A type of block for deriving the formula for the difference of two square numbers

Abstract

A type of block for deriving the formula for the difference of two squares, characterized in that it comprises ten block tablets, namely a first block tablet, a second block tablet, a third block tablet, a fourth block tablet, a fifth block tablet, a sixth block tablet, a seventh block tablet and an eighth block tablet, wherein the sixth block tablet and the eighth block tablet each have two block tablets; That the first block tablet is a square tablet with a side length of a and an area of ​​a 2 is; That the second block tablet is a square tablet with a side length of b and an area of ​​b 2 is; That the third block tablet is a rectangular tablet with a length of (a+b), a width of (ab) and an area of ​​(a+b)*(ab); That the fourth block tablet is a rectangular tablet with a length of a and a width of (ab); That the fifth block tablet is a rectangular tablet with a length of (ab) and a width of b; That the sixth block tablet is a right-angled trapezoidal tablet with an upper base of b, a lower base of a and a height of (ab); That the seventh block table is an isosceles right-angled triangular table with two right-angled sides of length (ab); That the eighth block tablet is an isosceles right-angled triangular tablet with two right-angled sides of length b and an area of ​​b 2 / 2 is; The first and third block panels are used as the base layer during the derivation, and the remaining eight block panels are placed on top of the base layer as the derivation layer.

Description

[0001] The present utility model relates to the field of mathematical demonstration, in particular to a type of block for deriving the formula for the difference between two square numbers. State of the art

[0002] The formula for the difference of two square numbers: (a+b)(ab)=a 2 -b 2is one of the most basic and important formulas in middle school mathematics textbooks and is mainly derived through the algebraic operation of multiplying polynomials by polynomials. The derivation process of the formula obtained in this way cannot reflect the principle of combining numbers and shapes in mathematics, lacks the support of perceptual activities, and is not conducive to students' in-depth understanding of the principles of mathematical operations. Therefore, it is necessary to design a demonstration teaching aid that is easy to use and can clearly derive the formula for the difference of two square numbers. Content of the utility model

[0003] The technical solution of the present utility model is as follows: a type of block for deriving the formula for the difference between two square numbers, characterized in that it comprises ten block panels, namely a first block panel, a second block panel, a third block panel, a fourth block panel, a fifth block panel, a sixth block panel, a seventh block panel and an eighth block panel, wherein the sixth block panel and the eighth block panel each have two block panels; That the first block panel is a square panel with a side length of a and an area of ​​a 2 is; That the second block panel is a square panel with a side length of b and an area of ​​b 2 is; That the third block panel is a rectangular panel with a length of (a+b), a width of (ab) and an area of ​​(a+b)*(ab); That the fourth block panel is a rectangular panel with a length of a and a width of (ab); That the fifth block panel is a rectangular panel with a length of (ab) and a width of b; That the sixth block panel is a rectangular trapezoidal panel with an upper base of b, a lower base of a and a height of (ab); That the seventh block panel is an isosceles right-angled triangular panel with two right-angled sides of length (ab); That the eighth block panel is an isosceles right-angled triangular panel with two right-angled sides of length b and an area of ​​b 2 / 2 is; The first block panel and the third block panel are used as the base layer in the derivation, and the remaining eight block panels are placed on the base layer as the derivation layer.

