JPH0617138Y2 - Mathematics teaching tool - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION [Industrial field of application] The present invention relates to a teaching tool for mathematics for allowing a student to easily understand the volume ratio of prisms and pyramids having the same bottom area and height.

[Prior Art] It is easy for a student to prove and understand that the area ratio of a rectangle and a triangle, which are two plane figures having the same length and height of the base, is 2: 1 just by showing them.

On the other hand, in the case of a solid figure, for example, it is not always easy to understand that the volume ratio is 3: 1 in the case of a prism and a pyramid having the same bottom area and height.

As a typical conventional teaching tool for the above-mentioned three-dimensional figure, as shown in FIG. 20, a wooden cube is divided into six quadrangular pyramids 10 and these are connected by hinges.

[Problems to be solved by the invention] Explaining while showing the above teaching tools to the students, the classroom suddenly turns quiet for a moment, and it becomes a situation where one can look into magic tricks and attract students so much.

However, this teaching tool has a disadvantage that it is relatively expensive, and even if the student is made of cardboard, it cannot be used in various ways despite the labor. In other words, it is only teaching tools to show, not teaching tools to operate.

Also, it is not easy to understand that "the volume ratio of two similar figures is equal to the cube of the similarity ratio".

Cubes and triangular prisms are easy to divide and understand, but if you try to explain that the volume ratio of a regular tetrahedron with a similarity ratio of 2: 1 will be 8: 1, you will not know which one is the larger tetrahedron. Even if it divides like this, it cannot be divided into eight smaller regular tetrahedrons. That is, when the plane is divided by the plane passing through the midpoint of the side, it is divided into four front bodies and one regular octahedron.

The present invention has been made in view of the above problems, and in order to understand that the volume ratio of a prism and a pyramid having the same base area and height is 3: 1 by using a plurality of triangular pyramids. Not only can it be used to understand that "the volume ratio of two similar figures is equal to the cube of the similarity ratio", but it can be easily made even by first-year middle school students.
The purpose is to provide a teaching tool for mathematics that aims to be a teaching tool that students can operate from the teaching tools they show.

[Means for Solving Problems] An explanation will be given with reference to FIGS. 1 to 5.

The teaching tool for mathematics according to this invention is a cube 1 (A, B, C,
D, E, F, G, H) three quadrangular pyramids 3 (A, E, F, G, H) sharing one diagonal line 2 (A, G), 4
(A, B, C, F, G), 5 (A, C, D, G, H) A vertical edge when a total of six pairs of three pyramids are formed, and insteps Z are arranged adjacent to each other in series, and adjacent pyramids of instep Z and Z are combined at the symmetry plane 6 to form a quadrangular pyramid 3. The vertical edges 9'and 9'of the instep Z corresponding to 9 and the triangular pyramid of Z are connected to each other in a freely bendable manner.

[Embodiment] An embodiment will be described with reference to FIGS. 1 to 19. Cube 1 (A, B, C, D, E, F, G, H) One congruent line 2 (AG) sharing three congruent quadrangular pyramids 3 (A,
E, F, G, H), 4 (A, B, C, F, G), 5
For (A, C, D, G, H), as shown in FIG. 3, for example, in a quadrangular pyramid 3 (A, E, F, G, H), the symmetry plane 6 (A, E, G) is 2 To divide. At this time, a pair of so-called mirror-symmetrical triangular pyramids 7 (A, E, F, G), which face the symmetry plane 6 (A, E, G) as mirrors,
8 (A, E, G, H) can be made. Here, the triangular pyramid 7 is
If the triangular pyramid 8 is called B, the above three quadrangular pyramids 3, 4, 5
There are 3 pairs of insteps and 6 triangular pyramids in total. When making a student make such a triangular pyramid, Z, cardboard is used, and a development view as shown in FIG. 4 is cut out and assembled.
At this time, to make a pair of mirror-symmetrical triangular pyramids, Z,
The triangular pyramid is made by folding the fold into a mountain fold and bending it toward the other side, and the triangular pyramid Z is made by folding the fold into a valley fold and bending toward the front.

In this way, a total of 6 triangular pyramids and 3 Z
Make an individual. It is good to color-code them so that they can be easily distinguished.

Then, the above six triangular pyramids are arranged in a row in the order of A, Z, A, Z, A, Z. In other words, the mirror-symmetrical triangular pyramid and B are arranged side by side.

Next, adjacent triangular pyramids, Z are connected to each other. In that case, two mirror-symmetrical triangular pyramids, Z are combined to form a quadrangular pyramid 3 as shown in FIG. Vertical edge 9 (A,
Vertical ridges 9 '(AE) and 9' (AE) of both of the above-mentioned two triangular pyramids corresponding to E) and Z are pasted with duct tape or scotch tape (registered trademark) and connected to each other in a bendable manner.

