The last reply in particular leads to the common misconception that the angle A below is larger than the angle B.
To clarify what angles measure I do a little pirouette and tell people this:
Angles measure turn.And as with all measures, we shouldn't jump in to teaching about standard units of measuring (degrees) until the students have had experience with non-standard units (e.g. full turns, half turns, right angles etc.)
I used to show students what a right angle is by pointing to the corner of a piece of paper. Now I get them to make their own right angle by doing the simplest Origami as shown below:
So to get students to really understand the notion that angles measure turn, I have them estimate angles using some cheap-and-cheerful angle measurers as shown:
Here I want students to actually turn the arms of the angle to create the angle. Here is a video of a student using them in a class to see if the angles in a quadrilateral are greater than or less than a right angle.
I find if they have experience estimating angles first, then when they come to measure angles with a protractor, they will not be confused by the two scales that most protractors have.
Finally, to counter the misconception that angles cannot be larger than 360 degrees I might ask students to either use the cardboard angle measurer above or to stand up and turn 180 degrees, then again, then again and ask "How many degrees have you turned now?" This idea of having angles beyond 360 degrees will be important in higher grades when they start learning about periodic functions and unit circles as this site shows.