middle school mathematics140
初中数学中常见的数学思想方法见解
初中数学中常见的数学思想方法见解作为一门基础学科，数学在我们的生活和学习中扮演着非常重要的角色。
在初中数学学习中，学生需要掌握许多基本概念、基本原理和方法。
除了常见的数学知识点之外，还有一些重要的数学思想方法，如数学归纳法、逆向思维、抽象思维等。
本文将针对初中数学中常见的数学思想方法进行探讨，重点分析其原理和实际应用，并给出具体的数学题例子。
一、数学归纳法数学归纳法是初中数学中常见的数学思想方法之一，它是证明自然数的某些性质时常用的一种方法。
数学归纳法的基本思想是：证明一个性质对于所有自然数都成立，只需证明当自然数 n = 1 时成立，且当自然数 n 成立时，自然数 n+1 也成立，即可推出该性质对于所有自然数都成立。
例如，我们要证明一个常见的命题：对于任意自然数 n，1+2+3+...+n = n(n+1)/2。
首先当 n=1 时，左侧等式为 1，右侧等式为 1×(1+1)/2=1，两边相等。
再假设对于自然数 n 成立，即1+2+3+...+n = n(n+1)/2，那么将 n+1 代入等式，得到：1+2+3+...+(n+1) = [1+2+3+...+n] + (n+1)由假设可得左侧等式为 n(n+1)/2 + (n+1)，经过化简得到：(n+1)(n+2)/2 = (n+1)(n+2)/2，由此证明了该命题对于任意自然数 n 成立。
数学归纳法还可以用于证明一些更复杂的命题，例如利用数学归纳法证明斐波那契数列的性质。
斐波那契数列是一个非常经典的数学问题，其定义为：对于自然数 n，斐波那契数列的第 n 项 F(n) 等于前两项的和，即 F(n) = F(n-1) + F(n-2)，其中 F(1)=1，F(2)=1。
利用数学归纳法可以证明：对于任意自然数 n，斐波那契数列的第 n 项 F(n) 满足 F(n) = (1/√5){[(1+√5)/2]^n - [(1-√5)/2]^n}。
高中数学课程描述（英文）
Mathematics Course DescriptionMathematics course in middle school has two parts: compulsory courses and optional courses. Compulsory courses content lots of modern mathematical knowledge and conceptions, such as calculus,statistics, analytic geometry, algorithm and vector. Optional courses are chosen by students which is according their interests.Compulsory Courses:Set TheoryCourse content:This course introduces a new vocabulary and set of rules that is foundational to the mathematical discussions. Learning the basics of this all-important branch of mathematics so that students are prepared to tackle and understand the concept of mathematical functions. Students learn about how entities are grouped into sets and how to conduct various operations of sets such as unions and intersections(i.e. the algebra of sets). We conclude with a brief introduction to the relationship between functions and sets to set the stage for the next stepKey Topics:The language of set theorySet membershipSubsets, supersets, and equalitySet theory and functionsFunctionsCourse content:This lesson begins with talking about the role of functions and look at the concept of mapping values between domain and range. From there student spend a good deal of time looking at how to visualize various kinds of functions using graphs. This course will begin with the absolute value function and then move on to discuss both exponential and logarithmic functions. Students get an opportunity to see how these functions can be used to model various kinds of phenomena.Key Topics:Single-variable functionsTwo –variable functionsExponential functionLogarithmic functionPower- functionCalculusCourse content:In the first step, the course introduces the conception of limit, derivative and differential. Then students can fully understand what is limit of number sequence and what is limit of function through some specific practices. Moreover, the method to calculate derivative is also introduced to students.Key Topics:Limit theoryDerivativeDifferentialAlgorithmCourse content:Introduce the conception of algorithm and the method to design algorithm. Then the figures of flow charts and the conception of logical structure, like sequential structure, contracture of condition and cycle structure are introduced to students. Next step students can use the knowledge of algorithm to make simple programming language, during this procedure, student also approach to grammatical rules and statements which is as similar as BASIC language.Key Topics:AlgorithmLogical structure of flow chart and algorithmOutput statementInput statementAssignment statementStatisticsCourse content:The course starts with basic knowledge of statistics, such as systematic sampling and group sampling. During the lesson students acquire the knowledge like how to estimate collectivity distribution according frequency distribution of samples, and how to compute numerical characteristics of collectivity by looking at numerical characteristics of samples. Finally, the relationship and the interdependency of two variables is introduced to make sure that students mastered in how to make scatterplot, how to calculate regression line, and what is Method of Square.Key Topics:Systematic samplingGroup samplingRelationship between two variablesInterdependency of two variablesBasic Trigonometry ICourse content:This course talks about the properties of triangles and looks at the relationship that exists between their internal angles and lengths of their sides. This leads to discussion of the most commonly used trigonometric functions that relate triangle properties to unit circles. This includes the sine, cosine and tangent functions. Students can use these properties and functions to solve a number of issues.Key Topics:Common AnglesThe polar coordinate systemTriangles propertiesRight trianglesThe trigonometric functionsApplications of basic trigonometryBasic Trigonometry IICourse content:This course will look at the very important inverse trig functions such as arcsin, arcos, and arctan, and see how they can be used to determine angle values. Students also learn core trig identities such as the reduction and double angle identities and use them as a means for deriving proofs. Key Topics:Derivative trigonometric functionsInverse