The origin of the region

The Nile River in ancient Egypt floods in July every year, and the flood gradually subsides in June165438+1October. The silt left after the flood formed fertile soil, which also brought the need for land retest. Geometry came into being in order to measure land. In fact, geometry originally meant "land survey". Land survey needs to make graphics the research object of mathematics. The amount of land, the size of the number is the area.

Regional teaching

① Build an area model in multiple experiences to understand the meaning of area.

Take a look: which is the bigger of the two pairs of footprints in the snow?

Touch: Find out which objects around you have faces and touch them with your hands. Comparing the surfaces of any two objects, we feel that the surfaces of the objects are large and small, and we feel that the surfaces cannot be separated from the body.

Painting: painting on the surface of an object, recognizing that the area is the size of the area.

Comparison: Compare the sizes of regular and irregular figures, and compare the enclosed and unsealed areas. Students realize that only a closed figure can have a definite area.

Spelling: Put the jigsaw puzzle together and make a square with seven pieces, so that students can understand the size of the surface and form a sense of unit.

② The understanding and application of area is gradually improved.

In the study of grades 3-6, students' understanding of plane, surface and surface size gradually deepens. (The area of rectangle and square is level 3-the area of parallelogram-the area of trapezoid-the area of triangle-the surface area of cuboid and cube is level 5-the area of circle-the side area of cylinder, and the surface area is level 6).

For the study of area, we need to understand and apply it in constant exploration, experience and practice.

intersecting surface

Cross-section includes cross-section, vertical cross-section, flat cross-section and inclined cross-section. The primary school stage is generally cross-sectional, that is, cut parallel to the bottom.

In normal teaching, teachers seldom organize a class. However, related problems often occur in practice, so it is still difficult for students to find the cross section. Teachers can design a series of mathematical activities to guide students to think deeply, experience and understand the meaning of sections in the activities.

Activity 1: concrete object, export part

Activity 2: Cut the cube and experience different sections formed by different cutting methods (cross cutting, longitudinal cutting and oblique cutting) of the same geometry.

Organize students to cut potatoes in groups, a few in each group. The question leads to, if you cut the cube at will, what shape will the cross section be after cutting? The cross section may be triangle, square, rectangle, trapezoid, pentagon and hexagon. The heptagon cannot be cut out because the cube has only six faces.

Guide students to find that the cross section obtained by cutting a cube from different angles may be a plane figure with different shapes, and the number of sides of the plane figure is determined by the number of faces on the cube surface through which the cross section passes.

superficial area

Definition: the quantity describing the surface area and its calculation formula.

The sum of the areas that all three-dimensional figures can touch is the surface area of this figure.

We often say that the surface area refers to a three-dimensional figure that can be touched in an ideal state, and the sum of the areas of each surface is calculated for each surface. After learning the surface areas of cuboids and cubes, students can expand their applications in the following situations.

(1) Find the sum of the surface areas you can see.

② Find the area combination of all exposed surfaces (several figures are stacked together).

(3) Cut a three-dimensional figure, and find the sum of the areas of the added surface and the sum of the areas of all three-dimensional figures after fireball cutting.

(4) Which way of packing is the most economical? (several identical objects are tied together)

Teaching concept of surface area;

① packaging teaching method

Can guide students to think like a three-dimensional figure and draw a bright coat. (You can draw or paste materials) How to wear this coat? In this process, students need to package three-dimensional graphics of several faces.

(2) The teaching design of turning solid into plane.

The plane development diagram of three-dimensional graphics is beneficial to the development of students' space concept. In class, it can help students understand the surface area of three-dimensional graphics in the mutual transformation between three-dimensional and two-dimensional, guide students to cut along the edge of three-dimensional graphics, transform three-dimensional graphics into plane graphics, guide students to observe graphics and find the three-dimensional graphics of unfolded plane graphics.

(3) Teaching design of changing plane into three-dimensional.

Provide some cardboard to the students, and then propose to make a dictation model of the original cuboid and cube together. In the process of doing it, students will find that they only need to prepare six rectangles with appropriate data to make a cuboid, and then enclose the six rectangles into a cuboid with tape in a certain way.

border area/region

Definition: ① the area of the side development diagram of a three-dimensional figure ② the size of the side of an object or the surface of a closed figure.

Cuboid, cube and cylinder are very common in primary schools. Generally, the total area of the front, back, left and right sides of a cuboid cube is called its side area. The lateral area of a cylinder is the surface area of the cylinder. The cone is cut along the generatrix, and the side development diagram of the cone is fan-shaped, which can be calculated by using the formula of fan-shaped area.

