The operation law is based onequationThe operation rules abstracted and generalized by observation, comparison and analysis.It is not only an important mathematical law, but also an inherent property of mathematical operations.includeAdditive commutative lawandAssociative law、Commutative law of multiplicationAnd associative law, and the distributive law of multiplication to addition, etc.
The operation law is not only an important mathematical law, but also an inherent property of mathematical operations.
1. The operation law can be derived according to the definition of operation.
The operation law is the operation law abstracted and generalized through the observation, comparison and analysis of some equations.This process belongs to the induction from concrete to abstract, from special to general, which reflects the basic characteristics of reasonable reasoning.However, from the perspective of knowledge logic, the definition of operation law and related operation are concomitant.While defining the four operations, mathematicians need to consider "whether the corresponding operation law can be logically deduced from the definition".
2. Operation definition and operation law are the basis for exploring relevant calculation methods.
The method, program or way to complete the operation and get the result is usually called the operation method or calculation method.The operation procedure and key points required by the operation method can be expressed in relatively accurate, standard and easy to understand text language, or the current operation can be summarized into the relevant operations that students have mastered earlier, which is called "operation rules".[1]
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Commutative law
The commutative law is a widely used oneMathematical nounIt refers to the ability to change the order of something without changing its final result.The commutative law is the majoritymathematicsThe fundamental properties of a branch, and manyMathematical proofNeed to rely onCommutative law。That is, given the binary computation on the set S, if a+b=b+a is satisfied for any a and b in S, then the commutative law is said to be satisfied.
For example, inFour arithmetic operationsMedium,additionandmultiplicationBoth satisfy the commutative law.The addition commutative law means that two numbers are added and exchangedAddendThe position of and their sum remain unchanged.That is, a+b=b+a.Commutative law of multiplicationIt refers to the multiplication and exchange of two numbersfactorAnd their product remains the same.That is, axb=bxa.In addition, in theaggregateIn operation, intersection, unionSymmetry differenceThe commutative law is satisfied in all operations.[2]
For example, in the common four operations, addition andmultiplicationBoth satisfy the law of association.Associative law of additionIt refers to the addition of three numbers. First add the first two numbers and then add the third number, or first add the last two numbers and then add the first number. Their sum remains unchanged.That is, (a+b)+c=a+(b+c);Associative law of multiplicationIt means to multiply three numbers, multiply the first two numbers and then the third number, or multiply the last two numbers and then the first number, and their product will not change.That is, (axb) xc=ax (bxc).In addition, in set operations, the intersection and union operations of sets meetAssociative law。[2]
Distributive law
Two on a given set SBinary operationX and+, if there is cx (a+b)=(cxa)+(cxb) for a, b, c in any S, then the operation x pair operation+satisfies the leftDistributive law。If there is (a+b) xc=(axc)+(bxc) for a, b, c in any S, then the operation x pair operation+satisfies the rightDistributive law。
The teaching of operation law in primary school mathematics has not only explicit knowledge and skill value, but also implicit process and method value.From the aspect of dominance, the operation law isnumberAndAlgebraPart of the important knowledge, using the operation law to carry out simple calculation will help students to continuously improve their operation ability;From the recessive aspect, the teaching of the operation law can help students enrich and deepen their understanding of the operation itself, feel the basic mathematical ideas such as abstraction, reasoning, and models, and also gain some insights that are very useful for mental growth.
1. Feel the value of reasonable reasoning in the process of exploring the operation law.
For primary school students, the operation law is the law abstracted and generalized from the observation, comparison and analysis of some similar operation phenomena. It is an important discovery obtained through exploration activities.The essence of this process is the induction from special to general, from concrete to abstract.Therefore, the process of exploring and discovering the operation law will help students feel the value of reasonable reasoning and develop their preliminary ability of reasonable reasoning.
2. Feel the simple model idea in the process of representing the operation law.
After students have preliminarily summarized the basic content of the operation law with the help of examples, the next key is to guide them to express the discovered law in an appropriate way.This process is also known as "representation operation law".Generally speaking, there are three main representations of the operation law. One isLinguistic representationThe second is graphic representation, and the third isSymbolic representation。Among them, graphic representation and symbolic representation have the meaning of preliminary mathematical modeling. Through this process, students can initially feel the idea of models, understand the characteristics and value of mathematical expression, and improve their interest in learning mathematics and applying mathematics.
3. Deepen the understanding of the operation itself in the process of applying the operation law to explain the calculation method.
Operation definition and operation law are the premise and basis for understanding relevant operation methods.However, primary school students are limited by age characteristics, knowledge and experience, and cognitive ability. When they explore relevant computing methods, they cannot logically reason from the definition and law of operations, nor are they aware of the role of the law of operations in the exploration of computing methods.In fact, when pupils explore and understand calculation methods, they will think more with the help of facts in real life and relevant life experience.
4. Feeling in the process of simple calculation by using the operation lawConversion thought。
Primary school students' exploration and discovery of operation laws mainly rely on reasonable reasoning, while simple calculation by applying operation laws is deductive reasoning.At the same time, the simple calculation process also reflects the transformation idea of "equivalent deformation".That is, for a more complex calculation problem, the effective calculation strategy is to first apply the operation law or other operation properties and operation laws to transform it into a relatively simple and familiar problem, so as to make the calculation approach more reasonable and simple.In teaching, on the one hand, we should attach importance to guiding students to analyze and think about the basis of each step of operation, so as to think in a reasonable and orderly way;On the other hand, it is also necessary to properly guide students to realize the strategic value of turning difficulty into ease, complexity into simplicity, and unfamiliar into familiarity, so as to be enlightened by the thought of transformation.[1]