trig functionsIdentities●Pythagorean identities●Reduction identities●Angle sum/Difference identities●Double-angle identitiesAnalytic Geometry ICourse content:This course introduces analytic geometry as the means for using functions and polynomials to mathematically represent points, lines, planes and ellipses. All of these concepts are vital in student’s mathematical development since they are used in rendering and optimization, collision detection, response and other critical areas. Students look at intersection formulas and distance formulas with respect to lines, points, planes and also briefly talk about ellipsoidal intersections. Key Topics:Parametric representationParallel and perpendicular linesIntersection of two linesDistance from a point to a lineAngles between linesAnalytic Geometry IICourse content:Students look at how analytic geometry plays an important role in a number of different areas of class design. Students continue intersection discussion by looking at a way to detect collision between two convex polygons. Then students can wrap things up with a look at the Lambertian Diffuse Lighting model to see how vector dot products can be used to determine the lighting and shading of points across a surface.Key Topics:ReflectionsPolygon/polygon intersectionLightingSequence of NumberCourse content:This course begin with introducing several conceptions of sequence of number, such as, term, finite sequence of number, infinite sequence of number, formula of general term and recurrence formula.Then, the conception of geometric sequence and arithmetic sequence is introduced to students. Through practices and mathematical games, students gradually understand and utilizethe knowledge of sequence of number, eventually students are able to solve mathematical questions.Key Topics:Sequence of numberGeometric sequenceArithmetic sequenceInequalityThis course introduces conception of inequality as well as its properties. In the following lessons students learn the solutions and arithmetic of one-variable quadratic inequality, two variables inequality, fundamental inequality as well how to solve simple linear programming problems. Key Topics:Unequal relationship and InequalityOne-variable quadratic inequality and its solutionTwo-variable inequality and linear programmingFundamental inequalityVector MathematicsCourse content:After an introduction to the concept of vectors, students look at how to perform various important mathematical operations on them. This includes addition and subtraction, scalar multiplication, and the all-important dot and cross products. After laying this computational foundation, students engage in games and talk about their relationship with planes and the plane representation, revisit distance calculations using vectors and see how to rotate and scale geometry using vector representations of mesh vertices.Key Topics:Linear combinationsVector representationsAddition/ subtractionScalar multiplication/ divisionThe dot productVector projectionThe cross productOptional CoursesMatrix ICourse content:In this course, students are introduced to the concept of a matrix like vectors, matrices and so on. In the first two lessons, student look at matrices from a purely mathematical perspective. The course talks about what matrices are and what problems they are intended to solve and then looks at various operations that can be performed using them. This includes topics like matrix addition and subtraction and multiplication by scalars or by other matrices. At the end, students can conclude this course with an overview of the concept of using matrices to solve system of linear equations.Key Topics:Matrix relationsMatrix operations●Addition/subtraction●Scalar multiplication●Matrix Multiplication●Transpose●Determinant●InversePolynomialsCourse content:This course begins with an examination of the algebra of polynomials and then move on to look at the graphs for various kinds of polynomial functions. The course starts with linear interpolation using polynomials that is commonly used to draw polygons on display. From there students are asked to look at how to take complex functions that would be too costly to compute in a relatively relaxed studying environment and use polynomials to approximate the behavior of the function to produce similar results. Students can wrap things up by looking at how polynomials can be used as means for predicting the future values of variables.Key Topics:Polynomial algebra ( single variable)●addition/subtraction●multiplication/divisionQuadratic equationsGraphing polynomialsLogical Terms in MathematicsCourse content:This course introduces the relationships of four kinds of statements, necessary and sufficient conditions, basic logical conjunctions, existing quantifier and universal quantifier. By learning mathematical logic terms, students can be mastered in the usage of common logical terms and can self-correct logical mistakes. At the end of this course, students can deeply understand the mathematical expression is not only accurate but also concise.Key Topics:Statement and its relationshipNecessary and sufficient conditionsBasic logical conjunctionsExisting quantifier and universal quantifierConic Sections and EquationCourse content:By using the knowledge of coordinate method which have been taught in the lesson of linear and circle, in this lesson students learn how to set an equation according the character of conic sections. Students is able to find out the property of conic sections during establishing equations. The aim of this course is to make students understand the idea of combination of number and shape by using the method of coordinate to solve simple geometrical problems which are related to conic sections.Key Topics:Curve and equation OvalHyperbolaParabola。