[0004] Compared to existing technology, the positive effects of this utility model are as follows: This solution uses the area changes between multiple block panels to calculate (a+b)(ab)=a 2 -b 2 It combines numbers and shapes to reflect the mathematical principle of the formula for the difference between two square numbers and the geometric meaning of the formula for the difference between two square numbers. The entire operation method is simple and clear, and the derivation process is clear. Experiencing the discovery process of mathematical inferences can help students understand the principles of mathematical operations and deepen their learning of the relevant knowledge. The block boards used in this solution are inexpensive to produce and therefore easy to promote and use. Brief description of the drawings Fig. 1 is a schematic representation of the individual block panels of the present utility model; Fig. Figure 2 is a schematic diagram of the arrangement of the block panels of step 1 of the derivative combination 1 of the present utility model; Fig. 3 is a schematic diagram of the arrangement of the block boards of method 1 of step 2 of derivation combination 1 of the present utility model; Fig. 4 is a schematic diagram of the arrangement of the block boards of method 2 of step 2 of derivation combination 1 of the present utility model; Fig. 5 is a schematic diagram of the arrangement of the block boards of method 3 of step 2 of derivation combination 1 of the present utility model; Fig. 6 is a schematic representation of an alternative arrangement of the block panels of method 3 of step 2 of derivative combination 1 of the present utility model; Fig. 7 is a schematic diagram of an arrangement of the block boards of method 4 of step 2 of derivation combination 1 of the present utility model; Fig. Figure 8 is a schematic representation of an alternative arrangement of the block panels of method 4 of step 2 of derivative combination 1 of the present utility model; Fig. 9 is a schematic diagram of the arrangement of the block panels of step 1 of the derivative combination 2 of the present utility model; Fig. 10 is a schematic diagram of the arrangement of the block panels of step 2 of the derivative combination 2 of the present utility model; Fig. 11 is a schematic diagram of the arrangement of the block panels of step 1 of the derivative combination 3 of the present utility model; Fig. Figure 12 is a schematic diagram of the arrangement of the block panels of step 2 of the derivative combination 3 of the present utility model. List of reference symbols:

[0005] 1. first block panel; 2. second block panel; 3. third block panel; 4. fourth block panel; 5. fifth block panel; 6. sixth block panel; 7. seventh block panel; 8. eighth block panel. Description of the preferred embodiments

[0006] In order to clarify the purpose, technical solutions, and advantages of the embodiments of the present utility model, the technical solutions in the embodiments of the present utility model are described clearly and completely below in conjunction with the drawings in the embodiments of the present utility model. Obviously, the described embodiments represent a part of the embodiments of the present utility model and not all embodiments. Starting from the embodiments in the utility model, all other embodiments that an average person skilled in the art can produce without creative effort fall within the scope of the present utility model.

[0007] A type of block for deriving the formula for the difference of two square numbers, as in Fig. 1, comprises ten block panels, namely a first block panel 1, a second block panel 2, a third block panel 3, a fourth block panel 4, a fifth block panel 5, a sixth block panel 6, a seventh block panel 7 and an eighth block panel 8, wherein the sixth block panel 6 and the eighth block panel 8 each have two block panels.

[0008] The first block panel 1 is a square panel with a side length of a and an area of ​​a 2 ; the second block panel 2 is a square panel with a side length of b and an area of ​​b 2; the third block panel 3 is a rectangular panel with a length of (a+b), a width of (ab), and an area of ​​(a+b)*(ab); the fourth block panel 4 is a rectangular panel with a length of a and a width of (ab); the fifth block panel 5 is a rectangular panel with a length of (ab) and a width of b; the sixth block panel 6 is a rectangular trapezoidal panel with an upper base of b, a lower base of a, and a height of (ab); the seventh block panel 7 is an isosceles right-angled triangular panel with two right-angled sides of length (ab); the eighth block panel 8 is an isosceles right-angled triangular panel with two right-angled sides of length b and an area of ​​b 2 / 2.

[0009] Please note that in the description of this scheme, the letters a and b represent the side lengths. If b>a is also true, simply swap a and b. There are no restrictions on the specific lengths of a and b, and they can be adjusted according to the actual situation.

[0010] During the demonstration, this derivation model is divided into two layers. The first block table 1 and the third block table 3 are used as the base layer for the derivation, and the remaining eight block tables are placed on top of the base layer as a derivation layer. The derivation can be based on the square first block table 1 or the rectangular third block table 3.

