Recently, I saw a wonderful teaching record fragment of "the volume of a cylinder" in "Primary School Teaching Design", which explained the idea advocated by the new curriculum standard from a brand-new angle and left a deep impression on me. Now take it down and reward it with your peers.
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Teacher: There are big cylinders and small cylinders. How do you think the volume of a cylinder should be calculated?
Student: (Most students raise their hands) The bottom area times the height.
Teacher: Then how do you understand this calculation method?
Health 1: I read it in a book.
More than half of the raised hands are put down. Obviously, most students saw or heard this conclusion and didn't understand the real meaning. However, some students, hands in the air, are eager to try, and the expression on their faces tells the teacher that they have a better answer. The teacher pushed the boat with the current and let them speak. )
Health 2: I think so: Cubes, cubes and cylinders are all three-dimensional figures, and volume refers to the size of the space they occupy. The volume of cuboids and cubes can be calculated by multiplying the height by the bottom area, so I think it should be possible to multiply the height by the bottom area when calculating the volume of cylinders!
Teacher: You can quickly connect the cylinder with the cuboid and cube you have learned before, and then connect it with the volume calculation method of the cylinder. Great! Of course, this is just your guess, if only it could be proved again.
S3: I can prove it. When we derived the cuboid volume formula, we used the unit of volume pendulum method. Now we can calculate the cylinder volume by the number of layers (bottom area) × the number of layers (height). According to this idea, we can also put an appropriate unit of volume inside the cylinder, using the number of layers × the number of layers. The number of layers is also its bottom area, and the number of layers of the pendulum is also high. Doesn't that prove that the formula for calculating the volume of a cylinder is the height multiplied by the bottom area?
There was warm applause in the classroom immediately, and many students were impressed by his wonderful speech, and rational thinking exuded attractive charm. )
Teacher: You are so clever that you can solve today's problems with what you have learned before! (At this time, more hands are raised. )
Health 4: I have an idea, whether it is feasible. When deducing the calculation method of circular area, we converted a circle into a rectangle, and the bottom of the cylinder was a circle, so I thought, can we convert the cylinder into a cuboid?
Teacher: (thumbs up) Your idea is very interesting! You can try it later and think about how to divide a cylinder into approximate cuboids.
Health 5: I have another idea: we can imagine a cylinder as a superposition of countless disks of the same size. Then the volume of the cylinder should be the area of each disk × the number of turns. The number of circles is equivalent to the height of a cylinder. So I think the volume of a cylinder can be multiplied by the area (bottom area) of each circle.
Teacher: What a great idea! The teacher can't help clapping his hands. )
Health 6: I have seen my parents "stick chopsticks". Ten identical chopsticks in Shuang Yi are tied together to form an approximate cylinder. We can regard each chopstick as a cuboid, so the volume of the approximate cylinder should be the sum of the volumes of these twenty small cuboids. Because they have the same height, using the law of multiplication and distribution, it becomes the sum of the bottom areas of these twenty small cuboids × height.
Teacher: You really think!
Health 7: I have another idea: when studying the area of a circle, we know that when the radius of the circle is equal to the side length of a square, the area of the circle is about 3. 14 times that of the square. Divide the countless circles stacked into this cylinder in this way, then isn't the volume of the cylinder about 3. 14 times that of this cuboid? The volume of a cuboid is multiplied by its bottom area × height, and the volume of a cylinder is multiplied by 3. 14, that is, the bottom area × height of the cylinder is used.
Health 8: Knead cylindrical plasticine into cuboid plasticine with the same height. The volume of a cuboid is calculated by multiplying the bottom area by the height, so the volume of a cylinder is also calculated by multiplying the bottom area by the height!
Teacher: I didn't expect a plasticine to have such a function. You are really not simple!
……
Throughout the class, the children and teachers did not give warm applause.
In the past, mathematics classroom teaching was loyal to the subject, but it turned its back on students, embodied rights, but forgot democracy, pursued efficiency, but forgot significance. This clip reflects that under the new curriculum standard concept, it is no longer a monologue of teachers' wishful thinking, but a sincere dialogue among students, mathematics materials and teachers.
Now let's enjoy this wonderful clip from the perspective of "dialogue".
First, "dialogue" calls for enthusiasm for learning.
The New Curriculum Standard points out that meaningful mathematics learning must be based on students' subjective wishes and knowledge and experience. In such an atmosphere, students can think positively. In today's digital and information-based society, students have many ways to receive information and acquire knowledge. The method of calculating the volume of a cylinder is no stranger to students. If teachers follow the traditional teaching procedure (creating situations-studying and discussing-drawing conclusions), students will easily have a wrong understanding of this part of knowledge and lose enthusiasm for the learning process. At the beginning of this class, the teacher asked "How to calculate the volume of a cylinder", so that students can present the existing knowledge conclusions first, and then pass "How do you understand this formula?" Guide students to pay attention to the understanding of the meaning of the formula, and students actively engage in thinking activities to arouse their enthusiasm for learning.
Second, "dialogue" produces a spark of wisdom.
