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如何发挥教学过程中习题的潜能?来看这个习题:已知a,b,c是△ABC的三条边,比较(a+b+c)、4(ab+bc+ca)的大小。这道题的解答可以用特殊值法。取a=b=c=1,得(a+b+c)2=9,4(ab+bc+ca)=12,所以(a+b+c)2<4(ab+bc+ca)。将这道题稍微变形,就是高中数学教材第二册(上)不等式B组题的第6题:设a,b,c为△ABC的三边,求证:a2+b2+c2<2(ab+bc+ca)。这道题的解法需围绕三角形的特征,结合不等式证明的方法,可以得到许多不同的证明方法。并且依据已经证明的结论,还可以进行变式引申。
How to play the potential of the exercises in the teaching process? To see this exercise: Know that a, b, c are the three sides of △ABC, compare the size of (a+b+c) and 4 (ab+bc+ca). The answer to this question can be used special value method. Take a = b = c = 1, get (a + b + c) 2 = 9, 4 (ab + bc + ca) = 12, so (a + b + c) 2 <4 (ab + bc + ca) . This will be a slight distortion of the title, is the second volume of high school mathematics textbooks (on) Inequality B Group Question 6: Let a, b, c for the three sides of △ ABC, verify: a2+b2+c2<2 (ab +bc+ca). The solution to this problem needs to be centered around the characteristics of the triangles. Combined with the method of inequality proof, many different methods of proof can be obtained. And based on the already proven conclusions, it is also possible to carry out variant extensions.