In the past, mathematics courses in primary and secondary schools generally only discussed the symmetry of figures in plane geometry and solid geometry. In the coordinate transformation of analytic geometry, the translation transformation and rotation transformation of graphics are discussed. In the past, coordinate transformation was ignored as a higher requirement. Most math classes in secondary normal schools are handled in this way. On-the-job learning of mathematics by college-educated teachers usually begins directly with spatial analytic geometry or mathematical analysis. Therefore, the knowledge about the translation and rotation of plane graphics has become the blind spot of most primary school teachers' mathematics knowledge. Therefore, although the strict definition of geometric transformation is not required to study the nature of transformation in the whole compulsory education stage, it is necessary for teachers to thoroughly understand the related concepts of graphic transformation in order to do a good job in this part of teaching. Generally speaking, the so-called translation is to move a graphic to a certain distance; The so-called rotation is to rotate a figure around a vertex by a certain angle. This description is more suitable for students' cognitive level, but it is definitely not enough for teachers. Please look at a case. [Case] In an open class teaching Translation and Rotation, the teacher created a situation of playing on the playground. When discussing the movement of the Ferris wheel, at first, the students thought it was rotation. Unexpectedly, a classmate insisted on speaking. He said: I have been on the Ferris wheel. I always sit on it with my head up and my feet down, so I think it is translation, not rotation. Everyone froze for a moment, and the teacher's countermeasure was to let the students discuss in groups. This is hilarious, and some people agree that the direction of people has not changed; Some people object on the grounds that people are circling. I didn't know whether it was translation, rotation or neither until after class. After class, the teachers who came to watch were also talking about it. Most people think that the movement between people sitting on the Ferris wheel and the cabin is not translation, and a few think it is translation. Is it spinning? There are also two opinions. This shows that it is very necessary for teachers to understand the concepts themselves. Here, the most important concepts and attributes are described as follows in as simple a way as possible. 1, what is transformation? Generally speaking, the so-called transformation refers to the corresponding law that meets certain requirements in an upper set. As far as graphic transformation is concerned, because geometric figures are all sets of points, graphic transformation can be realized through point transformation. If every point of a plane figure corresponds to a point of a new figure in the plane, and every point in the new figure only corresponds to a point in the original figure, such correspondence is called transformation. The most important geometric transformations are congruence transformation and similarity transformation. A transformation that can keep the shape and size of a graph unchanged is an congruent transformation. In congruence transformation, the distance between any two points in the original graph is equal to the distance between the corresponding two points in the new graph, so it is also called distance-preserving transformation. A transformation that can keep the shape of a graphic unchanged and only change the size of the graphic is a similar transformation. In similarity transformation, the sizes of all corners in the original graph remain unchanged, so it is also called conformal transformation. Primary school mathematics mainly introduces translation transformation, rotation transformation and axial symmetry transformation, which are congruent transformations. Similar transformation only permeates in the second learning period. For example, when learning scale, two figures are scaled up or down, which is actually a similar transformation. 2. What are translation transformation, rotation transformation and axial symmetry transformation? Let's talk about translation and rotation first. If the connecting line from any point in the original drawing to the corresponding point in the new drawing has the same direction and length, such congruence transformation is called translation transformation, which is called translation for short. That is to say, the basic feature of translation is that "the straight lines between each point and its corresponding point are parallel (or coincident) and equal" before and after the graphic moves. Obviously, determining translation transformation requires two elements: one is direction, and the other is distance. If every point in the new figure is obtained by rotating a point in the original figure by an equal angle around a fixed point (called the rotation center), such congruent transformation is called rotation transformation, which is called rotation for short. That is to say, the basic feature of rotation is that "the distance between the corresponding points and the rotation center is equal, and the included angle of the connecting line between each group of corresponding points and the rotation center is equal to the rotation angle" before and after the graphic rotation. Obviously, determining the rotation transformation requires three elements: rotation center, rotation direction and rotation angle. Now you can answer the previous question about the Ferris wheel cockpit. The Ferris wheel is spinning, but the cockpit above and the people inside are always head up and feet down. Is it a translator? According to the basic characteristics of translation, we can draw the connecting line between the upper and lower points of the cockpit at any two positions during the movement (as shown in figure 1). They are parallel and equal, so they are translation. So are the cockpit and the people inside spinning? According to the basic characteristics of rotation, draw the connecting line between the lower midpoint of the cockpit and the rotation center of the Ferris wheel (as shown in Figure 2), and their lengths are obviously unequal. It is clear that the Ferris wheel is rotating, but the cockpit and the people inside are not