JP2000047572A - Teaching tool for mathematics - Google Patents

Abstract

PROBLEM TO BE SOLVED: To enable a learner to understand not only the shape of a target polyhedron, but also the mathematical formation principle of the polyhedron and to actualize efficient storage by providing six blocks each consisting of a rectangular cone consisting of a square base surface part and four isosceles triangular tilted surface parts. SOLUTION: The external shape of each block A is a rectangular cone shape and the base surface is square. Isosceles triangular tilted surface parts 2 including the sides of the square base surface part 1 as bases are formed at the four sides (a) of the square bottom surface part 1 respectively and constitute a rectangular cone A1. The square base surface part 1 corresponds to the base surface of the rectangular cone A1 and the angle between each tilted surface part 2 and square base surface part 1 of the rectangular cone A1 is 45 deg. when viewed from the side of one side (a) of the square base surface part 1. Six blocks like this are formed in total and connected to adjacent blocks A on the sides (a) of the square base surface parts 1 and adjacent blocks A can mutually be folded back.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention comprises a plurality of blocks formed according to a predetermined definition. By changing the combination of these blocks, it is possible to form a polyhedron from a rectangular parallelepiped or a rectangular parallelepiped from a polyhedron. The present invention relates to a mathematical teaching tool which can be used as an aid for a learner to understand a specific law of a polyhedron, and which can be applied to furniture such as a chair having a quaint design.

[0002]

2. Description of the Related Art In learning mathematics, textbooks for learning polyhedrons having complicated shapes first have texts for understanding the contents. In addition,
Actually make a model, etc., and show it to the learner to assist understanding. In recent years, many teaching materials using visuals such as computer graphics have been developed.

[0003]

SUMMARY OF THE INVENTION Polyhedrons of complex shape,
For example, it is difficult to understand a polyhedron made up of a large number of polygons such as diamonds and pentagons only with explanatory texts, drawings, and pictures. Furthermore, although a polyhedron created by computer graphics can be made realistic, a special programmer is required and the production is expensive. Further, devices such as a personal computer and a display are required, and it is not easy to prepare a learning environment equipped with these devices.

[0004] Further, even if a polyhedron model is manufactured,
In the case of cutting out the material and producing only one shape,
The learner will be able to understand the shape,
We cannot expect the possibility of evoking new mathematical ideas from the polyhedral model. Further, when the polyhedron is manufactured in a large number, restrictions are imposed on the storage means due to the complexity of its shape, and the storage method may be difficult.

[0005]

The inventors of the present invention have conducted intensive studies to solve the above-mentioned problems, and as a result, have found that the present invention sets the angle between the square bottom surface and the square bottom surface to 4 degrees.
Six blocks each consisting of a quadrangular pyramid consisting of four slopes of an isosceles triangle with 5 degrees are provided, and the blocks are connected to each other so as to be freely foldable, and the entire surface of the square bottom portion is positioned outward. A mathematics teaching tool that forms a rectangular parallelepiped, or a rectangular parallelepiped with the entire bottom surface of the square positioned on the inner side, or a square bottom portion, and one side and the opposite side of the square bottom portion are defined as the bottom side. Two triangular slopes with both side angles of 36 degrees, the other side of the square bottom part and its opposite side as the lower base, both side angles of 72 degrees, and both hypotenuses and the upper base are adjacent triangular slope parts. There are six blocks each consisting of a hypotenuse and a roof-like body composed of two trapezoidal slopes of the same length. Each block is freely foldable and the top ridge of the roof-like body of the adjacent block is And an array at right angles to The purpose is simply to form a rectangular parallelepiped by positioning the entire bottom surface of the square on the outside and forming a rectangular parallelepiped by positioning the entire bottom of the square on the inside. Not only does the learner understand the shape of the polyhedron, but also the understanding of the mathematical formation of the polyhedron, and the storage can be performed efficiently, thus solving the above problem.

[0006]

Embodiments of the present invention will be described below with reference to the drawings. First, a first embodiment of the present invention will be described with reference to FIGS. In the first embodiment, two types of polyhedrons, a rectangular parallelepiped and a rhomboid dodecahedron, can be assembled from six blocks A, A,... Connected in a predetermined arrangement (FIG. 1, FIG. 1). 4 and 5).

