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在中学学生的数学作业中,經常出現下面錯誤的等式: lg(a+b)=lga+lgb。发生这类錯誤的不是个別学生,有时甚至达到全班学生半数以上,并且在练习中会不止一次地以不同的形式重复出現。类似的錯誤还有: (a~(1/2)+b~(1/2))~2=a+b; (a-b x~(1/2))~2=a~2-b~2x; [c(a-b)~2]~n=c~na~(3n)-c~b~(2n); (x~2+y~2)~(1/2)=x+y; 0.4~(1/2)=0.2; (-2~)(1/2)·-3~(1/2)=6~(1/2); lg(a/b)=lga/lgb; 1g 1.46=16.44;(正确答案:1g 1.46=0.1644) 从lgx=0.6351,得出x=1.4316;(正确答案:x=4.316) 从x~2/6=24,得出x~2=4。 x,x+1和x+3是三个連續的奇数(x是正整数); sin(A+B)=sin A+sin B。教师們详细地探討这类错誤产生的根源,分析它們形成的过程,进一步寻求預防和克服这类錯誤的办法,对学生学习质量的提高有着重大的意义。
In mathematics work for middle school students, the following erroneous equation often appears: lg(a+b)=lga+lgb. It is not individual students who have made such mistakes, sometimes even reaching more than half of the students in the class, and will appear repeatedly in different forms during the exercise. Similar errors are: (a~(1/2)+b~(1/2))~2=a+b; (ab x~(1/2))~2=a~2-b~2x ; [c(ab)~2]~n=c~na~(3n)-c~b~(2n); (x~2+y~2)~(1/2)=x+y; 0.4~ (1/2)=0.2; (-2~)(1/2)·-3~(1/2)=6~(1/2); lg(a/b)=lga/lgb; 1g 1.46= 16.44; (correct answer: 1g 1.46=0.1644) From lgx=0.6351, we get x=1.4316; (correct answer: x=4.316) From x~2/6=24, we get x~2=4. x, x+1 and x+3 are three consecutive odd numbers (x is a positive integer); sin(A+B)=sin A+sin B. Teachers explore in detail the root causes of such mistakes, analyze the process of their formation, and seek further ways to prevent and overcome such mistakes, which are of great significance to the improvement of students’ learning quality.