八年級數(shù)學(xué)上冊（蘇科版）第二章《實(shí)數(shù)》第1課時(shí)：無理數(shù)的引入與實(shí)數(shù)的初步認(rèn)識教學(xué)設(shè)計(jì)一、教學(xué)內(nèi)容分析從《義務(wù)教育數(shù)學(xué)課程標(biāo)準(zhǔn)（2022年版）》審視，本節(jié)課屬于“數(shù)與代數(shù)”領(lǐng)域，是學(xué)生數(shù)系擴(kuò)張歷程中的關(guān)鍵一步。知識技能圖譜上，它要求學(xué)生從已知的有理數(shù)出發(fā)，經(jīng)歷無理數(shù)的發(fā)現(xiàn)過程，理解其“無限不循環(huán)”的本質(zhì)特征，并初步建立實(shí)數(shù)的概念框架，明確實(shí)數(shù)與數(shù)軸上的點(diǎn)一一對應(yīng)的關(guān)系。這既是前一階段“數(shù)的開方”知識的自然延伸，也為后續(xù)學(xué)習(xí)二次根式、函數(shù)、解析幾何等奠基，起著承上啟下的樞紐作用。過程方法路徑上，課標(biāo)強(qiáng)調(diào)通過具體實(shí)例，讓學(xué)生經(jīng)歷從具體到抽象、從特殊到一般的認(rèn)知過程，體驗(yàn)數(shù)學(xué)探究的基本方法。本課可設(shè)計(jì)“探究發(fā)現(xiàn)歸納應(yīng)用”的活動(dòng)鏈，引導(dǎo)學(xué)生像數(shù)學(xué)家一樣，通過操作、計(jì)算、質(zhì)疑、論證，主動(dòng)建構(gòu)無理數(shù)的概念，發(fā)展科學(xué)探究精神和嚴(yán)謹(jǐn)?shù)倪壿嬐评砟芰?。素養(yǎng)價(jià)值滲透方面，本課是培育學(xué)生數(shù)感、符號意識、推理能力和模型觀念的絕佳載體。無理數(shù)的發(fā)現(xiàn)過程蘊(yùn)含著數(shù)學(xué)發(fā)展中的理性精神與批判意識，數(shù)軸模型的構(gòu)建則深刻體現(xiàn)了“數(shù)形結(jié)合”這一核心思想，對于培養(yǎng)學(xué)生用數(shù)學(xué)的眼光觀察現(xiàn)實(shí)世界，用數(shù)學(xué)的思維思考現(xiàn)實(shí)世界具有重要意義。基于“以學(xué)定教”原則進(jìn)行學(xué)情研判。學(xué)生的已有基礎(chǔ)與障礙在于：他們已熟練掌握有理數(shù)的概念、運(yùn)算及在數(shù)軸上的表示，并初步接觸了平方根、立方根。然而，從“可表示為分?jǐn)?shù)”的有理數(shù)思維定勢，跨越到“無限不循環(huán)”的無理數(shù)認(rèn)知，存在顯著的認(rèn)知沖突和抽象思維挑戰(zhàn)。部分學(xué)生可能難以真正理解“無限不循環(huán)”的意涵，或?qū)?\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">?這類數(shù)是否“真實(shí)存在”心存疑惑。針對此，教學(xué)調(diào)適策略是：充分利用幾何直觀（如拼圖、單位正方形對角線）和計(jì)算探索（如用計(jì)算器進(jìn)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">?的十進(jìn)制展開），化抽象為具體。在過程評估設(shè)計(jì)上，將通過追問（如“你能找到一個(gè)平方等于2的分?jǐn)?shù)嗎？”）、觀察小組討論、分析隨堂生成的錯(cuò)例等方式，動(dòng)態(tài)診斷學(xué)生的理解層次，并為不同認(rèn)知風(fēng)格的學(xué)生提供多元的表征支持（如幾何模型、數(shù)值計(jì)算、邏輯推理），實(shí)施分層引導(dǎo)。二、教學(xué)目標(biāo)知識目標(biāo)：學(xué)生通過操作探究與推理分析，能準(zhǔn)確陳述無理數(shù)的定義（無限不循環(huán)小數(shù)），并列舉常見的無理數(shù)類型（如開方開不盡的數(shù)、圓周率π\(zhòng)piπ等）；能清晰界定實(shí)數(shù)的概念，并初步對實(shí)數(shù)進(jìn)行合理分類；理解實(shí)數(shù)與數(shù)軸上的點(diǎn)之間的一一對應(yīng)關(guān)系。能力目標(biāo)：學(xué)生能運(yùn)用計(jì)算、反證等方法，論證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">?