INDUCTION AND APPLICATION RESEARCH ON CORE TRANSFORMATION TECHNIQUES IN ELEMENTARY MATHEMATICS PROBLEM-SOLVING

Danish Scientific Journal No102, 2025 27 MATHEMATICAL SCIENCES INDUCTION AND APPLICATION RESEARCH ON CORE TRANSFORMATION TECHNIQUES IN ELEMENTARY MATHEMATICS PROBLEM-SOLVING He Xiaoxia JiNing Normal University, School of Mathematics and statistic, Ulanqab, Inner Mongolia, P. R. China https://doi.org/10.5281/zenodo.17740982 Abstract Problem-solving in elementary mathematics is flexible, and the process often relies on various algebraic or geometric transformation techniques. As the core of mathematics learning and problem-solving, these techniques come in diverse forms and are systematically sorted out in this paper. The purpose is to summarize the common core transformation techniques in elementary mathematics and clarify their application and value in problemsolving. Keywords: Elementary Mathematics; Transformation Techniques; Problem-Solving; Application 1 Introduction In the process of mathematical problem-solving, appropriate transformation and flexible application of techniques are crucial for improving problem-solving efficiency and deepening understanding. When facing complex mathematical problems, rational use of transformation techniques can quickly clarify the problemsolving direction, simplify the problem structure, thereby efficiently obtaining clear and complete solutions, and making mathematical learning fulfilling and meaningful. Mathematical transformation (or algebraic/formal conversion) refers to a method of adjusting the form of the original problem to derive conclusions more conveniently during problem-solving. It is not only a preparatory stage for realizing advanced mathematical thinking such as "reduction to a known problem", "transformation" and "association", but also an important bridge connecting known conditions and unknown conclusions. When the conditions given in the problem are insufficient or obscure, transforming the form can establish connections between scattered elements, unclear conditions and relevant concepts, theorems and properties, and convert the problem to be solved into an easy-to-handle standard form—this is the fundamental significance of transformation [1]. In elementary mathematics, there is a rich variety of transformation methods, including but not limited to analytical method, scaling method, mathematical induction, constant separation method, variable substitution method, term splitting and complementing method, etc. These transformation techniques play a pivotal role in mathematical problem-solving. Proficiency in mastering and flexible application of these techniques can not only effectively solve various mathematical problems, but also stimulate students' strong interest in mathematics learning, and effectively exercise and improve their logical thinking ability and innovative thinking ability. 2 Application of Transformation Techniques in Elementary Mathematics In solving elementary mathematics problems, the appropriate use of mathematical transformation is the key to "simplifying complexity" and improving problem-solving efficiency. These techniques can effectively convert the form of complex problems, helping us quickly lock in the problem-solving direction and simplify the calculation process. For this reason, this paper will systematically summarize the common core transformation techniques in elementary mathematics. 2.1 Scaling Method The scaling method is a core transformation technique closely related to inequality theory. Based on the properties of inequalities, it appropriately enlarges or reduces the algebraic terms or conditions in the problem to simplify the structure of the expression, construct the target inequality relationship, or determine its value range. When applying the scaling method, strict attention must be paid to its directionality and magnitude: the choice of scaling operation (enlargement or reduction) must be determined according to the proof goal or conclusion requirement of the specific problem, which is the key to ensuring the logical rigor of the reasoning process and the accuracy of the conclusion [2]. Example 1: Prove that ( ) 2 22 1 1 1 3 13 5 2 21n + + + +  − . When facing such problems where it is difficult to directly judge the size relationship between expressions or values, the core problem-solving strategy is to apply appropriate mathematical transformation techniques. Specifically, when comparing sizes, we can consider using the scaling method: by purposefully and appropriately enlarging the left-hand side expression, convert it into a form that is easy to compare with the right-hand side expression, thereby clearly determining the size relationship between the two. Proof:  ( ) ( )( ) 2 11 2 3 2 1 21 nn n−− − = 1 1 1 2 2 3 2 1nn  −  −−  ( ) 2n 28 Danish Scientific Journal No102, 2025  ( ) 2 22 1 1 1 135 21n + + + + − 1 1 1 1 1 1 1 12 1 3 3 5 2 3 2 1nn         + − + − + + −        −−        = 11 11 2 2 1n  +−  −  13 122  + = . 