Ok, here's a fairly typical SAT-type of question that requires application of fundamental ideas in geometry. Some students 'see' a way to find the value of x in less than 15 seconds. These students have strong conceptual ability and are confident of their knowledge and reasoning. They are not thrown by a question that is somewhat different from the textbook problems they've seen.
Some of the issues as I see them are:
(1) How do we raise the knowledge/experiential base and confidence of those students who cannot seem to find a solution path and inevitably give up?
(2) How do we extend the thinking of those talented students who solve the problem in short order and just sit there complacently?
Perhaps, beyond these considerations and the instructional strategies employed is the bigger issue of PROVIDING FREQUENT CHALLENGES FOR OUR STUDENTS THAT GO BEYOND NORMAL TEXTBOOK FARE.
Where does one normally find these types of challenges in school curricula? Embedded in a natural way in the regular set of textbook exercises that can be routinely assigned? OR are they labeled as Standardized Test Practice at the end of a section or in a separate part of the chapter or text? OR are they found primarily in ancillary materials provided by the publisher? Can you guess where I think they should be and if they should be labeled? Should they be stand-alone multiple-choice questions or more open-ended with several parts that go beyond the 'answer?'
Before I suggest an activity based on this innocent-looking problem, I invite readers to consider a variety of methods of solution (so far I've observed about 6-7 'different' approaches) and how one might go beyond this standardized test question to deepen student reasoning. To remove the element of surprise and focus attention on process, the 'answer' is 90... (sorry!).
Have fun but if you 'see' the answer in 15 seconds or less, don't stop there!