Scope of formula

Description of Bottleneck

When using a mathematical formula students don’t check whether the prerequisites for the applicability of this formula apply.

Intended Learning Outcome

This bottleneck relates to the following intended learning outcome: Students always check whether a given formula comes with prerequisites for its applicability and whether these prerequisites hold in a given situation.

Examples

1. Square root and square are inverse to each other for nonnegative real numbers, i.e. for <math>a \ge 0</math>: <math>\sqrt{a^2}=a</math>. Without the prerequisite <math>a \ge 0</math> the formula would read <math>\sqrt{a^2}=|a|</math>. In fact, many students use <math>\sqrt{a^2}=a</math> without checking the applicability/validity of this prerequisite. Of course, the fomula then leads to wrong results if the ignored prerequisite does not hold. This is e.g. the case for <math>a=-2</math>. There <math>\sqrt{a^2}=\sqrt{(-2)^2}=\sqrt{4}=2</math> and, hence, does not equal <math>a=-2</math>.
2. Scope of quadratic formula as described in DecodingWork:Scope of quadratic formula