the sixth grade of compulsory education or who have finished learning but need to be consolidated.
The content of this section is based on students' understanding of the characteristics of cones and their mastery of the calculation method of cylindrical volume. Some students may already know the volume formula of the cone through preview, but the formula is familiar and the principle is abstract. How is the volume formula of cone derived? How to understand the principle through formulas? It is difficult for students, so I did this lesson for this learning content.
Through the study of this micro-course, students can break through the difficulty of how to deduce the volume of a cone, and can explain the origin of the volume formula in a scientific way, so as to better understand, master and apply the volume formula of a cone and lay a solid foundation for learning the relevant knowledge of solid geometry in the future.
Teaching demand analysis
Applicable object analysis
This lesson is suitable for students who are about to learn "the volume of a cone" or who have already learned it but need to consolidate it. The content of this section is based on students' understanding of the characteristics of cones and their mastery of the calculation method of cylindrical volume.
The ability of senior high school students to analyze and solve problems is gradually enhanced, which creates favorable conditions for students to explore independently and study cooperatively. They have mastered some geometric knowledge and learned some transformation methods between geometric figures. However, the students' concept of three-dimensional space has not been fully developed, and the transformation between forms is still difficult. In view of students' reality, I mainly use observation, guessing and operation in teaching, so that students can experience the generation and formation of knowledge personally.
Learning content analysis
This lesson is the key and difficult part of geometry knowledge in primary schools, and it is a leap in learning volume calculation of three-dimensional graphics in primary schools. Through the teaching of this part of knowledge, we can develop students' spatial concept and imagination, deeply understand the new field of geometric volume derivation, and lay a good foundation for students to further learn geometric knowledge. Attach importance to analogy in teaching, change the infiltration of ideas, intuitively guide students to experience the exploration process of "guessing, analogy, observation, experiment, inquiry, reasoning and summary", and understand and master the derivation process and calculation formula of cone volume.
Analysis of teaching objectives
1. Make students carry out mathematical activities such as operation, guess, estimation, verification, discussion and induction on the basis of understanding cylinders and equal-height cones, and deduce the volume formula of cones; Master the calculation formula of cone volume, and can apply the formula to solve related practical problems.
2. Enable students to further accumulate learning experience of space and graphics in activities, enhance the concept of space and develop mathematical thinking.
Teaching process design
(1) orientation method.
1, talk: There are many conical objects in life.
Health: This year, my family had a bumper harvest of grain. Dad and others piled the rice in piles, just big cones. However, how do you find the volume of these cones?
Teacher: Think about whether you can help Ma Xiaolan solve this problem! ?
2. Reveal the topic.
(2) Experimental verification
Teacher: Recall: How did we discuss the formula of cylindrical volume before (transforming a cylinder into a cuboid)?
Teacher: Think about it. How do we explore the volume of a cone?
Teacher: Oh, yes. Maybe we can convert the volume of a cone into the volume of a cylinder!
1, estimate the volume relationship between a cone and a cylinder.
Displays a direct view of cylinders and cones.
Teacher: Please estimate the relationship between the volume of a cylinder and the volume of a cone.
Q: This is only our estimate. What method can be used to verify our estimate?
Teacher: To test our conjecture, let's do an experiment together!
2. Clarify the experimental method.
(1) experimental idea: fill a conical container with sand, then pour it into an empty cylindrical container, and look at it several times to find out the relationship between conical volume and cylindrical volume.
(2) experimental considerations:
(1) sand filling, but not more;
Be careful not to spill it when you pour it;
3. Summary of the report.
What are the characteristics of (1) compared with the original cylindrical and conical containers?
(2) Conclusion: When the bottoms are equal and the heights are equal:
① The volume of a cylinder is three times that of a cone;
② The volume of a cone is one third of that of a cylinder.
(3) Summarize the formula of cone volume: cone volume = bottom area × height.
(3) class summary.
Teacher: Students, after today's study, do you know how the cone volume formula is derived? Will you find the volume of a conical object in the future?
(4) Consolidation after class.
A pile of rice, almost conical, with a bottom area of 18 square decimeter and a height of 5 decimeters. How many cubic centimeters is its volume?
Study Guide
Please use this video when previewing or reviewing the "Volume of Cone" in Unit 2 "Cylinders and Cones" in the sixth grade mathematics textbook published by Jiangsu Education Publishing House, and try to solve practical problems by using what you have learned after watching it. In addition, there are many related materials, which can be consolidated by searching more on the Internet.
Auxiliary learning materials
Jiangsu education printing plate mathematics textbook sixth grade volume 2
Introduction of production technology
Make PPT courseware, then use screen recording software to record the process, use camera to shoot the experimental process, and finally use non-editing software to integrate.
