Complete square formula derivation building block

Abstract

The utility model relates to a complete square formula derivation building block which comprises a first building block board, a second building block board, a third building block board, a fourth building block board, a fifth building block board, a sixth building block board, a seventh building block board and an eighth building block board. According to the utility model, the complete square formulas (a + b) 2 = a2 + 2a + b2 and (a-b) 2 = a2-2a + b2 and the relationship between the two can be vividly deduced by utilizing the area change after the movement and recombination among the plurality of building block plates, the mathematical principle of the complete square formulas and the geometric significance of the complete square formulas are reflected through the combination of numbers and forms, the whole operation steps are simple and clear, the deduction process is clear, and the calculation efficiency is high. A mathematical conclusion discovery process is experienced, students can be helped to understand mathematical operation principles, and learning of related knowledge is deepened. The building block board used in the scheme is low in manufacturing cost, so that the building block board is easy to popularize and apply.

Description

Technical Field

[0001] The utility model relates to the technical field of educational appliances, in particular to a building block for deriving perfect square formulas. Background Technique

[0002] The perfect square formula is one of the most basic and important formulas in mathematics, including two formulas, namely:

[0003] The square of the sum of two numbers: (a + b)2 = a 2 + 2ab + b 2

[0004] The square of the difference of two numbers: (a - b)2 = a 2 - 2ab + b 2

[0005] In junior high school mathematics textbooks, the formula is mainly derived through algebraic operations such as multiplying polynomials by polynomials. The derivation process of the obtained formula cannot reflect the principle of the combination of numbers and shapes in mathematics, lacks the understanding of perceptual activities, and is not conducive to students' profound understanding of the principles of mathematical operations.

[0006] Therefore, a building block for deriving perfect square formulas is proposed, which can solve the above-mentioned drawbacks through an intuitive way of building blocks. Content of the Utility Model

[0007] The utility model provides a building block for deriving perfect square formulas to solve the problems of the prior art.

[0008] To solve the above technical problems, the utility model is realized through the following technical solutions: A building block for deriving perfect square formulas is composed of 12 building blocks of 8 types.

[0009] The first building block board is a square with a side length of (a + b) and an area of (a + b)2, and there is 1 piece in total;

[0010] The second building block board is a square with a side length of a and an area of a 2 , and there is 1 piece in total;

[0011] The third building block board is a square with a side length of b and an area of b 2 , and there are 2 pieces in total;

[0012] The fourth building block board is a square with a side length of (a - b) and an area of (a - b)2, and there are 2 pieces in total;

[0013] The fifth and sixth building block boards are rectangles with a length of a and a width of b, and the area is ab, and there is 1 piece each;

[0014] The seventh and eighth building block boards are rectangles with a length of (a - b) and a width of b, and the area is (ab - b2 ), two pieces each.

[0015] Among the above-mentioned wooden building blocks, the seventh (or eighth) rectangular wooden building block and the third square wooden building block can be combined to form the fifth (or sixth) rectangular wooden building block.

[0016] It should be noted that in the description of this solution, the letters a and b represent side lengths. Generally, a > b. If b > a also holds, just swap the positions of a and b in the expressions in this article.

[0017] This utility model uses the combination of numbers and shapes, adopts various derivation methods, and demonstrates two perfect square formulas: (a ± b)² = a 2 ± 2ab + b 2 The derivation process and the relationship between these two perfect square formulas are reflected, the mathematical principle is clear, the operation is simple and straightforward, the derivation process is clear, the production cost is low, and it is easy to promote.

[0018] The usage principle of the perfect square formula derivation building blocks:

[0019] The perfect square formula derivation building blocks can derive the square of the sum of two numbers and the square of the difference of two numbers in the perfect square formula, as well as the relationship between the two perfect square formulas.

[0020] (I) Perfect square formula (square of the sum of two numbers) (a + b)² = a 2 + 2ab + b 2 The derivation building blocks are composed of 5 blocks in two layers:

[0021] The first basic layer: one block, a square wooden building block with side length (a + b), the first wooden building block;

[0022] The second derivation layer: four blocks, a square wooden building block with side length a, the second wooden building block, a square wooden building block with side length b, the third wooden building block, and two rectangular wooden building blocks with length a and width b, the fifth and sixth wooden building blocks;

[0023] For the usage method, see the specific embodiments below.

