What is a Variable?
©Alan Selby, August 2000.
Goal: Master the mathematical use of the word variable
Previous: Chapter 9, How to Talk or Describe
Numbers and Quantities
What is a Variable, Sections: [Introduction] [ Variation between Examples ] [ Variation of Letters ] [ When does a letter denote a variable ] [ Cases of Double Variation ] [ Three Notions of a Variable ] [ Constants ] [ Talking about numbers ] [ Dependent or Independent Variables ]
Introduction
Look in a dictionary, encyclopedia and a mathematics text for a definition of
what is a variable, an introduction that is understandable to you and easily
explained to others. If you find such a definition or introduction clear enough
to help in mathematics after arithmetic, the rest of this essay need not be
read.
Alice in Wonderland if she could speak today, would observe
that the view of a variable as a function begs the question of how
to explain the notion of a function without using the concept of a variable.
The essay or chapter before put the concepts of what is a variable first and
before the use of symbols and notation in mathematics for numbers, amounts,
quantities and functions.
Variation in a Single Example
variation = amount of change
The next diagram shows the height of a bird during its journey from one tree
to another. The flight is over the ground intervals
[a,b], [b,c], [c,d], [d,e], [e,f]


Letters on horizontal axis end ground intervals where the height
behavior changes. If height is measured above or below sea level, and the
tops of both trees were below sea level, then increasing height would
correspond to make the height relative to sea level less negative.
Identify the intervals where the height of the bird is constant, where this
height is increasing (becoming more positive or less negative) and where this
height is decreasing (becoming less positive or more negative). The height may
have different behaviors on different ground or time intervals. This exercise
could be redone on a graph of height versus time. In this case, the ground
intervals would correspond to time intervals.
To vary means to change. Identify the ground intervals where the height of
the bird is constant (not variable) and where it is variable.
Conclusion: Whether or not a number or quantity is constant or not,
variable may depend on the interval in which is observed or examined or
remembered. We can talk about numbers and quantities being variable without or
before the use of letters to represent them.
The following diagram shows the speed of a car along a straight
road.
Piecewise linear graph of speed versus time
Identify the time intervals where the speed of the car is constant and where
it is variable.
Challenge (a hard exercise): From the above
diagram, how would you find the distance traveled by the car in a constant-speed
interval and in the variable speed intervals. Find a solution without the use of
calculus. Hint: See an old high school physic text.
What is a Variable Sections: [ Variation between Examples ] [ Variation of Letters ] [ When does a letter denote a variable ] [ Cases of Double Variation ] [ Three Notions of a Variable ] [ Constants ] [ Talking about numbers ] [ Dependent or Independent Variables ]
Next Section: Variation
between Examples or Chapter 10, Describing &
Changing Calculations
This webpage and its sections on What is a Variable (not part of
Three Skills for Algebra) is a postscript originally posted online in August
2000. There is a hidden curriculum in mathematics in which talking about
numbers and quantities, the first skill for algebra at this website, is not
discussed.
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