[Operation] As shown in FIG. 5, the above-mentioned series of six triangular pyramids are arranged so that the bases of all the triangular pyramids are isosceles triangles, and from the left, Oto, Ko, Oto, Ko, Oto, Ko So that

As a first learning example, first, by moving the triangular pyramids B and A on both ends in the arrow direction, three congruent quadrangular pyramids (also referred to as Yanma) X, Y, and Z are formed as shown in FIG.

Further, if the quadrangular pyramids X and Z at both ends are moved in the arrow direction and placed on the quadrangular pyramid Y in the center, a cube C is formed as shown in FIG.

Here, by comparing the quadrangular pyramids X, Y, and Z with the cube C, the following can be immediately found.

Height of square pyramid = Height of cube Square area of square pyramid = Bottom area of cube Square pyramid volume = (1/3) × Volume of cube Volume ratio of cube and square pyramid whose bottom area and height are equal Is confirmed to be 3: 1.

As a second learning example, as shown in FIG. 8, the cube C can be double-sided with one ridgeline AE as an axis to form congruent triangular prisms P and Q on the left and right. Comparing this triangular prism and a cube, the height of the triangular prism = the height of the cube The bottom area of the triangular prism = (1/2) × the bottom area of the cube The volume of the triangular prism = (1/2) × the volume of the cube. , For two prisms of equal height,
It is assumed that the volume ratio is equal to the bottom area ratio.

As a third learning example, as shown in FIG. 9, two prisms P,
When one of the Q, for example, the right triangular prism Q is pulled in the direction of the arrow, the triangular prism Q is divided into three triangular pyramids q ₁ , q ₂ , and q ₃ .

By comparing these triangular pyramids with the triangular prisms P left on the left side, the following can be immediately understood.

The height of the triangular pyramid = the height of the triangular prism The bottom area of the triangular pyramid = the bottom area of the triangular prism The volume of the triangular pyramid = (1/3) × the volume of the triangular prism It is easily understood that it is 3: 1.

As a fourth learning example, the three triangular pyramids formed by extending to the right side of FIG. 9 are returned to the original triangular prism Q, and as shown in FIG. 10, the left and right triangular prisms P and Q are centered on the ridge line AE. Rotate backward in the direction of the arrow and attach the back side to form one triangular prism R as shown in FIG.

B Here, when comparing this triangular prism R with one of the original two triangular prisms P and Q, the height of the original triangular prism = the height of the triangular prism just created = the bottom area of the original triangular prism = (1/2 ) × bottom area of the new triangular prism The volume of the original triangular prism = (1/2) × the volume of the new triangular prism.

From this, it is understood that, in a triangular prism having the same height, the volume ratio is equal to the ratio of the bottom area.

As a fifth learning example, when the AC line of the triangular prism R shown in FIG. 11 is divided and pushed in the direction of the arrow, it is deformed as shown in FIG. 12 to form three triangular pyramids.

Comparing this triangular pyramid with the triangular prism R, the height of the triangular pyramid = the height of the triangular prism R is the bottom area of the triangular pyramid = the bottom area of the triangular prism R, the volume of the triangular pyramid = (1/3) × the volume of the triangular prism R. I understand.

From this, it can be understood that the volume ratio of the triangular prism and the triangular pyramid having the same bottom area and height is 3: 1.

In the first to fifth learning examples described above, a rectangular prism and a quadrangular pyramid whose bottom surface is a square, and a triangular prism and a triangular pyramid whose bottom surface are right-angled isosceles triangles are sequentially assembled, whereby a prism having the same base area and height is formed. It is understood that and the pyramidal volume ratio becomes 3: 1.

The following learning example shows two triangular pyramids corresponding to vertical edges when two triangular pyramids and Z are combined to form a quadrangular pyramid in addition to the teaching tool (1) consisting of the above six triangular pyramids. No. 13 which was made by connecting both vertical ridges with duct tape or cellophane tape
Various learning is performed by adding a teaching tool (2) composed of two triangular pyramids as shown in the figure.

As a sixth learning example, first, a side formed by connecting the triangular pyramid on the right side of one triangular prism R formed by the teaching tool (1) in the fourth learning example as shown in FIG. 14 with duct tape. Rotate around AD as an axis and place it on the left side.

Next, as shown in FIG. 15, the above teaching tool (2) was assembled into a triangular pyramid whose base is an isosceles right triangle. Of the above-mentioned teaching tools (1) To the surface of.

When you observe the assembled three-dimensional solid, you can see that it is a triangular pyramid similar to one triangular pyramid.

Then, comparing the corresponding sides, it can be seen that the similarity ratio is 2: 1.

Also, when comparing the volumes, the above-mentioned assembled solid consists of 8 triangular pyramids and Z, so the volume ratio is 8: 1.
It's easy to see that

From this, the following theorem can be inferred.

