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]
Flight of a Bird
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:
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.
Three Skills
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Chapters and Appendices
Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice