First, based on teaching practice, grasp the cognitive characteristics of students
1. From perceptual phenomena to experiential characteristics
The textbook first guides students to perceive translation, rotation, symmetry and axisymmetric figures, and provides rich materials for students to understand translation, rotation and symmetry with the help of many phenomena in life, such as flag raising, propeller rotation, buildings, plants (such as maple leaves) and animals (such as butterflies). Make use of students' existing life experience, such as origami, spinning windmill, looking in the mirror, etc., to gain translation, rotation, symmetry and other experiences. Through observation, operation, imagination, thinking, communication and other activities, we can initially perceive the transformation phenomenon and feel its characteristics as a whole. The textbook then guides students to understand translation, rotation and axisymmetric graphics, mainly to learn the operations of translation, rotation and axisymmetric graphics in grid paper, so that students can experience the process and methods in the process of hands-on, with the focus on guiding students to play a positive transfer role in learning, from perceiving phenomena to experiencing characteristics.
2. From single to comprehensive application.
Whenever we talk about "graphics and transformation", teachers will let students appreciate some beautiful patterns and think about the formation of patterns, and then inspire students to try to make some simple patterns by translation, rotation or axial symmetry. On this basis. Let students flexibly use translation, rotation or symmetry to design and make patterns. On the one hand, it is the comprehensive application of mathematics, aesthetics and handwork, on the other hand, it is the combination of students' innovative spirit and practical ability.
3. Links to other content
Transformation is closely related to the understanding of graphics. For example, parallelograms are directly obtained by translation transformation: two groups of parallelograms with parallel opposite sides are parallelograms. Students can use the parallel movement test of ruler and triangle to experience the characteristics of graphic transformation.
Transformation is also closely related to the measurement of graphics. In primary school, the area formulas of squares, parallelograms, triangles and trapeziums were all derived with the idea of translation and rotation.
These connections are hidden. Only by grasping them as a whole first, and then observing and thinking, can we find the dynamic connections between them.
Second, the infiltration of mathematical concepts, breakthrough teaching difficulties
As for the teaching difficulties of "graphics and transformation", on the one hand, we should pay attention to understanding the mathematical connotation of the content of "graphics and transformation", on the other hand, we should pay attention to the connection between "graphics and transformation" and related knowledge.
1. Pay attention to understand the mathematical connotation of "graphics and transformation"
The first is understanding transformation. If every point of a plane figure. Both correspond to a point of a new figure on the plane. However, each point in the new diagram only corresponds to one point in the original diagram, so this correspondence is called transformation. The most important geometric transformations are congruence transformation and similarity transformation, and primary school mathematics mainly introduces translation transformation, rotation transformation and axisymmetric transformation. These three transformations are congruent transformations.
The second is to understand translation transformation, rotation transformation and axial symmetry transformation. 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 connecting lines of each point and its corresponding point are parallel (or coincident) and equal" before and after graphic movement. Obviously, determining the translation transformation requires two elements, namely direction and 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. In other words, the basic feature of rotation is that "the distance between the corresponding points and the rotation center is equal, and the included angle between each group of corresponding points and the connecting line is equal to the rotation angle" before and after the rotation of the figure. Obviously, determining the rotation transformation requires three elements, namely, rotation center, rotation direction and rotation angle.
Symmetry is a term used in many disciplines, and the discussion of primary school mathematics is limited to the symmetry of graphics. And only refers to the symmetry of a plane figure about a straight line. If the line segment connecting each group of corresponding points in the new drawing and the original drawing is perpendicular to the same straight line and is divided into two by the straight line, such congruent transformation is called axisymmetric transformation, and each group of corresponding points is symmetrical to each other. The straight line that bisects the line connecting the symmetrical points vertically is called the symmetry axis. Axisymmetric graphics can also be regarded as semi-basic axisymmetric transformation. We can use more popular language to describe the axisymmetric figure intuitively: fold a figure in half. If the figures on both sides of the crease are completely coincident, this figure is called an axisymmetric figure, and the crease is called an axis of symmetry.