The outer shape of each block A is shown in FIG.
As shown in (B), (C), and the like, it has a quadrangular pyramid shape and the bottom surface is a square, which is referred to as a square bottom portion 1. A slope 2 of an isosceles triangle having the sides a of the square bottom 1 as bases is formed by four sides a, a,.
.. Are respectively formed.

[0008] The four inclined surface portion 2, 2, by ..., quadrangular pyramid body A ₁ is constructed [see FIG. 3 (A)]. The square bottom 1 are indicated by the bottom of the quadrangular pyramid body A _1. Then, the angle of the inclined surface portion 2, 2, ... and the square bottom part 1 of the quadrangular pyramid body A ₁ is 45 degrees as viewed from the side a side of the square bottom portion 1 [FIG 3 (A) , (C)].

FIG. 2 is a development view of the block A. Each side a, a,... Of the square bottom portion 1 is formed with an isosceles triangle having the base as the base. The ratio of the length of the base to the slope of the slope 2 is as follows. (Slope portion 2 of the base) hypotenuse :( slope portion 2) = 2: √3 Thus, each ridge line m which oblique sides of the inclined surface portions 2 adjacent pyramids body A ₁ constitutes a square bottom part 1 Is the same as the above ratio, and is as shown below. (The length of one side a of the square bottom part 1): (the length of the ridge line m)
= 2: √3.

A total of six blocks A are formed (see FIG. 1). In each of the six adjacent blocks A, A of the six blocks A, A, sides a, a of the square bottom portions 1, 1 are connected to each other, and the adjacent blocks A, A can be folded back to each other. As a specific structure of the connecting means, the sides a, a of the blocks A, A are connected by a sheet-like material (see FIGS. 1A to 1C).

[0011] As a connection arrangement row of the six blocks A, A, ..., square bottom portions 1, 1 of all blocks A, A, ...
.. Are arranged so that a rectangular parallelepiped can be assembled with the whole surface of 1,. Specifically, first, four blocks A, A,...
, Two blocks A, A are arranged in a direction orthogonal to the above-mentioned one row [see FIGS. 1 (A), (B), (C)].

The six blocks A, A,... Are not limited to the above, and are formed by directing the entire surfaces of the respective square bottom portions 1, 1,. Are not limited to only one block A, A,... Other arrangement examples of blocks A, A,... Are described in FIGS. 14 (A), (B), (C), (D).

The six blocks A, A,...
As shown in FIG. 6 or the like, square bottom portion 1,1, a ... an outer side, and a quadrangular pyramid body A ₁ side and the inner side, six quadrangular pyramid body A
_1, A _1, rectangular parallelepiped constituted combine to concentrate the ... vertex of the one point [Fig 4 (A) refer to Fig. In this cuboid, quadrangular pyramids A ₁ , A ₁ ,... Are in contact with each other without any space inside. FIG. 4 (B), a P ₅ -P ₅ cross-sectional view taken along FIG. 4 (A), the block A,
A, ... quadrangular pyramid body A ₁ each other shows a state in which contact with each other.

Next, the six blocks A, A,...
8, as shown in FIG. 9, four pyramid A ₁ side and outer side, the square bottom portion 1,1, by assembling in a ... inwardly, to form a rhombic dodecahedron (See FIG. 5A). That is, adjacent quadrangular pyramids A ₁ , A ₁
In FIG. 5, the adjacent slope portions 2 and 2 are on the same plane, and can form a rhombic surface q [FIG.
reference〕.

FIG. 5 (B) is a graph showing P ₆ − in FIG. 5 (A).
A P ₆ cross-sectional view taken, as shown in this, when assembled blocks A, A, a rhombic dodecahedron at ... is
Inside, a rectangular parallelepiped hollow space having six square bottom surfaces 1, 1,... As inner walls is formed.

Thus, a total of twelve rhombic surfaces q, q,
A rhombic dodecahedron is formed. This rhombic dodecahedron has the same shape on all rhombic surfaces, and in this state, a space can be filled in a state where a plurality of them are stacked. That is, the rhombic dodecahedrons can be efficiently stacked so that no space exists.

Next, a second embodiment of the present invention will be described.
A description will be given based on FIGS. 10 to 13. In the second embodiment, two types of polyhedrons, a rectangular parallelepiped and a regular pentagonal dodecahedron, can be assembled from the six connected blocks B, B,... (See FIG. 13). First, the block B in the second embodiment has a substantially roof shape.
It is composed of a square bottom 1, two triangular slopes 3, 3 and two trapezoidal slopes 4, 4 [FIGS.
(A) etc.].