大学数学与中学数学的关系及其对中学数学教学的作用
大学数学与中学数学的关系及其对中学数学教学的作用【摘要】大学数学专业的主要任务是培养合格的中学数学教师,然而在大学数学的教学活动中,常常有学生向教师提出:“大学数学在中学数学教学中用不上”,甚至有的中学教师也持此种看法。
这不仅影响了大学数学专业学生学习大学数学的主动性也挫伤了一些在职教师教授、进修大学数学的积极性。
让此看法漫延,无疑将影响我国的数学教育工作。
我们认为,持此类看法的大学学生和在职教师,恰恰是对数学的理解比较肤浅,对大学数学课对中学数学教学工作的指导作用认识不够所造成的;另一方面也使我们大学教师认识到,应当努力改革大学数学课的教学工作,提高学生对大学数学课对中学数学教学的指导工作的认识。
【关键词】大学数学中学数学联系指导作用.University mathematics relationship with the middle school mathematics and its effect on middle school mathematicsteaching【Abstract】The main task of mathematics in normal universities is to cultivate qualified middle school mathematics teachers, in college mathematics teaching activity, however, often have a student asked the teacher: \"not in the middle school mathematics teaching in higher mathematics\", and even some middle school teachers also hold this view. This not only affects the initiative of student learning of mathematics in normal universities of higher mathematics professor also dampened some in-service teachers, study the enthusiasm of higher mathematics. Let this view, will undoubtedly affect our country's mathematics education work. We believe that with the view of college students and teachers, it is the understanding of mathematics is superficial and math in middle school mathematics teaching in the normal universities work caused by the guidance to know enough; On the other hand also to make our college teachers realize that should strive to reform college mathematics teaching, improve students' math in middle school mathematics teaching in the normal universities guidance work【Key words】University mathematics middle school mathematics guiding function connection.目录1. 引言 (5)2 初等数学与高等数学的联系 (5)2.1初等数学是高等数学的基础，二者有本质的联系 (6)2.2 知识方面的联系 (8)2.3 思想方面的联系 (8)3 大学数学教学与中学数学教学的主要差异 (9)3.1内容上的差异 (9)3.2教师教学方法上的差异 (9)3.3学生学习方法上的差异 (9)4 高师数学课对中学数学教学的指导作用 (10)4.1从初等数学与高等数学的联系看高等数学对中学数学教学的指导作用 (10)4.2从教师素质看高等数学对中学数学教学的指导作用 (10)4.3从数学教育教学的研究看高等数学对中学数学教学的指导作用 (11)4.4从中学数学的教学过程看高等数学对中学数学教学的指导作用 (12)5 数学分析课程对中学数学教学的指导作用 (12)5.1 数学分析为中学数学中的一些问题和方法提供了理论依据 (12)5.2数学分析的学习有助于记忆公式，证明等式，研究变量关 (13)5.3 用高观点分析和处理中学数学中的一些问题 (13)5.4用数学分析的理论和思想指导，编拟中学数学练习题 (13)6 总结 (13)参考文献 (14)1 引言近几年来大学师范院校数学系的不少大学生对学习大学数学存在不少看法如“现在学的大学数学好像与中学数学没有多大联系”,“学习大学数学对今后当中学数学教师作用不大”,有的甚至提出“大学数学在中学教学里根本用不上”等等.这些看法正如著名数学家克莱因早已指出的那样“新的大学生一入学就发现他面对的问题好像和中学里学过的东西一点也没有联系似的，但是毕业以后当了老师，他们又突然发现要他们按老师的教法来教传统的中学数学，却由于缺乏指导，他们很难辨明当前数学内容和所受大学数学训练之间的联系，于是很快坠入相沿成习的教学方法，而他们所受的大学训练至多成为一种愉快的回忆，却对他们对教学毫无影响.”然而现在在新的数学教材中已经出现了一些基础的高等数学知识，这可以说是数学发展的一种必然趋势，所以现在的中学数学教师必须掌握大学数学的基础知识以适应数学发展和教材改革.所以大学数学知识在开阔视野、指导数学解题、指导数学教学、对初等数学问题加以诠释等方面的作用就尤为突出了.2 中学数学与大学数学的联系一般说来,数学史家把数学的发展分成四个阶段(萌芽时期、初等数学时期、古典高等数学时期、现代高等数学时期)或五个时期(再加上“当代时期”).无论何种方法都把第二发展时期叫做“初等数学时期”这个时期的数学知识和经验就是“初等数学”,而把第三、第四或第三、四、五阶段叫做“高等数学时期”,这些阶段的数学知识和经验就是“高等数学”理论意义下的初等数学和高等数学是按照恩格斯(Engles)的经典分法所谓初等数学就是指常量数学,高等数学就是指变量数学,并把笛卡尔(RDescartes)1637年发明的解析几何看成为出现高等数学或进入高等数学时期的标志,而教育意义下的初等数学和高等数学是依据教育的发展历程和教育的等级加以区分的即视普通初等、中等教育(即中、小学教育)阶段的数学主要内容为初等数学,视高等教育阶段的数学主要内容为高等数学.当然由于社会和教育的思想、方法、手段尤其是教育内容都在不断发展“初等数学”和“高等数学”也是一个变化的客体对象两者没有严格的概念区别.事实上,数学科学是一个不可分割的整体,它的生命力在于各部分之间的有机联系,只从学科表面上看难以看清两者之间的内在联系,这就需要深入研究初等数学,理清其中最基本的思想和方法,努力寻求初等数学和高等数学的结合点.2.1 初等数学是高等数学的基础，二者有本质的联系将高等数学的理论应用于初等数学,使其内在的本质联系得以体现，进而去指导初等数学的教学工作是一个值得研究的课题.俗话说，站得高才能看得远.因此笔者认为,作为中学教师除掌握中学数学各种类型题的已熟知的初等方法外,还应善于用高等数学方法解决中学数学问题,特别是一些用初等数学方法难以解决或虽能解决但显得难、繁而用高等数学方法则易于解决的中学数学问题,从而拓广解题思路和技巧,提高教师专业水平,促进中学数学教学.下面略举几例说明.例1.证明:当0,,>c b a 时，有不等式abc c b a 3333≥++.证明 :设333()3,(0,),f x x b c bcx x =++-∈+∞bc x x f 33)(2-=' 令 0)(='x f ，即0332=-bc x ， 解得驻点bc x =，且),0(bc x ∈∀，有),(;0)(+∞∈∀<'bc x x f ，有0)(>'x f ，知函数)(x f 在点bc x =取极小值，其极小值为 bc bc c b bc bc f 3)()(333-++=332c bc bc b ++-=.0)(33≥-=c b由于)(x f 在),0(∞上连续，且只有一个极小点，因此这个极小点就是最小点，则),0(+∞∈∀x ，有 0)(3)(233333≥-≥-++=c b bcx c b x x f .令a x =，于是，,03333≥-++abc c b a即 .3333abc c b a ≥++例2.已知数列.,12,1}{11n n n n n a a a a a 求数列通项满足-+==+解：设 12)()1(,1)1(),,1[,)(-+=+=+∞∈=x x x f x f f x a x f 且.（1）显然 当时，有)(N n x ∈=1212)()1(1-+=-+=++n n n n a a n f n f 或. 当212)1()11(1=-+=+=f f x 时，有.对（1）式两边关于x 求导，得2ln 2)()1(x x f x f +'=+'.从而 2ln 2)1()(1-+-'='n n f n f 21211(2)2ln 22ln 2(1)2ln 22ln 22ln 22(21)(1)ln 221(1)2ln 22ln 2,n n n n n n f n f f f -----'=-++=⋅⋅⋅⋅⋅⋅⋅'=++⋅⋅⋅++-'=+-'=+-故2ln 22ln 2)1()(-+'='x f x f 的原函数为 ⎰⎰⎰⎰-+'='=dx dx dx f dx x f x f x 2ln 22ln 2)1()()(.（2） 将）式，得方程组代入（22)2(,1)1(==f f2ln 21(1),4ln 222(1)f c f c'-=+⎧⎨'-=+⎩解此方程组，得0,12ln 2)1(=-='c f 并将其代入（2），且令n x =，有()(2ln 21)22ln 22,n n f n n n n =-+-⋅=- 即 01.121n n n n C C C n ++⋅⋅⋅++ 高等数学的许多方法和技巧都能直接应用于中学数学解题,它常能起到以简驭繁并能使问题得以深化和拓广的作用.以上只是给出两个实例说明高等数学能指导中学数学解决初等代数和初等几何且收到了很好的效果.在教学过程中结合具体内容不失时机地介绍给学生对于丰富学生的解题方法特别是作为教师在将来的数学教学中用它来预测答案确定初等解法的路线构造习题检验结果都有重要的作用。
