初中数学探索规律型题目解法探析

近年全国各地的中考数学试题一般会设计若干道有关发现规律的题型.通常情况下,对于上述这一类数学问题求解的过程如下:将问题所呈现出的特殊形式作为切入点,经由猜测和验证等方法的运用,找出问题的一般性规律,梳理解题的具体思路,应用于其它一般性问题的解决.结合具体例子分析发现如下4种探索规律型题目的解法,即常规型数列的规律、变式数列的规律、根据算式探寻内在的规律以及图形的规律.     

自然数求和公式在初中数学规律题中的应用初探

自然数列在数列中有广泛的运用.这是因为所有的数列中,各项的序号都组成自然数列.任何数列的通项公式都可以看作：数列各项的数与它的序号之间固定的数量关系.自然数求和公式在初中数学规律题中展示出特殊的魅力.     

例谈三角恒等变换中的求值问题

我们知道变换是数学的重要工具,也是数学学习的主要对象之一,三角恒等变换就是其中一种,在三角函数的化简、求值、证明中都离不开三角恒等变换.而三角函数求值是三角恒等变换公式的基本应用,也是考查的重点,要掌握求值问题的解题规律和途径,应正确选用公式,并掌握公式的变形应用技巧,具体说来主要有下面三种题型：     

计算期望的变形公式及其应用

本文介绍了一组用于计算随机变量数学期望的变形公式.这些公式在有的情形下,会给期望计算带来方便,且公式证明中所涉及的方法和结论还可以用来解决其它一些问题.     

由特殊到一般解题两例

众所周知,通常解决数学问题是借助题意条件,凭借定义、定理或性质,按照运算的一般规律进行求解.事实上,有些问题的处理可以打破惯例,从特殊出发寻找问题的着眼点得到所求,然后对一般进行验证,达到解决问题的     

特殊化数学思想及其应用

特殊化思想是重要的数学思想之一．应用特殊化思想解决数学问题，遵循了由特殊到一般的认识规律，是数学发现的重要途径．特别地，运用特殊化思想解某些数学选择题，可以快捷地得到问题的答案．但是，如果对特殊化数学思想缺乏正确理解，有可能对正确的选择产生怀疑或可能犯“特殊代替一般”的逻辑错误，导致错误的选择.     

规律教学的关键：观察、概括、证明与运用——以八年级“探究两位数相乘的规律”教学为例

初中数学中规律教学无处不在,包括从具体的情境中抽象概括出新的数学概念,运算法则(规律)的总结,再到很多习题中的规律问题,等等.如何开展规律问题的教学值得深入研究.在进行八年级“探究两位数相乘的规律”的教学时,教师要辨析小学与初中的学段特征,引导学生观察、发现并概括规律,用含字母的符号语言进行描述,接着运用所学的整式乘法、因式分解等知识进行证明,最后运用规律解决新的问题.这样的解题教学过程就是带领学生“深度思考”的过程,同时,在解题教学的对话过程中,教师也要修炼和精进“善于聆听”的基本功.     

新课程数学建模教学探讨——探究极值一例

各种数学概念、公式、法则、定理、推论等等,都是一些具体的数学模型,《数学课程标准》指出:数学教学活动不仅要考虑数学自身的特点,更要遵循学生学习数学的心理规律,强调从学生已有的生活经验出发,让学生亲自经历实际问题抽象成数学模型并进行解释与应用的过程,并倡导“问题情     

从一道中考题学习解"找规律题"

数学题,可以分为两大类,一类是应用数学规律题,一类是发现数学规律题.应用数学规律题,指的是需要学生应用以前学习过的数学规律解答的题目.发现数学规律题,指的是与学生以前学习的数学规律没有什么关系,需要学生先从已知的事物中找出规律,才能够解答的题目.学生所做数学题,绝大多数属于第一类.     