The square bottom portion 1 is a square as in the first embodiment, and the same reference numerals are used. Triangular slope portions 3 and 3 are formed on one side a and the opposite side a of the square bottom portion 1, respectively. Further, trapezoidal slope portions 4 and 4 are formed on the other side a of the square bottom portion 1 and the side a opposite thereto. FIG. 11 is a development view of the block B.

First, the base of the triangular slope 3 is the same as one side a of the square bottom 1, and the angles on both sides sandwiching the base,
That is, the angle between the bottom side and the hypotenuse is 36 degrees. Thus, the apex angle of the triangular slope 3 becomes 108 degrees. Next, the trapezoidal slope 4 is positioned at the bottom (square bottom 1).
(Same as the side a), that is, the angle formed between the lower base and both hypotenuses is 72 degrees, and the angle formed between both hypotenuses and the upper base is 108 degrees (see FIG. 11). .

[0020] Then, mate and the hypotenuse of the hypotenuse and the trapezoidal inclined surface 4 of the triangular inclined surface portion 3, also roof-like body B ₁ of the combine and the upper base between the trapezoid slant portions 4 hipped roof type is configured [See FIG. 12 (A)]. Its roof-like body B _1, 4 pieces of the inclined ridge line j, j, is ... and one of the top ridge k exists.
The inclined ridge line j is formed by joining the oblique sides of the triangular slope 3 and the trapezoidal slope 4 together. The top ridge line k is formed by joining the upper bases of the opposing trapezoidal slope portions 4 and 4 together. The lengths of the inclined ridge j and the top ridge k are equal.

Here, the bottom of the triangular slope 3 and the lower base of the trapezoidal slope 4 are common to one side a of the square bottom 1 and have the same length. The hypotenuse of the triangular slope part 3 and the hypotenuse and the upper base of the trapezoidal slope part 4 are all the same length. Therefore, the ratio of the length of the oblique side of the triangular slope portion 3 and the length of the upper base and the oblique side of the trapezoidal slope portion 4 to the length of one side a of the square bottom portion 1 is shown below.

(The hypotenuse of the triangular slope 3 or the trapezoidal slope 4
: (One side a of the square bottom part 1) = 1:
τ Here, τ = (1 + √5) ÷ 2 ≒ 1.618 That is, the length of one side “a” of the square bottom part 1 is equal to the triangular slope part 3
Τ of the length of the hypotenuse or the hypotenuse and upper base of the trapezoidal slope 4
Double. This is the so-called golden ratio.

The square bottom surface 1 has a plurality of blocks B, B,.
As in the first embodiment described above, sides a, a of the square bottom portions 1, 1 are connected to each other. In the connection structure, adjacent blocks B, B can be folded back to each other [FIG.
(A), (B), (C)). The connecting means is the same as in the first embodiment. That is, the sides a, a of the blocks B, B may be connected by a sheet-like connecting member.

The arrangement of the six blocks B, B,... Is similar to that of the first embodiment, and
To be able to assemble a rectangular parallelepiped with 1, ... Specifically, first, four blocks B, B,... Are connected in a row, and two blocks B, B are arranged from both sides of any one of the blocks B in a direction orthogonal to the row. .

Further, the longitudinal direction of the top ridge lines k, k of the adjacent blocks B, B is arranged so as to be perpendicular to each other [FIG.
0 (A)]. Then, the six blocks B, B,...
Are arranged so that a rectangular parallelepiped can be assembled by turning the entire surface of each square bottom portion 1, 1,... Toward the outside or the inside, if the blocks B, B,. The sequence is not limited to only one. An example of an array configuration of other blocks B, B,.
5 (A), (B), and (C).

[0026] The six blocks B, B, ... square bottom portion 1,1, ... was on the inner surface side, it is projected roof-like body B ₁ outward, the square bottom portion 1,1, ... a When assembled into a rectangular parallelepiped, the triangular slopes 3 and trapezoidal slopes 4 of the adjacent blocks B, B constitute a regular pentagonal surface p (FIG. 13A).
reference〕.