基于核心素养的初中数学可视化教学研究
1. 核心素养的定义强调学生的学科能力与综合能力的结合，旨在培养学生的学习兴趣和创造力，能够主动掌握知识，灵活运用知识，解决实际问题。2. 核心素养：学科思维与方法助力个性与创新 核心素养是通过培养学生的学科思维能力和方法能力来促进学生的个性发展和创新能力的提升，使学生在学科学习中能够思维敏捷、批判思考、逻辑清晰，并能够将学科知识应用于实际生活中。
为了评价可视化教学在初中数学中的应用效果，教师需要采用多种评价方法其中，学生参与度、课堂表现、作业完成情况等是常用的评价指标同时，教师还可以利用可视化教学的方式，设计一些评价题目，让学生通过绘制图形、制作动画等方式，展示自己的学习成果和思考过程
核心素养在初中数学中的应用
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初中数学课程的核心素养目标
初中数学课程的核心素养实现方式
初中数学课程的核心素养应用案例
初中数学课程的核心素养评价方法
在初中数学教育中，培养学生的核心素养已经成为课程目标的重要组成部分这包括数学思维能力、问题解决能力、数学交流能力、数学应用能力和数学创新能力等多个方面其中，数学思维能力是核心素养的核心，包括逻辑推理、抽象思维、数学想象等能力
可视化教学方法及应用
Visual teaching methods and applications
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数学可视化教学
背景
意义
应用范围
发展
趋势
个性化
需求
数学可视化教学背景及意义
数学可视化教学是一种直观且生动的教学方式，帮助学生更好地理解抽象的数学概念
背景是内容的基础，深入了解背景才能更好地理解内容
1.通过活动设计引发学生好奇心和求知欲设计生动有趣的数学问题和活动，引发学生的好奇心和求知欲。例如，可以通过游戏、实例等方式将抽象的数学概念呈现出来，让学生在参与中感受到数学的乐趣。
统计学与数学的区别 (6)
统计学与数学的区别Introduction统计学与数学有这些相似之处，是因为两者都需要学习数学中的概率方面，但是它们之间有很多不同点。
统计学是一种应用数学，它主要关注收集和分析数据以做出推论和。
而数学更注重理论和公式，主要关注证明和推导。
本文旨在比较统计学和数学之间的不同之处，并提供具体事例来证明它们的区别。
不同点1：目的不同统计学和数学的主要区别在于目的的不同。
数学专注于解决已知规律下的公式和证明问题，而统计学专注于从收集的数据中得出。
例如，一位数学家可以使用公式解决一个已知的几何问题。
他可以使用类似于勾股定理的公式计算数字，并证明答案是正确的。
然而，一个统计学家可能会收集患者的健康数据来分析他们是否更容易患上心脏病。
在这种情况下，数学公式并不适用，因为他们正在处理现实世界中的未知数。
统计学家将使用数据分析来确定是否存在一个模式或趋势，以及如何应对这些趋势。
不同点2：数据和证据的使用统计学和数学也使用不同类型的数据和证据。
数学家使用基本的数字、变量和公式来证明问题，而统计学家会收集和分析现实世界中的实际数据。
例如，一个数学家可以使用一组数字来证明一个公式的正确性。
他们可以把数字代入公式中，证明答案是正确的。
统计学家则可能会收集大量的数据，例如消费者行为数据或公司销售数据来确定产品或市场的趋势。
他们会对这些数据进行统计分析，以得出有关销售趋势、产品受欢迎程度等方面的。
不同点3：主题和关注的领域统计学和数学的关注点也不相同。
数学家在其工作中关注的主要领域包括代数、几何和计算。
统计学家则更关注数据分析、民意调查和实验设计。
例如，数学家可能会研究线性代数或微积分来研究数学基本理论。
而统计学家可能会研究假设检验、方差分析和回归分析等统计学基础概念，并将其应用于实际问题中，例如医疗研究或制造业。
不同点4：解决问题的方式数学家和统计学家也使用不同的方法来解决问题。
数学家经常使用演绎法来证明他们的。
他们从一个已知的公理或假设出发，然后使用推理和逻辑来建立证明。
毕业论文《不等式在中学数学的应用》
摘要：在我们的一般生活和生产中，量有相等关系，也有不等关系，凡是比较量大小有关的问题，都要用到不等式的知识，在中学数学中初看起来不等式的内容涉及并不多，但事实上只有不等式关系才使绝对的。
不等式在中学数学算是一个比较难的知识，但近年高考对不等式颇为重视，所以不等式在中学数学中算是一个很重要的内容。
所以不等式的内容是中学数学必不可少的。
本文通过理解掌握均值不等式、绝对值不等式来说明不等式在中学数学中的重要性，研究均值不等式、绝对值不等式所得相关结果，用于解决最值问题、不等式证明以及实际生活中的实际问题，具有极为重要的意义。
关键词：不等式；均值不等式；绝对值不等式Inequality in middle school mathematics applicationUndergraduate: yu hongSupervisor: Wang Yuan LunAbstract: In our normal life and production .quantity is equal relations, also has the relation of inequality, normally have a size related problems, must use the inequality of knowledge. In the middle school mathematics at first seems inequality involves not much,but in fact only the inequality relationship that absolute. Inequality in middle school mathematics is a difficult knowledge,but in recent years the college entrance examination for inequality is quite seriously.So the inequality in middle school mathematics is a very important content. So the content of middle school mathematics inequality is essential. This article through the understanding of mean value inequality and absolute value inequality to illustrate the importance of inequality in middle school mathematics ,study of mean inequality, absolute value inequality of income related results, For solving the most value problem, proof of inequality and the actual life of the practical problems have very important significance.Key words:an inequality; the mean inequality; absolute value inequality目录绪论 (1)1 不等式 (1)1.1 不等式的由来 (1)1.2 不等式的定义 (1)1.3 不等式的基本性质 (1)1.4不等式解法 (4)2 .均值不等式和绝对值不等式 (6)2.1 均值不等式 (6)2.1.1 利用均值不等式证明不等式 (6)2.1.2 抓条件“一正、二定、三等”求最值 (8)2.1.3 抓“当且仅当……等号成立”的条件，实现相等与不等的转化.92.1.4 利用均值不等式解应用题 (10)2.2 绝对值不等式 (13)2.2.1 几何意义 (13)2.2.2 应用举例 (13)总结 (18)参考文献 (19)致谢 (20)绪论均值不等式是高中数学中的重要知识点之一，应用均值不等式求最值是历年高考考查的重要知识点之一。
《中学数学课程标准与教材研究》课程教学大纲-(7.26)
《中学数学课程标准与教材研究》课程教学大纲课程中英文名称：中学数学课程标准与教材研究（Research on the Curriculum Standards and Teaching Materials of MiddleSchool Mathematics）课程代码139121402适用专业数学与应用数学（师范）课程类型必修课开设学期 5学分数 2一、编写说明（一）课程的性质和目的《中学数学课程标准与教材研究》是一门理论性与实践性相结合的交叉性、综合性学科。
它以一般教学论为基础，广泛地应用现代教育学、心理学、数学教育等方面的有关理论、思想和方法，结合我国数学教育课程改革的现状，综合研究数学教育基本理念与数学教学活动的内容、过程、方法之间的关系。
该课程是普通高等师范院校数学与应用数学本科生必修专业基础课，是培养中学数学教师的主干课程。
本课程的教学目的是：（1）熟悉中学数学课程标准的基本理念、课程目标及内容标准；准确掌握课程标准的核心思想，明确其对数学教学的指导意义。
掌握新课改以来两个课标的基本内容、特点、价值，树立正确的数学课程与教学观。
（2）了解教材的编制原理和使用原则，获得全面分析中学数学教材特点的技能，能够剖析教材内容体系中的重点、难点，达到能用、会用中学数学教材的目标，提高分析、处理和使用教材的能力，主要是数学教学设计的能力。
（3）树立课程资源的意识，有能力实施课程标准所倡导的理念，有能力驾驭数学教材，并能合理地开发与整合各种课程资源，灵活运用数学教材。
培养职前教师数学教学的信念，形成热爱数学教学的情感态度。
（二）大纲制定的依据根据本专业人才培养的目标所需要的基本理论和基本技能的要求，根据本课程的教学性质、条件和教学实践而制定。
(三) 大纲内容选编原则（1）本大纲所列各部分与高等师范学校对职前教师培养的基本要求相符，同时依托于《教师教育课程标准（试行）》的基本理念和课程目标进行选择。
（2）贯彻师范性与学术性的统一、理论与实践的统一，注重内容宽、新、实相结合，力求理论观点高，结构严谨，层次分明，较系统地体现数学教学的主要理论，突出反映现代数学教学的研究成果，并密切联系我国数学教育实际与发展趋势，具有中国特色。
中学数学中极限的应用
中学数学中极限的应用摘要极限在中学数学中是十分重要的内容，从中学的数学角度看，则应着重从直观上考察无穷运动，从思维与想象结合中加以辩证理解。
对于中学数学中极限的研究主要集中在极限思想在解题中的运用，重点放在用极限解题的技巧上。
本文介绍了极限在中学数学中的五个模块（即解析几何、立体几何、数列、三角函数、不等式）中的应用，以及通过与常规解答方法的对比，突出极限的思想方法在中学数学的解题中的重要性，不仅降低了某些问题的解题难度，在寻找解题思路、探索发现新结论有着重大作用，而且开阔学生的视野，对培养学生的创新思维、探索能力和学生解题的技巧也大有益处。
