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1、初中数学化归思想方法|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 1 页，共 4 页第 2 页初中数学化归思想方法一位日本的数学教育家曾经提出：无论科技工作者，教育工作者，或是社会的其他人才，最重要的是要有数学的精神与思想方法，而数学知识那么是第二位的。这与我国古代教育家提出的“授之以鱼，不如授之以渔的思想实质上是一致的。在具体的数学思想方法中，“化归思想又是世界数学家们都十分重视的数学思想方法，因为，在解决问题的过程中，数学家往往不是直接对问题展开攻击，而是对问题进行变形、转化，直至把它化归为某个些已经解决的问题，或容易解决的问题。匈牙利著名
2、数学家P•罗莎曾用以下比喻十分生动地说明了化归思想的实质。她写道：“假设在你面前有煤气灶、水龙头、水壶和火柴，现在的任务是要烧水，你应当怎样去做？正确的答复是：“在水壶中放上水，点燃煤气，再把水壶放到煤气灶上。接着，罗莎又提出第二个问题：“假设所有的条件都不变，只是水壶中已有了足够的水，这时你应该怎样去做？对此，人们往往答复说：“点燃煤气，再把壶放到煤气灶上。但罗莎却认为这并不是最好的答复，因为，“只有物理学家才会这样做，而数学家那么会倒去壶中的水，并且声称我已经把后一问题化归成先前的问题了，而先前的问题我已答复。“把水倒掉这是多么简洁的答复呀！比喻有点夸|精.|品.|可.|编.|辑
3、.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 2 页，共 4 页文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y
4、9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6
5、R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y
6、3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P
7、2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6
8、W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I
9、7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10文档编码:CV4P2Y9H4Z4 HS6O6W6R5L8 ZR9W2I7Y3C10第 3 页张，但它确实说明了数学家思考与解决问题的一个特点，与其他应用科学家相比，他们更善于使用化归思想。下面还是让我们用一些例题来说明。例 1 鸡兔同笼，笼中有头50，有足 1
10、40，问鸡、兔各有几只？分析 化归的实质是不断变更问题，这里可以先对成分进行变形。每只鸡有2 只脚，每只兔有4 只脚，这是问题中不言而喻的成分。现在对问题中的成分进行变形：“一声令下，要求每只鸡悬起一只脚呈金鸡独立状，又要求每只兔悬起两只前脚呈玉兔拜月状。那么，笼中仍有头50，而脚只剩下70 只了，并且，这时鸡的头数与足数相等，而兔的足数与兔的头数不等有一头兔，就多出一只脚，现在有头 50，有足 70，这就说明有兔20 头，有鸡 30 头。我国古代算法书上就是这样解的以上是从变更题设条件来寻找化归方法的。下例那么是从变更任务来实现化归目的。例 2 有 18 瓶牛奶分放在4×6=24
11、个方格内，每格只能放一瓶，在数牛奶瓶时要求横数的瓶数为偶数，竖数的瓶数也为偶数，这件事能办到吗？注 这个问题是1984 年国际数学教育与计算机教育会议上，一位英国朋友在小组讨论时提出的。当时有两个不相容的答案：“这件事能办到与“这件事不能办到。|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 3 页，共 4 页文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4
12、B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ
13、4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 Z
14、Z4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5
15、ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5
16、 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q
17、5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7
18、Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4第 4 页分析 不妨试放一下请用铅笔在小方格内打上试放符号“，并给出你的答案。可能屡试不成瓶太多了，很难照顾全面。因此，能否用化归思想变更题目的任务：在4×6=24个方格内打上 2418=6 个不放牛奶瓶的符号用“×表示。余下工作请读者自己完成，这时你的结论一定是：这件事不仅一定能办到，而
19、且放置奶瓶的方法有多种，请你至少给出3 种。|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 4 页，共 4 页文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:C
20、S8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:
21、CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码
22、:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编
23、码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档
24、编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文
25、档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4文档编码:CS8F9M10V5X3 HK1Y10R10F7Q5 ZZ4B1I2T3A4