A demonstration tool for constructing convex hull and minimum spanning tree

Abstract

The utility model discloses a demonstration teaching aid of constructing and determining the maximum and minimum Zhao Shuangxian diagram, and the demonstration teaching aid of constructing and determining the maximum and minimum Zhao Shuangxian diagram includes bottom plate, a pair of horizontal sliding ruler, a pair of vertical sliding ruler, four cross connecting blocks and four laser sliding blocks one, is equipped with four arc-shaped sliding groove and a pair of linear inclined sliding groove of cross arrangement on the bottom plate, can realize the linkage between each component and form the multiple square one and right triangle one required by the maximum and minimum Zhao Shuangxian diagram of determining the accumulation. The demonstration teaching aid of constructing and determining the maximum Zhao Shuangxian diagram includes bottom plate, square reflector plate and four laser sliding blocks two, is equipped with a pair of linear inclined sliding groove of cross arrangement on the bottom plate, can form the multiple square two and right triangle two required by the maximum Zhao Shuangxian diagram of determining the accumulation. The demonstration teaching aid is through the intuitive and image display mode, and students can more intuitively understand the construction principle of Zhao Shuangxian diagram and the mathematical rule behind it.

Description

Technical Field

[0001] This utility model relates to the field of mathematics education and geometry teaching aids, specifically to a demonstration teaching tool for constructing and defining the maximum and minimum product of a Zhao Shuang chord diagram. Background Technology

[0002] In the study of chord diagrams, existing publications indicate that the Pythagorean theorem inequality can be verified using Pythagorean squares and large squares; and that Zhao Shuang's large square can visually illustrate the problems of "maximum product for a fixed sum" and "minimum sum for a fixed product." In teaching practice, teachers often use information technology and mathematical software to demonstrate the quantitative relationship between "maximum product for a fixed sum" and "minimum sum for a fixed product," and how to adjust the values ​​of various variables to maximize the product or minimize the sum of the perimeters. However, beginners and students often find these problems difficult to understand and master. In recent years, the development of educational technology has led to the emergence of many teaching aids, such as mathematical software, interactive whiteboards, and physical models. While these tools improve teaching effectiveness to some extent, they often lack hands-on operation and interactivity, which is not conducive to students' in-depth understanding and mastery.

[0003] Addressing the shortcomings of existing teaching tools in solving specific problems such as "maximum product with a fixed sum" and "minimum sum with a fixed product" using Zhao Shuang's large square graph, the patent background technology aims to provide an intuitive and vivid teaching tool through innovative design to demonstrate the principle of minimizing the sum when the product of the large square graph is fixed, thereby improving teaching effectiveness and learning efficiency. Utility Model Content

[0004] To overcome the shortcomings of the existing technology, the purpose of this utility model is to provide a demonstration teaching aid for constructing and defining the maximum, minimum, and maximum product of a chord graph, so as to solve the problem that there is no demonstration teaching aid in the existing technology that can intuitively and vividly demonstrate the maximum and minimum product of a chord graph.

[0005] To this end, this utility model proposes a demonstration teaching tool for constructing a fixed product and a minimum Zhao Shuang chord diagram, including a base plate, a pair of horizontal sliding rulers, a pair of vertical sliding rulers, four cross connecting blocks and four laser sliders; the base plate is provided with four arc-shaped sliding grooves distributed in a ring array and a pair of straight oblique sliding grooves arranged in a cross shape; A pair of horizontal sliding rulers and a pair of vertical sliding rulers are movably connected by the cross connecting block to form a large square that can be proportionally reduced or enlarged; the bottom of the laser slider is slidably connected to the corresponding arc-shaped sliding groove, and the upper end of the laser slider is slidably engaged with the corresponding horizontal sliding ruler or vertical sliding ruler; The laser slider has a laser head, and the laser line emitted by the laser head is always perpendicular to the horizontal or vertical sliding ruler on the opposite side. The four laser lines intersect to form a small square, the line connecting the center points of the four laser sliders forms an oblique square, and the intersection of two adjacent laser sliders and two laser lines forms a right triangle, thus constructing a definite sum and minimum chord diagram.

[0006] As a preferred embodiment of this utility model, the horizontal sliding ruler is provided with a first linear sliding groove; the vertical sliding ruler is provided with a second linear sliding groove; the cross connecting block is provided with an upper boss, a lower boss and a first sliding guide post from top to bottom; the first sliding guide post is slidably engaged with the first linear oblique sliding groove, the lower boss is slidably engaged with the second linear sliding groove on the vertical sliding ruler, and the upper boss is slidably engaged with the first linear sliding groove on the horizontal sliding ruler.

