EFFECTIVE ASPECTS OF LEARNING VECTORS AND OPERATIONS ON THEM USING THE SUDOKU INTERACTIVE METHOD
Abstract
This work is a project study. Sudoku is a game that develops logical thinking, which first appeared in the USA in 1979. The idea of Sudoku goes back to the concept of "Latin squares" created by the Swiss mathematician Leonard Euler. In the study, operations are performed on vectors based on a 4×4 Sudoku field and their interrelationships are analyzed. Using this method, students have the opportunity to combine mathematical concepts with logical thinking in the study of vectors. As a result, the importance of the "Sudoku" method in the educational process and its role in increasing the effectiveness of mastering vectors are substantiated.
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RESEARCH AND EDUCATION ISSN: 2181-3191 VOLUME 4 | ISSUE 9 | 2025 Multidisciplinary Scientific Journal October, 2025 79 DOI: https://doi.org/10.5281/zenodo.17589905 EFFECTIVE ASPECTS OF LEARNING VECTORS AND OPERATIONS ON THEM USING THE SUDOKU INTERACTIVE METHOD Azimova Dilnoza Orifovna Teacher of the Department of Mathematics and Informatics, Bukhara State Pedagogical Institute [email protected] https://orcid.org/ 0009-0006-0277-1513 Abasova Shakhnoza Shukhratovna Bukhara State Pedagogical Institute 2st year student, Physics department [email protected] Abstract: This work is a project study. Sudoku is a game that develops logical thinking, which first appeared in the USA in 1979. The idea of Sudoku goes back to the concept of "Latin squares" created by the Swiss mathematician Leonard Euler. In the study, operations are performed on vectors based on a 4×4 Sudoku field and their interrelationships are analyzed. Using this method, students have the opportunity to combine mathematical concepts with logical thinking in the study of vectors. As a result, the importance of the "Sudoku" method in the educational process and its role in increasing the effectiveness of mastering vectors are substantiated. Keywords: Vector, Sudoku, Crossword, Scalar Product, Method. In the modern education system, the use of innovative and interactive methods in teaching mathematics is of particular importance. In particular, the use of various game and visual methods to convey to students the section on vectors, which is one of the most complex topics, in an understandable way gives effective results. One of such methods is the game "Sudoku". Sudoku, by its nature, is an exercise that develops logical thinking, concentration and memory. It became popular in the USA in 1979, and its idea is actually based on the concept of "Latin squares" created by the Swiss mathematician Leonard Euler.Dastlab gazetalarda krossvord o‘rniga raqamli Sudoku, presented in the form of squares, is widely used today not only as a game, but also as an educational tool.
RESEARCH AND EDUCATION ISSN: 2181-3191 VOLUME 4 | ISSUE 9 | 2025 Multidisciplinary Scientific Journal October, 2025 80 This work examines the process of performing operations on vectors using the example of a 4×4 Sudoku field. According to the rules for solving Sudoku, each row, column, and 2×2 block contains the numbers 1 to 4 once. This structure allows you to place vector elements, compare them, and analyze their combinations. As a result, students will master the concept of vectors not only theoretically, but also in a practical and playful way. Thus, using the Sudoku method, the process of studying the vector section becomes more interesting, interactive and effective. Crossword is an English word that means "word intersection". Crossword is one of the popular intellectual word games in many countries of the world, which helps to develop a person's thinking skills, broaden their horizons, strengthen their memory, and form grammatical literacy. A vector is a quantity that has a direction and a specific length. Its designation is: 𝑎, 𝑏, 𝐴𝐵 A(x, y) Athe beginning of the vector B(x, y) Bthe end of the vector 𝐴𝐵 = (𝑥2−𝑥1)−(𝑦2−𝑦1) Vector sum (triangle rule) Vectors have the property of parallel displacement. 𝑎 𝑏 𝑎 +𝑏 =𝑐 If the beginning of the first vector coincides with the end of the second vector, this is called the triangle rule. If the vectors intersect at a point, this is called the parallelogram rule. Now, since vectors have the property of parallel movement, we will intersect them at a point. Scalar product of vectors
