八年级数学上册（苏科版）第二章《实数》第1课时：无理数的引入与实数的初步认识教学设计

八年级数学上册（苏科版）第二章《实数》第1课时：无理数的引入与实数的初步认识教学设计一、教学内容分析从《义务教育数学课程标准（2022年版）》审视，本节课属于“数与代数”领域，是学生数系扩张历程中的关键一步。知识技能图谱上，它要求学生从已知的有理数出发，经历无理数的发现过程，理解其“无限不循环”的本质特征，并初步建立实数的概念框架，明确实数与数轴上的点一一对应的关系。这既是前一阶段“数的开方”知识的自然延伸，也为后续学习二次根式、函数、解析几何等奠基，起着承上启下的枢纽作用。过程方法路径上，课标强调通过具体实例，让学生经历从具体到抽象、从特殊到一般的认知过程，体验数学探究的基本方法。本课可设计“探究发现归纳应用”的活动链，引导学生像数学家一样，通过操作、计算、质疑、论证，主动建构无理数的概念，发展科学探究精神和严谨的逻辑推理能力。素养价值渗透方面，本课是培育学生数感、符号意识、推理能力和模型观念的绝佳载体。无理数的发现过程蕴含着数学发展中的理性精神与批判意识，数轴模型的构建则深刻体现了“数形结合”这一核心思想，对于培养学生用数学的眼光观察现实世界，用数学的思维思考现实世界具有重要意义。基于“以学定教”原则进行学情研判。学生的已有基础与障碍在于：他们已熟练掌握有理数的概念、运算及在数轴上的表示，并初步接触了平方根、立方根。然而，从“可表示为分数”的有理数思维定势，跨越到“无限不循环”的无理数认知，存在显著的认知冲突和抽象思维挑战。部分学生可能难以真正理解“无限不循环”的意涵，或对2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​这类数是否“真实存在”心存疑惑。针对此，教学调适策略是：充分利用几何直观（如拼图、单位正方形对角线）和计算探索（如用计算器进行2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​的十进制展开），化抽象为具体。在过程评估设计上，将通过追问（如“你能找到一个平方等于2的分数吗？”）、观察小组讨论、分析随堂生成的错例等方式，动态诊断学生的理解层次，并为不同认知风格的学生提供多元的表征支持（如几何模型、数值计算、逻辑推理），实施分层引导。二、教学目标知识目标：学生通过操作探究与推理分析，能准确陈述无理数的定义（无限不循环小数），并列举常见的无理数类型（如开方开不尽的数、圆周率π\piπ等）；能清晰界定实数的概念，并初步对实数进行合理分类；理解实数与数轴上的点之间的一一对应关系。能力目标：学生能运用计算、反证等方法，论证2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​不是有理数，发展逻辑推理和批判性思维能力；能通过构造直角三角形等几何方式，在数轴上找到表示无理数的点，强化数形结合与动手操作能力；能在具体情境中辨识无理数，并初步进行实数的大小比较。情感态度与价值观目标：学生在重温无理数发现史的过程中，感受数学知识源于实践又不断超越直观的理性魅力，体会数学的确定性与发展性；在小组协作探究中，养成乐于分享、敢于质疑、严谨求实的科学态度。科学（学科）思维目标：重点发展学生的抽象思维与演绎推理思维。通过从具体数值（如2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​的计算结果）中抽象出“无限不循环”这一普遍属性，形成无理数概念；通过“假设2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​是有理数”导出矛盾，体验反证法的逻辑力量，提升思维的严密性。评价与元认知目标：引导学生运用“定义”作为标尺，评价一个数是否为无理数；在课堂小结环节，鼓励学生反思探索无理数概念时遇到的困难及克服策略，比较几何探究与代数推理的不同路径，提升对数学学习方法的元认知水平。三、教学重点与难点教学重点：无理数概念的建立。其确立依据在于，从课程标准看，无理数是实数概念形成的核心基石，是贯穿初等数学的“大概念”；从学科体系看，它是数系从有理数扩张到实数的质变点，不理解无理数，后续关于实数的所有运算与性质都将成为无源之水。从中考视角分析，对无理数概念的辨识与理解是基础且高频的考点。教学难点：对“无限不循环小数”本质的理解及2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​是无理数的证明。其预设依据源于学情：学生首次接触“无限”且“不循环”这种超越有限经验的数学对象，认知跨度大；证明2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​不是有理数需要运用反证法，逻辑链条较长，且涉及对“互质”概念的灵活运用，是思维上的难点，也是作业和考试中的典型失分点。突破方向在于，先用计算器展示其“算不尽”的直观，再用几何拼图引发认知冲突，最后通过精巧设问引导学生一步步完成推理。四、教学准备清单1.教师准备1.1媒体与教具：多媒体课件（含无理数发现史微视频、2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​小数点后多位展示动画）；几何画板动态演示“在数轴上找点”；两个全等的等腰直角三角形纸板。1.2学习材料：设计分层学习任务单（含探究引导、分层练习题）；准备实物投影仪用于展示学生作品。2.