关键词中学数学极限极限思想极限的应用AbstractLimit is very important in middle school mathematics content, from the perspective of middle school mathematics should focus on the visual inspection from the endless campaign, from the combination of thinking and imagination to be understood dialectically. Limits on high school mathematics studies have focused on extreme thinking in problem solving in the use of a focus on problem-solving skills with a limit on. This article describes the limits of mathematics in secondary schools of the five modules (ie, analytic geometry, solid geometry, series, trigonometry, inequalities) application, and by comparison with the conventional solution methods, highlighting the limits of thinking in middle school mathematics solution question, not only reduces the difficulty of solving certain problems, looking for problem-solving ideas, explore new findings have a major role in discovery and broaden their vision, innovative thinking on the students to explore the capabilities and skills of students in problem solving also great benefit.KeywordSecondary School Mathematics limit limit thought the application of limit目录摘要 (I)关键词 (I)Abstract (II)Keyword (II)第一章引言 (1)1.1 对极限发展的认识 (1)1.2 极限具有丰富的现实背景 (1)第二章中学数学中极限的概念 (2)2.1 极限的引入——割圆术 (2)2.2 极限的定义 (3)第三章中学数学中极限的应用 (3)3.1 在解析几何中的应用 (4)3.2 在立体几何中的应用 (7)3.3 在数列中的应用 (10)3.4 在三角函数中的应用 (13)3.5 在不等式中的应用 (14)第四章总结 (16)致谢语 (16)引用文献 (17)第一章引言提到极限，大家并不陌生，“没有最好只有更好”、“大江东去浪淘尽”、“孤帆远影碧空尽，唯见长江天际流”，这些就是文学意境与数学概念的相通，即极限概念正是这种孤帆远影等的现实精确化、形式化的解释，也就是无限远处的逼近于0。
高中数学课程描述(英文)
Mathematics Course DescriptionMathematics course in middle school has two parts: compulsory courses and optional courses. Compulsory courses content lots of modern mathematical knowledge and conceptions, such as calculus, statistics, analytic geometry, algorithm and vector. Optional courses are chosen by students which is according their interests.Compulsory Courses:Set TheoryCourse content:This course introduces a new vocabulary and set of rules that is foundational to the mathematical discussions. Learning the basics of this all-important branch of mathematics so that students are prepared to tackle and understand the concept of mathematical functions. Students learn about how entities are grouped into sets and how to conduct various operations of sets such as unions and intersections (i.e. the algebra of sets). We conclude with a brief introduction to the relationship between functions and sets to set the stage for the next stepKey Topics:➢The language of set theory➢Set membership➢Subsets, supersets, and equality➢Set theory and functionsFunctionsCourse content:This lesson begins with talking about the role of functions and look at the concept of mapping values between domain and range. From there student spend a good deal of time looking at how to visualize various kinds of functions using graphs. This course will begin with the absolute value function and then move on to discuss both exponential and logarithmic functions. Students get an opportunity to see how these functions can be used to model various kinds of phenomena. Key Topics:➢Single-variable functions➢Two –variable functions➢Exponential function➢ Logarithmic function➢Power- functionCalculusCourse content:In the first step, the course introduces the conception of limit, derivative and differential. Then students can fully understand what is limit of number sequence and what is limit of function through some specific practices. Moreover, the method to calculate derivative is also introduced to students.Key Topics:➢Limit theory➢Derivative➢DifferentialAlgorithmCourse content:Introduce the conception of algorithm and the method to design algorithm. Then the figures of flow charts and the conception of logical structure, like sequential structure, contracture of condition and cycle structure are introduced to students. Next step students can use the knowledge of algorithm to make simple programming language, during this procedure, student also approach to grammatical rules and statements which is as similar as BASIC language.Key Topics:➢Algorithm➢Logical structure of flow chart and algorithm➢Output statement➢Input statement➢Assignment statementStatisticsCourse content:The course starts with basic knowledge of statistics, such as systematic sampling and group sampling. During the lesson students acquire the knowledge like how to estimate collectivity distribution according frequency distribution of samples, and how to compute numerical characteristics of collectivity by looking at numerical characteristics of samples. Finally, the relationship and the interdependency of two variables is introduced to make sure that students mastered in how to make scatterplot, how to calculate regression line, and what is Method of Square.Key Topics:➢Systematic sampling➢Group sampling➢Relationship between two variables➢Interdependency of two variablesBasic Trigonometry ICourse content:This course talks about the properties of triangles and looks at the relationship that exists between their internal angles and lengths of their sides. This leads to discussion of the most commonly used trigonometric functions that relate triangle properties to unit circles. This includes the sine, cosine and tangent functions. Students can use these properties and functions to solve a number of issues.Key Topics:➢Common Angles➢The polar coordinate system➢Triangles properties➢Right triangles➢The trigonometric functions➢Applications of basic