[0007] As a preferred technical solution of this utility model, the laser slider one further includes a fixing block one, a strip-shaped boss one and a sliding guide post two, which are fixedly connected from top to bottom; the sliding guide post two is slidably engaged with the arc-shaped slide groove, the strip-shaped boss one is slidably engaged with the straight slide groove one or the straight slide groove two; and the laser head one is fixedly installed on one side of the fixing block.

[0008] As a preferred technical solution of this utility model, it also includes a flexible rope, the two ends of each flexible rope being connected to the center position of the fixed block on the adjacent two laser sliders, thereby forming a slanted square.

[0009] In a preferred embodiment of this utility model, the upper boss and the lower boss are arranged in a cross shape and are fixedly connected by a connecting plate.

[0010] As a preferred embodiment of this utility model, the base plate also has a first scale line arranged in a cross shape; the horizontal sliding ruler and the vertical sliding ruler are both provided with a second scale line.

[0011] This utility model also proposes a demonstration teaching aid for constructing and determining the maximum volume of the Zhao Shuang chord diagram, including: a base plate, a square reflector and four laser sliders; the base plate is provided with a pair of straight oblique grooves arranged in a cross shape; The square reflector includes four vertically extending side plates, forming a large square of fixed size. The laser slider includes a fixed block, a laser head, and a strip-shaped protrusion. The strip-shaped protrusion slides in conjunction with the straight inclined groove. The fixed block is fixedly installed on the upper end of the strip-shaped protrusion. The laser head is fixedly installed on the side wall of the fixed block. The laser line emitted by the laser head forms an angle of ° with the straight inclined groove. The laser line passes through the transparent material of the laser head on the adjacent laser slider and shines on the side plate of the square reflector at a distance. Among them, the line connecting the center points of the fixed block 2 forms a small square 2, and the line connecting the intersection points of the four laser lines 2 and the four side plates of the square reflector forms an oblique square 2; the two adjacent intersection points of the laser lines 2 and the side plates of the square reflector form a right triangle 2 with the adjacent fixed block 2, thus constructing the chord diagram with the maximum sum and volume.

[0012] As a preferred technical solution of this utility model, the side plate of the square reflector is provided with scale lines three and a large number of fixing nails; flexible ropes two are suspended in sequence on the fixing nails corresponding to the laser line two to form an oblique square two.

[0013] As a preferred technical solution of this utility model, the laser slider two also includes a locking screw, which runs through the entire laser slider two from top to bottom, and the locking screw abuts against the straight inclined groove.

[0014] This utility model also proposes a demonstration teaching aid for constructing and determining the maximum, minimum and maximum product of the Zhao Shuang chord diagram, including a base plate, a pair of horizontal sliding rulers, a pair of vertical sliding rulers, four cross connecting blocks and four laser sliders; it also includes a square reflector, four laser sliders and a flexible rope; the base plate is provided with four arc-shaped grooves distributed in a ring array and a pair of straight oblique grooves arranged in a cross shape; A pair of horizontal sliding rulers and a pair of vertical sliding rulers are movably connected by the cross connecting block to form a large square that can be proportionally reduced or enlarged; the bottom of the laser slider is slidably connected to the corresponding arc-shaped sliding groove, and the upper end of the laser slider is slidably engaged with the corresponding horizontal sliding ruler or vertical sliding ruler; The laser slider has a laser head, and the laser line emitted by the laser head is always perpendicular to the horizontal sliding ruler or the vertical sliding ruler on the opposite side. The four laser lines intersect to form a small square. The line connecting the center points of the four laser sliders forms an oblique square. The intersection of two adjacent laser sliders and two laser lines forms a right triangle, thereby constructing the integral definite sum minimum Zhao Shuang chord diagram. The square reflector includes four vertically extending side plates, forming a large square of fixed size. The laser slider includes a fixed block, a laser head, and a strip-shaped protrusion. The strip-shaped protrusion slides in conjunction with the straight inclined groove. The fixed block is fixedly installed on the upper end of the strip-shaped protrusion. The laser head is fixedly installed on the side wall of the fixed block. The laser line emitted by the laser head forms an angle of ° with the straight inclined groove. The laser line passes through the transparent material of the laser head on the adjacent laser slider and shines on the side plate of the square reflector at a distance. Among them, the center point of the fixed block 2 is connected to form a small square 2, and the line connecting the intersection points of the four laser lines 2 and the four side plates of the square reflector forms an oblique square 2; the two ends of the flexible rope 2 and the adjacent fixed block 2 form a right triangle 2, thus constructing the Zhao Shuang chord diagram with the maximum sum and volume.