不是有理數(shù)，發(fā)展邏輯推理和批判性思維能力；能通過構(gòu)造直角三角形等幾何方式，在數(shù)軸上找到表示無理數(shù)的點(diǎn)，強(qiáng)化數(shù)形結(jié)合與動(dòng)手操作能力；能在具體情境中辨識無理數(shù)，并初步進(jìn)行實(shí)數(shù)的大小比較。情感態(tài)度與價(jià)值觀目標(biāo)：學(xué)生在重溫?zé)o理數(shù)發(fā)現(xiàn)史的過程中，感受數(shù)學(xué)知識源于實(shí)踐又不斷超越直觀的理性魅力，體會數(shù)學(xué)的確定性與發(fā)展性；在小組協(xié)作探究中，養(yǎng)成樂于分享、敢于質(zhì)疑、嚴(yán)謹(jǐn)求實(shí)的科學(xué)態(tài)度。科學(xué)（學(xué)科）思維目標(biāo)：重點(diǎn)發(fā)展學(xué)生的抽象思維與演繹推理思維。通過從具體數(shù)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">?的計(jì)算結(jié)果）中抽象出“無限不循環(huán)”這一普遍屬性，形成無理數(shù)概念；通過“假設(shè)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">?是有理數(shù)”導(dǎo)出矛盾，體驗(yàn)反證法的邏輯力量，提升思維的嚴(yán)密性。評價(jià)與元認(rèn)知目標(biāo)：引導(dǎo)學(xué)生運(yùn)用“定義”作為標(biāo)尺，評價(jià)一個(gè)數(shù)是否為無理數(shù)；在課堂小結(jié)環(huán)節(jié)，鼓勵(lì)學(xué)生反思探索無理數(shù)概念時(shí)遇到的困難及克服策略，比較幾何探究與代數(shù)推理的不同路徑，提升對數(shù)學(xué)學(xué)習(xí)方法的元認(rèn)知水平。三、教學(xué)重點(diǎn)與難點(diǎn)教學(xué)重點(diǎn)：無理數(shù)概念的建立。其確立依據(jù)在于，從課程標(biāo)準(zhǔn)看，無理數(shù)是實(shí)數(shù)概念形成的核心基石，是貫穿初等數(shù)學(xué)的“大概念”；從學(xué)科體系看，它是數(shù)系從有理數(shù)擴(kuò)張到實(shí)數(shù)的質(zhì)變點(diǎn)，不理解無理數(shù)，后續(xù)關(guān)于實(shí)數(shù)的所有運(yùn)算與性質(zhì)都將成為無源之水。從中考視角分析，對無理數(shù)概念的辨識與理解是基礎(chǔ)且高頻的考點(diǎn)。教學(xué)難點(diǎn)：對“無限不循環(huán)小數(shù)”本質(zhì)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">?是無理數(shù)的證明。其預(yù)設(shè)依據(jù)源于學(xué)情：學(xué)生首次接觸“無限”且“不循環(huán)”這種超越有限經(jīng)驗(yàn)的數(shù)學(xué)對象，認(rèn)知跨度大；證明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">?不是有理數(shù)需要運(yùn)用反證法，邏輯鏈條較長，且涉及對“互質(zhì)”概念的靈活運(yùn)用，是思維上的難點(diǎn)，也是作業(yè)和考試中的典型失分點(diǎn)。突破方向在于，先用計(jì)算器展示其“算不盡”的直觀，再用幾何拼圖引發(fā)認(rèn)知沖突，最后通過精巧設(shè)問引導(dǎo)學(xué)生一步步完成推理。四、教學(xué)準(zhǔn)備清單1.教師準(zhǔn)備1.1媒體與教具：多媒體課件（含無理數(shù)發(fā)現(xiàn)史微視頻、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">?小數(shù)點(diǎn)后多位展示動(dòng)畫）；幾何畫板動(dòng)態(tài)演示“在數(shù)軸上找點(diǎn)”；兩個(gè)全等的等腰直角三角形紙板。1.2學(xué)習(xí)材料：設(shè)計(jì)分層學(xué)習(xí)任務(wù)單（含探究引導(dǎo)、分層練習(xí)題）；準(zhǔn)備實(shí)物投影儀用于展示學(xué)生作品。2.學(xué)生準(zhǔn)備2.1課前預(yù)習(xí)：復(fù)習(xí)有理數(shù)的定義與分類；了解畢達(dá)哥拉斯學(xué)派與希帕索斯的故事（閱讀簡史材料）。2.2學(xué)具攜帶：計(jì)算器、直尺、圓規(guī)、練習(xí)本。五、教學(xué)過程第一、導(dǎo)入環(huán)節(jié)1.