八年级数学（上）·二次根式的混合运算与综合应用-基于运算能力与模型意识培养的教学设计

八年级数学（上）·二次根式的混合运算与综合应用——基于运算能力与模型意识培养的教学设计一、教学内容分析

本节课选自北师大版《数学》八年级上册第二章“实数”中“二次根式”单元的第三课时。从《义务教育数学课程标准（2022年版）》审视，本课处于“数与代数”领域的“数与式”主题脉络中。在知识技能图谱上，它上承二次根式的乘除法则与最简形式，下启实数运算体系的完整建构与后续代数式的复杂变形，是学生从单一运算规则走向综合运算能力的关键枢纽。其认知要求已从“理解”“掌握”具体法则，跃升至在复杂情境中“运用”法则解决问题，强调运算的准确性、灵活性与策略性。在过程方法层面，课标倡导的“模型观念”与“运算能力”在此交汇。本节课通过设计解决实际问题的任务，引导学生经历“从实际情境中抽象出数学问题—运用二次根式运算建立数学模型—求解并解释”的完整过程，这正是数学建模思想的微型化实践。在素养价值渗透上，二次根式作为实数家族的重要成员，其运算的严谨性有助于培养学生理性求真的科学精神；而综合应用中所涉及的优化、估算等问题，则能引导学生体会数学的工具价值，发展其有条理、重逻辑的思维品质。

立足“以学定教”原则，需进行立体学情研判。学生已有的基础是：掌握了二次根式的概念、性质及乘除运算的基本法则，具备了初步的代数式变形能力和有理数混合运算的经验。潜在的认知障碍在于：第一，面对包含加、减、乘、除、乘方、括号等多种运算的混合式子时，容易混淆运算顺序或遗忘法则的适用条件；第二，在化简与运算的交织过程中，对“先化简后运算”或“边运算边化简”的策略选择感到困惑；第三，将运算技能应用于实际问题时，从文字语言到数学符号语言的转化仍是难点。因此，教学调适策略应着眼于搭建“脚手架”：通过设计由浅入深、从程序性到策略性的任务链，辅以清晰的操作步骤提示卡（脚手架），帮助学生厘清运算逻辑。同时，预设贯穿课堂的形成性评价，如观察学生板演步骤、聆听小组讨论中的典型错误、分析随堂练习的即时反馈，以便动态捕捉学情，对普遍性难点进行定点突破，并对理解较快的学生提供更具挑战性的变式问题。二、教学目标

知识目标：学生能准确叙述二次根式混合运算的顺序，理解并能在具体算式中识别运算的层级；能在包含加、减、乘、除及乘方的混合算式中，正确、熟练地综合运用二次根式的性质和运算法则进行化简与计算，最终将结果化为最简二次根式或整式。

能力目标：学生能够从简单的实际问题（如几何图形中的长度、面积计算）中抽象出二次根式的运算式，并予以求解，初步体验数学建模的过程；在运算过程中，能根据式子的结构特征，灵活选择“先化简后运算”或“运用乘法公式”等优化策略，发展运算的策略性思维和简捷求解能力。

情感态度与价值观目标：在解决与实际生活或几何图形相关的问题过程中，感受数学的应用价值，增强学习兴趣；在小组合作探究与交流中，敢于发表自己的见解，也能认真倾听、理性辨析同伴的思路，体验合作学习的效能感与严谨求实的数学态度之重要性。

科学（学科）思维目标：重点发展学生的“模型意识”与“有序思维”。通过将实际问题转化为二次根式运算模型的任务，强化从具体到抽象的建模思维；通过严格遵循运算顺序和步步有据的推理，培养思维的条理性和逻辑性，克服运算中的随意性。

评价与元认知目标：引导学生建立“运算自查清单”（如：运算顺序对吗？每一步运用了哪个法则？结果是最简形式吗？），学会在完成运算后进行自主检验与反思；鼓励学生在解决综合应用问题时，尝试评价不同解法的优劣，初步形成策略优化的元认知意识。三、教学重点与难点