[0024] (II) Perfect square formula (square of the difference of two numbers) (a - b)² = a 2 - 2ab + b 2 The derivation building blocks have two combinations and three methods:

[0025] Combination 1: Composed of five blocks, a square wooden building block with side length a, the second wooden building block, with an area of a 2 , a square wooden building block with side length b, the third wooden building block, with an area of b 2, a fourth building block of a square with side length (a - b), the area of the fourth building block is (a - b)^2, two rectangular building blocks of the fifth and sixth building blocks with length a and width b respectively, and the area is ab;

[0026] For the usage method, see the specific embodiments below.

[0027] Combination 2: Consists of 11 building blocks in four layers:

[0028] The third basic layer: One building block, a square building block of the second building board with side length a;

[0029] The fourth auxiliary layer: Four building blocks, a square building block of the third building board with side length b (b < a), a square building block of the fourth building board with side length (a - b), and two rectangular building blocks of the seventh and eighth building blocks with length (a - b) and width b respectively;

[0030] The fifth derivation layer: Four building blocks, a square building block of the third building board with side length b, a square building block of the fourth building board with side length (a - b), and two rectangular building blocks of the seventh and eighth building blocks with length b and width (a - b) respectively;

[0031] The sixth control layer: Two building blocks, rectangular building blocks of the fifth and sixth building blocks with length a and width b respectively. The building blocks of the seventh and third building blocks in the derivation layer and the auxiliary layer can be combined into a sixth building block, and the building blocks of the eighth and third building blocks can be combined into a fifth building block. The area of the fifth building block or the sixth building block is equal to the sum of the areas of a square building block with side length b and two figures of a rectangular building block with length b and width (a - b);

[0032] For the usage method, see the specific embodiments below.

[0033] (III) Derivation of the relationship between two perfect square formulas, that is, the relationship between the square of the sum of two numbers and the square of the difference of two numbers: (a + b)^2 - (a - b)^2 = 4ab, or (a + b)^2 = (a - b)^2 + 4ab or (a + b)^2 - 4ab = (a - b)^2, which consists of two layers of building blocks.

[0034] The seventh basic layer: One square building block, a square building block of the first building board with side length (a + b), and the area is (a + b)^2;

[0035] The eighth derivation layer: Consists of seven building blocks: a square building block of the fourth building board with side length (a - b), two square building blocks of the third building board with side length b, two rectangular building blocks of the fifth and sixth building blocks with length a and width b respectively, and two rectangular building blocks of the seventh and eighth building blocks with length (a - b) and width b respectively;

[0036] There are ten derivation methods: For the usage method, please refer to the specific embodiments below.

[0037] The beneficial effects of the present utility model are as follows:

[0038] 1. By using the area change after the movement and recombination between multiple product wood boards, the present utility model can vividly derive the perfect square formula (a + b)2 = a 2 + 2ab + b 2 and (a - b)2 = a 2 - 2ab + b 2 and the relationship between them, combining numbers and shapes, reflecting the mathematical principle and geometric meaning of the perfect square formula. The whole operation steps are simple and clear, the derivation process is clear, and the process of discovering mathematical conclusions can be experienced, which can help students understand the principle of mathematical operations and deepen the learning of relevant knowledge. The product wood boards used in this solution have low production costs, so they are easy to promote and apply. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 are the schematic diagrams of the respective building block structures of the present utility model.

[0040] Figure 2 is a schematic diagram of an embodiment of the present utility model Figure 1 ;

[0041] Figure 3 is a schematic diagram of an embodiment of the present utility model Figure 2 ;

[0042] Figure 4 is a schematic diagram of an embodiment of the present utility model Figure 3 ;

[0043] Figure 5 is a schematic diagram of an embodiment of the present utility model Figure 4 ;

[0044] Figure 6 is a schematic diagram of an embodiment of the present utility model Figure 5 ;

[0045] Figure 7 is a schematic diagram of an embodiment of the present utility model Figure 6 ;

[0046] Figure 8 is a schematic diagram of an embodiment of the present utility model Figure 7 ;