RESEARCH AND EDUCATION ISSN: 2181-3191 VOLUME 4 | ISSUE 9 | 2025 Multidisciplinary Scientific Journal October, 2025 81 𝑎 ×𝑏 =𝑥1×𝑥2+𝑦1+𝑦2+𝑧1×𝑧2 𝑎 ×𝑏 =|𝑎 |×|𝑏 |×cos𝑥 Before we can work out the examples of vectors, we need to know the geometric meaning of vectors. Adding a vector to a vector, subtracting a vector from a vector, and multiplying a vector by a vector all produce a vector. However, multiplying vectors by scalar numbers produces a number. Now let's look at the mixed product of vectors:𝑐 =|𝑎 |×|𝑏 |×sin𝑥 [𝑎×𝑏]=|𝑖 𝑗 𝑘 𝑥1𝑦1𝑧1 𝑥2𝑦2𝑧2|= 𝑦1×𝑧2×𝑖 +𝑥1×𝑦2×𝑘 +𝑥2×𝑧1×𝑗 −𝑦1× 𝑥2×𝑘 −𝑧1×𝑦2×𝑖 −𝑥1×𝑧2×𝑗 =(𝑦1×𝑧2−𝑦2×𝑧1)×𝑖 +(𝑥2×𝑧1−𝑥1× 𝑧2)×𝑗 +(𝑥1×𝑦2−𝑦1×𝑥2)×𝑘 =𝑙𝑖 ×𝑚𝑗 ×𝑛𝑘 We enter the markup: 𝑦1×𝑧2−𝑦2×𝑧1=𝑙 𝑥2×𝑧1−𝑥1×𝑧2=𝑚 𝑥1×𝑦2−𝑦1×𝑥2=𝑛 There is also the concept of the length of a vector. Geometrically, the length of a vector indicates the point from which it starts and the point to which it ends. The general formula for calculating the length of a vector is as follows: |𝑎 |=√𝑥2+𝑦2+𝑧2 Examples 1) |𝑎| =2 |𝑏| =1 cos𝑥=𝜋 3 𝑎 ×𝑏 =|𝑎 |×|𝑏 |cos𝑥 𝑎 ×𝑏 =2×1 2=1 2) |𝑎| =2 |𝑏| =−2 cos𝑥=𝜋 𝑎 ×𝑏 =−2×2×(−1)=4 3) |𝑎| =3 |𝑏| =1 sin𝑥=𝜋 2 2 2 3 4 1 2 1 4 3 2 3 2 4 1 2 3 1 4 4 1 2 3
RESEARCH AND EDUCATION ISSN: 2181-3191 VOLUME 4 | ISSUE 9 | 2025 Multidisciplinary Scientific Journal October, 2025 82 𝑐 =|𝑎| ×|𝑏| ×sin𝑥 𝑐 =3×1×1=3 4) |𝑎| =6 |𝑏| =1 sin𝑥=𝜋 6 𝑐 =6×1×1 2=3 5) |𝑎| =16 |𝑏| =1 4 sin𝑥=𝜋 2 𝑐 =16×1 4×1=4 6) |𝑎| =√3 𝑏 =1 √3 sin𝑥=𝜋 2 𝑐 =√3×1 √3×1=1 7) 𝑎 =(1; 2;1) 𝑏 =(1; 1;1) 𝑎 ×𝑏 =𝑥1×𝑥2+𝑦1×𝑦2×𝑧1×𝑧2 𝑎 ×𝑏 =1×1+2×1+1×1=4 8) 𝑎 =(1;0;0) 𝑏 =(1;0;0) 𝑎 ×𝑏 =1×1+0×0+0=1 9) 𝑎 =( 2;1;0) 𝑏 =( 1 2;1;0) 𝑎 ×𝑏 =2×1 2+1×1+0×0=2 10) 𝑎 =(18;0;0) 𝑏 =(1 6;0;0) 𝑎 ×𝑏 =18×1 6×0×0+0×0=3 y o’ n a l t i r u v c h i 2 6 k o l l i n e a r 3 b i r l i k 5 4 t e n g n o l s k a l y a r
RESEARCH AND EDUCATION ISSN: 2181-3191 VOLUME 4 | ISSUE 9 | 2025 Multidisciplinary Scientific Journal October, 2025 83 Questions 1. If a vector 𝑎 makes angles 𝛼 ,𝛽,𝑎𝑛𝑑 ℒ with the coordinate axes, respectively, then cos𝛼, cos𝛽, cosℒ,𝑎 , are called the cosines of the vector …….. 2. If the vector is aligned with the coordinate axes 𝛼 ,𝛽,𝑎𝑛𝑑 ℒ If it forms angles, thencos𝛼, cos𝛽, cosℒ,𝑎 the cosines of the vector are called... 3. Vectors that lie on a straight line or parallel straight lines are called... vectors. 4. Vectors with length equal to one are called ….. vectors 5. If two non-zero vectors have equal lengths and the same direction, then such vectors are called ..... vectors. 6. . A vector whose beginning and end coincide is called a ….. vector. The product of the lengths of two given vectors and the cosine of the angle between them is called the . product of these two vectors. This work analyzes the process of solving practical examples of vectors using the Sudoku method. The Sudoku game, with its logical structure, can be used as an effective tool for teaching many concepts in mathematics, including vectors and their interrelationships. Using the example of a 4×4 Sudoku grid, the work demonstrates the mathematical and logical foundations of placing numbers from 1 to 4 in each row and column, analyzing their combinations, and the relationships between vectors. By using the Sudoku method, students will not only be able to master the topic of vectors, but also develop skills such as logical thinking, decision-making, and concentration. This method makes the learning process interesting and interactive, while providing a better understanding of mathematical concepts. It was also noted that the use of the Sudoku method in teaching is more effective compared to traditional approaches to studying the topic of vectors. This shows the importance of introducing creative and innovative methods in mathematical education. In the future, further improving this method and connecting it with other mathematical concepts will provide students with the opportunity to gain even broader knowledge [2-10].
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