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Appetizers and Lessons for Mathematics and Reason
  online logic chapters  - the best starting point for further site exploration.  Bon Appetite.

10 Find a Number
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Book Entrance ]


Three Skills
For 
Algebra
Volume 2

Chapters and Appendices

Book Entrance

10  Formullas & Shorthand Notation
10  Changing Calculations
10. Replacement & Substitution
10 Find a Number
10 Formulas as Potential Calculations

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

Chapter 10
Describing & Changing Calculations

Previous Section: 10. Replacement Principle


3  To Find A Rectangle's Dimensions

The rectangle area formula is easy to compute if you are given the width W and the length L. But can we use the area formula A = L·W to find the width W when the area A and length L are given? The rectangle area formula says
A = L·W
From this formula we can directly compute A provided the other two quantities L and W are given or known.

If you multiply W by a non-zero number called it, and then divide by the same number called it you get back your original quantity W. This is a rule or property of arithmetic with whole numbers and fractions, etc. A description of these properties will be given later. So multiplying and then dividing W by the number L gives the same result as doing nothing to W. This suggests the expression [( W·L)/(L)] when calculated, gives you the width W. Now we can write W = [( W·L)/(L)]. The equality sign is used to signal that the expression on either side of it gives the same result. But the expression W ·L whenever computed is the same as the area A. So we replace the computation of W ·L by A with the understanding that A = W ·L always represents this product W ·L.

Now W can be obtained by calculating [(( W ·L))/(L)]. The latter gives the same result as [(A)/(L)]. So we have a new width formula W = [(A)/(L)] for computing W whenever L and A are given. This formula is correct if A = W ·L for every rectangle you meet. Similarly, the length formula L = [(A)/(W)] can be obtained by interchanging the roles of the actors L and W above.


Chapter Sections:  10  Formullas & Shorthand Notation ] 10  Changing Calculations ] 10. Replacement & Substitution ] [ 10 Find a Number ] 10 Formulas as Potential Calculations ]

Next Section 10 Formulas as Potential Calculations

Next Chapter: 11 Why Shorthand

 

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Parents: Help your Child/Teen Learn covers  Speaking Skills, Reading & Writing Preparing for Science Having Patience, etc

Math How-TOs
1. Arithmetic   2. Algebra   3.  More Algebra  4.  Geometry 5 More Geometry 6.  Calculus
>> densely written 
>> use as skill checklists

Online Volumes (orders)
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

Skill & Concept 
Review or  Development 

 1. Decimal Arith - Video Based ]
2   Fractions  
3.  Fractions  with Units  
3. Solving Linear Equations  - 
making alg easier
4. Formulas forwards & Backwards - unifying theme for Algebra
5.  Proportionality, Back- & For-wards - theme at work.
6.  Logic - Math Free, good for precision in  work & studies 
7. Euclidean-Geometry  (leanly)
8. Slopes and Lines 
9. Why Study Slopes - a context 
10.  Quadratics
11  Polynomials
12  Factored Polys - a context
13 Functions - For-& Back -wards
14  Number Theory, Richly
15. Exponents, Radicals & logs.  
16   Calculus - Examples & Advice 
17.   Real  Analysis 
18  Electric Circuits Etc (So So)
19 Maps, Similarity & Trig, (alt view)
20 Complex numbers  

21 Logic with Symbols+truth tables

22  Consistent Story Telling
23. Even More Logic

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