情境創(chuàng)設(shè)與認(rèn)知沖突1.1（教師展示兩個(gè)全等的等腰直角三角形）同學(xué)們，如果每個(gè)直角三角形的腰長都是1，那么拼成的這個(gè)正方形的面積是多少？（學(xué)生易答：2。）很好，面積是2。那么它的邊長呢？1.2我們設(shè)邊長為aaa，則有a2=2a^2=2a2=2。aaa是幾？你能找到一個(gè)確切的分?jǐn)?shù)或小數(shù)來表示它嗎？大家拿出計(jì)算器，試著算算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......結(jié)果是1.414213562...而且好像永遠(yuǎn)算不完。）“老師，它是不是循環(huán)小數(shù)？我們多算幾位看看...”2.提出核心問題2.1大家發(fā)現(xiàn)了一個(gè)“怪?jǐn)?shù)”：它的平方等于2，但它本身卻寫不成一個(gè)有限小數(shù)或循環(huán)小數(shù)。它到底是不是一個(gè)“數(shù)”？在我們學(xué)過的有理數(shù)家族里，能找到它的位置嗎？今天，我們就一起來揭開這類神秘?cái)?shù)字的面紗。3.明晰學(xué)習(xí)路徑3.1本節(jié)課，我們將首先像一位偵探一樣，用推理證明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">?不是有理數(shù)；然后為這類“新數(shù)”正式命名——無理數(shù)；最后，把有理數(shù)和無理數(shù)統(tǒng)合到一個(gè)更龐大的家族——實(shí)數(shù)中，并探索它們?nèi)绾卧跀?shù)軸上“安家落戶”。第二、新授環(huán)節(jié)任務(wù)一：追本溯源——證明√2不是有理數(shù)教師活動(dòng)：首先，引導(dǎo)學(xué)生明確論證目標(biāo)：證明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">?不能表示為兩個(gè)整數(shù)之比（分?jǐn)?shù)）。提出反證法思路：“我們先假設(shè)一個(gè)相反的情況成立，看看會導(dǎo)致什么結(jié)果?！睅ьI(lǐng)學(xué)生一步步書寫：假設(shè)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互質(zhì)，且q≠0），兩邊平方得2=p2q22=\frac{p^2}{q^2}2=q2p2?，即p2=2q2p^2=2q^2p2=2q2。關(guān)鍵性提問：“從這個(gè)式子，你能判斷p的奇偶性嗎？為什么？”引導(dǎo)學(xué)生得出p是偶數(shù)，設(shè)p=2k。代入得4k2=2q24k^2=2q^24k2=2q2，即q2=2k2q^2=2k^2q2=2k2。繼續(xù)追問：“現(xiàn)在，q又是什么數(shù)？這與我們最初的什么假設(shè)矛盾了？”最后總結(jié)：“看，這個(gè)矛盾說明我們最初的假設(shè)‘√2是有理數(shù)’是錯(cuò)誤的?！睂W(xué)生活動(dòng)：跟隨教師的引導(dǎo)，理解反證法的邏輯起點(diǎn)。積極思考關(guān)鍵問題，嘗試推理p和q的奇偶性變化。在教師引導(dǎo)下，完成整個(gè)推導(dǎo)過程，并理解“互質(zhì)”假設(shè)在推導(dǎo)中的核心作用。最終理解矛盾所在，認(rèn)同“√2不能寫成分?jǐn)?shù)形式”的結(jié)論。即時(shí)評價(jià)標(biāo)準(zhǔn)：1.邏輯跟隨：能否理解每一步推導(dǎo)的目的，而非機(jī)械記憶。2.關(guān)鍵點(diǎn)突破：能否獨(dú)立或經(jīng)提示后，分析出p為偶數(shù)是推理的轉(zhuǎn)折點(diǎn)。3.矛盾闡釋：能否清晰說出最終得出的矛盾是什么（p和q都含有因子2，與“互質(zhì)”假設(shè)矛盾）。形成知識、思維、方法清單：★核心結(jié)論：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">?不是有理數(shù)。它不能表示為兩個(gè)整數(shù)的比?！P(guān)鍵方法：反證法。當(dāng)要證明某個(gè)結(jié)論“不是”什么時(shí)，可以先假設(shè)“它是”，然后推導(dǎo)出邏輯矛盾，從而證明原結(jié)論?！袼季S要點(diǎn)：論證依賴于“互質(zhì)”的設(shè)定和整數(shù)的奇偶性分析，體現(xiàn)了數(shù)學(xué)推理的嚴(yán)謹(jǐn)性。“同學(xué)們，這個(gè)證明就像一場精彩的辯論，我們設(shè)下‘圈套’，最終讓假設(shè)不攻自破?！