教学重点：二次根式混合运算的顺序与综合运算法则的正确应用。确立依据源于课标对“运算能力”的核心要求，以及本单元知识的结构性地位：混合运算是二次根式四则运算知识的综合集成与高阶应用，是检验学生是否真正构建起二次根式运算认知网络的关键节点，也是后续学习函数、解直角三角形等内容时处理代数式的重要基础。从中考视角看，二次根式的混合运算常作为基础计算题或复杂代数题中的一环出现，其准确性是基本要求。

教学难点：一是根据算式的具体特征灵活选择合理的运算策略进行简化计算；二是将实际问题准确转化为二次根式运算模型。难点成因在于，策略选择需要学生对运算律、乘法公式及二次根式性质有融会贯通的理解，并具备一定的观察力和分析力，这对八年级学生的思维灵活性提出了挑战。实际问题的转化则涉及数学阅读、信息提取与数学建模能力，学生容易因对情境理解偏差或等量关系寻找困难而受阻。预设突破方向是通过典型例题的对比剖析，暴露思维过程，归纳策略选择的原则；并通过搭建“问题拆解”脚手架，引导学生分步完成从“实际问题”到“数学表达式”的转化。四、教学准备清单1.教师准备

1.1媒体与教具：交互式多媒体课件（内含问题情境动画、分层例题与练习、课堂小结思维导图框架）；几何图形模具（用于展示面积、周长问题）。

1.2文本与工具：设计并打印《分层学习任务单》（含“基础闯关”“能力攀升”“挑战自我”三个梯度）；准备课堂板书的思维导图主框架。2.学生准备

复习二次根式的乘除运算法则及最简二次根式的概念；每人准备课堂练习本、草稿纸。3.环境预设

学生按4人异质小组就座，便于开展合作讨论与互评。五、教学过程第一、导入环节

1.情境创设：同学们，想象一下，我们正在为一间长方形的“数学工作室”设计地面铺装方案。已知工作室的长为(32+6)(3\sqrt{2}+\sqrt{6})(32<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">​+6<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">​)米，宽为(22−6)(2\sqrt{2}\sqrt{6})(22<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">​−6<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">​)米。我们不仅要计算铺设地面所需的总面积，还想在四周贴上踢脚线，需要计算周长。看看这间神奇的“数学工作室”，你能发现哪些我们学过的“老朋友”？对，它的边长是用我们刚学的二次根式表示的。那么，它的面积和周长该如何表示和计算呢？

1.1问题提出：面对这样既有加法又有乘法，还有括号的复杂二次根式表达式，我们该如何进行运算？运算的顺序又该如何确定？这就是我们今天要攻克的核心问题：二次根式的混合运算与综合应用。

1.2路径明晰：这节课，我们将像一位熟练的工程师，首先回顾并确认我们的“运算工具箱”里有哪些工具（法则），然后学习如何按照正确的“施工顺序”（运算顺序）来使用它们。我们会从简单的算式组合开始练手，逐步升级到解决像“工作室装修”这样的实际问题。最后，大家还要学会评估哪种运算路径更快捷。准备好了吗？让我们开始今天的探索之旅。第二、新授环节任务一：回顾“工具箱”——运算规则再确认

教师活动：首先，我会通过课件快速呈现一组基础问题：“计算12×3\sqrt{12}\times\sqrt{3}12<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">​×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">​”、“计算62÷226\sqrt{2}\div2\sqrt{2}62<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">​÷22<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">​”、“将18\sqrt{18}18<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">​化为最简二次根式”。不急于让学生计算，而是提问：“看到这些题目，你脑海中首先调用的‘工具’是什么？能说出它的名字和使用注意吗？”通过追问，引导学生不仅说出法则内容，更强调每一步的依据。例如，在学生回答乘法法则后，我会追问：“这里被开方数12和3相乘得36，36开方得6，这个过程本质上运用了我们学过的哪个运算性质？”从而将学生的注意力引向a⋅b=ab\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}a<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<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">​=ab<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">​(a≥0,b≥0)(a\geq0,b\geq0)(a≥0,b≥0)这一核心性质。同时，我会在黑板的“工具箱”区域板书：工具1：乘除法法则（依据：性质）；工具2：化简原则（化为最简）。

