? -Reading Notes on "How to Teach Math Well in Primary Schools" 13

? Rotate a rectangle around an edge to get a cylinder. Similarly, divide this rectangle in two to get a right triangle, and rotate it around its right edge to get a cone. According to our intuitive experience, the triangle before rotation is half the area of the rectangle, and they rotate synchronously. Therefore, the volume of a cone should be half that of a cylinder. But in fact, why not half? It turns out that causality is involved here. Some phenomena are one cause and one effect, while others are multiple causes and one effect. So, how to describe causality in mathematics?

? In a sense, mathematics is a knowledge of "relationship". For example, knowing the natural number "2" is actually knowing its relationship with "1". The so-called rectangular area formula is actually to express the relationship between the area and the edge of a rectangle.

Generally speaking, relationships fall into two categories. The first category is called the relationship between "combination and separation", and the relationship between species and genus of concepts belongs to this category. The second category is called "dependence and restriction", and causality belongs to this category. There are often two ways of thinking when studying causality. One is to find the result by reason, which is called "seeking the result by reason"; The other is to find the cause through the result, which is called "finding the cause by holding the fruit".

? For example, for a square, if the side length is called "finding the cause and finding the result", then we know that finding the side length with the perimeter is "finding the cause and finding the result". These two problems are not difficult to solve, because it is a problem of "one cause and one effect". Because all four sides of a square are equal. )

? For a rectangle, if the length and width of the rectangle are known, then the perimeter can be calculated; On the other hand, if we know the perimeter of a rectangle, if we want to find the length and width of this rectangle, there will be a lot of uncertainty in the answer, because this is a "two causes and one effect" problem.

? From this, I think of the introduction stage of Fujian Luo's lecture on "Understanding the Circle": first show the plane figures of rectangles and squares, and then ask how much data is needed to describe their sizes; By the way, how much data do you think is needed to describe the size of a circle? This involves the relationship between several reasons and several results. In the usual practice, such judgment questions often appear, which makes children feel very confused. For example:

1. The greater the perimeter of the square, the greater the area. (√)

2. The greater the circumference of a rectangle, the greater its area. (×)

The larger the circumference, the larger the area. (√)

? Sometimes the area and perimeter will be interchanged to make a judgment question. It can be found that there are also several causes and effects, which require children's mathematical understanding ability to be very high. In practical teaching, children who don't learn through understanding often get confused. Of course, children who really understand will say, "a circle and a square can be determined by one condition, while a rectangle needs two factors to adjust each other to solve the problem." From this piece, it is once again explained that learning mathematics focuses on understanding, and only after understanding mathematics can we have a clear idea.

? Back to the question of why the volume of a cone with equal base and equal height is not half that of a cylinder, which is essentially a problem of two causes and one effect. In other words, we should not only consider the equal product before rotation, but also consider the distance from the rotating body to the rotating shaft.

1. Preliminary explanation

? As shown in Figure 7-24 above, the BC side length of the rectangle is 5cm, and the AB side length is 3cm. Assuming that the rectangle rotates around the AB side to get a cylinder, it is not difficult to calculate the volume of the cylinder is 75π. Assuming that the rectangle ABCD rotates around the BC side in Figure 7-24 to get a cylinder, the volume of the cylinder can be calculated to be 45π. We find that the same rectangle rotates around different sides and has different volumes. This fully shows that the size of the area before rotation is not the only reason that restricts the size of the volume after rotation, and the surface of rotation is also related to the distance from the axis of rotation. If the rotated volume is regarded as the result of causality, then this result is not one reason, at least the above two reasons.

? Pappus, a mathematician in the late ancient Greece, recorded a theorem in his Mathematical Compilation: If a plane closed figure rotates once around a straight line of the figure, the volume of the rotating body is equal to the area of the initial surface multiplied by the perimeter of the center of gravity. This theorem shows that the volume of a rotating body is the product of the following two quantities: the first is the area before rotation; The second is the circumferential length of the rotation of the center of gravity of the rotating surface. The length of the circumference is determined by the length of the radius, so it can be said that the volume of the rotating body is determined by the area of the rotating surface and the distance from the center of gravity of the rotating surface to the rotating shaft.

2. The center of gravity of the chart

The center of gravity of plane figure mentioned here is a concept in physics. Take the rectangular ABCD in the above figure as an example (AB=3, BC=5). The distance from the BC side of the center of gravity to 0.5cm is 1.5cm, and the distance from the center of gravity to the AB side is 2.5cm. If this rectangle rotates around the BC side, the circumference of the rotation of the center of gravity mentioned in Pappus Theorem is "2× 1.5×π=3π" and the area of the rectangle is ". If this rectangle rotates around the AB side, the volume of the rotated cylinder is also (2×2.5×π)×(3×5)=75π. The reason why the two answers are different is that the distance from the center of gravity to the axis of rotation is different.

? A cone is formed by a right triangle rotating around a right side, so let's look at the center of gravity of the triangle.

After a series of transformations (children in grade six can understand this area transformation process), the triangle center of gravity can be summarized as two sentences:

(1) The center of gravity of an arbitrary triangle is the intersection of three midlines;

(2) The center of gravity of any triangle is located on the bisector of each middle line near the bottom.

3. Interpretation of questions

? As shown in the above figure, the upper diagonal AC, the middle line DF of the triangle ABD and the middle line BE of the triangle BDC are added to the rectangle. In this way, the center of gravity n of triangle BCD and the center of gravity m of triangle ABD are obtained respectively. The distance from the center of gravity n to the rotation axis BC is the length of line segment NH, and the distance from the center of gravity m to the rotation axis BC is the length of line segment MG. These two distances are different.

? Using the knowledge of "similar triangles's corresponding sides are proportional" in junior high school, we can know that the length of line segment NH is one third of that of line segment AB, and the length of line segment MG is two thirds of that of AB, so the distance from the center of gravity of triangle ABD to the rotation axis is twice that of triangle BDC.

? Since the circumference of any circle is proportional to the radius of this circle, the rotation circumference of the center of gravity of triangle ABD is twice that of triangle BDC. The areas of the two triangles are equal, and the radius of rotation of the center of gravity is twice. According to Pappus theorem, the volume of triangle ABD is twice that of triangle BDC. If the rotational volume of triangle BDC is 1, the rotational volume of triangle ABD is 2 and the total number of copies is 3. Therefore, the volume of the cylinder rotated by the rectangular ABCD is three times that of the cone rotated by the triangular BDC, and conversely, the volume of the cone is one third of that of the cylinder. So far, it not only explains the question of "why not half", but also explains the question of "why not one third".

Because the gravity center of graphics used in physics is similar to that of junior high school, it is still difficult for primary school students to fully understand it. Of course, in college, we can better understand the volume formula of cone with calculus. However, the lack of knowledge does not prevent pupils from exploring the truth behind knowledge. Mathematics should encourage children to explore and study. ...