无尺作图的基础作图体系的简化

简化了《无尺作图》的原基础作图体系中七个作图命题的作图过程,便得：1.两个基本命题实际作图过程中使用圆规的次数从原来的约300次和200次都减少到100次以下;2.简化了的那些命题的逻辑推理更加简明精巧;3.整个体系中的命题个数减少两个,而且其逻辑结构与更加优美。     

基本轨迹与几何作图复习研究

复习目标 了解命题的组成、互逆命题的概念以及反证法证明的基本步骤；了解轨迹的概念及五种基本轨迹，并能根据五种基本轨迹写出一些简单的轨迹；掌握教材所涉及的几种基本作图，能正确而熟练地进行尺规作图．     

多面体截面的投影作图法

多面体截面的作图较难,方法也不只高中《立体几何》课本中的几种基本作图法。探讨、研究多面体作图问题有利于激发学生的学习兴趣,开阔学生的视野,提高学生的几何作图能力。但《立体几何》中的作图易受《平面几何》中尺规作图习惯定势的干扰。原因是没有弄清两种作图     

尺规作图的回归:为何,何为

<正>尺规作图在初中平面几何中的地位可以说是“几经沉浮”.改革开放前对几何作图要求较高,改革开放后因为义务教育的逐步普及,一段时间内对几何作图的要求逐步弱化,至2001年《全日制义务教育数学课程标准(实验稿)》的版本,尺规作图的要求已经降至最低.《义务教育数学课程标准(2011年版)》开始逐步提高对尺规作图的要求,重新要求了解作图的道理;《义务教育数学课程标准(2022年版)》对尺规作图的要求进一步提高,小学阶段就开始增加尺规作图,初中阶段基于基本作图的简单几何作图要求有所提升,要求经历尺规作图的过程,理解尺规作图的基本原理与方法.     

明理得法 水到渠成——以“作一个角的平分线”为例浅析初中尺规作图教学

一、引言尺规作图,指用没有刻度的直尺和圆规作图.与用刻度尺、量角器等工具作图相比,尺规作图显得更加客观、精准.观察尺规作图所得几何图形,我们可以将一些结论由"特殊"引向"一般",并归纳出几何的一般性结论.在初     

大道至简 大简至美——南京市2017年中考数学尺规作图题赏析和思考

尺规作图,顾名思义,是指用没有刻度的直尺和圆规来作图,它起源于古希腊的数学课题.尺规作图,题型多样,对于培养学生的动手操作能力有着不可替代的作用.南京市2017年初中毕业学业考试数学中呈现了一道这样的题,仅用尺规,用两种不同的方法判断一个角是否为直角.考生的奇思妙想精彩纷呈,笔者有幸参与此题批阅,现摘其解法,与大家分享,同时,将自己的思考奉上与各位交流.     

两种类型的平移作图问题解决思路探析

“图形的平移”考查比较频繁的是作图,不过这方面的中考命题大多不以尺规作图呈现,而是以另两种类型为主.在介绍小正方形网格中平移作图与平面直角坐标系中平移作图两种类型的基础上,简要说明了平移作图的步骤,再以例题分析的形式探究了这两种类型的平移作图问题的解决思路.     

二次曲线的利用不变性作图法

二次曲线的利用不变性作图法刘德金（山东德州师专２２１０００）［１］和［２］两文各自提出一种不经过坐标变换作出二次曲线图形的直接作图法，避免了坐标平移与旋转等复杂的计算．本文将提出另一种二次曲线的直接作图法—二次曲线的利用不变性作图法，以使我们对“数”...     

趣谈无尺作图

所谓的无尺作图指的是不用直尺、三角板,只用圆规作图．无尺作图不需要很高深的知识,但有挑战性,能吸引人去探索,培养数学思维和数学能力．现举例说明．一、作已知线段的倍数点     

尺规作图的过往

<正>尺规作图是中学几何证明学习的良好工具,它亦能培养逻辑思维能力.尺规作图的起源不仅仅为培养思维,更是要解决数学问题.尺规作图是由几何作图发展而来,而几何作图是几何学产生、发展的产物.我们今天就来一起追溯尺规作图的过往.1几何作图与尺规作图几何作图兴起于希腊数学史上的雅典时期(公元前5世纪—公元前3世纪).为几何作图的兴起奠定思想基础的,首推阿那克萨哥拉(Anaxagoras,公元前500-前428).他是希腊     

The construction fo meanings for trend in active graphing

The development of increased and accessible computing power has been a major agent in the current emphasis placed upon the presentation of data in graphical form as a means of informing or persuading. However research in Science and Mathematics Education has shown that skills in the interpretation and production of graphs are relatively difficult for Secondary school pupils. Exploratory studies have suggested that the use of spreadsheets might have the potential to change fundamentally how children learn graphing skills. We describe research using a pedagogic strategy developed during this exploratory work, which we call Active Graphing, in which access to spread sheets allows graphs to be used as analytic tools within practical experiments. Through a study of pairs of 8 and 9 year old pupils working on such tasks, we have been able to identify aspects of their interaction with the experiment itself, the data collected and the graphs, and so trace the emergence of meanings for trend. This revised version was published online in July 2006 with corrections to the Cover Date.     