trigonometryBasic Trigonometry IICourse content:This course will look at the very important inverse trig functions such as arcsin, arcos, and arctan, and see how they can be used to determine angle values. Students also learn core trig identities such as the reduction and double angle identities and use them as a means for deriving proofs. Key Topics:➢Derivative trigonometric functions➢Inverse trig functions➢Identities●Pythagorean identities●Reduction identities●Angle sum/Difference identities●Double-angle identitiesAnalytic Geometry ICourse content:This course introduces analytic geometry as the means for using functions and polynomials to mathematically represent points, lines, planes and ellipses. All of these concepts are vital in student’s mathematical development since they are used in rendering and optimization, collision detection, response and other critical areas. Students look at intersection formulas and distance formulas with respect to lines, points, planes and also briefly talk about ellipsoidal intersections. Key Topics:➢Parametric representation➢Parallel and perpendicular lines➢Intersection of two lines➢Distance from a point to a line➢Angles between linesAnalytic Geometry IICourse content:Students look at how analytic geometry plays an important role in a number of different areas of class design. Students continue intersection discussion by looking at a way to detect collision between two convex polygons. Then students can wrap things up with a look at the Lambertian Diffuse Lighting model to see how vector dot products can be used to determine the lighting and shading of points across a surface.Key Topics:➢Reflections➢Polygon/polygon intersection➢LightingSequence of NumberCourse content:This course begin with introducing several conceptions of sequence of number, such as, term, finite sequence of number, infinite sequence of number, formula of general term and recurrence formula. Then, the conception of geometric sequence and arithmetic sequence is introduced to students. Through practices and mathematical games, students gradually understand and utilizethe knowledge of sequence of number, eventually students are able to solve mathematical questions.Key Topics:➢Sequence of number➢Geometric sequence➢Arithmetic sequenceInequalityThis course introduces conception of inequality as well as its properties. In the following lessons students learn the solutions and arithmetic of one-variable quadratic inequality, two variables inequality, fundamental inequality as well how to solve simple linear programming problems.Key Topics:➢Unequal relationship and Inequality➢One-variable quadratic inequality and its solution➢Two-variable inequality and linear programming➢Fundamental inequalityVector MathematicsCourse content:After an introduction to the concept of vectors, students look at how to perform various important mathematical operations on them. This includes addition and subtraction, scalar multiplication, and the all-important dot and cross products. After laying this computational foundation, students engage in games and talk about their relationship with planes and the plane representation, revisit distance calculations using vectors and see how to rotate and scale geometry using vector representations of mesh vertices.Key Topics:➢Linear combinations➢Vector representations➢Addition/ subtraction➢Scalar multiplication/ division➢The dot product➢Vector projection➢The cross productOptional CoursesMatrix ICourse content:In this course, students are introduced to the concept of a matrix like vectors, matrices and so on. In the first two lessons, student look at matrices from a purely mathematical perspective. The course talks about what matrices are and what problems they are intended to solve and then looks at various operations that can be performed using them. This includes topics like matrix addition and subtraction and multiplication by scalars or by other matrices. At the end, students can conclude this course with an overview of the concept of using matrices to solve system of linear equations.Key Topics:➢Matrix relations➢Matrix operations●Addition/subtraction●Scalar multiplication●Matrix Multiplication●Transpose●Determinant●InversePolynomialsCourse content:This course begins with an examination of the algebra of polynomials and then move on to look at the graphs for various kinds of polynomial functions. The course starts with linear interpolation using polynomials that is commonly used to draw polygons on display. From there students are asked to look at how to take complex functions that would be too costly to compute in a relatively relaxed studying environment and use polynomials to approximate the behavior of the function to produce similar results. Students can wrap things up by looking at how polynomials can be used as means for predicting the future values of variables.Key Topics:➢Polynomial algebra ( single variable)●addition/subtraction●multiplication/division➢Quadratic equations➢Graphing polynomialsLogical Terms in MathematicsCourse content:This course introduces the relationships of four kinds of statements, necessary and sufficient conditions, basic logical