Draft II of Lecture Notes on Cone Volume Dear leaders and teachers,
Hello, guys. The content of the lesson I'm going to tell you today is the first unit of sixth grade mathematics of Beijing Normal University Press-The Volume of a Cone. I will elaborate on teaching material analysis, the choice of teaching methods, the guidance of learning methods and the teaching process.
I. teaching material analysis
The volume of a cone is learned on the basis of students' mastery of the calculation, application and understanding of the basic characteristics of a cone, which is the content of the last lesson of geometry learning in primary schools. Cone is a kind of shape that people often encounter in production and life. Teaching this part well is conducive to further developing students' concept of space and laying a foundation for further solving some practical problems.
Mathematics curriculum standards require teachers to be organizers, guides and collaborators of students' mathematics activities. Teachers should actively use all kinds of teaching resources, creatively use teaching materials and design a teaching process suitable for students' development. According to the concept of new curriculum standards, the characteristics of teaching materials and the reality of students, I have formulated the following teaching objectives and teaching difficulties.
1, teaching objectives:
(1) Understand the derivation process of the cone volume formula, master the calculation formula of the cone volume, and be able to use the volume formula to calculate the volume of the cone.
(2) Cultivate students' observation ability, understanding ability and spatial concept, and apply what they have learned to solve practical problems.
(3) Enable students to have a successful experience in the experience and experience the connection between mathematics and life.
2. Teaching emphasis: master the formula for calculating the volume of the cone, and use the formula to calculate the volume of the cone to solve some practical problems.
3. Difficulties in teaching: Understanding the multiple relationship between cylindrical volume and conical volume under the conditions of equal bottom and equal height.
4. Preparation of teaching AIDS:
(1) multimedia courseware.
(2) Several groups of cones and cylinders with equal bottom, equal height, unequal bottom and unequal bottom, sand and experimental reports; A graduated ruler, rope, etc.
Second, oral teaching methods
Mr. Ye Shengtao, a famous educator in China, pointed out that teaching is for not teaching. There are methods in teaching, but there is no fixed method in teaching. What is important is that the methods are correct. According to the concept of new curriculum standards, the characteristics of teaching materials and students' cognitive rules, I mainly used the following teaching methods in this class.
1, review the import method. By reviewing the volume calculation formula and derivation process of cuboid, cube and cylinder, students can learn new knowledge and communicate the relationship between old and new knowledge.
2. Situational teaching method. By asking students to guess the relationship between cylinder volume and cone volume, we can induce students to verify their guesses, integrate knowledge with interest, stimulate interest with emotion and promote knowledge.
3. Heuristic analysis. Through the analysis and comparison of the results of the three experiments, students' problem consciousness is cultivated, their thinking is inspired and their intelligence is developed.
The learning method of independent inquiry runs through the whole process of teaching materials. Proper use of multimedia teaching methods can enhance the novelty of teaching, thus stimulating the enthusiasm of students to participate in learning, making them show their individuality in the learning state of seeking knowledge and experience the fun of learning and using mathematics.
Third, theoretical study.
Teaching and learning are inseparable, and teaching is to learn better. Teaching method is the navigation of learning method, and learning method is the epitome of teaching method. Tao Xingzhi, a famous educator, pointed out that a good teacher is not teaching, not teaching students, but teaching students to learn. In view of this understanding, while emphasizing teaching methods, we should pay more attention to the guidance of learning methods. In the learning process of this lesson, I mainly guide students to learn the following learning methods:
1, the method of transformation and migration. By reviewing the derivation process of cylindrical volume, students can learn to discover and grasp the internal relationship between knowledge, and promote the formation of cognitive level and the internalization of new knowledge.
2. Comparative analysis methods. Through the comparative analysis of the results of three experiments, students' horizons can be broadened, knowledge confusion can be prevented, and students' ability to analyze and solve problems can be improved.
3. Cooperative inquiry method. Through the interaction of students in group experiments, we can establish group consciousness and promote common improvement.
Fourth, talk about procedures.
The new curriculum regards the teaching process as a process of communication, positive interaction and common development between teachers and students. According to the new curriculum concept and
(A) create a situation, causing problems
Show me cuboids, cubes, cylinders and cones and ask:
1. What methods have we learned to calculate the volume of an object? What is their calculation formula?
2. How is the volume calculation method of cylinder derived? In this lesson, we will learn the volume of a cone. (blackboard writing: the volume of a cone)
3. Which method of calculating the volume of an object do you think is related to the cone? Why?
4. Guess what is the relationship between cylinder volume and cone volume? (blackboard writing: V cylinder =3v cone? Guess)
By creating a scene where the volume of a cone is more closely related to whose volume, this link naturally leads to a new lesson, attracts students' attention, stimulates students' enthusiasm for exploring knowledge, and lays a good foundation for the study of the new lesson. )
5. How to verify your guess? (blackboard writing: verification)
(2) Cooperative inquiry and problem solving.