The third is to understand the relationship between translation transformation, rotation transformation and axial symmetry transformation. First of all, these three transformations can keep the shape and size of the graph unchanged. This is their main similarity. Secondly. If two consecutive axisymmetric transformations are carried out, in general, when the two symmetrical axes are parallel, the final results of these two axisymmetric transformations are equivalent to a translation transformation, the direction of which is perpendicular to the symmetrical axis, and the translation distance is twice the distance between the two symmetrical axes. In short, two folds (symmetry axes parallel to each other) are equivalent to one translation. When two symmetry axes intersect. Then the final result of these two axisymmetric transformations is equivalent to a rotation transformation, with the rotation center as the intersection of the symmetrical axes and the rotation angle twice as much as the included angle between the two symmetrical axes. In short, two folds (the intersection of symmetry axes) are equivalent to one rotation.
Fourth, pay attention to specific situations and operational activities and experience the characteristics of transformation. Students' understanding of translation, rotation and axisymmetric graphics is not obtained from concepts, but from their perception of relevant specific situations and hands-on experience. Therefore, teachers should create valuable situational activities and operational activities to help students understand the characteristics of transformation.
Fifth, pay attention to cultivating students' spatial concept in the process of transformation. The main purpose of "graphic transformation" is to guide students to explore and understand space and graphics from the perspective of movement change, and to develop students' spatial concept. Graphic transformation is an intuitive and abstract knowledge, which requires a certain spatial imagination and is a brand-new way of thinking for students. It is not easy to master and use it well. Therefore, in the teaching activities of graphics and transformation, we should strive to develop students' spatial concept in the process of combining operation, thinking and language expression. For example, when learning translation, teachers should pay attention to guiding students to describe it in mathematical language. Encourage students to simulate and represent the motion mode of objects through actions or symbols, and consciously and gradually improve students' thinking level. At the beginning, students can move by hand or use multimedia presentations. Then, teachers should encourage students to gradually break away from physical operation and intuitive demonstration, so that students can try to "translate in their minds" and develop their spatial imagination.
2. Pay attention to the connection between "graphics and transformation" and related knowledge.
One is to understand graphics from the perspective of transformation. In the teaching process of understanding graphics, the attributes of graphics can be portrayed dynamically and intuitively with the help of transformation. Such as rectangle, square, triangle, etc., when we know their characteristics, we can find the hidden characteristics of graphics clearly and intuitively through translation, rotation and symmetrical transformation.
The second is to understand measurement from the perspective of transformation. In primary school, in the process of deriving the area and volume formulas of plane geometry and solid geometry. You can always feel the important role of transformation. In the process of deriving the area formulas of triangle, parallelogram, trapezoid and circle, many methods such as splicing and cutting are used, and the essence of these methods is the transformation of graphics.
Third, strengthen teaching reflection and optimize classroom generation.
The process of teaching reflection can not only make teachers consolidate their professional quality, accumulate teaching and research materials, but also optimize classroom generation. Therefore, in the process of reflection on Graphics and Transformation, the author noticed that the role of graphic transformation in understanding graphics and understanding measurement is irreplaceable.
By studying Graphics and Transformation, students can improve their cognitive ability of graphics. Through this part of the preliminary study, teachers realize that the arrangement level of the field of "graphics and transformation" in textbooks has gradually risen from perceptual intuitive understanding to rational essence understanding, and from static understanding to moving state understanding. They can also find that the relationship between "graphics and transformation" and "graphics understanding" and "measurement" is implicit, but they are closely related.
The transformation of graphics not only provides strong support for students to feel and understand abstract concepts, but also helps students to acquire corresponding knowledge and skills. Moreover, it provides convenience for students to explore the essence of graphics independently, and is helpful to cultivate students' intuitive perception, operation technology, geometric intuition, spatial concept and independent innovation consciousness developed from it.
By exploring the best way of teaching "Graphics and Transformation", teachers can help students understand graphics knowledge, develop the concept of space, and make transformation an effective way of thinking for students to analyze and solve problems.
Author: Beijing Zhongguancun No.3 Primary School
(Editor Huang Shuhong)