What Do Angles Measure?

I've had a lot of fun asking this question both to educators and students recently. Typical replies are; "They measure degrees"; "They measure the size of the vertex/point."; "They measure the distance between the two lines."
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:

A question which I'm often asked is why is a right angle 90 degrees (and not, say, 100 degrees)? Well the answer lies in how many degrees are in a full turn and there will always be some students who know this, especially if they are into skateboarding or snowboarding: 360 degrees. So why 360 degrees? Well the ancient Babylonians were the first folk to consider breaking the full turn into smaller standard parts. They knew that the Earth took 365 days to go around the Sun (long, long before Copernicus) but they also knew that 365 was not exactly a friendly number to work with. they chose 360 instead as they used a base 60 for their numbers. Good job they did otherwise we would be saying that a right angle is 91¼ degrees!
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.