Exploration is the lifeline of mathematics. Advocating inquiry learning and guiding students to experience the process of knowledge formation is the concept of current primary school mathematics reform. Understanding the formula for calculating the volume of a cone is the focus of this lesson. I designed the following links for students to discover the volume of a cone through group cooperation, independent exploration and hands-on operation.
1, show me the experimental record sheet.
Number of experiments
Comparing cylinders and cones, we find that
Experimental results: the relationship between their volumes
first time
second time
the third time
2. Teachers guide students to understand the experimental list, do experiments according to the experimental record list, and teachers visit and guide them.
3. Ask students to introduce the experimental process and results. (remove? )
4. Q: Why are the results different after three experiments?
5. What is the relationship between the volume of a cylinder with equal bottom and equal height and the volume of a cone? (blackboard writing: V cone =v column =sh)
6. In this formula, what do S and H stand for respectively? What did Sh get? Why take it?
7. What are the conditions for finding the volume of a cone?
Teacher's summary: V-cone =sh is obtained through guessing and experimental verification.
Through this design, students can experience the process of knowledge formation, constantly improve and optimize their knowledge structure through exchanges and comparisons with peers, and highlight key points and break through difficulties through independent inquiry and cooperative exchanges. )
(c) Migration and application, and improvement at different levels.
Practice is an important link in mastering knowledge, forming skills and developing intelligence. According to students' age characteristics and cognitive rules, from easy to difficult, from shallow to deep, I try my best to reflect the vertical and horizontal connection of knowledge. I have designed the following groups of exercises, please see:
1, try to answer.
Show three sets of data and ask students to choose one set of answers.
The bottom radius is 4 cm and the height is 6 cm.
The bottom surface is 4 cm in diameter and 5 cm in height.
The bottom circumference is 25. 12cm, and the height is 4cm.
After answering, please ask a classmate to write on the blackboard.
Q: Why did you choose the bottom radius and height?
Summary: To find the volume of the cone, first find the bottom area of the cone, and then find the volume of the cone according to the formula.
2. Example 1: (The courseware shows the scene of the textbook) On the threshing floor, there is a conical wheat pile. The radius of the bottom is 2m and the height is 1. 5 meters. Can you calculate the volume of the wheat pile?
(Students communicate with the whole class in independent column calculation)
judge
(1) The volume of a cone is equal to the volume of a cylinder.
(2) The volume of a cylinder is larger than that of a cone with equal bottom and equal height.
(3) The height of the cone is three times that of the cylinder, and the volume of the cone is equal to the volume of the cylinder.
Step 4 fill in the blanks
(1) The volume of a cylinder is 6 cubic meters, and the volume of a cone with equal bottom and equal height is ().
(2) The bottom radius and height of cylinder and cone are the same, the volume of cone is 18 cubic meter, and the volume of cylinder is ().
(The design of this link, 1 and 2 questions mainly highlight the focus of this lesson. You can use the volume formula to calculate the volume of the cone and solve some practical problems; The third and fourth questions are to break through the difficulties in this lesson and understand the multiple relationship between cylindrical volume and conical volume under the conditions of equal bottom and equal height. The design of these exercises has played a role in consolidating and improving. Reflect that mathematics comes from life and is applied to life. )
(4) Summarize the evaluation and encourage development.
Classroom summary is to summarize and summarize the knowledge learned in this class and evaluate the students' learning situation, so I designed the following questions:
1. What have you learned from these classes?
2. Do you have any new ideas? Is there a problem?
(This will not only help students to consolidate new knowledge, improve their knowledge structure and improve their ability to organize knowledge, but also enable them to experience the pleasure of exploring success and establish confidence in learning mathematics well. )
Five, say blackboard writing design
Cone volume
Guess V-shaped cylinder with equal base and equal height =3v cone
↓
confirm
V cone =v cylinder /3=sh/3
The design of blackboard writing strives to embody knowledge and conciseness, so that students can see at a glance and play the role of making the finishing point.
The above is just my overall thinking and teaching presupposition for this course. In the actual teaching process, I will attach great importance to the generation of classroom resources, constantly reflect on the classroom, adjust the teaching process in time, and achieve the best teaching effect.
