I have heard, from a colleague who works with prospective elementary teachers, that many of them are not good at proportional thinking. My students (in pre-calc) seem to be fine at it, but ... they're using it even when it doesn't apply. My question for you is how to help them see why proportional reasoning is not always a sensible choice.
#54. Determining a Distance: A woman standing on a hill sees a flagpole
she knows is 60 feet tall [yeah, right]. The angle of depression to the
bottom of the pole is 14 degrees, and the angle of elevation to the top
of the pole is 18 degrees. Find her distance x from the pole.
student wanted to average the two angles at 16 degrees
each. Another said the observer could stand on a stool to be a little
the angles would be 16 degrees each. Their answers were very close.
There were other good (but wrong) methods that all came down to assuming
this relationship was linear in a way that it's not. Since their
answers were very close, it was hard to help them see what was wrong
with their reasoning.
Can anyone help me here?