[0015] The demonstration teaching aid provided by this utility model for constructing and determining the maximum, minimum, and maximum product chord diagrams of Zhao Shuang is achieved by setting up a base plate containing an arc-shaped slide and a pair of straight oblique slides, a pair of horizontal sliding rulers, a pair of vertical sliding rulers, four cross connecting blocks, and four laser sliders. This enables the linkage between the components and simultaneously forms multiple squares and right triangles required for determining the minimum and maximum product chord diagrams of Zhao Shuang.

[0016] By setting up a base plate containing a cross-shaped arrangement of straight inclined slides, a square reflector, four laser sliders, and a flexible rope, the linkage between the components is achieved, simultaneously forming multiple squares and right triangles required for the maximum volume of the chord diagram.

[0017] This demonstration tool uses an intuitive and visual approach to help students understand the construction principles of Zhao Shuang's string diagram and the underlying mathematical rules more clearly. Therefore, the promotion and application of this tool will have a positive impact on mathematics education and the field of geometric figure demonstration tools.

[0018] In addition to the purposes, features, and advantages described above, this application has other purposes, features, and advantages. A further detailed description of this application will be provided below with reference to the figures. Attached Figure Description

[0019] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a three-dimensional structural diagram of the demonstration teaching aid for constructing the integral and minimum Zhao Shuang chord diagram of this utility model; Figure 2 This is a schematic diagram of the forward structure of the demonstration teaching aid for constructing the integral definite and minimum Zhao Shuang chord diagram of this utility model; Figure 3 This is a structural diagram of the cross-connecting block in the demonstration teaching aid for constructing the integral definiteness and minimum Zhao Shuang chord diagram of this utility model; Figure 4 This is a structural diagram of the laser slider in the demonstration teaching aid for constructing the integral definite and minimum Zhao Shuang chord diagram of this utility model; Figure 5 This is a demonstration teaching aid for constructing the integral definiteness and minimum Zhao Shuang chord diagram of this utility model. Figure 1 ; Figure 6 This is a demonstration teaching aid for constructing the integral definiteness and minimum Zhao Shuang chord diagram of this utility model. Figure 2 ; Figure 7 This is a demonstration teaching aid for constructing the integral definiteness and minimum Zhao Shuang chord diagram of this utility model. Figure 3 ; Figure 8 This is a three-dimensional structural diagram of the teaching aid for constructing and demonstrating the maximum volume Zhao Shuang chord diagram of this utility model; Figure 9 This is a schematic diagram of the forward structure of the teaching aid for constructing and demonstrating the maximum volume Zhao Shuang chord diagram of this utility model; Figure 10 This is a structural diagram of the laser slider two in the demonstration teaching aid for constructing and displaying the maximum volume Zhao Shuang chord diagram of this utility model; Figure 11 This is a demonstration teaching aid for constructing and displaying the maximum definite volume Zhao Shuang chord diagram of this utility model. Figure 1 ; Figure 12 This is a demonstration teaching aid for constructing and displaying the maximum definite volume Zhao Shuang chord diagram of this utility model. Figure 2 ; Figure 13 This is a demonstration teaching aid for constructing and displaying the maximum definite volume Zhao Shuang chord diagram of this utility model. Figure 3 ; Figure 14 This is a three-dimensional structural diagram of the demonstration teaching aid for constructing and defining the maximum, minimum, and maximum product Zhao Shuang chord diagrams of this utility model; Figure 15 This is a forward structural diagram of the demonstration teaching aid for constructing and defining the maximum, minimum, and maximum product Zhao Shuang chord diagrams of this utility model; Explanation of reference numerals in the attached figures 1. Base plate; 2. Horizontal sliding ruler; 3. Vertical sliding ruler; 4. Cross connecting block; 5. Laser slider one; 6. Flexible rope one; 7. Square reflector; 8. Laser slider two; 9. Flexible rope two; 11. Arc-shaped slide groove; 12. Straight oblique slide groove; 13. Scale line one; 21. Scale line two; 22. Straight slide groove one; 31. Straight slide groove two; 41. Connecting plate; 42. Upper boss; 43. Lower boss; 44. Sliding guide post one; 51. Laser circuit one; 52. Laser head one; 53. Fixing block one; 54. Strip-shaped boss one; 55. Sliding guide post two; 71. Scale line three; 72. Fixing nail; 81. Laser circuit two; 82. Fixing block two; 83. Laser head two; 84. Strip-shaped boss two; 85. Locking screw; 101. Large square one; 102. Small square one; 103. Oblique square one; 104. Right triangle one; 105. Large square two; 106. Small square two; 107. Oblique square two; 108. Right triangle two. Detailed Implementation

[0020] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0021] Example

[0022] like Figures 1-4 As shown, the demonstration teaching aid for constructing the integral and minimum chord diagram of this utility model includes a base plate 1, a pair of horizontal sliding rulers 2, a pair of vertical sliding rulers 3, four cross-shaped connecting blocks 4, and four laser sliders 5. The base plate 1 is provided with four arc-shaped sliding grooves 11 arranged in a ring array and a pair of straight oblique sliding grooves 12 arranged in a cross shape. The base plate 1 also has scale lines 13 arranged in a cross shape. The center line of the straight oblique sliding groove 12 passes through the diagonal of the square-shaped base plate 1, and the center line of the arc-shaped sliding groove 11 is a hyperbola, and the XY coordinate axis of the hyperbola passes through the diagonal of the square-shaped base plate 1.