"Cone Volume" Lecture Note 3 The content of the class I am talking about is "Cone Volume", Unit 2, Section 2, Grade 6 Primary School Mathematics (PEP). This lesson is a method to calculate the volume of a cone by comparing a cylinder with a cone after learning the first lesson "Understanding of a cone". Next, I will elaborate from five aspects: teaching materials, teaching methods, learning methods, teaching models and the cultivation of three books.
First of all, talk about textbooks.
Mathematics curriculum standards emphasize that starting from students' existing life experience, students can experience the process of abstracting practical problems into mathematical models and explaining and applying them, so that students can gain an understanding of mathematics and further develop their thinking ability, emotional attitude and so on. "The volume of a cone" is taught on the basis of learning the circumference and area of a circle, the volume calculation of cuboids, cubes and cylinders, and a preliminary understanding of the characteristics of a cone. Is the focus of this unit. Through the teaching of this course, we can cultivate students' operational ability and practical ability, cultivate innovative spirit, and lay a good foundation for students' in-depth study and independent development in the future. The sixth grade is the last school year of primary school. Students have a certain foundation in mathematics knowledge and a certain development in logical thinking ability. The students' abilities of acceptance, problem analysis and language expression have been obviously improved, which provides a powerful condition for understanding the knowledge of this lesson. However, there are some obstacles in the teaching of this course because of the great personality differences among students.
According to the requirements of curriculum standards, the arrangement characteristics of teaching materials and the actual situation of students, I have determined the teaching objectives as follows:
1. Emotional goal: to cultivate students' spirit of exploration and cooperation.
2. Knowledge goal: Understand the derivation process of the cone volume formula, master the cone volume formula, and use the formula to solve problems in life.
3. Ability goal: to cultivate students' spatial imagination, cooperative communication ability, innovative thinking and hands-on operation ability.
Key points: understand the derivation process of cone volume formula and master the calculation formula of cone volume.
Difficulty: the derivation process of cone volume calculation formula.
Key: In the process of formula derivation, the cylinder and the cone must have equal bases and equal heights, so there is an inevitable relationship between them.
Second, oral teaching methods
In order to make students learn mathematics in situations and experience mathematics in activities, I design teaching methods according to the characteristics of this class and the cognitive rules of primary school students, and adopt the following teaching methods: focusing on lectures, experiments and observations, supplemented by discussion and practice, to achieve teaching objectives. In teaching, we should not only give full play to students' main role, but also mobilize students to actively participate in the whole process of teaching.
This lesson introduces multimedia demonstration into the classroom, giving students a vivid, vivid and intuitive understanding and revealing the inherent law of knowledge clearly and enlighteningly. In addition, students' practical operation, teachers' guidance and questions make the teaching process organically combined, which fully demonstrates the advantages of audio-visual teaching and makes it easier to optimize the teaching process than other teaching means and methods.
Third, theoretical study.
Teaching methods and learning methods are interrelated. "Teaching" is to "learn" better. The main role of students is fully reflected in teaching. Students should try their best to practice, think and speak. Teachers should inspire and guide students to think and discover from different angles. Create a certain problem situation, so that students can observe, discuss, experiment, understand and summarize the whole learning process around the problem.
The ancients said: "Give people fish, just one meal; And people's catches will be inexhaustible for life. " The new curriculum requires students not only to "learn" but also to "learn". This lesson adopts teaching activities suitable for students to observe, guess, operate, compare, communicate, discuss and summarize. In order to better guide the learning methods, I organize teaching in the form of group cooperation. In this way, on the one hand, students can discover, experience and create new knowledge, on the other hand, they can also enhance their sense of cooperation, and generate can think creatively in activities.
Fourth, talk about the teaching mode.
This lesson adopts the situation of primary school mathematics-inquiry teaching mode.
(A), create a situation to reveal the problem
The so-called creation of situations means that teachers should create an inquiry atmosphere that can mobilize students' previous experience and promote students' thinking participation before class. In this lesson, I created two kinds of ice cream, how to buy more economical scenes. The purpose of doing this is not only to stimulate interest, but also to enable students to gradually form a mathematical vision and actively seek mathematical solutions when facing practical problems.
(2) Explore and discover, and build a model.
This is an important step for students to build new knowledge. Through observation, practice, exploration, thinking, communication and other activities, students should be helped to explain the basic strategies for solving problems and establish basic mathematical models.
1, intuitive introduction, intuitive guess
In teaching, I first ask students to recall the calculation of the volume of objects they have learned before, and then guess which object this cone may be related to. Then guess what kind of relationship they have. The purpose of this link is to enable students to link existing knowledge information with new knowledge, and lay a foundation for students to adjust their cognitive structure and build new knowledge.