比蝿?wù)二：概念生成——概括無理數(shù)的特征與定義教師活動(dòng)：引導(dǎo)發(fā)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">?，還有哪些數(shù)也有這種‘算不盡、不循環(huán)’的特性？請舉例。”學(xué)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">?、圓周率π\(zhòng)piπ等。教師利用課件動(dòng)態(tài)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\(zhòng)pi0.1010010001...0.1010010001...（每兩個(gè)1之間0依次多一個(gè)）等數(shù)的十進(jìn)制展開。提問：“這些數(shù)和我們熟悉的有理數(shù)（有限小數(shù)、無限循環(huán)小數(shù)）根本區(qū)別在哪里？”引導(dǎo)學(xué)生對比歸納，提煉出“無限不循環(huán)”這一核心特征。最后，給出無理數(shù)的規(guī)范定義：無限不循環(huán)小數(shù)叫做無理數(shù)。并強(qiáng)調(diào)：“注意，是‘無限’且‘不循環(huán)’，兩個(gè)條件缺一不可?！睂W(xué)生活動(dòng)：積極舉例，并觀察教師展示的各種例子。對比有理數(shù)的特征，小組討論后嘗試用自己的語言描述這類“新數(shù)”的特點(diǎn)。最終理解并記憶無理數(shù)的定義。嘗試判斷教師給出的新例子（如0.3?，1....）是否屬于無理數(shù)。即時(shí)評價(jià)標(biāo)準(zhǔn)：1.舉例遷移：能否舉出除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\(zhòng)piπ外的其他無理數(shù)實(shí)例。2.特征概括：能否準(zhǔn)確歸納出“無限不循環(huán)”這一本質(zhì)特征，而非僅停留在“算不盡”的直觀。3.概念辨析：能否運(yùn)用定義正確判斷簡單的小數(shù)是否為無理數(shù)。形成知識、思維、方法清單：★無理數(shù)定義：無限不循環(huán)小數(shù)?！癯Ｒ婎愋停海?）開方開不盡的數(shù)，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）圓周率π\(zhòng)piπ及某些含有π\(zhòng)piπ的數(shù)；（3）人為構(gòu)造的無限不循環(huán)小數(shù)，如0.1010010001...。▲概念辨析：判斷一個(gè)數(shù)是否為無理數(shù)，關(guān)鍵看其小數(shù)部分是否“無限”且“不循環(huán)”。無限循環(huán)小數(shù)可以化成分?jǐn)?shù)，是有理數(shù)?！按蠹矣涀。瑹o理數(shù)并不是‘沒有道理’的數(shù)，而是‘不能表示為整數(shù)比的數(shù)’，這個(gè)名字有點(diǎn)歷史誤會。”任務(wù)三：體系建構(gòu)——初識實(shí)數(shù)及其分類教師活動(dòng)：提問：“現(xiàn)在，我們認(rèn)識了有理數(shù)和無理數(shù)。它們之間是什么關(guān)系？能把我們學(xué)過的所有‘?dāng)?shù)’放在一起，給個(gè)總稱嗎？”引出實(shí)數(shù)的概念：有理數(shù)和無理數(shù)統(tǒng)稱為實(shí)數(shù)。通過韋恩圖或樹狀圖，與學(xué)生一起構(gòu)建實(shí)數(shù)的分類體系。強(qiáng)調(diào)分類標(biāo)準(zhǔn)：按定義（是否為無限不循環(huán)小數(shù)）分為有理數(shù)和無理數(shù)；按符號分為正實(shí)數(shù)、0、負(fù)實(shí)數(shù)。進(jìn)行辨析練習(xí)：“請判斷‘實(shí)數(shù)不是有理數(shù)就是無理數(shù)’、‘帶根號的數(shù)都是無理數(shù)’這些說法對嗎？”學(xué)生活動(dòng)：理解“統(tǒng)稱”的含義，知曉實(shí)數(shù)是有理數(shù)和無理數(shù)的并集。參與分類圖的構(gòu)建，理清從屬關(guān)系。對辨析問題進(jìn)行思考和討論，加深對概念外延的理解，例如認(rèn)識到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是有理數(shù)，從而明確“帶根號的數(shù)不一定無理”。即時(shí)評價(jià)標(biāo)準(zhǔn)：1.體系理解：能否理解實(shí)數(shù)、有理數(shù)、無理數(shù)三個(gè)概念之間的包含關(guān)系。2.