《正数和负数》导学案
重点:(1)正数、负数的概念及意义;(2)具有相反意义的量的表示法及其应用。
一、正数、负数的概念
1.什么是正数、负数?
在现实生活中,人们为了表示物体的个数(比如捕获得到的猎物)或者事物的顺序,便用上了自然数0,1,2,3,…,这些数也称为整数,其中数“0”表示“没有”、“无”或“空”等意思。后来由于分配或度量的结果不是整数,又用上了分数和小数,如
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
,3.25,0.98,…。那么,有了这些数,是否就满足我们生活、生产的需要了呢?显然不是,如果大家平时关注天气预报,就会发现一类新的数,比如“今晚到明天北京的气温是-4~7°℃”,它表示气温是零下4度到零上7度。
一般地,我们把像2,
![](data:image/png;base64,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)
,0.98这样大于 0的数叫做正数,像-4,-
![](data:image/png;base64,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)
,3.25 这样在正数前面加上“-”的数叫做负数,其中“-”念“负号”,-4念“负 4”,有时为了对比或强调,也会在正数前面加上符号“+”(念“正号”),如+
![](data:image/png;base64,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)
,念“正
![](data:image/png;base64,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)
”。
2.数0的意义
(1)0既不是正数,也不是负数:
(2)0是正数和负数的分界,而且+0、-0都还是 0;
(3)0不仅可以表示没有,还可以表示某种量的基准,如0°
C表示实际温度为冰点时的计量结果。
例1 下列数中哪些是正数?哪些是负数?
-13, 0.2, 3, -1
![](data:image/png;base64,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)
,+
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAkCAYAAACAGLraAAAAAXNSR0IArs4c6QAAAa9JREFUSEvV1D9IVXEUB/CPUkGUBCI0BG6hzZUgTg7pEujQUIMQGkQUNAQNjhk4BDVEDoGiixS1NbY4OCQ4FiFKQ9DQ0iKB0t976Pfk9ui+93u8HDxwufDeOd/7Ped8z7dDm9HRZr19BTiPYXzBCOawUs+4isFpXMRjfMcAFnANb8ogVQDX8QGvU3IXnmATMzkAtzCGq/iEo3iEH7iDnRpI7hB7sIgXWMphUD+ryxjHbXxuFeAM7mMaG7lbqOWdxD08/FdxJDWawTHcxXIqPo7BpIVvzYZ4CFNYxbuU3JcEFdrYiyoGMbSniP2XI35/ngOQfWO5OqgE/K8Andm8/yT+rF/jiUJlNzJBdtNt7K+hZJFpNMQwkQtJhWEucY3ZjtSLSTzA18JYhpKhXMH7HCFFwTwuFc9bhIxDgbO5SoyVhoy307rOJiMJh1rPYVDOicOKq4wLDFsLk216TJFwGP2p/+7k0MHor8iV8rnUfwhtK5dBOa9m67GBGGTDFo4k7z+VpB20W7L1aOtmoYOPeIVfCLAQUqz2WTMG8X98cSK9w4lHsYaXrWyh7Vs4IAC/AaPfSiVBfKXEAAAAAElFTkSuQmCC)
,1, -3.14。
正数:
;负数:
。
例2 下列说法中正确的是()。
A.不带负号的数都是正数; B.0°
C是一个确定的温度;
C.一个数前面加上“-”,这个数就是负数; D.若
a是正数,则-
a不一定是负数;
例3 下列各数:2023,-29,15%,-0,-
![](data:image/png;base64,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)
,
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,
![](data:image/png;base64,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)
,-5.5,+
![](data:image/png;base64,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)
,-1.
正数有:
;
负数有:
;
非正数有:
;
非负数有:
;
自然数有:
;
二、用正、负数表示具有相反意义的量
在现实中我们会遇到许多具有相反意义的量,比如股票价格的上升与下跌,财务上的收入与支出,商业经营中的盈利与亏损,物体运动方向的向东与向西等等,为了方便统计和计算,对于这样一些具有相反意义的量,我们就可以使用正数和负数来表示,在用正、负数表示相反意义的量时,哪种意义为正是可以任意选择的,但一般地,把具有积极向上的意义,如“前进、上升、收入、零上”等规定为正,而对于运动方向来说,通常规定向东、向南为正,向西、向北为负,例如,排球比赛或乒乓球比赛中,胜局数用正数表示,那么败局数就用负数表示了:股市指数上涨的点数用正数表示,那么下跌的点数就用负数表示。
例4 用正数和负数分别表示下列各组量:
(1)向南走2千米和向北走3千米;
(2)收入 5500 元和支出 3800 元;
(3)增产 15%和减产 8%;
(4)比海平面高 600 米和比海平面低 200 米
例5 A,B,C,D四位同学的立定跳远成绩分别是 1.65 米,1.70 米,2米,1.80米。若以B同学的成绩为基准,记作0米,则A同学的成绩记作
米,C同学的成绩记作
米。
三、综合应用实例
例6体育课上,某中学对七年级男生进行引体向上测试,以能做5个为标准,超过的个数用正数表示,离达标还缺的个数用负数表示,其中8名男生的成绩如下:2,-1,0,3,-2,-3,1,0.
(1)这8名男生达到标准要求的合格率是多少?
(2)这8名男生共做了多少个引体向上?