First, the area formula derivation teaching of rectangle and square, counting squares can strengthen students' understanding of the area and realize that the area is the result of tile measurement of area units.
In the teaching of deducing the formula of rectangular area, a rectangle of 5 cm×3 cm is given first, so that students can estimate the area, and then students are guided to put it on a square piece of paper with a side length of 1 cm (area unit). How many units of area can be put in this rectangle? How many areas are there? So a rectangle with an area of 1 cm2 is presented (as shown on the right). Students can get it by counting squares (area units): the long side of the rectangle has 5 area units, the wide side has 3 area units, and the total number of area units is 5×3= 15 (units). Then let the students use 12 small squares with an area of 1 cm2 to spell out different rectangles and draw a schematic diagram (as shown below).
Then observe and count the number of long-side pendulums and wide-side pendulums, and find that the area of a rectangle = the number of area units placed on the long side (that is, the number of area units placed on the wide side) × the number of area units placed on the wide side (that is, the number of rows), and find that the number of area units per row is exactly the number of long scales of the rectangle.
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The number of rows is exactly the number of wide scales, and the rectangular area = number of long scales × number of wide scales = length × width. In the teaching process of deducing the formula of rectangular area, the author transforms the area into a square, so that students can understand that the calculation of area is to calculate the number of area units, and the process of counting squares is the process of students actively exploring and discovering the relationship between length, width and area units.
Second, in the teaching of the derivation of parallelogram area formula, let students feel the idea of transformation in the process of counting squares.
In the teaching of the derivation of parallelogram area formula, the teaching bottleneck and students' confusion are: why should parallelogram be transformed into rectangle? This is also the difference between the derivation of parallelogram area formula and rectangular area formula. The teaching material is to count the area of a parallelogram and a rectangle (with equal base length and equal height and width) drawn with squares (and note: a square represents 1 cm2, and less than a square is half a square) to experience the base length, equal height and width, and equal area. Experience parallelogram can be transformed into rectangle with the same area by cutting and splicing, and the area can be calculated. However, the author believes that this number does not really make students realize the idea of transformation, and in order to let students calculate the area, the textbook also specifically indicates that "a square represents 1 cm2, and less than a square is half a square", which obviously cannot solve the students' confusion and the bottleneck of teaching, and does not really play the role of calculating the square. In the author's view, the process of counting squares is to make students realize that in the process of counting squares, the area of a figure that cannot be directly measured by standard area units can be accurately obtained by "cutting and spelling", and its method is "transformation". In order to achieve this goal, we can expand it like this.
Link 1: Introduction of estimated area. In the lead-in session, the teacher first takes out a piece of parallelogram paper and asks the students to touch its area, and then asks the students to estimate its area.
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Link 2: Draw a few squares. In order to verify whose estimate is more accurate, let students think: Is there any way to know the area of this parallelogram accurately? Some students said to measure the length of the bottom and adjacent sides, and then multiply them. Some students said to put it in squares. The teacher drew a parallelogram on the paper and told the students that "every square is a square with an area of 1 cm2". Students operate independently on the grid paper, and the teacher puts forward the operation requirements: please clearly show the counting process on the grid paper, so that people can see it at a glance.
Link 3: Students' operation, feedback and communication. When students have their own methods and answers, we exchange and find that the effect of counting squares is highlighted.
In addition to getting 20 full lattices first, students can also find 20 and a half, and 2 1 half gets 24. Most students use the transformation method, as shown in figure 1, and spell out a full lattice with left and right unsatisfied lattices. Figs. 2 and 3 are cut and transformed by students as a whole, with an area of 24 cm2. The students in Figure 2 have found that it is a transformed rectangle, which is calculated by multiplying the length by the width, that is, multiplying the bottom by the height.
In the above teaching, we get the following conclusion: Let students count squares, not only to count the results, but more importantly, let students experience and realize that parallelograms can be transformed into rectangles and find them by themselves. Under the guidance of the students in Figure 2, most students are open-minded and know that "as long as the area of a rectangle is calculated, the area of a parallelogram can be known". The teacher followed the trend and asked the students to think again: Can any parallelogram be cut and spliced into rectangles in this way? Can the area of parallelogram be obtained by calculating the area of spliced rectangles?
It can be seen that the students found that parallelogram can be transformed into rectangle by cutting and splicing, and the surface of parallelogram can be obtained by calculating the area of spliced rectangle.