[0023] Specifically, a pair of horizontal sliding rulers 2 and a pair of vertical sliding rulers 3 are movably connected by a cross connecting block 4 to form a large square 101 that can be proportionally reduced or enlarged; the bottom of the laser slider 5 is slidably connected to the corresponding arc-shaped sliding groove 11, and the upper end of the laser slider 5 is slidably engaged with the corresponding horizontal sliding ruler 2 or vertical sliding ruler 3.

[0024] Among them, such as Figure 3As shown, the horizontal sliding ruler 2 is provided with a straight groove 22 and a scale line 21; the vertical sliding ruler 3 is provided with a straight groove 31 and a scale line 21; the cross connecting block 4 is provided with an upper boss 42, a lower boss 43 and a sliding guide post 44 from top to bottom; the sliding guide post 44 is slidably engaged with the straight oblique groove 12, the lower boss 43 is slidably engaged with the straight groove 31 on the vertical sliding ruler 3, and the upper boss 42 is slidably engaged with the straight groove 22 on the horizontal sliding ruler 2; the upper boss 42 and the lower boss 43 are arranged in a cross shape and are also fixedly connected by a connecting plate 41; at the same time, the connecting plate 41 separates the horizontal sliding ruler 2 and the vertical sliding ruler 3, reducing the mutual friction between the two.

[0025] like Figure 4 As shown, the laser slider 5 includes a laser head 52, a fixing block 53, a strip-shaped boss 54, and a sliding guide post 55. The fixing block 53, the strip-shaped boss 54, and the sliding guide post 55 are fixedly connected from top to bottom. The sliding guide post 55 is slidably engaged with the arc-shaped slide groove 11, and the strip-shaped boss 54 is slidably engaged with the straight slide groove 22 or the straight slide groove 31. The laser head 52 is fixedly installed on the side of the fixing block 53, so that the laser line 51 of the laser head 52 is always perpendicular to the horizontal sliding ruler 2 or the vertical sliding ruler 3 on the opposite side.

[0026] In addition, the demonstration teaching aid for constructing the integral and minimum Zhao Shuang chord diagram of this utility model also includes a flexible rope 6, the two ends of each flexible rope 6 are connected to the center position of the fixed block 53 on the two adjacent laser sliders 5, thereby forming a slanted square 1.

[0027] like Figures 1-7 As shown, during the demonstration and teaching aid of the present invention for constructing the integral and minimum Zhao Shuang chord diagram, one laser slider 5 is first manually pushed to move inward. The sliding guide post 2 55 of the laser slider 5 moves inward along the arc-shaped slide groove 11. At the same time, the strip boss 54 drives the corresponding horizontal sliding ruler 2 or vertical sliding ruler 3 to move in a straight line inward. Then, the cross connecting block 4 drives the other horizontal sliding rulers 2 or vertical sliding rulers 3 to move synchronously, so that the four laser sliders 5 are linked together.

[0028] Among them, the four laser lines 51 intersect to form a small square 102, the line connecting the center points of the four laser sliders 5 forms an oblique square 103, and the intersection of two adjacent laser sliders 5 and two laser lines 51 forms a right triangle 104. In addition, under the constraint of the arc-shaped slide groove 11, the area of ​​the right triangle 104 remains unchanged. At the same time, by observing the scale and calculating the area, we found that the area of ​​the oblique square 103 formed by the four flexible ropes 6 is equal to the area of ​​the four right triangles 104 plus the area of ​​the small square 102. However, as the area of ​​the large square 101 formed by the connection of a pair of horizontal sliding rulers 2 and a pair of vertical sliding rulers 3 becomes smaller and smaller, the area of ​​the oblique square 103 formed by the four flexible ropes 6 also becomes smaller and smaller.

[0029] like Figure 7 As shown, when the length and width of the four right triangles 104 are equal, the area of ​​the small square 102 formed by the intersection of the four laser lines 51 is zero, and the area of ​​the oblique square 103 formed by the four flexible ropes 6 reaches the minimum value; that is, the "product is constant and the sum is minimum" is achieved.