分類操作：能否根據(jù)定義對給定的實(shí)數(shù)（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\(zhòng)piπ）進(jìn)行正確分類。3.誤區(qū)澄清：能否識別并解釋關(guān)于實(shí)數(shù)分類的常見錯(cuò)誤說法。形成知識、思維、方法清單：★實(shí)數(shù)概念：有理數(shù)和無理數(shù)統(tǒng)稱為實(shí)數(shù)。這是目前我們所學(xué)的最大的數(shù)集?！飳?shí)數(shù)分類（按定義）：實(shí)數(shù)分為有理數(shù)（有限小數(shù)或無限循環(huán)小數(shù)）和無理數(shù)（無限不循環(huán)小數(shù)）?！裰匾吻澹号袛嘁粋€(gè)數(shù)是不是無理數(shù)，必須化簡或分析到最后形式。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">?化簡后為2，是有理數(shù)；而π2\frac{\pi}{2}2π?雖然含有π\(zhòng)piπ，但它本身就是一個(gè)確定的無限不循環(huán)小數(shù)，也是無理數(shù)。“我們把數(shù)域從有理數(shù)‘?dāng)U軍’到了實(shí)數(shù)，現(xiàn)在我們的‘武器庫’更加強(qiáng)大了！”任務(wù)四：形數(shù)統(tǒng)一——探究實(shí)數(shù)與數(shù)軸上的點(diǎn)教師活動(dòng)：回顧舊知：“我們知道，每一個(gè)有理數(shù)都可以用數(shù)軸上的一個(gè)點(diǎn)來表示。那么，無理數(shù)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">?，能在數(shù)軸上找到它的位置嗎？”引導(dǎo)學(xué)生利用幾何直觀：構(gòu)造一個(gè)兩直角邊均為1的直角三角形，斜邊即為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">?。演示如何以原點(diǎn)為圓心，斜邊長為半徑畫弧，與數(shù)軸正半軸的交點(diǎn)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">?。追問：“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？”推廣結(jié)論：“事實(shí)上，每一個(gè)實(shí)數(shù)（無論有理還是無理）都可以用數(shù)軸上的一個(gè)點(diǎn)來表示；反過來，數(shù)軸上的每一個(gè)點(diǎn)都表示一個(gè)實(shí)數(shù)。”即實(shí)數(shù)與數(shù)軸上的點(diǎn)一一對應(yīng)。學(xué)生活動(dòng)：觀察教師的幾何作圖演示，理解如何在數(shù)軸上“構(gòu)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">?的點(diǎn)。在任務(wù)單上嘗試模仿，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">?的點(diǎn)。思考并回答教師關(guān)于負(fù)無理數(shù)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">?的提問。最終理解并認(rèn)同“一一對應(yīng)”的結(jié)論。即時(shí)評價(jià)標(biāo)準(zhǔn)：1.操作理解：能否理解利用勾股定理在數(shù)軸上定位無理數(shù)的幾何原理。2.結(jié)論歸納：能否從特殊（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">?）推廣到一般，說出實(shí)數(shù)與數(shù)軸點(diǎn)的一一對應(yīng)關(guān)系。3.逆向思維：當(dāng)看到數(shù)軸上一個(gè)任意點(diǎn)時(shí)，能否意識到它一定對應(yīng)一個(gè)確定的實(shí)數(shù)（可能是有理數(shù)，也可能是無理數(shù)）。形成知識、思維、方法清單：★核心關(guān)系：實(shí)數(shù)與數(shù)軸上的點(diǎn)一一對應(yīng)。這是實(shí)數(shù)完備性的直觀體現(xiàn)。●幾何方法：利用勾股定理，可以在數(shù)軸上作出表示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為正整數(shù)）的點(diǎn)。這是“以形助數(shù)”的典范?！鴶?shù)學(xué)思想：數(shù)形結(jié)合思想。