学生准备2.1课前预习：复习有理数的定义与分类；了解毕达哥拉斯学派与希帕索斯的故事（阅读简史材料）。2.2学具携带：计算器、直尺、圆规、练习本。五、教学过程第一、导入环节1.情境创设与认知冲突1.1（教师展示两个全等的等腰直角三角形）同学们，如果每个直角三角形的腰长都是1，那么拼成的这个正方形的面积是多少？（学生易答：2。）很好，面积是2。那么它的边长呢？1.2我们设边长为aaa，则有a2=2a^2=2a2=2。aaa是几？你能找到一个确切的分数或小数来表示它吗？大家拿出计算器，试着算算2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">1.414213562......结果是1.414213562...而且好像永远算不完。）“老师，它是不是循环小数？我们多算几位看看...”2.提出核心问题2.1大家发现了一个“怪数”：它的平方等于2，但它本身却写不成一个有限小数或循环小数。它到底是不是一个“数”？在我们学过的有理数家族里，能找到它的位置吗？今天，我们就一起来揭开这类神秘数字的面纱。3.明晰学习路径3.1本节课，我们将首先像一位侦探一样，用推理证明2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​不是有理数；然后为这类“新数”正式命名——无理数；最后，把有理数和无理数统合到一个更庞大的家族——实数中，并探索它们如何在数轴上“安家落户”。第二、新授环节任务一：追本溯源——证明√2不是有理数教师活动：首先，引导学生明确论证目标：证明2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​不能表示为两个整数之比（分数）。提出反证法思路：“我们先假设一个相反的情况成立，看看会导致什么结果。”带领学生一步步书写：假设2=pq\sqrt{2}=\frac{p}{q}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​=qp​（p,q互质，且q≠0），两边平方得2=p2q22=\frac{p^2}{q^2}2=q2p2​，即p2=2q2p^2=2q^2p2=2q2。关键性提问：“从这个式子，你能判断p的奇偶性吗？为什么？”引导学生得出p是偶数，设p=2k。代入得4k2=2q24k^2=2q^24k2=2q2，即q2=2k2q^2=2k^2q2=2k2。继续追问：“现在，q又是什么数？这与我们最初的什么假设矛盾了？”最后总结：“看，这个矛盾说明我们最初的假设‘√2是有理数’是错误的。”学生活动：跟随教师的引导，理解反证法的逻辑起点。积极思考关键问题，尝试推理p和q的奇偶性变化。在教师引导下，完成整个推导过程，并理解“互质”假设在推导中的核心作用。最终理解矛盾所在，认同“√2不能写成分数形式”的结论。即时评价标准：1.逻辑跟随：能否理解每一步推导的目的，而非机械记忆。2.关键点突破：能否独立或经提示后，分析出p为偶数是推理的转折点。3.矛盾阐释：能否清晰说出最终得出的矛盾是什么（p和q都含有因子2，与“互质”假设矛盾）。形成知识、思维、方法清单：★核心结论：2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​不是有理数。它不能表示为两个整数的比。▲关键方法：反证法。当要证明某个结论“不是”什么时，可以先假设“它是”，然后推导出逻辑矛盾，从而证明原结论。●思维要点：论证依赖于“互质”的设定和整数的奇偶性分析，体现了数学推理的严谨性。“同学们，这个证明就像一场精彩的辩论，我们设下‘圈套’，最终让假设不攻自破。”任务二：概念生成——概括无理数的特征与定义教师活动：引导发散：“除了2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​，还有哪些数也有这种‘算不尽、不循环’的特性？请举例。”学生可能提到3\sqrt{3}3<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​、圆周率π\piπ等。教师利用课件动态展示3\sqrt{3}3<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​、π\pi0.1010010001...0.1010010001...（每两个1之间0依次多一个）等数的十进制展开。提问：“这些数和我们熟悉的有理数（有限小数、无限循环小数）根本区别在哪里？”引导学生对比归纳，提炼出“无限不循环”这一核心特征。最后，给出无理数的规范定义：无限不循环小数叫做无理数。并强调：“注意，是‘无限’且‘不循环’，两个条件缺一不可。”学生活动：积极举例，并观察教师展示的各种例子。对比有理数的特征，小组讨论后尝试用自己的语言描述这类“新数”的特点。最终理解并记忆无理数的定义。尝试判断教师给出的新例子（如0.3̇，1....）是否属于无理数。即时评价标准：1.举例迁移：能否举出除2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​、π\piπ外的其他无理数实例。2.特征概括：能否准确归纳出“无限不循环”这一本质特征，而非仅停留在“算不尽”的直观。3.概念辨析：能否运用定义正确判断简单的小数是否为无理数。形成知识、思维、方法清单：★无理数定义：无限不循环小数。