Horizontal regional teaching design

Prepare all kinds of physical models of straight cylinders before class.

① Know a straight cylinder and perceive what a side is.

Let the students observe these three-dimensional figures carefully first, then point out the bottom of these three-dimensional figures as required and color them. What are the common features of the top and bottom surfaces? Finally, I know that every three-dimensional figure is a side except the upper and lower bottom surfaces. I touch the sides with my hand and find out what they have in common.

(2) Understand the side development diagram of the straight cylinder.

Let the students guess first. If you cut the edge with eye scissors and the height reaches the cylinder, what figure will you get?

③ Know the side area of a straight cylinder.

What is the connection between the side of the main AC straight tube and this rectangle? The length of a rectangle is the perimeter of the bottom of a right cylinder, and the width is the height of the right cylinder. Because the area of a rectangle is equal to the length times the width, the lateral area of these right cylinders can be calculated by multiplying the perimeter of the bottom by the height.

This teaching design provides students with sufficient practical opportunities. Students can discover the understanding of the relationship between cylindrical side and plane figure and the calculation method of side area through hands-on operation and the change of drawing value.

Jianping

The bottom surface is two parallel surfaces in the prism, and there are two bottom surfaces in the prism. Different placement methods will change the bottom and side of the cuboid. Both sides of a cylinder are round, and a cone has only one round bottom.

Bottom area teaching

(1) Enrich students' understanding of the bottom surface from a dynamic perspective.

(1) Straight: Cut the column with a plane perpendicular to the generatrix of the column, and the cross-sectional area obtained is equal to the bottom area.

② Ray: Rectangular, triangular, regular polygon, circular and other plane figures are orthographically projected, and the formed projection plane refers to the bottom surface of a straight prism, and the distance between the two bottom surfaces is the height of the prism.

③ Translation: The rectangle is perpendicular to the plane and moves along the plane. From the starting point to the end point, the part swept by this edge is the bottom surface of this prism.

④ Rotation

(2) A base plane is the link to communicate knowledge, and the method of calculating the volume of prism is mastered to form a knowledge system.

① Find common ground and communicate.

(2) knowledge migration and construction system

roll

Professor Zhang Dianzhou pointed out that the size of the space occupied by an object is called its volume, which is not a strict definition, but an explanation. Volume is a measure of the size of an object. After the object moves, the volume remains the same, and the sum of the volumes of two non-overlapping objects is the sum of the original two volumes.

The teaching of volume generally adopts the way of concept formation ① Experiment 1, realizing that objects occupy space. Put the small stone into the water and find that the water level has risen. ② Experiment 2 realized that the space occupied by objects is different. Put two stones of different sizes in the same cup and find that the big stone takes up a lot of space. Intuitively judge and verify the size of the space occupied by objects, and show the familiar items such as matchbox, pencil box and shoe box, which box occupies more space, to help students understand the meaning of the space occupied by objects. (4) It means volume. Find the object around you and talk about its volume.

Graphic conversion

Two most important transformation forms of graphic transformation: congruence transformation and similarity transformation. There are two forms in the transformation process of primary school graphics. One is that the shape and size of the graph are unchanged before and after the graph transformation, but the position has changed, which is called congruent transformation; The other is that the shape remains the same, but the size has changed. This change is called similarity transformation. Congruence transformation mainly studies translation transformation, rotation transformation and axisymmetric transformation; The zoom-in and zoom-out time of sixth grade graphics is also a similar transformation.

The transformation of graphics can guide students to understand the changing phenomenon in specific situations in teaching, and then experience the changing characteristics by observing the activities of operation classification. Grade three students draw a figure on grid paper by translation, and the transformed figure is a difficult point in teaching. Many students think that the space between two figures is the number of translation squares. Therefore, teachers can guide students to find that the distances of corresponding points are equal before and after translation through animation demonstration, and determine the translation distance through the distance of points, so as to guide students to draw points before connecting lines.

After learning scales in the sixth grade, we learned two figures. When scaling up or scaling down, teachers can introduce it from life, so that students can feel that there are many phenomena of scaling up and scaling down in life. In the process of exploration, students find that the aspect ratio of enlarged or reduced graphics is exactly the same as that of the original graphics. But at this time, the learning of enlarging or reducing is not the learning of similar transformation, but mainly the intuitive perception, that is, the enlarged or reduced figure has the same shape as the original figure, but the size is different.

Radial

There is no strict definition of rotation, but it is described intuitively with the help of graphics. The angle at which a plane figure rotates around a point o on the plane like this is called the rotation of the figure.