[0030] Example

[0031] like Figures 8-10 As shown, the demonstration teaching aid for constructing and displaying the maximum volume of the Zhao Shuang chord diagram of this utility model includes a base plate 1, a square reflector 7, four laser sliders 8 and a flexible rope 9; wherein, the base plate 1 is provided with a pair of straight oblique grooves 12 arranged in a cross shape; and the center line of the straight oblique grooves 12 passes through the diagonal of the square-shaped base plate 1, and the base plate 1 is also provided with scale lines 13 arranged in a cross shape.

[0032] Specifically, such as Figure 10 As shown, the square reflector 7 is composed of four vertically extending side plates, forming a large square 105 with a fixed size; the side plates of the square reflector 7 are provided with scale lines 71, and the top of the side plates are provided with a large number of fixing nails 72, which are used to suspend the flexible rope 9.

[0033] like Figure 10As shown, the laser slider 2 8 includes a fixing block 2 82, a laser head 2 83, a strip-shaped boss 2 84, and a locking screw 85; the strip-shaped boss 2 84 slides in conjunction with the straight inclined groove 12, the fixing block 2 82 is fixedly installed on the upper end of the strip-shaped boss 2 84, the laser head 2 83 is fixedly installed on the side wall of the fixing block 2 82, and the laser line 2 81 emitted by the laser head 2 83 forms a 45° angle with the straight inclined groove 12. The laser line 2 81 passes through the transparent material of the laser head 2 83 on the adjacent laser slider 2 8 and further illuminates the side plate of the square reflector 7 in the distance. The lines connecting the intersection points of the four laser lines 2 81 and the four side plates of the square reflector 7 form an oblique square 2 107; however, the flexible rope 2 9 is suspended in sequence on the fixing nail 72 corresponding to the laser lines 2 81. The locking screw 85 runs through the entire laser slider 8 from top to bottom. The locking screw 85 abuts against the straight inclined slide groove 12 and is used to temporarily fix the laser slider 8 located on the straight inclined slide groove 12 so that its relative position remains stable.

[0034] like Figures 8-13 As shown, during the demonstration and teaching aid of the present invention for constructing the integral and minimum Zhao Shuang chord diagram, firstly, one laser slider 28 is manually pushed to slide in the straight inclined slide groove 12 and locked by the locking screw 85. Then, the remaining laser slider 28 is pushed in sequence, so that the laser line 281 in the laser slider 28 passes through the center of the fixing block 282 in the next laser slider 28, so that all four laser slider 28 are moved to the corresponding positions, and the positions of the four laser slider 28 are distributed in a ring array about the center point of the base plate 1. Then, the flexible rope 29 is suspended in sequence on the fixing nail 72 corresponding to the laser line 281, forming an oblique square 2107.

[0035] Among them, the center points of fixed block 2 82 are connected to form a small square 2 106, and the flexible rope 2 9 is connected in sequence to form an oblique square 2 107; the two ends of the flexible rope 2 9 and the adjacent fixed block 2 82 form a right triangle 2 108; in addition, under the constraint of the straight oblique groove 12, the sum of the two legs of the right triangle remains unchanged. At the same time, by observing the scale and calculating the area, we found that the area of ​​the large square 2 105 formed by the square reflector 7 remains unchanged. Meanwhile, twice the total area of ​​the four right triangles 2 108 is equal to the area of ​​the large square 2 105 minus the area of ​​the small square 2 106 formed by the center points of fixed block 2 82; however, the small square 2 106 formed by the center points of fixed block 2 82 becomes smaller and smaller, that is, the total area of ​​the four right triangles 2 108 becomes larger and larger.

[0036] like Figure 13As shown, when the four laser sliders 28 move inward to the center of the large square 2, the length and width of the right triangle 2108 are the same. At this time, the small square 2106 formed by connecting the center positions of the four fixed blocks 282 is zero, and the total area of ​​the four equilateral triangles reaches the maximum value; that is, the "maximum sum and maximum area" is achieved.

[0037] Example

[0038] like Figures 1-15 As shown, the demonstration teaching aid for constructing and determining the maximum, minimum and maximum volume of the Zhao Shuang chord diagram of this utility model includes a base plate 1, a pair of horizontal sliding rulers 2, a pair of vertical sliding rulers 3, four cross connecting blocks 4 and four laser sliders 5; it also includes a square reflector 7, four laser sliders 8 and a flexible rope 9. The base plate 1 has four arc-shaped sliding grooves 11 arranged in a ring array and a pair of straight oblique sliding grooves 12 arranged in a cross shape. The base plate 1 also has scale lines 13 arranged in a cross shape. The center line of the straight oblique sliding groove 12 passes through the diagonal of the square-shaped base plate 1. The center line of the arc-shaped sliding groove 11 is a hyperbola, and the XY coordinate axis of the hyperbola passes through the diagonal of the square-shaped base plate 1.

