Let the equation of the circle be: x 2+y 2 = r 2 (x, y is the coordinate of the circle in the plane rectangular coordinate system, and r is the radius. )
Take a quarter circle of the first quadrant and integrate * 4 = r 2 to get the area of 1/4 circle.
Teaching content: Nine-year compulsory education, Six-year primary school mathematics Volume 11, pages 1 15 to 1 16.
Teaching purpose:
1. Through the operation, students are guided to deduce the calculation formula of circular area, and some simple practical problems can be solved by using the formula.
2. Stimulate students' interest in participating in the whole classroom teaching activities, cultivate students' ability of analysis, observation and generalization, and develop students' concept of space.
3. Mathematical thought and limit thought of infiltration transformation.
Teaching emphasis: derivation of circle area formula.
The key to teaching is to find out the relationship between the circle and the transformed approximate figure.
Teaching aids: multimedia computers, slides.
Learning tools: 16 and 32 circles, scissors, scales, a round piece of paper.
Teaching process:
First, introduce questions.
1. Inspire students to recall the derivation process of parallelogram, triangle and trapezoid area calculation formulas. (Microcomputer demonstration)
The microcomputer displays a circle and then colors it red. Q: What is this number? What do you think of when you see a circle? What is the size of the plane part surrounded by a circle? (Circular area) Displays the theme. How to calculate the area of a circle? Please think about it.
[Comment: By recalling old knowledge, students can be interested in exploring new knowledge from old knowledge and decide their own ideological direction, which is conducive to the cultivation of students' imagination. ]
Second, the new curriculum teaching
1. Guess the size of the circle area by measurement.
Use a small square transparent plastic sheet with a side length equal to the radius to directly measure the circular area.
(as shown in the figure) after observation, it is concluded that the circular area is less than four small squares, which seems to be less than three.
A small square is bigger. Preliminary guess: The area of the circle is more than three times that of r2.
It can be seen that it is impossible to get the exact area of a circle by measurement. When we learn to derive the area formula of geometric figures, we always turn new figures into familiar ones by splitting and splicing. Can we still deduce the calculation formula of circular area in this way today?
[Comment: This exploratory question makes students have suspense and introduces deep thinking. It echoes and blends with the verification after obtaining the calculation formula of circular area. Students can get a very vivid representation of the relationship between the area of a circle and the multiple of r2, which is helpful to avoid confusion with the calculation formula of the circumference of a circle (C=2πr). ]
2. Student operation.
(1) The students cut out circles with equal parts of 16 and 32 respectively and put them together to form two approximate rectangles. (microcomputer display) The teacher asked:
(1) Is the mosaic rectangular? It is an approximate rectangle because its upper and lower sides are not line segments. )
What is the relationship between a circle and an approximate rectangle? (The shape has changed, but the area is equal)
③ What is the difference between the numbers after bisecting circle 16 and 32? (The figure after 32 equal parts is closer to rectangle)
If you divide a circle into 64 parts, what will happen if 128 parts ... form a rectangle? The more copies of a circle are equally divided, the closer the figure is to a rectangle. )
④ What part of the circle is the approximate rectangle? How to express it in letters? (Half of the circumference, C/2=πr), which part of the circle is its width? (radius r)
⑤ Can you deduce the formula for calculating the area of a circle?
[Comment: Instruct students to do it themselves, cut a circle into an approximate rectangle through microcomputer demonstration, and derive the formula for calculating the area of a circle from the formula of rectangular area. In this way, students' initial spatial imagination can be cultivated, and the dialectical materialism viewpoint of direct substitution can also be infiltrated. ]
(2) Divide the circle 16 into an approximate parallelogram. The base of the parallelogram is equivalent to a quarter of the circumference of the circle (C/4=πr/2), which is higher than twice the radius of the circle (2r), so S = π r/2.2r = π r2 (see figure 1).
(3) The circle 16 can be spliced into an approximate isosceles triangle. The base of a triangle
It is equivalent to 1/4 of the circumference and 4 times of the radius, so S = 1/2.2π r/4r = π r2.
(See Figure 2).
(4) After dividing the circle, it can be spliced into an approximate isosceles trapezoid. The sum of the upper bottom and the lower bottom of the trapezoid is half of the circumference of the circle, which is more than twice the radius of the circle, so S = 1/2 π r 2r = π r2 (see Figure 3).
3. Summary: No matter what approximate figure we put the circles together, we can deduce the formula of circle area S=πr2, which verifies the correctness of the original guess. Explain that when calculating the area of a circle, you must know the radius.
4. Compare the calculation formulas of the length and area of the circle, find out the connection and difference, and strengthen the memory. Both formulas are related to π, but the circumference of a circle is equal to π times the diameter length, the area of a circle is equal to π times the area of a square with a radius as its side length, that is, π times r2, and so on.
5. Self-study example 1. Pay attention to the order of writing and operation.
[Comment: Guide students to transform the circular area into approximate rectangle, isosceles triangle and isosceles trapezoid through many different experiments and adopt the transformation method, thus deriving the calculation formula of the circular area. At the same time, computer demonstration is used to turn static into dynamic and virtual into reality, which helps students to concretize the abstract content and further deepen their understanding of the derivation process of circular area formula.
Third, reading questions
Fourth, consolidate practice.
1. Look at the picture and calculate the area of the circle.
2. Find the area of the circle according to the following conditions.
R=6 cm d =0.8 cm r= 1.5 decimeter.
3. The radius of a circular iron plate is 3 decimeters, and how many square decimeters is its area?
4. What relevant data is needed to measure the area of a circular paper? Let's see who finishes first and who has more ideas.
(1) Measure the radius of the circle and calculate the area according to S=πr2.
(2) The diameter of a circle can be measured and the area can be calculated according to S=π(d/2)2.
(3) The circumference of a circle can be measured, and the area can be calculated according to S = π (c/2π) 2.
[General Comment: This lesson has two characteristics:
First, always put students in the main position of learning and purposefully cultivate students' ability to acquire knowledge.
Learning is an internal activity of students. Therefore, in classroom teaching, we should not only pay attention to the learning results, but also pay attention to the learning process to cultivate students' ability to explore and acquire knowledge. In the teaching of this class, we should firmly grasp the key point of "the derivation of the formula of circular area" and dare to let students do it themselves and make induction and reasoning. Through students' different cutting and splicing, using the methods of hypothesis, transformation and imagination, the circular area is transformed into other plane figures through equal product deformation, and the calculation method of circular area is gradually summarized. Such multi-level operation and multi-angle thinking not only communicate the connection between old and new knowledge, but also stimulate students' curiosity to the maximum extent. Students are full of interest in learning, and the classroom atmosphere is very active, so that students not only know what it is, but also know why.
(B) the use of modern teaching methods to assist classroom teaching has improved teaching efficiency.
Computer-aided classroom teaching has its intuitive, vivid and vivid characteristics. It can make the static picture dynamic and abstract content visible, and it is not limited by time and space. In this class, microcomputer demonstration is properly used, which fully arouses students' interest in learning and improves classroom teaching efficiency, which is incomparable to other teaching methods. ]
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