rotating, but translating. What's going on here? It turns out that while the Ferris wheel drives the cockpit to rotate clockwise, the gravity of the earth makes the cockpit hanging on the hook rotate slightly counterclockwise, so that the cockpit and the people inside always keep the upward direction, and the cockpit moves the same distance from every point on the human body. In fact, rotation and translation in mathematics mainly focus on the relationship between the corresponding points of two static figures at the beginning and end of the movement, which is different from the focus of studying "rotation" and "translation" of objects in physics. First of all, symmetry symmetry is a term that will be used in many disciplines and occupies a very important position in mathematics. Concepts related to symmetry, such as symmetric polynomial, symmetric space and symmetric principle, are all important concepts in mathematics. Primary school mathematics only discusses the symmetry of graphics, and only refers to the symmetry of plane graphics about a straight line. As for other symmetries of graphs, such as rotational symmetry and its special case central symmetry, it is beyond our discussion. However, when students mention such phenomena as parallelogram (central symmetry) and fan blades (rotational symmetry), the teacher should not categorically deny their symmetry, just point out that they are not axisymmetric figures. If the line segments connecting each group of corresponding points in the new image and the original image are perpendicular to the same line and bisected by this line, such congruence transformation is called axisymmetric transformation, each group of corresponding points is symmetric point, and the line vertically bisecting the line segments connected by the symmetric points is called symmetric axis. In other words, the basic feature of axial symmetry is that "line segments connecting any group of corresponding points are vertically bisected by the axis of symmetry". Obviously, the key to determine the axisymmetric transformation is to find the axis of symmetry. The figure that constitutes the axis symmetry can be one, which is usually called the axis symmetry figure (as shown in Figure 3); It can also be two, which is usually called that these two figures are symmetrical about a straight line (as shown in Figure 4). Either of the two axisymmetric figures can be regarded as the result of the axisymmetric transformation of the other figure. Axisymmetric graphics can also be regarded as semi-basic axisymmetric transformation. We can also use more popular language to describe the axisymmetric figure intuitively: if a figure is folded in half, if the figures on both sides of the crease are completely coincident, this figure is called an axisymmetric figure, and the crease (straight line) is called an axis of symmetry. Of course, this description focuses on the characterization of graphic attributes, and the penetration of motion transformation viewpoint is not so prominent. In mathematics, in order to describe the direction and distance of translation, directed line segments or vectors are usually used and discussed in a specific coordinate system. In order to describe rotating elements, the simplest method is to use polar coordinates. Because the transformation of graphics is the corresponding relationship between points, it is inseparable from the coordinate system to describe it accurately. If we regard the transformation of graphics as a kind of movement, we also need a frame of reference. In fact, this is also the main reason why translation and rotation were put together in analytic geometry in the past. In elementary school mathematics, it is also true that lattice paper is often used when discussing translation and rotation. (3) What is the relationship between translation transformation, rotation transformation and axial symmetry transformation? First of all, these three transformations can keep the shape and size of graphics unchanged, which is their main similarity. Secondly, if two consecutive axisymmetric transformations are carried out, in general, when the two symmetrical axes are equal, the final results of these two axisymmetric transformations are equivalent to a translation transformation, the direction of translation is perpendicular to the symmetrical axis, and the distance of translation is twice that of two symmetrical objects. In short, two folds (symmetry axes parallel to each other) are equivalent to one translation. When two symmetrical axes intersect, the final result of these two symmetrical transformations is equivalent to a rotation transformation, the rotation center is the intersection point of the symmetrical axes, and the rotation angle is twice the included angle of the two symmetrical axes. In short, two folds (the intersection of symmetry axes) are equivalent to one rotation. The above two conclusions are aimed at the general situation of graphics. Through axisymmetric transformation, some special graphics may be translated or rotated only once. For example, the "house with chimney" in Figure 5 has undergone two axisymmetric transformations (the symmetry axes are parallel and separated by 4 grids), which is equivalent to a translation of 8 grids to the right at a time. The "house without chimney" in Figure 6 is equivalent to translation as long as it undergoes axisymmetric transformation. In addition, the above two conclusions are equally true in turn. That is, a translation transformation can be replaced by two axisymmetric transformations (the symmetry axes are parallel to each other); A rotation transformation can also be replaced by two axisymmetric transformations (the axes of symmetry intersect). They move in different ways, but the effect is the same. In primary school mathematics textbooks, some patterns can be generated by different transformations. For example, in the four-leaf mode of fig. 7, each leaf can be obtained by axisymmetric transformation of adjacent leaves, 90-degree rotation of adjacent leaves, or translation of leaves on the same row. Understanding the relationship between the three congruent transformations is also helpful for us to understand the key points of studying graphic transformation in mathematics, mainly lies in the relative position relationship of graphics before and after transformation and the relationship of corresponding points.