[0047] Figure 9 is a schematic diagram of an embodiment of the present utility model Figure 8 ;

[0048] Figure 10Schematic diagram of an embodiment of the present utility model Figure 9 ;

[0049] Figure 11 Schematic diagram of an embodiment of the present utility model Figure 10 ;

[0050] Figure 12 Schematic diagram of an embodiment of the present utility model Figure 10 One;

[0051] Figure 13 Schematic diagram of an embodiment of the present utility model Figure 10 Two;

[0052] Figure 14 Schematic diagram of an embodiment of the present utility model Figure 10 Three;

[0053] Figure 15 Schematic diagram of an embodiment of the present utility model Figure 10 Four;

[0054] Figure 16 Schematic diagram of an embodiment of the present utility model Figure 10 Five.

[0055] Figures 1-16 Among them: 1. First product board; 2. Second product board; 3. Third product board; 4. Fourth product board; 5. Fifth product board; 6. Sixth product board; 7. Seventh product board; 8. Eighth product board. Specific implementation manner

[0056] To make the objectives, technical solutions, and advantages of the embodiments of the present utility model clearer, the technical solutions in the embodiments of the present utility model will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present utility model. Obviously, the described embodiments are part of the embodiments of the present utility model, rather than all of the embodiments. Based on the embodiments of the present utility model, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present utility model.

[0057] Embodiment 1: As Figures 1~16 shown, a kind of perfect square formula derivation building block is composed of 12 building blocks of 8 types:

[0058] The first product board is a square with a side length of (a + b) and an area of (a + b)2, and there is 1 piece in total;

[0059] The second product board is a square with a side length of a and an area of a 2 , and there is 1 piece in total;

[0060] The third product board is a square with a side length of b and an area of b 2 , and there are 2 pieces in total;

[0061] The fourth building block for multiplication is square with side length of (a - b), and its area is (a - b)^2. There are 2 pieces in total.

[0062] The fifth and sixth building blocks for multiplication are rectangles with length of a and width of b, and their area is ab. There is 1 piece for each.

[0063] The seventh and eighth building blocks for multiplication are rectangles with length of (a - b) and width of b, and their area is (ab - b 2 ), and there are 2 pieces for each.

[0064] Among the above building blocks for multiplication, the seventh rectangular building block and the third square building block can be assembled into the fifth rectangular building block; the seventh rectangular building block and the third square building block can be assembled into the sixth rectangular building block; the eighth rectangular building block and the third square building block can be assembled into the fifth rectangular building block; the eighth rectangular building block and the third square building block can be assembled into the sixth rectangular building block.

[0065] It should be noted that in the description of this solution, the letters a and b represent side lengths. Generally, a > b. If b > a also holds, just swap the positions of a and b in the expressions in this article.

[0066] II. Usage method of the building blocks for deriving the perfect square formula:

[0067] The building blocks for deriving the perfect square formula can derive the square of the sum of two numbers and the square of the difference of two numbers in the perfect square formula, as well as the relationship between the two perfect square formulas.

[0068] (I) Perfect square formula (square of the sum of two numbers) (a + b)^2 = a 2 + 2ab + b 2 The building blocks for derivation consist of 5 building blocks in two layers:

[0069] The first layer (basic layer): one building block, which is the first building block for multiplication, a square with side length of (a + b).

[0070] The second layer (derivation layer): four building blocks, one second building block for multiplication, a square with side length of a,

[0071] one third building block for multiplication, a square with side length of b, and two fifth and sixth building blocks for multiplication, rectangles with length and width of a and b respectively.

[0072] Usage method:

[0073] (1) Such as Figure 2 、 Figure 3As shown in the figure, assemble 5 building blocks on the first layer and the second layer respectively, stack the two layers of building blocks together, and the top view is a square with a side length of (a + b);

[0074] (2) There is only the first building block board on the first layer. On the second layer's derivation layer, the second building block board, the third building block board, the fifth building block board, and the sixth building block board can be placed. Adjust the positions of the second building block board, the third building block board, the fifth building block board, and the sixth building block board until the second layer of building blocks completely coincides with the first layer of building blocks up and down;

[0075] (3) Such as Figure 2 、 Figure 3 two derivation methods;

[0076] (4) The above derivation shows that: (a + b)2 = a 2 + 2ab + b 2 .

