From 44835a8f3040953072350573bc750de4d37adb5e Mon Sep 17 00:00:00 2001 From: HuangHai <10402852@qq.com> Date: Wed, 16 Jul 2025 15:22:51 +0800 Subject: [PATCH] 'commit' --- .../media/image1.jpeg | Bin 0 -> 16763 bytes .../media/image10.png | Bin 0 -> 8457 bytes .../media/image11.png | Bin 0 -> 80760 bytes .../media/image12.png | Bin 0 -> 48234 bytes .../media/image18.png | Bin 0 -> 157030 bytes .../media/image19.png | Bin 0 -> 17983 bytes .../media/image2.png | Bin 0 -> 17987 bytes .../media/image20.png | Bin 0 -> 62786 bytes .../media/image21.png | Bin 0 -> 19717 bytes .../media/image22.png | Bin 0 -> 100400 bytes .../media/image23.jpeg | Bin 0 -> 8761 bytes .../media/image24.jpeg | Bin 0 -> 9588 bytes .../media/image25.png | Bin 0 -> 37766 bytes .../media/image26.png | Bin 0 -> 120817 bytes .../media/image27.png | Bin 0 -> 216098 bytes .../media/image28.png | Bin 0 -> 54031 bytes .../media/image29.png | Bin 0 -> 90283 bytes .../media/image3.jpeg | Bin 0 -> 27354 bytes 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a/dsLightRag/static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image4.wmf b/dsLightRag/static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image4.wmf new file mode 100644 index 0000000000000000000000000000000000000000..f8068789c5f929787eea23537ca3bbb5bcaa6335 GIT binary patch literal 834 zcmah{J!n%=6#njgNo&rv{Qn@l!KH%*2ZxGp(jk85zNBOkZ*tH1?sv|;-+kwv&mZ5tfEkAxmYNrM zTu~sf_7vd5o{9QtwITsQN{`25gRZDE6E36)3>W4}hdRO4#XtmS!Q3&hfhd``#0Q-F z%s$`@O|rr$Ryz;N{YMTk#2ne*jE3nsw2$U8#YKBT?&$wdruUSDSx%?Zq9j^Ucd1`WO)ImtwW$0Gew{BB%d0@* zR&6I!ujiWE&Sv$2(?-lc`M{V}>NoB->(i8MwOzY6lNZP}yRDkjN^Uqk!~*47!)Xgn z-fT9iJBay;~JwLL76K+}m!}U5sSlcIi^b=Q5k#vfy5D2iXb((MV3* zpx`c5*4=?&Z2Bkb3}u!o`AWf+e-4#H@+o=Ne{N z7U>zoD$RD0^qTkFIO!sjLNk3`uREeRWdd)ccb@Vp=S%%|$6 ![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image3.jpeg) +> ![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image3.jpeg) > height="6.254861111111111in"} 话题2 数量的本质 数量的本质是多与少,因为动物也能够分辨出多与少,比如,一只狗对一只狼的反应与对一群狼的反应是不一样的,这就说明狗知道狼数量的多与少。在一本书中描述了一个故事,这个故事表明动物对于数量的多少具有相当强的分辨能力[^31]: @@ -1107,7 +1107,7 @@ P(1),P(2),..., P(k),... 人们长期以来习惯于用分数来表示有理数。据记载,最初是荷兰数学家、工程师斯蒂芬(Simon Stevin,1548-1620)开始用小数来表示有理数的,但与现在的形式有所不同,他用 24 3(1)7(2)5(3) -来表示有理数24![](./static/Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image4.wmf)。直到十八世纪,一个稳定的十进位小数的表达形式才逐渐形成,即把前面的分数表示为24.375,这种表示方法一直沿用至今。 +来表示有理数24![](./static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image4.wmf)。直到十八世纪,一个稳定的十进位小数的表达形式才逐渐形成,即把前面的分数表示为24.375,这种表示方法一直沿用至今。 后来,人们尝试用小数来表示无理数。