conjunctions, existing quantifier and universal quantifier. By learning mathematical logic terms, students can be mastered in the usage of common logical terms and can self-correct logical mistakes. At the end of this course, students can deeply understand the mathematical expression is not only accurate but also concise.Key Topics:➢Statement and its relationship➢Necessary and sufficient conditions➢Basic logical conjunctions➢Existing quantifier and universal quantifierConic Sections and EquationCourse content:By using the knowledge of coordinate method which have been taught in the lesson of linear and circle, in this lesson students learn how to set an equation according the character of conic sections. Students is able to find out the property of conic sections during establishing equations. The aim of this course is to make students understand the idea of combination of number and shape by using the method of coordinate to solve simple geometrical problems which are related to conic sections.Key Topics:➢Curve and equation ➢Oval➢Hyperbola➢Parabola。
中美初中数学教材几何习题的比较研究——以《勾股定理》为例
中美初中数学教材几何习题的比较研究——以《勾股定理》为例研究生姓名：李悦导师姓名：周莹教授学科专业：学科教学·数学研究方向：数学课程与教学论年级：2016级中文摘要随着课程和教育的改革，比较研究始终是教育教学研究的热点。
几何内容是数学教学的主要内容之一，各界对它的重视与日俱增。
教材习题作为教材内容的重要部分，引导学生理解知识内容、构建知识体系、深入理解知识、提高解题能力、发展数学思维、增强数学素养。
因此，国际间几何习题的比较研究对于借鉴别国优势有重要的意义和作用。
现有研究中，专门针对几何习题的比较研究寥寥无几，故有必要通过中外对比研究为我国初中数学教材几何习题的编写建言献策。
本研究选取美国CM版和中国人教版教材，以几何部分“勾股定理”一章的习题为例进行比较研究。
首先用文献分析法对“数学教材习题的相关研究”和“数学教材几何习题的比较研究”进行综述，了解已有研究，得到习题的表层研究趋于完善、几何习题深层比较维度有待挖掘、范希尔几何思维水平理论日益凸显等关键启示，为本文的比较研究奠定基础。
其次是中美两版数学教材的整体比较。
用内容分析法结合比较研究法从“整体结构”、“内容编排”和“所选章节内容设置”三个方面对中美两版教材进行整体比较分析，明确两版教材主体内容的异同，旨在为习题的比较研究作铺垫。
再次对中美两版初中数学教材几何习题进行比较研究。
从编排方式、数量、文字特征、插图、解题策略等维度对两版教材“勾股定理”习题进行表层比较分析，再基于范希尔几何思维水平理论，结合统计知识，从习题的背景、几何思维水平两个维度进行深层比较分析，得到两版教材几何习题的共同点：两版教材均分层设置习题；注重独立解题；习题的几何思维水平较高；但忽略了信息技术；缺乏合作意识；缺乏与其他学科的交互融合。
差异点：CM版重探索过程，人教版重解题能力；CM版注重强调关联知识内容；CM版更注重一题多用，注重还原现实生活中的原有情境；CM版插图类型更丰富，人教版较缺乏“实物”意识；人教版习题的几何思维水平更高。
数学与应用数学课程
数学与应用数学专业03013001数学分析Mathematical Analysis【300—16—1、2、3、4】内容提要：实数、极限理论、一元微积分理论、级数、多元函数的微积分、曲线与曲面积分。
修读对象：数学与应用数学专业本科生教材：《数学分析讲义》（第四版）(上、下册)刘玉琏等编高等教育出版社参考书目：《数学分析》(第二版)上、下册华东师范大学数学系编高等教育出版社《微积分教程》上、下册韩云瑞扈志明主编清华大学出版社03013002 高等代数 Higher Algebra 【198—11—2、3】内容提要：多项式理论、行列式、矩阵、线性方程组、向量空间、线性变换、欧氏空间、正交变换、二次型。
修读对象：数学与应用数学专业本科生教材：《高等代数》(第四版)张禾瑞郝炳新编高等教育出版社参考书目：《高等代数》(上、下册)钮佩琨等编哈尔滨出版社03013003 解析几何 Analytical Geometry 【70—4—1】内容提要：向量代数、直线与平面、常见二次曲面、二次曲面的一般理论。
修读对象：数学与应用数学专业本科生教材：《解析几何》吕林根许子道等主编高等教育出版社参考书目：《空间解析几何》陈希英主编哈尔滨工业大学出版社《空间解析几何引论》(第二版)南开大学吴大任等编高等教育出版社03013004 常微分方程 Ordinary Differential Equation 【72—4—4】先修课程：数学分析、高等代数内容提要：一阶方程的初等积分法、解的存在唯一性定理、高阶线性方程与一阶线性方程组的基本理论、高阶常系数线性方程和一阶常系数线性方程组的解法。
修读对象：数学与应用数学专业本科生教材：《常微分方程》王高雄编高等教育出版社参考书目：《常微分方程》中山大学数学力学系常微分方程组编人民教育出版社《常微分方程》东北师范大学数学系微分方程教研室编高等教育出版社03013005 复变函数 Complex Variable Function 【72—4—5】先修课程：数学分析内容提要：复数、复变函数、解析函数、复变函数积分、调和函数、柯西积分理论、幂级数展开、孤立奇点的分类与特征、函数与亚纯函数、残数理论、保形变换。
利用mathematica软件提高中学数学的教学效果
第39卷增刊黔南民族师范学院学报Vol.39No.Supplement 2019年6月JOURNAL OF QIANNAN NORMAL UNIVERSITY FOR NATIONALITIES Jun.2019利用Mathematica软件提高中学数学的教学效果董银峰（黔南民族师范学院数学与统计学院，贵州都匀558000）摘要：在中学数学教学活动中引入Mathematica软件可以提高教学效果。
有利因素有:可以降低教学难度和激发学生的学习兴趣;培养学生的自学能力;培养学生的维。
教学表明引入Mathemahca软件辅教学良好的教学效果。
关键词：中学数学;教学效果；Mathemahca［中图分类号］G633.6［文献标识码］A［文章编号#1674-2389(2019)增刊-0072-03 Enhancing Teaching Effect ic MiCdle School Mathematics with MathematicaDONG Yin-feng(School of Mathematicsand Statistics$Qiannan Normal University for Nationalities$8uyun558000$Guizhou$China) Abstract:Introducing Mathematica th middle school mathematics can enhanca the teaching effect.This paper suggestc that there are three beneficial factorc.Teaching with Mathematica can lower study difficulty and inspire study enthusiasm.Studying mathematics with Mathematica,studentc can cultivate the self-study capacity and cultivate computational thinking capacity.The given teaching cases suggest that teaching with Mathematica will have a fruitful result.Key wordt:middle school mathematics;teaching effect;Mathematica数学学科作为基础学科,不仅在自然科学和工程技术的研究中极为重要，并经济和管理等社会学科的研究大显身手。
中学数学的英文
中学数学的英文English:"Middle school mathematics, often referred to as junior high school mathematics or secondary school mathematics, encompasses a wide range of fundamental mathematical concepts and skills essential for building a solid foundation in mathematics education. It typically covers topics such as arithmetic, algebra, geometry, trigonometry, and basic statistics and probability. In middle school mathematics, students learn how to perform operations with integers, fractions, and decimals, solve equations and inequalities, work with geometric shapes and their properties, and apply trigonometric functions to solve basic problems. Moreover, they develop critical thinking skills through problem-solving and logical reasoning exercises, which are integral components of mathematical education. Middle school mathematics not only aims to equip students with mathematical knowledge and techniques but also fosters their ability to analyze problems, think creatively, and communicate mathematically. This stage of mathematical education lays the groundwork for more advanced topics in high school and beyond, preparing students forfurther studies in mathematics, science, engineering, and various other fields."Translated content:"中学数学通常被称为初中数学或中学数学，涵盖了一系列基础的数学概念和技能，对建立扎实的数学教育基础至关重要。
