2022年人教版小学数学六年级上册《鸡兔同笼》教学设计 .pdf

人教版小学数学六年级上册鸡兔同笼教学设计教学目标：1了解“鸡兔同笼”问题，感受古代数学问题的趣味性。2尝试用不同的方法解决“鸡兔同笼”问题，使学生体会假设和列方程的一般性。3在解决问题的过程中，培养学生的思维能力，并向学生渗透转化、函数等数学思想和方法。教学重点：尝试用不同的方法解决“鸡兔同笼”问题，体会用假设法和方程法解决问题的优越性。教学难点：理解用假设法解决“鸡兔同笼”问题的算理教学过程一、创设情境，生成问题(1).出示原题师：同学们，我们国家有着几千年的悠久文化，在我国古代更是产生了许多位数学家和许多部数学著作，孙子算经就是其中一部，大约产生于一千五百年前，书中记载着这样一道有名的数学趣题（课件出示孙子算经中的原题）：今有雉兔同笼，上有三十五头，下有九十四足，问雉兔各几何？（2）理解题意师：同学们知道这道题的意思吗？请试着说一说。生：这道题的意思是现在，鸡和兔在一个笼子里，从上面数有35 个头，从下面数有 94 只脚，问鸡和兔各有多少只？师：这道题的意思正如同学们所想的一样，也就是：（课件出示）笼子里有若干只鸡和兔，从上面数有35 个头，从下面数有94 只脚，鸡和兔各有多少只？（3）揭示课题师：这就是著名的“鸡兔同笼”问题，也正是这节课要研究的问题。二、探究交流、解决问题1出示例 1 师：为便于研究，我们可先从简单问题入手，把题中的“35 个头”和“94只脚”分别换成“8 个头”和“26 只脚”，就变成了例1：笼子里有若干只鸡兔。从上面数，有8 个头，从下面数，有26 只脚，鸡和兔各有几只？2理解题意.我们一起来看看被关在同一个笼子里的鸡和兔给我们带来了什么信息？学生理解：鸡和兔共8 只。鸡和兔共有 26 条腿。鸡有 2条腿。兔有 4 条腿。（课件出示）3探索策略（一）猜想验证，1、我们先来猜猜，笼子中可能会有几只鸡几只兔呢？学生猜测，在猜测时要抓住哪个条件呢？（鸡和兔一共是8 只）那是不是抓住了这个条件就一定能猜对呢？学生猜测，老师板书2、怎样才能确定同学们猜的对不对？（把鸡的腿和兔的腿加起来看等不等于26。）3、和学生一起验证，找出正确的答案。列表法：鸡 的只 数876543210兔 的只 数012345678共 有腿 数1 61820222426283032先假设有8 只鸡，0只兔子，腿就有16 条。腿太少，然后又假设有7 只鸡，1 只兔子，腿还是太少了。这样试下去就得到了有3 只鸡，5 只兔子。师：学生说出“7只鸡，1 只兔子”，问“怎样计算出的腿数？”72+14=14+4=18 问“3 只鸡，5 只兔子是26 条腿吗？”32+54=6+20=26 师：谁和他的方法一样？能再讲讲吗？师：追问“有些同学在填表时写出的腿数特别快，让我们采访一下有什么秘诀？”4、“像你们这样，采用列表的方法，不重复、不遗漏的写出所有可能的答案。这种逐一列举的方法在数学中也称为“列举法”5、你们觉得用猜想列表法解决鸡兔同笼问题怎么样？（生：麻烦，而且当头和脚的只数越多时，越不容易找出答案。）文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N36、那我们还有研究新方法的必要。（二）尝试假设法1、为了研究老师把所有的可能按顺序列出来了，我们先看表格中左起的第一列，8 和 0是什么意思？（就是有8 只鸡和 0 只兔，也就是假设笼子里全是鸡，）那笼子里是不是全是鸡呢？（不是）那就是把里面的兔也看成鸡来计算了，那把一只4 条腿的兔当成一只2条腿的鸡来算会有什么结果呢？（就会少算两条腿）（课件出示：把一只兔当成一只鸡算，就少了两条腿。）2、假设全是鸡一共就有16 条腿。实际有26 条腿，这样笼子里就少了10 条腿，为什么会少了 10 条腿呢？（把兔当了鸡在算。一只兔当成一只鸡算少两条腿，那把几只兔当成了鸡算就会少算10 条腿呢？即10 里面有几个2。就把几兔当成了鸡算，5 个 2，用五只兔当成了鸡算，这个五就表示应该有5只兔）3、上面的过程能用算式表示出来吗？请同学们试试看。（学生试着列算式，请一个学生到黑板上去板演。）4、假设全是鸡：（板书）82=16（条）（如果把兔全当成鸡一共就有82=16 条腿）26-16=10（条）（把兔看成鸡来算，4 条腿兔有当成两条腿的鸡算，每只兔就少了两条腿，10 条腿是少算了兔的腿）4-2=2（假设全是鸡，是把4 条腿的兔有当成两条腿的鸡。所以 4-2 表示是一只兔当成一只鸡就要少算2 条腿。）