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Writing a Good Definition - Concept

16,141 views
Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Writing a definition is a common exercise during the early stages of Geometry. An excellent geometry definition will classify, quantify, and not have a counterexample. Once a term is defined, it can be used in subsequent definitions; for example, once parallel lines are defined, they can be used in the definition of a parallelogram.

In geometry, it's imperative that you
can write a good definition, because
it will help you to understand the properties
of whatever it is you're talking about.


The three key components of a good definition.


The first one, it uses previously
defined terms.
So if you've already defined what parallel
lines are, you can use that to define
a parallelogram.


Secondly, it classifies and quantifies.
That is, by classifying
it, is it a polygon?
Is it a line?
What is it?
And quantifies how many.
So if you're talking about a polygon, you're
going to want to say how many sides.


And, last, it has no counter-example.
But what is a counter-example?
A counter-example is something, an example,
that will make a definition or
conjecture incorrect.
So if you can find a counter-example to your
definition, you haven't written a good one.


So a short example is let's say I had a
square and I said that a square is a
quadrilateral.
Which means that it has four sides.
And I just left my definition like that.
Turned it into Mr. McCall.


Well, I'm going to say a quadrilateral,
well that could be a trapezoid, where
I could draw in one pair
of parallel sides.
It could be a kite where we have two pairs
of congruent consecutive sides.
I could draw in a rhombus.
I could draw in a parallelogram.
I could draw in lots of counter-examples
that would make this definition not
true or it wouldn't make it specific
enough for just a square.


Let's look at two other ones.
Let's say something that's not related
to geometry, directly, a skateboard.
Let's say I define a skateboard as something
with wheels that you ride.
Well, that's not very descriptive.
This is not a good definition.
First and foremost because I could say
that this could be a bike, because a
bike is something that has
wheels that you ride.


What about a good definition?
A good definition for a parallelogram is a
quadrilateral with two pairs of parallel
congruent sides.
Notice that we're using words that
we probably already defined.
So quadrilateral, we would have defined before
we started defining a parallelogram.
Quadrilateral has four sides.
Parallel lines we say never intersect.
Two lines in the same plane
that never intersect.
And congruent means having the
same measure or same length.


Notice I was able to write this definition
of a parallelogram using three words
that I've already previously defined
and there's no other counter-example
I could draw or come up with that would
make this not apply to a parallelogram.


So keep that in mind when you're writing
good definitions and it will help you
even on your test and quizzes.

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