浅析构造在解题中的应用

构造能力是数学中的一种重要的创新能力,需要在深刻理解概念、公式、定理、法则的基础上,注意条件之间的内在联系,恰当对式子结构进行变形,找到构造的生发点,注意公式的逆用;能够根据式子特点,找到模型,探寻规律,从而使问题的解决变得流畅、简洁、有效、和谐.     

统计方法在研究微积分与后续课程相关性中的应用与实证分析

着眼于本科生后期专业课程的学习效果,探讨微积分课程学习效果的有效性.以北京理工大学某经管类专业全部学生大一大二两学年的学习成绩为依据,分析微积分课程学习与后续理科课程学习的相关性,提出评估微积分课程学习效果的量化指标θ值的概念,并基于回归分析和相关性分析给出其算法.在实例分析中,通过研究,发现学生微积分课程学习效果的好坏会影响他们对后续相关课程的学习,同时也发现学生将微积分知识运用到间接相关科目的能力比运用到直接相关的科目的能力薄弱.     

Singular Value Decomposition and Its Visualization

Singular value decomposition (SVD) is a useful tool in functional data analysis (FDA). Compared to principal component analysis (PCA), SVD is more fundamental, because SVD simultaneously provides the PCAs in both row and column spaces. We compare SVD and PCA from the FDA view point, and extend the usual SVD to variations by considering different centerings. A generalized scree plot is proposed to select an appropriate centering in practice. Several useful matrix views of the SVD components are introduced to explore different features in data, including SVD surface plots, image plots, curve movies, and rotation movies. These methods visualize both column and row information of a two-way matrix simultaneously, relate the matrix to relevant curves, show local variations, and highlight interactions between columns and rows. Several toy examples are designed to compare the different variations of SVD, and real data examples are used to illustrate the usefulness of the visualization methods.     

Tracking customer portfolio composition: A factor analysis approach

In this paper, we address the changing composition of a customer portfolio taking into account actions undertaken by the company to adapt its service offer to market conditions and/or technological innovations. We present a specific methodology to identify clusters of customers in different periods and then compare them over time. The classification process takes into account both qualitative and quantitative aspects of the consumption levels of the services or products offered by the company. The possibility of period‐to‐period variation in the customer portfolio and the service or product offer is also considered, in order to achieve a more realistic scenario. The core of the proposed methodology is related to the family of exploratory factorial and cluster techniques. The customers are classified by using a bicriterial clustering methodology based on ‘tandem’ analysis (multiple factor analysis+cluster analysis of the main factors). The bicriterial approach allows for a compromise between customers' consumption levels (a quantitative criterion) and their consumption/non‐consumption pattern (a qualitative criterion). The evolution of the customer portfolio composition is explored through multiple correspondence analysis. This technique allows visual comparison of the position of different clusters against time and the identification of key changes in customer consumption behavior. The methodology is tested on realistic customer portfolio scenarios for a major telecommunication company. We simulate various scenarios to show the strengths of our proposal. Copyright © 2009 John Wiley & Sons, Ltd.     

What drives value creation in investment projects? An application of sensitivity analysis to project finance transactions

Evaluating the economic attractiveness of large projects often requires the development of large and complex financial models. Model complexity can prevent management from obtaining crucial information, with the risk of a suboptimal exploitation of the modelling efforts. We propose a methodology based on the so-called “differential importance measure (D)$(D)$” to enhance the managerial insights obtained from financial models. We illustrate our methodology by applying it to a project finance case study. We show that the additivity property of D grants analysts and managers full flexibility in combining parameters into any group and at the desired aggregation level. We analyze investment criteria related to both the investors’s and lenders’ perspectives. Results indicate that exogenous factors affect investors (sponsors and lenders) in different ways, whether exogenous variables are considered individually or by groups.     

When does data envelopment analysis outperform a naïve efficiency measurement model?

