Date: May 7, 2013 7:57 PM
Author: Robert Hansen
Subject: Re: Based on the quadrilateral tree

On May 7, 2013, at 12:46 PM, Joe Niederberger <niederberger@comcast.net> wrote:

>> The restrictive definition allows its advocates to be certain that an isosceles trapezoid is not a parallelogram.

>

> *Absolutely* certain.

>

> It would be nice if these things could be taught in way that lets the students see when things are, for lack of a better term, "conventional", and when they are not.

> This reminds me of the discussion a while back about the conventionality of the area measure of the unit square,

> which, it became clear, is not universally recognized as such.

>

> Cheers,

> Joe N

I would definitely place 0.9999... with the "unit square" example.

I would place "whole numbers" with trapezoid v parallelogram. You get around the former lack of accepted strict definition by being explicit, like "a whole number including zero" or "a whole number not including zero". You can also drop "whole" altogether by saying "non negative integer" or "positive integer" although I am not sure younger minds have trained themselves to respect the the subtle difference. In those cases I would stick with whole numbers with or without zero, when such a distinction is even necessary.

Likewise, one can say "a trapezoid that is/is not a parallelogram" or "a quadrilateral with at least one pair of parallel sides" or "a quadrilateral with only one pair of parallel sides".

And if a teacher states that a trapezoid is never a parallelogram, then that is certainly their prerogative, as long as they mention that this is not universally accepted, but is now the norm for this class.

Bob Hansen