[0077] (2) The perfect square formula (the square of the difference between two numbers) (a - b)2 = a 2 - 2ab + b 2 Derivation building blocks, there are two combinations and three methods:

[0078] Combination 1: Consists of five building blocks, a second building block board which is a square with a side length of a, with an area of a 2 , a third building block board which is a square with a side length of b, with an area of b 2 , a fourth building block board which is a square with a side length of (a - b), with an area of (a - b)2, and two fifth building block boards and sixth building block boards which are rectangles with lengths and widths of a and b respectively, with an area of ab.

[0079] Derivation method 1, such as Figure 4 :

[0080] The first layer (basic layer): Two square building blocks, one is a second building block board which is a square with a side length of a, with an area of a 2 , and the other is a third building block board which is a square with a side length of b, with an area of b 2 ;

[0081] The second layer (derivation layer): Consists of three building blocks: a fourth building block board which is a square with a side length of (a - b), and two fifth building block boards and sixth building block boards which are rectangles with lengths and widths of a and b respectively;

[0082] The area of the first layer is (a 2 + b 2 ). In the second layer's derivation layer, remove the fifth building block board and the sixth building block board which are rectangles. What remains is the fourth building block board, with an area of (a - b)2. Therefore, it can be deduced that:

[0083] (a - b)2 = (a 2 + b 2 ) - 2ab = a 2 - 2ab + b 2 。

[0084] Derivation method 2, such as Figure 5

[0085] The first layer (basic layer): a square building block, a second building block board which is a square building block with side length a, and the area is a 2 ;

[0086] The second layer (derivation layer): composed of three building blocks: a fourth building block board which is a square with side length (a - b), and two fifth and sixth building block boards which are rectangles with length a and width b respectively;

[0087] The third layer (supplementary layer): a third building block board which is a square building block with side length b, and the area is b 2 。

[0088] The area of the first layer is a 2 , in the second layer of the derivation layer, remove the fifth and sixth building block boards of the rectangular building blocks. The overlapping part removed twice is the third building block board. Supplement it in the third layer. What remains is the fourth building block board with an area of (a - b)2. Therefore, it can be deduced that: (a - b)2 = (a 2 + b 2 ) - 2ab = a 2 - 2ab + b 2 。

[0089] The above derivation shows that: (a - b)2 = a 2 - 2ab + b 2 。

[0090] Combination 2: Composed of 11 building blocks in four layers:

[0091] The first layer (basic layer): a building block, the second building block board which is a square building block with side length a;

[0092] The second layer (auxiliary layer): four building blocks, a third building block board which is a square building block with side length b (b < a), a fourth building block board which is a square building block with side length (a - b), and two seventh and eighth building block boards which are rectangles with length (a - b) and width b respectively;

[0093] The third layer (derivation layer): four building blocks, a third building block board which is a square building block with side length b, a fourth building block board which is a square building block with side length (a - b), and two seventh and eighth building block boards which are rectangles with length b and width (a - b) respectively;

[0094] For the two building blocks in the fourth layer (control layer), the rectangular building blocks with lengths and widths of a and b respectively, the fifth building block board, the sixth building block board, the building blocks in the derivation layer and the auxiliary layer, the seventh building block board and the third building block board can be assembled into a sixth building block board, the eighth building block board and the third building block board can be assembled into a fifth building block board, and the area of the fifth building block board or the sixth building block board is equal to the sum of the areas of two figures, namely a square building block with side length b and a rectangular building block with lengths and widths of b and (a - b) respectively.

[0095] Derivation method 3, as Figure 6 , the second layer (auxiliary layer) and the third layer (derivation layer) are arranged in the same way up and down.

