Now, combining with the teaching practice of solid geometry, I will talk about my own experience in cultivating students' spatial imagination ability, so as to discuss with my colleagues.
First, we should pay attention to the formation of the concept of space.
The transformation from number to shape, from plane to space is a leap in understanding, and there must be a gradual training process.
1, using physical models and other means for intuitive teaching.
By using physical model for intuitive teaching, students can form the overall image of space concept in their minds. One-dimensional and two-dimensional graphics are basically consistent with physical shapes and human visual images. Therefore, the intuitive ability of plane geometry is easier for students to master. The physical object of three-dimensional space is painted on two-dimensional plane, and the formation of graphics, physical object and human vision are not completely consistent, so the intuitive imagination of spatial shape becomes particularly difficult. In mathematics teaching, teachers should guide students to concretize the spatial form in their minds by observing and analyzing physical models, making models and measuring objects on the spot. In this way, over time, we gradually leave physical objects, models and graphics to think about spatial forms. The deeper the image, the richer the imagination. For example, when teaching the concept of prism, I instruct students to observe a series of different physical models of prism, summarize the similarities of these physical models, and then get the concept of prism. Therefore, it is an indispensable and effective way to cultivate students' spatial imagination with the help of visual teaching AIDS such as physical models.
2. Strengthen the cultivation of drawing ability and reading ability.
By drawing sketches or schematic diagrams, the space concept formed in students' minds is "visualized" and "visualized". Spatial imagination is a thinking process in which image thinking and logical thinking alternate, and geometric language, namely geometric figure, is the best language to express this kind of thinking. First of all, we should draw an intuitive view, be familiar with the drawing method of basic graphics, keep the shape of basic graphics in our mind, and then analyze the positional relationship and measurement relationship of basic elements in graphics. It is feasible to draw a direct view with cubes and cuboids as guides, so special attention should be paid to it. Secondly, on the basis of being familiar with the basic graphics, it develops into the spatial shape described by physical objects or common language, and analyzes the positional relationship and measurement relationship between the basic elements.
Example: 1: a regular triangular pyramid, the length of its side is 1, and the three sides are perpendicular to each other. Find the volume of a regular triangular pyramid.
Analysis: Many students will draw graphs in the form of 1, which brings great inconvenience to solving this problem. The position of the spatial graph in the graph 1 is not conducive to the solution of this problem. The title means "three sides are perpendicular to each other", so it can be placed as shown in Figure 2.
Solution: As shown in Figure 2, the volume of the regular triangular pyramid is:
V=？ PBC？ PA=×× 1=
3. Study the composition and properties of graphics.
By deeply understanding the internal structure and characteristics of spatial form, we can "get" basic graphics from complex graphics, and then analyze the relationship between basic graphics and basic elements.
Example 2: As shown in Figure 3, it is a straight triangular prism, ABC-A 1B 1 c, ∠ BCA = 90, and points D and E are the midpoint of A1b1respectively. if
Analysis: The cosine value of the angle formed by a straight line BD and AE is required, and the internal structure of a straight triangular prism must be known. We can take the midpoint f of BC, connect EF, connect DE and AF, and find the cosine of ∠AEF in △AEF.
Solution: take the midpoint f of BC, connect EF, connect DE and AF.
In △A 1B 1C 1, point d and point e are the midpoint of A 1B 1 and a1respectively.
de∨= b 1c 1。
If BC ∨= b 1c 1, then DE ∨= BC is DE ∨= BF.
The quadrilateral BDEF is a parallelogram, DB∨EF.
Then the angle formed by the straight line BD and AE is ∠AEF.
Let BC=AC=CC 1= 1.
At Rt△ACB, AB==
AF==
In Rt△BB 1B, BD==
Then EF=BD=
In Rt△AA 1E, AE==
At △AEF,
cos∠AEF==
Second, master the expression method of spatial form
1, using traditional drawing tools.
In order to meet the needs of study, life and work, people express space graphics into various plane graphics according to human visual laws. For example, drawing a straight view of a "cube" requires students to master the drawing method of three-dimensional graphics on the basis of mastering the making method of plane graphics.