The angle formed between the trapezoidal slope 4 and the square bottom 1 of the block B is 31.7 degrees, and the angle formed between the triangle slope 3 and the square bottom 1 is 58.3 degrees.
Therefore, since the corner angles of the adjacent square bottom portions 1 and 1 of the adjacent blocks B and B are right angles, a regular pentagon formed by the adjacent triangular slope portions 3 and trapezoidal slope portions 4 of the adjacent blocks B and B The surface p is a flat surface (see the part A in FIG. 13B).

FIG. 13 (B) is a diagram showing P in FIG. 13 (A).
FIG. ₁₁ is a sectional view taken along the line _11- P11. As shown in the figure, when a regular pentagonal dodecahedron is assembled with blocks B, B,. A rectangular parallelepiped hollow space is formed as the inner wall.

[0029]

According to the first aspect of the present invention, a square bottom portion 1 is provided.
The square bottom part 1 and of four inclined surface portion 2,2 of the angle was 45 degrees isosceles triangle, ... quadrangular pyramid body A ₁ Metropolitan block A consisting of a 6 Kosonae consisting blocks A, A together Are connected to each other so that they can be folded back,
Blocks are formed by forming a rectangular parallelepiped by positioning the entire surface on the outer side, or forming a rectangular parallelepiped by positioning the entire bottom surface of the square on the inner side. A rectangular parallelepiped and a rhombic dodecahedron can be formed by assembling them from A, A,... According to different procedures, which can help the learner to understand the specific rules of the polyhedron. Further, storage can be performed efficiently. Further, the present invention can be applied to furniture such as chairs having a quaint design.

First, the blocks A, A,... Are formed in accordance with a predetermined definition. When the blocks A, A,. , A rectangular parallelepiped can be formed. The block A, A, ... can be configured assembling the rhombic dodecahedron as outward pyramids body A ₁ side.

Each diamond is represented by an adjacent block A, A
Are formed at the stage of assembling the blocks A, A,..., And the learner confirms the process of forming the rhombic dodecahedron by forming the rhombus. Thus, knowledge acquisition of three-dimensional learning can be efficiently performed. As described above, it is possible not only to make the learner understand the shape of the polyhedron but also to understand the mathematical structure of the polyhedron. This is a learning effect that cannot be obtained from a rhombic dodecahedral molding alone.

Further, storage can be performed efficiently. That is, in the present invention, the blocks A, A,... Can be assembled into a rectangular parallelepiped, and the storage state can be space-saving. Further, the present invention can be applied to furniture such as a chair having a quaint design by applying these.

Next, a second aspect of the present invention is the square bottom portion 1.
And two triangular slope portions 3 and 3 having one side and the opposite side of the square bottom portion 1 as a base and an angle between both sides of the base being 36 degrees.
And two trapezoidal slopes having the other side and the opposite side of the square bottom part 1 as a lower base, an angle at both sides thereof of 72 degrees, and both hypotenuses and an upper base having the same length as the hypotenuse of the adjacent triangular slope part 3. part 4,4 6 blocks B consisting configured roof-like body B ₁ Tokyo at Kosonae, each block B, B each other block B adjacent and freely folded to one another, the roof-like body B of B
The top ridge lines k, k of ₁ and B ₁ are arranged at right angles to each other, and a rectangular parallelepiped is formed by positioning the entire surface of the square bottoms 1, 1,. ... A mathematics teaching tool in which the entire surface is positioned on the inner side to form a rectangular parallelepiped, and a regular dodecahedron consisting of a rectangular parallelepiped and a regular pentagon is constructed by assembling the blocks B, B, ... by different procedures. This can assist the learner in understanding the specific rules of the polyhedron. Further, storage can be performed efficiently.

First, the blocks B, B,... Are formed in accordance with a predetermined definition, and the blocks B, B,. Thus, a rectangular parallelepiped can be formed. The block B, B, the roof-like body B _1, B ₁ of ..., ... can be configured pentagon dodecahedron When assembling the side as outer side.

Each of the regular pentagons is constituted by a triangular slope 3 and a trapezoidal slope 4 of adjacent blocks B, B. At the stage of assembling the blocks B, B,. By being formed, the learner can confirm the process of forming a regular pentagonal dodecahedron,
Knowledge acquisition of three-dimensional learning can be performed efficiently. As described above, it is possible not only to make the learner understand the shape of the polyhedron but also to understand the mathematical structure of the polyhedron. This is a learning effect that cannot be obtained from a molded article having only a regular pentagonal dodecahedron.

