Dependent or Independent Variables,
A Matter of Choice
The area A of a rectangle is given by the product of its dimensions, say
width W and length L. In algebraic shorthand notation
A = WL
That being said, if we choose to calculate A from the values of W and L, then
A will be called the dependent quantity or variable while W and L will be called
the independent variables or quantities. On the other hand, we choose to compute
W in the equation A = WL from the values of A and L, then W = A/L and we will
call W the dependent variable and will we call A and L the independent
variables. So which variable is dependent or not depends on how we choose to use
a formula, or which ones we choose to give or calculate. If we are situation
where we know A = WL but we do not know which will be given and which will
be given, we cannot say which one will be dependent, albeit the form A = WL
suggests A will be the dependent variable. What we can say for sure or
certain is the the equation A = WL links or relates the area, width and area of
a given rectangle. The set theoretic view or codification of a relation
will be given below.
Postscript - Another Reading Guide
Arithmetic and algebraic expressions and formulas are like pictures better
read in silence than spoken aloud. That situation has led to a lack of words
in mathematics. The Discussion of Three Skills for Algebra in Chapter
8 to 12 provide a wordy, or too wordy remedy. See how much you swallow
today, and return for the rest to perfect your skills and comprehension later.
Note too, further skills for algebra, a fourth and/or fifth appear in Chapter
14. That chapter introduces two unifying themes for secondary school
mathematics:
- The forward and backward use of almost all formulas and equations (those
in function form y = something)
- The connection between arithmetic (or numerical) solution and algebraic
solution methods in the backward use of formulas and equations. If you
understand the algebraic solution, you are halfway to understanding the full
strength use of algebra in senior high school mathematics and
calculus.
So now, read on and get a headstart or catch-up for
high school mathematics and/or calculus. Good luck.
See too Chapter 7 with its Arithmetic Skill Check
for Calculus and Pre-calculus Students - Is your arithmetic precise and
efficient?
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Three Skills
For
Algebra
understanding & explaining
Reason and Math
Volume 2
Printed in Canada
ISBN 0-9697564-2-9
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Chapters and Appendices
Home Postscript: The 4-th Skill For Algebra Foreword 1. Introduction 2. Implication Rules 3. Chains of Reason 4. Romeo and Juliet 4. Induction Mathematical 5 Knowledge Islands 6 Old Language 7 Arith Skill Check 7. The Next Chapters 8 The Three Skills 8 VNR-Concise-Encyclopedia PS. What is a Variable 9. Algebra Talk 10 Two More Skills 11 Why Shorthand 12 Shorthand Usage 13 What's Next 14 Compound Interest 15 Linear Equations PS I. Distributive Law PS II. Polynomials 16 Painless Proofs 17 Pythagoras 18 Rules of Algebra 19 Functions & Sets 20 Degrees & Radians 21 What's Next 22. Arith & Geometric Sums 23 Summation Notation 24 Your Money 25 Induction & Recursion 26 What's Next 27 Pronouns in Logic 28 Occurrence Tables 29 Contrapositive 30 Truth Tables 31 Indirect Reason A. Advice For Learning
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
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