Quandaries and Queries


I am a teacher.  In an FCAT sixth grade review test, there was a question to the students to draw a square and then they referred to it as a rectangle.

What is the definition that makes a rectangle a square that can be taught to the students without confusing them.

Thank you.




This points to an important 'big idea' in mathematics:
inclusive definitions, and how mathematicians talk and reason.

Consider any property, formula, measurement, construction that you might do for a rectangle. With the property, formula and measurement also hold for a square? Will the construction you have in mind sometimes create a square, as on particular example of a rectangle?

Think for example of something in Geometer's Sketchpad. when you construct a rectangle, and drag your points around, sometimes you pass an example which happens to also be a square. Do you want to say the construction is wrong? Would any student say the construction is wrong?

So the mathematical idea is to give definitions by the properties (often the transformations) that they do have, not be properties they don't have. This gives an 'inclusive definition' where objects of one type may be a subset of objects of a related type. A rectangle is a quadrilateral which has four equal angles. (That is enough, the rest that I say will follow from that if you are in the plane!) It also has opposite sides parallel. It has opposite sides of equal length.

We can reason with all those properties. When we do, say creating a formula for the area, we get facts which hold true for all objects with these properties. Squares happen to have the properties, so they also satisfy the formulas.

It is not sensible to redo all this reasoning, separately, to find out that squares have all these properties. In fact, it is not humanely feasible to reason so carefully that you always exclude squares when you say: "Take a rectangle. Then .... "

I particularly like to do this reasoning with a focus on 'symmetries' (mirrors and rotations). As we add more symmetries, we have fewer examples. So a rectangle has a couple of mirrors (joining midpoints of opposite sides). A square has these mirrors and more. As we 'include' more symmetries, we work down to fewer examples. This works pretty well.



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