Teacher: What did you learn from today's study? Do you have any questions?
A classmate raised his hand and raised such a question after exchanging learning gains.
Student: Why do we learn the volume of a cylindrical cone through experiments? Is there any other way to get the cone volume formula?
Suddenly hearing such a question really stumped me and I didn't know how to answer it.
Analysis:
The study of cone volume belongs to the field of "graphics and geometry", but the geometry of "cone" does not begin to contact as early as the first grade as other geometries (cuboid, cube and cylinder), but it is not known until the next semester of the sixth grade. At the same time, the learning of cone volume is also different from the exploration of other geometric volumes.
1. Sort out the textbooks and how to explore the "quantity" of primary school geometry.
As early as the first grade, I had a preliminary understanding of cuboids, cubes, cylinders and spheres. This is because in daily life, students actually touch the most objects, so the perceptual body is more real. However, the further study of these geometries has spanned the sixth grade. This is because the study of geometric structure characteristics, area and volume is abstract. Therefore, the arrangement of "geometric volume" is all concentrated in the sixth grade, and the textbooks are studied in the order of "cuboid volume-cube volume-cylinder volume-cone volume".
First of all, the textbook arranges "Exploration of cuboid volume formula". The exploration process of cuboid volume formula is similar to that of rectangular area formula, which is obtained by measuring volume. Measure the volume of a cuboid with a suitable unit of volume. The length of a cuboid is several centimeters, and several unit of volume can be placed in each row along the length. The width of a cuboid is several centimeters, and each layer along the width can be arranged in this way; The height of a cuboid is several centimeters, so you can put several layers along the height. The unit of volume number contained in a cuboid is the product of its length, width and height. So the volume of a cuboid is explored by measurement. The volume of a cube is based on the fact that the cube is a special cuboid, and its volume formula can be derived from the volume of the cuboid.
The exploration of the cylinder volume formula is to convert the cylinder into a cuboid with the same volume after proper division and splicing, and the bottom area and height of the cylinder and the converted cuboid are equal, so the volume of the cylinder is calculated by "bottom area × height".
However, the textbook does not explore the cone volume formula according to the above formula, but adopts special experimental operation. Through observation, speculation and verification, it is pointed out that its volume formula is "bottom area× height×1/3".
Then, why don't the textbooks follow the experience of learning cuboids, cubes and cylinders, but use cylinders and cones with equal bottoms and equal heights to do operation experiments?
2. Think about two special examples.
Example 1: Can you calculate the volume of a plumb hammer with a cylindrical container filled with a proper amount of water?
This problem is not difficult to solve. First, observe the height of water in the cylindrical container, put the plumb hammer into the cylindrical container filled with water (plumb hammer must not be in water, and water cannot overflow the cylindrical container), then observe the current height of water, and then we can convert the volume of plumb hammer into the volume of cylinder formed after the water level rises.