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Products. In the following research, as long as it is verified by operation that any parallelogram can be spliced into a rectangle or a square as long as it is cut along the height, and the equal relationship between the spliced rectangle and the parallelogram can be found, it can be concluded that the parallelogram area = bottom × height.
The above teaching shows that the students' transformation idea stems from the intuitive counting of squares, and they want to complete squares and implement this simple transformation method. Therefore, in the teaching of derivation of parallelogram area formula, our teacher's teaching foothold should be to let students experience the progressive transformation from square cutting and repairing to graphic cutting and repairing in several squares, so as to realize the seamless connection between book knowledge and student experience.
Third, the derivation teaching of triangular and trapezoidal area formulas, counting squares allows students to expand their thinking, establish spatial relations, and realize assimilation thinking by different routes.
After learning the derivation of parallelogram area formula, the textbook does not use square in the derivation of triangle and parallelogram area formula, but lets students make a parallelogram with two identical triangles or trapezoids. If you think about it from the students' point of view, how do students know that two identical triangles or trapezoid can be combined into a parallelogram? It is basically difficult for students to remember.
In the author's opinion, students should make full use of their intuitive perception of squares and know the mystery by counting them. The derivation of triangle area formula can transfer the cutting and spelling of parallelogram, but at the same time it has its own transformation method, that is, adding spelling, which requires more space imagination. Therefore, the teaching of triangle area formula derivation should highlight the key points on this point. For example, in the teaching of triangle area, the teacher first provides students with a parallelogram supported by squares (the side length of each square is 1 cm), calculates the area of the parallelogram, and then makes students think again, "Can you know the area of any figure from the figure?" Some students think about it a little, and suddenly feel that the area of the triangle is 12 cm2. The method is to divide the parallelogram into two parts by diagonal.
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Two identical triangles (Figure 5), realize that the areas of these two triangles are equal, which is equal to half the area of a parallelogram with equal base and equal height. At the same time, I vaguely realized that two identical triangles can be combined into a parallelogram. On this basis, the teacher once again presents a triangle with a square (Figure 6), so that students can continue to explore and cultivate students' personalized and diversified transformation ideas.
With this experience, we can use squares more boldly when teaching the derivation of trapezoidal area formula. Let students give full play to their intelligence, form a multi-angle exploration and discovery trapezoidal area calculation method, and let students' wisdom be displayed (as shown in figure 10, 13).
Square number makes students think clearly, and many transformation methods are derived from it. Make the transformation relationship between graphics visually presented to students. "Two identical triangles or trapezoids can be combined into a parallelogram." At this time, the appearance of addition spelling is so natural and accords with the characteristics of students' thinking. When the area is in a square, students are more likely to have changing ideas, which contain a variety of changing ideas, so that students can truly experience and explore the true meaning of knowledge and know why. The function of squaring is reflected incisively and vividly at this time.
Fourth, the derivation teaching of the formula of circular area, in which several squares arouse students' association, break through Fiona Fang, and understand the problem-solving principle of turning curves into straight lines.
As a curve figure, the number of circles and squares seems a bit far-fetched. In fact, we can use the same way of thinking, put it in a square and calculate the area of the circle by counting the number of squares occupied by a quarter of the circle, as shown in figure 14. And we can guess the multiple of the circular area and the small square (the square of the radius), so we can get the guess that the circular area = the square of the radius ×3 times more, which echoes the actual operation formula.
Then guide the students: Can you convert a circle into a figure with an area that can be calculated, provide students with eight eighth circles, as shown in figure 15, organize students to operate, and so on, and draw the following conclusions.
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This process. By observing the relationship between spliced rectangles (parallelograms), it is verified that the circular area obtained by several squares is equal to the square of radius ×3 times more, and the specific value of "3 times more" is "π".
In short, in the formula derivation teaching of plane graphics, counting square can not only be used as the basic method of measuring area, but also reflect the transformation strategy in counting square, which naturally helps students to establish the conjecture of transformation method and formula, and can also be used as a typical example to sort out the relationship and review and summarize the formula derivation after the students verify it. However, the number of squares is not without defects. In many cases, it is necessary to have a specific shape and a specific arrangement to adapt to the students' operation. However, this does not affect the function of counting square in the teaching of derivation of plane figure area formula. In teaching, teachers can use special cases to find problems, use general graphics to verify operations, and finally return to typical examples to sort out the relationship between deductive process and graphics.