You need to use double quotes. Single quotes prevent the shell variable from being interpolated by the shell. You use single quotes to prevent the shell from doing interpolation which you may have to do if your regular expression used $ as part of the pattern. You can also use a backslash to quote a $ if you’re using double quotes.
Also, you may need to put your variable in curly braces ${var} in order to help separate it from the rest of the pattern.
Example:
$ string="test this"
$ var="test"
$ echo $string | grep "^${var}"