On Jul 9, 2013, at 3:47 PM, Richard Strausz <Richard.Strausz@farmington.k12.mi.us> wrote:
> There are 2 intersecting chords in a circle whose lengths are 10 and 7. If the longer chord is divided into parts of 1 and 9, what are the lengths of the 2 parts of the shorter chord?
As Wayne already pointed out, in the context of geometry, do they understand the proof of the theorem behind the solution of this problem?
Regarding the quadratic formula, why not simply print it at the top of the exam?
I would be MUCH more interested in hiding the chord relationship in the problem (since that is the topic) than hiding the quadratic formula. I wouldn't give them a problem just like the one I worked out on the board.
Generally, the sequence would begin with a discussion and proof of this chord theorem, then its application. In the application phase, I would start by using a few problems like the one you presented. I would walk them up to the point of seeing the algebraic relationships in an actual circle with two intersecting chords (direct teaching) and then change to a Socratic mode to get them to realize that this results in a pair of equations and (with guidance) that this results in a quadratic to be solved. I would continue to press them for the solution to this quadratic, hoping that at least some of them get it right away, some of them catch up right away, and some of them, well, barely hang on. At this point we have a (proven) theorem of chords, the realization that this results in a pair of simultaneous expressions that further reduce to a quadratic expression that can be solved using the quadratic formula (which is a generalized form of completing the square). The next and final task is to look at a few problems that involve the same pieces except do not involve the full diagram, like the arch over a doorway.
A few of those types of problems would be on the exam. Putting the quadratic formula on the board during the exam will be the least of their task.:)