2, experimental exploration, found the law.
This link is cooperative learning, which guides students to do experiments in groups and concludes that the volume of a cone is one-third of that of a cylinder under the condition of equal bottom and equal height. Finally, according to the calculation method of cylinder volume, students are guided to try to summarize the calculation formula of cone volume. In this way, students have personally experienced the process of knowledge formation, thus improving their thinking ability, hands-on operation ability, summing-up ability and cooperation consciousness.
3. Enlighten and guide, and deduce the formula.
In this session, first, let the students deduce the calculation method of cone volume according to the calculation method of cylinder volume, and then guide the students to say, what does sh mean? Why take the third one? This will give students a deeper understanding. Throughout this process, I have always been guided by the important idea of guiding students to actively construct knowledge, guiding students to truly master what they have learned and develop their mathematical ability through independent exploration, cooperation and communication, and solving problems, so as to truly achieve "hands-on operation and experience success".
(3) Understand the application and strengthen the experience.
Because the mathematical knowledge created by students in the process of exploring, discovering and establishing models and the mathematical methods discovered must have an internalization process, I designed four levels of exercises to pay attention to each child.
Basic exercises
First of all, solve the problem in the situation, which ice cream is more economical to buy. Then calculate the volumes of cone-shaped ice unicorn and cylindrical ice cream. When calculating the volume of cone ice cream, students are allowed to finish it selectively, so that the number and difficulty of students are open, which not only pays attention to students with learning difficulties, but also promotes the development of top students and special students.
Variant exercise
This is a set of judgment questions.
Application exercise
Let students solve problems in life. It can help students deepen their understanding of what they have learned and cultivate their ability to solve life problems.
Comprehensive exercise
Machining the cylinder into the largest conical part. Find the trimmed volume.
This is a question of thinking expansion. First, guide students to think independently, then solve problems, and finally draw a conclusion. This not only pays attention to the structure of new knowledge, but also enables students to further expand and extend their knowledge.
In this way, students can fully understand, deepen their experience and further strengthen the newly established mathematical knowledge in the application. Let everyone learn valuable mathematics and let different people get different development in mathematics.
(4) Summarize and improve experience.
This link is mainly to guide students to systematically summarize the knowledge of this lesson, and also to sort out the process, methods and experiences of exploration and discovery.
After this class, I gave the students a practical assignment, making a cone and cylinder out of cardboard. The requirement is that the volume of cone and cylinder is equal.
Operation practice is a process of using hands and brains, and it is an effective means to cultivate skills and promote the development of thinking. It is also an extended learning activity, which allows students to continuously acquire knowledge and improve their learning skills; Cultivate students' thirst for knowledge; Consolidate the knowledge learned, expand the field of knowledge and produce knowledge transfer; Cultivate students' sense of cooperation; Let students understand that there is no time limit and space limit in learning, so as to cultivate students' good study habits.
Five, said sansheng training
In the whole teaching process, I try to take care of all students' learning feelings and teach them in accordance with their aptitude. Students with learning difficulties learn the most basic content. On the basis of meeting the requirements of curriculum standards, top students should appropriately expand their knowledge and expand their thinking. In teaching, simple questions are reserved for students with learning difficulties and difficult questions are reserved for top students. There are strong and weak links in experimental operation. Finally, hierarchical exercise, basic exercise and variant exercise mainly focus on students with learning difficulties and promote the development of top students. Practical exercises and thinking development are mainly aimed at top students and special students. Let different students get different development in this class.
In short, in this class, teaching materials are the main source, teachers are the dominant, students are the theme, training is the main line, and thinking is the core. With the development of each child as the purpose, students can learn mathematics in situations and experience mathematics in activities. This not only emphasizes the formation process of knowledge, but also emphasizes the development process of students' thinking, so that each child can improve his ability and develop his thinking in the process of acquiring new knowledge.
The requirement of this teaching competition is isomorphism of the same topic, with the aim of improving together. Three math teachers in our six-year group really cooperated, choosing courses, preparing lessons, making courseware, writing lesson plan design and lecture materials later. Although we have made careful preparations, there are still many regrets in teaching.
1, the application of multimedia courseware is not perfect.
2. In the training of "Three Books", the attention to poor students is not in place.
There is waste in the classroom, resulting in insufficient teaching time.
4. In group cooperation, students' participation needs to be improved.
In the future work, we must attend more lectures, study more, study more, summarize more and reflect more, so that every point in the next 40 minutes of math class will be effective.