Rotation is a common phenomenon in life, but it is not absolute in life, but in mathematics. Teachers should guide students to pay attention to whether the size and shape of graphics have changed before and after rotation with the help of relevant life experience. Are the distances from the corresponding points to the center of rotation equal? Whether the included angle between the corresponding point and the rotation center is equal to the rotation angle. Distinguishing rotary motion requires grasping the three elements of rotation, the direction and angle of the center of rotation.

The study of rotation can be divided into two periods. In the first stage, students are required to feel and understand the rotation by observing and visualizing the graphic movement phenomenon in daily life. In the second learning stage, it is required to draw a graph on a square paper as required, and use the rotation motion of the graph to design and appreciate the rotated graph.

In the first stage, when teaching translation and rotation, we can start with the amusement parks that students are familiar with, observe the relevant videos of the amusement parks, and encourage students to classify according to their different movement modes, so as to further understand the characteristic points of translation and rotation.

In the second phase of teaching, first of all, the teacher should make clear the specific requirements of this phase and ask the students to draw a figure on the grid paper. The rotated graphics only require students to rotate the simple graphics 90 degrees on the grid paper, and do not require the graphics to rotate at any angle around a point. Secondly, we should pay attention to the course of graphic appreciation and design in teaching. When designing or appreciating a pattern, teachers should encourage students to express their feelings and explanations, and allow students to express their opinions, but let students clearly express the movement relationship in the pattern in their own language.

symmetrical

There is no clear definition, but students are required to understand symmetry with concrete examples, in which axisymmetric graphics are very important. Symmetry can be understood as a figure or an object. For a certain point, line or plane, there is a one-to-one correspondence in size, shape and arrangement.

Symmetric figures can be divided into center, axis, axis and rotation. Parallelogram is a figure with central symmetry. A circle is a symmetrical figure. Not only axial symmetry, but also central symmetry and rotational symmetry. All centrally symmetric figures are rotationally symmetric.

The difference between axisymmetric graphics and centrally symmetric graphics;

After the axisymmetric figure is folded along a straight line, the parts on both sides of the straight line will definitely overlap each other. A centrally symmetric figure is a figure that rotates 180 degrees around a certain point and then coincides with the original figure.

It is axially symmetric and centrally symmetric, with rectangles, squares and circular diamonds. It is an equilateral triangular trapezoid with angles and triangles. Only centrally symmetric figures have parallelograms, which are neither axisymmetric nor centrally symmetric, as well as equilateral triangles and non-isosceles trapezoid.

Axial symmetry

There is no strict definition. In primary school, we can understand the axial symmetry by observing and intuitively feeling the graphic movement phenomenon in daily life, and we can understand the characteristics of the axial symmetry graphic by completing an axial symmetry graphic on grid paper.

Axisymmetric and axisymmetric graphics are two concepts that are both related and easily confused. Axisymmetric refers to the symmetry of two figures about a line or a point, which reveals the special position relationship between the two figures, while axisymmetric figures reveal the special properties of a figure itself. It can also be understood that axisymmetric and axisymmetric figures are symmetrical about a line, the former is a symmetrical figure, and the latter refers to two parts of a symmetrical figure.

Axisymmetric graphics are learned in two stages:

In the first period, students mainly compare and summarize a large number of axisymmetric phenomena in their lives, find the common features between these figures, and describe them in their own language.

In the second stage, students are required to complete an axisymmetric figure on grid paper, design and appreciate the characteristics of axisymmetric figures, and pay attention to the relationship between corresponding points.

In teaching, we should start with the symmetrical phenomena that students are familiar with, find their common ground through observation, and then let students start to stack them one by one. By comparison, we can find that the left and right sides are completely coincident after folding, and find out the characteristics of axisymmetric graphics and know the axis of symmetry. Teachers can start teaching from the food that students are familiar with, and draw food along the outline of food with the help of courseware, so that students can see the process of abstracting objects into plane graphics, such as butterfly diagrams. Then ask students to choose an axisymmetric figure from a set of plane figures or patterns, and explain or verify their choice.

axis of symmetry

If a plane figure is folded along a straight line, the parts on both sides of the straight line can overlap each other. This figure is called an axisymmetric figure, and this straight line is its axis of symmetry.

Symmetry axis should know the following points:

① The symmetry axis is a straight line, not a line segment or a ray. ② Finding the symmetry axis is the key to determine the symmetry axis. ③ The distance from the symmetry point to the symmetry axis is equal. ④ There is not necessarily only one axis of symmetry, but also two, three or countless axes of symmetry.