显然,要用小数表示所有的无理数,首先要用小数表示所有的有理数。正如在前几个话题中谈到的那样,在历史上,人们在很长的一段时间是用分数m/n的形式来表示有理数的,其中m,n ∈ N,n≠0,并且称不能表示为分数形式的数为无理数。 这样,为了用小数表示有理数,就需要讨论小数与分数之间的关系。并且只需要讨论区间(0,1)中的数,因为其余的数可以通过平移得到。区间(0,1)中的数可以用小数表示为 @@ -1604,14 +1604,14 @@ m/n。可是,如果这个同学只投了1次并且投中了,因为1/1 从上面的要求可以知道,小学两个学段的内容都涉及数位。理解数位的核心是理解"十进制计数法"的准则,准确地把握数位的概念不仅对于认识数是重要的,对于数的运算也是非常重要的,这个概念贯穿小学"数与代数"学习的始终。 教学片断设计:通过计数单位认识数位"万" 1.拿出一个千位的第纳斯木块:数小正方体的个数 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image10.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image10.png) 教师带领学生一起数小正方体的个数,启发学生:"正面最下一行有几个?"学生回答"十"以后,教师总结:"10个个是十"。教师接着启发学生:"正面有几个小正方体?"并引导学生通过列来数:二十、三十、......。学生回答"一百"以后,教师总结:"10个十是百"。然后教师启发学生:"一共有多少小正方体?"并引导学生通过纵向来数:二百、三百、......。学生回答"一千"以后,教师总结:"10个百是千"。 2.拿出十个千位的第纳斯木块:数小正方体的个数 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image11.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image11.png) 教师给出"万"的定义:"这里有十个木块模型,表示有一万个小正方体。"然后问学生:"万是多少个千?"然后逐一数木块模型:二千、三千、......。学生回答"10个一千"以后,教师总结:"10个千是万"。 教师总结说:"在个、十、百、千的基础上,今天我们又知道了万。我们把个、十、百、千、万叫做计数单位,也叫数位。"然后,教师启发学生回答这些计数单位之间的关系,一定要让学生自己得到答案:数位相依差10倍。最后告诉学生问题的核心:数位依次相差10倍,就是十进制计数法。在教学过程中可以说一些轻松的话题,比如:人们采用十进位制计数法大概是因为人有十个手指头。 3.拿出零乱的第纳斯木块:数小正方体的个数 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image12.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image12.png) 教师启发学生如何用计数单位来数小正方体的个数。学生自然而然地就会把一样单位的第纳斯放在一起,然后数出各个单位的个数。如果学生的摆放是杂乱的,比如,3个百放到一起、4个十放到一起、2个千放到一起,教师要启发学生按照数位的顺序摆放,最后计算出小正方体的个数。 4.用第纳斯木块表示给出的数 教师提问:"如何用木块表示2342这个数?"然后引导学生操作,在操作过程中引发学生思考:个位的数是2,千位的数也是2,可以用一样的木块表示吗? @@ -1808,7 +1808,7 @@ m(m﹤n)个1份表示为m/n、读作"n分之m"。让每一个学生说出一 教学片断设计:乘法的意义 1. 发现生活中的乘法 借助下面的图画,教师讲故事。某一个班级根据学生的兴趣,分两个小组活动:一个小组进行体育活动,活动内容是学习轮滑;一个小组进行文艺活动,活动内容是排练合唱。现在请同学们帮助老师数一数,这两个小组各有多少同学。 -> ![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image18.png) +> ![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image18.png) > height="1.8006944444444444in"} 教师提出问题:"容易计算的是哪个小组的人数?" 学生能够回答:"合唱小组。" @@ -1816,8 +1816,8 @@ m(m﹤n)个1份表示为m/n、读作"n分之m"。让每一个学生说出一 学生的回答可能是五花八门的,比如,排队整齐,看得清楚等等。 然后,教师要引发学生抽象出问题的本质:每一行的人数相等、或者每一列的人数相等。比如,教师可以启发学生说:"排队整齐就会怎么样啊?"引导学生自己得到结论。 教师给出下面抽象了的图,解释什么是行的人数相等、列的人数相等。 -![图片2.png](./static/Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image19.png){width="2.115972222222222in" -![图片1.png](./static/Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image20.png){width="2.7083333333333335in" +![图片2.png](./static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image19.png){width="2.115972222222222in" +![图片1.png](./static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image20.png){width="2.7083333333333335in" 对于第一个图,教师引领学生读出列数、并在黑板上书写: 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 20 引导学生认识这个加法的特征:加数都是一样的。