"Water has no brilliance and swings; There is no fire in the stone, and the impact begins with light. " The activation of thinking and the eruption of spirituality stem from the enlightenment and collision of dialogue. If this lesson is designed according to the textbook: by transforming a cylinder into a cuboid, study the relationship between the cylinder and the cuboid, and get the calculation formula: bottom area × height. After this learning process, students' thinking is the same and their development is limited. The teacher unfolded the textbook accordingly, presented the formula first, and then asked, "How to understand this formula?" Let students' thinking "explode" along their own unique understanding.
Third, "dialogue" wins the opening and communication of the mind.
"That's great! Of course, this is just your guess. If only it could be proved again. " "You are so smart! You can use the knowledge you have learned before to solve today's problems! " "You this idea is very interesting! You can try it later and think about how to divide a cylinder into approximate cuboids. " ..... Teachers constantly affirm every point of view of students and ignite every spark that students find; At the same time, like a program host, be calm and sincere, listen to and accept students' voices. In class, the students were really fascinated. It was strange that they said one method after another, and even the teacher couldn't help applauding. In this situation, it is not difficult to see that teachers can pay attention to crouching down to communicate with students, pay attention to seeking students' voices, let students open their hearts and let go of their thoughts in a "zero distance" and positive psychological state, and have a real dialogue with teachers and students in a way of "horizon integration" to win the opening and communication of their hearts.
Mathematics teaching is carried out in dialogue, which embodies democracy and equality and highlights creation and generation. In effective dialogue, there is not only the transmission of information, but also the sublimation of thinking; It can not only enhance students' understanding, but also promote teachers' reflection; There is not only the joy of inheritance, but also the passion of creation. There are many wonderful things in this teaching clip that deserve our appreciation and admiration. I want to say: I am very encouraged in my heart. I will learn from this teacher and make my class a wonderful moment!
The teaching content of the second lesson of the volume of a cylinder: P 19-20 pages of example 5, example 6 and supplementary examples, and complete the "doing" and exercise 3No. 1 ~ 4.
Teaching objectives:
1. With the help of cuboid volume formula, the volume formula of cylinder is derived by cutting and assembling, and the volume and volume of cylinder can be calculated correctly by using this formula.
2. Initially learn the ability to solve practical problems with transformed mathematical ideas and methods.
Infiltrate and transform ideas and cultivate students' consciousness of independent exploration.
Teaching emphasis: master the calculation formula of cylinder volume.
Teaching difficulty: derivation of the formula for calculating the volume of a cylinder.
Teaching process:
First, review.
What is the volume formula of 1. cuboid? (cuboid volume = length× width× height, and the unified formula of cuboid and cube volume is "bottom area× height", that is, cuboid volume = bottom area× height)
2. Take out a cylindrical object and ask the students to point out what the bottom, height, side and surface of the cylinder are and how to find it.
3. Review the derivation process of the formula for calculating the area of a circle: cut the circle into approximate rectangles, find out the relationship between the circle and the rectangle, and then derive the formula for calculating the area of the circle by using the formula for calculating the area of the rectangle.
Second, the new lesson
1. Derivation of the formula for calculating cylinder volume.
(1) Find the area of a circle by converting it into a rectangle, and then deduce the volume of a cylinder. (Cut the cylinder along the sector at the bottom of the cylinder and the height of the cylinder, and you can get 16 blocks of the same size, and put them together to form a three-dimensional figure similar to a cuboid.
The volume of a cylinder teaching plan 3 teaching content:
P 19-20 Example 5, Example 6 and supplementary examples, and do the problem and exercise 3 No.65438 +0 ~ 4.
Teaching objectives:
1. With the help of cuboid volume formula, the volume formula of cylinder is derived by cutting and assembling, and the volume and volume of cylinder can be calculated correctly by using this formula.
2. Initially learn the ability to solve practical problems with transformed mathematical ideas and methods.
3. Infiltrate and transform ideas to cultivate students' awareness of independent exploration.
Teaching focus:
Master the calculation formula of cylinder volume.
Teaching difficulties:
Derivation of formula for calculating cylinder volume.
Teaching process:
First, review.
What is the volume formula of 1. cuboid? What about the cube? (cuboid volume = length, width and height, and the unified formula of cuboid and cube volume has high bottom area, that is, cuboid volume = high bottom area)
2. Take out a cylindrical object and ask the students to point out what the bottom, height, side and surface of the cylinder are and how to find it. (Delete)
3. Review the derivation process of the formula for calculating the area of a circle: cut the circle into approximate rectangles, find out the relationship between the circle and the rectangle, and then derive the formula for calculating the area of the circle by using the formula for calculating the area of the rectangle.
Teacher's summary: The derivation of the area formula of a circle is to transform a surface figure into a rectangle that we have learned before. Today, we will also use the idea of transformation to derive the volume formula of the cylinder. Guess what shape it will become?