●常见类型：（1）开方开不尽的数，如2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​,53\sqrt[3]{5}35<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​；（2）圆周率π\piπ及某些含有π\piπ的数；（3）人为构造的无限不循环小数，如0.1010010001...。▲概念辨析：判断一个数是否为无理数，关键看其小数部分是否“无限”且“不循环”。无限循环小数可以化成分数，是有理数。“大家记住，无理数并不是‘没有道理’的数，而是‘不能表示为整数比的数’，这个名字有点历史误会。”任务三：体系建构——初识实数及其分类教师活动：提问：“现在，我们认识了有理数和无理数。它们之间是什么关系？能把我们学过的所有‘数’放在一起，给个总称吗？”引出实数的概念：有理数和无理数统称为实数。通过韦恩图或树状图，与学生一起构建实数的分类体系。强调分类标准：按定义（是否为无限不循环小数）分为有理数和无理数；按符号分为正实数、0、负实数。进行辨析练习：“请判断‘实数不是有理数就是无理数’、‘带根号的数都是无理数’这些说法对吗？”学生活动：理解“统称”的含义，知晓实数是有理数和无理数的并集。参与分类图的构建，理清从属关系。对辨析问题进行思考和讨论，加深对概念外延的理解，例如认识到4=2\sqrt{4}=24<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​=2是有理数，从而明确“带根号的数不一定无理”。即时评价标准：1.体系理解：能否理解实数、有理数、无理数三个概念之间的包含关系。2.分类操作：能否根据定义对给定的实数（如−13\frac{1}{3}−31​,7\sqrt{7}7<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​,0,3.14,π\piπ）进行正确分类。3.误区澄清：能否识别并解释关于实数分类的常见错误说法。形成知识、思维、方法清单：★实数概念：有理数和无理数统称为实数。这是目前我们所学的最大的数集。★实数分类（按定义）：实数分为有理数（有限小数或无限循环小数）和无理数（无限不循环小数）。●重要澄清：判断一个数是不是无理数，必须化简或分析到最后形式。例如，4\sqrt{4}4<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​化简后为2，是有理数；而π2\frac{\pi}{2}2π​虽然含有π\piπ，但它本身就是一个确定的无限不循环小数，也是无理数。“我们把数域从有理数‘扩军’到了实数，现在我们的‘武器库’更加强大了！”任务四：形数统一——探究实数与数轴上的点教师活动：回顾旧知：“我们知道，每一个有理数都可以用数轴上的一个点来表示。那么，无理数呢？比如2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​，能在数轴上找到它的位置吗？”引导学生利用几何直观：构造一个两直角边均为1的直角三角形，斜边即为2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​。演示如何以原点为圆心，斜边长为半径画弧，与数轴正半轴的交点即表示2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​。追问：“那−2\sqrt{2}−2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​呢？3\sqrt{3}3<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​呢？”推广结论：“事实上，每一个实数（无论有理还是无理）都可以用数轴上的一个点来表示；反过来，数轴上的每一个点都表示一个实数。”即实数与数轴上的点一一对应。学生活动：观察教师的几何作图演示，理解如何在数轴上“构造”出表示2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​的点。在任务单上尝试模仿，找到表示2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​的点。思考并回答教师关于负无理数和3\sqrt{3}3<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​的提问。最终理解并认同“一一对应”的结论。即时评价标准：1.操作理解：能否理解利用勾股定理在数轴上定位无理数的几何原理。2.结论归纳：能否从特殊（2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​）推广到一般，说出实数与数轴点的一一对应关系。3.逆向思维：当看到数轴上一个任意点时，能否意识到它一定对应一个确定的实数（可能是有理数，也可能是无理数）。形成知识、思维、方法清单：★核心关系：实数与数轴上的点一一对应。这是实数完备性的直观体现。●几何方法：利用勾股定理，可以在数轴上作出表示n\sqrt{n}n<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​（n为正整数）的点。这是“以形助数”的典范。▲数学思想：数形结合思想。将抽象的数与直观的图形（数轴）联系起来，使得实数的存在性和顺序性变得可视、可感。