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 $a\geq 0$: ${\sqrt {a^{2}}}=a$. Without the prerequisite $a\geq 0$ the formula would read ${\sqrt {a^{2}}}=|a|$. In fact, many students use ${\sqrt {a^{2}}}=a$ 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 $a=-2$. There ${\sqrt {a^{2}}}={\sqrt {(-2)^{2}}}={\sqrt {4}}=2$ and, hence, does not equal $a=-2$.
2. Scope of quadratic formula as described in DecodingWork:Scope of quadratic formula