[0011] And the remaining eight block panels located on the derivation layer are divided into three usage combinations: in combination 1, a second block panel 2, a fifth block panel 5, a sixth block panel 6, and a seventh block panel 7 are combined; in combination 2, a second block panel 2, a fourth block panel 4, and a fifth block panel 5 are combined; and in combination 3, two sixth block panels 6 and two eighth block panels 8 are combined.

[0012] The use of this derivation model is presented in detail below using the attached drawings.

[0013] When deriving the formula for the difference between two square numbers using the block for deriving the formula for the difference between two square numbers, one can start from the square first block table 1 and the square second block table 2 to derive the result from the rectangular third block table 3; likewise, one can start from the rectangular third block table 3 to derive the result of subtracting the area of ​​the square first block table 1 from the area of ​​the square second block table 2; or one can start with the square first block table 1, the square second block table 2 and the rectangular third block table 3 at the same time to establish a connection between the two.

[0014] (1) Starting from the square first block panel 1, the second block panel 2, the fifth block panel 5, the sixth block panel 6 and the seventh block panel 7 have the following arrangements: 1. As in Fig. As shown in Figure 2, the square second block panel 2, the rectangular fifth block panel 5, the rectangular trapezoidal sixth block panel 6, and the isosceles right-angled triangular seventh block panel 7 are placed on the square first block panel 1, and the square second block panel 2, the rectangular fifth block panel 5, the rectangular trapezoidal sixth block panel 6, and the isosceles right-angled triangular seventh block panel 7 can be assembled into a square first block panel 1. Through the puzzle, it can be seen that the sum of the areas of the rectangular fifth block panel 5, the rectangular trapezoidal sixth block panel 6, and the isosceles right-angled triangular seventh block panel 7 = Area of ​​the square first block panel 1 - Area of ​​the square second block panel 2 = (a 2 - b 2 ), 2. As in Fig. 3, then adjust the position of the rectangular fifth block panel 5, the right-angled trapezoidal sixth block panel 6, the isosceles right-angled triangular seventh block panel 7 so that they can be assembled into an isosceles trapezoid, the area of ​​which is (a + b) (a - b); 3. If, as in Fig. 4, adjusts the position of the rectangular fifth block panel 5, the rectangular trapezoidal sixth block panel 6 and the isosceles right-angled triangular seventh block panel 7, they can be assembled into a parallelogram, the area being (a + b)(a - b); 4. As shown in FIGS. 5 and 6, the rectangular trapezoidal sixth block panel 6 can be composed of the rectangular fifth block panel 5 and the isosceles right-angled triangular seventh block panel 7, so that the area of ​​the rectangular trapezoidal sixth block panel 6 is equal to the sum of the areas of the rectangular fifth block panel 5 and the isosceles right-angled triangular seventh block panel 7, which is equal to (a + b)(ab) / 2. Therefore, the sum of the areas of the rectangular fifth block panel 5, the rectangular trapezoidal sixth block panel 6, and the isosceles right-angled seventh block panel 7 is equal to (a + b)(a - b); 5. As in the Fig. 7 and Fig. 8, adjust the positions of the rectangular fifth block panel 5, the rectangular trapezoidal sixth block panel 6 and the isosceles right-angled triangular seventh block panel 7 so that they can be assembled into a rectangle with an area of ​​(a + b)(a - b); (ii) Starting from the square first block panel 1, the second block panel 2, the fourth block panel 4 and the fifth block panel 5 have the following arrangements: 6. As in Fig. As shown in Figure 9, the square second block panel 2, the rectangular fourth block panel 4, and the rectangular fifth block panel 5 lie on the square first block panel 1. The square second block panel 2, the rectangular fourth block panel 4, and the rectangular fifth block panel 5 can be combined to form a square first block panel 1. From the puzzle, it can be seen that the sum of the areas of the rectangular fourth block panel 4 and the rectangular fifth block panel 5 = Area of ​​the square first block panel 1 - Area of ​​the square second block panel 2 = (a 2 - b 2 ); 7. As in Fig. 10, adjust the position of the rectangular fourth block panel 4, the rectangular fifth block panel 5 so that they can be reassembled into a rectangle, this time the area is (a + b) (a - b); (iii) Starting from the rectangular third block panel 3, the second block panel 2, two rectangular trapezoidal sixth block panels 6 and two isosceles right-angled triangular eighth block panels 8 have the following arrangements: 8. As in Fig. 11, the two rectangular trapezoidal sixth block panels 6 are placed on the rectangular third block panel 3, the two rectangular trapezoidal sixth block panels 6 can be put together to form a rectangular third block panel 3, from the puzzle it can be seen that the sum of the areas of the two rectangular trapezoidal sixth block panels 6 = the area of ​​the rectangular third block panel 3 = (a + b)(a - b); 9. As in Fig. 12, the two rectangular trapezoidal sixth block panels 6 are placed on the square first block panel 1, the position of the two rectangular trapezoidal sixth block panels 6 is adjusted, and the two isosceles right-angled triangular eighth block panels 8 are inserted, and it can be seen that the sum of the areas of the two rectangular trapezoidal sixth block panels 6 = the area of ​​the square first block panel 1 - the area of ​​the two isosceles right-angled triangular eighth block panels 8 = (a 2 - b 2 ); 10. As in the Fig. 5 and Fig. As shown in Figure 6, the sixth block panel 6 can be assembled with the fifth block panel 5 and the seventh block panel 7. When building with blocks, the sixth block panel 6 can be used instead of the fifth block panel 5 and the seventh block panel 7 if necessary.

