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Dependent or Independent Variables
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Three Skills
For 
Algebra
Volume 2

Vol 2, Three Skills for Algebra covers many  topics in algebra and logic that students starting calculus should have mastered or will have to master. Also includes arithmetic review problems to catch common mistakes of students starting calculus.  A fourth skill in this misnamed volume gives a unifying theme for high school maths.

Chapters and Appendices

Book Entrance

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

Foreword
1. Introduction
2. Implication Rules [4]
3. Chains of Reason [3]
4. Induction Mathematical
4. Romeo and Juliet
6  Old Language
5 Knowledge Islands [2]
7  Arith Skill Check [4 X 2]
Arith Webvideos
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable [8]
9. Algebra Talk [7]
10 Two More Skills[5]
11 Why Shorthand
12 Shorthand Usage [10]
13 What's Next
PS: The 4-th Skill For Algebra
14 Compound Interest [6]
15 Linear Equations [5]
16 Painless Proofs
17 Pythagoras
PS I.  Distributive Law
PS II. Polynomials
18 Rules of Algebra [20]
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums [2]
23 Summation Notation
24 Your Money [3]
25 Induction & Recursion [4]
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Pathways for Learning

Would you like to show yourself or others how to be  algebra power users?

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

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:

  1. The forward and backward use of almost all formulas and equations (those in function form y = something)
  2. 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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Online Volumes
 
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1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3 .Why.Slopes.&
.More.Math.1995

Math How-TOs etc  2008
1. Arithmetic
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus

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2.  Algebra Videos (to appear)
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Solving Linear Equations  04-2005
5. Euclidean-Geometry To Complex No.s 
6.  Analytic Geometry/Functions 2006
7.  Number Theory. 2006-7
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9 Calculus  2005
10..Real  Analysis 1995
11 Electric Circuits Etc  2007
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1. K0-11Applied Math Program Outline  
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