[0039] Specifically, such as Figures 1-4 As shown, a pair of horizontal sliding rulers 2 and a pair of vertical sliding rulers 3 are movably connected by a cross connecting block 4 to form a large square 101 that can be proportionally reduced or enlarged; the bottom of the laser slider 5 is slidably connected to the corresponding arc-shaped sliding groove 11, and the upper end of the laser slider 5 is slidably engaged with the corresponding horizontal sliding ruler 2 or vertical sliding ruler 3.

[0040] Among them, such as Figure 3 As shown, the horizontal sliding ruler 2 is provided with a straight groove 22 and a scale line 21; the vertical sliding ruler 3 is provided with a straight groove 31 and a scale line 21; the cross connecting block 4 is provided with an upper boss 42, a lower boss 43 and a sliding guide post 44 from top to bottom; the sliding guide post 44 is slidably engaged with the straight oblique groove 12, the lower boss 43 is slidably engaged with the straight groove 31 on the vertical sliding ruler 3, and the upper boss 42 is slidably engaged with the straight groove 22 on the horizontal sliding ruler 2; the upper boss 42 and the lower boss 43 are arranged in a cross shape and are also fixedly connected by a connecting plate 41; at the same time, the connecting plate 41 separates the horizontal sliding ruler 2 and the vertical sliding ruler 3, reducing the mutual friction between the two.

[0041] like Figure 4As shown, the laser slider 5 includes a laser head 52, a fixing block 53, a strip-shaped boss 54, and a sliding guide post 55. The fixing block 53, the strip-shaped boss 54, and the sliding guide post 55 are fixedly connected from top to bottom. The sliding guide post 55 is slidably engaged with the arc-shaped slide groove 11, and the strip-shaped boss 54 is slidably engaged with the straight slide groove 22 or the straight slide groove 31. The laser head 52 is fixedly installed on the side of the fixing block 53, so that the laser line 51 of the laser head 52 is always perpendicular to the horizontal sliding ruler 2 or the vertical sliding ruler 3 on the opposite side.

[0042] In addition, the demonstration teaching aid for constructing the integral and minimum Zhao Shuang chord diagram of this utility model also includes a flexible rope 6, the two ends of each flexible rope 6 are connected to the center position of the fixed block 53 on the two adjacent laser sliders 5, thereby forming a slanted square 1.

[0043] Specifically, such as Figures 8-10 As shown, the square reflector 7 is composed of four vertically extending side plates, forming a large square 105 with a fixed size; the side plates of the square reflector 7 are provided with scale lines 71, and the top of the side plates are provided with a large number of fixing nails 72, which are used to suspend the flexible rope 9.

[0044] like Figure 10 As shown, the laser slider 2 8 includes a laser line 2 81, a fixing block 2 82, a laser head 2 83, a strip-shaped boss 2 84, and a locking screw 85. The strip-shaped boss 2 84 slides in conjunction with the straight inclined slide groove 12. The fixing block 2 82 is fixedly installed on the upper end of the strip-shaped boss 2 84. The laser head 2 83 is fixedly installed on the side wall of the fixing block 2 82, and the laser line 2 81 emitted by the laser head 2 83 forms a 45° angle with the straight inclined slide groove 12. The laser line 2 81 passes through the transparent material of the laser head 2 83 on the adjacent laser slider 2 8 and further illuminates the side plate of the square reflector 7 at a distance. The flexible rope 2 9 is suspended sequentially on the fixing nails 72 corresponding to the laser line 2 81, forming a slanted square 2 107. The locking screw 85 runs through the entire laser slider 2 8 from top to bottom and abuts against the straight inclined slide groove 12 to temporarily fix the laser slider 2 8 located on the straight inclined slide groove 12, so as to keep its relative position stable.

[0045] like Figures 1-7 As shown, the process of constructing the universal teaching aid of this utility model and the minimum Zhao Shuang chord diagram is as follows: First, manually push a laser slider 5 to move inward. The sliding guide post 55 of the laser slider 5 moves inward along the arc-shaped slide groove 11. At the same time, the strip boss 54 drives the corresponding horizontal sliding ruler 2 or vertical sliding ruler 3 to move in a straight line inward. Then, the cross connecting block 4 drives the other horizontal sliding rulers 2 or vertical sliding rulers 3 to move synchronously, so that the four laser sliders 5 are linked.