学生活动：学生快速口答或心算基础问题，并大声说出所运用的法则名称及注意事项。在教师追问下，思考并回答法则背后的数学原理（二次根式的性质）。他们会意识到，熟练、准确地调用这些基础工具是进行复杂混合运算的前提。

即时评价标准：1.准确性：能否快速、正确地得出基础题目的答案。2.表述清晰度：能否用规范的数学语言描述运算法则，而非仅仅给出结果。3.原理关联：能否在教师提示下，将法则与更基本的二次根式性质联系起来。

形成知识、思维、方法清单：★核心工具包：进行二次根式运算，必须熟练掌握两个核心工具：乘法法则a⋅b=ab\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}a<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<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">​=ab<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">​(a≥0,b≥0)(a\geq0,b\geq0)(a≥0,b≥0)与除法法则ab=ab\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}b<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">​a<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">​​=ba​<pathd="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,704.7,510.7,1060.3,512,1067l00c4.7,7.3,11,11,19,11H40000v40H1012.3s271.3,567,271.3,567c38.7,80.7,84,175,136,283c52,108,89.167,185.3,111.5,232c22.3,46.7,33.8,70.3,34.5,71c4.7,4.7,12.3,7,23,7s12,1,12,1s109,253,109,253c72.7,168,109.3,252,110,252c10.7,8,22,16.7,34,26c22,17.3,33.3,26,34,26s26,26,26,26s76,59,76,59s76,60,76,60zMhv40hz">​(a≥0,b>0)(a\geq0,b>0)(a≥0,b>0)。▲化简先行原则：在大多数混合运算中，先将各项化为最简二次根式，能为后续计算扫清障碍，这是一个重要的策略意识。◆思维起点：遇到复杂的式子不要慌，先“拆解”，识别其中包含哪些基本的运算类型（加、减、乘、除、乘方），这有助于规划运算路径。任务二：厘清“施工图”——运算顺序再建立

教师活动：出示一道混合运算题：(12+18)×3−50÷2(\sqrt{12}+\sqrt{18})\times\sqrt{3}\sqrt{50}\div\sqrt{2}(12<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">​+18<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">​)×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">​−50<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<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">​。提问：“这个算式里，出现了我们小学就学过的加、减、乘、除，还有括号。面对这样的‘综合工程’，我们应该先做什么，后做什么？运算的顺序规则有没有因为变成了二次根式而改变呢？”给予学生片刻思考后，请一位学生阐述顺序。随后，我会引导全班共同将其拆解为几个步骤：第一步，处理括号内（若有）；第二步，处理乘除（同级运算从左到右）；第三步，处理加减（合并同类二次根式）。我会强调：“数学世界的运算顺序是通用的‘交通规则’，不会因为车辆从整数变成二次根式就改变。请大家务必遵守规则，否则会‘计算拥堵’甚至‘得出错解’。”接着，我带领学生按照这个顺序，一步步板演，并在每一步旁标注所依据的“工具”和“顺序规则”。

学生活动：学生类比有理数的混合运算顺序，尝试说出二次根式混合运算的顺序。观察教师的规范板演，对照自己的思路，确认顺序规则的一致性。部分学生会主动在草稿纸上跟随计算。

即时评价标准：1.迁移能力：能否将有理数运算的顺序规则顺利迁移到二次根式运算中。2.程序性知识掌握：在叙述顺序时，能否清晰说明“先乘除，后加减，有括号先算括号里”的具体含义。3.步骤跟随度：在教师示范时，能否专注观察并理解每一步的意图。

