Students always seem to get confused about the difference between the length of an arc and the measure of an arc. When I teach this concept, I try to relate it to sports. The vast majority of the kids in my class have run around the track at some point. So I start asking them questions. :)
If you have to run one lap around the track which lane do you want to run in?
The inside one!
Why?
It's shorter!
Why?
The circle is smaller.
Interesting... You have still run one lap though. All the way around the circle would be 360 degrees. So, it should be the same distance. If you run the outside lane, that's the same distance as the inside lane.
[At this point, controlled chaos ensues. Most of the athletes become indignant.]
Then, I tell them that the measure of an arc is the degree measure. The length of the arc is the actual distance you would have to run. Most of them get it by this point.
I made a handy little "cheat sheet" for students to glue in their notebooks.