[0046] Among them, the four laser lines 51 intersect to form a small square 102, the line connecting the center points of the four laser sliders 5 forms an oblique square 103, and the intersection of two adjacent laser sliders 5 and two laser lines 51 forms a right triangle 104. In addition, under the constraint of the arc-shaped slide groove 11, the area of ​​the right triangle 104 remains unchanged. At the same time, by observing the scale and calculating the area, we found that the area of ​​the oblique square 103 formed by the four flexible ropes 6 is equal to the area of ​​the four right triangles 104 plus the area of ​​the small square 102. However, as the area of ​​the large square 101 formed by the connection of a pair of horizontal sliding rulers 2 and a pair of vertical sliding rulers 3 becomes smaller and smaller, the area of ​​the oblique square 103 formed by the four flexible ropes 6 also becomes smaller and smaller.

[0047] like Figure 7 As shown, when the length and width of the four right triangles 104 are equal, the area of ​​the small square 102 formed by the intersection of the four laser lines 51 is zero, and the area of ​​the oblique square 103 formed by the four flexible ropes 6 reaches the minimum value; that is, the "product is constant and the sum is minimum" is achieved.

[0048] like Figures 8-13 As shown, the process of constructing the universal teaching aid of this utility model and the minimum Zhao Shuang chord diagram is as follows: First, manually push one laser slider 28 to slide in the straight inclined slide groove 12 and lock it with the locking screw 85. Then, push the remaining laser slider 28 in sequence so that the laser line 281 in the laser slider 28 passes through the center of the fixing block 282 in the next laser slider 28, so that all four laser slider 28 move to the corresponding positions, and the positions of the four laser slider 28 are distributed in a ring array about the center point of the base plate 1. Then, the flexible rope 29 is suspended in sequence on the fixing nail 72 corresponding to the laser line 281 to form an oblique square 2107.

[0049] Among them, the center points of fixed block 2 82 are connected to form a small square 2 106, and the flexible rope 2 9 is connected in sequence to form an oblique square 2 107; the two ends of the flexible rope 2 9 and the adjacent fixed block 2 82 form a right triangle 2 108; in addition, under the constraint of the straight oblique groove 12, the sum of the two legs of the right triangle remains unchanged. At the same time, by observing the scale and calculating the area, we found that the area of ​​the large square 2 105 formed by the square reflector 7 remains unchanged. Meanwhile, twice the total area of ​​the four right triangles 2 108 is equal to the area of ​​the large square 2 105 minus the area of ​​the small square 2 106 formed by the center points of fixed block 2 82; however, the small square 2 106 formed by the center points of fixed block 2 82 becomes smaller and smaller, that is, the total area of ​​the four right triangles 2 108 becomes larger and larger.

[0050] like Figure 13As shown, when the four laser sliders 28 move inward to the center of the large square 2, the length and width of the right triangle 2108 are the same. At this time, the small square 2106 formed by connecting the center positions of the four fixed blocks 282 is zero, and the total area of ​​the four equilateral triangles reaches the maximum value; that is, the "maximum sum and maximum area" is achieved.

[0051] The following is a brief description, with reference to the accompanying drawings, of the construction and working principle of the demonstration teaching aid of the Zhao Shuang chord diagram with maximum, minimum and maximum product.

[0052] The demonstration teaching aid provided by this utility model for constructing and determining the maximum, minimum, and maximum product chord diagrams of Zhao Shuang is achieved by setting up a base plate containing an arc-shaped slide and a pair of straight oblique slides, a pair of horizontal sliding rulers, a pair of vertical sliding rulers, four cross connecting blocks, and four laser sliders. This enables the linkage between the components and simultaneously forms multiple squares and right triangles required for determining the minimum and maximum product chord diagrams of Zhao Shuang.

[0053] By setting up a base plate with a cross-shaped arrangement of straight inclined slides, a square reflector, four laser sliders, and a flexible rope, the various components are linked together to simultaneously form multiple squares and right triangles required for the Zhao Shuang string diagram with maximum volume. This demonstration tool, through an intuitive and visual presentation, enables students to more intuitively understand the construction principles of the Zhao Shuang string diagram and the mathematical rules behind it.

[0054] This patent relates to the field of mathematics education and geometric figure demonstration teaching aids, specifically a demonstration teaching aid for intuitively displaying the "Zhao Shuang string diagram with maximum fixed product and minimum fixed product sum". The design and application of this teaching aid have significant beneficial effects on improving students' understanding of the Zhao Shuang string diagram and deepening their understanding of geometric figures and mathematical principles.

[0055] First, this demonstration tool uses an intuitive and visual approach to help students understand the construction principles of Zhao Shuang's string diagram and the underlying mathematical rules more directly. Traditional mathematics teaching often relies on text and formulas, which students often find difficult to understand and master. This tool, however, transforms abstract mathematical concepts and formulas into concrete graphics and dynamic demonstrations, thereby reducing the difficulty of learning and increasing students' interest and efficiency.