一数初中数学重难点方法合集
一数初中数学重难点方法合集Middle school mathematics can be a challenging subject for many students, as it often introduces more complex concepts and requires a deeper understanding of foundational principles. In order to help students navigate these difficulties, it is important to have a collection of effective methods for addressing key points in middle school math. These methods can help guide students towards a more comprehensive understanding of the subject and improve their problem-solving skills. By breaking down difficult concepts into smaller, more digestible pieces, students can approach the material with confidence and overcome their fears of challenging problems.初中数学对许多学生来说可能是一个具有挑战性的学科，因为它往往引入更复杂的概念，并要求对基础原理有更深入的理解。
为了帮助学生应对这些困难，拥有一套有效的方法来解决初中数学中的重难点是至关重要的。
这些方法可以帮助学生更全面地理解学科并提高他们的解决问题的能力。
通过将困难的概念分解为更小、更易理解的部分，学生可以更有信心地对待材料并克服对具有挑战性问题的恐惧。
2024年初中数学新课程标准英文版
2024年初中数学新课程标准英文版2024 Middle School Mathematics New Curriculum StandardsIn 2024, a new set of mathematics curriculum standards will be implemented for middle schools across the country. These standards aim to enhance the quality of math education and better prepare students for future academic and professional pursuits.The updated curriculum will focus on fostering critical thinking skills, problem-solving abilities, and a deeper understanding of mathematical concepts. Students will be encouraged to actively engage with the material through hands-on activities, real-world applications, and collaborative learning experiences.One of the key changes in the new curriculum is the integration of technology into math instruction. Students will have access to digital tools and resources that can support their learning and help them visualize complex mathematical ideas. This shift towards a more tech-savvy approach is meant to reflect the increasing importance of technology in the modern world.Another important aspect of the new curriculum is the emphasis on personalized learning. Teachers will be encouraged to tailor their instruction to meet the diverse needs and learning styles of individual students. This personalized approach is designed to ensure that all students have the opportunity to succeed and reach their full potential in mathematics.Overall, the 2024 middle school mathematics curriculum standards represent a forward-thinking approach to math education that prioritizes critical thinking, problem-solving, and personalized learning. By implementing these standards, educators hope to inspire a new generation of mathematically literate and confident individuals who are well-equipped to tackle the challenges of the 21st century.。
初中数学中的跨学科项目化教学研究
初中数学中的跨学科项目化教学研究摘要随着教育内容的不断发展和改变，现代教育注重学生发展全面素质，扩大了教育的视野和方法。
跨学科项目化教学法在这种情况下应运而生，其通过激发学生的学习兴趣和主动性，引导学生积极参与探究过程，有效地促进了学生知识和技能的发展。
本文针对此，结合初中数学教育，探讨跨学科项目化教学在初中数学中的应用。
具体包括跨学科项目化教学的基本理论、教学设计、教学实施效果等方面进行阐述，证明了跨学科项目化教学在初中数学教育中的实践应用是一种有效的教学方法。
初中数学；跨学科项目化教学；教学设计；实施效果AbstractWith the continuous development and change of education content, modern education attaches importance to the comprehensive development of students' qualities, and expands the vision and methods of education. Interdisciplinary project-based teaching method has emerged under such circumstances. By stimulating students' interest in learning and initiative, guiding students to actively participate in the exploration process, it effectively promotes the development of students' knowledge and skills. This paper explores the application of interdisciplinary project-based teaching in middle school mathematics education, combined with basic theories, teaching design, and teaching implementation effects. This paper proves that interdisciplinary project-based teaching is an effective teaching method in middle school mathematics education.middle school mathematics; interdisciplinary project-based teaching; teaching design; implementation effect一、跨学科项目化教学法起源于20世纪80年代中期，是指将不同学科知识与技能结合起来，通过一个项目来促进学生对不同学科的理解和应用能力。
初中数学难点重点大盘点
04
解答题中的关键技巧
相似三角形的判定与性质
一、相似三角形的判定方法： 二、相似三角形的性质： 三、 面积之比等于边长之比的平方
1. 解决实际问题如影子定理、旗杆定理等。 2. 推导勾股定理。 3. 推导正弦定理和余弦定理。
例子
已知两个直角三角形，其中一个三角形的两条直角边分别为6cm和8cm,另一个三角形 的斜边长为10cm,求这两个三角形的相似比。解答过程如下：利用勾股定理计算斜边长， 判断两三角形相似，根据相似三角形对应边成比例求解相似比。
例子
(1)√3 + √5 = (√3 + √5) × (√3 - √5)/2√10 = (√3 + √5) × (√3 −√5)/2√(50) = 5√2(1)√3 + √5 = (√3 + √5) × (√3 - √5)/2√10 = (√3 + √5) × (√3 −√5)/2√(50) = 5√2
02 总结与建议
总结与建议
1. 总结：通过对中考数学卷子的分析，发现近年来数学卷子越来越注重考查学生对基础知识的掌握和应用能力的提高。因此，重新复习巩固基础知识是提高数 学成绩的关键。
2. 建议：每日都要安排适量的时间进行数学练习，加强对基础知识的掌握，并通过相关题型的练习提高自 己犯错的原因和解题思路，不断总结经验和教训，提高自己的数学能力。
初中数学重难点大盘点
Analysis of Key and Difficult Points in Middle School Mathematics
Contents
01 目录 02 选择题中的易错点 03 填空题中的难点 04 解答题中的关键技巧 05 总结与建议
选择题中的易错点
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7、初次申领驾驶证的驾驶人在实习期内可以单独驾驶机动车上高速公路行驶。答案：错
8、行车中遇到执行紧急任务的消防车、救车，驾驶人的视野受限，很难观察到灯光照射区域以外的交通情况，因此要减速行驶。答案：对
10、驾驶机动车掉头、转弯、下陡坡时的最高速度不能超过每小时40公里。答案：错
1、驾驶机动车在道路上违反交通安全法规的行为属于违法行为。（对）
2、造成交通事故后逃逸构成犯罪的人不能申请机动车驾驶证。（对）
3、驾驶机动车在路口遇到这种信号灯禁止通行。（错、绿灯行）
4、立交桥上一般都是单向行驶，车辆不必减速行驶。（错、太绝对了、应看情况）
5、将点火开关转到ACC位置起动机工作。（错）