2.2 Analytical Method The analytical method is a process that starts from the conclusion or goal to be proved, and through the gradual transformation and equivalent reasoning of the conclusion, finally deduces or relies on the known conditions or axioms. The analytical method has the advantages of clear objectives and direct thinking. It provides a problemsolving path based on reverse thinking, which can effectively guide us to find the necessary intermediate links connecting the conclusion and the known conditions. Therefore, in certain proof or derivation problems, the analytical method is usually more convenient and efficient. Example 2: Given that ,,abc represents the lengths of the three sides of ABC , and 0m , prove that: a b c a m b m c m + + + + . This problem will be solved using the analytical method. The specific steps are to start from the conclusion to be proved, conduct reasonable algebraic transformation and simplification of it, and gradually reason backwards to the known conditions. This process will ensure the rigor of the reasoning chain and ultimately successfully derive or verify the result we need to prove. Proof: To prove a b c a m b m c m + + + + , it suffices to prove 0 a b c a m b m c m + −  + + + .  a b c a m b m c m +− + + + = ( )( ) ( )( ) ( )( ) ( )( )( ) a b m c m b a m c m c a m b m a m b m c m + + + + + − + + + + + 0a , 0b , 0c , 0m ( )( )( ) 0a m b m c m + + +  also ( )( ) ( )( ) ( )( ) a b m c m b a m c m c a m b m+ + + + + − + + = 2 2 2 abc abm acm am abc abm bcm bm abc bcm acm cm+ + + + + + + − − − − = 2 2 2 2abm am abc bm cm+ + + − = ( ) 2 2abm abc a b c m+ + + − Since the sum of any two sides of ABC is greater than the third side,  0a b c+ −   ( ) 20a b c m+ −  ( ) 2 20abm abc a b c m + + + −  a b c a m b m c m  +  + + + . The analytical method is a thinking approach centered on reverse reasoning, whose uniqueness lies in starting from the target conclusion and reversely seeking the necessary conditions. This method is particularly suitable for problems where the known conditions are scattered or the relationship between the known conditions and the conclusion to be proved is unclear. In such cases, we can adopt the analytical method: starting from the final conclusion or target expression, conduct appropriate transformation and logical analysis on it, and deduce step by step until tracing back to the known conditions or axiomatic basis given in the problem. This idea of seeking the cause from the effect can efficiently construct the problem-solving logical chain, thereby obtaining the final result. 2.3 Substitution Method Substitution method is a core mathematical transformation technique. It refers to introducing new variables to replace part or the whole of the original algebraic formula or expression during problem-solving. Based on the principle of equivalent substitution, this method aims to convert complex or obscure research Danish Scientific Journal No102, 2025 29 objects into new ones with simple structures and standardized forms. Applying the substitution method can effectively simplify the expression structure of the problem, hide complexity, thereby making the problem-solving process clearer and the thinking more straightforward. Example 3: Given ( ) 12f x x x− = + , find the analytical formula of ( ) fx . Solution: Let 1xt−= , then ( ) 2 1xt=+ , and 1t− , substitute x with the expression containing t , we have ( ) ( ) ( ) 22 1 2 1 2 1 2 2f t t t t t t= + + + = + + + + = 243tt++ ( ) 1t− , replace t with x , then ( ) 243f x x x= + + ( ) 1x− , therefore, the analytical formula of ( ) fx is ( ) 243f x x x= + + ( ) 1x− . When applying the substitution method to solve problems, two key elements must be focused on: (1) Appropriateness of variable selection: It is necessary to carefully choose a suitable "new variable" to replace the numbers or expressions in the problem, so as to maximize the simplification of the problem. (2) Limitation of the range of values: After substitution, the range of values of the new variable must be strictly determined. Only by meeting the above two points simultaneously can the equivalence of the problem-solving process and the accuracy of the conclusion be ensured, thereby effectively optimizing the thinking process and broadening the problem-solving ideas [3]. 2.4 Separation of Constants Method Example 4: Solve the equation 6 2 3 5 10 6 7 9 x x x x x x x x − − − − + = + − − − − . Analyzing this problem, if conventional methods are adopted—such as direct common denominator reduction or eliminating denominators—the calculation process will be lengthy and complex. At this point, the optimal strategy is to appropriately transform the original expression. By applying the core technique of separating constants, the structure of the expression can be significantly simplified, the computational difficulty reduced, and thus more efficient and convenient solution can be achieved. Solution: The original equation can be transformed into 4 4 4 4 1 1 1 1 10 6 7 9x x x x + + + = + + + − − − − , that is, 1 1 1 1 10 6 7 9x x x x +=+ − − − − , further transformation yields 1 1 1 1 10 9 7 6x x x x − = − − − − − , 22 11 19 90 13 42x x x x = − + − + , 22 19 90 13 42x x x x− + = − + , 8x= . 