启发学生:"一共有几个加数?"当学生回答"10个"以后,教师写出乘法的算式: @@ -1827,8 +1827,8 @@ m(m﹤n)个1份表示为m/n、读作"n分之m"。让每一个学生说出一 然后,教师利用第二个图,解释乘法:10 + 10 = 10 × 2 = 20。 2. 用图形理解乘法 教师进一步给出图形,让学生画出被乘数(不需要指明这个概念)、写出上面的通过加法得到乘法的算式、计算结果。比如,通过下面两个图形 -![图片3.png](./static/Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image21.png){width="1.5229166666666667in" -![图片4.png](./static/Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image22.png){width="1.4354166666666666in" +![图片3.png](./static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image21.png){width="1.5229166666666667in" +![图片4.png](./static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image22.png){width="1.4354166666666666in" 分别得到算式: 5 + 5 + 5 + 5 = 5 × 4 = 20; 4 + 4 + 4 + 4 + 4 = 4 × 5 = 20。 @@ -1856,7 +1856,7 @@ a ÷ b = a × (1/b)。 对于后一种表示方法,称1/b为b的倒数,因此,后一种表示方法可以用语言叙述为:"除以一个数等于乘以这个数的倒数"。这种表示方法更多地应用于分数的除法:"除以一个分数等于乘以这个分数的倒数"。因此,在学生最初认识倒数时,更多地是关注分数的倒数。 教学片断设计:认识倒数 > 1. 通过分数认识1 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image23.jpeg) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image23.jpeg) 教师通过媒体演示,把一个月饼分为六份(如上第一个图所示)。教师指着其中的一份、并以回忆的口气询问学生: "每份月饼是原来月饼的多少?" 当学生回答1/6以后,教师在黑板上书写:1×1/6 = @@ -1878,7 +1878,7 @@ B: 3/4 × 4/3 = ﹏ , 7/2 × 2/7 = ﹏ , 5/9 × 9/5 = ﹏ ,7/6 × 6/7 = a × b = 1, 称a和b互为倒数。然后再补充说:因为任何数乘以0到不能为1,所以0没有倒数。 3. 通过倒数计算除数为分数时的除法 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image24.jpeg) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image24.jpeg) 仍然回到分月饼的媒体演示。如上面第一个图所示:把一个月饼分成六份。教师提出问题: "半块月饼是一个月饼的多少?" 当学生回答1/2以后,教师在黑板上书写"1/2"。然后利用媒体演示,如上第二个图所示:把六份月饼分为相等的两堆(两堆各三份),教师引导学生思考: @@ -1979,8 +1979,8 @@ t。 2 × 3 = 6,3 × 2 = 6 → 2 × 3 = 3 × 2, 7 × 8 = 56, 8 × 7 = 56 → 7 × 8 = 8 × 7, 教师提出问题:"是不是对所有的数,乘法的两个因子都可以交换呢?"学生的回答很可能是肯定的,既便如此,教师仍然进一步提出问题:"为什么会这样呢?"一般来说,学生回答不了这个问题。于是教师引导学生思考:"我们来回忆一下乘法是什么",然后用媒体显示下面的图 -![图片3.png](./static/Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image21.png){width="1.5229166666666667in" -![图片4.png](./static/Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image22.png){width="1.4354166666666666in" +![图片3.png](./static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image21.png){width="1.5229166666666667in" +![图片4.png](./static/Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image22.png){width="1.4354166666666666in" 启发学生回答"左边的算式是什么",当学生回答5×4以后,教师在黑板上写出算式;进一步启发学生回答"右边的算式是什么",当学生回答4×5以后,教师在黑板上写出算式。教师问:"这两个算式是不是相等的?"学生回答相等之后,教师一边在两个式子之间写上等号,一边问学生:"为什么相等啊?"