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    高中數學解題必須掌握的數學思想!

    高中數學解題必須掌握的數學思想!

      要學好數學,學會解題是關鍵。在進行解題的過程中,不僅需要加強必要的訓練,還要掌握一定的解題規律與技巧。下面具體介紹一下數學思想。
      數學思想方法在解題中有不可忽視的作用
      解題過程通常的程序是:閱讀數學知識,理解概念;在對例題和老師的講解進行反思,思考例題的方法、技巧和解題的規范過程;然后做數學練習題。
      1.函數與方程的思想
      函數與方程的思想是中學數學最基本的思想。所謂函數的思想是指用運動變化的觀點去分析和研究數學中的數量關系,建立函數關系或構造函數,再運用函數的圖像與性質去分析、解決相關的問題。而所謂方程的思想是分析數學中的等量關系,去構建方程或方程組,通過求解或利用方程的性質去分析解決問題。
      2.數形結合的思想
      數與形在一定的條件下可以轉化。如某些代數問題、三角問題往往有幾何背景,可以借助幾何特征去解決相關的代數三角問題;而某些幾何問題也往往可以通過數量的結構特征用代數的方法去解決。因此數形結合的思想對問題的解決有舉足輕重的作用。
      3.分類討論的思想
      分類討論的思想之所以重要,原因一是因為它的邏輯性較強,原因二是因為它的知識點的涵蓋比較廣,原因三是因為它可培養學生的分析和解決問題的能力。原因四是實際問題中常常需要分類討論各種可能性。
      解決分類討論問題的關鍵是化整為零,在局部討論降低難度。常見的類型:
      類型1:由數學概念引起的的討論,如實數、有理數、絕對值、點(直線、圓)與圓的位置關系等概念的分類討論;
      類型2:由數學運算引起的討論,如不等式兩邊同乘一個正數還是負數的問題;
      類型3 :由性質、定理、公式的限制條件引起的討論,如一元二次方程求根公式的應用引起的討論;
      類型4:由圖形位置的不確定性引起的討論,如直角、銳角、鈍角三角形中的相關問題引起的討論。
      類型5:由某些字母系數對方程的影響造成的分類討論,如二次函數中字母系數對圖象的影響,二次項系數對圖象開口方向的影響,一次項系數對頂點坐標的影響,常數項對截距的影響等。
      分類討論思想是對數學對象進行分類尋求解答的一種思想方法,其作用在于克服思維的片面性,全面考慮問題。分類的原則:分類不重不漏。分類的步驟:
      ①確定討論的對象及其范圍;
      ②確定分類討論的分類標準;
      ③按所分類別進行討論;
      ④歸納小結、綜合得出結論。注意動態問題一定要先畫動態圖。
      4.轉化與化歸的思想
      轉化與化歸市中學數學最基本的數學思想之一,是一切數學思想方法的核心.數形結合的思想體現了數與形的轉化;函數與方程的思想體現了函數、方程、不等式之間的相互轉化;分類討論思想體現了局部與整體的相互轉化,所以以上三種思想也是轉化與化歸思想的具體呈現。
      轉化包括等價轉化和非等價轉化,等價轉化要求在轉化的過程中前因和后果是充分的也是必要的;不等價轉化就只有一種情況,因此結論要注意檢驗、調整和補充。轉化的原則是將不熟悉和難解的問題轉為熟知的、易解的和已經解決的問題,將抽象的問題轉為具體的和直觀的問題;將復雜的轉為簡單的問題;將一般的轉為特殊的問題;將實際的問題轉為數學的問題等等使問題易于解決。
      轉化與化歸的指導思想
      ①把什么問題進行轉化,即化歸對象 .
      ②化歸到何處去,即化歸目標 .
      ③如何進行化歸,即化歸方法 .
      常見的轉化方法有
      ①直接轉化法:把原問題直接轉化為基本定理、基本公式或基本圖形問題
      ②換元法:運用“換元”把式子轉化為有理式或使整式降冪等,把較復雜的函數、方程、不等式問題轉化為易于解決的基本問題
      ③數形結合法:研究原問題中數量關系(解析式)與空間形式(圖形)關系,通過互相變換獲得轉化途徑
      ④等價轉化法:把原問題轉化為一個易于解決的等價命題,達到化歸的目的
      ⑤特殊化方法:把原問題的形式向特殊化形式轉化,并證明特殊化后的問題,使結論適合原問題
      ⑥構造法:“構造”一個合適的數學模型,把問題變為易于解決的問題
      ⑦坐標法:以坐標系為工具,用計算方法解決幾何問題也是轉化方法的一個重要途徑
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