102=5（只）兔（那把多少只兔当成鸡算就会少10 条腿呢？就看10 里面有几个2 就是把几只兔当成了鸡来算，所以102=5 就是兔的只数。）8-5=3（只）鸡（用鸡兔的总只数减去兔的只数就是鸡的只数，8-5=3 只鸡）5、算出来后，我们还要检验算的对不对，谁愿意口头检验。生：32+54=26（只），5+3=8（只）。师：看来做对了，最后写上答语。6、假设全是兔7、我们再回到表格中，看看右起第一列中的8 和 0 是什么意思？（笼子里全是兔）那是不是全都是兔呢？（不是）也就是假设笼子里全是兔。那把兔当了鸡在算。那就是把里面的鸡也当成兔来计算了，那把一只2 条腿的鸡当成一只4 条腿的兔来算会有什么结果呢？（就会多算两条腿）（课件出示：把一只鸡当成一只兔算，就多了两条腿）8、先用假设全是鸡的办法解决了这个问题，现在假设全是兔又应该怎么分析和解决这个问题呢？同学们能自己解决吗？如果有困难可以同桌或小组讨论。（学生讨论写算式，然后指名板演。）小结：刚才我们假设都是鸡或都是兔，所以把这种方法叫做假设法。这是解答鸡兔同笼问文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3题的一种基本方法。（板书：假设法）（三）画图法给每只动物先安上2 条腿（也就是都看成鸡），这样一共用16 条腿，还剩下10 条腿。一次增加 2 条腿，一只鸡就变成了一只兔，要把10 条安完，要把5 只鸡变成兔。问：谁听懂他的方法了？能再说说吗？你觉得这样做怎么样？（结合课件演示）师：画图的方法非常便于观察、非常容易理解，但如果鸡兔只数很多时，就会不太适合。（四）列方程解在解决鸡兔同笼问题时，除了假设法外，还有别的方法吗？（方程的方法）要用列方程的方法就必须找到等量关系式。（兔的只数+鸡的只数=8；兔的腿+鸡的腿=26 条腿）（课件出示）这里我们需要求兔的只数和鸡的只数，共有两个未知数。那我们可以设一个未知数为X，再把另一个表示出来。这道题我们可以设兔的知数为X只，根据兔和鸡共有8 只。那鸡的只数就可以表示成：（8-X）只），因为一只鸡有2 条腿，所以X只鸡就共有2X条腿。一只兔有 4 只脚，（8-X）只兔就有4（8-X）只脚。又因为鸡和兔共有26 只脚，所以2X+4（8-X）=26 1、解：设鸡有X只，兔有（8-X）只。2X+4（8-X）=26 在解的时候可以根据等式的性质将减变成加，分别加上4X，再来解。2、解：设有兔X只，鸡有（8-X）只。4X+2（8-X）=26 同样抽生说出自己想法。那种方程好解一点，（设兔的只数为X好解点）所以我们可以设脚数多的兔为X，在解的时候容易一点。列方程的重点是找出等量关系：设头数，以脚数相等来列出方程；3小结方法（1）请同学们回忆一下，在解决鸡兔同笼问题时，用到了哪些方法？（猜想法，列表法，假设法和方程法。）（2）要你们解决孙子算经中原题，你现在会选用哪种方法呢？（有的选择假设法，有的选择代数法。）师：下面同学们就用自己喜欢的方法解决这个问题。课件出示孙子算经中原题学生解答并集体讲评请同学们想一想，在日常生活中还有哪些情况类似于鸡兔同笼问题？学生举出实例：(略）师：可见生活中类似于鸡兔同笼的问题有很多，这些问题都可用不同的数学方法来解文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3决，课后可用我们喜欢的方法解决这些问题。三、巩固应用，内化提高1.“做一做1”鸡兔同笼问题传到日本时就变成了“龟鹤问题”，你认为“龟鹤问题”与“鸡兔同笼”有什么相似之处？课件出示（龟相当于兔，鹤相当于鸡）展示学生作业，并抽生说说思路。师：看来鸡兔问题这类问题我们不只局限算鸡和兔的只数问题上，只要能用“鸡兔同笼”问题来解答的问题都可以统一叫做“鸡兔同笼”问题。下面我们就用刚才学到的“鸡兔同笼”方法，来帮我们解决生活中遇到的一些实际问题。2、“考考你”自行车和三轮车共10 辆，总共有26 个轮子。自行车和三轮车各有多少辆？信封里有2 元和 5 元 的钞票共8 张，34 元。两种钞票各多少张？3、那你知道早在一千五百年前的古人又是怎么解决鸡兔同笼问题的？课件出示“阅读资料”四、回顾整理，反思提升本节课你有什么收获？板书设计：鸡兔同笼1，列表法2，画图法3，假设法4、列方程法文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3文档编码:CC8V7R6T4X1 HK4H6L6Z6P5 ZH10V5K9B10N3