Further, storage can be performed efficiently. That is, in the present invention, the blocks B, B,... Can be assembled into a rectangular parallelepiped, and the storage state can be space-saving.

[Brief description of the drawings]

1 (A) is P ₂ perspective view (A) is P ₁ -P ₁ cross-sectional view along a line (A) of the 6 blocks connected in the first embodiment (B) is (A) -P ₂ sectional view taken along the line

FIG. 2 is an exploded view of a single block.

3 (A) is a perspective view of a block alone (B) is P ₄ -P ₄ cross-sectional view along a line P ₃ -P ₃ arrow sectional view (C) is (A) of (A)

[4] (A) is a perspective view of the formation of the rectangular parallelepiped of the first embodiment of the present invention (B) is P ₅ -P ₅ cross-sectional view along a line (A)

FIG. 5A is a perspective view in which a rhombic dodecahedron is formed from the first embodiment of the present invention. FIG. 5B is a cross-sectional view taken along the line P ₆ -P _{6 in} FIG.

FIG. 6 is a schematic view showing a process of forming a rectangular parallelepiped.

FIG. 7 is a perspective view showing a state in which a rectangular parallelepiped is being formed;

FIG. 8 is a schematic view showing a process of forming a rhombic dodecahedron.

FIG. 9 is a perspective view showing a state in which a rhombic dodecahedron is being formed;

FIG. 10 (A) is a cross-sectional view of 6 connected in the second embodiment.
Perspective view of the blocks (B) is P ₈ -P ₈ cross-sectional view along a line P ₇ -P ₇ cross-sectional view along a line (A) (C) is (A)

FIG. 11 is an exploded view of a block unit according to the second embodiment.

12A is a perspective view of a block unit according to the second embodiment, FIG. 12B is a cross-sectional view taken along the line P ₉ -P _{9 in} FIG. 12A, and FIG. 12C is a cross-sectional view taken along the line P ₁₀ -P _{10 in} FIG. Sectional view

13A is a perspective view showing a regular pentagonal dodecahedron formed from the second embodiment of the present invention. FIG. 13B is a sectional view taken along the line P ₁₁ -P _{11 of} FIG. Schematic diagram showing a state during the formation of a dihedron

14A is a schematic plan view showing an example of an arrangement of blocks according to the first embodiment. FIG. 14B is a schematic plan view showing an example of an arrangement of other blocks according to the first embodiment. FIG. 4D is a schematic plan view illustrating an example of an arrangement of another block in the embodiment. FIG.

FIG. 15A is a schematic plan view showing an example of the arrangement of blocks in the second embodiment; FIG. 15B is a schematic plan view showing an example of the arrangement of another block in the second embodiment; FIG. 4 is a schematic plan view illustrating an example of an arrangement of another block in the embodiment.

[Explanation of symbols]

A: Block A ₁ : Square pyramid 1: Square bottom 2 ... Slope m: Ridge B ... Block B ₁ ... Roof 3 ... Triangular slope 4 ... Trapezoidal slope k ... Top ridge

Claims (2)

[Claims] 1. A system comprising six blocks each comprising a square bottom portion and a quadrangular pyramid comprising four slopes of an isosceles triangle having an angle of 45 degrees with the square bottom portion. A mathematics characterized by forming a rectangular parallelepiped by arranging the entire surface of the square bottom portion on the outer side, or forming a rectangular parallelepiped by positioning the entire surface of the square bottom portion on the inner side. Teaching aids. 2. A square bottom portion, two triangular slope portions having one side and opposite side of the square bottom portion as bases and both sides of the bottom side having an angle of 36 degrees, and the other side and opposite sides of the square bottom portion. A block consisting of a roof-like body composed of two trapezoidal slopes having the same length as the hypotenuses of the adjacent triangular slopes on both hypotenuses and the upper base with an angle of 72 degrees on both sides is set as the lower base. Each block, each block can be folded back to each other, and the top ridge line of the roof-like body of the adjacent block is arranged at right angles to each other, forming a rectangular parallelepiped by positioning the entire surface of the square bottom surface on the outer side, Alternatively, a teaching tool for mathematics characterized by forming a rectangular parallelepiped with the entire bottom surface of a square positioned inside.