Creating Good Mathematical Definitions
In order to clearly communicate yourself to others, you need to use words whose definitions are understood by everybody in
the same way. Picture a young man saying to his girlfriend, “I love you.” And his girlfriend replying, “I love you, too.” Are
they using the word love in the same way? Do they understand precisely what the other one means when they use the word
love? Probably not. Love, as a word and as a concept, can carry an enormous variety of meanings and implications and
generate very different ideas in the minds of different users of the word. It may well be that it is a good thing for a word like
love to be so vague; we probably need powerful-but-vague concepts in our world.
Developing good mathematical definitions for a word or concept is a three-step process:
1. Classify the concept: What is it? To what larger category of objects or concepts does it belong?
2. Differentiate the new word. How does it differ from other closely related objects or concepts? What makes it unique?
3. Test your definition by trying to find counterexamples.
Example: Looking for a counterexample. A counterexample is a specific example that satisfies the given part of a statement,
but which doesn’t satisfy the conclusion.
A. Complete the following sentence:
Vertical angles, formed by _____________________, are____________________________________________
B. Molly’s answer for A above was “Vertical angles, formed by crossing lines, are those
across from each other.” For Molly’s statement, a counterexample would be a
diagram of angles “formed by crossing lines, across from each other” that are NOT
vertical angles. The angles indicated in this diagram are “across from each other,”
but are not vertical angles.
C. Improve on Molly’s definition for vertical angles so that it is precise enough to have no counterexamples, but
broad enough to include all true examples of vertical angles. How about,
Vertical angles, formed by two intersecting lines, are congruent and have only the vertex in common.
Part 1: Create a definition of “Geobug” that will allow you to successfully distinguish between “true” Geobugs and figures
that are not truly Geobugs.
A. Use the Geobug Definition-Builder form to identify essential and non-essential (i.e. possible distracting)
characteristics and then craft a definition.
B. Check your definition with the others in your group to see if all agree that your definition will clearly distinguish
between Geobugs and non-Geobugs.
C. Use your definition to determine which of the following are Geobugs.
D.
EXAMPLES
NON-EXAMPLES
GEOBUGS
ESSENTIAL CHARACTERISTICS
DEFINITION
NON-ESSENTIAL CHARACTERISTICS
Part 2: Complete the templates for each term listed. Remember that essential characteristics are those that all true examples
share, while non-essential characteristics are those that true examples may or may not have.
EXAMPLE
ESSENTIAL
NON-EXAMPLES
Congruent
Angles
CHARACTERISTICS
NON-EXAMPLES
EXAMPLE
Perpendicular
Lines
NON-ESSENTIAL
ESSENTIAL
CHARACTERISTICS
CHARACTERISTICS
NON-ESSENTIAL
CHARACTERISTIC
S
DEFINITION
EXAMPLES
DEFINITION
NON-EXAMPLES
NON-EXAMPLES
EXAMPLE
S
ESSENTIAL
CHARACTERISTICS
DEFINITION
Parallel
Lines
Corresponding
Angles
NON-ESSENTIAL
ESSENTIAL
CHARACTERISTICS
CHARACTERISTICS
DEFINITION
NON-ESSENTIAL
CHARACTERISTICS
EXAMPLES
ESSENTIAL
NON-EXAMPLES
Alternate
Interior Angles
CHARACTERISTICS
NON-ESSENTIAL
ESSENTIAL
CHARACTERISTICS
CHARACTERISTICS
DEFINITION
EXAMPLES
ESSENTIAL
CHARACTERISTICS
DEFINITION
EXAMPLES
NON-EXAMPLES
Vertical
Angles
NON-ESSENTIAL
CHARACTERISTICS
DEFINITION
NON-EXAMPLES
Perpendicular
Bisector
EXAMPLES
NON-EXAMPLES
Parallelogram
NON-ESSENTIAL
ESSENTIAL
NON-ESSENTIAL
CHARACTERISTICS
CHARACTERISTICS
CHARACTERISTICS
DEFINITION
EXAMPLES
NON-EXAMPLES
EXAMPLES
NON-EXAMPLES
ESSENTIAL
NON-ESSENTIAL
ESSENTIAL
NON-ESSENTIAL
CHARACTERISTICS
CHARACTERISTICS
CHARACTERISTICS
CHARACTERISTICS
DEFINITION
EXAMPLES
ESSENTIAL
CHARACTERISTICS
DEFINITION
DEFINITION
NON-EXAMPLES
EXAMPLES
NON-EXAMPLES
NON-ESSENTIAL
ESSENTIAL
NON-ESSENTIAL
CHARACTERISTICS
CHARACTERISTICS
CHARACTERISTICS
DEFINITION