这时,学生的回答可能是多样的,教师要归纳出下面的结论 行数 × 列数 = 列数 × 行数 并且总结说:"乘法是加法的简便运算,先算行、还是先算列,结果都是一样的。"教师进一步提问:"如果要用字母表示这个性质,应当如何表示呢?"启发学生用a表示"行"、用b表示"列",于是可以得到一般表达式 @@ -2029,23 +2029,23 @@ x的含义也是必要的。 1. 创设情境引发思考,体会路程、速度、时间之间的关系 在学习之前,学生是对速度、路程、时间这些概念是知道的,但不一定真正理解。事实上,只有通过三者之间的关系,才能真正理解这些概念的含义;反之,只有真正理解了这些概念的含义,才可能准确表达三者之间的关系。可以通过缺失信息的方法来理解概念,本片段的重点是理解速度,并且通过速度来认识路程模型。 什么是速度。教师给出下面的情境讲故事:早晨,小丽和小强在学校见面,分别询问对方上学所需要的时间。同学们是否能帮他们比一比,谁走得更快些? -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image25.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image25.png) 可能会有一部分学生觉得小丽走的快一些,因为花费的时间少;也可能会有一部分学生认为没有办法进行比较,因为不知道谁家远。这时,教师要引发学生思考:如果再知道了什么,就可以知道谁的速度更快一些。然后,让学生举例说明。 学生举的例子可能是五花八门的,比如,小丽和小强的家离的很近、小丽的家离学校更近、小强的家离学校更近、等等。无论如何,教师要引发学生思考:应当如何判断速度的快慢。引导学生注意:速度的快慢不仅与时间有关,而且与距离有关。让学生感悟:速度 = 距离/时间。 如何度量速度。学生已经知道,通过距离和时间可以度量速度。教师通过下面的实际例子,希望学生知道如何计算速度。教师利用下面的图,讲述关于速度的故事:左边的图是神舟飞船在轨道上运行,右边的图是自行车运动员在野外练习。由图上看,每个单位的速度都是8千米,那么,神舟飞船和自行车运动员速度是一样的吗? -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image26.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image26.png) 让学生回答,应当如何表示速度的单位:米/秒,米/小时;千米/秒,千米/小时。其中,分子表示的是距离单位,分母表示的是时间单位。让学生理解:只有单位一样才能比较速度的快慢。如果学生有较好的理解能力,可以让学生知道,我们所说的时间是一个平均数:物体行驶一段距离之后,是用这段距离除以所用时间得到的。 让学生询问家长、或者查阅与速度有关的资料,在课堂上讲述与速度有关的故事。在询问或查阅过程中要掌握两个基本要素:交通方式,单位时间的速度、即单位时间的距离。让学生知道:速度 = 距离/时间,反之,距离 = 速度×时间。比如,得到下面的一些图、以及关于速度的说明: -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image27.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image27.png) 并且,可以提出相关的问题: 成人步行10分钟大约行走多少距离?飞机飞行1小时大约飞行距离?声音传播从操场的一边到另一边大约需要时间?从太阳到地球光行走大约需要多少时间? 2. 通过算式理解模型的变化 教师可以通过图画等,讲述模型变化的故事。比如,利用下面的图和算式: -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image28.png) -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image29.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image28.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image29.png) height="1.5972222222222223in"}小 教师可以提出问题:同学们能不能帮助这几位同学解释一下算式的意义? 教师要引导学生知道路程模型的基本形态:路程 = @@ -2074,7 +2074,7 @@ height="1.5972222222222223in"}小 进一步,让学生利用身边的物体说明什么是角。比如,桌面上有四个角,数学教科书上有角,红领巾上有角,等等。让学生感悟角的特征:边是直的,前端是尖的,角的大小与边长无关。 教师利用角的大小讲解什么是直角、锐角、钝角。然后进入具体操作阶段:发给每一位同学一张长方形的纸,根据教师的要求,让学生动手折出一个角、并说明是什么角。教师再一次总结:"如果两个线段的一个端点重合,我们把两个线段所夹部分叫做角,把这两个线段叫做角的边。" 2. 通过画图抽象出角 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image2.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image2.png) 引导学生理解"角"的有效方法是让学生利用"尺规作图"画出"角"的图,因为几何作图是思维抽象的具体表达。比如,让学生利用铅笔和直尺,在纸上画出如上面图(a)那样角,同桌讨论:图中的什么是角。还可以画出直角、钝角,探讨如何画直角。 然后可以如图(b)那样,通过画图比较两个角的大小:因为∠2包含∠1,所以∠2大于∠1,从而理解什么是角的大小。对于理解力比较好的学生,还可以演示图(c),启发学生思考:角的大小是由什么决定的? 