Example 3: In a cube AC 1 with a side length of A, points E and F are the midpoint of AB and BC, respectively. Find the area of section A 1EF.
Analysis: Draw the front view of the cube, as shown in Figure 4, and find out the three sides of the section A 1EF.
Solution: connect AF
In Rt△AA 1E, a1e = = a = a.
In Rt△EBF, EF==a = a = a.
In Rt△ABF, AF==a = a = a.
In Rt△A 1AF, a1f = = a = a.
In △A 1EF,
cos∠A 1EF==-
Then sin∠A 1EF=
Then s △ a1ef =× a× a× sin ∠ a1ef =× a× a× = a2.
2. Computer-assisted instruction
The graphics drawn by hand with paper, pen, compass, ruler and other conventional drawing tools are static, which is easy to cover up its extremely important geometric laws. Using the computer software "Geometer's Sketchpad" to assist in teaching, the geometric laws can be kept dynamically. The graphics drawn by the geometric sketchpad can be moved: lock the target with the mouse and drag it at will; Animation motion can be defined to express the motion of geometric figures, such as cutting off a corner of a cube and defining its separation and merger through control buttons; It can make geometric figures rotate or dynamically present the formation of geometric figures, thus cultivating students' spatial imagination ability.
For some three-dimensional abstract spatial graphics or spatial imagination, it is difficult for students to establish correct spatial concepts by using traditional teaching methods. Using multimedia technology in computer to express various graphics and deepen students' understanding of such graphics; Using computer two-dimensional and three-dimensional image technology to process three-dimensional space graphics, so that students can systematically and intuitively establish the concept of space. For example, when teaching the "Three Vertical Theorem", I use the teaching software "Geometer's Sketchpad" to display "Plane, Straight Lines in Plane, Vertical Lines in Plane, Slant Lines in Plane and Projections of Slant Lines in Plane" on the screen one by one, which can be split and combined. By dragging the mouse or the control button, I can observe the projection of straight lines and oblique lines in the plane from any angle.
Third, several points that should be paid attention to in cultivating spatial imagination ability
1, based on the teaching of basic concepts of spatial graphics, to prevent errors caused by vague concepts.
Example 4: As shown in Figure 5, in a cube AC 1 with a length of 3, e, f, g and h are bisectors on sides A 1D 1 and B1respectively, and the volume of geometric body ABCD-EFGH is calculated.
Analysis: Some students mistakenly think that geometry ABCD-EFGH is a quadrangular prism, which leads to mistakes in solving problems. If BCGH is taken as the bottom surface, the geometric body ABCD-EFGH is a regular quadrangular prism.
Solution: According to the meaning of the question, the geometric body ABCD-EFGH is a regular quadrangular prism, and its volume is:
VABCD-EFGH=SBCGH×AB=×AB
=×3= 18
2. Overcome the negative effects of mental constraints.
It is necessary to guide students to be good at drawing and recognizing pictures, draw pictures properly according to known conditions or use their spatial imagination ability to recognize pictures according to given pictures to prevent some negative effects in drawing and recognizing pictures. It is particularly important to remember that "seeing is believing" may not be applicable when spatial graphics are presented as plane intuitive graphics.
Example 5: As shown in Figure 6, in the cube AC 1 with a side length of A, find the degree of the angle formed by the straight line AD 1 and A1B..
Analysis: To find the degree of the angle formed by the out-of-plane straight line AD 1 and A 1B, it is necessary to connect BC 1 and a1,and then calculate △ a/from △A 1BC 1.
Solution: connect BC 1, A 1C 1.
In △A 1BC 1,
Yizhi a1b = BC1= a1c1= a.
Then △A 1BC 1 is a regular triangle, then △ A 1bc 1 = 60.
The existence and movement of anything involves its spatial form. Spatial form is reflected by human mind, which produces the concept of space. How to understand the space on which people live is an important intelligence, so it is very important to cultivate students' spatial imagination ability in mathematics teaching.