Big floor symbols

You could use \left...\right for stretchable delimiters, or perhaps one of the pairs of the \bigl...\bigr family of commands:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\[
\Bigl\lfloor\dfrac{1}{2}\Bigr\rfloor\qquad
\left\lfloor\dfrac{1}{2}\right\rfloor
\]

\end{document}

A LaTeX-y way to handle this issue would be to define a macro called, say, \floor, using the \DeclarePairedDelimiter device of the mathtools package. With such a setup, you can pass an optional explicit sizing instruction -- \Big and \bigg in the example code below -- or you can use the "starred" version of the macro -- \floor* -- to autosize the left and right hand brackets. Both possibilities are pursued in the following code.

\documentclass{article}
\usepackage{mathtools} % for "\DeclarePairedDelimiter" macro
\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
\begin{document}
\[
\floor[\Big]{\frac{1}{2}}
\qquad
\floor[\bigg]{\frac{1}{2}}
\qquad
\floor*{\frac{1}{2}}  % autosize vertical dim. of brackets
\]
\end{document}