教学设计分析:对于小学低年级学生,认识角要比认识线段更困难一些,因为在开始阶段,角的指向并不很明显。因此,通过教具"表"认识"角"是有意义的,只有通过角的大小的变化才能让学生对角的指向有直观感受。特别是,教师由角的具体描述过度到角的一般定义,可以让学生感悟定义的抽象过程。第二阶段的教学,通过学生自己的几何作图认识角、感悟角的大小也是重要的,有利于学生清晰地把握角的概念。 @@ -2103,7 +2103,7 @@ height="1.5972222222222223in"}小 当学生学习了平移、旋转和轴对称的内容之后,安排"综合与实践"的课程是必要的,可以让学生感悟三种运动的共性与差异,从而加深对三种运动的理解,加深对图形刚体运动的理解。在教学的过程之中,引导学生学会记录图形运动的过程是必要的,这是一种数学的基本素养。 教学片断设计:平移、旋转的综合运用 1. 在图画变化的过程中感悟图形的运动 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image30.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image30.png) 分小组活动,要求学生吧上面左边的图变化为右边的图,并用平移和旋转记录运动的过程。在这个过程中一定要帮助学生积累思维的经验和实践的经验:思维清晰、层次鲜明。比如,首先用符号确定图的坐标,这个坐标可以是一维的、也可以是二维的。 一维坐标就是对每个小方块标上1个数字,如果以"行"为基准,就是每行4个数字,总体从1到16,而图所占坐标是6、7、10、11。 二维坐标就是对每个小方块标上2个数字,如果还是以"行"为基准,就是每行4对数字,总体从16对数字,而图所占坐标是 @@ -2139,11 +2139,11 @@ height="1.5972222222222223in"}小 俄罗斯 24 26 32 82 教师引导学生分6个小组讨论、画条形统计图。教师引发学生思考:这个统计表的数据告诉我们很多信息,现在,同学们能不能用图表示第30届伦敦奥运会中国获奖情况?在这个阶段,学生画出的图可能是各种各样的。 教师帮助学生理清思路,在黑板上画出横轴与纵轴,启发学生思考:是不是要分出金牌、银牌、铜牌?得到学生的认可之后,教师在横轴上画出三段四个标记,在每段下面表明金、银、铜。再启发学生思考:金牌是多少?根据学生的回答,在纵轴的适当位置标出38。同样的方法,得到27和23。然后横轴与纵轴的标记连线,得到下面的条形统计图。 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image31.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image31.png) 然后,让每两个小组分别画出美国、英国、俄罗斯的获奖情况,提醒学生画直方图注意事项:横轴与纵轴刻度比例合理,图形大小适当。比较各小组画出的图,让学生知道如何画直方图。通过图形直观分析各个国家获金牌的情况:美国金牌较多、但银牌铜牌较少;中国比较适度;英国的图形结构与美国相似;俄罗斯银牌较多、但金牌较少。让学生从中领悟使用直方图分析问题的便捷。 2. 利用条形统计图分析数据 教师启发学生思考:能不能用一个图,把四个国家获奖牌的情况都放在一起?让学生出主意:如果要放在一起,应当怎么放?对于学生的各种回答,教师要点出问题的要点:分类。可以以奖牌分类,也就是分成:金牌、银牌、铜牌;还可以以国家分类。然后启发学生想象:如果以国家分类,得到的图就是把以前做过的四个图合并在一起。于是尝试以奖牌分类、做直方图,让每一个同学画出下面的图。 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image32.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image32.png) 引导学生利用上面的图,预测第31届里约热内卢奥运会的情况。学生的回答可以是多种多样的,对于这样的问题,教师的主要工作是引导学生"论述有据",也就是说,引导学生能够依据数据回答问题,比如,美国与中国获取金牌数量还可能是前两名,因为英国与俄罗斯的差距比较大;再比如,俄罗斯可能会超过英国,因为俄罗斯的奖牌总量比英国多许多,因此有更大的潜力;等等。最后,教师总结说,同学们的说法都有一定的道理,但只通过一届奥运会的成绩进行预测,可靠性还是不够的,应当利用更多的历史数据,比如连续五年的,这就是下次课我们要讲折线统计图。 教学设计分析:制作统计图的目的是为了直观地表示数据、直观地分析数据,统计图已经成为现代生活中不可缺少的表达信息的工具。因此,统计图的教学不仅是为了向学生传授数学知识,也是为了培养学生的数学素养。 教学设计选取了刚刚闭幕的伦敦奥运会的资料,这样的背景现实有趣,能够引发学生的学习兴趣。教学设计的第一步是从统计表到条形统计图,可以让学生切实感受条形统计图的直观性,教师在学生的能够在讨论中归纳总结出画条形统计图的要点,帮助学生理清思路。在这个基础上,启发学生把四个国家的数据归纳到一个条形统计图中,进行数据比较,进一步帮助学生理解统计图的作用。最后,通过预测的故事,引出下一节课要讲述的内容。 @@ -2158,7 +2158,7 @@ height="1.5972222222222223in"}小 教学片断设计:估计投篮命中率 1. 通过样本感悟随机性 教师讲述小明投篮的故事。小强非常喜欢打篮球,长大了想当一名篮球运动员。小强让体育老师测验自己投篮的命中率,体育老师在一个星期的5天里,每天测验一次,每次定点投篮10次,投中球数的记录如下:第一天3个、第二天4个、第三天3个、第四天6个、第五天4个。如下图所示,教师要求学生用图表示小强投篮的情况,帮助小强计算平均每天投中几个球,并且利用平均数帮助小强估计投篮的命中率。 