形成知识、思维、方法清单：★运算顺序铁律：二次根式的混合运算顺序与有理数、整式运算顺序完全相同：先乘方，再乘除，最后加减；有括号，先算括号内的。这是保证运算正确性的根本法则。◆程序化思维：面对复杂运算，养成“先观全局定顺序，再分步执行细计算”的思维习惯，是避免顺序错误的关键。▲步骤可视化：在初学时，建议像老师板演那样，将每一步运算单独写成一行，并简要注明理由（如“化简”、“乘法运算”、“合并同类项”），这有利于自我检查和他人审阅。任务三：实战“基础工程”——规范步骤演练

教师活动：出示两道递进式例题：例1：8+32−18\sqrt{8}+3\sqrt{2}\sqrt{18}8<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">​+32<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">​−18<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">​；例2：(5+2)(5−3)(\sqrt{5}+2)(\sqrt{5}3)(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">​+2)(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">​−3)。对于例1，我会提问：“同学们，拿到这道题，你的第一步打算做什么？为什么？”引导学生得出“先化简各项”的策略。然后请一位学生上台板演，要求其边写边讲解。对于例2，我会问：“这个算式像我们以前学过的什么形式？它能否直接运用某种公式简化计算？”启发学生联想到多项式乘法，并思考是否适用乘法公式。在学生尝试后，我会对比展示直接逐项相乘和视作多项式相乘两种方法，让学生体会后者在有些情况下的便捷性。我会小结：“看，在‘工具箱’里，我们不仅有基本的乘除法则，还有从整式那里继承来的‘乘法公式’这个‘高级工具’，关键是要能识别出使用它的时机。”

学生活动：独立或小组讨论思考教师提出的引导性问题。对于例1，学生积极思考“化简先行”策略，并观察同伴板演，评价其步骤规范性。对于例2，学生尝试计算，在教师引导下发现其与多项式乘法的联系，并对比不同方法的优劣。他们开始意识到，运算是需要观察和选择的。

即时评价标准：1.策略应用主动性：面对题目，是否能主动思考“先化简”或“寻找简便方法”。2.表达逻辑性：上台板演的学生能否清晰、有条理地解释每一步的运算依据。3.方法对比意识：在例2的讨论中，能否对不同方法进行比较，并简单说明看法。

形成知识、思维、方法清单：★策略一：化简优先：当算式中含有可化简的二次根式时，优先将其化为最简形式，往往能简化计算，尤其是便于后续的合并同类项。★策略二：公式助攻：二次根式的乘法满足分配律，遇到形如(a+b)(c+d)(\sqrt{a}+\sqrt{b})(\sqrt{c}+\sqrt{d})(a<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<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<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<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">​)或类似平方差公式(m+n)(m−n)(m+n)(mn)(m+n)(m−n)的结构时，应积极运用多项式乘法的法则或乘法公式进行计算，这能大大提高效率和准确性。◆观察先行：动手计算前，花几秒钟整体观察算式的结构特征，决定运算策略，这是一种高效的数学思维习惯。任务四：挑战“综合项目”——实际应用建模

教师活动：回归导入时的“数学工作室”问题。将问题分解为两个子任务递进出示。子任务A：请列出工作室面积S的表达式。子任务B：请计算面积S的具体数值（结果保留根号）。首先，我会说：“大家先把问题静静地读两遍，尝试着用我们工具箱里的‘零件’自己搭一搭，画一画。”给予学生独立审题和列式的时间。随后，我邀请一位学生分享其列出的面积表达式：S=(32+6)(22−6)S=(3\sqrt{2}+\sqrt{6})(2\sqrt{2}\sqrt{6})S=(32<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">​+6<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">​)(22<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">​−6<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">​)。我会追问：“你是怎么想到用乘法来求面积的？长和宽这两个量是什么关系？”以此强化“矩形面积=长×宽”这一数学模型。接着，聚焦到计算这个表达式：“观察这个式子，它让你联想到任务三中的哪道例题？你打算用什么方法计算？”引导学生将其识别为多项式乘法，并选择运用分配律或视为(a+b)(c−d)(a+b)(cd)(a+b)(c−d)型展开。组织学生以小组为单位进行计算，并巡视指导，特别关注是否有学生尝试识别平方差公式（此处不是标准平方差，但可观察结构）。计算完毕后，请小组展示并对比不同展开方式。