“The volume ratio of a similar solid having a similarity ratio of 1: K is 1: K ³ ”. Next, as shown in FIG. 14, a triangular pyramid Z on the upper left side of one triangular prism R is connected with a gum tape to connect the sides. While rotating around AD, place it on the right side as shown in FIG.

Then, if the teaching tool (2) is attached to this, a triangular pyramid similar to the triangular pyramid Z is formed.

Since the triangular pyramid and Otsu are mirror images, you can assemble them into mirror images by simply changing the assembly method. By combining the triangular pyramid and B, an expanded triangular pyramid can be made, so by combining the expanded triangular pyramid, a larger triangular pyramid can be made.

Combining four triangular pyramids with an enlarged triangular pyramid and four triangular pyramids with an enlarged triangular pyramid Z makes a triangular pyramid with a similarity ratio of 4: 1.

As a seventh learning example, first, one triangular prism R shown in FIG. 11 of the fourth learning example is made with the teaching tool (1). Next, it is extruded toward the opposite side to form three triangular pyramids as shown in FIG. Then, as shown in FIG. 17, the innermost triangular pyramid is rotated to the right or left to be attached to the side of the second triangular pyramid.

Next, the triangular pyramid of the foremost virtual line is turned around the edge CD as an arrow and attached to the bottom surface of the second triangular pyramid.

Finally, the vertices C where the three triangular pyramids are gathered are placed upside down and replaced as shown in FIG.
When the teaching tool (2) is fitted in, a square pyramid with a diamond-shaped bottom as shown in FIG. 19 is completed. This quadrangular pyramid has a diagonal length of 2a on the bottom surface, Height is Therefore, according to the formula On the other hand, since the volume of triangular pyramid or B is (1/6) a ³ , the volume of quadrangular pyramid = 8 × the volume of triangular pyramid. This is easily confirmed by the fact that the quadrangular pyramid is assembled from eight triangular pyramids. From the confirmation by calculation and the confirmation by the model, it can be understood as more accurate that the volume of the cone = (1/3) x base area x height.

[Effects of the Invention] According to this invention, a teaching tool is used which is composed of an instep made by dividing a cube, six triangular pyramids of two kinds of Z, and an inward bendable connection so that Zs are adjacent to each other. It is possible to easily assemble a quadrangular prism and a quadrangular pyramid, or a triangular prism and a triangular pyramid, and to confirm that the volume ratio thereof is 3: 1 by the operation of the students themselves.

It is also easy to prove that "the volume ratio of two similar figures is equal to the cube of the similarity ratio" by adding another teaching tool in which two triangular pyramids of A and Z are connected to the above teaching tools. It is possible to provide various learning examples not found in conventional teaching tools.

Since the triangular pyramid of the teaching tool can be easily made by a student by cutting out cardboard, the teaching tool can be manufactured at a low cost.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a perspective view of a cube, which is a basic figure required to make a teaching tool for mathematics according to the present invention, and FIG. 2 is a perspective view of a quadrangular pyramid formed by the cube. Fig. 3 is a perspective view of a triangular pyramid of instep Z made from the same quadrangular pyramid as above, Fig. 4 is a development view of the same triangular pyramid, and Fig. 5 is a perspective view of a teaching tool made by connecting six triangular pyramids as above. Figures 6, 6 and 7 are perspective views of a first learning example made by using the above teaching tool, FIG. 8 is a perspective view of a second learning example, and FIG. 9 is a perspective view of a third learning example. Figure, 10th and 11th
The figure is a perspective view of the fourth learning example, FIG. 12 is a perspective view of the fifth learning example, and FIG. 13 is a perspective view of another teaching tool made by connecting triangular pyramids and Z, 14th and 15th. , 16 are perspective views of the sixth learning example, and FIGS. 17, 18, and 19 are explanatory views of the seventh learning example. FIG. 20 is a perspective view of a conventional teaching tool. 1 ... Cube 2 ... Diagonal line 3, 4, 5 ... Square pyramid 6 ... Symmetrical plane 7,8 ... Triangular pyramid 9, 9 '... Vertical edge A, Z ... Reflective symmetrical triangular pyramid

Claims (1)

[Scope of utility model registration request] 1. A total of six pairs of three triangular pyramids each sharing one diagonal of a cube, each of which has a mirror-symmetrical instep, and two pairs of triangular pyramids formed by dividing the three quadrangular pyramids in the plane of symmetry. These triangular pyramids are arranged in series and the instep A and the second B are arranged adjacent to each other, and the adjacent instep and the second B pyramid are united with the above symmetry plane to form a quadrangular pyramid. A teaching tool for mathematics, characterized in that both vertical edges are connected to each other in a freely bendable manner.