Moving Away From “THE” Answer

In my collection of problem resources, I have a book full of MENSA problems. One particular puzzle caught my eye a couple years ago.

TrapezoidProblem

 

Exhibit A: An innocent looking trapezoid, just waiting to be cut up.

Forgive the distortion caused by the scanned page. The intent of the question author is to offer what appears to be, for all practical purposes, an isosceles trapezoid. Showing this problem to students seldom leads students to uncover what the author of the book claims is the solution.

TrapezoidBookSolution

 

 

 

 

 

Exhibit B: "THE SOLUTION"

This problem struck a chord with me. I wondered whether it would be possible to find other solutions. I wanted my students to investigate the possibilities and, if there were no other ways to do this, to rule out cases with mathematical reasoning. Several of my students came up with what appeared to be a solution, but it seemed like an accident of distorting the sketch a student made. I challenged the students to rigorously show whether or not the solution works in all cases, or if there were certain dimensional restrictions or conditions under which the students' solution worked. One of my students went on to model this problem in Geogebra.

IsoscelesTrapScreenShot

 Exhibit C: A student models a solution in Geogebra. Is this solution a special case?

We should be careful about the language we use in class as math teachers. Had I told my students to find "THE" answer, they may not have uncovered the possibilities of the colored trapezoid above. My students went on to write inequality statements under which the second solution exists or does not exist. When I posed the problem to my students, I asked for "AN ANSWER," a far cry different from "THE" answer.

Amazing how definite a difference an indefinite article can make.

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