学生活动：学生独立阅读问题，将文字语言转化为数学符号语言，列出面积表达式。在小组内，他们合作讨论如何计算这个复杂的乘法表达式，尝试运用分配律进行运算。他们可能会争论是先化简括号内的项（此处已最简）还是直接相乘。在计算过程中，体验多项式乘法与二次根式运算的结合。最终，他们需要将计算结果化简，并可能惊喜地发现结果是一个整数（6）。

即时评价标准：1.建模准确性：能否正确地将实际问题中的“长”“宽”与数学表达式对应，并选择正确的运算（乘法）建立模型。2.计算坚韧性：面对步骤稍多的计算，能否保持耐心和细致，一步步准确完成。3.合作有效性：在小组讨论中，是否积极参与，贡献思路，并倾听他人意见。

形成知识、思维、方法清单：★应用核心：数学建模：解决实际问题的关键步骤是“建模”——将实际问题中的数量关系，用数学符号和表达式正确地表示出来。例如，矩形面积问题建模为“S=长×宽”。▲运算综合性：在实际应用中，运算往往是法则、顺序、策略的全面检验。计算(32+6)(22−6)(3\sqrt{2}+\sqrt{6})(2\sqrt{2}\sqrt{6})(32<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">​+6<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">​)(22<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">​−6<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">​)需要综合运用分配律、二次根式乘法法则及合并同类项。◆结果检验：当运算结果异常简洁（如得到整数）时，这本身就是一个很好的成功反馈和检验信号，鼓励学生享受数学的简洁之美。任务五：策略“优化研讨”——方法对比提炼

教师活动：提出一个对比性问题：“计算(3+1)2(\sqrt{3}+1)^2(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">​+1)2，你有哪些不同的方法？比比看，谁的方法更巧妙？”给学生12分钟独立思考与尝试。预计学生会出现两种主要方法：方法一：直接利用(a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2(a+b)2=a2+2ab+b2公式；方法二：写成(3+1)(3+1)(\sqrt{3}+1)(\sqrt{3}+1)(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">​+1)(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">​+1)后按多项式乘法展开。请持不同方法的学生代表上台展示。随后，我引导学生对比：“两种方法都正确，但在计算过程、步骤多少和出错概率上，有没有差异？你更喜欢哪一种？为什么？”通过讨论，让学生自己体会到乘法公式在特定结构下带来的简洁性和可靠性。我会顺势总结：“看来，我们的‘工具箱’里，工具没有好坏之分，但有‘更合适’之说。学会根据‘工程’（算式）的特点，选择最合适的工具和工艺，才是高手。”

学生活动：学生积极尝试不同的计算方法，展示自己的解题过程。在对比讨论中，他们分析两种方法的异同，从步骤、计算量、易错点等角度评价方法的优劣。他们开始形成这样的观念：数学解题不仅追求答案正确，也追求过程优化。

即时评价标准：1.方法多样性：能否想出至少一种正确的计算方法。2.批判性思维：能否对不同的方法进行比较，并给出有理由的偏好选择。3.归纳能力：能否从具体例子的对比中，初步归纳出选择运算策略的一般性原则（如：遇平方形式，优先考虑公式）。

形成知识、思维、方法清单：★策略优化意识：数学运算追求准确与效率的统一。当算式符合乘法公式结构时，主动运用公式通常是更优策略。▲完全平方公式应用：(a+b)2=a+2ab+b(\sqrt{a}+\sqrt{b})^2=a+2\sqrt{ab}+b(a<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<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=a+2ab<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和(a−b)2=a−2ab+b(\sqrt{a}\sqrt{b})^2=a2\sqrt{ab}+b(a<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,