YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes your attention. |
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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Chapter 10
Describing & Changing Calculations
Previous Section: 10 Finding Number, Using formulas
backwards
4 Formulas as Potential Calculations
We have discussed or described two recipes or formulas for calculating areas
and volumes without doing any arithmetic. Given the heights, lengths and widths
involved, we could compute the areas and volumes. That is easy to do by hand. It
is also easy or easier to use a calculator to do the arithmetic for us. Think in
terms of potential calculations: formulas describe calculations that could be
done (or avoided) as needed. We can postpone calculations, unless we need to do
them. Note that when you see a formula for the first time, you may need to
practice using it.
5 Further Readings
The following books (and others) cover ideas not included above.
- Mathematics Made Simple by A. Sperling and M. Stuart,
Doubleday 1981 edition, ISBN 0-385-17481-0.
- Algebra, the Easy Way by D. Downing, 1989, Barron's
Educational Series, Inc, 250 Wireless Boulevard, Hauppauge, New York 11788.
ISBN 0-8120-4194-1.
- How to Solve Algebra Word Problems by W. A. Nardi, Simon
& Shuster Inc, Gulf+Western Building, One Gulf + Western Plaza, New
York, NY 10023. ISBN 0-6680-06574-5.
6 Two Notions of What is a Variable
6.1 With and Without Symbols
Numbers and quantities which may change or vary are said to be variables.
This first notion of a variable does not involve or require the presence of
shorthand notation (symbols) to represent the number or quantity in question.
But there is a second notion of a variable employed in mathematics. A symbol
or letter which represents a number or quantity is also be called a variable if
the number or quantity concerned may change or vary, that is if the number or
quantity represented is a variable according to the first notion. While a symbol
or letter may be called a variable, not all variables are given or represented
letters or symbols. We can talk about numbers and quantities without employing a
written symbol for each one.
Remark. A change may be required in mathematics texts and
dictionaries to recognize both notions and not just the second.
Chapter Sections: [ 10 Formullas & Shorthand Notation ] [ 10 Changing Calculations ] [ 10. Replacement & Substitution ] [ 10 Find a Number ] [ 10 Formulas as Potential Calculations ]
Next Chapter: 11 Why Shorthand
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www.whyslopes.com
2. Three Skills for Algebra
Foreword, Chapters
& Appendices
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
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
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