[0096] (1) According to the schematic diagram of the patent solution Figure 6 , assemble 11 building blocks on the first layer, the second layer, the third layer and the fourth layer respectively, stack the four layers of building blocks together, and the top view is a square with side length a;

[0097] (2) Cover the sixth building block board of the fourth layer (control layer) on the seventh building block board and the third building block board of the third layer to prove that the sum of the areas of the seventh building block board and the third building block board is equal to ab, and cover the fifth building block board on the eighth building block board and the third building block board of the third layer to prove that the sum of the areas of the eighth building block board and the third building block board is equal to ab;

[0098] (3) Remove the fifth building block board and the sixth building block board of the fourth layer;

[0099] (4) Conduct derivation on the third layer (derivation layer), remove the seventh building block board and the third building block board, which is equivalent to subtracting an ab from the first building block board;

[0100] (5) Continue to conduct derivation on the third layer (derivation layer) and the second layer (auxiliary layer), remove the eighth building block board and the third building block board, which is equivalent to subtracting another ab from the first building block board;

[0101] (6) During the derivation on the third layer (derivation layer), the third building block board of the second layer (auxiliary layer) is removed, and the third building block board needs to be placed back on the auxiliary layer again. Therefore, b should be added 2 , in this way, the second layer (auxiliary layer) returns to its original state;

[0102] (7) Only the fourth building block board remains on the derivation layer, and its area is (a - b)2;

[0103] (8) The above derivation shows that: (a - b)2 = a 2 - 2ab + b 2 .

[0104] (3) Derivation of the relationship between two perfect square formulas, that is, the relationship between the square of the sum of two numbers and the square of the difference of two numbers: \((a + b)^2-(a - b)^2 = 4ab\), or \((a + b)^2=(a - b)^2 + 4ab\) or \((a + b)^2-4ab=(a - b)^2\), which consists of two layers of building blocks.

[0105] The first layer (basic layer): A square building block, the first product board of a square building block with side length \((a + b)\), and the area is \((a + b)^2\).

[0106] The second layer (derivation layer): It consists of seven building blocks: the fourth product board of a square building block with side length \((a - b)\), the third product board of two square building blocks with side length \(b\), the fifth and sixth product boards of two rectangular building blocks with length \(a\) and width \(b\), and the seventh and eighth product boards of two rectangular building blocks with length \((a - b)\) and width \(b\).

[0107] There are the following ten derivation methods:

[0108] Such as Figure 7 、Such as Figure 8 、Such as Figure 9 、Such as Figure 10 、Such as Figure 11 、Such as Figure 12 、Such as Figure 13 、Such as Figure 14 、Such as Figure 15 、Such as Figure 16 。

[0109] The above derivation shows that \((a + b)^2-(a - b)^2 = 4ab\), or \((a + b)^2=(a - b)^2 + 4ab\) or \((a + b)^2-4ab=(a - b)^2\).

[0110] The above embodiments can be combined with each other.

[0111] The utility model can vividly derive two perfect square formulas \((a\pm b)^2 = a\) 2 \(\pm2ab + b\) 2 and the relationship between the two formulas by using the area change after the movement and recombination of multiple product boards. The utility model conducts the derivation of the perfect square formula, combines numbers and shapes, reflects the mathematical principle and geometric meaning of the perfect square formula, has a simple and clear operation, a clear derivation process, and experiences the discovery process of mathematical conclusions, which can help students understand the mathematical operation principle and deepen the learning of relevant knowledge. The utility model has a low production cost and is easy to promote.

[0112] Although the embodiments of the present invention have been shown and described above, those of ordinary skill in the art can understand that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the claims and their equivalents.

[0113] In the present invention, the building blocks for the derivation of the perfect square formula are not limited to the production materials. The building blocks for the derivation of the perfect square formula can be made of wooden boards, acrylic boards, plexiglass boards, or plastic boards and can be used as learning tools to help students learn the perfect square formula; they can also be made of magnetic tiles as teaching aids for teachers' teaching demonstrations.

[0114] It should be noted that the terms "first", "second", and "first product wooden board", "second product wooden board", etc. in the specification, claims, and the above-mentioned drawings of this application are used to distinguish similar objects and do not necessarily need to describe a specific order or sequence. It should be understood that such used data can be interchanged under appropriate circumstances so that the embodiments of this application described here can be implemented in an order other than those illustrated or described here.

