数学史融入高中数学教学的设计研究——以“三角函数”为例

数学史作为数学文化中不可缺少的关键部分,有利于学生对数学的深入理解和认识,也能帮助学生了解人类文明的发展历程.新课标对数学史融入高中数学教学较为关注,需要教师在知识讲解期间合理渗透数学史,保证学生能对知识学习产生浓厚兴趣,加深学生对知识的印象与理解,让学生的视野拓宽,为学生学科素养的发展打下良好基础.因此,为提高教学有效性,教师应该将知识讲解与数学史深度融合.在高中数学中,“三角函数”是教学重点,在高考中占有较大比例.基于此,本文围绕“三角函数”教学设计展开,深入探究数学史在其中的融入对策.

基于认知学习理论的高中数学教学模式探究——以“三角函数”为例

随着教育改革的深入,认知学习理论在高中数学教学中的应用越来越受到重视.“三角函数”作为高中数学的重要组成部分,其教学效果对学生数学能力的提升具有重要意义.当前高中数学教学中,传统的教学模式难以满足学生个性化学习需求.因此,文章基于认知学习理论,以奠定认知学习基础、提升学生学习体验及增强学生学习动机等为教学目标,探究高中数学教学中“三角函数”章节的教学策略,以期提高学生对三角函数知识的理解和掌握程度,激发学生的学习兴趣,提升学生的数学思维能力.

《全日制普通高级中学教科书(必修)·数学》第一册(下) 第四章“三角函数”简介

三角函数是中学数学的重要内容，它的基础主要是几何中的相似形和圆，研究方法主要是代数变形和图象分析，因此三角函数的研究已经初步把几何与代数联系起来了。本章所介绍的知识，既是解决生产实际问题的工具，又是学习中学后续内容和高等数学的基础。

高中数学教学策略探讨——以“三角函数”教学为例

文章首先阐述了高中数学教学现状,然后提出了高中数学教学的优化策略,包括联系生活实际,培养学生的探究能力;利用多媒体技术,激发学生的学习兴趣;开展趣味数学活动,调动学生主观能动性。

On Triangle Interpolation Approximation of the Double Function

本文以两组不同的节点构造了一个组合型的二元三角插值多项式算子Lmn(f;x,y)，并且研究了二元连续周期函数对这个算子的收敛性及收敛阶的估计等问题。

New Travelling Wave Solutions to Compound KdV-Burgers Equation

The compound KdV-Burgers equation and combined KdV-mKdV equation are real physical models concerning many branches in physics.In this paper,applying the improved trigonometric function method to these equations,rich explicit and exact travelling wave solutions,which contain solitary-wave solutions,periodic solutions,and combined formal solitary-wave solutions,are obtained.

Coupled Nonlinear Schrodinger Equation: Symmetries and Exact Solutions

The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger （CNLS） equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.

Linear Support Vector Machine Based Inter-harmonic Modeling and Parameter Estimation

There are often system. The high measure many inter-harmonics in power t accuracy of inter-harmonics order, amplitude and initial phase is needed. A new approach is presented for inter-harmonic modeling and parameter estimation based on linear support vector machine （SVM）. Firstly, parameter estimation of linear model is realized based on standard linear SVM. Then, interharmonic model is transformed to a linear model according to trigonometric functions. The approach obtains order of inter-harmonic model with windowed Blackman-Tukey （BT） spectrum analysis, and gets number and frequency of harmonics. Finally, the linear SVM is applied to estimate the inter-harmonic parameters, amplitude and initial phase. The simulation results show that the proposed approach has high precision and good antinoise. The accuracy of three parameters are all higher than 98%.

Jacobian Elliptic Function Method and Solitary Wave Solutions for Hybrid Lattice Equation

In this paper, we have successfully extended the Jacobian elliptic function expansion approach to nonlinear differential-difference equations. The Hybrid lattice equation is chosen to illustrate this approach. As a consequence, twelve families of Jacobian elliptic function solutions with different parameters of the Hybrid lattice equation are obtained. When the modulus m → 1 or O, doubly-periodic solutions degenerate to solitonic solutions and trigonometric function solutions, respectively.

New Exact Solutions of the Integrable Broer-Kaup Equations in （2＋ 1）-DimensionalSpaces

In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.