Affect and graphing calculator use

This article reports on a qualitative study of six high school calculus students designed to build an understanding about the affect associated with graphing calculator use in independent situations. DeBellis and Goldin's (2006) framework for affect as a representational system was used as a lens through which to understand the ways in which graphing calculator use impacted students’ affective pathways. It was found that using the graphing calculator helped students maintain productive affective pathways for problem solving as long as they were using graphing calculator capabilities for which they had gone through a process of instrumental genesis (Artigue, 2002) with respect to the mathematical task they were working on. Furthermore, graphing calculator use and the affect that is associated with its use may be influenced by the perceived values of others, including parents and teachers (past, present and future).     

Three women's understandings of algebra in a precalculus course integrated with the graphing calculator

This study investigated the role of function in a precalculus classroom which incorporated the graphing calculator in the instructional process. Perspectives were taken from students, teachers, and textbooks. Emphasis was placed on choice of functional symbol system when thinking and problem solving, connections across symbol systems, the role of the instructor and the textbook in learning, affective components, and the effect of the graphing calculator.The study starts with a defination of the concept of structure as it relates to function. The account of a semester-long qualitative study on students' concept images of function and its role in problem solving follows. It was found that the students involved in the study entered the graph-intensive course with predominantly symbolic notions of algebra, in part due to prior instruction. The students also possessed highly procedural views of algebraic content. These preconceptions and expectations resulted in the students' inability to effectively coordinate graphic and symbolic notions of algebra, both in procedural and conceptual realms. Implications and curricular suggestions are provided.     

The role of the graphic calculator in mediating graphing activity

The PIGMI (Portable Information Technologies for supporting Graphical Mathematics Investigations) Project ¹ investigated the role of portable technologies in facilitating development of students' graphing skills and concepts. This paper examines the impact of a recent shift towards calculating and computing tools as increasingly accessible, everyday technologies on the nature of learning in a traditionally difficult curriculum area. The paper focuses on the use of graphic calculators by undergraduates taking an innovative new mathematics course at the Open University. A questionnaire survey of both students and tutors was employed to investigate perceptions of the graphic calculator and the features which facilitate graphing and linking between representations. Key features included visualization of functions, immediate feedback and rapid graph plotting. A follow-up observational case study of a pair of students illustrated how the calculator can shape mathematical activity, serving a catalytic, facilitating and checking role. The features of technology-based activities which can structure and support collaborative problem solving were also examined. In sum, the graphic calculator technology acted as a critical mediator in both the students' collaboration and in their problem solving. The pedagogic implications of using portables are considered, including the tension between using and over-using portables to support mathematical activity.     

Why sin 80x looks like sin x on some graphing calculators

In rare cases, the use of advanced mathematical calculators can give incorrect results. One such error occurs with graphing calculators because the screen is not continuous, but a rectangular array of pixels. On some frequently used calculators, the graph of sin?80x looks like sin?x. We also study other related examples.     

Bunny hops: using multiplicities of zeroes in calculus for graphing

Students learn a lot of material in each mathematics course they take. However, they are not always able to make meaningful connections between content in successive mathematics courses. This paper reports on a technique to address a common topic in calculus I courses (intervals of increase/decrease and concave up/down) while also making use of students’ pre-existing knowledge about the behaviour of functions around zeroes based on multiplicities.     

Filling in the gaps: modelling incomplete CBL data using a graphing calculator

This article discusses the real-world problem-solving lesson that emerged when a high school math teacher used a motion detector with a CBL and graphing calculator to obtain the 'bounce' data of a ping-pong ball. While practising the 'bounce' data collection--a series of diminishing parabolas--the teacher accidentally pulled the motion detector away and then, realizing his mistake, pulled it back. The resulting data showed a series of parabolas, but one was missing. The teacher used this opportunity to create a lesson in which his students collect 'bad data' and then fill in the missing parabola using critical components of parabolas, such as the latus rectum and the vertex, and using matrices. The article provides all the necessary directions, formulas, and names of resources needed to replicate the lesson. The creation of this lesson demonstrates that a serendipitous error can create a genuine and authentic problem-solving activity for math students.     

Covariational reasoning and invariance among coordinate systems

Researchers continue to emphasize the importance of covariational reasoning in the context of students’ function concept, particularly when graphing in the Cartesian coordinate system (CCS). In this article, we extend the body of literature on function by characterizing two pre-service teachers’ thinking during a teaching experiment focused on graphing in the polar coordinate system (PCS). We illustrate how the participants engaged in covariational reasoning to make sense of graphing in the PCS and make connections with graphing in the CCS. By foregrounding covariational relationships, the students came to understand graphs in different coordinate systems as representative of the same relationship despite differences in the perceptual shapes of these graphs. In synthesizing the students’ activity, we provide remarks on instructional approaches to graphing and how the PCS forms a potential context for promoting covariational reasoning.