[0115] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by orientation words such as "base layer", "derivation layer", "control layer", "auxiliary layer", "front, back, up, down, left, right", "horizontal, vertical, perpendicular, horizontal", and "top, bottom" is usually based on the orientation or positional relationship shown in the drawings. It is only for the convenience of describing the present invention and simplifying the description. Without contrary instructions, these orientation words do not indicate and imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation. Therefore, it should not be understood as a limitation on the protection scope of the present invention; the orientation words "inside, outside" refer to the inside and outside relative to the contour of each component itself.

Claims

1. A perfect square formula derivation building block, comprising a first building block board (1), a second building block board (2), a third building block board (3), a fourth building block board (4), a fifth building block board (5), a sixth building block board (6), a seventh building block board (7) and an eighth building block board (8), characterized in that: The number of the first building board (1), the second building board (2), the fifth building board (5) and the sixth building board (6) is set to one, and the number of the third building board (3), the fourth building board (4), the seventh building board (7) and the eighth building board (8) is set to two; The first building block (1) is a square, with a side length of (a+b) and an area of ​​(a+b). 2 ; The second building block (2) is a square with a side length of a and an area of ​​a. 2 ; The third building block (3) is a square with a side length of b and an area of ​​b. 2 ; The fourth building block (4) is a square, with a side length of (ab) and an area of ​​(ab). 2 ; The fifth building board (5) and the sixth building board (6) are rectangular, with a length of a, a width of b, and an area of ​​ab; The seventh building block board (7) and the eighth building block board (8) are rectangular, with a length of (ab), a width of b, and an area of ​​(ab-b 2 ).

2. A perfect square formula derivation building block according to claim 1, characterized in that: Perfect square formula, (a+b)2=a 2 +2ab+b 2 The derivation building blocks consist of two layers of five building blocks, including the first basic layer and the second derivation layer; The first base layer is configured as a first building block board (1); The second derivation layer: four building blocks, including a second building block board (2), a third building block board (3), a fifth building block board (5) and a sixth building block board (6).

3. A perfect square formula derivation building block according to claim 1, characterized in that: Perfect Square Formula (ab) 2 =a 2 -2ab+b 2 There are two combinations of derivation blocks: combination one and combination two.

4. A perfect square formula derivation building block according to claim 3, characterized in that: The combination 1 is composed of two layers of five building boards, including a second building board (2), a third building board (3), a fourth building board (4), a fifth building board (5) and a sixth building board (6).

5. A perfect square formula derivation building block according to claim 3, characterized in that: The combination 2 is composed of four layers and eleven building blocks, including a third basic layer, a fourth auxiliary layer, a fifth derivation layer and a sixth control layer; The third base layer is set as a second building block board (2); A fourth auxiliary layer, four building blocks, including a third building block board (3), a fourth building block board (4), a seventh building block board (7) and an eighth building block board (8); The fifth derivation layer, four building blocks, including a third building block board (3), a fourth building block board (4), a seventh building block board (7) and an eighth building block board (8); The sixth comparison layer comprises two building blocks, including a fifth building board (5) and a sixth building board (6); the seventh building board (7) and the third building board (3) can be assembled into the sixth building board (6); the eighth building board (8) and the third building board (3) can be assembled into the fifth building board (5); the seventh building board (7) and the third building board (3) can be assembled into the fifth building board (5); the seventh building board (8) and the third building board (3) can be assembled into the sixth building board (6); the area of ​​the third building board (3) and the seventh building board (7) or the third building board (3) and the eighth building board (8) is equal to the area of ​​the fifth building board (5) or the sixth building board (6).

6. A perfect square formula derivation building block according to claim 1, characterized in that: The derivation of the relationship between two perfect squares, that is, the derivation of the relationship between the square of the sum of two numbers and the square of the difference between the two numbers: (a+b) 2 -(ab) 2 =4ab, or (a+b) 2 =(ab) 2 +4ab or (a+b) 2 -4ab=(ab) 2 , consists of two layers of building blocks, including the seventh basic layer and the eighth derivation layer; The seventh base layer is configured as a first building block board (1); The eighth derivation layer is composed of seven building blocks, including two third building blocks (3), a fourth building block (4), a fifth building block (5), a sixth building block (6), a seventh building block (7) and an eighth building block (8).

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