[0015] The block products of this solution can be processed into learning aids for students to use for their own learning, and they can also be processed into teaching aids for teachers to use for demonstration purposes; there are no restrictions here. And in specific applications, it can also be manufactured and processed using various materials, including but not limited to wood panels, acrylic panels, plexiglass panels, magnets, etc. The specific material can be determined according to needs, and details are not required here.

[0016] In summary, this solution illustrates in a vivid and imaginative way the formula (a+b)(ab)=a 2 -b 2can be derived by using the change in area between multiple block boards, combining numbers and shapes, reflecting the mathematical principle of the formula for the difference of two square numbers and the geometric meaning of the formula for the difference of two square numbers, and that the entire operation method is simple and clear, and the derivation process is unambiguous. Experiencing the discovery process of mathematical inferences can help students understand the principle of mathematical operations and deepen the learning of related knowledge. The block boards used in this solution are inexpensive to produce and therefore easy to promote and use.

Claims

[1] A type of block for deriving the formula for the difference of two square numbers, characterized by that it comprises ten block panels, namely a first block panel, a second block panel, a third block panel, a fourth block panel, a fifth block panel, a sixth block panel, a seventh block panel and an eighth block panel, the sixth block panel and the eighth block panel each having two block panels; That the first block panel is a square panel with a side length of a and an area of a 2 is; That the second block panel is a square panel with a side length of b and an area of b 2 is; That the third block panel is a rectangular panel with a length of (a+b), a width of (ab) and an area of (a+b)*(ab); That the fourth block panel is a rectangular panel with a length of a and a width of (ab); That the fifth block panel is a rectangular panel with a length of (ab) and a width of b; That the sixth block panel is a rectangular trapezoidal panel with an upper base of b, a lower base of a and a height of (ab); That the seventh block panel is an isosceles right-angled triangular panel with two right-angled sides of length (ab); That the eighth block panel is an isosceles right-angled triangular panel with two right-angled sides of length b and an area of b 2 / 2 is; The first block panel and the third block panel are used as the base layer in the derivation, and the remaining eight block panels are placed on the base layer as the derivation layer. [2] Block for deriving the formula for the difference between two square numbers according to claim 1, characterized bythat the remaining eight block panels are divided into three usage combinations: in combination 1, a second block panel, a fifth block panel, a sixth block panel and a seventh block panel are combined; in combination 2, a second block panel, a fourth block panel and a fifth block panel are combined; and in combination 3, two sixth block panels and two eighth block panels are combined.

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