Continuous Spectrum of Trigonometric Rosen-Morse and Eckart Potentials from Free Particle Spectrum

The shape invariant symmetry of the Trigonometric Rosen-Morse and Eckart potentials has been studied through realization of so（3） and so（2, 1） Lie algebras respectively. In this work, by using the free particle eigenfunction, we obtain continuous spectrum of these potentials by means of their shape invariance symmetry in an algebraic method.

Objective Reduction Solutions to Higher-Order Boussinesq System in (2+1)-Dimensions

With the help of an objective reduction approach （ORA）, abundant exact solutions of （2＋1）-dimensional higher-order Boussinesq system （including some hyperboloid function solutions, trigonometric function solutions, and a rational function solution） are obtained. It is shown that some novel soliton structures, like single linearity soliton structure, breath soliton structure, single linearity y-periodic solitary wave structure, libration dromion structure, and kink-like multisoliton structure with actual physical meaning exist in the （2＋1）-dimensional higher-order Boussinesq system.

Incremental learning of the triangular membership functions based on single-pass FCM and CHC genetic model

In order to improve the efficiency of learning the triangular membership functions( TMFs) for mining fuzzy association rule( FAR) in dynamic database,a single-pass fuzzy c means( SPFCM)algorithm is combined with the real-coded CHC genetic model to incrementally learn the TMFs. The cluster centers resulting from SPFCM are regarded as the midpoint of TMFs. The population of CHC is generated randomly according to the cluster center and constraint conditions among TMFs. Then a new population for incremental learning is composed of the excellent chromosomes stored in the first genetic process and the chromosomes generated based on the cluster center adjusted by SPFCM. The experiments on real datasets show that the number of generations converging to the solution of the proposed approach is less than that of the existing batch learning approach. The quality of TMFs generated by the approach is comparable to that of the batch learning approach. Compared with the existing incremental learning strategy,the proposed approach is superior in terms of the quality of TMFs and time cost.

Traveling Wave Solutions for Generalized Bretherton Equation

This paper studies the Generalized Bretherton equation using trigonometric function method including the sech-function method, the sine-cosine function method, and the tanh-function method, and He＇s semi-inverse method （He＇s variational method）. Various traveling wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency, and wave speed.

Exact Solutions of (2+1)-Dimensional Boiti-Leon-Pempinelle Equation with (G'/G)-Expansion Method

In this paper,we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation byusing the (G'/G)-expansion method,and with the help of Maple.As a result,non-travelling wave solutions with threearbitrary functions are obtained including hyperbolic function solutions,trigonometric function solutions,and rationalsolutions.This method can be applied to other higher-dimensional nonlinear partial differential equations.

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Rieatti equation expansion method, we present explicit solutions of the （2＋1）-dimensional variable coefficients Broer-Kaup （VCBK） equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.

Exact Solutions to Triple sine-Gordon Equation

In this paper, a new transformation is introduced to solve triple sine-Gordon equation. It is shown that this intermediate transformation method is powerful to solve complex special type nonlinear evolution equation.

Study on productivity of horizontal wells with segmental perforation based on pseudo steady-state flow

Segmental perforation is widely used for horizontal wells. However,the flow of fluid in porous media is a complex problem. Using the Fourier transform,principle of potential superposition,trigonometric function transform,asymptotic analyses,a pressure solution of a pseudo steady-state flow model in 3D circular-boxed media has been established. Comparing with the productivity of vertical wells,an equivalence radius model can be obtained. Based on the model,a method of evaluating the productivity of segmental perforation horizontal well is presented by means of principle of superposition. It shows that the equivalence radius is different for various positions of horizontal wells; the output of both ends of horizontal wells is greater than the others under the same length of perforation interval; it is more important to obtain high productivity by increasing the length of perforation interval than enlarging the spacing between perforation intervals. The result of this research can be used to ascertain the yield of each perforated interval.

A Teaching Experiment in Modelling through Scale Reduction Methods： A Bridge to Later Trigonometric Methods

In a teaching experiment, Japanese Grade 9 students investigated how to measure the height of an aerial balloon using different models involving angles and distances, and also to evaluate the models they developed. As novices to mathematical modelling, they needed to decide which of several possible models were both valid and practicable, and the errors in measurement that are likely to arise. Opportunities to construct and use paper models, as scale reductions of the real situation, and discussing their results in small groups were effective in moving forward the thinking of many students on the dimensions mentioned above. While students were less able to identify different sources of errors, many came to appreciate the need to learn trigonometric techniques that are more suitable in dealing with problems of this kind.