This paper compares the results from data envelopment analysis (DEA) to a naïve efficiency measurement model, which generates a scalar efficiency score by averaging all output–input ratios. Random data and real-life data are used to test the relative performance of the naïve model against various DEA models. The results suggest that the proposed the naïve model replicates DEA efficiency scores almost perfectly for constant return-to-scales and low heterogeneity in output–input data. It is therefore concluded that heterogeneity in output–input data is important to take advantage of the capability of DEA. It is also shown that heterogeneity is more relevant to efficiency measurement than the number of dimensions.     

Composite multilinearity,epistemic uncertainty and risk achievement worth

Risk achievement worth is one of the most widely utilized importance measures. RAW is defined as the ratio of the risk metric value attained when a component has failed over the base case value of the risk metric. Traditionally, both the numerator and denominator are point estimates. Relevant literature has shown that inclusion of epistemic uncertainty (i) induces notable variability in the point estimate ranking and (ii) causes the expected value of the risk metric to differ from its nominal value. We investigate the conditions under which the equality of the nominal and expected values of a reliability risk metric holds. We then study how the presence of epistemic uncertainty affects RAW and the associated ranking. We propose an extension of RAW (called ERAW) which allows one to obtain a ranking robust to epistemic uncertainty. We discuss the properties of ERAW and the conditions under which it coincides with RAW. We apply our findings to a probabilistic risk assessment model developed for the safety analysis of NASA lunar space missions.     

A sensitivity analysis algorithm for hierarchical decision models

In this paper, a comprehensive algorithm is developed to analyze the sensitivity of hierarchical decision models (HDM), including the analytic hierarchy process and its variants, to single and multiple changes in the local contribution matrices at any level of the decision hierarchy. The algorithm is applicable to all HDM that use an additive function to derive the overall contribution vector. It is independent of pairwise comparison scales, judgment quantification techniques and group opinion combining methods. The allowable range/region of perturbations, contribution tolerance, operating point sensitivity coefficient, total sensitivity coefficient and the most critical decision element at a certain level are identified in the HDM SA algorithm. An example is given to demonstrate the application of the algorithm and show that HDM SA can reveal information more significant and useful than simply knowing the rank order of the decision alternatives.     

Contribution Biplots

To interpret the biplot, it is necessary to know which points—usually variables—are the ones that are important contributors to the solution, especially when there are many variables involved. This information can be calculated separately as part of the biplot's numerical results, but this means that a table has to be consulted along with the graphical display. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic information directly into the display itself, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses, such as correspondence analysis, principal component analysis, log-ratio analysis, and various forms of discriminant analysis, and, in fact, to any method based on dimension reduction through the singular value decomposition. In the contribution biplot, one set of points, usually the rows of a data matrix, optimally represents the spatial positions of the cases or sample units, according to an appropriate distance measure. The other set of points, usually the columns of the data matrix, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that often only one common scale for the row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, the contribution biplot also solves the problem in correspondence analysis and log-ratio analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution. This article has supplementary materials online.     

Improved impedance conditions for a thin layer problem in a nonsmooth domain

In this article, we consider the model problem of the Laplace equation in a domain with a thin layer on a part of its boundary. The singularities appearing where boundary conditions change deteriorate the efficiency of the classical impedance condition used to replace the layer. Modified impedance conditions are proposed, which lead to some improvements in the error estimates.     

Actor analysis methods and their use for public policy analysts

Public policy analysts use methods rooted in OR and systems analysis to support policy makers in their judgement. In doing so, most policy analysts recognize the value of a certain understanding of the role of actors in policy making processes. Different methods are available to aid such understanding and, although they all focus on actors, there are important differences between them. Insight into the range of available methods and their characteristics will thus help policy analysts to learn more about the potential and limitations involved in analyzing multi-actor processes. This article provides such an overview, based on the main requirements these methods should meet. This overview is used to discuss some of the implications for policy analysts who are interested in analyzing multi-actor processes, focusing specifically on trade-offs between analytic quality and practical usability.