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数学是一门严谨的学科,作为教师,我们在教学过程中一定要注意考虑全面,要培养学生敢于质疑的精神,提高学生的逻辑思维能力,提升学生的数学素养。教师在讲解直线与椭圆的位置关系时,时常会有结论:若直线l:y=kx+b与椭圆(x~2)/(a~2)+(y~2)/(b~2)=1(a>b>0)相切,将直线方程与椭圆联立,则Δ=0(?)方程组有一解(?)直线与椭圆有且只有一个公共点(?)直线与椭圆相切。此种情况在讲解直线与圆相切的代数判定时也会用到,导致学生错误地理解为直线与曲线相切和直线与曲线有且只有一个公共点等价,事实真的是
Mathematics is a rigorous discipline. As a teacher, we must pay attention to comprehensiveness in the teaching process, cultivate students’ spirit of daring to question, improve students’ logical thinking ability, and improve students’ mathematical accomplishment. When explaining the position relationship between a straight line and an ellipse, the teacher will often conclude that if the line l: y = kx + b and the ellipse (x ~ 2) / (a ~ 2) + (y ~ 2) / (b ~ 2) = 1 (a> b> 0), the linear equation is set up to be an ellipse, then there is a solution to the system of Δ = 0 (?). The straight line and the ellipse have one and only one common point cut. This situation is also used when explaining the algebraic determination of tangent lines and circles, which leads students to mistakenly understand that the line is tangent to the curve and the line is equivalent to the curve with one and only one common point. The fact is