[0056] Secondly, the application of this teaching aid helps cultivate students' spatial imagination and innovative thinking. Through observation and manipulation of this teaching aid, students can not only deepen their understanding of Zhao Shuang's string diagram, but also cultivate their spatial imagination and innovative thinking in the process of exploration and innovation. This has positive significance for improving students' mathematical literacy and comprehensive abilities.

[0057] In addition, this teaching aid also has excellent teaching support functions. Teachers can use it in the classroom to help students understand and master the relevant knowledge of Zhao Shuang's string diagram through demonstration and explanation. At the same time, students can also use this teaching aid for independent learning and exploration, thereby improving their learning outcomes and abilities.

[0058] In conclusion, the demonstration teaching aid for constructing and determining the maximum and minimum product of Zhao Shuang's chord diagrams as described in this patent has significant beneficial effects. It not only improves students' understanding and mastery of Zhao Shuang's chord diagrams but also cultivates their spatial imagination and innovative thinking. Furthermore, it provides teachers with a powerful auxiliary tool for instruction. Therefore, the promotion and application of this teaching aid will have a positive impact on mathematics education and the field of geometric figure demonstration teaching aids.

[0059] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A demonstration teaching aid for constructing a definite product and a minimum Zhao Shuang chord diagram, characterized in that, include: The base plate (1), a pair of horizontal sliding rulers (2), a pair of vertical sliding rulers (3), four cross connecting blocks (4) and four laser sliders (5); the base plate (1) is provided with four arc-shaped sliding grooves (11) distributed in a ring array and a pair of straight oblique sliding grooves (12) arranged in a cross shape. A pair of horizontal sliding rulers (2) and a pair of vertical sliding rulers (3) are movably connected by the cross connecting block (4) to form a large square (101) that can be proportionally reduced or enlarged; the bottom of the laser slider (5) is slidably connected to the corresponding arc-shaped sliding groove (11), and the upper end of the laser slider (5) is slidably engaged with the corresponding horizontal sliding ruler (2) or the vertical sliding ruler (3); Among them, the laser slider (5) has a laser head (52), and the laser line (51) emitted by the laser head (52) is always perpendicular to the horizontal sliding ruler (2) or the vertical sliding ruler (3) on the opposite side. The four laser lines (51) intersect to form a small square (102), the line connecting the center points of the four laser sliders (5) forms an oblique square (103), and the intersection of two adjacent laser sliders (5) and two laser lines (51) forms a right triangle (104), thereby constructing the integral and minimum Zhao Shuang chord diagram.

2. The demonstration teaching aid for constructing the integral definite and minimum Zhao Shuang chord diagram according to claim 1, characterized in that, The transverse sliding ruler (2) is provided with a straight sliding groove 1 (22); the longitudinal sliding ruler (3) is provided with a straight sliding groove 2 (31); the cross connecting block (4) is provided with an upper boss (42), a lower boss (43) and a sliding guide post 1 (44) from top to bottom; the sliding guide post 1 (44) is slidably engaged with the straight oblique sliding groove (12), the lower boss (43) is slidably engaged with the straight sliding groove 2 (31) on the longitudinal sliding ruler (3), and the upper boss (42) is slidably engaged with the straight sliding groove 1 (22) on the transverse sliding ruler (2).

3. The demonstration teaching aid for constructing the integral definite and minimum Zhao Shuang chord diagram according to claim 2, characterized in that, The laser slider (5) further includes a fixing block (53), a strip boss (54), and a sliding guide post (55) that are fixedly connected from top to bottom; the sliding guide post (55) is slidably engaged with the arc-shaped slide groove (11), the strip boss (54) is slidably engaged with the straight slide groove (22) or the straight slide groove (31); and the laser head (52) is fixedly installed on the side of the fixing block (53).

4. The demonstration teaching aid for constructing the integral definite and minimum Zhao Shuang chord diagram according to claim 3, characterized in that, It also includes a flexible rope (6), with the two ends of each flexible rope (6) connected to the center of the fixed block (53) on the two adjacent laser sliders (5), thus forming a slanted square.

5. The demonstration teaching aid for constructing the integral definite and minimum Zhao Shuang chord diagram according to claim 2, characterized in that, The upper boss (42) and the lower boss (43) are arranged in a cross shape and are also fixedly connected by a connecting plate (41).

6. The demonstration teaching aid for constructing the integral definite and minimum Zhao Shuang chord diagram according to claim 1, characterized in that, The base plate (1) also has a cross-shaped scale line (13); the horizontal sliding ruler (2) and the vertical sliding ruler (3) are both provided with a scale line (21).

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