In this module, students will learn derivatives and their applications, reasoning and proof, the expansion of number system and the introduction of complex numbers.
The establishment of calculus is a milestone in the development of mathematics, and its development and wide application have initiated a new period of transition to modern mathematics, providing important methods and means for studying variables and functions. The concept of derivative is one of the core concepts of calculus, which has extremely rich practical background and wide application. In this module, students will go through the process of describing real problems from average change rate to instantaneous change rate through a large number of examples, understand the concept of derivative, understand the role of derivative in studying monotonicity and extremum of functions, and initially understand the concept of definite integral, so as to lay the foundation for further study of calculus. Through the study of this module, students will understand the idea of derivative and its rich connotation, feel the role of derivative in solving practical problems and understand the cultural value of calculus.
"Reasoning and proof" is the basic thinking process of mathematics, and it is also a way of thinking that people often use in their study and life. Reasoning generally includes reasonable reasoning and deductive reasoning. Reasonable reasoning is the reasoning process of inferring a certain result according to the existing facts and correct conclusions (including definitions, axioms, theorems, etc.). ), the results of experiments and practice, as well as personal experience and intuition. Induction and analogy are common thinking methods of rational reasoning. In the process of solving problems, perceptual reasoning has the functions of guessing and finding conclusions, exploring and providing ideas, which is conducive to the cultivation of innovative consciousness. Deductive reasoning is a reasoning process that draws new conclusions according to existing facts and correct conclusions (including definitions, axioms and theorems). ) and strict logical rules. Rational reasoning and deductive reasoning are closely related and complement each other. Proof usually includes logical proof, experimental proof and practical proof, and the correctness of mathematical conclusions must be guaranteed by logical proof, that is, on the basis of correct premise, conclusions can be drawn by correctly using reasoning rules. In this module, students will further understand rational reasoning, deductive reasoning and their connections and differences by reviewing what they have learned; Understand the characteristics and basic methods of mathematical proof, including direct proof methods (such as analysis, synthesis and mathematical induction) and indirect proof methods (such as reduction to absurdity); Feel the role of logical proof in mathematics and daily life, and develop the habit of reasoning and argumentation.
The process of number system expansion embodies the process of mathematical discovery and creation, and also reflects the objective needs and background of mathematical development. The introduction of complex numbers is the last expansion of the number system in middle schools. In this module, students will understand the process of solving the expansion of number system and the necessity of introducing complex numbers, learn some basic knowledge of complex numbers and experience the role of human rational thinking in the expansion of number system.
Contents and requirements