Tuesday, November 23, 2010

Another Cone in a Sphere Problem? - A Guide for the rest of us...

Students who have been out of geometry for a year or so and are preparing for standardized test like Math I Subject Test or SATs/ACTS need occasional review. The following is similar to several other cone problems I've posed before but even our strongest Algebra 2 through Calculus students lose their "edge" when it comes to "solid" geometry questions (yes, believe it or not, my terminal course in high school was called Sold Geometry and we covered topics like spherical trigonometry!).

A right circular cone of height 16 is inscribed in a sphere of diameter 20. What is the diameter of the base of the cone?


1)  Are these kinds of problems somewhat hard merely because students forget? I can think of several more reasons:

  • The problem itself is somewhat challenging, however it's far from over their heads!
  • The student never experienced a question like this in Geometry; perhaps questions like these were in the B or C or D exercises in the text and were never assigned or only for the "honors" students? Do you recall seeing a problem similar to this in the textbook from which you taught?
  • The student did not take a formal course in geometry
  • The topic was covered in a cursory manner or perhaps not at all because of time crunch. That's the whole point of a standardized curriculum, isn't it? To know what is needed to be covered and plan accordingly. Of course, I'm  a realist enough to know the myriad of reasons why the best laid plans oft go .........
  • Students don't remember how to start because key geometry strategies were not explicitly stated and reiterated ad nauseam. Were your students asked daily to begin by reciting the key strategies such as those for circle and sphere problems? Were they placed on index cards or blocked out in a particular section of their notebook?:
    • DRAW THE BEST DIAGRAM YOU CAN (and believe me, I'm no artist!)
    • Always locate the CENTER of circles, spheres and label the point
    • Label the measurements of all segments (angles) - I know, everyone does that!
    • Successful problem-solving in mathematics is based on finding relationships! Were guiding/leading questions asked 
      • What do the cone and sphere have in common? 
      • TRUE  FALSE  The height of the cone is the same as the diameter of the sphere.  EXPLAIN!
    • Was the student exposed to the strategy of comparing the 2-dimensional analogue of the 3-D problem? Would it be a right triangle in a circle? Equilateral triangle inscribed in a circl or???  
    • Oh and yes... 
      • Draw the radius of the sphere (or circle) so that it is the hypotenuse of some right triangle!

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

1 comment:

Anonymous said...

The trickiest piece is reducing this to 2 dimensions. Without that insight, I think this problem gets stopped in its tracks.