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激活猜想：
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图形编号 ① ②
多边形的面积/平 方厘米
多边形边上的钉子 数/枚
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闵嗣鹤.著
乔治·皮克 （1859～1943) Georg Pick 奥地利数学 家
皮克定理
该定理被誉为有时以来 “最重要的100个数学 定理”之一。
乔治· 皮克（奥地利）
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2 3 3.5 4
4 6 7 8
激活猜想：
图形编号
① ② ③ ④ 多边形的面积/平 (s) 方厘米 多边形边上的钉子 (n) 数/枚
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4 6
3.5 4
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S=n÷2
数学是研究千变万化中不变的规律 ——开普勒
要求： 1.画两个内部有1枚钉子的多边形。 2.分别标上序号。 3.在表1中填出所画多边形的面积和 边上的钉子数。
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n=8
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当a=0 时：s= n÷2-1
充 实 猜 想 ：
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多边形的面积/平 (s) 方厘米
多边形边上的钉子 (n) 数/枚
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图形中间点的个数 用a来表示
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S=n÷2
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充 实 猜 想 ：
图形编号 ① ② ③ ④
多边形的面积/平 (s) 方厘米
多边形边上的钉子 (n) 数/枚
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自主探究：
图形 编号 ① ② ③
多边形内部的钉 多边形的面积/ 子数/枚 (a) 平方厘米 (s)
多边形边上的 钉子数/枚 (n)
苏 教 版 五 年 级 数 学（上）
引出问题：
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10cm²
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当a=1 时：s=n÷2 当a=2 时：s=n÷2+1 当a=3 时：s=n÷2+2 当a=4 时：s=n÷2+3
……
当a=1时，S=n÷2
当a=2时，S=n÷2+1 当a=3时，S=n÷2+2 …… 你能用一个式子，表示出s与n之 间的关系吗？
引出问题—— 激活猜想—— 进行验证—— 修正猜想—— 自主探究—— 概括规律
10 9 4 8
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自主探究：
请先在小组内说说自己的想法，再通过围一围、算一算进行验证。