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image33.jpeg) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image33.jpeg) 因为小强五天一共投中3 + 4 + 3 + 6 + 4 = 20个球,以此平均每天投中20/5 = 4个球。计算模式是:平均数 = 进球数/天数。在计算平均数的过程中,教师应当让学生感悟到:虽然小强每天都投球10次,但进球数却是不确定的;虽然不确定,但进球数相对稳定在平均数附近。这就是对随机性感悟,对统计意义上的平均数的感悟。 @@ -2185,14 +2185,14 @@ height="1.5972222222222223in"}小 三角形三边关系的证明 证明方法如下: 作下图所示的三角形ABC。在三角形ABC中,[三角不等式](https://zhida.zhihu.com/search?content_id=248217850&content_type=Article&match_order=1&q=%E4%B8%89%E8%A7%92%E4%B8%8D%E7%AD%89%E5%BC%8F&zd_token=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJ6aGlkYV9zZXJ2ZXIiLCJleHAiOjE3NTIzNzg0NDAsInEiOiLkuInop5LkuI3nrYnlvI8iLCJ6aGlkYV9zb3VyY2UiOiJlbnRpdHkiLCJjb250ZW50X2lkIjoyNDgyMTc4NTAsImNvbnRlbnRfdHlwZSI6IkFydGljbGUiLCJtYXRjaF9vcmRlciI6MSwiemRfdG9rZW4iOm51bGx9.rH6r8SvGmu-I9piEsmZfg2HjbXzUduYclZ2jfA3jZRs&zhida_source=entity)可以表示为\|AB\|+\|BC\|>\|AC\|。 -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image34.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image34.png) height="1.91044072615923in"}\ ①延长直线AB至点D,并使\|BD\|=\|BC\|,连接\|DC\|,那么三角形BCD为等腰三角形。所以∠BDC=∠BCD。 ②记它们均为α,根据[欧几里得第五公理](https://zhida.zhihu.com/search?content_id=248217850&content_type=Article&match_order=1&q=%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%AC%AC%E4%BA%94%E5%85%AC%E7%90%86&zd_token=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJ6aGlkYV9zZXJ2ZXIiLCJleHAiOjE3NTIzNzg0NDAsInEiOiLmrKflh6Dph4zlvpfnrKzkupTlhaznkIYiLCJ6aGlkYV9zb3VyY2UiOiJlbnRpdHkiLCJjb250ZW50X2lkIjoyNDgyMTc4NTAsImNvbnRlbnRfdHlwZSI6IkFydGljbGUiLCJtYXRjaF9vcmRlciI6MSwiemRfdG9rZW4iOm51bGx9.ltcWsMYJv-ZzcuBaSjYN69JC8hnIyPMFsfhIlum4yqc&zhida_source=entity),∠ACD大于角∠ADC(α)。 ③由于∠ACD的对边为AD,∠ADC(α)的对边为AC,所以根据大角对大边([几何原本](https://zhida.zhihu.com/search?content_id=248217850&content_type=Article&match_order=1&q=%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC&zd_token=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJ6aGlkYV9zZXJ2ZXIiLCJleHAiOjE3NTIzNzg0NDAsInEiOiLlh6DkvZXljp_mnKwiLCJ6aGlkYV9zb3VyY2UiOiJlbnRpdHkiLCJjb250ZW50X2lkIjoyNDgyMTc4NTAsImNvbnRlbnRfdHlwZSI6IkFydGljbGUiLCJtYXRjaF9vcmRlciI6MSwiemRfdG9rZW4iOm51bGx9.Q1rCY0S2bj5Dwp3Fg7xb_VSFESz2_pCUETDybnHANvo&zhida_source=entity)中的命题19)就可以得到\|AB\|+\|BC\|=\|AB\|+\|BD\|=\|AD\|>\|AC\|。 求证:在三角形ABC中,P为其内部任意一点。请证明:∠BPC \> ∠A。 证明过程: -![](./Images/aaa7ee7bd3f44617a10b3c3de120faee/media/image35.png) +![](./Images/fbf8f3feebaf486aac25f04fe6702a2d/media/image35.png) 延长BP交AC于D ∵∠BPC是△PCD的一个外角,∠PDC是△BAD的一个外角 ∴∠BPC=∠PCD+∠PDC,∠PDC=∠DBA+∠A

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