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YOU are better than YOU think. Show
yourself how:
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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 12
Shorthand Usage Guide
Previous Section: 12 Symbols &
Numbers, How to Choose Letters to be pronouns or placeholders for numbers and
Quantities
8 Offspring Naming Conventions
Parents and mathematicians both may have difficulty naming their children.
The names of grandparents and uncles, etc, are often used along with numerals,
for instance, Richard III, Frederick IV etc, to name children. Similarly
mathematicians and you may lack imagination when naming numbers and quantities.
Typically we, a teacher or s textbook may use, reuse and recycle endlessly the
symbols x, y, z, x1, x2,
x99, a, b, c as shorthand (pronouns) for
numbers and quantities.
In any situation, the shorthand we use for a quantity or number is
unimportant. We just have to employ different symbols for different numbers and
quantities. Letters by themselves from the language of your choice, or letters
with subscripts can serve as shorthand symbols.
Traditionally, letters at the start of an alphabet stand for numbers or
quantities that will be given or will remain constant17.
In the same tradition, letters at the end of the alphabet have been used for
numbers or quantities which are unknown or variable. This tradition or guideline
is often broken. You are allowed to break and introduce other conventions as
convenient, for each problem you meet. Here there is only one rule or
requirement. Statements like
- let L be the length of my foot,
- let A be the area of the circle or
- let M be a whole number
are needed to explain the shorthand. The meaning and roles of shorthand symbols
can also be described using pictures or oral explanations. Each symbol we see
raises the questions: What does it stand for? What does it refer to? The
role of each letter should be stated in words, suggested by diagrams or
otherwise shown.
Chapter Sections: [ 12 Symbols & Pronouns ] [ 12 Pronouns ] [ 12 Shorthand Usage Guide ] [ 12 Pronouns in Mathematics ] [ 12 Big and Small Letters ] [ 12 Subscripts Etc ] [ 12 An Exercise ] [ 12 Symbols & Numbers ] [ 12 Offspring Naming Conventions ] [ 12 A Review & Answers to Exercise ]
Next Chapter: 13 What's Next
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www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
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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