將抽象的數(shù)與直觀的圖形（數(shù)軸）聯(lián)系起來，使得實(shí)數(shù)的存在性和順序性變得可視、可感。“看，這個(gè)點(diǎn)（指著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">?對應(yīng)的點(diǎn)）雖然我們不能用有限小數(shù)寫出來，但它確確實(shí)實(shí)、獨(dú)一無二地存在于數(shù)軸上，這就是數(shù)學(xué)的確定性美?！钡谌?、當(dāng)堂鞏固訓(xùn)練設(shè)計(jì)核心：構(gòu)建分層、變式的訓(xùn)練體系，并提供即時(shí)反饋。基礎(chǔ)層（全體必做，直接應(yīng)用核心概念）：1.判斷下列說法是否正確，并說明理由：（1）無理數(shù)都是無限小數(shù)。（2）無限小數(shù)都是無理數(shù)。（3）帶根號的數(shù)都是無理數(shù)。2.將下列各數(shù)填入相應(yīng)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˙。有理數(shù)集合：{…}；無理數(shù)集合：{…}；正實(shí)數(shù)集合：{…}。綜合層（多數(shù)學(xué)生嘗試，在新情境中綜合運(yùn)用）：3.如圖，數(shù)軸上點(diǎn)A表示的數(shù)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?。?.已知邊長為1的正方形的對角線長為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">?，請利用這個(gè)結(jié)論，在數(shù)軸上準(zhǔn)確標(biāo)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">?的點(diǎn)（要求保留作圖痕跡）。挑戰(zhàn)層（學(xué)有余力者選做，開放探究）：5.我們知道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是有理數(shù)，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">?是無理數(shù)。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是正整數(shù)）在什么情況下是有理數(shù)？什么情況下是無理數(shù)？你能提出一個(gè)猜想并嘗試解釋嗎？反饋機(jī)制：基礎(chǔ)層題目通過全班齊答或快速互查解決，教師點(diǎn)評易錯(cuò)點(diǎn)（如第1題第2小句）。綜合層題目請學(xué)生上臺展示第4題作圖過程，并講解第3題的估算策略，教師補(bǔ)充優(yōu)化。挑戰(zhàn)層題目作為思考題，邀請有想法的學(xué)生分享其猜想，教師給予肯定并引導(dǎo)課下繼續(xù)探究，不做統(tǒng)一要求?！暗?題把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">?放進(jìn)無理數(shù)集合的同學(xué)要小心啦，化簡是第一步！”“誰來展示一下你是怎么在數(shù)軸上‘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，先構(gòu)造√2，再以它為直角邊…思路很清晰！”第四、課堂小結(jié)知識整合：引導(dǎo)學(xué)生以“實(shí)數(shù)”為中心，用思維導(dǎo)圖的形式回顧本節(jié)課的核心概念鏈：從證明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">?不是有理數(shù)→發(fā)現(xiàn)并定義“無理數(shù)”（無限不循環(huán)小數(shù)）→有理數(shù)與無理數(shù)“統(tǒng)稱”為實(shí)數(shù)→實(shí)數(shù)與數(shù)軸上的點(diǎn)“一一對應(yīng)”。鼓勵(lì)學(xué)生上臺繪制并講解。方法提煉：“今天我們用了哪些重要的數(shù)學(xué)方法攻克了‘無理數(shù)’這個(gè)堡壘？”師生共同回顧：反證法（邏輯推理）、從特殊到一般（歸納定義）、數(shù)形結(jié)合（在數(shù)軸上找點(diǎn)）、估算與辨析。作業(yè)布置：必做題（鞏固基礎(chǔ)）：1.完成課本對應(yīng)練習(xí)，重點(diǎn)辨識無理數(shù)與實(shí)數(shù)分類。2.在數(shù)軸上標(biāo)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