2.5 Method of Splitting and Supplementing Terms Polynomial multiplication is a basic algebraic operation. Its process usually involves combining like terms to simplify the expression (including combining or canceling terms). As the inverse operation of polynomial multiplication, factorization aims to restore the polynomial to its product form. To achieve this reverse process, we often need to use a special transformation technique—the method of splitting and supplementing terms. The core idea of the method of splitting and supplementing terms is to break the existing structure of the original expression. It creates conditions for applicable grouping factorization or formula-based factorization by splitting existing terms, or adding (supplementing) appropriate terms (i.e., adding and then subtracting to keep the original value unchanged). By reversing the process of combining like terms in polynomial multiplication, this method effectively provides a breakthrough for the factorization of complex polynomials. Example 5: Factorize the expression 3 41mm+− Solution: 3 41mm+− = 3 2 2 4 2 2 2 1m m m m m− + − + − = ( ) ( ) ( ) 2 2 2 1 2 1 2 1m m m m m− + − + − = ( ) ( ) 2 2 1 2 1m m m− + + . The application of the method of splitting and supplementing terms in factorization does not rely on fixed rules or formulas, but is a highly flexible skill dependent on experience. Specifically, how to select the terms to be split and the terms to be added (supplemented) mainly depends on the solver's keen observation and analysis of the internal structure and characteristics of the problem. This method requires solvers to have strong structural transformation ability. Therefore, only through a large number of problem-solving practices and repeated exercises can one truly grasp the essence of this skill and flexibly carry out transformations in practical applications [4]. 2.6 Mathematical Induction Mathematical Induction (MI for short) is a rigorous and powerful mathematical proof method. It is mainly used to prove that Proposition ()Pn involving natural number n holds for the entire set of natural numbers (or starting from a certain initial value). Its core logic relies on two key steps: first, prove that the proposition holds at the initial value (base step); second, assume that the proposition holds for any natural number k , and then deduce that it also holds for 1k+ (inductive step). With this logical structure, Mathematical Induction is widely used in fields such as number 30 Danish Scientific Journal No102, 2025 theory to prove that a given mathematical theorem is universally valid for infinitely many discrete cases (i.e., 1,2,3,n= , etc.) [5]. Example 6: Prove by Mathematical Induction: 1 1 1 1 1 3 3 5 5 7 (2 1)(2 1) 2 1 n n n n + + + + =    − + + . Proof: When 1n= , the left-hand side = the right-hand side = 1 3 , so the equation holds.Assume that the equation holds when nk= , i.e., 1 1 1 1 1 3 3 5 5 7 (2 1)(2 1) 2 1 k k k k + + + + =    − + + . Then when 1nk=+ , we have 1 1 1 1 1 1 3 3 5 5 7 (2 1)(2 1) (2 1)(2 3)k k k k + + + + +    − + + + 1 2 1 (2 1)(2 3) k k k k =+ + + + 2 2 3 1 (2 1)(2 3) kk kk ++ =++ (2 1)( 1) (2 1)(2 3) kk kk ++ =++ 1 2( 1) 1 k k + =++ . This shows that when 1nk=+ , the equation also holds. The standard Mathematical Induction is widely applied in the following core mathematical fields: Proof of the universal validity of propositions: Prove that a given algebraic expression, equation, or property holds universally for all natural numbers (or an infinite sequence starting from a certain initial value). Verification of sequence relations: Verify the correctness of the sum formula for the first n terms (summation formula) of a sequence or its general term formula. Proof of inequalities related to natural numbers: Prove various inequality relations involving natural number variables, including the determination and proof of boundary conditions. 3 Summary and Outlook The system of transformation methods and techniques in mathematics is extensive, and this paper only discusses several core branches among them. However, as the most basic and commonly used means in mathematical learning activities, transformation, with its high flexibility and diverse forms of expression, has irreplaceable value for improving students' examinationtaking ability and practical problem-solving ability under the background of current educational reforms and social development needs. This paper systematically sorts out and summarizes some core transformation techniques in middle school mathematics, and elaborates on them through specific examples. Research confirms that proficient mastery of these basic transformation techniques can significantly improve our efficiency and ease in solving problems, make the problem-solving process handy, and fully experience the inherent fun of mathematical learning. It should be emphasized that the most prominent feature of transformation techniques lies in their flexibility and variability, which determines that they cannot be mastered through rote memorization. True proficiency depends on repeated practice and experience accumulation of problem solvers in practical learning. Only through this practice-driven learning method can these techniques be internalized into thinking habits, and finally achieve flexible application of them, thus truly realizing the goal of efficient and accurate problem-solving. References: 1. Wan Lidan. A Brief Discussion on Transformation Techniques in Junior High School Mathematics[J]. Chi Zi (Red Child), 2016(21):276-276. 2. Xu Jiasheng, Dai Jiarong. Proving Inequalities by the Scaling Method[J]. Mathematics, Physics and Chemistry Learning, 2006(19):7-9. 3. Ran Zhonghua. The Application of the Substitution Method in Mathematics Teaching[J]. Scientific Consult, 2010(12):61-61. 4. Jiao Yuhua. Basic Methods and Techniques of Factorization[J]. Journal of Mathematics Teaching Communication, 2002(8):67-68. 5. Cheng Keling. Mathematical Induction and Its Applications[J]. Journal of Chifeng University, 2011(3):28-29.