“看，这个点（指着2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​对应的点）虽然我们不能用有限小数写出来，但它确确实实、独一无二地存在于数轴上，这就是数学的确定性美。”第三、当堂巩固训练设计核心：构建分层、变式的训练体系，并提供即时反馈。基础层（全体必做，直接应用核心概念）：1.判断下列说法是否正确，并说明理由：（1）无理数都是无限小数。（2）无限小数都是无理数。（3）带根号的数都是无理数。2.将下列各数填入相应的集合：−52,9,0,3.14159,π,7,0.12˙3˙\frac{5}{2},\sqrt{9},0,3.14159,\pi,\sqrt{7},0.1\dot{2}\dot{3}−25​,9<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​,0,3.14159,π,7<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​,0.12˙3˙。有理数集合：{…}；无理数集合：{…}；正实数集合：{…}。综合层（多数学生尝试，在新情境中综合运用）：3.如图，数轴上点A表示的数可能是（）A.5\sqrt{5}5<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​B.10\sqrt{10}10<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​C.15\sqrt{15}15<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​D.20\sqrt{20}20<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​（需估算大小）。4.已知边长为1的正方形的对角线长为2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​，请利用这个结论，在数轴上准确标出表示2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​和3\sqrt{3}3<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​的点（要求保留作图痕迹）。挑战层（学有余力者选做，开放探究）：5.我们知道4=2\sqrt{4}=24<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​=2是有理数，2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​是无理数。那么，n\sqrt{n}n<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​（n是正整数）在什么情况下是有理数？什么情况下是无理数？你能提出一个猜想并尝试解释吗？反馈机制：基础层题目通过全班齐答或快速互查解决，教师点评易错点（如第1题第2小句）。综合层题目请学生上台展示第4题作图过程，并讲解第3题的估算策略，教师补充优化。挑战层题目作为思考题，邀请有想法的学生分享其猜想，教师给予肯定并引导课下继续探究，不做统一要求。“第2题把9\sqrt{9}9<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​放进无理数集合的同学要小心啦，化简是第一步！”“谁来展示一下你是怎么在数轴上‘搭建’出3\sqrt{3}3<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​的？哦，先构造√2，再以它为直角边…思路很清晰！”第四、课堂小结知识整合：引导学生以“实数”为中心，用思维导图的形式回顾本节课的核心概念链：从证明2\sqrt{2}2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47,65,47zM83480Hv40hz">​不是有理数→发现并定义“无理数”（无限不循环小数）→有理数与无理数“统称”为实数→实数与数轴上的点“一一对应”。鼓励学生上台绘制并讲解。方法提炼：“今天我们用了哪些重要的数学方法攻克了‘无理数’这个堡垒？”师生共同回顾：反证法（逻辑推理）、从特殊到一般（归纳定义）、数形结合（在数轴上找点）、估算与辨析。作业布置：必做题（巩固基础）：1.完成课本对应练习，重点辨识无理数与实数分类。2.在数轴上标出表示−2\sqrt{2}−2<pathd="M95,702c2.7,0,7.17,2.7,13.5,8c5.8,5.3,9.5,10,9.5,14c0,2,0.3,3.3,1,4c1.3,2.7,23.83,20.7,67.5,54c44.2,33.3,65.8,50.3,66.5,51c1.3,1.3,3,2,5,2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,71,104,213c68.7,142,137.5,285,206.5,429c69,144,104.5,217.7,106.5,221l00c5.3,9.3,12,14,20,14Hv40H845.2724s225.272,467,225.272,467s235,486,235,486c2.7,4.7,9,7,19,7c6,0,10,1,12,3s194,422,194,422s65,47