Second, the new lesson
1. Derivation of the formula for calculating cylinder volume.
(1) Find the area of a circle by converting it into a rectangle, and then deduce the volume of a cylinder. (Cut the cylinder along the fan at the bottom of the cylinder and the height of the cylinder, and you can get 16 blocks of the same size, which together is a three-dimensional graphic courseware demonstration of an approximate cuboid. )
(2) Because our division is not fine enough, it doesn't look like a cuboid; If it is divided into more sectors, the three-dimensional figure will be closer to a cuboid. (The courseware demonstrates subdividing a cylinder into cuboids. )
Play this process repeatedly, guide students to observe and think, and discuss: in the process of change, which ones have changed and which ones have not?
What is the relationship between the bottom area and volume of a cuboid and a cylinder?
Students tell the demonstration process and summarize the formula.
(3) Through observation, make students clear that the bottom area of a cuboid is equal to the bottom area of a cylinder, and the height of a cuboid is the height of the cylinder. (cuboid volume = high base area, so cylinder volume = high base area, v = sh)
"The Volume of a Cylinder" lesson plan 4 "Mathematics Curriculum Standard" points out that "mathematics teaching should let students experience the formation process of knowledge, learn to observe and analyze the real society by using mathematical thinking mode, solve problems in daily life and subject study, and increase the awareness of applied mathematics". The new curriculum standard not only pays attention to let students grasp the conclusion in learning, but also pays attention to individual experience. Let students experience in activities and apply in practice, that is, let students actively participate, communicate in practice, explore in cooperation and experience the process of knowledge formation. By constantly discovering, asking, analyzing and solving problems, we can accumulate life experience, cultivate mathematics application ability, experience the fun of mathematics and feel the application value of mathematics in life.
The volume of cylinder is learned on the basis of students' preliminary understanding of volume and its meaning and mastering the calculation method of cuboid and cube volume. This section includes the derivation of cylinder volume calculation formula, which can be used to solve practical problems in life.
The teaching situation is as follows:
One: situational introduction, perceptual knowledge
Teacher: (takes out plasticine) Do you know its volume? How did you know? Tell us about it
Health: Knead into a cuboid or cube, measure the length, width and height, and then use the formula: length × width × height to calculate the volume.
Teacher: Can you make other figures that we have learned? (Student's operation: Knead into a cylinder)
Teacher: Can you calculate its volume now? Guess what I should do? (Student's operation: Knead the cylinder into a cuboid)
Teacher: What did you find?
Health: the shape has changed, but the volume has not changed.
Teacher: We have learned what graphics can be converted into and how to find the area.
Health: The circle is cut into an approximate rectangle.
Teacher: The volume of cylindrical plasticine will be obtained. What should I do if I find the volume of water in a cylindrical container?
Health: Pour water into a cuboid container, and then measure and calculate.
Teacher: What is the volume of cylindrical iron?
Health: Immerse in the water and find out the amount of water discharged.
Teacher: What is the volume of the column at the entrance of the shopping mall? The students looked at each other and were at a loss.
Second, independent inquiry, migration and transformation
1, guide
Teacher: Some students convert the cylinder into the three-dimensional figure we have learned and calculate its volume.
Let the students discuss with each other how to reform, and then organize the class to report.
Health: Divide the bottom of the cylinder into many equal sectors, then cut the cylinder and put it together, and it becomes an approximate cuboid.
Step 2 operate
Students take out carrots (cylindrical molds) and knives prepared in advance and ask them to cut everything and put it together.
3. Impression: On-site demonstration of cylindrical die (cutting).
(1) Let a student take the cut half and fork it;
(2) another student pieced together the cut half;
③ What shape can be observed? What did you find at the same time?
Explore, discuss and summarize in groups of four.
Panel report:
Health: cuboids and cylinders remain unchanged: volume, bottom area, height, etc. Lateral area, surface area and bottom circumference have all changed.
4, courseware demonstration, let students understand: the more sectors are divided, the closer the three-dimensional figure is to the cuboid.
5. Discussion: What is the connection between cylinder and assembled approximate cuboid? What did you find?
6. Report:
Cylinder → approximate cuboid
① Equal volume, equal bottom area, equal height and unequal surface area,
According to the students' answers, the blackboard is as follows:
Volume of cuboid = bottom area × height
↓ ↓ ↓
Volume of cylinder = bottom area × height
Guide the students to express the calculation formula with letters: V=Sh.
Teacher: What conditions must be known to calculate the volume of a cylinder with this formula?
Health: bottom area and height.
Teacher: If you are given the diameter (radius or circumference) and height of a cylinder, how do you find the volume of the cylinder?
Health: According to the formula, calculate the radius first, then the bottom area …
Teaching reflection:
Make full use of what students have learned to pave the way for teaching, and guide students to turn cylinders into three-dimensional figures by transfer method. Then through observation, practice and comparison, find out the relationship between the two numbers, and deduce the calculation formula of cylinder volume. The intuitive and effective teaching process does not need complicated explanations from teachers, and the sparks of students' thinking naturally erupt in the learning space of independent exploration and interactive discussion. The teaching content, key points and difficulties are not only implemented and solved, but more importantly, the improvement of students' comprehensive ability.
In practical teaching, only by constantly inducing students' desire for active thinking, creating unrestrained thinking space and letting students experience the process of knowledge discovery, exploration and creation can teachers cultivate students' innovative ability more effectively and let students discover the concept of "from life to life" in mathematics learning.