There are numerous instances in mathematics where common errors occur. In my opinion, the most frequent in my career occurred when 0 divided a number. Division by 0 occurs in elementary mathematics, coordinate geometry (finding slope) and calculus (finding limits). Explaining why division by 0 is undefined is something students should be introduced to when division of whole numbers is first taught. In general, students learn that division is another form of multiplication. In the problem, 28 / 4 = 7, 28 is the dividend, 4 is the divisor and 7 is the quotient. Students learn that the product of the quotient and the divisor is equal to the dividend. Thus, 7 x 4 = 28. If I wanted students to discover why you can’t divide by 0, I would ask them a series of basic division problems.
For example, in the problem 63 / 9, students need to find a number that when multiplied by 9 will equal 63. That number is 7, since 9 x 7 = 63. After a student confirms that the product of the quotient and the divisor is equal to the dividend, he should be asked the following problem. 7 / 0 is equal to what? Students will try to think of a number that will equal to 7 when multiplied by 0. Since the product of 0 and any number is 0, they will conclude that there is no number that can be multiplied by 0 that will equal 7. At this point, you inform them that division by zero is undefined. In other words, there is no way that you can divide by 0. The previous problem should be followed by the following question. 0 / 7 is equal to what? In this problem, you want to find a number that when multiplied by 7 is 0. The answer to this is 0, since 0 / 7 = 0. The concept that division by 0 is undefined should be reinforced at every grade level in every math course.